AUTOMATES APODEIXEIS GEWMETRIKWN JEWRHMATWN. Metaptuqiak ergasða

Tm ma Majhmatikì Panepist mio Kr thc AUTOMATES APODEIXEIS GEWMETRIKWN JEWRHMATWN Metaptuqiak ergasða KaterÐna Lagoudˆkh Dekèmbrioc 2007 Hrˆkleio, Kr th

Perieqìmena 1 EISAGWGH 3 2 H MEJODOS TWN EMBADWN 6 2.1 JEMELIWDHS GEWMETRIKES ENNOIES (MEROS I) 7 2.2 BASIKES PROTASEIS I................. 13 2.3 EFARMOGES....................... 17 2.4 H MEJODOS APALOIFHS SHMEIWN........ 22 2.5 ELEUJERA SHMEIA................... 39 2.6 ALGORIJMOS....................... 49 2.7 EIDIKES TEQNIKES APALOIFHS SHMEIWN-SUSTHMA (GIB)............................ 51 2.8 EFARMOGES TOU ALGORIJMOU........... 58 3 AUTOMATES APODEIXEIS STHN EPIPEDH GEWME- TRIA 75 3.1 JEMELIWDHS GEWMETRIKES ENNOIES (MEROS II) 76 3.2 BASIKES PROTASEIS II................ 87 3.3 GRAMMIKES KATASKEUASTIKES GEWMETRIKES PROTASEIS........................ 101 3.4 DHMIOURGIA ELAQISTOU SUNOLOU KATASKEUWN106 3.5 TEQNIKES APALOIFHS SHMEIWN.......... 109 3.6 ALGORIJMOS....................... 120 3.7 EMBADIKES SUNTETAGMENES........... 121 3.8 EIDIKES TEQNIKES APALOIFHS SHMEIWN- SUSTH- MA (GIB).......................... 124 3.9 EFARMOGES ALGORIJMOU I............. 129 3.10 EMBADIKES SUNTETAGMENES KAI EIDIKA SHMEIA TRIGWNWN......................... 145 3.11 EFARMOGES ALGORIJMOU II............. 149 1

PERIEQŸOMENA 2 4 SUMPLHRWSH THS MEJODOU ME QRHSH G- WNIWN 157 4.1 TRIGWNOMETRIKES SUNARTHSEIS......... 157 4.2 KUKLOI.......................... 165 4.3 TEQNIKES APALOIFHS OMOKUKLIKWN SHMEIWN 175 4.4 PLHREIS GWNIES.................... 186 4.5 EFAPTOMENIKH SUNARTHSH............ 190 4.6 MEJODOS TWN PLHRWN GWNIWN- SUSTHMA (GIB) 201 4.7 AUTOMATES APODEIXEIS GEWMETRIKWN PROTASEWN THS KATHGORIAS C.................. 208 4.8 TEQNIKES APALOIFHS SHMEIWN MESW GEWMETRIK- WN POSOTHTWN..................... 209

Kefˆlaio 1 EISAGWGH H ergasða aut anafèretai sthn autìmath paragwg apodeðxewn gewmetrik n jewrhmˆtwn me ènan eidikì trìpo pou sthrðzetai se mia beltiwmènh mèjodo kai thn efarmog thc se èna prìgramma, to opoðo parˆgei se pollèc peript seic mikrèc, euanˆgnwstec kai komyèc apodeðxeic ekatontˆdwn gewmetrik n jewrhmˆtwn. Apì tic arqèc tou 1930, o A.T arski eis gage thn mèjodo apaloif - c shmeðwn mèsw gewmetrik n posot twn. H mèjodoc aut basðsthke sthn algebrik prosèggish thc apìdeixhc jewrhmˆtwn thc stoiqei dhc gewmetrðac. 'Epeita o W en T sun W u eis gage thn algebrik mèjodo, h opoða gia pr th tìte forˆ qrhsimopoi jhke sthn autìmath apìdeixh ekatontˆdwn gewmetrik n jewrhmˆtwn. Mèqri thn ergasða tou W u, pollèc epituqhmènec algebrikoð mèjodoi eðqan beltiwjeð me skopì thn autìmath apìdeixh gewmetrik n jewrhmˆtwn. Pollˆ upologistikˆ progrˆmmata basðsthkan sthn mèjodo aut kai h megalôterh beltðwsh thc mejìdou pro lje apì to Panepist mio tou Tèxac, ìpou to prìgramma qrhsimopoi jhke gia thn apìdeixh problhmˆtwn thc EukleÐdeiac kai mh- EukleÐdeiac gewmetrðac. 'Etsi loipìn pollˆ dôskola probl mata twn opoðwn oi paradosiakèc apodeðxeic apaitoôsan megˆlh eufuða gia thn lôsh touc, mporoôn plèon na apodeiqjoôn se deuterìlepta me thn qr sh twn upologistik n programmˆtwn. H algebrik mèjodoc, h opoða eðnai polô diaforetik apì tic paradosiakèc mejìdouc apodeðxewn, qrhsimopoi jhke apì polloôc gewmètrec prokeimènou na exetˆsoun an mia prìtash eðnai alhj c ìqi. Oi apodeðxeic pou parˆgontai apì ton upologist qrhsimopoioôn upologismoôc poluwnômwn. Ta polu numa autˆ mporeð na perilambˆnoun ekatontˆdec ìrouc kai dekˆdec metablhtèc. Gia ton lìgo autì, oi apodeðxeic pou parˆgontai apì ton upologist eðnai euanˆgnwstec kai mikrèc. To basikì ergaleðo pou qrhsimopoieðtai se aut th mèjodo gia thn epðlush gewmetrik n problhmˆtwn eðnai to embadìn trig nwn tetrapleôr- 3

KEFŸALAIO 1. EISAGWGH 4 wn. H mèjodoc twn embad n prokeimènou na parˆgei mikrèc apodeðxeic epilègei tic swstèc gewmetrikèc posìthtec kai ta katˆllhla l mmata pou anafèrontai se autèc. Ta kôria qarakthristikˆ thc mejìdou eðnai ta akìlouja. 1. Oi apodeðxeic twn problhmˆtwn pou parˆgontai sômfwna me th mèjodo eðnai genikˆ mikrèc se èktash. Apì ta l mmata pou qrhsimopoioôntai katˆ thn diˆrkeia twn apodeðxewn, o upologist c mporeð na parˆgei gr gora ènan megˆlo arijmì apodeðxewn kai na epilèxei thn pio sôntomh apìdeixh. To gegonìc autì apoteleð thn bˆsh paragwg c poikðlwn trìpwn apìdeixhc miac gewmetrik c prìtashc. 2. H mèjodoc eðnai tìso apodotik ste na parˆgei apodeðxeic dôskolwn jewrhmˆtwn dðqwc na prosjètei bohjhtikˆ shmeða kai eujeðec. 3. Dedomènou ìti h sqhmatik anaparˆstash miac gewmetrik c prìtashc mporeð na èqei pˆnw apì mia pijanèc ekdoqèc, oi apodeðxeic pou prosfèrei h mèjodoc eðnai anexˆrthtec apo tic sqhmatikèc anaparastˆseic twn gewmetrik n protˆsewn. 4. Oi apodeðxeic pou parˆgontai èqoun kajarì gewmetrikì nìhma kai eðnai euanˆgnwstec. H mèjodoc twn embad n efarmìzetai se kataskeuastikˆ gewmetrikˆ jewr mata, ìpou to sq ma twn gewmetrik n aut n protˆsewn mporeð na sqediasteð me qr sh tou kanìna kai tou diab th. Oi basikèc gewmetrikèc posìthtec pou qrhsimopoioôntai eðnai to embadìn, oi Pujagìreiec diaforèc kai oi pl reic gwnðec. To embadìn qrhsimopoieðtai gia thn apìdeixh gewmetrik n protˆsewn pou perilambˆnoun ènnoiec, ìpwc h parallhlða eujôgrammwn tmhmˆtwn, h sômptwsh kai h suggrammikìthta shmeðwn. H ènnoia thc Pujagìreiac diaforˆc trig nwn, qrhsimopoieðtai gia thn apìdeixh gewmetrik n protˆsewn pou perilambˆnoun kôklouc, kajetìthta kai analogðec eujugrˆmmwn tmhmˆtwn, en oi pl reic gwnðec qrhsimopoioôntai se protˆseic pou perièqoun kôklouc kai gwnðec. H mèjodoc twn embad n èqei genikˆ jewrhjeð wc èna sônolo eidik n teqnasmˆtwn pou epilôoun pollˆ gewmetrikˆ probl mata. O J.Z.Zhang meletoôse thn mèjodo twn embad n apì to 1975. Anagn rise thn genikìthta thc mejìdou kai thn beltðwse se mia susthmatik mèjodo epðlushc gewmetrik n problhmˆtwn, poikðlwn epipèdwn duskolðac, pou anafèrontai se basikèc gewmetrikèc protˆseic pou perilambˆnontai sta sqolikˆ biblða allˆ kai se protˆseic pou èqoun tejeð se majhmatikoôc diagwnismoôc. H paragwg automatopoihmènwn apodeðxewn gewmetrik n jewrhmˆtwn susqetðzetai me thn duskolða mˆjhshc kai didaskalðac thc gewmetrðac. Basizìmenoi ston ˆjlo pou èqei katafèrei to prìgramma pou perilambˆnei thn mèjodo twn embad n, pisteôoume ìti h mèjodoc aut mporeð na suneisfèrei pollˆ sthn ekpaðdeush thc gewmetrðac. To gegonìc ìti oi apodeðxeic pou parˆgontai apì ton upologist eðnai mikrèc se èktash

KEFŸALAIO 1. EISAGWGH 5 kai èqoun sq ma pou oi majhtèc mporoôn na to sqediˆsoun me qartð kai molôbi, mporeð na suntelèsei drastikˆ sthn antimet pish twn duskoli n pou parousiˆzoun oi majhtèc sto mˆjhma thc gewmetrðac. Akìmh den eðnai lðgec oi forèc pou oi kajhghtèc zhtˆne apì touc majhtèc enallaktikoôc kalôterouc trìpouc apìdeixhc gewmetrik n jewrhmˆtwn. Etsi loipìn, h mèjodoc aut mporeð na dunam sei thn ikanìthta twn majht n sthn epðlush gewmetrik n problhmˆtwn kai na prosfèrei thn dunatìthta stouc majhtèc na diamorf noun dikèc touc eikasðec kai na rwtˆne ton upologist gia thn orjìthtˆ touc. Pio sugkekrimèna, wc apˆnthsh sth duskolða aut, o J.Z.Zhang doôleye me paidiˆ gumnasðou kai dhmioôrghse èna nèo sôsthma axiwmˆtwn sthn gewmetrða basismèno sthn ènnoia tou embadoô. Qrhsimopoi ntac to nèo tou sôsthma, o Zhang katèbale megˆlh prospˆjeia gia na prowj sei mia nèa metarrôjmish sta sqoleða thc deuterobˆjmiac ekpaðdeushc thc KÐnac, me skopì thn apodotikìterh ekpaðdeush thc gewmetrðac. H epituqhmènh efarmog thc mejìdou tou od ghse sthn qr sh thc kai se panepisthmiakì epðpedo. Epiprosjètwc, h mèjodoc tou embadoô qrhsimopoieðtai kai sthn ekpaðdeush twn Kinèzwn majht n gia thn summetoq touc stic Ejnikèc Majhmatikèc Olumpiˆdec.

Kefˆlaio 2 H MEJODOS TWN EMBADWN Ja parousiˆsoume thn mèjodo thc autìmathc paragwg c twn paradosiak n apodeðxewn twn eukleðdeiwn gewmetrik n jewrhmˆtwn, pou an k- oun sthn kathgorða twn jewrhmˆtwn twn shmeðwn tom c tou Hilbert. Oi gewmetrikèc protˆseic pou epilôontai sômfwna me th mèjodo twn embad n eðnai kataskeuastikèc gewmetrikèc protˆseic pou an koun sthn sqetik gewmetrða. H sqetik gewmetrða meletˆ probl mata parallhlðac, sômptwshc kai suggrammikìthtac. H idèa kleidð thc mejìdou eðnai h apaloif twn shmeðwn pou kataskeuˆzontai sthn prìtash qrhsimopoi ntac èxi basikèc gewmetrikèc protˆseic, l mmata, pou aforoôn to embadìn trig nwn kai tetrapleôrwn. Ta shmeða ta apaloðfoume me seirˆ antðjeth apì ekeðnh pou kataskeuˆsthkan. 'Ena gewmetrikì je rhma eðnai alhjèc mìno kˆtw apì sugkekrimènec ikanèc kai anagkaðec sunj kec, oi opoðec den dðnontai me saf neia sthn perigraf tou probl matoc. An to je rhma mporeð na perigrafeð kataskeuastikˆ, to prìgramma mporeð na parˆgei me ènan susthmatikì trìpo tic sunj kec autèc. Sthn parakˆtw enìthta ja parousiˆsoume tic èxi basikèc protˆseic kai èpeita ja d soume trða paradeðgmata me skopì na epexhg soume th mèjodo. 6

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 7 2.1 JEMELIWDHS GEWMETRIKES EN- NOIES (MEROS I) Sthn ergasða aut ìlec oi diadikasðec pragmatopoioôntai sto epðpedo. Ta basikˆ gewmetrikˆ antikeðmena sto epðpedo eðnai ta shmeða kai oi eujeðec. Gia na dhl soume ta shmeða pˆnw sto epðpedo qrhsimopoioôme kefalaða grˆmmata, ìpwc A, B, C,... EÐnai gnwstì ìti apì duo diaforetikˆ shmeða A, B pernˆ mða kai mìno mða eujeða. Gia na dhl soume thn eujeða aut qrhsimopoioôme ton sumbolismì AB BA. Autì èqei wc skopì na d soume sthn eujeða mia apì tic duo kateujônseic, ètsi ste na anaferìmaste plèon se prosanatolismènec eujeðec. 'Etsi loipìn, h prosanatolismènh eujeða AB èqei kateôjunsh apì to shmeðo A sto shmeðo B, en h prosanatolismènh eujeða BA èqei thn antðjeth kateôjunsh, apì to shmeðo B sto shmeðo A. DÔo shmeða A kai B pˆnw se mia prosanatolismènh eujeða orðzoun ta prosanatolismèna eujôgramma tm mata, twn opoðwn to m koc AB eðnai jetikì eˆn h kateôjunsh apì to A prìc to B eðnai Ðdia me thn kateujunsh thc eujeðac kai eðnai arnhtikì eˆn h kateôjunsh apì to A proc to B eðnai antðjeth apì thn kateôjunsh thc eujeðac. SÔmfwna me ta parapˆnw èqoume oti AB = BA kai AB = 0 an kai mìno an A = B. 'Estw ìti èqw tèssera shmeða A, B, C kai D pˆnw se mia eujeða tètoia - ste A B. An o lìgoc twn mhk n twn prosanatolismènwn eujugrˆmmwn tmhmˆtwn AB kai CD eðnai t, tìte ja èqoume AB CD = t AB = tcd. An ta duo prosanatolismèna eujôgramma tm mata AB kai CD èqoun thn Ðdia kateôjunsh tìte o lìgoc touc eðnai jetikìc, t 0. An èqoun antðjeth kateôjunsh tìte o lìgoc touc eðnai arnhtikìc, t 0. AparaÐthth proupìjesh gia na isqôoun ta parapˆnw eðnai oti ta shmeða C kai D prèpei na an koun pˆnw sthn eujeða pou orðzoun ta shmeða A kai B.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 8 An pˆrw t ra shmeðo C pˆnw sthn eujeða AB tìte AB = AC + CB AC + CB=1. AB AB Onomˆzoume touc lìgouc AC CB kai, lìgoi jèsewc suntetagmènec jèsewc tou shmeðou C se sqèsh me to AB AB AB. EÐnai fanerì ìti gia opoiousd pote duo pragmatikoôc arijmoôc s kai t pou ikanopoioôn thn sqèsh s + t = 1, upˆrqei monadikì shmeðo C pˆnw sthn AB tètoio ste AC CB = s kai = t. AB AB Eidik perðptwsh eðnai h epilog shmeðou M pˆnw sto AB tètoio ste na eðnai to mèso tou tm matoc AB, opìte ja èqoume AM AB = MB AB = 1 2 DÔo diaforetikˆ shmeða pˆnta orðzoun mia eujeða. TrÐa shmeða den brðskontai pˆnta pˆnw se mia eujeða, an ìmwc sumbaðnei autì tìte ta shmeða autˆ onomˆzontai suggrammikˆ. ORISMOS 2.1.1 EujeÐa eðnai èna sônolo suggrammik n shmeðwn. An l eðnai mia eujeða kai shmeðo A tètoio ste A l, tìte lème ìti to A brðsketai pˆnw sthn eujeða l. PROTASH 2.1.2 TrÐa shmeða A, B, C eðnai suggrammikˆ an kai mìno an S ABC = 0. Apìdeixh An S ABC = 0 tìte apì to axðwma A 3 (selðda 215) ta shmeða A, B, C eðnai suggrammikˆ. Gia to antðstrofo t ra upojètoume ìti ta shmeða A, B, C eðnai suggrammikˆ.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 9 An A = C tìte BC = tcc = 0 apì axðwma A 6 (selðda 217): S ABC = ts ACC = 0 An A C tìte t = AB AC apì axðwma A 6: S ABC = ts ACC = 0. PORISMA 2.1.3 Duo diaforetikˆ shmeða A kai B orðzoun monadik eujeða AB h opoða apoteleð to sônolo ìlwn twn shmeðwn C gia ta opoða isqôei S ABC = 0. Apìdeixh 'Estw P, Q, R trða diaforetikˆ shmeða pˆnw sthn eujeða AB. Jèloume na deðxoume ìti: S P QR = 0. Apì thn prìtash 2.1.2 paðrnoume: S AQR = AQ AB S ABR = 0. 'Ara ta shmeða A, Q, R suggrammikˆ. Akìmh apì thn prìtash 2.1.2 paðrnoume: S P QA = AP AR S AQR = 0. 'Ara ta shmeða A, Q, P suggrammikˆ. Apì thn prìtash 2.1.2 pˆli paðrnoume: S P QR = QR QA S P QA = 0. To epìmeno gewmetrikì antikeðmeno pou ja melet soume eðnai to trðgwno. 'Opwc gnwrðzoume trða mh-suggrammikˆ shmeða A, B kai C sqhmatðzoun trðgwno, to opoðo sumbolðzetai wc ex c ABC. To embadìn tou ABC sumbolðzetai wc ex c ABC. GnwrÐzoume ìti ABC = 1 hbc, ìpou h to Ôyoc tou trig nou pˆnw sthn pleurˆ BC. 2 Sthn mèjodo twn embad n o parapˆnw tôpoc den ja jewreðtai wc basikì stoiqeðo, antðjeta ja qrhsimopoi soume wc basikèc protˆseic ˆlla aplˆ stoiqeða twn embad n. An ta shmeða eðnai suggrammikˆ tìte to ABC onomˆzetai ekfulismèno kai isqôei ìti ABC=0. 'Estw ìti èqw tèssera opoiad pote shmeða A, B, C kai R pˆnw sto

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 10 epðpedo, tìte parathroôme ìti sqhmatðzontai tèssera trðgwna ABC, RBC, RAB kai RAC kai to embadìn twn tessˆrwn trig nwn ikanopoioôn eftˆ diaforetikèc sqèseic, oi opoðec exart ntai apì thn jèsh pou èqoun metaxô touc ta shmeða A, B, C kai R. An to shmeðo R brðsketai sto eswterikì tou trig nou ABC, ìpwc sto parakˆtw sq ma, ja èqoume: ABC= RAB+ RBC+ RCA. A R B C Sq ma 2.1: Gia tic treðc parakˆtw peript seic omoðwc ja èqoume R C A B C R C A B A B Sq ma 2.2: ABC= RAB+ RBC- RCA, an ABCR kurtì tetrˆpleuro ABC= RBC+ RCA- RAB, an ABCR kurtì tetrˆpleuro ABC= RAB+ RCA- RBC, an ABCR kurtì tetrˆpleuro R

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 11 T ra gia tic treðc parakˆtw peript seic ja èqoume R C C C A B R A B A B R Sq ma 2.3: ABC= RAB- RBC- RCA, an to C eswterikì shmeðo tou RAB ABC= RBC- RAB- RCA, an to A eswterikì shmeðo tou RBC ABC= RCA- RBC- RAB, an to B eswterikì shmeðo tou RCA H eisagwg t ra thc ènnoiac tou proshmasmènou embadoô prosanatolismènou trig nou ja mac bohj sei na sumperilˆboume kai tic eftˆ parapˆnw sqèseic se mða mìno sqèsh. 'Ena trðgwno ABC èqei duo prosanatolismoôc. An A B C èqei thn antðjeth forˆ twn deikt n tou rologioô tìte to trðgwno ABC èqei jetikì prosanatolismì, alli c to trðgwno ABC èqei arnhtikì prosanatolismì. 'Etsi loipìn, ta trðgwna ABC, BCA, CAB èqoun ton Ðdio prosanatolismì, en ta trðgwna ACB, CBA, BAC èqoun antðjeto prosanatolismì. To proshmasmèno embadìn prosanatolismènou trig nou ABC, sumbolðzetai S ABC kai èqei thn Ðdia apìluth tim me to ABC kai eðnai jetikì an o prosanatolismìc tou trig nou eðnai jetikìc, alli c to S ABC eðnai arnhtikì. Epomènwc ja èqoume ìti S ABC = S BCA = S CAB = S ACB = S BAC = S CBA

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 12 T ra oi eftˆ parapˆnw sqèseic twn embad n ABC, RAB, RBC, RCA sunoyðzontai mìno se mða S ABC = S RBC + S RCA + S RAB. (I) 'Omoia mporoôme na orðsoume ta prosanatolismèna tetrapleura. Dojèntwn tessˆrwn A, B, C, DshmeÐwn kajorðzoume to prosanatolismèno tetrˆpleuro ABCD,sÔmfwna me thn akìloujh kateôjunsh twn shmeðwn A B C D. Ta tetrˆpleura BCDA, CDAB, DABC è- qoun ton Ðdio prosanatolismì me to tetrˆpleuro ABCD, epeid ta shmeða touc akoloujoôn kuklikèc metajèseic. D C D C A B A B D C D C A D C B A D C B A B A B Sq ma 2.4: Tèssera shmeða mporoôn na akolouj soun èxi, 6 = 4! 4, diaforetikèc diadromèc, ABCD, ADCB, ACBD, ADBC, ACDB, ABDC. T ra mporoôme na orðsoume to embadìn prosanatolismènou tetrapleôrou ABCD wc ex c S ABCD = S ABC + S ACD. Genikˆ mporoôme na orðsoume to proshmasmèno embadìn prosanatolismènou n polug nou A 1 A 2...A n me n 3 na eðnai: S A1 A 2...A n = n i=3 S A 1 A i 1 A i

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 13 D C A B Sq ma 2.5: A 6 A 5 A n _- 1 A 4 A n A 3 A 1 A 2 Sq ma 2.6: 2.2 BASIKES PROTASEIS I Sthn enìthta aut ja parousiˆsoume tic èxi protˆseic pou apoteloôn thn bˆsh thc mejìdou twn embad n. Thn mèjodo twn embad n ja thn qrhsimopoi soume kai gia na apodeðxoume merikèc apì autèc. PROTASH B.1 An ta shmeða C kai D eðnai pˆnw sthn Ðdia eujeða AB me A B kai P na eðnai opoiod pote shmeðo pou den an kei pˆnw sthn eujeða AB, tìte: S P CD S P AB = CD AB

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 14 P A D B C Sq ma 2.7: H apìdeixh thc prìtashc eðnai profan c. PROTASH B.2 (JEWRHMA TWN EPIPLEURWN TRIG- WNWN) 'Estw M eðnai h tom twn eujei n AB kai P Q kai Q M. Tìte èqoume: S P AB S QAB = P M QM, P M P Q = S P AB S P AQB, QM P Q = S QAB S P AQB Ta parakˆtw sq mata deðqnoun tèsseric pijanèc peript seic efarmog c tou jewr matoc twn epðpleurwn trig nwn. P P A M B N Q Q A M B P P Q A B M A B M Q Apìdeixh Sq ma 2.8: Pr toc trìpoc

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 15 'Estw N eðnai shmeðo pˆnw sthn eujeða AB tètoio ste MN = AB. Tìte me qr sh thc prìtashc B.1 (selðda 13), ja pˆroume: S P AB S QAB = S P MN S QMN DeÔteroc trìpoc = P M QM Me qr sh pˆli thc prìtashc B.1 (selðda 13), ja pˆroume: S P AB S QAB = S P AB S P AM S P AM S QAM S QAM S QAB PROTASH B.3 = AB P M AM AM QM AB = P M QM. Ac eðnai P AB kai QAB duo trðgwna kai R shmeðo thc P Q tètoio ste P R = tp Q, tìte isqôei ìti: S RAB = ts QAB + (1 t)s P AB. Q R P A B Apìdeixh Sq ma 2.9: 'Estw m = S ABP Q tìte S RAB = m S ARQ S BP R = m (1 t)s AP Q ts BP Q = m (1 t)(m S P AB ) t(m S QAB ) = ts QAB + (1 t)s P AB PROTASH B.4 Ac eðnai A, B, C kai D tèssera shmeða. Tìte AB CD an kai mìno an S ADBC = 0 S ABC = S ABD.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 16 A D B C Sq ma 2.10: H apìdeixh thc prìtashc eðnai profan c. PROTASH B.5 'Estw to parallhlìgrammo ABCD kai P aujaðreto shmeðo, (Sq ma 2.8). Tìte S ABC = S P AB + S P CD kai S P AB = S P DAC = S P DBC Apìdeixh P S D C A B Sq ma 2.11: ProekteÐnoume thn BC kai paðrnoume shmeðo S tètoio ste h P S na eðnai parˆllhlh sth CD. Apì thn prìtash B.4 (selðda 15), S ABC = S DBC, S P DC = S SDC kai S P AB = S SAB = S DBS. Opìte S P AB + S P CD = S DBS S DCS = S DBC = S ABC. O deôteroc tôpoc eðnai sunèpeia tou pr tou. PROTASH B.6 'Estw to paralhlìgrammo ABCD kai P, Q duo shmeða. Tìte S AP Q +

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 17 S CP Q = S BP Q + S DP Q kai S P AQB = S P DQC. Apìdeixh 'Estw O to shmeðo tom c twn diagwnðwn AC kai BD, (Sq ma 2.10). D C P O Q A B Sq ma 2.12: Oi diag nioi parallhlogrˆmmou gn rðzoume ìti diqotomoôntai, ˆra to O eðnai to mèso tou AC kai apì thn prìtash B.3 (selðda 15) ja pˆroume: S OP Q = CO CA S AP Q + AO CA S CP Q = 1 2 S AP Q + 1 2 S CP Q 'Ara 2S OP Q = S AP Q + S CP Q. To O eðnai mèso tou BD. Opìte omoðwc ja pˆroume: 2S OP Q = S BP Q + S DP Q Katal goume: S AP Q + S CP Q = S BP Q + S DP Q 2.3 EFARMOGES A. JEWRHMA TOU CEVA

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 18 'Estw trðgwno ABC kai P tuqìn shmeðo tou epipèdou (entìc ektìc tou trig nou). 'Estw epðshc D to shmeðo tom c twn eujei n AP kai CB kaj c kai E to shmeðo tom c twn eujei n BP kai AC kai tèloc F to shmeðo tom c twn eujei n CP kai AB. Tìte isqôei: AF BD CE = 1 F B DC EA Apìdeixh C E P E E P D C P D C D A B A B F A F B Sq ma 2.13: Ta parapˆnw sq mata deðqnoun mìno treðc apì tic pijanèc jèseic tou shmeðou P. H qrhsimìthta twn prosanatolismènwn embad n sthn mèjodo eðnai ìti kˆnei thn apìdeixh twn gewmetrik n protˆsewn sunoptik kai pio akrib c. Skopìc mac eðnai na apaleðyoume ta shmeða D, E kai F apì touc lìgouc AF, BD, CE F B DC EA. Autì ja to epitôqoume me thn efarmog tou jewr matoc twn epðpleurwn trig nwn se kajèna apì touc treðc lìgouc. 'Etsi loipìn ja pˆroume: AF BD CE = S AP C S BP A S CP B F B DC EA S BCP S CAP S ABP = 1 B.JEWRHMA TOU PAPPOU 'Estw trða shmeða A, B kai C pˆnw se mia eujeða kai trða shmeða A 1, B 1

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 19 kai C 1 pˆnw se mia ˆllh eujeða. 'Estw P to shmeðo tom c twn eujei n AB 1, A 1 B, Q to shmeðo tom c twn eujei n AC 1, A 1 C kai S to shmeðo tom c twn eujei n BC 1, B 1 C. Ta shmeða P, Q kai S eðnai suggrammikˆ. Apìdeixh C 1 A 1 B 1 P Q S A B C Sq ma 2.14: Genikˆ èna prìblhma suggrammikìthtac to metatrèpoume se prìblhma lìgwn wc ex c: 'Estw X 1 to shmeðo tom c twn eujei n P Qkai BC 1 kai X 2 to shmeðo tom c twn eujei n P Q kai B 1 C. ArkeÐ na deðxoume ìti X 1 = X 2 to opoðo eðnai isodônamo me thn akìloujh isìthta: P X 1 QX 1 QX 2 P X 2 = 1 (1) Skopìc mac eðnai na apaloðyoume apì thn sqèsh (1) ta shmeða X 1 kai X 2 qrhsimopoi ntac to je rhma twn epðpleurwn trig nwn. Opìte ja pˆroume: P X 1 QX 1 QX 2 P X 2 = S P BC 1 S QBC1 S QCB1 S P CB1 (2) T ra prèpei na apaloðyoume apì thn sqèsh (2) ta shmeða P kai Q. Skopìc mac eðnai h telik mac sqèsh na mhn perilambˆnei shmeða ta opoða

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 20 kataskeuˆsthkan apì tomèc eujei n, allˆ na apoteleðtai mìno apì ta arqikˆ shmeða epð twn eujei n A, B, C, A 1, B 1, C 1. 'Etsi loipìn me qr sh thc prìtashc B.2 (selðda 14), tou jewr matoc dhlad twn epipleôrwn trig nwn, ja pˆroume: S QBC1 = QC 1 AC 1 S ABC1 = S A 1 CC 1 S ABC1 S ACC1 A 1 S P CB1 = P B 1 AB 1 S ACB1 = S A 1 BB 1 S ACB1 S ABB1 A 1 S P BC1 = P B A 1 B S A 1 BC 1 = S ACC 1 S A1 CB 1 S ACC1 A 1 S QCB1 = QC A 1 C S A 1 CB 1 = S ABB 1 S A1 BC S ABB1 A 1 Antikajist ntac tic parapˆnw sqèseic sth sqèsh (2) kai efarmìzontac thn prìtash B.1 (selðda 13), ja pˆroume: S P BC1 S QBC1 S QCB1 S P CB1 = S ABB 1 S ACB1 S A1 BC 1 S A1 BB 1 S ACC1 S ABC1 S A1 CB 1 S A1 CB 1 = AB A 1 C 1 AC A 1 B 1 = 1 AC A 1 B 1 AB A 1 C 1 G.JEWRHMA TOU MENELAOU 'Estw trðgwno ABC. 'Estw F, D kai E shmeða pˆnw stic pleurèc AB, BC kai CA antðstoiqa. Ta shmeða E, F kai D eðnai suggrammikˆ an kai mìno an AF BD = 1 F B DC Apìdeixh CE EA = S AP C S BCP S BP A S CAP S CP B S ABP H diatèmnousa mporeð na pernˆ apì tic duo pleurèc enìc trig nou kai apì thn proèktash thc trðthc na pernˆ kai apì tic treðc proektˆseic twn pleur n enìc trig nou. To pleonèkthma thc mejìdou eðnai ìti den exartˆtai apì thn jèsh twn shmeðwn sthn sqhmatik anaparˆstash tou probl matoc. To gegonìc autì ofeiletai sto ìti h mèjodoc eðnai apokleistikˆ algebrik. 'Etsi loipìn den qreiˆzetai na diakrðnoume peript seic anˆloga me thn jèsh twn shmeðwn kai na d soume diaforetikèc apodeðxeic

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 21 D C E D C E A B F A B F Sq ma 2.15: anˆ perðptwsh. Epomènwc h akìloujh apìdeixh perilambˆnei kai tic duo peript seic. An ta E, F kai D eðnai suggrammikˆ tìte me qr sh thc prìtashc B.1 (selðda 13) kai B.2 (selðda 14), gia touc treðc lìgouc ja èqoume: AF F B = S AEF S BEF, BD DC = S BEF S CEF, CE EA = S CEF S AEF Opìte: AF BD CE = S AP C S BP A S CP B F B DC EA S BCP S CAP S ABP = 1 Upˆrqoun ìmwc kai ˆllec epilogèc trig nwn pou ja mac odhgoôsan me th bo jeia tou jewr matoc twn epðpleurwn trig nwn sthn epðlush tou probl matoc, ìpwc AF F B = S ADE S BDE, BD DC = S EBD S ECD, CE EA = S DEC S DEA 'Opou kai pˆli mac dðnei: AF BD CE = 1 F B DC EA Se endeqìmenh algorijmik ulopoðhsh thc mejìdou ja prèpei na protimhjeð o pr toc trìpoc pou sthrðzetai sta arqikˆ dedomèna kai den qrhsi-

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 22 mopoieð prìsjetec kataskeuèc (shmeða, eujeðec k.l.p). Gia thn apìdeixh t ra tou antistrìfou, èqoume ìti gia ta shmeða E, F, D pou brðskontai pˆnw stic pleurèc AC, AB, BC isqôei ìti: AF BD CE = 1 F B DC EA kai jèloume na deðxoume ìti ta E, F kai D eðnai suggrammikˆ. Ergazìmaste wc ex c: Upojètoume ìti h EF sunantˆ thn BC sto H, tìte arkeð na deðxoume ìti D H.IsqÔei ìmwc kai gia ta shmeða E, F kai H ìti: AF BH CE = 1 F B HC EA BH = BD HC DC 'Ara D H 2.4 H MEJODOS APALOIFHS SHMEI- WN Sthn enìthta aut parajètoume tic apodeðxeic tri n gewmetrik n jewrhmˆtwn, oi opoðec akoloujoôn mia genik mejodologða. Kajèna apì ta jewr mata autˆ mporoôn na perigrafoôn san mia akoloujða kataskeu c shmeðwn kai diatôpwshc miac arijmhtik c sqèshc metaxô aut n twn shmeðwn. Prin proqwrðsoume sthn parousðash thc mejìdou prèpei na anafèroume ìti poujenˆ sthn mèjodo den anafèrontai eujôgramma tm mmata. Ja doôme sthn sunèqeia kai sta paradeðgmata pou ja parajèsoume ìti sthn kataskeuastik perigraf twn protˆsewn ta eujôgramma tm mata (SEGMENT) dhl nontai wc eujeðec (LINE). O lìgoc pou sumbaðnei autì ofeðletai apokleistikˆ sth dhmiourgða thc mejìdou. H mèjodoc è-

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 23 qei kataskeuasteð me tètoio trìpo ste na metaqeirðzetai mìno isìthtec metaxô gewmetrik n posot twn kai katˆ sunèpeia adunateð na epexergasteð aniswtikèc sqèseic. 'Etsi loipìn epeid metaxô eujôgrammwn tmhmˆtwn kuriarqoôn aniswtikèc sqèseic gia ton lìgo autì apofeôgoume na grˆfoume sthn kataskeuastik perigraf thc prìtashc: (SEGM EN T A B). A. JEWRHMA TOU Ceva C E P E E P D C P D C D A B A B F A F B Sq ma 2.16: P 1. 'Estw tèssera aujaðreta shmeða A, B, C, D P 2. 'Estw D to shmeðo tom c twn eujei n AP kai CB. P 3. 'Estw E to shmeðo tom c twn eujei n BP kai AC. P 4. 'Estw F to shmeðo tom c twn eujei n CP kai AB. Sumpèrasma: z= BD CE AF = 1 DC EA F B H apìdeixh akoloujeð ta parakˆtw b mata. Efarmìzoume sthn proc apìdeixh sqèsh kai se kˆje lìgo pou summetèqei se aut n mia apì tic protˆseic B.1-B.6 (selðda 13-16). 'Etsi apaloðfoume stadiakˆ ta shmeða pou eisˆgoun oi kataskeuèc P 2, P 3, P 4 kai ta antikajistoôme me ta katˆllhla embadˆ trig nwn.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 24 'Etsi loipìn qrhsimopoi ntac kai stic treðc peript seic apaloif c twn shmeðwn F, E, D, thn prìtash B.2 (selðda 14), èqoume: Apaloif shmeðou F (kataskeu P 4 ). z= BD CE AF = BD CE S AP C DC EA F B DC EA S BCP Apaloif shmeðou E (kataskeu P 3 ). z= BD CE S AP C DC EA S BCP Apaloif shmeðou D (kataskeu P 2 ). z= S BP A S CAP S CP B S ABP S AP C S BCP = 1 B. JEWRHMA TOU PAPPOU = BD S CP B S AP C DC S ABP S BCP C 1 A 1 B 1 P Q S A B C Sq ma 2.17: To je rhma autì mporeð na perigrafeð sômfwna me tic akìloujec kataskeuèc. P 1. 'Estw tèssera aujaðreta shmeða A, B, A 1, B 1 P 2. 'Estw C shmeðo pˆnw sthn AB. P 3. 'Estw C 1 shmeðo pˆnw sthn A 1 B 1. P 4. 'Estw P to shmeðo tom c twn eujei n AB 1 kai A 1 B. P 5. 'Estw Q to shmeðo tom c twn eujei n AC 1 kai A 1 C. P 6. 'Estw S to shmeðo tom c twn eujei n BC 1 kai B 1 C.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 25 P 7. 'Estw X 1 to shmeðo tom c twn eujei n P Q kai BC 1. P 8. 'Estw X 2 to shmeðo tom c twn eujei n P Q kai B 1 C. Sumpèrasma: z= P X 1 QX 1 QX 2 P X 2 = 1 C 1 A 1 B 1 P Q X 1 X 2 A B C Sq ma 2.18: H apìdeixh akoloujeð ta parakˆtw b mata. Apaloif shmeðou X 2 (kataskeu P 8 ) Qr sh prìtashc B.2 (selðda 14). z= P X 1 QX 1 QX 2 P X 2 = P X 1 QX 1 SQCB 1 S P CB1 Apaloif shmeðou X 1 (kataskeu P 7 ) Qr sh prìtashc B.2. z= P X 1 QX 1 SQCB 1 S P CB1 = S P BC 1 S QBC1 SQCB 1 S P CB1 Apaloif shmeðou S (kataskeu P 6 ). Den upˆrqei poujenˆ sthn teleutaða sqèsh gewmetik posìthta pou na perilambˆnei to shmeðo S. Epomènwc h mèjodoc proqwrˆei sthn amèswc epìmenh apaloif shmeðou. Apaloif shmeðou Q (kataskeu P 5 ) Qr sh prìtashc B.1 kai B.2, (selðda 13).

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 26 z= S P BC 1 S P CB1 SACC 1 S A1 CC 1 SACB 1 S ABC1 Apaloif shmeðou P (kataskeu P 4 ) Qr sh prìtashc B.1 kai B.2. z= S ABB 1 S ACB1 SA 1 BC 1 S A1 BB 1 SACC 1 S ABC1 SA 1 CB 1 S A1 CC 1 = = AB AC A1C 1 A 1 B 1 AC AB A1B 1 A 1 C 1 = 1 G. JEWRHMA TOU MENELAOU D C E D C E A B F A B F Sq ma 2.19: To je rhma autì mporeð na perigrafeð sômfwna me tic akìloujec kataskeuastikèc protˆseic. P 1. 'Estw pènte aujaðreta shmeða A, B, C, K, L P 2. 'Estw D to shmeðo tom c twn BC kai KL. P 3. 'Estw E to shmeðo tom c twn AC kai KL P 4. 'Estw F to shmeðo tom c twn AB kai KL. Sumpèrasma: z= AF F B BD DC CE EA = 1

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 27 H apìdeixh akoloujeð ta parakˆtw b mata. 'Etsi loipìn qrhsimopoi ntac kai stic treðc peript seic apaloif c twn shmeðwn F, E, D, thn prìtash B.2 (selðda 14), èqoume: Apaloif shmeðou F (kataskeu P 4 ). z= AF BD CE = S AKL BF CD AE S BKL BD CE CD AE Apaloif shmeðou E (kataskeu P 3 ). z= S AKL S BKL BD CD CE AE = S AKL S BKL BD CD SCKL S AKL Apaloif shmeðou D (kataskeu P 2 ). z= S AKL S BKL SCKL S AKL BD CD = S AKL S BKL SCKL S AKL SBKL S CKL = 1 PERIGRAFH THS MEJODOU Oi protˆseic twn shmeðwn tom c tou Hilbert: C H. Proc to parìn h mèjodoc eðnai periorismènh sthn epðlush protˆsewn pou an koun sthn kathgorða C H. Oi gewmetrikèc protˆseic pou an koun sthn kathgorða aut ja oristoôn parakˆtw, 29. Gia na perigrˆyoume me saf neia tic protˆseic twn shmeðwn tom c tou Hilbert qreiazìmaste tic ènnoiec: gewmetrikèc posìthtec kai kataskeuèc. Oi gewmetrikèc posìthtec pou an koun sthn kathgorða aut eðnai duo eid n: lìgoc duo prosanatolismènwn eujugrˆmmwn tmhmˆtwn pˆnw se mia eujeða pˆnw se duo parˆllhlec eujeðec kai embadìn prosanatolismènwn trig nwn tetrapleôrwn. Mia kataskeu qrhsimopoieðtai gia na eisˆgei èna nèo shmeðo apì kˆpoia dh gnwstˆ shmeða. Oi kataskeuèc èqoun thn parakˆtw morf :

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 28 K 1 (P OINT A). 'Estw èna aujaðreto shmeðo A pˆnw sto epðpedo. To shmeðo A eðnai eleôjero shmeðo. K 2 (ON LINE A X Y ). 'Estw ìti to shmeðo A an kei pˆnw sthn eujeða XY. To shmeðo A eðnai hmi-eleôjero shmeðo. K 3 (ON P LINE A B X Y ). 'Estw ìti to A eðnai shmeðo pˆnw sthn eujeða pou pernˆ apì to shmeðo B kai eðnai parˆllhlh sthn eujeða XY. To shmeðo A eðnai hmi-eleôjero shmeðo. K 4 (LRAT IO A X Y r). 'Estw ìti to A eðnai shmeðo pˆnw sthn eujeða XY tètoio ste XA = rxy, ìpou r eðnai ènac rhtìc arijmìc, rht èkfrash kˆpoiwn gewmetrik n posot twn metablht. Stic duo pr tec peript seic to shmeðo A eðnai stajerì shmeðo, en sthn trðth eðnai hmieleôjero shmeðo. K 5 (P RAT IO A B X Y r). 'Estw ìti to shmeðo A an kei pˆnw sthn eujeða pou pernˆei apì to shmeðo B kai eðnai parˆllhlh sthn eujeða XY tètoia ste BA = rxy, ìpou r eðnai ènac rhtìc arijmìc, rht èkfrash kˆpoiwn gewmetrik n posot twn metablht. Stic duo pr tec peript seic to shmeðo A eðnai stajerì shmeðo, en sthn trðth eðnai hmi-eleôjero shmeðo. K 6 (INT ER A (LINE X Y )(LINE P Q)). 'Estw ìti to A eðnai to shmeðo tom c thc eujeðac XY kai thc eujeðac P Q. To shmeðo A eðnai stajerì shmeðo. K 7 (INT ER A (LINE X Y )(P LINE W P Q))). 'Estw ìti to A eðnai to shmeðo tom c twn eujei n XY kai thc eujeðac pou pernˆ apì to shmeðo W kai eðnai parˆllhlh sthn eujeða P Q. To shmeðo A eðnai stajerì shmeðo. K 8 (INT ER A (P LINE Z X Y )(P LINE W P Q))). 'Estw ìti to A eðnai to shmeðo tom c thc eujeðac pou pernˆ apì to shmeðo Z kai eðnai

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 29 parˆllhlh sthn eujeða XY kai thc eujeðac pou pernˆ apì to shmeðo W kai eðnai parˆllhlh sthn eujeða P Q. To shmeðo A eðnai stajerì shmeðo. ORISMOS 2.4.1 Oi protˆseic twn shmeðwn tom c Hilbert èqoun thn parakˆtw morf : Prìtash=(K 1, K 2,..., K n, S) Kˆje prìtash aut c thc kathgorðac perigrˆfetai apì èna peperasmèno pl joc kataskeu n, oi opoðec èqoun thn morf twn kataskeu n K 1, K 2,..., K 8 (selðda 28-28). Oi kataskeuèc entˆssontai katˆ diatetagmèno trìpo wc ex c: arqikˆ eisˆgontai oi kataskeuèc tôpou K 1, oi opoðec èqoun thn morf thc K 1 kai eisˆgoun aujaðreta shmeða sto epðpedo. SuneqÐzoume me thn prosj kh ˆllwn kataskeu n K 2,..., K n, oi opoðec è- qoun thn morf twn kataskeu n K 2,..., K 8. Kajemiˆ apì tic parapˆnw kataskeuèc eisˆgei nèa shmeða pou proèrqontai apì tomèc eujei n kai apì lìgouc parˆllhlwn eujôgrammwn tmhmˆtwn. Oi eujeðec èqoun kataskeuasteð apì ta shmeða pou èqoun eisˆgei oi kataskeuèc tôpou K 1 kai apì ta shmeða pou èqoun dh kataskeuasteð mèsw twn kataskeu n K 2,..., K n. Tèloc h prìtash katal gei sto sumpèrasma (S), to opoðo apoteleðtai apì thn isìthta duo poluwnumik n ekfrˆsewn gewmetrik n posot twn. To zhtoômeno kˆje forˆ eðnai h apìdeixh thc isìthtac twn duo aut n poluwnumik n ekfrˆsewn. Oi gewmetrikèc autèc posìthtec perilambˆnoun shmeða pou èqoun eisaqjeð apì tic kataskeuèc K i, i = 1,..., n. Oi gewmetrikèc idiìthtec pou kaloômaste na deðxoume pwc èqoun oi parapˆnw gewmetrikèc posìthtec eðnai h suggrammikìthta, h parallhlða kaj c kai sqèseic metaxô lìgwn eujugrˆmmwn tmhmˆtwn. 'Ola ta paradeðgmata pou dìjhkan prohgoumènwc an koun sthn kathgorða C H. ParathroÔme ìti oi apodeðxeic twn paradeigmˆtwn prèpei na exasfalðzoun ìti poujenˆ mèsa sthn prìtash den parousiˆzontai mhdenikoð paronomastèc, epeid alli c h epðlush thc prìtashc ja apotôgqane. To gegonìc autì ofeðletai sthn mèqri t ra apousða perioristik n sunjhk n

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 30 sthn perigraf thc prìtashc. Genikˆ èna gewmetrikì je rhma eðnai alhjèc mìno kˆtw apì kˆpoiec perioristikèc sunj kec. T ra gia tic protˆseic pou an koun sthn kathgorða C H oi perioristikèc sunj kec parˆgontai me susthmatikì trìpo. ORISMOS 2.4.2 'Estw prìtash P=(K 1, K 2,..., K n, S), tìte oi perioristikèc sunj kec gia na eðnai h P prìtash thc kathgorðac C H eðnai oi akìloujec. 1. An K i eðnai h kataskeu K 1 (selðda 28), tìte den qreiˆzontai perioristikèc sunj kec. 2. An K i eðnai mia apì tic kataskeuèc K 2,..., K 5 (selðda 28-28), tìte oi perioristikèc sunj kec gia thn K i eðnai X Y. 3. An K i eðnai mia apì tic kataskeuèc K 6,..., K 8 (selðda 28), tìte oi perioristikèc sunj kec gia thn K i eðnai XY P Q. 4. Gia to sumpèrasma (S) apaitoôme oi paronomastèc twn gewmetrik n posot twn na mhn eðnai mhdèn. PERIORISTIKES SUNJHKES GIA TA PARADEIGMA- TA 1.Gia to je rhma tou CEV A (selðda 23): F B, D C, E A, C B, C A, A B. 2.Gia to je rhma tou PAPPOU (selðda 24): A B, C D, AB 1 A 1 B, AC 1 A 1 C, P Q BC 1, P Q B 1 C, Q X 1, P X 2. 3.Gia to je rhma tou MENELAOU (selðda 26): F B, D C, E A, A C, B C, DE AB. To monadikì pou apomènei gia na perigrˆyoume th mèjodo eðnai na d soume kˆpoiec teqnikèc apaloif c shmeðwn apì ta embadˆ kai apì touc lìgouc mhk n kaj c kai thn teqnik apaloif c shmeðwn pou brðskontai pˆnw se duo eujeðec.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 31 TEQNIKES APALOIFHS SHMEIWN MESW TOU L- OGOU MHKWN Oi teqnikèc apaloif c shmeðwn mèsw tou lìgou mhk n kaj c kai mèsw twn embad n, ìpwc ja doôme parakˆtw, perilambˆnoun mia gewmetrik posìthta G kai mia kataskeu K. LHMMA 2.4.3 An G= P Y kai K=(LRAT IO Y U V t) me perioristik QR sunj kh U V. An to shmeðo P brðsketai pˆnw sthn eujeða UV tìte: Y U P V Q R Sq ma 2.20: P Y = P U+UY QR QR = P U + UY UV UV QR UV = = P U +t UV QR UV. An to shmeðo P den brðsketai pˆnw sthn eujeða UV tìte: paðrnoume shmeðo S tètoio ste P S = QR, opìte to Y ja eðnai to shmeðo tom c twn P S kai UV kai P S QR. Apì thn prìtash B.2 (selðda 14) kai B.6 (selðda 16) paðrnoume: P Y = P Y = S P UV QR P S S P USV = S P UV S QURV

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 32 R S Q P Y U V Sq ma 2.21: LHMMA 2.4.4 An G= P Y kai K=(ON LINE Y U V ) me perioristik sunj kh U V. An jèsoume t = UY tìte èqoume to parapˆnw QR UV l mma. kai K=(P RAT IO Y W U V t) me perioris- LHMMA 2.4.5 An G= P Y QR tik sunj kh U V. Q R Y W P U V Sq ma 2.22: An to shmeðo P brðsketai pˆnw sth W Y tìte prokôptei polô eôkola ìti: P Y = P W +t UV QR QR UV. An to shmeðo P den brðsketai pˆnw sth W Y tìte: paðrnoume shmeða K kai L tètoia ste W K UV prìtash B.2 (selðda 14) èqoume = 1 kai P L QR = 1. Apì thn P Y = P Y = S P W K QR P L S P W KL = S P UW V S QURV

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 33 R Q L P Y W K U V kai K=(ON P LINE Y U V ) me perior- LHMMA 2.4.6 An G= P Y QR istik sunj kh U V. Sq ma 2.23: An jèsoume t = W Y UV tìte èqoume to parapˆnw l mma. LHMMA 2.4.7 An G= P Y kai K=(INT ER Y (LINE A B)(LINE C D)) QR me perioristik sunj kh AB CD. P Y = QR S P CD S QCRD, S P AB S QARB, an P / CD; an P / AB; 0, alli c. D C Y A P B S T Q R Sq ma 2.24:

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 34 An to P den eðnai pˆnw sthn eujeða CD, pˆre duo shmeða S kai T pˆnw sto P Y tètoia ste ST = QR, tìte to Y eðnai h tom twn ST kai CD. Apì thn prìtash B.2 (selðda 14), èqoume: P Y = S P CD ST S SCT D. Epeid P Y QR apì thn prìtash B.6 (selðda 16), èqoume: S SCT D = S QCRD. Akìma epeid to P den an kei sthn eujeða CD kai ST QR, èqoume: QR CD. Apì thn prìtash B.4 (selðda 15) paðrnoume: S QCRD 0. Oi ˆllec peript seic apodeiknôontai ìmoia. LHMMA 2.4.8 An G= P Y kai K=(INT ER Y (LINE C D)(P LINE W A B)) QR me perioristik sunj kh AB CD. P Y = QR S P CD S QCRD, S P AW B an P / CD; S QARB, an P CD kai P Y ; 0, alli c. An to P den an kei sthn eujeða CD, tìte h apìdeixh eðnai Ðdia me to l mma 2.4.7 (selðda 33). D P C W S A B Q Sq ma 2.25: An to P an kei pˆnw sthn eujeða CD, tìte dialègoume shmeðo S pˆnw sthn eujeða pou dièrqetai apì to shmeðo W tètoio ste W S = AB. Apì to l mma 2.4.7 (selðda 33), èqoume P Y = S P W S QR S QW RS. Apì thn prìtash B.5 (selðda 16), paðrnoume S P W S = S P AB S W AB = S P AW B. AfoÔ W S = AB, apì thn prìtash B.6 (selðda 16), paðrnoume S QW RS = S QARB, to opoðo apodeiknôei to zhtoômeno. Epeid AB CD, èqoume S QARB 0. R LHMMA 2.4.9 An G= P Y QR kai K=(INT ER Y (P LINE R C D)(P LINE W A B)) me perioristik sunj kh AB CD.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 35 P Y = QR { SP W AB S QCRD, S P CRD S QCRD, an QR AB; alli c; D C R Y W A B Sq ma 2.26: Autì to l mma apodeiknôetai ìmoia me to l mma 2.4.8 (selðda 34). APALOIFES SHMEIWN MESW TWN EMBADWN kai K=(LRAT IO Y P Q t) me perior- LHMMA 2.4.10 An G=S ABY istik sunj kh P Q. S ABY = ts ABQ + (1 t)s ABP A B Y P Q Sq ma 2.27: H apìdeixh thc prìtashc eðnai polô eôkolh. Efarmìzoume thn prìtash B.3 (selðda 15).

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 36 kai K=(ON LINE Y P Q) me peri- LHMMA 2.4.11 An G=S ABY oristik sunj kh P Q. An epilèxoume t = P Y P Q parapˆnw perðptwsh. tìte èqoume akrib c to Ðdio apotèlesma me thn kai K=(P RAT IO Y R P Q t) me peri- LHMMA 2.4.12 An G=S ABY oristik sunj kh P Q. S ABY = S ABR + ts AP BQ Q P S Y R A B Sq ma 2.28: Gia na apodeðxoume thn perðptwsh aut kˆnoume thn parakˆtw diadikasða. PaÐrnoume èna shmeðo S tètoio ste RS = P Q. Apì to l mma 2.4.10 (selðda 35), paðrnoume S ABY = ts ABS + (1 t)s ABR. Apì thn prìtash B.6 (selðda 16), paðrnoume S ABS = S ABR + S ABQ S ABP = S ABR + S AP BQ. Me mia apl antikatˆstash paðrnoume to zhtoômeno. LHMMA 2.4.13 An G=S ABY perioristik sunj kh P Q. kai K=(ON P LINE Y R P Q) me An epilèxoume t = P Y P Q parapˆnw l mma. tìte èqoume akrib c to Ðdio apotèlesma me to

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 37 ApodeiknÔetai ìmoia me to parapˆnw l mma. LHMMA 2.4.14 An G=S ABY kai K=(INT ER Y (LINE P Q)(LINE U V )) me perioristik sunj kh P Q UV. S ABY = 1 S P UQV (S P UV S ABQ + S QV U S ABP ) V U Y P Q A B Sq ma 2.29: Gia thn apìdeixh thc prìtashc aut c paðrnoume: t = P Y tìte 1 t = Y Q P Q P Q Apì thn prìtash B.3 (selðda 15), èqoume: S ABY = P Y P Q S ABQ + Y Q P Q S ABP Apì thn prìtash B.2 (selðda 14), èqoume: P Y P Q = S P UV S P UQV, Y Q P Q = S QV U S P UQV Epeid P Q UV èqoume ìti S P UQV 0 Me mia apl antikatˆstash paðrnoume to zhtoômeno. LHMMA 2.4.15 An G=S ABY kai K=(INT ER Y (LINE U V )(P LINE R P Q)) me perioristik sunj kh P Q UV. S ABY = 1 S P UQV (S P UQR S ABV + S P RQV S ABU )

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 38 Y U V S P R Q Sq ma 2.30: Gia na apodeðxoume thn prìtash aut paðrnoume shmeðo S tètoio ste 1 RS = P Q. Tìte apì to l mma 2.4.14 (selðda 37) èqoume S RUSV (S USR S ABV + S V RS S ABU ). 'Eqoume epðshc ta akìlouja: apì thn prìtash B.6 (selðda 16): S RUSV = S P UQV apì thn prìtash B.5 (selðda 16): S USR = S UQP S RQP = S P UQR apì thn prìtash B.5: S V SR = S V QP S RP Q = S P RQV Me mia apl antikatˆstash paðrnoume to zhtoômeno. LHMMA 2.4.16 An G=S ABY kai K=(INT ER Y (P LINE W U V )(P LINE R P Q)) me perioristik sunj kh P Q UV. S ABY = S P W QR S P UQV S AUBV + S ABW Gia na apodeðxoume thn prìtash aut paðrnoume shmeðo X tètoio ste 1 W X = UV. Tìte apì to l mma 2.4.15 (selðda 37) èqoume S P W QX (S P W QR S ABX + S P RQX S ABW ). 'Eqoume apì thn prìtash B.6 (selðda 16): S P W QX = S P UQV S ABX = S ABW + S ABV S ABU = S ABW + S AUBV

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 39 X W V U A B Y R P Q Sq ma 2.31: S P RQX = S P RQ + S P QX = S P RQ + S P QW + S P QV S P QU = S P W QR + S P UQV Me mia apl antikatˆstash paðrnoume to zhtoômeno. Mènei t ra na exetˆsoume thn perðptwsh: An G=S ABY kai K=(P OINT Y ). Aut h perðptwsh ja analujeð me thn bo jeia thc parakˆtw enìthtac (bl. l mma 2.5.3 selðda 41). 2.5 ELEUJERA SHMEIA Skopìc mac gia tic gewmetrikèc protˆseic P=(K 1, K 2,..., K n, S) pou an koun sthn kathgorða C H, eðnai na apaloðyoume ìla ta bohjhtikˆ shmeða ètsi ste oi nèec gewmetrikèc posìthtec pou ja prokôyoun na perièqoun mìno eleôjera shmeða. Oi gewmetrikèc autèc posìthtec den eðnai genikˆ anexˆrthtec, afou gia parˆdeigma an gia opoiad pote tèssera shmeða A, B, C, D pˆroume to S ABC, tìte autì, sômfwna me ìsa dh è- qoume anafèrei sthn arq, ja isoôtai me S ABC = S ABD + S ADC + S DBC.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 40 EÐnai anˆgkh epomènwc oi telikèc gewmetrikèc posìthtec na ekfrastoôn sunart sei kˆpoiwn anexˆrthtwn metablht n. Gia na to epitôqoume qreiazìmaste thn ènnoia twn embadik n suntetagmènwn. ORISMOS 2.5.1 'Estw ìti A, O, U, V eðnai tèssera shmeða tètoia - ste ta O, U, V na mhn eðnai suggrammikˆ. Oi embadikèc suntetagmènec gia to shmeðo A wc proc to OUV ja eðnai: x A = S OUA S OUV y A = S OAV S OUV z A = S AUV S OUV EÐnai fanerì ìti x A + y A + z A = 1. V O A T U Sq ma 2.32: PROTASH 2.5.2 Ta shmeða tou epipèdou eðnai se èna proc èna antistoqða me ta shmeða (x, y, z) pou ikanopoioôn thn sqèsh x + y + z = 1. Apìdeixh 'Estw O, U, V eðnai trða mh-suggrammikˆ shmeða. Tìte gia kˆje shmeðo A, oi embadikèc suntetagmènec ja ikanopoioôn thn sqèsh x A +y A +z A = 1. AntÐstrofa, gia opoiad pote x, y kai z tètoia ste: x + y + z = 1 mporoôme na broôme shmeðo A tou opoðou oi embadikèc suntetagmènec ja eðnai oi x, y kai z.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 41 An z = 1, pˆre shmeðo A tètoio ste x = OA UV. Tìte apì to l mma 2.4.12 (selðda 36), paðrnoume: x A = S OUA S OUV, y A = x = y kai z A = 1. An z 1, pˆre shmeðo B pˆnw sthn UV tètoio ste x 1 z = UB UV. PaÐrnoume shmeðo B pˆnw sthn OB tètoio ste z = AB OB. Apì to je rhma twn epig niwn trig nwn èqoume: z A = S AUV S OUV = AB OB = z x A = S OUA S OUV = (1 z) S OUB S OUV = x S OUV S OUV = x ìmoia y A = y. To l mma pou akoloujeð ekfrˆzei to embadìn eleôjerwn shmeðwn sunart - sei twn embadik n suntetagmènwn kai twn tri n eleôjerwn shmeðwn wc proc ta trða shmeða anaforˆc pou epilèxame. LHMMA 2.5.3 An G(Y )=S ABY kai K=(P OINT Y ). QwrÐc blˆbh thc genikìthtac èqoume upojèsei ìti ìla ta eleôjera shmeða thc prìtashc P èqoun eisaqjeð sthn arq thc prìtashc. Gia na qrhsimopoioôme to l mma autì, upojètoume epðshc ìti upˆrqoun toulˆqiston tèssera eleôjera shmeða O, U, V kai Y sthn prìtash P. Ta shmeða O, U, V ja eðnai ta shmeða anaforˆc. To l mma autì ekfrˆzei to embadìn eleôjerwn shmeðwn se mia sqèsh pou perilambˆnei tic embadikèc suntetagmènec kai twn tri n eleôjerwn shmeðwn wc proc ta shmeða anaforˆc. S ABY = 1 S OUV S OUA S OV A 1 S OUB S OV B 1 S OUY S OV Y 1

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 42 IsodÔnama grˆfetai: S ABY = 1 S OUV (S OAUB S OV Y + S OBV A S OUY ) + S OAB. H perioristik sunj kh thc perðptwshc aut c eðnai ìti ta shmeða O, U, V den eðnai suggrammikˆ. Apìdeixh O Y U W V Sq ma 2.33: 'Opwc eðqame apodeðxei sthn arq thc enìthtac 2.1, gia tèssera eleôjera shmeða A, B, Y, O isqôei S ABY = S ABO + S AOY + S OBY dhlad S ABY = S OAB + S OBY S OAY. T ra qreiˆzetai na upologðsoume mìno ta S OBY kai S OAY. PaÐrnoume W na eðnai to shmeðo tom c twn UV kai OY. Tìte apì to l mma 2.4.14 (selðda 37) èqoume: S OBW = 1 S OUY V (S OBV S OUY + S OBU S OY V ). Apì thn basik prìtash B.2 (selðda 14) paðrnoume: S OBY S OBW Epomènwc èqoume: S OBY = 1 S OUV (S OBV S OUY + S OBU S OY V ) (1) = S OUY V S OUV. An t ra OY UV, h parapˆnw apìdeixh apotugqˆnei. Parìlo autˆ mporoôme na upologðsoume thn (1) c ex c: Epeid OY UV, S OUY = S OV Y = S OY V.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 43 H (1) gðnetai: S OBY = 1 S OUV S OUY (S OBV S OBU ) = S OUY S OUV S OUBV to opoðo eðnai isodônamo me S OBY S OUY = S OUBV S OUV. An OY BU, tìte kai oi duo merièc tou parapˆnw tôpou isoôntai me thn monˆda. An OY kai BU tèmnontai sto shmeðo M, tìte apì tic protˆseic B.1, B.2 (selðda 14), B.3 (selðda 15) paðrnoume: S OBY S OUY = BM UM = S V BM S V UM = S V BU S V MU S V UO = S V BU S V OU S OUV = S OUBV S OUV ApodeÐxame dhlad ìti h (1) eðnai alhj c kˆtw apì thn sunj kh ìti ta O, U, V den eðnai suggrammikˆ. 'Omoia èqoume: S OAY = 1 S OUV (S OAV S OUY + S OAU S OY V ) (2) Antikajist ntac thn (1) kai (2) sthn S ABY = S OAB + S OBY S OAY kai shmei nontac ìti S OAU S OBU = S OAUB kai S OBV S OAV = S OBV A prokôptei to zhtoômeno. EFARMOGES TWN TEQNIKWN APALOIFHS BOHJHTIK- WN SHMEIWN PARADEIGMA 1 'Estw trapèzio ABCD. H parˆllhlh proc th bˆsh tou trapezðou tèmnei tic duo pleurèc tou kai tic diagwnðouc tou sta shmeða H, G, F kai E. IsqÔei ìti: EF = HG. AB AB MporoÔme na perigrˆyoume thn prìtash wc ex c: K 1 : (P OINT A) 'Estw aujaðreto shmeðo A. K 2 : (P OINT B) 'Estw aujaðreto shmeðo B.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 44 K 3 : (P OINT C) 'Estw aujaðreto shmeðo C. K 4 : (ON P LINE D C A B) 'Estw D shmeðo pˆnw sthn eujeða pou dièrqetai apì to C kai eðnai parˆllhlh sthn eujeða AB. K 5 : (ON LINE E B C) 'Estw E shmeðo pˆnw sthn eujeða BC. K 6 : (INT ER H (LINE A D)(P LINE E A B)) 'Estw H to shmeðo tom c twn eujei n AD kai thc eujeðac pou pernˆ apì to shmeðo E kai eðnai parˆllhlh sthn AB. K 7 : (INT ER F (LINE B D)(LINE E H)) 'Estw F to shmeðo tom c twn eujei n BD kai EH. K 8 : (INT ER G (LINE A C)(LINE E F )) 'Estw G to shmeðo tom c twn eujei n AC kai EF. Sumpèrasma: EF AB HG AB = 1 D C H G F E A B Sq ma 2.34: H apìdeixh pragmatopoieðtai wc ex c: Apaloif shmeðou G (Kataskeu K 8 )(Qr sh l mmatoc 2.4.7 (selðda 33): HG = S ACH AB S ABC ) EF AB HG AB = EF S ABC AB S ACH

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 45 Apaloif shmeðou F (Kataskeu K 7 )(Qr sh l mmatoc 2.4.7: EF S BDE S ABD ) AB = EF S ABC AB S ACH = S BDES ABC S ACH S ABD Apaloif shmeðou H (Kataskeu K 6 )(Qr sh l mmatoc 2.4.15 (selðda 37): S ACH = S ACDS ABE S ABD ) S BDE S ABC S ACH S ABD = S BDES ABC ( S ABD ) S ACD S ABE S ABD = S BDES ABC S ACD S ABE Apaloif shmeðou E (Kataskeu K 5 )(Qr sh prìtashc B.2 (selðda 14): S ABE = BE BC S ABC, S BDE = (S BCD BE BC )) S BDE S ABC S ACD S ABE = ( S BE BCD )S BC ABC BE S ACD S ABC BC = S BCD S ACD Apaloif shmeðou D (Kataskeu K 4 )(Qr sh prìtashc B.2: S ACD = ( CD AB S ABC), S BCD = (S ABC CD AB )) S BCD S ACD = S CD ABC AB CD S ABC AB = 1 PARADEIGMA 2 An to trðgwno LMN eðnai to trðgwno tou Ceva wc proc to shmeðo S tou trig nou ABC, tìte isqôei ìti: S AMLS BNM S CLN S ANL S BLM S CNM = 1. SHMEIWSH: 'Estw ABC trðgwno kai S to shmeðo tom c twn eujei n pou en noun tic korufèc tou trig nou me tic pleurèc pou brðskontai apènanti twn koruf n. 'Estw M to shmeðo tom c twn eujei n BS kai AC, L to shmeðo tom c twn eujei n AS kai BC kai N to shmeðo tom c twn eujei n CS kai AB, tìte to eujôgrammo tm ma ML onomˆzetai eujôgrammo tm ma tou Ceva kai to trðgwno MLN onomˆzetai trðgwno tou Ceva wc proc to shmeðo S gia to trðgwno ABC. MporoÔme na perigrˆyoume thn prìtash wc ex c:

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 46 A N M S B L C Sq ma 2.35: K 1 : (P OINT A) 'Estw aujaðreto shmeðo A. K 2 : (P OINT B) 'Estw aujaðreto shmeðo B. K 3 : (P OINT C) 'Estw aujaðreto shmeðo C. K 4 : (P OINT S) 'Estw aujaðreto shmeðo S. K 5 : (INT ER L (LINE B C)(LINE S A)) 'Estw L to shmeðo tom c twn eujei n BC kai SA. K 6 : (INT ER M (LINE A C)(LINE S B)) 'Estw M to shmeðo tom c twn eujei n AC kai SB. K 7 : (INT ER N (LINE A B)(LINE S C)) 'Estw N to shmeðo tom c twn eujei n AB kai SC. Sumpèrasma: S AMLS BNM S CLN S ANL S BLM S CNM = 1 H apìdeixh pragmatopoieðtai me efarmog tou l mmatoc 2.4.14 (selðda 37) kai stic treðc peript seic apaloif c bohjhtik n shmeðwn wc ex c: Apaloif shmeðou N (Kataskeu K 7 ): (S ALN S CSM S ABC S ACBS, S BMN = S BCSS ABM S ACBS, S CLN = S CSLS ABC S ACBS ) = S ACSS ABL S ACBS, S CMN = S AML S BNM S CLN S ANL S BLM S CNM = ( S CSLS ABC )( S BCS S ABM )S ALM S ACBS ( S ACBS ) ( S CSM S ABC )S BLM ( S ACS S ABL )S ACBS ( S ACBS ) = S CSLS BCS S ABM S ALM S CSM S BLM S ACS S ABL

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 47 Apaloif shmeðou M (Kataskeu K 6 ): (S BLM S BCS S ACS S ABCS, S ALM = S ACLS ABS S ABCS, S ABM = S ABSS ABC S ABCS ) = S BSLS ABC S ABCS, S CSM = S CSL S BCS S ABM S ALM S CSM S BLM S ACS S ABL = S CSLS BCS S ABS S ABC ( S ACL S ABS )( S ABCS )S ABCS S BCS S ACS S BSL S ABC S ACS S ABL (S ABCS ) 2 = S CSL(S ABS ) 2 S ACL S BSL (S ACS ) 2 S ABL Apaloif shmeðou L (Kataskeu K 5 ): (S ABL S BCS S ABS S ABSC, S ACL = S ACSS ABC S ABSC, S CSL = S BCSS ACS S ABSC ) = S ABSS ABC S ABSC, S BSL = S CSL (S ABS ) 2 S ACL S BSL (S ACS ) 2 S ABL = S BCSS ACS (S ABS ) 2 ( S ACS S ABC )( S ABSC ) 2 (S ACS ) 2 S BCS S ABS ( S ABS S ABC )( S ABSC = 1 ) 2 APALOIFES SHMEIWN POU BRISKONTAI PANW SE DUO EUJEIES H teqnik aut efarmìzetai mìno ìtan upˆrqoun toulˆqiston pènte eleôjera hmi-eleôjera shmeða pˆnw se duo eujeðec L 1 kai L 2. 'Otan èqoume mia gewmetrik posìthta G kai L 1, L 2 duo eujeðec, duo sônola dhlad pou perièqoun suggrammikˆ shmeða, tìte qrhsimopoioôme ta parakˆtw l mmata. LHMMA 2.5.4 An G=S ABC S ABC = 0, an A, B, C suggrammikˆ; 1 h 2 ABC, an A L 1 (L 2 ) kai B, C L 2 (L 1 ) ; 1 h 2 BCA, an B L 1 (L 2 ) kai A, C L 2 (L 1 ); 1 h 2 CAB, an C L 1 (L 2 ) kai B, A L 2 (L 1 ). 'Opou h A eðnai h prosanatolismènh apìstash apì to A sto BC kai anˆloga ta h B, h C Perioristikèc sunj kec: Gia thn pr th perðptwsh den upˆrqoun. Gia thn deôterh perðptwsh B C.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 48 Gia thn trðth perðptwsh A C. Gia thn tètarth perðptwsh A B. LHMMA 2.5.5 An G= AB CD G=[ AB CD ] = AB CD Spˆzoume mia gewmetrik posìthta ston lìgo duo posot twn. Perioristikèc sunj kec: C D. LHMMA 2.5.6 An G=AB AB = OB OA, an L 1, L 2 tèmnontai sto O; O 1 B O 1 A, an L 1 L 2 kai A L 1 kai B L 1 ; O 2 B O 2 A, an L 1 L 2 kai A L 2 kai B L 2. 'Opou O 1 kai O 2 eðnai stajerˆ shmeða pˆnw stic L 1 kai L 2 antðstoiqa. Perioristikèc sunj kec: Gia thn pr th perðptwsh L 1 L 2, O B, O A, A B. Gia thn deôterh perðptwsh O 1 B, O 1 A, A B. Gia thn trðth perðptwsh O 2 B, O 2 A, A B. LHMMA 2.5.7 An G=h A h A = { a, an L 1 L 2 ; boa, an L 1 L 2.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 49 'Opou a eðnai h apìstash metaxô twn L 1 kai L 2 kai b=sin( (L 1, L 2 )). Ta a kai b eðnai eleôjeroi parˆmetroi. Perioristikèc sunj kec: Gia thn pr th perðptwsh den upˆrqoun. Gia thn deôterh perðptwsh O A. 2.6 ALGORIJMOS Eisˆgoume thn gewmetrik prìtash P, h opoða an kei sthn kathgorða C H kai eðnai diatupwmènh wc ex c: P=(K 1, K 2,..., K n, S). O algìrijmoc exetˆzei an h prìtash eðnai alhj c yeud c kai an eðnai alhj c parˆgei thn apìdeixh akolouj ntac ta parakˆtw b mata, alli c termatðzei. BHMA 1. Gia i = n,..., 1, xekin ntac dhlad apì to shmeðo pou èqei kataskeuasteð teleutaða katal gwntac sta shmeða pou èqoun eisaqjeð aujaðreta mèsw thc K 1 efarmìzoume to epìmeno b ma. BHMA 2. 'Elegqoc perioristik n sunjhk n twn K i. An o algìrijmoc epibebai nei thn prìtash P, tìte h P eðnai alhj c kˆtw apì tic perioristikèc sunj kec pou autìmata parˆgontai apì ton orismì 2.4.2 (selðda 30), alli c h P eðnai yeud c. BHMA 3. Apì to (S) sumpèrasma thc prìtashc paðrnoume tic gewmetrikèc posìthtec, G 1, G 2,..., G l kai tic fèrnoume sto pr to mèloc me skopì to deôtero mèloc na eðnai monˆda. Gia kajemiˆ apì autèc tic gewmetrikèc posìthtec efarmìzoume to epìmeno b ma. BHMA 4. Gia kajèna apì ta G j,j = 1,..., l, apaloðfoume ta bohjhtikˆ shmeða pou kataskeuˆsthkan apì tic K i.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 50 Gia thn apaloif twn shmeðwn aut n qrhsimopoioôme ta parakˆtw ergaleða: tic protˆseic B.1 (selðda 13) èwc B.6 (selðda 16), tic teqnikèc apaloif c shmeðwn mèsw lìgwn mhk n (selðda 31), tic teqnikèc apaloif c shmeðwn mèsw twn embad n (selðda 34) kai thn teqnik apaloif c shmeðwn pou brðskontai pˆnw se duo eujeðec (selðda 47). Kˆnoume tic apaitoômenec aplopoi seic kai an to apotèlesma thc a- paloif c eðnai mia gewmetrik posìthta G' j apoteloômenh mìno apì eleôjera shmeða, tìte efarmìzoume to l mma 2.5.3 (selðda 41), me skopì na prokôyei mia rht èkfrash anexˆrthtwn metablht n, alli c efarmìzoume apì thn arq to b ma 4. BHMA 5. Katal xame sto na exis soume ton arqikì lìgo gewmetrik n posot twn me mia rht èkfrash anexˆrthtwn metablht n. GÐnontai oi apaitoômenec aplopoi seic kai an katal xoume sthn isìthta thc ekfrashc aut c me th monˆda tìte h prìtash P eðnai alhj c alli c h P eðnai yeud c. Apìdeixh orjìthtac algorðjmou. To sumpèrasma thc prìtashc perilambˆnei thn isìthta gewmetrik n posot twn. Ta duo mèlh thc isìthtac ta sumbolðzoume: A kai B. Opìte h prìtash P mporeð na grafteð isodônama: P=(K 1, K 2,..., K n, (A, B)). 'Etsi loipìn qrhsimopoi ntac ton parapˆnw algìrijmo sthn prìtash P, katal goume sthn tropopoi sh tou sumperˆsmatoc thc prìtashc se mia rht ekfrash gewmetrik n posot twn r = A. Parapˆnw ex ghsh B qreiˆzetai to teleutaðo b ma tou algorðjmou. 'Etsi loipìn an deðxoume ìti: r = 1 tìte h prìtash P eðnai alhj c kˆtw apì tic perioristikèc sunj kec pou parˆgontai mèsw tou orismoô 2.4.2 (selðda 30), kai autì epeid to b ma 4 tou algorðjmou, thc apaloif c dhlad twn bohjhtik n shmeðwn apì tic gewmetrikèc posìthtec, eðnai orjì mìno kˆtw apì autèc tic perioristikèc sunj kec. Na anafèroume ìti oi gewmetrikèc posìthtec pou perilambˆnontai sto r perièqoun eleôjerec paramètrouc, mporoôn e- pomènwc na pˆroun aujaðretec timèc. An broôme timèc gia tic gewmetrikèc posìthtec tètoiec ste ìtan ja tic antikatast soume sto r na d soun apotèlesma r 1, tìte brðskoume èna antiparˆdeigma sômfwna me to opoðo den isqôei h prìtash.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 51 SHMEIWSH: An h prìtash P eðnai yeud c tìte to sumpèrasma thc prìtashc eðnai yeudèc, ektìc apì èna uposônolo thc pou èqei mikrìterh diˆstash. Pio sugkekrimèna, h P mporeð na gðnei alhj c prosjètwntac mia epiplèon sunj kh r = 1 stic eleôjerec gewmetrikèc paramètrouc thc P. Gia parˆdeigma, h prìtash << oi duo diˆmesoi enìc trig nou eðnai Ðsec >>, eðnai yeud c. An ìmwc prosjèsoume mia epiplèon sunj kh stic eleôjerec paramètrouc h prìtash metatrèpetai se alhj c:<< oi duo diˆmesoi enìc trig nou eðnai Ðsec an oi duo apènanti pleurèc tou trig nou eðnai Ðsec >>. Me autìn ton trìpo probˆlontai teqnikèc eôreshc nèwn jewrhmˆtwn. 2.7 EIDIKES TEQNIKES APALOIFH- S SHMEIWN-SUSTHMA (GIB) Ta l mmata 2.4.3-2.5.3 (selðdec 31-41) twn teqnik n apaloif c pou parousiˆsame sthn prohgoômenh enìthta, apoteloôn tic genikèc peript seic apaloif n. An basizìmastan mìno se autˆ ta l mmata, tìte oi apodeðxeic twn gewmetrik n jewrhmˆtwn ja tan polô megˆlec se èktash. Me skopì na mei soume thn èktash twn apodeðxewn ja eisˆgoume tic teqnikèc apaloif c shmeðwn se eidikèc peript seic. Prin parousiˆsoume merikèc apì tic teqnikèc autèc eðnai anˆgkh na shmei soume ìti prin proboôme sthn efarmog twn eidik n teqnik n a- paloif c prèpei pr ta na sulleqjoôn apì tic kataskeuèc ìla ta suggrammikˆ shmeða kai oi parˆllhlec eujeðec, ètsi ste na mporeð o algìrijmoc na efarmìsei tic teqnikèc autìmata. Gia ton skopì autì eisˆgoume èna eidikì sôsthma, thn gewmetrik bˆsh plhrofori n (GIB), h opoða basðzetai sthn kataskeuastik perigraf thc prìtashc kai saf c emploutðzetai me ìla ta suggrammikˆ shmeða kai tic parˆllhlec eujeðec pou perièqei h prìtash. To (GIB) eðnai èna polô isqurì sôsthma sthn autìmath paragwg gewmetrik n protˆsewn. Eidikèc peript seic tou l mmatoc 2.4.14 (selðda 37). An Γ = S ABY kai K=(INT ER Y (LINE P Q)(LINE U V )) PERIPTWSH 1. An AB UV tìte S ABY = S ABU.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 52 P P P B U A Y B V Q U A Y V Q U Y A V Q B P A B A P U Y V U Y V Q B Q Sq ma 2.36: Apìdeixh: AfoÔ to Y brðsketai pˆnw sthn UV tìte AB UY ˆra apì thn prìtash B.4 (selðda 15) isqôei S ABY = S ABU. PERIPTWSH 2. An AB P Q tìte S ABY = S ABP. Apìdeixh: 'Omoia me thn perðptwsh 1. PERIPTWSH 3. An U, V, A suggrammikˆ tìte S ABY = S UBV S AP Q S UP V Q. Apìdeixh: PaÐrnoume ton lìgo S ABY S UBV. AfoÔ ta U, V, A eðnai suggrammikˆ apì thn prìtash B.1 (selðda 13)ja èqoume S ABY S UBV = AY kai tèloc apì UV to l mma 2.4.7 (selðda 33) èqoume ìti AY = S AP Q UV S UV P Q. H zhtoômenh sqèsh plèon eðnai emfan c. PERIPTWSH 4. An U, V, B suggrammikˆ tìte S ABY = S AUV S BP Q S UP V Q. Apìdeixh: 'Omoia me thn perðptwsh 3. PERIPTWSH 5. An P, Q, A suggrammikˆ tìte S ABY = S P BQS AUV S P UQV. Apìdeixh: PaÐrnoume ton lìgo S ABY S P BQ. AfoÔ ta P, Q, A eðnai suggrammikˆ apì thn prìtash B.1 (selðda 13)ja èqoume S ABY S P BQ = Y A kai tèloc apì P Q

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 53 to l mma 2.4.7 (selðda 33) èqoume ìti Y A UV plèon eðnai emfan c. = S AUV S P UQV. H zhtoômenh sqèsh PERIPTWSH 6. An P, Q, B suggrammikˆ tìte S ABY = S AP QS BUV S P UQV. Apìdeixh: AkoloujoÔme thn Ðdia diadikasða me thn perðptwsh 5. PERIPTWSH 7. An to U to V brðskontai pˆnw sthn AB tìte S ABY = S UP QS ABV S V P Q S ABU S UP V Q. Apìdeixh: An U brðsketai pˆnw sthn AB tìte apì thn prìtash B.3 (selðda 15) paðrnoume: S ABY = V Y S UV ABU + Y U S UV ABV. 'Omwc apì prìtash B.2 (selðda 14) èqoume: UY V Y UV UV = S UP Q S UP V Q = S V P Q S UP V Q PERIPTWSH 8. Se kˆje ˆllh perðptwsh S ABY = S P UV S ABQ +S QV U S ABP S UP V Q Genik perðptwsh (bl. apìdeixh l mmatoc 2.4.14). Eidikèc peript seic tou l mmatoc 2.4.15 (selðda 37). An Γ = S ABY kai K=(INT ER Y (LINE U V )(P LINE R P Q)) U U A V V Y R B Y R P Q P Q U A U B R V Y B R V S Y B A P Q A P Q Sq ma 2.37: PERIPTWSH 1. An AB UV tìte S ABY = S ABU.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 54 Apìdeixh: AfoÔ to Y brðsketai pˆnw sthn UV tìte AB UY ˆra apì thn prìtash B.4 (selðda 15) isqôei S ABY = S ABU. PERIPTWSH 2. An AB P Q tìte S ABY = S ABR. Apìdeixh: AfoÔ to Y brðsketai pˆnw sthn eujeða R kai AB P Q tìte RY AB. Opìte apì thn prìtash B.4 (selðda 15) isqôei S ABY = S ABU. PERIPTWSH 3. An U, V, A suggrammikˆ tìte S ABY = S UBV S AP RQ S UP V Q. Apìdeixh: PaÐrnoume ton lìgo S ABY S UBV. AfoÔ ta U, V, A eðnai suggrammikˆ apì thn prìtash B.1 (selðda 13) èqoume S ABY S UBV = AY kai tèloc apì UV to l mma 2.4.8 (selðda 34) èqoume ìti AY = S AP RQ UV S UP V Q. H zhtoômenh sqèsh plèon eðnai emfan c. PERIPTWSH 4. An U, V, B suggrammikˆ tìte S ABY = S AUV S BP RQ S UP V Q. Apìdeixh: 'Omoia me thn perðptwsh 3. PERIPTWSH 5. An AY P Q tìte S ABY = S BQRP S AUV S P UQV. Apìdeixh: PaÐrnoume RS = P Q sunep c to SRP Q eðnai parallhlìgrammo. Apì thn prìtash B.5 (selðda 16 èqoume S BQRP = S BQP S RQP = S BRS. PaÐrnoume ton lìgo S ABY S BRS, sômfwna me thn prìtash B.1 (selðda 13) èqoume S ABY. Tèloc apì to l mma 2.4.8 (selðda 34) èqoume ìti Y A RS = S AUV S RUSV S BRS = AY RS = S AUV S P UQV. PERIPTWSH 6. An BY P Q tìte S ABY = S AP RQS BUV S P UQV. Apìdeixh: AkoloujoÔme thn Ðdia diadikasða me thn perðptwsh 5. PERIPTWSH 7. Se kˆje ˆllh perðptwsh S ABY = S P UQRS ABV +S P RQV S ABU S P UQV Genik perðptwsh (bl. apìdeixh l mmatoc 2.4.15).

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 55 Eidikèc peript seic tou l mmatoc 2.4.16 (selðda 38). R W P U A Y Q V B W R A P U Q B V Y W U R B A S Y V P Q Sq ma 2.38: An Γ = S ABY kai K=(INT ER Y (LINE R P Q)(P LINE W U V )) PERIPTWSH 1. An AB UV tìte S ABY = S ABW. Apìdeixh: AfoÔ to Y brðsketai pˆnw sthn U V kai AB U V tìte W Y AB. 'Ara apì thn prìtash B.4 (selðda 15) isqôei S ABY = S ABW. PERIPTWSH 2. An AB P Q tìte S ABY = S ABR. Apìdeixh: AfoÔ to Y brðsketai pˆnw sthn eujeða R kai AB P Q tìte RY AB. Opìte apì thn prìtash B.4 (selðda 15) isqôei S ABY = S ABU. PERIPTWSH 3. An AY P Q tìte S ABY = S AUW V S BQRP S P UQV. Apìdeixh: PaÐrnoume RS = P Q sunep c to SRP Q eðnai parallhlìgrammo. Apì thn prìtash B.5 (selðda 16) èqoume S BQRP = S BQP S RQP = S BRS. PaÐrnoume ton lìgo S ABY S BRS, sômfwna me thn prìtash B.1 (selðda 13) èqoume S ABY S BRS = AY. Tèloc apì to l mma 2.4.9 (selðda 34) RS èqoume ìti Y A = S AUW V RS S RUSV = S AUV S P UQV. PERIPTWSH 4. An BY P Q tìte S ABY = S BUW V S AP RQ S P UQV.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 56 Apìdeixh: 'Omoia me thn perðptwsh 3. PERIPTWSH 5. An AY UV tìte S ABY = S BV W U S AP RQ S UP V Q. Apìdeixh: 'Omoia me thn perðptwsh 3. PERIPTWSH 6. An BY UV tìte S ABY = S AUW V S BP RQ S UP V Q. Apìdeixh: AkoloujoÔme thn Ðdia diadikasða me thn perðptwsh 3. PERIPTWSH 7. Se kˆje ˆllh perðptwsh S ABY = S P W QRS AUBV S P UQV +S ABW Genik perðptwsh (bl. apìdeixh l mmatoc 2.4.16). EFARMOGH TOU SUSTHMATOS GIB. PARADEIGMA 'Estw duo parˆllhlec eujeðec AB kai CD. 'Estw P to shmeðo tom c twn eujei n AC kai BD. 'Estw Q to shmeðo tom c twn eujei n AD kai BC. 'Estw M to shmeðo tom c twn eujei n P Q kai AB. IsqÔei ìti to shmeðo M eðnai mèson thc AB. DIATUPWSH K 1 : (P OINT S A B P ): 'Estw trða aujaðreta shmeða A, B, P K 2 : (ON LINE C A P ): 'Estw C shmeðo pˆnw sthn eujeða AP. K 3 : (INT ER D (LINE B P )(P LINE C A B)): 'Estw D to shmeðo tom c thc eujeðac BP kai thc eujeðac pou pernˆ apì to shmeðo C kai eðnai parˆllhlh sthn AB. K 4 : (INT ER Q (LINE A D)(LINE B C)): 'Estw Q to shmeðo tom c twn eujei n AD kai BC.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 57 K 5 : (INT ER M (LINE A B)(LINE P Q)): 'Estw M to shmeðo tom - c twn eujei n AB kai P Q. S: IsqÔei: AM = BM. Sullog ìlwn twn suggrammik n shmeðwn kai twn parˆllhlwn eujei n: (M, P, Q)(Q, A, D)(Q, B, C)(M, A, B)(D, P, B)(C, A, P ) kai DC MAB. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, M, Q, D, C, P, B, A ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: AD BC, AB P Q kai A P, B M.. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = AM BM = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou M, qrhsimopoioôme thn prìtash B.2 (selðda 14) ) ( AM BM = S AP Q S BP Q ) AM BM = S AP Q S BP Q (Apaloif shmeðou Q, qrhsimopoioôme thn trðth kai thn èkth eidik perðptwsh teqn. apaloif c tou l mmatoc 2.4.14 (selðda 37) ) (S BP Q = S BP CS ABD S ABCD, S AP Q = S AP DS ABC S ABDC )

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 58 = S AP DS ABC ( S ABDC ) ( S BP C S ABD )S ABDC = S AP DS ABC S BP C S ABD (Apaloif shmeðou D, qrhsimopoioôme thn pr th kai thn tètarth eidik perðptwsh teqn. apaloif c tou l mmatoc 2.4.15 (selðda 37) ) (S ABD = S ABC, S AP D = S ABP S P ACB S BAP B = S P ACB, S P ACB = S P AC + S P CB = S BP C, epeid (C, A, P ) suggrammikˆ) = S BP CS ABC S BP C S ABC BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. =1. 2.8 EFARMOGES TOU ALGORIJMOU PARADEIGMA 1 'Estw A 1, B 1, C 1 kai D 1 shmeða pˆnw stic pleurèc CD, DA, AB kai BC antðstoiqa enìc parallhlogrˆmmou ABCD tètoia ste CA 1 = DB 1 = CD DA AC 1 = BD 1 = 1. Tìte to embadìn tou trig nou ABA AB BC 3 2 eðnai ta trða dekatotrðta tou embadoô tou parallhlogrˆmmou ABCD. DIATUPWSH PROTASHS K 1 (P OINT A): 'Estw A aujaðreto shmeðo. K 2 (P OINT B): 'Estw B aujaðreto shmeðo. K 3 (P OINT C): 'Estw C aujaðreto shmeðo. K 4 (P RAT IO D C A B 1): 'Estw shmeðo D pˆnw sthn eujeða pou pernˆ apì to C kai eðnai parˆllhlh sthn eujeða AB tètoia ste CD AB = 1.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 59 D A 1 C B 1 A2 D 2 C 2 D 1 A C 1 B 2 B Sq ma 2.39: K 5 (LRAT IO A 1 C D 1 3 ): 'Estw shmeðo A 1 pˆnw sthn CD tètoio ste 1 = A 1C. 3 CD K 6 (LRAT IO B 1 D A 1 3 ): 'Estw shmeðo B 1 pˆnw sthn DA tètoio ste 1 = DB. 1 3 DA K 7 (INT ER A 2 (LINE A A 1 )(LINE B B 1 )): 'Estw A 2 to shmeðo tom - c twn eujei n AA 1 kai BB 1. S: S ABA 2 S ABCD = 3 13 ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, A 2, B 1, A 1, D, C, B, A ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n:c D, D A, A A 1, B B 1, AA 1 BB 1, A B. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = 3S ABCD 13S ABA2 kai ektel ta parakˆtw b mata.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 60 BHMA 4. Apaloif bohjhtik n shmeðwn apì tic gewmetrikèc posìthtec. (Apaloif shmeðou A 2, qrhsimopoioôme th prìtash B.2 (selðda 14) ) (S ABA2 = S ABB 1 S ABA1 S ABA1 B 1 ) S ABA2 S ABCD = 3S ABCDS ABA1 B 1 13S ABA1 S ABB1 (Apaloif shmeðou B 1, qrhsimopoioôme th prìtash B.3 (selðda 15) ) (S ABB1 = 2 3 S ABD, S ABA1 B 1 = 2S ADA 1 3S ABA1 3 ) 3S ABCD S ABA1 B 1 13S ABA1 S ABB1 = 3 13 S ABCD(S ABA1 2 3 S ADA 1 ) 2 3 S ABDS ABA1 (Apaloif shmeðou A 1, qrhsimopoioôme th prìtash B.3 ) (S ABA1 = 1 3 (S ABD + 2S ABC ), S ADA1 = 2 3 S ACD) 3 13 S ABCD(S ABA1 2 3 S ADA 1 ) 2 = ( 2 3 S ABC+ 1 3 S ABD+ 4 9 S ACD)( 9 26 S ABCD) 3 S ABDS ABA1 S ABD ( 2 3 S ABC+ 1 3 S ABD) (Apaloif shmeðou D, qrhsimopoioôme th prìtash B.4 (selðda 15) ) (S ABD = S ABC, S ACD = S ABC, S ABCD = 2S ABC ) ( 2 3 S ABC+ 1 3 S ABD+ 4 9 S ACD)( 9 26 S ABCD) S ABD ( 2 3 S ABC+ 1 3 S ABD) = ( 9 13 S ABC)( 13 6 S ABC) S ABC ( 3 2 S ABC) BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. ( 9 13 S ABC)( 13 6 S ABC) S ABC ( 3 2 S ABC) =1 PARADEIGMA 2

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 61 C D K E A F B G Sq ma 2.40: 'Estw trða shmeða A, B, C sto epðpedo kai E, D shmeða pˆnw stic eujeðec BC kai AC antðstoiqa.'estw K, F kai G ta shmeða tom c twn eujei n AE me BD, CK me AB kai DE me AB antðstoiqa. Tìte AF = AG BF GB DIATUPWSH K 1 (P OINT A): 'Estw A aujaðreto shmeðo. K 2 (P OINT B): 'Estw B aujaðreto shmeðo. K 3 (P OINT C): 'Estw C aujaðreto shmeðo. K 4 (ON LINE E B C): 'Estw E shmeðo pˆnw sthn BC K 5 (ON LINE D A C): 'Estw D shmeðo pˆnw sthn AC K 6 (INT ER K (LINE A E)(LINE B D): 'Estw K to shmeðo tom c twn eujei n AE kai BD. K 7 (INT ER F (LINE C K)(LINE A B)): 'Estw F to shmeðo tom c twn eujei n CK kai AB. K 8 (INT ER G (LINE D E)(LINE A B)): 'Estw G to shmeðo tom c twn eujei n DE kai AB.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 62 S: AF BF AG BG = 1. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, G, F, K, D, E, C, B, A ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: DE AB, CK AB, AE BD, A C, B C, A G, B F. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = AF BF AG BG = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou G, qrhsimopoioôme th prìtash B.2 (selðda 14) ) ( AG BG = S AED S BED ) AF BF AG BG = AF BF S AED S BED (Apaloif shmeðou F, qrhsimopoioôme th prìtash B.2) ( AF BF = S ACK S BCK ) AF BF S AED S BED = S BED S AED AF BF = S BEDS ACK S AED S BCK (Apaloif shmeðou K, qrhsimopoioôme tic protˆseic B.2 kai B.3 (selðda 15) ) (S BCK = S BCDS ABE S ABED, S ACK = S ACES ABD S ABED )

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 63 S BEDS ACK S AED S BCK (Aplopoi seic) = S S ACE S ABD BED S ABED S S BCD S ABE AED S ABED S S ACE S ABD BED S ABED S S BCD S ABE AED S ABED = S BEDS ACES ABD S AED S BCD S ABE (Efarmog thc teqnik c apaloif c shmeðwn pˆnw se duo eujeðec (selðda 48) ) (S ABE = 1 2 (BEACb), S BCD = 1 2 (CDBCb) S AED = 1 2 (CEADb), S BED = 1 2 (CDBEb) S ABD = 1 2 (BCADb), S ACE = 1 2 (CEACb)) S BEDS ACE S ABD S AED S BCD S ABE = ( 1 ( 1 2 2 CDBEb)( 1 2 BCADb)( 1 2 CEACb) 1 1 BEACb)( CDBCb)( 2 2 CEADb) BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. =1. PARADEIGMA 3 'Estw ABCD parallhlìgrammo kai èstw P, Q, R, S shmeða pˆnw stic pleurèc AB, BC, CD, DA tètoia ste: AP = CR kai BQ = DS. IsqÔei ìti to P QRS eðnai epðshc parallhlìgrammo. DIATUPWSH K 1 (P OINT A): 'Estw A aujaðreto shmeðo. K 2 (P OINT B): 'Estw B aujaðreto shmeðo. K 3 (P OINT C): 'Estw C aujaðreto shmeðo.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 64 A S D R P B Q C Sq ma 2.41: K 4 (P RAT IO D A B, C, 1): 'Estw ìti to shmeðo D an kei pˆnw sthn eujeða pou pernˆ apì to shmeðo A kai eðnai parˆllhlh thn eujeða BC tètoio ste AD = BC. K 5 (LRAT IO S D A r 2 ): 'Estw S shmeðo pˆnw sthn eujeða DA tètoio ste SD = r 2 DA. K 6 (LRAT IO P A B r 1 ): 'Estw P shmeðo pˆnw sthn eujeða AB tètoio ste P A = r 1 AB. K 7 (LRAT IO R C D r 1 ): 'Estw R shmeðo pˆnw sthn eujeða CD tètoio ste RC = r 1 CD. K 8 (LRAT IO Q B C r 2 ): 'Estw Q shmeðo pˆnw sthn eujeða BC tètoio ste QB = r 2 BC. S: RQ = SP. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, P, Q, R, S, D, C, B, A ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n:

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 65 C D, B C, D A, A B. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = RQ SP = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou Q, qrhsimopoioôme to l mma 2.4.3 (selðda 31) ) ( RQ SP = S BCR S BSCP ) RQ SP = S BCR S BSCP (Apaloif shmeðou R, qrhsimopoioôme to l mma 2.4.10 (selðda 35) ) (S BCR = r 1 S BCD ) = r 1S BCD S BSCP (Apaloif shmeðou P, qrhsimopoioôme to l mma 2.4.10 (selðda 35) ) (S BSCP = (S BCS + r 1 S ABC S ABC )) = r 1 S BCD S BCS r 1 S ABC +S ABC (Apaloif shmeðou S, qrhsimopoioôme to l mma 2.4.10 (selðda 35) ) (S BCS = (r 2 S BCD S BCD r 2 S ABC )) = r 1 S BCD r 2 S BCD +S BCD +r 2 S ABC +r 1 S ABC S ABC (Apaloif shmeðou D, qrhsimopoioôme th prìtash B.4 (selðda 15) ) (S BCD = S ABC )

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 66 = r 1S ABC r 1 S ABC BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. =1. PARADEIGMA 4 (JEWRHMA TOU PASCAL) 'Estw A, B, C, D, F, E èxi shmeða tou epipèdou. 'Estw P, Q, S trða suggrammikˆ shmeða. 'Estw P to shmeðo tom c twn eujei n AB, DE, Q to shmeðo tom c twn eujei n BC, EF kai S to shmeðo tom c twn eujei n CD, F A. 'Estw P 1 to shmeðo tom c twn eujei n AC kai DE, Q 1 to shmeðo tom c twn eujei n BE, CF kai S 1 to shmeðo tom c twn eujei n AB, F D. Ta shmeða P 1, Q 1, S 1 eðnai suggrammikˆ. E Q 1 S F B A P 1 C Q D S 1 Sq ma 2.42: DIATUPWSH PROTASHS K 1 (P OINT A): 'Estw A aujaðreto shmeðo. K 2 (P OINT B): 'Estw B aujaðreto shmeðo. K 3 (P OINT D): 'Estw D aujaðreto shmeðo. P

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 67 E Z 2 Z 1 B F C S Q A P 1 D S 1 P Sq ma 2.43: K 4 (P OINT E): 'Estw E aujaðreto shmeðo. K 5 (P OINT S): 'Estw S aujaðreto shmeðo. K 6 (ON LINE C D S): 'Estw C shmeðo pˆnw sthn DS K 7 (INT ER P (LINE A B)(LINE D E): 'Estw P to shmeðo tom c twn eujei n AB kai DE. K 8 (INT ER Q (LINE B C)(LINE S P )): 'Estw Q to shmeðo tom c twn eujei n BC kai SP. K 9 (INT ER F (LINE E Q)(LINE A S)): 'Estw F to shmeðo tom c twn eujei n EQ kai AS. K 10 (INT ER S 1 (LINE D F )(LINE A B)): 'Estw S 1 to shmeðo tom - c twn eujei n DF kai AB. K 11 (INT ER P 1 (LINE A C)(LINE D E)): 'Estw P 1 to shmeðo tom - c twn eujei n AC kai DE.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 68 K 12 (INT ER Z 2 (LINE B E)(LINE S 1 P 1 )): 'Estw Z 2 to shmeðo tom - c twn eujei n BE kai S 1 P 1. K 13 (INT ER Z 1 (LINE C F )(LINE S 1 P 1 )): 'Estw Z 1 to shmeðo tom - c twn eujei n CF kai S 1 P 1. S: S 1Z 1 P 1 Z 1 = S 1Z 2 P 1 Z 2. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, Z 1, Z 2, P 1, S 1, F, Q, P, C, S, D, E, A, B ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: DE AB, BC SP, EQ AS, DF BA, AC DE, CF BE, BE S 1 P 1, CF S 1 P 1, D S, P 1 Z 1, S 1 Z 2. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = S 1Z 1 P 1 Z 1 P 1 Z 2 S 1 Z 2 = 1. kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou Z 1, qrhsimopoioôme th prìtash B.2 (selðda 14) ) ( S 1Z 1 P 1 Z 1 = S CF S 1 S CF P1 ) S 1 Z 1 P 1 Z 1 P 1 Z 2 S 1 Z 2 = S CF S 1 S CF P1 P 1 Z 2 S 1 Z 2 ) (Apaloif shmeðou Z 2, qrhsimopoioôme th prìtash B.2, B.3 (selðda 15) ( P 1Z 2 S 1 Z 2 = S BEP 1 S BES1 ) S CF S1 S CF P1 P 1 Z 2 S 1 Z 2 = S CF S 1 S BEP1 S BES1 S CF P1

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 69 (Apaloif shmeðou P 1, qrhsimopoioôme th prìtash B.2, B.3) (S CF P1 = S DECS ACF S ADCE, S BEP1 = S AECS BDE S ADCE ) = (S CF S 1 S BDE S AEC ) S ADCE S ADCE S BES1 S ACF S DEC (Aplopoi seic) = S BDE S CF S1 S AEC S BES1 S ACF S DEC (Apaloif shmeðou S 1, qrhsimopoioôme th prìtash B.2, B.3) (S BES1 = S BDF S BAE S BDAF, S CF S1 = S DCF S BAF S BDAF ) = ( S BDE S AEC S BAF S DCF )S BDAF S BDAF ( S ACF S DEC S BAE S BDF ) (Aplopoi seic) = S BDE S AEC S BAF S DCF S ACF S DEC S BAE S BDF (Apaloif shmeðou F, qrhsimopoioôme th prìtash B.2, B.3) (S BDF = S ACQS BDE +S AES S BDQ S AESQ, S ACF = S ASCS AEQ S AESQ S BAF = S AEQS BAS S AESQ, S DCF = S ESQS ADC S AESQ ) = S BDE S AEC S BAS S AEQ S ADC S ESQ S AESQ S AESQ S AESQ S AESQ ( S BAE S DEC S AEQ S ASC ) (S BDE S SASQ +S BDQ S AES ) (Aplopoi seic) = S BDE S AEC S BAS S ADC S ESQ S BAE S DEC S ASC (S BDE S ASQ +S BDQ S AES ) (Apaloif shmeðou Q, qrhsimopoioôme th prìtash B.2, B.3)

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 70 (S BDQ = S BSP S BDC S BSCP, S ASQ = S ASP S BSC S BSCP, S ESQ = S ESP S BSC S BSCP ) = ( S BDE S AEC S BAS S ADC S ESP S BSC )S BSCP S BSCP S BAE S DEC S ASC (S BDE S BSC S ASP +S BDC S BSP S AES ) (Aplopoi seic) = S BDE S AEC S BAS S ADC S ESP S BSC S BAE S DEC S ASC (S BDE S BSC S ASP +S BDC S BSP S AES ) (Apaloif shmeðou P, qrhsimopoioôme th prìtash B.2, B.3) (S BSP = S BDES BAS S BDAE, S ASP = S ADES BAS S BDAE, S ESP = S DESS BAE S BDAE ) = ( S BDE S AEC S BAS S ADC S BAE S DES S BSC )S BDAE S BDAE S BAS S BDE S BAE S DEC S ASC ( S BDC S AES S BSC S ADE ) (Aplopoi seic) = S BSC S ADC S AEC S DES S ASC S DEC (S BDC S AES +S BSC S ADE ) (Apaloif shmeðou C, qrhsimopoioôme th prìtash B.3) (S BDC = S BDS DC DS, S ASC = ( DC DS 1)S ADS S DEC = S DES DC DS, S AEC = S AES DC DS + S ADE DC DS S ADE S ADC = S ADS DC DS, S BSC = ( DC DS 1)S BDS) = S DECS BDS ( 1+ DC ) DC S DS DS ADS( S ADE + DC S DS ADE+ DC S DS AES) S ADS ( 1+ DC ) DC S DS DS DESS BDS ( S ADE + DC S DS ADE+ DC S DS AES) BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. =1. PARADEIGMA 5 (JEWRHMA DESARGUES)

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 71 'Estw duo trðgwna ABC kai A 1 B 1 C 1. 'Estw ìti oi eujeðec AA 1, BB 1, CC 1 tèmnontai sto shmeðo S. 'Estw P to shmeðo tom c twn eujei n BC, B 1 C 1, Q to shmeðo tom c twn eujei n CA kai C 1 A 1 kai R to shmeðo tom c twn eujei n AB kai A 1 B 1. Ta shmeða P, Q, R eðnai suggrammikˆ. Q C P C 1 S B B 1 A A 1 R DIATUPWSH Sq ma 2.44: K 1 (P OINT A): 'Estw A aujaðreto shmeðo. K 2 (P ONT B): 'Estw B aujaðreto shmeðo. K 3 (P OINT C): 'Estw C aujaðreto shmeðo. K 4 (P OINT S): 'Estw S aujaðreto shmeðo. K 5 (ON LINE A 1 S A): 'Estw A 1 shmeðo pˆnw sthn SA K 6 (ON LINE B 1 S B): 'Estw B 1 shmeðo pˆnw sthn SB K 7 (ON LINE C 1 S C): 'Estw C 1 shmeðo pˆnw sthn SC K 8 (INT ER P (LINE B 1 C 1 )(LINE B C)): 'Estw P to shmeðo tom - c twn eujei n B 1 C 1 kai BC.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 72 Q C P C 1 S B B 1 A A 1 Z Z 1 2 Sq ma 2.45: K 9 (INT ER Q (LINE A 1 C 1 )(LINE A C)): 'Estw Q to shmeðo tom c twn eujei n A 1 C 1 kai AC. K 10 (INT ER R (LINE A 1 B 1 )(LINE A B)): 'Estw R to shmeðo tom - c twn eujei n A 1 B 1 kai AB. K 11 (INT ER Z 2 (LINE A 1 B 1 )(LINE P Q)): 'Estw Z 2 to shmeðo tom - c twn eujei n A 1 B 1 kai P Q. K 12 (INT ER Z 1 (LINE A B)(LINE P Q)): 'Estw Z 1 to shmeðo tom c twn eujei n AB kai P Q. S: P Z 1 QZ 1 = P Z 2 QZ 2. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, Z 1, Z 2, Q, P, C 1, B 1, A 1, S, C, B, A ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc ikan n kai anagkaðwn sunjhk n twn parapˆnw kataskeu n: BC B 1 C 1, AC A 1 C 1, AB A 1 B 1, AB P Q, A 1 B 1 P Q, Q Z 1, P Z 2, S A, S B, S C.

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 73 BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = P Z 1 QZ 1 QZ 2 P Z 2 = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou Z 1, qrhsimopoioôme th prìtash B.2 (selðda 14) ) ( P Z 1 QZ 1 = S ABP S ABQ ) P Z 1 QZ 1 QZ 2 P Z 2 = S ABP S ABQ QZ 2 P Z 2 (Apaloif shmeðou Z 2, qrhsimopoioôme th prìtash B.2) ( QZ 2 P Z 2 = S A 1 B 1 Q S A1 B 1 P ) S ABP S ABQ QZ 2 P Z 2 = S ABP S ABQ S A1 B 1 Q S A1 B 1 P ) (Apaloif shmeðou Q, qrhsimopoioôme th prìtash B.2, B.6 (selðda 16) (S ABQ = S AA 1 C 1 S ABC S AA1, S A1 B CC 1 Q = S A 1 B 1 C 1 S ACA1 1 S AA1 BB 1 ) = S S A1 B 1 C S 1 ACA1 ABP S AA1 CC 1 S AA1 C S S 1 ABC A1 B 1 P S AA1 CC 1 (Aplopoi seic) = S ABP S A1 B 1 C 1 S ACA1 S A1 B 1 P S AA1 C 1 S ABC (Apaloif shmeðou P, qrhsimopoioôme th prìtash B.2) (S ABP = S BB 1 C 1 S ABC S BB1, S A1 B CC 1 P = S A 1 B 1 C 1 S BCB1 1 S BB1 CC 1 )

KEFŸALAIO 2. H MEJODOS TWN EMBADWN 74 = S S BB1 C S A 1 B 1 C 1 S 1 ABC ACA1 S BB1 CC 1 S A1 B S AA1 C 1 S 1 C S 1 BCB1 ABC S BB1 CC 1 (Aplopoi seic) = S ACA 1 S BB1 C 1 S AA1 C 1 S BCB1 (Apaloif shmeðou C 1, qrhsimopoioôme th prìtash B.2) (S AA1 C 1 = (S ACA1 SC 1 SC ), S BB 1 C 1 = (S BCB1 SC 1 SC )) = S ACA 1 ( SC 1 SC S BCB 1 ) S BCB1 ( SC 1 SC S ACA 1 ) BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. =1.

Kefˆlaio 3 AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA Se autì to kefˆlaio ja parousiˆsoume thn autìmath apìdeixh kataskeuastik n protˆsewn pou perièqoun kˆjetec eujeðec kai kôklouc. To ergaleðo me to opoðo antimetwpðzoume probl mata kajetìthtac eðnai h Pujagìreia diaforˆ. H mèjodoc pou anafèretai se autì to kefˆlaio aforˆ kataskeuastikèc protˆseic sthn metrik gewmetrða. Ja parousiˆsoume thn mèjodo thc autìmathc paragwg c twn apodeðxewn twn eukleðdeiwn gewmetrik n jewrhmˆtwn, pou an koun sthn klˆsh twn grammik n kataskeuastik n gewmetrik n protˆsewn. Mia gewmetrik prìtash eðnai grammik kataskeuastik ìtan h prìtash aut mporeð na perigrafjeð san mia akoloujða shmeðwn, ètsi ste kˆje shmeðo thc akoloujðac na mporeð katˆ monadikì trìpo na kataskeuasteð apì ta prohgoômena shmeða thc akoloujðac. Pio sugkekrimèna, mia gewmetrik prìtash lègetai grammik kataskeuastik an ta shmeða thc prìtashc mporoôn na perigrafoôn sômfwna me tic akìloujec kataskeuèc: - èstw èna eleôjero shmeðo. - èstw aujaðreto shmeðo pˆnw se mia eujeða. - èstw h tom duo eujei n. 75

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA76 - èstw h tom miac eujeðac kai enìc kôklou h tom duo kôklwn ìtan to to ˆllo shmeðo tom c touc èqei dh kataskeuasteð. AxÐzei na shmei soume epðshc ìti h kathgorða C H (selðda 29),twn kataskeuastik n protˆsewn shmeðwn tom c tou Hilbert eðnai upokathgorða thc kathgorðac C L (selðda 103), twn grammik n kataskeuastik n protˆsewn. H apaloif twn shmeðwn pou kataskeuˆzontai stic protˆseic autèc pragmatopoieðtai me thn qr sh èxi basik n gewmetrik n protˆsewn, pou aforoôn tic Pujagìreiec diaforèc trig nwn kai tetrapleôrwn. Sthn parakˆtw enìthta ja parousiˆsoume tic èxi basikèc protˆseic kai èpeita ja d soume trða paradeðgmata me skopì na epexhg soume th mèjodo. 3.1 JEMELIWDHS GEWMETRIKES EN- NOIES (MEROS II) H eisagwg miac nèac gewmetrik c posìthtac, thc Pujagìreiac diaforˆc, ja mac bohj sei sthn epðlush problhmˆtwn pou perilambˆnoun zht mata, ìpwc thn kajetìthta kai thn analogða eujugrˆmmwn tmhmˆtwn. Arqikˆ ja eisˆgoume to je rhma twn epig niwn trig nwn pou ja mac bohj sei stic apodeðxeic jewrhmˆtwn pou aforoôn Pujagìreiec diaforèc kaj c kai sthn metatrop sqèsewn pou perilambˆnoun isìthta gwni n se sqèsh pou perilambˆnei isìthta lìgwn metaxô Pujagìreiwn diafor n kai embad n prosanatolismènwn trig nwn. Prèpei ìmwc na shmei soume ìti to je rhma autì den èqei sumperilhfjeð sto prìgramma tou upologist. 'Estw Ox kai Oy duo hmieujeðec pou den èqoun koinì forèa. JewroÔme to hmiepðpedo p me akm ton forèa thc Ox pou perièqei thn Oy kai to hmiepðpedo q me akm ton forèa thc Oy pou perièqei thn Ox. To sônolo twn koin n shmeðwn twn hmiepipèdwn p kai q onomˆzetai gwnða me koruf to O kai pleurèc tic hmieujeðec Ox, Oy. To sômbolo thc gwnðac eðnai:. Kˆje gwnða èqei monadikì mètro, pou eðnai ènac arijmìc m, me 0 m 180.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA77 O parakˆtw tôpoc upologðzei to embadìn trig nou qrhshmopoi ntac to mètro twn gwni n tou: ABC = 1 AB BC sin( B) 2 = 1 2 AC BC sin( C) = 1 2 AB AC sin( A) An ABC = XY Z ABC + XY Z = 180 tìte onomˆzoume ta trðgwna ABC kai XY Z wc epig nia. PROTASH 3.1.1 (JEWRHMA TWN EPIGWNIWN TRIG- WNWN) An ABC = XY Z ABC + XY Z = 180 tìte èqoume: ABC = AB BC. XY Z XY Y Z Apìdeixh X A B Z C Sq ma 3.1: QwrÐc blˆbh thc genikìthtac, upojètoume ìti to shmeðo B tautðzetai me to shmeðo Y kai ìti to shmeðo Z eðnai pˆnw sthn eujeða BC. Epomènwc afoô ABC = XY Z, tìte to shmeðo X ja eðnai pˆnw sthn AB. Opìte èqoume to apodeiktèo:

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA78 ABC = ABC ABZ = BC AB. XY Z ABZ XY Z BZ XY An t ra èqoume thn perðptwsh: ABC + XY Z = 180 tìte to shmeðo Z ja brðsketai aristerˆ tou shmeðou B pˆnw sthn eujeða BC. X A Z B C Opìte èqoume to apodeiktèo: ABC = ABC XY C = AB BC. XY Z XY C XY Z XY BZ Sq ma 3.2: EFARMOGES THS PROTASHS 1 A. Pˆre tèsseric hmieujeðec pou pernˆne apì to O kai duo eujeðec pou kìboun tic hmieujeðec autèc sta shmeða A, B, C, D kai P, Q, R, S. DeÐxte ìti: AB CD = P Q RS AD BC P S QR Apìdeixh ArkeÐ na deðxoume ìti: AB CD P S QR AD BC P Q RS = 1 Opìte paðrnw: AB CD P S QR = AB CD P S QR AD BC P Q RS AD BC P Q RS

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA79 O A B C D P Q R S Sq ma 3.3: Me qr sh t ra tou jewr matoc twn epðpleurwn trig nwn, oi lìgoi mhk n isoôntai me lìgouc embad n. 'Ara AB CD P S QR = OAB OCD OP S OQR AD BC P Q RS OAD OBC OP Q ORS T ra mèsw tou jewr matoc twn epig niwn trig nwn, h teleutaða sqèsh gðnetai: OAB OCD OP S OQR OP Q ORS OAD OBS = OA OB OC OD OP OS OQ OR OP OQ OR OS OA OD OB OS = 1 PROTASH 3.1.2 (H ANISOTHTA TWN EPIGWNIWN TRIG- WNWN) An ABC > XY Z kai ABC + XY Z < 180 tìte: ABC > AB BC XY Z XY Y Z Apìdeixh Ftiˆqnoume èna isoskelèc trðgwno KLM me KL = KM kai LKM = ABC XY Z. ProekteÐnoume thn LM mèqri to N ètsi ste: M KN = XY Z.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA80 K L M N Sq ma 3.4: Opìte ja èqoume: LKN = ABC. Apì to je rhma twn epig niwn trig nwn ja pˆroume: ABC = ABC LKN MKN XY Z LKN MKN XY Z = AB BC LKN MK KN LK KN MKN XY Y Z = AB BC LKN XY Y Z MKN > AB BC XY Y Z. PORISMA 3.1.3 a. An ABC > XY Z kai ABC + XY Z > 180 tìte ABC XY Z < AB BC XY Y Z b. (AntÐstrofo tou jewr matoc twn epig niwn trig nwn) An ABC XY Z = AB BC XY Y Z tìte ABC = XY Z ABC + XY Z = 180. EFARMOGES THS ANISOTHTAS PARADEIGMA 1.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA81 Dedomènou trig nou ABC, an B > C tìte AC > AB Apìdeixh A B C Sq ma 3.5: Me bˆsh thn prìtash 3.1.2 (selðda 79), epeid B > C kai B+ C < 180 tìte 1 = ABC > AB BC = AB ACB AC CB AC 'Ara AC > AB PARADEIGMA 2 To ˆjroisma duo opoiond pote pleur n enìc trig nou eðnai megalôtero apì thn trðth. Apìdeixh ProekteÐnw thn BC mèqri to D, ètsi ste CD = AC. Epomènwc: CAD = CDA. Akìmh BDA = CAD = BAD BAC < BAD Apì thn prìtash 3.1.2, ja pˆroume: 1 = BDA > BD DA = BD BAD BA AD AB

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA82 A B C D Sq ma 3.6: 'Ara BD < AB, dhlad BC + CD < AB to zhtoômeno. H ènnoia thc Pujagìreiac diaforˆc ja eisaqjeð mèsw thc ènnoiac twn epiembadik n trig nwn. Thn ènnoia tou epiembadikoô trig nou thn sumbolðzoume C BAC kai thn orðzoume wc ex c: Sthn pleurˆ AB enìc trig nou ABC sqediˆzoume tetrˆgwno ABEF tètoio ste ta prosanatolismèna embadˆ S ABC kai S ABEF na èqoun to Ðdio prìshmo. F E C A B Sq ma 3.7: To C BAC eðnai ènac pragmatikìc arijmìc tètoioc ste: C BAC = 'Omoia: { ACF, A 90 ; ACF, A 90.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA83 C ABC = kai C ACB = { BEC, B 90 ; BEC, B 90. { CF E, C 90 ; CF E, C 90. Genikˆ ta C BAC, C ABC kai C ACB eðnai diaforetikˆ. IsqÔei ìmwc ìti: C BAC = C CAB, C ABC = C CBA kai C ACB = C BCA. PROTASH 3.1.4 Gia èna trðgwno ABC èqoume ìti: C ABC + C BAC = AB 2 2 Apìdeixh 'Opwc blèpoume sto parapˆnw sq ma, an h A kai h B eðnai oxeðec tìte: C ABC + C BAC = BEC + ACF = ABEF = AB2 2 2 An h A eðnai ambleða kai h B eðnai oxeða tìte: C ABC + C BAC = BEC ACF = ABEF = AB2 2 2 An h A eðnai oxeða kai h B eðnai ambleða tìte: C ABC + C BAC = BEC + ACF = ABEF = AB2 2 2 Epomènwc gia ìlec tic peript seic isqôei ìti: C ABC + C BAC = AB2 2 An ergastoôme anˆloga ja pˆroume:

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA84 C BCA + C ABC = BC2 2 kai C BAC + C BCA = CA2 2 SumperaÐnoume apì tic parapˆnw exis seic ìti: C ABC = (AB2 +BC 2 AC 2 ) 4 C BAC = (AB2 +AC 2 BC 2 ) 4 C ACB = (AC2 +BC 2 AB 2 ) 4 Me skopì t ra na sundèsoume tic duo autèc ènnoiec dðnoume ton akìloujo orismì. ORISMOS 3.1.5 Pujagìreia diaforˆ trig nou ABC wc proc to B, onomˆzoume thn posìthta AB 2 +BC 2 AC 2, thn opoða sumbolðzoume wc ex c: P ABC kai isoôtai me: P ABC = 4C ABC = AB 2 + BC 2 AC 2 PROTASH 3.1.6 (PUJAGOREIO JEWRHMA) A. P ABC = 0 an kai mìno an ABC = 90 B. P ABC > 0 an kai mìno an ABC < 90 G. P ABC < 0 an kai mìno an ABC > 90 Apìdeixh Gia to (A): an koitˆxoume to sq ma ja doôme ìti BEC = 0 an kai mìno an h ABC = 90. Epomènwc P ABC = 4C ABC = 0 Gia to (B): an ABC < 90 tìte C ABC = BEC ˆra P ABC > 0.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA85 Kai antðstrofa gia na isqôei P ABC > 0 ja prèpei to C ABC na eðnai jetikì, ˆra h gwnða ABC na eðnai oxeða. Gia to (G): an ABC > 90 tìte C ABC = BEC ˆra P ABC < 0. Kai antðstrofa gia na isqôei P ABC < 0 ja prèpei to C ABC na eðnai arnhtikì, ˆra h gwnða ABC na eðnai ambleða. PROTASH 3.1.7 (JEWRHMA TWN PUJAGOREIWN DI- AFORWN) An ABC 90 èqoume: 1. ABC = XY Z an kai mìno an P ABC P XY Z = AB BC XY Y Z 2. ABC + XY Z = 180 an kai mìno an P ABC P XY Z = AB BC XY Y Z Apìdeixh 1. An isqôei ABC = XY Z, opìte kai oi duo gwnðec oxeðec kai oi duo gwnðec ambleðec, ˆra ja èqoume: P ABC P XY Z = 4C ABC 4C XY Z = BEC Y EZ = BE BC Y E Y Z = AB BC XY Y Z. An t ra pˆroume to antðstrofo tou jewr matoc twn epig niwn trig nwn: h parakˆtw sqèsh ja isqôei ìtan kai oi duo gwnðec eðnai Ðsec kai eðnai kai oi duo oxeðec kai oi duo ambleðec. P ABC P XY Z = 4C ABC 4C XY Z = BEC Y EZ = BE BC Y E Y Z = AB BC XY Y Z = ABC XY Z. 2. Anˆloga ergazìmaste kai gia thn perðptwsh pou isqôei ìti: ABC + XY Z = 180 ˆra h mia gwnða eðnai oxeða kai h ˆllh ambleða.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA86 OrÐzoume t ra thn Pujagìreia diaforˆ enìc prosanatolismènou tetrapleôrou wc ex c: P ABCD = AB 2 BC 2 + CD 2 DA 2. D C A B Sq ma 3.8: Idiìthtec twn Pujagìreiwn diafor n gia ta tetrˆpleura 1. P ABCD = P CDAB = P BADC = P DCBA. 2. P ABCD = P BCDA = P DABC = P ADCB = P CBAD. 3.P ABCD = P BAC P DAC = P ABD P CBD = P DCA P BCA = P CDB P ADB. 4. P ABBC = P ABC, P AABC = P BAC, P ABCC = P ACB, P ABCA = P BAC. 5. P ABAC = 0, P ABCB = 0. 6. P AP BQ + P BP CQ = P AP CQ. PROTASH 3.1.8 An ta shmeða A, B kai C eðnai suggrammikˆ tìte isqôei: P ABC = 2 BA BC. Apìdeixh

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA87 A C B Sq ma 3.9: IsqÔei: AC = AB CB opìte: P ABC = AB 2 + BC 2 AC 2 = AB 2 + CB 2 (AB CB) 2 = 2 BA BC ORISMOS 3.1.9 QrhsimopoioÔme ton sumbolismì AB CD gia na dhl soume ìti ta shmeða A, B, C kai D ikanopoioôn mia apì tic parakˆtw sunj kec: 1. h eujeða AB eðnai kˆjeth sthn eujeða CD 2. A = B 3. C = D 3.2 BASIKES PROTASEIS II PROTASH B.7 AC BD an kai mìno an P ABCD = P ABD P CBD = 0. Apìdeixh 'Estw P kai Q eðnai ta shmeða tom c twn eujei n pou pernˆne apì ta A kai C antðstoiqa kai eðnai kˆjetec sthn eujeða BD. Apì to Pujagìreio je rhma paðrnoume: P ABD = AB 2 + BD 2 AD 2 = AP 2 + BP 2 + BD 2 AP 2 P D 2

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA88 A C B P D Sq ma 3.10: =BP 2 + BD 2 (BD BP ) 2 =2 BP BD OmoÐwc ergazìmaste kai gia to P CBD kai brðskoume: P CBD = 2 BQ BD. 'Ara P ABD = P CBD an kai mìno an BP = BQ, dhlad P = Q pou eðnai isodônamo me to AC BD. 'Amesh sunèpeia thc prohgoômenhc prìtashc eðnai h prìtash pou akoloujeð. PROTASH 3.2.1 'Estw P kai Q eðnai ta shmeða tom c twn eujei n pou pernˆne apì ta shmeða A kai C antðstoiqa kai eðnai kˆjetec sthn eujeða BD. Tìte isqôei: P ABCD = 2 QP BD Apìdeixh Qrhsimopoi ntac tic protˆseic B.7 (selðda 87) kai 3.1.8 (selðda 86)ja pˆroume: P ABCD = P ABD P CBD = P P BD P QBD = 2 BP BD 2 BQ BD =

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA89 2 QP BD. PROTASH B.8 'Estw D eðnai to shmeðo tom c thc eujeðac pou pernˆ apì to P kai eðnai kˆjeth sthn AB me (A B). Tìte èqoume: AD DB = P P AB P P BA, AD AB = P P AB 2 AB 2, AD AB = P P BA 2 AB 2. P A D B Apìdeixh Sq ma 3.11: Apì thn prìtash B.7 (selðda 87) paðrnoume: P P AB = P DAB = 2 AB AD, P P BA = P DBA = 2 BA BD, 'Ara AD DB = P P AB P P BA, OmoÐwc kai gia tic ˆllec peript seic ja èqoume:

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA90 P ABA = 2 AB 2 ìpou eðnai plèon xekˆjarec oi parakˆtw sqèseic P P AB 2 AB 2 P P BA 2 AB 2 = AD AB, = AD AB. PROTASH B.9 'Estw AB kai P Q na eðnai duo eujeðec pou den tèmnontai kˆjeta kai Y eðnai to shmeðo tom c thc eujeðac P Q kai thc eujeðac pou dièrqetai apì to A kai h opoða eðnai kˆjeth sthn AB. Tìte P Y QY = P P AB P QAB, P Y P Q = P P AB P P AQB, QY P Q = P QAB P P AQB. Apìdeixh P Y Q M A N B Sq ma 3.12:

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA91 Gia thn pr th sqèsh. PaÐrnoume M kai N na eðnai oi orjog niec probolèc twn shmeðwn P kai Q antðstoiqa pˆnw sthn AB. Apì thn prìtash B.7 (selðda 87), ja pˆroume: P P AB P QAB = P MAB P NAB = AM AB = AM AN AB AN Gia thn deôterh sqèsh. = P Y QY. P P AB P P AQB = P MAB P P AQB = AM AB MN AB = AM MN = P Y P Q. 'Omoia kai gia thn trðth sqèsh. PROTASH B.10 Pˆre R na eðnai shmeðo pˆnw sthn eujeða P Q me lìgouc jèsewc r 1 = P R P Q me anaforˆ to P Q. Tìte gia opoiad pote shmeða A kai B kai r 2 = RQ P Q èqoume: P RAB = r 1 P QAB + r 2 P P AB kai P ARB = r 1 P AQB + r 2 P AP B r 1 r 2 P P QP. H apìdeixh thc prìtashc ja gðnei me thn bo jeia thc parakˆtw prìtashc. PROTASH 3.2.2 'Estw A, B kai C trða suggrammikˆ shmeða. Tìte gia opoiod pote shmeðo P an P P AC 0 èqoume: P P AB P P AC = AB AC Apìdeixh 'Estw Q eðnai h orjog nia probol tou shmeðou P pˆnw sthn eujeða AB. Apì thn prìtash B.7 (selðda 87), P P AB = P QAB = 2AQ AB P P AC = P QAC = 2AQ AC 'Ara eðnai fanerì plèon ìti: P P AB P P AC = AB AC Apìdeixh prìtashc B.10

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA92 P A B C Q Sq ma 3.13: R Q P A P1 R1 Q1 B Sq ma 3.14: ArkeÐ na deðxoume ìti: RA 2 = r 1 QA 2 + r 2 P A 2 r 1 r 2 P Q 2 (1) RB 2 = r 1 QB 2 + r 2 P B 2 r 1 r 2 P Q 2 (2) Opìte P RAB = RA 2 +AB 2 RB 2 = r 1 (QA 2 +AB 2 QB 2 )+r 2 (P A 2 + AB 2 P B 2 ) = r 1 P QAB + r 2 P P AB Gia na deðxoume thn (1), qrhsimopoioôme thn prohgoômenh prìtash kai paðrnoume: P AP R P AP Q = P R = r P Q 1 Tìte: r 1 QA 2 + r 2 P A 2 r 1 r 2 P Q 2 = r 1 QA 2 + (1 r 1 )P A 2 r 1 (1 r 1 )P Q 2

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA93 =P A 2 + r 1 (QA 2 P A 2 P Q 2 ) + r 2 1P Q 2 =P A 2 + P R 2 r 1 P AP Q = P A 2 + P R 2 P AP R = AR 2. 'Omoia ergazìmaste kai gia thn apìdeixh thc (2). PROTASH B.11 'Estw ABCD parallhlìgrammo. Gia opoiad pote shmeða P kai Q è- qoume: P AP BQ = P DP CQ P AP Q + P CP Q = P BP Q + P DP Q P P AQ + P P CQ = P P BQ + P P DQ + 2P BAD Apìdeixh Gia thn apìdeixh thc prìtashc B.11 ja qreiastoôme thn prohgoômenh prìtash: PaÐrnoume shmeðo O na eðnai h tom twn AC kai BD. Apì thn pr th exðswsh thc prìtashc B.10 (selðda 91), paðrnoume: 2P OP Q = P AP Q + P CP Q = P BP Q + P DP Q. Apì thn deôterh exðswsh thc prìtashc B.10 paðrnoume: 2P OP Q = P P AQ + P P CQ 1 2 P ACA = P BP Q + P DP Q 1 2 P BDB. Mènei na deðxoume ìti 2P BAD = 1 2 (P ACA P BDB ) (1). Gia na to deðxoume ja qrhsimopoi soume thn parakˆtw prìtash. PROTASH 3.2.3 Gia èna parallhlìgrammo ABCD èqoume: AC 2 + BD 2 = 2AB 2 + 2BC 2 alli c P ABC = P BAD. Apìdeixh PaÐrnoume shmeðo O na eðnai h tom twn AC kai BD. Apì thn prìtash

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA94 B.10 (selðda 91) paðrnoume: AC 2 = 4AO 2 = 4( 1 2 AB2 + 1 2 AD2 1 4 BD2 ) = 2AB 2 + 2AD 2 BD 2 Apìdeixh thc prìtashc B.11(Sunèqeia) Me thn qr sh thc prohgoômenhc prìtashc eðnai plèon faner h isìthta (1). PROTASH B.12 'Estw ABCD parallhlìgrammo kai P opoiod pote shmeðo. Tìte: P P AB = P P DC P ADC = P P DAC kai P AP B = P AP A P P DAC. Apìdeixh P A B D C Sq ma 3.15: Apì thn prìtash B.11 (selðda 93), paðrnoume: P P AB = P P AC P P AD = P CADP = P P DAC = P P DC P ADC T ra gia thn deôterh exðswsh paðrnoume: P AP B = P AP A + P AP C + P AP D = P AP A + P CP DA = P AP A P P DAC O lìgoc parˆllhlwn eujugrˆmmwn tmhmˆtwn mporeð na ekfrasteð apì ton lìgo Pujagìreiwn diafor n, sômfwna me tic parakˆtw protˆseic.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA95 PROTASH 3.2.4 An AB CD kai EF KL tìte P AEBF P CKDL Apìdeixh = AB EF. CD KL 'Estw ìti ta M kai N eðnai shmeða tètoia ste AB = CM kai EF = KN. Apì tic protˆseic B.11 (selðda 93) kai 3.2.2 (selðda 91), èqoume: P AEBF = P AKBN = P AKN P BKN = EF KL (P AKL P BKL ) = EF KL P AKBL. 'Omoia apodeiknôoume ìti: P AKBL = AB CD P CKDL. PROTASH 3.2.5 An AB CD tìte: AB CD = P ADBC 2CD 2. Apìdeixh Apì thn prìtash 3.2.4 (selðda 94) paðrnoume: AB = AB CD = P ACBD CD CD CD P CCDD = P ADBC 2CD 2. ORISMOS 3.2.6 PaÐrnoume S na eðnai h orjog nia probol tou shmeðou R pˆnw sthn eujeða P Q. H prosanatolismènh apìstash tou R apì th P Q sumbolðzetai me h R,P Q kai eðnai ènac pragmatikìc arijmìc o opoðoc èqei to Ðdio prìshmo me to S RP Q kai isqôei h R,P Q = RS. R P S Q Sq ma 3.16: PROTASH 3.2.7 Gia opoiad pote dôo trðgwna ABC kai KLM, me S h A = h A,BC kai h K,LM isqôei ìti: ABC BC h A = S KLM P Q h K

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA96 Apìdeixh K A E B C L F M Sq ma 3.17: QwrÐc blˆbh thc genikìthtac, upojètoume ìti ta shmeða B, C, L kai M eðnai pˆnw sthn Ðdia eujeða. Fèrnoume to Ôyoc KF tou trig nou KLM kai E na eðnai shmeðo pˆnw sthn KF tètoio ste AE BC. Tìte S ABC = S EBC. Apì tic protˆseic B.1 kai B.2 (selðda 13) paðrnoume: S ABC S ALM = BC LM S ELM S KLM = EF KF Pollaplasiˆzontac tic duo parapˆnw sqèseic paðrnoume: S ABC BC h A = S KLM P Q h K. Na shmei soume ìti: ta h A, h K èqoun to Ðdio prìshmo me ta S ABC, S KLM. PORISMA 3.2.8 Gia èna trðgwno ABC èqoume: h A,BC BC = h B,CA AC = h C,AB AB. Apìdeixh An sthn prìtash 3.2 (selðda 95) bˆloume ìpou KLM to BCA kai CAB ja pˆroume to zhtoômeno.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA97 PROTASH 3.2.9 Se èna trðgwno ABC, èqoume: S ABC = 1 2 h A BC = 1 2 h B AC = 1 2 h C AB. PROTASH 3.2.10 Se èna trðgwno ABC, èqoume: 16S 2 ABC = 4AB2 AC 2 P 2 BAC. Apìdeixh A B N C Sq ma 3.18: 'Estw N h orjog nia probol tou shmeðou A pˆnw sthn BC. Apì thn prìtash 3.2.2 (selðda 91) paðrnoume: P ABC P ABN Tìte: = BC BN P ABC = BC BN P ABN = BC BN P NBN = 2BCBN 'Ara: 16S 2 ABC = 4AN 2 BC 2 = 4(AB 2 BN 2 )BC 2 = 4AB 2 AC 2 P 2 BAC. PROTASH 3.2.11 'Alloi Herron Qin tôpoi gia trðgwna eðnai oi akìloujoi.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA98 16S 2 ABC = P ACBP ABC + P BCB P BAC 16S 2 ABC = P BACP ACB + P ACA P ABC 16S 2 ABC = P CABP CBA + P ABA P ACB H apìdeixh touc eðnai eôkolh kai gia autì thn paraleðpoume. PROTASH 3.2.12 Se èna tetrˆpleuro ABCD, èqoume: 16S 2 ABCD = 4AC2 BD 2 P 2 ABCD. Apìdeixh A B D C S Sq ma 3.19: 'Estw shmeðo S tètoio ste to CSDB na eðnai parallhlìgrammo. Tìte: CS = BD. Apì tic protˆseic B.6 (selðda 16), B.11 (selðda 93) kai 3.2.10 (selðda 97) paðrnoume: SABCD 2 = S2 AACS = S2 XAC = 1 16 (4AS2 AC 2 PSAC 2 ) = 1 16 (4BD2 AC 2 PSAAC 2 ) = 1 16 (4BD2 AC 2 PBADC 2 ). EFARMOGES THS PUJAGOREIAS DIAFORAS SE

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA99 GEWMETRIKA PROBLHMATA. PARADEIGMA 1. (TO JEWRHMA TOU ORJOKEN- TROU) Ta trða Ôyh enìc trig nou dièrqontai apì to Ðdio shmeðo. Apìdeixh M A L C N B Sq ma 3.20: Fèrnoume ta duo Ôyh AN kai BM tou trig nou ABC, ta opoða tèmnontai sto shmeðo L. ArkeÐ na deðxoume ìti CL AB, dhlad P ACL = P BCL. Apì thn prìtash B.7 (selðda 87), èqoume: epeid BL AC tìte P ACL = P ACB kai AL BC tìte P BCL = P BCA 'Ara P ACL = P ACB = P BCA = P BCL. PARADEIGMA 2. 'Estw L shmeðo pˆnw sto Ôyoc AD, trig nou ABC. Oi eujeðec BL kai CL tèmnoun tic eujeðec AB kai AC sta shmeða M kai N antðstoiqa.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA100 IsqÔei ìti M DA = ADN Apìdeixh M A L N B D C Sq ma 3.21: Gia na apodeðxoume thn isìthta gwni n, apì to je rhma twn epig niwn trig nwn kai thn prìtash??, arkeð na deðxoume ìti: P MDA S MDA = P ADN S ADN. Apì thn prìtash B.2 (selðda 14), to je rhma twn epðpleurwn trig nwn, èqoume: S MDA = AM AB S BDA = S ALC S ALBC S BDA S ADN = AN AC S ADC = S ABL S ABCL S ADC T ra me qr sh thc prìtashc B.10 (selðda 91), ja pˆroume: P ADN = NC AC P ADA = S BCL S ABCL P ADA P MDA = MB AB P ADA = S BCL S ALBC P ADA Opìte P ADN S ADN SMDA P MDA = S BDA S ADC SALC S ABL = BD DC DC BD = 1.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA101 3.3 GRAMMIKES KATASKEUASTIKES GEWMETRIKES PROTASEIS Se aut n thn kathgorða upˆrqoun treðc gewmetrikèc posìthtec: 1. to embadìn trig nwn tetrapleôrwn 2. h Pujagìreia diaforˆ trig nwn tetrapleôrwn 3. o lìgoc parˆllhlwn eujugrˆmmwn tmhmˆtwn. Ta shmeða eðnai ta basikˆ gewmetrikˆ antikeðmena apì ta opoða mporoôme na eisˆgoume duo ˆlla gewmetrikˆ antikeðmena, tic eujeðec kai touc kôklouc. Mia eujeða mporeð na an kei se mia apì tic parakˆtw kathgorðec: (LINE U V ): eðnai h eujeða pou pernˆ apì ta shmeða U kai V. (P LINE W U V ): eðnai h eujeða pou pernˆ apì to shmeðo W kai eðnai parallhlh sthn (LINE U V ). (T LINE W U V ): eðnai h eujeða pou pernˆ apì to shmeðo W kai eðnai kˆjeth sthn (LINE U V ). (BLINE U V ): eðnai h eujeða pou eðnai mesokˆjetoc thc UV. O kôkloc pou pernˆ apì to shmeðo U kai èqei kèntro to O sumbolðzetai: (CIR O U). Gia tic protˆseic pou an koun sthn kathgorða C L, mporoôme na perigrˆyoume tic kataskeuèc touc me kˆpoion apì touc akìloujouc trìpouc: K 1 (P OINT (S) Y 1,..., Y n ) : 'Estw aujaðreta shmeða Y 1,..., Y n sto epðpedo. Kajèna apì ta Y i èqei duo bajmoôc eleujerðac.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA102 K 2 (ON Y B C) : 'Estw shmeðo Y pˆnw sthn eujeða BC. To shmeðo Y èqei ènan bajmì eleujerðac. K 3 (ON Y (CIR O U)) : 'Estw shmeðo Y pˆnw ston kôklo (CIR O U). To shmeðo Y èqei ènan bajmì eleujerðac. K 4 (INT ER Y ln 1 ln 2 ) : 'Estw shmeðo Y to shmeðo tom c twn duo eujei n. To shmeðo Y eðnai stajerì shmeðo. Pio sugkekrimèna: 1. An ln 1 eðnai thc morf c (LINE U V ) (P LINE W U V ) kai ln 2 eðnai thc morf c (LINE A B) (P LINE R A B). 2. An ln 1 eðnai thc morf c (LINE U V ) (P LINE W U V ) kai ln 2 eðnai thc morf c (BLINE A B) (T LINE R A B). 3. An ln 1 eðnai thc morf c (BLINE U V ) (T LINE W U V ) kai ln 2 eðnai thc morf c (BLINE A B) (T LINE R A B). K 5 (INT ER Y ln (CIR O A)) : 'Estw shmeðo Y na eðnai to shmeðo tom c thc eujeðac kai tou kôklou, diaforetikì tou shmeðou A. To shmeðo Y eðnai stajerì shmeðo. H eujeða ln mporeð na èqei mia apì tic parakˆtw morfèc: (LINE A V ), (P LINE A U V ), (T LINE A U V ). K 6 (INT ER Y (CIR O 1 A)(CIR O 2 A)) : 'Estw shmeðo Y na eðnai to shmeðo tom c twn duo kôklwn, diaforetikì tou shmeðou A. To shmeðo Y eðnai stajerì shmeðo. K 7 (P RAT IO Y A B C r) : 'Estw shmeðo Y pˆnw sthn eujeða pou pernˆei apì to shmeðo A kai eðnai parˆllhlh sthn eujeða BC, (P LINE A B C), tètoia ste AY = rbc, ìpou r eðnai ènac rhtìc arijmìc, rht èkfrash kˆpoiwn gewmetrik n posot twn metablht. Stic duo pr tec peript seic to shmeðo Y eðnai stajerì shmeðo, en sthn trðth to shmeðo Y èqei èna bajmì eleujerðac. K 8 (T RAT IO Y B C r) : 'Estw shmeðo Y pˆnw sthn eujeða (T LINE B B C) tètoia ste r = 4S BCY P BCB = BY, ìpou r eðnai ènac rhtìc arijmìc, rht èkfrash kˆpoiwn gewmetrik n posot twn metablht. Stic duo pr tec BC peript seic to shmeðo Y eðnai stajerì shmeðo, en sthn trðth to shmeðo Y èqei èna bajmì eleujerðac.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA103 Sto sônolo oi parapˆnw kataskeuèc eðnai 22, epeid lambˆnoume upìyhn kai ta tèssera eðdh eujei n. 'Etsi loipìn h kataskeu K 2 anafèretai sthn tuqaða epilog shmeðwn pˆnw se eujeðec pou an koun se èna apì ta tèssera eðdh eujei n, h kataskeu K 4 anafèretai sthn dhmiourgða shmeðou pou prokôptei apì thn tom eujei n, o sundiasmìc twn opoðwn gðnetai me 10 diaforetikoôc trìpouc. Tèloc h kataskeu K 5 anafèretai sthn dhmiourgða shmeðou pou prokôptei apì thn tom kôklou kai eujeðac, h opoða èqei treic pijanèc morfèc. ORISMOS 3.3.1 Oi protˆseic pou an koun sthn kathgorða C L èqoun thn parakˆtw morf : Prìtash=(K 1, K 2,..., K n, S) Kˆje prìtash aut c thc kathgorðac perigrˆfetai apì èna peperasmèno pl joc kataskeu n, oi opoðec èqoun thn morf twn kataskeu n K 1, K 2,..., K 8 (selðda 102). Oi kataskeuèc entˆssontai katˆ diatetagmèno trìpo wc ex c: arqikˆ eisˆgetai h kataskeu K 1, h opoða èqei thn morf thc K 1 kai eisˆgei aujaðreta shmeða sto epðpedo. SuneqÐzoume me thn prosj kh ˆllwn kataskeu n K 2,..., K n, oi opoðec èqoun thn morf twn kataskeu n K 2,..., K 8. Kajemiˆ apì tic parapˆnw kataskeuèc eisˆgei nèa shmeða pou proèrqontai apì tomèc eujei n kai kôklwn kai apì lìgouc parˆllhlwn eujôgrammwn tmhmˆtwn. Oi eujeðec kai oi kôkloi èqoun kataskeuasteð apì ta shmeða pou èqei eisˆgei h K 1 kai apì ta shmeða pou èqoun dh kataskeuasteð mèsw twn kataskeu n K 2,..., K n. Tèloc h prìtash katal gei sto sumpèrasma (S), to opoðo apoteleðtai apì thn isìthta duo poluwnumik n ekfrˆsewn gewmetrik n posot twn. To zhtoômeno kˆje forˆ eðnai h apìdeixh thc isìthtac twn duo aut n poluwnumik n ekfrˆsewn. Oi gewmetrikèc autèc posìthtec perilambˆnoun shmeða pou èqoun eisaqjeð apì tic kataskeuèc K i, i = 1,..., n. Oi gewmetrikèc idiìthtec pou kaloômaste na deðxoume pwc èqoun oi parapˆnw gewmetrikèc posìthtec eðnai h suggrammikìthta, h parallhlða, h kajetìthta, h armonikìthta, isìthta gwni n, isìthta twn mhk n duo euju-

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA104 grˆmmwn tmhmˆtwn, thn isìthta ginomènou eujugrˆmmwn tmhmˆtwn. Akìmh sthn klˆsh aut qrhsimopoioônte ènnoiec ìpwc orjìkentro, barôkentro, perigegrammènos-egegrammènoc kôkloc. 'Ola ta paradeðgmata pou dìjhkan prohgoumènwc an koun sthn kathgorða C L. EpÐshc kai gia tic protˆseic pou an koun sthn kathgorða C L oi perioristikèc sunj kec parˆgontai me susthmatikì trìpo. Oi entolèc twn grammik n idiot twn eðnai oi parakˆtw: (COLLINEAR A B C) : Ta shmeða A, B kai C eðnai suggrammikˆ an kai mìno an S ABC = 0. (P ARALLEL A B C D) : AB eðnai parˆllhlh sthn CD an kai mìno an S ACD = S BCD. (P ERP ENDICULAR A B C D) : AB eðnai kˆjeth sthn CD an kai mìno an P ACD = P BCD. (MIDP OINT M A B) : To M eðnai to mèson thc AB an kai mìno an AM MB = 1. (EQDIST ANCE A B C D) : To AB èqei Ðso m koc me to CD an kai mìno an P ABA = P CDC. (HARMONIC A B C D) : Ta A, B kai C, D eðnai armonikˆ shmeða an kai mìno an AC = DA. CB DB (EQ P RODUCT A B C D P Q R S) : To ginìmeno twn AB kai CD eðnai Ðso me to ginìmeno twn P Q kai RS, to opoðo eðnai isodônamo me 1. AB P Q = ± RS CD an AB P Q kai RS CD,

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA105 2.P ACBD = ±P P RQS an AB CD kai RS P Q 3.P ABA P CDC = P P QP P RSR alli c (T ANGENT O 1 A O 2 B) : O kôkloc (CIRCLE O 1 A) eðnai efaptomenikìc ston kôklo (CIRCLE O 2 B) an kai mìno an d 2 + r 2 1 + r 2 2 2dr 1 2dr 2 2r 1 r 2 = 0 ìpou d = O 1 O 2 2, r1 = O 1 A 2, r 2 = O 2 B 2. ORISMOS 3.3.2 'Estw prìtash P=(K 1, K 2,..., K n,s), tìte oi periorismoð gia na eðnai h P prìtash thc kathgorðac C L eðnai oi akìloujec. 1.An K i eðnai h kataskeu K 1 tìte den qreiˆzontai perioristikèc sunj kec. 2. An K i eðnai mia apì tic kataskeuèc K 2, K 7, K 8, selðda 102, tìte h perioristik sunj kh gia thn K i eðnai B C. 3. An K i eðnai mia apì tic treðc peript seic thc kataskeu c K 4, selðda 102, tìte h perioristik sunj kh gia thn K i gia thn pr th kai trðth perðptwsh eðnai: UV AB kai gia thn deôterh perðptwsh eðnai: UV AB. 4. An K i eðnai h kataskeu K 3, selðda 102, tìte h perioristik sunj kh gia thn K i eðnai O P. 5. An K i eðnai h kataskeu K 5, selðda 102, tìte h perioristik sunj kh gia thn K i eðnai O P, Y P. 6.An K i eðnai h kataskeu K 6, selðda 102, tìte h perioristik sunj kh

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA106 gia thn K i eðnai O 1, O 2, A mh suggrammikˆ. 3.4 DHMIOURGIA ELAQISTOU SUNOLOU KATASKEUWN 'Opwc eðdame parapˆnw gia tic protˆseic pou an koun sthn kathgorða C L, to pl joc twn kataskeu n kai twn gewmetrik n posot twn eðnai parapˆnw sugkritikˆ me tic kataskeuèc kai tic gewmetrikèc posìthtec pou upˆrqoun sthn kathgorða C H. 'Etsi loipìn gia to epìmeno b ma, thn dhmiourgða dhlad teqnik n apaloif c shmeðwn, ja prèpei na skeftoôme 66 peript seic apaloif c gia kˆje kataskeu kai kˆje gewmetrik posìthta. To gegonìc autì mporeð na aplousteuteð me thn dhmioiurgða enìc elˆqistou sunìlou kataskeu n. Parakˆtw ja deðxoume ìti oi parapˆnw kataskeuèc mporoôn na antikatastajoôn apì 5 mìno basikèc kataskeuèc. Oi kataskeuèc autèc eðnai: K 1 (P OINT (S) Y 1,..., Y n ): 'Estw aujaðreta shmeða Y 1,..., Y n sto epðpedo. Kajèna apì ta Y i èqei duo bajmoôc eleujerðac. K 7 (P RAT IO Y A B C r). 'Estw shmeðo Y pˆnw sthn eujeða pou pernˆei apì to shmeðo A kai eðnai parˆllhlh sthn eujeða BC, (P LINE A B C), tètoia ste AY = rbc, ìpou r eðnai ènac rhtìc arijmìc, rht èkfrash kˆpoiwn gewmetrik n posot twn metablht. Stic duo pr tec peript seic to shmeðo Y eðnai stajerì shmeðo, en sthn trðth to shmeðo Y èqei èna bajmì eleujerðac. O periorismìc eðnai B C. K 8 (T RAT IO Y A B r). 'Estw shmeðo Y pˆnw sthn eujeða (T LINE A A B) tètoia ste r = 4S ABY P ABA = AY, ìpou r eðnai ènac rhtìc arijmìc, rht èkfrash kˆpoiwn gewmetrik n posot twn metablht. Stic duo pr tec AB peript seic to shmeðo Y eðnai stajerì shmeðo, en sthn trðth to shmeðo Y èqei èna bajmì eleujerðac. O periorismìc eðnai B A. K 41 (INT ER Y (LINE U V )(LINE P Q)). 'Estw shmeðo Y to shmeðo tom c twn duo eujei n. To shmeðo Y eðnai stajerì shmeðo. O periorismìc thc kataskeu c aut c eðnai UV P Q. K 42 (F OOT Y P U V ) h opoða eðnai isodônamh me thn kataskeu (INT ER Y (LINE U V )(T LINE P U

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA107 'Estw shmeðo Y to shmeðo tom c thc eujeðac UV kai thc kˆjet c thc dierqìmenhc apì to shmeðo P. To shmeðo Y eðnai stajerì shmeðo. O periorismìc thc kataskeu c aut c eðnai U V. Se autì to elˆqisto sônolo kataskeu n mporoôme na antikatast soume kai ta tèssera eðdh eujei n apì èna mìno eðdoc: (LINE U V ). 1. H eujeða (P LINE W U V ) mporeð na antikatastajeð apì thn eujeða (LINE W N). To shmeðo N eisˆgetai apì thn kataskeu (P RAT IO N W U V 1). 2. H eujeða (T LINE W U V ) mporeð na antikatastajeð apì thn eujeða (LINE N W ). An ta W, U, V eðnai suggrammikˆ to shmeðo N eisˆgetai apì thn kataskeu (T RAT IO N W U 1), alloi c eisˆgetai apì thn kataskeu (F OOT N W U V ). 3. H eujeða (BLINE U V ) mporeð na antikatastajeð apì thn eujeða (LINE N M). Ta shmeða M kai N eisˆgontai apì tic kataskeuèc (T RAT IO N M U 1) kai (P RAT IO M U U V 1 2 antðstoiqa. ) (MIDP OINT M U V ) T ra ja suneqðsoume me thn antikatˆstash twn 22 kataskeu n apì tic 5 basikèc kataskeuèc, antikajist ntac tautìqrona kai ta tèssera eðdh twn eujei n apì thn (LINE UV ). 1. H kataskeu K 2 (ON Y (LINE U V )) eðnai isodônamh me thn kataskeu K 7 (P RAT IO Y U U V r)), to r eðnai aìristo. 2. H kataskeu K 5 (INT ER Y (LINE U V )(CIR O U)) eðnai isodônamh me tic kataskeuèc (F OOT N O U V ) kai (P RAT IO Y N N U 1) 3. H kataskeu K 3 (ON Y (CIR O P )) eðnai isodônamh me thn kataskeu (INT ER Y (LINE P Q)(CIR O P ), gia tuqaða epilog shmeðou Q. 4. H kataskeu K 6 (INT ER Y (CIR O 1 P )(CIR O 2 P )) eðnai isodônamh me tic kataskeuèc (F OOT N P O 1 O 2 ) kai (P RAT IO Y N N P 1) Akìmh sth diatôpwsh twn protˆsewn pou an koun sthn klˆsh C L ja qrhsimopoihjoôn perissìterec kataskeuèc lìgwn mhk n, tic opoðec ja entˆxoume stic duo basikèc kataskeuèc lìgwn pou èqoume sunant sei,

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA108 P RAT IO kai T RAT IO. K 9 (MIDP OINT Y U V ) DiatÔpwsh kataskeu c: èstw shmeðo Y to mèson thc eujeðac UV. H kataskeu aut eðnai isodônamh me thn (P RAT IO Y U U V 1 2 ). K 10 (SY MMET RY Y U V ) DiatÔpwsh kataskeu c: èstw shmeðo Y to summetrikì tou shmeðou V wc proc to shmeðo U. H kataskeu aut eðnai isodônamh me thn (P RAT IO Y U U V 1). UY UV K 11 (LRAT IO Y U V r) DiatÔpwsh kataskeu c: èstw shmeðo Y pˆnw sthn U V tètoio ste: = r H kataskeu aut eðnai isodônamh me thn (P RAT IO Y U U V r). UY Y V K 12 (MRAT IO Y U V r) DiatÔpwsh kataskeu c: èstw shmeðo Y pˆnw sthn U V tètoio ste: = r H kataskeu aut eðnai isodônamh me thn (P RAT IO Y U U V K 13 (HARMONIC Y A B C) r ). 1+r DiatÔpwsh kataskeu c: èstw shmeðo Y suggrammikì me ta trða suggrammikˆ shmeða A, B, C, ètsi ste: = Y A. CB Y CA B K 14 (INV ERSION Y U O R)

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA109 DiatÔpwsh kataskeu c: èstw ìti to shmeðo Y eðnai h antistrof tou shmeðou U wc proc ton kôklo (CIR O R). H kataskeu aut eðnai isodônamh me: (LRAT IO Y O R OR ) an U OR OU (LRAT IO Y O U P ORO P OUO ) alli c (ShmeÐwsh: h antistrof eðnai ènac metasqhmatismìc tou epipèdou ston eautì tou, pou orðzetai me thn bo jeia enìc kôklou. An O to kèntro tou kôklou kai r h aktðna tou tìte se kˆje shmeðo A diaforetikì tou O, antistoiqoôme to shmeðo B pˆnw sthn hmieujeða OA, ètsi ste: OB OA = ρ 2. O kôkloc (O, r) lègetai kôkloc thc antistrof c, to shmeðo O lègetai kèntro antistrof c kai h aktðna r lègetai dônamh thc antistrof c.) K 15 (CONST ANT p(r)) H kataskeu aut eisˆgei ènan algebrikì arijmì r o opoðoc eðnai rðza tou anˆgwgou poluwnômou p(r), dhlad p(r) = 0. Ed na shmei soume ìti h kataskeu (CONST ANT ),mèsa se mia gewmetrik prìtash, mac bohjˆ sthn kataskeu isìpleurwn trig nwn kai h kataskeu (T RAT IO) sthn kataskeu tetrag nwn. 3.5 TEQNIKES APALOIFHS SHMEI- WN 'Etsi loipìn me thn dhmiourgða tou elˆqistou sunìlou kataskeu n arkoômaste na deðxoume teqnikèc apaloif c bohjhtik n shmeðwn mìno gia tic pènte kataskeuèc: K 1, K 7, K 8, K 41, K 42. APALOIFES SHMEIWN MESW GEWMETRIKWN POSOTHTWN

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA110 Onomˆzoume G(Y ) mia grammik gewmetrik posìthta me metablht Y kai eðnai mia apì tic akìloujec gewmetrikèc posìthtec: S ABY, S ABCY, P ABY, P ABCY, gia A, B, C, Y diaforetikˆ shmeða. Gia trða suggrammikˆ shmeða Y, P, Q, apì tic protˆseic B.3, selðda 15 kai B.10, selðda 91, paðrnoume: G(Y )= P Y G(Q)+ Y Q G(P) (A) P Q P Q EpÐshc an G(Y )=P AY B tìte apì thn prìtash B.10 gia trða suggrammikˆ shmeða Y, P, Q paðrnoume: G(Y )= P Y G(Q)+ Y QG(P)- P Y Y Q P P Q P Q P Q P Q P QP (B) Onomˆzoume P AY B mia tetragwnik gewmetrik posìthta me metablht Y. LHMMA 3.5.1 'Estw G(Y ) eðnai mia grammik gewmetrik posìthta kai shmeðo Y eisˆgetai apì thn kataskeu (P RAT IO Y W P Q r) tìte èqoume: { G(Y )= ( P W + r)γ(q) + ( W Q P Q P Q G(W ) + r(γ(q) Γ(P )), r)γ(p ), an W an kei pˆnw sthn eujeða P Q; alli c. Apìdeixh An ta W, P, Q eðnai suneujeiakˆ èqoume: Y W P Q Sq ma 3.22: P Y = P W + r P Q P Q Y Q P Q = W Q P Q r

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA111 An tic antikatast soume sthn (A), (selðda 110) ja pˆroume thn zhtoômenh sqèsh. Y W S P Q Sq ma 3.23: Gia thn deôterh t ra sqèsh, an pˆroume shmeðo S tètoio ste W S = P Q. Apì thn (A) (selðda 110) paðrnoume: G(Y )= W Y G(S)+ Y S W S r)g(w)(1) W S G(W)=rG(S)+(1 Apì tic protˆseic B.6, selðda 16 kai B.11, selðda 93, paðrnoume: G(S)=G(W)+G(Q)- G(P)(2) Antikajist ntac thn (2) sthn (1) paðrnoume thn zhtoômenh sqèsh. LHMMA 3.5.2 'Estw G(Y ) eðnai mia grammik gewmetrik posìthta kai shmeðo Y eisˆgetai apì thn kataskeu (INT ER Y (LINE P Q)(LINE X Z)) tìte èqoume: G(Y )= S P XZΓ(Q) S QXZ Γ(P ) S P XQZ Apìdeixh Apì to je rhma twn epig niwn trig nwn, P Y P Q = S P XZ S P XQZ Y Q P Q = S QXZ S P XQZ

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA112 P Q X Z Y Sq ma 3.24: Antikajist ntac tic parapˆnw sqèseic sthn (A) (selðda 110) paðrnoume thn zhtoômenh sqèsh. LHMMA 3.5.3 'Estw G(Y ) eðnai mia grammik gewmetrik posìthta kai shmeðo Y eisˆgetai apì thn kataskeu (F OOT Y W P Q) tìte èqoume: G(Y )= P W P QΓ(Q)+P W QP Γ(P ) 2P Q 2 Apìdeixh W P Y Q Sq ma 3.25: Apì thn prìtash B.8 (selðda 89), paðrnoume: P Y P Q = P W P Q P P QP Y Q P Q = P W QP P P QP

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA113 An antikatast soume tic parapˆnw sqèseic sthn (A) (selðda 110) paðrnoume thn zhtoômenh. LHMMA 3.5.4 'Estw shmeðo Y eisˆgetai apì thn kataskeu (P RAT IO Y W P Q r) tìte èqoume: P AY B = P AW B + r(p AQB P AP B + P W P Q ) r(1 r)p P QP Apìdeixh An pˆroume shmeðo S tètoio ste: W S = P Q. Tìte apì thn (B)(selÐda 110) paðrnoume: G(Y )= W Y W S G(S)+ Y S W S G(W)- W Y W S Y S P W S P QP=rG(S)+(1 r)g(w)-r(1 r)p P QP (1) Y W S P Q Sq ma 3.26: Apì thn prìtash B.11, selðda 93, thn deôterh sqèsh paðrnoume: G(S)=G(W)+G(Q)-G(P)+P W P Q (2) An t ra sthn (1) antikajist soume thn sqèsh (2) ja pˆroume thn zhtoômenh sqèsh. LHMMA 3.5.5 'Estw shmeðo Y eisˆgetai apì thn kataskeu (T RAT IO Y P Q r) tìte èqoume:

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA114 S ABY = S ABP r 4 P P AQB Apìdeixh Y B A P A 1 Q Sq ma 3.27: PaÐrnoume shmeðo A 1 na eðnai h orjog nia probol tou shmeðou A pˆnw sthn P Q. Tìte apì tic protˆseic B.4 (selðda 15) kai B.8 (selðda 89), ja pˆroume: S P AY S P QY = S P A 1 Y S P QY = P A 1 P Q = P A 1 P Q P QP Q = P AP Q P QP Q 'Ara S P AY = P AP Q P QP Q S P QY = r 4 P AP Q. An ergastoôme ìmoia gia to shmeðo B, ja pˆroume: S P BY = P BP Q P QP Q S P QY = r 4 P BP Q. T ra S ABY = S ABP + S P BY S P AY = S ABP r 4 P P AQB. LHMMA 3.5.6 'Estw shmeðo Y eisˆgetai apì thn kataskeu (T RAT IO Y P Q r) tìte èqoume:

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA115 P ABY = P ABP 4rS P AQB Apìdeixh Y B 1 B A A 1 P Q Sq ma 3.28: PaÐrnoume shmeða A 1, B 1 na eðnai h orjog niec probolèc twn shmeðwn A, B antðstoiqa pˆnw sthn P Y. Tìte P BP AY P Y P Y = P B 1 P A 1 Y P Y P Y = A 1B 1 P Y = S P A 1 QB 1 S P QY = S P AQB S P QY. Epeid P Y P Q, S 2 P QY = 1 4 P Q2 P Y 2. Akìmh P Y P Y = 2P Y 2 = 4rS P QY. 'Etsi loipìn P ABY = P ABP P BP AY = P ABP 4rS P AQB. LHMMA 3.5.7 'Estw shmeðo Y eisˆgetai apì thn kataskeu (T RAT IO Y P Q r) tìte èqoume: P AY B = P AP B + r 2 P P QP 4r(S AP Q + S BP Q ) Apìdeixh Apì to prohgoômeno l mma ja pˆroume:

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA116 Y B A P Q Sq ma 3.29: P AP Y = 4rS AP Q kai P BP Y = 4rS BP Q Akìmh P Y P Y = 2P Y 2 = 4rS P QY = r 2 P P QP Tìte P AY B = P AP B P AP Y P BP Y + P Y P Y = P AP B + r 2 P P QP 4r(S AP Q + S BP Q ). APALOIFES SHMEIWN MESW LOGWN EUJUGRAMMWN TMHMATWN LHMMA 3.5.8 'Estw shmeðo Y eisˆgetai apì thn kataskeu (F OOT Y W P Q), upojètoume A P tìte èqoume: AY = CD { PW CAD P CDC, an A P Q; S AP Q S CP DQ, an A / P Q. Apìdeixh An A P Q, paðrnoume shmeðo S tètoio ste: AS = CD. Apì tic protˆseic B.8 (selðda 89) kai B.12 (selðda 94), tìte: AY = AY = P W AS CD AS P ASA = P W CAD P CDC.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA117 W C D S P Y Q A Sq ma 3.30: An A / P Q tìte h deôterh sqèsh prokôptei ˆmesa apì to je rhma twn epðpleurwn trig nwn. LHMMA 3.5.9 'Estw shmeðo Y eisˆgetai apì thn kataskeu (T RAT IO Y P Q r) tìte èqoume: AY = CD { PAP Q P CP DQ, an A / P Y ; S AP Q r 4 P P QP S CP DQ, an A P Y. Apìdeixh An A / P Y tìte h pr th sqèsh prokôptei apì thn prìtash B.9 (selðda 90). Y A C P Q D An A P Y tìte Sq ma 3.31:

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA118 AY = AP Y P.(1) CD CD CD Apì to je rhma twn epðpleurwn trig nwn èqoume: AP CD = S AP Q S CP DQ (2) Y P = S Y P Q CD S CP DQ = r 4 P P QP S CP DQ (3) Apì tic (2) kai (3) me antikatastash sthn (1) paðrnoume thn zhtoômenh sqèsh. Me thn efarmog twn parapˆnw lhmmˆtwn epitugqˆnoume thn apaloif twn bohjhtik n shmeðwn apì tic gewmetrikèc posìthtec thc prìtashc. Skopìc mac eðnai oi gewmetrikèc posìthtec pou ja prokôyoun na apìteloôntai mìno apì eleôjera shmeða. Na shmei soume ed ìti oi gewmetrikèc posìthtec tou embadoô kai thc Pujagìreiac diaforˆc den eðnai anexˆrthtec, to gegonìc autì epibebai netai mèsw tou tôpou Herron Qin: 16S 2 ABC = 4AB2 AC 2 P 2 BAC. Gia ton lìgo autì epijumoôme na ekfrˆsoume tic gewmetrikèc autèc posìthtec se mia sqèsh pou perilambˆnei anexˆrthtec metablhtèc. 'Etsi eisˆgoume trða nèa shmeða O, U, V tètoia ste UO OV. Katal goume epomènwc na ekfrˆsoume tic gewmetrikèc autèc posìthtec sunart sh twn embadik n suntetagmènwn twn eleôjerwn shmeðwn thc prìtashc wc proc to OUV. Sunèpeia twn parapˆnw eðnai to l mma pou akoloujeð. LHMMA 3.5.10 Gia opoiad pote eleôjera shmeða A, B, C èqoume: 1. S ABC = (S OV B S OV C )S OUA +(S OV C S OV A )S OUB +(S OV A S OV B )S OUC S OUV 2. P ABC = AB 2 + CB 2 AC 2 3. AB 2 = OU 2 (S OV A S OV B ) S 2 OUV + OV 2 (S OUA S OUB ) S 2 OUV

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA119 4. S 2 OUV = OU 2 OV 2 4 Apìdeixh V B M A O U Sq ma 3.32: 1. Anatrèxte sthn apìdeixh tou l mmatoc 2.5.3 (selðda 41), twn teqnik n apaloif c shmeðwn apì ta embadˆ. 2. EÐnai o orismìc thc Pujagìreiac diaforˆc. 3. Eisˆgoume shmeðo M apì thn kataskeu (INT ER M (P LINE A O U)(P LINE B O V ). Apì to Pujagìreio je rhma: AB 2 = AM 2 + BM 2 Apì to l mma 2.4.5 (selðda 32) twn teqnik n apaloif c shmeðwn apì lìgouc eujugrˆmmwn tmhmˆtwn, gia A / M B kai B / M A, paðrnoume: AM OU = S AOBV S OOUV = S AOV S BOV S OUV BM OV = S BOAU S OOUV = S BOU S AOU S OUV Apì ta parapˆnw prokôptei h zhtoômenh sqèsh. 4. EÐnai sunèpeia gnwst c prìtashc.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA120 3.6 ALGORIJMOS Eisˆgoume thn gewmetrik prìtash P, h opoða an kei sthn kathgorða C L kai eðnai diatupwmènh wc ex c: P=(K 1, K 2,..., K n, S). O algìrijmoc exetˆzei an h prìtash eðnai alhj c yeud c kai an eðnai alhj c parˆgei thn apìdeixh akolouj ntac ta parakˆtw b mata, alli c termatðzei. BHMA 1. Gia i = n,..., 1, xekin ntac dhlad apì to shmeðo pou èqei kataskeuasteð teleutaða katal gwntac sta shmeða pou èqoun eisaqjeð aujaðreta mèsw thc K 1 efarmìzoume to epìmeno b ma. BHMA 2. 'Elegqoc perioristik n sunjhk n twn K i. An h perioristik sunj kh eðnai: an A B èlegxe an sumbaðnei P ABA = 0, an AB P Q èlegxe an sumbaðnei S AP Q = S BP Q kai tèloc an AB /P Q èlegxe an P AP Q = P BP Q. An o algìrijmoc epibebai nei thn prìtash P, tìte h P eðnai alhj c kˆtw apì tic perioristikèc sunj kec pou autìmata parˆgontai apì ton orismì 3.3.2 (selðda 105), alli c h P eðnai yeud c. BHMA 3. Apì to (S) sumpèrasma thc prìtashc paðrnoume tic gewmetrikèc posìthtec, G 1, G 2,..., G l kai tic fèrnoume sto pr to mèroc me skopì to deôtero mèloc na eðnai monˆda. Gia kajemiˆ apì tic gewmetrikèc posìthtec efarmìzoume to epìmeno b ma. BHMA 4. Gia kajèna apì ta G j,j = 1,..., l, apaloðfoume ta bohjhtikˆ shmeða pou kataskeuˆsthkan apì tic K i. Gia thn apaloif twn shmeðwn aut n qrhsimopoioôme ta parakˆtw ergaleða: tic protˆseic B.1 èwc B.6 (selðdec 13)- 16), tic protˆseic B.7 èwc B.12 (selðdec 87-94), tic teqnikèc apaloif c shmeðwn mèsw lìgwn mhk n (selðda 116), tic teqnikèc apaloif c shmeðwn mèsw twn embad n kai twn Pujagìreiwn diafor n (selðda 110). An to apotèlesma thc apaloif c eðnai mia gewmetrik posìthta G' j apoteloômenh mìno apì shmeða pou epilèqjhkan aujaðreta, tìte efarmìzoume to l mma 3.5.10 (selðda 118), me skopì na prokôyei mia rht

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA121 èkfrash anexˆrthtwn metablhtwn, alli c efarmìzoume apì thn arq to b ma 4. BHMA 5. Katal xame sto na exis soume touc arqikoôc lìgouc gewmetrik n posot twn me lìgouc gewmetrik n posot twn pou perilambˆnoun mìno anexˆrthtec metˆblhtèc. GÐnontai oi apaitoômenec aplopoi seic kai an katal xoume sthn isìthta me th monˆda tìte h prìtash P eðnai alhj c alli c h P eðnai yeud c. Apìdeixh orjìthtac algorðjmou. Parapˆnw ex ghsh qreiˆzetai to teleutaðo b ma. 'Etsi loipìn an katal x- oume sthn exðswsh tou lìgou gewmetrik n posot twn me thn monˆda, dhlad r = 1 tìte h prìtash eðnai alhj c. Alli c afoô oi gewmetrikèc posìthtec apoteloôntai apì eleôjerec paramètrouc, mporoôme na broôme timèc gia tic gewmetrikèc posìthtec tètoiec ste r 1. Sunep c brðskoume èna antiparˆdeigma gia to opoðo h prìtash den isqôei. 3.7 EMBADIKES SUNTETAGMENES EÐqame eisˆgei sto l mma 3.5.10 (selðda 118) to orjog nio sôsthma embadik n suntetagmènwn. EÐdame pwc gia na qrhsimopoi soume to l mma autì èprepe na eisˆgoume trða aujaðreta shmeða O, U kai V, tètoia ste OU OV. T ra ja belti soume to l mma 3.5.10 kai katˆ sunèpeia kai ton algìrijmo me thn eisagwg tou loxoô sust matoc embadik n suntetagmènwn, katˆ to opoðo opoiad pote trða shmeða mporoôn na qrhsimopoihjoôn wc shmeða anaforˆc. GnwrÐzoume dh ìti an O, U kai V eðnai trða mh suneujeiakˆ shmeða kai A eðnai èna opoiod pote shmeðo tìte oi embadikèc suntetagmènec tou shmeðou A wc proc to OUV eðnai oi akìloujec: x A = S OUA S OUV, y A = S V OA S OUV kai z A = S UV A S OUV. EÐnai fanerì ìti isqôei: x A + y A + z A = 1. PROTASH 3.7.1 Ta shmeða tou epipèdou eðnai se èna proc èna anti-

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA122 stoiqða me ta shmeða (x, y, z) ètsi ste x A + y A + z A = 1. PROTASH 3.7.2 Gia opoiad pote shmeða A, B kai C èqoume: S ABC = S OUV x A y A z A x B y B z B x C y C z C = S OUV x A y A 1 x B y B 1 x C y C 1 Na shmei soume ed ìti gia an èna shmeðo R an kei pˆnw sthn eujeða AB, tìte oi embadikèc suntetagmènec tou shmeðou R prèpei na ikanopoioôn thn parakˆtw sqèsh, h opoða eðnai h exðswsh thc eujeðac AB : x A y A 1 x B y B 1 x R y R 1 = 0 PROTASH 3.7.3 'Estw R shmeðo pˆnw sthn eujeða P Q kai r 1 = P R, r P Q 2 = RQ P Q. Tìte: x R = r 1 x Q + r 2 x P, y R = r 1 y Q + r 2 y P, z R = r 1 z Q + r 2 z P Apìdeixh H prìtash aut eðnai ˆmesh sunèpeia thc B.3 (selðda 15). PROTASH 3.7.4 Gia duo shmeða A kai B èqoume: AB 2 = OV 2 (x B x A ) 2 + OU 2 (y B y A ) 2 + P UOV (x B x A )(y B y A ). Apìdeixh 'Estw K kai L na eðnai shmeða tètoia ste KA OU, KB OV, LA OV kai LB OU.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA123 V L B A K O U Sq ma 3.33: Tìte apì prìtash 3.2.3 (selðda 93) paðrnoume: AB 2 = AK 2 + BK 2 P AKB = OU 2 ( AK OU )2 + OV 2 ( BK OV )2 + P LAK (1) Apì to l mma 2.4.9 (selðda 34), paðrnoume: AK = S BOAV BK OU S UOV = y B y A OV = S BOAU S OV U = x A x B (2). 'Omwc apì thn prìtash 3.2.4 (selðda 94) epeid AL OV kai AK OU paðrnoume: P LAK = AL AK P OV OU UOV = (y B y A )(x B x A )P UOV. (3) Antikajist ntac thn (2) kai (3) sthn (1) paðrnoume thn zhtoômenh sqèsh. PORISMA 3.7.5 IsqÔei ìti: 2AB 2 = P OV U (x B x A ) 2 + P OUV (y B y A ) 2 + P UOV (z B z A ) 2 Apìdeixh Apì thn prìtash 3.7.1 (selðda 121):

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA124 z A z B = (x B x A ) + (y B y A ) kai apì thn prohgoômenh prìtash paðrnoume thn zhtoômenh sqèsh. SHMEIWSH: 'Enac enallaktikìc trìpoc metatrop c twn gewmetrik n posot twn se mia èkfrash pou perilambˆnei mìno anexˆrthtec metablhtèc eðnai o akìloujoc. An sumbolðsoume me A thn èkfrash pou perilambˆnei embadˆ kai Pujagìreiec diaforèc eleôjerwn shmeðwn, tìte an oi posìthtec autèc perièqoun ligìtera apì trða shmeða den qreiˆzetai na kˆnoume tðpota. 'Omwc an oi posìthtec autèc perièqoun pˆnw apì duo shmeða tìte epilègoume trða eleôjera shmeða O, U, V apì ta shmeða pou perilambˆnei h A, kai efarmìzoume tic protˆseic 3.7.2 kai 3.7.4 (selðda 122), me skopì na metatrèyoume ta embadˆ kai tic Pujagìreiec diaforèc se embadikèc suntetagmènec wc proc to OUV. T ra h A perièqei embadikèc suntetagmènec eleôjerwn shmeðwn, OU 2, OV 2, UV 2 kai S OUV. H mình algebrik èkfrash pou en nei tic parapˆnw posìthtec eðnai o tôpoc Herron Qin 16SOUV 2 = 4OU 2 OV 2 PUOV 2.(1) Antikajist ntac thn (1) sthn A paðrnoume mia èkfrash pou perilambˆnei mìno anexˆrthtec metablhtèc. 3.8 EIDIKES TEQNIKES APALOIFH- S SHMEIWN- SUSTHMA (GIB) Sto prohgoômeno kefˆlaio anafèrame to sôsthma (GIB) twn autìmatwn apodeðxewn gewmetrik n protˆsewn pou ak koun sthn kathgorða C H. Sthn enìthta aut ja parousiˆsoume thn diadikasða tou sust matoc (GIB) stic autìmatec apodeðxeic gewmetrik n protˆsewn pou ak koun sthn kathgorða C L. H mèjodoc pou parousiˆsame sto parìn kefˆlaio, anafèretai sthn epðlush gewmetrik n protˆsewn pou an koun sthn kathgorða C L. H mèjodoc sthrðzetai se èna elˆqisto sônolo kataskeu n (sunolikˆ 5 kataskeuèc), gegonìc pou epirreˆzei thn èktash twn apodeðxewn twn gewmetrik n protˆsewn. To elˆqisto sônolo kataskeu n apì th mia pleurˆ bohjˆei sthn apofug dhmiourgðac epiplèon teqnik n apaloif c shmeðwn, allˆ apì thn ˆllh sumbˆlei sthn eisagwg megˆlou pl jouc bohjhtik n shmeðwn. Endeiktikˆ anafèroume to ex c parˆdeigma, h kataskeu (P LINE W U V ) èqei antikatastajeð apì thn kataskeu (LINE W S), ìpou S eðnai èna bo-

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA125 hjhtikì shmeðo pou eisˆgetai mèsw thc kataskeu c (P RAT IO S W U V 1). Sthn enìthta aut ja anafèroume eidikèc peript seic teqnik n apaloif - c gia mia apì tic kataskeuèc pou eisˆgame sto parìn kefˆlaio me skopì thn meðwsh thc èktashc twn apodeðxewn. Sth sunèqeia ja d soume èna parˆdeigma gia na exhg soume thn leitourgða tou (GIB) sthn autìmath paragwg apodeðxewn gewmetrik n protˆsewn pou ak koun sthn kathgorða C L. Eidikèc peript seic teqnik n apaloif c thc kataskeu c (F OOT Y P U V ). P A P P A B U P Y V U B Y P=A=B V U Y P A V U Y B V U Y V U=A=B Y V P U=A Y V=B Sq ma 3.34: PERIPTWSH 1. An AB UV tìte S ABY = S ABU. Apìdeixh: AfoÔ AB UV ˆra AB UY epomènwc apì thn prìtash B.4 (selðda 15) paðrnoume thn zhtoômenh sqèsh. PERIPTWSH 2. An AB UV tìte S ABY = S ABP. Apìdeixh: AfoÔ AB UV ˆra AB P Y epomènwc apì thn prìtash B.4 (selðda 15) paðrnoume thn zhtoômenh sqèsh. PERIPTWSH 3. An A, U, V suggrammikˆ tìte S ABY = S UBV P P UAV P UV U.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA126 Apìdeixh: PaÐrnoume ton lìgo S ABY S UBV. Efarmìzoume thn prìtash B.1 (selðda 13) kai paðrnoume: S ABY S UBV = AY. T ra apì to l mmma 3.5.8 (selðda UV AY 116) èqoume: = P P UAV UV P UV U. PERIPTWSH 4. An B, U, V suggrammikˆ tìte S ABY = S AUV P P UBV P UV U. Apìdeixh: ApodeiknÔetai ìmoia me thn perðptwsh 3. PERIPTWSH 5. An AB UV tìte P ABY = P ABP. Apìdeixh: AfoÔ AB UV tìte AB Y P ˆra apì thn prìtash B.7 (selðda 87) paðrnoume P ABY = P ABP. PERIPTWSH 6. An AB UV tìte P ABY = P ABU. Apìdeixh: 'Amesh apìrroia thc prìtashc B.7. PERIPTWSH 7. An B, U, V suggrammikˆ tìte P ABY = P ABU P P BU P UBU. Apìdeixh: PaÐrnoume ton lìgo P ABY P ABU. Efarmìzoume thn prìtash 3.2.2 (selðda 91) kai paðrnoume: P ABY P ABU = BY. T ra apì to l mmma 3.5.8 (selðda BU 116) èqoume: BY = P P BU BU P UBU. PERIPTWSH 8. An A = B = P tìte P AY B = 16S2 P UV P UV U. Apìdeixh: Apì ton tôpo (B)(selÐda 110) twn teqnik n apaloif c kai qrhsimopoi ntac to gegonìc ìti A = B = P paðrnoume ton tôpo P P Y P = P P UV P P V P +P P V U P P UP P P UV P P V U P UV U Herron Qin (selðda 97) paðrnoume:. Me qr sh t ra tou tôpou P P Y P = P P UV P P V P +P P V U P P UP P P UV P P V U P UV U = 16S2 P UV P UP V P P V U +16S 2 P UV P V P U P V UP 16S 2 P UV +P UV U P UP V P UV U

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA127 = 16S2 P UV +P UP V (P UV U P P V U P V UP ) P UV U = 16S2 P UV +P UP V (P V UP P V UP ) P UV U = 16S2 P UV P UV U. PERIPTWSH 9. An A = B = U tìte P AY B = P 2 P UV P UV U. Apìdeixh: Apì ton tôpo (B)(selÐda 110) twn teqnik n apaloif c kai qrhsimopoi ntac to gegonìc ìti A = B = U paðrnoume: P UY U = P P UV P UV U P P UV P P V U P UV U = P P UV (P UV U P P V U ) P UV U = P P UV P UV P U P UV U = P P UV P V UP P UV U = P 2 P UV P UV U. PERIPTWSH 10. An A = B = V tìte P AY B = P 2 P V U P UV U. Apìdeixh: ApodeiknÔetai ìmoia me thn perðptwsh 9. PERIPTWSH 11. An A = U, B = V tìte P AY B = P P V U P P UV P UV U. Apìdeixh: Efarmìzontac ton tôpo (B)(selÐda 110) twn teqnik n a- paloif c gia A = B = U paðrnoume apeujeðac thn zhtoômenh sqèsh. Efarmog tou sust matoc (GIB) se gewmetrikèc protˆseic thc kathgorðac C L. PARADEIGMA: JEWRHMA TOU ORJOKENTROU DIATUPWSH K 1 : (P OINT S A B C): 'Estw trða aujaðreta shmeða A, B, C K 2 : (F OOT E B A C): 'Estw shmeðo E to shmeðo tom c thc eujeðac AC kai thc eujeðac pou pernˆ apì to shmeðo B kai eðnai kˆjeth sthn AC.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA128 E A H C F B Sq ma 3.35: K 3 : (F OOT F A B C): 'Estw shmeðo F to shmeðo tom c thc eujeðac BC kai thc eujeðac pou pernˆ apì to shmeðo A kai eðnai kˆjeth sthn BC. K 4 : (INT ER H (LINE A F )(LINE B E): 'Estw H to shmeðo tom c twn eujei n AF kai BE. S: IsqÔei: P ACH = P BCH. Sullog suggrammik n shmeðwn kai kˆjetwn eujei n: (H, A, F )(F, C, B)(H, E, B)(E, A, C) akìmh HAF BCF, HEB EAC. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, H, F, E ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: AF BE kai A C, B C. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = P ACH P BCH = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA129 (Apaloif shmeðou H, qrhsimopoioôme thn èkth eidik perðptwsh twn teqnik n apaloif c thc kataskeu c (F OOT) (selðda 125) ) (P BCH = P ACB, P ACH = P ACB, afoô BH AC kai CH AB.) P ACH P BCH = P ACB P ACB BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. =1. 3.9 EFARMOGES ALGORIJMOU I PARADEIGMA 1. Stic duo pleurèc AC kai BC enìc trig nou ABC, sqediˆzoume duo tetrˆgwna ACDE kai BCF G. To M eðnai to mèson thc AB. IsqÔei ìti h CM eðnai kˆjeth sthn DF. D F E C G A M B DIATUPWSH Sq ma 3.36: K 1 (P OINT S A B C): 'Estw trða aujaðreta shmeða A, B, C K 2 (T RAT IO D C A 1): 'Estw D shmeðo pˆnw sthn eujeða (T LINE C C A)

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA130 tètoio ste: r = CD CA = 1. K 3 (T RAT IO F C B 1): 'Estw F shmeðo pˆnw sthn eujeða (T LINE C C B) tètoio ste: r = CF CB = 1. K 4 (MIDP OINT M A B): 'Estw shmeðo M to mèson thc eujeðac AB. S: P DCM = P F CM. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, M, F, D ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: A B, C B, C A. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = P DCM P F CM = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou M, qrhsimopoioôme to l mma 3.5.1 (selðda 110) ) (P F CM = 1 2 P ACF, P DCM = 1 2 P BCD) P DCM P F CM = 1 2 P BCD 1 2 P ACF (Apaloif shmeðou F, qrhsimopoioôme to l mma 3.5.6 (selðda 114) ) (P ACF = 4S ABC ) 1 2 P BCD 1 2 P ACF = P BCD 4S ABC

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA131 (Apaloif shmeðou D, qrhsimopoioôme to l mma 3.5.6) (P BCD = 4S ABC ) P BCD 4S ABC = 4S ABC 4S ABC BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. 4S ABC 4S ABC = 1. PARADEIGMA 2. 'Estw A, B, C, D tèssera shmeða, pou apoteloôn armonik akoloujða kai O tuqaðo shmeðo ektìc thc eujeðac AB. Tìte kˆje diatèmnousa tèmnei tic eujeðec OA, OB, OC, OD se tèssera armonikˆ shmeða. Apìdeixh O X P R Q S Y A C B D DIATUPWSH Sq ma 3.37: K 1 (P OINT S A B X Y O): 'Estw pènte aujaðreta shmeða A, B, X, Y, O K 2 (MRAT IO C A B r): 'Estw C shmeðo pˆnw sthn AB tètoio ste: r = AC CB K 3 (MRAT IO D A B r): 'Estw D shmeðo pˆnw sthn AB tètoio ste: r = AD DB

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA132 K 4 (INT ER P (LINE O A)(LINE X Y )): 'Estw P to shmeðo tom c twn eujei n OA me XY. K 5 (INT ER Q (LINE O B)(LINE X Y )): 'Estw Q to shmeðo tom c twn eujei n OB me XY. K 6 (INT ER R (LINE O C)(LINE X Y )): 'Estw R to shmeðo tom c twn eujei n OC me XY. K 7 (INT ER S (LINE O D)(LINE X Y )): 'Estw S to shmeðo tom c twn eujei n OD me XY. S: IsqÔei P S QS = P R QR. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, S, R, Q, P, D, C ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: OA XY, OB XY, OC XY, OD XY, A B. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = P S QS P R QR = 1. kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou S, qrhsimopoioôme thn prìtash B.2 (selðda 14), to je rhma twn epðpleurwn trig nwn) ( P S QS = S ODP S OCQ )

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA133 P S QS P R QR = S ODP P R QR S ODQ (Apaloif shmeðou R, qrhsimopoioôme thn prìtash B.2 ) ( P R QR = S OCP S OCQ ) S ODP P R QR S ODQ = S ODP S OCQ S OCP S ODQ (Apaloif shmeðou Q, qrhsimopoioôme to l mma 3.5.2 (selðda 111) ) (S ODQ = S OXY S OBD S OXBY, S OCQ = S OXY S OBC S OXBY ) S ODP S OCQ S OCP S ODQ = S S ODP ( OXY S OBC ) S OXBY S OCP ( S OXY S OBD ) S OXBY (Aplopoi seic) S ODP ( S OXY S OBC S OXBY ) S OCP ( S OXY S OBD S OXBY ) = S ODP S OBC S OCP S OBD (Apaloif shmeðou P, qrhsimopoioôme to l mma 3.5.2) (S OCP = S OXY S OAC S OXAY, S ODP = S OXY S OAD S OXAY ) S ODP S OBC S OCP S OBD = ( S OXY S OAD )S S OBC OXAY S OXY S OAC S S OBD OXAY (Aplopoi seic) ( S OXY S OAD )S S OBC OXAY S OXY S OAC = S OADS OBC S S OBD S OAC S OBD OXAY (Apaloif shmeðou D, qrhsimopoioôme thn prìtash B.2 ) (S OBD = S OAB r 1, S OAD = rs OAB r 1 )

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA134 S OAD S OBC S OAC S OBD = S rs OBC( OAB ) r 1 S OAC ( S OAB r 1 ) (Aplopoi seic) S OBC ( rs OAB ) r 1 = rs S OAC ( S OBC OAB r 1 ) S OAC (Apaloif shmeðou C, qrhsimopoioôme thn prìtash B.2 ) (S OAC = rs OAB r+1, S OBC = S OAB r+1 ) rs OBC S OAC = r( S OAB ) r+1 rs OAB r+1 BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. = 1 PARADEIGMA 3. 'Estw AD kai AA 1 eðnai to Ôyoc kai to mèson tou trig nou ABC. Fèrnoume apì to shmeðo A 1 tic parˆllhlec stic eujeðec AB kai AC, oi opoðec tèmnoun thn AD sta shmeða P kai Q. Tìte isqôei ìti ta shmeða A, D, P, Q eðnai armonikˆ. C A 1 A Q D P B Sq ma 3.38:

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA135 DIATUPWSH K 1 : (P OINT S A B C) 'Estw trða aujaðreta shmeða A, B, C K 2 : (F OOT D A B C) (INT ER D (LINE B C)(T LINE A B C)) 'Estw D to shmeðo tom c thc eujeðac pou dièrqetai apì to A kai eðnai kˆjeth sthn BC. K 3 : (MIDP OINT A 1 B C) (P RAT IO A 1 B B C 1 ) 'Estw shmeðo 2 A 1 to mèson thc BC. K 4 : (INT ER P (LINE A D)(P LINE A 1 A B)) 'Estw P to shmeðo tom c thc eujeðac AD kai thc parˆllhlhc apì to shmeðo A 1 sthn AC. K 5 : (INT ER Q (LINE A D)(P LINE A 1 A C)) 'Estw Q to shmeðo tom c thc eujeðac AD kai thc parˆllhlhc apì to shmeðo A 1 sthn AC. S: IsqÔei AP DP = AQ DQ. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, Q, P, A 1, D ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: B C kai AD AC. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = AP DP AQ DQ = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou Q, qrhsimopoioôme to l mma 2.4.15 (selðda 37) )

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA136 ( AQ DQ = S ACA 1 S ADA1 ) AP DP AQ DQ = ( AP DP )( S ADA 1 S ACA1 ) (Apaloif shmeðou P, qrhsimopoioôme to l mma 2.4.15 ) ( AP DP = S ABA 1 S ADA1 ) ( S ABA 1 S ADA1 )( S ADA 1 S ACA1 ) = S ABA 1 S ACA1 (Apaloif shmeðou A 1, qrhsimopoioôme to l mma 3.5.1 (selðda 110) ) (S ACA1 = 1 2 S ABC, S ABA1 = 1 2 S ABC) 1 2 S ABC 1 2 S ABC BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. =1. PARADEIGMA 4. 'Estw F to mèson thc pleurˆc BC tou trig nou ABC. 'Estw D kai E oi orjog niec probolèc twn shmeðwn C kai B pˆnw stic pleurèc AB kai AC antðstoiqa. To shmeðo G eðnai to mèson thc DE. IsqÔei ìti h eujeða F G tèmnei kˆjeta thn DE sto shmeðo G. Apìdeixh DIATUPWSH K 1 : (P OINT S A B C) 'Estw trða aujaðreta shmeða A, B, C K 2 : (F OOT D C A B) (INT ER D (LINE A B)(T LINE C A B))

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA137 A D G E B F C Sq ma 3.39: 'Estw D to shmeðo tom c thc eujeðac pou dièrqetai apì to C kai eðnai kˆjeth sthn AB. K 3 : (F OOT E B A C) (INT ER E (LINE A C)(T LINE B A C)) 'Estw E to shmeðo tom c thc eujeðac pou dièrqetai apì to B kai eðnai kˆjeth sthn AC. K 4 : (MIDP OINT F B C) (P RAT IO F B B C 1 ) 'Estw shmeðo F 2 to mèson thc BC. K 5 : (MIDP OINT G D E) (P RAT IO G D D E 1 ) 'Estw shmeðo G 2 to mèson thc DE. S: IsqÔei P EDG = P EDF. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, G, F, E, D ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: A B, A C, B C, D E. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA138 mèloc. 'Etsi èqoume: G 1 = P EDG P EDF = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou G, qrhsimopoioôme to l mma 3.5.1 (selðda 110) ) (P EDG = 1 2 P DED) P EDG P EDF = 1 2 P DED P EDF (Apaloif shmeðou F, qrhsimopoioôme to l mma 3.5.1 ) (P EDF = 1 2 (P CDE + P BDE )) P DED 2( 1 2 P CDE+ 1 2 P BDE) (Apaloif shmeðou E, qrhsimopoioôme to l mma 3.5.3 (selðda 112) ) (P BDE = P ADBP ACB P ACA, P CDE = P CDCP BAC P ACA, P DED = P CDCP BAC P BAC P ACB +P ADA P ACB P ACA ) P CDC P BAC P BAC P ACB +P ADA P ACB P ACA 2( 1 P CDC P BAC + 1 P ADB P ACB ) 2 P ACA 2 P ACA = P CDCP BAC P BAC P ACB +P ADA P ACB P CDC P BAC +P ADB P ACB (Apaloif shmeðou D, qrhsimopoioôme to l mma 3.5.3 (selðda 112) ) (P ADB = P BACP ABC P ABA, P ADA = P BAC 2 P ABA, P CDC = 16S2 ABC P ABA ) 16S ABC 2 P P P BAC P BAC P ACB +P BAC 2 ACB ABA P ABA 16S ABC 2 P P P BAC +P BAC P ABC ACB ABA P ABA = PBACPACB PACBPABA+16S2 ABC (P ACB P ABC 16S 2 ABC ) (Herron Qin : 16SABC 2 = P BACP ACB + P ACA P ABC kai P ABC = 1 (P 2 BCB P ACA + P ABA ), P ACB = 1(P 2 BCB + P ACA P ABA ), P BAC = 1(P 2 BCB P ACA P ABA ))

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA139 16(32P BAC P ACB 16P ACB P ABA +16P ACA P ABC ) = 24 ( 2PBCB 2 +2P BCBP ACA +2P BCB P ABA ) 16(16P BAC P ACB 16P ACB P ABC +16P ACA P ABC ) 2 3 ( 4PBCB 2 +4P BCBP ACA +4P BCB P ABA ) BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. =1. PARADEIGMA 5. Stic pleurèc enìc trig nou ABC sqediˆzoume trða isìpleura trðgwna A 1 BC, AB 1 C kai ABC 1. Tìte to CA 1 C 1 B 1 eðnai parallhlìgrammo. (ShmeÐwsh: sthn diatôpwsh thc prìtashc apaitoôme: r = 3 kai autì epeid gnwrðzoume ìti to embadìn isopleôrou trig nou isoôtai: AB 1 C = AC 2 3, dhlad 1ACB 4 2 1E = 3AC 2, ìmwc to Ôyoc isìpleurou trig nou eðnai kai diˆmesoc, ètsi loipìn prokôptei ìti: 1 E = 3. Gia autìn ton 4 B AE lìgo akoloujoôme thn parakˆtw perigraf thc prìtashc.) Apìdeixh B 1 C 1 C E F A 1 A G B DIATUPWSH Sq ma 3.40: K 1 (P OINT S A B C): 'Estw trða aujaðreta shmeða A, B, C K 2 (CONST ANT r 2 3).

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA140 K 3 (MIDP OINT E A C) (P RAT IO E A A C 1 ): 'Estw E to mèson thc eujeðac 2 AC. K 4 (T RAT IO B 1 E A r): 'Estw B 1 shmeðo pˆnw sthn eujeða (T LINE E E A) tètoio ste: r = B 1E EA. K 5 (MIDP OINT F B C) (P RAT IO F B B C 1 ): 'Estw F to mèson thc eujeðac 2 BC. K 6 (T RAT IO A 1 F C r): 'Estw A 1 shmeðo pˆnw sthn eujeða (T LINE F F C) tètoio ste: r = A 1F F C. K 7 (MIDP OINT G A B) (P RAT IO G A A B 1 ): 'Estw G to mèson thc eujeðac 2 AB. K 8 (T RAT IO C 1 G B r): 'Estw C 1 shmeðo pˆnw sthn eujeða (T LINE G G B) tètoio ste: r = C 1G GB. S: S DB1 A 1 = S CB1 C 1. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, C 1, G, A 1, F, B 1, E ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: B C, A E, G A, F C, A B, G B. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = S CB 1 A 1 S CB1 C 1 = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou G 1, qrhsimopoioôme to l mma 3.5.6 (selðda 114) )

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA141 (S CB1 C 1 = 1 4 (rp B 1 BCG + 4S CB1 G)) S CB1 A 1 S CB1 C 1 = S CB1 A 1 1 4 rp B 1 BCG+S CB1 G (Apaloif shmeðou G, qrhsimopoioôme to l mma 3.5.1 (selðda 110) ) (S CB1 G = 1 2 (S BCB 1 +S ACB1 ), P B1 BCG = 1 2 (P BCB 1 P ACB1 )) 4S CB1 A 1 1 2 rp BCB 1 + 1 2 rp ACB 1 +2S BCB1 +2S ACB1 (Apaloif shmeðou A 1, qrhsimopoioôme to l mma 3.5.6 kai 3.5.7 (selðda 114) ) (S CB1 A 1 = 1 4 (rp B 1 CF 4S CB1 F )) 2rP B1 CF 8S CB1 F rp BCB1 rp ACB1 4S BCB1 4S ACB1 (Apaloif shmeðou F, qrhsimopoioôme to l mma 3.5.1) (S CB1 F = 1 2 S BCB 1, P B1 CF = 1 2 P BCB 1 ) rp BCB1 4S BCB1 rp BCB1 rp ACB1 4S BCB1 4S ACB1 (Apaloif shmeðou B 1, qrhsimopoioôme to l mma 3.5.6 kai 3.5.7) (S ACB1 = 1 4 rp CAE, P ACB1 = P ACE S BCB1 = 1 4 (rp CABE 4S BCE, P BCB1 = P BCE + 4rS ABE )) rp CABE +rp BCE 4S BCE +4r 2 S ABE rp CABE rp CAE +rp BCE rp ACE 4S BCE +4r 2 S ABE (Apaloif shmeðou E, qrhsimopoioôme to l mma 3.5.1) (P ACE = 1 2 P ACA, P CAE = 1 2 P ACA

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA142 S ABE = 1 2 S ABC, S BCE = 1 2 S ABC P BCE = 1 2 P ACB, P CABE = 1 2 (P BCB P ABC )) 1 2 rp BCB+ 1 2 rp ACB 1 2 rp ABC+2r 2 S ABC 2S ABC 1 2 rp BCB+ 1 2 rp ACB 1 2 rp ABC+2r 2 S ABC 2S ABC BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. = 1. PARADEIGMA 6. To antðstrofo enìc kôklou pou pernˆ apì kèntro antistrof c eðnai eujeða. SHMEIWSH: H diatôpwsh pou ja akolouj sei sumbolðzei thn eujeða (e) wc: QG. H prìtash katal gei sto sumpèrasma ìti h eujeða aut eðnai kˆjeth sthn OA kai autì giatð an Q h probol tou O pˆnw sthn eujeða (e) kai P to antðstrofo tou Q kai pˆroume akìmh shmeðo G pˆnw sthn (e), opìte ja isqôei ìti: GQ QP. O P R Q G DIATUPWSH Sq ma 3.41: K 1 (P OINT S O A X): 'Estw trða aujaðreta shmeða O, A, X

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA143 X G R O Q A U P Sq ma 3.42: K 2 (LRAT IO P O A r 1 ): 'Estw P shmeðo pˆnw sthn OA tètoio ste r 1 = P O OA. K 3 (INV ERSION Q P O A) (LRAT IO Q O A OA antðstrofo tou P wc proc ton kôklo (CIR O A). OP ): 'Estw Q to K 4 (MIDP OINT U P O r) (P RAT IO U P P O 1 ): 'Estw U to mèson thc eujeðac 2 OP. K 5 (INT ER R (LINE O X)(CIR U O): 'Estw R to shmeðo tom c thc eujeðac OX kai tou kôklou (CIR U O). K 6 (INV ERSION G R O A) (LRAT IO G O R P OAO P ORO ): 'Estw G to antðstrofo tou R wc proc ton kôklo (CIR O A). S: P AOG = P AOQ. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, G, R, U, Q, P, X, A, O ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: O A, O P, O U, O R.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA144 BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = P AOG P AOQ = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou G, qrhsimopoioôme to l mma 3.5.1 (selðda 110) ) (P AOG = P AORP OAO P ORO ) P AOG P AOQ = P AORP OAO P AOQ P ORO (Apaloif shmeðou R, qrhsimopoioôme to l mma 3.5.4 (selðda 113), thn perðptwsh 9 twn eidik n teqn. apal. kataskeu c (F OOT)(selÐda 127) to l mma 3.5.1 (selðda 113) kai to l mma 3.5.3 (selðda 112). (P ORO = 4P 2 XOU P OXO, P AOR = 2P XOU P AOX P OXO ) = 2P XOU P AOX P OAO P OXO P AOQ P OXO (4P 2 XOU ) = P AOXP OAO 2P AOQ P XOU (Apaloif shmeðou U, qrhsimopoioôme to l mma 3.5.1 (selðda 110) ) (P AOQ = 1 2 P XOP) = P AOXP OAO 2P AOQ ( 1 2 P XOP ) (Apaloif shmeðou Q, qrhsimopoioôme to l mma 3.5.1) (P AOQ = OA OP P OAO) = P AOXP OAO = P AOX OA P OA OP OAOP XOP P OP XOP (Apaloif shmeðou P, qrhsimopoioôme to l mma 3.5.1)

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA145 (P XOP = r 1 P AOX, OA OP = 1 r 1 ) = P AOXr 1 P AOX r 1 BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. = 1. 3.10 EMBADIKES SUNTETAGMENES KAI EIDIKA SHMEIA TRIGWN- WN Eisˆgoume mia nèa kataskeu : (ARAT IO A O U V r O r U r V ): 'Estw shmeðo A tètoio ste: r O = S AUV S OUV, r U = S OAV S OUV, r V = S OUA S OUV, oi opoðec eðnai oi embadikèc suntetagmènec tou A wc proc to OUV. Ta r O r U r V mporeð na eðnai rhtèc ekfrˆseic gewmetrik n posot twn, rhtoð algebrikoð arijmoð. H perioristik sunj kh eðnai: ta shmeða O, U, V mh suneujeiakˆ. O bajmìc eleujerðac tou A exartˆtai apì ton arijmì twn aprosdioristi n sta r O r U r V. LHMMA 3.10.1 'Estw G(Y ) mia grammik gewmetrik posìthta kai to shmeðo Y eisˆgetai apì thn kataskeu :

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA146 (ARAT IO Y O U V r O r U r V ) Tìte: Γ(Y ) = r O Γ(O) + r U Γ(U) + r V Γ(V ). Apìdeixh O Y U S V Sq ma 3.43: QwrÐc blˆbh thc genikìthtac, upojètoume ìti to shmeðo tom c twn eujei n OY kai UV eðnai to S. An h OY eðnai parˆllhlh sthn Y V tìte prèpei eðte h UY na tèmnetai me thn OV eðte h V Y na tèmnetai me thn OU. Apì thn prìtash B.4 (selðda 15), èqoume: Γ(Y ) = OY Γ(S) + Y S OY Γ(O) = OS OS ( US OS UV Apì to je rhma twn epðpleurwn trig nwn èqoume: SV Γ(V ) + Γ(U)) + Y S Γ(O) (1). UV OS r O = Y S, OY = S OUY V US OS OS S OUV, UV = S OUY S OUY V, SV UV = S OY V S OUY V Me mia apl antikatˆstash twn parapˆnw sqèsewn sthn sqèsh (1) paðrnoume thn zhtoômenh sqèsh. LHMMA 3.10.2 'Estw G(Y )=P AY B thn kataskeu : kai to shmeðo Y eisˆgetai apì (ARAT IO Y O U V r O r U r V ). Tìte:

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA147 Γ(Y ) = r O Γ(O)+r U Γ(U)+r V Γ(V ) 2(r O r U OU 2 +r O r V OV 2 +r U r V UV 2 ). Apìdeixh O Y U S V Sq ma 3.44: SuneqÐzoume apì thn apìdeixh tou parapˆnw l mmatoc. QrhsimopoioÔme thn sqèsh (B)(selÐda 110) twn teqnik n apaloif c: Γ(Y ) = OY Γ(S) + Y S OY Y S Γ(O) P OS OS OS OS OSO Γ(S) = US SV US SV Γ(V ) + Γ(U) P UV UV UV UV UV U Me mia apl antikatˆstash tou Γ(S) sto Γ(Y ) paðrnoume: Γ(Y ) r = OY US SV P OS UV UV UV U OY Y S Y S P SV OS OS OS OSO = r V P OY UV UV U r O P OS OSO. ìpou r = r O Γ(O) + r U Γ(U) + r V Γ(V ). Apì thn sqèsh (B)(selÐda 110) paðrnoume: P OSO = US P UV OV O + SV P UV OUO US SV P UV UV UV U. Tìte: Γ(Y ) r = SV = r V P OY US UV UV U r O P OY SV OS UV OV O r O P OY US SV OS UV OUO + r O P OS UV UV UV U = r O r V P OV O r O r U P OUO r U r V ( S Y UV S OUY V + S OUV S OUY V )P UV U

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA148 = r O r V P OV O r O r U P OUO r U r V P UV U. Tèloc gia tic peript seic ìpou to shmeðo Y eisˆgetai apì thn kataskeu (ARAT IO Y O U V r O r U r V ) kai jèloume na apaloðyoume to Y apì thn gewmetrik posìthta Γ = AY ja akolouj soume thn parakˆtw logik. CD 'Ena apì ta shmeða O, U, V, èstw to O, ikanopoiheð thn sunj kh ìti ta A, Y, O den eðnai suneujeiakˆ, sunep c ja èqoume: Γ = S OAY S OCAD. T ra loipìn mporoôme na qrhsimopoi soume to l mma 3.10.1 (selðda 145) gia na apaloðyoume to Y. Qrhsimopoi ntac thn kataskeu ARAT IO mporoôme na qrhsimopoioôme me eukolða tic parakˆtw kataskeuèc eidik n shmeðwn twn trig nwn: 1.(CENT ROID G A B C): To shmeðo G eðnai to barôkentro tou trig nou ABC. H kataskeu aut eðnai isodônamh thc kataskeu c (ARAT IO G A B C 1 3 1 3 1 ) 3 O periorismìc eðnai ìti ta shmeða A, B, C den eðnai suggrammikˆ. 2.(ORT HOCENT ER H A B C): To shmeðo H eðnai to orjìkentro tou trig nou ABC. H kataskeu aut eðnai isodônamh thc kataskeu c (ARAT IO H A B C P ABCP ACB 16S 2 ABC P BAC P BCA 16S 2 ABC P CAB P CBA ) 16SABC 2 O periorismìc eðnai ìti ta shmeða A, B, C den eðnai suggrammikˆ. 3.(CIRCUMCENT ER O A B C): To shmeðo O eðnai to perðkentro

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA149 tou trig nou ABC. H kataskeu aut eðnai isodônamh thc kataskeu c (ARAT IO O A B C P BCBP BAC 16S 2 ABC P ACA P ABC 16S 2 ABC P ABA P ACB ) 16SABC 2 O periorismìc eðnai ìti ta shmeða A, B, C den eðnai suggrammikˆ. 4.(INCENT ER C I A B): To shmeðo I eðnai to ègkentro tou trig nou ABC. H kataskeu aut eðnai isodônamh thc kataskeu c (ARAT IO I A B C 2P IABP IBA P AIB P ABA P IAB P IBI P AIB P ABA P IBA P IAI P AIB P ABA ) Oi periorismoð eðnai A B kai IA IB. SHMEIWSH: ParathroÔme ìti sthn teleutaða kataskeu tou egegrammˆnou kôklou to shmeðo pou kataskeuˆsame eðnai h mia koruf tou trig nou ABC. Autì ofeðletai sto gegonìc ìti oi embadikèc suntetagmènec tou ègkentrou eðnai tetˆrtou bajmoô, ìsec dhlad kai oi aprosdioristðec stic embadikèc suntetagmènec. Oi aprosdiristðec autèc ofeðlontai ston lìgo ìti to prìgramma den mporeð na diaqwrðsei tic embadikèc suntetagmènec tou ègkentrou I apì ekeðnec twn tri n parakèntrwn I a, I b, I c tou trig nou ABC dðqwc na qrhsimopoihjoôn anis thtec metaxô twn embadik n suntetagmènwn. 'Etsi loipìn, autì pou kˆnoume eðnai na antistrèyoume to prìblhma. Oi apodeðxeic twn parapˆnw isodunami n brðskontai sto parˆrthma G, selðda 224. 3.11 EFARMOGES ALGORIJMOU II PARADEIGMA 1.(JEWRHMA EULER) 'Estw O to kèntro perigeggrammènou kôklou sto trðgwno ABC. 'Estw

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA150 M to kèntro bˆrouc kai H to orjìkentro tou trig nou ABC. Tìte to shmeðo M brðsketai pˆnw sto eujôgrammo tm ma OH kai qwrðzei to tm ma autì se lìgo 1 2. Apìdeixh C H A O M B DIATUPWSH Sq ma 3.45: K 1 : (P OINT S A B C): 'Estw trða aujaðreta shmeða A, B, C K 2 : (CIRCUMCENT ER O A B C) (ARAT IO O A B C P BCBP BAC 16S 2 ABC to perðkentro tou trig nou ABC. P ACA P ABC 16S 2 ABC P ABA P ACB )): 'Estw shmeðo O 16SABC 2 K 3 : (CENT ROID M A B C) (ARAT IO M A B C 1 3 shmeðo M to barôkentro tou trig nou ABC. 1 3 1): 'Estw 3 K 4 : (LRAT IO H M O 2) (P RAT IO H M M O 2): 'Estw H shmeðo pˆnw sthn eujeða M O tètoio ste: HM = 2M O. S: IsqÔei: P ABC = P CBH. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, H, M, O ekteloôme ta parakˆtw b mata.

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA151 BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: A, B, C mh suggrammikˆ, O M. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = P ABC P CBH = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou H, qrhsimopoioôme to l mma 3.5.1 (selðda 110) ) (P CBH = 3P CBM 2P CBO ) P ABC P CBH = P ABC 3P CBM 2P CBO (Apaloif shmeðou M, qrhsimopoioôme to l mma 3.10.1 (selðda 145) ) (P CBM = 1 3 (P BCB + P ABC )) P ABC 3( 1 3 P BCB+P ABC ) 2P CBO = 3P ABC 6P CBO +3P BCB +3P ABC (Apaloif shmeðou O, qrhsimopoioôme to l mma 3.10.1) (P CBO = 1 2 P BCB) 3P ABC 6( 1 2 P BCB)+3P BCB +3P ABC = 4S ABC 4S ABC = 2P ABC 2P ABC BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. =1. PARADEIGMA 2. 'Estw trðgwno ABC. Fèrnoume parˆllhlh proc thn mia pleurˆ tou

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA152 trig nou h opoða pernˆ apì to kèntro bˆrouc tou trig nou G. Tìte h parˆllhlh aut qwrðzei to trðgwno se duo mèrh, se lìgo 4 : 5. Apìdeixh A P G Q B C DIATUPWSH Sq ma 3.46: K 1 : (P OINT S A B C): 'Estw trða aujaðreta shmeða A, B, C K 2 : (CENT ROID G A B C) (ARAT IO G A B C 1 3 shmeðo G to barôkentro tou trig nou ABC. K 3 : (INT ER P (LINE A B)(P LINE G B C) 1 3 1): 'Estw 3 (INT ER P (LINE A B)(LINE G N), (P RAT IO N G B C 1): 'Estw P to shmeðo tom c thc eujeðac AB kai thc eujeðac pou pernˆ apì to shmeðo G kai eðnai parˆllhlh sthn BC. K 4 : (INT ER Q (LINE A C)(P LINE G B C) (INT ER Q (LINE A C)(LINE G N), (P RAT IO N G B C 1): 'Estw Q to shmeðo tom c thc eujeðac AC kai thc eujeðac pou pernˆ apì to shmeðo G kai eðnai parˆllhlh sthn BC. S: IsqÔei: 4S BCQP = 5S AP Q. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, Q, P, G

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA153 ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: AB GN, AC GN, B C. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = 4S BCQP 5S AP Q = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou Q, qrhsimopoioôme ta l mmata 3.5.1 kai 3.5.2 (selðda 110) ) (S AP Q = S AGN S AP C S AGCN, S AGN = S AGC S AGB = S AGC +S ABG = S ABGC, S AGCN = S AGC + S ACN, S ACN = S ACG S ACB, S AGCN = S ABC S BCQP = S P BCQ = S AGN S P BC S CGN S P BCA S AGCN, S AGN = S ABGC = S BGC S BAC, S CGN = S CGB ˆra S AP Q = S ABGCS ACP S ABC kai S BCQP = S BCP S ABC S BCG S ACP S ABC ) 4S BCQP 5S AP Q = 4(S BCP S ABC S BCG S ACP )S ABC 5( S ABGC S ACP )S ABC = 4(S BCP S ABC S BCG S ACP ) 5S ABGC S ACP (Apaloif shmeðou P, qrhsimopoioôme ta l mmata 3.5.1 kai 3.5.2) (S ACP = S AGN S ACB S AGBN, S AGN = S AGC S AGB, S AGBN = S AGB + S ABN = S AGB + S ABG + S ABC = S ABC S BCP = S BGN S BCA S AGBN, S AGBN = S ABC, S BGN = S BGC ˆra S ACP = S ABGC, S BCP = S BCG ) = 4(S ABGCS BCG S ABC +S BGC S 2 ABC )S ABC 5S ABGC ( S ABGC S ABC )S ABC = 4(S ABGC+S ABC )S BCG 5(S ABGC ) 2

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA154 (Apaloif shmeðou G, qrhsimopoioôme to l mma 3.10.1 (selðda 145) ) (S BCG = 1 3 S ABC, S ABGC = 2 3 S ABC) = 4(5S ABC)3 2 S ABC 5(2S ABC ) 2 3 2 BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtès- aplopoi seic. =1. PARADEIGMA 3. 'Estw A, B, C, D tèssera shmeða pˆnw ston kôklo kèntrou O. Tìte oi kˆjetec apì ta barôkentra twn tessˆrwn trig nwn ABC, ABD, ACD kai BCD stic efaptomènec tou kôklou O sta shmeða D, C, B, A tèmnontai. Apìdeixh D B L P O N G1 M C A DIATUPWSH Sq ma 3.47:

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA155 K 1 : (P OINT S A B C, D): 'Estw tèssera aujaðreta shmeða A, B, C, D K 2 : (CIRCUMCENT ER O A B C): 'Estw shmeðo O to perðkentro tou trig nou ABC. K 3 : (CENT ROID M A B C): 'Estw shmeðo D to barôkentro tou trig nou ABC. K 4 : (CENT ROID N A B D): 'Estw shmeðo E to barôkentro tou trig nou ABD. K 5 : (CENT ROID L B C D): 'Estw shmeðo F to barôkentro tou trig nou BCD. K 6 : (INT ER P (P LINE M O D)(P LINE N O C)): 'Estw P to shmeðo tom c thc parˆllhlhc apì to shmeðo M sthn OD kai thc parˆllhlhc apì to shmeðo N sthn OC. S: IsqÔei: S AOP = S AOL. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, P, L, N, M ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: A, B, C mh suggrammikˆ, A, B, D mh suggrammikˆ, B, C, D mh suggrammikˆ, A, B, C mh suggrammikˆ kai OD OC. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = S AOP S AOL = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou P, qrhsimopoioôme to l mma 2.4.16 (selðda 38) )

KEFŸALAIO 3. AUTOMATES APODEIXEIS STHN EPIPEDH GEWMETRIA156 (S AOP = S CMON S ADO +S CDO S AOM S CDO ) S CMCN S ADO +S CDO S AOM S AOL ( S CDO ) (Apaloif shmeðou L, qrhsimopoioôme to l mma 3.10.1 (selðda 145) ) (S AOL = 1 3 (S ADO + S ACO + S ABO ) 3(S CMNO S ADO +S CDO S AOM ) (S ADO S ACO S ABO )S CDO (Apaloif shmeðou N, qrhsimopoioôme to l mma 3.10.1) (S CMON = (S COM + 1 3 S CDO 1 3 S BCO 1 3 S ACO)) ( 3)( 3S COM S ADO +3S CDO S AOM S CDO S ADO +S BCO S ADO +S ADO S ACO ) 3S CDO (S ADO +S ACO +S ABO ) (Apaloif shmeðou M, qrhsimopoioôme to l mma 3.10.1) (S AOM = 1 3 (S ACO + S ABO ), S COM = 1 3 (S BCO + S ACO )) 3(3S CDO S ADO +3S CDO S ACO +3S CDO S ABO ) 3 2 (S ADO +S ACO +S ABO )S CDO BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. =1.

Kefˆlaio 4 SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN 4.1 TRIGWNOMETRIKES SUNARTH- SEIS Se autì to kefˆlaio ja eisˆgoume thn ènnoia twn trigwnometrik n sunart sewn, me skopì oi tôpoi tou embadoô trig nou kai thc Pujagìreiac diaforˆc trig nou na ekfrastoôn sunart sei twn trigwnometrik n sunart sewn. EpÐshc qrhsimopoioôme thn ènnoia tou embadoô trig nou gia na orðsoume tic trigwnometrikèc sunart seic. Na shmei soume ìti oi sunart seic tou hmitìnou kai sunhmitìnou orðzontai me ton sun jh trìpo. ORISMOS 4.1.1 To hmðtono miac gwnðac A eðnai duo forèc to prosanatolismèno embadìn tou trig nou ABC ìtan: AB = AC = 1. Idiìthtec hmitìnou. 1. sin(0 ) = sin(180 ) = 0, sin(90 ) = 1 157

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN158 2. sin( Q) = sin(180 Q). 3. sin( A) = sin( B)an kai mìno an A = B A + B = 180. 4. A + B 180 kai A < B sunepˆgetai sin( A) < sin( B) (Sunèpeia thc anis thtac epig niwn trig nwn) Oi parapˆnw idiìthtec aporrèoun apeujeðac apì ton orismì tou hmitìnou. PROTASH 4.1.2 S ABC = 1 2 Apìdeixh AB AC sin( A). P A Q B C Sq ma 4.1: 'Estw duo shmeða P kai Q pˆnw stic pleurèc AB kai AC antðstoiqa ètsi ste AP = AQ = 1. Apì to je rhma twn epig niwn trig nwn ja pˆroume ABC = AP Q AB AC = AP Q AB AC = 1 AB AC sin( A). AP AQ 2 Efarmìzontac thn parapˆnw prìtash kai gia tic treðc gwnðec tou trig nou ABC ja pˆroume: ABC = 1 2 b c sin( A) = 1 2 a c sin( B) = 1 2 a b sin( C) Wc sunèpeia twn parapˆnw èqoume tic akìloujec protˆseic. PROTASH 4.1.3 (Nìmoc hmitìnwn)

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN159 Se èna trðgwno ABC, an BC = a, CA = b kai AB = c tìte ja isqôei ìti: sin( A) a = sin( B) b = sin( C) c PROTASH 4.1.4 'Estw P shmeðo tou epipèdou kai P A, P B, P C treðc hmieujeðec tètoiec ste: AP C = y, CP B = z kai AP B = x = y + z < 180. Tìte ta shmeða A, B kai C eðnai suggrammikˆ an kai mìno an sin( x) P C = sin( y) P B = sin( z) P A. Apìdeixh P y x z A C B Sq ma 4.2: An ta shmeða A, B kai C eðnai suggrammikˆ tìte èqoume: P AB = P AC + P CB dhlad P A P B sin(x) = P A P C sin(y) + P B P C sin(z) Diair ntac t ra kai tic duo pleurèc thc isìthtac me to P A P B P C paðrnoume to zhtoômeno.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN160 AntÐstrofa, apì thn sqèsh: sin( x) P C = sin( y) P B isqôei ìmwc ìti = sin( z) P A prokôptei ìti P AB = P AC + P CB ABC = P AB P AC P CB = 0 Epomènwc ta shmeða A, B kai C eðnai suggrammikˆ. ORISMOS 4.1.5 OrÐzoume to sunhmðtono thc gwnðac A wc ex - c: cos( A) = { sin(90 A), an A 90 ; sin( A 90 ), an A > 90. PROTASH 4.1.6 (Nìmoc sunhmitìnwn) Se èna trðgwno ABC èqoume: P ABC = 2 AB CB cos( B) 1 2 Apìdeixh Apì to parapˆnw sq ma parathroôme ìti: an h B 90 tìte C ABC = BEC = 1 2 AB BC sin(90 B) = AB BC cos( B). 'Etsi loipìn: P ABC = 4C ABC = 2 AB BC cos( B) 'Omoia apodeiknôetai kai h perðptwsh: B > 90. ORISMOS 4.1.7 H prosanatolismènh gwnða ABC eðnai ènac pragmatikìc arijmìc tètoioc ste:

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN161 F E C A B Sq ma 4.3: 1. h apìluth tim thc gwnðac ABC eðnai Ðsh me thn tim thc kanonik c gwnðac ABC kai 2. h ABC èqei Ðdio prìshmo me to S ABC. ORISMOS 4.1.8 EpekteÐnoume ton orismì tou sunhmitìnou kai tou hmitìnou stic prosanatolismènec gwnðec wc ex c: 'Estw A eðnai mia arnhtik gwnða tìte: sin( A) = sin( A), cos( A) = cos( A) MporoÔme plèon na orðsoume to prosanatolismèno embadìn kai th Pujagìreia diaforˆ trig nou qrhsimopoi ntac thn ènnoia thc prosanatolismènhc gwnðac. PROTASH 4.1.9 S ABC = 1 2 AB BC sin( ABC), P ABC = 2 AB BC cos( ABC) ORISMOS 4.1.10 H prosanatolismènh gwnða anˆmesa se duo kateujunìmenec eujeðec P Q kai AB, sumbolðzetai wc ex c: (P Q, AB) kai orðzetai ìpwc parakˆtw.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN162 Q Y P O X A B Sq ma 4.4: 'Estw shmeða O, X kai Y tètoia ste ta OY P Q kai OXBA na eðnai parallhlìgramma. Tìte orðzoume: (P Q, AB) = (XOY ) Idiìthtec prosanatolismènwn gwni n 1. (P Q, AB) = (AB, P Q) 2. (P Q, AB) = 180 + (QP, AB) = 180 + (P Q, BA) 3. (P Q, AB) = (QP, BA) 4. (P Q, AB) + (AB, CD) = (P Q, CD) Oi parapˆnw idiìthtec aporrèoun apeujeðac apì ton orismì twn prosanatolismènwn gwni n. T ra mporoôme na orðsoume to prosanatolismèno embadìn kai thn Pujagìreia diaforˆ tetrpleôrwn qrhsimopoi ntac tic trigwnometrikèc sunart seic. PROTASH 4.1.11 a. S ABCD = 1 2 AC BD sin( (AC, BD)) b. P ABCD = 2 AC BD cos( (AC, DB))

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN163 Apìdeixh a. 'Estw shmeðo X tètoio ste to AXDB na eðnai parallhlìgrammo. Tìte ja èqoume: AX = BD. Apì thn prìtash B.6 (selðda 16), paðrnoume: S ABCD = S AACX = S XAC = 1XAAC sin( XAC) = 1 ACBD sin( (AC, BD)). 2 2 B D C A X Sq ma 4.5: b. 'Estw shmeðo X tètoio ste to CXDB na eðnai parallhlìgrammo. Tìte ja èqoume: CX = BD. Apì thn prìtash B.11 (selðda 93), paðrnoume: P ABCD = P CBAD = P CCAX = P XCA = 2 XA AC cos( XCA) = 2 AC BD cos( (AC, DB)) B D A C X Sq ma 4.6: EFARMOGES A. (O tôpoc Herron Qin) Se èna trðgwno ABC isqôei: 16S 2 ABC = 4 AB2 CB 2 P 2 ABC.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN164 Apìdeixh Apì thn prìtash 4.1.9 paðrnoume: sin( ABC) = 2S ABC AB BC cos( ABC) = P ABC 2AB BC QrhsimopoioÔme thn tautìthta: sin( ABC) 2 + cos( ABC) 2 = 1 èqoume: 4S 2 ABC AB 2 BC 2 + P 2 ABC 4AB 2 BC 2 = 1 To opoðo mac dðnei to zhtoômeno. B. (O tôpoc Herron Qin gia tetrˆpleura) Gia èna tetrˆpleuro ABCD isqôei: 16S ABCD = 4AC 2 BD 2 P 2 ABCD. Apìdeixh Apì th prìtash 4.1.11 paðrnoume: sin( (AC, BD)) = 2S ABCD AC BD cos( (AC, DB)) = P ABCD 2AC BD 1 kai qrhsimopoi ntac pˆli thn tautìthta: sin( (AC, BD)) 2 +cos( (AC, DB)) 2 = èqoume: 4S 2 ABCD AC 2 BD 2 + P 2 ABCD 4AC 2 BD 2 = 1 To opoðo mac dðnei to zhtoômeno.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN165 4.2 KUKLOI Se aut thn enìthta ja melet soume ektenèstera èna gewmetrikì antikeðmeno pou sunant same kai sto kefˆlaio 3, touc kôklouc. ShmeÐa pou brðskontai pˆnw ston Ðdio kôklo onomˆzontai omokuklikˆ. P1 B A P Q Sq ma 4.7: 'Estw P shmeðo pˆnw ston kôklo kai shmeðo P 1 to antidiametrikì tou. Fèrnoume thn efaptomènh tou kôklou sto P kai paðrnoume shmeðo Q pˆnw sthn efaptomènh tètoio ste S P P1 Q > 0. Gia opoiod pote shmeðo A pˆnw ston kôklo, orðzoume thn A na eðnai h prosanatolismènh gwnða AP Q. To tìxo P A pou perièqetai sthn AP Q onomˆzetai antðstoiqo tìxo thc. Akìmh gia duo shmeða A kai B pˆnw ston kôklo isqôei ìti: B A = BP A. T ra ja orðsoume thn prosanatolismènh qord. H prosanatolismènh qord ÃB èqei apìluth tim thn tim tou eujôgrammou tm matoc AB kai prìshmo Ðdio me to prìshmo thc prosanatolismènhc

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN166 gwnðac BP A isodônama to Ðdio prìshmo me to S BP A. 'Opwc blèpoume sto parapˆnw sq ma, ìtan S BP A > 0 tìte ÃB > 0. H prosanatolismènh qord P A eðnai pˆnta jetik. PROTASH 4.2.1 SÔmfwna me ta parapˆnw paðrnoume: ÃB = d sin( BP A) = d sin( B A) P A = d sin( AP Q) ìpou d eðnai h diˆmetroc tou kôklou. Apìdeixh Ta prìshma kai stic duo pleurèc twn exis sewn eðnai ta Ðdia. Mènei na elègxoume thn apìluth tim twn exis sewn. 'Etsi loipìn, apì ton nìmo twn hmitìnwn sto trðgwno BAP èqoume: AB = sin( BP A) BP sin( BAP ) 'Omwc anˆloga apì th jèsh tou shmeðou A pˆnw ston kôklo diakrðnoume tic parakˆtw peript seic: an oi gwnðec BAP, BP 1 P eðnai eggegrammènec pou baðnoun sto Ðdio tìxo tìte: BAP = BP 1 P an oi gwnðec BAP, BP 1 P eðnai eggegrammènec kai baðnoun se tìxo pou dèqetai h mia thn ˆllh tìte: BAP + BP 1 P = 180. Se kˆje perðptwsh paðrnoume: BP = sin( BAP ) BP sin( BP 1 P ) Akìmh sin( BP 1 P ) = BP P P 1. 'Ara BP = P P sin( BP 1 P ) 1. Katal goume sunep c: AB = sin( BP A) P P 1, ìpou d = P P 1

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN167 T ra sin( AP Q) = sin( π 2 P 1P A) = cos( P 1 P A) = P A P P 1. PROTASH 4.2.2 'Estw d na eðnai h diˆmetroc kôklou, ston opoðo eðnai eggegrammèno trðgwno ABC. Tìte: S ABC = g AB gbc gac 2d Apìdeixh B A P C Sq ma 4.8: 'Opwc faðnetai apì to sq ma, ja èqoume: ABC = CP A ABC + CP A = 180. SÔmfwna me thn prìtash 4.2.1 (selðda 166) kai thn prìtash 4.1.2 (selðda 158) èqoume: 1 2d ABC = 1AB BC sin( ABC) = 1 AB BC sin( CP A) = 2 2 AB BC CA Gia ta prìshma t ra thc exðswshc, diapist noume polô eôkola pwc an enallˆxoume th jèsh duo koruf n tou trig nou, ta prìshma kai stic duo pleurèc thc exðswshc ja allˆxoun. Gia autì ton lìgo arkeð na elègxoume mìno thn perðptwsh pou deðqnei to sq ma. 'Etsi loipìn ja pˆroume: S ABC 0, S CP B 0, S BP A 0, S CP A 0

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN168 tìte: ÃB 0, BC 0, ÃC 0. PROTASH 4.2.3 (OMOKUKLIKO JEWRHMA) An oi kôkloi pou perièqoun ta eggegrammèna trðgwna ABC, XY Z eðnai Ðsoi tìte: S ABC S XY Z = g AB gbc gca gxy gy Z gzx. Apìdeixh H apìdeixh eðnai ˆmesh apìrroia thc prohgoômenhc prìtashc. PROTASH 4.2.4 'Estw d na eðnai h diˆmetroc tou eggegrammènou trig nou ABC. Tìte: P ABC = 2ÃB CB cos( CP A). Apìdeixh B A B P C O P C O A Sq ma 4.9: Apì thn prìtash 4.1.6(selÐda 160) paðrnoume:

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN169 P ABC = 2AB BC cos( ABC) Apì ta parapˆnw sq mata parathroôme ìti: ABC = CP A ABC + CP A = 180 tìte: P ABC = 2AB BC cos( CP A). Mènei na elègxoume pìte ta prìshma kai stic duo pleurèc thc exðswshc eðnai Ðdia. Katarq n an enallˆxoume th jèsh twn shmeðwn A kai C ta prìshma kai stic duo pleurèc thc exðswshc den allˆzoun. Mènoun loipìn oi parakˆtw peript seic: an to P brðsketai pˆnw sto tìxo AC an to P brðsketai pˆnw sto tìxo AB Sthn pr th perðptwsh èqoume ìti ÃB 0, CB 0 Epeid ABC + CP A = 180 ta P ABC kai cos( CP A) èqoun pˆnta antðjeta prìshma. H prìtash eðnai alhj c se aut thn perðptwsh. Sthn deôterh perðptwsh èqoume ìti ÃB 0, CB 0 Epeid ABC = CP A ta P ABC prìshma. kai cos( CP A) èqoun pˆnta Ðdia PROTASH 4.2.5 (OMOKUKLIKO JEWRHMA GIA TH PUJAGOREIA DIAFORA)

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN170 An oi kôkloi stouc opoðouc eðnai eggegrammèna ta trðgwna ABC, XY Z eðnai Ðdioi kai P ABC = 0 tìte: P ABC AB P XY Z = g BC g cos( AP C) gxy Y g. Z cos( XP Z) Apìdeixh H apìdeixh eðnai ˆmesh apìrroia thc prohgoômenhc prìtashc. 'Estw BC h prosanatolismènh qord tou kôklou me kèntro to shmeðo O kai diˆmetro BB. OrÐzoume thn epiqord thc prosanatolismènhc qord c BC na eðnai h BC thc opoðac h apìluth tim eðnai Ðsh me CB kai èqei to Ðdio prìshmo me to prìshmo thc pujagìreiac diaforˆc P BP C. B O B' C Sq ma 4.10: EÐnai faner plèon h sqèsh: BC 2 + BC 2 = d 2. PROTASH 4.2.6 'Estw A, B, C trða omokuklikˆ shmeða. Tìte isqôei: P ABC = 2g AB g CB d CA d.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN171 Apìdeixh Apì ton tôpo Herron Qin: 16S 2 ABC = 4AB2 AC 2 P 2 BAC kai thn prìtash 4.2.2 (selðda 167), paðrnoume: P ABC = 4AB 2 AC 2 16S 2 ABC = 4AB2 CB 2 (d 2 AC 2 ) d 2 = 2g AB g CB d CA d. P ABC = 2 g AB g CB d CA d Apì thn prìtash 4.2.4 (selðda 168), mporoôme eôkola na elègxoume ìti ta prìshma kai sta duo mèlh thc exðswshc eðnai Ðdia. PROTASH 4.2.7 'Estw A, B, C, D tèssera omokuklikˆ shmeða. 'Estw E to shmeðo tom c tou kôklou kai thc eujeðac pou pernˆ apì to shmeðo AC gbd geb D kai eðnai parˆllhlh sthn AC. Tìte isqôei: S ABCD = g, ìpou 2d d h diˆmetroc tou kôklou. Apìdeixh An AC BD èqoume E = B kai S ABCD = 0. 'Etsi loipìn, epeid DE AC èqoume: S ABCD = g AC ged S EBD = g AC g BD g EB 2d. PROTASH 4.2.8 (TO JEWRHMA TOU PTOLEMAIOU) 'Estw tèssera shmeða A, B, C, D omokuklikˆ. Tìte isqôei: ÃB CD + ÃD BC = ÃC BD. Apìdeixh

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN172 D E O A C B Sq ma 4.11: Fèrnoume apì to shmeðo D parˆllhlh sthn AC pou tèmnei ton kôklo sto shmeðo E. Apì thn prìtash 4.2.7 (selðda 171), paðrnoume: gac g BD g EB 2d = S ABCD = S BCE + S EAB (1) H sqèsh (1) me efarmog thc prìtashc 4.2.2 (selðda 167), gðnetai: S BCE + S EAB = g BC g EC g EB+ g EA g BA g BE 2d (2) An pˆroume to B wc shmeðo anaforˆc ja èqoume: ẼB = BE, ÃE = CD, CE = ÃD. Me mia apl antikatˆstash twn parapˆnw sqèsewn sthn (2) ja pˆroume thn zhtoômenh sqèsh. PROTASH 4.2.9 'Estw AB = d h diˆmetroc enìc kôklou kai P, Q duo shmeða pˆnw ston kôklo. Tìte isqôei: d P Q = ÃQÂP ÃP ÂQ.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN173 D E O A C B Sq ma 4.12: Apìdeixh B O J P Q A Sq ma 4.13: Efarmìzoume to je rhma tou PtolemaÐou gia ta shmeða A, B, P, Q: ÃB P Q + ÃQ BP = ÃP BQ Epilègoume J wc shmeðo anaforˆc. An S JAB < 0 èqoume:

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN174 ÃB = d, ÂQ = BQ, ÂP = BP. An S JAB > 0 J = A èqoume: ÃB = d, ÂQ = QB, ÂP = P B. H zhtoômenh sqèsh eðnai plèon faner. PROTASH 4.2.10 'Estw AB = d h diˆmetroc enìc kôklou kai P, Q duo shmeða pˆnw ston kôklo. Tìte isqôei: d P Q = ÃQÃP + ÂP ÂQ. Apìdeixh B S O J P Q A Sq ma 4.14: Fèrnoume to antidiametrikì tou shmeðou Q. Efarmìzoume to je rhma tou PtolemaÐou gia ta shmeða A, B, P, S: ÃB P B + ÃS P B = ÃB P S Epilègoume J wc shmeðo anaforˆc. An ÃB kai QS èqoun to Ðdio prìshmo tìte èqoume: ÃB P S = d P Q

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN175 BS = ÃQ ÃS P B = ÂQÂP An t ra ÃB kai QS èqoun diaforetikì prìshmo tìte èqoume: ÃB P S = d P Q BS = ÃQ ÃS P B = ÂQÂP H zhtoômenh sqèsh eðnai alhj c kai gia tic duo peript seic. 4.3 TEQNIKES APALOIFHS OMOKUK- LIKWN SHMEIWN Eisˆgoume thn kataskeu twn omokuklik n shmeðwn. K 17 (CIRCLE Y 1,..., Y m ), (m 3). Ta shmeða Y 1,..., Y m brðskontai pˆnw ston Ðdio kôklo. O bajmìc eleujerðac thc kataskeu c eðnai: m + 3. Ikanèc kai anagkaðec sunj kec den upˆrqoun. To parakˆtw l mma eðnai ˆmesh sunèpeia twn protˆsewn: 4.2.2, 4.2.6, 4.2.7, 4.2.8, 4.2.9, 4.2.10 (selðdec 167-174). LHMMA 4.3.1 'Estw A, B, C, D tèssera shmeða pˆnw se kôklo kèntrou O kai diamètrou d. 'Estw A to shmeðo anaforˆc. SumbolÐzoume: B

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN176 thn gwnða AOB 2. Tìte isqôoun ta parakˆtw: S BCD = g BC g DC g DB 2d P BCD = 2 BC DC DB BD = d sin(bd) BD = d cos(bd) To l mma 4.3.1 (selðda 175) mac bohjˆei na ekfrˆsoume tic Pujagìreiec diaforèc omokuklik n shmeðwn se sqèseic pou apoteloôntai mìno apì thn diˆmetro tou kôklou kai tic trigwnometrikèc sunart seic anexˆrthtwn gwni n. EFARMOGES PARADEIGMA 1.(TO JEWRHMA SIMSON) 'Estw D shmeðo tou perigegrammènou kôklou se trðgwno ABC. Apì to shmeðo D fèrnoume kˆjetec stic pleurèc tou trig nou ABC. 'Estw E to shmeðo tom c thc kajètou apì to shmeðo D sthn pleurˆ BC, F to shmeðo tom c thc kajètou apì to shmeðo D sthn pleurˆ AC, G to shmeðo tom c thc kajètou apì to shmeðo D sthn pleurˆ AB. Ta shmeða E, F, G eðnai suggrammikˆ. DIATUPWSH

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN177 D E C F O A G B Sq ma 4.15: K 1 (CIRCLE A, B, C, D): 'Estw tèssera aujaðreta omokuklikˆ shmeða A, B, C, D. K 2 (F OOT E D B C): 'Estw E to shmeðo tom c thc eujeðac B, C kai thc eujeðac (T LINE D B C). K 3 (F OOT F D A C): 'Estw F to shmeðo tom c thc eujeðac AC kai thc eujeðac (T LINE D B C). K 4 (F OOT G D A B): 'Estw G to shmeðo tom c thc eujeðac AB kai thc eujeðac (T LINE D A B).. K 5 (INT ER H (LINE E F )(LINE A B)): 'Estw H to shmeðo tom c twn eujei n EF kai AB. S: IsqÔei: AG BG = AH BH. ALGORIJMOS

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN178 BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, H, G, F, E, D, C, B, A ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: C B, A C, A B, EF AB, G B, B H. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = AG BG AH BH = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou H, qrhsimopoioôme thn prìtash B.2 (selðda 14),(je rhma epðpleurwn trig nwn) ) ( AH BH = S AEF S BEF ) AG BG AH BH = AG BG S AEF S BEF (Apaloif shmeðou G, qrhsimopoioôme thn prìtash B.8 (selðda 89) ) ( AG BG = P BAD P ABD ) P BAD S BEF S AEF ( P ABD ) (Apaloif shmeðou F, qrhsimopoioôme to l mma 3.5.3 (selðda 112) )

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN179 (S AEF = P CADS ACE P ACA, S BEF = P ACDS ABE P ACA ) P BAD P ACD P ACA S ABE P ABD ( P CAD S ACE )P ACA = P BADP ACD S ABE P ABD P CAD S ACE (Apaloif shmeðou E, qrhsimopoioôme to l mma 3.5.3) (S ACE = P BCDS ABC P BCB, S ABE = P CBDS ABC P BCB ) P BAD P ACD P CBD P BCB S ABC P CAD ( P BCD S ABC )P ABD P BCB = P BADP ACD P CBD P CAD P BCD P ABD (Apaloif shmeðou A, B, C, D, qrhsimopoioôme to l mma 4.3.1 (selðda 175) ) (P ABD = 2( BDÃB cos(ad)), P BCD = 2( CD BC cos(bd)) P CAD = 2(ÃDÃC cos(cd)), P CBD = 2( BD BC cos(cd)) P ACD = 2( CDÃC cos(ad)), P BAD = 2(ÃDÃB cos(bd))) (2 g AD g AB cos(bd))( 2 g CD g AC cos(ad))(2 g BD g BC cos(cd)) (2 g AD g AC cos(cd))( 2 g CD g BC cos(bd))( 2 g BD g AB cos(ad)) BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. =1 PARADEIGMA 2 (JEWRHMA TOU PASCAL PANW SE KUKLO)

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN180 'Estw A, B, C, D, E, F èxi shmeða pˆnw ston kôklo. 'Estw P to shmeðo tom c twn eujei n AB kai DF, Q to shmeðo tom c twn eujei n AB kai DF, S to shmeðo tom c twn eujei n AB kai DF. Ta shmeða P, Q, S eðnai sugrammikˆ. C S E D Q A P B F DIATUPWSH Sq ma 4.16: K 1 (CIRCLE A B C D F E): 'Estw èxi aujaðreta omokuklikˆ shmeða A, B, C, D, F, E. K 2 (INT ER P (LINE D F )(LINE A B)): 'Estw P to shmeðo tom c twn eujei n DF kai AB. K 3 (INT ER Q (LINE F E)(LINE B C)): 'Estw Q to shmeðo tom c twn eujei n F E kai BC. K 4 (INT ER S (LINE E A)(LINE C D)): 'Estw S to shmeðo tom c twn eujei n EA kai CD. K 5 (INT ER H (LINE P Q)(LINE C D)): 'Estw H to shmeðo tom c

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN181 twn eujei n P Q kai CD. S: IsqÔei: CS DS = CH DH. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, H, S, Q, P, E, F, D, C, B, A ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: DF AB, EF BC, AE CD, P Q CD, D S, D H. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = CS DS CH DH = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou H, qrhsimopoioôme thn prìtash B.2 (selðda 14),(je rhma epðpleurwn trig nwn) ) ( CH DH = S CP Q S DP Q ) CS DS CH DH = CS DS S CP Q S DP Q (Apaloif shmeðou S, qrhsimopoioôme thn prìtash B.2)

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN182 ( CS DS = S ACE S ADE ) S DP Q S ACE S CP Q S ADE (Apaloif shmeðou Q, qrhsimopoioôme to l mma 2.4.14 (selðda 37) ) (S CP Q = S CF ES BCP S BF CE, S DP Q = S DEP S BCF S BF CE ) S ACE ( S DEP S BCF )S BF CE ( S CF E S BCP )S ADE ( S BF CE ) = S ACES DEP S BCF S CF E S BCP )S ADE (Apaloif shmeðou P, qrhsimopoioôme to l mma 2.4.14) (S BCP = S BDF S ABC S ADBF, S DEP = S DF ES ABD S ADBF ) S ACE ( S DF E S ABD )S BCF S ADBF S CF E ( S BDF S ABC )S ADE ( S ADBF ) = S ACES DF E S ABD S BCF S CF E S BDF S ABC S ADE (Apaloif shmeðou A, B, C, D, E, F, qrhsimopoioôme to l mma 4.3.1 (selðda 175) ) DE (S ADE = g AE g AD g BC, S 2d ABC = g AC g AB g 2d DF S BDF = g BF g BD g F, S 2d CF E = g ECE g CF g 2d CF S BCF = g BF g BC g BD, S 2d ABD = g AD g AB g 2d F S DF E = g EDE g DF g CE, S 2d ACE = g AE g AC g ) 2d

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN183 ( g CE g AE g AC)( g F E g DE g DF )( g BD g AD g AB)( g CF g BF g BC)(2d) 4 ( g F E g CE g CF )( g DF g BF g BD)( g BC g AC g AB)( g DE g AE g AD)(2d) 4 BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. =1 PARADEIGMA 3.(TO GENIKO JEWRHMA THS PETALOUDAS) 'Estw èxi omokuklikˆ shmeða, A, B, C, D, E, F. Oi eujeðec CD kai EF tèmnoun thn AB sta shmeða M kai N antðstoiqa. Oi eujeðec CF kai DE tèmnoun thn AB sta shmeða G kai H antðstoiqa. IsqÔei ìti: MG BH = BM AG NH AN Apìdeixh D F A M G H N O B E C DIATUPWSH Sq ma 4.17:

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN184 K 1 (CIRCLE A B C D E F ): 'Estw èxi aujaðreta omokuklikˆ shmeða A, B, C, D, E, F. K 2 (INT ER M (LINE D C)(LINE A B)): 'Estw P to shmeðo tom c twn eujei n DF kai AB. K 3 (INT ER N (LINE E F )(LINE A B)): 'Estw Q to shmeðo tom c twn eujei n F E kai BC. K 4 (INT ER G (LINE A B)(LINE C F )): 'Estw S to shmeðo tom c twn eujei n EA kai CD. K 5 (INT ER H (LINE D E)(LINE A B)): 'Estw H to shmeðo tom c twn eujei n P Q kai CD. S: IsqÔei: MG BH = BM BA. AG NH AB AN ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, H, S, Q, P, E, F, D, C, B, A ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: DE AB, AB CF, EF AB, AB CD, A B, N H, N A, A G. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = MG BH AG NH BM AB AB AN = 1 kai ektel ta parakˆtw b mata.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN185 BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou H, qrhsimopoioôme thn prìtash B.2 (selðda 14),(je rhma epðpleurwn trig nwn} ( BH NH = S BDE S DEN ) BM AB S BDE AB S AN DEN MG AG (Apaloif shmeðou G, qrhsimopoioôme thn prìtash B.2) ( MG AG = S CF M S ACF ) S CF M S BDE BM AB S DEN S ACF AB AN (Apaloif shmeðou N, qrhsimopoioôme to l mma 2.4.14 (selðda 37) ) ( AB = S AEBF AN S AEF, S DEN = S DEF S ABE S AEBF ) S CF M S BDE S AEF ( S AEBF ) = S CF M S BDES AEF BM S BM AB AEBF S DEF S ABE S ACF S DEF S ABE S ACF AB (Apaloif shmeðou M, qrhsimopoioôme to l mma 2.4.14) ( BM AB = S BCD S ACBD, S CF M = S CDF S ABC S ACBD ) ( S CDF S ABC )S BDE S AEF ( S ACBD ) ( S BCD )S DEF S ABE S ACF ( S ACBD ) = S CDF S ABC S BDE S AEF S BCD S DEF S ABE S ACF

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN186 (Apaloif shmeðou A, B, C, D, E, F, qrhsimopoioôme to l mma 4.3.1 (selðda 175} AC (S ACF = g CF g F g A AB, S 2d ABE = g BE g AE g 2d EF S DEF = g DF g DE g CD, S 2d BCD = g BD g BC g 2d EF S AEF = g AF g AE g DE, S 2d BDE = g BE g BD g 2d BC S ABC = g AC g AB g DF, S 2d CDF = g CF g CD g ) 2d ( g DF g CF g CD)( g BC g AC g AB)( g DE g BE g BD)( g EF g AF g AE)(2d 4 ) ( g CD g BD g BC)( g EF g DF g DE)( g BE g AE g AB)( g CF g AF g AC)(2d) 4 BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. =1 4.4 PLHREIS GWNIES Se aut n thn enìthta ja eisˆgoume èna nèo eðdoc gwnðac, thn pl rh gwnða, thc opoðac h perigraf den exartˆtai apì thn seirˆ twn shmeðwn pˆnw stic eujeðec. GwnÐec autoô tou eðdouc qrhsimopoioôntai gia na aplopoi soun pollèc apodeðxeic gewmetrik n jewrhmˆtwn. ORISMOS 4.4.1 Mia pl rh gwnða apoteleðtai apì èna diatetagmèno zeôgoc eujei n l kai m kai sumbolðzetai wc ex c: [l, m]. Mia

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN187 pl rh gwnða [l, m] peristèfetai antðjeta apì th forˆ tou rologioô me skopì na fèrei thn l parˆllhla proc thn m. Gia parˆdeigma, an èqw na metr sw thn [AB, CD] : C A B D Sq ma 4.18: ORISMOS 4.4.2 DÔo pl rhc gwnðec [l, m] kai [u, v] eðnai Ðsec an upˆrqei peristrof R tètoia ste: R(l) u kai R(m) v. Na shmei soume epðshc ìti gia tèssera diakekrimèna shmeða A, B kai C, D pˆnw stic eujeðec l kai m antðstoiqa, h [l, m] sumbolðzetai epðshc wc ex c: [AB, CD], [BA, CD], [AB, DC], [l, DC], [AB, m]. Gia treða shmeða A, B kai C isqôei [ABC] = [AB, BC] ORISMOS 4.4.3 An l m, tìte h [l, m] onomˆzetai orj pl rh gwnða kai sumbolðzetai wc ex c: [1]. An l m tìte h [l, m] onomˆzetai epðpedh pl rh gwnða kai sumbolðzetai wc ex c: [0].

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN188 ORISMOS 4.4.4 'Estw a, b, c kai d na eðnai tèsseric eujeðec. 'Estw R na eðnai mia peristrof tètoia ste R(a) d. OrÐzoume [c, d] + [a, b] = [c, R(b)]. Basikèc idiìthtec twn pl rwn gwni n. 1. [c, d] = [0] an kai mìno an c d 2. [c, d] = [1] an kai mìno an c d 3. [u, v] = [v, u] 4. [1] + [1] = [0] 5. [c, d] + [0] = [c, d] 6. [c, d] + [a, b] = [a, b] + [c, d] 7. [c, d] + ( [a, b] + [e, f]) = ( [c, d] + [a, b]) + [e, f] 8. [c, e] + [e, d] = [c, d] 9. [AB, CD] = [BA, CD] = [AB, DC] = [BA, DC] 10. An [c, d] = [0] tìte gia opoiod pote eujeða p èqoume [c, p] = [d, p]. Antistrìfwc an gia mia eujeða p èqoume [c, p] = [d, p] tìte [c, d] = [0]

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN189 11. An AB = AC èqoume [AB, BC] = [BC, AC]. Antistrìfwc an [AB, BC] = [BC, AC] tìte AB = AC ta A, B, C eðnai suggrammikˆ. 12. Ta shmeða A, B, C kai D eðnai omokuklikˆ suggrammikˆ an kai mìno an [AB, BC] = [AD, DC] 13. An AB eðnai h diˆmetroc tou kôklou ston opoðo eðnai eggegrammèno to trðgwno ABC tìte [AC, BC] = [1] 14. An O eðnai to kèntro tou kôklou ston opoðo eðnai eggegrammèno to trðgwno ABC tìte [BO, OC] = 2 [AB, AC] Oi apodeðxeic twn idiot twn 1 mèqri 9 eðnai profan c, mènei loipìn na apodeðxoume tic upìloipec idiìthtec. H idiìthta 10 apodeiknôetai ˆmesa qrhshmopoi ntac tic idiìthtec 1, 4, 5 kai 7. H idiìthta 11 apodeiknôetai wc ex c: An isqôei: AB = AC tìte tan( [ABC]) = tan( [AB, BC]) = 4S ABC P ABC tan( [BCA]). Opìte: [ABC] = [BCA]. = 4S BCA BC 2 = 4S BCA P BCA = tan( [BC, CA]) = AntÐstrofa an [ABC] = [BCA] kai S ABC 0, apì ton orismì thc efaptomènhc ja èqoume: P ABC = P BCA dhlad AB 2 = AC 2. Gia tic idiìthtec 12, 13, 14 anatrèxte sto parˆrthma D. Sth sunèqeia ja parousiˆsoume duo trìpouc epðlushc gewmetrik n jewrhmˆtwn me thn qr sh pl rwn gwni n.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN190 O ènac trìpoc eðnai na eisˆgoume sthn mèjodo mia nèa gewmetrik ènnoia, thn efaptomenik sunˆrthsh. 'Opwc ja doôme sthn epìmenh enìthta, h nèa aut gewmetrik posìthta ja sunodeuteð apì èna pl joc orism n kai protˆsewn pou aforoôn isìthtec metaxô pl rwn gwni n. To jetikì sthn epilog thc diadikasðac aut c eðnai ìti den qreiˆzetai na eisˆgoume nèec teqnikèc apaloif c shmeðwn apì tic gewmetrikèc posìthtec afoô ìpwc ja doôme h efaptomenik sunˆrthsh miac pl rhc gwnðac ekfrˆzetai sunart sei tou embadoô kai thc Pujagìreiac diaforˆc trig nwn tetrapleôrwn. 'Omwc me thn epilog aut c thc diasikasðac qˆnoume thn monadikìthta pou mac prosfèroun oi paradosiakèc apodeðxeic pou qrhsimopoioôn sthn apodeiktik touc diadikasða mìno ekfrˆseic metaxô pl rwn gwni n. Gia ton skopì autì ja parousiˆsoume kai ènan deôtero trìpo epðlushc twn gewmetrik n protˆsewn. O trìpoc autìc sthrðzetai sto gn rimo plèon sôsthma (GIB) pou sunant same xanˆ stic enìthtec 2.7 kai 3.8. Gia na upenjumðsoume, to sôsthma autì basðzetai ex olokl rou sth kataskeuastik perigraf thc prìtashc. Sullègei se omˆdec, prin thn apodeiktik diadikasða, ìlec tic idiìthtec pou mporeð na sundèoun ta shmeða kai tic eujeðec metaxô touc (suggrammikìthta, parallhlða, kajetìthta, klp.). Skopìc tou eðnai h autìmath apaloif twn bohjhtik n shmeðwn pou perièqontai stic pl reic gwnðec. Meionèkthma thc epilog c aut c thc diadikasðac eðnai h dhmiourgða nèwn teqnik n apaloif c shmeðwn pou perièqontai stic pl reic gwnðec. 4.5 EFAPTOMENIKH SUNARTHSH ORISMOS 4.5.1 H efaptomenik sunˆrthsh gia thn pl rh gwnða [P Q, AB] orðzetai wc ex c: tan([p Q, AB]) = sin( (P Q,AB)). cos( (P Q,AB)) Gia na elègxoume an o orismìc eðnai kalˆ orismènoc arkeð na exetˆsoume an me thn enallag twn shmeðwn P, Q kai A, B h isìthta isqôei. ParathroÔme ìti

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN191 sin( (P Q, AB)) = sin( (P Q, BA)) = sin( (QP, AB)) cos( (P Q, AB)) = cos( (P Q, BA)) = cos( (QP, AB)) PROTASH 4.5.2 [AB, P Q] = [XY, UV ] an kai mìno an (AB, P Q) = (XY, UV ) (AB, P Q) (XY, UV ) = 180. Apìdeixh QwrÐc blˆbh thc genikìthtac, upojètoume ìti AB XY. Tìte [AB, P Q] = [XY, UV ] an kai mìno an P Q UV, to opoðo, ìpwc dh gnwrðzoume apì tic idiìthtec twn prosanatolismènwn gwni n, isqôei an kai mìno an: (AB, P Q) = (XY, UV ) (AB, P Q) (XY, UV ) = 180. PROTASH 4.5.3 tan( [AB, P Q]) = 4S AP BQ P AQBP Apìdeixh EÐnai ˆmesh apìrroia thc prìtashc 4.1.11 (selðda 162). PROTASH 4.5.4 [AB, P Q] = [XY, UV ] an kai mìno an tan( [AB, P Q]) = tan( [XY, UV ]) Apìdeixh An [AB, P Q] = [XY, UV ] apì thn prìtash 4.5.2 (selðda 191) paðrnoume: (AB, P Q) = (XY, UV ) (AB, P Q) (XY, UV ) = 180. EÐnai profan c h isìthta kai gia tic duo peript seic: tan( [AB, P Q]) = tan( [XY, UV ]). AntÐstrofa, an tan( [AB, P Q]) = tan( [XY, U V ]) tìte èqoume:

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN192 sin( (AB,P Q)) cos( (AB,P Q)) = sin( (XY,UV )) cos( (XY,UV )). h parapˆnw sqèsh isqôei an kai mìno an (AB, P Q) = (XY, UV ) (AB, P Q) (XY, UV ) = 180 ìpou sômfwna apì thn prìtash 4.5.2 (selðda 191) paðrnoume: [AB, P Q] = [XY, UV ]. PROTASH 4.5.5 [AB, P Q] = [XY, UV ] an kai mìno an S AP BQ P XV Y U = S XUY V P AQBP Apìdeixh H prìtash aut eðnai ˆmesh sunèpeia twn protˆsewn 4.5.3 kai 4.5.4 (selðda 191). PROTASH 4.5.6 (TO JEWRHMA TWN EPIGWNIWN TRIG- WNWN) 'Estw duo trðgwna ABC kai XY Z. An [ABC] = [XY Z], [ABC]/ne [1] kai [ABC]/ne [0] tìte: S ABC S XY Z = P ABC P XY Z = t, ìpou t 2 = AB2 BC 2 XY 2 ZY 2. Apìdeixh Apì thn prìtash 4.5.5 (selðda 192): an [ABC] = [XY Z] tìte: S ABC S XY Z = P ABC P XY Z = t.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN193 Apì ton tôpo Herron Qin gia to trðgwno ABC: 16SABC 2 + P ABC 2 = 4AB2 CB 2 (1) Apì ton tôpo Herron Qin gia to trðgwno XY Z: 16S 2 XY Z + P 2 XY Z = 4XY 2 ZY 2 EpÐshc S ABC = ts XY Z kai P ABC = tp XY Z (2) Me mia apl antikatˆstash thc (2) sthn (1) paðrnoume thn zhtoômenh sqèsh. ParathroÔme ìti h sumbol thc eisagwg c thc efaptomenik c sunˆrthshc sthn mèjodo epðlushc gewmetrik n jewrhmˆtwn eðnai pwc plèon mporoôme na apodeðxoume isqurismoôc ìpwc [AB, CD] = [EF, GH] kai [AB, CD] = [EF, GH]+ [XY, W Z]. UpenjumÐzoume ìti mèqri t ra eðqame tèsseric trìpouc anaparˆstashc eujei n: (LINE U V ), (P LINE P U V ), (T LINE P U V ), (BLINE U V ), touc opoðouc antikatast same katˆ thn dhmiourgða tou elˆqistou sunìlou kataskeu n me èna mìno eðdoc eujeðac,(line U V ). T ra me thn ènnoia twn pl rwn gwni n prostðjetai akìmh ènac trìpoc anaparˆstashc eujei n, ton opoðo kai ja antikatast soume me ton (LINE U V ), ètsi ste na mhn qreiasteð na eisˆgoume nèec teqnikèc apaloif c bohjhtik n shmeðwn. H nèa anaparˆstash eujei n eðnai h parakˆtw: (ALINE P Q U W V ):eðnai h eujeða l pou pernˆ apì to shmeðo P tètoia ste [P Q, l] = [UW, W V ]. Me thn eisagwg thc nèac anaparˆstashc eujei n parˆgontai kai eptˆ nèec kataskeuèc.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN194 1.(ON Y (ALINE P Q L M N)). 'Estw aujaðreto shmeðo Y pˆnw sthn (ALINE P Q L M N)). Perioristikèc sunj kec: P Q, L M, N M. 2.(INT ER Y (LINE U V )(ALINE P Q L M N)). 'Estw Y to shmeðo tom c thc eujeðac (LINE U V ) kai thc (ALINE P Q L M N)). Perioristik sunj kh: [P Q, UV ] [LM, MN] 3.(INT ER Y (P LINE W U V )(ALINE P Q L M N)). 'Estw Y to shmeðo tom c thc eujeðac (P LINE W U V ) kai thc (ALINE P Q L M N)). Perioristik sunj kh: [P Q, UV ] [LM, MN] 4.(INT ER Y (T LINE W U V )(ALINE P Q L M N)). 'Estw Y to shmeðo tom c thc eujeðac (T LINE W U V ) kai thc (ALINE P Q L M N)). Perioristik sunj kh: [UV, P Q] + [LM, MN] [1] 5.(INT ER Y (BLINE U V )(ALINE P Q L M N)). 'Estw Y to shmeðo tom c thc eujeðac (BLINE U V ) kai thc (ALINE P Q L M N)). Perioristik sunj kh: [UV, P Q] + [LM, MN] [1] 6.(INT ER Y (ALINE U V X Y Z)(ALINE P Q L M N)). 'Estw Y to shmeðo tom c thc eujeðac (ALINE U V X Y Z) kai thc (ALINE P Q L M N)). Perioristik sunj kh: [UV, P Q] [NM, ML] + [XY, Y Z]

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN195 7.(INT ER Y (CIR O P )(ALINE P Q L M N)). 'Estw Y to shmeðo tom c thc eujeðac (ALINE P Q L M N)) kai tou kôklou (CIR O P ). Perioristikèc sunj kec: Y P, P O, P Q, L M, N M Mènei t ra na deðxoume ìti h perigraf eujeðac ALINE mporeð na antikatastajeð apì thn perigraf LINE. PROTASH 4.5.7 An UW / W V tìte h perigraf l = (ALINE P Q U W V ) eðnai Ðdia me thn perigraf (LINE P R), ìpou R to shmeðo pou eisˆgetai mèsw thc kataskeu c (T RAT IO R Q P 4S UW V P UW V ). (ShmeÐwsh: eðnai anagkaðo na upojèsoume ìti U W / W V epeid an Ðsque h metaxô touc kajetìthta h eujeða l ja perigrafìtan wc ex c: (T LINE P P Q)). Apìdeixh R V P Q W U Sq ma 4.19: 'Estw ìti h kˆjeth pou pernˆ apì to Q kai eðnai kˆjeth sthn P Q tèmnei thn eujeða l sto shmeðo R. Dhlad to shmeðo R eisˆgetai apì thn kataskeu

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN196 (T RAT IO R Q P r), ìpou r = 4S RP Q P QP Q. Epeid apì thn prìtash B.7 (selðda 87): RQ QP tìte P RQP = P QP Q Opìte: r = 4S RP Q P QP Q = 4S QP R P QP R = tan( [RP Q]) = tan( [V W U]) = 4S UW V P UW V. EFARMOGES PARADEIGMA 1. 'Estw N, M shmeða pˆnw stic pleurèc AC, AB trig nou ABC kai R to shmeðo tom c twn eujei n BN, CM. To shmeðo R brðsketai pˆnw sto Ôyoc AD tou trig nou ABC. IsqÔei ìti h AD eðnai h diqotìmoc thc gwnðac MDN. Apìdeixh A M R N B D C DIATUPWSH Sq ma 4.20: K 1 (P OINT S A B C): 'Estw trða aujaðreta shmeða A, B, C K 2 (F OOT D A B C): 'Estw D h orjog nia probol tou shmeðou A pˆnw sthn BC.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN197 K 3 (ON R (LINE A D)): 'Estw R shmeðo pˆnw sthn AD. K 4 (INT ER M (LINE A B)(LINE C R)): 'Estw M to shmeðo tom c twn eujei n AB kai CR. K 5 (INT ER N (LINE A C)(LINE B R)): 'Estw N to shmeðo tom c twn eujei n AC kai BR. S: IsqÔei S ADM P ADN = S ADN P ADM. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, N, M, R ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: A D, C B, AB CR, AC BR. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = ( S ADM P ADN ) S ADN P ADM = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou N, qrhsimopoioôme to l mma 3.5.2 (selðda 111) ) (S ADN = S ACDS ABR S ABCR, P ADN = P ADRS ABC S ABCR ) ( S ADM ) P ADR S ABC S ABCR P ADM ( S ACD S ABR ) S ABCR = S ADM P ADRS ABC S ACD S ABR P ADM (Apaloif shmeðou M, qrhsimopoioôme to l mma 3.5.2) (P ADM = P ADRS ABC S ACBR, S ADM = S ACRS ABD S ACBR )

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN198 P ADR S ABC ( S ACR )S ABD S ACBR S ABR S ACD P ADR S ABC S ACBR = S ACRS ABD S ACD S ABR (Apaloif shmeðou R, qrhsimopoioôme thn prìtash B.2 (selðda 14) ) (S ABR = S ABD AR AD, S ACR = S ACD AR AD ) S ABD (S ACD AR AD ) S ACD (S ABD AR AD ) BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. =1. PARADEIGMA 2. Sthn upoteðnousa AB enìc orjogwnðou trig nou ABC sqediˆzoume èna tetrˆgwno ABF E. 'Estw P to shmeðo tom c twn diagwnðwn AF kai BE tou tetrag nou ABF E. IsqÔei ìti: [ACP ] = [P CB]. Apìdeixh DIATUPWSH K 1 (P OINT S B C): 'Estw duo aujaðreta shmeða B, C K 2 (T RAT IO A C B r): 'Estw A shmeðo pˆnw sthn eujeða (T LINE C C B) tètoio ste: r = CA CB = r. K 3 (T RAT IO F B A 1): 'Estw F shmeðo pˆnw sthn eujeða (T LINE B B A) tètoio ste: r = BF BA = 1. K 4 (T RAT IO E A B 1): 'Estw shmeðo E pˆnw sthn eujeða (T LINE A A B) tètoio ste: r = AE AB = 1

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN199 E A P F C B Sq ma 4.21: K 5 (INT ER P (LINE B E)(LINE A F ): 'Estw P to shmeðo tom c twn eujei n BE kai AF. S: IsqÔei [ACP ] = [P CB] S CAP P BCP = S BCP )P ACP. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, P, E, F, A ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: A B, C B, BE AF. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = ( S CAP P BCP ) ( S BCP )P ACP = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN200 (Apaloif shmeðou P, qrhsimopoioôme to l mma 3.5.2 (selðda 111) ) (P ACP = P ACES BAF S BAEF, S BCP = S BAF S BCE S BAEF, P BCP = P BCF S BAE S BAEF, S CAP = S CAF S BAE S BAEF ) ( S CAP P BCP ) ( S BCP )P ACP = ( S CAF S BAE )( P BCF S BAE ) S BAEF S BAEF = S CAF (S BAE)2 P BCF ( S BAF S BCE )( P ACE S BAF ) (S BAF ) 2 S BCE P ACE S BAEF S BAEF ) (Apaloif shmeðou E, qrhsimopoioôme to l mma 3.5.6, 3.5.5 (selðda 113) (P ACE = P CAC 4S BCA, S BCE = 1 4 (P CBA 4S BCA ), S BAE = 1 4 P BAB) S CAF P BCF ( 1 4 P BAB) (S BAF ) 2 ( 1 4 (P CBA 4S BCA ))(P CAC 4S BCA ) (Apaloif shmeðou F, qrhsimopoioôme to l mma 3.5.6, 3.5.5) (S BAF = 1 4 P BAB, P BCF = P BCB 4S BCA, S CAF = 1 4 (P BAC 4S BCA )) ( 1 4 P BAB) 2 ( 1 4 (P BAC 4S BCA ))(P BCF 4S BCA ) = (P BAC 4S BCA )(P BCB 4S BCA ) ( 1 4 P CBA+S BCA )(P CAC 4S BCA )( 1 4 P BAB) 2 (P CBA 4S BCA )(P CAC 4S BCA ) (Apaloif shmeðou A, qrhsimopoioôme to l mma 3.5.5, 3.5.6, 3.5.7) (P CAC = r 2 P BCB, P CBA = P BCB, S BCA = 1 4 rp BCB, P BAC = r 2 P BCB ) (P BCB r 2 +P BCB r)(p BCB r+p BCB ) (P BCB r+p BCB )(P BCB r 2 +P BCB r) BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec apì anexˆrthtec metablhtèc - aplopoi seic. =1.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN201 4.6 MEJODOS TWN PLHRWN GWNIWN- SUSTHMA (GIB) H mèjodoc twn pl rwn gwni n dhmiourg jhke me skopì thn autìmath apìdeixh ekatontˆdwn gewmetrik n jewrhmˆtwn pou perilambˆnoun sqèseic metaxô gwni n. Ta pleonekt mata thc mejìdou eðnai shmantikˆ. Prosfèrei mikrèc se èktash kai euanˆgnwstec apodeðxeic. Na shmeiwjeð pwc èna gewmetrikì je rhma ìtan apodeiknôetai me thn qr sh thc mejìdou twn pl rwn gwni n èqei thn idiìthta na eðnai anexˆrthto apì to diˆgramma tou jewr matoc. To gegonìc autì eðnai exairetikì an sullogistoôme thn antðstoiqh diadikasða epðlushc tou jewr matoc me thn qr sh ìmwc apl n gwni n. Se aut n thn perðptwsh ja apaitoôntan diaqwrismìc peript sewn anˆloga me thn jèsh twn shmeðwn pˆnw sto diˆgramma kai oi apodeðxeic ja dièferan anˆ perðptwsh. AntÐjeta h mèjodoc twn pl rwn gwni n basðzetai sto pleonèkthma pou thc prosfèrei h kataskeuastik seirˆ twn shmeðwn. H upìjesh tou gewmetrikoô jewr matoc eisˆgei me thn seirˆ ta shmeða me tètoion trìpo ste kˆje shmeðo pou eisˆgetai na mporeð na kataskeuasteð apì ta d- h upˆrqonta shmeða. Gia parˆdeigma, an d soume thn plhroforða ìti to shmeðo M eðnai to mèson thc AB, ìpou A, B eleôjera shmeða, tìte h kataskeuastik seirˆ ja eðnai: kataskeuˆzoume pr ta to shmeðo A èpeita to shmeðo B kai tèloc to shmeðo M. 'Etsi loipìn katˆ thn apodeiktik diadikasða h mèjodoc twn pl rwn gwni n antikajistˆ pl reic gwnðec me gwnðec pou perilambˆnoun qamhlìterhc seirˆc shmeða. Autì ofeðletai sthn kataskeuastik seirˆ twn shmeðwn, ìpou h jèsh twn shmeðwn aut n (qamhlìterhc seirˆc) sto diˆgramma den exartˆtai apì thn jèsh twn shmeðwn (uyhlìterhc seirˆc). BASIKA BHMATA THS MEJODOU TWN PLHRWN GWNIWN. H mèjodoc twn pl rwn gwni n akoloujeð ta parakˆtw b mata gia thn apìdeixh gewmetrik n protˆsewn. BHMA 1. H upìjesh thc prìtashc eisˆgetai sto sôsthma (GIB). To sôsthma autì perilambˆnei ìla ta gewmetrikˆ stoiqeða pou aporrèoun apì tic kataskeuèc pou qrhsimopoieð h prìtash.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN202 BHMA 2. To sumpèrasma thc prìtashc metatrèpetai se isìthta metaxô pl rwn gwni n thc morf c [0] = f i n i, ìpou f i eðnai mia pl rh gwnða kai n i ènac akèraioc suntelest c. BHMA 3. H mèjodoc èqei efodiasteð me èna epiplèon sôsthma, to sôsthma (GKB) twn genik n gn sewn, to opoðo apoteleð thn bˆsh tou sust matoc (GIB). H apodeiktik diadikasða qrhsimopoieð touc kanìnec tou sust matoc (GKB) se sundiasmì me ta stoiqeða pou perilambˆnei to sôsthma (GIB). Skopìc tou eðnai h antikatˆstash twn pl rwn g- wni n, pou upˆrqoun sto sumpèrasma thc prìtashc, apì isodônamec ekfrˆseic pl rwn gwni n. Gia parˆdeigma, h [u, v] mporeð na grafteð wc [1] an to (GIB) perilambˆnei stoiqeðo ìpwc u v. EpÐshc efarmìzontai oi idiìthtec twn pl rwn gwni n, me skopì thn aplopoðhsh twn ekfrˆsewn pou proèrqontai èpeita apì kˆje antikatˆstash. Gia parˆdeigma, [u, v] + [v, u] = [u, v] [u, v] = [0] kai [1] + [1] = [0]. BHMA 4. H apodeiktik diadikasða telei nei ìtan brèjei mia seirˆ apì katˆllhlouc kanìnec, oi opoðoi ìtan efarmostoôn sthn exðswsh tou sumperˆsmatoc ja thn fèroun sthn morf [0] = [0]. To sôsthma (GKB) èqei efodiasteð me touc parakˆtw kanìnec: R 1. Duo shmeða A, B orðzoun mia eujeða. Perioristik sunj kh A B. R 2. TrÐa shmeða A, B, C orðzoun ènan kôklo. Perioristik sunj kh S ABC 0. R 3. [AB, CD] = [EF, CD] an kai mìno an [AB, EF ] = [0]. Perioristik sunj kh C D. R 4. [AB, CD] = [AB, EF ]+ [1] an kai mìno an [CD, EF ] = [1]. Perioristik sunj kh C, D, E, F mh suggrammikˆ. R 5. Tèssera shmeða A, B, C, D eðnai omokuklikˆ an kai mìno an [AC, BC] = [AD, BD]. Perioristik sunj kh A, B, C, D mh suggrammikˆ. R 6. AB = AC an kai mìno an [AB, BC] = [BC, AC]. Perioristik sunj kh S ABC 0

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN203 R 7. An O eðnai to kèntro tou perigegrammènou kôklou sto trðgwno ABC tìte [OA, AB] = [1] + [AC, BC]. Perioristik sunj kh S ABC 0 R 8. An AB BC an kai mìno an AC eðnai h diˆmetroc tou perigegrammènou kôklou sto trðgwno ABC. Perioristik sunj kh S ABC 0 Autì pou mènei t ra na kˆnoume eðnai na dhmiourg soume èna nèo sôsthma teqnik n apaloif c shmeðwn, basismèno ìqi stic kataskeuèc shmeðwn, ìpwc gnwrðzame mèqri t ra, allˆ sta stoiqeða pou perilambˆnei to (GIB). Teqnikèc apaloif c shmeðwn apì tic pl reic gwnðec. 'Estw [AB, CY ] mia pl rh gwnða kai Y to shmeðo pou jèloume na apaloðyoume. SÔmfwna me ìsa dh èqoume anafèrei gia na apaloðyoume apì tic sqèseic pou akoloujoôn to shmeðo Y prèpei na upojèsoume ìti ìla ta upìloipa shmeða èqoun kataskeuasteð pio prðn apì to Y (shmeða qamhlìterhc seirˆc). T 1. An to shmeðo Y brðsketai pˆnw sth CD tìte [AB, CY ] = [AB, CD]. T 2. An CY EF tìte [AB, CY ] = [AB, EF ]. T 3. An CY EF tìte [AB, CY ] = [AB, EF ] + [1]. T 4. An to shmeðo Y brðsketai pˆnw sth EF kai E, Y, C, D omokuklikˆ tìte [AB, CY ] = [AB, EF ] + [ED, CD]. T 5. An to shmeðo Y brðsketai pˆnw sth EF kai isqôei CY = CE tìte [AB, CY ] = [AB, EF ] + [CE, EF ]. T 6. An to shmeðo Y eðnai to kèntro tou perigegrammˆnou kôklou sto trðgwno CDE tìte [AB, CY ] = [AB, CD] + [ED, EC] + [1]. T 7. An [EF, CY ] = [u] kai [u] gnwst gwnða tìte [AB, CY ] = [AB, EF ] + [u]. ApodeÐxeic

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN204 Oi treðc pr tec teqnikèc kai h teleutaða eðnai profaneðc. T 4. [AB, CY ] = [AB, EF ] + [EF, CY ], apì T 1 : [EF, CY ] = [EY, CY ] kai apì R 5 : [EY, CY ] = [EF, CF ]. 'Ara [EF, CY ] = [EF, CF ] T 5. Apì T 1 : [CE, EF ] = [CE, EY ] = [EY, CY ] = [EF, CY ]. Opìte me mia apl antikatˆstash parathroôme ìti epalhjeôetai h idiìthta 8. T 6. Apì thn R 7 : [CD, CY ] = [ED, EC] + [1]. Opìte me mia apl antikatˆstash parathroôme ìti epalhjeôetai h idiìthta 8. EFARMOGES THS MEJODOU PARADEIGMA 1 'Estw A, B, C trðgwno. 'Estw AD to Ôyoc tou trig nou pˆnw sthn BC kai èstw E, F, G ta mèsa twn pleur n AB, BC, AC antðstoiqa. IsqÔei ìti ta shmeða D, E, F, G eðnai omokuklikˆ. A E G B D F C DIATUPWSH Sq ma 4.22: (P OINT S A B C) (F OOT D A B C) (MIDP OINT E A B) (MIDP OINT F B C) (MIDP OINT G A C) S: [GE, GD] + [F E, F D] = [0] To sôsthma (GIB) perilambˆnei se omˆdec ìlec tic sqèseic pou sundèoun ta shmeða kai tic eujeðec metaxô touc (suggrammikìthta, parallhlða, kajetìthta klp.).

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN205 O 1 : Perièqei ìla ta shmeða thc prìtashc me thn seirˆ pou kataskeuˆsthkan, (A, B, C, D, E, F, G) O 2 : Perièqei ìla ta eleôjera shmeða thc prìtashc, (A, B, C) O 3 : Perièqei ìlec tic eujeðec kai ta suggrammikˆ shmeða thc prìtashc, ((E, A, B)(G, A, C)(D, F, B, C)) p.q:(g, A, C) shmaðnei ìti ta G, A, C eðnai suggrammikˆ. O 4 : Perièqei ìlec tic parˆllhlec eujeðec thc prìtashc, ((EG)(BDF C), (F G)(AEB), (EF )(AGC)) p.q: (EG)(BDF C) shmaðnei ìti oi eujeðec EG kai BDF C eðnai metaxô touc parˆllhlec. O 5 : Perièqei ìlec tic kˆjetec eujeðec thc prìtashc, ((AD)(BDF C)) p.q:(ad)(bdf C) shmaðnei ìti oi eujeðec AD kai BDF C eðnai metaxô touc kˆjetec. O 6 : Perièqei ìlouc touc kôklouc thc prìtashc, ((BA)(DBA(E)), (CA)(DCA(G)), (BC)(BC(F ))) p.q:(ba)(dba(e)) shmaðnei ìti oi BA eðnai h diˆmetroc kai E eðnai to kèntro perigegrammènou kôklou sto trðgwno DBA. Qrhsimopoi ntac merikèc apì tic parapˆnw plhroforðec parˆgetai h apìdeixh. Apìdeixh [GE, GD] + [F E, F D] (apì R 3 epeid GE DC, paðrnoume [GE, GD] = [GD, DC]) = [GD, DC] + [F E, F D] (apì R 7 epeid G kèntro perigegrammènou kôklou sto trðgwno DCA, paðrnoume [GD, DC] = [DA, CA] + [1]) = [F E, F D] + [DA, CA] + [1] (apì R 3 epeid F E CA, paðrnoume [F E, F D] = [F D, CA])

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN206 = [F D, CA] + [DA, CA] + [1] (apì R 4 epeid F D DA, paðrnoume [F D, CA] = [DA, CA] + [1]) = [0]. PARADEIGMA 2 (JEWRHMA SIMSON) 'Estw D shmeðo pˆnw ston perigegrammèno kôklo tou trig nou ABC. Fèrnoume apì to D tic treðc kajètouc stic pleurèc BC, AC, AB. 'Estw E, F, G ta pìdia twn kajètwn pˆnw stic pleurèc BC, AC, AB antðstoiqa. IsqÔei ìti ta E, F, G eðnai suggrammikˆ. E C D F O A G B DIATUPWSH Sq ma 4.23: (CIRCLE A B C D) (F OOT E D B C) (F OOT F D A C) (F OOT G D A B) S: [EF, F G] = [0] To sôsthma (GIB) perilambˆnei se omˆdec ìlec tic sqèseic pou sundèoun ta shmeða kai tic eujeðec metaxô touc (suggrammikìthta, parallhlða, kajetìthta klp.). O 1 : Perièqei ìla ta shmeða thc prìtashc me thn seirˆ pou kataskeuˆsthkan, (A, B, C, D, E, F, G) O 2 : Perièqei ìla ta eleôjera shmeða thc prìtashc, (A, B, C, D)

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN207 O 3 : Perièqei ìlec tic eujeðec kai ta suggrammikˆ shmeða thc prìtashc, ((G, A, B), (E, C, B), (A, F, C), (D, F ), (D, G), (D, E)) O 4 : Perièqei ìlec tic parˆllhlec eujeðec thc prìtashc, ( ) O 5 : Perièqei ìlec tic kˆjetec eujeðec thc prìtashc, ((DF )(AF C), (DE)(ECB), (DG)(AGB)) O 6 : Perièqei ìlouc touc kôklouc thc prìtashc, ((AD)(AGF D), (DB)(BEGD), (CD)(DEF C)) Qrhsimopoi ntac merikèc apì tic parapˆnw plhroforðec parˆgetai h apìdeixh. Apìdeixh [EF, GF ] (apì idiìthta 8 paðrnoume [F E, GF ] = [EF, DF ] + [DF, GF ]) = [EF, DF ] + [DF, GF ] (apì idiìthta 12(A, D G F kai D, C, E, F omokuklikˆ) paðrnoume [EF, DF ] = [EC, DC], [DF, GF ] = [DA, GA]) = [EC, DC] + [DA, GA] (apì T 1 epeid E BC kai G AB paðrnoume [EC, DC] = [BC, DC], [DA, GA] = [DA, BA]) = [BC, DC] + [DA, BA] (apì idiìthta 12(A, D B C omokuklikˆ) paðrnoume [BC, DC] = [BA, DA]) = [BA, DA] + [DA, BA] (apì idiìthta 8 paðrnoume [BA, DA] + [DA, BA] = [BA, BA]) = [BA, BA] = [0].

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN208 4.7 AUTOMATES APODEIXEIS GEWMETRIK- WN PROTASEWN THS KATHGO- RIAS C To sônolo ìlwn twn kataskeuastik n protˆsewn to sumbolðzoume me C. Na shmei soume ìti kataskeuastikèc onomˆzoume tic protˆseic oi opoðec mporoôn na apodeiqjoôn me thn qr sh kanìna kai diab th. Gia na perigrˆyoume tic protˆseic pou an koun sthn kathgorða C, prèpei na eisˆgoume duo nèec kataskeuèc. Ja xekin soume me thn eisagwg enìc nèou eðdouc kôklou:(circle O r), o opoðoc perigrˆfetai me bˆsh to kèntro tou O kai thn aktðna tou r. To r mporeð na eðnai ènac algebrikìc arijmìc, mia rht èkfrash gewmetrik n posot twn metablht. Oi nèec kataskeuèc pou eisˆgoume eðnai oi ex c: K 22 (INT ER Y (LINE A B)(CIRCLE O r)) DiatÔpwsh kataskeu c: to Y eðnai to shmeðo tom c thc eujeðac (LINE A B) kai tou kôklou (CIRCLE O r)). Oi periorismoð thc kataskeu c aut c eðnai: r 0, A B. To shmeðo Y eðnai stajerì shmeðo kai èqei duo dunatìthtec. K 23 (INT ER Y (CIRCLE O 1 r 1 )(CIRCLE O 2 r 2 )) DiatÔpwsh kataskeu c: to Y eðnai to shmeðo tom c tou kôklou (CIRCLE O 1 r 1 ) kai tou kôklou (CIRCLE O 2 r 2 )). Oi periorismoð thc kataskeu c aut c eðnai: r 1 0, r 2 0, O 1 O 2. To shmeðo Y eðnai stajerì shmeðo kai èqei duo dunatìthtec.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN209 4.8 TEQNIKES APALOIFHS SHMEI- WN MESW GEWMETRIKWN POSOTHTWN PROTASH 4.8.1 'Estw Y to shmeðo tom c thc eujeðac UV kai tou kôklou (CIRCLE O r). Tìte èqoume thn parakˆtw sqèsh: (UY UV )2 P OUV UY + OU 2 r UV 2 UV UV 2 = 0 (C) Apìdeixh O V U Y 1 M Y Sq ma 4.24: 'Estw Y 1 èna akìmh shmeðo tom c thc eujeðac UV kai tou kôklou (CIRCLE O r) kai èstw M to mèson thc Y 1 Y. Apì thn prìtash B.8 (selðda 89), paðrnoume: P OUV UV 2 = P MUV UV 2 = 2 UM UV = UY UV + UY 1 UV (1). Apì thn prìtash B.10 (selðda 91), paðrnoume: OU 2 = Y 1U OY 2 + UY Y 1 Y Y 1 Y OX2 Y 1U UY Y Y 1 Y Y 1 Y 1Y 2 2 = OY 1 + UY UY1 = r + UY UY 1 (2).

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN210 Apì tic (1) kai (2) paðrnoume thn sqèsh (C) (selðda 209). 'Etsi loipìn me thn bo jeia thc sqèshc (C) èqoume tic parakˆtw teqnikèc apaloif c. LHMMA 4.8.2 'Estw Γ(Y ) mia apì tic parakˆtw gewmetrikèc posìthtec kai to shmeðo Y eisˆgetai apì thn kataskeu (INT ER Y (LINE P Q)(CIRCLE O r)). Tìte: S ABY = P Y P Q S QAP B + S ABP P ABY = P Y P Q P QAP B + P ABP P AY B = P Y P Q P QAP B + P ABP P Y P Q (1 P Y P Q )P P QP O lìgoc P Y P Q (Sq ma 4.27) ikanopoieð thn sqèsh (C) (selðda 209) Apìdeixh Apì thn prìtash B.3 (selðda 15), paðrnoume: S ABY = ( P Y P Q )S ABQ + (1 P Y P Q )S ABP = P Y P Q S QAP B + S ABP Apì thn prìtash B.10 (selðda 91), thn pr th sqèsh, paðrnoume: P ABY = P Y P Q P QAB + (1 P Y P Q )P P AB = P Y P Q P QAP B + P P AB Apì thn prìtash B.10 (selðda 91), thn deôterh sqèsh, paðrnoume: P AY B = P Y P Q P QAB + (1 P Y P Q )P P AB P Y P Q (1 P Y P Q )P P QP = P Y P Q P QAP B + P P AB P Y P Q (1 P Y P Q )P P QP

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN211 LHMMA 4.8.3 'Estw Γ(Y ) = AY CD kataskeu (INT ER Y (LINE P Q)(CIRCLE O r)). kai to shmeðo Y eisˆgetai apì thn Tìte AY CD = { AP + P Y P Q CD P Q CD S AP Q S CP DQ,, an A P Q; alli c. (Sq ma 4.27) Apìdeixh H pr th perðptwsh eðnai tetrimmènh. Gia thn deôterh perðptwsh paðrnoume shmeðo S tètoio ste AS = CD. Apì thn prìtash B.2 (selðda 14) paðrnoume: AY S AP Q S CP DQ CD = AY AS = S AP Q S AP SQ = PROTASH 4.8.4 H kataskeu, (INT ER Y (CIRCLE O 1 r 1 )(CIRCLE O 2 r 2 )) eðnai isodônamh me tic akìloujec duo kataskeuèc: (LRAT IO O O 1 O 2 r) (T RAT IO Y O O 1 s) ìpou r = O 1O 2 2 +r1 r 2 2O 1 O 2 2, s 2 = r 1 r 2 O 1 O 2 2 1 Apìdeixh Y O 1 O O 2 Sq ma 4.25:

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN212 'Estw O h orjog nia probol tou shmeðou Y pˆnw sthn O 1 O 2. Apì thn prìtash B.8 (selðda 89), èqoume: r = O 1O O 1 O 2 = P Y O 1 O 2 kai P O1 = O 2 1O 2 +r1 r 2 O 2 O 1 2O 1 O 2 s 2 = OY 2 OO 1 2 = r 1 OO 1 2 1 = r 1 r 2 O 1 O 2 2 1 LHMMA 4.8.5 'Estw Γ(Y ) = S ABY kai to shmeðo Y eisˆgetai apì thn kataskeu (INT ER Y (CIRCLE O 1 r 1 )(CIRCLE O 2 r 2 )). S ABY = S ABO1 + rs O2 AO 1 B rs 4 P O 2 AO 1 B. (Sq ma 4.28) Apìdeixh 'Estw O h orjog nia probol tou shmeðou Y pˆnw sthn O 1 O 2. Apì to l mmma 3.5.5 (selðda 113) kai thn prìtash B.3 (selðda 15), èqoume: S ABY = S ABO s 4 P OAO 1 B = rs ABO2 + (1 r)s ABO1 s 4 (rp O 2 AB + (1 r)p O1 AB P O1 AB) = S ABO1 + rs O2 AO 1 B rs 4 P O 2 AO 1 B. LHMMA 4.8.6 'Estw Γ(Y ) = P ABY kai to shmeðo Y eisˆgetai apì thn kataskeu (INT ER Y (CIRCLE O 1 r 1 )(CIRCLE O 2 r 2 )). P ABY = P ABO1 + rp O2 AO 1 B 4rsS O2 AO 1 B.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN213 (Sq ma 4.28) Apìdeixh 'Estw O h orjog nia probol tou shmeðou Y pˆnw sthn O 1 O 2. Apì to l mmma 3.5.6 (selðda 114), thn prìtash B.10, (selðda 91) kai thn prìtash B.3 (selðda 15), èqoume: P ABY = P ABO 4sS OAO1 B = rp ABO2 + (1 r)p ABO1 4s(rS O2 AB + (1 r)s O1 AB S O1 AB) = P ABO1 + rp O2 AO 1 B 4rsS O2 AO 1 B.

PARARTHMATA 214

PARARTHMA A H sqetik gewmetrða meletˆ probl mata pou perilambˆnoun zht mata suggrammikìthtac, parallhlðac, sômptwshc. Sto parˆrthma autì ja melet - soume eidikˆ thn ènnoia thc suggrammmikìthtac. O akrib c orismìc thc ènnoiac aut c aporrèei apì ta akìlouja èxi axi mata. AxÐwma A 1 : Gia trða suggrammikˆ shmeða A, B kai C tètoia ste AC A B, o lìgoc twn eujôgrammwn tmhmˆtwn AC kai AB: eðnai ènac AB pragmatikìc arijmìc pou ikanopoieð thn parakˆtw sqèsh. AC = CA = CA = AC. AB AB BA BA EpÐshc isqôei: CA AB = 0 an kai mìno an C = A. AxÐwma A 2 : 'Estw A kai B duo diaforetikˆ shmeða. Gia ènan pragmatikì arijmì s, upˆrqei monadikì shmeðo C to opoðo eðnai suggrammikì me ta shmeða A kai B kai ikanopoieð tic parakˆtw sqèseic. 1. AC AB = s kai 2. AC AB + CB AB = 1. T ra gia to prosanatolismèno embadìn trig nou ABC: S ABC isqôoun oi parakˆtw basikèc idiìthtec. AxÐwma A 3 : S ABC = S CAB = S BCA = S BAC = S CBA = S ACB. 215

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN216 An A, B, C trða mh suggrammikˆ shmeða tìte isqôei S ABC 0. AxÐwma A 4 : Upˆrqoun toulˆqiston trða shmeða A, B, C gia ta opoða isqôei: S ABC 0. (To axðwma autì exasfalðzei ìti ìla ta shmeða den eðnai suggrammikˆ.) AxÐwma A 5 : Gia opoiad pote tèssera shmeða A, B, C, D isqôei ìti: S ABC = S ABD + S ADC + S DBC. (To axðwma autì exasfalðzei ìti ìla ta shmeða an koun sto Ðdio epðpedo.) Sunèpeia tou axi matoc 5 eðnai o orismìc pou d same gia to embadìn prosanatolismènou tetrapleôrou. Pio sugkekrimèna, orðsame to embadìn prosanatolismènou tetrapleôrou ABCD wc ex c: S ABCD = S ABC + S ACD. Me skopì na dikaiolog soume ìti to S ABCD eðnai kalˆ orismèno, qreiˆzetai na apodeðxoume ìti apì ton parapˆnw orismì prokôptei S ABCD = S BCDA = S CDAB = S DABC 'Etsi loipìn, me qr sh thc sqèshc S ABCD = S ABC + S ACD parathroôme ìti S ABCD = S CDAB = S ABC + S ACD kai S BCDA = S DABC = S BCD + S BDA. ArkeÐ t ra na deðxoume ìti S ABCD = S BCDA. Me qr sh t ra thc sqèshc (I), selðda 12 (blèpe sq mata selðdac 10-11). S ABC = S RBC + S RCA + S RAB

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN217 èqoume: S ABCD = S ABC + S ACD =S RBC + S RCA + S RAB + S RCD + S RDA + S RAC =S RBC + S RAB + S RCD + S RDA. S BCDA = S BCD + S BDA =S RCD + S RDB + S RBC + S RDA + S RBD + S RAB =S RCD + S RBC + S RDA + S RAB. EÐnai faner plèon h isìthta. Oi parapˆnw sqèseic isqôoun gia opoiad pote jèsh tou R sto epðpedo. AxÐwma A 6 : 'Estw A, B kai C trða suggrammikˆ shmeða tètoia ste AB = tac. Tìte gia opoiod pote shmeðo P isqôei: S P AB = ts P AC. SHMEIWSH: To axðwma 6 dhl nei mia apì tic pio shmantikèc idiìthtec twn embad n. Apì autì prokôptoun ìlec oi basikèc protˆseic thc mejìdou twn embad n.

PARARTHMA B H sqetik gewmetrða, ìpwc anafèrame sto prohgoômeno parˆrthma, asqoleðtai me thn melèth problhmˆtwn parallhlðac, sômptwshc kai suggrammikìthtac. Ta akìlouja apoteloôn mia omˆda axiwmˆtwn thc sqetik - c gewmetrðac, ta opoða ìpwc ja deðxoume aporrèoun apì ta axi mata A 1,...,A 6, selðda 215-217. AxÐwma 1. 'Estw duo diaforetikˆ shmeða A kai B. Tìte upˆrqei monadik eujeða h opoða dièrqetai apì ta shmeða autˆ. AxÐwma 2. 'Estw eujeða e. Apì shmeðo ektìc thc eujeðac e upˆrqei mða kai mìno mða eujeða e 1 h opoða dièrqetai apì to shmeðo autì kai eðnai parˆllhlh sthn eujeða e. AxÐwma 3. Upˆrqoun trða diaforetikˆ shmeða, A, B, C, tètoia ste to shmeðo C na mhn an kei sthn eujeða pou orðzoun ta shmeða A, B. AxÐwma 4.AxÐwma tou Desargues: 'Estw l 1, l 2, l 3 treðc diaforetikèc eujeðec oi opoðec eðte eðnai parˆllhlec eðte tèmnontai sto shmeðo S. 'Estw A, A 1 shmeða pˆnw sthn eujeða l 1, B, B 1 shmeða pˆnw sthn eujeða l 2 kai C, C 1 shmeða pˆnw sthn eujeða l 3, ta shmeða autˆ eðnai diaforetikˆ apì to shmeðo S an oi treðc eujeðec 218

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN219 tèmnontai. An AB A 1 B 1 kai BC B 1 C 1 tìte AC A 1 C 1. AxÐwma 5.AxÐwma tou Pascal: 'Estw duo diaforetikèc eujeðec l 1, l 2 kai A, B, C kai A 1, B 1, C 1 eðnai shmeða pˆnw stic eujeðec l 1 kai l 2 antðstoiqa. An BC 1 B 1 C kai AB 1 A 1 B tìte AC 1 A 1 C. Ja deðxoume t ra ìti ta axi mata pou mìlic anafèrame eðnai sunèpeia twn axiwmˆtwn A 1,...,A 6. Arqikˆ to axðwma 1 aporrèei apì to pìrisma 2.1.3. To axðwma 3 aporrèei apì ta axi mata A 3,A 4. Sthn sunèqeia ja apodeðxoume ìti kai ta axi mata 2,4 kai 5 aporrèoun apì ta axi mata A 1,...,A 6. Apìdeixh axi matoc 2. 'Estw eujeða AB kai C shmeðo ektìc eujeðac. Apì to axðwma A 2,mporoÔme na epilèxoume shmeða D kai E tètoia ste to D na eðnai to mèson tou tm - matoc CA kai to E summetrikì tou B wc proc to D, opìte: ED = DB. Apì thn prìtash B.2 paðrnoume: S EAB = 2S DAB = S CAB Apì thn prìtash B.4 paðrnoume: CE AB Gia thn monadikìthta t ra paðrnoume shmeðo F tètoio ste F C AB. Apì thn prìtash B.5 èqoume: S F CE = S F AB S CAB = 0 'Ara to shmeðo F brðsketai pˆnw sthn eujeða CE. Oi apodeðxeic twn axiwmˆtwn 4 kai 5 dðnontai mèsw thc algorijmik c mejìdou. AxÐwma tou Desargues.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN220 C 1 C S B B 1 A A 1 Sq ma 4.26: Apìdeixh DIATUPWSH K 1 : (P OINT S A B C, S) 'Estw tèssera aujaðreta shmeða A, B, C, S K 2 : (ON LINE A 1 S A) 'Estw A 1 shmeðo pˆnw sthn eujeða SA. K 3 : (INT ER B 1 (LINE S B)(P LINE A 1 A B)) 'Estw B 1 to shmeðo tom c twn eujei n SB kai thc parˆllhlhc apì to shmeðo A 1 sthn eujeða AB. K 4 : (INT ER C 1 (LINE S C)(P LINE A 1 A C)) 'Estw C 1 to shmeðo tom c twn eujei n SC kai thc parˆllhlhc apì to shmeðo A 1 sthn eujeða AC. S: IsqÔei ìti S B1 BC = S C1 BC. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, C 1, B 1, A 1 ekteloôme ta parakˆtw b mata.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN221 BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: S A, AB SB kai AC SC. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc. 'Etsi èqoume: G 1 = S BCB 1 S BCC1 = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou C 1, qrhsimopoioôme to l mma 2.4.15, selðda 37) (S BCC1 = S ACA 1 S SBC S SAC ) S BCB1 S BCC1 = S BCB 1 S SAC S ACA1 S SBC (Apaloif shmeðou B 1, qrhsimopoioôme to l mma 2.4.15) (S BCB1 = S ABA 1 S SBC S SAB ) S ABA1 S SBC S SAC S ACA1 S SBC S SAB = S ABA 1 S SAC S ACA1 S SAB (Apaloif shmeðou A 1, qrhsimopoioôme to l mma 2.4.11, selðda 36) (S ACA1 = (( SA 1 SA )S SAC), S ABA1 = (( SA 1 SA )S SAB)) ( S SAB SA 1 SA +S SAB)S SAC ( S SAC SA 1 SA +S SAC)S SAB BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec mìno apì anexˆrthtec metablhtèc. = 1. AxÐwma tou Pascal.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN222 A 1 C 1 B 1 A B C Apìdeixh DIATUPWSH Sq ma 4.27: K 1 : (P OINT S A B A 1 ) 'Estw trða aujaðreta shmeða A, B, A 1 K 2 : (ON LINE C A B) 'Estw C shmeðo pˆnw sthn eujeða AB. K 3 : (ON P LINE B 1 A B A 1 ) 'Estw shmeðo B 1 pˆnw sthn eujeða pou dièrqetai apì to shmeðo A kai eðnai parˆllhlh sthn BA 1. K 4 : (INT ER C 1 (LINE A 1 B 1 )(P LINE A C A 1 )) 'Estw C 1 to shmeðo tom c twn eujei n SC kai thc parˆllhlhc apì to shmeðo A 1 sthn eujeða AC. S: IsqÔei ìti S BCB1 = S C1 CB 1. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, C 1, B 1 ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc perioristik n sunjhk n twn parapˆnw kataskeu n: B A, B A 1, A 1 B 1 CA 1. BHMA 3. Metaforˆ ìlwn twn gewmetrik n posot twn sto pr to mèloc.

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN223 'Etsi èqoume: G 1 = S BCB 1 S CB1 C 1 = 1 kai ektel ta parakˆtw b mata. BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou C 1, qrhsimopoioôme to l mma 2.4.15, selðda 37) (S CB1 C 1 = S AA1 B 1 C) S BCB1 S CB1 C 1 = S BCB 1 S A1 CB 1 S AA1 B 1 CS A1 CB 1 = S BCB 1 S AA1 B 1 C (Apaloif shmeðou B 1, qrhsimopoioôme to l mma 2.4.13, selðda 36) (S AA1 B 1 C = AB 1 BA 1 S BA1 C, S BCB1 = ( AB 1 BA 1 S BA1 C)) S BA1 C AB 1 BA 1 S BA1 C AB 1 BA 1 BHMA 5. Nèec gewmetrikèc posìthtec apoteloômenec mìno apì anexˆrthtec metablhtèc. = 1.

PARARTHMA G Sto kefˆlaio 4, parousiˆsame tic kataskeuèc twn eidik n shmeðwn twn trig nwn, ìpwc to kèntro bˆrouc, to orjìkentro, to ègkentro kai to perðkentro kaj c kai thn isodunamða twn kataskeu n aut n me thn kataskeu (ARAT IO Y A B C r O r U r V ). Sto parˆrthma autì ja apodeðxoume tic isodunamðec autèc. 1.BarÔkentro G trig nou ABC. CENT ROID G A B C (ARAT IO G A B C 1 3 Apìdeixh 1 1 ) 3 3 'Estw G to kèntro bˆrouc trig nou ABC kai M to mèson thc pleurˆc BC. Epeid to shmeðo M eðnai to mèson thc BC èqoume apì prìtash B.2: S ABM = S AMC kai S GBM = S GMC. Tìte S GAB = S ABM S GBM = S AMC S GMC = S GCA. 'Omoia S GBC = S GAB = S GCA = 1 3 S ABC. 'Ara r A = S GBC S ABC = 1 3 r B = S AGC S ABC = 1 3, r C = S ACG S ABC = 1 3 224

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN225 2.Orjìkentro H trig nou ABC. ORT HOCENT ER H A B C (ARAT IO H A B C P ABCP ACB 16S 2 ABC Apìdeixh P BAC P BCA 16S 2 ABC P CAB P CBA ) 16SABC 2 'Estw H to orjìkentro trig nou ABC. 'Etsi loipìn h tom twn uy n CD kai AE eðnai to shmeðo H. Opìte me qr sh twn protˆsewn B.2, selðda 14 kai B.8, selðda 89, ja èqoume: r B ra = S AHC S ABC S HBC S ABC = S AHC S HBC = AD DB = P CAB P ABC r B rc = S AHC S ABH S ABH S ABC = S AHC S ABH = CE EB = P BCA P ABC Akìmh r A : r B : r C = P ABC P BCA : P CAB P BCA : P ABC P CAB Apì ton tôpo Herron Qin. P ABC P BCA + P CAB P BCA + P ABC P CAB = 2AB 2 P BCA + P ABC P CAB = 16S 2 ABC EÐnai faner plèon h sqèsh. 3.PerÐkentro O trig nou ABC. CIRCUMCENT ER O A B C (ARAT IO H A B C P BCBP BAC 16S 2 ABC Apìdeixh P ACA P ABC 16S 2 ABC P ABA P ACB ) 16SABC 2 Jèloume na upologðsoume touc lìgouc: r A = S OBC S ABO S ABC S ABC, r B = S AOC S ABC, r C =

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN226 A a H 1 O B D E a O 1 R C Sq ma 4.28: ParathroÔme ìti oi parapˆnw lìgoi aforoôn embadˆ trig nwn pou èqoun koin bˆsh. Sunep c antð na upologðsoume touc lìgouc twn embad n aut n, ja upologðsoume touc lìgouc twn uy n touc. SumbolÐzoume me O 1 to tm ma OE kai H 1 to tm ma AD. Ja apodeðxoume ton pr to lìgo r A = S OBC S ABC. Oi upìloipoi lìgoi prokôptoun ìmoia. Jètoume: a = BC, b = AC, c = AB Opìte: r A = O 1 H 1 Akìmh: O 1 = R cos( (A)) GnwrÐzoume ìti to embadìn trig nou eggegrammènou se kôklo dðnetai apì ton akìloujo tôpo:s ABC = abc 4R Apì ton nìmo twn hmitìnwn: 2R = a = sin( (A)) b = sin( (B)) c sin( (C))

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN227 EpÐshc 8R 3 = abc sin( A) sin( B) sin( C) Apì ton nìmo twn sunhmitìnwn: cos(a) = b2 +c 2 a 2 2bc Tèloc H 1 = c sin( (B)) = b sin( (C)) = 2S ABC a 'Ara O 1 H 1 = (b2 +c 2 a 2 )a 2 16S 2 ABC 4.'Egkentro I trig nou ABC. INCENT ER C I B C (ARAT IO I A B C 2P IABP IBA P AIB P ABA Apìdeixh P IAB P IBI P AIB P ABA P IBA P IAI P AIB P ABA ) A H r I B E Q C Sq ma 4.29:

PARARTHMA D Sto parˆrthma autì ja parousiˆsoume tic apodeðxeic twn idiot twn twn pl rwn gwni n 10, 11, 12, selðda 189, efarmìzwntac ton algìrijmo. 'Estw ìti ta A, B, C kai D eðnai tèssera shmeða pˆnw se kôklo kèntrou O. Tìte [ACB] = [ADB] kai [AOB] = 2 [ACB]. Apìdeixh C P D O A N B Sq ma 4.30: Pr ta qrhsimopoioôme ton algìrijmo gia na upologðsoume thn gwnða [ACB]. 'Etsi loipìn zhtˆme apì ton upologist na upologðsei to tan( [ACB]). DIATUPWSH 228

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN229 K 1 (P OINT S A B): 'Estw duo aujaðreta shmeða A, B K 2 (ON O (BLINE A B)): 'Estw O shmeðo eujeðac pou eðnai kˆjeth sthn AB. K 3 (T RAT IO P B A r): 'Estw shmeðo P pˆnw sthn eujeða (T LINE B B A) tètoio ste: r = BP BA. K 4 (INT ER C (LINE A P )(CIR O A) (F OOT N O A P ), (P RAT IO C N N A): 'Estw C to shmeðo tom c thc eujeðac AP kai tou kôklou kèntrou O kai aktðnac OA. S: Upolìgise: ( 4)S ABC P ACB. ALGORIJMOS BHMA 1. Gia kajèna apì ta shmeða, me thn akìloujh seirˆ, C, P, O ekteloôme ta parakˆtw b mata. BHMA 2. 'Elegqoc ikan n kai anagkaðwn sunjhk n twn parapˆnw kataskeu n: A C, A O, A B. BHMA 3. - BHMA 4. Apaloif bohjhtik n shmeðwn apì thn G 1. (Apaloif shmeðou C, qrhsimopoioôme to l mma 2, KEF.3) ( 4)S ABC P ACB = ( 4)( 2P OAP S ABP ) P AP A 2(P OP O P AP B P AOA )P OAP = P AP A 4S ABP P OP O P AP B P AOA (Apaloif shmeðou P, qrhsimopoioôme to l mma 2, KEF.3) 4( 1 4 P ABAr) P BOB +P ABA r 2 +8S ABO r P AP B r 2 P AOA = 4( 1 4 P ABAr) P BOB P AOA +8S ABO r (Apaloif shmeðou O, apì ton orismì thc Pujagìreiac diaforˆc kai thc idiìthtac twn shmeðwn pˆnw ston kôklo, OA = OB.)

KEFŸALAIO 4. SUMPLHRWSH THS MEJODOU ME QRHSH GWNIWN230 P ABA r 8S ABO r (Aplopoi seic) P ABA 8S ABO. Apì ton upologismì thc efaptomènhc thc gwnðac [ACB], diapist noume ìti h tan( [ACB]) den exartˆtai apì ta shmeða P, C, epomènwc ja isqôei: [ACB] = [ADB]. Mènei na apodeðxoume ìti: [AOB] = 2 [ACB], dhlad tan( [AOB]) = tan(2 [ACB]). Me bˆsh ton akìloujo trigwnometrikì tôpo, thn tim thc tan( [ACB]) pou mìlic upologðsame kai ton tôpo tou Herron Qin (16SAOB 2 = 4AO2 AB 2 PAOB 2 ) ja pˆroume: tan(2 [ACB]) = 4S AOB 2AO 2 AB 2 = 4S AOB P AOB 2 tan( [ACB]) = 8AB2 S AOB = 1 tan( [ACB]) 2 16SAOB 2 AB4 = tan( [AOB]) Epomènwc [AOB] = 2 [ACB]. 8AB 2 S AOB 4OA 2 OB 2 P 2 AOB AB4 =