If two triangles are perspective from a point, then they are also perspective from a line.

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1 Mth 487 hter 4 Prtie Prolem Solutions 1. Give the definition of eh of the following terms: () omlete qudrngle omlete qudrngle is set of four oints, no three of whih re olliner, nd the six lines inident with eh ir of these oints. The four oints re lled verties nd the six lines re lled sides of the qudrngle. () omlete qudrilterl omlete qudrilterl is set of four lines, no three of whih re onurrent, nd the six oints inident with eh ir of these lines. The four lines re lled sides nd the six oints re lled verties of the qudrilterl. () ersetivity etween enils of oints one-to-one ming etween two enils of oints is lled ersetivity if the lines inident with the orresonding oints of the two enils re onurrent. The oint where the lines interset is lled the enter of the ersetivity. (d) ersetivity etween enils of lines one-to-one ming etween two enils of lines is lled ersetivity if the oints of intersetion of the orresonding lines of the two enils re olliner. The line ontining the oints of intersetion is lled the xis of the ersetivity. (e) rojetivity etween enils of oints one-to-one ming etween two enils of oints is lled rojetivity if the ming is omosition of finitely mny elementry orresondenes or ersetivities. (f) The hrmoni onjugte of oint with reset to oints nd. Four olliner oints,,, form hrmoni set, denoted H(,), if nd re digonl oints of qudrngle nd nd re on the sides determined y the third digonl oint. The oint is the hrmoni onjugte of with reset to nd. (g) oint oni oint oni is the set of oints of intersetion of orresonding lines of two rojetively, ut not ersetively, relted enils of lines with distint enters. (h) line oni line oni is the set of lines tht join orresonding oints of two rojetively, ut not ersetively, relted enils of oints with distint xes. 2. Stte eh of the following: () esrgues Theorem If two tringles re ersetive from oint, then they re lso ersetive from line. () The Fundmentl Theorem of Projetive Geometry rojetivity etween two enils of oints is uniquely determined y three irs of orresonding oints.

2 3. True or Flse () In lne rojetive geometry, if two tringles re ersetive from oint, then they re lso ersetive from line. True. This is onsequene of esrgues Theorem () In the Poinré Hlf Plne, if two tringles re ersetive from oint, then they re lso ersetive from line. Flse. See Homework xerise #4.18 [Hint: ik ir of tringles with ir of orresonding sides tht re rllel.] () In lne rojetive geometry, if two tringles re ersetive from line, then they re lso ersetive from oint. True. This is onsequene of the dul of esrgues Theorem. (d) very oint in lne rojetive geometry is inident with t lest 4 distint lines. True. This is onsequene of the dul of Theorem 4.4, whih is true sine Plne Projetive Geometries stisfy the rinile of dulity. (e) If H(, ) then H(, ). True. This is onsequene of Theorem 4.8. (f) If H(,) nd H(, ) then = True. This is onsequene of the Fundmentl Theorem 4.7. (g) If,, nd,, re distint elements in enils of oints with distint xes nd, there there exists ersetivity suh tht o ˆ Flse. Theorem 4.10 gurntees tht there is rojetivity suh tht, ut this rojetivity is not neessrily ersetivity (for exmle, the onstrution we did in lss to rove this theorem required two ersetivities). 4. Prove tht xiom 3 in indeendent of xiom 1 nd xiom 2. onsider the following model: l m n In this model,,, nd re oints, nd l,m, nd n re lines. Notie tht ny ir of distint oints re on extly one line [ nd re on m, nd re on l, nd nd re on n]. lso notie tht ny two distint lines re inident with t lest one oint [in ft, lotm =, lotn =, nd motn = ]. However, sine there re only 3 oints in this model, xiom 3 is not stisfied.

3 5. () Stte nd rove the dul of xiom 3. Rell xiom 3 sttes: There exist t lest four oints, no three of whih re olliner. Then the ul of xiom 3 is: There exist t lest four lines, no three of whih re onurrent. Proof: Let,,, nd e four distint oints, no three of whih re olliner ( we know these oints exist y xiom 3). Using xiom 1, the lines,,,,, nd ll exist. Sine no three of the oints,,, nd re olliner, these six lines must e distint. onsider the four lines,,, nd. To show tht no three of these lines re onurrent, we roeed y ontrdition. Suose not. Then three of these lines would e onurrent. For exmle, suose tht,, nd re onurrent. Using the ul of xiom 1, is the only oint of intersetion of nd. Therefore, must e the oint of onurreny for the three lines,, nd. ut then is on. This ontrdits our ssumtion tht,, nd re nonolliner. The other ses re similr. Therefore, there exist t lest four lines, no three of whih re onurrent.. () Stte nd rove the dul of xiom 4. Rell tht xiom 4 sttes: The three digonl oints of omlete qudrngle re never olliner. Then the ul of xiom 4 is: The three digonl lines of omlete qudrilterl re never onurrent. Proof: Let d e omlete qudrilterl (we know tht suh qudrilterl exists from the ul of xiom 3). Let =, F =, G = d, H = d, I = nd J = d. These oints exist y xiom 2, nd re unique y the ul of xiom 1. Using xiom 1, the digonl lines G, FH, nd IJ exist. lim: The digonl lines G, FH, nd IJ re not onurrent. We will rove this lim using roof y ontrdition. Suose tht the lines G, FH, nd I re onurrent. Then G FH must e the oint of onurreny etween these lines. Therefore, the oints I, J, nd G FH re olliner. Sine d is omlete qudrilterl, no three of the lines = H, = F, = FG, nd d = GH re onurrent. Thus, (using the dul of the rgument in the roof of the ul of xiom 3), F, G, nd H re four oints, no three of whih re olliner. Hene, FGH is omlete qudrngle with digonl oints F GH = d = J, G FH, nd H FG = = I. Hene, using xiom 4, then the oints I, J, nd G F H re nonolliner, whih ontrdits our revious ssumtion tht they re olliner. Therefore, the digonl lines of the omlete qudrilterl d re not onurrent.. 6. () Prove tht omlete qudrngle exists. Proof: y xiom 3, there re 4 distint oints no three of whih re olliner. ll these oints,,, nd. y xiom 1, the lines,,,,, nd ll exist. We lim tht these six lines re ll distint. To see this, first suose tht =. This would use,, nd to e olliner, whih ontrdits our erlier ssumtion. The other ses re similr (note tht in the se where we ssume = we hve tht,,, nd re ll olliner.) onsequently, omlete qudrngle exists..

4 () rw model for omlete qudrngle FGH. H F G The oints, F, G, nd H, long with the lines F, G, H, FG, FH, nd GH form omlete qudrngle. () Identify the irs of oosite sides in the qudrngle F GH. There re 3 irs of oosite sides in the qudrngle: F nd GH H nd FG G nd FH (d) onstrut nd identify the digonl oints of the qudrngle F GH. I H K F G J Let I = F GH Let J = H FG Let K = G FH Then I,J nd K re the digonl oints of this omlete qudrngle. 7. () Prove tht omlete qudrilterl exists. Proof: y xiom 3, there re 4 distint oints no three of whih re olliner. ll these oints,,, nd. y xiom 1, the lines,,,,, nd ll exist. s in the roof of the existene of omlete qudrngle, these six lines re ll distint, otherwise, three of the originl oints would e olliner ontrry to our revious ssumtion. onsider the lines,,, nd. Using the dul of xiom 1, let = nd let F =. Notie tht nd F must e distint from,,, nd, otherwise this would one gin fore 3 of our originl oints to e olliner, ontrry to our revious ssumtion. From this, we see tht no three of the lines,,, nd re onurrent. Hene the oints,,,,, nd F long with the lines,,, nd form omlete qudrilterl..

5 () rw model for omlete qudrilterl d. J N I U T S d The lines,,, nd d long with the oints J, U, S, T, I, nd N form omlete qudrilterl. () Identify the irs of oosite oints in the qudrilterl d. There re three irs of oosite oints in this qudrilterl: J nd I; U nd T; S nd N. (d) onstrut nd identify the digonl lines of the qudrilterl d. J N I U T S d The digonl lines in this qudrilterl re JI, UT, nd SN. 8. () onstrut n exmle of two tringles tht re ersetive from oint. e sure to identify the oint tht the tringles re ersetive from. In the digrm ove, nd re ersetive from the oint. () re these two tringles lso ersetive from line? If so, identify the line tht the tringles re ersetive from. If not, exlin why they nnot e ersetive from line. P Q R From the digrm ove, if we let = P, = Q, nd = R, notie tht R is inident with the line PQ, so nd re ersetive from the line PQ.

6 9. Illustrte rojetivity from enil of lines,, with enter to enil of lines,, with enter. 10. Prove eh of the following: () The dul of esrgues Theorem ul of esrgues Theorem: If two tringles re ersetive from line, then they re lso ersetive from oint. Proof: Suose nd re ersetive from line. Let P =, Q = nd R =. y the definition of ersetivity from line, the oints P, Q nd R re olliner. Let =. To show tht, nd re onurrent, we must show tht is on the line. onsider the tringles R nd Q. Sine P, Q, R re olliner, P is on line QR. Sine P =, P is on line nd line. Hene tringles R nd Q re ersetive from oint P, y the definition of ersetive from oint. Hene y xiom 5 (esrgues Theorem), tringles R nd Q re ersetive from line. y definition of ersetivity from line, the oints = R Q, = R Q nd = re olliner. Hene is on the line. Therefore, nd re onurrent. Therefore, nd re ersetive from oint.. P R Q () Theorem 4.6 Theorem: If,, nd re three distint olliner oints, then hrmoni onjugte of with reset to nd exists. Proof: Let,, nd e three distint olliner oints. y xiom 3, there is oint suh tht, nd re non-olliner. y Theorem 4.3, there is oint F on tht is distint from nd. Let G = F nd let H = G. lim: The oints,f,g, nd H nd the lines F, G, H, FG, FH, nd GH determine omlete qudrngle.

7 To see this, notie tht the oints,f,g nd H re distint. nd F re distint y onstrution. For the others, first suose tht G = F. Sine is inident to F nd G = F is inident to, then,,g = F, is olliner set, ontrry to our revious ssumtions. The other ses re similr. Next, Suose tht,f nd G re olliner. Sine G is inident to F, F is inident to, nd is inident to, then,, nd re olliner, ontrry to our revious ssumtions. The other ses re similr. This roves the lim. Notie tht FH is the remining side of the omlete qudrngle. Then if we tke = FH, then we hve onstruted the hrmoni set H(, ).. G H F () The Fundmentl Theorem of Projetive Geometry Theorem: rojetivity etween two enils of oints is uniquely determined y three irs of orresonding oints. Proof: We must show tht if,,, nd re in enil of oints with xis nd,, re in enil of oints with xis, then there exists unique oint on suh tht. ssume,,, nd re in enil of oints with xis nd tht,, nd re in enil of oints with xis. Reell tht there exists oint on suh tht (to find, we find d the imge of under the first elementry orresondne, nd then find the imge of d under the seond elementry oresondne, nd ontinue through eh of the finitely mny elementry orresondnes in the rojetivity). Suose there is rojetivity nd oint suh tht. Sine nd, we hve. Therefore, using xiom 6, = The frequeny rtio 3 : 4 : 5 is lso equivlent to the rtio 3 2 : 15 8 : 9 8, whih gives the hord G,, lled the dominnt of the mjor trid of the exmle ove. Show H(G,) where G = ( 2 3 ), = ( 8 15 ), nd = (8 9 ). In the digrm ove, we hve onstruted the hrmoni set H(G, ).

8 12. nswer the following questions sed on the following digrm: () Find, the hrmoni onjugte of with reset to nd. To find the hrmoni onjugte of with reset to nd, we onstrut m rorite qudrngle (one with nd s digonl oints nd the interstion of one of the remining ir oosite sides) we then onstrut to omlete the hrmoni set y finding the oint tht the remining oosite side intersets the line. G H F () Pik oint not on nd onstrut n elementry orresondene etween the oints,,, nd enil of lines with enter. d The digrm given ove illustrtes the elementry orresondne d

9 () Find line distint from = nd extend the elementry orresondene you onstruted in rt () to ersetivity etween,,, nd orresonding oints on. d The digrm given ove illustrtes the ersetivity ˆ (d) xtend this ersetivity to rojetivity. The digrm shown ove illustrtes rojetivity.

10 13. Given the following rojetivity: q " " " " " " " r () Identify eh elementry orresondne in this rojetivity. The elementry orresondnes re s follows: () Find the imge of under this rojetivity. The imge of the oint d under this rojetivity is the line d s illustrted in the following digrm: q d " d " " " " " d " r