Orthopoles and the Pappus Theorem


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1 Forum Geometriorum Volume 4 (2004) FORUM GEOM ISSN Orthopoles n the Pppus Theorem tul Dixit n Drij Grinerg strt. If the verties of tringle re projete onto given line, the perpeniulrs from the projetions to the orresponing sielines of the tringle interset t one point, the orthopole of the line with respet to the tringle. We prove severl theorems on orthopoles using the Pppus theorem, funmentl result of projetive geometry. 1. Introution Theorems on orthopoles re often prove with the help of oorintes or omplex numers. In this note we prove some theorems on orthopoles y using wellknown result from projetive geometry, the Pppus theorem. Notly, we nee not even use it in the generl se. Wht we nee is simple ffine theorem whih is speil se of the Pppus theorem. We enote the intersetion of two lines g n g y g g. Here is the Pppus theorem in the generl se. Theorem 1. Given two lines in plne, let,, e three points on one line n,, three points on the other line. The three points,, re olliner. ' ' ' Figure 1 Theorem 1 remins vli if some of the points,,,,, re projete to infinity, even if one of the two lines is the line t infinity. In this pper, the only se we nee is the speil se if the points,, re points t infinity. For the ske of ompleteness, we give proof of the Pppus theorem for this se. Pulition Dte: Mrh 30, ommuniting Eitor: Pul Yiu.
2 54.. Dixit n D. Grinerg Y Figure 2 Let =, Y =, =. The points,, eing infinite points, we hve Y,, n Y. See Figure 2. We ssume the lines n interset t point P, n leve the esy se to the reer. In Figure 3, let Y = Y. We show tht Y = Y. Sine Y, wehve P P = PY in signe lengths. Sine P,wehve P P = P P. From these, P P = PY P, n Y. Sine Y, the point Y lies on the line Y. Thus, Y = Y, n the points, Y, re olliner. Y' P Figure 3
3 Orthopoles n the Pppus theorem The orthoenters of fourline We enote y the tringle oune y three lines,,. omplete qurilterl, or, simply, fourline is set of four lines in plne. The fourline onsisting of lines,,,, is enote y. Ifg is line, then ll lines perpeniulr to g hve n infinite point in ommon. This infinite point will e lle g. With this nottion, P g is the perpeniulr from P to g. Now, we estlish the wellknown Steiner s theorem. Theorem 2 (Steiner). If,,, re ny four lines, the orthoenters of,,, re olliner. K N M F D E L Figure 4 Proof. Let D, E, F e the intersetions of with,,, n K, L, M, N the orthoenters of,,, n. Note tht K = E F, eing the intersetion of the perpeniulrs from E to n from F to. Similrly, L = F D n M = D E. The points D, E, F eing olliner n the points,, eing infinite, we onlue from the Pppus theorem tht K, L, M re olliner. Similrly, L, M, N re olliner. The four orthoenters lie on the sme line. The line KLMN is lle the Steiner line of the fourline D. Theorem 2 is usully ssoite with Miquel points [6, 9] n prove using ril xes. onsequene of suh proofs is the ft tht the Steiner line of the fourline is the ril xis of the irles with imeters D, E, F, where =, =, =, D =, E =, F =. lso, the Steiner line is the iretrix of the prol touhing the four lines,,,. The Steiner line is lso lle fourorthoenter line in [6, 11] or the orthoentri line in [5], where it is stuie using ryentri oorintes.
4 56.. Dixit n D. Grinerg 3. The orthopole n the fourline We prove the theorem tht gives rise to the notion of orthopole. Theorem 3. Let e tringle n line. If,, re the pels of,, on, then the perpeniulrs from,, to the lines,, interset t one point. This point is the orthopole of the line with respet to. M W D ' ' L Figure 5 Proof. Denote y,, the lines,,. y Theorem 2, the orthoenters K, L, M, N of tringles,,, lie on line. Let D =, n W =. The orthoenter L of is the intersetion of the perpeniulrs from D to n from to. Sine the perpeniulr from to is lso the perpeniulr from to, L = D. nlogously, M = D. y the Pppus theorem, the points W, M, L re olliner. Hene, W lies on the line KLMN. Sine W =, the intersetion W of the lines KLMN n lies on. Similrly, this intersetion W lies on. Hene, the point W is the ommon point of the four lines,,, n KLMN. Sine,, re the perpeniulrs from,, to,, respetively, the perpeniulrs from,, to,, n the line KLMN interset t one point. This lrey shows more thn the sttement of the theorem. In ft, we onlue tht the orthopole of with respet to tringle lies on the Steiner line of the omplete qurilterl. The usul proof of Theorem 3 involves similr tringles ([1], [10, hpter 11]) n oes not iretly le to the fourline. Theorem 4 origintes from R. Goormghtigh, pulishe s prolem [7]. It ws lso mentione in [5, Proposition 6], with referene to [2]. The following orollry is immeite.
5 Orthopoles n the Pppus theorem 57 orollry 4. Given fourline, the orthopoles of,,, with respet to,,, lie on the Steiner line of the fourline. K W N W M W W L Figure 6 4. Two theorems on the ollinrity of quruples of orthopoles Theorem 5. If,,, D re four points n e is line, then the orthopoles of e with respet to tringles D, D, D, re olliner. W Y e D Figure 7
6 58.. Dixit n D. Grinerg Proof. Denote these orthopoles y, Y,, W respetively. If,,, D re the pels of,,, D on e, then = D D. Similrly, Y = D D, = D D. Now,,, lie on one line, n D, D, D lie on the line t infinity. y Pppus theorem, the points, Y, re olliner. Likewise, Y,, W re olliner. We onlue tht ll four points, Y,, W re olliner. Theorem 5 ws lso prove using oorintes y N. Dergies in [3] n y R. Goormghtigh in [8, p.178]. speil se of Theorem 5 ws shown in [11] using the Desrgues theorem. 1 nother theorem surprisingly similr to Theorem 5 ws shown in [9] using omplex numers. Theorem 6. Given five lines,,,, e, the orthopoles of e with respet to,,, re olliner. W e Y Figure 8 Proof. Denote these orthopoles y, Y,, W respetively. Let the line interset,, t D, E, F, n let D, E, F e the pels of D, E, F on e. Sine E = n F = re two verties of tringle, n E n F re the pels of these verties on e, the orthopole = E F. Similrly, Y = F D, n = D E. Sine D, E, F lie on one line, n,, lie on the line t infinity, the Pppus theorem yiels the ollinerity of the points, Y,. nlogously, the points Y,, W re olliner. The four points, Y,, W re on the sme line. 1 In [11], Witzyński proves Theorem 5 for the se when,,, D lie on one irle n the line e rosses this irle. Inste of orthopoles, he equivlently onsiers Simson lines. The Simson lines of two points on the irumirle of tringle interset t the orthopole of the line joining the two points.
7 Orthopoles n the Pppus theorem 59 Referenes [1]. ogomolny, Orthopole, [2] J. W. lwson, The omplete qurilterl, nnls of Mthemtis, Ser. 2, 20 (1919) [3] N. Dergies, Hyinthos messge 3352, ugust 5, [4]. Dixit, Hyinthos messge 3340, ugust 4, [5] J.P. Ehrmnn, Steiner s theorems on the omplete qurilterl, Forum Geom., 4 (2004) [6] W. Glltly, The Moern Geometry of the Tringle, 2n e. 1913, Frnis Hogson, Lonon. [7] R. Goormghtigh, Question 2388, Nouvelles nnles e Mthémtiques, Série 4, 19 (1919) 39. [8] R. Goormghtigh, stuy of qurilterl insrie in irle, mer. Mth. Monthly, 49 (1942) [9]. Hsu, On ertin ollinerity of orthopoles, n of isopoles, Soohow J. Mth., 10 (1984) [10] R. Honserger, Episoes of 19th n 20th entury Eulien Geometry, Mth. sso. meri, [11] K. Witzyński, On olliner Griffiths points, Journl of Geometry, 74 (2002) tul hy Dixit: 32, Snehnhn Soiety, Kelkr Ro, Rmngr, Domivli (Est) , Mumi, Mhrshtr, Ini Emil ress: tul Drij Grinerg: Gerolsäkerweg 7, D Krlsruhe, Germny Emil ress: rij
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