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The Radii by R, Ra, Rb,Rc

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The Radii by R, Ra, Rb,Rc The Orthopole, by R. GOORMAGHTIGH,La Louviero, Belgium. Introduction. 1. In this little work are presented the more important resear- ches on one of the newest chapters in the modern Geometry of the triangle, and whose origin is, the following theorem due to. Professor J. Neuberg: Let ƒ¿, ƒÀ, y be the projections of A, B, C on a straight line m; the perpendiculars from a on BC, ƒÀ on CA, ƒÁ on ABare concurrent(1). The point of concurrence. M is called the orthopole of m for the triangle ABC. The following general conventions will now be observed : the sides of the triangle of reference ABC will be denoted by a, b, c; the centre and radius of the circumcircle by 0 and R; the orthocentre by H; the centroid by G; the in- and ex-centresby I, Ia 1b, Ic and the radii by r, ra, rb,rc;the mid-points of BC, CA, AB by Am, Bm, Cm;the feet of the perpendiculars AH, BH, OH from A, B, C on BO, CA, AB by An,Bn, Cn; the contact points of the incircle with BC, CA, AB by D, E, F; the corresponding points for the circles Ia, Ib, Ic, by Da, Ea,Fa, Dc, Eb, F0, Dc, Ec, Fc; the area of ABC by 4; the centre of the nine-point circle by 0g; the Lemoine . point by K; the Brocard points by theGergonne and Nagel points by and N(2); further 2s=a+b+c. Two points whose joins to A, B, C are isogonal conjugate in the angles A, B, C are counterpoints. The images, through Am, Bm, Cm,of the points where a straight line m meets the 'sides, are on another straight line, which ' will be called reciprocal line to m, (1 Neubeitg, NCM,1875, p. 189; H. van Aubel, NCH,1875, p, 316; Des- boves Questionsdo Geometric,1875, p. 241;Neuberg, NOM1,1878,p. 379; Lemoino, ME, 1884,p. 50; Cwojdzihski, AMPh,3u-ser., voi 1, p. 174; Neuberg,J AMPh,sd ser.vol.3, p.89, Surles projections et eontreprojections d'untriangle, 1890, p. 74-76,M, 1911, p.244,M,1012, p.238; Gallatly, Themodern geometry of thetriangle, 2fla-ed., p. 46. (2) 78 R. GOORMAGHTIGH : The straight line passing through the projections of any point Q of the circumeircle on the sides will be called Simson line of Q; Q is the pole of the Simson line. AmBm, Cm is the medial triangle of ABC; the parallels to BC, CA, AB drawn respectively through A, B, C form the antimedial triangle of A BC. The following abbreviations will further be used for mathematical journals referred to : AFAS, Association frangaiso pour l'avancement des Sciences AG, Annales do Mathematiques (Gergonne), AMPh Archiv der Mathematik and Physik, BB, Bulletin de l'Acadbmie de Belgique (Classe des Sciences), BME, Bulletin de Math6matiques 616mentaires, ET, Educational Times, IM, Interm6diaire des Math6maticiens, JME, Journal do Math6matiques 616mentaires (Bourget, de Longchamps), JV, Journal de Math6matiques 616mentaires (Vuibert), M, Mathesis, MB, Mtmoires in-8° de l'Acad6mie royal de Belgique, NA, Nouvelles Annales do Mathematiques, NCM, Nouvelle Correspondance Math6matique, QJ, Quarterly Journal, SL, M6moires de la SoclLte royale des sciences de Liege, TMJ, Tohoku Mathematical Journal, , WT, Wiskundig Tijdschrift. I. General theorems on the orthopole. 2. The ortlaopole theorem. Draw m' through A parallel to. m and let ƒÀ•L, ƒÁ•L be the projections of B and C on m•L, p and q the projections of ƒÀ•L on AC and of ƒÁ on AB; ƒÀ•LƒÏ and ƒÁ•Lq meet AAn, at Mb and Mc.Then MbpAnC, ƒÀ•LpƒÁ•LC, are cyclic. Hence (Fig. 1) Therefore . If ƒ¿, ƒÀ, ƒÁ are the projections , of A, B, G on m, the perpendiculars from a on BC,ƒÀon CA, ƒÁon AB are con- current at M and AM is equipollent to Aoc. 3. .Distance from M to m. When m moves parallelto. itself, the figure ƒ¿ƒÀƒÁM remains -unchanged in shape andsize, H sliding along a straight line 8 perpendicular to m; the distance S from H to m remains unchanged. Let ƒÆa ƒÆb, ƒÆc be the angles which .the sidesmake with m, these angles being measured from m as axis, and in the same sense. Then THE ORTHOPOLE: 79 4. The ortlaopde M lies on the Simson line perpendicular to m. If. T be the point where Aa cuts the circumcircle and T•Lits projection on BC, then the Angle TAC is equal to 90•Ž-ƒÆb, and the angles BCT and BAT are equal to 90•‹-ƒÆc. Since TC =2Rsin TAC=2RcosƒÆb, TT=TOsin BCT =2Rcos ƒÆbcos ƒÆc=AMb Hence MT•L is parallel to AT, and, by a known property, T•L is the Simsonline M per- pendicular to m. 5. Other Simson lines passing througla M. Let Q. and Q•L be the points where m meets the circumcircle; the perpendicular QQ, from Q on BC meets again the circumcircle at Q2. If Q" is the projection of T on QQ2, Qƒ¿TQ" is cyclic; hence the anglesAQ2Q,ƒ¿TQ, ƒ¿Q"Q are equal and AQ2 is parallel to ƒ¿Q" and also to MQ1, since ƒ¿M= TT=Q"Q1. Therefore MQ1 is the Simson line of Q. When m meets the circumcircle at Q and Q', the-Simson lines of Q and Q' pass through M(2); when m is tangent at Q to the circumcircle, the orthopole of m is the point where the Sum son line of Q' touches its envelope (S to z ne r s tricuep). 6. Orthopole and pedal triangles. Let XYZ and X'Y'Z' ,be any two triangles; then a certain point V, called of the two triangles has the same relative barydentric coordinates in ., XYZ and X 'Y'Z' . if. X, p, v are these coordinates, V isthe. centre of gravity (1) Neuberg, Sur lea cercles podaires, BB, 1910; de Lepiney, M, 1914, p. 177. (2) Neuberg, Sur lea projections et contre-profa Lions d'un triangle, MB, 1890; goons M, 1896, p. 57. Hence, 11 is the orthopole of 130, CA, AB. (8) Neuberg, Sur lea equicentres de deux systentes de n points, SL, 1918. R. GOORMAGHTIGH: of masses ƒÁ, ƒÊ, v at X, Y, Z and at X', Y', Z', Therefore, V being any point, Hence, the vectors being in the same plane, Let now PaPbPc and SaSbSc be the pedal triangles(1) of two points P and S on m, in the triangle ABC and .denote by M equicentre of PaPbPc, SaSbSc; then We will provo that this new point M is the orthopole of m. If 3 is the pedal triangle of a, ƒ¿1ƒ¿2ƒ¿ 3:ƒ¿ƒ¿1ƒ¿3= ƒ¿ƒ¿3sin B :ƒ¿ƒ¿2Sin C=ƒ¿ƒ¿1ƒ¿ b•Aƒ¿cosƒÆc : c•AƒÆcosƒÆb =b secƒÆb : c secƒÆc: Hence .M lies on ƒ¿ƒ¿1; and so for the perpendiculars from ƒÀ on CA and from ƒÁ on AB. The orthopole M of m has the same relative barycentric coordinates a see ƒÆa, bseeƒÆb, cseeƒÆc in the pedal triangle of any point P on m (2). In section 27, we will find that, for any straight line m, . Hence the absolute harycentric coordinates(3) of the orthopoteM in the pedal triangle. of any point -on m are (4 ) 7. Orthopoleand pedal circles. Lemoyne's Theorem. Analytical proof(5), Let Pa, Pb, Pc be the lengths of thesides of the pedal triangle PaPbPc of a point P on m; the barycentric equation to the circle PaPbPc in the "triangle PaPbPc is (1) Trianglesformed by the projectionsPa, Pb,Pcand Ba,Bb, Be of Pand.s on BC,CA, AB respectively. (2) Neuberg, Fiurleg cercies poZalres, 1313, 1910. (3) For absolutebarycontric coordinates, , 4) Goor'masrhtigh. M. 1925.p.197. ( (5) Lemoyne's theoremhas beenpublished without proof in riA, 1904,p: 400. Proof3were given by N e u b e r g (anal.,)138, 1910, Lienard (anal.),M, 1912,p. 288, 1014p. 38,dQ Lepiney (geom.),M, 1914, p. ,177,Gallatly (geom.),Modern geometry of thetriangle, Sd ed, p. 51, Bouvaist (geom.),NA, 1915, p.'555, Thebault (geom.), NA, 1918,p. 293,TMJ, Vol19, Goormaght1gh (anal.),M. 1925,p. 196(the here givenproof). 80 THE ORTHOPOLE. 81 and it is known that the power, for the circle PaPbPc, of 'a point M, whose absolute barycentric coordinates in PAPBPC are ƒÁ, ƒÊ, v, is But Pa,Pb,Pc being the tripolar coordinates PA,PB,PC of P, d being the distance from m to 0. Hence The orthopole J of m has the same power witli regard to the pedal circle of any point P on m; this power is twicethe product of the distances from m to 0 and to N(2). d and 8 will be taken in the same sense when 0 and . Mare on the same side of m. 8. Geometrical proof of Lemoyne' s theorem (3 ). Let J be one of the points where the pedal circle of P meets m;the perpendicular g at J to m is a tangent to the conic 1 inscribedin ABC, having a focus at P, the pedal circle being the auxiliary circle of X. Hence, if g cuts the sides of ABC in a1, b1, c1,-'the mid-points a1, b1', c1' of a1, Bbl, Cc1, are on a straight line going through. the centre of A2 (Newton); this centre is also the centre of thepedal circle of P (Fig. 2). When g moves, parallel to itself) a1 bl' c1' remains - tangent to a parabola _(7r), inscribed in the medial triangle, the directrix being the circumdiameter t parallel to m; in section 27, we will prove "that the focus of is the orthopole of t.
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