X3066

2nd LEMOINE HOMOTHETIC CENTER

This is a correction for trilinears given in TCCT, p. 188.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 6, 110 }: Central Line L(6, 110) { 25, 182 }: Central Line L(25, 182) { 3, 373 }: Central Line L(3, 373) { 51, 394 }: Central Line L(51, 394) { 125, 381 }: Central Line L(125, 381) { 107, 458 }: Central Line L(107, 458) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6329, 0.4497) X_182 = MIDPOINT OF BROCARD DIAMETER (0.5500, 0.1846) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_373 = CENTROID OF THE PEDAL TRIANGLE OF THE CENTROID (0.5614, 0.2395) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6093, 0.3186) X_394 = -CEVA CONJUGATE OF X(3) (0.3778, 0.0284) X_125 = CENTER OF JERABEK HYPERBOLA (0.6803, 0.0260) X_381 = MIDPOINT OF X(2) AND X(4) (0.5748, 0.2979) X_107 = Ψ(SYMMEDIAN POINT, ORTHOCENTER) (0.5733, 0.6071) X_458 = EULER LINE INTERCEPT OF LINE X(76)X(275) (0.5800, 0.3115) X_3066 = 2nd LEMOINE HOMOTHETIC CENTER (0.5807, 0.2828) A B C X6 X110 X25 X182 X3 X373 X51 X394 X125 X381 X107 X458 X3066 (a, b, c) = (5, 6, 7) | AB = 1

X3066 (2nd LEMOINE HOMOTHETIC CENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a(a^{2} + 3b^{2} - c^{2})(a^{2} - b^{2} + 3c^{2})\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(0.950789516177\)
▸ Incident Central Lines7 lines

{2, 1350}, {3, 373}, {6, 110}, {25, 182}, {51, 394}, {107, 458}, {125, 381}

▸ Functional Relations1 relation

  • X3066 = barycentric product of vertices of submedial triangle

Notes

Suppose X is a point, with isogonal conjugate \(X^{- 1}\). It is well known that the pedal triangle of X is homothetic to the antipedal triangle of \(X^{- 1}\). The Lemoine homothetic center, X1285, is the homothetic center when X = X6, and X3066 is the homothetic center when X = X2. See also X1285. (Peter Moses, Dec. 7, 2005)

In general if X =\(x : y : z\) (trilinears), then the homothetic center is given by

a(x + y cos C)(x + z cos B) : b(y + z cos A)(y + x cos C): c(z + x cos B)(z + y cos A).