X477

TIXIER ANTIPODE

X477 = reflection of X476 in X3
< 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) { 476, 3 }: Central Line L(476, 3) { 30, 110 }: Central Line L(30, 110) { 50, 112 }: Central Line L(50, 112) { 107, 186 }: Central Line L(107, 186) { 74, 523 }: Central Line L(74, 523) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_476 = TIXIER POINT (0.9824, 0.2685) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_50 = -CEVA CONJUGATE OF X(184) (2.1559, 2.8368) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.5162, 0.6121) X_107 = Ψ(SYMMEDIAN POINT, ORTHOCENTER) (0.5733, 0.6071) X_186 = INVERSE-IN-CIRCUMCIRCLE OF X(4) (0.7957, 0.8758) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.7484, -0.3437) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0856, -0.0096) X_477 = TIXIER ANTIPODE (0.0176, -0.0644) A B C X476 X3 X30 X110 X50 X112 X107 X186 X74 X523 X477 (a, b, c) = (5, 6, 7) | AB = 1

X477 (TIXIER ANTIPODE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(1/[4 \cos A + \cos(A - 2B) + \cos(A - 2C) - 3 \cos(B - C)] : :\)
Barycentrics: \((a^{8} + a^{6} b^{2} - 4 a^{4} b^{4} + a^{2} b^{6} + b^{8} - 3 a^{6} c^{2} + 2 a^{4} b^{2} c^{2} + 2 a^{2} b^{4} c^{2} - 3 b^{6} c^{2} + 3 a^{4} c^{4} - 2 a^{2} b^{2} c^{4} + 3 b^{4} c^{4} - a^{2} c^{6} - b^{2} c^{6}) \cdot (a^{8} - 3 a^{6} b^{2} + 3 a^{4} b^{4} - a^{2} b^{6} + a^{6} c^{2} + 2 a^{4} b^{2} c^{2} - 2 a^{2} b^{4} c^{2} - b^{6} c^{2} - 4 a^{4} c^{4} + 2 a^{2} b^{2} c^{4} + 3 b^{4} c^{4} + a^{2} c^{6} - 3 b^{2} c^{6} + c^{8})\)
Search Value (6, 9, 13): \(13.591244760485\)
▸ Alternate Coordinates1 form

  • Barycentrics: \(1/(a^{6} (b^{2} + c^{2}) - a^{4} (3 b^{4} - 2 b^{2} c^{2} + 3 c^{4}) + a^{2} (3 b^{6} - 2 b^{4} c^{2} - 2 b^{2} c^{4} + 3 c^{6}) - (b^{2} - c^{2})^{2} (b^{4} + 3 b^{2} c^{2} + c^{4})) : :\)

▸ Incident Central Lines69 lines

{2, 9060}, {3, 476}, {4, 1304}, {5, 16166}, {20, 10420}, {22, 16167}, {23, 1302}, {24, 22239}, {30, 110}, {50, 112}, {74, 523}, {99, 3260}, {107, 186}, {109, 15228}, {111, 5915}, {125, 5627}, {140, 18319}, {265, 14851}, {376, 691}, {378, 935}, {403, 1301}, {468, 9064}, {511, 9160}, {512, 9161}, {526, 16169}, {550, 1291}, {631, 22104}, {759, 23226}, {842, 2411}, {925, 2071}, {930, 18859}, {933, 13619}, {1138, 12244}, {1141, 23286}, {1290, 3651}, {1300, 15470}, {1499, 9184}, {2222, 22342}, {2452, 2713}, {2549, 2715}, {2687, 7429}, {2688, 7440}, {2689, 7421}, {2690, 7430}, {2695, 7454}, {2752, 7425}, {2758, 7444}, {2766, 7414}, {2770, 7418}, {2782, 20404}, {3154, 14644}, {3233, 15034}, {5473, 9203}, {5474, 9202}, {5663, 14480}, {5994, 15743}, {5995, 11586}, {6080, 12290}, {6236, 11594}, {7417, 10102}, {7471, 15035}, {9781, 12052}, {10423, 18533}, {10990, 14536}, {12030, 14127}, {12084, 15112}, {13398, 16386}, {14094, 14611}, {15396, 17511}

▸ Functional Relations16 relations