X3779

X3056com[T(-bc, b²)]

X3779 = homothetic center of extangents triangle and reflection of intangents triangle in X6
< 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) { 7, 8 }: Gergonne–Nagel Line { 6, 31 }: Central Line L(6, 31) { 181, 200 }: Central Line L(181, 200) { 40, 511 }: Central Line L(40, 511) { 210, 966 }: Central Line L(210, 966) { 269, 1362 }: Central Line L(269, 1362) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_7 = GERGONNE POINT (0.5903, 0.2769) X_8 = NAGEL POINT (0.4694, 0.1333) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_181 = APOLLONIUS POINT (0.6163, 0.3093) X_200 = -CEVA CONJUGATE OF X(9) (0.4392, 0.1037) X_40 = BEVAN POINT (0.4286, -0.0292) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2335, -0.3020) X_210 = -CEVA CONJUGATE OF X(37) (0.4925, 0.1466) X_966 = INTERSECTION OF LINES X(2)X(6) AND X(4)X(9) (0.5191, 0.1803) X_269 = ISOGONAL CONJUGATE OF X(200) (0.6369, 0.3847) X_1362 = BRISSE TRANSFORM OF X(105) (0.4408, 0.4266) X_3779 = com[T(-bc, b2)] (0.6743, 0.3766) A B C X7 X8 X6 X31 X181 X200 X40 X511 X210 X966 X269 X1362 X3779 (a, b, c) = (5, 6, 7) | AB = 1

X3779 (X3056com[T(-bc, b²)]) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Barycentrics: \(a^{2}(b^{3} + c^{3} - ab^{2} - ac^{2} - abc) : :\)
Search Value (6, 9, 13): \(-1.024025285509\)
▸ Incident Central Lines24 lines

{1, 3688}, {6, 31}, {7, 8}, {19, 1843}, {40, 511}, {56, 1818}, {181, 200}, {210, 966}, {269, 1362}, {291, 1740}, {579, 2223}, {869, 2277}, {968, 3690}, {1400, 2340}, {1402, 3190}, {1486, 2911}, {1631, 2174}, {1654, 3681}, {1721, 2808}, {1743, 3271}, {1964, 2275}, {2093, 2810}, {2294, 3728}, {3169, 3174}

▸ Functional Relations3 relations

  • X3779 = homothetic center of extangents triangle and reflection of intangents triangle in X6

  • X3779 = perspector of extangents triangle and Mandart-excircles triangle

  • X3779 = X7-of-extangents-triangle if ABC is acute

Notes

Let A'B'C' be the extangents triangle. Let A″ be the trilinear pole, wrt A'B'C', of line BC, and define B″ and C″ cyclically. Let A* be the trilinear pole, wrt A'B'C', of line B″C″, and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X3779. (Randy Hutson, December 2, 2017)