X3419

ISOGONAL CONJUGATE OF X3418

X3419 is the isogonal conjugate of X3418.
< 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) { 8, 3434 }: Fuhrmann Line { 4, 8 }: Fuhrmann Line { 5, 78 }: Central Line L(5, 78) { 9, 80 }: Central Line L(9, 80) { 30, 63 }: Central Line L(30, 63) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_8 = NAGEL POINT (0.4694, 0.1333) X_3434 = ISOGONAL CONJUGATE OF X(3433) (0.5142, 0.2156) X_4 = ORTHOCENTER (0.6122, 0.3958) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_78 = ISOGONAL CONJUGATE OF X(34) (0.3835, 0.0491) X_9 = MITTENPUNKT (0.5110, 0.1615) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.8980, -0.1000) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4566, 0.0823) X_3419 = ISOGONAL CONJUGATE OF X(3418) (0.4918, 0.1745) A B C X8 X3434 X4 X5 X78 X9 X80 X30 X63 X3419 (a, b, c) = (5, 6, 7) | AB = 1

X3419 (ISOGONAL CONJUGATE OF X3418) 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 \(f(a,b,c) = 1/x(3418)\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(3.562328451872\)
▸ Alternate Coordinates1 form

  • Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a^{4} - a^{3}(b + c) + a(b + c)(b^{2} + c^{2}) - (b^{2} - c^{2})^{2} (E\). Danneels)

▸ Incident Central Lines20 lines

{1, 442}, {4, 8}, {5, 78}, {9, 80}, {11, 997}, {30, 63}, {40, 1726}, {100, 1006}, {150, 2893}, {226, 519}, {381, 908}, {392, 497}, {405, 950}, {443, 938}, {474, 1210}, {960, 1479}, {1060, 2000}, {1104, 1714}, {1376, 1737}, {2475, 2894}

▸ Functional Relations4 relations