X432

EULER X155-6th-SUBSTITUTION POINT

< 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) { 2, 3 }: Euler Line A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_4 = ORTHOCENTER (0.6122, 0.3958) X_432 = EULER X(155)-6th-SUBSTITUTION POINT (0.6045, 0.3755) A B C X2 X3 X4 X432 (a, b, c) = (5, 6, 7) | AB = 1

X432 (EULER X155-6th-SUBSTITUTION POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((v^{2} + w^{2})\sec A : (w^{2} + u^{2})\sec B : (u^{2} + v^{2})\sec C\), where \(u : v : w\) = X155; e.g., \(u = u(A,B,C) = (\cos A)(\cos^{2} B + \cos^{2} C - \cos^{2} A)\)
Barycentrics: \((v^{2} + w^{2})\tan A : (w^{2} + u^{2})\tan B : (u^{2} + v^{2})\tan C\)
Search Value (6, 9, 13): \(6.350209277246\)
▸ Alternate Coordinates1 form

  • Barycentrics: \((a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) \cdot (a^{16} b^{2} - 8 a^{14} b^{4} + 28 a^{12} b^{6} - 56 a^{10} b^{8} + 70 a^{8} b^{10} - 56 a^{6} b^{12} + 28 a^{4} b^{14} - 8 a^{2} b^{16} + b^{18} + a^{16} c^{2} - 4 a^{12} b^{4} c^{2} + 8 a^{10} b^{6} c^{2} - 34 a^{8} b^{8} c^{2} + 80 a^{6} b^{10} c^{2} - 84 a^{4} b^{12} c^{2} + 40 a^{2} b^{14} c^{2} - 7 b^{16} c^{2} - 8 a^{14} c^{4} - 4 a^{12} b^{2} c^{4} + 32 a^{10} b^{4} c^{4} - 20 a^{8} b^{6} c^{4} - 40 a^{6} b^{8} c^{4} + 100 a^{4} b^{10} c^{4} - 80 a^{2} b^{12} c^{4} + 20 b^{14} c^{4} + 28 a^{12} c^{6} + 8 a^{10} b^{2} c^{6} - 20 a^{8} b^{4} c^{6} + 32 a^{6} b^{6} c^{6} - 44 a^{4} b^{8} c^{6} + 88 a^{2} b^{10} c^{6} - 28 b^{12} c^{6} - 56 a^{10} c^{8} - 34 a^{8} b^{2} c^{8} - 40 a^{6} b^{4} c^{8} - 44 a^{4} b^{6} c^{8} - 80 a^{2} b^{8} c^{8} + 14 b^{10} c^{8} + 70 a^{8} c^{10} + 80 a^{6} b^{2} c^{10} + 100 a^{4} b^{4} c^{10} + 88 a^{2} b^{6} c^{10} + 14 b^{8} c^{10} - 56 a^{6} c^{12} - 84 a^{4} b^{2} c^{12} - 80 a^{2} b^{4} c^{12} - 28 b^{6} c^{12} + 28 a^{4} c^{14} + 40 a^{2} b^{2} c^{14} + 20 b^{4} c^{14} - 8 a^{2} c^{16} - 7 b^{2} c^{16} + c^{18}) : :\)

▸ Incident Central Lines2 lines

{2, 3}, {135, 12359}

Notes

As a point on the Euler line, X432 has Shinagawa coefficients \(((E^{2} + 4EF - 4S^{2})F, -E^{3} - 3E^{2}F - 4EF^{2} + 4(E - F)S^{2})\).