X414

EULER X29-3rd-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_414 = EULER X(29)-3rd-SUBSTITUTION POINT (0.6058, 0.3789) A B C X2 X3 X4 X414 (a, b, c) = (5, 6, 7) | AB = 1

X414 (EULER X29-3rd-SUBSTITUTION POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(u^{3}(v^{2} + w^{2} - vw) : v^{3}(w^{2} + u^{2} - wu) : w^{3}(u^{2} + v^{2} - uv)\), where \(u : v : w\) = X29; e.g., \(u = u(A,B,C) = (\sec A)1/(\cos B + \cos C)\)
Barycentrics: \(au^{3}(v^{2} + w^{2} - vw) : bv^{3}(w^{2} + u^{2} - wu) : cw^{3}(u^{2} + v^{2} - uv)\)
Search Value (6, 9, 13): \(3.791063862784\)
▸ Alternate Coordinates1 form

  • Barycentrics: \((a + b) \cdot (a - b - c)^{3} (a + c) \cdot (a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) \cdot (a^{8} b^{2} + 2 a^{7} b^{3} - a^{6} b^{4} - 4 a^{5} b^{5} - a^{4} b^{6} + 2 a^{3} b^{7} + a^{2} b^{8} - a^{8} b c - a^{7} b^{2} c + 3 a^{6} b^{3} c + 3 a^{5} b^{4} c - 3 a^{4} b^{5} c - 3 a^{3} b^{6} c + a^{2} b^{7} c + a b^{8} c + a^{8} c^{2} - a^{7} b c^{2} - 3 a^{6} b^{2} c^{2} + a^{5} b^{3} c^{2} + 4 a^{4} b^{4} c^{2} + a^{3} b^{5} c^{2} - 3 a^{2} b^{6} c^{2} - a b^{7} c^{2} + b^{8} c^{2} + 2 a^{7} c^{3} + 3 a^{6} b c^{3} + a^{5} b^{2} c^{3} - a^{2} b^{5} c^{3} - 3 a b^{6} c^{3} - 2 b^{7} c^{3} - a^{6} c^{4} + 3 a^{5} b c^{4} + 4 a^{4} b^{2} c^{4} + 4 a^{2} b^{4} c^{4} + 3 a b^{5} c^{4} - b^{6} c^{4} - 4 a^{5} c^{5} - 3 a^{4} b c^{5} + a^{3} b^{2} c^{5} - a^{2} b^{3} c^{5} + 3 a b^{4} c^{5} + 4 b^{5} c^{5} - a^{4} c^{6} - 3 a^{3} b c^{6} - 3 a^{2} b^{2} c^{6} - 3 a b^{3} c^{6} - b^{4} c^{6} + 2 a^{3} c^{7} + a^{2} b c^{7} - a b^{2} c^{7} - 2 b^{3} c^{7} + a^{2} c^{8} + a b c^{8} + b^{2} c^{8}) : :\)

▸ Functional Relations1 relation

  • X414 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {21, 29, 410}

Notes

As a point on the Euler line, X414 has Shinagawa coefficients \(([(2E-F)F-S^{2}]FS^{4} -\$abS_{A}S_{B}\$F^{2}S^{2} +\$ab\$F^{2}S^{4}, -[(3E-F)F-S^{2}]FS^{4} -\$ab(S_{A}S_{B})^{3}\$ +2\$abS_{C}^{3}\$FS^{2} -4\$abS_{A}S_{B}\$F^{2}S^{2} -2\$abS_{C}\$[(E+F)^{2}-2S^{2}]FS^{2} -\$ab\$F^{2}S^{4})\).