X865

INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X512

X865 = crossdifference of every pair of points on line X99X647
< 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 { 351, 888 }: Central Line L(351, 888) 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_351 = CENTER OF THE PARRY CIRCLE (1.3537, 1.1619) X_888 = ISOTOMIC CONJUGATE OF X(886) (0.2587, -0.3165) X_865 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(512) (-0.1837, -1.6869) A B C X2 X3 X4 X351 X888 X865 (a, b, c) = (5, 6, 7) | AB = 1

X865 (INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X512) 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 \(f(A,B,C) = \sin 2B \csc^{2} C \sin(C - A) \csc(A - B) - \sin 2C \csc^{2} B \sin(B - A) \csc(A - C)\)
Barycentrics: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = (\sin A)f(A,B,C)\)
Search Value (6, 9, 13): \(8.781960940271\)
▸ Incident Central Lines2 lines

{2, 3}, {351, 888}

▸ Functional Relations1 relation

  • X865 = crossdifference of every pair of points on line X99X647

Notes

As a point on the Euler line, X865 has Shinagawa coefficients \(((E + F)^{3}F - (7E - 2F)FS^{2} + S^{4}, -(E + F)^{4} + 2(E + F)(2E - F)S^{2} - S^{4})\).