X854

INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X56

X854 = crossdifference of every pair of points on line X8X647
< 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_854 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(56) (0.7792, 0.8325) A B C X2 X3 X4 X854 (a, b, c) = (5, 6, 7) | AB = 1

X854 (INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X56) 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 \(f(A,B,C) = \csc^{2}(C/2) \sin 2B \sin(C - A) + \csc^{2}(B/2) \sin 2C \sin(B - A)\)
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): \(0.087118721786\)
▸ Incident Central Lines1 line

{2, 3}

▸ Functional Relations1 relation

  • X854 = crossdifference of every pair of points on line X8X647

Notes

As a point on the Euler line, X854 has Shinagawa coefficients \(([(2E-F)F-S^{2}]S^{2}-\$abS_{C}^{3}\$ -\$abS_{A}S_{B}\$(E-2F) +\$abS_{C}\$[(E+F)^{2}-2S^{2}], [(E+F)F+S^{2}]S^{2}-\$abS_{A}S_{B}\$(E+F) +\$abS_{C}\$S^{2})\).