X853

INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X55

X853 = crossdifference of every pair of points on line X7X647
< 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 { 657, 663 }: Central Line L(657, 663) 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_657 = CROSSDIFFERENCE OF X(1) AND X(7) (10.1538, -2.2611) X_663 = CROSSDIFFERENCE OF X(2) AND X(7) (8.6667, -3.2660) X_853 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(55) (-4.0788, -11.8784) A B C X2 X3 X4 X657 X663 X853 (a, b, c) = (5, 6, 7) | AB = 1

X853 (INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X55) 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 \(f(A,B,C) = \sec^{2}(C/2) \sin 2B \sin(C - A) + \sec^{2}(B/2) \sin 2C \sin(B - A)\)
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): \(6.730258608066\)
▸ Incident Central Lines2 lines

{2, 3}, {657, 663}

▸ Functional Relations1 relation

  • X853 = crossdifference of every pair of points on line X7X647

Notes

As a point on the Euler line, X853 has Shinagawa coefficients \(([(2E-F)F-S^{2}]S^{2}-\$bcS_{A}^{2}\$F +\$abS_{C}\$[(E+F)F+S^{2}]-\$ab\$(E+F)S^{2}, [(E+F)F+S^{2}]S^{2}+\$bcS_{A}^{2}\$(E+F) -\$abS_{C}\$[(E+F)^{2}+S^{2}]+\$ab\$(3E+F)S^{2})\).