X855

INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X57

X855 = reflection of X851 in X859
< 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) { 851, 859 }: Euler Line { 2, 3 }: Euler Line { 513, 663 }: Central Line L(513, 663) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_851 = -HIRST INVERSE OF X(73) (1.4149, 2.4958) X_859 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(81) (1.1279, 1.7448) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0529, -0.0806) X_663 = CROSSDIFFERENCE OF X(2) AND X(7) (8.6667, -3.2660) X_855 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(57) (0.8408, 0.9939) A B C X851 X859 X2 X3 X513 X663 X855 (a, b, c) = (5, 6, 7) | AB = 1

X855 (INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X57) 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 \(f(A,B,C) = \cot(C/2) \sin 2B \sin(C - A) + \cot(B/2) \sin 2C \sin(B - A)\)
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): \(0.008868750603\)
▸ Incident Central Lines2 lines

{2, 3}, {513, 663}

▸ Functional Relations2 relations

Notes

As a point on the Euler line, X855 has Shinagawa coefficients \((ES^{2}+\$abS_{C}^{2}\$+\$abS_{A}S_{B}\$ -\$abS_{C}\$(E+F), -3ES^{2}-3\$abS_{A}S_{B}\$ +2\$ab\$S^{2})\).