X856

INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X63

X856 = crossdifference of every pair of points on line X19X647
< 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 { 521, 656 }: Central Line L(521, 656) 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_521 = ISOGONAL CONJUGATE OF X(108) (1.1224, 0.1534) X_656 = CROSSDIFFERENCE OF X(1) AND X(19) (7.5714, -0.8554) X_856 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(63) (0.3538, -0.2804) A B C X2 X3 X4 X521 X656 X856 (a, b, c) = (5, 6, 7) | AB = 1

X856 (INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X63) 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 \(f(A,B,C) = \tan C \sin 2B \sin(C - A) + \tan B \sin 2C \sin(B - A)\)
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): \(2.540881951041\)
▸ Incident Central Lines2 lines

{2, 3}, {521, 656}

▸ Functional Relations1 relation

  • X856 = crossdifference of every pair of points on line X19X647

Notes

As a point on the Euler line, X856 has Shinagawa coefficients \((\$abS_{A}S_{B}\$+\$abS_{C}\$F-\$ab\$(E+F)F, -\$abS_{A}S_{B}\$)\).