X860

INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X92

X860 = crossdifference of every pair of points on line X48X647
< 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 { 10, 201 }: Central Line L(10, 201) { 240, 522 }: Central Line L(240, 522) { 34, 997 }: Central Line L(34, 997) { 8, 1068 }: Central Line L(8, 1068) 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_10 = SPIEKER CENTER (0.5204, 0.1833) X_201 = -CEVA CONJUGATE OF X(12) (0.4897, 0.1062) X_240 = -LINE CONJUGATE OF X(48) (0.7457, 0.8594) X_522 = ISOGONAL CONJUGATE OF X(109) (-0.0318, 0.3430) X_34 = -BETH CONJUGATE OF X(4) (0.6332, 0.4792) X_997 = MIDPOINT OF X(1) AND X(200) (0.5053, 0.1685) X_8 = NAGEL POINT (0.4694, 0.1333) X_1068 = -CROSS CONJUGATE OF X(4) (0.6478, 0.5373) X_860 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(92) (0.7856, 0.8494) A B C X2 X3 X4 X10 X201 X240 X522 X34 X997 X8 X1068 X860 (a, b, c) = (5, 6, 7) | AB = 1

X860 (INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X92) 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 \(f(A,B,C) = (\tan A + \tan B) \sin 2B \sin(C - A) + (\tan A + \tan C) \sin 2C \sin(B - A)\)
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): \(1.652613633134\)
▸ Incident Central Lines5 lines

{2, 3}, {8, 1068}, {10, 201}, {34, 997}, {240, 522}

▸ Functional Relations2 relations

  • X860 = crossdifference of every pair of points on line X48X647

  • X860 = inverse-in-polar-circle of X3109

Notes

As a point on the Euler line, X860 has Shinagawa coefficients \((\$abS_{C}\$F-\$ab\$(E+F)F, -\$abS_{A}S_{B}\$+\$ab\$S^{2})\).