X453

POINT ALSHAIN

< 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 { 46, 1800 }: Central Line L(46, 1800) { 1014, 1454 }: Central Line L(1014, 1454) 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_46 = -CEVA CONJUGATE OF X(1) (0.6917, 0.4543) X_1800 = INVERSE MIMOSA TRANSFORM OF X(155) (0.7749, 0.2677) X_1014 = ISOGONAL CONJUGATE OF X(210) (0.6222, 0.3285) X_1454 = -BETH CONJUGATE OF X(498) (0.6215, 0.3253) X_453 = POINT ALSHAIN (0.6610, 0.5233) A B C X2 X3 X4 X46 X1800 X1014 X1454 X453 (a, b, c) = (5, 6, 7) | AB = 1

X453 (POINT ALSHAIN) 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 \(f(A,B,C) = (\cos B + \cos C - \cos A)^{2}/(\cos B + \cos C)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(1.182719776028\)
▸ Incident Central Lines3 lines

{2, 3}, {46, 1800}, {1014, 1454}

Notes

As a point on the Euler line, X453 has Shinagawa coefficients \((2\$aS_{B}S_{C}\$+\$aS_{A}\$(E+2F)-2\$a\$S^{2}-2abcF, -\$aS_{A}\$E+2\$a\$S^{2}+abcE)\).