X379

EULER LINE INTERCEPT OF LINE X6X_7

X379 = complement of X31015
< 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 { 6, 7 }: Central Line L(6, 7) { 63, 169 }: Central Line L(63, 169) { 41, 226 }: Central Line L(41, 226) { 264, 823 }: Central Line L(264, 823) 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_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_7 = GERGONNE POINT (0.5903, 0.2769) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4566, 0.0823) X_169 = -CEVA CONJUGATE OF X(1) (0.5497, 0.2510) X_41 = -CEVA CONJUGATE OF X(31) (0.5823, 0.2244) X_226 = -CEVA CONJUGATE OF X(65) (0.5778, 0.2588) X_264 = ISOTOMIC CONJUGATE OF CIRCUMCENTER (0.5720, 0.3293) X_823 = ISOGONAL CONJUGATE OF X(822) (0.5623, 0.6035) X_379 = EULER LINE INTERCEPT OF LINE X(6)X(7) (0.5733, 0.2937) A B C X2 X3 X4 X6 X7 X63 X169 X41 X226 X264 X823 X379 (a, b, c) = (5, 6, 7) | AB = 1

X379 (EULER LINE INTERCEPT OF LINE X6X_7) 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 \(f(a,b,c) = bc[a^{5} + (b + c)(bca^{2} - (bc + ca + ab)(b - c)^{2})]\)
Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a^{5} + (b + c)(bca^{2} - (bc + ca + ab)(b - c)^{2})\)
Search Value (6, 9, 13): \(0.232161131211\)
▸ Incident Central Lines5 lines

{2, 3}, {6, 7}, {41, 226}, {63, 169}, {264, 823}

▸ Functional Relations4 relations

Notes

As a point on the Euler line, X379 has Shinagawa coefficients \((\$a S_{B}S_{C}\$, \$a\$S^{2})\).