X1004

INTERCEPT OF EULER LINE AND LINE X7X_100

< 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 { 7, 100 }: Central Line L(7, 100) { 46, 200 }: Central Line L(46, 200) { 63, 210 }: Central Line L(63, 210) { 65, 224 }: Central Line L(65, 224) 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_7 = GERGONNE POINT (0.5903, 0.2769) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_46 = -CEVA CONJUGATE OF X(1) (0.6917, 0.4543) X_200 = -CEVA CONJUGATE OF X(9) (0.4392, 0.1037) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4566, 0.0823) X_210 = -CEVA CONJUGATE OF X(37) (0.4925, 0.1466) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_224 = -CEVA CONJUGATE OF X(63) (1.2479, 0.4419) X_1004 = INTERCEPT OF EULER LINE AND LINE X(7)X(100) (0.5701, 0.2854) A B C X2 X3 X4 X7 X100 X46 X200 X63 X210 X65 X224 X1004 (a, b, c) = (5, 6, 7) | AB = 1

X1004 (INTERCEPT OF EULER LINE AND LINE X7X_100) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a^{5} - 2a^{4}(b + c) + 2a^{2}(b^{3} + c^{3}) - a(b^{2} + c^{2})^{2} + 2bc(b - c)(b^{2} - c^{2}) : :\)
Search Value (6, 9, 13): \(1.158137454279\)
▸ Incident Central Lines5 lines

{2, 3}, {7, 100}, {46, 200}, {63, 210}, {65, 224}

Notes

As a point on the Euler line, X1004 has Shinagawa coefficients \(((E+F)S^{2}-\$abS_{A}S_{B}\$, -(E+F)S^{2} +\$ab\$S^{2})\).