X3030

INTERSECTION X10X_11∩X43X_57

X3030 = reflection of X1357 in X1054
< 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) { 1357, 1054 }: Central Line L(1357, 1054) { 10, 11 }: Central Line L(10, 11) { 43, 57 }: Central Line L(43, 57) { 106, 386 }: Central Line L(106, 386) { 373, 2177 }: Central Line L(373, 2177) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1357 = BRISSE TRANSFORM OF X(100) (0.8000, 0.1866) X_1054 = 6th SHARYGIN POINT (1.4286, -0.2333) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_43 = -CEVA CONJUGATE OF X(1) (0.6358, 0.2963) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_386 = INVERSE-IN-BROCARD-CIRCLE OF X(58) (0.5912, 0.2526) X_373 = CENTROID OF THE PEDAL TRIANGLE OF THE CENTROID (0.5614, 0.2395) X_2177 = -ISOCONJUGATE OF X(89) (0.5755, 0.2311) X_3030 = INTERSECTION X(10)X(11)∩X(43)X(57) (2.0571, -0.6532) A B C X1357 X1054 X10 X11 X43 X57 X106 X386 X373 X2177 X3030 (a, b, c) = (5, 6, 7) | AB = 1

X3030 (INTERSECTION X10X_11∩X43X_57) 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 \(f(a,b,c) = a[(2bc - ab - ac)^{2} - (b^{2} - c^{2})^{2}]\)
Search Value (6, 9, 13): \(-1.977975392540\)

X3030 lies on the Apollonius circle.

▸ Incident Central Lines8 lines

{10, 11}, {43, 57}, {106, 386}, {373, 2177}, {1293, 2291}, {2796, 3029}, {2832, 3034}, {2842, 3031}

▸ Functional Relations4 relations

Notes

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle. X3030 = X107 of A'B'C'. (Randy Hutson, August 29, 2018))