X1122

1st GRINBERG POINT

Let AB be the touchpoint of the A-excircle and line AB, let AC be the touchpoint of the A-excircle and line AC, and let MA be the midpoint of segment ABAC. Define MB and MC cyclically. Let A', B', C' be the touchpoints of the incircle with lines BC, CA, AB, respectively. The triangles MAMBMC and A'B'C' are perspective, and the perspector is X1122. (Darij Grinberg, 12/28/02)
< 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) { 7, 8 }: Gergonne–Nagel Line { 56, 269 }: Central Line L(56, 269) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_7 = GERGONNE POINT (0.5903, 0.2769) X_269 = ISOGONAL CONJUGATE OF X(200) (0.6369, 0.3847) X_8 = NAGEL POINT (0.4694, 0.1333) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_1122 = 1st GRINBERG POINT (0.6289, 0.3227) A B C X7 X269 X8 X56 X1122 (a, b, c) = (5, 6, 7) | AB = 1

X1122 (1st GRINBERG POINT) 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 \(f(a,b,c) = [a(b + c) + (b - c)^{2}]/(b + c - a)\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(0.277169929693\)
▸ Incident Central Lines2 lines

{7, 8}, {56, 269}

▸ Functional Relations3 relations