X118

MIDPOINT OF X4 AND X101

X118 is the midpoint of X4 and X101.
< 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) { 4, 101 }: Central Line L(4, 101) { 2, 103 }: Central Line L(2, 103) { 5, 116 }: Central Line L(5, 116) { 11, 226 }: Central Line L(11, 226) { 122, 440 }: Central Line L(122, 440) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_4 = ORTHOCENTER (0.6122, 0.3958) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.2143, 0.5249) X_2 = CENTROID (0.5374, 0.2000) X_103 = ANTIPODE OF X(101) (0.7857, -0.3208) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_116 = MIDPOINT OF X(4) AND X(103) (0.6990, 0.0375) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_226 = -CEVA CONJUGATE OF X(65) (0.5778, 0.2588) X_122 = -OF-MEDIAL-TRIANGLE (0.5195, -0.0036) X_440 = COMPLEMENT OF X(27) (0.4970, 0.0943) X_118 = MIDPOINT OF X(4) AND X(101) (0.4133, 0.4603) A B C X4 X101 X2 X103 X5 X116 X11 X226 X122 X440 X118 (a, b, c) = (5, 6, 7) | AB = 1

X118 (MIDPOINT OF X4 AND X101) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(g(b,c,a) + g(c,b,a) : :\), where \(g(b,c,a) = b^{3}c/[(b - c) \cot A + (b - a) \cot C]\)
Search Value (6, 9, 13): \(-2.178056337287\)

X118 lies on the nine-point circle.

▸ Incident Central Lines9 lines

{2, 103}, {4, 101}, {5, 116}, {11, 226}, {117, 928}, {122, 440}, {136, 430}, {381, 544}, {516, 910}

▸ Functional Relations10 relations

Notes

Let HaHbHc be the orthic triangle. Let La be the Gergonne line of triangle AHbHc, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines HaA″, HbB″, HcC″ concur in X118, which is X3-of-A″B″C″. (Randy Hutson, July 31 2018)