X117

MIDPOINT OF X4 AND X109

X117 is the midpoint of X4 and X109.
< 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, 109 }: Central Line L(4, 109) { 11, 65 }: Central Line L(11, 65) { 2, 102 }: Central Line L(2, 102) { 5, 124 }: Central Line L(5, 124) { 10, 123 }: Central Line L(10, 123) 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_109 = Ψ(INCENTER, CIRCUMCENTER) (0.3636, 0.5938) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_2 = CENTROID (0.5374, 0.2000) X_102 = Λ(INCENTER, ORTHOCENTER) (0.6364, -0.3897) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_124 = -OF-MEDIAL-TRIANGLE (0.6243, 0.0030) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_123 = -OF-MEDIAL-TRIANGLE (0.5401, -0.0057) X_117 = MIDPOINT OF X(4) AND X(109) (0.4879, 0.4948) A B C X4 X109 X11 X65 X2 X102 X5 X124 X10 X123 X117 (a, b, c) = (5, 6, 7) | AB = 1

X117 (MIDPOINT OF X4 AND X109) 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 \(f(a,b,c) = g(b,c,a) + g(c,b,a)\), and, \(g(b,c,a) = b^{2}c/[c(\sec B - \sec C) + a(\sec B - \sec A)]\)
Barycentrics: \(h(a,b,c) : h(b,c,a) : h(c,a,b)\), where \(h(a,b,c) = af(a,b,c)\)
Search Value (6, 9, 13): \(-2.757846626874\)

X117 lies on the nine-point circle

▸ Incident Central Lines7 lines

{2, 102}, {4, 109}, {5, 124}, {10, 123}, {11, 65}, {118, 928}, {136, 407}

▸ Functional Relations10 relations