X3680

X2136com(EXTOUCH TRIANGLE)

X3680 = X4-of-Garcia-reflection-triangle
< 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) { 40, 104 }: Central Line L(40, 104) { 7, 145 }: Central Line L(7, 145) { 1, 474 }: Central Line L(1, 474) { 4, 519 }: Central Line L(4, 519) { 84, 517 }: Central Line L(84, 517) { 10, 1000 }: Central Line L(10, 1000) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1420 = -BETH CONJUGATE OF X(145) (0.6753, 0.4242) X_40 = BEVAN POINT (0.4286, -0.0292) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_7 = GERGONNE POINT (0.5903, 0.2769) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.6735, 0.3333) X_1 = INCENTER (0.5714, 0.2333) X_474 = EULER LINE INTERCEPT OF LINE X(10)X(56) (0.5441, 0.2176) X_4 = ORTHOCENTER (0.6122, 0.3958) X_519 = ISOGONAL CONJUGATE OF X(106) (0.9566, 0.6107) X_84 = ISOGONAL CONJUGATE OF X(40) (0.6071, 0.6816) X_517 = ISOGONAL CONJUGATE OF X(104) (0.8180, 0.7154) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_1000 = ISOGONAL CONJUGATE OF X(999) (0.5463, 0.1961) X_3680 = com(EXTOUCH TRIANGLE) (0.2679, 0.0583) A B C X1420 X40 X104 X7 X145 X1 X474 X4 X519 X84 X517 X10 X1000 X3680 (a, b, c) = (5, 6, 7) | AB = 1

X3680 (X2136com(EXTOUCH TRIANGLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Barycentrics: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a(a - b - c)/(3a - b - c)\)
Search Value (6, 9, 13): \(16.226961690788\)

X3680 lies on the Feuerbach hyperbola.

▸ Incident Central Lines20 lines

{1, 474}, {4, 519}, {7, 145}, {8, 3452}, {9, 3057}, {10, 1000}, {21, 1697}, {40, 104}, {57, 1476}, {78, 1320}, {84, 517}, {200, 2098}, {518, 3062}, {728, 2170}, {941, 3169}, {1449, 2298}, {1482, 3577}, {2099, 2900}, {2320, 3601}, {3189, 3244}

▸ Functional Relations6 relations

Notes

Let A'B'C' be Triangle T(-1,3) (aka excenters-reflections triangle). Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X3680. (Randy Hutson, March 21, 2019)

Let A' be the orthocenter of BCX1, and define B' and C' cyclically. A'B'C' is also the anticevian triangle, wrt intouch triangle, of X1, and also the Garcia reflection triangle (Gemini triangle 8). X3680 is the orthocenter of A'B'C'. (see also X1699) (Randy Hutson, March 21, 2019)