X3340

INTERSECTION OF LINES X1X_3 AND X7X_145

X3340 = reflection of X1697 in X1
< 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) { 1697, 1 }: OI Line (Incenter–Circumcenter Line) { 1, 3 }: OI Line (Incenter–Circumcenter Line) { 7, 145 }: Central Line L(7, 145) { 8, 226 }: Central Line L(8, 226) { 388, 519 }: Central Line L(388, 519) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1697 = INSIMILICENTER(BEVAN CIRCLE, INCIRCLE) (0.5449, 0.1845) X_1 = INCENTER (0.5714, 0.2333) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_7 = GERGONNE POINT (0.5903, 0.2769) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.6735, 0.3333) X_8 = NAGEL POINT (0.4694, 0.1333) X_226 = -CEVA CONJUGATE OF X(65) (0.5778, 0.2588) X_388 = ∩X(7)X(8) (0.5790, 0.2635) X_519 = ISOGONAL CONJUGATE OF X(106) (0.9566, 0.6107) X_3340 = INTERSECTION OF LINES X(1)X(3) AND X(7)X(145) (0.5980, 0.2821) A B C X1697 X1 X3 X7 X145 X8 X226 X388 X519 X3340 (a, b, c) = (5, 6, 7) | AB = 1

X3340 (INTERSECTION OF LINES X1X_3 AND X7X_145) 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 \(f(a,b,c) = (a - b + c)(a + b - c)(a + b + c) + (4r/R)abc\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(0.648455949920\)
▸ Incident Central Lines19 lines

{1, 3}, {7, 145}, {8, 226}, {9, 1405}, {34, 1126}, {78, 1706}, {84, 1389}, {109, 1468}, {221, 2003}, {278, 1869}, {386, 1457}, {388, 519}, {595, 1451}, {950, 962}, {1125, 1788}, {1404, 1449}, {1419, 2263}, {1871, 1887}, {1953, 2257}

▸ Functional Relations6 relations

Notes

See the note at X3333.

The reflection of the excentral triangle in X1 and the intouch triangle are homothetic from X(3340. (Randy Hutson, January 29, 2015)