X3487

INTERSECTION OF LINES X1X_4 AND X3X_7

Let A' be the midpoint of X1 and the A-intouch point. Define B' and C' cyclically. The triangle A'B'C' is homothetic to the 2nd extouch triangle, and the center of homothety is X3487. (Randy Hutson, September 14, 2016)
< 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) { 1, 4 }: Incenter–Orthocenter Line { 3, 7 }: Central Line L(3, 7) { 2, 72 }: Central Line L(2, 72) { 8, 442 }: Central Line L(8, 442) { 78, 443 }: Central Line L(78, 443) { 57, 631 }: Central Line L(57, 631) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_4 = ORTHOCENTER (0.6122, 0.3958) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_7 = GERGONNE POINT (0.5903, 0.2769) X_2 = CENTROID (0.5374, 0.2000) X_72 = ISOGONAL CONJUGATE OF X(28) (0.4367, 0.0733) X_8 = NAGEL POINT (0.4694, 0.1333) X_442 = COMPLEMENT OF SCHIFFLER POINT (0.5418, 0.2114) X_78 = ISOGONAL CONJUGATE OF X(34) (0.3835, 0.0491) X_443 = COMPLEMENT OF X(452) (0.5427, 0.2138) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_631 = 3/5*OG (0.5224, 0.1608) X_3487 = INTERSECTION OF LINES X(1)X(4) AND X(3)X(7) (0.5749, 0.2471) A B C X1 X4 X3 X7 X2 X72 X8 X442 X78 X443 X57 X631 X3487 (a, b, c) = (5, 6, 7) | AB = 1

X3487 (INTERSECTION OF LINES X1X_4 AND X3X_7) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(1 + \cos A + \cos B + \cos C + \cos B \cos C : :\)
Search Value (6, 9, 13): \(1.331820532796\)
▸ Incident Central Lines13 lines

{1, 4}, {2, 72}, {3, 7}, {5, 938}, {8, 442}, {9, 1125}, {57, 631}, {78, 443}, {142, 936}, {281, 1148}, {329, 405}, {386, 2140}, {474, 1260}