X280

X1-CROSS CONJUGATE OF X8

X280 = isogonal conjugate of X221
< 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) { 8, 20 }: Central Line L(8, 20) { 7, 253 }: Central Line L(7, 253) { 21, 268 }: Central Line L(21, 268) { 2, 318 }: Central Line L(2, 318) { 78, 282 }: Central Line L(78, 282) { 75, 309 }: Central Line L(75, 309) 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_8 = NAGEL POINT (0.4694, 0.1333) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_7 = GERGONNE POINT (0.5903, 0.2769) X_253 = -CROSS CONJUGATE OF X(2) (3.5087, 32.5688) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_268 = ISOGONAL CONJUGATE OF X(196) (0.7710, 1.5678) X_2 = CENTROID (0.5374, 0.2000) X_318 = ISOTOMIC CONJUGATE OF X(77) (0.5514, 0.2839) X_78 = ISOGONAL CONJUGATE OF X(34) (0.3835, 0.0491) X_282 = -CROSS CONJUGATE OF X(9) (0.6199, 0.7507) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4988, 0.1682) X_309 = ISOTOMIC CONJUGATE OF X(40) (0.6237, 0.7391) X_280 = -CROSS CONJUGATE OF X(8) (0.6379, 0.8041) A B C X1 X8 X20 X7 X253 X21 X268 X2 X318 X78 X282 X75 X309 X280 (a, b, c) = (5, 6, 7) | AB = 1

X280 (X1-CROSS CONJUGATE OF X8) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((\csc^{2} A/2)/(-1 - \cos A + \cos B + \cos C)\)
Search Value (6, 9, 13): \(5.034628048323\)
▸ Incident Central Lines8 lines

{2, 318}, {7, 253}, {8, 20}, {21, 268}, {75, 309}, {78, 282}, {285, 1043}, {341, 345}

▸ Functional Relations8 relations

Notes

Let Ea be the ellipse passing through A, and with foci at B and C (the A-Soddy ellipse). Let La be the polar of X20 with respect to Ea. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The lines AA', BB', CC' concur in X280. (Randy Hutson, September 14, 2016)