X528

DIRECTION OF VECTOR AX + BX + CX, where X = X11

X528 = X519-Hirst inverse of X522
< 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) { 2, 11 }: Central Line L(2, 11) { 9, 80 }: Central Line L(9, 80) { 8, 190 }: Central Line L(8, 190) { 142, 214 }: Central Line L(142, 214) { 104, 376 }: Central Line L(104, 376) { 119, 381 }: Central Line L(119, 381) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_9 = MITTENPUNKT (0.5110, 0.1615) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.8980, -0.1000) X_8 = NAGEL POINT (0.4694, 0.1333) X_190 = YFF PARABOLIC POINT (0.0748, 0.3999) X_142 = COMPLEMENT OF X(9) (0.5506, 0.2192) X_214 = -CEVA CONJUGATE OF X(44) (0.3571, 0.3499) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_376 = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2) (0.4626, 0.0042) X_119 = FEUERBACH ANTIPODE (0.3776, 0.4312) X_381 = MIDPOINT OF X(2) AND X(4) (0.5748, 0.2979) X_528 = DIRECTION OF VECTOR AX + BX + CX, where X = X(11) (1.0237, -0.1286) A B C X2 X11 X9 X80 X8 X190 X142 X214 X104 X376 X119 X381 X528 (a, b, c) = (5, 6, 7) | AB = 1

X528 (DIRECTION OF VECTOR AX + BX + CX, where X = X11) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((2ax - by - cz)/a\), where \(x = x(A,B,C) = 1 - \cos(B-C)\), \(y = x(B,C,A)\), \(z = x(C,A,B)\)

X528 lies on the line at infinity.

▸ Incident Central Lines23 lines

{1, 1086}, {2, 11}, {7, 664}, {8, 190}, {9, 80}, {30, 511}, {104, 376}, {119, 381}, {142, 214}, {153, 3543}, {377, 3303}, {428, 1824}, {549, 1484}, {962, 3189}, {1279, 1738}, {1329, 1479}, {1537, 3174}, {1699, 3158}, {1750, 2900}, {1770, 3555}, {2094, 3474}, {3008, 3246}, {3032, 3034}

▸ Functional Relations6 relations

Notes

Let Ia be the reflection of X1 in the perpendicular bisector of BC, and define Ib and Ic cyclically. X528 = X518 of IaIbIc. (Randy Hutson, January 29, 2018)