X1936

CROSSDIFFERENCE OF PU(15)

X1936 = perspector of hyperbola {{A, B, C,X21,X651,PU(16)}}
< 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, 3 }: OI Line (Incenter–Circumcenter Line) { 33, 63 }: Central Line L(33, 63) { 2, 212 }: Central Line L(2, 212) { 11, 238 }: Central Line L(11, 238) { 4, 255 }: Central Line L(4, 255) { 29, 270 }: Central Line L(29, 270) 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_1 = INCENTER (0.5714, 0.2333) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_33 = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES (0.6019, 0.3546) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4566, 0.0823) X_212 = -CEVA CONJUGATE OF X(41) (0.4719, 0.0786) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_238 = -LINE CONJUGATE OF X(37) (1.6190, 1.4775) X_4 = ORTHOCENTER (0.6122, 0.3958) X_255 = ISOGONAL CONJUGATE OF X(158) (0.4217, 0.0365) X_29 = CEVAPOINT OF INCENTER AND ORTHOCENTER (0.5903, 0.3384) X_270 = ISOGONAL CONJUGATE OF X(201) (0.6168, 0.3881) X_1936 = CROSSDIFFERENCE OF PU(15) (-1.1948, -3.0115) A B C X2 X1 X3 X33 X63 X212 X11 X238 X4 X255 X29 X270 X1936 (a, b, c) = (5, 6, 7) | AB = 1

X1936 (CROSSDIFFERENCE OF PU(15)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\cos^{2} A - \cos B \cos C : :\)
Search Value (6, 9, 13): \(5.480685124509\)
▸ Alternate Coordinates1 form

  • Trilinears: \((a - b - c) (a^{4} - a^{2} (b^{2} - b c + c^{2}) - b c (b - c)^{2}) : :\)

▸ Incident Central Lines28 lines

{1, 3}, {2, 212}, {4, 255}, {11, 238}, {20, 603}, {29, 270}, {31, 497}, {33, 63}, {47, 1479}, {58, 950}, {73, 411}, {100, 1818}, {109, 516}, {225, 412}, {243, 1430}, {388, 1496}, {495, 738}, {511, 1364}, {521, 650}, {580, 1210}, {750, 1253}, {896, 1776}, {908, 1331}, {938, 1451}, {1044, 1406}, {1046, 1858}, {1058, 1497}, {1762, 1859}

▸ Functional Relations10 relations

Notes

X1936 is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(16) and U(16) of bicentric points; This conic also passes through X21 and X651; see the notes just before X1908. (Randy Hutson, 9/23/2011)

Let L be the line X1X3 = trilinear polar of X651. Let V be the trilinear polar of the cevapoint of X1 and X3, so that V = X521X650; let M = X3157 = X1-Ceva conjugate of X3, and let N = X1745 = X3-Ceva conjugate of X1. The lines L, V, MN concur in X1936. (Randy Hutson, December 26, 2015)

Let A1B1C1 and A2B2C2 be the 1st and 2nd bicentrics of the orthic triangle. The six vertices A1, B1, C1, A2, B2, C2 lie on a conic, denoted here by H. Let A' be the intersection of the tangents to H at A1 and A2. Define B' and C' cyclically. The lines AA', BB', CC' concur in X1936. Let A″ be the intersection of the tangents to H at B2 and C1. Define B' and C' cyclically. The lines AA″, BB″, CC″ concur in X1936. (Randy Hutson, March 25, 2016)