X3162

X2-CEVA CONJUGATE OF X25

X3162 = complement of the isogonal conjugate of X159
< 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) { 25, 32 }: Central Line L(25, 32) { 6, 66 }: Central Line L(6, 66) { 22, 112 }: Central Line L(22, 112) { 19, 614 }: Central Line L(19, 614) { 204, 612 }: Central Line L(204, 612) { 305, 648 }: Central Line L(305, 648) 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_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6329, 0.4497) X_32 = 3rd POWER POINT (0.6400, 0.3332) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_66 = ISOGONAL CONJUGATE OF X(22) (0.5912, 1.0593) X_22 = EXETER POINT (0.4143, -0.1221) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.5162, 0.6121) X_19 = CLAWSON POINT (0.6246, 0.4244) X_614 = INTERSECTION OF LINES X(1)X(2) AND X(11)X(33) (0.6310, 0.2916) X_204 = -CEVA CONJUGATE OF X(19) (0.6683, 3.3982) X_612 = INTERSECTION OF LINES X(1)X(2) AND X(9)X(31) (0.5582, 0.2203) X_305 = ISOTOMIC CONJUGATE OF X(25) (0.3258, 0.0397) X_648 = TRILINEAR POLE OF EULER LINE (0.4563, 0.5855) X_3162 = -CEVA CONJUGATE OF X(25) (0.5902, 1.1453) A B C X2 X25 X32 X6 X66 X22 X112 X19 X614 X204 X612 X305 X648 X3162 (a, b, c) = (5, 6, 7) | AB = 1

X3162 (X2-CEVA CONJUGATE OF X25) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\), where \(x : y : z\) = X2 and \(u : v : w\) = X25
Barycentrics: \(a^{2} (a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) \cdot (a^{6} + a^{4} b^{2} - a^{2} b^{4} - b^{6} + a^{4} c^{2} - 2 a^{2} b^{2} c^{2} + b^{4} c^{2} - a^{2} c^{4} + b^{2} c^{4} - c^{6}) : :\)
Search Value (6, 9, 13): \(0.227368078389\)

X3162 lies on hyperbola {{X2,X6,X216,X233,X1249,X1560,X3162}}. This hyperbola is a circumconic of the medial triangle, and the locus of perspectors of circumconics centered at a point on the Euler line. The hyperbola is tangent to Euler line at X2. (Randy Hutson, June 7, 2019)

▸ Incident Central Lines63 lines

{2, 2138}, {3, 19615}, {4, 5359}, {6, 66}, {19, 614}, {22, 112}, {24, 1627}, {25, 32}, {132, 19149}, {154, 22075}, {197, 4548}, {204, 612}, {216, 1033}, {230, 8746}, {232, 1609}, {233, 7539}, {305, 648}, {378, 1180}, {394, 15595}, {426, 9475}, {468, 1611}, {607, 40962}, {608, 40961}, {653, 30918}, {1181, 37074}, {1194, 1968}, {1249, 7386}, {1370, 23115}, {1435, 40180}, {1498, 1529}, {1560, 30771}, {1783, 10327}, {1993, 41363}, {2211, 42295}, {3053, 21213}, {3163, 31152}, {5064, 5309}, {5094, 8792}, {5354, 6995}, {5523, 7391}, {6353, 8744}, {6997, 41370}, {7378, 34482}, {7396, 45245}, {7485, 39575}, {7735, 8745}, {8770, 44467}, {8891, 36879}, {9308, 11324}, {15487, 36103}, {16063, 41366}, {17903, 24605}, {21621, 40940}, {22120, 34609}, {23050, 40181}, {26283, 36414}, {28701, 30737}, {31101, 40583}, {34131, 39071}, {35477, 38862}, {39951, 40124}, {40191, 40192}, {40825, 44077}

▸ Functional Relations20 relations

  • X3162 = reflection of Xi in Xj for these {i,j}: {25, 42820}, {15255, 46737}

  • X3162 = complement of X13575

  • X3162 = polar conjugate of X40009

  • X3162 = complement of the isogonal conjugate of X159

  • X3162 = complement of the isotomic conjugate of X1370

  • X3162 = isogonal conjugate of the isotomic conjugate of X41361

  • X3162 = isogonal conjugate of the polar conjugate of X41766

  • X3162 = polar conjugate of the isotomic conjugate of X159

  • X3162 = medial-isogonal conjugate of X23300

  • X3162 = crosspoint of Xi and Xj for these (i,j): {2, 1370}, {8793, 40357}, {41361, 41766}

  • X3162 = crosssum of Xi and Xj for these (i,j): {6, 34207}, {69, 28708}

  • X3162 = crossdifference of every pair of points on line {3265, 8673}

  • X3162 = Xi-complementary conjugate of Xj for these (i,j): {1, 23300}, {6, 36907}, {31, 25}, {48, 14376}, {159, 10}, {1370, 2887}, {3162, 226}, {8793, 1215}, {18596, 141}, {18629, 17046}, {21582, 626}, {23115, 18589}, {41361, 20305}, {47125, 21253}

  • X3162 = Xi-Ceva conjugate of Xj for these (i,j): {2, 25}, {1370, 455}, {2138, 17807}, {8743, 6}, {40357, 17407}, {41361, 159}

  • X3162 = Xi-isoconjugate of Xj for these (i,j): {3, 39733}, {48, 40009}, {63, 13575}, {304, 34207}, {34055, 39129}, {39172, 46244}

  • X3162 = barycentric product Xi*Xj for these {i,j}: {3, 41766}, {4, 159}, {6, 41361}, {19, 18596}, {20, 33584}, {22, 17407}, {25, 1370}, {112, 47125}, {393, 23115}, {427, 8793}, {455, 13575}, {607, 18629}, {1973, 21582}, {2207, 28419}, {40357, 40938}

  • X3162 = barycentric quotient Xi/Xj for these {i,j}: {4, 40009}, {19, 39733}, {25, 13575}, {159, 69}, {455, 1370}, {1184, 40185}, {1370, 305}, {1843, 39129}, {1974, 34207}, {8793, 1799}, {17407, 18018}, {17409, 40358}, {18596, 304}, {20968, 39172}, {21582, 40364}, {23115, 3926}, {33578, 33579}, {33584, 253}, {36417, 40144}, {41361, 76}, {41766, 264}, {46766, 40404}, {47125, 3267}

  • X3162 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {25, 3172, 17409}, {427, 16318, 13854}, {1184, 2207, 25}, {1196, 14581, 36417}, {1196, 36417, 25}, {8743, 8879, 40144}, {14580, 17409, 25}, {15255, 15264, 25}

  • X3162 = perspector of circumconic centered at X25

  • X3162 = center of circumconic that is locus of trilinear poles of lines passing through X25; this conic is the polar conjugate of the de Longchamps line.