X418

X3-CEVA-CONJUGATE OF X216

X418 = isotomic conjugate of the polar conjugate of X217
< 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, 3 }: Euler Line { 97, 110 }: Central Line L(97, 110) { 51, 216 }: Central Line L(51, 216) { 154, 160 }: Central Line L(154, 160) { 157, 161 }: Central Line L(157, 161) { 184, 577 }: Central Line L(184, 577) 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_3 = CIRCUMCENTER (0.5000, 0.1021) X_4 = ORTHOCENTER (0.6122, 0.3958) X_97 = ISOGONAL CONJUGATE OF X(53) (0.4798, 0.0756) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6093, 0.3186) X_216 = -CEVA CONJUGATE OF X(51) (0.5201, 0.1353) X_154 = -CEVA CONJUGATE OF X(6) (0.4807, -0.3954) X_160 = -OF-TANGENTIAL-TRIANGLE (0.4841, -0.0647) X_157 = -OF-TANGENTIAL-TRIANGLE (0.5266, 0.3812) X_161 = -OF-TANGENTIAL-TRIANGLE (0.2845, -1.4852) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.5788, 0.1496) X_577 = HARMONIC OF X(216) (0.4663, 0.0464) X_418 = -CEVA-CONJUGATE OF X(216) (0.4854, 0.0640) A B C X2 X3 X4 X97 X110 X51 X216 X154 X160 X157 X161 X184 X577 X418 (a, b, c) = (5, 6, 7) | AB = 1

X418 (X3-CEVA-CONJUGATE OF X216) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((\cos A)(\csc 2B + \csc 2C) : :\)
Barycentrics: \((\sin 2A)(\csc 2B + \csc 2C) : :\)
Search Value (6, 9, 13): \(0.440187399170\)
▸ Incident Central Lines93 lines

{2, 3}, {6, 26898}, {9, 26901}, {19, 26908}, {33, 26904}, {34, 26903}, {48, 23199}, {49, 19210}, {51, 216}, {52, 42441}, {53, 23607}, {54, 2055}, {57, 26900}, {97, 110}, {125, 50209}, {143, 46025}, {154, 160}, {157, 161}, {184, 577}, {212, 23198}, {217, 46394}, {268, 26867}, {275, 26902}, {324, 32428}, {343, 23181}, {476, 32439}, {511, 13409}, {682, 23209}, {1495, 22052}, {1614, 14152}, {1624, 10192}, {1629, 46760}, {1799, 6394}, {1974, 26899}, {2052, 42329}, {2972, 3917}, {3060, 26895}, {3284, 13366}, {3564, 23158}, {3567, 26896}, {3690, 35072}, {3819, 44436}, {5158, 15004}, {5562, 31388}, {5650, 46831}, {6146, 31381}, {6389, 22062}, {7011, 26866}, {7999, 14059}, {8266, 34828}, {8798, 31504}, {8884, 40448}, {8901, 35098}, {9407, 44078}, {9475, 40938}, {9967, 51335}, {10003, 12012}, {10184, 42862}, {10316, 34396}, {10979, 34417}, {11197, 39530}, {11402, 15905}, {11433, 26870}, {11442, 18437}, {11515, 44122}, {11516, 44083}, {13450, 51888}, {13567, 26905}, {14569, 42459}, {14855, 40948}, {15649, 17810}, {15653, 19467}, {16089, 46724}, {22075, 22391}, {22341, 22344}, {22401, 23210}, {23208, 42671}, {23613, 32320}, {23635, 47328}, {31353, 44088}, {31626, 38833}, {34983, 39201}, {34987, 37084}, {35225, 37813}, {35360, 42453}, {40681, 50645}, {40947, 41523}, {44711, 51243}, {44712, 51242}, {44716, 52032}, {46093, 52128}, {47409, 47426}, {51477, 52153}

▸ Functional Relations22 relations

  • X418 = reflection of X13409 in X46832

  • X418 = isogonal conjugate of X8795

  • X418 = circumcircle-inverse of X41202

  • X418 = isotomic conjugate of the isogonal conjugate of X44088

  • X418 = isogonal conjugate of the isotomic conjugate of X5562

  • X418 = isotomic conjugate of the polar conjugate of X217

  • X418 = isogonal conjugate of the polar conjugate of X216

  • X418 = orthic-isogonal conjugate of X31353

  • X418 = X43679-complementary conjugate of X20305

  • X418 = Xi-Ceva conjugate of Xj for these (i,j): {3, 216}, {4, 31353}, {95, 41334}, {110, 32320}, {216, 217}, {14587, 32661}, {14941, 3289}, {23181, 17434}, {40448, 6}

  • X418 = Xi-cross conjugate of Xj for these (i,j): {42556, 3}, {44088, 217}, {46394, 216}

  • X418 = Danneels point of X3

  • X418 = X1764-of-orthic-triangle if ABC is acute

  • X418 = Xi-isoconjugate of Xj for these (i,j): {1, 8795}, {4, 40440}, {19, 276}, {63, 8794}, {75, 8884}, {92, 275}, {95, 158}, {97, 6521}, {264, 2190}, {661, 42405}, {799, 15422}, {821, 19166}, {822, 42401}, {823, 15412}, {1096, 34384}, {1577, 16813}, {1969, 8882}, {2052, 2167}, {2148, 18027}, {2616, 6528}, {3112, 19174}, {6520, 34386}, {8901, 23999}, {18831, 24006}, {36119, 43752}, {42300, 51315}

  • X418 = Xi-Dao conjugate of Xj for these (i,j): {3, 8795}, {5, 264}, {6, 276}, {95, 1147}, {130, 523}, {206, 8884}, {216, 18027}, {275, 22391}, {389, 45198}, {850, 2972}, {1511, 43752}, {2052, 40588}, {3162, 8794}, {6503, 34384}, {14618, 15450}, {14767, 42368}, {15422, 38996}, {16089, 52128}, {18022, 52032}, {19174, 34452}, {20572, 39171}, {21243, 23295}, {23962, 35441}, {34386, 37867}, {36033, 40440}, {36830, 42405}

  • X418 = crosspoint of Xi and Xj for these (i,j): {3, 577}, {6, 22261}, {184, 2351}, {216, 5562}, {14587, 32661}

  • X418 = crosssum of Xi and Xj for these (i,j): {2, 5889}, {4, 2052}, {264, 317}, {275, 8884}

  • X418 = crossdifference of every pair of points on line {647, 14165}

  • X418 = X2-line conjugate of X44893

  • X418 = barycentric product Xi*Xj for these {i,j}: {3, 216}, {5, 577}, {6, 5562}, {32, 52347}, {48, 44706}, {51, 394}, {53, 1092}, {69, 217}, {71, 44709}, {76, 44088}, {95, 46394}, {99, 42293}, {110, 17434}, {184, 343}, {212, 44708}, {219, 30493}, {222, 44707}, {228, 16697}, {248, 44716}, {255, 1953}, {311, 14585}, {324, 23606}, {326, 2179}, {520, 1625}, {647, 23181}, {822, 2617}, {1154, 50433}, {1393, 2289}, {1568, 18877}, {1636, 36831}, {2181, 6507}, {2351, 52032}, {3199, 3964}, {3284, 44715}, {3926, 40981}, {3990, 18180}, {4055, 17167}, {4558, 15451}, {6368, 32661}, {6798, 45800}, {7069, 7125}, {7117, 44710}, {8798, 15905}, {9247, 18695}, {9380, 34900}, {10316, 41168}, {14213, 52430}, {14379, 42459}, {14570, 39201}, {14575, 28706}, {14576, 16391}, {14587, 39019}, {18315, 34983}, {18604, 21807}, {19210, 36412}, {20806, 27372}, {23357, 35442}, {23582, 41219}, {31388, 41891}, {31626, 32078}, {32320, 35360}, {36296, 44711}, {36297, 44712}, {40441, 42445}, {40799, 42353}, {41334, 42487}, {44713, 46112}, {44714, 46113}

  • X418 = barycentric quotient Xi/Xj for these {i,j}: {3, 276}, {5, 18027}, {6, 8795}, {25, 8794}, {32, 8884}, {48, 40440}, {51, 2052}, {107, 42401}, {110, 42405}, {184, 275}, {216, 264}, {217, 4}, {343, 18022}, {394, 34384}, {577, 95}, {669, 15422}, {1092, 34386}, {1576, 16813}, {1625, 6528}, {2179, 158}, {2181, 6521}, {3051, 19174}, {3199, 1093}, {3284, 43752}, {5562, 76}, {6528, 42369}, {6751, 41760}, {9247, 2190}, {14575, 8882}, {14585, 54}, {15451, 14618}, {17434, 850}, {23181, 6331}, {23606, 97}, {26880, 19188}, {27372, 43678}, {27374, 27376}, {28706, 44161}, {28724, 41488}, {30493, 331}, {31388, 26166}, {32078, 40684}, {32661, 18831}, {34983, 18314}, {35442, 23962}, {36433, 19210}, {39201, 15412}, {40981, 393}, {41219, 15526}, {42293, 523}, {42353, 40822}, {42556, 14767}, {44088, 6}, {44706, 1969}, {44707, 7017}, {44709, 44129}, {44716, 44132}, {46394, 5}, {50433, 46138}, {52347, 1502}, {52430, 2167}

  • X418 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {2, 6638, 852}, {2, 26874, 3}, {2, 30506, 5}, {3, 4, 26897}, {3, 25, 6641}, {3, 441, 14096}, {3, 6617, 7484}, {3, 6638, 2}, {3, 15329, 44891}, {3, 15781, 35921}, {3, 23246, 1590}, {3, 23256, 1589}, {3, 26874, 34003}, {3, 45842, 3520}, {4, 26876, 3}, {5, 13322, 4}, {6, 26909, 26898}, {25, 3135, 237}, {25, 26865, 3}, {51, 26907, 216}, {110, 39243, 97}, {184, 577, 23606}, {184, 23195, 20775}, {216, 26907, 32078}, {237, 3148, 27369}, {436, 8613, 41202}, {577, 26880, 184}, {1113, 1114, 41202}, {3129, 3130, 3518}, {3131, 3132, 186}, {3155, 3156, 24}, {3917, 6509, 2972}, {46600, 46601, 40853}

Notes

As a point on the Euler line, X418 has Shinagawa coefficients \((F^{2} + S^{2},-(E + F)F - S^{2})\).

X418 is the intersection of the isogonal conjugate of polar conjugate of Brocard axis (i.e., line X184X418) and the polar conjugate of isogonal conjugate of Brocard axis (i.e., line X2X3) (Randy Hutson, March 14, 2018)