X411

EULER X21-2nd-SUBSTITUTION POINT

X411 = ABC-to-excentral barycentric image of X12
< 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 { 40, 78 }: Central Line L(40, 78) { 35, 516 }: Central Line L(35, 516) { 81, 581 }: Central Line L(81, 581) { 255, 651 }: Central Line L(255, 651) { 56, 938 }: Central Line L(56, 938) 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_40 = BEVAN POINT (0.4286, -0.0292) X_78 = ISOGONAL CONJUGATE OF X(34) (0.3835, 0.0491) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5373, 0.1706) X_516 = ISOGONAL CONJUGATE OF X(103) (0.7703, 0.7386) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.5849, 0.2510) X_581 = ORTHOHARMONIC OF X(284) (0.5657, 0.2106) X_255 = ISOGONAL CONJUGATE OF X(158) (0.4217, 0.0365) X_651 = TRILINEAR POLE OF LINE X(1)X(3) (0.3048, 0.5599) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_938 = ANTICOMPLEMENT OF X(936) (0.5846, 0.2462) X_411 = EULER X(21)-2nd-SUBSTITUTION POINT (0.4414, -0.0514) A B C X2 X3 X4 X40 X78 X35 X516 X81 X581 X255 X651 X56 X938 X411 (a, b, c) = (5, 6, 7) | AB = 1

X411 (EULER X21-2nd-SUBSTITUTION POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((1/v)^{2} + (1/w)^{2} - (1/u)^{2} : (1/w)^{2} + (1/u)^{2} - (1/v)^{2} : (1/u)^{2} + (1/v)^{2} - (1/w)^{2}\), where \(u : v : w\) = X21; e.g., \(u = u(A,B,C) = 1/(\cos B + \cos C)\)
Barycentrics: \(a[(1/v)^{2} + (1/w)^{2} - (1/u)^{2}] : b[(1/w)^{2} + (1/u)^{2} - (1/v)^{2}] : c[(1/u)^{2} + (1/v)^{2} - (1/w)^{2}]\)
Search Value (6, 9, 13): \(15.340293988322\)
▸ Alternate Coordinates2 forms

  • Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = \cos B \cos C - (\cos A + \cos B + \cos C)\cos A\)

  • Barycentrics: \(a (a^{6} - a^{5} b - 2 a^{4} b^{2} + 2 a^{3} b^{3} + a^{2} b^{4} - a b^{5} - a^{5} c - a^{4} b c + a b^{4} c + b^{5} c - 2 a^{4} c^{2} + 2 a^{2} b^{2} c^{2} + 2 a^{3} c^{3} - 2 b^{3} c^{3} + a^{2} c^{4} + a b c^{4} - a c^{5} + b c^{5}) : :\)

▸ Linear Combinations1 combination

  • X411 = \(3\) X2 - \(4\) X52265, \(4\) X3 - \(3\) X17549, \(2\) X4 - \(3\) X17577, \(5\) X3091 - \(6\) X17530, \(5\) X3522 - \(3\) X37299, \(7\) X3523 - \(6\) X37298, \(4\) X6842 - \(3\) X17577, X6895 - \(4\) X52265, \(2\) X6906 - \(3\) X17549

▸ Incident Central Lines261 lines

{1, 1254}, {2, 3}, {8, 3428}, {10, 44425}, {32, 40129}, {35, 516}, {36, 1210}, {40, 78}, {46, 12520}, {54, 34800}, {55, 962}, {56, 938}, {57, 10393}, {63, 1490}, {64, 34259}, {65, 45230}, {73, 1936}, {81, 581}, {84, 4652}, {90, 1156}, {104, 18481}, {108, 1895}, {145, 22770}, {165, 936}, {170, 2108}, {171, 4300}, {185, 970}, {191, 31803}, {198, 27382}, {212, 37694}, {225, 45231}, {243, 821}, {255, 651}, {273, 17134}, {285, 46021}, {329, 1259}, {386, 1754}, {388, 26357}, {391, 46012}, {496, 41345}, {497, 37579}, {515, 2975}, {517, 3871}, {573, 2287}, {580, 32911}, {774, 1758}, {934, 34059}, {942, 18444}, {943, 11374}, {944, 11249}, {946, 1621}, {950, 37583}, {965, 37499}, {971, 3916}, {991, 37522}, {993, 5691}, {1014, 5738}, {1035, 18623}, {1044, 9316}, {1064, 3072}, {1071, 3218}, {1125, 15931}, {1155, 1858}, {1158, 9961}, {1193, 37570}, {1290, 2695}, {1292, 1311}, {1376, 5584}, {1437, 34148}, {1445, 5732}, {1446, 5088}, {1476, 22767}, {1479, 36152}, {1617, 14986}, {1699, 5248}, {1742, 37603}, {1750, 31424}, {1764, 3430}, {1766, 27396}, {1768, 12845}, {1776, 1898}, {1780, 3216}, {1788, 11502}, {1792, 4417}, {1804, 5932}, {1861, 34851}, {1897, 20222}, {1935, 2635}, {2077, 5057}, {2078, 12053}, {2801, 6763}, {2829, 4996}, {2886, 6253}, {2897, 41005}, {2951, 5506}, {3035, 50031}, {3062, 51576}, {3075, 4303}, {3085, 40292}, {3086, 7677}, {3100, 17102}, {3219, 5777}, {3303, 5734}, {3304, 15933}, {3306, 8726}, {3341, 37141}, {3465, 44706}, {3474, 11509}, {3476, 10966}, {3561, 34035}, {3576, 5253}, {3579, 5887}, {3585, 14794}, {3616, 22753}, {3624, 38150}, {3647, 31871}, {3648, 9809}, {3652, 31828}, {3681, 17857}, {3746, 4301}, {3811, 41338}, {3817, 5259}, {3868, 5709}, {3873, 12704}, {3874, 5536}, {3876, 5720}, {3913, 34711}, {4255, 37537}, {4265, 29181}, {4293, 8071}, {4294, 8069}, {4295, 11507}, {4296, 46974}, {4299, 14793}, {4305, 22766}, {4316, 14792}, {4421, 34632}, {4511, 14110}, {4640, 12688}, {4855, 6282}, {5080, 11827}, {5096, 44882}, {5122, 31805}, {5131, 16767}, {5132, 5327}, {5172, 6284}, {5204, 5704}, {5218, 37601}, {5250, 10268}, {5251, 19925}, {5260, 5587}, {5267, 28164}, {5284, 8227}, {5288, 28236}, {5303, 5450}, {5396, 46441}, {5438, 37551}, {5440, 31793}, {5441, 14804}, {5443, 14799}, {5453, 45931}, {5493, 5537}, {5535, 5884}, {5603, 10267}, {5657, 11499}, {5687, 20007}, {5690, 18524}, {5693, 11684}, {5694, 16139}, {5698, 11495}, {5706, 19767}, {5715, 31266}, {5721, 24883}, {5736, 10446}, {5744, 9799}, {5752, 5889}, {5759, 11517}, {5812, 31053}, {5818, 18491}, {5819, 32561}, {5842, 15908}, {5907, 22076}, {5927, 31445}, {6198, 37565}, {6224, 22775}, {6361, 11248}, {6516, 40702}, {6684, 7688}, {6690, 31936}, {6700, 12512}, {7280, 10090}, {7288, 37578}, {7330, 21165}, {7354, 15844}, {7360, 52345}, {7967, 10680}, {7991, 8715}, {8273, 25524}, {8666, 38669}, {9342, 31423}, {9535, 19763}, {9778, 10310}, {9812, 11496}, {9940, 27003}, {10031, 12776}, {10058, 10724}, {10129, 41869}, {10164, 12617}, {10167, 37582}, {10171, 25542}, {10246, 45977}, {10306, 20070}, {10595, 16202}, {10860, 35242}, {11220, 41854}, {11424, 13323}, {11508, 30305}, {11531, 25439}, {11680, 48482}, {11849, 28174}, {12245, 20013}, {12699, 32613}, {12702, 32141}, {12705, 35258}, {13243, 24467}, {13369, 26877}, {13397, 32706}, {13464, 34486}, {14798, 30384}, {14829, 51978}, {15323, 29325}, {15489, 46850}, {15622, 16678}, {15852, 37539}, {16113, 48698}, {16193, 30284}, {17100, 24466}, {17613, 31663}, {17757, 31799}, {18219, 38399}, {18861, 38761}, {19716, 19769}, {19788, 19850}, {20171, 30273}, {21363, 51290}, {21635, 35204}, {22765, 34773}, {22791, 37621}, {23692, 40602}, {25406, 36741}, {26321, 28186}, {28146, 33862}, {29473, 37202}, {32760, 39599}, {34195, 37625}, {34461, 34462}, {35250, 37821}, {36740, 51212}, {37469, 48897}, {37562, 48363}, {37584, 37700}

▸ Functional Relations14 relations

  • X411 = midpoint of X20 and X37437

  • X411 = reflection of Xi in Xj for these {i,j}: {1, 51717}, {4, 6842}, {2975, 11012}, {3871, 11491}, {6831, 52265}, {6895, 6831}, {6906, 3}, {34772, 33597}, {52367, 15908}

  • X411 = complement of X6895

  • X411 = anticomplement of X6831

  • X411 = orthocentroidal-circle-inverse of X6828

  • X411 = isotomic conjugate of the isogonal conjugate of X44087

  • X411 = Xi-Ceva conjugate of Xj for these (i,j): {1098, 1}, {24032, 651}

  • X411 = X1630-cross conjugate of X34035

  • X411 = X411-Dao conjugate of X6895

  • X411 = crosssum of X661 and X3270

  • X411 = ABC-to-excentral barycentric image of X12

  • X411 = barycentric product Xi*Xj for these {i,j}: {8, 34035}, {75, 1630}, {76, 44087}, {92, 3561}

  • X411 = barycentric quotient Xi/Xj for these {i,j}: {1630, 1}, {3561, 63}, {34035, 7}, {44087, 6}

  • X411 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {2, 3, 6986}, {2, 4, 6828}, {2, 20, 6836}, {2, 3149, 6915}, {2, 3522, 37423}, {2, 4188, 37282}, {2, 6835, 6991}, {2, 6836, 6943}, {2, 6870, 6855}, {2, 6894, 5}, {2, 6895, 6831}, {2, 7538, 27378}, {2, 14953, 37389}, {2, 20846, 21}, {2, 35979, 404}, {2, 50695, 4}, {2, 50700, 6835}, {3, 4, 21}, {3, 5, 1006}, {3, 20, 6909}, {3, 382, 6914}, {3, 405, 37106}, {3, 474, 3523}, {3, 550, 37403}, {3, 1012, 4189}, {3, 2915, 7488}, {3, 3149, 2}, {3, 3560, 6875}, {3, 3651, 7411}, {3, 4192, 37399}, {3, 6905, 404}, {3, 6906, 17549}, {3, 6911, 631}, {3, 6924, 6940}, {3, 6942, 13587}, {3, 6985, 4}, {3, 7420, 4225}, {3, 7489, 5428}, {3, 7580, 20}, {3, 9122, 7520}, {3, 11479, 37246}, {3, 13743, 7508}, {3, 16117, 550}, {3, 19513, 19649}, {3, 19540, 13732}, {3, 19541, 405}, {3, 19543, 37431}, {3, 19548, 4220}, {3, 20420, 37306}, {3, 21669, 17574}, {3, 36002, 6912}, {3, 36558, 13731}, {3, 37034, 17928}, {3, 37251, 140}, {3, 37400, 37402}, {3, 37411, 1012}, {3, 37412, 37254}, {3, 37426, 3522}, {3, 45976, 549}, {3, 49127, 37328}, {4, 21, 6912}, {4, 376, 6868}, {4, 631, 6824}, {4, 2476, 7548}, {4, 3090, 6866}, {4, 6824, 10883}, {4, 6825, 2476}, {4, 6828, 52269}, {4, 6838, 6932}, {4, 6842, 17577}, {4, 6852, 6841}, {4, 6853, 5}, {4, 6855, 6870}, {4, 6856, 3091}, {4, 6857, 6837}, {4, 6868, 11114}, {4, 6874, 381}, {4, 6875, 3560}, {4, 6876, 3}, {4, 6942, 52270}, {4, 6985, 36002}, {4, 6988, 2}, {5, 1006, 5047}, {5, 28459, 6902}, {20, 6838, 4}, {20, 6960, 6840}, {20, 6962, 6943}, {20, 7580, 33557}, {20, 37421, 6925}, {21, 36002, 4}, {40, 6261, 3869}, {40, 6326, 31806}, {40, 6796, 100}, {40, 52026, 78}, {63, 1490, 12528}, {73, 1936, 3562}, {140, 6841, 6852}, {140, 6946, 17535}, {140, 37251, 6946}, {140, 37374, 6972}, {255, 1745, 651}, {376, 6927, 6865}, {376, 6942, 3}, {381, 21161, 16858}, {382, 6914, 21669}, {404, 7411, 3}, {404, 35976, 35977}, {405, 19541, 3091}, {431, 1885, 4}, {442, 20420, 6839}, {474, 16293, 2}, {474, 37284, 6857}, {546, 5428, 7489}, {550, 6922, 37428}, {550, 37406, 7491}, {580, 37732, 32911}, {581, 37530, 81}, {631, 6864, 2}, {631, 6911, 17531}, {946, 10902, 1621}, {1012, 37411, 3146}, {1071, 37623, 3218}, {1155, 1858, 7098}, {1158, 50528, 9961}, {1532, 31789, 5046}, {1532, 44238, 31789}, {1745, 51281, 255}, {2041, 2042, 6845}, {2478, 6848, 6945}, {2635, 22361, 1935}, {2915, 7515, 36018}, {3075, 4303, 17074}, {3086, 7742, 7677}, {3090, 6883, 17536}, {3091, 37106, 405}, {3145, 27622, 4224}, {3146, 4189, 1012}, {3146, 6871, 4}, {3149, 11344, 6835}, {3149, 20846, 6828}, {3428, 11500, 8}, {3522, 4188, 3}, {3522, 35986, 37426}, {3523, 6837, 6857}, {3523, 37105, 3}, {3545, 28466, 16861}, {3560, 6875, 21}, {3560, 6918, 6855}, {3627, 7508, 13743}, {3651, 6905, 3}, {3832, 16865, 6913}, {4186, 52271, 28376}, {4188, 35986, 3522}, {5004, 5005, 4224}, {5535, 16132, 5884}, {5709, 18446, 3868}, {6675, 8226, 6884}, {6825, 6869, 4}, {6826, 6889, 4197}, {6827, 6834, 4193}, {6830, 6863, 7504}, {6831, 52265, 2}, {6836, 6932, 52269}, {6836, 6962, 2}, {6840, 6960, 5}, {6848, 6987, 2478}, {6850, 6934, 17579}, {6851, 6954, 6833}, {6855, 6870, 6828}, {6857, 37284, 21}, {6865, 6927, 2}, {6865, 6942, 37301}, {6865, 37301, 6986}, {6868, 52270, 21}, {6876, 6985, 21}, {6878, 6896, 6887}, {6880, 6891, 17566}, {6880, 6899, 6891}, {6888, 37433, 8727}, {6903, 6949, 6882}, {6905, 6940, 6924}, {6906, 17577, 6912}, {6907, 37468, 2475}, {6908, 50701, 377}, {6909, 6915, 6943}, {6909, 6932, 6912}, {6909, 33557, 20}, {6910, 10431, 6847}, {6915, 6986, 2}, {6917, 6937, 6175}, {6919, 37313, 25875}, {6924, 6940, 404}, {6928, 6941, 37375}, {6953, 6992, 5084}, {6986, 36002, 52269}, {6988, 20846, 6986}, {6988, 50695, 6828}, {7411, 36002, 1005}, {7412, 37380, 29}, {7420, 7580, 37420}, {7466, 35981, 1005}, {7483, 8727, 6888}, {7491, 37406, 4}, {7513, 23512, 6836}, {11112, 37424, 37163}, {11344, 37229, 2}, {11499, 35239, 5657}, {11827, 18242, 5080}, {12511, 25440, 165}, {13738, 28077, 33849}, {13747, 50206, 2}, {14110, 37837, 4511}, {14782, 14783, 6861}, {16440, 16441, 1817}, {18481, 26286, 104}, {19520, 37240, 4208}, {27653, 35981, 21}, {28452, 37438, 6901}, {31793, 40262, 5440}, {31952, 37328, 37048}, {35986, 36003, 7411}, {37034, 37264, 11349}, {37229, 50700, 6915}, {37282, 37423, 6986}, {42789, 42790, 37117}

Notes

As a point on the Euler line, X411 has Shinagawa coefficients \((\$aS_{A}\$, -\$aS_{A}\$ - abc)\).