X439

EULER X193-9th-SUBSTITUTION POINT

X439 = reflection of X32972 in X32970
< 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 { 230, 2996 }: Central Line L(230, 2996) { 193, 3053 }: Central Line L(193, 3053) { 187, 3926 }: Central Line L(187, 3926) 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_230 = -CEVA CONJUGATE OF X(114) (0.9746, 0.6698) X_2996 = CD(X(25)) (-0.6196, -0.1183) X_193 = -CEVA CONJUGATE OF X(2) (0.7878, 0.4690) X_3053 = INTERSECTION X(3)X(6)∩X(4)X(230) (0.7332, 0.4873) X_187 = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER) (1.1986, 1.2559) X_3926 = com[INVERSE(n(TANGENTIAL TRIANGLE))] (0.2953, 0.0167) X_439 = EULER X(193)-9th-SUBSTITUTION POINT (0.6570, 0.5128) A B C X2 X3 X4 X230 X2996 X193 X3053 X187 X3926 X439 (a, b, c) = (5, 6, 7) | AB = 1

X439 (EULER X193-9th-SUBSTITUTION POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(au^{2} : bv^{2} : cw^{2}\), where \(u : v : w\) = X193; e.g., \(u(A,B,C) = (\csc A)(\cot B + \cot C - \cot A)\)
Barycentrics: \((au)^{2} : (bv)^{2} : (cw)^{2}\)
Search Value (6, 9, 13): \(0.915650269390\)
▸ Alternate Coordinates1 form

  • Barycentrics: \((3 a^{2} - b^{2} - c^{2})^{2} : :\)

▸ Linear Combinations1 combination

  • X439 = \(3\) X2 - \(4\) X32970

▸ Incident Central Lines70 lines

{2, 3}, {32, 51170}, {69, 5023}, {99, 6392}, {115, 38259}, {187, 3926}, {193, 3053}, {230, 2996}, {620, 32816}, {754, 32825}, {1611, 18287}, {1975, 37667}, {1992, 22331}, {2482, 7758}, {3618, 15815}, {3619, 5585}, {3620, 5210}, {3767, 32456}, {3785, 5206}, {3815, 5395}, {3972, 31400}, {4366, 5265}, {4558, 40320}, {5007, 37809}, {5013, 51171}, {5032, 11147}, {5171, 39141}, {5281, 6645}, {5286, 7782}, {5304, 7783}, {5475, 18845}, {5921, 39647}, {6423, 6462}, {6424, 6463}, {6461, 40318}, {7603, 32884}, {7618, 7772}, {7737, 32829}, {7746, 32826}, {7751, 32824}, {7754, 46453}, {7759, 32837}, {7763, 7926}, {7767, 15655}, {7779, 32841}, {7780, 32836}, {7785, 32835}, {7793, 32830}, {7795, 15513}, {7800, 8588}, {7816, 21843}, {7821, 47102}, {7854, 8182}, {7857, 43448}, {7862, 43618}, {7886, 43619}, {7891, 37668}, {7922, 14907}, {9292, 51427}, {9742, 41400}, {13356, 14930}, {19118, 30558}, {20065, 32458}, {23357, 34161}, {26658, 35292}, {29585, 37552}, {34286, 36426}, {34511, 35007}, {39143, 44381}, {41139, 41895}

▸ Functional Relations9 relations

  • X439 = reflection of X32972 in X32970

  • X439 = anticomplement of X32972

  • X439 = orthocentroidal-circle-inverse of X52250

  • X439 = X6353-Ceva conjugate of X193

  • X439 = Xi-isoconjugate of Xj for these (i,j): {2996, 38252}, {8769, 8770}

  • X439 = Xi-Dao conjugate of Xj for these (i,j): {69, 6340}, {115, 3566}, {2996, 51579}

  • X439 = barycentric product Xi*Xj for these {i,j}: {193, 193}, {1707, 18156}, {4590, 15525}, {6337, 6353}, {10607, 21447}

  • X439 = barycentric quotient Xi/Xj for these {i,j}: {193, 2996}, {1707, 8769}, {3053, 8770}, {3167, 6391}, {6337, 6340}, {6353, 34208}, {15525, 115}, {19118, 14248}, {41584, 47730}, {41588, 27364}

  • X439 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {2, 4, 52250}, {2, 20, 32982}, {2, 3146, 32980}, {2, 3522, 33023}, {2, 3552, 32981}, {2, 5059, 14063}, {2, 6658, 3832}, {2, 6872, 33050}, {2, 14035, 32991}, {2, 14068, 5068}, {2, 17576, 33051}, {2, 17578, 32966}, {2, 19691, 33290}, {2, 21734, 32965}, {2, 32981, 32979}, {2, 32997, 33200}, {2, 33014, 3522}, {2, 33018, 15022}, {2, 33214, 33019}, {2, 33244, 3146}, {2, 33260, 33025}, {2, 50689, 32963}, {2, 50693, 6655}, {3, 7819, 33215}, {3, 8369, 16043}, {3, 14001, 32990}, {3, 32973, 2}, {3, 32985, 32973}, {3, 33242, 8359}, {4, 32989, 2}, {4, 33233, 32988}, {4, 33235, 35927}, {4, 35297, 32989}, {20, 16925, 2}, {20, 33199, 33017}, {20, 33203, 5025}, {140, 14033, 32987}, {140, 32987, 2}, {193, 51579, 6337}, {376, 7807, 32974}, {376, 32951, 33234}, {384, 3523, 2}, {548, 32954, 32986}, {550, 11288, 14064}, {550, 14064, 33272}, {631, 1003, 32971}, {631, 32971, 2}, {3053, 6337, 193}, {3053, 32459, 6337}, {3091, 7907, 2}, {3522, 33025, 33260}, {3522, 33205, 2}, {3528, 33191, 6656}, {3530, 11286, 32978}, {3534, 8361, 33238}, {3552, 16044, 33187}, {3552, 32964, 2}, {3552, 33259, 14035}, {5025, 13586, 33254}, {5025, 16925, 33203}, {5025, 33203, 2}, {5025, 33254, 20}, {5056, 33000, 2}, {5129, 33054, 2}, {6337, 32459, 51579}, {6655, 33208, 50693}, {6857, 33039, 2}, {7483, 33037, 2}, {7486, 16923, 2}, {7791, 33181, 2}, {7791, 33246, 33181}, {7791, 33276, 10304}, {7807, 32974, 2}, {7807, 33219, 33195}, {7807, 33234, 32951}, {7816, 21843, 32828}, {7824, 33198, 2}, {7824, 33255, 33198}, {7866, 8703, 33226}, {7887, 8598, 3529}, {7892, 33008, 33202}, {7892, 33202, 2}, {7901, 33253, 33210}, {7907, 14042, 32998}, {7907, 33007, 3091}, {9855, 33279, 49140}, {10299, 14039, 11285}, {10303, 16924, 2}, {10304, 33181, 7791}, {10304, 33246, 2}, {11001, 32955, 33229}, {11111, 17694, 33038}, {11288, 33272, 2}, {11291, 11292, 1656}, {11361, 33000, 5056}, {13586, 16925, 20}, {13586, 33245, 33268}, {14001, 32990, 2}, {14035, 32964, 33259}, {14035, 32991, 32979}, {14035, 33204, 33002}, {14035, 33259, 2}, {14037, 33004, 2}, {14042, 32998, 3091}, {14063, 33265, 5059}, {14069, 21735, 8356}, {15692, 33198, 7824}, {15692, 33255, 2}, {15696, 33184, 33247}, {15717, 33201, 2}, {16044, 33206, 2}, {16917, 17558, 2}, {16923, 33016, 7486}, {16924, 33274, 10303}, {16925, 33017, 33245}, {16925, 33254, 5025}, {16925, 33268, 33199}, {17538, 33189, 7841}, {17694, 33038, 2}, {19689, 33258, 2}, {19696, 33006, 50688}, {21495, 21511, 37269}, {21508, 21537, 11340}, {27088, 32985, 35287}, {32951, 33234, 32974}, {32959, 33703, 33228}, {32961, 33257, 3543}, {32964, 33187, 33206}, {32964, 33266, 3552}, {32965, 33225, 2}, {32966, 33193, 17578}, {32966, 33262, 2}, {32967, 33280, 3839}, {32970, 32972, 2}, {32973, 32990, 14001}, {32973, 35287, 3}, {32981, 32991, 14035}, {32985, 33227, 32990}, {32985, 35287, 2}, {32988, 32989, 33233}, {32988, 33233, 2}, {32989, 35927, 4}, {32998, 33007, 14042}, {33002, 33204, 2}, {33002, 33259, 33204}, {33014, 33205, 33023}, {33017, 33245, 33199}, {33017, 33268, 20}, {33019, 33214, 15683}, {33020, 33188, 2}, {33025, 33260, 33023}, {33186, 44245, 5077}, {33187, 33206, 16044}, {33193, 33262, 32966}, {33199, 33245, 2}, {33216, 33239, 5}, {33222, 33247, 33184}, {33224, 33226, 7866}, {33233, 33235, 33250}, {33233, 33250, 4}, {33235, 35297, 4}, {33245, 33268, 33017}, {33246, 33276, 7791}, {33250, 35297, 33233}, {33252, 33283, 33264}, {35297, 35927, 2}, {35303, 35304, 15685}, {37172, 37173, 381}, {37340, 37341, 15703}

Notes

As a point on the Euler line, X439 has Shinagawa coefficients \(((E + F) ^{2} - 4S^{2},4S^{2})\)