X₁₀₄

ANTIPODE OF X₁₀₀

X₁₀₄ is the antipode of X₁₀₀.

5, 6, 7 3, 4, 5 Kepler Kimberling

X₁₀₄ (ANTIPODE OF X₁₀₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {20, 149}

• Related Centers: X₂₀ (DE LONGCHAMPS POINT), X₁₄₉ (REFLECTION OF X₂₀ IN X₁₀₄)

Trilinears: \(1/(-1 + \cos B + \cos C) : 1/(-1 + \cos C + \cos A) : 1/(-1 + \cos C + \cos B)\)

Barycentrics: \(a/(-1 + \cos B + \cos C) : b/(-1 + \cos C + \cos A) : c/(-1 + \cos C + \cos B)\)

Tripolars: \(|-1 + \cos B + \cos C| : :\)

Search Value (6, 9, 13): \(10.962889653417\)

▸ Alternate Coordinates1 form

• Trilinears: \(1/[b^{3} + c^{3} - (a^{2} + bc)(b + c) + 2abc] : :\)

▸ Linear Combinations1 combination

• X₁₀₄ = \(2R\) X₁ - \(3R\) X₂ + \((2R - 2r)\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines565 lines

{1, 109}, {2, 119}, {3, 8}, {4, 11}, {5, 5253}, {6, 10759}, {7, 934}, {9, 48}, {10, 6940}, {12, 6952}, {20, 149}, {21, 110}, {24, 3435}, {25, 3420}, {28, 107}, {30, 1290}, {32, 12199}, {35, 5559}, {36, 80}, {40, 1293}, {55, 1000}, {57, 3577}, {63, 37611}, {65, 1389}, {74, 7429}, {79, 946}, {84, 1420}, {90, 6261}, {98, 2787}, {99, 314}, {102, 3738}, {103, 3887}, {105, 885}, {106, 1769}, {111, 2830}, {112, 1108}, {140, 5260}, {145, 11248}, {150, 6516}, {165, 4900}, {177, 13444}, {182, 13194}, {186, 2766}, {238, 9365}, {256, 1064}, {294, 919}, {355, 404}, {371, 19113}, {372, 19112}, {376, 528}, {378, 3433}, {381, 22799}, {382, 22938}, {386, 34458}, {388, 6833}, {402, 12752}, {405, 5811}, {411, 18481}, {474, 5818}, {476, 1325}, {484, 13143}, {493, 12765}, {494, 12766}, {497, 6938}, {499, 6256}, {511, 2703}, {512, 2699}, {513, 953}, {514, 2717}, {516, 1308}, {517, 901}, {518, 2742}, {519, 2077}, {520, 2744}, {521, 2745}, {522, 2716}, {523, 2687}, {524, 2746}, {525, 2747}, {545, 38531}, {549, 38762}, {550, 38754}, {551, 3255}, {573, 3032}, {614, 9088}, {631, 958}, {651, 34586}, {667, 2726}, {675, 7445}, {691, 37960}, {693, 2861}, {739, 2423}, {758, 6596}, {813, 2196}, {825, 2344}, {835, 37399}, {840, 3309}, {898, 13136}, {900, 14127}, {912, 4511}, {925, 16049}, {927, 2481}, {929, 14198}, {932, 7155}, {935, 37961}, {941, 32693}, {942, 17097}, {943, 2646}, {962, 10680}, {963, 18283}, {971, 1156}, {972, 6366}, {983, 8685}, {991, 29067}, {1001, 5851}, {1039, 1472}, {1056, 6935}, {1061, 36076}, {1066, 9363}, {1125, 6920}, {1147, 3045}, {1155, 17636}, {1181, 34046}, {1201, 3073}, {1251, 18471}, {1294, 2803}, {1295, 2804}, {1296, 2805}, {1297, 2806}, {1301, 30733}, {1302, 4228}, {1304, 2074}, {1305, 2997}, {1309, 37305}, {1310, 19262}, {1311, 7443}, {1319, 2720}, {1350, 9024}, {1376, 3036}, {1381, 3308}, {1382, 3307}, {1392, 1482}, {1415, 11998}, {1436, 7003}, {1455, 1870}, {1458, 1937}, {1468, 28624}, {1470, 12832}, {1476, 12672}, {1477, 35355}, {1478, 6830}, {1479, 5533}, {1483, 11849}, {1490, 38271}, {1499, 2721}, {1503, 2722}, {1519, 5193}, {1532, 15325}, {1593, 1862}, {1602, 9912}, {1603, 9798}, {1604, 37310}, {1610, 15654}, {1617, 3427}, {1621, 2320}, {1656, 34126}, {1698, 38133}, {1699, 5561}, {1709, 7284}, {1764, 35636}, {1783, 7117}, {1791, 8707}, {1811, 6079}, {1837, 20118}, {2078, 17010}, {2099, 14497}, {2217, 26704}, {2256, 2335}, {2291, 23893}, {2298, 8687}, {2346, 12755}, {2475, 26470}, {2550, 6955}, {2551, 6967}, {2648, 2701}, {2691, 7464}, {2715, 5006}, {2718, 3667}, {2719, 6000}, {2728, 3220}, {2734, 23224}, {2777, 10767}, {2794, 10768}, {2808, 14947}, {2818, 3937}, {2841, 38513}, {2865, 3263}, {2886, 6951}, {3057, 20586}, {3062, 13462}, {3065, 34921}, {3068, 13913}, {3069, 13977}, {3085, 6977}, {3090, 6667}, {3091, 18761}, {3098, 13235}, {3100, 8759}, {3149, 5704}, {3241, 10679}, {3295, 7320}, {3296, 3304}, {3333, 5665}, {3337, 31870}, {3338, 17098}, {3359, 3872}, {3413, 36736}, {3414, 36735}, {3417, 7412}, {3418, 4219}, {3434, 6948}, {3436, 6891}, {3448, 19478}, {3476, 8069}, {3486, 8071}, {3520, 34441}, {3522, 10993}, {3523, 38760}, {3524, 6174}, {3525, 20400}, {3528, 5584}, {3533, 38758}, {3560, 3616}, {3579, 28218}, {3600, 6847}, {3601, 7160}, {3612, 7162}, {3617, 38128}, {3618, 38119}, {3623, 37622}, {3624, 15017}, {3651, 4297}, {3653, 16858}, {3655, 10031}, {3658, 6740}, {3746, 13606}, {3813, 11826}, {3816, 6965}, {3817, 33709}, {3843, 38141}, {3869, 24467}, {3871, 26285}, {3873, 37533}, {3880, 11256}, {3916, 31786}, {4180, 20779}, {4183, 23204}, {4188, 11499}, {4189, 10267}, {4193, 26492}, {4220, 9070}, {4222, 15617}, {4223, 9057}, {4224, 9056}, {4231, 5322}, {4233, 9064}, {4294, 10806}, {4295, 26437}, {4305, 26357}, {4311, 6245}, {4313, 37287}, {4317, 6845}, {4341, 8809}, {4571, 6790}, {4853, 10270}, {4855, 5534}, {4861, 37562}, {4866, 5531}, {4881, 18857}, {4973, 5535}, {4999, 6853}, {5056, 38319}, {5080, 6882}, {5151, 37391}, {5204, 6942}, {5217, 7317}, {5248, 28162}, {5251, 5316}, {5258, 6684}, {5265, 6848}, {5267, 10902}, {5284, 7489}, {5288, 11362}, {5429, 29038}, {5433, 6949}, {5434, 7680}, {5537, 26726}, {5542, 34917}, {5552, 6961}, {5555, 10531}, {5556, 38038}, {5557, 11551}, {5558, 7373}, {5560, 5691}, {5587, 6702}, {5597, 12462}, {5598, 12463}, {5606, 5901}, {5663, 13868}, {5693, 30144}, {5720, 35262}, {5732, 28291}, {5734, 12001}, {5759, 5856}, {5776, 37519}, {5777, 17614}, {5779, 35272}, {5841, 6840}, {5842, 15326}, {5844, 35000}, {5848, 6776}, {5854, 8668}, {5881, 15863}, {5886, 6912}, {5903, 21398}, {6068, 21168}, {6073, 22102}, {6135, 13454}, {6136, 13426}, {6200, 35882}, {6284, 13274}, {6361, 22770}, {6396, 35883}, {6560, 35784}, {6561, 35785}, {6584, 6595}, {6594, 21153}, {6597, 16132}, {6705, 10106}, {6763, 31806}, {6770, 22773}, {6773, 22774}, {6796, 7280}, {6797, 37582}, {6831, 18990}, {6834, 7288}, {6850, 10527}, {6852, 25466}, {6879, 10590}, {6889, 30478}, {6890, 20076}, {6897, 19843}, {6903, 11827}, {6911, 18519}, {6915, 18480}, {6923, 11680}, {6924, 18525}, {6937, 26363}, {6943, 10526}, {6945, 18516}, {6956, 10599}, {6958, 11681}, {6968, 10589}, {6972, 20060}, {6973, 10584}, {6975, 10200}, {6976, 26105}, {6986, 8701}, {7091, 12705}, {7133, 12768}, {7161, 37616}, {7330, 19861}, {7411, 9963}, {7414, 30250}, {7458, 9084}, {7459, 9083}, {7469, 9060}, {7487, 22479}, {7580, 12690}, {7581, 19014}, {7582, 19013}, {7587, 13267}, {7588, 8104}, {7589, 12748}, {7676, 12730}, {7686, 16615}, {7707, 12760}, {7709, 22680}, {7966, 35445}, {7978, 31523}, {7982, 8699}, {7991, 12653}, {8074, 32625}, {8075, 8097}, {8076, 8098}, {8077, 8103}, {8107, 12733}, {8108, 12734}, {8109, 13260}, {8110, 13261}, {8224, 12744}, {8225, 13262}, {8227, 32557}, {8583, 13227}, {8609, 32685}, {8686, 23836}, {8698, 31663}, {8982, 26325}, {9059, 19649}, {9081, 28475}, {9104, 16434}, {9342, 38042}, {9540, 13922}, {9623, 21164}, {9655, 38163}, {9657, 10894}, {9778, 9802}, {9799, 37302}, {9840, 38470}, {9845, 28226}, {9862, 22504}, {9946, 18443}, {9951, 10860}, {9956, 17531}, {10039, 12749}, {10057, 12616}, {10072, 26333}, {10073, 10572}, {10101, 10295}, {10222, 28222}, {10299, 35023}, {10303, 38763}, {10306, 25416}, {10308, 12688}, {10309, 12246}, {10427, 21151}, {10429, 37252}, {10434, 12550}, {10476, 38484}, {10702, 35014}, {10769, 23698}, {10778, 17702}, {10779, 23699}, {10783, 22756}, {10784, 22757}, {10788, 22520}, {10864, 33576}, {10882, 13244}, {10942, 27529}, {11041, 37541}, {11240, 34629}, {11411, 22659}, {11414, 13222}, {11415, 30240}, {11492, 11844}, {11493, 11843}, {11822, 13228}, {11823, 13230}, {11824, 13269}, {11825, 13270}, {11828, 13275}, {11829, 13276}, {11845, 22755}, {11846, 22761}, {11847, 22762}, {12000, 20057}, {12032, 37998}, {12243, 22565}, {12244, 22583}, {12249, 22777}, {12250, 22778}, {12251, 22779}, {12252, 22780}, {12253, 19159}, {12254, 22781}, {12255, 22782}, {12256, 22595}, {12257, 22624}, {12383, 22586}, {12519, 30238}, {12650, 15803}, {12680, 37605}, {12771, 15997}, {12772, 18456}, {13172, 22514}, {13200, 19162}, {13396, 31394}, {13587, 23961}, {13674, 22783}, {13750, 18260}, {13794, 22784}, {13886, 22763}, {13935, 13991}, {13939, 22764}, {14513, 38707}, {14800, 37710}, {14882, 37734}, {15016, 30147}, {15071, 15446}, {15173, 18398}, {15175, 37525}, {15178, 28184}, {15179, 20323}, {15910, 34600}, {15931, 17009}, {15952, 34594}, {16371, 34627}, {16417, 38074}, {17501, 31673}, {17502, 28210}, {17579, 37820}, {18230, 38131}, {18469, 33653}, {18473, 28899}, {18492, 38161}, {18493, 38044}, {18524, 28224}, {19709, 38084}, {19860, 37534}, {20050, 35251}, {21495, 26594}, {22531, 22771}, {22532, 22772}, {22533, 22776}, {23361, 37115}, {23850, 37116}, {24016, 38459}, {24390, 31775}, {24475, 34195}, {24817, 24826}, {24852, 29057}, {25919, 37017}, {26287, 37733}, {26319, 26381}, {26320, 26405}, {26322, 26439}, {26323, 26440}, {26324, 26441}, {26705, 36009}, {26712, 35921}, {28148, 30389}, {28160, 36002}, {28170, 30392}, {28180, 31662}, {28206, 37624}, {28214, 31666}, {28444, 38314}, {28469, 30269}, {28486, 30272}, {30576, 36069}, {31509, 35242}, {31659, 37291}, {32198, 38455}, {32247, 32270}, {32337, 32363}, {32558, 37234}, {32669, 32689}, {32688, 32702}, {33667, 33857}, {33812, 34486}, {33994, 38307}, {36167, 38711}, {36476, 36484}, {36529, 36543}, {36560, 36575}

▸ Functional Relations36 relations

• X₁₀₄ = midpoint of X₂₀ and X₁₄₉

• X₁₀₄ = reflection of X_i in X_j for these (i,j): (4, 11), (100, 3), (153, 119), (1537, 1387)

• X₁₀₄ = isogonal conjugate of X₅₁₇

• X₁₀₄ = isotomic conjugate of X₃₂₆₂

• X₁₀₄ = complement of X₁₅₃

• X₁₀₄ = anticomplement of X₁₁₉

• X₁₀₄ = cevapoint of X_i and X_j for these (i,j): (1, 36), (44, 55)

• X₁₀₄ = X₂₁-beth conjugate of X₁₀₉

• X₁₀₄ = circumcircle-antipode of X₁₀₀

• X₁₀₄ = point of intersection, other than A, B, and C, of the circumcircle and Feuerbach hyperbola

• X₁₀₄ = Λ(X₁, X₃)

• X₁₀₄ = Ψ(X₁₀₁, X₉)

• X₁₀₄ = X₁₂₅-of-the-hexyl-triangle

• X₁₀₄ = X₁₁₃-of-excentral-triangle

• X₁₀₄ = X₁₁₀-of-2nd-circumperp-triangle

• X₁₀₄ = trilinear pole of line X₆X₆₅₀

• X₁₀₄ = Ψ(X₆, X₆₅₀)

• X₁₀₄ = Ψ(X₁₉₀, X₆₃)

• X₁₀₄ = Ψ(X₁, X₅₂₂)

• X₁₀₄ = Ψ(X₁₉, X₆₄₉)

• X₁₀₄ = Ψ(X₁₀₉, X₁)

• X₁₀₄ = Ψ(X₆₅₁, X₂)

• X₁₀₄ = Feuerbach hyperbola antipode of X₄

• X₁₀₄ = trilinear pole wrt 2nd circumperp triangle of line X₃X₁₄₂

• X₁₀₄ = reflection of X₂₆₈₇ in the Euler line

• X₁₀₄ = reflection of X₂₆₉₉ in the Brocard axis

• X₁₀₄ = reflection of X₉₅₃ in line X₁X₃

• X₁₀₄ = crossdifference of every pair of points on line X₁₇₆₉X₃₃₁₀

• X₁₀₄ = Thomson-isogonal conjugate of X₅₁₃

• X₁₀₄ = Lucas-isogonal conjugate of X₅₁₃

• X₁₀₄ = Cundy-Parry Phi transform of X₈

• X₁₀₄ = Cundy-Parry Psi transform of X₅₆

• X₁₀₄ = trilinear product of circumcircle intercepts of line X₁X₅₂₂

• X₁₀₄ = trilinear pole, wrt Thomson triangle, of line X₄₀X₃₉₂

• X₁₀₄ = areal center of pedal triangles of PU(3)

• X₁₀₄ = areal center of pedal triangles of PU(6)

Notes

Let L_A be the reflection of the line X₁X₅₁₃ in line BC, and define L_B and L_C cyclically. Let A' = L_B∩L_C, B' = L_C∩L_A, C' = L_A∩L_B. The lines AA', BB', CC' concur in X₁₀₄. (Randy Hutson, 9/23/2011)

Let A', B', C' be the intersections of the antiorthic axis and lines BC, CA, AB, resp. Let O_a, O_b, O_c be the circumcenters of AB'C', BC'A', CA'B', resp. The lines AO_a, BO_b, CO_c conucr in X₁₀₄. (Randy Hutson, December 2, 2017)

Let J_aJ_bJ_c be the excentral triangle (i.e. the antiorthic triangle). Let A″B″C″ be the triangle bounded by the orthic axes of J_aBC, J_bCA, J_cAB. Then A″B″C″ is perspective to ABC at X₁₀₄; c.f. X₈₀₆₈. (Randy Hutson, December 2, 2017)

Let J_aJ_bJ_c be the excentral triangle. The de Longchamps lines of triangles J_aBC, J_bCA, J_cAB bound a triangle perspective to ABC at X₁₀₄. (Randy Hutson, June 27, 2018)

Let J_aJ_bJ_c be the excentral triangle. The Hatzipolakis axes of triangles J_aBC, J_bCA, J_cAB bound a triangle perspective to ABC at X₁₀₄. (Randy Hutson, June 27, 2018)