X₅₆₈

ORTHOHARMONIC OF X₅₀

X₅₆₈ = Brocard-circle-inverse of X₅₆₇

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

X₅₆₈ (ORTHOHARMONIC OF X₅₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {24, 49}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₂₄ (PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE), X₄₉ (CENTER OF SINE-TRIPLE-ANGLE CIRCLE)

Trilinears: \(\sin A + \cos A \cot D/2\), where where cot \(D/2 = (6R^{2} - a^{2} - b^{2} - c^{2})/(4 \operatorname{area})\) and \(R = abc/(4 \operatorname{area})\)

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

Search Value (6, 9, 13): \(1.764876474634\)

▸ Alternate Coordinates2 forms

• Trilinears: \(a[3 R^{2}(b^{2} + c^{2} - a^{2}) + a^{2}b^{2} + a^{2}c^{2} - b^{4} - c^{4}]\), where \(R = abc/(4 \operatorname{area})\) (Wimalasiri Perera, August 29, 2011)

• Trilinears: \(\cos A - 2 \cos 2A \cos(B - C) : :\)

▸ Linear Combinations1 combination

• 3 X₂ - 4 X₁₃₃₆₃, 3 X₅₉₄₆ - 2 X₁₃₃₆₃, 3 X₅₉₄₆ - X₁₅₀₆₇, 4 X₅₉₄₆ - X₂₃₀₃₉, 8 X₁₃₃₆₃ - 3 X₂₃₀₃₉, 4 X₁₅₀₆₇ - 3 X₂₃₀₃₉, X₃ + 2 X₅₂, X₃ - 4 X₃₈₉, 2 X₃ + X₆₂₄₃, 5 X₃ - 8 X₉₇₂₉, 5 X₃ - 2 X₁₀₆₂₅, 11 X₃ - 8 X₁₃₃₄₈, 7 X₃ - 16 X₁₅₀₁₂, 7 X₃ - 4 X₁₅₆₄₄, X₃ + 8 X₁₆₆₂₅, 3 X₃ - 4 X₁₆₈₃₆, 13 X₃ - 16 X₁₇₇₀₄, 2 X₃ - 5 X₃₇₄₈₁, 4 X₃ - X₃₇₄₈₄, 2 X₃ - 3 X₄₀₂₈₀, 4 X₆ - X₁₈₄₃₈, X₅₂ + 2 X₃₈₉, 4 X₅₂ - X₆₂₄₃, 5 X₅₂ + 4 X₉₇₂₉, 5 X₅₂ + X₁₀₆₂₅, 4 X₅₂ + X₁₃₃₄₀, 11 X₅₂ + 4 X₁₃₃₄₈, 7 X₅₂ + 8 X₁₅₀₁₂, 7 X₅₂ + 2 X₁₅₆₄₄, X₅₂ - 4 X₁₆₆₂₅, 3 X₅₂ + 2 X₁₆₈₃₆, 13 X₅₂ + 8 X₁₇₇₀₄, 4 X₅₂ + 5 X₃₇₄₈₁, 8 X₅₂ + X₃₇₄₈₄, 4 X₅₂ + 3 X₄₀₂₈₀, 8 X₃₈₉ + X₆₂₄₃, 5 X₃₈₉ - 2 X₉₇₂₉, 10 X₃₈₉ - X₁₀₆₂₅, 8 X₃₈₉ - X₁₃₃₄₀, 11 X₃₈₉ - 2 X₁₃₃₄₈, 7 X₃₈₉ - 4 X₁₅₀₁₂, 7 X₃₈₉ - X₁₅₆₄₄, X₃₈₉ + 2 X₁₆₆₂₅, 3 X₃₈₉ - X₁₆₈₃₆, 13 X₃₈₉ - 4 X₁₇₇₀₄, 8 X₃₈₉ - 5 X₃₇₄₈₁, 16 X₃₈₉ - X₃₇₄₈₄, 8 X₃₈₉ - 3 X₄₀₂₈₀, 2 X₅₇₆ + X₃₇₄₇₃, X₁₃₅₁ + 2 X₁₉₁₆₁, 2 X₃₃₁₃ - 5 X₁₂₀₁₇, 4 X₅₀₉₇ - X₄₄₄₃₉, 5 X₆₂₄₃ + 16 X₉₇₂₉, X₆₂₄₃ + 4 X₉₇₃₀, 5 X₆₂₄₃ + 4 X₁₀₆₂₅, 11 X₆₂₄₃ + 16 X₁₃₃₄₈, 7 X₆₂₄₃ + 32 X₁₅₀₁₂, 7 X₆₂₄₃ + 8 X₁₅₆₄₄, X₆₂₄₃ - 16 X₁₆₆₂₅, 3 X₆₂₄₃ + 8 X₁₆₈₃₆, 13 X₆₂₄₃ + 32 X₁₇₇₀₄, and many more

X₅₆₈ lies on the cubics K885 and K1139.

▸ Incident Central Lines304 lines

{2, 1154}, {3, 6}, {4, 94}, {5, 3567}, {20, 10263}, {23, 15032}, {24, 49}, {25, 10540}, {26, 6800}, {30, 3060}, {51, 381}, {54, 1658}, {68, 973}, {110, 12106}, {140, 7998}, {155, 7506}, {156, 3518}, {184, 2070}, {185, 382}, {186, 1994}, {195, 1147}, {343, 37347}, {355, 31732}, {373, 1656}, {376, 13391}, {378, 39522}, {399, 7545}, {542, 9971}, {546, 9781}, {547, 11451}, {549, 2979}, {550, 10574}, {631, 6101}, {632, 7999}, {974, 20127}, {1092, 43809}, {1181, 7517}, {1199, 7488}, {1204, 15041}, {1216, 3526}, {1263, 13505}, {1352, 16776}, {1353, 6403}, {1493, 6242}, {1503, 7540}, {1511, 11004}, {1614, 37440}, {1657, 40647}, {1843, 21852}, {1899, 31723}, {1992, 14984}, {1993, 6644}, {1995, 15068}, {2072, 13567}, {2262, 40263}, {2777, 7729}, {2781, 20126}, {2842, 31825}, {2854, 23236}, {2937, 35268}, {3090, 11591}, {3091, 5876}, {3146, 13491}, {3270, 9642}, {3292, 43586}, {3357, 17823}, {3522, 13421}, {3523, 10627}, {3525, 32142}, {3527, 34801}, {3529, 16982}, {3534, 21969}, {3545, 13364}, {3564, 11188}, {3574, 5449}, {3575, 13292}, {3627, 6241}, {3628, 11444}, {3819, 15694}, {3830, 6000}, {3832, 45959}, {3843, 10110}, {3845, 13451}, {3850, 15058}, {3851, 5907}, {3853, 12290}, {3855, 45958}, {3861, 11439}, {3917, 5054}, {5012, 7502}, {5055, 5891}, {5056, 14128}, {5067, 32205}, {5068, 18874}, {5070, 11793}, {5072, 45187}, {5073, 10575}, {5076, 11381}, {5422, 7514}, {5447, 15720}, {5448, 36518}, {5576, 12359}, {5609, 14002}, {5654, 14643}, {5655, 12824}, {5722, 18330}, {5777, 31819}, {5944, 32136}, {6033, 39835}, {6090, 6642}, {6240, 12370}, {6293, 18381}, {6321, 39806}, {6515, 18420}, {6688, 15703}, {6756, 39871}, {6759, 18378}, {6971, 39271}, {7401, 31810}, {7403, 38136}, {7528, 11411}, {7529, 12164}, {7530, 11456}, {7550, 15018}, {7553, 18914}, {7555, 15080}, {7556, 11003}, {7575, 11422}, {7576, 45968}, {7687, 22584}, {7689, 11424}, {7691, 43651}, {7720, 12803}, {7721, 12804}, {7723, 11746}, {7730, 32423}, {7731, 10264}, {7785, 15536}, {8254, 32338}, {8541, 39562}, {8550, 8705}, {8584, 44265}, {8703, 20791}, {9019, 11179}, {9544, 47485}, {9703, 34986}, {9704, 10282}, {9714, 19347}, {9777, 9818}, {9792, 19176}, {9826, 37645}, {9833, 41589}, {9969, 18440}, {9970, 37827}, {10024, 12233}, {10111, 15473}, {10124, 44299}, {10125, 22051}, {10224, 20424}, {10254, 18388}, {10255, 23515}, {10272, 12273}, {10311, 22146}, {10539, 13621}, {10594, 32139}, {10605, 44413}, {10620, 11806}, {10628, 23325}, {10653, 36978}, {10654, 36980}, {10937, 12897}, {10982, 12163}, {10984, 13564}, {11202, 37922}, {11225, 18400}, {11250, 15055}, {11264, 34799}, {11402, 14070}, {11423, 12107}, {11433, 18531}, {11435, 18453}, {11436, 18455}, {11441, 13861}, {11442, 11818}, {11455, 15687}, {11539, 44324}, {11548, 33523}, {11555, 16461}, {11556, 16462}, {11561, 12383}, {11562, 11800}, {11585, 31802}, {11624, 40693}, {11626, 40694}, {11675, 43461}, {11695, 15082}, {11745, 12134}, {11801, 12281}, {11802, 12307}, {11807, 38790}, {11819, 34224}, {12026, 13504}, {12083, 33586}, {12121, 14708}, {12188, 39846}, {12219, 20304}, {12226, 13472}, {12228, 15035}, {12235, 12429}, {12237, 22809}, {12238, 22810}, {12241, 18563}, {12242, 22815}, {12254, 32196}, {12278, 45971}, {12289, 45970}, {12324, 44544}, {12358, 37643}, {12699, 31728}, {13013, 22813}, {13014, 22814}, {13188, 39817}, {13366, 18475}, {13371, 26879}, {13403, 18562}, {13423, 36966}, {13561, 16880}, {14216, 18382}, {14269, 16194}, {14448, 36253}, {14516, 31830}, {14561, 14787}, {14790, 18916}, {14791, 18911}, {14845, 19709}, {14855, 15681}, {14865, 32138}, {15004, 15038}, {15033, 18570}, {15053, 43574}, {15061, 18281}, {15073, 47277}, {15083, 18369}, {15091, 15801}, {15317, 18532}, {15360, 44262}, {15688, 36987}, {15760, 41588}, {15800, 18569}, {16227, 44214}, {16266, 17928}, {16270, 18931}, {16618, 47582}, {16867, 38534}, {16980, 18526}, {17505, 43949}, {17800, 46850}, {17810, 18451}, {18121, 39235}, {18390, 18403}, {18404, 39571}, {18434, 21400}, {18447, 19366}, {18462, 19410}, {18463, 19411}, {18494, 47328}, {18559, 30522}, {18583, 41716}, {18952, 31815}, {19043, 19051}, {19044, 19052}, {19136, 45016}, {19362, 32322}, {19456, 44080}, {19552, 32409}, {20379, 31857}, {21661, 38594}, {21844, 43394}, {22530, 22808}, {22971, 38789}, {23049, 34146}, {23061, 43584}, {23293, 39504}, {25739, 44288}, {26876, 46025}, {26881, 37936}, {26913, 37938}, {31745, 31763}, {32062, 38335}, {32063, 41580}, {32165, 45731}, {32210, 35475}, {32246, 32272}, {32326, 34779}, {34117, 41613}, {34484, 43605}, {34545, 35921}, {34775, 34780}, {37649, 44201}, {38321, 44665}, {40266, 42450}, {41171, 41724}, {43816, 44056}, {45780, 47391}

▸ Functional Relations9 relations

• X₅₆₈ = midpoint of X_i and X_j for these {i,j}: {51, 14831}, {52, 9730}, {1986, 45237}, {3060, 5890}, {5889, 11459}, {6243, 13340}, {6403, 15531}, {7576, 45968}

• X₅₆₈ = reflection of X_i in X_j for these {i,j}: {2, 5946}, {3, 9730}, {265, 45237}, {381, 51}, {1352, 16776}, {2979, 549}, {3845, 13451}, {3917, 5892}, {5054, 16226}, {5562, 10170}, {5655, 12824}, {5891, 5943}, {9730, 389}, {10170, 5462}, {11455, 15687}, {11459, 5}, {12162, 46847}, {13340, 3}, {14643, 16222}, {15060, 13364}, {15061, 46430}, {15067, 13363}, {15072, 45956}, {15305, 3845}, {15531, 1353}, {15681, 14855}, {16776, 32191}, {18435, 381}, {18436, 11459}, {23039, 2}, {32063, 41580}, {32609, 16223}, {37484, 13340}, {41330, 15544}, {44214, 16227}, {45237, 12236}, {46847, 10110}

• X₅₆₈ = anticomplement of X₁₅₀₆₇

• X₅₆₈ = Brocard-circle-inverse of X₅₆₇

• X₅₆₈ = crosssum of X₃ and X₁₄₈₅₂

• X₅₆₈ = Cundy-Parry Phi transform of X₅₀

• X₅₆₈ = Cundy-Parry Psi transform of X₉₄

• X₅₆₈ = homothetic center of Ehrmann side-triangle and 2nd anti-Conway triangle

• X₅₆₈ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {3, 6, 567}, {3, 52, 6243}, {3, 389, 37481}, {3, 567, 14805}, {3, 6243, 37484}, {3, 9730, 40280}, {3, 11432, 36753}, {3, 14627, 578}, {3, 15037, 182}, {3, 36747, 37495}, {3, 36749, 37472}, {3, 36752, 37471}, {3, 36753, 13353}, {3, 37489, 3581}, {3, 37493, 36749}, {3, 37496, 37480}, {4, 265, 18430}, {4, 3448, 34514}, {4, 6102, 34783}, {4, 18951, 25738}, {4, 34783, 18439}, {5, 5889, 18436}, {5, 16881, 3567}, {5, 31834, 15056}, {6, 1351, 18449}, {6, 3581, 14805}, {6, 18578, 566}, {6, 37489, 3}, {24, 12161, 49}, {25, 18445, 10540}, {52, 389, 3}, {52, 37481, 37484}, {143, 6102, 4}, {155, 7506, 18350}, {182, 37478, 3}, {185, 5446, 382}, {185, 18396, 10938}, {195, 45735, 1147}, {371, 372, 50}, {373, 5562, 10170}, {373, 10170, 1656}, {381, 13321, 51}, {389, 15644, 15012}, {389, 16625, 52}, {399, 7545, 46261}, {567, 3581, 3}, {567, 6243, 18438}, {567, 19129, 13353}, {569, 46730, 3}, {576, 11438, 13352}, {576, 44490, 6}, {1199, 7488, 32046}, {1493, 32171, 9545}, {1986, 12236, 265}, {1993, 6644, 22115}, {2070, 15087, 184}, {2979, 15045, 549}, {3557, 3558, 18114}, {3567, 5889, 5}, {3567, 11459, 5640}, {3575, 13292, 44076}, {3581, 18449, 37477}, {3853, 45957, 12290}, {3917, 5892, 5054}, {3917, 16226, 5892}, {5102, 9786, 37497}, {5102, 37497, 36747}, {5462, 5562, 1656}, {5462, 10170, 373}, {5640, 5889, 11459}, {5640, 11459, 5}, {5876, 10095, 3091}, {5890, 15072, 45956}, {5891, 5943, 5055}, {5946, 15067, 13363}, {6101, 12006, 631}, {6102, 6746, 25738}, {6243, 37481, 3}, {6243, 40280, 13340}, {7689, 11424, 14130}, {7999, 15028, 632}, {9545, 44879, 32171}, {9729, 10625, 3}, {9781, 12111, 546}, {9786, 36747, 3}, {10110, 12162, 3843}, {10115, 32352, 195}, {10224, 26917, 45622}, {10263, 13630, 20}, {10575, 13598, 5073}, {11002, 37644, 45237}, {11412, 15043, 140}, {11430, 32110, 3}, {11432, 37489, 19129}, {11438, 13352, 3}, {11444, 15024, 3628}, {11477, 37475, 37483}, {11557, 21649, 399}, {11562, 11800, 12902}, {11591, 15026, 3090}, {11806, 13417, 10620}, {11819, 43588, 34224}, {12233, 41587, 10024}, {12359, 45089, 5576}, {13321, 14831, 18435}, {13336, 46728, 3}, {13340, 37481, 40280}, {13340, 40280, 3}, {13358, 38898, 3448}, {13363, 15067, 2}, {13364, 15060, 3545}, {13382, 13598, 10575}, {17834, 36752, 3}, {18396, 40909, 382}, {18404, 39571, 43821}, {18952, 31815, 37444}, {31728, 31757, 12699}, {31732, 31760, 355}, {31830, 32358, 14516}, {34417, 46261, 7545}, {36749, 37490, 3}, {37470, 37480, 3}, {37470, 37517, 37496}, {37475, 37483, 3}, {37481, 40280, 9730}, {37486, 37514, 3}, {37490, 37493, 37472}, {40647, 45186, 1657}

Notes

Let (A) be the circle centered at A that cuts off a segment of line BC equal to the radius of (A). Define (B) and (C) cyclically. Then X₅₆₈ is the radical center of circles (A), (B), (C). (Randy Hutson, December 2, 2017)