X₃₇₂

{X₃,X₆}-HARMONIC CONJUGATE OF X₃₇₁

X₃₇₂ is the {X₃,X₆}-harmonic conjugate of X₃₇₁.

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

X₃₇₂ ({X₃,X₆}-HARMONIC CONJUGATE OF X₃₇₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}

• Related Centers: X₃₇₁ (KENMOTU POINT), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT)

Trilinears: \(\cos(A + \pi /4) : :\)

Barycentrics: \(\sin A \cos(A + \pi /4) : :\)

Search Value (6, 9, 13): \(-1.871135541073\)

▸ Alternate Coordinates3 forms

• Trilinears: \(\cos A - \sin A : :\)

• Trilinears: \(a(S_{A} - S) : :\)

• Trilinears: \(a(b^{2} + c^{2} - a^{2} - 2S) : :\)

▸ Linear Combinations1 combination

• X₃₇₂ = \(3\) X₂ + X₄₃₁₃₃, \(2\) X₆₄₀ + X₄₃₁₃₃, \(3\) X₆ - \(4\) X₄₄₄₈₁, \(3\) X₆ - \(2\) X₄₄₅₀₁, X₁₁₅₁ - \(3\) X₁₂₉₆₈, \(2\) X₁₅₀₅ + X₄₀₂₇₅, \(2\) X₆₂₂₁ - \(5\) X₄₁₄₁₁, X₁₁₈₂₄ - \(4\) X₄₃₁₂₁, X₁₁₈₂₄ + \(2\) X₄₅₄₈₈, \(3\) X₃₅₈₄₁ - \(8\) X₄₄₄₈₁, \(3\) X₃₅₈₄₁ - \(4\) X₄₄₅₀₁, \(2\) X₄₃₁₂₁ + X₄₅₄₈₈, X₃₀₇₁ - \(3\) X₃₂₇₈₈, \(2\) X₃₀₇₁ - \(3\) X₃₅₈₂₃, X₃₀₇₁ + \(3\) X₄₁₉₄₆, \(2\) X₃₀₇₁ + X₄₂₂₆₇, \(3\) X₃₀₇₁ - X₄₂₂₇₁, \(5\) X₃₀₇₁ + \(3\) X₄₃₂₀₉, X₃₀₇₁ + \(6\) X₅₂₀₄₈, \(2\) X₇₅₈₄ - \(3\) X₃₂₇₈₈, \(4\) X₇₅₈₄ - X₃₅₈₂₁, \(4\) X₇₅₈₄ - \(3\) X₃₅₈₂₃, \(2\) X₇₅₈₄ + \(3\) X₄₁₉₄₆, \(2\) X₇₅₈₄ + X₄₂₂₅₉, \(4\) X₇₅₈₄ + X₄₂₂₆₇, \(6\) X₇₅₈₄ - X₄₂₂₇₁, \(10\) X₇₅₈₄ + \(3\) X₄₃₂₀₉, X₇₅₈₄ + \(3\) X₅₂₀₄₈, \(6\) X₃₂₇₈₈ - X₃₅₈₂₁, \(3\) X₃₂₇₈₈ + X₄₂₂₅₉, \(6\) X₃₂₇₈₈ + X₄₂₂₆₇, \(9\) X₃₂₇₈₈ - X₄₂₂₇₁, \(5\) X₃₂₇₈₈ + X₄₃₂₀₉, X₃₂₇₈₈ + \(2\) X₅₂₀₄₈, X₃₅₈₂₁ - \(3\) X₃₅₈₂₃, X₃₅₈₂₁ + \(6\) X₄₁₉₄₆, X₃₅₈₂₁ + \(2\) X₄₂₂₅₉, \(3\) X₃₅₈₂₁ - \(2\) X₄₂₂₇₁, \(5\) X₃₅₈₂₁ + \(6\) X₄₃₂₀₉, X₃₅₈₂₁ + \(12\) X₅₂₀₄₈, X₃₅₈₂₃ + \(2\) X₄₁₉₄₆, \(3\) X₃₅₈₂₃ + \(2\) X₄₂₂₅₉, \(3\) X₃₅₈₂₃ + X₄₂₂₆₇, \(9\) X₃₅₈₂₃ - \(2\) X₄₂₂₇₁, \(5\) X₃₅₈₂₃ + \(2\) X₄₃₂₀₉, X₃₅₈₂₃ + \(4\) X₅₂₀₄₈, \(3\) X₄₁₉₄₆ - X₄₂₂₅₉, \(6\) X₄₁₉₄₆ - X₄₂₂₆₇, \(9\) X₄₁₉₄₆ + X₄₂₂₇₁, \(5\) X₄₁₉₄₆ - X₄₃₂₀₉, \(3\) X₄₂₂₅₉ + X₄₂₂₇₁, \(5\) X₄₂₂₅₉ - \(3\) X₄₃₂₀₉, X₄₂₂₅₉ - \(6\) X₅₂₀₄₈, \(3\) X₄₂₂₆₇ + \(2\) X₄₂₂₇₁, \(5\) X₄₂₂₆₇ - \(6\) X₄₃₂₀₉, X₄₂₂₆₇ - \(12\) X₅₂₀₄₈, \(5\) X₄₂₂₇₁ + \(9\) X₄₃₂₀₉, X₄₂₂₇₁ + \(18\) X₅₂₀₄₈, X₄₃₂₀₉ - \(10\) X₅₂₀₄₈, X₄₈₉ - \(3\) X₃₅₉₄₈, X₄₈₉ + \(3\) X₄₅₄₂₁, \(2\) X₇₉₆₈ + X₃₅₆₁₁, X₃₅₆₄₂ + \(2\) X₄₉₂₂₇, X₃₅₈₅₇ + \(2\) X₄₈₇₁₅, X₃₅₈₈₃ + \(2\) X₄₈₇₀₁, X₃₅₈₂₅ + \(2\) X₄₉₂₆₇, X₃₅₈₇₉ + \(2\) X₄₉₂₁₃, X₁₂₃₇₆ + \(2\) X₄₉₂₁₇, X₃₅₈₂₇ + \(2\) X₄₉₂₆₉, \(3\) X₁₁₂₄₂ - X₁₂₉₇₀, \(6\) X₁₁₂₄₂ + X₃₅₈₆₅, \(3\) X₁₁₂₄₂ + X₄₉₂₅₁, \(2\) X₁₂₉₇₀ + X₃₅₈₆₅, X₃₅₈₃₅ - \(4\) X₄₆₆₈₉, X₃₅₈₂₉ + \(2\) X₄₉₂₇₁, X₃₅₈₈₁ + \(2\) X₄₉₂₁₉

X₃₇₂ lies on the cubics K006, K233, K250, K337, K415, K424b, K633, K897, K1197, K1257, the curve Q168.

▸ Incident Central Lines779 lines

{1, 1703}, {2, 485}, {3, 6}, {4, 486}, {5, 615}, {8, 35842}, {10, 13975}, {11, 13977}, {12, 13958}, {13, 3392}, {14, 3367}, {17, 3366}, {18, 3391}, {20, 1588}, {24, 5412}, {25, 494}, {30, 3071}, {35, 2066}, {36, 2067}, {40, 5416}, {46, 8942}, {51, 3155}, {54, 6413}, {55, 1124}, {56, 1335}, {64, 6416}, {67, 19396}, {68, 35836}, {69, 11291}, {74, 19110}, {76, 35866}, {79, 35854}, {80, 35852}, {81, 16440}, {83, 22716}, {84, 35844}, {86, 21992}, {90, 42013}, {98, 13938}, {99, 19055}, {100, 19081}, {104, 19112}, {110, 7599}, {112, 19093}, {113, 13990}, {114, 6290}, {115, 13967}, {119, 13991}, {125, 13969}, {127, 13985}, {132, 13992}, {140, 590}, {155, 8943}, {157, 30399}, {165, 1702}, {184, 3156}, {185, 21641}, {186, 10880}, {193, 487}, {194, 22718}, {212, 3076}, {230, 49028}, {235, 13937}, {238, 31546}, {254, 41515}, {256, 8330}, {262, 10849}, {265, 35834}, {291, 8346}, {315, 492}, {355, 13973}, {376, 6459}, {378, 11473}, {381, 13847}, {382, 13785}, {394, 1584}, {395, 18585}, {396, 15765}, {397, 14813}, {398, 14814}, {399, 44593}, {402, 35790}, {405, 31473}, {427, 8281}, {489, 32419}, {490, 7388}, {491, 642}, {493, 35804}, {498, 31472}, {499, 44623}, {515, 13936}, {517, 7968}, {542, 32292}, {546, 13993}, {547, 43212}, {549, 8981}, {550, 19116}, {589, 3060}, {591, 754}, {601, 605}, {602, 606}, {603, 3077}, {626, 11313}, {631, 3068}, {632, 32789}, {637, 11293}, {671, 35698}, {736, 49351}, {760, 45713}, {940, 16432}, {944, 19065}, {946, 13971}, {952, 35843}, {958, 1377}, {971, 35845}, {993, 31453}, {997, 30557}, {999, 3298}, {1092, 8963}, {1131, 5056}, {1132, 1328}, {1147, 10665}, {1154, 12971}, {1181, 19356}, {1199, 26891}, {1296, 11836}, {1297, 19114}, {1327, 3317}, {1344, 2465}, {1345, 2466}, {1352, 37342}, {1376, 1378}, {1385, 7969}, {1478, 13963}, {1479, 13962}, {1482, 35811}, {1490, 19067}, {1498, 17820}, {1503, 36714}, {1511, 49268}, {1583, 5406}, {1586, 11547}, {1589, 11427}, {1590, 11433}, {1593, 3092}, {1594, 26950}, {1598, 13943}, {1599, 5417}, {1600, 1993}, {1656, 8252}, {1657, 18510}, {1658, 11265}, {1672, 2007}, {1673, 2008}, {1676, 2009}, {1677, 2010}, {1680, 2013}, {1681, 2014}, {1699, 13942}, {1700, 2017}, {1701, 2018}, {1707, 6212}, {1708, 8953}, {1742, 31545}, {1743, 32556}, {1843, 19359}, {1916, 8314}, {2015, 2471}, {2016, 2472}, {2041, 40694}, {2042, 40693}, {2043, 10654}, {2044, 10653}, {2045, 42152}, {2046, 42149}, {2077, 26459}, {2099, 38235}, {2544, 2546}, {2545, 2547}, {2777, 13288}, {2782, 35825}, {2794, 13748}, {3008, 31540}, {3085, 31408}, {3090, 23267}, {3091, 13941}, {3146, 22615}, {3167, 13068}, {3235, 3237}, {3236, 3238}, {3295, 3297}, {3316, 3533}, {3357, 49250}, {3373, 42783}, {3374, 42784}, {3428, 19013}, {3515, 5410}, {3518, 26886}, {3522, 9541}, {3523, 7585}, {3524, 19054}, {3525, 13886}, {3526, 8253}, {3528, 9681}, {3529, 23273}, {3530, 31454}, {3534, 43338}, {3563, 39383}, {3564, 32494}, {3567, 26916}, {3574, 13986}, {3576, 18991}, {3579, 49226}, {3590, 43558}, {3591, 12818}, {3618, 11292}, {3627, 42226}, {3628, 18538}, {3629, 40289}, {3664, 31541}, {3788, 11316}, {3832, 23253}, {3843, 45385}, {3845, 42418}, {3859, 41947}, {3972, 35938}, {4297, 49547}, {4383, 16433}, {4769, 45444}, {4845, 30336}, {5054, 13846}, {5066, 42607}, {5067, 31414}, {5204, 18996}, {5217, 19038}, {5254, 12967}, {5286, 44595}, {5318, 42175}, {5321, 42177}, {5334, 42173}, {5335, 35732}, {5339, 42278}, {5340, 42279}, {5405, 31561}, {5415, 10902}, {5419, 5446}, {5432, 9646}, {5433, 9661}, {5461, 13783}, {5473, 19073}, {5474, 19075}, {5478, 13982}, {5479, 13981}, {5480, 13972}, {5587, 13947}, {5590, 7376}, {5597, 35778}, {5598, 35780}, {5603, 13959}, {5640, 35299}, {5657, 19066}, {5663, 12376}, {5690, 49232}, {5860, 26620}, {5861, 6397}, {5871, 7000}, {5889, 26922}, {5890, 6457}, {5989, 8305}, {6000, 11242}, {6102, 44613}, {6145, 35858}, {6179, 6316}, {6201, 7374}, {6202, 13949}, {6229, 44364}, {6241, 11463}, {6245, 13974}, {6246, 13976}, {6247, 13980}, {6248, 13983}, {6249, 13984}, {6250, 13880}, {6278, 13771}, {6284, 19029}, {6291, 45841}, {6300, 33351}, {6301, 33353}, {6399, 9756}, {6401, 11975}, {6402, 11978}, {6415, 14528}, {6515, 11091}, {6525, 22839}, {6642, 10961}, {6680, 11315}, {6684, 13883}, {6699, 46688}, {6750, 13960}, {6759, 10534}, {6771, 49208}, {6774, 49210}, {6776, 39893}, {6811, 9753}, {6813, 9744}, {7160, 35862}, {7354, 19027}, {7387, 35777}, {7488, 11417}, {7494, 18289}, {7592, 8911}, {7598, 7602}, {7691, 19095}, {7709, 32470}, {7738, 12124}, {7759, 51401}, {7762, 35684}, {7773, 32435}, {7834, 11314}, {7851, 32436}, {7887, 32432}, {7980, 45720}, {7987, 9583}, {8149, 49352}, {8196, 13944}, {8203, 13945}, {8212, 13956}, {8213, 13957}, {8276, 19006}, {8301, 8337}, {8304, 8309}, {8307, 8316}, {8320, 8325}, {8321, 8327}, {8323, 8332}, {8336, 8341}, {8339, 8348}, {8703, 41945}, {8884, 19184}, {8909, 47391}, {8939, 15805}, {8948, 14248}, {8962, 43650}, {8964, 41588}, {8972, 10303}, {8980, 38737}, {8983, 10165}, {8988, 38133}, {8991, 23328}, {8992, 15819}, {8994, 38727}, {8997, 38748}, {8998, 38793}, {9441, 31544}, {9582, 16192}, {9616, 35242}, {9647, 15326}, {9660, 15338}, {9680, 15717}, {9682, 17928}, {9683, 10323}, {9751, 19092}, {9752, 9758}, {9880, 13968}, {9921, 45401}, {9927, 13970}, {9932, 12424}, {9993, 13946}, {10109, 43434}, {10113, 13979}, {10132, 11402}, {10164, 13912}, {10246, 44635}, {10266, 35870}, {10267, 19050}, {10269, 19048}, {10282, 10533}, {10299, 43509}, {10304, 43512}, {10310, 18999}, {10525, 35797}, {10526, 35799}, {10531, 13964}, {10532, 13965}, {10606, 19088}, {10610, 49256}, {10666, 13754}, {10669, 35807}, {10673, 12978}, {10679, 35817}, {10680, 35819}, {10799, 12841}, {10818, 15055}, {10819, 15035}, {10839, 10844}, {10842, 10851}, {10893, 13952}, {10894, 13953}, {10895, 13954}, {10896, 13955}, {11001, 42413}, {11012, 26458}, {11202, 11241}, {11248, 19047}, {11249, 19049}, {11251, 35791}, {11252, 35781}, {11253, 35779}, {11257, 19089}, {11414, 19005}, {11447, 11449}, {11448, 12111}, {11462, 11464}, {11496, 13940}, {11539, 43888}, {11542, 35738}, {11543, 42252}, {11812, 43211}, {11822, 19007}, {11823, 19009}, {11826, 19023}, {11827, 19025}, {11828, 19031}, {11829, 19033}, {11834, 38698}, {11835, 38716}, {11897, 13948}, {11941, 11964}, {11942, 11969}, {11943, 11966}, {11944, 11968}, {11960, 11979}, {11962, 11982}, {12041, 49216}, {12042, 49212}, {12100, 52045}, {12110, 22723}, {12117, 19057}, {12118, 19061}, {12119, 19077}, {12120, 19085}, {12121, 19051}, {12122, 19091}, {12123, 19104}, {12160, 51905}, {12164, 19493}, {12257, 14912}, {12296, 22617}, {12359, 49224}, {12425, 35837}, {12514, 30556}, {12556, 19097}, {12584, 32291}, {12591, 19588}, {12599, 13978}, {12600, 13987}, {12602, 13881}, {12619, 49240}, {12787, 49079}, {12819, 14226}, {12835, 12840}, {12836, 12839}, {12837, 12838}, {12891, 12893}, {12892, 17702}, {12960, 12972}, {12961, 12973}, {12966, 13048}, {13045, 13049}, {13046, 13050}, {13132, 26523}, {13133, 26522}, {13287, 13289}, {13367, 21640}, {13388, 45126}, {13630, 43868}, {13638, 45576}, {13666, 19099}, {13687, 13988}, {13712, 26619}, {13749, 36655}, {13763, 45441}, {13786, 19101}, {13794, 45024}, {13807, 13849}, {13835, 26289}, {13885, 22722}, {13889, 19408}, {13893, 31423}, {13901, 31499}, {13903, 15720}, {13911, 26446}, {13913, 21154}, {13915, 34128}, {13922, 38760}, {14230, 15847}, {14241, 43342}, {14561, 37343}, {14853, 21736}, {14893, 41951}, {14996, 21568}, {14997, 21565}, {15022, 43791}, {15043, 26912}, {15061, 19052}, {15141, 19385}, {15640, 43343}, {15686, 43210}, {15690, 42417}, {15692, 42522}, {15702, 43254}, {15704, 42225}, {15712, 41962}, {15716, 43259}, {15723, 42578}, {15764, 43228}, {15803, 51841}, {16029, 16030}, {16034, 16035}, {16113, 19079}, {16125, 16149}, {16202, 44645}, {16203, 44643}, {16419, 34516}, {16441, 32911}, {16962, 36453}, {16963, 36470}, {16964, 42236}, {16965, 42235}, {17277, 21909}, {17818, 19021}, {17819, 17821}, {18400, 32385}, {18581, 42248}, {18582, 42249}, {18909, 18924}, {18916, 26945}, {18923, 18925}, {18980, 19439}, {18981, 19437}, {19041, 31988}, {19056, 34473}, {19063, 22676}, {19068, 52027}, {19069, 22843}, {19071, 22890}, {19074, 21156}, {19076, 21157}, {19082, 38693}, {19083, 22951}, {19090, 22712}, {19094, 38717}, {19105, 26521}, {19109, 21166}, {19113, 34474}, {19115, 38699}, {19183, 19185}, {19355, 19357}, {19360, 44588}, {19384, 19389}, {19387, 19398}, {19436, 19440}, {19438, 19441}, {19548, 36587}, {19708, 42525}, {19843, 31413}, {19925, 49619}, {21445, 33371}, {21492, 37633}, {21547, 37679}, {21548, 37674}, {21550, 37682}, {21552, 37687}, {21553, 37680}, {21567, 37685}, {22466, 35860}, {22521, 33434}, {22536, 22553}, {22592, 45079}, {22596, 26438}, {22682, 22721}, {22753, 22764}, {22831, 22877}, {22832, 22922}, {22833, 22977}, {22960, 22962}, {22961, 35861}, {23046, 42640}, {23275, 33703}, {23312, 32490}, {23358, 32384}, {23698, 39824}, {24243, 49387}, {24813, 24818}, {24828, 24843}, {25406, 39876}, {26290, 26384}, {26291, 26408}, {26292, 26454}, {26293, 26455}, {26294, 26456}, {26295, 26457}, {26326, 45366}, {26327, 45367}, {26328, 45606}, {26329, 45608}, {26330, 49026}, {26332, 45651}, {26333, 45653}, {26363, 31484}, {26398, 44582}, {26422, 44584}, {26451, 44610}, {26463, 46453}, {26465, 37561}, {26487, 44620}, {26492, 44618}, {26498, 45597}, {26507, 45595}, {26516, 44594}, {26879, 26951}, {30427, 45840}, {31400, 31403}, {31401, 31411}, {31421, 31427}, {31422, 31437}, {31424, 31438}, {31425, 31440}, {31439, 31663}, {31448, 31459}, {31449, 31464}, {31450, 31465}, {31451, 31471}, {31452, 31475}, {31455, 31481}, {31456, 31482}, {31457, 31483}, {31458, 31486}, {31687, 31692}, {31688, 31690}, {32169, 32171}, {32233, 32252}, {32274, 32304}, {32330, 32342}, {32369, 32400}, {32497, 38110}, {33347, 33453}, {33349, 33455}, {33367, 33437}, {33369, 33439}, {33370, 37334}, {33373, 33431}, {33392, 34509}, {33395, 34508}, {33699, 43341}, {33813, 49266}, {33814, 48714}, {34089, 43568}, {34091, 43570}, {34200, 52047}, {34862, 49234}, {35404, 42573}, {35434, 43384}, {35699, 49215}, {35733, 42990}, {35742, 47864}, {35754, 35847}, {35829, 35881}, {35839, 35869}, {35849, 35851}, {35853, 49241}, {35855, 49243}, {35863, 49249}, {35871, 49259}, {35873, 49261}, {35875, 49263}, {36436, 36447}, {36437, 36449}, {36438, 41943}, {36454, 36465}, {36455, 36467}, {36456, 41944}, {36474, 36491}, {36477, 36556}, {36489, 36552}, {36510, 36584}, {36526, 36550}, {36557, 36582}, {36656, 45440}, {36712, 36990}, {36998, 45406}, {38602, 48700}, {38608, 49270}, {38624, 49218}, {39823, 39825}, {39852, 39854}, {40108, 49230}, {41491, 45420}, {41948, 42639}, {41958, 43410}, {41960, 41969}, {41966, 43316}, {42022, 45414}, {42085, 42247}, {42086, 42246}, {42093, 42205}, {42094, 42203}, {42101, 42209}, {42102, 42207}, {42117, 42251}, {42118, 42250}, {42133, 42185}, {42134, 42183}, {42154, 51924}, {42159, 42189}, {42160, 42188}, {42161, 42187}, {42162, 42190}, {42163, 42213}, {42164, 42212}, {42165, 42211}, {42166, 42214}, {42167, 42202}, {42169, 42200}, {42228, 42999}, {42230, 42998}, {42558, 45384}, {42568, 43319}, {42572, 47599}, {42574, 43792}, {43320, 44245}, {43336, 43382}, {43337, 43787}, {43376, 46935}, {43377, 43508}, {43411, 43518}, {43433, 50690}, {43503, 43519}, {43506, 43514}, {43507, 43515}, {43560, 43569}, {43561, 43571}, {43601, 43825}, {43831, 43864}, {44392, 49355}, {44583, 45360}, {44585, 45358}, {44600, 45620}, {44602, 45621}, {44627, 45623}, {44629, 45624}, {44647, 49104}, {45376, 45543}, {45423, 45587}, {45425, 45585}, {45427, 45716}, {45439, 48659}, {45545, 45575}, {45547, 49348}, {45555, 49356}, {45573, 45714}, {45596, 45600}, {45598, 45602}, {48906, 49228}, {49086, 49317}, {49102, 49214}, {49105, 49236}, {49106, 49238}, {49107, 49242}, {49108, 49244}, {49109, 49246}, {49110, 49248}, {49111, 49252}, {49112, 49254}, {49113, 49258}, {49114, 49260}, {49115, 49262}, {49116, 49264}, {49948, 51925}

▸ Functional Relations47 relations

• X₃₇₂ = midpoint of X_i and X_j for these {i,j}: {3, 45488}, {4, 8982}, {637, 20065}, {638, 43133}, {1152, 12969}, {3071, 42259}, {7968, 49227}, {12376, 35827}, {12970, 49251}, {32788, 41946}, {35611, 35642}, {35821, 42267}, {35825, 35879}, {35829, 35881}, {35857, 35883}, {35948, 45421}, {48701, 48715}, {49213, 49267}, {49217, 49269}, {49219, 49271}

• X₃₇₂ = reflection of X_i in X_j for these {i,j}: {3, 43121}, {315, 639}, {371, 32}, {638, 640}, {3071, 7584}, {10960, 42864}, {11824, 3}, {12376, 49269}, {12971, 49257}, {35611, 49227}, {35642, 7968}, {35699, 49215}, {35754, 49209}, {35821, 3071}, {35823, 32788}, {35825, 49213}, {35827, 49217}, {35829, 49219}, {35831, 49221}, {35833, 44648}, {35835, 49223}, {35837, 49225}, {35839, 49231}, {35841, 6}, {35843, 49233}, {35845, 49235}, {35847, 49239}, {35849, 49237}, {35851, 49211}, {35853, 49241}, {35855, 49243}, {35857, 48701}, {35859, 49245}, {35861, 49247}, {35863, 49249}, {35865, 49251}, {35867, 49253}, {35869, 49255}, {35871, 49259}, {35873, 49261}, {35875, 49263}, {35877, 49265}, {35879, 49267}, {35881, 49271}, {35883, 48715}, {39894, 49229}, {41946, 52048}, {42060, 642}, {42267, 42259}, {44485, 575}, {44501, 44481}, {49223, 46689}, {49602, 13936}

• X₃₇₂ = isogonal conjugate of X₄₈₆

• X₃₇₂ = isotomic conjugate of X₃₄₃₉₂

• X₃₇₂ = complement of X₆₃₈

• X₃₇₂ = anticomplement of X₆₄₀

• X₃₇₂ = X₃₇₂ = circumcircle-inverse of X(2460

• X₃₇₂ = Brocard-circle-inverse of X₃₇₁

• X₃₇₂ = 1st-Lemoine-circle-inverse of X₂₄₆₂

• X₃₇₂ = Schoutte-circle-inverse of X₆₃₉₆

• X₃₇₂ = 2nd-Brocard-circle-inverse of X₃₁₀₂

• X₃₇₂ = isogonal conjugate of the anticomplement of X₆₄₂

• X₃₇₂ = isogonal conjugate of the complement of X₄₈₇

• X₃₇₂ = isogonal conjugate of the isotomic conjugate of X₄₉₁

• X₃₇₂ = isotomic conjugate of the polar conjugate of X₅₄₁₂

• X₃₇₂ = isogonal conjugate of the polar conjugate of X₁₅₈₆

• X₃₇₂ = polar conjugate of the isotomic conjugate of X₅₄₀₉

• X₃₇₂ = polar conjugate of the isogonal conjugate of X₂₆₉₂₀

• X₃₇₂ = Thomson-isogonal conjugate of X₁₃₈₃₅

• X₃₇₂ = orthic-isogonal conjugate of X₃₇₁

• X₃₇₂ = psi-transform of X₇₅₉₈

• X₃₇₂ = X₃₁-complementary conjugate of X₁₀₉₆₀

• X₃₇₂ = X_i-Ceva conjugate of X_j for these (i,j): {2, 10960}, {4, 371}, {491, 5409}, {589, 6}, {1586, 5412}, {3069, 8855}, {5413, 8854}, {5419, 6419}, {13439, 26875}, {18820, 8576}

• X₃₇₂ = X_i-cross conjugate of X_j for these (i,j): {1147, 371}, {12240, 4}, {26920, 5409}

• X₃₇₂ = X_i-isoconjugate of X_j for these (i,j): {1, 486}, {19, 11091}, {31, 34392}, {63, 41516}, {75, 8576}, {91, 371}, {92, 6414}, {158, 26922}, {485, 3378}, {1577, 39384}, {1585, 1820}, {1953, 16037}, {8769, 8940}, {19217, 24245}

• X₃₇₂ = X_i-Dao conjugate of X_j for these (i,j): {2, 10960}, {2, 34392}, {3, 486}, {6, 11091}, {69, 5408}, {206, 8576}, {371, 34116}, {372, 638}, {577, 5408}, {1147, 26922}, {1505, 42060}, {3162, 41516}, {5392, 24246}, {5409, 13430}, {6414, 22391}, {10962, 13428}

• X₃₇₂ = cevapoint of X₆ and X₁₀₁₃₃

• X₃₇₂ = crosspoint of X_i and X_j for these (i,j): {4, 13440}, {491, 1586}, {3373, 3374}

• X₃₇₂ = crosssum of X_i and X_j for these (i,j): {3, 10666}, {6, 3156}, {615, 8036}, {3371, 3372}, {6414, 8576}

• X₃₇₂ = crossdifference of every pair of points on line {523, 17432}

• X₃₇₂ = perspector of ABC and the Lucas(-1:1) tangents triangle

• X₃₇₂ = perspector of the tangential triangle and the Lucas(-1) central triangle

• X₃₇₂ = perspector of the Lucas(-1:1) inner tangential triangle and the Lucas central triangle

• X₃₇₂ = perspector of the Lucas(-4:3) central triangle and the circumcevian triangle of X₆

• X₃₇₂ = perspector of the Lucas(-1:1) central triangle and cevian triangle of X₅₈₉

• X₃₇₂ = radical center of the Lucas(-2:1) circles

• X₃₇₂ = perspector of ABC and 2nd Lucas secondary tangents triangle

• X₃₇₂ = perspector of tangential triangle and Lucas(-1) secondary central triangle

• X₃₇₂ = perspector of Lucas(-1) inner triangle and Lucas tangents triangle

• X₃₇₂ = inverse-in-2nd-Brocard-circle of X₃₁₀₂

• X₃₇₂ = inverse-in-Lucas(-1)-radical-circle of X₂₄₅₉

• X₃₇₂ = perspector of ABC and the free vertices of the Kenmotu squares (described at X₃₇₁)

• X₃₇₂ = insimilicenter of circumcircle and Lucas(-1) radical circle

• X₃₇₂ = barycentric product of circumcircle intercepts of inner Vecten circle

• X₃₇₂ = barycentric product X_i*X_j for these {i,j}: {3, 1586}, {4, 5409}, {6, 491}, {24, 11090}, {32, 45806}, {52, 16032}, {56, 13461}, {69, 5412}, {264, 26920}, {317, 6413}, {371, 13439}, {485, 1993}, {486, 1600}, {492, 44192}, {494, 39388}, {571, 34391}, {589, 642}, {1307, 14326}, {1585, 10665}, {5408, 13440}, {5413, 13441}, {6563, 39383}, {7763, 8577}, {9723, 41515}

• X₃₇₂ = barycentric quotient X_i/X_j for these {i,j}: {2, 34392}, {3, 11091}, {6, 486}, {24, 1585}, {25, 41516}, {32, 8576}, {54, 16037}, {184, 6414}, {371, 13428}, {485, 5392}, {491, 76}, {571, 371}, {577, 26922}, {1147, 5408}, {1576, 39384}, {1586, 264}, {1600, 491}, {1993, 492}, {3053, 8940}, {5058, 8036}, {5408, 13430}, {5409, 69}, {5410, 1322}, {5412, 4}, {5413, 13429}, {6413, 68}, {7763, 45805}, {8577, 2165}, {8911, 10666}, {10133, 24245}, {10665, 11090}, {10960, 638}, {11090, 20563}, {13439, 34391}, {13461, 3596}, {16032, 34385}, {26875, 637}, {26920, 3}, {32575, 45473}, {39383, 925}, {41410, 21464}, {41515, 847}, {44077, 5413}, {44192, 485}, {45806, 1502}

• X₃₇₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 1703, 35774}, {1, 5414, 35809}, {1, 6502, 35769}, {1, 35641, 35810}, {1, 35774, 35641}, {1, 45526, 45586}, {1, 45528, 45584}, {1, 45530, 45715}, {1, 45640, 35818}, {1, 45642, 35816}, {1, 45715, 45572}, {2, 485, 10576}, {2, 638, 640}, {2, 1587, 485}, {2, 13935, 5420}, {2, 43133, 638}, {2, 45508, 641}, {3, 6, 371}, {3, 182, 45553}, {3, 371, 6200}, {3, 1152, 6396}, {3, 1160, 1350}, {3, 1161, 12306}, {3, 1351, 9732}, {3, 1505, 35841}, {3, 3102, 40274}, {3, 3103, 45564}, {3, 3311, 1151}, {3, 3312, 6}, {3, 3594, 6420}, {3, 5050, 43119}, {3, 5093, 12313}, {3, 6199, 6449}, {3, 6221, 6409}, {3, 6395, 3312}, {3, 6398, 1152}, {3, 6407, 6451}, {3, 6408, 6450}, {3, 6417, 6221}, {3, 6418, 3311}, {3, 6419, 6453}, {3, 6420, 6419}, {3, 6421, 3102}, {3, 6424, 2460}, {3, 6426, 6454}, {3, 6427, 6425}, {3, 6428, 3592}, {3, 6430, 6485}, {3, 6431, 6480}, {3, 6432, 35771}, {3, 6445, 6496}, {3, 6446, 6456}, {3, 6448, 6426}, {3, 6449, 6411}, {3, 6450, 6410}, {3, 6456, 6412}, {3, 6475, 43415}, {3, 6481, 6487}, {3, 6501, 6199}, {3, 9732, 45499}, {3, 9733, 11825}, {3, 9739, 45498}, {3, 10898, 11514}, {3, 11917, 1161}, {3, 12314, 9733}, {3, 18459, 10898}, {3, 26341, 5085}, {3, 30435, 39648}, {3, 43118, 45552}, {3, 45410, 182}, {3, 45411, 43120}, {3, 45489, 9738}, {3, 45578, 9739}, {3, 45579, 43144}, {4, 486, 6565}, {4, 3069, 486}, {4, 5413, 35765}, {4, 6460, 6560}, {4, 6560, 35820}, {4, 6565, 35787}, {4, 10881, 5413}, {4, 13939, 42561}, {4, 42227, 42231}, {4, 42229, 42233}, {4, 42561, 42268}, {4, 45510, 48466}, {4, 45525, 48734}, {4, 46622, 6414}, {5, 615, 10577}, {5, 3070, 6564}, {5, 13966, 615}, {5, 42216, 3070}, {5, 48772, 45554}, {6, 39, 45512}, {6, 371, 6419}, {6, 1151, 3311}, {6, 1152, 3}, {6, 1160, 35793}, {6, 1161, 35795}, {6, 3053, 6424}, {6, 3311, 35771}, {6, 3312, 6420}, {6, 3592, 6417}, {6, 3594, 3312}, {6, 5013, 6422}, {6, 5062, 45515}, {6, 5085, 19145}, {6, 5210, 8375}, {6, 6199, 6435}, {6, 6396, 6200}, {6, 6398, 6396}, {6, 6409, 3592}, {6, 6410, 1151}, {6, 6411, 6199}, {6, 6412, 6221}, {6, 6420, 35770}, {6, 6421, 45513}, {6, 6424, 45514}, {6, 6425, 6431}, {6, 6426, 1152}, {6, 6430, 6410}, {6, 6431, 6427}, {6, 6432, 6418}, {6, 6434, 6411}, {6, 6438, 6398}, {6, 6452, 6480}, {6, 6468, 6441}, {6, 6469, 6412}, {6, 6471, 6432}, {6, 6489, 6496}, {6, 6566, 45498}, {6, 8406, 43118}, {6, 9732, 45463}, {6, 9733, 3102}, {6, 11477, 9975}, {6, 12963, 5058}, {6, 12968, 32}, {6, 12969, 1505}, {6, 36748, 26868}, {6, 45578, 45565}, {10, 48764, 45546}, {10, 49601, 35788}, {15, 16, 6396}, {15, 61, 371}, {15, 62, 3390}, {15, 3365, 3}, {16, 61, 3365}, {16, 62, 371}, {16, 3390, 3}, {16, 51728, 11481}, {20, 1588, 6561}, {20, 6561, 42266}, {20, 7586, 1588}, {20, 42523, 7586}, {25, 3093, 35764}, {32, 1505, 6}, {32, 12968, 41411}, {32, 40275, 6200}, {32, 45488, 35841}, {35, 3299, 2066}, {36, 3301, 2067}, {39, 182, 371}, {39, 1152, 45498}, {39, 5062, 6}, {39, 6566, 9739}, {39, 9739, 45565}, {40, 18992, 35775}, {40, 35775, 35610}, {50, 568, 371}, {52, 571, 371}, {55, 1124, 35808}, {55, 18995, 1124}, {56, 1335, 35768}, {56, 19037, 1335}, {58, 573, 371}, {61, 62, 6420}, and many more

Notes

X₃₇₂ is the external center of similitude of the circumcircle and the 2nd Lemoine circle (cosine circle) (Peter J. C. Moses, 5/9/03). Also, X₃₇₂ is the external center of similitude of the Gallatly circle (defined just before X₂₀₀₇) and the 1st Lemoine circle (Peter J. C. Moses, 9/10/03).

Let A'B'C' be the Lucas(-1) central triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₇₂. (Randy Hutson, July 23, 2015)