X₄₈₆

INNER VECTEN POINT

X₄₈₆ = complement of X₄₈₇

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

X₄₈₆ (INNER VECTEN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {5, 6}, {2, 371}

• Related Centers: X₅ (NINE-POINT CENTER), X₆ (SYMMEDIAN POINT), X₂ (CENTROID), X₃₇₁ (KENMOTU POINT)

Trilinears: \(\sec(A + \pi /4) : \sec(B + \pi /4) : \sec(C + \pi /4)\)

Barycentrics: \(\sin A \sec(A + \pi /4) : \sin B \sec(B + \pi /4) : \sin C \sec(C + \pi /4)\)

Search Value (6, 9, 13): \(1.854520701879\)

▸ Alternate Coordinates3 forms

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

• Trilinears: \(\sin A - \cos(B - C) : \sin B - \cos(C - A) : \sin C - \cos(A - B)\) (Peter J. C. Moses, 8/22/03)

• Barycentrics: \(1/(S_{A} - S) : :\)

▸ Linear Combinations1 combination

• X₄₈₆ = X₇₉₈₀ + \(2\) X₉₉₀₆, X₇₉₈₀ - \(4\) X₁₂₂₆₈, X₉₉₀₆ + \(2\) X₁₂₂₆₈, \(3\) X₂ - \(4\) X₆₁₁₉, \(3\) X₂ + X₁₂₂₂₁, \(2\) X₂ + X₂₂₄₈₄, X₄₈₇ - \(4\) X₆₁₁₉, \(2\) X₄₈₇ + \(3\) X₂₂₄₈₄, \(2\) X₆₄₂ + X₁₂₂₂₁, \(4\) X₆₄₂ + \(3\) X₂₂₄₈₄, \(4\) X₆₁₁₉ + X₁₂₂₂₁, \(8\) X₆₁₁₉ + \(3\) X₂₂₄₈₄, \(2\) X₆₃₁₅ - \(3\) X₉₇₆₇, \(2\) X₁₂₂₂₁ - \(3\) X₂₂₄₈₄, \(3\) X₃₅₈₂₃ - X₃₅₈₃₃, \(3\) X₃ + X₂₂₈₀₉, X₁₂₁₂₃ + \(2\) X₁₂₆₀₁, \(3\) X₁₂₁₂₃ + \(2\) X₂₂₈₀₉, X₁₂₁₂₃ - \(4\) X₄₉₁₀₃, \(3\) X₁₂₆₀₁ - X₂₂₈₀₉, X₁₂₆₀₁ + \(2\) X₄₉₁₀₃, \(2\) X₁₃₉₃₄ + X₂₂₆₁₇, X₂₂₈₀₉ + \(6\) X₄₉₁₀₃, X₃₂₄₉₂ + \(2\) X₃₂₄₉₄, X₄ + \(2\) X₄₈₇₃₄, \(2\) X₆₂₅₁ + X₁₂₂₅₆, \(2\) X₁₃₉₃₃ + X₃₂₄₉₈, X₃₅₈₂₀ - \(3\) X₃₅₈₃₀, \(4\) X₅ + X₆₂₈₀, \(4\) X₅ - X₆₂₈₁, \(2\) X₅ + X₄₉₃₁₇, X₆₂₈₀ + \(2\) X₆₂₉₀, X₆₂₈₁ + \(2\) X₄₉₃₁₇, \(2\) X₇₅₈₃ - \(3\) X₄₉₂₂₀, \(2\) X₁₃₈₈₁ + X₂₂₅₉₁, \(4\) X₁₃₈₈₁ - X₂₂₅₉₂, \(3\) X₁₉₁₀₄ - \(4\) X₁₉₁₁₆, \(5\) X₁₉₁₀₅ - \(4\) X₁₉₁₁₇, \(2\) X₂₂₅₉₁ + X₂₂₅₉₂, X₄₉₅₅₁ + \(2\) X₄₉₆₂₃, \(2\) X₁₃₉₂₉ + X₂₂₆₀₁, \(2\) X₁₃₉₂₈ + X₂₂₆₀₃, X₂₂₅₉₉ + \(2\) X₂₂₉₂₆, X₂₂₅₉₇ + \(2\) X₂₂₈₈₁, \(2\) X₁₁₃₂ + X₄₅₀₇₇, \(2\) X₁₁₅₂ + \(3\) X₁₃₂₈, X₁₁₅₂ - \(3\) X₁₃₈₄₇, and many others

X₄₈₆ lies on the Kiepert circumhyperbola, the cubics K006, K070a, K120, K122, K168, K250, K336, K424b, K690, K906, K1192, K1195, K1197, K1202, K1203, K1225, the curve Q115.

▸ Incident Central Lines654 lines

{1, 7980}, {2, 371}, {3, 615}, {4, 372}, {5, 6}, {8, 35642}, {10, 1685}, {11, 1335}, {12, 1124}, {13, 2044}, {14, 2043}, {15, 2042}, {16, 2041}, {17, 2046}, {18, 2045}, {20, 1132}, {22, 35777}, {25, 9921}, {30, 1152}, {32, 9986}, {39, 22719}, {40, 13947}, {46, 3378}, {55, 12343}, {56, 13955}, {61, 3366}, {62, 3391}, {64, 22839}, {69, 640}, {76, 492}, {83, 7389}, {95, 19199}, {96, 8911}, {98, 6813}, {100, 35773}, {110, 49223}, {113, 46689}, {115, 1505}, {119, 19047}, {140, 1151}, {141, 11314}, {145, 35811}, {146, 35827}, {147, 35825}, {148, 35879}, {149, 35883}, {153, 35857}, {182, 12229}, {194, 35867}, {226, 482}, {230, 6424}, {235, 3093}, {262, 3103}, {265, 49269}, {275, 1585}, {355, 7968}, {376, 14226}, {381, 1327}, {382, 6398}, {387, 36690}, {388, 35769}, {394, 1592}, {395, 18587}, {396, 18586}, {402, 12799}, {427, 3092}, {442, 31473}, {459, 3536}, {488, 2996}, {489, 2460}, {490, 671}, {491, 7752}, {493, 13002}, {494, 11210}, {495, 3297}, {496, 3298}, {497, 35803}, {498, 2066}, {499, 2067}, {511, 12237}, {515, 13971}, {516, 13975}, {517, 13973}, {524, 9975}, {542, 13782}, {546, 3594}, {547, 6431}, {548, 6412}, {549, 6409}, {550, 6410}, {590, 1656}, {591, 42024}, {607, 16027}, {608, 16033}, {616, 35754}, {617, 35851}, {627, 35847}, {628, 35849}, {631, 3317}, {632, 6425}, {638, 32488}, {641, 5490}, {847, 41515}, {925, 13521}, {944, 13959}, {946, 13936}, {952, 44636}, {962, 35611}, {1038, 19473}, {1040, 9631}, {1131, 1587}, {1270, 35793}, {1271, 35795}, {1321, 5962}, {1329, 1378}, {1346, 2465}, {1347, 2466}, {1348, 1669}, {1349, 1668}, {1368, 1579}, {1377, 2886}, {1478, 6502}, {1479, 5414}, {1482, 49233}, {1503, 13972}, {1504, 1506}, {1578, 6823}, {1586, 2052}, {1591, 10601}, {1657, 6450}, {1659, 8978}, {1666, 2039}, {1667, 2040}, {1670, 2010}, {1671, 2009}, {1676, 1687}, {1677, 1688}, {1686, 2048}, {1689, 5403}, {1690, 5404}, {1698, 1702}, {1699, 1703}, {1737, 16232}, {1916, 33340}, {1993, 15233}, {2047, 13332}, {2351, 3156}, {2353, 45428}, {2362, 12047}, {2459, 14234}, {2549, 48772}, {2560, 2566}, {2561, 2567}, {2777, 13969}, {2781, 32304}, {2782, 32471}, {2794, 13967}, {2800, 13976}, {2829, 13977}, {2888, 12971}, {2896, 35783}, {2975, 35785}, {3035, 9679}, {3054, 8375}, {3068, 3090}, {3085, 35808}, {3086, 35768}, {3089, 35764}, {3095, 49253}, {3146, 6454}, {3299, 7951}, {3301, 7741}, {3371, 3373}, {3372, 3374}, {3385, 3388}, {3386, 3387}, {3424, 5870}, {3434, 35797}, {3436, 35799}, {3448, 12376}, {3522, 43408}, {3523, 3591}, {3524, 42638}, {3525, 6453}, {3526, 6221}, {3528, 51911}, {3529, 42637}, {3530, 6411}, {3533, 43412}, {3534, 6456}, {3541, 11473}, {3542, 5412}, {3543, 43407}, {3545, 7581}, {3547, 11513}, {3549, 10897}, {3575, 13937}, {3589, 11313}, {3590, 5056}, {3592, 3628}, {3614, 19028}, {3616, 35763}, {3618, 39876}, {3624, 9583}, {3627, 6426}, {3634, 13912}, {3648, 35855}, {3652, 49243}, {3815, 6422}, {3817, 49547}, {3830, 41946}, {3832, 23249}, {3839, 23253}, {3843, 6395}, {3845, 43562}, {3850, 6432}, {3851, 6418}, {3853, 6438}, {3854, 43432}, {3855, 23267}, {3858, 6471}, {3859, 6442}, {3917, 21653}, {4240, 35791}, {4999, 9678}, {5012, 9677}, {5020, 8276}, {5050, 49228}, {5054, 6449}, {5055, 6417}, {5058, 7746}, {5059, 6485}, {5062, 5475}, {5066, 43340}, {5067, 32785}, {5070, 6199}, {5071, 13886}, {5072, 6428}, {5073, 6408}, {5076, 6448}, {5079, 6427}, {5204, 9647}, {5217, 9660}, {5254, 6421}, {5286, 45513}, {5318, 42187}, {5321, 42188}, {5392, 11090}, {5408, 6504}, {5411, 7507}, {5419, 13439}, {5422, 15234}, {5462, 12239}, {5466, 14334}, {5476, 44502}, {5480, 36656}, {5552, 45643}, {5587, 18992}, {5590, 7375}, {5597, 12484}, {5598, 12485}, {5601, 35781}, {5602, 35779}, {5603, 19065}, {5613, 49211}, {5617, 49209}, {5657, 35610}, {5663, 13979}, {5691, 13942}, {5705, 31438}, {5790, 49232}, {5818, 19066}, {5840, 13991}, {5871, 13949}, {5878, 49251}, {5886, 7969}, {5901, 44635}, {5921, 39894}, {5972, 10819}, {6033, 49213}, {6146, 19356}, {6193, 35837}, {6194, 35839}, {6202, 7374}, {6213, 7348}, {6223, 35845}, {6224, 35853}, {6225, 35865}, {6250, 22646}, {6256, 45653}, {6259, 49235}, {6265, 49241}, {6284, 13958}, {6287, 49255}, {6288, 49257}, {6321, 49267}, {6399, 13748}, {6407, 15694}, {6423, 7745}, {6429, 11539}, {6436, 31414}, {6437, 16239}, {6445, 43882}, {6446, 17800}, {6452, 15696}, {6455, 15720}, {6462, 35807}, {6469, 43336}, {6470, 15699}, {6473, 43796}, {6476, 9692}, {6479, 50690}, {6480, 43314}, {6481, 33703}, {6484, 15702}, {6487, 11001}, {6496, 15693}, {6500, 45384}, {6501, 19709}, {6522, 43790}, {6568, 35947}, {6569, 19055}, {6639, 18457}, {6642, 9682}, {6643, 11514}, {6667, 13913}, {6721, 8997}, {6722, 8980}, {6723, 8994}, {6776, 14244}, {6809, 8954}, {6810, 40448}, {6812, 45300}, {6814, 13380}, {7173, 19030}, {7354, 18966}, {7376, 18841}, {7386, 12320}, {7392, 8854}, {7395, 19005}, {7401, 10963}, {7484, 12169}, {7486, 8972}, {7505, 10880}, {7529, 44598}, {7603, 31481}, {7612, 9757}, {7618, 49786}, {7694, 13749}, {7697, 49231}, {7728, 49217}, {7736, 45512}, {7737, 12968}, {7749, 9675}, {7787, 35767}, {7795, 45472}, {7988, 19004}, {7989, 19003}, {7998, 12274}, {7999, 12285}, {8200, 44601}, {8207, 44603}, {8220, 44628}, {8221, 44630}, {8227, 18991}, {8591, 35699}, {8681, 9824}, {8703, 43212}, {8724, 49215}, {8781, 13873}, {8901, 16034}, {8909, 9820}, {8944, 8948}, {8945, 8949}, {8953, 30324}, {8968, 11427}, {8970, 14826}, {8995, 32396}, {8998, 12900}, {9612, 51842}, {9616, 31423}, {9632, 37696}, {9646, 19038}, {9661, 18996}, {9691, 43523}, {9732, 49029}, {9739, 12124}, {9833, 10534}, {9834, 13944}, {9835, 13945}, {9838, 13956}, {9839, 13957}, {9873, 13946}, {9874, 35863}, {9956, 13911}, {9970, 49265}, {9974, 45861}, {9996, 44605}, {10008, 42009}, {10104, 44586}, {10124, 42568}, {10138, 35400}, {10175, 13883}, {10201, 11265}, {10303, 43512}, {10319, 12662}, {10356, 19011}, {10358, 18993}, {10527, 45641}, {10528, 35817}, {10529, 35819}, {10531, 45642}, {10532, 45640}, {10590, 31408}, {10591, 35802}, {10594, 35776}, {10595, 35810}, {10596, 35816}, {10597, 35818}, {10598, 35796}, {10599, 35798}, {10653, 18585}, {10654, 15765}, {10738, 48715}, {10742, 48701}, {10749, 49271}, {10796, 44587}, {10820, 13990}, {10895, 18995}, {10896, 19037}, {10898, 18531}, {10942, 44644}, {10943, 44646}, {11061, 35877}, {11178, 44473}, {11231, 31439}, {11266, 18569}, {11291, 12322}, {11294, 45508}, {11315, 15293}, {11316, 45872}, {11418, 37444}, {11463, 25739}, {11484, 13889}, {11488, 42174}, {11489, 42172}, {11499, 44591}, {11500, 13940}, {11511, 12597}, {11515, 12980}, {11516, 12981}, {11548, 34515}, {11835, 40556}, {12103, 42569}, {12110, 13938}, {12113, 13948}, {12114, 13952}, {12115, 13964}, {12116, 13965}, {12232, 44665}, {12240, 13754}, {12323, 32421}, {12383, 35835}, {12384, 35829}, {12699, 49227}, {12849, 35871}, {12856, 49249}, {12918, 49219}, {12919, 49259}, {12970, 14216}, {13219, 35881}, {13333, 13478}, {13390, 40149}, {13403, 43864}, {13440, 51833}, {13678, 35873}, {13692, 49261}, {13712, 41490}, {13763, 14233}, {13783, 37809}, {13789, 35305}, {13794, 13988}, {13798, 35875}, {13812, 49263}, {13835, 15294}, {13880, 14229}, {13915, 15088}, {13925, 35018}, {13980, 15311}, {13981, 41023}, {13982, 41022}, {13984, 29012}, {13986, 18400}, {14061, 19056}, {14227, 14243}, {14238, 45407}, {14245, 45462}, {14269, 43503}, {14568, 45421}, {14639, 19108}, {14643, 19051}, {14644, 19110}, {14651, 35824}, {14813, 42152}, {14814, 42149}, {14853, 35840}, {14912, 39893}, {15022, 42605}, {15059, 19060}, {15061, 49216}, {15235, 17811}, {15236, 17825}, {15561, 49266}, {15682, 42414}, {15684, 43209}, {15687, 52048}, {15688, 43210}, {15691, 51849}, {15692, 43257}, {15701, 42417}, {15709, 43506}, {15884, 38426}, {16267, 36449}, {16268, 36467}, {16626, 49239}, {16627, 49237}, {16772, 42195}, {16773, 42196}, {17538, 42413}, {18404, 18459}, {18440, 49229}, {18553, 44481}, {18927, 45608}, {18940, 44518}, {19049, 26470}, {19057, 23234}, {19073, 36765}, {19082, 31272}, {19088, 40686}, {19126, 19143}, {19130, 45542}, {19422, 19490}, {19710, 43435}, {21616, 30557}, {21736, 45102}, {21737, 45552}, {22235, 42234}, {22236, 35738}, {22237, 42232}, {22645, 48779}, {22647, 35861}, {22758, 44607}, {22877, 44667}, {22922, 44666}, {22955, 49247}, {23515, 46688}, {24246, 34853}, {24843, 29243}, {25555, 44656}, {25639, 31484}, {26362, 49025}, {26363, 31453}, {26386, 44583}, {26394, 45360}, {26410, 44585}, {26418, 45358}, {26446, 49226}, {26466, 45596}, {26467, 45598}, {26468, 44595}, {26469, 44597}, {26494, 45600}, {26503, 45602}, {26521, 35944}, {26618, 33457}, {26619, 33364}, {30399, 30427}, {30428, 32734}, {31403, 31404}, {31413, 31418}, {31427, 31428}, {31432, 31434}, {31437, 31441}, {31459, 31460}, {31464, 31466}, {31471, 31476}, {31474, 31479}, {31482, 31488}, {31485, 31493}, {31981, 49351}, {32354, 35859}, {32379, 49245}, {32490, 45473}, {32495, 45862}, {32502, 32508}, {32807, 39387}, {33347, 33349}, {33355, 33437}, {33356, 33439}, {33359, 33441}, {33361, 33443}, {33367, 33369}, {33371, 33373}, {34507, 44501}, {36445, 36447}, {36463, 36465}, {36473, 36552}, {36474, 36550}, {36491, 36526}, {36495, 36584}, {36496, 36582}, {36657, 36990}, {36709, 45441}, {37640, 51855}, {37641, 51854}, {38224, 49212}, {38752, 48714}, {40330, 45576}, {41949, 41963}, {41962, 49133}, {41966, 43786}, {42085, 42199}, {42086, 42201}, {42125, 42189}, {42128, 42190}, {42159, 42280}, {42162, 42281}, {42163, 42248}, {42166, 42249}, {42167, 42684}, {42169, 42685}, {42183, 42813}, {42185, 42814}, {42191, 42817}, {42192, 42818}, {42522, 46936}, {42992, 50245}, {43256, 50687}, {43414, 43793}, {43415, 49134}, {43449, 45434}, {43508, 43524}, {43522, 46333}, {43816, 43868}, {43884, 50688}, {44392, 45488}, {44596, 45514}, {45106, 45440}, {45366, 48454}, {45367, 48455}, {45499, 49039}, {45606, 48468}, {45651, 48482}, {48476, 49026}

▸ Functional Relations42 relations

• X₄₈₆ = midpoint of X_i and X_j for these {i,j}: {1, 9906}, {3, 12601}, {4, 12256}, {20, 12296}, {485, 22591}, {487, 12221}, {488, 2996}, {1152, 23261}, {3071, 32494}, {6251, 48734}, {6280, 6281}, {6290, 49317}, {13926, 50719}, {22615, 42261}, {26618, 33457}, {49155, 49156}

• X₄₈₆ = reflection of X_i in X_j for these {i,j}: {1, 12268}, {3, 49103}, {4, 6251}, {485, 13881}, {487, 642}, {642, 6119}, {1152, 13966}, {6280, 49317}, {6281, 6290}, {6290, 5}, {6300, 33447}, {6301, 33445}, {6337, 641}, {7980, 1}, {10820, 13990}, {12123, 3}, {12256, 48734}, {12787, 10}, {12799, 402}, {13132, 10067}, {13133, 10083}, {13966, 13993}, {13975, 49619}, {19146, 13972}, {22501, 50719}, {22592, 485}, {22598, 33451}, {22600, 33449}, {22615, 23261}, {32492, 3071}, {36655, 45860}, {42060, 640}, {42261, 1152}, {44648, 7584}, {48659, 22596}, {49791, 615}

• X₄₈₆ = isogonal conjugate of X₃₇₂

• X₄₈₆ = isotomic conjugate of X₄₉₁

• X₄₈₆ = complement of X₄₈₇

• X₄₈₆ = anticomplement of X₆₄₂

• X₄₈₆ = polar conjugate of X₁₅₈₆

• X₄₈₆ = complement of the isogonal conjugate of X₈₉₄₆

• X₄₈₆ = complement of the isotomic conjugate of X₂₄₂₄₃

• X₄₈₆ = isogonal conjugate of the anticomplement of X₆₄₀

• X₄₈₆ = isogonal conjugate of the complement of X₆₃₈

• X₄₈₆ = isotomic conjugate of the anticomplement of X₆₁₅

• X₄₈₆ = isotomic conjugate of the isogonal conjugate of X₈₅₇₆

• X₄₈₆ = isogonal conjugate of the isotomic conjugate of X₃₄₃₉₂

• X₄₈₆ = isotomic conjugate of the polar conjugate of X₄₁₅₁₆

• X₄₈₆ = polar conjugate of the isotomic conjugate of X₁₁₀₉₁

• X₄₈₆ = polar conjugate of the isogonal conjugate of X₆₄₁₄

• X₄₈₆ = psi-transform of X₁₃₅₂₀

• X₄₈₆ = X_i-complementary conjugate of X_j for these (i,j): {31, 24245}, {494, 18589}, {1973, 13934}, {8946, 10}, {19217, 141}, {24243, 2887}, {26461, 1214}

• X₄₈₆ = X_i-Ceva conjugate of X_j for these (i,j): {2, 24245}, {847, 485}, {16037, 6414}, {34392, 11091}, {41516, 8940}

• X₄₈₆ = X_i-cross conjugate of X_j for these (i,j): {3, 485}, {615, 2}, {3071, 4}, {5420, 10195}, {6414, 11091}, {6561, 1327}, {8576, 41516}, {8911, 1585}, {13785, 1328}, {13951, 10194}, {26950, 11090}, {32494, 5491}, {32575, 491}, {42258, 1131}, {42260, 43570}, {43880, 3317}, {45385, 43569}

• X₄₈₆ = X_i-isoconjugate of X_j for these (i,j): {1, 372}, {19, 5409}, {31, 491}, {47, 485}, {48, 1586}, {63, 5412}, {92, 26920}, {371, 3377}, {560, 45806}, {604, 13461}, {1748, 6413}, {2180, 16032}, {8577, 44179}

• X₄₈₆ = X_i-Dao conjugate of X_j for these (i,j): {2, 491}, {2, 24245}, {3, 372}, {6, 5409}, {485, 34853}, {486, 487}, {1249, 1586}, {1600, 10960}, {1993, 10962}, {3161, 13461}, {3162, 5412}, {5409, 9723}, {6374, 45806}, {8576, 44199}, {8577, 37864}, {13439, 24246}, {13934, 42060}, {22391, 26920}, {33365, 39388}

• X₄₈₆ = cevapoint of X_i and X_j for these (i,j): {3, 10666}, {6, 3156}, {615, 8036}, {3371, 3372}, {6414, 8576}

• X₄₈₆ = crosspoint of X₂ and X₂₄₂₄₃

• X₄₈₆ = crosssum of X₆ and X₁₀₁₃₃

• X₄₈₆ = trilinear pole of line {523, 17432}

• X₄₈₆ = external center of similitude of nine-point circle and 2nd Lemoine circle

• X₄₈₆ = pole wrt polar circle of trilinear polar of X₁₅₈₆

• X₄₈₆ = X₄₈-isoconjugate (polar conjugate) of X₁₅₈₆

• X₄₈₆ = Kosnita(X₄₈₆,X₄₈₆) point

• X₄₈₆ = trilinear product of vertices of inner Vecten triangle

• X₄₈₆ = X₄ of inner Vecten triangle

• X₄₈₆ = perspector of ABC and outer-squares triangle

• X₄₈₆ = perspector of 4th tri-squares triangle and 4th tri-squares central triangle

• X₄₈₆ = 4th-tri-squares-isogonal conjugate of X₃₂₄₉₄

• X₄₈₆ = 3rd-anti-tri-squares-isogonal conjugate of X₃₂₄₉₂

• X₄₈₆ = {X₅,X₆}-harmonic conjugate of X₄₈₅

• X₄₈₆ = inner-Vecten-isogonal conjugate of X₆₄₂

• X₄₈₆ = barycentric product X_i*X_j for these {i,j}: {4, 11091}, {5, 16037}, {6, 34392}, {68, 1585}, {69, 41516}, {76, 8576}, {264, 6414}, {371, 5392}, {485, 13428}, {492, 2165}, {847, 5408}, {850, 39384}, {2052, 26922}, {2996, 8940}, {5413, 20563}, {11090, 13429}, {13430, 41515}, {18820, 45473}, {21464, 42024}, {24243, 24245}, {34391, 44193}

• X₄₈₆ = barycentric quotient X_i/X_j for these {i,j}: {2, 491}, {3, 5409}, {4, 1586}, {6, 372}, {8, 13461}, {25, 5412}, {68, 11090}, {76, 45806}, {96, 16032}, {184, 26920}, {371, 1993}, {372, 1600}, {485, 13439}, {492, 7763}, {615, 642}, {1322, 3536}, {1505, 32575}, {1585, 317}, {2165, 485}, {2351, 6413}, {3069, 39388}, {3155, 26875}, {3156, 10960}, {5392, 34391}, {5408, 9723}, {5413, 24}, {6413, 10665}, {6414, 3}, {8036, 615}, {8576, 6}, {8577, 44192}, {8911, 1147}, {8940, 193}, {10666, 5408}, {10881, 15208}, {11090, 13441}, {11091, 69}, {11474, 15217}, {13428, 492}, {13429, 1585}, {14593, 41515}, {16037, 95}, {21464, 45421}, {24245, 487}, {26922, 394}, {32734, 39383}, {34392, 76}, {35765, 15187}, {39384, 110}, {41515, 13440}, {41516, 4}, {44193, 371}, {45473, 42060}

• X₄₈₆ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 49155, 13133}, {1, 49156, 13132}, {2, 371, 5418}, {2, 487, 642}, {2, 1588, 371}, {2, 12221, 487}, {2, 13428, 11091}, {2, 43134, 45509}, {3, 615, 5420}, {3, 3071, 6561}, {3, 6561, 42260}, {3, 13785, 3071}, {3, 13934, 49791}, {3, 13943, 8277}, {3, 13951, 615}, {3, 45385, 13951}, {4, 372, 6560}, {4, 3069, 372}, {4, 6460, 35820}, {4, 6560, 22644}, {4, 6565, 42268}, {4, 13429, 41516}, {4, 13939, 3069}, {4, 42561, 6565}, {5, 6, 485}, {5, 485, 42277}, {5, 1352, 10514}, {5, 5874, 6215}, {5, 6214, 10516}, {5, 6289, 10515}, {5, 7583, 42265}, {5, 7584, 6}, {5, 18762, 42262}, {5, 19116, 7583}, {5, 19117, 18538}, {5, 42262, 42274}, {5, 49317, 6281}, {5, 49355, 1352}, {6, 13881, 49220}, {6, 18762, 42274}, {6, 31411, 19103}, {6, 42262, 5}, {6, 42265, 7583}, {6, 42274, 42277}, {6, 44648, 19104}, {6, 49220, 19105}, {11, 19027, 1335}, {12, 19029, 1124}, {15, 3392, 2042}, {16, 3367, 2041}, {20, 1132, 23259}, {20, 13935, 6396}, {20, 13941, 13935}, {20, 23259, 35821}, {20, 35821, 42275}, {68, 2165, 485}, {69, 5491, 42060}, {140, 42215, 1151}, {155, 9722, 485}, {262, 19089, 3103}, {371, 10577, 2}, {371, 35823, 1588}, {372, 6565, 4}, {372, 13939, 43431}, {372, 35787, 35820}, {372, 35820, 6460}, {372, 42268, 22644}, {372, 42561, 42268}, {381, 3070, 42269}, {381, 3312, 3070}, {381, 48659, 22596}, {382, 6398, 42259}, {382, 13961, 6398}, {382, 42259, 42276}, {485, 42274, 5}, {546, 42216, 23251}, {550, 35256, 6410}, {590, 42583, 1656}, {615, 3071, 3}, {615, 13785, 6561}, {615, 32494, 13934}, {615, 43880, 13951}, {631, 3317, 32786}, {631, 6459, 6200}, {631, 23273, 6459}, {642, 6119, 2}, {944, 13959, 35762}, {946, 13936, 35774}, {1132, 13935, 35821}, {1132, 13941, 20}, {1132, 35813, 42275}, {1151, 8252, 140}, {1152, 13847, 13966}, {1322, 41516, 4}, {1328, 22615, 23261}, {1328, 42261, 22615}, {1352, 3767, 485}, {1352, 49355, 6278}, {1478, 13962, 6502}, {1479, 13963, 5414}, {1504, 1506, 31463}, {1587, 3091, 6564}, {1587, 7586, 6420}, {1588, 10577, 5418}, {1588, 12221, 35833}, {1656, 3311, 590}, {1656, 18510, 3311}, and many others

Notes

X₄₈₆ is a perspector of triangles associated with squares that circumscribe ABC. For details and references, see X₄₈₅. (Floor van Lamoen, 4/26/98)