X₄₈₅

VECTEN POINT

X₄₈₅ = complement of X₄₈₈

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

X₄₈₅ (VECTEN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

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): \(0.817644849614\)

▸ Alternate Coordinates3 forms

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

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

• Barycentrics: \(1/(S_{A} + S) : :\)

▸ Linear Combinations1 combination

• X₄₈₅ = X₇₉₈₁ + \(2\) X₉₉₀₇, X₇₉₈₁ - \(4\) X₁₂₂₆₉, X₉₉₀₇ + \(2\) X₁₂₂₆₉, \(3\) X₂ - \(4\) X₆₁₁₈, \(3\) X₂ + X₁₂₂₂₂, \(2\) X₂ + X₂₂₄₈₅, X₄₈₈ - \(4\) X₆₁₁₈, \(2\) X₄₈₈ + \(3\) X₂₂₄₈₅, \(2\) X₆₄₁ + X₁₂₂₂₂, \(4\) X₆₄₁ + \(3\) X₂₂₄₈₅, \(4\) X₆₁₁₈ + X₁₂₂₂₂, \(8\) X₆₁₁₈ + \(3\) X₂₂₄₈₅, \(2\) X₆₃₁₁ - \(3\) X₉₇₆₈, \(2\) X₁₂₂₂₂ - \(3\) X₂₂₄₈₅, \(3\) X₃₅₈₂₂ - X₃₅₈₃₂, \(3\) X₃ + X₂₂₈₁₀, X₁₂₁₂₄ + \(2\) X₁₂₆₀₂, \(3\) X₁₂₁₂₄ + \(2\) X₂₂₈₁₀, X₁₂₁₂₄ - \(4\) X₄₉₁₀₄, \(3\) X₁₂₆₀₂ - X₂₂₈₁₀, X₁₂₆₀₂ + \(2\) X₄₉₁₀₄, \(2\) X₁₃₈₈₂ + X₂₂₆₄₆, X₂₂₈₁₀ + \(6\) X₄₉₁₀₄, X₃₂₄₉₅ + \(2\) X₃₂₄₉₇, X₄ + \(2\) X₄₈₇₃₅, \(2\) X₆₂₅₀ + X₁₂₂₅₇, \(2\) X₁₃₈₇₉ + X₃₂₄₉₉, X₃₅₈₂₁ - \(3\) X₃₅₈₃₁, \(4\) X₅ - X₆₂₇₈, \(4\) X₅ + X₆₂₇₉, \(2\) X₅ + X₄₉₃₁₈, X₆₂₇₈ + \(2\) X₄₉₃₁₈, X₆₂₇₉ + \(2\) X₆₂₈₉, \(2\) X₇₅₈₄ - \(3\) X₄₉₂₂₁, \(4\) X₁₃₈₈₁ - X₂₂₅₉₁, \(2\) X₁₃₈₈₁ + X₂₂₅₉₂, \(5\) X₁₉₁₀₂ - \(4\) X₁₉₁₁₆, \(3\) X₁₉₁₀₃ - \(4\) X₁₉₁₁₇, X₂₂₅₉₁ + \(2\) X₂₂₅₉₂, X₄₉₅₅₂ + \(2\) X₄₉₆₂₂, \(2\) X₁₃₈₇₆ + X₂₂₆₃₀, \(2\) X₁₃₈₇₅ + X₂₂₆₃₂, X₂₂₆₂₈ + \(2\) X₂₂₉₂₅, X₂₂₆₂₆ + \(2\) X₂₂₈₈₀, \(2\) X₁₁₃₁ + X₄₅₀₇₆, \(2\) X₁₁₅₁ + \(3\) X₁₃₂₇, X₁₁₅₁ - \(3\) X₁₃₈₄₆, and many others

X₄₈₅ lies on the Kiepert circumhyperbola, the cubics K006, K070b, K120, K122, K168, K250, K336, K424a, K690, K906, K1192, K1195, K1197, K1202, K1203, K1225, the curve Q115, and thewe lines: {1, 7981}, {2, 372}, {3, 590}, {4, 371}, {5, 6}, {8, 35641}, {10, 1686}, {11, 1124}, {12, 1335}, {13, 2043}, {14, 2044}, {15, 2041}, {16, 2042}, {17, 2045}, {18, 2046}, {20, 1131}, {22, 35776}, {25, 9922}, {30, 1151}, {32, 9987}, {39, 22717}, {40, 13893}, {46, 3377}, {55, 9646}, {56, 9661}, {61, 3367}, {62, 3392}, {64, 22838}, {69, 639}, {76, 491}, {83, 7388}, {95, 19200}, {96, 26920}, {98, 6811}, {100, 35772}, {110, 49222}, {113, 46688}, {115, 1504}, {119, 19048}, {140, 1152}, {141, 11313}, {145, 35810}, {146, 35826}, {147, 35824}, {148, 35878}, {149, 35882}, {153, 35856}, {182, 12230}, {194, 35866}, {226, 481}, {230, 6423}, {235, 3092}, {262, 3102}, {265, 49268}, {275, 1586}, {355, 7969}, {376, 14241}, {381, 1328}, {382, 6221}, {387, 36691}, {388, 35768}, {394, 1591}, {395, 18586}, {396, 18587}, {402, 12800}, {427, 3093}, {459, 3535}, {487, 2996}, {489, 671}, {490, 2459}, {492, 7752}, {493, 11209}, {494, 13005}, {495, 3298}, {496, 3297}, {497, 35802}, {498, 5414}, {499, 6502}, {511, 8992}, {515, 8983}, {516, 13912}, {517, 13911}, {524, 9974}, {542, 13662}, {546, 3592}, {547, 6432}, {548, 6411}, {549, 6410}, {550, 6409}, {607, 16033}, {608, 16027}, {615, 1656}, {616, 35753}, {617, 35850}, {627, 35848}, {628, 35846}, {631, 3316}, {632, 6426}, {637, 32489}, {642, 5491}, {847, 41516}, {925, 13520}, {944, 13902}, {946, 13883}, {952, 44635}, {962, 35610}, {1038, 19474}, {1040, 12911}, {1132, 1588}, {1270, 35794}, {1271, 35792}, {1322, 5962}, {1329, 1377}, {1346, 2466}, {1347, 2465}, {1348, 1668}, {1349, 1669}, {1368, 1578}, {1378, 2886}, {1478, 2067}, {1479, 2066}, {1482, 49232}, {1503, 13910}, {1505, 1506}, {1579, 6823}, {1585, 2052}, {1592, 10601}, {1614, 9677}, {1657, 6449}, {1659, 8953}, {1666, 2040}, {1667, 2039}, {1670, 2009}, {1671, 2010}, {1676, 1688}, {1677, 1687}, {1685, 2051}, {1689, 5404}, {1690, 5403}, {1698, 1703}, {1699, 1702}, {1737, 2362}, {1916, 33341}, {1991, 42023}, {1993, 15234}, {2048, 13332}, {2351, 3155}, {2353, 45429}, {2460, 14238}, {2549, 48773}, {2560, 2567}, {2561, 2566}, {2777, 8994}, {2781, 32303}, {2782, 32470}, {2794, 8980}, {2800, 8988}, {2829, 13913}, {2888, 12965}, {2896, 35782}, {2975, 35784}, {3054, 8376}, {3069, 3090}, {3085, 35809}, {3086, 31408}, {3089, 35765}, {3095, 49252}, {3146, 6453}, {3299, 7741}, {3301, 7951}, {3371, 3374}, {3372, 3373}, {3385, 3387}, {3386, 3388}, {3424, 5871}, {3434, 35796}, {3436, 35798}, {3448, 12375}, {3522, 43407}, {3523, 3590}, {3524, 42637}, {3525, 6454}, {3526, 6398}, {3528, 51910}, {3529, 42638}, {3530, 6412}, {3533, 43411}, {3534, 6455}, {3541, 11474}, {3542, 5413}, {3543, 43408}, {3545, 7582}, {3547, 11514}, {3549, 10898}, {3575, 13884}, {3589, 11314}, {3591, 5056}, {3594, 3628}, {3614, 19027}, {3616, 35762}, {3618, 39875}, {3627, 6425}, {3634, 13975}, {3648, 35854}, {3652, 49242}, {3815, 6421}, {3817, 49548}, {3830, 41945}, {3832, 23259}, {3839, 23263}, {3843, 6199}, {3845, 43563}, {3850, 6431}, {3851, 6417}, {3853, 6437}, {3854, 43433}, {3855, 23273}, {3858, 6470}, {3859, 6441}, {3917, 21654}, {4187, 31473}, {4240, 35790}, {5020, 8277}, {5050, 49229}, {5054, 6450}, {5055, 6418}, {5058, 5475}, {5059, 6484}, {5062, 7746}, {5066, 43341}, {5067, 32786}, {5070, 6395}, {5071, 13939}, {5072, 6427}, {5073, 6407}, {5076, 6447}, {5079, 6428}, {5217, 31499}, {5254, 6422}, {5286, 31403}, {5318, 42189}, {5321, 42190}, {5392, 11091}, {5409, 6504}, {5410, 7507}, {5417, 13428}, {5422, 15233}, {5462, 12240}, {5466, 14333}, {5476, 44501}, {5480, 36655}, {5552, 45642}, {5587, 18991}, {5591, 7376}, {5597, 12486}, {5598, 12487}, {5601, 35778}, {5602, 35780}, {5603, 19066}, {5613, 49210}, {5617, 49208}, {5657, 35611}, {5663, 13915}, {5691, 9583}, {5790, 49233}, {5818, 19065}, {5840, 13922}, {5870, 8975}, {5878, 49250}, {5886, 7968}, {5901, 44636}, {5921, 39893}, {5972, 10820}, {6033, 49212}, {6146, 19355}, {6193, 35836}, {6194, 35838}, {6201, 7000}, {6212, 7347}, {6222, 13749}, {6223, 35844}, {6224, 35852}, {6225, 35864}, {6251, 22617}, {6256, 45652}, {6259, 49234}, {6265, 49240}, {6284, 13901}, {6287, 49254}, {6288, 49256}, {6321, 49266}, {6408, 15694}, {6424, 7745}, {6430, 11539}, {6438, 16239}, {6445, 17800}, {6446, 43881}, {6451, 15696}, {6456, 15720}, {6463, 35806}, {6468, 43337}, {6471, 15699}, {6472, 43795}, {6476, 9693}, {6478, 50690}, {6480, 33703}, {6481, 43315}, {6485, 15702}, {6486, 11001}, {6497, 15693}, {6500, 19709}, {6501, 45385}, {6519, 43789}, {6568, 19056}, {6569, 35946}, {6639, 18459}, {6643, 11513}, {6667, 13977}, {6721, 13989}, {6722, 13967}, {6723, 13969}, {6748, 26868}, {6776, 14229}, {6809, 40448}, {6810, 13599}, {6812, 13380}, {6814, 45300}, {7173, 19029}, {7354, 18965}, {7375, 18841}, {7386, 12321}, {7387, 9683}, {7392, 8855}, {7395, 19006}, {7401, 10961}, {7484, 12170}, {7486, 13941}, {7505, 10881}, {7529, 44599}, {7612, 9758}, {7618, 49787}, {7694, 13748}, {7697, 49230}, {7728, 49216}, {7736, 45513}, {7737, 12963}, {7747, 9675}, {7756, 9674}, {7765, 31465}, {7787, 35766}, {7795, 45473}, {7988, 19003}, {7989, 19004}, {7998, 12275}, {7999, 12286}, {8200, 44600}, {8207, 44602}, {8220, 44627}, {8221, 44629}, {8227, 18992}, {8591, 35698}, {8681, 9823}, {8703, 43211}, {8724, 49214}, {8781, 13926}, {8901, 16029}, {8909, 12231}, {8940, 8946}, {8941, 8947}, {8991, 15311}, {8993, 29012}, {8995, 18400}, {8997, 9738}, {8998, 10819}, {9542, 43507}, {9612, 51841}, {9614, 31432}, {9616, 41869}, {9647, 12943}, {9660, 12953}, {9669, 31474}, {9676, 34148}, {9690, 49134}, {9691, 15684}, {9692, 50692}, {9733, 49028}, {9833, 10533}, {9834, 13890}, {9835, 13891}, {9838, 13899}, {9839, 13900}, {9873, 13892}, {9874, 35862}, {9956, 13973}, {9970, 49264}, {9975, 45860}, {9996, 44604}, {10008, 42060}, {10104, 44587}, {10124, 42569}, {10137, 35400}, {10175, 13936}, {10201, 11266}, {10303, 43511}, {10319, 12663}, {10356, 19012}, {10358, 18994}, {10527, 45640}, {10528, 35816}, {10529, 35818}, {10531, 45643}, {10532, 45641}, {10590, 35801}, {10591, 35803}, {10594, 35777}, {10595, 35811}, {10596, 35817}, {10597, 35819}, {10598, 35797}, {10599, 35799}, {10653, 15765}, {10654, 18585}, {10738, 48714}, {10742, 48700}, {10749, 49270}, {10796, 44586}, {10895, 18996}, {10896, 19038}, {10897, 18531}, {10942, 44643}, {10943, 44645}, {11061, 35876}, {11178, 44474}, {11265, 18569}, {11292, 12323}, {11293, 45509}, {11315, 45871}, {11316, 15294}, {11417, 37444}, {11462, 25739}, {11484, 13943}, {11488, 42173}, {11489, 42171}, {11499, 44590}, {11500, 13887}, {11511, 12598}, {11515, 12982}, {11516, 12983}, {11548, 34516}, {11824, 21737}, {11836, 40556}, {12047, 16232}, {12103, 42568}, {12110, 13885}, {12113, 13894}, {12114, 13895}, {12115, 13906}, {12116, 13907}, {12239, 13754}, {12322, 32419}, {12383, 35834}, {12384, 35828}, {12699, 49226}, {12849, 35870}, {12856, 49248}, {12900, 13990}, {12918, 49218}, {12919, 49258}, {12964, 14216}, {13219, 35880}, {13333, 43531}, {13403, 43863}, {13429, 51833}, {13644, 14230}, {13663, 37809}, {13669, 35306}, {13674, 13848}, {13678, 35872}, {13692, 49260}, {13712, 15293}, {13798, 35874}, {13812, 49262}, {13835, 41491}, {13916, 41023}, {13917, 41022}, {13921, 14244}, {13979, 15088}, {13986, 32396}, {13993, 35018}, {14061, 19055}, {14228, 14242}, {14231, 45463}, {14234, 45406}, {14269, 43504}, {14568, 45420}, {14639, 19109}, {14643, 19052}, {14644, 19111}, {14651, 35825}, {14813, 42149}, {14814, 42152}, {14853, 35841}, {14912, 39894}, {15022, 42604}, {15059, 19059}, {15061, 49217}, {15235, 17825}, {15236, 17811}, {15561, 49267}, {15682, 42413}, {15687, 52047}, {15688, 43209}, {15691, 51850}, {15692, 43256}, {15701, 42418}, {15709, 43505}, {15883, 38425}, {16267, 36468}, {16268, 36450}, {16626, 49238}, {16627, 49236}, {16772, 42197}, {16773, 42198}, {16964, 51728}, {17538, 42414}, {18404, 18457}, {18440, 49228}, {18553, 44482}, {18926, 45607}, {18939, 44518}, {19050, 26470}, {19058, 23234}, {19074, 36765}, {19081, 31272}, {19087, 40686}, {19126, 19144}, {19130, 45543}, {19423, 19491}, {19710, 43434}, {21616, 30556}, {22235, 42233}, {22237, 42231}, {22238, 35738}, {22616, 48778}, {22647, 35860}, {22758, 44606}, {22793, 31439}, {22876, 44667}, {22921, 44666}, {22955, 49246}, {23515, 46689}, {24245, 34853}, {24842, 29243}, {25555, 44657}, {26361, 49024}, {26386, 44582}, {26394, 45357}, {26410, 44584}, {26418, 45359}, {26446, 49227}, {26466, 45597}, {26467, 45595}, {26468, 44594}, {26469, 44596}, {26494, 45601}, {26503, 45599}, {26516, 35945}, {26617, 33456}, {26620, 33365}, {30398, 30428}, {30427, 32734}, {31475, 37720}, {31981, 49352}, {32354, 35858}, {32379, 49244}, {32491, 45472}, {32492, 45863}, {32503, 32509}, {33346, 33348}, {33354, 33438}, {33357, 33436}, {33358, 33442}, {33360, 33440}, {33366, 33368}, {33370, 33372}, {34507, 44502}, {35731, 36437}, {36445, 36446}, {36463, 36464}, {36473, 36553}, {36474, 36549}, {36492, 36526}, {36495, 36585}, {36496, 36581}, {36658, 36990}, {36714, 45440}, {37640, 51853}, {37641, 51852}, {38224, 49213}, {38235, 40663}, {38752, 48715}, {40330, 45577}, {41950, 41964}, {41961, 49133}, {41965, 43785}, {42085, 42200}, {42086, 42202}, {42125, 42187}, {42128, 42188}, {42159, 42281}, {42162, 42280}, {42163, 42246}, {42166, 42247}, {42168, 42684}, {42170, 42685}, {42184, 42813}, {42186, 42814}, {42193, 42817}, {42194, 42818}, {42236, 51727}, {42523, 46936}, {43257, 50687}, {43413, 43794}, {43449, 45435}, {43521, 46333}, {43816, 43867}, {43883, 50688}, {44394, 45489}, {44595, 45515}, {45101, 45544}, {45107, 45441}, {45365, 48454}, {45368, 48455}, {45498, 49038}, {45605, 48469}, {45650, 48482}, {48477, 49027}

▸ Functional Relations43 relations

• X₄₈₅ = midpoint of X_i and X_j for these {i,j}: {1, 9907}, {3, 12602}, {4, 12257}, {20, 12297}, {486, 22592}, {487, 2996}, {488, 12222}, {1151, 23251}, {3070, 32497}, {6250, 48735}, {6278, 6279}, {6289, 49318}, {13873, 50720}, {22644, 42260}, {26617, 33456}, {49157, 49158}

• X₄₈₅ = reflection of X_i in X_j for these {i,j}: {1, 12269}, {3, 49104}, {4, 6250}, {486, 13881}, {488, 641}, {641, 6118}, {1151, 8981}, {6278, 6289}, {6279, 49318}, {6289, 5}, {6304, 33446}, {6305, 33444}, {6337, 642}, {7981, 1}, {8981, 13925}, {10819, 8998}, {12124, 3}, {12257, 48735}, {12788, 10}, {12800, 402}, {13134, 10068}, {13135, 10084}, {13912, 49618}, {19145, 13910}, {22502, 50720}, {22591, 486}, {22627, 33450}, {22629, 33448}, {22644, 23251}, {32495, 3070}, {36656, 45861}, {42009, 639}, {42260, 1151}, {44647, 7583}, {48660, 22625}, {49790, 590}

• X₄₈₅ = isogonal conjugate of X₃₇₁

• X₄₈₅ = isotomic conjugate of X₄₉₂

• X₄₈₅ = complement of X₄₈₈

• X₄₈₅ = anticomplement of X₆₄₁

• X₄₈₅ = polar conjugate of X₁₅₈₅

• X₄₈₅ = complement of the isogonal conjugate of X₈₉₄₈

• X₄₈₅ = complement of the isotomic conjugate of X₂₄₂₄₄

• X₄₈₅ = isogonal conjugate of the anticomplement of X₆₃₉

• X₄₈₅ = isogonal conjugate of the complement of X₆₃₇

• X₄₈₅ = isotomic conjugate of the anticomplement of X₅₉₀

• X₄₈₅ = isotomic conjugate of the isogonal conjugate of X₈₅₇₇

• X₄₈₅ = isogonal conjugate of the isotomic conjugate of X₃₄₃₉₁

• X₄₈₅ = isotomic conjugate of the polar conjugate of X₄₁₅₁₅

• X₄₈₅ = polar conjugate of the isotomic conjugate of X₁₁₀₉₀

• X₄₈₅ = polar conjugate of the isogonal conjugate of X₆₄₁₃

• X₄₈₅ = psi-transform of X₁₃₅₂₁

• X₄₈₅ = X_i-complementary conjugate of X_j for these (i,j): {31, 24246}, {493, 18589}, {1973, 13882}, {8948, 10}, {19218, 141}, {24244, 2887}, {26454, 1214}

• X₄₈₅ = X_i-Ceva conjugate of X_j for these (i,j): {2, 24246}, {847, 486}, {16032, 6413}, {34391, 11090}, {41515, 8944}

• X₄₈₅ = X_i-cross conjugate of X_j for these (i,j): {3, 486}, {590, 2}, {3070, 4}, {5418, 10194}, {6413, 11090}, {6560, 1328}, {8577, 41515}, {8976, 10195}, {13665, 1327}, {26920, 1586}, {26951, 11091}, {32497, 5490}, {32568, 492}, {42259, 1132}, {42261, 43571}, {43879, 3316}, {45384, 43568}

• X₄₈₅ = X_i-isoconjugate of X_j for these (i,j): {1, 371}, {19, 5408}, {31, 492}, {47, 486}, {48, 1585}, {63, 5413}, {92, 8911}, {372, 3378}, {560, 45805}, {605, 13457}, {1748, 6414}, {2180, 16037}, {8576, 44179}

• X₄₈₅ = X_i-Dao conjugate of X_j for these (i,j): {2, 492}, {2, 24246}, {3, 371}, {6, 5408}, {485, 488}, {486, 34853}, {1249, 1585}, {1599, 10962}, {1993, 10960}, {3162, 5413}, {5408, 9723}, {6374, 45805}, {8576, 37864}, {8577, 44196}, {8911, 22391}, {13428, 24245}, {13882, 42009}, {33364, 39387}

• X₄₈₅ = cevapoint of X_i and X_j for these (i,j): {3, 10665}, {6, 3155}, {590, 8035}, {3385, 3386}, {6413, 8577}

• X₄₈₅ = crosspoint of X₂ and X₂₄₂₄₄

• X₄₈₅ = crosssum of X_i and X_j for these (i,j): {3, 8909}, {6, 10132}, {3311, 8826}

• X₄₈₅ = trilinear pole of line {523, 17431}

• X₄₈₅ = internal center of similitude of nine-point circle and 2nd Lemoine circle

• X₄₈₅ = pole wrt polar circle of trilinear polar of X₁₅₈₅

• X₄₈₅ = X₄₈-isoconjugate (polar conjugate) of X₁₅₈₅

• X₄₈₅ = trilinear product of vertices of outer Vecten triangle

• X₄₈₅ = X₄-of-outer-Vecten-triangle

• X₄₈₅ = X₄-of-X₂-quadsquares-triangle

• X₄₈₅ = perspector of ABC and inner-squares triangle

• X₄₈₅ = perspector of 3rd tri-squares triangle and 3rd tri-squares central triangle

• X₄₈₅ = Kosnita(X₄₈₅,X₄₈₅) point

• X₄₈₅ = 3rd-tri-squares-isogonal conjugate of X₃₂₄₉₇

• X₄₈₅ = 4th-anti-tri-squares-isogonal conjugate of X₃₂₄₉₅

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

• X₄₈₅ = outer-Vecten-isogonal conjugate of X₆₄₁

• X₄₈₅ = barycentric product X_i*X_j for these {i,j}: {4, 11090}, {5, 16032}, {6, 34391}, {68, 1586}, {69, 41515}, {76, 8577}, {85, 13455}, {264, 6413}, {372, 5392}, {486, 13439}, {491, 2165}, {847, 5409}, {850, 39383}, {2996, 8944}, {5412, 20563}, {11091, 13440}, {13441, 41516}, {18819, 45472}, {21463, 42023}, {24244, 24246}, {34392, 44192}

• X₄₈₅ = barycentric quotient X_i/X_j for these {i,j}: {2, 492}, {3, 5408}, {4, 1585}, {6, 371}, {25, 5413}, {68, 11091}, {76, 45805}, {96, 16037}, {184, 8911}, {371, 1599}, {372, 1993}, {486, 13428}, {491, 7763}, {590, 641}, {1123, 13457}, {1321, 3535}, {1504, 32568}, {1586, 317}, {2165, 486}, {2351, 6414}, {3068, 39387}, {3070, 8968}, {3155, 10962}, {5392, 34392}, {5409, 9723}, {5412, 24}, {6413, 3}, {6414, 10666}, {8035, 590}, {8576, 44193}, {8577, 6}, {8944, 193}, {10665, 5409}, {10880, 15207}, {11090, 69}, {11091, 13430}, {11473, 15214}, {13439, 491}, {13440, 1586}, {13455, 9}, {14593, 41516}, {16032, 95}, {21463, 45420}, {24246, 488}, {26454, 8950}, {26920, 1147}, {32734, 39384}, {34391, 76}, {35764, 15188}, {39383, 110}, {41515, 4}, {41516, 13429}, {44192, 372}, {45472, 42009}

• X₄₈₅ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 49157, 13135}, {1, 49158, 13134}, {2, 372, 5420}, {2, 488, 641}, {2, 1587, 372}, {2, 12222, 488}, {2, 13439, 11090}, {2, 43133, 45508}, {3, 590, 5418}, {3, 3070, 6560}, {3, 6560, 42261}, {3, 8276, 9682}, {3, 8976, 590}, {3, 13665, 3070}, {3, 13882, 49790}, {3, 13889, 8276}, {3, 45384, 8976}, {4, 371, 6561}, {4, 3068, 371}, {4, 6459, 35821}, {4, 6561, 22615}, {4, 6564, 42269}, {4, 13440, 41515}, {4, 13886, 3068}, {4, 31412, 6564}, {5, 6, 486}, {5, 486, 42274}, {5, 1352, 10515}, {5, 5875, 6214}, {5, 6215, 10516}, {5, 6290, 10514}, {5, 7583, 6}, {5, 7584, 42262}, {5, 18538, 42265}, {5, 19116, 18762}, {5, 19117, 7584}, {5, 42265, 42277}, {5, 49318, 6278}, {5, 49356, 1352}, {6, 13881, 49221}, {6, 18538, 42277}, {6, 42262, 7584}, {6, 42265, 5}, {6, 42277, 42274}, {6, 44647, 19103}, {6, 49221, 19102}, {11, 19028, 1124}, {12, 19030, 1335}, {15, 3391, 2041}, {16, 3366, 2042}, {20, 1131, 23249}, {20, 8972, 9540}, {20, 9540, 6200}, {20, 23249, 35820}, {20, 35812, 9680}, {20, 35820, 42276}, and many others

Notes

Erect a square outwardly from each side of triangle ABC. Let A'B'C' be the triangle formed by the respective centers of the squares. The lines AA', BB', CC' concur in X₄₈₅. For details, visit Floor van Lamoen’s site, Vierkanten in een driehoek: 1. Omgeschreven vierkanten (van Lamoen, 4/26/98) and his article “Friendship Among Triangle Centers,” :cite:`Forum Geometricorum <forum-geometricorum>`, 1 (2001) 1-6. See also Paul Yiu’s papers, “Squares Erected on the Sides of a Triangle”, and “On the Squares Erected Externally on the Sides of a Triangle”.