X₃₅

{X₁,X₃}-HARMONIC CONJUGATE OF X₃₆

Let A' be the inverse-in-circumcircle of the A-excenter, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X₃₅.

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

X₃₅ ({X₁,X₃}-HARMONIC CONJUGATE OF X₃₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 3}

• Related Centers: X₃₆ (INVERSE-IN-CIRCUMCIRCLE OF INCENTER), X₁ (INCENTER), X₃ (CIRCUMCENTER)

Trilinears: \(1 + 2 \cos A : 1 + 2 \cos B : 1 + 2 \cos C\)

Barycentrics: \(\sin A + \sin 2A : :\)

Search Value (6, 9, 13): \(3.284689530119\)

▸ Alternate Coordinates3 forms

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

• Trilinears: \(\sin(3A/2) \csc(A/2) : :\)

• Barycentrics: \(a^{2}(b^{2} + c^{2} - a^{2} + bc) : :\)

▸ Linear Combinations1 combination

• X₃₅ = \(3\) X₁ - \(2\) X₁₁₀₁₁, \(3\) X₁ - X₁₁₂₈₀, \(3\) X₂ + X₂₀₀₆₆, \(6\) X₂ - \(5\) X₃₁₂₆₂, \(2\) X₁₀ + X₁₁₀₁₅, \(2\) X₁₂ - \(3\) X₃₅₈₄, \(2\) X₁₂ + X₄₃₂₄, X₁₂ - \(3\) X₄₉₉₅, \(3\) X₂₃₃₀ - X₁₉₃₆₉, \(4\) X₂₆₄₆ - X₁₁₀₀₉, \(2\) X₂₆₄₆ + X₁₁₀₁₀, \(3\) X₂₆₄₆ - X₁₁₀₁₁, \(6\) X₂₆₄₆ - X₁₁₂₈₀, X₂₉₇₅ - \(3\) X₁₇₅₄₉, \(3\) X₃₅₇₆ - X₁₁₀₁₄, \(3\) X₃₅₈₄ - X₃₅₈₅, \(3\) X₃₅₈₄ + X₄₃₂₄, \(3\) X₃₅₈₄ + \(2\) X₁₅₃₃₈, X₃₅₈₅ - \(6\) X₄₉₉₅, X₃₅₈₅ + \(2\) X₁₅₃₃₈, X₃₈₇₁ + \(2\) X₅₂₆₇, \(2\) X₃₈₇₁ + X₅₂₈₈, X₃₈₇₁ + \(3\) X₁₇₅₄₉, X₄₃₂₄ + \(6\) X₄₉₉₅, \(3\) X₄₉₉₅ + X₁₅₃₃₈, \(4\) X₅₂₆₇ - X₅₂₈₈, \(2\) X₅₂₆₇ - \(3\) X₁₇₅₄₉, X₅₂₈₈ - \(6\) X₁₇₅₄₉, \(3\) X₅₉₀₂ - \(4\) X₁₃₇₅₀, \(4\) X₆₆₆₈ - \(3\) X₁₇₅₃₀, \(3\) X₇₆₇₆ + X₈₅₄₃, X₁₁₀₀₉ + \(2\) X₁₁₀₁₀, \(3\) X₁₁₀₀₉ - \(4\) X₁₁₀₁₁, \(3\) X₁₁₀₀₉ - \(2\) X₁₁₂₈₀, \(3\) X₁₁₀₁₀ + \(2\) X₁₁₀₁₁, \(3\) X₁₁₀₁₀ + X₁₁₂₈₀, X₁₁₀₁₂ + \(2\) X₁₁₈₄₉, \(3\) X₁₅₀₁₅ - \(2\) X₃₃₅₉₈, X₁₅₉₀₈ - \(3\) X₂₁₁₅₅, X₂₀₀₆₆ + \(2\) X₂₅₆₃₉, \(2\) X₂₀₀₆₆ + \(3\) X₃₁₁₅₉, \(2\) X₂₀₀₆₆ + \(5\) X₃₁₂₆₂, \(4\) X₂₅₆₃₉ - \(3\) X₃₁₁₅₉, \(4\) X₂₅₆₃₉ - \(5\) X₃₁₂₆₂, \(3\) X₃₁₁₅₉ - \(5\) X₃₁₂₆₂

X₃₅ lies on the cubics K073, K434, K679, K1056.

▸ Incident Central Lines964 lines

{1, 3}, {2, 1479}, {4, 498}, {5, 3583}, {6, 5312}, {8, 993}, {9, 90}, {10, 21}, {11, 140}, {12, 30}, {15, 1250}, {16, 5357}, {19, 14017}, {20, 1478}, {22, 612}, {23, 5297}, {24, 33}, {25, 1900}, {28, 1869}, {31, 386}, {32, 2276}, {34, 378}, {37, 267}, {38, 7293}, {39, 1914}, {41, 3730}, {42, 58}, {43, 1011}, {47, 212}, {54, 6286}, {60, 5127}, {61, 7127}, {63, 3811}, {71, 284}, {72, 191}, {73, 74}, {75, 21410}, {77, 7163}, {78, 3422}, {79, 226}, {81, 4278}, {84, 7162}, {86, 25599}, {98, 10086}, {99, 1909}, {101, 1334}, {103, 1803}, {104, 5559}, {106, 28218}, {110, 501}, {112, 13116}, {125, 12896}, {145, 8666}, {149, 24387}, {172, 187}, {181, 35203}, {182, 3056}, {183, 3760}, {186, 1825}, {192, 7793}, {197, 8185}, {198, 3731}, {199, 1961}, {200, 1259}, {201, 1725}, {202, 5237}, {203, 5238}, {210, 31445}, {213, 17735}, {214, 3884}, {218, 4258}, {221, 10076}, {225, 7414}, {228, 846}, {238, 3216}, {255, 991}, {259, 34201}, {270, 2073}, {291, 8671}, {306, 24632}, {319, 34016}, {350, 1078}, {355, 6914}, {371, 3301}, {372, 2066}, {376, 388}, {377, 10198}, {380, 1723}, {381, 12953}, {382, 10895}, {384, 27020}, {385, 25264}, {389, 11429}, {390, 3086}, {392, 2932}, {404, 1125}, {405, 1376}, {411, 516}, {442, 6690}, {474, 1001}, {495, 550}, {496, 549}, {497, 499}, {500, 1154}, {511, 2330}, {515, 6906}, {518, 3916}, {519, 2975}, {528, 4999}, {535, 20060}, {538, 4400}, {546, 3614}, {548, 4325}, {551, 5253}, {572, 1404}, {573, 1405}, {574, 2241}, {575, 8540}, {578, 11436}, {580, 14547}, {590, 31499}, {595, 902}, {596, 32923}, {603, 10623}, {609, 3053}, {611, 1350}, {613, 5085}, {614, 7485}, {632, 10593}, {650, 11247}, {672, 4251}, {674, 5135}, {692, 1437}, {741, 29151}, {748, 17749}, {750, 16451}, {758, 20612}, {813, 2711}, {849, 1326}, {851, 29640}, {910, 16601}, {920, 10393}, {936, 4512}, {944, 5450}, {946, 5443}, {947, 1795}, {950, 1006}, {953, 23153}, {954, 4312}, {956, 3632}, {958, 3679}, {960, 5440}, {968, 975}, {971, 15837}, {976, 4414}, {978, 8616}, {983, 15315}, {984, 3220}, {995, 3915}, {997, 4855}, {1005, 6745}, {1009, 29633}, {1010, 19842}, {1012, 5691}, {1018, 2329}, {1035, 34033}, {1036, 34446}, {1056, 3528}, {1058, 3524}, {1064, 22072}, {1066, 22053}, {1071, 1768}, {1089, 7081}, {1094, 3165}, {1095, 3166}, {1100, 5124}, {1110, 14887}, {1124, 1152}, {1145, 32157}, {1147, 6238}, {1151, 1335}, {1158, 10093}, {1191, 21000}, {1210, 4314}, {1212, 5540}, {1215, 24850}, {1220, 4234}, {1255, 6186}, {1297, 13311}, {1329, 11113}, {1386, 5096}, {1398, 11410}, {1425, 21663}, {1428, 5092}, {1442, 7279}, {1444, 3879}, {1447, 7264}, {1464, 18360}, {1468, 2177}, {1469, 3098}, {1475, 5030}, {1476, 13606}, {1486, 31318}, {1490, 1709}, {1498, 10060}, {1511, 3024}, {1587, 13905}, {1588, 13963}, {1593, 11398}, {1599, 3084}, {1600, 3083}, {1609, 3553}, {1626, 23206}, {1656, 9668}, {1657, 9654}, {1658, 8144}, {1699, 3149}, {1707, 7085}, {1708, 10399}, {1714, 13726}, {1718, 2955}, {1728, 10382}, {1734, 1946}, {1738, 7523}, {1742, 1745}, {1743, 4254}, {1749, 17637}, {1761, 22021}, {1776, 26878}, {1782, 18673}, {1784, 1940}, {1785, 7412}, {1788, 3488}, {1807, 10260}, {1816, 17188}, {1819, 2328}, {1824, 20832}, {1836, 11374}, {1837, 18395}, {1838, 4219}, {1858, 31837}, {1870, 3520}, {1891, 4227}, {1918, 3736}, {1935, 4551}, {1950, 2197}, {1975, 3761}, {2067, 6200}, {2071, 4296}, {2080, 10799}, {2136, 8668}, {2161, 26744}, {2174, 17454}, {2175, 3781}, {2178, 3247}, {2192, 17821}, {2209, 5145}, {2218, 16290}, {2222, 2687}, {2242, 5206}, {2252, 2302}, {2267, 4266}, {2270, 11434}, {2278, 2323}, {2280, 4253}, {2289, 15830}, {2291, 20219}, {2292, 15618}, {2293, 13329}, {2300, 5110}, {2309, 4279}, {2320, 21398}, {2324, 15817}, {2346, 5542}, {2361, 2964}, {2475, 3822}, {2477, 22115}, {2478, 26364}, {2548, 31497}, {2550, 6857}, {2551, 11111}, {2653, 20666}, {2664, 16372}, {2782, 15452}, {2800, 21740}, {2802, 4861}, {2886, 7483}, {2887, 25645}, {2901, 17763}, {2933, 3185}, {2937, 9628}, {2948, 13204}, {3011, 7465}, {3023, 33813}, {3027, 12042}, {3028, 12041}, {3035, 4187}, {3052, 4255}, {3065, 7161}, {3070, 9646}, {3071, 9660}, {3090, 5225}, {3099, 11494}, {3100, 7488}, {3101, 25080}, {3120, 24160}, {3146, 10590}, {3157, 12163}, {3159, 32936}, {3207, 32625}, {3218, 3874}, {3219, 3647}, {3230, 21008}, {3237, 8161}, {3238, 8160}, {3244, 5303}, {3270, 13367}, {3286, 4649}, {3293, 4267}, {3294, 35342}, {3297, 6410}, {3298, 6409}, {3311, 19037}, {3312, 19038}, {3357, 7355}, {3419, 26066}, {3434, 6910}, {3454, 29846}, {3474, 3487}, {3485, 6361}, {3486, 5657}, {3501, 16788}, {3509, 3970}, {3515, 7071}, {3522, 4293}, {3525, 10589}, {3526, 9669}, {3529, 5229}, {3530, 15172}, {3534, 9655}, {3541, 11393}, {3560, 5587}, {3600, 10304}, {3616, 4188}, {3626, 17574}, {3627, 10592}, {3628, 5326}, {3633, 12513}, {3634, 5047}, {3648, 17484}, {3649, 5719}, {3652, 12738}, {3683, 5044}, {3684, 16552}, {3688, 7193}, {3689, 34790}, {3693, 17744}, {3695, 3712}, {3697, 5302}, {3705, 4894}, {3720, 4210}, {3723, 21773}, {3724, 3743}, {3751, 12329}, {3753, 5426}, {3754, 27086}, {3767, 9598}, {3779, 5138}, {3792, 19624}, {3797, 30167}, {3814, 5046}, {3816, 13747}, {3817, 6915}, {3828, 16858}, {3864, 18265}, {3868, 4880}, {3869, 4867}, {3870, 4652}, {3877, 30144}, {3878, 4511}, {3881, 3957}, {3885, 22837}, {3887, 8648}, {3897, 14923}, {3898, 4881}, {3899, 5730}, {3911, 5442}, {3912, 21511}, {3920, 5322}, {3925, 6675}, {3944, 19548}, {3947, 33557}, {3961, 16064}, {3987, 34868}, {3991, 5525}, {4018, 16126}, {4044, 26243}, {4056, 7179}, {4067, 11684}, {4084, 34195}, {4191, 26102}, {4193, 31263}, {4203, 6685}, {4216, 10448}, {4218, 24443}, {4220, 30362}, {4225, 4653}, {4260, 19133}, {4261, 5301}, {4292, 7411}, {4295, 5703}, {4297, 6909}, {4300, 22350}, {4306, 9316}, {4313, 18391}, {4319, 17928}, {4323, 34632}, {4326, 15299}, {4333, 9579}, {4347, 17080}, {4366, 7824}, {4384, 16367}, {4386, 5283}, {4396, 7780}, {4413, 11108}, {4423, 16408}, {4428, 16371}, {4429, 19846}, {4432, 25079}, {4471, 7301}, {4533, 15481}, {4557, 24436}, {4570, 9273}, {4645, 25650}, {4647, 32932}, {4674, 8683}, {4714, 16824}, {4760, 16720}, {4781, 17164}, {4816, 8168}, {4848, 21161}, {4868, 17016}, {4870, 28198}, {4920, 33870}, {5009, 21035}, {5011, 17451}, {5013, 16502}, {5015, 30171}, {5023, 9331}, {5054, 11238}, {5070, 9671}, {5080, 15680}, {5082, 30478}, {5120, 16667}, {5160, 7575}, {5171, 10802}, {5174, 17515}, {5195, 17084}, {5219, 6985}, {5223, 6600}, {5252, 18481}, {5263, 19270}, {5265, 15692}, {5272, 7484}, {5274, 10303}, {5276, 25092}, {5284, 17531}, {5287, 11340}, {5291, 20691}, {5298, 12100}, {5300, 24587}, {5332, 7772}, {5370, 7492}, {5393, 16441}, {5405, 16440}, {5424, 17097}, {5427, 15174}, {5428, 10543}, {5434, 8703}, {5438, 31435}, {5473, 10062}, {5474, 10061}, {5475, 31501}, {5506, 5528}, {5531, 14872}, {5533, 6713}, {5541, 10914}, {5550, 17572}, {5552, 6872}, {5588, 11498}, {5589, 11497}, {5603, 6942}, {5690, 7508}, {5722, 24914}, {5727, 22760}, {5732, 10042}, {5775, 12536}, {5836, 19525}, {5840, 6842}, {5842, 6831}, {5881, 22758}, {5886, 6924}, {5887, 6326}, {5889, 9637}, {5951, 26700}, {5974, 11990}, {6000, 26888}, {6001, 33597}, {6019, 14650}, {6042, 35205}, {6146, 26956}, {6147, 11246}, {6221, 18996}, {6253, 8727}, {6256, 6938}, {6283, 12974}, {6285, 6759}, {6292, 8299}, {6396, 6502}, {6398, 18995}, {6405, 12975}, {6423, 31459}, {6533, 16823}, {6560, 31472}, {6642, 9817}, {6645, 13586}, {6656, 26629}, {6668, 17530}, {6693, 29631}, {6771, 13076}, {6774, 13075}, {6825, 10320}, {6834, 26333}, {6841, 18406}, {6853, 13199}, {6863, 10525}, {6880, 10531}, {6882, 8070}, {6883, 9581}, {6894, 12558}, {6907, 10523}, {6908, 10321}, {6911, 8227}, {6912, 19925}, {6913, 7989}, {6916, 10629}, {6918, 7988}, {6920, 10175}, {6921, 10200}, {6923, 10953}, {6934, 26332}, {6940, 10090}, {6977, 12116}, {7051, 10645}, {7080, 17576}, {7098, 15556}, {7160, 7284}, {7176, 7278}, {7191, 15246}, {7235, 24640}, {7270, 17512}, {7292, 7496}, {7320, 15180}, {7330, 17857}, {7352, 7689}, {7353, 7692}, {7356, 7691}, {7362, 7690}, {7416, 15622}, {7428, 23361}, {7489, 9956}, {7504, 20104}, {7506, 9673}, {7512, 22347}, {7580, 9612}, {7583, 13901}, {7584, 13958}, {7587, 30411}, {7588, 30423}, {7589, 30408}, {7675, 18412}, {7677, 30331}, {7713, 11383}, {7737, 9596}, {7745, 31460}, {7746, 9664}, {7747, 31476}, {7807, 26590}, {7992, 12330}, {8075, 30370}, {8076, 30420}, {8077, 11013}, {8167, 16862}, {8188, 11503}, {8189, 11504}, {8356, 26561}, {8572, 16486}, {8580, 13615}, {8583, 30294}, {8614, 23071}, {8669, 11688}, {8685, 29096}, {8725, 12944}, {8731, 33138}, {8981, 19030}, {9342, 17536}, {9538, 10298}, {9540, 13904}, {9574, 16780}, {9578, 11501}, {9599, 31401}, {9638, 11461}, {9645, 14070}, {9656, 17800}, {9657, 15696}, {9658, 12083}, {9665, 31455}, {9666, 32046}, {9708, 17571}, {9709, 16418}, {9712, 15076}, {9751, 10080}, {9780, 16865}, {9818, 19372}, {9821, 12837}, {9860, 12178}, {9875, 12326}, {9896, 12328}, {9897, 12331}, {9898, 12333}, {9899, 12335}, {9900, 12336}, {9901, 12337}, {9902, 12338}, {9903, 12339}, {9904, 12327}, {9905, 12341}, {9906, 12343}, {9907, 12344}, {9931, 9932}, {9938, 19471}, {9943, 17613}, {10037, 11414}, {10040, 11824}, {10041, 11825}, {10054, 12117}, {10055, 12118}, {10057, 12119}, {10059, 12120}, {10063, 11257}, {10064, 12122}, {10067, 12123}, {10068, 12124}, {10069, 34473}, {10077, 21157}, {10078, 21156}, {10079, 22712}, {10081, 15055}, {10089, 21166}, {10091, 15035}, {10106, 21578}, {10118, 13289}, {10122, 12432}, {10149, 15646}, {10179, 35271}, {10197, 17579}, {10282, 10535}, {10479, 32916}, {10527, 20075}, {10605, 19349}, {10610, 13079}, {10646, 19373}, {10647, 30433}, {10648, 30434}, {10651, 30300}, {10652, 30301}, {10707, 34649}, {10789, 11490}, {10806, 15868}, {10860, 30290}, {10877, 26316}, {10927, 26341}, {10928, 26348}, {10944, 34773}, {10947, 26492}, {10954, 11827}, {10974, 22080}, {11063, 21864}, {11101, 21674}, {11102, 33116}, {11110, 16828}, {11112, 25466}, {11114, 11681}, {11115, 26115}, {11171, 12836}, {11189, 11202}, {11194, 19704}, {11204, 32065}, {11219, 12750}, {11231, 17606}, {11239, 20076}, {11250, 32047}, {11263, 20292}, {11319, 26030}, {11329, 16831}, {11343, 17284}, {11349, 29571}, {11350, 17022}, {11375, 12699}, {11392, 18533}, {11430, 19365}, {11438, 19366}, {11446, 11449}, {11454, 19367}, {11468, 19368}, {11522, 22753}, {11545, 12104}, {11570, 18444}, {11571, 12515}, {11587, 35201}, {11715, 18861}, {11753, 35207}, {11762, 35208}, {11771, 35209}, {11780, 35210}, {11828, 11951}, {11829, 11952}, {11848, 11852}, {11909, 26451}, {12005, 26877}, {12054, 12835}, {12121, 12903}, {12185, 15561}, {12334, 12407}, {12340, 12408}, {12342, 12409}, {12350, 14830}, {12359, 12428}, {12373, 20127}, {12374, 14643}, {12556, 13128}, {12584, 32286}, {12608, 12775}, {12609, 16155}, {12619, 12743}, {12653, 22560}, {12680, 34862}, {12888, 12893}, {12901, 19469}, {12904, 15061}, {12910, 12972}, {12911, 12973}, {12940, 20427}, {12984, 19473}, {12985, 19474}, {13043, 13049}, {13044, 13050}, {13061, 19475}, {13062, 19476}, {13080, 18244}, {13089, 13146}, {13173, 13174}, {13206, 13221}, {13253, 22775}, {13278, 26726}, {13293, 19505}, {13541, 34139}, {13588, 25526}, {13602, 15179}, {13665, 13897}, {13666, 13714}, {13675, 13679}, {13723, 29674}, {13743, 17663}, {13785, 13954}, {13786, 13837}, {13795, 13799}, {13887, 13888}, {13935, 13962}, {13940, 13942}, {13966, 19029}, {14100, 31658}, {14118, 24025}, {14377, 30949}, {14873, 33329}, {14986, 15717}, {15104, 21165}, {15462, 32290}, {15815, 16781}, {15908, 21155}, {15950, 22791}, {16058, 16569}, {16059, 25502}, {16061, 16818}, {16113, 16152}, {16117, 16118}, {16139, 33857}, {16154, 21077}, {16163, 18968}, {16286, 17123}, {16299, 33174}, {16342, 19858}, {16346, 19859}, {16405, 29825}, {16436, 29573}, {16453, 17122}, {16468, 20992}, {16469, 21002}, {16471, 19764}, {16475, 16688}, {16484, 20470}, {16496, 22769}, {16583, 34872}, {16783, 17754}, {16817, 28611}, {16819, 33047}, {16825, 23407}, {16826, 19308}, {16842, 19872}, {16845, 26040}, {16850, 19856}, {16857, 19876}, {16859, 19877}, {16915, 27255}, {16916, 27091}, {16917, 31996}, {16930, 27048}, {16931, 27026}, {16992, 32092}, {17023, 21495}, {17030, 17684}, {17104, 35193}, {17126, 19767}, {17316, 21508}, {17532, 34626}, {17534, 31253}, {17535, 19878}, {17558, 19855}, {17567, 26105}, {17588, 19874}, {17605, 22793}, {17638, 22935}, {17647, 24987}, {17660, 26201}, {17674, 24542}, {17692, 26752}, {17696, 27299}, {17717, 19543}, {17719, 24851}, {17729, 17758}, {17734, 21935}, {17760, 33952}, {17784, 19843}, {18180, 22300}, {18357, 31649}, {18491, 18492}, {18515, 18526}, {18518, 18761}, {18589, 24780}, {18915, 18931}, {18922, 18925}, {18954, 35243}, {18957, 35248}, {18958, 35241}, {18959, 35246}, {18960, 35247}, {18961, 35249}, {18962, 35250}, {18965, 35255}, {18966, 35256}, {18999, 19003}, {19000, 19004}, {19175, 19192}, {19182, 19185}, {19237, 29610}, {19314, 26241}, {19354, 19357}, {19370, 19454}, {19371, 19455}, {19434, 19440}, {19435, 19441}, {19472, 22978}, {19513, 21321}, {19649, 24239}, {19864, 32942}, {20107, 31272}, {20108, 32944}, {20459, 23530}, {20475, 23370}, {20677, 21004}, {20831, 20989}, {20963, 33863}, {20999, 22344}, {21319, 33099}, {21477, 29598}, {21516, 29596}, {21518, 29602}, {21537, 26626}, {21616, 27385}, {21669, 31673}, {22060, 32913}, {22267, 27248}, {22556, 22650}, {22557, 22651}, {22558, 22652}, {22559, 22653}, {22676, 22729}, {22843, 22884}, {22890, 22929}, {22951, 22980}, {22954, 22962}, {23226, 35057}, {23358, 32378}, {24036, 33950}, {24046, 28082}, {24068, 32927}, {24161, 24715}, {24248, 24309}, {24424, 24684}, {24466, 31775}, {24813, 24845}, {24820, 24821}, {24953, 31419}, {25582, 27127}, {26298, 26493}, {26299, 26502}, {26300, 26512}, {26301, 26513}, {26321, 28204}, {26353, 26498}, {26354, 26507}, {26355, 26516}, {26356, 26521}, {26562, 30131}, {26686, 35297}, {27143, 28410}, {27324, 33816}, {27622, 33109}, {27802, 31320}, {28606, 30142}, {28761, 31058}, {29055, 29300}, {29574, 35276}, {30107, 33819}, {30110, 33830}, {30295, 30424}, {30296, 30425}, {30297, 30426}, {30385, 30431}, {30386, 30432}, {30435, 31461}, {30944, 33140}, {31140, 31493}, {32143, 32210}, {32168, 32171}, {32233, 32307}, {32256, 32261}, {32259, 32305}, {32330, 32403}, {32347, 32356}, {32350, 32401}, {33635, 33671}, {33866, 33949}, {34927, 34931}

▸ Functional Relations47 relations

• X₃₅ = midpoint of X_i and X_j for these {i,j}: {1, 11010}, {3, 11849}, {12, 15338}, {2975, 3871}, {3585, 4324}, {5086, 11015}, {6906, 11491}

• X₃₅ = reflection of X_i in X_j for these {i,j}: {1, 2646}, {79, 14526}, {2975, 5267}, {3584, 4995}, {3585, 12}, {4324, 15338}, {5086, 10}, {5288, 2975}, {6763, 3916}, {6842, 31659}, {11009, 1}, {11012, 3}, {11280, 11011}, {12047, 13411}, {24390, 4999}, {31159, 2}

• X₃₅ = isogonal conjugate of X₇₉

• X₃₅ = isotomic conjugate of X₂₀₅₆₅

• X₃₅ = isogonal conjugate of the anticomplement of X₃₆₄₇

• X₃₅ = isogonal conjugate of the complement of X₃₆₄₈

• X₃₅ = isogonal conjugate of the isotomic conjugate of X₃₁₉

• X₃₅ =Thomson-isogonal conjugate of X₅₆₅₉

• X₃₅ =excentral-isogonal conjugate of X₂₉₄₉

• X₃₅ = complement of isogonal conjugate of X₃₄₄₄₁

• X₃₅ = anticomplement of X₂₅₆₃₉

• X₃₅ = orthocenter of cross-triangle of ABC and inner Yff triangle

• X₃₅ = insimilicenter of circumcircles of ABC and inner Yff triangle; the exsimilicenter is X₁

• X₃₅ = homothetic center of Trinh triangle and anti-tangential midarc triangle

• X₃₅ = circumcircle-inverse of X₄₈₄

• X₃₅ = isogonal conjugate of the anticomplement of X₃₆₄₇

• X₃₅ = isogonal conjugate of the complement of X₃₆₄₈

• X₃₅ = isogonal conjugate of the isotomic conjugate of X₃₁₉

• X₃₅ = Thomson isogonal conjugate of X₅₆₅₉

• X₃₅ = excentral isogonal conjugate of X₂₉₄₉

• X₃₅ = complement of isotomic conjugate of isogonal conjugate of X₂₀₉₈₈

• X₃₅ = complement of polar conjugate of isogonal conjugate of X₂₂₁₂₂

• X₃₅ = complement of complement of X₂₀₀₆₆

• X₃₅ = Cundy-Parry Psi transform of X₁₅₁₇₅

• X₃₅ = X₃₄₄₄₁-complementary conjugate of X₁₀

• X₃₅ = X_i-Ceva conjugate of X_j for these (i,j): {1, 35197}, {21, 35194}, {943, 1}, {1255, 6}, {1442, 2003}, {5951, 484}, {11107, 6198}, {15168, 1757}, {18359, 2323}, {33670, 33669}

• X₃₅ = X_i-cross conjugate of X_j for these (i,j): {500, 1}, {2174, 2003}, {2594, 6198}, {6149, 7343}

• X₃₅ = X_i-isoconjugate of X_j for these (i,j): {1, 79}, {2, 2160}, {4, 7100}, {6, 30690}, {7, 7073}, {31, 20565}, {36, 2166}, {57, 7110}, {58, 6757}, {65, 3615}, {75, 6186}, {81, 8818}, {94, 7113}, {191, 30602}, {225, 1789}, {265, 1870}, {273, 8606}, {513, 6742}, {522, 26700}, {523, 13486}, {554, 1251}, {649, 15455}, {1081, 33653}, {1127, 10230}, {1989, 3218}, {3466, 34301}, {3468, 34303}, {4707, 32678}, {7004, 34922}, {11060, 20924}, {11076, 21739}, {13610, 14844}, {21044, 35049}, {21828, 32680}

• X₃₅ = cevapoint of X_i and X_j for these (i,j): {55, 1030}, {2594, 22342}

• X₃₅ = crosspoint of X_i and X_j for these (i,j): {1, 3467}, {21, 35196}, {59, 8701}, {100, 4570}, {1442, 3219}, {11107, 35193}

• X₃₅ = crosssum of X_i and X_j for these (i,j): {1, 3336}, {6, 20988}, {11, 4977}, {481, 482}, {513, 3120}, {1086, 23729}, {2160, 7073}, {3122, 23751}, {4466, 23727}

• X₃₅ = trilinear pole of line {2605, 9404}

• X₃₅ = crossdifference of every pair of points on line {650, 4802}

• X₃₅ = barycentric product X_i*X_j for these {i,j}: {1, 3219}, {6, 319}, {8, 2003}, {9, 1442}, {21, 16577}, {31, 33939}, {42, 34016}, {50, 20566}, {55, 17095}, {57, 4420}, {58, 3969}, {63, 6198}, {75, 2174}, {80, 323}, {81, 3678}, {100, 14838}, {101, 4467}, {110, 7265}, {190, 2605}, {219, 7282}, {226, 35193}, {249, 21054}, {261, 21794}, {304, 14975}, {312, 1399}, {314, 21741}, {321, 17104}, {333, 2594}, {445, 1794}, {593, 7206}, {651, 35057}, {664, 9404}, {692, 18160}, {765, 7202}, {943, 16585}, {1126, 3578}, {1214, 11107}, {1255, 3647}, {1268, 17454}, {1441, 35192}, {1812, 1825}, {2167, 35194}, {2611, 4567}, {2982, 31938}, {4557, 16755}, {4570, 8287}, {4600, 20982}, {6149, 18359}, {6187, 7799}, {6335, 23226}, {7110, 7279}, {7186, 17743}, {7343, 17484}, {8652, 23883}, {19620, 33670}, {21824, 24041}, {22342, 31623}

• X₃₅ = barycentric quotient X_i/X_j for these {i,j}: {1, 30690}, {2, 20565}, {6, 79}, {31, 2160}, {32, 6186}, {37, 6757}, {41, 7073}, {42, 8818}, {48, 7100}, {50, 36}, {55, 7110}, {80, 94}, {100, 15455}, {101, 6742}, {163, 13486}, {186, 17923}, {284, 3615}, {319, 76}, {323, 320}, {500, 5249}, {526, 4707}, {1399, 57}, {1415, 26700}, {1442, 85}, {2003, 7}, {2161, 2166}, {2174, 1}, {2193, 1789}, {2307, 554}, {2477, 2003}, {2594, 226}, {2605, 514}, {2611, 16732}, {3219, 75}, {3444, 30602}, {3578, 1269}, {3647, 4359}, {3678, 321}, {3969, 313}, {4420, 312}, {4467, 3261}, {6149, 3218}, {6187, 1989}, {6198, 92}, {7115, 34922}, {7186, 3662}, {7202, 1111}, {7206, 28654}, {7265, 850}, {7279, 17095}, {7282, 331}, {7343, 21739}, {8287, 21207}, {9404, 522}, {11107, 31623}, {14270, 21828}, {14838, 693}, {14975, 19}, {16577, 1441}, {17095, 6063}, {17104, 81}, {17190, 16709}, {17454, 1125}, {18755, 14844}, {20566, 20573}, {20982, 3120}, {21054, 338}, {21741, 65}, {21794, 12}, {21824, 1109}, {22094, 4466}, {22115, 22128}, {22342, 1214}, {23226, 905}, {33939, 561}, {34016, 310}, {35057, 4391}, {35192, 21}, {35193, 333}, {35194, 14213}, {35197, 17483}

• X₃₅ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 3, 36}, {1, 36, 5563}, {1, 40, 5903}, {1, 46, 5902}, {1, 55, 3746}, {1, 57, 18398}, {1, 65, 5425}, {1, 165, 46}, {1, 484, 65}, {1, 3336, 942}, {1, 3337, 354}, {1, 3550, 5264}, {1, 3576, 21842}, {1, 5010, 3}, {1, 5119, 5697}, {1, 5131, 3337}, {1, 7280, 56}, {1, 7991, 25415}, {1, 10902, 14798}, {1, 11280, 11011}, {1, 14794, 11012}, {1, 15803, 3338}, {1, 16192, 15803}, {1, 17596, 3670}, {1, 17700, 30274}, {1, 30282, 3612}, {1, 32613, 14795}, {2, 1479, 7741}, {2, 4294, 1479}, {2, 5248, 5259}, {2, 5259, 25542}, {2, 25639, 31262}, {3, 55, 1}, {3, 56, 7280}, {3, 999, 5204}, {3, 1482, 26286}, {3, 3295, 56}, {3, 3579, 7688}, {3, 3612, 14803}, {3, 3746, 5563}, {3, 5217, 5010}, {3, 6244, 5584}, {3, 10246, 32612}, {3, 10267, 3576}, {3, 10306, 3428}, {3, 10310, 165}, {3, 10679, 11249}, {3, 10902, 15931}, {3, 11248, 40}, {3, 11507, 46}, {3, 14882, 484}, {3, 15931, 35202}, {3, 16202, 10269}, {3, 26285, 2077}, {3, 26357, 14793}, {3, 31393, 13370}, {3, 32613, 10902}, {3, 32760, 14798}, {3, 35000, 3579}, {3, 35238, 35242}, {3, 35251, 35238}, {4, 498, 7951}, {4, 5218, 498}, {5, 6284, 3583}, {8, 993, 5258}, {8, 4189, 993}, {10, 21, 5251}, {10, 4304, 10572}, {10, 10572, 80}, {11, 15171, 4857}, {15, 1250, 5353}, {15, 7006, 2307}, {16, 10638, 5357}, {20, 1478, 10483}, {20, 3085, 1478}, {20, 5281, 3085}, {21, 100, 10}, {31, 386, 1203}, {32, 2276, 5280}, {32, 31451, 2276}, {36, 3746, 1}, {36, 5537, 3245}, {36, 14795, 14798}, {36, 14799, 15931}, {36, 14803, 14800}, {39, 1914, 5299}, {40, 3601, 1}, {40, 5903, 3245}, {40, 11248, 5537}, {41, 3730, 5526}, {55, 56, 3295}, {55, 1470, 31393}, {55, 5010, 36}, {55, 5172, 24929}, {55, 5204, 3303}, {55, 5217, 3}, {55, 10310, 11507}, {55, 10966, 26358}, {55, 11492, 8186}, {55, 11493, 8187}, {55, 14794, 11009}, {55, 16678, 3750}, {55, 26357, 1697}, {55, 32613, 32760}, {55, 34879, 2078}, {56, 3295, 1}, {56, 7280, 36}, {58, 33771, 42}, {63, 3811, 5904}, {65, 3579, 484}, {65, 14882, 3256}, {65, 24929, 1}, {72, 4640, 191}, {73, 109, 34043}, {74, 10088, 19470}, {75, 21586, 21410}, {78, 12514, 5692}, {78, 35258, 12514}, {79, 15228, 1770}, {80, 5441, 10572}, {100, 10058, 80}, {104, 10087, 7972}, {110, 10065, 7727}, {140, 15171, 11}, {165, 8069, 36}, {172, 1500, 16785}, {187, 1500, 172}, {197, 13730, 8185}, {212, 601, 47}, {221, 10606, 10076}, {226, 1770, 79}, {226, 31730, 1770}, {226, 32610, 8606}, {267, 1717, 33642}, {371, 5414, 3301}, {372, 2066, 3299}, {376, 388, 4299}, {381, 12953, 18514}, {382, 10895, 18513}, {382, 31479, 10895}, {390, 3523, 3086}, {404, 1621, 1125}, {405, 1376, 1698}, {474, 1001, 3624}, {484, 5172, 36}, {484, 15932, 46}, {484, 24929, 5425}, {484, 34871, 3}, {495, 550, 7354}, {495, 7354, 5270}, {496, 549, 5433}, {496, 5433, 3582}, {496, 10386, 3058}, {497, 631, 499}, {498, 4302, 4}, {499, 4309, 497}, {548, 15888, 4325}, {548, 18990, 15326}, {549, 3058, 3582}, {549, 10386, 496}, {550, 7354, 4316}, {574, 2241, 2275}, {574, 10987, 16784}, {595, 1193, 5315}, {595, 4256, 1193}, {672, 4251, 17745}, {902, 1193, 595}, {902, 4256, 5315}, {942, 1155, 3336}, {942, 31663, 1155}, {943, 3651, 226}, {944, 6950, 5450}, {950, 6684, 1737}, {954, 11495, 4312}, {956, 3913, 3632}, {958, 4421, 5687}, {958, 5687, 3679}, {968, 975, 27785}, {988, 3749, 1}, {993, 8715, 8}, {999, 3303, 1}, {1011, 19763, 1724}, {1012, 11500, 5691}, {1058, 3524, 7288}, {1058, 7288, 10072}, {1125, 10624, 30384}, {1158, 18446, 15071}, {1250, 2307, 7006}, {1259, 20835, 31424}, {1319, 9957, 1}, {1381, 1382, 484}, {1385, 3057, 1}, {1385, 3579, 13145}, {1385, 14792, 36}, {1385, 26086, 3}, {1388, 34880, 5193}, {1399, 2594, 2003}, {1420, 1470, 13370}, {1420, 31393, 1}, {1466, 1617, 3361}, {1470, 11510, 1420}, {1478, 31452, 3085}, {1482, 34471, 1}, {1656, 9668, 10896}, {1657, 9654, 12943}, {1697, 3576, 1}, {1698, 3586, 10826}, {1737, 6684, 5445}, {1770, 31730, 15228}, {1837, 26446, 18395}, {2077, 10902, 3}, {2077, 32613, 15931}, {2077, 32760, 36}, {2078, 34879, 15931}, {2098, 10246, 1}, {2177, 4257, 16474}, {2241, 2275, 16784}, {2275, 10987, 2241}, {2307, 7006, 5353}, {2550, 6857, 19854}, {2594, 6149, 35197}, {2646, 11010, 11009}, {2646, 14794, 36}, {2915, 20872, 9591}, {2975, 17549, 5267}, {3052, 4255, 16466}, {3057, 26086, 14792}, {3058, 5433, 496}, {3085, 5281, 31452}, {3149, 11496, 1699}, {3219, 4420, 3678}, {3256, 7688, 484}, {3295, 35239, 3340}, {3303, 5204, 999}, {3304, 6767, 1}, {3333, 10389, 1}, {3428, 10306, 7991}, {3434, 6910, 26363}, {3486, 5657, 10573}, {3515, 7071, 11399}, {3524, 10385, 10072}, {3529, 8164, 5229}, {3530, 15172, 15325}, {3560, 11499, 5587}, {3576, 14793, 36}, {3579, 24929, 65}, {3583, 4330, 6284}, {3584, 3585, 12}, {3584, 4324, 3585}, {3612, 5119, 1}, {3612, 5697, 24926}, {3624, 9614, 23708}, {3647, 3678, 3219}, {3666, 5266, 1}, {3730, 4262, 41}, {3746, 14799, 14798}, {3746, 14803, 24926}, {3748, 5045, 1}, {3748, 32636, 5045}, {3869, 22836, 4867}, {3871, 5267, 5288}, {3871, 17549, 2975}, {3920, 6636, 5322}, {3976, 17715, 1}, {4189, 8715, 5258}, {4251, 24047, 672}, {4255, 16466, 5313}, {4261, 5301, 16470}, {4292, 13405, 13407}, {4299, 10056, 388}, {4302, 5218, 7951}, {4304, 10572, 5441}, {4314, 10164, 1210}, {4316, 5270, 7354}, {4326, 21153, 15299}, {4366, 7824, 26959}, {4421, 16370, 3679}, {4423, 16408, 34595}, {4428, 16371, 25055}, {4855, 5250, 997}, {4995, 15338, 12}, {5010, 11010, 14794}, {5010, 32613, 14799}, {5010, 32760, 15931}, {5015, 32851, 30171}, {5045, 5122, 32636}, {5046, 27529, 3814}, {5048, 15178, 1}, {5085, 10387, 613}, {5126, 31792, 20323}, {5127, 15792, 60}, {5132, 8053, 238}, {5132, 16287, 3216}, {5172, 14882, 65}, {5172, 35000, 484}, {5217, 11849, 14794}, {5248, 25440, 2}, {5263, 19270, 19863}, {5268, 7298, 25}, {5284, 17531, 19862}, {5300, 33113, 30172}, {5326, 7173, 3628}, {5432, 6284, 5}, {5563, 15931, 34890}, {5597, 26423, 1}, {5598, 26399, 1}, {5687, 16370, 958}, {5691, 31434, 10827}, {5697, 14803, 5563}, {5703, 9778, 4295}, {5711, 19765, 1}, {5919, 24928, 1}, {6398, 31474, 18995}, {6656, 26629, 30104}, {6769, 10383, 1}, {6914, 32141, 355}, {6938, 10786, 6256}, {7081, 7283, 1089}, {7288, 10385, 1058}, {7691, 10066, 7356}, {7742, 15803, 36}, {7807, 26590, 30103}, {7982, 13384, 1}, {7987, 8071, 36}, {8069, 10310, 46}, {8069, 11507, 1}, {8071, 11508, 1}, {8186, 8187, 57}, {8666, 25439, 145}, {9627, 18447, 1}, {9819, 30389, 1}, {9957, 13624, 1319}, {10088, 19470, 6126}, {10267, 14793, 21842}, {10267, 26357, 1}, {10269, 26358, 1}, {10306, 22766, 25415}, {10679, 11249, 7982}, {10679, 22768, 1}, {10902, 32760, 14795}, {10965, 16203, 1}, {10966, 16202, 1}, {10966, 26358, 7962}, {11011, 11280, 11009}, {11375, 12699, 18393}, {11461, 11464, 9638}, {11849, 33862, 11012}, {12100, 15170, 5298}, {12512, 13405, 4292}, {12515, 12739, 11571}, {13743, 18524, 18480}, {14795, 14799, 10902}, {14796, 14797, 1}, {14796, 14802, 36}, {14797, 14801, 36}, {14801, 14802, 3}, {15326, 15888, 18990}, {15326, 18990, 4325}, {15696, 31480, 9657}, {17637, 22937, 1749}, {17735, 18755, 213}, {18518, 28444, 18761}, {20323, 31792, 1}, {24929, 35000, 3256}, {24953, 34612, 31419}, {25414, 26287, 1}, {26285, 32613, 3}, {30282, 31508, 5119}, {30478, 34607, 5082}, {31159, 31262, 25639}, {32622, 32623, 15931}

• X₃₅ = X₉₄₃-aleph conjugate of X₃₅

• X₃₅ = X_i-beth conjugate of X_j for these (i,j): (100, 35), (643, 10)

• X₃₅ = perspector of ABC and orthic triangle of incentral triangle

• X₃₅ = X₂₉₇₅ of X₁-Brocard triangle

• X₃₅ = crossdifference of every pair of points on line X₆₅₀X₄₈₀₂

• X₃₅ = homothetic center of intangents and Kosnita triangles

• X₃₅ = perspector of ABC and extraversion triangle of X₃₆

• X₃₅ = Hofstadter 3/2 point

• X₃₅ = homothetic center of 2nd isogonal triangle of X₁ and cevian triangle of X₃; see X₃₆

• X₃₅ = insimilicenter of circumcircle and circumcircle of reflection triangle of X₁; exsimilicenter is X₃₆

• X₃₅ = Cundy-Parry Phi transform of X₅₉₀₂

Notes

Let H_aH_bH_c be the orthic triangle. Let H_bH_cA″ be the triangle similar to ABC such that segment H_aA″ crosses the line H_bH_c, and define B″ and C″ cyclically. (Equivalently, A″ is the reflection of A in H_bH_c.) Let I_a be the incenter of H_bH_cA″, and define I_b and I_c cyclically. The lines AI_a, BI_b, CI_c concur in X₃₅. (Randy Hutson, November 18, 2015)