X₁₄₅

ANTICOMPLEMENT OF NAGEL POINT

Let O_A be the circle tangent to side BC at its midpoint and to the circumcircle on the same side of BC as A. Define O_B, O_C cyclically. Then X₁₄₅ is the trilinear pole of the line of the exsimilicenters (the Monge line) of O_A, O_B, O_C. See the reference at X₁₀₀₁.

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

X₁₄₅ (ANTICOMPLEMENT OF NAGEL POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Anticomplementary Triangle △K_aK_bK_c — vertices K_a, K_b, K_c (Formed by lines through A, B, C parallel to opposite sides)

• Central Lines: {2, 8}

• Related Centers: X₂ (CENTROID), X₈ (NAGEL POINT)

Trilinears: \(bc(3a - b - c) : :\)

Barycentrics: \(3a - b - c : :\)

Search Value (6, 9, 13): \(-1.126872339638\)

▸ Alternate Coordinates2 forms

• Trilinears: \(-1 + \csc A/2 \sin B/2 \sin C/2 : :\)

• Trilinears: \(2(r/R) - \sin B \sin C : :\)

▸ Linear Combinations1 combination

• X₁₄₅ = \(4\) X₁ - \(3\) X₂, \(3\) X₁ - \(2\) X₁₀, \(7\) X₁ - \(6\) X₅₅₁, \(5\) X₁ - \(4\) X₁₁₂₅, \(7\) X₁ - \(5\) X₁₆₉₈, \(2\) X₁ - \(3\) X₃₂₄₁, \(6\) X₁ - \(5\) X₃₆₁₆, \(8\) X₁ - \(5\) X₃₆₁₇, \(4\) X₁ - X₃₆₂₁, \(8\) X₁ - \(7\) X₃₆₂₂, \(4\) X₁ - \(5\) X₃₆₂₃, \(9\) X₁ - \(7\) X₃₆₂₄, \(5\) X₁ - \(2\) X₃₆₂₅, \(7\) X₁ - \(4\) X₃₆₂₆, \(3\) X₁ - X₃₆₃₂, \(11\) X₁ - \(8\) X₃₆₃₄, \(3\) X₁ - \(4\) X₃₆₃₅, \(9\) X₁ - \(8\) X₃₆₃₆, \(5\) X₁ - \(3\) X₃₆₇₉, \(17\) X₁ - \(12\) X₃₈₂₈, \(9\) X₁ - \(5\) X₄₆₆₈, \(11\) X₁ - \(6\) X₄₆₆₉, \(7\) X₁ - \(3\) X₄₆₇₇, \(12\) X₁ - \(7\) X₄₆₇₈, \(13\) X₁ - \(8\) X₄₆₉₁, \(9\) X₁ - \(4\) X₄₇₀₁, \(19\) X₁ - \(12\) X₄₇₄₅, \(15\) X₁ - \(8\) X₄₇₄₆, \(11\) X₁ - \(5\) X₄₈₁₆, \(14\) X₁ - \(11\) X₅₅₅₀, \(10\) X₁ - \(7\) X₉₇₈₀, \(17\) X₁ - \(14\) X₁₅₈₀₈, \(13\) X₁ - \(10\) X₁₉₈₆₂, \(23\) X₁ - \(17\) X₁₉₈₇₂, \(13\) X₁ - \(9\) X₁₉₈₇₅, \(29\) X₁ - \(21\) X₁₉₈₇₆, \(18\) X₁ - \(13\) X₁₉₈₇₇, \(21\) X₁ - \(16\) X₁₉₈₇₈, \(23\) X₁ - \(18\) X₁₉₈₈₃, \(4\) X₁ + X₂₀₀₁₄, \(4\) X₁ + \(3\) X₂₀₀₄₉, \(2\) X₁ + X₂₀₀₅₀, \(12\) X₁ - \(5\) X₂₀₀₅₂, \(6\) X₁ - X₂₀₀₅₃, \(8\) X₁ - X₂₀₀₅₄, \(6\) X₁ - \(7\) X₂₀₀₅₇, \(53\) X₁ - \(38\) X₂₂₂₆₆, \(11\) X₁ - \(9\) X₂₅₀₅₅, \(8\) X₁ - \(3\) X₃₁₁₄₅, \(27\) X₁ - \(20\) X₃₁₂₅₃, \(17\) X₁ - \(13\) X₃₄₅₉₅, \(13\) X₁ - \(6\) X₃₄₆₄₁, and many others

X₁₄₅ lies on on the Jerabek circumhyperbola of the intouch triangle, the Feuerbach circumhyperbola of the anticomplementary triangle, the cubics K201, K295, K360, K365, K372, K571, K830, K1011, K1078, K1083, K1086.

▸ Incident Central Lines1493 lines

{1, 2}, {3, 1483}, {4, 149}, {5, 10247}, {6, 346}, {7, 1266}, {9, 4029}, {11, 5154}, {12, 3813}, {20, 517}, {21, 956}, {22, 8192}, {23, 9798}, {30, 8148}, {31, 37588}, {35, 8666}, {36, 8715}, {37, 391}, {38, 37598}, {40, 3218}, {44, 49714}, {45, 4969}, {46, 23958}, {55, 2975}, {56, 100}, {57, 2136}, {58, 17539}, {63, 1697}, {65, 3189}, {69, 3242}, {72, 452}, {75, 3945}, {80, 1392}, {81, 1043}, {85, 17158}, {86, 17589}, {92, 7518}, {99, 28531}, {101, 9083}, {104, 11248}, {125, 27715}, {140, 37624}, {141, 26104}, {142, 4402}, {144, 192}, {146, 7978}, {147, 7970}, {148, 7983}, {150, 10695}, {151, 10696}, {152, 10697}, {165, 21734}, {174, 7057}, {186, 47476}, {188, 1488}, {190, 1992}, {194, 7976}, {213, 27523}, {218, 644}, {226, 4323}, {238, 27549}, {244, 24440}, {263, 27809}, {278, 5174}, {279, 664}, {280, 1433}, {291, 38247}, {304, 31130}, {312, 4696}, {318, 34231}, {319, 5232}, {320, 4346}, {321, 4673}, {322, 17863}, {329, 950}, {330, 1002}, {333, 17588}, {335, 27480}, {341, 4358}, {344, 1279}, {345, 3744}, {347, 2897}, {350, 20943}, {354, 3893}, {355, 3091}, {376, 12702}, {377, 1056}, {381, 37705}, {382, 28224}, {385, 50776}, {388, 2099}, {392, 3876}, {402, 16211}, {404, 999}, {405, 6767}, {411, 22770}, {443, 15934}, {474, 7373}, {485, 35810}, {486, 35811}, {487, 7980}, {488, 7981}, {491, 26515}, {492, 26514}, {495, 2476}, {496, 4193}, {497, 2098}, {511, 50419}, {514, 38371}, {515, 962}, {516, 5059}, {522, 14812}, {523, 36171}, {524, 4419}, {527, 30332}, {528, 4440}, {529, 6284}, {535, 34719}, {536, 4454}, {537, 24695}, {542, 50883}, {543, 50888}, {594, 16884}, {595, 1331}, {597, 17269}, {599, 17395}, {616, 7975}, {617, 7974}, {621, 51689}, {622, 51691}, {627, 22912}, {628, 22867}, {631, 5690}, {651, 34040}, {659, 25574}, {668, 18135}, {672, 3208}, {704, 40907}, {718, 40906}, {726, 4788}, {730, 20081}, {740, 1278}, {752, 49708}, {754, 34685}, {758, 3648}, {760, 20065}, {856, 38284}, {858, 47536}, {860, 38295}, {872, 27291}, {894, 3886}, {900, 24097}, {901, 44873}, {902, 35284}, {908, 4345}, {912, 23340}, {940, 3996}, {942, 3889}, {946, 3832}, {957, 39698}, {958, 1621}, {959, 39694}, {960, 3681}, {961, 15375}, {964, 19717}, {966, 16777}, {982, 4642}, {984, 4704}, {986, 4392}, {993, 3746}, {996, 1126}, {1001, 5260}, {1006, 16202}, {1010, 4720}, {1018, 4253}, {1046, 4427}, {1058, 2478}, {1066, 26050}, {1086, 50120}, {1100, 2345}, {1104, 17776}, {1108, 27396}, {1145, 6921}, {1146, 27541}, {1150, 16347}, {1155, 34711}, {1158, 12703}, {1159, 11112}, {1191, 1265}, {1220, 2334}, {1270, 5604}, {1271, 5605}, {1284, 12642}, {1319, 1788}, {1329, 36972}, {1334, 21384}, {1376, 3304}, {1385, 3523}, {1386, 17280}, {1387, 6931}, {1388, 7288}, {1389, 6839}, {1398, 35974}, {1400, 3169}, {1420, 3158}, {1434, 7268}, {1449, 2321}, {1453, 3710}, {1468, 5255}, {1469, 25304}, {1470, 38901}, {1475, 3501}, {1478, 11009}, {1479, 5080}, {1484, 6971}, {1500, 16975}, {1587, 35641}, {1588, 35642}, {1616, 4383}, {1617, 37301}, {1656, 10283}, {1657, 28212}, {1699, 16189}, {1706, 3306}, {1724, 40091}, {1738, 23675}, {1743, 3161}, {1757, 49696}, {1829, 6995}, {1834, 3936}, {1837, 5048}, {1854, 37781}, {1870, 4200}, {1909, 4441}, {1938, 25271}, {1997, 6556}, {2049, 19740}, {2089, 12643}, {2093, 4311}, {2094, 34701}, {2102, 14807}, {2103, 14808}, {2176, 3780}, {2177, 32919}, {2238, 16969}, {2241, 5291}, {2256, 2287}, {2257, 3692}, {2275, 17756}, {2276, 17448}, {2280, 2329}, {2292, 7226}, {2320, 5559}, {2325, 16670}, {2403, 3667}, {2550, 11038}, {2551, 5289}, {2646, 3769}, {2771, 17652}, {2784, 5992}, {2802, 3874}, {2809, 17489}, {2810, 42448}, {2886, 15888}, {2888, 7979}, {2895, 26117}, {2896, 7977}, {2899, 26791}, {2979, 50586}, {3036, 31272}, {3052, 4884}, {3060, 16980}, {3061, 3930}, {3068, 44635}, {3069, 44636}, {3090, 5790}, {3160, 9436}, {3194, 14954}, {3212, 3598}, {3219, 5250}, {3246, 49702}, {3247, 3686}, {3263, 18156}, {3305, 51779}, {3307, 24648}, {3308, 24649}, {3315, 17054}, {3332, 29016}, {3333, 27003}, {3336, 5541}, {3339, 4315}, {3416, 3620}, {3419, 3487}, {3448, 7984}, {3452, 37723}, {3475, 4863}, {3485, 5086}, {3524, 34718}, {3525, 38028}, {3526, 38112}, {3529, 28174}, {3543, 11278}, {3545, 18357}, {3560, 12000}, {3576, 11362}, {3578, 49728}, {3579, 3655}, {3586, 51423}, {3589, 17309}, {3601, 5744}, {3614, 3829}, {3618, 17233}, {3619, 17295}, {3629, 17262}, {3630, 17255}, {3631, 17323}, {3640, 30333}, {3641, 30334}, {3644, 28582}, {3650, 15678}, {3653, 15721}, {3654, 13624}, {3656, 3839}, {3662, 3755}, {3663, 21296}, {3664, 17151}, {3666, 37655}, {3671, 50725}, {3673, 30806}, {3678, 3898}, {3684, 9310}, {3685, 3751}, {3689, 20323}, {3693, 26690}, {3695, 17526}, {3696, 4699}, {3698, 3742}, {3699, 6552}, {3701, 4737}, {3702, 4385}, {3704, 33089}, {3707, 16676}, {3714, 46897}, {3716, 23057}, {3717, 7290}, {3723, 17275}, {3726, 3959}, {3727, 49509}, {3730, 45751}, {3736, 17178}, {3739, 4371}, {3750, 10448}, {3753, 5045}, {3754, 3892}, {3756, 31227}, {3758, 49484}, {3790, 16475}, {3791, 17715}, {3812, 17609}, {3814, 37720}, {3816, 21031}, {3817, 37714}, {3823, 17241}, {3834, 4000}, {3836, 49677}, {3851, 38138}, {3854, 9779}, {3855, 38034}, {3878, 4127}, {3881, 5902}, {3883, 7174}, {3884, 3988}, {3887, 21222}, {3894, 4084}, {3897, 24929}, {3899, 4067}, {3900, 17496}, {3901, 4302}, {3902, 4968}, {3905, 21285}, {3907, 4804}, {3915, 5247}, {3927, 11111}, {3940, 5084}, {3946, 17296}, {3952, 17460}, {3974, 32942}, {3976, 24443}, {3977, 35261}, {3983, 4711}, {3984, 5795}, {3987, 4694}, {3991, 25082}, {3993, 49448}, {3994, 25253}, {4007, 5750}, {4021, 17272}, {4025, 45290}, {4026, 17238}, {4030, 17599}, {4033, 41316}, {4034, 5257}, {4050, 17474}, {4051, 17451}, {4065, 17461}, {4083, 31291}, {4085, 33087}, {4090, 4903}, {4101, 26580}, {4115, 24049}, {4160, 31290}, {4162, 4462}, {4194, 6198}, {4195, 37685}, {4197, 31419}, {4201, 32863}, {4202, 48847}, {4205, 27081}, {4225, 23853}, {4231, 11401}, {4232, 11363}, {4240, 11910}, {4248, 20818}, {4295, 10052}, {4297, 7990}, {4298, 5586}, {4301, 5691}, {4305, 5119}, {4310, 4645}, {4314, 9819}, {4318, 8271}, {4339, 32929}, {4349, 17116}, {4352, 17137}, {4353, 17288}, {4356, 17247}, {4357, 32099}, {4361, 4648}, {4363, 4747}, {4364, 28337}, {4366, 16920}, {4388, 32928}, {4390, 41239}, {4395, 17313}, {4399, 15668}, {4405, 49738}, {4421, 5204}, {4423, 45085}, {4429, 4966}, {4431, 7229}, {4433, 21010}, {4439, 16477}, {4445, 17045}, {4449, 21302}, {4470, 4665}, {4474, 48172}, {4478, 17327}, {4487, 4849}, {4504, 4729}, {4514, 5016}, {4527, 4649}, {4535, 49473}, {4547, 10176}, {4552, 12848}, {4555, 34342}, {4643, 4725}, {4646, 4850}, {4647, 41819}, {4657, 17372}, {4658, 26860}, {4659, 4667}, {4660, 49464}, {4662, 10179}, {4663, 4676}, {4664, 49515}, {4670, 28329}, {4675, 50125}, {4686, 7222}, {4690, 4748}, {4693, 32935}, {4695, 9335}, {4716, 50288}, {4723, 46937}, {4727, 16666}, {4732, 40328}, {4740, 49483}, {4741, 11160}, {4754, 50257}, {4764, 49525}, {4771, 21921}, {4772, 24325}, {4774, 47834}, {4780, 24231}, {4806, 4879}, {4814, 48242}, {4821, 49474}, {4831, 10385}, {4862, 33800}, {4865, 4892}, {4873, 4982}, {4883, 19804}, {4887, 50108}, {4894, 36974}, {4900, 5558}, {4901, 17353}, {4902, 36606}, {4922, 50343}, {4923, 4967}, {4930, 51409}, {4943, 30719}, {4973, 37572}, {4981, 30711}, {4986, 33942}, {4996, 10087}, {5011, 17736}, {5014, 7270}, {5015, 30991}, {5047, 9708}, {5049, 5439}, {5051, 31037}, {5054, 50823}, {5056, 5818}, {5057, 12701}, {5067, 38042}, {5068, 5587}, {5081, 7952}, {5083, 39776}, {5090, 7378}, {5091, 16380}, {5100, 5300}, {5189, 47537}, {5208, 41723}, {5217, 5303}, {5218, 34471}, {5220, 47357}, {5221, 39777}, {5223, 17261}, {5226, 9578}, {5229, 13463}, {5245, 30414}, {5246, 30415}, {5248, 5258}, {5259, 17544}, {5263, 17379}, {5266, 33168}, {5273, 5837}, {5284, 8162}, {5295, 31025}, {5372, 19278}, {5376, 11607}, {5434, 34502}, {5440, 24928}, {5442, 21842}, {5542, 48627}, {5543, 25719}, {5563, 25440}, {5564, 17394}, {5597, 5602}, {5598, 5601}, {5640, 23841}, {5695, 28503}, {5711, 14996}, {5719, 6856}, {5720, 5804}, {5722, 6919}, {5728, 25243}, {5737, 19333}, {5748, 9581}, {5761, 6844}, {5768, 37531}, {5770, 33596}, {5772, 17368}, {5774, 19270}, {5802, 22021}, {5815, 31018}, {5828, 37704}, {5847, 6646}, {5856, 12730}, {5880, 30340}, {5883, 50190}, {5920, 12533}, {5921, 39898}, {5933, 39773}, {6018, 33551}, {6144, 17334}, {6168, 34497}, {6172, 50110}, {6175, 16137}, {6193, 9933}, {6194, 22713}, {6225, 7973}, {6264, 9803}, {6265, 6979}, {6327, 49454}, {6366, 47695}, {6376, 25278}, {6462, 8210}, {6463, 8211}, {6548, 44314}, {6554, 6603}, {6600, 7677}, {6601, 17097}, {6630, 35313}, {6636, 8193}, {6645, 16919}, {6653, 33823}, {6701, 47033}, {6702, 32558}, {6763, 37563}, {6796, 32905}, {6826, 10597}, {6827, 10806}, {6830, 10943}, {6838, 21740}, {6840, 12116}, {6842, 32213}, {6847, 37533}, {6848, 37700}, {6850, 10805}, {6867, 14497}, {6875, 37621}, {6882, 32214}, {6893, 10596}, {6903, 35457}, {6905, 10680}, {6906, 10679}, {6909, 10306}, {6911, 12001}, {6924, 12331}, {6933, 8164}, {6938, 14988}, {6940, 16203}, {6941, 10942}, {6942, 22765}, {6948, 24475}, {6950, 11849}, {6953, 45770}, {6958, 19914}, {6959, 19907}, {6960, 10786}, {6972, 10785}, {6981, 11729}, {6987, 14054}, {7046, 11109}, {7052, 36930}, {7184, 25573}, {7408, 49542}, {7426, 47493}, {7486, 9956}, {7487, 41722}, {7492, 37546}, {7500, 44662}, {7504, 31479}, {7520, 37547}, {7580, 8158}, {7585, 7969}, {7586, 7968}, {7672, 7674}, {7681, 37725}, {7760, 30225}, {7787, 10800}, {7925, 50772}, {7951, 24387}, {7987, 43174}, {7989, 38155}, {7995, 18452}, {8000, 9874}, {8055, 8834}, {8056, 35577}, {8071, 37293}, {8094, 8422}, {8113, 12633}, {8114, 12634}, {8125, 8242}, {8126, 11924}, {8168, 25524}, {8186, 49556}, {8187, 49555}, {8197, 11367}, {8204, 11366}, {8227, 15022}, {8239, 11687}, {8240, 11688}, {8241, 11690}, {8243, 12638}, {8275, 35258}, {8390, 11685}, {8392, 11686}, {8581, 12448}, {8591, 9884}, {8668, 11509}, {8726, 12658}, {8732, 34489}, {8972, 13902}, {9316, 9363}, {9331, 25092}, {9352, 32636}, {9451, 17136}, {9535, 44039}, {9538, 27505}, {9540, 35763}, {9541, 35610}, {9543, 9616}, {9575, 14930}, {9588, 30392}, {9589, 28164}, {9624, 10175}, {9654, 17577}, {9669, 37375}, {9709, 17531}, {9776, 11518}, {9782, 11524}, {9783, 11527}, {9787, 11528}, {9789, 11532}, {9791, 11533}, {9793, 11534}, {9795, 11899}, {9799, 12650}, {9800, 12651}, {9801, 12652}, {9802, 11280}, {9804, 12654}, {9807, 12656}, {9809, 13253}, {9845, 10860}, {9848, 12125}, {9859, 17616}, {9897, 21630}, {9911, 12087}, {9945, 37545}, {9961, 12680}, {10025, 30695}, {10026, 23903}, {10031, 10609}, {10044, 17647}, {10074, 17100}, {10164, 30389}, {10167, 31798}, {10267, 37106}, {10303, 15178}, {10371, 33075}, {10404, 20292}, {10405, 14942}, {10427, 14151}, {10436, 32087}, {10441, 50702}, {10446, 11521}, {10465, 10890}, {10473, 35634}, {10480, 35614}, {10481, 32098}, {10483, 34690}, {10531, 13729}, {10569, 12128}, {10570, 44765}, {10572, 11415}, {10588, 15950}, {10590, 37710}, {10593, 43734}, {10624, 20214}, {10699, 20344}, {10700, 21290}, {10701, 34186}, {10702, 34188}, {10703, 33650}, {10704, 14360}, {10705, 13219}, {10707, 10896}, {10711, 45631}, {10866, 11678}, {10895, 11235}, {10965, 22760}, {11008, 17347}, {11024, 11525}, {11061, 32298}, {11108, 15935}, {11113, 15172}, {11114, 15171}, {11200, 28870}, {11230, 46936}, {11234, 18258}, {11249, 11491}, {11252, 11844}, {11253, 11843}, {11281, 37703}, {11365, 13595}, {11376, 32537}, {11499, 13279}, {11501, 18967}, {11517, 37313}, {11535, 11891}, {11541, 28190}, {11544, 15679}, {11604, 21398}, {11875, 32147}, {11876, 32146}, {11900, 45289}, {12019, 50890}, {12047, 37708}, {12114, 38669}, {12115, 37437}, {12383, 12898}, {12384, 13099}, {12389, 12400}, {12391, 12395}, {12438, 16212}, {12527, 12575}, {12528, 12672}, {12529, 12709}, {12530, 12721}, {12532, 12758}, {12534, 12876}, {12535, 12877}, {12542, 12655}, {12543, 12657}, {12641, 41554}, {12667, 18243}, {12751, 25485}, {12763, 13271}, {12782, 23473}, {12849, 13100}, {12854, 37544}, {13161, 33134}, {13541, 17777}, {13601, 17625}, {13678, 13702}, {13745, 49718}, {13798, 13822}, {13869, 36154}, {13935, 35762}, {13941, 13959}, {14077, 17494}, {14210, 33937}, {14552, 28606}, {14555, 34064}, {14584, 51562}, {14759, 25272}, {14774, 24068}, {14872, 45776}, {14997, 16483}, {15015, 33812}, {15325, 17566}, {15558, 46685}, {15569, 27268}, {15570, 38053}, {15600, 17312}, {15640, 28208}, {15672, 18253}, {15674, 21677}, {15676, 35016}, {15683, 28194}, {15688, 50809}, {15702, 38066}, {15703, 38081}, {15705, 50817}, {15707, 50832}, {15708, 50821}, {15709, 50830}, {15863, 16173}, {15971, 48909}, {16062, 19823}, {16210, 51712}, {16284, 26563}, {16394, 50042}, {16397, 37540}, {16466, 17697}, {16468, 49685}, {16469, 17339}, {16478, 33166}, {16486, 37680}, {16597, 26147}, {16610, 21896}, {16667, 17355}, {16672, 17330}, {16705, 33297}, {16706, 17386}, {16781, 33854}, {16930, 20132}, {16971, 17750}, {17037, 20212}, {17056, 46875}, {17063, 46190}, {17079, 32007}, {17103, 32004}, {17117, 17391}, {17119, 17392}, {17141, 24282}, {17143, 34284}, {17160, 17378}, {17163, 49598}, {17166, 29298}, {17236, 49511}, {17237, 50076}, {17243, 37650}, {17246, 40341}, {17256, 50077}, {17276, 28566}, {17287, 17396}, {17290, 50112}, {17301, 17374}, {17305, 21356}, {17311, 17366}, {17320, 17360}, {17338, 35227}, {17343, 17772}, {17358, 38047}, {17369, 50087}, {17387, 37756}, {17441, 50698}, {17469, 33163}, {17491, 24851}, {17579, 18990}, {17597, 18141}, {17614, 51788}, {17615, 17622}, {17619, 51362}, {17690, 43993}, {17721, 28808}, {17724, 25529}, {17728, 37828}, {17740, 37539}, {17742, 33950}, {17766, 24248}, {17774, 17783}, {17781, 41864}, {17794, 21219}, {17800, 28216}, {18230, 24393}, {18231, 30478}, {18283, 37420}, {18444, 37108}, {18446, 37421}, {18467, 37736}, {18519, 21669}, {18600, 21281}, {18623, 34039}, {19692, 51710}, {19738, 48862}, {19741, 27797}, {19743, 48863}, {19796, 26729}, {19824, 23537}, {19945, 24418}, {19954, 27918}, {20061, 20074}, {20089, 28850}, {20173, 30807}, {20257, 30949}, {20293, 48303}, {20330, 38149}, {20347, 36854}, {20530, 24761}, {20533, 32029}, {20669, 23559}, {20760, 28376}, {20895, 44735}, {20911, 39731}, {20970, 27040}, {21075, 27131}, {21077, 30384}, {21283, 33112}, {21301, 48333}, {21343, 46403}, {21616, 37721}, {21620, 31019}, {21746, 50577}, {21870, 30861}, {21933, 27395}, {21935, 33141}, {22370, 27624}, {22647, 22969}, {22758, 37622}, {22793, 50688}, {23528, 48380}, {23536, 33131}, {24004, 24485}, {24416, 33910}, {24473, 50193}, {24474, 50701}, {24914, 36920}, {25048, 34434}, {25241, 43161}, {25244, 27340}, {25261, 27288}, {25269, 51196}, {25270, 33888}, {25280, 30963}, {25466, 33108}, {25592, 33298}, {25639, 37719}, {25723, 31721}, {25875, 42884}, {25962, 51416}, {26051, 37635}, {26062, 35262}, {26064, 43990}, {26098, 32920}, {26137, 30543}, {26242, 41015}, {26394, 26395}, {26418, 26419}, {26487, 33281}, {26494, 26495}, {26503, 26504}, {26777, 48285}, {26818, 27334}, {26824, 29066}, {26853, 29350}, {27065, 31435}, {27086, 37579}, {27484, 31342}, {27506, 38955}, {28029, 42461}, {28146, 49140}, {28160, 49135}, {28178, 49138}, {28186, 33703}, {28292, 47676}, {28522, 49532}, {28600, 41836}, {28610, 34716}, {28629, 44840}, {28633, 28640}, {28634, 28639}, {28635, 28641}, {28646, 44447}, {28849, 40868}, {29366, 48143}, {30283, 37022}, {30312, 41555}, {30424, 51101}, {30588, 43533}, {30720, 40621}, {30852, 50443}, {30962, 34063}, {31162, 31673}, {31246, 33559}, {31313, 50293}, {31423, 38127}, {31447, 31662}, {31547, 31567}, {31548, 31568}, {31551, 31559}, {31552, 31560}, {31730, 34632}, {31786, 37423}, {31992, 32212}, {32095, 41838}, {32354, 32394}, {32760, 45392}, {32784, 50315}, {32846, 49472}, {32859, 50065}, {33100, 38456}, {33118, 42378}, {33655, 36931}, {33930, 49779}, {34607, 36004}, {34610, 37299}, {34628, 51082}, {34629, 50910}, {34648, 50871}, {34689, 49736}, {34707, 36005}, {34744, 37568}, {34753, 50843}, {34930, 34932}, {35058, 51223}, {35104, 39780}, {35239, 37105}, {35242, 51705}, {35448, 37403}, {35659, 35669}, {35661, 35665}, {35979, 39783}, {36037, 36944}, {36404, 50026}, {36855, 37191}, {36991, 43166}, {37339, 37592}, {37407, 37615}, {37462, 40587}, {37532, 48363}, {37553, 38000}, {37631, 49734}, {37760, 47321}, {37901, 47538}, {37907, 47472}, {37909, 47540}, {37913, 49553}, {38057, 42819}, {38071, 50797}, {38074, 51709}, {38087, 51006}, {38089, 50953}, {38176, 46935}, {38514, 47274}, {39349, 39363}, {40420, 44794}, {41135, 50885}, {41792, 41794}, {41821, 41921}, {42045, 49745}, {43511, 49227}, {43512, 49226}, {43816, 43824}, {44367, 50250}, {45508, 45572}, {45509, 45573}, {46716, 50626}, {47359, 51146}, {47490, 51725}, {47721, 47869}, {47724, 50761}, {47793, 48294}, {47796, 48287}, {47805, 48327}, {47824, 48344}, {47825, 48289}, {47836, 48328}, {47840, 48347}, {48037, 48338}, {48164, 48332}, {48208, 48290}, {48798, 48819}, {48800, 48820}, {48806, 48824}, {48877, 48903}, {48936, 50422}, {49456, 49503}, {49457, 49689}, {49508, 51035}, {49700, 49712}, {49716, 49735}, {49743, 50171}, {49746, 50074}, {50041, 51591}, {50043, 50054}, {50044, 50061}, {50050, 50068}, {50055, 50067}, {50154, 50164}, {50155, 50259}, {50175, 50183}, {50179, 50279}, {50247, 50248}, {50398, 51715}, {50690, 51118}, {50794, 51706}, {50949, 51149}, {50950, 51193}, {51156, 51169}, {1, 3633}, {2, 20049}, {7, 12630}, {8, 20050}, {3242, 49679}, {3621, 20014}, {3868, 3885}, {7972, 26726}, {8148, 18526}, {12536, 12541}, {34631, 50818}, {34747, 51093}, {34748, 50805}, {45719, 45720}, {49469, 49498}, {49490, 49678}, {1, 3244}, {2, 3241}, {3, 1483}, {4, 1482}, {6, 51147}, {7, 3243}, {8, 1}, {10, 3635}, {20, 944}, {40, 5882}, {65, 34791}, {69, 3242}, {72, 9957}, {75, 49478}, {100, 1317}, {144, 390}, {146, 7978}, {147, 7970}, {148, 7983}, {149, 1320}, {150, 10695}, {151, 10696}, {152, 10697}, {153, 10698}, {192, 49470}, {193, 51192}, {194, 7976}, {238, 49691}, {239, 49771}, {329, 7962}, {355, 10222}, {487, 7980}, {488, 7981}, {551, 51091}, {597, 51145}, {599, 50998}, {616, 7975}, {617, 7974}, {621, 51689}, {622, 51691}, {627, 22912}, {628, 22867}, {944, 37727}, {956, 37728}, {962, 7982}, {984, 49471}, {1145, 12735}, {1278, 24349}, {1320, 25416}, {1757, 49696}, {1992, 51000}, {2136, 12437}, {2475, 34195}, {2550, 42871}, {2888, 7979}, {2896, 7977}, {2975, 37734}, {3146, 962}, {3241, 51093}, {3416, 49465}, {3419, 50194}, {3434, 2099}, {3436, 2098}, {3448, 7984}, {3578, 49739}, {3579, 32900}, {3617, 3623}, {3621, 8}, {3625, 1125}, {3632, 10}, {3644, 49461}, {3648, 5441}, {3655, 51087}, {3679, 51071}, {3681, 5919}, {3751, 49684}, {3828, 51095}, {3868, 3555}, {3869, 3057}, {3893, 5836}, {3952, 17460}, {4240, 12626}, {4295, 12559}, {4307, 50284}, {4419, 17318}, {4440, 24841}, {4454, 4644}, {4461, 4344}, {4462, 4162}, {4645, 49675}, {4659, 4667}, {4660, 49464}, {4661, 3877}, {4664, 50778}, {4677, 551}, {4678, 20057}, {4701, 3636}, {4729, 4504}, {4740, 51055}, {4764, 49525}, {4814, 48325}, {4924, 4856}, {4929, 3950}, {5086, 11011}, {5176, 5048}, {5223, 30331}, {5541, 33337}, {5691, 4301}, {5881, 946}, {5903, 3874}, {5904, 3878}, {5921, 39898}, {6193, 9933}, {6194, 22713}, {6223, 7971}, {6224, 7972}, {6225, 7973}, {6327, 49454}, {6361, 18481}, {6764, 12629}, {6796, 32905}, {7057, 12646}, {7426, 47493}, {7672, 15185}, {7991, 4297}, {8591, 9884}, {9797, 12127}, {9799, 12650}, {9800, 12651}, {9801, 12652}, {9802, 12653}, {9803, 6264}, {9804, 12654}, {9807, 12656}, {9809, 13253}, {9812, 11224}, {9859, 17644}, {9874, 8000}, {9897, 21630}, {9961, 12680}, {10449, 50637}, {10914, 942}, {11041, 36867}, {11061, 32298}, {11160, 50999}, {11362, 13607}, {11415, 30323}, {11525, 14563}, {11684, 10543}, {11691, 8422}, {12125, 9848}, {12245, 3}, {12247, 12737}, {12383, 12898}, {12384, 13099}, {12389, 12400}, {12391, 12395}, {12526, 4314}, {12527, 12575}, {12528, 12672}, {12529, 12709}, {12530, 12721}, {12531, 11}, {12532, 12758}, {12533, 5920}, {12534, 12876}, {12535, 12877}, {12541, 3680}, {12542, 12655}, {12543, 12657}, {12625, 21627}, {12632, 3189}, {12645, 5}, {12648, 3870}, {12649, 36846}, {12672, 13600}, {12699, 11278}, {12702, 34773}, {12751, 25485}, {12849, 13100}, {13219, 10705}, {13678, 13702}, {13798, 13822}, {14360, 10704}, {14450, 16126}, {14807, 2102}, {14808, 2103}, {14872, 45776}, {14923, 65}, {15971, 48909}, {16017, 8241}, {17164, 2650}, {17299, 49467}, {17363, 3883}, {17494, 47729}, {17777, 13541}, {17784, 3476}, {18525, 22791}, {20008, 9797}, {20012, 20037}, {20014, 20050}, {20016, 50015}, {20017, 49687}, {20050, 3633}, {20051, 20040}, {20052, 3616}, {20053, 3632}, {20054, 3621}, {20055, 36534}, {20070, 20}, {20072, 49709}, {20076, 36977}, {20085, 149}, {20095, 6224}, {20293, 48303}, {20344, 10699}, {21290, 10700}, {21301, 48333}, {21302, 4449}, {22647, 22969}, {24248, 49455}, {24349, 49490}, {25304, 1469}, {25413, 24475}, {25722, 8581}, {26117, 41813}, {26824, 48304}, {28610, 34716}, {31145, 2}, {31162, 51077}, {31298, 17794}, {31302, 192}, {31547, 31567}, {31548, 31568}, {31551, 31559}, {31552, 31560}, {31888, 15680}, {32354, 32394}, {32850, 4864}, {33090, 17015}, {33650, 10703}, {33703, 48661}, {34186, 10701}, {34188, 10702}, {34605, 34749}, {34611, 34699}, {34627, 3656}, {34628, 51082}, {34631, 50805}, {34632, 50811}, {34641, 51103}, {34689, 49736}, {34718, 50824}, {34747, 51096}, {34748, 50831}, {34790, 31792}, {34932, 34930}, {35659, 35669}, {35661, 35665}, {36154, 13869}, {36972, 1329}, {36991, 43166}, {38514, 47274}, {39351, 14942}, {39776, 5083}, {42020, 38496}, {46403, 21343}, {46685, 15558}, {47321, 47491}, {47490, 51725}, {47724, 50761}, {47745, 13464}, {48304, 50767}, {48798, 48819}, {48800, 48820}, {48806, 48824}, {48877, 48903}, {49060, 49329}, {49061, 49330}, {49168, 22837}, {49169, 22836}, {49447, 49462}, {49448, 3993}, {49450, 37}, {49459, 24325}, {49470, 49475}, {49474, 49479}, {49493, 49491}, {49501, 49523}, {49503, 49456}, {49532, 49535}, {49677, 3836}, {49688, 1386}, {49689, 49457}, {49690, 49524}, {49698, 1279}, {49702, 3246}, {49704, 49695}, {49707, 238}, {49709, 49699}, {49712, 49700}, {49714, 44}, {49719, 5434}, {50043, 50070}, {50055, 50072}, {50074, 49746}, {50107, 50130}, {50154, 50182}, {50155, 50259}, {50172, 42045}, {50248, 50247}, {50277, 49735}, {50279, 50179}, {50289, 3879}, {50295, 50281}, {50343, 4922}, {50810, 3655}, {50818, 34748}, {50835, 47357}, {50864, 31162}, {50871, 34648}, {50872, 34631}, {51192, 49681}, {51515, 10283}

▸ Functional Relations40 relations

• X₁₄₅ = midpoint of X_i and X_j for these {i,j}: {1, 3633}, {2, 20049}, {7, 12630}, {8, 20050}, {3242, 49679}, {3621, 20014}, {3868, 3885}, {7972, 26726}, {8148, 18526}, {12536, 12541}, {34631, 50818}, {34747, 51093}, {34748, 50805}, {45719, 45720}, {49469, 49498}, {49490, 49678}

• X₁₄₅ = reflection of X_i in X_j for these {i,j}: {1, 3244}, {2, 3241}, {3, 1483}, {4, 1482}, {6, 51147}, {7, 3243}, {8, 1}, {10, 3635}, {20, 944}, {40, 5882}, {65, 34791}, {69, 3242}, {72, 9957}, {75, 49478}, {100, 1317}, {144, 390}, {146, 7978}, {147, 7970}, {148, 7983}, {149, 1320}, {150, 10695}, {151, 10696}, {152, 10697}, {153, 10698}, {192, 49470}, {193, 51192}, {194, 7976}, {238, 49691}, {239, 49771}, {329, 7962}, {355, 10222}, {487, 7980}, {488, 7981}, {551, 51091}, {597, 51145}, {599, 50998}, {616, 7975}, {617, 7974}, {621, 51689}, {622, 51691}, {627, 22912}, {628, 22867}, {944, 37727}, {956, 37728}, {962, 7982}, {984, 49471}, {1145, 12735}, {1278, 24349}, {1320, 25416}, {1757, 49696}, {1992, 51000}, {2136, 12437}, {2475, 34195}, {2550, 42871}, {2888, 7979}, {2896, 7977}, {2975, 37734}, {3146, 962}, {3241, 51093}, {3416, 49465}, {3419, 50194}, {3434, 2099}, {3436, 2098}, {3448, 7984}, {3578, 49739}, {3579, 32900}, {3617, 3623}, {3621, 8}, {3625, 1125}, {3632, 10}, {3644, 49461}, {3648, 5441}, {3655, 51087}, {3679, 51071}, {3681, 5919}, {3751, 49684}, {3828, 51095}, {3868, 3555}, {3869, 3057}, {3893, 5836}, {3952, 17460}, {4240, 12626}, {4295, 12559}, {4307, 50284}, {4419, 17318}, {4440, 24841}, {4454, 4644}, {4461, 4344}, {4462, 4162}, {4645, 49675}, {4659, 4667}, {4660, 49464}, {4661, 3877}, {4664, 50778}, {4677, 551}, {4678, 20057}, {4701, 3636}, {4729, 4504}, {4740, 51055}, {4764, 49525}, {4814, 48325}, {4924, 4856}, {4929, 3950}, {5086, 11011}, {5176, 5048}, {5223, 30331}, {5541, 33337}, {5691, 4301}, {5881, 946}, {5903, 3874}, {5904, 3878}, {5921, 39898}, {6193, 9933}, {6194, 22713}, {6223, 7971}, {6224, 7972}, {6225, 7973}, {6327, 49454}, {6361, 18481}, {6764, 12629}, {6796, 32905}, {7057, 12646}, {7426, 47493}, {7672, 15185}, {7991, 4297}, {8591, 9884}, {9797, 12127}, {9799, 12650}, {9800, 12651}, {9801, 12652}, {9802, 12653}, {9803, 6264}, {9804, 12654}, {9807, 12656}, {9809, 13253}, {9812, 11224}, {9859, 17644}, {9874, 8000}, {9897, 21630}, {9961, 12680}, {10449, 50637}, {10914, 942}, {11041, 36867}, {11061, 32298}, {11160, 50999}, {11362, 13607}, {11415, 30323}, {11525, 14563}, {11684, 10543}, {11691, 8422}, {12125, 9848}, {12245, 3}, {12247, 12737}, {12383, 12898}, {12384, 13099}, {12389, 12400}, {12391, 12395}, {12526, 4314}, {12527, 12575}, {12528, 12672}, {12529, 12709}, {12530, 12721}, {12531, 11}, {12532, 12758}, {12533, 5920}, {12534, 12876}, {12535, 12877}, {12541, 3680}, {12542, 12655}, {12543, 12657}, {12625, 21627}, {12632, 3189}, {12645, 5}, {12648, 3870}, {12649, 36846}, {12672, 13600}, {12699, 11278}, {12702, 34773}, {12751, 25485}, {12849, 13100}, {13219, 10705}, {13678, 13702}, {13798, 13822}, {14360, 10704}, {14450, 16126}, {14807, 2102}, {14808, 2103}, {14872, 45776}, {14923, 65}, {15971, 48909}, {16017, 8241}, {17164, 2650}, {17299, 49467}, {17363, 3883}, {17494, 47729}, {17777, 13541}, {17784, 3476}, {18525, 22791}, {20008, 9797}, {20012, 20037}, {20014, 20050}, {20016, 50015}, {20017, 49687}, {20050, 3633}, {20051, 20040}, {20052, 3616}, {20053, 3632}, {20054, 3621}, {20055, 36534}, {20070, 20}, {20072, 49709}, {20076, 36977}, {20085, 149}, {20095, 6224}, {20293, 48303}, {20344, 10699}, {21290, 10700}, {21301, 48333}, {21302, 4449}, {22647, 22969}, {24248, 49455}, {24349, 49490}, {25304, 1469}, {25413, 24475}, {25722, 8581}, {26117, 41813}, {26824, 48304}, {28610, 34716}, {31145, 2}, {31162, 51077}, {31298, 17794}, {31302, 192}, {31547, 31567}, {31548, 31568}, {31551, 31559}, {31552, 31560}, {31888, 15680}, {32354, 32394}, {32850, 4864}, {33090, 17015}, {33650, 10703}, {33703, 48661}, {34186, 10701}, {34188, 10702}, {34605, 34749}, {34611, 34699}, {34627, 3656}, {34628, 51082}, {34631, 50805}, {34632, 50811}, {34641, 51103}, {34689, 49736}, {34718, 50824}, {34747, 51096}, {34748, 50831}, {34790, 31792}, {34932, 34930}, {35659, 35669}, {35661, 35665}, {36154, 13869}, {36972, 1329}, {36991, 43166}, {38514, 47274}, {39351, 14942}, {39776, 5083}, {42020, 38496}, {46403, 21343}, {46685, 15558}, {47321, 47491}, {47490, 51725}, {47724, 50761}, {47745, 13464}, {48304, 50767}, {48798, 48819}, {48800, 48820}, {48806, 48824}, {48877, 48903}, {49060, 49329}, {49061, 49330}, {49168, 22837}, {49169, 22836}, {49447, 49462}, {49448, 3993}, {49450, 37}, {49459, 24325}, {49470, 49475}, {49474, 49479}, {49493, 49491}, {49501, 49523}, {49503, 49456}, {49532, 49535}, {49677, 3836}, {49688, 1386}, {49689, 49457}, {49690, 49524}, {49698, 1279}, {49702, 3246}, {49704, 49695}, {49707, 238}, {49709, 49699}, {49712, 49700}, {49714, 44}, {49719, 5434}, {50043, 50070}, {50055, 50072}, {50074, 49746}, {50107, 50130}, {50154, 50182}, {50155, 50259}, {50172, 42045}, {50248, 50247}, {50277, 49735}, {50279, 50179}, {50289, 3879}, {50295, 50281}, {50343, 4922}, {50810, 3655}, {50818, 34748}, {50835, 47357}, {50864, 31162}, {50871, 34648}, {50872, 34631}, {51192, 49681}, {51515, 10283}

• X₁₄₅ = isogonal conjugate of X₃₄₄₅

• X₁₄₅ = isotomic conjugate of X₄₃₇₃

• X₁₄₅ = complement of X₃₆₂₁

• X₁₄₅ = anticomplement of X₈

• X₁₄₅ = incircle-inverse of X₃₇₇₄₃

• X₁₄₅ = Steiner-circumellipse-inverse of X₃₀₀₈

• X₁₄₅ = orthoptic-circle-of-Steiner-circumellipse-inverse of X₅₁₂₁

• X₁₄₅ = anticomplement of the isogonal conjugate of X₅₆

• X₁₄₅ = anticomplement of the isotomic conjugate of X₇

• X₁₄₅ = complement of the isotomic conjugate of X₃₆₆₀₆

• X₁₄₅ = isogonal conjugate of the anticomplement of X₂₈₈₅

• X₁₄₅ = isogonal conjugate of the complement of X₄₂₀₂₀

• X₁₄₅ = isotomic conjugate of the anticomplement of X₃₁₆₁

• X₁₄₅ = isotomic conjugate of the isogonal conjugate of X₃₀₅₂

• X₁₄₅ = polar conjugate of the isogonal conjugate of X₂₀₈₁₈

• X₁₄₅ = anticomplementary isogonal conjugate of X₃₄₃₆

• X₁₄₅ = X_i-anticomplementary conjugate of X_j for these (i,j): {1, 3436}, {2, 21286}, {6, 329}, {7, 6327}, {25, 5942}, {31, 144}, {32, 3177}, {34, 4}, {41, 30695}, {56, 8}, {57, 69}, {58, 3869}, {59, 3952}, {65, 1330}, {77, 1370}, {81, 20245}, {85, 315}, {87, 20557}, {101, 4462}, {106, 5176}, {108, 20293}, {109, 513}, {221, 6223}, {222, 4329}, {226, 21287}, {241, 20552}, {251, 20248}, {269, 3434}, {273, 11442}, {278, 21270}, {279, 21285}, {284, 18750}, {513, 33650}, {560, 21218}, {593, 21273}, {603, 20}, {604, 2}, {608, 5905}, {649, 37781}, {651, 20295}, {664, 21301}, {667, 39351}, {692, 4468}, {738, 6604}, {757, 35614}, {849, 2975}, {934, 21302}, {951, 7270}, {961, 17751}, {1014, 17135}, {1037, 10327}, {1042, 2475}, {1088, 21280}, {1106, 145}, {1118, 5906}, {1262, 21272}, {1319, 21290}, {1333, 63}, {1393, 2888}, {1394, 6225}, {1395, 193}, {1396, 17220}, {1397, 192}, {1398, 12649}, {1399, 3648}, {1400, 2895}, {1401, 21289}, {1402, 1654}, {1404, 30578}, {1407, 7}, {1408, 1}, {1409, 3151}, {1410, 3152}, {1411, 5080}, {1412, 75}, {1413, 962}, {1414, 512}, {1415, 514}, {1416, 518}, {1417, 519}, {1418, 2890}, {1420, 42020}, {1422, 21279}, {1424, 32548}, {1427, 2893}, {1428, 17794}, {1429, 20345}, {1431, 4388}, {1434, 17137}, {1436, 189}, {1438, 30807}, {1447, 20554}, {1455, 151}, {1456, 152}, {1457, 153}, {1458, 20344}, {1459, 34188}, {1461, 693}, {1462, 20347}, {1474, 92}, {1477, 32850}, {1973, 30694}, {2006, 21277}, {2099, 21291}, {2149, 190}, {2162, 20348}, {2194, 45738}, {2199, 20211}, {2215, 26872}, {3213, 14361}, {3248, 17036}, {3450, 25253}, {3451, 312}, {3669, 150}, {3676, 21293}, {4017, 3448}, {4554, 21304}, {4564, 668}, {4565, 7192}, {4573, 17217}, {4617, 46402}, {4625, 44445}, {4637, 4374}, {6063, 21275}, {6611, 5932}, {6614, 3900}, {7023, 36845}, {7045, 3888}, {7099, 347}, {7121, 20535}, {7125, 6527}, {7153, 21281}, {7175, 30660}, {7178, 21294}, {7180, 21221}, {7251, 21215}, {7316, 17491}, {7341, 17140}, {7366, 4452}, {8059, 4397}, {8809, 32064}, {9447, 46706}, {9456, 908}, {16945, 3621}, {16947, 17147}, {18268, 1959}, {20567, 33796}, {23971, 35312}, {23979, 4552}, {24027, 100}, {28607, 5744}, {28615, 17781}, {32636, 2891}, {32669, 3904}, {32674, 4391}, {32675, 3762}, {32714, 46400}, {34079, 14206}, {36049, 20296}, {36059, 20294}, {36098, 6371}, {36110, 8677}, {36141, 30565}, {36146, 3766}, {38828, 4106}, {40151, 21296}, {41280, 17486}, {41526, 21219}, {42467, 8048}, {43924, 149}, {51640, 13219}, {51641, 34186}, {51642, 148}, {51649, 12384}, {51653, 147}, {51654, 34547}, {51655, 14360}, {51656, 146}, {51658, 34548}, {51662, 34550}

• X₁₄₅ = X_i-complementary conjugate of X_j for these (i,j): {8699, 513}, {36603, 141}, {36606, 2887}, {36621, 17046}, {38255, X₁₄₅ = 21244}, {40026, 626}

• X₁₄₅ = X_i-Ceva conjugate of X_j for these (i,j): {7, 2}, {190, 31182}, {664, 30719}, {1016, 30720}, {1266, 30577}, {3875, 3210}, {3879, 17778}, {4076, 100}, {4998, 25737}, {9312, 3177}, {10106, 5484}, {18743, 3161}, {35577, 3622}, {39126, 5435}, {40420, 9}, {41629, 1743}, {43290, 3667}, {44724, 43290}

• X₁₄₅ = X_i-cross conjugate of X_j for these (i,j): {1, 44301}, {1743, 5435}, {3158, 3161}, {3161, 2}, {3667, 43290}, {3756, 3667}, {3950, 18743}, {4534, 4462}, {4849, 1743}, {4856, 41629}, {4943, 30720}, {4953, 4521}, {6555, 24150}, {12640, 44720}, {31182, 190}, {40621, 30719}, {44720, 39701}, {45219, 1420}, {51658, 100}

• X₁₄₅ = X_i-isoconjugate of X_j for these (i,j): {1, 3445}, {2, 38266}, {6, 8056}, {8, 16945}, {9, 40151}, {31, 4373}, {32, 40014}, {41, 27818}, {55, 19604}, {56, 3680}, {513, 1293}, {514, 34080}, {604, 6557}, {607, 27832}, {649, 27834}, {650, 38828}, {900, 36042}, {1015, 5382}, {1022, 2429}, {1106, 6556}, {1261, 46367}, {1333, 4052}, {1420, 33963}, {3158, 16079}, {3762, 32645}, {31343, 43924}

• X₁₄₅ = X_i-Dao conjugate of X_j for these (i,j): {1, 3680}, {1, 45036}, {2, 4373}, {3, 3445}, {9, 8056}, {37, 4052}, {145, 3621}, {223, 19604}, {478, 40151}, {514, 40621}, {522, 3756}, {900, 5516}, {1086, 4521}, {1293, 39026}, {1358, 3669}, {1420, 34039}, {1743, 3928}, {3052, 36641}, {3160, 27818}, {3161, 6557}, {3667, 3756}, {3950, 22034}, {4162, 36639}, {4849, 22312}, {4859, 24392}, {4862, 30827}, {5375, 27834}, {5435, 36640}, {6376, 40014}, {6552, 6556}, {10029, 36905}, {10563, 11530}, {14321, 17058}, {21255, 21342}, {32664, 38266}, {39048, 51839}

• X₁₄₅ = cevapoint of X_i and X_j for these (i,j): {1, 2136}, {8, 8834}, {200, 45193}, {1743, 3158}, {3052, 20818}, {3161, 15519}, {3667, 3756}, {3950, 4849}, {4162, 4534}, {4404, 4939}, {4521, 4953}, {4943, 40621}, {12640, 45219}, {12643, 12646}

• X₁₄₅ = crosspoint of X_i and X_j for these (i,j): {1, 39969}, {2, 36606}, {7, 5435}, {664, 1016}, {18743, 39126}, {43290, 44724}

• X₁₄₅ = crosssum of X_i and X_j for these (i,j): {1, 21214}, {6, 21000}, {244, 6615}, {663, 1015}, {3271, 17424}

• X₁₄₅ = trilinear pole of line {2976, 3667}

• X₁₄₅ = crossdifference of every pair of points on line {649, 6363}

• X₁₄₅ = X₆₄₃-beth conjugate of X₅₆

• X₁₄₅ = exsimilicenter of incircle and AC-incircle

• X₁₄₅ = X₆₄-of-intouch-triangle

• X₁₄₅ = trilinear pole of line X₂₉₇₆X₃₆₆₇ (radical axis of incircle and AC-incircle, and the pole of X₂ wrt the Spieker circle)

• X₁₄₅ = X₈-of-anticomplementary-triangle

• X₁₄₅ = X₁₁₃₈₁-of-excentral-triangle

• X₁₄₅ = perspector of intouch triangle and Gemini triangle 29

• X₁₄₅ = {X₃₇₇₉₄,X₃₇₇₉₅}-harmonic conjugate of X₂

• barycentric product X_i*X_j for these {i,j}: {1, 18743}, {7, 3161}, {8, 5435}, {9, 39126}, {10, 41629}, {43, 27496}, {56, 44723}, {57, 44720}, {75, 1743}, {76, 3052}, {83, 4884}, {85, 3158}, {86, 3950}, {88, 4487}, {92, 4855}, {99, 14321}, {100, 4462}, {190, 3667}, {222, 44721}, {264, 20818}, {274, 4849}, {278, 44722}, {279, 6555}, {306, 4248}, {312, 1420}, {313, 33628}, {321, 16948}, {333, 4848}, {514, 43290}, {519, 31227}, {646, 51658}, {658, 4546}, {662, 4404}, {664, 4521}, {666, 4925}, {668, 4394}, {673, 4899}, {799, 4729}, {1016, 3756}, {1086, 44724}, {1088, 4936}, {1222, 45204}, {1268, 4856}, {1275, 4953}, {1978, 8643}, {2403, 17780}, {3210, 39701}, {3676, 30720}, {3699, 30719}, {4076, 40617}, {4162, 4554}, {4452, 24150}, {4504, 27805}, {4534, 4998}, {4555, 14425}, {4564, 4939}, {4573, 44729}, {4600, 21950}, {4881, 18359}, {4891, 32009}, {4898, 30598}, {4918, 14534}, {4935, 39962}, {4949, 32042}, {6049, 6557}, {6357, 44727}, {6632, 23764}, {8055, 44301}, {12640, 40420}, {15519, 27818}, {32017, 45219}, {34234, 51433}

• barycentric quotient X_i/X_j for these {i,j}: {1, 8056}, {2, 4373}, {6, 3445}, {7, 27818}, {8, 6557}, {9, 3680}, {10, 4052}, {31, 38266}, {56, 40151}, {57, 19604}, {75, 40014}, {77, 27832}, {100, 27834}, {101, 1293}, {109, 38828}, {346, 6556}, {604, 16945}, {644, 31343}, {692, 34080}, {765, 5382}, {1122, 45205}, {1200, 45229}, {1279, 51839}, {1420, 57}, {1743, 1}, {2136, 24151}, {2403, 6548}, {2441, 23345}, {2976, 6084}, {3052, 6}, {3158, 9}, {3161, 8}, {3210, 27835}, {3434, 27826}, {3667, 514}, {3731, 10563}, {3756, 1086}, {3870, 27819}, {3873, 27827}, {3875, 27813}, {3879, 27820}, {3891, 27817}, {3896, 27823}, {3905, 27821}, {3907, 27831}, {3950, 10}, {4162, 650}, {4248, 27}, {4394, 513}, {4404, 1577}, {4449, 27837}, {4452, 27828}, {4453, 27836}, {4462, 693}, {4487, 4358}, {4504, 4369}, {4521, 522}, {4534, 11}, {4546, 3239}, {4729, 661}, {4848, 226}, {4849, 37}, {4855, 63}, {4856, 1125}, {4881, 3218}, {4884, 141}, {4891, 3739}, {4898, 1698}, {4899, 3912}, {4917, 3305}, {4918, 1211}, {4924, 29571}, {4925, 918}, {4929, 17284}, {4936, 200}, {4939, 4858}, {4943, 4521}, {4949, 4802}, {4952, 17279}, {4953, 1146}, {4964, 25666}, {5435, 7}, {6049, 5435}, {6555, 346}, {8643, 649}, {9312, 27829}, {9436, 10029}, {12640, 3452}, {14100, 45202}, {14284, 21120}, {14321, 523}, {14350, 28161}, {14351, 28225}, {14425, 900}, {15519, 3161}, {16948, 81}, {17780, 2415}, {18211, 16726}, {18743, 75}, {20818, 3}, {21950, 3120}, {23344, 2429}, {23511, 47636}, {23764, 6545}, {24150, 6553}, {27496, 6384}, {27818, 16078}, {30719, 3676}, {30720, 3699}, {31182, 3667}, {31227, 903}, {32665, 36042}, {32719, 32645}, {33628, 58}, {37764, 17951}, {39126, 85}, {39701, 39694}, {40151, 16079}, {40617, 1358}, {40621, 3756}, {41629, 86}, {43290, 190}, {44301, 8051}, {44720, 312}, {44721, 7017}, {44722, 345}, {44723, 3596}, {44724, 1016}, {44729, 3700}, {45036, 3928}, {45204, 3663}, {45219, 3752}, {51433, 908}, {51658, 3669}

• {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 2, 3622}, {1, 8, 2}, {1, 10, 3616}, {1, 43, 1201}, {1, 200, 19861}, {1, 978, 1149}, {1, 1125, 38314}, {1, 1698, 551}, {1, 1722, 28011}, {1, 3214, 28370}, {1, 3241, 3623}, {1, 3244, 3241}, {1, 3293, 995}, {1, 3617, 46934}, {1, 3621, 3617}, {1, 3624, 3636}, {1, 3625, 9780}, {1, 3626, 5550}, {1, 3632, 10}, {1, 3635, 20057}, {1, 3679, 1125}, {1, 3811, 4511}, {1, 3870, 34772}, {1, 3938, 36565}, {1, 4668, 3624}, {1, 4677, 1698}, {1, 4701, 19877}, {1, 4816, 3634}, {1, 4853, 19860}, {1, 4882, 8583}, {1, 5262, 17024}, {1, 5312, 50604}, {1, 5554, 10586}, {1, 6738, 10580}, {1, 6765, 78}, {1, 7080, 24558}, {1, 10573, 3086}, {1, 11519, 4853}, {1, 12629, 3872}, {1, 12647, 3085}, {1, 12648, 10528}, {1, 12649, 10529}, {1, 16830, 29570}, {1, 18391, 14986}, {1, 20014, 31145}, {1, 20018, 20036}, {1, 20049, 20054}, {1, 20050, 3621}, {1, 20052, 46932}, {1, 20053, 4678}, {1, 31145, 46933}, {1, 31397, 5703}, {1, 34595, 51105}, {1, 34747, 3633}, {1, 36846, 38460}, {1, 39584, 38475}, {1, 39587, 29624}, {1, 41575, 12649}, {1, 41684, 499}, {1, 49168, 10527}, {1, 49169, 5552}, {1, 49458, 36534}, {1, 49476, 17316}, {1, 49495, 239}, {1, 49762, 29579}, {1, 49772, 16020}, {1, 50016, 16825}, {1, 50286, 29569}, {1, 50581, 1193}, {1, 50582, 29840}, {1, 51093, 3244}, {2, 8, 3617}, {2, 10, 46932}, {2, 239, 24599}, {2, 3006, 30762}, {2, 3617, 46933}, {2, 3621, 8}, {2, 3622, 46934}, {2, 3623, 1}, {2, 3912, 30833}, {2, 4393, 17014}, {2, 4678, 10}, {2, 6542, 29616}, {2, 9780, 46930}, {2, 17316, 29621}, {2, 20008, 12649}, {2, 20011, 20012}, {2, 20013, 20007}, {2, 20014, 3621}, {2, 20040, 20036}, {2, 20045, 26245}, {2, 20048, 19998}, {2, 20050, 20054}, {2, 20052, 4678}, {2, 20056, 7172}, {2, 26773, 27112}, {2, 26820, 27165}, {2, 29585, 29624}, {2, 29588, 29585}, {2, 29824, 30948}, {2, 46933, 46931}, {2, 51093, 51092}, {3, 1483, 7967}, {5, 10247, 10595}, {6, 17314, 346}, {6, 17388, 17314}, {7, 3875, 4452}, {7, 4452, 4373}, {7, 4460, 3875}, {7, 9312, 43983}, {7, 25718, 9312}, {8, 10, 4678}, {8, 938, 5554}, {8, 3241, 1}, {8, 3244, 3623}, {8, 3616, 10}, {8, 3621, 31145}, {8, 3622, 46933}, {8, 3623, 3622}, {8, 3632, 20052}, {8, 3633, 20014}, {8, 3957, 10587}, {8, 4511, 7080}, {8, 9780, 3679}, {8, 9797, 36845}, {8, 10453, 17751}, {8, 10578, 24987}, {8, 20011, 20051}, {8, 20014, 20054}, {8, 20018, 20012}, {8, 20037, 20036}, {8, 20053, 3632}, {8, 20057, 3616}, {8, 27383, 6735}, {8, 34772, 10528}, {8, 36845, 12649}, {8, 38314, 9780}, {8, 38460, 10529}, {8, 39567, 24599}, {8, 49771, 39567}, {10, 551, 19878}, {10, 3244, 3635}, {10, 3616, 2}, {10, 3624, 19877}, {10, 3625, 4746}, {10, 3632, 8}, {10, 3635, 1}, {10, 3636, 3624}, {10, 4062, 27558}, {10, 4678, 3617}, {10, 4701, 4668}, {10, 4746, 3679}, {10, 19878, 1698}, {10, 20053, 20052}, {10, 46932, 46933}, {11, 11681, 5154}, {11, 12607, 11681}, {12, 3813, 11680}, {12, 11680, 5141}, {20, 3868, 9965}, {20, 20075, 20066}, {20, 20076, 20067}, {37, 5839, 391}, {40, 5731, 3522}, {40, 5882, 5731}, {42, 10453, 2}, {42, 50001, 30942}, {45, 4969, 37654}, {55, 2975, 4189}, {55, 12513, 2975}, {56, 100, 4188}, {56, 3913, 100}, {63, 4313, 17576}, {65, 3189, 17784}, {65, 3476, 3600}, {65, 3600, 21454}, {65, 34791, 3873}, {65, 37738, 3476}, {69, 4360, 3672}, {72, 3488, 452}, {72, 9957, 3877}, {75, 21605, 21432}, {78, 938, 2}, {78, 6765, 3935}, {81, 1043, 11115}, {149, 20060, 4}, {192, 193, 144}, {192, 3177, 25237}, {192, 20072, 20073}, {192, 49704, 390}, {193, 20073, 20072}, {200, 10580, 2}, {226, 12625, 5175}, {239, 17316, 2}, {239, 17389, 17316}, {239, 29575, 29628}, {239, 29619, 17244}, {279, 6604, 51351}, {319, 17321, 5232}, {319, 17393, 17321}, {320, 50101, 4346}, {344, 3759, 37681}, {344, 49698, 10005}, {354, 3893, 5836}, {355, 5603, 3091}, {355, 10222, 5603}, {377, 5082, 33110}, {388, 3434, 2475}, {392, 34790, 3876}, {495, 24390, 2476}, {496, 17757, 4193}, {497, 3436, 5046}, {499, 27529, 2}, {499, 45701, 27529}, {551, 1698, 5550}, {551, 3626, 1698}, {664, 6604, 279}, {899, 21214, 27625}, {899, 30947, 2}, {944, 3868, 20076}, {944, 3885, 20075}, {944, 6361, 18481}, {946, 16200, 5734}, {950, 7962, 9785}, {950, 11523, 329}, {956, 3295, 21}, {958, 1621, 16865}, {958, 3303, 1621}, {960, 5919, 3890}, {978, 1149, 28370}, {995, 50575, 3293}, {999, 5687, 404}, {1001, 5260, 16859}, {1056, 5082, 377}, {1058, 3421, 2478}, {1100, 17299, 2345}, {1125, 3625, 3679}, {1125, 3626, 51069}, {1125, 3679, 9780}, {1125, 4746, 10}, {1125, 9780, 2}, {1125, 51071, 1}, {1149, 3214, 978}, {1150, 19765, 16347}, {1193, 35633, 29824}, {1193, 50581, 3240}, {1210, 6735, 25005}, {1210, 27383, 2}, {1319, 1788, 5265}, {1319, 41687, 1788}, {1376, 3304, 5253}, {1376, 5253, 17572}, {1385, 5657, 3523}, {1388, 40663, 7288}, {1420, 3158, 4855}, {1420, 4848, 5435}, {1420, 4855, 4881}, {1449, 2321, 5749}, {1468, 5255, 17126}, {1482, 18525, 22791}, {1697, 6762, 63}, {1698, 4677, 3626}, {1698, 5550, 2}, {1698, 51069, 9780}, {1722, 28011, 7292}, {1743, 3950, 3161}, {1743, 4898, 3950}, {1743, 4929, 4899}, {1829, 7718, 6995}, {1837, 32049, 5176}, {1909, 17144, 4441}, {1998, 3870, 3935}, {1999, 50582, 12649}, {2098, 10950, 497}, {2098, 41711, 12635}, {2099, 10944, 388}, {2176, 3780, 37657}, {2275, 20691, 17756}, {2340, 28124, 28057}, {2550, 42871, 11038}, {3006, 26228, 2}, {3008, 29573, 29627}, {3008, 29627, 2}, {3011, 30741, 2}, {3057, 3486, 390}, {3057, 37740, 3486}, {3085, 3086, 10320}, {3085, 10527, 2}, {3085, 34625, 10527}, {3086, 5552, 2}, {3086, 34619, 5552}, {3160, 32003, 9436}, {3177, 20111, 144}, {3189, 3873, 4190}, {3189, 34791, 3600}, {3240, 29824, 2}, {3241, 3616, 20057}, {3241, 3621, 3622}, {3241, 3633, 3621}, {3241, 9797, 36846}, {3241, 15519, 37743}, {3241, 20014, 3617}, {3241, 20049, 31145}, {3241, 20050, 8}, {3241, 20053, 3616}, {3241, 20054, 46934}, {3241, 20057, 3635}, {3241, 34747, 20049}, {3241, 36845, 38460}, {3241, 38314, 51071}, {3241, 49495, 39567}, {3243, 3340, 11520}, {3243, 3680, 3340}, {3244, 3625, 51071}, {3244, 3632, 20057}, {3244, 3633, 8}, {3244, 3634, 51097}, {3244, 15808, 51095}, {3244, 20014, 3622}, {3244, 20049, 3617}, {3244, 20050, 2}, {3244, 34747, 20050}, {3244, 41575, 38460}, {3247, 3686, 5296}, {3340, 10106, 7}, {3419, 3487, 5177}, {3434, 3891, 30699}, {3476, 14923, 4190}, {3485, 5086, 6871}, {3485, 5252, 5261}, {3486, 3869, 6872}, {3555, 3868, 4430}, {3555, 37727, 36977}, {3579, 32900, 3655}, {3579, 51087, 32900}, {3600, 12632, 17784}, {3600, 17784, 4190}, {3601, 24391, 5744}, {3616, 3632, 4678}, {3616, 4678, 46932}, {3616, 19877, 3624}, {3616, 20050, 20053}, {3616, 20052, 3617}, {3616, 20053, 8}, {3616, 20057, 1}, {3616, 27558, 24946}, {3617, 3622, 2}, {3617, 20054, 31145}, {3617, 31145, 8}, {3617, 46932, 10}, {3617, 46934, 46931}, {3617, 51092, 3623}, {3621, 3623, 2}, {3621, 4678, 20052}, {3621, 20013, 20015}, {3621, 20018, 20051}, {3621, 20040, 20012}, {3621, 20041, 20036}, {3621, 20049, 20014}, {3621, 20052, 3632}, {3621, 20057, 46932}, {3621, 29588, 39587}, {3621, 51071, 46930}, {3622, 20054, 8}, {3622, 31145, 3617}, {3622, 46930, 1125}, {3623, 3633, 20054}, {3623, 20011, 20036}, {3623, 20014, 8}, {3623, 20049, 3621}, {3623, 20050, 31145}, {3623, 20052, 3616}, {3623, 20053, 46932}, {3623, 20054, 46933}, {3623, 31145, 46934}, {3624, 3632, 4668}, {3624, 3636, 3616}, {3624, 4668, 10}, {3624, 19877, 2}, {3625, 3679, 8}, {3625, 51071, 1125}, {3626, 4677, 8}, {3626, 19878, 10}, {3632, 3635, 3616}, {3632, 4668, 4701}, {3632, 20052, 31145}, {3632, 20053, 3621}, {3632, 20057, 2}, {3633, 3635, 20053}, {3633, 20039, 20051}, {3633, 20050, 20049}, {3633, 51093, 1}, {3634, 41150, 1125}, {3635, 4701, 3636}, {3635, 20050, 20052}, {3635, 20052, 3622}, {3635, 20053, 2}, {3635, 20057, 3623}, {3636, 4668, 19877}, {3636, 4701, 10}, {3648, 5441, 15680}, {3655, 50810, 10304}, {3656, 34627, 3839}, {3661, 26626, 2}, {3661, 29584, 26626}, {3663, 21296, 45789}, {3664, 17151, 31995}, {3679, 38314, 2}, {3679, 51071, 38314}, {3681, 3890, 960}, {3685, 3751, 17350}, {3693, 40133, 26690}, {3699, 42020, 6552}, {3702, 4385, 4671}, 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29832, 2}, {20048, 20052, 20047}, {20050, 20057, 3632}, {20050, 51092, 46933}, {20050, 51093, 3623}, {20053, 20057, 10}, {20053, 38314, 4746}, {20054, 31145, 3621}, {20054, 46932, 20052}, {20054, 51092, 1}, {20064, 20068, 20078}, {20066, 20067, 20}, {20072, 20073, 144}, {20072, 49704, 49709}, {20075, 20076, 20}, {20247, 21272, 3212}, {21075, 41012, 27131}, {21627, 37709, 5175}, {22765, 32141, 6942}, {22836, 49169, 34619}, {22837, 49168, 34625}, {23891, 29400, 29715}, {23891, 29438, 29400}, {24349, 50284, 20090}, {24393, 38316, 18230}, {24599, 29621, 2}, {24883, 25650, 2}, {24936, 25446, 2}, {24968, 25479, 2}, {25415, 45287, 4295}, {25459, 25663, 2}, {25719, 40719, 31994}, {25746, 25805, 2}, {25773, 25843, 2}, {25879, 25965, 2}, {26029, 26093, 2}, {26038, 26102, 2}, {26395, 48493, 26394}, {26419, 48494, 26418}, {26495, 49402, 26494}, {26504, 49401, 26503}, {26514, 49060, 492}, {26515, 49061, 491}, {26531, 26658, 2}, {26626, 50079, 3661}, {26757, 26805, 2}, {26964, 27096, 2}, {26965, 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29621}, {39581, 48856, 16830}, {42871, 49486, 32922}, {44635, 49232, 3068}, {44636, 49233, 3069}, {44720, 44722, 6555}, {45476, 49329, 492}, {45477, 49330, 491}, {45572, 48746, 45508}, {45573, 48747, 45509}, {46933, 46934, 2}, {48856, 50310, 2}, {49447, 49462, 192}, {49447, 49470, 49462}, {49447, 49709, 5698}, {49451, 49466, 8}, {49470, 49681, 49704}, {49470, 51192, 390}, {49476, 49495, 8}, {49476, 49771, 1}, {49490, 49493, 49491}, {49491, 49493, 24349}, {49497, 49534, 49707}, {49695, 49709, 49699}, {49699, 49709, 49704}, {49765, 50114, 17284}, {50017, 50286, 16816}, {50043, 50070, 51592}, {50805, 50818, 50872}, {50805, 50831, 50818}, {51068, 51105, 2}, {51069, 51091, 51071}, {51072, 51103, 2}

Notes

Let A' be the reflection of the midpoint of segment BC in X₁, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₁₄₅. Let A″ be the reflection of the A in X₁, and define B″ and C″ cyclically. Let A’’’B’’’C’’’ be the intouch triangle. The lines A’’A’’’, B’’B’’’, C’’C’’’ concur in X₁₄₅. (Randy Hutson, 9/23/2011)

Let H_a be the hyperbola passing through A, with foci at B and C. This is called the A-Soddy hyperbola in (Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 12.4 The Soddy hyperbolas, p. 143). Let L_a be the polar of X₂ with respect to H_a. Define L_b and L_c cyclically. The lines L_a, L_b, L_c concur in X₁₄₅. (Randy Hutson, September 5, 2015)

Let A'B'C' and A″B″C″ be the intouch and extouch triangles. X₁₄₅ is the radical center of the circumcircles of AA'A″, BB'B″, CC'C″. (Randy Hutson, September 14, 2016)