Clark Kimberling's

Encyclopedia of Triangle Centers - ETC

X(68001) = PERSPECTOR OF THESE TRIANGLES: CTR28-8 AND CTR13-9.1

X(68001) lies on these lines: {1, 84}, {4, 519}, {8, 63989}, {10, 6969}, {33, 53530}, {40, 997}, {56, 54156}, {57, 2800}, {80, 1537}, {145, 67999}, {200, 517}, {226, 64322}, {388, 54198}, {392, 30503}, {515, 7962}, {518, 54135}, {551, 6935}, {758, 68032}, {944, 12575}, {946, 3340}, {962, 11682}, {971, 64897}, {1000, 5658}, {1158, 1420}, {1319, 52027}, {1389, 33576}, {1482, 9856}, {1490, 3057}, {1512, 62218}, {1519, 5587}, {1532, 3679}, {1538, 5790}, {1697, 6261}, {1864, 2099}, {2093, 22753}, {2096, 4315}, {2098, 12650}, {2136, 17857}, {2801, 3243}, {2950, 12740}, {3149, 7991}, {3158, 6326}, {3333, 64021}, {3576, 6950}, {3600, 54199}, {3601, 40257}, {3656, 8727}, {3869, 68036}, {3872, 67998}, {3877, 7411}, {3884, 12520}, {3890, 10884}, {3899, 41338}, {4311, 64190}, {4853, 5777}, {4915, 18908}, {5119, 52026}, {5250, 37106}, {5289, 6282}, {5450, 63208}, {5534, 23340}, {5573, 32486}, {5603, 11019}, {5657, 20103}, {5691, 30323}, {5693, 6762}, {5697, 63988}, {5727, 26333}, {5730, 6769}, {5734, 11520}, {5768, 63993}, {5886, 31249}, {5887, 57279}, {5903, 67880}, {6256, 37709}, {6830, 38021}, {6831, 11522}, {6833, 9624}, {6847, 11518}, {6848, 11362}, {6879, 8227}, {6883, 31435}, {6906, 64953}, {6927, 43174}, {6938, 50811}, {6941, 11530}, {7308, 64733}, {7489, 61146}, {7967, 30331}, {8583, 31788}, {8726, 58679}, {9578, 12608}, {9589, 37468}, {9613, 64119}, {9836, 11534}, {9845, 64358}, {9948, 64703}, {10039, 63966}, {10106, 63962}, {10157, 40587}, {10270, 17614}, {10396, 64042}, {10703, 34036}, {10860, 37611}, {10914, 67881}, {10944, 12679}, {11224, 41702}, {11249, 54290}, {11373, 33899}, {11499, 63138}, {11525, 59388}, {12526, 22770}, {12528, 36846}, {12559, 21628}, {12565, 31786}, {12616, 50443}, {12629, 14872}, {12701, 64261}, {12767, 37587}, {14647, 44675}, {15733, 43166}, {16126, 37447}, {16189, 18452}, {16670, 52431}, {17638, 30223}, {17652, 66062}, {18446, 31393}, {19861, 37560}, {28194, 50701}, {31159, 37714}, {33597, 53053}, {37252, 54422}, {37526, 66019}, {37533, 48667}, {37708, 41698}, {37712, 64203}, {37738, 64000}, {37837, 61763}, {41556, 64192}, {45770, 49163}, {46917, 63132}, {61762, 63399}, {63391, 67886}, {63987, 64120}, {64162, 64324}, {66107, 68057}

X(68001) = reflection of X(i) in X(j) for these {i,j}: {8, 67874}, {40, 997}, {2093, 22753}, {2096, 4315}, {5727, 26333}, {5768, 63993}, {6282, 5289}, {10860, 37611}, {18391, 946}, {31146, 3656}, {41556, 64192}, {63137, 5720}, {63430, 1}, {66226, 45776}
X(68001) = perspector of circumconic {{A, B, C, X(37141), X(65337)}}
X(68001) = pole of line {56, 12650} with respect to the Feuerbach hyperbola
X(68001) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(1433), X(3680)}}, {{A, B, C, X(4052), X(52037)}}
X(68001) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 12672, 12705}, {1, 6001, 63430}, {1, 7995, 12114}, {517, 5720, 63137}, {1699, 13253, 25415}, {1699, 25415, 3577}, {2098, 12688, 12650}, {3680, 68000, 5881}, {5881, 7982, 3680}, {6001, 45776, 66226}, {10698, 67988, 6264}

X(68002) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR5-2.2 WRT CTR28-8

X(68002) lies on these lines: {2, 84}, {3, 5328}, {4, 4313}, {5, 5658}, {7, 6848}, {8, 6932}, {20, 5748}, {40, 27525}, {140, 12246}, {329, 6838}, {376, 22792}, {381, 64144}, {498, 64130}, {515, 3622}, {549, 48664}, {631, 6259}, {908, 37421}, {938, 1532}, {944, 1387}, {946, 7966}, {962, 10528}, {971, 3090}, {1058, 1538}, {1071, 5704}, {1158, 27065}, {1389, 4323}, {1490, 3091}, {1519, 9785}, {1698, 54227}, {1750, 18219}, {3085, 67999}, {3088, 28836}, {3146, 52026}, {3452, 37108}, {3522, 68003}, {3525, 34862}, {3529, 40262}, {3545, 5787}, {3616, 6957}, {3617, 7971}, {3628, 12684}, {3634, 7992}, {3817, 63981}, {3839, 64261}, {4305, 41698}, {4308, 12115}, {5045, 8166}, {5046, 5731}, {5056, 6245}, {5070, 61556}, {5218, 12679}, {5219, 37434}, {5260, 18237}, {5261, 63992}, {5273, 5811}, {5281, 66992}, {5342, 50442}, {5435, 6834}, {5450, 16859}, {5550, 12114}, {5657, 54199}, {5693, 5775}, {5705, 59687}, {5709, 64143}, {5714, 19541}, {5744, 6960}, {5768, 6941}, {5780, 6907}, {5842, 10248}, {5927, 6856}, {6001, 9780}, {6261, 6871}, {6264, 20085}, {6668, 16112}, {6831, 36991}, {6860, 12671}, {6889, 18230}, {6908, 18228}, {6919, 10884}, {6925, 27383}, {6931, 11220}, {6933, 9942}, {6943, 10430}, {6949, 31188}, {6953, 9776}, {6979, 62773}, {6981, 13369}, {6986, 56889}, {6988, 37822}, {7288, 12678}, {7485, 9910}, {7681, 10580}, {7682, 11036}, {8164, 9856}, {8165, 30503}, {8236, 10531}, {8889, 12136}, {8972, 19068}, {9612, 50700}, {9778, 64119}, {9812, 11500}, {9842, 25525}, {9940, 67992}, {9948, 54447}, {10303, 52027}, {10588, 12688}, {10589, 12680}, {10590, 63988}, {11491, 30332}, {12536, 37700}, {13941, 19067}, {14647, 18243}, {17527, 21151}, {26364, 63971}, {31018, 40256}, {32785, 49234}, {32786, 49235}, {54198, 59417}, {54445, 64120}, {59333, 61012}, {59385, 64156}, {60954, 63437}, {63399, 64114}, {64108, 64190}, {66465, 68057}

X(68002) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6260, 6223}, {2, 67994, 6705}, {5, 5658, 9799}, {631, 6259, 54052}, {1071, 6969, 5704}, {5219, 67048, 37434}, {6260, 63966, 2}, {6260, 6705, 67993}, {6705, 67993, 67994}, {12608, 64148, 962}, {14647, 18243, 54228}, {19877, 54228, 14647}, {64813, 67889, 4}

X(68003) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR20-2.7 WRT CTR28-8

X(68003) lies on these lines: {1, 6927}, {2, 515}, {3, 3452}, {4, 30282}, {5, 40262}, {10, 37837}, {20, 30852}, {35, 63989}, {40, 27383}, {55, 946}, {72, 40249}, {78, 6962}, {84, 3305}, {140, 6245}, {142, 6911}, {226, 6905}, {411, 2077}, {517, 59584}, {548, 22792}, {549, 971}, {550, 64813}, {631, 1490}, {908, 63438}, {936, 6261}, {938, 13607}, {950, 6834}, {993, 67874}, {997, 64315}, {1006, 5316}, {1125, 6918}, {1158, 61122}, {1210, 1319}, {1385, 9843}, {1532, 4304}, {1750, 6935}, {2800, 6174}, {3035, 65404}, {3090, 64261}, {3428, 6745}, {3522, 68002}, {3524, 5658}, {3525, 64144}, {3526, 5787}, {3530, 34862}, {3560, 9842}, {3579, 54198}, {3586, 6969}, {3601, 6848}, {3614, 6831}, {3814, 4297}, {3817, 5842}, {3911, 6880}, {3947, 65387}, {4314, 7681}, {4848, 21740}, {4855, 6838}, {5044, 9942}, {5217, 66992}, {5218, 63992}, {5219, 50701}, {5436, 6964}, {5438, 6908}, {5691, 6956}, {5703, 13464}, {5720, 5745}, {5768, 31231}, {5837, 45770}, {5919, 63287}, {5927, 37298}, {6001, 10164}, {6223, 15717}, {6256, 6865}, {6692, 6970}, {6734, 47745}, {6825, 57284}, {6826, 58463}, {6833, 63998}, {6864, 26105}, {6891, 64706}, {6906, 67048}, {6915, 34486}, {6921, 10884}, {6953, 62829}, {6960, 57287}, {6961, 41854}, {6972, 64707}, {6987, 30827}, {7682, 24929}, {7987, 12667}, {8726, 17567}, {9799, 10303}, {10106, 10786}, {10299, 12246}, {10902, 54348}, {11012, 21075}, {11218, 63259}, {11227, 17564}, {11374, 64001}, {11491, 12053}, {12108, 61556}, {12512, 64119}, {12617, 58404}, {12679, 63756}, {12684, 61811}, {12688, 52793}, {13405, 22753}, {15692, 54052}, {16192, 64190}, {16293, 25893}, {18483, 50700}, {19862, 63980}, {20206, 40555}, {21151, 60972}, {21154, 63432}, {21484, 67974}, {22770, 59722}, {22835, 51783}, {24391, 37700}, {30478, 67881}, {31190, 64317}, {31788, 59675}, {34772, 64279}, {35242, 63962}, {37251, 55108}, {37623, 67850}, {37713, 66465}, {37732, 40958}, {38150, 47357}, {51755, 64310}, {54227, 64118}, {60942, 66051}, {61804, 67994}, {63168, 68032}, {64154, 64188}

X(68003) = midpoint of X(i) and X(j) for these {i,j}: {2, 52026}, {3, 67889}, {3576, 64148}, {5658, 52027}, {11218, 64280}, {54052, 67993}
X(68003) = reflection of X(i) in X(j) for these {i,j}: {6260, 67889}, {66465, 37713}
X(68003) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 52026, 515}, {140, 64804, 6245}, {631, 1490, 6705}, {1210, 33597, 5882}, {3149, 13411, 946}, {3524, 5658, 52027}, {5219, 50701, 67877}, {5703, 67880, 13464}, {6880, 18446, 3911}, {6961, 41854, 67041}, {6970, 18443, 6692}, {30503, 59572, 6684}

X(68004) = PARALLELOGIC CENTER OF THESE TRIANGLES: CTR22-4.1 WRT CTR28-8

X(68004) lies on these lines: {1, 475}, {3, 34703}, {4, 519}, {8, 33}, {10, 451}, {19, 3189}, {20, 64932}, {24, 8715}, {25, 3913}, {27, 50292}, {28, 12437}, {29, 4102}, {34, 145}, {108, 4848}, {225, 1897}, {232, 20691}, {235, 12607}, {264, 17144}, {273, 17158}, {281, 4007}, {318, 4673}, {378, 8666}, {406, 3679}, {427, 3813}, {468, 64123}, {469, 50306}, {518, 1902}, {521, 67893}, {522, 1770}, {528, 3575}, {529, 1885}, {535, 18560}, {551, 52252}, {607, 4513}, {944, 1753}, {952, 1872}, {958, 7071}, {962, 52849}, {1038, 52365}, {1172, 2321}, {1210, 15500}, {1452, 63130}, {1593, 12513}, {1594, 24387}, {1824, 1891}, {1825, 57287}, {1826, 36934}, {1828, 1862}, {1829, 3880}, {1841, 17388}, {1848, 5090}, {1869, 7009}, {1870, 3244}, {1876, 34791}, {1883, 33895}, {1887, 10944}, {1890, 5853}, {1905, 10914}, {2136, 7713}, {2212, 3717}, {2299, 3710}, {2329, 2332}, {2356, 49476}, {2802, 41722}, {2907, 36797}, {3088, 34625}, {3089, 34619}, {3100, 34823}, {3169, 44103}, {3192, 50581}, {3208, 41320}, {3214, 61226}, {3241, 4200}, {3303, 62972}, {3434, 11392}, {3486, 40971}, {3515, 4421}, {3516, 11194}, {3541, 45700}, {3542, 45701}, {3555, 67965}, {3632, 65128}, {3695, 56178}, {3871, 52427}, {3900, 22300}, {4186, 64744}, {4198, 12536}, {4212, 42057}, {4213, 4685}, {4219, 24391}, {4222, 12640}, {4347, 45281}, {5101, 10912}, {5125, 23710}, {5178, 30687}, {5247, 8750}, {5338, 64146}, {5687, 11399}, {5882, 37305}, {6197, 64117}, {6737, 7046}, {6738, 63965}, {6744, 17917}, {6995, 12632}, {7282, 50563}, {7412, 11362}, {7507, 11235}, {7952, 64163}, {8144, 60427}, {8668, 11383}, {8756, 37055}, {10573, 51359}, {11236, 37197}, {11363, 56176}, {12528, 64875}, {15149, 29574}, {16785, 56832}, {18719, 64002}, {24524, 54412}, {26020, 37722}, {29573, 37382}, {31623, 60730}, {34822, 66593}, {35974, 62837}, {37441, 43174}, {37468, 64930}, {38462, 56814}, {39579, 66251}, {41789, 41863}, {48696, 54428}, {55431, 64314}, {57808, 65206}, {64003, 64858}

X(68004) = X(i)-isoconjugate-of-X(j) for these {i, j}: {48, 66633}
X(68004) = X(i)-Dao conjugate of X(j) for these {i, j}: {1249, 66633}, {3686, 4001}, {17058, 4025}
X(68004) = pole of line {3583, 3667} with respect to the polar circle
X(68004) = intersection, other than A, B, C, of circumconics {{A, B, C, X(145), X(64068)}}, {{A, B, C, X(1039), X(11363)}}, {{A, B, C, X(3189), X(42360)}}, {{A, B, C, X(3680), X(3879)}}, {{A, B, C, X(4052), X(4102)}}
X(68004) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 56876, 1861}, {8, 33, 46878}, {1824, 12135, 1891}, {1897, 5174, 225}, {6198, 56877, 10}

X(68005) = X(5) OF CTR28-8

X(68005) lies on these lines: {5, 3812}, {72, 48482}, {84, 17616}, {405, 6261}, {515, 960}, {912, 63980}, {942, 63963}, {946, 5173}, {971, 5450}, {1012, 1898}, {1071, 11375}, {1155, 1158}, {1484, 58611}, {1490, 15931}, {1709, 59327}, {1728, 63992}, {1837, 12672}, {1858, 6831}, {2478, 67998}, {2779, 5908}, {2800, 6797}, {3256, 12705}, {3359, 17646}, {3427, 5811}, {5044, 6796}, {5692, 64261}, {5730, 14872}, {5842, 31837}, {5881, 17615}, {5887, 6928}, {5927, 6256}, {6245, 21616}, {6260, 47510}, {6675, 6705}, {6684, 18251}, {6830, 13750}, {6834, 14647}, {6835, 20292}, {6962, 9961}, {7082, 37302}, {9943, 52265}, {10157, 63964}, {10395, 63989}, {11499, 62357}, {12047, 67919}, {12114, 40263}, {12671, 25917}, {12675, 37737}, {15071, 37692}, {15297, 18237}, {16471, 57276}, {23961, 31828}, {26878, 64280}, {31775, 41871}, {31788, 64763}, {31806, 64171}, {31870, 64157}, {37700, 42843}, {37730, 45776}, {38043, 58608}, {40256, 58660}, {44229, 64119}, {50195, 67856}, {52027, 59319}

X(68005) = midpoint of X(i) and X(j) for these {i,j}: {72, 48482}, {1158, 12688}, {6245, 31803}, {6261, 12664}, {12114, 40263}, {31828, 34862}
X(68005) = reflection of X(i) in X(j) for these {i,j}: {942, 63963}, {1071, 18260}, {6796, 5044}, {31788, 64763}, {32159, 5777}, {40256, 58660}
X(68005) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {515, 5777, 32159}

X(68006) = TRIPOLE OF PERSPECTIVITY AXIS OF THESE TRIANGLES: EULER0 AND ABC

X(68006) lies on the Wallace hyperbola and on these lines: {1, 280}, {2, 253}, {4, 55304}, {20, 394}, {22, 46944}, {63, 347}, {69, 41914}, {110, 23608}, {147, 7396}, {148, 43670}, {194, 63092}, {275, 54746}, {487, 55898}, {488, 55894}, {651, 55114}, {2060, 3183}, {3079, 15312}, {3091, 59424}, {3101, 60784}, {3343, 62346}, {3523, 46832}, {3543, 51892}, {6194, 10565}, {6525, 34147}, {6527, 31956}, {6617, 36413}, {11348, 11427}, {14362, 40839}, {15238, 32064}, {27382, 56943}, {27402, 52676}, {30265, 53087}, {32973, 46625}, {37187, 42352}, {40138, 46831}, {44436, 45245}, {44440, 60114}, {45200, 56013}, {51952, 55888}, {51953, 55883}, {55119, 62798}

X(68006) = isogonal conjugate of X(31956)
X(68006) = anticomplement of X(459)
X(68006) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 31956}, {19, 3348}, {48, 42465}, {1973, 56594}, {2155, 14365}, {2184, 28781}
X(68006) = X(i)-Dao conjugate of X(j) for these {i, j}: {3, 31956}, {6, 3348}, {459, 459}, {1249, 42465}, {3344, 3346}, {6337, 56594}, {45245, 14365}
X(68006) = X(i)-Ceva conjugate of X(j) for these {i, j}: {6527, 20}, {37669, 2}, {56593, 3183}
X(68006) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {1, 32001}, {20, 21270}, {48, 3146}, {63, 32064}, {154, 5905}, {163, 8057}, {184, 18663}, {204, 6515}, {255, 253}, {610, 4}, {1249, 5906}, {1394, 56927}, {1437, 18655}, {1895, 317}, {4100, 57451}, {4575, 3265}, {8057, 21294}, {15905, 8}, {18750, 11442}, {19614, 54111}, {23995, 41678}, {24027, 36118}, {35200, 40996}, {35602, 4329}, {36841, 21300}, {37669, 6327}, {42658, 21221}, {52948, 66914}
X(68006) = X(i)-cross conjugate of X(j) for these {i, j}: {3183, 14362}, {40839, 2}
X(68006) = pole of line {3265, 8057} with respect to the DeLongchamps circle
X(68006) = pole of line {107, 53639} with respect to the Kiepert parabola
X(68006) = pole of line {1498, 3348} with respect to the Stammler hyperbola
X(68006) = pole of line {8057, 15427} with respect to the Steiner circumellipse
X(68006) = pole of line {6527, 31956} with respect to the Wallace hyperbola
X(68006) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(2060)}}, {{A, B, C, X(20), X(14361)}}, {{A, B, C, X(253), X(1032)}}, {{A, B, C, X(280), X(63877)}}, {{A, B, C, X(394), X(46351)}}, {{A, B, C, X(459), X(3183)}}, {{A, B, C, X(1073), X(3350)}}, {{A, B, C, X(1249), X(3356)}}, {{A, B, C, X(13157), X(54746)}}, {{A, B, C, X(31956), X(41489)}}, {{A, B, C, X(41081), X(41082)}}, {{A, B, C, X(41083), X(41084)}}, {{A, B, C, X(41514), X(46355)}}
X(68006) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 17037, 14361}, {2, 20213, 253}, {2, 20217, 51358}, {2, 51358, 14572}, {1073, 1249, 2}, {1498, 3346, 20}, {47848, 47850, 63}

X(68007) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTI-AQUILA WRT EULER0

X(68007) lies on these lines: {1, 3346}, {3, 36908}, {10, 59361}, {515, 33546}, {517, 20329}, {962, 54053}, {1125, 6523}, {1385, 15312}, {1394, 59345}, {3183, 3576}, {3817, 51342}, {4297, 15311}, {11363, 68008}, {41402, 66932}, {51118, 64505}, {52384, 55044}

X(68007) = midpoint of X(i) and X(j) for these {i,j}: {1, 3346}
X(68007) = reflection of X(i) in X(j) for these {i,j}: {10, 59361}, {6523, 1125}

X(68008) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTI-ARA WRT EULER0

X(68008) lies on these lines: {4, 253}, {24, 20329}, {25, 3346}, {185, 1885}, {235, 33546}, {427, 6523}, {468, 59361}, {1593, 3183}, {3088, 42452}, {11363, 68007}, {23047, 51342}

X(68009) = PERSPECTOR OF THESE TRIANGLES: CTR28-69 AND 2ND ANTI-CONWAY

X(68009) lies on these lines: {2, 27082}, {3, 6723}, {4, 1192}, {5, 11425}, {6, 3091}, {20, 15752}, {24, 16013}, {25, 32340}, {51, 16879}, {64, 10151}, {68, 63838}, {115, 46829}, {125, 5895}, {185, 58544}, {235, 23300}, {338, 14249}, {381, 389}, {382, 45622}, {403, 20303}, {546, 9786}, {578, 5072}, {599, 11444}, {973, 7547}, {974, 6241}, {1498, 19360}, {1593, 2929}, {1620, 3146}, {1853, 11381}, {1885, 61735}, {1899, 10019}, {1902, 44545}, {3070, 19039}, {3071, 19040}, {3089, 23324}, {3357, 44872}, {3517, 18376}, {3542, 18405}, {3545, 12241}, {3627, 37487}, {3763, 6816}, {3832, 11469}, {3839, 13568}, {3843, 18488}, {3850, 18356}, {3851, 6288}, {3854, 11433}, {3855, 11431}, {3857, 52163}, {5055, 13403}, {5068, 23292}, {5073, 44673}, {5079, 11430}, {5159, 41427}, {5876, 12236}, {5893, 23291}, {5902, 5927}, {6247, 68010}, {6622, 41362}, {6623, 15811}, {7507, 9969}, {9707, 12254}, {10297, 17834}, {10516, 14913}, {10625, 64689}, {10733, 39084}, {10821, 11456}, {11432, 61955}, {11438, 15432}, {11439, 52003}, {11449, 15044}, {11576, 32395}, {11704, 35490}, {11746, 12111}, {11801, 32139}, {12061, 23049}, {12163, 23323}, {12235, 67878}, {13160, 47355}, {13851, 17845}, {14216, 37984}, {15010, 32392}, {15118, 19153}, {15153, 34781}, {15431, 37643}, {16252, 18918}, {17810, 23047}, {17814, 58726}, {18388, 61953}, {18418, 68022}, {18551, 61968}, {19357, 35487}, {21659, 61680}, {22647, 23308}, {23251, 44634}, {23261, 44633}, {26937, 61721}, {31383, 45004}, {33537, 44920}, {36752, 63671}, {37444, 48872}, {37476, 63674}, {51797, 52525}, {51998, 64726}, {59349, 59411}, {61506, 63662}

X(68009) = midpoint of X(i) and X(j) for these {i,j}: {4, 58378}, {15077, 32605}
X(68009) = reflection of X(i) in X(j) for these {i,j}: {3532, 58378}, {58378, 43592}
X(68009) = inverse of X(5895) in Jerabek hyperbola
X(68009) = complement of X(27082)
X(68009) = X(i)-Ceva conjugate of X(j) for these {i, j}: {52913, 523}
X(68009) = pole of line {5895, 16879} with respect to the Jerabek hyperbola
X(68009) = pole of line {20, 1249} with respect to the Kiepert hyperbola
X(68009) = pole of line {53496, 59652} with respect to the Orthic inconic
X(68009) = pole of line {1620, 37672} with respect to the Stammler hyperbola
X(68009) = pole of line {58759, 59652} with respect to the Steiner inellipse
X(68009) = pole of line {32831, 54111} with respect to the Wallace hyperbola
X(68009) = pole of line {35018, 40138} with respect to the 1st Terzic hyperbola
X(68009) = pole of line {59652, 59662} with respect to the dual conic of DeLongchamps circle
X(68009) = intersection, other than A, B, C, of circumconics {{A, B, C, X(15077), X(38253)}}, {{A, B, C, X(34286), X(45245)}}
X(68009) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 43592, 3532}, {3146, 47296, 1620}, {46473, 46476, 45245}, {64024, 67903, 64080}

X(68010) = PERSPECTOR OF THESE TRIANGLES: CTR28-69 AND ANTI-EULER

X(68010) lies on these lines: {2, 11472}, {3, 11469}, {4, 3426}, {20, 1216}, {30, 5921}, {64, 3089}, {74, 4232}, {113, 30769}, {125, 6623}, {146, 31099}, {185, 11431}, {193, 5663}, {376, 11820}, {378, 38396}, {381, 15431}, {541, 3448}, {1499, 2394}, {1503, 49670}, {1902, 64021}, {2777, 32247}, {3088, 6225}, {3091, 4846}, {3146, 11411}, {3522, 15052}, {3523, 4550}, {3545, 44750}, {3549, 33541}, {3832, 66749}, {3839, 7706}, {5059, 16659}, {5702, 38920}, {6000, 6776}, {6247, 68009}, {6353, 35450}, {6622, 61540}, {6756, 32601}, {6995, 11455}, {7398, 16194}, {7426, 41428}, {7487, 11381}, {7519, 46431}, {7699, 52284}, {8717, 10304}, {10293, 11738}, {10605, 68027}, {10606, 15448}, {11745, 22334}, {12112, 35485}, {12244, 12292}, {12254, 12300}, {12324, 18396}, {13596, 63030}, {14216, 22533}, {15066, 46349}, {15105, 15811}, {15305, 54013}, {15311, 36990}, {15717, 32620}, {16655, 64726}, {17578, 40909}, {18925, 58795}, {26864, 35483}, {26882, 62067}, {31861, 51171}, {32063, 60765}, {32337, 32340}, {35254, 50693}, {35492, 41450}, {35513, 48876}, {37460, 54050}, {37689, 45723}, {47457, 63420}, {49140, 64032}, {49669, 66755}, {51993, 62003}, {52101, 61982}, {53780, 62029}, {61088, 68014}, {62174, 64097}, {63031, 67925}, {63092, 66717}, {64096, 66742}

X(68010) = reflection of X(i) in X(j) for these {i,j}: {4, 3426}, {5059, 41465}, {35512, 64}, {65563, 4}
X(68010) = inverse of X(67894) in Jerabek hyperbola
X(68010) = pole of line {32062, 61506} with respect to the Jerabek hyperbola
X(68010) = intersection, other than A, B, C, of circumconics {{A, B, C, X(64), X(58082)}}, {{A, B, C, X(35512), X(52452)}}
X(68010) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11381, 12250, 7487}, {12112, 35485, 64059}

X(68011) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR28-69 WRT HATZIPOLAKIS-MOSES

X(68011) lies on these lines: {4, 973}, {185, 32393}, {1154, 12293}, {1498, 7526}, {1853, 7564}, {5576, 6000}, {5921, 41726}, {7488, 68028}, {7729, 15431}, {10024, 49108}, {10628, 12295}, {11381, 32332}, {11469, 32354}, {12162, 12606}, {12292, 16655}, {14118, 32391}, {15432, 41725}, {32359, 49669}

X(68011) = reflection of X(i) in X(j) for these {i,j}: {185, 32393}, {32369, 63728}, {67915, 32369}

X(68012) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR28-69 WRT 3RD HATZIPOLAKIS

X(68012) lies on these lines: {4, 18936}, {64, 2929}, {125, 22970}, {185, 22968}, {2071, 22966}, {2072, 18488}, {11381, 22483}, {11469, 22647}, {11472, 12084}, {12290, 43616}, {12295, 13474}, {15305, 57648}, {22529, 46372}, {22538, 68020}, {48670, 66717}

X(68012) = midpoint of X(i) and X(j) for these {i,j}: {12290, 43616}
X(68012) = reflection of X(i) in X(j) for these {i,j}: {185, 22968}, {67916, 22833}

X(68013) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR28-69 WRT MIXTILINEAR

X(68013) lies on these lines: {51, 7682}, {57, 185}, {329, 5907}, {517, 5562}, {1902, 23154}, {2093, 2807}, {2095, 13754}, {2096, 6000}, {2097, 34146}, {3917, 6282}, {7956, 18180}, {9729, 62773}, {9730, 61535}, {9965, 12111}, {10373, 11573}, {12294, 34371}, {12688, 42549}, {15030, 37822}, {42448, 51490}

X(68013) = midpoint of X(i) and X(j) for these {i,j}: {9965, 12111}
X(68013) = reflection of X(i) in X(j) for these {i,j}: {185, 57}, {329, 5907}

X(68014) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR28-69 WRT WALSMITH

X(68014) lies on these lines: {3, 38851}, {4, 67}, {6, 5663}, {64, 1177}, {110, 68017}, {125, 15126}, {185, 15118}, {206, 32607}, {389, 16003}, {511, 7723}, {542, 12162}, {578, 52098}, {895, 12111}, {1112, 53023}, {1205, 11381}, {1350, 12358}, {1351, 22584}, {1503, 12292}, {1593, 15141}, {1986, 5480}, {2393, 32250}, {2854, 5921}, {3618, 66734}, {5085, 44573}, {5169, 12824}, {5181, 5907}, {5622, 6241}, {5876, 12293}, {6000, 35371}, {6593, 7527}, {7526, 15462}, {7687, 15432}, {7722, 14853}, {9019, 10296}, {9517, 65612}, {9818, 19376}, {10733, 41716}, {10752, 12281}, {11061, 11469}, {11425, 56568}, {11746, 15431}, {11799, 49116}, {12133, 36990}, {12219, 51212}, {12270, 52699}, {12294, 21650}, {13416, 31884}, {14094, 32245}, {14448, 68020}, {14561, 14708}, {14644, 67922}, {15138, 18374}, {15305, 41737}, {16222, 19130}, {18125, 22466}, {18382, 44795}, {18435, 63700}, {19149, 19457}, {31860, 54376}, {32233, 49669}, {32251, 64031}, {47336, 61543}, {61088, 68010}

X(68014) = midpoint of X(i) and X(j) for these {i,j}: {895, 12111}, {1205, 11381}, {1351, 22584}, {10733, 41716}, {10752, 12281}, {12219, 51212}, {12294, 21650}
X(68014) = reflection of X(i) in X(j) for these {i,j}: {67, 15738}, {185, 15118}, {1350, 12358}, {1986, 5480}, {5181, 5907}, {6593, 63723}, {19161, 7687}, {36990, 12133}, {37473, 32246}, {40949, 4}, {67917, 32274}
X(68014) = inverse of X(14983) in polar circle
X(68014) = perspector of circumconic {{A, B, C, X(9060), X(65356)}}
X(68014) = pole of line {9517, 14983} with respect to the polar circle
X(68014) = pole of line {5523, 11799} with respect to the Kiepert hyperbola
X(68014) = pole of line {40112, 58357} with respect to the Stammler hyperbola
X(68014) = intersection, other than A, B, C, of circumconics {{A, B, C, X(64), X(39269)}}, {{A, B, C, X(9517), X(14983)}}, {{A, B, C, X(34802), X(46105)}}
X(68014) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 2781, 40949}, {1205, 11381, 36201}, {2781, 15738, 67}, {2781, 32246, 37473}, {2781, 32274, 67917}, {9970, 34802, 16010}, {32274, 67917, 61665}

X(68015) = ORTHOLOGY CENTER OF THESE TRIANGLES: GEMINI 111 WRT CTR28-69

X(68015) lies on circumconic {{A, B, C, X(15740), X(52452)}} and on these lines: {2, 64}, {4, 3426}, {8, 9899}, {20, 2979}, {23, 9914}, {30, 64756}, {107, 58797}, {154, 21734}, {185, 63031}, {193, 34146}, {376, 12315}, {390, 7355}, {631, 35450}, {1204, 4232}, {1498, 3522}, {1503, 5059}, {1559, 41425}, {1587, 35864}, {1588, 35865}, {1593, 63030}, {1660, 43813}, {1853, 50689}, {2777, 49135}, {3090, 61540}, {3091, 5878}, {3146, 6515}, {3357, 3523}, {3516, 64058}, {3528, 32063}, {3529, 64758}, {3532, 15448}, {3543, 14216}, {3575, 32601}, {3600, 6285}, {3617, 12779}, {3620, 41735}, {3622, 12262}, {3623, 7973}, {3832, 6247}, {3839, 22802}, {3854, 5893}, {5032, 64031}, {5056, 65151}, {5261, 12940}, {5274, 12950}, {5894, 11206}, {5895, 17578}, {5907, 54039}, {5921, 52071}, {5925, 15683}, {6001, 20015}, {6241, 68020}, {6293, 63012}, {6353, 34469}, {6526, 51892}, {6623, 26917}, {6759, 10304}, {6776, 64029}, {6995, 11381}, {7398, 11439}, {7408, 13568}, {7486, 61749}, {7487, 12290}, {7585, 49250}, {7586, 49251}, {8567, 35260}, {9543, 17819}, {10076, 14986}, {10192, 61804}, {10282, 62067}, {10303, 67890}, {10528, 49186}, {10529, 49185}, {10565, 11440}, {10606, 15717}, {11001, 64033}, {11202, 58188}, {11204, 61788}, {12086, 46373}, {12162, 61113}, {12163, 34621}, {12964, 43512}, {12970, 43511}, {14530, 21735}, {15022, 40686}, {15692, 64027}, {15721, 64063}, {15811, 52301}, {16252, 61820}, {16704, 68016}, {17821, 62063}, {17845, 62152}, {18381, 50688}, {18383, 62007}, {18400, 49140}, {19087, 63016}, {19088, 63015}, {22334, 66531}, {22948, 67925}, {23328, 61834}, {23329, 46936}, {25563, 61863}, {31304, 64102}, {31978, 41715}, {33703, 34780}, {33748, 34779}, {34007, 41736}, {34109, 59361}, {34224, 49670}, {34782, 62120}, {35512, 52404}, {35711, 52448}, {38282, 43903}, {41362, 50690}, {41435, 63431}, {41603, 43695}, {41819, 63371}, {44762, 62124}, {49349, 62987}, {49350, 62986}, {50687, 51491}, {51170, 68019}, {51171, 63420}, {51403, 58378}, {52028, 63123}, {52102, 61982}, {58758, 59424}, {61088, 66755}, {61747, 61856}, {61914, 67868}, {62032, 68058}, {66747, 68026}

X(68015) = midpoint of X(i) and X(j) for these {i,j}: {49080, 49081}
X(68015) = reflection of X(i) in X(j) for these {i,j}: {2, 68027}, {4, 13093}, {8, 9899}, {20, 12250}, {1498, 15105}, {3146, 12324}, {3529, 64758}, {5059, 64726}, {6225, 64}, {12279, 30443}, {33703, 34780}, {34781, 20427}, {49135, 64034}, {54211, 4}, {58795, 5894}, {64187, 14216}
X(68015) = anticomplement of X(6225)
X(68015) = X(i)-Dao conjugate of X(j) for these {i, j}: {6225, 6225}
X(68015) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {34426, 8}, {42468, 21270}
X(68015) = pole of line {26937, 32062} with respect to the Jerabek hyperbola
X(68015) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {64, 6225, 2}, {64, 64714, 6696}, {1498, 15105, 54050}, {1498, 3522, 64059}, {1498, 54050, 3522}, {1503, 64726, 5059}, {2777, 64034, 49135}, {3357, 5656, 3523}, {5878, 67894, 3091}, {5894, 11206, 50693}, {5894, 58795, 11206}, {5895, 32064, 17578}, {6000, 20427, 34781}, {6000, 30443, 12279}, {6225, 68024, 64714}, {6225, 68027, 64}, {6247, 66752, 3832}, {6696, 64714, 68024}, {8567, 35260, 61791}, {8567, 68025, 35260}, {12250, 34781, 20427}, {12324, 15311, 3146}, {14216, 64187, 3543}, {20427, 34781, 20}, {49080, 49081, 34146}

X(68016) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND ANTI-PAVLOV WRT CTR28-69

X(68016) lies on these lines: {58, 9899}, {64, 81}, {333, 6225}, {1498, 64376}, {1503, 68054}, {1812, 12111}, {2883, 5235}, {3193, 49185}, {3357, 64393}, {4184, 12335}, {4225, 22778}, {5333, 6696}, {5878, 64405}, {6000, 64720}, {6001, 68031}, {6247, 64400}, {6266, 64404}, {6267, 64403}, {6285, 64382}, {7355, 64414}, {7973, 64415}, {8991, 64417}, {9914, 64395}, {10060, 64420}, {10076, 64421}, {11381, 64378}, {12202, 64381}, {12250, 64384}, {12262, 64377}, {12468, 64396}, {12469, 64397}, {12502, 64398}, {12779, 64401}, {12791, 64402}, {12920, 64406}, {12930, 64407}, {12940, 64408}, {12950, 64409}, {13093, 64419}, {13094, 64422}, {13095, 64423}, {13980, 64418}, {15311, 67852}, {16704, 68015}, {19087, 64385}, {19088, 64386}, {22802, 64399}, {34146, 41610}, {35864, 64412}, {35865, 64413}, {40571, 64025}, {41629, 68027}, {48513, 64379}, {48514, 64380}, {48672, 64383}, {48766, 64389}, {48767, 64390}, {49080, 64391}, {49081, 64392}, {49186, 64394}, {49250, 64410}, {49251, 64411}, {49349, 64387}, {49350, 64388}, {54340, 68059}, {64024, 64425}, {64424, 64714}

X(68017) = PERSPECTOR OF THESE TRIANGLES: CTR28-69 AND INVERSE-OF-X(32)-CIRCUMCONCEVIAN-OF-X(6)

X(68017) lies on these lines: {2, 34146}, {4, 69}, {5, 67922}, {6, 12111}, {22, 7998}, {23, 54374}, {64, 1176}, {110, 68014}, {141, 15056}, {182, 6241}, {185, 3618}, {206, 14118}, {524, 67266}, {568, 7403}, {895, 12825}, {1204, 19137}, {1350, 11444}, {1351, 5876}, {1503, 15305}, {1593, 20806}, {1619, 6800}, {1885, 13562}, {2063, 6090}, {2781, 10516}, {2807, 59406}, {2979, 29181}, {3060, 53023}, {3091, 19161}, {3098, 7999}, {3146, 3313}, {3547, 10170}, {3564, 18435}, {3567, 19130}, {3589, 10574}, {3619, 52520}, {3832, 9969}, {3917, 34608}, {4550, 19131}, {5050, 5622}, {5093, 8548}, {5133, 5640}, {5157, 9968}, {5480, 5889}, {5596, 11469}, {5650, 7494}, {5890, 14561}, {5891, 10519}, {5921, 50649}, {6000, 25406}, {6593, 12270}, {6776, 12162}, {7387, 15067}, {7395, 64716}, {7404, 9730}, {7495, 33879}, {7500, 33884}, {7512, 55649}, {7553, 13340}, {7723, 10752}, {7731, 32271}, {9729, 63119}, {9970, 12281}, {11061, 21650}, {11180, 34382}, {11284, 54376}, {11381, 11574}, {11439, 12220}, {11455, 29012}, {11468, 43811}, {11514, 26918}, {11591, 33878}, {11704, 52989}, {12017, 13491}, {12225, 54334}, {12272, 44439}, {12279, 44882}, {12290, 46264}, {12300, 44492}, {12324, 41256}, {13160, 18504}, {13754, 14853}, {14216, 41257}, {14457, 18124}, {14855, 33750}, {15045, 16223}, {15059, 41670}, {15073, 18440}, {15074, 48662}, {15100, 51941}, {15102, 19140}, {15531, 46442}, {15751, 61676}, {17508, 35921}, {17928, 34778}, {18436, 21850}, {18438, 39884}, {18439, 48906}, {18534, 55593}, {18583, 34783}, {19124, 22151}, {19459, 68022}, {20819, 31952}, {22467, 63431}, {26206, 63420}, {32142, 55629}, {32444, 44716}, {33523, 33586}, {33537, 68019}, {34380, 44804}, {34775, 66733}, {35904, 40917}, {36983, 61088}, {37473, 67865}, {37511, 40330}, {37925, 55603}, {39874, 44479}, {41614, 68023}, {43605, 64028}, {44668, 47353}, {45957, 51732}, {48910, 64050}, {51171, 64025}, {61734, 67222}, {63425, 67882}

X(68017) = midpoint of X(i) and X(j) for these {i,j}: {15305, 66750}
X(68017) = reflection of X(i) in X(j) for these {i,j}: {568, 38136}, {3060, 53023}, {5890, 14561}, {10519, 5891}, {15072, 5085}, {55610, 15067}, {66736, 10516}
X(68017) = pole of line {1899, 14927} with respect to the Jerabek hyperbola
X(68017) = pole of line {5254, 22240} with respect to the Kiepert hyperbola
X(68017) = pole of line {184, 7667} with respect to the Stammler hyperbola
X(68017) = intersection, other than A, B, C, of circumconics {{A, B, C, X(64), X(1235)}}, {{A, B, C, X(1176), X(14615)}}, {{A, B, C, X(51508), X(52578)}}
X(68017) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1843, 44870, 51537}, {5907, 12294, 69}, {7503, 19149, 1176}, {11381, 11574, 14927}, {11439, 12220, 36990}, {15062, 66730, 44883}, {15305, 66750, 1503}, {66736, 66756, 10516}

X(68018) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR28-69 WRT ANTICEVIAN-OF-X(235)

X(68018) lies on these lines: {2, 11425}, {3, 68}, {4, 394}, {5, 1092}, {6, 6815}, {20, 64}, {22, 34782}, {24, 32269}, {30, 5562}, {51, 9825}, {52, 31833}, {54, 58357}, {110, 16252}, {125, 16196}, {140, 12370}, {141, 7503}, {154, 59349}, {184, 6823}, {185, 3564}, {235, 9306}, {265, 37452}, {315, 37200}, {316, 19169}, {323, 34007}, {376, 34224}, {382, 16654}, {389, 61658}, {403, 59659}, {427, 13346}, {428, 13598}, {511, 3575}, {524, 5889}, {539, 40647}, {542, 46850}, {546, 1568}, {548, 12041}, {550, 32138}, {569, 43595}, {578, 7399}, {631, 12022}, {801, 34170}, {858, 58922}, {974, 12421}, {1060, 12428}, {1062, 18970}, {1105, 53481}, {1147, 15760}, {1181, 6193}, {1192, 64060}, {1204, 44241}, {1209, 52262}, {1216, 12358}, {1330, 37420}, {1352, 1593}, {1368, 43652}, {1370, 64037}, {1498, 37201}, {1511, 10020}, {1514, 15068}, {1531, 3853}, {1594, 43574}, {1657, 64036}, {1812, 6840}, {1885, 5907}, {1907, 3818}, {1941, 6530}, {1993, 12233}, {1995, 15873}, {2071, 2888}, {2883, 11441}, {2979, 12225}, {3060, 11745}, {3089, 35259}, {3091, 37669}, {3146, 16621}, {3289, 7745}, {3292, 43831}, {3357, 63441}, {3410, 12086}, {3521, 63720}, {3522, 34799}, {3523, 53050}, {3529, 16659}, {3530, 45970}, {3541, 37497}, {3543, 16656}, {3547, 13394}, {3548, 14852}, {3549, 47391}, {3561, 51368}, {3580, 22467}, {3589, 13434}, {3631, 34005}, {3796, 7400}, {3917, 12362}, {4292, 62402}, {4846, 9936}, {5446, 67237}, {5449, 10257}, {5480, 7544}, {5576, 37495}, {5892, 58806}, {5893, 50009}, {5944, 25337}, {5999, 45201}, {6090, 37197}, {6240, 11412}, {6241, 44458}, {6243, 38321}, {6247, 11413}, {6288, 37477}, {6515, 9786}, {6643, 18396}, {6644, 41587}, {6676, 13367}, {6756, 45186}, {6776, 10996}, {6803, 10601}, {6816, 17811}, {6960, 28754}, {7283, 23983}, {7383, 37476}, {7386, 18945}, {7401, 10982}, {7486, 62708}, {7487, 33586}, {7493, 17821}, {7512, 12383}, {7527, 62382}, {7528, 44413}, {7542, 12038}, {7550, 43818}, {7553, 45286}, {7576, 64051}, {7667, 13348}, {7689, 44240}, {7691, 15138}, {8550, 41614}, {8703, 45731}, {9729, 10112}, {9730, 13292}, {9777, 9815}, {9781, 67319}, {9820, 10024}, {9833, 11414}, {9927, 11585}, {10128, 27355}, {10263, 31830}, {10304, 27082}, {10516, 28419}, {10574, 45968}, {10605, 11411}, {10606, 30552}, {10619, 22352}, {10627, 30522}, {10984, 31804}, {11206, 52404}, {11225, 15012}, {11250, 67926}, {11440, 16386}, {11444, 52069}, {11459, 18560}, {11572, 51360}, {11591, 52070}, {11645, 34614}, {11793, 13403}, {11799, 18350}, {11819, 13391}, {12024, 15717}, {12111, 15311}, {12161, 50008}, {12254, 67321}, {12293, 18531}, {12294, 13562}, {13160, 23292}, {13372, 32410}, {13383, 51393}, {13470, 67336}, {13488, 15030}, {13561, 15122}, {13567, 17928}, {13630, 32358}, {13754, 43577}, {14118, 37636}, {14216, 21312}, {14531, 34380}, {14709, 62592}, {14710, 62593}, {14788, 15033}, {14790, 37483}, {14913, 39871}, {15067, 52073}, {15153, 31101}, {15341, 23128}, {15559, 41171}, {15644, 18400}, {15740, 53021}, {15761, 51425}, {16165, 16618}, {16238, 63735}, {16658, 33703}, {17834, 18533}, {18381, 37480}, {18420, 36747}, {18440, 67885}, {18474, 23335}, {18475, 34002}, {18563, 23039}, {18909, 61113}, {18914, 64100}, {18916, 37475}, {20299, 47090}, {21663, 44247}, {22109, 34153}, {22416, 63548}, {22466, 23308}, {23336, 34826}, {26926, 52520}, {29181, 41716}, {31383, 39568}, {31832, 67893}, {32275, 35240}, {32819, 57008}, {33523, 52397}, {34483, 34802}, {34622, 50955}, {34781, 35513}, {34785, 44239}, {36989, 37485}, {37198, 46264}, {37347, 37472}, {37444, 41362}, {37648, 39571}, {37814, 63734}, {40111, 61608}, {40196, 54211}, {41724, 43601}, {41738, 43813}, {43604, 52104}, {43844, 64179}, {43957, 44862}, {44246, 63392}, {44704, 46700}, {44870, 62962}, {45248, 61680}, {52385, 64003}, {52398, 64034}, {53414, 62361}, {58434, 58805}, {62391, 63146}, {63722, 67896}

X(68018) = midpoint of X(i) and X(j) for these {i,j}: {20, 14516}, {1657, 64036}, {3529, 16659}, {6240, 11412}, {12111, 52071}, {12225, 12278}
X(68018) = reflection of X(i) in X(j) for these {i,j}: {4, 64035}, {52, 31833}, {185, 31829}, {1885, 5907}, {3146, 16621}, {5889, 13568}, {6146, 3}, {7553, 45286}, {10112, 9729}, {10263, 31830}, {12162, 31831}, {12294, 13562}, {12370, 140}, {12605, 1216}, {13142, 9825}, {13403, 11793}, {16655, 12134}, {21659, 12362}, {26926, 52520}, {32358, 13630}, {32410, 13372}, {39871, 14913}, {44829, 13348}, {45186, 6756}, {45970, 3530}, {52070, 11591}, {61658, 66614}, {67893, 31832}
X(68018) = inverse of X(22834) in Jerabek hyperbola
X(68018) = anticomplement of X(12241)
X(68018) = perspector of circumconic {{A, B, C, X(44326), X(65309)}}
X(68018) = X(i)-Dao conjugate of X(j) for these {i, j}: {12241, 12241}, {65809, 13567}
X(68018) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {45301, 8}
X(68018) = pole of line {22089, 65694} with respect to the circumcircle
X(68018) = pole of line {1368, 5562} with respect to the Jerabek hyperbola
X(68018) = pole of line {2165, 5063} with respect to the Kiepert hyperbola
X(68018) = pole of line {24, 154} with respect to the Stammler hyperbola
X(68018) = pole of line {20, 317} with respect to the Wallace hyperbola
X(68018) = pole of line {6563, 8057} with respect to the dual conic of polar circle
X(68018) = pole of line {136, 46658} with respect to the dual conic of Wallace hyperbola
X(68018) = intersection, other than A, B, C, of circumconics {{A, B, C, X(64), X(2351)}}, {{A, B, C, X(68), X(253)}}, {{A, B, C, X(1300), X(6146)}}, {{A, B, C, X(1899), X(22261)}}, {{A, B, C, X(15394), X(16391)}}, {{A, B, C, X(18848), X(20477)}}, {{A, B, C, X(26937), X(45838)}}, {{A, B, C, X(34403), X(52350)}}
X(68018) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 12429, 1899}, {3, 44665, 6146}, {3, 68, 67902}, {5, 1092, 11064}, {20, 14516, 1503}, {20, 5894, 20725}, {20, 5921, 12324}, {30, 12134, 16655}, {30, 31831, 12162}, {68, 12118, 12301}, {323, 34007, 66727}, {343, 63631, 3}, {489, 490, 20477}, {524, 13568, 5889}, {578, 7399, 37649}, {631, 12022, 64038}, {1216, 17702, 12605}, {1350, 17845, 20}, {2979, 12278, 12225}, {3292, 43831, 61607}, {3541, 67878, 45303}, {3547, 19357, 13394}, {3547, 66735, 19357}, {3564, 31829, 185}, {3917, 21659, 12362}, {5889, 38323, 13568}, {6776, 10996, 66608}, {6823, 66762, 184}, {7400, 18925, 3796}, {9729, 10112, 11245}, {9825, 13142, 51}, {10024, 22115, 9820}, {11413, 11442, 6247}, {11441, 44440, 2883}, {11793, 13403, 34664}, {12111, 52071, 15311}, {13348, 44829, 7667}, {15761, 61753, 51425}, {16196, 61544, 125}, {18420, 36747, 45089}, {34785, 46728, 44239}, {37198, 64717, 46264}, {37497, 67878, 3541}, {39571, 66607, 37648}, {40111, 61750, 61608}, {61113, 64756, 18909}

X(68019) = ORTHOLOGY CENTER OF THESE TRIANGLES: X(3)-CIRCUMCONCEVIAN-OF-X(6) WRT CTR28-69

X(68019) lies on these lines: {3, 19132}, {6, 64}, {20, 34774}, {22, 110}, {66, 52518}, {69, 2883}, {141, 64024}, {155, 44544}, {159, 9968}, {182, 10606}, {193, 1503}, {206, 15748}, {221, 3056}, {511, 1498}, {518, 7973}, {524, 41735}, {599, 67870}, {648, 34808}, {1151, 19134}, {1152, 19135}, {1177, 43713}, {1192, 1974}, {1204, 19118}, {1351, 6000}, {1353, 64096}, {1469, 2192}, {1598, 21851}, {1619, 37672}, {1657, 34776}, {1843, 15811}, {1853, 5480}, {2393, 55722}, {2777, 32264}, {2935, 9970}, {3098, 17821}, {3162, 12145}, {3357, 5050}, {3516, 21637}, {3556, 62245}, {3564, 5878}, {3618, 6696}, {3620, 68024}, {3629, 68021}, {5032, 68027}, {5039, 12202}, {5085, 8567}, {5093, 13093}, {5102, 8549}, {5596, 17845}, {5656, 63428}, {5663, 32276}, {5847, 12779}, {5894, 25406}, {5921, 66752}, {5925, 46264}, {6001, 64084}, {6247, 14853}, {6467, 12174}, {6759, 33878}, {6776, 15311}, {7169, 62207}, {7716, 37473}, {8550, 61088}, {9019, 66723}, {9786, 67922}, {9914, 19459}, {10249, 55711}, {10282, 55610}, {10519, 16252}, {10541, 41593}, {11202, 55629}, {11206, 61044}, {11381, 12167}, {11425, 67898}, {11432, 17822}, {11598, 52699}, {11744, 55977}, {12017, 64027}, {12087, 15580}, {12250, 14912}, {12262, 16475}, {12315, 44456}, {12324, 15583}, {12940, 39897}, {12950, 39873}, {13094, 45729}, {13095, 45728}, {13142, 31670}, {13293, 45016}, {14216, 21850}, {14528, 34207}, {14530, 55593}, {14561, 40686}, {14913, 68022}, {15139, 41424}, {15585, 62174}, {17810, 19161}, {17811, 41580}, {17814, 37511}, {18405, 48901}, {18440, 22802}, {18583, 65151}, {19123, 35477}, {19139, 37497}, {19153, 53094}, {20079, 41362}, {20300, 38072}, {20427, 48906}, {22151, 58762}, {23041, 55646}, {32063, 55584}, {33537, 68017}, {34815, 42671}, {35228, 55626}, {35450, 53091}, {36201, 64104}, {36989, 48872}, {36990, 39871}, {37648, 61735}, {39874, 64187}, {39899, 48672}, {40318, 64025}, {40330, 67868}, {41427, 46374}, {41602, 64060}, {41719, 44882}, {41729, 43273}, {41737, 64587}, {43813, 55676}, {44883, 53093}, {47355, 63699}, {48873, 64719}, {48876, 67890}, {50414, 55595}, {51170, 68015}, {52703, 63421}, {54131, 61658}, {54173, 61610}, {54211, 66742}, {59399, 61540}, {63371, 63385}, {64726, 66755}, {66608, 66750}

X(68019) = midpoint of X(i) and X(j) for these {i,j}: {193, 6225}, {12315, 44456}, {39874, 64187}, {39899, 48672}
X(68019) = reflection of X(i) in X(j) for these {i,j}: {3, 34779}, {6, 64031}, {20, 34774}, {64, 6}, {69, 2883}, {159, 9968}, {1350, 19149}, {1498, 64716}, {1657, 34776}, {2935, 9970}, {5925, 46264}, {9924, 1498}, {12324, 15583}, {14216, 21850}, {17845, 5596}, {17847, 51941}, {18440, 22802}, {20079, 41362}, {20427, 48906}, {33878, 6759}, {34778, 34117}, {41737, 64587}, {48872, 36989}, {48873, 64719}, {53097, 159}, {55582, 34787}, {61088, 8550}, {64037, 31670}, {67888, 1351}, {68021, 3629}
X(68019) = pole of line {684, 42658} with respect to the circumcircle
X(68019) = pole of line {8673, 62176} with respect to the cosine circle
X(68019) = pole of line {25, 52028} with respect to the Jerabek hyperbola
X(68019) = pole of line {235, 63533} with respect to the Kiepert hyperbola
X(68019) = pole of line {30211, 62176} with respect to the MacBeath circumconic
X(68019) = pole of line {1503, 7396} with respect to the Stammler hyperbola
X(68019) = intersection, other than A, B, C, of circumconics {{A, B, C, X(64), X(64975)}}, {{A, B, C, X(154), X(51437)}}, {{A, B, C, X(1073), X(52028)}}, {{A, B, C, X(1297), X(41489)}}
X(68019) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 64, 52028}, {185, 68023, 6}, {193, 6225, 1503}, {511, 1498, 9924}, {511, 64716, 1498}, {1350, 19149, 154}, {1351, 6000, 67888}, {1351, 67888, 17813}, {2781, 19149, 1350}, {2781, 51941, 17847}, {5085, 34778, 8567}, {19153, 63431, 53094}, {20079, 51538, 41362}, {34117, 34778, 5085}, {46373, 64031, 11470}, {49250, 49349, 64}

X(68020) = PERSPECTOR OF THESE TRIANGLES: CTR28-69 AND UCFT-OF-2ND ANTI-EXTOUCH

X(68020) lies on these lines: {3, 12058}, {4, 52}, {5, 44084}, {6, 64}, {20, 9967}, {24, 1216}, {25, 5562}, {26, 45118}, {51, 7507}, {143, 66728}, {155, 44080}, {161, 26883}, {184, 68026}, {186, 5447}, {187, 62271}, {235, 343}, {237, 31388}, {378, 569}, {389, 427}, {403, 1209}, {468, 11793}, {511, 3575}, {578, 41725}, {858, 22834}, {973, 1112}, {1154, 6756}, {1495, 2917}, {1594, 5462}, {1595, 6102}, {1596, 5876}, {1597, 34783}, {1598, 18436}, {1843, 14531}, {1885, 6000}, {1902, 2807}, {1986, 11806}, {2393, 61139}, {2781, 13568}, {3088, 5890}, {3089, 11459}, {3515, 3917}, {3516, 37476}, {3517, 23039}, {3520, 37513}, {3541, 9730}, {3542, 5891}, {3567, 63081}, {3574, 58550}, {3850, 9827}, {5064, 14831}, {5094, 64854}, {5198, 45187}, {5449, 45179}, {5663, 13292}, {5892, 37119}, {5921, 12282}, {6101, 37458}, {6145, 13851}, {6241, 68015}, {6243, 18494}, {6353, 11444}, {6403, 20080}, {6467, 64717}, {6622, 15056}, {6623, 15058}, {6696, 52003}, {6746, 16625}, {6815, 37511}, {7487, 11412}, {7503, 19131}, {7505, 10170}, {8889, 15043}, {9729, 37649}, {9826, 32144}, {9937, 18451}, {10151, 13446}, {10574, 63085}, {10625, 18533}, {11245, 46363}, {11381, 44438}, {11557, 33547}, {11562, 15472}, {11591, 21841}, {11695, 62958}, {11750, 14915}, {12038, 34116}, {12160, 17836}, {12173, 45186}, {12301, 36747}, {12825, 32263}, {13348, 37931}, {13434, 30100}, {13598, 66725}, {13630, 64474}, {14128, 37942}, {14448, 68014}, {14516, 34382}, {14641, 35481}, {15010, 27355}, {15028, 52299}, {15030, 37197}, {15060, 44960}, {15100, 18947}, {15115, 25711}, {15125, 43831}, {15473, 31830}, {15809, 31802}, {16198, 66604}, {16226, 62980}, {16238, 64689}, {17845, 44439}, {18475, 52432}, {19128, 37126}, {19467, 50649}, {21213, 46728}, {22538, 68012}, {23292, 41589}, {31807, 65376}, {31834, 64471}, {32062, 34751}, {32142, 37935}, {37777, 43614}, {37984, 45958}, {44226, 45959}, {44479, 46850}, {45286, 45780}, {46443, 61713}, {61544, 63709}, {61658, 62962}, {62966, 64060}, {63012, 64025}, {63662, 67067}, {67883, 67915}

X(68020) = perspector of circumconic {{A, B, C, X(1301), X(30450)}}
X(68020) = pole of line {25, 61139} with respect to the Jerabek hyperbola
X(68020) = pole of line {235, 14576} with respect to the Kiepert hyperbola
X(68020) = pole of line {520, 6753} with respect to the Orthic inconic
X(68020) = pole of line {1147, 6643} with respect to the Stammler hyperbola
X(68020) = intersection, other than A, B, C, of circumconics {{A, B, C, X(64), X(5392)}}, {{A, B, C, X(68), X(14642)}}, {{A, B, C, X(847), X(41489)}}
X(68020) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 52, 47328}, {6, 6293, 185}, {52, 12162, 68}, {52, 18474, 12235}, {185, 12294, 1593}, {378, 35603, 569}, {1112, 23047, 10110}, {1594, 52000, 5462}, {1595, 6102, 67923}, {5907, 64820, 235}

X(68021) = ORTHOLOGY CENTER OF THESE TRIANGLES: UCFT-OF-2ND EHRMANN WRT CTR28-69

X(68021) lies on these lines: {2, 1660}, {3, 8263}, {4, 6}, {20, 2393}, {24, 54149}, {64, 524}, {66, 5921}, {69, 11413}, {141, 52028}, {146, 13248}, {154, 40132}, {159, 17928}, {193, 34146}, {206, 43815}, {376, 34787}, {511, 20427}, {542, 2892}, {575, 67890}, {576, 5878}, {597, 64024}, {599, 6696}, {631, 10249}, {858, 32064}, {895, 3146}, {1192, 41585}, {1351, 48672}, {1352, 3546}, {1353, 64716}, {1619, 11433}, {1853, 14826}, {1885, 10602}, {1992, 6225}, {1995, 11206}, {2777, 34788}, {2781, 12250}, {3091, 23327}, {3523, 61683}, {3542, 5622}, {3564, 12085}, {3618, 67870}, {3629, 68019}, {4663, 12779}, {5621, 32241}, {5894, 53097}, {5895, 17813}, {6000, 63722}, {6247, 15069}, {6353, 64656}, {6622, 62375}, {6623, 15125}, {6642, 39879}, {6643, 54183}, {6759, 11179}, {6995, 58483}, {7464, 63422}, {7529, 64719}, {8540, 12950}, {8584, 64714}, {9716, 31099}, {9729, 9833}, {9924, 44882}, {9972, 18400}, {10192, 10541}, {10250, 61749}, {10519, 44883}, {11180, 34118}, {11188, 36989}, {11216, 66752}, {11427, 41602}, {11477, 15311}, {11585, 18440}, {11821, 54334}, {12017, 61610}, {12324, 46373}, {12940, 19369}, {13203, 52124}, {13488, 54218}, {14927, 52071}, {15074, 49669}, {15126, 30769}, {15585, 53094}, {15740, 38323}, {16252, 53093}, {17821, 51737}, {17845, 64196}, {18537, 44503}, {18913, 63129}, {18934, 64066}, {20423, 22802}, {22401, 59363}, {29959, 58492}, {31383, 44079}, {31725, 39562}, {32284, 64096}, {32605, 41737}, {33748, 41593}, {33750, 35228}, {34507, 65151}, {34777, 51212}, {34778, 63428}, {34782, 43273}, {34785, 46264}, {35471, 67917}, {36203, 51938}, {36983, 66742}, {37201, 41614}, {37460, 38885}, {38064, 64063}, {39899, 47527}, {40680, 63419}, {41580, 63031}, {41715, 63012}, {47586, 60317}, {51024, 68058}, {51491, 54131}, {54132, 64187}, {54173, 64027}, {55724, 64758}, {58378, 62376}, {59373, 63699}, {62174, 63431}, {63064, 68027}, {64033, 67237}

X(68021) = midpoint of X(i) and X(j) for these {i,j}: {55724, 64758}, {63064, 68027}
X(68021) = reflection of X(i) in X(j) for these {i,j}: {4, 8549}, {69, 63420}, {146, 13248}, {1498, 8550}, {5596, 6776}, {5878, 576}, {5921, 66}, {6225, 64031}, {9924, 44882}, {12779, 4663}, {15069, 6247}, {17845, 64196}, {39879, 48906}, {51212, 34777}, {53097, 5894}, {63428, 34778}, {64714, 8584}, {64716, 1353}, {66752, 11216}, {68019, 3629}
X(68021) = pole of line {394, 41580} with respect to the Stammler hyperbola
X(68021) = pole of line {3926, 37201} with respect to the Wallace hyperbola
X(68021) = intersection, other than A, B, C, of circumconics {{A, B, C, X(287), X(41735)}}, {{A, B, C, X(1249), X(56268)}}, {{A, B, C, X(8743), X(57648)}}, {{A, B, C, X(15740), X(41370)}}, {{A, B, C, X(43695), X(60428)}}
X(68021) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1503, 6776, 5596}, {1503, 8549, 4}, {1503, 8550, 1498}, {1992, 6225, 64031}, {59373, 68024, 63699}

X(68022) = PERSPECTOR OF THESE TRIANGLES: CTR28-69 AND CTR12-3.6

X(68022) lies on these lines: {2, 12174}, {3, 64}, {4, 193}, {5, 18909}, {6, 44870}, {20, 62217}, {25, 12111}, {26, 64097}, {52, 18535}, {110, 3516}, {113, 3851}, {141, 68025}, {155, 1597}, {156, 56516}, {182, 33537}, {185, 5020}, {381, 11432}, {382, 12134}, {394, 11381}, {399, 45015}, {511, 15811}, {546, 3527}, {550, 11820}, {1092, 54992}, {1147, 11472}, {1181, 5050}, {1204, 35259}, {1352, 2883}, {1368, 12324}, {1593, 3167}, {1596, 11411}, {1598, 13754}, {1614, 54994}, {1625, 9605}, {1906, 6515}, {1993, 11403}, {1995, 64025}, {3088, 61607}, {3091, 18950}, {3426, 12085}, {3517, 12163}, {3526, 51425}, {3832, 9777}, {3843, 18474}, {5056, 5544}, {5064, 66727}, {5093, 46847}, {5198, 5889}, {5447, 35237}, {5562, 33878}, {5651, 64029}, {5656, 6823}, {5663, 6642}, {5876, 7387}, {5878, 64035}, {5893, 64031}, {5894, 60746}, {6090, 11413}, {6193, 13488}, {6225, 14826}, {6241, 66607}, {6243, 58764}, {6247, 30771}, {6623, 61544}, {6677, 18913}, {6696, 59543}, {7393, 15060}, {7395, 11456}, {7484, 15056}, {7509, 55682}, {7529, 34783}, {7592, 14094}, {7689, 55572}, {7723, 9919}, {7959, 67968}, {8718, 55648}, {8889, 32605}, {9715, 14157}, {9730, 11484}, {9818, 19347}, {9909, 26883}, {10323, 12112}, {10574, 11284}, {10982, 11482}, {11402, 43605}, {11410, 15062}, {11412, 55580}, {11414, 11459}, {11426, 18445}, {11440, 15750}, {11442, 37197}, {11444, 37198}, {11457, 16072}, {11591, 35243}, {12082, 55595}, {12241, 39899}, {12250, 44241}, {12279, 15066}, {12290, 21312}, {12293, 22538}, {12294, 19588}, {12310, 12825}, {12316, 62004}, {12362, 34781}, {12605, 64033}, {13474, 37498}, {13562, 41735}, {13598, 44456}, {14118, 26864}, {14128, 64098}, {14516, 44438}, {14913, 68019}, {15041, 20771}, {15043, 62209}, {15052, 17928}, {15054, 20772}, {15063, 32285}, {15083, 44413}, {16194, 36747}, {16195, 63425}, {16196, 67894}, {16266, 32137}, {16419, 66608}, {16656, 31670}, {16774, 18358}, {18381, 22808}, {18418, 68009}, {18436, 18534}, {18531, 34780}, {18537, 18914}, {19459, 68017}, {20850, 46730}, {21243, 64024}, {22467, 34469}, {26918, 51946}, {31833, 64094}, {32272, 38791}, {32621, 63723}, {33586, 45187}, {34622, 63631}, {35253, 50693}, {35265, 38438}, {37412, 48917}, {37484, 44454}, {37514, 67891}, {41369, 59655}, {43894, 61753}, {43895, 61701}, {44247, 54050}, {44762, 46264}, {45186, 55724}, {46372, 63420}, {47391, 55575}, {48662, 64037}, {48876, 52404}, {50963, 67883}, {52069, 64717}, {55701, 67879}, {59659, 65151}, {61749, 67878}, {66609, 66756}

X(68022) = reflection of X(i) in X(j) for these {i,j}: {3, 17814}, {18909, 5}
X(68022) = pole of line {520, 44680} with respect to the circumcircle
X(68022) = pole of line {1204, 33586} with respect to the Jerabek hyperbola
X(68022) = pole of line {2451, 58796} with respect to the MacBeath circumconic
X(68022) = pole of line {40494, 58757} with respect to the MacBeath inconic
X(68022) = pole of line {57071, 65656} with respect to the Orthic inconic
X(68022) = pole of line {20, 3167} with respect to the Stammler hyperbola
X(68022) = pole of line {14341, 52613} with respect to the Steiner inellipse
X(68022) = pole of line {6337, 14615} with respect to the Wallace hyperbola
X(68022) = intersection, other than A, B, C, of circumconics {{A, B, C, X(64), X(34208)}}, {{A, B, C, X(1073), X(2996)}}, {{A, B, C, X(3426), X(52566)}}, {{A, B, C, X(6391), X(14379)}}, {{A, B, C, X(8798), X(27364)}}, {{A, B, C, X(14248), X(33581)}}, {{A, B, C, X(44704), X(59707)}}
X(68022) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 18439, 13093}, {4, 12164, 1351}, {394, 11381, 67885}, {1147, 11472, 55571}, {1181, 15030, 11479}, {1593, 11441, 3167}, {1993, 11439, 11403}, {5562, 39568, 33878}, {5921, 68023, 6391}, {6225, 14826, 31829}, {9818, 32139, 19347}, {11441, 15305, 1593}, {11456, 15058, 7395}, {12162, 18451, 3}, {12163, 46261, 3517}, {14094, 65095, 45016}, {15083, 46849, 44413}, {17811, 58795, 46850}, {32139, 45959, 9818}, {46372, 68028, 63420}

X(68023) = PERSPECTOR OF THESE TRIANGLES: CTR28-69 AND CTR13-3.6

X(68023) lies on these lines: {3, 19118}, {4, 193}, {6, 64}, {24, 33878}, {25, 394}, {69, 235}, {74, 41616}, {159, 44439}, {182, 3516}, {186, 55610}, {378, 5050}, {399, 32240}, {427, 9777}, {428, 54132}, {468, 10519}, {524, 62966}, {576, 11403}, {611, 7071}, {613, 1398}, {895, 12133}, {1112, 10752}, {1181, 34779}, {1350, 1974}, {1352, 37197}, {1353, 13488}, {1498, 6467}, {1503, 10602}, {1595, 18917}, {1596, 34380}, {1597, 5093}, {1598, 6403}, {1829, 64084}, {1843, 5198}, {1862, 10759}, {1885, 6225}, {1902, 3751}, {1986, 48679}, {1992, 62962}, {2065, 35908}, {2207, 5028}, {2211, 45141}, {2935, 34470}, {3088, 11432}, {3089, 41584}, {3092, 35840}, {3093, 35841}, {3098, 15750}, {3517, 55584}, {3518, 55580}, {3520, 12017}, {3527, 16774}, {3541, 18583}, {3542, 48876}, {3575, 51212}, {3620, 6622}, {5020, 66736}, {5064, 20423}, {5085, 11410}, {5094, 14561}, {5095, 12165}, {5102, 8541}, {5185, 10758}, {5186, 10753}, {5476, 62980}, {5480, 7507}, {5544, 62960}, {5596, 64717}, {6353, 62174}, {6530, 57533}, {7387, 18438}, {7714, 51028}, {7716, 55722}, {8778, 40825}, {9715, 64052}, {9924, 26883}, {9967, 11414}, {9968, 32366}, {9970, 19504}, {10594, 55724}, {10754, 12131}, {10755, 12138}, {10766, 12145}, {11284, 44084}, {11380, 13355}, {11381, 67888}, {11402, 41715}, {11413, 63069}, {11425, 21637}, {11441, 19588}, {11482, 35502}, {11574, 37198}, {11820, 49670}, {12111, 40318}, {12173, 31670}, {12220, 39568}, {12308, 32234}, {12315, 39874}, {12370, 39899}, {12825, 64214}, {13367, 19132}, {14865, 53092}, {14912, 67899}, {15073, 39879}, {15463, 45016}, {17506, 55648}, {17811, 44079}, {17813, 32062}, {17814, 67920}, {18386, 53023}, {18451, 34382}, {19125, 34117}, {19131, 54994}, {19149, 19459}, {19467, 34774}, {20806, 57648}, {21650, 32276}, {21844, 55639}, {22538, 34777}, {29181, 37196}, {31884, 55576}, {32534, 55629}, {35325, 40126}, {35472, 55643}, {35473, 55682}, {35475, 55701}, {35479, 55602}, {37199, 39141}, {37491, 41716}, {37511, 66607}, {37981, 47571}, {38317, 52298}, {41614, 68017}, {44091, 55582}, {44281, 52238}, {44879, 55595}, {44960, 61545}, {47740, 62953}, {50955, 62974}, {50963, 62982}, {50967, 62978}, {51538, 66725}, {53091, 55571}, {53097, 55578}, {54173, 62965}, {54174, 62979}, {55570, 55604}, {55572, 55593}, {55574, 55616}, {55575, 55705}, {59399, 64474}, {66771, 66807}, {66790, 66805}

X(68023) = reflection of X(i) in X(j) for these {i,j}: {26869, 14853}
X(68023) = pole of line {42658, 44680} with respect to the circumcircle
X(68023) = pole of line {3569, 6753} with respect to the cosine circle
X(68023) = pole of line {3566, 30735} with respect to the polar circle
X(68023) = pole of line {25, 67888} with respect to the Jerabek hyperbola
X(68023) = pole of line {235, 44518} with respect to the Kiepert hyperbola
X(68023) = pole of line {2451, 30211} with respect to the MacBeath circumconic
X(68023) = pole of line {520, 57071} with respect to the Orthic inconic
X(68023) = pole of line {1368, 3167} with respect to the Stammler hyperbola
X(68023) = pole of line {6337, 62698} with respect to the Wallace hyperbola
X(68023) = intersection, other than A, B, C, of circumconics {{A, B, C, X(3), X(45207)}}, {{A, B, C, X(64), X(2996)}}, {{A, B, C, X(6391), X(14642)}}, {{A, B, C, X(34208), X(40801)}}
X(68023) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1351, 12167}, {4, 193, 39871}, {6, 12294, 1593}, {6, 68019, 185}, {394, 64820, 25}, {1350, 1974, 3515}, {1351, 12164, 193}, {1351, 68022, 6391}, {1597, 5093, 39588}, {1598, 44456, 6403}, {1885, 46444, 6776}, {3089, 63428, 41584}, {5093, 39588, 11405}, {5095, 51941, 12165}, {6391, 68022, 5921}, {6776, 64716, 12174}, {11470, 12294, 6}, {19149, 50649, 19459}

X(68024) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR5-2.2 WRT CTR28-69

X(68024) lies on these lines: {2, 64}, {3, 61606}, {4, 54}, {5, 5544}, {20, 11064}, {69, 64031}, {107, 6621}, {113, 6643}, {140, 12250}, {154, 3146}, {159, 51538}, {185, 6622}, {206, 14927}, {221, 5274}, {235, 11433}, {376, 22802}, {381, 34781}, {394, 32605}, {403, 18909}, {459, 57517}, {546, 32063}, {549, 48672}, {631, 5878}, {632, 35450}, {1147, 67201}, {1181, 6623}, {1192, 62973}, {1204, 38282}, {1249, 59424}, {1498, 3091}, {1503, 3832}, {1514, 19357}, {1568, 52398}, {1596, 3527}, {1597, 43841}, {1619, 63664}, {1656, 67894}, {1853, 5068}, {1906, 14853}, {2192, 5261}, {2777, 3528}, {2937, 41465}, {3090, 6000}, {3357, 3525}, {3522, 5895}, {3523, 15311}, {3524, 20427}, {3529, 10282}, {3530, 64758}, {3541, 32111}, {3543, 34782}, {3545, 14216}, {3616, 12779}, {3617, 7973}, {3618, 41735}, {3619, 15056}, {3620, 68019}, {3627, 14530}, {3628, 13093}, {3634, 9899}, {3839, 64037}, {3845, 64033}, {3850, 34780}, {3854, 44762}, {3855, 18381}, {4232, 13568}, {5056, 6247}, {5059, 61721}, {5067, 65151}, {5070, 61540}, {5071, 20299}, {5218, 12950}, {5225, 26888}, {5229, 10535}, {5260, 22778}, {5448, 34938}, {5550, 12262}, {5596, 51537}, {5894, 15717}, {5907, 41715}, {5925, 10304}, {6001, 68034}, {6285, 10588}, {6526, 56296}, {6616, 52448}, {6624, 14249}, {6776, 37197}, {6803, 64179}, {6815, 43614}, {6816, 41736}, {7288, 12940}, {7355, 10589}, {7378, 15811}, {7409, 16656}, {7485, 9914}, {7486, 40686}, {7505, 18931}, {7506, 66749}, {7712, 32391}, {8567, 58434}, {8797, 17703}, {8889, 11381}, {8972, 19088}, {9812, 40660}, {10151, 18945}, {10182, 61138}, {10193, 61836}, {10303, 10606}, {10574, 22967}, {10675, 42139}, {10676, 42142}, {10996, 43813}, {11202, 17538}, {11204, 61814}, {11411, 15761}, {11449, 27082}, {11451, 58492}, {11563, 18951}, {11799, 64048}, {12087, 15577}, {12111, 32392}, {12174, 23291}, {12964, 42561}, {12970, 31412}, {13941, 19087}, {14790, 67869}, {15022, 23332}, {15105, 61856}, {15305, 68026}, {15585, 61044}, {15682, 34785}, {15683, 68058}, {15751, 19132}, {16051, 46850}, {17578, 17845}, {17704, 30443}, {17826, 43466}, {17827, 43465}, {18383, 41099}, {18405, 61982}, {18916, 44958}, {18918, 35488}, {18928, 41602}, {19347, 44226}, {20079, 67865}, {21663, 32601}, {23325, 61945}, {23328, 55864}, {23329, 61886}, {25406, 64061}, {25563, 61867}, {26869, 45004}, {30402, 42140}, {30403, 42141}, {30552, 40196}, {31978, 54039}, {32321, 35500}, {32767, 61921}, {32785, 49250}, {32786, 49251}, {32903, 46333}, {33522, 59349}, {34117, 37784}, {34469, 52297}, {34787, 51212}, {36983, 64100}, {37126, 64759}, {37201, 37669}, {38443, 43697}, {40330, 64716}, {40658, 59387}, {41362, 50689}, {41589, 64025}, {41719, 58922}, {43903, 52292}, {44960, 67899}, {50414, 62028}, {50709, 62149}, {52071, 53050}, {59373, 63699}, {59659, 61113}, {61735, 61914}, {62947, 66729}, {63119, 63420}

X(68024) = pole of line {5562, 10606} with respect to the Stammler hyperbola
X(68024) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(8884), X(16251)}}, {{A, B, C, X(15740), X(38808)}}, {{A, B, C, X(17703), X(61348)}}, {{A, B, C, X(37878), X(59424)}}
X(68024) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2883, 6225}, {2, 68015, 6696}, {3, 66752, 64726}, {5, 5656, 12324}, {20, 16252, 35260}, {154, 5893, 3146}, {185, 6622, 37643}, {631, 5878, 54050}, {1498, 3091, 32064}, {1498, 67868, 3091}, {2883, 64024, 2}, {2883, 6696, 64714}, {5894, 61680, 15717}, {5895, 10192, 3522}, {6696, 64714, 68015}, {8567, 58434, 61820}, {9833, 67890, 14862}, {10303, 54211, 10606}, {16252, 51491, 17821}, {17821, 51491, 20}, {36982, 45979, 10574}, {61749, 67890, 4}, {63699, 68021, 59373}

X(68025) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR9-2.6 WRT CTR28-69

X(68025) lies on circumconic {{A, B, C, X(393), X(51348)}} and on these lines: {2, 58795}, {3, 15105}, {4, 6}, {20, 51261}, {24, 15152}, {30, 41597}, {64, 3523}, {140, 6000}, {141, 68022}, {154, 3522}, {468, 36982}, {524, 9968}, {546, 18128}, {548, 50414}, {550, 6759}, {1204, 15448}, {1619, 3516}, {1656, 6247}, {1657, 5878}, {1660, 46374}, {1853, 5068}, {1885, 10619}, {2777, 62144}, {3357, 15712}, {3533, 67894}, {3589, 44870}, {3628, 52102}, {3850, 14864}, {3851, 14216}, {3854, 32064}, {3858, 18381}, {5045, 6001}, {5056, 12324}, {5059, 5895}, {5073, 9833}, {5493, 40660}, {5663, 41674}, {5882, 40658}, {5925, 62127}, {7395, 15579}, {8567, 35260}, {9914, 15577}, {10182, 61813}, {10282, 33923}, {10299, 10606}, {10540, 22955}, {10575, 59659}, {10990, 15647}, {11064, 12279}, {11202, 62069}, {11204, 61789}, {11381, 23292}, {11414, 15582}, {11439, 37649}, {11803, 18400}, {12162, 34002}, {12174, 13567}, {12242, 13474}, {12250, 17821}, {12791, 46472}, {13093, 15720}, {13382, 41589}, {13568, 26883}, {14157, 43617}, {14530, 20427}, {14531, 47094}, {15153, 35488}, {15581, 39568}, {15585, 34146}, {15717, 68027}, {15873, 67899}, {16619, 41725}, {17845, 49135}, {18282, 44158}, {18325, 48669}, {18920, 43592}, {20299, 35018}, {21841, 68026}, {22802, 62036}, {23324, 34780}, {23329, 55859}, {26888, 63273}, {29181, 52016}, {31166, 64196}, {32184, 45979}, {32269, 64025}, {32767, 44904}, {34779, 64067}, {34785, 62159}, {35450, 61803}, {37669, 61150}, {40285, 50008}, {40341, 46207}, {40686, 61886}, {41963, 49250}, {41964, 49251}, {46219, 65151}, {46850, 53415}, {48672, 62131}, {50691, 61721}, {51425, 64030}, {54050, 62067}, {54211, 62110}, {55856, 61747}, {61680, 61834}, {61792, 64027}, {62023, 64033}, {62107, 64758}, {62124, 64059}, {62147, 64187}

X(68025) = midpoint of X(i) and X(j) for these {i,j}: {1498, 2883}, {5878, 34782}, {5894, 6225}, {6247, 12315}, {9833, 51491}, {17845, 68058}
X(68025) = reflection of X(i) in X(j) for these {i,j}: {140, 14862}, {548, 50414}, {5893, 2883}, {6696, 16252}, {14864, 3850}, {52102, 3628}, {61540, 64063}
X(68025) = pole of line {51, 5894} with respect to the Jerabek hyperbola
X(68025) = pole of line {394, 12279} with respect to the Stammler hyperbola
X(68025) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {140, 14862, 16252}, {154, 6225, 5894}, {1498, 2883, 1503}, {1498, 5656, 2883}, {1503, 2883, 5893}, {5878, 32063, 34782}, {6000, 14862, 140}, {6000, 16252, 6696}, {6000, 64063, 61540}, {6696, 16252, 58434}, {12315, 67890, 6247}, {12324, 64024, 23332}, {14864, 61749, 3850}, {17845, 66752, 68058}, {35260, 68015, 8567}

X(68026) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR9-4.6 WRT CTR28-69

X(68026) lies on these lines: {3, 206}, {4, 14542}, {5, 2883}, {20, 41715}, {25, 185}, {26, 6759}, {30, 66754}, {51, 64037}, {52, 2393}, {64, 7395}, {66, 7401}, {154, 5562}, {155, 1660}, {159, 12166}, {184, 68020}, {389, 1503}, {511, 34774}, {546, 32393}, {550, 44544}, {568, 64033}, {569, 41593}, {578, 34117}, {1181, 1619}, {1216, 7502}, {1853, 64854}, {2165, 15575}, {2777, 14641}, {2781, 15644}, {2807, 40658}, {3313, 59346}, {3357, 7514}, {3542, 5656}, {3549, 12162}, {3567, 64034}, {3917, 17821}, {5446, 10115}, {5447, 11202}, {5462, 11818}, {5596, 7487}, {5609, 10628}, {5663, 13383}, {5878, 10575}, {5889, 11206}, {5890, 34781}, {5891, 47525}, {5893, 13474}, {5907, 6676}, {6225, 6816}, {6644, 44679}, {6696, 16836}, {6697, 7405}, {6997, 10574}, {7493, 12111}, {7507, 11381}, {7528, 9730}, {7529, 12315}, {7539, 40686}, {7564, 46849}, {7568, 10170}, {7715, 13382}, {7729, 58795}, {8549, 11432}, {8550, 46363}, {9714, 32063}, {9934, 11562}, {10095, 63714}, {10110, 41362}, {10192, 11793}, {10540, 48669}, {11424, 63422}, {11425, 12294}, {11444, 35260}, {12058, 35602}, {12233, 15809}, {12235, 22663}, {12241, 64820}, {12279, 66752}, {12362, 15311}, {13289, 15132}, {13346, 19139}, {13562, 52520}, {14128, 61606}, {14530, 18436}, {14531, 34750}, {14576, 41373}, {14786, 65151}, {14855, 20427}, {14915, 18569}, {15030, 64024}, {15043, 32064}, {15305, 68024}, {15577, 46728}, {15818, 64759}, {16072, 64714}, {16223, 63716}, {16621, 66713}, {16655, 67923}, {17704, 23328}, {17824, 43844}, {17845, 45186}, {18376, 44863}, {18383, 63672}, {18909, 41735}, {18925, 41719}, {19153, 37476}, {19467, 65654}, {21841, 68025}, {25711, 36201}, {26879, 41603}, {31305, 36989}, {31804, 32366}, {31867, 44924}, {32321, 64049}, {32352, 32359}, {32379, 45118}, {32534, 43896}, {34118, 61676}, {34224, 52000}, {34382, 61751}, {34780, 37481}, {37498, 64031}, {37514, 63420}, {37515, 44883}, {39571, 58483}, {40285, 46261}, {41602, 67902}, {43581, 44108}, {44084, 67903}, {46847, 63728}, {47328, 61139}, {52093, 64726}, {66606, 67894}, {66747, 68015}, {66758, 67263}, {67891, 68028}

X(68026) = midpoint of X(i) and X(j) for these {i,j}: {52, 9833}, {185, 1498}, {550, 44544}, {5562, 6293}, {5596, 19161}, {5878, 10575}, {6241, 36982}, {6759, 41725}, {9934, 11562}, {17845, 45186}, {32352, 32359}
X(68026) = reflection of X(i) in X(j) for these {i,j}: {389, 41589}, {1216, 10282}, {5907, 16252}, {6247, 9729}, {13474, 5893}, {14216, 58492}, {18381, 5462}, {18383, 63697}, {18569, 58545}, {31978, 40647}, {32366, 41729}, {32392, 41725}, {41362, 10110}, {51756, 58547}
X(68026) = pole of line {1593, 5925} with respect to the Jerabek hyperbola
X(68026) = pole of line {800, 27371} with respect to the Kiepert hyperbola
X(68026) = pole of line {1370, 64718} with respect to the Stammler hyperbola
X(68026) = intersection, other than A, B, C, of circumconics {{A, B, C, X(17703), X(34207)}}, {{A, B, C, X(52041), X(56345)}}
X(68026) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {52, 9833, 2393}, {154, 6293, 5562}, {389, 6756, 9969}, {5656, 6241, 36982}, {6000, 40647, 31978}, {6000, 9729, 6247}, {6759, 41725, 13754}, {7528, 14216, 51756}, {9730, 14216, 58492}, {13754, 41725, 32392}, {64719, 65376, 34782}

X(68027) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR13-2.2 WRT CTR28-69

X(68027) lies on these lines: {2, 64}, {4, 13399}, {20, 15105}, {30, 11411}, {154, 62063}, {376, 3917}, {381, 66749}, {459, 51892}, {519, 9899}, {541, 13203}, {549, 5656}, {1204, 62979}, {1498, 10304}, {1503, 11160}, {1853, 61985}, {1992, 34146}, {2777, 62042}, {3357, 3524}, {3522, 58795}, {3534, 34781}, {3543, 15311}, {3545, 5878}, {3830, 64187}, {3839, 6247}, {3845, 48672}, {5032, 68019}, {5055, 61540}, {5071, 65151}, {5893, 61954}, {5894, 62120}, {5895, 50687}, {5925, 62160}, {6001, 34632}, {6759, 19708}, {7355, 10385}, {7714, 11381}, {8567, 15705}, {8703, 12315}, {9833, 62130}, {10192, 61806}, {10282, 15710}, {10605, 68010}, {10606, 15692}, {11001, 20427}, {11202, 62058}, {11204, 15715}, {11239, 49186}, {11240, 49185}, {11433, 62962}, {11442, 40196}, {12262, 38314}, {12279, 33523}, {12379, 37645}, {12779, 53620}, {13445, 37669}, {14216, 15682}, {14530, 45759}, {14862, 61814}, {14864, 62028}, {14927, 66615}, {15640, 64037}, {15697, 34782}, {15698, 64027}, {15702, 67890}, {15708, 16252}, {15717, 68025}, {15721, 23328}, {17821, 62059}, {17845, 62148}, {18381, 62017}, {18383, 62009}, {18400, 62161}, {18913, 62966}, {18925, 66720}, {18931, 62961}, {19053, 49251}, {19054, 49250}, {19087, 63058}, {19088, 63059}, {19710, 64033}, {20299, 41106}, {21356, 41735}, {22802, 41099}, {23324, 61994}, {23325, 61973}, {23329, 61895}, {23332, 61944}, {25563, 61861}, {30443, 34608}, {32063, 34200}, {32321, 37948}, {34469, 62978}, {34785, 62135}, {34801, 35512}, {37940, 64759}, {40686, 61924}, {41362, 62032}, {41629, 68016}, {44762, 50693}, {45185, 62113}, {45420, 49080}, {45421, 49081}, {50414, 62066}, {50975, 64719}, {51028, 67888}, {51358, 58758}, {51491, 62007}, {52028, 63127}, {59373, 63420}, {61088, 64014}, {61606, 61829}, {61680, 61825}, {61721, 62005}, {61735, 61927}, {61747, 61859}, {61749, 61899}, {61833, 64063}, {61912, 67868}, {62030, 68058}, {62081, 64059}, {63022, 64031}, {63064, 68021}, {66372, 66723}

X(68027) = midpoint of X(i) and X(j) for these {i,j}: {2, 68015}
X(68027) = reflection of X(i) in X(j) for these {i,j}: {2, 64}, {5656, 35450}, {6225, 2}, {11001, 20427}, {11206, 54050}, {12315, 8703}, {15640, 64037}, {15682, 14216}, {34781, 3534}, {48672, 3845}, {51028, 67888}, {62160, 5925}, {63064, 68021}, {64014, 61088}, {64033, 19710}, {64187, 3830}, {66752, 67894}
X(68027) = anticomplement of X(64714)
X(68027) = X(i)-Dao conjugate of X(j) for these {i, j}: {64714, 64714}
X(68027) = pole of line {13474, 18931} with respect to the Jerabek hyperbola
X(68027) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {64, 68015, 6225}, {6000, 54050, 11206}, {12250, 12324, 64726}, {12250, 13093, 12324}, {12250, 64034, 64758}

X(68028) = X(5) OF CTR28-69

X(68028) lies on these lines: {4, 67}, {5, 32364}, {30, 63728}, {52, 23324}, {64, 15305}, {110, 32345}, {125, 52003}, {140, 6000}, {143, 10628}, {154, 15056}, {185, 23332}, {378, 63658}, {974, 23294}, {1154, 18383}, {1498, 7509}, {1503, 5907}, {1593, 46374}, {1594, 66713}, {1853, 12111}, {2777, 32137}, {2883, 7399}, {3091, 6293}, {3153, 6145}, {3357, 12041}, {3850, 63737}, {4550, 44679}, {5066, 63697}, {5159, 31978}, {5448, 5663}, {5562, 41362}, {5876, 18381}, {5891, 34782}, {5893, 9822}, {5894, 11381}, {5895, 11439}, {5925, 11455}, {5943, 32392}, {5944, 32401}, {6101, 34786}, {6102, 23325}, {6241, 40686}, {6247, 11585}, {6640, 14643}, {6759, 15060}, {7395, 64061}, {7488, 68011}, {7514, 40285}, {7547, 63659}, {7691, 56924}, {8567, 12279}, {9968, 32154}, {10255, 25711}, {10263, 18376}, {10282, 14128}, {10574, 61735}, {10575, 23328}, {10606, 12290}, {11262, 32393}, {11412, 18405}, {11444, 17845}, {11459, 64037}, {11479, 34117}, {11591, 18400}, {11598, 12292}, {12022, 15739}, {12233, 20300}, {12278, 41673}, {13368, 19506}, {13434, 17824}, {13491, 23329}, {13630, 32767}, {14118, 15139}, {14216, 18435}, {15067, 34785}, {15311, 31833}, {15331, 64027}, {15811, 34778}, {16194, 51491}, {19149, 33537}, {20376, 58447}, {21650, 23315}, {30739, 36982}, {31724, 32369}, {32062, 68058}, {32379, 34864}, {32391, 35921}, {32903, 54044}, {37119, 67921}, {40916, 58795}, {40928, 52293}, {44235, 63695}, {45958, 61749}, {46372, 63420}, {54384, 63662}, {61940, 63714}, {62982, 67915}, {64024, 66756}, {67891, 68026}

X(68028) = midpoint of X(i) and X(j) for these {i,j}: {5562, 41362}, {5876, 18381}, {5894, 11381}, {6101, 34786}, {6247, 12162}, {11598, 12292}, {21650, 23315}
X(68028) = reflection of X(i) in X(j) for these {i,j}: {185, 32184}, {5893, 44870}, {10282, 14128}, {11262, 32393}, {13630, 32767}, {41589, 5}, {61749, 45958}
X(68028) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {185, 23332, 32184}, {34146, 44870, 5893}, {63420, 68022, 46372}

X(68029) = PERSPECTOR OF THESE TRIANGLES: CTR28-189 AND BEVAN ANTIPODAL

X(68029) lies on these lines: {1, 6611}, {4, 57}, {33, 41403}, {40, 221}, {46, 1743}, {65, 17831}, {942, 55115}, {1394, 37305}, {1422, 5908}, {1697, 53557}, {1698, 5514}, {1753, 47848}, {3359, 5909}, {4295, 60634}, {5930, 68036}, {5932, 56544}, {6260, 40212}, {7013, 37421}, {8809, 40396}, {9612, 54009}, {10374, 37550}, {10980, 11022}, {13539, 38674}, {13737, 15803}, {20324, 31393}, {52117, 64761}

X(68029) = perspector of circumconic {{A, B, C, X(65159), X(65330)}}
X(68029) = pole of line {285, 1819} with respect to the Stammler hyperbola
X(68029) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(1103)}}, {{A, B, C, X(40), X(40836)}}, {{A, B, C, X(84), X(7078)}}, {{A, B, C, X(223), X(45818)}}, {{A, B, C, X(7008), X(7074)}}, {{A, B, C, X(8809), X(52097)}}
X(68029) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6611, 40953, 1}

X(68030) = PERSPECTOR OF THESE TRIANGLES: CTR28-189 AND 3RD EXTOUCH

X(68030) lies on circumconic {{A, B, C, X(7003), X(7078)}} and on these lines: {4, 1903}, {40, 221}, {208, 5706}, {1422, 3182}, {1498, 2270}, {5908, 8808}, {5909, 6907}, {6611, 64347}, {13737, 40658}, {14557, 15239}, {15498, 37414}, {37528, 53557}

X(68031) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND ANTI-PAVLOV WRT CTR28-189

X(68031) lies on these lines: {1, 64376}, {3, 14996}, {4, 64401}, {10, 64400}, {20, 20019}, {21, 517}, {27, 56887}, {40, 81}, {46, 1014}, {58, 7991}, {65, 64414}, {145, 7415}, {165, 4658}, {333, 962}, {411, 56181}, {412, 56014}, {511, 33557}, {515, 66212}, {516, 64072}, {859, 8158}, {946, 5235}, {970, 6915}, {1010, 59417}, {1408, 5183}, {1412, 5128}, {1702, 64386}, {1703, 64385}, {1817, 3193}, {1836, 64408}, {1902, 64378}, {2800, 66005}, {2802, 66004}, {3057, 64382}, {3562, 24310}, {3579, 64393}, {4184, 10306}, {4220, 48917}, {4221, 12702}, {4225, 22770}, {4278, 5537}, {4653, 11531}, {4720, 12245}, {4921, 28194}, {5119, 64420}, {5323, 37567}, {5333, 6684}, {5584, 18185}, {5603, 17557}, {5657, 14005}, {5752, 12111}, {5762, 31902}, {5812, 64407}, {6001, 68016}, {6197, 14014}, {6361, 64384}, {6769, 54356}, {6986, 10441}, {7957, 18178}, {7982, 64415}, {8227, 64425}, {9537, 16049}, {9911, 64395}, {10164, 28619}, {12197, 64381}, {12458, 64396}, {12459, 64397}, {12497, 64398}, {12696, 64402}, {12697, 64403}, {12698, 64404}, {12699, 64405}, {12700, 64406}, {12701, 64409}, {12703, 64422}, {12704, 64423}, {13912, 64417}, {13975, 64418}, {16704, 20070}, {17531, 33879}, {17551, 26446}, {18206, 63985}, {22793, 64399}, {24556, 26062}, {25526, 43174}, {28618, 58441}, {31162, 64424}, {31774, 37163}, {32475, 57093}, {34632, 41629}, {35610, 64412}, {35611, 64413}, {37062, 37685}, {37418, 37584}, {37421, 56020}, {37559, 56048}, {38329, 53412}, {41338, 62843}, {45923, 48924}, {48487, 64379}, {48488, 64380}, {48661, 64383}, {48740, 64389}, {48741, 64390}, {49054, 64391}, {49055, 64392}, {49163, 64394}, {49226, 64410}, {49227, 64411}, {49323, 64387}, {49324, 64388}, {50810, 51669}

X(68031) = midpoint of X(i) and X(j) for these {i,j}: {66212, 68054}
X(68031) = reflection of X(i) in X(j) for these {i,j}: {21, 64720}, {67852, 64072}
X(68031) = pole of line {1385, 7330} with respect to the Stammler hyperbola
X(68031) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 81, 37402}, {46, 64421, 1014}, {516, 64072, 67852}, {517, 64720, 21}, {16704, 20070, 37422}, {66212, 68054, 515}

X(68032) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(57) WRT CTR28-189

X(68032) lies on these lines: {1, 3}, {4, 6762}, {8, 67880}, {9, 5603}, {20, 62832}, {84, 962}, {101, 2270}, {104, 60968}, {145, 54051}, {200, 22753}, {329, 946}, {347, 36984}, {376, 43175}, {390, 63438}, {515, 15239}, {516, 2096}, {518, 54159}, {527, 11372}, {551, 67962}, {573, 47299}, {758, 68001}, {934, 56544}, {944, 68057}, {958, 64669}, {1006, 38316}, {1012, 43166}, {1058, 64004}, {1320, 66058}, {1490, 3555}, {1519, 28609}, {1537, 60965}, {1699, 37822}, {1706, 12245}, {1768, 50891}, {2094, 10860}, {2951, 63432}, {3091, 63135}, {3149, 6765}, {3158, 6905}, {3241, 7966}, {3243, 18446}, {3306, 59417}, {3421, 4847}, {3452, 8227}, {3577, 3872}, {3616, 61122}, {3624, 11218}, {3646, 9624}, {3656, 3929}, {3753, 64325}, {3820, 54447}, {3868, 7971}, {3870, 52026}, {3873, 64150}, {3881, 12520}, {3889, 10884}, {4221, 18164}, {4301, 12705}, {4342, 62839}, {4345, 67120}, {4853, 7686}, {5082, 64001}, {5231, 7680}, {5250, 5734}, {5290, 15908}, {5437, 5657}, {5703, 7160}, {5705, 63257}, {5715, 24390}, {5758, 10396}, {5762, 10384}, {5763, 11373}, {5795, 5804}, {5812, 9614}, {5853, 50701}, {5881, 66251}, {5886, 7308}, {6001, 62823}, {6173, 54158}, {6261, 41863}, {6361, 9841}, {6684, 62773}, {6692, 31423}, {6764, 50700}, {6844, 24386}, {6854, 38200}, {6868, 41864}, {6911, 46917}, {6927, 59722}, {6987, 64162}, {7171, 28174}, {7330, 22791}, {7983, 24469}, {8257, 61275}, {8583, 63976}, {9589, 10085}, {9785, 62836}, {10595, 55104}, {10698, 66068}, {10864, 41869}, {11019, 64111}, {11038, 54206}, {11496, 62824}, {11522, 41229}, {11523, 63986}, {11827, 66682}, {12114, 12651}, {12120, 18241}, {12526, 45776}, {12565, 12675}, {12672, 54422}, {12687, 64003}, {12842, 12864}, {12848, 63993}, {13374, 64673}, {13464, 31435}, {14217, 63974}, {15185, 50528}, {17718, 55300}, {18444, 62815}, {19854, 20196}, {26446, 61535}, {28228, 64129}, {28234, 63137}, {30305, 66239}, {31142, 38021}, {34371, 64084}, {34498, 52384}, {34631, 48363}, {34791, 64077}, {35514, 60955}, {37106, 62856}, {37407, 51723}, {38030, 58813}, {38036, 52457}, {38053, 54205}, {39542, 60937}, {42871, 65404}, {51423, 56545}, {54135, 61705}, {62812, 64449}, {63168, 68003}, {63399, 67886}, {64047, 67047}, {64138, 64372}

X(68032) = midpoint of X(i) and X(j) for these {i,j}: {962, 9965}
X(68032) = reflection of X(i) in X(j) for these {i,j}: {40, 57}, {200, 22753}, {329, 946}, {2093, 2095}, {3421, 7682}, {6282, 999}, {58808, 63430}, {64111, 11019}
X(68032) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(56), X(61121)}}, {{A, B, C, X(1295), X(7994)}}, {{A, B, C, X(3577), X(64106)}}, {{A, B, C, X(3680), X(31786)}}, {{A, B, C, X(8726), X(51497)}}, {{A, B, C, X(9940), X(51498)}}
X(68032) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3428, 3576}, {516, 63430, 58808}, {517, 2095, 2093}, {517, 999, 6282}, {962, 62874, 84}, {1482, 5709, 1697}, {1697, 5709, 40}, {3421, 7682, 5587}, {4301, 62858, 12705}

X(68033) = ORTHOLOGY CENTER OF THESE TRIANGLES: INVERSE-OF-X(10)-CIRCUMCONCEVIAN-OF-X(37) WRT CTR28-189

X(68033) lies on these lines: {1, 7175}, {3, 2938}, {4, 740}, {5, 21926}, {10, 67853}, {37, 40}, {75, 946}, {84, 54344}, {192, 962}, {200, 22014}, {355, 49459}, {376, 50111}, {381, 50086}, {511, 64134}, {515, 49470}, {516, 3993}, {517, 984}, {518, 5693}, {536, 31162}, {551, 51044}, {581, 2667}, {726, 4301}, {742, 64085}, {944, 49471}, {952, 49678}, {986, 15488}, {1071, 64546}, {1351, 9355}, {1482, 49490}, {1695, 17038}, {1699, 49474}, {1721, 63442}, {1953, 5698}, {1959, 24280}, {1962, 37400}, {2171, 64168}, {2550, 21801}, {2783, 48902}, {2800, 66067}, {2802, 66057}, {2805, 6326}, {3061, 3923}, {3149, 64727}, {3241, 51064}, {3543, 51054}, {3545, 50096}, {3576, 15569}, {3644, 68035}, {3656, 31178}, {3679, 51038}, {3696, 5587}, {3739, 8227}, {3781, 24341}, {3842, 5657}, {4032, 4295}, {4192, 17592}, {4307, 17452}, {4664, 28194}, {4687, 6684}, {4688, 38021}, {4698, 31423}, {4699, 68034}, {4704, 20070}, {4709, 19925}, {4732, 5818}, {5480, 49531}, {5603, 24325}, {5691, 49469}, {5844, 49689}, {5881, 28581}, {5886, 40328}, {6001, 67978}, {6996, 24257}, {7146, 24248}, {7377, 27474}, {7406, 27480}, {7611, 48886}, {8148, 49503}, {9943, 58620}, {9965, 21328}, {10306, 34247}, {10446, 29057}, {10863, 27489}, {11224, 49498}, {11496, 54410}, {11531, 49448}, {12245, 49457}, {12699, 29010}, {17444, 64016}, {17768, 18161}, {17860, 22000}, {18492, 49468}, {19647, 46904}, {20718, 33536}, {21033, 36695}, {21068, 49653}, {22791, 49493}, {26446, 61522}, {27804, 50694}, {28174, 51046}, {28212, 61623}, {28234, 49450}, {29309, 31395}, {32857, 41777}, {37529, 67887}, {37569, 44670}, {38034, 61549}, {38035, 49481}, {39551, 55004}, {39573, 60634}, {41869, 49462}, {44671, 61705}, {49461, 52852}, {49475, 61296}, {50094, 50810}

X(68033) = midpoint of X(i) and X(j) for these {i,j}: {192, 962}, {3241, 51064}, {3543, 51054}, {5691, 49469}, {11531, 49448}, {49461, 52852}, {49470, 51063}
X(68033) = reflection of X(i) in X(j) for these {i,j}: {40, 37}, {75, 946}, {376, 50111}, {944, 49471}, {984, 20430}, {1071, 64546}, {3679, 51038}, {3696, 67858}, {4709, 19925}, {9943, 58620}, {12245, 49457}, {30271, 15569}, {30273, 3993}, {31178, 3656}, {49459, 355}, {49474, 64088}, {49490, 1482}, {49531, 5480}, {50086, 381}, {50810, 50094}, {51044, 551}, {61296, 49475}, {63427, 24325}
X(68033) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {192, 962, 29054}, {516, 3993, 30273}, {517, 20430, 984}, {1699, 49474, 64088}, {3696, 67858, 5587}, {5603, 63427, 24325}, {5886, 64728, 40328}, {7982, 11372, 64084}, {15569, 30271, 3576}, {49470, 51063, 515}

X(68034) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR5-2.2 WRT CTR28-189

X(68034) lies on these lines: {1, 3091}, {2, 40}, {3, 5284}, {4, 1385}, {5, 8}, {7, 90}, {10, 5056}, {11, 938}, {12, 6945}, {20, 1125}, {21, 22753}, {30, 10248}, {35, 30332}, {46, 64114}, {55, 6915}, {56, 6912}, {65, 5704}, {69, 38035}, {78, 64669}, {100, 6918}, {104, 32558}, {140, 6361}, {142, 64696}, {144, 38036}, {145, 5068}, {147, 38220}, {149, 16174}, {150, 5543}, {153, 16173}, {165, 10303}, {226, 11037}, {278, 52248}, {329, 6846}, {354, 12528}, {355, 3241}, {376, 22793}, {377, 22835}, {381, 944}, {382, 38028}, {388, 6957}, {390, 9614}, {392, 5806}, {404, 11496}, {411, 1001}, {412, 17917}, {442, 7956}, {452, 5715}, {474, 59412}, {496, 3487}, {497, 5703}, {498, 6979}, {499, 3336}, {515, 3622}, {516, 3523}, {517, 3090}, {519, 7989}, {546, 10246}, {547, 50810}, {549, 48661}, {551, 3839}, {581, 29814}, {631, 9778}, {908, 5815}, {942, 47743}, {952, 3851}, {997, 6993}, {999, 5714}, {1012, 5253}, {1056, 11373}, {1058, 7743}, {1071, 64149}, {1158, 27003}, {1319, 5229}, {1387, 9654}, {1478, 4308}, {1479, 4313}, {1483, 5066}, {1490, 4666}, {1519, 6847}, {1537, 31272}, {1565, 32086}, {1621, 3149}, {1656, 5657}, {1698, 4301}, {1702, 8972}, {1703, 13941}, {1768, 33709}, {1829, 6622}, {1836, 6974}, {1902, 8889}, {2051, 19853}, {2077, 17572}, {2094, 3652}, {2095, 11684}, {2098, 3614}, {2099, 7173}, {2475, 26333}, {2476, 7681}, {2550, 25681}, {2551, 5087}, {2646, 5225}, {2800, 66063}, {2801, 50190}, {2802, 66045}, {2807, 15043}, {2886, 6991}, {2975, 6913}, {3057, 10588}, {3062, 38054}, {3070, 13959}, {3071, 13902}, {3085, 6953}, {3146, 3576}, {3153, 51693}, {3189, 11235}, {3219, 12704}, {3244, 37714}, {3305, 68036}, {3306, 12705}, {3428, 5047}, {3434, 6864}, {3436, 6939}, {3474, 5433}, {3475, 37722}, {3476, 10895}, {3486, 7548}, {3488, 9669}, {3522, 10165}, {3525, 3579}, {3526, 28174}, {3528, 28146}, {3529, 13624}, {3543, 4297}, {3544, 10222}, {3555, 10157}, {3582, 65384}, {3583, 4305}, {3600, 9612}, {3617, 7982}, {3618, 64085}, {3620, 64084}, {3621, 16200}, {3623, 5881}, {3625, 16189}, {3626, 11224}, {3628, 12702}, {3632, 61264}, {3633, 38155}, {3634, 7991}, {3636, 61274}, {3648, 16617}, {3653, 15682}, {3654, 61899}, {3655, 41099}, {3656, 5071}, {3679, 61924}, {3742, 12688}, {3813, 6764}, {3816, 6943}, {3828, 61906}, {3829, 12635}, {3830, 38022}, {3838, 6925}, {3843, 34773}, {3845, 51700}, {3850, 10283}, {3854, 5882}, {3855, 7967}, {3857, 28224}, {3858, 61273}, {3868, 13374}, {3869, 5775}, {3873, 5777}, {3877, 6933}, {3889, 14872}, {3890, 6969}, {3916, 59386}, {3957, 17857}, {4193, 7680}, {4197, 15908}, {4208, 8583}, {4292, 5265}, {4300, 26102}, {4317, 61703}, {4323, 7741}, {4339, 33106}, {4342, 51784}, {4345, 7951}, {4423, 6986}, {4430, 63967}, {4511, 5175}, {4678, 28234}, {4699, 68033}, {4816, 16191}, {4870, 15933}, {4881, 31295}, {4928, 38329}, {4999, 5698}, {5045, 5927}, {5046, 26332}, {5055, 5690}, {5067, 26446}, {5070, 61524}, {5072, 10247}, {5076, 58230}, {5079, 8148}, {5080, 6893}, {5082, 64083}, {5141, 67857}, {5154, 5554}, {5177, 19861}, {5180, 6862}, {5218, 12701}, {5219, 12053}, {5222, 36662}, {5231, 54398}, {5249, 37434}, {5259, 37106}, {5260, 22770}, {5273, 5536}, {5281, 10624}, {5308, 7377}, {5333, 37422}, {5437, 63985}, {5439, 9856}, {5493, 19878}, {5506, 18230}, {5552, 6964}, {5558, 13257}, {5584, 8167}, {5713, 33107}, {5744, 6824}, {5758, 6832}, {5768, 6841}, {5805, 6857}, {5809, 7678}, {5817, 20330}, {5880, 6966}, {5889, 58469}, {5921, 16475}, {6001, 68024}, {6049, 45287}, {6172, 60895}, {6191, 30415}, {6192, 30414}, {6223, 10586}, {6224, 11729}, {6244, 16862}, {6253, 49736}, {6261, 6870}, {6282, 37436}, {6557, 46937}, {6667, 64189}, {6762, 66465}, {6763, 60911}, {6796, 61155}, {6826, 7704}, {6827, 26127}, {6833, 62773}, {6836, 26105}, {6844, 63986}, {6849, 12116}, {6866, 21740}, {6867, 10598}, {6886, 18228}, {6890, 12609}, {6894, 40259}, {6896, 10596}, {6900, 37820}, {6901, 10525}, {6904, 64078}, {6909, 25524}, {6919, 19860}, {6920, 11249}, {6924, 64792}, {6931, 45776}, {6932, 25466}, {6935, 20292}, {6944, 27529}, {6946, 11248}, {6956, 10584}, {6960, 10198}, {6965, 10526}, {6972, 10200}, {6973, 10599}, {6978, 37562}, {6990, 26470}, {7080, 30852}, {7373, 38669}, {7379, 16020}, {7384, 26626}, {7402, 29627}, {7406, 17397}, {7407, 16823}, {7485, 9911}, {7503, 11365}, {7507, 7718}, {7609, 17257}, {7613, 11512}, {7682, 24987}, {7968, 31412}, {7969, 42561}, {7970, 23514}, {7972, 38161}, {7973, 23332}, {7978, 23515}, {7983, 36519}, {7984, 36518}, {8164, 9957}, {8165, 9623}, {8236, 37701}, {8273, 33557}, {8834, 26719}, {9535, 19858}, {9581, 64160}, {9589, 10164}, {9619, 43448}, {9620, 31404}, {9671, 10543}, {9782, 26492}, {9799, 10883}, {9809, 12611}, {9940, 9961}, {10031, 38077}, {10109, 34718}, {10172, 46932}, {10269, 21669}, {10304, 64005}, {10310, 17531}, {10394, 16193}, {10430, 37447}, {10446, 19863}, {10449, 10886}, {10516, 51192}, {10529, 31053}, {10582, 10884}, {10587, 64148}, {10592, 64897}, {10698, 23513}, {10724, 34123}, {10728, 38032}, {10738, 64473}, {10742, 38044}, {10863, 21620}, {10893, 17577}, {10894, 37375}, {10915, 66243}, {11012, 16865}, {11019, 11036}, {11038, 63970}, {11110, 64400}, {11231, 61886}, {11240, 67855}, {11263, 64130}, {11281, 52269}, {11362, 46933}, {11372, 62778}, {11444, 67967}, {11451, 58487}, {11523, 24386}, {11541, 31666}, {11723, 14644}, {11724, 14639}, {11737, 50798}, {11827, 66099}, {12000, 38665}, {12005, 61705}, {12111, 64662}, {12162, 64663}, {12247, 60759}, {12262, 66752}, {12512, 15692}, {12536, 22836}, {12541, 34619}, {12632, 59722}, {12645, 19709}, {12669, 58564}, {12811, 37705}, {12812, 38112}, {13253, 59419}, {13373, 64358}, {13405, 51785}, {13607, 61954}, {13743, 61552}, {13888, 42522}, {13942, 42523}, {14561, 39898}, {14647, 54199}, {14869, 28216}, {15017, 21630}, {15071, 58565}, {15178, 61964}, {15305, 64661}, {15640, 51109}, {15672, 16113}, {15674, 49177}, {15677, 16125}, {15683, 50828}, {15684, 50819}, {15698, 28202}, {15700, 50813}, {15702, 28198}, {15717, 31730}, {15721, 50808}, {15808, 28164}, {16496, 38146}, {17018, 37732}, {17127, 37530}, {17188, 37113}, {17502, 17538}, {17558, 40998}, {17784, 27385}, {18225, 33593}, {18240, 66002}, {18398, 31803}, {18440, 38040}, {18444, 63988}, {18491, 64173}, {18526, 61278}, {19065, 42262}, {19066, 42265}, {19582, 30741}, {19872, 63468}, {19875, 50872}, {19876, 61897}, {20053, 61263}, {21075, 46873}, {21077, 34625}, {21297, 38324}, {21454, 64124}, {22758, 45977}, {23841, 27355}, {24349, 67853}, {24473, 31821}, {24703, 30478}, {24954, 26040}, {25507, 37402}, {25525, 37421}, {26103, 37365}, {26725, 37433}, {27138, 28292}, {27268, 29054}, {27382, 54324}, {27525, 63137}, {28150, 50693}, {28154, 62127}, {28168, 62021}, {28172, 50690}, {28182, 62100}, {28186, 61984}, {28190, 62008}, {28204, 41106}, {28208, 50807}, {28232, 61848}, {29648, 50698}, {29666, 50699}, {30290, 67051}, {30340, 64197}, {31145, 61930}, {31671, 38043}, {31673, 50689}, {31738, 62187}, {31870, 64047}, {32064, 40658}, {32557, 34789}, {32785, 49226}, {32786, 49227}, {33597, 62870}, {33748, 39878}, {34036, 66593}, {34628, 51108}, {34631, 51072}, {34638, 62059}, {34640, 67959}, {34648, 51105}, {34748, 61246}, {35242, 61820}, {35262, 37435}, {35514, 61595}, {35641, 42274}, {35642, 42277}, {35762, 42269}, {35763, 42268}, {36991, 38053}, {37105, 52769}, {37126, 49553}, {37229, 54348}, {37298, 38073}, {37522, 64013}, {37542, 37691}, {37623, 62838}, {37719, 67046}, {37727, 38140}, {38023, 51023}, {38041, 60884}, {38059, 63974}, {38066, 61910}, {38072, 50999}, {38076, 51093}, {38083, 61913}, {38138, 61940}, {38315, 67865}, {38316, 67866}, {38513, 67216}, {40257, 64281}, {40333, 43166}, {42270, 44635}, {42273, 44636}, {43174, 46935}, {44431, 48900}, {44841, 68000}, {48571, 62434}, {48899, 50420}, {50687, 51705}, {50796, 61296}, {50799, 61951}, {50800, 61949}, {50805, 61931}, {50809, 61865}, {50811, 61985}, {50812, 61778}, {50815, 62166}, {50818, 61284}, {50821, 61895}, {50823, 61922}, {50825, 61872}, {50832, 62015}, {50833, 62088}, {50862, 61994}, {50863, 51085}, {50867, 62005}, {51068, 61920}, {51071, 61943}, {51074, 61972}, {51084, 62058}, {51103, 61958}, {51110, 61989}, {51723, 67048}, {51792, 66247}, {52412, 56887}, {54052, 64119}, {54392, 63992}, {55858, 61614}, {58221, 62097}, {58224, 62119}, {58383, 64071}, {58421, 64136}, {58441, 61863}, {58588, 67992}, {59372, 64699}, {59374, 63971}, {59380, 67986}, {59415, 64192}, {59420, 62083}, {59503, 61267}, {59591, 62710}, {60926, 60995}, {61245, 61942}, {61262, 61937}, {61266, 61921}, {61281, 61948}, {61286, 61946}, {61580, 66008}, {62830, 64731}, {62858, 64143}, {62864, 64131}, {63962, 64762}, {64008, 64138}, {65452, 66515}

X(68034) = midpoint of X(i) and X(j) for these {i,j}: {3622, 3832}
X(68034) = reflection of X(i) in X(j) for these {i,j}: {3090, 61268}, {3523, 3624}, {3622, 9624}, {9588, 51073}, {9780, 3090}, {30389, 15808}, {50800, 61949}, {50813, 15700}, {50867, 62005}, {61980, 50807}, {62088, 50833}
X(68034) = anticomplement of X(31423)
X(68034) = X(i)-Dao conjugate of X(j) for these {i, j}: {31423, 31423}
X(68034) = X(i)-complementary conjugate of X(j) for these {i, j}: {24680, 10}
X(68034) = pole of line {4962, 21188} with respect to the incircle
X(68034) = pole of line {3776, 4778} with respect to the orthoptic circle of the Steiner Inellipse
X(68034) = pole of line {77, 2999} with respect to the dual conic of Yff parabola
X(68034) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1385), X(3940)}}, {{A, B, C, X(6684), X(58009)}}, {{A, B, C, X(7318), X(7319)}}, {{A, B, C, X(38306), X(60634)}}
X(68034) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3091, 59387}, {1, 3817, 3091}, {2, 20070, 6684}, {2, 946, 962}, {4, 3616, 5731}, {4, 5886, 3616}, {4, 9955, 9779}, {5, 1482, 5818}, {5, 18493, 5603}, {5, 63257, 11681}, {8, 11681, 5828}, {8, 5603, 5734}, {10, 7988, 5056}, {11, 3485, 938}, {20, 1125, 54445}, {65, 10589, 5704}, {140, 6361, 64108}, {145, 5068, 5587}, {226, 14986, 11037}, {226, 50443, 14986}, {355, 51709, 10595}, {381, 38314, 50864}, {381, 5901, 944}, {496, 3487, 10580}, {497, 11375, 5703}, {499, 18393, 4295}, {499, 4295, 5435}, {515, 9624, 3622}, {516, 3624, 3523}, {517, 61268, 3090}, {551, 12571, 5691}, {551, 30308, 3839}, {631, 12699, 9778}, {908, 64081, 5815}, {944, 5901, 38314}, {946, 6684, 31162}, {946, 8227, 2}, {1058, 11374, 10578}, {1125, 1699, 20}, {1125, 51118, 7987}, {1482, 5818, 8}, {1519, 6847, 67999}, {1656, 22791, 5657}, {1656, 5657, 19877}, {1698, 10171, 7486}, {1698, 4301, 59417}, {1699, 7987, 51118}, {2886, 7958, 6991}, {3086, 12047, 7}, {3086, 38037, 6837}, {3146, 46934, 3576}, {3544, 59388, 61261}, {3545, 10595, 355}, {3545, 51709, 3241}, {3576, 18483, 3146}, {3616, 9779, 4}, {3617, 15022, 10175}, {3622, 3832, 515}, {3623, 54448, 5881}, {3656, 9956, 12245}, {3813, 25568, 6764}, {3850, 10283, 18525}, {3855, 7967, 18480}, {4301, 10171, 1698}, {4423, 64077, 6986}, {5071, 12245, 9956}, {5072, 10247, 18357}, {5079, 8148, 38042}, {5261, 18220, 1}, {5587, 13464, 145}, {5603, 5818, 1482}, {5691, 30308, 12571}, {5715, 24541, 64079}, {6684, 31162, 20070}, {6826, 10531, 52367}, {6847, 55108, 9776}, {7743, 11374, 1058}, {7988, 11522, 10}, {8227, 38021, 946}, {9589, 34595, 10164}, {9612, 44675, 3600}, {9614, 13411, 390}, {9669, 37737, 3488}, {10165, 41869, 3522}, {10222, 61261, 59388}, {11230, 12699, 631}, {11362, 54447, 46933}, {11376, 17605, 388}, {11729, 59391, 6224}, {12047, 23708, 3086}, {12645, 19709, 61259}, {12645, 61259, 38074}, {15808, 28164, 30389}, {16173, 67876, 153}, {19843, 21616, 18228}, {19883, 50865, 15692}, {22791, 61269, 1656}, {25055, 50802, 3543}, {28208, 50807, 61980}, {28228, 51073, 9588}, {30384, 37692, 3085}, {38034, 61272, 3}, {38053, 42356, 36991}, {46933, 61914, 54447}, {63988, 64675, 18444}

X(68035) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR9-2.2 WRT CTR28-189

X(68035) lies on these lines: {1, 3529}, {2, 40}, {3, 15808}, {4, 3632}, {5, 28228}, {10, 3851}, {20, 64952}, {30, 13607}, {79, 66228}, {165, 61814}, {355, 14269}, {381, 38098}, {382, 515}, {497, 17706}, {516, 550}, {517, 546}, {519, 15687}, {527, 49600}, {547, 50814}, {549, 51075}, {551, 15688}, {553, 65988}, {944, 50865}, {952, 62013}, {1125, 3530}, {1483, 51095}, {1537, 6154}, {1699, 3855}, {1836, 66230}, {1902, 52285}, {2800, 66065}, {2802, 66052}, {3090, 63468}, {3091, 38127}, {3146, 16200}, {3336, 30384}, {3338, 4031}, {3428, 19526}, {3522, 61275}, {3528, 5603}, {3533, 61271}, {3534, 61277}, {3543, 51077}, {3544, 7991}, {3576, 62097}, {3579, 14869}, {3616, 62067}, {3624, 61836}, {3627, 11278}, {3644, 68033}, {3653, 62076}, {3654, 61933}, {3655, 62163}, {3656, 4297}, {3671, 40270}, {3679, 61967}, {3817, 5079}, {3828, 47478}, {3832, 63143}, {3843, 38155}, {3858, 38176}, {3880, 31822}, {4292, 20323}, {4295, 63993}, {4342, 57282}, {4669, 61977}, {4681, 29054}, {4745, 61259}, {4746, 38138}, {5057, 64201}, {5073, 61287}, {5076, 61244}, {5180, 64369}, {5493, 5886}, {5657, 61921}, {5690, 12571}, {5691, 34747}, {5714, 9819}, {5731, 62149}, {5748, 63138}, {5881, 20054}, {5882, 20057}, {5901, 12512}, {6261, 43166}, {6361, 10165}, {6705, 37532}, {6796, 61153}, {6999, 29625}, {7373, 30424}, {7982, 9812}, {7989, 50810}, {8148, 62004}, {9624, 9778}, {9955, 10172}, {9956, 11737}, {10164, 18493}, {10171, 61524}, {10222, 28164}, {10246, 62134}, {10247, 62053}, {10248, 62003}, {10595, 51705}, {10624, 37080}, {11008, 64084}, {11009, 66247}, {11012, 17574}, {11230, 61853}, {11372, 60957}, {11415, 63135}, {11496, 17571}, {11551, 36946}, {11813, 63990}, {12047, 37563}, {12101, 61246}, {12245, 50796}, {12563, 15172}, {12575, 39542}, {12645, 34648}, {12651, 57000}, {12701, 63999}, {12705, 67334}, {13624, 28216}, {14563, 66682}, {14893, 50801}, {15178, 28178}, {15682, 51094}, {15684, 51082}, {15686, 51085}, {15700, 50808}, {15715, 25055}, {17504, 51709}, {17538, 30392}, {17563, 64001}, {17573, 22753}, {17624, 31391}, {18481, 49139}, {19746, 37062}, {19829, 37088}, {19862, 61850}, {19875, 61928}, {19883, 61829}, {21075, 63142}, {21620, 30305}, {26446, 61905}, {28146, 62151}, {28158, 34773}, {28168, 61286}, {28190, 32900}, {28202, 51103}, {28204, 62022}, {28208, 61597}, {28224, 58240}, {29311, 64532}, {31253, 61269}, {31399, 59417}, {31662, 44245}, {31837, 67866}, {34638, 62109}, {34647, 64117}, {34718, 61969}, {35242, 61798}, {35403, 50804}, {35404, 51087}, {37571, 64160}, {38022, 61800}, {38028, 62062}, {38314, 62122}, {40341, 64085}, {44903, 51080}, {46932, 61265}, {50689, 61256}, {50693, 64954}, {50811, 62166}, {50817, 61985}, {50821, 61916}, {50831, 50870}, {50868, 62015}, {50871, 62011}, {50872, 61994}, {51071, 62046}, {51074, 53620}, {51108, 62057}, {51423, 63146}, {51700, 62101}, {54370, 60942}, {54447, 67096}, {55109, 63984}, {58244, 61253}, {58245, 59388}, {58441, 61272}, {60933, 64277}, {61257, 61975}, {61268, 61892}, {61276, 62105}, {61280, 62144}, {61292, 62034}, {62125, 64953}

X(68035) = midpoint of X(i) and X(j) for these {i,j}: {381, 51120}, {382, 3244}, {946, 962}, {1482, 51118}, {3543, 51077}, {3627, 11278}, {4297, 48661}, {4301, 12699}, {5882, 41869}, {7982, 31673}, {9589, 31730}, {11531, 47745}, {15684, 51082}, {35404, 51087}, {61292, 62034}
X(68035) = reflection of X(i) in X(j) for these {i,j}: {549, 51075}, {550, 3636}, {3626, 546}, {5690, 12571}, {6684, 946}, {12512, 5901}, {13464, 22791}, {15686, 51085}, {19925, 40273}, {35404, 51119}, {43174, 9955}, {44903, 51080}, {50801, 14893}, {50814, 547}, {50827, 381}, {50868, 62015}, {68037, 6684}
X(68035) = pole of line {28229, 39226} with respect to the circumcircle
X(68035) = pole of line {2999, 17365} with respect to the dual conic of Yff parabola
X(68035) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 11531, 47745}, {516, 22791, 13464}, {516, 3636, 550}, {517, 40273, 19925}, {517, 546, 3626}, {946, 28194, 6684}, {946, 962, 28194}, {962, 31162, 946}, {1482, 12699, 51118}, {1482, 51118, 515}, {3627, 11278, 28236}, {3656, 48661, 4297}, {4301, 51118, 1482}, {5603, 9589, 31730}, {5882, 41869, 28172}, {5901, 12512, 50828}, {5901, 28198, 12512}, {6684, 28194, 68037}, {7982, 9812, 31673}, {9812, 20050, 50688}, {9955, 28212, 43174}, {9955, 43174, 10172}, {10595, 64005, 51705}, {11531, 47745, 28234}, {19925, 40273, 18483}

X(68036) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR12-1.2 WRT CTR28-189

X(68036) lies on these lines: {1, 3}, {4, 4847}, {8, 50700}, {9, 946}, {10, 6864}, {19, 56887}, {20, 36845}, {30, 10864}, {58, 61086}, {63, 962}, {72, 63992}, {84, 516}, {144, 67999}, {200, 3149}, {219, 2270}, {223, 64069}, {278, 1753}, {329, 63989}, {347, 7177}, {390, 62836}, {405, 64669}, {411, 3870}, {497, 10396}, {515, 6762}, {518, 1490}, {527, 63962}, {580, 7290}, {602, 38857}, {610, 22153}, {758, 7971}, {936, 22753}, {1012, 12651}, {1058, 5759}, {1071, 12565}, {1103, 1465}, {1108, 5022}, {1125, 61122}, {1158, 3928}, {1210, 64111}, {1419, 37498}, {1435, 37417}, {1445, 14986}, {1496, 2263}, {1630, 15836}, {1699, 5812}, {1706, 11362}, {1708, 12053}, {1709, 6763}, {1728, 9614}, {1750, 14872}, {1763, 9121}, {1766, 2257}, {2057, 51378}, {2136, 28234}, {2328, 17560}, {2550, 64001}, {2551, 7682}, {2800, 66068}, {2802, 66058}, {2814, 53395}, {2886, 5715}, {2947, 55311}, {3158, 6796}, {3218, 20070}, {3220, 9911}, {3244, 7966}, {3296, 21151}, {3305, 68034}, {3434, 64003}, {3555, 7580}, {3586, 11827}, {3646, 5763}, {3679, 64291}, {3753, 19521}, {3811, 52026}, {3813, 38454}, {3827, 22778}, {3868, 64150}, {3869, 68001}, {3873, 10884}, {3874, 12520}, {3881, 12511}, {3889, 7411}, {3927, 9856}, {3929, 31162}, {3951, 67998}, {4294, 63438}, {4298, 6916}, {4300, 62819}, {4301, 12514}, {4314, 59345}, {4666, 6986}, {4863, 6253}, {5144, 62340}, {5223, 5777}, {5227, 64085}, {5231, 6831}, {5234, 6913}, {5250, 17558}, {5290, 6907}, {5437, 6684}, {5493, 64129}, {5534, 6985}, {5587, 6849}, {5603, 16845}, {5657, 17582}, {5659, 54447}, {5705, 7680}, {5722, 31799}, {5731, 62832}, {5732, 12675}, {5762, 7330}, {5805, 31419}, {5806, 9708}, {5815, 67874}, {5837, 64322}, {5840, 66059}, {5842, 49170}, {5850, 54227}, {5904, 63988}, {5930, 68029}, {6001, 54422}, {6245, 24477}, {6261, 11523}, {6361, 10860}, {6737, 12245}, {6745, 6927}, {6765, 11500}, {6835, 25006}, {6836, 26015}, {6848, 21075}, {6865, 11019}, {6887, 7308}, {6908, 21620}, {6915, 67097}, {6918, 8580}, {6926, 64124}, {6939, 18250}, {6987, 63999}, {6988, 7160}, {7162, 37701}, {7289, 51490}, {7397, 40940}, {7686, 9623}, {8583, 50203}, {9612, 15908}, {9785, 67120}, {9841, 31730}, {9845, 18481}, {9947, 18529}, {9961, 62235}, {10085, 58808}, {10165, 60985}, {10526, 18492}, {10580, 37423}, {10624, 62810}, {11037, 37108}, {11349, 39592}, {11415, 56545}, {11495, 58567}, {11496, 31424}, {12512, 43175}, {12513, 12650}, {12526, 12672}, {12573, 35514}, {12575, 62839}, {12608, 28609}, {12609, 60895}, {12667, 15239}, {12671, 15733}, {12687, 64075}, {12701, 30223}, {12717, 44421}, {12777, 12842}, {14217, 64372}, {15298, 55300}, {15299, 51785}, {15829, 31806}, {15954, 33811}, {17580, 59417}, {17728, 50031}, {17831, 44661}, {18163, 62843}, {18206, 37422}, {18444, 62861}, {18446, 41863}, {18540, 22793}, {19541, 34790}, {20008, 64321}, {20078, 67043}, {20223, 23528}, {21077, 63966}, {22791, 26921}, {22991, 49183}, {23072, 34033}, {24392, 48482}, {24467, 28174}, {25524, 58637}, {29054, 35635}, {30330, 51489}, {31146, 37428}, {31418, 67877}, {31789, 66682}, {31870, 64733}, {33137, 36670}, {33633, 37483}, {34498, 47848}, {35658, 37469}, {37411, 63981}, {37426, 64679}, {37861, 51957}, {37862, 51955}, {38036, 55108}, {38324, 53396}, {45036, 54192}, {48363, 63138}, {48883, 67849}, {50701, 63146}, {52027, 64074}, {56176, 66244}, {57287, 64079}, {59387, 63135}, {60957, 67065}, {63259, 64346}, {63967, 68000}, {64116, 66469}, {64117, 66215}

X(68036) = midpoint of X(i) and X(j) for these {i,j}: {40, 6766}, {6762, 68057}
X(68036) = reflection of X(i) in X(j) for these {i,j}: {40, 5709}, {84, 62858}, {1490, 64077}, {5534, 6985}, {5758, 946}, {6765, 11500}, {6769, 3}, {9589, 12700}, {11523, 6261}, {12650, 12513}, {63981, 37411}, {67886, 1158}
X(68036) = inverse of X(13528) in Bevan circle
X(68036) = pole of line {513, 13528} with respect to the Bevan circle
X(68036) = pole of line {21, 63430} with respect to the Stammler hyperbola
X(68036) = pole of line {672, 2270} with respect to the Gheorghe circle
X(68036) = intersection, other than A, B, C, of circumconics {{A, B, C, X(40), X(6601)}}, {{A, B, C, X(84), X(1617)}}, {{A, B, C, X(1295), X(6769)}}, {{A, B, C, X(3333), X(51512)}}, {{A, B, C, X(3680), X(14110)}}, {{A, B, C, X(7994), X(54226)}}, {{A, B, C, X(8726), X(51498)}}, {{A, B, C, X(37560), X(56287)}}
X(68036) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 517, 6769}, {20, 62874, 63430}, {40, 3333, 3}, {40, 5535, 5128}, {40, 6766, 517}, {56, 7957, 6282}, {63, 962, 12705}, {516, 62858, 84}, {517, 5709, 40}, {518, 64077, 1490}, {1125, 67962, 61122}, {1158, 28194, 67886}, {3218, 20070, 63985}, {3245, 5536, 5535}, {3928, 67886, 1158}, {5603, 55104, 31435}, {6361, 63399, 10860}, {6762, 68057, 515}, {6763, 9589, 1709}, {6918, 58643, 8580}, {7330, 12699, 11372}, {10085, 64005, 58808}, {10624, 62810, 66239}, {12651, 62824, 1012}, {19541, 34790, 67881}, {31424, 43166, 11496}, {62832, 63141, 5731}

X(68037) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR18-2 WRT CTR28-189

X(68037) lies on these lines: {1, 21735}, {2, 40}, {3, 61159}, {8, 28172}, {10, 3843}, {20, 20053}, {165, 10595}, {355, 15684}, {484, 64124}, {515, 1657}, {516, 3627}, {517, 548}, {519, 15686}, {551, 15706}, {553, 37563}, {944, 3633}, {950, 3245}, {952, 62141}, {1125, 12108}, {1155, 64703}, {1385, 45759}, {1482, 14093}, {1483, 62108}, {2093, 17706}, {3528, 11224}, {3579, 13464}, {3626, 28146}, {3634, 61907}, {3654, 38335}, {3656, 15718}, {3679, 62029}, {3817, 61911}, {3828, 14892}, {3850, 9956}, {4114, 5119}, {4292, 45081}, {4297, 15689}, {4301, 61811}, {4668, 5691}, {4669, 62050}, {4701, 28186}, {4726, 29054}, {4745, 62010}, {5072, 12699}, {5128, 63993}, {5183, 10624}, {5603, 61817}, {5818, 61983}, {5882, 9778}, {5886, 61840}, {5901, 41983}, {6705, 37584}, {7982, 62083}, {7987, 62058}, {7989, 61959}, {9589, 10175}, {9812, 31399}, {10164, 61832}, {10165, 61807}, {10172, 12812}, {11010, 11552}, {11376, 63215}, {11531, 51705}, {12245, 46333}, {12563, 51787}, {12571, 50821}, {14891, 31663}, {14893, 19925}, {15688, 61284}, {16192, 61780}, {18481, 62128}, {18525, 58207}, {19875, 61951}, {19878, 31447}, {22791, 61837}, {22793, 23046}, {28164, 62164}, {28202, 61510}, {28224, 58201}, {30305, 41348}, {31673, 50691}, {31797, 50243}, {32900, 44245}, {33179, 46332}, {34648, 62025}, {34718, 62167}, {35242, 61783}, {36279, 40270}, {37567, 63999}, {37568, 64110}, {37624, 62071}, {37727, 59420}, {38127, 41869}, {44675, 63206}, {45760, 58441}, {47745, 50810}, {49135, 61250}, {49137, 61247}, {49163, 49184}, {50693, 61291}, {50796, 62011}, {50802, 61922}, {50822, 51119}, {50865, 61973}, {50872, 64952}, {59372, 66050}, {61254, 62028}, {61274, 61814}, {63073, 64084}

X(68037) = midpoint of X(i) and X(j) for these {i,j}: {946, 20070}, {1657, 3625}, {5493, 12702}, {6361, 11362}, {7991, 31730}, {47745, 64005}
X(68037) = reflection of X(i) in X(j) for these {i,j}: {3627, 4691}, {3635, 548}, {6684, 40}, {13464, 3579}, {13607, 12512}, {18483, 43174}, {32900, 44245}, {68035, 6684}
X(68037) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 20070, 946}, {40, 28194, 6684}, {946, 20070, 28194}, {1657, 3625, 515}, {3579, 28228, 13464}, {6361, 11362, 28150}, {6361, 63468, 11362}, {6684, 28194, 68035}, {7991, 31730, 28234}, {28174, 43174, 18483}, {50810, 64005, 47745}

X(68038) = PERSPECTOR OF THESE TRIANGLES: EULER53 AND ANTI-EULER

X(68038) lies on these lines: {4, 16251}, {20, 33702}, {30, 41374}, {122, 376}, {631, 16253}, {1075, 15005}, {3529, 5656}, {5922, 15311}, {6525, 15682}, {6624, 23240}

X(68038) = reflection of X(i) in X(j) for these {i,j}: {4, 16251}, {33702, 20}

X(68039) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST ANTI-AURIGA WRT EULER53

X(68039) lies on these lines: {1, 6354}, {3, 26326}, {4, 26290}, {20, 5597}, {30, 45696}, {382, 26386}, {511, 48489}, {515, 48487}, {516, 45711}, {517, 48455}, {550, 26398}, {962, 26395}, {1151, 45365}, {1152, 45366}, {1503, 48513}, {1657, 45369}, {1885, 26371}, {2777, 48535}, {2794, 48474}, {2829, 48533}, {3146, 26394}, {3428, 26360}, {3529, 26381}, {3627, 45355}, {4297, 26365}, {4299, 45373}, {4302, 45371}, {5073, 18496}, {5691, 26382}, {5840, 48464}, {6284, 26380}, {6459, 26385}, {6460, 26384}, {6836, 26425}, {7354, 26351}, {12203, 26379}, {12943, 26388}, {12953, 26387}, {17702, 48472}, {23698, 48462}, {26296, 64005}, {26302, 39568}, {26310, 68049}, {26319, 26413}, {26334, 68051}, {26344, 68052}, {26383, 68050}, {26389, 68053}, {26390, 64725}, {26393, 64074}, {26396, 68045}, {26397, 68046}, {26399, 64075}, {26400, 64076}, {26401, 64079}, {26402, 64078}, {26406, 41338}, {28164, 48511}, {29181, 45724}, {42258, 44582}, {42259, 44583}, {42266, 45357}, {42267, 45360}, {45345, 68041}, {45348, 68042}, {45349, 68043}, {45352, 68044}, {45354, 68048}, {48537, 64509}, {63386, 64354}, {64379, 68054}

X(68039) = reflection of X(i) in X(j) for these {i,j}: {48454, 48460}, {48493, 48487}, {68040, 1}
X(68039) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 26290, 26359}, {30, 48460, 48454}, {515, 48487, 48493}, {48454, 48460, 45696}

X(68040) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND ANTI-AURIGA WRT EULER53

X(68040) lies on these lines: {1, 6354}, {3, 26327}, {4, 26291}, {20, 5598}, {30, 45697}, {382, 26410}, {511, 48490}, {515, 48488}, {516, 45712}, {517, 48454}, {550, 26422}, {962, 26419}, {1151, 45368}, {1152, 45367}, {1503, 48514}, {1657, 45370}, {1885, 26372}, {2777, 48536}, {2794, 48475}, {2829, 48534}, {3146, 26418}, {3428, 26359}, {3529, 26405}, {3627, 45356}, {4297, 26366}, {4299, 45374}, {4302, 45372}, {5073, 18498}, {5691, 26406}, {5840, 48465}, {6284, 26404}, {6459, 26409}, {6460, 26408}, {6836, 26401}, {7354, 26352}, {12203, 26403}, {12943, 26412}, {12953, 26411}, {17702, 48473}, {23698, 48463}, {26297, 64005}, {26303, 39568}, {26311, 68049}, {26320, 26389}, {26335, 68051}, {26345, 68052}, {26382, 41338}, {26407, 68050}, {26413, 68053}, {26414, 64725}, {26417, 64074}, {26420, 68045}, {26421, 68046}, {26423, 64075}, {26424, 64076}, {26425, 64079}, {26426, 64078}, {28164, 48512}, {29181, 45725}, {42258, 44584}, {42259, 44585}, {42266, 45359}, {42267, 45358}, {45346, 68042}, {45347, 68041}, {45350, 68044}, {45351, 68043}, {45353, 68047}, {48538, 64509}, {63386, 64355}, {64380, 68054}

X(68040) = reflection of X(i) in X(j) for these {i,j}: {48455, 48461}, {48494, 48488}, {68039, 1}
X(68040) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {30, 48461, 48455}, {515, 48488, 48494}, {48455, 48461, 45697}

X(68041) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST ANTI-KENMOTU CENTERS WRT EULER53

X(68041) lies on these lines: {3, 45440}, {4, 12305}, {6, 20}, {30, 591}, {382, 6289}, {489, 53097}, {490, 36990}, {492, 3146}, {511, 49325}, {515, 49323}, {516, 45713}, {550, 43119}, {625, 36655}, {962, 45476}, {1151, 35947}, {1152, 45487}, {1503, 49038}, {1657, 45488}, {1885, 45400}, {2777, 49369}, {2794, 49315}, {2829, 48703}, {3102, 42267}, {3529, 45406}, {3534, 45411}, {3627, 45438}, {4297, 45398}, {4299, 45492}, {4302, 45490}, {5059, 62987}, {5073, 45375}, {5691, 45444}, {5840, 48684}, {5875, 61096}, {6284, 45404}, {6290, 35002}, {7000, 7778}, {7354, 45470}, {7374, 26294}, {7761, 11825}, {7795, 10514}, {8982, 48905}, {9766, 48477}, {11293, 31884}, {11294, 53023}, {11477, 26441}, {12124, 36709}, {12203, 45402}, {12297, 42284}, {12943, 45458}, {12953, 45460}, {12963, 54996}, {13758, 42637}, {14227, 26288}, {15683, 45421}, {15684, 49361}, {17702, 49313}, {17800, 22809}, {19924, 44654}, {23698, 49309}, {28164, 49347}, {36656, 45498}, {39568, 45428}, {39679, 42261}, {42266, 45462}, {42271, 44392}, {42839, 48476}, {43133, 64080}, {43134, 55722}, {45345, 68039}, {45347, 68040}, {45416, 64074}, {45422, 64075}, {45424, 64076}, {45426, 64005}, {45430, 68047}, {45432, 68048}, {45434, 68049}, {45436, 64077}, {45446, 68050}, {45454, 64725}, {45456, 68053}, {45494, 64078}, {45496, 64079}, {49363, 62163}, {49371, 64509}, {63300, 63386}, {64387, 68054}

X(68041) = midpoint of X(i) and X(j) for these {i,j}: {20, 68052}, {3146, 68045}, {49038, 64638}
X(68041) = reflection of X(i) in X(j) for these {i,j}: {3, 68043}, {13748, 9733}, {49329, 49323}, {68042, 20}
X(68041) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {20, 29181, 68042}, {20, 51212, 42258}, {20, 6460, 44882}, {20, 68052, 29181}, {30, 9733, 13748}, {49038, 64638, 1503}

X(68042) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND ANTI-KENMOTU CENTERS WRT EULER53

X(68042) lies on these lines: {3, 45441}, {4, 12306}, {6, 20}, {30, 1991}, {382, 6290}, {489, 36990}, {490, 53097}, {491, 3146}, {511, 49326}, {515, 49324}, {516, 45714}, {550, 43118}, {625, 36656}, {962, 45477}, {1151, 45486}, {1152, 35946}, {1503, 49039}, {1657, 45489}, {1885, 45401}, {2777, 49370}, {2794, 49316}, {2829, 48704}, {3103, 42266}, {3529, 45407}, {3534, 45410}, {3627, 45439}, {4297, 45399}, {4299, 45493}, {4302, 45491}, {5059, 62986}, {5073, 45376}, {5691, 45445}, {5840, 48685}, {5874, 61097}, {6284, 45405}, {6289, 35002}, {7000, 26295}, {7354, 45471}, {7374, 7778}, {7761, 11824}, {7795, 10515}, {8982, 11477}, {9766, 48476}, {11293, 53023}, {11294, 31884}, {12123, 36714}, {12203, 45403}, {12296, 42283}, {12943, 45459}, {12953, 45461}, {12968, 54996}, {13638, 42638}, {14242, 26289}, {15683, 45420}, {15684, 49364}, {17702, 49314}, {17800, 22810}, {19924, 44655}, {21736, 45472}, {23698, 49310}, {26441, 48905}, {28164, 49348}, {36655, 45499}, {39568, 45429}, {39648, 42260}, {42267, 45463}, {42272, 44394}, {42841, 48477}, {43133, 55722}, {43134, 64080}, {45346, 68040}, {45348, 68039}, {45417, 64074}, {45423, 64075}, {45425, 64076}, {45427, 64005}, {45431, 68047}, {45433, 68048}, {45435, 68049}, {45437, 64077}, {45447, 68050}, {45455, 64725}, {45457, 68053}, {45495, 64078}, {45497, 64079}, {49362, 62163}, {49372, 64509}, {63301, 63386}, {64388, 68054}

X(68042) = midpoint of X(i) and X(j) for these {i,j}: {20, 68051}, {3146, 68046}, {49039, 64639}
X(68042) = reflection of X(i) in X(j) for these {i,j}: {3, 68044}, {13749, 9732}, {49330, 49324}, {68041, 20}
X(68042) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {20, 29181, 68041}, {20, 51212, 42259}, {20, 6459, 44882}, {20, 68051, 29181}, {30, 9732, 13749}, {9732, 13749, 1991}, {49039, 64639, 1503}

X(68043) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST ANTI-KENMOTU-FREE-VERTICES WRT EULER53

X(68043) lies on these lines: {3, 45440}, {4, 641}, {5, 7690}, {20, 372}, {30, 9739}, {39, 42259}, {182, 550}, {371, 35947}, {376, 45553}, {382, 45554}, {511, 48742}, {515, 48740}, {516, 45715}, {639, 12305}, {642, 18860}, {962, 45572}, {1151, 45574}, {1152, 45577}, {1160, 32419}, {1503, 48766}, {1657, 12601}, {1885, 45502}, {2777, 48786}, {2794, 48732}, {2829, 48705}, {3071, 6566}, {3146, 45508}, {3529, 45510}, {3534, 45410}, {3627, 45542}, {3830, 48778}, {4297, 45500}, {4299, 45582}, {4302, 45580}, {5062, 6781}, {5073, 45377}, {5691, 45546}, {5840, 48686}, {6284, 45506}, {6315, 8721}, {6459, 45515}, {6460, 45512}, {7354, 45570}, {9733, 49086}, {9737, 48467}, {10483, 65145}, {11165, 13749}, {12203, 45504}, {12297, 42269}, {12943, 45560}, {12953, 45562}, {15683, 33457}, {17538, 45550}, {17702, 48730}, {18993, 63548}, {19924, 44475}, {23698, 48726}, {28164, 48764}, {32421, 49038}, {35946, 51910}, {39568, 45532}, {42260, 51212}, {42267, 45565}, {45349, 68039}, {45351, 68040}, {45520, 64074}, {45525, 62147}, {45526, 64075}, {45528, 64076}, {45530, 64005}, {45534, 68047}, {45536, 68048}, {45538, 68049}, {45540, 64077}, {45548, 68050}, {45556, 64725}, {45558, 68053}, {45584, 64078}, {45586, 64079}, {48735, 49028}, {48788, 64509}, {61097, 64638}, {63302, 63386}, {64389, 68054}

X(68043) = midpoint of X(i) and X(j) for these {i,j}: {3, 68041}, {49038, 61096}, {61097, 64638}
X(68043) = reflection of X(i) in X(j) for these {i,j}: {48466, 9739}, {48746, 48740}, {68044, 550}
X(68043) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 45440, 64691}, {4, 45498, 641}, {30, 9739, 48466}, {550, 29181, 68044}, {9739, 48466, 41490}

X(68044) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND ANTI-KENMOTU-FREE-VERTICES WRT EULER53

X(68044) lies on these lines: {3, 45441}, {4, 642}, {5, 7692}, {20, 371}, {30, 9738}, {39, 42258}, {182, 550}, {372, 35946}, {376, 45552}, {382, 45555}, {511, 48743}, {515, 48741}, {516, 45716}, {639, 21736}, {640, 12306}, {641, 18860}, {962, 45573}, {1151, 45576}, {1152, 45575}, {1161, 32421}, {1503, 48767}, {1657, 12602}, {1885, 45503}, {2777, 48787}, {2794, 48733}, {2829, 48706}, {3070, 6567}, {3146, 45509}, {3529, 45511}, {3534, 45411}, {3627, 45543}, {3830, 48779}, {4297, 45501}, {4299, 45583}, {4302, 45581}, {5058, 6781}, {5073, 45378}, {5691, 45547}, {5840, 48687}, {6284, 45507}, {6311, 8721}, {6459, 45513}, {6460, 45514}, {7354, 45571}, {9732, 49087}, {9737, 48466}, {10483, 65146}, {11165, 13748}, {12203, 45505}, {12296, 42268}, {12943, 45561}, {12953, 45563}, {15683, 33456}, {17538, 45551}, {17702, 48731}, {18994, 63548}, {19924, 44476}, {21737, 64691}, {23698, 48727}, {28164, 48765}, {32419, 49039}, {35947, 51911}, {39568, 45533}, {42261, 51212}, {42266, 45564}, {45350, 68040}, {45352, 68039}, {45521, 64074}, {45524, 62147}, {45527, 64075}, {45529, 64076}, {45531, 64005}, {45535, 68047}, {45537, 68048}, {45539, 68049}, {45541, 64077}, {45549, 68050}, {45557, 64725}, {45559, 68053}, {45585, 64078}, {45587, 64079}, {48734, 49029}, {48789, 64509}, {61096, 64639}, {63303, 63386}, {64390, 68054}

X(68044) = midpoint of X(i) and X(j) for these {i,j}: {3, 68042}, {49039, 61097}, {61096, 64639}
X(68044) = reflection of X(i) in X(j) for these {i,j}: {48467, 9738}, {48747, 48741}, {68043, 550}
X(68044) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 45499, 642}, {30, 9738, 48467}, {515, 48741, 48747}, {550, 29181, 68043}, {9738, 48467, 41491}, {49039, 61097, 32419}

X(68045) = ORTHOLOGY CENTER OF THESE TRIANGLES: 3RD ANTI-TRI-SQUARES-CENTRAL WRT EULER53

X(68045) lies on these lines: {3, 26330}, {4, 641}, {20, 1151}, {30, 1160}, {193, 5059}, {382, 26468}, {492, 3146}, {511, 49056}, {515, 49054}, {516, 45719}, {550, 26516}, {962, 26514}, {1152, 49026}, {1503, 49080}, {1657, 49028}, {1885, 26375}, {2549, 5062}, {2777, 49098}, {2794, 49046}, {2829, 48711}, {3529, 10783}, {3627, 49016}, {4297, 26369}, {4299, 49032}, {4302, 49030}, {5073, 18539}, {5691, 26444}, {5840, 48692}, {6284, 26435}, {6459, 26462}, {7354, 26355}, {12203, 26429}, {12297, 13882}, {12943, 26479}, {12953, 26473}, {15683, 45420}, {17702, 49044}, {23698, 49040}, {26300, 64005}, {26306, 39568}, {26314, 68049}, {26324, 64077}, {26331, 39809}, {26396, 68039}, {26420, 68040}, {26449, 68050}, {26485, 68053}, {26490, 64725}, {26512, 64074}, {26517, 64075}, {26518, 64076}, {26519, 64079}, {26520, 64078}, {28164, 49078}, {42258, 44594}, {42259, 44595}, {42266, 49018}, {42413, 51212}, {45524, 62147}, {48477, 64638}, {49012, 68047}, {49014, 68048}, {49048, 62171}, {49100, 64509}, {63305, 63386}, {64391, 68054}

X(68045) = reflection of X(i) in X(j) for these {i,j}: {3146, 68041}, {48476, 49038}, {48477, 64638}, {49060, 49054}, {68046, 5059}, {68051, 3529}
X(68045) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 12124, 33364}, {4, 26294, 26361}, {30, 49038, 48476}, {5059, 29181, 68046}, {12297, 35947, 31412}, {48476, 49038, 5860}

X(68046) = ORTHOLOGY CENTER OF THESE TRIANGLES: 4TH ANTI-TRI-SQUARES-CENTRAL WRT EULER53

X(68046) lies on these lines: {3, 26331}, {4, 642}, {20, 1152}, {30, 1161}, {193, 5059}, {382, 26469}, {491, 3146}, {511, 49057}, {515, 49055}, {516, 45720}, {550, 26521}, {962, 26515}, {1151, 49027}, {1503, 49081}, {1657, 49029}, {1885, 26376}, {2549, 5058}, {2777, 49099}, {2794, 49047}, {2829, 48712}, {3529, 8982}, {3627, 49017}, {4297, 26370}, {4299, 49033}, {4302, 49031}, {5073, 26438}, {5691, 26445}, {5840, 48693}, {6284, 26436}, {6460, 26457}, {7354, 26356}, {12203, 26430}, {12296, 13934}, {12943, 26480}, {12953, 26474}, {15683, 45421}, {17702, 49045}, {23698, 49041}, {26301, 64005}, {26307, 39568}, {26315, 68049}, {26325, 64077}, {26330, 39809}, {26397, 68039}, {26421, 68040}, {26450, 68050}, {26486, 68053}, {26491, 64725}, {26513, 64074}, {26522, 64075}, {26523, 64076}, {26524, 64079}, {26525, 64078}, {28164, 49079}, {42258, 44596}, {42259, 44597}, {42267, 49019}, {42414, 51212}, {45525, 62147}, {48476, 64639}, {49013, 68047}, {49015, 68048}, {49049, 62171}, {49101, 64509}, {63306, 63386}, {64392, 68054}

X(68046) = reflection of X(i) in X(j) for these {i,j}: {3146, 68042}, {48476, 64639}, {48477, 49039}, {49061, 49055}, {68045, 5059}, {68052, 3529}
X(68046) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 12123, 33365}, {4, 26295, 26362}, {30, 49039, 48477}, {5059, 29181, 68045}, {12296, 35946, 42561}, {48477, 49039, 5861}

X(68047) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST AURIGA WRT EULER53

X(68047) lies on these lines: {3, 8196}, {4, 5599}, {5, 35244}, {20, 5597}, {30, 9834}, {40, 5600}, {55, 226}, {382, 8200}, {511, 39880}, {515, 12454}, {517, 9835}, {962, 5598}, {1151, 13890}, {1152, 13944}, {1503, 12468}, {1657, 11875}, {1770, 26393}, {1885, 11384}, {2777, 13208}, {2794, 12478}, {2829, 13228}, {3146, 5601}, {3529, 11843}, {3627, 18495}, {4297, 11366}, {4299, 11879}, {4301, 11367}, {4302, 11877}, {5073, 45379}, {5602, 20070}, {5690, 18497}, {5691, 8197}, {5840, 12462}, {6284, 18955}, {6361, 11823}, {6459, 19008}, {6460, 19007}, {7354, 11873}, {7991, 8204}, {8186, 64005}, {8187, 9589}, {8190, 39568}, {8198, 68051}, {8199, 68052}, {8203, 12699}, {8207, 12702}, {10483, 65123}, {11208, 12459}, {11253, 28174}, {11492, 64074}, {11493, 64077}, {11837, 12203}, {11861, 68049}, {11863, 68050}, {11865, 64725}, {11867, 68053}, {11869, 12943}, {11871, 12953}, {11872, 37567}, {11881, 64078}, {11883, 64079}, {12179, 23698}, {12365, 12415}, {12452, 29181}, {13229, 64509}, {28164, 49555}, {28212, 32147}, {28228, 49556}, {35778, 42266}, {35781, 42267}, {42258, 44600}, {42259, 44601}, {43577, 43850}, {45353, 68040}, {45430, 68041}, {45431, 68042}, {45534, 68043}, {45535, 68044}, {45625, 64075}, {45627, 64076}, {49012, 68045}, {49013, 68046}, {63312, 63386}, {64396, 68054}

X(68047) = reflection of X(i) in X(j) for these {i,j}: {9834, 11252}, {12454, 12458}, {12455, 9835}, {68048, 55}
X(68047) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {30, 11252, 9834}, {55, 516, 68048}, {517, 9835, 12455}

X(68048) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND AURIGA WRT EULER53

X(68048) lies on these lines: {3, 8203}, {4, 5600}, {5, 35245}, {20, 5598}, {30, 9835}, {40, 5599}, {55, 226}, {382, 8207}, {511, 39881}, {515, 12455}, {517, 9834}, {962, 5597}, {1151, 13891}, {1152, 13945}, {1503, 12469}, {1657, 11876}, {1770, 26417}, {1885, 11385}, {2777, 13209}, {2794, 12479}, {2829, 13230}, {3146, 5602}, {3529, 11844}, {3627, 18497}, {4297, 11367}, {4299, 11880}, {4301, 11366}, {4302, 11878}, {5073, 45380}, {5601, 20070}, {5690, 18495}, {5691, 8204}, {5840, 12463}, {6284, 18956}, {6361, 11822}, {6459, 19010}, {6460, 19009}, {7354, 11874}, {7991, 8197}, {8186, 9589}, {8187, 64005}, {8191, 39568}, {8196, 12699}, {8200, 12702}, {8205, 68051}, {8206, 68052}, {10483, 65124}, {11207, 12458}, {11252, 28174}, {11492, 64077}, {11493, 64074}, {11838, 12203}, {11862, 68049}, {11864, 68050}, {11866, 64725}, {11868, 68053}, {11870, 12943}, {11871, 37567}, {11872, 12953}, {11882, 64078}, {11884, 64079}, {12180, 23698}, {12366, 12416}, {12453, 29181}, {13231, 64509}, {28164, 49556}, {28212, 32146}, {28228, 49555}, {35779, 42267}, {35780, 42266}, {42258, 44602}, {42259, 44603}, {43577, 43851}, {45354, 68039}, {45432, 68041}, {45433, 68042}, {45536, 68043}, {45537, 68044}, {45626, 64075}, {45628, 64076}, {49014, 68045}, {49015, 68046}, {63313, 63386}, {64397, 68054}

X(68048) = reflection of X(i) in X(j) for these {i,j}: {9835, 11253}, {12454, 9834}, {12455, 12459}, {68047, 55}
X(68048) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {30, 11253, 9835}, {55, 516, 68047}, {515, 12459, 12455}, {9835, 11253, 11208}

X(68049) = ORTHOLOGY CENTER OF THESE TRIANGLES: 5TH BROCARD WRT EULER53

X(68049) lies on circumconic {{A, B, C, X(3098), X(7767)}} and on these lines: {2, 34616}, {3, 7846}, {4, 3096}, {5, 35248}, {20, 32}, {30, 76}, {39, 60651}, {83, 31670}, {147, 7916}, {194, 34624}, {316, 35456}, {376, 7803}, {382, 9996}, {511, 7877}, {515, 12495}, {516, 9941}, {546, 42787}, {548, 66096}, {550, 26316}, {576, 12252}, {962, 9997}, {1151, 13892}, {1152, 13946}, {1350, 7879}, {1503, 12502}, {1513, 7940}, {1657, 9301}, {1885, 11386}, {2076, 44518}, {2777, 13210}, {2794, 8782}, {2829, 13235}, {2896, 3146}, {3091, 7914}, {3094, 7745}, {3099, 64005}, {3522, 10583}, {3529, 9862}, {3534, 7827}, {3543, 7865}, {3627, 18500}, {3972, 44251}, {4297, 11368}, {4299, 10047}, {4302, 10038}, {5073, 18503}, {5188, 7904}, {5691, 9857}, {5840, 12499}, {5999, 7746}, {6284, 18957}, {6392, 15683}, {6459, 19012}, {6460, 19011}, {6656, 48881}, {6683, 60654}, {7354, 10877}, {7470, 7790}, {7754, 48905}, {7760, 46264}, {7768, 33878}, {7770, 48910}, {7796, 43460}, {7812, 19924}, {7836, 30270}, {7878, 21850}, {7894, 48906}, {7915, 13862}, {7932, 67854}, {9923, 9984}, {9994, 68051}, {9995, 68052}, {10348, 12110}, {10483, 65127}, {10723, 43449}, {10828, 39568}, {10871, 64725}, {10872, 68053}, {10873, 12943}, {10874, 12953}, {10878, 64078}, {10879, 64079}, {11054, 15685}, {11494, 64074}, {11861, 68047}, {11862, 68048}, {11885, 68050}, {12122, 37242}, {12251, 29012}, {13236, 64509}, {22744, 64077}, {26164, 67339}, {26310, 68039}, {26311, 68040}, {26314, 68045}, {26315, 68046}, {26317, 64075}, {26318, 64076}, {28164, 49561}, {31401, 37182}, {32027, 54173}, {32829, 60658}, {35782, 42266}, {35783, 42267}, {40278, 47618}, {40814, 52397}, {42258, 44604}, {42259, 44605}, {43453, 48879}, {43577, 43854}, {45434, 68041}, {45435, 68042}, {45538, 68043}, {45539, 68044}, {63315, 63386}, {64398, 68054}

X(68049) = reflection of X(i) in X(j) for these {i,j}: {9873, 9821}, {12495, 12497}
X(68049) = pole of line {3098, 7767} with respect to the Wallace hyperbola
X(68049) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 9993, 7846}, {4, 10357, 10356}, {30, 9821, 9873}, {3098, 10356, 10357}, {9821, 9873, 7811}, {10356, 10357, 3096}

X(68050) = ORTHOLOGY CENTER OF THESE TRIANGLES: GOSSARD WRT EULER53

X(68050) lies on these lines: {2, 3}, {40, 16210}, {511, 39886}, {515, 12626}, {516, 12438}, {944, 16211}, {962, 11910}, {1151, 13894}, {1152, 13948}, {1503, 12791}, {2777, 13212}, {2794, 12796}, {2829, 13268}, {4297, 11831}, {4299, 11913}, {4302, 11912}, {5691, 11900}, {5731, 51712}, {5840, 12752}, {6284, 18958}, {6459, 19018}, {6460, 19017}, {7354, 11909}, {9033, 13202}, {10483, 65121}, {10721, 22337}, {11839, 12203}, {11848, 64074}, {11852, 64005}, {11863, 68047}, {11864, 68048}, {11885, 68049}, {11901, 68051}, {11902, 68052}, {11903, 64725}, {11904, 68053}, {11905, 12943}, {11906, 12953}, {11914, 64078}, {11915, 64079}, {12181, 23698}, {12369, 12418}, {12583, 29181}, {13281, 64509}, {22755, 64077}, {25406, 51741}, {26383, 68039}, {26407, 68040}, {26449, 68045}, {26450, 68046}, {26452, 64075}, {26453, 64076}, {28164, 49585}, {34601, 44985}, {35790, 42266}, {35791, 42267}, {42258, 44610}, {42259, 44611}, {43577, 43849}, {45446, 68041}, {45447, 68042}, {45548, 68043}, {45549, 68044}, {52945, 66360}, {55141, 62350}, {63320, 63386}, {64402, 68054}, {64510, 66797}

X(68050) = reflection of X(i) in X(j) for these {i,j}: {20, 402}, {1650, 4}, {12626, 12696}
X(68050) = pole of line {523, 10152} with respect to the polar circle
X(68050) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {4, 10721, 14989}, {10733, 34549, 44967}, {10745, 38790, 66772}
X(68050) = intersection, other than A, B, C, of circumconics {{A, B, C, X(20), X(9033)}}, {{A, B, C, X(265), X(35241)}}, {{A, B, C, X(4240), X(38956)}}, {{A, B, C, X(12113), X(47111)}}, {{A, B, C, X(18508), X(34334)}}, {{A, B, C, X(27089), X(57290)}}
X(68050) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 30, 1650}, {30, 402, 20}, {515, 12696, 12626}

X(68051) = ORTHOLOGY CENTER OF THESE TRIANGLES: INNER-GREBE WRT EULER53

X(68051) lies on these lines: {3, 6202}, {4, 640}, {5, 35246}, {6, 20}, {30, 1161}, {376, 45552}, {382, 6215}, {489, 1271}, {490, 61044}, {511, 39887}, {515, 6258}, {516, 3641}, {550, 26341}, {962, 5605}, {1151, 8974}, {1152, 13949}, {1503, 6267}, {1587, 8396}, {1657, 11916}, {1885, 11388}, {2777, 7732}, {2794, 6319}, {2829, 13269}, {3529, 10783}, {3627, 18509}, {4297, 11370}, {4299, 10048}, {4302, 10040}, {5073, 26336}, {5589, 64005}, {5590, 36709}, {5595, 39568}, {5689, 5691}, {5840, 12753}, {5860, 61097}, {5870, 40268}, {6227, 23698}, {6279, 49138}, {6281, 12509}, {6284, 18959}, {7000, 12306}, {7354, 10927}, {7725, 9929}, {8198, 68047}, {8205, 68048}, {9994, 68049}, {10483, 65125}, {10792, 12203}, {10919, 64725}, {10921, 68053}, {10923, 12943}, {10925, 12953}, {10929, 64078}, {10931, 64079}, {11293, 51538}, {11497, 64074}, {11901, 68050}, {13282, 64509}, {13690, 62169}, {13810, 62049}, {14227, 32419}, {14927, 43134}, {17538, 45550}, {19924, 44471}, {21736, 33364}, {22756, 64077}, {26334, 68039}, {26335, 68040}, {26342, 64075}, {26343, 64076}, {26362, 36655}, {28164, 49586}, {29317, 42858}, {35792, 42266}, {35795, 42267}, {35946, 42637}, {36701, 45545}, {42561, 53480}, {43577, 43852}, {52667, 53512}, {63321, 63386}, {64403, 68054}

X(68051) = reflection of X(i) in X(j) for these {i,j}: {20, 68042}, {5871, 1161}, {12627, 12697}, {68045, 3529}, {68052, 20}
X(68051) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 10517, 10514}, {20, 29181, 68052}, {20, 51212, 6460}, {30, 1161, 5871}, {1161, 5871, 5861}, {7000, 12306, 33365}, {10514, 11824, 10517}, {29181, 68042, 20}

X(68052) = ORTHOLOGY CENTER OF THESE TRIANGLES: OUTER-GREBE WRT EULER53

X(68052) lies on these lines: {3, 6201}, {4, 639}, {5, 35247}, {6, 20}, {30, 1160}, {376, 45553}, {382, 6214}, {489, 61044}, {490, 1270}, {511, 39888}, {515, 6257}, {516, 3640}, {550, 26348}, {962, 5604}, {1151, 8975}, {1152, 13950}, {1503, 6266}, {1588, 8416}, {1657, 11917}, {1885, 11389}, {2777, 7733}, {2794, 6320}, {2829, 13270}, {3529, 8982}, {3627, 18511}, {4297, 11371}, {4299, 10049}, {4302, 10041}, {5073, 26346}, {5588, 64005}, {5591, 36714}, {5594, 39568}, {5688, 5691}, {5840, 12754}, {5861, 61096}, {5871, 40268}, {6226, 23698}, {6278, 12510}, {6280, 49138}, {6284, 18960}, {7354, 10928}, {7374, 12305}, {7726, 9930}, {8199, 68047}, {8206, 68048}, {9995, 68049}, {10483, 65126}, {10793, 12203}, {10920, 64725}, {10922, 68053}, {10924, 12943}, {10926, 12953}, {10930, 64078}, {10932, 64079}, {11294, 51538}, {11498, 64074}, {11902, 68050}, {13283, 64509}, {13691, 62049}, {13811, 62169}, {14242, 32421}, {14927, 43133}, {17538, 45551}, {19924, 44472}, {21736, 26294}, {22757, 64077}, {26344, 68039}, {26345, 68040}, {26349, 64075}, {26350, 64076}, {26361, 36656}, {28164, 49587}, {29317, 42859}, {31412, 53479}, {34112, 64500}, {35793, 42267}, {35794, 42266}, {35947, 42638}, {36703, 45544}, {43577, 43853}, {52666, 53515}, {63322, 63386}, {64404, 68054}

X(68052) = reflection of X(i) in X(j) for these {i,j}: {20, 68041}, {5870, 1160}, {12628, 12698}, {68046, 3529}, {68051, 20}
X(68052) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 10518, 10515}, {20, 29181, 68051}, {20, 51212, 6459}, {30, 1160, 5870}, {1160, 5870, 5860}, {10515, 10518, 5590}, {10515, 11825, 10518}, {29181, 68041, 20}

X(68053) = ORTHOLOGY CENTER OF THESE TRIANGLES: OUTER-JOHNSON WRT EULER53

X(68053) lies on these lines: {3, 3822}, {4, 958}, {5, 35250}, {8, 36999}, {11, 64079}, {12, 20}, {30, 4421}, {56, 6840}, {72, 5691}, {165, 50239}, {355, 382}, {376, 10599}, {411, 10895}, {511, 39890}, {515, 5812}, {518, 64261}, {550, 26487}, {962, 10950}, {1001, 26332}, {1151, 13896}, {1152, 13953}, {1259, 5080}, {1329, 50701}, {1376, 37468}, {1503, 12930}, {1657, 11929}, {1837, 64003}, {1885, 11391}, {2475, 5584}, {2646, 40271}, {2777, 13214}, {2794, 12935}, {2829, 6851}, {3091, 24953}, {3146, 3436}, {3434, 52837}, {3522, 10585}, {3529, 10786}, {3543, 34606}, {3585, 7580}, {3614, 6962}, {3627, 18517}, {3754, 52682}, {3913, 5842}, {4127, 18525}, {4190, 50031}, {4297, 9655}, {4299, 10523}, {4301, 9668}, {4302, 10954}, {4333, 17613}, {4428, 63257}, {4999, 6844}, {5073, 18518}, {5130, 12173}, {5204, 6943}, {5302, 5587}, {5450, 34620}, {5731, 9657}, {5758, 44669}, {5762, 49168}, {5790, 16139}, {5791, 19925}, {5794, 64004}, {5840, 12762}, {5841, 12114}, {6284, 18962}, {6459, 19026}, {6460, 19025}, {6690, 59345}, {6825, 65949}, {6827, 25524}, {6833, 30264}, {6836, 7354}, {6850, 11495}, {6868, 7680}, {6869, 18242}, {6876, 59392}, {6890, 15326}, {6892, 63754}, {6925, 65631}, {6928, 22753}, {6987, 25466}, {7491, 11496}, {7548, 31245}, {9579, 9943}, {9589, 37711}, {9612, 65404}, {9780, 59356}, {9812, 64754}, {10431, 10522}, {10483, 37022}, {10724, 62616}, {10728, 13199}, {10795, 12203}, {10827, 64005}, {10830, 39568}, {10872, 68049}, {10921, 68051}, {10922, 68052}, {10955, 64078}, {11194, 63980}, {11235, 22770}, {11867, 68047}, {11868, 68048}, {11904, 68050}, {12183, 23698}, {12372, 12423}, {12433, 60895}, {12513, 48482}, {12587, 29181}, {12678, 64707}, {12738, 38756}, {13295, 64509}, {15682, 34746}, {17532, 59320}, {18480, 26921}, {21077, 28164}, {21677, 52841}, {26066, 63438}, {26389, 68039}, {26413, 68040}, {26485, 68045}, {26486, 68046}, {28160, 37700}, {28194, 34700}, {28534, 54156}, {28628, 67877}, {35798, 42266}, {35799, 42267}, {38945, 66249}, {40272, 67047}, {41229, 68057}, {42258, 44620}, {42259, 44621}, {43577, 43860}, {45456, 68041}, {45457, 68042}, {45558, 68043}, {45559, 68044}, {52851, 63138}, {56998, 59326}, {63325, 63386}, {64407, 68054}

X(68053) = reflection of X(i) in X(j) for these {i,j}: {6869, 18242}, {11500, 10526}, {12513, 48482}, {12635, 5812}, {64075, 5}, {64077, 4}, {64725, 382}
X(68053) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 11827, 958}, {30, 10526, 11500}, {382, 516, 64725}, {3146, 3436, 6253}, {6836, 7354, 63991}, {10526, 11500, 11236}

X(68054) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND ANTI-PAVLOV WRT EULER53

X(68054) lies on circumconic {{A, B, C, X(15909), X(51502)}} and on these lines: {3, 5333}, {4, 5235}, {20, 81}, {21, 516}, {30, 4921}, {58, 64005}, {86, 3522}, {165, 14005}, {333, 3146}, {376, 42025}, {382, 64405}, {411, 54972}, {515, 66212}, {550, 64393}, {962, 7415}, {1010, 9778}, {1043, 20070}, {1151, 64417}, {1152, 64418}, {1503, 68016}, {1657, 64419}, {1699, 17557}, {1885, 64378}, {2287, 50695}, {2829, 66005}, {3091, 64425}, {3193, 64075}, {3529, 64384}, {3543, 64424}, {3627, 64399}, {4184, 64074}, {4225, 64077}, {4297, 64377}, {4299, 64421}, {4302, 64420}, {4653, 9589}, {4720, 7991}, {5059, 16704}, {5073, 64383}, {5691, 64401}, {5840, 66004}, {6284, 64382}, {6459, 64386}, {6460, 64385}, {6869, 37783}, {6904, 24557}, {7354, 64414}, {8025, 50693}, {9441, 27660}, {9812, 11110}, {10164, 17551}, {12203, 64381}, {12512, 25526}, {12943, 64408}, {12953, 64409}, {14007, 64108}, {15683, 41629}, {15717, 25507}, {15852, 25060}, {16948, 37422}, {17185, 63141}, {17553, 50865}, {18206, 63984}, {24556, 37267}, {26637, 37256}, {26638, 37435}, {26860, 62124}, {27644, 50702}, {28164, 64072}, {28620, 58221}, {29181, 41610}, {31730, 37402}, {37537, 61409}, {39568, 64395}, {42028, 62120}, {42258, 64410}, {42259, 64411}, {42266, 64412}, {42267, 64413}, {64076, 64394}, {64078, 64422}, {64079, 64423}, {64379, 68039}, {64380, 68040}, {64387, 68041}, {64388, 68042}, {64389, 68043}, {64390, 68044}, {64391, 68045}, {64392, 68046}, {64396, 68047}, {64397, 68048}, {64398, 68049}, {64402, 68050}, {64403, 68051}, {64404, 68052}, {64406, 64725}, {64407, 68053}

X(68054) = reflection of X(i) in X(j) for these {i,j}: {66212, 68031}, {67852, 64720}
X(68054) = pole of line {15931, 37057} with respect to the Stammler hyperbola
X(68054) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 64400, 5333}, {30, 64720, 67852}, {962, 7415, 64415}

X(68055) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1ST ANTI-PARRY WRT EULER53

X(68055) lies on these lines: {2, 13479}, {3, 68056}, {4, 64880}, {20, 2847}, {83, 12039}, {98, 36207}, {99, 523}, {107, 110}, {147, 64923}, {512, 48960}, {514, 45709}, {522, 47747}, {524, 50641}, {525, 48971}, {599, 38361}, {670, 35136}, {671, 34518}, {895, 41254}, {1296, 20187}, {1350, 34808}, {1351, 43976}, {1632, 5467}, {2407, 35278}, {2799, 48947}, {2804, 48690}, {2854, 38664}, {3800, 45722}, {4563, 53367}, {5486, 7790}, {8547, 12203}, {8681, 38294}, {9003, 15342}, {9131, 9216}, {9146, 18012}, {11185, 63646}, {11443, 36794}, {11636, 59098}, {13398, 53862}, {17983, 54395}, {18860, 40888}, {23878, 48972}, {28161, 48970}, {34473, 40879}, {34574, 65610}, {36898, 63719}, {47618, 64927}, {48709, 55126}, {48948, 55129}, {48951, 55122}, {48953, 48958}, {48975, 64877}, {53490, 59561}, {55141, 66774}, {55226, 57216}, {64090, 64924}, {65324, 65353}

X(68055) = reflection of X(i) in X(j) for these {i,j}: {98, 36207}, {38664, 48540}, {47747, 48959}, {48539, 9145}, {68056, 3}
X(68055) = trilinear pole of line {16051, 24855}
X(68055) = perspector of circumconic {{A, B, C, X(23582), X(52940)}}
X(68055) = X(i)-isoconjugate-of-X(j) for these {i, j}: {656, 63181}, {798, 63179}, {810, 10603}
X(68055) = X(i)-Dao conjugate of X(j) for these {i, j}: {16051, 1499}, {31998, 63179}, {39062, 10603}, {40596, 63181}, {62702, 7652}
X(68055) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {162, 66869}, {37216, 13219}, {65353, 21294}
X(68055) = pole of line {1624, 11634} with respect to the circumcircle
X(68055) = pole of line {9529, 34186} with respect to the DeLongchamps circle
X(68055) = pole of line {125, 53992} with respect to the polar circle
X(68055) = pole of line {20, 524} with respect to the Kiepert parabola
X(68055) = pole of line {351, 520} with respect to the Stammler hyperbola
X(68055) = pole of line {648, 5468} with respect to the Steiner circumellipse
X(68055) = pole of line {11053, 23583} with respect to the Steiner inellipse
X(68055) = pole of line {690, 3265} with respect to the Wallace hyperbola
X(68055) = pole of line {17907, 37803} with respect to the dual conic of Jerabek hyperbola
X(68055) = pole of line {5489, 33919} with respect to the dual conic of Wallace hyperbola
X(68055) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {20, 146, 36173}, {7728, 23240, 38797}
X(68055) = intersection, other than A, B, C, of circumconics {{A, B, C, X(99), X(43448)}}, {{A, B, C, X(107), X(892)}}, {{A, B, C, X(691), X(32713)}}, {{A, B, C, X(4240), X(16051)}}, {{A, B, C, X(9182), X(24855)}}, {{A, B, C, X(20187), X(35179)}}
X(68055) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 53351, 648}, {522, 48959, 47747}, {523, 9145, 48539}, {648, 61181, 107}, {2854, 48540, 38664}, {9145, 48539, 99}, {53350, 53351, 110}, {53350, 61182, 61181}

X(68056) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2ND ANTI-PARRY WRT EULER53

X(68056) lies on these lines: {3, 68055}, {4, 2847}, {6, 1632}, {20, 64880}, {74, 1294}, {98, 523}, {111, 2374}, {385, 64921}, {512, 48991}, {514, 45710}, {522, 48990}, {525, 49003}, {648, 5191}, {925, 52124}, {1316, 13479}, {2452, 35278}, {2799, 48980}, {2804, 48691}, {2854, 23235}, {3800, 45723}, {9215, 9979}, {9301, 64927}, {9862, 64923}, {11177, 64924}, {20975, 41254}, {23878, 49004}, {28161, 49002}, {33878, 64882}, {34473, 36207}, {43291, 62237}, {47283, 66459}, {47323, 60119}, {47325, 60317}, {48710, 55126}, {48981, 55129}, {48982, 55122}, {48984, 48989}, {49007, 64877}, {53350, 54439}, {55141, 66775}, {61102, 64781}

X(68056) = reflection of X(i) in X(j) for these {i,j}: {23235, 48539}, {48540, 9142}, {48993, 48990}, {68055, 3}
X(68056) = pole of line {146, 14698} with respect to the DeLongchamps circle
X(68056) = pole of line {126, 133} with respect to the polar circle
X(68056) = pole of line {42743, 56437} with respect to the Stammler hyperbola
X(68056) = pole of line {14919, 41909} with respect to the Steiner circumellipse
X(68056) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {20, 34186, 36174}, {10745, 20127, 22338}, {14360, 34549, 64102}
X(68056) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(1294), X(53866)}}, {{A, B, C, X(9307), X(14223)}}
X(68056) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {522, 48990, 48993}, {523, 9142, 48540}, {2854, 48539, 23235}, {9142, 48540, 98}, {20975, 47285, 41254}

X(68057) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR12-1.2 WRT EULER53

X(68057) lies on these lines: {1, 1427}, {2, 37551}, {3, 5436}, {4, 9}, {5, 3587}, {8, 50696}, {20, 57}, {30, 84}, {34, 7070}, {46, 2955}, {55, 12651}, {63, 3146}, {65, 10382}, {72, 1750}, {78, 36002}, {90, 59324}, {142, 37108}, {165, 405}, {171, 35658}, {200, 7957}, {208, 44695}, {219, 15811}, {223, 66249}, {226, 962}, {329, 20070}, {354, 64679}, {376, 37526}, {382, 7330}, {411, 3601}, {442, 1699}, {452, 9778}, {484, 1728}, {495, 7160}, {515, 6762}, {517, 1490}, {518, 63981}, {519, 6766}, {527, 6223}, {550, 37534}, {738, 5088}, {774, 4907}, {936, 19541}, {942, 5732}, {944, 68032}, {946, 6908}, {954, 53053}, {971, 54422}, {986, 1721}, {1005, 19860}, {1006, 35242}, {1046, 64741}, {1158, 28150}, {1254, 4319}, {1394, 1936}, {1419, 3562}, {1446, 18655}, {1449, 5706}, {1453, 1754}, {1482, 7966}, {1498, 2323}, {1593, 5285}, {1657, 7171}, {1698, 8226}, {1708, 5128}, {1709, 54290}, {1722, 9441}, {1768, 12690}, {1834, 2257}, {1864, 37567}, {1906, 21015}, {1948, 52578}, {2093, 44547}, {2829, 66068}, {2951, 3339}, {2999, 37537}, {3057, 64152}, {3091, 7308}, {3149, 5438}, {3158, 6769}, {3218, 5059}, {3219, 17578}, {3220, 39568}, {3247, 37528}, {3295, 43166}, {3305, 3832}, {3306, 3522}, {3332, 5717}, {3333, 3488}, {3338, 5441}, {3340, 10393}, {3359, 31789}, {3361, 63991}, {3419, 5691}, {3487, 4301}, {3529, 58808}, {3534, 37612}, {3543, 3929}, {3576, 3651}, {3579, 6913}, {3627, 18540}, {3646, 3817}, {3673, 10444}, {3680, 56273}, {3755, 52223}, {3781, 44870}, {3812, 11495}, {3854, 35595}, {3927, 64197}, {4208, 59385}, {4293, 7091}, {4302, 59335}, {4316, 17437}, {4324, 17700}, {4330, 17699}, {4512, 37224}, {4654, 55109}, {5119, 9589}, {5129, 59418}, {5177, 5250}, {5198, 26935}, {5219, 6838}, {5221, 5918}, {5227, 36990}, {5255, 12652}, {5290, 63974}, {5314, 63664}, {5435, 67041}, {5439, 10857}, {5536, 10085}, {5537, 11517}, {5541, 13257}, {5584, 13615}, {5658, 28228}, {5687, 7994}, {5705, 8727}, {5708, 31805}, {5715, 6907}, {5720, 37585}, {5734, 51779}, {5735, 57282}, {5745, 37434}, {5758, 6260}, {5762, 6259}, {5763, 67889}, {5768, 60968}, {5776, 68059}, {5777, 12702}, {5786, 21384}, {5804, 60985}, {5805, 37424}, {5811, 63132}, {5812, 10942}, {5815, 61003}, {5837, 45039}, {5840, 66058}, {5851, 28646}, {5927, 63468}, {6173, 37427}, {6261, 64316}, {6264, 54441}, {6284, 37550}, {6684, 6846}, {6734, 10431}, {6832, 31423}, {6836, 9581}, {6843, 18483}, {6848, 30827}, {6865, 7682}, {6889, 8227}, {6890, 31231}, {6916, 64001}, {6925, 9579}, {6926, 31190}, {6953, 20196}, {6985, 37531}, {6987, 31730}, {6990, 54447}, {7085, 11403}, {7289, 29181}, {7290, 37570}, {7293, 33524}, {7354, 54408}, {7383, 56468}, {7400, 56452}, {7411, 54392}, {7686, 30503}, {7951, 59341}, {7982, 18446}, {7992, 15726}, {8158, 12629}, {8273, 10582}, {8557, 66104}, {8580, 58637}, {8583, 37240}, {8726, 37426}, {8728, 38150}, {9312, 62385}, {9799, 24391}, {9842, 18228}, {9844, 10398}, {10164, 16845}, {10268, 11496}, {10483, 65129}, {10724, 64372}, {10864, 28164}, {10884, 11518}, {10980, 58567}, {11001, 26877}, {11108, 21153}, {11362, 51781}, {11381, 26893}, {11529, 12520}, {12053, 54366}, {12437, 54051}, {12511, 54318}, {12526, 12688}, {12649, 64707}, {12650, 22770}, {12664, 54156}, {12680, 62823}, {12701, 57285}, {12704, 37002}, {12953, 30223}, {14022, 50031}, {14054, 15071}, {14110, 15829}, {15803, 37022}, {15972, 48919}, {16117, 37615}, {16863, 33575}, {17284, 19542}, {17532, 50865}, {18529, 58631}, {19753, 37078}, {19861, 35990}, {20195, 37407}, {21165, 21669}, {22792, 52684}, {23698, 24469}, {23958, 62152}, {24474, 41854}, {26001, 37185}, {27003, 50693}, {27065, 50689}, {28146, 59318}, {28610, 67994}, {33576, 38271}, {34746, 51102}, {35445, 54430}, {36279, 54159}, {36991, 54398}, {36999, 42012}, {37112, 41867}, {37244, 64112}, {37249, 59326}, {37284, 59320}, {37420, 40212}, {37423, 63413}, {37556, 63274}, {37581, 67885}, {37623, 52027}, {37625, 50528}, {38036, 51706}, {38316, 64669}, {40661, 67998}, {41229, 68053}, {43161, 63999}, {43173, 59677}, {43577, 43856}, {45036, 50371}, {49135, 67334}, {50692, 67335}, {50695, 57287}, {50700, 57284}, {52404, 56445}, {52423, 66608}, {52819, 64696}, {53056, 64128}, {56317, 57276}, {58798, 63138}, {59333, 64076}, {60982, 63971}, {62218, 63976}, {66107, 68001}, {66465, 68002}

X(68057) = reflection of X(i) in X(j) for these {i,j}: {1, 64077}, {84, 5709}, {1490, 37411}, {5758, 6260}, {6762, 68036}, {6769, 11500}, {9799, 24391}, {10864, 62858}, {11523, 1490}, {12629, 8158}, {12650, 22770}, {37531, 6985}
X(68057) = pole of line {514, 66520} with respect to the Bevan circle
X(68057) = pole of line {3218, 9536} with respect to the Gheorghe circle
X(68057) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(19), X(22334)}}, {{A, B, C, X(57), X(11471)}}, {{A, B, C, X(281), X(5665)}}, {{A, B, C, X(1427), X(1869)}}, {{A, B, C, X(6598), X(55116)}}, {{A, B, C, X(18249), X(56139)}}
X(68057) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 63141, 37551}, {3, 67880, 5437}, {4, 5759, 12572}, {5, 3587, 61122}, {5, 61122, 51780}, {8, 50696, 63998}, {20, 57, 9841}, {20, 938, 64706}, {30, 5709, 84}, {40, 11372, 12514}, {40, 41869, 12705}, {46, 3586, 10396}, {46, 64005, 10860}, {72, 1750, 68000}, {84, 5709, 3928}, {382, 37584, 7330}, {515, 68036, 6762}, {517, 1490, 11523}, {517, 37411, 1490}, {946, 6908, 25525}, {962, 37421, 226}, {1657, 37532, 7171}, {1750, 7991, 72}, {2270, 8804, 9}, {2951, 3339, 9943}, {3218, 5059, 63984}, {3529, 63399, 58808}, {3579, 31822, 6913}, {3627, 26921, 18540}, {5493, 12572, 5759}, {5493, 54286, 40}, {5691, 41338, 57279}, {5758, 6260, 28609}, {6260, 28194, 5758}, {6284, 37550, 66239}, {6907, 12699, 5715}, {6925, 64003, 9579}, {6985, 37531, 52026}, {12514, 51118, 11372}, {28164, 62858, 10864}

X(68058) = X(3) OF EULER53

X(68058) lies on these lines: {4, 1192}, {5, 10193}, {20, 5893}, {30, 156}, {64, 3543}, {154, 5059}, {193, 1503}, {235, 13202}, {382, 13093}, {389, 40928}, {546, 23328}, {550, 14156}, {1204, 13473}, {1249, 22049}, {1498, 33703}, {1514, 35471}, {1531, 63441}, {1559, 45844}, {1657, 16252}, {1853, 17578}, {1885, 5480}, {2777, 3627}, {2781, 11381}, {2892, 41585}, {2935, 3518}, {3357, 3853}, {3522, 58434}, {3830, 20427}, {3832, 8567}, {3845, 64027}, {3850, 11204}, {3858, 25563}, {3861, 23329}, {5073, 5878}, {5076, 65151}, {5656, 11541}, {5972, 39084}, {6000, 10263}, {6240, 10721}, {8991, 42284}, {9833, 49136}, {9934, 37495}, {10117, 12086}, {10182, 62104}, {10282, 62155}, {11001, 17821}, {11202, 62144}, {11206, 50692}, {11403, 44883}, {11425, 49670}, {11469, 47353}, {12102, 23325}, {12103, 61747}, {12173, 34118}, {12233, 15033}, {12250, 18405}, {12315, 62040}, {12324, 50691}, {13371, 34584}, {13568, 44438}, {13980, 42283}, {14379, 47030}, {14530, 62170}, {15089, 15800}, {15105, 18381}, {15578, 63664}, {15583, 51163}, {15585, 48872}, {15640, 64714}, {15682, 64037}, {15683, 68024}, {15684, 48672}, {15687, 20299}, {15704, 32903}, {17800, 67890}, {17819, 42413}, {17820, 42414}, {17845, 49135}, {18376, 61540}, {18400, 62041}, {18504, 44280}, {18848, 44704}, {18912, 35490}, {19087, 52666}, {19088, 52667}, {21849, 22967}, {26883, 46374}, {27082, 59551}, {28158, 40660}, {28172, 40658}, {31670, 38263}, {32062, 68028}, {32063, 49133}, {32064, 50690}, {32767, 61988}, {33524, 35228}, {34780, 62035}, {34781, 62042}, {34786, 62034}, {35260, 62152}, {35450, 62016}, {36982, 44668}, {37197, 51998}, {37201, 48881}, {38790, 64036}, {40196, 61150}, {41587, 64891}, {44882, 63699}, {46265, 62062}, {47527, 64759}, {48879, 61610}, {49250, 53518}, {49251, 53519}, {50688, 54050}, {50689, 61735}, {50693, 61680}, {51024, 68021}, {51737, 67339}, {57584, 67902}, {61606, 62136}, {62021, 67894}, {62023, 64758}, {62030, 68027}, {62032, 68015}, {64587, 66762}

X(68058) = midpoint of X(i) and X(j) for these {i,j}: {1498, 33703}, {3146, 5895}, {5073, 5878}, {9833, 49136}, {15640, 64714}, {17845, 49135}, {64037, 64187}
X(68058) = reflection of X(i) in X(j) for these {i,j}: {20, 5893}, {1657, 16252}, {2883, 51491}, {3357, 3853}, {5894, 4}, {5925, 6696}, {6247, 3627}, {10192, 61721}, {15105, 18381}, {15583, 51163}, {15704, 61749}, {17845, 68025}, {18381, 62026}, {23315, 13202}, {34782, 22802}, {34786, 62034}, {41362, 382}, {44762, 5878}, {48872, 15585}, {48879, 61610}, {61540, 62013}, {62155, 10282}
X(68058) = pole of line {10019, 41580} with respect to the Jerabek hyperbola
X(68058) = pole of line {1249, 63533} with respect to the Kiepert hyperbola
X(68058) = pole of line {8567, 12111} with respect to the Stammler hyperbola
X(68058) = intersection, other than A, B, C, of circumconics {{A, B, C, X(15749), X(33893)}}, {{A, B, C, X(37878), X(38253)}}
X(68058) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 5894, 23332}, {4, 5925, 6696}, {20, 5893, 10192}, {20, 61721, 5893}, {30, 22802, 34782}, {382, 15311, 41362}, {2777, 3627, 6247}, {3146, 5895, 1503}, {3357, 3853, 23324}, {5893, 50709, 20}, {5925, 6696, 5894}, {12250, 62028, 18405}, {17845, 66752, 68025}, {22802, 34782, 2883}, {34782, 51491, 22802}, {61540, 62013, 18376}

X(68059) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR28-329 WRT ABC

X(68059) lies on circumconic {{A, B, C, X(7003), X(57671)}} and on these lines: {1, 15498}, {4, 1903}, {6, 11471}, {25, 22778}, {33, 7973}, {34, 64}, {40, 2182}, {65, 185}, {84, 15237}, {198, 9121}, {208, 2192}, {227, 40945}, {478, 35889}, {497, 10361}, {515, 52097}, {517, 5924}, {950, 8807}, {962, 5928}, {1118, 17832}, {1204, 40985}, {1426, 3270}, {1436, 3345}, {1490, 5909}, {1593, 12335}, {1697, 34032}, {1828, 11381}, {1829, 5895}, {1842, 55120}, {1851, 12679}, {1864, 1902}, {1870, 12262}, {1875, 6285}, {1888, 11436}, {1891, 13568}, {2269, 15852}, {2883, 46878}, {3182, 6611}, {3429, 6003}, {4219, 64722}, {5130, 12930}, {5514, 47441}, {5776, 68057}, {6245, 51490}, {7412, 40658}, {7957, 26893}, {9799, 34371}, {10605, 49185}, {12053, 51365}, {12680, 17441}, {14557, 63998}, {16388, 26932}, {16389, 34048}, {21871, 64004}, {24474, 34783}, {34434, 64332}, {37046, 45126}, {52026, 64818}, {52384, 53557}, {54340, 68016}

X(68059) = reflection of X(i) in X(j) for these {i,j}: {1490, 5909}, {40953, 4}, {51490, 6245}
X(68059) = pole of line {225, 1857} with respect to the Feuerbach hyperbola
X(68059) = pole of line {652, 6129} with respect to the orthic inconic
X(68059) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6245, 51490, 61671}

X(68060) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTI-AQUILA WRT CTR28-329

X(68060) lies on these lines: {1, 196}, {3, 12335}, {34, 38983}, {102, 1420}, {515, 47441}, {962, 54054}, {1319, 68061}, {1385, 37818}, {2360, 37418}, {3182, 3576}, {4297, 6261}, {7412, 45126}, {18443, 40657}

X(68060) = midpoint of X(i) and X(j) for these {i,j}: {1, 3345}
X(68060) = reflection of X(i) in X(j) for these {i,j}: {37818, 1385}

X(68061) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND ANTI-CIRCUMPERP-TANGENTIAL WRT CTR28-329

X(68061) lies on these lines: {1, 19904}, {11, 47441}, {55, 3182}, {56, 3345}, {354, 44696}, {1319, 68060}, {2646, 37818}, {3057, 7355}, {6284, 12680}, {8811, 19614}, {17603, 40657}, {52384, 53557}, {55307, 67949}

X(68062) = EULER LINE INTERCEPT OF X(7298)X(38458)

As a point on the Euler line, X(68062) has Shinagawa coefficients: {-e (e + f) + (e + f)^2 + 5 R^4, (-(e/4) - f) (e + f)}

See David Nguyen, euclid 8188.

X(68062) lies on these lines: {2, 3}, {7298, 38458}

X(68062) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7492, 21213}, {22, 7394, 23}, {22, 7517, 37913}, {6636, 13595, 7488}

X(68063) = MIDPOINT OF X(52) AND X(137)

X(68063) lies on the nine-point circle of the orthic triangle and these lines: {51, 128}, {52, 137}, {53, 1263}, {143, 25150}, {187, 65517}, {511, 34837}, {930, 3567}, {1112, 6756}, {1141, 3060}, {1154, 61594}, {1216, 58432}, {1994, 58068}, {3518, 58062}, {5462, 13372}, {5562, 23516}, {5890, 44976}, {5943, 58429}, {5946, 38615}, {6243, 57324}, {7731, 34308}, {9730, 63412}, {10095, 61587}, {10263, 38618}, {11002, 67091}, {12026, 14449}, {12236, 45147}, {13321, 13512}, {15043, 38706}, {23320, 64095}, {38710, 64051}, {45186, 63409}

X(68063) = midpoint of X(i) and X(j) for these {i,j}: {52, 137}, {10263, 38618}, {12026, 14449}, {45186, 63409}
X(68063) = reflection of X(i) in X(j) for these {i,j}: {1216, 58432}, {13372, 5462}, {61587, 10095}

X(68064) = MIDPOINT OF X(52) AND X(130)

X(68064) lies on the nine-point circle of the orthic triangle and these lines: {51, 129}, {52, 130}, {143, 27359}, {511, 34838}, {1112, 32438}, {1154, 61589}, {1298, 3060}, {1303, 3567}, {1994, 58069}, {3518, 58065}, {5462, 34839}, {5890, 44991}, {6243, 57333}, {10095, 61588}, {10110, 39835}, {13321, 67822}

X(68064) = midpoint of X(52) and X(130)
X(68064) = reflection of X(i) in X(j) for these {i,j}: {34839, 5462}, {61588, 10095}, {65500, 143}

X(68065) = MIDPOINT OF X(52) AND X(136)

X(68065) lies on the nine-point circle of the orthic triangle and these lines: {51, 131}, {52, 136}, {143, 53802}, {511, 34840}, {568, 13556}, {571, 5961}, {925, 3567}, {1112, 3575}, {1147, 34338}, {1154, 61593}, {1300, 3060}, {1994, 58066}, {3518, 58061}, {5462, 34844}, {5890, 44974}, {6243, 57334}, {10095, 61590}, {12236, 55121}, {15043, 67842}, {38718, 64051}, {50387, 65517}

X(68065) = midpoint of X(52) and X(136)
X(68065) = reflection of X(i) in X(j) for these {i,j}: {34844, 5462}, {61590, 10095}

X(68066) = MIDPOINT OF X(52) AND X(127)

X(68066) lies on the nine-point circle of the orthic triangle and these lines: {5, 58528}, {6, 3425}, {51, 125}, {52, 127}, {112, 3567}, {143, 11437}, {389, 2794}, {511, 34841}, {568, 10749}, {578, 34217}, {973, 65500}, {1154, 61586}, {1216, 58428}, {1297, 3060}, {1994, 58064}, {2799, 39806}, {3518, 58049}, {5446, 64509}, {5462, 6720}, {5890, 10735}, {5943, 58430}, {5946, 38608}, {6102, 19163}, {6243, 57332}, {6746, 13166}, {8779, 44668}, {9517, 12236}, {9530, 21849}, {9730, 14689}, {9753, 37473}, {10095, 61591}, {10263, 38624}, {11002, 12384}, {11432, 11641}, {13310, 13321}, {13754, 66594}, {14900, 16225}, {15043, 38699}, {16222, 53760}, {22391, 37813}, {38717, 64051}, {45186, 63410}, {46430, 53719}, {58515, 65093}

X(68066) = midpoint of X(i) and X(j) for these {i,j}: {52, 127}, {6102, 19163}, {10263, 38624}, {45186, 63410}
X(68066) = reflection of X(i) in X(j) for these {i,j}: {5, 58528}, {132, 58529}, {1216, 58428}, {6720, 5462}, {58515, 65093}, {61591, 10095}
X(68066) = Taylor-circle-inverse of X(67279)
X(68066) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {51, 132, 58529}, {112, 3567, 16224}

X(68067) = MIDPOINT OF X(52) AND X(122)

X(68067) lies on the nine-point circle of the orthic triangle and these lines: {5, 58524}, {6, 14703}, {51, 133}, {52, 122}, {107, 3567}, {143, 53803}, {389, 974}, {511, 34842}, {568, 10745}, {1154, 61583}, {1216, 58424}, {1294, 3060}, {1994, 58067}, {2790, 39835}, {2797, 39806}, {3184, 9730}, {3518, 58048}, {5446, 64505}, {5462, 6716}, {5562, 36520}, {5890, 10152}, {5943, 58431}, {5946, 38605}, {6102, 49117}, {6243, 57329}, {9033, 12236}, {10095, 61592}, {10263, 38621}, {11002, 34549}, {11432, 14673}, {13321, 38577}, {13352, 40082}, {15043, 23239}, {16222, 53757}, {23240, 37481}, {32411, 62501}, {34980, 45960}, {38714, 64051}, {45186, 63411}, {46430, 53716}, {58511, 65093}

X(68067) = midpoint of X(i) and X(j) for these {i,j}: {52, 122}, {6102, 49117}, {10263, 38621}, {45186, 63411}
X(68067) = reflection of X(i) in X(j) for these {i,j}: {5, 58524}, {133, 58530}, {1216, 58424}, {6716, 5462}, {58511, 65093}, {61592, 10095}
X(68067) = {X(51),X(133)}-harmonic conjugate of X(58530)

X(68068) = MIDPOINT OF X(52) AND X(131)

X(68068) lies on the nine-point circle of the orthic triangle and these lines: {6, 13558}, {51, 136}, {52, 131}, {143, 53802}, {389, 11800}, {511, 34844}, {578, 5961}, {925, 3060}, {1112, 55121}, {1154, 61590}, {1300, 3567}, {1994, 58061}, {3518, 58066}, {5462, 34840}, {5890, 44990}, {6243, 57314}, {10095, 61593}, {11438, 13496}, {15043, 38718}, {18390, 22823}, {39118, 39571}, {39806, 56304}, {64051, 67842}, {65517, 65656}

X(68068) = midpoint of X(52) and X(131)
X(68068) = reflection of X(i) in X(j) for these {i,j}: {34840, 5462}, {61593, 10095}

X(68069) = MIDPOINT OF X(52) AND X(128)

X(68069) lies on the nine-point circle of the orthic triangle and these lines: {6, 15959}, {30, 32409}, {51, 129}, {52, 128}, {143, 25150}, {511, 13372}, {568, 31656}, {578, 23320}, {930, 3060}, {973, 12236}, {1112, 45147}, {1141, 3567}, {1154, 61587}, {1216, 58429}, {1994, 58062}, {3518, 58068}, {3575, 32410}, {3580, 14769}, {5462, 34837}, {5640, 13504}, {5890, 44981}, {5943, 58432}, {5946, 38618}, {6152, 27423}, {6243, 57316}, {6592, 14449}, {9730, 63409}, {9781, 13505}, {10095, 61594}, {10263, 38615}, {11002, 11671}, {11432, 15960}, {11746, 45258}, {12077, 65517}, {13321, 38587}, {13567, 23319}, {14071, 32196}, {14652, 34545}, {14656, 18315}, {15043, 38710}, {15366, 37649}, {38706, 64051}, {39839, 41222}, {45186, 63412}

X(68069) = midpoint of X(i) and X(j) for these {i,j}: {52, 128}, {3575, 32410}, {6152, 27423}, {6592, 14449}, {10263, 38615}, {14071, 32196}, {45186, 63412}
X(68069) = reflection of X(i) in X(j) for these {i,j}: {1216, 58429}, {34837, 5462}, {45258, 11746}, {61594, 10095}
X(68069) = crosssum of X(3) and X(55073)

X(68070) = MIDPOINT OF X(52) AND X(25641)

X(68070) lies on the nine-point circle of the orthic triangle and these lines: {30, 974}, {51, 3258}, {52, 25641}, {143, 16168}, {250, 13558}, {389, 64510}, {476, 3060}, {477, 3567}, {511, 11657}, {523, 1112}, {568, 66781}, {1553, 21649}, {1986, 34150}, {2781, 12079}, {3154, 11746}, {3233, 14984}, {5462, 31379}, {5627, 7731}, {5640, 66801}, {5890, 14989}, {5946, 38610}, {6070, 13417}, {6102, 66778}, {6243, 57305}, {6746, 66771}, {9826, 47084}, {9971, 66813}, {10263, 38609}, {10419, 14703}, {11002, 14731}, {11412, 66787}, {11432, 66777}, {11807, 32417}, {12068, 41673}, {12077, 65500}, {12824, 14611}, {13321, 38581}, {14934, 16222}, {15043, 38701}, {16319, 44084}, {20403, 58900}, {31945, 41670}, {32411, 62501}, {36164, 46430}, {38700, 64051}, {39806, 47143}, {39835, 62489}, {41671, 55308}, {44668, 47351}, {47208, 66165}, {47222, 65586}, {55319, 58498}, {63659, 63715}, {65516, 65856}

X(68070) = midpoint of X(i) and X(j) for these {i,j}: {52, 25641}, {476, 16978}, {1553, 21649}, {1986, 34150}, {6070, 13417}, {6102, 66778}, {10263, 38609}
X(68070) = reflection of X(i) in X(j) for these {i,j}: {3154, 11746}, {3258, 12052}, {16319, 44084}, {31379, 5462}, {41673, 12068}, {47084, 9826}, {55308, 41671}, {55319, 58498}
X(68070) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {51, 3258, 12052}, {476, 3060, 16978}

X(68071) = MIDPOINT OF X(52) AND X(133)

X(68071) lies on the nine-point circle of the orthic triangle and these lines: {5, 58530}, {51, 122}, {52, 133}, {107, 3060}, {143, 53803}, {185, 38956}, {389, 64505}, {511, 6716}, {568, 22337}, {1112, 9033}, {1154, 61592}, {1216, 58431}, {1294, 3567}, {1994, 58048}, {2777, 5446}, {2790, 39806}, {2797, 39835}, {3184, 45186}, {3518, 58067}, {5462, 34842}, {5890, 44985}, {5943, 58424}, {5946, 38621}, {6243, 57301}, {9037, 58598}, {9047, 58668}, {9530, 21849}, {9730, 63411}, {10095, 61583}, {10263, 38605}, {11002, 34186}, {11732, 58469}, {13321, 38591}, {14449, 61569}, {15043, 38714}, {23239, 64051}

X(68071) = midpoint of X(i) and X(j) for these {i,j}: {52, 133}, {185, 38956}, {3184, 45186}, {10263, 38605}, {14449, 61569}
X(68071) = reflection of X(i) in X(j) for these {i,j}: {5, 58530}, {122, 58524}, {1216, 58431}, {6716, 58511}, {11732, 58469}, {34842, 5462}, {61583, 10095}
X(68071) = {X(51),X(122)}-harmonic conjugate of X(58524)

X(68072) = MIDPOINT OF X(52) AND X(132)

X(68072) lies on the nine-point circle of the orthic triangle and these lines: {3, 16224}, {5, 58529}, {6, 41382}, {51, 127}, {52, 132}, {112, 3060}, {143, 11437}, {389, 64509}, {511, 6720}, {568, 12918}, {1112, 9517}, {1154, 61591}, {1216, 58430}, {1297, 3567}, {1994, 58049}, {2781, 10264}, {2794, 5446}, {2799, 39835}, {3518, 58064}, {5462, 34841}, {5890, 44988}, {5943, 58428}, {5946, 38624}, {6102, 19160}, {6243, 57304}, {6746, 12145}, {9037, 58603}, {9047, 58673}, {9730, 63410}, {10095, 61586}, {10110, 66594}, {10263, 38608}, {11002, 13219}, {11432, 12413}, {13115, 13321}, {14449, 61573}, {14689, 16225}, {15043, 38717}, {34217, 64095}, {35431, 40121}, {38699, 64051}

X(68072) = midpoint of X(i) and X(j) for these {i,j}: {52, 132}, {6102, 19160}, {10263, 38608}, {14449, 61573}, {14689, 45186}
X(68072) = reflection of X(i) in X(j) for these {i,j}: {5, 58529}, {127, 58528}, {1216, 58430}, {6720, 58515}, {34841, 5462}, {61586, 10095}, {66594, 10110}
X(68072) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {51, 127, 58528}, {16225, 45186, 14689}

X(68073) = MIDPOINT OF X(52) AND X(45180)

X(68073) lies on the nine-point circle of the orthic triangle and these lines: {51, 46439}, {52, 45180}, {143, 476}, {389, 34145}, {568, 14980}, {1112, 1510}, {1154, 2072}, {1291, 3060}, {2501, 65500}, {3567, 14979}, {6243, 57326}, {6746, 16221}, {14984, 43969}, {32409, 32411}, {39835, 65485}

X(68073) = midpoint of X(52) and X(45180)
X(68073) = reflection of X(32409) in X(32411)

X(68074) = MIDPOINT OF X(52) AND X(16978)

X(68074) lies on the nine-point circle of the orthic triangle and these lines: {3, 16978}, {5, 12052}, {30, 1112}, {51, 25641}, {52, 3258}, {143, 16168}, {476, 3567}, {477, 3060}, {511, 31379}, {523, 12236}, {568, 20957}, {1986, 36184}, {3284, 7575}, {5446, 64510}, {5462, 22104}, {5640, 66787}, {5663, 52219}, {5890, 44967}, {5946, 38609}, {6102, 66795}, {6243, 57306}, {6746, 10223}, {7471, 16222}, {9971, 66810}, {10263, 38610}, {11002, 34193}, {11412, 66801}, {11432, 66794}, {11806, 32417}, {13321, 38580}, {15043, 38700}, {15544, 18907}, {36169, 58516}, {38701, 64051}, {39806, 62489}, {39835, 62490}, {46430, 46632}, {53809, 65516}, {63659, 63708}

X(68074) = midpoint of X(i) and X(j) for these {i,j}: {3, 16978}, {52, 3258}, {1986, 36184}, {6102, 66795}, {10263, 38610}
X(68074) = reflection of X(i) in X(j) for these {i,j}: {5, 12052}, {22104, 5462}, {36169, 58516}, {66790, 10223}

X(68075) = MIDPOINT OF X(52) AND X(20625)

X(68075) lies on the nine-point circle of the orthic triangle and these lines: {6, 54067}, {51, 18402}, {52, 20625}, {143, 53808}, {933, 3567}, {973, 1112}, {3060, 18401}, {5890, 44977}, {5946, 38616}, {6243, 57369}, {6748, 10214}, {13321, 38585}, {32409, 32411}

X(68075) = midpoint of X(52) and X(20625)

X(68076) = MIDPOINT OF X(52) AND X(2679)

X(68076) lies on the nine-point circle of the orthic triangle and these lines: {3, 16979}, {51, 33330}, {52, 2679}, {143, 53797}, {460, 1112}, {511, 620}, {512, 39806}, {568, 66837}, {805, 3567}, {2698, 3060}, {5462, 22103}, {5890, 44971}, {5946, 67833}, {6102, 66836}, {6243, 57347}, {10263, 66821}, {11002, 66822}, {13321, 66840}, {15043, 38703}, {64051, 67840}

X(68076) = midpoint of X(i) and X(j) for these {i,j}: {3, 16979}, {52, 2679}, {6102, 66836}, {10263, 66821}
X(68076) = reflection of X(i) in X(j) for these {i,j}: {22103, 5462}, {65517, 143}