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 {{OF, 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 {{OF, 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 {{OF, 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}

X(68077) = X(2)X(13476)∩X(354)X(40504)

See Ivan Pavlov, euclid 8194.

X(68077) lies on these lines: {2, 13476}, {354, 40504}, {1962, 58571}

X(68078) = (name pending)

As a point on the Euler line, X(68078) has Shinagawa coefficients: {-(63/4) e (e + f) + 8 (e + f)^2 + 120 R^4, 1/4 e (e + f)}

See David Nguyen, euclid 8201.

X(68078) lies on this line: {2, 3}

X(68078) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2072, 7542, 7575}, {5159, 6676, 858}, {5159, 16977, 1368}, {6639, 10297, 44282}

X(68079) = EULER LINE INTERCEPT OF X(54)X(1353)

As a point on the Euler line, X(68079) has Shinagawa coefficients: {1/3 (-12 e + 15 (e + f)), 2 e - 3 (e + f)}

See David Nguyen, euclid 8201.

X(68079) lies on these lines: {2, 3}, {54, 1353}, {141, 32348}, {155, 44683}, {185, 13394}, {343, 13367}, {389, 31807}, {578, 41588}, {599, 45248}, {973, 9967}, {1040, 52793}, {1078, 41005}, {1092, 19131}, {1147, 44201}, {1176, 43617}, {1352, 17821}, {3284, 9606}, {3564, 19357}, {3567, 18438}, {3796, 26937}, {5562, 59553}, {5690, 24301}, {5889, 61690}, {5907, 10192}, {6102, 45118}, {6697, 44882}, {7763, 41008}, {8981, 10898}, {9722, 44523}, {10116, 12359}, {10165, 37613}, {10182, 11793}, {10316, 31406}, {10541, 47558}, {10634, 42121}, {10635, 42124}, {10897, 13966}, {11202, 64035}, {11424, 32269}, {11425, 13142}, {11431, 53092}, {11449, 37636}, {12233, 58447}, {12241, 61646}, {13416, 25711},{13562, 23041}, {14528, 64060}, {15072, 43903}, {15738, 38727}, {15873, 32223}, {15905, 31400}, {16310, 36751}, {18457, 19116}, {18459, 19117}, {19126, 64061}, {19467, 37638}, {20300, 20376}, {21167, 52520}, {21243, 34782},{22660, 44516}, {23292, 31802}, {23328, 46850}, {23332, 44829}, {25406, 58378}, {35602, 43653}, {37476, 45298}, {39575, 59657}, {40686, 46264}, {43595, 63734}, {44158, 64049}, {45187, 64064}, {45303, 61139}, {45946, 54320},{50649, 63709}, {64992, 65090}

X(68079) = midpoint of X(i) and X(j) for these {i,j}: {3, 3549}, {3541, 9715}
X(68079) = complement of X(7507)
X(68079) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 12362}, {2, 3515, 9825}, {2, 3575, 5}, {3, 140, 1368}, {3, 631, 16196}, {3, 3523, 16976}, {3, 3526, 6643}, {3, 3546, 10691}, {3, 3547, 31829}, {3, 3549, 30}, {3, 5054, 3546}, {3, 6639, 12605}, {3, 6642, 15818}, {3, 6676, 6823}, {3, 6823, 44241}, {3, 7542, 5}, {3, 10024, 44249}, {3, 15760, 550}, {5, 1658, 37458}, {140, 548, 32144}, {140, 1658, 5}, {140, 3530, 7516}, {140, 9825, 2}, {427, 7488, 65376}, {468, 7503, 5}, {631, 7512, 37118}, {2070, 7403, 7715}, {3147, 7395, 6677}, {3523, 7494, 3}, {3524, 7400, 3}, {3526, 6643, 5159}, {3530, 16197, 3}, {3541, 9715, 30}, {6639, 12605, 5}, {7526, 13383, 1596}, {7542, 12605, 6639}, {10996, 15717, 3}, {12359, 18475, 31804}, {19467, 37638, 61544}, {23292, 46730, 31802}

X(68080) = EULER LINE INTERCEPT OF X(1619)X(45303)

As a point on the Euler line, X(68080) has Shinagawa coefficients: {-4 e (e + f) + (e + f)^2 + 40 R^4, -f (e + f)}

See David Nguyen, euclid 8201.

X(68080) lies on these lines: {2, 3}, {1619, 45303}, {11451, 26206}

X(68080) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 13595, 7503}

X(68081) = (name pending)

As a point on the Euler line, X(68081) has Shinagawa coefficients: {-7 e (e + f) + 5 (e + f)^2 + 40 R^4, 4 (-(e/4) - f) (e + f)}

See David Nguyen, euclid 8201.

X(68081) lies on this line: {2, 3}

X(68081) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6995, 10565, 7493}

X(68082) = EULER LINE INTERCEPT OF X(69)X(13367)

As a point on the Euler line, X(68082) has Shinagawa coefficients: {4 (e/4 + f), 2 e - 3 (e + f)}

See David Nguyen, euclid 8201.

X(68082) lies on these lines: {2, 3}, {69, 13367}, {524, 14528}, {3796, 18913}, {4549, 44516}, {5562, 64177}, {5907, 35260}, {6193, 44201}, {7691, 37645}, {9967, 15043}, {10519, 35602}, {10610, 18951}, {10984, 18931}, {11064, 11821}, {11411, 18475}, {11427, 46730}, {11487, 51393}, {12164, 64058}, {12245, 24301}, {13346, 33522}, {14376, 55732}, {14826, 17821}, {15448, 33537}, {15740, 21663}, {18840, 54075}, {18945, 37638}, {19126, 43652}, {19131, 34148}, {19357, 63174}, {21167, 41719}, {25406, 26937}, {32348, 32354}, {32379, 44833}, {32831, 41008}, {37613, 54445}

X(68082) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20, 7507}, {3, 631, 7386}, {3, 3547, 376}, {3, 6676, 20}, {3, 6823, 3522}, {3, 7494, 10996}, {3, 16197, 61113}, {186, 631, 6803}, {376, 631, 37119}, {3088, 9715, 34608}, {3523, 17928, 631}, {7493, 14118, 4}

X(68083) = EULER LINE INTERCEPT OF X(13562)X(47447)

As a point on the Euler line, X(68083) has Shinagawa coefficients: {1/3 (-54 e (e + f) + 33 (e + f)^2 + 360 R^4), (e + f) (7 e - 9 (e + f))}

See David Nguyen, euclid 8201.

X(68083) lies on these lines: {2, 3}, {13562, 47447}

X(68084) = X(4)X(93)∩X(30)X(143)

See Ivan Pavlov, euclid 8205.

X(68084) lies on these lines: {2, 18874}, {3, 5640}, {4, 93}, {5, 3917}, {20, 5946}, {23, 5944}, {26, 11425}, {30, 143}, {51, 550}, {52, 3627}, {54, 5899}, {125, 63474}, {140, 6688}, {156, 7530}, {185, 62036}, {186, 43823}, {195, 14157}, {235, 61574}, {323, 26863}, {373, 14869}, {381, 6101}, {382, 3060}, {399, 15801}, {511, 546}, {539, 67322}, {547, 5447}, {548, 5462}, {549, 32205}, {567, 12088}, {568, 3146}, {578, 17714}, {632, 11592}, {973, 61744}, {1112, 6240}, {1173, 15037}, {1192, 12084}, {1199, 37945}, {1216, 3850}, {1351, 32139}, {1493, 1614}, {1503, 11264}, {1511, 3518}, {1595, 63734}, {1598, 11387}, {1656, 44299}, {1657, 3567}, {1885, 66604}, {2070, 43394}, {2777, 13358}, {2781, 18383}, {2854, 38632}, {2937, 10610}, {2979, 3851}, {3090, 13340}, {3091, 15067}, {3153, 43865}, {3313, 38136}, {3448, 64757}, {3527, 35243}, {3529, 11002}, {3530, 5943}, {3534, 15043}, {3543, 34783}, {3544, 33884}, {3574, 61750}, {3581, 14865}, {3628, 15082}, {3819, 35018}, {3830, 5889}, {3832, 23039}, {3843, 11412}, {3845, 5562}, {3853, 13754}, {3855, 62188}, {3858, 5891}, {3859, 13570}, {3861, 5907}, {5066, 11793}, {5072, 7999}, {5073, 5890}, {5076, 12111}, {5079, 7998}, {5198, 15068}, {5609, 51882}, {5650, 61900}, {5878, 34751}, {5892, 33923}, {6000, 62026}, {6146, 61299}, {6746, 18560}, {7387, 11426}, {7502, 11424}, {7517, 9707}, {7526, 33586}, {7553, 12370}, {8703, 64854}, {8718, 37949}, {9019, 16511}, {9729, 12103}, {9730, 15704}, {9827, 37478}, {9927, 48901}, {10096, 13446}, {10113, 31724}, {10125, 32223}, {10170, 12811}, {10574, 13321}, {10575, 62041}, {10594, 61753}, {10628, 63728}, {10733, 38898}, {11188, 55724}, {11250, 64095}, {11430, 12107}, {11432, 64098}, {11439, 38335}, {11451, 15720}, {11455, 62016}, {11459, 61984}, {11465, 61811}, {11565, 12022}, {11572, 18555}, {11695, 12100}, {11743, 13565}, {11803, 14862}, {11807, 13419}, {11817, 15052}, {11819, 30522}, {12041, 12086}, {12046, 14845}, {12061, 14984}, {12082, 36753}, {12087, 36153}, {12101, 46849}, {12106, 13346}, {12161, 18534}, {12162, 15687}, {12236, 34584}, {12239, 42225}, {12240, 42226}, {12279, 15684}, {12290, 62023}, {12605, 15807}, {13154, 52518}, {13292, 45732}, {13339, 16661}, {13352, 32171}, {13367, 37936}, {13371, 15465}, {13383, 58407}, {13434, 13564}, {13474, 62013}, {13482, 37956}, {13488, 47328}, {13561, 41587}, {13596, 63392}, {13621, 43574}, {13861, 37498}, {14269, 15058}, {14531, 16194}, {14627, 37924}, {14805, 38435}, {14810, 58532}, {14831, 33699}, {14855, 62159}, {14893, 31834}, {14915, 16625}, {15012, 58203}, {15030, 61988}, {15038, 47748}, {15045, 15696}, {15056, 54048}, {15062, 32608}, {15072, 49136}, {15074, 20423}, {15083, 37517}, {15305, 62008}, {15559, 34826}, {15681, 66606}, {15682, 64030}, {15699, 27355}, {15711, 55320}, {15712, 36987}, {16168, 36160}, {16226, 19710}, {16261, 61991}, {16655, 32358}, {16776, 52987}, {16836, 44245}, {16964, 36978}, {16965, 36980}, {16978, 66778}, {16979, 66826}, {16981, 50688}, {17538, 40280}, {17578, 18439}, {17702, 63683}, {17834, 31861}, {18128, 45969}, {18350, 52294}, {18369, 37496}, {18378, 34148}, {18379, 44288}, {18400, 44056}, {18488, 41586}, {18569, 31670}, {18952, 34938}, {19211, 38577}, {20424, 21660}, {20791, 62131}, {21850, 50649}, {22115, 34484}, {22948, 53779}, {23060, 37505}, {23292, 64472}, {25338, 63737}, {28146, 31760}, {28154, 65423}, {28160, 31757}, {29012, 58806}, {29317, 32191}, {31663, 58474}, {31732, 33697}, {31737, 38140}, {32136, 36749}, {32138, 37489}, {32140, 64048}, {33532, 37514}, {33533, 37486}, {33879, 61892}, {34200, 58470}, {34469, 47527}, {34553, 42279}, {34555, 42278}, {34603, 44076}, {35007, 61675}, {35452, 43601}, {37477, 44802}, {37494, 63664}, {38848, 43809}, {43615, 58559}, {43816, 60466}, {43821, 46450}, {44324, 61940}, {44407, 45970}, {44457, 66609}, {44493, 63673}, {44544, 64037}, {44668, 61749}, {44862, 60749}, {44879, 48912}, {45237, 51522}, {45956, 62044}, {46219, 54041}, {46728, 49671}, {49139, 52093}, {49140, 61136}, {54047, 61919}, {61858, 63632}, {61975, 66756}, {62021, 64025}, {62024, 66748}, {62144, 65093}, {62155, 64100}, {62171, 66747}, {63693, 63727}

X(68084) = midpoint of X(i) and X(j) for these {i,j}: {4, 10263}, {5, 45186}, {52, 3627}, {185, 62036}, {382, 6102}, {3146, 13491}, {3853, 14449}, {5446, 13598}, {5876, 6243}, {6101, 64051}, {7553, 12370}, {10575, 62041}, {10733, 38898}, {13421, 45959}, {14831, 33699}, {15687, 21969}, {15800, 32196}, {16655, 32358}, {16978, 66778}, {16979, 66826}, {22948, 53779}, {31732, 33697}, {43893, 48914}, {44544, 64037}
X(68084) = reflection of X(i) in X(j) for these {i,j}: {3, 10095}, {52, 16982}, {125, 63474}, {140, 10110}, {143, 5446}, {548, 5462}, {550, 12006}, {1216, 3850}, {5562, 45958}, {5907, 3861}, {6101, 14128}, {10096, 13446}, {10110, 12002}, {10625, 32142}, {10627, 5}, {11561, 1112}, {11591, 546}, {11793, 44863}, {12103, 9729}, {12605, 15807}, {13421, 10263}, {13470, 12241}, {13474, 62013}, {13565, 11743}, {13630, 143}, {14810, 58532}, {15644, 3628}, {31663, 58474}, {31834, 44870}, {32137, 3853}, {32171, 63738}, {33923, 58531}, {34200, 58470}, {40647, 16881}, {43615, 58559}, {44829, 43575}, {45732, 13292}, {45959, 4}, {54044, 13364}, {63414, 140}
X(68084) = pole of line {3850, 18914} with respect to the Jerabek hyperbola
X(68084) = pole of line {5421, 7765} with respect to the Kiepert hyperbola
X(68084) = pole of line {49, 549} with respect to the Stammler hyperbola
X(68084) = pole of line {43459, 44148} with respect to the Wallace hyperbola
X(68084) = center of circles {{OF, X(i), X(j), X(k)}} for these {i, j, k}: {52, 3627, 36160}, {185, 36179, 62036}
X(68084) = intersection, other than A, B, C, of circumconics {{A, B, C, X(93), X(14483)}}, {{A, B, C, X(11140), X(55982)}}
X(68084) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 10095, 13363}, {3, 9781, 15026}, {4, 1154, 45959}, {4, 62187, 18436}, {4, 6243, 5876}, {5, 45186, 13391}, {23, 37472, 5944}, {30, 12241, 13470}, {30, 143, 13630}, {30, 16881, 40647}, {30, 43575, 44829}, {52, 3627, 5663}, {140, 10110, 13364}, {140, 63414, 54044}, {381, 6101, 14128}, {381, 64051, 6101}, {511, 546, 11591}, {548, 13451, 5462}, {568, 3146, 13491}, {1154, 10263, 13421}, {1216, 67067, 3850}, {1656, 64050, 54042}, {2937, 15033, 10610}, {3843, 11412, 15060}, {3845, 5562, 45958}, {3853, 13754, 32137}, {3853, 14449, 13754}, {3858, 5891, 11017}, {5446, 13598, 30}, {5446, 40647, 21849}, {5663, 16982, 52}, {5876, 10263, 6243}, {7530, 36747, 156}, {9781, 15026, 10095}, {10625, 32142, 10627}, {11793, 44863, 5066}, {13321, 17800, 10574}, {13391, 32142, 10625}, {13421, 45959, 1154}, {14374, 14375, 50476}, {14627, 37924, 52525}, {14845, 55856, 12046}, {14893, 31834, 44870}, {15038, 47748, 61134}, {16881, 21849, 143}, {20424, 43893, 43831}, {21849, 40647, 16881}, {37949, 43845, 8718}, {54048, 61970, 15056}

X(68085) = EULER LINE INTERCEPT OF X(110)X(343)

See David Nguyen and Ivan Pavlov, euclid 8206.

X(68085) lies on these lines: {2, 3}, {49, 63734}, {51, 14389}, {52, 44516}, {54, 41587}, {68, 9707}, {69, 19153}, {98, 42410}, {110, 343}, {111, 53949}, {114, 14103}, {141, 18374}, {154, 11442}, {182, 61645}, {183, 26269}, {184, 3580}, {206, 46442}, {230, 22240}, {323, 59553}, {325, 60694}, {351, 65972}, {394, 59551}, {542, 44110}, {568, 61619}, {590, 11418}, {597, 11416}, {615, 11417}, {827, 1799}, {925, 5966}, {933, 64992}, {1125, 64039}, {1141, 53958}, {1194, 5355}, {1287, 53929}, {1297, 53957}, {1302, 18401}, {1352, 35264}, {1495, 21243}, {1503, 23293}, {1614, 12359}, {1627, 41336}, {1629, 66707}, {1899, 6800}, {1993, 61655}, {1994, 41588}, {2697, 16166}, {2770, 11635}, {2883, 11440}, {2979, 11064}, {3060, 23292}, {3100, 5432}, {3164, 17004}, {3167, 45794}, {3292, 61681}, {3313, 58450}, {3410, 35265}, {3448, 45082}, {3564, 9544}, {3589, 9971}, {3629, 13622}, {3796, 18911}, {3815, 10313}, {3818, 44082}, {3917, 5972}, {3920, 9627}, {4296, 5433}, {5012, 13394}, {5097, 61659}, {5272, 38458}, {5422, 61506}, {5449, 34224}, {5562, 64063}, {5640, 37649}, {5650, 12058}, {5921, 61610}, {5944, 44076}, {6030, 15059}, {6053, 12825}, {6101, 58435}, {6241, 44158}, {6390, 37808}, {6563, 10190}, {6690, 20243}, {6696, 12279}, {6699, 14855}, {6720, 20410}, {7664, 8024}, {7917, 33651}, {7998, 53415}, {8254, 12226}, {8718, 43608}, {9060, 53959}, {9064, 67740}, {9306, 37636}, {9704, 32358}, {9777, 21970}, {9820, 11412}, {10160, 11226}, {10182, 51394}, {10263, 58407}, {10272, 12219}, {10282, 14516}, {10420, 53935}, {10540, 67926}, {10575, 20191}, {10625, 43839}, {11003, 11245}, {11204, 50434}, {11225, 44109}, {11402, 37644}, {11420, 23303}, {11421, 23302}, {11422, 61658}, {11444, 59659}, {11449, 68018}, {11454, 15311}, {11459, 44201}, {11464, 44665}, {12022, 18475}, {12111, 16252}, {12134, 26882}, {12272, 15585}, {13219, 51240}, {13366, 32225}, {13398, 53963}, {13445, 23328}, {14569, 37766}, {14979, 16167}, {15066, 43653}, {15080, 26913}, {15360, 51132}, {16243, 62722}, {16789, 22151}, {18018, 40022}, {18435, 46817}, {18436, 61608}, {19126, 26156}, {19127, 62376}, {19130, 44106}, {20477, 58436}, {20806, 31267}, {22165, 32244}, {22352, 61691}, {23096, 53953}, {26233, 45201}, {26703, 26711}, {26879, 64049}, {28408, 37485}, {29681, 60359}, {30737, 37688}, {31383, 61700}, {34417, 42785}, {34782, 58922}, {34799, 61544}, {34826, 64036}, {34986, 41586}, {35254, 64101}, {35266, 50958}, {35345, 37647}, {35360, 56297}, {36414, 52905}, {37671, 65711}, {38463, 63611}, {39431, 58975}, {40588, 45215}, {40938, 64646}, {41362, 41482}, {41464, 51126}, {41578, 41714}, {41593, 41594}, {41614, 61683}, {43697, 47558}, {44111, 61677}, {44673, 64100}, {44683, 61606}, {45016, 48876}, {45118, 52000}, {46730, 66727}, {47204, 59529}, {47582, 59771}, {52525, 67902}, {53944, 67799}, {56292, 64066}, {57486, 66125}, {59701, 62838}, {60358, 66632}, {61657, 63076}, {61747, 63425}

X(68085) = midpoint of X(23293) and X(26881)
X(68085) = inverse of X(60455) in DeLongchamps circle
X(68085) = inverse of X(10296) in 2nd DrozFarny circle
X(68085) = inverse of X(31236) in orthocentroidal circle
X(68085) = inverse of X(18403) in orthoptic circle of the Steiner Inellipse
X(68085) = inverse of X(31236) in Yff hyperbola
X(68085) = complement of X(31074)
X(68085) = anticomplement of X(62958)
X(68085) = X(i)-Dao conjugate of X(j) for these {i, j}: {62958, 62958}
X(68085) = X(i)-anticomplementary conjugate of X(j) for these {i,j}: {53109, 21270}
X(68085) = pole of line {523, 60455} with respect to the DeLongchamps circle
X(68085) = pole of line {523, 10296} with respect to the 2nd DrozFarny circle
X(68085) = pole of line {523, 31236} with respect to the orthocentroidal circle
X(68085) = pole of line {523, 18403} with respect to the orthoptic circle of the Steiner Inellipse
X(68085) = pole of line {427, 44420} with respect to the Parry circle
X(68085) = pole of line {185, 34005} with respect to the Jerabek hyperbola
X(68085) = pole of line {6, 23293} with respect to the Kiepert hyperbola
X(68085) = pole of line {3, 9972} with respect to the Stammler hyperbola
X(68085) = pole of line {525, 13196} with respect to the Steiner inellipse
X(68085) = pole of line {523, 31236} with respect to the Yff hyperbola
X(68085) = pole of line {69, 6697} with respect to the Wallace hyperbola
X(68085) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(22), X(64982)}}, {{A, B, C, X(24), X(5966)}}, {{A, B, C, X(25), X(41593)}}, {{A, B, C, X(98), X(6240)}}, {{A, B, C, X(262), X(7547)}}, {{A, B, C, X(264), X(31236)}}, {{A, B, C, X(297), X(42410)}}, {{A, B, C, X(305), X(30744)}}, {{A, B, C, X(378), X(18401)}}, {{A, B, C, X(403), X(53935)}}, {{A, B, C, X(427), X(2373)}}, {{A, B, C, X(550), X(34168)}}, {{A, B, C, X(827), X(46592)}}, {{A, B, C, X(842), X(37970)}}, {{A, B, C, X(858), X(1799)}}, {{A, B, C, X(1105), X(34005)}}, {{A, B, C, X(1141), X(18533)}}, {{A, B, C, X(1297), X(3520)}}, {{A, B, C, X(1594), X(15319)}}, {{A, B, C, X(1989), X(62978)}}, {{A, B, C, X(2409), X(53957)}}, {{A, B, C, X(2697), X(13619)}}, {{A, B, C, X(3147), X(3459)}}, {{A, B, C, X(3542), X(53963)}}, {{A, B, C, X(4235), X(53949)}}, {{A, B, C, X(4244), X(26711)}}, {{A, B, C, X(5094), X(18018)}}, {{A, B, C, X(5133), X(40413)}}, {{A, B, C, X(6643), X(57746)}}, {{A, B, C, X(6997), X(10603)}}, {{A, B, C, X(7482), X(11635)}}, {{A, B, C, X(7526), X(40801)}}, {{A, B, C, X(7576), X(39431)}}, {{A, B, C, X(7607), X(32534)}}, {{A, B, C, X(8770), X(21213)}}, {{A, B, C, X(8889), X(13575)}}, {{A, B, C, X(10295), X(53959)}}, {{A, B, C, X(12173), X(15619)}}, {{A, B, C, X(13573), X(60455)}}, {{A, B, C, X(13622), X(62958)}}, {{A, B, C, X(15392), X(44214)}}, {{A, B, C, X(16166), X(37937)}}, {{A, B, C, X(17928), X(65090)}}, {{A, B, C, X(18386), X(38305)}}, {{A, B, C, X(18403), X(60590)}}, {{A, B, C, X(18859), X(67730)}}, {{A, B, C, X(21284), X(53929)}}, {{A, B, C, X(22466), X(23047)}}, {{A, B, C, X(23096), X(37951)}}, {{A, B, C, X(37913), X(56306)}}, {{A, B, C, X(39436), X(52397)}}, {{A, B, C, X(44061), X(57602)}}, {{A, B, C, X(46591), X(58975)}}
X(68085) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 22, 858}, {2, 23, 427}, {51, 58447, 14389}, {154, 11442, 46818}, {154, 37638, 11442}, {184, 3580, 45968}, {184, 61646, 3580}, {343, 10192, 110}, {10575, 20191, 43607}, {13394, 13567, 5012}, {18475, 63735, 12022}, {23292, 32269, 3060}, {23293, 26881, 1503}, {32223, 58447, 51}, {34986, 41586, 41628}, {41586, 64064, 34986}, {41588, 61690, 1994}, {44201, 51425, 11459}

   Points on the Moses HK-parabola, X(68086) - X(68089)  

Contributed by Clark Kimberling, based on notes and data from Peter Moses, March 26, 2025.

The Moses HK-parabola is introduced here as the inscribed parabola having focus X(112) and directrix the line HK = X(4)X(6). The Moses HK-parabola passes through X(i) for these i: 525, 2501, 14401, 15639, 17925, 17926, 23090, 32320, 43925, 52131, 52132, 57195, 57201, 57202, 57203, 57204, 58760, 58780, 58812, 60505, 68086, 68087, 68088, 68089

X(68086) = X(110)X(677)∩X(112)X(6078)

X(68086) lies on the Moses HK-parabola and these lines: {110, 677}, {112, 6078}, {162, 660}, {518, 5089}, {525, 61197}, {648, 53227}, {1783, 17925}, {1897, 17926}, {2501, 61180}, {4238, 63743}, {8693, 56183}, {34337, 39686}, {57202, 61201}, {57204, 61205}

X(68086) = X(i)-Ceva conjugate of X(j) for these (i,j): {648, 4238}, {5379, 37908}
X(68086) = X(i)-isoconjugate of X(j) for these (i,j): {525, 51838}, {656, 6185}, {673, 10099}, {810, 57537}, {14208, 41934}, {23696, 66941}, {31637, 55261}, {51664, 62715}
X(68086) = X(i)-Dao conjugate of X(j) for these (i,j): {518, 525}, {39062, 57537}, {40596, 6185}
X(68086) = crosspoint of X(648) and X(4238)
X(68086) = crosssum of X(647) and X(10099)
X(68086) = trilinear pole of line {6184, 20776}
X(68086) = barycentric product X(i)*X(j) for these {i,j}: {29, 66978}, {99, 42071}, {107, 65744}, {110, 34337}, {112, 4437}, {162, 4712}, {518, 4238}, {648, 6184}, {811, 42079}, {883, 37908}, {1026, 54407}, {1362, 36797}, {1783, 16728}, {1861, 54353}, {2284, 15149}, {3126, 5379}, {6331, 39686}, {6528, 20776}
X(68086) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 6185}, {648, 57537}, {1362, 17094}, {2223, 10099}, {4238, 2481}, {4437, 3267}, {4712, 14208}, {6184, 525}, {16728, 15413}, {20776, 520}, {32676, 51838}, {34337, 850}, {37908, 885}, {39686, 647}, {42071, 523}, {42079, 656}, {54353, 31637}, {61206, 41934}, {65744, 3265}, {66978, 307}

X(68087) = X(110)X(525)∩X(112)X(6082)

X(68087) lies on the Moses HK-parabola and these lines: {2, 60505}, {4, 54607}, {110, 525}, {112, 6082}, {249, 57216}, {297, 9141}, {394, 36823}, {468, 524}, {648, 892}, {1499, 32729}, {1560, 11064}, {2799, 3233}, {3580, 10552}, {4235, 5468}, {6090, 63464}, {7471, 64919}, {8115, 52132}, {8116, 52131}, {9209, 67106}, {9514, 47122}, {10553, 51405}, {10554, 32234}, {14401, 34211}, {26864, 60704}, {32661, 65306}, {34336, 39689}, {35325, 39195}, {41672, 67398}, {57202, 61199}, {57203, 61198}

X(68087) = X(i)-Ceva conjugate of X(j) for these (i,j): {648, 4235}, {18020, 468}
X(68087) = X(i)-cross conjugate of X(j) for these (i,j): {1649, 34336}, {58780, 5095}
X(68087) = X(i)-isoconjugate of X(j) for these (i,j): {656, 10630}, {661, 15398}, {810, 57539}, {895, 23894}, {897, 10097}, {923, 14977}, {3708, 34574}, {5466, 36060}, {14208, 41936}, {36142, 51258}
X(68087) = X(i)-Dao conjugate of X(j) for these (i,j): {468, 65609}, {524, 525}, {1560, 5466}, {1648, 125}, {2482, 14977}, {6593, 10097}, {23992, 51258}, {36830, 15398}, {39062, 57539}, {40596, 10630}, {48317, 64258}, {66127, 66124}
X(68087) = cevapoint of X(i) and X(j) for these (i,j): {690, 44915}, {1649, 39689}, {5095, 58780}
X(68087) = crosspoint of X(i) and X(j) for these (i,j): {468, 14052}, {648, 4235}
X(68087) = crosssum of X(i) and X(j) for these (i,j): {647, 10097}, {895, 14060}
X(68087) = trilinear pole of line {2482, 5095}
X(68087) = barycentric product X(i)*X(j) for these {i,j}: {99, 5095}, {107, 65747}, {110, 34336}, {112, 36792}, {162, 24038}, {250, 52629}, {468, 5468}, {524, 4235}, {648, 2482}, {811, 42081}, {935, 62661}, {1366, 36797}, {1649, 18020}, {1783, 16733}, {2418, 15471}, {3266, 61207}, {4232, 66963}, {4590, 58780}, {5467, 44146}, {6331, 39689}, {7664, 60503}, {8030, 65350}, {23992, 55270}
X(68087) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 15398}, {112, 10630}, {187, 10097}, {250, 34574}, {468, 5466}, {524, 14977}, {648, 57539}, {690, 51258}, {1366, 17094}, {1560, 65609}, {1649, 125}, {2482, 525}, {4235, 671}, {5095, 523}, {5467, 895}, {5468, 30786}, {5642, 66124}, {7067, 52355}, {8030, 14417}, {14273, 64258}, {15471, 2408}, {16733, 15413}, {23106, 45807}, {24038, 14208}, {34336, 850}, {36792, 3267}, {39689, 647}, {42081, 656}, {44102, 9178}, {44146, 52632}, {47443, 34539}, {52068, 4064}, {52629, 339}, {54274, 20975}, {55270, 57552}, {58780, 115}, {60503, 10415}, {61206, 41936}, {61207, 111}, {65747, 3265}
X(68087) = {X(468),X(5095)}-harmonic conjugate of X(52467)

X(68088) = X(99)X(249)∩X(110)X(2501)

X(68088) lies on the Moses HK-parabola and these lines: {99, 249}, {110, 2501}, {113, 38970}, {114, 230}, {476, 59116}, {1625, 57204}, {2420, 58780}, {3580, 10552}, {6792, 64177}, {14984, 65517}, {32320, 61199}, {35325, 58760}, {58812, 61206}

X(68088) = X(i)-Ceva conjugate of X(j) for these (i,j): {648, 4226}, {60504, 56389}
X(68088) = X(i)-isoconjugate of X(j) for these (i,j): {810, 57553}, {36051, 60338}
X(68088) = X(i)-Dao conjugate of X(j) for these (i,j): {114, 60338}, {3564, 525}, {35067, 62645}, {39062, 57553}, {51610, 115}
X(68088) = crosspoint of X(648) and X(4226)
X(68088) = crosssum of X(647) and X(35364)
X(68088) = crossdifference of every pair of points on line {35364, 44114}
X(68088) = barycentric product X(i)*X(j) for these {i,j}: {110, 2974}, {648, 35067}, {3564, 4226}, {51481, 56389}, {60504, 62590}
X(68088) = barycentric quotient X(i)/X(j) for these {i,j}: {230, 60338}, {648, 57553}, {2974, 850}, {3564, 62645}, {4226, 35142}, {35067, 525}, {52144, 35364}, {56389, 2987}, {61213, 3563}

X(68089) = X(2)X(2501)∩X(4)X(525)

X(68089) = polar conjugate of X(41173)
X(68089) = polar conjugate of the isotomic conjugate of X(62555)
X(68089) = polar conjugate of the isogonal conjugate of X(41167)
X(68089) = X(i)-Ceva conjugate of X(j) for these (i,j): {264, 868}, {648, 297}, {36426, 35088}
X(68089) = X(i)-cross conjugate of X(j) for these (i,j): {35088, 36426}, {41167, 62555}, {59805, 2967}
X(68089) = X(i)-isoconjugate of X(j) for these (i,j): {48, 41173}, {163, 47388}, {248, 36084}, {293, 2715}, {810, 57562}, {1910, 43754}, {4575, 41932}, {4592, 67167}, {14600, 36036}, {17974, 36104}
X(68089) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 47388}, {132, 2715}, {136, 41932}, {232, 60506}, {511, 32661}, {868, 65726}, {1249, 41173}, {2679, 14600}, {2799, 525}, {5139, 67167}, {5976, 17932}, {11672, 43754}, {35088, 287}, {38970, 98}, {38987, 248}, {39000, 17974}, {39039, 36084}, {39062, 57562}, {41172, 3}, {46413, 41175}, {55267, 879}, {57294, 14585}, {61505, 15407}, {62431, 41009}, {62595, 2966}
X(68089) = crosspoint of X(i) and X(j) for these (i,j): {297, 648}, {4230, 56307}
X(68089) = crosssum of X(i) and X(j) for these (i,j): {248, 647}, {879, 1899}
X(68089) = trilinear pole of line {35088, 66939}
X(68089) = crossdifference of every pair of points on line {248, 8779}
X(68089) = barycentric product X(i)*X(j) for these {i,j}: {4, 62555}, {99, 66939}, {264, 41167}, {297, 2799}, {325, 16230}, {525, 36426}, {648, 35088}, {850, 2967}, {868, 877}, {2501, 32458}, {2970, 15631}, {3267, 51334}, {3569, 44132}, {4230, 62431}, {4240, 65974}, {6331, 59805}, {6333, 6530}, {14618, 36790}, {18022, 58262}, {34765, 54380}, {46052, 60179}, {65973, 67406}
X(68089) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 41173}, {132, 60506}, {232, 2715}, {240, 36084}, {297, 2966}, {325, 17932}, {511, 43754}, {523, 47388}, {648, 57562}, {684, 17974}, {868, 879}, {877, 57991}, {2489, 67167}, {2491, 14600}, {2501, 41932}, {2799, 287}, {2967, 110}, {3569, 248}, {4230, 57742}, {6333, 6394}, {6530, 685}, {11672, 32661}, {14618, 34536}, {16230, 98}, {17994, 1976}, {23290, 60594}, {23996, 4575}, {32458, 4563}, {34854, 32696}, {35088, 525}, {36426, 648}, {36790, 4558}, {40703, 36036}, {41167, 3}, {44114, 878}, {44132, 43187}, {51334, 112}, {52492, 53691}, {54380, 34761}, {55267, 65726}, {55275, 51963}, {58262, 184}, {59805, 647}, {62555, 69}, {65754, 35912}, {65974, 34767}, {66939, 523}, {67070, 15391}, {67173, 66879}, {67406, 65776}

X(68090) = X(1)X(34912)∩X(40)X(176)

See Benjamin Lee Warren and César Lozada, euclid 8208.

X(68090) lies on these lines: {1, 34912}, {40, 176}, {2951, 67180}, {10578, 63904}

X(68091) = X(1)X(34911)∩X(40)X(175)

See Benjamin Lee Warren and César Lozada, euclid 8208.

X(68091) lies on these lines: {1, 34911}, {40, 175}, {2951, 64623}, {10578, 63904}

X(68092) = X(1)X(84)∩X(4)X(1336)

See Benjamin Lee Warren and César Lozada, euclid 8208.

X(68092) lies on these lines: {1, 84}, {4, 1336}, {40, 30557}, {198, 6213}, {946, 13389}, {962, 55397}, {1158, 13388}, {1486, 8234}, {2262, 6212}, {3084, 63985}, {5405, 63989}, {7090, 38015}, {9799, 30334}, {12514, 61094}, {13390, 63962}, {31574, 37426}, {37560, 65083}

X(68092) = reflection of X(63380) in X(1498)
X(68092) = pole of the line {56, 58896} with respect to the Feuerbach circumhyperbola
X(68092) = (X(1), X(12705))-harmonic conjugate of X(63380)

X(68093) = (name pending)

As a point on the Euler line, X(68093) has Shinagawa coefficients: {1/3 (-(267/2) e (e + f)^2 + 75 (e + f)^3 + 1092 (e + f) R^4 - 480 R^6), -((e + f) (4 e - 5 (e + f)) (2 e - 3 (e + f)))}

See David Nguyen, euclid 8214.

X(68093) lies on this line: {2, 3}

X(68094) = (name pending)

As a point on the Euler line, X(68094) has Shinagawa coefficients: {-((-(e/4) - f) (10 e - 11 (e + f)) ((9 e)/2 - 5 (e + f))), 365/4 e (e + f)^2 - 45 (e + f)^3 - 934 (e + f) R^4 + 720 R^6}

See David Nguyen, euclid 8214.

X(68094) lies on this line: {2, 3}

X(68095) = (name pending)

As a point on the Euler line, X(68095) has Shinagawa coefficients: {-13 e (e + f) + 5 (e + f)^2 + 120 R^4, (e + f) (-8 e + 9 (e + f))}

See David Nguyen, euclid 8214.

X(68095) lies on this line: {2, 3}

X(68096) = (name pending)

As a point on the Euler line, X(68096) has Shinagawa coefficients: {1/3 (-429 e (e + f)^2 + 141 (e + f)^3 + 6744 (e + f) R^4 - 8640 R^6), (e + f) (-44 e (e + f) + 27 (e + f)^2 + 288 R^4)}

See David Nguyen, euclid 8214.

X(68096) lies on this line: {2, 3}

X(68097) = (name pending)

As a point on the Euler line, X(68097) has Shinagawa coefficients: {-((-(e/4) - f) (11/2 e (e + f)^2 + 5 (e + f)^3 - 320 (e + f) R^4 + 640 R^6)), (2 e - 3 (e + f)) (1/4 e (e + f)^2 + (e + f)^3 - 38 (e + f) R^4 + 80 R^6)}

See David Nguyen, euclid 8221.

X(68097) lies on this line: {2, 3}

X(68098) = (name pending)

As a point on the Euler line, X(68098) has Shinagawa coefficients: {-(697/2) e (e + f)^3 + 115 (e + f)^4 + 6268 (e + f)^2 R^4 - 12352 (e + f) R^6 + 8960 R^8, -((2 e - 3 (e + f)) (45 e (e + f)^2 - 23 (e + f)^3 - 440 (e + f) R^4 + 320 R^6))}

See David Nguyen, euclid 8221.

X(68098) lies on this line: {2, 3}

X(68099) = (name pending)

As a point on the Euler line, X(68099) has Shinagawa coefficients: {1/3 (-(819/4) e (e + f)^3 + 66 (e + f)^4 + 3516 (e + f)^2 R^4 - 5928 (e + f) R^6 + 2880 R^8), (e + f) (227/4 e (e + f)^2 - 18 (e + f)^3 - 912 (e + f) R^4 + 1184 R^6)}

See David Nguyen, euclid 8221.

X(68099) lies on this line: {2, 3}

X(68100) = (name pending)

As a point on the Euler line, X(68100) has Shinagawa coefficients: {-(253/4) e (e + f)^4 + 22 (e + f)^5 + 774 (e + f)^3 R^4 + 880 (e + f)^2 R^6 - 7760 (e + f) R^8 + 9600 R^10, -((e + f) (-(263/4) e (e + f)^3 + 18 (e + f)^4 + 1382 (e + f)^2 R^4 - 3120 (e + f) R^6 + 2560 R^8))}

See David Nguyen, euclid 8221.

X(68100) lies on this line: {2, 3}

   Points on the Moses X(4)X(8)-parabola, X(68101) - X(68125)  

Contributed by Clark Kimberling, based on notes and data from Peter Moses, March 29, 2025.

The Moses X(4)X(8)-parabola is introduced here as the parabola having focus X(100) and directrix the line HK = X(4)X(8). The Moses X(4)X(8)-parabola passes through X(i) for these i: 513, 693, 4036, 4397, 14434, 15632, 25142, 27855, 50487, 62430, and 68101--68125.

X(68101) = X(8)X(513)∩X(10)X(522)

X(68101) lies on the Moses X(4)X(8)-parabola and these lines: {8, 513}, {10, 522}, {75, 693}, {100, 65639}, {499, 48230}, {514, 4793}, {519, 46781}, {521, 64744}, {523, 764}, {536, 62552}, {650, 17281}, {900, 1145}, {1000, 3900}, {1227, 62430}, {1643, 17369}, {1647, 23757}, {1734, 4443}, {2517, 28205}, {2968, 42769}, {3632, 14812}, {3667, 11362}, {3679, 23838}, {3728, 64868}, {3887, 36923}, {4010, 14434}, {4086, 28221}, {4358, 34764}, {4391, 25030}, {4397, 4926}, {4404, 28217}, {4448, 17780}, {4543, 39771}, {4714, 66285}, {6161, 49998}, {7046, 43933}, {9001, 49688}, {14507, 52627}, {15632, 56881}, {16732, 57035}, {17278, 31250}, {17279, 31287}, {17280, 27115}, {17342, 31209}, {22072, 42312}, {22271, 50487}, {22837, 48283}, {23814, 28161}, {25025, 28601}, {25036, 25038}, {26078, 59837}, {26364, 48181}, {28183, 30591}, {29188, 57052}, {30144, 48302}, {32941, 48285}, {35175, 36240}, {35353, 47975}, {38462, 53157}, {42757, 56893}, {43082, 52344}, {63217, 65867}

X(68101) = midpoint of X(3632) and X(14812)
X(68101) = reflection of X(i) in X(j) for these {i,j}: {6161, 62323}, {24457, 10}
X(68101) = isotomic conjugate of X(4618)
X(68101) = isotomic conjugate of the isogonal conjugate of X(3251)
X(68101) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {8046, 150}, {41529, 66862}, {53656, 21282}
X(68101) = X(i)-Ceva conjugate of X(j) for these (i,j): {668, 4358}, {693, 3762}, {36791, 35092}
X(68101) = X(i)-cross conjugate of X(j) for these (i,j): {4542, 4370}, {35092, 36791}
X(68101) = X(i)-isoconjugate of X(j) for these (i,j): {6, 4638}, {31, 4618}, {88, 32665}, {101, 2226}, {106, 901}, {109, 1318}, {163, 30575}, {190, 41935}, {679, 692}, {902, 39414}, {903, 32719}, {1919, 57564}, {3257, 9456}, {6551, 43922}, {9268, 23345}, {23344, 59150}, {31227, 32645}, {32659, 65336}, {32739, 54974}, {41461, 53634}
X(68101) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 4618}, {9, 4638}, {11, 1318}, {115, 30575}, {214, 901}, {519, 100}, {900, 513}, {1015, 2226}, {1086, 679}, {1647, 1}, {2087, 52206}, {4370, 3257}, {6544, 1022}, {9296, 57564}, {35092, 88}, {35587, 52478}, {36912, 52925}, {38979, 106}, {40594, 39414}, {40619, 54974}, {51402, 1320}, {53985, 36125}, {55053, 41935}, {55055, 9456}, {62571, 4555}
X(68101) = cevapoint of X(4543) and X(6544)
X(68101) = crosspoint of X(i) and X(j) for these (i,j): {75, 24004}, {668, 4358}, {693, 3762}, {17780, 36944}
X(68101) = crosssum of X(i) and X(j) for these (i,j): {667, 9456}, {692, 32665}, {14260, 23345}
X(68101) = crossdifference of every pair of points on line {2251, 7113}
X(68101) = barycentric product X(i)*X(j) for these {i,j}: {1, 52627}, {44, 65867}, {75, 6544}, {76, 3251}, {85, 4543}, {312, 39771}, {513, 36791}, {514, 4738}, {519, 3762}, {523, 16729}, {646, 14027}, {668, 35092}, {678, 3261}, {693, 4370}, {900, 4358}, {905, 65585}, {1017, 40495}, {1022, 58254}, {1111, 53582}, {1317, 4391}, {1635, 3264}, {1647, 24004}, {1978, 42084}, {3911, 4768}, {4120, 30939}, {4152, 24002}, {4542, 4554}, {4723, 30725}, {4858, 66979}, {4908, 63217}, {7035, 14442}, {15413, 42070}, {17924, 65742}, {20568, 33922}, {21821, 52619}, {46109, 53532}
X(68101) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4638}, {2, 4618}, {44, 901}, {88, 39414}, {513, 2226}, {514, 679}, {519, 3257}, {523, 30575}, {650, 1318}, {667, 41935}, {668, 57564}, {678, 101}, {693, 54974}, {900, 88}, {902, 32665}, {1017, 692}, {1022, 59150}, {1023, 9268}, {1317, 651}, {1635, 106}, {1639, 1320}, {1647, 1022}, {1960, 9456}, {2087, 23345}, {2251, 32719}, {3251, 6}, {3261, 57929}, {3689, 5548}, {3762, 903}, {4120, 4674}, {4152, 644}, {4358, 4555}, {4370, 100}, {4530, 23838}, {4542, 650}, {4543, 9}, {4723, 4582}, {4738, 190}, {4768, 4997}, {4791, 36594}, {4895, 2316}, {4908, 52925}, {6544, 1}, {8028, 1023}, {14027, 3669}, {14442, 244}, {16704, 4622}, {16729, 99}, {17780, 5376}, {21821, 4557}, {22086, 36058}, {22371, 906}, {23757, 52031}, {24004, 62536}, {30583, 52900}, {30725, 56049}, {30939, 4615}, {33920, 51908}, {33922, 44}, {34764, 64459}, {35092, 513}, {36791, 668}, {36924, 65235}, {38462, 65336}, {39771, 57}, {42070, 1783}, {42084, 649}, {46050, 2087}, {52627, 75}, {52680, 4591}, {53532, 1797}, {53535, 40215}, {53582, 765}, {58254, 24004}, {61047, 1415}, {61062, 57181}, {63217, 40833}, {65585, 6335}, {65742, 1332}, {65867, 20568}, {66962, 5382}, {66979, 4564}

X(68102) = X(8)X(693)∩X(513)X(3681)

X(68102) lies on the Moses X(4)X(8)-parabola and these lines: {8, 693}, {100, 65646}, {513, 3681}, {518, 63742}, {3126, 62236}, {3887, 30565}, {3900, 55954}, {4036, 4651}, {35348, 67097}, {42455, 62725}, {47787, 50095}

X(68102) = X(668)-Ceva conjugate of X(17264)
X(68102) = X(1308)-isoconjugate of X(67146)
X(68102) = X(i)-Dao conjugate of X(j) for these (i,j): {3887, 513}, {35125, 34578}
X(68102) = crosspoint of X(668) and X(17264)
X(68102) = barycentric product X(i)*X(j) for these {i,j}: {646, 47007}, {668, 35125}, {3887, 17264}, {3935, 30565}
X(68102) = barycentric quotient X(i)/X(j) for these {i,j}: {3887, 34578}, {3935, 37143}, {5526, 1308}, {17264, 35171}, {22108, 67146}, {35125, 513}, {47007, 3669}

X(68103) = X(8)X(521)∩X(513)X(3869)

X(68103) lies on the Moses X(4)X(8)-parabola and these lines: {8, 521}, {69, 693}, {100, 35011}, {513, 3869}, {3738, 3904}, {3900, 56094}, {4036, 17751}, {4585, 53535}, {6003, 66995}, {7253, 42455}, {15632, 17780}, {21189, 30144}, {22301, 50487}, {26641, 63088}, {42757, 62826}

X(68103) = X(47645)-anticomplementary conjugate of X(149)
X(68103) = X(668)-Ceva conjugate of X(32851)
X(68103) = X(i)-isoconjugate of X(j) for these (i,j): {109, 63750}, {649, 23592}, {1411, 2222}, {1415, 34535}, {1919, 57568}, {2006, 32675}, {43924, 46649}
X(68103) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 63750}, {1146, 34535}, {3738, 513}, {5375, 23592}, {6149, 109}, {9296, 57568}, {35128, 2006}, {35204, 2222}, {38984, 1411}, {40624, 57645}, {57434, 80}
X(68103) = crosspoint of X(668) and X(32851)
X(68103) = barycentric product X(i)*X(j) for these {i,j}: {312, 66968}, {646, 3025}, {668, 35128}, {3596, 57174}, {3738, 32851}, {3904, 4511}, {4391, 4996}, {20924, 53285}, {34544, 35519}, {53045, 56757}
X(68103) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 23592}, {215, 1415}, {522, 34535}, {644, 46649}, {650, 63750}, {654, 1411}, {668, 57568}, {2323, 2222}, {2361, 32675}, {3025, 3669}, {3738, 2006}, {3904, 18815}, {4391, 57645}, {4511, 655}, {4736, 4605}, {4996, 651}, {5081, 65329}, {32851, 35174}, {34544, 109}, {35128, 513}, {53285, 2161}, {56757, 53811}, {57174, 56}, {66968, 57}

X(68104) = X(100)X(476)∩X(513)X(53349)

X(68104) lies on the Moses X(4)X(8)-parabola and these lines: {30, 14206}, {100, 476}, {321, 54527}, {513, 53349}, {668, 16077}, {693, 17136}, {4240, 24001}, {4397, 4427}, {5080, 15632}, {6062, 36789}, {9141, 42703}, {42721, 66084}

X(68104) = X(668)-Ceva conjugate of X(42716)
X(68104) = X(i)-isoconjugate of X(j) for these (i,j): {514, 40353}, {649, 40384}, {1919, 31621}
X(68104) = X(i)-Dao conjugate of X(j) for these (i,j): {30, 513}, {1650, 18210}, {5375, 40384}, {9296, 31621}
X(68104) = crosspoint of X(668) and X(42716)
X(68104) = trilinear pole of line {1099, 3163}
X(68104) = barycentric product X(i)*X(j) for these {i,j}: {30, 42716}, {100, 36789}, {190, 1099}, {321, 3233}, {646, 1354}, {668, 3163}, {1332, 34334}, {1978, 42074}, {4554, 6062}, {4567, 58263}, {4601, 58346}, {5379, 52624}, {6335, 16163}, {6386, 9408}, {42703, 65777}
X(68104) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 40384}, {668, 31621}, {692, 40353}, {1099, 514}, {1354, 3669}, {3081, 14399}, {3163, 513}, {3233, 81}, {5379, 34568}, {6062, 650}, {9408, 667}, {14401, 18210}, {16163, 905}, {16240, 6591}, {34334, 17924}, {36789, 693}, {42074, 649}, {42716, 1494}, {58263, 16732}, {58344, 3121}, {58346, 3125}, {58347, 14419}

X(68105) = X(100)X(805)∩X(513)X(65205)

X(68105) lies on the Moses X(4)X(8)-parabola and these lines: {100, 805}, {511, 1959}, {513, 65205}, {668, 53196}, {693, 3909}, {1332, 65305}, {4036, 4553}, {4397, 53338}, {7062, 36790}, {14966, 23997}, {15632, 56878}, {42717, 63741}

X(68105) = X(i)-Ceva conjugate of X(j) for these (i,j): {668, 42717}, {4567, 42702}
X(68105) = X(i)-isoconjugate of X(j) for these (i,j): {514, 41932}, {649, 34536}, {1919, 57541}, {3120, 41173}, {3261, 67167}, {7649, 47388}, {21131, 57562}
X(68105) = X(i)-Dao conjugate of X(j) for these (i,j): {511, 513}, {5375, 34536}, {9296, 57541}, {41172, 16732}
X(68105) = crosspoint of X(668) and X(42717)
X(68105) = trilinear pole of line {11672, 23996}
X(68105) = barycentric product X(i)*X(j) for these {i,j}: {37, 15631}, {100, 36790}, {190, 23996}, {511, 42717}, {646, 1355}, {668, 11672}, {692, 32458}, {877, 42702}, {1332, 2967}, {1978, 42075}, {2396, 5360}, {3952, 16725}, {4554, 7062}, {4567, 41167}, {4601, 58262}, {6335, 65748}, {6386, 9419}, {14966, 42703}
X(68105) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 34536}, {668, 57541}, {692, 41932}, {906, 47388}, {1355, 3669}, {2967, 17924}, {5360, 2395}, {7062, 650}, {9419, 667}, {11672, 513}, {15631, 274}, {16725, 7192}, {23996, 514}, {32458, 40495}, {36425, 1980}, {36790, 693}, {41167, 16732}, {42075, 649}, {42702, 879}, {42717, 290}, {46888, 14296}, {58262, 3125}, {65748, 905}

X(68106) = X(100)X(4394)∩X(190)X(513)

X(68106) lies on the Moses X(4)X(8)-parabola and these lines: {8, 56850}, {100, 4394}, {190, 513}, {518, 3717}, {644, 36086}, {668, 36803}, {677, 765}, {692, 46973}, {693, 3952}, {883, 62430}, {1016, 3900}, {1026, 2284}, {3006, 15632}, {3264, 64223}, {3309, 32094}, {3887, 53582}, {4033, 4397}, {6633, 14077}, {14434, 61176}, {14839, 40538}, {16594, 26015}, {17780, 68102}, {23354, 27855}, {24482, 62706}, {25142, 61166}, {29824, 49714}, {42720, 63743}, {49693, 59690}, {50487, 61172}, {52778, 58989}

X(68106) = reflection of X(43921) in X(40538)
X(68106) = X(i)-Ceva conjugate of X(j) for these (i,j): {668, 42720}, {1016, 3693}
X(68106) = X(3126)-cross conjugate of X(4712)
X(68106) = X(i)-isoconjugate of X(j) for these (i,j): {105, 1027}, {513, 51838}, {514, 41934}, {649, 6185}, {673, 43929}, {884, 56783}, {885, 1416}, {1024, 1462}, {1438, 62635}, {1919, 57537}, {2195, 43930}, {21143, 57536}, {36086, 43921}, {43924, 62715}
X(68106) = X(i)-Dao conjugate of X(j) for these (i,j): {518, 513}, {5375, 6185}, {6184, 62635}, {9296, 57537}, {17435, 1086}, {38989, 43921}, {39026, 51838}, {39046, 1027}, {39063, 43930}, {40609, 885}
X(68106) = cevapoint of X(3126) and X(4712)
X(68106) = crosspoint of X(668) and X(42720)
X(68106) = crosssum of X(i) and X(j) for these (i,j): {667, 43929}, {764, 43921}
X(68106) = trilinear pole of line {4712, 6184}
X(68106) = crossdifference of every pair of points on line {43921, 43929}
X(68106) = X(14839)-line conjugate of X(43921)
X(68106) = barycentric product X(i)*X(j) for these {i,j}: {100, 4437}, {190, 4712}, {312, 66978}, {518, 42720}, {646, 1362}, {666, 23102}, {668, 6184}, {765, 53583}, {883, 3693}, {1016, 3126}, {1025, 3717}, {1026, 3912}, {1252, 62430}, {1332, 34337}, {1978, 42079}, {2284, 3263}, {3952, 16728}, {6335, 65744}, {6386, 39686}, {17060, 52778}, {20336, 68086}, {20683, 55260}, {23612, 36803}, {35094, 57731}, {35505, 57950}
X(68106) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 6185}, {101, 51838}, {241, 43930}, {518, 62635}, {644, 62715}, {665, 43921}, {668, 57537}, {672, 1027}, {692, 41934}, {883, 34018}, {1025, 56783}, {1026, 673}, {1362, 3669}, {2223, 43929}, {2283, 1462}, {2284, 105}, {2340, 1024}, {3126, 1086}, {3693, 885}, {4437, 693}, {4712, 514}, {6184, 513}, {16728, 7192}, {20683, 55261}, {20776, 22383}, {23102, 918}, {23612, 665}, {34337, 17924}, {35505, 764}, {39686, 667}, {42071, 6591}, {42079, 649}, {42720, 2481}, {53583, 1111}, {54325, 1438}, {57731, 57536}, {61055, 57181}, {62430, 23989}, {65744, 905}, {66978, 57}, {68086, 28}

X(68107) = X(100)X(4076)∩X(513)X(3952)

X(68107) lies on the Moses X(4)X(8)-parabola and these lines: {100, 4076}, {350, 62296}, {513, 3952}, {519, 3992}, {537, 43922}, {650, 61402}, {667, 53685}, {668, 693}, {899, 17793}, {1227, 3263}, {3699, 4397}, {3701, 49998}, {4036, 61174}, {4103, 47874}, {4152, 36791}, {4448, 17780}, {4562, 48548}, {4568, 48557}, {9059, 59096}, {14434, 23354}, {25142, 25312}, {27855, 41314}, {29824, 49714}, {31209, 61406}, {32927, 49997}, {48107, 54099}

X(68107) = X(i)-Ceva conjugate of X(j) for these (i,j): {668, 24004}, {7035, 4358}
X(68107) = X(i)-cross conjugate of X(j) for these (i,j): {3251, 4370}, {6544, 16729}, {68101, 4738}
X(68107) = X(i)-isoconjugate of X(j) for these (i,j): {106, 23345}, {514, 41935}, {649, 2226}, {667, 679}, {901, 43922}, {1015, 4638}, {1022, 9456}, {1318, 43924}, {1417, 23838}, {1919, 54974}, {1960, 59150}, {1980, 57929}, {3248, 4618}, {3249, 57564}, {6549, 32719}, {30575, 57129}
X(68107) = X(i)-Dao conjugate of X(j) for these (i,j): {214, 23345}, {519, 513}, {900, 764}, {1647, 244}, {4370, 1022}, {5375, 2226}, {6631, 679}, {9296, 54974}, {17780, 3315}, {36912, 23352}, {38979, 43922}, {52871, 23838}, {52872, 55244}, {62571, 6548}
X(68107) = cevapoint of X(i) and X(j) for these (i,j): {3251, 4370}, {4738, 68101}
X(68107) = crosspoint of X(668) and X(24004)
X(68107) = trilinear pole of line {4370, 4738}
X(68107) = barycentric product X(i)*X(j) for these {i,j}: {75, 53582}, {100, 36791}, {190, 4738}, {312, 66979}, {519, 24004}, {646, 1317}, {668, 4370}, {670, 21821}, {678, 1978}, {765, 52627}, {1016, 68101}, {1017, 6386}, {1023, 3264}, {1332, 65585}, {2415, 4487}, {3251, 31625}, {3257, 58254}, {3943, 55243}, {3952, 16729}, {4152, 4554}, {4169, 30939}, {4358, 17780}, {4543, 67038}, {4723, 62669}, {6335, 65742}, {6544, 7035}, {18743, 66962}, {21805, 55262}, {35092, 57950}
X(68107) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 23345}, {100, 2226}, {190, 679}, {519, 1022}, {644, 1318}, {668, 54974}, {678, 649}, {692, 41935}, {765, 4638}, {1016, 4618}, {1017, 667}, {1023, 106}, {1317, 3669}, {1635, 43922}, {1978, 57929}, {2325, 23838}, {3251, 1015}, {3257, 59150}, {3762, 6549}, {3943, 55244}, {3952, 30575}, {3992, 4049}, {4152, 650}, {4169, 4674}, {4358, 6548}, {4370, 513}, {4487, 2403}, {4543, 2170}, {4723, 60480}, {4738, 514}, {4768, 60578}, {4908, 23352}, {5376, 39414}, {6544, 244}, {8028, 1635}, {16729, 7192}, {17780, 88}, {21805, 55263}, {21821, 512}, {22371, 22383}, {23344, 9456}, {24004, 903}, {30731, 1320}, {33922, 2087}, {35092, 764}, {36791, 693}, {39771, 53538}, {40522, 60809}, {42070, 6591}, {42084, 21143}, {52627, 1111}, {53582, 1}, {57950, 57564}, {58254, 3762}, {61047, 57181}, {61210, 1417}, {62669, 56049}, {65585, 17924}, {65742, 905}, {66962, 8056}, {66979, 57}, {68101, 1086}

X(68108) = X(100)X(6081)∩X(513)X(20294)

X(68108) lies on the Moses X(4)X(8)-parabola and these lines: {8, 30201}, {78, 57101}, {100, 6081}, {345, 63744}, {513, 20294}, {520, 3265}, {521, 6332}, {677, 765}, {693, 20293}, {1259, 58253}, {1459, 24562}, {2968, 34949}, {3699, 15632}, {3737, 58333}, {3869, 34414}, {3900, 4397}, {4163, 35057}, {7250, 65409}, {14298, 57197}, {14434, 62584}, {15313, 50333}, {23090, 57055}, {35518, 63245}

X(68108) = reflection of X(7250) in X(65409)
X(68108) = isotomic conjugate of the isogonal conjugate of X(58340)
X(68108) = isotomic conjugate of the polar conjugate of X(57055)
X(68108) = isogonal conjugate of the polar conjugate of X(15416)
X(68108) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1034, 33650}, {3345, 149}, {7037, 39351}, {7152, 4440}, {8064, 9965}, {8806, 3448}, {13138, 34162}, {41514, 150}, {47850, 37781}, {56596, 21293}, {57643, 34188}, {58995, 12649}
X(68108) = X(i)-Ceva conjugate of X(j) for these (i,j): {668, 345}, {1264, 23983}, {1332, 3692}, {4561, 3998}, {4571, 1259}, {15416, 57055}
X(68108) = X(58340)-cross conjugate of X(57055)
X(68108) = X(i)-isoconjugate of X(j) for these (i,j): {19, 32714}, {25, 36118}, {34, 108}, {56, 36127}, {107, 1042}, {109, 1118}, {162, 1426}, {207, 58995}, {278, 32674}, {393, 1461}, {513, 24033}, {514, 23985}, {604, 54240}, {608, 653}, {649, 23984}, {658, 2207}, {664, 7337}, {667, 24032}, {934, 1096}, {1020, 5317}, {1119, 8750}, {1254, 52920}, {1395, 18026}, {1397, 52938}, {1398, 1897}, {1410, 36126}, {1427, 24019}, {1435, 1783}, {1474, 52607}, {1842, 59090}, {1851, 59128}, {1857, 6614}, {1875, 36110}, {1880, 65232}, {1919, 57538}, {1973, 13149}, {3209, 65330}, {3668, 32713}, {3924, 52775}, {4605, 36420}, {4626, 6059}, {6525, 36079}, {6529, 52373}, {6591, 7128}, {7012, 43923}, {7103, 32691}, {7143, 52921}, {8747, 53321}, {36044, 51399}, {36417, 46406}
X(68108) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 36127}, {6, 32714}, {11, 1118}, {125, 1426}, {521, 513}, {656, 7649}, {2968, 158}, {3161, 54240}, {3239, 17924}, {5375, 23984}, {6337, 13149}, {6338, 4569}, {6503, 934}, {6505, 36118}, {6631, 24032}, {7358, 4}, {9296, 57538}, {11517, 108}, {14714, 1096}, {17421, 7103}, {24031, 6245}, {26932, 1119}, {34467, 1398}, {35071, 1427}, {35072, 278}, {35508, 393}, {35580, 51399}, {38966, 6524}, {38983, 34}, {38985, 1042}, {39004, 1875}, {39006, 1435}, {39025, 7337}, {39026, 24033}, {40618, 1847}, {40626, 273}, {46093, 1410}, {51574, 52607}, {55063, 196}, {55068, 8747}, {57055, 59935}, {62573, 1446}, {62584, 18026}, {62585, 52938}, {62647, 653}
X(68108) = crosspoint of X(i) and X(j) for these (i,j): {326, 1332}, {345, 668}, {1265, 4571}
X(68108) = crosssum of X(i) and X(j) for these (i,j): {608, 667}, {1096, 6591}, {1398, 43923}
X(68108) = trilinear pole of line {24031, 35072}
X(68108) = crossdifference of every pair of points on line {608, 1426}
X(68108) = barycentric product X(i)*X(j) for these {i,j}: {3, 15416}, {9, 52616}, {69, 57055}, {72, 15411}, {75, 57057}, {76, 58340}, {78, 6332}, {100, 23983}, {190, 24031}, {200, 30805}, {219, 35518}, {255, 52622}, {271, 57245}, {304, 57108}, {305, 65102}, {306, 57081}, {312, 57241}, {326, 3239}, {332, 8611}, {341, 4091}, {345, 521}, {346, 4131}, {394, 4397}, {522, 3719}, {525, 1792}, {646, 1364}, {650, 1264}, {652, 3718}, {668, 35072}, {905, 1265}, {1021, 52396}, {1043, 24018}, {1231, 58338}, {1259, 4391}, {1260, 15413}, {1332, 2968}, {1459, 52406}, {1812, 52355}, {1946, 57919}, {1978, 2638}, {2287, 3265}, {2289, 35519}, {2327, 14208}, {3596, 36054}, {3692, 4025}, {3900, 3926}, {3952, 16731}, {3998, 7253}, {4086, 6514}, {4130, 7055}, {4143, 4183}, {4163, 7183}, {4176, 65103}, {4561, 34591}, {4571, 26932}, {4587, 17880}, {4612, 7068}, {6386, 39687}, {7058, 57109}, {7259, 17216}, {10397, 57783}, {19611, 57045}, {20336, 23090}, {23224, 59761}, {40071, 57134}, {44189, 57101}, {46102, 58253}, {52565, 58329}, {55112, 61040}
X(68108) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 32714}, {8, 54240}, {9, 36127}, {63, 36118}, {69, 13149}, {72, 52607}, {78, 653}, {100, 23984}, {101, 24033}, {190, 24032}, {212, 32674}, {219, 108}, {255, 1461}, {271, 65330}, {283, 65232}, {312, 52938}, {326, 658}, {345, 18026}, {394, 934}, {520, 1427}, {521, 278}, {647, 1426}, {650, 1118}, {652, 34}, {657, 1096}, {668, 57538}, {692, 23985}, {822, 1042}, {905, 1119}, {1021, 8747}, {1043, 823}, {1098, 52919}, {1259, 651}, {1260, 1783}, {1264, 4554}, {1265, 6335}, {1331, 7128}, {1332, 55346}, {1364, 3669}, {1459, 1435}, {1792, 648}, {1802, 8750}, {1804, 4617}, {1809, 65331}, {1946, 608}, {2287, 107}, {2289, 109}, {2322, 36126}, {2327, 162}, {2328, 24019}, {2522, 7103}, {2638, 649}, {2968, 17924}, {3063, 7337}, {3239, 158}, {3265, 1446}, {3270, 6591}, {3682, 1020}, {3692, 1897}, {3694, 61178}, {3710, 65207}, {3718, 46404}, {3719, 664}, {3900, 393}, {3926, 4569}, {3964, 65296}, {3990, 53321}, {3998, 4566}, {4025, 1847}, {4091, 269}, {4130, 1857}, {4131, 279}, {4183, 6529}, {4397, 2052}, {4571, 46102}, {4587, 7012}, {6056, 1415}, {6332, 273}, {6514, 1414}, {7054, 52920}, {7055, 36838}, {7117, 43923}, {7125, 6614}, {7183, 4626}, {7358, 59935}, {8611, 225}, {8641, 2207}, {10397, 208}, {14418, 1877}, {15411, 286}, {15416, 264}, {16731, 7192}, {21789, 5317}, {22383, 1398}, {23090, 28}, {23189, 1396}, {23224, 1407}, {23614, 7117}, {23983, 693}, {24018, 3668}, {24031, 514}, {30805, 1088}, {32320, 1410}, {33572, 30691}, {34406, 42381}, {34591, 7649}, {35072, 513}, {35518, 331}, {36054, 56}, {36197, 58757}, {39687, 667}, {51640, 62192}, {52307, 1875}, {52355, 40149}, {52387, 4605}, {52613, 1439}, {52616, 85}, {52622, 57806}, {56003, 52775}, {57045, 1895}, {57049, 47372}, {57055, 4}, {57057, 1}, {57081, 27}, {57101, 196}, {57108, 19}, {57109, 6354}, {57134, 1474}, {57180, 6059}, {57241, 57}, {57245, 342}, {58253, 26932}, {58329, 8748}, {58331, 60428}, {58335, 27376}, {58338, 1172}, {58340, 6}, {58796, 40933}, {59759, 54948}, {61040, 55110}, {61054, 57181}, {65102, 25}, {65103, 6524}, {65302, 65537}, {65575, 36419}, {65752, 65103}, {66898, 7252}
X(68108) = {X(78),X(57111)}-harmonic conjugate of X(57101)

X(68109) = X(100)X(6082)∩X(513)X(53332)

X(68109) lies on the Moses X(4)X(8)-parabola and these lines: {100, 6082}, {321, 54607}, {513, 53332}, {524, 14210}, {668, 892}, {693, 65161}, {874, 68101}, {1332, 17708}, {4553, 50487}, {4585, 62430}, {5468, 24039}, {7067, 36792}, {9141, 42703}, {16733, 52068}

X(68109) = X(i)-Ceva conjugate of X(j) for these (i,j): {668, 42721}, {4601, 42713}
X(68109) = X(i)-isoconjugate of X(j) for these (i,j): {111, 66945}, {514, 41936}, {649, 10630}, {1919, 57539}, {3122, 34574}, {32740, 62626}
X(68109) = X(i)-Dao conjugate of X(j) for these (i,j): {524, 513}, {1648, 3125}, {5375, 10630}, {9296, 57539}
X(68109) = crosspoint of X(668) and X(42721)
X(68109) = trilinear pole of line {2482, 24038}
X(68109) = barycentric product X(i)*X(j) for these {i,j}: {100, 36792}, {190, 24038}, {524, 42721}, {646, 1366}, {668, 2482}, {799, 52068}, {1332, 34336}, {1649, 4601}, {1978, 42081}, {3952, 16733}, {4062, 24039}, {4554, 7067}, {4567, 52629}, {5380, 23106}, {5468, 42713}, {6335, 65747}, {6386, 39689}, {20336, 68087}, {42724, 66963}
X(68109) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 10630}, {668, 57539}, {692, 41936}, {896, 66945}, {1332, 15398}, {1366, 3669}, {1649, 3125}, {2482, 513}, {4062, 23894}, {4567, 34574}, {5095, 6591}, {7067, 650}, {8030, 14419}, {14210, 62626}, {16702, 43926}, {16733, 7192}, {21839, 9178}, {24038, 514}, {34336, 17924}, {36792, 693}, {39689, 667}, {42081, 649}, {42713, 5466}, {42721, 671}, {52068, 661}, {52629, 16732}, {54274, 3121}, {65747, 905}, {68087, 28}

X(68110) = X(100)X(2867)∩X(513)X(16085)

X(68110) lies on the Moses X(4)X(8)-parabola and these lines: {10, 21174}, {100, 2867}, {306, 57186}, {321, 43673}, {513, 16085}, {525, 14208}, {668, 16077}, {693, 15416}, {850, 4036}, {1332, 17708}, {3265, 57109}, {3267, 63220}, {3998, 53173}, {7068, 36793}, {14434, 62614}, {20336, 34767}, {35518, 63245}, {52396, 52616}

X(68110) = isotomic conjugate of X(52920)
X(68110) = isotomic conjugate of the isogonal conjugate of X(57109)
X(68110) = X(668)-Ceva conjugate of X(20336)
X(68110) = X(i)-isoconjugate of X(j) for these (i,j): {27, 61206}, {28, 32676}, {31, 52920}, {32, 52919}, {58, 32713}, {101, 36420}, {107, 2206}, {112, 1474}, {162, 2203}, {163, 5317}, {514, 41937}, {649, 23964}, {667, 24000}, {1333, 24019}, {1395, 52914}, {1397, 52921}, {1576, 8747}, {1919, 23582}, {1980, 23999}, {2189, 32674}, {2204, 65232}, {2207, 4556}, {4091, 23975}, {4610, 36417}, {4636, 7337}, {7649, 57655}, {23224, 24022}, {32715, 52954}, {32739, 36419}, {36131, 52955}, {44162, 55229}
X(68110) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 52920}, {10, 32713}, {37, 24019}, {115, 5317}, {125, 2203}, {525, 513}, {647, 6591}, {1015, 36420}, {3265, 16757}, {4858, 8747}, {5375, 23964}, {6338, 52935}, {6376, 52919}, {6631, 24000}, {9296, 23582}, {14401, 14399}, {15526, 28}, {17434, 22383}, {23285, 17924}, {34591, 1474}, {35071, 1333}, {35072, 2189}, {38985, 2206}, {39008, 52955}, {40591, 32676}, {40603, 107}, {40619, 36419}, {40626, 270}, {51574, 112}, {55065, 1096}, {62564, 162}, {62565, 65232}, {62573, 81}, {62584, 52914}, {62585, 52921}, {62604, 55231}, {62614, 648}
X(68110) = crosspoint of X(668) and X(20336)
X(68110) = crosssum of X(667) and X(2203)
X(68110) = trilinear pole of line {15526, 17879}
X(68110) = barycentric product X(i)*X(j) for these {i,j}: {37, 52617}, {72, 3267}, {76, 57109}, {100, 36793}, {190, 17879}, {304, 4064}, {305, 55232}, {306, 14208}, {312, 66980}, {313, 24018}, {321, 3265}, {326, 52623}, {339, 1332}, {520, 27801}, {525, 20336}, {646, 1367}, {656, 40071}, {668, 15526}, {850, 3998}, {1089, 30805}, {1231, 52355}, {1577, 52396}, {1978, 2632}, {3261, 52387}, {3269, 6386}, {3682, 20948}, {3695, 15413}, {3718, 57243}, {3926, 4036}, {3990, 44173}, {4025, 52369}, {4033, 17216}, {4086, 52565}, {4131, 28654}, {4143, 41013}, {4554, 7068}, {4561, 20902}, {4601, 5489}, {5379, 23107}, {6332, 57807}, {6356, 15416}, {6358, 52616}, {14638, 52345}, {21046, 55202}, {26942, 35518}, {40364, 55230}, {40495, 52386}, {42698, 62428}, {42703, 53173}, {42706, 63220}, {56189, 60597}
X(68110) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 52920}, {10, 24019}, {37, 32713}, {71, 32676}, {72, 112}, {75, 52919}, {100, 23964}, {125, 6591}, {190, 24000}, {201, 32674}, {228, 61206}, {305, 55231}, {306, 162}, {307, 65232}, {312, 52921}, {313, 823}, {321, 107}, {326, 4556}, {339, 17924}, {345, 52914}, {513, 36420}, {520, 1333}, {521, 2189}, {523, 5317}, {525, 28}, {647, 2203}, {656, 1474}, {668, 23582}, {692, 41937}, {693, 36419}, {822, 2206}, {906, 57655}, {1264, 4612}, {1332, 250}, {1367, 3669}, {1577, 8747}, {1650, 14399}, {1978, 23999}, {2632, 649}, {2972, 22383}, {3265, 81}, {3267, 286}, {3269, 667}, {3682, 163}, {3695, 1783}, {3710, 65201}, {3719, 4636}, {3926, 52935}, {3949, 8750}, {3990, 1576}, {3998, 110}, {4024, 1096}, {4036, 393}, {4064, 19}, {4086, 8748}, {4091, 849}, {4131, 593}, {4143, 1444}, {4158, 906}, {4397, 36421}, {4466, 57200}, {4605, 24033}, {4705, 2207}, {5360, 34859}, {5379, 59153}, {5489, 3125}, {6332, 270}, {6335, 32230}, {6356, 32714}, {6358, 36127}, {7066, 1415}, {7068, 650}, {8611, 2299}, {9033, 52955}, {14208, 27}, {15416, 59482}, {15526, 513}, {17094, 1396}, {17216, 1019}, {17879, 514}, {18210, 43925}, {20336, 648}, {20902, 7649}, {21107, 4211}, {23616, 18210}, {23974, 4131}, {23983, 65575}, {24018, 58}, {24020, 4091}, {26942, 108}, {27801, 6528}, {30805, 757}, {34388, 54240}, {35518, 46103}, {36793, 693}, {40071, 811}, {40364, 55229}, {41013, 6529}, {41077, 51420}, {42698, 35360}, {42699, 52913}, {42700, 52917}, {42701, 53176}, {50487, 36417}, {51367, 4246}, {51640, 16947}, {52345, 57219}, {52355, 1172}, {52369, 1897}, {52385, 4565}, {52386, 692}, {52387, 101}, {52396, 662}, {52565, 1414}, {52609, 5379}, {52613, 1437}, {52616, 2185}, {52617, 274}, {52623, 158}, {55230, 1973}, {55232, 25}, {55234, 1395}, {56189, 16813}, {56235, 15384}, {57109, 6}, {57185, 7337}, {57241, 2150}, {57243, 34}, {57807, 653}, {59163, 61197}, {60597, 18180}, {61058, 57181}, {62573, 16757}, {63235, 36077}, {66980, 57}

X(68111) = X(350)X(62296)∩X(513)X(668)

X(68111) lies on the Moses X(4)X(8)-parabola and these lines: {350, 62296}, {513, 668}, {536, 6381}, {693, 4033}, {3264, 64223}, {4103, 64867}, {4583, 28151}, {14434, 41314}, {24004, 27855}, {25142, 40521}, {33908, 40552}, {39360, 46796}

X(68111) = midpoint of X(i) and X(j) for these {i,j}: {668, 66535}, {39360, 46796}
X(68111) = X(i)-Ceva conjugate of X(j) for these (i,j): {668, 41314}, {31625, 536}
X(68111) = X(14434)-cross conjugate of X(13466)
X(68111) = X(i)-isoconjugate of X(j) for these (i,j): {739, 23892}, {1919, 57542}, {23349, 37129}
X(68111) = X(i)-Dao conjugate of X(j) for these (i,j): {536, 513}, {891, 8027}, {1646, 1015}, {9296, 57542}, {13466, 43928}, {40614, 23892}, {41314, 27195}, {52882, 62619}
X(68111) = cevapoint of X(13466) and X(14434)
X(68111) = crosspoint of X(i) and X(j) for these (i,j): {536, 36957}, {668, 41314}
X(68111) = crosssum of X(667) and X(23349)
X(68111) = trilinear pole of line {8031, 13466}
X(68111) = barycentric product X(i)*X(j) for these {i,j}: {536, 41314}, {646, 61078}, {668, 13466}, {889, 8031}, {1978, 42083}, {6381, 23891}, {6386, 59797}, {14434, 31625}, {23343, 35543}
X(68111) = barycentric quotient X(i)/X(j) for these {i,j}: {536, 43928}, {668, 57542}, {899, 23892}, {3230, 23349}, {6381, 62619}, {8031, 891}, {13466, 513}, {14434, 1015}, {23343, 739}, {23891, 37129}, {39011, 8027}, {41314, 3227}, {42083, 649}, {59797, 667}, {61049, 57181}, {61078, 3669}
X(68111) = {X(668),X(31625)}-harmonic conjugate of X(66535)

X(68112) = X(101)X(795)∩X(513)X(21225)

X(68112) lies on the Moses X(4)X(8)-parabola and these lines: {101, 795}, {513, 21225}, {650, 2978}, {693, 20983}, {788, 3250}, {798, 9231}, {1912, 3572}, {3789, 14434}, {4083, 27855}, {9008, 53581}, {9010, 19586}, {9286, 20979}, {25142, 47760}

X(68112) = X(668)-Ceva conjugate of X(2276)
X(68112) = X(i)-isoconjugate of X(j) for these (i,j): {789, 870}, {871, 1492}, {985, 46132}, {4817, 5388}, {14621, 37133}, {37207, 63242}, {40746, 52611}, {41072, 63230}
X(68112) = X(i)-Dao conjugate of X(j) for these (i,j): {788, 513}, {3789, 46132}, {19584, 52611}, {38995, 871}
X(68112) = crosspoint of X(668) and X(2276)
X(68112) = crosssum of X(i) and X(j) for these (i,j): {667, 14621}, {871, 40495}, {27855, 63242}
X(68112) = crossdifference of every pair of points on line {871, 14621}
X(68112) = barycentric product X(i)*X(j) for these {i,j}: {668, 55049}, {669, 4469}, {692, 62414}, {788, 2276}, {798, 4476}, {824, 18900}, {869, 3250}, {984, 46386}, {1491, 40728}, {3063, 12837}, {3661, 8630}, {3862, 58864}
X(68112) = barycentric quotient X(i)/X(j) for these {i,j}: {869, 37133}, {984, 52611}, {2276, 46132}, {3250, 871}, {4469, 4609}, {4476, 4602}, {8630, 14621}, {18900, 4586}, {40728, 789}, {46386, 870}, {55049, 513}, {58864, 63242}, {62414, 40495}

X(68113) = X(513)X(47662)∩X(693)X(8678)

X(68113) lies on the Moses X(4)X(8)-parabola and these lines: {513, 47662}, {693, 8678}, {830, 47660}, {4036, 47694}, {4160, 47652}, {4397, 47697}, {4705, 18108}, {4885, 31096}, {7192, 62430}, {9013, 68103}, {27855, 58784}, {48324, 57096}

X(68113) = X(668)-Ceva conjugate of X(17289)
X(68113) = X(830)-Dao conjugate of X(513)
X(68113) = crosspoint of X(668) and X(17289)
X(68113) = barycentric product X(i)*X(j) for these {i,j}: {830, 17289}, {2483, 33941}, {3920, 47660}
X(68113) = barycentric quotient X(i)/X(j) for these {i,j}: {5280, 831}, {17289, 57975}

X(68114) = X(100)X(65637)∩X(513)X(53358)

X(68114) lies on the Moses X(4)X(8)-parabola and these lines: {8, 56850}, {100, 65637}, {190, 68101}, {513, 53358}, {668, 62430}, {693, 17780}, {3434, 15632}, {3952, 68102}, {4397, 24004}, {11607, 36802}, {35171, 36236}, {42722, 63745}, {65198, 68103}

X(68114) = X(668)-Ceva conjugate of X(42722)
X(68114) = X(i)-Dao conjugate of X(j) for these (i,j): {528, 513}, {52884, 840}
X(68114) = crosspoint of X(668) and X(42722)
X(68114) = barycentric product X(i)*X(j) for these {i,j}: {312, 66981}, {528, 42722}, {646, 3322}, {668, 35113}
X(68114) = barycentric quotient X(i)/X(j) for these {i,j}: {3322, 3669}, {35113, 513}, {42722, 18821}, {52985, 840}, {66981, 57}

X(68115) = X(144)X(513)∩X(693)X(3681)

X(68115) lies on the Moses X(4)X(8)-parabola and these lines: {144, 513}, {210, 28143}, {693, 3681}, {1253, 65102}, {3900, 28132}, {4036, 22271}, {4105, 8012}, {4110, 4397}, {4171, 6607}, {4524, 6608}, {6602, 8641}, {14434, 40609}, {22276, 50487}, {23612, 33570}

X(68115) = X(i)-Ceva conjugate of X(j) for these (i,j): {668, 3693}, {3900, 52614}
X(68115) = X(i)-isoconjugate of X(j) for these (i,j): {658, 6185}, {927, 56783}, {1416, 46135}, {1438, 65847}, {1461, 57537}, {1462, 34085}, {4569, 51838}, {4626, 62715}, {34018, 36146}, {39293, 43930}, {41934, 46406}, {57536, 58817}
X(68115) = X(i)-Dao conjugate of X(j) for these (i,j): {518, 4569}, {926, 513}, {6184, 65847}, {17435, 57792}, {35508, 57537}, {39014, 34018}, {40609, 46135}
X(68115) = crosspoint of X(i) and X(j) for these (i,j): {668, 3693}, {3900, 52614}
X(68115) = crosssum of X(i) and X(j) for these (i,j): {279, 43930}, {667, 1462}
X(68115) = crossdifference of every pair of points on line {1462, 6185}
X(68115) = barycentric product X(i)*X(j) for these {i,j}: {220, 3126}, {312, 66982}, {518, 52614}, {646, 15615}, {657, 4712}, {668, 39014}, {926, 3693}, {1253, 53583}, {1362, 4130}, {3119, 66978}, {3239, 42079}, {3717, 46388}, {3900, 6184}, {4397, 39686}, {4437, 8641}, {4524, 16728}, {4578, 35505}, {6602, 66967}, {14827, 62430}, {23612, 28132}, {33570, 59269}, {34337, 65102}, {42071, 57055}, {65103, 65744}
X(68115) = barycentric quotient X(i)/X(j) for these {i,j}: {518, 65847}, {926, 34018}, {1362, 36838}, {2340, 34085}, {3126, 57792}, {3693, 46135}, {3900, 57537}, {4712, 46406}, {6184, 4569}, {8638, 1462}, {8641, 6185}, {15615, 3669}, {20776, 65296}, {35505, 59941}, {39014, 513}, {39686, 934}, {42071, 13149}, {42079, 658}, {46388, 56783}, {52614, 2481}, {57180, 62715}, {61055, 4617}, {66982, 57}

X(68116) = X(100)X(65642)∩X(513)X(4468)

X(68116) lies on the Moses X(4)X(8)-parabola and these lines: {2, 17427}, {100, 65642}, {200, 57055}, {210, 14298}, {513, 4468}, {3239, 3900}, {3681, 4131}, {4036, 57232}, {4105, 4130}, {4397, 62725}, {4767, 15632}, {6552, 14434}, {35518, 62430}, {42341, 43932}

X(68116) = reflection of X(43932) in X(65499)
X(68116) = anticomplement of X(17427)
X(68116) = isotomic conjugate of the anticomplement of X(17426)
X(68116) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2125, 37781}, {8917, 149}, {42483, 150}, {63904, 33650}
X(68116) = X(i)-Ceva conjugate of X(j) for these (i,j): {668, 346}, {3699, 45791}, {4163, 4130}, {4578, 728}, {5423, 23970}, {57928, 3693}
X(68116) = X(i)-cross conjugate of X(j) for these (i,j): {17426, 2}, {65752, 480}
X(68116) = X(i)-isoconjugate of X(j) for these (i,j): {7, 6614}, {56, 4626}, {57, 4617}, {109, 479}, {269, 934}, {279, 1461}, {513, 24013}, {514, 23971}, {604, 36838}, {649, 23586}, {651, 738}, {658, 1407}, {664, 7023}, {667, 24011}, {1042, 4616}, {1106, 4569}, {1262, 58817}, {1358, 59151}, {1397, 52937}, {1415, 23062}, {1427, 4637}, {1435, 65296}, {1919, 57581}, {3598, 58998}, {3676, 7339}, {4554, 7366}, {4573, 62192}, {7045, 43932}, {7053, 36118}, {7099, 13149}, {7177, 32714}, {10481, 65540}, {24027, 59941}, {30682, 32674}, {43924, 59457}, {46406, 52410}, {61376, 65545}
X(68116) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 4626}, {11, 479}, {522, 59941}, {1146, 23062}, {2968, 1088}, {3161, 36838}, {3900, 513}, {5375, 23586}, {5452, 4617}, {6552, 4569}, {6600, 934}, {6608, 3676}, {6631, 24011}, {7358, 7056}, {9296, 57581}, {14714, 269}, {17115, 43932}, {23050, 36118}, {24010, 63973}, {24771, 658}, {35072, 30682}, {35508, 279}, {38966, 1119}, {38991, 738}, {39025, 7023}, {39026, 24013}, {40624, 57880}, {62585, 52937}
X(68116) = crosspoint of X(i) and X(j) for these (i,j): {346, 668}, {728, 4578}
X(68116) = crosssum of X(i) and X(j) for these (i,j): {513, 18725}, {667, 1407}, {738, 43932}
X(68116) = trilinear pole of line {24010, 35508}
X(68116) = crossdifference of every pair of points on line {738, 1407}
X(68116) = barycentric product X(i)*X(j) for these {i,j}: {8, 4130}, {9, 4163}, {100, 23970}, {190, 24010}, {200, 3239}, {220, 4397}, {312, 4105}, {341, 657}, {346, 3900}, {480, 4391}, {522, 728}, {644, 4081}, {646, 3022}, {650, 5423}, {663, 30693}, {668, 35508}, {1021, 4082}, {1043, 4171}, {1146, 4578}, {1253, 52622}, {1265, 65103}, {1978, 24012}, {2310, 6558}, {2321, 58329}, {3119, 3699}, {3596, 57180}, {3700, 56182}, {4515, 7253}, {4572, 52064}, {6335, 65752}, {6602, 35519}, {6607, 63239}, {7046, 57055}, {7071, 15416}, {7101, 57108}, {7256, 36197}, {7259, 52335}, {8641, 59761}, {18344, 30681}, {41798, 65448}, {45791, 62725}
X(68116) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 36838}, {9, 4626}, {41, 6614}, {55, 4617}, {100, 23586}, {101, 24013}, {190, 24011}, {200, 658}, {220, 934}, {312, 52937}, {341, 46406}, {346, 4569}, {480, 651}, {521, 30682}, {522, 23062}, {644, 59457}, {650, 479}, {657, 269}, {663, 738}, {668, 57581}, {692, 23971}, {728, 664}, {1043, 4635}, {1146, 59941}, {1253, 1461}, {1260, 65296}, {2287, 4616}, {2310, 58817}, {2328, 4637}, {3022, 3669}, {3059, 61241}, {3063, 7023}, {3119, 3676}, {3239, 1088}, {3900, 279}, {4081, 24002}, {4105, 57}, {4130, 7}, {4163, 85}, {4171, 3668}, {4391, 57880}, {4397, 57792}, {4515, 4566}, {4524, 1427}, {4578, 1275}, {5423, 4554}, {6555, 62532}, {6602, 109}, {6605, 65545}, {6607, 1418}, {7046, 13149}, {7071, 32714}, {7079, 36118}, {8641, 1407}, {14427, 62789}, {14936, 43932}, {17426, 17427}, {23970, 693}, {24010, 514}, {24012, 649}, {28070, 65188}, {30693, 4572}, {34409, 42388}, {35508, 513}, {41798, 65553}, {45791, 35312}, {51418, 23973}, {52064, 663}, {52614, 34855}, {56182, 4573}, {57055, 7056}, {57064, 50561}, {57108, 7177}, {57180, 56}, {58329, 1434}, {58835, 9533}, {59141, 65540}, {61050, 57181}, {63461, 62192}, {65102, 7053}, {65103, 1119}, {65448, 37780}, {65752, 905}
X(68116) = {X(200),X(57055)}-harmonic conjugate of X(58835)

X(68117) = X(100)X(4394)∩X(513)X(47663)

X(68117) lies on the Moses X(4)X(8)-parabola and these lines: {8, 30199}, {100, 4394}, {513, 47663}, {693, 3900}, {926, 4131}, {3239, 3887}, {3309, 4468}, {3434, 4106}, {3870, 43049}, {3873, 43932}, {4162, 25925}, {4380, 17784}, {4397, 53343}, {4855, 30234}, {9048, 25304}, {15313, 50333}, {15636, 16184}, {28475, 57287}, {52365, 62430}

X(68117) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {42361, 33650}, {53653, 3436}, {53888, 144}, {64242, 149}, {66196, 150}
X(68117) = X(668)-Ceva conjugate of X(344)
X(68117) = X(i)-isoconjugate of X(j) for these (i,j): {109, 55013}, {1292, 2191}, {36041, 57469}, {37206, 57656}
X(68117) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 55013}, {3309, 513}, {4904, 6601}, {5519, 57469}
X(68117) = crosspoint of X(i) and X(j) for these (i,j): {344, 668}, {32008, 53653}
X(68117) = crosssum of X(667) and X(57656)
X(68117) = crossdifference of every pair of points on line {20229, 57656}
X(68117) = barycentric product X(i)*X(j) for these {i,j}: {344, 3309}, {666, 58281}, {1445, 44448}, {3870, 4468}, {15636, 42720}, {31605, 55337}, {38375, 65199}
X(68117) = barycentric quotient X(i)/X(j) for these {i,j}: {218, 1292}, {344, 54987}, {650, 55013}, {3309, 277}, {3870, 37206}, {8642, 57656}, {15636, 62635}, {43049, 40154}, {51652, 17107}, {58281, 918}
X(68117) = {X(21302),X(62725)}-harmonic conjugate of X(693)

X(68118) = X(100)X(4076)∩X(513)X(49698)

X(68118) lies on the Moses X(4)X(8)-parabola and these lines: {8, 30198}, {75, 23819}, {100, 4076}, {145, 51656}, {513, 49698}, {521, 66517}, {522, 693}, {900, 4397}, {2517, 28221}, {3667, 4404}, {3900, 56323}, {4036, 4811}, {4391, 4962}, {4925, 14284}, {4943, 58858}, {15313, 68103}, {15632, 61185}, {25020, 59968}, {25142, 48080}, {26078, 59972}

X(68118) = reflection of X(i) in X(j) for these {i,j}: {145, 51656}, {14284, 4925}
X(68118) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2137, 149}, {6553, 33650}, {8051, 150}, {44301, 34548}, {53630, 329}
X(68118) = X(668)-Ceva conjugate of X(18743)
X(68118) = X(61079)-cross conjugate of X(145)
X(68118) = X(i)-isoconjugate of X(j) for these (i,j): {109, 33963}, {1293, 3445}, {1919, 57578}, {3939, 16079}, {8056, 34080}, {16945, 31343}, {27834, 38266}
X(68118) = X(i)-Dao conjugate of X(j) for these (i,j): {8, 31343}, {11, 33963}, {3667, 513}, {3756, 3680}, {4521, 58794}, {9296, 57578}, {40617, 16079}, {40621, 8056}, {45036, 1293}
X(68118) = cevapoint of X(4943) and X(31182)
X(68118) = crosspoint of X(668) and X(18743)
X(68118) = crosssum of X(667) and X(38266)
X(68118) = crossdifference of every pair of points on line {41, 34543}
X(68118) = barycentric product X(i)*X(j) for these {i,j}: {75, 31182}, {85, 4943}, {145, 4462}, {312, 58858}, {646, 61079}, {668, 40621}, {2403, 4487}, {3257, 58282}, {3667, 18743}, {4391, 6049}, {4404, 41629}, {4521, 39126}, {4953, 62532}, {15519, 24002}, {15637, 24004}, {30719, 44720}, {44723, 51656}
X(68118) = barycentric quotient X(i)/X(j) for these {i,j}: {145, 27834}, {650, 33963}, {668, 57578}, {1420, 38828}, {1743, 1293}, {2976, 51839}, {3052, 34080}, {3161, 31343}, {3667, 8056}, {3669, 16079}, {3756, 58794}, {4394, 3445}, {4404, 4052}, {4462, 4373}, {4487, 2415}, {4521, 3680}, {4943, 9}, {5435, 65173}, {6049, 651}, {8643, 38266}, {14321, 56174}, {14350, 10563}, {15519, 644}, {15637, 1022}, {18743, 53647}, {24002, 16078}, {30719, 19604}, {31182, 1}, {40621, 513}, {43290, 5382}, {51656, 40151}, {58282, 3762}, {58811, 1122}, {58858, 57}, {61079, 3669}

X(68119) = X(10)X(23809)∩X(513)X(3762)

X(68119) lies on the Moses X(4)X(8)-parabola and these lines: {10, 23809}, {513, 3762}, {693, 21606}, {2517, 28199}, {3679, 57178}, {3900, 52356}, {4036, 28165}, {4145, 25142}, {4391, 25030}, {4397, 28205}, {4770, 4777}, {14434, 48090}, {15632, 40521}, {31250, 59837}

X(68119) = isotomic conjugate of the isogonal conjugate of X(4825)
X(68119) = X(i)-Ceva conjugate of X(j) for these (i,j): {668, 4671}, {36804, 4908}
X(68119) = X(i)-isoconjugate of X(j) for these (i,j): {89, 34073}, {1415, 30607}, {2163, 4588}, {4604, 28607}
X(68119) = X(i)-Dao conjugate of X(j) for these (i,j): {1146, 30607}, {4777, 513}, {36911, 4604}, {36912, 52924}, {40587, 4588}, {55045, 2163}, {61073, 89}
X(68119) = crosspoint of X(668) and X(4671)
X(68119) = crosssum of X(667) and X(28607)
X(68119) = barycentric product X(i)*X(j) for these {i,j}: {75, 53584}, {76, 4825}, {312, 66984}, {668, 61073}, {1577, 4803}, {3679, 4791}, {4125, 47683}, {4671, 4777}, {4767, 4957}, {4908, 63216}
X(68119) = barycentric quotient X(i)/X(j) for these {i,j}: {45, 4588}, {522, 30607}, {2177, 34073}, {3679, 4604}, {3711, 5549}, {4671, 4597}, {4767, 5385}, {4770, 28658}, {4775, 28607}, {4777, 89}, {4791, 39704}, {4803, 662}, {4814, 2364}, {4825, 6}, {4893, 2163}, {4908, 52924}, {4931, 53114}, {4944, 2320}, {4957, 52620}, {28603, 52901}, {53584, 1}, {61073, 513}, {63216, 40833}, {66984, 57}

X(68120) = X(512)X(693)∩X(513)X(47664)

X(68120) lies on the Moses X(4)X(8)-parabola and these lines: {8, 30207}, {512, 693}, {513, 47664}, {2978, 38238}, {4036, 48080}, {6005, 47666}, {21727, 48081}, {24948, 50508}, {25142, 50497}, {27855, 48079}, {29350, 47675}, {31150, 50500}, {47939, 50481}, {48107, 50520}, {48548, 50487}

X(68120) = reflection of X(i) in X(j) for these {i,j}: {47666, 50483}, {47939, 50481}, {48107, 50520}
X(68120) = X(668)-Ceva conjugate of X(4687)
X(68120) = X(6013)-isoconjugate of X(10013)
X(68120) = X(6005)-Dao conjugate of X(513)
X(68120) = crosspoint of X(668) and X(4687)
X(68120) = crosssum of X(667) and X(64845)
X(68120) = crossdifference of every pair of points on line {1185, 64845}
X(68120) = barycentric product X(i)*X(j) for these {i,j}: {4687, 6005}, {17018, 47666}
X(68120) = barycentric quotient X(i)/X(j) for these {i,j}: {6005, 56051}, {8655, 64845}, {50483, 56236}

X(68121) = X(7)X(513)∩X(693)X(18743)

X(68121) lies on the Moses X(4)X(8)-parabola and these lines: {7, 513}, {75, 23819}, {693, 18743}, {3434, 4106}, {3766, 62430}, {4036, 56253}, {4397, 4408}, {6084, 16593}, {20900, 68101}, {21297, 68102}, {22278, 50487}, {40154, 43932}

X(68121) = X(i)-Dao conjugate of X(j) for these (i,j): {5853, 4578}, {6084, 513}, {39048, 6078}, {61074, 1280}
X(68121) = barycentric product X(i)*X(j) for these {i,j}: {668, 61074}, {1279, 65869}, {3021, 24002}, {35111, 59941}
X(68121) = barycentric quotient X(i)/X(j) for these {i,j}: {1279, 6078}, {3021, 644}, {6084, 1280}, {35111, 4578}, {52210, 39272}, {61074, 513}

X(68122) = X(514)X(4397)∩X(693)X(4806)

X(68122) lies on the Moses X(4)X(8)-parabola and these lines: {514, 4397}, {693, 4806}, {1459, 47974}, {2517, 28213}, {4036, 4462}, {4391, 28229}, {4684, 4742}, {4802, 68101}, {7650, 28220}, {21146, 25142}, {30024, 47963}, {48108, 50487}, {48109, 62430}

X(68122) = reflection of X(i) in X(j) for these {i,j}: {4811, 4801}, {47974, 1459}
X(68122) = X(668)-Ceva conjugate of X(19804)
X(68122) = X(i)-isoconjugate of X(j) for these (i,j): {2334, 8694}, {25430, 34074}
X(68122) = X(i)-Dao conjugate of X(j) for these (i,j): {4778, 513}, {51576, 8694}, {55056, 56237}, {62608, 4606}
X(68122) = crosspoint of X(668) and X(19804)
X(68122) = barycentric product X(i)*X(j) for these {i,j}: {75, 53586}, {799, 52332}, {3616, 4801}, {4673, 30723}, {4778, 19804}, {4811, 21454}, {4815, 42028}
X(68122) = barycentric quotient X(i)/X(j) for these {i,j}: {1449, 8694}, {3616, 4606}, {4765, 4866}, {4778, 25430}, {4790, 2334}, {4801, 5936}, {4811, 56086}, {4815, 60267}, {4841, 56237}, {19804, 53658}, {42028, 4614}, {48580, 56048}, {52332, 661}, {53586, 1}

X(68123) = X(513)X(4801)∩X(523)X(764)

X(68123) lies on the Moses X(4)X(8)-parabola and these lines: {513, 4801}, {514, 4036}, {523, 764}, {693, 18158}, {1577, 28213}, {2517, 28199}, {2605, 5625}, {4397, 4802}, {4705, 40086}, {4815, 28209}, {4966, 4975}, {7199, 16709}, {7650, 28220}, {14434, 62588}, {21106, 40166}, {21146, 50487}, {23789, 57099}, {25142, 48098}, {28229, 50327}, {29186, 48283}, {42455, 52569}, {47842, 48406}, {47965, 48230}, {48119, 48342}, {48152, 62430}

X(68123) = midpoint of X(48119) and X(48342)
X(68123) = reflection of X(i) in X(j) for these {i,j}: {4705, 40086}, {30591, 4978}, {47842, 48406}, {57099, 23789}
X(68123) = X(668)-Ceva conjugate of X(4359)
X(68123) = X(i)-isoconjugate of X(j) for these (i,j): {163, 30582}, {1126, 8701}, {1576, 30594}, {4629, 52555}, {28615, 37212}
X(68123) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 30582}, {1213, 37212}, {3647, 8701}, {4858, 30594}, {4977, 513}, {16726, 40438}, {35076, 1255}, {62588, 6540}
X(68123) = crosspoint of X(668) and X(4359)
X(68123) = crosssum of X(667) and X(28615)
X(68123) = crossdifference of every pair of points on line {28615, 33882}
X(68123) = barycentric product X(i)*X(j) for these {i,j}: {75, 53587}, {514, 6533}, {553, 4985}, {668, 35076}, {1125, 4978}, {1269, 4979}, {3702, 30724}, {4359, 4977}, {4983, 52572}, {4988, 16709}, {7199, 8040}, {8025, 30591}
X(68123) = barycentric quotient X(i)/X(j) for these {i,j}: {523, 30582}, {1100, 8701}, {1125, 37212}, {1577, 30594}, {4359, 6540}, {4976, 32635}, {4977, 1255}, {4978, 1268}, {4979, 1126}, {4983, 52555}, {4985, 4102}, {6533, 190}, {8025, 4596}, {8040, 1018}, {16709, 4632}, {30581, 6578}, {30591, 6539}, {30593, 62535}, {35076, 513}, {50512, 28615}, {53587, 1}

X(68124) = X(513)X(26824)∩X(661)X(764)

X(68124) lies on the Moses X(4)X(8)-parabola and these lines: {513, 26824}, {514, 50487}, {661, 764}, {693, 29198}, {2978, 48133}, {4036, 21146}, {4397, 48108}, {6372, 47672}, {7192, 8027}, {8672, 48148}, {14404, 47917}, {23789, 65152}, {25142, 47666}, {27674, 48618}, {48141, 50521}

X(68124) = reflection of X(i) in X(j) for these {i,j}: {2978, 48133}, {50497, 47672}, {50521, 48141}
X(68124) = X(i)-Ceva conjugate of X(j) for these (i,j): {668, 3739}, {54118, 21820}
X(68124) = X(8708)-isoconjugate of X(40433)
X(68124) = X(6372)-Dao conjugate of X(513)
X(68124) = crosspoint of X(668) and X(3739)
X(68124) = crosssum of X(667) and X(57397)
X(68124) = crossdifference of every pair of points on line {1206, 57397}
X(68124) = barycentric product X(i)*X(j) for these {i,j}: {3720, 47672}, {3739, 6372}, {16748, 50497}, {18166, 48393}
X(68124) = barycentric quotient X(i)/X(j) for these {i,j}: {6372, 32009}, {20963, 8708}

X(68125) = X(42)X(649)∩X(192)X(513)

X(68125) lies on the Moses X(4)X(8)-parabola and these lines: {2, 20983}, {42, 649}, {192, 513}, {667, 40735}, {693, 17149}, {798, 23551}, {3056, 23464}, {3221, 3728}, {4083, 62638}, {4397, 45242}, {4928, 14434}, {6373, 20681}, {9010, 19586}, {19581, 27855}, {20979, 63504}, {21191, 25140}

X(68125) = X(668)-Ceva conjugate of X(1575)
X(68125) = X(i)-isoconjugate of X(j) for these (i,j): {190, 57535}, {727, 54985}, {3226, 8709}
X(68125) = X(i)-Dao conjugate of X(j) for these (i,j): {726, 6386}, {6373, 513}, {17793, 54985}, {55053, 57535}
X(68125) = crosspoint of X(668) and X(1575)
X(68125) = crosssum of X(667) and X(20332)
X(68125) = crossdifference of every pair of points on line {239, 20332}
X(68125) = barycentric product X(i)*X(j) for these {i,j}: {513, 20671}, {667, 20532}, {1575, 6373}, {3063, 59806}, {3733, 20690}, {3837, 21760}, {6591, 20759}, {40155, 62558}
X(68125) = barycentric quotient X(i)/X(j) for these {i,j}: {667, 57535}, {1575, 54985}, {6373, 32020}, {20532, 6386}, {20671, 668}, {20690, 27808}, {21760, 8709}, {38367, 3253}, {65498, 62421}

X(68126) = X(101)X(476)∩X(514)X(14543)

X(68126) lies on the Yff parabola and these lines: {10, 54527}, {30, 2173}, {101, 476}, {190, 16077}, {514, 14543}, {3234, 5134}, {3239, 35342}, {4240, 56829}, {6544, 13589}, {16086, 53582}, {37168, 62652}, {57057, 61233}

X(68126) = X(i)-isoconjugate of X(j) for these (i,j): {513, 40384}, {667, 31621}, {693, 40353}, {14399, 59145}, {18210, 34568}
X(68126) = X(i)-Dao conjugate of X(j) for these (i,j): {30, 514}, {1650, 4466}, {6631, 31621}, {39026, 40384}
X(68126) = trilinear pole of line {3163, 6062}
X(68126) = barycentric product X(i)*X(j) for these {i,j}: {1, 68104}, {10, 3233}, {100, 1099}, {101, 36789}, {190, 3163}, {664, 6062}, {668, 42074}, {1331, 34334}, {1354, 3699}, {1897, 16163}, {1978, 9408}, {2173, 42716}, {4561, 16240}, {4570, 58263}, {4600, 58346}
X(68126) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 40384}, {190, 31621}, {1099, 693}, {1354, 3676}, {3081, 11125}, {3163, 514}, {3233, 86}, {6062, 522}, {9408, 649}, {14401, 4466}, {16163, 4025}, {16240, 7649}, {32739, 40353}, {34334, 46107}, {36789, 3261}, {42074, 513}, {42716, 33805}, {58263, 21207}, {58343, 53521}, {58344, 3122}, {58346, 3120}, {58347, 4750}, {68104, 75}

X(68127) = X(101)X(649)∩X(514)X(4552)

X(68127) lies on the Yff parabola and these lines: {101, 649}, {514, 4552}, {517, 2183}, {644, 57057}, {928, 66978}, {1018, 3239}, {2149, 57061}, {2427, 23981}, {3234, 5011}, {4024, 61168}, {4091, 4564}, {4559, 43060}, {6073, 55153}, {6544, 61239}, {31182, 61237}, {57015, 64139}

X(68127) = X(4564)-Ceva conjugate of X(22350)
X(68127) = X(i)-isoconjugate of X(j) for these (i,j): {104, 2401}, {513, 59196}, {667, 57550}, {693, 41933}, {2423, 18816}, {13136, 15635}, {34051, 43728}, {43933, 65302}, {55943, 57468}
X(68127) = X(i)-Dao conjugate of X(j) for these (i,j): {517, 514}, {6631, 57550}, {35014, 4858}, {39026, 59196}, {40613, 2401}, {57293, 3942}
X(68127) = trilinear pole of line {23980, 42078}
X(68127) = barycentric product X(i)*X(j) for these {i,j}: {1, 15632}, {8, 66977}, {59, 66969}, {100, 24028}, {101, 26611}, {109, 55016}, {190, 23980}, {668, 42078}, {765, 42757}, {908, 2427}, {1331, 21664}, {1361, 3699}, {1897, 65743}, {1978, 59800}, {2183, 2397}, {4561, 42072}, {4564, 60339}, {4619, 55153}, {6735, 23981}, {21801, 64828}, {22350, 53151}, {23101, 36037}, {23706, 51379}
X(68127) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 59196}, {190, 57550}, {1361, 3676}, {2183, 2401}, {2427, 34234}, {15632, 75}, {21664, 46107}, {23101, 36038}, {23980, 514}, {24028, 693}, {26611, 3261}, {32739, 41933}, {42072, 7649}, {42078, 513}, {42757, 1111}, {55016, 35519}, {59800, 649}, {60339, 4858}, {61057, 43924}, {65743, 4025}, {66969, 34387}, {66977, 7}

X(68128) = X(101)X(6065)∩X(514)X(1018)

X(68128) lies on the Yff parabola and these lines: {100, 649}, {101, 6065}, {190, 51560}, {514, 1018}, {518, 672}, {657, 765}, {677, 1252}, {883, 1025}, {919, 3939}, {1023, 38379}, {1026, 63743}, {2284, 54325}, {3218, 31020}, {3239, 3952}, {4024, 61163}, {4358, 17755}, {4375, 53337}, {6017, 28879}, {6184, 35505}, {17460, 43065}, {32739, 65208}, {45751, 53582}, {47676, 54118}, {53581, 61168}, {57015, 64139}

X(68128) = X(i)-Ceva conjugate of X(j) for these (i,j): {190, 1026}, {765, 2340}, {67038, 56714}
X(68128) = X(i)-isoconjugate of X(j) for these (i,j): {105, 62635}, {294, 43930}, {513, 6185}, {514, 51838}, {666, 43921}, {667, 57537}, {673, 1027}, {693, 41934}, {764, 57536}, {884, 34018}, {885, 1462}, {1024, 56783}, {2481, 43929}, {3669, 62715}
X(68128) = X(i)-Dao conjugate of X(j) for these (i,j): {518, 514}, {918, 23100}, {2284, 24203}, {6631, 57537}, {17435, 1111}, {39026, 6185}, {39046, 62635}
X(68128) = crosspoint of X(190) and X(1026)
X(68128) = crosssum of X(649) and X(1027)
X(68128) = trilinear pole of line {6184, 42079}
X(68128) = crossdifference of every pair of points on line {1027, 27846}
X(68128) = barycentric product X(i)*X(j) for these {i,j}: {1, 68106}, {8, 66978}, {100, 4712}, {101, 4437}, {190, 6184}, {306, 68086}, {518, 1026}, {668, 42079}, {672, 42720}, {765, 3126}, {883, 2340}, {1018, 16728}, {1025, 3693}, {1110, 62430}, {1252, 53583}, {1331, 34337}, {1362, 3699}, {1897, 65744}, {1978, 39686}, {2283, 3717}, {2284, 3912}, {3263, 54325}, {3932, 54353}, {4561, 42071}, {6065, 66967}, {6632, 35505}, {23102, 36086}, {23612, 51560}, {35094, 59149}, {39258, 55260}
X(68128) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 6185}, {190, 57537}, {672, 62635}, {692, 51838}, {1025, 34018}, {1026, 2481}, {1362, 3676}, {1458, 43930}, {2223, 1027}, {2283, 56783}, {2284, 673}, {2340, 885}, {3126, 1111}, {3939, 62715}, {4437, 3261}, {4712, 693}, {6184, 514}, {9454, 43929}, {16728, 7199}, {20776, 1459}, {23612, 2254}, {32739, 41934}, {34337, 46107}, {35094, 23100}, {35505, 6545}, {39258, 55261}, {39686, 649}, {42071, 7649}, {42079, 513}, {42720, 18031}, {53583, 23989}, {54325, 105}, {59149, 57536}, {61055, 43924}, {65744, 4025}, {66978, 7}, {66982, 3271}, {68086, 27}, {68106, 75}, {68115, 2310}

X(68129) = X(101)X(6082)∩X(514)X(4427)

X(68129) lies on the Yff parabola and these lines: {10, 54607}, {101, 6082}, {190, 892}, {514, 4427}, {524, 896}, {649, 3882}, {1331, 17708}, {3570, 6544}, {3936, 24628}, {5468, 23889}, {6542, 53582}, {6651, 16704}, {46148, 53581}

X(68129) = X(4600)-Ceva conjugate of X(4062)
X(68129) = X(i)-isoconjugate of X(j) for these (i,j): {513, 10630}, {667, 57539}, {693, 41936}, {897, 66945}, {923, 62626}, {3125, 34574}, {6591, 15398}
X(68129) = X(i)-Dao conjugate of X(j) for these (i,j): {524, 514}, {690, 21131}, {1648, 3120}, {2482, 62626}, {6593, 66945}, {6631, 57539}, {39026, 10630}
X(68129) = crosssum of X(649) and X(66945)
X(68129) = trilinear pole of line {2482, 7067}
X(68129) = barycentric product X(i)*X(j) for these {i,j}: {1, 68109}, {99, 52068}, {100, 24038}, {101, 36792}, {190, 2482}, {306, 68087}, {664, 7067}, {668, 42081}, {896, 42721}, {1018, 16733}, {1331, 34336}, {1366, 3699}, {1649, 4600}, {1897, 65747}, {1978, 39689}, {4062, 5468}, {4561, 5095}, {4570, 52629}, {21839, 24039}, {23889, 42713}
X(68129) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 10630}, {187, 66945}, {190, 57539}, {524, 62626}, {1331, 15398}, {1366, 3676}, {1649, 3120}, {2482, 514}, {4062, 5466}, {4570, 34574}, {5095, 7649}, {7067, 522}, {8030, 4750}, {16733, 7199}, {21839, 23894}, {23992, 21131}, {24038, 693}, {32739, 41936}, {34336, 46107}, {36792, 3261}, {39689, 649}, {42081, 513}, {42721, 46277}, {52068, 523}, {52629, 21207}, {54274, 3122}, {62661, 21205}, {65747, 4025}, {68087, 27}, {68109, 75}

X(68130) = X(101)X(2867)∩X(514)X(16086)

X(68130) lies on the Yff parabola and these lines: {10, 43673}, {72, 64885}, {101, 2867}, {190, 16077}, {306, 34767}, {514, 16086}, {525, 656}, {850, 1577}, {1331, 17708}, {3234, 4115}, {3239, 7265}, {3265, 24018}, {3682, 53173}, {4091, 52616}, {4561, 17932}, {6332, 57057}, {6544, 62564}, {14208, 63220}, {15416, 20336}, {22037, 31182}, {23875, 53583}, {23876, 57111}

X(68130) = isotomic conjugate of X(52919)
X(68130) = isotomic conjugate of the polar conjugate of X(4064)
X(68130) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {34440, 149}, {64958, 21294}
X(68130) = X(i)-Ceva conjugate of X(j) for these (i,j): {190, 306}, {3265, 57109}, {4561, 3682}
X(68130) = X(i)-cross conjugate of X(j) for these (i,j): {122, 6356}, {57109, 66980}
X(68130) = X(i)-isoconjugate of X(j) for these (i,j): {6, 52920}, {27, 32676}, {28, 112}, {31, 52919}, {58, 24019}, {81, 32713}, {100, 36420}, {107, 1333}, {108, 2189}, {110, 5317}, {162, 1474}, {163, 8747}, {250, 6591}, {270, 32674}, {286, 61206}, {513, 23964}, {604, 52921}, {608, 52914}, {648, 2203}, {649, 24000}, {667, 23582}, {692, 36419}, {693, 41937}, {823, 2206}, {1096, 4556}, {1304, 52955}, {1437, 6529}, {1919, 23999}, {1974, 55231}, {2150, 36127}, {2207, 52935}, {2299, 65232}, {4091, 24022}, {4131, 23975}, {4612, 7337}, {4623, 36417}, {5379, 43925}, {17924, 57655}, {18210, 59153}, {22383, 32230}, {23224, 23590}, {23985, 65575}, {32695, 51420}, {36131, 52954}
X(68130) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 52919}, {9, 52920}, {10, 24019}, {37, 107}, {115, 8747}, {125, 1474}, {226, 65232}, {244, 5317}, {525, 514}, {647, 7649}, {1086, 36419}, {2968, 36421}, {3161, 52921}, {3265, 21178}, {4466, 59186}, {5375, 24000}, {6338, 4610}, {6503, 4556}, {6587, 21172}, {6631, 23582}, {6741, 8748}, {7358, 2326}, {8054, 36420}, {9296, 23999}, {14401, 11125}, {15526, 27}, {17434, 1459}, {23285, 46107}, {34591, 28}, {35071, 58}, {35072, 270}, {35441, 21102}, {38983, 2189}, {38985, 1333}, {39008, 52954}, {39020, 44698}, {39026, 23964}, {40586, 32713}, {40591, 112}, {40603, 823}, {40626, 46103}, {51574, 162}, {55065, 393}, {55066, 2203}, {56325, 36127}, {62564, 648}, {62573, 86}, {62604, 55229}, {62614, 811}, {62647, 52914}
X(68130) = crosspoint of X(i) and X(j) for these (i,j): {190, 306}, {4561, 40071}
X(68130) = crosssum of X(649) and X(1474)
X(68130) = trilinear pole of line {2632, 7068}
X(68130) = crossdifference of every pair of points on line {1474, 2206}
X(68130) = barycentric product X(i)*X(j) for these {i,j}: {1, 68110}, {8, 66980}, {10, 3265}, {12, 52616}, {42, 52617}, {69, 4064}, {71, 3267}, {72, 14208}, {75, 57109}, {100, 17879}, {101, 36793}, {125, 4561}, {190, 15526}, {201, 35518}, {304, 55232}, {305, 55230}, {306, 525}, {307, 52355}, {313, 520}, {321, 24018}, {326, 4036}, {339, 1331}, {345, 57243}, {394, 52623}, {521, 57807}, {523, 52396}, {594, 30805}, {647, 40071}, {656, 20336}, {664, 7068}, {668, 2632}, {693, 52387}, {822, 27801}, {850, 3682}, {905, 52369}, {1089, 4131}, {1231, 8611}, {1264, 66287}, {1332, 20902}, {1367, 3699}, {1577, 3998}, {1826, 4143}, {1978, 3269}, {3261, 52386}, {3695, 4025}, {3700, 52565}, {3710, 17094}, {3926, 4024}, {3949, 15413}, {3952, 17216}, {3990, 20948}, {4055, 44173}, {4086, 52385}, {4091, 28654}, {4158, 46107}, {4466, 52609}, {4563, 21046}, {4600, 5489}, {4605, 23983}, {6332, 26942}, {7066, 35519}, {8804, 14638}, {15414, 21011}, {15416, 37755}, {34388, 57241}, {55234, 57919}, {56246, 60597}
X(68130) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 52920}, {2, 52919}, {8, 52921}, {10, 107}, {12, 36127}, {37, 24019}, {42, 32713}, {71, 112}, {72, 162}, {78, 52914}, {100, 24000}, {101, 23964}, {122, 21172}, {125, 7649}, {190, 23582}, {201, 108}, {228, 32676}, {304, 55231}, {305, 55229}, {306, 648}, {313, 6528}, {321, 823}, {326, 52935}, {339, 46107}, {394, 4556}, {514, 36419}, {520, 58}, {521, 270}, {523, 8747}, {525, 27}, {647, 1474}, {649, 36420}, {652, 2189}, {656, 28}, {661, 5317}, {668, 23999}, {810, 2203}, {822, 1333}, {1214, 65232}, {1259, 4636}, {1331, 250}, {1367, 3676}, {1650, 11125}, {1826, 6529}, {1897, 32230}, {2197, 32674}, {2200, 61206}, {2525, 17171}, {2631, 52955}, {2632, 513}, {2972, 1459}, {3239, 36421}, {3265, 86}, {3267, 44129}, {3269, 649}, {3682, 110}, {3690, 8750}, {3694, 65201}, {3695, 1897}, {3700, 8748}, {3708, 6591}, {3710, 36797}, {3719, 4612}, {3926, 4610}, {3949, 1783}, {3990, 163}, {3998, 662}, {4010, 34856}, {4024, 393}, {4036, 158}, {4055, 1576}, {4064, 4}, {4079, 2207}, {4086, 1896}, {4091, 593}, {4131, 757}, {4143, 17206}, {4158, 1331}, {4466, 17925}, {4561, 18020}, {4605, 23984}, {4705, 1096}, {5489, 3120}, {6332, 46103}, {6356, 36118}, {6358, 54240}, {7066, 109}, {7068, 522}, {8057, 44698}, {8611, 1172}, {8804, 57219}, {9033, 52954}, {9391, 1430}, {14208, 286}, {14429, 37168}, {15523, 46151}, {15526, 514}, {17216, 7192}, {17879, 693}, {18210, 57200}, {20336, 811}, {20902, 17924}, {21011, 61193}, {21012, 61217}, {21043, 58757}, {21046, 2501}, {21134, 2969}, {23224, 849}, {23616, 4466}, {23974, 30805}, {24018, 81}, {24020, 4131}, {24031, 65575}, {24459, 31905}, {26942, 653}, {27801, 57973}, {30805, 1509}, {32656, 57655}, {32739, 41937}, {34388, 52938}, {35442, 21102}, {35518, 57779}, {36054, 2150}, {36793, 3261}, {37754, 22383}, {37755, 32714}, {39201, 2206}, {40071, 6331}, {40152, 4565}, {41013, 36126}, {41077, 18653}, {51366, 4241}, {51640, 1408}, {51664, 1396}, {52355, 29}, {52369, 6335}, {52385, 1414}, {52386, 101}, {52387, 100}, {52396, 99}, {52565, 4573}, {52613, 1790}, {52616, 261}, {52617, 310}, {52623, 2052}, {53010, 40117}, {53012, 1301}, {53581, 36417}, {55230, 25}, {55232, 19}, {55234, 608}, {56246, 16813}, {57055, 2326}, {57057, 7054}, {57109, 1}, {57241, 60}, {57243, 278}, {57807, 18026}, {57919, 55233}, {59163, 61220}, {60597, 17167}, {61058, 43924}, {62573, 21178}, {66287, 1118}, {66928, 7337}, {66980, 7}, {68108, 1098}, {68110, 75}

X(68131) = X(190)X(649)∩X(514)X(3952)

X(68131) lies on the Yff parabola and these lines: {190, 649}, {239, 53582}, {514, 3952}, {536, 899}, {812, 68107}, {3239, 25268}, {3807, 53584}, {3835, 61402}, {4024, 65191}, {4103, 47790}, {4358, 17755}, {4375, 17780}, {4562, 48544}, {4659, 9458}, {6544, 42720}, {14433, 23891}, {17318, 46126}, {24403, 64178}, {30835, 61406}, {36863, 57050}, {48141, 54099}, {53581, 61163}

X(68131) = X(i)-Ceva conjugate of X(j) for these (i,j): {190, 23891}, {7035, 899}
X(68131) = X(i)-isoconjugate of X(j) for these (i,j): {667, 57542}, {739, 43928}, {3227, 23349}, {23892, 37129}
X(68131) = X(i)-Dao conjugate of X(j) for these (i,j): {536, 514}, {891, 21143}, {1646, 244}, {6631, 57542}, {13466, 62619}, {40614, 43928}
X(68131) = crosspoint of X(190) and X(23891)
X(68131) = crosssum of X(649) and X(23892)
X(68131) = trilinear pole of line {13466, 42083}
X(68131) = barycentric product X(i)*X(j) for these {i,j}: {1, 68111}, {190, 13466}, {536, 23891}, {668, 42083}, {899, 41314}, {1978, 59797}, {3699, 61078}, {4607, 8031}, {6381, 23343}, {7035, 14434}
X(68131) = barycentric quotient X(i)/X(j) for these {i,j}: {190, 57542}, {536, 62619}, {899, 43928}, {3230, 23892}, {3994, 35353}, {8031, 4728}, {13466, 514}, {14434, 244}, {23343, 37129}, {23891, 3227}, {39011, 21143}, {41314, 31002}, {42083, 513}, {59797, 649}, {61049, 43924}, {61078, 3676}, {68111, 75}

X(68132) = X(101)X(65635)∩X(514)X(4115)

X(68132) lies on the Yff parabola and these lines: {101, 65635}, {190, 4589}, {514, 4115}, {649, 4427}, {740, 2238}, {874, 3570}, {1018, 53581}, {3239, 61165}, {3952, 4024}, {6651, 16704}, {39916, 62755}, {50016, 53582}, {57050, 61168}

X(68132) = X(4601)-Ceva conjugate of X(4039)
X(68132) = X(i)-isoconjugate of X(j) for these (i,j): {667, 57554}, {3669, 62714}, {37128, 66937}
X(68132) = X(i)-Dao conjugate of X(j) for these (i,j): {740, 514}, {6631, 57554}, {39786, 17205}
X(68132) = crosssum of X(649) and X(66937)
X(68132) = trilinear pole of line {4094, 35068}
X(68132) = barycentric product X(i)*X(j) for these {i,j}: {190, 35068}, {668, 4094}, {3027, 3699}, {3570, 4037}, {3952, 4368}, {4103, 4366}, {4375, 61402}, {27853, 66878}, {39044, 40521}
X(68132) = barycentric quotient X(i)/X(j) for these {i,j}: {190, 57554}, {3027, 3676}, {3747, 66937}, {3939, 62714}, {4037, 4444}, {4094, 513}, {4103, 40098}, {4154, 17212}, {4368, 7192}, {4375, 61403}, {35068, 514}, {40521, 30663}, {61059, 43924}, {66878, 3572}

X(68133) = X(514)X(4088)∩X(522)X(4375)

X(68133 lies on the Yff parabola and these lines: {514, 4088}, {522, 4375}, {649, 2786}, {824, 1491}, {3239, 21196}, {3716, 64859}, {3835, 4024}, {4036, 63814}, {4391, 9237}, {4444, 62423}, {4791, 62556}, {6544, 27481}, {7265, 53581}, {9508, 28898}, {21212, 30764}, {23596, 62415}, {23879, 42661}, {25381, 30519}, {27929, 48185}, {29945, 29955}, {40459, 49279}, {40774, 47828}, {45661, 53584}, {47656, 53585}, {48082, 49282}, {48235, 64862}

X(68133) = reflection of X(4486) in X(4522)
X(68133) = X(i)-Ceva conjugate of X(j) for these (i,j): {190, 3661}, {51614, 3783}
X(68133) = X(i)-isoconjugate of X(j) for these (i,j): {825, 985}, {1492, 40746}, {14621, 34069}
X(68133) = X(i)-Dao conjugate of X(j) for these (i,j): {824, 514}, {3789, 825}, {19584, 1492}, {27481, 4586}, {33568, 4809}, {38995, 40746}, {61065, 14621}
X(68133) = crosspoint of X(190) and X(3661)
X(68133) = crosssum of X(649) and X(40746)
X(68133) = crossdifference of every pair of points on line {21764, 40746}
X(68133) = barycentric product X(i)*X(j) for these {i,j}: {190, 61065}, {824, 3661}, {850, 4476}, {869, 30870}, {984, 62415}, {1491, 33931}, {1577, 4469}, {1928, 68112}, {1978, 62414}, {3783, 63219}, {3797, 23596}, {4122, 30966}, {4475, 4505}, {4486, 63234}, {4522, 7179}, {12837, 35519}, {30665, 63228}
X(68133) = barycentric quotient X(i)/X(j) for these {i,j}: {824, 14621}, {869, 34069}, {984, 1492}, {1491, 985}, {2276, 825}, {3250, 40746}, {3661, 4586}, {3773, 4613}, {3799, 5384}, {3864, 30664}, {4122, 40718}, {4469, 662}, {4476, 110}, {4486, 63237}, {4522, 52133}, {12837, 109}, {30870, 871}, {33931, 789}, {61065, 514}, {62414, 649}, {62415, 870}, {63228, 41072}, {63234, 37207}, {68112, 560}

X(68134) = X(1)X(649)∩X(192)X(514)

X(68134) lies on the Yff parabola and these lines: {1, 649}, {37, 4083}, {192, 514}, {512, 2667}, {519, 62558}, {875, 58173}, {891, 3768}, {3159, 4024}, {3239, 19582}, {3735, 17458}, {4065, 53587}, {4375, 21385}, {6544, 21832}, {14433, 23891}, {17475, 21343}, {26752, 27138}

X(68134) = X(i)-Ceva conjugate of X(j) for these (i,j): {190, 899}, {649, 3768}, {59797, 39011}
X(68134) = X(39011)-cross conjugate of X(59797)
X(68134) = X(i)-isoconjugate of X(j) for these (i,j): {100, 57542}, {667, 57572}, {739, 889}, {898, 3227}, {4607, 37129}, {5381, 43928}, {31002, 34075}
X(68134) = X(i)-Dao conjugate of X(j) for these (i,j): {536, 1978}, {891, 514}, {1646, 75}, {6631, 57572}, {8054, 57542}, {14434, 62619}, {39011, 31002}, {40614, 889}, {52882, 57994}
X(68134) = crosspoint of X(i) and X(j) for these (i,j): {1, 23891}, {190, 899}, {649, 3768}
X(68134) = crosssum of X(i) and X(j) for these (i,j): {1, 23892}, {190, 4607}, {649, 37129}
X(68134) = crossdifference of every pair of points on line {899, 4607}
X(68134) = barycentric product X(i)*X(j) for these {i,j}: {1, 14434}, {190, 39011}, {513, 42083}, {514, 59797}, {522, 61049}, {536, 3768}, {649, 13466}, {663, 61078}, {890, 6381}, {891, 899}, {1646, 23891}, {3230, 4728}, {3248, 68111}, {3699, 47016}, {4526, 52896}, {7035, 14441}, {8031, 23892}, {14404, 62755}, {14430, 62739}, {14431, 62740}, {14437, 52900}, {19945, 23343}
X(68134) = barycentric quotient X(i)/X(j) for these {i,j}: {190, 57572}, {649, 57542}, {890, 37129}, {891, 31002}, {899, 889}, {1646, 62619}, {3230, 4607}, {3768, 3227}, {6381, 57994}, {13466, 1978}, {14404, 41683}, {14434, 75}, {14441, 244}, {39011, 514}, {42083, 668}, {47016, 3676}, {59797, 190}, {61049, 664}, {61078, 4572}

X(68135) = X(101)X(65642)∩X(514)X(40872)

X(68135) lies on the Yff parabola and these lines: {101, 65642}, {220, 57108}, {514, 40872}, {649, 3309}, {657, 3900}, {663, 52614}, {728, 4163}, {1018, 3234}, {3239, 28058}, {3730, 4091}, {4024, 57049}, {4105, 57180}, {4936, 38379}, {6332, 53583}, {6544, 24771}, {24010, 52064}, {25924, 53357}

X(68135) = reflection of X(45755) in X(62747)
X(68135) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {57641, 150}, {63905, 33650}
X(68135) = X(i)-Ceva conjugate of X(j) for these (i,j): {190, 200}, {220, 65752}, {728, 24010}, {4130, 4105}, {56235, 42}
X(68135) = X(i)-isoconjugate of X(j) for these (i,j): {7, 4617}, {56, 36838}, {57, 4626}, {85, 6614}, {108, 30682}, {109, 23062}, {269, 658}, {279, 934}, {479, 651}, {513, 23586}, {514, 24013}, {604, 52937}, {649, 24011}, {664, 738}, {667, 57581}, {693, 23971}, {1042, 4635}, {1088, 1461}, {1106, 46406}, {1119, 65296}, {1262, 59941}, {1275, 43932}, {1407, 4569}, {1415, 57880}, {1418, 65545}, {1427, 4616}, {3668, 4637}, {3669, 59457}, {4554, 7023}, {4572, 7366}, {4625, 62192}, {6610, 65553}, {7045, 58817}, {7053, 13149}, {7056, 32714}, {7177, 36118}, {7339, 24002}, {31615, 41292}, {59181, 65540}, {61241, 61373}, {63178, 65188}
X(68135) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 36838}, {11, 23062}, {1146, 57880}, {2968, 57792}, {3119, 53242}, {3161, 52937}, {3900, 514}, {5375, 24011}, {5452, 4626}, {6552, 46406}, {6600, 658}, {6608, 24002}, {6631, 57581}, {14714, 279}, {17115, 58817}, {23050, 13149}, {24771, 4569}, {35508, 1088}, {38966, 1847}, {38983, 30682}, {38991, 479}, {39025, 738}, {39026, 23586}
X(68135) = crosspoint of X(i) and X(j) for these (i,j): {190, 200}, {4130, 68116}
X(68135) = crosssum of X(i) and X(j) for these (i,j): {269, 649}, {479, 58817}
X(68135) = trilinear pole of line {24012, 35508}
X(68135) = crossdifference of every pair of points on line {269, 479}
X(68135) = barycentric product X(i)*X(j) for these {i,j}: {1, 68116}, {8, 4105}, {9, 4130}, {55, 4163}, {100, 24010}, {101, 23970}, {190, 35508}, {200, 3900}, {210, 58329}, {220, 3239}, {312, 57180}, {341, 8641}, {346, 657}, {480, 522}, {644, 3119}, {650, 728}, {663, 5423}, {668, 24012}, {1021, 4515}, {1043, 4524}, {1253, 4397}, {1897, 65752}, {2287, 4171}, {2310, 4578}, {3022, 3699}, {3063, 30693}, {3692, 65103}, {3939, 4081}, {4041, 56182}, {4082, 21789}, {4391, 6602}, {4554, 52064}, {4845, 65448}, {6065, 23615}, {6066, 23104}, {6558, 14936}, {6559, 52614}, {6607, 56118}, {7046, 57108}, {7079, 57055}, {7101, 65102}, {7259, 36197}, {7367, 57049}, {14827, 52622}, {45791, 62747}, {53008, 58338}
X(68135) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 52937}, {9, 36838}, {41, 4617}, {55, 4626}, {100, 24011}, {101, 23586}, {190, 57581}, {200, 4569}, {220, 658}, {346, 46406}, {480, 664}, {522, 57880}, {650, 23062}, {652, 30682}, {657, 279}, {663, 479}, {692, 24013}, {728, 4554}, {1253, 934}, {1802, 65296}, {2175, 6614}, {2287, 4635}, {2310, 59941}, {2328, 4616}, {3022, 3676}, {3063, 738}, {3119, 24002}, {3239, 57792}, {3900, 1088}, {3939, 59457}, {4081, 52621}, {4105, 7}, {4130, 85}, {4163, 6063}, {4171, 1446}, {4524, 3668}, {4845, 65553}, {4936, 62532}, {5423, 4572}, {6066, 59151}, {6559, 65847}, {6602, 651}, {6607, 10481}, {6608, 53242}, {7071, 36118}, {7079, 13149}, {8012, 61241}, {8551, 63203}, {8641, 269}, {10482, 65545}, {14827, 1461}, {14936, 58817}, {23970, 3261}, {24010, 693}, {24012, 513}, {32739, 23971}, {35508, 514}, {51418, 24015}, {52064, 650}, {52614, 62786}, {55965, 42388}, {56182, 4625}, {57064, 67148}, {57108, 7056}, {57180, 57}, {58329, 57785}, {58835, 50561}, {61050, 43924}, {65102, 7177}, {65103, 1847}, {65752, 4025}, {68116, 75}

X(68136) = X(101)X(6065)∩X(514)X(657)

X(68136) lies on the Yff parabola and these lines: {101, 6065}, {169, 4498}, {218, 51652}, {514, 657}, {649, 42325}, {652, 66514}, {3234, 61237}, {3239, 60366}, {3900, 22108}, {4024, 45755}, {4091, 46388}, {16572, 30719}, {30199, 59979}, {38379, 53583}, {52594, 65659}

X(68136) = midpoint of X(21390) and X(62747)
X(68136) = X(190)-Ceva conjugate of X(3870)
X(68136) = X(i)-isoconjugate of X(j) for these (i,j): {277, 1292}, {651, 55013}, {2191, 37206}, {54987, 57656}
X(68136) = X(i)-Dao conjugate of X(j) for these (i,j): {3309, 514}, {38991, 55013}
X(68136) = crosspoint of X(i) and X(j) for these (i,j): {190, 3870}, {1445, 65208}
X(68136) = crosssum of X(649) and X(2191)
X(68136) = crossdifference of every pair of points on line {2191, 2293}
X(68136) = barycentric product X(i)*X(j) for these {i,j}: {1, 68117}, {218, 4468}, {1026, 15636}, {1617, 44448}, {3309, 3870}, {4904, 65208}, {6600, 31605}, {7719, 24562}, {36086, 58281}, {43049, 55337}
X(68136) = barycentric quotient X(i)/X(j) for these {i,j}: {218, 37206}, {663, 55013}, {3870, 54987}, {4468, 57791}, {7719, 65339}, {8642, 2191}, {21059, 1292}, {51652, 40154}, {65208, 63906}, {68117, 75}

X(68137) = X(9)X(514)∩X(101)X(65646)

X(68137) lies on the Yff parabola and these lines: {9, 514}, {101, 65646}, {649, 3730}, {657, 23893}, {672, 52228}, {1023, 38379}, {2516, 7644}, {3234, 61239}, {3239, 55337}, {3294, 4024}, {3309, 65405}, {3887, 6594}, {14077, 15254}, {23100, 32008}, {28292, 60912}, {42462, 62747}, {44827, 52614}, {45755, 53584}, {56244, 66995}

X(68137) = X(190)-Ceva conjugate of X(3935)
X(68137) = X(i)-isoconjugate of X(j) for these (i,j): {1308, 34578}, {37143, 67146}
X(68137) = X(3887)-Dao conjugate of X(514)
X(68137) = crosspoint of X(190) and X(3935)
X(68137) = crosssum of X(649) and X(67146)
X(68137) = barycentric product X(i)*X(j) for these {i,j}: {1, 68102}, {190, 35125}, {3699, 47007}, {3887, 3935}, {5526, 30565}, {17264, 22108}
X(68137) = barycentric quotient X(i)/X(j) for these {i,j}: {3935, 35171}, {5526, 37143}, {8645, 67146}, {19624, 1308}, {22108, 34578}, {35125, 514}, {47007, 3676}, {68102, 75}

X(68138) = X(100)X(31182)∩X(514)X(644)

X(68138) lies on the Yff parabola and these lines: {100, 31182}, {514, 644}, {516, 14506}, {649, 35341}, {1023, 38371}, {2348, 3021}, {3239, 3699}, {3667, 6078}, {3911, 16593}, {16572, 56646}, {39760, 43065}, {58817, 63906}

X(68138) = X(63906)-Ceva conjugate of X(3008)
X(68138) = X(1477)-isoconjugate of X(37626)
X(68138) = X(5853)-Dao conjugate of X(514)
X(68138) = barycentric product X(i)*X(j) for these {i,j}: {190, 35111}, {3021, 3699}
X(68138) = barycentric quotient X(i)/X(j) for these {i,j}: {2348, 37626}, {3021, 3676}, {23704, 43760}, {35111, 514}

X(68139) = X(100)X(514)∩X(101)X(65242)

X(68139) lies on the Yff parabola and these lines: {100, 514}, {101, 65242}, {190, 3239}, {527, 1155}, {649, 21362}, {908, 3234}, {918, 66979}, {1252, 7658}, {3218, 31020}, {3676, 14589}, {3911, 16593}, {3912, 24593}, {4024, 22003}, {4025, 43986}, {4564, 65165}, {6544, 53337}, {9318, 64112}, {17780, 38376}, {21183, 51357}, {23890, 56543}, {25737, 31182}, {46919, 52985}, {53586, 65168}, {57057, 65233}

X(68139) = X(4998)-Ceva conjugate of X(6745)
X(68139) = X(i)-isoconjugate of X(j) for these (i,j): {667, 57565}, {2291, 35348}, {23351, 34056}, {34068, 60479}, {36141, 60579}
X(68139) = X(i)-Dao conjugate of X(j) for these (i,j): {527, 514}, {6366, 42462}, {6594, 23893}, {6631, 57565}, {33573, 11}, {35091, 60579}, {35110, 60479}
X(68139) = trilinear pole of line {6068, 35110}
X(68139) = barycentric product X(i)*X(j) for these {i,j}: {8, 66983}, {190, 35110}, {664, 6068}, {668, 42082}, {1331, 65587}, {1978, 59798}, {3321, 3699}, {4998, 62579}, {6745, 56543}
X(68139) = barycentric quotient X(i)/X(j) for these {i,j}: {190, 57565}, {527, 60479}, {1155, 35348}, {3321, 3676}, {3328, 21132}, {6068, 522}, {6366, 60579}, {6603, 23893}, {6745, 63748}, {23890, 34056}, {35091, 42462}, {35110, 514}, {42082, 513}, {56543, 62723}, {59798, 649}, {62579, 11}, {65587, 46107}, {66983, 7}

X(68140) = X(10)X(514)∩X(101)X(65647)

X(68140) lies on the Yff parabola and these lines: {2, 4024}, {10, 514}, {101, 65647}, {649, 3219}, {1268, 21131}, {2786, 9508}, {3239, 56078}, {3730, 53581}, {4375, 50343}, {4608, 6545}, {6544, 41841}, {6546, 31290}, {8774, 58699}, {14838, 28594}, {20315, 23886}, {21135, 32025}, {21200, 28653}, {21204, 45746}, {27486, 45684}, {40774, 47828}, {55343, 57068}

X(68140) = X(190)-Ceva conjugate of X(6542)
X(68140) = X(i)-isoconjugate of X(j) for these (i,j): {667, 57560}, {1929, 2702}, {9278, 17940}, {17962, 37135}
X(68140) = X(i)-Dao conjugate of X(j) for these (i,j): {2786, 514}, {6631, 57560}, {35080, 6650}, {39041, 37135}, {41841, 35148}, {57461, 1125}
X(68140) = crosspoint of X(190) and X(6542)
X(68140) = crosssum of X(649) and X(17962)
X(68140) = crossdifference of every pair of points on line {1914, 17962}
X(68140) = barycentric product X(i)*X(j) for these {i,j}: {190, 35080}, {2786, 6542}, {9508, 20947}, {17731, 18004}
X(68140) = barycentric quotient X(i)/X(j) for these {i,j}: {190, 57560}, {1326, 17940}, {1757, 37135}, {2786, 6650}, {5029, 17962}, {6541, 66283}, {6542, 35148}, {9508, 1929}, {17731, 17930}, {17735, 2702}, {17990, 2054}, {18004, 11599}, {27929, 40725}, {35080, 514}, {38348, 40767}

X(68141) = X(7)X(514)∩X(101)X(65637)

X(68141) lies on the Yff parabola and these lines: {7, 514}, {101, 65637}, {169, 649}, {513, 23840}, {1023, 38371}, {2826, 10427}, {3239, 56937}, {3309, 17668}, {3762, 53583}, {3970, 4024}, {5784, 30199}, {23100, 63218}

X(68141) = X(190)-Ceva conjugate of X(26015)
X(68141) = X(i)-Dao conjugate of X(j) for these (i,j): {2826, 514}, {65947, 51567}
X(68141) = crosspoint of X(190) and X(26015)
X(68141) = barycentric product X(i)*X(j) for these {i,j}: {190, 65947}, {2826, 26015}, {5580, 52621}
X(68141) = barycentric quotient X(i)/X(j) for these {i,j}: {2826, 51567}, {5580, 3939}, {65947, 514}

X(68142) = X(100)X(53582)∩X(514)X(1635)

X(68142) lies on the Yff parabola and these lines: {100, 53582}, {514, 1635}, {650, 53586}, {900, 2490}, {2516, 28220}, {3667, 6544}, {4024, 43061}, {4375, 45684}, {4765, 28169}, {4786, 31992}, {6006, 52593}, {14435, 47766}, {31182, 47765}, {48576, 53587}

X(68142) = midpoint of X(i) and X(j) for these {i,j}: {4786, 31992}, {14435, 47766}
X(68142) = X(190)-Ceva conjugate of X(3241)
X(68142) = X(i)-isoconjugate of X(j) for these (i,j): {6014, 39963}, {41436, 65235}
X(68142) = X(6006)-Dao conjugate of X(514)
X(68142) = crosspoint of X(190) and X(3241)
X(68142) = crosssum of X(649) and X(41436)
X(68142) = crossdifference of every pair of points on line {2177, 8162}
X(68142) = barycentric product X(i)*X(j) for these {i,j}: {3241, 6006}, {30829, 66524}
X(68142) = barycentric quotient X(i)/X(j) for these {i,j}: {3241, 53659}, {6006, 36588}, {8656, 41436}, {16670, 65235}, {66524, 39963}

X(68143) = X(514)X(3241)∩X(649)X(3306)

X(68143) lies on the Yff parabola and these lines: {514, 3241}, {649, 3306}, {812, 6544}, {900, 53583}, {3239, 31992}, {4024, 47664}, {6545, 52620}, {6546, 53584}, {31182, 45684}, {47652, 53587}, {49298, 53586}, {53337, 53582}

X(68143) = X(190)-Ceva conjugate of X(41140)
X(68143) = X(6017)-isoconjugate of X(55935)
X(68143) = X(6009)-Dao conjugate of X(514)
X(68143) = crosspoint of X(190) and X(41140)
X(68143) = barycentric product X(6009)*X(41140)

X(68144) = X(57)X(649)∩X(145)X(514)

X(68144) lies on the Yff parabola and these lines: {2, 31182}, {57, 649}, {145, 514}, {169, 4498}, {812, 53583}, {2496, 59835}, {2976, 3021}, {3175, 4024}, {3239, 4382}, {4728, 6009}, {10196, 59752}, {21129, 53582}, {30719, 63574}, {48147, 53587}, {49296, 53586}

X(68144) = X(190)-Ceva conjugate of X(3008)
X(68144) = X(1280)-isoconjugate of X(6078)
X(68144) = X(i)-Dao conjugate of X(j) for these (i,j): {5853, 6558}, {6084, 514}, {61074, 36807}
X(68144) = crosspoint of X(190) and X(3008)
X(68144) = barycentric product X(i)*X(j) for these {i,j}: {1, 68121}, {190, 61074}, {3008, 6084}, {3021, 3676}, {35111, 58817}
X(68144) = barycentric quotient X(i)/X(j) for these {i,j}: {3021, 3699}, {6084, 36807}, {35111, 6558}, {48032, 1280}, {61074, 514}, {68121, 75}

X(68145) = X(190)X(51560)∩X(514)X(48304)

X(68145) lies on the Yff parabola and these lines: {190, 51560}, {514, 48304}, {522, 4659}, {649, 693}, {657, 17335}, {3239, 17494}, {4024, 4468}, {4375, 48172}, {4384, 45755}, {4498, 53581}, {4702, 4724}, {6544, 47787}, {7192, 53586}, {28843, 44550}, {29627, 54264}, {30565, 53584}, {31182, 48008}, {47658, 53585}, {47832, 62552}, {48141, 53587}

X(68145) = X(190)-Ceva conjugate of X(4384)
X(68145) = X(i)-isoconjugate of X(j) for these (i,j): {1002, 8693}, {2279, 37138}, {32724, 62622}
X(68145) = X(i)-Dao conjugate of X(j) for these (i,j): {4762, 514}, {33570, 2254}, {55059, 60677}, {61076, 27475}
X(68145) = crosspoint of X(190) and X(4384)
X(68145) = crosssum of X(649) and X(2279)
X(68145) = crossdifference of every pair of points on line {869, 2279}
X(68145) = barycentric product X(i)*X(j) for these {i,j}: {8, 66987}, {190, 61076}, {4384, 4762}, {4441, 4724}, {21615, 66513}, {45755, 60720}
X(68145) = barycentric quotient X(i)/X(j) for these {i,j}: {1001, 37138}, {2280, 8693}, {4384, 32041}, {4724, 1002}, {4762, 27475}, {45755, 40779}, {61076, 514}, {63229, 53227}, {66513, 2279}, {66987, 7}

X(68146) = X(40)X(649)∩X(144)X(514)

X(68146) lies on the Yff parabola, the Mandart hyperbola, and these lines: {8, 3239}, {40, 649}, {72, 4024}, {144, 514}, {1023, 3234}, {1145, 6544}, {3057, 14298}, {3059, 3900}, {3588, 53581}, {3650, 53587}, {3904, 53583}, {6068, 6366}, {8611, 21677}, {18239, 30199}, {23890, 56543}, {36922, 53584}, {41852, 53585}

X(68146) = X(190)-Ceva conjugate of X(6745)
X(68146) = X(i)-isoconjugate of X(j) for these (i,j): {667, 57563}, {14733, 34056}, {18889, 65553}, {34068, 60487}, {36141, 62723}
X(68146) = X(i)-Dao conjugate of X(j) for these (i,j): {527, 658}, {2968, 57565}, {6366, 514}, {6594, 37139}, {6631, 57563}, {33573, 7}, {35091, 62723}, {35110, 60487}, {52870, 65553}, {62579, 60479}
X(68146) = crosspoint of X(190) and X(6745)
X(68146) = barycentric product X(i)*X(j) for these {i,j}: {8, 62579}, {190, 35091}, {522, 6068}, {1323, 65448}, {3239, 35110}, {3321, 4163}, {3328, 3699}, {4081, 66983}, {4397, 42082}, {6366, 6745}, {14392, 30806}, {52622, 59798}, {57108, 65587}
X(68146) = barycentric quotient X(i)/X(j) for these {i,j}: {190, 57563}, {527, 60487}, {1323, 65553}, {3239, 57565}, {3321, 4626}, {3328, 3676}, {6068, 664}, {6366, 62723}, {6603, 37139}, {6745, 35157}, {14392, 1156}, {33573, 60479}, {35091, 514}, {35110, 658}, {42082, 934}, {52333, 21132}, {59798, 1461}, {60431, 65335}, {62579, 7}, {65680, 34056}, {66983, 59457}

X(68147) = X(110)X(901)∩X(523)X(7477)

X(68147) lies on the Kiepert parabola and these lines: {100, 7253}, {110, 901}, {517, 859}, {523, 7477}, {643, 23181}, {664, 7192}, {669, 16680}, {1325, 3233}, {2397, 4246}, {2783, 66294}, {2803, 68104}, {3265, 53332}, {3658, 23832}, {4557, 51562}, {35259, 57520}, {48382, 67454}, {48391, 53743}, {62555, 62643}

X(68147) = X(99)-Ceva conjugate of X(64828)
X(68147) = X(i)-isoconjugate of X(j) for these (i,j): {661, 59196}, {798, 57550}, {1577, 41933}, {2250, 2401}, {34234, 55259}
X(68147) = X(i)-Dao conjugate of X(j) for these (i,j): {517, 523}, {31998, 57550}, {36830, 59196}, {57293, 18210}
X(68147) = crosspoint of X(99) and X(64828)
X(68147) = crosssum of X(512) and X(55259)
X(68147) = trilinear pole of line {23980, 59800}
X(68147) = barycentric product X(i)*X(j) for these {i,j}: {81, 15632}, {99, 23980}, {110, 26611}, {333, 66977}, {517, 64828}, {645, 1361}, {648, 65743}, {662, 24028}, {670, 59800}, {799, 42078}, {859, 2397}, {2427, 17139}, {4558, 21664}, {4563, 42072}, {4565, 55016}, {4567, 42757}, {52378, 66969}, {61057, 62534}
X(68147) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57550}, {110, 59196}, {859, 2401}, {1361, 7178}, {1576, 41933}, {2397, 57984}, {2427, 38955}, {4246, 16082}, {15632, 321}, {21664, 14618}, {23980, 523}, {24028, 1577}, {26611, 850}, {42072, 2501}, {42078, 661}, {42746, 52499}, {42757, 16732}, {59800, 512}, {61057, 7180}, {64828, 18816}, {65743, 525}, {66977, 226}

X(68148) = X(110)X(6078)∩X(523)X(4436)

X(68148) lies on the Kiepert parabola and these lines: {99, 36802}, {100, 7192}, {110, 6078}, {190, 7253}, {518, 2223}, {523, 4436}, {660, 662}, {669, 53280}, {677, 4558}, {883, 2283}, {2284, 54353}, {2414, 4238}, {2795, 66290}, {2878, 68105}, {3233, 7469}, {3952, 63918}, {4437, 20776}, {4567, 21789}, {53322, 58766}

X(68148) = isotomic conjugate of the polar conjugate of X(68086)
X(68148) = X(i)-isoconjugate of X(j) for these (i,j): {523, 51838}, {661, 6185}, {673, 55261}, {798, 57537}, {1024, 66941}, {1027, 13576}, {1577, 41934}, {4017, 62715}, {10099, 36124}, {18785, 62635}
X(68148) = X(i)-Dao conjugate of X(j) for these (i,j): {518, 523}, {17435, 16732}, {31998, 57537}, {34961, 62715}, {36830, 6185}
X(68148) = crosssum of X(512) and X(55261)
X(68148) = trilinear pole of line {6184, 39686}
X(68148) = crossdifference of every pair of points on line {39786, 55261}
X(68148) = barycentric product X(i)*X(j) for these {i,j}: {69, 68086}, {81, 68106}, {99, 6184}, {100, 16728}, {110, 4437}, {333, 66978}, {645, 1362}, {648, 65744}, {662, 4712}, {670, 39686}, {799, 42079}, {1026, 18206}, {2223, 55260}, {2284, 30941}, {3126, 4567}, {3286, 42720}, {3912, 54353}, {4238, 25083}, {4558, 34337}, {4563, 42071}, {4570, 53583}, {6331, 20776}, {18157, 54325}, {42747, 52502}, {61055, 62534}
X(68148) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57537}, {110, 6185}, {163, 51838}, {1362, 7178}, {1576, 41934}, {2223, 55261}, {2283, 66941}, {2284, 13576}, {3126, 16732}, {3286, 62635}, {4238, 54235}, {4437, 850}, {4712, 1577}, {5546, 62715}, {6184, 523}, {15615, 63462}, {16728, 693}, {20683, 66282}, {20752, 10099}, {20776, 647}, {23612, 24290}, {24290, 66290}, {34337, 14618}, {39686, 512}, {42071, 2501}, {42079, 661}, {42747, 46784}, {53583, 21207}, {54325, 18785}, {54353, 673}, {61055, 7180}, {65744, 525}, {66978, 226}, {68086, 4}, {68106, 321}, {68115, 36197}

X(68149) = X(110)X(6079)∩X(523)X(4427)

X(68149) lies on the Kiepert parabola and these lines: {8, 56950}, {99, 901}, {100, 3733}, {110, 6079}, {519, 902}, {523, 4427}, {643, 765}, {669, 53268}, {1649, 62644}, {2415, 46541}, {2796, 66288}, {3233, 7478}, {3939, 9059}, {4062, 51578}, {4237, 56797}, {4557, 53685}, {8683, 34594}, {17145, 54391}, {17780, 23344}, {20045, 62740}, {21290, 61479}, {22371, 36791}, {39766, 40091}, {52924, 55243}

X(68149) = X(4600)-Ceva conjugate of X(16704)
X(68149) = X(i)-isoconjugate of X(j) for these (i,j): {88, 55263}, {106, 55244}, {512, 679}, {649, 30575}, {661, 2226}, {669, 57929}, {798, 54974}, {1318, 4017}, {1577, 41935}, {3122, 4618}, {3125, 4638}, {4049, 9456}, {4674, 23345}, {4730, 59150}, {36125, 66924}
X(68149) = X(i)-Dao conjugate of X(j) for these (i,j): {214, 55244}, {519, 523}, {1647, 3120}, {4370, 4049}, {5375, 30575}, {31998, 54974}, {34961, 1318}, {36830, 2226}, {39054, 679}, {52872, 66285}
X(68149) = crosssum of X(512) and X(55263)
X(68149) = trilinear pole of line {1017, 4370}
X(68149) = barycentric product X(i)*X(j) for these {i,j}: {44, 55243}, {81, 68107}, {86, 53582}, {99, 4370}, {100, 16729}, {110, 36791}, {333, 66979}, {645, 1317}, {648, 65742}, {662, 4738}, {670, 1017}, {678, 799}, {902, 55262}, {1023, 30939}, {3251, 4601}, {3977, 46541}, {4152, 4573}, {4542, 55194}, {4543, 4620}, {4558, 65585}, {4563, 42070}, {4567, 68101}, {4570, 52627}, {4591, 58254}, {4600, 6544}, {4615, 8028}, {4623, 21821}, {6331, 22371}, {16704, 17780}, {24004, 52680}, {41629, 66962}, {61047, 62534}
X(68149) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 55244}, {99, 54974}, {100, 30575}, {110, 2226}, {519, 4049}, {662, 679}, {678, 661}, {799, 57929}, {902, 55263}, {1017, 512}, {1023, 4674}, {1317, 7178}, {1576, 41935}, {3251, 3125}, {3285, 23345}, {3689, 61179}, {3943, 66285}, {4120, 66288}, {4152, 3700}, {4169, 4013}, {4370, 523}, {4542, 55195}, {4543, 21044}, {4567, 4618}, {4570, 4638}, {4591, 59150}, {4738, 1577}, {5546, 1318}, {6544, 3120}, {8028, 4120}, {16704, 6548}, {16729, 693}, {17780, 4080}, {21821, 4705}, {22356, 66924}, {22371, 647}, {36791, 850}, {39771, 53545}, {42070, 2501}, {46541, 6336}, {52627, 21207}, {52680, 1022}, {53582, 10}, {55243, 20568}, {55262, 57995}, {61047, 7180}, {65585, 14618}, {65742, 525}, {66962, 4052}, {66979, 226}, {68101, 16732}, {68107, 321}

X(68150) = X(110)X(6080)∩X(523)X(2071)

X(68150) lies on the Kiepert parabola and these lines: {3, 2416}, {110, 6080}, {160, 56306}, {520, 4091}, {523, 2071}, {525, 15781}, {669, 684}, {1092, 23103}, {1649, 6503}, {2797, 66299}, {2972, 34950}, {3233, 7480}, {3265, 15414}, {3964, 4143}, {4558, 65305}, {5489, 16391}, {6368, 41077}, {9723, 57069}, {10607, 38354}, {11413, 46612}, {38942, 62173}, {46616, 59744}, {53263, 58766}

X(68150) = midpoint of X(684) and X(14329)
X(68150) = isotomic conjugate of the polar conjugate of X(32320)
X(68150) = isogonal conjugate of the polar conjugate of X(52613)
X(68150) = X(34287)-anticomplementary conjugate of X(21294)
X(68150) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 394}, {16391, 2972}, {52613, 32320}
X(68150) = X(i)-isoconjugate of X(j) for these (i,j): {4, 36126}, {19, 15352}, {92, 6529}, {107, 158}, {112, 6521}, {162, 1093}, {393, 823}, {523, 24021}, {648, 6520}, {661, 34538}, {798, 57556}, {799, 36434}, {811, 6524}, {850, 24022}, {1096, 6528}, {1577, 23590}, {1896, 36127}, {2052, 24019}, {2179, 42401}, {2181, 52779}, {2207, 57973}, {8748, 54240}, {20948, 23975}, {23999, 58757}, {24000, 66299}, {24006, 32230}, {32713, 57806}, {36043, 51385}, {36119, 58071}, {52439, 57968}
X(68150) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 15352}, {125, 1093}, {130, 14569}, {520, 523}, {1147, 107}, {1511, 58071}, {2972, 13450}, {6503, 6528}, {14390, 65181}, {17423, 6524}, {17434, 14618}, {22391, 6529}, {31998, 57556}, {34591, 6521}, {35071, 2052}, {35579, 51385}, {36033, 36126}, {36830, 34538}, {37867, 648}, {38985, 158}, {38996, 36434}, {38999, 52661}, {46093, 4}, {55066, 6520}, {62573, 18027}, {62603, 42401}, {66896, 35709}
X(68150) = crosspoint of X(i) and X(j) for these (i,j): {99, 394}, {57414, 59077}
X(68150) = crosssum of X(i) and X(j) for these (i,j): {393, 512}, {520, 59361}, {523, 6247}, {1093, 66299}, {2501, 14569}
X(68150) = crossdifference of every pair of points on line {393, 800}
vbarycentric product X(i)*X(j) for these {i,j}: {3, 52613}, {69, 32320}, {99, 35071}, {100, 16730}, {163, 24020}, {184, 4143}, {255, 24018}, {326, 822}, {394, 520}, {418, 15414}, {525, 1092}, {577, 3265}, {645, 1363}, {647, 3964}, {656, 6507}, {799, 42080}, {810, 1102}, {1576, 23974}, {2430, 62347}, {2972, 4558}, {3049, 4176}, {3267, 23606}, {3682, 4091}, {3719, 51640}, {3926, 39201}, {3990, 4131}, {3998, 23224}, {4055, 30805}, {4100, 14208}, {4158, 7254}, {4563, 34980}, {4573, 7065}, {4574, 7215}, {4592, 37754}, {6331, 66896}, {14379, 20580}, {14585, 52617}, {15394, 58796}, {16391, 52584}, {18604, 57109}, {19210, 60597}, {23103, 23582}, {23616, 47390}, {34386, 58305}, {36054, 52385}, {36433, 44173}, {40152, 57241}, {46088, 52347}, {51394, 62665}, {57414, 58763}
X(68150) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 15352}, {48, 36126}, {95, 42401}, {97, 52779}, {99, 57556}, {110, 34538}, {163, 24021}, {184, 6529}, {255, 823}, {326, 57973}, {394, 6528}, {417, 41678}, {418, 61193}, {520, 2052}, {577, 107}, {647, 1093}, {656, 6521}, {669, 36434}, {810, 6520}, {822, 158}, {1092, 648}, {1102, 57968}, {1363, 7178}, {1576, 23590}, {1636, 52661}, {2972, 14618}, {3049, 6524}, {3265, 18027}, {3269, 66299}, {3284, 58071}, {3964, 6331}, {4100, 162}, {4143, 18022}, {5562, 65183}, {6507, 811}, {7065, 3700}, {14379, 65181}, {14574, 23975}, {14585, 32713}, {15414, 57844}, {16391, 30450}, {16730, 693}, {17434, 13450}, {19210, 16813}, {22341, 54240}, {23103, 15526}, {23286, 8794}, {23606, 112}, {23613, 3269}, {23974, 44173}, {24018, 57806}, {24020, 20948}, {32320, 4}, {32661, 32230}, {34384, 42369}, {34386, 54950}, {34980, 2501}, {35071, 523}, {36054, 1896}, {36433, 1576}, {37754, 24006}, {39201, 393}, {40152, 52938}, {41219, 12077}, {42080, 661}, {42293, 14569}, {46088, 8884}, {52430, 24019}, {52584, 59139}, {52613, 264}, {58305, 53}, {58310, 2207}, {58796, 14249}, {59176, 65176}, {60597, 62275}, {61355, 61217}, {66896, 647}

X(68151) = X(110)X(6081)∩X(523)X(57590)

X(68151) lies on the Kiepert parabola and these lines: {21, 2417}, {110, 6081}, {521, 1946}, {523, 57590}, {643, 23181}, {677, 4558}, {1812, 63744}, {2798, 66297}, {3233, 37966}, {3265, 63245}, {3733, 48383}, {4131, 23224}, {4990, 7253}, {7192, 65868}, {7254, 52613}, {16731, 66898}, {17898, 37228}, {53308, 57242}, {53309, 58766}, {58340, 68108}

X(68151) = isotomic conjugate of the polar conjugate of X(23090)
X(68151) = isogonal conjugate of the polar conjugate of X(15411)
X(68151) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 1812}, {4558, 2327}, {4592, 394}, {15411, 23090}
X(68151) = X(i)-cross conjugate of X(j) for these (i,j): {24031, 1259}, {34591, 6512}, {57241, 57081}
X(68151) = X(i)-isoconjugate of X(j) for these (i,j): {19, 52607}, {34, 61178}, {65, 36127}, {107, 1254}, {108, 225}, {158, 53321}, {393, 1020}, {512, 24032}, {523, 24033}, {608, 65207}, {653, 1880}, {661, 23984}, {798, 57538}, {1096, 4566}, {1118, 4551}, {1400, 54240}, {1402, 52938}, {1425, 36126}, {1426, 1897}, {1577, 23985}, {1824, 36118}, {1826, 32714}, {2333, 13149}, {2501, 7128}, {4605, 5317}, {6354, 24019}, {6520, 52610}, {6529, 37755}, {7045, 58757}, {8736, 65232}, {18026, 57652}, {21935, 52775}, {24027, 66299}, {32674, 40149}, {46102, 55208}, {52577, 66952}
X(68151) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 52607}, {521, 523}, {522, 66299}, {656, 24006}, {1147, 53321}, {3239, 14618}, {6503, 4566}, {7358, 41013}, {11517, 61178}, {17115, 58757}, {31998, 57538}, {34467, 1426}, {35071, 6354}, {35072, 40149}, {36830, 23984}, {37867, 52610}, {38983, 225}, {38985, 1254}, {39054, 24032}, {40582, 54240}, {40602, 36127}, {40605, 52938}, {40626, 57809}, {46093, 1425}, {55068, 158}, {62647, 65207}
X(68151) = cevapoint of X(57057) and X(58340)
X(68151) = crosspoint of X(99) and X(1812)
X(68151) = crosssum of X(512) and X(1880)
X(68151) = trilinear pole of line {35072, 39687}
X(68151) = crossdifference of every pair of points on line {1880, 52577}
X(68151) = barycentric product X(i)*X(j) for these {i,j}: {3, 15411}, {63, 57081}, {69, 23090}, {81, 68108}, {86, 57057}, {99, 35072}, {100, 16731}, {110, 23983}, {271, 57213}, {274, 58340}, {283, 6332}, {284, 52616}, {304, 57134}, {314, 36054}, {326, 1021}, {332, 652}, {333, 57241}, {345, 23189}, {348, 58338}, {394, 7253}, {520, 7058}, {521, 1812}, {522, 6514}, {645, 1364}, {662, 24031}, {670, 39687}, {799, 2638}, {905, 1792}, {1043, 4091}, {1098, 24018}, {1259, 4560}, {1260, 15419}, {1264, 7252}, {1265, 7254}, {1437, 15416}, {1444, 57055}, {2193, 35518}, {2287, 4131}, {2289, 18155}, {2327, 4025}, {2328, 30805}, {2968, 4558}, {3265, 7054}, {3270, 4563}, {3719, 3737}, {3926, 21789}, {3964, 17926}, {3998, 65575}, {4397, 18604}, {4554, 66898}, {4587, 17219}, {4592, 34591}, {7183, 58329}, {17206, 57108}, {52355, 65568}, {52613, 59482}, {61054, 62534}
X(68151) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 52607}, {21, 54240}, {78, 65207}, {99, 57538}, {110, 23984}, {163, 24033}, {219, 61178}, {255, 1020}, {283, 653}, {284, 36127}, {332, 46404}, {333, 52938}, {394, 4566}, {520, 6354}, {521, 40149}, {577, 53321}, {652, 225}, {662, 24032}, {822, 1254}, {1021, 158}, {1092, 52610}, {1098, 823}, {1146, 66299}, {1259, 4552}, {1364, 7178}, {1437, 32714}, {1444, 13149}, {1576, 23985}, {1790, 36118}, {1792, 6335}, {1793, 65329}, {1812, 18026}, {1946, 1880}, {2193, 108}, {2289, 4551}, {2326, 36126}, {2327, 1897}, {2638, 661}, {2968, 14618}, {3270, 2501}, {3682, 4605}, {4091, 3668}, {4131, 1446}, {4558, 55346}, {4575, 7128}, {6056, 4559}, {6332, 57809}, {6514, 664}, {7054, 107}, {7058, 6528}, {7252, 1118}, {7253, 2052}, {7254, 1119}, {8611, 56285}, {14936, 58757}, {15411, 264}, {16731, 693}, {17926, 1093}, {18604, 934}, {21789, 393}, {22383, 1426}, {23090, 4}, {23189, 278}, {23224, 1427}, {23614, 53560}, {23983, 850}, {24031, 1577}, {32320, 1425}, {34591, 24006}, {35072, 523}, {35518, 52575}, {36054, 65}, {39687, 512}, {51640, 7147}, {52613, 6356}, {52616, 349}, {53560, 66297}, {57055, 41013}, {57057, 10}, {57081, 92}, {57108, 1826}, {57134, 19}, {57213, 342}, {57241, 226}, {58338, 281}, {58340, 37}, {59482, 15352}, {60794, 65175}, {61054, 7180}, {65102, 1824}, {66898, 650}, {68108, 321}
X(68151) = {X(23189),X(58338)}-harmonic conjugate of X(57081)

X(68152) = X(99)X(523)∩X(110)X(6082)

X(68152) lies on the Kiepert parabola and these lines: {2, 5914}, {6, 47047}, {32, 51999}, {99, 523}, {110, 6082}, {126, 230}, {187, 524}, {325, 3233}, {385, 31128}, {526, 15631}, {543, 40553}, {620, 40486}, {669, 1634}, {1576, 58766}, {1649, 5467}, {2407, 18311}, {2418, 4235}, {2528, 61219}, {3053, 34161}, {3265, 4558}, {3933, 14357}, {3964, 65712}, {5007, 40517}, {5181, 39072}, {5201, 35298}, {6189, 30508}, {6190, 30509}, {6722, 66353}, {8029, 62672}, {8591, 17948}, {9009, 67367}, {9155, 47550}, {9170, 31614}, {9888, 42007}, {10418, 67396}, {11053, 41176}, {13162, 44386}, {14360, 16092}, {15300, 35087}, {18310, 50941}, {22329, 62299}, {23342, 34245}, {23991, 31274}, {33875, 51322}, {34760, 62662}, {36883, 51894}, {37803, 47241}, {39356, 44372}, {40429, 62427}, {44369, 62338}, {47238, 62310}, {48947, 67566}, {53136, 67400}, {53274, 55271}, {61444, 64212}

X(68152) = midpoint of X(i) and X(j) for these {i,j}: {99, 9182}, {4590, 14588}, {8591, 17948}, {15300, 35087}
X(68152) = reflection of X(i) in X(j) for these {i,j}: {2, 9164}, {23991, 36953}, {44398, 620}, {64258, 40553}
X(68152) = isotomic conjugate of the polar conjugate of X(68087)
X(68152) = X(i)-complementary conjugate of X(j) for these (i,j): {40429, 21256}, {57728, 4892}
X(68152) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 5468}, {4590, 524}, {57552, 38239}
X(68152) = X(i)-cross conjugate of X(j) for these (i,j): {1649, 2482}, {33915, 524}
X(68152) = X(i)-isoconjugate of X(j) for these (i,j): {111, 23894}, {661, 10630}, {798, 57539}, {897, 9178}, {923, 5466}, {1577, 41936}, {2643, 34574}, {10097, 36128}, {36142, 64258}
X(68152) = X(i)-Dao conjugate of X(j) for these (i,j): {187, 10561}, {524, 523}, {690, 8029}, {1648, 115}, {2482, 5466}, {5468, 14061}, {6593, 9178}, {14961, 65609}, {23992, 64258}, {31998, 57539}, {36830, 10630}, {41177, 44398}, {52881, 14977}, {62563, 10555}, {62594, 51258}
X(68152) = cevapoint of X(1649) and X(2482)
X(68152) = crosspoint of X(i) and X(j) for these (i,j): {99, 5468}, {524, 36953}
X(68152) = crosssum of X(i) and X(j) for these (i,j): {111, 39024}, {512, 9178}
X(68152) = trilinear pole of line {2482, 8030}
X(68152) = crossdifference of every pair of points on line {9178, 21906}
X(68152) = barycentric product X(i)*X(j) for these {i,j}: {69, 68087}, {81, 68109}, {99, 2482}, {100, 16733}, {110, 36792}, {249, 52629}, {524, 5468}, {645, 1366}, {648, 65747}, {662, 24038}, {670, 39689}, {691, 23106}, {799, 42081}, {892, 8030}, {896, 24039}, {1649, 4590}, {1992, 66963}, {2418, 27088}, {3266, 5467}, {4235, 6390}, {4558, 34336}, {4563, 5095}, {4573, 7067}, {4610, 52068}, {9146, 20380}, {14210, 23889}, {14444, 64460}, {16702, 42721}, {17708, 62661}, {23992, 31614}, {33915, 52940}, {34537, 54274}, {42370, 46049}, {47389, 58780}, {66625, 66626}
X(68152) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57539}, {110, 10630}, {187, 9178}, {249, 34574}, {524, 5466}, {690, 64258}, {896, 23894}, {1366, 7178}, {1576, 41936}, {1641, 18007}, {1649, 115}, {2482, 523}, {3266, 52632}, {3292, 10097}, {4235, 17983}, {4558, 15398}, {5095, 2501}, {5181, 65609}, {5467, 111}, {5468, 671}, {6390, 14977}, {6593, 10561}, {6629, 62626}, {7067, 3700}, {8030, 690}, {9155, 8430}, {14417, 51258}, {14443, 61339}, {14444, 33919}, {16733, 693}, {18311, 10555}, {20380, 8599}, {23106, 35522}, {23889, 897}, {23992, 8029}, {24038, 1577}, {24039, 46277}, {27088, 2408}, {30454, 20578}, {30455, 20579}, {31614, 57552}, {33915, 1648}, {34336, 14618}, {36792, 850}, {39689, 512}, {39785, 23288}, {42081, 661}, {46049, 42344}, {50567, 62629}, {52068, 4024}, {52629, 338}, {54274, 3124}, {58780, 8754}, {59152, 34539}, {59801, 22260}, {61207, 8753}, {62656, 9134}, {62661, 9979}, {65747, 525}, {66963, 5485}, {68087, 4}, {68109, 321}
X(68152) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {99, 4590, 9182}, {187, 2482, 47077}, {325, 7664, 46986}, {9182, 14588, 99}, {44397, 64258, 40553}

X(68153) = X(100)X(669)∩X(190)X(523)

X(68153) lies on the Kiepert parabola and these lines: {45, 46912}, {99, 65250}, {100, 669}, {110, 6064}, {190, 523}, {238, 239}, {512, 660}, {850, 36803}, {874, 3573}, {1018, 4613}, {3700, 3952}, {3712, 8299}, {3733, 4436}, {4151, 32094}, {4427, 7192}, {4595, 23861}, {6654, 17163}, {7253, 53338}, {8639, 9266}, {20356, 59720}

X(68153) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 3570}, {1016, 2238}, {24037, 385}
X(68153) = X(i)-isoconjugate of X(j) for these (i,j): {291, 66937}, {741, 876}, {798, 57554}, {875, 18827}, {1019, 52205}, {3572, 37128}, {3733, 30663}, {4017, 62714}, {4444, 18268}, {7199, 51856}, {18267, 52619}, {40098, 57129}
X(68153) = X(i)-Dao conjugate of X(j) for these (i,j): {740, 523}, {1966, 7199}, {4155, 22260}, {8299, 876}, {31998, 57554}, {34961, 62714}, {35068, 4444}, {39029, 66937}, {39786, 1086}, {62553, 66286}
X(68153) = crosspoint of X(99) and X(3570)
X(68153) = crosssum of X(512) and X(3572)
X(68153) = trilinear pole of line {4368, 35068}
X(68153) = barycentric product X(i)*X(j) for these {i,j}: {99, 35068}, {190, 4368}, {645, 3027}, {740, 3570}, {799, 4094}, {874, 2238}, {1018, 39044}, {3573, 3948}, {3747, 27853}, {3952, 4366}, {4033, 8300}, {4154, 27805}, {4557, 56660}, {27808, 51328}, {27926, 66283}, {61059, 62534}
X(68153) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57554}, {740, 4444}, {874, 40017}, {1018, 30663}, {1914, 66937}, {2238, 876}, {3027, 7178}, {3570, 18827}, {3573, 37128}, {3747, 3572}, {3802, 4481}, {3948, 66286}, {3952, 40098}, {3985, 60577}, {4037, 35352}, {4094, 661}, {4154, 4369}, {4366, 7192}, {4368, 514}, {4375, 17205}, {4557, 52205}, {5546, 62714}, {8300, 1019}, {27855, 16727}, {27919, 23829}, {35068, 523}, {39044, 7199}, {41333, 875}, {51328, 3733}, {53681, 17212}, {56660, 52619}, {61059, 7180}
X(68153) = {X(4094),X(4154)}-harmonic conjugate of X(4366)

X(68154) = X(100)X(523)∩X(110)X(6083)

X(68154) lies on the Kiepert parabola and these lines: {36, 214}, {100, 523}, {110, 6083}, {512, 901}, {644, 23084}, {647, 1252}, {669, 23861}, {850, 4998}, {3233, 42746}, {3265, 6516}, {3712, 8299}, {3733, 53280}, {3952, 23067}, {4132, 67445}, {4427, 7253}, {4557, 55246}, {6163, 65313}, {7192, 17136}, {8672, 14513}, {15635, 56751}, {16598, 65739}, {17145, 54391}, {17757, 36195}, {31296, 43986}, {41405, 42664}, {50557, 54110}, {51562, 64868}

X(68154) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 4585}, {4998, 3936}, {24041, 323}
X(68154) = X(i)-isoconjugate of X(j) for these (i,j): {759, 66284}, {798, 57555}, {3737, 63750}, {4017, 62713}, {7252, 34535}, {34079, 60074}, {36069, 66289}
X(68154) = X(i)-Dao conjugate of X(j) for these (i,j): {758, 523}, {3738, 56283}, {6149, 3737}, {6370, 23105}, {31998, 57555}, {34586, 66284}, {34961, 62713}, {35069, 60074}, {38982, 66289}
X(68154) = crosspoint of X(99) and X(4585)
X(68154) = trilinear pole of line {35069, 65746}
X(68154) = barycentric product X(i)*X(j) for these {i,j}: {99, 35069}, {645, 3028}, {648, 65746}, {662, 4736}, {758, 4585}, {1983, 35550}, {4552, 4996}, {27808, 52059}, {61060, 62534}
X(68154) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57555}, {215, 7252}, {758, 60074}, {1983, 759}, {2245, 66284}, {2610, 66289}, {3028, 7178}, {4551, 34535}, {4552, 57645}, {4559, 63750}, {4585, 14616}, {4736, 1577}, {4996, 4560}, {5546, 62713}, {34544, 3737}, {35069, 523}, {35128, 56283}, {52059, 3733}, {57174, 18191}, {61060, 7180}, {65746, 525}, {66968, 17197}

X(68155) = X(2)X(669)∩X(6)X(523)

X(68155) lies on the Kiepert parabola and these lines: {2, 669}, {6, 523}, {83, 23099}, {99, 14606}, {187, 46302}, {384, 14824}, {512, 7804}, {804, 4107}, {880, 17941}, {1576, 57991}, {1649, 7711}, {2528, 11123}, {2782, 39501}, {2793, 32135}, {3265, 10190}, {3733, 59631}, {4226, 41337}, {5092, 32472}, {5108, 59786}, {5466, 62889}, {5468, 38366}, {5489, 51244}, {7253, 38348}, {8029, 58784}, {8289, 60226}, {9479, 19571}, {11182, 14318}, {18092, 18105}, {21006, 41328}, {23342, 34245}, {37912, 58780}, {38382, 53272}, {41761, 56739}, {45662, 62173}, {51510, 60863}, {60028, 60855}

X(68155) = midpoint of X(6) and X(46778)
X(68155) = reflection of X(66267) in X(62688)
X(68155) = X(i)-Ceva conjugate of X(j) for these (i,j): {83, 2086}, {99, 385}, {4027, 35078}, {46294, 4027}
X(68155) = X(35078)-cross conjugate of X(4027)
X(68155) = X(i)-isoconjugate of X(j) for these (i,j): {662, 41517}, {694, 37134}, {798, 57558}, {805, 1581}, {1934, 17938}, {1967, 18829}, {4602, 66998}, {43763, 46161}, {65351, 66942}
X(68155) = X(i)-Dao conjugate of X(j) for these (i,j): {804, 523}, {1084, 41517}, {2086, 47648}, {2491, 67070}, {8290, 18829}, {19576, 805}, {31998, 57558}, {35078, 1916}, {36213, 46161}, {39043, 37134}, {39786, 40099}, {41178, 141}, {62649, 882}
X(68155) = crosspoint of X(i) and X(j) for these (i,j): {99, 385}, {4027, 46294}, {17941, 40820}
X(68155) = crosssum of X(i) and X(j) for these (i,j): {512, 694}, {882, 40810}
X(68155) = crossdifference of every pair of points on line {511, 694}
X(68155) = barycentric product X(i)*X(j) for these {i,j}: {99, 35078}, {115, 46294}, {385, 804}, {419, 24284}, {523, 4027}, {850, 51318}, {880, 2086}, {1577, 51903}, {1691, 14295}, {2533, 53681}, {2643, 46295}, {3114, 58779}, {3978, 5027}, {4010, 27982}, {4039, 4107}, {4154, 4369}, {11183, 60863}, {36897, 58850}
X(68155) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57558}, {385, 18829}, {419, 65351}, {512, 41517}, {804, 1916}, {1580, 37134}, {1691, 805}, {2086, 882}, {2679, 67070}, {4027, 99}, {4154, 27805}, {5027, 694}, {8623, 46161}, {9426, 66998}, {14295, 18896}, {14602, 17938}, {17941, 39292}, {24284, 40708}, {27982, 4589}, {35078, 523}, {40820, 39291}, {42652, 51494}, {46294, 4590}, {46295, 24037}, {51318, 110}, {51903, 662}, {53681, 4594}, {56976, 41209}, {58752, 42061}, {58779, 3094}, {58850, 5976}, {62649, 47648}

X(68156) = X(110)X(65636)∩X(523)X(4360)

X(68156) lies on the Kiepert parabola and these lines: {81, 6654}, {86, 3253}, {110, 65636}, {523, 4360}, {669, 1621}, {812, 4366}, {874, 3573}, {1019, 14621}, {2528, 4467}, {3572, 20132}, {4375, 6652}, {10566, 21007}, {18057, 18155}, {18108, 20295}, {20142, 27854}, {20147, 20981}, {24286, 66286}, {30940, 59488}, {33295, 47070}

X(68156) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 33295}, {4593, 385}
X(68156) = X(i)-isoconjugate of X(j) for these (i,j): {798, 57566}, {1018, 52205}, {4033, 51856}, {4557, 30663}, {18267, 27808}, {30657, 56257}, {34067, 43534}
X(68156) = X(i)-Dao conjugate of X(j) for these (i,j): {812, 523}, {1966, 4033}, {31998, 57566}, {35119, 43534}, {39786, 594}, {40620, 40098}, {62552, 35352}
X(68156) = crosspoint of X(i) and X(j) for these (i,j): {83, 3570}, {99, 33295}
X(68156) = crosssum of X(39) and X(3572)
X(68156) = crossdifference of every pair of points on line {20683, 21830}
X(68156) = barycentric product X(i)*X(j) for these {i,j}: {81, 27855}, {86, 4375}, {99, 35119}, {350, 50456}, {659, 30940}, {812, 33295}, {1019, 39044}, {3733, 56660}, {4366, 7192}, {5009, 65101}, {7199, 8300}, {51328, 52619}, {57129, 64222}, {61061, 62534}
X(68156) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57566}, {812, 43534}, {1019, 30663}, {3733, 52205}, {4366, 3952}, {4368, 4103}, {4375, 10}, {5009, 813}, {7192, 40098}, {8300, 1018}, {12835, 4559}, {16737, 30642}, {27855, 321}, {27918, 35352}, {30940, 4583}, {31905, 65338}, {33295, 4562}, {35119, 523}, {39044, 4033}, {50456, 291}, {51328, 4557}, {56660, 27808}, {61061, 7180}

X(68157) = X(523)X(17217)∩X(669)X(4467)

X(68157) lies on the Kiepert parabola and these lines: {523, 17217}, {669, 4467}, {768, 21123}, {824, 3250}, {826, 52619}, {850, 21441}, {918, 7192}, {3261, 21121}, {3733, 57214}, {4122, 33931}, {7199, 62423}, {18155, 21614}, {23829, 30519}, {30870, 62415}

X(68157) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 30966}, {4602, 3314}
X(68157) = X(i)-isoconjugate of X(j) for these (i,j): {825, 40747}, {34069, 40718}
X(68157) = X(i)-Dao conjugate of X(j) for these (i,j): {788, 9426}, {824, 523}, {27481, 4613}, {61065, 40718}
X(68157) = crosspoint of X(99) and X(30966)
X(68157) = trilinear pole of line {61065, 62414}
X(68157) = barycentric product X(i)*X(j) for these {i,j}: {99, 61065}, {670, 62414}, {693, 4469}, {824, 30966}, {3261, 4476}, {4481, 33931}, {40773, 62415}
X(68157) = barycentric quotient X(i)/X(j) for these {i,j}: {824, 40718}, {1491, 40747}, {3661, 4613}, {3736, 825}, {4469, 100}, {4476, 101}, {4481, 985}, {12837, 4559}, {30966, 4586}, {40773, 1492}, {55049, 9426}, {61065, 523}, {62414, 512}, {68112, 2205}

X(68158) = X(110)X(1649)∩X(523)X(2407)

X(68158) lies on the Kiepert parabola and these lines: {2, 3233}, {3, 5967}, {99, 6035}, {110, 1649}, {476, 8029}, {523, 2407}, {542, 5191}, {669, 15329}, {1316, 15928}, {1576, 14559}, {1634, 62173}, {1640, 23968}, {2528, 52603}, {3265, 5468}, {5489, 14366}, {5502, 58766}, {7468, 44821}, {7471, 8371}, {7473, 50941}, {11123, 14611}, {14356, 57598}, {14999, 34761}, {15000, 18553}, {15919, 65770}, {30221, 44010}, {39295, 58346}, {46048, 65750}, {47200, 57607}, {51820, 65783}

X(68158) = isotomic conjugate of the polar conjugate of X(60505)
X(68158) = X(99)-Ceva conjugate of X(14999)
X(68158) = X(798)-isoconjugate of X(57547)
X(68158) = X(i)-Dao conjugate of X(j) for these (i,j): {542, 523}, {23967, 14223}, {31998, 57547}, {35582, 67082}, {57464, 62551}
X(68158) = crosspoint of X(99) and X(14999)
X(68158) = crosssum of X(512) and X(14998)
X(68158) = trilinear pole of line {23967, 65750}
X(68158) = barycentric product X(i)*X(j) for these {i,j}: {69, 60505}, {99, 23967}, {524, 66958}, {542, 14999}, {648, 65750}, {4558, 38552}, {5649, 58252}, {6035, 46048}, {7473, 65722}, {39295, 60340}, {42743, 46786}, {45662, 50941}, {51227, 64607}, {51474, 60511}
X(68158) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57547}, {542, 14223}, {5191, 14998}, {14999, 5641}, {23967, 523}, {23968, 54554}, {38552, 14618}, {42743, 46787}, {45662, 50942}, {46048, 1640}, {58252, 18312}, {60340, 62551}, {60505, 4}, {64607, 51228}, {65723, 65727}, {65750, 525}, {66354, 23350}, {66958, 671}

X(68159) = X(110)X(65638)∩X(523)X(5468)

X(68159) lies on the Kiepert parabola and these lines: {2, 5914}, {99, 1649}, {110, 65638}, {523, 5468}, {543, 1641}, {669, 11634}, {892, 8029}, {3233, 67536}, {5108, 35606}, {7804, 41939}, {8371, 9182}, {9146, 62555}

X(68159) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 9182}, {52940, 1641}
X(68159) = X(798)-isoconjugate of X(57561)
X(68159) = X(i)-Dao conjugate of X(j) for these (i,j): {543, 523}, {31998, 57561}, {33921, 14443}, {35087, 9180}, {41176, 1648}, {52883, 18823}
X(68159) = crosspoint of X(99) and X(9182)
X(68159) = trilinear pole of line {35087, 59803}
X(68159) = barycentric product X(i)*X(j) for these {i,j}: {99, 35087}, {543, 9182}, {670, 59803}, {1641, 34760}, {9181, 45809}
X(68159) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57561}, {543, 9180}, {1641, 34763}, {9181, 843}, {9182, 18823}, {35087, 523}, {59803, 512}
X(68159) = {X(9182),X(34760)}-harmonic conjugate of X(8371)

X(68160) = X(1)X(523)∩X(110)X(65639)

X(68160) lies on the Kiepert parabola and these lines: {1, 523}, {21, 3733}, {86, 4833}, {110, 65639}, {140, 6003}, {214, 900}, {512, 42285}, {513, 1125}, {524, 27929}, {551, 55244}, {662, 57021}, {669, 8053}, {1647, 4448}, {2267, 14321}, {2309, 55969}, {2490, 7252}, {3251, 68101}, {3589, 9013}, {3658, 23832}, {3667, 13624}, {3884, 4132}, {4106, 18650}, {4187, 31946}, {4369, 49738}, {4472, 40459}, {7253, 28221}, {8689, 9002}, {9820, 30212}, {16704, 34764}, {17195, 64913}, {17245, 24924}, {17780, 23344}, {28213, 47844}, {30939, 59487}, {37168, 50943}, {61637, 67418}

X(68160) = midpoint of X(1) and X(62323)
X(68160) = X(39699)-anticomplementary conjugate of X(21294)
X(68160) = X(99)-Ceva conjugate of X(16704)
X(68160) = X(42084)-cross conjugate of X(4370)
X(68160) = X(i)-isoconjugate of X(j) for these (i,j): {37, 4638}, {42, 4618}, {101, 30575}, {679, 4557}, {798, 57564}, {901, 4674}, {1018, 2226}, {1318, 4551}, {4033, 41935}, {4080, 32665}, {5376, 55263}, {9268, 55244}, {21805, 39414}
X(68160) = X(i)-Dao conjugate of X(j) for these (i,j): {519, 3952}, {900, 523}, {1015, 30575}, {1647, 10}, {6544, 4049}, {31998, 57564}, {35092, 4080}, {38979, 4674}, {40589, 4638}, {40592, 4618}, {40620, 54974}
X(68160) = cevapoint of X(3251) and X(6544)
X(68160) = crosspoint of X(99) and X(16704)
X(68160) = crosssum of X(42) and X(55263)
X(68160) = crossdifference of every pair of points on line {2245, 52963}
X(68160) = barycentric product X(i)*X(j) for these {i,j}: {58, 52627}, {81, 68101}, {86, 6544}, {99, 35092}, {274, 3251}, {333, 39771}, {513, 16729}, {645, 14027}, {678, 7199}, {799, 42084}, {900, 16704}, {1017, 52619}, {1019, 4738}, {1317, 4560}, {1434, 4543}, {1635, 30939}, {2087, 55243}, {3285, 65867}, {3733, 36791}, {3762, 52680}, {4152, 17096}, {4370, 7192}, {4542, 4573}, {4600, 14442}, {7254, 65585}, {15419, 42070}, {16726, 68107}, {17197, 66979}, {17205, 53582}, {17925, 65742}, {30572, 30606}, {52337, 55194}, {61062, 62534}
X(68160) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 4638}, {81, 4618}, {99, 57564}, {513, 30575}, {678, 1018}, {900, 4080}, {1017, 4557}, {1019, 679}, {1317, 4552}, {1635, 4674}, {1647, 4049}, {2087, 55244}, {3251, 37}, {3285, 901}, {3733, 2226}, {4120, 4013}, {4152, 30730}, {4370, 3952}, {4542, 3700}, {4543, 2321}, {4738, 4033}, {6544, 10}, {7192, 54974}, {7199, 57929}, {7252, 1318}, {8028, 4169}, {14027, 7178}, {14442, 3120}, {16704, 4555}, {16729, 668}, {21821, 40521}, {22371, 4574}, {30576, 4622}, {33922, 3943}, {35092, 523}, {36791, 27808}, {37168, 65336}, {39771, 226}, {42084, 661}, {47683, 36594}, {52337, 55195}, {52627, 313}, {52680, 3257}, {61047, 4559}, {61062, 7180}, {65742, 52609}, {68101, 321}

X(68161) = X(75)X(523)∩X(669)X(4897)

X(68161) lies on the Kiepert parabola and these lines: {75, 523}, {86, 2400}, {99, 59021}, {274, 56283}, {333, 7192}, {665, 918}, {669, 4897}, {850, 40216}, {883, 2283}, {1444, 3733}, {2402, 4560}, {3004, 8034}, {3126, 62430}, {3666, 4025}, {3675, 62429}, {4467, 17140}, {6362, 52619}, {6586, 48070}, {7203, 8270}, {16708, 18155}, {17069, 24631}, {18157, 59489}, {24560, 53415}, {30941, 63742}, {50333, 63223}

X(68161) = X(99)-Ceva conjugate of X(30941)
X(68161) = X(i)-isoconjugate of X(j) for these (i,j): {798, 57536}, {919, 18785}, {1018, 41934}, {4557, 51838}, {13576, 32666}, {36086, 56853}
X(68161) = X(i)-Dao conjugate of X(j) for these (i,j): {518, 4557}, {918, 523}, {17435, 37}, {31998, 57536}, {35094, 13576}, {38980, 18785}, {38989, 56853}, {40620, 6185}, {40625, 62715}, {62429, 53510}
X(68161) = cevapoint of X(3126) and X(53583)
X(68161) = crosspoint of X(99) and X(30941)
X(68161) = crosssum of X(i) and X(j) for these (i,j): {512, 56853}, {16583, 55261}
X(68161) = trilinear pole of line {35094, 35505}
X(68161) = crossdifference of every pair of points on line {41333, 51436}
X(68161) = barycentric product X(i)*X(j) for these {i,j}: {81, 62430}, {86, 53583}, {99, 35094}, {274, 3126}, {333, 66967}, {645, 3323}, {670, 35505}, {693, 16728}, {918, 30941}, {2254, 18157}, {3675, 55260}, {3912, 23829}, {4437, 7192}, {4712, 7199}, {6184, 52619}, {15419, 34337}, {16727, 68106}, {52304, 55194}, {61056, 62534}
X(68161) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57536}, {665, 56853}, {918, 13576}, {1019, 51838}, {1362, 4559}, {2254, 18785}, {3126, 37}, {3286, 919}, {3323, 7178}, {3675, 55261}, {3733, 41934}, {4437, 3952}, {4560, 62715}, {4712, 1018}, {6184, 4557}, {7192, 6185}, {15149, 65333}, {16728, 100}, {18157, 51560}, {18206, 36086}, {23829, 673}, {30941, 666}, {35094, 523}, {35505, 512}, {43042, 66941}, {50357, 14625}, {52304, 55195}, {52619, 57537}, {53583, 10}, {61056, 7180}, {62430, 321}, {65744, 4574}, {66967, 226}

X(68162) = X(110)X(65640)∩X(186)X(523)

X(68162) lies on the Kiepert parabola and these lines: {3, 65694}, {110, 65640}, {186, 523}, {250, 53776}, {512, 37084}, {520, 14270}, {669, 1510}, {924, 12095}, {1576, 47390}, {1609, 57071}, {1624, 3233}, {3265, 53263}, {5489, 59162}, {8907, 46616}, {14703, 15470}, {22089, 59744}, {35225, 65610}, {37814, 62339}, {38354, 62555}, {41203, 44451}, {42660, 65390}

X(68162) = isotomic conjugate of the polar conjugate of X(58760)
X(68162) = isogonal conjugate of the polar conjugate of X(15423)
X(68162) = X(i)-Ceva conjugate of X(j) for these (i,j): {24, 34338}, {99, 1993}, {15423, 58760}, {57065, 30451}, {63835, 39013}
X(68162) = X(i)-cross conjugate of X(j) for these (i,j): {6754, 571}, {39013, 63835}, {55072, 3133}
X(68162) = X(i)-isoconjugate of X(j) for these (i,j): {91, 925}, {1820, 30450}, {2165, 65251}, {2168, 65845}, {5392, 36145}, {20571, 32734}, {55215, 60501}
X(68162) = X(i)-Dao conjugate of X(j) for these (i,j): {134, 5}, {135, 847}, {577, 65309}, {924, 523}, {34116, 925}, {39013, 5392}
X(68162) = crosspoint of X(i) and X(j) for these (i,j): {54, 13398}, {99, 1993}
X(68162) = crosssum of X(i) and X(j) for these (i,j): {5, 65694}, {512, 2165}, {520, 3548}, {523, 12359}
X(68162) = crossdifference of every pair of points on line {216, 2165}
X(68162) = barycentric product X(i)*X(j) for these {i,j}: {3, 15423}, {24, 52584}, {47, 63827}, {69, 58760}, {99, 39013}, {317, 30451}, {523, 63835}, {525, 52432}, {571, 6563}, {647, 55551}, {924, 1993}, {1147, 57065}, {1748, 63832}, {3133, 15412}, {3265, 36416}, {4558, 34338}, {4563, 6754}, {6753, 9723}, {7763, 34952}, {18315, 55072}, {18883, 44808}, {34948, 42700}, {44179, 55216}, {57484, 63959}
X(68162) = barycentric quotient X(i)/X(j) for these {i,j}: {24, 30450}, {47, 65251}, {52, 65845}, {571, 925}, {924, 5392}, {1147, 65309}, {1993, 46134}, {3133, 14570}, {6563, 57904}, {6753, 847}, {6754, 2501}, {14533, 52932}, {15423, 264}, {30451, 68}, {34338, 14618}, {34948, 66954}, {34952, 2165}, {36416, 107}, {39013, 523}, {41213, 12077}, {44077, 65176}, {44179, 55215}, {44808, 37802}, {52317, 56272}, {52432, 648}, {52436, 32734}, {52584, 20563}, {55072, 18314}, {55216, 91}, {55551, 6331}, {57065, 55553}, {58760, 4}, {63827, 20571}, {63835, 99}, {63959, 39116}

X(68163) = X(110)X(54049)∩X(523)X(2070)

X(68163) lies on the Kiepert parabola and these lines: {3, 13152}, {110, 54049}, {512, 11810}, {523, 2070}, {924, 10282}, {1510, 6150}, {1576, 14587}, {2413, 3518}, {2528, 53263}, {3233, 64485}, {5926, 59744}, {14270, 20188}, {20184, 39481}, {39180, 39201}

X(68163) = X(99)-Ceva conjugate of X(1994)
X(68163) = X(i)-isoconjugate of X(j) for these (i,j): {930, 2962}, {11140, 36148}
X(68163) = X(i)-Dao conjugate of X(j) for these (i,j): {1510, 523}, {35591, 66883}, {39018, 11140}, {53986, 93}
X(68163) = crosspoint of X(99) and X(1994)
X(68163) = crosssum of X(i) and X(j) for these (i,j): {512, 2963}, {523, 21230}
X(68163) = crossdifference of every pair of points on line {570, 2963}
X(68163) = barycentric product X(i)*X(j) for these {i,j}: {49, 67102}, {99, 39018}, {1166, 58828}, {1510, 1994}, {2965, 41298}, {3459, 58876}, {3518, 63830}, {20577, 25044}, {30529, 44809}, {57135, 57489}, {57137, 63172}
X(68163) = barycentric quotient X(i)/X(j) for these {i,j}: {1510, 11140}, {1994, 46139}, {2965, 930}, {3518, 38342}, {32002, 55217}, {39018, 523}, {57137, 25043}, {58828, 1225}, {58876, 45799}, {63172, 55283}, {67102, 20572}

X(68164) = X(21)X(523)∩X(110)X(65644)

X(68164) lies on the Kiepert parabola and these lines: {1, 30222}, {3, 3733}, {21, 523}, {110, 65644}, {1649, 53309}, {3233, 3658}, {3746, 8702}, {7192, 56934}, {7253, 56946}, {8562, 35193}, {8674, 16164}, {11101, 46611}, {15175, 21789}, {23226, 34435}, {44814, 53315}, {53295, 60342}, {53306, 62173}

X(68164) = X(99)-Ceva conjugate of X(37783)
X(68164) = X(i)-isoconjugate of X(j) for these (i,j): {1290, 5620}, {4551, 55012}
X(68164) = X(8674)-Dao conjugate of X(523)
X(68164) = crosspoint of X(99) and X(37783)
X(68164) = crosssum of X(523) and X(13605)
X(68164) = barycentric product X(i)*X(j) for these {i,j}: {99, 35090}, {8674, 37783}, {17796, 65669}, {32849, 42741}
X(68164) = barycentric quotient X(i)/X(j) for these {i,j}: {5127, 65238}, {7252, 55012}, {17796, 66280}, {19622, 1290}, {35090, 523}, {37783, 35156}, {42741, 21907}

X(68165) = X(110)X(6082)∩X(351)X(523)

X(68165) lies on the Kiepert parabola and these lines: {2, 67566}, {110, 6082}, {351, 523}, {669, 35298}, {690, 3265}, {1499, 4786}, {1649, 3566}, {2408, 4232}, {3233, 7472}, {9168, 38918}, {9215, 36168}, {15638, 35133}, {30217, 35283}, {32472, 59982}, {47194, 58349}

X(68165) = midpoint of X(i) and X(j) for these {i,j}: {8644, 9125}, {9123, 59927}, {24976, 46001}
X(68165) = reflection of X(44552) in X(9185)
X(68165) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 1992}, {52141, 35234}
X(68165) = X(i)-isoconjugate of X(j) for these (i,j): {798, 57569}, {1296, 55923}, {21448, 37216}
X(68165) = X(i)-Dao conjugate of X(j) for these (i,j): {1499, 523}, {11147, 35179}, {31998, 57569}, {35133, 5485}
X(68165) = crosspoint of X(99) and X(1992)
X(68165) = crosssum of X(512) and X(21448)
X(68165) = crossdifference of every pair of points on line {574, 21448}
X(68165) = barycentric product X(i)*X(j) for these {i,j}: {99, 35133}, {691, 58283}, {1499, 1992}, {2408, 27088}, {5468, 15638}, {6082, 35234}, {8644, 11059}, {9125, 52141}, {14207, 36277}, {30234, 42724}, {61345, 62568}
X(68165) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57569}, {1384, 1296}, {1499, 5485}, {1992, 35179}, {4232, 65353}, {8644, 21448}, {15638, 5466}, {27088, 2418}, {35133, 523}, {35234, 65870}, {36277, 37216}, {58283, 35522}, {65469, 17952}
X(68165) = {X(9123),X(15724)}-harmonic conjugate of X(59927)

X(68166) = X(110)X(2867)∩X(250)X(523)

X(68166) lies on the Kiepert parabola and these lines: {107, 59652}, {110, 2867}, {250, 523}, {441, 1503}, {669, 1624}, {858, 3233}, {1649, 5502}, {2409, 23977}, {2794, 40542}, {4226, 62555}, {5181, 39072}, {10991, 65734}, {15448, 66123}, {23582, 58342}, {34369, 57261}, {39838, 65765}, {51431, 66939}

X(68166) = isotomic conjugate of the polar conjugate of X(15639)
X(68166) = X(99)-Ceva conjugate of X(34211)
X(68166) = X(60341)-cross conjugate of X(65749)
X(68166) = X(i)-isoconjugate of X(j) for these (i,j): {798, 57549}, {2435, 8767}
X(68166) = X(i)-Dao conjugate of X(j) for these (i,j): {1503, 523}, {15595, 2419}, {23976, 43673}, {31998, 57549}, {39071, 2435}, {57296, 15526}
X(68166) = cevapoint of X(60341) and X(65749)
X(68166) = crosspoint of X(i) and X(j) for these (i,j): {99, 34211}, {2409, 60506}
X(68166) = crosssum of X(512) and X(34212)
X(68166) = trilinear pole of line {23976, 65749}
X(68166) = crossdifference of every pair of points on line {34212, 41172}
X(68166) = barycentric product X(i)*X(j) for these {i,j}: {69, 15639}, {99, 23976}, {441, 2409}, {648, 65749}, {662, 24023}, {1503, 34211}, {15595, 60506}, {23582, 60341}, {34156, 66076}
X(68166) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57549}, {441, 2419}, {1503, 43673}, {2409, 6330}, {2445, 43717}, {8779, 2435}, {15639, 4}, {23976, 523}, {24023, 1577}, {34211, 35140}, {39473, 66964}, {42671, 34212}, {58256, 66161}, {60341, 15526}, {60506, 9476}, {65749, 525}

X(68167) = X(20)X(523)∩X(110)X(53881)

X(68167) lies on the Kiepert parabola and these lines: {3, 2416}, {20, 523}, {22, 46612}, {30, 53159}, {69, 3265}, {110, 53881}, {159, 669}, {520, 5489}, {526, 12825}, {577, 55269}, {684, 1649}, {924, 36982}, {1553, 23097}, {1650, 57290}, {2407, 3233}, {2528, 3313}, {3184, 9033}, {5664, 15774}, {6148, 62555}, {8029, 15328}, {8907, 46616}, {11064, 47071}, {11413, 46613}, {16251, 58342}, {22089, 46608}, {24974, 55130}, {34291, 58766}, {51394, 53235}, {66073, 67182}

X(68167) = reflection of X(i) in X(j) for these {i,j}: {14380, 38401}, {62172, 57128}, {62350, 3}
X(68167) = isotomic conjugate of the isogonal conjugate of X(58345)
X(68167) = isotomic conjugate of the polar conjugate of X(14401)
X(68167) = isogonal conjugate of the polar conjugate of X(52624)
X(68167) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 11064}, {3233, 16163}, {3265, 41077}, {51254, 1650}, {52624, 14401}, {62569, 47414}
X(68167) = X(58345)-cross conjugate of X(14401)
X(68167) = X(i)-isoconjugate of X(j) for these (i,j): {19, 34568}, {798, 57570}, {823, 40353}, {1304, 36119}, {2159, 15459}, {2349, 32695}, {8749, 65263}, {16080, 36131}, {24019, 40384}, {36117, 52493}
X(68167) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 34568}, {30, 107}, {1511, 1304}, {1650, 4}, {3163, 15459}, {9033, 523}, {14401, 2394}, {31998, 57570}, {35071, 40384}, {38999, 74}, {39008, 16080}, {57295, 18808}, {62569, 16077}, {62573, 31621}, {62613, 42308}, {66130, 52475}
X(68167) = cevapoint of X(14401) and X(58352)
X(68167) = crosspoint of X(i) and X(j) for these (i,j): {69, 2407}, {99, 11064}, {3233, 16163}, {3265, 41077}, {18557, 66073}
X(68167) = crosssum of X(i) and X(j) for these (i,j): {25, 2433}, {512, 8749}, {32695, 32713}
X(68167) = crossdifference of every pair of points on line {8749, 14581}
X(68167) = barycentric product X(i)*X(j) for these {i,j}: {3, 52624}, {30, 41077}, {69, 14401}, {76, 58345}, {99, 39008}, {394, 58263}, {520, 36789}, {525, 16163}, {1099, 24018}, {1511, 18557}, {1636, 3260}, {1650, 2407}, {3163, 3265}, {3233, 15526}, {3284, 66073}, {3926, 58346}, {4143, 16240}, {5664, 51254}, {6148, 18558}, {6394, 58351}, {9033, 11064}, {9408, 52617}, {20580, 38956}, {23097, 62665}, {34334, 52613}, {34403, 58352}, {41079, 51394}
X(68167) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 34568}, {30, 15459}, {99, 57570}, {520, 40384}, {1099, 823}, {1495, 32695}, {1636, 74}, {1650, 2394}, {2407, 42308}, {2631, 36119}, {3163, 107}, {3233, 23582}, {3265, 31621}, {3284, 1304}, {9033, 16080}, {9408, 32713}, {9409, 8749}, {11064, 16077}, {14345, 10152}, {14401, 4}, {16163, 648}, {16240, 6529}, {18558, 5627}, {34334, 15352}, {36789, 6528}, {38956, 65181}, {39008, 523}, {39201, 40353}, {41077, 1494}, {42074, 24019}, {50433, 67756}, {51254, 39290}, {51394, 44769}, {52624, 264}, {58257, 65753}, {58263, 2052}, {58343, 58070}, {58344, 2207}, {58345, 6}, {58346, 393}, {58348, 35907}, {58349, 60428}, {58351, 6530}, {58352, 1249}, {62665, 59145}, {66123, 52475}, {66899, 1637}
X(68167) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14380, 38401, 65723}, {57128, 62172, 65754}

X(68168) = X(86)X(523)∩X(110)X(65647)

X(68168) lies on the Kiepert parabola and these lines: {2, 661}, {86, 523}, {110, 65647}, {669, 4184}, {1649, 53333}, {2786, 5029}, {3733, 56934}, {4467, 10190}, {4608, 8029}, {11123, 17161}, {11183, 53335}, {14838, 40776}, {17731, 28602}, {17934, 17943}, {31308, 64859}, {45693, 50556}, {51314, 66286}

X(68168) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {9510, 3448}, {39718, 21294}
X(68168) = X(99)-Ceva conjugate of X(17731)
X(68168) = X(i)-isoconjugate of X(j) for these (i,j): {798, 57560}, {2054, 37135}, {2702, 9278}
X(68168) = X(i)-Dao conjugate of X(j) for these (i,j): {2786, 523}, {27929, 18014}, {31998, 57560}, {35080, 11599}, {39042, 37135}, {41841, 66283}, {57461, 1213}
X(68168) = crosspoint of X(99) and X(17731)
X(68168) = crosssum of X(512) and X(2054)
X(68168) = crossdifference of every pair of points on line {2054, 3747}
X(68168) = barycentric product X(i)*X(j) for these {i,j}: {99, 35080}, {2786, 17731}, {9508, 52137}
X(68168) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57560}, {1326, 2702}, {1931, 37135}, {2786, 11599}, {5029, 2054}, {6542, 66283}, {9508, 9278}, {17731, 35148}, {18004, 6543}, {35080, 523}

X(68169) = X(110)X(65638)∩X(523)X(1992)

X(68169) lies on the Kiepert parabola and these lines: {2, 67566}, {110, 65638}, {523, 1992}, {669, 1995}, {690, 62555}, {804, 1649}, {1499, 11159}, {2793, 9135}, {3265, 9168}, {8029, 8599}, {11336, 59982}, {17937, 34245}, {58766, 67588}

X(68169) = reflection of X(14327) in X(65469)
X(68169) = X(99)-Ceva conjugate of X(22329)
X(68169) = X(i)-Dao conjugate of X(j) for these (i,j): {2793, 523}, {61071, 5503}, {62578, 46144}
X(68169) = crosspoint of X(99) and X(22329)
X(68169) = barycentric product X(i)*X(j) for these {i,j}: {99, 61071}, {2793, 22329}, {17952, 65469}
X(68169) = barycentric quotient X(i)/X(j) for these {i,j}: {2030, 2709}, {2793, 5503}, {22329, 46144}, {61071, 523}

X(68170) = X(522)X(3733)∩X(523)X(49274)

X(68170) lies on the Kiepert parabola and these lines: {45, 4931}, {522, 3733}, {523, 49274}, {669, 50339}, {900, 7192}, {3737, 28205}, {4363, 64859}, {4557, 51562}, {4693, 4775}, {4825, 68119}, {4840, 4926}, {7253, 28183}, {15175, 21789}, {23352, 63216}, {23809, 48189}, {28221, 47844}, {48291, 64868}

X(68170) = reflection of X(4840) in X(5214)
X(68170) = X(39705)-anticomplementary conjugate of X(21294)
X(68170) = X(99)-Ceva conjugate of X(5235)
X(68170) = X(i)-isoconjugate of X(j) for these (i,j): {4559, 30607}, {4588, 53114}, {4604, 28658}, {30588, 34073}
X(68170) = X(i)-Dao conjugate of X(j) for these (i,j): {4777, 523}, {53167, 30587}, {55045, 53114}, {55067, 30607}, {61073, 30588}
X(68170) = cevapoint of X(4825) and X(53584)
X(68170) = crosspoint of X(99) and X(5235)
X(68170) = crosssum of X(512) and X(28658)
X(68170) = barycentric product X(i)*X(j) for these {i,j}: {81, 68119}, {86, 53584}, {99, 61073}, {274, 4825}, {333, 66984}, {514, 4803}, {3679, 47683}, {4653, 4791}, {4671, 4833}, {4720, 43052}, {4777, 5235}
X(68170) = barycentric quotient X(i)/X(j) for these {i,j}: {3737, 30607}, {4273, 4588}, {4653, 4604}, {4775, 28658}, {4777, 30588}, {4802, 30587}, {4803, 190}, {4825, 37}, {4833, 89}, {4893, 53114}, {5235, 4597}, {47683, 39704}, {53584, 10}, {61073, 523}, {66984, 226}, {68119, 321}

X(68171) = X(2)X(525)∩X(298)X(523)

X(68171) lies on the Kiepert parabola and these lines: {2, 525}, {298, 523}, {299, 45808}, {530, 9141}, {669, 34008}, {850, 20579}, {1649, 30472}, {3233, 35315}, {3268, 6138}, {4467, 66490}, {5466, 60252}, {6035, 36839}, {6563, 19772}, {8029, 62631}, {9200, 25187}, {18311, 37785}, {30468, 62572}, {52268, 65754}

X(68171) = isotomic conjugate of X(36840)
X(68171) = isotomic conjugate of the isogonal conjugate of X(57123)
X(68171) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3441, 21221}, {19777, 21294}
X(68171) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 299}, {11128, 43962}, {40707, 62551}
X(68171) = X(43962)-cross conjugate of X(11128)
X(68171) = X(i)-isoconjugate of X(j) for these (i,j): {31, 36840}, {163, 11085}, {798, 57580}, {2152, 5619}, {2154, 5994}, {10218, 32676}, {11086, 32678}, {14560, 51806}, {39381, 56829}
X(68171) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 36840}, {115, 11085}, {5664, 23284}, {11126, 110}, {15526, 10218}, {15610, 11088}, {18334, 11086}, {23871, 523}, {30468, 8015}, {30472, 23896}, {31998, 57580}, {35444, 20579}, {38994, 3458}, {40579, 5619}, {40581, 5994}, {43961, 36210}, {43962, 14}, {47899, 8738}, {62572, 11092}, {66262, 61}
X(68171) = crosspoint of X(99) and X(299)
X(68171) = crosssum of X(512) and X(3458)
X(68171) = crossdifference of every pair of points on line {1495, 3458}
X(68171) = barycentric product X(i)*X(j) for these {i,j}: {76, 57123}, {99, 43962}, {299, 23871}, {523, 11128}, {850, 11130}, {1095, 20948}, {3267, 56515}, {3268, 11078}, {7799, 23283}, {23965, 36839}
X(68171) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 36840}, {14, 5619}, {16, 5994}, {99, 57580}, {299, 23896}, {471, 36309}, {523, 11085}, {525, 10218}, {526, 11086}, {1095, 163}, {3268, 11092}, {6138, 3458}, {8552, 50466}, {9205, 52040}, {11078, 476}, {11081, 14560}, {11128, 99}, {11130, 110}, {14380, 39381}, {14446, 61371}, {23283, 1989}, {23870, 36210}, {23871, 14}, {23873, 11582}, {30460, 20578}, {30468, 20579}, {32679, 51806}, {35444, 8015}, {36208, 5995}, {36839, 23588}, {37850, 16807}, {43962, 523}, {44719, 38413}, {50465, 32662}, {51805, 32678}, {52342, 6137}, {55223, 16464}, {56515, 112}, {57123, 6}, {60009, 36297}, {62551, 23284}, {66873, 9207}

X(68172) = X(2)X(525)∩X(299)X(523)

X(68172) lies on the Kiepert parabola and these lines: {2, 525}, {298, 45808}, {299, 523}, {531, 9141}, {669, 34009}, {850, 20578}, {1649, 30471}, {3233, 35314}, {3268, 6137}, {4467, 66489}, {5466, 60253}, {6035, 36840}, {6563, 19773}, {8029, 62632}, {9201, 25183}, {18311, 37786}, {30465, 62572}, {52267, 65754}

X(68172) = isotomic conjugate of X(36839)
X(68172) = isotomic conjugate of the isogonal conjugate of X(57122)
X(68172) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3440, 21221}, {19776, 21294}
X(68172) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 298}, {11129, 43961}, {40706, 62551}
X(68172) = X(43961)-cross conjugate of X(11129)
X(68172) = X(i)-isoconjugate of X(j) for these (i,j): {31, 36839}, {163, 11080}, {798, 57579}, {2151, 5618}, {2153, 5995}, {10217, 32676}, {11081, 32678}, {14560, 51805}, {39380, 56829}
X(68172) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 36839}, {115, 11080}, {5664, 23283}, {11127, 110}, {15526, 10217}, {15609, 11083}, {18334, 11081}, {23870, 523}, {30465, 8014}, {30471, 23895}, {31998, 57579}, {35443, 20578}, {38993, 3457}, {40578, 5618}, {40580, 5995}, {43961, 13}, {43962, 36211}, {47898, 8737}, {62572, 11078}, {66263, 62}
X(68172) = crosspoint of X(99) and X(298)
X(68172) = crosssum of X(512) and X(3457)
X(68172) = crossdifference of every pair of points on line {1495, 3457}
X(68172) = barycentric product X(i)*X(j) for these {i,j}: {76, 57122}, {99, 43961}, {298, 23870}, {523, 11129}, {850, 11131}, {1094, 20948}, {3267, 56514}, {3268, 11092}, {7799, 23284}, {23965, 36840}
X(68172) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 36839}, {13, 5618}, {15, 5995}, {99, 57579}, {298, 23895}, {470, 36306}, {523, 11080}, {525, 10217}, {526, 11081}, {1094, 163}, {3268, 11078}, {6137, 3457}, {8552, 50465}, {9204, 52039}, {11086, 14560}, {11092, 476}, {11129, 99}, {11131, 110}, {14380, 39380}, {14447, 61370}, {23284, 1989}, {23870, 13}, {23871, 36211}, {23872, 11581}, {30463, 20579}, {30465, 20578}, {32679, 51805}, {35443, 8014}, {36209, 5994}, {36840, 23588}, {37848, 16806}, {43961, 523}, {44718, 38414}, {50466, 32662}, {51806, 32678}, {52343, 6138}, {55221, 16463}, {56514, 112}, {57122, 6}, {60010, 36296}, {62551, 23283}, {66872, 9206}

X(68173) = X(6)X(669)∩X(194)X(523)

X(68173) lies on the Kiepert parabola and these lines: {6, 669}, {39, 3221}, {194, 523}, {524, 62649}, {688, 23099}, {887, 888}, {1649, 38998}, {8029, 66271}, {9023, 58752}, {9046, 23642}, {9491, 66886}, {14972, 62173}, {23342, 63747}, {38382, 53272}, {44821, 66145}

X(68173) = isogonal conjugate of the isotomic conjugate of X(62611)
X(68173) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 3231}, {669, 887}
X(68173) = X(i)-isoconjugate of X(j) for these (i,j): {798, 57571}, {799, 57540}, {886, 37132}, {34087, 36133}
X(68173) = X(i)-Dao conjugate of X(j) for these (i,j): {538, 4609}, {888, 523}, {1645, 76}, {31998, 57571}, {35073, 57993}, {38996, 57540}, {38998, 886}, {39010, 34087}, {62611, 60028}
X(68173) = crosspoint of X(i) and X(j) for these (i,j): {6, 23342}, {99, 3231}, {669, 887}
X(68173) = crosssum of X(i) and X(j) for these (i,j): {2, 63749}, {512, 3228}, {670, 886}
X(68173) = crossdifference of every pair of points on line {538, 886}
X(68173) = barycentric product X(i)*X(j) for these {i,j}: {6, 62611}, {99, 39010}, {512, 52067}, {538, 887}, {669, 35073}, {888, 3231}, {1645, 23342}, {5118, 52625}, {9148, 33875}, {30736, 65497}
X(68173) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57571}, {538, 57993}, {669, 57540}, {887, 3228}, {888, 34087}, {1645, 60028}, {3231, 886}, {33875, 9150}, {35073, 4609}, {39010, 523}, {52067, 670}, {52625, 66278}, {62611, 76}, {65497, 729}

X(68174) = X(4)X(523)∩X(157)X(669)

X(68174) lies on the Kiepert parabola and these lines: {3, 65694}, {4, 523}, {5, 62339}, {113, 131}, {157, 669}, {512, 43083}, {526, 12825}, {924, 12162}, {1510, 31976}, {1576, 14559}, {1649, 6132}, {1879, 55278}, {3233, 5502}, {3265, 34291}, {3566, 38401}, {5467, 38359}, {8029, 10412}, {15328, 15928}, {16171, 24974}, {16178, 16221}, {35522, 62555}, {45147, 53247}, {46616, 59744}, {52487, 58757}, {55265, 56403}

X(68174) = reflection of X(62339) in X(5)
X(68174) = X(99)-Ceva conjugate of X(3580)
X(68174) = X(i)-isoconjugate of X(j) for these (i,j): {5504, 36114}, {10420, 36053}, {14910, 65262}
X(68174) = X(i)-Dao conjugate of X(j) for these (i,j): {113, 10420}, {16178, 1300}, {34834, 18878}, {39005, 5504}, {39021, 2986}, {55121, 523}, {56792, 10419}, {65753, 52552}, {65905, 43755}, {67191, 687}
X(68174) = crosspoint of X(i) and X(j) for these (i,j): {99, 3580}, {403, 41512}, {65614, 65972}
X(68174) = crosssum of X(i) and X(j) for these (i,j): {512, 14910}, {523, 23306}, {5504, 15470}
X(68174) = crossdifference of every pair of points on line {3284, 14910}
X(68174) = barycentric product X(i)*X(j) for these {i,j}: {99, 39021}, {113, 65614}, {403, 6334}, {686, 44138}, {2394, 34104}, {3003, 65972}, {3580, 55121}, {5627, 58790}, {14264, 65757}, {14618, 34333}, {47236, 62338}, {55265, 65715}, {57486, 60342}, {62361, 65473}
X(68174) = barycentric quotient X(i)/X(j) for these {i,j}: {403, 687}, {686, 5504}, {1725, 65262}, {2433, 39379}, {3003, 10420}, {3580, 18878}, {6334, 57829}, {13754, 43755}, {15329, 18879}, {21731, 14910}, {34104, 2407}, {34333, 4558}, {39021, 523}, {44084, 32708}, {44138, 57932}, {47236, 1300}, {55121, 2986}, {55265, 15454}, {58790, 6148}, {65614, 40423}, {65715, 55264}, {65757, 52552}, {65972, 40832}

X(68175) = X(25)X(669)∩X(193)X(523)

X(68175) lies on the Kiepert parabola and these lines: {2, 6562}, {25, 669}, {114, 126}, {193, 523}, {804, 62555}, {2528, 19568}, {3265, 11123}, {3566, 62645}, {5477, 42663}, {24974, 55142}, {53272, 62173}, {53274, 55271}, {57071, 63535}

X(68175) = orthic-isogonal conjugate of X(51613)
X(68175) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 51613}, {99, 230}
X(68175) = X(i)-isoconjugate of X(j) for these (i,j): {8773, 10425}, {36051, 65277}, {36105, 43705}
X(68175) = X(i)-Dao conjugate of X(j) for these (i,j): {114, 65277}, {39001, 43705}, {39072, 10425}, {51610, 69}, {55122, 523}, {55152, 8781}, {56788, 40428}
X(68175) = crosspoint of X(99) and X(230)
X(68175) = crosssum of X(512) and X(2987)
X(68175) = crossdifference of every pair of points on line {1570, 2987}
X(68175) = barycentric product X(i)*X(j) for these {i,j}: {99, 55152}, {230, 55122}, {2489, 2974}, {8754, 68088}, {14384, 38359}, {35067, 58757}, {42663, 51481}, {51820, 55267}
X(68175) = barycentric quotient X(i)/X(j) for these {i,j}: {230, 65277}, {460, 65354}, {1692, 10425}, {2974, 52608}, {42663, 2987}, {44099, 32697}, {51820, 55266}, {55122, 8781}, {55152, 523}, {58757, 57553}, {68088, 47389}

X(68176) = X(99)X(669)∩X(523)X(4576)

X(68176) lies on the Kiepert parabola and these lines: {99, 669}, {385, 31128}, {523, 4576}, {538, 3231}, {1649, 2396}, {3233, 56430}, {3265, 65171}, {3266, 5976}, {5108, 35606}, {5468, 38366}, {5969, 66293}, {8716, 59785}, {23342, 63747}

X(68176) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 23342}, {34537, 3231}
X(68176) = X(i)-isoconjugate of X(j) for these (i,j): {798, 57540}, {37132, 63749}
X(68176) = X(i)-Dao conjugate of X(j) for these (i,j): {538, 523}, {888, 23099}, {1645, 3124}, {31998, 57540}, {35073, 60028}, {38998, 63749}
X(68176) = crosspoint of X(99) and X(23342)
X(68176) = crosssum of X(i) and X(j) for these (i,j): {512, 63749}, {3228, 31639}
X(68176) = trilinear pole of line {35073, 52067}
X(68176) = crossdifference of every pair of points on line {1645, 63749}
X(68176) = barycentric product X(i)*X(j) for these {i,j}: {99, 35073}, {538, 23342}, {670, 52067}, {3231, 63747}, {5118, 30736}, {34537, 62611}
X(68176) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57540}, {538, 60028}, {3231, 63749}, {5118, 729}, {9148, 66293}, {23342, 3228}, {30736, 66278}, {35073, 523}, {39010, 23099}, {52067, 512}, {62611, 3124}, {63747, 34087}

X(68177) = EULER LINE INTERCEPT OF X(76)X(3793)

As a point on the Euler line, X(68177) has Shinagawa coefficients: {2 (E+F)^2-3 S^2,7 S^2}

See Juan José Isach Mayo, euclid 8245.

X(68177) lies on these lines: {2, 3}, {32, 63923}, {76, 3793}, {698, 5052}, {736, 15480}, {1506, 32459}, {1975, 18907}, {3053, 64093}, {3589, 7756}, {3734, 7767}, {3788, 53418}, {3933, 7737}, {3972, 5305}, {5034, 42421}, {5215, 12815}, {5475, 59545}, {5503, 60146}, {5943, 58211}, {6337, 15484}, {6390, 7745}, {6392, 21309}, {6645, 15172}, {6680, 53419}, {7747, 7789}, {7753, 59546}, {7760, 52229}, {7768, 63945}, {7783, 53489}, {7784, 43618}, {7787, 63633}, {7794, 63941}, {7804, 63548}, {7812, 32820}, {7839, 47287}, {7858, 59634}, {7863, 14537}, {15491, 15515}, {15513, 58446}, {17130, 63928}, {18501, 39141}, {18844, 60262}, {19661, 34505}, {22253, 32822}, {22331, 63955}, {30103, 65632}, {30104, 65631}, {30435, 32815}, {31664, 31665}, {32520, 61624}, {32836, 63936}, {39590, 44377}, {40894, 40895}, {53106, 60186}, {60209, 62912}

X(68177) = midpoint of X(i) and X(j) for these {i,j}: {384, 19687}, {6656, 6658}, {6661, 66328}, {19686, 66319}, {19695, 19696}
X(68177) = reflection of X(i) in X(j) for these {i,j}: {6655, 8364}, {6656, 19697}, {7819, 384}, {8357, 7819}, {19695, 66347}, {66318, 66321}, {66321, 66319}, {66326, 66318}, {66335, 6661}, {66349, 66340}
X(68177) = complement of X(19695)
X(68177) = anticomplement of X(66347)
X(68177) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 19695, 66347}, {2, 19696, 19695}, {2, 33250, 548}, {3, 32981, 66391}, {4, 8361, 37350}, {4, 8369, 8361}, {4, 33181, 11318}, {4, 33201, 32954}, {5, 14035, 66409}, {20, 8362, 8354}, {20, 11286, 8362}, {140, 3552, 27088}, {382, 14001, 33184}, {384, 6655, 6661}, {384, 6656, 19697}, {384, 6658, 6656}, {384, 7819, 66318}, {384, 7924, 19692}, {384, 7948, 66317}, {384, 19686, 19687}, {384, 19688, 19677}, {384, 19691, 19702}, {384, 19693, 66319}, {384, 19696, 2}, {384, 33256, 19689}, {384, 66328, 6655}, {439, 32983, 3526}, {546, 66393, 7807}, {550, 7770, 8359}, {1003, 14035, 5}, {1656, 32979, 3363}, {3146, 7866, 66392}, {3146, 14039, 7866}, {3529, 33198, 11287}, {3530, 66412, 32992}, {3552, 8370, 140}, {3552, 14034, 8370}, {3830, 33242, 14064}, {3853, 8368, 5025},{3972, 32819, 5305}, {5025, 66408, 3853}, {5059, 32956, 5077}, {5073, 33237, 32974}, {6655, 6661, 8364}, {6655, 8364, 66335}, {6655, 19694, 6656}, {6655, 66320, 384}, {6655, 66335, 8357}, {6656, 6661, 19694}, {6656, 7819, 66343}, {6656, 19687, 6658}, {6656, 19689, 66344}, {6656, 19694, 8364}, {6656, 19697, 7819}, {6656, 66343, 66326}, {6658, 19689, 33256}, {6661, 8364, 7819}, {6661, 66319, 66320}, {7439, 21490, 19280},{7745, 7816, 6390}, {7770, 33007, 550}, {7791, 66387, 15704}, {7807, 11361, 546}, {7819, 8357, 66326}, {7819, 66321, 384}, {7819, 66335, 8364}, {7824, 8598, 33923}, {7841, 14037, 33185}, {7841, 33280, 62036}, {7887, 14068, 3845}, {7892, 33229, 8360}, {7892, 66419, 33229}, {7924, 19692, 66342}, {7924, 66342, 66346}, {7948, 19691, 66349}, {7948, 19702, 66340}, {7948, 66317, 19702}, {8356, 33257, 12103}, {8357, 66318, 7819}, {8357, 66343, 6656}, {8358, 62123, 33260}, {8360, 62026, 33229}, {8363, 33019, 66394}, {8363, 66423, 33019}, {8366, 33283, 33212}, {8367, 33923, 7824}, {11285, 33244, 8703}, {11317, 32961, 3858}, {13586, 32992, 3530}, {14031, 33007, 7770}, {14033, 32981, 3}, {14033, 33239, 32971}, {14033, 66391, 66415}, {14036, 33019, 8363}, {14036, 66423, 66394}, {14037, 33280, 7841}, {14038, 66405, 7933}, {14042, 33225, 33228}, {14042, 33228, 3861}, {14063, 33220, 33186}, {14068, 33255, 7887}, {15687, 33186, 14063}, {16044, 35297, 3628}, {16898, 33193, 33234}, {16924, 33187, 33235}, {16924, 33235, 549}, {16925, 33016, 33270}, {16925, 33270, 33233}, {19670, 66320, 19696}, {19686, 19693, 384}, {19686, 66320, 66328}, {19687, 66319, 384}, {19687, 66320, 8364}, {19687, 66321, 8357}, {19689, 33256, 6656}, {19689, 66344, 7819}, {19690, 66322, 66341}, {19690, 66341, 66334}, {19691, 66317, 7948}, {19692, 66346, 7819}, {19697, 66344, 19689}, {19702, 66340, 7819}, {19702, 66349, 7948}, {32954, 33201, 8369}, {32964, 44543, 632}, {32968, 35927, 3}, {32971, 32981, 33239}, {32971, 33239, 3}, {32975, 35287, 15720}, {32979, 32985, 1656}, {32991, 33216, 5070}, {32997, 66395, 62159}, {33016, 33233, 5}, {33018, 33246, 33249}, {33018, 33249, 5066}, {33019, 66423, 62034}, {33183, 50687, 33292}, {33185, 62036, 7841}, {33193, 33234, 62155}, {33214, 35955, 550}, {33229, 35954, 7892}, {33229, 66419, 62026}, {35954, 66419, 8360}, {37060, 38071, 32963}, {62034, 66394, 33019}, {66317, 66349, 66340}, {66320, 66328, 6661}

X(68178) = X(110)X(669)∩X(523)X(1634)

X(68178) lies on the Kiepert parabola and these lines: {3, 5967}, {23, 3233}, {99, 6037}, {110, 669}, {237, 511}, {249, 39201}, {523, 1634}, {877, 2396}, {1576, 47390}, {1624, 58766}, {1649, 15329}, {2421, 63741}, {2528, 50947}, {3003, 47406}, {3265, 4576}, {3266, 5976}, {3292, 56393}, {4226, 41337}, {4558, 65305}, {5201, 35298}, {5467, 38354}, {8115, 53385}, {8116, 53384}, {9181, 42660}, {11328, 46124}, {11332, 35259}, {30508, 46600}, {30509, 46601}, {33884, 33927}, {34834, 46094}, {37916, 64607}, {44135, 65975}, {46127, 66886}, {51440, 60525}

X(68178) = isogonal conjugate of the isotomic conjugate of X(15631)
X(68178) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 2421}, {249, 3289}
X(68178) = X(58262)-cross conjugate of X(11672)
X(68178) = X(i)-isoconjugate of X(j) for these (i,j): {336, 53149}, {661, 34536}, {798, 57541}, {879, 36120}, {1109, 41173}, {1577, 41932}, {1821, 2395}, {1910, 43665}, {2422, 46273}, {2616, 60594}, {20948, 67167}, {24006, 47388}, {36036, 51441}
X(68178) = X(i)-Dao conjugate of X(j) for these (i,j): {511, 523}, {2679, 51441}, {11672, 43665}, {31998, 57541}, {36830, 34536}, {40601, 2395}, {41172, 338}, {46094, 879}, {57294, 20975}, {62596, 66459}
X(68178) = cevapoint of X(11672) and X(58262)
X(68178) = crosspoint of X(99) and X(2421)
X(68178) = crosssum of X(512) and X(2395)
X(68178) = trilinear pole of line {9419, 11672}
X(68178) = crossdifference of every pair of points on line {2395, 62562}
X(68178) = barycentric product X(i)*X(j) for these {i,j}: {6, 15631}, {81, 68105}, {99, 11672}, {100, 16725}, {110, 36790}, {237, 2396}, {249, 41167}, {325, 14966}, {511, 2421}, {645, 1355}, {648, 65748}, {662, 23996}, {670, 9419}, {799, 42075}, {805, 46888}, {877, 3289}, {1576, 32458}, {1959, 23997}, {2966, 23098}, {2967, 4558}, {4230, 36212}, {4573, 7062}, {4590, 58262}, {4609, 36425}, {23357, 62555}, {23611, 43187}, {42743, 46787}, {47390, 68089}, {51386, 58070}, {59152, 59805}
X(68178) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57541}, {110, 34536}, {237, 2395}, {511, 43665}, {877, 60199}, {1355, 7178}, {1576, 41932}, {1625, 60594}, {2211, 53149}, {2396, 18024}, {2421, 290}, {2491, 51441}, {2967, 14618}, {3289, 879}, {4230, 16081}, {7062, 3700}, {9418, 2422}, {9419, 512}, {11672, 523}, {14574, 67167}, {14966, 98}, {15631, 76}, {16725, 693}, {23098, 2799}, {23357, 41173}, {23611, 3569}, {23996, 1577}, {23997, 1821}, {32458, 44173}, {32661, 47388}, {33569, 66459}, {36425, 669}, {36790, 850}, {39469, 51404}, {41167, 338}, {42075, 661}, {42743, 46786}, {46888, 14295}, {51334, 66299}, {58262, 115}, {59805, 23105}, {62555, 23962}, {65748, 525}, {68105, 321}

  Harmonic pencil and harmonic lines: X(68179) - X(68232)  

Let r₁, r₂, r₃, r₄ be four distinct lines concurrent in a point P. These four lines are said to be an harmonic pencil (or harmonic bundle) if there exists a line ρ, intersecting them at Q₁, Q₂, Q₃, Q₄, respectively, and such that (Q₁, Q₂; Q₃, Q₄) are in harmonic range.

The most important fact in the above definition is that if such harmonic range occurs for a line intersecting the pencil of lines, it occurs for any other line intersecting that pencil.

In this section, three concurrent lines r₁, r₂, r₃ are given and the tripole of the fourth line r₄ is calculated, such that r₁, r₂, r₃, r₄ form an harmonic pencil. This fourth line r₄ is denoted here as the {r₁, r₂}-harmonic line of-r₃.

The appearance of (r₁, r₂, r₃)→n in the following lists means that the tripole of the {r₁, r₂}-harmonic line of-r₃ is X(n):

• For some lines concurrent in X(1):
  ({1, 2}, {1, 3}, {1, 4}) → 65226; ({1, 2}, {1, 3}, {1, 6}) → 27834; ({1, 2}, {1, 3}, {1, 7}) → 68179; ({1, 2}, {1, 4}, {1, 3}) → 655; ({1, 2}, {1, 4}, {1, 6}) → 65236; ({1, 2}, {1, 4}, {1, 7}) → 68180; ({1, 2}, {1, 6}, {1, 3}) → 4606; ({1, 2}, {1, 6}, {1, 4}) → 68181; ({1, 2}, {1, 6}, {1, 7}) → 68182; ({1, 2}, {1, 7}, {1, 3}) → 653; ({1, 2}, {1, 7}, {1, 4}) → 651; ({1, 2}, {1, 7}, {1, 6}) → 37206; ({1, 3}, {1, 4}, {1, 2}) → 658; ({1, 3}, {1, 4}, {1, 6}) → 68183; ({1, 3}, {1, 4}, {1, 7}) → 190; ({1, 3}, {1, 6}, {1, 2}) → 662; ({1, 3}, {1, 6}, {1, 4}) → 65217; ({1, 3}, {1, 6}, {1, 7}) → 65222; ({1, 3}, {1, 7}, {1, 2}) → 68184; ({1, 3}, {1, 7}, {1, 4}) → 68185; ({1, 3}, {1, 7}, {1, 6}) → 68186; ({1, 4}, {1, 6}, {1, 2}) → 68187; ({1, 4}, {1, 6}, {1, 3}) → 68188; ({1, 4}, {1, 6}, {1, 7}) → 68189; ({1, 4}, {1, 7}, {1, 2}) → 68190; ({1, 4}, {1, 7}, {1, 3}) → 38340; ({1, 4}, {1, 7}, {1, 6}) → 68191; ({1, 6}, {1, 7}, {1, 2}) → 68192; ({1, 6}, {1, 7}, {1, 3}) → 61240; ({1, 6}, {1, 7}, {1, 4}) → 68193
• For some lines concurrent in X(2):
  ({1, 2}, {2, 3}, {2, 6}) → 68194; ({1, 2}, {2, 3}, {2, 7}) → 68195; ({1, 2}, {2, 6}, {2, 3}) → 68196; ({1, 2}, {2, 6}, {2, 7}) → 58135; ({1, 2}, {2, 7}, {2, 3}) → 54970; ({1, 2}, {2, 7}, {2, 6}) → 668; ({2, 3}, {2, 6}, {1, 2}) → 68197; ({2, 3}, {2, 6}, {2, 7}) → 65274; ({2, 3}, {2, 7}, {1, 2}) → 68198; ({2, 3}, {2, 7}, {2, 6}) → 68199; ({2, 6}, {2, 7}, {1, 2}) → 58132; ({2, 6}, {2, 7}, {2, 3}) → 68200
• For some lines concurrent in X(3):
  ({1, 3}, {2, 3}, {3, 6}) → 68201; ({1, 3}, {3, 6}, {2, 3}) → 68202; ({1, 3}, {3, 6}, {3, 7}) → 68208; ({1, 3}, {3, 7}, {2, 3}) → 68203; ({1, 3}, {3, 7}, {3, 6}) → 68204; ({2, 3}, {3, 6}, {1, 3}) → 68205; ({2, 3}, {3, 7}, {1, 3}) → 68206; ({3, 6}, {3, 7}, {1, 3}) → 68207
• For some lines concurrent in X(4):
  ({1, 4}, {2, 4}, {4, 6}) → 68209; ({1, 4}, {4, 6}, {2, 4}) → 68210; ({2, 4}, {4, 6}, {1, 4}) → 68211; ({2, 4}, {4, 7}, {1, 4}) → 65330
• For some lines concurrent in X(6):
  ({1, 6}, {2, 6}, {3, 6}) → 43356; ({1, 6}, {2, 6}, {4, 6}) → 68212; ({1, 6}, {2, 6}, {6, 7}) → 68213; ({1, 6}, {3, 6}, {2, 6}) → 931; ({1, 6}, {3, 6}, {4, 6}) → 68214; ({1, 6}, {3, 6}, {6, 7}) → 68215; ({1, 6}, {4, 6}, {2, 6}) → 68216; ({1, 6}, {4, 6}, {3, 6}) → 68217; ({1, 6}, {6, 7}, {2, 6}) → 68218; ({1, 6}, {6, 7}, {3, 6}) → 68219; ({2, 6}, {3, 6}, {1, 6}) → 34594; ({2, 6}, {3, 6}, {4, 6}) → 68220; ({2, 6}, {4, 6}, {1, 6}) → 59097; ({2, 6}, {4, 6}, {3, 6}) → 925; ({2, 6}, {4, 6}, {6, 7}) → 68221; ({2, 6}, {6, 7}, {1, 6}) → 43349; ({3, 6}, {4, 6}, {1, 6}) → 53683; ({3, 6}, {4, 6}, {2, 6}) → 59038; ({3, 6}, {6, 7}, {1, 6}) → 68222; ({3, 6}, {6, 7}, {2, 6}) → 68223
• For some lines concurrent in X(7):
  ({1, 7}, {2, 7}, {3, 7}) → 13149; ({1, 7}, {2, 7}, {4, 7}) → 68224; ({1, 7}, {2, 7}, {6, 7}) → 68225; ({1, 7}, {3, 7}, {2, 7}) → 68226; ({1, 7}, {4, 7}, {2, 7}) → 50392; ({1, 7}, {4, 7}, {3, 7}) → 68227; ({1, 7}, {6, 7}, {2, 7}) → 68228; ({1, 7}, {6, 7}, {3, 7}) → 68229; ({2, 7}, {4, 7}, {1, 7}) → 68230; ({2, 7}, {6, 7}, {1, 7}) → 68231; ({3, 7}, {4, 7}, {1, 7}) → 664; ({3, 7}, {4, 7}, {2, 7}) → 658; ({3, 7}, {6, 7}, {2, 7}) → 68232

Note: These results come from a specific application of a more general theorem involving the conservation of cross-ratios. For more information, see this link.

X(68179) = TRIPOLE OF THE {X(1)X(2), X(1)X(3)}-HARMONIC LINE OF X(1)X(7)

X(68179) lies on these lines: {88, 21454}, {658, 21362}, {664, 27834}, {673, 60937}, {1020, 68184}, {1025, 68192}, {1156, 3486}, {3305, 34234}, {3732, 68180}, {4552, 65235}, {4606, 62669}, {24029, 68185}, {36101, 56200}, {43760, 60941}

X(68179) = trilinear pole of the line {1, 3523} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68179) = barycentric product X(i)*X(j) for these {i, j}: {190, 44794}, {658, 56200}, {664, 7320}
X(68179) = trilinear product X(i)*X(j) for these {i, j}: {100, 44794}, {651, 7320}, {934, 56200}
X(68179) = trilinear quotient X(i)/X(j) for these (i, j): (190, 4853), (651, 3304), (658, 7271), (664, 5437), (4552, 3698), (4554, 31995), (4569, 43983), (7320, 650)

X(68180) = TRIPOLE OF THE {X(1)X(2), X(1)X(4)}-HARMONIC LINE OF X(1)X(7)

X(68180) lies on these lines: {88, 65046}, {664, 65259}, {673, 41441}, {1020, 68190}, {1156, 7319}, {3732, 68179}, {5748, 36100}, {20059, 36101}, {20223, 34234}, {37131, 60993}

X(68180) = trilinear pole of the line {1, 3091} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68180) = barycentric product X(i)*X(j) for these {i, j}: {190, 65046}, {651, 65047}, {664, 7319}, {4554, 41441}
X(68180) = trilinear product X(i)*X(j) for these {i, j}: {100, 65046}, {109, 65047}, {651, 7319}, {664, 41441}
X(68180) = trilinear quotient X(i)/X(j) for these (i, j): (100, 62245), (651, 5204), (664, 3928), (4552, 3962), (4554, 21296), (4572, 21605), (6516, 23140), (7319, 650), (18026, 17917)

X(68181) = TRIPOLE OF THE {X(1)X(2), X(1)X(6)}-HARMONIC LINE OF X(1)X(4)

X(68181) lies on these lines: {162, 3939}, {651, 4574}, {653, 1018}, {658, 65233}, {662, 4587}, {673, 60958}, {823, 65160}, {3882, 37206}, {4552, 68183}, {4606, 14543}, {8814, 43760}, {35338, 65227}, {61233, 68187}

X(68181) = trilinear pole of the line {1, 2318} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68181) = pole of the line {17554, 54392} with respect to the Yff parabola
X(68181) = barycentric product X(i)*X(j) for these {i, j}: {3699, 8814}, {8813, 65160}
X(68181) = trilinear product X(i)*X(j) for these {i, j}: {644, 8814}, {8813, 56183}
X(68181) = trilinear quotient X(i)/X(j) for these (i, j): (100, 54358), (101, 54321), (190, 54392), (644, 13615), (8814, 3669)

X(68182) = TRIPOLE OF THE {X(1)X(2), X(1)X(6)}-HARMONIC LINE OF X(1)X(7)

X(68182) lies on these lines: {651, 65194}, {658, 1018}, {664, 68186}, {673, 7308}, {799, 6558}, {1025, 68184}, {1156, 6154}, {35341, 68192}, {37223, 65166}, {43760, 65384}

X(68182) = trilinear pole of the line {1, 4924} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68182) = pole of the line {37681, 62856} with respect to the Kiepert parabola
X(68182) = trilinear quotient X(i)/X(j) for these (i, j): (190, 10582), (664, 60955), (4554, 32086)

X(68183) = TRIPOLE OF THE {X(1)X(3), X(1)X(4)}-HARMONIC LINE OF X(1)X(6)

X(68183) lies on these lines: {100, 1020}, {162, 32714}, {190, 4566}, {658, 14543}, {662, 934}, {673, 60939}, {799, 4569}, {1156, 5665}, {1461, 65259}, {4552, 68181}, {4606, 65233}, {9776, 34234}, {37142, 63157}, {61180, 65213}, {65159, 65217}

X(68183) = trilinear pole of the line {1, 1427} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68183) = pole of the line {20007, 63141} with respect to the Yff parabola
X(68183) = barycentric product X(i)*X(j) for these {i, j}: {9, 50392}, {664, 5665}, {934, 43533}, {1427, 68194}, {1446, 59079}, {4566, 63157}
X(68183) = trilinear product X(i)*X(j) for these {i, j}: {55, 50392}, {651, 5665}, {1020, 63157}, {1042, 68194}, {1461, 43533}, {3668, 59079}
X(68183) = trilinear quotient X(i)/X(j) for these (i, j): (190, 20007), (651, 3601), (658, 3945), (664, 5273), (934, 62812), (1461, 4252), (5665, 650)

X(68184) = TRIPOLE OF THE {X(1)X(3), X(1)X(7)}-HARMONIC LINE OF X(1)X(2)

X(68184) lies on these lines: {100, 59125}, {190, 63203}, {664, 4606}, {673, 60955}, {1020, 68179}, {1025, 68182}, {1156, 3485}, {1461, 65222}, {3732, 68190}

X(68184) = trilinear pole of the line {1, 3522} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68184) = barycentric product X(i)*X(j) for these {i, j}: {75, 59125}, {664, 5558}
X(68184) = trilinear product X(i)*X(j) for these {i, j}: {2, 59125}, {651, 5558}
X(68184) = trilinear quotient X(i)/X(j) for these (i, j): (7, 47921), (190, 4882), (651, 3303), (658, 4328), (664, 7308), (4552, 3983), (4554, 32087), (5558, 650)

X(68185) = TRIPOLE OF THE {X(1)X(3), X(1)X(7)}-HARMONIC LINE OF X(1)X(4)

X(68185) lies on these lines: {88, 65028}, {100, 63782}, {653, 63203}, {664, 37212}, {673, 60938}, {1020, 65226}, {1156, 3296}, {1461, 65217}, {1813, 65222}, {2349, 5325}, {3732, 38340}, {4566, 68188}, {5437, 34234}, {18230, 36101}, {24029, 68179}, {30679, 36100}

X(68185) = trilinear pole of the line {1, 376} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68185) = barycentric product X(i)*X(j) for these {i, j}: {190, 65028}, {651, 64995}, {653, 30679}, {664, 3296}, {4572, 61375}
X(68185) = trilinear product X(i)*X(j) for these {i, j}: {100, 65028}, {108, 30679}, {109, 64995}, {651, 3296}, {4554, 61375}
X(68185) = trilinear quotient X(i)/X(j) for these (i, j): (7, 47965), (57, 48340), (65, 58299), (85, 48268), (651, 3295), (653, 65128), (658, 7190), (664, 3305), (668, 42032), (934, 52424), (3296, 650), (4552, 3697), (4554, 42696), (4569, 52422), (4573, 63158), (6516, 55466)

X(68186) = TRIPOLE OF THE {X(1)X(3), X(1)X(7)}-HARMONIC LINE OF X(1)X(6)

X(68186) lies on these lines: {100, 58103}, {190, 35312}, {664, 68182}, {673, 21454}, {934, 65222}, {1025, 4606}, {1156, 5083}, {3305, 36101}, {4566, 68189}, {34234, 56054}, {34821, 37129}, {43762, 60941}, {56543, 68192}

X(68186) = trilinear pole of the line {1, 1418} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68186) = barycentric product X(i)*X(j) for these {i, j}: {75, 58103}, {100, 56348}, {651, 56054}, {664, 10390}, {668, 34821}
X(68186) = trilinear product X(i)*X(j) for these {i, j}: {2, 58103}, {101, 56348}, {109, 56054}, {190, 34821}, {651, 10390}
X(68186) = trilinear quotient X(i)/X(j) for these (i, j): (651, 10389), (664, 18230), (1275, 65194), (10390, 650)

X(68187) = TRIPOLE OF THE {X(1)X(4), X(1)X(6)}-HARMONIC LINE OF X(1)X(2)

X(68187) lies on these lines: {88, 63078}, {100, 53288}, {162, 35281}, {190, 61237}, {651, 61212}, {664, 37141}, {673, 58001}, {937, 37129}, {1156, 9780}, {2255, 20332}, {4552, 68188}, {4606, 25268}, {5271, 34234}, {6335, 65213}, {14543, 27834}, {37223, 65200}, {61233, 68181}, {65226, 65233}

X(68187) = isotomic conjugate of the complement of X(60492)
X(68187) = trilinear pole of the line {1, 329} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68187) = pole of the line {936, 37267} with respect to the Yff parabola
X(68187) = barycentric product X(i)*X(j) for these {i, j}: {75, 58991}, {100, 58001}, {664, 66940}, {668, 937}, {1978, 2255}
X(68187) = trilinear product X(i)*X(j) for these {i, j}: {2, 58991}, {101, 58001}, {190, 937}, {322, 58957}, {651, 66940}, {668, 2255}
X(68187) = trilinear quotient X(i)/X(j) for these (i, j): (100, 2256), (190, 936), (651, 1466), (937, 649), (1783, 11406), (2255, 667)

X(68188) = TRIPOLE OF THE {X(1)X(4), X(1)X(6)}-HARMONIC LINE OF X(1)X(3)

X(68188) lies on these lines: {651, 61237}, {658, 68230}, {673, 62775}, {1156, 38271}, {4552, 68187}, {4566, 68185}, {14543, 65226}, {27834, 65233}, {34234, 56943}, {36101, 36629}, {37203, 67985}, {40577, 65218}

X(68188) = trilinear pole of the line {1, 1864} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68188) = pole of the line {84, 27383} with respect to the Yff parabola
X(68188) = barycentric product X(i)*X(j) for these {i, j}: {9, 68230}, {658, 36629}, {664, 38271}, {934, 36624}
X(68188) = trilinear product X(i)*X(j) for these {i, j}: {55, 68230}, {651, 38271}, {934, 36629}, {1461, 36624}
X(68188) = trilinear quotient X(i)/X(j) for these (i, j): (7, 65412), (109, 37519), (190, 27383), (226, 65414), (651, 15803), (664, 9965), (1813, 23072), (4551, 21866), (4552, 67850)

X(68189) = TRIPOLE OF THE {X(1)X(4), X(1)X(6)}-HARMONIC LINE OF X(1)X(7)

X(68189) lies on these lines: {658, 61237}, {662, 65194}, {4566, 68186}, {9965, 36101}

X(68189) = trilinear pole of the line {1, 5809} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68189) = trilinear quotient X(664)/X(60990)

X(68190) = TRIPOLE OF THE {X(1)X(4), X(1)X(7)}-HARMONIC LINE OF X(1)X(2)

X(68190) lies on these lines: {57, 43759}, {673, 67698}, {1020, 68180}, {1156, 5221}, {3732, 68184}, {23707, 37523}, {23890, 37211}, {36101, 62778}

X(68190) = trilinear pole of the line {1, 3146} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68190) = barycentric product X(i)*X(j) for these {i, j}: {664, 5556}, {4554, 67698}, {4561, 10977}
X(68190) = trilinear product X(i)*X(j) for these {i, j}: {651, 5556}, {664, 67698}, {1332, 10977}
X(68190) = trilinear quotient X(i)/X(j) for these (i, j): (651, 5217), (664, 3929), (934, 62207), (4552, 4005), (4554, 32099), (5556, 650), (10977, 6591)

X(68191) = TRIPOLE OF THE {X(1)X(4), X(1)X(7)}-HARMONIC LINE OF X(1)X(6)

X(68191) lies on these lines: {662, 35312}, {1156, 52819}, {4566, 65222}, {5249, 36101}, {61180, 65218}

X(68191) = trilinear pole of the line {1, 52023} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68191) = trilinear quotient X(664)/X(61024)

X(68192) = TRIPOLE OF THE {X(1)X(6), X(1)X(7)}-HARMONIC LINE OF X(1)X(2)

X(68192) lies on these lines: {88, 23587}, {190, 62533}, {651, 65165}, {664, 61240}, {673, 5437}, {1025, 68179}, {1156, 3035}, {4624, 4765}, {5325, 65261}, {35341, 68182}, {37138, 61222}, {37139, 65194}, {37223, 43290}, {43762, 60938}, {56543, 68186}

X(68192) = isotomic conjugate of the complement of X(4765)
X(68192) = trilinear pole of the line {1, 144} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68192) = pole of the line {8580, 61006} with respect to the Yff parabola
X(68192) = barycentric product X(i)*X(j) for these {i, j}: {100, 56074}, {190, 56043}
X(68192) = trilinear product X(i)*X(j) for these {i, j}: {100, 56043}, {101, 56074}
X(68192) = trilinear quotient X(i)/X(j) for these (i, j): (99, 24557), (190, 8580), (658, 62793), (664, 60937), (668, 4461), (4554, 31994)

X(68193) = TRIPOLE OF THE {X(1)X(6), X(1)X(7)}-HARMONIC LINE OF X(1)X(4)

X(68193) lies on these lines: {662, 65165}, {1156, 8232}, {4566, 61240}, {5273, 36101}, {65193, 65218}

X(68193) = trilinear pole of the line {1, 5759} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line

X(68194) = TRIPOLE OF THE {X(1)X(2), X(2)X(3)}-HARMONIC LINE OF X(2)X(6)

X(68194) lies on the Steiner circumellipse and these lines: {99, 14543}, {190, 7256}, {643, 68195}, {645, 664}, {662, 58132}, {668, 7258}, {671, 43533}, {799, 4569}, {3227, 63157}, {5665, 35176}, {35136, 57060}, {36841, 54951}, {53655, 57216}, {59646, 64985}, {65205, 68200}

X(68194) = isogonal conjugate of the Gibert circumtangential conjugate of X(59079)
X(68194) = trilinear pole of the line {2, 1043} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68194) = pole of the the tripolar of X(4252) with respect to the Stammler hyperbola
X(68194) = pole of the the tripolar of X(3945) with respect to the Steiner-Wallace hyperbola
X(68194) = barycentric product X(i)*X(j) for these {i, j}: {76, 59079}, {99, 43533}, {668, 63157}, {5665, 7257}
X(68194) = trilinear product X(i)*X(j) for these {i, j}: {75, 59079}, {190, 63157}, {645, 5665}, {662, 43533}, {1043, 68183}
X(68194) = trilinear quotient X(i)/X(j) for these (i, j): (99, 62812), (645, 3601), (662, 4252), (799, 3945), (811, 7490), (5665, 7180), (6335, 1869), (7257, 5273), (7258, 20007)

X(68195) = TRIPOLE OF THE {X(1)X(2), X(2)X(3)}-HARMONIC LINE OF X(2)X(7)

X(68195) lies on the Steiner circumellipse and these lines: {99, 59061}, {190, 65206}, {643, 68194}, {662, 68200}, {664, 14543}, {903, 28610}, {1332, 53647}, {18026, 65170}, {58132, 65205}

X(68195) = isogonal conjugate of the Gibert circumtangential conjugate of X(59061)
X(68195) = trilinear pole of the line {2, 950} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68195) = barycentric product X(i)*X(j) for these {i, j}: {76, 59061}, {190, 67941}
X(68195) = trilinear product X(i)*X(j) for these {i, j}: {75, 59061}, {100, 67941}
X(68195) = trilinear quotient X(i)/X(j) for these (i, j): (2, 7655), (190, 11523)

X(68196) = TRIPOLE OF THE {X(1)X(2), X(2)X(6)}-HARMONIC LINE OF X(2)X(3)

X(68196) lies on the Steiner circumellipse and these lines: {643, 68198}, {645, 32042}, {648, 4427}, {662, 58135}, {6540, 52609}, {65205, 68199}

X(68196) = trilinear pole of the line {2, 41014} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68196) = pole of the the tripolar of X(63014) with respect to the Steiner-Wallace hyperbola
X(68196) = trilinear quotient X(i)/X(j) for these (i, j): (99, 62808), (799, 63014), (811, 6994)

X(68197) = TRIPOLE OF THE {X(2)X(3), X(2)X(6)}-HARMONIC LINE OF X(1)X(2)

X(68197) lies on the Steiner circumellipse and these lines: {2, 57891}, {99, 4556}, {110, 190}, {290, 57704}, {313, 40589}, {643, 54970}, {645, 4629}, {662, 668}, {664, 4565}, {670, 4610}, {671, 43531}, {799, 54957}, {903, 42028}, {1043, 57888}, {1494, 57876}, {2214, 18827}, {4555, 4591}, {4558, 65275}, {4569, 4637}, {4627, 53658}, {6528, 52919}, {14570, 54951}, {17940, 35148}, {18026, 65232}, {57977, 65168}, {65205, 65274}, {65282, 65850}

X(68197) = reflection of X(57891) in X(2)
X(68197) = isotomic conjugate of X(23879)
X(68197) = isogonal conjugate of X(42664)
X(68197) = cross-difference of every pair of points on the line X(52327)X(52328)
X(68197) = trilinear pole of the line {2, 58} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68197) = perspector of the inconic with center X(23879)
X(68197) = pole of the line {1010, 5224} with respect to the Kiepert parabola
X(68197) = pole of the line {834, 42664} with respect to the Stammler hyperbola
X(68197) = pole of the line {14349, 23879} with respect to the Steiner-Wallace hyperbola
X(68197) = barycentric product X(i)*X(j) for these {i, j}: {58, 57977}, {76, 58951}, {81, 37218}, {86, 835}, {99, 43531}, {110, 57824}, {190, 56047}, {593, 65850}, {648, 57876}, {799, 2214}, {4600, 43927}, {6331, 57704}
X(68197) = trilinear product X(i)*X(j) for these {i, j}: {58, 37218}, {75, 58951}, {81, 835}, {99, 2214}, {100, 56047}, {162, 57876}, {163, 57824}, {662, 43531}, {811, 57704}, {849, 65850}, {1333, 57977}, {4567, 43927}
X(68197) = trilinear quotient X(i)/X(j) for these (i, j): (2, 47842), (6, 50488), (75, 23879), (81, 834), (86, 14349), (99, 28606), (100, 56926), (162, 44103), (274, 45746), (321, 23282), (645, 3876), (662, 386), (668, 56810), (670, 33935), (757, 52615), (799, 5224), (811, 469), (835, 37), (1333, 8637), (1978, 42714)

X(68198) = TRIPOLE OF THE {X(2)X(3), X(2)X(7)}-HARMONIC LINE OF X(1)X(2)

X(68198) lies on the Steiner circumellipse and these lines: {190, 14544}, {643, 68196}, {662, 68199}, {1332, 53658}, {58135, 65205}

X(68198) = trilinear pole of the line {2, 4292} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line

X(68199) = TRIPOLE OF THE {X(2)X(3), X(2)X(7)}-HARMONIC LINE OF X(2)X(6)

X(68199) lies on the Steiner circumellipse and these lines: {99, 14544}, {643, 58135}, {648, 68210}, {662, 68198}, {668, 55241}, {671, 60170}, {811, 65270}, {3228, 14553}, {4573, 53642}, {35136, 57249}, {65205, 68196}

X(68199) = isotomic conjugate of the polar conjugate of X(68210)
X(68199) = trilinear pole of the line {2, 1901} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68199) = pole of the the tripolar of X(37504) with respect to the Stammler hyperbola
X(68199) = pole of the the tripolar of X(14552) with respect to the Steiner-Wallace hyperbola
X(68199) = barycentric product X(i)*X(j) for these {i, j}: {69, 68210}, {99, 60170}, {670, 14553}
X(68199) = trilinear product X(i)*X(j) for these {i, j}: {63, 68210}, {662, 60170}, {799, 14553}
X(68199) = trilinear quotient X(i)/X(j) for these (i, j): (99, 31424), (662, 37504), (799, 14552), (811, 7498), (14553, 798)

X(68200) = TRIPOLE OF THE {X(2)X(6), X(2)X(7)}-HARMONIC LINE OF X(2)X(3)

X(68200) lies on the Steiner circumellipse and these lines: {643, 58132}, {648, 17136}, {662, 68195}, {65205, 68194}

X(68200) = isotomic conjugate of the polar conjugate of X(68209)
X(68200) = trilinear pole of the line {2, 56020} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68200) = barycentric product X(69)*X(68209)
X(68200) = trilinear product X(63)*X(68209)
X(68200) = trilinear quotient X(i)/X(j) for these (i, j): (99, 62829), (811, 7518)

X(68201) = TRIPOLE OF THE {X(1)X(3), X(2)X(3)}-HARMONIC LINE OF X(3)X(6)

X(68201) lies on the MacBeath circumconic and these lines: {4558, 57194}, {14543, 68202}

X(68201) = trilinear pole of the line {3, 4653} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68201) = orthocorrespondent of X(53829)
X(68201) = pole of the the tripolar of X(4189) with respect to the Stammler hyperbola
X(68201) = pole of the the tripolar of X(34282) with respect to the Steiner-Wallace hyperbola
X(68201) = trilinear quotient X(i)/X(j) for these (i, j): (662, 4189), (799, 34282)

X(68202) = TRIPOLE OF THE {X(1)X(3), X(3)X(6)}-HARMONIC LINE OF X(2)X(3)

X(68202) lies on the MacBeath circumconic and these lines: {99, 1332}, {110, 36077}, {287, 57831}, {648, 61197}, {662, 1331}, {895, 16428}, {1414, 1813}, {4558, 52935}, {4563, 4623}, {4616, 65296}, {14543, 68201}, {14597, 40412}, {17708, 63220}, {52610, 68224}

X(68202) = isotomic conjugate of the polar conjugate of X(36077)
X(68202) = isogonal conjugate of the complement of X(50557)
X(68202) = trilinear pole of the line {3, 81} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68202) = inverse Mimosa transform of X(46382)
X(68202) = orthocorrespondent of X(i) for these i: {36077, 38967}
X(68202) = pole of the the tripolar of X(405) with respect to the Stammler hyperbola
X(68202) = pole of the line {36077, 43356} with respect to the Steiner circumellipse
X(68202) = pole of the the tripolar of X(44140) with respect to the Steiner-Wallace hyperbola
X(68202) = barycentric product X(i)*X(j) for these {i, j}: {69, 36077}, {81, 54970}, {86, 65227}, {99, 51223}, {110, 57831}, {250, 63220}, {274, 36080}, {799, 2215}, {2335, 4573}
X(68202) = trilinear product X(i)*X(j) for these {i, j}: {58, 54970}, {63, 36077}, {81, 65227}, {86, 36080}, {99, 2215}, {163, 57831}, {662, 51223}, {1414, 2335}
X(68202) = trilinear quotient X(i)/X(j) for these (i, j): (3, 46382), (81, 46385), (86, 23882), (99, 5271), (163, 5320), (190, 5295), (648, 39585), (653, 1882), (662, 405), (799, 44140), (873, 15417), (1414, 37543), (2215, 512), (2335, 4041), (4561, 42706), (4565, 1451)

X(68203) = TRIPOLE OF THE {X(1)X(3), X(3)X(7)}-HARMONIC LINE OF X(2)X(3)

X(68203) lies on the MacBeath circumconic and these lines: {14545, 68204}, {52610, 68208}

X(68203) = trilinear pole of the line {3, 11036} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line

X(68204) = TRIPOLE OF THE {X(1)X(3), X(3)X(7)}-HARMONIC LINE OF X(3)X(6)

X(68204) lies on the MacBeath circumconic and these lines: {4566, 68208}, {14545, 68203}

X(68204) = intersection, other than {A, B, C}, of the circumconics through X(i), X(j) for these {i, j}: {110, 287}, {658, 68219}

X(68205) = TRIPOLE OF THE {X(2)X(3), X(3)X(6)}-HARMONIC LINE OF X(1)X(3)

X(68205) lies on the MacBeath circumconic and these lines: {81, 1797}, {287, 57830}, {651, 1625}, {653, 68207}, {895, 57666}, {1172, 2989}, {1331, 57151}, {1332, 56248}, {3193, 60049}, {4552, 7252}, {40571, 46638}

X(68205) = trilinear pole of the line {3, 1724} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68205) = touchpoint of MacBeath circumconic and line {14543, 68205}
X(68205) = pole of the line {18133, 35998} with respect to the Kiepert parabola
X(68205) = pole of the line {48281, 57042} with respect to the Stammler hyperbola
X(68205) = pole of the the tripolar of X(44139) with respect to the Steiner-Wallace hyperbola
X(68205) = barycentric product X(i)*X(j) for these {i, j}: {81, 56248}, {99, 57666}, {110, 57830}, {1414, 44040}, {4592, 60816}
X(68205) = trilinear product X(i)*X(j) for these {i, j}: {29, 40518}, {58, 56248}, {163, 57830}, {662, 57666}, {4558, 60816}, {4565, 44040}
X(68205) = trilinear quotient X(i)/X(j) for these (i, j): (10, 21721), (60, 57212), (81, 48281), (86, 47796), (99, 32939), (163, 44085), (190, 56318), (283, 57042), (284, 48387), (333, 20293), (662, 404), (799, 44139), (1897, 56319), (2193, 57103), (4560, 44311), (4561, 42705), (23189, 39006)

X(68206) = TRIPOLE OF THE {X(2)X(3), X(3)X(7)}-HARMONIC LINE OF X(1)X(3)

X(68206) lies on the MacBeath circumconic and these lines: {110, 653}, {648, 52938}, {664, 4558}, {938, 54972}, {1305, 58987}, {1331, 4552}, {1813, 4566}, {1814, 2219}, {4563, 4572}, {14545, 68208}

X(68206) = isotomic conjugate of the anticomplement of X(60494)
X(68206) = trilinear pole of the line {3, 226} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68206) = perspector of the inconic with center X(60494)
X(68206) = barycentric product X(i)*X(j) for these {i, j}: {109, 57911}, {349, 58987}, {664, 54972}, {2219, 4554}, {13395, 56727}
X(68206) = trilinear product X(i)*X(j) for these {i, j}: {651, 54972}, {664, 2219}, {1415, 57911}, {1441, 58987}
X(68206) = trilinear quotient X(i)/X(j) for these (i, j): (100, 15830), (651, 581), (664, 62857), (2219, 663)

X(68207) = TRIPOLE OF THE {X(3)X(6), X(3)X(7)}-HARMONIC LINE OF X(1)X(3)

X(68207) lies on the MacBeath circumconic and these lines: {653, 68205}, {2989, 62798}

X(68207) = trilinear pole of the line {3, 6180} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68207) = touchpoint of MacBeath circumconic and line {4566, 68207}
X(68207) = trilinear quotient X(i)/X(j) for these (i, j): (7, 23806), (1813, 23171), (4552, 22027), (4554, 18738)

X(68208) = TRIPOLE OF THE {X(1)X(3), X(3)X(6)}-HARMONIC LINE OF X(3)X(7)

X(68208) lies on the MacBeath circumconic and these lines: {4566, 68204}, {14545, 68206}, {52610, 68203}, {61197, 68224}

X(68208) = intersection, other than {A, B, C}, of the circumconics through X(i), X(j) for these {i, j}: {100, 68229}, {110, 287}

X(68209) = TRIPOLE OF THE {X(1)X(4), X(2)X(4)}-HARMONIC LINE OF X(4)X(6)

X(68209) lies on these lines: {648, 17136}, {1897, 22003}, {14543, 68210}

X(68209) = polar conjugate of the isotomic conjugate of X(68200)
X(68209) = trilinear pole of the line {4, 17056} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68209) = barycentric product X(4)*X(68200)
X(68209) = trilinear product X(19)*X(68200)
X(68209) = trilinear quotient X(i)/X(j) for these (i, j): (648, 62829), (823, 7518)

X(68210) = TRIPOLE OF THE {X(1)X(4), X(4)X(6)}-HARMONIC LINE OF X(2)X(4)

X(68210) lies on these lines: {648, 68199}, {14543, 68209}, {14553, 16081}, {16080, 60170}

X(68210) = polar conjugate of the isotomic conjugate of X(68199)
X(68210) = trilinear pole of the line {4, 41083} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68210) = barycentric product X(i)*X(j) for these {i, j}: {4, 68199}, {648, 60170}, {6331, 14553}
X(68210) = trilinear product X(i)*X(j) for these {i, j}: {19, 68199}, {162, 60170}, {811, 14553}
X(68210) = trilinear quotient X(i)/X(j) for these (i, j): (162, 37504), (648, 31424), (811, 14552), (823, 7498), (14553, 810)

X(68211) = TRIPOLE OF THE {X(2)X(4), X(4)X(6)}-HARMONIC LINE OF X(1)X(4)

X(68211) lies on these lines: {107, 58986}, {112, 653}, {272, 52781}, {648, 65274}, {1751, 16080}, {6335, 51566}, {6336, 40574}, {13149, 65232}, {24019, 54240}, {57732, 66951}

X(68211) = polar conjugate of the isogonal conjugate of X(58986)
X(68211) = polar conjugate of the isotomic conjugate of X(65274)
X(68211) = trilinear pole of the line {4, 580} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68211) = barycentric product X(i)*X(j) for these {i, j}: {4, 65274}, {28, 51566}, {29, 1305}, {92, 65254}, {112, 40011}, {162, 2997}, {190, 40574}, {264, 58986}, {272, 1897}, {648, 1751}, {811, 2218}, {5546, 58074}, {6528, 66951}, {8750, 57784}
X(68211) = trilinear product X(i)*X(j) for these {i, j}: {4, 65254}, {19, 65274}, {92, 58986}, {100, 40574}, {112, 2997}, {162, 1751}, {272, 1783}, {648, 2218}, {823, 66951}, {1172, 1305}, {1474, 51566}
X(68211) = trilinear quotient X(i)/X(j) for these (i, j): (27, 23800), (28, 43060), (100, 51574), (108, 66918), (112, 2352), (162, 579), (272, 905), (278, 51658), (648, 3868), (811, 18134), (823, 5125), (1172, 8676), (1305, 1214), (1751, 656), (1783, 209), (1896, 57043), (1897, 22021), (2218, 647), (2322, 58333), (2997, 525)

X(68212) = TRIPOLE OF THE {X(1)X(6), X(2)X(6)}-HARMONIC LINE OF X(4)X(6)

X(68212) lies on the circumcircle and these lines: {107, 4436}, {4427, 59079}

X(68212) = trilinear pole of the line {6, 16845} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line

X(68213) = TRIPOLE OF THE {X(1)X(6), X(2)X(6)}-HARMONIC LINE OF X(6)X(7)

X(68213) lies on the circumcircle and these lines: {109, 65194}, {190, 28879}, {664, 58103}, {927, 4436}, {28903, 53337}

X(68213) = trilinear pole of the line {6, 5308} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line

X(68214) = TRIPOLE OF THE {X(1)X(6), X(3)X(6)}-HARMONIC LINE OF X(4)X(6)

X(68214) lies on the circumcircle and these lines: {101, 54442}, {107, 53280}, {643, 58992}, {927, 68147}, {14543, 68217}, {36077, 65177}, {59084, 65201}

X(68214) = trilinear pole of the line {6, 1006} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68214) = Collings transform of X(6829)
X(68214) = trilinear quotient X(162)/X(7497)

X(68215) = TRIPOLE OF THE {X(1)X(6), X(3)X(6)}-HARMONIC LINE OF X(6)X(7)

X(68215) lies on the circumcircle and these lines: {105, 20967}, {927, 53280}, {3573, 43344}, {6575, 54440}, {9057, 65313}

X(68215) = intersection, other than {A, B, C}, of the circumconics through X(i), X(j) for these {i, j}: {74, 98}, {643, 666}

X(68216) = TRIPOLE OF THE {X(1)X(6), X(4)X(6)}-HARMONIC LINE OF X(2)X(6)

X(68216) lies on the circumcircle and these lines: {99, 53761}, {662, 8059}, {36797, 40117}, {58945, 65201}

X(68216) = trilinear pole of the line {6, 452} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68216) = Collings transform of X(5177)
X(68216) = trilinear quotient X(i)/X(j) for these (i, j): (162, 37245), (1897, 1868)

X(68217) = TRIPOLE OF THE {X(1)X(6), X(4)X(6)}-HARMONIC LINE OF X(3)X(6)

X(68217) lies on the circumcircle and these lines: {74, 7580}, {98, 35988}, {104, 1817}, {110, 53761}, {915, 4183}, {934, 3658}, {953, 52889}, {1300, 37441}, {4246, 40117}, {4588, 54442}, {13398, 57119}, {13589, 67743}, {14543, 68214}, {35360, 36077}, {39439, 56374}

X(68217) = trilinear pole of the line {6, 6913} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68217) = Collings transform of X(6907)
X(68217) = trilinear quotient X(162)/X(7501)

X(68218) = TRIPOLE OF THE {X(1)X(6), X(6)X(7)}-HARMONIC LINE OF X(2)X(6)

X(68218) lies on the circumcircle and these lines: {100, 62533}, {103, 43182}, {109, 65165}, {658, 58985}, {662, 59067}, {664, 53622}, {668, 6574}, {813, 21362}, {874, 8706}, {2291, 5316}, {57928, 65642}

X(68218) = isotomic conjugate of the complement of X(50347)
X(68218) = trilinear pole of the line {6, 144} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68218) = Collings transform of X(31043)
X(68218) = trilinear quotient X(i)/X(j) for these (i, j): (190, 44798), (664, 8581)

X(68219) = TRIPOLE OF THE {X(1)X(6), X(6)X(7)}-HARMONIC LINE OF X(3)X(6)

X(68219) lies on the circumcircle and these lines: {103, 41853}

X(68219) = intersection, other than {A, B, C}, of the circumconics through X(i), X(j) for these {i, j}: {74, 98}, {651, 68229}

X(68220) = TRIPOLE OF THE {X(2)X(6), X(3)X(6)}-HARMONIC LINE OF X(4)X(6)

X(68220) lies on the circumcircle and these lines: {74, 3528}, {98, 7484}, {107, 1634}, {111, 46952}, {112, 65177}, {648, 58950}, {691, 30221}, {842, 37899}, {907, 35278}, {1297, 59343}, {1300, 35502}, {1302, 50947}, {1632, 53862}, {3563, 7714}, {4226, 58116}, {9064, 52913}, {23181, 59038}

X(68220) = trilinear pole of the line {6, 631} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68220) = Collings transform of X(i) for these i: {3090, 13341}
X(68220) = pole of the line {3523, 10601} with respect to the Kiepert parabola
X(68220) = pole of the the tripolar of X(10601) with respect to the Stammler hyperbola
X(68220) = pole of the the tripolar of X(32828) with respect to the Steiner-Wallace hyperbola
X(68220) = barycentric product X(i)*X(j) for these {i, j}: {99, 46952}, {4558, 66596}
X(68220) = trilinear product X(i)*X(j) for these {i, j}: {662, 46952}, {4575, 66596}
X(68220) = trilinear quotient X(i)/X(j) for these (i, j): (110, 1497), (162, 1598), (662, 10601), (799, 32828), (4575, 10984)

X(68221) = TRIPOLE OF THE {X(2)X(6), X(4)X(6)}-HARMONIC LINE OF X(6)X(7)

X(68221) lies on the circumcircle and these lines: {75, 26702}, {648, 36071}, {927, 1632}

X(68221) = trilinear pole of the line {6, 857} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68221) = Collings transform of X(379)

X(68222) = TRIPOLE OF THE {X(3)X(6), X(6)X(7)}-HARMONIC LINE OF X(1)X(6)

X(68222) lies on the circumcircle and these lines: {81, 59074}, {103, 1621}, {6078, 57151}, {28535, 66199}, {53337, 53627}

X(68222) = intersection, other than {A, B, C}, of the circumconics through X(i), X(j) for these {i, j}: {74, 98}, {81, 658}

X(68223) = TRIPOLE OF THE {X(3)X(6), X(6)X(7)}-HARMONIC LINE OF X(2)X(6)

X(68223) lies on the circumcircle and these lines: {86, 103}, {648, 40116}, {753, 17189}, {4616, 24016}, {55284, 65642}

X(68223) = trilinear pole of the line {6, 14953} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68223) = Collings transform of X(31014)

X(68224) = TRIPOLE OF THE {X(1)X(7), X(2)X(7)}-HARMONIC LINE OF X(4)X(7)

X(68224) lies on the MacBeath circumconic and these lines: {110, 4566}, {651, 13149}, {658, 1813}, {664, 1331}, {677, 14544}, {895, 52560}, {927, 15439}, {943, 62744}, {1332, 4554}, {1814, 2982}, {1815, 18652}, {4558, 4573}, {6056, 45253}, {36838, 65296}, {39796, 65810}, {40573, 60025}, {52610, 68202}, {56320, 60487}, {60041, 60047}, {61197, 68208}, {65301, 65847}

X(68224) = isotomic conjugate of the isogonal conjugate of X(32651)
X(68224) = isogonal conjugate of X(33525)
X(68224) = isotomic conjugate of the polar conjugate of X(58993)
X(68224) = polar conjugate of the anticomplement of X(59990)
X(68224) = trilinear pole of the line {3, 7} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68224) = orthocorrespondent of X(i) for these i: {15607, 58993}
X(68224) = pole of the the tripolar of X(8021) with respect to the Stammler hyperbola
X(68224) = pole of the line {43344, 58993} with respect to the Steiner circumellipse
X(68224) = pole of the the tripolar of X(51978) with respect to the Steiner-Wallace hyperbola
X(68224) = barycentric product X(i)*X(j) for these {i, j}: {7, 54952}, {69, 58993}, {75, 36048}, {76, 32651}, {85, 65217}, {99, 52560}, {348, 65334}, {658, 40435}, {664, 60041}, {934, 40422}, {943, 4569}, {1275, 56320}, {2259, 46406}, {2982, 4554}, {4566, 40412}, {4573, 60188}, {6063, 15439}
X(68224) = trilinear product X(i)*X(j) for these {i, j}: {2, 36048}, {7, 65217}, {57, 54952}, {63, 58993}, {75, 32651}, {77, 65334}, {85, 15439}, {651, 60041}, {658, 943}, {662, 52560}, {664, 2982}, {934, 40435}, {1020, 40412}, {1414, 60188}, {1461, 40422}, {1794, 13149}, {2259, 4569}, {4552, 63193}, {6516, 40573}, {7045, 56320}
X(68224) = trilinear quotient X(i)/X(j) for these (i, j): (77, 52306), (190, 64171), (279, 50354), (651, 14547), (653, 1859), (658, 942), (662, 8021), (664, 40937), (799, 51978), (934, 2260), (943, 657), (1020, 40952), (1275, 61220), (1414, 46882), (1446, 23752), (1461, 40956), (1794, 65102), (1813, 23207), (2259, 8641), (2982, 663)

X(68225) = TRIPOLE OF THE {X(1)X(7), X(2)X(7)}-HARMONIC LINE OF X(6)X(7)

X(68225) lies on these lines: {7, 1002}, {100, 4573}, {226, 67140}, {651, 927}, {658, 4551}, {664, 1018}, {883, 3952}, {4566, 36838}, {4569, 53227}, {4674, 21314}, {5219, 27475}, {13149, 61178}, {18793, 52161}, {31526, 56717}, {31618, 64206}, {36905, 52156}, {38955, 59255}, {40779, 62705}, {42290, 43063}, {43035, 67143}

X(68225) = isotomic conjugate of the complement of X(50356)
X(68225) = trilinear pole of the line {7, 37} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68225) = pole of the line {5223, 64304} with respect to the Yff parabola
X(68225) = barycentric product X(i)*X(j) for these {i, j}: {7, 32041}, {85, 37138}, {100, 62946}, {190, 62784}, {226, 51563}, {241, 53227}, {651, 59255}, {658, 60668}, {664, 27475}, {668, 42290}, {927, 62622}, {1002, 4554}, {2279, 4572}, {4569, 40779}, {4617, 59260}, {4625, 60677}, {6063, 8693}
X(68225) = trilinear product X(i)*X(j) for these {i, j}: {7, 37138}, {57, 32041}, {65, 51563}, {85, 8693}, {100, 62784}, {101, 62946}, {109, 59255}, {190, 42290}, {651, 27475}, {658, 40779}, {664, 1002}, {934, 60668}, {1458, 53227}, {2279, 4554}, {4552, 42302}, {4569, 60673}, {4573, 60677}, {4626, 59269}, {6614, 59260}
X(68225) = trilinear quotient X(i)/X(j) for these (i, j): (2, 45755), (7, 4724), (57, 66513), (85, 4762), (109, 60722), (190, 37658), (651, 2280), (658, 5228), (664, 1001), (668, 3886), (934, 1471), (1002, 663), (1441, 4804), (1897, 28044), (1978, 28809), (2279, 3063), (4552, 59207), (4554, 4384), (4566, 42289), (4569, 40719)

X(68226) = TRIPOLE OF THE {X(1)X(7), X(3)X(7)}-HARMONIC LINE OF X(2)X(7)

X(68226) lies on these lines: {651, 50392}, {5738, 62723}, {13149, 65170}

X(68226) = trilinear pole of the line {7, 3522} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68226) = trilinear quotient X(658)/X(11518)

X(68227) = TRIPOLE OF THE {X(1)X(7), X(4)X(7)}-HARMONIC LINE OF X(3)X(7)

X(68227) lies on these lines: {2407, 4573}, {4554, 42716}

X(68227) = trilinear pole of the line {7, 30} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68227) = trilinear quotient X(i)/X(j) for these (i, j): (658, 24929), (4569, 54357)

X(68228) = TRIPOLE OF THE {X(1)X(7), X(6)X(7)}-HARMONIC LINE OF X(2)X(7)

X(68228) lies on these lines: {7, 1360}, {100, 4624}, {658, 23973}, {664, 32040}, {927, 26716}, {4554, 65165}, {15511, 55937}, {42317, 62744}

X(68228) = trilinear pole of the line {7, 1419} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68228) = pole of the line {63975, 64304} with respect to the Yff parabola
X(68228) = barycentric product X(i)*X(j) for these {i, j}: {7, 32040}, {85, 65243}, {109, 59259}, {651, 55983}, {664, 55937}, {4569, 42317}, {4573, 54668}, {6063, 26716}
X(68228) = trilinear product X(i)*X(j) for these {i, j}: {7, 65243}, {57, 32040}, {85, 26716}, {109, 55983}, {651, 55937}, {658, 42317}, {1414, 54668}, {1415, 59259}
X(68228) = trilinear quotient X(i)/X(j) for these (i, j): (651, 42316), (658, 59215), (664, 5223), (4554, 29616), (24002, 61673), (26716, 41)

X(68229) = TRIPOLE OF THE {X(1)X(7), X(6)X(7)}-HARMONIC LINE OF X(3)X(7)

X(68229) lies on these lines: {5249, 52156}

X(68229) = trilinear pole of the line {7, 62183} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line

X(68230) = TRIPOLE OF THE {X(2)X(7), X(4)X(7)}-HARMONIC LINE OF X(1)X(7)

X(68230) lies on these lines: {658, 68188}, {664, 61185}, {38271, 62744}

X(68230) = trilinear pole of the line {7, 1210} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68230) = barycentric product X(i)*X(j) for these {i, j}: {85, 68188}, {4569, 38271}, {4626, 36624}
X(68230) = trilinear product X(i)*X(j) for these {i, j}: {7, 68188}, {658, 38271}, {4617, 36624}, {4626, 36629}
X(68230) = trilinear quotient X(i)/X(j) for these (i, j): (658, 15803), (934, 37519), (1088, 65412), (1446, 65414), (4554, 27383), (4566, 21866), (4569, 9965)

X(68231) = TRIPOLE OF THE {X(2)X(7), X(6)X(7)}-HARMONIC LINE OF X(1)X(7)

X(68231) lies on these lines: {7, 3021}, {190, 68138}, {664, 53337}, {4554, 43290}, {26007, 31188}, {34018, 60666}, {52156, 52164}, {60487, 67580}

X(68231) = isotomic conjugate of the anticomplement of X(54261)
X(68231) = trilinear pole of the line {7, 1743} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68231) = perspector of the inconic with center X(54261)
X(68231) = pole of the line {37681, 60963} with respect to the Kiepert parabola
X(68231) = barycentric product X(i)*X(j) for these {i, j}: {658, 56088}, {664, 42318}, {668, 42315}, {4554, 60666}
X(68231) = trilinear product X(i)*X(j) for these {i, j}: {190, 42315}, {651, 42318}, {664, 60666}, {934, 56088}
X(68231) = trilinear quotient X(i)/X(j) for these (i, j): (190, 59216), (658, 51302), (664, 3243), (668, 10005), (934, 42314), (1978, 59201), (4554, 29627), (4569, 51351)

X(68232) = TRIPOLE OF THE {X(3)X(7), X(6)X(7)}-HARMONIC LINE OF X(2)X(7)

X(68232) lies on these lines: {86, 52156}, {110, 658}, {162, 13149}, {643, 4554}, {664, 5546}, {1414, 36838}, {3019, 56144}, {4573, 4636}

X(68232) = trilinear pole of the line {7, 284} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68232) = barycentric product X(i)*X(j) for these {i, j}: {99, 63149}, {4565, 58024}, {4573, 56144}
X(68232) = trilinear product X(i)*X(j) for these {i, j}: {662, 63149}, {1414, 56144}
X(68232) = trilinear quotient X(i)/X(j) for these (i, j): (99, 41228), (1414, 991), (4573, 24635)

X(68233) = X(1)X(474)∩X(36)X(1293)

X(68233) lies on the cubic K086 and these lines: {1, 474}, {3, 67147}, {36, 1293}, {56, 33963}, {80, 53618}, {106, 65650}, {519, 22942}, {999, 40151}, {1149, 6065}, {1319, 51769}, {1323, 65173}, {1457, 38828}, {1785, 61478}, {1795, 10428}, {3086, 6556}, {3669, 30198}, {5563, 14261}, {16489, 56795}, {16784, 17967}, {27834, 54391}, {38460, 63774}, {56798, 57287}

X(68233) = reflection of X(36) in X(43081)
X(68233) = incircle-inverse of X(5836)
X(68233) = Conway-circle-inverse of X(35634)
X(68233) = X(i)-isoconjugate of X(j) for these (i,j): {1420, 12641}, {2743, 3667}
X(68233) = barycentric product X(i)*X(j) for these {i,j}: {2827, 27834}, {3445, 37758}, {3680, 37789}, {5193, 6557}, {8056, 38460}
X(68233) = barycentric quotient X(i)/X(j) for these {i,j}: {2827, 4462}, {5193, 5435}, {34080, 2743}, {37789, 39126}, {38460, 18743}, {58369, 4521}

X(68234) = X(1)X(142)∩X(36)X(1292)

X(68234) lies on the cubic K086 and these lines: {1, 142}, {36, 1292}, {57, 55013}, {1086, 60375}, {1785, 61480}, {1795, 36041}, {3338, 14268}, {3660, 5580}, {3676, 4905}, {5570, 18413}, {10980, 40154}, {41555, 43065}

X(68234) = incircle-inverse of X(142)
X(68234) = X(i)-isoconjugate of X(j) for these (i,j): {1617, 34894}, {2742, 3309}, {6600, 15728}, {21059, 51567}, {54236, 61491}
X(68234) = X(i)-Dao conjugate of X(j) for these (i,j): {10427, 3870}, {65930, 55337}, {65947, 4468}
X(68234) = barycentric product X(i)*X(j) for these {i,j}: {277, 26015}, {2191, 37788}, {2826, 37206}, {6601, 30379}
X(68234) = barycentric quotient X(i)/X(j) for these {i,j}: {277, 51567}, {2826, 4468}, {3660, 1445}, {15733, 55337}, {17107, 15728}, {26015, 344}, {30379, 6604}, {38468, 21609}, {40154, 43762}, {43065, 3870}, {56850, 31638}

X(68235) = X(1)X(512)∩X(36)X(741)

X(68235) lies on the cubic K086 and these lines: {1, 512}, {36, 741}, {80, 334}, {106, 805}, {511, 18792}, {519, 56154}, {1326, 1911}, {2311, 5526}, {3009, 17209}, {5006, 18268}, {61433, 65864}

X(68235) = X(65941)-Dao conjugate of X(3948)

X(68236) = X(1)X(1145)∩X(36)X(2743)

X(68236) lies on the cubics K086 and K338 and these lines: {1, 1145}, {36, 2743}, {40, 2137}, {80, 61484}, {644, 23617}, {1358, 4862}, {1420, 61079}, {1743, 40621}, {1785, 36125}, {1795, 61478}, {3241, 56314}, {4738, 6555}, {5697, 38515}, {15637, 37743}, {30236, 59326}, {30725, 64145}, {56939, 61481}, {63774, 64743}

X(68236) = reflection of X(22942) in X(1)
X(68236) = X(i)-isoconjugate of X(j) for these (i,j): {1293, 2827}, {3445, 38460}, {3680, 5193}, {37758, 38266}, {58369, 65173}
X(68236) = X(45036)-Dao conjugate of X(38460)
X(68236) = barycentric product X(i)*X(j) for these {i,j}: {2743, 4462}, {5435, 12641}
X(68236) = barycentric quotient X(i)/X(j) for these {i,j}: {145, 37758}, {1420, 37789}, {1743, 38460}, {2743, 27834}, {4394, 2827}, {12641, 6557}, {67843, 5193}

X(68237) = X(1)X(3939)∩X(36)X(1477)

X(68237) lies on the cubic K086 and these lines: {1, 3939}, {36, 1477}, {165, 15728}, {1174, 64446}, {1785, 36124}, {1795, 61480}, {3423, 8069}, {5119, 61491}, {10056, 62901}, {10482, 43762}, {13405, 51567}, {43050, 61230}

X(68237) = X(i)-isoconjugate of X(j) for these (i,j): {277, 43065}, {1292, 2826}, {2191, 26015}, {3660, 6601}, {15733, 40154}, {37788, 57656}, {56850, 57469}
X(68237) = barycentric product X(i)*X(j) for these {i,j}: {218, 51567}, {1445, 34894}, {2742, 4468}, {6600, 43762}, {15728, 55337}
X(68237) = barycentric quotient X(i)/X(j) for these {i,j}: {218, 26015}, {1445, 38468}, {1617, 30379}, {2742, 37206}, {3870, 37788}, {21059, 43065}, {51567, 57791}

X(68238) = X(1)X(905)∩X(36)X(103)

X(68238) lies on the cubic K086 and these lines: {1, 905}, {36, 103}, {80, 1323}, {106, 65642}, {241, 971}, {677, 1815}, {911, 32625}, {1262, 38666}, {1785, 34578}, {1795, 2338}, {36101, 41700}

X(68238) = X(516)-isoconjugate of X(2717)
X(68238) = X(35116)-Dao conjugate of X(30807)
X(68238) = crossdifference of every pair of points on line {910, 42756}
X(68238) = barycentric product X(2801)*X(36101)
X(68238) = barycentric quotient X(i)/X(j) for these {i,j}: {911, 2717}, {2801, 30807}, {36101, 35164}, {45144, 61437}, {57442, 58259}, {61435, 63851}
X(68238) = {X(2338),X(36039)}-harmonic conjugate of X(5526)

X(68239) = TRILINEAR POLE OF X(513)X(614)

X(68239) lies on the conic {{A,B,C,X(1),X(2)}}, the curve Q063, and these lines: {1, 1633}, {2, 1565}, {20, 6553}, {28, 16726}, {88, 7291}, {190, 17170}, {278, 1358}, {517, 1280}, {527, 34892}, {957, 2097}, {1219, 5082}, {1257, 18732}, {2401, 6084}, {2809, 39959}, {2832, 35348}, {3227, 5088}, {5540, 8056}, {9710, 59760}

X(68239) = isogonal conjugate of X(41391)
X(68239) = X(i)-isoconjugate of X(j) for these (i,j): {1, 41391}, {6, 49991}, {71, 14954}
X(68239) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 41391}, {9, 49991}
X(68239) = trilinear pole of line {513, 614}
X(68239) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 49991}, {6, 41391}, {28, 14954}, {16726, 46537}

X(68240) = TRILINEAR POLE OF X(652)X(22063)

X(68240) lies on the conic {{A,B,C,X(2),X(3)}}, the cubic K681 and these lines: {1, 1361}, {29, 18180}, {77, 7215}, {78, 5562}, {109, 947}, {222, 38579}, {517, 10538}, {945, 38573}, {1437, 35196}, {1872, 65213}, {2814, 60569}, {2817, 10570}, {3345, 52824}, {3362, 64760}, {5088, 60046}, {8677, 37628}

X(68240) = isogonal conjugate of X(45766)
X(68240) = X(2183)-cross conjugate of X(222)
X(68240) = X(i)-isoconjugate of X(j) for these (i,j): {1, 45766}, {1897, 53304}
X(68240) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 45766}, {34467, 53304}
X(68240) = cevapoint of X(1364) and X(8677)
X(68240) = trilinear pole of line {652, 22063}
X(68240) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 45766}, {22383, 53304}

X(68241) = TRILINEAR POLE OF X(650)X(21105)

X(68241) lies on the Feuerbach circumhyperbola and these lines: {1, 53529}, {4, 64155}, {7, 18240}, {8, 2801}, {9, 1768}, {11, 3062}, {21, 18645}, {36, 55966}, {79, 66020}, {80, 971}, {84, 20418}, {100, 43182}, {103, 61437}, {144, 46684}, {165, 6068}, {515, 24297}, {516, 1320}, {518, 12641}, {527, 5537}, {528, 3680}, {676, 23838}, {885, 2827}, {943, 66055}, {952, 4900}, {1000, 2800}, {1156, 11219}, {1387, 24644}, {1389, 61021}, {1392, 60984}, {1476, 53055}, {1537, 59372}, {1699, 55922}, {1709, 56262}, {2346, 60936}, {2802, 56090}, {2826, 23893}, {2829, 3577}, {2950, 15298}, {2951, 5856}, {3091, 38207}, {3254, 15726}, {3255, 10177}, {4866, 37424}, {5281, 55920}, {5561, 5805}, {5732, 56101}, {5843, 12515}, {5850, 64189}, {5853, 56097}, {6001, 64330}, {6006, 46041}, {6596, 17768}, {7091, 11522}, {7284, 11372}, {7319, 45043}, {8227, 38124}, {9579, 62178}, {10308, 63989}, {10398, 38308}, {10483, 56152}, {10728, 14496}, {10993, 41854}, {12248, 14497}, {12611, 59380}, {12619, 60884}, {12667, 43734}, {13464, 15179}, {15528, 41861}, {15587, 17661}, {15909, 31391}, {17649, 64265}, {21151, 64012}, {21398, 36971}, {26105, 34919}, {30304, 64264}, {30308, 38095}, {30424, 55924}, {30513, 60896}, {31231, 41706}, {31272, 64699}, {35514, 64056}, {36991, 64836}, {37714, 38202}, {38123, 64008}, {38693, 51090}, {43736, 62789}, {45393, 60885}, {50371, 56117}, {54370, 55961}, {56263, 64130}, {59391, 67871}, {60961, 66199}, {60997, 66021}, {62778, 67876}, {64329, 67995}

X(68241) = reflection of X(i) in X(j) for these {i,j}: {100, 43182}, {144, 46684}, {3062, 11}, {17661, 15587}, {34789, 7}, {60884, 12619}, {64056, 35514}, {64765, 43177}
X(68241) = isogonal conjugate of X(5537)
X(68241) = antigonal image of X(3062)
X(68241) = symgonal image of X(43182)
X(68241) = X(i)-cross conjugate of X(j) for these (i,j): {2291, 34578}, {3660, 1}, {33573, 514}
X(68241) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5537}, {6, 60935}, {36052, 66021}
X(68241) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 5537}, {9, 60935}, {119, 66021}
X(68241) = trilinear pole of line {650, 21105}
X(68241) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 60935}, {6, 5537}, {8609, 66021}

X(68242) = X(1)X(523)∩X(104)X(476)

X(68242) lies on these lines: {1, 523}, {104, 476}, {517, 6740}, {759, 2689}, {2074, 62713}, {2075, 45766}, {5088, 14616}, {12331, 68147}, {13746, 18115}, {34209, 56845}, {36195, 45926}

X(68243) = X(1)X(512)∩X(104)X(805)

X(68243) lies on these lines: {1, 512}, {104, 805}, {292, 5006}, {295, 4584}, {511, 18206}, {517, 56154}, {741, 2701}, {1326, 2223}, {1931, 5360}, {2311, 5060}, {5088, 18827}, {15148, 62714}

X(68244) = X(1)X(21)∩X(10)X(409)

X(68244) lies on these lines: {1, 21}, {3, 18134}, {5, 52360}, {8, 11101}, {10, 409}, {28, 1792}, {30, 67820}, {35, 3178}, {36, 49676}, {55, 17512}, {60, 34772}, {72, 1098}, {99, 5088}, {100, 1325}, {104, 6083}, {110, 4511}, {145, 54313}, {163, 57015}, {229, 404}, {250, 2074}, {295, 4584}, {333, 36011}, {345, 14015}, {515, 56951}, {517, 643}, {519, 759}, {662, 5440}, {859, 34179}, {908, 1793}, {932, 53970}, {943, 1791}, {1006, 6176}, {1010, 25466}, {1014, 17179}, {1019, 6003}, {1043, 37227}, {1259, 13739}, {1283, 38456}, {1330, 3145}, {1376, 11116}, {1444, 3418}, {2064, 7009}, {2150, 22021}, {2185, 24929}, {2311, 23531}, {2363, 5266}, {2894, 37113}, {3109, 17757}, {3509, 5060}, {3685, 62342}, {3705, 4228}, {3771, 13588}, {3936, 37311}, {3940, 56440}, {4184, 29839}, {4188, 25663}, {4189, 63056}, {4203, 25689}, {4218, 18139}, {4234, 5434}, {4260, 37306}, {4267, 11102}, {4276, 29671}, {4417, 11334}, {4592, 51369}, {5047, 25441}, {5080, 7424}, {5176, 6740}, {5179, 27415}, {5224, 19287}, {5279, 7054}, {5432, 52244}, {5562, 6906}, {5731, 62389}, {6224, 60452}, {7259, 41391}, {7411, 25664}, {10026, 21004}, {10449, 13733}, {11110, 24953}, {11112, 52361}, {11681, 13746}, {11813, 39136}, {12579, 34920}, {14956, 60448}, {15952, 34773}, {16370, 17378}, {17104, 22836}, {17531, 25669}, {17549, 40592}, {21495, 25665}, {25440, 35991}, {25507, 56770}, {26141, 27086}, {26702, 65882}, {27529, 37158}, {30117, 37791}, {30941, 51607}, {34172, 37375}, {34594, 65875}, {36797, 45766}, {37369, 52367}, {39435, 65883}, {40980, 56018}, {53707, 65885}

X(68244) = reflection of X(i) in X(j) for these {i,j}: {1325, 12030}, {2651, 5127}, {39136, 11813}
X(68244) = isotomic conjugate of the polar conjugate of X(56830)
X(68244) = X(320)-Ceva conjugate of X(37783)
X(68244) = X(i)-isoconjugate of X(j) for these (i,j): {4, 43693}, {25, 40715}, {42, 16099}, {512, 35169}, {3120, 57741}, {3122, 57990}
X(68244) = X(i)-Dao conjugate of X(j) for these (i,j): {6505, 40715}, {35122, 1577}, {36033, 43693}, {39054, 35169}, {40592, 16099}
X(68244) = crossdifference of every pair of points on line {661, 40977}
X(68244) = barycentric product X(i)*X(j) for these {i,j}: {58, 42709}, {63, 447}, {69, 56830}, {81, 16086}, {304, 56919}, {645, 51643}, {799, 42662}, {867, 4567}
X(68244) = barycentric quotient X(i)/X(j) for these {i,j}: {48, 43693}, {63, 40715}, {81, 16099}, {447, 92}, {662, 35169}, {867, 16732}, {4567, 57990}, {16086, 321}, {42662, 661}, {42709, 313}, {51643, 7178}, {56830, 4}, {56919, 19}
X(68244) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 37816, 409}, {1793, 51382, 15776}, {5440, 51420, 662}

X(68245) = X(1)X(2)∩X(3)X(44720)

X(68245) lies on these lines: {1, 2}, {3, 44720}, {20, 6552}, {36, 4738}, {40, 21227}, {100, 2757}, {104, 1811}, {190, 59586}, {242, 4076}, {341, 5687}, {484, 62222}, {515, 21290}, {517, 3699}, {668, 5088}, {944, 42020}, {1222, 17614}, {1311, 2748}, {1329, 5100}, {1376, 4737}, {1447, 4986}, {2370, 2743}, {2726, 52778}, {3667, 4063}, {3685, 3992}, {3732, 40883}, {3820, 4514}, {3871, 52353}, {3913, 46937}, {3921, 17277}, {3952, 63136}, {4103, 5011}, {4487, 54391}, {4695, 32927}, {4997, 7743}, {5015, 21031}, {5081, 55016}, {5119, 27538}, {5179, 27546}, {5440, 43290}, {5844, 65742}, {6073, 14507}, {6555, 59417}, {6558, 41391}, {6767, 30829}, {8715, 56311}, {9369, 25440}, {12245, 44722}, {14155, 67723}, {17072, 48281}, {17682, 59525}, {17757, 32850}, {23850, 38901}, {32937, 54286}, {37829, 42378}, {38455, 52871}, {40091, 59669}, {48363, 65197}, {49782, 61730}, {59717, 62300}

X(68245) = midpoint of X(8) and X(47624)
X(68245) = reflection of X(i) in X(j) for these {i,j}: {38460, 67348}, {47622, 6789}, {53618, 50535}
X(68245) = orthoptic-circle-of-the-Steiner-inellipse-inverse of X(62673)
X(68245) = orthoptic-circle-of-the-Steiner-circumellipse-inverse of X(10327)
X(68245) = incircle-of-anticomplementary-triangle-inverse of X(78)}
X(68245) = psi-transform of X(4011)
X(68245) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 60367, 1737}, {341, 5687, 7283}, {3679, 7081, 16821}, {3992, 48696, 3685}, {6735, 49991, 16086}

X(68246) = X(1)X(88)∩X(3)X(56938)

X(68246) lies on these lines: {1, 88}, {3, 56938}, {104, 65649}, {121, 64141}, {190, 1145}, {901, 63136}, {1417, 14122}, {2827, 21385}, {4555, 5088}, {4738, 36237}, {9456, 21888}, {10774, 59415}, {14664, 38693}, {17100, 34139}, {20098, 64743}, {38513, 53790}, {52755, 64695}

X(68246) = midpoint of X(20098) and X(64743)
X(68246) = reflection of X(i) in X(j) for these {i,j}: {1320, 106}, {13541, 214}, {21290, 1145}
X(68246) = X(1635)-isoconjugate of X(46118)
X(68246) = barycentric quotient X(901)/X(46118)
X(68246) = {X(1320),X(14193)}-harmonic conjugate of X(106)

X(68247) = X(1)X(1283)∩X(10)X(125)

X(68247) lies on these lines: {1, 1283}, {10, 125}, {19, 3125}, {37, 3708}, {65, 43693}, {75, 150}, {225, 1365}, {758, 34895}, {759, 30117}, {942, 2363}, {1247, 21381}, {2166, 34301}, {5088, 18827}, {5497, 56149}, {5902, 13610}, {18481, 34860}, {29656, 42285}, {57847, 57990}

X(68247) = X(i)-isoconjugate of X(j) for these (i,j): {3, 447}, {58, 16086}, {63, 56830}, {69, 56919}, {99, 42662}, {643, 51643}, {867, 4570}, {1333, 42709}
X(68247) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 16086}, {37, 42709}, {3162, 56830}, {36103, 447}, {38986, 42662}, {50330, 867}, {55060, 51643}
X(68247) = trilinear pole of line {661, 40977}
X(68247) = barycentric product X(i)*X(j) for these {i,j}: {19, 40715}, {37, 16099}, {92, 43693}, {661, 35169}, {3125, 57990}, {16732, 57741}
X(68247) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 42709}, {19, 447}, {25, 56830}, {37, 16086}, {798, 42662}, {1973, 56919}, {3125, 867}, {7180, 51643}, {16099, 274}, {35169, 799}, {40715, 304}, {43693, 63}, {57741, 4567}, {57990, 4601}

X(68248) = X(1)X(651)∩X(101)X(41798)

X(68248) lies on these lines: {1, 651}, {101, 41798}, {104, 65646}, {150, 62723}, {944, 56665}, {952, 1121}, {5088, 35157}, {5731, 52746}, {20096, 62731}, {28236, 60579}

X(68249) = X(1)X(5)∩X(3)X(51975)

X(68249) lies on these lines: {1, 5}, {3, 51975}, {30, 36909}, {100, 43655}, {104, 65639}, {515, 38954}, {517, 51562}, {900, 12515}, {2222, 53877}, {2771, 15343}, {2800, 18342}, {4557, 12331}, {5088, 35174}, {10260, 38722}, {10742, 25436}, {11499, 59283}, {12531, 52479}, {12619, 18341}, {12773, 53303}, {18359, 38665}, {25437, 57298}, {28204, 36590}, {36910, 65808}, {38903, 62395}

X(68249) = reflection of X(i) in X(j) for these {i,j}: {10742, 25436}, {12515, 56756}, {18341, 12619}
X(68249) = barycentric product X(52351)*X(67467)
X(68249) = barycentric quotient X(67467)/X(17923)
X(68249) = {X(80),X(56417)}-harmonic conjugate of X(11)

X(68250) = X(1)X(30)∩X(3)X(6757)

X(68250) lies on these lines: {1, 30}, {3, 6757}, {4, 58740}, {36, 2166}, {92, 186}, {104, 476}, {265, 515}, {355, 36195}, {517, 6742}, {1141, 1290}, {1385, 3615}, {2070, 23850}, {2072, 17073}, {2217, 37976}, {2695, 26700}, {2975, 52344}, {5088, 65292}, {5196, 26201}, {5899, 51621}, {6224, 63642}, {10412, 46610}, {30690, 54093}, {34922, 55017}, {37406, 41496}

X(68250) = reflection of X(i) in X(j) for these {i,j}: {355, 36195}, {7424, 1385}, {51883, 52200}
X(68250) = barycentric product X(94)*X(67402)
X(68250) = barycentric quotient X(67402)/X(323)
X(68250) = {X(50148),X(51883)}-harmonic conjugate of X(52200)

X(68251) = X(1)X(84)∩X(3)X(271)

X(68251) lies on the cubic K436 and these lines: {1, 84}, {3, 271}, {4, 41084}, {57, 15237}, {104, 6081}, {189, 5768}, {280, 944}, {285, 1437}, {515, 1359}, {517, 13138}, {521, 4091}, {1256, 1490}, {1439, 56972}, {1440, 41004}, {2733, 8059}, {3149, 3341}, {3220, 34187}, {5088, 53642}, {6905, 37141}, {7335, 12680}, {8886, 37302}, {9376, 11500}, {15405, 52663}, {18283, 46355}, {18446, 41081}, {33597, 52389}, {34162, 37417}, {37468, 52078}, {43724, 66921}, {45766, 65213}, {51490, 63399}, {52407, 65179}, {56939, 64191}

X(68251) = X(i)-isoconjugate of X(j) for these (i,j): {40, 36121}, {102, 7952}, {196, 15629}, {198, 52780}, {2331, 36100}, {3195, 34393}, {8058, 36067}, {32677, 64211}, {36055, 47372}, {53152, 57118}
X(68251) = X(i)-Dao conjugate of X(j) for these (i,j): {23986, 64211}, {51221, 47372}
X(68251) = crossdifference of every pair of points on line {2331, 14298}
X(68251) = barycentric product X(i)*X(j) for these {i,j}: {189, 46974}, {271, 34050}, {285, 51368}, {515, 41081}, {1433, 64194}, {1455, 44189}, {2406, 61040}, {14304, 65179}, {34400, 51361}, {37141, 39471}, {46391, 53642}
X(68251) = barycentric quotient X(i)/X(j) for these {i,j}: {84, 52780}, {515, 64211}, {1433, 36100}, {1436, 36121}, {1455, 196}, {2182, 7952}, {2188, 15629}, {8755, 47372}, {34050, 342}, {37141, 65295}, {41081, 34393}, {46391, 8058}, {46974, 329}, {51361, 55116}, {51368, 57810}, {53522, 59935}, {61040, 2399}
X(68251) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {84, 46881, 1071}, {222, 63430, 1071}

X(68252) = X(1)X(2841)∩X(34)X1357)

X(68252) lies on the circumconic {{A,B,C,X(1),X(6)}} and these lines: {1, 2841}, {34, 1357}, {46, 39969}, {295, 5378}, {517, 1120}, {979, 1054}, {996, 3980}, {1222, 10914}, {1878, 15635}, {2802, 56145}, {2810, 56179}, {2832, 60580}, {3226, 5088}, {3445, 53303}, {8679, 34893}, {9268, 36058}, {12014, 43531}, {26892, 37999}, {32913, 56150}, {53790, 56113}

X(68252) = isogonal conjugate of X(68245)
X(68252) = X(1)-isoconjugate of X(68245)
X(68252) = X(3)-Dao conjugate of X(68245)
X(68252) = barycentric quotient X(6)/X(68245)

X(68253) = X(1)X(21)∩X(103)X(6083)

X(68253) lies on these lines: {1, 21}, {103, 6083}, {110, 58328}, {516, 643}, {2249, 65885}, {6745, 54442}, {23692, 53388}, {26702, 65881}, {59074, 65886}

X(68254) = X(1)X(6)∩X(103)X(1810)

X(68254) lies on the circumconic {{A,B,C,X(1),X(6)}} and these lines: {1, 6}, {103, 1810}, {144, 28981}, {165, 38876}, {516, 644}, {2371, 2742}, {3309, 17410}, {4936, 5732}, {35341, 44425}, {58328, 65208}

X(68255) = X(1)X(2)∩X(103)X(6079)

X(68255) lies on these lines: {1, 2}, {103, 6079}, {165, 6555}, {516, 3699}, {1155, 4152}, {3667, 11067}, {3717, 43290}, {3749, 59686}, {4082, 64135}, {4437, 67643}, {4578, 5537}, {4767, 63145}, {24177, 67066}, {28234, 65742}, {30681, 59678}, {30829, 43179}, {34607, 59599}, {51380, 58670}

X(68255) = reflection of X(i) in X(j) for these {i,j}: {5121, 52907}, {51615, 50535}
X(68255) = incircle-of-anticomplementary-triangle-inverse of X(64083)
X(68255) = {X(17780),X(49991)}-harmonic conjugate of X(6745)

X(68256) = X(1)X(4)∩X(103)X(1309)

X(68256) lies on these lines: {1, 4}, {20, 64931}, {35, 38870}, {103, 1309}, {108, 44425}, {165, 7046}, {318, 4297}, {516, 1897}, {519, 37420}, {971, 1364}, {1324, 3515}, {1360, 44044}, {1709, 40971}, {1753, 10085}, {1861, 34589}, {1872, 12680}, {2222, 51762}, {2730, 20624}, {3522, 10538}, {3679, 37410}, {5081, 28236}, {5537, 56183}, {15614, 39535}, {15626, 34142}, {21664, 28160}, {23711, 60062}, {32534, 54090}, {35455, 55571}, {37441, 53008}, {52781, 67643}, {53151, 63145}

X(68256) = reflection of X(1785) in X(45766)
X(68256) = polar-circle-inverse of X(1699)

X(68257) = X(1)X(30)∩X(103)X(476)

X(68257) lies on these lines: {1, 30}, {20, 6757}, {63, 50144}, {103, 476}, {265, 28160}, {382, 58740}, {516, 6742}, {1325, 1789}, {2166, 4316}, {3615, 4297}, {5691, 52388}, {37411, 41496}

X(68258) = X(1)X(513)∩X(3)X(521)

X(68258) lies on these lines: {1, 513}, {3, 521}, {73, 1459}, {514, 64323}, {520, 56839}, {522, 5882}, {523, 64283}, {900, 64191}, {944, 43728}, {1364, 3270}, {1388, 6129}, {2975, 68103}, {3319, 35013}, {3667, 59998}, {3738, 11713}, {3900, 7966}, {6075, 45950}, {8674, 15626}, {9001, 22769}, {21189, 21842}, {35050, 63820}, {39199, 55362}, {42772, 52242}, {53314, 53551}

X(68258) = midpoint of X(944) and X(43728)
X(68258) = reflection of X(i) in X(j) for these {i,j}: {35050, 63820}, {42757, 1}
X(68258) = reflection of X(42757) in the OI line
X(68258) = X(i)-isoconjugate of X(j) for these (i,j): {101, 65345}, {953, 1897}, {1309, 61482}, {1783, 65249}, {1785, 35011}, {5081, 59018}, {7012, 46041}, {8750, 46136}, {9268, 53157}
X(68258) = X(i)-Dao conjugate of X(j) for these (i,j): {1015, 65345}, {26932, 46136}, {34467, 953}, {35587, 38462}, {39006, 65249}, {61066, 6335}
X(68258) = crosspoint of X(952) and X(67453)
X(68258) = crosssum of X(i) and X(j) for these (i,j): {100, 34151}, {953, 46041}
X(68258) = crossdifference of every pair of points on line {44, 1783}
X(68258) = barycentric product X(i)*X(j) for these {i,j}: {521, 43043}, {905, 952}, {1332, 6075}, {2265, 4025}, {26932, 67453}, {35013, 65302}
X(68258) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 65345}, {905, 46136}, {952, 6335}, {1459, 65249}, {2087, 53157}, {2265, 1897}, {6075, 17924}, {7117, 46041}, {14578, 35011}, {22086, 52479}, {22383, 953}, {43043, 18026}, {52478, 65336}, {61481, 65223}, {67453, 46102}

X(68259) = X(40)X(14077)∩X(63)X(693)

X(68259) lies on the Kiepert parabola of the excentral triangle and these lines: {1, 1938}, {7, 23806}, {9, 4885}, {40, 14077}, {46, 9373}, {57, 650}, {63, 693}, {84, 64787}, {165, 9443}, {226, 28834}, {239, 514}, {513, 5536}, {522, 28589}, {523, 53300}, {652, 3676}, {654, 21104}, {657, 7658}, {824, 1764}, {884, 982}, {894, 27346}, {905, 43932}, {928, 48352}, {1445, 27417}, {1491, 2473}, {1638, 9404}, {1699, 15280}, {1730, 47886}, {1999, 25271}, {2487, 22108}, {2488, 4040}, {2820, 6608}, {3004, 39470}, {3219, 26985}, {3306, 31209}, {3835, 10025}, {3928, 4762}, {3929, 45320}, {4105, 15599}, {4369, 21390}, {4394, 53396}, {4467, 55125}, {4477, 42341}, {4524, 9511}, {5119, 50767}, {5437, 31287}, {5709, 8760}, {6139, 48282}, {6182, 41338}, {6763, 47724}, {7203, 43060}, {7289, 9001}, {7308, 31250}, {7991, 9366}, {8142, 9841}, {8642, 53326}, {11068, 62748}, {11679, 21438}, {11934, 54408}, {12514, 48295}, {14349, 64885}, {14829, 21611}, {16574, 25667}, {18164, 57130}, {18199, 23189}, {18200, 23187}, {20980, 43051}, {21173, 53539}, {22383, 62812}, {23725, 62240}, {23958, 26777}, {24627, 27014}, {26114, 38000}, {26824, 67335}, {27003, 27115}, {27064, 27139}, {28006, 56547}, {29066, 62858}, {29362, 53403}, {29427, 29529}, {30910, 56525}, {47729, 62874}, {48125, 67334}, {48304, 56288}, {50336, 53400}, {56543, 56742}, {58322, 65697}

X(68259) = reflection of X(i) in X(j) for these {i,j}: {1019, 4091}, {4105, 15599}
X(68259) = X(i)-Ceva conjugate of X(j) for these (i,j): {4569, 1}, {65575, 905}
X(68259) = X(i)-isoconjugate of X(j) for these (i,j): {37, 53683}, {692, 62914}
X(68259) = X(i)-Dao conjugate of X(j) for these (i,j): {657, 3900}, {1086, 62914}, {40589, 53683}
X(68259) = cevapoint of X(44408) and X(57237)
X(68259) = crosspoint of X(i) and X(j) for these (i,j): {81, 658}, {9503, 34085}
X(68259) = crosssum of X(i) and X(j) for these (i,j): {37, 657}, {513, 21346}, {649, 40133}, {650, 17451}, {798, 40977}, {9502, 46388}
X(68259) = crossdifference of every pair of points on line {42, 41339}
X(68259) = barycentric product X(i)*X(j) for these {i,j}: {1, 46402}, {75, 44408}, {85, 57237}, {514, 37659}, {1019, 45744}, {4025, 4219}, {4569, 14714}, {6063, 57175}
X(68259) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 53683}, {514, 62914}, {4219, 1897}, {14714, 3900}, {37659, 190}, {44408, 1}, {45744, 4033}, {46402, 75}, {57175, 55}, {57237, 9}
X(68259) = {X(654),X(21104)}-harmonic conjugate of X(58324)

X(68260) = X(40)X(14077)∩X(63)X(68102)

X(68260) lies on the Kiepert parabola of the excentral triangle and these lines: {40, 14077}, {63, 68102}, {103, 15731}, {165, 513}, {514, 45290}, {649, 3730}, {901, 1022}, {1155, 35348}, {3887, 68248}, {7280, 8641}

X(68260) = crosssum of X(1155) and X(23057)

X(68261) = X(63)X(15632)∩X(104)X(517)

X(68261) lies on the Kiepert parabola of the excentral triangle and these lines: {36, 53292}, {57, 33646}, {63, 15632}, {103, 1155}, {104, 517}, {513, 1768}, {971, 67515}, {1319, 10703}, {1537, 55314}, {2800, 66843}, {2807, 3025}, {3326, 43043}, {4926, 56423}, {5537, 53297}, {6073, 64193}, {6075, 13226}, {7004, 67453}, {11714, 67434}, {13243, 14513}, {13257, 55317}, {22102, 66628}, {24201, 41166}, {36040, 67521}, {37374, 53792}, {39756, 64761}, {46684, 67445}, {64128, 67436}, {64129, 67435}

X(68261) = midpoint of X(i) and X(j) for these {i,j}: {13243, 14513}, {14511, 64189}
X(68261) = reflection of X(i) in X(j) for these {i,j}: {1537, 55314}, {6073, 64193}, {6075, 13226}, {13257, 55317}, {52478, 104}, {67445, 46684}

X(68262) = X(40)X(30199)∩X(63)X(68117)

X(68262) lies on the Kiepert parabola of the excentral triangle and these lines: {1, 43932}, {3, 8642}, {20, 28475}, {40, 30199}, {55, 43049}, {63, 68117}, {165, 4394}, {513, 5537}, {516, 4106}, {649, 3309}, {650, 2820}, {905, 8641}, {918, 28589}, {1499, 3265}, {2473, 48136}, {2487, 59835}, {2488, 2821}, {3667, 11067}, {4219, 6591}, {4380, 9778}, {6003, 7659}, {6972, 64832}, {9511, 17115}, {14300, 58322}, {15313, 53300}, {28473, 49296}, {30198, 48032}, {42322, 46684}, {59320, 65481}, {62432, 64787}, {65413, 65664}

X(68262) = reflection of X(i) in X(j) for these {i,j}: {650, 15599}, {42322, 46684}
X(68262) = X(4578)-Ceva conjugate of X(1)
X(68262) = X(58817)-Dao conjugate of X(59941)
X(68262) = crosspoint of X(100) and X(56359)
X(68262) = crosssum of X(513) and X(4319)
X(68262) = crossdifference of every pair of points on line {614, 40126}

X(68263) = X(40)X(513)∩X(63)X(68101)

X(68263) lies on the Kiepert parabola of the excentral triangle and these lines: {40, 513}, {46, 24457}, {63, 68101}, {484, 23838}, {900, 12515}, {3667, 40256}, {4063, 16558}, {5119, 53535}, {7280, 48307}, {11010, 14812}, {16139, 28217}, {26286, 48390}, {30323, 48283}, {37618, 48302}

X(68264) = X(40)X(521)∩X(63)X(4397)

X(68264) lies on the Kiepert parabola of the excentral triangle and these lines: {1, 23224}, {40, 521}, {46, 21189}, {57, 6129}, {63, 4397}, {109, 57179}, {484, 513}, {517, 23187}, {522, 45766}, {523, 53300}, {610, 65102}, {656, 2849}, {822, 17898}, {3336, 42757}, {4091, 8058}, {4132, 44408}, {4151, 14294}, {8677, 21173}, {11010, 68258}, {20367, 21187}, {23874, 57121}, {39199, 48307}, {40256, 43728}, {48281, 53305}, {56288, 57091}, {57198, 64878}

X(68264) = reflection of X(i) in X(j) for these {i,j}: {1, 23224}, {48281, 53305}, {48307, 39199}
X(68264) = X(4091)-Dao conjugate of X(4131)
X(68264) = crosspoint of X(i) and X(j) for these (i,j): {100, 775}, {823, 40438}
X(68264) = crosssum of X(i) and X(j) for these (i,j): {513, 774}, {656, 42440}, {822, 1962}
X(68264) = crossdifference of every pair of points on line {820, 836}

X(68265) = X(40)X(8702)∩X(63)X(4036)

X(68265) lies on the Kiepert parabola of the excentral triangle and these lines: {40, 8702}, {46, 57099}, {57, 31947}, {63, 4036}, {513, 5535}, {523, 68250}, {1768, 62492}, {1938, 34948}, {4063, 28195}, {4414, 48303}

X(68266) = X(40)X(521)∩X(63)X(68103)

X(68266) lies on the Kiepert parabola of the excentral triangle and these lines: {1, 513}, {40, 521}, {63, 68103}, {102, 104}, {514, 64147}, {515, 43728}, {522, 944}, {900, 64145}, {1437, 3737}, {1459, 10571}, {2222, 2720}, {2849, 10702}, {3667, 64120}, {6006, 59998}, {6129, 63208}, {7280, 23224}, {7289, 9001}, {11715, 46041}, {14266, 56424}, {21172, 28080}, {21189, 37618}, {23187, 26286}, {23696, 32735}, {23730, 49296}

X(68266) = reflection of X(i) in X(j) for these {i,j}: {1, 68258}, {46041, 11715}
X(68266) = crosssum of X(i) and X(j) for these (i,j): {2183, 53285}, {46393, 51361}

X(68267) = X(57)X(650)∩X(103)X(15731)

X(68267) lies on the Kiepert parabola of the excentral triangle and these lines: {57, 650}, {103, 15731}, {144, 57064}, {165, 58835}, {513, 3062}, {514, 2400}, {1024, 9503}, {7658, 9533}, {15634, 33573}, {23730, 60581}, {43736, 65680}

X(68267) = X(i)-isoconjugate of X(j) for these (i,j): {2398, 11051}, {2426, 10405}, {23972, 65642}, {40869, 53622}, {41339, 61240}, {51436, 55284}
X(68267) = X(13609)-Dao conjugate of X(30807)
X(68267) = crosspoint of X(36101) and X(65245)
X(68267) = crosssum of X(910) and X(46392)
X(68267) = crossdifference of every pair of points on line {41339, 42077}
X(68267) = barycentric product X(i)*X(j) for these {i,j}: {165, 2400}, {2424, 16284}, {7658, 36101}, {13609, 65245}, {58835, 67128}
X(68267) = barycentric quotient X(i)/X(j) for these {i,j}: {144, 42719}, {165, 2398}, {2400, 44186}, {2424, 3062}, {7658, 30807}, {9533, 24015}, {17106, 23973}, {43736, 53640}

  Co-normal points and co-normals hyperbolas: X(68268) - X(68324)  

Let 𝒞 be a conic (not a circle) and 𝒩 a point on the plane of 𝒞 and not on it. The points on 𝒞 whose normals concur at 𝒩 are called the 𝒩-co-normal points of-𝒞.

Let 𝒞 be a conic (not a circle) with Cartesian general equation 𝒞(x, y) = a*x^2 + 2*h*x*y + b*y^2 + 2*g*x + 2*f*y + c = 0 and 𝒩(X, Y) a point not on 𝒞. The points on 𝒞 whose normals concur in 𝒩 are the intersections, real or imaginary, of 𝒞(x, y) and the rectangular hyperbola ℛ(x, y, X, Y) with Cartesian equation:

Source: Robert Frederick Davis, The Mathematical Gazette, Vol. 3, No. 48 (Dec., 1904), p. 108.

Translating the previous result into trilinear coordinates, making 𝒞 = F_A*u^2 + F_B*v^2 + F_C*w^2 + 2*G_A*v*w + 2*G_B*u*w + 2*G_C*u*v = 0 and 𝒩 = U : V : W, equation (1) is:

The rectangular hyperbola ℛ(𝒞, 𝒩) is named here the 𝒩-co-normals hyperbola of 𝒞.

Some properties of ℛ(𝒞, 𝒩):

1. ℛ(𝒞, 𝒩) passes through 𝒩 and through the center of 𝒞.
2. If 𝒩 is the center of 𝒞 or any of the drawn normals passes through this center, ℛ(𝒞, 𝒩) degenerates to a pair of lines, obviously, the axes of 𝒞.
3. For a given central 𝒞 and variable 𝒩, relative to a triangle ABC, all ℛ are homothetic with the ABC-circumscribed rectangular hyperbola ℛ₀(𝒞) with trilinear equation
   ∑( ((b^2-c^2)*G_A+a*((cos(B)*F_C-G_B)*b-(cos(C)*F_B-G_C)*c))*v*w ) = 0. This circum-rectangular hyperbola ℛ₀(𝒞) is named here the basic co-normals hyperbola of 𝒞.

   In the simplest case when 𝒞 is the circumconic with perspector P, ℛ₀(𝒞) is the circum-rectangular hyperbola with perspector P₀=PolarConjugate(IsotomicConjugate(IdealOfTripolar(P))).

4. More properties can be seen in documents about parabola, ellipse and hyperbolas in MasterJee.

Note: A list of 𝒩-co-normal hyperbolas of some named conics and for 𝒩 in {X(1)..X(6)} can be seen here.

X(68268) = CENTER OF THE X(1)-CO-NORMAL HYPERBOLA OF BROCARD INELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {1, 39, 511, 512, 2223}

X(68268) lies on these lines: {805, 3110}

X(68269) = CENTER OF THE X(2)-CO-NORMAL HYPERBOLA OF BROCARD INELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {2, 39, 237, 511, 512, 9292}

X(68269) lies on these lines: {51, 41330}, {107, 419}, {187, 68270}, {230, 67561}, {511, 5182}, {1691, 9217}, {2080, 36212}, {2679, 67540}, {3111, 44562}, {3568, 67366}, {5642, 44215}, {7827, 67630}, {14113, 67551}, {63544, 63556}, {65751, 67560}, {68069, 68076}

X(68269) = pole of the line {2782, 35279} with respect to the Thomson-Gibert-Moses hyperbola
X(68269) = (X(187), X(68271))-harmonic conjugate of X(68270)

X(68270) = CENTER OF THE X(4)-CO-NORMAL HYPERBOLA OF BROCARD INELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {4, 39, 237, 511, 512, 3978, 44125, 44126, 63554}

X(68270) lies on these lines: {6, 67549}, {99, 511}, {113, 2679}, {115, 67561}, {187, 68269}, {512, 52446}, {805, 35060}, {2387, 67560}, {14113, 67550}, {18322, 66832}, {31850, 67859}

X(68270) = midpoint of X(18322) and X(66832)
X(68270) = reflection of X(i) in X(j) for these (i, j): (187, 68271), (805, 35060), (5167, 2679), (67550, 14113), (67561, 115)
X(68270) = pole of the line {5661, 67630} with respect to the Brocard inellipse
X(68270) = pole of the line {35901, 43765} with respect to the Jerabek circumhyperbola
X(68270) = (X(187), X(68271))-harmonic conjugate of X(68269)

X(68271) = CENTER OF THE X(5)-CO-NORMAL HYPERBOLA OF BROCARD INELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {5, 39, 237, 511, 512}

X(68271) lies on these lines: {115, 512}, {187, 68269}, {511, 5026}, {805, 3111}, {2698, 12110}, {9427, 24973}, {10272, 67539}, {11272, 53797}, {14962, 38382}, {31848, 57347}, {38611, 50664}, {41330, 66832}, {46046, 67366}, {64687, 67215}, {66827, 67220}, {66834, 67630}, {67352, 67840}, {67376, 67833}

X(68271) = midpoint of X(i) and X(j) for these (i, j): {187, 68270}, {2698, 31850}, {46046, 67366}
X(68271) = reflection of X(67833) in X(67376)
X(68271) = X(68323)-of-orthic triangle (ABC acute)
X(68271) = (X(68269), X(68270))-harmonic conjugate of X(187)

X(68272) = CENTER OF THE X(2)-CO-NORMAL HYPERBOLA OF CIRCUMHYPERBOLA DUAL OF YFF PARABOLA

This co-normal hyperbola passes through centers X(n) for these n: {2, 1086, 66486}

X(68272) lies on these lines: {31380, 62675}

X(68273) = CENTER OF THE X(2)-CO-NORMAL HYPERBOLA OF DE LONGCHAMPS ELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {1, 2, 513, 517, 859}

X(68273) lies on these lines: {36, 45022}, {513, 3582}, {517, 4881}, {551, 34583}, {3025, 56884}, {5642, 61638}

X(68273) = (X(36), X(68275))-harmonic conjugate of X(68274)

X(68274) = CENTER OF THE X(4)-CO-NORMAL HYPERBOLA OF DE LONGCHAMPS ELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {1, 4, 513, 517, 859}

X(68274) lies on these lines: {36, 45022}, {100, 517}, {113, 3259}, {484, 20470}, {513, 3583}, {901, 35059}, {946, 31849}, {10572, 41682}

X(68274) = reflection of X(i) in X(j) for these (i, j): (36, 68275), (901, 35059), (56884, 3259)
X(68274) = (X(36), X(68275))-harmonic conjugate of X(68273)

X(68275) = CENTER OF THE X(5)-CO-NORMAL HYPERBOLA OF DE LONGCHAMPS ELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {1, 5, 513, 517, 859, 49993}

X(68275) lies on these lines: {11, 513}, {36, 45022}, {214, 517}, {244, 2605}, {901, 1621}, {953, 31849}, {3884, 67418}, {5330, 67416}, {5901, 53800}, {6690, 22102}, {8674, 51402}, {10272, 61638}, {13756, 67419}, {20718, 68154}, {28346, 38472}, {31847, 57320}, {35016, 66858}, {37702, 66865}, {38614, 67456}, {38707, 67420}, {38954, 67216}, {46044, 67449}, {46101, 65450}, {55317, 61166}, {58572, 67442}, {61731, 66845}, {64548, 67425}, {64688, 67213}, {65739, 67445}, {65854, 68160}, {66862, 67627}

X(68275) = midpoint of X(i) and X(j) for these (i, j): {36, 68274}, {953, 31849}, {3025, 3259}, {46044, 67449}
X(68275) = reflection of X(i) in X(j) for these (i, j): (38614, 67456), (61166, 55317)
X(68275) = intersection, other than {A, B, C}, of the circumconics through X(i), X(j) for these {i, j}: {6075, 34434}, {17101, 38390}
X(68275) = pole of the line {3025, 53525} with respect to the de Longchamps ellipse
X(68275) = (X(68273), X(68274))-harmonic conjugate of X(36)

X(68276) = CENTER OF THE X(6)-CO-NORMAL HYPERBOLA OF DE LONGCHAMPS ELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {1, 6, 513, 517, 3286}

X(68276) lies on these lines: {517, 5096}, {1386, 5091}

X(68277) = CENTER OF THE X(2)-CO-NORMAL HYPERBOLA OF EXCENTRAL-HEXYL ELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {1, 2, 3, 3307, 3308, 5506, 15015, 21153, 26446, 56177, 56203}

X(68277) lies on these lines: {1, 13144}, {2, 5426}, {3, 16128}, {8, 33812}, {10, 140}, {11, 3841}, {20, 15017}, {35, 63917}, {80, 3634}, {100, 1125}, {119, 4297}, {149, 3624}, {153, 7987}, {404, 11263}, {474, 64342}, {515, 38752}, {516, 1519}, {519, 66641}, {528, 19883}, {535, 35271}, {549, 2771}, {551, 2802}, {631, 6326}, {946, 33814}, {1145, 3244}, {1317, 3625}, {1320, 3636}, {1387, 15808}, {1537, 5493}, {1698, 6224}, {1768, 3523}, {2800, 10164}, {2801, 21154}, {2829, 68003}, {2842, 34583}, {2932, 5248}, {3065, 5506}, {3109, 11814}, {3216, 64710}, {3526, 62354}, {3530, 3647}, {3616, 5541}, {3622, 12653}, {3626, 7972}, {3635, 64056}, {3678, 17660}, {3722, 23869}, {3814, 28160}, {3817, 5840}, {3828, 59415}, {3874, 58591}, {3911, 12739}, {3918, 17636}, {3956, 31157}, {4065, 58397}, {4067, 11570}, {4084, 64139}, {4301, 11729}, {4304, 39692}, {4315, 10956}, {4669, 50843}, {4691, 12531}, {4745, 10031}, {4973, 5852}, {4996, 27385}, {5044, 47320}, {5047, 46816}, {5083, 41538}, {5087, 28154}, {5267, 5660}, {5316, 51636}, {5433, 41541}, {5440, 6681}, {5542, 6594}, {5550, 20095}, {5587, 6952}, {5657, 11014}, {5691, 66045}, {5731, 26364}, {5854, 50841}, {5856, 38054}, {5883, 17564}, {5886, 11849}, {6246, 58421}, {6265, 6684}, {6667, 9945}, {6702, 10609}, {6831, 67046}, {6881, 23513}, {6921, 22836}, {7280, 66012}, {7288, 37736}, {8227, 13199}, {8256, 61283}, {8674, 24920}, {8983, 48715}, {9024, 38049}, {9780, 9897}, {9802, 46934}, {9803, 10303}, {9809, 15717}, {10087, 44675}, {10090, 13411}, {10171, 59391}, {10172, 17647}, {10427, 51090}, {10698, 43174}, {10724, 12571}, {10993, 16174}, {11274, 34641}, {11362, 19907}, {11599, 53729}, {11698, 13624}, {12108, 18253}, {12119, 19925}, {12247, 31423}, {12248, 67706}, {12512, 34789}, {12611, 31730}, {13146, 15674}, {13605, 53743}, {13607, 64140}, {13922, 49548}, {13971, 48714}, {13991, 49547}, {16371, 61716}, {17567, 30143}, {17572, 33593}, {17638, 52793}, {19077, 49619}, {19078, 49618}, {19877, 20085}, {19878, 31272}, {20107, 57287}, {21636, 53720}, {23340, 31870}, {24036, 51406}, {24466, 51118}, {24914, 41558}, {25436, 56749}, {26725, 36006}, {28619, 66005}, {31673, 61580}, {31793, 58613}, {31806, 66047}, {33598, 58404}, {34595, 66063}, {34600, 37291}, {35016, 52264}, {37828, 61287}, {43151, 64765}, {43176, 66023}, {45310, 50395}, {46685, 58698}, {48680, 61268}, {49511, 51157}, {50117, 51062}, {50842, 51096}, {50845, 51103}, {50890, 51069}, {50891, 51108}, {50893, 51070}, {50894, 51107}, {50906, 51705}, {51003, 51199}, {51004, 51008}, {51005, 51158}, {51007, 51196}, {51714, 64123}, {54286, 61275}, {55317, 66858}

X(68277) = midpoint of X(i) and X(j) for these (i, j): {2, 15015}, {100, 16173}, {5440, 61649}, {5660, 38693}, {6174, 34123}, {59415, 64011}
X(68277) = reflection of X(i) in X(j) for these (i, j): (551, 34123), (10164, 38760), (15015, 50844), (16173, 1125), (21630, 16173), (50889, 59419), (59391, 10171), (59415, 3828), (59419, 2), (61649, 6681)
X(68277) = complement of X(37718)
X(68277) = X(10176)-of-anti-inner-Garcia triangle
X(68277) = X(37718)-of-medial triangle
X(68277) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (3, 68278, 21635), (10, 214, 33337), (11, 58453, 19862), (100, 1125, 21630), (100, 64012, 1125), (140, 22935, 10265), (149, 3624, 33709), (214, 3035, 10), (6265, 38762, 6684), (6702, 31235, 51073), (7972, 64141, 3626), (10609, 31235, 6702), (12119, 64008, 19925), (24466, 67876, 51118), (47742, 51111, 10)

X(68278) = CENTER OF THE X(5)-CO-NORMAL HYPERBOLA OF EXCENTRAL-HEXYL ELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {3, 5, 10, 3307, 3308, 6265, 10222, 18233, 18243, 21635, 22836, 22935, 22936, 22991, 23015, 31658, 36865, 37837, 60911, 64731}

X(68278) lies on these lines: {1, 66008}, {2, 6326}, {3, 16128}, {4, 15015}, {5, 22935}, {10, 6265}, {11, 13411}, {21, 66017}, {30, 50844}, {36, 66012}, {80, 10175}, {100, 946}, {104, 5251}, {119, 214}, {140, 2771}, {149, 8227}, {153, 3576}, {226, 10090}, {355, 33337}, {516, 12611}, {517, 58613}, {519, 19907}, {528, 16174}, {551, 12737}, {631, 1768}, {758, 66047}, {908, 4996}, {912, 6681}, {950, 39692}, {952, 1125}, {960, 2800}, {1006, 17009}, {1145, 6745}, {1210, 12739}, {1317, 44675}, {1385, 11698}, {1387, 13405}, {1484, 11230}, {1532, 54192}, {1537, 6174}, {1656, 59419}, {1698, 12247}, {1699, 13199}, {1737, 41558}, {2801, 6666}, {2802, 11729}, {2816, 53740}, {2842, 67414}, {3086, 37736}, {3090, 37718}, {3244, 64140}, {3452, 51506}, {3523, 9809}, {3616, 6264}, {3624, 5531}, {3634, 12619}, {3636, 64742}, {3655, 50906}, {3678, 61551}, {3817, 10738}, {3881, 61534}, {3911, 11570}, {4297, 10742}, {4304, 12764}, {4311, 12763}, {4757, 61530}, {4999, 64693}, {5055, 50889}, {5083, 64124}, {5432, 17638}, {5433, 17660}, {5440, 67857}, {5528, 38037}, {5541, 5603}, {5587, 6224}, {5657, 13253}, {5693, 17566}, {5719, 58587}, {5745, 9946}, {5818, 9897}, {5840, 18483}, {5854, 59722}, {5882, 12751}, {5884, 13747}, {5886, 12331}, {6073, 66843}, {6154, 38038}, {6246, 10609}, {6260, 48695}, {6691, 12005}, {6705, 58461}, {6796, 25681}, {6797, 37737}, {6881, 38062}, {6901, 56790}, {6905, 35204}, {6911, 42843}, {6920, 46816}, {6940, 33860}, {6946, 33593}, {6952, 34600}, {6959, 22836}, {6975, 37571}, {7972, 47745}, {7987, 12248}, {7993, 25055}, {9945, 65948}, {9964, 54357}, {10087, 12053}, {10164, 12515}, {10171, 60759}, {10595, 12653}, {10698, 11362}, {10711, 51705}, {10902, 63917}, {10956, 66230}, {11263, 45976}, {11700, 52659}, {11715, 17575}, {12119, 31673}, {12532, 59491}, {12571, 22938}, {12608, 66630}, {12736, 64110}, {12738, 19862}, {12740, 31397}, {12747, 61261}, {12749, 63987}, {12755, 61016}, {12775, 63989}, {13257, 21154}, {13271, 64117}, {15950, 17636}, {16173, 38665}, {16200, 64743}, {17546, 38669}, {18481, 38755}, {19878, 34126}, {19925, 61580}, {20095, 68034}, {22799, 28164}, {24466, 28150}, {24541, 31399}, {25436, 56754}, {25438, 59584}, {25440, 64762}, {26364, 40257}, {26446, 48667}, {26725, 66011}, {28172, 52836}, {31235, 38133}, {31272, 49176}, {31730, 34474}, {31760, 58504}, {32557, 37726}, {33594, 37251}, {33598, 67856}, {35023, 68035}, {35638, 43223}, {37732, 64710}, {38053, 66010}, {38127, 64141}, {38760, 46684}, {39870, 66030}, {41012, 65739}, {45770, 64763}, {45944, 63365}, {50796, 64011}, {50908, 64189}, {51077, 64746}, {51517, 61268}, {54445, 64009}, {54447, 64278}, {58631, 61566}, {59691, 63964}, {61269, 61601}, {61648, 63270}, {62710, 64322}, {63990, 64745}, {64154, 64188}, {64160, 67945}, {64267, 64733}, {66007, 66515}

X(68278) = midpoint of X(i) and X(j) for these (i, j): {3, 21635}, {5, 22935}, {10, 6265}, {100, 946}, {119, 214}, {355, 33337}, {1145, 25485}, {1385, 11698}, {1532, 54192}, {3244, 64140}, {3655, 50906}, {4297, 10742}, {5440, 67857}, {5660, 10165}, {5882, 12751}, {6073, 66843}, {6246, 10609}, {6260, 48695}, {6326, 10265}, {6713, 66051}, {7972, 47745}, {9945, 65948}, {9946, 18254}, {10698, 11362}, {10711, 51705}, {11715, 37725}, {12119, 31673}, {12331, 21630}, {12611, 33814}, {13271, 64117}, {17660, 63967}, {25436, 56754}, {31730, 34789}, {33598, 67856}, {34600, 67873}, {39870, 66030}, {50796, 64011}, {51077, 64746}
X(68278) = reflection of X(i) in X(j) for these (i, j): (1484, 33709), (6684, 3035), (6702, 58421), (6713, 58453), (12005, 58591), (12619, 3634), (13464, 11729), (18483, 67876), (19925, 61580), (22938, 12571), (31760, 58504), (64742, 3636)
X(68278) = complement of X(10265)
X(68278) = X(1568)-of-K798i triangle
X(68278) = X(10265)-of-medial triangle
X(68278) = X(11557)-of-Wasat triangle
X(68278) = X(11562)-of-4th Euler triangle
X(68278) = X(11806)-of-2nd circumperp triangle
X(68278) = X(20117)-of-anti-inner-Garcia triangle
X(68278) = X(21635)-of-anti-X3-ABC reflections triangle
X(68278) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (2, 6326, 10265), (80, 64008, 10175), (104, 64012, 10165), (1484, 11230, 33709), (1656, 62354, 59419), (5660, 64012, 104), (5886, 12331, 21630), (6224, 66045, 5587), (6265, 38752, 10), (6702, 58421, 10172), (12515, 38762, 10164), (15015, 15017, 4), (21635, 68277, 3), (34123, 37725, 11715), (34474, 34789, 31730)

X(68279) = CENTER OF THE X(1)-CO-NORMAL HYPERBOLA OF JOHNSON CIRCUMCONIC

This co-normal hyperbola passes through centers X(n) for these n: {1, 5, 942, 2574, 2575, 43165, 45776, 51759}

X(68279) lies on these lines: {3, 67568}, {110, 37113}, {1387, 2771}, {2779, 11727}, {5886, 67346}, {5901, 61638}, {5972, 52259}, {7687, 15252}, {9033, 67847}

X(68280) = CENTER OF THE X(2)-CO-NORMAL HYPERBOLA OF JOHNSON CIRCUMCONIC

This co-normal hyperbola passes through centers X(n) for these n: {2, 5, 2574, 2575, 14561, 18504, 36518, 43841, 64179, 67868}

X(68280) lies on these lines: {2, 2777}, {4, 15036}, {5, 1511}, {30, 48375}, {74, 5067}, {110, 5056}, {113, 1656}, {125, 3090}, {140, 34584}, {146, 15029}, {265, 5079}, {373, 46430}, {381, 38723}, {389, 68293}, {399, 61911}, {541, 15699}, {542, 5050}, {546, 38726}, {547, 5663}, {631, 13202}, {632, 1539}, {974, 11695}, {1112, 11793}, {1216, 58516}, {1352, 32300}, {1533, 30745}, {1995, 22109}, {2072, 29012}, {2771, 38319}, {3091, 16163}, {3448, 61914}, {3525, 10721}, {3526, 16111}, {3545, 15035}, {3564, 44912}, {3614, 46683}, {3628, 6699}, {3832, 15051}, {3851, 12295}, {5066, 68316}, {5068, 10733}, {5070, 7728}, {5071, 5642}, {5072, 12121}, {5095, 40330}, {5651, 15463}, {5654, 18951}, {5655, 61908}, {5891, 16222}, {5907, 9826}, {6677, 55292}, {6721, 67479}, {6816, 19506}, {7173, 46687}, {7395, 13289}, {7486, 15059}, {7505, 15473}, {7579, 67884}, {8998, 42262}, {9140, 61912}, {9813, 14561}, {9934, 37515}, {9956, 11723}, {10109, 32423}, {10110, 41673}, {10117, 64585}, {10264, 61907}, {10272, 12812}, {10628, 41670}, {10706, 61895}, {10990, 61886}, {11801, 44904}, {11807, 13416}, {12041, 55856}, {12227, 17814}, {12244, 60781}, {12358, 41671}, {12383, 61921}, {12825, 64854}, {12902, 61923}, {13293, 66607}, {13990, 42265}, {14156, 46031}, {14677, 55861}, {14683, 15025}, {15040, 61937}, {15041, 61887}, {15042, 61990}, {15081, 24981}, {15113, 16836}, {15694, 38788}, {15703, 38789}, {16003, 61905}, {16278, 64089}, {16534, 20304}, {17855, 65095}, {17856, 66606}, {17928, 25564}, {20126, 61901}, {20127, 46219}, {20397, 61900}, {24206, 32257}, {24930, 58431}, {25565, 29959}, {29181, 37942}, {30714, 61919}, {32223, 68319}, {32396, 50139}, {32609, 61920}, {33511, 61575}, {33512, 61576}, {38638, 61931}, {38728, 55857}, {38790, 55860}, {41737, 63119}, {42582, 46688}, {42583, 46689}, {42786, 49116}, {44573, 44870}, {44673, 44911}, {61548, 61894}, {61925, 64182}, {64183, 67096}, {66734, 66756}

X(68280) = midpoint of X(i) and X(j) for these (i, j): {2, 36518}, {113, 15061}, {381, 38793}, {5642, 14644}, {5891, 16222}, {14643, 23515}, {38792, 45311}
X(68280) = reflection of X(i) in X(j) for these (i, j): (15061, 6723), (20417, 15061), (68281, 14561), (68317, 36518)
X(68280) = complement of X(38727)
X(68280) = X(38727)-of-medial triangle
X(68280) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (5, 5972, 7687), (5, 12900, 5972), (113, 1656, 6723), (113, 6723, 20417), (140, 46686, 37853), (1656, 15046, 15061), (3090, 64101, 125), (3628, 61574, 6699), (3851, 38794, 12295), (5055, 14643, 23515), (6699, 61574, 38791), (10272, 12812, 15088), (10272, 15088, 36253), (15029, 46936, 38729), (15046, 15061, 113), (15059, 15063, 65092)

X(68281) = CENTER OF THE X(6)-CO-NORMAL HYPERBOLA OF JOHNSON CIRCUMCONIC

This co-normal hyperbola passes through centers X(n) for these n: {5, 6, 1597, 2574, 2575, 3527, 5480}

X(68281) lies on these lines: {4, 10250}, {6, 13}, {25, 23048}, {110, 7398}, {125, 8889}, {389, 2781}, {403, 21639}, {511, 10257}, {575, 13403}, {576, 32257}, {578, 9815}, {597, 11430}, {895, 45011}, {1351, 6723}, {1568, 37784}, {1594, 32285}, {1596, 23326}, {2393, 51742}, {2777, 5622}, {3542, 34788}, {3618, 48378}, {5093, 23515}, {5095, 39571}, {5462, 15115}, {5480, 10169}, {5621, 35501}, {5642, 11427}, {5943, 5972}, {6593, 37505}, {6699, 21851}, {9140, 63031}, {9777, 12828}, {9813, 14561}, {9972, 22830}, {10602, 61747}, {10733, 63123}, {10752, 65092}, {11426, 30714}, {11432, 16003}, {11438, 20423}, {11443, 62947}, {11458, 44958}, {11470, 20299}, {11482, 32275}, {12099, 65402}, {12140, 21637}, {12233, 38791}, {12242, 32246}, {12295, 53091}, {12358, 58555}, {13567, 45311}, {15073, 64063}, {15088, 61624}, {15465, 16534}, {16163, 51171}, {17702, 59399}, {18400, 44102}, {18449, 63735}, {18919, 67890}, {21850, 37853}, {25555, 50649}, {29012, 64891}, {32155, 44235}, {32271, 44489}, {38110, 48375}, {38726, 51732}, {39569, 53507}, {40135, 44231}, {41616, 45089}, {44439, 58445}, {54218, 67868}, {61749, 67904}, {63127, 66740}

X(68281) = midpoint of X(i) and X(j) for these (i, j): {113, 39562}, {403, 21639}, {1568, 37784}, {5093, 23515}, {18449, 63735}
X(68281) = reflection of X(i) in X(j) for these (i, j): (15462, 32300), (44673, 62375), (48375, 38110), (68280, 14561)
X(68281) = pole of the line {690, 32276} with respect to the 2nd Lemoine (or cosine) circle
X(68281) = pole of the line {30, 40349} with respect to the Kiepert circumhyperbola
X(68281) = pole of the line {39232, 55121} with respect to the orthic inconic
X(68281) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (6, 5476, 18388), (7687, 18388, 68317)

X(68282) = CENTER OF THE X(1)-CO-NORMAL HYPERBOLA OF KIEPERT PARABOLA

This co-normal hyperbola passes through centers X(n) for these n: {1, 30, 523, 11101, 18661}

X(68282) lies on these lines: {1, 31522}, {30, 50921}, {110, 476}, {1290, 13589}, {3109, 66789}, {16332, 47324}, {36155, 66796}, {47274, 66793}, {62509, 67596}

X(68282) = midpoint of X(47274) and X(66793)
X(68282) = reflection of X(i) in X(j) for these (i, j): (3109, 66789), (14985, 7471), (47324, 16332), (66796, 36155)
X(68282) = cross-difference of every pair of points on the line X(2088)X(35090)
X(68282) = pole of the line {1290, 15329} with respect to the circumcircle
X(68282) = pole of the line {30, 2948} with respect to the Kiepert parabola
X(68282) = pole of the line {526, 68164} with respect to the Stammler hyperbola

X(68283) = CENTER OF THE X(6)-CO-NORMAL HYPERBOLA OF KIEPERT PARABOLA

This co-normal hyperbola passes through centers X(n) for these n: {6, 30, 523, 1995, 5201, 5914, 14995, 16310, 22329, 45279}

X(68283) lies on these lines: {6, 53793}, {23, 9142}, {30, 11579}, {110, 476}, {112, 32229}, {182, 50149}, {691, 5467}, {842, 46127}, {1316, 68310}, {1976, 6094}, {3014, 36173}, {3258, 22112}, {5118, 66111}, {5651, 50146}, {6088, 32729}, {6795, 16168}, {7468, 9145}, {11003, 66820}, {11007, 66812}, {16187, 22104}, {16324, 47324}, {32224, 62490}, {34312, 50147}, {35522, 66115}, {62509, 67597}

X(68283) = reflection of X(i) in X(j) for these (i, j): (34312, 50147), (47324, 16324), (66812, 11007)
X(68283) = cross-difference of every pair of points on the line X(2088)X(23992)
X(68283) = intersection, other than {A, B, C}, of the circumconics through X(i), X(j) for these {i, j}: {476, 34574}, {691, 14559}
X(68283) = perspector of the circumconic through X(34539) and X(39295)
X(68283) = inverse of X(50149) in 1st Brocard circle
X(68283) = pole of the line {691, 9060} with respect to the circumcircle
X(68283) = pole of the line {30, 2930} with respect to the Kiepert parabola
X(68283) = pole of the line {526, 1649} with respect to the Stammler hyperbola
X(68283) = pole of the line {3268, 52629} with respect to the Steiner-Wallace hyperbola

X(68284) = CENTER OF THE X(3)-CO-NORMAL HYPERBOLA OF LEMOINE INELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {3, 524, 597, 1499, 8352}

X(68284) lies on these lines: {524, 14360}, {5108, 50983}, {13234, 31654}, {62293, 68285}

X(68284) = reflection of X(68285) in X(68286)
X(68284) = (X(62293), X(68285))-harmonic conjugate of X(68286)

X(68285) = CENTER OF THE X(4)-CO-NORMAL HYPERBOLA OF LEMOINE INELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {4, 524, 597, 1499, 8352}

X(68285) lies on these lines: {265, 34169}, {524, 20099}, {6792, 50959}, {31654, 50934}, {62293, 68284}

X(68285) = reflection of X(68284) in X(68286)
X(68285) = (X(68284), X(68286))-harmonic conjugate of X(62293)

X(68286) = CENTER OF THE X(5)-CO-NORMAL HYPERBOLA OF LEMOINE INELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {5, 524, 597, 1499, 8352, 14086}

X(68286) lies on these lines: {111, 524}, {597, 20381}, {62293, 68284}

X(68286) = midpoint of X(68284) and X(68285)
X(68286) = pole of the line {11053, 40544} with respect to the Lemoine inellipse
X(68286) = (X(62293), X(68285))-harmonic conjugate of X(68284)

X(68287) = CENTER OF THE X(1)-CO-NORMAL HYPERBOLA OF LOZADA-SODDY CONIC

This co-normal hyperbola passes through centers X(n) for these n: {1, 13601, 52803}

X(68287) lies on these lines: {55, 1293}, {2802, 12577}, {2810, 58577}, {2841, 11700}, {3664, 59812}, {21362, 28393}, {31792, 53790}, {58696, 68288}

X(68288) = CENTER OF THE X(2)-CO-NORMAL HYPERBOLA OF LOZADA-SODDY CONIC

This co-normal hyperbola passes through centers X(n) for these n: {2, 7419, 52803}

X(68288) lies on these lines: {1357, 3683}, {2842, 5642}, {5510, 68289}, {58696, 68287}

X(68288) = (X(68289), X(68290))-harmonic conjugate of X(5510)

X(68289) = CENTER OF THE X(3)-CO-NORMAL HYPERBOLA OF LOZADA-SODDY CONIC

This co-normal hyperbola passes through centers X(n) for these n: {3, 7419, 13601, 52803}

X(68289) lies on these lines: {1511, 2842}, {5510, 68288}

X(68289) = reflection of X(5510) in X(68290)
X(68289) = (X(5510), X(68288))-harmonic conjugate of X(68290)

X(68290) = CENTER OF THE X(5)-CO-NORMAL HYPERBOLA OF LOZADA-SODDY CONIC

This co-normal hyperbola passes through centers X(n) for these n: {5, 7419, 52803}

X(68290) lies on these lines: {2842, 10272}, {5510, 68288}

X(68290) = midpoint of X(5510) and X(68289)
X(68290) = (X(5510), X(68288))-harmonic conjugate of X(68289)

X(68291) = CENTER OF THE X(1)-CO-NORMAL HYPERBOLA OF MACBEATH CIRCUMCONIC

This co-normal hyperbola passes through centers X(n) for these n: {1, 6, 942, 1439, 2574, 2575, 9724, 66760}

X(68291) lies on these lines: {1, 2778}, {81, 105}, {125, 18635}, {518, 11064}, {942, 1511}, {1387, 2771}, {2779, 16193}, {3024, 3321}, {6126, 50190}, {11018, 44403}, {12826, 64377}, {18839, 51881}, {58654, 63259}, {58671, 61663}

X(68291) = midpoint of X(i) and X(j) for these (i, j): {15904, 65524}, {18839, 51881}
X(68291) = pole of the line {3100, 10149} with respect to the Feuerbach circumhyperbola
X(68291) = X(14769)-of-inverse-in-incircle triangle
X(68291) = X(15366)-of-intouch triangle
X(68291) = X(15367)-of-incircle-circles triangle
X(68291) = (X(354), X(65524))-harmonic conjugate of X(15904)

X(68292) = CENTER OF THE X(2)-CO-NORMAL HYPERBOLA OF MACBEATH CIRCUMCONIC

This co-normal hyperbola passes through centers X(n) for these n: {2, 5, 6, 2574, 2575, 5943, 5972, 15113, 15740, 15751, 16836, 38398, 41670, 45979, 59553, 61676}

X(68292) lies on these lines: {2, 2781}, {5, 63685}, {74, 59777}, {110, 10601}, {113, 15151}, {125, 37439}, {141, 12828}, {154, 35904}, {182, 20772}, {373, 597}, {468, 9019}, {542, 6688}, {547, 5663}, {631, 16105}, {974, 15045}, {1112, 3917}, {1656, 25711}, {1995, 16165}, {2777, 67263}, {3060, 41673}, {3066, 33851}, {3090, 15738}, {3796, 15647}, {5020, 15462}, {5181, 64692}, {5544, 11579}, {5622, 17825}, {5943, 5972}, {6593, 11284}, {6723, 10219}, {7399, 63695}, {7571, 15059}, {8681, 32300}, {9140, 10516}, {9517, 44564}, {9729, 65095}, {9826, 12900}, {10127, 17702}, {11064, 51742}, {11451, 45237}, {11695, 16270}, {12045, 44321}, {12827, 37648}, {13391, 14156}, {13416, 41671}, {14708, 15060}, {14924, 15054}, {15036, 44878}, {15051, 31860}, {15113, 45979}, {15303, 44838}, {15305, 61735}, {15806, 32205}, {16042, 62516}, {16194, 44573}, {16222, 23039}, {16238, 63659}, {16776, 47597}, {16836, 68317}, {32246, 40132}, {34990, 44889}, {36518, 64100}, {37935, 48378}, {38793, 44211}, {48154, 63684}, {50140, 61574}, {52293, 63723}, {61676, 68318}

X(68292) = midpoint of X(i) and X(j) for these (i, j): {2, 41670}, {1112, 3917}, {3060, 41673}, {5642, 12099}, {5943, 5972}, {14708, 15060}, {15113, 45979}, {16194, 44573}, {16836, 68317}, {61676, 68318}
X(68292) = reflection of X(i) in X(j) for these (i, j): (5943, 68293), (6723, 10219), (11746, 5943)
X(68292) = pole of the line {29181, 47277} with respect to the Jerabek circumhyperbola
X(68292) = pole of the line {524, 47237} with respect to the Thomson-Gibert-Moses hyperbola
X(68292) = X(11219)-of-submedial triangle (ABC acute)
X(68292) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (373, 5642, 12099), (5972, 68293, 11746)

X(68293) = CENTER OF THE X(5)-CO-NORMAL HYPERBOLA OF MACBEATH CIRCUMCONIC

This co-normal hyperbola passes through centers X(n) for these n: {5, 6, 2574, 2575, 3589, 4846, 5462, 5893, 9729, 9815, 9820, 9822, 9826, 9827, 12900, 15119, 15805, 22973, 32300, 32396, 43594, 58547}

X(68293) lies on these lines: {2, 1112}, {5, 113}, {51, 41673}, {52, 21971}, {74, 11465}, {110, 5020}, {140, 58516}, {143, 61506}, {182, 15647}, {265, 7401}, {381, 44573}, {389, 68280}, {399, 11484}, {511, 21847}, {542, 10128}, {569, 20771}, {1154, 44911}, {1316, 53796}, {1511, 6642}, {1539, 13203}, {1656, 12358}, {1986, 3090}, {2777, 11695}, {2781, 6688}, {2854, 6329}, {3024, 9817}, {3028, 19372}, {3047, 15018}, {3066, 6644}, {3091, 12133}, {3448, 7392}, {3545, 12292}, {5050, 35904}, {5055, 7723}, {5056, 13148}, {5068, 66734}, {5071, 7722}, {5159, 44084}, {5462, 12900}, {5544, 9818}, {5640, 21968}, {5892, 44920}, {5943, 5972}, {6102, 37643}, {6593, 19137}, {6699, 63679}, {7404, 15061}, {7486, 12219}, {7526, 63128}, {7535, 52831}, {7728, 18537}, {9825, 15465}, {9934, 37514}, {10095, 16238}, {10110, 48378}, {10113, 18420}, {10117, 17825}, {10263, 21970}, {10601, 13198}, {10961, 49268}, {10963, 49269}, {11284, 19504}, {11424, 59495}, {11591, 37638}, {12039, 40670}, {12052, 12068}, {12236, 15026}, {12284, 15024}, {12295, 14845}, {12362, 15473}, {12824, 15059}, {12825, 15043}, {13391, 32269}, {13598, 48375}, {13754, 44912}, {14915, 63821}, {15045, 17854}, {15074, 61680}, {15151, 38791}, {15472, 66607}, {15807, 18874}, {16105, 38727}, {17855, 68317}, {18438, 52290}, {18570, 22112}, {22462, 54073}, {22584, 61919}, {32142, 60780}, {32227, 40132}, {34417, 37814}, {40949, 47355}, {44212, 63475}, {46431, 66606}, {46682, 66529}, {52070, 64730}, {53795, 57588}, {64821, 66961}, {64822, 66960}

X(68293) = midpoint of X(i) and X(j) for these (i, j): {5, 9826}, {113, 16270}, {140, 58516}, {974, 65095}, {1112, 13416}, {5159, 44084}, {5462, 12900}, {5943, 68292}, {5972, 11746}, {6723, 41671}, {9822, 32300}, {10110, 48378}, {12052, 12068}, {12362, 15473}, {15151, 38791}
X(68293) = complement of X(13416)
X(68293) = pole of the line {16976, 34380} with respect to the Stammler hyperbola
X(68293) = X(11)-of-submedial triangle (ABC acute)
X(68293) = X(13416)-of-medial triangle
X(68293) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (2, 1112, 13416), (974, 36518, 65095), (1656, 16222, 12358), (5943, 5972, 11746), (6688, 41671, 6723), (11746, 68292, 5972), (12824, 15059, 54376), (15024, 64101, 46430), (36518, 64854, 974)

X(68294) = CENTER OF THE X(1)-CO-NORMAL HYPERBOLA OF MACBEATH INCONIC

This co-normal hyperbola passes through centers X(n) for these n: {1, 5, 30, 523, 3007}

X(68294) lies on these lines: {476, 953}, {51701, 62496}, {62493, 67596}

X(68294) = pole of the line {34464, 65856} with respect to the MacBeath inconic

X(68295) = CENTER OF THE X(6)-CO-NORMAL HYPERBOLA OF MACBEATH INCONIC

This co-normal hyperbola passes through centers X(n) for these n: {5, 6, 30, 523, 6530}

X(68295) lies on these lines: {1316, 18583}, {12052, 47324}, {51733, 62509}

X(68296) = CENTER OF THE X(1)-CO-NORMAL HYPERBOLA OF MANDART INELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {1, 9, 950, 3057, 3307, 3308, 9898, 14100, 14749, 15558, 60545, 60961, 62333, 66193, 66194, 66195, 66197, 66198, 66200, 66201, 66202, 66203, 66204, 66205, 66206, 66207, 66210, 66211, 66214, 66219, 66224, 66234, 66242, 66992}

X(68296) lies on these lines: {1, 1537}, {2, 11}, {9, 68301}, {30, 15368}, {56, 38759}, {80, 7160}, {104, 1058}, {119, 3295}, {214, 4314}, {388, 52836}, {495, 67864}, {496, 6713}, {516, 3660}, {529, 5048}, {938, 64189}, {944, 40290}, {950, 952}, {999, 38761}, {1056, 10728}, {1145, 1697}, {1210, 64193}, {1317, 3486}, {1320, 9785}, {1329, 26358}, {1385, 1387}, {1388, 5731}, {1479, 18242}, {1768, 41556}, {1776, 51463}, {1837, 3036}, {1864, 46685}, {2098, 57288}, {2478, 13278}, {2800, 63999}, {2802, 12575}, {3057, 5854}, {3086, 21154}, {3303, 10956}, {3488, 10698}, {3586, 7966}, {3601, 34123}, {3612, 16173}, {3746, 20400}, {3813, 62333}, {4294, 24466}, {4309, 10090}, {4326, 10427}, {4342, 64137}, {4345, 11114}, {4857, 8068}, {5083, 10391}, {5225, 59390}, {5533, 63281}, {5554, 18802}, {5732, 9580}, {5842, 30384}, {5851, 14100}, {5853, 46694}, {5856, 66203}, {6767, 10742}, {7373, 38753}, {7681, 11508}, {7962, 11113}, {8071, 10058}, {9581, 34122}, {9614, 38038}, {9668, 64186}, {9669, 23513}, {9670, 13273}, {9809, 14151}, {9819, 64056}, {9848, 17638}, {10164, 65388}, {10284, 37730}, {10382, 13257}, {10384, 64372}, {10386, 33814}, {10387, 51007}, {10543, 39778}, {10572, 17622}, {10593, 38319}, {10609, 41864}, {10624, 12736}, {10629, 12761}, {10679, 32554}, {10738, 16202}, {10786, 59391}, {10866, 12743}, {10965, 12607}, {11019, 46684}, {11193, 15914}, {11373, 38032}, {11376, 35262}, {11570, 12711}, {11715, 63993}, {11729, 24929}, {11849, 55297}, {12619, 18527}, {12701, 64150}, {12735, 15170}, {12751, 31393}, {12758, 66226}, {13226, 41166}, {13405, 67876}, {14740, 40998}, {14986, 38693}, {15528, 50196}, {16541, 54065}, {17115, 64440}, {17724, 35015}, {17768, 18839}, {18254, 64131}, {21635, 30331}, {26476, 64123}, {33646, 64489}, {37734, 66259}, {38754, 67261}, {40270, 46681}, {40296, 58587}, {40615, 67571}, {41426, 64076}, {43175, 51783}, {61146, 64138}

X(68296) = midpoint of X(i) and X(j) for these (i, j): {950, 15558}, {1387, 15171}, {3057, 66206}, {10624, 12736}, {66203, 66210}
X(68296) = reflection of X(i) in X(j) for these (i, j): (14740, 68298), (24465, 18240), (46681, 40270)
X(68296) = cross-difference of every pair of points on the line X(665)X(55334)
X(68296) = pole of the line {2804, 15914} with respect to the incircle
X(68296) = pole of the line {518, 6735} with respect to the Feuerbach circumhyperbola
X(68296) = pole of the line {654, 1768} with respect to the Mandart inellipse
X(68296) = X(974)-of-Hutson intouch triangle
X(68296) = X(3035)-of-Mandart-incircle triangle
X(68296) = X(11746)-of-Ursa-minor triangle
X(68296) = X(38759)-of-2nd anti-circumperp-tangential triangle
X(68296) = X(41673)-of-intouch triangle
X(68296) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (11, 55, 3035), (11, 6154, 60782), (11, 13274, 66065), (55, 497, 15845), (2478, 13278, 55016), (3303, 12764, 10956), (10956, 12764, 38757), (14740, 40998, 68298)

X(68297) = CENTER OF THE X(2)-CO-NORMAL HYPERBOLA OF MANDART INELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {2, 9, 3307, 3308, 40998, 66205, 66940}

X(68297) lies on these lines: {55, 46694}, {5083, 26105}, {6174, 60986}, {66021, 66515}, {68298, 68299}

X(68297) = (X(68298), X(68300))-harmonic conjugate of X(68299)

X(68298) = CENTER OF THE X(3)-CO-NORMAL HYPERBOLA OF MANDART INELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {3, 9, 960, 3057, 3307, 3308, 5795, 12572, 54464}

X(68298) lies on these lines: {2, 24465}, {9, 11}, {10, 38161}, {100, 18228}, {119, 49183}, {124, 4422}, {149, 38211}, {513, 64444}, {528, 18227}, {908, 5857}, {936, 24466}, {952, 960}, {958, 1387}, {1145, 2551}, {1317, 15829}, {1329, 61524}, {2802, 18250}, {2829, 12572}, {2886, 61511}, {3035, 3452}, {3036, 5837}, {3715, 13274}, {4011, 41883}, {5044, 5840}, {5087, 5762}, {5123, 28174}, {5234, 16173}, {5250, 55016}, {5273, 31272}, {5289, 12735}, {5325, 45310}, {5745, 6667}, {5791, 23513}, {5795, 5854}, {5833, 38152}, {6702, 18249}, {6713, 31445}, {8165, 64141}, {9708, 64138}, {9809, 10427}, {11344, 45393}, {12514, 64193}, {13226, 55869}, {13244, 15479}, {13257, 64154}, {14740, 40998}, {15253, 26611}, {18233, 60759}, {19843, 38038}, {21154, 31424}, {22775, 51506}, {24703, 67962}, {28915, 38390}, {30294, 34687}, {30827, 31235}, {31852, 46663}, {37560, 52116}, {38357, 65824}, {46435, 61122}, {68297, 68299}

X(68298) = midpoint of X(14740) and X(68296)
X(68298) = reflection of X(68299) in X(68300)
X(68298) = complement of X(24465)
X(68298) = inverse of X(66068) in Mandart inellipse
X(68298) = pole of the line {53573, 53574} with respect to the Spieker circle
X(68298) = pole of the line {6366, 66068} with respect to the Mandart inellipse
X(68298) = X(1112)-of-2nd Zaniah triangle
X(68298) = X(24465)-of-medial triangle
X(68298) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (11, 6068, 66068), (14740, 40998, 68296), (18227, 58663, 46694), (68297, 68299, 68300)

X(68299) = CENTER OF THE X(4)-CO-NORMAL HYPERBOLA OF MANDART INELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {4, 9, 950, 3307, 3308, 10395, 44547, 52819}

X(68299) lies on these lines: {4, 12736}, {9, 14740}, {11, 118}, {72, 15558}, {100, 10382}, {104, 10396}, {149, 5809}, {214, 10393}, {452, 64139}, {950, 2802}, {1387, 5777}, {1708, 46684}, {1728, 10058}, {1768, 10398}, {1837, 3754}, {2800, 13601}, {3306, 10394}, {3586, 67945}, {3678, 62333}, {5729, 41166}, {6154, 14100}, {6260, 15528}, {6692, 10391}, {6702, 10395}, {8000, 10698}, {9581, 41562}, {9844, 12690}, {10399, 11570}, {11715, 57278}, {12758, 18397}, {13226, 64157}, {13464, 64131}, {26476, 58565}, {46685, 53055}, {46694, 64171}, {64372, 66015}, {68297, 68298}

X(68299) = reflection of X(68298) in X(68300)
X(68299) = pole of the line {516, 17757} with respect to the Feuerbach circumhyperbola
X(68299) = pole of the line {3887, 64372} with respect to the Mandart inellipse
X(68299) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (5728, 13257, 5083), (14740, 66203, 66199), (68298, 68300, 68297)

X(68300) = CENTER OF THE X(5)-CO-NORMAL HYPERBOLA OF MANDART INELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {5, 9, 3307, 3308, 64738, 66206}

X(68300) lies on these lines: {100, 36868}, {517, 58475}, {971, 6667}, {6690, 58683}, {15733, 46694}, {34122, 66248}, {68297, 68298}

X(68300) = midpoint of X(68298) and X(68299)
X(68300) = (X(68297), X(68299))-harmonic conjugate of X(68298)

X(68301) = CENTER OF THE X(6)-CO-NORMAL HYPERBOLA OF MANDART INELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {6, 9, 3307, 3308}

X(68301) lies on these lines: {6, 2829}, {9, 68296}, {528, 50115}, {3035, 4254}, {3553, 64192}, {5120, 38759}, {20400, 37503}

X(68302) = CENTER OF THE X(2)-CO-NORMAL HYPERBOLA OF MOSES HK-PARABOLA

This co-normal hyperbola passes through centers X(n) for these n: {2, 297, 525, 1503, 41370}

X(68302) lies on these lines: {112, 525}, {9140, 44216}

X(68302) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (68303, 68304, 2867), (68303, 68305, 68304)

X(68303) = CENTER OF THE X(3)-CO-NORMAL HYPERBOLA OF MOSES HK-PARABOLA

This co-normal hyperbola passes through centers X(n) for these n: {3, 297, 525, 1503, 8743, 36823}

X(68303) lies on these lines: {112, 525}, {1503, 10749}, {10264, 54074}, {33504, 67217}, {36471, 43389}

X(68303) = midpoint of X(2867) and X(68304)
X(68303) = reflection of X(68304) in X(68305)
X(68303) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (2867, 68302, 68304), (68302, 68304, 68305)

X(68304) = CENTER OF THE X(4)-CO-NORMAL HYPERBOLA OF MOSES HK-PARABOLA

This co-normal hyperbola passes through centers X(n) for these n: {4, 297, 525, 1503, 63856}

X(68304) lies on these lines: {112, 525}, {265, 5523}, {879, 935}, {1503, 10735}, {2451, 39190}, {6794, 18809}, {18338, 39575}, {20031, 44427}, {33885, 67663}, {41377, 45031}, {50718, 67222}, {57086, 67667}, {57655, 67491}

X(68304) = reflection of X(i) in X(j) for these (i, j): (2867, 68303), (53912, 18338), (68303, 68305)
X(68304) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (2867, 68302, 68303), (68303, 68305, 68302)

X(68305) = CENTER OF THE X(5)-CO-NORMAL HYPERBOLA OF MOSES HK-PARABOLA

This co-normal hyperbola passes through centers X(n) for these n: {5, 297, 525, 1503}

X(68305) lies on these lines: {112, 525}

X(68305) = midpoint of X(68303) and X(68304)
X(68305) = (X(68302), X(68304))-harmonic conjugate of X(68303)

X(68306) = CENTER OF THE X(1)-CO-NORMAL HYPERBOLA OF NEUBERG-GIBERT HYPERBOLA

This co-normal hyperbola passes through centers X(n) for these n: {1, 30, 523, 11101, 60603, 61272}

X(68306) lies on these lines: {3109, 66789}, {11734, 62496}

X(68306) = midpoint of X(3109) and X(66789)

X(68307) = CENTER OF THE X(3)-CO-NORMAL HYPERBOLA OF NEUBERG-GIBERT HYPERBOLA

This co-normal hyperbola passes through centers X(n) for these n: {3, 30, 523, 38609, 60603}

X(68307) lies on these lines: {2, 66795}, {3, 476}, {20, 57305}, {30, 6699}, {125, 47852}, {186, 66790}, {376, 66781}, {382, 66787}, {548, 64510}, {549, 3258}, {550, 25641}, {631, 20957}, {1511, 14611}, {1553, 14677}, {1656, 44967}, {3233, 5663}, {3520, 66771}, {3523, 57306}, {3524, 14731}, {3528, 34193}, {3529, 66815}, {3530, 31379}, {3534, 14989}, {3579, 66789}, {5010, 33965}, {5054, 66791}, {5122, 59823}, {5946, 16978}, {6070, 34153}, {7280, 33964}, {7471, 12041}, {8703, 18319}, {9179, 38623}, {10272, 32417}, {10304, 66773}, {10620, 60605}, {11749, 17504}, {12006, 68074}, {12017, 66807}, {12042, 53738}, {12052, 13363}, {12068, 61574}, {12295, 21315}, {13391, 68070}, {14480, 15040}, {15055, 36193}, {15688, 66786}, {15692, 66802}, {15693, 34312}, {15698, 66820}, {15710, 66817}, {16163, 34209}, {16340, 38727}, {17502, 66770}, {17511, 38728}, {18324, 59231}, {18571, 62490}, {33813, 53728}, {34128, 36184}, {34577, 63715}, {34584, 36169}, {36172, 38788}, {37950, 47327}, {38613, 66111}, {52056, 65086}, {54173, 66813}, {55610, 66805}, {62067, 66788}, {62087, 66816}, {62100, 66772}, {66793, 67706}

X(68307) = midpoint of X(i) and X(j) for these (i, j): {3, 38609}, {20, 66778}, {476, 38610}, {550, 25641}, {1511, 46632}, {1553, 14677}, {3579, 66789}, {6070, 34153}, {7471, 12041}, {9179, 38623}, {12042, 53738}, {16163, 34209}, {33813, 53728}, {37950, 47327}, {38613, 66111}
X(68307) = reflection of X(i) in X(j) for these (i, j): (31379, 3530), (61574, 12068), (68074, 12006), (68308, 68309)
X(68307) = complement of X(66795)
X(68307) = X(38609)-of-anti-X3-ABC reflections triangle
X(68307) = X(66795)-of-medial triangle
X(68307) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (3, 476, 38610), (3, 38580, 38701), (3, 38700, 38609), (20, 57305, 66778), (3523, 66792, 57306), (5054, 66791, 66801), (14677, 64652, 1553), (22104, 68308, 68309), (38609, 38610, 476)

X(68308) = CENTER OF THE X(4)-CO-NORMAL HYPERBOLA OF NEUBERG-GIBERT HYPERBOLA

This co-normal hyperbola passes through centers X(n) for these n: {4, 30, 523, 18279, 25641, 60603}

X(68308) lies on these lines: {2, 14989}, {4, 476}, {5, 31379}, {20, 66787}, {30, 6699}, {113, 14611}, {146, 5627}, {265, 1553}, {381, 3258}, {382, 57305}, {477, 3091}, {523, 46686}, {541, 12079}, {546, 16168}, {1511, 21269}, {1539, 32417}, {1699, 66779}, {3090, 38701}, {3146, 38700}, {3233, 17702}, {3545, 66773}, {3627, 38609}, {3817, 66770}, {3832, 34193}, {3839, 14731}, {3843, 20957}, {3845, 18319}, {3850, 66818}, {3851, 57306}, {3854, 66788}, {5663, 21316}, {6070, 7728}, {6756, 63708}, {7471, 12295}, {10110, 68074}, {10151, 66790}, {10895, 66783}, {10896, 66782}, {11479, 66777}, {11558, 24043}, {11749, 23046}, {12041, 21315}, {12068, 38726}, {12900, 47084}, {13202, 46632}, {13754, 68070}, {14269, 66791}, {14508, 15081}, {14644, 36172}, {14934, 36518}, {18323, 47327}, {18492, 66796}, {18535, 66794}, {23047, 66771}, {23323, 62501}, {23515, 36164}, {31378, 57471}, {31673, 66789}, {34312, 41099}, {38580, 61984}, {38677, 50689}, {38678, 61964}, {39491, 55141}, {39809, 53738}, {39838, 53728}, {47336, 62490}, {47353, 66813}, {53023, 66809}, {55308, 61574}, {59385, 66804}, {59387, 66784}, {61954, 66817}, {61985, 66802}, {61989, 66820}, {62489, 67862}, {62491, 65948}, {62492, 67864}, {62496, 66592}, {62509, 66594}

X(68308) = midpoint of X(i) and X(j) for these (i, j): {4, 25641}, {5, 66778}, {113, 34150}, {265, 1553}, {1511, 21269}, {1539, 34209}, {3258, 66781}, {3627, 38609}, {6070, 7728}, {7471, 12295}, {13202, 46632}, {18319, 66795}, {18323, 47327}, {31378, 57471}, {31673, 66789}, {39809, 53738}, {39838, 53728}
X(68308) = reflection of X(i) in X(j) for these (i, j): (31379, 5), (38726, 12068), (47084, 12900), (55308, 61574), (55319, 20304), (68074, 10110), (68307, 68309)
X(68308) = pole of the line {55130, 66773} with respect to the polar circle
X(68308) = X(3258)-of-Ehrmann-mid triangle
X(68308) = X(24201)-of-Ehrmann-vertex triangle (ABC acute)
X(68308) = X(25641)-of-Euler triangle
X(68308) = X(31379)-of-Johnson triangle
X(68308) = X(66843)-of-orthic triangle (ABC acute)
X(68308) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (381, 66781, 3258), (3146, 66815, 38700), (3545, 66773, 66801), (3845, 18319, 66795), (3851, 66772, 57306), (68307, 68309, 22104)

X(68309) = CENTER OF THE X(5)-CO-NORMAL HYPERBOLA OF NEUBERG-GIBERT HYPERBOLA

This co-normal hyperbola passes through centers X(n) for these n: {5, 30, 523, 14993, 60603, 61272}

X(68309) lies on these lines: {2, 18319}, {3, 66815}, {5, 476}, {30, 6699}, {140, 25641}, {265, 64652}, {477, 632}, {546, 38609}, {547, 3258}, {548, 66778}, {549, 66781}, {1553, 64642}, {1656, 66819}, {3090, 38580}, {3233, 32423}, {3523, 66772}, {3526, 34193}, {3530, 64510}, {3533, 66788}, {3545, 66791}, {3627, 38700}, {3628, 16168}, {3851, 66792}, {3858, 44967}, {5054, 66773}, {5055, 14731}, {5066, 66795}, {5071, 66802}, {7471, 11801}, {8703, 14989}, {10272, 14611}, {11539, 66786}, {11749, 15699}, {13451, 16978}, {14869, 38701}, {14993, 64101}, {16239, 31379}, {18357, 66789}, {34312, 61910}, {37942, 66790}, {38028, 66779}, {38110, 66809}, {38111, 66804}, {38112, 66784}, {38677, 61900}, {38678, 55861}, {44278, 59231}, {55856, 57306}, {58531, 68074}, {61268, 66793}, {61858, 66816}, {61864, 66817}, {61920, 66820}

X(68309) = midpoint of X(i) and X(j) for these (i, j): {140, 25641}, {546, 38609}, {548, 66778}, {7471, 11801}, {10272, 34209}, {18357, 66789}, {68307, 68308}
X(68309) = reflection of X(i) in X(j) for these (i, j): (31379, 16239), (68074, 58531)
X(68309) = X(38586)-of-submedial triangle (ABC acute)
X(68309) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (7471, 21315, 11801), (11749, 15699, 66801), (22104, 68308, 68307), (57305, 66787, 5)

X(68310) = CENTER OF THE X(6)-CO-NORMAL HYPERBOLA OF NEUBERG-GIBERT HYPERBOLA

This co-normal hyperbola passes through centers X(n) for these n: {6, 30, 523, 1995, 44401, 60603}

X(68310) lies on these lines: {1316, 68283}, {20113, 34094}, {47457, 62508}

X(68311) = CENTER OF THE X(2)-CO-NORMAL HYPERBOLA OF PRIVALOV CONIC

This co-normal hyperbola passes through centers X(n) for these n: {2, 3136, 5452}

X(68311) lies on these lines: {2772, 45311}, {2808, 10219}, {6710, 68312}

X(68311) = (X(6710), X(68313))-harmonic conjugate of X(68312)

X(68312) = CENTER OF THE X(4)-CO-NORMAL HYPERBOLA OF PRIVALOV CONIC

This co-normal hyperbola passes through centers X(n) for these n: {4, 3136, 5452, 40960, 61669}

X(68312) lies on these lines: {101, 14547}, {2772, 7687}, {2809, 63999}, {6710, 68311}

X(68312) = reflection of X(6710) in X(68313)
X(68312) = (X(6710), X(68313))-harmonic conjugate of X(68311)

X(68313) = CENTER OF THE X(5)-CO-NORMAL HYPERBOLA OF PRIVALOV CONIC

This co-normal hyperbola passes through centers X(n) for these n: {5, 3136, 5452}

X(68313) lies on these lines: {2772, 20304}, {6710, 68311}

X(68313) = midpoint of X(6710) and X(68312)
X(68313) = (X(68311), X(68312))-harmonic conjugate of X(6710)

X(68314) = CENTER OF THE X(6)-CO-NORMAL HYPERBOLA OF PRIVALOV CONIC

This co-normal hyperbola passes through centers X(n) for these n: {1, 6, 1643, 5452, 20970}

X(68314) lies on these lines: {1, 28345}, {6, 2801}, {101, 1449}, {1100, 52969}, {1386, 2809}, {2348, 11028}, {2784, 68315}, {14760, 30621}, {16667, 44858}

X(68315) = CENTER OF THE X(6)-CO-NORMAL HYPERBOLA OF STEINER CIRCUMELLIPSE

This co-normal hyperbola passes through centers X(n) for these n: {2, 4, 6, 39, 115, 1640, 3413, 3414, 8105, 8106, 12815, 13881, 14482, 14537, 16601, 17355, 65612}

X(68315) lies on these lines: {2, 5503}, {4, 60140}, {5, 44499}, {6, 13}, {30, 2030}, {39, 25555}, {51, 11226}, {69, 14061}, {98, 5039}, {99, 3618}, {111, 5642}, {114, 7736}, {125, 6792}, {141, 6722}, {147, 14930}, {148, 5182}, {182, 2549}, {187, 19924}, {206, 39834}, {230, 511}, {385, 51396}, {518, 11725}, {524, 625}, {538, 44380}, {543, 597}, {574, 10168}, {575, 5254}, {576, 3767}, {599, 14971}, {620, 3589}, {671, 18800}, {690, 2492}, {698, 14148}, {1084, 5661}, {1350, 38737}, {1351, 38224}, {1352, 23514}, {1353, 38229}, {1384, 54131}, {1503, 67863}, {1550, 57430}, {1562, 5622}, {1569, 13331}, {1570, 5965}, {1648, 45311}, {1656, 10542}, {1691, 6781}, {1692, 29012}, {1743, 66678}, {1916, 7875}, {1976, 51431}, {1992, 9166}, {2482, 5024}, {2682, 34169}, {2782, 18583}, {2784, 68314}, {2794, 5480}, {2795, 51747}, {2799, 14316}, {3068, 13773}, {3069, 13653}, {3091, 50641}, {3094, 58445}, {3098, 21843}, {3124, 5972}, {3564, 61576}, {3751, 38220}, {3981, 58447}, {5028, 24206}, {5032, 11161}, {5033, 48892}, {5034, 32135}, {5038, 7765}, {5050, 6321}, {5085, 36772}, {5092, 38736}, {5097, 38735}, {5107, 15993}, {5286, 10358}, {5305, 11623}, {5334, 41060}, {5335, 41061}, {5523, 44102}, {5913, 13857}, {5967, 52450}, {6054, 37665}, {6055, 7735}, {6114, 14136}, {6115, 14137}, {6230, 44594}, {6231, 44597}, {6388, 6723}, {6531, 67406}, {6593, 50711}, {6719, 11053}, {6721, 31489}, {6776, 14639}, {6794, 16278}, {7603, 25565}, {7738, 10992}, {7739, 42852}, {7746, 40107}, {7749, 44453}, {7755, 13330}, {7756, 39560}, {7786, 50640}, {7806, 22486}, {7809, 50249}, {7835, 18906}, {7845, 50253}, {7868, 32458}, {7925, 51371}, {7983, 59406}, {8430, 52038}, {8593, 41135}, {8791, 12828}, {8980, 13926}, {9167, 36764}, {9478, 50251}, {9605, 14981}, {9745, 61743}, {9830, 36523}, {9880, 11179}, {9969, 58518}, {10516, 64091}, {10723, 25406}, {10753, 14651}, {10991, 30435}, {11054, 41137}, {11064, 16317}, {11177, 63005}, {11178, 43620}, {11477, 38740}, {11488, 51013}, {11489, 51010}, {11599, 59408}, {11645, 53419}, {11711, 38049}, {12017, 38730}, {12042, 21850}, {12584, 44533}, {13178, 16475}, {13640, 19054}, {13760, 19053}, {13873, 13967}, {13881, 34507}, {14389, 30516}, {14482, 64090}, {14568, 39099}, {14830, 21309}, {14931, 63020}, {15018, 62298}, {15092, 18358}, {15342, 25320}, {15595, 66163}, {15903, 25072}, {16092, 52233}, {16303, 47581}, {16315, 47574}, {16325, 47572}, {16990, 36849}, {18553, 63534}, {19055, 26456}, {19056, 26463}, {19109, 66474}, {19136, 39840}, {20190, 63548}, {20399, 31406}, {20415, 44511}, {20416, 44512}, {20774, 40065}, {22246, 48657}, {22247, 48310}, {22505, 38136}, {22510, 51206}, {22511, 51207}, {22515, 48906}, {22579, 37640}, {22580, 37641}, {23004, 36757}, {23005, 36758}, {25562, 31415}, {29181, 38747}, {31274, 47355}, {31400, 38751}, {31670, 38749}, {31695, 51012}, {31696, 51015}, {32300, 32740}, {33813, 38110}, {33878, 38739}, {34127, 48876}, {34366, 51429}, {34369, 65755}, {34473, 46453}, {34481, 61681}, {35021, 44531}, {36519, 36771}, {37637, 50977}, {37689, 54132}, {38119, 53733}, {38732, 53091}, {38733, 55705}, {39022, 67690}, {39023, 67679}, {39809, 46264}, {39835, 58471}, {39838, 53023}, {40112, 67553}, {40825, 48901}, {41134, 63109}, {41145, 41254}, {41148, 63115}, {44377, 51397}, {44501, 49220}, {44502, 49221}, {44529, 49116}, {45018, 63011}, {46124, 47200}, {46980, 67397}, {47184, 47571}, {47240, 47585}, {47326, 47455}, {47550, 51258}, {51185, 51798}, {51963, 52472}, {51980, 52672}, {52471, 65751}, {53792, 67495}, {54173, 62992}, {58610, 58694}, {58621, 58682}, {59399, 67268}, {60496, 60498}, {64490, 67377}, {66706, 66763}

X(68315) = midpoint of X(i) and X(j) for these (i, j): {6, 115}, {148, 14928}, {187, 53505}, {385, 51396}, {671, 18800}, {1570, 53475}, {5107, 15993}, {5477, 11646}, {6055, 20423}, {7845, 50253}, {8430, 52038}, {9880, 11179}, {10754, 50567}, {12042, 21850}, {16315, 47574}, {22515, 48906}, {31670, 38749}, {31695, 51012}, {31696, 51015}, {34369, 65755}, {39809, 46264}, {47550, 51258}, {52471, 65751}, {58610, 58694}, {58621, 58682}
X(68315) = reflection of X(i) in X(j) for these (i, j): (141, 6722), (620, 3589), (9969, 58518), (18358, 15092), (19662, 5461), (38736, 5092), (39835, 58471), (41672, 6), (51397, 44377), (67377, 64490), (67862, 19130)
X(68315) = complement of X(50567)
X(68315) = cross-difference of every pair of points on the line X(526)X(2930)
X(68315) = intersection, other than {A, B, C}, of the circumconics through X(i), X(j) for these {i, j}: {265, 60140}, {1989, 5503}
X(68315) = center of the inconic with perspector X(9154)
X(68315) = perspector of the circumconic through X(476) and X(9473)
X(68315) = inverse of X(18907) in orthosymmedial circle
X(68315) = inverse of X(44468) in Moses-Parry circle
X(68315) = pole of the line {125, 57618} with respect to the Dao-Moses-Telv circle
X(68315) = pole of the line {542, 44468} with respect to the Moses-Parry circle
X(68315) = pole of the line {2799, 18907} with respect to the orthosymmedial circle
X(68315) = pole of the line {549, 7853} with respect to the Evans conic
X(68315) = pole of the line {30, 114} with respect to the Kiepert circumhyperbola
X(68315) = pole of the line {67, 35364} with respect to the orthic inconic
X(68315) = pole of the line {323, 2030} with respect to the Stammler hyperbola
X(68315) = pole of the line {2793, 5984} with respect to the Steiner circumellipse
X(68315) = pole of the line {98, 111} with respect to the Steiner inellipse
X(68315) = pole of the line {7799, 22329} with respect to the Steiner-Wallace hyperbola
X(68315) = X(187)-of-1st orthosymmedial triangle
X(68315) = X(2030)-of-these triangles: 4th Brocard, orthocentroidal
X(68315) = X(6390)-of-1st Brocard triangle
X(68315) = X(7767)-of-6th anti-Brocard triangle
X(68315) = X(28345)-of-orthic triangle (ABC acute)
X(68315) = X(50567)-of-medial triangle
X(68315) = X(50983)-of-anti-McCay triangle
X(68315) = X(59813)-of-anti-Honsberger triangle (ABC acute)
X(68315) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (2, 10754, 50567), (6, 6034, 115), (6, 11646, 5477), (115, 5477, 11646), (148, 5182, 14928), (148, 51171, 5182), (671, 59373, 18800), (3124, 41939, 10418), (10418, 41939, 5972), (13670, 13790, 6055), (41135, 63127, 8593)

X(68316) = CENTER OF THE X(3)-CO-NORMAL HYPERBOLA OF THOMSON-GIBERT-MOSES HYPERBOLA

This co-normal hyperbola passes through centers X(n) for these n: {3, 5642, 9177, 47049}

X(68316) lies on these lines: {2, 14644}, {3, 541}, {20, 38795}, {30, 5972}, {74, 15692}, {99, 11656}, {110, 3524}, {113, 376}, {125, 5054}, {140, 20396}, {141, 542}, {146, 62063}, {165, 50878}, {265, 15694}, {381, 12900}, {399, 15700}, {511, 18579}, {543, 33509}, {547, 7687}, {548, 38791}, {597, 33851}, {631, 9140}, {690, 9126}, {1539, 15686}, {2482, 53725}, {2777, 8703}, {2854, 50983}, {3448, 15708}, {3523, 9143}, {3528, 15023}, {3530, 20417}, {3534, 14643}, {3543, 64101}, {3545, 12295}, {3548, 33556}, {3582, 46687}, {3584, 46683}, {3653, 12778}, {3796, 15693}, {3819, 5663}, {3830, 36518}, {4550, 64606}, {5055, 12121}, {5066, 68280}, {5071, 10733}, {5085, 5648}, {5181, 11179}, {5447, 25711}, {5476, 6644}, {5504, 17040}, {5609, 15712}, {5731, 50877}, {5892, 14984}, {6053, 13392}, {6055, 53735}, {6174, 53753}, {6593, 54169}, {6689, 15115}, {6723, 11539}, {7426, 10564}, {7502, 25566}, {7575, 19924}, {7728, 15688}, {9033, 45681}, {9144, 21166}, {9167, 67221}, {10113, 15699}, {10124, 20304}, {10168, 15118}, {10182, 44262}, {10272, 34200}, {10299, 15054}, {10304, 10706}, {10519, 41720}, {10620, 15706}, {10721, 62120}, {11005, 64019}, {11064, 47333}, {11178, 18580}, {11202, 36201}, {11645, 15122}, {11723, 28194}, {11801, 47598}, {11812, 32423}, {12017, 32114}, {12038, 33563}, {12041, 17504}, {12108, 20379}, {12244, 15710}, {12317, 61809}, {12383, 15702}, {12827, 49672}, {12828, 35486}, {12893, 38396}, {12898, 38066}, {12901, 54994}, {12902, 61864}, {13202, 15681}, {13857, 44265}, {14093, 15042}, {14094, 15717}, {14641, 65095}, {14677, 15714}, {14683, 61812}, {14915, 35266}, {15021, 61138}, {15025, 61867}, {15027, 55863}, {15029, 33703}, {15039, 61803}, {15041, 15716}, {15044, 61886}, {15046, 62040}, {15055, 15698}, {15057, 61814}, {15059, 15709}, {15061, 15701}, {15081, 61859}, {15088, 47599}, {15113, 18400}, {15303, 15462}, {15361, 51733}, {15535, 26614}, {15690, 34584}, {15695, 38789}, {15707, 24981}, {15713, 34128}, {15715, 20125}, {15720, 23236}, {16165, 43586}, {16222, 21969}, {16532, 41674}, {18281, 45286}, {21356, 32275}, {21358, 32233}, {22265, 52695}, {23583, 32162}, {25487, 44213}, {32110, 40112}, {32225, 44214}, {37909, 43576}, {38323, 43839}, {38724, 61843}, {38788, 62073}, {38790, 62088}, {40685, 61839}, {41134, 67641}, {41982, 61598}, {41983, 61548}, {44211, 59495}, {44573, 64689}, {44682, 51522}, {49268, 52046}, {49269, 52045}, {50967, 52699}, {50979, 64880}, {54131, 64764}, {54132, 64095}, {61778, 64102}, {61846, 64183}

X(68316) = midpoint of X(i) and X(j) for these (i, j): {3, 5642}, {74, 56567}, {99, 11656}, {113, 376}, {125, 64182}, {381, 16163}, {549, 1511}, {597, 33851}, {1539, 15686}, {2482, 53725}, {3524, 11693}, {5181, 11179}, {6055, 53735}, {6174, 53753}, {6593, 54169}, {7426, 10564}, {9140, 30714}, {9143, 16003}, {10272, 34200}, {10706, 16111}, {11064, 47333}, {11694, 12100}, {13202, 15681}, {13392, 14891}, {13857, 44265}, {15035, 38793}, {15303, 54173}, {16165, 44218}, {25487, 44213}, {32110, 40112}, {32609, 38727}, {36518, 38723}, {44211, 59495}, {44214, 51394}
X(68316) = reflection of X(i) in X(j) for these (i, j): (381, 12900), (549, 48378), (6699, 549), (7687, 547), (9140, 20397), (15118, 10168), (16534, 5642), (20304, 10124), (36253, 45311), (37853, 34200), (40685, 61839), (45311, 140)
X(68316) = pole of the line {9003, 32254} with respect to the circumcircle
X(68316) = pole of the line {7464, 15055} with respect to the Stammler hyperbola
X(68316) = pole of the line {14915, 44265} with respect to the Thomson-Gibert-Moses hyperbola
X(68316) = X(5642)-of-anti-X3-ABC reflections triangle
X(68316) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (631, 15020, 30714), (631, 30714, 20397), (1511, 48378, 6699), (3523, 15034, 16003), (5054, 15040, 64182), (5054, 64182, 125), (5972, 38726, 46686), (10304, 10706, 16111), (10706, 15036, 10304), (15462, 54173, 15303), (15693, 20126, 38727), (15693, 32609, 20126), (15720, 23236, 38729), (16163, 38794, 12900)

X(68317) = CENTER OF THE X(4)-CO-NORMAL HYPERBOLA OF THOMSON-GIBERT-MOSES HYPERBOLA

This co-normal hyperbola passes through centers X(n) for these n: {4, 5642, 52451}

X(68317) lies on these lines: {2, 2777}, {4, 5642}, {5, 541}, {6, 13}, {20, 15023}, {30, 5972}, {74, 5071}, {110, 3839}, {125, 3545}, {146, 61936}, {376, 13202}, {382, 38795}, {389, 65095}, {403, 32225}, {511, 47332}, {524, 37984}, {546, 16534}, {547, 6699}, {549, 1539}, {1503, 68318}, {1511, 15687}, {1514, 47097}, {1531, 7426}, {1533, 10989}, {1551, 2682}, {1561, 36194}, {2781, 61676}, {2854, 50959}, {3090, 10990}, {3091, 9140}, {3448, 61954}, {3524, 10721}, {3534, 15046}, {3543, 16163}, {3817, 50876}, {3830, 14643}, {3832, 9143}, {3843, 30714}, {3845, 17702}, {3850, 36253}, {3851, 16003}, {3855, 14094}, {3858, 5609}, {3860, 32423}, {5054, 16111}, {5055, 6723}, {5056, 38729}, {5066, 5663}, {5068, 15054}, {5095, 11180}, {5181, 54131}, {5587, 50878}, {5603, 50877}, {5640, 54037}, {5648, 53023}, {5878, 15751}, {6000, 41670}, {6054, 16278}, {6623, 54132}, {7486, 15021}, {7978, 38074}, {8703, 48375}, {8994, 42602}, {9813, 9970}, {10113, 23046}, {10250, 51023}, {10264, 61942}, {10272, 14893}, {10297, 11645}, {10620, 61933}, {10719, 14500}, {10720, 14499}, {10733, 61985}, {11064, 47310}, {11178, 32257}, {11179, 32300}, {11430, 20772}, {11693, 12121}, {11694, 12101}, {11720, 34648}, {11723, 28204}, {11737, 20304}, {11793, 16105}, {11799, 19924}, {11801, 61957}, {11897, 24930}, {12041, 15699}, {12100, 34584}, {12244, 61899}, {12295, 14269}, {12317, 61951}, {12358, 58536}, {12368, 38021}, {12383, 61980}, {12811, 20379}, {12824, 15030}, {12825, 14831}, {12902, 61971}, {13289, 54994}, {13392, 61999}, {13969, 42603}, {14448, 15058}, {14644, 18950}, {14677, 61885}, {14683, 61962}, {14892, 15088}, {14971, 53709}, {14984, 67067}, {15020, 17578}, {15022, 15057}, {15027, 61946}, {15035, 15682}, {15036, 62130}, {15040, 62020}, {15041, 61908}, {15045, 17853}, {15051, 15683}, {15059, 61924}, {15061, 61920}, {15078, 25564}, {15081, 61947}, {15113, 15311}, {15153, 63861}, {15473, 62961}, {15681, 38794}, {15694, 20127}, {15701, 38788}, {15703, 38790}, {16836, 68292}, {17855, 68293}, {18504, 64063}, {19709, 20126}, {20125, 61973}, {20396, 61940}, {20423, 64880}, {22251, 62014}, {22802, 38398}, {23236, 61970}, {24981, 61967}, {25711, 44870}, {30308, 50921}, {32223, 47334}, {32609, 61993}, {34128, 61910}, {34153, 61995}, {36196, 57431}, {36201, 51737}, {38626, 41989}, {38723, 62040}, {38724, 61950}, {38725, 61934}, {38728, 61887}, {40685, 61922}, {41737, 59373}, {43573, 67869}, {44275, 51993}, {45958, 63684}, {45979, 66758}, {47478, 61548}, {53715, 59376}, {61927, 64102}, {64014, 67890}

X(68317) = midpoint of X(i) and X(j) for these (i, j): {4, 5642}, {113, 381}, {125, 10706}, {265, 56567}, {376, 13202}, {549, 1539}, {1511, 15687}, {1514, 47097}, {1531, 7426}, {1533, 10989}, {1551, 2682}, {1561, 36194}, {3543, 16163}, {5095, 11180}, {5181, 54131}, {6033, 11656}, {6054, 16278}, {9140, 15063}, {10272, 14893}, {10719, 14500}, {10720, 14499}, {11064, 47310}, {11178, 32271}, {11693, 38335}, {11694, 12101}, {11720, 34648}, {12295, 64182}, {12824, 15030}, {12825, 14831}, {13392, 61999}, {15303, 47353}, {23515, 38789}, {36196, 57431}, {38791, 45311}, {47334, 58885}
X(68317) = reflection of X(i) in X(j) for these (i, j): (376, 48378), (549, 12900), (6699, 547), (7687, 381), (11179, 32300), (16836, 68292), (20304, 11737), (20417, 45311), (32223, 47334), (32257, 11178), (37853, 549), (40685, 61922), (45311, 5), (68280, 36518)
X(68317) = pole of the line {1648, 66266} with respect to the Hutson-Parry circle
X(68317) = pole of the line {9003, 66498} with respect to the nine-point circle
X(68317) = pole of the line {9033, 9185} with respect to the orthoptic circle of Steiner inellipse
X(68317) = pole of the line {323, 15055} with respect to the Stammler hyperbola
X(68317) = X(5642)-of-Euler triangle
X(68317) = X(45311)-of-Johnson triangle
X(68317) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (5, 38791, 20417), (113, 7687, 6053), (1539, 12900, 37853), (3545, 10706, 125), (7687, 18388, 68281), (13202, 64101, 48378), (14269, 64182, 12295), (19709, 20126, 23515), (19709, 38789, 20126), (46686, 61574, 5972)

X(68318) = CENTER OF THE X(6)-CO-NORMAL HYPERBOLA OF THOMSON-GIBERT-MOSES HYPERBOLA

This co-normal hyperbola passes through centers X(n) for these n: {6, 5642, 9177}

X(68318) lies on these lines: {2, 9769}, {5, 542}, {6, 5642}, {110, 59373}, {113, 11179}, {125, 25336}, {182, 541}, {373, 61665}, {511, 18579}, {524, 5972}, {599, 5095}, {690, 9188}, {895, 63127}, {1177, 15740}, {1503, 68317}, {1992, 5181}, {2393, 35266}, {2777, 51737}, {2781, 16836}, {2794, 50147}, {2854, 5943}, {2930, 30734}, {3018, 57618}, {3292, 47458}, {3524, 10752}, {3589, 10173}, {3618, 9140}, {3849, 16324}, {5050, 5655}, {5182, 9144}, {5465, 18800}, {5476, 17702}, {6699, 10168}, {6723, 25329}, {8584, 59553}, {8705, 32267}, {9027, 47459}, {9143, 51171}, {9970, 38064}, {10541, 10990}, {11061, 63109}, {11064, 47545}, {11178, 12900}, {11180, 64101}, {11284, 16510}, {11402, 51185}, {11579, 56567}, {11645, 46686}, {11656, 12177}, {11694, 14984}, {13352, 15462}, {13658, 13778}, {13857, 44102}, {14763, 25488}, {14848, 64182}, {15035, 54132}, {15051, 54170}, {15063, 53093}, {15116, 38398}, {15360, 22151}, {15471, 19510}, {15534, 61645}, {15751, 31166}, {16163, 54131}, {16278, 51798}, {19153, 36201}, {19924, 32217}, {20126, 45016}, {20417, 63694}, {20582, 32257}, {20583, 65430}, {21358, 64104}, {23327, 51023}, {26255, 64606}, {32225, 47455}, {32233, 38072}, {32278, 38023}, {32298, 38087}, {34155, 44493}, {35486, 50967}, {36518, 47353}, {37196, 51024}, {37907, 41583}, {38793, 54173}, {38795, 63722}, {40112, 53777}, {45237, 64692}, {46512, 50187}, {48378, 54169}, {50150, 51372}, {50959, 51744}, {51130, 51745}, {61676, 68292}

X(68318) = midpoint of X(i) and X(j) for these (i, j): {2, 15303}, {6, 5642}, {113, 11179}, {125, 34319}, {597, 6593}, {599, 5095}, {1992, 5181}, {5465, 18800}, {9140, 56565}, {10168, 25556}, {11064, 47545}, {11579, 56567}, {11656, 12177}, {16163, 54131}, {16278, 51798}, {20582, 41595}, {40112, 53777}, {50150, 51372}
X(68318) = reflection of X(i) in X(j) for these (i, j): (597, 32300), (6699, 10168), (11178, 12900), (15118, 597), (32257, 20582), (45311, 3589), (54169, 48378), (61676, 68292)
X(68318) = cross-difference of every pair of points on the line X(2780)X(39232)
X(68318) = pole of the line {187, 47097} with respect to the Kiepert circumhyperbola
X(68318) = pole of the line {23061, 41617} with respect to the Stammler hyperbola
X(68318) = pole of the line {9027, 47465} with respect to the Thomson-Gibert-Moses hyperbola
X(68318) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (2, 25321, 13169), (2, 52699, 15303), (6593, 32300, 15118), (34319, 47352, 125)

X(68319) = CENTER OF THE X(5)-CO-NORMAL HYPERBOLA OF WALSMITH RECTANGULAR HYPERBOLA

This co-normal hyperbola passes through centers X(n) for these n: {5, 468, 47208}

X(68319) lies on these lines: {2, 3}, {113, 61691}, {125, 46817}, {141, 47581}, {230, 52951}, {355, 47476}, {511, 12900}, {575, 61619}, {576, 51742}, {952, 51725}, {973, 30531}, {1352, 47455}, {1353, 47459}, {1495, 23515}, {1503, 20304}, {1514, 12041}, {2493, 43291}, {3292, 38795}, {3564, 6593}, {3580, 14643}, {5158, 47144}, {5476, 47556}, {5480, 47569}, {5523, 57319}, {5609, 51425}, {5663, 47296}, {5690, 47471}, {5886, 47321}, {5913, 57355}, {6000, 20397}, {6102, 16227}, {6722, 62490}, {6723, 14915}, {7624, 47256}, {7746, 16306}, {8705, 63475}, {9970, 62376}, {10175, 51693}, {10222, 47492}, {10516, 47453}, {10540, 15027}, {11178, 47544}, {11579, 38851}, {11704, 64036}, {11793, 58481}, {11801, 64498}, {12241, 58435}, {13374, 58639}, {14561, 32113}, {14881, 47568}, {14984, 35370}, {15025, 25739}, {15034, 50435}, {15061, 32111}, {15081, 35265}, {15088, 15448}, {15362, 40112}, {15582, 61612}, {16309, 52200}, {16310, 33505}, {16511, 25488}, {16625, 58551}, {16659, 45622}, {16760, 67872}, {16776, 47449}, {18358, 47454}, {18809, 34109}, {18883, 34209}, {19918, 32204}, {21850, 47468}, {23514, 47326}, {32110, 36518}, {32223, 68280}, {32269, 51391}, {32275, 44102}, {32423, 62516}, {34128, 51548}, {34380, 47558}, {34507, 47549}, {38022, 47593}, {38724, 46818}, {38728, 50434}, {38729, 51403}, {38791, 44673}, {38796, 47325}, {39663, 46634}, {40330, 52238}, {43588, 61608}, {45019, 61598}, {47277, 59399}, {47322, 61315}, {47323, 57306}, {47324, 57305}, {47458, 63722}, {47474, 48906}, {47496, 51709}, {47552, 64067}, {47571, 48876}, {47579, 49111}, {54215, 64764}, {61591, 62509}, {61606, 64061}

X(68319) = midpoint of X(i) and X(j) for these (i, j): {2, 47334}, {3, 47336}, {4, 47335}, {5, 468}, {23, 47341}, {125, 46817}, {140, 44961}, {141, 47581}, {186, 23323}, {355, 47476}, {381, 18579}, {403, 44452}, {546, 18571}, {549, 47332}, {550, 47309}, {858, 16619}, {1514, 12041}, {3627, 47308}, {3845, 47333}, {3850, 22249}, {5476, 47556}, {5480, 47569}, {5690, 47471}, {7574, 47342}, {7575, 10297}, {8703, 47310}, {10011, 64966}, {10151, 15646}, {10222, 47492}, {10257, 11563}, {11178, 47544}, {11793, 58481}, {11799, 15122}, {13374, 58639}, {14120, 37459}, {14881, 47568}, {15687, 47031}, {16309, 52200}, {16760, 67872}, {19918, 32204}, {21850, 47468}, {32110, 58885}, {32269, 51391}, {34209, 47148}, {34507, 47549}, {37935, 63838}, {37938, 37971}, {37942, 44911}, {37967, 46517}, {43893, 47090}, {44234, 46031}, {44266, 47097}, {47474, 48906}, {47496, 51709}, {47552, 64067}, {47571, 48876}, {47579, 49111}, {51425, 63839}
X(68319) = reflection of X(i) in X(j) for these (i, j): (140, 37911), (5159, 3628), (12105, 47316), (16531, 44234), (37934, 22249), (37935, 44900), (44911, 15350)
X(68319) = complement of X(15122)
X(68319) = intersection, other than {A, B, C}, of the circumconics through X(i), X(j) for these {i, j}: {250, 51519}, {7519, 60590}
X(68319) = inverse of X(3830) in 2nd Droz-Farny circle
X(68319) = inverse of X(7519) in orthoptic circle of Steiner inellipse
X(68319) = inverse of X(51519) in circumcircle
X(68319) = pole of the line {523, 51519} with respect to the circumcircle
X(68319) = pole of the line {523, 3830} with respect to the 2nd Droz-Farny circle
X(68319) = pole of the line {44467, 53419} with respect to the Moses-Parry circle
X(68319) = pole of the line {523, 7519} with respect to the orthoptic circle of Steiner inellipse
X(68319) = pole of the line {6, 15061} with respect to the Kiepert circumhyperbola
X(68319) = pole of the line {525, 37644} with respect to the Steiner inellipse
X(68319) = pole of the line {5650, 21649} with respect to the Thomson-Gibert-Moses hyperbola
X(68319) = X(15122)-of-medial triangle
X(68319) = X(47335)-of-Euler triangle
X(68319) = X(47336)-of-anti-X3-ABC reflections triangle
X(68319) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (3, 403, 47336), (32110, 36518, 58885), (38795, 63735, 3292), (47492, 51713, 10222)

X(68320) = CENTER OF THE X(1)-CO-NORMAL HYPERBOLA OF YFF HYPERBOLA

This co-normal hyperbola passes through centers X(n) for these n: {1, 30, 381, 523, 18115}

X(68320) lies on these lines: {30, 13464}, {381, 47274}, {517, 36155}, {523, 9955}, {946, 52200}, {3109, 51709}, {3656, 36154}, {5603, 36171}, {5901, 62496}, {13869, 18480}, {18493, 47270}, {38021, 47273}, {40273, 62493}, {42422, 63257}

X(68320) = midpoint of X(i) and X(j) for these (i, j): {946, 52200}, {13869, 18480}

X(68321) = CENTER OF THE X(6)-CO-NORMAL HYPERBOLA OF YFF HYPERBOLA

This co-normal hyperbola passes through centers X(n) for these n: {6, 30, 381, 523, 18114}

X(68321) lies on these lines: {30, 575}, {51, 47348}, {262, 16188}, {373, 67611}, {511, 11007}, {523, 19130}, {858, 44422}, {1316, 5476}, {1995, 55308}, {2452, 3818}, {3258, 5640}, {5169, 6070}, {5480, 62490}, {5946, 16340}, {6795, 48901}, {7533, 14480}, {7684, 58913}, {7685, 58912}, {7706, 64510}, {14389, 47327}, {14537, 15860}, {14853, 36181}, {15019, 17511}, {18114, 57603}, {18388, 36169}, {18583, 62509}, {19924, 50147}, {20423, 36163}, {21849, 36190}, {22104, 61743}, {25555, 36177}, {30499, 56925}, {31861, 32417}, {38072, 47284}, {38613, 66096}, {42785, 47285}, {51451, 65093}, {57583, 67406}, {57592, 58470}

X(68321) = midpoint of X(i) and X(j) for these (i, j): {2452, 3818}, {5476, 16279}, {6795, 48901}
X(68321) = reflection of X(36177) in X(25555)
X(68321) = pole of the line {38581, 38583} with respect to the Yff hyperbola

X(68322) = CENTER OF THE X(2)-CO-NORMAL HYPERBOLA OF YFF PARABOLA

This co-normal hyperbola passes through centers X(n) for these n: {2, 514, 516, 3008, 50573, 56746, 62705}

X(68322) lies on these lines: {2, 53801}, {101, 514}, {516, 50895}, {23890, 60487}

X(68322) = X(13170)-of-1st circumperp triangle
X(68322) = (X(68323), X(68324))-harmonic conjugate of X(927)

X(68323) = CENTER OF THE X(3)-CO-NORMAL HYPERBOLA OF YFF PARABOLA

This co-normal hyperbola passes through centers X(n) for these n: {3, 514, 516, 3730, 20367}

X(68323) lies on these lines: {101, 514}, {516, 10739}, {1566, 67212}, {2140, 67726}, {14732, 61730}, {20328, 40554}, {31851, 33331}

X(68323) = midpoint of X(927) and X(68324)
X(68323) = reflection of X(31851) in X(33331)
X(68323) = pole of the line {516, 38572} with respect to the Yff parabola
X(68323) = X(68271)-of-excentral triangle
X(68323) = (X(927), X(68322))-harmonic conjugate of X(68324)

X(68324) = CENTER OF THE X(4)-CO-NORMAL HYPERBOLA OF YFF PARABOLA

This co-normal hyperbola passes through centers X(n) for these n: {4, 514, 516, 63851}

X(68324) lies on these lines: {101, 514}, {220, 18329}, {265, 5134}, {516, 10725}, {526, 66274}, {655, 65680}, {919, 21132}, {929, 2702}, {1566, 31841}, {2690, 35182}, {2724, 24047}, {4253, 59808}, {5011, 12515}, {10695, 61436}, {14732, 56746}, {24045, 33331}, {40554, 67625}, {48900, 67726}, {49304, 52334}

X(68324) = reflection of X(i) in X(j) for these (i, j): (101, 34805), (927, 68323), (2724, 31852), (10695, 61436), (67568, 33331)
X(68324) = pole of the line {516, 63416} with respect to the Yff parabola
X(68324) = X(64687)-of-1st circumperp triangle
X(68324) = (X(927), X(68322))-harmonic conjugate of X(68323)

X(68325) = X(2)X(187)∩X(4)X(7622)

See Benjamin Lee Warren, Francisco Javier García Capitán and Ercole Suppa, euclid 8286.

X(68325) lies on these lines: {2, 187}, {4, 7622}, {5, 7615}, {115, 63025}, {376, 7619}, {381, 7618}, {524, 5055}, {538, 61924}, {543, 3545}, {547, 7610}, {671, 63083}, {754, 61899}, {1506, 32984}, {1656, 15597}, {2482, 34803}, {3055, 5077}, {3090, 7775}, {3091, 53142}, {3363, 18584}, {3767, 63028}, {3832, 34504}, {3839, 32479}, {3845, 63647}, {5066, 12040}, {5067, 34506}, {5068, 47617}, {5071, 7617}, {5079, 7758}, {5461, 7736}, {5485, 60192}, {5611, 22491}, {5615, 22492}, {7486, 63930}, {7620, 61936}, {7739, 9166}, {7753, 63107}, {7759, 61914}, {7764, 61921}, {7817, 31404}, {7827, 32963}, {7840, 53127}, {7843, 46936}, {7870, 32962}, {7883, 32999}, {7891, 33013}, {8355, 42849}, {8667, 61910}, {8716, 11737}, {9167, 14033}, {9740, 61912}, {9741, 18546}, {9761, 18582}, {9763, 18581}, {9766, 10109}, {10554, 63036}, {11054, 64809}, {11163, 43620}, {11165, 19709}, {11317, 37647}, {11485, 33475}, {11486, 33474}, {13468, 61908}, {13663, 61389}, {13783, 61388}, {14971, 59373}, {15484, 44401}, {15699, 63945}, {19130, 64942}, {22489, 36758}, {22490, 36757}, {31401, 33006}, {31417, 32967}, {31489, 37350}, {32833, 32994}, {33016, 41134}, {33017, 55801}, {33240, 48310}, {35287, 39590}, {39601, 63077}, {40727, 61920}, {41133, 44543}, {44904, 63933}, {46935, 63935}, {50571, 66391}, {51122, 61929}, {51123, 61934}, {53141, 61944}, {53144, 61928}, {55823, 61888}, {59546, 61935}, {60781, 63931}, {61887, 63941}, {61894, 63938}, {61903, 63928}, {61905, 63950}, {61906, 63942}, {61907, 63932}, {61909, 63940}

X(68325) = midpoint of X(8176) and X(67292)
X(68325) = reflection of X(i) in X(j) for these (i, j): (2, 67292), (67292, 66511)
X(68325) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (2, 5475, 37809), (2, 8176, 66466), (2, 23334, 5569), (5, 11184, 7615), (381, 9771, 7618), (1153, 63956, 66699), (3845, 63647, 66616), (5066, 12040, 66587), (5071, 9770, 7617), (5569, 23334, 47102), (5569, 47102, 8182), (7615, 11184, 34511), (7617, 9770, 63955), (11165, 19709, 20112)

   Points on Warren Q-circles: X(68326)-X(68333)  

Contributed by Clark Kimberling, April 18, 2025. This preamble is based on notes from Benjamin Warren, April 10, 2025. The points X(68326)-X(68333) and list of Q-circles were contributed by Peter Moses, April 18, 2025.

In the plane of a triangle ABC, let P be a point. Let
A'B'C' = circumcevian triangle of P
Ma = midpoint of segment BC
A'' = reflection of A' in Ma, and define B'' and C'' cyclically
P' = anticomplement of P.
The circle {{A'',B'',C''}}, here introduced as the Warren Q-circle, has diameter P'X(4). See the note at X(4).

The appearance of (n, (name), {i(1), i(2),..., i(k)}, m) in the following list means that if X(n) = Q, then the points X(i(1)), X(i(2)),..., X(i(k)) lie on the circle, which has center X(m).

(2, (orthocentroidal circle), {2, 4, 6235, 6324, 6785, 6787, 6788, 6792, 6794, 8426, 8427, 9144, 10773, 11005, 13522, 13524, 13531, 14700, 15924, 22540, 31862, 31863, 34235, 61729, 61730, 61731, 61732, 67224}, 381)

(3, {3, 4, 15098, 18338, 18341, 18342, 18347, 18348, 31847, 31848, 31849, 31850, 31851, 31852, 31853, 31854, 31864, 31865, 31866, 43389, 43395, 43396, 53716, 53727, 67225, 67226, 67227, 67228, 67229, 67230, 67231, 67232, 67233, 67234}, 5)

(6, (orthosymmedial circle), {4, 6, 1316, 6792, 12508, 13239, 23322, 31850, 52465, 52466, 52471, 67382, 67478}, 5480)

(8, (Fuhrmann circle), {4, 8, 6788, 10774, 13498, 13514, 13543, 13545, 13547, 13549, 18328, 18339, 18340, 18341, 18343, 18865, 36154}, 355)

(20, {4, 20, 18337, 18339, 32616, 32617, 67464, 67568, 67662, 67721}, 3)

(69, {4, 69, 6792, 18331, 18335, 18337, 18343, 18347, 35902, 36163}, 1352)

(99, {4, 99, 112, 7472, 12833, 15342, 18331, 46046, 67224, 67232, 67667}, 114)

(100, {4, 100, 108, 10773, 15343, 18341, 34151, 36167, 46044, 67474}, 119)

(149, {4, 149, 10767, 10768, 10769, 10770, 10771, 10772, 10773, 10774, 10775, 10776, 10777, 10778, 10779, 10780, 10781, 10782, 31512, 36175, 67477}, 10738)

(2394, {4, 13, 14, 2132, 2394, 6794, 22265, 34298, 46341, 57472}, 42733)

(6334, {4, 113, 115, 6334, 10297, 15341, 15738, 31854, 44933, 44934, 46341, 52475, 64628}, 44921)

(9979, {4, 107, 111, 671, 5523, 7426, 9979, 20410, 24007, 24008, 41125, 45237, 46339, 52125}, 44203)

(14618, {4, 403, 5523, 5962, 6116, 6117, 6761, 14618, 35718, 50718}, 16229)

(14977, {2, 4, 111, 935, 11188, 14833, 14977, 35902, 50718, 61494}, 68326)

(15412, {4, 15, 16, 186, 3484, 11674, 13509, 15412, 47064, 49124, 62341, 66134, 66135, 66136, 66137, 66138, 66139, 66140}, 15451)

(16230, {4, 107, 132, 136, 468, 1112, 3563, 12131, 16230, 52476}, 68327)

(18312, {4, 5, 115, 3818, 18312, 19163, 32274, 34235, 42426, 50718, 66171, 67670}, 68328)

(21222, {4, 1320, 5011, 10697, 21222, 38670, 38674, 51896, 64234, 64446, 64616}, 68329)

(24978, {4, 1263, 2079, 5523, 7575, 10214, 12236, 24978, 38734, 65500}}, 68330)

(35522, {4, 67, 115, 858, 1560, 13219, 14360, 14981, 14982, 35522}}, 68331)

(44427, {4, 112, 1300, 1986, 5523, 5667, 6110, 6111, 10295, 44427, 53769, 53772}, 16230)

(50333, {4,11, 858, 15343, 16870, 17615, 20344, 20621, 34188, 37725, 50333}, 68332)

(50351, {4, 10, 72, 1083, 3109, 6790, 11607, 14887, 50351, 56951}, 68333)

(53345, {4, 23, 98, 107, 2592, 2593, 3448, 12384, 34239, 34240, 38664, 38672, 41377, 51939, 52076, 53345, 53769}, 41079)

X(68326) = X(3)X(18310)∩X(4)X(14977)

X(68326) lies on these lines: {3, 18310}, {4, 14977}, {5, 18311}, {351, 2793}, {381, 523}, {382, 44206}, {525, 1352}, {542, 1640}, {1499, 23288}, {2395, 15928}, {2780, 14277}, {2799, 9880}, {5512, 14672}, {10278, 21732}, {11179, 45327}, {11180, 53374}, {11615, 11619}, {11622, 14272}, {15451, 33752}, {16188, 18312}, {16229, 39530}, {18440, 45801}, {23878, 65754}, {29959, 30209}, {34952, 44823}, {39491, 64919}, {53378, 54132}, {62384, 66167}

X(68326) = midpoint of X(i) and X(j) for these {i,j}: {4, 14977}, {382, 44206}, {11180, 53374}, {53378, 54132}
X(68326) = reflection of X(i) in X(j) for these {i,j}: {3, 18310}, {11179, 45327}, {14272, 11622}, {18311, 5}, {21732, 10278}, {65723, 18312}
X(68326) = X(i)-Dao conjugate of X(j) for these (i,j): {5512, 842}, {23967, 65324}, {42426, 30247}, {65728, 5486}
X(68326) = barycentric product X(i)*X(j) for these {i,j}: {1640, 11185}, {1995, 18312}, {16092, 55135}, {30209, 60502}, {44203, 51227}
X(68326) = barycentric quotient X(i)/X(j) for these {i,j}: {542, 65324}, {1640, 5486}, {1995, 5649}, {6103, 30247}, {11185, 6035}, {30209, 65308}, {44203, 51228}, {55135, 52094}

X(68327) = X(4)X(690)∩X(460)X(512)

X(68327) lies on these lines: {4, 690}, {24, 39477}, {25, 14270}, {30, 65758}, {107, 53972}, {113, 133}, {115, 2971}, {235, 39509}, {381, 6334}, {460, 512}, {523, 16231}, {525, 42399}, {546, 44921}, {648, 67489}, {826, 16229}, {1495, 65615}, {1593, 44826}, {1595, 53567}, {1596, 66498}, {1597, 53247}, {1598, 53263}, {1637, 9409}, {2491, 3199}, {2799, 67862}, {2848, 6130}, {4232, 9189}, {4240, 64607}, {5099, 66939}, {6089, 39534}, {6140, 47230}, {6529, 23977}, {6531, 8753}, {7687, 55121}, {7927, 14618}, {9134, 32121}, {9185, 52301}, {10151, 52475}, {12077, 67534}, {15451, 55280}, {15475, 18384}, {17983, 18007}, {23347, 41392}, {32478, 57065}, {36898, 48982}, {41077, 65754}, {42736, 47204}, {44564, 44818}, {45687, 66163}, {45688, 47206}, {53386, 65784}, {53563, 65128}, {54659, 60338}

X(68327) = midpoint of X(i) and X(j) for these {i,j}: {4, 16230}, {1637, 58346}, {12077, 67534}, {16229, 59932}, {44705, 58757}
X(68327) = reflection of X(i) in X(j) for these {i,j}: {44921, 546}, {51513, 58757}
X(68327) = polar circle inverse of X(18331)
X(68327) = polar conjugate of the isotomic conjugate of X(1637)
X(68327) = polar conjugate of the isogonal conjugate of X(14398)
X(68327) = X(i)-Ceva conjugate of X(j) for these (i,j): {523, 55276}, {4240, 1990}, {15475, 51513}, {18384, 8754}, {18808, 2501}, {52485, 65755}
X(68327) = X(14398)-cross conjugate of X(1637)
X(68327) = X(i)-isoconjugate of X(j) for these (i,j): {63, 44769}, {69, 36034}, {74, 4592}, {99, 35200}, {255, 16077}, {304, 32640}, {326, 1304}, {394, 65263}, {662, 14919}, {799, 18877}, {1101, 34767}, {1102, 32695}, {1494, 4575}, {2159, 4563}, {2315, 55264}, {2349, 4558}, {2433, 62719}, {3926, 36131}, {6507, 15459}, {14380, 24041}, {32661, 33805}, {36831, 62277}, {40352, 55202}
X(68327) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 99}, {136, 1494}, {523, 34767}, {1084, 14919}, {1637, 45792}, {3005, 14380}, {3162, 44769}, {3163, 4563}, {3258, 69}, {5139, 74}, {6523, 16077}, {14401, 4143}, {15259, 1304}, {16178, 65715}, {38986, 35200}, {38996, 18877}, {38999, 3964}, {39008, 3926}, {48317, 36890}, {53989, 46751}, {57295, 3265}, {62598, 305}, {62613, 47389}, {63463, 44715}, {65757, 52617}, {65763, 6333}
X(68327) = crosspoint of X(i) and X(j) for these (i,j): {1990, 4240}, {2501, 18808}
X(68327) = crosssum of X(i) and X(j) for these (i,j): {3, 8552}, {3265, 62338}, {14380, 14919}, {51394, 52613}
X(68327) = crossdifference of every pair of points on line {394, 4558}
X(68327) = barycentric product X(i)*X(j) for these {i,j}: {4, 1637}, {19, 36035}, {25, 41079}, {30, 2501}, {112, 58261}, {115, 4240}, {158, 2631}, {225, 14400}, {264, 14398}, {338, 23347}, {393, 9033}, {403, 65615}, {460, 65758}, {512, 46106}, {523, 1990}, {647, 52661}, {661, 1784}, {685, 65755}, {847, 14397}, {850, 14581}, {1093, 1636}, {1109, 56829}, {1294, 55276}, {1300, 55265}, {1495, 14618}, {1568, 15422}, {1650, 6529}, {1826, 11125}, {1989, 62172}, {2052, 9409}, {2173, 24006}, {2207, 66073}, {2394, 16240}, {2395, 67406}, {2407, 8754}, {2420, 2970}, {2433, 34334}, {2489, 3260}, {2643, 24001}, {2682, 65350}, {3163, 18808}, {3269, 58071}, {3284, 66299}, {4024, 52954}, {4036, 52955}, {5664, 18384}, {6110, 20578}, {6111, 20579}, {6344, 52743}, {6524, 41077}, {6526, 14345}, {6531, 65754}, {8749, 58263}, {8753, 66122}, {8884, 14391}, {9214, 14273}, {10412, 39176}, {11064, 58757}, {14254, 47230}, {14399, 41013}, {14583, 44427}, {14920, 15475}, {15454, 47236}, {16080, 58346}, {16230, 35906}, {17994, 60869}, {23977, 65759}, {32646, 57424}, {32713, 65753}, {34854, 65778}, {35235, 41392}, {43752, 55219}, {43768, 51513}, {47228, 53178}, {51389, 53149}, {51431, 60338}, {51965, 55121}, {52945, 66300}, {52949, 66297}, {52951, 66943}, {52956, 66287}, {60428, 66124}, {62519, 67405}
X(68327) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 44769}, {30, 4563}, {115, 34767}, {393, 16077}, {512, 14919}, {669, 18877}, {798, 35200}, {1096, 65263}, {1300, 55264}, {1495, 4558}, {1636, 3964}, {1637, 69}, {1650, 4143}, {1784, 799}, {1973, 36034}, {1974, 32640}, {1990, 99}, {2173, 4592}, {2207, 1304}, {2407, 47389}, {2489, 74}, {2501, 1494}, {2631, 326}, {2682, 14417}, {2971, 2433}, {3124, 14380}, {3199, 36831}, {3258, 45792}, {3260, 52608}, {4240, 4590}, {6524, 15459}, {6529, 42308}, {8754, 2394}, {9033, 3926}, {9406, 4575}, {9407, 32661}, {9409, 394}, {11125, 17206}, {14206, 55202}, {14273, 36890}, {14391, 52347}, {14397, 9723}, {14398, 3}, {14399, 1444}, {14400, 332}, {14581, 110}, {14583, 60053}, {16240, 2407}, {17994, 35910}, {18384, 39290}, {18808, 31621}, {20975, 62665}, {23347, 249}, {24001, 24037}, {24006, 33805}, {35906, 17932}, {36035, 304}, {36417, 32715}, {39176, 10411}, {41077, 4176}, {41079, 305}, {43752, 55218}, {44203, 66767}, {46106, 670}, {47236, 65715}, {51513, 62722}, {51965, 18878}, {52439, 32695}, {52661, 6331}, {52743, 52437}, {52954, 4610}, {52955, 52935}, {55206, 44693}, {55219, 44715}, {55265, 62338}, {56829, 24041}, {57204, 40352}, {58261, 3267}, {58344, 3284}, {58346, 11064}, {58757, 16080}, {62172, 7799}, {65478, 56576}, {65615, 57829}, {65753, 52617}, {65754, 6393}, {65755, 6333}, {65758, 57872}, {67406, 2396}

X(68328) = X(2)X(47442)∩X(4)X(18312)

X(68328) lies on these lines: {2, 47442}, {4, 18312}, {381, 23878}, {512, 48889}, {520, 18553}, {523, 546}, {525, 67865}, {647, 5169}, {804, 11620}, {850, 7533}, {868, 62688}, {1995, 30476}, {2485, 39565}, {2799, 39491}, {3091, 62307}, {3448, 58900}, {3566, 61542}, {3850, 47256}, {5133, 47258}, {6248, 42733}, {13595, 47264}, {14002, 47255}, {16229, 39530}, {29012, 40550}, {31861, 64788}, {32472, 44823}, {44560, 66376}, {46990, 67237}, {47004, 67223}, {59742, 66529}, {62937, 66122}

X(68328) = midpoint of X(4) and X(18312)
X(68328) = reflection of X(11620) in X(39509)

X(68329) = X(3)X(2814)∩X(4)X(21222)

X(68329) lies on these lines: {3, 2814}, {4, 21222}, {5, 3762}, {514, 39212}, {517, 2254}, {764, 2826}, {812, 24833}, {900, 64138}, {918, 24828}, {1385, 14413}, {1482, 3887}, {1491, 28537}, {2530, 28473}, {2815, 53527}, {2827, 12773}, {2832, 38324}, {3309, 3777}, {3654, 45328}, {3716, 5886}, {3738, 6265}, {4301, 23795}, {4895, 10222}, {5398, 22384}, {5603, 53343}, {5818, 53364}, {8648, 26286}, {9956, 14430}, {10679, 53278}, {10680, 53286}, {14419, 44805}, {20430, 64862}, {22765, 52726}, {23057, 33179}, {23141, 36754}, {23800, 32475}, {25380, 26446}, {30212, 48281}, {44824, 47893}

X(68329) = midpoint of X(i) and X(j) for these {i,j}: {4, 21222}, {4301, 23795}
X(68329) = reflection of X(i) in X(j) for these {i,j}: {3, 3960}, {3654, 45328}, {3762, 5}, {4895, 10222}

X(68330) = X(4)X(24978)∩X(30)X(45259)

X(68330) lies on these lines: {4, 24978}, {30, 45259}, {382, 42731}, {523, 19918}, {525, 16229}, {1637, 44810}, {3091, 41078}, {6130, 44204}, {11801, 45147}, {33294, 65467}, {39481, 59937}, {41079, 65871}

X(68330) = midpoint of X(4) and X(24978)
X(68330) = reflection of X(39481) in X(59937)

X(68331) = X(2)X(44820)∩X(4)X(35522)

X(68331) lies on these lines: {2, 44820}, {4, 35522}, {5, 2492}, {30, 44813}, {114, 804}, {523, 6334}, {526, 1352}, {690, 18313}, {1503, 24284}, {2780, 18309}, {2793, 14279}, {2799, 39491}, {2881, 10749}, {3569, 10516}, {5480, 9035}, {5926, 16235}, {6088, 10748}, {6134, 65390}, {9148, 9775}, {9517, 18312}, {13449, 20403}, {14417, 47627}, {33752, 64920}, {44918, 59843}, {44932, 59900}, {47354, 59775}, {51212, 53369}, {53345, 66161}

X(68331) = midpoint of X(4) and X(35522)
X(68331) = reflection of X(i) in X(j) for these {i,j}: {2492, 5}, {59843, 44918}, {59900, 44932}, {65390, 6134}
X(68331) = anticomplement of X(44820)

X(68332) = X(2)X(44819)∩X(4)X(9521)

X(68332) lies on these lines: {2, 44819}, {3, 53573}, {4, 9521}, {5, 676}, {119, 900}, {355, 6366}, {517, 4528}, {523, 6334}, {928, 5777}, {2804, 44929}, {2826, 38757}, {2827, 14285}, {3091, 47695}, {3309, 14321}, {3817, 48286}, {3904, 59387}, {5055, 45318}, {5587, 10015}, {5720, 53285}, {6084, 10743}, {6087, 10746}, {7330, 53300}, {14425, 44805}, {14872, 53550}, {19925, 23887}, {28217, 59899}, {29278, 39212}, {53532, 60005}, {58679, 65840}

X(68332) = midpoint of X(i) and X(j) for these {i,j}: {4, 50333}, {14872, 53550}
X(68332) = reflection of X(i) in X(j) for these {i,j}: {3, 53573}, {676, 5}, {65840, 58679}
X(68332) = anticomplement of X(44819)

X(68333) = X(4)X(50351)∩X(512)X(5887)

X(68333) lies on these lines: on lines {4, 50351}, {512, 5887}, {513, 5777}, {514, 19925}, {520, 63707}, {523, 946}, {826, 39212}, {900, 62434}, {944, 30580}, {2814, 48056}, {2821, 50333}, {3091, 49303}, {3309, 31837}, {3667, 12512}, {3678, 6003}, {5691, 62634}, {6684, 28602}, {7178, 17606}, {18908, 48047}, {37829, 44729}, {50349, 66106}

X(68333) = midpoint of X(4) and X(50351)

Antiproducts: X(68334)-X(68355)

X(68334) = ANTIPRODUCT OF X(1) AND X(20)

X(68334) lies on these lines: {2, 610}, {4, 7}, {8, 2893}, {9, 5232}, {20, 307}, {69, 189}, {72, 32099}, {75, 5175}, {77, 1490}, {84, 7013}, {144, 8804}, {150, 2823}, {226, 1419}, {269, 1750}, {279, 63998}, {347, 515}, {390, 66685}, {440, 5273}, {452, 4357}, {497, 58906}, {651, 5776}, {857, 27382}, {944, 41007}, {950, 3672}, {962, 12324}, {1440, 66090}, {1441, 59387}, {1442, 18446}, {1713, 27624}, {1864, 24471}, {1901, 4644}, {2184, 54111}, {2293, 44431}, {3146, 18655}, {3160, 6356}, {3419, 32087}, {3486, 41003}, {3487, 18631}, {3586, 3663}, {3664, 9612}, {3668, 5691}, {4293, 53596}, {4313, 13442}, {4341, 63988}, {5177, 10436}, {5222, 5802}, {5226, 5736}, {5296, 37169}, {5435, 5740}, {5744, 51414}, {5749, 37445}, {5758, 11411}, {5811, 11487}, {5819, 25964}, {5933, 33867}, {7319, 17895}, {8048, 58009}, {8232, 64701}, {9436, 50696}, {9776, 19802}, {9778, 20291}, {9812, 17220}, {10445, 12848}, {10446, 52673}, {10591, 24179}, {12528, 52385}, {12572, 17272}, {12664, 62402}, {13577, 30501}, {15936, 18633}, {15982, 30547}, {17481, 21221}, {18230, 30810}, {18634, 24604}, {20061, 48381}, {20305, 24683}, {22464, 64261}, {24162, 28080}, {24682, 26130}, {30809, 59681}, {32064, 68352}, {36844, 36845}, {37421, 64700}, {41874, 64584}, {53997, 67267}, {62787, 67048}

X(68334) = reflection of X(i) in X(j) for these {i,j}: {347, 41010}
X(68334) = inverse of X(38948) in anticomplementary circle
X(68334) = anticomplement of X(610)
X(68334) = X(i)-Ceva conjugate of X(j) for these {i, j}: {57921, 2}
X(68334) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {2, 6225}, {3, 68006}, {4, 14361}, {6, 17037}, {64, 2}, {253, 69}, {275, 57517}, {459, 4}, {1073, 20}, {1301, 525}, {2052, 59424}, {2155, 192}, {2184, 8}, {3345, 63877}, {3346, 1032}, {5896, 11064}, {5931, 20245}, {6526, 6515}, {8809, 7}, {13157, 2888}, {14379, 46717}, {14642, 3164}, {15384, 648}, {15394, 6527}, {16080, 51892}, {19611, 4329}, {19614, 6360}, {30457, 144}, {33581, 194}, {34403, 1370}, {36079, 4025}, {41088, 20211}, {41489, 193}, {41530, 315}, {44326, 512}, {44692, 329}, {46639, 523}, {46968, 41077}, {52158, 63}, {52559, 253}, {52581, 11442}, {53012, 3151}, {53639, 850}, {53886, 3265}, {56235, 513}, {57414, 394}, {57780, 68347}, {57921, 6327}, {58759, 3448}, {59077, 20580}, {60803, 9965}, {61349, 6392}, {64987, 12324}, {65181, 520}, {65224, 7253}, {65374, 6332}, {65574, 2895}, {66492, 67092}, {67118, 5596}, {67119, 32354}
X(68334) = pole of line {3900, 17896} with respect to the anticomplementary circle
X(68334) = pole of line {3900, 54247} with respect to the polar circle
X(68334) = pole of line {4397, 14208} with respect to the Steiner circumellipse
X(68334) = pole of line {1792, 1817} with respect to the Wallace hyperbola
X(68334) = pole of line {64885, 68108} with respect to the dual conic of polar circle
X(68334) = pole of line {3361, 3668} with respect to the dual conic of Yff parabola
X(68334) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(7), X(44189)}}, {{A, B, C, X(69), X(14256)}}, {{A, B, C, X(189), X(1119)}}, {{A, B, C, X(273), X(34404)}}, {{A, B, C, X(309), X(1847)}}, {{A, B, C, X(312), X(342)}}, {{A, B, C, X(1426), X(1903)}}, {{A, B, C, X(5932), X(52392)}}, {{A, B, C, X(18623), X(35510)}}, {{A, B, C, X(28786), X(59608)}}, {{A, B, C, X(34408), X(56084)}}
X(68334) = barycentric product X(i)*X(j) for these (i, j): {18678, 69}
X(68334) = barycentric quotient X(i)/X(j) for these (i, j): {18678, 4}
X(68334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 41004, 7}, {7, 5932, 14256}, {515, 41010, 347}, {3146, 68349, 18655}, {4329, 21270, 8}

X(68335) = ANTIPRODUCT OF X(3) AND X(7)

X(68335) lies on these lines: {2, 222}, {4, 5906}, {7, 11433}, {8, 6001}, {20, 63436}, {33, 64875}, {63, 573}, {69, 189}, {72, 52366}, {92, 1947}, {144, 2895}, {152, 2807}, {193, 62798}, {253, 55114}, {255, 27379}, {306, 56545}, {320, 20921}, {343, 28739}, {347, 20211}, {377, 67901}, {394, 27540}, {406, 3157}, {475, 8757}, {497, 37516}, {527, 20223}, {908, 21621}, {914, 17776}, {971, 52365}, {1211, 55406}, {1330, 3436}, {1407, 26005}, {1419, 18652}, {1654, 26053}, {1905, 3868}, {1935, 24538}, {3219, 6350}, {3562, 4194}, {3869, 44662}, {4419, 18662}, {5342, 55109}, {5744, 14555}, {5748, 18141}, {5779, 65684}, {5781, 45802}, {6180, 13567}, {6508, 24316}, {7017, 54451}, {7046, 61185}, {7078, 27505}, {8679, 36844}, {9370, 20306}, {9776, 18928}, {10327, 17615}, {11411, 41013}, {11415, 20220}, {11678, 33078}, {12534, 64527}, {13386, 31552}, {13387, 31551}, {14213, 34932}, {14544, 63965}, {14557, 64122}, {17257, 62857}, {17294, 17732}, {17347, 54107}, {17483, 37644}, {17484, 45794}, {17778, 26119}, {17862, 53994}, {17950, 18663}, {18909, 23661}, {18915, 24537}, {19811, 20348}, {20995, 28122}, {21270, 32001}, {22117, 33305}, {23528, 63962}, {26118, 26892}, {26668, 56456}, {27184, 63070}, {27539, 63068}, {30807, 32859}, {31143, 55913}, {31600, 34052}, {32782, 55910}, {32858, 60935}, {32911, 55905}, {36991, 68352}, {36996, 68345}, {40263, 56876}, {40880, 64082}, {50442, 65045}, {55399, 63009}, {55907, 63037}, {56869, 64988}, {56875, 64048}, {59491, 63003}

X(68335) = reflection of X(i) in X(j) for these {i,j}: {20, 63436}, {222, 41883}, {3868, 1905}, {36850, 5928}
X(68335) = isotomic conjugate of X(54451)
X(68335) = anticomplement of X(222)
X(68335) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 42464}, {31, 54451}
X(68335) = X(i)-Dao conjugate of X(j) for these {i, j}: {2, 54451}, {9, 42464}, {222, 222}
X(68335) = X(i)-Ceva conjugate of X(j) for these {i, j}: {7017, 2}
X(68335) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {1, 347}, {2, 52365}, {4, 7}, {8, 4329}, {9, 20}, {10, 2897}, {19, 145}, {21, 17134}, {25, 3210}, {27, 3873}, {28, 3875}, {29, 75}, {33, 2}, {34, 4452}, {37, 3152}, {41, 3164}, {42, 18667}, {55, 6360}, {75, 68351}, {78, 6527}, {84, 55119}, {92, 3434}, {101, 66520}, {108, 4025}, {158, 56927}, {162, 4467}, {200, 56943}, {210, 3151}, {212, 46717}, {264, 21285}, {270, 4360}, {273, 6604}, {275, 68345}, {278, 36845}, {281, 8}, {282, 280}, {284, 20222}, {286, 20244}, {294, 3100}, {312, 1370}, {314, 18659}, {318, 69}, {333, 20243}, {342, 56872}, {346, 52366}, {393, 12649}, {459, 68352}, {461, 41915}, {607, 192}, {608, 17480}, {643, 6563}, {644, 20294}, {651, 59926}, {653, 3900}, {765, 68339}, {811, 4374}, {1039, 3672}, {1041, 279}, {1096, 30699}, {1098, 68340}, {1172, 1}, {1320, 3007}, {1334, 18666}, {1395, 46716}, {1783, 522}, {1785, 36918}, {1824, 17778}, {1826, 2475}, {1857, 5905}, {1870, 41803}, {1896, 17220}, {1897, 693}, {1969, 21280}, {2052, 68336}, {2190, 68344}, {2204, 17148}, {2212, 194}, {2298, 4296}, {2299, 17147}, {2321, 52364}, {2322, 3869}, {2326, 2975}, {3064, 149}, {3239, 34188}, {3596, 68347}, {3718, 68355}, {4041, 39352}, {4086, 13219}, {4183, 63}, {4876, 62314}, {5089, 52164}, {5379, 17136}, {6059, 21216}, {6198, 41808}, {6335, 21302}, {6591, 58371}, {7003, 962}, {7008, 9965}, {7012, 664}, {7017, 6327}, {7020, 21279}, {7046, 329}, {7070, 68006}, {7071, 3177}, {7079, 144}, {7101, 3436}, {7133, 175}, {7156, 17037}, {7719, 7674}, {7952, 5932}, {8121, 7048}, {8611, 34186}, {8748, 3868}, {8750, 17496}, {8756, 64743}, {13426, 31552}, {13454, 31551}, {13455, 55883}, {14493, 9312}, {15742, 21272}, {18026, 46402}, {18344, 4440}, {24019, 65099}, {28044, 27484}, {31623, 17135}, {32085, 20247}, {32635, 20291}, {34894, 62386}, {36119, 41804}, {36121, 22464}, {36122, 9436}, {36125, 1266}, {36126, 23683}, {36127, 17896}, {36128, 4442}, {36797, 7192}, {40117, 8058}, {40396, 77}, {40446, 39126}, {40573, 16465}, {40838, 9799}, {40971, 20211}, {41013, 2893}, {41083, 20221}, {42013, 176}, {43742, 64694}, {44130, 17137}, {44426, 150}, {44687, 20477}, {44690, 68341}, {44691, 68342}, {44692, 253}, {46102, 68350}, {46103, 17140}, {46110, 21293}, {52663, 10538}, {52914, 17161}, {53008, 2895}, {55116, 6223}, {55206, 148}, {55346, 35312}, {56183, 514}, {56245, 22}, {57492, 189}, {57779, 17143}, {59482, 21273}, {61427, 44354}, {63965, 31527}, {65103, 39351}, {65160, 513}, {65201, 523}, {65213, 4131}, {65333, 53357}, {67181, 10529}
X(68335) = X(i)-cross conjugate of X(j) for these {i, j}: {1158, 31600}
X(68335) = pole of line {46389, 58894} with respect to the polar circle
X(68335) = pole of line {2804, 4397} with respect to the Steiner circumellipse
X(68335) = pole of line {109, 14544} with respect to the Yff parabola
X(68335) = pole of line {1817, 54451} with respect to the Wallace hyperbola
X(68335) = pole of line {24171, 44675} with respect to the dual conic of Yff parabola
X(68335) = intersection, other than A, B, C, of circumconics {{A, B, C, X(189), X(1158)}}, {{A, B, C, X(222), X(54451)}}, {{A, B, C, X(309), X(8048)}}, {{A, B, C, X(2994), X(44189)}}, {{A, B, C, X(4391), X(26871)}}, {{A, B, C, X(6925), X(56545)}}, {{A, B, C, X(28788), X(56293)}}, {{A, B, C, X(30513), X(55400)}}, {{A, B, C, X(34234), X(34277)}}
X(68335) = barycentric product X(i)*X(j) for these (i, j): {264, 56293}, {312, 34052}, {1158, 75}, {31600, 8}
X(68335) = barycentric quotient X(i)/X(j) for these (i, j): {1, 42464}, {2, 54451}, {1158, 1}, {10692, 36052}, {31600, 7}, {34052, 57}, {56293, 3}
X(68335) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 37781, 26871}, {144, 2895, 26872}, {144, 56943, 68343}, {144, 68348, 56943}, {189, 329, 64194}, {222, 41883, 2}, {2994, 5905, 48380}, {5905, 5942, 92}, {5905, 6515, 56927}, {5928, 34371, 36850}, {18750, 33066, 329}, {18928, 63152, 9776}

X(68336) = ANTIPRODUCT OF X(3) AND X(9)

X(68336) lies on circumconic {{A, B, C, X(28788), X(54457)}} and on these lines: {2, 212}, {4, 5906}, {7, 2897}, {8, 2894}, {63, 33536}, {69, 674}, {85, 20292}, {100, 18134}, {189, 9812}, {278, 14544}, {333, 11680}, {860, 60691}, {962, 52366}, {1253, 25970}, {1330, 5175}, {1331, 28776}, {1726, 29307}, {1830, 5905}, {3006, 3719}, {3562, 5125}, {3868, 5174}, {4388, 14552}, {4645, 17784}, {5057, 18750}, {5706, 23542}, {5735, 20223}, {5762, 65684}, {6604, 56872}, {7270, 14923}, {10431, 26871}, {11442, 20242}, {17093, 23973}, {18203, 24248}, {21279, 32064}, {24552, 26543}, {24984, 66249}, {27504, 63840}, {36845, 36918}, {37826, 38462}, {41723, 54457}, {49687, 57287}, {52390, 54125}

X(68336) = reflection of X(i) in X(j) for these {i,j}: {68343, 65684}
X(68336) = anticomplement of X(212)
X(68336) = X(i)-Ceva conjugate of X(j) for these {i, j}: {57787, 2}
X(68336) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {2, 56943}, {4, 144}, {7, 20}, {19, 3177}, {25, 21218}, {27, 63}, {29, 45738}, {34, 192}, {56, 3164}, {57, 6360}, {65, 18666}, {75, 52366}, {85, 4329}, {92, 329}, {108, 17494}, {158, 5942}, {196, 20211}, {222, 46717}, {225, 1654}, {226, 3151}, {264, 3436}, {273, 8}, {275, 68343}, {278, 2}, {279, 347}, {281, 30695}, {286, 3869}, {329, 55114}, {331, 69}, {342, 6223}, {348, 6527}, {393, 30694}, {459, 68348}, {607, 46706}, {608, 194}, {653, 514}, {664, 20294}, {693, 34188}, {934, 66520}, {1014, 20222}, {1041, 25242}, {1088, 52365}, {1118, 193}, {1119, 145}, {1275, 68339}, {1395, 17486}, {1396, 17147}, {1427, 18667}, {1434, 17134}, {1435, 3210}, {1440, 280}, {1441, 52364}, {1446, 2897}, {1509, 68340}, {1659, 46421}, {1847, 7}, {1874, 39367}, {1876, 39350}, {1877, 17487}, {1880, 1655}, {1897, 4468}, {1969, 21286}, {2052, 68335}, {2969, 17036}, {3668, 3152}, {4573, 6563}, {4626, 59926}, {5236, 20533}, {6063, 1370}, {6335, 4462}, {6336, 908}, {7012, 65195}, {7017, 54113}, {7115, 46725}, {7128, 4552}, {7149, 20212}, {7178, 39352}, {7233, 62314}, {7282, 3648}, {7649, 39351}, {8736, 46707}, {8810, 20213}, {11546, 4461}, {13149, 693}, {13390, 46422}, {16082, 64194}, {17094, 34186}, {17924, 37781}, {18026, 513}, {18623, 68006}, {20567, 68347}, {23984, 651}, {24032, 61185}, {31623, 18750}, {31643, 64039}, {32674, 21225}, {32714, 17496}, {34398, 44447}, {36118, 522}, {36124, 10025}, {36127, 25259}, {36419, 62798}, {37790, 30578}, {39267, 20110}, {40149, 2895}, {40444, 56545}, {40446, 3729}, {40573, 3219}, {41284, 19121}, {41514, 41514}, {43762, 62386}, {43923, 9263}, {44129, 20245}, {44696, 17037}, {46102, 190}, {46103, 54107}, {46107, 33650}, {46404, 20295}, {52575, 21287}, {52938, 20293}, {54235, 30807}, {54240, 4391}, {55110, 9965}, {55208, 21220}, {55346, 100}, {56783, 3100}, {57538, 68337}, {57785, 20243}, {57787, 6327}, {57792, 68351}, {57809, 1330}, {57918, 68355}, {61178, 31290}, {63186, 40}, {64984, 3101}, {64988, 189}, {65232, 4560}, {65270, 20296}, {65329, 3762}, {65330, 6332}, {65331, 3904}, {65332, 661}, {65335, 30565}, {65352, 1959}, {65537, 2804}
X(68336) = pole of line {3261, 46110} with respect to the Steiner circumellipse
X(68336) = barycentric product X(i)*X(j) for these (i, j): {1729, 75}
X(68336) = barycentric quotient X(i)/X(j) for these (i, j): {1729, 1}
X(68336) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 52365, 68345}, {7, 68352, 52365}, {5762, 65684, 68343}, {11442, 20242, 21270}

X(68337) = ANTIPRODUCT OF X(3) AND X(11)

X(68337) lies on these lines: {2, 7117}, {8, 21403}, {76, 21285}, {80, 21207}, {99, 33637}, {100, 1305}, {101, 1577}, {110, 811}, {150, 34387}, {190, 65236}, {264, 21270}, {286, 20289}, {311, 21276}, {349, 5086}, {651, 24035}, {664, 693}, {668, 891}, {2893, 34388}, {2975, 18738}, {3260, 21277}, {3436, 68351}, {3732, 4391}, {4554, 17136}, {4560, 27135}, {4566, 18026}, {5080, 44150}, {5176, 35517}, {5303, 29477}, {6335, 14543}, {7124, 26654}, {14615, 21286}, {16749, 21935}, {17134, 18749}, {20954, 56252}, {21302, 61184}, {22131, 28962}, {26653, 59619}, {42719, 57192}, {57976, 65282}

X(68337) = anticomplement of X(7117)
X(68337) = trilinear pole of line {3772, 17861}
X(68337) = perspector of circumconic {{A, B, C, X(31625), X(57538)}}
X(68337) = X(i)-isoconjugate-of-X(j) for these {i, j}: {649, 56003}, {667, 40436}, {1459, 56305}, {1919, 59759}, {2638, 52775}, {3248, 65370}, {22383, 55994}
X(68337) = X(i)-Dao conjugate of X(j) for these {i, j}: {3772, 521}, {5375, 56003}, {6631, 40436}, {9296, 59759}, {17861, 46396}, {53849, 36054}
X(68337) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {19, 17036}, {34, 54102}, {59, 6360}, {108, 4440}, {190, 34188}, {250, 18662}, {653, 149}, {765, 56943}, {1016, 52366}, {1275, 52365}, {1783, 39351}, {1897, 37781}, {2149, 3164}, {4551, 39352}, {4564, 20}, {4619, 66520}, {4620, 20243}, {4998, 4329}, {5379, 63}, {6335, 33650}, {7012, 2}, {7045, 347}, {7115, 192}, {7128, 145}, {15742, 329}, {18020, 21273}, {18026, 150}, {23984, 12649}, {23999, 68338}, {24000, 62798}, {24032, 56927}, {24033, 30699}, {24041, 68340}, {31615, 20294}, {32674, 9263}, {32714, 58371}, {34922, 17483}, {39294, 517}, {46102, 8}, {46254, 35614}, {46404, 21293}, {52378, 20222}, {55346, 7}, {57538, 68336}, {57756, 64694}, {61178, 21221}, {65207, 3448}, {65232, 17154}, {65233, 34186}, {65573, 3151}, {67038, 1370}
X(68337) = pole of line {3270, 11918} with respect to the polar circle
X(68337) = pole of line {17147, 62798} with respect to the Kiepert parabola
X(68337) = pole of line {668, 18026} with respect to the Steiner circumellipse
X(68337) = pole of line {192, 3100} with respect to the Yff parabola
X(68337) = pole of line {3733, 48383} with respect to the Wallace hyperbola
X(68337) = pole of line {2975, 11683} with respect to the Moses HK-parabola
X(68337) = pole of line {18135, 67124} with respect to the dual conic of Feuerbach hyperbola
X(68337) = pole of line {1332, 4033} with respect to the dual conic of Hofstadter ellipse
X(68337) = intersection, other than A, B, C, of circumconics {{A, B, C, X(668), X(1305)}}, {{A, B, C, X(1978), X(2864)}}, {{A, B, C, X(3772), X(41314)}}, {{A, B, C, X(3924), X(23354)}}, {{A, B, C, X(3952), X(33637)}}, {{A, B, C, X(16749), X(27853)}}, {{A, B, C, X(18026), X(44765)}}, {{A, B, C, X(35174), X(51566)}}, {{A, B, C, X(53332), X(57976)}}
X(68337) = barycentric product X(i)*X(j) for these (i, j): {646, 65688}, {1837, 4554}, {1978, 3924}, {3772, 668}, {16749, 3952}, {17189, 4033}, {17861, 190}, {21935, 799}, {40968, 4572}, {41004, 6335}, {53279, 76}, {64654, 65282}
X(68337) = barycentric quotient X(i)/X(j) for these (i, j): {100, 56003}, {190, 40436}, {668, 59759}, {1016, 65370}, {1783, 56305}, {1837, 650}, {1897, 55994}, {3772, 513}, {3924, 649}, {4554, 34399}, {6335, 34406}, {16749, 7192}, {17189, 1019}, {17861, 514}, {21935, 661}, {23984, 52775}, {26934, 1459}, {36570, 43924}, {40968, 663}, {40980, 7252}, {41004, 905}, {53279, 6}, {53850, 36054}, {57538, 54948}, {64654, 6371}, {65445, 14936}, {65688, 3669}
X(68337) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {264, 21270, 68338}

X(68338) = ANTIPRODUCT OF X(3) AND X(12)

X(68338) lies on these lines: {2, 2197}, {7, 23989}, {75, 1444}, {76, 21286}, {92, 1172}, {110, 57779}, {264, 21270}, {286, 17220}, {311, 21277}, {313, 5176}, {314, 17135}, {572, 40564}, {2286, 26622}, {3260, 21276}, {3436, 44140}, {4360, 14616}, {10447, 64365}, {11681, 18147}, {14213, 37793}, {14615, 21285}, {17016, 17863}, {17143, 21273}, {17861, 49487}, {20174, 24633}, {20244, 20246}, {21279, 68351}, {22132, 28917}, {60970, 64194}

X(68338) = anticomplement of X(2197)
X(68338) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {19, 56291}, {21, 3151}, {27, 2475}, {28, 17778}, {29, 2895}, {33, 46707}, {58, 18667}, {60, 6360}, {81, 3152}, {86, 2897}, {250, 4552}, {261, 4329}, {270, 2}, {284, 18666}, {286, 2893}, {333, 52364}, {757, 347}, {873, 68351}, {1098, 56943}, {1172, 1654}, {1509, 52365}, {2150, 3164}, {2185, 20}, {2189, 192}, {2212, 46714}, {2299, 1655}, {2326, 144}, {3737, 39352}, {4556, 66520}, {4612, 20294}, {5379, 3882}, {7058, 52366}, {18020, 21272}, {18021, 68347}, {18155, 13219}, {23582, 61185}, {23999, 68337}, {24000, 651}, {24041, 68339}, {31623, 1330}, {36419, 12649}, {36421, 5942}, {39177, 44003}, {44130, 21287}, {46103, 8}, {46254, 3888}, {52379, 1370}, {52914, 514}, {52919, 521}, {52921, 4391}, {55231, 21302}, {55233, 21301}, {57215, 3448}, {57779, 69}, {59482, 329}, {64457, 4296}, {65015, 5174}, {65201, 31290}
X(68338) = pole of line {651, 68339} with respect to the Kiepert parabola
X(68338) = pole of line {3869, 16678} with respect to the Wallace hyperbola
X(68338) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1746), X(40574)}}, {{A, B, C, X(2217), X(55035)}}
X(68338) = barycentric product X(i)*X(j) for these (i, j): {1746, 75}
X(68338) = barycentric quotient X(i)/X(j) for these (i, j): {1746, 1}
X(68338) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {264, 21270, 68337}, {17143, 54109, 21273}

X(68339) = ANTIPRODUCT OF X(4) AND X(11)

X(68339) lies on circumconic {{A, B, C, X(46964), X(62333)}} and on these lines: {2, 8735}, {20, 68351}, {22, 23402}, {99, 13397}, {100, 1305}, {190, 53652}, {347, 4373}, {664, 65290}, {668, 54110}, {693, 6516}, {811, 3658}, {906, 23882}, {925, 1310}, {927, 46964}, {934, 41906}, {1632, 1633}, {4329, 20477}, {4427, 21272}, {10058, 21207}, {10538, 62386}, {17136, 68350}, {17905, 28963}, {18689, 67725}, {43349, 67734}

X(68339) = anticomplement of X(8735)
X(68339) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {48, 17036}, {59, 5905}, {603, 54102}, {765, 68335}, {906, 39351}, {1101, 62798}, {1110, 30694}, {1252, 5942}, {1262, 12649}, {1275, 68336}, {1331, 37781}, {1332, 33650}, {1813, 149}, {2149, 193}, {4564, 4}, {4570, 92}, {4619, 521}, {4620, 20242}, {4998, 21270}, {6516, 150}, {7012, 6515}, {7045, 56927}, {23067, 21221}, {24027, 30699}, {24041, 68338}, {31615, 20293}, {32660, 9263}, {36059, 4440}, {44717, 8}, {46102, 5906}, {47390, 18662}, {52378, 3868}, {55194, 21300}, {59151, 17896}, {62719, 35614}, {65164, 21293}, {65233, 3448}, {67038, 11442}
X(68339) = pole of line {918, 4440} with respect to the DeLongchamps circle
X(68339) = pole of line {92, 1172} with respect to the Kiepert parabola
X(68339) = pole of line {4561, 4571} with respect to the Steiner circumellipse
X(68339) = pole of line {193, 5942} with respect to the Yff parabola
X(68339) = barycentric product X(i)*X(j) for these (i, j): {190, 24179}, {4554, 62333}
X(68339) = barycentric quotient X(i)/X(j) for these (i, j): {24179, 514}, {62333, 650}
X(68339) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4329, 20477, 68340}

X(68340) = ANTIPRODUCT OF X(4) AND X(12)

X(68340) lies on these lines: {2, 8736}, {20, 29207}, {75, 1444}, {92, 27174}, {99, 54109}, {332, 20245}, {1950, 64780}, {4329, 20477}, {5361, 7560}, {6527, 68351}, {17140, 18654}, {17147, 62798}, {20076, 56927}, {20243, 35614}, {21286, 51612}, {37095, 62857}

X(68340) = anticomplement of X(8736)
X(68340) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {48, 56291}, {60, 5905}, {212, 46707}, {249, 61185}, {261, 21270}, {270, 6515}, {283, 2895}, {332, 21287}, {593, 12649}, {757, 56927}, {849, 30699}, {1098, 68335}, {1101, 651}, {1437, 17778}, {1444, 2893}, {1509, 68336}, {1790, 2475}, {1812, 1330}, {2150, 193}, {2185, 4}, {2193, 1654}, {4556, 521}, {4612, 20293}, {4636, 4391}, {6514, 52364}, {7054, 5942}, {18604, 3152}, {23189, 21221}, {24041, 68337}, {46103, 5906}, {47390, 4552}, {52379, 11442}, {52935, 46400}, {55196, 21300}, {57779, 317}, {62719, 3888}, {65568, 8}
X(68340) = pole of line {651, 24035} with respect to the Kiepert parabola
X(68340) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4329, 20477, 68339}

X(68341) = ANTIPRODUCT OF X(4) AND X(15)

X(68341) lies on these lines: {2, 8739}, {20, 617}, {22, 299}, {66, 69}, {298, 858}, {302, 30744}, {303, 68085}, {616, 2071}, {621, 3153}, {622, 44440}, {628, 7488}, {633, 37444}, {2992, 3180}, {7493, 63105}, {11420, 40901}, {11421, 34541}, {44239, 52193}, {47090, 52194}

X(68341) = anticomplement of X(8739)
X(68341) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {13, 5905}, {63, 616}, {300, 21270}, {2153, 193}, {3457, 21216}, {5995, 17498}, {23895, 7253}, {36061, 23871}, {36296, 192}, {38414, 4560}, {39377, 18668}, {40709, 8}, {44690, 68335}, {65570, 12383}, {66926, 4391}
X(68341) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 1370, 68342}

X(68342) = ANTIPRODUCT OF X(4) AND X(16)

X(68342) lies on these lines: {2, 8740}, {20, 616}, {22, 298}, {66, 69}, {299, 858}, {302, 68085}, {303, 30744}, {617, 2071}, {621, 44440}, {622, 3153}, {627, 7488}, {634, 37444}, {2993, 3181}, {7493, 63106}, {11420, 34540}, {11421, 40900}, {44239, 52194}, {47090, 52193}

X(68342) = anticomplement of X(8740)
X(68342) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {14, 5905}, {63, 617}, {301, 21270}, {2154, 193}, {3458, 21216}, {5994, 17498}, {23896, 7253}, {36061, 23870}, {36297, 192}, {38413, 4560}, {39378, 18668}, {40710, 8}, {44691, 68335}, {65569, 12383}, {66927, 4391}
X(68342) = pole of line {11127, 23870} with respect to the DeLongchamps circle
X(68342) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 1370, 68341}

X(68343) = ANTIPRODUCT OF X(5) AND X(7)

X(68343) lies on these lines: {3, 56254}, {6, 18662}, {8, 5842}, {9, 20223}, {63, 321}, {92, 3219}, {144, 2895}, {189, 4102}, {190, 329}, {192, 62798}, {201, 24537}, {212, 64858}, {255, 42456}, {394, 4552}, {651, 6360}, {894, 62857}, {908, 33113}, {914, 32859}, {1708, 17862}, {1993, 17479}, {2406, 7011}, {3869, 7283}, {5278, 14213}, {5744, 32939}, {5759, 52365}, {5762, 65684}, {5905, 6350}, {7078, 20222}, {7580, 61185}, {11433, 41563}, {14544, 22117}, {17147, 55399}, {17262, 39767}, {17336, 20921}, {17781, 50105}, {18607, 28968}, {18736, 42718}, {18750, 20920}, {20078, 26871}, {20905, 55871}, {21271, 68346}, {23661, 55104}, {25734, 45738}, {26591, 55869}, {26611, 28826}, {26921, 41013}, {27287, 33761}, {27378, 44706}, {27472, 55987}, {34234, 67335}, {37584, 38462}, {37787, 54284}, {48380, 55873}

X(68343) = reflection of X(i) in X(j) for these {i,j}: {68336, 65684}
X(68343) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {9, 2888}, {41, 17035}, {54, 7}, {95, 21285}, {97, 52365}, {275, 68336}, {2148, 145}, {2167, 3434}, {2169, 347}, {2190, 56927}, {8882, 12649}, {35196, 75}, {36078, 36038}, {36134, 4467}, {44687, 69}, {54034, 3210}, {56254, 2893}, {62265, 329}, {62268, 30699}, {62276, 21280}, {62277, 68351}
X(68343) = pole of line {52355, 57091} with respect to the Steiner circumellipse
X(68343) = pole of line {4551, 14544} with respect to the Yff parabola
X(68343) = pole of line {23536, 64163} with respect to the dual conic of Yff parabola
X(68343) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2995), X(34393)}}, {{A, B, C, X(4417), X(64194)}}, {{A, B, C, X(6796), X(13478)}}
X(68343) = barycentric product X(i)*X(j) for these (i, j): {6796, 75}
X(68343) = barycentric quotient X(i)/X(j) for these (i, j): {6796, 1}
X(68343) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {144, 56943, 68335}, {190, 54107, 329}, {1993, 17479, 68344}, {5762, 65684, 68336}, {25734, 45738, 56545}

X(68344) = ANTIPRODUCT OF X(5) AND X(8)

X(68344) lies on these lines: {1, 1441}, {2, 17043}, {3, 17221}, {6, 4552}, {7, 528}, {8, 40999}, {75, 1442}, {77, 3875}, {85, 7269}, {145, 347}, {192, 651}, {221, 64071}, {222, 17147}, {226, 54744}, {241, 4852}, {264, 40440}, {273, 34772}, {307, 519}, {321, 45126}, {322, 4511}, {326, 20895}, {379, 1953}, {394, 18662}, {517, 17134}, {536, 28968}, {648, 56948}, {944, 4329}, {952, 21270}, {1014, 33296}, {1214, 3187}, {1443, 39126}, {1445, 16834}, {1456, 49462}, {1458, 32921}, {1471, 49477}, {1482, 17220}, {1617, 17150}, {1943, 28606}, {1992, 41563}, {1993, 17479}, {1999, 17080}, {2003, 32933}, {2006, 30834}, {2256, 25255}, {2398, 6600}, {2594, 34388}, {3007, 37727}, {3057, 52385}, {3160, 4460}, {3173, 25254}, {3175, 28997}, {3188, 34791}, {3210, 17074}, {3217, 49759}, {3240, 14594}, {3244, 3668}, {3262, 44179}, {3672, 53997}, {3759, 37787}, {3811, 57810}, {3879, 22464}, {3895, 7013}, {3896, 8270}, {3969, 56366}, {3995, 34048}, {4318, 49470}, {4331, 50284}, {4336, 28850}, {4358, 56418}, {4361, 17077}, {4464, 9436}, {4664, 29007}, {4967, 25723}, {4970, 9316}, {5226, 34064}, {5278, 16577}, {5564, 17095}, {5723, 17243}, {5740, 10573}, {5882, 18650}, {6180, 17318}, {6358, 19684}, {6360, 62798}, {6505, 17862}, {6510, 26651}, {6542, 17086}, {7190, 9312}, {7225, 43040}, {7982, 18655}, {8148, 18661}, {10944, 41003}, {12575, 50563}, {14543, 20818}, {15253, 29830}, {15500, 65581}, {16091, 34611}, {17073, 48381}, {17075, 17388}, {17233, 28780}, {17234, 37771}, {17262, 62669}, {17314, 28739}, {17316, 37800}, {17317, 61008}, {17319, 25726}, {17438, 24315}, {17452, 24268}, {17858, 18689}, {18481, 20291}, {18525, 20289}, {20017, 26942}, {20905, 53996}, {20946, 28982}, {21617, 29574}, {24173, 32577}, {25243, 55432}, {25717, 64739}, {27396, 40863}, {27547, 39351}, {28996, 35652}, {29584, 41246}, {30145, 45196}, {30806, 55391}, {33298, 41808}, {34748, 53380}, {36589, 50132}, {37541, 64161}, {37732, 57830}, {37756, 60988}, {41010, 61296}, {41140, 61016}, {41226, 59771}, {41823, 65384}, {42289, 50281}, {45222, 52424}, {49455, 53531}, {49492, 54292}, {50102, 64115}, {50109, 60992}, {55119, 56936}

X(68344) = reflection of X(i) in X(j) for these {i,j}: {21270, 41007}
X(68344) = X(i)-Dao conjugate of X(j) for these {i, j}: {40688, 3813}, {47794, 44311}
X(68344) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {54, 329}, {57, 2888}, {95, 21286}, {97, 52366}, {604, 17035}, {2148, 144}, {2167, 3436}, {2169, 56943}, {2190, 68335}, {8882, 5942}, {35196, 18750}, {36078, 3762}, {44687, 54113}, {54034, 3177}, {62264, 7}, {62268, 30694}, {62269, 21218}
X(68344) = pole of line {29013, 30725} with respect to the incircle
X(68344) = pole of line {4453, 17094} with respect to the Steiner circumellipse
X(68344) = pole of line {307, 4887} with respect to the dual conic of Yff parabola
X(68344) = pole of line {306, 3911} with respect to the dual conic of Moses-Feuerbach circumconic
X(68344) = intersection, other than A, B, C, of circumconics {{A, B, C, X(519), X(17579)}}, {{A, B, C, X(903), X(2997)}}, {{A, B, C, X(1320), X(8715)}}, {{A, B, C, X(3204), X(34230)}}, {{A, B, C, X(36887), X(47794)}}, {{A, B, C, X(40440), X(52553)}}
X(68344) = barycentric product X(i)*X(j) for these (i, j): {85, 8715}, {145, 27814}, {3204, 6063}, {4554, 48302}, {39478, 46405}, {47794, 664}
X(68344) = barycentric quotient X(i)/X(j) for these (i, j): {3204, 55}, {8715, 9}, {27814, 4373}, {39478, 654}, {47794, 522}, {48302, 650}
X(68344) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 17885, 24202}, {85, 17393, 7269}, {145, 347, 56927}, {347, 56927, 41804}, {664, 4360, 7}, {952, 41007, 21270}, {1993, 17479, 68343}, {3875, 25716, 77}, {5723, 17243, 28741}, {17221, 21271, 3}

X(68345) = ANTIPRODUCT OF X(5) AND X(9)

X(68345) lies on these lines: {2, 7004}, {3, 56254}, {7, 2897}, {8, 5884}, {92, 11220}, {100, 17165}, {222, 14544}, {991, 18662}, {1071, 23661}, {1441, 17616}, {1897, 17074}, {2979, 21271}, {3434, 17140}, {3616, 56940}, {3873, 39126}, {4303, 20222}, {4855, 56318}, {5732, 20223}, {10167, 64194}, {10202, 38462}, {10391, 17862}, {10394, 54284}, {10538, 18444}, {11680, 53566}, {12675, 23528}, {13369, 41013}, {17221, 20477}, {17784, 24349}, {22053, 64858}, {24840, 28364}, {26910, 53151}, {31657, 65684}, {36996, 68335}

X(68345) = anticomplement of X(7069)
X(68345) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {7, 2888}, {54, 144}, {56, 17035}, {95, 3436}, {97, 56943}, {275, 68335}, {2148, 3177}, {2167, 329}, {2190, 5942}, {8882, 30694}, {35196, 45738}, {36078, 47772}, {54034, 21218}, {62264, 145}, {62276, 21286}, {62277, 52366}
X(68345) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 52365, 68336}, {17140, 68350, 3434}

X(68346) = ANTIPRODUCT OF X(5) AND X(20)

X(68346) lies on these lines: {2, 42459}, {3, 95}, {4, 6527}, {5, 40680}, {6, 64781}, {20, 32000}, {25, 30737}, {26, 44135}, {30, 69}, {53, 6389}, {76, 39568}, {99, 54992}, {183, 9909}, {253, 3146}, {268, 1948}, {273, 10538}, {290, 67186}, {297, 20208}, {309, 5088}, {311, 7387}, {316, 34725}, {317, 382}, {322, 7283}, {325, 34609}, {338, 1609}, {339, 18534}, {340, 5073}, {381, 45198}, {393, 441}, {401, 9308}, {458, 3164}, {523, 34777}, {550, 52710}, {648, 38292}, {1033, 60516}, {1073, 35061}, {1078, 16195}, {1235, 11414}, {1272, 31181}, {1316, 19118}, {1494, 15684}, {1598, 41009}, {1632, 33582}, {1657, 44134}, {1947, 7011}, {1975, 14615}, {2052, 6617}, {2070, 55561}, {2450, 63535}, {2453, 20987}, {2782, 19588}, {2968, 55393}, {3167, 57275}, {3260, 3964}, {3575, 68355}, {3618, 59649}, {3627, 63155}, {3628, 8797}, {3629, 64915}, {3663, 64931}, {3729, 64780}, {3785, 65376}, {3830, 32002}, {3875, 64930}, {3933, 34938}, {5020, 62698}, {6090, 41202}, {6144, 64923}, {6356, 55394}, {7395, 44142}, {7517, 44138}, {7767, 31305}, {7776, 14790}, {8573, 41760}, {9530, 48910}, {9723, 12084}, {10154, 34229}, {11250, 44180}, {11432, 41481}, {12108, 36948}, {13219, 52842}, {14767, 36751}, {15312, 51212}, {15589, 34608}, {15703, 40410}, {15811, 59527}, {15851, 36794}, {16089, 38283}, {16199, 40022}, {16655, 44141}, {18569, 62338}, {20563, 46200}, {21271, 68343}, {23335, 40697}, {26166, 37198}, {32816, 50572}, {34621, 52713}, {34815, 37200}, {35930, 59566}, {35941, 56290}, {37188, 43981}, {37668, 44442}, {37928, 67606}, {39099, 64926}, {40318, 65767}, {40341, 64783}, {41244, 46832}, {41489, 41678}, {43988, 52253}, {46717, 62953}, {50692, 52711}, {51888, 64585}, {52559, 53639}, {52843, 68354}, {53795, 64023}, {54105, 61970}, {55391, 64054}, {55392, 64053}, {55474, 55885}, {55480, 55890}, {55958, 61882}, {57008, 68022}, {57822, 62137}, {57897, 62046}, {58408, 61315}, {61843, 63173}, {62275, 62334}

X(68346) = reflection of X(i) in X(j) for these {i,j}: {37921, 2453}
X(68346) = isotomic conjugate of X(52441)
X(68346) = anticomplement of X(42459)
X(68346) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {2148, 17037}, {2155, 17035}, {2167, 6225}, {2169, 68006}, {2184, 2888}, {2190, 14361}
X(68346) = pole of line {418, 26864} with respect to the Stammler hyperbola
X(68346) = pole of line {30474, 62428} with respect to the Steiner circumellipse
X(68346) = pole of line {376, 5562} with respect to the Wallace hyperbola
X(68346) = pole of line {3090, 32000} with respect to the dual conic of Moses HK-parabola
X(68346) = intersection, other than A, B, C, of circumconics {{A, B, C, X(95), X(15400)}}, {{A, B, C, X(3426), X(8884)}}, {{A, B, C, X(8795), X(36889)}}, {{A, B, C, X(19185), X(43918)}}, {{A, B, C, X(35061), X(35510)}}, {{A, B, C, X(42459), X(45249)}}
X(68346) = barycentric product X(i)*X(j) for these (i, j): {26883, 76}
X(68346) = barycentric quotient X(i)/X(j) for these (i, j): {2, 52441}, {26883, 6}, {45249, 42459}
X(68346) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 6527, 41005}, {53, 6389, 52251}, {253, 32001, 40996}, {264, 20477, 3}, {382, 40995, 317}, {401, 40896, 9308}, {401, 9308, 15905}, {37188, 43981, 65809}

X(68347) = ANTIPRODUCT OF X(6) AND X(19)

X(68347) lies on these lines: {2, 1973}, {8, 2893}, {10, 26260}, {19, 26153}, {20, 28845}, {347, 56928}, {607, 857}, {858, 29829}, {1370, 17135}, {1441, 5090}, {2172, 28404}, {3101, 3661}, {6360, 21217}, {7357, 20243}, {17137, 21280}, {17904, 33313}, {18589, 26203}, {18639, 24605}, {20552, 52366}, {24995, 27532}, {56883, 56943}

X(68347) = anticomplement of X(1973)
X(68347) = X(i)-Ceva conjugate of X(j) for these {i, j}: {40364, 2}
X(68347) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {1, 21216}, {2, 193}, {3, 194}, {4, 6392}, {7, 30699}, {8, 30694}, {25, 46712}, {39, 10340}, {48, 17486}, {63, 192}, {67, 19577}, {69, 2}, {72, 1655}, {75, 5905}, {76, 4}, {77, 3210}, {78, 3177}, {83, 7754}, {85, 12649}, {86, 3187}, {95, 1993}, {99, 525}, {125, 54104}, {141, 8878}, {183, 47740}, {184, 8264}, {190, 25259}, {193, 18287}, {219, 21218}, {249, 41676}, {261, 62798}, {264, 6515}, {265, 19570}, {274, 3868}, {275, 56017}, {276, 5889}, {279, 11851}, {287, 385}, {290, 51481}, {295, 19565}, {304, 8}, {305, 69}, {306, 1654}, {307, 17778}, {308, 3060}, {310, 17220}, {312, 5942}, {314, 92}, {315, 41361}, {325, 40867}, {326, 6360}, {328, 37779}, {332, 63}, {337, 6542}, {343, 17035}, {345, 144}, {348, 145}, {394, 3164}, {487, 6463}, {488, 6462}, {524, 7665}, {525, 148}, {561, 21270}, {647, 25054}, {648, 33294}, {656, 21220}, {662, 17498}, {668, 4391}, {670, 850}, {671, 47286}, {683, 54412}, {799, 7253}, {801, 9308}, {892, 9979}, {905, 9263}, {1016, 3732}, {1176, 8267}, {1231, 2475}, {1236, 34163}, {1241, 1843}, {1260, 46706}, {1264, 56943}, {1265, 30695}, {1275, 1897}, {1331, 21225}, {1332, 17494}, {1444, 17147}, {1459, 21224}, {1494, 3580}, {1502, 11442}, {1565, 54102}, {1790, 17148}, {1799, 6}, {1969, 5906}, {1978, 20293}, {2366, 15014}, {2373, 37784}, {2407, 45292}, {2525, 39346}, {2996, 2996}, {3222, 2451}, {3260, 66914}, {3265, 39352}, {3267, 3448}, {3504, 2998}, {3596, 68335}, {3620, 8892}, {3690, 46714}, {3695, 46707}, {3718, 329}, {3917, 52637}, {3926, 20}, {3933, 2896}, {3964, 46717}, {3977, 17487}, {3998, 18666}, {4025, 4440}, {4143, 34186}, {4176, 6527}, {4554, 521}, {4555, 10015}, {4558, 31296}, {4561, 514}, {4563, 523}, {4569, 17896}, {4572, 46400}, {4573, 65099}, {4580, 25047}, {4590, 110}, {4592, 4560}, {4600, 14543}, {4601, 53349}, {4602, 21300}, {4615, 53352}, {4620, 14544}, {4998, 651}, {5490, 12222}, {5491, 12221}, {5641, 54395}, {6035, 65714}, {6063, 56927}, {6331, 520}, {6332, 39351}, {6333, 39359}, {6340, 20080}, {6385, 20242}, {6390, 8591}, {6393, 147}, {6394, 401}, {6516, 17496}, {7019, 6646}, {7053, 46716}, {7055, 347}, {7056, 4452}, {7177, 17480}, {7182, 7}, {7763, 6193}, {7767, 51860}, {7769, 11271}, {7799, 12383}, {8781, 3564}, {8858, 698}, {10008, 9742}, {10159, 7762}, {10217, 46708}, {10218, 46709}, {11064, 39358}, {11090, 62986}, {11091, 62987}, {12215, 8782}, {14208, 21221}, {14376, 20065}, {14417, 39356}, {14534, 56019}, {14575, 40382}, {14615, 14361}, {14616, 62305}, {14977, 45291}, {15164, 2592}, {15165, 2593}, {15413, 149}, {15414, 44003}, {15419, 17154}, {17206, 1}, {17932, 2799}, {17970, 19566}, {18020, 648}, {18021, 68338}, {18022, 317}, {18023, 41724}, {18025, 48381}, {18816, 48380}, {18829, 3569}, {18830, 21438}, {18878, 44427}, {19611, 18663}, {20336, 2895}, {20563, 45794}, {20567, 68336}, {20570, 2994}, {20769, 30667}, {22370, 41840}, {24243, 26503}, {24244, 26494}, {25083, 39350}, {26932, 17036}, {26942, 56291}, {28706, 2888}, {30680, 20073}, {30786, 524}, {31617, 15801}, {31621, 10733}, {31624, 64878}, {31637, 239}, {32014, 56018}, {32830, 11469}, {33297, 17911}, {34055, 17489}, {34254, 5596}, {34384, 264}, {34385, 5392}, {34386, 3}, {34391, 13439}, {34392, 13428}, {34399, 278}, {34403, 3146}, {34405, 393}, {34409, 281}, {34410, 459}, {34411, 7952}, {34412, 1249}, {34537, 53350}, {34767, 62639}, {34897, 14712}, {35136, 2501}, {35139, 41079}, {35140, 297}, {35518, 37781}, {36212, 39355}, {36214, 40858}, {36952, 7785}, {37214, 37782}, {37215, 23874}, {37669, 17037}, {37804, 11061}, {40032, 11433}, {40050, 315}, {40071, 1330}, {40360, 33796}, {40364, 6327}, {40373, 40381}, {40405, 25}, {40410, 41628}, {40412, 40571}, {40413, 40318}, {40423, 2986}, {40428, 2987}, {40708, 7779}, {40709, 3180}, {40710, 3181}, {40711, 62983}, {40712, 62984}, {40824, 5921}, {40827, 41723}, {40829, 7703}, {40830, 12111}, {40832, 13754}, {41530, 32001}, {42287, 63042}, {42313, 7774}, {42333, 324}, {43187, 53345}, {43714, 20081}, {44181, 46639}, {44182, 41909}, {44183, 44766}, {44326, 8057}, {44877, 56021}, {45792, 14731}, {46134, 14618}, {46139, 18314}, {46140, 44146}, {46142, 57257}, {46144, 39905}, {46746, 6504}, {46810, 2575}, {46813, 2574}, {47388, 36849}, {47389, 99}, {47390, 46726}, {49280, 39364}, {52351, 20072}, {52385, 18667}, {52392, 37759}, {52396, 3151}, {52437, 18301}, {52565, 3152}, {52608, 512}, {52609, 31290}, {52617, 13219}, {52940, 53351}, {54911, 56022}, {54960, 52744}, {54986, 26545}, {54987, 26546}, {54988, 46106}, {55023, 6339}, {55202, 7192}, {55264, 18808}, {55388, 37881}, {55965, 25239}, {55972, 37174}, {56053, 65097}, {56267, 37667}, {57738, 62636}, {57750, 43190}, {57751, 2988}, {57752, 2989}, {57753, 2990}, {57754, 2991}, {57755, 46638}, {57756, 46640}, {57757, 44765}, {57758, 57647}, {57759, 42405}, {57760, 43756}, {57761, 287}, {57762, 14919}, {57763, 4558}, {57764, 18315}, {57765, 63763}, {57775, 2052}, {57780, 68334}, {57783, 189}, {57784, 2997}, {57798, 539}, {57799, 511}, {57800, 394}, {57801, 17950}, {57819, 37644}, {57822, 37645}, {57825, 6994}, {57829, 323}, {57833, 333}, {57845, 50248}, {57846, 44363}, {57847, 44370}, {57848, 20536}, {57849, 19623}, {57852, 141}, {57853, 81}, {57854, 86}, {57859, 15983}, {57865, 20086}, {57872, 325}, {57873, 62999}, {57876, 17379}, {57878, 63009}, {57903, 52}, {57904, 68}, {57918, 3434}, {57919, 3436}, {57928, 39470}, {57985, 16704}, {57987, 30941}, {57991, 34211}, {58005, 56559}, {59482, 46713}, {60101, 1351}, {60114, 43981}, {60202, 18440}, {60217, 21850}, {60235, 56020}, {60241, 27377}, {60729, 31308}, {60872, 63093}, {62277, 17479}, {62719, 6758}, {63179, 4232}, {63182, 6353}, {63195, 6995}, {64982, 1992}, {64985, 27}, {65032, 3618}, {65164, 522}, {65278, 14316}, {65279, 24978}, {65287, 54262}, {65301, 47695}, {65307, 4580}, {65328, 55974}, {65354, 16230}, {66767, 63646}, {66933, 17493}, {67038, 61185}
X(68347) = pole of line {824, 4560} with respect to the DeLongchamps circle
X(68347) = barycentric product X(i)*X(j) for these (i, j): {18683, 69}
X(68347) = barycentric quotient X(i)/X(j) for these (i, j): {18683, 4}
X(68347) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 17492, 21270}, {8, 21274, 17492}, {1370, 68351, 18659}

X(68348) = ANTIPRODUCT OF X(7) AND X(20)

X(68348) lies on these lines: {2, 77}, {8, 12565}, {63, 63001}, {92, 10405}, {144, 2895}, {145, 9539}, {279, 13567}, {280, 67994}, {329, 2391}, {346, 54113}, {459, 36118}, {2184, 54111}, {3146, 68352}, {4869, 20921}, {6223, 39130}, {6515, 9965}, {11433, 60939}, {17778, 30694}, {18663, 68349}, {20007, 52366}, {20059, 20223}, {21454, 26871}, {27540, 53997}, {30699, 39351}, {54107, 64015}

X(68348) = anticomplement of X(18623)
X(68348) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 54226}
X(68348) = X(i)-Dao conjugate of X(j) for these {i, j}: {9, 54226}
X(68348) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {9, 6225}, {33, 14361}, {41, 17037}, {64, 7}, {212, 68006}, {253, 21285}, {459, 68336}, {1073, 52365}, {2155, 145}, {2184, 3434}, {5931, 17137}, {7037, 63877}, {8809, 6604}, {19611, 68351}, {19614, 347}, {30457, 8}, {33581, 3210}, {41088, 5932}, {41489, 12649}, {44692, 69}, {52158, 75}, {53012, 2897}, {56235, 21302}, {57921, 21280}, {60799, 55119}, {65374, 4131}, {65574, 2893}
X(68348) = pole of line {4163, 8058} with respect to the Steiner circumellipse
X(68348) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1422), X(7992)}}, {{A, B, C, X(10405), X(41081)}}
X(68348) = barycentric product X(i)*X(j) for these (i, j): {75, 7992}
X(68348) = barycentric quotient X(i)/X(j) for these (i, j): {1, 54226}, {7992, 1}
X(68348) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {56943, 68335, 144}

X(68349) = ANTIPRODUCT OF X(8) AND X(20)

X(68349) lies on these lines: {2, 7}, {8, 3668}, {20, 41004}, {29, 58786}, {69, 279}, {75, 1446}, {77, 34772}, {78, 269}, {85, 5232}, {145, 347}, {241, 4869}, {273, 4373}, {280, 6355}, {314, 61413}, {320, 62787}, {348, 3945}, {391, 948}, {936, 7271}, {938, 3663}, {962, 41010}, {966, 52023}, {1119, 5125}, {1122, 24797}, {1210, 4862}, {1427, 3965}, {1439, 3868}, {1440, 39695}, {1441, 3617}, {1788, 65688}, {2287, 6180}, {3146, 18655}, {3160, 3879}, {3522, 18650}, {3664, 5703}, {3672, 5738}, {3718, 40704}, {3875, 20008}, {4021, 15933}, {4328, 32098}, {4329, 20070}, {4341, 4511}, {4346, 17863}, {4419, 18635}, {4452, 5932}, {4887, 5704}, {4888, 13411}, {5930, 20019}, {6356, 37180}, {6515, 20211}, {6734, 31995}, {6895, 21279}, {7282, 7518}, {7365, 37655}, {8809, 68352}, {9312, 32099}, {10481, 17272}, {14986, 53596}, {17014, 17086}, {17270, 31994}, {18663, 68348}, {19611, 54111}, {20013, 53997}, {20076, 55119}, {20212, 37781}, {21255, 51302}, {24599, 37800}, {24607, 56020}, {25015, 54398}, {25887, 45227}, {27383, 62789}, {29616, 45744}, {30712, 60041}, {31185, 59594}, {32001, 36118}, {33673, 37780}, {38459, 55391}, {40663, 63575}, {40999, 46933}, {43983, 62779}, {50700, 64122}, {54425, 62985}, {57866, 63235}

X(68349) = anticomplement of X(27382)
X(68349) = X(i)-isoconjugate-of-X(j) for these {i, j}: {41, 67941}, {3063, 68195}
X(68349) = X(i)-Dao conjugate of X(j) for these {i, j}: {3160, 67941}, {10001, 68195}
X(68349) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {34, 14361}, {57, 6225}, {64, 329}, {253, 21286}, {603, 68006}, {604, 17037}, {1073, 52366}, {2155, 144}, {2184, 3436}, {8809, 69}, {19614, 56943}, {33581, 3177}, {36079, 693}, {41489, 5942}, {44692, 54113}, {52158, 18750}, {60803, 189}, {65374, 20296}
X(68349) = pole of line {3064, 4162} with respect to the polar circle
X(68349) = pole of line {522, 17094} with respect to the Steiner circumellipse
X(68349) = pole of line {651, 25736} with respect to the Hutson-Moses hyperbola
X(68349) = pole of line {521, 43923} with respect to the dual conic of Spieker circle
X(68349) = pole of line {7, 33116} with respect to the dual conic of Moses-Feuerbach circumconic
X(68349) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(58005)}}, {{A, B, C, X(9), X(1257)}}, {{A, B, C, X(63), X(4373)}}, {{A, B, C, X(75), X(5273)}}, {{A, B, C, X(253), X(27413)}}, {{A, B, C, X(273), X(5435)}}, {{A, B, C, X(329), X(39695)}}, {{A, B, C, X(672), X(7655)}}, {{A, B, C, X(903), X(28610)}}, {{A, B, C, X(5226), X(60041)}}, {{A, B, C, X(5249), X(30712)}}, {{A, B, C, X(5328), X(40424)}}, {{A, B, C, X(9965), X(36606)}}, {{A, B, C, X(10436), X(56264)}}, {{A, B, C, X(18228), X(58002)}}, {{A, B, C, X(25527), X(34399)}}, {{A, B, C, X(27382), X(35510)}}
X(68349) = barycentric product X(i)*X(j) for these (i, j): {4554, 7655}, {11523, 85}
X(68349) = barycentric quotient X(i)/X(j) for these (i, j): {7, 67941}, {109, 59061}, {664, 68195}, {7655, 650}, {11523, 9}
X(68349) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 307, 2}, {347, 56927, 145}, {18655, 68334, 3146}, {32003, 34059, 20008}, {39126, 40702, 17863}, {41804, 56927, 347}

X(68350) = ANTIPRODUCT OF X(9) AND X(11)

X(68350) lies on these lines: {2, 2310}, {7, 57036}, {8, 2801}, {75, 25722}, {99, 43344}, {100, 190}, {109, 65206}, {144, 28057}, {145, 53531}, {192, 42079}, {522, 4552}, {651, 2398}, {662, 7253}, {664, 23973}, {883, 926}, {1026, 25268}, {1310, 46964}, {1419, 65957}, {1441, 17668}, {1770, 67848}, {1897, 32714}, {2406, 40576}, {2765, 9070}, {2951, 45738}, {2975, 53296}, {3434, 17140}, {3667, 21362}, {3939, 62669}, {4319, 26651}, {4440, 36221}, {4454, 17165}, {4459, 13576}, {4499, 53358}, {4569, 30704}, {4579, 32735}, {4858, 53382}, {6327, 52365}, {6606, 35157}, {7192, 53350}, {9016, 25304}, {9961, 52346}, {11680, 53564}, {12530, 20245}, {14100, 20905}, {14923, 49499}, {15587, 25001}, {15726, 30807}, {17136, 68339}, {17164, 57287}, {21273, 64709}, {23529, 59688}, {24799, 63589}, {26006, 45275}, {26031, 53524}, {26669, 65952}, {28058, 60935}, {30628, 39126}, {34085, 61184}, {41906, 58992}, {43353, 53606}, {49719, 49722}, {53397, 63130}, {57928, 65642}, {62789, 66225}, {64741, 67059}, {66280, 68104}, {68106, 68118}

X(68350) = reflection of X(i) in X(j) for these {i,j}: {145, 53531}, {192, 42079}, {4552, 35338}, {66225, 62789}
X(68350) = anticomplement of X(2310)
X(68350) = trilinear pole of line {11019, 21049}
X(68350) = perspector of circumconic {{A, B, C, X(1016), X(57581)}}
X(68350) = X(i)-isoconjugate-of-X(j) for these {i, j}: {663, 63192}, {667, 56026}, {1459, 14493}, {3063, 23618}
X(68350) = X(i)-Dao conjugate of X(j) for these {i, j}: {6631, 56026}, {10001, 23618}, {11019, 3900}, {40133, 7658}, {41006, 4885}, {43182, 513}, {59573, 522}
X(68350) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {56, 17036}, {59, 144}, {109, 39351}, {249, 54107}, {651, 37781}, {658, 150}, {664, 33650}, {934, 149}, {1016, 54113}, {1020, 21221}, {1252, 30695}, {1262, 2}, {1275, 69}, {1407, 54102}, {1461, 4440}, {2149, 3177}, {2283, 14732}, {4564, 329}, {4566, 3448}, {4567, 18750}, {4569, 21293}, {4570, 45738}, {4590, 54109}, {4619, 514}, {4620, 20245}, {4998, 3436}, {6516, 34188}, {6614, 58371}, {7012, 5942}, {7045, 8}, {7115, 30694}, {7128, 5905}, {7339, 145}, {7340, 35614}, {14733, 45293}, {23586, 6604}, {23964, 46713}, {23971, 4452}, {23979, 194}, {23984, 6515}, {23985, 6392}, {23990, 46706}, {24013, 36845}, {24027, 192}, {24032, 5906}, {31615, 4462}, {35049, 14213}, {44717, 56943}, {46102, 68335}, {52378, 63}, {52610, 39352}, {53243, 44005}, {53321, 148}, {55346, 4}, {57538, 317}, {59105, 6366}, {59151, 522}, {59457, 3434}, {67038, 21286}
X(68350) = pole of line {1, 25255} with respect to the Kiepert parabola
X(68350) = pole of line {3733, 53300} with respect to the Stammler hyperbola
X(68350) = pole of line {190, 658} with respect to the Steiner circumellipse
X(68350) = pole of line {4422, 40537} with respect to the Steiner inellipse
X(68350) = pole of line {2, 85} with respect to the Yff parabola
X(68350) = pole of line {6, 10580} with respect to the Hutson-Moses hyperbola
X(68350) = pole of line {7192, 65674} with respect to the Wallace hyperbola
X(68350) = pole of line {1565, 65752} with respect to the dual conic of polar circle
X(68350) = pole of line {344, 34019} with respect to the dual conic of Feuerbach hyperbola
X(68350) = intersection, other than A, B, C, of circumconics {{A, B, C, X(100), X(68241)}}, {{A, B, C, X(513), X(53278)}}, {{A, B, C, X(883), X(59573)}}, {{A, B, C, X(3570), X(26818)}}, {{A, B, C, X(3699), X(52937)}}, {{A, B, C, X(4557), X(43344)}}, {{A, B, C, X(4569), X(4578)}}, {{A, B, C, X(6606), X(41006)}}, {{A, B, C, X(11019), X(17780)}}, {{A, B, C, X(20905), X(42720)}}, {{A, B, C, X(23343), X(40133)}}, {{A, B, C, X(23845), X(32735)}}, {{A, B, C, X(23973), X(43182)}}, {{A, B, C, X(50333), X(56323)}}, {{A, B, C, X(53337), X(60992)}}
X(68350) = barycentric product X(i)*X(j) for these (i, j): {100, 20905}, {1200, 4572}, {1978, 20978}, {3699, 60992}, {10167, 6335}, {11019, 190}, {14100, 4554}, {21049, 99}, {26818, 3952}, {30610, 59573}, {40133, 668}, {41006, 664}, {45203, 53640}, {65174, 8}
X(68350) = barycentric quotient X(i)/X(j) for these (i, j): {190, 56026}, {651, 63192}, {664, 23618}, {1200, 663}, {1783, 14493}, {10167, 905}, {11019, 514}, {14100, 650}, {20905, 693}, {20978, 649}, {21049, 523}, {22088, 1459}, {26818, 7192}, {40133, 513}, {41006, 522}, {43182, 7658}, {59573, 4885}, {60992, 3676}, {65174, 7}, {65804, 14936}
X(68350) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {100, 61185, 3952}, {190, 65200, 4578}, {522, 35338, 4552}, {4578, 65200, 17780}

X(68351) = ANTIPRODUCT OF X(9) AND X(19)

X(68351) lies on these lines: {2, 607}, {8, 20235}, {20, 68339}, {69, 3827}, {145, 347}, {281, 26157}, {1370, 17135}, {1783, 28736}, {1951, 26655}, {3436, 68337}, {3873, 6604}, {4872, 20220}, {5738, 17016}, {6360, 6542}, {6527, 68340}, {7493, 29830}, {11396, 41007}, {16049, 30941}, {17137, 20243}, {20347, 37201}, {21279, 68338}, {29616, 56943}

X(68351) = reflection of X(i) in X(j) for these {i,j}: {607, 18639}
X(68351) = anticomplement of X(607)
X(68351) = X(i)-Ceva conjugate of X(j) for these {i, j}: {57918, 2}
X(68351) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {1, 30694}, {2, 5942}, {3, 3177}, {7, 5905}, {34, 6392}, {48, 21218}, {56, 21216}, {57, 193}, {63, 144}, {69, 329}, {73, 1655}, {75, 68335}, {77, 2}, {78, 30695}, {85, 4}, {86, 92}, {201, 46707}, {212, 46706}, {222, 192}, {269, 30699}, {270, 46713}, {273, 6515}, {279, 12649}, {304, 3436}, {305, 21286}, {307, 2895}, {326, 56943}, {331, 5906}, {332, 18750}, {337, 56883}, {348, 8}, {603, 194}, {651, 25259}, {658, 521}, {664, 4391}, {738, 11851}, {757, 62798}, {873, 68338}, {905, 39351}, {1014, 3187}, {1088, 56927}, {1214, 1654}, {1231, 1330}, {1275, 61185}, {1332, 4468}, {1395, 46712}, {1414, 525}, {1434, 3868}, {1439, 17778}, {1444, 63}, {1799, 20248}, {1803, 25237}, {1804, 6360}, {1812, 45738}, {1813, 17494}, {1814, 10025}, {3718, 54113}, {3926, 52366}, {3942, 17036}, {4025, 37781}, {4554, 20293}, {4561, 4462}, {4564, 3732}, {4565, 17498}, {4569, 46400}, {4573, 7253}, {4620, 53349}, {4625, 850}, {4626, 17896}, {4637, 65099}, {6063, 21270}, {6183, 64886}, {6516, 514}, {7013, 20211}, {7045, 651}, {7053, 3210}, {7055, 4329}, {7056, 7}, {7125, 3164}, {7177, 145}, {7182, 69}, {7183, 20}, {7210, 41361}, {7318, 2994}, {13436, 31551}, {13453, 31552}, {15413, 33650}, {17094, 21221}, {17206, 3869}, {19611, 68348}, {20567, 11442}, {27832, 3621}, {30682, 36845}, {30805, 34188}, {31637, 30807}, {33673, 14361}, {34400, 962}, {36059, 21225}, {37755, 56291}, {40152, 18666}, {40443, 9}, {43736, 48381}, {44708, 17035}, {44717, 65195}, {47487, 43989}, {51653, 7665}, {51664, 148}, {52385, 3151}, {52392, 17484}, {52411, 17486}, {52565, 52364}, {53642, 64885}, {55205, 512}, {56005, 25239}, {56382, 2475}, {56972, 9965}, {57479, 5932}, {57701, 41839}, {57738, 1959}, {57785, 17220}, {57787, 317}, {57792, 68336}, {57873, 32099}, {57918, 6327}, {57985, 14206}, {59457, 4566}, {60716, 47740}, {62277, 68343}, {63193, 40571}, {63194, 56020}, {65082, 13387}, {65164, 513}, {65232, 33294}, {65233, 31290}, {65296, 522}, {65299, 47772}, {65301, 53343}
X(68351) = pole of line {918, 3287} with respect to the DeLongchamps circle
X(68351) = pole of line {17094, 35518} with respect to the Steiner circumellipse
X(68351) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {18659, 68347, 1370}

X(68352) = ANTIPRODUCT OF X(9) AND X(20)

X(68352) lies on these lines: {2, 7070}, {4, 8}, {7, 2897}, {100, 11347}, {145, 5930}, {189, 10431}, {223, 3870}, {253, 18655}, {280, 64003}, {306, 17784}, {516, 56943}, {518, 10374}, {1439, 3873}, {2270, 3692}, {2475, 10368}, {2975, 37046}, {3146, 68348}, {3182, 62874}, {3616, 18641}, {5759, 65684}, {5809, 11433}, {5906, 6223}, {5932, 20221}, {6350, 9778}, {7532, 9780}, {7672, 46017}, {8808, 26015}, {8809, 68349}, {9799, 12324}, {10365, 12649}, {10400, 20292}, {10430, 26871}, {14544, 18624}, {20262, 25006}, {29641, 62343}, {32064, 68334}, {32850, 55112}, {36991, 68335}, {39130, 41869}, {61185, 64143}

X(68352) = anticomplement of X(7070)
X(68352) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {7, 6225}, {56, 17037}, {64, 144}, {222, 68006}, {253, 3436}, {278, 14361}, {459, 68335}, {1073, 56943}, {2155, 3177}, {2184, 329}, {8809, 8}, {8810, 1032}, {19611, 52366}, {30457, 30695}, {33581, 21218}, {36079, 522}, {41082, 64583}, {41489, 30694}, {52158, 45738}, {56235, 4462}, {57921, 21286}, {60800, 20212}
X(68352) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 9812, 92}, {52365, 68336, 7}

X(68353) = ANTIPRODUCT OF X(13) AND X(16)

X(68353) lies on circumconic {{A, B, C, X(3440), X(14373)}} and on these lines: {2, 11081}, {30, 298}, {265, 301}, {299, 60474}, {302, 3130}, {633, 13340}, {3181, 11085}, {11064, 40665}, {17403, 19773}, {41887, 41889}

X(68353) = anticomplement of X(11081)
X(68353) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {1094, 18301}, {2154, 3180}, {2166, 16770}, {3376, 51271}, {11086, 192}, {11092, 8}, {23284, 21221}, {50466, 6360}, {51806, 2}, {65569, 617}

X(68354) = ANTIPRODUCT OF X(15) AND X(16)

X(68354) lies on these lines: {2, 50}, {4, 69}, {23, 325}, {26, 9723}, {30, 1272}, {94, 11071}, {95, 57805}, {99, 1273}, {183, 5169}, {290, 18125}, {297, 22151}, {328, 3153}, {524, 53416}, {631, 22463}, {850, 924}, {892, 60034}, {1007, 7493}, {1238, 11819}, {1369, 37671}, {1494, 57471}, {3001, 36163}, {3146, 52864}, {3432, 7488}, {3448, 13207}, {3933, 7540}, {3964, 7517}, {4558, 60524}, {4577, 5641}, {6148, 7809}, {7519, 37668}, {7527, 7750}, {7530, 7776}, {7552, 7752}, {7556, 7763}, {7565, 59635}, {7788, 62963}, {9145, 45918}, {9970, 51396}, {10024, 41008}, {10296, 13219}, {11061, 39099}, {11063, 18122}, {11416, 53569}, {11433, 62335}, {14118, 45198}, {14451, 40705}, {14570, 40853}, {14859, 35139}, {16310, 41626}, {18020, 41203}, {18323, 40996}, {18354, 52347}, {18365, 24975}, {18563, 41005}, {18878, 57890}, {20573, 33565}, {35296, 44388}, {37496, 39235}, {41586, 58943}, {43697, 54124}, {46439, 61490}, {46571, 62376}, {48913, 57822}, {49669, 64018}, {51481, 53507}, {52843, 68346}, {54913, 60256}, {62899, 62926}

X(68354) = reflection of X(i) in X(j) for these {i,j}: {50, 34827}, {1272, 52149}, {3146, 52864}, {4558, 60524}, {61490, 46439}
X(68354) = isogonal conjugate of X(34448)
X(68354) = isotomic conjugate of X(33565)
X(68354) = anticomplement of X(50)
X(68354) = perspector of circumconic {{A, B, C, X(6331), X(57903)}}
X(68354) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 34448}, {31, 33565}, {810, 52998}, {9247, 9381}, {34900, 62268}
X(68354) = X(i)-Dao conjugate of X(j) for these {i, j}: {2, 33565}, {3, 34448}, {50, 50}, {39062, 52998}, {40604, 51256}, {46439, 512}, {52032, 34900}, {62576, 9381}
X(68354) = X(i)-Ceva conjugate of X(j) for these {i, j}: {20573, 2}
X(68354) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {1, 18301}, {75, 1272}, {92, 12383}, {94, 8}, {265, 6360}, {328, 4329}, {476, 4560}, {1141, 17479}, {1577, 14731}, {1821, 65770}, {1989, 192}, {2166, 2}, {5627, 18668}, {6344, 5905}, {10412, 21221}, {11060, 17486}, {14206, 67092}, {14213, 67091}, {15475, 21220}, {18359, 3648}, {18384, 21216}, {18815, 41808}, {18817, 21270}, {20573, 6327}, {30690, 6224}, {32678, 31296}, {32680, 523}, {35139, 7192}, {36096, 62307}, {36129, 525}, {39295, 6758}, {43082, 4440}, {46138, 21271}, {46456, 7253}, {57716, 39118}, {63759, 69}, {66922, 1}
X(68354) = X(i)-cross conjugate of X(j) for these {i, j}: {2070, 37766}, {11597, 2}
X(68354) = pole of line {512, 11442} with respect to the anticomplementary circle
X(68354) = pole of line {512, 14592} with respect to the circumcircle of the Johnson triangle
X(68354) = pole of line {512, 47328} with respect to the polar circle
X(68354) = pole of line {1899, 44135} with respect to the Jerabek hyperbola
X(68354) = pole of line {5254, 14389} with respect to the Kiepert hyperbola
X(68354) = pole of line {249, 14570} with respect to the Kiepert parabola
X(68354) = pole of line {184, 566} with respect to the Stammler hyperbola
X(68354) = pole of line {311, 850} with respect to the Steiner circumellipse
X(68354) = pole of line {3, 2888} with respect to the Wallace hyperbola
X(68354) = pole of line {520, 1216} with respect to the dual conic of polar circle
X(68354) = pole of line {3267, 7769} with respect to the dual conic of Brocard inellipse
X(68354) = pole of line {6331, 15415} with respect to the dual conic of Jerabek hyperbola
X(68354) = pole of line {5664, 52032} with respect to the dual conic of Orthic inconic
X(68354) = pole of line {20975, 58908} with respect to the dual conic of Wallace hyperbola
X(68354) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(2070)}}, {{A, B, C, X(50), X(11597)}}, {{A, B, C, X(264), X(7578)}}, {{A, B, C, X(311), X(18020)}}, {{A, B, C, X(340), X(13582)}}, {{A, B, C, X(511), X(9380)}}, {{A, B, C, X(1235), X(5641)}}, {{A, B, C, X(1352), X(43697)}}, {{A, B, C, X(5562), X(47390)}}, {{A, B, C, X(9141), X(44138)}}, {{A, B, C, X(24978), X(44146)}}, {{A, B, C, X(34405), X(44135)}}, {{A, B, C, X(44134), X(44175)}}
X(68354) = barycentric product X(i)*X(j) for these (i, j): {2070, 76}, {11557, 40832}, {11597, 20573}, {18022, 9380}, {19552, 7769}, {24978, 99}, {37766, 69}, {58733, 7799}
X(68354) = barycentric quotient X(i)/X(j) for these (i, j): {2, 33565}, {6, 34448}, {264, 9381}, {323, 51256}, {343, 34900}, {648, 52998}, {1994, 34418}, {2070, 6}, {9380, 184}, {11557, 3003}, {11597, 50}, {19552, 2963}, {24978, 523}, {37766, 4}, {37779, 38542}, {38539, 14579}, {46439, 10413}, {58733, 1989}, {58924, 14910}
X(68354) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 69, 44135}, {30, 52149, 1272}, {50, 34827, 2}, {316, 340, 3260}, {317, 44128, 69}, {15164, 15165, 311}, {40853, 44363, 14570}, {52220, 52221, 37779}, {60524, 64783, 4558}

X(68355) = ANTIPRODUCT OF X(19) AND X(19)

X(68355) lies on these lines: {2, 2207}, {4, 30737}, {20, 64}, {22, 3785}, {76, 37201}, {99, 30552}, {112, 28696}, {183, 59349}, {264, 6815}, {304, 4329}, {305, 315}, {393, 26154}, {401, 6515}, {441, 3172}, {858, 32816}, {1078, 7493}, {1593, 41005}, {1968, 6389}, {2071, 6337}, {2138, 46741}, {2896, 46717}, {3164, 7791}, {3575, 68346}, {3926, 11413}, {3933, 21312}, {6000, 44141}, {6816, 62698}, {7396, 62310}, {7401, 44142}, {7503, 40680}, {7767, 11414}, {7787, 63084}, {8743, 28406}, {8879, 28412}, {10996, 26166}, {11348, 18928}, {12225, 64018}, {13219, 32006}, {15589, 52404}, {16043, 22240}, {16096, 31942}, {17137, 20243}, {17138, 18659}, {28432, 56832}, {28695, 41370}, {32548, 56376}, {32815, 52071}, {32818, 65711}, {32838, 63657}, {37198, 41008}, {40995, 67885}, {52398, 58846}

X(68355) = isotomic conjugate of X(42484)
X(68355) = anticomplement of X(2207)
X(68355) = X(i)-isoconjugate-of-X(j) for these {i, j}: {31, 42484}, {1973, 2139}, {2155, 40186}
X(68355) = X(i)-Dao conjugate of X(j) for these {i, j}: {2, 42484}, {2207, 2207}, {6337, 2139}, {45245, 40186}
X(68355) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {1, 6392}, {3, 21216}, {31, 46712}, {63, 193}, {69, 5905}, {75, 6515}, {76, 5906}, {77, 30699}, {78, 30694}, {255, 194}, {304, 4}, {305, 21270}, {326, 2}, {332, 92}, {336, 51481}, {345, 5942}, {348, 12649}, {394, 192}, {520, 21220}, {561, 317}, {577, 17486}, {662, 33294}, {799, 520}, {822, 25054}, {1098, 46713}, {1102, 20}, {1259, 3177}, {1264, 329}, {1332, 25259}, {1444, 3187}, {1804, 3210}, {2167, 56017}, {2289, 21218}, {2632, 54104}, {3265, 21221}, {3682, 1655}, {3718, 68335}, {3719, 144}, {3926, 8}, {3964, 6360}, {3998, 1654}, {4020, 10340}, {4091, 9263}, {4131, 4440}, {4176, 4329}, {4558, 17498}, {4561, 4391}, {4563, 7253}, {4592, 525}, {4620, 61180}, {4625, 23683}, {6507, 3164}, {6517, 17496}, {7055, 7}, {7177, 11851}, {7182, 56927}, {7183, 145}, {15394, 18663}, {17206, 3868}, {18604, 17148}, {23224, 21224}, {24018, 148}, {24041, 648}, {28724, 17489}, {30805, 149}, {33805, 50435}, {34055, 7754}, {40364, 11442}, {46254, 35360}, {52385, 17778}, {52387, 46707}, {52396, 2895}, {52430, 8264}, {52565, 2475}, {52608, 21300}, {52616, 37781}, {52617, 21294}, {55202, 850}, {57780, 32001}, {57918, 68336}, {57955, 2052}, {57985, 62305}, {57998, 6504}, {62276, 5889}, {62277, 1993}, {62719, 110}, {65164, 521}
X(68355) = X(i)-cross conjugate of X(j) for these {i, j}: {1619, 46741}, {15259, 2}
X(68355) = pole of line {525, 2451} with respect to the DeLongchamps circle
X(68355) = pole of line {648, 56008} with respect to the Kiepert parabola
X(68355) = pole of line {3265, 52617} with respect to the Steiner circumellipse
X(68355) = pole of line {20, 159} with respect to the Wallace hyperbola
X(68355) = pole of line {5020, 40995} with respect to the dual conic of Moses HK-parabola
X(68355) = intersection, other than A, B, C, of circumconics {{A, B, C, X(64), X(1619)}}, {{A, B, C, X(253), X(40009)}}, {{A, B, C, X(2207), X(15259)}}, {{A, B, C, X(32830), X(34415)}}, {{A, B, C, X(39434), X(61088)}}
X(68355) = barycentric product X(i)*X(j) for these (i, j): {1619, 76}, {2138, 305}, {46741, 69}
X(68355) = barycentric quotient X(i)/X(j) for these (i, j): {2, 42484}, {20, 40186}, {69, 2139}, {1619, 6}, {2138, 25}, {15259, 2207}, {46741, 4}
X(68355) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {20, 253, 32830}, {20, 51884, 69}, {1370, 13575, 40123}, {1968, 6389, 26204}, {7750, 20477, 20}

  Similar inscribed & circumscribed triangles: X(68356) - X(68369)  

1) Given a triangle ABC, to find points A', B', C' such that A'B'C' is inscribed in and directly similar to ABC.

2) Given a triangle ABC, to find points A", B", C" such that A"B"C" is circumscribed to and directly similar to ABC.

and, for barycentrics of vertices of T" = A"B"C", let's take the parallel lines to the sidelines of T' through A, B, C. In this way, barycentrics of vertices of T" can be expressed as functions of the same parameter t, as:

 A" = 1 : (t-1)*(-a^2+b^2+c^2)/((a^2-b^2+c^2)*t-a^2+b^2) : t*(-a^2+b^2+c^2)/((a^2+b^2-c^2)*t-b^2)
 B" = ((a^2-b^2+c^2)*t-a^2+b^2)/(-a^2+b^2+c^2) : t-1 : ((a^2-b^2+c^2)*t-c^2)*(t-1)/((a^2+b^2-c^2)*t-a^2+c^2)
 C" = ((a^2+b^2-c^2)*t-b^2)/(-a^2+b^2+c^2) : t*((a^2+b^2-c^2)*t-a^2+c^2)/((a^2-b^2+c^2)*t-c^2) : t

1. When t varies, P' moves on a line with tripole Q'(P), equivalent to:
    Q'(P) = IsotomicConjugate(Anticomplement(IsotomicConjugate(PolarConjugate(Orthoassociate(P)))))
   or, algebraically but simpler:
    Q'(P) = BarycentricQuotient(AntigonalConjugate(P), P)
   this is, for P = x : y : z, normalized barycentrics:
    Q'(P) = ((x+1)*x*SA+(y-1)*y*SB+(z-1)*z*SC)^-1 : :
2. When t varies, P" moves on a circle through X(4) and center O"(P) = reflection of P in X(5).
3. As constructed, triangles T' and T" are homothetic for any t, and, as t varies, their homothetic center moves on the Yff hyperbola (see Wolfram Mathworld's Yff Hyperbola).

((1, 18359), (2, 671), (3, 94), (5, 13582), (6, 18019), (7, 41798), (8, 88), (9, 68356), (10, 6650), (11, 68357), (12, 68358), (13, 11092), (14, 11078), (15, 68359), (16, 68360), (17, 68361), (18, 68362), (20, 16080), (21, 68363), (22, 46105), (23, 76), (27, 68364), (64, 52516), (65, 52500), (66, 52513), (67, 23), (68, 52505), (69, 111), (72, 21907), (74, 46106), (76, 694), (80, 3218), (83, 17949), (98, 297), (99, 2501), (100, 17924), (101, 46107), (105, 46108), (106, 46109), (107, 525), (108, 4391), (109, 46110), (110, 14618), (111, 44146), (112, 850), (115, 13485), (125, 15351), (136, 54453), (146, 40384), (147, 34536), (148, 34537), (149, 1016), (150, 1252), (152, 59195), (153, 59196), (186, 5392), (226, 17947), (242, 39700), (264, 60039), (265, 323), (315, 1976), (316, 6), (321, 17946), (329, 34056), (376, 58268), (381, 55957), (382, 56063), (403, 6504), (468, 2996), (476, 44427), (550, 18366), (621, 6151), (622, 2981), (671, 524), (842, 60502), (858, 4), (879, 46787), (895, 3266), (915, 48380), (917, 48381), (925, 57065), (930, 67102), (933, 18314), (935, 9979), (962, 56234), (1113, 2592), (1114, 2593), (1117, 40604), (1141, 14918), (1156, 37780), (1157, 53028), (1177, 52512), (1263, 37779), (1289, 33294), (1294, 51358), (1297, 60516), (1300, 3580), (1304, 41079), (1305, 57043), (1309, 10015), (1312, 13581), (1313, 13580), (1316, 1916), (1320, 4358), (1325, 321), (1337, 41000), (1338, 41001), (1370, 60133), (1657, 66768), (1785, 2994), (1878, 39696), (1916, 3978), (1995, 55973), (2070, 11140), (2071, 2052), (2072, 13579), (2074, 43675), (2373, 5523), (2374, 47286), (2394, 51228), (2697, 50188), (3109, 4080), (3146, 44877), (3153, 275), (3154, 12066), (3254, 3935), (3436, 34051), (3448, 249), (3465, 30690), (3484, 324), (3563, 51481), (3681, 34578), (3869, 2006), (4226, 14223))

(1, 355), (2, 381), (3, 4), (4, 3), (5, 5), (6, 1352), (7, 5779), (8, 1482), (9, 5805), (10, 946), (11, 119), (12, 26470), (13, 5617), (14, 5613), (15, 20428), (16, 20429), (17, 16626), (18, 16627), (20, 382), (21, 37230), (22, 31723), (23, 7574), (24, 18404), (25, 18531), (26, 18569), (27, 68365), (32, 54393), (35, 68366), (36, 68367), (37, 64088), (39, 6248), (40, 12699), (49, 58922), (51, 5891), (52, 5562), (53, 42353), (54, 6288), (55, 37820), (56, 37821), (57, 37822), (58, 37823), (61, 37824), (62, 37825), (63, 37826), (64, 5878), (65, 5887), (66, 19149), (67, 9970), (68, 155), (69, 1351), (72, 24474), (74, 7728), (75, 20430), (76, 3095), (79, 3652), (80, 6265), (83, 6287), (84, 6259), (90, 41688), (95, 42350), (98, 6033), (99, 6321), (100, 10738), (101, 10739), (102, 10740), (103, 10741), (104, 10742), (105, 10743), (106, 10744), (107, 10745), (108, 10746), (109, 10747), (110, 265), (111, 10748), (112, 10749), (113, 125), (114, 115), (115, 114), (116, 118), (117, 124), (118, 116), (119, 11), (120, 5511), (121, 5510), (122, 133), (123, 25640), (124, 117), (125, 113), (126, 5512), (127, 132), (128, 137), (129, 130), (130, 129), (131, 136), (132, 127), (133, 122), (136, 131), (137, 128), (140, 546), (141, 5480), (142, 63970), (143, 11591), (144, 60922), (145, 12645), (146, 10620), (147, 12188), (148, 13188), (149, 12331), (150, 38572), (151, 38573), (152, 38574), (153, 12773), (155, 68), (175, 68368), (176, 68369), (182, 3818), (184, 18474), (185, 12162), (186, 18403), (187, 13449), (190, 24833), (191, 16159), (193, 11898), (194, 13108), (195, 2888), (206, 51756), (214, 6246), (216, 39530), (226, 51755), (235, 11585)

X(68356) = TRIPOLE OF THE LINE-LOCUS OF X(9) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

X(68356) lies on the circumhyperbola dual of Yff parabola and these lines: {2, 1111}, {7, 149}, {75, 23989}, {85, 63166}, {86, 16727}, {664, 3957}, {673, 3218}, {675, 1308}, {903, 35171}, {1223, 60969}, {2400, 36038}, {3935, 30806}, {4358, 36807}, {4373, 17154}, {4671, 39749}, {5744, 42318}, {6548, 53240}, {8024, 57925}, {9436, 18815}, {14154, 61157}, {15728, 65637}, {16750, 52394}, {17121, 43190}, {18359, 62429}, {20568, 62536}, {21296, 39695}, {24203, 29817}, {27475, 27493}, {34018, 39293}, {37206, 60970}, {37780, 38468}, {39704, 62697}, {47871, 62619}, {48571, 60479}

X(68356) = isotomic conjugate of X(3935)
X(68356) = isogonal conjugate of X(19624)
X(68356) = polar conjugate of X(60355)
X(68356) = trilinear pole of the line {142, 514} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68356) = perspector of the inconic with center X(26015)
X(68356) = touchpoint of circumhyperbola dual of Yff parabola and line {26015, 68356}
X(68356) = pole of the the tripolar of X(60355) with respect to the polar circle
X(68356) = pole of the line {3254, 3887} with respect to the Steiner circumellipse
X(68356) = pole of the line {3935, 19624} with respect to the Steiner-Wallace hyperbola
X(68356) = barycentric product X(i)*X(j) for these {i, j}: {75, 34578}, {76, 67146}, {85, 3254}, {514, 35171}, {664, 60489}, {693, 37143}, {1308, 3261}, {24002, 60488}
X(68356) = trilinear product X(i)*X(j) for these {i, j}: {2, 34578}, {7, 3254}, {75, 67146}, {513, 35171}, {514, 37143}, {651, 60489}, {693, 1308}, {1088, 42064}, {3676, 60488}, {15734, 37780}
X(68356) = trilinear quotient X(i)/X(j) for these (i, j): (2, 5526), (7, 2078), (75, 3935), (76, 17264), (85, 37787), (92, 60355), (513, 8645), (514, 22108), (693, 3887), (1088, 38459), (1308, 692), (3254, 55), (3261, 30565), (15734, 18889), (20880, 61030), (24002, 43050)

X(68357) = TRIPOLE OF THE LINE-LOCUS OF X(11) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

X(68357) lies on these lines: {516, 5080}, {908, 37798}, {1146, 34529}, {1262, 55153}, {3436, 35313}, {4391, 37781}, {20920, 30807}

X(68357) = cyclocevian conjugate of X(100)
X(68357) = isotomic conjugate of X(37781)
X(68357) = polar conjugate of X(60356)
X(68357) = trilinear pole of the line {676, 2804} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68357) = perspector of the inconic with center X(651)
X(68357) = pole of the the tripolar of X(60356) with respect to the polar circle
X(68357) = barycentric product X(75)*X(29374)
X(68357) = trilinear product X(2)*X(29374)
X(68357) = trilinear quotient X(i)/X(j) for these (i, j): (2, 1768), (75, 37781), (92, 60356), (908, 34345), (4564, 57105), (29374, 6)

X(68358) = TRIPOLE OF THE LINE-LOCUS OF X(12) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

X(68358) lies on these lines: {149, 39630}, {5249, 5483}, {6734, 11604}

X(68358) = intersection, other than {A, B, C}, of the circumconics through X(i), X(j) for these {i, j}: {2, 331}, {1029, 2185}
X(68358) = trilinear product X(15910)*X(21907)
X(68358) = trilinear quotient X(i)/X(j) for these (i, j): (15910, 17796), (21907, 15932)

X(68359) = TRIPOLE OF THE LINE-LOCUS OF X(15) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

X(68359) lies on the cubic K342b and these lines: {3, 252}, {14, 11140}, {17, 94}, {76, 55220}, {264, 11127}, {301, 302}, {324, 473}, {338, 11130}, {622, 11582}, {11126, 51268}, {16806, 40855}, {40667, 53474}, {41478, 41760}, {60502, 65346}

X(68359) = polar conjugate of X(10632)
X(68359) = isotomic conjugate of X(11126)
X(68359) = isogonal conjugate of X(11135)
X(68359) = trilinear pole of the line {623, 18314} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68359) = perspector of the inconic with center X(33530)
X(68359) = pole of the the tripolar of X(10632) with respect to the polar circle
X(68359) = pole of the line {3201, 11135} with respect to the Stammler hyperbola
X(68359) = pole of the line {11126, 11135} with respect to the Steiner-Wallace hyperbola
X(68359) = barycentric product X(i)*X(j) for these {i, j}: {14, 34389}, {17, 301}, {76, 11087}, {264, 52203}, {300, 11600}, {8603, 20573}, {8836, 11140}, {20572, 50469}, {20579, 55220}
X(68359) = trilinear product X(i)*X(j) for these {i, j}: {75, 11087}, {92, 52203}, {2154, 34389}, {2962, 8836}, {8603, 63759}
X(68359) = trilinear quotient X(i)/X(j) for these (i, j): (2, 35199), (17, 2152), (63, 64465), (75, 11126), (92, 10632), (301, 65571), (561, 11132), (2166, 11083), (2962, 8604), (8836, 2964), (11087, 31), (11600, 2151), (19779, 1095), (23994, 66262)
X(68359) = (X(11092), X(11144))-harmonic conjugate of X(52203)

X(68360) = TRIPOLE OF THE LINE-LOCUS OF X(16) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

X(68360) lies on the cubic K342a and these lines: {3, 252}, {13, 11140}, {18, 94}, {76, 55222}, {264, 11126}, {300, 303}, {324, 472}, {338, 11131}, {621, 11581}, {11127, 51275}, {16807, 40854}, {40668, 53474}, {41477, 41760}, {60502, 65347}

X(68360) = polar conjugate of X(10633)
X(68360) = isotomic conjugate of X(11127)
X(68360) = isogonal conjugate of X(11136)
X(68360) = trilinear pole of the line {624, 18314} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68360) = perspector of the inconic with center X(33529)
X(68360) = pole of the the tripolar of X(10633) with respect to the polar circle
X(68360) = pole of the line {3200, 11136} with respect to the Stammler hyperbola
X(68360) = pole of the line {11127, 11136} with respect to the Steiner-Wallace hyperbola
X(68360) = barycentric product X(i)*X(j) for these {i, j}: {13, 34390}, {18, 300}, {76, 11082}, {264, 52204}, {301, 11601}, {8604, 20573}, {8838, 11140}, {20572, 50468}, {20578, 55222}
X(68360) = trilinear product X(i)*X(j) for these {i, j}: {75, 11082}, {92, 52204}, {2153, 34390}, {2962, 8838}, {8604, 63759}
X(68360) = trilinear quotient X(i)/X(j) for these (i, j): (2, 35198), (18, 2151), (63, 64464), (75, 11127), (92, 10633), (300, 65572), (561, 11133), (2166, 11088), (2962, 8603), (8838, 2964), (11082, 31), (11601, 2152), (19778, 1094), (23994, 66263)
X(68360) = (X(11078), X(11143))-harmonic conjugate of X(52204)

X(68361) = TRIPOLE OF THE LINE-LOCUS OF X(17) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

X(68361) lies on the cubic K185 and these lines: {2, 5469}, {524, 11117}, {8838, 41999}, {19712, 62984}, {23302, 32036}

X(68361) = reflection of X(32036) in X(23302)
X(68361) = antitomic conjugate of X(302)
X(68361) = isotomic conjugate of the antitomic conjugate of X(44361)
X(68361) = trilinear pole of the line {629, 23872} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68361) = inverse of X(11602) in Steiner circumellipse
X(68361) = barycentric product X(302)*X(11602)
X(68361) = trilinear product X(11602)*X(65571)

X(68362) = TRIPOLE OF THE LINE-LOCUS OF X(18) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

X(68362) lies on the cubic K185 and these lines: {2, 5470}, {524, 11118}, {8836, 42000}, {19713, 62983}, {23303, 32037}

X(68362) = reflection of X(32037) in X(23303)
X(68362) = antitomic conjugate of X(303)
X(68362) = isotomic conjugate of the antitomic conjugate of X(44362)
X(68362) = trilinear pole of the line {630, 23873} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68362) = inverse of X(11603) in Steiner circumellipse
X(68362) = barycentric product X(303)*X(11603)
X(68362) = trilinear product X(11603)*X(65572)

X(68363) = TRIPOLE OF THE LINE-LOCUS OF X(21) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

X(68363) lies on the Kiepert hyperbola and these lines: {2, 16732}, {4, 2771}, {10, 1109}, {92, 60246}, {94, 48380}, {98, 1290}, {226, 24086}, {321, 338}, {648, 40395}, {671, 35156}, {1029, 17483}, {1751, 21376}, {3218, 24624}, {4358, 60251}, {4359, 60235}, {4440, 54119}, {4552, 60188}, {4707, 60074}, {5080, 55012}, {6539, 42708}, {6742, 57710}, {13576, 66280}, {20905, 36789}, {30588, 53510}, {39295, 66922}, {46105, 46108}, {59491, 66634}

X(68363) = polar conjugate of X(2074)
X(68363) = isogonal conjugate of X(19622)
X(68363) = isotomic conjugate of X(37783)
X(68363) = trilinear pole of the line {442, 523} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68363) = touchpoint of Kiepert circumhyperbola and line {62305, 68363}
X(68363) = pole of the line {8674, 47235} with respect to the polar circle
X(68363) = pole of the line {8674, 11604} with respect to the Steiner circumellipse
X(68363) = pole of the line {19622, 37783} with respect to the Steiner-Wallace hyperbola
X(68363) = barycentric product X(i)*X(j) for these {i, j}: {75, 5620}, {321, 21907}, {523, 35156}, {693, 66280}, {850, 1290}, {1441, 11604}, {1577, 65238}
X(68363) = trilinear product X(i)*X(j) for these {i, j}: {2, 5620}, {10, 21907}, {226, 11604}, {514, 66280}, {523, 65238}, {661, 35156}, {1290, 1577}
X(68363) = trilinear quotient X(i)/X(j) for these (i, j): (2, 5127), (10, 17796), (75, 37783), (92, 2074), (226, 5172), (313, 32849), (514, 42741), (661, 42670), (1290, 163), (1577, 8674), (3261, 65669), (3936, 35204), (5620, 6), (7178, 51646), (11604, 284), (14206, 16164), (21907, 58), (24006, 47235)

X(68364) = TRIPOLE OF THE LINE-LOCUS OF X(27) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

X(68364) lies on the Kiepert hyperbola and these lines: {4, 14543}, {98, 53925}, {1751, 21044}, {16080, 48381}, {21017, 60135}, {24624, 37774}

X(68364) = polar conjugate of X(57591)
X(68364) = trilinear pole of the line {440, 523} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68364) = pole of the the tripolar of X(57591) with respect to the polar circle
X(68364) = barycentric product X(850)*X(53925)
X(68364) = trilinear product X(1577)*X(53925)
X(68364) = trilinear quotient X(92)/X(57591)

X(68365) = CENTER OF THE CIRCLE-LOCUS OF X(27) IN THE SIMILAR-TO-ABC CIRCUMSCRIBED TRIANGLES

X(68365) lies on these lines: {2, 3}, {71, 265}, {79, 41393}, {113, 40589}, {572, 18388}, {573, 18390}, {1568, 1790}, {1762, 8251}, {2193, 45926}, {2328, 18400}, {2822, 10741}, {3017, 3284}, {3095, 46182}, {3583, 23207}, {5886, 51697}, {7687, 37508}, {8680, 20430}, {10246, 51721}, {10319, 18540}, {13851, 22080}, {14547, 18455}, {14561, 51731}, {18591, 45924}, {22139, 44665}, {40263, 41340}, {55010, 63323}

X(68365) = midpoint of X(i) and X(j) for these (i, j): {4, 3151}, {30266, 52844}
X(68365) = reflection of X(i) in X(j) for these (i, j): (3, 440), (27, 5)
X(68365) = intersection, other than {A, B, C}, of the circumconics through X(i), X(j) for these {i, j}: {27, 265}, {68, 7554}
X(68365) = anticomplement of X(15762) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(68365) = anticomplement of X(44243) with respect to the Moses-Steiner osculatory triangle
X(68365) = X(15762)-of-anti-Ehrmann-mid triangle
X(68365) = X(3151)-of-Euler triangle
X(68365) = X(440)-of-X3-ABC reflections triangle
X(68365) = X(27)-of-Johnson triangle

X(68366) = CENTER OF THE CIRCLE-LOCUS OF X(35) IN THE SIMILAR-TO-ABC CIRCUMSCRIBED TRIANGLES

X(68366) lies on these lines: {2, 33862}, {3, 25639}, {4, 8}, {5, 35}, {10, 36865}, {11, 61534}, {30, 11012}, {65, 62354}, {79, 24475}, {119, 546}, {140, 31262}, {149, 10222}, {214, 40259}, {381, 4421}, {382, 22758}, {388, 61287}, {497, 61276}, {528, 65949}, {944, 33281}, {946, 6265}, {952, 3585}, {1012, 45630}, {1056, 61282}, {1058, 61279}, {1154, 48937}, {1352, 9047}, {1385, 2475}, {1478, 11011}, {1479, 2646}, {1483, 5270}, {1484, 5563}, {1656, 58404}, {1699, 45770}, {2476, 32613}, {2779, 7728}, {2886, 7491}, {3073, 45926}, {3091, 20066}, {3560, 12953}, {3579, 6840}, {3584, 61512}, {3652, 51118}, {3655, 12116}, {3656, 26332}, {3822, 37621}, {3825, 45976}, {3838, 24299}, {3871, 59392}, {3913, 11929}, {4190, 26492}, {4294, 6867}, {4297, 47032}, {4302, 6862}, {4857, 5901}, {5046, 9956}, {5225, 6826}, {5434, 32214}, {5499, 15931}, {5587, 11010}, {5691, 11014}, {5722, 13750}, {5787, 65998}, {5805, 41688}, {5840, 6831}, {5841, 24390}, {5842, 6842}, {5881, 11280}, {5885, 20292}, {6101, 38474}, {6224, 11567}, {6253, 37406}, {6583, 11604}, {6796, 6980}, {6830, 26285}, {6839, 9955}, {6864, 61266}, {6868, 31418}, {6871, 26487}, {6885, 10591}, {6890, 35249}, {6893, 18782}, {6895, 28146}, {6901, 11230}, {6902, 11231}, {6903, 31663}, {6911, 10896}, {6914, 14794}, {6915, 59391}, {6923, 18481}, {6924, 7741}, {6928, 26446}, {6929, 37568}, {6951, 13624}, {6952, 26086}, {6971, 25440}, {6985, 36999}, {7330, 24468}, {7354, 10943}, {7681, 28452}, {7951, 32141}, {8227, 64473}, {10267, 17532}, {10269, 50239}, {10483, 32153}, {10599, 20075}, {10679, 10894}, {10680, 11235}, {10707, 45977}, {10724, 21669}, {10785, 31295}, {10915, 13272}, {11374, 64086}, {11491, 17577}, {11500, 18499}, {11680, 26286}, {11826, 37356}, {11928, 22753}, {12047, 37733}, {12114, 18544}, {12515, 12616}, {12737, 13273}, {13729, 38140}, {13743, 16761}, {14217, 64291}, {16128, 40263}, {16159, 24474}, {17502, 37163}, {17530, 31659}, {17579, 32612}, {18444, 49107}, {18447, 51751}, {18524, 63964}, {18527, 58569}, {18990, 37726}, {19925, 24042}, {22765, 24387}, {23961, 37256}, {26333, 33596}, {26386, 26413}, {26389, 26410}, {28160, 37437}, {28204, 62969}, {37251, 67857}, {37290, 65632}, {37573, 45944}, {37702, 61541}, {52383, 54350}, {61277, 65991}, {61533, 63273}

X(68366) = midpoint of X(i) and X(j) for these (i, j): {4, 52367}, {5691, 11014}, {5881, 11280}, {15908, 52837}
X(68366) = reflection of X(i) in X(j) for these (i, j): (3, 25639), (35, 5), (944, 33281), (11849, 67856), (37727, 11011), (37733, 12047)
X(68366) = anticomplement of X(33862)
X(68366) = pole of the line {1837, 37826} with respect to the Feuerbach circumhyperbola
X(68366) = isogonal conjugate of X(3652) with respect to the Johnson triangle
X(68366) = anticomplement of X(33862) with respect to these triangles: anti-Artzt, 1st anti-Brocard, anti-McCay, anticomplementary, Artzt, 1st Brocard, 1st Brocard-reflected, inner-Fermat, outer-Fermat, 1st half-diamonds, 2nd half-diamonds, 1st half-squares, 2nd half-squares, inverse-in-excircles, McCay, medial, 1st Neuberg, 2nd Neuberg, inner-Vecten, outer-Vecten
X(68366) = anticomplement of X(67856) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(68366) = complement of X(11010) with respect to the Fuhrmann triangle
X(68366) = complement of X(11849) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(68366) = X(35)-of-Johnson triangle
X(68366) = X(11849)-of-Ehrmann-mid triangle
X(68366) = X(25639)-of-X3-ABC reflections triangle
X(68366) = X(30420)-of-Ehrmann-vertex triangle (ABC acute)
X(68366) = X(33862)-of-anticomplementary triangle
X(68366) = X(52367)-of-Euler triangle
X(68366) = X(67856)-of-anti-Ehrmann-mid triangle
X(68366) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (4, 355, 68367), (4, 3434, 10526), (4, 10525, 12699), (79, 49176, 24475), (381, 11849, 67856), (1479, 6917, 5886), (6871, 37000, 26487), (6923, 48482, 18481), (10738, 37230, 946)

X(68367) = CENTER OF THE CIRCLE-LOCUS OF X(36) IN THE SIMILAR-TO-ABC CIRCUMSCRIBED TRIANGLES

X(68367) lies on these lines: {2, 23961}, {3, 3814}, {4, 8}, {5, 36}, {10, 36866}, {12, 32760}, {30, 119}, {79, 61541}, {80, 14988}, {104, 37375}, {136, 64513}, {140, 31263}, {145, 23960}, {153, 28204}, {381, 535}, {382, 11499}, {388, 25405}, {484, 5587}, {497, 61287}, {515, 6265}, {516, 35460}, {519, 10738}, {529, 65948}, {546, 26470}, {758, 6246}, {912, 62354}, {952, 3583}, {993, 6980}, {1056, 61279}, {1058, 61282}, {1155, 6917}, {1319, 1478}, {1352, 9037}, {1385, 5046}, {1479, 5048}, {1483, 4857}, {1484, 65140}, {1532, 5841}, {1656, 6681}, {1737, 13273}, {1837, 53615}, {2392, 37823}, {2475, 9956}, {2478, 18857}, {2829, 6882}, {3058, 32213}, {3091, 20067}, {3149, 45631}, {3245, 18357}, {3560, 5172}, {3579, 37437}, {3582, 60759}, {3627, 5537}, {3652, 19925}, {3655, 12115}, {3656, 26333}, {3822, 7489}, {3825, 37535}, {3843, 62318}, {3851, 67710}, {4192, 30981}, {4193, 32612}, {4293, 6973}, {4299, 6959}, {4316, 61580}, {4640, 5123}, {4973, 38161}, {5087, 6256}, {5122, 6826}, {5126, 5229}, {5131, 7989}, {5183, 61258}, {5187, 26492}, {5193, 18990}, {5270, 5901}, {5450, 6971}, {5535, 7330}, {5538, 5720}, {5570, 5722}, {5691, 45770}, {5816, 67501}, {5840, 17757}, {5842, 38757}, {5844, 22938}, {5881, 18514}, {6001, 16128}, {6259, 41688}, {6284, 10942}, {6288, 18330}, {6684, 47032}, {6713, 17533}, {6838, 35250}, {6839, 38140}, {6840, 28160}, {6841, 33961}, {6842, 57288}, {6872, 26487}, {6895, 33697}, {6902, 13624}, {6909, 10728}, {6911, 12943}, {6912, 59392}, {6913, 41345}, {6914, 7951}, {6924, 10483}, {6930, 10590}, {6939, 61266}, {6941, 26286}, {6951, 11231}, {6965, 11230}, {7491, 18242}, {7686, 16159}, {7741, 32153}, {9955, 13729}, {10113, 61638}, {10222, 20060}, {10269, 17556}, {10572, 37733}, {10598, 20076}, {10679, 11236}, {10680, 10893}, {10744, 40100}, {10747, 38954}, {10826, 24467}, {10894, 37234}, {11114, 32613}, {11496, 11929}, {11500, 18542}, {11681, 26285}, {11827, 37406}, {11928, 12513}, {12515, 12761}, {12737, 12764}, {13391, 48937}, {13587, 64008}, {13743, 67856}, {15325, 23513}, {15680, 33862}, {16118, 27247}, {18455, 51889}, {18524, 38755}, {18838, 57282}, {22835, 26332}, {24475, 37702}, {26086, 27529}, {26321, 63963}, {30144, 40264}, {31649, 61512}, {31659, 57002}, {31673, 35459}, {31835, 47033}, {32141, 65134}, {33110, 38176}, {34586, 56825}, {35448, 64725}, {36001, 44982}, {36004, 66045}, {37251, 67046}, {37356, 64000}, {37725, 65632}, {38455, 64138}, {51518, 64792}, {54391, 59391}, {56790, 61553}

X(68367) = midpoint of X(i) and X(j) for these (i, j): {4, 5080}, {382, 35000}, {2077, 52851}, {5881, 64896}, {6909, 10728}, {36001, 44982}, {37725, 65632}
X(68367) = reflection of X(i) in X(j) for these (i, j): (3, 3814), (36, 5), (145, 23960), (1532, 67864), (10225, 9956), (10738, 24042), (12737, 30384), (22765, 67857), (37727, 5048), (41698, 22799)
X(68367) = anticomplementary conjugate of the anticomplement of X(23959)
X(68367) = anticomplement of X(23961)
X(68367) = orthoassociate of X(41722)
X(68367) = inverse of X(355) in Johnson triangle circumcircle
X(68367) = inverse of X(12245) in anticomplementary circle
X(68367) = inverse of X(41722) in polar circle
X(68367) = pole of the line {513, 12245} with respect to the anticomplementary circle
X(68367) = pole of the line {355, 513} with respect to the Johnson triangle circumcircle
X(68367) = pole of the line {513, 41722} with respect to the polar circle
X(68367) = pole of the line {1837, 6797} with respect to the Feuerbach circumhyperbola
X(68367) = isogonal conjugate of X(6265) with respect to the Johnson triangle
X(68367) = anticomplement of X(23961) with respect to these triangles: anti-Artzt, 1st anti-Brocard, anti-McCay, anticomplementary, Artzt, 1st Brocard, 1st Brocard-reflected, inner-Fermat, outer-Fermat, 1st half-diamonds, 2nd half-diamonds, 1st half-squares, 2nd half-squares, inverse-in-excircles, McCay, medial, 1st Neuberg, 2nd Neuberg, inner-Vecten, outer-Vecten
X(68367) = anticomplement of X(67857) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(68367) = complement of X(484) with respect to the Fuhrmann triangle
X(68367) = complement of X(22765) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(68367) = X(36)-of-Johnson triangle
X(68367) = X(3814)-of-X3-ABC reflections triangle
X(68367) = X(5080)-of-Euler triangle
X(68367) = X(22765)-of-Ehrmann-mid triangle
X(68367) = X(23961)-of-anticomplementary triangle
X(68367) = X(30370)-of-Ehrmann-vertex triangle (ABC acute)
X(68367) = X(32760)-of-outer-Johnson triangle
X(68367) = X(46031)-of-2nd Conway triangle
X(68367) = X(54073)-of-Fuhrmann triangle
X(68367) = X(67857)-of-anti-Ehrmann-mid triangle
X(68367) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {4, 5080, 34172}, {100, 36167, 64688}
X(68367) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (4, 355, 68366), (4, 3436, 10525), (4, 10526, 12699), (381, 22765, 67857), (1478, 6929, 5886), (5187, 37002, 26492), (6256, 6928, 18481), (31160, 52851, 2077)

X(68368) = CENTER OF THE CIRCLE-LOCUS OF X(175) IN THE SIMILAR-TO-ABC CIRCUMSCRIBED TRIANGLES

X(68368) lies on these lines: {3, 14121}, {5, 175}, {355, 382}, {952, 30334}, {1656, 31534}, {5587, 51763}, {7330, 51955}

X(68368) = reflection of X(i) in X(j) for these (i, j): (3, 14121), (175, 5)
X(68368) = complement of X(51763) with respect to the Fuhrmann triangle
X(68368) = X(175)-of-Johnson triangle
X(68368) = X(12277)-of-4th Euler triangle
X(68368) = X(12288)-of-3rd Euler triangle
X(68368) = X(14121)-of-X3-ABC reflections triangle
X(68368) = (X(355), X(5779))-harmonic conjugate of X(68369)

X(68369) = CENTER OF THE CIRCLE-LOCUS OF X(176) IN THE SIMILAR-TO-ABC CIRCUMSCRIBED TRIANGLES

X(68369) lies on these lines: {3, 7090}, {5, 176}, {355, 382}, {952, 30333}, {1656, 31535}, {3579, 8986}, {5587, 51764}, {7330, 51957}

X(68369) = reflection of X(i) in X(j) for these (i, j): (3, 7090), (176, 5), (8986, 3579)
X(68369) = anticomplement of X(8984) with respect to the Moses-Steiner osculatory triangle
X(68369) = complement of X(51764) with respect to the Fuhrmann triangle
X(68369) = X(176)-of-Johnson triangle
X(68369) = X(7090)-of-X3-ABC reflections triangle
X(68369) = X(12276)-of-4th Euler triangle
X(68369) = X(12287)-of-3rd Euler triangle
X(68369) = (X(355), X(5779))-harmonic conjugate of X(68368)

X(68370) = X(6)X(474)∩X(511)X(3953)

See Antreas Hatzipolakis and César Lozada, euclid 8301.

X(68370) lies on these lines: {1, 10108}, {6, 474}, {171, 35197}, {221, 37610}, {511, 3953}, {519, 2650}, {1201, 51714}, {1480, 48897}, {3157, 5264}, {3670, 11573}, {3909, 52564}, {4424, 67968}, {5255, 6126}, {5724, 49743}, {10106, 53530}, {12575, 66659}, {18139, 37693}

X(68371) = X(3)X(1724)∩X(35)X(10108)

See Antreas Hatzipolakis and César Lozada, euclid 8301.

X(68371) lies on these lines: {3, 1724}, {30, 12571}, {35, 10108}, {1385, 48915}, {3579, 48909}, {5010, 49557}, {13624, 48894}, {15178, 48919}, {16192, 48907}, {17502, 37425}, {31663, 48893}, {31666, 48903}, {33697, 50416}, {39578, 64534}

X(68371) = midpoint of X(i) and X(j) for these (i, j): {15178, 48919}, {31663, 48893}

X(68372) = X(7)X(349)∩X(31)X(56)

See Antreas Hatzipolakis and César Lozada, euclid 8301.

X(68372) lies on these lines: {7, 349}, {12, 1463}, {31, 56}, {57, 3216}, {65, 519}, {73, 59173}, {244, 42450}, {942, 49745}, {982, 50617}, {1104, 3937}, {1122, 1439}, {1319, 17705}, {1355, 1358}, {1357, 2842}, {1412, 7342}, {1417, 13370}, {1428, 8614}, {1431, 52372}, {1469, 3214}, {2099, 63460}, {2213, 40151}, {2392, 24167}, {2594, 62739}, {3600, 20041}, {3670, 11573}, {3752, 23154}, {3784, 37549}, {4292, 45022}, {5298, 28389}, {8679, 24443}, {9433, 50513}, {11509, 15625}, {16610, 29958}, {17054, 26892}, {18360, 41346}, {18838, 20617}, {24046, 67893}, {26910, 62802}, {34502, 39780}, {37591, 41777}, {40959, 64132}, {41682, 49487}, {42448, 52541}, {56412, 64115}

X(68372) = pole of the line {1428, 3733} with respect to the incircle
X(68372) = pole of the line {3782, 41007} with respect to the circumhyperbola dual of Yff parabola
X(68372) = pole of the line {12053, 63997} with respect to the Feuerbach circumhyperbola
X(68372) = pole of the line {47795, 52595} with respect to the Steiner inellipse
X(68372) = barycentric product X(i)*X(j) for these {i,j}: {56, 17184}, {57, 3670}, {65, 18601}, {226, 52564}, {278, 11573}, {331, 23197}, {1014, 4016}, {1408, 20896}, {1412, 3454}, {1434, 20966}, {3669, 3909}, {4565, 21121}, {7203, 61167}, {7341, 20654}
X(68372) = trilinear product X(i)*X(j) for these {i,j}: {34, 11573}, {56, 3670}, {65, 52564}, {273, 23197}, {604, 17184}, {1014, 20966}, {1396, 22073}, {1400, 18601}, {1408, 3454}, {1412, 4016}, {1434, 40986}, {3909, 43924}, {16947, 20896}
X(68372) = trilinear quotient X(i)/X(j) for these (i,j): (57, 40394), (1408, 3453), (1441, 59138), (3454, 3701), (3670, 8), (3909, 3699), (4016, 2321), (11573, 78), (17184, 312), (18601, 333), (20896, 30713), (20966, 210), (21121, 4086), (22073, 3694), (23197, 212)
X(68372) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (65, 20615, 4298), (1401, 17114, 65), (1401, 63580, 50626), (17114, 50626, 63580), (50626, 63580, 65)

X(68373) = X(107)X(110)∩X(402)X(525)

See Antreas Hatzipolakis and Peter Moses, euclid 8302.

X(68373) lies on these lines: {107, 110}, {402, 525}, {1650, 14401}, {15183, 52720}, {15351, 45289}

X(68373) = tripolar centroid of X(23582)
X(68373) = crossdifference of every pair of points on line {1304, 3269}

X(68374) = X(107)X(110)∩X(389)X(974)

See Antreas Hatzipolakis and Peter Moses, euclid 8302.

X(68374) lies on these lines: {107, 110}, {133, 2970}, {389, 974}, {24930, 47202}, {47179, 62501}, {47236, 61204}

X(68374) = midpoint of X(1112) and X(67281)

   Points on with the Warren G-circles: X(68375) - X(69385)  

Contributed by Clark Kimberling and Peter Moses, based on notes from Benjamin Warren, April 22, 2025.

Let P be a point in the plane of a triangle ABC, and let
A'B'C' = medial triangle
Ab = reflection of A in line B'P, and define Bc and Ca cyclically
Ac = reflection of A in line C'P, and define Ba and Cb cyclically
Pa = circumcenter of ABcCb, and definer Pb and Pc cyclically.
The centroids of the following four points lie on a circle here named the Warren G(P)-circle: ABC, APbPc, BPcPa, CPaPb.

If P = p:q:r, then barycentrics for the center of the Warren G(P)-circle are given by

(a^2 - b^2 - c^2)^2*p^5 + (a^4 - 2*a^2*b^2 + b^4 + 4*a^2*c^2 - 10*b^2*c^2 - 3*c^4)*p^4*q - 2*(a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 - 2*b^2*c^2 - 5*c^4)*p^3*q^2 - 2*(a^4 - 2*a^2*b^2 + b^4 - 6*b^2*c^2 + 7*c^4)*p^2*q^3 + (a^2 - b^2 + c^2)*(a^2 - b^2 + 5*c^2)*p*q^4 + (a^4 - 2*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4)*q^5 + (a^4 + 4*a^2*b^2 - 3*b^4 - 2*a^2*c^2 - 10*b^2*c^2 + c^4)*p^4*r + 4*(2*a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 4*b^2*c^2 + c^4)*p^3*q*r - 2*(a^4 - b^4 - 10*a^2*c^2 + 2*b^2*c^2 + 5*c^4)*p^2*q^2*r - 4*(2*a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 - c^4)*p*q^3*r + (a^4 - 4*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*q^4*r - 2*(a^4 + 2*a^2*b^2 - 5*b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*p^3*r^2 - 2*(a^4 - 10*a^2*b^2 + 5*b^4 + 2*b^2*c^2 - c^4)*p^2*q*r^2 + 2*(7*a^4 - 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 6*b^2*c^2 - 3*c^4)*p*q^2*r^2 - 2*(a^4 - 2*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4)*q^3*r^2 - 2*(a^4 + 7*b^4 - 2*a^2*c^2 - 6*b^2*c^2 + c^4)*p^2*r^3 - 4*(2*a^4 + 2*a^2*b^2 - b^4 - 2*a^2*c^2 + c^4)*p*q*r^3 - 2*(a^4 - 4*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*q^2*r^3 + (a^2 + b^2 - c^2)*(a^2 + 5*b^2 - c^2)*p*r^4 + (a^4 - 2*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4)*q*r^4 + (a^4 - 4*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*r^5 : :

For example, the center of the Warren G(X(10))-circle is X(5886), and the squared radius is R*(R - 2*r) / 9.

The appearance of i in the following list of 16 points means that X(i) lies on the Warren G(X(10))-circle:

2, 5603, 32631, 61732, 67625, and 68375, 68376, ..., 68385.

X(68375) = X(1)X(2)∩X(106)X(528)

X(68375) lies on these lines: {1, 2}, {106, 528}, {121, 1120}, {244, 50891}, {376, 41343}, {537, 50915}, {946, 67723}, {952, 57300}, {999, 1308}, {2099, 14027}, {2743, 10679}, {3304, 13744}, {3476, 39752}, {3656, 67718}, {3667, 5603}, {3699, 11731}, {3756, 10700}, {5434, 56421}, {5881, 13625}, {10247, 53799}, {11238, 18340}, {11274, 66643}, {15170, 60687}, {17301, 67625}, {17724, 38026}, {24222, 59377}, {32577, 34719}, {46914, 50107}, {50101, 52759}

X(68375) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1647, 24864}, {1, 23869, 6788}, {36444, 36462, 1644}

X(68376) = X(2)X(512)∩X(76)X(61421)

X(68376) lies on these lines: {2, 512}, {76, 61421}, {99, 65546}, {115, 3903}, {316, 63822}, {511, 5603}, {962, 67358}, {1621, 67360}, {2703, 5263}, {3110, 3616}, {4983, 24504}, {5170, 17127}, {7983, 50440}, {11725, 56154}, {14061, 40608}

X(68377) = X(2)X(210)∩X(7)X(840)

X(68377) lies on these lines: {1, 18343}, {2, 210}, {7, 840}, {11, 3315}, {105, 5845}, {120, 1280}, {644, 11730}, {946, 67724}, {948, 39754}, {1083, 3616}, {1125, 67385}, {1647, 5219}, {1836, 60061}, {2802, 50913}, {3309, 5603}, {3487, 67382}, {4310, 24403}, {4419, 24359}, {4904, 10699}, {5525, 29660}, {5542, 9318}, {5901, 14661}, {15636, 63498}, {22769, 46586}, {33143, 61732}, {37633, 37703}, {38375, 63521}, {44675, 67387}

X(68377) = crossdifference of every pair of points on line {14411, 66513}

X(68378) = X(2)X(523)∩X(30)X(5603)

X(68378) lies on these lines: {1, 36154}, {2, 523}, {3, 52200}, {8, 13869}, {10, 47274}, {23, 16332}, {30, 5603}, {55, 36167}, {86, 57589}, {125, 6742}, {329, 67326}, {495, 30447}, {858, 16272}, {946, 67722}, {952, 38724}, {962, 36158}, {1109, 24955}, {1125, 47270}, {1290, 1621}, {1316, 19684}, {1699, 62493}, {2452, 5278}, {2453, 19701}, {2486, 21907}, {2611, 24916}, {3109, 3616}, {3576, 62496}, {3622, 36171}, {3624, 47273}, {3816, 5520}, {4934, 24624}, {5159, 50144}, {5196, 25557}, {5249, 67325}, {5253, 38570}, {5333, 67328}, {5432, 31522}, {5550, 47272}, {5883, 61699}, {5886, 62491}, {6739, 7984}, {6740, 11735}, {6741, 15059}, {6914, 46636}, {7424, 15950}, {10269, 46618}, {11007, 18139}, {12699, 68320}, {14731, 68282}, {14844, 37701}, {16304, 37911}, {19785, 67324}, {24145, 53564}, {25055, 62500}, {30745, 67601}, {33100, 64484}, {38028, 53809}, {47404, 59297}, {52002, 64345}, {55017, 63171}, {57325, 59382}

X(68378) = orthoptic-circle-of-the-Steiner-inellipse-inverse of X(47799)
X(68378) = orthoptic-circle-of-the-Steiner-circumellipse-inverse of X(48203)
X(68378) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3616, 38514, 3109}, {13869, 36155, 8}, {16332, 67596, 23}

X(68379) = X(2)X(165)∩X(11)X(109)

X(68379) lies on these lines: {1, 18328}, {2, 165}, {7, 15634}, {11, 109}, {103, 62674}, {118, 14942}, {514, 5603}, {517, 61730}, {664, 11728}, {946, 67568}, {952, 15735}, {962, 31852}, {1025, 54370}, {1125, 67574}, {1146, 10697}, {1566, 2717}, {1836, 60060}, {2099, 5532}, {2723, 35184}, {3086, 67577}, {3091, 67583}, {3234, 5819}, {3616, 67567}, {5219, 15737}, {5657, 67212}, {5886, 67625}, {7416, 20988}, {12699, 67721}, {17718, 61732}, {40554, 62383}, {44675, 67576}, {46344, 61086}, {56144, 62715}, {67570, 68034}

X(68379) = reflection of X(i) in X(j) for these {i,j}: {5657, 67212}, {67625, 5886}
X(68379) = {X(946),X(67726)}-harmonic conjugate of X(67568)

X(68380) = X(2)X(3738)∩X(11)X(109)

X(68380) lies on these lines: {1, 18341}, {2, 3738}, {3, 10771}, {4, 53752}, {11, 109}, {80, 11700}, {100, 6718}, {102, 6713}, {104, 117}, {124, 31272}, {149, 53742}, {151, 53748}, {499, 38507}, {631, 53740}, {900, 61732}, {952, 51408}, {1387, 10703}, {1484, 61571}, {1638, 67625}, {1795, 8068}, {2222, 57446}, {2800, 5603}, {2804, 67628}, {2818, 57298}, {5131, 62496}, {5433, 38565}, {5840, 38697}, {6702, 13532}, {7972, 47115}, {10698, 11727}, {10716, 45310}, {10724, 38785}, {10726, 38761}, {10732, 65948}, {10738, 38607}, {10740, 38602}, {10742, 61578}, {10747, 60759}, {10767, 53717}, {10768, 53724}, {10769, 53734}, {10778, 53758}, {10779, 53759}, {11715, 50899}, {12736, 34242}, {14217, 14690}, {17660, 58600}, {21154, 38691}, {22938, 38777}, {26492, 38501}, {33650, 66063}, {34126, 38776}, {38693, 64507}, {39270, 56638}, {58419, 64008}, {59391, 64501}, {67453, 67477}

X(68380) = reflection of X(i) in X(j) for these {i,j}: {38691, 21154}, {38776, 34126}
X(68380) = reflection of X(61732) in the IN line
X(68380) = {X(11),X(109)}-harmonic conjugate of X(10777)

X(68381) = X(2)X(51)∩X(4)X(3110)

X(68381) lies on these lines: {2, 51}, {4, 3110}, {114, 56154}, {512, 5603}, {631, 67358}, {3903, 11724}, {5883, 67625}, {6905, 67360}, {7970, 40608}, {50440, 64089}

X(68382) = X(1)X(18339)∩X(2)X(521)

X(68382) lies on these lines: {1, 18339}, {2, 521}, {55, 28347}, {57, 67568}, {123, 13138}, {354, 5603}, {497, 2720}, {1364, 60356}, {5722, 15524}, {10584, 63757}, {12115, 64512}, {14257, 41084}, {18391, 67426}

X(68383) = X(2)X(6003)∩X(5)X(580)

X(68383) lies on these lines: {2, 6003}, {5, 580}, {643, 42425}, {758, 5603}, {867, 2328}, {2606, 14680}, {2886, 65881}, {6912, 68244}, {6929, 34172}, {13478, 56835}

X(68384) = X(1)X(18339)∩X(2)X(515)

X(68384) lies on these lines: {1, 18339}, {2, 515}, {4, 67714}, {5, 18340}, {117, 51565}, {355, 67476}, {381, 57320}, {517, 67628}, {522, 5603}, {944, 67226}, {946, 67725}, {952, 51408}, {999, 67461}, {1012, 2716}, {1125, 67466}, {1385, 67464}, {1478, 61481}, {1785, 23708}, {1897, 11727}, {2222, 6911}, {2968, 10696}, {3086, 67471}, {3616, 31866}, {5886, 61732}, {6830, 56690}, {8227, 51889}, {34231, 39762}, {37820, 67477}, {44675, 67470}, {51422, 60758}, {67460, 68034}

X(68384) = reflection of X(i) in X(j) for these {i,j}: {61732, 5886}, {67635, 3576}

X(68385) = X(2)X(3)∩X(523)X(5603)

X(68385) lies on these lines: {2, 3}, {113, 6740}, {523, 5603}, {946, 47270}, {1125, 67722}, {1699, 62496}, {3576, 62493}, {3656, 50145}, {5520, 7680}, {5627, 56402}, {5886, 62491}, {6739, 64101}, {6741, 7978}, {6742, 11723}, {10176, 67634}, {10595, 13869}, {11522, 47273}, {13464, 47274}, {18493, 52200}, {34301, 51654}, {36518, 53794}, {37701, 61732}, {37735, 50148}, {38021, 62500}, {38034, 53809}, {38514, 68034}, {45924, 61479}, {47471, 50144}

X(68385) = reflection of X(44280) in X(28462)

X(68386) = X(1)X(3024)∩X(12)X(8287)

See Antreas Hatzipolakis and Peter Moses, euclid 8304.

X(68386) lies on these lines: {1, 3024}, {12, 8287}, {56, 40214}, {57, 2940}, {354, 56849}, {942, 39751}, {1319, 2392}, {1365, 13751}, {3649, 5045}, {7144, 16577}, {11553, 22461}, {17637, 59818}, {18398, 33642}, {39791, 44913}

X(68386) = X(58565)-Dao conjugate of X(8)
X(68386) = crosspoint of X(7) and X(16577)
X(68386) = barycentric product X(16577)*X(58565)

X(68387) = X(2)X(65240)∩X(36)X(63868)

See Antreas Hatzipolakis and Peter Moses, euclid 8315.

X(68387) lies on the Mandart circumellipse and these lines: {2, 65240}, {36, 63868}, {88, 18593}, {100, 34921}, {162, 37966}, {190, 57066}, {411, 46037}, {514, 38340}, {651, 14838}, {653, 65100}, {655, 14147}, {673, 19302}, {1156, 3065}, {2349, 3218}, {3911, 18653}, {4564, 37212}, {21739, 34234}, {27003, 65249}, {36101, 60989}, {37131, 60948}, {37787, 65261}

X(68387) = isogonal conjugate of X(68388)
X(68387) = X(i)-cross conjugate of X(j) for these (i,j): {650, 63868}, {4120, 1476}, {5131, 7045}, {8674, 7}, {14400, 21}, {53283, 99}, {61708, 52377}, {62359, 55346}
X(68387) = X(i)-isoconjugate of X(j) for these (i,j): {2, 42657}, {9, 59837}, {37, 35055}, {484, 650}, {522, 19297}, {663, 17484}, {3063, 17791}, {3064, 23071}, {3709, 56935}, {3737, 21864}, {4560, 58285}, {4895, 47058}, {9404, 50148}, {11076, 35057}, {26744, 66284}, {50462, 65105}, {52356, 66970}, {61214, 66012}
X(68387) = X(i)-Dao conjugate of X(j) for these (i,j): {478, 59837}, {10001, 17791}, {32664, 42657}, {40589, 35055}
X(68387) = cevapoint of X(i) and X(j) for these (i,j): {3, 14395}, {36, 650}, {513, 62211}
X(68387) = crosssum of X(661) and X(58900)
X(68387) = trilinear pole of line {1, 399}
X(68387) = barycentric product X(i)*X(j) for these {i,j}: {75, 34921}, {109, 40716}, {651, 21739}, {664, 3065}, {4554, 19302}, {4564, 60486}, {4585, 26743}, {7343, 65292}, {14147, 17078}
X(68387) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 42657}, {56, 59837}, {58, 35055}, {109, 484}, {651, 17484}, {664, 17791}, {1414, 56935}, {1415, 19297}, {1983, 26744}, {3065, 522}, {4559, 21864}, {7343, 35057}, {14147, 36910}, {19302, 650}, {21739, 4391}, {26700, 50148}, {26743, 60074}, {34921, 1}, {36059, 23071}, {40716, 35519}, {59837, 31522}, {60486, 4858}, {61231, 66012}, {63868, 52356}

X(68388) = X(2)X(9034)∩X(44)X(513)

See Antreas Hatzipolakis and Peter Moses, euclid 8315.

X(68388) lies on these lines: {2, 9034}, {6, 14399}, {44, 513}, {51, 2878}, {101, 26700}, {521, 4944}, {526, 1637}, {926, 11193}, {1639, 3738}, {2170, 38358}, {2316, 2341}, {2775, 5540}, {2827, 4773}, {2850, 14395}, {3196, 53527}, {3700, 35057}, {3887, 66026}, {4120, 8674}, {4131, 31250}, {4931, 8702}, {8774, 47784}, {9001, 47881}, {21297, 40166}, {26744, 35055}, {34151, 52985}, {40584, 57174}, {59817, 65707}

X(68388) = isogonal conjugate of X(68387)
X(68388) = X(i)-complementary conjugate of X(j) for these (i,j): {42, 3258}, {476, 3741}, {1918, 18334}, {1989, 116}, {2166, 21252}, {6186, 51402}, {6187, 6741}, {8750, 1511}, {11060, 1086}, {14560, 1125}, {32678, 3739}, {32680, 21240}, {32739, 34834}, {39295, 52602}, {52153, 2968}, {56193, 31845}
X(68388) = X(i)-Ceva conjugate of X(j) for these (i,j): {651, 6126}, {1290, 55}, {4591, 3057}, {35055, 42657}, {62928, 11}
X(68388) = X(42657)-cross conjugate of X(59837)
X(68388) = X(i)-isoconjugate of X(j) for these (i,j): {2, 34921}, {59, 60486}, {109, 21739}, {651, 3065}, {664, 19302}, {1415, 40716}, {1443, 14147}, {7343, 38340}
X(68388) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 21739}, {1146, 40716}, {6615, 60486}, {32664, 34921}, {38991, 3065}, {39025, 19302}
X(68388) = cevapoint of X(661) and X(58900)
X(68388) = crosspoint of X(80) and X(651)
X(68388) = crosssum of X(i) and X(j) for these (i,j): {3, 14395}, {36, 650}, {513, 62211}
X(68388) = crossdifference of every pair of points on line {1, 399}
X(68388) = barycentric product X(i)*X(j) for these {i,j}: {8, 59837}, {10, 35055}, {75, 42657}, {484, 522}, {650, 17484}, {663, 17791}, {1639, 47058}, {4041, 56935}, {4391, 19297}, {4560, 21864}, {6126, 52356}, {11076, 57066}, {18155, 58285}, {23071, 44426}, {26744, 60074}, {35057, 50148}
X(68388) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 34921}, {484, 664}, {522, 40716}, {650, 21739}, {663, 3065}, {2170, 60486}, {3063, 19302}, {11076, 38340}, {17484, 4554}, {17791, 4572}, {19297, 651}, {21864, 4552}, {23071, 6516}, {26744, 4585}, {35055, 86}, {42657, 1}, {50148, 65292}, {56935, 4625}, {58285, 4551}, {59837, 7}
X(68388) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {654, 46393, 650}, {661, 9404, 650}, {2590, 2591, 9404}, {2610, 3013, 21894}, {13401, 40137, 650}, {14298, 46389, 650}, {14298, 65450, 9404}, {46393, 65680, 654}

X(68389) = X(3)X(49)∩X(4)X(96)

X(68389) lies on the cubic K1398 and these lines: {3, 49}, {4, 96}, {30, 8800}, {32, 45089}, {50, 64037}, {52, 2351}, {97, 18925}, {569, 52435}, {570, 7592}, {577, 6146}, {578, 54034}, {1157, 64032}, {2986, 60166}, {3003, 35603}, {3133, 10539}, {3135, 6759}, {3564, 8905}, {4558, 64756}, {5889, 66602}, {7399, 54393}, {11411, 52350}, {12160, 44200}, {12241, 61748}, {12420, 15478}, {14516, 63835}, {14806, 15032}, {15512, 17834}, {34799, 63762}, {44665, 66482}

X(68390) = X(1)X(1073)∩X(4)X(1903)

X(68390) lies on the cubic K033 and these lines: {1, 1073}, {4, 1903}, {8, 1032}, {40, 3348}, {65, 7157}, {271, 37228}, {280, 62864}, {282, 5706}, {1712, 3343}, {6617, 8886}, {15324, 40117}, {44547, 57492}

X(68390) = X(8)-Ceva conjugate of X(39130)
X(68390) = X(i)-isoconjugate of X(j) for these (i,j): {2360, 3346}, {3194, 47849}, {28783, 41083}
X(68390) = X(i)-Dao conjugate of X(j) for these (i,j): {1073, 41082}, {8808, 7}, {13613, 64885}
X(68390) = barycentric product X(i)*X(j) for these {i,j}: {280, 8807}, {321, 8886}, {1712, 56944}, {1903, 6527}, {8803, 34404}, {14361, 52389}, {41086, 47435}
X(68390) = barycentric quotient X(i)/X(j) for these {i,j}: {1033, 3194}, {1498, 1817}, {1712, 41083}, {1903, 3346}, {3343, 41082}, {8803, 223}, {8807, 347}, {8886, 81}, {41086, 3344}, {41087, 47849}, {52384, 8810}, {52389, 1032}, {53013, 8805}, {58895, 6129}

X(68391) = X(1)X(3344)∩X(8)X(1032)

X(68391) lies on the cubic K033 and these lines: {1, 3344}, {8, 1032}, {10, 64987}, {40, 3182}, {72, 3176}, {962, 46353}, {11529, 63866}, {40836, 55063}

X(68391) = isogonal conjugate of X(8886)
X(68391) = X(1032)-Ceva conjugate of X(8805)
X(68391) = X(i)-cross conjugate of X(j) for these (i,j): {65, 40}, {41088, 7952}
X(68391) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8886}, {84, 1498}, {285, 8803}, {1033, 41081}, {1433, 1712}, {2208, 6527}, {6616, 60799}, {6617, 7129}
X(68391) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 8886}, {281, 14361}, {3350, 41084}
X(68391) = crosspoint of X(41080) and X(63877)
X(68391) = barycentric product X(i)*X(j) for these {i,j}: {329, 3346}, {347, 8805}, {1032, 7952}, {7080, 8810}, {41088, 47633}, {47849, 64211}
X(68391) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8886}, {198, 1498}, {227, 8807}, {329, 6527}, {2331, 1712}, {3195, 1033}, {3344, 41084}, {3346, 189}, {7078, 6617}, {7952, 14361}, {8805, 280}, {8810, 1440}, {28783, 1433}, {41088, 3343}, {47849, 41081}

X(68392) = X(1)X(3342)∩X(4)X(8805)

X(68392) lies on the cubic K033 and these lines: {1, 3342}, {4, 8805}, {8, 1034}, {40, 219}, {72, 5930}, {1073, 47849}, {1259, 6617}, {1264, 57782}, {10373, 60802}, {47441, 55063}

X(68392) = isogonal conjugate of X(8885)
X(68392) = X(1034)-Ceva conjugate of X(8806)
X(68392) = X(i)-cross conjugate of X(j) for these (i,j): {65, 72}, {41087, 1214}, {53012, 52389}
X(68392) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8885}, {21, 207}, {27, 3197}, {28, 1490}, {29, 1035}, {34, 13614}, {58, 3176}, {204, 47637}, {284, 40837}, {1172, 47848}, {1474, 56943}, {2203, 33672}, {2299, 5932}, {3194, 3341}, {3737, 57117}
X(68392) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 8885}, {10, 3176}, {226, 5932}, {3343, 47637}, {3351, 41083}, {11517, 13614}, {40590, 40837}, {40591, 1490}, {40611, 207}, {51574, 56943}, {62564, 33672}
X(68392) = cevapoint of X(65) and X(8811)
X(68392) = crosspoint of X(i) and X(j) for these (i,j): {1032, 19611}, {1034, 57643}
X(68392) = crosssum of X(i) and X(j) for these (i,j): {204, 1033}, {207, 1035}
X(68392) = barycentric product X(i)*X(j) for these {i,j}: {63, 8806}, {71, 56596}, {72, 41514}, {226, 57643}, {306, 3345}, {307, 47850}, {321, 66932}, {345, 8811}, {1034, 1214}, {1231, 7037}, {1400, 57782}, {3342, 56944}, {3998, 7149}, {7007, 52565}, {7152, 20336}, {40838, 52385}, {41087, 47634}, {42699, 60800}, {52389, 63877}
X(68392) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8885}, {37, 3176}, {65, 40837}, {71, 1490}, {72, 56943}, {73, 47848}, {219, 13614}, {228, 3197}, {306, 33672}, {1034, 31623}, {1073, 47637}, {1214, 5932}, {1400, 207}, {1409, 1035}, {3342, 41083}, {3345, 27}, {4559, 57117}, {7007, 8748}, {7037, 1172}, {7152, 28}, {8611, 14302}, {8806, 92}, {8811, 278}, {40838, 1896}, {41087, 3341}, {41514, 286}, {47850, 29}, {56596, 44129}, {56944, 47436}, {57454, 3194}, {57643, 333}, {57782, 28660}, {66932, 81}
X(68392) = {X(1034),X(63877)}-harmonic conjugate of X(7149)

X(68393) = X(1)X(19611)∩X(7)X(253)

X(68393) lies on the cubic K034 and these lines: {1, 19611}, {2, 63877}, {7, 253}, {8, 14362}, {92, 2184}, {610, 65224}, {3692, 56235}

X(68393) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {58, 63877}, {8885, 14361}, {46639, 68108}, {47637, 69}
X(68393) = X(75)-Ceva conjugate of X(19611)
X(68393) = X(i)-isoconjugate of X(j) for these (i,j): {2, 47439}, {6, 3344}, {32, 47633}, {154, 3346}, {184, 46353}, {204, 47849}, {1032, 3172}, {1249, 28783}, {2131, 28782}, {3350, 28781}, {58342, 59077}
X(68393) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 3344}, {1073, 1}, {3343, 47849}, {6376, 47633}, {8808, 52078}, {32664, 47439}, {62605, 46353}
X(68393) = barycentric product X(i)*X(j) for these {i,j}: {1, 47435}, {75, 3343}, {92, 46351}, {304, 41085}, {561, 47437}, {1033, 57780}, {1498, 57921}, {1712, 34403}, {2184, 6527}, {5931, 8807}, {14361, 19611}
X(68393) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3344}, {31, 47439}, {75, 47633}, {92, 46353}, {1033, 204}, {1073, 47849}, {1498, 610}, {1712, 1249}, {2184, 3346}, {3343, 1}, {6527, 18750}, {8803, 30456}, {8807, 5930}, {8809, 8810}, {14361, 1895}, {19611, 1032}, {19614, 28783}, {41085, 19}, {44692, 8805}, {46351, 63}, {47435, 75}, {47437, 31}

X(68394) = X(1)X(280)∩X(7)X(92)

X(68394) lies on the cubic K034 and these lines: {1, 280}, {2, 19611}, {7, 92}, {8, 1032}, {63, 47851}, {75, 44189}, {309, 14548}, {345, 44327}, {394, 13138}, {938, 7020}, {3341, 46350}, {7011, 37141}, {11529, 39130}, {15466, 65270}, {18623, 65330}, {20223, 55119}, {23681, 24213}, {44695, 65213}, {53642, 54107}

X(68394) = isogonal conjugate of X(57454)
X(68394) = isotomic conjugate of X(63877)
X(68394) = polar conjugate of the isogonal conjugate of X(46881)
X(68394) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {58, 19611}, {84, 68334}, {154, 20211}, {189, 32064}, {610, 6223}, {1394, 5932}, {1413, 68349}, {1422, 68352}, {1433, 253}, {1436, 3146}, {2192, 68348}, {2208, 18663}, {8886, 14362}, {40836, 32001}, {41084, 69}, {41086, 2895}, {52078, 2893}, {60803, 54111}
X(68394) = X(i)-Ceva conjugate of X(j) for these (i,j): {75, 280}, {44189, 189}, {47436, 46350}
X(68394) = X(i)-cross conjugate of X(j) for these (i,j): {3176, 5932}, {47848, 56943}
X(68394) = X(i)-isoconjugate of X(j) for these (i,j): {1, 57454}, {6, 3342}, {31, 63877}, {32, 47634}, {40, 7152}, {41, 46352}, {198, 3345}, {221, 47850}, {223, 7037}, {1034, 2199}, {2187, 41514}, {2331, 66932}, {3209, 57643}, {3351, 34167}, {7007, 7011}, {7114, 40838}, {10397, 58995}, {41080, 47440}
X(68394) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 63877}, {3, 57454}, {9, 3342}, {278, 196}, {282, 1}, {3160, 46352}, {3341, 47850}, {6376, 47634}, {13612, 14298}, {14302, 55063}
X(68394) = cevapoint of X(1) and X(47851)
X(68394) = barycentric product X(i)*X(j) for these {i,j}: {1, 47436}, {7, 46350}, {75, 3341}, {84, 33672}, {189, 56943}, {207, 57783}, {264, 46881}, {280, 5932}, {309, 1490}, {561, 47438}, {1035, 57793}, {3197, 44190}, {14302, 53642}, {34404, 47848}, {40837, 44189}, {46355, 66090}
X(68394) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3342}, {2, 63877}, {6, 57454}, {7, 46352}, {75, 47634}, {84, 3345}, {189, 41514}, {207, 208}, {271, 57643}, {280, 1034}, {282, 47850}, {309, 56596}, {1035, 221}, {1433, 66932}, {1436, 7152}, {1490, 40}, {2192, 7037}, {3176, 7952}, {3197, 198}, {3341, 1}, {5932, 347}, {7003, 40838}, {7008, 7007}, {8885, 3194}, {14302, 8058}, {33672, 322}, {39130, 8806}, {40836, 7149}, {40837, 196}, {46350, 8}, {46355, 66091}, {46881, 3}, {47436, 75}, {47438, 31}, {47637, 41082}, {47848, 223}, {47851, 3351}, {52384, 8811}, {56943, 329}, {57783, 57782}, {60803, 60800}, {66090, 55015}
X(68394) = {X(7003),X(52037)}-harmonic conjugate of X(189)

X(68395) = X(65)X(495)∩X(942)X(5453)

See Antreas Hatzipolakis and Peter Moses, euclid 8318.

X(68395) lies on these lines: {65, 495}, {517, 43915}, {942, 5453}, {1243, 1439}, {3028, 55010}, {20617, 31794}

X(68396) = X(323)X(14491)∩X(381)X(33884)

See Antreas Hatzipolakis and Peter Moses, euclid 8318.

X(68396) lies on these lines: {323, 14491}, {381, 33884}, {3167, 5645}, {15024, 45184}, {24206, 31074}

X(68397) = X(51)X(3631)∩X(141)X(2979)

See Antreas Hatzipolakis and Peter Moses, euclid 8318.

X(68397) lies on these lines: {2, 44323}, {51, 3631}, {141, 2979}, {373, 20583}, {511, 11737}, {524, 5943}, {599, 11002}, {2393, 34573}, {2854, 12045}, {3589, 40673}, {3629, 11451}, {3819, 41579}, {5650, 8705}, {6329, 6688}, {9019, 51143}, {9730, 50958}, {9973, 44299}, {14913, 51127}, {16776, 21969}, {29181, 44804}, {41149, 64692}, {43129, 54044}, {61667, 63124}

X(68397) = midpoint of X(i) and X(j) for these {i,j}: {51, 3631}, {3819, 41579}, {9730, 50958}, {16776, 50991}, {20582, 29959}, {43129, 54044}, {61667, 63124}
X(68397) = reflection of X(6329) in X(6688)
X(68397) = complement of X(44323)

X(68398) = X(1)X(1292)∩X(105)X(165)

Suppose that P is a point on the circumcircle. Then the combo Q(P) = 3*X(165)-2*P lies on the Bevan circle. Four examples are indicated here: X(68398) = Q(X(105))
X(68399) = Q(X(106))
X(68400) = Q(X(111))
X(68401) = Q(X(15731))

X(68398) lies on on the Bevan circle and these lines: {1, 1292}, {10, 34547}, {40, 5540}, {55, 53540}, {57, 3021}, {105, 165}, {120, 1699}, {516, 20344}, {517, 38589}, {528, 1768}, {1282, 2820}, {1358, 1697}, {1695, 3034}, {1698, 5511}, {1706, 3039}, {2775, 2948}, {2788, 13174}, {2795, 9860}, {2809, 7991}, {2814, 64761}, {2826, 5541}, {2835, 7994}, {2836, 9904}, {2838, 12408}, {2941, 21381}, {3339, 59814}, {3576, 38619}, {3579, 38575}, {4859, 56796}, {5119, 51770}, {5537, 34464}, {5587, 15521}, {5691, 50911}, {7982, 67834}, {7987, 11716}, {8227, 57327}, {9523, 13221}, {9586, 58053}, {9587, 58055}, {9778, 20097}, {10699, 11531}, {10712, 50865}, {10743, 41869}, {11010, 67715}, {16192, 38694}, {26296, 48541}, {26297, 48542}, {31423, 57299}, {31435, 34124}, {35242, 38603}, {37551, 52826}, {38670, 63469}, {62497, 66793}, {62498, 66776}

X(68398) = reflection of X(i) in X(j) for these {i,j}: {1, 1292}, {5540, 40}, {5691, 50911}, {11531, 10699}, {34547, 10}, {38575, 3579}, {41869, 10743}, {50865, 10712}
X(68398) = excentral-isogonal conjugate of X(518)
X(68398) = X(5853)-Ceva conjugate of X(1)
X(68398) = X(43760)-Dao conjugate of X(35160)
X(68398) = {X(11716),X(38712)}-harmonic conjugate of X(7987)

X(68399) = X(1)X(1293)∩X(106)X(165)

X(68399) lies on on the Bevan circle and these lines: {1, 1293}, {10, 34548}, {40, 1054}, {57, 6018}, {106, 165}, {121, 1699}, {516, 21290}, {517, 13541}, {962, 11814}, {1282, 2821}, {1357, 1697}, {1695, 3030}, {1698, 5510}, {1706, 3038}, {1766, 3973}, {1768, 2802}, {2136, 64249}, {2776, 2948}, {2789, 13174}, {2796, 9860}, {2810, 39156}, {2815, 64761}, {2827, 5541}, {2841, 16389}, {2842, 9904}, {2844, 12408}, {2938, 5539}, {3339, 59812}, {3576, 38620}, {3579, 38576}, {5119, 51765}, {5587, 15522}, {5691, 50914}, {7982, 67835}, {7987, 11717}, {8227, 57328}, {9527, 13221}, {9586, 58052}, {9587, 58054}, {9778, 20098}, {10700, 11531}, {10713, 50865}, {10744, 41869}, {11010, 38515}, {12702, 21381}, {13462, 63774}, {14664, 38671}, {16192, 38695}, {17777, 20070}, {31423, 57300}, {35242, 38604}, {37551, 52827}, {62499, 66793}, {62500, 66776}

X(68399) = midpoint of X(17777) and X(20070)
X(68399) = reflection of X(i) in X(j) for these {i,j}: {1, 1293}, {962, 11814}, {1054, 40}, {5691, 50914}, {11531, 10700}, {34548, 10}, {38576, 3579}, {38671, 14664}, {41869, 10744}, {50865, 10713}
X(68399) = excentral-isogonal conjugate of X(519)
X(68399) = X(3880)-Ceva conjugate of X(1)
X(68399) = {X(11717),X(38713)}-harmonic conjugate of X(7987)

X(68400) = X(1)X(1296)∩X(111)X(165)

X(68400) lies on on the Bevan circle and these lines: {1, 1296}, {10, 66869}, {40, 33962}, {57, 6019}, {111, 165}, {126, 1699}, {516, 14360}, {517, 38593}, {518, 37751}, {519, 37749}, {543, 9860}, {1054, 2938}, {1282, 2824}, {1697, 3325}, {1698, 5512}, {1768, 2805}, {2780, 2948}, {2793, 13174}, {2813, 39156}, {2819, 64761}, {2830, 5541}, {2852, 64760}, {2854, 9904}, {2941, 5540}, {3048, 9586}, {3339, 59819}, {3576, 38623}, {3579, 11258}, {3624, 40556}, {5119, 51814}, {5587, 22338}, {5691, 50924}, {7982, 67838}, {7987, 11721}, {7991, 38688}, {8227, 57331}, {9583, 11835}, {9584, 11833}, {9587, 58059}, {9591, 14657}, {9778, 20099}, {9956, 38799}, {10704, 11531}, {10717, 50865}, {10748, 41869}, {11010, 38518}, {13221, 62506}, {14650, 35242}, {14654, 31730}, {14688, 16475}, {16192, 38698}, {18480, 38800}, {23699, 64005}, {31423, 38796}, {31663, 52698}, {33535, 35447}, {37551, 52832}, {38675, 63469}, {62507, 66793}, {62508, 66776}

X(68400) = reflection of X(i) in X(j) for these {i,j}: {1, 1296}, {5691, 50924}, {11258, 3579}, {11531, 10704}, {14654, 31730}, {33535, 35447}, {38800, 18480}, {41869, 10748}, {50865, 10717}, {66869, 10}
X(68400) = excentral-isogonal conjugate of X(524)
X(68400) = X(24394)-Ceva conjugate of X(1)
X(68400) = {X(11721),X(38716)}-harmonic conjugate of X(7987)

X(68401) = X(1)X(2291)∩X(101)X(165)

X(68401) lies on on the Bevan circle and these lines: {1, 2291}, {9, 1768}, {43, 38486}, {57, 1358}, {101, 165}, {219, 9904}, {610, 64760}, {910, 16554}, {1054, 1743}, {1155, 5526}, {1282, 1635}, {1781, 21381}, {2170, 10980}, {2448, 2590}, {2449, 2591}, {2801, 41798}, {3218, 67657}, {5011, 5536}, {5537, 6603}, {11407, 52705}, {15734, 65522}, {15855, 15931}, {34522, 55163}, {41338, 45721}, {53056, 64446}, {56632, 60905}, {66524, 66863}

X(68401) = reflection of X(1) in X(14074)
X(68401) = excentral-isogonal conjugate of X(15726)
X(68401) = X(527)-Ceva conjugate of X(1)
X(68401) = X(1156)-Dao conjugate of X(1121)

X(68402) = X(1)X(53529)∩X(57)X(934)

X(68402) llies on these lines: {1, 53529}, {57, 934}, {223, 31142}, {651, 2124}, {1086, 62793}, {1419, 15730}, {3160, 52457}, {5514, 20196}, {5526, 43064}, {5723, 23511}, {6282, 53804}, {7994, 8916}, {16572, 64980}, {34492, 61007}, {47057, 62705}, {47621, 60017}, {47623, 52161}
X(68402) = reflection of X(i) in X(j) for these {i,j}: {57, 934}, {61493, 52879}
X(68402) = X(527)-Ceva conjugate of X(57)
X(68402) = X(34056)-Dao conjugate of X(1121)
X(68402) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {934, 61493, 52879}, {52879, 61493, 57}

X(68403) = X(1)X(5072)∩X(5)X(10)

See Benjamin Lee Warren and Ercole Suppa, euclid 8330.

X(68403) lies on these lines: {1, 5072}, {2, 28146}, {4, 17502}, {5, 10}, {11, 5049}, {30, 10171}, {40, 5079}, {140, 12571}, {143, 65435}, {165, 1656}, {354, 7741}, {355, 3623}, {381, 3576}, {382, 58221}, {499, 31776}, {515, 5066}, {516, 547}, {519, 14892}, {546, 13624}, {548, 19878}, {549, 28154}, {551, 61270}, {632, 51118}, {942, 7173}, {952, 11737}, {962, 61921}, {1125, 3850}, {1385, 3091}, {1482, 4816}, {1538, 6830}, {1657, 34595}, {1698, 61919}, {1699, 5055}, {1829, 35487}, {2771, 23513}, {2801, 58604}, {3090, 3579}, {3241, 61257}, {3533, 10248}, {3545, 5886}, {3614, 9957}, {3616, 61945}, {3624, 3843}, {3627, 19862}, {3628, 18483}, {3632, 58238}, {3634, 12812}, {3653, 61954}, {3654, 61926}, {3655, 61944}, {3656, 38176}, {3679, 61931}, {3824, 3825}, {3828, 28212}, {3832, 33697}, {3833, 3838}, {3845, 10165}, {3851, 8227}, {3855, 18481}, {3857, 31673}, {3858, 4297}, {3860, 28190}, {4870, 37718}, {5045, 10593}, {5056, 12699}, {5070, 41869}, {5071, 9779}, {5076, 67706}, {5131, 65141}, {5219, 18527}, {5439, 31828}, {5550, 31666}, {5587, 10247}, {5603, 31145}, {5657, 38083}, {5691, 58230}, {5731, 41106}, {5790, 11224}, {5818, 11278}, {5844, 61934}, {5901, 28236}, {5902, 17605}, {5919, 7743}, {5927, 6990}, {6361, 61914}, {6684, 28216}, {6841, 11227}, {6873, 67998}, {6912, 23961}, {6915, 33862}, {6982, 64659}, {7987, 61984}, {7989, 16200}, {8703, 51074}, {9519, 61581}, {9590, 21308}, {9620, 18584}, {9669, 10389}, {9778, 61899}, {10095, 31751}, {10109, 10172}, {10124, 28182}, {10164, 15699}, {10283, 50796}, {10590, 51788}, {10592, 31792}, {10595, 61258}, {10886, 39550}, {10896, 31795}, {12100, 28158}, {12101, 51076}, {12512, 16239}, {12702, 61923}, {12811, 15178}, {13374, 56762}, {13464, 61259}, {14128, 31757}, {14893, 28172}, {15682, 51084}, {15704, 58219}, {15726, 61595}, {15759, 50869}, {15808, 58232}, {16192, 55858}, {17527, 61029}, {17606, 67977}, {18357, 33179}, {18492, 30392}, {18874, 31760}, {19872, 61903}, {19883, 23046}, {20057, 61248}, {20070, 67096}, {25055, 61948}, {26201, 58615}, {28194, 47478}, {28208, 38028}, {30308, 50821}, {31162, 61925}, {31423, 61905}, {31439, 42582}, {31447, 48661}, {31730, 55856}, {32205, 65423}, {34627, 61279}, {34638, 61869}, {34648, 61949}, {35242, 55857}, {35271, 62969}, {37606, 51792}, {37613, 63674}, {37712, 67241}, {38068, 61909}, {38076, 38138}, {38155, 61260}, {41099, 54445}, {41990, 51078}, {44904, 61524}, {46219, 64005}, {50799, 61943}, {50800, 51105}, {50808, 61898}, {50811, 61950}, {50812, 61854}, {50815, 61997}, {50816, 61823}, {50820, 62025}, {50833, 61998}, {50862, 61963}, {50865, 61908}, {50873, 61838}, {51071, 61251}, {51073, 61907}, {51086, 62138}, {51087, 61247}, {51705, 61956}, {54448, 61287}, {58218, 62128}, {58240, 61510}, {59417, 61927}, {59503, 61929}, {61271, 61941}, {61648, 65140}, {61649, 61703}, {61895, 64108}, {63963, 64813}, {65399, 67867}

X(68403) = midpoint of X(i) and X(j) for these {i,j}: {4, 17502}, {5, 3817}, {165, 22793}, {381, 11230}, {946, 38042}, {1699, 11231}, {3579, 9812}, {3656, 38176}, {3845, 10165}, {5066, 61269}, {5587, 51709}, {5886, 38140}, {10172, 50802}, {10175, 38034}, {10222, 59388}, {10246, 18480}, {10283, 50796}, {18483, 58441}, {19883, 23046}, {22791, 38127}, {51071, 61251}, {51087, 61247}
X(68403) = reflection of X(i) in X(j) for these {i,j}: {9955, 3817}, {10171, 61267}, {10172, 10109}, {26201, 58615}, {31662, 1125}, {31663, 58441}, {58441, 3628}
X(68403) = pole of line {4802, 47805} with respect to the orthoptic circle of the Steiner inellipse
X(68403) = pole of line {10950, 28212} with respect to the Feuerbach hyperbola
X(68403) = center of circles {{X(i), X(j), X(k)}} for these {i, j, k}: {4, 17502, 31849}, {381, 11230, 67216}
X(68403) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5,3817,517},{5,9955,9956},{5,38034,10175},{381,7988,11230},{381,11230,28160},{946,38042,517},{1125,28186,31662},{1699,5055,11231},{1699,11231,28198},{1699,61265,5055},{3091,61268,1385},{3545,5886,38140},{3628,18483,31663},{3628,28178,58441},{3817,10175,38034},{3845,10165,28168},{3851,8227,18480},{5066,61269,515},{5071,9779,26446},{5603,61936,61263},{5790,61933,61264},{5886,38140,28204},{10109,28174,10172},{10172,50802,28174},{10175,38034,517},{12811,61272,19925},{12812,40273,3634},{18483,58441,28178},{18493,61937,7989},{19925,61272,15178},{22791,38127,517},{30308,61920,50821},{38021,61264,5790},{48661,61911,64850},{48661,64850,31447},{61261,68034,10222}

X(68404) = X(5)X(13)∩X(30)X(48311)

X(68404) lies on these lines: {5, 13}, {30, 48311}, {530, 47478}, {542, 14892}, {546, 6669}, {549, 59393}, {616, 5079}, {618, 35018}, {3091, 47610}, {3545, 59383}, {3628, 5478}, {3845, 21156}, {3850, 6771}, {3857, 36961}, {5055, 59394}, {5056, 13103}, {5066, 41022}, {5068, 48655}, {5072, 6770}, {5459, 11737}, {5463, 61916}, {5473, 55856}, {10109, 36769}, {11543, 47855}, {11705, 61259}, {12811, 22796}, {15092, 20253}, {19709, 36318}, {22489, 38071}, {25154, 61910}, {35749, 61920}, {36770, 61900}, {41042, 61939}, {42137, 52266}, {42628, 47861}, {43401, 52650}, {51482, 61925}, {59378, 61933}

X(68404) = midpoint of X(i) and X(j) for these {i,j}: {5, 59401}, {549, 59393}, {3845, 21156}, {22489, 38071}
X(68404) = reflection of X(20252) in X(59401)

X(68405) = X(5)X(14)∩X(30)X(48312)

X(68405) lies on these lines: {5, 14}, {30, 48312}, {531, 47478}, {542, 14892}, {546, 6670}, {549, 59395}, {617, 5079}, {619, 35018}, {3091, 47611}, {3545, 59384}, {3628, 5479}, {3845, 21157}, {3850, 6774}, {3857, 36962}, {5055, 59396}, {5056, 13102}, {5066, 41023}, {5068, 48656}, {5072, 6773}, {5460, 11737}, {5464, 61916}, {5474, 55856}, {10109, 47867}, {11542, 47856}, {11706, 61259}, {12811, 22797}, {15092, 20252}, {19709, 36320}, {22490, 38071}, {25164, 61910}, {36327, 61920}, {36765, 38229}, {41043, 61939}, {42136, 52263}, {42627, 47862}, {43402, 44223}, {51483, 61925}, {59379, 61933}

X(68405) = midpoint of X(i) and X(j) for these {i,j}: {5, 59402}, {549, 59395}, {3845, 21157}, {22490, 38071}, {36765, 38229}
X(68405) = reflection of X(20253) in X(59402)

X(68406) = MIDPOINT OF X(68404) AND X(68405)

X(68406) lies on these lines: {2, 38731}, {5, 39}, {98, 5072}, {99, 5079}, {148, 61921}, {381, 34127}, {542, 14892}, {543, 47478}, {546, 6722}, {547, 22247}, {620, 35018}, {632, 39809}, {671, 61925}, {1656, 21166}, {2482, 61916}, {2794, 5066}, {3090, 33813}, {3091, 12042}, {3544, 51523}, {3545, 38224}, {3628, 67863}, {3832, 38739}, {3839, 26614}, {3845, 38737}, {3850, 6036}, {3851, 14061}, {3855, 38741}, {3857, 39838}, {3858, 38749}, {3861, 38747}, {5055, 14639}, {5056, 6321}, {5067, 38730}, {5068, 6033}, {5070, 10723}, {5071, 8591}, {5461, 11737}, {6054, 61931}, {6055, 61942}, {6721, 12812}, {7486, 38750}, {8724, 61926}, {9166, 38743}, {9167, 61909}, {9880, 61910}, {10109, 36521}, {10722, 38634}, {11591, 58518}, {11632, 61932}, {11725, 61259}, {12117, 61901}, {12131, 35487}, {12188, 61935}, {12811, 67862}, {13172, 61914}, {13188, 61923}, {14128, 39806}, {14651, 22566}, {14830, 61944}, {14971, 38071}, {15022, 35369}, {15088, 67479}, {15699, 38748}, {16239, 38736}, {18874, 39835}, {19709, 49102}, {20094, 67096}, {20399, 61600}, {31274, 61900}, {38220, 61263}, {38627, 61599}, {38635, 61905}, {38732, 61920}, {38733, 61911}, {38735, 61934}, {38738, 55856}, {38745, 41989}, {44904, 61561}, {46031, 62490}, {61560, 61940}, {61919, 64089}

X(68406) = midpoint of X(i) and X(j) for these {i,j}: {5, 23514}, {381, 34127}, {3839, 26614}, {3845, 38737}, {14651, 22566}, {14971, 38071}, {21166, 22515}, {36519, 38229}, {68404,68405}
X(68406) = reflection of X(i) in X(j) for these {i,j}: {23514, 15092}, {61576, 23514}
X(68406) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 15092, 61576}, {5, 38229, 36519}, {5, 61576, 61575}, {3851, 14061, 22505}, {23514, 36519, 38229}

X(68407) = (name pending)

See David Nguyen and Ercole Suppa, euclid 8386.

X(68407) lies on this line: {8825, 9739}

X(68407) = isogonal conjugate of the anticomplement of X(15885)

X(68408) = X(6)X(1163)∩X(20)X(1160)

See David Nguyen and Ercole Suppa, euclid 8388.

X(68408) lies on these lines: {6, 1163}, {20, 1160}, {3156, 8904}, {5870, 11381}, {6276, 10619}, {17816, 44609}

X(68409) = X(3)X(6)∩X(230)X(58189)

See David Nguyen and Juan José Isach Mayo, euclid 8390.

X(68409) lies on these lines: {3, 6}, {230, 58189}, {11614, 61974}, {31415, 61797}, {43291, 62070}, {53419, 62080}

X(68409) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 5585, 21309}, {5585, 53095, 15513}

X(68410) = X(5)X(523)∩X(6)X(2065)

See Elias Hagos and Peter Moses, euclid 8395.

X(68410) lies on these lines: {5, 523}, {6, 2065}, {32, 28343}, {576, 2871}, {648, 58734}, {2023, 52672}, {2781, 59290}, {3001, 23098}, {3095, 59363}, {14251, 23635}, {14966, 44668}, {20975, 34156}, {34157, 34990}, {34349, 52727}

X(68410) = reflection of X(52727) in X(34349)

X(68411) = X(5)X(523)∩X(1482)X(59366)

See Elias Hagos and Peter Moses, euclid 8395.

X(68411) lies on these lines: {5, 523}, {1482, 59366}, {2778, 59293}, {14260, 45776}, {18210, 39175}, {22835, 42753}, {23101, 45916}, {34345, 52731}, {34977, 39173}

X(68411) = reflection of X(52731) in X(34345)

X(68412) = X(4)X(525)∩X(5)X(523)

See Elias Hagos and Peter Moses, euclid 8395.

X(68412) lies on these lines: {2, 58273}, {3, 45681}, {4, 525}, {5, 523}, {20, 5664}, {107, 61500}, {155, 8057}, {381, 5489}, {382, 58346}, {512, 40647}, {520, 5446}, {546, 39491}, {631, 18556}, {850, 44142}, {1656, 65723}, {2394, 3832}, {2501, 8743}, {2548, 62384}, {2799, 38745}, {2848, 44810}, {3090, 53383}, {3150, 58261}, {3800, 10279}, {3843, 42733}, {5466, 18841}, {6368, 35719}, {7762, 33294}, {7776, 62555}, {7927, 40550}, {8673, 66754}, {8675, 43130}, {9007, 64067}, {9033, 16534}, {9517, 16230}, {10278, 57588}, {10280, 53266}, {14246, 62629}, {16104, 53178}, {17578, 63248}, {18808, 18855}, {30474, 31857}, {32204, 34291}, {33754, 45801}, {37814, 39228}, {39201, 45735}, {41079, 65871}, {44235, 59745}, {44817, 55129}, {45327, 46512}, {46371, 58757}, {47262, 62662}, {57128, 61757}, {58070, 61181}

X(68412) = midpoint of X(41079) and X(65871)
X(68412) = reflection of X(i) in X(j) for these {i,j}: {24978, 16230}, {41079, 68330}, {45681, 65754}
X(68412) = crosspoint of X(648) and X(13485)
X(68412) = crosssum of X(647) and X(7669)
X(68412) = crossdifference of every pair of points on line {50, 8779}

X(68413) = X(5)X(523)∩X(10)X(514)

See Elias Hagos and Peter Moses, euclid 8395.

X(68413) lies on these lines: {5, 523}, {10, 514}, {12, 60074}, {56, 14838}, {495, 56283}, {513, 3579}, {522, 21077}, {661, 3730}, {1019, 37405}, {1482, 39212}, {1577, 11681}, {2812, 52726}, {2850, 44812}, {3737, 57708}, {3887, 12738}, {4041, 5903}, {4086, 30172}, {4122, 63826}, {4129, 27553}, {4560, 20060}, {6003, 11500}, {9013, 43146}, {11236, 64934}, {13589, 14513}, {16117, 42325}, {21201, 42758}, {28161, 42757}, {34605, 45671}, {37050, 47809}, {47842, 51572}, {50346, 56289}

X(68413) = crosssum of X(i) and X(j) for these (i,j): {215, 46384}, {513, 65524}
X(68413) = crossdifference of every pair of points on line {50, 1914}

X(68414) = X(3)X(512)∩X(5)X(523)

See Elias Hagos and Peter Moses, euclid 8395.

X(68414) lies on these lines: {1, 2608}, {3, 512}, {4, 62489}, {5, 523}, {6, 8574}, {24, 47221}, {32, 647}, {49, 57136}, {61, 57123}, {62, 57122}, {110, 39138}, {114, 51232}, {155, 520}, {403, 46371}, {525, 22660}, {526, 5607}, {576, 8675}, {669, 23208}, {684, 690}, {826, 3574}, {850, 7752}, {868, 6328}, {924, 6759}, {1112, 55383}, {1510, 62173}, {1649, 3005}, {1656, 53266}, {1657, 53275}, {2126, 3733}, {2395, 2548}, {2971, 36955}, {3095, 3906}, {3527, 14380}, {3737, 56289}, {3851, 39482}, {3933, 52629}, {4108, 52300}, {5007, 6041}, {5013, 10097}, {6132, 9517}, {6140, 53263}, {6368, 68174}, {7746, 47229}, {7772, 10567}, {7775, 23878}, {7785, 31296}, {7808, 62688}, {7812, 36900}, {7862, 30476}, {7927, 34347}, {8552, 44826}, {8562, 14809}, {8651, 20993}, {9033, 34104}, {9168, 38526}, {9177, 14824}, {9213, 14246}, {9409, 39477}, {9426, 18796}, {9737, 30209}, {10190, 11007}, {10684, 46245}, {11183, 15000}, {14264, 32112}, {14366, 33803}, {14618, 16868}, {15329, 44830}, {15359, 20975}, {16229, 35488}, {16837, 61196}, {17414, 37338}, {18308, 42733}, {22260, 46127}, {30510, 52630}, {32204, 62510}, {32816, 62642}, {32832, 53347}, {34964, 49673}, {38987, 65728}, {39509, 41079}, {54003, 67534}

X(68414) = midpoint of X(i) and X(j) for these {i,j}: {684, 21731}, {23109, 23110}
X(68414) = reflection of X(i) in X(j) for these {i,j}: {{9409, 39477}, {14270, 6132}, {14809, 8562}, {23105, 5}, {41079, 39509}, {42733, 18308}, {44814, 34291}, {44826, 8552}, {51232, 114}, {53263, 6140}
X(68414) = reflection of X(23105) in the Euler line
X(68414) = X(i)-Ceva conjugate of X(j) for these (i,j): {34990, 55384}, {39295, 2088}, {40173, 6}, {40511, 3124}, {44549, 3269}, {66162, 3569}
X(68414) = X(55384)-cross conjugate of X(34990)
X(68414) = X(162)-isoconjugate of X(46087)
X(68414) = X(i)-Dao conjugate of X(j) for these (i,j): {125, 46087}, {15295, 14781}, {34990, 850}
X(68414) = crosspoint of X(110) and X(523)
X(68414) = crosssum of X(110) and X(523)
X(68414) = crossdifference of every pair of points on line {50, 230}
X(68414) = barycentric product X(i)*X(j) for these {i,j}: {99, 55384}, {523, 34990}, {525, 1112}, {2799, 47635}, {4705, 16734}, {14618, 23217}
X(68414) = barycentric quotient X(i)/X(j) for these {i,j}: {647, 46087}, {1112, 648}, {11060, 14781}, {16734, 4623}, {23217, 4558}, {34990, 99}, {47635, 2966}, {55384, 523}

X(68415) = X(5)X(523)∩X(514)X(1125)

See Elias Hagos and Peter Moses, euclid 8395.

X(68415) lies on these lines: {5, 523}, {451, 48209}, {513, 13369}, {514, 1125}, {522, 18483}, {867, 42753}, {928, 62435}, {1577, 27555}, {3004, 23100}, {3700, 24045}, {4223, 47797}, {4560, 37369}, {4802, 33528}, {4977, 66968}, {6362, 16160}, {7178, 10571}, {14377, 17069}, {21789, 44253}, {28473, 40257}, {30172, 52355}, {34958, 59285}, {38330, 64934}

X(68415) = crosspoint of X(693) and X(38340)
X(68415) = crosssum of X(692) and X(9404)
X(68415) = crossdifference of every pair of points on line {50, 17735}

X(68416) = X(5)X(523)∩X(10)X(14260)

See Elias Hagos and Peter Moses, euclid 8395.

X(68416) lies on these lines: {5, 523}, {10, 14260}, {3006, 58254}, {3814, 42753}, {16173, 61768}, {62703, 66865}

X(68417) = X(4)X(543)∩X(5)X(523)

See Elias Hagos and Peter Moses, euclid 8395.

X(68417) lies on these lines: {2, 8877}, {4, 543}, {5, 523}, {32, 10418}, {115, 59422}, {325, 23106}, {538, 57604}, {576, 16534}, {620, 34161}, {754, 7417}, {2482, 52483}, {2548, 35606}, {5099, 5968}, {5181, 51980}, {5461, 9214}, {7752, 31857}, {7812, 14002}, {14357, 53136}, {14995, 52533}, {37760, 52630}, {51999, 67396}

X(68418) = X(5)X(523)∩X(522)X(6260)

See Elias Hagos and Peter Moses, euclid 8395.

X(68418) lies on these lines: {5, 523}, {514, 12616}, {522, 6260}, {3900, 64804}, {4041, 10571}, {11500, 35057}, {14838, 59285}, {35100, 66968}

X(68419) = X(5)X(523)∩X(522)X(6684)

See Elias Hagos and Peter Moses, euclid 8395.

X(68419) lies on these lines: {5, 523}, {451, 48204}, {514, 19925}, {522, 6684}, {928, 62434}, {1577, 27687}, {2774, 18004}, {3700, 3730}, {3884, 49290}, {3900, 31837}, {4223, 47809}, {4560, 37158}, {4777, 33528}, {5499, 6362}, {14887, 51562}, {23104, 50333}, {28602, 62494}, {56283, 59283}