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)