leftri rightri

This is PART 35: Centers X(68001) - X(70000)

PART 1: Introduction and Centers X(1) - X(1000) PART 2: Centers X(1001) - X(3000) PART 3: Centers X(3001) - X(5000)
PART 4: Centers X(5001) - X(7000) PART 5: Centers X(7001) - X(10000) PART 6: Centers X(10001) - X(12000)
PART 7: Centers X(12001) - X(14000) PART 8: Centers X(14001) - X(16000) PART 9: Centers X(16001) - X(18000)
PART 10: Centers X(18001) - X(20000) PART 11: Centers X(20001) - X(22000) PART 12: Centers X(22001) - X(24000)
PART 13: Centers X(24001) - X(26000) PART 14: Centers X(26001) - X(28000) PART 15: Centers X(28001) - X(30000)
PART 16: Centers X(30001) - X(32000) PART 17: Centers X(32001) - X(34000) PART 18: Centers X(34001) - X(36000)
PART 19: Centers X(36001) - X(38000) PART 20: Centers X(38001) - X(40000) PART 21: Centers X(40001) - X(42000)
PART 22: Centers X(42001) - X(44000) PART 23: Centers X(44001) - X(46000) PART 24: Centers X(46001) - X(48000)
PART 25: Centers X(48001) - X(50000) PART 26: Centers X(50001) - X(52000) PART 27: Centers X(52001) - X(54000)
PART 28: Centers X(54001) - X(56000) PART 29: Centers X(56001) - X(58000) PART 30: Centers X(58001) - X(60000)
PART 31: Centers X(60001) - X(62000) PART 32: Centers X(62001) - X(64000) PART 33: Centers X(64001) - X(66000)
PART 34: Centers X(66001) - X(68000) PART 35: Centers X(68001) - X(70000) PART 36: Centers X(70001) - X(72000)

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

Barycentrics    a*(a^6-4*a^5*(b+c)+a^4*(b^2+10*b*c+c^2)+(b^2-c^2)^2*(3*b^2-2*b*c+3*c^2)-a^2*(b-c)^2*(5*b^2+18*b*c+5*c^2)-4*a*(b-c)^2*(b^3-2*b^2*c-2*b*c^2+c^3)+4*a^3*(2*b^3-3*b^2*c-3*b*c^2+2*c^3)) : :
X(68001) = -3*X[5603]+2*X[11019], -3*X[5657]+4*X[20103], -6*X[5886]+5*X[31249], -5*X[16189]+X[18452], -3*X[46917]+2*X[63132]

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

Barycentrics    a^7-3*a^6*(b+c)-3*a^5*(b+c)^2+3*(b-c)^4*(b+c)^3-a*(b^2-c^2)^2*(b^2-10*b*c+c^2)+a^3*(b-c)^2*(3*b^2+2*b*c+3*c^2)+3*a^4*(3*b^3+b^2*c+b*c^2+3*c^3)-3*a^2*(b-c)^2*(3*b^3+5*b^2*c+5*b*c^2+3*c^3) : :
X(68002) = -9*X[2]+2*X[84], 4*X[5]+3*X[5658], -8*X[140]+X[12246], 3*X[376]+4*X[22792], 6*X[381]+X[64144], 6*X[549]+X[48664], 5*X[631]+2*X[6259], 2*X[1490]+5*X[3091], 5*X[1698]+2*X[54227], X[3146]+6*X[52026], -5*X[3522]+12*X[68003], -11*X[3525]+4*X[34862], -X[3529]+8*X[40262], -9*X[3545]+2*X[5787], 5*X[3617]+2*X[7971], -8*X[3628]+X[12684], -8*X[3634]+X[7992], 6*X[3817]+X[63981], -9*X[3839]+2*X[64261], -11*X[5056]+4*X[6245]

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

Barycentrics    4*a^7-5*a^6*(b+c)+(b-c)^4*(b+c)^3+a^5*(-8*b^2+4*b*c-8*c^2)+4*a*b*c*(b^2-c^2)^2+4*a^3*(b-c)^2*(b^2+c^2)-a^2*(b-c)^2*(7*b^3+9*b^2*c+9*b*c^2+7*c^3)+a^4*(11*b^3+b^2*c+b*c^2+11*c^3) : :
X(68003) = X[5]+2*X[40262], X[10]+2*X[37837], X[72]+2*X[40249], -4*X[140]+X[6245], 2*X[548]+X[22792], X[550]+2*X[64813], 5*X[631]+X[1490], -7*X[3090]+X[64261], 5*X[3522]+7*X[68002], 3*X[3524]+X[5658], 11*X[3525]+X[64144], -7*X[3526]+X[5787], -4*X[3530]+X[34862], 2*X[3579]+X[54198], 2*X[5044]+X[9942], X[6223]+11*X[15717], 5*X[7987]+X[12667], -X[9799]+13*X[10303], -13*X[10299]+X[12246], -4*X[12108]+X[61556]

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

Barycentrics    (a-b-c)*(a^2+b^2-c^2)*(2*a^2-b^2-c^2+a*(b+c))*(a^2-b^2+c^2) : :

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

Barycentrics    a*(a^8*(b+c)-2*a^6*(b-c)^2*(b+c)-(b-c)^4*(b+c)^5-2*a^7*(b^2+b*c+c^2)+2*a^5*(b+c)^2*(3*b^2-5*b*c+3*c^2)+4*a^4*b*c*(-2*b^3+b^2*c+b*c^2-2*c^3)+2*a*(b^2-c^2)^2*(b^4-b^3*c+2*b^2*c^2-b*c^3+c^4)-2*a^3*(b-c)^2*(3*b^4+5*b^3*c+2*b^2*c^2+5*b*c^3+3*c^4)+2*a^2*(b-c)^2*(b^5+5*b^4*c+4*b^3*c^2+4*b^2*c^3+5*b*c^4+c^5)) : :
X(68005) = 3*X[5692]+X[64261], -3*X[5927]+X[6256], -3*X[10157]+2*X[63964], -X[12671]+5*X[25917]

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

Barycentrics    5*a^12+(b^2-c^2)^6-10*a^10*(b^2+c^2)+36*a^6*(b^2-c^2)^2*(b^2+c^2)+a^8*(-9*b^4+34*b^2*c^2-9*c^4)-a^4*(b^2-c^2)^2*(29*b^4+54*b^2*c^2+29*c^4)+2*a^2*(b^2-c^2)^2*(3*b^6+13*b^4*c^2+13*b^2*c^4+3*c^6) : :

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

Barycentrics    4*a^16-3*a^15*(b+c)+a^14*(-5*b^2+6*b*c-5*c^2)+(b^2-c^2)^8-a^12*(b-c)^2*(37*b^2+78*b*c+37*c^2)-a^9*(b-c)^2*(b+c)^3*(55*b^2-38*b*c+55*c^2)+a^10*(b^2-c^2)^2*(119*b^2-38*b*c+119*c^2)+a^13*(5*b^3-b^2*c-b*c^2+5*c^3)+a^11*(b-c)^2*(17*b^3+55*b^2*c+55*b*c^2+17*c^3)-a^2*(b-c)^6*(b+c)^4*(3*b^4+10*b^2*c^2+3*c^4)+a*(b-c)^6*(b+c)^5*(3*b^4+10*b^2*c^2+3*c^4)-a^3*(b-c)^6*(b+c)^3*(5*b^4+16*b^3*c+6*b^2*c^2+16*b*c^3+5*c^4)+a^7*(b-c)^2*(b+c)^3*(55*b^4-72*b^3*c+146*b^2*c^2-72*b*c^3+55*c^4)-a^8*(b^2-c^2)^2*(145*b^4-72*b^3*c+350*b^2*c^2-72*b*c^3+145*c^4)-a^5*(b-c)^2*(b+c)^3*(17*b^6-38*b^5*c+127*b^4*c^2-148*b^3*c^3+127*b^2*c^4-38*b*c^5+17*c^6)+a^6*(b^2-c^2)^2*(81*b^6-38*b^5*c+319*b^4*c^2-148*b^3*c^3+319*b^2*c^4-38*b*c^5+81*c^6)-a^4*(b^2-c^2)^2*(15*b^8+4*b^7*c+76*b^6*c^2-68*b^5*c^3+202*b^4*c^4-68*b^3*c^5+76*b^2*c^6+4*b*c^7+15*c^8) : :
X(68007) = X[962]+3*X[54053], -X[3183]+3*X[3576], -3*X[3817]+2*X[51342]

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

Barycentrics    (a^2+b^2-c^2)*(a^2-b^2+c^2)*(2*a^18-7*a^16*(b^2+c^2)-12*a^12*(b^2-c^2)^2*(b^2+c^2)+(b^2-c^2)^8*(b^2+c^2)+8*a^14*(b^4+c^4)+2*a^2*(b^2-c^2)^6*(b^4+6*b^2*c^2+c^4)+4*a^10*(b^2-c^2)^2*(11*b^4+38*b^2*c^2+11*c^4)-2*a^8*(b^2-c^2)^2*(41*b^6+151*b^4*c^2+151*b^2*c^4+41*c^6)+8*a^6*(b^2-c^2)^2*(9*b^8+26*b^6*c^2+58*b^4*c^4+26*b^2*c^6+9*c^8)-4*a^4*(b^2-c^2)^2*(7*b^10+11*b^8*c^2+46*b^6*c^4+46*b^4*c^6+11*b^2*c^8+7*c^10)) : :

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

Barycentrics    3*a^10-6*a^6*(b^2-c^2)^2-2*a^8*(b^2+c^2)-6*(b^2-c^2)^4*(b^2+c^2)+a^2*(b^2-c^2)^2*(11*b^4-6*b^2*c^2+11*c^4) : :
X(68009) = -3*X[2]+X[27082], -3*X[51]+X[16879], X[10733]+2*X[39084]

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

Barycentrics    3*a^10-26*a^6*(b^2-c^2)^2+3*a^8*(b^2+c^2)+30*a^4*(b^2-c^2)^2*(b^2+c^2)-(b^2-c^2)^4*(b^2+c^2)-a^2*(b^2-c^2)^2*(9*b^4+46*b^2*c^2+9*c^4) : :
X(68010) = -3*X[2]+4*X[11472], -3*X[376]+2*X[11820], -5*X[3091]+4*X[4846], -7*X[3523]+8*X[4550], -3*X[3545]+2*X[44750], -7*X[3832]+6*X[66749], -9*X[3839]+8*X[7706], -8*X[8717]+9*X[10304], -11*X[15717]+12*X[32620], -5*X[17578]+4*X[40909], -8*X[31861]+7*X[51171], -8*X[35254]+7*X[50693], -3*X[35513]+4*X[48876], -4*X[49669]+3*X[66755], -8*X[51993]+9*X[62003], -16*X[52101]+13*X[61982], -2*X[53780]+3*X[62029], -3*X[62174]+4*X[64097], -4*X[64096]+3*X[66742]

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

Barycentrics    a^2*(a^18*(b^2+c^2)-5*a^16*(b^4+c^4)-(b^2-c^2)^6*(b^2+c^2)^2*(b^4+5*b^2*c^2+c^4)+a^14*(8*b^6-3*b^4*c^2-3*b^2*c^4+8*c^6)-a^10*(b^2-c^2)^2*(14*b^6-3*b^4*c^2-3*b^2*c^4+14*c^6)-a^6*b^2*c^2*(b^2-c^2)^2*(25*b^6-9*b^4*c^2-9*b^2*c^4+25*c^6)+a^12*(-11*b^6*c^2+28*b^4*c^4-11*b^2*c^6)+a^8*(b^2-c^2)^2*(14*b^8+17*b^6*c^2-14*b^4*c^4+17*b^2*c^6+14*c^8)+a^2*(b^2-c^2)^4*(5*b^10+16*b^8*c^2+11*b^6*c^4+11*b^4*c^6+16*b^2*c^8+5*c^10)-a^4*(b^2-c^2)^2*(8*b^12-7*b^10*c^2-2*b^8*c^4+34*b^6*c^6-2*b^4*c^8-7*b^2*c^10+8*c^12)) : :

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

Barycentrics    a^2*(a^18*(b^2+c^2)-5*a^16*(b^4+c^4)-(b^2-c^2)^6*(b^2+c^2)^2*(b^4+5*b^2*c^2+c^4)+a^14*(8*b^6+5*b^4*c^2+5*b^2*c^4+8*c^6)-a^6*b^2*c^2*(b^2-c^2)^2*(65*b^6-97*b^4*c^2-97*b^2*c^4+65*c^6)+a^12*(-43*b^6*c^2+52*b^4*c^4-43*b^2*c^6)+a^8*(b^2-c^2)^2*(14*b^8+17*b^6*c^2-134*b^4*c^4+17*b^2*c^6+14*c^8)+a^2*(b^2-c^2)^4*(5*b^10+8*b^8*c^2-41*b^6*c^4-41*b^4*c^6+8*b^2*c^8+5*c^10)-a^10*(14*b^10-71*b^8*c^2+45*b^6*c^4+45*b^4*c^6-71*b^2*c^8+14*c^10)-a^4*(b^2-c^2)^2*(8*b^12-39*b^10*c^2-34*b^8*c^4+162*b^6*c^6-34*b^4*c^8-39*b^2*c^10+8*c^12)) : :
X(68012) = -3*X[15305]+X[57648]

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

Barycentrics    a^2*(a^9*(b^2+c^2)-(b-c)^4*(b+c)^3*(b^2+c^2)^2-2*a^7*(b-c)^2*(2*b^2+b*c+2*c^2)-a^8*(b^3+b^2*c+b*c^2+c^3)+2*a^2*(b-c)^2*(b+c)^3*(2*b^4-3*b^3*c+4*b^2*c^2-3*b*c^3+2*c^4)+2*a^5*(b-c)^2*(3*b^4-3*b^3*c-4*b^2*c^2-3*b*c^3+3*c^4)+2*a^6*(2*b^5+b^4*c-b^3*c^2-b^2*c^3+b*c^4+2*c^5)+a*(b^2-c^2)^2*(b^6-6*b^5*c+7*b^4*c^2-20*b^3*c^3+7*b^2*c^4-6*b*c^5+c^6)-2*a^4*(3*b^7-4*b^5*c^2+b^4*c^3+b^3*c^4-4*b^2*c^5+3*c^7)-2*a^3*(2*b^8-9*b^7*c+6*b^6*c^2+b^5*c^3+b^3*c^5+6*b^2*c^6-9*b*c^7+2*c^8)) : :
X(68013) = -3*X[51]+4*X[7682], -3*X[3917]+2*X[6282], -4*X[9729]+5*X[62773], -3*X[9730]+4*X[61535], -3*X[15030]+2*X[37822]

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

Barycentrics    a^2*(a^14*(b^2+c^2)-3*a^12*(b^4+c^4)-(b^2-c^2)^4*(b^2+c^2)^2*(b^4+3*b^2*c^2+c^4)+a^10*(b^6+2*b^4*c^2+2*b^2*c^4+c^6)-a^6*(b^2-c^2)^2*(5*b^6+9*b^4*c^2+9*b^2*c^4+5*c^6)-a^4*(b^2-c^2)^2*(b^8-8*b^6*c^2-4*b^4*c^4-8*b^2*c^6+c^8)+a^8*(5*b^8-9*b^6*c^2+6*b^4*c^4-9*b^2*c^6+5*c^8)+a^2*(b^2-c^2)^2*(3*b^10+2*b^8*c^2-b^6*c^4-b^4*c^6+2*b^2*c^8+3*c^10)) : :
X(68014) = -X[110]+3*X[68017], -2*X[1112]+3*X[53023], -5*X[3618]+3*X[66734], -3*X[5085]+2*X[44573], -3*X[5622]+X[6241], -X[7722]+3*X[14853], -X[12270]+3*X[52699], -4*X[13416]+3*X[31884]

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

Barycentrics    3*a^10+3*a^8*(b^2+c^2)+30*a^4*(b^2-c^2)^2*(b^2+c^2)-(b^2-c^2)^4*(b^2+c^2)+a^6*(-26*b^4+44*b^2*c^2-26*c^4)-a^2*(b^2-c^2)^2*(9*b^4+38*b^2*c^2+9*c^4) : :
X(68015) = -3*X[2]+4*X[64], -12*X[154]+13*X[21734], -3*X[376]+2*X[12315], -5*X[631]+6*X[35450], -4*X[1498]+5*X[3522], -12*X[1853]+11*X[50689], -7*X[3090]+8*X[61540], -5*X[3091]+4*X[5878], -8*X[3357]+7*X[3523], -7*X[3528]+6*X[32063], -3*X[3543]+4*X[14216], -5*X[3617]+4*X[12779], -5*X[3620]+4*X[41735], -7*X[3622]+8*X[12262], -5*X[3623]+4*X[7973], -7*X[3832]+8*X[6247], -9*X[3839]+8*X[22802], -17*X[3854]+16*X[5893]

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

Barycentrics    a*(a+b)*(a+c)*(a^10-2*a^9*(b+c)+12*a^5*(b-c)^2*(b+c)^3-3*a^8*(b^2+c^2)-16*a^3*(b-c)^2*(b+c)^3*(b^2+c^2)+2*a^4*(b^2-c^2)^2*(b^2+c^2)+(b^2-c^2)^4*(b^2+c^2)+2*a^6*(b^2+c^2)^2-a^2*(b^2-c^2)^2*(3*b^4+2*b^2*c^2+3*c^4)+2*a*(b-c)^2*(b+c)^3*(3*b^4+10*b^2*c^2+3*c^4)) : :

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)

Barycentrics    a^2*(a^8*(b^2+c^2)+a^4*b^2*c^2*(b^2+c^2)+a^6*(-2*b^4+b^2*c^2-2*c^4)-(b^2-c^2)^2*(b^6+4*b^4*c^2+4*b^2*c^4+c^6)+a^2*(2*b^8-b^6*c^2+6*b^4*c^4-b^2*c^6+2*c^8)) : :
X(68017) = -4*X[5]+X[67922], 2*X[6]+X[12111], X[110]+2*X[68014], -4*X[141]+7*X[15056], -4*X[182]+X[6241], -2*X[185]+5*X[3618], X[895]+2*X[12825], -2*X[1350]+5*X[11444], X[1351]+2*X[5876], X[1885]+2*X[13562], -5*X[3091]+2*X[19161], -4*X[3098]+7*X[7999], X[3146]+2*X[3313], -5*X[3567]+8*X[19130], -8*X[3589]+5*X[10574], -7*X[3619]+4*X[52520], -7*X[3832]+4*X[9969], -4*X[5480]+X[5889]

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)

Barycentrics    (a^2-b^2-c^2)*(2*a^8+(b^2-c^2)^4-3*a^6*(b^2+c^2)-a^2*(b^2-c^2)^2*(b^2+c^2)+a^4*(b^4+6*b^2*c^2+c^4)) : :
X(68018) = -3*X[51]+4*X[9825], -3*X[376]+X[34224], -2*X[382]+3*X[16654], -4*X[389]+3*X[61658], -3*X[428]+2*X[13598], -5*X[631]+3*X[12022], -3*X[2979]+X[12225], -3*X[3060]+4*X[11745], -5*X[3522]+X[34799], -3*X[3543]+4*X[16656], -3*X[3917]+2*X[12362], -2*X[5446]+3*X[67237], -3*X[5892]+2*X[58806]

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

Barycentrics    a^2*(a^10-7*a^8*(b^2+c^2)+2*a^4*(b^2-c^2)^2*(b^2+c^2)+2*a^6*(5*b^4-2*b^2*c^2+5*c^4)-a^2*(b^2-c^2)^2*(11*b^4+10*b^2*c^2+11*c^4)+(b^2-c^2)^2*(5*b^6+11*b^4*c^2+11*b^2*c^4+5*c^6)) : :
X(68019) = -4*X[3]+5*X[19132], -4*X[141]+5*X[64024], -4*X[182]+3*X[10606], -3*X[599]+4*X[67870], -3*X[1853]+4*X[5480], -4*X[3098]+5*X[17821], -2*X[3357]+3*X[5050], -5*X[3618]+4*X[6696], -5*X[3620]+7*X[68024], -3*X[5032]+X[68027], -6*X[5085]+5*X[8567], -3*X[5093]+X[13093], -3*X[5102]+2*X[8549]

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

Barycentrics    a^2*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^8*(b^2+c^2)+(b^2-c^2)^2*(b^2+c^2)^3-2*a^6*(2*b^4+b^2*c^2+2*c^4)+2*a^4*(3*b^6+b^4*c^2+b^2*c^4+3*c^6)-2*a^2*(2*b^8+b^6*c^2+2*b^4*c^4+b^2*c^6+2*c^8)) : :
X(68020) = -3*X[11245]+4*X[46363]

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

Barycentrics    3*a^12-6*a^10*(b^2+c^2)-(b^2-c^2)^4*(b^2+c^2)^2+a^8*(b^4+26*b^2*c^2+c^4)-a^4*(b^2-c^2)^2*(3*b^4+2*b^2*c^2+3*c^4)+2*a^2*(b^2-c^2)^2*(b^6-5*b^4*c^2-5*b^2*c^4+c^6)+4*a^6*(b^6-3*b^4*c^2-3*b^2*c^4+c^6) : :
X(68021) = -2*X[141]+3*X[52028], -3*X[376]+2*X[34787], -4*X[575]+3*X[67890], -6*X[597]+5*X[64024], -3*X[599]+4*X[6696], -5*X[631]+6*X[10249], -3*X[1351]+X[48672], -3*X[1992]+X[6225], -5*X[3091]+6*X[23327], -7*X[3523]+6*X[61683], -5*X[3618]+4*X[67870], -X[5895]+3*X[17813]

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

Barycentrics    a^2*(a^8-6*a^6*(b^2+c^2)+4*a^4*(3*b^4-b^2*c^2+3*c^4)+(b^2-c^2)^2*(3*b^4+14*b^2*c^2+3*c^4)-2*a^2*(5*b^6-b^4*c^2-b^2*c^4+5*c^6)) : :
X(68022) = -5*X[3091]+3*X[18950]

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

Barycentrics    a^2*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^6-3*b^6-5*b^4*c^2-5*b^2*c^4-3*c^6-5*a^4*(b^2+c^2)+a^2*(7*b^4+2*b^2*c^2+7*c^4)) : :

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

Barycentrics    a^10-9*a^8*(b^2+c^2)-10*a^4*(b^2-c^2)^2*(b^2+c^2)+3*(b^2-c^2)^4*(b^2+c^2)-a^2*(b^2-c^2)^2*(3*b^4-14*b^2*c^2+3*c^4)+6*a^6*(3*b^4-2*b^2*c^2+3*c^4) : :
X(68024) = -9*X[2]+2*X[64], -5*X[3]+12*X[61606], 3*X[69]+4*X[64031], -8*X[140]+X[12250], 6*X[154]+X[3146], 4*X[159]+3*X[51538], -8*X[206]+X[14927], 3*X[376]+4*X[22802], 6*X[381]+X[34781], 4*X[546]+3*X[32063], 6*X[549]+X[48672], 5*X[631]+2*X[5878], -10*X[632]+3*X[35450], 4*X[1147]+3*X[67201], 2*X[1498]+5*X[3091], -10*X[1656]+3*X[67894], -6*X[1853]+13*X[5068], -4*X[3357]+11*X[3525], 5*X[3522]+2*X[5895], -9*X[3524]+2*X[20427]

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

Barycentrics    2*a^10-11*a^8*(b^2+c^2)-14*a^4*(b^2-c^2)^2*(b^2+c^2)+(b^2-c^2)^4*(b^2+c^2)+2*a^2*(b^2-c^2)^2*(b^4+10*b^2*c^2+c^4)+4*a^6*(5*b^4-4*b^2*c^2+5*c^4) : :
X(68025) = 3*X[2]+X[58795], -3*X[3]+X[15105], -3*X[64]+7*X[3523], -9*X[154]+5*X[3522], -X[550]+3*X[6759], -5*X[1656]+3*X[6247], X[1657]+3*X[5878], -9*X[1853]+13*X[5068], -3*X[3357]+5*X[15712], -17*X[3533]+9*X[67894], -7*X[3851]+3*X[14216], -17*X[3854]+9*X[32064], -5*X[3858]+3*X[18381], -11*X[5056]+3*X[12324], X[5059]+3*X[5895], X[5073]+3*X[9833], -X[5493]+3*X[40660]

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

Barycentrics    a^2*(a^12*(b^2+c^2)+4*a^6*b^2*c^2*(b^2+c^2)^2-(b^2-c^2)^4*(b^2+c^2)^3+a^10*(-4*b^4+2*b^2*c^2-4*c^4)+a^8*(5*b^6-9*b^4*c^2-9*b^2*c^4+5*c^6)-a^4*(b^2-c^2)^2*(5*b^6+3*b^4*c^2+3*b^2*c^4+5*c^6)+2*a^2*(b^2-c^2)^2*(2*b^8+b^6*c^2+2*b^4*c^4+b^2*c^6+2*c^8)) : :
X(68026) = X[20]+3*X[41715], -3*X[51]+X[64037], -3*X[154]+X[5562], 3*X[568]+X[64033], -3*X[1853]+5*X[64854], -5*X[3567]+X[64034], -3*X[3917]+5*X[17821], -2*X[5447]+3*X[11202], X[5889]+3*X[11206], 3*X[5890]+X[34781], -2*X[6696]+3*X[16836], 3*X[7729]+X[58795], -2*X[10095]+3*X[63714], -3*X[10192]+2*X[11793], -5*X[11444]+9*X[35260]

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

Barycentrics    5*a^10+3*a^8*(b^2+c^2)+46*a^4*(b^2-c^2)^2*(b^2+c^2)-(b^2-c^2)^4*(b^2+c^2)+a^6*(-38*b^4+68*b^2*c^2-38*c^4)-a^2*(b^2-c^2)^2*(15*b^4+58*b^2*c^2+15*c^4) : :
X(68027) = -X[20]+4*X[15105], -6*X[154]+7*X[62063], -4*X[549]+3*X[5656], -2*X[1498]+3*X[10304], -6*X[1853]+5*X[61985], -4*X[3357]+3*X[3524], -5*X[3522]+2*X[58795], -3*X[3545]+2*X[5878], -3*X[3839]+4*X[6247], -3*X[5032]+2*X[68019], -3*X[5055]+4*X[61540], -5*X[5071]+6*X[65151], -8*X[5893]+9*X[61954], -4*X[5894]+3*X[62120], -2*X[5895]+3*X[50687]

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

Barycentrics    a^2*(a^12*(b^2+c^2)-4*a^10*(b^4+c^4)-8*a^6*b^2*c^2*(b^4-b^2*c^2+c^4)+5*a^8*(b^6+c^6)-(b^2-c^2)^4*(b^6+6*b^4*c^2+6*b^2*c^4+c^6)-a^4*(b^2-c^2)^2*(5*b^6+b^4*c^2+b^2*c^4+5*c^6)+4*a^2*(b^12-3*b^8*c^4+4*b^6*c^6-3*b^4*c^8+c^12)) : :
X(68028) = -X[52]+3*X[23324], X[64]+3*X[15305], -3*X[154]+7*X[15056], -X[185]+3*X[23332], 3*X[1853]+X[12111], -5*X[3091]+X[6293], -4*X[3850]+3*X[63737], -3*X[5066]+2*X[63697], -3*X[5891]+X[34782], -X[5895]+5*X[11439], X[5925]+3*X[11455], -3*X[5943]+X[32392], -X[6102]+3*X[23325], -X[6241]+5*X[40686], -X[6759]+3*X[15060], -5*X[8567]+X[12279]

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

Barycentrics    a*(a^3+a^2*(b+c)-(b-c)^2*(b+c)-a*(b+c)^2)*(a^6-a^4*(b-c)^2+2*a^5*(b+c)-4*a^3*(b-c)^2*(b+c)+2*a*(b-c)^4*(b+c)+(b-c)^2*(b+c)^4-a^2*(b-c)^2*(b^2+6*b*c+c^2)) : :

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

Barycentrics    a*(a^3+a^2*(b+c)-(b-c)^2*(b+c)-a*(b+c)^2)*(a^11*(b+c)-a^9*(b+c)^3+3*a^10*(b^2+c^2)+(b-c)^4*(b+c)^6*(b^2+c^2)+a*(b-c)^4*(b+c)^5*(3*b^2-2*b*c+3*c^2)-2*a^7*(b-c)^2*(3*b^3+5*b^2*c+5*b*c^2+3*c^3)+a^8*(-11*b^4+2*b^3*c+2*b^2*c^2+2*b*c^3-11*c^4)-a^2*(b-c)^4*(b+c)^2*(b^4+10*b^3*c+10*b^2*c^2+10*b*c^3+c^4)-2*a^4*(b^2-c^2)^2*(3*b^4-6*b^3*c+10*b^2*c^2-6*b*c^3+3*c^4)+2*a^6*(b-c)^2*(7*b^4+10*b^3*c+10*b^2*c^2+10*b*c^3+7*c^4)-a^3*(b-c)^2*(b+c)^3*(11*b^4-8*b^3*c+10*b^2*c^2-8*b*c^3+11*c^4)+2*a^5*(b-c)^2*(7*b^5+15*b^4*c+18*b^3*c^2+18*b^2*c^3+15*b*c^4+7*c^5)) : :

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

Barycentrics    a*(a+b)*(a+c)*(a^4-2*a^3*(b+c)+2*a*(b+c)^3+3*(b^2-c^2)^2-4*a^2*(b^2+b*c+c^2)) : :

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

Barycentrics    a*(a^6-8*a^3*b*c*(b+c)+8*a*b*(b-c)^2*c*(b+c)-(b-c)^4*(b+c)^2+a^4*(-3*b^2+2*b*c-3*c^2)+a^2*(3*b^4-4*b^3*c+18*b^2*c^2-4*b*c^3+3*c^4)) : :
X(68032) = -3*X[1699]+2*X[37822], -4*X[3452]+5*X[8227], -8*X[3820]+9*X[54447], -4*X[6684]+5*X[62773], -8*X[6692]+7*X[31423], -4*X[6911]+3*X[46917], -3*X[11038]+X[54206], -3*X[26446]+4*X[61535], -2*X[31142]+3*X[38021], -3*X[38036]+2*X[52457], -3*X[38053]+2*X[54205]

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

Barycentrics    a*(a^3*(b^2+3*b*c+c^2)+a^2*(b^3+c^3)-(b-c)^2*(b^3+4*b^2*c+4*b*c^2+c^3)-a*(b^4+b^3*c+b*c^3+c^4)) : :
X(68033) = -2*X[10]+3*X[67853], -3*X[1699]+X[49474], -3*X[3545]+2*X[50096], -3*X[3576]+4*X[15569], X[3644]+4*X[68035], -2*X[3696]+3*X[5587]

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

Barycentrics    a^4-2*a^3*(b+c)+2*a*(b-c)^2*(b+c)+3*(b^2-c^2)^2-4*a^2*(b^2-b*c+c^2) : :
X(68034) = 2*X[1]+5*X[3091], -9*X[2]+2*X[40], 3*X[4]+4*X[1385], -8*X[5]+X[8], -4*X[10]+11*X[5056], -X[20]+8*X[1125], X[69]+6*X[38035], -2*X[104]+9*X[32558], -8*X[140]+X[6361], -8*X[142]+X[64696], X[144]+6*X[38036], X[145]+13*X[5068], X[147]+6*X[38220], -X[149]+8*X[16174], X[153]+6*X[16173], -6*X[165]+13*X[10303], 6*X[354]+X[12528], 4*X[355]+3*X[3241], 3*X[376]+4*X[22793], 6*X[381]+X[944]

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

Barycentrics    2*a^4+5*a^3*(b+c)-5*a*(b-c)^2*(b+c)-3*(b^2-c^2)^2+a^2*(b^2-10*b*c+c^2) : :
X(68035) = -5*X[1]+X[3529], -9*X[2]+5*X[40], -5*X[3]+7*X[15808], -5*X[4]+X[3632], -5*X[10]+7*X[3851], -3*X[20]+7*X[64952], -15*X[165]+19*X[61814], -5*X[355]+9*X[14269], -5*X[381]+3*X[38098], -5*X[551]+3*X[15688], X[944]+3*X[50865], -5*X[1125]+4*X[3530], -5*X[1483]+6*X[51095], -5*X[1537]+X[6154], -15*X[1699]+11*X[3855], -7*X[3090]+3*X[63468], -5*X[3091]+3*X[38127], X[3146]+3*X[16200]

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

Barycentrics    a*(a^6-4*a^3*b*c*(b+c)+4*a*b*(b-c)^2*c*(b+c)-(b-c)^4*(b+c)^2-a^4*(3*b^2+2*b*c+3*c^2)+a^2*(3*b^4+10*b^2*c^2+3*c^4)) : :
X(68036) = -2*X[1158]+3*X[3928], -3*X[1699]+2*X[5812], -3*X[3158]+4*X[6796], -2*X[3811]+3*X[52026], -2*X[6245]+3*X[24477], -4*X[10526]+5*X[18492], -4*X[12608]+3*X[28609], -4*X[21077]+5*X[63966], -3*X[24392]+2*X[48482], -3*X[52027]+2*X[64074], -4*X[64116]+3*X[66469]

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

Barycentrics    6*a^4+7*a^3*(b+c)-7*a*(b-c)^2*(b+c)-(b^2-c^2)^2-a^2*(5*b^2+14*b*c+5*c^2) : :
X(68037) = -7*X[1]+11*X[21735], -3*X[2]+7*X[40], -7*X[10]+5*X[3843], 7*X[20]+X[20053], -9*X[165]+5*X[10595], -7*X[355]+3*X[15684], -7*X[551]+9*X[15706], -7*X[944]+3*X[3633], -7*X[1125]+8*X[12108], -7*X[1385]+9*X[45759], -7*X[1482]+15*X[14093], -7*X[1483]+15*X[62108], -7*X[3528]+3*X[11224], -14*X[3634]+13*X[61907], -7*X[3654]+3*X[38335], -7*X[3656]+11*X[15718], -7*X[3679]+3*X[62029], -21*X[3817]+23*X[61911], -7*X[3828]+6*X[14892]

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

Barycentrics    (3*a^4-(b^2-c^2)^2-2*a^2*(b^2+c^2))*(15*a^12-77*a^8*(b^2-c^2)^2-12*a^10*(b^2+c^2)+128*a^6*(b^2-c^2)^2*(b^2+c^2)+(b^2-c^2)^4*(25*b^4+62*b^2*c^2+25*c^4)-a^4*(b^2-c^2)^2*(27*b^4+250*b^2*c^2+27*c^4)-4*a^2*(b^2-c^2)^2*(13*b^6-29*b^4*c^2-29*b^2*c^4+13*c^6)) : :
X(68038) = -5*X[631]+4*X[16253]

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

Barycentrics    -2*a^7-8*a^3*b^2*c^2+2*a^6*(b+c)-3*a^4*(b-c)^2*(b+c)-4*a^2*b*(b-c)^2*c*(b+c)+(b-c)^4*(b+c)^3+3*a^5*(b^2+c^2)-a*(b^2-c^2)^2*(b^2+c^2)+4*a*(a-b-c)*(a+b-c)*(a-b+c)*sqrt(R*(r+4*R))*S : :

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

Barycentrics    2*a^7+8*a^3*b^2*c^2-2*a^6*(b+c)+3*a^4*(b-c)^2*(b+c)+4*a^2*b*(b-c)^2*c*(b+c)-(b-c)^4*(b+c)^3-3*a^5*(b^2+c^2)+a*(b^2-c^2)^2*(b^2+c^2)+4*a*(a-b-c)*(a+b-c)*(a-b+c)*sqrt(R*(r+4*R))*S : :

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

Barycentrics    2*a^6+3*a^4*(b^2+c^2)-(b^2-c^2)^2*(b^2+c^2)-4*a^2*(b^4+c^4)+2*(3*a^4-(b^2-c^2)^2-2*a^2*(b^2+c^2))*S : :

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

Barycentrics    2*a^6+3*a^4*(b^2+c^2)-(b^2-c^2)^2*(b^2+c^2)-4*a^2*(b^4+c^4)-2*(3*a^4-(b^2-c^2)^2-2*a^2*(b^2+c^2))*S : :

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

Barycentrics    2*a^6+3*a^4*(b^2+c^2)-(b^2-c^2)^2*(b^2+c^2)-4*a^2*(b^4+c^4)+2*(4*a^4-(b^2-c^2)^2-3*a^2*(b^2+c^2))*S : :

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

Barycentrics    2*a^6+3*a^4*(b^2+c^2)-(b^2-c^2)^2*(b^2+c^2)-4*a^2*(b^4+c^4)-2*(4*a^4-(b^2-c^2)^2-3*a^2*(b^2+c^2))*S : :

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

Barycentrics    2*a^6+3*a^4*(b^2+c^2)-(b^2-c^2)^2*(b^2+c^2)-4*a^2*(b^4+c^4)+(11*a^4-5*(b^2-c^2)^2-6*a^2*(b^2+c^2))*S : :

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

Barycentrics    2*a^6+3*a^4*(b^2+c^2)-(b^2-c^2)^2*(b^2+c^2)-4*a^2*(b^4+c^4)+(-11*a^4+5*(b^2-c^2)^2+6*a^2*(b^2+c^2))*S : :

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

Barycentrics    -(a^2*(a-b-c)^2*(a+b-c)*(a-b+c))+4*(2*a^3-a^2*(b+c)-(b-c)^2*(b+c))*sqrt(R*(r+4*R))*S : :

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

Barycentrics    a^2*(a-b-c)^2*(a+b-c)*(a-b+c)+4*(2*a^3-a^2*(b+c)-(b-c)^2*(b+c))*sqrt(R*(r+4*R))*S : :

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

Barycentrics    2*a^8-b^8+b^6*c^2+b^2*c^6-c^8+3*a^6*(b^2+c^2)-a^4*(b^4-3*b^2*c^2+c^4)-3*a^2*(b^6+b^4*c^2+b^2*c^4+c^6) : :
X(68049) = -2*X[39]+3*X[60651], -2*X[194]+3*X[34624], -6*X[5188]+5*X[7904], -8*X[6683]+9*X[60654]

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

Barycentrics    (2*a^4-(b^2-c^2)^2-a^2*(b^2+c^2))*(2*a^12-2*a^10*(b^2+c^2)+16*a^6*(b^2-c^2)^2*(b^2+c^2)+a^8*(-9*b^4+20*b^2*c^2-9*c^4)-4*a^4*(b^2-c^2)^2*(b^4+8*b^2*c^2+c^4)+(b^2-c^2)^4*(3*b^4+8*b^2*c^2+3*c^4)-2*a^2*(b^2-c^2)^2*(3*b^6-7*b^4*c^2-7*b^2*c^4+3*c^6)) : :
X(68050) = -2*X[40]+3*X[16210], -2*X[944]+3*X[16211], -2*X[4297]+3*X[11831], -3*X[5731]+4*X[51712], -3*X[11852]+X[64005], -3*X[25406]+4*X[51741], X[34601]+2*X[44985]

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

Barycentrics    2*a^6+3*a^4*(b^2+c^2)-(b^2-c^2)^2*(b^2+c^2)-4*a^2*(b^4+c^4)+(-3*a^4+(b^2-c^2)^2+2*a^2*(b^2+c^2))*S : :

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

Barycentrics    2*a^6+3*a^4*(b^2+c^2)-(b^2-c^2)^2*(b^2+c^2)-4*a^2*(b^4+c^4)+(3*a^4-(b^2-c^2)^2-2*a^2*(b^2+c^2))*S : :

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

Barycentrics    3*a^7-3*a^6*(b+c)+4*a^4*(b-c)^2*(b+c)-2*(b-c)^4*(b+c)^3+a^5*(-4*b^2+2*b*c-4*c^2)+2*a*(b^2-c^2)^2*(b^2-b*c+c^2)+a^2*(b-c)^2*(b^3+7*b^2*c+7*b*c^2+c^3)-a^3*(b^4-10*b^2*c^2+c^4) : :
X(68053) = -4*X[5450]+3*X[34620], -3*X[11194]+4*X[63980], -3*X[11235]+2*X[22770]

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

Barycentrics    (a+b)*(a+c)*(a^5-6*a^4*(b+c)+2*(b-c)^2*(b+c)^3+a*(b^2-c^2)^2-2*a^3*(b^2+c^2)+4*a^2*(b^3+b^2*c+b*c^2+c^3)) : :

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

Barycentrics    (a-b)*(a+b)*(a-c)*(a+c)*(a^4-3*(b^2-c^2)^2-2*a^2*(b^2+c^2)) : :
X(68055) = -2*X[18860]+3*X[40888], -3*X[34473]+4*X[40879]

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

Barycentrics    a^8+3*b^2*c^2*(b^2-c^2)^2-a^6*(b^2+c^2)-5*a^2*(b^2-c^2)^2*(b^2+c^2)+a^4*(5*b^4-9*b^2*c^2+5*c^4) : :
X(68056) = -3*X[34473]+2*X[36207], -4*X[43291]+3*X[62237]

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

Barycentrics    a*(a^6-(b^2-c^2)^2*(b^2-6*b*c+c^2)+a^2*(b+c)^2*(3*b^2-2*b*c+3*c^2)-a^4*(3*b^2+10*b*c+3*c^2)) : :
X(68057) = -3*X[3158]+2*X[6769], -2*X[5763]+3*X[67889], -2*X[12437]+3*X[54051], -3*X[28610]+X[67994], -4*X[37623]+3*X[52027], -6*X[66465]+7*X[68002]

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

Barycentrics    8*a^10-24*a^2*b^2*c^2*(b^2-c^2)^2-9*a^8*(b^2+c^2)+22*a^4*(b^2-c^2)^2*(b^2+c^2)-5*(b^2-c^2)^4*(b^2+c^2)-8*a^6*(2*b^4-5*b^2*c^2+2*c^4) : :
X(68058) = -7*X[5]+6*X[10193], -X[64]+3*X[3543], -3*X[154]+X[5059], -5*X[382]+X[13093], -4*X[389]+3*X[40928], -4*X[546]+3*X[23328], -3*X[1853]+5*X[17578], -5*X[3522]+6*X[58434], -3*X[3830]+X[20427], -7*X[3832]+5*X[8567], -3*X[3845]+2*X[64027], -4*X[3850]+3*X[11204], -5*X[3858]+4*X[25563], -4*X[3861]+3*X[23329], -5*X[5076]+3*X[65151]

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

Barycentrics    a*(2*a^7*(b-c)^2+a^8*(b+c)-2*a^6*(b-c)^2*(b+c)-4*a^4*b*(b-c)^2*c*(b+c)+2*a^2*(b-c)^4*(b+c)^3-(b-c)^6*(b+c)^3-2*a*(b-c)^2*(b+c)^4*(b^2+c^2)+2*a^3*(b^2-c^2)^2*(3*b^2+2*b*c+3*c^2)-2*a^5*(b-c)^2*(3*b^2+4*b*c+3*c^2)) : :
X(68059) = -3*X[52026]+4*X[64818]

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

Barycentrics    a*(a^12+2*a^9*b*c*(b+c)-8*a^7*b*(b-c)^2*c*(b+c)+2*a*b*(b-c)^4*c*(b+c)^5+a^10*(-6*b^2+8*b*c-6*c^2)+(b-c)^6*(b+c)^4*(b^2+c^2)+3*a^8*(b-c)^2*(5*b^2+4*b*c+5*c^2)-8*a^3*b*c*(b+c)*(b^3-b^2*c+b*c^2-c^3)^2+4*a^5*b*(b-c)^2*c*(3*b^3+b^2*c+b*c^2+3*c^3)-4*a^6*(b-c)^2*(5*b^4+8*b^3*c+10*b^2*c^2+8*b*c^3+5*c^4)+a^4*(b^2-c^2)^2*(15*b^4+4*b^3*c+26*b^2*c^2+4*b*c^3+15*c^4)-2*a^2*(b^2-c^2)^2*(3*b^6+5*b^4*c^2+5*b^2*c^4+3*c^6)) : :
X(68060) = X[962]+3*X[54054], -X[3182]+3*X[3576]

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

Barycentrics    a*(a-b-c)*(2*a^9*(b-c)^2+a^10*(b+c)-3*a^8*(b-c)^2*(b+c)+(b-c)^8*(b+c)^3+2*a*(b-c)^6*(b+c)^2*(b^2+c^2)-4*a^7*(b-c)^2*(2*b^2+3*b*c+2*c^2)+4*a^5*(b^2-c^2)^2*(3*b^2-b*c+3*c^2)+2*a^6*(b-c)^2*(b^3-11*b^2*c-11*b*c^2+c^3)-4*a^3*(b^2-c^2)^2*(2*b^4-3*b^3*c+6*b^2*c^2-3*b*c^3+2*c^4)+2*a^4*(b-c)^2*(b^5+19*b^4*c+12*b^3*c^2+12*b^2*c^3+19*b*c^4+c^5)-a^2*(b-c)^2*(3*b^7+13*b^6*c+7*b^5*c^2+41*b^4*c^3+41*b^3*c^4+7*b^2*c^5+13*b*c^6+3*c^7)) : :
X(68061) = -3*X[354]+2*X[44696]

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)

Barycentrics    a^2*(a^10 - b^10 + b^6*c^4 + b^4*c^6 - c^10 - a^8*(b^2 + c^2) - a^6*(2*b^4 + b^2*c^2 + 2*c^4) + a^4*(2*b^6 + b^4*c^2 + b^2*c^4 + 2*c^6) + a^2*(b^8 + b^6*c^2 + 6*b^4*c^4 + b^2*c^6 + c^8)) : :

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)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^16 - 6*a^14*b^2 + 16*a^12*b^4 - 26*a^10*b^6 + 30*a^8*b^8 - 26*a^6*b^10 + 16*a^4*b^12 - 6*a^2*b^14 + b^16 - 6*a^14*c^2 + 24*a^12*b^2*c^2 - 38*a^10*b^4*c^2 + 27*a^8*b^6*c^2 + 2*a^6*b^8*c^2 - 22*a^4*b^10*c^2 + 18*a^2*b^12*c^2 - 5*b^14*c^2 + 16*a^12*c^4 - 38*a^10*b^2*c^4 + 35*a^8*b^4*c^4 - 18*a^6*b^6*c^4 + 16*a^4*b^8*c^4 - 24*a^2*b^10*c^4 + 13*b^12*c^4 - 26*a^10*c^6 + 27*a^8*b^2*c^6 - 18*a^6*b^4*c^6 - 2*a^4*b^6*c^6 + 12*a^2*b^8*c^6 - 23*b^10*c^6 + 30*a^8*c^8 + 2*a^6*b^2*c^8 + 16*a^4*b^4*c^8 + 12*a^2*b^6*c^8 + 28*b^8*c^8 - 26*a^6*c^10 - 22*a^4*b^2*c^10 - 24*a^2*b^4*c^10 - 23*b^6*c^10 + 16*a^4*c^12 + 18*a^2*b^2*c^12 + 13*b^4*c^12 - 6*a^2*c^14 - 5*b^2*c^14 + c^16) : :
X(68063) = 3 X[51] - X[128], X[930] - 5 X[3567], X[1141] + 3 X[3060], X[5562] - 3 X[23516], 3 X[5890] + X[44976], 3 X[5943] - 2 X[58429], 3 X[5946] - X[38615], X[6243] + 3 X[57324], X[7731] + 3 X[34308], 3 X[9730] - X[63412], 9 X[11002] - X[67091], 9 X[13321] - X[13512], 7 X[15043] - 3 X[38706], 3 X[38710] + X[64051]

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)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^20 - 6*a^18*b^2 + 15*a^16*b^4 - 20*a^14*b^6 + 16*a^12*b^8 - 12*a^10*b^10 + 16*a^8*b^12 - 20*a^6*b^14 + 15*a^4*b^16 - 6*a^2*b^18 + b^20 - 6*a^18*c^2 + 26*a^16*b^2*c^2 - 44*a^14*b^4*c^2 + 37*a^12*b^6*c^2 - 16*a^10*b^8*c^2 - 5*a^8*b^10*c^2 + 28*a^6*b^12*c^2 - 37*a^4*b^14*c^2 + 22*a^2*b^16*c^2 - 5*b^18*c^2 + 15*a^16*c^4 - 44*a^14*b^2*c^4 + 43*a^12*b^4*c^4 - 14*a^10*b^6*c^4 - a^8*b^8*c^4 - 8*a^6*b^10*c^4 + 29*a^4*b^12*c^4 - 30*a^2*b^14*c^4 + 10*b^16*c^4 - 20*a^14*c^6 + 37*a^12*b^2*c^6 - 14*a^10*b^4*c^6 - 2*a^8*b^6*c^6 - 11*a^4*b^10*c^6 + 18*a^2*b^12*c^6 - 8*b^14*c^6 + 16*a^12*c^8 - 16*a^10*b^2*c^8 - a^8*b^4*c^8 + 8*a^4*b^8*c^8 - 4*a^2*b^10*c^8 - 3*b^12*c^8 - 12*a^10*c^10 - 5*a^8*b^2*c^10 - 8*a^6*b^4*c^10 - 11*a^4*b^6*c^10 - 4*a^2*b^8*c^10 + 10*b^10*c^10 + 16*a^8*c^12 + 28*a^6*b^2*c^12 + 29*a^4*b^4*c^12 + 18*a^2*b^6*c^12 - 3*b^8*c^12 - 20*a^6*c^14 - 37*a^4*b^2*c^14 - 30*a^2*b^4*c^14 - 8*b^6*c^14 + 15*a^4*c^16 + 22*a^2*b^2*c^16 + 10*b^4*c^16 - 6*a^2*c^18 - 5*b^2*c^18 + c^20) : :
X(68064) = 3 X[51] - X[129], X[1298] + 3 X[3060], X[1303] - 5 X[3567], 3 X[5890] + X[44991], X[6243] + 3 X[57333], 9 X[13321] - X[67822]

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)

Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4)*(a^14*b^2 - 4*a^12*b^4 + 6*a^10*b^6 - 5*a^8*b^8 + 5*a^6*b^10 - 6*a^4*b^12 + 4*a^2*b^14 - b^16 + a^14*c^2 - 4*a^12*b^2*c^2 + 7*a^10*b^4*c^2 - 8*a^8*b^6*c^2 + a^6*b^8*c^2 + 14*a^4*b^10*c^2 - 17*a^2*b^12*c^2 + 6*b^14*c^2 - 4*a^12*c^4 + 7*a^10*b^2*c^4 + 2*a^8*b^4*c^4 - 2*a^6*b^6*c^4 - 14*a^4*b^8*c^4 + 27*a^2*b^10*c^4 - 16*b^12*c^4 + 6*a^10*c^6 - 8*a^8*b^2*c^6 - 2*a^6*b^4*c^6 + 12*a^4*b^6*c^6 - 14*a^2*b^8*c^6 + 26*b^10*c^6 - 5*a^8*c^8 + a^6*b^2*c^8 - 14*a^4*b^4*c^8 - 14*a^2*b^6*c^8 - 30*b^8*c^8 + 5*a^6*c^10 + 14*a^4*b^2*c^10 + 27*a^2*b^4*c^10 + 26*b^6*c^10 - 6*a^4*c^12 - 17*a^2*b^2*c^12 - 16*b^4*c^12 + 4*a^2*c^14 + 6*b^2*c^14 - c^16) : :
X(68065) = 3 X[51] - X[131], 3 X[568] + X[13556], X[925] - 5 X[3567], X[1300] + 3 X[3060], 3 X[5890] + X[44974], X[6243] + 3 X[57334], 7 X[15043] - 3 X[67842], 3 X[38718] + X[64051]

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)

Barycentrics    a^2*(a^16*b^2 - 4*a^14*b^4 + 6*a^12*b^6 - 4*a^10*b^8 + 4*a^6*b^12 - 6*a^4*b^14 + 4*a^2*b^16 - b^18 + a^16*c^2 - 4*a^14*b^2*c^2 + 5*a^12*b^4*c^2 - 2*a^10*b^6*c^2 - a^8*b^8*c^2 + 7*a^4*b^12*c^2 - 10*a^2*b^14*c^2 + 4*b^16*c^2 - 4*a^14*c^4 + 5*a^12*b^2*c^4 - 4*a^10*b^4*c^4 + 3*a^8*b^6*c^4 - a^4*b^10*c^4 + 8*a^2*b^12*c^4 - 7*b^14*c^4 + 6*a^12*c^6 - 2*a^10*b^2*c^6 + 3*a^8*b^4*c^6 - 8*a^6*b^6*c^6 - 6*a^2*b^10*c^6 + 7*b^12*c^6 - 4*a^10*c^8 - a^8*b^2*c^8 + 8*a^2*b^8*c^8 - 3*b^10*c^8 - a^4*b^4*c^10 - 6*a^2*b^6*c^10 - 3*b^8*c^10 + 4*a^6*c^12 + 7*a^4*b^2*c^12 + 8*a^2*b^4*c^12 + 7*b^6*c^12 - 6*a^4*c^14 - 10*a^2*b^2*c^14 - 7*b^4*c^14 + 4*a^2*c^16 + 4*b^2*c^16 - c^18) : :
X(68066) = 3 X[51] - X[132], 3 X[51] - 2 X[58529], X[112] - 5 X[3567], X[112] - 3 X[16224], 5 X[3567] - 3 X[16224], 3 X[568] + X[10749], X[1297] + 3 X[3060], 3 X[5890] + X[10735], 3 X[5943] - 2 X[58430], 3 X[5946] - X[38608], X[6243] + 3 X[57332], 3 X[9730] - X[14689], 9 X[11002] - X[12384], X[13310] - 9 X[13321], X[14900] - 3 X[16225], 7 X[15043] - 3 X[38699], 3 X[16222] - X[53760], 3 X[38717] + X[64051], 3 X[46430] - X[53719]

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)

Barycentrics    a^2*(a^18*b^2 - 4*a^16*b^4 + a^14*b^6 + 21*a^12*b^8 - 49*a^10*b^10 + 49*a^8*b^12 - 21*a^6*b^14 - a^4*b^16 + 4*a^2*b^18 - b^20 + a^18*c^2 - 4*a^16*b^2*c^2 + 10*a^14*b^4*c^2 - 26*a^12*b^6*c^2 + 46*a^10*b^8*c^2 - 36*a^8*b^10*c^2 - 6*a^6*b^12*c^2 + 30*a^4*b^14*c^2 - 19*a^2*b^16*c^2 + 4*b^18*c^2 - 4*a^16*c^4 + 10*a^14*b^2*c^4 - 6*a^12*b^4*c^4 + 5*a^10*b^6*c^4 - 39*a^8*b^8*c^4 + 84*a^6*b^10*c^4 - 76*a^4*b^12*c^4 + 29*a^2*b^14*c^4 - 3*b^16*c^4 + a^14*c^6 - 26*a^12*b^2*c^6 + 5*a^10*b^4*c^6 + 52*a^8*b^6*c^6 - 57*a^6*b^8*c^6 + 50*a^4*b^10*c^6 - 13*a^2*b^12*c^6 - 12*b^14*c^6 + 21*a^12*c^8 + 46*a^10*b^2*c^8 - 39*a^8*b^4*c^8 - 57*a^6*b^6*c^8 - 6*a^4*b^8*c^8 - a^2*b^10*c^8 + 36*b^12*c^8 - 49*a^10*c^10 - 36*a^8*b^2*c^10 + 84*a^6*b^4*c^10 + 50*a^4*b^6*c^10 - a^2*b^8*c^10 - 48*b^10*c^10 + 49*a^8*c^12 - 6*a^6*b^2*c^12 - 76*a^4*b^4*c^12 - 13*a^2*b^6*c^12 + 36*b^8*c^12 - 21*a^6*c^14 + 30*a^4*b^2*c^14 + 29*a^2*b^4*c^14 - 12*b^6*c^14 - a^4*c^16 - 19*a^2*b^2*c^16 - 3*b^4*c^16 + 4*a^2*c^18 + 4*b^2*c^18 - c^20) : :
X(68067) = 3 X[51] - X[133], 3 X[51] - 2 X[58530], X[107] - 5 X[3567], 3 X[568] + X[10745], X[1294] + 3 X[3060], X[3184] - 3 X[9730], X[5562] - 3 X[36520], 3 X[5890] + X[10152], 3 X[5943] - 2 X[58431], 3 X[5946] - X[38605], X[6243] + 3 X[57329], 9 X[11002] - X[34549], 9 X[13321] - X[38577], 7 X[15043] - 3 X[23239], 3 X[16222] - X[53757], X[23240] - 5 X[37481], 3 X[38714] + X[64051], 3 X[46430] - X[53716]

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)

Barycentrics    a^2*(a^14*b^2 - 4*a^12*b^4 + 6*a^10*b^6 - 5*a^8*b^8 + 5*a^6*b^10 - 6*a^4*b^12 + 4*a^2*b^14 - b^16 + a^14*c^2 - 4*a^12*b^2*c^2 + 7*a^10*b^4*c^2 - 6*a^8*b^6*c^2 - 3*a^6*b^8*c^2 + 14*a^4*b^10*c^2 - 13*a^2*b^12*c^2 + 4*b^14*c^2 - 4*a^12*c^4 + 7*a^10*b^2*c^4 - 2*a^8*b^4*c^4 + 2*a^6*b^6*c^4 - 12*a^4*b^8*c^4 + 15*a^2*b^10*c^4 - 6*b^12*c^4 + 6*a^10*c^6 - 6*a^8*b^2*c^6 + 2*a^6*b^4*c^6 + 8*a^4*b^6*c^6 - 6*a^2*b^8*c^6 + 4*b^10*c^6 - 5*a^8*c^8 - 3*a^6*b^2*c^8 - 12*a^4*b^4*c^8 - 6*a^2*b^6*c^8 - 2*b^8*c^8 + 5*a^6*c^10 + 14*a^4*b^2*c^10 + 15*a^2*b^4*c^10 + 4*b^6*c^10 - 6*a^4*c^12 - 13*a^2*b^2*c^12 - 6*b^4*c^12 + 4*a^2*c^14 + 4*b^2*c^14 - c^16) : :
X(68068) = 3 X[51] - X[136], X[925] + 3 X[3060], X[1300] - 5 X[3567], 3 X[5890] + X[44990], X[6243] + 3 X[57314], 7 X[15043] - 3 X[38718], X[64051] + 3 X[67842]

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)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^12 - 4*a^10*b^2 + 7*a^8*b^4 - 8*a^6*b^6 + 7*a^4*b^8 - 4*a^2*b^10 + b^12 - 4*a^10*c^2 + 8*a^8*b^2*c^2 - 4*a^6*b^4*c^2 - 3*a^4*b^6*c^2 + 6*a^2*b^8*c^2 - 3*b^10*c^2 + 7*a^8*c^4 - 4*a^6*b^2*c^4 + a^4*b^4*c^4 - 2*a^2*b^6*c^4 + 4*b^8*c^4 - 8*a^6*c^6 - 3*a^4*b^2*c^6 - 2*a^2*b^4*c^6 - 4*b^6*c^6 + 7*a^4*c^8 + 6*a^2*b^2*c^8 + 4*b^4*c^8 - 4*a^2*c^10 - 3*b^2*c^10 + c^12) : :
X(68069) = 3 X[51] - X[137], 3 X[568] + X[31656], X[930] + 3 X[3060], X[1141] - 5 X[3567], 9 X[5640] - X[13504], 3 X[5890] + X[44981], 3 X[5943] - 2 X[58432], 3 X[5946] - X[38618], X[6243] + 3 X[57316], 3 X[9730] - X[63409], 7 X[9781] + X[13505], 9 X[11002] - X[11671], 9 X[13321] - X[38587], 7 X[15043] - 3 X[38710], 3 X[38706] + X[64051]

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)

Barycentrics    a^2*(a^14*b^2 - 3*a^12*b^4 + a^10*b^6 + 5*a^8*b^8 - 5*a^6*b^10 - a^4*b^12 + 3*a^2*b^14 - b^16 + a^14*c^2 - 2*a^12*b^2*c^2 + 5*a^10*b^4*c^2 - 10*a^8*b^6*c^2 + 2*a^6*b^8*c^2 + 13*a^4*b^10*c^2 - 12*a^2*b^12*c^2 + 3*b^14*c^2 - 3*a^12*c^4 + 5*a^10*b^2*c^4 + 2*a^8*b^4*c^4 + 4*a^6*b^6*c^4 - 22*a^4*b^8*c^4 + 15*a^2*b^10*c^4 - b^12*c^4 + a^10*c^6 - 10*a^8*b^2*c^6 + 4*a^6*b^4*c^6 + 20*a^4*b^6*c^6 - 6*a^2*b^8*c^6 - 7*b^10*c^6 + 5*a^8*c^8 + 2*a^6*b^2*c^8 - 22*a^4*b^4*c^8 - 6*a^2*b^6*c^8 + 12*b^8*c^8 - 5*a^6*c^10 + 13*a^4*b^2*c^10 + 15*a^2*b^4*c^10 - 7*b^6*c^10 - a^4*c^12 - 12*a^2*b^2*c^12 - b^4*c^12 + 3*a^2*c^14 + 3*b^2*c^14 - c^16) : :
X(68070) = 3 X[51] - X[3258], 3 X[51] - 2 X[12052], X[476] + 3 X[3060], 3 X[3060] - X[16978], X[477] - 5 X[3567], 3 X[568] + X[66781], 3 X[5627] + X[7731], 9 X[5640] - 5 X[66801], 3 X[5890] + X[14989], 3 X[5946] - X[38610], X[6243] + 3 X[57305], 3 X[9971] + X[66813], 9 X[11002] - X[14731], X[11412] - 5 X[66787], 3 X[12824] - X[14611], 9 X[13321] - X[38581], X[14934] - 3 X[16222], 7 X[15043] - 3 X[38701], 2 X[31945] - 3 X[41670], X[36164] - 3 X[46430], 3 X[38700] + X[64051]

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)

Barycentrics    a^2*(a^14*b^2 - 2*a^12*b^4 - 4*a^10*b^6 + 15*a^8*b^8 - 15*a^6*b^10 + 4*a^4*b^12 + 2*a^2*b^14 - b^16 + a^14*c^2 + 5*a^10*b^4*c^2 - 18*a^8*b^6*c^2 + 7*a^6*b^8*c^2 + 16*a^4*b^10*c^2 - 13*a^2*b^12*c^2 + 2*b^14*c^2 - 2*a^12*c^4 + 5*a^10*b^2*c^4 + 6*a^8*b^4*c^4 + 8*a^6*b^6*c^4 - 40*a^4*b^8*c^4 + 19*a^2*b^10*c^4 + 4*b^12*c^4 - 4*a^10*c^6 - 18*a^8*b^2*c^6 + 8*a^6*b^4*c^6 + 40*a^4*b^6*c^6 - 8*a^2*b^8*c^6 - 18*b^10*c^6 + 15*a^8*c^8 + 7*a^6*b^2*c^8 - 40*a^4*b^4*c^8 - 8*a^2*b^6*c^8 + 26*b^8*c^8 - 15*a^6*c^10 + 16*a^4*b^2*c^10 + 19*a^2*b^4*c^10 - 18*b^6*c^10 + 4*a^4*c^12 - 13*a^2*b^2*c^12 + 4*b^4*c^12 + 2*a^2*c^14 + 2*b^2*c^14 - c^16) : :
X(68071) = 3 X[51] - X[122], 3 X[51] - 2 X[58524], X[107] + 3 X[3060], 3 X[568] + X[22337], X[1294] - 5 X[3567], 3 X[5890] + X[44985], 3 X[5943] - 2 X[58424], 3 X[5946] - X[38621], X[6243] + 3 X[57301], 3 X[9730] - X[63411], 9 X[11002] - X[34186], 9 X[13321] - X[38591], 7 X[15043] - 3 X[38714], 3 X[23239] + X[64051]

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)

Barycentrics    a^2*(a^12*b^2 - 2*a^10*b^4 + a^8*b^6 - a^4*b^10 + 2*a^2*b^12 - b^14 + a^12*c^2 - 2*a^6*b^6*c^2 + a^4*b^8*c^2 - 2*a^2*b^10*c^2 + 2*b^12*c^2 - 2*a^10*c^4 + 4*a^6*b^4*c^4 - 2*b^10*c^4 + a^8*c^6 - 2*a^6*b^2*c^6 + b^8*c^6 + a^4*b^2*c^8 + b^6*c^8 - a^4*c^10 - 2*a^2*b^2*c^10 - 2*b^4*c^10 + 2*a^2*c^12 + 2*b^2*c^12 - c^14) : :
X(68072) = X[3] - 3 X[16224], 3 X[51] - X[127], 3 X[51] - 2 X[58528], X[112] + 3 X[3060], 3 X[568] + X[12918], X[1297] - 5 X[3567], 3 X[5890] + X[44988], 3 X[5943] - 2 X[58428], 3 X[5946] - X[38624], X[6243] + 3 X[57304], 3 X[9730] - X[63410], 9 X[11002] - X[13219], X[13115] - 9 X[13321], X[14689] - 3 X[16225], 3 X[16225] + X[45186], 7 X[15043] - 3 X[38717], 3 X[38699] + X[64051]

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)

Barycentrics    a^2*(a^20*b^2 - 6*a^18*b^4 + 15*a^16*b^6 - 20*a^14*b^8 + 14*a^12*b^10 - 14*a^8*b^14 + 20*a^6*b^16 - 15*a^4*b^18 + 6*a^2*b^20 - b^22 + a^20*c^2 - 8*a^18*b^2*c^2 + 24*a^16*b^4*c^2 - 38*a^14*b^6*c^2 + 38*a^12*b^8*c^2 - 32*a^10*b^10*c^2 + 38*a^8*b^12*c^2 - 54*a^6*b^14*c^2 + 53*a^4*b^16*c^2 - 28*a^2*b^18*c^2 + 6*b^20*c^2 - 6*a^18*c^4 + 24*a^16*b^2*c^4 - 36*a^14*b^4*c^4 + 27*a^12*b^6*c^4 - 10*a^10*b^8*c^4 - 14*a^8*b^10*c^4 + 52*a^6*b^12*c^4 - 77*a^4*b^14*c^4 + 56*a^2*b^16*c^4 - 16*b^18*c^4 + 15*a^16*c^6 - 38*a^14*b^2*c^6 + 27*a^12*b^4*c^6 - a^8*b^8*c^6 - 24*a^6*b^10*c^6 + 58*a^4*b^12*c^6 - 62*a^2*b^14*c^6 + 25*b^16*c^6 - 20*a^14*c^8 + 38*a^12*b^2*c^8 - 10*a^10*b^4*c^8 - a^8*b^6*c^8 + 12*a^6*b^8*c^8 - 19*a^4*b^10*c^8 + 42*a^2*b^12*c^8 - 24*b^14*c^8 + 14*a^12*c^10 - 32*a^10*b^2*c^10 - 14*a^8*b^4*c^10 - 24*a^6*b^6*c^10 - 19*a^4*b^8*c^10 - 28*a^2*b^10*c^10 + 10*b^12*c^10 + 38*a^8*b^2*c^12 + 52*a^6*b^4*c^12 + 58*a^4*b^6*c^12 + 42*a^2*b^8*c^12 + 10*b^10*c^12 - 14*a^8*c^14 - 54*a^6*b^2*c^14 - 77*a^4*b^4*c^14 - 62*a^2*b^6*c^14 - 24*b^8*c^14 + 20*a^6*c^16 + 53*a^4*b^2*c^16 + 56*a^2*b^4*c^16 + 25*b^6*c^16 - 15*a^4*c^18 - 28*a^2*b^2*c^18 - 16*b^4*c^18 + 6*a^2*c^20 + 6*b^2*c^20 - c^22) : :
X(68073) = 3 X[51] - X[46439], 3 X[568] + X[14980], X[1291] + 3 X[3060], 5 X[3567] - X[14979], X[6243] + 3 X[57326]

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)

Barycentrics    a^2*(a^18*b^2 - 5*a^16*b^4 + 8*a^14*b^6 - 14*a^10*b^10 + 14*a^8*b^12 - 8*a^4*b^16 + 5*a^2*b^18 - b^20 + a^18*c^2 - 6*a^16*b^2*c^2 + 16*a^14*b^4*c^2 - 28*a^12*b^6*c^2 + 35*a^10*b^8*c^2 - 21*a^8*b^10*c^2 - 14*a^6*b^12*c^2 + 34*a^4*b^14*c^2 - 22*a^2*b^16*c^2 + 5*b^18*c^2 - 5*a^16*c^4 + 16*a^14*b^2*c^4 - 16*a^12*b^4*c^4 + 8*a^10*b^6*c^4 - 19*a^8*b^8*c^4 + 44*a^6*b^10*c^4 - 54*a^4*b^12*c^4 + 36*a^2*b^14*c^4 - 10*b^16*c^4 + 8*a^14*c^6 - 28*a^12*b^2*c^6 + 8*a^10*b^4*c^6 + 28*a^8*b^6*c^6 - 28*a^6*b^8*c^6 + 28*a^4*b^10*c^6 - 26*a^2*b^12*c^6 + 10*b^14*c^6 + 35*a^10*b^2*c^8 - 19*a^8*b^4*c^8 - 28*a^6*b^6*c^8 + 7*a^2*b^10*c^8 - 5*b^12*c^8 - 14*a^10*c^10 - 21*a^8*b^2*c^10 + 44*a^6*b^4*c^10 + 28*a^4*b^6*c^10 + 7*a^2*b^8*c^10 + 2*b^10*c^10 + 14*a^8*c^12 - 14*a^6*b^2*c^12 - 54*a^4*b^4*c^12 - 26*a^2*b^6*c^12 - 5*b^8*c^12 + 34*a^4*b^2*c^14 + 36*a^2*b^4*c^14 + 10*b^6*c^14 - 8*a^4*c^16 - 22*a^2*b^2*c^16 - 10*b^4*c^16 + 5*a^2*c^18 + 5*b^2*c^18 - c^20) : :
X(68074) = 3 X[51] - X[25641], X[476] - 5 X[3567], X[477] + 3 X[3060], 3 X[568] + X[20957], 9 X[5640] - 5 X[66787], 3 X[5890] + X[44967], 3 X[5946] - X[38609], X[6243] + 3 X[57306], X[7471] - 3 X[16222], 3 X[9971] + X[66810], 9 X[11002] - X[34193], X[11412] - 5 X[66801], 9 X[13321] - X[38580], 7 X[15043] - 3 X[38700], 3 X[38701] + X[64051], 3 X[46430] - X[46632]

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)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^22 - 6*a^20*b^2 + 14*a^18*b^4 - 15*a^16*b^6 + 6*a^14*b^8 + 6*a^8*b^14 - 15*a^6*b^16 + 14*a^4*b^18 - 6*a^2*b^20 + b^22 - 6*a^20*c^2 + 28*a^18*b^2*c^2 - 49*a^16*b^4*c^2 + 35*a^14*b^6*c^2 + 2*a^12*b^8*c^2 - 19*a^10*b^10*c^2 + 2*a^8*b^12*c^2 + 29*a^6*b^14*c^2 - 40*a^4*b^16*c^2 + 23*a^2*b^18*c^2 - 5*b^20*c^2 + 14*a^18*c^4 - 49*a^16*b^2*c^4 + 67*a^14*b^4*c^4 - 44*a^12*b^6*c^4 + 6*a^10*b^8*c^4 + 17*a^8*b^10*c^4 - 25*a^6*b^12*c^4 + 34*a^4*b^14*c^4 - 30*a^2*b^16*c^4 + 10*b^18*c^4 - 15*a^16*c^6 + 35*a^14*b^2*c^6 - 44*a^12*b^4*c^6 + 44*a^10*b^6*c^6 - 25*a^8*b^8*c^6 + 3*a^6*b^10*c^6 - 2*a^4*b^12*c^6 + 14*a^2*b^14*c^6 - 10*b^16*c^6 + 6*a^14*c^8 + 2*a^12*b^2*c^8 + 6*a^10*b^4*c^8 - 25*a^8*b^6*c^8 + 16*a^6*b^8*c^8 - 6*a^4*b^10*c^8 - 4*a^2*b^12*c^8 + 5*b^14*c^8 - 19*a^10*b^2*c^10 + 17*a^8*b^4*c^10 + 3*a^6*b^6*c^10 - 6*a^4*b^8*c^10 + 6*a^2*b^10*c^10 - b^12*c^10 + 2*a^8*b^2*c^12 - 25*a^6*b^4*c^12 - 2*a^4*b^6*c^12 - 4*a^2*b^8*c^12 - b^10*c^12 + 6*a^8*c^14 + 29*a^6*b^2*c^14 + 34*a^4*b^4*c^14 + 14*a^2*b^6*c^14 + 5*b^8*c^14 - 15*a^6*c^16 - 40*a^4*b^2*c^16 - 30*a^2*b^4*c^16 - 10*b^6*c^16 + 14*a^4*c^18 + 23*a^2*b^2*c^18 + 10*b^4*c^18 - 6*a^2*c^20 - 5*b^2*c^20 + c^22) : :
X(68075) = 3 X[51] - X[18402], X[933] - 5 X[3567], 3 X[3060] + X[18401], 3 X[5890] + X[44977], 3 X[5946] - X[38616], X[6243] + 3 X[57369], 9 X[13321] - X[38585]

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)

Barycentrics    a^2*(a^14*b^2 - 5*a^12*b^4 + 13*a^10*b^6 - 21*a^8*b^8 + 21*a^6*b^10 - 13*a^4*b^12 + 5*a^2*b^14 - b^16 + a^14*c^2 - 6*a^12*b^2*c^2 + 11*a^10*b^4*c^2 - 8*a^8*b^6*c^2 - 8*a^6*b^8*c^2 + 19*a^4*b^10*c^2 - 14*a^2*b^12*c^2 + 5*b^14*c^2 - 5*a^12*c^4 + 11*a^10*b^2*c^4 - 14*a^8*b^4*c^4 + 16*a^6*b^6*c^4 - 24*a^4*b^8*c^4 + 19*a^2*b^10*c^4 - 13*b^12*c^4 + 13*a^10*c^6 - 8*a^8*b^2*c^6 + 16*a^6*b^4*c^6 + 12*a^4*b^6*c^6 - 8*a^2*b^8*c^6 + 21*b^10*c^6 - 21*a^8*c^8 - 8*a^6*b^2*c^8 - 24*a^4*b^4*c^8 - 8*a^2*b^6*c^8 - 24*b^8*c^8 + 21*a^6*c^10 + 19*a^4*b^2*c^10 + 19*a^2*b^4*c^10 + 21*b^6*c^10 - 13*a^4*c^12 - 14*a^2*b^2*c^12 - 13*b^4*c^12 + 5*a^2*c^14 + 5*b^2*c^14 - c^16) : :
X(68076) = 3 X[51] - X[33330], 3 X[568] + X[66837], X[805] - 5 X[3567], X[2698] + 3 X[3060], 3 X[5890] + X[44971], 3 X[5946] - X[67833], X[6243] + 3 X[57347], 9 X[11002] - X[66822], 9 X[13321] - X[66840], 7 X[15043] - 3 X[38703], X[64051] + 3 X[67840]

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)

Barycentrics    a*(-3*b^2*(b-c)^2*c^2*(b+c)+14*a^3*b*c*(b+c)^2+3*a^4*(b+c)^3-2*a*b*c*(3*b^4+b^3*c-12*b^2*c^2+b*c^3+3*c^4)-a^2*(3*b^5+5*b^4*c-28*b^3*c^2-28*b^2*c^3+5*b*c^4+3*c^5)) : :

See Ivan Pavlov, euclid 8194.

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

X(68078) = (name pending)

Barycentrics    (a^2 - b^2 - c^2)*(2*a^10 - 4*a^6*(b^2 - c^2)^2 - 2*a^8*(b^2 + c^2) - 2*(b^2 - c^2)^4*(b^2 + c^2) + a^2*(b^2 - c^2)^2*(2*b^4 - 3*b^2*c^2 + 2*c^4) + a^4*(4*b^6 - 3*b^4*c^2 - 3*b^2*c^4 + 4*c^6)) : :

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)

Barycentrics    -((a^2 - b^2 - c^2)*(4*a^8 - (b^2 - c^2)^4 - 5*a^6*(b^2 + c^2) + 5*a^2*(b^2 - c^2)^2*(b^2 + c^2) - a^4*(3*b^4 + 2*b^2*c^2 + 3*c^4))) : :

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)

Barycentrics    -a^12 + a^10*(b^2 + c^2) + 2*a^8*(b^4 - 6*b^2*c^2 + c^4) + a^2*(b^2 - c^2)^2*(b^6 - 9*b^4*c^2 - 9*b^2*c^4 + c^6) - 2*a^6*(b^6 - 5*b^4*c^2 - 5*b^2*c^4 + c^6) - a^4*(b^8 - 12*b^6*c^2 + 6*b^4*c^4 - 12*b^2*c^6 + c^8) : :

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)

Barycentrics    9*a^12 - 10*a^10*(b^2 + c^2) + (b^2 - c^2)^4*(b^2 + c^2)^2 + a^8*(-17*b^4 + 6*b^2*c^2 - 17*c^4) + 4*a^6*(5*b^6 - b^4*c^2 - b^2*c^4 + 5*c^6) - 2*a^2*(b^2 - c^2)^2*(5*b^6 + 3*b^4*c^2 + 3*b^2*c^4 + 5*c^6) + a^4*(7*b^8 - 4*b^6*c^2 + 26*b^4*c^4 - 4*b^2*c^6 + 7*c^8) : :

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)

Barycentrics    (a^2 - b^2 - c^2)*(7*a^8 - (b^2 - c^2)^4 - 8*a^6*(b^2 + c^2) + 8*a^2*(b^2 - c^2)^2*(b^2 + c^2) - 2*a^4*(3*b^4 + 2*b^2*c^2 + 3*c^4)) : :

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)

Barycentrics    10*a^12 - 11*a^10*(b^2 + c^2) + (b^2 - c^2)^4*(b^2 + c^2)^2 + a^8*(-19*b^4 + 18*b^2*c^2 - 19*c^4) + 2*a^6*(11*b^6 - 7*b^4*c^2 - 7*b^2*c^4 + 11*c^6) - a^2*(b^2 - c^2)^2*(11*b^6 - 3*b^4*c^2 - 3*b^2*c^4 + 11*c^6) + 8*a^4*(b^8 - 2*b^6*c^2 + 4*b^4*c^4 - 2*b^2*c^6 + c^8) : :

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)

Barycentrics    a^2*(a^6*(b^2+c^2)-3*a^4*(b^2+c^2)^2-(b^2-c^2)^2*(b^4-3*b^2*c^2+c^4)+3*a^2*(b^6+c^6)) : :
X(68084) = -3*X[2]+4*X[18874], -5*X[3]+9*X[5640], -5*X[5]+3*X[3917], -X[20]+3*X[5946], -3*X[51]+X[550], -5*X[140]+6*X[6688], -9*X[373]+7*X[14869], -3*X[381]+X[6101], X[382]+3*X[3060], -3*X[547]+2*X[5447], -3*X[549]+4*X[32205], 3*X[568]+X[3146], -5*X[632]+4*X[11592], -25*X[1656]+21*X[44299], -X[1657]+5*X[3567]

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)

Barycentrics    2*a^6-a^4*(b^2+c^2)+(b^2-c^2)^2*(b^2+c^2)-2*a^2*(b^4-b^2*c^2+c^4) : :

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}





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

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

underbar



X(68086) = X(110)X(677)∩X(112)X(6078)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a^2 + b^2 - c^2)*(a*b - b^2 + a*c - c^2)^2*(a^2 - b^2 + c^2) : :

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)

Barycentrics    (a^2-b^2)*(a^2-c^2)*(2*a^2-b^2-c^2)^2*(a^2+b^2-c^2)*(a^2-b^2+c^2) : :
X(68087) = 3 X[110] + X[17708], 3 X[5642] - X[62594]

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)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^2 - b^2 - c^2)*(2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)^2 : :

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)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-(a^2*b^2) + b^4 - a^2*c^2 + c^4)^2 : :
X(68089) = X[43673] + 3 X[65714]

X(68089) lies on the Moses HK-parabola and these lines: {2, 2501}, {4, 525}, {112, 65648}, {114, 132}, {264, 43665}, {297, 34765}, {324, 850}, {338, 60500}, {458, 1640}, {523, 9756}, {524, 53156}, {648, 14223}, {877, 2421}, {1235, 14618}, {1990, 18311}, {1993, 32320}, {4235, 5664}, {6248, 16229}, {8743, 50437}, {9979, 14401}, {14977, 52710}, {18121, 65610}, {24978, 57203}, {35088, 65974}, {36471, 38970}, {37174, 65710}, {38652, 44817}, {39931, 44427}, {42441, 63829}, {45327, 52288}, {46052, 62555}, {46942, 62307}, {48466, 54029}, {48467, 54028}, {53374, 62950}

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)

Barycentrics    a*(16*c*b*(3*a^3-5*(b+c)*a^2+a*(b-c)^2+(b^2-c^2)*(b-c))*S+a^7-3*(b+c)*a^6+(b^2+18*c*b+c^2)*a^5+(b+c)*(5*b^2+14*c*b+5*c^2)*a^4-(5*b^4+5*c^4+2*c*b*(34*b^2+7*c*b+34*c^2))*a^3-(b^2-c^2)*(b-c)*(b^2-18*c*b+c^2)*a^2+3*(b^2-c^2)^2*(b^2+6*c*b+c^2)*a-(b^2-c^2)*(b-c)^3*(b^2+6*c*b+c^2)) : :

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)

Barycentrics    a*(-16*c*b*(3*a^3-5*(b+c)*a^2+a*(b-c)^2+(b^2-c^2)*(b-c))*S+a^7-3*(b+c)*a^6+(b^2+18*c*b+c^2)*a^5+(b+c)*(5*b^2+14*c*b+5*c^2)*a^4-(5*b^4+5*c^4+2*c*b*(34*b^2+7*c*b+34*c^2))*a^3-(b^2-c^2)*(b-c)*(b^2-18*c*b+c^2)*a^2+3*(b^2-c^2)^2*(b^2+6*c*b+c^2)*a-(b^2-c^2)*(b-c)^3*(b^2+6*c*b+c^2)) : :

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)

Barycentrics    a*(-8*S^3+a^6-(3*b^2-2*c*b+3*c^2)*a^4+4*(b+c)*b*c*a^3+3*(b^2-c^2)^2*a^2-4*(b^2-c^2)*(b-c)*b*c*a-(b^2-c^2)^2*(b+c)^2) : :

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)

Barycentrics    20*a^18 - 45*a^16*(b^2 + c^2) + 5*(b^2 - c^2)^6*(b^2 + c^2)^3 + a^14*(-30*b^4 + 26*b^2*c^2 - 30*c^4) - 30*a^2*(b^2 - c^2)^4*(b^2 + c^2)^2*(b^4 + b^2*c^2 + c^4) + 2*a^12*(65*b^6 + 47*b^4*c^2 + 47*b^2*c^4 + 65*c^6) - 2*a^10*(15*b^8 + 11*b^6*c^2 - 28*b^4*c^4 + 11*b^2*c^6 + 15*c^8) + 2*a^4*(b^2 - c^2)^2*(15*b^10 + 47*b^8*c^2 + 62*b^6*c^4 + 62*b^4*c^6 + 47*b^2*c^8 + 15*c^10) - 4*a^8*(30*b^10 + 17*b^8*c^2 + 15*b^6*c^4 + 15*b^4*c^6 + 17*b^2*c^8 + 30*c^10) + a^6*(70*b^12 - 34*b^10*c^2 - 86*b^8*c^4 - 92*b^6*c^6 - 86*b^4*c^8 - 34*b^2*c^10 + 70*c^12) : :

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)

Barycentrics    50*a^22 - 205*a^20*(b^2 + c^2) + 5*(b^2 - c^2)^8*(b^2 + c^2)^3 - 2*a^2*(b^2 - c^2)^6*(b^2 + c^2)^2*(35*b^4 + b^2*c^2 + 35*c^4) + 2*a^18*(85*b^4 + 324*b^2*c^2 + 85*c^4) + a^16*(385*b^6 - 507*b^4*c^2 - 507*b^2*c^4 + 385*c^6) - 2*a^14*(370*b^8 + 179*b^6*c^2 - 442*b^4*c^4 + 179*b^2*c^6 + 370*c^8) + a^12*(70*b^10 + 912*b^8*c^2 - 854*b^6*c^4 - 854*b^4*c^6 + 912*b^2*c^8 + 70*c^10) + a^4*(b^2 - c^2)^4*(215*b^10 + 193*b^8*c^2 + 120*b^6*c^4 + 120*b^4*c^6 + 193*b^2*c^8 + 215*c^10) - 2*a^6*(b^2 - c^2)^2*(55*b^12 - 81*b^10*c^2 + 117*b^8*c^4 + 122*b^6*c^6 + 117*b^4*c^8 - 81*b^2*c^10 + 55*c^12) + 2*a^10*(350*b^12 - 475*b^10*c^2 - 22*b^8*c^4 + 566*b^6*c^6 - 22*b^4*c^8 - 475*b^2*c^10 + 350*c^12) + a^8*(-470*b^14 + 492*b^12*c^2 + 688*b^10*c^4 - 646*b^8*c^6 - 646*b^6*c^8 + 688*b^4*c^10 + 492*b^2*c^12 - 470*c^14) : :

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)

Barycentrics    -2*a^12 - 5*a^10*(b^2 + c^2) + 11*a^8*(b^2 + c^2)^2 + 7*(b^2 - c^2)^4*(b^2 + c^2)^2 - a^2*(b^2 - c^2)^2*(5*b^6 - 13*b^4*c^2 - 13*b^2*c^4 + 5*c^6) + 2*a^6*(5*b^6 - 9*b^4*c^2 - 9*b^2*c^4 + 5*c^6) - 8*a^4*(2*b^8 + b^6*c^2 - 4*b^4*c^4 + b^2*c^6 + 2*c^8) : :

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)

Barycentrics    10*a^18 - 57*a^16*(b^2 + c^2) + 37*(b^2 - c^2)^6*(b^2 + c^2)^3 + 6*a^14*(9*b^4 + 46*b^2*c^2 + 9*c^4) - 4*a^2*(b^2 - c^2)^4*(b^2 + c^2)^2*(21*b^4 - 34*b^2*c^2 + 21*c^4) + 2*a^12*(67*b^6 - 149*b^4*c^2 - 149*b^2*c^4 + 67*c^6) - 2*a^10*(111*b^8 + 124*b^6*c^2 - 318*b^4*c^4 + 124*b^2*c^6 + 111*c^8) - 4*a^8*(15*b^10 - 164*b^8*c^2 + 121*b^6*c^4 + 121*b^4*c^6 - 164*b^2*c^8 + 15*c^10) - 2*a^4*(b^2 - c^2)^2*(27*b^10 + 149*b^8*c^2 - 120*b^6*c^4 - 120*b^4*c^6 + 149*b^2*c^8 + 27*c^10) + 2*a^6*(121*b^12 - 166*b^10*c^2 - 209*b^8*c^4 + 572*b^6*c^6 - 209*b^4*c^8 - 166*b^2*c^10 + 121*c^12) : :

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)

Barycentrics    (a^2 - b^2 - c^2)*(4*a^26 - 17*a^24*(b^2 + c^2) - (b^2 - c^2)^10*(b^2 + c^2)^3 + 8*a^2*(b^2 - c^2)^8*(b^2 + c^2)^2*(b^4 + 6*b^2*c^2 + c^4) + 2*a^22*(6*b^4 - 23*b^2*c^2 + 6*c^4) + a^20*(46*b^6 + 312*b^4*c^2 + 312*b^2*c^4 + 46*c^6) - 16*a^18*(5*b^8 + 12*b^6*c^2 + 45*b^4*c^4 + 12*b^2*c^6 + 5*c^8) - 2*a^4*(b^2 - c^2)^6*(9*b^10 + 134*b^8*c^2 + 277*b^6*c^4 + 277*b^4*c^6 + 134*b^2*c^8 + 9*c^10) - a^16*(15*b^10 + 695*b^8*c^2 - 126*b^6*c^4 - 126*b^4*c^6 + 695*b^2*c^8 + 15*c^10) - 2*a^6*(b^2 - c^2)^4*(2*b^12 - 157*b^10*c^2 - 350*b^8*c^4 - 478*b^6*c^6 - 350*b^4*c^8 - 157*b^2*c^10 + 2*c^12) + 4*a^14*(30*b^12 + 213*b^10*c^2 + 118*b^8*c^4 - 234*b^6*c^6 + 118*b^4*c^8 + 213*b^2*c^10 + 30*c^12) - 4*a^12*(15*b^14 - 94*b^12*c^2 + 110*b^10*c^4 - 287*b^8*c^6 - 287*b^6*c^8 + 110*b^4*c^10 - 94*b^2*c^12 + 15*c^14) + a^8*(b^2 - c^2)^2*(65*b^14 + 307*b^12*c^2 - 215*b^10*c^4 - 1181*b^8*c^6 - 1181*b^6*c^8 - 215*b^4*c^10 + 307*b^2*c^12 + 65*c^14) + a^10*(-60*b^16 - 944*b^14*c^2 + 992*b^12*c^4 + 560*b^10*c^6 - 584*b^8*c^8 + 560*b^6*c^10 + 992*b^4*c^12 - 944*b^2*c^14 - 60*c^16)) : :

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)

Barycentrics    92*a^28 - 483*a^26*(b^2 + c^2) + 23*(b^2 - c^2)^10*(b^2 + c^2)^4 - a^2*(b^2 - c^2)^8*(b^2 + c^2)^3*(207*b^4 + 112*b^2*c^2 + 207*c^4) + a^24*(667*b^4 + 1888*b^2*c^2 + 667*c^4) + 2*a^22*(391*b^6 - 822*b^4*c^2 - 822*b^2*c^4 + 391*c^6) + 2*a^4*(b^2 - c^2)^6*(b^2 + c^2)^2*(299*b^8 + 269*b^6*c^2 + 532*b^4*c^4 + 269*b^2*c^6 + 299*c^8) - 2*a^20*(1449*b^8 + 1187*b^6*c^2 - 656*b^4*c^4 + 1187*b^2*c^6 + 1449*c^8) + a^18*(1495*b^10 + 4745*b^8*c^2 - 1496*b^6*c^4 - 1496*b^4*c^6 + 4745*b^2*c^8 + 1495*c^10) + 4*a^12*b^2*c^2*(329*b^12 - 1510*b^10*c^2 - 449*b^8*c^4 + 1084*b^6*c^6 - 449*b^4*c^8 - 1510*b^2*c^10 + 329*c^12) + a^16*(3105*b^12 - 1938*b^10*c^2 + 623*b^8*c^4 + 7588*b^6*c^6 + 623*b^4*c^8 - 1938*b^2*c^10 + 3105*c^12) - 2*a^6*(b^2 - c^2)^4*(161*b^14 + 486*b^12*c^2 + 1432*b^10*c^4 + 2177*b^8*c^6 + 2177*b^6*c^8 + 1432*b^4*c^10 + 486*b^2*c^12 + 161*c^14) - 4*a^14*(1035*b^14 + 116*b^12*c^2 - 1206*b^10*c^4 + 903*b^8*c^6 + 903*b^6*c^8 - 1206*b^4*c^10 + 116*b^2*c^12 + 1035*c^14) - a^8*(b^2 - c^2)^2*(1587*b^16 + 74*b^14*c^2 - 3320*b^12*c^4 - 2378*b^10*c^6 - 1654*b^8*c^8 - 2378*b^6*c^10 - 3320*b^4*c^12 + 74*b^2*c^14 + 1587*c^16) + a^10*(2875*b^18 - 3393*b^16*c^2 + 320*b^14*c^4 + 2552*b^12*c^6 - 3378*b^10*c^8 - 3378*b^8*c^10 + 2552*b^6*c^12 + 320*b^4*c^14 - 3393*b^2*c^16 + 2875*c^18) : :

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)

Barycentrics    10*a^24 - 31*a^22*(b^2 + c^2) + (b^2 - c^2)^8*(b^2 + c^2)^4 - 7*a^20*(b^4 - 16*b^2*c^2 + c^4) - a^2*(b^2 - c^2)^6*(b^2 + c^2)^3*(13*b^4 + 19*b^2*c^2 + 13*c^4) + a^18*(111*b^6 - 50*b^4*c^2 - 50*b^2*c^4 + 111*c^6) + a^4*(b^2 - c^2)^4*(b^2 + c^2)^2*(29*b^8 - 8*b^6*c^2 + 6*b^4*c^4 - 8*b^2*c^6 + 29*c^8) - a^16*(71*b^8 + 258*b^6*c^2 - 146*b^4*c^4 + 258*b^2*c^6 + 71*c^8) - 2*a^14*(67*b^10 - 158*b^8*c^2 + 9*b^6*c^4 + 9*b^4*c^6 - 158*b^2*c^8 + 67*c^10) + 2*a^12*(77*b^12 + 53*b^10*c^2 - 137*b^8*c^4 + 198*b^6*c^6 - 137*b^4*c^8 + 53*b^2*c^10 + 77*c^12) + a^6*(b^2 - c^2)^2*(21*b^14 + 125*b^12*c^2 + 23*b^10*c^4 - 57*b^8*c^6 - 57*b^6*c^8 + 23*b^4*c^10 + 125*b^2*c^12 + 21*c^14) + a^10*(46*b^14 - 338*b^12*c^2 + 230*b^10*c^4 - 50*b^8*c^6 - 50*b^6*c^8 + 230*b^4*c^10 - 338*b^2*c^12 + 46*c^14) - 2*a^8*(58*b^16 - 55*b^14*c^2 - 70*b^12*c^4 + 127*b^10*c^6 - 56*b^8*c^8 + 127*b^6*c^10 - 70*b^4*c^12 - 55*b^2*c^14 + 58*c^16) : :

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)

Barycentrics    10*a^30 - 41*a^28*(b^2 + c^2) + (b^2 - c^2)^10*(b^2 + c^2)^5 + 2*a^26*(7*b^4 + 61*b^2*c^2 + 7*c^4) - a^2*(b^2 - c^2)^8*(b^2 + c^2)^4*(14*b^4 + 53*b^2*c^2 + 14*c^4) + a^24*(159*b^6 + 56*b^4*c^2 + 56*b^2*c^4 + 159*c^6) + a^4*(b^2 - c^2)^6*(b^2 + c^2)^3*(41*b^8 + 165*b^6*c^2 - 120*b^4*c^4 + 165*b^2*c^6 + 41*c^8) - a^22*(206*b^8 + 481*b^6*c^2 + 446*b^4*c^4 + 481*b^2*c^6 + 206*c^8) + a^20*(-181*b^10 + 208*b^8*c^2 + 661*b^6*c^4 + 661*b^4*c^6 + 208*b^2*c^8 - 181*c^10) + a^6*(b^2 - c^2)^4*(b^2 + c^2)^2*(6*b^12 + 7*b^10*c^2 + 430*b^8*c^4 - 966*b^6*c^6 + 430*b^4*c^8 + 7*b^2*c^10 + 6*c^12) + a^18*(470*b^12 + 707*b^10*c^2 + 334*b^8*c^4 - 942*b^6*c^6 + 334*b^4*c^8 + 707*b^2*c^10 + 470*c^12) - a^16*(45*b^14 + 577*b^12*c^2 + 1719*b^10*c^4 - 1013*b^8*c^6 - 1013*b^6*c^8 + 1719*b^4*c^10 + 577*b^2*c^12 + 45*c^14) - 2*a^14*(225*b^16 + 229*b^14*c^2 - 682*b^12*c^4 - 541*b^10*c^6 + 2114*b^8*c^8 - 541*b^6*c^10 - 682*b^4*c^12 + 229*b^2*c^14 + 225*c^16) - a^8*(b^2 - c^2)^2*(179*b^18 + 584*b^16*c^2 + 3*b^14*c^4 - 1639*b^12*c^6 + 1449*b^10*c^8 + 1449*b^8*c^10 - 1639*b^6*c^12 + 3*b^4*c^14 + 584*b^2*c^16 + 179*c^18) + a^12*(245*b^18 + 543*b^16*c^2 + 626*b^14*c^4 - 3762*b^12*c^6 + 2924*b^10*c^8 + 2924*b^8*c^10 - 3762*b^6*c^12 + 626*b^4*c^14 + 543*b^2*c^16 + 245*c^18) + 2*a^10*(85*b^20 + 56*b^18*c^2 - 923*b^16*c^4 + 1184*b^14*c^6 + 1286*b^12*c^8 - 3120*b^10*c^10 + 1286*b^8*c^12 + 1184*b^6*c^14 - 923*b^4*c^16 + 56*b^2*c^18 + 85*c^20) : :

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}




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

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.

underbar



X(68101) = X(8)X(513)∩X(10)X(522)

Barycentrics    b*(b - c)*c*(-2*a + b + c)^2 : :
X(68101) = 3 X[10] - 2 X[23808], 9 X[4036] - 8 X[4791], 5 X[4036] - 4 X[50327], 10 X[4791] - 9 X[50327], 4 X[23808] - 3 X[24457], 3 X[42455] - 4 X[52356], 3 X[36848] - 2 X[55244], X[3762] - 3 X[4768], 3 X[1734] - 2 X[23809], 3 X[3679] - X[23838], 3 X[4448] - 2 X[57051], 3 X[4543] + X[39771], 3 X[26078] - 2 X[59837], 3 X[48181] - 2 X[59972], 2 X[48285] - 3 X[55969]

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)

Barycentrics    a*(b - c)*(a^2 - 2*a*b + b^2 - 2*a*c + b*c + c^2)^2 : :
X(68102) = 2 X[3681] + X[30613]

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)

Barycentrics    a*(a - b - c)^2*(b - c)*(a^2 - b^2 + b*c - c^2)^2 : :

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)

Barycentrics    (a - b)*b*(a - c)*c*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)^2 : :

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)

Barycentrics    a^3*(a - b)*(a - c)*(a^2*b^2 - b^4 + a^2*c^2 - c^4)^2 : :

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)

Barycentrics    a*(a - b)*(a - c)*(a*b - b^2 + a*c - c^2)^2 : :
X(68106) = X[660] + 3 X[3799]

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)

Barycentrics    (a - b)*b*(a - c)*(2*a - b - c)^2*c : :

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)

Barycentrics    a*(a - b - c)^2*(b - c)*(a^2 - b^2 - c^2)^2 : :

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)

Barycentrics    (a - b)*b*(a - c)*c*(2*a^2 - b^2 - c^2)^2 : :

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)

Barycentrics    b*(b - c)*c*(b + c)^2*(-a^2 + b^2 + c^2)^2 : :

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)

Barycentrics    (a - b)*b*(a - c)*c*(a*b + a*c - 2*b*c)^2 : :
X(68111) = 3 X[668] + X[889], 5 X[668] + 3 X[9296], X[668] + 3 X[31625], 5 X[889] - 9 X[9296], X[889] - 9 X[31625], X[889] - 3 X[66535], X[9296] - 5 X[31625], 3 X[9296] - 5 X[66535], 3 X[31625] - X[66535], 3 X[13466] - X[39011], 5 X[40552] - 3 X[66546]

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)

Barycentrics    a^5*(b - c)*(b^2 + b*c + c^2)^2 : :

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)

Barycentrics    a*(b - c)*(a^2 + b^2 + b*c + c^2)^2 : :

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)

Barycentrics    (a - b)*b*(a - c)*c*(2*a^3 - 2*a^2*b + a*b^2 - b^3 - 2*a^2*c + b^2*c + a*c^2 + b*c^2 - c^3)^2 : :

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)

Barycentrics    a^3*(a - b - c)^2*(b - c)*(a*b - b^2 + a*c - c^2)^2 : :
X(68115) = X[28132] - 3 X[59269]

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}

"934";

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

Barycentrics    a*(a - b - c)^4*(b - c) : :
X(68116) = 3 X[210] - X[14298], 3 X[3681] + X[4131]

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)

Barycentrics    a*(b - c)*(a^2 - 2*a*b + b^2 - 2*a*c + c^2)^2 : :
X(68117) = 3 X[3873] - 4 X[43932]

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)

Barycentrics    b*(b - c)*c*(-3*a + b + c)^2 : :
X(68118) = 9 X[693] - 8 X[4815], 4 X[4404] - 3 X[4462], 4 X[4036] - 3 X[4811], 3 X[25020] - 2 X[59968], 3 X[26078] - 2 X[59972]

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)

Barycentrics    b*(b - c)*c*(-a + 2*b + 2*c)^2 : :
X(68119) = 3 X[10] - X[23809], X[3762] - 9 X[4086], 3 X[3679] - X[57178], 3 X[4391] + X[68101], 5 X[31250] - 3 X[59837]

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)

Barycentrics    a*(b - c)*(2*a*b + 2*a*c + b*c)^2 : :
X(68120) = 5 X[2978] - 6 X[38238], 4 X[25142] - 3 X[50497], 3 X[31150] - 4 X[50500], 3 X[48548] - 4 X[50487]

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)

Barycentrics    b*(b - c)*c*(2*a^2 - a*b + b^2 - a*c - 2*b*c + c^2)^2 : :

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)

Barycentrics    b*(b - c)*c*(3*a + b + c)^2 : :
X(68122) = 9 X[4397] - 8 X[4404], 5 X[693] - 4 X[50327], 4 X[4036] - 3 X[4462], 3 X[4801] - 2 X[4815], 3 X[4811] - 4 X[4815]

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)

Barycentrics    b*(b - c)*c*(2*a + b + c)^2 : :
X(68123) = 3 X[4036] - 4 X[50334], 3 X[4978] - X[4985], 2 X[4985] - 3 X[30591], 2 X[47965] - 3 X[48230]

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)

Barycentrics    a*(b - c)*(a*b + a*c + 2*b*c)^2 : :
X(68124) = 8 X[661] - 9 X[14434], 4 X[7192] - 3 X[8027], 3 X[14404] - 2 X[47917], 4 X[25142] - 3 X[47666]

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)

Barycentrics    a^3*(b - c)*(a*b^2 - b^2*c + a*c^2 - b*c^2)^2 : :
X(68125) = 4 X[3572] - 3 X[8027], 9 X[14434] - 8 X[27854]

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)

Barycentrics    (a - b)*(a - c)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)^2 : :

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)

Barycentrics    a^2*(a - b)*(a - c)*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3)^2 : :

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)

Barycentrics    a^2*(a - b)*(a - c)*(a*b - b^2 + a*c - c^2)^2 : :
X(68128) = 3 X[14439] - X[38980]

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)

Barycentrics    (a - b)*(a - c)*(2*a^2 - b^2 - c^2)^2 : :

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)

Barycentrics    (b - c)*(b + c)^2*(-a^2 + b^2 + c^2)^2 : :
X(68130) = 3 X[14429] - 2 X[52599]

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)

Barycentrics    (a - b)*(a - c)*(a*b + a*c - 2*b*c)^2 : :

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)

Barycentrics    (a - b)*(a - c)*(b + c)^2*(a^2 - b*c)^2 : :

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)

Barycentrics    (b - c)*(b^2 + b*c + c^2)^2 : :
X(68133) = 2 X[25381] - 3 X[47808], 2 X[27929] - 3 X[48185]

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)

Barycentrics    a^2*(b - c)*(a*b + a*c - 2*b*c)^2 : :

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)

Barycentrics    a^2*(a - b - c)^4*(b - c) : :
X(68135) = 3 X[14427] - 2 X[59979]

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)

Barycentrics    a^2*(b - c)*(a^2 - 2*a*b + b^2 - 2*a*c + c^2)^2 : :

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)

Barycentrics    a^2*(b - c)*(a^2 - 2*a*b + b^2 - 2*a*c + b*c + c^2)^2 : :

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)

Barycentrics    (a - b)*(a - c)*(a - b - c)^2*(2*a^2 - a*b + b^2 - a*c - 2*b*c + c^2)^2 : :

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)

Barycentrics    (a - b)*(a - c)*(2*a^2 - a*b - b^2 - a*c + 2*b*c - c^2)^2 : :
X(68139) = 3 X[100] + X[37143], 3 X[6174] - X[40629]

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)

Barycentrics    (b - c)*(-a^2 - a*b + b^2 - a*c + b*c + c^2)^2 : :
X(68140) = X[27929] - 3 X[28602]

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)

Barycentrics    (b - c)*(a^2*b - 2*a*b^2 + b^3 + a^2*c + 2*a*b*c - b^2*c - 2*a*c^2 - b*c^2 + c^3)^2 : :

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)

Barycentrics    (5*a - b - c)^2*(b - c) : :
X(68142) = 5 X[1635] + X[47768], 13 X[1635] - X[47878], 7 X[1635] - X[47883], 5 X[44551] - 2 X[48422], 13 X[47768] + 5 X[47878], 7 X[47768] + 5 X[47883], 7 X[47878] - 13 X[47883], 8 X[650] + X[53586], 16 X[2490] - 7 X[3239], 2 X[2490] + 7 X[4394], 25 X[2490] - 7 X[59589], X[3239] + 8 X[4394], 25 X[3239] - 16 X[59589], 25 X[4394] + 2 X[59589], X[4024] - 10 X[43061], X[4765] + 2 X[47767], 7 X[4765] + 2 X[48397], 7 X[47767] - X[48397], X[52593] + 3 X[66524], 3 X[14435] + X[53584], 3 X[47766] - X[53584], 4 X[31182] - X[47765], 7 X[48576] - X[53587]

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)

Barycentrics    (b - c)*(4*a^2 - a*b + b^2 - a*c - 4*b*c + c^2)^2 : :

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)

Barycentrics    (b - c)*(2*a^2 - a*b + b^2 - a*c - 2*b*c + c^2)^2 : :
X(68144) = 3 X[6545] - 4 X[62635], 9 X[6544] - 8 X[62552]

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)

Barycentrics    (b - c)*(-a^2 + a*b + a*c + 2*b*c)^2 : :
X(68145) = 3 X[649] - 4 X[4817], 4 X[4375] - 3 X[48572], 4 X[28843] - 3 X[44550], 3 X[47832] - 2 X[62552]

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)

Barycentrics    (a - b - c)^2*(b - c)*(2*a^2 - a*b - b^2 - a*c + 2*b*c - c^2)^2 : :

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)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3)^2 : :

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)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a*b - b^2 + a*c - c^2)^2 : :

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)

Barycentrics    (a^2 - b^2)*(2*a - b - c)^2*(a^2 - c^2) : :

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)

Barycentrics    a^4*(b^2 - c^2)*(a^2 - b^2 - c^2)^4 : :
X(68150) = 4 X[37084] - 3 X[39201]

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)

Barycentrics    a^2*(a + b)*(a - b - c)^2*(b - c)*(a + c)*(a^2 - b^2 - c^2)^2 : :

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)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(2*a^2 - b^2 - c^2)^2 : :
X(68152) = 5 X[2] - 4 X[9165], 5 X[9164] - 2 X[9165], 3 X[99] + X[892], X[99] + 3 X[4590], X[99] - 3 X[14588], 5 X[99] + 3 X[31998], 7 X[99] - 3 X[33799], X[892] - 9 X[4590], X[892] - 3 X[9182], X[892] + 9 X[14588], 5 X[892] - 9 X[31998], 7 X[892] + 9 X[33799], 3 X[4590] - X[9182], 5 X[4590] - X[31998], 7 X[4590] + X[33799], X[9182] + 3 X[14588], and many others

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)

Barycentrics    (a - b)*(a - c)*(b + c)*(a^2 - b*c)^2 : :

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)

Barycentrics    a^2*(a - b)*(a - c)*(b + c)*(a^2 - b^2 + b*c - c^2)^2 : :

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)

Barycentrics    (b^2 - c^2)*(-a^2 + b*c)^2*(a^2 + b*c)^2 : :
X(68155) = X[2395] + 3 X[5652]

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)

Barycentrics    (a + b)*(b - c)*(a + c)*(a^2 - b*c)^2 : :

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)

Barycentrics    (a + b)*(b - c)*(a + c)*(b^2 + b*c + c^2)^2 : :

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)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6)^2 : :
X(68158) = 3 X[4226] + X[65716], 3 X[45662] - X[65728]

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)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(2*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 4*b^2*c^2 - c^4)^2 : :
X(68159) = 3 X[5468] + X[62672], 3 X[1641] - X[62655]

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)

Barycentrics    (a + b)*(2*a - b - c)^2*(b - c)*(a + c) : :
X(68160) = 7 X[3737] + X[5214], 9 X[3737] - X[47683], 9 X[5214] + 7 X[47683], 3 X[30580] + X[66284], X[48291] + 3 X[50349], 3 X[4833] + X[7192], X[4833] + 3 X[47845], X[7192] - 9 X[47845], 3 X[551] - X[55244], 3 X[4448] + X[53535], 3 X[3251] + X[68101]

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)

Barycentrics    (a + b)*(b - c)*(a + c)*(a*b - b^2 + a*c - c^2)^2 : :

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)

Barycentrics    a^4*(b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4)^2 : :
X(68162) = X[39201] - 4 X[44809], X[34952] + 2 X[44808]

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)

Barycentrics    a^4*(b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - b^2*c^2 + c^4)^2 : :
X(68163) = 2 X[37084] - 3 X[44809]

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)

Barycentrics    a^2*(a + b)*(b - c)*(a + c)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2 + c^3)^2 : :

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)

Barycentrics    (b^2 - c^2)*(-5*a^2 + b^2 + c^2)^2 : :
X(68165) = 9 X[351] - X[17436], X[8599] + 3 X[9123], X[8599] - 9 X[15724], X[8599] - 3 X[59927], X[9123] + 3 X[15724], 3 X[15724] - X[59927], X[3265] + 8 X[8651], 3 X[8644] + X[62568], 3 X[9125] - X[62568]

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)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(2*a^6 - a^4*b^2 - b^6 - a^4*c^2 + b^4*c^2 + b^2*c^4 - c^6)^2 : :
X(68166) = X[685] + 3 X[35278]

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)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 + c^2)^2*(-2*a^4 + a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4)^2 : :
X(68167) = 2 X[14380] - 3 X[65723], 4 X[38401] - 3 X[65723], 3 X[5489] - 4 X[43083], 9 X[1649] - 8 X[60342], 4 X[58263] - 3 X[58346], 4 X[57128] - 3 X[65754], 2 X[62172] - 3 X[65754], 3 X[8029] - 4 X[15328]

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)

Barycentrics    (a + b)*(b - c)*(a + c)*(a^2 + a*b - b^2 + a*c - b*c - c^2)^2 : :

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)

Barycentrics    (b^2 - c^2)*(4*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 4*b^2*c^2 + c^4)^2 : :

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)

Barycentrics    (a + b)*(a - 2*b - 2*c)^2*(b - c)*(a + c) : :
X(68170) = 3 X[4833] - 2 X[47683], 3 X[4825] - 4 X[68119], 2 X[23809] - 3 X[48189]

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)

Barycentrics    1/((a^2 - b^2)*(a^2 - c^2)*(Sqrt[3]*(a^2 + b^2 - c^2) - 2*S)^2*(Sqrt[3]*(a^2 - b^2 + c^2) - 2*S)^2) : :
Barycentrics    (sqrt(3)*(-a^2+b^2+c^2)-2*S)^2*(b^2-c^2) : :
X(68171) = 3 X[9205] - X[35444]

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)

Barycentrics    1/((a^2 - b^2)*(a^2 - c^2)*(Sqrt[3]*(a^2 + b^2 - c^2) + 2*S)^2*(Sqrt[3]*(a^2 - b^2 + c^2) + 2*S)^2) : :
Barycentrics    (sqrt(3)*(-a^2+b^2+c^2)+2*S)^2*(b^2-c^2) : :
X(68172) = 3 X[9204] - X[35443]

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)

Barycentrics    a^4*(b^2 - c^2)*(a^2*b^2 + a^2*c^2 - 2*b^2*c^2)^2 : :
X(68173) = 3 X[8029] - 4 X[66271]

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)

Barycentrics    (b^2 - c^2)*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6)^2 : :
X(68174) = 2 X[46608] - 3 X[65723], 9 X[1649] - 8 X[8562], 3 X[8029] - 4 X[10412]

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)

Barycentrics    (b^2 - c^2)*(2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)^2 : :
X(68175) = 4 X[2395] - 3 X[8029], 3 X[11123] - 2 X[62642]

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)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^2*b^2 + a^2*c^2 - 2*b^2*c^2)^2 : :

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)

Barycentrics    -6 a^4+a^2 (b^2+c^2)+b^4-6 b^2 c^2+c^4 : :

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)

Barycentrics    a^4*(a^2 - b^2)*(a^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4)^2 : :
X(68178) = 3 X[9155] - X[38987]

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}





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

This preamble and centers X(68179)-X(68232) were contributed by César Eliud Lozada, April 3, 2025.

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):

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.

underbar

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

Barycentrics    (a-b)*(a-c)*(a+b-c)*(a-b+c)*(a^2-6*b*a+b^2-c^2)*(a^2-6*c*a-b^2+c^2) : :

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)

Barycentrics    (a-b)*(a-c)*(a+b-c)*(a-b+c)*(3*a^2-2*b*a+3*b^2-3*c^2)*(3*a^2-2*c*a-3*b^2+3*c^2) : :

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)

Barycentrics    (a-b)*(a-c)*(a^3-(3*b+c)*a^2-(b+c)*(3*b+c)*a+(b^2-c^2)*(b-c))*(a^3-(b+3*c)*a^2-(b+3*c)*(b+c)*a+(b^2-c^2)*(b-c)) : :

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)

Barycentrics    (a-b)*(a-c)*(a^2-2*(3*b+c)*a+(b-c)^2)*(a^2-2*(b+3*c)*a+(b-c)^2) : :

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)

Barycentrics    a*(a-b)*(a-c)*(a+b-c)^2*(a-b+c)^2*(a^2-2*(b+c)*a+(b+c)*(b-3*c))*(a^2-2*(b+c)*a-(b+c)*(3*b-c)) : :

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)

Barycentrics    (a-b)*(a-c)*(a+b-c)*(a-b+c)*(a^2+6*b*a+b^2-c^2)*(a^2+6*c*a-b^2+c^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)

Barycentrics    (a-b)*(a-c)*(a+b-c)*(a-b+c)*(a^2+4*b*a+b^2-c^2)*(a^2+4*c*a-b^2+c^2) : :

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)

Barycentrics    a*(a-b)*(a-c)*(a+b-c)*(a-b+c)*(a^2-2*(2*b+c)*a+(3*b-c)*(b-c))*(a^2-2*(b+2*c)*a+(b-c)*(b-3*c)) : :

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)

Barycentrics    (a-b)*(a-c)*(a^3+(3*b-c)*a^2+(b+c)*(3*b-c)*a+(b^2-c^2)*(b-c))*(a^3-(b-3*c)*a^2-(b+c)*(b-3*c)*a+(b^2-c^2)*(b-c)) : :

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)

Barycentrics    a*(a-b)*(a-c)*(a+b-c)*(a-b+c)*(a^3+(3*b-c)*a^2-(b+c)^2*a-(b^2-c^2)*(3*b+c))*(a^3-(b-3*c)*a^2-(b+c)^2*a+(b^2-c^2)*(b+3*c)) : :

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)

Barycentrics    (a-b)*(a-c)*(a+b-c)*(a-b+c)*(a^4-2*(2*b+c)*a^3+2*b*(3*b-c)*a^2-2*(b^2-c^2)*(2*b+c)*a+(b^2-c^2)*(b-c)^2)*(a^4-2*(b+2*c)*a^3-2*c*(b-3*c)*a^2+2*(b^2-c^2)*(b+2*c)*a-(b^2-c^2)*(b-c)^2) : :

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)

Barycentrics    (a-b)*(a-c)*(a+b-c)*(a-b+c)*(3*a^2+2*b*a+3*b^2-3*c^2)*(3*a^2+2*c*a-3*b^2+3*c^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)

Barycentrics    (a-b)*(a-c)*(a+b-c)*(a-b+c)*(a^4-(2*b+c)*a^3-2*b*c*a^2+(b^2-c^2)*(2*b+c)*a-(b^2-c^2)*(b-c)^2)*(a^4-(b+2*c)*a^3-2*b*c*a^2-(b^2-c^2)*(b+2*c)*a+(b^2-c^2)*(b-c)^2) : :

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)

Barycentrics    (a-b)*(a-c)*(a^2+2*(3*b-c)*a+(b-c)^2)*(a^2-2*(b-3*c)*a+(b-c)^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)

Barycentrics    (a-b)*(a-c)*(a+b-c)*(a-b+c)*(a^4+2*(b-c)*a^3-2*b*(3*b+c)*a^2+2*(b^2-c^2)*(b-c)*a+(b^2-c^2)*(b-c)^2)*(a^4-2*(b-c)*a^3-2*c*(b+3*c)*a^2+2*(b^2-c^2)*(b-c)*a-(b^2-c^2)*(b-c)^2) : :

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)

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^2-2*(b+c)*a+(b+c)*(b-3*c))*(a^2-2*(b+c)*a-(b+c)*(3*b-c)) : :

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)

Barycentrics    (a-b)*(a-c)*(3*a^3+(b-c)*a^2+(b+c)*(b-3*c)*a+(b^2-c^2)*(3*b-c))*(3*a^3-(b-c)*a^2-(b+c)*(3*b-c)*a+(b^2-c^2)*(b-3*c)) : :

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)

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^2+4*(b+c)*a+(b+3*c)*(b+c))*(a^2+4*(b+c)*a+(b+c)*(3*b+c)) : :

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)

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^2+(b+c)*a+b*(b+c))*(a^2+(b+c)*a+c*(b+c)) : :

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)

Barycentrics    (a-b)*(a-c)*(3*a^3+(5*b+c)*a^2+(b+c)*(5*b-3*c)*a+(b^2-c^2)*(3*b+c))*(3*a^3+(b+5*c)*a^2-(b+c)*(3*b-5*c)*a-(b^2-c^2)*(b+3*c)) : :

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)

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^3+3*(b+c)*a^2+(b+c)*(3*b-c)*a+(b^2-c^2)*(b+3*c))*(a^3+3*(b+c)*a^2-(b+c)*(b-3*c)*a-(b^2-c^2)*(3*b+c)) : :

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)

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^3-3*(b+c)*a^2-(b+3*c)*(b+c)*a+(b^2-c^2)*(3*b-c))*(a^3-3*(b+c)*a^2-(b+c)*(3*b+c)*a+(b^2-c^2)*(b-3*c)) : :

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)

Barycentrics    a*(a^2-b^2)*(a^2-c^2)*((2*b+c)*a^2+(b+c)*c*a-2*(b^2-c^2)*b)*((b+2*c)*a^2+(b+c)*b*a+2*(b^2-c^2)*c) : :

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)

Barycentrics    a*(a^2-b^2)*(a^2-c^2)*((2*b+c)*a^2+2*(b+c)*b*a+(b^2-c^2)*c)*((b+2*c)*a^2+2*(b+c)*c*a-(b^2-c^2)*b) : :

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)

Barycentrics    (a-b)*(a-c)*(a+b-c)*(a-b+c)*(a^5+3*b*a^4-2*(2*b^2+2*b*c+c^2)*a^3-4*b*(b+c)^2*a^2+(b^2-c^2)*(3*b^2-4*b*c-c^2)*a+(b^2-c^2)^2*b)*(a^5+3*c*a^4-2*(b^2+2*b*c+2*c^2)*a^3-4*c*(b+c)^2*a^2+(b^2-c^2)*(b^2+4*b*c-3*c^2)*a+(b^2-c^2)^2*c) : :

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)

Barycentrics    a*(a-b)*(a-c)*(a+b-c)*(a-b+c)*((2*b+c)*a^4-(2*b^2+c^2)*a^3-(2*b^3+c^3+b*c*(3*b+4*c))*a^2+(b^2-c^2)*(2*b^2-c^2)*a+2*(b^2-c^2)*(b-c)*b*c)*((b+2*c)*a^4-(b^2+2*c^2)*a^3-(b^3+2*c^3+b*c*(4*b+3*c))*a^2+(b^2-c^2)*(b^2-2*c^2)*a+2*(b^2-c^2)*(b-c)*b*c) : :

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)

Barycentrics    a*(a^2-b^2)*(a^2-c^2)*((b-c)*a^2+(b+c)*b*a-(b^2-c^2)*c)*((b-c)*a^2-(b+c)*c*a-(b^2-c^2)*b) : :

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)

Barycentrics    (a-b)*(a-c)*(a+b-c)*(a-b+c)*(a^5-(2*b^2+b*c+c^2)*a^3-c*(b+c)^2*a^2+(b^2-c^2)*(b+c)*b*a+(b^2-c^2)^2*c)*(a^5-(b^2+b*c+2*c^2)*a^3-b*(b+c)^2*a^2-(b^2-c^2)*(b+c)*c*a+(b^2-c^2)^2*b) : :

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)

Barycentrics    a*(a-b)*(a-c)*(a+b-c)*(a-b+c)*((b-c)*a^4-(b^2-c^2)*a^3-(b^3+2*b*c^2-c^3)*a^2+(b^4-c^4)*a+(b^2-c^2)*(b-c)*b*c)*((b-c)*a^4-(b^2-c^2)*a^3-(b^3-2*b^2*c-c^3)*a^2+(b^4-c^4)*a-(b^2-c^2)*(b-c)*b*c) : :

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)

Barycentrics    a*(a-b)*(a-c)*(a+b-c)*(a-b+c)*((2*b+c)*a^4-(2*b^2+c^2)*a^3-(2*b^3+c^3+b*c*(2*b+3*c))*a^2+(b^2-c^2)*(2*b^2-c^2)*a+(b^2-c^2)*(b-c)*b*c)*((b+2*c)*a^4-(b^2+2*c^2)*a^3-(b^3+2*c^3+b*c*(3*b+2*c))*a^2+(b^2-c^2)*(b^2-2*c^2)*a+(b^2-c^2)*(b-c)*b*c) : :

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)

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^3-3*(b+c)*a^2-(b+3*c)*(b+c)*a+(b^2-c^2)*(3*b-c))*(a^3-3*(b+c)*a^2-(b+c)*(3*b+c)*a+(b^2-c^2)*(b-3*c)) : :

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)

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^3+3*(b+c)*a^2+(b+c)*(3*b-c)*a+(b^2-c^2)*(b+3*c))*(a^3+3*(b+c)*a^2-(b+c)*(b-3*c)*a-(b^2-c^2)*(3*b+c)) : :

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)

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^3-(b+c)*b*a-(b^2-c^2)*c)*(a^3-(b+c)*c*a+(b^2-c^2)*b) : :

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)

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^4-2*(b+3*c)*(b+c)*a^2-8*b*c*(b+c)*a+(b^2-c^2)^2)*(a^4-2*(b+c)*(3*b+c)*a^2-8*b*c*(b+c)*a+(b^2-c^2)^2) : :

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)

Barycentrics    (a-b)*(a-c)*(a^3-(2*b+3*c)*a^2+(b^2-6*b*c-3*c^2)*a+c*(b-c)^2)*(a^3-(3*b+2*c)*a^2-(3*b^2+6*b*c-c^2)*a+b*(b-c)^2) : :

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)

Barycentrics    a*(a^2-b^2)*(a^2-c^2)*((2*b+c)*a^4-2*(b+c)*b*a^3-2*(b+c)*(b^2+c^2)*a^2+2*(b^2-c^2)*(b-c)*b*a+(b^2-c^2)^2*c)*((b+2*c)*a^4-2*(b+c)*c*a^3-2*(b+c)*(b^2+c^2)*a^2+2*(b^2-c^2)*(b-c)*c*a+(b^2-c^2)^2*b) : :

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)

Barycentrics    a*(a-b)*(a-c)*((2*b+c)*a^4-(2*b^2+2*b*c+c^2)*a^3-(2*b^3+c^3+b*c*(6*b+c))*a^2+(2*b^2+2*b*c+c^2)*(b-c)^2*a+(b^2-c^2)*(b-c)*b*c)*((b+2*c)*a^4-(b^2+2*b*c+2*c^2)*a^3-(b^3+2*c^3+b*c*(b+6*c))*a^2+(b^2+2*b*c+2*c^2)*(b-c)^2*a+(b^2-c^2)*(b-c)*b*c) : :

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)

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^4+2*(b+c)*(3*b-c)*a^2+4*b*c*(b+c)*a+(b^2-c^2)^2)*(a^4-2*(b+c)*(b-3*c)*a^2+4*b*c*(b+c)*a+(b^2-c^2)^2) : :

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)

Barycentrics    a*(a^2-b^2)*(a^2-c^2)*((2*b+c)*a^4-c*(b+c)*a^3-(b+c)*(4*b^2-3*b*c+c^2)*a^2+(b^2-c^2)*(b-c)*c*a+2*(b^2-c^2)^2*b)*((b+2*c)*a^4-b*(b+c)*a^3-(b+c)*(b^2-3*b*c+4*c^2)*a^2+(b^2-c^2)*(b-c)*b*a+2*(b^2-c^2)^2*c) : :

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)

Barycentrics    (a-b)*(a-c)*(a^3+(3*b-2*c)*a^2+(3*b^2+c^2)*a+b*(b-c)^2)*(a^3-(2*b-3*c)*a^2+(b^2+3*c^2)*a+c*(b-c)^2) : :

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)

Barycentrics    a*(a-b)*(a-c)*((2*b+c)*a^4-(2*b^2-2*b*c+c^2)*a^3-(2*b^3+5*b^2*c+c^3)*a^2+(2*b^2+4*b*c+c^2)*(b-c)^2*a+2*(b^2-c^2)*(b-c)*b*c)*((b+2*c)*a^4-(b^2-2*b*c+2*c^2)*a^3-(b^3+5*b*c^2+2*c^3)*a^2+(b^2+4*b*c+2*c^2)*(b-c)^2*a+2*(b^2-c^2)*(b-c)*b*c) : :

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)

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^4-2*(3*b^2+c^2)*a^2+(b^2-c^2)^2)*(a^4-2*(b^2+3*c^2)*a^2+(b^2-c^2)^2) : :

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)

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^5+(b^2-c^2)*a^3+(b+c)*(b^2+b*c-c^2)*a^2+(b^2-c^2)*(b^3-c^3))*(a^5-(b^2-c^2)*a^3-(b+c)*(b^2-b*c-c^2)*a^2+(b^2-c^2)*(b^3-c^3)) : :

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)

Barycentrics    a*(a-b)*(a-c)*((b-c)*a^4-(b^2+4*b*c-c^2)*a^3-(b^3-c^3+2*b*c*(3*b-2*c))*a^2+(b^2-2*b*c-c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)*b*c)*((b-c)*a^4-(b^2-4*b*c-c^2)*a^3-(b^3-c^3+2*b*c*(2*b-3*c))*a^2+(b^2+2*b*c-c^2)*(b-c)^2*a+(b^2-c^2)*(b-c)*b*c) : :

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)

Barycentrics    (a^2-b^2)*(a^2-c^2)*((b+c)*a^4+(3*b^2-c^2)*a^3+(b+c)*(3*b^2-b*c-c^2)*a^2+(b^2-c^2)^2*a+(b^2-c^2)*(b-c)*b*c)*((b+c)*a^4-(b^2-3*c^2)*a^3-(b+c)*(b^2+b*c-3*c^2)*a^2+(b^2-c^2)^2*a+(b^2-c^2)*(b-c)*b*c) : :

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)

Barycentrics    (a-b)*(a-c)*(a+b-c)^2*(a-b+c)^2*(a^3-b*a^2-(b+c)^2*a+b*(b^2-c^2))*(a^3-c*a^2-(b+c)^2*a-(b^2-c^2)*c) : :

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)

Barycentrics    (a-b)*(a-c)*(a+b-c)*(a-b+c)*((2*b+c)*a+b*c-c^2)*((b+2*c)*a-b^2+b*c) : :

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)

Barycentrics    (a-b)*(a-c)*(a+b-c)^2*(a-b+c)^2*(3*a^3-(3*b+c)*a^2-3*(b+c)^2*a+(b^2-c^2)*(3*b-c))*(3*a^3-(b+3*c)*a^2-3*(b+c)^2*a+(b^2-c^2)*(b-3*c)) : :

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)

Barycentrics    (a-b)*(a-c)*(a+b-c)^2*(a-b+c)^2*(a^3-(2*b+c)*a^2-(b+c)^2*a+(b^2-c^2)*(2*b-c))*(a^3-(b+2*c)*a^2-(b+c)^2*a+(b^2-c^2)*(b-2*c)) : :

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)

Barycentrics    (a-b)*(a-c)*(a+b-c)*(a-b+c)*(3*a^2+2*(b-c)*a+(3*b+c)*(b-c))*(3*a^2-2*(b-c)*a-(b+3*c)*(b-c)) : :

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)

Barycentrics    (a-b)*(a-c)*(a+b-c)^2*(a-b+c)^2*(a^5+(2*b+c)*a^4-2*(2*b^2+b*c+c^2)*a^3-2*(b^3+2*b^2*c+c^3)*a^2+(b^2-c^2)*(b+c)*(3*b-c)*a+(b^2-c^2)*(b-c)*c*(3*b+c))*(a^5+(b+2*c)*a^4-2*(b^2+b*c+2*c^2)*a^3-2*(b^3+2*b*c^2+c^3)*a^2+(b^2-c^2)*(b+c)*(b-3*c)*a+(b^2-c^2)*(b-c)*b*(b+3*c)) : :

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)

Barycentrics    (a-b)*(a-c)*(a+b-c)^2*(a-b+c)^2*(a^3+(3*b-c)*a^2-(b+c)^2*a-(b^2-c^2)*(3*b+c))*(a^3-(b-3*c)*a^2-(b+c)^2*a+(b^2-c^2)*(b+3*c)) : :

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)

Barycentrics    (a-b)*(a-c)*(a+b-c)*(a-b+c)*(3*a^2-2*(2*b+c)*a+(b-c)*(b-3*c))*(3*a^2-2*(b+2*c)*a+(3*b-c)*(b-c)) : :

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)

Barycentrics    (a+b-c)*(a-b+c)*(a^2-b^2)*(a^2-c^2)*(a^4-(b+c)*a^3-(b+c)*b*a^2+(b^2-c^2)*(b+c)*a+(b^2-c^2)*(b-c)*c)*(a^4-(b+c)*a^3-(b+c)*c*a^2-(b^2-c^2)*(b+c)*a+(b^2-c^2)*(b-c)*b) : :

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)

Barycentrics    a^2*(a + b - 3*c)*(a - 3*b + c)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c + 5*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2 + c^3) : :

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)

Barycentrics    (a^2 - 2*a*b + b^2 - 2*b*c + c^2)*(a^2 + b^2 - 2*a*c - 2*b*c + c^2)*(a^2*b - 2*a*b^2 + b^3 + a^2*c + 2*a*b*c - b^2*c - 2*a*c^2 - b*c^2 + c^3) : :

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)

Barycentrics    a^2*(a + b)*(a + c)*(-b^2 + a*c)*(a*b - c^2)*(a^3*b^2 - a*b^4 - a^2*b^2*c + a^3*c^2 - a^2*b*c^2 + b^3*c^2 + b^2*c^3 - a*c^4) : :

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)

Barycentrics    (3*a - b - c)*(a^3 - a^2*b - a*b^2 + b^3 - 2*a^2*c + 5*a*b*c - b^2*c - 2*a*c^2 - b*c^2 + c^3)*(a^3 - 2*a^2*b - 2*a*b^2 + b^3 - a^2*c + 5*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3) : :

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)

Barycentrics    a^2*(a^2 - 2*a*b + b^2 - 2*a*c + c^2)*(a^3 - a^2*b - a*b^2 + b^3 - 2*a^2*c + 2*a*b*c - 2*b^2*c + a*c^2 + b*c^2)*(a^3 - 2*a^2*b + a*b^2 - a^2*c + 2*a*b*c + b^2*c - a*c^2 - 2*b*c^2 + c^3) : :

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)

Barycentrics    a^2*(a^3 - a^2*b - a*b^2 + b^3 + a*c^2 + b*c^2 - 2*c^3)*(a^3 + a*b^2 - 2*b^3 - a^2*c + b^2*c - a*c^2 + c^3)*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5 + a^4*c + a^2*b^2*c - 2*a*b^3*c - 2*a^3*c^2 + a^2*b*c^2 + b^3*c^2 - 2*a*b*c^3 + b^2*c^3 + 2*a*c^4 - c^5) : :

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)

Barycentrics    a*(a^3 + a^2*b + a*b^2 + b^3 - 2*a^2*c - 4*a*b*c - 2*b^2*c + 3*a*c^2 + 3*b*c^2 - 2*c^3)*(a^3 - 2*a^2*b + 3*a*b^2 - 2*b^3 + a^2*c - 4*a*b*c + 3*b^2*c + a*c^2 - 2*b*c^2 + c^3) : :

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)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^5*c - 2*a^4*b*c + 2*a^2*b^3*c - a*b^4*c + 2*a^3*b*c^2 - 2*a^2*b^2*c^2 + 2*a*b^3*c^2 - 2*b^4*c^2 - 2*a^3*c^3 + 2*a^2*b*c^3 - 2*a*b*c^4 + b^2*c^4 + a*c^5)*(a^5*b - 2*a^3*b^3 + a*b^5 - 2*a^4*b*c + 2*a^3*b^2*c + 2*a^2*b^3*c - 2*a*b^4*c + a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + 2*a^2*b*c^3 + 2*a*b^2*c^3 - 2*a^2*c^4 - a*b*c^4 - 2*b^2*c^4 + c^6) : :

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)

Barycentrics    (a^4 + a^3*b - 4*a^2*b^2 + a*b^3 + b^4 - 2*a^3*c + 4*a^2*b*c + 4*a*b^2*c - 2*b^3*c - 7*a*b*c^2 + 2*a*c^3 + 2*b*c^3 - c^4)*(a^4 - 2*a^3*b + 2*a*b^3 - b^4 + a^3*c + 4*a^2*b*c - 7*a*b^2*c + 2*b^3*c - 4*a^2*c^2 + 4*a*b*c^2 + a*c^3 - 2*b*c^3 + c^4) : :
X(68241) = 2 X[4] - 3 X[64155], 4 X[3035] - 5 X[64698], 3 X[5660] - 4 X[10427], 4 X[20418] - 3 X[51768], 3 X[165] - 2 X[6068], 2 X[1156] - 3 X[11219], 4 X[1387] - 3 X[24644], 2 X[1537] - 3 X[59372], 5 X[3091] - 6 X[38207], 5 X[11522] - 6 X[38055], 2 X[11372] - 3 X[16173], 5 X[8227] - 6 X[38124], 2 X[12611] - 3 X[59380], 4 X[15528] - 3 X[41861], 6 X[21151] - 5 X[64012], 5 X[30308] - 6 X[38095], 5 X[31272] - 4 X[64699], 5 X[37714] - 6 X[38202], 6 X[38123] - 5 X[64008], 3 X[38693] - 2 X[51090], 3 X[59391] - 2 X[67871], 5 X[62778] - 4 X[67876]

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)

Barycentrics    (a + b)*(a + c)*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^6*b - 2*a^4*b^3 + a^2*b^5 + a^6*c - 2*a^5*b*c + a^4*b^2*c + a^3*b^3*c - a^2*b^4*c + a*b^5*c - b^6*c + a^4*b*c^2 - b^5*c^2 - 2*a^4*c^3 + a^3*b*c^3 - 2*a*b^3*c^3 + 2*b^4*c^3 - a^2*b*c^4 + 2*b^3*c^4 + a^2*c^5 + a*b*c^5 - b^2*c^5 - b*c^6) : :

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)

Barycentrics    a^2*(a + b)*(a + c)*(-b^2 + a*c)*(a*b - c^2)*(a^4*b - a^3*b^2 + a*b^4 - b^5 + a^4*c - a^2*b^2*c - a^3*c^2 - a^2*b*c^2 + b^3*c^2 + b^2*c^3 + a*c^4 - c^5) : :

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(68243) = X(740)-isoconjugate of X(65876)
X(68243) = X(65939)-Dao conjugate of X(3948)
X(68243) = barycentric quotient X(18268)/X(65876)

X(68244) = X(1)X(21)∩X(10)X(409)

Barycentrics    a*(a + b)*(a + c)*(a^4 - a^3*b + a*b^3 - b^4 - a^3*c + a^2*b*c + a*b^2*c - b^3*c + a*b*c^2 + a*c^3 - b*c^3 - c^4) : :
X(68244) = 2 X[34172] - 3 X[37375]

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)

Barycentrics    a^4 - a^2*b^2 - 3*a^2*b*c + 4*a*b^2*c - b^3*c - a^2*c^2 + 4*a*b*c^2 - 2*b^2*c^2 - b*c^3 : :
X(68245) = 3 X[5205] - X[38475], 5 X[5205] - 2 X[47626], X[5205] + 2 X[67343], 5 X[38475] - 6 X[47626], X[38475] + 6 X[67343], X[47626] + 5 X[67343]

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)

Barycentrics    a*(a + b - 2*c)*(a - 2*b + c)*(a^4 + a^3*b - 2*a^2*b^2 - a*b^3 + b^4 + a^3*c - 7*a^2*b*c + 7*a*b^2*c - b^3*c - 2*a^2*c^2 + 7*a*b*c^2 - 4*b^2*c^2 - a*c^3 - b*c^3 + c^4) : :
X(68246) = 3 X[1] - 4 X[66502], 2 X[106] - 3 X[14193], X[1320] - 3 X[14193], 4 X[121] - 5 X[64141], 2 X[10774] - 3 X[59415], 4 X[14664] - 3 X[38693]

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)

Barycentrics    a*(b + c)*(a^4 + a^3*b + a*b^3 + b^4 - a^3*c - a^2*b*c - a*b^2*c - b^3*c - a*b*c^2 + a*c^3 + b*c^3 - c^4)*(a^4 - a^3*b + a*b^3 - b^4 + a^3*c - a^2*b*c - a*b^2*c + b^3*c - a*b*c^2 + a*c^3 - b*c^3 + c^4) : :

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)

Barycentrics    a*(a^2 - 2*a*b + b^2 + a*c + b*c - 2*c^2)*(a^2 + a*b - 2*b^2 - 2*a*c + b*c + c^2)*(3*a^5 - 4*a^4*b - a^3*b^2 + 3*a^2*b^3 - 2*a*b^4 + b^5 - 4*a^4*c + 9*a^3*b*c - 4*a^2*b^2*c - a*b^3*c - a^3*c^2 - 4*a^2*b*c^2 + 6*a*b^2*c^2 - b^3*c^2 + 3*a^2*c^3 - a*b*c^3 - b^2*c^3 - 2*a*c^4 + c^5) : :

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)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^6 - 2*a^4*b^2 + a^2*b^4 - a^4*b*c + 3*a^3*b^2*c - 3*a*b^4*c + b^5*c - 2*a^4*c^2 + 3*a^3*b*c^2 - 4*a^2*b^2*c^2 + 3*a*b^3*c^2 + 3*a*b^2*c^3 - 2*b^3*c^3 + a^2*c^4 - 3*a*b*c^4 + b*c^5) : :
X(68249) = 2 X[25437] - 3 X[57298]

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)

Barycentrics    (a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + a*c + c^2)*(a^6 - 2*a^4*b^2 + a^2*b^4 - a^4*b*c + a^3*b^2*c - a*b^4*c + b^5*c - 2*a^4*c^2 + a^3*b*c^2 + a*b^3*c^2 + a*b^2*c^3 - 2*b^3*c^3 + a^2*c^4 - a*b*c^4 + b*c^5) : :
X(68250) = 3 X[50148] - X[51883], 3 X[50148] - 2 X[52200]

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)

Barycentrics    a*(a^2 - b^2 - c^2)*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c + 2*a*b*c + b^2*c - a*c^2 - b*c^2 - c^3)*(a^3 + a^2*b - a*b^2 - b^3 - a^2*c + 2*a*b*c - b^2*c - a*c^2 + b*c^2 + c^3)*(2*a^4 - a^3*b - a^2*b^2 + a*b^3 - b^4 - a^3*c + 2*a^2*b*c - a*b^2*c - a^2*c^2 - a*b*c^2 + 2*b^2*c^2 + a*c^3 - c^4) : :

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)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^3*c - 4*a^2*b*c + 3*a*b^2*c + 2*a^2*c^2 - 4*a*b*c^2 + b^2*c^2 + a*c^3)*(a^3*b + 2*a^2*b^2 + a*b^3 - 4*a^2*b*c - 4*a*b^2*c + a^2*c^2 + 3*a*b*c^2 + b^2*c^2 - c^4) : :

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)

Barycentrics    a^2*(a + b)*(a + c)*(a^5 - 3*a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5 - 3*a^4*c + 5*a^3*b*c - 2*a^2*b^2*c - 3*a*b^3*c + 3*b^4*c + 2*a^3*c^2 - 2*a^2*b*c^2 + 2*a^2*c^3 - 3*a*b*c^3 - 3*a*c^4 + 3*b*c^4 + c^5) : :

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)

Barycentrics    a^2*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4 - 4*a^3*c + 7*a^2*b*c - 6*a*b^2*c + 3*b^3*c + 6*a^2*c^2 - 6*a*b*c^2 - 4*a*c^3 + 3*b*c^3 + c^4) : :

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)

Barycentrics    6*a^3 - 9*a^2*b + 4*a*b^2 - b^3 - 9*a^2*c + 12*a*b*c - 3*b^2*c + 4*a*c^2 - 3*b*c^2 - c^3 : :

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)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(4*a^5 - 5*a^4*b - 2*a^3*b^2 + 4*a^2*b^3 - 2*a*b^4 + b^5 - 5*a^4*c + 10*a^3*b*c - 4*a^2*b^2*c - 2*a*b^3*c + b^4*c - 2*a^3*c^2 - 4*a^2*b*c^2 + 8*a*b^2*c^2 - 2*b^3*c^2 + 4*a^2*c^3 - 2*a*b*c^3 - 2*b^2*c^3 - 2*a*c^4 + b*c^4 + c^5) : :
X(68256) = 2 X[4] - 3 X[1785], X[4] - 3 X[45766], 3 X[1699] - 4 X[44901], 5 X[11522] - 6 X[51616], 5 X[3522] - 3 X[10538]

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)

Barycentrics    (a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + a*c + c^2)*(3*a^5 - 3*a^4*b - 2*a^3*b^2 + 2*a^2*b^3 - a*b^4 + b^5 - 3*a^4*c + 3*a^3*b*c - a*b^3*c + b^4*c - 2*a^3*c^2 + 4*a*b^2*c^2 - 2*b^3*c^2 + 2*a^2*c^3 - a*b*c^3 - 2*b^2*c^3 - a*c^4 + b*c^4 + c^5) : :
X(68257) = 3 X[50148] - 2 X[51883], 9 X[50148] - 8 X[52200], 3 X[51883] - 4 X[52200]

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)

Barycentrics    a*(b - c)*(a^2 - b^2 - c^2)*(2*a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3 - b^4 - 2*a^3*c + 4*a^2*b*c - 2*a*b^2*c - a^2*c^2 - 2*a*b*c^2 + 2*b^2*c^2 + 2*a*c^3 - c^4) : :

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)

Barycentrics    a*(b - c)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + a^2*b*c + a*b^2*c - b^3*c - a^2*c^2 + a*b*c^2 + 2*b^2*c^2 + a*c^3 - b*c^3) : :
X(68259) = 3 X[1699] - 4 X[15280]

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)

Barycentrics    a*(b - c)*(11*a^4 - 17*a^3*b + 3*a^2*b^2 + a*b^3 + 2*b^4 - 17*a^3*c + 29*a^2*b*c - 7*a*b^2*c - 5*b^3*c + 3*a^2*c^2 - 7*a*b*c^2 + 6*b^2*c^2 + a*c^3 - 5*b*c^3 + 2*c^4) : :
X(68260) = 5 X[165] - 2 X[11124]

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)

Barycentrics    a*(2*a^8 - 4*a^7*b - 3*a^6*b^2 + 11*a^5*b^3 - 2*a^4*b^4 - 10*a^3*b^5 + 5*a^2*b^6 + 3*a*b^7 - 2*b^8 - 4*a^7*c + 18*a^6*b*c - 15*a^5*b^2*c - 23*a^4*b^3*c + 34*a^3*b^4*c - 15*a*b^6*c + 5*b^7*c - 3*a^6*c^2 - 15*a^5*b*c^2 + 52*a^4*b^2*c^2 - 24*a^3*b^3*c^2 - 31*a^2*b^4*c^2 + 23*a*b^5*c^2 - 2*b^6*c^2 + 11*a^5*c^3 - 23*a^4*b*c^3 - 24*a^3*b^2*c^3 + 52*a^2*b^3*c^3 - 11*a*b^4*c^3 - 5*b^5*c^3 - 2*a^4*c^4 + 34*a^3*b*c^4 - 31*a^2*b^2*c^4 - 11*a*b^3*c^4 + 8*b^4*c^4 - 10*a^3*c^5 + 23*a*b^2*c^5 - 5*b^3*c^5 + 5*a^2*c^6 - 15*a*b*c^6 - 2*b^2*c^6 + 3*a*c^7 + 5*b*c^7 - 2*c^8) : :
X(68261) = 3 X[104] - X[66853], 3 X[52478] - 2 X[66853], 3 X[1768] + X[34464], 2 X[66843] - 3 X[67449], 2 X[22102] - 3 X[66628]

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)

Barycentrics    a*(b - c)*(3*a^4 - 6*a^3*b + 4*a^2*b^2 - 2*a*b^3 + b^4 - 6*a^3*c + 10*a^2*b*c - 2*a*b^2*c - 2*b^3*c + 4*a^2*c^2 - 2*a*b*c^2 + 2*b^2*c^2 - 2*a*c^3 - 2*b*c^3 + c^4) : :
X(68262) = 4 X[3] - 3 X[30234], 3 X[165] - 2 X[4394], X[4380] - 3 X[9778]

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)

Barycentrics    a*(b - c)*(a^6 + a^5*b - 2*a^4*b^2 - 2*a^3*b^3 + a^2*b^4 + a*b^5 + a^5*c - a^4*b*c - 2*a^3*b^2*c + 2*a^2*b^3*c + a*b^4*c - b^5*c - 2*a^4*c^2 - 2*a^3*b*c^2 + 12*a^2*b^2*c^2 - 6*a*b^3*c^2 - 2*a^3*c^3 + 2*a^2*b*c^3 - 6*a*b^2*c^3 + 2*b^3*c^3 + a^2*c^4 + a*b*c^4 + a*c^5 - b*c^5) : :

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)

Barycentrics    a*(b - c)*(a^6 + a^5*b - 2*a^4*b^2 - 2*a^3*b^3 + a^2*b^4 + a*b^5 + a^5*c - a^4*b*c - 2*a^3*b^2*c + 2*a^2*b^3*c + a*b^4*c - b^5*c - 2*a^4*c^2 - 2*a^3*b*c^2 + 6*a^2*b^2*c^2 - 2*a*b^3*c^2 - 2*a^3*c^3 + 2*a^2*b*c^3 - 2*a*b^2*c^3 + 2*b^3*c^3 + a^2*c^4 + a*b*c^4 + a*c^5 - b*c^5) : :

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)

Barycentrics    a*(b - c)*(a^6 + a^5*b - 2*a^4*b^2 - 2*a^3*b^3 + a^2*b^4 + a*b^5 + a^5*c - a^4*b*c - 2*a^3*b^2*c + 2*a^2*b^3*c + a*b^4*c - b^5*c - 2*a^4*c^2 - 2*a^3*b*c^2 + 4*a^2*b^2*c^2 + 2*a*b^3*c^2 - 2*a^3*c^3 + 2*a^2*b*c^3 + 2*a*b^2*c^3 + 2*b^3*c^3 + a^2*c^4 + a*b*c^4 + a*c^5 - b*c^5) : :

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)

Barycentrics    a*(b - c)*(3*a^6 - 3*a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - a^2*b^4 - 3*a*b^5 + 2*b^6 - 3*a^5*c + 7*a^4*b*c - 2*a^3*b^2*c - 6*a^2*b^3*c + 5*a*b^4*c - b^5*c - 4*a^4*c^2 - 2*a^3*b*c^2 + 10*a^2*b^2*c^2 - 2*a*b^3*c^2 - 2*b^4*c^2 + 6*a^3*c^3 - 6*a^2*b*c^3 - 2*a*b^2*c^3 + 2*b^3*c^3 - a^2*c^4 + 5*a*b*c^4 - 2*b^2*c^4 - 3*a*c^5 - b*c^5 + 2*c^6) : : <
X(68266) = 3 X[1] - 2 X[42757], X[42757] - 3 X[68258]

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)

Barycentrics    a*(b - c)*(3*a^2 - 2*a*b - b^2 - 2*a*c + 2*b*c - c^2)*(a^3 - a^2*b - a*b^2 + b^3 + a*c^2 + b*c^2 - 2*c^3)*(a^3 + a*b^2 - 2*b^3 - a^2*c + b^2*c - a*c^2 + c^3) : :

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}





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

This preamble and centers X(68268)-X(68324) were contributed by César Eliud Lozada, April 11, 2025.

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:

ℛ(x, y, X, Y) = ( a*x + h*y + g )*( y - Y ) - ( h*x + b*y + f )*( x - X ) = 0     (1)

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

Translating the previous result into trilinear coordinates, making 𝒞 = FA*u^2 + FB*v^2 + FC*w^2 + 2*GA*v*w + 2*GB*u*w + 2*GC*u*v = 0 and 𝒩 = U : V : W, equation (1) is:

ℛ(𝒞, 𝒩) = ∑( ((GB-cos(A)*GC-cos(B)*FA)*V - (GC-cos(A)*GB-cos(C)*FA)*W)*u^2
+ ((FB-FC+cos(B)*GB-cos(C)*GC)*U + (cos(B)*GA-GC+cos(C)*FB)*V - (cos(C)*GA-GB+cos(B)*FC)*W)*v*w ) = 0   (2)

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)*GA+a*((cos(B)*FC-GB)*b-(cos(C)*FB-GC)*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.

underbar

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

Barycentrics    a^2*(2*b*c*a^7-(b+c)*(b^2+c^2)*a^6+(b^2+c^2)*(b^2-3*b*c+c^2)*a^5+(b+c)*(3*b^4+3*c^4-(3*b^2-4*b*c+3*c^2)*b*c)*a^4-(3*b^6+3*c^6-(3*b^4+3*c^4-(b^2+c^2)*b*c)*b*c)*a^3-(b+c)*(b^6+c^6-(b^2-c^2)^2*b*c)*a^2+(b^8+c^8-(b^4+c^4+(3*b^2+5*b*c+3*c^2)*b*c)*(b-c)^2*b*c)*a+(b^6-c^6)*(b-c)*b*c) : :

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

Barycentrics    a^2*((b^4-4*b^2*c^2+c^4)*a^8-(b^2+c^2)*(4*b^4-11*b^2*c^2+4*c^4)*a^6+(4*b^8+4*c^8-(9*b^4-4*b^2*c^2+9*c^4)*b^2*c^2)*a^4-(b^2+c^2)*(b^8+c^8-(5*b^4-7*b^2*c^2+5*c^4)*b^2*c^2)*a^2-(b^6-c^6)*(b^2-c^2)*b^2*c^2) : :
X(68269) = 2*X(187)+X(68270) = X(187)+2*X(68271) = X(68270)-4*X(68271)

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

Barycentrics    a^2*((b^4+c^4)*a^8-(b^2+c^2)*(4*b^4-5*b^2*c^2+4*c^4)*a^6+(4*b^8+4*c^8-(3*b^4-4*b^2*c^2+3*c^4)*b^2*c^2)*a^4-(b^2+c^2)*(b^4+c^4+(b^2-b*c-c^2)*b*c)*(b^4+c^4-(b^2+b*c-c^2)*b*c)*a^2-(b^6-c^6)*(b^2-c^2)*b^2*c^2) : :
X(68270) = 2*X(187)-3*X(68269) = 3*X(68269)-4*X(68271)

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

Barycentrics    a^2*(b^2-c^2)^2*(a^8-4*(b^2+c^2)*a^6+2*(2*b^4+b^2*c^2+2*c^4)*a^4-(b^4-c^4)*(b^2-c^2)*a^2-(b^4+b^2*c^2+c^4)*b^2*c^2) : :
X(68271) = X(187)-3*X(68269) = 3*X(2698)+X(64686) = 3*X(14113)-X(15630) = 3*X(31850)-X(64686) = 3*X(68269)+X(68270)

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

Barycentrics    2*a^6-2*(b+c)*a^5-6*(b^2-b*c+c^2)*a^4+(b+c)*(13*b^2-18*b*c+13*c^2)*a^3-(9*b^4+9*c^4-5*(b^2+c^2)*b*c)*a^2-3*(b^4-c^4)*(b-c)*a+(b-c)^2*(5*b^4+5*c^4+b*c*(3*b^2-8*b*c+3*c^2)) : :

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

Barycentrics    a*((b^2-4*b*c+c^2)*a^7+2*(b+c)*b*c*a^6-(3*b^2-4*b*c+3*c^2)*(b-c)^2*a^5-2*(b+c)*(3*b^2-5*b*c+3*c^2)*b*c*a^4+(3*b^6+3*c^6-(9*b^4+9*c^4-2*(9*b^2-11*b*c+9*c^2)*b*c)*b*c)*a^3+(b^2-c^2)*(b-c)*(5*b^2-2*b*c+5*c^2)*b*c*a^2-(b^2-c^2)^2*(b^4+c^4-(3*b^2-7*b*c+3*c^2)*b*c)*a-(b^2-c^2)^3*(b-c)*b*c) : :
X(68273) = 2*X(36)+X(68274) = X(36)+2*X(68275) = X(68274)-4*X(68275)

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

Barycentrics    a*((b^2+c^2)*a^7-2*(b+c)*b*c*a^6-(3*b^2+4*b*c+3*c^2)*(b-c)^2*a^5+2*(b^3+c^3)*b*c*a^4+(3*b^6+3*c^6-(5*b^4+5*c^4-2*(b^2-b*c+c^2)*b*c)*b*c)*a^3+(b^2-c^2)^2*(b+c)*b*c*a^2-(b^2-c^2)^2*(b^4+c^4-(3*b^2-7*b*c+3*c^2)*b*c)*a-(b^2-c^2)^3*(b-c)*b*c) : :
X(68274) = 2*X(36)-3*X(68273) = 3*X(68273)-4*X(68275)

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

Barycentrics    a*(b-c)^2*(a^6-(b+c)*a^5-2*(b^2-b*c+c^2)*a^4+2*(b^2-c^2)*(b-c)*a^3+(b^4+c^4-b*c*(b^2-9*b*c+c^2))*a^2-(b+c)*(b^4+c^4-b*c*(4*b^2-9*b*c+4*c^2))*a-(b^2-c^2)^2*b*c) : :
X(68275) = X(36)-3*X(68273) = X(901)-3*X(34583) = 3*X(14115)-X(15635) = 3*X(68273)+X(68274)

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

Barycentrics    a*(a^8-(b+c)*a^7-2*(b^2-b*c+c^2)*a^6+(b+c)*(2*b^2-b*c+2*c^2)*a^5+(b^4+c^4-(2*b^2-b*c+2*c^2)*b*c)*a^4-(b+c)*(b^4+c^4)*a^3-(b^4+c^4-2*(3*b^2-4*b*c+3*c^2)*b*c)*b*c*a^2+(b^3+c^3)*(b-c)^2*b*c*a+(b^2-c^2)^2*(b-c)^2*b*c) : :

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

Barycentrics    4*a^4-(b+c)*a^3-(5*b^2-2*b*c+5*c^2)*a^2+(b+c)*(b^2+b*c+c^2)*a+(b^2-c^2)^2 : :
X(68277) = 5*X(1)+X(64743) = 5*X(631)+X(6326) = 2*X(1145)+X(3244) = 4*X(1387)-7*X(15808) = X(1387)+2*X(35023) = 2*X(3626)+X(7972) = 2*X(3626)-5*X(64141) = 2*X(3828)+X(64011) = X(5440)+2*X(6681) = X(6265)+2*X(6684) = X(6265)+5*X(38762) = 2*X(6684)-5*X(38762) = X(7972)+5*X(64141) = 2*X(10427)+X(51090) = X(12119)+2*X(19925) = X(12119)+5*X(64008) = 2*X(19925)-5*X(64008) = 2*X(24466)+X(51118) = X(24466)+2*X(67876) = X(51118)-4*X(67876)

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

Barycentrics    2*a^7-3*(b+c)*a^6-2*(2*b^2-3*b*c+2*c^2)*a^5+7*(b^3+c^3)*a^4+(2*b^4+2*c^4-3*b*c*(3*b^2-2*b*c+3*c^2))*a^3-(b^2-c^2)*(b-c)*(5*b^2+b*c+5*c^2)*a^2+3*b*c*(b^2-c^2)^2*a+(b^2-c^2)^3*(b-c) : :
X(68278) = X(4)+3*X(15015) = X(4)-5*X(15017) = X(80)-3*X(10175) = X(80)-5*X(64008) = 5*X(631)-X(1768) = 5*X(1656)-3*X(59419) = 5*X(1656)-X(62354) = 3*X(5587)+X(6224) = 3*X(5587)-7*X(66045) = X(6224)+7*X(66045) = 3*X(10164)-X(12515) = 3*X(10164)-5*X(38762) = 3*X(10175)-5*X(64008) = 3*X(10711)+X(64145) = X(12515)-5*X(38762) = X(12738)+5*X(19862) = X(12738)+3*X(57298) = 3*X(15015)+5*X(15017) = 3*X(51705)-X(64145) = 3*X(59419)-X(62354)

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

Barycentrics    2*a^10-2*(b+c)*a^9-2*(3*b^2-2*b*c+3*c^2)*a^8+2*(b+c)*(2*b^2-b*c+2*c^2)*a^7+(5*b^4+5*c^4-2*(3*b^2-4*b*c+3*c^2)*b*c)*a^6-4*(b+c)*b^2*c^2*a^5+(b^4+c^4+2*(b^2-b*c+c^2)*b*c)*(b-c)^2*a^4-2*(b^2-c^2)*(b-c)*(2*b^4+2*c^4+(b^2+c^2)*b*c)*a^3-(b^2-c^2)^2*(3*b^4+3*c^4-2*(b^2+c^2)*b*c)*a^2+2*(b^4-c^4)*(b^2-c^2)^2*(b-c)*a+(b^4-c^4)*(b^2-c^2)^3 : :

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

Barycentrics    2*a^10-10*(b^2+c^2)*a^8+(13*b^4+8*b^2*c^2+13*c^4)*a^6+(b^2+c^2)*(b^4-10*b^2*c^2+c^4)*a^4-(b^2-c^2)^2*(11*b^4-4*b^2*c^2+11*c^4)*a^2+5*(b^4-c^4)*(b^2-c^2)^3 : :
X(68280) = X(4)+2*X(48378) = 4*X(140)-X(37853) = 2*X(140)+X(46686) = X(146)+5*X(38729) = X(265)+5*X(38795) = 3*X(381)+X(38723) = 4*X(3628)-X(6699) = 8*X(3628)+X(38791) = 2*X(3628)+X(61574) = 7*X(3851)-X(12295) = 5*X(5071)+X(5642) = 5*X(5071)-X(14644) = 2*X(6699)+X(38791) = X(6699)+2*X(61574) = X(12295)+5*X(38794) = 7*X(15036)-10*X(48378) = X(16534)+2*X(20304) = X(16534)+8*X(35018) = X(20304)-4*X(35018) = X(37853)+2*X(46686)

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

Barycentrics    2*a^12-8*(b^2+c^2)*a^10+(7*b^4+12*b^2*c^2+7*c^4)*a^8+6*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^6-2*(b^2-c^2)^2*(5*b^4+b^2*c^2+5*c^4)*a^4+2*(b^4-c^4)*(b^2-c^2)*(b^4-3*b^2*c^2+c^4)*a^2+(b^4-c^4)^2*(b^2-c^2)^2 : :
X(68281) = 2*X(6)+X(7687) = 2*X(576)+X(32257) = X(1351)+2*X(6723) = X(5622)+3*X(14853) = X(5972)-4*X(18583) = 4*X(15118)-X(20417) = X(32246)+2*X(44495)

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

Barycentrics    (a-b)*(a-c)*(2*a^10+2*(b+c)*a^9-2*(b^2-b*c+c^2)*a^8-(b+c)*(3*b^2-2*b*c+3*c^2)*a^7-(b^4+c^4+2*(b-c)^2*b*c)*a^6+(b+c)*(b^4+c^4-2*(b-c)^2*b*c)*a^5-(b^6+c^6-(b+c)^2*b^2*c^2)*a^4-(b^2-c^2)*(b-c)*(b^4+c^4+b*c*(4*b^2+7*b*c+4*c^2))*a^3+(b^2-c^2)^2*(3*b^4+3*c^4-b*c*(2*b^2+7*b*c+2*c^2))*a^2+(b^2-c^2)^3*(b-c)*(b^2+4*b*c+c^2)*a-(b^2-c^2)^4*(b-c)^2) : :
X(68282) = 3*X(3109)-4*X(68306) = 3*X(66789)-2*X(68306)

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

Barycentrics    a^2*(a^2-b^2)*(a^2-c^2)*(a^8-(2*b^4-b^2*c^2+2*c^4)*a^4+b^2*c^2*(b^2+c^2)*a^2+(b^2-c^2)^2*(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2)) : :

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

Barycentrics    4*a^12-9*(b^2+c^2)*a^10+(31*b^4-68*b^2*c^2+31*c^4)*a^8+(b^2+c^2)*(6*b^4+23*b^2*c^2+6*c^4)*a^6-6*(5*b^8+5*c^8+b^2*c^2*(15*b^4-31*b^2*c^2+15*c^4))*a^4+(b^2+c^2)*(7*b^8+7*c^8+5*b^2*c^2*(3*b^2-5*c^2)*(5*b^2-3*c^2))*a^2-(b^2+c^2)^2*(b^8+c^8+2*b^2*c^2*(4*b^4-11*b^2*c^2+4*c^4)) : :
X(68284) = 3*X(62293)-X(68285) = 3*X(62293)-2*X(68286)

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

Barycentrics    12*a^12-39*(b^2+c^2)*a^10+(95*b^4-52*b^2*c^2+95*c^4)*a^8-(b^2+c^2)*(16*b^4+b^2*c^2+16*c^4)*a^6-6*(18*b^8+18*c^8-31*b^2*c^2*(b^4-b^2*c^2+c^4))*a^4+(b^2+c^2)*(47*b^8+47*c^8-b^2*c^2*(69*b^4-56*b^2*c^2+69*c^4))*a^2-(b^4-c^4)^2*(7*b^4-4*b^2*c^2+7*c^4) : :
X(68285) = 3*X(62293)-2*X(68284) = 3*X(62293)-4*X(68286)

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

Barycentrics    4*a^12-15*(b^2+c^2)*a^10+8*(4*b^4+b^2*c^2+4*c^4)*a^8-(b^2+c^2)*(11*b^4+12*b^2*c^2+11*c^4)*a^6-3*(13*b^8+13*c^8-2*b^2*c^2*(23*b^4-31*b^2*c^2+23*c^4))*a^4+(b^2+c^2)*(20*b^8+20*c^8-b^2*c^2*(72*b^4-113*b^2*c^2+72*c^4))*a^2-(b^2+c^2)^2*(3*b^8+3*c^8-b^2*c^2*(13*b^4-22*b^2*c^2+13*c^4)) : :
X(68286) = 3*X(62293)-X(68284) = 3*X(62293)+X(68285)

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

Barycentrics    a^2*((b^2+c^2)*a^5-(b+c)*(b^2+4*b*c+c^2)*a^4-2*(b^4+c^4-b*c*(7*b^2+b*c+7*c^2))*a^3+2*(b+c)*(b^4+c^4-b*c*(b^2+6*b*c+c^2))*a^2+(b^6+c^6-(14*b^4+14*c^4-b*c*(41*b^2-48*b*c+41*c^2))*b*c)*a-(b^2-c^2)*(b-c)^5) : :

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

Barycentrics    a^2*((b^2+c^2)*a^5-(b+c)*(b^2+3*b*c+c^2)*a^4-2*(b^4+c^4-3*b*c*(2*b^2-b*c+2*c^2))*a^3+(b+c)*(2*b^4+2*c^4-b*c*(2*b^2-b*c+2*c^2))*a^2+(b^6+c^6-3*(4*b^4+4*c^4-b*c*(13*b^2-24*b*c+13*c^2))*b*c)*a-(b+c)*(b^6+c^6-(5*b^4+5*c^4-2*b*c*(7*b^2-12*b*c+7*c^2))*b*c)) : :
X(68288) = X(5510)+2*X(68289) = X(5510)-4*X(68290) = X(68289)+2*X(68290)

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

Barycentrics    a^2*((b^2+c^2)*a^8-(b+c)*(2*b^2+3*b*c+2*c^2)*a^7-(2*b^4+2*c^4-19*(b^2+c^2)*b*c)*a^6+(b+c)*(6*b^4+6*c^4-5*b*c*(3*b^2-b*c+3*c^2))*a^5-b*c*(3*b^2-2*b*c+3*c^2)*(11*b^2-17*b*c+11*c^2)*a^4-(b+c)*(2*b^2-9*b*c+2*c^2)*(3*b^4+3*c^4-2*b*c*(3*b^2-5*b*c+3*c^2))*a^3+(2*b^2-3*b*c+2*c^2)*(b^6+c^6+(6*b^4+6*c^4-5*b*c*(5*b^2-4*b*c+5*c^2))*b*c)*a^2+(b^2-c^2)*(b-c)*(2*b^6+2*c^6-b*c*(17*b^2-7*b*c+17*c^2)*(b-c)^2)*a+(b-c)^2*(b^2-c^2)^2*(-b^4-c^4+3*b*c*(b^2-b*c+c^2))) : :
X(68289) = X(5510)-3*X(68288) = 3*X(68288)-2*X(68290)

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

Barycentrics    a^2*((b^2+c^2)*a^8-(b+c)*(2*b^2+3*b*c+2*c^2)*a^7-(2*b^4+2*c^4-19*(b^2+c^2)*b*c)*a^6+(b+c)*(6*b^4+6*c^4-5*b*c*(3*b^2-b*c+3*c^2))*a^5-b*c*(3*b^2-2*b*c+3*c^2)*(11*b^2-17*b*c+11*c^2)*a^4-(b+c)*(6*b^6+6*c^6-(39*b^4+39*c^4-2*b*c*(52*b^2-81*b*c+52*c^2))*b*c)*a^3+(2*b^8+2*c^8+(9*b^6+9*c^6-(58*b^4+58*c^4-b*c*(199*b^2-320*b*c+199*c^2))*b*c)*b*c)*a^2+(b^2-c^2)*(b-c)*(2*b^6+2*c^6-(17*b^4+17*c^4-b*c*(65*b^2-144*b*c+65*c^2))*b*c)*a-(b^2-c^2)^2*(b^6+c^6-(5*b^4+5*c^4-18*b*c*(b-c)^2)*b*c)) : :
X(68290) = X(5510)+3*X(68288) = 3*X(68288)-X(68289)

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

Barycentrics    a*((b+c)*a^7-(b^2-4*b*c+c^2)*a^6-(b+c)*(b^2+b*c+c^2)*a^5+(b^4+c^4-4*b*c*(b^2-b*c+c^2))*a^4-(b+c)*(b^4-4*b^2*c^2+c^4)*a^3+(b-c)^2*(b^4-4*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b-c)*(b^2+3*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2*(b^2+c^2)) : :
X(68291) = 3*X(354)-X(15904) = 3*X(354)+X(65524)

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

Barycentrics    a^2*((b^2+c^2)*a^8-2*(b^4-4*b^2*c^2+c^4)*a^6-7*b^2*c^2*(b^2+c^2)*a^4+2*(b^8+c^8-b^2*c^2*(5*b^4-12*b^2*c^2+5*c^4))*a^2-(b^4-c^4)*(b^2-c^2)*((b^2+c^2)^2-9*b^2*c^2)) : :
X(68292) = 3*X(2)+X(12824) = 3*X(373)+X(5642) = 3*X(373)-X(12099) = X(974)-3*X(15045) = X(974)+5*X(64101) = 5*X(5943)-X(11800) = 2*X(5972)+X(11746) = 5*X(5972)+X(11800) = X(5972)+2*X(68293) = 7*X(6723)+2*X(13402) = X(9826)+2*X(12900) = 7*X(10219)+X(13402) = 5*X(11746)-2*X(11800) = X(11746)-4*X(68293) = X(11800)-10*X(68293) = X(12824)-3*X(41670) = 3*X(15045)+5*X(64101) = X(32246)+5*X(64764) = 3*X(36518)+X(64100) = X(45311)-3*X(63632)

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

Barycentrics    a^2*((b^2+c^2)*a^8-2*(b^2-c^2)^2*a^6-3*b^2*c^2*(b^2+c^2)*a^4+2*(b^8+c^8-4*(b^2-c^2)^2*b^2*c^2)*a^2-(b^4-c^4)*(b^2-c^2)*(b^4-5*b^2*c^2+c^4)) : :
X(68293) = 3*X(2)+X(1112) = 5*X(1656)-X(12358) = 5*X(1656)+3*X(16222) = 3*X(5892)+X(46686) = 3*X(5943)+X(5972) = 3*X(5943)-X(11746) = 9*X(5943)-X(11800) = 3*X(6688)-X(6723) = 3*X(6688)+X(41671) = 5*X(6688)-X(44321) = 5*X(6723)-3*X(44321) = X(12236)-5*X(15026) = X(12284)-9*X(46430) = X(12358)+3*X(16222) = 3*X(12824)+5*X(15059) = 3*X(12824)+X(54376) = 11*X(15024)-3*X(46430) = 5*X(15059)-X(54376) = 5*X(41671)+3*X(44321) = 3*X(46430)+5*X(64101)

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

Barycentrics    (b^2+c^2)*a^13-2*b*c*(b+c)*a^12-(3*b^4+3*c^4-2*(b^2+c^2)*b*c)*a^11-(b+c)*(b^4+c^4-b*c*(5*b^2-4*b*c+5*c^2))*a^10+(2*b^6+2*c^6-(2*b^4+2*c^4-b*c*(3*b^2-8*b*c+3*c^2))*b*c)*a^9+(b^3+c^3)*(3*b^4+3*c^4-2*b*c*(b+c)^2)*a^8+2*(b^8+c^8-(2*b^6+2*c^6+(2*b^4+2*c^4-b*c*(5*b^2-3*b*c+5*c^2))*b*c)*b*c)*a^7-(b^2-c^2)*(b-c)*(2*b^6+2*c^6+(2*b^4+2*c^4-b*c*(b^2-5*b*c+c^2))*b*c)*a^6-(b-c)^2*(3*b^8+3*c^8+(2*b^4+2*c^4+b*c*(7*b^2+12*b*c+7*c^2))*(b-c)^2*b*c)*a^5-2*(b^2-c^2)*(b-c)*(b^8+c^8-2*(2*b^4+2*c^4+b*c*(b^2-3*b*c+c^2))*b^2*c^2)*a^4+(b^2-c^2)^2*(b-c)^2*(b^6+c^6+(4*b^4+4*c^4+b*c*(7*b^2+2*b*c+7*c^2))*b*c)*a^3+(b^2-c^2)^3*(b-c)*(3*b^6+3*c^6-(b^4+c^4+b*c*(4*b^2-b*c+4*c^2))*b*c)*a^2-2*(b^4-c^4)*(b^2-c^2)^3*b*c*(b-c)^2*a-(b-c)^2*(b^2-c^2)^4*(b^2+c^2)*(b^3+c^3) : :

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

Barycentrics    (b^2+c^2)*a^16-6*b^2*c^2*a^14-(b^2+c^2)*(10*b^4-23*b^2*c^2+10*c^4)*a^12+(16*b^8+16*c^8-(9*b^4+16*b^2*c^2+9*c^4)*b^2*c^2)*a^10-18*(b^4-c^4)*(b^2-c^2)*b^2*c^2*a^8-2*(b^2-c^2)^2*(8*b^8+8*c^8-(5*b^4-2*b^2*c^2+5*c^4)*b^2*c^2)*a^6+(b^4-c^4)*(b^2-c^2)*(10*b^8+10*c^8-3*(5*b^4-6*b^2*c^2+5*c^4)*b^2*c^2)*a^4-3*b^2*c^2*(b^2-c^2)^4*(b^4+c^4)*a^2-(b^2-c^2)^6*(b^2+c^2)*(b^4+c^4) : :

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

Barycentrics    (-a+b+c)*(2*a^5-2*(b+c)*a^4-(b^2-4*b*c+c^2)*a^3+(b^2-c^2)*(b-c)*a^2-(b-c)^2*(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c)^3) : :
X(68296) = X(11)+3*X(3058) = X(3035)-3*X(49736) = X(5083)-3*X(64162) = 3*X(11113)+X(25416) = X(14740)-3*X(40998) = 3*X(40998)-2*X(68298)

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

Barycentrics    a*(-a+b+c)*((b+c)*a^6-2*(b-c)^2*a^5-(b+c)*(b^2-3*b*c+c^2)*a^4+4*(b^4+c^4-b*c*(3*b^2-b*c+3*c^2))*a^3-(b+c)*(b^4+c^4+2*b*c*(b^2-5*b*c+c^2))*a^2-2*(b-c)^2*(b^4+c^4-2*b*c*(b^2+5*b*c+c^2))*a+(b^2-c^2)*(b-c)^3*(b^2+3*b*c+c^2)) : :
X(68297) = 2*X(68298)+X(68299) = X(68298)+2*X(68300) = X(68299)-4*X(68300)

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

Barycentrics    (-a+b+c)*(2*a^5-3*(b^2+c^2)*a^3-(b+c)*(b^2-4*b*c+c^2)*a^2+(b-c)^2*(b^2+6*b*c+c^2)*a+(b^2-c^2)*(b-c)^3) : :
X(68298) = X(14740)+3*X(40998) = 3*X(40998)-X(68296) = 3*X(68297)-X(68299) = 3*X(68297)-2*X(68300)

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

Barycentrics    a*(-a+b+c)*((b+c)*a^6-2*(b^2+c^2)*a^5-(b+c)*(b^2-3*b*c+c^2)*a^4+2*(b-c)^2*(2*b^2+b*c+2*c^2)*a^3-(b^2-c^2)^2*(b+c)*a^2-2*(b^2-c^2)^2*(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c)^2*(b^3-c^3)) : :
X(68299) = 3*X(68297)-2*X(68298) = 3*X(68297)-4*X(68300)

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

Barycentrics    a*(-a+b+c)*((b+c)*a^6-2*(b^2-b*c+c^2)*a^5-(b+c)*(b^2-3*b*c+c^2)*a^4+(4*b^4+4*c^4-b*c*(9*b^2-4*b*c+9*c^2))*a^3-(b+c)*(b^4+c^4+b*c*(b^2-6*b*c+c^2))*a^2-(b^2-3*b*c-2*c^2)*(2*b^2+3*b*c-c^2)*(b-c)^2*a+(b^2-c^2)^3*(b-c)) : :
X(68300) = 3*X(68297)-X(68298) = 3*X(68297)+X(68299)

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

Barycentrics    (-a+b+c)*(2*a^7+2*(b+c)*a^6-(3*b^2+8*b*c+3*c^2)*a^5-(b-3*c)*(3*b-c)*(b+c)*a^4+16*b*(b-c)^2*c*a^3+(b^4-c^4)*(b^2-c^2)*a+(b^2-c^2)^3*(b-c)) : :

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

Barycentrics    (a^2-b^2)*(a^2-c^2)*(3*a^16-3*(b^2+c^2)*a^14-(2*b^4-7*b^2*c^2+2*c^4)*a^12-5*(b^4-c^4)*(b^2-c^2)*a^10+2*(b^2-c^2)^2*(5*b^4+9*b^2*c^2+5*c^4)*a^8+(b^4-c^4)*(b^2-c^2)*(3*b^4-22*b^2*c^2+3*c^4)*a^6-(b^2-c^2)^4*(10*b^4+13*b^2*c^2+10*c^4)*a^4+(b^4-c^4)*(b^2-c^2)^3*(5*b^4-2*b^2*c^2+5*c^4)*a^2-(-4*b^2*c^2+(b^2-c^2)^2)*(b^4-c^4)^2*(b^2-c^2)^2) : :
X(68302) = X(2867)-4*X(68303) = X(2867)+2*X(68304) = X(2867)+8*X(68305) = 2*X(68303)+X(68304) = X(68303)+2*X(68305) = X(68304)-4*X(68305)

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

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^16-(b^2+c^2)*a^14-(b^4-3*b^2*c^2+c^4)*a^12-(b^4-c^4)*(b^2-c^2)*a^10+(b^2-c^2)^2*(3*b^4+5*b^2*c^2+3*c^4)*a^8+(b^4-c^4)*(b^2-c^2)*(b^2-3*b*c+c^2)*(b^2+3*b*c+c^2)*a^6-(b^2-c^2)^4*(3*b^4+4*b^2*c^2+3*c^4)*a^4+(b^2-c^2)^4*(b^6+c^6)*a^2+2*(b^4-c^4)^2*(b^2-c^2)^2*b^2*c^2) : :
X(68303) = X(2867)+3*X(68302) = X(2867)+2*X(68305) = 3*X(68302)-X(68304) = 3*X(68302)-2*X(68305)

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

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^12-(b^2+c^2)*a^10+(b^4-b^2*c^2+c^4)*a^8-4*(b^4-c^4)*(b^2-c^2)*a^6+(b^2-c^2)^2*(5*b^4+7*b^2*c^2+5*c^4)*a^4-3*(b^8-c^8)*a^2*(b^2-c^2)+(b^4-c^4)^2*(b^2-c^2)^2) : :
X(68304) = 3*X(112)-4*X(15639) = X(2867)-3*X(68302) = X(2867)-4*X(68305) = 3*X(6794)-2*X(33504) = 3*X(68302)-2*X(68303) = 3*X(68302)-4*X(68305)

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

Barycentrics    (a^2-b^2)*(a^2-c^2)*(2*a^16-2*(b^2+c^2)*a^14-(b^4-4*b^2*c^2+c^4)*a^12-4*(b^4-c^4)*(b^2-c^2)*a^10+(b^2-c^2)^2*(7*b^4+13*b^2*c^2+7*c^4)*a^8+(b^4-c^4)*(b^2-c^2)*(2*b^4-15*b^2*c^2+2*c^4)*a^6-(b^2-c^2)^4*(7*b^4+9*b^2*c^2+7*c^4)*a^4+(b^4-c^4)*(b^2-c^2)^3*(2*b^2-3*b*c+2*c^2)*(2*b^2+3*b*c+2*c^2)*a^2-(b^4-c^4)^2*(b^2-c^2)^2*(b^4-4*b^2*c^2+c^4)) : :
X(68305) = X(2867)-9*X(68302) = X(2867)-3*X(68303) = X(2867)+3*X(68304) = 3*X(68302)-X(68303) = 3*X(68302)+X(68304)

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

Barycentrics    4*a^13+(b+c)*a^12-8*(b^2+c^2)*a^11-(b+c)^3*a^10-(b^4-26*b^2*c^2+c^4)*a^9-(b+c)*(b^2+b*c+c^2)*(b^2-4*b*c+c^2)*a^8+(b^2+c^2)*(9*b^4-26*b^2*c^2+9*c^4)*a^7-(b+c)*(b^6+c^6-2*(b^4+c^4+b*c*(b^2-5*b*c+c^2))*b*c)*a^6-(b^8+c^8+b^2*c^2*(17*b^4-40*b^2*c^2+17*c^4))*a^5+(b+c)*(4*b^8+4*c^8-(4*b^6+4*c^6+(7*b^4+7*c^4-5*b*c*(b^2+b*c+c^2))*b*c)*b*c)*a^4-(b^4-c^4)*(b^2-c^2)*(5*b^4-14*b^2*c^2+5*c^4)*a^3-(b^2-c^2)^2*(b+c)*(2*b^6+2*c^6+b^2*c^2*(b^2-5*b*c+c^2))*a^2+(b^2-c^2)^4*(b^2+2*c^2)*(2*b^2+c^2)*a+(b^2-c^2)^3*(b+c)^2*b*c*(b^3-c^3) : :
X(68306) = 3*X(3109)+X(68282) = 3*X(66789)-X(68282)

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

Barycentrics    4*a^16-13*(b^2+c^2)*a^14+(9*b^4+46*b^2*c^2+9*c^4)*a^12+2*(b^2+c^2)*(5*b^4-29*b^2*c^2+5*c^4)*a^10-2*(5*b^8+5*c^8+b^2*c^2*(5*b^4-42*b^2*c^2+5*c^4))*a^8-(b^2+c^2)*(9*b^8+9*c^8-b^2*c^2*(61*b^4-109*b^2*c^2+61*c^4))*a^6+(b^2-c^2)^2*(13*b^8+13*c^8-b^2*c^2*(7*b^4+27*b^2*c^2+7*c^4))*a^4-(b^4-c^4)*(b^2-c^2)^3*(4*(b^2+c^2)^2-b^2*c^2)*a^2+(b^2-c^2)^6*b^2*c^2 : :
X(68307) = 3*X(3)+X(476) = 5*X(3)-X(477) = 7*X(3)+X(38580) = 9*X(3)-X(38581) = 3*X(3)-X(38610) = X(3)+3*X(38700) = 7*X(3)-3*X(38701) = X(20)+3*X(57305) = 5*X(631)-X(20957) = 15*X(631)-7*X(66819) = 3*X(1511)-X(14611) = X(1553)-3*X(64652) = 7*X(3523)-3*X(57306) = 7*X(3523)+X(66792) = 9*X(5054)-X(66791) = 9*X(5054)-5*X(66801) = X(14611)+3*X(46632) = X(14677)+3*X(64652) = 3*X(20957)-7*X(66819) = 3*X(22104)-X(68308)

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

Barycentrics    2*a^16-2*(b^2+c^2)*a^14-4*(3*b^4-5*b^2*c^2+3*c^4)*a^12+(b^2+c^2)*(23*b^4-38*b^2*c^2+23*c^4)*a^10-(5*b^8+5*c^8+2*b^2*c^2*(25*b^4-48*b^2*c^2+25*c^4))*a^8-(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*(12*b^4-23*b^2*c^2+12*c^4)*a^6+(b^2-c^2)^2*(2*b^8+2*c^8+b^2*c^2*(19*b^4-54*b^2*c^2+19*c^4))*a^4+(b^4-c^4)*(b^2-c^2)^3*(7*b^4-17*b^2*c^2+7*c^4)*a^2-(3*b^4+7*b^2*c^2+3*c^4)*(b^2-c^2)^6 : :
X(68308) = 3*X(113)-X(14611) = X(477)-5*X(3091) = 3*X(477)-7*X(66819) = 15*X(3091)-7*X(66819) = X(3146)+3*X(38700) = X(3146)+7*X(66815) = 9*X(3545)-X(66773) = 9*X(3545)-5*X(66801) = 9*X(3839)-X(14731) = 3*X(3839)+X(66786) = 3*X(3845)+X(18319) = 3*X(3845)-X(66795) = 7*X(3851)-3*X(57306) = 7*X(3851)+X(66772) = X(14611)+3*X(34150) = X(14731)+3*X(66786) = 3*X(31378)-5*X(64101) = 3*X(38700)-7*X(66815) = 3*X(57306)+X(66772) = X(66773)-5*X(66801)

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

Barycentrics    2*a^16-11*(b^2+c^2)*a^14+(21*b^4+26*b^2*c^2+21*c^4)*a^12-(b^2+c^2)*(13*b^4+20*b^2*c^2+13*c^4)*a^10-(5*b^8+5*c^8-4*b^2*c^2*(10*b^4-3*b^2*c^2+10*c^4))*a^8+(b^2+c^2)*(3*b^8+3*c^8-b^2*c^2*(10*b^4-7*b^2*c^2+10*c^4))*a^6+(b^2-c^2)^2*(11*b^8+11*c^8-b^2*c^2*(26*b^4-27*b^2*c^2+26*c^4))*a^4-(b^4-c^4)*(b^2-c^2)^3*(11*b^4-10*b^2*c^2+11*c^4)*a^2+(3*b^4+8*b^2*c^2+3*c^4)*(b^2-c^2)^6 : :
X(68309) = 3*X(2)+X(18319) = 9*X(2)-X(38581) = 3*X(5)+X(476) = 5*X(5)-X(20957) = X(5)+3*X(57305) = X(5)-5*X(66787) = 3*X(140)-X(38610) = 5*X(476)+3*X(20957) = X(476)-9*X(57305) = X(7471)+3*X(21315) = 3*X(10272)-X(14611) = X(11749)-9*X(15699) = X(11749)-5*X(66801) = X(11801)-3*X(21315) = X(14611)+3*X(34209) = 9*X(15699)-5*X(66801) = 3*X(18319)+X(38581) = 3*X(22104)-X(68307) = 3*X(22104)+X(68308) = 3*X(25641)+X(38610)

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

Barycentrics    4*a^14-7*(b^2+c^2)*a^12-2*(b^4-10*b^2*c^2+c^4)*a^10+2*(b^2+c^2)*(2*b^2+b*c-2*c^2)*(2*b^2-b*c-2*c^2)*a^8-2*(b^8+c^8+6*(b^2-c^2)^2*b^2*c^2)*a^6-(b^2+c^2)*(b^8+c^8-3*b^2*c^2*(3*b^4-5*b^2*c^2+3*c^4))*a^4-2*(b^2-c^2)^2*b^2*c^2*(b^4-b^2*c^2+c^4)*a^2+3*(b^4-c^4)*(b^2-c^2)^3*b^2*c^2 : :

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

Barycentrics    a^2*((b^2+c^2)*a^6-(b+c)*(2*b^2-b*c+2*c^2)*a^5+(b^4+c^4+b*c*(b^2+10*b*c+c^2))*a^4+b*c*(b+c)*(2*b^2-13*b*c+2*c^2)*a^3-(b^6+c^6+2*(b^4+c^4-b*c*(b^2+5*b*c+c^2))*b*c)*a^2+(b^2-c^2)*(b-c)*(2*b^4+2*c^4+b*c*(b^2-5*b*c+c^2))*a-(b^6+c^6+(b^4+c^4-2*b*c*(3*b^2+4*b*c+3*c^2))*b*c)*(b-c)^2) : :
X(68311) = 2*X(6710)+X(68312) = X(6710)+2*X(68313) = X(68312)-4*X(68313)

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

Barycentrics    a^2*((b^2+c^2)*a^6-(b+c)*(2*b^2-b*c+2*c^2)*a^5+(b^2+c^2)*(b^2+b*c+c^2)*a^4+b*c*(2*b-c)*(b-2*c)*(b+c)*a^3-(b^3-c^3)*(b-c)*(b^2+3*b*c+c^2)*a^2+(b^2-c^2)*(b-c)*(2*b^4+2*c^4+b*c*(b^2-b*c+c^2))*a-(b^2-c^2)^2*(b^4+c^4-b*c*(b^2+b*c+c^2))) : :
X(68312) = 2*X(6710)-3*X(68311) = 3*X(68311)-4*X(68313)

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

Barycentrics    a^2*((b^2+c^2)*a^6-(b+c)*(2*b^2-b*c+2*c^2)*a^5+(b^4+c^4+b*c*(b^2+6*b*c+c^2))*a^4+b*c*(b+c)*(2*b^2-9*b*c+2*c^2)*a^3-(b^4-3*b^2*c^2+c^4)*(b+c)^2*a^2+(b^2-c^2)*(b-c)*(2*b^4+2*c^4+b*c*(b^2-3*b*c+c^2))*a-(b^6+c^6+(b^4+c^4-2*b*c*(2*b^2+3*b*c+2*c^2))*b*c)*(b-c)^2) : :
X(68313) = X(6710)-3*X(68311) = 3*X(68311)+X(68312)

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

Barycentrics    a*(4*a^5-5*(b+c)*a^4+2*(b+c)^2*a^3-(b+c)*(4*b^2-7*b*c+4*c^2)*a^2+2*(b^3-c^3)*(b-c)*a+(b^2-c^2)*(b^3-c^3)) : :

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

Barycentrics    2*a^6-2*(b^2+c^2)*a^4+(3*b^4-4*b^2*c^2+3*c^4)*a^2+(b^4-c^4)*(b^2-c^2) : :
X(68315) = X(99)-5*X(3618) = X(114)-3*X(14561) = X(148)+3*X(5182) = X(148)+7*X(51171) = 2*X(575)+X(38734) = X(576)+2*X(20398) = X(671)+3*X(59373) = X(2482)-3*X(47352) = 3*X(5085)-X(38738) = 3*X(5182)-X(14928) = 3*X(5182)-7*X(51171) = 2*X(6721)-3*X(38317) = X(8593)+3*X(41135) = X(8593)-5*X(63127) = X(14928)-7*X(51171) = X(18800)-3*X(59373) = 3*X(34473)+X(51212) = 3*X(36696)+X(67224) = 3*X(41135)+5*X(63127) = X(52472)+3*X(65616)

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

Barycentrics    10*a^10-24*(b^2+c^2)*a^8+11*(b^4+4*b^2*c^2+c^4)*a^6+(b^2+c^2)*(11*b^4-36*b^2*c^2+11*c^4)*a^4-(b^2-c^2)^2*(9*b^4+10*b^2*c^2+9*c^4)*a^2+(b^4-c^4)*(b^2-c^2)^3 : :
X(68316) = X(74)-5*X(15692) = X(113)+5*X(15051) = X(376)-5*X(15051) = X(3830)-3*X(36518) = X(3830)+3*X(38723) = X(6053)-4*X(13392) = X(6053)+4*X(14891) = 2*X(6723)-3*X(11539) = 2*X(6723)+X(34153) = 2*X(10272)+X(37853) = 3*X(11539)+X(34153) = X(12041)-3*X(17504) = X(12041)+5*X(22251) = 5*X(14093)-X(20127) = 3*X(15061)-7*X(15701) = X(15061)+3*X(38638) = 5*X(15692)+X(56567) = 9*X(15707)+X(24981) = 9*X(15707)-5*X(38728) = 3*X(17504)+5*X(22251)

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

Barycentrics    2*a^10+6*(b^2+c^2)*a^8-(23*b^4-16*b^2*c^2+23*c^4)*a^6+(b^2+c^2)*(13*b^4-18*b^2*c^2+13*c^4)*a^4+(b^2-c^2)^2*(9*b^4-20*b^2*c^2+9*c^4)*a^2-7*(b^4-c^4)*(b^2-c^2)^3 : :
X(68317) = X(125)-3*X(3545) = X(146)+7*X(61936) = X(146)+2*X(65092) = 5*X(3091)-X(9140) = 5*X(3091)+X(15063) = X(3534)-9*X(15046) = X(3534)-3*X(38793) = 3*X(3545)+X(10706) = X(3830)+3*X(14643) = 3*X(5055)-2*X(6723) = 3*X(5055)+X(7728) = 2*X(6723)+X(7728) = 3*X(11693)-X(12121) = X(12121)+3*X(38335) = X(12295)-3*X(14269) = 7*X(14643)-3*X(38638) = 3*X(15046)-X(38793) = X(32257)+2*X(32271) = X(51023)+3*X(52699) = 7*X(61936)-2*X(65092)

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

Barycentrics    10*a^8-6*(b^2+c^2)*a^6-(11*b^4-16*b^2*c^2+11*c^4)*a^4+2*(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^2+(b^4-c^4)^2 : :
X(68318) = 3*X(6)+X(5648) = 5*X(6)+X(32114) = 5*X(125)+X(25336) = X(125)-3*X(47352) = X(599)-5*X(64764) = 5*X(3618)-X(9140) = 5*X(3618)+X(56565) = X(5095)+5*X(64764) = 3*X(5642)-X(5648) = 5*X(5642)-X(32114) = X(6699)+2*X(25556) = 2*X(6723)+X(25329) = 2*X(6723)-3*X(48310) = 3*X(15462)+X(20423) = X(25329)+3*X(48310) = X(25336)-5*X(34319) = X(32257)+2*X(41595) = 3*X(34155)+X(50977) = X(34319)+3*X(47352) = X(51023)+3*X(66740)

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

Barycentrics    2*a^10-7*(b^2+c^2)*a^8+2*(3*b^4+2*b^2*c^2+3*c^4)*a^6+2*(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^4-2*(b^2-c^2)^2*(2*b^2-3*b*c+2*c^2)*(2*b^2+3*b*c+2*c^2)*a^2+3*(b^4-c^4)*(b^2-c^2)^3 : :
X(68319) = 9*X(2)-X(7464) = 3*X(2)+X(11799) = X(3)+3*X(403) = 7*X(3)-3*X(16386) = 5*X(3)+3*X(31726) = X(3)-3*X(44452) = 3*X(3)+X(62288) = X(3292)-5*X(38795) = X(3292)+3*X(63735) = X(3580)+3*X(14643) = 3*X(3580)+X(63720) = X(5609)-3*X(51425) = X(5609)+3*X(63839) = X(10222)-3*X(51713) = 9*X(14643)-X(63720) = 5*X(15034)+3*X(50435) = 5*X(15034)-9*X(59648) = X(32110)+3*X(36518) = 3*X(36518)-X(58885) = 5*X(38795)+3*X(63735)

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

Barycentrics    (b+c)*a^9+2*(b^2-b*c+c^2)*a^8-(b+c)*(b^2+c^2)*a^7-2*(2*b^4+2*c^4-(b^2+c^2)*b*c)*a^6-(b+c)*(3*b^4+3*c^4-(3*b^2+b*c+3*c^2)*b*c)*a^5+(3*b^4+3*c^4+2*(b^2-4*b*c+c^2)*b*c)*b*c*a^4+(b^3+c^3)*(b-c)^2*(5*b^2+9*b*c+5*c^2)*a^3+2*(b^2-c^2)^2*(2*b^4+2*c^4-b*c*(2*b^2+b*c+2*c^2))*a^2-(b^2-c^2)^3*(b-c)*(2*b^2+b*c+2*c^2)*a-(b^2-c^2)^4*(2*b^2-b*c+2*c^2) : :

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

Barycentrics    2*(b^2+c^2)*a^10-2*(b^4+b^2*c^2+c^4)*a^8-5*(b^4-c^4)*(b^2-c^2)*a^6+(7*b^8+7*c^8-2*b^2*c^2*(5*b^4-4*b^2*c^2+5*c^4))*a^4-(b^6+c^6)*(b^2-c^2)^2*a^2-(b^6-c^6)*(b^2-c^2)^3 : :
X(68321) = X(1316)-3*X(5476) = X(1316)+3*X(16279)

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

Barycentrics    (a-b)*(a-c)*(3*a^6-3*(b+c)*a^5+(2*b^2-b*c+2*c^2)*a^4-2*(b^2-c^2)*(b-c)*a^3-(3*b^2-2*b*c+3*c^2)*(b-c)^2*a^2+(b^2-c^2)*(b-c)^3*a+(b+2*c)*(2*b+c)*(b-c)^4) : :
X(68322) = X(927)-4*X(68323) = X(927)+2*X(68324) = 4*X(34805)-X(60065) = 2*X(68323)+X(68324)

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

Barycentrics    (a-b)*(a-c)*(2*a^6-2*(b+c)*a^5+(b^2+c^2)*a^4-(b^2-c^2)*(b-c)*a^3-(b-c)^2*(2*b^2-b*c+2*c^2)*a^2+(b^2-c^2)*(b-c)^3*a+(b-c)^4*(b^2+3*b*c+c^2)) : :
X(68323) = X(927)+3*X(68322) = 3*X(68322)-X(68324)

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

Barycentrics    (a-b)*(a-c)*(a^6-(b+c)*a^5+(b^2-b*c+c^2)*a^4-(b^2-c^2)*(b-c)*a^3-(b-c)^2*(b^2-b*c+c^2)*a^2+(b^2-c^2)^2*(b-c)^2) : :
X(68324) = 3*X(101)-4*X(3234) = X(927)-3*X(68322) = 2*X(1566)-3*X(61730) = 2*X(3234)-3*X(34805) = 3*X(68322)-2*X(68323)

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

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

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

X(68325) = X(2)X(187)∩X(4)X(7622)

Barycentrics    a^4-16*a^2*b^2+13*b^4-16*a^2*c^2-22*b^2*c^2+13*c^4 : :
X(68325) = 7*X(2)-4*X(1153), 5*X(2)-2*X(5569), X(2)+2*X(8176), 4*X(2)-X(8182), 5*X(2)+X(23334), 8*X(2)+X(44678), 13*X(2)-4*X(46893), 11*X(2)-2*X(47101) 10*X(2)-X(47102), 7*X(2)+2*X(63956), 2*X(2)+X(66466), X(2)-4*X(66511), 7*X(2)-X(66699), 10*X(1153)-7*X(5569), 2*X(1153)+7*X(8176), 16*X(1153)-7*X(8182) 13*X(1153)-7*X(46893), 2*X(1153)+X(63956), X(1153)-7*X(66511), 4*X(1153)-X(66699)

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

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

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





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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

underbar



X(68326) = X(3)X(18310)∩X(4)X(14977)

Barycentrics    (b^2 - c^2)*(-a^4 + b^4 - 4*b^2*c^2 + c^4)*(-2*a^6 + 2*a^4*b^2 - a^2*b^4 + b^6 + 2*a^4*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6) : :
X(68326) = 3 X[8371] - 2 X[9175], X[18440] + 2 X[45801]

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

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

X(68327) = X(4)X(690)∩X(460)X(512)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-2*a^4 + a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4) : :
X(68327) = 3 X[4] + X[44427], 3 X[16230] - X[44427], X[41079] - 3 X[44203], X[14273] - 3 X[17994], 3 X[381] - X[6334], X[2501] + 3 X[44705], 2 X[2501] - 3 X[51513], X[2501] - 3 X[58757], 2 X[44705] + X[51513], 3 X[1637] - X[9409], X[9409] + 3 X[58346], 3 X[10151] - X[52475], 3 X[15451] - X[55280], X[41077] - 3 X[65754], 3 X[44564] - 2 X[44818]

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

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

X(68328) = X(2)X(47442)∩X(4)X(18312)

Barycentrics    (b^2 - c^2)*(-a^10 + 2*a^6*b^4 - a^2*b^8 - 3*a^6*b^2*c^2 + a^4*b^4*c^2 + 2*b^8*c^2 + 2*a^6*c^4 + a^4*b^2*c^4 - 2*b^6*c^4 - 2*b^4*c^6 - a^2*c^8 + 2*b^2*c^8) : :
X(68328) = 3 X[381] - X[33752], X[37742] - 3 X[39482], 5 X[3091] - X[62307]

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

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

X(68329) = X(3)X(2814)∩X(4)X(21222)

Barycentrics    a*(b - c)*(a^3*b^2 - a^2*b^3 - a*b^4 + b^5 - 2*a^3*b*c + 2*a^2*b^2*c + 2*a*b^3*c - 2*b^4*c + a^3*c^2 + 2*a^2*b*c^2 - 4*a*b^2*c^2 + b^3*c^2 - a^2*c^3 + 2*a*b*c^3 + b^2*c^3 - a*c^4 - 2*b*c^4 + c^5) : :
X(68329) = 2 X[1385] - 3 X[14413], 2 X[3716] - 3 X[5886], 3 X[5603] - X[53343], 5 X[5818] - 3 X[53364], 4 X[9956] - 3 X[14430], 3 X[14419] - 2 X[44805], 3 X[23057] - 4 X[33179], 4 X[25380] - 3 X[26446], 2 X[44824] - 3 X[47893]

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

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

X(68330) = X(4)X(24978)∩X(30)X(45259)

Barycentrics    (b^2 - c^2)*(a^8 - 2*a^4*b^4 + b^8 + 5*a^4*b^2*c^2 - a^2*b^4*c^2 - 4*b^6*c^2 - 2*a^4*c^4 - a^2*b^2*c^4 + 6*b^4*c^4 - 4*b^2*c^6 + c^8) : :
X(68330) = X[382] + 3 X[42731], X[16230] + 3 X[44203], 3 X[1637] - X[44810], 5 X[3091] - X[41078], X[6130] - 3 X[44204], 3 X[41079] + X[65871]

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

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

X(68331) = X(2)X(44820)∩X(4)X(35522)

Barycentrics    (b^2 - c^2)*(-a^10 + 2*a^6*b^4 - a^2*b^8 - 3*a^6*b^2*c^2 + 4*a^4*b^4*c^2 - 3*a^2*b^6*c^2 + 2*b^8*c^2 + 2*a^6*c^4 + 4*a^4*b^2*c^4 - 2*b^6*c^4 - 3*a^2*b^2*c^6 - 2*b^4*c^6 - a^2*c^8 + 2*b^2*c^8) : :
X(68331) = X[3569] - 3 X[10516], X[51212] + 3 X[53369], X[53345] - 3 X[66161]

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

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

X(68332) = X(2)X(44819)∩X(4)X(9521)

Barycentrics    (b - c)*(-2*a^6 + 3*a^5*b - 3*a^4*b^2 + 2*a^3*b^3 + 4*a^2*b^4 - 5*a*b^5 + b^6 + 3*a^5*c - 6*a^4*b*c + 6*a^3*b^2*c - 4*a^2*b^3*c - a*b^4*c + 2*b^5*c - 3*a^4*c^2 + 6*a^3*b*c^2 - 8*a^2*b^2*c^2 + 6*a*b^3*c^2 - b^4*c^2 + 2*a^3*c^3 - 4*a^2*b*c^3 + 6*a*b^2*c^3 - 4*b^3*c^3 + 4*a^2*c^4 - a*b*c^4 - b^2*c^4 - 5*a*c^5 + 2*b*c^5 + c^6) : :
X(68332) = 5 X[3091] - X[47695], 3 X[3817] - X[48286], X[3904] + 3 X[59387], 3 X[5055] - 2 X[45318], 3 X[5587] - X[10015], 3 X[14425] - 2 X[44805]

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

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

X(68333) = X(4)X(50351)∩X(512)X(5887)

Barycentrics    (b - c)*(-2*a^4*b^2 + 3*a^3*b^3 + a^2*b^4 - 3*a*b^5 + b^6 + a^3*b^2*c - 2*a^2*b^3*c + a*b^4*c - 2*a^4*c^2 + a^3*b*c^2 - 2*a^2*b^2*c^2 + 2*a*b^3*c^2 - b^4*c^2 + 3*a^3*c^3 - 2*a^2*b*c^3 + 2*a*b^2*c^3 + a^2*c^4 + a*b*c^4 - b^2*c^4 - 3*a*c^5 + c^6) : :
X(68333) = X[944] - 3 X[30580], 5 X[3091] - X[49303], X[5691] + 3 X[62634], 2 X[6684] - 3 X[28602]

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

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



leftri

Antiproducts: X(68334)-X(68355)

rightri

This preamble and centers X(68334)-X(68355) were contributed by Ivan Pavlov on Apr 22, 2025.

If P=(u:v:w) and Q=(p:q:r) in barycentric coordinates, we define the antiproduct of P and Q as the point (-p u + q v + r w : p u - q v + r w : p u + q v - r w).
The antiproduct of two points can be conveniently constructed as the anticomplement of their barycentric product. Note that the antiproduct of P and the isogonal conjugate of P is always X(69). The antiproduct of P and the isotomic conjugate of P is always X(2). Of course, the antiproduct of any point P and the centroid is the anticomplement of P.
The following relations also holds:
(1) II-Caph point of P = antiproduct of complement and anticomplement of P. See X(32001) for definition of II-Caph.
(2) Vijay 6th parallel transform of P = antiproduct of complement and isotomic conjugate of P.
(3) M(P) = antiproduct of G and (KP2(G) of P and P), see the preambles of X(40896) and X(55917) for definitions of M and KP2. G denotes the centroid.
(4) The antiproduct of the crosspoint and the cevian product of P and Q coincides with the antiproduct of P and Q.
(5) The antiproduct of G and the infinity point of the tripolar of G coincides with the perspector of the conic {A,B,C,G,P}.

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

Barycentrics    3*a^5+a^4*(b+c)+2*a^2*(b-c)^2*(b+c)-a*(b^2-c^2)^2-2*a^3*(b^2+c^2)-(b-c)^2*(3*b^3+5*b^2*c+5*b*c^2+3*c^3) : :

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

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

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

Barycentrics    a^6+2*a^3*b*c*(b+c)-2*a*b*(b-c)^2*c*(b+c)+a^4*(-3*b^2+2*b*c-3*c^2)-(b^2-c^2)^2*(b^2+c^2)+a^2*(b-c)^2*(3*b^2+4*b*c+3*c^2) : :

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

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

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

Barycentrics    a^6-a^5*(b+c)+a^3*b*c*(b+c)-a^4*(b^2+c^2)-(b^2-c^2)^2*(b^2-b*c+c^2)+a^2*(b-c)^2*(b^2+b*c+c^2)+a*(b^5-b^3*c^2-b^2*c^3+c^5) : :

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

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

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

Barycentrics    (a-b)*b*(a-c)*c*(a^3+(b-c)^2*(b+c)) : :

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

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

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

Barycentrics    b*c*(-a^6+b*c*(b^2-c^2)^2+a^4*(b^2-b*c+c^2)-a^3*(b^3+c^3)+a*(b^5-b^3*c^2-b^2*c^3+c^5)) : :

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

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

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

Barycentrics    (a-b)*(a-c)*(a^5+2*b*(b-c)^2*c*(b+c)+a*(b^2-c^2)^2-2*a^3*(b^2-b*c+c^2)) : :

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

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

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

Barycentrics    a^8-a^5*b*c*(b+c)+2*a^3*b^2*c^2*(b+c)+2*b^2*c^2*(b^2-c^2)^2-a^2*(b^2-c^2)^2*(b^2-b*c+c^2)-a^6*(3*b^2+b*c+3*c^2)+3*a^4*(b^4+c^4)+a*b*c*(b^5-b^4*c-b*c^4+c^5) : :

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

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

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

Barycentrics    sqrt(3)*(a^2-b^2-c^2)^2*(a^2+b^2-c^2)*(a^2-b^2+c^2)-2*(a^6+a^4*(b^2+c^2)-(b^2-c^2)^2*(b^2+c^2)-a^2*(b^2+c^2)^2)*S : :

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

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

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

Barycentrics    sqrt(3)*(a^2-b^2-c^2)^2*(a^2+b^2-c^2)*(a^2-b^2+c^2)+2*(a^6+a^4*(b^2+c^2)-(b^2-c^2)^2*(b^2+c^2)-a^2*(b^2+c^2)^2)*S : :

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

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

X(68343) = ANTIPRODUCT OF X(5) AND X(7)

Barycentrics    a^6-a^5*(b+c)+b*c*(b^2-c^2)^2-2*a^4*(b^2+c^2)+a^2*(b-c)^2*(b^2+b*c+c^2)+a^3*(2*b^3+b^2*c+b*c^2+2*c^3)-a*(b^5-b^3*c^2-b^2*c^3+c^5) : :

X(68343) lies on these lines: {3, 56254}, {6, 18662}, {8, 5842}, {9, 20223}, {63, 321}, {92, 3219}, {144, 2895}, {189, 4102}, {190, 329}, {192, 62798}, {201, 24537}, {212, 64858}, {255, 42456}, {394, 4552}, {651, 6360}, {894, 62857}, {908, 33113}, {914, 32859}, {1708, 17862}, {1993, 17479}, {2406, 7011}, {3869, 7283}, {5278, 14213}, {5744, 32939}, {5759, 52365}, {5762, 65684}, {5905, 6350}, {7078, 20222}, {7580, 61185}, {11433, 41563}, {14544, 22117}, {17147, 55399}, {17262, 39767}, {17336, 20921}, {17781, 50105}, {18607, 28968}, {18736, 42718}, {18750, 20920}, {20078, 26871}, {20905, 55871}, {21271, 68346}, {23661, 55104}, {25734, 45738}, {26591, 55869}, {26611, 28826}, {26921, 41013}, {27287, 33761}, {27378, 44706}, {27472, 55987}, {34234, 67335}, {37584, 38462}, {37787, 54284}, {48380, 55873}

X(68343) = reflection of X(i) in X(j) for these {i,j}: {68336, 65684}
X(68343) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {9, 2888}, {41, 17035}, {54, 7}, {95, 21285}, {97, 52365}, {275, 68336}, {2148, 145}, {2167, 3434}, {2169, 347}, {2190, 56927}, {8882, 12649}, {35196, 75}, {36078, 36038}, {36134, 4467}, {44687, 69}, {54034, 3210}, {56254, 2893}, {62265, 329}, {62268, 30699}, {62276, 21280}, {62277, 68351}
X(68343) = pole of line {52355, 57091} with respect to the Steiner circumellipse
X(68343) = pole of line {4551, 14544} with respect to the Yff parabola
X(68343) = pole of line {23536, 64163} with respect to the dual conic of Yff parabola
X(68343) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2995), X(34393)}}, {{A, B, C, X(4417), X(64194)}}, {{A, B, C, X(6796), X(13478)}}
X(68343) = barycentric product X(i)*X(j) for these (i, j): {6796, 75}
X(68343) = barycentric quotient X(i)/X(j) for these (i, j): {6796, 1}
X(68343) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {144, 56943, 68335}, {190, 54107, 329}, {1993, 17479, 68344}, {5762, 65684, 68336}, {25734, 45738, 56545}

X(68344) = ANTIPRODUCT OF X(5) AND X(8)

Barycentrics    (a+b-c)*(a-b+c)*(a^3+b*c*(b+c)-a*(b+c)^2) : :
X(68344) = -3*X[2]+4*X[17043]

X(68344) lies on these lines: {1, 1441}, {2, 17043}, {3, 17221}, {6, 4552}, {7, 528}, {8, 40999}, {75, 1442}, {77, 3875}, {85, 7269}, {145, 347}, {192, 651}, {221, 64071}, {222, 17147}, {226, 54744}, {241, 4852}, {264, 40440}, {273, 34772}, {307, 519}, {321, 45126}, {322, 4511}, {326, 20895}, {379, 1953}, {394, 18662}, {517, 17134}, {536, 28968}, {648, 56948}, {944, 4329}, {952, 21270}, {1014, 33296}, {1214, 3187}, {1443, 39126}, {1445, 16834}, {1456, 49462}, {1458, 32921}, {1471, 49477}, {1482, 17220}, {1617, 17150}, {1943, 28606}, {1992, 41563}, {1993, 17479}, {1999, 17080}, {2003, 32933}, {2006, 30834}, {2256, 25255}, {2398, 6600}, {2594, 34388}, {3007, 37727}, {3057, 52385}, {3160, 4460}, {3173, 25254}, {3175, 28997}, {3188, 34791}, {3210, 17074}, {3217, 49759}, {3240, 14594}, {3244, 3668}, {3262, 44179}, {3672, 53997}, {3759, 37787}, {3811, 57810}, {3879, 22464}, {3895, 7013}, {3896, 8270}, {3969, 56366}, {3995, 34048}, {4318, 49470}, {4331, 50284}, {4336, 28850}, {4358, 56418}, {4361, 17077}, {4464, 9436}, {4664, 29007}, {4967, 25723}, {4970, 9316}, {5226, 34064}, {5278, 16577}, {5564, 17095}, {5723, 17243}, {5740, 10573}, {5882, 18650}, {6180, 17318}, {6358, 19684}, {6360, 62798}, {6505, 17862}, {6510, 26651}, {6542, 17086}, {7190, 9312}, {7225, 43040}, {7982, 18655}, {8148, 18661}, {10944, 41003}, {12575, 50563}, {14543, 20818}, {15253, 29830}, {15500, 65581}, {16091, 34611}, {17073, 48381}, {17075, 17388}, {17233, 28780}, {17234, 37771}, {17262, 62669}, {17314, 28739}, {17316, 37800}, {17317, 61008}, {17319, 25726}, {17438, 24315}, {17452, 24268}, {17858, 18689}, {18481, 20291}, {18525, 20289}, {20017, 26942}, {20905, 53996}, {20946, 28982}, {21617, 29574}, {24173, 32577}, {25243, 55432}, {25717, 64739}, {27396, 40863}, {27547, 39351}, {28996, 35652}, {29584, 41246}, {30145, 45196}, {30806, 55391}, {33298, 41808}, {34748, 53380}, {36589, 50132}, {37541, 64161}, {37732, 57830}, {37756, 60988}, {41010, 61296}, {41140, 61016}, {41226, 59771}, {41823, 65384}, {42289, 50281}, {45222, 52424}, {49455, 53531}, {49492, 54292}, {50102, 64115}, {50109, 60992}, {55119, 56936}

X(68344) = reflection of X(i) in X(j) for these {i,j}: {21270, 41007}
X(68344) = X(i)-Dao conjugate of X(j) for these {i, j}: {40688, 3813}, {47794, 44311}
X(68344) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {54, 329}, {57, 2888}, {95, 21286}, {97, 52366}, {604, 17035}, {2148, 144}, {2167, 3436}, {2169, 56943}, {2190, 68335}, {8882, 5942}, {35196, 18750}, {36078, 3762}, {44687, 54113}, {54034, 3177}, {62264, 7}, {62268, 30694}, {62269, 21218}
X(68344) = pole of line {29013, 30725} with respect to the incircle
X(68344) = pole of line {4453, 17094} with respect to the Steiner circumellipse
X(68344) = pole of line {307, 4887} with respect to the dual conic of Yff parabola
X(68344) = pole of line {306, 3911} with respect to the dual conic of Moses-Feuerbach circumconic
X(68344) = intersection, other than A, B, C, of circumconics {{A, B, C, X(519), X(17579)}}, {{A, B, C, X(903), X(2997)}}, {{A, B, C, X(1320), X(8715)}}, {{A, B, C, X(3204), X(34230)}}, {{A, B, C, X(36887), X(47794)}}, {{A, B, C, X(40440), X(52553)}}
X(68344) = barycentric product X(i)*X(j) for these (i, j): {85, 8715}, {145, 27814}, {3204, 6063}, {4554, 48302}, {39478, 46405}, {47794, 664}
X(68344) = barycentric quotient X(i)/X(j) for these (i, j): {3204, 55}, {8715, 9}, {27814, 4373}, {39478, 654}, {47794, 522}, {48302, 650}
X(68344) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 17885, 24202}, {85, 17393, 7269}, {145, 347, 56927}, {347, 56927, 41804}, {664, 4360, 7}, {952, 41007, 21270}, {1993, 17479, 68343}, {3875, 25716, 77}, {5723, 17243, 28741}, {17221, 21271, 3}

X(68345) = ANTIPRODUCT OF X(5) AND X(9)

Barycentrics    a^5*(b+c)-a^3*b*c*(b+c)+b*c*(b^2-c^2)^2-2*a^4*(b^2-b*c+c^2)+a^2*(b-c)^2*(2*b^2+b*c+2*c^2)-a*(b^5-b^3*c^2-b^2*c^3+c^5) : :

X(68345) lies on these lines: {2, 7004}, {3, 56254}, {7, 2897}, {8, 5884}, {92, 11220}, {100, 17165}, {222, 14544}, {991, 18662}, {1071, 23661}, {1441, 17616}, {1897, 17074}, {2979, 21271}, {3434, 17140}, {3616, 56940}, {3873, 39126}, {4303, 20222}, {4855, 56318}, {5732, 20223}, {10167, 64194}, {10202, 38462}, {10391, 17862}, {10394, 54284}, {10538, 18444}, {11680, 53566}, {12675, 23528}, {13369, 41013}, {17221, 20477}, {17784, 24349}, {22053, 64858}, {24840, 28364}, {26910, 53151}, {31657, 65684}, {36996, 68335}

X(68345) = anticomplement of X(7069)
X(68345) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {7, 2888}, {54, 144}, {56, 17035}, {95, 3436}, {97, 56943}, {275, 68335}, {2148, 3177}, {2167, 329}, {2190, 5942}, {8882, 30694}, {35196, 45738}, {36078, 47772}, {54034, 21218}, {62264, 145}, {62276, 21286}, {62277, 52366}
X(68345) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 52365, 68336}, {17140, 68350, 3434}

X(68346) = ANTIPRODUCT OF X(5) AND X(20)

Barycentrics    a^8+4*b^2*c^2*(b^2-c^2)^2-3*a^6*(b^2+c^2)-a^2*(b^2-c^2)^2*(b^2+c^2)+a^4*(3*b^4-2*b^2*c^2+3*c^4) : :
X(68346) = -5*X[3618]+4*X[59649]

X(68346) lies on these lines: {2, 42459}, {3, 95}, {4, 6527}, {5, 40680}, {6, 64781}, {20, 32000}, {25, 30737}, {26, 44135}, {30, 69}, {53, 6389}, {76, 39568}, {99, 54992}, {183, 9909}, {253, 3146}, {268, 1948}, {273, 10538}, {290, 67186}, {297, 20208}, {309, 5088}, {311, 7387}, {316, 34725}, {317, 382}, {322, 7283}, {325, 34609}, {338, 1609}, {339, 18534}, {340, 5073}, {381, 45198}, {393, 441}, {401, 9308}, {458, 3164}, {523, 34777}, {550, 52710}, {648, 38292}, {1033, 60516}, {1073, 35061}, {1078, 16195}, {1235, 11414}, {1272, 31181}, {1316, 19118}, {1494, 15684}, {1598, 41009}, {1632, 33582}, {1657, 44134}, {1947, 7011}, {1975, 14615}, {2052, 6617}, {2070, 55561}, {2450, 63535}, {2453, 20987}, {2782, 19588}, {2968, 55393}, {3167, 57275}, {3260, 3964}, {3575, 68355}, {3618, 59649}, {3627, 63155}, {3628, 8797}, {3629, 64915}, {3663, 64931}, {3729, 64780}, {3785, 65376}, {3830, 32002}, {3875, 64930}, {3933, 34938}, {5020, 62698}, {6090, 41202}, {6144, 64923}, {6356, 55394}, {7395, 44142}, {7517, 44138}, {7767, 31305}, {7776, 14790}, {8573, 41760}, {9530, 48910}, {9723, 12084}, {10154, 34229}, {11250, 44180}, {11432, 41481}, {12108, 36948}, {13219, 52842}, {14767, 36751}, {15312, 51212}, {15589, 34608}, {15703, 40410}, {15811, 59527}, {15851, 36794}, {16089, 38283}, {16199, 40022}, {16655, 44141}, {18569, 62338}, {20563, 46200}, {21271, 68343}, {23335, 40697}, {26166, 37198}, {32816, 50572}, {34621, 52713}, {34815, 37200}, {35930, 59566}, {35941, 56290}, {37188, 43981}, {37668, 44442}, {37928, 67606}, {39099, 64926}, {40318, 65767}, {40341, 64783}, {41244, 46832}, {41489, 41678}, {43988, 52253}, {46717, 62953}, {50692, 52711}, {51888, 64585}, {52559, 53639}, {52843, 68354}, {53795, 64023}, {54105, 61970}, {55391, 64054}, {55392, 64053}, {55474, 55885}, {55480, 55890}, {55958, 61882}, {57008, 68022}, {57822, 62137}, {57897, 62046}, {58408, 61315}, {61843, 63173}, {62275, 62334}

X(68346) = reflection of X(i) in X(j) for these {i,j}: {37921, 2453}
X(68346) = isotomic conjugate of X(52441)
X(68346) = anticomplement of X(42459)
X(68346) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {2148, 17037}, {2155, 17035}, {2167, 6225}, {2169, 68006}, {2184, 2888}, {2190, 14361}
X(68346) = pole of line {418, 26864} with respect to the Stammler hyperbola
X(68346) = pole of line {30474, 62428} with respect to the Steiner circumellipse
X(68346) = pole of line {376, 5562} with respect to the Wallace hyperbola
X(68346) = pole of line {3090, 32000} with respect to the dual conic of Moses HK-parabola
X(68346) = intersection, other than A, B, C, of circumconics {{A, B, C, X(95), X(15400)}}, {{A, B, C, X(3426), X(8884)}}, {{A, B, C, X(8795), X(36889)}}, {{A, B, C, X(19185), X(43918)}}, {{A, B, C, X(35061), X(35510)}}, {{A, B, C, X(42459), X(45249)}}
X(68346) = barycentric product X(i)*X(j) for these (i, j): {26883, 76}
X(68346) = barycentric quotient X(i)/X(j) for these (i, j): {2, 52441}, {26883, 6}, {45249, 42459}
X(68346) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 6527, 41005}, {53, 6389, 52251}, {253, 32001, 40996}, {264, 20477, 3}, {382, 40995, 317}, {401, 40896, 9308}, {401, 9308, 15905}, {37188, 43981, 65809}

X(68347) = ANTIPRODUCT OF X(6) AND X(19)

Barycentrics    a^7-b^7+b^4*c^3+b^3*c^4-c^7-2*a^2*b^2*c^2*(b+c)-a^3*(b^2-c^2)^2+a^4*(b^3+c^3) : :

X(68347) lies on these lines: {2, 1973}, {8, 2893}, {10, 26260}, {19, 26153}, {20, 28845}, {347, 56928}, {607, 857}, {858, 29829}, {1370, 17135}, {1441, 5090}, {2172, 28404}, {3101, 3661}, {6360, 21217}, {7357, 20243}, {17137, 21280}, {17904, 33313}, {18589, 26203}, {18639, 24605}, {20552, 52366}, {24995, 27532}, {56883, 56943}

X(68347) = anticomplement of X(1973)
X(68347) = X(i)-Ceva conjugate of X(j) for these {i, j}: {40364, 2}
X(68347) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {1, 21216}, {2, 193}, {3, 194}, {4, 6392}, {7, 30699}, {8, 30694}, {25, 46712}, {39, 10340}, {48, 17486}, {63, 192}, {67, 19577}, {69, 2}, {72, 1655}, {75, 5905}, {76, 4}, {77, 3210}, {78, 3177}, {83, 7754}, {85, 12649}, {86, 3187}, {95, 1993}, {99, 525}, {125, 54104}, {141, 8878}, {183, 47740}, {184, 8264}, {190, 25259}, {193, 18287}, {219, 21218}, {249, 41676}, {261, 62798}, {264, 6515}, {265, 19570}, {274, 3868}, {275, 56017}, {276, 5889}, {279, 11851}, {287, 385}, {290, 51481}, {295, 19565}, {304, 8}, {305, 69}, {306, 1654}, {307, 17778}, {308, 3060}, {310, 17220}, {312, 5942}, {314, 92}, {315, 41361}, {325, 40867}, {326, 6360}, {328, 37779}, {332, 63}, {337, 6542}, {343, 17035}, {345, 144}, {348, 145}, {394, 3164}, {487, 6463}, {488, 6462}, {524, 7665}, {525, 148}, {561, 21270}, {647, 25054}, {648, 33294}, {656, 21220}, {662, 17498}, {668, 4391}, {670, 850}, {671, 47286}, {683, 54412}, {799, 7253}, {801, 9308}, {892, 9979}, {905, 9263}, {1016, 3732}, {1176, 8267}, {1231, 2475}, {1236, 34163}, {1241, 1843}, {1260, 46706}, {1264, 56943}, {1265, 30695}, {1275, 1897}, {1331, 21225}, {1332, 17494}, {1444, 17147}, {1459, 21224}, {1494, 3580}, {1502, 11442}, {1565, 54102}, {1790, 17148}, {1799, 6}, {1969, 5906}, {1978, 20293}, {2366, 15014}, {2373, 37784}, {2407, 45292}, {2525, 39346}, {2996, 2996}, {3222, 2451}, {3260, 66914}, {3265, 39352}, {3267, 3448}, {3504, 2998}, {3596, 68335}, {3620, 8892}, {3690, 46714}, {3695, 46707}, {3718, 329}, {3917, 52637}, {3926, 20}, {3933, 2896}, {3964, 46717}, {3977, 17487}, {3998, 18666}, {4025, 4440}, {4143, 34186}, {4176, 6527}, {4554, 521}, {4555, 10015}, {4558, 31296}, {4561, 514}, {4563, 523}, {4569, 17896}, {4572, 46400}, {4573, 65099}, {4580, 25047}, {4590, 110}, {4592, 4560}, {4600, 14543}, {4601, 53349}, {4602, 21300}, {4615, 53352}, {4620, 14544}, {4998, 651}, {5490, 12222}, {5491, 12221}, {5641, 54395}, {6035, 65714}, {6063, 56927}, {6331, 520}, {6332, 39351}, {6333, 39359}, {6340, 20080}, {6385, 20242}, {6390, 8591}, {6393, 147}, {6394, 401}, {6516, 17496}, {7019, 6646}, {7053, 46716}, {7055, 347}, {7056, 4452}, {7177, 17480}, {7182, 7}, {7763, 6193}, {7767, 51860}, {7769, 11271}, {7799, 12383}, {8781, 3564}, {8858, 698}, {10008, 9742}, {10159, 7762}, {10217, 46708}, {10218, 46709}, {11064, 39358}, {11090, 62986}, {11091, 62987}, {12215, 8782}, {14208, 21221}, {14376, 20065}, {14417, 39356}, {14534, 56019}, {14575, 40382}, {14615, 14361}, {14616, 62305}, {14977, 45291}, {15164, 2592}, {15165, 2593}, {15413, 149}, {15414, 44003}, {15419, 17154}, {17206, 1}, {17932, 2799}, {17970, 19566}, {18020, 648}, {18021, 68338}, {18022, 317}, {18023, 41724}, {18025, 48381}, {18816, 48380}, {18829, 3569}, {18830, 21438}, {18878, 44427}, {19611, 18663}, {20336, 2895}, {20563, 45794}, {20567, 68336}, {20570, 2994}, {20769, 30667}, {22370, 41840}, {24243, 26503}, {24244, 26494}, {25083, 39350}, {26932, 17036}, {26942, 56291}, {28706, 2888}, {30680, 20073}, {30786, 524}, {31617, 15801}, {31621, 10733}, {31624, 64878}, {31637, 239}, {32014, 56018}, {32830, 11469}, {33297, 17911}, {34055, 17489}, {34254, 5596}, {34384, 264}, {34385, 5392}, {34386, 3}, {34391, 13439}, {34392, 13428}, {34399, 278}, {34403, 3146}, {34405, 393}, {34409, 281}, {34410, 459}, {34411, 7952}, {34412, 1249}, {34537, 53350}, {34767, 62639}, {34897, 14712}, {35136, 2501}, {35139, 41079}, {35140, 297}, {35518, 37781}, {36212, 39355}, {36214, 40858}, {36952, 7785}, {37214, 37782}, {37215, 23874}, {37669, 17037}, {37804, 11061}, {40032, 11433}, {40050, 315}, {40071, 1330}, {40360, 33796}, {40364, 6327}, {40373, 40381}, {40405, 25}, {40410, 41628}, {40412, 40571}, {40413, 40318}, {40423, 2986}, {40428, 2987}, {40708, 7779}, {40709, 3180}, {40710, 3181}, {40711, 62983}, {40712, 62984}, {40824, 5921}, {40827, 41723}, {40829, 7703}, {40830, 12111}, {40832, 13754}, {41530, 32001}, {42287, 63042}, {42313, 7774}, {42333, 324}, {43187, 53345}, {43714, 20081}, {44181, 46639}, {44182, 41909}, {44183, 44766}, {44326, 8057}, {44877, 56021}, {45792, 14731}, {46134, 14618}, {46139, 18314}, {46140, 44146}, {46142, 57257}, {46144, 39905}, {46746, 6504}, {46810, 2575}, {46813, 2574}, {47388, 36849}, {47389, 99}, {47390, 46726}, {49280, 39364}, {52351, 20072}, {52385, 18667}, {52392, 37759}, {52396, 3151}, {52437, 18301}, {52565, 3152}, {52608, 512}, {52609, 31290}, {52617, 13219}, {52940, 53351}, {54911, 56022}, {54960, 52744}, {54986, 26545}, {54987, 26546}, {54988, 46106}, {55023, 6339}, {55202, 7192}, {55264, 18808}, {55388, 37881}, {55965, 25239}, {55972, 37174}, {56053, 65097}, {56267, 37667}, {57738, 62636}, {57750, 43190}, {57751, 2988}, {57752, 2989}, {57753, 2990}, {57754, 2991}, {57755, 46638}, {57756, 46640}, {57757, 44765}, {57758, 57647}, {57759, 42405}, {57760, 43756}, {57761, 287}, {57762, 14919}, {57763, 4558}, {57764, 18315}, {57765, 63763}, {57775, 2052}, {57780, 68334}, {57783, 189}, {57784, 2997}, {57798, 539}, {57799, 511}, {57800, 394}, {57801, 17950}, {57819, 37644}, {57822, 37645}, {57825, 6994}, {57829, 323}, {57833, 333}, {57845, 50248}, {57846, 44363}, {57847, 44370}, {57848, 20536}, {57849, 19623}, {57852, 141}, {57853, 81}, {57854, 86}, {57859, 15983}, {57865, 20086}, {57872, 325}, {57873, 62999}, {57876, 17379}, {57878, 63009}, {57903, 52}, {57904, 68}, {57918, 3434}, {57919, 3436}, {57928, 39470}, {57985, 16704}, {57987, 30941}, {57991, 34211}, {58005, 56559}, {59482, 46713}, {60101, 1351}, {60114, 43981}, {60202, 18440}, {60217, 21850}, {60235, 56020}, {60241, 27377}, {60729, 31308}, {60872, 63093}, {62277, 17479}, {62719, 6758}, {63179, 4232}, {63182, 6353}, {63195, 6995}, {64982, 1992}, {64985, 27}, {65032, 3618}, {65164, 522}, {65278, 14316}, {65279, 24978}, {65287, 54262}, {65301, 47695}, {65307, 4580}, {65328, 55974}, {65354, 16230}, {66767, 63646}, {66933, 17493}, {67038, 61185}
X(68347) = pole of line {824, 4560} with respect to the DeLongchamps circle
X(68347) = barycentric product X(i)*X(j) for these (i, j): {18683, 69}
X(68347) = barycentric quotient X(i)/X(j) for these (i, j): {18683, 4}
X(68347) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 17492, 21270}, {8, 21274, 17492}, {1370, 68351, 18659}

X(68348) = ANTIPRODUCT OF X(7) AND X(20)

Barycentrics    a^6+2*a^5*(b+c)-4*a^3*(b-c)^2*(b+c)+2*a*(b-c)^4*(b+c)+a^4*(-5*b^2+6*b*c-5*c^2)-(b^2-c^2)^2*(3*b^2+2*b*c+3*c^2)+a^2*(b-c)^2*(7*b^2+10*b*c+7*c^2) : :

X(68348) lies on these lines: {2, 77}, {8, 12565}, {63, 63001}, {92, 10405}, {144, 2895}, {145, 9539}, {279, 13567}, {280, 67994}, {329, 2391}, {346, 54113}, {459, 36118}, {2184, 54111}, {3146, 68352}, {4869, 20921}, {6223, 39130}, {6515, 9965}, {11433, 60939}, {17778, 30694}, {18663, 68349}, {20007, 52366}, {20059, 20223}, {21454, 26871}, {27540, 53997}, {30699, 39351}, {54107, 64015}

X(68348) = anticomplement of X(18623)
X(68348) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 54226}
X(68348) = X(i)-Dao conjugate of X(j) for these {i, j}: {9, 54226}
X(68348) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {9, 6225}, {33, 14361}, {41, 17037}, {64, 7}, {212, 68006}, {253, 21285}, {459, 68336}, {1073, 52365}, {2155, 145}, {2184, 3434}, {5931, 17137}, {7037, 63877}, {8809, 6604}, {19611, 68351}, {19614, 347}, {30457, 8}, {33581, 3210}, {41088, 5932}, {41489, 12649}, {44692, 69}, {52158, 75}, {53012, 2897}, {56235, 21302}, {57921, 21280}, {60799, 55119}, {65374, 4131}, {65574, 2893}
X(68348) = pole of line {4163, 8058} with respect to the Steiner circumellipse
X(68348) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1422), X(7992)}}, {{A, B, C, X(10405), X(41081)}}
X(68348) = barycentric product X(i)*X(j) for these (i, j): {75, 7992}
X(68348) = barycentric quotient X(i)/X(j) for these (i, j): {1, 54226}, {7992, 1}
X(68348) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {56943, 68335, 144}

X(68349) = ANTIPRODUCT OF X(8) AND X(20)

Barycentrics    (a+b-c)*(a-b+c)*(a^3+3*b^3+b^2*c+b*c^2+3*c^3-3*a^2*(b+c)-a*(b+c)^2) : :

X(68349) lies on these lines: {2, 7}, {8, 3668}, {20, 41004}, {29, 58786}, {69, 279}, {75, 1446}, {77, 34772}, {78, 269}, {85, 5232}, {145, 347}, {241, 4869}, {273, 4373}, {280, 6355}, {314, 61413}, {320, 62787}, {348, 3945}, {391, 948}, {936, 7271}, {938, 3663}, {962, 41010}, {966, 52023}, {1119, 5125}, {1122, 24797}, {1210, 4862}, {1427, 3965}, {1439, 3868}, {1440, 39695}, {1441, 3617}, {1788, 65688}, {2287, 6180}, {3146, 18655}, {3160, 3879}, {3522, 18650}, {3664, 5703}, {3672, 5738}, {3718, 40704}, {3875, 20008}, {4021, 15933}, {4328, 32098}, {4329, 20070}, {4341, 4511}, {4346, 17863}, {4419, 18635}, {4452, 5932}, {4887, 5704}, {4888, 13411}, {5930, 20019}, {6356, 37180}, {6515, 20211}, {6734, 31995}, {6895, 21279}, {7282, 7518}, {7365, 37655}, {8809, 68352}, {9312, 32099}, {10481, 17272}, {14986, 53596}, {17014, 17086}, {17270, 31994}, {18663, 68348}, {19611, 54111}, {20013, 53997}, {20076, 55119}, {20212, 37781}, {21255, 51302}, {24599, 37800}, {24607, 56020}, {25015, 54398}, {25887, 45227}, {27383, 62789}, {29616, 45744}, {30712, 60041}, {31185, 59594}, {32001, 36118}, {33673, 37780}, {38459, 55391}, {40663, 63575}, {40999, 46933}, {43983, 62779}, {50700, 64122}, {54425, 62985}, {57866, 63235}

X(68349) = anticomplement of X(27382)
X(68349) = X(i)-isoconjugate-of-X(j) for these {i, j}: {41, 67941}, {3063, 68195}
X(68349) = X(i)-Dao conjugate of X(j) for these {i, j}: {3160, 67941}, {10001, 68195}
X(68349) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {34, 14361}, {57, 6225}, {64, 329}, {253, 21286}, {603, 68006}, {604, 17037}, {1073, 52366}, {2155, 144}, {2184, 3436}, {8809, 69}, {19614, 56943}, {33581, 3177}, {36079, 693}, {41489, 5942}, {44692, 54113}, {52158, 18750}, {60803, 189}, {65374, 20296}
X(68349) = pole of line {3064, 4162} with respect to the polar circle
X(68349) = pole of line {522, 17094} with respect to the Steiner circumellipse
X(68349) = pole of line {651, 25736} with respect to the Hutson-Moses hyperbola
X(68349) = pole of line {521, 43923} with respect to the dual conic of Spieker circle
X(68349) = pole of line {7, 33116} with respect to the dual conic of Moses-Feuerbach circumconic
X(68349) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(58005)}}, {{A, B, C, X(9), X(1257)}}, {{A, B, C, X(63), X(4373)}}, {{A, B, C, X(75), X(5273)}}, {{A, B, C, X(253), X(27413)}}, {{A, B, C, X(273), X(5435)}}, {{A, B, C, X(329), X(39695)}}, {{A, B, C, X(672), X(7655)}}, {{A, B, C, X(903), X(28610)}}, {{A, B, C, X(5226), X(60041)}}, {{A, B, C, X(5249), X(30712)}}, {{A, B, C, X(5328), X(40424)}}, {{A, B, C, X(9965), X(36606)}}, {{A, B, C, X(10436), X(56264)}}, {{A, B, C, X(18228), X(58002)}}, {{A, B, C, X(25527), X(34399)}}, {{A, B, C, X(27382), X(35510)}}
X(68349) = barycentric product X(i)*X(j) for these (i, j): {4554, 7655}, {11523, 85}
X(68349) = barycentric quotient X(i)/X(j) for these (i, j): {7, 67941}, {109, 59061}, {664, 68195}, {7655, 650}, {11523, 9}
X(68349) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 307, 2}, {347, 56927, 145}, {18655, 68334, 3146}, {32003, 34059, 20008}, {39126, 40702, 17863}, {41804, 56927, 347}

X(68350) = ANTIPRODUCT OF X(9) AND X(11)

Barycentrics    (a-b)*(a-c)*(-2*a*(b-c)^2+a^2*(b+c)+(b-c)^2*(b+c)) : :
X(68350) = -3*X[2]+2*X[2310], -4*X[4858]+3*X[53382]

X(68350) lies on these lines: {2, 2310}, {7, 57036}, {8, 2801}, {75, 25722}, {99, 43344}, {100, 190}, {109, 65206}, {144, 28057}, {145, 53531}, {192, 42079}, {522, 4552}, {651, 2398}, {662, 7253}, {664, 23973}, {883, 926}, {1026, 25268}, {1310, 46964}, {1419, 65957}, {1441, 17668}, {1770, 67848}, {1897, 32714}, {2406, 40576}, {2765, 9070}, {2951, 45738}, {2975, 53296}, {3434, 17140}, {3667, 21362}, {3939, 62669}, {4319, 26651}, {4440, 36221}, {4454, 17165}, {4459, 13576}, {4499, 53358}, {4569, 30704}, {4579, 32735}, {4858, 53382}, {6327, 52365}, {6606, 35157}, {7192, 53350}, {9016, 25304}, {9961, 52346}, {11680, 53564}, {12530, 20245}, {14100, 20905}, {14923, 49499}, {15587, 25001}, {15726, 30807}, {17136, 68339}, {17164, 57287}, {21273, 64709}, {23529, 59688}, {24799, 63589}, {26006, 45275}, {26031, 53524}, {26669, 65952}, {28058, 60935}, {30628, 39126}, {34085, 61184}, {41906, 58992}, {43353, 53606}, {49719, 49722}, {53397, 63130}, {57928, 65642}, {62789, 66225}, {64741, 67059}, {66280, 68104}, {68106, 68118}

X(68350) = reflection of X(i) in X(j) for these {i,j}: {145, 53531}, {192, 42079}, {4552, 35338}, {66225, 62789}
X(68350) = anticomplement of X(2310)
X(68350) = trilinear pole of line {11019, 21049}
X(68350) = perspector of circumconic {{A, B, C, X(1016), X(57581)}}
X(68350) = X(i)-isoconjugate-of-X(j) for these {i, j}: {663, 63192}, {667, 56026}, {1459, 14493}, {3063, 23618}
X(68350) = X(i)-Dao conjugate of X(j) for these {i, j}: {6631, 56026}, {10001, 23618}, {11019, 3900}, {40133, 7658}, {41006, 4885}, {43182, 513}, {59573, 522}
X(68350) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {56, 17036}, {59, 144}, {109, 39351}, {249, 54107}, {651, 37781}, {658, 150}, {664, 33650}, {934, 149}, {1016, 54113}, {1020, 21221}, {1252, 30695}, {1262, 2}, {1275, 69}, {1407, 54102}, {1461, 4440}, {2149, 3177}, {2283, 14732}, {4564, 329}, {4566, 3448}, {4567, 18750}, {4569, 21293}, {4570, 45738}, {4590, 54109}, {4619, 514}, {4620, 20245}, {4998, 3436}, {6516, 34188}, {6614, 58371}, {7012, 5942}, {7045, 8}, {7115, 30694}, {7128, 5905}, {7339, 145}, {7340, 35614}, {14733, 45293}, {23586, 6604}, {23964, 46713}, {23971, 4452}, {23979, 194}, {23984, 6515}, {23985, 6392}, {23990, 46706}, {24013, 36845}, {24027, 192}, {24032, 5906}, {31615, 4462}, {35049, 14213}, {44717, 56943}, {46102, 68335}, {52378, 63}, {52610, 39352}, {53243, 44005}, {53321, 148}, {55346, 4}, {57538, 317}, {59105, 6366}, {59151, 522}, {59457, 3434}, {67038, 21286}
X(68350) = pole of line {1, 25255} with respect to the Kiepert parabola
X(68350) = pole of line {3733, 53300} with respect to the Stammler hyperbola
X(68350) = pole of line {190, 658} with respect to the Steiner circumellipse
X(68350) = pole of line {4422, 40537} with respect to the Steiner inellipse
X(68350) = pole of line {2, 85} with respect to the Yff parabola
X(68350) = pole of line {6, 10580} with respect to the Hutson-Moses hyperbola
X(68350) = pole of line {7192, 65674} with respect to the Wallace hyperbola
X(68350) = pole of line {1565, 65752} with respect to the dual conic of polar circle
X(68350) = pole of line {344, 34019} with respect to the dual conic of Feuerbach hyperbola
X(68350) = intersection, other than A, B, C, of circumconics {{A, B, C, X(100), X(68241)}}, {{A, B, C, X(513), X(53278)}}, {{A, B, C, X(883), X(59573)}}, {{A, B, C, X(3570), X(26818)}}, {{A, B, C, X(3699), X(52937)}}, {{A, B, C, X(4557), X(43344)}}, {{A, B, C, X(4569), X(4578)}}, {{A, B, C, X(6606), X(41006)}}, {{A, B, C, X(11019), X(17780)}}, {{A, B, C, X(20905), X(42720)}}, {{A, B, C, X(23343), X(40133)}}, {{A, B, C, X(23845), X(32735)}}, {{A, B, C, X(23973), X(43182)}}, {{A, B, C, X(50333), X(56323)}}, {{A, B, C, X(53337), X(60992)}}
X(68350) = barycentric product X(i)*X(j) for these (i, j): {100, 20905}, {1200, 4572}, {1978, 20978}, {3699, 60992}, {10167, 6335}, {11019, 190}, {14100, 4554}, {21049, 99}, {26818, 3952}, {30610, 59573}, {40133, 668}, {41006, 664}, {45203, 53640}, {65174, 8}
X(68350) = barycentric quotient X(i)/X(j) for these (i, j): {190, 56026}, {651, 63192}, {664, 23618}, {1200, 663}, {1783, 14493}, {10167, 905}, {11019, 514}, {14100, 650}, {20905, 693}, {20978, 649}, {21049, 523}, {22088, 1459}, {26818, 7192}, {40133, 513}, {41006, 522}, {43182, 7658}, {59573, 4885}, {60992, 3676}, {65174, 7}, {65804, 14936}
X(68350) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {100, 61185, 3952}, {190, 65200, 4578}, {522, 35338, 4552}, {4578, 65200, 17780}

X(68351) = ANTIPRODUCT OF X(9) AND X(19)

Barycentrics    a^7-a^6*(b+c)+a^4*(b-c)^2*(b+c)+a^2*(b-c)^2*(b+c)^3-a^5*(b^2+c^2)+a*(b^2-c^2)^2*(b^2+c^2)-(b+c)*(b^3-b^2*c+b*c^2-c^3)^2-a^3*(b^4-6*b^2*c^2+c^4) : :

X(68351) lies on these lines: {2, 607}, {8, 20235}, {20, 68339}, {69, 3827}, {145, 347}, {281, 26157}, {1370, 17135}, {1783, 28736}, {1951, 26655}, {3436, 68337}, {3873, 6604}, {4872, 20220}, {5738, 17016}, {6360, 6542}, {6527, 68340}, {7493, 29830}, {11396, 41007}, {16049, 30941}, {17137, 20243}, {20347, 37201}, {21279, 68338}, {29616, 56943}

X(68351) = reflection of X(i) in X(j) for these {i,j}: {607, 18639}
X(68351) = anticomplement of X(607)
X(68351) = X(i)-Ceva conjugate of X(j) for these {i, j}: {57918, 2}
X(68351) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {1, 30694}, {2, 5942}, {3, 3177}, {7, 5905}, {34, 6392}, {48, 21218}, {56, 21216}, {57, 193}, {63, 144}, {69, 329}, {73, 1655}, {75, 68335}, {77, 2}, {78, 30695}, {85, 4}, {86, 92}, {201, 46707}, {212, 46706}, {222, 192}, {269, 30699}, {270, 46713}, {273, 6515}, {279, 12649}, {304, 3436}, {305, 21286}, {307, 2895}, {326, 56943}, {331, 5906}, {332, 18750}, {337, 56883}, {348, 8}, {603, 194}, {651, 25259}, {658, 521}, {664, 4391}, {738, 11851}, {757, 62798}, {873, 68338}, {905, 39351}, {1014, 3187}, {1088, 56927}, {1214, 1654}, {1231, 1330}, {1275, 61185}, {1332, 4468}, {1395, 46712}, {1414, 525}, {1434, 3868}, {1439, 17778}, {1444, 63}, {1799, 20248}, {1803, 25237}, {1804, 6360}, {1812, 45738}, {1813, 17494}, {1814, 10025}, {3718, 54113}, {3926, 52366}, {3942, 17036}, {4025, 37781}, {4554, 20293}, {4561, 4462}, {4564, 3732}, {4565, 17498}, {4569, 46400}, {4573, 7253}, {4620, 53349}, {4625, 850}, {4626, 17896}, {4637, 65099}, {6063, 21270}, {6183, 64886}, {6516, 514}, {7013, 20211}, {7045, 651}, {7053, 3210}, {7055, 4329}, {7056, 7}, {7125, 3164}, {7177, 145}, {7182, 69}, {7183, 20}, {7210, 41361}, {7318, 2994}, {13436, 31551}, {13453, 31552}, {15413, 33650}, {17094, 21221}, {17206, 3869}, {19611, 68348}, {20567, 11442}, {27832, 3621}, {30682, 36845}, {30805, 34188}, {31637, 30807}, {33673, 14361}, {34400, 962}, {36059, 21225}, {37755, 56291}, {40152, 18666}, {40443, 9}, {43736, 48381}, {44708, 17035}, {44717, 65195}, {47487, 43989}, {51653, 7665}, {51664, 148}, {52385, 3151}, {52392, 17484}, {52411, 17486}, {52565, 52364}, {53642, 64885}, {55205, 512}, {56005, 25239}, {56382, 2475}, {56972, 9965}, {57479, 5932}, {57701, 41839}, {57738, 1959}, {57785, 17220}, {57787, 317}, {57792, 68336}, {57873, 32099}, {57918, 6327}, {57985, 14206}, {59457, 4566}, {60716, 47740}, {62277, 68343}, {63193, 40571}, {63194, 56020}, {65082, 13387}, {65164, 513}, {65232, 33294}, {65233, 31290}, {65296, 522}, {65299, 47772}, {65301, 53343}
X(68351) = pole of line {918, 3287} with respect to the DeLongchamps circle
X(68351) = pole of line {17094, 35518} with respect to the Steiner circumellipse
X(68351) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {18659, 68347, 1370}

X(68352) = ANTIPRODUCT OF X(9) AND X(20)

Barycentrics    3*a^6-4*a^5*(b+c)+4*a^3*b*c*(b+c)-a^4*(b+c)^2+a^2*(b^2-c^2)^2-(b^2-c^2)^2*(3*b^2-2*b*c+3*c^2)+4*a*(b^5-b^3*c^2-b^2*c^3+c^5) : :

X(68352) lies on these lines: {2, 7070}, {4, 8}, {7, 2897}, {100, 11347}, {145, 5930}, {189, 10431}, {223, 3870}, {253, 18655}, {280, 64003}, {306, 17784}, {516, 56943}, {518, 10374}, {1439, 3873}, {2270, 3692}, {2475, 10368}, {2975, 37046}, {3146, 68348}, {3182, 62874}, {3616, 18641}, {5759, 65684}, {5809, 11433}, {5906, 6223}, {5932, 20221}, {6350, 9778}, {7532, 9780}, {7672, 46017}, {8808, 26015}, {8809, 68349}, {9799, 12324}, {10365, 12649}, {10400, 20292}, {10430, 26871}, {14544, 18624}, {20262, 25006}, {29641, 62343}, {32064, 68334}, {32850, 55112}, {36991, 68335}, {39130, 41869}, {61185, 64143}

X(68352) = anticomplement of X(7070)
X(68352) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {7, 6225}, {56, 17037}, {64, 144}, {222, 68006}, {253, 3436}, {278, 14361}, {459, 68335}, {1073, 56943}, {2155, 3177}, {2184, 329}, {8809, 8}, {8810, 1032}, {19611, 52366}, {30457, 30695}, {33581, 21218}, {36079, 522}, {41082, 64583}, {41489, 30694}, {52158, 45738}, {56235, 4462}, {57921, 21286}, {60800, 20212}
X(68352) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 9812, 92}, {52365, 68336, 7}

X(68353) = ANTIPRODUCT OF X(13) AND X(16)

Barycentrics    -(sqrt(3)*b^2*c^2*(-2*a^4+(b^2-c^2)^2+a^2*(b^2+c^2)))+2*(a^6-a^2*b^2*c^2-(b^2-c^2)^2*(b^2+c^2))*S : :

X(68353) lies on circumconic {{A, B, C, X(3440), X(14373)}} and on these lines: {2, 11081}, {30, 298}, {265, 301}, {299, 60474}, {302, 3130}, {633, 13340}, {3181, 11085}, {11064, 40665}, {17403, 19773}, {41887, 41889}

X(68353) = anticomplement of X(11081)
X(68353) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {1094, 18301}, {2154, 3180}, {2166, 16770}, {3376, 51271}, {11086, 192}, {11092, 8}, {23284, 21221}, {50466, 6360}, {51806, 2}, {65569, 617}

X(68354) = ANTIPRODUCT OF X(15) AND X(16)

Barycentrics    a^8+a^4*b^2*c^2-2*a^6*(b^2+c^2)-(b^2-c^2)^2*(b^4+c^4)+a^2*(2*b^6-b^4*c^2-b^2*c^4+2*c^6) : :
X(68354) = -3*X[2]+2*X[50], -5*X[631]+4*X[22463], -4*X[16310]+3*X[41626]

X(68354) lies on these lines: {2, 50}, {4, 69}, {23, 325}, {26, 9723}, {30, 1272}, {94, 11071}, {95, 57805}, {99, 1273}, {183, 5169}, {290, 18125}, {297, 22151}, {328, 3153}, {524, 53416}, {631, 22463}, {850, 924}, {892, 60034}, {1007, 7493}, {1238, 11819}, {1369, 37671}, {1494, 57471}, {3001, 36163}, {3146, 52864}, {3432, 7488}, {3448, 13207}, {3933, 7540}, {3964, 7517}, {4558, 60524}, {4577, 5641}, {6148, 7809}, {7519, 37668}, {7527, 7750}, {7530, 7776}, {7552, 7752}, {7556, 7763}, {7565, 59635}, {7788, 62963}, {9145, 45918}, {9970, 51396}, {10024, 41008}, {10296, 13219}, {11061, 39099}, {11063, 18122}, {11416, 53569}, {11433, 62335}, {14118, 45198}, {14451, 40705}, {14570, 40853}, {14859, 35139}, {16310, 41626}, {18020, 41203}, {18323, 40996}, {18354, 52347}, {18365, 24975}, {18563, 41005}, {18878, 57890}, {20573, 33565}, {35296, 44388}, {37496, 39235}, {41586, 58943}, {43697, 54124}, {46439, 61490}, {46571, 62376}, {48913, 57822}, {49669, 64018}, {51481, 53507}, {52843, 68346}, {54913, 60256}, {62899, 62926}

X(68354) = reflection of X(i) in X(j) for these {i,j}: {50, 34827}, {1272, 52149}, {3146, 52864}, {4558, 60524}, {61490, 46439}
X(68354) = isogonal conjugate of X(34448)
X(68354) = isotomic conjugate of X(33565)
X(68354) = anticomplement of X(50)
X(68354) = perspector of circumconic {{A, B, C, X(6331), X(57903)}}
X(68354) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 34448}, {31, 33565}, {810, 52998}, {9247, 9381}, {34900, 62268}
X(68354) = X(i)-Dao conjugate of X(j) for these {i, j}: {2, 33565}, {3, 34448}, {50, 50}, {39062, 52998}, {40604, 51256}, {46439, 512}, {52032, 34900}, {62576, 9381}
X(68354) = X(i)-Ceva conjugate of X(j) for these {i, j}: {20573, 2}
X(68354) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {1, 18301}, {75, 1272}, {92, 12383}, {94, 8}, {265, 6360}, {328, 4329}, {476, 4560}, {1141, 17479}, {1577, 14731}, {1821, 65770}, {1989, 192}, {2166, 2}, {5627, 18668}, {6344, 5905}, {10412, 21221}, {11060, 17486}, {14206, 67092}, {14213, 67091}, {15475, 21220}, {18359, 3648}, {18384, 21216}, {18815, 41808}, {18817, 21270}, {20573, 6327}, {30690, 6224}, {32678, 31296}, {32680, 523}, {35139, 7192}, {36096, 62307}, {36129, 525}, {39295, 6758}, {43082, 4440}, {46138, 21271}, {46456, 7253}, {57716, 39118}, {63759, 69}, {66922, 1}
X(68354) = X(i)-cross conjugate of X(j) for these {i, j}: {2070, 37766}, {11597, 2}
X(68354) = pole of line {512, 11442} with respect to the anticomplementary circle
X(68354) = pole of line {512, 14592} with respect to the circumcircle of the Johnson triangle
X(68354) = pole of line {512, 47328} with respect to the polar circle
X(68354) = pole of line {1899, 44135} with respect to the Jerabek hyperbola
X(68354) = pole of line {5254, 14389} with respect to the Kiepert hyperbola
X(68354) = pole of line {249, 14570} with respect to the Kiepert parabola
X(68354) = pole of line {184, 566} with respect to the Stammler hyperbola
X(68354) = pole of line {311, 850} with respect to the Steiner circumellipse
X(68354) = pole of line {3, 2888} with respect to the Wallace hyperbola
X(68354) = pole of line {520, 1216} with respect to the dual conic of polar circle
X(68354) = pole of line {3267, 7769} with respect to the dual conic of Brocard inellipse
X(68354) = pole of line {6331, 15415} with respect to the dual conic of Jerabek hyperbola
X(68354) = pole of line {5664, 52032} with respect to the dual conic of Orthic inconic
X(68354) = pole of line {20975, 58908} with respect to the dual conic of Wallace hyperbola
X(68354) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(2070)}}, {{A, B, C, X(50), X(11597)}}, {{A, B, C, X(264), X(7578)}}, {{A, B, C, X(311), X(18020)}}, {{A, B, C, X(340), X(13582)}}, {{A, B, C, X(511), X(9380)}}, {{A, B, C, X(1235), X(5641)}}, {{A, B, C, X(1352), X(43697)}}, {{A, B, C, X(5562), X(47390)}}, {{A, B, C, X(9141), X(44138)}}, {{A, B, C, X(24978), X(44146)}}, {{A, B, C, X(34405), X(44135)}}, {{A, B, C, X(44134), X(44175)}}
X(68354) = barycentric product X(i)*X(j) for these (i, j): {2070, 76}, {11557, 40832}, {11597, 20573}, {18022, 9380}, {19552, 7769}, {24978, 99}, {37766, 69}, {58733, 7799}
X(68354) = barycentric quotient X(i)/X(j) for these (i, j): {2, 33565}, {6, 34448}, {264, 9381}, {323, 51256}, {343, 34900}, {648, 52998}, {1994, 34418}, {2070, 6}, {9380, 184}, {11557, 3003}, {11597, 50}, {19552, 2963}, {24978, 523}, {37766, 4}, {37779, 38542}, {38539, 14579}, {46439, 10413}, {58733, 1989}, {58924, 14910}
X(68354) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 69, 44135}, {30, 52149, 1272}, {50, 34827, 2}, {316, 340, 3260}, {317, 44128, 69}, {15164, 15165, 311}, {40853, 44363, 14570}, {52220, 52221, 37779}, {60524, 64783, 4558}

X(68355) = ANTIPRODUCT OF X(19) AND X(19)

Barycentrics    a^10+a^2*(b^2-c^2)^4-a^8*(b^2+c^2)-(b^2-c^2)^2*(b^2+c^2)^3-2*a^6*(b^4-6*b^2*c^2+c^4)+2*a^4*(b^6-3*b^4*c^2-3*b^2*c^4+c^6) : :

X(68355) lies on these lines: {2, 2207}, {4, 30737}, {20, 64}, {22, 3785}, {76, 37201}, {99, 30552}, {112, 28696}, {183, 59349}, {264, 6815}, {304, 4329}, {305, 315}, {393, 26154}, {401, 6515}, {441, 3172}, {858, 32816}, {1078, 7493}, {1593, 41005}, {1968, 6389}, {2071, 6337}, {2138, 46741}, {2896, 46717}, {3164, 7791}, {3575, 68346}, {3926, 11413}, {3933, 21312}, {6000, 44141}, {6816, 62698}, {7396, 62310}, {7401, 44142}, {7503, 40680}, {7767, 11414}, {7787, 63084}, {8743, 28406}, {8879, 28412}, {10996, 26166}, {11348, 18928}, {12225, 64018}, {13219, 32006}, {15589, 52404}, {16043, 22240}, {16096, 31942}, {17137, 20243}, {17138, 18659}, {28432, 56832}, {28695, 41370}, {32548, 56376}, {32815, 52071}, {32818, 65711}, {32838, 63657}, {37198, 41008}, {40995, 67885}, {52398, 58846}

X(68355) = isotomic conjugate of X(42484)
X(68355) = anticomplement of X(2207)
X(68355) = X(i)-isoconjugate-of-X(j) for these {i, j}: {31, 42484}, {1973, 2139}, {2155, 40186}
X(68355) = X(i)-Dao conjugate of X(j) for these {i, j}: {2, 42484}, {2207, 2207}, {6337, 2139}, {45245, 40186}
X(68355) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {1, 6392}, {3, 21216}, {31, 46712}, {63, 193}, {69, 5905}, {75, 6515}, {76, 5906}, {77, 30699}, {78, 30694}, {255, 194}, {304, 4}, {305, 21270}, {326, 2}, {332, 92}, {336, 51481}, {345, 5942}, {348, 12649}, {394, 192}, {520, 21220}, {561, 317}, {577, 17486}, {662, 33294}, {799, 520}, {822, 25054}, {1098, 46713}, {1102, 20}, {1259, 3177}, {1264, 329}, {1332, 25259}, {1444, 3187}, {1804, 3210}, {2167, 56017}, {2289, 21218}, {2632, 54104}, {3265, 21221}, {3682, 1655}, {3718, 68335}, {3719, 144}, {3926, 8}, {3964, 6360}, {3998, 1654}, {4020, 10340}, {4091, 9263}, {4131, 4440}, {4176, 4329}, {4558, 17498}, {4561, 4391}, {4563, 7253}, {4592, 525}, {4620, 61180}, {4625, 23683}, {6507, 3164}, {6517, 17496}, {7055, 7}, {7177, 11851}, {7182, 56927}, {7183, 145}, {15394, 18663}, {17206, 3868}, {18604, 17148}, {23224, 21224}, {24018, 148}, {24041, 648}, {28724, 17489}, {30805, 149}, {33805, 50435}, {34055, 7754}, {40364, 11442}, {46254, 35360}, {52385, 17778}, {52387, 46707}, {52396, 2895}, {52430, 8264}, {52565, 2475}, {52608, 21300}, {52616, 37781}, {52617, 21294}, {55202, 850}, {57780, 32001}, {57918, 68336}, {57955, 2052}, {57985, 62305}, {57998, 6504}, {62276, 5889}, {62277, 1993}, {62719, 110}, {65164, 521}
X(68355) = X(i)-cross conjugate of X(j) for these {i, j}: {1619, 46741}, {15259, 2}
X(68355) = pole of line {525, 2451} with respect to the DeLongchamps circle
X(68355) = pole of line {648, 56008} with respect to the Kiepert parabola
X(68355) = pole of line {3265, 52617} with respect to the Steiner circumellipse
X(68355) = pole of line {20, 159} with respect to the Wallace hyperbola
X(68355) = pole of line {5020, 40995} with respect to the dual conic of Moses HK-parabola
X(68355) = intersection, other than A, B, C, of circumconics {{A, B, C, X(64), X(1619)}}, {{A, B, C, X(253), X(40009)}}, {{A, B, C, X(2207), X(15259)}}, {{A, B, C, X(32830), X(34415)}}, {{A, B, C, X(39434), X(61088)}}
X(68355) = barycentric product X(i)*X(j) for these (i, j): {1619, 76}, {2138, 305}, {46741, 69}
X(68355) = barycentric quotient X(i)/X(j) for these (i, j): {2, 42484}, {20, 40186}, {69, 2139}, {1619, 6}, {2138, 25}, {15259, 2207}, {46741, 4}
X(68355) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {20, 253, 32830}, {20, 51884, 69}, {1370, 13575, 40123}, {1968, 6389, 26204}, {7750, 20477, 20}





leftri  Similar inscribed & circumscribed triangles: X(68356) - X(68369)  rightri

This preamble and centers X(68356)-X(68369) were contributed by César Eliud Lozada, April 22, 2025.

1) Given a triangle ABC, to find points A', B', C' such that A'B'C' is inscribed in and directly similar to ABC.

2) Given a triangle ABC, to find points A", B", C" such that A"B"C" is circumscribed to and directly similar to ABC.

Starting from A' on BC, barycentrics of vertices of T' = A'B'C' can be expressed as functions of a real parameter t, as:

 A' = 0 : 1-t : t
 B' = (a^2-b^2+c^2)*t-a^2+b^2 : 0 : -(a^2-b^2+c^2)*t+c^2
 C' = -(a^2+b^2-c^2)*t+b^2 : (a^2+b^2-c^2)*t-a^2+c^2 : 0

and, for barycentrics of vertices of T" = A"B"C", let's take the parallel lines to the sidelines of T' through A, B, C. In this way, barycentrics of vertices of T" can be expressed as functions of the same parameter t, as:

 A" = 1 : (t-1)*(-a^2+b^2+c^2)/((a^2-b^2+c^2)*t-a^2+b^2) : t*(-a^2+b^2+c^2)/((a^2+b^2-c^2)*t-b^2)
 B" = ((a^2-b^2+c^2)*t-a^2+b^2)/(-a^2+b^2+c^2) : t-1 : ((a^2-b^2+c^2)*t-c^2)*(t-1)/((a^2+b^2-c^2)*t-a^2+c^2)
 C" = ((a^2+b^2-c^2)*t-b^2)/(-a^2+b^2+c^2) : t*((a^2+b^2-c^2)*t-a^2+c^2)/((a^2-b^2+c^2)*t-c^2) : t

Now, let P=X(n) be a chosen ETC center of ABC and let P' = P-of-T' and P" = P-of-T". Then:
  1. When t varies, P' moves on a line with tripole Q'(P), equivalent to:
     Q'(P) = IsotomicConjugate(Anticomplement(IsotomicConjugate(PolarConjugate(Orthoassociate(P)))))
    or, algebraically but simpler:
     Q'(P) = BarycentricQuotient(AntigonalConjugate(P), P)
    this is, for P = x : y : z, normalized barycentrics:
     Q'(P) = ((x+1)*x*SA+(y-1)*y*SB+(z-1)*z*SC)-1 : :
  2. When t varies, P" moves on a circle through X(4) and center O"(P) = reflection of P in X(5).
  3. As constructed, triangles T' and T" are homothetic for any t, and, as t varies, their homothetic center moves on the Yff hyperbola (see Wolfram Mathworld's Yff Hyperbola).

The appearance of (i, j) in the following list means that, for P = X(i) (i<=5000), the point Q'(P) is X(j):
((1, 18359), (2, 671), (3, 94), (5, 13582), (6, 18019), (7, 41798), (8, 88), (9, 68356), (10, 6650), (11, 68357), (12, 68358), (13, 11092), (14, 11078), (15, 68359), (16, 68360), (17, 68361), (18, 68362), (20, 16080), (21, 68363), (22, 46105), (23, 76), (27, 68364), (64, 52516), (65, 52500), (66, 52513), (67, 23), (68, 52505), (69, 111), (72, 21907), (74, 46106), (76, 694), (80, 3218), (83, 17949), (98, 297), (99, 2501), (100, 17924), (101, 46107), (105, 46108), (106, 46109), (107, 525), (108, 4391), (109, 46110), (110, 14618), (111, 44146), (112, 850), (115, 13485), (125, 15351), (136, 54453), (146, 40384), (147, 34536), (148, 34537), (149, 1016), (150, 1252), (152, 59195), (153, 59196), (186, 5392), (226, 17947), (242, 39700), (264, 60039), (265, 323), (315, 1976), (316, 6), (321, 17946), (329, 34056), (376, 58268), (381, 55957), (382, 56063), (403, 6504), (468, 2996), (476, 44427), (550, 18366), (621, 6151), (622, 2981), (671, 524), (842, 60502), (858, 4), (879, 46787), (895, 3266), (915, 48380), (917, 48381), (925, 57065), (930, 67102), (933, 18314), (935, 9979), (962, 56234), (1113, 2592), (1114, 2593), (1117, 40604), (1141, 14918), (1156, 37780), (1157, 53028), (1177, 52512), (1263, 37779), (1289, 33294), (1294, 51358), (1297, 60516), (1300, 3580), (1304, 41079), (1305, 57043), (1309, 10015), (1312, 13581), (1313, 13580), (1316, 1916), (1320, 4358), (1325, 321), (1337, 41000), (1338, 41001), (1370, 60133), (1657, 66768), (1785, 2994), (1878, 39696), (1916, 3978), (1995, 55973), (2070, 11140), (2071, 2052), (2072, 13579), (2074, 43675), (2373, 5523), (2374, 47286), (2394, 51228), (2697, 50188), (3109, 4080), (3146, 44877), (3153, 275), (3154, 12066), (3254, 3935), (3436, 34051), (3448, 249), (3465, 30690), (3484, 324), (3563, 51481), (3681, 34578), (3869, 2006), (4226, 14223))

The appearance of (i, j) in the following list means that, for P = X(i) (i<=250), the point O"(P) is X(j):
(1, 355), (2, 381), (3, 4), (4, 3), (5, 5), (6, 1352), (7, 5779), (8, 1482), (9, 5805), (10, 946), (11, 119), (12, 26470), (13, 5617), (14, 5613), (15, 20428), (16, 20429), (17, 16626), (18, 16627), (20, 382), (21, 37230), (22, 31723), (23, 7574), (24, 18404), (25, 18531), (26, 18569), (27, 68365), (32, 54393), (35, 68366), (36, 68367), (37, 64088), (39, 6248), (40, 12699), (49, 58922), (51, 5891), (52, 5562), (53, 42353), (54, 6288), (55, 37820), (56, 37821), (57, 37822), (58, 37823), (61, 37824), (62, 37825), (63, 37826), (64, 5878), (65, 5887), (66, 19149), (67, 9970), (68, 155), (69, 1351), (72, 24474), (74, 7728), (75, 20430), (76, 3095), (79, 3652), (80, 6265), (83, 6287), (84, 6259), (90, 41688), (95, 42350), (98, 6033), (99, 6321), (100, 10738), (101, 10739), (102, 10740), (103, 10741), (104, 10742), (105, 10743), (106, 10744), (107, 10745), (108, 10746), (109, 10747), (110, 265), (111, 10748), (112, 10749), (113, 125), (114, 115), (115, 114), (116, 118), (117, 124), (118, 116), (119, 11), (120, 5511), (121, 5510), (122, 133), (123, 25640), (124, 117), (125, 113), (126, 5512), (127, 132), (128, 137), (129, 130), (130, 129), (131, 136), (132, 127), (133, 122), (136, 131), (137, 128), (140, 546), (141, 5480), (142, 63970), (143, 11591), (144, 60922), (145, 12645), (146, 10620), (147, 12188), (148, 13188), (149, 12331), (150, 38572), (151, 38573), (152, 38574), (153, 12773), (155, 68), (175, 68368), (176, 68369), (182, 3818), (184, 18474), (185, 12162), (186, 18403), (187, 13449), (190, 24833), (191, 16159), (193, 11898), (194, 13108), (195, 2888), (206, 51756), (214, 6246), (216, 39530), (226, 51755), (235, 11585)

underbar

X(68356) = TRIPOLE OF THE LINE-LOCUS OF X(9) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

Barycentrics    (a^2+(b-2*c)*a+(b-c)^2)*(a^2-(2*b-c)*a+(b-c)^2)/a : :

X(68356) lies on the circumhyperbola dual of Yff parabola and these lines: {2, 1111}, {7, 149}, {75, 23989}, {85, 63166}, {86, 16727}, {664, 3957}, {673, 3218}, {675, 1308}, {903, 35171}, {1223, 60969}, {2400, 36038}, {3935, 30806}, {4358, 36807}, {4373, 17154}, {4671, 39749}, {5744, 42318}, {6548, 53240}, {8024, 57925}, {9436, 18815}, {14154, 61157}, {15728, 65637}, {16750, 52394}, {17121, 43190}, {18359, 62429}, {20568, 62536}, {21296, 39695}, {24203, 29817}, {27475, 27493}, {34018, 39293}, {37206, 60970}, {37780, 38468}, {39704, 62697}, {47871, 62619}, {48571, 60479}

X(68356) = isotomic conjugate of X(3935)
X(68356) = isogonal conjugate of X(19624)
X(68356) = polar conjugate of X(60355)
X(68356) = trilinear pole of the line {142, 514} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68356) = perspector of the inconic with center X(26015)
X(68356) = touchpoint of circumhyperbola dual of Yff parabola and line {26015, 68356}
X(68356) = pole of the the tripolar of X(60355) with respect to the polar circle
X(68356) = pole of the line {3254, 3887} with respect to the Steiner circumellipse
X(68356) = pole of the line {3935, 19624} with respect to the Steiner-Wallace hyperbola
X(68356) = barycentric product X(i)*X(j) for these {i, j}: {75, 34578}, {76, 67146}, {85, 3254}, {514, 35171}, {664, 60489}, {693, 37143}, {1308, 3261}, {24002, 60488}
X(68356) = trilinear product X(i)*X(j) for these {i, j}: {2, 34578}, {7, 3254}, {75, 67146}, {513, 35171}, {514, 37143}, {651, 60489}, {693, 1308}, {1088, 42064}, {3676, 60488}, {15734, 37780}
X(68356) = trilinear quotient X(i)/X(j) for these (i, j): (2, 5526), (7, 2078), (75, 3935), (76, 17264), (85, 37787), (92, 60355), (513, 8645), (514, 22108), (693, 3887), (1088, 38459), (1308, 692), (3254, 55), (3261, 30565), (15734, 18889), (20880, 61030), (24002, 43050)

X(68357) = TRIPOLE OF THE LINE-LOCUS OF X(11) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

Barycentrics    (a^5-(b+c)*a^4+c*(3*b-2*c)*a^3-2*c*(2*b+c)*(b-c)*a^2-(b-c)*(b^3+c^3-2*b*c*(b+2*c))*a+(b^2-c^2)^2*(b-c))*(a^5-(b+c)*a^4-b*(2*b-3*c)*a^3+2*b*(b+2*c)*(b-c)*a^2+(b-c)*(b^3+c^3-2*b*c*(2*b+c))*a-(b^2-c^2)^2*(b-c)) : :

X(68357) lies on these lines: {516, 5080}, {908, 37798}, {1146, 34529}, {1262, 55153}, {3436, 35313}, {4391, 37781}, {20920, 30807}

X(68357) = cyclocevian conjugate of X(100)
X(68357) = isotomic conjugate of X(37781)
X(68357) = polar conjugate of X(60356)
X(68357) = trilinear pole of the line {676, 2804} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68357) = perspector of the inconic with center X(651)
X(68357) = pole of the the tripolar of X(60356) with respect to the polar circle
X(68357) = barycentric product X(75)*X(29374)
X(68357) = trilinear product X(2)*X(29374)
X(68357) = trilinear quotient X(i)/X(j) for these (i, j): (2, 1768), (75, 37781), (92, 60356), (908, 34345), (4564, 57105), (29374, 6)

X(68358) = TRIPOLE OF THE LINE-LOCUS OF X(12) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

Barycentrics    (-a+b+c)*(a^3+(b-c)*a^2+(b^2-b*c-c^2)*a+(b^2-c^2)*(b-c))*(a^3-(b-c)*a^2-(b^2+b*c-c^2)*a+(b^2-c^2)*(b-c))*(a^4-(2*b^2+b*c+2*c^2)*a^2-b*c*(3*b+c)*a+(b^2-c^2)^2)*(a^4-(2*b^2+b*c+2*c^2)*a^2-b*c*(b+3*c)*a+(b^2-c^2)^2) : :

X(68358) lies on these lines: {149, 39630}, {5249, 5483}, {6734, 11604}

X(68358) = intersection, other than {A, B, C}, of the circumconics through X(i), X(j) for these {i, j}: {2, 331}, {1029, 2185}
X(68358) = trilinear product X(15910)*X(21907)
X(68358) = trilinear quotient X(i)/X(j) for these (i, j): (15910, 17796), (21907, 15932)

X(68359) = TRIPOLE OF THE LINE-LOCUS OF X(15) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

Barycentrics    Sec[2*A - Pi/6] : :
Barycentrics    1/(a^2*(a^2 - b^2 - c^2 - 2*Sqrt[3]*S)*(3*(a^2 - b^2 - c^2) + 2*Sqrt[3]*S)) : :

X(68359) lies on the cubic K342b and these lines: {3, 252}, {14, 11140}, {17, 94}, {76, 55220}, {264, 11127}, {301, 302}, {324, 473}, {338, 11130}, {622, 11582}, {11126, 51268}, {16806, 40855}, {40667, 53474}, {41478, 41760}, {60502, 65346}

X(68359) = polar conjugate of X(10632)
X(68359) = isotomic conjugate of X(11126)
X(68359) = isogonal conjugate of X(11135)
X(68359) = trilinear pole of the line {623, 18314} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68359) = perspector of the inconic with center X(33530)
X(68359) = pole of the the tripolar of X(10632) with respect to the polar circle
X(68359) = pole of the line {3201, 11135} with respect to the Stammler hyperbola
X(68359) = pole of the line {11126, 11135} with respect to the Steiner-Wallace hyperbola
X(68359) = barycentric product X(i)*X(j) for these {i, j}: {14, 34389}, {17, 301}, {76, 11087}, {264, 52203}, {300, 11600}, {8603, 20573}, {8836, 11140}, {20572, 50469}, {20579, 55220}
X(68359) = trilinear product X(i)*X(j) for these {i, j}: {75, 11087}, {92, 52203}, {2154, 34389}, {2962, 8836}, {8603, 63759}
X(68359) = trilinear quotient X(i)/X(j) for these (i, j): (2, 35199), (17, 2152), (63, 64465), (75, 11126), (92, 10632), (301, 65571), (561, 11132), (2166, 11083), (2962, 8604), (8836, 2964), (11087, 31), (11600, 2151), (19779, 1095), (23994, 66262)
X(68359) = (X(11092), X(11144))-harmonic conjugate of X(52203)

X(68360) = TRIPOLE OF THE LINE-LOCUS OF X(16) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

Barycentrics    Sec[2*A + Pi/6] : :
Barycentrics    1/(a^2*(3*(a^2 - b^2 - c^2) - 2*Sqrt[3]*S)*(a^2 - b^2 - c^2 + 2*Sqrt[3]*S)) : :

X(68360) lies on the cubic K342a and these lines: {3, 252}, {13, 11140}, {18, 94}, {76, 55222}, {264, 11126}, {300, 303}, {324, 472}, {338, 11131}, {621, 11581}, {11127, 51275}, {16807, 40854}, {40668, 53474}, {41477, 41760}, {60502, 65347}

X(68360) = polar conjugate of X(10633)
X(68360) = isotomic conjugate of X(11127)
X(68360) = isogonal conjugate of X(11136)
X(68360) = trilinear pole of the line {624, 18314} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68360) = perspector of the inconic with center X(33529)
X(68360) = pole of the the tripolar of X(10633) with respect to the polar circle
X(68360) = pole of the line {3200, 11136} with respect to the Stammler hyperbola
X(68360) = pole of the line {11127, 11136} with respect to the Steiner-Wallace hyperbola
X(68360) = barycentric product X(i)*X(j) for these {i, j}: {13, 34390}, {18, 300}, {76, 11082}, {264, 52204}, {301, 11601}, {8604, 20573}, {8838, 11140}, {20572, 50468}, {20578, 55222}
X(68360) = trilinear product X(i)*X(j) for these {i, j}: {75, 11082}, {92, 52204}, {2153, 34390}, {2962, 8838}, {8604, 63759}
X(68360) = trilinear quotient X(i)/X(j) for these (i, j): (2, 35198), (18, 2151), (63, 64464), (75, 11127), (92, 10633), (300, 65572), (561, 11133), (2166, 11088), (2962, 8603), (8838, 2964), (11082, 31), (11601, 2152), (19778, 1094), (23994, 66263)
X(68360) = (X(11078), X(11143))-harmonic conjugate of X(52204)

X(68361) = TRIPOLE OF THE LINE-LOCUS OF X(17) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

Barycentrics    (2*sqrt(3)*S-a^2+b^2+c^2)*(2*sqrt(3)*(a^2-2*b^2+c^2)*S+a^4-b^2*a^2-(b^2-c^2)*c^2)*(2*sqrt(3)*(a^2+b^2-2*c^2)*S+a^4-c^2*a^2+(b^2-c^2)*b^2) : :

X(68361) lies on the cubic K185 and these lines: {2, 5469}, {524, 11117}, {8838, 41999}, {19712, 62984}, {23302, 32036}

X(68361) = reflection of X(32036) in X(23302)
X(68361) = antitomic conjugate of X(302)
X(68361) = isotomic conjugate of the antitomic conjugate of X(44361)
X(68361) = trilinear pole of the line {629, 23872} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68361) = inverse of X(11602) in Steiner circumellipse
X(68361) = barycentric product X(302)*X(11602)
X(68361) = trilinear product X(11602)*X(65571)

X(68362) = TRIPOLE OF THE LINE-LOCUS OF X(18) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

Barycentrics    (-2*sqrt(3)*S-a^2+b^2+c^2)*(-2*sqrt(3)*(a^2-2*b^2+c^2)*S+a^4-b^2*a^2-(b^2-c^2)*c^2)*(-2*sqrt(3)*(a^2+b^2-2*c^2)*S+a^4-c^2*a^2+(b^2-c^2)*b^2) : :

X(68362) lies on the cubic K185 and these lines: {2, 5470}, {524, 11118}, {8836, 42000}, {19713, 62983}, {23303, 32037}

X(68362) = reflection of X(32037) in X(23303)
X(68362) = antitomic conjugate of X(303)
X(68362) = isotomic conjugate of the antitomic conjugate of X(44362)
X(68362) = trilinear pole of the line {630, 23873} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68362) = inverse of X(11603) in Steiner circumellipse
X(68362) = barycentric product X(303)*X(11603)
X(68362) = trilinear product X(11603)*X(65572)

X(68363) = TRIPOLE OF THE LINE-LOCUS OF X(21) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

Barycentrics    (b+c)*(a^3+(b-c)*a^2+(b^2-b*c-c^2)*a+(b^2-c^2)*(b-c))*(a^3-(b-c)*a^2-(b^2+b*c-c^2)*a+(b^2-c^2)*(b-c))/a : :

X(68363) lies on the Kiepert hyperbola and these lines: {2, 16732}, {4, 2771}, {10, 1109}, {92, 60246}, {94, 48380}, {98, 1290}, {226, 24086}, {321, 338}, {648, 40395}, {671, 35156}, {1029, 17483}, {1751, 21376}, {3218, 24624}, {4358, 60251}, {4359, 60235}, {4440, 54119}, {4552, 60188}, {4707, 60074}, {5080, 55012}, {6539, 42708}, {6742, 57710}, {13576, 66280}, {20905, 36789}, {30588, 53510}, {39295, 66922}, {46105, 46108}, {59491, 66634}

X(68363) = polar conjugate of X(2074)
X(68363) = isogonal conjugate of X(19622)
X(68363) = isotomic conjugate of X(37783)
X(68363) = trilinear pole of the line {442, 523} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68363) = touchpoint of Kiepert circumhyperbola and line {62305, 68363}
X(68363) = pole of the line {8674, 47235} with respect to the polar circle
X(68363) = pole of the line {8674, 11604} with respect to the Steiner circumellipse
X(68363) = pole of the line {19622, 37783} with respect to the Steiner-Wallace hyperbola
X(68363) = barycentric product X(i)*X(j) for these {i, j}: {75, 5620}, {321, 21907}, {523, 35156}, {693, 66280}, {850, 1290}, {1441, 11604}, {1577, 65238}
X(68363) = trilinear product X(i)*X(j) for these {i, j}: {2, 5620}, {10, 21907}, {226, 11604}, {514, 66280}, {523, 65238}, {661, 35156}, {1290, 1577}
X(68363) = trilinear quotient X(i)/X(j) for these (i, j): (2, 5127), (10, 17796), (75, 37783), (92, 2074), (226, 5172), (313, 32849), (514, 42741), (661, 42670), (1290, 163), (1577, 8674), (3261, 65669), (3936, 35204), (5620, 6), (7178, 51646), (11604, 284), (14206, 16164), (21907, 58), (24006, 47235)

X(68364) = TRIPOLE OF THE LINE-LOCUS OF X(27) IN THE SIMILAR-TO-ABC INSCRIBED TRIANGLES

Barycentrics    (b+c)*(a^6-(2*b^2+c^2)*a^4+b^2*(b-c)*a^3+c*(b+c)*(b^2+b*c-c^2)*a^2-(b^2-c^2)*(b-c)*b^2*a+(b^2-c^2)*(b-c)*(b^3+b*c^2+c^3))*(a^6-(b^2+2*c^2)*a^4-c^2*(b-c)*a^3-b*(b+c)*(b^2-b*c-c^2)*a^2-(b^2-c^2)*(b-c)*c^2*a+(b^2-c^2)*(b-c)*(b^3+b^2*c+c^3)) : :

X(68364) lies on the Kiepert hyperbola and these lines: {4, 14543}, {98, 53925}, {1751, 21044}, {16080, 48381}, {21017, 60135}, {24624, 37774}

X(68364) = polar conjugate of X(57591)
X(68364) = trilinear pole of the line {440, 523} and intersection, other than {A, B, C}, of every pair of circumconics with perspectors on this line
X(68364) = pole of the the tripolar of X(57591) with respect to the polar circle
X(68364) = barycentric product X(850)*X(53925)
X(68364) = trilinear product X(1577)*X(53925)
X(68364) = trilinear quotient X(92)/X(57591)

X(68365) = CENTER OF THE CIRCLE-LOCUS OF X(27) IN THE SIMILAR-TO-ABC CIRCUMSCRIBED TRIANGLES

Barycentrics    (-a^2+b^2+c^2)*(a^8+(b+c)*a^7+b*c*a^6-2*(b^2+c^2)*(b+c)*a^5-2*(b^3+c^3)*(b+c)*a^4+(b^2-c^2)^2*(b+c)*a^3+(b^2-c^2)^2*b*c*a^2+(b^2-c^2)^4) : :
X(68365) = 3*X(3)-2*X(44243) = 4*X(140)-3*X(21162) = 4*X(140)-5*X(31256) = 3*X(381)-2*X(15762)

X(68365) lies on these lines: {2, 3}, {71, 265}, {79, 41393}, {113, 40589}, {572, 18388}, {573, 18390}, {1568, 1790}, {1762, 8251}, {2193, 45926}, {2328, 18400}, {2822, 10741}, {3017, 3284}, {3095, 46182}, {3583, 23207}, {5886, 51697}, {7687, 37508}, {8680, 20430}, {10246, 51721}, {10319, 18540}, {13851, 22080}, {14547, 18455}, {14561, 51731}, {18591, 45924}, {22139, 44665}, {40263, 41340}, {55010, 63323}

X(68365) = midpoint of X(i) and X(j) for these (i, j): {4, 3151}, {30266, 52844}
X(68365) = reflection of X(i) in X(j) for these (i, j): (3, 440), (27, 5)
X(68365) = intersection, other than {A, B, C}, of the circumconics through X(i), X(j) for these {i, j}: {27, 265}, {68, 7554}
X(68365) = anticomplement of X(15762) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(68365) = anticomplement of X(44243) with respect to the Moses-Steiner osculatory triangle
X(68365) = X(15762)-of-anti-Ehrmann-mid triangle
X(68365) = X(3151)-of-Euler triangle
X(68365) = X(440)-of-X3-ABC reflections triangle
X(68365) = X(27)-of-Johnson triangle

X(68366) = CENTER OF THE CIRCLE-LOCUS OF X(35) IN THE SIMILAR-TO-ABC CIRCUMSCRIBED TRIANGLES

Barycentrics    a^7-(b+c)*a^6-(b^2-b*c+c^2)*a^5+(b^3+c^3)*a^4-(b^4+c^4)*a^3+(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)*(b-c)*(b^3+c^3)*a-(b^2-c^2)^3*(b-c) : :
X(68366) = 3*X(5)-2*X(61520) = 3*X(35)-4*X(61520) = 3*X(381)-X(11849) = 3*X(381)-2*X(67856) = 4*X(546)-3*X(52850) = 2*X(2646)-3*X(5886) = 3*X(3584)-4*X(61512) = 3*X(5587)-X(11010) = 4*X(9955)-X(11015) = X(11012)-3*X(31159) = 2*X(37568)-5*X(61261)

X(68366) lies on these lines: {2, 33862}, {3, 25639}, {4, 8}, {5, 35}, {10, 36865}, {11, 61534}, {30, 11012}, {65, 62354}, {79, 24475}, {119, 546}, {140, 31262}, {149, 10222}, {214, 40259}, {381, 4421}, {382, 22758}, {388, 61287}, {497, 61276}, {528, 65949}, {944, 33281}, {946, 6265}, {952, 3585}, {1012, 45630}, {1056, 61282}, {1058, 61279}, {1154, 48937}, {1352, 9047}, {1385, 2475}, {1478, 11011}, {1479, 2646}, {1483, 5270}, {1484, 5563}, {1656, 58404}, {1699, 45770}, {2476, 32613}, {2779, 7728}, {2886, 7491}, {3073, 45926}, {3091, 20066}, {3560, 12953}, {3579, 6840}, {3584, 61512}, {3652, 51118}, {3655, 12116}, {3656, 26332}, {3822, 37621}, {3825, 45976}, {3838, 24299}, {3871, 59392}, {3913, 11929}, {4190, 26492}, {4294, 6867}, {4297, 47032}, {4302, 6862}, {4857, 5901}, {5046, 9956}, {5225, 6826}, {5434, 32214}, {5499, 15931}, {5587, 11010}, {5691, 11014}, {5722, 13750}, {5787, 65998}, {5805, 41688}, {5840, 6831}, {5841, 24390}, {5842, 6842}, {5881, 11280}, {5885, 20292}, {6101, 38474}, {6224, 11567}, {6253, 37406}, {6583, 11604}, {6796, 6980}, {6830, 26285}, {6839, 9955}, {6864, 61266}, {6868, 31418}, {6871, 26487}, {6885, 10591}, {6890, 35249}, {6893, 18782}, {6895, 28146}, {6901, 11230}, {6902, 11231}, {6903, 31663}, {6911, 10896}, {6914, 14794}, {6915, 59391}, {6923, 18481}, {6924, 7741}, {6928, 26446}, {6929, 37568}, {6951, 13624}, {6952, 26086}, {6971, 25440}, {6985, 36999}, {7330, 24468}, {7354, 10943}, {7681, 28452}, {7951, 32141}, {8227, 64473}, {10267, 17532}, {10269, 50239}, {10483, 32153}, {10599, 20075}, {10679, 10894}, {10680, 11235}, {10707, 45977}, {10724, 21669}, {10785, 31295}, {10915, 13272}, {11374, 64086}, {11491, 17577}, {11500, 18499}, {11680, 26286}, {11826, 37356}, {11928, 22753}, {12047, 37733}, {12114, 18544}, {12515, 12616}, {12737, 13273}, {13729, 38140}, {13743, 16761}, {14217, 64291}, {16128, 40263}, {16159, 24474}, {17502, 37163}, {17530, 31659}, {17579, 32612}, {18444, 49107}, {18447, 51751}, {18524, 63964}, {18527, 58569}, {18990, 37726}, {19925, 24042}, {22765, 24387}, {23961, 37256}, {26333, 33596}, {26386, 26413}, {26389, 26410}, {28160, 37437}, {28204, 62969}, {37251, 67857}, {37290, 65632}, {37573, 45944}, {37702, 61541}, {52383, 54350}, {61277, 65991}, {61533, 63273}

X(68366) = midpoint of X(i) and X(j) for these (i, j): {4, 52367}, {5691, 11014}, {5881, 11280}, {15908, 52837}
X(68366) = reflection of X(i) in X(j) for these (i, j): (3, 25639), (35, 5), (944, 33281), (11849, 67856), (37727, 11011), (37733, 12047)
X(68366) = anticomplement of X(33862)
X(68366) = pole of the line {1837, 37826} with respect to the Feuerbach circumhyperbola
X(68366) = isogonal conjugate of X(3652) with respect to the Johnson triangle
X(68366) = anticomplement of X(33862) with respect to these triangles: anti-Artzt, 1st anti-Brocard, anti-McCay, anticomplementary, Artzt, 1st Brocard, 1st Brocard-reflected, inner-Fermat, outer-Fermat, 1st half-diamonds, 2nd half-diamonds, 1st half-squares, 2nd half-squares, inverse-in-excircles, McCay, medial, 1st Neuberg, 2nd Neuberg, inner-Vecten, outer-Vecten
X(68366) = anticomplement of X(67856) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(68366) = complement of X(11010) with respect to the Fuhrmann triangle
X(68366) = complement of X(11849) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(68366) = X(35)-of-Johnson triangle
X(68366) = X(11849)-of-Ehrmann-mid triangle
X(68366) = X(25639)-of-X3-ABC reflections triangle
X(68366) = X(30420)-of-Ehrmann-vertex triangle (ABC acute)
X(68366) = X(33862)-of-anticomplementary triangle
X(68366) = X(52367)-of-Euler triangle
X(68366) = X(67856)-of-anti-Ehrmann-mid triangle
X(68366) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (4, 355, 68367), (4, 3434, 10526), (4, 10525, 12699), (79, 49176, 24475), (381, 11849, 67856), (1479, 6917, 5886), (6871, 37000, 26487), (6923, 48482, 18481), (10738, 37230, 946)

X(68367) = CENTER OF THE CIRCLE-LOCUS OF X(36) IN THE SIMILAR-TO-ABC CIRCUMSCRIBED TRIANGLES

Barycentrics    a^7-(b+c)*a^6-(b^2-3*b*c+c^2)*a^5+(b^2-3*b*c+c^2)*(b+c)*a^4-(b^4-4*b^2*c^2+c^4)*a^3+(b^2-c^2)*(b-c)*(b^2+3*b*c+c^2)*a^2+(b^2-c^2)^2*(b^2-3*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(68367) = 3*X(5)-2*X(61521) = 3*X(36)-4*X(61521) = 3*X(381)-X(22765) = 3*X(381)-2*X(67857) = X(484)-3*X(5587) = 2*X(1155)-5*X(61261) = 2*X(1319)-3*X(5886) = X(2077)-3*X(31160) = 4*X(5087)-X(18481) = 4*X(5122)-9*X(61263) = 4*X(5123)-3*X(26446) = 4*X(5126)-7*X(61268) = 3*X(5131)-7*X(7989) = X(5535)-5*X(18492) = 4*X(25405)-5*X(61276) = 3*X(31160)+X(52851) = 2*X(31673)+X(35459) = 3*X(32760)-4*X(61533) = 3*X(38140)-X(41347)

X(68367) lies on these lines: {2, 23961}, {3, 3814}, {4, 8}, {5, 36}, {10, 36866}, {12, 32760}, {30, 119}, {79, 61541}, {80, 14988}, {104, 37375}, {136, 64513}, {140, 31263}, {145, 23960}, {153, 28204}, {381, 535}, {382, 11499}, {388, 25405}, {484, 5587}, {497, 61287}, {515, 6265}, {516, 35460}, {519, 10738}, {529, 65948}, {546, 26470}, {758, 6246}, {912, 62354}, {952, 3583}, {993, 6980}, {1056, 61279}, {1058, 61282}, {1155, 6917}, {1319, 1478}, {1352, 9037}, {1385, 5046}, {1479, 5048}, {1483, 4857}, {1484, 65140}, {1532, 5841}, {1656, 6681}, {1737, 13273}, {1837, 53615}, {2392, 37823}, {2475, 9956}, {2478, 18857}, {2829, 6882}, {3058, 32213}, {3091, 20067}, {3149, 45631}, {3245, 18357}, {3560, 5172}, {3579, 37437}, {3582, 60759}, {3627, 5537}, {3652, 19925}, {3655, 12115}, {3656, 26333}, {3822, 7489}, {3825, 37535}, {3843, 62318}, {3851, 67710}, {4192, 30981}, {4193, 32612}, {4293, 6973}, {4299, 6959}, {4316, 61580}, {4640, 5123}, {4973, 38161}, {5087, 6256}, {5122, 6826}, {5126, 5229}, {5131, 7989}, {5183, 61258}, {5187, 26492}, {5193, 18990}, {5270, 5901}, {5450, 6971}, {5535, 7330}, {5538, 5720}, {5570, 5722}, {5691, 45770}, {5816, 67501}, {5840, 17757}, {5842, 38757}, {5844, 22938}, {5881, 18514}, {6001, 16128}, {6259, 41688}, {6284, 10942}, {6288, 18330}, {6684, 47032}, {6713, 17533}, {6838, 35250}, {6839, 38140}, {6840, 28160}, {6841, 33961}, {6842, 57288}, {6872, 26487}, {6895, 33697}, {6902, 13624}, {6909, 10728}, {6911, 12943}, {6912, 59392}, {6913, 41345}, {6914, 7951}, {6924, 10483}, {6930, 10590}, {6939, 61266}, {6941, 26286}, {6951, 11231}, {6965, 11230}, {7491, 18242}, {7686, 16159}, {7741, 32153}, {9955, 13729}, {10113, 61638}, {10222, 20060}, {10269, 17556}, {10572, 37733}, {10598, 20076}, {10679, 11236}, {10680, 10893}, {10744, 40100}, {10747, 38954}, {10826, 24467}, {10894, 37234}, {11114, 32613}, {11496, 11929}, {11500, 18542}, {11681, 26285}, {11827, 37406}, {11928, 12513}, {12515, 12761}, {12737, 12764}, {13391, 48937}, {13587, 64008}, {13743, 67856}, {15325, 23513}, {15680, 33862}, {16118, 27247}, {18455, 51889}, {18524, 38755}, {18838, 57282}, {22835, 26332}, {24475, 37702}, {26086, 27529}, {26321, 63963}, {30144, 40264}, {31649, 61512}, {31659, 57002}, {31673, 35459}, {31835, 47033}, {32141, 65134}, {33110, 38176}, {34586, 56825}, {35448, 64725}, {36001, 44982}, {36004, 66045}, {37251, 67046}, {37356, 64000}, {37725, 65632}, {38455, 64138}, {51518, 64792}, {54391, 59391}, {56790, 61553}

X(68367) = midpoint of X(i) and X(j) for these (i, j): {4, 5080}, {382, 35000}, {2077, 52851}, {5881, 64896}, {6909, 10728}, {36001, 44982}, {37725, 65632}
X(68367) = reflection of X(i) in X(j) for these (i, j): (3, 3814), (36, 5), (145, 23960), (1532, 67864), (10225, 9956), (10738, 24042), (12737, 30384), (22765, 67857), (37727, 5048), (41698, 22799)
X(68367) = anticomplementary conjugate of the anticomplement of X(23959)
X(68367) = anticomplement of X(23961)
X(68367) = orthoassociate of X(41722)
X(68367) = inverse of X(355) in Johnson triangle circumcircle
X(68367) = inverse of X(12245) in anticomplementary circle
X(68367) = inverse of X(41722) in polar circle
X(68367) = pole of the line {513, 12245} with respect to the anticomplementary circle
X(68367) = pole of the line {355, 513} with respect to the Johnson triangle circumcircle
X(68367) = pole of the line {513, 41722} with respect to the polar circle
X(68367) = pole of the line {1837, 6797} with respect to the Feuerbach circumhyperbola
X(68367) = isogonal conjugate of X(6265) with respect to the Johnson triangle
X(68367) = anticomplement of X(23961) with respect to these triangles: anti-Artzt, 1st anti-Brocard, anti-McCay, anticomplementary, Artzt, 1st Brocard, 1st Brocard-reflected, inner-Fermat, outer-Fermat, 1st half-diamonds, 2nd half-diamonds, 1st half-squares, 2nd half-squares, inverse-in-excircles, McCay, medial, 1st Neuberg, 2nd Neuberg, inner-Vecten, outer-Vecten
X(68367) = anticomplement of X(67857) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(68367) = complement of X(484) with respect to the Fuhrmann triangle
X(68367) = complement of X(22765) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(68367) = X(36)-of-Johnson triangle
X(68367) = X(3814)-of-X3-ABC reflections triangle
X(68367) = X(5080)-of-Euler triangle
X(68367) = X(22765)-of-Ehrmann-mid triangle
X(68367) = X(23961)-of-anticomplementary triangle
X(68367) = X(30370)-of-Ehrmann-vertex triangle (ABC acute)
X(68367) = X(32760)-of-outer-Johnson triangle
X(68367) = X(46031)-of-2nd Conway triangle
X(68367) = X(54073)-of-Fuhrmann triangle
X(68367) = X(67857)-of-anti-Ehrmann-mid triangle
X(68367) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {4, 5080, 34172}, {100, 36167, 64688}
X(68367) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (4, 355, 68366), (4, 3436, 10525), (4, 10526, 12699), (381, 22765, 67857), (1478, 6929, 5886), (5187, 37002, 26492), (6256, 6928, 18481), (31160, 52851, 2077)

X(68368) = CENTER OF THE CIRCLE-LOCUS OF X(175) IN THE SIMILAR-TO-ABC CIRCUMSCRIBED TRIANGLES

Barycentrics    a*(a^5-2*(2*b^2-b*c+2*c^2)*a^3+2*(b+c)*(b^2+c^2)*a^2+(3*b^2+4*b*c+3*c^2)*(b-c)^2*a-2*(b^2-c^2)^2*(b+c))-4*S*(a^4-(b+c)*a^3+2*b*c*a^2+(b^2-c^2)*(b-c)*a-(b^2-c^2)^2) : :

X(68368) lies on these lines: {3, 14121}, {5, 175}, {355, 382}, {952, 30334}, {1656, 31534}, {5587, 51763}, {7330, 51955}

X(68368) = reflection of X(i) in X(j) for these (i, j): (3, 14121), (175, 5)
X(68368) = complement of X(51763) with respect to the Fuhrmann triangle
X(68368) = X(175)-of-Johnson triangle
X(68368) = X(12277)-of-4th Euler triangle
X(68368) = X(12288)-of-3rd Euler triangle
X(68368) = X(14121)-of-X3-ABC reflections triangle
X(68368) = (X(355), X(5779))-harmonic conjugate of X(68369)

X(68369) = CENTER OF THE CIRCLE-LOCUS OF X(176) IN THE SIMILAR-TO-ABC CIRCUMSCRIBED TRIANGLES

Barycentrics    a*(a^5-2*(2*b^2-b*c+2*c^2)*a^3+2*(b+c)*(b^2+c^2)*a^2+(3*b^2+4*b*c+3*c^2)*(b-c)^2*a-2*(b^2-c^2)^2*(b+c))+4*S*(a^4-(b+c)*a^3+2*b*c*a^2+(b^2-c^2)*(b-c)*a-(b^2-c^2)^2) : :
X(68369) = 3*X(3)-2*X(8984) = 3*X(7090)-X(8984)

X(68369) lies on these lines: {3, 7090}, {5, 176}, {355, 382}, {952, 30333}, {1656, 31535}, {3579, 8986}, {5587, 51764}, {7330, 51957}

X(68369) = reflection of X(i) in X(j) for these (i, j): (3, 7090), (176, 5), (8986, 3579)
X(68369) = anticomplement of X(8984) with respect to the Moses-Steiner osculatory triangle
X(68369) = complement of X(51764) with respect to the Fuhrmann triangle
X(68369) = X(176)-of-Johnson triangle
X(68369) = X(7090)-of-X3-ABC reflections triangle
X(68369) = X(12276)-of-4th Euler triangle
X(68369) = X(12287)-of-3rd Euler triangle
X(68369) = (X(355), X(5779))-harmonic conjugate of X(68368)

X(68370) = X(6)X(474)∩X(511)X(3953)

Barycentrics    a*((b^2+c^2)*a+b^3+c^3)*(2*a^3+(b+c)*a^2-(b^2+c^2)*a+(b+c)*b*c) : :

See Antreas Hatzipolakis and César Lozada, euclid 8301.

X(68370) lies on these lines: {1, 10108}, {6, 474}, {171, 35197}, {221, 37610}, {511, 3953}, {519, 2650}, {1201, 51714}, {1480, 48897}, {3157, 5264}, {3670, 11573}, {3909, 52564}, {4424, 67968}, {5255, 6126}, {5724, 49743}, {10106, 53530}, {12575, 66659}, {18139, 37693}

X(68371) = X(3)X(1724)∩X(35)X(10108)

Barycentrics    a*(8*(b+c)*a^5+(7*b^2+18*b*c+7*c^2)*a^4-(b+c)*(9*b^2-8*b*c+9*c^2)*a^3-(7*b^4+7*c^4+b*c*(17*b^2+16*b*c+17*c^2))*a^2+(b+c)*(b^4+c^4-8*b*c*(b^2+c^2))*a-(b^2-c^2)^2*b*c) : :
X(68371) = 9*X(3)-X(48883) = 7*X(3)+X(48897) = 3*X(3)+X(48926) = 3*X(31663)-X(48924) = X(48883)+3*X(48926) = 3*X(48893)+X(48924) = 3*X(48897)-7*X(48926)

See Antreas Hatzipolakis and César Lozada, euclid 8301.

X(68371) lies on these lines: {3, 1724}, {30, 12571}, {35, 10108}, {1385, 48915}, {3579, 48909}, {5010, 49557}, {13624, 48894}, {15178, 48919}, {16192, 48907}, {17502, 37425}, {31663, 48893}, {31666, 48903}, {33697, 50416}, {39578, 64534}

X(68371) = midpoint of X(i) and X(j) for these (i, j): {15178, 48919}, {31663, 48893}

X(68372) = X(7)X(349)∩X(31)X(56)

Barycentrics    a^2*(a+b-c)*(a-b+c)*((b^2+c^2)*a+b^3+c^3) : :

See Antreas Hatzipolakis and César Lozada, euclid 8301.

X(68372) lies on these lines: {7, 349}, {12, 1463}, {31, 56}, {57, 3216}, {65, 519}, {73, 59173}, {244, 42450}, {942, 49745}, {982, 50617}, {1104, 3937}, {1122, 1439}, {1319, 17705}, {1355, 1358}, {1357, 2842}, {1412, 7342}, {1417, 13370}, {1428, 8614}, {1431, 52372}, {1469, 3214}, {2099, 63460}, {2213, 40151}, {2392, 24167}, {2594, 62739}, {3600, 20041}, {3670, 11573}, {3752, 23154}, {3784, 37549}, {4292, 45022}, {5298, 28389}, {8679, 24443}, {9433, 50513}, {11509, 15625}, {16610, 29958}, {17054, 26892}, {18360, 41346}, {18838, 20617}, {24046, 67893}, {26910, 62802}, {34502, 39780}, {37591, 41777}, {40959, 64132}, {41682, 49487}, {42448, 52541}, {56412, 64115}

X(68372) = pole of the line {1428, 3733} with respect to the incircle
X(68372) = pole of the line {3782, 41007} with respect to the circumhyperbola dual of Yff parabola
X(68372) = pole of the line {12053, 63997} with respect to the Feuerbach circumhyperbola
X(68372) = pole of the line {47795, 52595} with respect to the Steiner inellipse
X(68372) = barycentric product X(i)*X(j) for these {i,j}: {56, 17184}, {57, 3670}, {65, 18601}, {226, 52564}, {278, 11573}, {331, 23197}, {1014, 4016}, {1408, 20896}, {1412, 3454}, {1434, 20966}, {3669, 3909}, {4565, 21121}, {7203, 61167}, {7341, 20654}
X(68372) = trilinear product X(i)*X(j) for these {i,j}: {34, 11573}, {56, 3670}, {65, 52564}, {273, 23197}, {604, 17184}, {1014, 20966}, {1396, 22073}, {1400, 18601}, {1408, 3454}, {1412, 4016}, {1434, 40986}, {3909, 43924}, {16947, 20896}
X(68372) = trilinear quotient X(i)/X(j) for these (i,j): (57, 40394), (1408, 3453), (1441, 59138), (3454, 3701), (3670, 8), (3909, 3699), (4016, 2321), (11573, 78), (17184, 312), (18601, 333), (20896, 30713), (20966, 210), (21121, 4086), (22073, 3694), (23197, 212)
X(68372) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (65, 20615, 4298), (1401, 17114, 65), (1401, 63580, 50626), (17114, 50626, 63580), (50626, 63580, 65)

X(68373) = X(107)X(110)∩X(402)X(525)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 + c^2)*(-2*a^4 + a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4)*(-2*a^16 + 4*a^14*b^2 + 4*a^12*b^4 - 14*a^10*b^6 + 7*a^8*b^8 + 4*a^6*b^10 - 2*a^4*b^12 - 2*a^2*b^14 + b^16 + 4*a^14*c^2 - 20*a^12*b^2*c^2 + 18*a^10*b^4*c^2 + 22*a^8*b^6*c^2 - 32*a^6*b^8*c^2 + 10*a^2*b^12*c^2 - 2*b^14*c^2 + 4*a^12*c^4 + 18*a^10*b^2*c^4 - 60*a^8*b^4*c^4 + 28*a^6*b^6*c^4 + 30*a^4*b^8*c^4 - 18*a^2*b^10*c^4 - 2*b^12*c^4 - 14*a^10*c^6 + 22*a^8*b^2*c^6 + 28*a^6*b^4*c^6 - 56*a^4*b^6*c^6 + 10*a^2*b^8*c^6 + 10*b^10*c^6 + 7*a^8*c^8 - 32*a^6*b^2*c^8 + 30*a^4*b^4*c^8 + 10*a^2*b^6*c^8 - 14*b^8*c^8 + 4*a^6*c^10 - 18*a^2*b^4*c^10 + 10*b^6*c^10 - 2*a^4*c^12 + 10*a^2*b^2*c^12 - 2*b^4*c^12 - 2*a^2*c^14 - 2*b^2*c^14 + c^16) : :
X(68373) = X[1650] - 3 X[14401], 5 X[15183] - 3 X[52720], X[45289] - 3 X[47071]

See Antreas Hatzipolakis and Peter Moses, euclid 8302.

X(68373) lies on these lines: {107, 110}, {402, 525}, {1650, 14401}, {15183, 52720}, {15351, 45289}

X(68373) = tripolar centroid of X(23582)
X(68373) = crossdifference of every pair of points on line {1304, 3269}

X(68374) = X(107)X(110)∩X(389)X(974)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^12*b^4 - 9*a^10*b^6 + 15*a^8*b^8 - 10*a^6*b^10 + 3*a^2*b^14 - b^16 + 4*a^12*b^2*c^2 + a^10*b^4*c^2 - 20*a^8*b^6*c^2 + 10*a^6*b^8*c^2 + 20*a^4*b^10*c^2 - 19*a^2*b^12*c^2 + 4*b^14*c^2 + 2*a^12*c^4 + a^10*b^2*c^4 + 18*a^8*b^4*c^4 - 56*a^4*b^8*c^4 + 39*a^2*b^10*c^4 - 4*b^12*c^4 - 9*a^10*c^6 - 20*a^8*b^2*c^6 + 72*a^4*b^6*c^6 - 23*a^2*b^8*c^6 - 4*b^10*c^6 + 15*a^8*c^8 + 10*a^6*b^2*c^8 - 56*a^4*b^4*c^8 - 23*a^2*b^6*c^8 + 10*b^8*c^8 - 10*a^6*c^10 + 20*a^4*b^2*c^10 + 39*a^2*b^4*c^10 - 4*b^6*c^10 - 19*a^2*b^2*c^12 - 4*b^4*c^12 + 3*a^2*c^14 + 4*b^2*c^14 - c^16) : :

See Antreas Hatzipolakis and Peter Moses, euclid 8302.

X(68374) lies on these lines: {107, 110}, {133, 2970}, {389, 974}, {24930, 47202}, {47179, 62501}, {47236, 61204}

X(68374) = midpoint of X(1112) and X(67281)





leftri   Points on with the Warren G-circles: X(68375) - X(69385)  rightri

Contributed by Clark Kimberling and Peter Moses, based on notes from Benjamin Warren, April 22, 2025.

Let P be a point in the plane of a triangle ABC, and let
A'B'C' = medial triangle
Ab = reflection of A in line B'P, and define Bc and Ca cyclically
Ac = reflection of A in line C'P, and define Ba and Cb cyclically
Pa = circumcenter of ABcCb, and definer Pb and Pc cyclically.
The centroids of the following four points lie on a circle here named the Warren G(P)-circle: ABC, APbPc, BPcPa, CPaPb.

If P = p:q:r, then barycentrics for the center of the Warren G(P)-circle are given by

(a^2 - b^2 - c^2)^2*p^5 + (a^4 - 2*a^2*b^2 + b^4 + 4*a^2*c^2 - 10*b^2*c^2 - 3*c^4)*p^4*q - 2*(a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 - 2*b^2*c^2 - 5*c^4)*p^3*q^2 - 2*(a^4 - 2*a^2*b^2 + b^4 - 6*b^2*c^2 + 7*c^4)*p^2*q^3 + (a^2 - b^2 + c^2)*(a^2 - b^2 + 5*c^2)*p*q^4 + (a^4 - 2*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4)*q^5 + (a^4 + 4*a^2*b^2 - 3*b^4 - 2*a^2*c^2 - 10*b^2*c^2 + c^4)*p^4*r + 4*(2*a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 4*b^2*c^2 + c^4)*p^3*q*r - 2*(a^4 - b^4 - 10*a^2*c^2 + 2*b^2*c^2 + 5*c^4)*p^2*q^2*r - 4*(2*a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 - c^4)*p*q^3*r + (a^4 - 4*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*q^4*r - 2*(a^4 + 2*a^2*b^2 - 5*b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*p^3*r^2 - 2*(a^4 - 10*a^2*b^2 + 5*b^4 + 2*b^2*c^2 - c^4)*p^2*q*r^2 + 2*(7*a^4 - 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 6*b^2*c^2 - 3*c^4)*p*q^2*r^2 - 2*(a^4 - 2*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4)*q^3*r^2 - 2*(a^4 + 7*b^4 - 2*a^2*c^2 - 6*b^2*c^2 + c^4)*p^2*r^3 - 4*(2*a^4 + 2*a^2*b^2 - b^4 - 2*a^2*c^2 + c^4)*p*q*r^3 - 2*(a^4 - 4*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*q^2*r^3 + (a^2 + b^2 - c^2)*(a^2 + 5*b^2 - c^2)*p*r^4 + (a^4 - 2*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4)*q*r^4 + (a^4 - 4*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*r^5 : :

For example, the center of the Warren G(X(10))-circle is X(5886), and the squared radius is R*(R - 2*r) / 9.

The appearance of i in the following list of 16 points means that X(i) lies on the Warren G(X(10))-circle:

2, 5603, 32631, 61732, 67625, and 68375, 68376, ..., 68385.

underbar



X(68375) = X(1)X(2)∩X(106)X(528)

Barycentrics    a^4 - 3*a^3*b - 5*a^2*b^2 + b^4 - 3*a^3*c + 19*a^2*b*c - 2*a*b^2*c - 5*a^2*c^2 - 2*a*b*c^2 - 2*b^2*c^2 + c^4 : :
X(68375) = 2 X[1] + X[6788], X[1] + 2 X[23869], X[8] - 4 X[67340], 4 X[551] - X[67626], 4 X[1125] - X[67343], 5 X[3616] - 2 X[6789], 7 X[3622] - X[6790], X[6788] - 4 X[23869], X[47622] + 2 X[53618], 2 X[121] + X[1120], 2 X[946] + X[67723], 2 X[3656] + X[67718], X[3699] - 4 X[11731], 2 X[3756] + X[10700], X[5881] + 2 X[13625]

X(68375) lies on these lines: {1, 2}, {106, 528}, {121, 1120}, {244, 50891}, {376, 41343}, {537, 50915}, {946, 67723}, {952, 57300}, {999, 1308}, {2099, 14027}, {2743, 10679}, {3304, 13744}, {3476, 39752}, {3656, 67718}, {3667, 5603}, {3699, 11731}, {3756, 10700}, {5434, 56421}, {5881, 13625}, {10247, 53799}, {11238, 18340}, {11274, 66643}, {15170, 60687}, {17301, 67625}, {17724, 38026}, {24222, 59377}, {32577, 34719}, {46914, 50107}, {50101, 52759}

X(68375) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1647, 24864}, {1, 23869, 6788}, {36444, 36462, 1644}

X(68376) = X(2)X(512)∩X(76)X(61421)

Barycentrics    a*(a^3*b^3 - a^2*b^4 + a^4*b*c - a^3*b^2*c - a^2*b^3*c + a*b^4*c + b^5*c - a^3*b*c^2 + 2*a^2*b^2*c^2 - a*b^3*c^2 + a^3*c^3 - a^2*b*c^3 - a*b^2*c^3 - b^3*c^3 - a^2*c^4 + a*b*c^4 + b*c^5) : :
X(68376) = X[76] + 2 X[61421], X[99] + 2 X[65546], 2 X[115] + X[3903], X[316] + 2 X[63822], X[962] + 2 X[67358], 2 X[3110] - 5 X[3616], X[7983] + 2 X[50440], 4 X[11725] - X[56154], 5 X[14061] - 2 X[40608]

X(68376) lies on these lines: {2, 512}, {76, 61421}, {99, 65546}, {115, 3903}, {316, 63822}, {511, 5603}, {962, 67358}, {1621, 67360}, {2703, 5263}, {3110, 3616}, {4983, 24504}, {5170, 17127}, {7983, 50440}, {11725, 56154}, {14061, 40608}

X(68377) = X(2)X(210)∩X(7)X(840)

Barycentrics    a^5 - 3*a^4*b + 4*a^3*b^2 - 4*a^2*b^3 + a*b^4 + b^5 - 3*a^4*c + a^3*b*c + 2*a^2*b^2*c - 5*a*b^3*c - b^4*c + 4*a^3*c^2 + 2*a^2*b*c^2 + 8*a*b^2*c^2 - 4*a^2*c^3 - 5*a*b*c^3 + a*c^4 - b*c^4 + c^5 : :
X(68377) = 2 X[1] + X[18343], 2 X[120] + X[1280], X[644] - 4 X[11730], 2 X[946] + X[67724], 2 X[1083] - 5 X[3616], 4 X[1125] - X[67385], 2 X[4904] + X[10699], 4 X[5901] - X[14661]

X(68377) lies on these lines: {1, 18343}, {2, 210}, {7, 840}, {11, 3315}, {105, 5845}, {120, 1280}, {644, 11730}, {946, 67724}, {948, 39754}, {1083, 3616}, {1125, 67385}, {1647, 5219}, {1836, 60061}, {2802, 50913}, {3309, 5603}, {3487, 67382}, {4310, 24403}, {4419, 24359}, {4904, 10699}, {5525, 29660}, {5542, 9318}, {5901, 14661}, {15636, 63498}, {22769, 46586}, {33143, 61732}, {37633, 37703}, {38375, 63521}, {44675, 67387}

X(68377) = crossdifference of every pair of points on line {14411, 66513}

X(68378) = X(2)X(523)∩X(30)X(5603)

Barycentrics    a^6 - a^4*b^2 - a^3*b^3 - a^2*b^4 + a*b^5 + b^6 + a^3*b^2*c - b^5*c - a^4*c^2 + a^3*b*c^2 + 3*a^2*b^2*c^2 - a*b^3*c^2 - b^4*c^2 - a^3*c^3 - a*b^2*c^3 + 2*b^3*c^3 - a^2*c^4 - b^2*c^4 + a*c^5 - b*c^5 + c^6 : :
X(68378) = 2 X[1] + X[36154], 4 X[2] - X[50145], X[3] + 2 X[52200], X[8] + 2 X[13869], X[8] - 4 X[36155], X[13869] + 2 X[36155], 2 X[10] + X[47274], X[23] - 4 X[16332], X[23] + 2 X[67596], 2 X[16332] + X[67596], 2 X[125] + X[6742], X[858] + 2 X[16272], 2 X[946] + X[67722], X[962] + 2 X[36158], 4 X[1125] - X[47270], and many others

X(68378) lies on these lines: {1, 36154}, {2, 523}, {3, 52200}, {8, 13869}, {10, 47274}, {23, 16332}, {30, 5603}, {55, 36167}, {86, 57589}, {125, 6742}, {329, 67326}, {495, 30447}, {858, 16272}, {946, 67722}, {952, 38724}, {962, 36158}, {1109, 24955}, {1125, 47270}, {1290, 1621}, {1316, 19684}, {1699, 62493}, {2452, 5278}, {2453, 19701}, {2486, 21907}, {2611, 24916}, {3109, 3616}, {3576, 62496}, {3622, 36171}, {3624, 47273}, {3816, 5520}, {4934, 24624}, {5159, 50144}, {5196, 25557}, {5249, 67325}, {5253, 38570}, {5333, 67328}, {5432, 31522}, {5550, 47272}, {5883, 61699}, {5886, 62491}, {6739, 7984}, {6740, 11735}, {6741, 15059}, {6914, 46636}, {7424, 15950}, {10269, 46618}, {11007, 18139}, {12699, 68320}, {14731, 68282}, {14844, 37701}, {16304, 37911}, {19785, 67324}, {24145, 53564}, {25055, 62500}, {30745, 67601}, {33100, 64484}, {38028, 53809}, {47404, 59297}, {52002, 64345}, {55017, 63171}, {57325, 59382}

X(68378) = orthoptic-circle-of-the-Steiner-inellipse-inverse of X(47799)
X(68378) = orthoptic-circle-of-the-Steiner-circumellipse-inverse of X(48203)
X(68378) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3616, 38514, 3109}, {13869, 36155, 8}, {16332, 67596, 23}

X(68379) = X(2)X(165)∩X(11)X(109)

Barycentrics    a^8 - 3*a^7*b + a^6*b^2 + 2*a^5*b^3 + a^3*b^5 - 3*a^2*b^6 + b^8 - 3*a^7*c + 7*a^6*b*c - 4*a^5*b^2*c + 2*a^4*b^3*c - 7*a^3*b^4*c + 7*a^2*b^5*c - 2*a*b^6*c + a^6*c^2 - 4*a^5*b*c^2 - 4*a^4*b^2*c^2 + 6*a^3*b^3*c^2 - a^2*b^4*c^2 + 6*a*b^5*c^2 - 4*b^6*c^2 + 2*a^5*c^3 + 2*a^4*b*c^3 + 6*a^3*b^2*c^3 - 6*a^2*b^3*c^3 - 4*a*b^4*c^3 - 7*a^3*b*c^4 - a^2*b^2*c^4 - 4*a*b^3*c^4 + 6*b^4*c^4 + a^3*c^5 + 7*a^2*b*c^5 + 6*a*b^2*c^5 - 3*a^2*c^6 - 2*a*b*c^6 - 4*b^2*c^6 + c^8 : :
X(68379) = 2 X[1] + X[18328], X[103] - 4 X[62674], 2 X[118] + X[14942], X[664] - 4 X[11728], 4 X[946] - X[67568], 2 X[946] + X[67726], X[67568] + 2 X[67726], X[962] + 2 X[31852], 4 X[1125] - X[67574], 2 X[1146] + X[10697], 5 X[3091] + X[67583], 5 X[3616] - 2 X[67567], 2 X[12699] + X[67721], X[67570] - 7 X[68034]

X(68379) lies on these lines: {1, 18328}, {2, 165}, {7, 15634}, {11, 109}, {103, 62674}, {118, 14942}, {514, 5603}, {517, 61730}, {664, 11728}, {946, 67568}, {952, 15735}, {962, 31852}, {1025, 54370}, {1125, 67574}, {1146, 10697}, {1566, 2717}, {1836, 60060}, {2099, 5532}, {2723, 35184}, {3086, 67577}, {3091, 67583}, {3234, 5819}, {3616, 67567}, {5219, 15737}, {5657, 67212}, {5886, 67625}, {7416, 20988}, {12699, 67721}, {17718, 61732}, {40554, 62383}, {44675, 67576}, {46344, 61086}, {56144, 62715}, {67570, 68034}

X(68379) = reflection of X(i) in X(j) for these {i,j}: {5657, 67212}, {67625, 5886}
X(68379) = {X(946),X(67726)}-harmonic conjugate of X(67568)

X(68380) = X(2)X(3738)∩X(11)X(109)

Barycentrics    a^9 - 2*a^8*b - 2*a^7*b^2 + 6*a^6*b^3 - a^5*b^4 - 5*a^4*b^5 + 4*a^3*b^6 - 2*a*b^8 + b^9 - 2*a^8*c + 10*a^7*b*c - 8*a^6*b^2*c - 12*a^5*b^3*c + 19*a^4*b^4*c - 5*a^3*b^5*c - 7*a^2*b^6*c + 7*a*b^7*c - 2*b^8*c - 2*a^7*c^2 - 8*a^6*b*c^2 + 27*a^5*b^2*c^2 - 14*a^4*b^3*c^2 - 17*a^3*b^4*c^2 + 21*a^2*b^5*c^2 - 6*a*b^6*c^2 - b^7*c^2 + 6*a^6*c^3 - 12*a^5*b*c^3 - 14*a^4*b^2*c^3 + 36*a^3*b^3*c^3 - 14*a^2*b^4*c^3 - 7*a*b^5*c^3 + 5*b^6*c^3 - a^5*c^4 + 19*a^4*b*c^4 - 17*a^3*b^2*c^4 - 14*a^2*b^3*c^4 + 16*a*b^4*c^4 - 3*b^5*c^4 - 5*a^4*c^5 - 5*a^3*b*c^5 + 21*a^2*b^2*c^5 - 7*a*b^3*c^5 - 3*b^4*c^5 + 4*a^3*c^6 - 7*a^2*b*c^6 - 6*a*b^2*c^6 + 5*b^3*c^6 + 7*a*b*c^7 - b^2*c^7 - 2*a*c^8 - 2*b*c^8 + c^9 : :
X(68380) = X[1] - 4 X[29008], 2 X[3] + X[10771], X[4] + 2 X[53752], 2 X[11] + X[109], 4 X[11] - X[10777], 2 X[109] + X[10777], X[80] + 2 X[11700], X[100] - 4 X[6718], X[102] - 4 X[6713], X[104] + 2 X[117], 2 X[124] - 5 X[31272], X[149] + 2 X[53742], X[151] + 2 X[53748], 5 X[631] - 2 X[53740], 4 X[1387] - X[10703], X[1484] + 2 X[61571], and many others

X(68380) lies on these lines: {1, 18341}, {2, 3738}, {3, 10771}, {4, 53752}, {11, 109}, {80, 11700}, {100, 6718}, {102, 6713}, {104, 117}, {124, 31272}, {149, 53742}, {151, 53748}, {499, 38507}, {631, 53740}, {900, 61732}, {952, 51408}, {1387, 10703}, {1484, 61571}, {1638, 67625}, {1795, 8068}, {2222, 57446}, {2800, 5603}, {2804, 67628}, {2818, 57298}, {5131, 62496}, {5433, 38565}, {5840, 38697}, {6702, 13532}, {7972, 47115}, {10698, 11727}, {10716, 45310}, {10724, 38785}, {10726, 38761}, {10732, 65948}, {10738, 38607}, {10740, 38602}, {10742, 61578}, {10747, 60759}, {10767, 53717}, {10768, 53724}, {10769, 53734}, {10778, 53758}, {10779, 53759}, {11715, 50899}, {12736, 34242}, {14217, 14690}, {17660, 58600}, {21154, 38691}, {22938, 38777}, {26492, 38501}, {33650, 66063}, {34126, 38776}, {38693, 64507}, {39270, 56638}, {58419, 64008}, {59391, 64501}, {67453, 67477}

X(68380) = reflection of X(i) in X(j) for these {i,j}: {38691, 21154}, {38776, 34126}
X(68380) = reflection of X(61732) in the IN line
X(68380) = {X(11),X(109)}-harmonic conjugate of X(10777)

X(68381) = X(2)X(51)∩X(4)X(3110)

Barycentrics    a*(-(a^7*b^3) + a^6*b^4 + 2*a^5*b^5 - 2*a^4*b^6 - a^3*b^7 + a^2*b^8 + a^8*b*c - a^7*b^2*c - 3*a^6*b^3*c + a^5*b^4*c + 4*a^4*b^5*c - a^3*b^6*c - 3*a^2*b^7*c + a*b^8*c + b^9*c - a^7*b*c^2 + 2*a^6*b^2*c^2 + a^5*b^3*c^2 - 2*a^4*b^4*c^2 + a^3*b^5*c^2 - a*b^7*c^2 - a^7*c^3 - 3*a^6*b*c^3 + a^5*b^2*c^3 + a^4*b^3*c^3 - a^3*b^4*c^3 + a^2*b^5*c^3 - a*b^6*c^3 - 3*b^7*c^3 + a^6*c^4 + a^5*b*c^4 - 2*a^4*b^2*c^4 - a^3*b^3*c^4 + 2*a^2*b^4*c^4 + a*b^5*c^4 + 2*a^5*c^5 + 4*a^4*b*c^5 + a^3*b^2*c^5 + a^2*b^3*c^5 + a*b^4*c^5 + 4*b^5*c^5 - 2*a^4*c^6 - a^3*b*c^6 - a*b^3*c^6 - a^3*c^7 - 3*a^2*b*c^7 - a*b^2*c^7 - 3*b^3*c^7 + a^2*c^8 + a*b*c^8 + b*c^9) : :
X(68381) = X[4] + 2 X[3110], 2 X[114] + X[56154], 5 X[631] - 2 X[67358], X[3903] - 4 X[11724], X[7970] + 2 X[40608], 2 X[50440] - 5 X[64089]

X(68381) lies on these lines: {2, 51}, {4, 3110}, {114, 56154}, {512, 5603}, {631, 67358}, {3903, 11724}, {5883, 67625}, {6905, 67360}, {7970, 40608}, {50440, 64089}

X(68382) = X(1)X(18339)∩X(2)X(521)

Barycentrics    a^9 - a^8*b - 4*a^7*b^2 + 4*a^6*b^3 + 4*a^5*b^4 - 4*a^4*b^5 - a*b^8 + b^9 - a^8*c + 9*a^7*b*c - 4*a^6*b^2*c - 11*a^5*b^3*c + 6*a^4*b^4*c - a^3*b^5*c + 3*a*b^7*c - b^8*c - 4*a^7*c^2 - 4*a^6*b*c^2 + 14*a^5*b^2*c^2 - 2*a^4*b^3*c^2 - 8*a^3*b^4*c^2 + 8*a^2*b^5*c^2 - 2*a*b^6*c^2 - 2*b^7*c^2 + 4*a^6*c^3 - 11*a^5*b*c^3 - 2*a^4*b^2*c^3 + 18*a^3*b^3*c^3 - 8*a^2*b^4*c^3 - 3*a*b^5*c^3 + 2*b^6*c^3 + 4*a^5*c^4 + 6*a^4*b*c^4 - 8*a^3*b^2*c^4 - 8*a^2*b^3*c^4 + 6*a*b^4*c^4 - 4*a^4*c^5 - a^3*b*c^5 + 8*a^2*b^2*c^5 - 3*a*b^3*c^5 - 2*a*b^2*c^6 + 2*b^3*c^6 + 3*a*b*c^7 - 2*b^2*c^7 - a*c^8 - b*c^8 + c^9 : :
X(68382) = X[1] + 2 X[66066], 2 X[123] + X[13138]

X(68382) lies on these lines: {1, 18339}, {2, 521}, {55, 28347}, {57, 67568}, {123, 13138}, {354, 5603}, {497, 2720}, {1364, 60356}, {5722, 15524}, {10584, 63757}, {12115, 64512}, {14257, 41084}, {18391, 67426}

X(68383) = X(2)X(6003)∩X(5)X(580)

Barycentrics    a^9 - 2*a^8*b - 2*a^7*b^2 + 6*a^6*b^3 - a^5*b^4 - 5*a^4*b^5 + 4*a^3*b^6 - 2*a*b^8 + b^9 - 2*a^8*c + 6*a^7*b*c - 2*a^6*b^2*c - 6*a^5*b^3*c + 7*a^4*b^4*c - 3*a^3*b^5*c - a^2*b^6*c + 3*a*b^7*c - 2*b^8*c - 2*a^7*c^2 - 2*a^6*b*c^2 + 7*a^5*b^2*c^2 - 4*a^4*b^3*c^2 - 5*a^3*b^4*c^2 + 7*a^2*b^5*c^2 - b^7*c^2 + 6*a^6*c^3 - 6*a^5*b*c^3 - 4*a^4*b^2*c^3 + 12*a^3*b^3*c^3 - 6*a^2*b^4*c^3 - 3*a*b^5*c^3 + 5*b^6*c^3 - a^5*c^4 + 7*a^4*b*c^4 - 5*a^3*b^2*c^4 - 6*a^2*b^3*c^4 + 4*a*b^4*c^4 - 3*b^5*c^4 - 5*a^4*c^5 - 3*a^3*b*c^5 + 7*a^2*b^2*c^5 - 3*a*b^3*c^5 - 3*b^4*c^5 + 4*a^3*c^6 - a^2*b*c^6 + 5*b^3*c^6 + 3*a*b*c^7 - b^2*c^7 - 2*a*c^8 - 2*b*c^8 + c^9 : :
X(68383) = X[643] + 2 X[42425], X[2606] + 2 X[14680]

X(68383) lies on these lines: {2, 6003}, {5, 580}, {643, 42425}, {758, 5603}, {867, 2328}, {2606, 14680}, {2886, 65881}, {6912, 68244}, {6929, 34172}, {13478, 56835}

X(68384) = X(1)X(18339)∩X(2)X(515)

Barycentrics    a^10 - 3*a^9*b - 2*a^8*b^2 + 9*a^7*b^3 - a^6*b^4 - 9*a^5*b^5 + 5*a^4*b^6 + 3*a^3*b^7 - 4*a^2*b^8 + b^10 - 3*a^9*c + 13*a^8*b*c - 11*a^7*b^2*c - 17*a^6*b^3*c + 29*a^5*b^4*c - 5*a^4*b^5*c - 13*a^3*b^6*c + 9*a^2*b^7*c - 2*a*b^8*c - 2*a^8*c^2 - 11*a^7*b*c^2 + 36*a^6*b^2*c^2 - 20*a^5*b^3*c^2 - 25*a^4*b^4*c^2 + 29*a^3*b^5*c^2 - 6*a^2*b^6*c^2 + 2*a*b^7*c^2 - 3*b^8*c^2 + 9*a^7*c^3 - 17*a^6*b*c^3 - 20*a^5*b^2*c^3 + 50*a^4*b^3*c^3 - 19*a^3*b^4*c^3 - 9*a^2*b^5*c^3 + 6*a*b^6*c^3 - a^6*c^4 + 29*a^5*b*c^4 - 25*a^4*b^2*c^4 - 19*a^3*b^3*c^4 + 20*a^2*b^4*c^4 - 6*a*b^5*c^4 + 2*b^6*c^4 - 9*a^5*c^5 - 5*a^4*b*c^5 + 29*a^3*b^2*c^5 - 9*a^2*b^3*c^5 - 6*a*b^4*c^5 + 5*a^4*c^6 - 13*a^3*b*c^6 - 6*a^2*b^2*c^6 + 6*a*b^3*c^6 + 2*b^4*c^6 + 3*a^3*c^7 + 9*a^2*b*c^7 + 2*a*b^2*c^7 - 4*a^2*c^8 - 2*a*b*c^8 - 3*b^2*c^8 + c^10 : :
X(68384) = 2 X[1] + X[18339], 2 X[4] + X[67714], 4 X[5] - X[18340], 2 X[117] + X[51565], 2 X[355] + X[67476], X[944] + 2 X[67226], 2 X[946] + X[67725], 4 X[1125] - X[67466], 4 X[1385] - X[67464], X[1897] - 4 X[11727], 2 X[2968] + X[10696], 5 X[3616] - 2 X[31866], 5 X[8227] - 2 X[51889], X[51422] + 2 X[60758], 4 X[67460] - 7 X[68034]

X(68384) lies on these lines: {1, 18339}, {2, 515}, {4, 67714}, {5, 18340}, {117, 51565}, {355, 67476}, {381, 57320}, {517, 67628}, {522, 5603}, {944, 67226}, {946, 67725}, {952, 51408}, {999, 67461}, {1012, 2716}, {1125, 67466}, {1385, 67464}, {1478, 61481}, {1785, 23708}, {1897, 11727}, {2222, 6911}, {2968, 10696}, {3086, 67471}, {3616, 31866}, {5886, 61732}, {6830, 56690}, {8227, 51889}, {34231, 39762}, {37820, 67477}, {44675, 67470}, {51422, 60758}, {67460, 68034}

X(68384) = reflection of X(i) in X(j) for these {i,j}: {61732, 5886}, {67635, 3576}

X(68385) = X(2)X(3)∩X(523)X(5603)

Barycentrics    a^10 - 2*a^9*b - 3*a^8*b^2 + 5*a^7*b^3 + 2*a^6*b^4 - 3*a^5*b^5 + 2*a^4*b^6 - a^3*b^7 - 3*a^2*b^8 + a*b^9 + b^10 - 2*a^9*c + 4*a^8*b*c - a^7*b^2*c - 4*a^6*b^3*c + 6*a^5*b^4*c - 3*a^4*b^5*c - a^3*b^6*c + 2*a^2*b^7*c - 2*a*b^8*c + b^9*c - 3*a^8*c^2 - a^7*b*c^2 + 5*a^6*b^2*c^2 - 5*a^5*b^3*c^2 - 4*a^4*b^4*c^2 + 7*a^3*b^5*c^2 + 5*a^2*b^6*c^2 - a*b^7*c^2 - 3*b^8*c^2 + 5*a^7*c^3 - 4*a^6*b*c^3 - 5*a^5*b^2*c^3 + 10*a^4*b^3*c^3 - 5*a^3*b^4*c^3 - 2*a^2*b^5*c^3 + 5*a*b^6*c^3 - 4*b^7*c^3 + 2*a^6*c^4 + 6*a^5*b*c^4 - 4*a^4*b^2*c^4 - 5*a^3*b^3*c^4 - 4*a^2*b^4*c^4 - 3*a*b^5*c^4 + 2*b^6*c^4 - 3*a^5*c^5 - 3*a^4*b*c^5 + 7*a^3*b^2*c^5 - 2*a^2*b^3*c^5 - 3*a*b^4*c^5 + 6*b^5*c^5 + 2*a^4*c^6 - a^3*b*c^6 + 5*a^2*b^2*c^6 + 5*a*b^3*c^6 + 2*b^4*c^6 - a^3*c^7 + 2*a^2*b*c^7 - a*b^2*c^7 - 4*b^3*c^7 - 3*a^2*c^8 - 2*a*b*c^8 - 3*b^2*c^8 + a*c^9 + b*c^9 + c^10 : :
X(68385) = X[4] + 2 X[3109], 4 X[5] - X[36154], 5 X[631] - 2 X[36158], 7 X[3090] - 4 X[36155], 5 X[3091] + X[36171], X[36001] - 4 X[44910], 2 X[113] + X[6740], 2 X[946] + X[47270], 4 X[1125] - X[67722], 2 X[3656] + X[50145], 2 X[6739] - 5 X[64101], 2 X[6741] + X[7978], X[6742] - 4 X[11723], 5 X[10595] - 2 X[13869], 5 X[11522] + X[47273], 4 X[13464] - X[47274], 5 X[18493] - 2 X[52200], X[38514] - 7 X[68034], 2 X[47471] + X[50144]

X(68385) lies on these lines: {2, 3}, {113, 6740}, {523, 5603}, {946, 47270}, {1125, 67722}, {1699, 62496}, {3576, 62493}, {3656, 50145}, {5520, 7680}, {5627, 56402}, {5886, 62491}, {6739, 64101}, {6741, 7978}, {6742, 11723}, {10176, 67634}, {10595, 13869}, {11522, 47273}, {13464, 47274}, {18493, 52200}, {34301, 51654}, {36518, 53794}, {37701, 61732}, {37735, 50148}, {38021, 62500}, {38034, 53809}, {38514, 68034}, {45924, 61479}, {47471, 50144}

X(68385) = reflection of X(44280) in X(28462)

X(68386) = X(1)X(3024)∩X(12)X(8287)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(b + c)*(a^2 - b^2 - b*c - c^2)*(a^2*b - b^3 + a^2*c + 4*a*b*c + 2*b^2*c + 2*b*c^2 - c^3) : :
X(68386) = 5 X[18398] - X[33642]

See Antreas Hatzipolakis and Peter Moses, euclid 8304.

X(68386) lies on these lines: {1, 3024}, {12, 8287}, {56, 40214}, {57, 2940}, {354, 56849}, {942, 39751}, {1319, 2392}, {1365, 13751}, {3649, 5045}, {7144, 16577}, {11553, 22461}, {17637, 59818}, {18398, 33642}, {39791, 44913}

X(68386) = X(58565)-Dao conjugate of X(8)
X(68386) = crosspoint of X(7) and X(16577)
X(68386) = barycentric product X(16577)*X(58565)

X(68387) = X(2)X(65240)∩X(36)X(63868)

Barycentrics    a*(a - b)*(a - c)*(a + b - c)*(a - b + c)*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c + a*b*c + b^2*c - a*c^2 - b*c^2 - c^3)*(a^3 + a^2*b - a*b^2 - b^3 - a^2*c + a*b*c - b^2*c - a*c^2 + b*c^2 + c^3) : :

See Antreas Hatzipolakis and Peter Moses, euclid 8315.

X(68387) lies on the Mandart circumellipse and these lines: {2, 65240}, {36, 63868}, {88, 18593}, {100, 34921}, {162, 37966}, {190, 57066}, {411, 46037}, {514, 38340}, {651, 14838}, {653, 65100}, {655, 14147}, {673, 19302}, {1156, 3065}, {2349, 3218}, {3911, 18653}, {4564, 37212}, {21739, 34234}, {27003, 65249}, {36101, 60989}, {37131, 60948}, {37787, 65261}

X(68387) = isogonal conjugate of X(68388)
X(68387) = X(i)-cross conjugate of X(j) for these (i,j): {650, 63868}, {4120, 1476}, {5131, 7045}, {8674, 7}, {14400, 21}, {53283, 99}, {61708, 52377}, {62359, 55346}
X(68387) = X(i)-isoconjugate of X(j) for these (i,j): {2, 42657}, {9, 59837}, {37, 35055}, {484, 650}, {522, 19297}, {663, 17484}, {3063, 17791}, {3064, 23071}, {3709, 56935}, {3737, 21864}, {4560, 58285}, {4895, 47058}, {9404, 50148}, {11076, 35057}, {26744, 66284}, {50462, 65105}, {52356, 66970}, {61214, 66012}
X(68387) = X(i)-Dao conjugate of X(j) for these (i,j): {478, 59837}, {10001, 17791}, {32664, 42657}, {40589, 35055}
X(68387) = cevapoint of X(i) and X(j) for these (i,j): {3, 14395}, {36, 650}, {513, 62211}
X(68387) = crosssum of X(661) and X(58900)
X(68387) = trilinear pole of line {1, 399}
X(68387) = barycentric product X(i)*X(j) for these {i,j}: {75, 34921}, {109, 40716}, {651, 21739}, {664, 3065}, {4554, 19302}, {4564, 60486}, {4585, 26743}, {7343, 65292}, {14147, 17078}
X(68387) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 42657}, {56, 59837}, {58, 35055}, {109, 484}, {651, 17484}, {664, 17791}, {1414, 56935}, {1415, 19297}, {1983, 26744}, {3065, 522}, {4559, 21864}, {7343, 35057}, {14147, 36910}, {19302, 650}, {21739, 4391}, {26700, 50148}, {26743, 60074}, {34921, 1}, {36059, 23071}, {40716, 35519}, {59837, 31522}, {60486, 4858}, {61231, 66012}, {63868, 52356}

X(68388) = X(2)X(9034)∩X(44)X(513)

Barycentrics    a*(a - b - c)*(b - c)*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2 - c^3) : :
X(68388) = 5 X[650] - 2 X[13401], X[650] - 4 X[14298], 7 X[650] - 4 X[14300], 5 X[650] - 8 X[40137], X[650] + 2 X[46389], X[661] + 2 X[65450], X[13401] - 10 X[14298], 7 X[13401] - 10 X[14300], X[13401] - 4 X[40137], X[13401] + 5 X[46389], 7 X[14298] - X[14300], 5 X[14298] - 2 X[40137], 2 X[14298] + X[46389], 5 X[14300] - 14 X[40137], 2 X[14300] + 7 X[46389], 4 X[40137] + 5 X[46389], 2 X[4131] - 5 X[31250]

See Antreas Hatzipolakis and Peter Moses, euclid 8315.

X(68388) lies on these lines: {2, 9034}, {6, 14399}, {44, 513}, {51, 2878}, {101, 26700}, {521, 4944}, {526, 1637}, {926, 11193}, {1639, 3738}, {2170, 38358}, {2316, 2341}, {2775, 5540}, {2827, 4773}, {2850, 14395}, {3196, 53527}, {3700, 35057}, {3887, 66026}, {4120, 8674}, {4131, 31250}, {4931, 8702}, {8774, 47784}, {9001, 47881}, {21297, 40166}, {26744, 35055}, {34151, 52985}, {40584, 57174}, {59817, 65707}

X(68388) = isogonal conjugate of X(68387)
X(68388) = X(i)-complementary conjugate of X(j) for these (i,j): {42, 3258}, {476, 3741}, {1918, 18334}, {1989, 116}, {2166, 21252}, {6186, 51402}, {6187, 6741}, {8750, 1511}, {11060, 1086}, {14560, 1125}, {32678, 3739}, {32680, 21240}, {32739, 34834}, {39295, 52602}, {52153, 2968}, {56193, 31845}
X(68388) = X(i)-Ceva conjugate of X(j) for these (i,j): {651, 6126}, {1290, 55}, {4591, 3057}, {35055, 42657}, {62928, 11}
X(68388) = X(42657)-cross conjugate of X(59837)
X(68388) = X(i)-isoconjugate of X(j) for these (i,j): {2, 34921}, {59, 60486}, {109, 21739}, {651, 3065}, {664, 19302}, {1415, 40716}, {1443, 14147}, {7343, 38340}
X(68388) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 21739}, {1146, 40716}, {6615, 60486}, {32664, 34921}, {38991, 3065}, {39025, 19302}
X(68388) = cevapoint of X(661) and X(58900)
X(68388) = crosspoint of X(80) and X(651)
X(68388) = crosssum of X(i) and X(j) for these (i,j): {3, 14395}, {36, 650}, {513, 62211}
X(68388) = crossdifference of every pair of points on line {1, 399}
X(68388) = barycentric product X(i)*X(j) for these {i,j}: {8, 59837}, {10, 35055}, {75, 42657}, {484, 522}, {650, 17484}, {663, 17791}, {1639, 47058}, {4041, 56935}, {4391, 19297}, {4560, 21864}, {6126, 52356}, {11076, 57066}, {18155, 58285}, {23071, 44426}, {26744, 60074}, {35057, 50148}
X(68388) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 34921}, {484, 664}, {522, 40716}, {650, 21739}, {663, 3065}, {2170, 60486}, {3063, 19302}, {11076, 38340}, {17484, 4554}, {17791, 4572}, {19297, 651}, {21864, 4552}, {23071, 6516}, {26744, 4585}, {35055, 86}, {42657, 1}, {50148, 65292}, {56935, 4625}, {58285, 4551}, {59837, 7}
X(68388) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {654, 46393, 650}, {661, 9404, 650}, {2590, 2591, 9404}, {2610, 3013, 21894}, {13401, 40137, 650}, {14298, 46389, 650}, {14298, 65450, 9404}, {46393, 65680, 654}

X(68389) = X(3)X(49)∩X(4)X(96)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^12 - 5*a^10*b^2 + 10*a^8*b^4 - 10*a^6*b^6 + 5*a^4*b^8 - a^2*b^10 - 5*a^10*c^2 + 8*a^8*b^2*c^2 - 6*a^4*b^6*c^2 + 5*a^2*b^8*c^2 - 2*b^10*c^2 + 10*a^8*c^4 + 2*a^4*b^4*c^4 - 4*a^2*b^6*c^4 + 8*b^8*c^4 - 10*a^6*c^6 - 6*a^4*b^2*c^6 - 4*a^2*b^4*c^6 - 12*b^6*c^6 + 5*a^4*c^8 + 5*a^2*b^2*c^8 + 8*b^4*c^8 - a^2*c^10 - 2*b^2*c^10) : :

X(68389) lies on the cubic K1398 and these lines: {3, 49}, {4, 96}, {30, 8800}, {32, 45089}, {50, 64037}, {52, 2351}, {97, 18925}, {569, 52435}, {570, 7592}, {577, 6146}, {578, 54034}, {1157, 64032}, {2986, 60166}, {3003, 35603}, {3133, 10539}, {3135, 6759}, {3564, 8905}, {4558, 64756}, {5889, 66602}, {7399, 54393}, {11411, 52350}, {12160, 44200}, {12241, 61748}, {12420, 15478}, {14516, 63835}, {14806, 15032}, {15512, 17834}, {34799, 63762}, {44665, 66482}

X(68389) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 155, 52032}, {4, 8883, 571}, {15653, 51458, 1147}

X(68390) = X(1)X(1073)∩X(4)X(1903)

Barycentrics    a*(b + c)*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c + 2*a*b*c + b^2*c - a*c^2 - b*c^2 - c^3)*(a^3 + a^2*b - a*b^2 - b^3 - a^2*c + 2*a*b*c - b^2*c - a*c^2 + b*c^2 + c^3)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 - 4*a^4*b^2*c^2 + 4*a^2*b^4*c^2 + 4*b^6*c^2 + 6*a^4*c^4 + 4*a^2*b^2*c^4 - 10*b^4*c^4 - 4*a^2*c^6 + 4*b^2*c^6 + c^8) : :

X(68390) lies on the cubic K033 and these lines: {1, 1073}, {4, 1903}, {8, 1032}, {40, 3348}, {65, 7157}, {271, 37228}, {280, 62864}, {282, 5706}, {1712, 3343}, {6617, 8886}, {15324, 40117}, {44547, 57492}

X(68390) = X(8)-Ceva conjugate of X(39130)
X(68390) = X(i)-isoconjugate of X(j) for these (i,j): {2360, 3346}, {3194, 47849}, {28783, 41083}
X(68390) = X(i)-Dao conjugate of X(j) for these (i,j): {1073, 41082}, {8808, 7}, {13613, 64885}
X(68390) = barycentric product X(i)*X(j) for these {i,j}: {280, 8807}, {321, 8886}, {1712, 56944}, {1903, 6527}, {8803, 34404}, {14361, 52389}, {41086, 47435}
X(68390) = barycentric quotient X(i)/X(j) for these {i,j}: {1033, 3194}, {1498, 1817}, {1712, 41083}, {1903, 3346}, {3343, 41082}, {8803, 223}, {8807, 347}, {8886, 81}, {41086, 3344}, {41087, 47849}, {52384, 8810}, {52389, 1032}, {53013, 8805}, {58895, 6129}

X(68391) = X(1)X(3344)∩X(8)X(1032)

Barycentrics    (a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 2*a*b*c + b^2*c - a*c^2 + b*c^2 - c^3)*(a^8 + 4*a^6*b^2 - 10*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 4*a^4*b^2*c^2 + 4*a^2*b^4*c^2 - 4*b^6*c^2 + 6*a^4*c^4 - 4*a^2*b^2*c^4 + 6*b^4*c^4 - 4*a^2*c^6 - 4*b^2*c^6 + c^8)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 + 4*a^6*c^2 + 4*a^4*b^2*c^2 - 4*a^2*b^4*c^2 - 4*b^6*c^2 - 10*a^4*c^4 + 4*a^2*b^2*c^4 + 6*b^4*c^4 + 4*a^2*c^6 - 4*b^2*c^6 + c^8) : :

X(68391) lies on the cubic K033 and these lines: {1, 3344}, {8, 1032}, {10, 64987}, {40, 3182}, {72, 3176}, {962, 46353}, {11529, 63866}, {40836, 55063}

X(68391) = isogonal conjugate of X(8886)
X(68391) = X(1032)-Ceva conjugate of X(8805)
X(68391) = X(i)-cross conjugate of X(j) for these (i,j): {65, 40}, {41088, 7952}
X(68391) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8886}, {84, 1498}, {285, 8803}, {1033, 41081}, {1433, 1712}, {2208, 6527}, {6616, 60799}, {6617, 7129}
X(68391) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 8886}, {281, 14361}, {3350, 41084}
X(68391) = crosspoint of X(41080) and X(63877)
X(68391) = barycentric product X(i)*X(j) for these {i,j}: {329, 3346}, {347, 8805}, {1032, 7952}, {7080, 8810}, {41088, 47633}, {47849, 64211}
X(68391) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8886}, {198, 1498}, {227, 8807}, {329, 6527}, {2331, 1712}, {3195, 1033}, {3344, 41084}, {3346, 189}, {7078, 6617}, {7952, 14361}, {8805, 280}, {8810, 1440}, {28783, 1433}, {41088, 3343}, {47849, 41081}

X(68392) = X(1)X(3342)∩X(4)X(8805)

Barycentrics    a*(b + c)*(a^2 - b^2 - c^2)*(a^6 + 2*a^5*b - a^4*b^2 - 4*a^3*b^3 - a^2*b^4 + 2*a*b^5 + b^6 - 2*a^5*c + 2*a^4*b*c + 2*a*b^4*c - 2*b^5*c - a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 + 4*a^3*c^3 + 4*b^3*c^3 - a^2*c^4 - 2*a*b*c^4 - b^2*c^4 - 2*a*c^5 - 2*b*c^5 + c^6)*(a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - a^2*b^4 - 2*a*b^5 + b^6 + 2*a^5*c + 2*a^4*b*c - 2*a*b^4*c - 2*b^5*c - a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - 4*a^3*c^3 + 4*b^3*c^3 - a^2*c^4 + 2*a*b*c^4 - b^2*c^4 + 2*a*c^5 - 2*b*c^5 + c^6) : :

X(68392) lies on the cubic K033 and these lines: {1, 3342}, {4, 8805}, {8, 1034}, {40, 219}, {72, 5930}, {1073, 47849}, {1259, 6617}, {1264, 57782}, {10373, 60802}, {47441, 55063}

X(68392) = isogonal conjugate of X(8885)
X(68392) = X(1034)-Ceva conjugate of X(8806)
X(68392) = X(i)-cross conjugate of X(j) for these (i,j): {65, 72}, {41087, 1214}, {53012, 52389}
X(68392) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8885}, {21, 207}, {27, 3197}, {28, 1490}, {29, 1035}, {34, 13614}, {58, 3176}, {204, 47637}, {284, 40837}, {1172, 47848}, {1474, 56943}, {2203, 33672}, {2299, 5932}, {3194, 3341}, {3737, 57117}
X(68392) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 8885}, {10, 3176}, {226, 5932}, {3343, 47637}, {3351, 41083}, {11517, 13614}, {40590, 40837}, {40591, 1490}, {40611, 207}, {51574, 56943}, {62564, 33672}
X(68392) = cevapoint of X(65) and X(8811)
X(68392) = crosspoint of X(i) and X(j) for these (i,j): {1032, 19611}, {1034, 57643}
X(68392) = crosssum of X(i) and X(j) for these (i,j): {204, 1033}, {207, 1035}
X(68392) = barycentric product X(i)*X(j) for these {i,j}: {63, 8806}, {71, 56596}, {72, 41514}, {226, 57643}, {306, 3345}, {307, 47850}, {321, 66932}, {345, 8811}, {1034, 1214}, {1231, 7037}, {1400, 57782}, {3342, 56944}, {3998, 7149}, {7007, 52565}, {7152, 20336}, {40838, 52385}, {41087, 47634}, {42699, 60800}, {52389, 63877}
X(68392) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8885}, {37, 3176}, {65, 40837}, {71, 1490}, {72, 56943}, {73, 47848}, {219, 13614}, {228, 3197}, {306, 33672}, {1034, 31623}, {1073, 47637}, {1214, 5932}, {1400, 207}, {1409, 1035}, {3342, 41083}, {3345, 27}, {4559, 57117}, {7007, 8748}, {7037, 1172}, {7152, 28}, {8611, 14302}, {8806, 92}, {8811, 278}, {40838, 1896}, {41087, 3341}, {41514, 286}, {47850, 29}, {56596, 44129}, {56944, 47436}, {57454, 3194}, {57643, 333}, {57782, 28660}, {66932, 81}
X(68392) = {X(1034),X(63877)}-harmonic conjugate of X(7149)

X(68393) = X(1)X(19611)∩X(7)X(253)

Barycentrics    a*(a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 - 3*c^4)*(a^4 + 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 - 4*a^4*b^2*c^2 + 4*a^2*b^4*c^2 + 4*b^6*c^2 + 6*a^4*c^4 + 4*a^2*b^2*c^4 - 10*b^4*c^4 - 4*a^2*c^6 + 4*b^2*c^6 + c^8) : :

X(68393) lies on the cubic K034 and these lines: {1, 19611}, {2, 63877}, {7, 253}, {8, 14362}, {92, 2184}, {610, 65224}, {3692, 56235}

X(68393) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {58, 63877}, {8885, 14361}, {46639, 68108}, {47637, 69}
X(68393) = X(75)-Ceva conjugate of X(19611)
X(68393) = X(i)-isoconjugate of X(j) for these (i,j): {2, 47439}, {6, 3344}, {32, 47633}, {154, 3346}, {184, 46353}, {204, 47849}, {1032, 3172}, {1249, 28783}, {2131, 28782}, {3350, 28781}, {58342, 59077}
X(68393) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 3344}, {1073, 1}, {3343, 47849}, {6376, 47633}, {8808, 52078}, {32664, 47439}, {62605, 46353}
X(68393) = barycentric product X(i)*X(j) for these {i,j}: {1, 47435}, {75, 3343}, {92, 46351}, {304, 41085}, {561, 47437}, {1033, 57780}, {1498, 57921}, {1712, 34403}, {2184, 6527}, {5931, 8807}, {14361, 19611}
X(68393) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3344}, {31, 47439}, {75, 47633}, {92, 46353}, {1033, 204}, {1073, 47849}, {1498, 610}, {1712, 1249}, {2184, 3346}, {3343, 1}, {6527, 18750}, {8803, 30456}, {8807, 5930}, {8809, 8810}, {14361, 1895}, {19611, 1032}, {19614, 28783}, {41085, 19}, {44692, 8805}, {46351, 63}, {47435, 75}, {47437, 31}

X(68394) = X(1)X(280)∩X(7)X(92)

Barycentrics    (a^3 - a^2*b - a*b^2 + b^3 + a^2*c + 2*a*b*c + b^2*c - a*c^2 - b*c^2 - c^3)*(a^3 + a^2*b - a*b^2 - b^3 - a^2*c + 2*a*b*c - b^2*c - a*c^2 + b*c^2 + c^3)*(a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - a^2*b^4 - 2*a*b^5 + b^6 - 2*a^5*c - 2*a^4*b*c + 2*a*b^4*c + 2*b^5*c - a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 + 4*a^3*c^3 - 4*b^3*c^3 - a^2*c^4 + 2*a*b*c^4 - b^2*c^4 - 2*a*c^5 + 2*b*c^5 + c^6) : :

X(68394) lies on the cubic K034 and these lines: {1, 280}, {2, 19611}, {7, 92}, {8, 1032}, {63, 47851}, {75, 44189}, {309, 14548}, {345, 44327}, {394, 13138}, {938, 7020}, {3341, 46350}, {7011, 37141}, {11529, 39130}, {15466, 65270}, {18623, 65330}, {20223, 55119}, {23681, 24213}, {44695, 65213}, {53642, 54107}

X(68394) = isogonal conjugate of X(57454)
X(68394) = isotomic conjugate of X(63877)
X(68394) = polar conjugate of the isogonal conjugate of X(46881)
X(68394) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {58, 19611}, {84, 68334}, {154, 20211}, {189, 32064}, {610, 6223}, {1394, 5932}, {1413, 68349}, {1422, 68352}, {1433, 253}, {1436, 3146}, {2192, 68348}, {2208, 18663}, {8886, 14362}, {40836, 32001}, {41084, 69}, {41086, 2895}, {52078, 2893}, {60803, 54111}
X(68394) = X(i)-Ceva conjugate of X(j) for these (i,j): {75, 280}, {44189, 189}, {47436, 46350}
X(68394) = X(i)-cross conjugate of X(j) for these (i,j): {3176, 5932}, {47848, 56943}
X(68394) = X(i)-isoconjugate of X(j) for these (i,j): {1, 57454}, {6, 3342}, {31, 63877}, {32, 47634}, {40, 7152}, {41, 46352}, {198, 3345}, {221, 47850}, {223, 7037}, {1034, 2199}, {2187, 41514}, {2331, 66932}, {3209, 57643}, {3351, 34167}, {7007, 7011}, {7114, 40838}, {10397, 58995}, {41080, 47440}
X(68394) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 63877}, {3, 57454}, {9, 3342}, {278, 196}, {282, 1}, {3160, 46352}, {3341, 47850}, {6376, 47634}, {13612, 14298}, {14302, 55063}
X(68394) = cevapoint of X(1) and X(47851)
X(68394) = barycentric product X(i)*X(j) for these {i,j}: {1, 47436}, {7, 46350}, {75, 3341}, {84, 33672}, {189, 56943}, {207, 57783}, {264, 46881}, {280, 5932}, {309, 1490}, {561, 47438}, {1035, 57793}, {3197, 44190}, {14302, 53642}, {34404, 47848}, {40837, 44189}, {46355, 66090}
X(68394) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3342}, {2, 63877}, {6, 57454}, {7, 46352}, {75, 47634}, {84, 3345}, {189, 41514}, {207, 208}, {271, 57643}, {280, 1034}, {282, 47850}, {309, 56596}, {1035, 221}, {1433, 66932}, {1436, 7152}, {1490, 40}, {2192, 7037}, {3176, 7952}, {3197, 198}, {3341, 1}, {5932, 347}, {7003, 40838}, {7008, 7007}, {8885, 3194}, {14302, 8058}, {33672, 322}, {39130, 8806}, {40836, 7149}, {40837, 196}, {46350, 8}, {46355, 66091}, {46881, 3}, {47436, 75}, {47438, 31}, {47637, 41082}, {47848, 223}, {47851, 3351}, {52384, 8811}, {56943, 329}, {57783, 57782}, {60803, 60800}, {66090, 55015}
X(68394) = {X(7003),X(52037)}-harmonic conjugate of X(189)

X(68395) = X(65)X(495)∩X(942)X(5453)

Barycentrics    a*(a^2*b - b^3 + a^2*c + a*b*c + b^2*c + b*c^2 - c^3)*(a^5*b - 2*a^3*b^3 + a*b^5 + a^5*c - 2*a^3*b^2*c - a^2*b^3*c + a*b^4*c + b^5*c - 2*a^3*b*c^2 - 4*a^2*b^2*c^2 - 2*a*b^3*c^2 - 2*a^3*c^3 - a^2*b*c^3 - 2*a*b^2*c^3 - 2*b^3*c^3 + a*b*c^4 + a*c^5 + b*c^5) : :

See Antreas Hatzipolakis and Peter Moses, euclid 8318.

X(68395) lies on these lines: {65, 495}, {517, 43915}, {942, 5453}, {1243, 1439}, {3028, 55010}, {20617, 31794}

X(68396) = X(323)X(14491)∩X(381)X(33884)

Barycentrics    a^2*(2*a^4 - 2*b^4 + 11*b^2*c^2 - 2*c^4)*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 - 7*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6) : :

See Antreas Hatzipolakis and Peter Moses, euclid 8318.

X(68396) lies on these lines: {323, 14491}, {381, 33884}, {3167, 5645}, {15024, 45184}, {24206, 31074}

X(68397) = X(51)X(3631)∩X(141)X(2979)

Barycentrics    a^2*(2*a^4*b^2 - 2*b^6 + 2*a^4*c^2 + 2*a^2*b^2*c^2 + 11*b^4*c^2 + 11*b^2*c^4 - 2*c^6) : :
X(68397) = 5 X[141] - X[2979], 3 X[141] + X[9971], 11 X[141] + X[64023], 3 X[2979] + 5 X[9971], 11 X[2979] + 5 X[64023], 11 X[9971] - 3 X[64023], 3 X[373] - X[20583], 3 X[5943] - 5 X[40670], X[5943] - 5 X[61676], 9 X[5943] - 5 X[64599], X[40670] - 3 X[61676], 3 X[40670] - X[64599], 9 X[61676] - X[64599], 5 X[599] + 3 X[11002], 3 X[3589] - X[40673], X[3629] - 5 X[11451], 3 X[5650] - 5 X[20582], 3 X[5650] + 5 X[29959], X[9973] + 7 X[44299], X[14913] + 2 X[51127], 5 X[16776] - X[21969], X[21969] + 5 X[50991], X[41149] - 3 X[64692]

See Antreas Hatzipolakis and Peter Moses, euclid 8318.

X(68397) lies on these lines: {2, 44323}, {51, 3631}, {141, 2979}, {373, 20583}, {511, 11737}, {524, 5943}, {599, 11002}, {2393, 34573}, {2854, 12045}, {3589, 40673}, {3629, 11451}, {3819, 41579}, {5650, 8705}, {6329, 6688}, {9019, 51143}, {9730, 50958}, {9973, 44299}, {14913, 51127}, {16776, 21969}, {29181, 44804}, {41149, 64692}, {43129, 54044}, {61667, 63124}

X(68397) = midpoint of X(i) and X(j) for these {i,j}: {51, 3631}, {3819, 41579}, {9730, 50958}, {16776, 50991}, {20582, 29959}, {43129, 54044}, {61667, 63124}
X(68397) = reflection of X(6329) in X(6688)
X(68397) = complement of X(44323)

X(68398) = X(1)X(1292)∩X(105)X(165)

Barycentrics    a*(a^7 - a^6*b - a^5*b^2 + a^4*b^3 - a^3*b^4 + a^2*b^5 + a*b^6 - b^7 - a^6*c - 9*a^5*b*c + 19*a^4*b^2*c - 18*a^3*b^3*c + 9*a^2*b^4*c - 5*a*b^5*c + 5*b^6*c - a^5*c^2 + 19*a^4*b*c^2 - 6*a^3*b^2*c^2 - 2*a^2*b^3*c^2 - a*b^4*c^2 - 9*b^5*c^2 + a^4*c^3 - 18*a^3*b*c^3 - 2*a^2*b^2*c^3 + 10*a*b^3*c^3 + 5*b^4*c^3 - a^3*c^4 + 9*a^2*b*c^4 - a*b^2*c^4 + 5*b^3*c^4 + a^2*c^5 - 5*a*b*c^5 - 9*b^2*c^5 + a*c^6 + 5*b*c^6 - c^7) : :
X(68398) = 2 X[105] - 3 X[165], 4 X[120] - 3 X[1699], 5 X[1698] - 4 X[5511], X[7991] + 2 X[38684], 3 X[3576] - 4 X[38619], 3 X[5587] - 2 X[15521], X[7982] - 4 X[67834], 5 X[7987] - 4 X[11716], 5 X[7987] - 6 X[38712], 2 X[11716] - 3 X[38712], 5 X[8227] - 6 X[57327], 3 X[9778] - X[20097], 7 X[16192] - 6 X[38694], 7 X[31423] - 6 X[57299], 5 X[35242] - 4 X[38603], 2 X[38670] - 5 X[63469]

Suppose that P is a point on the circumcircle. Then the combo Q(P) = 3*X(165)-2*P lies on the Bevan circle. Four examples are indicated here: X(68398) = Q(X(105))
X(68399) = Q(X(106))
X(68400) = Q(X(111))
X(68401) = Q(X(15731))

X(68398) lies on on the Bevan circle and these lines: {1, 1292}, {10, 34547}, {40, 5540}, {55, 53540}, {57, 3021}, {105, 165}, {120, 1699}, {516, 20344}, {517, 38589}, {528, 1768}, {1282, 2820}, {1358, 1697}, {1695, 3034}, {1698, 5511}, {1706, 3039}, {2775, 2948}, {2788, 13174}, {2795, 9860}, {2809, 7991}, {2814, 64761}, {2826, 5541}, {2835, 7994}, {2836, 9904}, {2838, 12408}, {2941, 21381}, {3339, 59814}, {3576, 38619}, {3579, 38575}, {4859, 56796}, {5119, 51770}, {5537, 34464}, {5587, 15521}, {5691, 50911}, {7982, 67834}, {7987, 11716}, {8227, 57327}, {9523, 13221}, {9586, 58053}, {9587, 58055}, {9778, 20097}, {10699, 11531}, {10712, 50865}, {10743, 41869}, {11010, 67715}, {16192, 38694}, {26296, 48541}, {26297, 48542}, {31423, 57299}, {31435, 34124}, {35242, 38603}, {37551, 52826}, {38670, 63469}, {62497, 66793}, {62498, 66776}

X(68398) = reflection of X(i) in X(j) for these {i,j}: {1, 1292}, {5540, 40}, {5691, 50911}, {11531, 10699}, {34547, 10}, {38575, 3579}, {41869, 10743}, {50865, 10712}
X(68398) = excentral-isogonal conjugate of X(518)
X(68398) = X(5853)-Ceva conjugate of X(1)
X(68398) = X(43760)-Dao conjugate of X(35160)
X(68398) = {X(11716),X(38712)}-harmonic conjugate of X(7987)

X(68399) = X(1)X(1293)∩X(106)X(165)

Barycentrics    a*(a^6 - a^5*b - 7*a^4*b^2 - 2*a^3*b^3 + 7*a^2*b^4 + 3*a*b^5 - b^6 - a^5*c + 3*a^4*b*c + 22*a^3*b^2*c - 6*a^2*b^3*c - 21*a*b^4*c + 3*b^5*c - 7*a^4*c^2 + 22*a^3*b*c^2 - 46*a^2*b^2*c^2 + 26*a*b^3*c^2 + b^4*c^2 - 2*a^3*c^3 - 6*a^2*b*c^3 + 26*a*b^2*c^3 - 6*b^3*c^3 + 7*a^2*c^4 - 21*a*b*c^4 + b^2*c^4 + 3*a*c^5 + 3*b*c^5 - c^6) : :
X(68399) = 2 X[106] - 3 X[165], 4 X[121] - 3 X[1699], X[13541] - 4 X[38590], 5 X[1698] - 4 X[5510], X[7991] + 2 X[38685], 3 X[3576] - 4 X[38620], 3 X[5587] - 2 X[15522], X[7982] - 4 X[67835], 5 X[7987] - 4 X[11717], 5 X[7987] - 6 X[38713], 2 X[11717] - 3 X[38713], 5 X[8227] - 6 X[57328], 3 X[9778] - X[20098], 4 X[14664] - 5 X[63469], 2 X[38671] - 5 X[63469], 7 X[16192] - 6 X[38695], 7 X[31423] - 6 X[57300], 5 X[35242] - 4 X[38604]

X(68399) lies on on the Bevan circle and these lines: {1, 1293}, {10, 34548}, {40, 1054}, {57, 6018}, {106, 165}, {121, 1699}, {516, 21290}, {517, 13541}, {962, 11814}, {1282, 2821}, {1357, 1697}, {1695, 3030}, {1698, 5510}, {1706, 3038}, {1766, 3973}, {1768, 2802}, {2136, 64249}, {2776, 2948}, {2789, 13174}, {2796, 9860}, {2810, 39156}, {2815, 64761}, {2827, 5541}, {2841, 16389}, {2842, 9904}, {2844, 12408}, {2938, 5539}, {3339, 59812}, {3576, 38620}, {3579, 38576}, {5119, 51765}, {5587, 15522}, {5691, 50914}, {7982, 67835}, {7987, 11717}, {8227, 57328}, {9527, 13221}, {9586, 58052}, {9587, 58054}, {9778, 20098}, {10700, 11531}, {10713, 50865}, {10744, 41869}, {11010, 38515}, {12702, 21381}, {13462, 63774}, {14664, 38671}, {16192, 38695}, {17777, 20070}, {31423, 57300}, {35242, 38604}, {37551, 52827}, {62499, 66793}, {62500, 66776}

X(68399) = midpoint of X(17777) and X(20070)
X(68399) = reflection of X(i) in X(j) for these {i,j}: {1, 1293}, {962, 11814}, {1054, 40}, {5691, 50914}, {11531, 10700}, {34548, 10}, {38576, 3579}, {38671, 14664}, {41869, 10744}, {50865, 10713}
X(68399) = excentral-isogonal conjugate of X(519)
X(68399) = X(3880)-Ceva conjugate of X(1)
X(68399) = {X(11717),X(38713)}-harmonic conjugate of X(7987)

X(68400) = X(1)X(1296)∩X(111)X(165)

Barycentrics    a*(a^8 - 6*a^6*b^2 + 2*a^5*b^3 - 4*a^4*b^4 + 4*a^3*b^5 + 2*a^2*b^6 + 2*a*b^7 - b^8 - 2*a^6*b*c + 6*a^5*b^2*c - 2*a^4*b^3*c + 2*a^2*b^5*c - 6*a*b^6*c + 2*b^7*c - 6*a^6*c^2 + 6*a^5*b*c^2 + 37*a^4*b^2*c^2 - 20*a^3*b^3*c^2 - 15*a^2*b^4*c^2 - 8*a*b^5*c^2 + 2*b^6*c^2 + 2*a^5*c^3 - 2*a^4*b*c^3 - 20*a^3*b^2*c^3 + 10*a^2*b^3*c^3 + 20*a*b^4*c^3 - 6*b^5*c^3 - 4*a^4*c^4 - 15*a^2*b^2*c^4 + 20*a*b^3*c^4 + 6*b^4*c^4 + 4*a^3*c^5 + 2*a^2*b*c^5 - 8*a*b^2*c^5 - 6*b^3*c^5 + 2*a^2*c^6 - 6*a*b*c^6 + 2*b^2*c^6 + 2*a*c^7 + 2*b*c^7 - c^8) : :
X(68400) = 2 X[111] - 3 X[165], 4 X[126] - 3 X[1699], 5 X[1698] - 4 X[5512], 3 X[3576] - 4 X[38623], 7 X[3624] - 8 X[40556], 3 X[5587] - 2 X[22338], X[7982] - 4 X[67838], 5 X[7987] - 4 X[11721], 5 X[7987] - 6 X[38716], 2 X[11721] - 3 X[38716], X[7991] + 2 X[38688], 5 X[8227] - 6 X[57331], 3 X[9778] - X[20099], 4 X[9956] - 3 X[38799], 4 X[14650] - 5 X[35242], 4 X[14688] - 3 X[16475], 7 X[16192] - 6 X[38698], 7 X[31423] - 6 X[38796], 4 X[31663] - 3 X[52698], 2 X[38675] - 5 X[63469]

X(68400) lies on on the Bevan circle and these lines: {1, 1296}, {10, 66869}, {40, 33962}, {57, 6019}, {111, 165}, {126, 1699}, {516, 14360}, {517, 38593}, {518, 37751}, {519, 37749}, {543, 9860}, {1054, 2938}, {1282, 2824}, {1697, 3325}, {1698, 5512}, {1768, 2805}, {2780, 2948}, {2793, 13174}, {2813, 39156}, {2819, 64761}, {2830, 5541}, {2852, 64760}, {2854, 9904}, {2941, 5540}, {3048, 9586}, {3339, 59819}, {3576, 38623}, {3579, 11258}, {3624, 40556}, {5119, 51814}, {5587, 22338}, {5691, 50924}, {7982, 67838}, {7987, 11721}, {7991, 38688}, {8227, 57331}, {9583, 11835}, {9584, 11833}, {9587, 58059}, {9591, 14657}, {9778, 20099}, {9956, 38799}, {10704, 11531}, {10717, 50865}, {10748, 41869}, {11010, 38518}, {13221, 62506}, {14650, 35242}, {14654, 31730}, {14688, 16475}, {16192, 38698}, {18480, 38800}, {23699, 64005}, {31423, 38796}, {31663, 52698}, {33535, 35447}, {37551, 52832}, {38675, 63469}, {62507, 66793}, {62508, 66776}

X(68400) = reflection of X(i) in X(j) for these {i,j}: {1, 1296}, {5691, 50924}, {11258, 3579}, {11531, 10704}, {14654, 31730}, {33535, 35447}, {38800, 18480}, {41869, 10748}, {50865, 10717}, {66869, 10}
X(68400) = excentral-isogonal conjugate of X(524)
X(68400) = X(24394)-Ceva conjugate of X(1)
X(68400) = {X(11721),X(38716)}-harmonic conjugate of X(7987)

X(68401) = X(1)X(2291)∩X(101)X(165)

Barycentrics    a*(a^5 + a^4*b - 8*a^3*b^2 + 8*a^2*b^3 - a*b^4 - b^5 + a^4*c + 9*a^3*b*c - 6*a^2*b^2*c - 7*a*b^3*c + 3*b^4*c - 8*a^3*c^2 - 6*a^2*b*c^2 + 16*a*b^2*c^2 - 2*b^3*c^2 + 8*a^2*c^3 - 7*a*b*c^3 - 2*b^2*c^3 - a*c^4 + 3*b*c^4 - c^5) : :
X(68401) = 3 X[165] - 2 X[15731]

X(68401) lies on on the Bevan circle and these lines: {1, 2291}, {9, 1768}, {43, 38486}, {57, 1358}, {101, 165}, {219, 9904}, {610, 64760}, {910, 16554}, {1054, 1743}, {1155, 5526}, {1282, 1635}, {1781, 21381}, {2170, 10980}, {2448, 2590}, {2449, 2591}, {2801, 41798}, {3218, 67657}, {5011, 5536}, {5537, 6603}, {11407, 52705}, {15734, 65522}, {15855, 15931}, {34522, 55163}, {41338, 45721}, {53056, 64446}, {56632, 60905}, {66524, 66863}

X(68401) = reflection of X(1) in X(14074)
X(68401) = excentral-isogonal conjugate of X(15726)
X(68401) = X(527)-Ceva conjugate of X(1)
X(68401) = X(1156)-Dao conjugate of X(1121)

X(68402) = X(1)X(53529)∩X(57)X(934)

Barycentrics    a*(a + b - c)*(a - b + c)*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7 - 3*a^6*c + 15*a^5*b*c - 15*a^4*b^2*c - 14*a^3*b^3*c + 27*a^2*b^4*c - 9*a*b^5*c - b^6*c + a^5*c^2 - 15*a^4*b*c^2 + 46*a^3*b^2*c^2 - 26*a^2*b^3*c^2 - 15*a*b^4*c^2 + 9*b^5*c^2 + 5*a^4*c^3 - 14*a^3*b*c^3 - 26*a^2*b^2*c^3 + 42*a*b^3*c^3 - 7*b^4*c^3 - 5*a^3*c^4 + 27*a^2*b*c^4 - 15*a*b^2*c^4 - 7*b^3*c^4 - a^2*c^5 - 9*a*b*c^5 + 9*b^2*c^5 + 3*a*c^6 - b*c^6 - c^7) : :
X(68402) = 3 X[57] - 4 X[52879], 3 X[57] - 2 X[61493], 3 X[934] - 2 X[52879], 3 X[934] - X[61493], 4 X[5514] - 5 X[20196]

X(68402) llies on these lines: {1, 53529}, {57, 934}, {223, 31142}, {651, 2124}, {1086, 62793}, {1419, 15730}, {3160, 52457}, {5514, 20196}, {5526, 43064}, {5723, 23511}, {6282, 53804}, {7994, 8916}, {16572, 64980}, {34492, 61007}, {47057, 62705}, {47621, 60017}, {47623, 52161}
X(68402) = reflection of X(i) in X(j) for these {i,j}: {57, 934}, {61493, 52879}
X(68402) = X(527)-Ceva conjugate of X(57)
X(68402) = X(34056)-Dao conjugate of X(1121)
X(68402) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {934, 61493, 52879}, {52879, 61493, 57}

X(68403) = X(1)X(5072)∩X(5)X(10)

Barycentrics    a^3*(b+c)-a*(b-c)^2*(b+c)-6*(b^2-c^2)^2+a^2*(6*b^2-2*b*c+6*c^2) : :
X(68403) = X[1]+11*X[5072], X[4]+X[17502], -7*X[5]+X[10], -1*X[40]+13*X[5079], X[140]+2*X[12571], X[143]+2*X[65435], -1*X[165]+5*X[1656], 7*X[355]+5*X[3623], 3*X[381]+X[3576], X[382]+3*X[58221], 2*X[546]+X[13624], -1*X[548]+4*X[19878], -1*X[551]+3*X[61270], 5*X[632]+X[51118], X[962]+23*X[61921], X[1125]+2*X[3850], X[1385]+5*X[3091], 7*X[1482]+5*X[4816], -1*X[1657]+13*X[34595], -5*X[1698]+17*X[61919]

See Benjamin Lee Warren and Ercole Suppa, euclid 8330.

X(68403) lies on these lines: {1, 5072}, {2, 28146}, {4, 17502}, {5, 10}, {11, 5049}, {30, 10171}, {40, 5079}, {140, 12571}, {143, 65435}, {165, 1656}, {354, 7741}, {355, 3623}, {381, 3576}, {382, 58221}, {499, 31776}, {515, 5066}, {516, 547}, {519, 14892}, {546, 13624}, {548, 19878}, {549, 28154}, {551, 61270}, {632, 51118}, {942, 7173}, {952, 11737}, {962, 61921}, {1125, 3850}, {1385, 3091}, {1482, 4816}, {1538, 6830}, {1657, 34595}, {1698, 61919}, {1699, 5055}, {1829, 35487}, {2771, 23513}, {2801, 58604}, {3090, 3579}, {3241, 61257}, {3533, 10248}, {3545, 5886}, {3614, 9957}, {3616, 61945}, {3624, 3843}, {3627, 19862}, {3628, 18483}, {3632, 58238}, {3634, 12812}, {3653, 61954}, {3654, 61926}, {3655, 61944}, {3656, 38176}, {3679, 61931}, {3824, 3825}, {3828, 28212}, {3832, 33697}, {3833, 3838}, {3845, 10165}, {3851, 8227}, {3855, 18481}, {3857, 31673}, {3858, 4297}, {3860, 28190}, {4870, 37718}, {5045, 10593}, {5056, 12699}, {5070, 41869}, {5071, 9779}, {5076, 67706}, {5131, 65141}, {5219, 18527}, {5439, 31828}, {5550, 31666}, {5587, 10247}, {5603, 31145}, {5657, 38083}, {5691, 58230}, {5731, 41106}, {5790, 11224}, {5818, 11278}, {5844, 61934}, {5901, 28236}, {5902, 17605}, {5919, 7743}, {5927, 6990}, {6361, 61914}, {6684, 28216}, {6841, 11227}, {6873, 67998}, {6912, 23961}, {6915, 33862}, {6982, 64659}, {7987, 61984}, {7989, 16200}, {8703, 51074}, {9519, 61581}, {9590, 21308}, {9620, 18584}, {9669, 10389}, {9778, 61899}, {10095, 31751}, {10109, 10172}, {10124, 28182}, {10164, 15699}, {10283, 50796}, {10590, 51788}, {10592, 31792}, {10595, 61258}, {10886, 39550}, {10896, 31795}, {12100, 28158}, {12101, 51076}, {12512, 16239}, {12702, 61923}, {12811, 15178}, {13374, 56762}, {13464, 61259}, {14128, 31757}, {14893, 28172}, {15682, 51084}, {15704, 58219}, {15726, 61595}, {15759, 50869}, {15808, 58232}, {16192, 55858}, {17527, 61029}, {17606, 67977}, {18357, 33179}, {18492, 30392}, {18874, 31760}, {19872, 61903}, {19883, 23046}, {20057, 61248}, {20070, 67096}, {25055, 61948}, {26201, 58615}, {28194, 47478}, {28208, 38028}, {30308, 50821}, {31162, 61925}, {31423, 61905}, {31439, 42582}, {31447, 48661}, {31730, 55856}, {32205, 65423}, {34627, 61279}, {34638, 61869}, {34648, 61949}, {35242, 55857}, {35271, 62969}, {37606, 51792}, {37613, 63674}, {37712, 67241}, {38068, 61909}, {38076, 38138}, {38155, 61260}, {41099, 54445}, {41990, 51078}, {44904, 61524}, {46219, 64005}, {50799, 61943}, {50800, 51105}, {50808, 61898}, {50811, 61950}, {50812, 61854}, {50815, 61997}, {50816, 61823}, {50820, 62025}, {50833, 61998}, {50862, 61963}, {50865, 61908}, {50873, 61838}, {51071, 61251}, {51073, 61907}, {51086, 62138}, {51087, 61247}, {51705, 61956}, {54448, 61287}, {58218, 62128}, {58240, 61510}, {59417, 61927}, {59503, 61929}, {61271, 61941}, {61648, 65140}, {61649, 61703}, {61895, 64108}, {63963, 64813}, {65399, 67867}

X(68403) = midpoint of X(i) and X(j) for these {i,j}: {4, 17502}, {5, 3817}, {165, 22793}, {381, 11230}, {946, 38042}, {1699, 11231}, {3579, 9812}, {3656, 38176}, {3845, 10165}, {5066, 61269}, {5587, 51709}, {5886, 38140}, {10172, 50802}, {10175, 38034}, {10222, 59388}, {10246, 18480}, {10283, 50796}, {18483, 58441}, {19883, 23046}, {22791, 38127}, {51071, 61251}, {51087, 61247}
X(68403) = reflection of X(i) in X(j) for these {i,j}: {9955, 3817}, {10171, 61267}, {10172, 10109}, {26201, 58615}, {31662, 1125}, {31663, 58441}, {58441, 3628}
X(68403) = pole of line {4802, 47805} with respect to the orthoptic circle of the Steiner inellipse
X(68403) = pole of line {10950, 28212} with respect to the Feuerbach hyperbola
X(68403) = center of circles {{X(i), X(j), X(k)}} for these {i, j, k}: {4, 17502, 31849}, {381, 11230, 67216}
X(68403) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5,3817,517},{5,9955,9956},{5,38034,10175},{381,7988,11230},{381,11230,28160},{946,38042,517},{1125,28186,31662},{1699,5055,11231},{1699,11231,28198},{1699,61265,5055},{3091,61268,1385},{3545,5886,38140},{3628,18483,31663},{3628,28178,58441},{3817,10175,38034},{3845,10165,28168},{3851,8227,18480},{5066,61269,515},{5071,9779,26446},{5603,61936,61263},{5790,61933,61264},{5886,38140,28204},{10109,28174,10172},{10172,50802,28174},{10175,38034,517},{12811,61272,19925},{12812,40273,3634},{18483,58441,28178},{18493,61937,7989},{19925,61272,15178},{22791,38127,517},{30308,61920,50821},{38021,61264,5790},{48661,61911,64850},{48661,64850,31447},{61261,68034,10222}

X(68404) = X(5)X(13)∩X(30)X(48311)

Barycentrics    Sqrt[3]*(2*a^6 - 9*a^4*b^2 + 2*a^2*b^4 + 5*b^6 - 9*a^4*c^2 - 24*a^2*b^2*c^2 - 5*b^4*c^2 + 2*a^2*c^4 - 5*b^2*c^4 + 5*c^6) - 2*(2*a^4 + 17*a^2*b^2 - 19*b^4 + 17*a^2*c^2 + 38*b^2*c^2 - 19*c^4)*S : :
X(68404) = 5 X[5] + X[13], 7 X[5] - X[5617], 2 X[5] + X[20252], 3 X[5] - X[36765], 7 X[13] + 5 X[5617], 2 X[13] - 5 X[20252], 3 X[13] + 5 X[36765], X[13] - 5 X[59401], 2 X[5617] + 7 X[20252], 3 X[5617] - 7 X[36765], X[5617] + 7 X[59401], 3 X[20252] + 2 X[36765], X[36765] + 3 X[59401], X[546] + 2 X[6669], X[616] - 13 X[5079], X[618] - 4 X[35018], 5 X[3091] + X[47610], 3 X[3545] + X[59383], 2 X[3628] + X[5478], 2 X[3850] + X[6771], 7 X[3857] - X[36961], 3 X[5055] + X[59394], 11 X[5056] + X[13103], 13 X[5068] - X[48655], 11 X[5072] + X[6770], X[5459] + 2 X[11737], X[5463] - 7 X[61916], X[5473] - 7 X[55856], 10 X[10109] - X[36769], X[11705] + 2 X[61259], 4 X[12811] - X[22796], 4 X[15092] - X[20253], 35 X[19709] + X[36318], X[25154] + 5 X[61910], X[35749] + 35 X[61920], 5 X[36770] - 11 X[61900], X[41042] - 7 X[61939], X[42137] + 5 X[52266], X[43401] + 5 X[52650], X[51482] + 11 X[61925], X[59378] + 3 X[61933]

X(68404) lies on these lines: {5, 13}, {30, 48311}, {530, 47478}, {542, 14892}, {546, 6669}, {549, 59393}, {616, 5079}, {618, 35018}, {3091, 47610}, {3545, 59383}, {3628, 5478}, {3845, 21156}, {3850, 6771}, {3857, 36961}, {5055, 59394}, {5056, 13103}, {5066, 41022}, {5068, 48655}, {5072, 6770}, {5459, 11737}, {5463, 61916}, {5473, 55856}, {10109, 36769}, {11543, 47855}, {11705, 61259}, {12811, 22796}, {15092, 20253}, {19709, 36318}, {22489, 38071}, {25154, 61910}, {35749, 61920}, {36770, 61900}, {41042, 61939}, {42137, 52266}, {42628, 47861}, {43401, 52650}, {51482, 61925}, {59378, 61933}

X(68404) = midpoint of X(i) and X(j) for these {i,j}: {5, 59401}, {549, 59393}, {3845, 21156}, {22489, 38071}
X(68404) = reflection of X(20252) in X(59401)

X(68405) = X(5)X(14)∩X(30)X(48312)

Barycentrics    Sqrt[3]*(2*a^6 - 9*a^4*b^2 + 2*a^2*b^4 + 5*b^6 - 9*a^4*c^2 - 24*a^2*b^2*c^2 - 5*b^4*c^2 + 2*a^2*c^4 - 5*b^2*c^4 + 5*c^6) + 2*(2*a^4 + 17*a^2*b^2 - 19*b^4 + 17*a^2*c^2 + 38*b^2*c^2 - 19*c^4)*S : :
X(68405) = 5 X[5] + X[14], 7 X[5] - X[5613], 2 X[5] + X[20253], 7 X[14] + 5 X[5613], 2 X[14] - 5 X[20253], X[14] - 5 X[59402], 2 X[5613] + 7 X[20253], X[5613] + 7 X[59402], X[546] + 2 X[6670], X[617] - 13 X[5079], X[619] - 4 X[35018], 5 X[3091] + X[47611], 3 X[3545] + X[59384], 2 X[3628] + X[5479], 2 X[3850] + X[6774], 7 X[3857] - X[36962], 3 X[5055] + X[59396], 11 X[5056] + X[13102], 13 X[5068] - X[48656], 11 X[5072] + X[6773], X[5460] + 2 X[11737], X[5464] - 7 X[61916], X[5474] - 7 X[55856], 10 X[10109] - X[47867], X[11706] + 2 X[61259], 4 X[12811] - X[22797], 4 X[15092] - X[20252], 35 X[19709] + X[36320], X[25164] + 5 X[61910], X[36327] + 35 X[61920], X[41043] - 7 X[61939], X[42136] + 5 X[52263], X[43402] + 5 X[44223], X[51483] + 11 X[61925], X[59379] + 3 X[61933]

X(68405) lies on these lines: {5, 14}, {30, 48312}, {531, 47478}, {542, 14892}, {546, 6670}, {549, 59395}, {617, 5079}, {619, 35018}, {3091, 47611}, {3545, 59384}, {3628, 5479}, {3845, 21157}, {3850, 6774}, {3857, 36962}, {5055, 59396}, {5056, 13102}, {5066, 41023}, {5068, 48656}, {5072, 6773}, {5460, 11737}, {5464, 61916}, {5474, 55856}, {10109, 47867}, {11542, 47856}, {11706, 61259}, {12811, 22797}, {15092, 20252}, {19709, 36320}, {22490, 38071}, {25164, 61910}, {36327, 61920}, {36765, 38229}, {41043, 61939}, {42136, 52263}, {42627, 47862}, {43402, 44223}, {51483, 61925}, {59379, 61933}

X(68405) = midpoint of X(i) and X(j) for these {i,j}: {5, 59402}, {549, 59395}, {3845, 21157}, {22490, 38071}, {36765, 38229}
X(68405) = reflection of X(20253) in X(59402)

X(68406) = MIDPOINT OF X(68404) AND X(68405)

Barycentrics    5*a^6*b^2 - 11*a^4*b^4 + 12*a^2*b^6 - 6*b^8 + 5*a^6*c^2 + 2*a^4*b^2*c^2 - 7*a^2*b^4*c^2 + 19*b^6*c^2 - 11*a^4*c^4 - 7*a^2*b^2*c^4 - 26*b^4*c^4 + 12*a^2*c^6 + 19*b^2*c^6 - 6*c^8 : :
X(68406) = 5 X[2] - X[38731], 7 X[5] - X[114], 5 X[5] + X[115], 19 X[5] - X[14981], X[5] + 2 X[15092], 3 X[5] - X[36519], 3 X[5] + X[38229], 13 X[5] - X[51872], 4 X[5] - X[61575], 2 X[5] + X[61576], 11 X[5] + X[67268], 5 X[114] + 7 X[115], 19 X[114] - 7 X[14981], X[114] + 14 X[15092], X[114] + 7 X[23514], 3 X[114] - 7 X[36519], 3 X[114] + 7 X[38229], 13 X[114] - 7 X[51872], 4 X[114] - 7 X[61575], 2 X[114] + 7 X[61576], 11 X[114] + 7 X[67268], 19 X[115] + 5 X[14981], X[115] - 10 X[15092], X[115] - 5 X[23514], 3 X[115] + 5 X[36519], 3 X[115] - 5 X[38229], 13 X[115] + 5 X[51872], 4 X[115] + 5 X[61575], 2 X[115] - 5 X[61576], 11 X[115] - 5 X[67268], X[14981] + 38 X[15092], and many others

X(68406) lies on these lines: {2, 38731}, {5, 39}, {98, 5072}, {99, 5079}, {148, 61921}, {381, 34127}, {542, 14892}, {543, 47478}, {546, 6722}, {547, 22247}, {620, 35018}, {632, 39809}, {671, 61925}, {1656, 21166}, {2482, 61916}, {2794, 5066}, {3090, 33813}, {3091, 12042}, {3544, 51523}, {3545, 38224}, {3628, 67863}, {3832, 38739}, {3839, 26614}, {3845, 38737}, {3850, 6036}, {3851, 14061}, {3855, 38741}, {3857, 39838}, {3858, 38749}, {3861, 38747}, {5055, 14639}, {5056, 6321}, {5067, 38730}, {5068, 6033}, {5070, 10723}, {5071, 8591}, {5461, 11737}, {6054, 61931}, {6055, 61942}, {6721, 12812}, {7486, 38750}, {8724, 61926}, {9166, 38743}, {9167, 61909}, {9880, 61910}, {10109, 36521}, {10722, 38634}, {11591, 58518}, {11632, 61932}, {11725, 61259}, {12117, 61901}, {12131, 35487}, {12188, 61935}, {12811, 67862}, {13172, 61914}, {13188, 61923}, {14128, 39806}, {14651, 22566}, {14830, 61944}, {14971, 38071}, {15022, 35369}, {15088, 67479}, {15699, 38748}, {16239, 38736}, {18874, 39835}, {19709, 49102}, {20094, 67096}, {20399, 61600}, {31274, 61900}, {38220, 61263}, {38627, 61599}, {38635, 61905}, {38732, 61920}, {38733, 61911}, {38735, 61934}, {38738, 55856}, {38745, 41989}, {44904, 61561}, {46031, 62490}, {61560, 61940}, {61919, 64089}

X(68406) = midpoint of X(i) and X(j) for these {i,j}: {5, 23514}, {381, 34127}, {3839, 26614}, {3845, 38737}, {14651, 22566}, {14971, 38071}, {21166, 22515}, {36519, 38229}, {68404,68405}
X(68406) = reflection of X(i) in X(j) for these {i,j}: {23514, 15092}, {61576, 23514}
X(68406) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 15092, 61576}, {5, 38229, 36519}, {5, 61576, 61575}, {3851, 14061, 22505}, {23514, 36519, 38229}

X(68407) = (name pending)

Barycentrics    a*(a^11*(3*b^2+3*c^2+4*S)-4*a*(b^2-c^2)^2*(b^6*S+b^2*c^4*S+c^6*S+b^4*c^2*(c^2+S))-a^9*(6*b^4+6*c^4+10*c^2*S+b^2*(21*c^2+10*S))-a^3*(b^2-c^2)^2*(3*b^6+3*c^6+b^4*(c^2-10*S)-10*c^4*S+b^2*(c^4-8*c^2*S))+a^7*(6*c^4*S+b^4*(22*c^2+6*S)+b^2*(22*c^4+4*c^2*S))+a^5*(6*b^8+6*c^6*(c^2-S)-3*b^6*(3*c^2+2*S)+2*b^4*(5*c^4+9*c^2*S)-9*b^2*(c^6-2*c^4*S))) : :

See David Nguyen and Ercole Suppa, euclid 8386.

X(68407) lies on this line: {8825, 9739}

X(68407) = isogonal conjugate of the anticomplement of X(15885)

X(68408) = X(6)X(1163)∩X(20)X(1160)

Barycentrics    a^8-32*a^4*b^2*c^2-(b^2-c^2)^4+6*a^6*(b^2+c^2)-6*a^2*(b^2-c^2)^2*(b^2+c^2)-4*(2*a^6+2*a^2*(b^2-c^2)^2-3*a^4*(b^2+c^2)-(b^2-c^2)^2*(b^2+c^2))*S : :

See David Nguyen and Ercole Suppa, euclid 8388.

X(68408) lies on these lines: {6, 1163}, {20, 1160}, {3156, 8904}, {5870, 11381}, {6276, 10619}, {17816, 44609}

X(68409) = X(3)X(6)∩X(230)X(58189)

Barycentrics    -97*a^4 + 89*a^2*(b^2 + c^2) : :

See David Nguyen and Juan José Isach Mayo, euclid 8390.

X(68409) lies on these lines: {3, 6}, {230, 58189}, {11614, 61974}, {31415, 61797}, {43291, 62070}, {53419, 62080}

X(68409) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 5585, 21309}, {5585, 53095, 15513}

X(68410) = X(5)X(523)∩X(6)X(2065)

Barycentrics    a^2*(a^10*b^4 - 3*a^8*b^6 + 4*a^6*b^8 - 4*a^4*b^10 + 3*a^2*b^12 - b^14 - 2*a^6*b^6*c^2 + 3*a^4*b^8*c^2 - 4*a^2*b^10*c^2 + 3*b^12*c^2 + a^10*c^4 + 2*a^6*b^4*c^4 + a^2*b^8*c^4 - 4*b^10*c^4 - 3*a^8*c^6 - 2*a^6*b^2*c^6 + 2*b^8*c^6 + 4*a^6*c^8 + 3*a^4*b^2*c^8 + a^2*b^4*c^8 + 2*b^6*c^8 - 4*a^4*c^10 - 4*a^2*b^2*c^10 - 4*b^4*c^10 + 3*a^2*c^12 + 3*b^2*c^12 - c^14) : :

See Elias Hagos and Peter Moses, euclid 8395.

X(68410) lies on these lines: {5, 523}, {6, 2065}, {32, 28343}, {576, 2871}, {648, 58734}, {2023, 52672}, {2781, 59290}, {3001, 23098}, {3095, 59363}, {14251, 23635}, {14966, 44668}, {20975, 34156}, {34157, 34990}, {34349, 52727}

X(68410) = reflection of X(52727) in X(34349)

X(68411) = X(5)X(523)∩X(1482)X(59366)

Barycentrics    a*(a^9*b^3 - a^8*b^4 - 4*a^7*b^5 + 4*a^6*b^6 + 6*a^5*b^7 - 6*a^4*b^8 - 4*a^3*b^9 + 4*a^2*b^10 + a*b^11 - b^12 - 2*a^8*b^3*c + 6*a^7*b^4*c + 4*a^6*b^5*c - 18*a^5*b^6*c + 18*a^3*b^8*c - 4*a^2*b^9*c - 6*a*b^10*c + 2*b^11*c + a^7*b^3*c^2 - 9*a^6*b^4*c^2 + 4*a^5*b^5*c^2 + 20*a^4*b^6*c^2 - 11*a^3*b^7*c^2 - 13*a^2*b^8*c^2 + 6*a*b^9*c^2 + 2*b^10*c^2 + a^9*c^3 - 2*a^8*b*c^3 + a^7*b^2*c^3 + 8*a^5*b^4*c^3 - 8*a^4*b^5*c^3 - 19*a^3*b^6*c^3 + 16*a^2*b^7*c^3 + 9*a*b^8*c^3 - 6*b^9*c^3 - a^8*c^4 + 6*a^7*b*c^4 - 9*a^6*b^2*c^4 + 8*a^5*b^3*c^4 - 12*a^4*b^4*c^4 + 16*a^3*b^5*c^4 + 9*a^2*b^6*c^4 - 18*a*b^7*c^4 + b^8*c^4 - 4*a^7*c^5 + 4*a^6*b*c^5 + 4*a^5*b^2*c^5 - 8*a^4*b^3*c^5 + 16*a^3*b^4*c^5 - 24*a^2*b^5*c^5 + 8*a*b^6*c^5 + 4*b^7*c^5 + 4*a^6*c^6 - 18*a^5*b*c^6 + 20*a^4*b^2*c^6 - 19*a^3*b^3*c^6 + 9*a^2*b^4*c^6 + 8*a*b^5*c^6 - 4*b^6*c^6 + 6*a^5*c^7 - 11*a^3*b^2*c^7 + 16*a^2*b^3*c^7 - 18*a*b^4*c^7 + 4*b^5*c^7 - 6*a^4*c^8 + 18*a^3*b*c^8 - 13*a^2*b^2*c^8 + 9*a*b^3*c^8 + b^4*c^8 - 4*a^3*c^9 - 4*a^2*b*c^9 + 6*a*b^2*c^9 - 6*b^3*c^9 + 4*a^2*c^10 - 6*a*b*c^10 + 2*b^2*c^10 + a*c^11 + 2*b*c^11 - c^12) : :

See Elias Hagos and Peter Moses, euclid 8395.

X(68411) lies on these lines: {5, 523}, {1482, 59366}, {2778, 59293}, {14260, 45776}, {18210, 39175}, {22835, 42753}, {23101, 45916}, {34345, 52731}, {34977, 39173}

X(68411) = reflection of X(52731) in X(34345)

X(68412) = X(4)X(525)∩X(5)X(523)

Barycentrics    (b^2 - c^2)*(a^8 - 2*a^4*b^4 + b^8 + a^4*b^2*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 - 2*a^4*c^4 + a^2*b^2*c^4 + 2*b^4*c^4 - 2*b^2*c^6 + c^8) : :
X(68412) = 2 X[3] - 3 X[45681], X[3] - 3 X[65754], X[4] + 3 X[65714], 4 X[5] - 3 X[14566], X[20] - 3 X[5664], 3 X[381] - X[5489], X[382] - 3 X[58346], 4 X[546] - 3 X[39491], 5 X[631] - 3 X[18556], 5 X[1656] - 3 X[65723], 3 X[2394] - 7 X[3832], 7 X[3090] - 3 X[53383], 2 X[10279] - 3 X[65610], 5 X[3843] - 3 X[42733], 3 X[16230] - X[62438], 3 X[24978] - 2 X[62438], 4 X[10280] - 3 X[53266], 5 X[17578] + 3 X[63248], 2 X[32204] - 3 X[34291], X[65871] + 2 X[68330]

See Elias Hagos and Peter Moses, euclid 8395.

X(68412) lies on these lines: {2, 58273}, {3, 45681}, {4, 525}, {5, 523}, {20, 5664}, {107, 61500}, {155, 8057}, {381, 5489}, {382, 58346}, {512, 40647}, {520, 5446}, {546, 39491}, {631, 18556}, {850, 44142}, {1656, 65723}, {2394, 3832}, {2501, 8743}, {2548, 62384}, {2799, 38745}, {2848, 44810}, {3090, 53383}, {3150, 58261}, {3800, 10279}, {3843, 42733}, {5466, 18841}, {6368, 35719}, {7762, 33294}, {7776, 62555}, {7927, 40550}, {8673, 66754}, {8675, 43130}, {9007, 64067}, {9033, 16534}, {9517, 16230}, {10278, 57588}, {10280, 53266}, {14246, 62629}, {16104, 53178}, {17578, 63248}, {18808, 18855}, {30474, 31857}, {32204, 34291}, {33754, 45801}, {37814, 39228}, {39201, 45735}, {41079, 65871}, {44235, 59745}, {44817, 55129}, {45327, 46512}, {46371, 58757}, {47262, 62662}, {57128, 61757}, {58070, 61181}

X(68412) = midpoint of X(41079) and X(65871)
X(68412) = reflection of X(i) in X(j) for these {i,j}: {24978, 16230}, {41079, 68330}, {45681, 65754}
X(68412) = crosspoint of X(648) and X(13485)
X(68412) = crosssum of X(647) and X(7669)
X(68412) = crossdifference of every pair of points on line {50, 8779}

X(68413) = X(5)X(523)∩X(10)X(514)

Barycentrics    a*(b - c)*(a^3*b^2 - a^2*b^3 - a*b^4 + b^5 + a^3*b*c - 2*a^2*b^2*c + a*b^3*c + a^3*c^2 - 2*a^2*b*c^2 + 3*a*b^2*c^2 - b^3*c^2 - a^2*c^3 + a*b*c^3 - b^2*c^3 - a*c^4 + c^5) : :

See Elias Hagos and Peter Moses, euclid 8395.

X(68413) lies on these lines: {5, 523}, {10, 514}, {12, 60074}, {56, 14838}, {495, 56283}, {513, 3579}, {522, 21077}, {661, 3730}, {1019, 37405}, {1482, 39212}, {1577, 11681}, {2812, 52726}, {2850, 44812}, {3737, 57708}, {3887, 12738}, {4041, 5903}, {4086, 30172}, {4122, 63826}, {4129, 27553}, {4560, 20060}, {6003, 11500}, {9013, 43146}, {11236, 64934}, {13589, 14513}, {16117, 42325}, {21201, 42758}, {28161, 42757}, {34605, 45671}, {37050, 47809}, {47842, 51572}, {50346, 56289}

X(68413) = crosssum of X(i) and X(j) for these (i,j): {215, 46384}, {513, 65524}
X(68413) = crossdifference of every pair of points on line {50, 1914}

X(68414) = X(3)X(512)∩X(5)X(523)

Barycentrics    a^2*(b^2 - c^2)*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 - 2*a^2*c^4 + c^6) : :
X(68414) = X[3] - 3 X[34291], 2 X[3] - 3 X[44814], 3 X[46953] - X[65390], 3 X[5] - 2 X[59741], 3 X[23105] - 4 X[59741], 5 X[1656] - 3 X[53266], X[1657] - 3 X[53275], 7 X[3851] - 6 X[39482], 3 X[6132] - 2 X[65418], 3 X[14270] - 4 X[65418]

See Elias Hagos and Peter Moses, euclid 8395.

X(68414) lies on these lines: {1, 2608}, {3, 512}, {4, 62489}, {5, 523}, {6, 8574}, {24, 47221}, {32, 647}, {49, 57136}, {61, 57123}, {62, 57122}, {110, 39138}, {114, 51232}, {155, 520}, {403, 46371}, {525, 22660}, {526, 5607}, {576, 8675}, {669, 23208}, {684, 690}, {826, 3574}, {850, 7752}, {868, 6328}, {924, 6759}, {1112, 55383}, {1510, 62173}, {1649, 3005}, {1656, 53266}, {1657, 53275}, {2126, 3733}, {2395, 2548}, {2971, 36955}, {3095, 3906}, {3527, 14380}, {3737, 56289}, {3851, 39482}, {3933, 52629}, {4108, 52300}, {5007, 6041}, {5013, 10097}, {6132, 9517}, {6140, 53263}, {6368, 68174}, {7746, 47229}, {7772, 10567}, {7775, 23878}, {7785, 31296}, {7808, 62688}, {7812, 36900}, {7862, 30476}, {7927, 34347}, {8552, 44826}, {8562, 14809}, {8651, 20993}, {9033, 34104}, {9168, 38526}, {9177, 14824}, {9213, 14246}, {9409, 39477}, {9426, 18796}, {9737, 30209}, {10190, 11007}, {10684, 46245}, {11183, 15000}, {14264, 32112}, {14366, 33803}, {14618, 16868}, {15329, 44830}, {15359, 20975}, {16229, 35488}, {16837, 61196}, {17414, 37338}, {18308, 42733}, {22260, 46127}, {30510, 52630}, {32204, 62510}, {32816, 62642}, {32832, 53347}, {34964, 49673}, {38987, 65728}, {39509, 41079}, {54003, 67534}

X(68414) = midpoint of X(i) and X(j) for these {i,j}: {684, 21731}, {23109, 23110}
X(68414) = reflection of X(i) in X(j) for these {i,j}: {{9409, 39477}, {14270, 6132}, {14809, 8562}, {23105, 5}, {41079, 39509}, {42733, 18308}, {44814, 34291}, {44826, 8552}, {51232, 114}, {53263, 6140}
X(68414) = reflection of X(23105) in the Euler line
X(68414) = X(i)-Ceva conjugate of X(j) for these (i,j): {34990, 55384}, {39295, 2088}, {40173, 6}, {40511, 3124}, {44549, 3269}, {66162, 3569}
X(68414) = X(55384)-cross conjugate of X(34990)
X(68414) = X(162)-isoconjugate of X(46087)
X(68414) = X(i)-Dao conjugate of X(j) for these (i,j): {125, 46087}, {15295, 14781}, {34990, 850}
X(68414) = crosspoint of X(110) and X(523)
X(68414) = crosssum of X(110) and X(523)
X(68414) = crossdifference of every pair of points on line {50, 230}
X(68414) = barycentric product X(i)*X(j) for these {i,j}: {99, 55384}, {523, 34990}, {525, 1112}, {2799, 47635}, {4705, 16734}, {14618, 23217}
X(68414) = barycentric quotient X(i)/X(j) for these {i,j}: {647, 46087}, {1112, 648}, {11060, 14781}, {16734, 4623}, {23217, 4558}, {34990, 99}, {47635, 2966}, {55384, 523}

X(68415) = X(5)X(523)∩X(514)X(1125)

Barycentrics    (b - c)*(a^5*b - a^3*b^3 - a^2*b^4 + b^6 + a^5*c - 2*a^4*b*c + a^2*b^3*c + a*b^4*c - b^5*c + 2*a^2*b^2*c^2 - a*b^3*c^2 - b^4*c^2 - a^3*c^3 + a^2*b*c^3 - a*b^2*c^3 + 2*b^3*c^3 - a^2*c^4 + a*b*c^4 - b^2*c^4 - b*c^5 + c^6) : :

See Elias Hagos and Peter Moses, euclid 8395.

X(68415) lies on these lines: {5, 523}, {451, 48209}, {513, 13369}, {514, 1125}, {522, 18483}, {867, 42753}, {928, 62435}, {1577, 27555}, {3004, 23100}, {3700, 24045}, {4223, 47797}, {4560, 37369}, {4802, 33528}, {4977, 66968}, {6362, 16160}, {7178, 10571}, {14377, 17069}, {21789, 44253}, {28473, 40257}, {30172, 52355}, {34958, 59285}, {38330, 64934}

X(68415) = crosspoint of X(693) and X(38340)
X(68415) = crosssum of X(692) and X(9404)
X(68415) = crossdifference of every pair of points on line {50, 17735}

X(68416) = X(5)X(523)∩X(10)X(14260)

Barycentrics    a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 + 3*a^2*b^5 + 2*a*b^6 - b^7 - a^4*b^2*c + 6*a^3*b^3*c - 6*a*b^5*c + b^6*c + a^5*c^2 - a^4*b*c^2 - 4*a^2*b^3*c^2 - 2*a*b^4*c^2 + 3*b^5*c^2 - 2*a^4*c^3 + 6*a^3*b*c^3 - 4*a^2*b^2*c^3 + 12*a*b^3*c^3 - 3*b^4*c^3 - 3*a^3*c^4 - 2*a*b^2*c^4 - 3*b^3*c^4 + 3*a^2*c^5 - 6*a*b*c^5 + 3*b^2*c^5 + 2*a*c^6 + b*c^6 - c^7 : :

See Elias Hagos and Peter Moses, euclid 8395.

X(68416) lies on these lines: {5, 523}, {10, 14260}, {3006, 58254}, {3814, 42753}, {16173, 61768}, {62703, 66865}

X(68417) = X(4)X(543)∩X(5)X(523)

Barycentrics    a^8*b^2 - 4*a^6*b^4 + 4*a^2*b^8 - b^10 + a^8*c^2 + 2*a^6*b^2*c^2 + 3*a^4*b^4*c^2 - 11*a^2*b^6*c^2 + 2*b^8*c^2 - 4*a^6*c^4 + 3*a^4*b^2*c^4 + 12*a^2*b^4*c^4 - b^6*c^4 - 11*a^2*b^2*c^6 - b^4*c^6 + 4*a^2*c^8 + 2*b^2*c^8 - c^10 : :

See Elias Hagos and Peter Moses, euclid 8395.

X(68417) lies on these lines: {2, 8877}, {4, 543}, {5, 523}, {32, 10418}, {115, 59422}, {325, 23106}, {538, 57604}, {576, 16534}, {620, 34161}, {754, 7417}, {2482, 52483}, {2548, 35606}, {5099, 5968}, {5181, 51980}, {5461, 9214}, {7752, 31857}, {7812, 14002}, {14357, 53136}, {14995, 52533}, {37760, 52630}, {51999, 67396}

X(68418) = X(5)X(523)∩X(522)X(6260)

Barycentrics    a*(b - c)*(a^6*b^2 - 2*a^5*b^3 - a^4*b^4 + 4*a^3*b^5 - a^2*b^6 - 2*a*b^7 + b^8 + a^6*b*c - 2*a^5*b^2*c + 3*a^4*b^3*c - 5*a^2*b^5*c + 2*a*b^6*c + b^7*c + a^6*c^2 - 2*a^5*b*c^2 + 3*a^4*b^2*c^2 - 5*a^3*b^3*c^2 + 2*a^2*b^4*c^2 + 3*a*b^5*c^2 - 2*b^6*c^2 - 2*a^5*c^3 + 3*a^4*b*c^3 - 5*a^3*b^2*c^3 + 8*a^2*b^3*c^3 - 3*a*b^4*c^3 - b^5*c^3 - a^4*c^4 + 2*a^2*b^2*c^4 - 3*a*b^3*c^4 + 2*b^4*c^4 + 4*a^3*c^5 - 5*a^2*b*c^5 + 3*a*b^2*c^5 - b^3*c^5 - a^2*c^6 + 2*a*b*c^6 - 2*b^2*c^6 - 2*a*c^7 + b*c^7 + c^8) : :

See Elias Hagos and Peter Moses, euclid 8395.

X(68418) lies on these lines: {5, 523}, {514, 12616}, {522, 6260}, {3900, 64804}, {4041, 10571}, {11500, 35057}, {14838, 59285}, {35100, 66968}

X(68419) = X(5)X(523)∩X(522)X(6684)

Barycentrics    (b - c)*(a^5*b - 2*a^4*b^2 + a^3*b^3 + a^2*b^4 - 2*a*b^5 + b^6 + a^5*c - 2*a^4*b*c + 2*a^3*b^2*c - a^2*b^3*c - a*b^4*c + b^5*c - 2*a^4*c^2 + 2*a^3*b*c^2 - 2*a^2*b^2*c^2 + 3*a*b^3*c^2 - b^4*c^2 + a^3*c^3 - a^2*b*c^3 + 3*a*b^2*c^3 - 2*b^3*c^3 + a^2*c^4 - a*b*c^4 - b^2*c^4 - 2*a*c^5 + b*c^5 + c^6) : :

See Elias Hagos and Peter Moses, euclid 8395.

X(68419) lies on these lines: {5, 523}, {451, 48204}, {514, 19925}, {522, 6684}, {928, 62434}, {1577, 27687}, {2774, 18004}, {3700, 3730}, {3884, 49290}, {3900, 31837}, {4223, 47809}, {4560, 37158}, {4777, 33528}, {5499, 6362}, {14887, 51562}, {23104, 50333}, {28602, 62494}, {56283, 59283}

X(68420) = EULER LINE INTERCEPT OF X(148)X(6179)

Barycentrics    -7*a^4+3*a^2*(b^2+c^2)+3*b^4-7*b^2*c^2+3*c^4 : :

As a point on the Euler line, X(68420) has Shinagawa coefficients: {(e+f)^2-7 S^2,20 S^2}

See Juan José Isach Mayo, euclid 8408.

X(68420) lies on the curve Q106 and these lines: {2, 3}, {148, 6179}, {194, 43618}, {698, 11008}, {1007, 45017}, {7747, 63018}, {7748, 63019}, {7751, 14712}, {7754, 35369}, {7756, 55085}, {7758, 19569}, {7759, 8591}, {7787, 43619}, {7797, 65633}, {7802, 17130}, {7814, 52695}, {7823, 20094}, {31664, 31665}, {32479, 34604}, {50570, 63957}, {51224, 63922}, {53105, 60136}

X(68420) = reflection of X(i) in X(j) for these {i,j}: {6655, 6658}, {6658, 19696}, {19691, 384}, {33256, 19687}
X(68420) = anticomplement of X(33256)
X(68420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6658, 19687}, {2, 33256, 6655}, {2, 62149, 33243}, {2, 66321, 66317}, {3, 14066, 33024}, {4, 33254, 2}, {4, 33265, 33259}, {20, 14068, 33004}, {20, 66419, 16044}, {382, 33235, 14062}, {382, 33257, 2}, {382, 66395, 33257}, {384, 6655, 66345}, {384, 7924, 66342}, {384, 8357, 2}, {384, 19691, 6655}, {384, 19695, 19690}, {384, 33256, 8357}, {384, 66342, 19692}, {384, 66345, 19689}, {546, 33276, 2}, {550, 14042, 2}, {550, 66423, 14042}, {1657, 11361, 33260}, {3146, 33193, 3552}, {3529, 33280, 2}, {3543, 33244, 32966}, {3552, 66407, 3146}, {3627, 13586, 32993}, {3853, 8598, 32967}, {5059, 14035, 33264}, {5073, 66387, 5025}, {6655, 6658, 19686}, {6655, 19686, 19689}, {6655, 19689, 66337}, {6655, 66317, 6656}, {6656, 19687, 66321}, {6656, 19693, 66317}, {6656, 66328, 19693}, {6658, 19691, 384}, {6658, 19693, 66328}, {7770, 49137, 66406}, {7791, 49138, 66421}, {7807, 62041, 8597}, {8357, 19687, 384}, {8370, 62155, 33267}, {11317, 15696, 33015}, {11361, 33260, 33020}, {14033, 62171, 33271}, {14034, 33234, 2}, {14035, 33264, 33021}, {14062, 33235, 2}, {14062, 33257, 33235}, {14068, 33004, 16044}, {15704, 66408, 7824}, {17538, 33016, 33022}, {19686, 66345, 384}, {19687, 19695, 19702}, {19687, 33256, 2}, {19690, 19691, 19695}, {19690, 19695, 6655}, {19695, 19702, 8357}, {32971, 62160, 33209}, {32979, 62152, 33207}, {32981, 33192, 7933}, {32981, 50692, 33192}, {33004, 66419, 14068}, {33007, 33019, 33225}, {33007, 33703, 33019}, {33019, 33703, 40246}, {33225, 40246, 33019}, {33239, 33279, 2}, {33239, 62042, 33279}, {33250, 62036, 14041}, {33257, 66422, 382}, {35297, 62026, 14044}, {35927, 50691, 32996}, {66317, 66328, 19686}, {66389, 66420, 33265}, {66395, 66422, 2}

X(68421) = X(3)X(6)∩X(303)X(20428)

Barycentrics    a^2*(3*a^6 - 8*a^4*b^2 + 9*a^2*b^4 - 4*b^6 - 8*a^4*c^2 + 8*a^2*b^2*c^2 + 2*b^4*c^2 + 9*a^2*c^4 + 2*b^2*c^4 - 4*c^6 - 2*Sqrt[3]*(a^4 - 3*a^2*b^2 + 2*b^4 - 3*a^2*c^2 - 2*b^2*c^2 + 2*c^4)*S) : :
X(68421) = 4 X[21401] - X[47066]

This is the point obtained from k = 1 in the note (May 6, 2025) at X(15).

X(68421) lies on these lines: {3, 6}, {303, 20428}, {531, 8176}, {5476, 52022}, {8594, 9880}, {36993, 62984}, {44666, 63732}

X(68421) = reflection of X(i) in X(j) for these {i,j}: {9736, 13350}, {44488, 44477}

X(68422) = X(3)X(6)∩X(302)X(20429)

Barycentrics    a^2*(3*a^6 - 8*a^4*b^2 + 9*a^2*b^4 - 4*b^6 - 8*a^4*c^2 + 8*a^2*b^2*c^2 + 2*b^4*c^2 + 9*a^2*c^4 + 2*b^2*c^4 - 4*c^6 + 2*Sqrt[3]*(a^4 - 3*a^2*b^2 + 2*b^4 - 3*a^2*c^2 - 2*b^2*c^2 + 2*c^4)*S) : :
X(68422) = 4 X[21402] - X[47068]

This is the point obtained from k = 1 and S -> -S in the note (May 6, 2025) at X(15). It is assumed that this point is related to X(16) as X(68421) is to X(15).

X(68422) lies on these lines: {3, 6}, {302, 20429}, {530, 8176}, {5476, 52021}, {8595, 9880}, {36995, 62983}, {44667, 63731}

X(68422) = reflection of X(i) in X(j) for these {i,j}: {9735, 13349}, {44487, 44478}

X(68423) = X(3)X(26913)∩X(30)X(5892)

Barycentrics    2*a^10 - 4*a^8*b^2 + a^6*b^4 + a^4*b^6 + a^2*b^8 - b^10 - 4*a^8*c^2 - 2*a^6*b^2*c^2 + 8*a^4*b^4*c^2 - 5*a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 + 8*a^4*b^2*c^4 + 8*a^2*b^4*c^4 - 2*b^6*c^4 + a^4*c^6 - 5*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(68423) = X[20] + 2 X[15807], 3 X[13364] - 2 X[66531], 2 X[140] + X[13470], X[140] - 4 X[44862], X[13470] + 8 X[44862], X[143] - 4 X[64038], X[568] + 3 X[67338], 5 X[632] + X[11750], 2 X[1216] + X[11264], 11 X[3525] + X[65149], 2 X[3628] + X[44829], X[3853] + 2 X[17712], 11 X[5070] + X[64718], 2 X[5447] + X[45970], X[6146] + 2 X[32142], X[7553] - 4 X[18874], 4 X[11017] - X[16655], 2 X[11591] + X[45732], 4 X[11592] - X[68018], X[11819] - 4 X[32205], 2 X[12241] + X[63414], X[12278] - 13 X[61811], X[12289] + 11 X[15720], 2 X[12362] + X[13630], X[12897] + 2 X[44245], X[13403] + 2 X[33923], X[13419] - 4 X[35018], X[15644] + 2 X[43575], 5 X[15712] + X[21659], 4 X[16239] - X[45286], 2 X[18128] + X[31834], X[18564] + 3 X[20791], X[34798] - 7 X[66606], 13 X[46219] - X[64032], 7 X[55856] - X[61139], 5 X[61940] - 2 X[67322]

See Antreas Hatzipolakis and Peter Moses, euclid 8425.

X(68423) lies on these lines: {2, 34513}, {3, 26913}, {5, 22352}, {20, 15807}, {30, 5892}, {140, 13470}, {143, 64038}, {265, 15246}, {389, 44056}, {539, 44324}, {547, 44407}, {549, 30522}, {550, 61744}, {568, 67338}, {632, 11750}, {1154, 11245}, {1216, 11264}, {1853, 7514}, {1899, 33533}, {3153, 13339}, {3525, 65149}, {3628, 44829}, {3819, 32423}, {3853, 17712}, {5012, 51391}, {5066, 29012}, {5070, 64718}, {5092, 46029}, {5447, 45970}, {5944, 59648}, {6146, 32142}, {6643, 32046}, {6676, 20304}, {7502, 61645}, {7553, 18874}, {8703, 32225}, {10610, 37452}, {10691, 54044}, {10984, 67869}, {11017, 16655}, {11585, 58407}, {11591, 45732}, {11592, 68018}, {11801, 55674}, {11819, 32205}, {12022, 54042}, {12100, 17702}, {12241, 63414}, {12278, 61811}, {12289, 15720}, {12362, 13630}, {12897, 44245}, {13154, 64037}, {13353, 61715}, {13391, 67336}, {13403, 33923}, {13419, 35018}, {14805, 31101}, {15311, 52073}, {15644, 43575}, {15712, 21659}, {16239, 45286}, {16252, 23060}, {18128, 31834}, {18377, 37515}, {18564, 20791}, {25739, 54006}, {34798, 66606}, {34826, 37126}, {35268, 44270}, {36201, 61574}, {37513, 37938}, {37649, 47341}, {43650, 44288}, {43821, 67321}, {44882, 46030}, {46219, 64032}, {48889, 50134}, {51392, 61659}, {55856, 61139}, {61134, 67861}, {61940, 67322}

X(68423) = midpoint of X(i) and X(j) for these {i,j}: {550, 61744}, {12022, 54042}
X(68423) = reflection of X(54044) in X(10691)

X(68424) = X(3)X(26913)∩X(4)X(567)

Barycentrics    2*a^10 - 4*a^8*b^2 + a^6*b^4 + a^4*b^6 + a^2*b^8 - b^10 - 4*a^8*c^2 + 6*a^6*b^2*c^2 - 5*a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 + 8*a^2*b^4*c^4 - 2*b^6*c^4 + a^4*c^6 - 5*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(68424) = 3 X[4] + X[65149], 6 X[15807] + X[65149], X[12897] - 3 X[13403], 2 X[12897] + 3 X[13470], 2 X[13403] + X[13470], 3 X[13403] + X[44829], 3 X[13470] - 2 X[44829], 3 X[381] + X[12289], 5 X[1656] - X[12278], X[1885] + 2 X[11565], X[3627] - 3 X[61744], X[11750] + 3 X[61744], 3 X[3830] + X[64718], 5 X[3843] - X[64032], 3 X[3845] - X[61139], X[45732] + 2 X[52070], X[5876] - 3 X[52069], X[44076] + 3 X[52069], X[5889] + 3 X[18564], 3 X[5890] + X[18562], 3 X[5890] - X[34798], 3 X[5946] - X[6240], X[6101] - 3 X[67337], X[6102] - 3 X[12022], 3 X[12022] + X[18563], 5 X[10574] - X[18565], X[11819] - 3 X[16657], 3 X[16657] - 2 X[67867], 3 X[12100] - 4 X[44862], 3 X[13363] - 2 X[31833], 3 X[13364] - 2 X[31830], 3 X[13451] - 4 X[40240], 2 X[14128] - 3 X[34664], X[14516] - 3 X[15060], 3 X[14893] - 2 X[67322], 5 X[15026] - 3 X[38321], 7 X[15043] + X[40242], 3 X[18435] + X[34799], 4 X[18874] - 3 X[67237], X[34783] + 3 X[67339], X[34797] - 5 X[37481]

See Antreas Hatzipolakis and Peter Moses, euclid 8425.

X(68424) lies on these lines: {3, 26913}, {4, 567}, {5, 13367}, {6, 52843}, {20, 18952}, {30, 143}, {54, 18403}, {113, 10619}, {140, 6723}, {156, 19467}, {184, 67869}, {186, 43821}, {265, 14118}, {381, 9707}, {403, 5944}, {539, 31834}, {546, 8254}, {550, 18555}, {569, 44263}, {578, 18377}, {1154, 12370}, {1199, 10296}, {1493, 66727}, {1503, 32137}, {1656, 12278}, {1658, 18390}, {1885, 11565}, {1899, 32138}, {2070, 43835}, {2072, 43394}, {2777, 18128}, {3153, 37472}, {3521, 64890}, {3575, 10095}, {3627, 11750}, {3830, 64718}, {3843, 64032}, {3845, 61139}, {3850, 45286}, {3853, 44407}, {3861, 13419}, {5133, 22804}, {5462, 45971}, {5663, 6146}, {5876, 44076}, {5889, 18564}, {5890, 18562}, {5907, 32423}, {5946, 6240}, {6101, 67337}, {6102, 12022}, {6288, 35500}, {7514, 12293}, {7526, 18396}, {7564, 18405}, {7687, 44516}, {7706, 50006}, {7728, 67879}, {8718, 18325}, {10019, 12140}, {10024, 10113}, {10212, 40685}, {10224, 11430}, {10263, 12225}, {10282, 44235}, {10540, 12254}, {10574, 18565}, {10627, 12362}, {10733, 61134}, {11264, 13754}, {11424, 44288}, {11572, 33332}, {11591, 44665}, {11819, 16657}, {12038, 49673}, {12041, 35491}, {12100, 44862}, {12106, 34785}, {12134, 45958}, {12162, 45731}, {12295, 64179}, {12429, 64105}, {12902, 34864}, {13142, 13421}, {13353, 34007}, {13363, 31833}, {13364, 31830}, {13406, 18475}, {13451, 40240}, {13491, 18560}, {13561, 18570}, {13567, 44242}, {13619, 43816}, {13861, 17845}, {14128, 34664}, {14130, 25739}, {14516, 15060}, {14644, 51033}, {14893, 67322}, {15026, 38321}, {15033, 31724}, {15043, 40242}, {15516, 29012}, {15646, 43817}, {17712, 62144}, {18364, 38724}, {18378, 41482}, {18383, 39504}, {18388, 18567}, {18435, 34799}, {18445, 66733}, {18474, 63682}, {18874, 67237}, {18912, 66721}, {18945, 32140}, {31726, 52525}, {31861, 64037}, {32142, 68018}, {32210, 67902}, {34148, 51391}, {34782, 46030}, {34783, 67339}, {34797, 37481}, {35480, 36753}, {35603, 44438}, {36966, 43844}, {37490, 66711}, {44279, 64049}, {46031, 64063}, {52008, 67067}, {61574, 61608}

X(68424) = midpoint of X(i) and X(j) for these {i,j}: {5, 21659}, {3627, 11750}, {5876, 44076}, {6102, 18563}, {6146, 52070}, {10263, 12225}, {12162, 45731}, {12370, 12605}, {12897, 44829}, {13491, 18560}, {18562, 34798}
X(68424) = reflection of X(i) in X(j) for these {i,j}: {4, 15807}, {143, 12241}, {389, 43575}, {3575, 10095}, {10627, 12362}, {11264, 45970}, {11591, 52073}, {11819, 67867}, {12134, 45958}, {13419, 3861}, {13421, 13142}, {45286, 3850}, {45732, 6146}, {45971, 5462}, {62144, 17712}, {68018, 32142}
X(68424) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 13367, 58435}, {54, 18403, 67861}, {265, 14118, 34826}, {3153, 43818, 37472}, {5890, 18562, 34798}, {5944, 43865, 403}, {10113, 10610, 10024}, {11750, 61744, 3627}, {11819, 16657, 67867}, {12022, 18563, 6102}, {13403, 44829, 12897}, {18570, 67903, 13561}, {18945, 49669, 32140}, {44076, 52069, 5876}

X(68425) = X(3)X(26913)∩X(5)X(1495)

Barycentrics    2*a^10 - 4*a^8*b^2 + a^6*b^4 + a^4*b^6 + a^2*b^8 - b^10 - 4*a^8*c^2 + 2*a^6*b^2*c^2 + 4*a^4*b^4*c^2 - 5*a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 + 4*a^4*b^2*c^4 + 8*a^2*b^4*c^4 - 2*b^6*c^4 + a^4*c^6 - 5*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(68425) = 3 X[5] + X[11750], 5 X[5] - X[61139], X[11750] - 3 X[13470], 5 X[11750] + 3 X[61139], 5 X[13470] + X[61139], 5 X[15807] + 4 X[17712], 3 X[547] - X[45286], 3 X[549] + X[21659], 3 X[12362] + X[13292], 7 X[3090] + X[65149], 7 X[3526] + X[12289], X[3575] - 3 X[13363], 7 X[3851] + X[64718], 9 X[5054] - X[12278], 9 X[5055] - X[64032], 3 X[5066] - X[13419], 3 X[5891] + X[45731], 3 X[5892] - X[45971], 3 X[5946] + X[12225], X[6101] + 3 X[12022], X[6102] + 3 X[67337], 5 X[10574] + 3 X[18564], 5 X[10574] - X[34798], 3 X[18564] + X[34798], X[11819] - 3 X[13364], 4 X[12046] - 3 X[23410], X[13491] + 3 X[52069], 3 X[15060] + X[34224], 3 X[15067] + X[44076], X[15704] + 3 X[61744], X[18562] + 7 X[66606], X[18565] - 9 X[20791], 3 X[34664] - X[45959], X[34797] - 9 X[40280], X[37484] - 9 X[67338], X[63414] - 3 X[67336]

See Antreas Hatzipolakis and Peter Moses, euclid 8425.

X(68425) lies on these lines: {3, 26913}, {4, 37471}, {5, 1495}, {30, 5462}, {54, 51391}, {68, 33533}, {140, 30522}, {182, 18377}, {265, 37126}, {511, 43575}, {542, 45734}, {546, 44829}, {547, 45286}, {548, 13403}, {549, 21659}, {1154, 12362}, {1216, 45970}, {1503, 45958}, {2072, 10610}, {3090, 65149}, {3153, 13353}, {3292, 36966}, {3526, 12289}, {3530, 17702}, {3575, 13363}, {3581, 43816}, {3628, 18400}, {3850, 44407}, {3851, 64718}, {3856, 67322}, {3861, 29012}, {5012, 67861}, {5054, 12278}, {5055, 64032}, {5066, 13419}, {5562, 11264}, {5663, 52073}, {5876, 45732}, {5891, 45731}, {5892, 45971}, {5946, 12225}, {6101, 12022}, {6102, 67337}, {6146, 11591}, {6288, 7550}, {6756, 18874}, {7505, 34513}, {7512, 43821}, {7514, 67878}, {7516, 18396}, {7525, 18390}, {7542, 20304}, {7574, 13434}, {10116, 31834}, {10282, 50140}, {10574, 18564}, {10619, 40111}, {10627, 12370}, {11565, 14128}, {11645, 46852}, {11793, 32423}, {11801, 34004}, {11819, 13364}, {12041, 34005}, {12046, 23410}, {12103, 12897}, {12241, 13391}, {12605, 13630}, {13336, 44263}, {13339, 34007}, {13491, 52069}, {14788, 22804}, {15060, 34224}, {15067, 44076}, {15361, 43836}, {15704, 61744}, {15800, 34545}, {16266, 18536}, {18381, 49671}, {18403, 61134}, {18475, 49673}, {18531, 32046}, {18562, 66606}, {18565, 20791}, {19155, 48906}, {22352, 61750}, {25739, 34864}, {31724, 43651}, {31830, 32205}, {32142, 44665}, {32348, 36253}, {34664, 45959}, {34797, 40280}, {34826, 35921}, {37452, 43394}, {37477, 43818}, {37484, 67338}, {37514, 52843}, {63414, 67336}, {64049, 67869}

X(68425) = midpoint of X(i) and X(j) for these {i,j}: {5, 13470}, {546, 44829}, {548, 13403}, {1216, 45970}, {5562, 11264}, {5876, 45732}, {6146, 11591}, {10116, 31834}, {10627, 12370}, {11565, 14128}, {12103, 12897}, {12605, 13630}
X(68425) = reflection of X(i) in X(j) for these {i,j}: {3530, 44862}, {6756, 18874}, {12006, 64038}, {31830, 32205}, {67322, 3856}
X(68425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2072, 10610, 58407}, {10574, 18564, 34798}, {18475, 49673, 58435}

X(68426) = X(3)X(26913)∩X(343)X(550)

Barycentrics    2*a^10 - 4*a^8*b^2 + a^6*b^4 + a^4*b^6 + a^2*b^8 - b^10 - 4*a^8*c^2 + 14*a^6*b^2*c^2 - 8*a^4*b^4*c^2 - 5*a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 - 8*a^4*b^2*c^4 + 8*a^2*b^4*c^4 - 2*b^6*c^4 + a^4*c^6 - 5*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(68426) = 5 X[13419] - 7 X[45286], 3 X[550] - X[11750], 5 X[632] - 3 X[61744], 3 X[2979] + X[18565], 3 X[3534] + X[12278], X[6101] - 3 X[54040], 5 X[6102] - 3 X[41628], 3 X[8703] - X[21659], 2 X[10095] - 3 X[66614], X[10263] - 3 X[38323], X[12289] - 5 X[15696], 2 X[12362] - 3 X[54044], 2 X[13292] - 3 X[13630], X[13292] - 3 X[31829], 3 X[13340] + X[34797], X[13491] - 3 X[44458], 3 X[14855] - X[45731], 3 X[15067] - X[18560], 3 X[15681] + X[64032], 3 X[15690] - 2 X[17712], 3 X[16657] - 4 X[32205], 3 X[16836] - 2 X[43575], 5 X[17538] - X[65149], X[18563] - 3 X[54042], 4 X[44862] - 5 X[61790], 5 X[62131] - X[64718]

See Antreas Hatzipolakis and Peter Moses, euclid 8425.

X(68426) lies on these lines: {2, 15807}, {3, 26913}, {5, 10564}, {20, 41466}, {30, 1216}, {323, 3521}, {343, 550}, {382, 15066}, {546, 53415}, {548, 13470}, {632, 61744}, {1092, 67869}, {1885, 14128}, {2071, 34826}, {2777, 31834}, {2979, 18565}, {3530, 13403}, {3534, 12278}, {3628, 12897}, {3631, 29012}, {5663, 68018}, {5876, 52071}, {6101, 54040}, {6102, 41628}, {6288, 7464}, {7512, 12121}, {8703, 21659}, {10095, 66614}, {10113, 37452}, {10263, 38323}, {11264, 40647}, {11412, 34798}, {12084, 67878}, {12103, 18400}, {12289, 15696}, {12362, 54044}, {12605, 13416}, {13292, 13630}, {13339, 43818}, {13340, 34797}, {13491, 44458}, {13565, 44236}, {14855, 45731}, {15067, 18560}, {15681, 64032}, {15690, 17712}, {15704, 61299}, {15760, 58407}, {16196, 20304}, {16657, 32205}, {16836, 43575}, {17538, 65149}, {17845, 33532}, {18325, 43598}, {18350, 51548}, {18377, 37480}, {18563, 54042}, {18952, 61113}, {24981, 44866}, {31833, 68084}, {32142, 52070}, {32171, 63631}, {32423, 46850}, {34005, 62302}, {34007, 37477}, {37636, 64180}, {38448, 38723}, {41724, 43807}, {43574, 67861}, {44241, 63734}, {44245, 44829}, {44407, 62144}, {44440, 61753}, {44665, 45732}, {44862, 61790}, {51394, 58435}, {55631, 61543}, {61139, 62155}, {62038, 67322}, {62131, 64718}, {63441, 67926}

X(68426) = midpoint of X(i) and X(j) for these {i,j}: {5876, 52071}, {11412, 34798}, {61139, 62155}
X(68426) = reflection of X(i) in X(j) for these {i,j}: {1885, 14128}, {11264, 40647}, {12897, 3628}, {13403, 3530}, {13470, 548}, {13630, 31829}, {32137, 64035}, {44829, 44245}, {52070, 32142}, {62038, 67322}, {68084, 31833}
X(68426) = anticomplement of X(15807)
X(68426) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {343, 550, 32210}, {51394, 61750, 58435}

X(68427) = X(3)X(26913)∩X(5)X(156)

Barycentrics    a^10 - 2*a^8*b^2 + a^6*b^4 - a^4*b^6 + 2*a^2*b^8 - b^10 - 2*a^8*c^2 + 2*a^6*b^2*c^2 + 2*a^4*b^4*c^2 - 5*a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 + 2*a^4*b^2*c^4 + 6*a^2*b^4*c^4 - 2*b^6*c^4 - a^4*c^6 - 5*a^2*b^2*c^6 - 2*b^4*c^6 + 2*a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(68427) = X[6515] + 3 X[18531], X[15068] - 3 X[16072], X[31383] - 3 X[44275]

See Antreas Hatzipolakis and Peter Moses, euclid 8425.

X(68427) lies on these lines: {2, 265}, {3, 26913}, {4, 3521}, {5, 156}, {26, 13470}, {30, 11438}, {51, 44288}, {54, 10255}, {68, 11591}, {125, 18570}, {140, 9927}, {143, 18569}, {155, 11264}, {182, 11801}, {343, 33533}, {381, 5422}, {389, 18377}, {403, 61752}, {546, 2883}, {567, 7577}, {568, 3153}, {578, 10224}, {597, 3818}, {1147, 45970}, {1154, 6515}, {1181, 67869}, {1495, 44270}, {1503, 46030}, {1514, 3845}, {1568, 61713}, {1593, 15807}, {1656, 58407}, {1853, 31861}, {1899, 5663}, {1993, 51391}, {2072, 12022}, {3060, 7574}, {3090, 6288}, {3448, 18435}, {3518, 65149}, {3534, 15361}, {3542, 11565}, {3567, 31724}, {3580, 67337}, {5012, 10254}, {5055, 41171}, {5462, 18383}, {5876, 25738}, {5890, 18403}, {5907, 18356}, {5944, 7505}, {6102, 18404}, {6143, 45622}, {6639, 10610}, {6640, 43394}, {6643, 10627}, {6644, 18396}, {6759, 44235}, {6776, 19155}, {6816, 14128}, {7386, 54044}, {7502, 63735}, {7503, 34826}, {7507, 35603}, {7514, 14852}, {7526, 13561}, {7528, 18874}, {7530, 61299}, {7544, 22804}, {7547, 36753}, {7575, 61645}, {7592, 67861}, {7706, 18376}, {7728, 67925}, {8703, 44569}, {9306, 32423}, {9544, 14643}, {9730, 13851}, {9786, 52843}, {9818, 61702}, {10113, 12099}, {10192, 68319}, {10193, 20397}, {10263, 37444}, {10297, 11245}, {10540, 62947}, {10575, 44271}, {10938, 11561}, {10984, 61750}, {11202, 44234}, {11204, 61548}, {11250, 13403}, {11425, 31283}, {11430, 61736}, {11442, 15060}, {11487, 15077}, {11572, 63672}, {11585, 12370}, {11750, 37440}, {11818, 13364}, {12006, 18379}, {12041, 35481}, {12106, 18400}, {12121, 61128}, {12241, 13371}, {12278, 43809}, {12289, 45735}, {12359, 52073}, {12362, 63734}, {12900, 61681}, {13352, 37938}, {13363, 18420}, {13367, 60780}, {13391, 14791}, {13406, 64049}, {13421, 64048}, {13621, 64032}, {13861, 64037}, {14130, 23294}, {14216, 32137}, {14516, 50143}, {14708, 67921}, {14790, 68084}, {14864, 46849}, {14915, 58483}, {15043, 18394}, {15045, 18392}, {15053, 43836}, {15061, 35473}, {15068, 16072}, {15072, 31726}, {15133, 50008}, {16227, 44283}, {17714, 44829}, {18128, 61749}, {18324, 26958}, {18350, 34799}, {18378, 64718}, {18388, 43573}, {18559, 58789}, {18563, 26879}, {18565, 43601}, {18580, 20396}, {18583, 41729}, {19357, 58435}, {19467, 32171}, {20379, 23291}, {21243, 36253}, {21659, 37814}, {22352, 44262}, {22466, 34350}, {22660, 43588}, {23293, 38724}, {23325, 39504}, {23332, 44236}, {26937, 32210}, {31101, 37477}, {31181, 44413}, {31383, 44275}, {31830, 41362}, {32138, 52070}, {32139, 45732}, {32140, 45959}, {34477, 47296}, {34513, 68085}, {34545, 62982}, {34609, 64099}, {34664, 67926}, {34782, 44232}, {34783, 43808}, {34786, 45971}, {35493, 38728}, {39503, 57136}, {40647, 44279}, {44076, 61753}, {44665, 53415}, {44961, 51733}, {46031, 61747}, {48906, 62375}, {51548, 62974}

X(68427) = midpoint of X(6644) and X(18396)
X(68427) = reflection of X(9306) in X(50140)
X(68427) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 61701, 63839}, {4, 18952, 13630}, {4, 43816, 37481}, {5, 6146, 156}, {5, 31804, 61608}, {5, 45731, 10539}, {381, 25739, 34514}, {5012, 14644, 10254}, {9730, 13851, 44263}, {10224, 43575, 578}, {13491, 43865, 4}, {18404, 18912, 6102}, {18569, 39571, 143}, {21659, 43817, 37814}, {45970, 49673, 1147}, {52070, 67902, 32138}

X(68428) = X(3)X(26913)∩X(30)X(11695)

Barycentrics    2*a^10 - 4*a^8*b^2 + a^6*b^4 + a^4*b^6 + a^2*b^8 - b^10 - 4*a^8*c^2 - 6*a^6*b^2*c^2 + 12*a^4*b^4*c^2 - 5*a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 + 12*a^4*b^2*c^4 + 8*a^2*b^4*c^4 - 2*b^6*c^4 + a^4*c^6 - 5*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(68428) = 3 X[140] + X[44829], 5 X[140] - X[45286], 5 X[44829] + 3 X[45286], X[143] + 3 X[67336], 3 X[548] + X[12897], 3 X[549] + X[13470], 15 X[631] + X[65149], 3 X[3917] + X[11264], X[5446] + 3 X[60749], 5 X[6101] + 3 X[41628], 3 X[7667] + X[68084], 3 X[10109] - X[67322], X[10116] + 3 X[44324], 9 X[11539] - X[61139], X[11750] + 7 X[14869], X[11819] - 9 X[64730], X[12278] - 17 X[61803], X[12289] + 15 X[15693], X[12370] + 3 X[54044], X[13403] + 3 X[34200], X[13419] - 5 X[48154], 3 X[15067] + X[45732], 15 X[15694] + X[64718], 9 X[20791] - X[34798], X[21659] + 7 X[44682], 17 X[55863] - X[64032], 3 X[61744] + 5 X[62104]

See Antreas Hatzipolakis and Peter Moses, euclid 8425.

X(68428) lies on these lines: {3, 26913}, {30, 11695}, {140, 44829}, {143, 67336}, {265, 45308}, {548, 12897}, {549, 13470}, {550, 15807}, {631, 65149}, {3530, 30522}, {3628, 61299}, {3850, 17712}, {3917, 11264}, {5092, 10224}, {5446, 60749}, {6101, 41628}, {7667, 68084}, {10109, 67322}, {10116, 44324}, {11539, 61139}, {11592, 44665}, {11750, 14869}, {11819, 64730}, {12108, 18400}, {12278, 61803}, {12289, 15693}, {12370, 54044}, {13347, 18377}, {13348, 43575}, {13391, 64038}, {13403, 34200}, {13419, 48154}, {15067, 45732}, {15606, 32165}, {15694, 64718}, {16239, 44407}, {17702, 61792}, {20304, 34002}, {20791, 34798}, {21659, 44682}, {23060, 51425}, {29012, 35018}, {32046, 37645}, {37452, 58407}, {43821, 44832}, {51391, 61134}, {55863, 64032}, {61744, 62104}

X(68428) = midpoint of X(i) and X(j) for these {i,j}: {550, 15807}, {3850, 17712}, {13348, 43575}, {15606, 32165}

X(68429) = X(3)X(26913)∩X(30)X(6699)

Barycentrics    2*a^10 - 4*a^8*b^2 - a^6*b^4 + 7*a^4*b^6 - 5*a^2*b^8 + b^10 - 4*a^8*c^2 + 10*a^6*b^2*c^2 - 8*a^4*b^4*c^2 + 5*a^2*b^6*c^2 - 3*b^8*c^2 - a^6*c^4 - 8*a^4*b^2*c^4 + 2*b^6*c^4 + 7*a^4*c^6 + 5*a^2*b^2*c^6 + 2*b^4*c^6 - 5*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(68429) = 3 X[3] + X[50435], X[50435] - 3 X[63839], 5 X[6699] + X[32223], X[32223] - 5 X[44673], X[186] + 3 X[15061], X[265] + 3 X[37941], 3 X[549] - X[51394], 5 X[631] - X[22115], X[1495] - 3 X[16532], X[2071] - 5 X[38728], X[2072] - 3 X[34128], X[3292] - 7 X[14869], X[3581] + 3 X[65085], 9 X[5054] - X[50461], X[5609] - 3 X[59648], 3 X[44452] - X[51425], X[7575] + 5 X[38729], 13 X[10303] - X[63720], X[11563] - 3 X[61691], X[14157] + 7 X[15057], 3 X[15055] + X[31726], 5 X[15059] - X[18403], 11 X[15720] + X[41724], X[18571] + 2 X[20397], 2 X[20396] + X[47335], X[25739] + 3 X[37955], X[34152] - 3 X[38727], 3 X[38727] + X[63735], 3 X[35489] + X[58789], 3 X[37943] - X[51548], 3 X[38793] - X[40111], 3 X[44282] - X[51403], 3 X[46451] + X[64624]

See Antreas Hatzipolakis and Peter Moses, euclid 8425.

X(68429) lies on these lines: {3, 26913}, {5, 21663}, {30, 6699}, {125, 15646}, {140, 9729}, {143, 23336}, {156, 26937}, {185, 58435}, {186, 15061}, {265, 37941}, {343, 549}, {389, 5498}, {403, 12041}, {539, 48378}, {546, 25563}, {631, 9545}, {1154, 10257}, {1192, 31283}, {1204, 60780}, {1495, 16532}, {1503, 16531}, {1511, 62302}, {1620, 52843}, {2071, 38728}, {2072, 34128}, {2777, 46031}, {3292, 14869}, {3520, 15807}, {3523, 18952}, {3581, 65085}, {3631, 50983}, {3853, 44872}, {5054, 15066}, {5449, 43615}, {5609, 59648}, {5663, 44452}, {5946, 37118}, {6000, 44234}, {6696, 32137}, {7575, 38729}, {8254, 15012}, {10018, 13491}, {10096, 14915}, {10113, 44246}, {10125, 40647}, {10151, 34584}, {10212, 43575}, {10255, 34798}, {10264, 51393}, {10303, 63720}, {10627, 16196}, {11264, 12038}, {11438, 61736}, {11563, 61691}, {11704, 18565}, {12006, 37649}, {12106, 23329}, {12108, 20585}, {13358, 45780}, {13363, 52262}, {13364, 44236}, {13391, 15122}, {13406, 43604}, {13470, 15331}, {13561, 18474}, {14157, 15057}, {15055, 31726}, {15059, 18403}, {15088, 23323}, {15311, 68319}, {15720, 41724}, {16111, 44283}, {16238, 45959}, {17702, 37968}, {17704, 34004}, {18282, 46850}, {18388, 34331}, {18400, 18571}, {20396, 47335}, {22467, 34826}, {23328, 46030}, {25739, 37955}, {26879, 43394}, {32110, 37938}, {32140, 58378}, {32171, 45732}, {32767, 45971}, {34152, 38727}, {34797, 45622}, {35489, 58789}, {35491, 43865}, {37943, 51548}, {38793, 40111}, {43584, 48411}, {43608, 45735}, {44235, 64027}, {44282, 51403}, {44911, 61574}, {46451, 64624}, {47486, 64757}, {49116, 51733}, {50143, 64180}

X(68429) = midpoint of X(i) and X(j) for these {i,j}: {3, 63839}, {5, 21663}, {125, 15646}, {403, 12041}, {6699, 44673}, {10113, 44246}, {10264, 51393}, {16111, 44283}, {32110, 37938}, {34152, 63735}, {44234, 61548}, {49116, 51733}
X(68429) = reflection of X(i) in X(j) for these {i,j}: {3853, 44872}, {23323, 15088}, {61574, 44911}
X(68429) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {140, 13630, 58407}, {140, 44158, 11591}, {1204, 60780, 67869}, {6696, 44232, 32137}, {32171, 67902, 45732}, {38727, 63735, 34152}

X(68430) = X(1)X(30)∩X(57)X(2608)

Barycentrics    a*(a - b)*(a - c)*(a + b - c)*(a - b + c)*(b + c)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + b*c - c^2)*(a^2 - b^2 + a*c + c^2) : :

X(68430) lies on the cubics K130 and K407 and these lines: {1, 30}, {57, 2608}, {110, 9811}, {476, 59828}, {4551, 56193}, {6742, 61220}, {23706, 34922}, {36064, 53936}, {43692, 52390}

X(68430) = X(i)-cross conjugate of X(j) for these (i,j): {2610, 18593}, {42768, 6757}
X(68430) = X(i)-isoconjugate of X(j) for these (i,j): {476, 3024}, {526, 62713}, {759, 35057}, {1021, 65228}, {1793, 54244}, {2341, 14838}, {2605, 6740}, {3737, 56422}, {6741, 36069}, {7252, 41226}, {9404, 24624}, {17104, 52356}, {21789, 63778}, {34079, 57066}, {35192, 60074}, {35193, 66284}, {52380, 57099}
X(68430) = X(i)-Dao conjugate of X(j) for these (i,j): {34586, 35057}, {35069, 57066}, {38982, 6741}, {56847, 52356}
X(68430) = cevapoint of X(1464) and X(51663)
X(68430) = barycentric product X(i)*X(j) for these {i,j}: {758, 38340}, {860, 65300}, {1020, 63642}, {1464, 15455}, {2245, 65292}, {3028, 32680}, {3936, 26700}, {4242, 63171}, {4552, 56844}, {4585, 52382}, {6370, 35049}, {6742, 18593}, {17078, 56193}, {32679, 55017}, {52390, 65162}
X(68430) = barycentric quotient X(i)/X(j) for these {i,j}: {758, 57066}, {1020, 63778}, {1443, 16755}, {1464, 14838}, {1835, 65100}, {1983, 35193}, {2245, 35057}, {2610, 6741}, {2624, 3024}, {3028, 32679}, {3724, 9404}, {4551, 41226}, {4559, 56422}, {8818, 52356}, {18593, 4467}, {21828, 53524}, {26700, 24624}, {32678, 62713}, {35049, 65283}, {38340, 14616}, {41804, 18160}, {44113, 65105}, {51663, 8287}, {52375, 60571}, {52382, 60074}, {53321, 65228}, {55017, 32680}, {56193, 36910}, {56844, 4560}, {61060, 2624}, {65300, 57985}

X(68431) = X(5)X(523)∩X(30)X(2088)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^8*b^4 - 3*a^6*b^6 + 3*a^4*b^8 - a^2*b^10 - a^4*b^6*c^2 + a^2*b^8*c^2 + a^8*c^4 + 2*a^4*b^4*c^4 - a^2*b^6*c^4 - b^8*c^4 - 3*a^6*c^6 - a^4*b^2*c^6 - a^2*b^4*c^6 + 2*b^6*c^6 + 3*a^4*c^8 + a^2*b^2*c^8 - b^4*c^8 - a^2*c^10) : :

X(68431) lies on the cubic K166 and these lines: {2, 18333}, {3, 39295}, {5, 523}, {6, 64221}, {30, 2088}, {94, 66459}, {265, 2782}, {381, 54554}, {476, 1316}, {1989, 49102}, {2452, 60053}, {2493, 56396}, {5627, 56409}, {5968, 56400}, {6795, 14880}, {11657, 66125}, {12042, 41392}, {14881, 53771}, {46127, 56398}

X(68431) = midpoint of X(265) and X(66075)
X(68431) = X(6149)-isoconjugate of X(43654)
X(68431) = X(14993)-Dao conjugate of X(43654)
X(68431) = barycentric product X(20573)*X(56393)
X(68431) = barycentric quotient X(i)/X(j) for these {i,j}: {1989, 43654}, {56393, 50}
X(68431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14356, 15475, 14254}, {58912, 58913, 14356}

X(68432) = X(6)X(17)∩X(32)X(39171)

Barycentrics    (4*a^2 - b^2 - c^2)*(a^4 - a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4) : :

X(68432) lies on the cubic K055 and these lines: {6, 17}, {32, 39171}, {930, 67107}, {1487, 2548}, {5007, 60824}, {11140, 53102}, {19136, 32737}, {34565, 36304}, {35007, 67115}

X(68432) = X(2964)-isoconjugate of X(43676)
X(68432) = X(i)-Dao conjugate of X(j) for these (i,j): {21975, 43676}, {51581, 7769}
X(68432) = barycentric product X(i)*X(j) for these {i,j}: {17, 67116}, {18, 67115}, {930, 32478}, {2963, 3629}, {3519, 62978}, {11140, 35007}
X(68432) = barycentric quotient X(i)/X(j) for these {i,j}: {2963, 43676}, {3629, 7769}, {32478, 41298}, {32737, 53884}, {35007, 1994}, {62978, 32002}, {67115, 303}, {67116, 302}
X(68432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {17, 18, 56738}, {67107, 67108, 930}

X(68433) = X(4)X(6)∩X(5)X(41891)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^10 - 3*a^8*b^2 - 2*a^6*b^4 + 4*a^4*b^6 - b^10 - 3*a^8*c^2 + 2*a^6*b^2*c^2 - 2*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 + 4*a^2*b^4*c^4 - 2*b^6*c^4 + 4*a^4*c^6 - 2*a^2*b^2*c^6 - 2*b^4*c^6 + 3*b^2*c^8 - c^10) : :

X(68433) lies on the cubic K126 and these lines: {4, 6}, {5, 41891}, {112, 42459}, {324, 14389}, {1033, 9715}, {3313, 53772}, {3580, 56296}, {6676, 16318}, {8879, 10565}, {10317, 65376}, {14091, 64646}, {14361, 37638}, {18438, 59661}, {34782, 58736}, {39575, 65809}, {47093, 47183}

X(68433) = polar conjugate of the isotomic conjugate of X(12225)
X(68433) = X(60241)-Ceva conjugate of X(4)
X(68433) = X(3575)-Dao conjugate of X(23292)
X(68433) = barycentric product X(4)*X(12225)
X(68433) = barycentric quotient X(12225)/X(69)
X(68433) = {X(393),X(1249)}-harmonic conjugate of X(8743)

X(68434) = X(6)X(13)∩X(110)X(143)

Barycentrics    a^2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6)*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + a^4*b^2*c^2 - a^2*b^4*c^2 + 2*b^6*c^2 - a^2*b^2*c^4 - 2*b^4*c^4 + 2*a^2*c^6 + 2*b^2*c^6 - c^8) : :

X(68434) lies on the cubic K067 and these lines: {3, 7731}, {5, 33565}, {6, 13}, {24, 3043}, {30, 40640}, {110, 143}, {125, 15047}, {323, 13392}, {1173, 5609}, {1199, 46440}, {1511, 37922}, {1986, 32608}, {1994, 43704}, {2070, 11557}, {2914, 10272}, {3153, 11805}, {3448, 50138}, {3526, 17847}, {5663, 13434}, {7526, 10620}, {7527, 11559}, {7592, 12308}, {9706, 47117}, {9919, 34117}, {10216, 24573}, {10264, 15037}, {10540, 11692}, {10628, 34864}, {10721, 52100}, {11538, 59492}, {11562, 14130}, {11591, 43580}, {11804, 66765}, {11807, 37924}, {12383, 38322}, {13417, 13564}, {13512, 14570}, {14049, 23236}, {14627, 32423}, {14644, 32341}, {15040, 22109}, {15091, 32223}, {15101, 43651}, {15782, 40604}, {16223, 43809}, {18369, 41671}, {19506, 48669}, {20806, 54048}, {25336, 44494}, {34577, 65111}, {38539, 58733}, {38790, 67339}, {61598, 67879}

X(68434) = midpoint of X(1199) and X(46440)
X(68434) = reflection of X(3) in X(27866)
X(68434) = X(i)-Ceva conjugate of X(j) for these (i,j): {250, 47053}, {58733, 2070}
X(68434) = X(33565)-isoconjugate of X(51804)
X(68434) = X(46439)-Dao conjugate of X(64935)
X(68434) = crosssum of X(115) and X(64937)
X(68434) = barycentric product X(i)*X(j) for these {i,j}: {249, 46439}, {2070, 37779}, {11063, 68354}, {24978, 47053}, {37766, 50461}, {40604, 58733}
X(68434) = barycentric quotient X(i)/X(j) for these {i,j}: {2070, 13582}, {9380, 43704}, {11063, 33565}, {37943, 9381}, {46439, 338}
X(68434) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {399, 15038, 265}, {2914, 10272, 50461}, {11557, 11597, 2070}

X(68435) = X(1)X(7)∩X(883)X(926)

Barycentrics    (a - b)*(a - c)*(a + b - c)*(a - b + c)*(a^3*b^2 - a*b^4 - 4*a^3*b*c + a^2*b^2*c + b^4*c + a^3*c^2 + a^2*b*c^2 + 2*a*b^2*c^2 - b^3*c^2 - b^2*c^3 - a*c^4 + b*c^4) : :

X(68435) lies on the cubic K085 and these lines: {1, 7}, {883, 926}, {1633, 6516}, {4998, 23831}, {14543, 21362}, {20295, 56543}, {32735, 39293}

X(68435) = barycentric product X(664)*X(49759)
X(68435) = barycentric quotient X(49759)/X(522)

X(68436) = X(230)X(231)∩X(520)X(54259)

Barycentrics    (-b + c)*(b + c)*(a^6 + 3*a^4*b^2 - a^2*b^4 - 3*b^6 + 3*a^4*c^2 - 6*a^2*b^2*c^2 + 3*b^4*c^2 - a^2*c^4 + 3*b^2*c^4 - 3*c^6) : :
X(68436) = 3 X[1637] - X[57071], 2 X[2485] - 3 X[6587], X[2485] - 3 X[47125], X[2485] + 3 X[47138], X[6587] + 2 X[47138]

X(68436) lies on the cubic K079 and these lines: {230, 231}, {520, 54259}, {3267, 14638}, {34212, 51316}, {45245, 62612}, {50642, 62176}, {52585, 64919}

X(68436) = midpoint of X(47125) and X(47138)
X(68436) = reflection of X(i) in X(j) for these {i,j}: {2489, 59652}, {6587, 47125}, {58759, 66161}, {62176, 50642}
X(68436) = X(65060)-complementary conjugate of X(21253)
X(68436) = crosspoint of X(107) and X(2996)
X(68436) = crosssum of X(520) and X(3053)
X(68436) = crossdifference of every pair of points on line {3, 19132}
X(68436) = barycentric product X(i)*X(j) for these {i,j}: {523, 7396}, {850, 9924}
X(68436) = barycentric quotient X(i)/X(j) for these {i,j}: {7396, 99}, {9924, 110}

X(68437) = X(3)X(6)∩X(428)X(13468)

Barycentrics    a^2*(a^6 - 4*a^4*b^2 + 3*a^2*b^4 - 4*a^4*c^2 - 2*a^2*b^2*c^2 + 2*b^4*c^2 + 3*a^2*c^4 + 2*b^2*c^4) : :

X(68437) lies on the cubic K075 and these lines: {3, 6}, {428, 13468}, {6997, 34229}, {7500, 37667}, {37184, 64028}, {37439, 58446}, {50774, 66381}

X(68437) = {X(32),X(8266)}-harmonic conjugate of X(5157)

X(68438) = X(6)X(17)∩X(128)X(1154)

Barycentrics    (a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^4 - a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*(2*a^6 - 3*a^4*b^2 + b^6 - 3*a^4*c^2 - 2*a^2*b^2*c^2 - b^4*c^2 - b^2*c^4 + c^6) : :

X(68438) lies on the curve Q101 and these lines: {6, 17}, {128, 1154}, {252, 6689}, {539, 1141}, {562, 14918}, {3574, 25043}, {10610, 35888}, {18400, 19552}, {21975, 32348}

X(68438) = barycentric product X(14389)*X(66883)
X(68438) = barycentric quotient X(i)/X(j) for these {i,j}: {2439, 58975}, {64877, 2413}, {66883, 60225}

X(68439) = X(1)X(2)∩X(1089)X(20655)

Barycentrics    (b + c)^2*(a*b + b^2 + a*c + b*c + c^2)*(a*b^2 + b^3 + a*c^2 + c^3) : :

X(68439) lies on the curve Q121 and these lines: {1, 2}, {1089, 20655}, {3454, 4016}, {21695, 52576}

X(68439) = barycentric product X(i)*X(j) for these {i,j}: {3454, 56810}, {4016, 42714}, {5224, 20654}
X(68439) = barycentric quotient X(i)/X(j) for these {i,j}: {3454, 56047}, {20654, 43531}, {56926, 3453}, {65803, 1919}

X(68440) = X(2)X(1138)∩X(6)X(13)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 13*a^4*b^2*c^2 - 5*a^2*b^4*c^2 - 4*b^6*c^2 + 6*a^4*c^4 - 5*a^2*b^2*c^4 + 6*b^4*c^4 - 4*a^2*c^6 - 4*b^2*c^6 + c^8) : :

X(68440) lies on the curve Q166 and these lines: {2, 1138}, {3, 14583}, {5, 1117}, {6, 13}, {30, 47053}, {94, 54827}, {376, 52056}, {378, 18384}, {476, 549}, {547, 34209}, {3090, 53137}, {3545, 59428}, {3830, 51254}, {5055, 39170}, {5071, 51835}, {5649, 54554}, {14787, 58725}, {15475, 34291}, {15694, 66125}, {18316, 60191}, {20126, 41512}, {37347, 43090}

X(68440) = barycentric product X(94)*X(37496)
X(68440) = barycentric quotient X(37496)/X(323)
X(68440) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1138, 47055}, {2, 14254, 14993}, {3, 14583, 51345}, {5, 3471, 3470}, {13, 14, 56404}, {14356, 56400, 265}

X(68441) = X(4)X(6)∩X(30)X(39020)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 2*b^2*c^2 - c^4)*(2*a^8 - a^6*b^2 - 5*a^4*b^4 + 5*a^2*b^6 - b^8 - a^6*c^2 + 10*a^4*b^2*c^2 - 5*a^2*b^4*c^2 - 4*b^6*c^2 - 5*a^4*c^4 - 5*a^2*b^2*c^4 + 10*b^4*c^4 + 5*a^2*c^6 - 4*b^2*c^6 - c^8) : :

X(68441) lies on the curve Q087 and these lines: {4, 6}, {30, 39020}, {133, 23976}, {1559, 57219}, {6529, 15311}, {6587, 42658}, {7737, 20313}, {8779, 13613}, {23590, 51358}

X(68441) = X(19614)-isoconjugate of X(54975)
X(68441) = X(4)-Dao conjugate of X(54975)
X(68441) = crossdifference of every pair of points on line {520, 1073}
X(68441) = barycentric product X(14249)*X(34147)
X(68441) = barycentric quotient X(i)/X(j) for these {i,j}: {1249, 54975}, {34147, 15394}

X(68442) = X(2)X(3)∩X(156)X(146111)

Barycentrics    2*a^16 - 8*a^14*b^2 + 11*a^12*b^4 - 5*a^10*b^6 - 2*a^6*b^10 + 3*a^4*b^12 - a^2*b^14 - 8*a^14*c^2 + 18*a^12*b^2*c^2 - 15*a^10*b^4*c^2 + 11*a^8*b^6*c^2 - 4*a^6*b^8*c^2 - 6*a^4*b^10*c^2 + 3*a^2*b^12*c^2 + b^14*c^2 + 11*a^12*c^4 - 15*a^10*b^2*c^4 - 8*a^8*b^4*c^4 + 6*a^6*b^6*c^4 + 15*a^4*b^8*c^4 - 3*a^2*b^10*c^4 - 6*b^12*c^4 - 5*a^10*c^6 + 11*a^8*b^2*c^6 + 6*a^6*b^4*c^6 - 24*a^4*b^6*c^6 + a^2*b^8*c^6 + 15*b^10*c^6 - 4*a^6*b^2*c^8 + 15*a^4*b^4*c^8 + a^2*b^6*c^8 - 20*b^8*c^8 - 2*a^6*c^10 - 6*a^4*b^2*c^10 - 3*a^2*b^4*c^10 + 15*b^6*c^10 + 3*a^4*c^12 + 3*a^2*b^2*c^12 - 6*b^4*c^12 - a^2*c^14 + b^2*c^14 : :

X(68442) lies on the Hung circle and these lines: {2, 3}, {156, 14611}, {389, 47327}, {476, 52525}, {523, 1614}, {3258, 64063}, {5944, 16168}, {9820, 67611}, {11657, 26879}, {13367, 62501}, {13491, 46632}, {14934, 32171}, {15111, 26882}, {15112, 26881}, {18914, 47146}, {21659, 34150}, {46045, 61749}

X(68442) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 36178, 36161}, {14894, 47351, 3518}

X(68443) = X(1)X(3)∩X(758)X(10693)

Barycentrics    a*(b + c)*(a^4 - b^4 + a^2*b*c - a*b^2*c - a*b*c^2 + 2*b^2*c^2 - c^4)*(a^4 - 2*a^2*b^2 + b^4 + a^2*b*c - a*b^2*c - 2*b^3*c - 2*a^2*c^2 - a*b*c^2 + 2*b^2*c^2 - 2*b*c^3 + c^4) : :

X(68443) lies on the cubic K1238 and these lines: {1, 3}, {758, 10693}, {1325, 12826}, {2778, 36001}, {5080, 58076}, {32782, 41742}

X(68443) = reflection of X(10693) in X(30447)
X(68443) = barycentric product X(37798)*X(44782)
X(68443) = barycentric quotient X(44782)/X(52500)

X(68444) = X(230)X(231)∩X(3564)X(52144)

Barycentrics    (a^2 - b^2 - c^2)*(2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*(a^10 - 2*a^8*b^2 + 3*a^6*b^4 - 5*a^4*b^6 + 4*a^2*b^8 - b^10 - 2*a^8*c^2 - a^6*b^2*c^2 + 4*a^4*b^4*c^2 - 7*a^2*b^6*c^2 + 2*b^8*c^2 + 3*a^6*c^4 + 4*a^4*b^2*c^4 + 6*a^2*b^4*c^4 - b^6*c^4 - 5*a^4*c^6 - 7*a^2*b^2*c^6 - b^4*c^6 + 4*a^2*c^8 + 2*b^2*c^8 - c^10) : :

X(68444) lies on the cubic K593 and these lines: {230, 231}, {3564, 52144}, {32762, 35296}, {43754, 56006}, {44174, 51458}, {51613, 61506}

X(68445) = X(6)X(61745)∩X(69)X(148)

Barycentrics    a^2*(2*a^4*b^4 - 3*a^2*b^6 - 2*a^4*b^2*c^2 + a^2*b^4*c^2 + 3*b^6*c^2 + 2*a^4*c^4 + a^2*b^2*c^4 - 4*b^4*c^4 - 3*a^2*c^6 + 3*b^2*c^6) : :

X(68445) lies on the cubic K895 and these lines: {6, 61745}, {69, 148}, {187, 237}, {3124, 6787}, {7934, 8288}, {36214, 67007}, {36821, 42007}

X(68446) = X(76)X(868)∩X(94)X(1916)

Barycentrics    (b^2*c^2*(-(a^8*b^2) + a^6*b^4 - 2*a^4*b^6 + 2*a^2*b^8 - a^8*c^2 + 4*a^6*b^2*c^2 - a^4*b^4*c^2 - 2*a^2*b^6*c^2 - b^8*c^2 + a^6*c^4 - a^4*b^2*c^4 + 2*a^2*b^4*c^4 + b^6*c^4 - 2*a^4*c^6 - 2*a^2*b^2*c^6 + b^4*c^6 + 2*a^2*c^8 - b^2*c^8)) : :

X(68446) lies on the cubic K217 and these lines: {76, 868}, {94, 1916}, {98, 9150}, {325, 523}, {2970, 44132}, {3124, 3580}, {3448, 5207}, {3978, 7809}, {11059, 57607}, {31644, 53474}, {41724, 46303}

X(68447) = X(4)X(15003)∩X(325)X(523)

Barycentrics    b^2*c^2*(-4*a^8 + a^6*b^2 - 6*a^4*b^4 + 7*a^2*b^6 + 2*b^8 + a^6*c^2 + 14*a^4*b^2*c^2 - 7*a^2*b^4*c^2 - 8*b^6*c^2 - 6*a^4*c^4 - 7*a^2*b^2*c^4 + 12*b^4*c^4 + 7*a^2*c^6 - 8*b^2*c^6 + 2*c^8) : :

X(68447) lies on the cubic K923 and these lines: {4, 15003}, {325, 523}, {476, 14634}, {7464, 38580}, {9140, 53474}, {14458, 34289}, {23097, 50434}, {36789, 46818}

X(68448) = X(1)X(7)∩X(5)X(52372)

Barycentrics    a*(a^6 + a^5*b - a^4*b^2 - 2*a^3*b^3 - a^2*b^4 + a*b^5 + b^6 + a^5*c + 3*a^4*b*c + 2*a^3*b^2*c - 2*a^2*b^3*c - 3*a*b^4*c - b^5*c - a^4*c^2 + 2*a^3*b*c^2 + 4*a^2*b^2*c^2 + 2*a*b^3*c^2 - b^4*c^2 - 2*a^3*c^3 - 2*a^2*b*c^3 + 2*a*b^2*c^3 + 2*b^3*c^3 - a^2*c^4 - 3*a*b*c^4 - b^2*c^4 + a*c^5 - b*c^5 + c^6) : :

X(68448) lies on the cubic K364 and these lines: {1, 7}, {5, 52372}, {54, 3336}, {68, 17857}, {155, 1406}, {1079, 60786}, {1464, 37733}, {7100, 11246}

X(68448) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1044, 50527}, {1042, 4351, 1}, {3336, 56844, 3468}

X(68449) = X(1)X(2)∩X(5)X(7073)

Barycentrics    a*(a^6 + a^5*b - a^4*b^2 - 2*a^3*b^3 - a^2*b^4 + a*b^5 + b^6 + a^5*c - a^4*b*c - 2*a^3*b^2*c + 2*a^2*b^3*c + a*b^4*c - b^5*c - a^4*c^2 - 2*a^3*b*c^2 - 2*a*b^3*c^2 - b^4*c^2 - 2*a^3*c^3 + 2*a^2*b*c^3 - 2*a*b^2*c^3 + 2*b^3*c^3 - a^2*c^4 + a*b*c^4 - b^2*c^4 + a*c^5 - b*c^5 + c^6) : :

X(68449) lies on the cubic K502 and these lines: {1, 2}, {5, 7073}, {54, 3336}, {1718, 63982}, {1725, 32911}, {3670, 56535}, {4846, 50527}, {4880, 54301}, {5396, 33857}, {6149, 17596}, {16475, 17699}, {24443, 63339}, {67076, 67080}

X(68449) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 43, 56812}, {10056, 17017, 1}

X(68450) = X(1)X(2)∩X(6)X(17934)

Barycentrics    a^5*b - a^4*b^2 - a^3*b^3 + a^5*c - a^4*b*c - a^3*b^2*c + a^2*b^3*c - a^4*c^2 - a^3*b*c^2 + 4*a^2*b^2*c^2 + a*b^3*c^2 - b^4*c^2 - a^3*c^3 + a^2*b*c^3 + a*b^2*c^3 - b^3*c^3 - b^2*c^4 : :

X(68450) lies on the cubic K502 and these lines: {1, 2}, {6, 17934}, {58, 27926}, {99, 9509}, {334, 25685}, {3570, 5277}, {3730, 53581}, {4600, 5108}, {20888, 27912}, {35127, 35148}

X(68451) = X(1)X(2)∩X(121)X(36912)

Barycentrics    (2*a - b - c)*(a^2 - a*b + 7*b^2 - a*c - 13*b*c + 7*c^2) : :
X(68451) = 5 X[2] - 3 X[1644], X[2] - 3 X[1647], 7 X[2] - 3 X[17780], 5 X[2] + 3 X[20042], 19 X[2] - 3 X[20058], 4 X[2] - 3 X[62630], X[1644] - 5 X[1647], 9 X[1644] - 5 X[8028], 7 X[1644] - 5 X[17780], 19 X[1644] - 5 X[20058], 4 X[1644] - 5 X[62630], 9 X[1647] - X[8028], 7 X[1647] - X[17780], 5 X[1647] + X[20042], 19 X[1647] - X[20058], 4 X[1647] - X[62630], 7 X[8028] - 9 X[17780], 5 X[8028] + 9 X[20042], 19 X[8028] - 9 X[20058], 4 X[8028] - 9 X[62630], 3 X[14028] - 2 X[51103], 5 X[17780] + 7 X[20042], 19 X[17780] - 7 X[20058], 4 X[17780] - 7 X[62630], 19 X[20042] + 5 X[20058], 4 X[20042] + 5 X[62630], 4 X[20058] - 19 X[62630], 3 X[53620] - X[62666], 2 X[903] + X[62441], X[6546] - 3 X[34764], 4 X[33709] - X[36945]

X(68451) lies on the cubic K700 and these lines: {1, 2}, {121, 36912}, {514, 46050}, {900, 21204}, {903, 62441}, {6546, 34764}, {10196, 33920}, {24858, 36936}, {33709, 36945}

X(68451) = midpoint of X(1644) and X(20042)
X(68451) = reflection of X(62633) in X(3828)
X(68451) = complement of X(8028)
X(68451) = complement of the isogonal conjugate of X(59150)
X(68451) = X(i)-complementary conjugate of X(j) for these (i,j): {679, 121}, {2226, 16594}, {39414, 513}, {59150, 10}
X(68451) = crosspoint of X(903) and X(62413)
X(68451) = crosssum of X(902) and X(21781)

X(68452) = X(1)X(2)∩X(46)X(56881)

Barycentrics    a^7 - a^6*b - 2*a^5*b^2 + 2*a^4*b^3 + a^3*b^4 - a^2*b^5 - a^6*c + a^5*b*c + 4*a^4*b^2*c - 5*a^3*b^3*c - 2*a^2*b^4*c + 4*a*b^5*c - b^6*c - 2*a^5*c^2 + 4*a^4*b*c^2 - 4*a^3*b^2*c^2 + 5*a^2*b^3*c^2 - b^5*c^2 + 2*a^4*c^3 - 5*a^3*b*c^3 + 5*a^2*b^2*c^3 - 8*a*b^3*c^3 + 2*b^4*c^3 + a^3*c^4 - 2*a^2*b*c^4 + 2*b^3*c^4 - a^2*c^5 + 4*a*b*c^5 - b^2*c^5 - b*c^6 : :

X(68452) lies on the cubic K817 and these lines: {1, 2}, {46, 56881}, {104, 67723}, {355, 31841}, {1158, 3667}, {2222, 2757}, {4939, 25438}, {10260, 17100}, {14266, 17857}

X(68452) = midpoint of X(67343) and X(67346)
X(68452) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 68245, 67343}, {25440, 51975, 67725}

X(68453) = X(6)X(13)∩X(110)X(186)

Barycentrics    a^2*(a^14 - 5*a^12*b^2 + 9*a^10*b^4 - 5*a^8*b^6 - 5*a^6*b^8 + 9*a^4*b^10 - 5*a^2*b^12 + b^14 - 5*a^12*c^2 + 11*a^10*b^2*c^2 - 9*a^8*b^4*c^2 + 11*a^6*b^6*c^2 - 14*a^4*b^8*c^2 + 6*a^2*b^10*c^2 + 9*a^10*c^4 - 9*a^8*b^2*c^4 - 8*a^6*b^4*c^4 + 7*a^4*b^6*c^4 + 7*a^2*b^8*c^4 - 6*b^10*c^4 - 5*a^8*c^6 + 11*a^6*b^2*c^6 + 7*a^4*b^4*c^6 - 16*a^2*b^6*c^6 + 5*b^8*c^6 - 5*a^6*c^8 - 14*a^4*b^2*c^8 + 7*a^2*b^4*c^8 + 5*b^6*c^8 + 9*a^4*c^10 + 6*a^2*b^2*c^10 - 6*b^4*c^10 - 5*a^2*c^12 + c^14) : :
X(68453) = 3 X[14643] - 2 X[21243], 3 X[15061] - 4 X[58447]

X(68453) lies on the cubic K882 and these lines: {3, 16219}, {6, 13}, {26, 14448}, {54, 7527}, {74, 18475}, {110, 186}, {155, 30714}, {184, 5663}, {185, 15132}, {541, 11456}, {569, 15738}, {1181, 16003}, {1511, 63425}, {1986, 64095}, {1993, 17702}, {2393, 9970}, {2781, 35707}, {2914, 10733}, {2931, 12165}, {3043, 12270}, {3047, 15102}, {5449, 11441}, {5609, 6102}, {5642, 15068}, {5891, 15462}, {7592, 36253}, {7728, 18400}, {9140, 15032}, {10264, 61619}, {10272, 34330}, {10601, 23515}, {10706, 51882}, {11597, 22584}, {11799, 41731}, {12228, 21650}, {12295, 19504}, {12383, 63649}, {12584, 63720}, {12824, 46261}, {14643, 21243}, {14644, 34545}, {15027, 43845}, {15054, 51033}, {15061, 58447}, {15063, 21659}, {15066, 68316}, {15141, 54183}, {16111, 17847}, {16165, 37478}, {17811, 38793}, {17814, 38795}, {18323, 23043}, {20397, 66609}, {32136, 38632}, {32234, 37784}, {32365, 64036}, {37489, 45082}, {38790, 64714}, {40640, 64101}, {43273, 58770}, {44282, 44569}, {50461, 64182}, {58881, 66734}, {63700, 64883}

X(68453) = midpoint of X(399) and X(18445)
X(68453) = reflection of X(i) in X(j) for these {i,j}: {74, 18475}, {265, 18388}, {10264, 61619}, {12827, 16534}, {18474, 113}, {37478, 16165}, {63425, 1511}, {67926, 10272}
X(68453) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 7722, 12893}, {399, 45016, 18451}, {399, 56568, 56567}, {3043, 12270, 12901}, {5609, 25711, 10539}, {18451, 45016, 113}, {34319, 56568, 19140}

X(68454) = X(6)X(13)∩X(57)X(47054)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 + a^4*c + 3*a^3*b*c - 2*a^2*b^2*c - 3*a*b^3*c + b^4*c - 2*a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 - 2*a^2*c^3 - 3*a*b*c^3 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5) : :

X(68454) lies on the cubic K1243 and these lines: {2, 60021}, {6, 13}, {57, 47054}, {94, 54727}, {759, 51345}, {2166, 3582}, {4185, 18384}, {14254, 66922}, {18316, 55027}, {24883, 56845}, {50757, 68250}, {52056, 61479}, {58733, 66102}

X(68454) = barycentric product X(94)*X(51340)
X(68454) = barycentric quotient X(51340)/X(323)

X(68455) = X(2)X(6)∩X(99)X(1296)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(2*a^4 - 5*a^2*b^2 - b^4 - 5*a^2*c^2 + 10*b^2*c^2 - c^4) : :

X(68455) lies on the cubic K408 and these lines: {2, 6}, {99, 1296}, {316, 57624}, {892, 2396}, {4226, 68152}, {4590, 34245}, {9146, 9191}, {14929, 57610}, {45143, 52229}, {57614, 67536}

X(68455) = isotomic conjugate of X(43674)
X(68455) = X(9124)-anticomplementary conjugate of X(21221)
X(68455) = X(i)-isoconjugate of X(j) for these (i,j): {31, 43674}, {661, 52230}
X(68455) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 43674}, {36830, 52230}
X(68455) = trilinear pole of line {12036, 37745}
X(68455) = barycentric product X(i)*X(j) for these {i,j}: {99, 52229}, {670, 67553}, {892, 12036}, {35179, 37745}
X(68455) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 43674}, {110, 52230}, {5468, 63854}, {12036, 690}, {37745, 1499}, {45143, 9178}, {52229, 523}, {56429, 52235}, {67553, 512}
X(68455) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2418, 57599, 99}, {5468, 9182, 2407}, {6189, 6190, 56429}, {9182, 23342, 5468}

X(68456) = X(2)X(6)∩X(523)X(46049)

Barycentrics    (2*a^2 - b^2 - c^2)*(a^4 - a^2*b^2 + 7*b^4 - a^2*c^2 - 13*b^2*c^2 + 7*c^4) : :
X(68456) = 5 X[2] - 3 X[1641], X[2] - 3 X[1648], 7 X[2] - 3 X[5468], 4 X[2] - 3 X[11053], 5 X[2] + 3 X[45291], 13 X[2] - 3 X[62658], X[1641] - 5 X[1648], 7 X[1641] - 5 X[5468], 9 X[1641] - 5 X[8030], 4 X[1641] - 5 X[11053], 6 X[1641] - 5 X[38239], 13 X[1641] - 5 X[62658], 7 X[1648] - X[5468], 9 X[1648] - X[8030], and many others

X(684) lies on the cubic K700 and these lines: {2, 6}, {523, 46049}, {671, 14444}, {690, 10278}, {7665, 10488}, {10190, 33921}, {11123, 34763}, {33915, 62662}

X(68456) = midpoint of X(i) and X(j) for these {i,j}: {671, 14444}, {1641, 45291}, {1992, 44915}
X(68456) = reflection of X(i) in X(j) for these {i,j}: {38239, 2}, {62440, 14444}, {62664, 20582}
X(68456) = complement of X(8030)
X(68456) = X(10630)-complementary conjugate of X(16597)
X(68456) = X(14444)-Dao conjugate of X(33915)
X(68456) = crosspoint of X(671) and X(46275)
X(68456) = crosssum of X(187) and X(46276)
X(68456) = crossdifference of every pair of points on line {512, 41449}
X(68456) = {X(2),X(38239)}-harmonic conjugate of X(11053)

X(68457) = X(2)X(6)∩X(3)X(843)

Barycentrics    a^2*(a^8 - 8*a^6*b^2 + 18*a^4*b^4 - 8*a^2*b^6 + b^8 - 8*a^6*c^2 + 15*a^4*b^2*c^2 - 21*a^2*b^4*c^2 + 10*b^6*c^2 + 18*a^4*c^4 - 21*a^2*b^2*c^4 - 8*a^2*c^6 + 10*b^2*c^6 + c^8) : :

X(68457) lies on the cubic K795 and these lines: {2, 6}, {3, 843}, {110, 59795}, {1499, 63424}, {5968, 52198}, {7841, 44956}, {15596, 38650}, {26714, 39446}

X(68457) = midpoint of X(7774) and X(38940)
X(68457) = reflection of X(i) in X(j) for these {i,j}: {183, 5108}, {6792, 3815}
X(68457) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 46949, 14898}

X(68458) = X(1)X(5)∩X(900)X(2245)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^6*b - a^5*b^2 - a^4*b^3 + a^3*b^4 + a^6*c - 2*a^5*b*c + a^4*b^2*c + a^3*b^3*c - a^2*b^4*c - a^5*c^2 + a^4*b*c^2 - 2*a*b^4*c^2 + b^5*c^2 - a^4*c^3 + a^3*b*c^3 + 4*a*b^3*c^3 - b^4*c^3 + a^3*c^4 - a^2*b*c^4 - 2*a*b^2*c^4 - b^3*c^4 + b^2*c^5) : :

X(68458) lies on the cubic K359 and these lines: {1, 5}, {900, 2245}, {2787, 40109}, {3110, 24624}, {14665, 65639}, {14839, 51562}, {18341, 60112}, {25436, 60089}, {46649, 57568}

X(68459) = X(1)X(5)∩X(2)X(39270)

Barycentrics    (a + b - c)*(a - b + c)*(b + c)*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^3 - 2*a^2*b - a*b^2 + 2*b^3 - 2*a^2*c + 3*a*b*c - b^2*c - a*c^2 - b*c^2 + 2*c^3) : :

X(68459) lies on the cubic K974 and these lines: {1, 5}, {2, 39270}, {21, 2222}, {274, 35174}, {2475, 53742}, {3754, 52391}, {20566, 57905}, {52356, 56321}, {56950, 63750}

X(68459) = X(57788)-Ceva conjugate of X(60091)
X(68459) = X(40663)-Dao conjugate of X(214)
X(68459) = barycentric product X(60091)*X(62826)
X(68459) = {X(5),X(56416)}-harmonic conjugate of X(80)

X(68460) = X(1)X(6)∩X(5)X(7110)

Barycentrics    a*(a - b - c)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*b*c - a^3*b^2*c + a^2*b^3*c + a*b^4*c - a^4*c^2 - a^3*b*c^2 - 2*a^2*b^2*c^2 - a*b^3*c^2 - b^4*c^2 + a^2*b*c^3 - a*b^2*c^3 - a^2*c^4 + a*b*c^4 - b^2*c^4 + c^6) : :

X(68460) lies on the cubic K502 and these lines: {1, 6}, {5, 7110}, {378, 50528}, {2182, 31937}, {3058, 36910}, {3434, 54283}, {9672, 15817}, {12609, 52252}, {12699, 16548}, {18590, 56462}, {56418, 67085}, {63171, 67081}, {67077, 68449}

X(68460) = X(14389)-Ceva conjugate of X(68449)
X(68460) = {X(1),X(9)}-harmonic conjugate of X(56540)

X(68461) = X(1)X(6)∩X(65)X(3021)

Barycentrics    a*(a^2 - 2*a*b + b^2 - 2*a*c + c^2)*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4 + a^3*c + 2*a^2*b*c - 3*a*b^2*c + 4*b^3*c - 3*a^2*c^2 - 3*a*b*c^2 - 6*b^2*c^2 + 3*a*c^3 + 4*b*c^3 - c^4) : :

X(68461) lies on the cubic K666 and these lines: {1, 6}, {65, 3021}, {277, 15733}, {946, 38386}, {17668, 68234}, {24181, 24389}, {40460, 67937}, {63999, 64306}

X(68461) = X(24181)-Dao conjugate of X(8)
X(68461) = crosspoint of X(7) and X(3870)
X(68461) = crosssum of X(55) and X(2191)
X(68461) = barycentric product X(i)*X(j) for these {i,j}: {1445, 24389}, {3870, 24181}

X(68462) = X(1)X(6)∩X(515)X(38386)

Barycentrics    a*(a^2 - 2*a*b + b^2 - 2*a*c + c^2)*(2*a^4 - 3*a^3*b + a^2*b^2 - a*b^3 + b^4 - 3*a^3*c + 2*a^2*b*c + a*b^2*c - 4*b^3*c + a^2*c^2 + a*b*c^2 + 6*b^2*c^2 - a*c^3 - 4*b*c^3 + c^4) : :

X(68462) lies on the cubic K1160 and these lines: {1, 6}, {515, 38386}, {1317, 16184}, {1319, 59807}, {2348, 59814}, {2742, 3660}, {3309, 43049}, {11700, 39754}, {15524, 39759}, {21620, 66651}, {33902, 39753}, {39752, 63772}, {39757, 63770}, {41339, 66505}

X(68462) = incircle-inverse of X(15185)
X(68462) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5526, 60375}, {60373, 67386, 41391}

X(68463) = X(1)X(6)∩X(105)X(52018)

Barycentrics    a*(a^4 - 4*a^3*b + 2*a^2*b^2 + b^4 - 4*a^3*c - 6*a^2*b*c + 2*a*b^2*c + 2*a^2*c^2 + 2*a*b*c^2 + 6*b^2*c^2 + c^4) : :

X(68463) lies on the cubic K1292 and these lines: {1, 6}, {105, 52018}, {239, 20075}, {997, 67331}, {3966, 37326}, {4228, 18206}, {4384, 63134}, {8301, 58887}, {16823, 54429}, {16825, 66671}, {16833, 34612}, {24796, 52783}, {33070, 37111}, {48883, 64679}, {52015, 62874}

X(68464) = X(6)X(17)∩X(39)X(252)

Barycentrics    (a^4 - a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*(a^8 - 2*a^6*b^2 + a^4*b^4 - 2*a^6*c^2 - a^4*b^2*c^2 + a^2*b^4*c^2 - b^6*c^2 + a^4*c^4 + a^2*b^2*c^4 + 2*b^4*c^4 - b^2*c^6) : :

X(68464) lies on the cubic K222 and these lines: {6, 17}, {32, 25043}, {39, 252}, {93, 10312}, {115, 19552}, {187, 930}, {230, 66883}, {385, 46139}, {5254, 35888}, {7747, 31392}, {7749, 21975}, {19553, 39018}

X(68464) = crossdifference of every pair of points on line {1510, 62589}
X(68464) = barycentric product X(2963)*X(44376)
X(68464) = barycentric quotient X(44376)/X(7769)

X(68465) = X(2)X(288)∩X(6)X(17)

Barycentrics    (a^2 - b^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^4 - a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4) : :

X(68465) lies on the cubic K260 and these lines: {2, 288}, {3, 34433}, {6, 17}, {51, 3078}, {53, 25043}, {206, 32737}, {216, 60824}, {252, 18912}, {577, 11077}, {930, 51222}, {2052, 11140}, {2351, 51477}, {3481, 8174}, {10979, 39171}, {14582, 17434}, {18436, 50667}, {19778, 39407}, {19779, 39406}, {36412, 59142}, {38342, 57765}

X(68465) = isogonal conjugate of X(57489)
X(68465) = isogonal conjugate of the polar conjugate of X(25043)
X(68465) = polar conjugate of the isotomic conjugate of X(60824)
X(68465) = X(31)-complementary conjugate of X(39171)
X(68465) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 39171}, {11140, 25043}
X(68465) = X(i)-cross conjugate of X(j) for these (i,j): {24862, 6368}, {32078, 5}, {42445, 216}
X(68465) = X(i)-isoconjugate of X(j) for these (i,j): {1, 57489}, {19, 63172}, {92, 25044}, {275, 2964}, {1510, 65221}, {1994, 2190}, {2148, 32002}, {2167, 3518}, {2965, 40440}, {7769, 62268}, {8884, 63760}, {36134, 67102}
X(68465) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 57489}, {5, 1994}, {6, 63172}, {137, 67102}, {216, 32002}, {2972, 63830}, {6663, 14129}, {15450, 1510}, {21975, 275}, {22391, 25044}, {39019, 41298}, {39171, 2}, {40588, 3518}, {46604, 8882}, {52032, 7769}, {61504, 14918}
X(68465) = cevapoint of X(17434) and X(39019)
X(68465) = crosspoint of X(3519) and X(11140)
X(68465) = crosssum of X(i) and X(j) for these (i,j): {1510, 47424}, {2965, 3518}
X(68465) = trilinear pole of line {15451, 65785}
X(68465) = crossdifference of every pair of points on line {1510, 52417}
X(68465) = barycentric product X(i)*X(j) for these {i,j}: {3, 25043}, {4, 60824}, {5, 3519}, {93, 5562}, {216, 11140}, {265, 66883}, {311, 51477}, {343, 2963}, {418, 20572}, {930, 6368}, {2439, 14592}, {2962, 44706}, {5449, 12044}, {15451, 46139}, {17434, 38342}, {19552, 34900}, {24862, 57764}, {36300, 40711}, {36301, 40712}, {42293, 55217}, {57765, 61378}
X(68465) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 63172}, {5, 32002}, {6, 57489}, {51, 3518}, {93, 8795}, {184, 25044}, {216, 1994}, {217, 2965}, {343, 7769}, {418, 49}, {930, 18831}, {1487, 39286}, {2439, 14590}, {2962, 40440}, {2963, 275}, {3519, 95}, {5562, 44180}, {6368, 41298}, {11140, 276}, {12077, 67102}, {14582, 2413}, {15451, 1510}, {17434, 63830}, {20572, 57844}, {24862, 137}, {25043, 264}, {32078, 1493}, {32737, 933}, {34983, 57135}, {36148, 65221}, {36300, 472}, {36301, 473}, {36412, 14129}, {38342, 42405}, {42445, 62589}, {44716, 51440}, {51477, 54}, {57195, 20577}, {58305, 37084}, {60824, 69}, {61378, 143}, {62260, 14577}, {62266, 2964}, {66883, 340}

X(68466) = X(6)X(17)∩X(115)X(33333)

Barycentrics    (a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(2*a^10 - 7*a^8*b^2 + 10*a^6*b^4 - 8*a^4*b^6 + 4*a^2*b^8 - b^10 - 7*a^8*c^2 + 10*a^6*b^2*c^2 - a^4*b^4*c^2 - 5*a^2*b^6*c^2 + 3*b^8*c^2 + 10*a^6*c^4 - a^4*b^2*c^4 + 2*a^2*b^4*c^4 - 2*b^6*c^4 - 8*a^4*c^6 - 5*a^2*b^2*c^6 - 2*b^4*c^6 + 4*a^2*c^8 + 3*b^2*c^8 - c^10) : :

X(68466) lies on the cubic K277 and these lines: {6, 17}, {115, 33333}, {143, 36412}, {1879, 50667}, {2081, 2600}, {3003, 65906}, {3284, 39019}, {14918, 66914}

X(68466) = complement of the isotomic conjugate of X(1263)
X(68466) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 6592}, {1263, 2887}, {2179, 40604}, {14579, 21231}, {51804, 3819}
X(68466) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 6592}, {37779, 1154}
X(68466) = X(2167)-isoconjugate of X(33643)
X(68466) = X(i)-Dao conjugate of X(j) for these (i,j): {6592, 2}, {40588, 33643}
X(68466) = crosspoint of X(2) and X(1263)
X(68466) = crosssum of X(6) and X(1157)
X(68466) = crossdifference of every pair of points on line {54, 1510}
X(68466) = barycentric product X(i)*X(j) for these {i,j}: {5, 50708}, {1154, 58926}, {1263, 6592}, {6150, 25043}, {14071, 38899}
X(68466) = barycentric quotient X(i)/X(j) for these {i,j}: {51, 33643}, {6150, 63172}, {50708, 95}, {58926, 46138}
X(68466) = {X(6),X(34520)}-harmonic conjugate of X(233)

X(68467) = X(3)X(252)∩X(6)X(17)

Barycentrics    (a^4 - a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 - 3*a^6*c^2 + 3*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - 2*b^6*c^2 + 3*a^4*c^4 + 2*a^2*b^2*c^4 + 4*b^4*c^4 - a^2*c^6 - 2*b^2*c^6) : :

X(68467) lies on the cubic K297 and these lines: {2, 19553}, {3, 252}, {4, 35888}, {6, 17}, {24, 93}, {183, 46139}, {381, 16337}, {382, 31392}, {562, 7577}, {1487, 3851}, {1657, 18370}, {3526, 21975}, {7507, 14111}, {7514, 11140}, {14940, 35311}, {15720, 39171}, {19268, 67091}, {25044, 32551}

X(68467) = X(2964)-isoconjugate of X(9221)
X(68467) = X(21975)-Dao conjugate of X(9221)
X(68467) = barycentric product X(567)*X(11140)
X(68467) = barycentric quotient X(i)/X(j) for these {i,j}: {567, 1994}, {2963, 9221}, {11140, 57900}, {32737, 59003}, {56407, 30529}
X(68467) = {X(252),X(25043)}-harmonic conjugate of X(3)

X(68468) = X(4)X(50667)∩X(6)X(17)

Barycentrics    (a^4 - a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6) : :

X(68468) lies on the cubic K381 and these lines: {4, 50667}, {6, 17}, {340, 38342}, {930, 40667}, {8172, 8173}, {8553, 39171}, {11063, 15770}, {11071, 11600}, {11082, 36304}, {11087, 36305}, {11140, 60191}, {12077, 20184}, {13483, 19778}, {13484, 19779}, {18374, 32737}

X(68468) = X(i)-Ceva conjugate of X(j) for these (i,j): {13582, 18370}, {38342, 8562}
X(68468) = X(i)-isoconjugate of X(j) for these (i,j): {1994, 51804}, {2964, 13582}
X(68468) = X(i)-Dao conjugate of X(j) for these (i,j): {21975, 13582}, {46604, 14579}, {53989, 67102}
X(68468) = crosspoint of X(i) and X(j) for these (i,j): {11584, 37779}, {13582, 18368}
X(68468) = crosssum of X(i) and X(j) for these (i,j): {11063, 18369}, {14367, 14579}
X(68468) = crossdifference of every pair of points on line {1493, 1510}
X(68468) = barycentric product X(i)*X(j) for these {i,j}: {93, 50461}, {930, 45147}, {1157, 25043}, {1263, 32637}, {1749, 2962}, {2963, 37779}, {3519, 37943}, {6140, 46139}, {6592, 25148}, {11063, 11140}, {11584, 21975}, {19552, 38542}
X(68468) = barycentric quotient X(i)/X(j) for these {i,j}: {930, 65279}, {1157, 63172}, {2963, 13582}, {6140, 1510}, {11063, 1994}, {32737, 1291}, {37779, 7769}, {37943, 32002}, {45147, 41298}, {50461, 44180}, {51477, 43704}, {56404, 30529}
X(68468) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11600, 11601, 19552}, {40667, 40668, 930}

X(68469) = X(6)X(17)∩X(252)X(43842)

Barycentrics    (a^4 - a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*(2*a^6 - 5*a^4*b^2 + 4*a^2*b^4 - b^6 - 5*a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + 4*a^2*c^4 + b^2*c^4 - c^6) : :

X(68469) lies on the cubic K429 and these lines: {6, 17}, {252, 43842}, {1974, 32737}, {11140, 54907}, {11245, 35885}, {13345, 60824}, {15444, 15445}, {36300, 36305}, {36301, 36304}, {40167, 40168}, {41598, 41627}

X(68469) = polar conjugate of the isotomic conjugate of X(39171)
X(68469) = X(17713)-Dao conjugate of X(1994)
X(68469) = barycentric product X(i)*X(j) for these {i,j}: {4, 39171}, {93, 41597}, {930, 20184}, {1487, 13431}, {2963, 41628}
X(68469) = barycentric quotient X(i)/X(j) for these {i,j}: {20184, 41298}, {32737, 20185}, {39171, 69}, {41597, 44180}, {41628, 7769}

X(68470) = X(5)X(523)∩X(113)X(526)

Barycentrics    (b^2 - c^2)*(-(a^10*b^2) + 4*a^8*b^4 - 6*a^6*b^6 + 4*a^4*b^8 - a^2*b^10 - a^10*c^2 + 6*a^8*b^2*c^2 - 5*a^6*b^4*c^2 - 3*a^4*b^6*c^2 + 2*a^2*b^8*c^2 + b^10*c^2 + 4*a^8*c^4 - 5*a^6*b^2*c^4 + 8*a^4*b^4*c^4 - a^2*b^6*c^4 - 4*b^8*c^4 - 6*a^6*c^6 - 3*a^4*b^2*c^6 - a^2*b^4*c^6 + 6*b^6*c^6 + 4*a^4*c^8 + 2*a^2*b^2*c^8 - 4*b^4*c^8 - a^2*c^10 + b^2*c^10) : :
X(68470) = 3 X[65754] + X[68174]

X(68470) lies on the cubic K722 and these lines: {2, 46608}, {4, 46616}, {5, 523}, {30, 62173}, {110, 57210}, {113, 526}, {140, 14809}, {403, 62172}, {520, 5448}, {924, 61749}, {1510, 64179}, {2450, 23350}, {3134, 3258}, {3613, 32112}, {7577, 18808}, {8070, 62329}, {8675, 19130}, {9003, 39509}, {14611, 56398}, {15760, 38401}, {16536, 16537}, {23301, 23333}, {32743, 55121}, {34291, 66119}, {52743, 56403}, {65754, 68174}

X(68470) = midpoint of X(i) and X(j) for these {i,j}: {4, 46616}, {58263, 60342}
X(68470) = reflection of X(i) in X(j) for these {i,j}: {14809, 140}, {62364, 5}
X(68470) = complement of X(46608)
X(68470) = reflection of X(62364) in the Euler line

X(68471) = X(2)X(265)∩X(5)X(523)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(2*a^6*b^2 - 6*a^4*b^4 + 6*a^2*b^6 - 2*b^8 + 2*a^6*c^2 - 2*a^4*b^2*c^2 - a^2*b^4*c^2 + b^6*c^2 - 6*a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 + 6*a^2*c^6 + b^2*c^6 - 2*c^8) : :

X(68471) lies on the cubic K759 and these lines: {2, 265}, {4, 52056}, {5, 523}, {30, 66125}, {125, 41390}, {140, 51254}, {230, 56395}, {381, 476}, {547, 31945}, {1989, 43620}, {3090, 59428}, {3091, 58733}, {3545, 14993}, {3839, 51345}, {5055, 5627}, {5066, 14583}, {5071, 51835}, {5961, 18570}, {7577, 65586}, {7607, 39295}, {7699, 65317}, {10254, 34333}, {18576, 49669}, {23956, 35473}, {38609, 51349}, {41512, 61574}, {46029, 57482}, {49102, 56401}, {49671, 52153}, {51391, 62551}

X(68471) = X(6149)-isoconjugate of X(13530)
X(68471) = X(14993)-Dao conjugate of X(13530)
X(68471) = barycentric product X(328)*X(16328)
X(68471) = barycentric quotient X(i)/X(j) for these {i,j}: {1989, 13530}, {16328, 186}
X(68471) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 34209, 14356}, {5, 39170, 14254}, {265, 67304, 18316}, {14254, 39170, 53137}, {14254, 58926, 58725}, {14356, 34209, 14254}, {14356, 39170, 34209}

X(68472) = X(5)X(523)∩X(30)X(110)

Barycentrics    a^14*b^2 - 2*a^12*b^4 - 4*a^10*b^6 + 15*a^8*b^8 - 15*a^6*b^10 + 4*a^4*b^12 + 2*a^2*b^14 - b^16 + a^14*c^2 - 4*a^12*b^2*c^2 + 10*a^10*b^4*c^2 - 19*a^8*b^6*c^2 + 17*a^6*b^8*c^2 + 2*a^4*b^10*c^2 - 12*a^2*b^12*c^2 + 5*b^14*c^2 - 2*a^12*c^4 + 10*a^10*b^2*c^4 - a^6*b^6*c^4 - 21*a^4*b^8*c^4 + 24*a^2*b^10*c^4 - 10*b^12*c^4 - 4*a^10*c^6 - 19*a^8*b^2*c^6 - a^6*b^4*c^6 + 30*a^4*b^6*c^6 - 14*a^2*b^8*c^6 + 11*b^10*c^6 + 15*a^8*c^8 + 17*a^6*b^2*c^8 - 21*a^4*b^4*c^8 - 14*a^2*b^6*c^8 - 10*b^8*c^8 - 15*a^6*c^10 + 2*a^4*b^2*c^10 + 24*a^2*b^4*c^10 + 11*b^6*c^10 + 4*a^4*c^12 - 12*a^2*b^2*c^12 - 10*b^4*c^12 + 2*a^2*c^14 + 5*b^2*c^14 - c^16 : :
X(68472) = 4 X[5] - 3 X[21315], 3 X[21315] - 2 X[34209], 4 X[14934] - X[21317], 3 X[14934] - 4 X[67094], 3 X[20393] + X[61598], X[20957] + 2 X[33505], X[21317] + 4 X[46045], 3 X[21317] - 16 X[67094], 3 X[46045] + 4 X[67094], X[74] - 3 X[57306], 4 X[140] - 3 X[47852], 2 X[46632] - 3 X[47852], 3 X[381] - 2 X[21316], and many others

X(68472) lies on the cubic K799 and these lines: {4, 21269}, {5, 523}, {30, 110}, {74, 57306}, {113, 16168}, {140, 46632}, {265, 14480}, {381, 21316}, {399, 17511}, {476, 14643}, {546, 3470}, {550, 47084}, {1138, 34193}, {1511, 55308}, {1539, 64510}, {2777, 38610}, {3154, 10264}, {3258, 5663}, {3627, 52219}, {5972, 38609}, {6070, 20304}, {7471, 10272}, {7575, 16319}, {9970, 66812}, {10088, 66799}, {10091, 66798}, {10096, 14979}, {10223, 20424}, {10297, 67525}, {10620, 65086}, {11657, 44282}, {11799, 67620}, {12041, 31379}, {14508, 14851}, {14611, 32423}, {14731, 36193}, {14993, 66787}, {15061, 66801}, {15806, 36161}, {16279, 39487}, {16334, 47334}, {17702, 66795}, {18285, 30221}, {18319, 36169}, {20127, 38701}, {25641, 61574}, {32609, 66791}, {36172, 38581}, {37938, 67617}, {38700, 38794}, {38793, 68307}, {46686, 66778}, {46817, 62509}, {47146, 68319}, {47335, 67526}, {51391, 62490}, {51522, 55319}, {52056, 60605}, {52070, 58924}, {57305, 64101}, {63700, 66807}

X(68472) = midpoint of X(i) and X(j) for these {i,j}: {110, 20957}, {265, 14480}, {399, 17511}, {477, 7728}, {5655, 34312}, {9970, 66812}, {10297, 67525}, {11799, 67620}, {12121, 44967}, {14611, 36184}, {14731, 36193}, {14934, 46045}, {36172, 38581}, {63700, 66807}
X(68472) = reflection of X(i) in X(j) for these {i,j}: {110, 33505}, {550, 47084}, {1511, 55308}, {3627, 52219}, {6070, 20304}, {7471, 10272}, {7575, 16319}, {10264, 3154}, {12041, 31379}, {16340, 3258}, {18319, 36169}, {21269, 4}, {25641, 61574}, {34150, 546}, {34209, 5}, {38609, 5972}, {46632, 140}, {47146, 68319}, {51522, 55319}, {66778, 46686}
X(68472) = reflection of X(34209) in the Euler line
X(68472) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 34209, 21315}, {140, 46632, 47852}, {12041, 45694, 31379}, {14254, 59370, 5}, {38581, 38789, 36172}

X(68473) = X(5)X(523)∩X(94)X(690)

Barycentrics    b^2*(b - c)*c^2*(b + c)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(-a^2 + b^2 - a*c - c^2)*(-a^2 + b^2 + a*c - c^2)*(-a^8 + 2*a^6*b^2 - a^4*b^4 + 2*a^6*c^2 - 3*a^4*b^2*c^2 + a^2*b^4*c^2 + b^6*c^2 - a^4*c^4 + a^2*b^2*c^4 - 2*b^4*c^4 + b^2*c^6) : :

X(68473) lies on the cubic K979 and these lines: {5, 523}, {94, 690}, {879, 43707}, {2088, 14582}, {6344, 16230}, {14295, 20573}, {35139, 35316}, {53263, 56407}

X(68473) = X(6149)-isoconjugate of X(9160)
X(68473) = X(14993)-Dao conjugate of X(9160)
X(68473) = barycentric product X(i)*X(j) for these {i,j}: {94, 62489}, {338, 64221}, {850, 56396}, {10412, 40879}
X(68473) = barycentric quotient X(i)/X(j) for these {i,j}: {94, 53192}, {1989, 9160}, {32761, 52603}, {40879, 10411}, {56396, 110}, {62489, 323}, {64221, 249}
X(68473) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10412, 14254, 15475}, {10412, 14592, 23105}, {14592, 58725, 43083}

X(68474) = X(5)X(523)∩X(20)X(476)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(-a^2 + b^2 - a*c - c^2)*(-a^2 + b^2 + a*c - c^2)*(-(a^6*b^2) + 3*a^4*b^4 - 3*a^2*b^6 + b^8 - a^6*c^2 - 4*a^4*b^2*c^2 + 3*a^2*b^4*c^2 + 2*b^6*c^2 + 3*a^4*c^4 + 3*a^2*b^2*c^4 - 6*b^4*c^4 - 3*a^2*c^6 + 2*b^2*c^6 + c^8) : :
X(68474) = 2 X[5] - 3 X[18279], 4 X[5] - 3 X[59370], X[20] - 3 X[59291]

X(68474) lies on the cubic K846 and these lines: {3, 14993}, {4, 5627}, {5, 523}, {20, 476}, {24, 47204}, {64, 265}, {133, 39376}, {631, 51835}, {1093, 6344}, {1656, 68440}, {1657, 51345}, {3357, 50464}, {3526, 56400}, {3832, 52449}, {3843, 58733}, {6759, 14560}, {7796, 35139}, {8918, 8919}, {11438, 40355}, {11799, 58723}, {15063, 41512}, {15454, 22104}, {17800, 52056}, {18384, 37197}, {18400, 53319}, {23097, 67604}, {39174, 57424}, {51895, 62583}

X(68474) = reflection of X(59370) in X(18279)
X(68474) = X(5627)-Ceva conjugate of X(39376)
X(68474) = X(47433)-cross conjugate of X(51358)
X(68474) = X(i)-isoconjugate of X(j) for these (i,j): {1294, 6149}, {15404, 35201}, {52414, 59499}
X(68474) = X(i)-Dao conjugate of X(j) for these (i,j): {1990, 14920}, {14993, 1294}, {39170, 53789}, {44436, 6148}, {50937, 186}, {62583, 52437}, {65907, 323}
X(68474) = crosspoint of X(5627) and X(6344)
X(68474) = crosssum of X(1511) and X(22115)
X(68474) = barycentric product X(i)*X(j) for these {i,j}: {94, 6000}, {265, 51358}, {2404, 43083}, {5627, 62583}, {6344, 44436}, {14254, 57488}, {14592, 46587}, {39376, 46106}, {51895, 57486}, {52646, 57482}, {56403, 56577}
X(68474) = barycentric quotient X(i)/X(j) for these {i,j}: {94, 54988}, {133, 14920}, {1989, 1294}, {2442, 53176}, {5627, 66764}, {6000, 323}, {11079, 15404}, {14582, 43701}, {39376, 14919}, {43083, 2416}, {44436, 52437}, {46587, 14590}, {47433, 1511}, {51358, 340}, {51385, 14165}, {51964, 14385}, {52153, 59499}, {52646, 57487}, {55276, 62172}, {56399, 53789}, {56403, 56683}, {62583, 6148}
X(68474) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 34209, 53137}, {5, 53137, 39170}, {476, 59428, 51254}, {8918, 8919, 56399}, {14254, 34209, 39170}, {14254, 39170, 14356}, {14254, 43089, 58704}, {14254, 53137, 5}

X(68475) = X(5)X(523)∩X(131)X(45180)

Barycentrics    b^2*(b - c)*c^2*(b + c)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(-a^2 + b^2 - a*c - c^2)*(-a^2 + b^2 + a*c - c^2)*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6) : :
X(68475) = 2 X[10412] + X[51479]

X(68475) lies on the cubic K1152 and these lines: {5, 523}, {131, 45180}, {137, 2970}, {265, 924}, {476, 925}, {1141, 53930}, {1510, 10113}, {1989, 2489}, {5627, 14618}, {6344, 66299}, {6368, 46085}, {8675, 18557}, {15328, 59428}, {16003, 20184}, {31998, 35139}, {38609, 38615}, {39021, 61339}, {55121, 63735}

X(68475) = X(i)-complementary conjugate of X(j) for these (i,j): {1725, 46439}, {51804, 3134}
X(68475) = X(10412)-Ceva conjugate of X(55121)
X(68475) = X(68174)-cross conjugate of X(55121)
X(68475) = X(i)-isoconjugate of X(j) for these (i,j): {50, 65262}, {662, 52557}, {1101, 15470}, {2624, 18879}, {4575, 38936}, {6149, 10420}, {22115, 36114}, {36034, 39371}, {36053, 52603}
X(68475) = X(i)-Dao conjugate of X(j) for these (i,j): {113, 52603}, {136, 38936}, {523, 15470}, {1084, 52557}, {3258, 39371}, {14993, 10420}, {16178, 186}, {34834, 10411}, {39005, 22115}, {39021, 323}, {55121, 60342}, {56399, 4558}, {56792, 14385}, {65753, 6148}, {67191, 14590}
X(68475) = cevapoint of X(523) and X(57512)
X(68475) = crosssum of X(22115) and X(62173)
X(68475) = crossdifference of every pair of points on line {50, 52557}
X(68475) = barycentric product X(i)*X(j) for these {i,j}: {94, 55121}, {328, 47236}, {338, 41512}, {403, 14592}, {523, 57486}, {686, 18817}, {850, 56403}, {1989, 65972}, {3580, 10412}, {5627, 65757}, {6334, 6344}, {14254, 65614}, {14582, 44138}, {14618, 39170}, {20573, 21731}, {40427, 68174}, {43088, 52504}
X(68475) = barycentric quotient X(i)/X(j) for these {i,j}: {94, 18878}, {115, 15470}, {265, 43755}, {403, 14590}, {476, 18879}, {512, 52557}, {686, 22115}, {1637, 39371}, {1989, 10420}, {2166, 65262}, {2501, 38936}, {3003, 52603}, {3580, 10411}, {6334, 52437}, {6344, 687}, {10412, 2986}, {14582, 5504}, {14592, 57829}, {15475, 14910}, {18384, 32708}, {18817, 57932}, {21731, 50}, {39021, 60342}, {39170, 4558}, {41512, 249}, {43088, 52505}, {44084, 14591}, {47236, 186}, {55121, 323}, {55265, 1511}, {56403, 110}, {57486, 99}, {60498, 51478}, {65757, 6148}, {65972, 7799}, {68174, 34834}

X(68476) = X(2)X(3)∩X(6)X(46998)

Barycentrics    5*a^10 - 10*a^8*b^2 + 5*a^6*b^4 + 8*a^4*b^6 - 10*a^2*b^8 + 2*b^10 - 10*a^8*c^2 + 11*a^6*b^2*c^2 - 9*a^4*b^4*c^2 + 17*a^2*b^6*c^2 - 7*b^8*c^2 + 5*a^6*c^4 - 9*a^4*b^2*c^4 - 18*a^2*b^4*c^4 + 5*b^6*c^4 + 8*a^4*c^6 + 17*a^2*b^2*c^6 + 5*b^4*c^6 - 10*a^2*c^8 - 7*b^2*c^8 + 2*c^10 : :
X(68476) = 8 X[549] + X[16281]

X(68476) lies on the McCay circumcircle and these lines: {2, 3}, {6, 46998}, {111, 11628}, {182, 9169}, {183, 53136}, {230, 50149}, {523, 7610}, {599, 46986}, {671, 63719}, {842, 58043}, {1641, 47570}, {1648, 11179}, {2080, 51541}, {2395, 47001}, {2452, 22329}, {2453, 46992}, {2502, 19905}, {2770, 9829}, {5108, 16760}, {6036, 9172}, {6054, 45774}, {6055, 6795}, {6792, 50979}, {7606, 8705}, {7607, 15398}, {7664, 8724}, {7665, 12243}, {8585, 58831}, {8860, 16092}, {10717, 33813}, {11053, 54169}, {11168, 16320}, {11632, 66460}, {12117, 30786}, {12355, 31125}, {14916, 48876}, {15271, 47171}, {15597, 62508}, {16315, 23055}, {16316, 34229}, {20423, 41939}, {20481, 52232}, {21448, 65729}, {32525, 50983}, {34320, 38611}, {37637, 46980}, {37688, 47285}, {38940, 50978}, {43273, 65620}, {47169, 50150}, {47239, 50147}, {47240, 63107}, {50955, 65718}, {50984, 67399}

X(68476) = midpoint of X(842) and X(58043)
X(68476) = orthoptic-circle-of-the-Steiner-inellipse-inverse of X(8352)
X(68476) = orthoptic-circle-of-the-Steiner-circumellipse-inverse of X(40246)
X(68476) = psi-transform of X(8593)
X(68476) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 376, 57607}, {2, 7417, 381}, {2, 7426, 1316}, {2, 7493, 45662}, {2, 9832, 36194}, {2, 26255, 57618}, {2, 36163, 47097}, {2, 53161, 5094}, {8860, 59227, 16092}, {11168, 16320, 50146}

X(68477) = X(3)X(6)∩X(512)X(1614)

Barycentrics    a^2*(2*a^12 - 8*a^10*b^2 + 11*a^8*b^4 - 5*a^6*b^6 - a^4*b^8 + a^2*b^10 - 8*a^10*c^2 + 18*a^8*b^2*c^2 - 15*a^6*b^4*c^2 + 7*a^4*b^6*c^2 - a^2*b^8*c^2 - b^10*c^2 + 11*a^8*c^4 - 15*a^6*b^2*c^4 + 2*a^4*b^4*c^4 + 6*b^8*c^4 - 5*a^6*c^6 + 7*a^4*b^2*c^6 - 10*b^6*c^6 - a^4*c^8 - a^2*b^2*c^8 + 6*b^4*c^8 + a^2*c^10 - b^2*c^10) : :
X(68477) = X[37495] - 3 X[67359]

X(68477) lies on the Hung circle and these lines: {3, 6}, {512, 1614}, {691, 52525}, {5099, 64063}, {5944, 53793}, {26883, 61754}, {31848, 61753}, {32171, 64634}, {52471, 63663}

X(68478) = X(1)X(3)∩X(2)X(3125)

Barycentrics    a*(a*b^3 + b^4 + a*b^2*c + a*b*c^2 + a*c^3 + c^4) : :

X(68478) lies on the Kiepert circumhyperbola of the Brocard triangle and these lines: {1, 3}, {2, 3125}, {6, 16566}, {10, 37025}, {37, 18179}, {38, 4475}, {39, 1959}, {42, 760}, {63, 5291}, {76, 321}, {81, 5006}, {141, 4016}, {536, 599}, {572, 44302}, {758, 37676}, {762, 29593}, {984, 20694}, {1111, 3782}, {1352, 2783}, {1572, 5256}, {1760, 2273}, {2051, 60245}, {2896, 6542}, {3688, 17446}, {3721, 3912}, {3726, 29574}, {3727, 17023}, {3778, 17470}, {3938, 9997}, {3944, 7697}, {3959, 4384}, {4118, 21035}, {4414, 24264}, {4415, 21138}, {4443, 4516}, {4735, 35552}, {4920, 16603}, {5224, 21810}, {5336, 54404}, {9593, 51304}, {10800, 17017}, {12782, 51836}, {13881, 17308}, {16552, 49760}, {16831, 20271}, {17239, 56541}, {17389, 55164}, {17738, 24273}, {17762, 18148}, {17872, 64007}, {18167, 40773}, {18183, 64553}, {18189, 18206}, {20718, 39957}, {21951, 24603}, {23639, 26601}, {24214, 43040}, {24268, 24296}, {24271, 24291}, {24326, 39712}, {29069, 33869}, {29591, 50570}, {30547, 33867}, {37445, 60586}, {43677, 60084}

X(68478) = reflection of X(54282) in X(3666)
X(68478) = crossdifference of every pair of points on line {650, 1980}
X(68478) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {986, 7146, 980}, {13388, 13389, 5061}

X(68479) = X(1)X(7)∩X(3)X(25878)

Barycentrics    a*(a^6*b - 3*a^5*b^2 + 2*a^4*b^3 + 2*a^3*b^4 - 3*a^2*b^5 + a*b^6 + a^6*c - 3*a^4*b^2*c + 3*a^2*b^4*c - b^6*c - 3*a^5*c^2 - 3*a^4*b*c^2 + 4*a^3*b^2*c^2 - a*b^4*c^2 + 3*b^5*c^2 + 2*a^4*c^3 - 2*b^4*c^3 + 2*a^3*c^4 + 3*a^2*b*c^4 - a*b^2*c^4 - 2*b^3*c^4 - 3*a^2*c^5 + 3*b^2*c^5 + a*c^6 - b*c^6) : :
X(68479) = X[9589] - 4 X[43165], 2 X[71] - 3 X[165], 5 X[1698] - 4 X[51758], 3 X[1699] - 4 X[34830]

X(68479) lies on the Feuerbach circumhyperbola of the tangential triangle and these lines: {1, 7}, {3, 25878}, {6, 1754}, {40, 916}, {69, 58035}, {71, 165}, {195, 16117}, {219, 11495}, {376, 33810}, {411, 1780}, {674, 7289}, {1214, 5918}, {1418, 10167}, {1419, 7070}, {1474, 4219}, {1498, 37426}, {1536, 25964}, {1630, 1633}, {1698, 51758}, {1699, 34830}, {1714, 37421}, {1723, 64741}, {1818, 63413}, {1839, 41860}, {2328, 7411}, {2772, 2938}, {2915, 2929}, {3062, 24708}, {3190, 9778}, {3220, 23361}, {3682, 12512}, {5400, 37650}, {5706, 48897}, {5736, 56144}, {6985, 15805}, {7416, 8053}, {9028, 39878}, {9441, 60785}, {10431, 17194}, {12689, 57015}, {15726, 40937}, {17245, 37374}, {18698, 59620}, {20793, 63203}, {20978, 40940}, {22053, 40960}, {35986, 61220}, {37022, 50677}, {37482, 68036}, {37543, 67264}, {56714, 59595}, {56809, 59418}, {59217, 65452}

X(68479) = reflection of X(i) in X(j) for these {i,j}: {1, 63395}, {33536, 3}
X(68479) = excentral-isogonal conjugate of X(1762)
X(68479) = X(i)-Ceva conjugate of X(j) for these (i,j): {2328, 1}, {7411, 3}, {37659, 6}
X(68479) = crosspoint of X(662) and X(24011)
X(68479) = crosssum of X(i) and X(j) for these (i,j): {513, 14714}, {661, 24012}
X(68479) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {170, 1742, 2951}, {991, 3332, 1}, {4319, 4341, 1}

X(68480) = X(1)X(7)∩X(514)X(50016)

Barycentrics    a^5 + 3*a^4*b - 2*a^3*b^2 - a^2*b^3 - a*b^4 + 3*a^4*c + a^3*b*c - a^2*b^2*c - a*b^3*c - 2*b^4*c - 2*a^3*c^2 - a^2*b*c^2 + 4*a*b^2*c^2 + 2*b^3*c^2 - a^2*c^3 - a*b*c^3 + 2*b^2*c^3 - a*c^4 - 2*b*c^4 : :
X(68480) = 5 X[1698] - 4 X[5179], 7 X[3624] - 8 X[51775]

X(68480) lies on the Evans circle and these lines: {1, 7}, {514, 50016}, {927, 28842}, {1434, 66666}, {1441, 2938}, {1698, 5179}, {1768, 20367}, {2795, 3509}, {3339, 67654}, {3474, 24268}, {3624, 51775}, {5251, 24342}, {7112, 32092}, {17729, 67726}, {18206, 19642}, {40719, 64301}, {45749, 53801}, {54330, 60905}

X(68480) = reflection of X(i) in X(j) for these {i,j}: {1, 5088}, {5195, 67572}, {5527, 67574}, {67726, 17729}
X(68480) = {X(7176),X(66670)}-harmonic conjugate of X(1)

X(68481) = X(1)X(2)∩X(15)X(37146)

Barycentrics    a^4 + 4*a^3*b + 5*a^2*b^2 + 3*a*b^3 + b^4 + 4*a^3*c + 10*a^2*b*c + 9*a*b^2*c + 3*b^3*c + 5*a^2*c^2 + 9*a*b*c^2 + 4*b^2*c^2 + 3*a*c^3 + 3*b*c^3 + c^4 : :

X(68481) lies on the BG KHO conic and these lines: {1, 2}, {15, 37146}, {16, 37147}, {58, 37039}, {140, 5799}, {316, 1326}, {579, 25354}, {940, 52782}, {942, 25498}, {993, 16848}, {1656, 5786}, {2047, 6561}, {2049, 5110}, {2901, 17303}, {3454, 19701}, {3743, 4261}, {3931, 46838}, {3986, 59639}, {4067, 51223}, {4202, 41818}, {4205, 5019}, {4252, 50410}, {4472, 50067}, {4658, 5224}, {4798, 57282}, {6560, 63810}, {6693, 16343}, {6703, 50409}, {6707, 8728}, {6998, 9993}, {7380, 43460}, {7790, 13740}, {16290, 51687}, {16458, 48843}, {18134, 28620}, {24961, 29066}, {25526, 48835}, {28619, 32782}, {32776, 41812}, {36969, 37145}, {36970, 37144}, {37314, 48866}, {48863, 56985}, {50298, 62805}, {52785, 62795}

X(68481) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1698, 56810}, {2, 1714, 3634}, {4205, 17398, 43531}

X(68482) = X(1)X(2)∩X(484)X(39185)

Barycentrics    a^4 + 2*a^3*b - a*b^3 + 2*a^3*c + a^2*b*c + 2*a*b^2*c - 2*b^3*c + 2*a*b*c^2 - 4*b^2*c^2 - a*c^3 - 2*b*c^3 : :

X(68482) lies on the Evansvil circle and these lines: {1, 2}, {484, 39185}, {1046, 3701}, {1757, 3992}, {3667, 4581}, {3820, 32861}, {4037, 41322}, {4723, 32919}, {4880, 24821}, {7171, 20368}, {17757, 32846}, {26446, 33092}, {33079, 37715}, {36926, 38456}, {49712, 59586}

X(68482) = reflection of X(5529) in X(5205)
X(68482) = Conway-circle-inverse of X(50608)
X(68482) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1737, 32847, 60454}, {3831, 41261, 1}

X(68483) = X(1)X(564)∩X(6)X(13)

Barycentrics    (a + b)*(a + c)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2) : :

X(68483) lies on the circumconic {{A,B,C,X(1),X(6)}} and these lines: {1, 564}, {6, 13}, {56, 759}, {58, 79}, {80, 1126}, {86, 20565}, {94, 43531}, {106, 476}, {1141, 59006}, {1438, 32678}, {1474, 64834}, {2163, 23681}, {3226, 35139}, {3445, 56847}, {4013, 9268}, {5127, 36052}, {11599, 39295}, {15065, 39977}, {15475, 60050}, {16704, 63642}, {18316, 54586}, {24624, 56343}, {24880, 52388}, {32680, 37129}, {36123, 36129}, {54554, 55003}, {60049, 60053}, {63759, 66946}

X(68483) = X(i)-cross conjugate of X(j) for these (i,j): {1989, 66922}, {11076, 65048}, {62211, 81}
X(68483) = X(68483) = X(i)-isoconjugate of X(j) for these (i,j): {6, 42701}, {10, 6149}, {35, 758}, {36, 3678}, {37, 323}, {50, 321}, {71, 52414}, {72, 186}, {100, 526}, {101, 32679}, {190, 2624}, {213, 7799}, {228, 340}, {319, 3724}, {668, 14270}, {692, 3268}, {860, 52408}, {906, 44427}, {1154, 56254}, {1332, 47230}, {1464, 4420}, {1783, 8552}, {1824, 52437}, {1983, 7265}, {2088, 4567}, {2174, 3936}, {2245, 3219}, {2290, 56246}, {2323, 16577}, {2361, 40999}, {2594, 4511}, {3969, 7113}, {3990, 14165}, {3998, 52418}, {4036, 52603}, {4053, 40214}, {4585, 55210}, {4705, 10411}, {5081, 22342}, {5379, 16186}, {5380, 44814}, {14590, 55232}, {18593, 52405}, {19627, 27801}, {20336, 34397}, {21741, 32851}, {22115, 41013}, {42713, 52668}, {42717, 60777}, {53176, 57109}, {57268, 60723}, {66989, 68430}
X(68483) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 42701}, {1015, 32679}, {1086, 3268}, {4988, 62551}, {5190, 44427}, {6626, 7799}, {8054, 526}, {14993, 10}, {15295, 42}, {15898, 3678}, {39006, 8552}, {40589, 323}, {40618, 45792}, {40627, 2088}, {55053, 2624}
X(68483) = cevapoint of X(i) and X(j) for these (i,j): {3120, 11125}, {14158, 52375}
X(68483) = trilinear pole of line {649, 2160}
X(68483) = crossdifference of every pair of points on line {526, 2624}
X(68483) = barycentric product X(i)*X(j) for these {i,j}: {1, 66922}, {27, 265}, {58, 94}, {79, 24624}, {80, 52393}, {81, 2166}, {86, 1989}, {310, 11060}, {328, 1474}, {476, 514}, {513, 32680}, {649, 35139}, {662, 43082}, {693, 32678}, {759, 30690}, {905, 36129}, {1141, 17167}, {1333, 63759}, {1459, 46456}, {1790, 6344}, {2006, 3615}, {2160, 14616}, {2206, 20573}, {3120, 39295}, {3261, 14560}, {4556, 10412}, {4610, 15475}, {5627, 18653}, {6740, 52374}, {7649, 60053}, {10566, 46155}, {11125, 39290}, {13486, 60074}, {14559, 62626}, {17206, 18384}, {17924, 36061}, {18359, 52375}, {20565, 34079}, {21102, 64516}, {32662, 46107}, {43083, 52919}, {43682, 52380}, {44129, 52153}, {44709, 65360}, {55236, 65283}, {56401, 60679}, {57985, 64834}
X(68483) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 42701}, {27, 340}, {28, 52414}, {58, 323}, {79, 3936}, {80, 3969}, {86, 7799}, {94, 313}, {265, 306}, {328, 40071}, {476, 190}, {513, 32679}, {514, 3268}, {649, 526}, {667, 2624}, {759, 3219}, {1141, 56246}, {1333, 6149}, {1411, 16577}, {1459, 8552}, {1474, 186}, {1790, 52437}, {1919, 14270}, {1989, 10}, {2006, 40999}, {2160, 758}, {2161, 3678}, {2166, 321}, {2206, 50}, {2341, 4420}, {3120, 62551}, {3122, 2088}, {3615, 32851}, {4025, 45792}, {4556, 10411}, {4750, 45808}, {6186, 2245}, {6740, 42033}, {6757, 61410}, {7649, 44427}, {8747, 14165}, {10412, 52623}, {11060, 42}, {11125, 5664}, {13486, 4585}, {14158, 40612}, {14560, 101}, {14582, 4064}, {14616, 33939}, {15475, 4024}, {17167, 1273}, {17209, 51383}, {18384, 1826}, {18653, 6148}, {21102, 41078}, {24624, 319}, {30690, 35550}, {32662, 1331}, {32678, 100}, {32680, 668}, {34079, 35}, {35139, 1978}, {36061, 1332}, {36129, 6335}, {39295, 4600}, {43082, 1577}, {43083, 68130}, {46155, 4568}, {50433, 3682}, {52153, 71}, {52372, 18593}, {52374, 41804}, {52375, 3218}, {52380, 56440}, {52393, 320}, {52449, 21094}, {52954, 14920}, {52955, 35201}, {55236, 6370}, {56395, 4062}, {56401, 60737}, {60053, 4561}, {62746, 68103}, {63759, 27801}, {64834, 860}, {65283, 55235}, {66284, 7265}, {66922, 75}, {66925, 8804}, {66945, 9213}, {67166, 2174}
X(68483) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {265, 68454, 45926}, {45926, 56402, 265}, {50148, 61479, 476}

X(68484) = X(6)X(13)∩X(125)X(2904)

Barycentrics    a^2*(a^20 - 6*a^18*b^2 + 13*a^16*b^4 - 8*a^14*b^6 - 14*a^12*b^8 + 28*a^10*b^10 - 14*a^8*b^12 - 8*a^6*b^14 + 13*a^4*b^16 - 6*a^2*b^18 + b^20 - 6*a^18*c^2 + 27*a^16*b^2*c^2 - 47*a^14*b^4*c^2 + 40*a^12*b^6*c^2 - 15*a^10*b^8*c^2 - 14*a^8*b^10*c^2 + 43*a^6*b^12*c^2 - 48*a^4*b^14*c^2 + 25*a^2*b^16*c^2 - 5*b^18*c^2 + 13*a^16*c^4 - 47*a^14*b^2*c^4 + 64*a^12*b^4*c^4 - 43*a^10*b^6*c^4 + 32*a^8*b^8*c^4 - 55*a^6*b^10*c^4 + 66*a^4*b^12*c^4 - 39*a^2*b^14*c^4 + 9*b^16*c^4 - 8*a^14*c^6 + 40*a^12*b^2*c^6 - 43*a^10*b^4*c^6 + 4*a^8*b^6*c^6 + 20*a^6*b^8*c^6 - 36*a^4*b^10*c^6 + 27*a^2*b^12*c^6 - 4*b^14*c^6 - 14*a^12*c^8 - 15*a^10*b^2*c^8 + 32*a^8*b^4*c^8 + 20*a^6*b^6*c^8 + 10*a^4*b^8*c^8 - 7*a^2*b^10*c^8 - 10*b^12*c^8 + 28*a^10*c^10 - 14*a^8*b^2*c^10 - 55*a^6*b^4*c^10 - 36*a^4*b^6*c^10 - 7*a^2*b^8*c^10 + 18*b^10*c^10 - 14*a^8*c^12 + 43*a^6*b^2*c^12 + 66*a^4*b^4*c^12 + 27*a^2*b^6*c^12 - 10*b^8*c^12 - 8*a^6*c^14 - 48*a^4*b^2*c^14 - 39*a^2*b^4*c^14 - 4*b^6*c^14 + 13*a^4*c^16 + 25*a^2*b^2*c^16 + 9*b^4*c^16 - 6*a^2*c^18 - 5*b^2*c^18 + c^20) : :

X(68484) lies on the Moses-Jerabek conic and these lines: {6, 13}, {125, 2904}, {389, 15463}, {578, 12901}, {1147, 12236}, {1204, 1986}, {11438, 25487}, {12038, 12228}, {12234, 32226}, {13289, 38534}, {19457, 34469}

X(68484) = crosssum of X(2072) and X(44452)

X(68485) = X(6)X(13)∩X(32)X(39176)

Barycentrics    a^2*(a^12 - 2*a^10*b^2 - a^8*b^4 + 4*a^6*b^6 - a^4*b^8 - 2*a^2*b^10 + b^12 - 2*a^10*c^2 + 7*a^8*b^2*c^2 - 5*a^6*b^4*c^2 - 6*a^4*b^6*c^2 + 9*a^2*b^8*c^2 - 3*b^10*c^2 - a^8*c^4 - 5*a^6*b^2*c^4 + 14*a^4*b^4*c^4 - 7*a^2*b^6*c^4 + 3*b^8*c^4 + 4*a^6*c^6 - 6*a^4*b^2*c^6 - 7*a^2*b^4*c^6 - 2*b^6*c^6 - a^4*c^8 + 9*a^2*b^2*c^8 + 3*b^4*c^8 - 2*a^2*c^10 - 3*b^2*c^10 + c^12) : :

X(68485) lies on the Moses-Lemoine conic and these lines: {6, 13}, {32, 39176}, {111, 34570}, {112, 800}, {868, 34470}, {1637, 61216}, {1968, 67191}, {2079, 15905}, {2088, 18877}, {2493, 34481}, {3003, 38872}, {3284, 14910}, {5063, 63846}, {5158, 19220}, {8749, 14581}, {32761, 47405}, {46432, 47228}

X(68485) = crosspoint of X(8749) and X(14910)
X(68485) = crosssum of X(3580) and X(11064)
X(68485) = crossdifference of every pair of points on line {526, 12825}
X(68485) = {X(3163),X(6128)}-harmonic conjugate of X(5309)

X(68486) = X(6)X(13)∩X(10)X(98)

Barycentrics    a^7 + a^6*b - a^5*b^2 - a^4*b^3 + a^3*b^4 - a*b^6 + a^6*c - a^4*b^2*c + a*b^5*c - b^6*c - a^5*c^2 - a^4*b*c^2 - a^3*b^2*c^2 + a^2*b^3*c^2 + a*b^4*c^2 - b^5*c^2 - a^4*c^3 + a^2*b^2*c^3 - 2*a*b^3*c^3 + 2*b^4*c^3 + a^3*c^4 + a*b^2*c^4 + 2*b^3*c^4 + a*b*c^5 - b^2*c^5 - a*c^6 - b*c^6 : :

X(68486) lies on the 2nd Evans circle and these lines: {2, 54834}, {6, 13}, {10, 98}, {30, 50252}, {99, 14829}, {111, 21950}, {148, 37683}, {671, 41629}, {2321, 37508}, {2771, 21890}, {2782, 49129}, {2783, 12515}, {3430, 50606}, {6321, 46976}, {9860, 52679}, {11177, 66100}, {12513, 22514}, {14651, 36677}

X(68486) = crossdifference of every pair of points on line {526, 53521}

X(68487) = X(11)X(523)∩X(36)X(186)

Barycentrics    a*(a - b - c)*(b - c)^2*(a^2 - b^2 + b*c - c^2)*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3) : :

X(68487) lies on the de Longchamps ellipse and these lines: {1, 953}, {11, 523}, {34, 2716}, {36, 186}, {56, 10016}, {57, 2717}, {212, 67435}, {215, 26884}, {244, 1459}, {513, 7004}, {517, 1457}, {650, 2170}, {910, 17455}, {1155, 53547}, {1319, 67516}, {1393, 31849}, {1421, 37815}, {1456, 67521}, {1772, 67441}, {2632, 41211}, {3025, 53525}, {3259, 3326}, {3328, 3675}, {3942, 15635}, {5048, 17460}, {6075, 18210}, {7069, 67213}, {7962, 15737}, {13141, 53826}, {13999, 53047}, {14284, 38357}, {15558, 61476}, {17463, 23758}, {18340, 66865}, {18839, 53552}, {22072, 67436}, {24430, 61729}, {25485, 41553}, {31847, 44706}, {34036, 56411}, {36037, 65249}, {37754, 47411}, {43909, 53540}, {46398, 57434}, {51402, 57446}

X(68487) = reflection of X(53530) in X(39756)
X(68487) = reflection of X(7004) in the OI line
X(68487) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 1769}, {11, 35015}, {57, 654}, {92, 2600}, {514, 46384}, {2051, 2610}, {16586, 53046}, {34586, 42768}, {47645, 513}, {52031, 46393}, {52659, 60339}
X(68487) = X(i)-isoconjugate of X(j) for these (i,j): {59, 40437}, {101, 53811}, {104, 52377}, {655, 32641}, {2222, 36037}, {2720, 51562}, {4564, 67178}, {13136, 32675}, {32669, 36804}, {39294, 52431}
X(68487) = X(i)-Dao conjugate of X(j) for these (i,j): {908, 4998}, {1015, 53811}, {2245, 4564}, {3259, 2222}, {3310, 14628}, {3738, 56757}, {6615, 40437}, {10015, 75}, {13999, 1309}, {23757, 52409}, {35128, 13136}, {38981, 51562}, {38984, 36037}, {40613, 52377}, {46398, 35174}, {55153, 36804}, {60339, 52351}, {66508, 54953}
X(68487) = crosspoint of X(i) and X(j) for these (i,j): {1, 3738}, {11, 53525}, {36, 61043}, {514, 22464}, {1845, 53047}, {3737, 17515}
X(68487) = crosssum of X(i) and X(j) for these (i,j): {1, 2222}, {59, 52377}, {101, 2342}, {4551, 52391}
X(68487) = crossdifference of every pair of points on line {1983, 23703}
X(68487) = barycentric product X(i)*X(j) for these {i,j}: {1, 46398}, {11, 16586}, {57, 57434}, {273, 38353}, {513, 53045}, {654, 36038}, {693, 53046}, {905, 53047}, {908, 53525}, {1086, 64139}, {1769, 3904}, {1845, 26932}, {2804, 3960}, {3218, 35015}, {3738, 10015}, {4453, 46393}, {4511, 42754}, {4560, 42768}, {4858, 34586}, {6735, 53546}, {14010, 18593}, {17515, 42761}, {17923, 35014}, {32851, 42753}, {45950, 65249}, {51402, 52031}, {65104, 65868}
X(68487) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 53811}, {654, 36037}, {1769, 655}, {1845, 46102}, {1870, 39294}, {2170, 40437}, {2183, 52377}, {2804, 36804}, {3259, 14628}, {3271, 67178}, {3310, 2222}, {3738, 13136}, {3960, 54953}, {8648, 32641}, {8677, 65299}, {10015, 35174}, {16586, 4998}, {21758, 2720}, {34586, 4564}, {35014, 52351}, {35015, 18359}, {35128, 56757}, {36038, 46405}, {38353, 78}, {39534, 65329}, {42753, 2006}, {42754, 18815}, {42759, 60091}, {42768, 4552}, {44428, 65223}, {46384, 43728}, {46393, 51562}, {46398, 75}, {52316, 52356}, {53045, 668}, {53046, 100}, {53047, 6335}, {53314, 37136}, {53525, 34234}, {57434, 312}, {64139, 1016}, {65104, 1309}
X(68487) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 34464, 2222}, {3259, 3326, 35015}, {3326, 35014, 35065}, {35014, 42753, 35015}

X(68488) = X(1)X(6)∩X(2)X(98)

Barycentrics    a*(a^5 - a^3*b^2 - a^3*b*c - a^2*b^2*c - a*b^3*c - b^4*c - a^3*c^2 - a^2*b*c^2 - 2*a*b^2*c^2 - b^3*c^2 - a*b*c^3 - b^2*c^3 - b*c^4) : :
X(68488) = X[41741] - 3 X[52699]

X(68488) lies on on the Feuerbach circumhyperbola of the Brocard triangle and these lines: {1, 6}, {2, 98}, {10, 2330}, {21, 511}, {35, 17792}, {51, 37325}, {55, 61172}, {58, 28369}, {59, 19890}, {69, 261}, {141, 5135}, {193, 17558}, {274, 12215}, {284, 15984}, {377, 46264}, {386, 1178}, {404, 5092}, {406, 1974}, {442, 1503}, {443, 25406}, {451, 19128}, {452, 14853}, {474, 5085}, {475, 19124}, {517, 56970}, {524, 15670}, {572, 1009}, {573, 13723}, {575, 5047}, {576, 16865}, {582, 16299}, {692, 4026}, {943, 4158}, {993, 1469}, {1125, 1428}, {1176, 43712}, {1211, 2194}, {1213, 37047}, {1316, 59787}, {1333, 15989}, {1350, 16370}, {1351, 16418}, {1441, 56910}, {1495, 4239}, {1691, 5277}, {1692, 16589}, {1756, 8424}, {1992, 17561}, {1993, 59358}, {2030, 37675}, {2175, 50295}, {2328, 4199}, {2475, 29012}, {2476, 3818}, {2478, 14561}, {3056, 5248}, {3098, 4189}, {3178, 5847}, {3292, 26637}, {3416, 47373}, {3564, 6675}, {3573, 9791}, {3589, 4187}, {3618, 5084}, {3620, 25663}, {3705, 5278}, {3736, 15994}, {3763, 25669}, {3821, 5091}, {3915, 50629}, {3916, 24471}, {3923, 4112}, {4188, 17508}, {4193, 38317}, {4194, 67882}, {4208, 66755}, {4260, 37306}, {4265, 19525}, {4273, 15990}, {4357, 7193}, {4512, 64084}, {5046, 19130}, {5050, 11108}, {5093, 16866}, {5096, 19524}, {5097, 16858}, {5102, 19538}, {5129, 51171}, {5150, 24295}, {5275, 40825}, {5294, 26890}, {5320, 5739}, {5476, 66099}, {5480, 11113}, {5554, 38116}, {5752, 37317}, {5965, 15674}, {6175, 11645}, {6872, 31670}, {6998, 26671}, {7621, 9830}, {8582, 38118}, {8616, 50635}, {8728, 48906}, {10198, 12588}, {10381, 62843}, {10455, 19269}, {10541, 16862}, {11110, 25898}, {11111, 51212}, {11112, 44882}, {11114, 48901}, {11354, 28204}, {11477, 19526}, {12017, 16408}, {12514, 52567}, {12589, 26363}, {13323, 37176}, {13587, 55674}, {14810, 17549}, {14912, 16845}, {14994, 37670}, {15516, 16861}, {15671, 64802}, {15677, 19924}, {15680, 29317}, {16371, 53094}, {16429, 51420}, {16842, 53093}, {16853, 55705}, {16854, 55703}, {16855, 55701}, {16857, 53091}, {16859, 39561}, {16860, 53092}, {16863, 55697}, {16864, 55699}, {16998, 32451}, {17104, 24931}, {17185, 22139}, {17527, 38110}, {17530, 67865}, {17531, 20190}, {17532, 36990}, {17534, 55706}, {17535, 55695}, {17536, 50664}, {17542, 55711}, {17543, 22330}, {17544, 22234}, {17546, 55704}, {17547, 55709}, {17548, 55649}, {17554, 33748}, {17570, 55710}, {17571, 33878}, {17572, 55687}, {17573, 55682}, {17574, 55606}, {17577, 48889}, {17579, 48898}, {17977, 32779}, {19146, 31473}, {19313, 25878}, {19314, 26657}, {19527, 36745}, {19528, 36746}, {19535, 31884}, {19537, 55676}, {19539, 55582}, {19561, 25354}, {19684, 29634}, {19704, 55651}, {19705, 55671}, {19717, 29838}, {19742, 29840}, {19854, 39900}, {19861, 38029}, {20423, 31156}, {21511, 48886}, {21850, 50241}, {22080, 27174}, {24247, 24265}, {24248, 24264}, {24249, 24258}, {24432, 52531}, {24446, 42708}, {24953, 39873}, {25555, 37162}, {25906, 48894}, {25978, 56731}, {26256, 61506}, {26889, 54311}, {27559, 38049}, {29181, 57002}, {29511, 59222}, {31245, 39892}, {31445, 43216}, {32431, 37049}, {32911, 50595}, {33750, 37267}, {33751, 36004}, {33854, 64713}, {34122, 51157}, {35258, 54296}, {35935, 63402}, {36006, 55688}, {37023, 37508}, {37256, 48892}, {37299, 48885}, {37307, 55672}, {37659, 56775}, {41741, 52699}, {43146, 59406}, {43273, 44217}, {43621, 50244}, {48662, 50740}, {48872, 57006}, {48905, 50239}, {48910, 50242}, {50202, 50979}, {50739, 63428}, {50742, 61044}, {51437, 56832}, {54417, 65543}, {56998, 59411}, {63070, 63722}

X(68488) = midpoint of X(21) and X(15988)
X(68488) = reflection of X(26543) in X(6675)
X(68488) = complement of X(63470)
X(68488) = psi-transform of X(37959)
X(68488) = crossdifference of every pair of points on line {513, 3569}
X(68488) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3821, 24253, 5091}, {3923, 24267, 24271}

X(68489) = X(1)X(6)∩X(110)X(912)

Barycentrics    a*(a^8*b - 2*a^6*b^3 + 2*a^2*b^7 - b^9 + a^8*c - 2*a^7*b*c - a^6*b^2*c + a^5*b^3*c - 2*a^4*b^4*c + 4*a^3*b^5*c + 3*a^2*b^6*c - 3*a*b^7*c - b^8*c - a^6*b*c^2 + a^2*b^5*c^2 - 2*a^6*c^3 + a^5*b*c^3 - 8*a^3*b^3*c^3 - 2*a^2*b^4*c^3 + 3*a*b^5*c^3 - 2*a^4*b*c^4 - 2*a^2*b^3*c^4 + 2*b^5*c^4 + 4*a^3*b*c^5 + a^2*b^2*c^5 + 3*a*b^3*c^5 + 2*b^4*c^5 + 3*a^2*b*c^6 + 2*a^2*c^7 - 3*a*b*c^7 - b*c^8 - c^9) : :

X(68489) lies on the Walsmith rectangular hyperbola and these lines: {1, 6}, {74, 37959}, {110, 912}, {125, 517}, {758, 15904}, {1495, 2771}, {2778, 32125}, {2836, 32113}, {3746, 56280}, {5142, 7703}, {6001, 9625}, {7984, 41742}, {14017, 40914}, {24933, 31837}, {32114, 34381}

X(68489) = midpoint of X(7984) and X(41742)

X(68490) = X(1)X(6)∩X(111)X(2691)

Barycentrics    a*(a^5*b - a^4*b^2 - a*b^5 + b^6 + a^5*c - 4*a^4*b*c - a^3*b^2*c + 2*a^2*b^3*c - 2*a*b^4*c - a^4*c^2 - a^3*b*c^2 + 2*a^2*b^2*c^2 + 3*a*b^3*c^2 - b^4*c^2 + 2*a^2*b*c^3 + 3*a*b^2*c^3 - 2*a*b*c^4 - b^2*c^4 - a*c^5 + c^6) : :

X(68490) lies on the Moses-Parry circle and these lines: {1, 6}, {111, 2691}, {112, 2074}, {115, 3290}, {187, 47231}, {519, 25095}, {2079, 54095}, {2492, 65862}, {3309, 3569}, {5142, 50718}, {5523, 47232}, {13745, 25081}, {14017, 21397}, {25088, 63360}

X(68491) = X(1)X(181)∩X(3)X(9564)

Barycentrics    a*(a^3*b + 3*a^2*b^2 + a*b^3 - b^4 + a^3*c + 4*a^2*b*c + a*b^2*c + 3*a^2*c^2 + a*b*c^2 + 2*b^2*c^2 + a*c^3 - c^4)*(a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + a^4*c + 2*a^3*b*c - a^2*b^2*c - 2*a*b^3*c + a^3*c^2 - a^2*b*c^2 - 2*b^3*c^2 - a^2*c^3 - 2*a*b*c^3 - 2*b^2*c^3 - a*c^4) : :
X(68491) = 5 X[1698] - X[10825]

X(68491) lies on the Feuerbach circumhyperbola of the medial triangle and these lines: {1, 181}, {3, 9564}, {214, 34458}, {442, 39591}, {573, 34261}, {1698, 10825}, {2092, 64125}, {6684, 51575}, {9565, 40587}, {16453, 52148}, {39566, 51571}

X(68492) = X(1)X(181)∩X(7)X(17143)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(a*b + b^2 + a*c + 4*b*c + c^2)*(a^2*b^2 + a*b^3 + 4*a*b^2*c + b^3*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 + a*c^3 + b*c^3) : :

X(68492) lies on the Jerabek circumhyperbola of the intouch triangle and these lines: {1, 181}, {7, 17143}, {1401, 5586}, {4322, 52024}, {10473, 35616}, {17114, 65383}, {34791, 39774}, {65385, 68372}

X(68493) = X(1)X(181)∩X(3)X(1045)

Barycentrics    a*(a^7*b^2 + 3*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - 3*a^3*b^6 - a^2*b^7 + a^7*b*c + 7*a^6*b^2*c + 9*a^5*b^3*c - 2*a^4*b^4*c - 9*a^3*b^5*c - 5*a^2*b^6*c - a*b^7*c + a^7*c^2 + 7*a^6*b*c^2 + 9*a^5*b^2*c^2 + 5*a^4*b^3*c^2 - 7*a^3*b^4*c^2 - 13*a^2*b^5*c^2 - 3*a*b^6*c^2 + b^7*c^2 + 3*a^6*c^3 + 9*a^5*b*c^3 + 5*a^4*b^2*c^3 - 10*a^3*b^3*c^3 - 9*a^2*b^4*c^3 - 3*a*b^5*c^3 + b^6*c^3 + 2*a^5*c^4 - 2*a^4*b*c^4 - 7*a^3*b^2*c^4 - 9*a^2*b^3*c^4 - 2*a*b^4*c^4 - 2*b^5*c^4 - 2*a^4*c^5 - 9*a^3*b*c^5 - 13*a^2*b^2*c^5 - 3*a*b^3*c^5 - 2*b^4*c^5 - 3*a^3*c^6 - 5*a^2*b*c^6 - 3*a*b^2*c^6 + b^3*c^6 - a^2*c^7 - a*b*c^7 + b^2*c^7) : :

X(68493) lies on the Jerabek circumhyperbola of the excentral triangle and these lines: {1, 181}, {3, 1045}, {9, 10476}, {40, 11679}, {191, 1764}, {573, 17733}, {3646, 37035}, {3741, 39591}, {5247, 37620}, {5541, 13244}

X(68494) = X(1)X(2962)∩X(6)X(17)

Barycentrics    (a + b)*(a + c)*(a^4 - a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4) : :

X(68494) lies on the circumconic {{A,B,C,X(1),X(6)}} these lines: {1, 2962}, {6, 17}, {56, 37806}, {58, 5443}, {106, 930}, {1438, 36148}, {3226, 46139}, {11140, 43531}, {25043, 60033}

X(68494) = X(i)-isoconjugate of X(j) for these (i,j): {10, 2964}, {37, 1994}, {49, 41013}, {72, 3518}, {100, 1510}, {143, 56254}, {213, 7769}, {228, 32002}, {321, 2965}, {692, 41298}, {906, 67102}, {1783, 63830}, {1824, 44180}, {1826, 63760}, {21807, 63172}
X(68494) = X(i)-Dao conjugate of X(j) for these (i,j): {1086, 41298}, {5190, 67102}, {6626, 7769}, {8054, 1510}, {21975, 10}, {39006, 63830}, {40589, 1994}, {46604, 42}
X(68494) = cevapoint of X(3120) and X(21103)
X(68494) = barycentric product X(i)*X(j) for these {i,j}: {27, 3519}, {58, 11140}, {81, 2962}, {86, 2963}, {93, 1790}, {252, 17167}, {514, 930}, {649, 46139}, {693, 36148}, {1437, 63764}, {1459, 38342}, {1487, 17168}, {3261, 32737}, {44129, 51477}
X(68494) = barycentric quotient X(i)/X(j) for these {i,j}: {27, 32002}, {58, 1994}, {86, 7769}, {252, 56246}, {514, 41298}, {649, 1510}, {930, 190}, {1333, 2964}, {1437, 63760}, {1459, 63830}, {1474, 3518}, {1790, 44180}, {2206, 2965}, {2962, 321}, {2963, 10}, {3519, 306}, {7649, 67102}, {11140, 313}, {17167, 57805}, {17209, 51440}, {21102, 20577}, {32737, 101}, {36148, 100}, {44709, 63805}, {46139, 1978}, {51477, 71}

X(68495) = X(5)X(49)∩X(6)X(17)

Barycentrics    a^10 - 4*a^8*b^2 + 5*a^6*b^4 - a^4*b^6 - 2*a^2*b^8 + b^10 - 4*a^8*c^2 + 3*a^6*b^2*c^2 + 4*a^4*b^4*c^2 - 3*b^8*c^2 + 5*a^6*c^4 + 4*a^4*b^2*c^4 + 4*a^2*b^4*c^4 + 2*b^6*c^4 - a^4*c^6 + 2*b^4*c^6 - 2*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(68495) = 4 X[13363] - X[66739], X[3] + 2 X[3574], X[3] - 4 X[6689], 2 X[3] + X[15800], X[3574] + 2 X[6689], 4 X[3574] - X[15800], 8 X[6689] + X[15800], 2 X[40909] - 5 X[61116], X[4] + 2 X[10610], 2 X[5] + X[54], 4 X[5] - X[6288], X[5] + 2 X[8254], 7 X[5] - 4 X[20584], 11 X[5] + 4 X[20585], 5 X[5] + X[36966], X[5] - 4 X[64486], 2 X[54] + X[6288], X[54] - 4 X[8254], and many others

X(68495) lies on the Thomson-Gibert-Moses hyperbola and these lines: {2, 568}, {3, 3574}, {4, 7712}, {5, 49}, {6, 17}, {11, 10066}, {12, 10082}, {30, 6030}, {52, 58489}, {74, 11805}, {93, 30536}, {125, 15037}, {140, 3581}, {143, 66765}, {154, 381}, {354, 3582}, {355, 12266}, {485, 49257}, {486, 49256}, {498, 13079}, {499, 18984}, {539, 3167}, {546, 58789}, {547, 50708}, {549, 43804}, {569, 6145}, {575, 15027}, {632, 30531}, {826, 34291}, {858, 13339}, {973, 6639}, {1199, 34826}, {1216, 58557}, {1493, 2888}, {1503, 5576}, {1511, 36853}, {1594, 13353}, {1614, 50138}, {1986, 6143}, {1989, 9222}, {2070, 58447}, {2072, 37649}, {2777, 3521}, {2914, 15018}, {2917, 7506}, {2931, 43839}, {3091, 12254}, {3311, 8995}, {3312, 13986}, {3459, 46954}, {3462, 14978}, {3520, 38788}, {3526, 5646}, {3542, 11576}, {3549, 12363}, {3567, 32338}, {3580, 3628}, {3815, 15093}, {3830, 61771}, {3843, 32340}, {3850, 55156}, {3851, 10619}, {5012, 39504}, {5054, 61773}, {5067, 5645}, {5070, 5544}, {5133, 10540}, {5169, 61752}, {5397, 6881}, {5449, 61712}, {5462, 22815}, {5498, 43597}, {5562, 10115}, {5640, 7730}, {5644, 15703}, {5648, 29959}, {5654, 14787}, {5655, 15030}, {5901, 7979}, {5944, 66766}, {6101, 44056}, {6102, 41726}, {6152, 7505}, {6153, 21660}, {6242, 14940}, {6243, 41590}, {6640, 15739}, {6642, 32333}, {7404, 32605}, {7486, 11271}, {7517, 53023}, {7527, 7728}, {7528, 32346}, {7529, 9920}, {7540, 13394}, {7545, 19130}, {7550, 51391}, {7558, 37484}, {7564, 65149}, {7569, 12161}, {7570, 54434}, {7574, 37513}, {7577, 52417}, {7583, 19095}, {7584, 19096}, {7605, 45237}, {7741, 12956}, {7951, 12946}, {8227, 9905}, {9730, 10628}, {9777, 41594}, {9781, 63693}, {9820, 35283}, {9956, 12785}, {10024, 16657}, {10095, 12226}, {10104, 12208}, {10127, 59648}, {10182, 23358}, {10224, 43651}, {10545, 12380}, {10938, 63727}, {11003, 34514}, {11232, 13366}, {11423, 18356}, {11430, 12121}, {11563, 43579}, {11585, 19129}, {11675, 18322}, {11702, 20304}, {11802, 43581}, {11803, 55856}, {12013, 15699}, {12175, 41598}, {12233, 63392}, {12236, 32205}, {12300, 37119}, {12900, 43704}, {13160, 37472}, {13321, 61646}, {13364, 37943}, {13365, 13368}, {13371, 37471}, {13413, 25739}, {13469, 20413}, {13564, 29317}, {13621, 44515}, {13630, 15101}, {13632, 35270}, {13754, 48411}, {13856, 21975}, {14071, 61594}, {14076, 15047}, {14118, 18442}, {14530, 34775}, {14561, 16776}, {14786, 43841}, {14789, 59771}, {14912, 25738}, {14924, 55857}, {15024, 60780}, {15033, 46029}, {15038, 63735}, {15043, 32339}, {15045, 61736}, {15059, 43580}, {15072, 44287}, {15087, 21243}, {15088, 47117}, {15537, 57368}, {16042, 25714}, {16239, 54201}, {17824, 36752}, {17835, 25563}, {18114, 46155}, {18281, 40280}, {18282, 38848}, {18369, 56924}, {18381, 32359}, {18435, 60763}, {18436, 65094}, {18449, 18583}, {18874, 21451}, {19457, 38789}, {20126, 45956}, {20299, 32349}, {20376, 23328}, {22115, 23292}, {26869, 32341}, {31392, 50479}, {32321, 32345}, {32330, 32365}, {32393, 32402}, {33332, 52525}, {34380, 66731}, {34545, 41730}, {35197, 65141}, {35500, 67861}, {36153, 43808}, {37118, 38728}, {37452, 64730}, {37454, 63720}, {37481, 67915}, {37495, 54040}, {38323, 38723}, {38724, 43573}, {38793, 43809}, {44802, 58407}, {45971, 51033}, {46219, 54202}, {50135, 51425}, {51392, 54006}, {57304, 57333}, {57307, 67349}, {64474, 64624}

X(68495) = midpoint of X(i) and X(j) for these {i,j}: {2, 61715}, {10274, 23325}
X(68495) = reflection of X(i) in X(j) for these {i,j}: {6145, 23325}, {23328, 20376}, {23358, 10182}, {55039, 61659}
X(68495) = Thomson-isogonal conjugate of X(6636)
X(68495) = crossdifference of every pair of points on line {1510, 2081}
X(68495) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 3574, 15800}, {5, 54, 6288}, {5, 567, 265}, {5, 8254, 54}, {5, 13434, 43821}, {5, 14389, 567}, {5, 15806, 43598}, {140, 20424, 7691}, {195, 1209, 3519}, {195, 1656, 1209}, {567, 11597, 54}, {1209, 12242, 195}, {1209, 13431, 15605}, {1209, 32396, 1656}, {1493, 13565, 2888}, {1656, 12242, 3519}, {2888, 3090, 13565}, {3091, 12254, 22804}, {3526, 12307, 32348}, {3574, 6689, 3}, {3628, 22051, 21230}, {5133, 61619, 10540}, {7741, 47378, 12956}, {8254, 64486, 5}, {11702, 20304, 33565}, {12228, 13434, 567}, {12242, 32396, 1209}, {21230, 22051, 15801}, {23292, 37347, 22115}, {33332, 52525, 64757}, {34545, 54000, 63839}, {43581, 64854, 11802}

X(68496) = X(6)X(17)∩X(110)X(539)

Barycentrics    a^12*b^4 - 4*a^10*b^6 + 5*a^8*b^8 - 5*a^4*b^12 + 4*a^2*b^14 - b^16 - 2*a^12*b^2*c^2 + 10*a^10*b^4*c^2 - 21*a^8*b^6*c^2 + 19*a^6*b^8*c^2 - a^4*b^10*c^2 - 9*a^2*b^12*c^2 + 4*b^14*c^2 + a^12*c^4 + 10*a^10*b^2*c^4 - 16*a^8*b^4*c^4 - a^6*b^6*c^4 + 7*a^4*b^8*c^4 + 3*a^2*b^10*c^4 - 4*b^12*c^4 - 4*a^10*c^6 - 21*a^8*b^2*c^6 - a^6*b^4*c^6 - 2*a^4*b^6*c^6 + 2*a^2*b^8*c^6 - 4*b^10*c^6 + 5*a^8*c^8 + 19*a^6*b^2*c^8 + 7*a^4*b^4*c^8 + 2*a^2*b^6*c^8 + 10*b^8*c^8 - a^4*b^2*c^10 + 3*a^2*b^4*c^10 - 4*b^6*c^10 - 5*a^4*c^12 - 9*a^2*b^2*c^12 - 4*b^4*c^12 + 4*a^2*c^14 + 4*b^2*c^14 - c^16 : :

X(68496) lies on the Walsmith rectuangular hyperbola and these lines: {6, 17}, {74, 5189}, {110, 539}, {125, 1154}, {1495, 25338}, {2914, 41724}, {2931, 37920}, {3574, 5876}, {5064, 11472}, {5449, 15801}, {5889, 7703}, {5899, 10117}, {7699, 37353}, {9927, 10594}, {10628, 32125}, {11803, 34826}, {12316, 14076}, {15067, 43817}, {15350, 32226}, {21660, 63734}, {32608, 60462}

X(68496) = midpoint of X(2914) and X(41724)
X(68496) = reflection of X(15091) in X(12242)

X(68497) = X(6)X(17)∩X(110)X(34567)

Barycentrics    a^2*(3*a^8 - 12*a^6*b^2 + 18*a^4*b^4 - 12*a^2*b^6 + 3*b^8 - 12*a^6*c^2 + 11*a^4*b^2*c^2 + 15*a^2*b^4*c^2 - 14*b^6*c^2 + 18*a^4*c^4 + 15*a^2*b^2*c^4 + 22*b^4*c^4 - 12*a^2*c^6 - 14*b^2*c^6 + 3*c^8) : :

X(68497) lies on the Hatzipolakis-Suppa ellipse [see X(46440)] and these lines: {3, 13421}, {4, 11538}, {6, 17}, {110, 34567}, {140, 2889}, {323, 61877}, {381, 10116}, {389, 18364}, {399, 1199}, {550, 15037}, {575, 13564}, {1173, 5899}, {1176, 53091}, {1493, 12834}, {1657, 36753}, {1993, 55860}, {1994, 55856}, {2070, 65093}, {2937, 15004}, {3448, 46446}, {3533, 63040}, {3549, 63127}, {3567, 32196}, {3851, 11441}, {5073, 64098}, {5422, 46219}, {5462, 32609}, {5576, 11061}, {7506, 43908}, {7517, 53092}, {7545, 22234}, {7592, 61970}, {9143, 23409}, {10601, 61875}, {10620, 13382}, {10982, 62016}, {11063, 61242}, {11482, 32154}, {12002, 37924}, {12161, 61919}, {13339, 33542}, {13353, 34565}, {13366, 18369}, {13434, 32608}, {13621, 15019}, {15018, 55859}, {15026, 55039}, {15032, 61976}, {15041, 35478}, {15720, 36749}, {16266, 61855}, {18350, 44111}, {18583, 46448}, {21308, 32136}, {32205, 61775}, {35018, 50461}, {35452, 43600}, {36747, 61803}, {36752, 62100}, {37496, 61792}, {37505, 43809}, {37509, 52680}, {37514, 62074}, {37672, 61878}, {37956, 58533}, {38724, 66765}, {39522, 62131}, {45016, 46447}, {45967, 46443}, {46442, 59399}, {49139, 66609}, {52100, 62023}, {55857, 63094}, {56292, 61907}, {61886, 62990}

X(68497) = X(11703)-Ceva conjugate of X(5899)
X(68497) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1173, 36153, 5899}, {1493, 12834, 22462}, {3519, 25555, 1656}, {14627, 34545, 15047}

X(68498) = X(6)X(17)∩X(111)X(5189)

Barycentrics    2*a^8*b^2 - 3*a^6*b^4 - a^4*b^6 + 3*a^2*b^8 - b^10 + 2*a^8*c^2 - 10*a^6*b^2*c^2 + 7*a^4*b^4*c^2 - 11*a^2*b^6*c^2 + 3*b^8*c^2 - 3*a^6*c^4 + 7*a^4*b^2*c^4 + 16*a^2*b^4*c^4 - 2*b^6*c^4 - a^4*c^6 - 11*a^2*b^2*c^6 - 2*b^4*c^6 + 3*a^2*c^8 + 3*b^2*c^8 - c^10 : :

X(68498) lies on the Moses-Parry circle and these lines: {6, 17}, {111, 5189}, {112, 37943}, {115, 37938}, {187, 25338}, {230, 15350}, {2079, 5899}, {3569, 22889}, {6032, 37353}, {8428, 37920}, {10594, 21397}, {47226, 47322}, {50718, 52295}

X(68499) = X(1)X(74)∩X(2)X(3)

Barycentrics    a*(a^9 + a^8*b - 2*a^7*b^2 - 3*a^6*b^3 + 3*a^4*b^5 + 2*a^3*b^6 - a^2*b^7 - a*b^8 + a^8*c - 2*a^7*b*c + 2*a^5*b^3*c - 2*a^4*b^4*c + 2*a^3*b^5*c - 2*a*b^7*c + b^8*c - 2*a^7*c^2 + 5*a^5*b^2*c^2 + 3*a^4*b^3*c^2 - 2*a^3*b^4*c^2 - 2*a^2*b^5*c^2 - a*b^6*c^2 - b^7*c^2 - 3*a^6*c^3 + 2*a^5*b*c^3 + 3*a^4*b^2*c^3 - 4*a^3*b^3*c^3 + 3*a^2*b^4*c^3 + 2*a*b^5*c^3 - 3*b^6*c^3 - 2*a^4*b*c^4 - 2*a^3*b^2*c^4 + 3*a^2*b^3*c^4 + 4*a*b^4*c^4 + 3*b^5*c^4 + 3*a^4*c^5 + 2*a^3*b*c^5 - 2*a^2*b^2*c^5 + 2*a*b^3*c^5 + 3*b^4*c^5 + 2*a^3*c^6 - a*b^2*c^6 - 3*b^3*c^6 - a^2*c^7 - 2*a*b*c^7 - b^2*c^7 - a*c^8 + b*c^8) : :

X(68499) lies on these lines: {1, 74}, {2, 3}, {477, 35056}, {5441, 34301}, {6757, 67722}, {10404, 10623}, {10543, 40669}, {12699, 58740}, {20277, 37571}, {43576, 48883}

X(68499) = reflection of X(i) in X(j) for these {i,j}: {4, 27555}, {37405, 3}
X(68499) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3464, 52390}, {42789, 42790, 11107}

X(68500) = X(2)X(3)∩X(74)X(141)

Barycentrics    2*a^10 - 3*a^8*b^2 - 2*a^6*b^4 + 4*a^4*b^6 - b^10 - 3*a^8*c^2 + 22*a^6*b^2*c^2 - 20*a^4*b^4*c^2 - 2*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 20*a^4*b^2*c^4 + 4*a^2*b^4*c^4 - 2*b^6*c^4 + 4*a^4*c^6 - 2*a^2*b^2*c^6 - 2*b^4*c^6 + 3*b^2*c^8 - c^10 : :
X(68500) = 2 X[2] + X[44458], 8 X[3] + X[6240], 10 X[3] - X[12225], 2 X[3] + X[38323], 5 X[3] - 2 X[67336], 2 X[20] + X[34613], X[20] + 2 X[67237], 8 X[140] + X[52071], 2 X[376] + X[7576], X[376] + 2 X[66614], X[382] - 4 X[23410], 2 X[428] + X[11001], 2 X[548] + X[38322], 4 X[549] - X[52069], 2 X[550] + X[7540], 10 X[631] - X[18560], 5 X[631] + 4 X[31829], and many others

X(68500) lies on these lines:{2, 3}, {74, 141}, {524, 5890}, {541, 5891}, {542, 61667}, {597, 15033}, {599, 10605}, {1986, 37511}, {2777, 5650}, {3564, 61136}, {3580, 37470}, {4549, 21766}, {4550, 50434}, {4846, 15066}, {5092, 16163}, {5480, 43576}, {5622, 51737}, {5651, 32111}, {7811, 44133}, {9730, 54040}, {10564, 14389}, {10606, 21358}, {11179, 41614}, {11592, 34798}, {12022, 16836}, {12383, 48906}, {13394, 15035}, {13857, 18388}, {15053, 15360}, {15111, 62508}, {15131, 67868}, {16261, 35283}, {18481, 34668}, {19924, 36987}, {20126, 67926}, {20791, 44665}, {31730, 34657}, {32833, 62338}, {35254, 41462}, {37283, 44882}, {37784, 50979}, {46818, 64098}, {50974, 53021}, {50977, 62382}, {50983, 62375}, {54169, 54347}, {54774, 60130}, {56567, 67308}, {61134, 63631}, {61619, 62380}, {61737, 67894}, {66606, 68018}

X(68500) = midpoint of X(i) and X(j) for these {i,j}: {20, 62963}, {62120, 67323}
X(68500) = reflection of X(i) in X(j) for these {i,j}: {4, 67238}, {16261, 35283}, {34613, 62963}, {62963, 67237}, {67266, 5055}, {67338, 43934}
X(68500) = complement of X(37077)
X(68500) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 376, 378}, {2, 2071, 44218}, {2, 3524, 49672}, {2, 3543, 18537}, {2, 6623, 5071}, {2, 44440, 381}, {2, 61113, 376}, {2, 62947, 547}, {3, 50008, 858}, {5, 7464, 35484}, {20, 67237, 34613}, {376, 378, 54995}, {376, 35921, 44285}, {376, 44831, 3534}, {376, 44832, 8703}, {376, 66614, 7576}, {547, 47332, 62947}, {549, 15760, 2}, {549, 44241, 44285}, {549, 44285, 35921}, {631, 31829, 18560}, {6803, 35502, 34939}, {8703, 38321, 52397}, {8703, 44218, 2071}, {8703, 44239, 376}, {15078, 47596, 66584}, {15705, 67338, 43934}, {18559, 19708, 7667}, {37347, 44287, 53843}, {38708, 38709, 37970}, {44210, 44273, 186}, {44214, 44262, 68085}, {44239, 66614, 38321}, {44241, 44285, 376}, {47031, 66718, 376}, {61113, 66614, 44458}

X(68501) = X(2)X(3)∩X(74)X(76)

Barycentrics    a^12 - 4*a^8*b^4 + 2*a^6*b^6 + 3*a^4*b^8 - 2*a^2*b^10 + 4*a^8*b^2*c^2 + 3*a^6*b^4*c^2 - 7*a^4*b^6*c^2 + a^2*b^8*c^2 - b^10*c^2 - 4*a^8*c^4 + 3*a^6*b^2*c^4 - 8*a^4*b^4*c^4 + a^2*b^6*c^4 + 4*b^8*c^4 + 2*a^6*c^6 - 7*a^4*b^2*c^6 + a^2*b^4*c^6 - 6*b^6*c^6 + 3*a^4*c^8 + a^2*b^2*c^8 + 4*b^4*c^8 - 2*a^2*c^10 - b^2*c^10 : :

X(68501) lies on these lines: {2, 3}, {74, 76}, {477, 38526}, {2986, 15080}, {3098, 3260}, {5890, 7760}, {7878, 15033}, {10605, 63933}, {15107, 34289}, {43453, 67356}

X(68502) = X(2)X(3)∩X(32)X(74)

Barycentrics    a^2*(2*a^10 - 4*a^8*b^2 + 2*a^6*b^4 - 2*a^4*b^6 + 4*a^2*b^8 - 2*b^10 - 4*a^8*c^2 + 9*a^6*b^2*c^2 + 3*a^4*b^4*c^2 - 5*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 + 3*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + 5*b^6*c^4 - 2*a^4*c^6 - 5*a^2*b^2*c^6 + 5*b^4*c^6 + 4*a^2*c^8 - 3*b^2*c^8 - 2*c^10) : :

X(68502) lies on these lines: {2, 3}, {6, 14634}, {32, 74}, {2394, 42660}, {3398, 61136}, {5007, 5890}, {7772, 15033}, {9463, 38653}, {10606, 22331}, {11455, 42671}, {11464, 44437}, {14919, 44080}, {30270, 43576}, {34396, 67925}, {41335, 51739}

X(68502) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 20897, 186}, {3, 31861, 37335}, {35469, 35470, 376}, {42789, 42790, 11331}

X(68503) = X(2)X(3)∩X(8)X(514)

Barycentrics    2*a^6 - 3*a^5*b + 2*a^4*b^2 + 2*a^3*b^3 - 2*a^2*b^4 + a*b^5 - 2*b^6 - 3*a^5*c - 2*a^4*b*c + a^3*b^2*c + a^2*b^3*c + 2*a*b^4*c + b^5*c + 2*a^4*c^2 + a^3*b*c^2 - 2*a^2*b^2*c^2 - 3*a*b^3*c^2 + 2*b^4*c^2 + 2*a^3*c^3 + a^2*b*c^3 - 3*a*b^2*c^3 - 2*b^3*c^3 - 2*a^2*c^4 + 2*a*b*c^4 + 2*b^2*c^4 + a*c^5 + b*c^5 - 2*c^6 : :
X(68503) = 3 X[2] - 4 X[37165]

X(68503) lies on these lines: {2, 3}, {8, 514}, {3007, 64932}, {3938, 4347}, {5691, 67574}, {7270, 42720}, {9318, 9579}, {17780, 21075}, {24403, 50065}, {36205, 63851}, {38514, 62494}, {53337, 64002}

X(68503) = reflection of X(37009) in X(37165)
X(68503) = anticomplement of X(37009)
X(68503) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 37256, 16380}, {37009, 37165, 2}

X(68504) = EULER LINE INTERCEPT OF X(148)X(7811)

Barycentrics    -7*a^4+5*a^2*(b^2+c^2)+5*b^4-7*b^2*c^2+5*c^4 : :

As a point on the Euler line, X(68504) has Shinagawa coefficients: {1/3 (3 (E+F)^2+7 S^2),-8 S^2}

See Juan José Isach Mayo, euclid 8449.

X(68504) lies on these lines: {2, 3}, {69, 35369}, {148, 7811}, {698, 11160}, {2896, 65633}, {5309, 14712}, {7739, 19569}, {7748, 11057}, {7756, 7809}, {7779, 44526}, {7799, 7842}, {7802, 11648}, {7818, 8591}, {7847, 14537}, {7885, 59634}, {7898, 20094}, {7929, 32836}, {8716, 41136}, {50248, 64018}, {55164, 63957}

X(68504) = midpoint of X(2) and X(19691)
X(68504) = reflection of X(i) in X(j) for these {i,j}: {2, 6655}, {384, 66349}, {6658, 2}, {19686, 7924}, {19687, 66335}, {19696, 66319}, {66319, 8357}, {66328, 6656}
X(68504) = anticomplement of X(19686)
X(68504) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3839, 33024}, {2, 6658, 66320}, {2, 19693, 66317}, {2, 33209, 62120}, {2, 50687, 33018}, {2, 61954, 33010}, {2, 61992, 32995}, {2, 62032, 14068}, {2, 62120, 33014}, {2, 62148, 33244}, {2, 66320, 19692}, {2, 66337, 66348}, {20, 33278, 2}, {384, 7924, 66326}, {384, 66337, 2}, {384, 66341, 19689}, {384, 66349, 66337}, {3524, 32966, 2}, {3543, 33263, 2}, {3543, 33272, 33263}, {3545, 33004, 2}, {3552, 33251, 2}, {3830, 33234, 66414}, {3830, 66414, 16044}, {3839, 32965, 2}, {5077, 40246, 2}, {5077, 66396, 11361}, {6655, 6658, 19690}, {6655, 19686, 7924}, {6655, 19689, 8357}, {6655, 19691, 6658}, {6655, 33256, 19691}, {6655, 66337, 66349}, {6656, 66317, 2}, {6656, 66321, 66323}, {6656, 66328, 66317}, {6658, 19690, 19692}, {6658, 66348, 384}, {6661, 7924, 66336}, {6661, 66336, 2}, {7739, 19569, 20088}, {7748, 11057, 19570}, {7841, 15681, 33246}, {7841, 33265, 2}, {7841, 66406, 33265}, {7898, 43619, 20094}, {7924, 19686, 2}, {7924, 66323, 6656}, {7924, 66326, 66337}, {7924, 66328, 66323}, {7933, 33255, 2}, {8357, 19696, 19689}, {8357, 66319, 66324}, {10304, 14063, 2}, {11001, 33238, 33251}, {11001, 33251, 3552}, {11287, 66397, 66405}, {11361, 66396, 40246}, {15681, 33246, 33265}, {15685, 33220, 33257}, {15705, 32980, 2}, {15708, 32963, 2}, {16044, 66414, 2}, {17565, 66099, 2}, {19686, 66317, 66321}, {19686, 66321, 19693}, {19686, 66336, 6661}, {19687, 66335, 66322}, {19687, 66344, 384}, {19689, 66324, 2}, {19690, 66320, 2}, {19695, 33256, 6655}, {19696, 66324, 66319}, {19696, 66340, 19686}, {32974, 33187, 2}, {32974, 62160, 33187}, {32982, 33209, 33014}, {32982, 62120, 2}, {32986, 66390, 66419}, {32986, 66419, 2}, {32997, 33019, 33260}, {33011, 33019, 54097}, {33012, 61924, 2}, {33017, 33264, 2}, {33019, 33022, 32996}, {33019, 33260, 32993}, {33020, 66417, 2}, {33021, 66413, 2}, {33023, 50687, 2}, {33188, 61906, 2}, {33192, 33263, 3543}, {33192, 33272, 2}, {33193, 33210, 2}, {33219, 33225, 2}, {33223, 62161, 33007}, {33229, 33267, 33259}, {33238, 33271, 3552}, {33246, 66406, 15681}, {33247, 33279, 33004}, {33251, 33271, 11001}, {33270, 61844, 2}, {33699, 66416, 14042}, {52250, 61812, 2}, {66317, 66328, 19693}, {66319, 66324, 19689}, {66321, 66323, 66317}, {66321, 66328, 19686}, {66322, 66335, 66345}, {66322, 66345, 2}, {66323, 66328, 66321}, {66325, 66349, 66347}, {66326, 66340, 66341}, {66326, 66349, 7924}, {66392, 66424, 13586}

X(68505) = X(1)X(3)∩X(42)X(5720)

Barycentrics    a*(a^6 - a^4*(b - c)^2 + 2*a^5*(b + c) + (b - c)^2*(b + c)^4 - 4*a^3*(b^3 + 2*b^2*c + 2*b*c^2 + c^3) + 2*a*(b - c)^2*(b^3 + 5*b^2*c + 5*b*c^2 + c^3) - a^2*(b^4 + 4*b^3*c - 2*b^2*c^2 + 4*b*c^3 + c^4)) : :

See Tran Viet Hung and David Nguyen, euclid 8454.

X(68505) lies on these lines: {1, 3}, {9, 44414}, {42, 5720}, {223, 39542}, {500, 12565}, {515, 18506}, {1056, 7190}, {1100, 64449}, {1103, 11374}, {1386, 39877}, {1490, 37698}, {1519, 63008}, {1743, 39523}, {2324, 9708}, {2334, 14872}, {2999, 5886}, {3553, 4270}, {3755, 6826}, {4512, 5398}, {5256, 5603}, {5287, 5657}, {5396, 63992}, {6261, 59301}, {6765, 56317}, {11230, 23511}, {12705, 36742}, {17018, 18446}, {17019, 59417}, {17022, 26446}, {19767, 63986}, {19860, 57007}, {31435, 36754}, {39898, 56328}, {45955, 46475}, {55104, 62831}

X(68505) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 30503, 50317}, {1, 37529, 37531}, {1, 63982, 37611}, {3, 1482, 31779}

X(68506) = X(2)X(3)∩X(523)X(7751)

Barycentrics    2*a^10 - 2*a^8*b^2 + a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 - 2*a^8*c^2 - 4*a^6*b^2*c^2 + 2*a^4*b^4*c^2 + 7*a^2*b^6*c^2 - b^8*c^2 + a^6*c^4 + 2*a^4*b^2*c^4 - 12*a^2*b^4*c^4 + b^6*c^4 + 2*a^4*c^6 + 7*a^2*b^2*c^6 + b^4*c^6 - 3*a^2*c^8 - b^2*c^8 : :
X(68506) = 7 X[3526] - 9 X[68476]

X(68506) lies on these lines: {2, 3}, {523, 7751}, {3788, 47171}, {3933, 16316}, {5319, 50149}, {6179, 38526}, {7746, 67616}, {7793, 47291}, {7796, 53136}, {7801, 46992}, {7829, 50147}, {7888, 46986}, {20081, 47289}, {46993, 54393}, {51535, 53726}

X(68506) = midpoint of X(3534) and X(16281)
X(68506) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1657, 57599}, {23, 36165, 36156}, {9832, 37915, 36157}

X(68507) = X(2)X(3)∩X(523)X(626)

Barycentrics    a^6*b^4 - 2*a^4*b^6 - a^2*b^8 + 2*b^10 + 2*a^6*b^2*c^2 - 2*a^4*b^4*c^2 + 5*a^2*b^6*c^2 - 3*b^8*c^2 + a^6*c^4 - 2*a^4*b^2*c^4 - 4*a^2*b^4*c^4 + b^6*c^4 - 2*a^4*c^6 + 5*a^2*b^2*c^6 + b^4*c^6 - a^2*c^8 - 3*b^2*c^8 + 2*c^10 : :
X(68507) = 3 X[2] + X[36187], 9 X[33219] - X[37915]

X(68507) lies on these lines: {2, 3}, {39, 67615}, {76, 51258}, {523, 626}, {691, 7911}, {2452, 7776}, {5099, 40517}, {5969, 6698}, {7767, 16315}, {7775, 50147}, {7780, 47238}, {7810, 46980}, {7830, 40544}, {7836, 47293}, {7853, 67616}, {7874, 47326}, {7883, 16092}, {7886, 47239}, {7909, 47288}, {7932, 60695}, {7934, 38526}, {10104, 65729}, {15535, 61543}, {23105, 62577}, {32152, 46981}, {34517, 53569}, {39604, 60590}, {45284, 65608}

X(68507) = midpoint of X(36156) and X(36187)
X(68507) = complement of X(36156)
X(68507) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 36187, 36156}, {858, 6656, 36157}, {5025, 36165, 14120}

X(68508) = X(2)X(3)∩X(7)X(99)

Barycentrics    (a + b)*(a + c)*(2*a^5 - 5*a^4*b + 4*a^2*b^3 - 2*a*b^4 + b^5 - 5*a^4*c + 2*a^3*b*c - 2*a*b^3*c + b^4*c - 2*b^3*c^2 + 4*a^2*c^3 - 2*a*b*c^3 - 2*b^2*c^3 - 2*a*c^4 + b*c^4 + c^5) : :

X(68508) lies on these lines: {2, 3}, {7, 99}, {388, 56946}, {643, 3476}, {4293, 68244}, {4295, 56833}, {4653, 24248}, {5060, 24247}, {5249, 66692}, {7288, 52360}, {17126, 66680}, {17154, 63159}, {18391, 56951}, {27415, 40127}, {28629, 40430}, {35262, 51382}, {44669, 67125}, {56177, 67123}

X(68508) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {376, 17512, 35915}, {4234, 7415, 21}

X(68509) = X(2)X(3)∩X(99)X(239)

Barycentrics    (a + b)*(a + c)*(a^5 + 3*a^4*b - a^2*b^3 - a*b^4 + 3*a^4*c + a^3*b*c - 3*a^2*b^2*c - a*b^3*c - 3*a^2*b*c^2 + 2*b^3*c^2 - a^2*c^3 - a*b*c^3 + 2*b^2*c^3 - a*c^4) : :

X(68509) lies on these lines: {2, 3}, {99, 239}, {1326, 17738}, {1580, 13174}, {2134, 17397}, {14621, 38814}, {17103, 26626}, {19791, 19849}, {20172, 56934}, {24632, 49560}, {33296, 56834}

X(68510) = X(2)X(3)∩X(11)X(99)

Barycentrics    (a + b)*(a + c)*(-(a^3*b^3) + a^2*b^4 - a*b^5 + b^6 + 2*a^4*b*c + a^3*b^2*c - 2*a^2*b^3*c + a*b^4*c + a^3*b*c^2 - 4*a^2*b^2*c^2 + 2*a*b^3*c^2 - b^4*c^2 - a^3*c^3 - 2*a^2*b*c^3 + 2*a*b^2*c^3 + a^2*c^4 + a*b*c^4 - b^2*c^4 - a*c^5 + c^6) : :

X(68510) lies on these lines: {2, 3}, {11, 99}, {314, 14947}, {691, 5520}, {3110, 24250}, {14534, 43671}, {19839, 19849}, {31059, 31126}, {36815, 50314}, {51390, 56154}

X(68511) = X(2)X(3)∩X(55)X(523)

Barycentrics    a*(a^8-(b+c)*a^7-(b^2+c^2)*a^6+(b+c)*(b^2+c^2)*a^5-(b^4-3*b^2*c^2+c^4)*a^4+(b+c)*(b^4+c^4-b*c*(b^2+b*c+c^2))*a^3+(b^4-c^4)*(b^2-c^2)*a^2-(b^2-c^2)*(b-c)*(b^4+c^4+b*c*(b^2+b*c+c^2))*a-(b^2-c^2)^2*b^2*c^2) : :

X(68511) lies on these lines: {2, 3}, {35, 47270}, {55, 523}, {1030, 46408}, {1290, 1621}, {1768, 61221}, {3295, 13869}, {3746, 47274}, {5432, 5520}, {10246, 46636}, {10777, 53927}, {10902, 67722}, {10950, 34435}, {11246, 38530}, {13204, 14985}, {17601, 61432}, {23860, 53279}, {28534, 67422}, {32613, 62491}, {47272, 64951}

X(68511) = circumcircle-inverse of X(851) X(68511) = polar-circle-inverse of X(37371)}
X(68511) = crossdifference of every pair of points on line {647, 3002}
X(68511) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1113, 1114, 851}, {1290, 1621, 68378}, {2074, 36001, 36195}, {3109, 47401, 3}

X(68512) = X(2)X(3)∩X(99)X(308)

Barycentrics    a^2*(a^6*b^2 - a^2*b^6 + a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - b^6*c^2 - 2*a^2*b^2*c^4 - 4*b^4*c^4 - a^2*c^6 - b^2*c^6) : :

X(68512) lies on these lines: {2, 3}, {39, 52580}, {99, 308}, {112, 56921}, {3455, 5149}, {3972, 8266}, {5201, 12150}, {5621, 38661}, {7804, 51862}, {7831, 53273}, {9149, 9466}, {14134, 27374}, {14965, 53026}, {15109, 44532}, {19596, 24273}, {21766, 50672}, {32833, 66886}

X(68512) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 11286, 22}, {384, 13586, 35929}, {13586, 15246, 3}

X(68513) = X(2)X(3)∩X(99)X(7894)

Barycentrics    7*a^4 - 3*a^2*b^2 - 3*a^2*c^2 + 4*b^2*c^2 : :
X(68513) = 9 X[2] - 7 X[14064], 6 X[2] - 7 X[32954], 3 X[2] - 7 X[32973], 15 X[2] - 7 X[32982], 15 X[2] - 14 X[33186], 5 X[2] - 7 X[33224], 3 X[2] + 7 X[33239], 2 X[14064] - 3 X[32954], X[14064] - 3 X[32973], 5 X[14064] - 3 X[32982], 5 X[14064] - 6 X[33186], 5 X[14064] - 9 X[33224], X[14064] + 3 X[33239], 5 X[32954] - 2 X[32982], 5 X[32954] - 4 X[33186], 5 X[32954] - 6 X[33224], X[32954] + 2 X[33239], 5 X[32973] - X[32982], 5 X[32973] - 2 X[33186], 5 X[32973] - 3 X[33224], X[32982] - 3 X[33224], X[32982] + 5 X[33239], 2 X[33186] - 3 X[33224], 2 X[33186] + 5 X[33239], 3 X[33224] + 5 X[33239]

X(68513) lies on these lines: {2, 3}, {32, 22253}, {99, 7894}, {187, 17130}, {194, 21309}, {538, 22331}, {598, 62362}, {620, 65630}, {1078, 15655}, {1384, 1975}, {2548, 32459}, {3053, 7751}, {3630, 4048}, {3734, 5023}, {3793, 32830}, {3934, 5210}, {3972, 9605}, {4366, 67261}, {5007, 8716}, {5013, 32456}, {5017, 6144}, {5024, 7782}, {5182, 11482}, {6337, 18907}, {6680, 44526}, {6781, 7784}, {7618, 9606}, {7737, 59545}, {7760, 51122}, {7801, 63938}, {7804, 15815}, {7808, 53095}, {7829, 34504}, {7834, 44519}, {7863, 63932}, {7881, 14712}, {9655, 26629}, {9668, 26686}, {10352, 18501}, {10983, 33813}, {12215, 62996}, {15271, 15513}, {15533, 63930}, {22486, 53092}, {31457, 42849}, {31470, 63101}, {31859, 43136}, {32516, 55705}, {32826, 43291}, {32833, 63936}, {35007, 63933}, {39141, 44456}, {53107, 56064}

X(68513) = midpoint of X(32973) and X(33239)
X(68513) = reflection of X(i) in X(j) for these {i,j}: {32954, 32973}, {32982, 33186}
X(68513) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 33007, 66425}, {2, 33247, 66347}, {2, 33250, 1657}, {5, 32981, 11159}, {20, 7866, 5077}, {20, 8369, 7866}, {20, 14069, 8357}, {376, 33201, 7819}, {382, 7807, 11318}, {384, 3552, 33235}, {384, 13586, 33004}, {384, 33004, 7770}, {384, 33014, 11285}, {384, 33235, 3}, {439, 14033, 140}, {550, 8364, 33023}, {550, 14001, 11287}, {550, 66393, 14001}, {1003, 3552, 3}, {1003, 33235, 384}, {3146, 33191, 8361}, {3522, 14039, 8362}, {3528, 33198, 8359}, {3529, 33181, 33184}, {3534, 33242, 6656}, {5025, 66387, 5073}, {5059, 32951, 66392}, {6656, 33244, 3534}, {6656, 33255, 33242}, {6658, 7887, 3830}, {6658, 33246, 7887}, {7770, 13586, 3}, {7791, 8598, 15696}, {7807, 8352, 33248}, {7807, 33007, 382}, {7841, 33257, 17800}, {7892, 33265, 33234}, {8353, 33214, 62134}, {8356, 33254, 62100}, {8357, 8369, 14069}, {8357, 14069, 7866}, {8360, 62155, 33238}, {8364, 33023, 11287}, {8368, 15704, 32974}, {8370, 32964, 3526}, {8703, 19697, 16043}, {11001, 33180, 67390}, {11285, 33014, 3}, {11285, 33235, 33014}, {11317, 32967, 61953}, {11361, 33233, 3851}, {12103, 33185, 32986}, {13860, 61126, 3}, {14001, 33023, 8364}, {14001, 35927, 550}, {14034, 33259, 44543}, {14035, 35297, 1656}, {14037, 33254, 8356}, {14065, 19691, 7841}, {14065, 33257, 19691}, {14068, 16925, 33249}, {14068, 33249, 381}, {14869, 66412, 32975}, {15682, 33236, 33199}, {15686, 66347, 33247}, {15696, 33237, 7791}, {16925, 19687, 381}, {16925, 33187, 19687}, {17578, 32955, 37350}, {19687, 33249, 14068}, {19691, 33225, 14065}, {32952, 62147, 33210}, {32956, 50693, 8354}, {32961, 66408, 61984}, {32968, 35287, 3530}, {32973, 32982, 33224}, {32979, 33216, 3628}, {32981, 32985, 5}, {32982, 33224, 33186}, {32996, 66423, 62016}, {33019, 66395, 49134}, {33186, 33224, 32954}, {33193, 33229, 49136}, {33197, 49138, 33200}, {33218, 66395, 33019}, {33225, 33257, 7841}, {33227, 66415, 3523}, {33228, 33280, 5076}, {33234, 33265, 62131}, {33240, 49136, 33229}, {33241, 49137, 33017}, {33244, 33255, 6656}, {33259, 44543, 46219}, {33266, 66319, 5054}, {35927, 66393, 11287}, {35938, 35939, 54993}

X(68514) = X(2)X(3)∩X(76)X(45017)

Barycentrics    8*a^4 - 7*a^2*b^2 - 7*a^2*c^2 + b^2*c^2 : :

X(68514) lies on these lines: {2, 3}, {76, 45017}, {187, 7894}, {194, 15513}, {2076, 51170}, {5017, 63122}, {5023, 7766}, {5116, 63123}, {5182, 55637}, {5206, 7760}, {5210, 7783}, {7618, 13571}, {7755, 32480}, {7780, 7782}, {7781, 7793}, {7787, 15515}, {7814, 14976}, {7878, 37512}, {7904, 32459}, {7946, 47101}, {15815, 62994}, {22486, 55679}, {31276, 32456}, {32522, 47113}, {38259, 44530}, {39141, 55649}, {42522, 43153}, {42523, 43154}, {43148, 63722}, {43152, 62174}, {53164, 62991}

X(68514) = anticomplement of X(33011)
X(68514) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 50693, 19691}, {2, 62102, 33209}, {3, 3552, 33022}, {3, 13586, 33004}, {3, 33014, 2}, {3, 33235, 33273}, {3, 33276, 3552}, {3, 61126, 3523}, {20, 33206, 32993}, {140, 33010, 2}, {140, 33268, 66419}, {140, 66419, 33010}, {376, 32977, 33271}, {376, 33259, 33019}, {376, 40279, 20}, {548, 7907, 33264}, {549, 33257, 33002}, {550, 32966, 66421}, {550, 33274, 32966}, {631, 33018, 2}, {631, 33265, 33018}, {632, 14066, 32994}, {3523, 6658, 2}, {3523, 33208, 6658}, {3524, 33254, 16044}, {3552, 33004, 7770}, {3552, 33022, 2}, {3552, 33276, 33014}, {7770, 13586, 3552}, {7791, 32952, 66336}, {8598, 15712, 16921}, {10303, 33024, 2}, {10303, 33193, 33024}, {10304, 32964, 33260}, {12100, 33250, 33015}, {14068, 61820, 2}, {15688, 33233, 33267}, {15717, 33244, 2}, {19692, 33258, 2}, {32964, 33260, 2}, {32964, 33263, 33203}, {32972, 33243, 68504}, {32972, 62094, 33243}, {32976, 62117, 66390}, {32989, 62083, 33207}, {32993, 33206, 2}, {32995, 61834, 2}, {33014, 33022, 3552}, {33019, 33259, 2}, {33216, 62084, 33253}, {33249, 44245, 66406}, {33274, 66421, 2}, {33275, 35297, 7933}, {35287, 62067, 32965}, {35297, 46853, 33275}, {35927, 61788, 33012}, {61811, 66387, 16922}

X(68515) = X(2)X(3)∩X(99)X(22331)

Barycentrics    13*a^4 - 7*a^2*b^2 - 7*a^2*c^2 + 6*b^2*c^2 : :
X(68515) = 15 X[2] - 13 X[33248], 21 X[2] - 13 X[33290], 7 X[33248] - 5 X[33290]

X(68515) lies on these lines: {2, 3}, {99, 22331}, {1975, 63925}, {3972, 22332}, {7754, 35007}, {7772, 32456}, {7846, 44541}, {7894, 31859}, {10353, 38635}, {11164, 63924}, {15655, 17128}, {39141, 55580}

X(68515) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1657, 33246, 33218}, {3552, 33235, 1003}, {7770, 33235, 13586}, {7887, 33250, 66395}, {8352, 33203, 7887}, {8363, 8598, 17538}, {8363, 17538, 33234}, {8598, 32973, 33234}, {17538, 32973, 8363}, {32973, 33234, 8366}, {32985, 33250, 7887}, {33186, 33271, 7841}, {62069, 66318, 33258}

X(68516) = X(2)X(3)∩X(99)X(5023)

Barycentrics    7*a^4 - 5*a^2*b^2 - 5*a^2*c^2 + 2*b^2*c^2 : :
X(68516) = 9 X[2] - 7 X[32961], 3 X[2] - 7 X[32964], 15 X[2] - 7 X[32996], 9 X[2] + 7 X[33214], 3 X[2] - 14 X[33227], 6 X[2] - 7 X[33233], 3 X[2] + 7 X[33254], X[32961] - 3 X[32964], 5 X[32961] - 3 X[32996], X[32961] - 6 X[33227], 2 X[32961] - 3 X[33233], X[32961] + 3 X[33254], 5 X[32964] - X[32996], 3 X[32964] + X[33214], 3 X[32996] + 5 X[33214], X[32996] - 10 X[33227], 2 X[32996] - 5 X[33233], X[32996] + 5 X[33254], X[33214] + 6 X[33227], 2 X[33214] + 3 X[33233], X[33214] - 3 X[33254], 4 X[33227] - X[33233], 2 X[33227] + X[33254], X[33233] + 2 X[33254]

X(68516) lies on these lines: {2, 3}, {76, 5210}, {83, 53095}, {99, 5023}, {183, 15513}, {187, 7754}, {315, 32459}, {1384, 7783}, {1975, 5206}, {2076, 6144}, {2482, 32821}, {3053, 7760}, {3972, 15815}, {5013, 7878}, {5017, 32455}, {5182, 53097}, {5585, 43459}, {6179, 8716}, {6781, 7773}, {7755, 34504}, {7757, 22331}, {7793, 15655}, {7799, 63938}, {7816, 8588}, {7828, 44519}, {7845, 51581}, {7847, 44541}, {7855, 35022}, {7857, 44526}, {7860, 41134}, {7863, 47101}, {7881, 14907}, {10541, 22486}, {11164, 34506}, {11165, 13571}, {11174, 15515}, {12150, 22332}, {14023, 59634}, {17130, 46893}, {21843, 32819}, {30136, 63212}, {32818, 51579}, {36521, 63927}, {39141, 55610}, {39646, 47113}, {39785, 63937}, {51224, 63932}, {51580, 51585}, {52695, 63950}, {53106, 62880}, {53107, 60198}, {58448, 65633}, {60103, 60209}, {60146, 60211}

X(68516) = midpoint of X(i) and X(j) for these {i,j}: {32961, 33214}, {32964, 33254}
X(68516) = reflection of X(i) in X(j) for these {i,j}: {32964, 33227}, {33233, 32964}
X(68516) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 14066, 5072}, {2, 17538, 19695}, {2, 19696, 3843}, {2, 33268, 1657}, {2, 62110, 33247}, {3, 1003, 11285}, {3, 3552, 7770}, {3, 11286, 33004}, {3, 13586, 33235}, {3, 33235, 1003}, {5, 33244, 66387}, {20, 7887, 66388}, {20, 32970, 33229}, {20, 35297, 7887}, {376, 439, 7807}, {376, 7807, 33234}, {550, 8361, 32997}, {550, 16925, 7841}, {550, 27088, 16925}, {631, 19687, 44543}, {631, 35927, 19687}, {1656, 6658, 11317}, {2482, 63935, 32821}, {3053, 7782, 31859}, {3146, 33216, 33249}, {3522, 6656, 35955}, {3522, 32985, 6656}, {3522, 33181, 33226}, {3523, 33239, 8370}, {3524, 32981, 32992}, {3528, 32973, 8356}, {3529, 32958, 54097}, {3529, 32989, 33228}, {3530, 66391, 16924}, {3552, 7770, 1003}, {3552, 33014, 33276}, {3552, 33022, 384}, {3552, 33276, 3}, {5059, 32969, 8352}, {5206, 32456, 1975}, {6655, 11288, 33218}, {6658, 33274, 1656}, {7770, 33235, 3552}, {7807, 33234, 33219}, {7807, 66349, 32951}, {7819, 46853, 33008}, {7866, 15688, 33260}, {7907, 33265, 382}, {8356, 32973, 33217}, {8357, 62104, 33207}, {8361, 32997, 7841}, {8369, 33923, 32965}, {8703, 33266, 33220}, {9855, 32966, 5073}, {11159, 15720, 16921}, {11287, 62085, 33275}, {11288, 15696, 6655}, {11318, 62131, 33256}, {13586, 33014, 3}, {13586, 33276, 3552}, {14063, 15704, 66396}, {14064, 50693, 8353}, {14869, 66409, 32999}, {14907, 59545, 7881}, {15712, 68177, 2}, {16370, 17693, 33035}, {16923, 66419, 3851}, {16925, 32997, 8361}, {16925, 33208, 550}, {27088, 33208, 7841}, {32954, 62100, 7833}, {32958, 54097, 33228}, {32961, 33254, 33214}, {32963, 66389, 3853}, {32964, 33214, 32961}, {32970, 33229, 7887}, {32982, 62127, 66424}, {32985, 33226, 33181}, {32985, 35955, 8366}, {32989, 54097, 32958}, {33000, 33193, 546}, {33017, 33252, 12103}, {33023, 33191, 8363}, {33181, 33226, 6656}, {33184, 44245, 33253}, {33189, 62113, 33272}, {33191, 62092, 33023}, {33201, 62067, 33215}, {33203, 62120, 33238}, {33205, 62097, 32986}, {33206, 33244, 33280}, {33206, 33280, 5}, {33209, 33248, 66392}, {33225, 33275, 11287}, {33227, 33254, 33233}, {33229, 35297, 32970}, {33246, 33260, 7866}, {33257, 33259, 381}, {33270, 52942, 3858}, {33283, 67390, 7841}, {35302, 51350, 41235}, {44682, 66415, 33012}, {62123, 66392, 33209}

X(68517) = X(2)X(3)∩X(99)X(7772)

Barycentrics    4*a^4 - a^2*b^2 - a^2*c^2 + 3*b^2*c^2 : :
X(68517) = 3 X[2] - 4 X[7892], 9 X[2] - 8 X[8363], 3 X[7892] - 2 X[8363], 3 X[7933] - 4 X[8363]

X(68517) lies on these lines: {2, 3}, {32, 20081}, {76, 35007}, {83, 45017}, {99, 7772}, {187, 31276}, {193, 4048}, {194, 3972}, {316, 7945}, {385, 22331}, {543, 7856}, {576, 39141}, {1384, 17129}, {1975, 7766}, {2076, 3620}, {2549, 10583}, {3051, 3360}, {3053, 17128}, {3096, 6781}, {3231, 53164}, {3303, 6645}, {3304, 4366}, {3329, 22332}, {3734, 7793}, {3849, 7922}, {3926, 20088}, {4027, 23235}, {5017, 20080}, {5152, 32826}, {5182, 22234}, {5286, 20094}, {5332, 32005}, {5395, 35005}, {5921, 35424}, {5989, 35369}, {6179, 63925}, {6337, 63018}, {6390, 7921}, {7296, 32107}, {7735, 44539}, {7737, 7836}, {7738, 63020}, {7745, 7891}, {7747, 7835}, {7748, 7932}, {7756, 7846}, {7757, 41940}, {7758, 34604}, {7777, 59545}, {7781, 12150}, {7782, 7804}, {7783, 62994}, {7786, 32456}, {7789, 7823}, {7794, 9939}, {7795, 7929}, {7801, 7946}, {7802, 7820}, {7806, 32819}, {7812, 7863}, {7814, 14537}, {7832, 7898}, {7837, 32820}, {7842, 7930}, {7843, 7870}, {7849, 11057}, {7854, 51224}, {7860, 7880}, {7873, 14976}, {7875, 63548}, {7899, 62203}, {7906, 18907}, {7909, 63931}, {7910, 7915}, {7919, 65633}, {7923, 44526}, {7925, 65630}, {7940, 39590}, {8290, 37665}, {10000, 10332}, {10334, 12110}, {10351, 32134}, {10358, 21166}, {11152, 38628}, {11185, 50570}, {12215, 51170}, {14907, 46226}, {16984, 44518}, {22330, 22486}, {32830, 50248}, {34885, 64018}, {35375, 40330}, {37512, 60855}, {40344, 55738}, {59546, 63028}

X(68517) = reflection of X(7933) in X(7892)
X(68517) = anticomplement of X(7933)
X(68517) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3552, 33014}, {2, 6658, 33019}, {2, 14035, 33018}, {2, 14068, 32993}, {2, 16044, 33010}, {2, 17578, 33290}, {2, 19691, 32974}, {2, 19693, 14035}, {2, 32979, 33024}, {2, 32981, 6658}, {2, 32997, 19690}, {2, 33007, 66421}, {2, 33014, 33022}, {2, 33018, 33011}, {2, 33025, 66348}, {2, 33209, 33025}, {2, 33214, 33023}, {2, 33244, 33260}, {2, 54097, 33287}, {4, 33224, 33248}, {4, 33225, 2}, {4, 33255, 33225}, {5, 66319, 14034}, {20, 14037, 2}, {21, 16913, 2}, {376, 16898, 33021}, {376, 66317, 2}, {382, 33220, 7901}, {384, 1003, 3552}, {384, 3552, 2}, {384, 7824, 11286}, {384, 13586, 7770}, {384, 35925, 14037}, {384, 44224, 66317}, {550, 6661, 7876}, {631, 33020, 2}, {1657, 33217, 7924}, {1975, 7766, 20105}, {3523, 33269, 2}, {3552, 33004, 13586}, {3830, 33218, 14045}, {3972, 7816, 194}, {5025, 19687, 66419}, {5056, 33262, 2}, {6655, 14001, 2}, {6656, 33257, 33264}, {6656, 66391, 33257}, {6658, 33019, 66420}, {7737, 7836, 7900}, {7745, 7891, 63021}, {7747, 7835, 7912}, {7770, 13586, 33004}, {7770, 33004, 2}, {7789, 7823, 7897}, {7791, 14039, 19689}, {7791, 19689, 2}, {7791, 33239, 33265}, {7795, 14712, 7929}, {7802, 7820, 7938}, {7807, 11361, 32966}, {7807, 32966, 2}, {7807, 68177, 11361}, {7819, 33250, 7833}, {7819, 35954, 14038}, {7833, 14038, 7819}, {7841, 33242, 14043}, {7866, 66387, 33256}, {7876, 14040, 6661}, {7887, 11159, 14042}, {7892, 7933, 2}, {7901, 66328, 382}, {7907, 8370, 33002}, {7907, 14032, 8370}, {7907, 33002, 2}, {7948, 9855, 33234}, {8352, 33186, 5025}, {8356, 19697, 16895}, {8361, 66408, 14062}, {8362, 8598, 33275}, {8368, 33229, 14065}, {8369, 19687, 5025}, {8369, 66419, 2}, {10303, 32971, 33261}, {10303, 33261, 2}, {11286, 33235, 7824}, {14001, 33007, 6655}, {14033, 16925, 16044}, {14034, 33246, 5}, {14035, 32973, 2}, {14036, 33257, 6656}, {14036, 33264, 2}, {14036, 66391, 33264}, {14037, 33187, 20}, {14039, 33239, 7791}, {14039, 33265, 2}, {14043, 19696, 7841}, {14063, 33181, 2}, {14065, 66405, 33229}, {16044, 16925, 2}, {16895, 33268, 8356}, {16898, 33021, 2}, {16909, 33831, 2}, {16914, 16915, 2}, {16916, 33062, 2}, {16919, 17692, 2}, {16920, 17693, 2}, {16924, 32985, 33259}, {16924, 33259, 2}, {16925, 33005, 32977}, {16953, 35929, 2}, {17686, 33063, 2}, {19686, 33225, 4}, {19686, 33255, 2}, {19689, 33265, 7791}, {19692, 33260, 2}, {19693, 32973, 33018}, {19694, 33267, 11287}, {32962, 32989, 2}, {32963, 33203, 2}, {32964, 32971, 2}, {32964, 33261, 10303}, {32965, 33198, 2}, {32974, 33193, 19691}, {32979, 33205, 2}, {32981, 33201, 2}, {32987, 33206, 2}, {33021, 66317, 16898}, {33190, 33271, 6655}, {33198, 35927, 32965}, {33234, 33237, 7948}, {33250, 35954, 7819}, {33266, 33269, 3523}, {66393, 68177, 7807}

X(68518) = X(2)X(3)∩X(99)X(20105)

Barycentrics    8*a^4 - 5*a^2*b^2 - 5*a^2*c^2 + 3*b^2*c^2 : :

X(68518) lies on these lines: {2, 3}, {99, 20105}, {187, 20081}, {194, 32456}, {2076, 20080}, {3972, 31652}, {5007, 7782}, {5017, 63061}, {5152, 35369}, {5182, 55721}, {5210, 17128}, {5921, 35375}, {6645, 64950}, {6781, 7912}, {7758, 52695}, {7766, 22331}, {7787, 53096}, {7793, 17131}, {7814, 19569}, {7821, 14976}, {7823, 32459}, {7856, 32480}, {7863, 9939}, {7877, 35022}, {7897, 59545}, {7946, 51224}, {9463, 53164}, {10353, 21166}, {15513, 31276}, {16984, 44519}, {22332, 62994}, {22486, 55704}, {39141, 52987}, {41134, 63931}

X(68518) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 33214, 19691}, {3, 35951, 3091}, {20, 33283, 68504}, {194, 32456, 45017}, {384, 33022, 2}, {439, 33244, 2}, {550, 33246, 7933}, {1003, 33276, 33004}, {3528, 33255, 33021}, {3552, 13586, 33014}, {3552, 33004, 1003}, {3552, 33014, 2}, {6658, 32964, 2}, {7807, 33268, 33264}, {7907, 33011, 2}, {7907, 33250, 66419}, {7907, 66419, 33011}, {8363, 44245, 7833}, {13586, 33235, 3552}, {14031, 15717, 2}, {15692, 66320, 2}, {16925, 33019, 2}, {16925, 33265, 33019}, {16925, 66390, 32955}, {19687, 33227, 33274}, {19687, 33274, 33002}, {19690, 33181, 2}, {19692, 32990, 2}, {27088, 33250, 7907}, {32964, 35927, 6658}, {32966, 33257, 66420}, {32973, 33208, 33260}, {32973, 33260, 2}, {32985, 33254, 6655}, {32989, 32993, 2}, {32989, 33193, 32993}, {32997, 33205, 2}, {33007, 33259, 33018}, {33018, 33259, 2}, {33024, 33206, 2}, {33181, 33207, 19690}, {33218, 62131, 67270}, {33257, 35297, 32966}, {35297, 66420, 2}

X(68519) = X(2)X(3)∩X(99)X(5210)

Barycentrics    9*a^4 - 7*a^2*b^2 - 7*a^2*c^2 + 2*b^2*c^2 : :
X(68519) = 5 X[2] - 3 X[33006], X[2] + 3 X[33208], X[33006] + 5 X[33208]

X(68519) lies on these lines: {2, 3}, {99, 5210}, {183, 8588}, {187, 7798}, {325, 47102}, {385, 15655}, {538, 5206}, {671, 44532}, {1350, 5182}, {1384, 63038}, {1975, 14711}, {2076, 15534}, {2482, 7788}, {3053, 7757}, {3972, 53095}, {4048, 50991}, {5012, 50672}, {5013, 12150}, {5017, 8584}, {5023, 7754}, {5085, 22486}, {5116, 51185}, {5569, 11164}, {5585, 7771}, {5858, 8594}, {5859, 8595}, {5989, 15300}, {6781, 63956}, {7618, 41624}, {7622, 14537}, {7749, 63957}, {7763, 63941}, {7790, 44541}, {7793, 63954}, {7813, 63948}, {7837, 11165}, {7857, 44519}, {7879, 59545}, {7903, 63947}, {8182, 37671}, {8290, 51589}, {8589, 11174}, {8593, 15533}, {8719, 34473}, {8860, 18546}, {9734, 39656}, {9755, 38225}, {9766, 51224}, {11057, 41134}, {11147, 66699}, {11149, 48913}, {11151, 12156}, {12215, 50992}, {13188, 58765}, {13637, 49260}, {13757, 49263}, {14907, 32459}, {15515, 44562}, {17129, 45017}, {18362, 32479}, {21166, 33706}, {26613, 66616}, {33685, 51123}, {37647, 43618}, {37667, 47287}, {39141, 55629}, {47113, 63424}, {50659, 63124}, {50990, 60702}, {51580, 51584}, {53142, 63034}

X(68519) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 376, 8353}, {2, 3534, 66388}, {2, 8353, 7841}, {2, 8598, 66387}, {2, 8703, 35955}, {2, 9855, 3830}, {2, 11001, 8352}, {2, 33007, 66409}, {2, 33008, 8358}, {2, 33265, 66405}, {2, 52942, 5066}, {2, 62059, 47061}, {2, 66387, 11317}, {2, 66389, 3845}, {2, 66390, 37350}, {2, 66405, 381}, {2, 66409, 44543}, {2, 66411, 11287}, {3, 3552, 11285}, {3, 11286, 33273}, {3, 13586, 1003}, {3, 33235, 7770}, {20, 33216, 33228}, {20, 33228, 66396}, {376, 35287, 35297}, {376, 35297, 7841}, {381, 33265, 66395}, {439, 3528, 6656}, {548, 16925, 33234}, {548, 33184, 33207}, {548, 33227, 16925}, {549, 33007, 44543}, {549, 66409, 2}, {550, 32964, 7887}, {1003, 11285, 11286}, {1003, 13586, 33235}, {2482, 47101, 7788}, {3363, 11812, 2}, {3524, 35927, 8370}, {3552, 11286, 1003}, {3552, 33273, 11286}, {5023, 7782, 7754}, {7833, 11288, 33219}, {7866, 62082, 33275}, {8353, 35297, 2}, {8356, 32985, 33220}, {8358, 8369, 2}, {8361, 62104, 33253}, {8369, 34200, 33008}, {8588, 32456, 183}, {8703, 27088, 2}, {10304, 32985, 8356}, {10999, 11000, 54097}, {11159, 15693, 2}, {11286, 33273, 11285}, {11287, 33246, 8366}, {11288, 15688, 7833}, {11318, 15689, 33264}, {12100, 66391, 2}, {13586, 33273, 3552}, {14041, 15681, 66397}, {15717, 33239, 32992}, {16041, 62120, 66424}, {16925, 33207, 33184}, {16925, 33234, 33218}, {17538, 32989, 33229}, {17693, 19535, 33036}, {19710, 37350, 66390}, {32954, 62085, 33260}, {32961, 33252, 15704}, {32970, 50693, 19695}, {33000, 33214, 3627}, {33008, 33266, 8369}, {33014, 33276, 3}, {33184, 33207, 33234}, {33203, 62102, 33247}, {33205, 33226, 8363}, {33205, 62083, 33226}, {33216, 33228, 33233}, {33233, 66396, 33228}, {33243, 33248, 67390}, {33246, 66411, 2}, {33259, 33268, 382}, {33265, 33274, 381}, {33274, 66405, 2}, {44682, 68177, 33001}, {62106, 67390, 33243}

X(68520) = X(2)X(3)∩X(32)X(20105)

Barycentrics    8*a^4 - 3*a^2*b^2 - 3*a^2*c^2 + 5*b^2*c^2 : :
X(68520) = 9 X[2] - 8 X[14065]

X(68520) lies on these lines: {2, 3}, {32, 20105}, {39, 45017}, {194, 5008}, {3360, 9463}, {3972, 5041}, {4048, 20080}, {5319, 8591}, {6179, 7816}, {6781, 7938}, {7786, 15602}, {7793, 17130}, {7821, 19569}, {7829, 32480}, {7922, 14976}, {12215, 63061}, {37517, 39141}, {37689, 44539}, {44532, 63005}, {59545, 63021}

X(68520) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {384, 3552, 33014}, {384, 13586, 11285}, {384, 33014, 2}, {384, 33235, 33004}, {439, 14031, 2}, {548, 35954, 16895}, {3552, 33004, 33235}, {6658, 32973, 2}, {7892, 33250, 33264}, {7907, 33010, 2}, {7933, 33257, 66421}, {8369, 19695, 14067}, {8369, 33257, 7933}, {8369, 66421, 2}, {14001, 33253, 66345}, {14034, 35297, 33002}, {14035, 32989, 33024}, {14037, 33252, 33202}, {14037, 33260, 2}, {14037, 35927, 33260}, {14039, 33254, 33021}, {14064, 33007, 68420}, {14064, 68420, 33019}, {14067, 19695, 7933}, {14067, 33257, 19695}, {14068, 33205, 2}, {16925, 19686, 33018}, {16925, 33018, 2}, {19687, 33246, 32966}, {19692, 32965, 2}, {32973, 33187, 6658}, {32981, 33205, 14068}, {32989, 33024, 2}, {33004, 33235, 33014}, {33007, 33197, 40246}, {33007, 33225, 33019}, {33019, 33225, 2}, {33201, 33244, 2}, {33202, 33252, 33260}, {33202, 35927, 33252}, {33225, 68420, 14064}, {33239, 33255, 6655}, {33250, 66393, 7892}, {33265, 66345, 33253}, {33266, 66320, 2}

X(68521) = X(2)X(3)∩X(99)X(41748)

Barycentrics    12*a^4 - 7*a^2*b^2 - 7*a^2*c^2 + 5*b^2*c^2 : :
X(68521) = 2 X[3] - 5 X[61126]

X(68521) lies on these lines: {2, 3}, {32, 45017}, {99, 41748}, {2076, 11160}, {3053, 20105}, {5008, 7757}, {5017, 63027}, {5041, 7782}, {5152, 8596}, {5182, 37517}, {7766, 8716}, {7836, 47102}, {7870, 14976}, {7871, 63947}, {7891, 63941}, {11055, 35007}, {11180, 35375}, {15602, 44562}, {21166, 44434}, {22486, 50664}, {31276, 46893}, {32459, 63021}, {35369, 37689}, {39141, 55587}

X(68521) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 13586, 33014}, {2, 33016, 33010}, {2, 33265, 66421}, {3552, 13586, 2}, {6655, 33191, 2}, {8598, 33246, 33264}, {11286, 33004, 2}, {32973, 33207, 2}, {32985, 33265, 2}, {32986, 33225, 2}, {33016, 33259, 2}, {33187, 35287, 2}, {33215, 66317, 2}, {33228, 33257, 66407}, {33246, 33264, 2}, {33266, 35927, 2}, {35297, 66419, 2}

X(68522) = X(2)X(3)∩X(99)X(6683)

Barycentrics    a^4 + 2*a^2*b^2 + 2*a^2*c^2 + 3*b^2*c^2 : :

X(68522) lies on these lines: {2, 3}, {6, 17129}, {32, 60855}, {39, 17128}, {69, 7921}, {76, 3329}, {83, 385}, {99, 6683}, {115, 6704}, {141, 7785}, {183, 7787}, {194, 11174}, {230, 10583}, {315, 16986}, {316, 6292}, {325, 46226}, {538, 55085}, {598, 7936}, {599, 7946}, {625, 7944}, {894, 17048}, {1078, 7804}, {1207, 57016}, {1506, 7832}, {1975, 22332}, {2076, 34573}, {2548, 3314}, {2896, 7745}, {3096, 5475}, {3499, 20965}, {3589, 7797}, {3618, 7920}, {3619, 5017}, {3734, 7783}, {3746, 4366}, {3763, 7773}, {3767, 7875}, {3788, 17005}, {3815, 7836}, {3933, 63018}, {3972, 7815}, {4027, 51523}, {4048, 7851}, {5116, 51126}, {5149, 7852}, {5152, 6722}, {5182, 55704}, {5201, 18092}, {5305, 63020}, {5563, 6645}, {6651, 25073}, {6694, 22510}, {6695, 22511}, {7603, 7899}, {7697, 10334}, {7736, 7906}, {7737, 7904}, {7746, 7846}, {7747, 7831}, {7751, 7878}, {7752, 7822}, {7753, 7768}, {7754, 62994}, {7757, 17130}, {7758, 63028}, {7760, 9466}, {7762, 63044}, {7766, 14535}, {7767, 20088}, {7769, 7820}, {7775, 7922}, {7777, 7795}, {7780, 12150}, {7782, 15482}, {7793, 15271}, {7794, 7840}, {7799, 9698}, {7800, 7823}, {7806, 32832}, {7809, 7849}, {7812, 7854}, {7814, 7869}, {7825, 7937}, {7827, 63924}, {7828, 7889}, {7829, 14568}, {7834, 16987}, {7835, 31455}, {7843, 7883}, {7844, 7943}, {7845, 32027}, {7857, 17006}, {7860, 7865}, {7861, 15031}, {7862, 7930}, {7864, 11185}, {7868, 7912}, {7873, 31168}, {7879, 7900}, {7881, 63021}, {7884, 43527}, {7891, 31401}, {7893, 16990}, {7894, 17131}, {7896, 7926}, {7898, 65630}, {7909, 14762}, {7910, 62203}, {7911, 31268}, {7914, 7934}, {7919, 39565}, {7932, 13881}, {7942, 10000}, {8177, 44000}, {9300, 13571}, {9478, 32528}, {9605, 20081}, {9606, 32820}, {10351, 10359}, {10352, 38664}, {10353, 12188}, {10358, 22712}, {12054, 61102}, {12110, 15819}, {14603, 59249}, {15018, 43843}, {16989, 32828}, {19570, 51860}, {20142, 29433}, {20486, 32942}, {22486, 55721}, {24256, 44771}, {30998, 54416}, {31407, 32825}, {32834, 51171}, {32893, 60647}, {34604, 63928}, {34873, 53419}, {36811, 52034}, {38655, 66063}, {38657, 66045}, {38907, 53127}, {39141, 53093}, {39668, 39998}, {40043, 40425}, {40332, 42534}, {46311, 64711}, {50659, 63119}, {53033, 63083}

X(68522) = complement of X(33021)
X(68522) = orthocentroidal-circle-inverse of X(7876)
X(68522) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 7876}, {2, 5, 7901}, {2, 381, 66324}, {2, 384, 7824}, {2, 3552, 11285}, {2, 5025, 7948}, {2, 5056, 33248}, {2, 6655, 8362}, {2, 6656, 16897}, {2, 7770, 384}, {2, 7807, 16923}, {2, 7819, 14043}, {2, 7887, 14047}, {2, 7892, 33245}, {2, 11286, 33273}, {2, 14001, 7907}, {2, 14031, 33258}, {2, 14035, 16043}, {2, 14037, 631}, {2, 14063, 32956}, {2, 16044, 6656}, {2, 16045, 16895}, {2, 16895, 19694}, {2, 16898, 7892}, {2, 16913, 474}, {2, 16916, 33047}, {2, 16918, 16912}, {2, 16920, 17684}, {2, 16921, 32967}, {2, 16924, 5025}, {2, 16925, 33015}, {2, 17541, 16918}, {2, 17686, 16917}, {2, 19686, 66417}, {2, 19689, 7807}, {2, 19692, 33259}, {2, 32961, 14065}, {2, 32962, 14064}, {2, 32963, 32951}, {2, 32964, 32978}, {2, 32965, 32960}, {2, 32966, 7866}, {2, 32968, 16921}, {2, 32971, 7791}, {2, 32973, 33001}, {2, 32979, 33202}, {2, 32987, 32961}, {2, 32989, 33003}, {2, 32991, 33180}, {2, 32992, 16922}, {2, 32993, 66345}, {2, 32994, 33240}, {2, 33002, 7887}, {2, 33009, 32969}, {2, 33020, 5}, {2, 33030, 17670}, {2, 33181, 33000}, {2, 33198, 16925}, {2, 33225, 140}, {2, 33261, 3090}, {2, 33262, 3533}, {2, 33265, 66418}, {2, 33269, 4}, {2, 33270, 32955}, {2, 33283, 33221}, {2, 44543, 14046}, {2, 52250, 33182}, {2, 66317, 549}, {2, 66413, 7924}, {2, 66415, 14041}, {4, 7876, 7924}, {4, 33269, 66413}, {5, 66416, 33020}, {6, 31276, 17129}, {20, 14034, 66328}, {76, 3329, 7839}, {76, 7808, 3329}, {83, 3934, 385}, {115, 6704, 7859}, {115, 7859, 7923}, {140, 6661, 33225}, {141, 7785, 7939}, {316, 6292, 7928}, {381, 7933, 14045}, {384, 7824, 13586}, {384, 7901, 5999}, {384, 33273, 3552}, {384, 33276, 1003}, {384, 66324, 7470}, {547, 66325, 2}, {598, 7936, 63931}, {631, 14037, 33246}, {1003, 33004, 33276}, {1506, 7832, 7925}, {1656, 33217, 2}, {2548, 3314, 7941}, {3090, 32968, 33261}, {3090, 33261, 16921}, {3096, 5475, 7885}, {3533, 33224, 33262}, {3545, 33221, 33283}, {3552, 11285, 33273}, {3552, 11286, 384}, {3589, 24273, 12215}, {3589, 59635, 7797}, {3734, 7786, 7783}, {3763, 7773, 7938}, {3855, 33223, 33290}, {5025, 16924, 33013}, {6655, 8370, 14042}, {6655, 14042, 8597}, {6656, 16044, 14041}, {6656, 66415, 16044}, {6658, 8356, 33267}, {7603, 7915, 7899}, {7746, 7846, 16984}, {7751, 7878, 63038}, {7752, 7822, 7931}, {7767, 53489, 20088}, {7770, 11285, 11286}, {7777, 7795, 7947}, {7791, 11361, 33256}, {7791, 32971, 11361}, {7794, 7858, 7840}, {7804, 31239, 1078}, {7809, 10159, 7849}, {7814, 47005, 7869}, {7819, 8367, 32992}, {7819, 32992, 2}, {7833, 14035, 19696}, {7841, 33018, 14044}, {7860, 55738, 7865}, {7866, 32966, 14046}, {7866, 44543, 32966}, {7876, 37455, 7824}, {7876, 66413, 4}, {7879, 15484, 7900}, {7885, 16988, 3096}, {7892, 16898, 66322}, {7901, 16896, 2}, {7948, 33013, 5025}, {8359, 19687, 33260}, {8361, 66342, 2}, {8362, 8370, 6655}, {8364, 33024, 33284}, {9466, 41940, 63925}, {11285, 11286, 3552}, {11285, 33273, 7824}, {11356, 19689, 384}, {14001, 32957, 2}, {14030, 33268, 32981}, {14031, 33258, 376}, {14032, 33275, 33007}, {14033, 32960, 32965}, {14033, 32965, 33257}, {14034, 66414, 20}, {14035, 16043, 7833}, {14036, 33015, 16925}, {14037, 35930, 384}, {14038, 33274, 32973}, {14039, 32978, 32964}, {14041, 16897, 6656}, {14043, 16922, 2}, {14044, 67269, 7841}, {14045, 66324, 7933}, {14069, 32975, 2}, {16045, 16921, 19694}, {16045, 32968, 2}, {16895, 16921, 2}, {16895, 32968, 32967}, {16924, 33006, 32991}, {16925, 33198, 14036}, {19687, 33260, 9855}, {19692, 33259, 8369}, {19694, 32967, 2}, {32488, 32489, 3545}, {32952, 32976, 2}, {32956, 32983, 14063}, {32973, 33001, 33274}, {32974, 33016, 14062}, {32979, 33017, 14066}, {32979, 33202, 33017}, {32981, 33008, 33268}, {32990, 33007, 33275}, {32991, 33180, 33006}, {32993, 33184, 33289}, {32993, 66345, 33184}, {33006, 33180, 5025}, {33023, 33280, 66406}, {33185, 33249, 2}, {33222, 61886, 2}, {33245, 66322, 7892}, {33834, 33835, 442}, {41940, 63925, 7760}

X(68523) = X(2)X(3)∩X(99)X(63925)

Barycentrics    11*a^4 - 8*a^2*b^2 - 8*a^2*c^2 + 3*b^2*c^2 : :

X(68523) lies on these lines: {2, 3}, {99, 63925}, {5182, 55588}, {5206, 17129}, {7782, 7798}, {7783, 35007}, {7839, 22331}, {7882, 52886}, {7905, 51587}, {7932, 44541}, {7947, 32459}, {8588, 17128}, {15655, 20081}, {22486, 55694}, {39141, 55614}, {52695, 63928}

X(68523) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {550, 33245, 67270}, {3528, 33266, 7892}, {3552, 33273, 384}, {7824, 13586, 33235}, {7907, 9855, 14044}, {7907, 33254, 9855}, {8363, 62091, 33260}, {8598, 33259, 19696}, {11286, 33235, 3552}, {13586, 33014, 33276}, {13586, 33276, 384}, {16925, 33267, 14046}, {32964, 33268, 14041}, {32970, 33252, 66406}, {32970, 66406, 33289}, {32985, 33275, 14043}, {33214, 33216, 14062}, {33244, 33274, 14042}, {33245, 67270, 33286}, {33254, 35287, 7907}

X(68524) = X(2)X(3)∩X(99)X(5041)

Barycentrics    7*a^4 - 2*a^2*b^2 - 2*a^2*c^2 + 5*b^2*c^2 : :
X(68524) = 6 X[2] - 7 X[14043]

X(68524) lies on these lines: {2, 3}, {99, 5041}, {3972, 7781}, {4048, 6144}, {5008, 7760}, {5024, 45017}, {5102, 39141}, {5149, 52886}, {6781, 7928}, {7737, 7947}, {7780, 17128}, {7783, 7878}, {12215, 32455}, {20105, 21309}, {32820, 34604}, {42010, 60146}

X(68524) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3627, 33289}, {2, 33250, 33267}, {20, 14036, 7948}, {384, 3552, 7824}, {384, 33276, 7770}, {550, 35954, 19689}, {3552, 7770, 33276}, {3552, 7824, 13586}, {5025, 32981, 66328}, {6658, 7901, 8597}, {6658, 8369, 7901}, {6661, 33260, 16897}, {7770, 33276, 7824}, {7791, 14038, 66322}, {7807, 19686, 14042}, {7833, 14037, 19694}, {7892, 33007, 33256}, {8356, 19692, 16896}, {8369, 66425, 2}, {8598, 19697, 33021}, {11361, 32973, 33245}, {14001, 33187, 33257}, {14001, 33257, 7924}, {14031, 32985, 16921}, {14034, 16925, 33013}, {14035, 33246, 32967}, {14037, 33214, 32956}, {14037, 33239, 7833}, {14039, 33244, 7876}, {14040, 33268, 2}, {16898, 35927, 33275}, {19687, 33225, 14041}, {19687, 66393, 33225}, {32954, 66419, 14045}, {32956, 33214, 7833}, {32956, 33239, 33214}, {32981, 33181, 33280}, {32981, 33255, 5025}, {33007, 33201, 7892}, {33019, 33220, 14047}, {33181, 33280, 5025}, {33198, 33254, 66414}, {33217, 33264, 67269}, {33242, 66387, 7933}, {33255, 33280, 33181}

X(68525) = X(2)X(3)∩X(99)X(7808)

Barycentrics    2*a^4 + a^2*b^2 + a^2*c^2 + 3*b^2*c^2 : :
X(68525) = 3 X[2] + 2 X[14034], 3 X[7878] - 2 X[41940]

X(68525) lies on these lines: {2, 3}, {6, 17128}, {32, 31276}, {39, 15301}, {61, 42675}, {62, 42674}, {69, 20088}, {76, 5007}, {83, 194}, {99, 7808}, {115, 7846}, {141, 7823}, {148, 5149}, {172, 30998}, {183, 22331}, {315, 46226}, {316, 7822}, {538, 7878}, {575, 39141}, {598, 7843}, {625, 7930}, {671, 7902}, {698, 44000}, {1506, 7835}, {1975, 3329}, {2076, 3619}, {2548, 7836}, {2782, 10353}, {2896, 7737}, {2996, 62903}, {3096, 7747}, {3303, 4366}, {3304, 6645}, {3314, 7745}, {3499, 62991}, {3589, 7864}, {3618, 4048}, {3620, 5017}, {3763, 7928}, {3767, 10000}, {3815, 7891}, {3849, 7936}, {3926, 63018}, {3933, 7921}, {3934, 3972}, {4027, 38664}, {5116, 63119}, {5182, 55708}, {5254, 7875}, {5286, 63020}, {5319, 19570}, {5395, 37668}, {5475, 7832}, {6179, 9466}, {6194, 12110}, {6292, 7802}, {6392, 63045}, {6683, 7782}, {6704, 7756}, {7603, 7940}, {7735, 44530}, {7738, 8290}, {7739, 51860}, {7748, 7859}, {7750, 16986}, {7751, 12150}, {7752, 7820}, {7753, 7796}, {7760, 17130}, {7771, 31239}, {7773, 7931}, {7775, 7909}, {7777, 7789}, {7781, 55085}, {7783, 11174}, {7785, 7795}, {7786, 7816}, {7790, 7889}, {7794, 7812}, {7797, 11185}, {7800, 14712}, {7801, 7858}, {7806, 59635}, {7809, 7869}, {7814, 7880}, {7825, 7944}, {7839, 20105}, {7842, 7937}, {7844, 15031}, {7849, 7860}, {7854, 9939}, {7856, 63924}, {7861, 7943}, {7865, 10159}, {7868, 7885}, {7881, 7941}, {7883, 63931}, {7893, 18907}, {7895, 7926}, {7911, 7914}, {7915, 7934}, {7920, 47286}, {7923, 44518}, {7942, 39565}, {8596, 63109}, {8782, 10345}, {9300, 32820}, {9606, 59634}, {10333, 10796}, {10351, 13108}, {10352, 23235}, {10356, 10722}, {10788, 49111}, {11057, 55738}, {11842, 61550}, {12215, 51171}, {13571, 32833}, {13881, 16984}, {14023, 34604}, {14976, 31168}, {16995, 18135}, {17129, 30435}, {18501, 32521}, {18840, 59266}, {18841, 35369}, {18906, 42534}, {20065, 63044}, {20179, 41838}, {22486, 55718}, {25497, 29479}, {31407, 32837}, {32828, 63047}, {32830, 63017}, {32834, 63048}, {35423, 66755}, {35424, 40330}, {40908, 63075}, {43527, 54823}, {43843, 63040}, {53164, 62712}, {59546, 63101}, {60728, 64018}, {63038, 63933}

X(68525) = midpoint of X(7876) and X(14034)
X(68525) = anticomplement of X(7876)
X(68525) = orthocentroidal-circle-inverse of X(7933)
X(68525) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 7933}, {2, 20, 33021}, {2, 384, 3552}, {2, 439, 33012}, {2, 3522, 33258}, {2, 3543, 66337}, {2, 3552, 33004}, {2, 3832, 33283}, {2, 5068, 33277}, {2, 6658, 7791}, {2, 14031, 20}, {2, 14033, 33264}, {2, 14035, 6655}, {2, 14037, 33225}, {2, 14068, 32974}, {2, 16044, 32966}, {2, 16919, 33062}, {2, 16920, 16914}, {2, 16924, 33002}, {2, 17692, 33063}, {2, 19690, 32956}, {2, 19692, 14001}, {2, 19693, 33260}, {2, 32971, 16044}, {2, 32973, 33259}, {2, 32974, 66345}, {2, 32979, 14063}, {2, 32981, 32965}, {2, 32991, 32963}, {2, 32993, 14064}, {2, 32995, 32972}, {2, 32996, 33180}, {2, 32997, 33202}, {2, 33010, 32967}, {2, 33011, 7887}, {2, 33014, 7824}, {2, 33018, 5025}, {2, 33019, 6656}, {2, 33022, 11285}, {2, 33024, 32961}, {2, 33198, 19689}, {2, 33201, 32964}, {2, 33205, 33206}, {2, 33244, 32990}, {2, 33260, 16043}, {2, 33269, 33020}, {2, 33278, 66336}, {2, 33287, 33182}, {2, 66320, 376}, {2, 66420, 11287}, {4, 16898, 2}, {4, 33221, 33251}, {5, 6661, 7892}, {5, 7892, 2}, {6, 17128, 20081}, {20, 14031, 19686}, {76, 7787, 7766}, {76, 7804, 7787}, {83, 194, 62994}, {83, 3734, 194}, {115, 7846, 7932}, {141, 7823, 7929}, {316, 7822, 7938}, {381, 33217, 7901}, {381, 66322, 2}, {384, 7770, 2}, {384, 7824, 1003}, {384, 7892, 35925}, {384, 11356, 6658}, {384, 35930, 14031}, {384, 66322, 44224}, {546, 7819, 8363}, {546, 8363, 5025}, {598, 7922, 7843}, {1003, 7824, 33014}, {1003, 33014, 3552}, {1656, 33220, 33245}, {1656, 33245, 2}, {2548, 7836, 63021}, {3096, 7747, 7898}, {3314, 7745, 7900}, {3525, 35951, 3}, {3589, 32819, 7864}, {3843, 33219, 14045}, {3933, 53489, 7921}, {3934, 3972, 7793}, {5025, 7819, 2}, {5025, 8370, 33018}, {5192, 33826, 2}, {5475, 7832, 7912}, {6655, 14035, 66419}, {6656, 11361, 33019}, {6656, 16895, 2}, {6658, 7791, 33264}, {6661, 66413, 2}, {7752, 7820, 7945}, {7770, 11286, 384}, {7785, 7795, 7897}, {7791, 14033, 6658}, {7791, 16045, 2}, {7791, 66389, 33247}, {7794, 7812, 7946}, {7807, 16921, 2}, {7807, 19697, 14036}, {7807, 66415, 16921}, {7819, 8370, 5025}, {7833, 14032, 19687}, {7849, 14537, 7860}, {7860, 47005, 7849}, {7866, 19694, 2}, {7868, 65630, 7885}, {7881, 15484, 7941}, {7887, 14043, 2}, {7887, 33013, 33011}, {7887, 33237, 14043}, {7892, 66413, 5}, {7901, 33217, 2}, {7901, 66322, 33217}, {7907, 14038, 8369}, {7907, 32992, 2}, {7914, 62203, 7911}, {7915, 39590, 7934}, {7948, 14042, 7841}, {8356, 68177, 33257}, {8359, 33250, 33275}, {8361, 14067, 2}, {8362, 19687, 7833}, {8363, 8370, 546}, {8364, 66409, 33229}, {8367, 33274, 2}, {8367, 35954, 33274}, {8369, 32992, 7907}, {11108, 16911, 2}, {11159, 33234, 19696}, {11285, 13586, 33022}, {11321, 16918, 2}, {11361, 16895, 6656}, {13740, 33827, 2}, {13741, 33825, 2}, {14001, 16924, 2}, {14001, 32976, 33197}, {14030, 33257, 68177}, {14033, 16045, 7791}, {14036, 16921, 7807}, {14036, 66415, 2}, {14037, 33269, 2}, {14039, 32968, 16925}, {14041, 19694, 7866}, {14043, 33013, 7887}, {14064, 33016, 32993}, {14065, 33185, 2}, {14069, 32961, 2}, {14069, 32983, 32961}, {16043, 33007, 33260}, {16044, 19689, 2}, {16897, 33256, 11287}, {16915, 17541, 2}, {16916, 17686, 2}, {16922, 33233, 2}, {16925, 32968, 2}, {17681, 17688, 2}, {17682, 33817, 2}, {19686, 33021, 20}, {19689, 32971, 32966}, {19691, 33280, 66407}, {19693, 33260, 33007}, {19696, 33234, 66421}, {19697, 66415, 7807}, {19702, 33228, 33185}, {32954, 32967, 2}, {32954, 44543, 32967}, {32956, 33017, 19690}, {32957, 32985, 33001}, {32957, 33001, 2}, {32960, 33239, 33008}, {32961, 32983, 33024}, {32965, 32981, 33265}, {32967, 44543, 33010}, {32970, 32999, 2}, {32971, 33198, 2}, {32975, 33000, 2}, {32975, 33191, 33000}, {32986, 33280, 19691}, {32987, 33181, 2}, {32988, 33183, 2}, {32998, 33189, 2}, {33013, 33237, 2}, {33020, 33225, 2}, {33020, 66317, 33225}, {33025, 33192, 6655}, {33185, 33228, 14065}, {33187, 33258, 3522}, {33219, 66323, 2}, {33225, 66317, 14037}, {33229, 66409, 14066}, {33246, 66416, 2}, {33818, 33832, 377}, {35732, 42282, 37336}, {66318, 66416, 33246}

X(68526) = X(2)X(3)∩X(385)X(45017)

Barycentrics    10*a^4 - 7*a^2*b^2 - 7*a^2*c^2 + 3*b^2*c^2 : :

X(68526) lies on these lines: {2, 3}, {385, 45017}, {2076, 11008}, {2482, 7946}, {5017, 63026}, {5023, 20081}, {5182, 55583}, {7766, 7782}, {7783, 22331}, {7787, 31652}, {7793, 32456}, {7888, 14976}, {7916, 52886}, {7923, 44541}, {7929, 59545}, {8588, 31276}, {14023, 52695}, {15655, 17129}, {22486, 55698}, {39141, 55606}, {53096, 62994}

X(68526) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 33014, 33276}, {2, 33235, 3552}, {2, 33243, 6655}, {3, 35950, 10303}, {439, 33208, 6655}, {3522, 33266, 33225}, {3530, 14034, 2}, {8598, 33227, 7907}, {13586, 33014, 3552}, {13586, 33276, 33235}, {15717, 33187, 33020}, {16925, 33238, 2}, {32964, 33265, 32966}, {32981, 61804, 33261}, {33235, 33276, 2}, {33244, 33259, 66419}, {33244, 35287, 33259}, {33248, 62127, 68504}, {33250, 33274, 33018}, {33268, 35297, 33019}

X(68527) = X(2)X(3)∩X(99)X(7878)

Barycentrics    5*a^4 - a^2*b^2 - a^2*c^2 + 4*b^2*c^2 : :
X(68527) = 6 X[2] - 5 X[7866], 3 X[2] - 5 X[14001], 9 X[2] - 5 X[32974], 3 X[2] + 5 X[32981], 9 X[2] - 10 X[33185], 7 X[2] - 5 X[33223], 3 X[7866] - 2 X[32974], X[7866] + 2 X[32981], 3 X[7866] - 4 X[33185], 7 X[7866] - 6 X[33223], 5 X[7866] - 2 X[33238], 3 X[14001] - X[32974], 3 X[14001] - 2 X[33185], 7 X[14001] - 3 X[33223], 5 X[14001] - X[33238], X[32974] + 3 X[32981], 7 X[32974] - 9 X[33223], 5 X[32974] - 3 X[33238], 3 X[32981] + 2 X[33185], 7 X[32981] + 3 X[33223], 5 X[32981] + X[33238], 14 X[33185] - 9 X[33223], 10 X[33185] - 3 X[33238], 15 X[33223] - 7 X[33238]

X(68527) lies on these lines: {2, 3}, {6, 7781}, {32, 63933}, {76, 1384}, {83, 5024}, {99, 7878}, {194, 43136}, {599, 63935}, {1235, 8778}, {1285, 32830}, {1975, 3972}, {1992, 32824}, {2548, 59545}, {3053, 3734}, {3629, 4048}, {3763, 7830}, {3788, 65630}, {3849, 7869}, {3926, 18907}, {3934, 5023}, {4045, 44519}, {4366, 7373}, {5013, 7804}, {5017, 40341}, {5093, 32134}, {5149, 35022}, {5182, 51524}, {5206, 15271}, {5210, 7815}, {5304, 32822}, {5305, 32815}, {5503, 53102}, {6179, 63954}, {6645, 6767}, {6680, 44518}, {6683, 53095}, {6781, 7822}, {7737, 7776}, {7747, 7778}, {7751, 22331}, {7754, 21309}, {7755, 34505}, {7763, 15484}, {7772, 8716}, {7773, 7835}, {7782, 11174}, {7784, 7820}, {7786, 14535}, {7787, 31859}, {7794, 63938}, {7801, 63932}, {7802, 7868}, {7808, 15815}, {7812, 32821}, {7823, 7881}, {7827, 11164}, {7834, 44526}, {7852, 65633}, {7863, 9766}, {7874, 62203}, {7879, 14712}, {7880, 63931}, {7888, 14537}, {7920, 20094}, {8588, 31239}, {8667, 17130}, {8719, 37479}, {9654, 26629}, {9669, 26686}, {10159, 55164}, {10349, 18501}, {10796, 10983}, {11482, 22486}, {12150, 51122}, {12215, 62995}, {12943, 30104}, {12953, 30103}, {15533, 63937}, {18843, 60262}, {20179, 31468}, {26864, 46900}, {30123, 61716}, {31401, 32459}, {32448, 53091}, {32834, 46453}, {32836, 63926}, {34504, 47352}, {43676, 62912}, {53105, 60186}, {60219, 62931}, {63062, 63654}

X(68527) = midpoint of X(14001) and X(32981)
X(68527) = reflection of X(i) in X(j) for these {i,j}: {7866, 14001}, {32974, 33185}
X(68527) = complement of X(33238)
X(68527) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3529, 8357}, {2, 3552, 33276}, {2, 19687, 382}, {2, 33007, 66424}, {2, 33229, 33241}, {2, 33235, 3}, {2, 33239, 550}, {2, 33243, 7791}, {2, 33257, 33234}, {2, 33280, 33229}, {2, 50688, 33292}, {2, 66422, 7841}, {3, 384, 11286}, {4, 8369, 32954}, {4, 32954, 11318}, {4, 33181, 8361}, {4, 33197, 33199}, {4, 33199, 37350}, {4, 33201, 8369}, {4, 68177, 11159}, {5, 32973, 11288}, {5, 66393, 32973}, {20, 7819, 11287}, {20, 14039, 7819}, {376, 33198, 8362}, {382, 33241, 33229}, {382, 33242, 2}, {384, 1003, 3}, {384, 3552, 7770}, {384, 35950, 8369}, {439, 32968, 549}, {550, 19697, 2}, {632, 66412, 32987}, {1003, 7770, 3552}, {1657, 6656, 5077}, {1657, 33237, 6656}, {1975, 3972, 30435}, {1975, 30435, 22253}, {3146, 14069, 33184}, {3522, 16045, 8359}, {3552, 7770, 3}, {3552, 33022, 13586}, {3552, 33276, 33235}, {3627, 8368, 14064}, {3853, 33186, 16041}, {3855, 33236, 2}, {5059, 33190, 67390}, {5076, 33240, 14063}, {6655, 14036, 33217}, {6655, 66387, 17800}, {6656, 14037, 33237}, {6656, 33007, 1657}, {6656, 35954, 14037}, {6658, 7841, 5073}, {6658, 7892, 7841}, {6661, 33187, 3534}, {6661, 33250, 7791}, {7737, 7789, 7776}, {7791, 33187, 33250}, {7791, 33250, 3534}, {7807, 14035, 381}, {7807, 66319, 14035}, {7808, 32456, 15815}, {7819, 66391, 20}, {7833, 14040, 19689}, {7887, 11361, 3843}, {7907, 44543, 5070}, {7933, 19696, 66388}, {8356, 33244, 15696}, {8360, 62036, 32982}, {8361, 8369, 33181}, {8361, 33181, 32954}, {8362, 66318, 33198}, {8364, 15704, 32986}, {8367, 15712, 32978}, {8369, 11159, 11318}, {8369, 37350, 33197}, {8369, 68177, 4}, {8370, 16925, 1656}, {8598, 32965, 62100}, {10999, 11000, 33190}, {11159, 32954, 4}, {11285, 13586, 3}, {11286, 14532, 11356}, {11317, 32966, 61970}, {11321, 17692, 16418}, {11361, 33225, 7887}, {14001, 32974, 33185}, {14001, 33238, 2}, {14031, 16925, 8370}, {14032, 33246, 16044}, {14033, 32970, 32979}, {14033, 32973, 5}, {14033, 66393, 11288}, {14035, 33255, 7807}, {14037, 33007, 6656}, {14038, 33257, 2}, {14039, 66391, 11287}, {14043, 33019, 33219}, {14043, 66328, 33019}, {14063, 66408, 5076}, {14068, 33228, 61984}, {15682, 32952, 33200}, {16043, 35927, 548}, {16044, 33233, 5055}, {16044, 33246, 33233}, {16044, 66320, 14032}, {16898, 33244, 8356}, {16914, 33035, 16857}, {16924, 35297, 3526}, {17130, 35007, 8667}, {17538, 33202, 8354}, {17578, 33183, 33285}, {19686, 33220, 3830}, {19687, 33229, 33280}, {19692, 33265, 7876}, {19693, 33225, 11361}, {19695, 33193, 49137}, {19696, 66388, 49134}, {32964, 32992, 5054}, {32970, 32979, 5}, {32971, 32985, 140}, {32973, 32979, 32970}, {32974, 33185, 7866}, {32977, 32991, 547}, {32978, 35287, 15712}, {32983, 32989, 3628}, {32987, 33216, 632}, {33007, 33237, 5077}, {33007, 35954, 33237}, {33016, 33249, 5072}, {33021, 33268, 35955}, {33180, 33703, 66392}, {33194, 49138, 33210}, {33201, 68177, 32954}, {33211, 62041, 66394}, {33217, 66387, 6655}, {33219, 66328, 15684}, {33221, 33272, 66347}, {33229, 33280, 382}, {33234, 33257, 15681}, {33255, 66319, 381}, {33256, 66395, 49139}, {33260, 66317, 16895}, {33266, 66416, 15693}, {33268, 35955, 62107}, {33279, 66423, 382}, {62155, 66347, 33272}, {66396, 68420, 62053}

X(68528) = X(2)X(3)∩X(76)X(15655)

Barycentrics    11*a^4 - 7*a^2*b^2 - 7*a^2*c^2 + 4*b^2*c^2 : :
X(68528) = 3 X[2] - 11 X[439], 9 X[2] - 11 X[32970], 15 X[2] - 11 X[32972], 27 X[2] - 11 X[54097], 3 X[439] - X[32970], 5 X[439] - X[32972], 9 X[439] - X[54097], 5 X[32970] - 3 X[32972], 3 X[32970] - X[54097], 9 X[32972] - 5 X[54097]

X(68528) lies on these lines: {2, 3}, {76, 15655}, {187, 63933}, {1384, 7760}, {2482, 63932}, {3053, 7781}, {5023, 7780}, {5024, 7878}, {5182, 55724}, {5210, 7816}, {5585, 7815}, {6179, 51122}, {7782, 30435}, {7783, 21309}, {7834, 44541}, {7949, 52886}, {8716, 35007}, {22486, 55701}, {32820, 63950}, {32821, 51224}, {32825, 63945}, {39141, 55593}

X(68528) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 62127, 67390}, {20, 33189, 66392}, {140, 33239, 11159}, {548, 32973, 11287}, {548, 33185, 33226}, {550, 8360, 33247}, {550, 32954, 5077}, {550, 32985, 32954}, {1003, 33014, 3}, {1657, 16925, 11318}, {3534, 33241, 32997}, {3552, 7824, 1003}, {3552, 33014, 7824}, {3552, 33276, 7770}, {6656, 33208, 62100}, {7770, 33276, 3}, {7807, 32997, 33241}, {7807, 33254, 3534}, {7841, 33268, 62131}, {7887, 33265, 17800}, {7907, 66387, 3843}, {8353, 33252, 62119}, {8368, 62104, 33023}, {8598, 16925, 1657}, {8598, 33229, 33214}, {11317, 16923, 61919}, {13586, 33235, 3}, {14069, 62097, 8354}, {16925, 33214, 33229}, {17538, 33205, 33184}, {19697, 62069, 33215}, {21735, 33201, 8359}, {32952, 66335, 7866}, {32959, 49135, 37350}, {32963, 66423, 61990}, {32964, 33250, 381}, {32966, 66395, 62023}, {32973, 33226, 33185}, {33000, 66408, 5072}, {33185, 33226, 11287}, {33186, 62123, 33272}, {33191, 50693, 8357}, {33193, 33249, 5076}, {33214, 33229, 1657}, {33233, 33257, 3830}, {33236, 62117, 33210}, {33237, 62082, 32965}, {33239, 35287, 140}, {33240, 62134, 19695}, {33242, 62085, 8356}, {33244, 33262, 66389}, {33244, 35297, 382}, {33254, 33266, 7807}, {46853, 66393, 16043}

X(68529) = X(2)X(991)∩X(4)X(6)

Barycentrics    -a^6 - 2*a^5*(b + c) - (b - c)^4*(b + c)^2 + 2*a*(b - c)^2*(b + c)^3 - 3*a^2*(b^2 - c^2)^2 + a^4*(5*b^2 - 2*b*c + 5*c^2) : :

See David Nguyen, euclid 8459.

X(68529) lies on these lines: {1, 5809}, {2, 991}, {3, 37650}, {4, 6}, {5, 4648}, {7, 53599}, {8, 12618}, {10, 62948}, {12, 67264}, {20, 1724}, {37, 5817}, {44, 5759}, {51, 4196}, {58, 50700}, {69, 36652}, {86, 36660}, {118, 5274}, {141, 36682}, {142, 66660}, {238, 43161}, {269, 1210}, {278, 1864}, {346, 29016}, {347, 1736}, {381, 63054}, {386, 37434}, {390, 64013}, {461, 23292}, {497, 34048}, {500, 6887}, {511, 36670}, {515, 7290}, {516, 1743}, {572, 7390}, {581, 6846}, {631, 17337}, {944, 1279}, {948, 5728}, {971, 4000}, {990, 5222}, {1086, 36996}, {1104, 64144}, {1253, 3074}, {1350, 7397}, {1351, 36654}, {1419, 9581}, {1449, 59389}, {1453, 63998}, {1456, 1837}, {1458, 1745}, {1471, 4293}, {1714, 37421}, {1738, 63971}, {1750, 40940}, {1754, 50696}, {1861, 53994}, {2261, 54234}, {2263, 18391}, {2293, 3085}, {2478, 37659}, {2550, 55432}, {2635, 54366}, {2801, 4310}, {2947, 40958}, {3008, 5732}, {3019, 50689}, {3073, 21059}, {3090, 17245}, {3091, 3945}, {3545, 17392}, {3575, 44100}, {3618, 13727}, {3663, 64197}, {3664, 38150}, {3668, 10398}, {3755, 11372}, {3772, 5658}, {3820, 8147}, {3832, 5733}, {3842, 10186}, {3854, 45942}, {3855, 63401}, {3914, 64130}, {3946, 66661}, {4266, 33536}, {4300, 19855}, {4340, 6835}, {4344, 59387}, {4349, 19925}, {4356, 64699}, {4419, 5779}, {4644, 5805}, {4656, 30326}, {4667, 67866}, {4859, 43177}, {5046, 63088}, {5084, 25878}, {5120, 37412}, {5230, 20978}, {5587, 64174}, {5603, 49478}, {5691, 16469}, {5712, 8226}, {5816, 7407}, {5882, 35227}, {6223, 23537}, {6847, 37732}, {6849, 36742}, {6864, 36746}, {6893, 63318}, {6982, 45926}, {6996, 51212}, {7365, 64157}, {7490, 17810}, {7613, 60896}, {8727, 63089}, {8814, 52518}, {9355, 24248}, {10392, 43035}, {10394, 37800}, {10431, 32911}, {10883, 63008}, {10900, 64129}, {11427, 14004}, {11500, 21002}, {11745, 37379}, {12652, 49772}, {12953, 38293}, {13478, 14484}, {13567, 57534}, {14561, 48938}, {16112, 66071}, {16670, 52835}, {16885, 21168}, {17277, 36706}, {17278, 21151}, {17365, 59386}, {17582, 37501}, {19130, 36671}, {19541, 37642}, {19624, 37000}, {20423, 48902}, {23511, 64705}, {23681, 41561}, {24177, 30304}, {24220, 36694}, {24597, 36002}, {31400, 53425}, {34627, 50130}, {34937, 68000}, {36012, 36743}, {36473, 40330}, {36721, 37654}, {37374, 63126}, {37407, 48897}, {37817, 54051}, {39531, 56814}, {43175, 60846}, {51424, 64747}, {52969, 63416}, {54370, 64168}, {54532, 54883}, {54757, 60112}, {57719, 60157}, {57720, 60164}, {66683, 67877}

X(68529) = polar conjugate of isotomic conjugate of X(25932)
X(68529) = barycentric product X(4)*X(25932
X(68529) = barycentric quotient X(25932)/X(69)
X(68529) = trilinear product X(19)*X(25932)
X(68529) = trilinear quotient X(63)/X(25932)
X(68529) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 6, 3332}, {5, 62183, 4648}, {20, 37681, 13329}, {1738, 64741, 63971}, {5222, 36991, 990}, {5809, 54425, 1}, {17337, 50677, 631}

X(68530) = X(1)X(64516)∩X(2602)X(3615)

Barycentrics    (a+b)^2*(a-b-c)^2*(a+c)^2*(a^2+a*b+b^2-c^2)^2*(a^2-b^2+a*c+c^2)^2*(a^4-2*a^2*b^2+b^4-a^2*c^2-b^2*c^2)*(a^4-a^2*b^2-2*a^2*c^2-b^2*c^2+c^4) : :

See Francisco Javier García Capitán and Ercole Suppa, euclid 8468.

X(68530) lies on these lines: {1, 64516}, {2602, 3615}

X(68530) = X(62746)-cross conjugate of X(64516)
X(68530) = X(i)-isoconjugate of X(j) for these {i,j}: {1393, 7144}, {2081, 63202}, {2594, 2599}
X(68530) = intersection, other than A, B, C, of the circumconics: {{A, B, C, X(1), X(2596)}}, {{A, B, C, X(60), X(14587)}}
X(68530) = trilinear quotient X(i)/X(j) for these {i,j}: {2081, 62746}, {2599, 3615}, {7144, 44687}, {63202, 64516}

X(68531) = X(1169)X(5291)∩X(1220)X(27714)

Barycentrics    (a+b)*(a+c)*(a^2+b^2+a*c+b*c)^2*(a^2+a*b+b*c+c^2)^2 : :

See Francisco Javier García Capitán and Ercole Suppa, euclid 8468.

X(68531) lies on these lines: {1169, 5291}, {1220, 27714}, {1240, 5209}, {2363, 10457}, {5061, 5080}

X(68531) = X(i)-isoconjugate of X(j) for these {i,j}: {58, 6042}, {65, 1682}, {429, 22345}, {1193, 2292}, {1211, 2300}, {1829, 22076}, {2092, 3666}, {3725, 4357}, {4267, 52567}, {6371, 61172}, {20967, 41003}, {21033, 61412}, {21810, 40153}, {24471, 40966}, {48131, 61168}, {50330, 53280}, {61051, 65573}
X(68531) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 6042}, {40602, 1682}, {40625, 66986}
X(68531) = intersection, other than A, B, C, of the circumconics: {{A, B, C, X(1), X(4600)}}, {{A, B, C, X(8), X(18812)}}, {{A, B, C, X(10), X(27714)}}, {{A, B, C, X(28), X(14012)}
X(68531) = barycentric product X(i)*X(j) for these {i,j}: {1169, 1240}, {1220, 14534}, {2363, 30710}, {6648, 57161}
X(68531) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 6042}, {284, 1682}, {1169, 1193}, {1220, 1211}, {1240, 1228}, {1798, 22097}, {2298, 2292}, {2359, 22076}, {2363, 3666}, {4560, 66986}, {4581, 21124}, {8707, 65191}, {14534, 4357}, {14624, 20653}, {30710, 18697}, {31643, 45196}, {32736, 61168}, {36147, 61172}, {57161, 3910}, {62749, 50330}, {64457, 54308}, {64984, 41003}
X(68531) = trilinear product X(i)*X(j) for these {i,j}: {1169, 30710}, {1220, 2363}, {2298, 14534}, {14624, 64457}, {36098, 57161}, {57162, 65281}
X(68531) = trilinear quotient X(i)/X(j) for these {i,j}: {10, 6042}, {21, 1682}, {1169, 2300}, {1193, 2363}, {1211, 30710}, {1220, 2292}, {1240, 18697}, {1791, 22076}, {1798, 22345}, {2092, 2298}, {3666, 14534}, {4581, 50330}, {8707, 61172}, {14624, 21810}, {17420, 57161}, {18155, 66986}, {18191, 61051}, {20911, 40827}, {31643, 41003}, {36147, 61168}, {40153, 64457}, {42661, 57162}, {52567, 60086}, {65191, 65229}

X(68532) = X(1)X(50847)∩X(2)X(13)

Barycentrics    Sqrt[3]*(7*a^4 - 8*a^2*b^2 + b^4 - 8*a^2*c^2 - 2*b^2*c^2 + c^4) + 2*(a^2 - 5*b^2 - 5*c^2)*S : :
X(68532) = X[1] + 2 X[50847], 5 X[2] - 2 X[13], 2 X[2] + X[616], X[2] - 4 X[618], 7 X[2] - 4 X[5459], X[2] + 2 X[5463], 11 X[2] - 8 X[6669], 31 X[2] - 16 X[35019], 10 X[2] - X[35749], 8 X[2] + X[35750], 7 X[2] + 2 X[35751], 11 X[2] - 2 X[35752], X[2] - 10 X[36767], X[2] + 8 X[36768], 5 X[2] + 4 X[36769], 7 X[2] - 10 X[36770], 13 X[2] - 4 X[47865], 5 X[2] - 4 X[48311], 4 X[2] - X[51482], 4 X[13] + 5 X[616], X[13] - 10 X[618], 7 X[13] - 10 X[5459], X[13] + 5 X[5463], 11 X[13] - 20 X[6669], 3 X[13] - 5 X[22489], 31 X[13] - 40 X[35019], 4 X[13] - X[35749], 16 X[13] + 5 X[35750], 7 X[13] + 5 X[35751], 11 X[13] - 5 X[35752], X[13] - 25 X[36767], X[13] + 20 X[36768], X[13] + 2 X[36769], 7 X[13] - 25 X[36770], 13 X[13] - 10 X[47865], 8 X[13] - 5 X[51482], 4 X[13] - 5 X[59378], X[616] + 8 X[618], 7 X[616] + 8 X[5459], X[616] - 4 X[5463], 11 X[616] + 16 X[6669], 3 X[616] + 4 X[22489], 31 X[616] + 32 X[35019], 5 X[616] + X[35749], 4 X[616] - X[35750], 7 X[616] - 4 X[35751], 11 X[616] + 4 X[35752], X[616] + 20 X[36767], X[616] - 16 X[36768], 5 X[616] - 8 X[36769], 7 X[616] + 20 X[36770], and many others

In the plane of a triangle ABC, let
A'B'C' = 1st Napoleon triangle = outer Napoleon triangle
A'' = reflection of A in line X(2)A', and define B'' and C'' cyclically
Ga = centroid of AA'A'', and define Gb and Gc cyclically
Ma = midpoint of segment A'A'', and define Mb and Mc cyclically
gA = centroid of AB''C'', and define gB and gC cyclically
O* = circumcircle of the triangle gAgBgC
P = reflection of X(2) in Ma
g = centroid of GaGbGc
O** = circle having diameter Pg
Then
(1) Ma = Mb = MC.
(2) The triangle gAgBgC is equilateral.
(3) O* = O**.
(4) X(68532) = center of O*.
See X(68533). (Benjamin Lee Warren, May 15, 2025)

X(68532) lies on these lines: {1, 50847}, {2, 13}, {6, 51202}, {8, 50849}, {14, 8591}, {20, 41042}, {69, 51012}, {99, 51483}, {145, 50848}, {148, 5460}, {193, 51011}, {298, 6390}, {302, 11295}, {376, 5617}, {531, 52695}, {542, 3524}, {543, 59379}, {547, 13103}, {549, 6770}, {599, 51159}, {617, 2482}, {621, 35931}, {627, 9885}, {628, 13083}, {634, 9763}, {671, 43555}, {1992, 51010}, {3180, 45879}, {3241, 12781}, {3533, 20415}, {3543, 5473}, {3618, 22580}, {3679, 51114}, {3828, 9901}, {3839, 36765}, {5055, 59394}, {5067, 16001}, {5071, 25154}, {5464, 32553}, {5470, 48312}, {5478, 61936}, {5983, 42036}, {6671, 22495}, {6695, 41977}, {6771, 15702}, {6772, 11489}, {6773, 8724}, {6774, 12243}, {7975, 31145}, {8703, 48655}, {8716, 9761}, {9115, 37640}, {10304, 41022}, {10385, 12952}, {11050, 12793}, {11121, 54594}, {11160, 51200}, {11177, 61634}, {11303, 43193}, {11304, 33412}, {11485, 49961}, {11488, 41745}, {11539, 59383}, {12100, 36344}, {12142, 62975}, {12355, 20253}, {13172, 25164}, {14061, 31695}, {15682, 22796}, {15683, 36961}, {15693, 36318}, {15698, 36363}, {15703, 20252}, {15708, 21156}, {15717, 41020}, {15719, 36383}, {16267, 48313}, {20080, 51201}, {22237, 42035}, {22490, 41135}, {22491, 42160}, {22493, 67072}, {22892, 33616}, {23005, 42910}, {30471, 33619}, {31693, 42137}, {32907, 61859}, {33603, 40707}, {33608, 49813}, {33613, 36386}, {33627, 41101}, {34508, 41973}, {35691, 42503}, {35753, 42602}, {35754, 42603}, {36327, 36521}, {36764, 41620}, {37786, 52194}, {41108, 49855}, {42062, 43447}, {42141, 49880}, {43401, 44383}, {47611, 48657}, {48310, 59410}, {54581, 56055}, {59393, 61954}, {59401, 61899}

X(68532) = midpoint of X(i) and X(j) for these {i,j}: {616, 59378}, {36769, 48311}
X(68532) = reflection of X(i) in X(j) for these {i,j}: {13, 48311}, {3839, 36765}, {5470, 48312}, {16267, 48313}, {41135, 22490}, {51482, 59378}, {59378, 2}, {59383, 11539}, {59394, 5055}, {59410, 48310}
X(68532) = anticomplement of X(22489)
X(68532) = circumcircle-of-inner-Napoleon-triangle-inverse of X(36769)
X(68532) = psi-transform of X(47867)
X(68532) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 616, 51482}, {2, 5463, 616}, {2, 36769, 35749}, {13, 5463, 36769}, {13, 35749, 51482}, {298, 35304, 51484}, {616, 51482, 35750}, {618, 5463, 2}, {618, 36768, 5463}, {5459, 36770, 2}, {5460, 9116, 148}, {5463, 36767, 618}, {5463, 36770, 35751}, {5979, 52021, 51485}, {35751, 36770, 5459}, {36764, 41620, 63032}, {36767, 36768, 2}

X(68533) = X(1)X(50850)∩X(2)X(14)

Barycentrics    Sqrt[3]*(7*a^4 - 8*a^2*b^2 + b^4 - 8*a^2*c^2 - 2*b^2*c^2 + c^4) - 2*(a^2 - 5*b^2 - 5*c^2)*S : :
X(68533) = X[1] + 2 X[50850], 5 X[2] - 2 X[14], 2 X[2] + X[617], X[2] - 4 X[619], 7 X[2] - 4 X[5460], X[2] + 2 X[5464], 11 X[2] - 8 X[6670], 31 X[2] - 16 X[35020], 10 X[2] - X[36327], 7 X[2] + 2 X[36329], 11 X[2] - 2 X[36330], 8 X[2] + X[36331], 13 X[2] - 4 X[47866], 5 X[2] + 4 X[47867], 5 X[2] - 4 X[48312], 4 X[2] - X[51483], 4 X[14] + 5 X[617], X[14] - 10 X[619], 7 X[14] - 10 X[5460], X[14] + 5 X[5464], 11 X[14] - 20 X[6670], 3 X[14] - 5 X[22490], 31 X[14] - 40 X[35020], 4 X[14] - X[36327], 7 X[14] + 5 X[36329], 11 X[14] - 5 X[36330], 16 X[14] + 5 X[36331], 13 X[14] - 10 X[47866], X[14] + 2 X[47867], 8 X[14] - 5 X[51483], 4 X[14] - 5 X[59379], X[617] + 8 X[619], 7 X[617] + 8 X[5460], X[617] - 4 X[5464], 11 X[617] + 16 X[6670], 3 X[617] + 4 X[22490], 31 X[617] + 32 X[35020], 5 X[617] + X[36327], 7 X[617] - 4 X[36329], 11 X[617] + 4 X[36330], 4 X[617] - X[36331], 13 X[617] + 8 X[47866], 5 X[617] - 8 X[47867], 5 X[617] + 8 X[48312], 2 X[617] + X[51483], 7 X[619] - X[5460], 2 X[619] + X[5464], 11 X[619] - 2 X[6670], 6 X[619] - X[22490], 31 X[619] - 4 X[35020], 40 X[619] - X[36327], and many others

In the plane of a triangle ABC, let
A'B'C' = 2nd Napoleon triangle = inner Napoleon triangle
A'' = reflection of A in line X(2)A', and define B'' and C'' cyclically
Ga = centroid of AA'A'', and define Gb and Gc cyclically
Ma = midpoint of segment A'A'', and define Mb and Mc cyclically
gA = centroid of AB''C'', and define gB and gC cyclically
O* = circumcircle of the triangle gAgBgC
P = reflection of X(2) in Ma
g = centroid of GaGbGc
O** = circle having diameter Pg
Then
(1) Ma = Mb = MC.
(2) The triangle gAgBgC is equilateral.
(3) O* = O**.
(4) X(68533) = center of O*.
See X(68532). (Benjamin Lee Warren, May 15,2025)

X(68533) lies on these lines: {1, 50850}, {2, 14}, {6, 51205}, {8, 50852}, {13, 8591}, {20, 41043}, {69, 51015}, {99, 51482}, {145, 50851}, {148, 5459}, {193, 51014}, {299, 6390}, {303, 11296}, {376, 5613}, {530, 52695}, {542, 3524}, {543, 59378}, {547, 13102}, {549, 6773}, {599, 51160}, {616, 2482}, {622, 35932}, {627, 13084}, {628, 9886}, {633, 9761}, {671, 43554}, {1992, 51013}, {3181, 45880}, {3241, 12780}, {3533, 20416}, {3543, 5474}, {3618, 22579}, {3679, 51115}, {3828, 9900}, {5055, 59396}, {5067, 16002}, {5071, 25164}, {5463, 32552}, {5469, 48311}, {5479, 61936}, {5982, 42035}, {6672, 22496}, {6694, 41978}, {6770, 8724}, {6771, 12243}, {6774, 15702}, {6775, 11488}, {7974, 31145}, {8703, 48656}, {8716, 9763}, {9117, 37641}, {10304, 41023}, {10385, 12951}, {11050, 12792}, {11122, 54593}, {11160, 51203}, {11177, 36776}, {11303, 33413}, {11304, 43194}, {11486, 49962}, {11489, 41746}, {11539, 59384}, {12100, 36319}, {12141, 62975}, {12355, 20252}, {13172, 25154}, {14061, 31696}, {15682, 22797}, {15683, 36962}, {15693, 36320}, {15698, 36362}, {15703, 20253}, {15708, 21157}, {15717, 41021}, {15719, 36382}, {16268, 48314}, {20080, 51204}, {22235, 42036}, {22489, 41135}, {22492, 42161}, {22494, 67071}, {22848, 33617}, {23004, 42911}, {30472, 33618}, {31694, 42136}, {32909, 61859}, {33602, 40706}, {33609, 49812}, {33612, 36388}, {33626, 41100}, {34509, 41974}, {35695, 42502}, {35749, 36521}, {35850, 42602}, {35851, 42603}, {37785, 52193}, {41107, 49858}, {41621, 63033}, {42063, 43446}, {42140, 49879}, {43402, 44382}, {47610, 48657}, {54580, 56056}, {59395, 61954}, {59402, 61899}

X(68533) = midpoint of X(i) and X(j) for these {i,j}: {617, 59379}, {47867, 48312}
X(68533) = reflection of X(i) in X(j) for these {i,j}: {14, 48312}, {5469, 48311}, {16268, 48314}, {41135, 22489}, {51483, 59379}, {59379, 2}, {59384, 11539}, {59396, 5055}
X(68533) = anticomplement of X(22490)
X(68533) = circumcircle-of-outer-Napoleon-triangle-inverse of X(47867)
X(68533) = psi-transform of X(36769)
X(68533) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 617, 51483}, {2, 5464, 617}, {2, 47867, 36327}, {14, 5464, 47867}, {14, 36327, 51483}, {299, 35303, 51485}, {617, 51483, 36331}, {619, 5464, 2}, {5459, 9114, 148}, {5978, 52022, 51484}

X(68534) = ISOTOMIC CONJUGATE OF X(1745)

Barycentrics    b*c*(a^5*b - 2*a^3*b^3 + a*b^5 - a^5*c - a^4*b*c + a*b^4*c + b^5*c + 2*a^3*c^3 - 2*b^3*c^3 - a*b*c^4 - a*c^5 + b*c^5)*(-(a^5*b) + 2*a^3*b^3 - a*b^5 + a^5*c - a^4*b*c - a*b^4*c + b^5*c - 2*a^3*c^3 - 2*b^3*c^3 + a*b*c^4 + a*c^5 + b*c^5) : :

X(68534) lies on the cubic K1399 and these lines: {69, 18749}, {332, 3362}, {333, 18751}, {345, 7361}, {7049, 30479}, {7182, 46752}, {20930, 57801}, {24031, 57806}, {44130, 60801}

X(68534) = isotomic conjugate of X(1745)
X(68534) = isotomic conjugate of the anticomplement of X(14058)
X(68534) = isotomic conjugate of the isogonal conjugate of X(3362)
X(68534) = isotomic conjugate of the polar conjugate of X(40165)
X(68534) = X(i)-cross conjugate of X(j) for these (i,j): {264, 75}, {3362, 40165}, {14058, 2}
X(68534) = X(i)-isoconjugate of X(j) for these (i,j): {6, 21767}, {25, 20764}, {31, 1745}, {32, 6360}, {184, 1148}, {560, 18749}, {1333, 21854}, {1402, 1816}, {2206, 42456}, {9247, 62605}
X(68534) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 1745}, {9, 21767}, {37, 21854}, {6374, 18749}, {6376, 6360}, {6505, 20764}, {40603, 42456}, {40605, 1816}, {62576, 62605}, {62605, 1148}
X(68534) = cevapoint of X(4391) and X(24031)
X(68534) = trilinear pole of line {6332, 17894}
X(68534) = barycentric product X(i)*X(j) for these {i,j}: {69, 40165}, {75, 7361}, {76, 3362}, {304, 7049}, {561, 8761}, {3926, 60801}
X(68534) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 21767}, {2, 1745}, {10, 21854}, {63, 20764}, {75, 6360}, {76, 18749}, {92, 1148}, {264, 62605}, {321, 42456}, {333, 1816}, {3362, 6}, {7049, 19}, {7361, 1}, {8761, 31}, {40165, 4}, {60801, 393}

X(68535) = ISOTOMIC CONJUGATE OF X(13855)

Barycentrics    b^2*c^2*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-(a^10*b^2) + 4*a^8*b^4 - 6*a^6*b^6 + 4*a^4*b^8 - a^2*b^10 - a^10*c^2 + a^8*b^2*c^2 + 2*a^6*b^4*c^2 - 2*a^4*b^6*c^2 - a^2*b^8*c^2 + b^10*c^2 + 4*a^8*c^4 + 2*a^6*b^2*c^4 - 4*a^4*b^4*c^4 + 2*a^2*b^6*c^4 - 4*b^8*c^4 - 6*a^6*c^6 - 2*a^4*b^2*c^6 + 2*a^2*b^4*c^6 + 6*b^6*c^6 + 4*a^4*c^8 - a^2*b^2*c^8 - 4*b^4*c^8 - a^2*c^10 + b^2*c^10) : :

X(68535) lies on the cubic K1399 and these lines: {5, 264}, {69, 56271}, {92, 18161}, {95, 40800}, {317, 1899}, {343, 15466}, {3964, 6331}, {6528, 20477}, {19211, 43752}, {33808, 57812}, {44133, 56593}, {52581, 57909}

X(68535) = isotomic conjugate of X(13855)
X(68535) = isotomic conjugate of the anticomplement of X(14057)
X(68535) = isotomic conjugate of the isogonal conjugate of X(1075)
X(68535) = polar conjugate of the isogonal conjugate of X(46717)
X(68535) = X(69)-Ceva conjugate of X(264)
X(68535) = X(i)-cross conjugate of X(j) for these (i,j): {14057, 2}, {41481, 46717}
X(68535) = X(i)-isoconjugate of X(j) for these (i,j): {31, 13855}, {9247, 34287}, {52430, 56271}
X(68535) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 13855}, {2052, 4}, {46717, 38281}, {47600, 39201}, {62576, 34287}
X(68535) = cevapoint of X(1075) and X(46717)
X(68535) = barycentric product X(i)*X(j) for these {i,j}: {76, 1075}, {264, 46717}, {276, 41481}, {18022, 41373}
X(68535) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 13855}, {264, 34287}, {1075, 6}, {2052, 56271}, {41373, 184}, {41481, 216}, {46717, 3}
X(68535) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {18027, 45198, 264}, {41009, 59139, 264}

X(68536) = X(20)X(317)∩X(64)X(264)

Barycentrics    (a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^12+b^2*c^2*(b^2-c^2)^4-a^10*(b^2+c^2)+3*a^2*(b^2-c^2)^2*(b^2+c^2)^3+a^8*(-6*b^4+13*b^2*c^2-6*c^4)+2*a^6*(7*b^6-9*b^4*c^2-9*b^2*c^4+7*c^6)+a^4*(-11*b^8+2*b^6*c^2+34*b^4*c^4+2*b^2*c^6-11*c^8)) : :

See Francisco Javier García Capitán and Ivan Pavlov, euclid 8470.

X(68536) lies on these lines: {1, 55346}, {4, 40196}, {20, 317}, {64, 264}, {107, 43601}, {185, 648}, {275, 12086}, {1593, 36794}, {1629, 12279}, {2071, 51031}, {3091, 58758}, {5894, 8146}, {6696, 66707}, {7398, 35711}, {8884, 10575}, {9786, 52578}, {15740, 17907}, {22467, 52913}, {30716, 36179}, {37200, 61150}, {43806, 56303}, {43849, 47111}, {43995, 44458}

X(68536) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {185, 1105, 648}

X(68537) = X(1)X(1053)∩X(85)X(169)

Barycentrics    a*(a+b-c)*(a-b+c)*(a^4+b^4+c^4-2*a^3*(b+c)+2*a^2*(b^2+b*c+c^2)-2*a*(b^3+c^3)) : :

See Francisco Javier García Capitán and Ivan Pavlov, euclid 8470.

X(68537) lies on these lines: {1, 1053}, {9, 33298}, {46, 36652}, {57, 20602}, {63, 30854}, {65, 16048}, {85, 169}, {348, 33950}, {664, 2082}, {1445, 1760}, {1452, 5125}, {1708, 37086}, {1748, 26003}, {1788, 2478}, {2002, 3758}, {5228, 56517}, {5540, 9312}, {6516, 26690}, {6604, 30616}, {7079, 31640}, {7183, 16572}, {17352, 62770}, {40131, 55082}

X(68537) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 46395}
X(68537) = X(i)-Dao conjugate of X(j) for these {i, j}: {9, 46395}
X(68537) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1110), X(44178)}}, {{A, B, C, X(9311), X(41788)}}
X(68537) = barycentric quotient X(i)/X(j) for these (i, j): {1, 46395}
X(68537) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {169, 44178, 3732}, {2082, 7131, 664}

X(68538) = X(5)X(6)∩X(40)X(1419)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^8 + 2*a^7*(b + c) - 2*a*(b - c)^2*(b + c)^3*(b^2 + c^2) - 2*a^6*(b^2 - b*c + c^2) - (b - c)^4*(b + c)^2*(b^2 + 4*b*c + c^2) - 2*a^4*b*c*(3*b^2 - 4*b*c + 3*c^2) - 6*a^5*(b^3 + b^2*c + b*c^2 + c^3) + 2*a^2*(b - c)^2*(b^4 + 5*b^3*c + 6*b^2*c^2 + 5*b*c^3 + c^4) + a^3*(6*b^5 + 6*b^4*c - 4*b^3*c^2 - 4*b^2*c^3 + 6*b*c^4 + 6*c^5)) : :

See Tran Viet Hung and David Nguyen, euclid 8472.

X(68538) lies on these lines: {3, 57701}, {5, 6}, {40, 1419}, {991, 12163}, {1993, 57534}, {2263, 19471}, {3332, 44665}, {3945, 11411}, {4648, 12359}, {5762, 64069}, {6238, 67264}, {9820, 37650}, {10539, 44100}, {13329, 47391}, {13754, 62183}, {33536, 36742}, {37559, 56293}

X(68539) = X(3)X(6)∩X(5)X(3332)

Barycentrics   -(a^2*(3*a^4 - 2*a^3*(b + c) + (b - c)^2*(b^2 + 4*b*c + c^2) - 2*a^2*(2*b^2 + b*c + 2*c^2) + 2*a*(b^3 + b^2*c + b*c^2 + c^3))) : :

See Tran Viet Hung and David Nguyen, euclid 8472.

X(68539) lies on these lines: {1, 15837}, {3, 6}, {4, 37681}, {5, 3332}, {31, 6244}, {35, 67264}, {37, 59381}, {40, 16469}, {44, 990}, {46, 1456}, {54, 57701}, {55, 52423}, {56, 38293}, {57, 22117}, {140, 4648}, {184, 37269}, {212, 52424}, {218, 64156}, {269, 23072}, {404, 63088}, {474, 37659}, {517, 7290}, {549, 63054}, {602, 1253}, {631, 3945}, {971, 1743}, {999, 1471}, {1086, 60922}, {1155, 56418}, {1191, 8158}, {1203, 5584}, {1260, 55399}, {1279, 1482}, {1331, 55437}, {1386, 67962}, {1418, 37545}, {1419, 15803}, {1449, 21153}, {1453, 31793}, {1617, 61397}, {1656, 17337}, {1724, 37537}, {1738, 52682}, {1742, 16477}, {1754, 4383}, {1993, 37309}, {2263, 36279}, {2293, 64951}, {2328, 17825}, {2361, 37541}, {3008, 5805}, {3019, 5070}, {3052, 67711}, {3072, 9709}, {3100, 5729}, {3167, 16059}, {3358, 16572}, {3523, 62997}, {3526, 5733}, {3654, 50294}, {3664, 38122}, {3672, 21168}, {3973, 64198}, {4000, 5762}, {4191, 11402}, {4344, 5657}, {4349, 6684}, {4419, 64065}, {4644, 31657}, {5012, 11350}, {5054, 17392}, {5222, 5759}, {5422, 20835}, {5706, 11108}, {5713, 50726}, {5732, 16670}, {6282, 64166}, {6600, 45728}, {6987, 48847}, {6992, 64167}, {7070, 64157}, {7074, 55086}, {7365, 59613}, {7411, 63074}, {7580, 32911}, {7982, 16487}, {8147, 35238}, {9441, 16468}, {9777, 16064}, {10222, 35227}, {10246, 49478}, {10306, 21059}, {10601, 13615}, {10679, 19624}, {10680, 15287}, {11227, 62812}, {11248, 21002}, {12702, 61086}, {13727, 17349}, {14853, 49131}, {14997, 36002}, {15299, 41339}, {15720, 63401}, {16020, 20330}, {16408, 25878}, {16411, 17811}, {16418, 63982}, {16419, 22139}, {16466, 61399}, {16885, 51516}, {17278, 38107}, {17365, 59380}, {17613, 36277}, {18583, 36474}, {24597, 37374}, {25406, 49132}, {25934, 62756}, {26446, 64174}, {31183, 61595}, {31671, 53599}, {34718, 50130}, {36636, 53056}, {36652, 63051}, {36674, 48906}, {36706, 51171}, {37364, 37642}, {37407, 49743}, {37594, 61122}, {37787, 64750}, {38572, 52969}, {38599, 42314}, {45942, 55863}, {48934, 66308}, {56809, 62245}, {64124, 67026}

X(68539) = isogonal conjugate of X(45097)
X(68539) = barycentric product of X(6)*X(45097)
X(68539) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 62183}, {3, 5093, 48908}, {6, 13329, 3}, {44, 990, 5779}, {371, 372, 5022}, {573, 5085, 3}, {580, 36745, 3}, {582, 36754, 3}, {1754, 4383, 19541}, {3332, 37650, 5}, {3973, 66661, 64198}

X(68540) = X(4)X(3945)∩X(6)X(25)

Barycentrics    -(a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2 - 3*b^2 + 2*b*c - 3*c^2 + 2*a*(b + c))) : :

See Tran Viet Hung and David Nguyen, euclid 8472.

X(68540) lies on these lines: {4, 3945}, {6, 25}, {33, 67264}, {34, 1827}, {221, 42447}, {269, 1876}, {427, 4648}, {428, 63054}, {468, 37650}, {607, 40983}, {608, 2293}, {991, 1593}, {1253, 1395}, {1398, 1458}, {1456, 1898}, {3195, 20978}, {3332, 3575}, {3515, 13329}, {3516, 50677}, {4344, 7718}, {4349, 49542}, {4644, 60879}, {4869, 57534}, {5064, 17392}, {5090, 64174}, {5094, 17245}, {5185, 44858}, {6353, 37681}, {6995, 62997}, {7290, 11363}, {11396, 49478}, {14004, 17379}, {17337, 37453}, {25878, 54407}, {37659, 62972}, {62971, 63088}, {62976, 63401}

X(68540) = isogonal conjugate of isotomic conjugate of X(57534)
X(68540) = polar conjugate of isotomic conjugate of X(5022)
X(68540) = barycentric product X(i)*X(j) for these {i,j}: {4, 5022}, {6, 57534}, {19, 62823}, {25, 4869}
X(68540) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 60092}, {4869, 305}, {5022, 69}, {57534, 76}, {62823, 304}
X(68540) = trilinear product X(i)*X(j) for these {i,j}: {19, 5022}, {25, 62823}, {31, 57534}, {1973, 4869}
X(68540) = trilinear quotient X(i)/X(j) for these {i,j}: {19, 60092}, {63, 5022}, {69, 62823}, {75, 57534}, {304, 4869}
X(68540) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 25, 44100}, {608, 2356, 7071}, {1474, 7716, 25}

X(68541) = X(5)X(4648)∩X(1419)X(7308)

Barycentrics    -(a^2*(a^2 - 3*b^2 + 2*b*c - 3*c^2 + 2*a*(b + c))*(a^4 + 2*a^2*b*c + 2*a^3*(b + c) - (b - c)^2*(b^2 + 4*b*c + c^2) - 2*a*(b^3 + b^2*c + b*c^2 + c^3))) : :

See Tran Viet Hung and David Nguyen, euclid 8472.

X(68541) lies on these lines: {2, 57701}, {5, 4648}, {960, 53996}, {991, 33537}, {1419, 7308}, {4869, 57534}, {13329, 45248}

X(68541) = medial-isotomic conjugate of X(5022)
X(68541) = complement of isogonal conjugate of X(37269)
X(68541) = barycentric product X(4869)*X(37269)
X(68541) = barycentric quotient X(37269)/X(60092)
X(68541) = trilinear product X(37269)*X(62823)

X(68542) = X(6)X(31)∩X(65)X(269)

Barycentrics    -(a^2*(a^3 + a^2*(b + c) + 3*(b - c)^2*(b + c) - a*(5*b^2 + 6*b*c + 5*c^2))) : :

See Tran Viet Hung and David Nguyen, euclid 8472.

X(68542) lies on these lines: {1, 3059}, {6, 31}, {40, 62183}, {56, 52020}, {65, 269}, {198, 64751}, {220, 4343}, {991, 5584}, {1418, 62819}, {1419, 14521}, {1449, 3174}, {1468, 50677}, {1471, 4255}, {2099, 40965}, {2122, 8614}, {2550, 3945}, {3189, 4344}, {3303, 3688}, {3332, 6253}, {3925, 4648}, {3946, 41570}, {3957, 26657}, {4000, 63258}, {4349, 63146}, {5738, 50441}, {7290, 37080}, {7964, 62812}, {7994, 8147}, {10964, 49478}, {11269, 17245}, {11495, 62797}, {17018, 37659}, {17784, 62997}, {26671, 63168}, {34612, 63054}, {54358, 64739}

X(68542) = barycentric product X(6575)*X(14281)
X(68542) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {20978, 61358, 6}

X(68543) = X(6)X(1511)∩X(265)X(57701)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^14 + 2*a^13*(b + c) + a^12*(-5*b^2 + 2*b*c - 5*c^2) - 4*a^11*(b^3 + b^2*c + b*c^2 + c^3) + 2*a*(b - c)^4*(b + c)^5*(b^4 + c^4) - (b - c)^6*(b + c)^4*(3*b^4 + 4*b^3*c + 4*b^2*c^2 + 4*b*c^3 + 3*c^4) + a^10*(5*b^4 - 4*b^3*c + 15*b^2*c^2 - 4*b*c^3 + 5*c^4) - 2*a^9*(b^5 + b^4*c - 3*b^3*c^2 - 3*b^2*c^3 + b*c^4 + c^5) - 2*a^3*(b - c)^2*(b + c)^3*(2*b^6 - 3*b^4*c^2 - 3*b^2*c^4 + 2*c^6) + a^8*(7*b^6 - 2*b^5*c - 15*b^4*c^2 + 6*b^3*c^3 - 15*b^2*c^4 - 2*b*c^5 + 7*c^6) + a^2*(b - c)^4*(b + c)^2*(7*b^6 + 10*b^5*c + 4*b^4*c^2 + 4*b^3*c^3 + 4*b^2*c^4 + 10*b*c^5 + 7*c^6) + a^7*(8*b^7 + 8*b^6*c - 2*b^5*c^2 - 2*b^4*c^3 - 2*b^3*c^4 - 2*b^2*c^5 + 8*b*c^6 + 8*c^7) + a^6*(-13*b^8 + 8*b^7*c + 8*b^6*c^2 - 2*b^5*c^3 + 2*b^4*c^4 - 2*b^3*c^5 + 8*b^2*c^6 + 8*b*c^7 - 13*c^8) + a^4*(b - c)^2*(b^8 + 6*b^6*c^2 + 6*b^5*c^3 + 2*b^4*c^4 + 6*b^3*c^5 + 6*b^2*c^6 + c^8) - 2*a^5*(b^9 + b^8*c + 3*b^7*c^2 + 3*b^6*c^3 - 4*b^5*c^4 - 4*b^4*c^5 + 3*b^3*c^6 + 3*b^2*c^7 + b*c^8 + c^9)) : :

See Tran Viet Hung and David Nguyen, euclid 8472.

X(68543) lies on these lines: {6, 1511}, {265, 57701}, {991, 12302}, {3945, 12319}, {4648, 23306}, {12888, 67264}, {17702, 62183}

X(68544) = X(3)X(6)∩X(30)X(4648)

Barycentrics    -(a^2*(a^4 - 6*a^3*(b + c) + a^2*(4*b^2 - 6*b*c + 4*c^2) - (b - c)^2*(5*b^2 + 4*b*c + 5*c^2) + 6*a*(b^3 + b^2*c + b*c^2 + c^3))) : :

See Tran Viet Hung and David Nguyen, euclid 8472.

X(68544) lies on these lines: {1, 45834}, {2, 54712}, {3, 6}, {30, 4648}, {36, 67264}, {45, 60884}, {74, 5545}, {186, 44100}, {269, 24929}, {376, 3945}, {381, 17245}, {549, 37650}, {550, 3332}, {971, 3731}, {990, 3723}, {999, 2293}, {1253, 52407}, {1385, 35227}, {1418, 15934}, {1419, 30282}, {1456, 3612}, {1458, 3295}, {1742, 16484}, {2177, 6244}, {3019, 15689}, {3247, 5732}, {3524, 37681}, {3534, 17392}, {3576, 16487}, {3601, 33633}, {3620, 36706}, {3973, 31658}, {4300, 16483}, {4675, 31671}, {5054, 17337}, {5315, 8273}, {5584, 16474}, {5733, 15696}, {5779, 16814}, {7290, 13624}, {7373, 42314}, {7411, 14996}, {7580, 37633}, {8147, 10269}, {8692, 52769}, {8703, 63054}, {10246, 61086}, {10304, 62997}, {11108, 48897}, {11227, 62695}, {11350, 15107}, {12702, 49478}, {15066, 20835}, {15485, 38031}, {15492, 59381}, {15569, 43178}, {15688, 63401}, {16059, 62209}, {16064, 26864}, {16370, 37659}, {16411, 59777}, {16418, 25878}, {16469, 67706}, {16499, 30283}, {17194, 37271}, {17234, 36721}, {17549, 63088}, {18358, 36474}, {18481, 64174}, {29571, 31672}, {30392, 41436}, {34417, 37269}, {38601, 44858}, {41453, 64449}, {45942, 62107}

X(68544) = isogonal conjugate of X(54690)
X(68544) = barycentric quotient X(6)/X(54690)
X(68544) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 991, 62183}, {15, 16, 4258}, {991, 50677, 3}

X(68545) = X(2)X(40)∩X(3)X(8)

Barycentrics    -3*a^4 - 2*a^3*b + 4*a^2*b^2 + 2*a*b^3 - b^4 - 2*a^3*c + 4*a^2*b*c - 2*a*b^2*c + 4*a^2*c^2 - 2*a*b*c^2 + 2*b^2*c^2 + 2*a*c^3 - c^4 : :
X(68545) = 2*X[1]-7*X[3523], X[1]-6*X[10164], X[1]+4*X[43174], 4*X[1]-9*X[54445], 2*X[1]+3*X[59417], 7*X[3523]+8*X[43174], 7*X[3523]+3*X[59417], 3*X[10164]+2*X[43174], 8*X[10164]-3*X[54445], 4*X[10164]+X[59417], 8*X[43174]-3*X[59417], 3*X[54445]+2*X[59417], 3*X[2]+2*X[40], 9*X[2]-4*X[946], 6*X[2]-X[962], 3*X[2]-8*X[6684], 3*X[2]-2*X[8227], 9*X[2]+X[20070], 7*X[2]-2*X[31162, 4*X[2]+X[34632], 3*X[40]+2*X[946], 4*X[40]+X[962], X[40]+4*X[6684], X[40]+X[8227], 6*X[40]-X[20070], 7*X[40]+3*X[31162], 3*X[40]+7*X[31423], 8*X[40]-3*X[34632], 8*X[946]-3*X[962], X[946]-6*X[6684], 2*X[946]-3*X[8227], 4*X[946]+X[20070], 2*X[946]-7*X[31423], X[962]-4*X[8227], 3*X[962]+2*X[20070], 2*X[962]+3*X[34632], 4*X[6684]-X[8227], 6*X[8227]+X[20070], 7*X[8227]-3*X[31162], 3*X[8227]-7*X[31423], and many others

See Keita Miyamoto, Francisco Javier García Capitán and David Nguyen, euclid 8486 and euclid 8488.

X(68545) lies on these lines: {1, 3523}, {2, 40}, {3, 8}, {4, 2355}, {5, 6361}, {7, 46}, {9, 37421}, {10, 20}, {12, 3474}, {21, 10310}, {30, 5818}, {35, 4313}, {36, 4308}, {43, 4300}, {55, 938}, {56, 41824}, {57, 11037}, {63, 5815}, {65, 5218}, {71, 27382}, {72, 31787}, {77, 1103}, {78, 30503}, {86, 68031}, {92, 37417}, {140, 5550}, {144, 21075}, {145, 3576}, {153, 46684}, {191, 6172}, {200, 10884}, {210, 9943}, {226, 5128}, {227, 18623}, {281, 412}, {318, 37410}, {329, 3359}, {333, 37402}, {355, 376}, {377, 64111}, {381, 10248}, {382, 38042}, {388, 1155}, {390, 1210}, {392, 17567}, {404, 3428}, {405, 6244}, {411, 1376}, {442, 5759}, {452, 24982}, {484, 498}, {497, 5704}, {499, 11010}, {515, 3522}, {516, 1698}, {517, 631}, {519, 7987}, {547, 50826}, {548, 18525}, {549, 1482}, {550, 5790}, {551, 11531}, {573, 5749}, {580, 17126}, {581, 3240}, {632, 18493}, {653, 7952}, {730, 32522}, {908, 63144}, {936, 59675}, {942, 10578}, {950, 35445}, {958, 6909}, {960, 6962}, {971, 3697}, {991, 3293}, {993, 59326}, {1000, 24928}, {1006, 11248}, {1010, 64376}, {1012, 5260}, {1056, 37582}, {1071, 3681}, {1125, 7991}, {1151, 19065}, {1152, 19066}, {1158, 3219}, {1276, 30414}, {1277, 30415}, {1320, 21154}, {1329, 5698}, {1334, 40127}, {1350, 59406}, {1385, 3241}, {1478, 37163}, {1479, 5445}, {1483, 12100}, {1490, 67097}, {1496, 9364}, {1571, 5286}, {1572, 31400}, {1621, 10306}, {1656, 28174}, {1657, 18357}, {1695, 6685}, {1697, 3911}, {1699, 3634}, {1702, 7586}, {1703, 7585}, {1706, 5745}, {1737, 4294}, {1742, 6048}, {1753, 4194}, {1766, 5296}, {1770, 10590}, {1836, 10588}, {1837, 63211}, {1902, 6353}, {1995, 9911}, {2041, 34560}, {2071, 47321}, {2077, 4189}, {2093, 13411}, {2099, 52793}, {2345, 37499}, {2475, 16113}, {2478, 13528}, {2550, 6836}, {2551, 4640}, {2807, 11444}, {2829, 64141}, {2886, 6943}, {3035, 64189}, {3057, 7288}, {3060, 58487}, {3068, 49227}, {3069, 49226}, {3086, 5119}, {3090, 9779}, {3098, 38116}, {3146, 5587}, {3158, 24391}, {3160, 4566}, {3161, 46937}, {3189, 4421}, {3218, 10528}, {3242, 21167}, {3244, 30389}, {3245, 65142}, {3256, 54430}, {3295, 10580}, {3305, 12705}, {3306, 68036}, {3333, 64142}, {3336, 10056}, {3339, 11036}, {3416, 25406}, {3421, 3916}, {3427, 64280}, {3434, 6865}, {3436, 5828}, {3475, 5221}, {3476, 5204}, {3485, 5432}, {3486, 5217}, {3487, 36279}, {3488, 64951}, {3515, 7718}, {3525, 5886}, {3526, 22791}, {3528, 18481}, {3529, 18480}, {3530, 10246}, {3533, 11230}, {3534, 38074}, {3543, 19875}, {3545, 22793}, {3550, 4339}, {3555, 11227}, {3562, 7074}, {3587, 6847}, {3600, 15803}, {3601, 4848}, {3619, 64085}, {3621, 5882}, {3622, 7982}, {3623, 28234}, {3624, 4301}, {3625, 61783}, {3626, 62067}, {3632, 58221}, {3636, 11224}, {3648, 37401}, {3650, 64065}, {3651, 11499}, {3653, 15719}, {3655, 15698}, {3656, 15702}, {3678, 15071}, {3679, 4297}, {3683, 6957}, {3698, 6974}, {3730, 61237}, {3740, 12688}, {3746, 8236}, {3751, 62174}, {3753, 6857}, {3811, 18444}, {3812, 7957}, {3817, 7486}, {3823, 52858}, {3826, 6991}, {3828, 3839}, {3830, 61259}, {3832, 10175}, {3843, 28178}, {3868, 63168}, {3869, 6988}, {3870, 8726}, {3871, 36845}, {3872, 63133}, {3873, 9940}, {3874, 15104}, {3876, 6001}, {3877, 6921}, {3897, 50371}, {3913, 6764}, {3921, 9947}, {3925, 6828}, {3947, 4312}, {3983, 5918}, {4188, 11012}, {4192, 26038}, {4193, 15908}, {4197, 7680}, {4208, 64003}, {4220, 26264}, {4229, 67852}, {4292, 5261}, {4293, 10039}, {4298, 51784}, {4302, 18395}, {4305, 5010}, {4307, 5530}, {4314, 31508}, {4317, 5131}, {4323, 5903}, {4329, 9537}, {4344, 5264}, {4345, 5697}, {4413, 6915}, {4420, 18446}, {4430, 12005}, {4512, 5129}, {4533, 58688}, {4646, 37642}, {4652, 6735}, {4662, 10178}, {4668, 28236}, {4669, 62059}, {4677, 61781}, {4678, 5881}, {4699, 29054}, {4745, 34628}, {4746, 58186}, {4763, 38329}, {4816, 58217}, {4861, 37611}, {4882, 64679}, {5044, 67998}, {5047, 11496}, {5054, 5901}, {5055, 40273}, {5059, 31673}, {5067, 9955}, {5068, 18483}, {5070, 38034}, {5071, 28198}, {5076, 28182}, {5080, 6850}, {5084, 31658}, {5085, 51192}, {5086, 59345}, {5088, 31994}, {5174, 37028}, {5175, 6987}, {5177, 64004}, {5180, 6863}, {5183, 11375}, {5219, 41348}, {5223, 43151}, {5225, 17606}, {5232, 64700}, {5235, 37422}, {5248, 5537}, {5253, 22770}, {5258, 63983}, {5274, 10624}, {5278, 37062}, {5304, 9593}, {5328, 6960}, {5418, 35611}, {5420, 35610}, {5438, 5837}, {5499, 16150}, {5538, 30147}, {5542, 18217}, {5556, 10592}, {5558, 67301}, {5658, 54228}, {5686, 5732}, {5692, 66019}, {5709, 9776}, {5714, 31479}, {5716, 37540}, {5748, 6825}, {5758, 6889}, {5777, 9961}, {5804, 6883}, {5806, 17552}, {5809, 7676}, {5811, 26878}, {5812, 10585}, {5836, 6966}, {5844, 15712}, {5846, 53094}, {5850, 64698}, {5902, 31452}, {6049, 37618}, {6175, 10894}, {6194, 12782}, {6210, 26029}, {6211, 17257}, {6212, 30412}, {6213, 30413}, {6245, 25006}, {6253, 49732}, {6254, 10174}, {6259, 14646}, {6282, 19860}, {6284, 54361}, {6409, 49233}, {6410, 49232}, {6459, 13973}, {6460, 13911}, {6554, 42316}, {6690, 28629}, {6696, 64022}, {6713, 64136}, {6734, 10268}, {6736, 62824}, {6745, 12526}, {6765, 10857}, {6767, 34753}, {6769, 54392}, {6796, 7688}, {6826, 26060}, {6827, 52367}, {6831, 33108}, {6833, 48363}, {6835, 7964}, {6837, 19855}, {6838, 12514}, {6848, 67999}, {6872, 25005}, {6875, 26285}, {6888, 19854}, {6890, 19843}, {6902, 10525}, {6903, 37820}, {6904, 24987}, {6905, 35239}, {6906, 35238}, {6907, 11681}, {6910, 14110}, {6912, 64074}, {6918, 9342}, {6922, 11680}, {6926, 10527}, {6927, 12672}, {6940, 11249}, {6951, 10526}, {6954, 37562}, {6972, 26363}, {6989, 37584}, {7280, 12647}, {7308, 67886}, {7330, 37427}, {7354, 63212}, {7385, 28893}, {7406, 29576}, {7411, 9799}, {7488, 37557}, {7492, 9626}, {7513, 54294}, {7580, 9709}, {7582, 31439}, {7672, 67930}, {7682, 25011}, {7738, 31443}, {7793, 12197}, {7956, 17575}, {7965, 34501}, {7967, 10299}, {7970, 38748}, {7971, 68003}, {7973, 10192}, {7976, 21163}, {7978, 38793}, {7983, 38737}, {7984, 38727}, {7988, 46936}, {8148, 15720}, {8164, 57282}, {8165, 12572}, {8193, 17928}, {8270, 66610}, {8580, 9949}, {8703, 34627}, {8715, 15931}, {8722, 12195}, {9083, 53888}, {9122, 40435}, {9352, 37623}, {9441, 66313}, {9540, 35774}, {9541, 49602}, {9548, 26065}, {9578, 63207}, {9582, 43512}, {9590, 38435}, {9612, 60995}, {9616, 13936}, {9624, 61842}, {9708, 37022}, {9800, 36002}, {9809, 12515}, {9860, 51578}, {9957, 33575}, {10072, 37563}, {10167, 34790}, {10171, 19872}, {10172, 15022}, {10198, 55109}, {10222, 61814}, {10265, 20095}, {10283, 12108}, {10321, 37797}, {10430, 37426}, {10434, 10449}, {10529, 63132}, {10531, 26127}, {10583, 12497}, {10584, 12700}, {10586, 12703}, {10587, 12704}, {10589, 12701}, {10697, 38772}, {10698, 38760}, {10703, 38784}, {10724, 34122}, {10786, 26921}, {10902, 12649}, {10912, 34711}, {10915, 16209}, {10916, 16208}, {10950, 63756}, {11019, 53053}, {11023, 60926}, {11041, 63260}, {11112, 31799}, {11113, 31777}, {11114, 11826}, {11220, 14872}, {11239, 37534}, {11260, 67959}, {11278, 61817}, {11372, 38130}, {11495, 18253}, {11522, 19862}, {11523, 59584}, {11691, 31790}, {11715, 64743}, {11827, 17579}, {12053, 31231}, {12102, 61260}, {12103, 38138}, {12115, 56880}, {12135, 15750}, {12324, 40660}, {12511, 44425}, {12513, 32157}, {12536, 49168}, {12541, 34625}, {12571, 50865}, {12608, 27131}, {12619, 13199}, {12635, 34744}, {12648, 37561}, {12651, 17554}, {12667, 54052}, {12669, 40659}, {12717, 17260}, {12779, 54050}, {13253, 68277}, {13348, 16980}, {13464, 46934}, {13600, 62835}, {13935, 35775}, {14007, 64400}, {14093, 50819}, {14217, 38133}, {14450, 16139}, {14664, 21290}, {14690, 33650}, {14740, 66002}, {14891, 50823}, {14923, 31786}, {15028, 58469}, {15043, 67967}, {15056, 52796}, {15177, 22467}, {15178, 61807}, {15489, 31785}, {15640, 34638}, {15681, 50813}, {15683, 50796}, {15690, 38081}, {15693, 37624}, {15694, 50825}, {15697, 51066}, {15700, 50824}, {15705, 31145}, {15709, 51709}, {15714, 50822}, {15715, 50818}, {15716, 34748}, {15718, 50805}, {15721, 25055}, {15933, 37080}, {15971, 35203}, {16020, 24174}, {16200, 61816}, {16486, 64442}, {17074, 64069}, {17502, 20053}, {17531, 22753}, {17538, 28160}, {17558, 64673}, {17572, 64322}, {17578, 18492}, {17580, 64112}, {17613, 31445}, {17697, 35261}, {17757, 37424}, {17776, 37320}, {17923, 56887}, {18242, 64190}, {18913, 64040}, {18931, 67919}, {19003, 42523}, {19004, 42522}, {19278, 63423}, {19708, 28204}, {19822, 37088}, {19876, 61924}, {19883, 50814}, {20013, 64324}, {20014, 61291}, {21077, 64143}, {21155, 62830}, {21168, 58798}, {21445, 50247}, {21454, 21620}, {22072, 24806}, {22758, 37403}, {22937, 37821}, {24390, 37364}, {24466, 59415}, {25440, 59320}, {25568, 64123}, {25722, 51489}, {26015, 56936}, {26258, 61233}, {26333, 37162}, {26671, 59237}, {27013, 28292}, {27268, 68033}, {27804, 58392}, {28154, 62028}, {28158, 50691}, {28168, 62147}, {28172, 62152}, {28190, 62131}, {28202, 41099}, {28224, 46853}, {28232, 61914}, {28346, 39156}, {28610, 45701}, {29611, 36698}, {29667, 50699}, {29679, 50698}, {30264, 59416}, {30282, 64163}, {30308, 61906}, {30312, 36976}, {31018, 40256}, {31393, 64124}, {31396, 37665}, {31404, 31441}, {31419, 37374}, {31499, 38235}, {31738, 33884}, {31760, 62187}, {31806, 64047}, {31837, 64021}, {31852, 67570}, {31884, 49524}, {32049, 34610}, {32558, 64138}, {33110, 48482}, {33697, 49138}, {33748, 51196}, {33899, 64144}, {33923, 37705}, {34200, 50798}, {34595, 61863}, {34619, 62858}, {34648, 62160}, {34701, 66251}, {34789, 66045}, {35193, 54442}, {35202, 48696}, {35260, 40658}, {35262, 64315}, {35263, 56987}, {35404, 50800}, {35788, 42261}, {35789, 42260}, {36745, 57280}, {37400, 59296}, {37434, 54357}, {37435, 63438}, {37436, 64001}, {37441, 56876}, {37525, 64766}, {37526, 62874}, {37545, 64350}, {37600, 41687}, {37619, 61109}, {37757, 56929}, {38022, 61843}, {38035, 63119}, {38047, 51212}, {38058, 50239}, {38076, 61989}, {38083, 41106}, {38087, 50965}, {38098, 50815}, {38118, 51171}, {38144, 48881}, {38155, 62110}, {38176, 62092}, {38200, 63413}, {38204, 63974}, {38327, 47776}, {39581, 66314}, {39885, 66755}, {40417, 58009}, {40999, 68334}, {41084, 61229}, {41229, 64129}, {41325, 46835}, {41430, 48878}, {41539, 62864}, {43161, 59413}, {44802, 49553}, {45759, 61245}, {46219, 61272}, {47745, 50811}, {48849, 66302}, {48883, 62189}, {48886, 51558}, {48899, 50417}, {48924, 48941}, {48932, 53014}, {48937, 62185}, {50797, 62116}, {50799, 62011}, {50802, 61912}, {50803, 61994}, {50806, 61885}, {50807, 61928}, {50812, 62129}, {50827, 61778}, {50828, 61806}, {50862, 62166}, {50867, 62048}, {50873, 61985}, {51071, 61805}, {51093, 61796}, {51516, 67986}, {51696, 66584}, {51955, 66636}, {51957, 66635}, {52366, 54337}, {54295, 66593}, {54370, 61023}, {54422, 59722}, {55651, 59407}, {55857, 61269}, {55862, 61270}, {56879, 62827}, {58230, 61286}, {58240, 61279}, {59335, 67120}, {59374, 60895}, {59400, 62069}, {59491, 63130}, {60896, 60957}, {60984, 64830}, {61244, 62061}, {61246, 62073}, {61249, 62085}, {61251, 62091}, {61253, 62093}, {61255, 62121}, {61256, 62124}, {61257, 62146}, {61262, 61984}, {61263, 61964}, {61267, 61903}, {61268, 61886}, {61277, 61822}, {61278, 61818}, {61283, 61802}, {61287, 61795}, {61295, 61789}, {61705, 64693}, {61717, 63273}, {62854, 66599}, {63135, 63430}, {63391, 64733}, {63753, 64270}, {64199, 64735}, {64340, 67971}, {64574, 64575}, {66609, 67963}

X(68545) = midpoint of X(i) and X(j) for these {i,j}: {40, 8227}, {1698, 63469}, {3522, 3617}, {5071, 50809}, {15714, 50822}, {18481, 61248}, {50797, 62116}
X(68545) = reflection of X(i) in X(j) for these {i,j}: {4, 61261}, {145, 61288}, {3091, 1698}, {3522, 35242}, {3616, 631}, {3623, 64953}, {5734, 3616}, {11522, 19862}, {15694, 50825}, {17578, 18492}, {18492, 31399}, {18493, 632}, {37714, 10}, {50806, 61885}, {50819, 14093}, {50873, 61985}, {61266, 11231}, {61284, 1385}, {62011, 50799}, {62129, 50812}
X(68545) = anticomplement of X(8227)
X(68545) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3523, 54445}, {1, 10164, 3523}, {1, 43174, 59417}, {2, 40, 8}, {3, 5690, 944}, {3, 59503, 34773}, {3, 61524, 5657}, {4, 3579, 9778}, {4, 26446, 9780}, {5, 6361, 9812}, {8, 64108, 3}, {10, 20, 59387}, {10, 165, 20}, {10, 12512, 5691}, {35, 18391, 4313}, {40, 946, 20070}, {40, 962, 34632}, {40, 6684, 2}, {40, 8227, 28194}, {40, 31423, 946}, {40, 61122, 5250}, {46, 3085, 7}, {55, 1788, 938}, {63, 7080, 5815}, {65, 5218, 5703}, {72, 59591, 64083}, {140, 5603, 5550}, {140, 12702, 5603}, {144, 27525, 21075}, {145, 15717, 3576}, {165, 5691, 12512}, {165, 9588, 10}, {210, 9943, 12528}, {355, 31663, 376}, {376, 50821, 53620}, {376, 53620, 50864}, {484, 498, 4295}, {497, 24914, 5704}, {498, 4295, 5226}, {499, 11010, 30305}, {548, 38112, 18525}, {549, 50810, 38314}, {944, 5657, 5690}, {944, 5690, 8}, {946, 6684, 31423}, {946, 20070, 962}, {946, 31423, 2}, {1071, 58643, 3681}, {1158, 64148, 6223}, {1210, 61763, 390}, {1376, 5584, 411}, {1385, 3654, 12245}, {1385, 12245, 3241}, {1697, 3911, 14986}, {1698, 63469, 516}, {1699, 3634, 5056}, {1702, 13975, 7586}, {1703, 13912, 7585}, {1737, 59316, 4294}, {2550, 65405, 59418}, {2886, 50031, 6943}, {3086, 5119, 9785}, {3090, 12699, 9779}, {3146, 46933, 5587}, {3339, 13405, 11036}, {3359, 55104, 56288}, {3522, 3617, 515}, {3523, 59417, 1}, {3524, 3654, 3241}, {3524, 12245, 1385}, {3528, 59388, 18481}, {3576, 11362, 145}, {3579, 26446, 4}, {3622, 61820, 10165}, {3624, 58441, 55864}, {3624, 63468, 4301}, {3634, 5493, 1699}, {3679, 16192, 4297}, {3817, 64850, 7486}, {3828, 51118, 7989}, {3832, 46932, 10175}, {3913, 24477, 6764}, {4292, 31434, 5261}, {4297, 16192, 10304}, {4301, 58441, 3624}, {4413, 64077, 6915}, {4512, 8582, 5129}, {4640, 37828, 2551}, {4662, 10178, 12680}, {5010, 10573, 4305}, {5059, 54448, 31673}, {5068, 46931, 54447}, {5071, 50809, 28198}, {5217, 40663, 3486}, {5432, 37567, 3485}, {5552, 56288, 329}, {5587, 31730, 3146}, {5657, 64108, 5731}, {5691, 12512, 20}, {5709, 37407, 9776}, {5731, 5775, 9803}, {5882, 63143, 3621}, {6908, 55104, 329}, {6988, 64107, 27383}, {7967, 10299, 13624}, {7982, 10165, 3622}, {7988, 51073, 46936}, {7989, 51118, 3839}, {8148, 15720, 38028}, {8703, 38066, 34627}, {9589, 64850, 3817}, {9778, 9780, 4}, {9785, 64114, 3086}, {9812, 19877, 5}, {9961, 63961, 5777}, {10039, 58887, 4293}, {10164, 43174, 1}, {10164, 59417, 54445}, {10171, 19872, 46935}, {10175, 41869, 3832}, {11231, 12699, 3090}, {11362, 31425, 15717}, {11415, 27529, 5748}, {11495, 38057, 36991}, {11500, 14647, 9799}, {12667, 64118, 54052}, {14217, 38133, 66063}, {15022, 46930, 10172}, {15705, 31145, 51705}, {15719, 34631, 3653}, {15721, 50872, 25055}, {15803, 31397, 3600}, {18483, 54447, 5068}, {19875, 50808, 3543}, {19875, 64005, 19925}, {19925, 50808, 64005}, {19925, 64005, 3543}, {21075, 54290, 144}, {22791, 61614, 3526}, {24914, 37568, 497}, {24982, 35258, 452}, {25055, 50829, 15721}, {31162, 38068, 2}, {31508, 67931, 4314}, {31658, 35514, 52653}, {31663, 50821, 355}, {31788, 64107, 3869}, {34627, 38066, 51068}, {34632, 68034, 962}, {37421, 63985, 64696}, {38118, 64084, 51171}, {38327, 62432, 47776}, {51784, 53056, 4298}, {54357, 63141, 37434}, {59491, 63130, 64081}, {60912, 63971, 6172}, {63143, 67706, 5882}, {64951, 67980, 3488}

X(68546) = X(30)X(340)∩X(74)X(186)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)^2*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4)^2 : :
X(68546) = 3 X[4] - 2 X[43911]

See Antreas Hatzipolakis and Peter Moses, euclid 8489.

X(68546) lies on these lines: {4, 5627}, {25, 52646}, {30, 340}, {74, 186}, {185, 38933}, {250, 5663}, {378, 10419}, {403, 16080}, {562, 14989}, {1515, 47146}, {1524, 36308}, {1525, 36311}, {1552, 10151}, {1597, 9139}, {1835, 36119}, {2071, 14919}, {2693, 2972}, {3270, 60798}, {3426, 59145}, {3470, 3520}, {5668, 39378}, {5669, 39377}, {5962, 10152}, {7480, 15054}, {8749, 14581}, {9717, 11410}, {10605, 57488}, {11079, 39176}, {12028, 39290}, {12133, 40355}, {12292, 32710}, {13754, 44769}, {13851, 57472}, {14385, 35473}, {15318, 36162}, {18808, 53158}, {18859, 44715}, {20417, 57587}, {22455, 40352}, {47327, 48364}, {52403, 62722}, {54585, 61347}, {56686, 59434}, {57471, 64890}

X(68546) = reflection of X(i) in X(j) for these {i,j}: {74, 34329}, {10421, 17986}
X(68546) = isogonal conjugate of X(16163)
X(68546) = polar conjugate of X(36789)
X(68546) = isogonal conjugate of the anticomplement of X(7687)
X(68546) = isogonal conjugate of the complement of X(10733)
X(68546) = polar conjugate of the isotomic conjugate of X(40384)
X(68546) = polar conjugate of the isogonal conjugate of X(40353)
X(68546) = X(i)-cross conjugate of X(j) for these (i,j): {25, 8749}, {512, 32695}, {12133, 4}, {20975, 2433}, {40353, 40384}, {40355, 40388}, {56792, 18808} X(68546) = X(i)-isoconjugate of X(j) for these (i,j): {1, 16163}, {3, 1099}, {48, 36789}, {63, 3163}, {69, 42074}, {77, 6062}, {78, 1354}, {162, 68167}, {163, 52624}, {255, 34334}, {304, 9408}, {326, 16240}, {336, 58343}, {656, 3233}, {662, 14401}, {811, 58345}, {905, 68126}, {1459, 68104}, {1553, 36062}, {1636, 24001}, {1784, 51394}, {2173, 11064}, {2407, 2631}, {3284, 14206}, {4575, 58263}, {4592, 58346}, {23097, 35200}, {35201, 51254}, {41077, 56829}, {55202, 58344}
X(68546) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 16163}, {115, 52624}, {125, 68167}, {133, 23097}, {136, 58263}, {1084, 14401}, {1249, 36789}, {3162, 3163}, {5139, 58346}, {6523, 34334}, {14385, 59495}, {15259, 16240}, {17423, 58345}, {18809, 1553}, {36103, 1099}, {36896, 11064}, {40596, 3233}
X(68546) = cevapoint of X(i) and X(j) for these (i,j): {25, 8749}, {74, 14264}, {2433, 20975}
X(68546) = crosssum of X(i) and X(j) for these (i,j): {30, 15774}, {39008, 58345}
X(68546) = trilinear pole of line {2433, 8749}
X(68546) = crossdifference of every pair of points on line {14401, 68167}
X(68546) = barycentric product X(i)*X(j) for these {i,j}: {4, 40384}, {25, 31621}, {74, 16080}, {264, 40353}, {523, 34568}, {1138, 40391}, {1304, 2394}, {1494, 8749}, {1990, 59145}, {2349, 36119}, {2433, 16077}, {5627, 57487}, {14380, 15459}, {18808, 44769}, {20975, 57570}, {22455, 46808}, {32695, 34767}, {40388, 65715}, {44427, 67756}, {52646, 66764}
X(68546) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 36789}, {6, 16163}, {19, 1099}, {25, 3163}, {74, 11064}, {112, 3233}, {393, 34334}, {512, 14401}, {523, 52624}, {607, 6062}, {608, 1354}, {647, 68167}, {1304, 2407}, {1783, 68104}, {1973, 42074}, {1974, 9408}, {1990, 23097}, {2207, 16240}, {2211, 58343}, {2394, 66073}, {2433, 9033}, {2489, 58346}, {2501, 58263}, {3049, 58345}, {5627, 57482}, {8749, 30}, {8750, 68126}, {11079, 51254}, {12079, 65753}, {14264, 62569}, {14380, 41077}, {14581, 3081}, {16080, 3260}, {17994, 58351}, {18808, 41079}, {18877, 51394}, {20975, 39008}, {22455, 46809}, {31621, 305}, {32695, 4240}, {32696, 65777}, {32715, 2420}, {34568, 99}, {35908, 51389}, {36119, 14206}, {40351, 9407}, {40352, 3284}, {40353, 3}, {40354, 1495}, {40355, 56399}, {40384, 69}, {40388, 15454}, {40391, 1272}, {41489, 38956}, {44102, 58347}, {47228, 1553}, {51544, 1531}, {51821, 47405}, {51964, 40948}, {52475, 66122}, {52646, 62583}, {52933, 65323}, {57204, 58344}, {57487, 6148}, {62176, 58352}, {67756, 60053}
X(68546) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {74, 52493, 1304}, {378, 14264, 57487}, {403, 40630, 16080}, {38937, 57487, 378}

X(68547) = X(4)X(6)∩X(30)X(5972)

Barycentrics    6*a^10 - 3*a^8*b^2 - 20*a^6*b^4 + 18*a^4*b^6 + 6*a^2*b^8 - 7*b^10 - 3*a^8*c^2 + 40*a^6*b^2*c^2 - 18*a^4*b^4*c^2 - 40*a^2*b^6*c^2 + 21*b^8*c^2 - 20*a^6*c^4 - 18*a^4*b^2*c^4 + 68*a^2*b^4*c^4 - 14*b^6*c^4 + 18*a^4*c^6 - 40*a^2*b^2*c^6 - 14*b^4*c^6 + 6*a^2*c^8 + 21*b^2*c^8 - 7*c^10 : :
X(68547) = 7 X[4] + X[12112], 3 X[4] + X[32111], 7 X[1514] - X[12112], 3 X[1514] - X[32111], 3 X[12112] - 7 X[32111], X[15448] + 3 X[51998], X[125] - 3 X[10151], 3 X[403] + X[10721], 5 X[3091] - X[50434], X[12121] - 3 X[51425], X[12121] + 3 X[64891], 2 X[13473] + X[15152], 3 X[16227] - X[17854], 5 X[17578] + 3 X[35265], X[32110] - 3 X[47332]

See Antreas Hatzipolakis and Peter Moses, euclid 8489.

X(68547) lies on these lines: {2, 20725}, {4, 6}, {30, 5972}, {98, 54620}, {113, 47309}, {125, 10151}, {186, 50709}, {403, 10721}, {468, 13202}, {511, 65095}, {524, 46988}, {1531, 29181}, {1533, 47339}, {1539, 46085}, {1853, 68010}, {2777, 37984}, {2935, 37777}, {3066, 3839}, {3091, 50434}, {3542, 68058}, {3543, 35266}, {3564, 38791}, {3627, 9820}, {3845, 7706}, {3853, 67869}, {3861, 13474}, {5895, 18931}, {6000, 11746}, {6623, 61721}, {6696, 35488}, {9826, 14915}, {10706, 64104}, {11064, 62288}, {12102, 13419}, {12121, 51425}, {12250, 68009}, {12362, 33751}, {13293, 44272}, {13473, 15152}, {13567, 64094}, {13568, 43589}, {16227, 17854}, {17578, 35265}, {18418, 44241}, {18918, 64714}, {29317, 47338}, {32110, 47332}, {33556, 62041}, {34584, 68319}, {35481, 58434}, {36201, 47457}, {37197, 51491}, {37853, 37911}, {44267, 58885}, {44279, 64035}, {44283, 44665}, {44438, 67868}, {49669, 66588}, {49670, 61680}, {52219, 62509}

X(68547) = midpoint of X(i) and X(j) for these {i,j}: {4, 1514}, {113, 47309}, {468, 13202}, {1533, 47339}, {1539, 47336}, {3543, 35266}, {3627, 46817}, {11064, 62288}, {13473, 51403}, {44267, 58885}, {51425, 64891}
X(68547) = reflection of X(i) in X(j) for these {i,j}: {15152, 51403}, {37853, 37911}, {47296, 37984}
X(68547) = complement of X(20725)
X(68547) = complement of the isogonal conjugate of X(20726)
X(68547) = X(20726)-complementary conjugate of X(10)
X(68547) = {X(4),X(5893)}-harmonic conjugate of X(12241)

X(68548) = X(4)X(9)∩X(11)X(515)

Barycentrics    2*a^7 - 3*a^6*b + 3*a^4*b^3 - 6*a^3*b^4 + 3*a^2*b^5 + 4*a*b^6 - 3*b^7 - 3*a^6*c + 12*a^5*b*c - 7*a^4*b^2*c + 7*a^2*b^4*c - 12*a*b^5*c + 3*b^6*c - 7*a^4*b*c^2 + 12*a^3*b^2*c^2 - 10*a^2*b^3*c^2 - 4*a*b^4*c^2 + 9*b^5*c^2 + 3*a^4*c^3 - 10*a^2*b^2*c^3 + 24*a*b^3*c^3 - 9*b^4*c^3 - 6*a^3*c^4 + 7*a^2*b*c^4 - 4*a*b^2*c^4 - 9*b^3*c^4 + 3*a^2*c^5 - 12*a*b*c^5 + 9*b^2*c^5 + 4*a*c^6 + 3*b*c^6 - 3*c^7 : :
X(68548) = 3 X[4] + X[48363], 3 X[1512] - X[48363], 2 X[11715] - 3 X[44675], X[11715] - 3 X[67857], 3 X[1519] - X[10698], 3 X[1737] - X[1768], X[1768] + 3 X[41698], X[17613] - 3 X[34122]

See Antreas Hatzipolakis and Peter Moses, euclid 8489.

X(68548) lies on these lines: {1, 38307}, {4, 9}, {11, 515}, {84, 54361}, {119, 6745}, {153, 26015}, {226, 64731}, {355, 63989}, {381, 10863}, {497, 7966}, {499, 5691}, {517, 14740}, {518, 38757}, {519, 1519}, {899, 33810}, {912, 13227}, {944, 37704}, {946, 2098}, {950, 18242}, {952, 1538}, {971, 12019}, {1012, 4413}, {1125, 6941}, {1155, 52836}, {1210, 6256}, {1478, 7682}, {1537, 28234}, {1541, 28292}, {1699, 8275}, {1737, 1768}, {1750, 64320}, {1776, 66058}, {1785, 36127}, {1837, 6260}, {1877, 51375}, {2829, 3911}, {3091, 19861}, {3149, 5204}, {3419, 67874}, {3486, 63966}, {3576, 6969}, {3585, 64001}, {3586, 64148}, {3634, 6906}, {3715, 38127}, {3817, 6968}, {3872, 18529}, {4292, 67860}, {4297, 6834}, {4311, 15866}, {4314, 10786}, {4316, 50701}, {4679, 64315}, {4848, 64119}, {5175, 67881}, {5229, 67880}, {5261, 64669}, {5290, 5804}, {5603, 51782}, {5768, 64697}, {5795, 15908}, {5853, 37725}, {5884, 41560}, {6245, 10826}, {6264, 28236}, {6705, 17606}, {6713, 28160}, {6737, 45631}, {6847, 7989}, {6850, 8582}, {6905, 28164}, {6929, 40998}, {6935, 54447}, {6938, 10164}, {6950, 58441}, {6973, 37611}, {6977, 51073}, {7681, 10106}, {8727, 38140}, {9581, 12667}, {9856, 18357}, {10058, 44425}, {10573, 54198}, {10591, 12650}, {10624, 10953}, {10724, 63145}, {10805, 21625}, {10893, 12053}, {11019, 12115}, {11373, 52683}, {11525, 59388}, {11826, 63990}, {12527, 37821}, {12608, 64163}, {12678, 61717}, {12736, 34293}, {13411, 63964}, {13729, 24987}, {15558, 65948}, {16616, 65949}, {17010, 64188}, {17613, 34122}, {17647, 67046}, {18239, 67937}, {18516, 51755}, {19541, 22758}, {22792, 67980}, {22837, 63986}, {24914, 37001}, {24982, 37437}, {25011, 37163}, {26333, 31397}, {28172, 38133}, {31140, 50796}, {37730, 64813}, {37828, 64725}, {37837, 66247}, {64041, 67877}

X(68548) = midpoint of X(i) and X(j) for these {i,j}: {4, 1512}, {153, 26015}, {1155, 52836}, {1737, 41698}, {5691, 21578}, {10724, 63145}
X(68548) = reflection of X(i) in X(j) for these {i,j}: {6745, 119}, {44675, 67857}
X(68548) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 10, 66992}, {4, 5818, 12705}, {17606, 64000, 6705}

X(68549) = 104TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    2*a^8 + a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - 3*b^8 + a^6*c^2 + 2*a^4*b^2*c^2 - 3*a^2*b^4*c^2 + 8*b^6*c^2 - 3*a^4*c^4 - 3*a^2*b^2*c^4 - 10*b^4*c^4 + 3*a^2*c^6 + 8*b^2*c^6 - 3*c^8 : :
X(68549) = 3 X[4] + X[11676], X[20] - 3 X[35297], 3 X[381] - X[15980], 3 X[403] - X[36166], 3 X[403] - 2 X[64966], 3 X[1513] - X[11676], 7 X[3090] - 5 X[40336], 5 X[3091] - X[5999], 5 X[3091] - 3 X[33228], X[3146] + 3 X[13586], 3 X[3545] - 2 X[8355], 7 X[3832] - 3 X[14041], 7 X[3832] + X[40236], 3 X[3839] - X[8352], X[5999] - 3 X[33228], X[8597] - 5 X[61985], X[9855] + 3 X[50687], 3 X[14041] + X[40236], 2 X[15980] - 3 X[37350], 5 X[17578] + 3 X[33265], 3 X[27088] - 4 X[37459], X[40246] - 9 X[61992], 3 X[52403] + X[57616], X[98] - 3 X[39663], 3 X[39663] - 2 X[43291], X[3793] + 4 X[22505], 4 X[6721] - 3 X[10256], X[10722] + 3 X[38227], 3 X[14639] + X[43460], X[18860] - 3 X[36519], 3 X[36519] - 2 X[44377], 3 X[23514] - X[58849], 3 X[38737] - 4 X[44381], 3 X[53023] - X[53505]

See Antreas Hatzipolakis and Peter Moses, euclid 8489.

X(68549) lies on these lines: {2, 3}, {6, 7694}, {98, 39663}, {114, 6390}, {115, 1503}, {127, 52458}, {132, 60428}, {147, 47286}, {187, 39838}, {230, 2794}, {339, 44145}, {511, 46096}, {538, 38745}, {620, 63440}, {1352, 64093}, {1384, 9752}, {1499, 1514}, {1550, 32111}, {1555, 5512}, {2023, 53419}, {2549, 64711}, {3163, 15471}, {3424, 54859}, {3564, 6033}, {3793, 22505}, {5028, 5475}, {5031, 29181}, {5033, 18424}, {5099, 62509}, {5309, 8550}, {5355, 12007}, {6054, 52229}, {6529, 6530}, {6721, 10256}, {7615, 47353}, {7697, 18358}, {7710, 43448}, {7737, 53017}, {7745, 67854}, {7746, 36997}, {7767, 54393}, {8719, 43619}, {8721, 44518}, {9744, 15048}, {9753, 18907}, {9756, 43620}, {10008, 32827}, {10722, 38227}, {11180, 40727}, {13748, 49220}, {13749, 49221}, {13881, 59363}, {14160, 48895}, {14356, 47581}, {14639, 43460}, {14853, 15484}, {14995, 47545}, {15069, 63955}, {16324, 38393}, {16334, 62551}, {18860, 36519}, {22682, 43457}, {23514, 58849}, {24256, 67865}, {32459, 38738}, {32515, 38383}, {36990, 53499}, {37689, 53016}, {38737, 44381}, {38747, 58448}, {41005, 41762}, {42275, 66426}, {42535, 53418}, {47219, 62507}, {47474, 66167}, {48983, 51258}, {51441, 51943}, {53023, 53505}, {54131, 66466}

X(68549) = midpoint of X(i) and X(j) for these {i,j}: {4, 1513}, {147, 47286}, {187, 39838}, {1550, 32111}, {3543, 8598}, {3830, 37461}, {7472, 62288}, {36990, 53499}, {41016, 41017}, {41044, 41045}, {51212, 51438}
X(68549) = reflection of X(i) in X(j) for these {i,j}: {3, 10011}, {98, 43291}, {230, 67872}, {6390, 114}, {14120, 37984}, {18860, 44377}, {36166, 64966}, {37350, 381}, {38738, 32459}, {38747, 58448}, {53419, 67863}, {56370, 5}, {63440, 620}
X(68549) = complement of X(54996)
X(68549) = crossdifference of every pair of points on line {647, 6090}
X(68549) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 5, 8361}, {5, 3845, 40250}, {5, 33185, 3090}, {5, 37348, 66415}, {5, 44224, 3628}, {20, 32970, 3}, {98, 39663, 43291}, {381, 37348, 5}, {381, 57634, 3363}, {403, 36166, 64966}, {460, 2450, 441}, {468, 868, 44216}, {868, 50707, 468}, {1529, 10151, 4}, {2043, 2044, 11318}, {3091, 5999, 33228}, {3091, 7770, 5}, {3832, 40236, 14041}, {18860, 36519, 44377}, {31693, 31694, 8355}, {31693, 41016, 1080}, {31694, 41017, 383}, {37242, 37451, 8359}

X(68550) = X(4)X(69)∩X(30)X(5972)

Barycentrics    4*a^10 - 3*a^8*b^2 - 11*a^6*b^4 + 11*a^4*b^6 + 3*a^2*b^8 - 4*b^10 - 3*a^8*c^2 + 16*a^6*b^2*c^2 - 7*a^4*b^4*c^2 - 18*a^2*b^6*c^2 + 12*b^8*c^2 - 11*a^6*c^4 - 7*a^4*b^2*c^4 + 30*a^2*b^4*c^4 - 8*b^6*c^4 + 11*a^4*c^6 - 18*a^2*b^2*c^6 - 8*b^4*c^6 + 3*a^2*c^8 + 12*b^2*c^8 - 4*c^10 : :
X(68550) = X[41721] - 5 X[51537], X[32237] - 4 X[46686], 3 X[381] - X[32110], 3 X[2072] - X[16111], 5 X[3091] - 3 X[61691], X[3448] - 3 X[13851], X[3581] - 5 X[3843], 3 X[3830] + X[37477], 3 X[3839] - X[32225], X[7728] + 3 X[18403], 3 X[10151] - X[32269], X[10295] - 3 X[36518], 5 X[15059] - 3 X[21663], X[15360] - 5 X[61985], X[15361] - 3 X[23046], 3 X[15362] - 7 X[61974], X[37496] + 7 X[62008], 3 X[18449] + X[48662], X[20725] - 3 X[47097], 3 X[38789] + X[58789], 3 X[48375] - 2 X[66595], 3 X[53023] - X[53777], X[56369] - 5 X[64101]

See Antreas Hatzipolakis and Peter Moses, euclid 8489.

X(68550) lies on these lines: {4, 69}, {30, 5972}, {113, 18323}, {381, 32110}, {382, 10564}, {542, 47277}, {858, 13202}, {1495, 10296}, {1503, 38791}, {1514, 29012}, {1533, 29323}, {1539, 14915}, {1568, 57584}, {2072, 16111}, {2777, 10297}, {3091, 61691}, {3292, 10733}, {3448, 13851}, {3543, 13857}, {3581, 3843}, {3627, 51391}, {3830, 9306}, {3839, 32225}, {3845, 21243}, {5159, 37853}, {5893, 44829}, {5943, 7706}, {6000, 7728}, {8352, 15595}, {10113, 11800}, {10151, 32269}, {10295, 36518}, {11645, 32111}, {12102, 64035}, {12162, 18430}, {12900, 47335}, {13348, 50009}, {13473, 46682}, {13474, 18377}, {14826, 62007}, {15059, 21663}, {15082, 50008}, {15125, 32355}, {15360, 61985}, {15361, 23046}, {15362, 61974}, {15644, 44279}, {15682, 59543}, {16976, 50709}, {17702, 58885}, {17814, 37496}, {18404, 46850}, {18405, 18449}, {18418, 18533}, {18550, 40280}, {19510, 29181}, {19924, 47310}, {20725, 47097}, {31173, 34360}, {32223, 37984}, {34584, 58871}, {34664, 58445}, {38789, 58789}, {40647, 58546}, {44263, 67891}, {44668, 65095}, {44673, 63838}, {44872, 63735}, {47296, 63821}, {47308, 48378}, {48375, 66595}, {51360, 62288}, {51392, 64891}, {51394, 64890}, {52219, 62490}, {53023, 53777}, {56369, 64101}, {59659, 62026}, {61990, 67878}

X(68550) = midpoint of X(i) and X(j) for these {i,j}: {4, 1531}, {113, 18323}, {382, 10564}, {858, 13202}, {1495, 10296}, {1514, 47339}, {1539, 18572}, {1568, 57584}, {3292, 10733}, {3543, 13857}, {3627, 51391}, {51360, 62288}, {51392, 64891}, {51394, 64890}
X(68550) = reflection of X(i) in X(j) for these {i,j}: {32223, 37984}, {37853, 5159}, {44673, 63838}, {47296, 63821}, {47308, 48378}, {47335, 12900}, {63735, 44872}

X(68551) = X(103)X(2271)∩X(677)X(916)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^3 - a^2*b - a*b^2 + b^3 + a*c^2 + b*c^2 - 2*c^3)^2*(a^3 + a*b^2 - 2*b^3 - a^2*c + b^2*c - a*c^2 + c^3)^2 : :

See Antreas Hatzipolakis and Peter Moses, euclid 8489.

X(68551) lies on these lines: {4, 14505}, {103, 2272}, {677, 916}, {971, 7291}, {1876, 36122}, {7053, 56640}, {15380, 54232}, {18344, 30204}, {37908, 59195}, {45250, 63434}, {52305, 53150}

X(68551) = isogonal conjugate of X(65745)
X(68551) = polar conjugate of X(59206)
X(68551) = polar conjugate of the isotomic conjugate of X(59195)
X(68551) = X(i)-cross conjugate of X(j) for these (i,j): {3937, 2424}, {56787, 53150}
X(68551) = X(i)-isoconjugate of X(j) for these (i,j): {1, 65745}, {3, 24014}, {48, 59206}, {63, 23972}, {69, 42077}, {78, 1360}, {255, 21665}, {304, 59799}, {326, 42073}, {521, 66976}, {603, 55019}, {905, 3234}, {906, 58280}, {910, 26006}, {43035, 51376}
X(68553) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 65745}, {1249, 59206}, {3162, 23972}, {5190, 58280}, {6523, 21665}, {7952, 55019}, {15259, 42073}, {36103, 24014}
X(68551) = cevapoint of X(i) and X(j) for these (i,j): {103, 54232}, {2424, 3937}
X(68551) = barycentric product X(i)*X(j) for these {i,j}: {4, 59195}, {25, 57548}, {103, 52781}, {677, 53150}, {2400, 40116}, {36101, 36122}
X(68551) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 59206}, {6, 65745}, {19, 24014}, {25, 23972}, {103, 26006}, {281, 55019}, {393, 21665}, {608, 1360}, {1973, 42077}, {1974, 59799}, {2207, 42073}, {2424, 39470}, {7649, 58280}, {8750, 3234}, {32674, 66976}, {36122, 30807}, {40116, 2398}, {52781, 35517}, {54232, 62591}, {57548, 305}, {59195, 69}, {65218, 42719}

X(68552) = X(4)X(279)∩X(5)X(169)

Barycentrics    2*a^8 - 2*a^7*b + a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 + 2*a^3*b^5 + 3*a^2*b^6 + 2*a*b^7 - 3*b^8 - 2*a^7*c + 2*a^5*b^2*c + 4*a^4*b^3*c + 2*a^3*b^4*c - 8*a^2*b^5*c - 2*a*b^6*c + 4*b^7*c + a^6*c^2 + 2*a^5*b*c^2 - 2*a^4*b^2*c^2 - 4*a^3*b^3*c^2 + 5*a^2*b^4*c^2 - 6*a*b^5*c^2 + 4*b^6*c^2 - 2*a^5*c^3 + 4*a^4*b*c^3 - 4*a^3*b^2*c^3 + 6*a*b^4*c^3 - 4*b^5*c^3 - 3*a^4*c^4 + 2*a^3*b*c^4 + 5*a^2*b^2*c^4 + 6*a*b^3*c^4 - 2*b^4*c^4 + 2*a^3*c^5 - 8*a^2*b*c^5 - 6*a*b^2*c^5 - 4*b^3*c^5 + 3*a^2*c^6 - 2*a*b*c^6 + 4*b^2*c^6 + 2*a*c^7 + 4*b*c^7 - 3*c^8 : :
X(68552) = X[103] - 3 X[61673], 5 X[3091] - X[3732], 3 X[3817] - X[51435], X[10725] + 3 X[67625], 5 X[31273] - X[67721]

See Antreas Hatzipolakis and Peter Moses, euclid 8489.

X(68552) lies on these lines: {2, 65745}, {4, 279}, {5, 169}, {11, 57}, {103, 61673}, {105, 44431}, {116, 31851}, {118, 5845}, {226, 15252}, {381, 42048}, {515, 34929}, {516, 6712}, {946, 15251}, {952, 10695}, {971, 1541}, {1146, 53804}, {1352, 64884}, {1530, 9436}, {1536, 4872}, {1750, 56848}, {2826, 65948}, {3091, 3732}, {3817, 51435}, {5762, 8558}, {5806, 35650}, {5972, 6678}, {10725, 67625}, {17044, 64502}, {21665, 23989}, {31273, 67721}, {31852, 40483}, {33331, 35967}, {46017, 64157}, {58898, 64500}

X(68552) = midpoint of X(i) and X(j) for these {i,j}: {4, 1565}, {116, 31851}, {1146, 67568}, {1530, 9436}, {1536, 4872}
X(68552) = reflection of X(i) in X(j) for these {i,j}: {31852, 40483}, {65808, 5}
X(68552) = complement of X(65745)
X(68552) = X(i)-complementary conjugate of X(j) for these (i,j): {36122, 118}, {59195, 18589}
X(68552) = X(23973)-Ceva conjugate of X(514)

X(68553) = X(4)X(1562)∩X(132)X(2777)

Barycentrics    2*a^14 - 2*a^12*b^2 - 5*a^10*b^4 + 13*a^8*b^6 - 16*a^6*b^8 + 8*a^4*b^10 + 3*a^2*b^12 - 3*b^14 - 2*a^12*c^2 + 12*a^10*b^2*c^2 - 13*a^8*b^4*c^2 + 8*a^6*b^6*c^2 - 4*a^4*b^8*c^2 - 12*a^2*b^10*c^2 + 11*b^12*c^2 - 5*a^10*c^4 - 13*a^8*b^2*c^4 + 16*a^6*b^4*c^4 - 4*a^4*b^6*c^4 + 21*a^2*b^8*c^4 - 15*b^10*c^4 + 13*a^8*c^6 + 8*a^6*b^2*c^6 - 4*a^4*b^4*c^6 - 24*a^2*b^6*c^6 + 7*b^8*c^6 - 16*a^6*c^8 - 4*a^4*b^2*c^8 + 21*a^2*b^4*c^8 + 7*b^6*c^8 + 8*a^4*c^10 - 12*a^2*b^2*c^10 - 15*b^4*c^10 + 3*a^2*c^12 + 11*b^2*c^12 - 3*c^14 : :
X(68553) = 3 X[6793] - X[13200], 3 X[6794] + X[10735], 3 X[6794] - X[65749], X[14689] - 3 X[67217]

See Antreas Hatzipolakis and Peter Moses, euclid 8489.

X(68553) lies on these lines: {4, 1562}, {6, 39838}, {30, 46097}, {132, 2777}, {389, 46098}, {525, 66594}, {690, 7687}, {2794, 43278}, {5480, 10169}, {6793, 13200}, {6794, 10735}, {14689, 67217}, {44673, 67872}

X(68553) = midpoint of X(i) and X(j) for these {i,j}: {4, 1562}, {10735, 65749}
X(68553) = crosssum of X(3) and X(38608)
X(68553) = {X(6794),X(10735)}-harmonic conjugate of X(65749)

X(68554) = X(4)X(542)∩X(381)X(67204)

Barycentrics    2*a^10 - 6*a^8*b^2 - 23*a^6*b^4 + 21*a^4*b^6 + 9*a^2*b^8 - 3*b^10 - 6*a^8*c^2 + 64*a^6*b^2*c^2 - 25*a^4*b^4*c^2 - 82*a^2*b^6*c^2 + 17*b^8*c^2 - 23*a^6*c^4 - 25*a^4*b^2*c^4 + 146*a^2*b^4*c^4 - 14*b^6*c^4 + 21*a^4*c^6 - 82*a^2*b^2*c^6 - 14*b^4*c^6 + 9*a^2*c^8 + 17*b^2*c^8 - 3*c^10 : :

See Antreas Hatzipolakis and Peter Moses, euclid 8489.

X(68554) lies on these lines: {4, 542}, {381, 67204}, {5969, 38801}, {5972, 57594}, {7417, 37853}, {13168, 29317}, {19662, 57624}, {38803, 67399}, {39838, 67490}, {45311, 57604}, {47589, 50959}

X(68554) = midpoint of X(i) and X(j) for these {i,j}: {4, 14856}, {39838, 67490}
X(68554) = reflection of X(38803) in X(67399)

X(68555) = X(4)X(94)∩X(30)X(5972)

Barycentrics    4*a^10 - 3*a^8*b^2 - 11*a^6*b^4 + 11*a^4*b^6 + 3*a^2*b^8 - 4*b^10 - 3*a^8*c^2 + 22*a^6*b^2*c^2 - 10*a^4*b^4*c^2 - 21*a^2*b^6*c^2 + 12*b^8*c^2 - 11*a^6*c^4 - 10*a^4*b^2*c^4 + 36*a^2*b^4*c^4 - 8*b^6*c^4 + 11*a^4*c^6 - 21*a^2*b^2*c^6 - 8*b^4*c^6 + 3*a^2*c^8 + 12*b^2*c^8 - 4*c^10 : :
X(68555) = 7 X[4] + X[146], 5 X[4] - X[265], 9 X[4] - X[3448], 3 X[4] + X[7728], 3 X[4] - X[10113], 17 X[4] - X[12317], 5 X[146] + 7 X[265], X[146] - 7 X[1539], 9 X[146] + 7 X[3448], 3 X[146] - 7 X[7728], 3 X[146] + 7 X[10113], 17 X[146] + 7 X[12317], X[265] + 5 X[1539], 9 X[265] - 5 X[3448], 3 X[265] + 5 X[7728], 3 X[265] - 5 X[10113], 17 X[265] - 5 X[12317], 9 X[1539] + X[3448], 3 X[1539] - X[7728], 3 X[1539] + X[10113], 17 X[1539] + X[12317], X[3448] + 3 X[7728], X[3448] - 3 X[10113], 17 X[3448] - 9 X[12317], 17 X[7728] + 3 X[12317], 17 X[10113] - 3 X[12317], 3 X[5] - X[16111], 5 X[5] - 3 X[38727], 3 X[13202] + X[16111], 5 X[13202] + 3 X[38727], 5 X[16111] - 9 X[38727], 5 X[5972] - 3 X[38726], X[5972] - 3 X[46686], 2 X[5972] - 3 X[61574], 11 X[5972] - 9 X[68316], 5 X[5972] - 9 X[68317], X[38726] - 5 X[46686], 2 X[38726] - 5 X[61574], 11 X[38726] - 15 X[68316], X[38726] - 3 X[68317], 11 X[46686] - 3 X[68316], 5 X[46686] - 3 X[68317], 11 X[61574] - 6 X[68316], 5 X[61574] - 6 X[68317], 5 X[68316] - 11 X[68317], X[74] - 5 X[3843], 5 X[3843] - 2 X[20396], X[110] + 3 X[3830], 3 X[113] - X[34153], 3 X[3627] + X[34153], X[125] - 3 X[3845], 3 X[381] + X[10721], 3 X[381] - X[12041], 9 X[381] - 5 X[15059], 3 X[381] - 2 X[15088], 3 X[10721] + 5 X[15059], X[10721] + 2 X[15088], 3 X[12041] - 5 X[15059], 5 X[15059] - 6 X[15088], X[399] + 7 X[62008], X[11801] - 3 X[14893], 3 X[12101] + X[61598], 5 X[546] - 2 X[20397], 3 X[546] - X[61548], 5 X[20304] - 4 X[20397], 3 X[20304] - 2 X[61548], 6 X[20397] - 5 X[61548], X[550] - 3 X[36518], X[1657] - 5 X[64101], 7 X[3090] - 3 X[38788], 5 X[3091] - X[20127], 5 X[3091] - 3 X[34128], X[20127] - 3 X[34128], X[3146] + 3 X[14643], X[3529] - 5 X[38794], 2 X[3530] - 3 X[68280], 3 X[3543] + X[12121], 9 X[3545] - 5 X[38728], 7 X[3832] - 3 X[15061], 9 X[3839] - X[12244], 7 X[3851] - 3 X[15055], 5 X[3858] - X[14677], 5 X[3858] - 3 X[23515], X[14677] - 3 X[23515], 4 X[3861] - X[20379], 3 X[5066] - 2 X[6723], X[5073] + 3 X[15035], 5 X[5076] + X[5609], 5 X[5076] - X[10733], 5 X[5076] + 3 X[38789], X[5609] - 3 X[38789], X[10733] + 3 X[38789], X[5655] + 3 X[50687], 3 X[5946] - X[17854], X[9140] - 5 X[61993], X[9143] + 7 X[62009], X[10264] - 5 X[61988], X[38626] - 10 X[61988], 9 X[10272] - 7 X[22250], 7 X[22250] + 9 X[62026], X[10620] - 9 X[14269], 3 X[10706] + X[12902], X[10706] + 3 X[38335], X[12902] - 9 X[38335], X[10990] - 7 X[61976], 5 X[11439] - X[22584], X[11562] + 3 X[32062], X[11806] - 3 X[67067], 8 X[12102] + X[38632], 2 X[12102] + X[38791], X[38632] - 4 X[38791], X[12295] - 3 X[15687], 3 X[12295] + X[24981], 5 X[12295] + 3 X[56567], 9 X[15687] + X[24981], 5 X[15687] + X[56567], 5 X[24981] - 9 X[56567], X[12308] + 15 X[35403], X[12383] + 7 X[50688], X[13201] - 9 X[16261], 2 X[13392] - 3 X[38792], X[13417] + 3 X[16194], X[13491] - 3 X[16222], 3 X[16222] + X[46431], X[14094] + 11 X[62004], 3 X[14644] + X[38790], 3 X[14644] - X[51522], 3 X[14644] - 7 X[61984], X[38790] + 7 X[61984], X[51522] - 7 X[61984], X[14683] + 15 X[62007], X[14731] + 3 X[60741], 7 X[15020] + 5 X[62035], 5 X[15021] - 17 X[61968], 5 X[15027] - X[64102], 13 X[15029] - X[49137], 5 X[15034] + 7 X[62024], 7 X[15036] - 3 X[15681], 5 X[15040] + 3 X[15684], 3 X[15041] - 11 X[61970], 13 X[15042] - 5 X[62150], 9 X[15046] - 5 X[15051], 9 X[15046] - X[17800], 5 X[15051] - X[17800], X[15054] - 13 X[61991], X[15063] + 5 X[62006], 5 X[15081] - 13 X[61982], X[15704] - 3 X[38793], 3 X[16163] - 5 X[22251], 5 X[22251] + 3 X[62036], X[16534] + 2 X[62013], X[17856] - 3 X[45956], X[20126] - 5 X[61985], X[21317] - 3 X[31378], X[25329] - 3 X[32271], X[32269] - 3 X[47336], 3 X[32609] + 5 X[62023], X[33703] + 3 X[38723], 3 X[38141] - X[53715], 3 X[38724] - 11 X[61990], 5 X[38729] - 11 X[41991], 5 X[38795] + X[62044], 2 X[44245] - 3 X[48375], 3 X[51538] + X[63700], 3 X[62017] + X[64182]

See Antreas Hatzipolakis and Peter Moses, euclid 8489.

X(68555) lies on these lines: {4, 94}, {5, 13202}, {30, 5972}, {74, 3843}, {110, 3830}, {113, 3627}, {125, 3845}, {156, 15472}, {381, 10721}, {382, 1511}, {399, 62008}, {541, 11801}, {542, 12101}, {546, 2777}, {548, 12900}, {550, 36518}, {974, 10095}, {1154, 44283}, {1499, 12064}, {1531, 41673}, {1657, 64101}, {2781, 48889}, {2854, 48895}, {2935, 13861}, {3024, 18513}, {3028, 18514}, {3090, 38788}, {3091, 20127}, {3146, 14643}, {3529, 38794}, {3530, 68280}, {3543, 12121}, {3545, 38728}, {3628, 37853}, {3832, 15061}, {3839, 12244}, {3850, 6699}, {3851, 15055}, {3853, 17702}, {3856, 40685}, {3858, 14677}, {3860, 45311}, {3861, 7687}, {5066, 6723}, {5073, 15035}, {5076, 5609}, {5640, 18550}, {5642, 33699}, {5655, 50687}, {5893, 68424}, {5946, 17854}, {6593, 48884}, {6698, 67884}, {7706, 13364}, {8998, 42225}, {9140, 61993}, {9143, 62009}, {9934, 32046}, {10117, 31861}, {10263, 12825}, {10264, 38626}, {10272, 22250}, {10620, 14269}, {10628, 46849}, {10706, 12902}, {10990, 61976}, {11017, 34007}, {11439, 22584}, {11557, 13474}, {11561, 18567}, {11562, 32062}, {11591, 44279}, {11720, 33697}, {11723, 28186}, {11805, 32340}, {11806, 67067}, {12102, 32423}, {12103, 48378}, {12106, 13293}, {12295, 15687}, {12308, 35403}, {12358, 45958}, {12383, 50688}, {13201, 16261}, {13391, 31726}, {13392, 38792}, {13417, 16194}, {13473, 30522}, {13491, 16222}, {13630, 58516}, {13915, 42269}, {13979, 42268}, {13990, 42226}, {14094, 62004}, {14644, 38790}, {14683, 62007}, {14731, 60741}, {14915, 41671}, {14984, 48901}, {15020, 62035}, {15021, 61968}, {15027, 64102}, {15029, 49137}, {15034, 62024}, {15036, 15681}, {15040, 15684}, {15041, 61970}, {15042, 62150}, {15046, 15051}, {15054, 61991}, {15063, 62006}, {15081, 61982}, {15114, 15473}, {15704, 38793}, {16163, 22251}, {16168, 52219}, {16531, 50709}, {16534, 62013}, {17856, 45956}, {18323, 51548}, {18381, 64587}, {18400, 47117}, {18403, 44573}, {18572, 41670}, {18583, 36201}, {19051, 23263}, {19052, 23253}, {20126, 61985}, {21316, 32417}, {21317, 31378}, {23315, 46030}, {25329, 32271}, {32142, 50009}, {32171, 35490}, {32210, 35488}, {32269, 47336}, {32609, 62023}, {32743, 44235}, {33703, 38723}, {33851, 48904}, {35786, 49216}, {35787, 49217}, {38141, 53715}, {38724, 61990}, {38729, 41991}, {38795, 62044}, {44245, 48375}, {44271, 65095}, {44438, 64498}, {44863, 58498}, {46045, 66778}, {47309, 58885}, {51391, 62288}, {51538, 63700}, {54012, 68293}, {58536, 63684}, {62017, 64182}, {63716, 68427}

X(68555) = midpoint of X(i) and X(j) for these {i,j}: {4, 1539}, {5, 13202}, {113, 3627}, {382, 1511}, {1531, 44267}, {5609, 10733}, {5642, 33699}, {6593, 48884}, {7728, 10113}, {10263, 12825}, {10272, 62026}, {10721, 12041}, {11557, 13474}, {11720, 33697}, {11805, 32340}, {12292, 38898}, {13491, 46431}, {16163, 62036}, {18323, 51548}, {18381, 64587}, {19506, 64588}, {33851, 48904}, {38790, 51522}, {46045, 66778}, {47309, 58885}, {51391, 62288}
X(68555) = reflection of X(i) in X(j) for these {i,j}: {74, 20396}, {548, 12900}, {974, 10095}, {6698, 67884}, {6699, 3850}, {7687, 3861}, {12041, 15088}, {12103, 48378}, {12236, 67867}, {12358, 45958}, {13630, 58516}, {20304, 546}, {20379, 7687}, {37853, 3628}, {38626, 10264}, {40685, 3856}, {45311, 3860}, {58498, 44863}, {61574, 46686}, {63684, 58536}
X(68555) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 7728, 10113}, {381, 10721, 12041}, {381, 12041, 15088}, {1539, 10113, 7728}, {3091, 20127, 34128}, {3858, 14677, 23515}, {5076, 38789, 10733}, {10733, 38789, 5609}, {14644, 38790, 51522}, {15046, 17800, 15051}, {16222, 46431, 13491}, {38726, 46686, 68317}, {38790, 61984, 14644}

X(68556) = X(1)X(4)∩X(30)X(1538)

Barycentrics   2*a^7 - a^6*b - 2*a^5*b^2 - a^4*b^3 - 2*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6 - 3*b^7 - a^6*c + 4*a^5*b*c + a^4*b^2*c + 4*a^3*b^3*c - 3*a^2*b^4*c - 8*a*b^5*c + 3*b^6*c - 2*a^5*c^2 + a^4*b*c^2 - 4*a^3*b^2*c^2 - 2*a^2*b^3*c^2 - 2*a*b^4*c^2 + 9*b^5*c^2 - a^4*c^3 + 4*a^3*b*c^3 - 2*a^2*b^2*c^3 + 16*a*b^3*c^3 - 9*b^4*c^3 - 2*a^3*c^4 - 3*a^2*b*c^4 - 2*a*b^2*c^4 - 9*b^3*c^4 + 5*a^2*c^5 - 8*a*b*c^5 + 9*b^2*c^5 + 2*a*c^6 + 3*b*c^6 - 3*c^7 : :
X(68556) = 3 X[1699] - X[30384], 3 X[1699] + X[41698], X[3146] + 3 X[4881], 3 X[9812] + X[63136], 3 X[17533] - X[17613]

See Antreas Hatzipolakis and Peter Moses, euclid 8489.

X(68556) lies on these lines: {1, 4}, {5, 66992}, {30, 1538}, {57, 56941}, {84, 10591}, {117, 13539}, {165, 6969}, {496, 22792}, {499, 52860}, {516, 1532}, {517, 14740}, {519, 1537}, {546, 9856}, {962, 51433}, {1012, 3817}, {1210, 10893}, {1319, 52836}, {1512, 28194}, {1737, 34789}, {1836, 7682}, {1837, 54198}, {1878, 2817}, {2829, 22835}, {3091, 12705}, {3146, 4881}, {3149, 51118}, {3359, 6973}, {3434, 67874}, {3577, 38307}, {3667, 7661}, {3832, 67999}, {3847, 64128}, {3874, 41560}, {3880, 38757}, {3911, 67857}, {4292, 7681}, {4294, 63966}, {4302, 68003}, {4847, 37822}, {5187, 63985}, {5274, 63430}, {6001, 12736}, {6245, 7702}, {6259, 9669}, {6684, 6941}, {6700, 11826}, {6705, 7741}, {6736, 12700}, {6826, 10863}, {6830, 41858}, {6831, 12571}, {6834, 31730}, {6844, 8257}, {6848, 41869}, {6905, 28150}, {6907, 40998}, {6919, 37560}, {6927, 64005}, {6935, 7988}, {6938, 10165}, {6968, 10175}, {9580, 64148}, {9581, 63962}, {9668, 67889}, {9812, 63136}, {9814, 59386}, {10248, 50700}, {10309, 11023}, {10525, 63146}, {10589, 52027}, {10593, 34862}, {10598, 63399}, {10624, 18242}, {10698, 41702}, {11112, 17618}, {11238, 12678}, {11373, 40267}, {11376, 37001}, {12572, 15908}, {12672, 19925}, {15171, 64813}, {17533, 17613}, {21578, 52851}, {25681, 64725}, {28232, 48363}, {28236, 64137}, {30852, 64078}, {31224, 66759}, {34256, 46435}, {37437, 41012}, {37569, 66465}, {47745, 64203}, {50443, 64120}, {54156, 54361}, {59387, 68001}, {59391, 64155}

X(68556) = midpoint of X(i) and X(j) for these {i,j}: {4, 1519}, {962, 51433}, {1319, 52836}, {1737, 34789}, {21578, 52851}, {30384, 41698}
X(68556) = reflection of X(i) in X(j) for these {i,j}: {{3911, 67857}, {44675, 22835}
X(68556) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 63986, 31673}, {1478, 1699, 946}, {1699, 41698, 30384}, {1877, 35015, 51616}, {10525, 67855, 63146}, {10893, 64119, 1210}, {10896, 12679, 6245}, {18483, 67877, 1699}

X(68557) = (name pending)

Barycentrics    8 a^10 - 17 a^8 b^2 - 6 a^6 b^4 + 20 a^4 b^6 - 2 a^2 b^8 - 3 b^10 - 17 a^8 c^2 + 18 a^4 b^4 c^2 - 2 a^2 b^6 c^2 + 9 b^8 c^2 - 6 a^6 c^4 + 18 a^4 b^2 c^4 + 24 a^2 b^4 c^4 - 6 b^6 c^4 + 20 a^4 c^6 - 2 a^2 b^2 c^6 - 6 b^4 c^6 - 2 a^2 c^8 + 9 b^2 c^8 - 3 c^10 : :

As a point on the Euler line, X(68557) has Shinagawa coefficients: {(e + f)^3 - (9*e*S^2)/4 + 6*(e + f)*S^2, (27*e*S^2)/4 - (e + f)*((e + f)^2 + 12*S^2)}

See Tran Quang Hung and Francisco Javier García Capitán, euclid 8494.

X(68557) lies on this line: {2, 3}

X(68558) = X(42)X(72)∩X(181)X(201)

Barycentrics   a*(b+c)^2*((a+b)^2+c^2)*((a+c)^2+b^2) : :

See Jean-Louis Ayme and César Lozada, euclid 8498.

X(68058) lies on these lines: {1, 1472}, {10, 18697}, {37, 2333}, {38, 2221}, {42, 72}, {55, 51686}, {181, 201}, {291, 337}, {756, 3695}, {872, 6042}, {943, 968}, {976, 1036}, {984, 30479}, {1062, 10448}, {1126, 3751}, {1215, 43677}, {1254, 6356}, {1310, 28482}, {1500, 3949}, {2049, 17872}, {4424, 56192}, {5295, 23668}, {7148, 23928}, {18673, 42550}, {24443, 29647}, {32691, 43659}, {41340, 59305}

X(68058) = barycentric product X(i)*X(j) for these {i,j}: {10, 56219}, {12, 2339}, {42, 60197}, {313, 2281}, {321, 1245}, {594, 56328}, {1036, 6358}, {1039, 26942}, {1089, 2221}, {1310, 4024}, {1472, 28654}, {1500, 57923}, {2171, 30479}, {4064, 36099}, {4079, 54982}, {4705, 37215}, {51686, 52369}
X(68058) = trilinear product X(i)*X(j) for these {i,j}: {10, 1245}, {12, 1036}, {37, 56219}, {181, 30479}, {201, 1039}, {213, 60197}, {321, 2281}, {594, 2221}, {756, 56328}, {872, 57923}, {1089, 1472}, {1310, 4705}, {2171, 2339}, {3695, 51686}, {4064, 32691}, {4079, 37215}, {21813, 63195}, {36099, 55232}, {50487, 54982}
X(68058) = trilinear quotient X(i)/X(j) for these (i,j): (10, 1010), (12, 388), (37, 2303), (42, 44119), (65, 5323), (125, 26933), (181, 1460), (201, 1038), (313, 44154), (523, 47844), (594, 2345), (756, 612), (1036, 60), (1039, 270), (1089, 4385), (1245, 58), (1254, 4320), (1310, 52935), (1472, 849), (1500, 54416)
X(68058) = (X(1), X(2339))-harmonic conjugate of X(1472)

X(68559) = X(104)X(1041)∩X(106)X(997)

Barycentrics    a*(b-c)^2*((a-b)^2+c^2)*((a-c)^2+b^2) : :

See Jean-Louis Ayme, Elias Hagos and César Lozada, euclid 8502.

X(68059) lies on these lines: {1, 1416}, {11, 53990}, {38, 7123}, {104, 1041}, {106, 997}, {244, 2968}, {291, 3976}, {614, 1783}, {982, 8817}, {999, 1037}, {1015, 17435}, {1111, 21185}, {1292, 4319}, {1357, 2821}, {1565, 2310}, {2170, 4449}, {3122, 17216}, {3323, 38363}, {3673, 46406}, {4845, 56359}, {8816, 14986}, {17205, 17877}, {20269, 23667}, {21132, 21139}, {31002, 57925}

X(68059) = cross-difference of every pair of points on the line X(1633)X(61160)
X(68059) = pole of the line {3271, 4904} with respect to the incircle
X(68059) = pole of the line {3239, 15413} with respect to the circumhyperbola dual of Yff parabola
X(68059) = pole of the line {1565, 14936} with respect to the Steiner inellipse
X(68059) = barycentric product X(i)*X(j) for these {i,j}: {11, 7131}, {85, 14935}, {244, 30701}, {513, 48070}, {1015, 57925}, {1037, 4858}, {1041, 26932}, {1086, 56179}, {1111, 7123}, {1146, 56359}, {1358, 56243}, {2170, 8817}, {2310, 30705}, {3120, 40403}, {4081, 63178}, {6545, 52778}, {7084, 23989}, {8269, 42462}, {17205, 56260}, {18210, 40411}
X(68059) = trilinear product X(i)*X(j) for these {i,j}: {7, 14935}, {11, 1037}, {244, 56179}, {649, 48070}, {764, 52778}, {1015, 30701}, {1041, 7004}, {1086, 7123}, {1111, 7084}, {2170, 7131}, {2310, 56359}, {3119, 63178}, {3125, 40403}, {3248, 57925}, {3271, 8817}, {4466, 57386}, {8027, 54967}, {14936, 30705}, {16726, 56260}, {52305, 59133}
X(68059) = trilinear quotient X(i)/X(j) for these (i,j): (11, 497), (116, 17671), (125, 21015), (244, 614), (513, 1633), (514, 3732), (661, 61160), (1015, 16502), (1037, 59), (1041, 7012), (1086, 4000), (1111, 3673), (1146, 6554), (1358, 7195), (1565, 17170), (2170, 2082), (2310, 4319), (2969, 1851), (3119, 28070), (3120, 3914)
X(68059) = (X(1), X(7131))-harmonic conjugate of X(7084)

X(68560) = X(3)X(1014)∩X(5)X(6)

Barycentrics    -4*a^6+7*a^4*b^2+2*a^3*b^3-4*a^2*b^4-2*a*b^5+b^6+2*a^3*b^2*c+2*a^2*b^3*c-2*a*b^4*c-2*b^5*c+7*a^4*c^2+2*a^3*b*c^2+4*a^2*b^2*c^2+4*a*b^3*c^2-b^4*c^2+2*a^3*c^3+2*a^2*b*c^3+4*a*b^2*c^3+4*b^3*c^3-4*a^2*c^4-2*a*b*c^4-b^2*c^4-2*a*c^5-2*b*c^5+c^6 : :
X(68560) = X[3332]+X[62183], X[3332]+3*X[63054], X[62183]-3*X[63054]

David Nguyen and Ercole Suppa, euclid 8508.

X(68060) lies on the circumconic {{A,B,C,X(68),X(8813)}} and these lines: {1 ,5762}, {3, 1014}, {4, 62997}, {5, 6}, {30, 3332}, {37, 64065}, {81, 8727}, {140, 4648}, {226, 59613}, {235, 2906}, {238, 38043}, {269, 24470}, {442, 63088}, {498, 38293}, {517, 4349}, {549, 13329}, {550, 991}, {632, 17245}, {912, 66683}, {940, 37364}, {971, 4667}, {1279, 10283}, {1386, 20330}, {1419, 57282}, {1449, 5805}, {1456, 39542}, {1482, 4344}, {1483, 29010}, {1656, 37681}, {1743, 61511}, {1754, 37631}, {2263, 32047}, {2293, 10386}, {3008, 38171}, {3019, 3627}, {3167, 6678}, {3193, 47510}, {3628, 37650}, {3629, 48888}, {3664, 31657}, {3672, 60922}, {4000, 61509}, {4346, 51514}, {4644, 5843}, {4675, 38111}, {5045, 67026}, {5050, 19512}, {5093, 36526}, {5222, 38107}, {5308, 59381}, {5690, 64174}, {5706, 37424}, {5763, 37594}, {5886, 16469}, {5901, 7290}, {6147, 66760}, {6907, 45923}, {6922, 45931}, {7411, 41819}, {7522, 63174}, {8226, 37685}, {8550, 24220}, {8703, 50677}, {8728, 22136}, {10883, 63039}, {11246, 47057}, {11529, 53804}, {13727, 20090}, {14996, 37374}, {15171, 67264}, {15251, 16475}, {16487, 61276}, {16666, 38137}, {16670, 38108}, {17014, 59386}, {17337, 55856}, {19541, 63007}, {21168, 29624}, {21841, 44100}, {25878, 36754}, {29571, 38113}, {30424, 65415}, {34380, 36477}, {35227, 61278}, {36652, 37677}, {36659, 39899}, {36660, 51170}, {36662, 66742}, {36722, 63052}, {37438, 63374}, {37468, 63297}, {37543, 56294}, {48906, 48934}

X(68060) = midpoint of X(3332)X(62183)
X(68060) = pole of line {7490, 14593} with respect to the Huygens hyperbola
X(68060) = pole of line {1993, 13615} with respect to Stammler hyperbola}
X(68060) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3332,62183,30},{3332,63054,62183},{5706,49743,37424},{13329,45942,17392}





leftri   Points on the Huygens hyperbola, X(68561) - X(68581)  rightri

Contributed by Clark Kimberling, based on notes and data from Peter Moses, May 24, 2025.

The Hugens hyperbola is the isogonal conjugate of the line X(3)X(49). See

Alperin, Roger C., "The Poncelet Pencil of Rectangular Hyperbolas", Forum Geometricorum, 10 (2010), 15-20:

Sandor Nagydobai Kiss, "The Poncelet Pencil's Hyperbolas as Locus Geometric and Their Equations in Barycentric Coordinates": article.

The Huygens hyperbola, given by the barycentric equation

(a^2 - b^2)*(a^2 - b^2 + c^2)*(-a^2 + b^2 + c^2)*x*y + (a^2 + b^2 - c^2)*(-a^2 + c^2)*(-a^2 + b^2 + c^2)*x*z + (b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*y*z = 0 ,

has center X(136) and perspector X(2501). It passes through X(i) for these i:

4, 93, 225, 254, 264, 393, 847, 1093, 1105, 1179, 1217, 1300, 1826, 6344, 6526, 6531, 8737, 8738, 8741, 8742, 8801, 8884, 14860, 15424, 16263, 17983, 18808, 18846, 18847, 18848, 18849, 18850, 18851, 18852, 18853, 18854, 18855, 24243, 24244, 32085, 34208, 35142, 36611, 36612, 38427, 38428, 40402, 41013, 41515, 41516, 42377, 47735, 52487, 55031, 55972, 56340, 57931, 59278, 60836, 64844, 66596, and X(68561)-to-X(68581).

underbar



X(68561) = X(4)X(13)∩X(264)X(693)

Barycentrics    b*c*(b^2 - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^3 - a^2*b - a*b^2 + b^3 + 2*a*b*c - a*c^2 - b*c^2)*(-a^3 + a*b^2 + a^2*c - 2*a*b*c + b^2*c + a*c^2 - c^3) : :

X(68561) lies on the Huygens hyperbola and these lines: {4, 513}, {104, 1300}, {225, 4017}, {264, 693}, {393, 6591}, {523, 41013}, {661, 1826}, {1290, 1309}, {3657, 38955}, {5136, 53406}, {6344, 43082}, {16082, 17983}, {16230, 18015}, {18108, 32085}, {18808, 66294}, {18816, 35142}, {37135, 65223}, {66275, 66299}

X(68561) = polar conjugate of X(64828)
X(68561) = polar conjugate of the isogonal conjugate of X(55259)
X(68561) = X(i)-isoconjugate of X(j) for these (i,j): {48, 64828}, {110, 22350}, {255, 4246}, {283, 23981}, {517, 4575}, {859, 1331}, {908, 32661}, {1790, 2427}, {1795, 68147}, {2183, 4558}, {2193, 24029}, {4570, 8677}, {4600, 23220}, {9247, 55258}, {17139, 32656}, {52307, 52378}, {62402, 65375}
X(68561) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 517}, {244, 22350}, {1249, 64828}, {5521, 859}, {6523, 4246}, {6741, 51379}, {25640, 68147}, {40622, 62402}, {47345, 24029}, {50330, 8677}, {50497, 23220}, {62576, 55258}
X(68561) = trilinear pole of line {2501, 3125}
X(68561) = barycentric product X(i)*X(j) for these {i,j}: {104, 14618}, {264, 55259}, {321, 43933}, {523, 16082}, {648, 66294}, {1309, 16732}, {1577, 36123}, {2250, 46107}, {2401, 41013}, {2501, 18816}, {3120, 65223}, {6591, 57984}, {17924, 38955}, {24006, 34234}, {31623, 66275}, {40149, 43728}, {57809, 61238}, {65302, 66299}
X(68561) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 64828}, {104, 4558}, {225, 24029}, {264, 55258}, {393, 4246}, {661, 22350}, {909, 4575}, {1309, 4567}, {1824, 2427}, {1880, 23981}, {2250, 1331}, {2401, 1444}, {2423, 1437}, {2501, 517}, {3121, 23220}, {3125, 8677}, {3700, 51379}, {4036, 51367}, {4516, 52307}, {6591, 859}, {7178, 62402}, {14571, 68147}, {14618, 3262}, {15635, 7254}, {16082, 99}, {16732, 65868}, {17924, 17139}, {18816, 4563}, {24006, 908}, {34234, 4592}, {34858, 32661}, {36110, 52378}, {36123, 662}, {37628, 6514}, {38955, 1332}, {41013, 2397}, {43728, 1812}, {43933, 81}, {55195, 35014}, {55208, 1457}, {55244, 57478}, {55259, 3}, {58757, 14571}, {61238, 283}, {65223, 4600}, {66275, 1214}, {66294, 525}

X(68562) = X(4)X(519)∩X(264)X(3264)

Barycentrics    (a + b - 3*c)*(a - 3*b + c)*(b + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

X(68562) lies on the Huygens hyperbola and these lines: {4, 519}, {225, 40663}, {264, 3264}, {393, 8756}, {1222, 4054}, {1293, 1300}, {1826, 3943}, {3445, 23710}, {3992, 41013}, {17983, 52747}, {35142, 53647}

X(68562) = polar conjugate of X(41629)
X(68562) = polar conjugate of the isotomic conjugate of X(4052)
X(68562) = X(53008)-cross conjugate of X(1826)
X(68562) = X(i)-isoconjugate of X(j) for these (i,j): {3, 16948}, {48, 41629}, {58, 4855}, {63, 33628}, {81, 20818}, {145, 1437}, {255, 4248}, {283, 1420}, {603, 52352}, {849, 52354}, {1408, 44722}, {1444, 3052}, {1743, 1790}, {1812, 67843}, {2193, 5435}, {3667, 4575}, {4394, 4558}, {4462, 32661}, {4592, 8643}, {4881, 57736}, {7254, 57192}
X(68562) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 4855}, {136, 3667}, {1249, 41629}, {3162, 33628}, {4075, 52354}, {5139, 8643}, {6523, 4248}, {7952, 52352}, {24151, 1444}, {36103, 16948}, {40586, 20818}, {47345, 5435}, {53982, 4881}, {59577, 44722}, {62575, 17206}
X(68562) = trilinear pole of line {2501, 4120}
X(68562) = barycentric product X(i)*X(j) for these {i,j}: {4, 4052}, {92, 56174}, {225, 6557}, {523, 65337}, {1293, 14618}, {1824, 40014}, {1826, 4373}, {2501, 53647}, {3680, 40149}, {8056, 41013}, {17988, 34899}, {24006, 27834}, {27818, 53008}
X(68562) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 41629}, {19, 16948}, {25, 33628}, {37, 4855}, {42, 20818}, {225, 5435}, {281, 52352}, {393, 4248}, {430, 4856}, {594, 52354}, {1293, 4558}, {1824, 1743}, {1826, 145}, {1880, 1420}, {2321, 44722}, {2333, 3052}, {2489, 8643}, {2501, 3667}, {3445, 1790}, {3680, 1812}, {4052, 69}, {4120, 39472}, {4373, 17206}, {6557, 332}, {7140, 3950}, {8056, 1444}, {8736, 4848}, {8754, 21950}, {17988, 37792}, {21016, 4884}, {24006, 4462}, {27834, 4592}, {34080, 4575}, {38266, 1437}, {40149, 39126}, {41013, 18743}, {53008, 3161}, {53647, 4563}, {55206, 4162}, {55208, 51656}, {56174, 63}, {57652, 67843}, {65337, 99}

X(68563) = X(4)X(145)∩X(264)X(2973)

Barycentrics    (a + b - 2*c)*(a - 2*b + c)*(b + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

X(68563) lies on the Huygens hyperbola and these lines: {4, 145}, {106, 1068}, {225, 4674}, {264, 2973}, {901, 1300}, {1000, 4415}, {1417, 14257}, {1826, 3950}, {4555, 35142}, {14018, 62732}, {16230, 18011}, {17927, 17983}, {18808, 66285}, {38955, 42759}, {41013, 52353}

X(68563) = polar conjugate of X(16704)
X(68563) = polar conjugate of the isotomic conjugate of X(4080)
X(68563) = X(860)-cross conjugate of X(4)
X(68563) = X(i)-isoconjugate of X(j) for these (i,j): {3, 52680}, {44, 1790}, {48, 16704}, {58, 5440}, {63, 3285}, {71, 30576}, {81, 22356}, {86, 23202}, {110, 53532}, {184, 30939}, {214, 57736}, {255, 37168}, {283, 1319}, {519, 1437}, {662, 22086}, {900, 4575}, {902, 1444}, {1023, 7254}, {1333, 3977}, {1404, 1812}, {1409, 30606}, {1412, 52978}, {1635, 4558}, {1960, 4592}, {2193, 3911}, {2251, 17206}, {3762, 32661}, {4565, 14418}, {8756, 18604}, {16729, 32659}, {17191, 52431}, {23189, 23703}, {23224, 46541}, {52407, 56950}
X(68563) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 5440}, {37, 3977}, {136, 900}, {244, 53532}, {1084, 22086}, {1249, 16704}, {3162, 3285}, {5139, 1960}, {6523, 37168}, {9460, 17206}, {36103, 52680}, {40586, 22356}, {40594, 1444}, {40595, 1790}, {40599, 52978}, {40600, 23202}, {47345, 3911}, {52872, 65742}, {52877, 22371}, {53982, 214}, {53985, 68160}, {55064, 14418}, {55065, 14429}, {62582, 332}, {62605, 30939}
X(68563) = cevapoint of X(i) and X(j) for these (i,j): {523, 42759}, {1832, 1833}
X(68563) = crosssum of X(22356) and X(23202)
X(68563) = trilinear pole of line {1826, 2501}
X(68563) = barycentric product X(i)*X(j) for these {i,j}: {4, 4080}, {10, 6336}, {27, 4013}, {88, 41013}, {92, 4674}, {225, 4997}, {313, 8752}, {321, 36125}, {523, 65336}, {648, 66285}, {901, 14618}, {903, 1826}, {1320, 40149}, {1824, 20568}, {1897, 4049}, {2316, 57809}, {2333, 57995}, {2501, 4555}, {3257, 24006}, {6335, 55244}, {18026, 61179}, {23838, 65207}, {30575, 38462}, {60480, 61178}
X(68563) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 16704}, {10, 3977}, {19, 52680}, {25, 3285}, {28, 30576}, {29, 30606}, {37, 5440}, {42, 22356}, {88, 1444}, {92, 30939}, {106, 1790}, {210, 52978}, {213, 23202}, {225, 3911}, {393, 37168}, {430, 4969}, {512, 22086}, {661, 53532}, {860, 51583}, {901, 4558}, {903, 17206}, {1320, 1812}, {1824, 44}, {1826, 519}, {1832, 36669}, {1833, 36668}, {1840, 4434}, {1870, 17191}, {1880, 1319}, {2316, 283}, {2333, 902}, {2489, 1960}, {2501, 900}, {3257, 4592}, {3943, 65742}, {4013, 306}, {4024, 14429}, {4041, 14418}, {4049, 4025}, {4080, 69}, {4555, 4563}, {4674, 63}, {4997, 332}, {6335, 55243}, {6336, 86}, {6548, 15419}, {7140, 3943}, {8736, 40663}, {8752, 58}, {9456, 1437}, {14321, 39472}, {14618, 65867}, {17927, 31059}, {23345, 7254}, {24006, 3762}, {32665, 4575}, {32719, 32661}, {36058, 18604}, {36125, 81}, {38462, 16729}, {41013, 4358}, {42752, 47420}, {44113, 17455}, {52963, 22371}, {53008, 2325}, {55206, 4895}, {55208, 53528}, {55244, 905}, {55263, 1459}, {57652, 1404}, {60578, 17219}, {61178, 62669}, {61179, 521}, {64835, 56950}, {65336, 99}, {66285, 525}, {66288, 4466}, {66924, 4091}

X(68564) = X(4)X(524)∩X(264)X(3266)

Barycentrics    (a^2 + b^2 - 5*c^2)*(a^2 + b^2 - c^2)*(a^2 - 5*b^2 + c^2)*(a^2 - b^2 + c^2) : :

X(68564) lies on the Huygens hyperbola and these lines: {2, 17983}, {4, 524}, {225, 55923}, {254, 37118}, {264, 3266}, {393, 468}, {458, 17952}, {935, 62515}, {1093, 37778}, {1217, 64474}, {1296, 1300}, {1826, 4062}, {3087, 57467}, {4232, 51541}, {4590, 32815}, {5099, 62514}, {5486, 43448}, {5967, 6531}, {6344, 43084}, {6525, 51823}, {8542, 8753}, {8737, 52039}, {8738, 52040}, {8801, 62977}, {8884, 37460}, {11185, 63179}, {16183, 40727}, {16230, 18012}, {18808, 66126}, {18852, 43999}, {32085, 43981}, {32825, 40074}, {34208, 62960}, {35142, 35179}, {37765, 53857}, {41013, 42713}, {47735, 62213}, {52289, 60866}, {52710, 55972}

X(68564) = polar conjugate of X(1992)
X(68564) = isotomic conjugate of the anticomplement of X(43620)
X(68564) = polar conjugate of the isotomic conjugate of X(5485)
X(68564) = polar conjugate of the isogonal conjugate of X(21448)
X(68564) = X(i)-cross conjugate of X(j) for these (i,j): {5094, 4}, {21448, 5485}, {24855, 671}, {43620, 2}
X(68564) = X(i)-isoconjugate of X(j) for these (i,j): {3, 36277}, {48, 1992}, {63, 1384}, {255, 4232}, {906, 4786}, {1331, 30234}, {1499, 4575}, {4592, 8644}, {9126, 36061}, {9247, 11059}, {14207, 32661}, {27088, 36060}, {35200, 35266}, {52430, 58782}
X(68564) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 35266}, {136, 1499}, {1249, 1992}, {1560, 27088}, {3162, 1384}, {5139, 8644}, {5190, 4786}, {5521, 30234}, {6523, 4232}, {16221, 9126}, {36103, 36277}, {48317, 9125}, {53992, 68165}, {62576, 11059}, {62595, 51438}
X(68564) = cevapoint of X(4) and X(52290)
X(68564) = trilinear pole of line {690, 2501}
X(68564) = barycentric product X(i)*X(j) for these {i,j}: {4, 5485}, {92, 55923}, {264, 21448}, {523, 65353}, {671, 52477}, {1296, 14618}, {2052, 55977}, {2501, 35179}, {14262, 60266}, {18022, 39238}, {24006, 37216}, {46111, 57467}, {58754, 59762}
X(68564) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 1992}, {19, 36277}, {25, 1384}, {264, 11059}, {297, 51438}, {393, 4232}, {468, 27088}, {1296, 4558}, {1990, 35266}, {2052, 58782}, {2489, 8644}, {2501, 1499}, {5094, 11165}, {5485, 69}, {6591, 30234}, {7649, 4786}, {8754, 6791}, {14262, 41614}, {14273, 9125}, {17983, 52141}, {21448, 3}, {24006, 14207}, {27376, 41585}, {35179, 4563}, {37216, 4592}, {39238, 184}, {41013, 42724}, {47230, 9126}, {52290, 11147}, {52477, 524}, {53419, 53778}, {55923, 63}, {55977, 394}, {57467, 3292}, {60428, 15471}, {65353, 99}
X(68564) = {X(4),X(5485)}-harmonic conjugate of X(52477)

X(68565) = X(4)X(218)∩X(264)X(281)

Barycentrics    (b + c)*(a^2 + b^2 - a*c - b*c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-a^2 + a*b + b*c - c^2) : :

X(68565) lies on the Huygens hyperbola and these lines: {4, 218}, {105, 59083}, {225, 2333}, {264, 281}, {393, 6059}, {666, 35142}, {919, 1300}, {1826, 4878}, {1880, 52607}, {1886, 2356}, {3991, 41013}, {15149, 67197}, {17983, 65333}, {18808, 66282}

X(68565) = polar conjugate of X(30941)
X(68565) = polar conjugate of the isotomic conjugate of X(13576)
X(68565) = polar conjugate of the isogonal conjugate of X(56853)
X(68565) = X(i)-cross conjugate of X(j) for these (i,j): {862, 4}, {56853, 13576}
X(68565) = X(i)-isoconjugate of X(j) for these (i,j): {3, 18206}, {48, 30941}, {58, 25083}, {63, 3286}, {81, 1818}, {86, 20752}, {184, 18157}, {241, 283}, {255, 15149}, {332, 52635}, {394, 54407}, {518, 1790}, {662, 53550}, {665, 4592}, {672, 1444}, {799, 23225}, {905, 54353}, {906, 23829}, {918, 4575}, {1025, 23189}, {1026, 7254}, {1437, 3912}, {1458, 1812}, {1808, 34253}, {1861, 18604}, {1876, 6514}, {2193, 9436}, {2223, 17206}, {2254, 4558}, {2327, 34855}, {4091, 4238}, {7183, 37908}, {8638, 55205}, {15419, 54325}, {16728, 36057}, {20778, 37128}, {23090, 41353}
X(68565) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 25083}, {136, 918}, {1084, 53550}, {1249, 30941}, {3162, 3286}, {5139, 665}, {5190, 23829}, {6523, 15149}, {20621, 16728}, {36103, 18206}, {38996, 23225}, {40586, 1818}, {40600, 20752}, {47345, 9436}, {62554, 1444}, {62599, 17206}, {62605, 18157}
X(68565) = cevapoint of X(37) and X(21956)
X(68565) = crosspoint of X(36124) and X(54235)
X(68565) = crosssum of X(1818) and X(20752)
X(68565) = trilinear pole of line {1824, 2501}
X(68565) = crossdifference of every pair of points on line {20778, 23225}
X(68565) = barycentric product X(i)*X(j) for these {i,j}: {4, 13576}, {10, 36124}, {37, 54235}, {92, 18785}, {105, 41013}, {225, 14942}, {264, 56853}, {281, 66941}, {294, 40149}, {321, 8751}, {523, 65333}, {648, 66282}, {666, 2501}, {673, 1826}, {862, 67197}, {885, 61178}, {919, 14618}, {1024, 65207}, {1824, 2481}, {1874, 33676}, {1880, 36796}, {2195, 57809}, {2333, 18031}, {2489, 36803}, {5379, 66290}, {6335, 55261}, {24006, 36086}, {28132, 52607}, {34085, 55206}, {53008, 56783}
X(68565) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 30941}, {19, 18206}, {25, 3286}, {37, 25083}, {42, 1818}, {92, 18157}, {105, 1444}, {213, 20752}, {225, 9436}, {294, 1812}, {393, 15149}, {430, 4966}, {512, 53550}, {666, 4563}, {669, 23225}, {673, 17206}, {862, 8299}, {884, 23189}, {919, 4558}, {1096, 54407}, {1426, 34855}, {1438, 1790}, {1824, 518}, {1826, 3912}, {1874, 39775}, {1880, 241}, {2195, 283}, {2333, 672}, {2489, 665}, {2501, 918}, {3747, 20778}, {5089, 16728}, {6059, 37908}, {6335, 55260}, {7140, 3932}, {7649, 23829}, {8750, 54353}, {8751, 81}, {10099, 4131}, {13576, 69}, {14942, 332}, {18785, 63}, {20683, 65744}, {28071, 1792}, {28132, 15411}, {32658, 18604}, {32666, 4575}, {34085, 55205}, {36086, 4592}, {36124, 86}, {36803, 52608}, {40149, 40704}, {41013, 3263}, {43929, 7254}, {51560, 55202}, {52030, 57738}, {52577, 51400}, {53008, 3717}, {54235, 274}, {55208, 53544}, {55261, 905}, {56853, 3}, {57652, 1458}, {61178, 883}, {62635, 15419}, {64216, 1437}, {65333, 99}, {66282, 525}, {66930, 23067}, {66941, 348}, {67197, 57987}

X(68566) = X(4)X(575)∩X(264)X(468)

Barycentrics    (a^2 + b^2 - c^2)*(2*a^2 + 2*b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^2 - b^2 + 2*c^2) : :

X(68566) lies on the Huygens hyperbola and these lines: {4, 575}, {25, 17983}, {225, 55927}, {264, 468}, {340, 55972}, {393, 1383}, {1093, 64471}, {1105, 37196}, {1217, 37460}, {1300, 11636}, {1990, 30489}, {2052, 10511}, {2489, 33885}, {4232, 51541}, {5112, 57822}, {6344, 43976}, {6530, 16263}, {8599, 18808}, {8801, 17907}, {20380, 52282}, {32085, 62968}, {35138, 35142}, {47735, 62195}, {64474, 65006}

X(68566) = polar conjugate of X(599)
X(68566) = polar conjugate of the isotomic conjugate of X(598)
X(68566) = polar conjugate of the isogonal conjugate of X(1383)
X(68566) = X(i)-cross conjugate of X(j) for these (i,j): {1383, 598}, {10301, 4}
X(68566) = X(i)-isoconjugate of X(j) for these (i,j): {3, 36263}, {48, 599}, {63, 574}, {255, 5094}, {326, 8541}, {656, 9145}, {810, 9146}, {1459, 3908}, {3906, 4575}, {4020, 10130}, {4141, 36058}, {4592, 17414}, {9247, 9464}, {13857, 35200}, {36060, 39785}
X(68566) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 13857}, {136, 3906}, {1249, 599}, {1560, 39785}, {3162, 574}, {5139, 17414}, {6523, 5094}, {15259, 8541}, {20619, 4141}, {36103, 36263}, {39062, 9146}, {40596, 9145}, {53992, 62568}, {62576, 9464}, {62595, 51397}
X(68566) = cevapoint of X(i) and X(j) for these (i,j): {4, 4232}, {115, 65468}, {523, 38361}, {10552, 27088}
X(68566) = trilinear pole of line {2501, 8599}
X(68566) = barycentric product X(i)*X(j) for these {i,j}: {4, 598}, {25, 40826}, {92, 55927}, {264, 1383}, {393, 64982}, {468, 18818}, {648, 8599}, {2052, 43697}, {2501, 35138}, {6331, 46001}, {6528, 30491}, {8744, 10512}, {10511, 37765}, {11636, 14618}, {16081, 52692}, {17983, 51541}, {23287, 65350}, {23297, 32085}, {30489, 46104}, {65328, 66299}
X(68566) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 599}, {19, 36263}, {25, 574}, {112, 9145}, {264, 9464}, {297, 51397}, {393, 5094}, {468, 39785}, {598, 69}, {648, 9146}, {1383, 3}, {1783, 3908}, {1990, 13857}, {2207, 8541}, {2489, 17414}, {2501, 3906}, {4232, 11165}, {5523, 19510}, {8599, 525}, {8744, 10510}, {8753, 42007}, {8754, 8288}, {8756, 4141}, {10301, 15810}, {10511, 34897}, {11636, 4558}, {17983, 42008}, {18818, 30786}, {20380, 65747}, {23287, 14417}, {23297, 3933}, {30489, 3917}, {30491, 520}, {32085, 10130}, {35138, 4563}, {40826, 305}, {43697, 394}, {44102, 62657}, {46001, 647}, {51541, 6390}, {52692, 36212}, {55927, 63}, {64982, 3926}, {65007, 14961}, {65008, 45807}

X(68567) = X(4)X(572)∩X(27)X(264)

Barycentrics    (a^2 + a*b + b^2 + a*c + b*c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2 + a*b + a*c + b*c + c^2) : :

X(68567) lies on the Huygens hyperbola and these lines: {4, 572}, {19, 1759}, {25, 1631}, {27, 264}, {225, 608}, {835, 9085}, {1300, 58951}, {1865, 37398}, {1869, 57702}, {4185, 34278}, {4196, 8801}, {6994, 56047}, {11501, 36744}, {34265, 37390}, {35142, 68197}

X(68567) = polar conjugate of X(5224)
X(68567) = polar conjugate of the isotomic conjugate of X(43531)
X(68567) = X(i)-isoconjugate of X(j) for these (i,j): {3, 28606}, {48, 5224}, {63, 386}, {72, 61409}, {184, 33935}, {212, 33949}, {222, 3876}, {255, 469}, {326, 44103}, {834, 1332}, {905, 65313}, {906, 45746}, {1331, 14349}, {1437, 56810}, {1444, 56926}, {4558, 47842}, {4563, 50488}, {4575, 23879}, {4592, 42664}, {22383, 33948}, {23224, 65204}
X(68567) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 23879}, {1249, 5224}, {3162, 386}, {5139, 42664}, {5190, 45746}, {5521, 14349}, {6523, 469}, {15259, 44103}, {36103, 28606}, {40837, 33949}, {62605, 33935}
X(68567) = cevapoint of X(11998) and X(13401)
X(68567) = barycentric product X(i)*X(j) for these {i,j}: {4, 43531}, {25, 57824}, {92, 2214}, {393, 57876}, {835, 7649}, {1826, 56047}, {1897, 43927}, {2052, 57704}, {2501, 68197}, {4185, 34265}, {6591, 37218}, {14618, 58951}, {43925, 65850}
X(68567) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 5224}, {19, 28606}, {25, 386}, {33, 3876}, {92, 33935}, {278, 33949}, {393, 469}, {835, 4561}, {1474, 61409}, {1826, 56810}, {1897, 33948}, {2207, 44103}, {2214, 63}, {2333, 56926}, {2489, 42664}, {2501, 23879}, {2969, 65116}, {6591, 14349}, {7649, 45746}, {8750, 65313}, {41013, 42714}, {43531, 69}, {43925, 52615}, {43927, 4025}, {56047, 17206}, {57704, 394}, {57824, 305}, {57876, 3926}, {58951, 4558}, {68197, 4563}

X(68568) = X(4)X(391)∩X(225)X(281)

Barycentrics    (a^2 - 2*a*b + b^2 - 2*a*c - 2*b*c - 3*c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2 - 2*a*b - 3*b^2 - 2*a*c - 2*b*c + c^2) : :

X(68568) lies on the Huygens hyperbola and these lines: {4, 391}, {29, 33630}, {225, 281}, {393, 461}, {1300, 59079}, {1826, 4061}, {4213, 34208}, {5665, 7003}, {7101, 41013}, {7498, 63157}, {35142, 68194}

X(68568) = polar conjugate of X(3945)
X(68568) = polar conjugate of the isotomic conjugate of X(43533)
X(68568) = X(4207)-cross conjugate of X(4)
X(68568) = X(i)-isoconjugate of X(j) for these (i,j): {3, 62812}, {48, 3945}, {63, 4252}, {222, 3601}, {255, 7490}, {603, 5273}, {1869, 18604}, {7099, 20007}, {23224, 65170}
X(68568) = X(i)-Dao conjugate of X(j) for these (i,j): {1249, 3945}, {3162, 4252}, {6523, 7490}, {7952, 5273}, {36103, 62812}
X(68568) = cevapoint of X(1146) and X(45745)
X(68568) = trilinear pole of line {2501, 4843}
X(68568) = barycentric product X(i)*X(j) for these {i,j}: {4, 43533}, {318, 5665}, {2501, 68194}, {14618, 59079}, {41013, 63157}
X(68568) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 3945}, {19, 62812}, {25, 4252}, {33, 3601}, {281, 5273}, {393, 7490}, {5665, 77}, {7046, 20007}, {39585, 28627}, {43533, 69}, {59079, 4558}, {63157, 1444}, {68183, 65296}, {68194, 4563}

X(68569) = X(4)X(86)∩X(264)X(310)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 + 2*a*b + b^2 + 2*a*c + 2*b*c - c^2)*(a^2 - b^2 + c^2)*(a^2 + 2*a*b - b^2 + 2*a*c + 2*b*c + c^2) : :

X(68569) lies on the circumconic {{A,B,C,X(2),X(7)}}, the Huygens hyperbola, and these lines: {2, 1826}, {4, 86}, {7, 225}, {19, 39721}, {27, 393}, {69, 407}, {75, 14018}, {264, 310}, {272, 4198}, {278, 44733}, {286, 37384}, {387, 1246}, {388, 60041}, {1300, 59065}, {1444, 4185}, {4212, 56052}, {5815, 58002}, {6994, 56047}, {7318, 17081}, {10449, 57825}, {18815, 62771}, {27426, 27447}, {27472, 27483}, {30699, 39700}, {32085, 52394}, {54314, 58017}, {57527, 63014}, {57923, 58013}

X(68569) = polar conjugate of X(966)
X(68569) = isotomic conjugate of the anticomplement of X(5292)
X(68569) = polar conjugate of the isotomic conjugate of X(58012)
X(68569) = polar conjugate of the isogonal conjugate of X(967)
X(68569) = X(i)-cross conjugate of X(j) for these (i,j): {967, 58012}, {1889, 4}, {5292, 2}, {39579, 92}
X(68569) = X(i)-isoconjugate of X(j) for these (i,j): {3, 968}, {37, 4288}, {48, 966}, {55, 54320}, {63, 2271}, {212, 3485}, {228, 11110}, {255, 4207}, {906, 45745}, {1331, 48099}, {2318, 64382}, {7650, 32656}
X(68569) = X(i)-Dao conjugate of X(j) for these (i,j): {223, 54320}, {1249, 966}, {3162, 2271}, {5190, 45745}, {5521, 48099}, {6523, 4207}, {36103, 968}, {40589, 4288}, {40837, 3485}
X(68569) = cevapoint of X(4) and X(7490)
X(68569) = trilinear pole of line {514, 2501}
X(68569) = barycentric product X(i)*X(j) for these {i,j}: {4, 58012}, {19, 58013}, {92, 969}, {264, 967}, {14618, 59065}
X(68569) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 966}, {19, 968}, {25, 2271}, {27, 11110}, {57, 54320}, {58, 4288}, {278, 3485}, {393, 4207}, {967, 3}, {969, 63}, {1396, 64382}, {6591, 48099}, {7649, 45745}, {17924, 7650}, {58012, 69}, {58013, 304}, {59065, 4558}

X(68570) = X(4)X(386)∩X(264)X(5224)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b - b^3 + a^2*c - a*b*c + a*c^2 + b*c^2)*(a^2*b + a*b^2 + a^2*c - a*b*c + b^2*c - c^3) : :

X(68570) lies on the Huygens hyperbola and these lines: {4, 386}, {25, 52150}, {53, 1826}, {225, 1829}, {264, 5224}, {270, 37390}, {393, 44103}, {1105, 37420}, {1300, 59006}, {1785, 10039}, {4194, 46880}, {8747, 8884}, {35142, 65275}

X(68570) = polar conjugate of X(14829)
X(68570) = polar conjugate of the isotomic conjugate of X(2051)
X(68570) = X(44092)-cross conjugate of X(1826)
X(68570) = X(i)-isoconjugate of X(j) for these (i,j): {2, 22118}, {3, 2975}, {48, 14829}, {60, 65576}, {63, 572}, {69, 20986}, {100, 23187}, {219, 17074}, {255, 11109}, {283, 37558}, {905, 65203}, {906, 17496}, {1331, 21173}, {1437, 17751}, {1444, 52139}, {1790, 21061}, {1812, 55323}, {2193, 52358}, {4564, 38344}, {11998, 44717}, {22056, 31629}, {23067, 57125}, {32656, 57244}, {36059, 57091}, {56325, 65568}
X(68570) = X(i)-Dao conjugate of X(j) for these (i,j): {1249, 14829}, {3162, 572}, {5190, 17496}, {5521, 21173}, {6523, 11109}, {8054, 23187}, {20620, 57091}, {32664, 22118}, {36103, 2975}, {38966, 58339}, {47345, 52358}
X(68570) = trilinear pole of line {2501, 42664}
X(68570) = barycentric product X(i)*X(j) for these {i,j}: {4, 2051}, {19, 54121}, {25, 57905}, {27, 51870}, {92, 34434}, {225, 46880}, {331, 60817}, {1826, 20028}, {2501, 65275}, {6591, 56252}, {7649, 56188}, {14618, 59006}, {17924, 56194}, {24006, 65260}, {39534, 64824}, {41013, 53083}
X(68570) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 14829}, {19, 2975}, {25, 572}, {31, 22118}, {34, 17074}, {225, 52358}, {393, 11109}, {649, 23187}, {1824, 21061}, {1826, 17751}, {1880, 37558}, {1973, 20986}, {2051, 69}, {2171, 65576}, {2333, 52139}, {2969, 24237}, {3064, 57091}, {3271, 38344}, {6591, 21173}, {7649, 17496}, {8735, 34589}, {8736, 52357}, {8750, 65203}, {17924, 57244}, {20028, 17206}, {34434, 63}, {39534, 64825}, {40976, 46879}, {44092, 52087}, {46880, 332}, {51513, 52322}, {51870, 306}, {52150, 1790}, {53083, 1444}, {54121, 304}, {55208, 51662}, {56188, 4561}, {56194, 1332}, {56285, 65577}, {57652, 55323}, {57905, 305}, {59006, 4558}, {60817, 219}, {65103, 58339}, {65260, 4592}, {65275, 4563}

X(68571) = X(4)X(162)∩X(264)X(811)

Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 + c^2)*9- b^2 - a*c + c^2) : :

X(68571) lies on the Huygens hyperbola and these lines: {4, 162}, {28, 108}, {29, 1807}, {34, 68483}, {53, 2189}, {80, 1172}, {112, 8735}, {264, 811}, {265, 44225}, {393, 24019}, {403, 62713}, {447, 38462}, {1093, 36126}, {1175, 1834}, {1300, 36069}, {1411, 8747}, {1785, 2074}, {1793, 4183}, {1877, 26743}, {1989, 40570}, {2605, 36130}, {2906, 52639}, {3109, 15252}, {3418, 3772}, {3419, 56003}, {3449, 64172}, {5546, 65813}, {6344, 36129}, {6531, 36104}, {7054, 11113}, {11105, 59482}, {13746, 37696}, {18808, 36119}, {35142, 36105}, {37202, 46533}, {37371, 46785}, {40395, 54528}, {40950, 56405}, {56419, 56831}

X(68571) = polar conjugate of X(3936)
X(68571) = polar conjugate of the isotomic conjugate of X(24624)
X(68571) = polar conjugate of the isogonal conjugate of X(34079)
X(68571) = X(i)-cross conjugate of X(j) for these (i,j): {1411, 759}, {1884, 4}, {1990, 278}, {34079, 24624}, {39534, 107}, {44113, 19}, {58313, 112}
X(68571) = X(i)-isoconjugate of X(j) for these (i,j): {3, 758}, {10, 52407}, {36, 72}, {37, 22128}, {48, 3936}, {63, 2245}, {69, 3724}, {71, 3218}, {73, 4511}, {78, 1464}, {184, 35550}, {212, 41804}, {219, 18593}, {228, 320}, {255, 860}, {306, 7113}, {307, 2361}, {326, 44113}, {520, 4242}, {525, 1983}, {647, 4585}, {654, 65233}, {759, 65746}, {822, 65162}, {906, 4707}, {1214, 2323}, {1231, 52426}, {1259, 1835}, {1331, 53527}, {1332, 21828}, {1409, 32851}, {1439, 58328}, {1443, 2318}, {1790, 4053}, {1793, 3028}, {1797, 40988}, {1870, 3682}, {2200, 20924}, {2610, 4558}, {3710, 52440}, {3738, 23067}, {3952, 22379}, {3960, 4574}, {3990, 17923}, {3998, 52413}, {4282, 26942}, {4575, 6370}, {4592, 42666}, {4736, 57736}, {4996, 52391}, {5081, 22341}, {6149, 52388}, {6516, 53562}, {6739, 35200}, {6757, 22115}, {7066, 17515}, {7591, 63779}, {17078, 52370}, {20336, 52434}, {21758, 52609}, {22136, 39149}, {22342, 63642}, {27086, 43708}, {52385, 52427}
X(68571) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 6739}, {136, 6370}, {1249, 3936}, {3162, 2245}, {5139, 42666}, {5190, 4707}, {5521, 53527}, {6523, 860}, {14993, 52388}, {15259, 44113}, {15898, 72}, {34586, 65746}, {36103, 758}, {36909, 3710}, {39052, 4585}, {40589, 22128}, {40837, 41804}, {53982, 4736}, {62605, 35550}
X(68571) = cevapoint of X(i) and X(j) for these (i,j): {4, 37168}, {19, 44113}, {8735, 58313}
X(68571) = trilinear pole of line {19, 2501}
X(68571) = barycentric product X(i)*X(j) for these {i,j}: {4, 24624}, {19, 14616}, {27, 80}, {28, 18359}, {29, 2006}, {53, 39277}, {86, 64835}, {92, 759}, {162, 60074}, {264, 34079}, {270, 60091}, {273, 2341}, {278, 6740}, {286, 2161}, {393, 57985}, {476, 65100}, {648, 66284}, {1172, 18815}, {1396, 52409}, {1411, 31623}, {1474, 20566}, {1969, 67166}, {2052, 57736}, {2222, 57215}, {2501, 65283}, {2605, 46456}, {2970, 9273}, {3737, 65329}, {6187, 44129}, {6198, 66922}, {6336, 56950}, {6344, 40214}, {7649, 47318}, {8747, 52351}, {8748, 52392}, {14618, 36069}, {14838, 36129}, {16080, 56645}, {17515, 34535}, {17925, 51562}, {18384, 34016}, {24006, 37140}, {32680, 54244}, {36804, 57200}, {36815, 65352}, {40149, 52380}, {40395, 45926}, {44113, 57555}, {46103, 52383}, {52356, 65232}, {52412, 68483}, {60571, 61178}
X(68571) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 3936}, {19, 758}, {25, 2245}, {27, 320}, {28, 3218}, {29, 32851}, {34, 18593}, {58, 22128}, {80, 306}, {92, 35550}, {107, 65162}, {162, 4585}, {278, 41804}, {286, 20924}, {393, 860}, {608, 1464}, {759, 63}, {1172, 4511}, {1333, 52407}, {1396, 1443}, {1411, 1214}, {1474, 36}, {1793, 3719}, {1807, 3998}, {1824, 4053}, {1973, 3724}, {1989, 52388}, {1990, 6739}, {2006, 307}, {2161, 72}, {2203, 7113}, {2204, 2361}, {2207, 44113}, {2222, 65233}, {2245, 65746}, {2299, 2323}, {2332, 58328}, {2341, 78}, {2489, 42666}, {2501, 6370}, {2605, 8552}, {4467, 45792}, {5317, 1870}, {6187, 71}, {6198, 42701}, {6591, 53527}, {6740, 345}, {7649, 4707}, {8747, 17923}, {8748, 5081}, {14616, 304}, {15065, 52369}, {17925, 4453}, {18359, 20336}, {18384, 8818}, {18815, 1231}, {20566, 40071}, {20982, 16186}, {24019, 4242}, {24624, 69}, {32671, 4575}, {32675, 23067}, {32676, 1983}, {34079, 3}, {34857, 3949}, {36069, 4558}, {36129, 15455}, {36910, 3710}, {37140, 4592}, {37168, 51583}, {39277, 34386}, {40214, 52437}, {41013, 61410}, {43925, 53314}, {44113, 35069}, {44129, 40075}, {47318, 4561}, {51562, 52609}, {52351, 52396}, {52371, 3694}, {52380, 1812}, {52383, 26942}, {52392, 52565}, {52431, 3682}, {54244, 32679}, {55208, 51663}, {55238, 4064}, {56416, 51367}, {56645, 11064}, {56950, 3977}, {57129, 22379}, {57200, 3960}, {57736, 394}, {57985, 3926}, {60074, 14208}, {60091, 57807}, {64835, 10}, {65100, 3268}, {65283, 4563}, {66284, 525}, {66289, 20902}, {67166, 48}, {68483, 52381}

X(68572) = X(4)X(39)∩X(225)X(2186)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 - b^4 + 2*a^2*c^2 + b^2*c^2)*(2*a^2*b^2 + a^2*c^2 + b^2*c^2 - c^4) : :

X(68572) lies on the Huygens hyperbola and these lines: {4, 39}, {6, 32085}, {25, 6531}, {53, 141}, {225, 2186}, {263, 393}, {458, 59229}, {1093, 27376}, {1105, 37200}, {1179, 8743}, {1217, 15644}, {1249, 31506}, {1300, 26714}, {1629, 56364}, {1826, 21035}, {1990, 30489}, {2052, 51334}, {2207, 8884}, {2353, 42288}, {2501, 52631}, {3186, 34208}, {3954, 41013}, {5117, 59262}, {5286, 10110}, {5999, 40799}, {6464, 7754}, {6620, 47735}, {8801, 66974}, {8887, 18855}, {9766, 35142}, {14860, 27359}, {14957, 15355}, {16263, 60428}, {17983, 46154}, {18808, 66291}, {22240, 62949}, {24243, 45594}, {24244, 26347}, {31360, 33734}, {33971, 45031}, {34818, 42300}, {37174, 37668}, {42551, 54412}

X(68572) = polar conjugate of X(183)
X(68572) = polar conjugate of the isotomic conjugate of X(262)
X(68572) = polar conjugate of the isogonal conjugate of X(263)
X(68572) = X(i)-cross conjugate of X(j) for these (i,j): {263, 262}, {56920, 264}
X(68572) = X(i)-isoconjugate of X(j) for these (i,j): {3, 52134}, {48, 183}, {63, 182}, {184, 3403}, {219, 60716}, {255, 458}, {304, 34396}, {326, 10311}, {394, 60685}, {1092, 51315}, {1437, 60737}, {1444, 60726}, {1790, 60723}, {2169, 59197}, {3288, 4592}, {4575, 23878}, {6507, 33971}, {6784, 62719}, {9247, 20023}, {14096, 34055}, {35200, 51372}, {44144, 52430}, {59208, 62277}
X(68572) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 51372}, {136, 23878}, {1249, 183}, {3162, 182}, {5139, 3288}, {6523, 458}, {14363, 59197}, {15259, 10311}, {36103, 52134}, {40938, 14994}, {62576, 20023}, {62595, 51373}, {62605, 3403}, {67187, 69}
X(68572) = crosspoint of X(3424) and X(40815)
X(68572) = crosssum of X(1350) and X(40805)
X(68572) = trilinear pole of line {2501, 3005}
X(68572) = barycentric product X(i)*X(j) for these {i,j}: {4, 262}, {25, 327}, {53, 42300}, {92, 2186}, {263, 264}, {275, 66919}, {393, 42313}, {427, 42299}, {523, 65349}, {648, 66291}, {685, 67173}, {1093, 54032}, {1235, 42288}, {1826, 60679}, {1969, 3402}, {2052, 43718}, {2501, 65271}, {6037, 16230}, {6331, 52631}, {6344, 57268}, {6524, 59257}, {6530, 66879}, {6531, 46807}, {13450, 51444}, {14618, 26714}, {16081, 51543}, {17994, 53196}, {18022, 46319}, {24006, 65252}, {27371, 39283}, {40149, 66936}, {40803, 43976}, {42298, 51338}, {47735, 67175}, {55972, 61359}, {65310, 66299}
X(68572) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 183}, {19, 52134}, {25, 182}, {34, 60716}, {53, 59197}, {92, 3403}, {262, 69}, {263, 3}, {264, 20023}, {297, 51373}, {327, 305}, {393, 458}, {427, 14994}, {1096, 60685}, {1824, 60723}, {1826, 60737}, {1843, 14096}, {1974, 34396}, {1990, 51372}, {2052, 44144}, {2186, 63}, {2207, 10311}, {2333, 60726}, {2489, 3288}, {2501, 23878}, {2971, 6784}, {3199, 59208}, {3402, 48}, {6037, 17932}, {6520, 51315}, {6524, 33971}, {6531, 46806}, {6620, 9755}, {8754, 66459}, {14569, 39530}, {18384, 56401}, {26714, 4558}, {32716, 43754}, {41013, 42711}, {42288, 1176}, {42299, 1799}, {42300, 34386}, {42313, 3926}, {43718, 394}, {46319, 184}, {46807, 6393}, {51338, 59211}, {51543, 36212}, {52631, 647}, {52926, 23181}, {53149, 67172}, {54032, 3964}, {56920, 52658}, {57260, 51542}, {57268, 52437}, {59257, 4176}, {60679, 17206}, {61359, 6776}, {65005, 60702}, {65252, 4592}, {65271, 4563}, {65349, 99}, {66291, 525}, {66879, 6394}, {66919, 343}, {66936, 1812}, {67173, 6333}, {67175, 10008}
X(68572) = {X(43718),X(66919)}-harmonic conjugate of X(262)

X(68573) = X(4)X(580)∩X(29)X(264)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^3 + b^3 - a*b*c - a*c^2 - b*c^2)*(a^3 - a*b^2 - a*b*c - b^2*c + c^3) : :

X(68573) lies on the Huygens hyperbola and these lines: {4, 580}, {19, 2198}, {25, 225}, {29, 264}, {33, 2901}, {34, 4341}, {272, 4198}, {393, 28076}, {607, 1826}, {1039, 2997}, {1300, 58986}, {1305, 36007}, {1834, 5320}, {1865, 37398}, {1877, 7151}, {3433, 23536}, {3914, 57659}, {6284, 8735}, {11114, 65254}, {14860, 52248}, {17983, 68211}, {24159, 39267}, {35142, 65274}, {36124, 58074}

X(68573) = polar conjugate of X(18134)
X(68573) = polar conjugate of the isotomic conjugate of X(1751)
X(68573) = X(i)-cross conjugate of X(j) for these (i,j): {12, 60816}, {42, 19}, {1834, 225}, {5521, 7649}, {21935, 158}
X(68573) = X(i)-isoconjugate of X(j) for these (i,j): {3, 3868}, {48, 18134}, {63, 579}, {69, 2352}, {77, 3190}, {78, 4306}, {81, 51574}, {85, 57501}, {209, 1444}, {222, 27396}, {255, 5125}, {521, 65315}, {905, 57217}, {1214, 56000}, {1331, 23800}, {1332, 43060}, {1437, 57808}, {1790, 22021}, {1812, 66918}, {2193, 56559}, {2198, 17206}, {6516, 8676}, {6517, 57092}, {7183, 41320}, {18607, 40572}, {20294, 36059}
X(68573) = X(i)-Dao conjugate of X(j) for these (i,j): {1249, 18134}, {3162, 579}, {5521, 23800}, {6523, 5125}, {20620, 20294}, {36103, 3868}, {38966, 58333}, {40586, 51574}, {47345, 56559}
X(68573) = cevapoint of X(i) and X(j) for these (i,j): {4, 37055}, {663, 8735}
X(68573) = barycentric product X(i)*X(j) for these {i,j}: {4, 1751}, {10, 40574}, {19, 2997}, {25, 40011}, {27, 41506}, {55, 58074}, {92, 2218}, {272, 1826}, {278, 56146}, {523, 68211}, {607, 15467}, {653, 23289}, {1305, 3064}, {2052, 66951}, {2333, 57784}, {2501, 65274}, {6591, 51566}, {8747, 40161}, {8748, 28786}, {14618, 58986}, {24006, 65254}
X(68573) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 18134}, {19, 3868}, {25, 579}, {33, 27396}, {42, 51574}, {225, 56559}, {272, 17206}, {393, 5125}, {607, 3190}, {608, 4306}, {1305, 65164}, {1751, 69}, {1824, 22021}, {1826, 57808}, {1973, 2352}, {2175, 57501}, {2218, 63}, {2299, 56000}, {2333, 209}, {2969, 65118}, {2997, 304}, {3064, 20294}, {6059, 41320}, {6591, 23800}, {8750, 57217}, {15467, 57918}, {23289, 6332}, {28786, 52565}, {32674, 65315}, {40011, 305}, {40161, 52396}, {40574, 86}, {41506, 306}, {55208, 51658}, {56146, 345}, {57652, 66918}, {58074, 6063}, {58986, 4558}, {65103, 58333}, {65254, 4592}, {65274, 4563}, {66951, 394}, {68211, 99}

X(68574) = X(4)X(58)∩X(56)X(225)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^3 + b^3 + a*b*c - a*c^2 - b*c^2)*(a^3 - a*b^2 + a*b*c - b^2*c + c^3) : :

X(68574) lies on the circumconic {{A,B,C,X(1),(6)}}, the Huygens hyperbola, and these lines: {1, 5136}, {4, 58}, {6, 1826}, {25, 52150}, {29, 3192}, {56, 225}, {86, 264}, {93, 68494}, {106, 1068}, {312, 40436}, {393, 1474}, {407, 37646}, {1093, 8747}, {1167, 2551}, {1300, 59005}, {1724, 36050}, {1834, 37239}, {1842, 51686}, {2049, 57662}, {2215, 8557}, {2345, 2983}, {2899, 56112}, {2995, 17863}, {3011, 34430}, {3445, 23710}, {4696, 56179}, {6344, 68483}, {12649, 44765}, {35142, 54951}, {36124, 51845}, {39585, 57748}, {42019, 64087}, {49487, 56285}, {56814, 57709}

X(68574) = isotomic conjugate of X(51612)
X(68574) = polar conjugate of X(4417)
X(68574) = polar conjugate of the isotomic conjugate of X(13478)
X(68574) = X(i)-cross conjugate of X(j) for these (i,j): {11, 7649}, {37226, 4}, {57652, 19}
X(68574) = X(i)-isoconjugate of X(j) for these (i,j): {2, 22134}, {3, 3869}, {9, 56553}, {31, 51612}, {48, 4417}, {59, 34588}, {63, 573}, {69, 3185}, {72, 4225}, {78, 10571}, {101, 57184}, {109, 57111}, {219, 17080}, {255, 17555}, {326, 3192}, {662, 52310}, {692, 57242}, {1331, 21189}, {1332, 6589}, {1444, 22276}, {1790, 21078}, {1812, 40590}, {2149, 40626}, {4564, 47411}, {4574, 16754}, {22076, 40452}, {38345, 44717}
X(68574) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 51612}, {11, 57111}, {478, 56553}, {650, 40626}, {1015, 57184}, {1084, 52310}, {1086, 57242}, {1249, 4417}, {3162, 573}, {5521, 21189}, {6523, 17555}, {6615, 34588}, {15259, 3192}, {32664, 22134}, {36103, 3869}
X(68574) = cevapoint of X(4) and X(16066)
X(68574) = trilinear pole of line {649, 2501}
X(68574) = barycentric product X(i)*X(j) for these {i,j}: {4, 13478}, {19, 2995}, {25, 57906}, {27, 15232}, {29, 40160}, {92, 2217}, {225, 19607}, {278, 10570}, {514, 26704}, {2501, 54951}, {7649, 44765}, {8735, 57757}, {14618, 59005}, {17924, 36050}, {24006, 65253}, {32653, 46107}
X(68574) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 51612}, {4, 4417}, {11, 40626}, {19, 3869}, {25, 573}, {31, 22134}, {34, 17080}, {56, 56553}, {393, 17555}, {512, 52310}, {513, 57184}, {514, 57242}, {608, 10571}, {650, 57111}, {1474, 4225}, {1824, 21078}, {1973, 3185}, {2170, 34588}, {2207, 3192}, {2217, 63}, {2333, 22276}, {2995, 304}, {3271, 47411}, {6591, 21189}, {8735, 124}, {10570, 345}, {13478, 69}, {15232, 306}, {15386, 44717}, {19607, 332}, {26704, 190}, {32653, 1331}, {36050, 1332}, {40160, 307}, {44765, 4561}, {54951, 4563}, {57200, 16754}, {57652, 40590}, {57906, 305}, {59005, 4558}, {65253, 4592}

X(68575) = X(4)X(147)∩X(225)X(1581)

Barycentrics    (b^2 - a*c)*(b^2 + a*c)*(a*b - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a*b + c^2) : :

X(68575) lies on the Huygens hyperbola and these lines: {4, 147}, {225, 1581}, {232, 419}, {264, 5117}, {290, 44114}, {297, 67078}, {393, 694}, {420, 9477}, {458, 45146}, {648, 1843}, {733, 1289}, {805, 1300}, {882, 16230}, {1629, 51954}, {1826, 21080}, {3563, 18858}, {3981, 19222}, {8743, 8789}, {8801, 56977}, {8884, 17970}, {17983, 65351}, {18808, 66267}, {18829, 35142}, {22028, 41013}, {27376, 51982}, {34208, 52283}, {34238, 41204}, {42377, 62955}, {47734, 47735}, {59932, 67070}

X(68575) = isotomic conjugate of X(12215)
X(68575) = polar conjugate of X(385)
X(68575) = isotomic conjugate of the anticomplement of X(53475)
X(68575) = isotomic conjugate of the isogonal conjugate of X(17980)
X(68575) = polar conjugate of the isotomic conjugate of X(1916)
X(68575) = polar conjugate of the isogonal conjugate of X(694)
X(68575) = X(i)-cross conjugate of X(j) for these (i,j): {297, 4}, {694, 1916}, {17984, 37892}, {53475, 2}
X(68575) = X(i)-isoconjugate of X(j) for these (i,j): {3, 1580}, {19, 58354}, {31, 12215}, {48, 385}, {63, 1691}, {69, 1933}, {163, 24284}, {171, 7193}, {172, 20769}, {184, 1966}, {238, 3955}, {255, 419}, {293, 36213}, {304, 14602}, {326, 44089}, {394, 56828}, {647, 56982}, {656, 56980}, {804, 4575}, {810, 17941}, {906, 4107}, {1176, 2236}, {1331, 4164}, {1437, 4039}, {1926, 14575}, {2086, 62719}, {2169, 63736}, {2196, 53681}, {3573, 22093}, {3917, 56971}, {3978, 9247}, {4020, 56976}, {4027, 66942}, {4579, 22384}, {4592, 5027}, {5026, 36060}, {7116, 27982}, {8623, 34055}, {8766, 51343}, {10547, 67160}, {14296, 32656}, {17974, 56679}, {17984, 52430}, {18902, 40364}, {35200, 51430}, {36061, 39495}, {36214, 51903}, {51318, 66933}
X(68575) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 12215}, {6, 58354}, {115, 24284}, {132, 36213}, {133, 51430}, {136, 804}, {1249, 385}, {1560, 5026}, {3162, 1691}, {5139, 5027}, {5190, 4107}, {5521, 4164}, {6523, 419}, {9467, 184}, {9470, 3955}, {14363, 63736}, {15259, 44089}, {16221, 39495}, {36103, 1580}, {39052, 56982}, {39062, 17941}, {39092, 3}, {40596, 56980}, {40938, 732}, {47648, 36212}, {48317, 11183}, {62576, 3978}, {62595, 5976}, {62605, 1966}
X(68575) = cevapoint of X(i) and X(j) for these (i,j): {4, 420}, {232, 1843}, {523, 44114}, {694, 17980}, {8754, 16230}
X(68575) = crosspoint of X(2998) and X(9473)
X(68575) = crosssum of X(1613) and X(52162)
X(68575) = trilinear pole of line {427, 2501}
X(68575) = barycentric product X(i)*X(j) for these {i,j}: {4, 1916}, {19, 1934}, {25, 18896}, {76, 17980}, {92, 1581}, {112, 56981}, {158, 66933}, {264, 694}, {297, 36897}, {393, 40708}, {420, 9477}, {427, 14970}, {523, 65351}, {648, 66267}, {733, 1235}, {805, 14618}, {882, 6331}, {1967, 1969}, {1974, 44160}, {2052, 36214}, {2501, 18829}, {8754, 39292}, {8789, 44161}, {9468, 18022}, {14251, 60199}, {16081, 40810}, {16230, 39291}, {17970, 18027}, {17984, 41517}, {18858, 68089}, {18872, 46111}, {20883, 43763}, {22456, 67070}, {24006, 37134}, {32085, 56977}, {34238, 44132}, {35142, 47734}, {37892, 54130}, {46104, 56978}, {57806, 66942}, {60577, 65332}, {65327, 66299}
X(68575) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 12215}, {3, 58354}, {4, 385}, {19, 1580}, {25, 1691}, {53, 63736}, {92, 1966}, {112, 56980}, {162, 56982}, {232, 36213}, {242, 53681}, {256, 20769}, {264, 3978}, {292, 3955}, {297, 5976}, {393, 419}, {419, 4027}, {420, 8290}, {427, 732}, {460, 12829}, {468, 5026}, {523, 24284}, {648, 17941}, {694, 3}, {733, 1176}, {805, 4558}, {881, 3049}, {882, 647}, {893, 7193}, {1096, 56828}, {1235, 35540}, {1581, 63}, {1826, 4039}, {1843, 8623}, {1916, 69}, {1927, 9247}, {1934, 304}, {1967, 48}, {1969, 1926}, {1973, 1933}, {1974, 14602}, {1990, 51430}, {2052, 17984}, {2207, 44089}, {2489, 5027}, {2501, 804}, {2967, 46888}, {2971, 2086}, {3572, 22093}, {5117, 9865}, {6331, 880}, {6530, 39931}, {6531, 40820}, {6591, 4164}, {7009, 27982}, {7649, 4107}, {8789, 14575}, {8791, 36820}, {9468, 184}, {9477, 57845}, {11380, 51320}, {14251, 3289}, {14273, 11183}, {14604, 40373}, {14618, 14295}, {14970, 1799}, {15391, 17974}, {16081, 14382}, {17442, 2236}, {17924, 14296}, {17938, 32661}, {17970, 577}, {17980, 6}, {17983, 60863}, {18022, 14603}, {18829, 4563}, {18872, 3292}, {18896, 305}, {20883, 67160}, {27369, 56915}, {32085, 56976}, {34238, 248}, {34854, 51324}, {36214, 394}, {36897, 287}, {37134, 4592}, {37892, 54129}, {38947, 63464}, {39291, 17932}, {39292, 47389}, {39645, 50732}, {40708, 3926}, {40810, 36212}, {41517, 36214}, {43534, 4019}, {43717, 51343}, {43763, 34055}, {44089, 51318}, {44160, 40050}, {44161, 18901}, {44162, 18902}, {46104, 56979}, {46292, 22143}, {47230, 39495}, {47642, 20794}, {47734, 3564}, {52460, 51322}, {54130, 37894}, {56828, 51903}, {56977, 3933}, {56978, 3917}, {56981, 3267}, {58260, 47418}, {65338, 18047}, {65349, 39681}, {65351, 99}, {65352, 17103}, {66267, 525}, {66933, 326}, {66942, 255}, {67070, 684}

X(68576) = X(4)X(7)∩X(225)X(3668)

Barycentrics    b*(-a + b - c)^2*(a + b - c)^2*c*(b + c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2) : :

X(68576) lies on the Huygens hyperbola and these lines: {4, 7}, {9, 37805}, {19, 63149}, {34, 4341}, {65, 52560}, {72, 57810}, {158, 6526}, {225, 3668}, {226, 1826}, {264, 6063}, {278, 393}, {307, 860}, {331, 54314}, {347, 1068}, {442, 1441}, {934, 1300}, {1105, 5088}, {1118, 65582}, {1398, 37377}, {1410, 8884}, {1444, 6359}, {1865, 52023}, {1880, 52607}, {1881, 4466}, {2160, 40573}, {3218, 37279}, {4293, 40933}, {4566, 52673}, {4569, 35142}, {4605, 22021}, {5236, 60991}, {5758, 55015}, {6046, 57285}, {6354, 53417}, {6531, 32714}, {6829, 53821}, {7490, 44697}, {7580, 17134}, {13149, 17983}, {17924, 42462}, {26003, 60970}, {41003, 56827}, {60431, 61003}

X(68576) = isotomic conjugate of X(1792)
X(68576) = polar conjugate of X(2287)
X(68576) = isotomic conjugate of the isogonal conjugate of X(1426)
X(68576) = polar conjugate of the isotomic conjugate of X(1446)
X(68576) = polar conjugate of the isogonal conjugate of X(1427)
X(68576) = X(273)-Ceva conjugate of X(3668)
X(68576) = X(i)-cross conjugate of X(j) for these (i,j): {225, 40149}, {1427, 1446}, {3120, 17924}, {6046, 3668}, {18210, 7178}, {51658, 1020}, {57285, 226}
X(68576) = X(i)-isoconjugate of X(j) for these (i,j): {3, 2328}, {6, 2327}, {9, 2193}, {21, 212}, {29, 6056}, {31, 1792}, {41, 1812}, {48, 2287}, {55, 283}, {58, 1260}, {60, 2318}, {71, 7054}, {73, 6061}, {78, 2194}, {81, 1802}, {100, 57134}, {101, 23090}, {109, 58338}, {110, 57108}, {112, 57057}, {162, 58340}, {163, 57055}, {184, 1043}, {200, 1437}, {201, 23609}, {219, 284}, {220, 1790}, {228, 1098}, {255, 4183}, {332, 2175}, {333, 52425}, {345, 57657}, {394, 2332}, {521, 65375}, {577, 2322}, {603, 56182}, {607, 6514}, {643, 1946}, {652, 5546}, {657, 4558}, {662, 65102}, {692, 57081}, {906, 1021}, {1172, 2289}, {1253, 1444}, {1259, 2299}, {1265, 2206}, {1331, 21789}, {1333, 3692}, {1334, 65568}, {1793, 2361}, {1794, 8021}, {1800, 7072}, {1819, 2192}, {2150, 3694}, {2185, 52370}, {2200, 7058}, {2204, 3719}, {2326, 3990}, {2359, 46889}, {2638, 5379}, {3239, 32661}, {3270, 4570}, {3900, 4575}, {3939, 23189}, {4055, 59482}, {4587, 7252}, {4592, 8641}, {6066, 17219}, {7012, 66898}, {7079, 18604}, {7253, 32656}, {7259, 22383}, {8606, 35193}, {8750, 68151}, {14827, 17206}, {15411, 32739}, {16947, 30681}, {32676, 68108}, {35196, 44707}, {35200, 58337}, {36054, 65201}, {36059, 58329}, {44130, 62257}, {44709, 62265}, {47390, 52335}, {57736, 58328}
X(68576) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 1792}, {9, 2327}, {10, 1260}, {11, 58338}, {37, 3692}, {57, 1819}, {115, 57055}, {125, 58340}, {133, 58337}, {136, 3900}, {223, 283}, {226, 1259}, {244, 57108}, {278, 13614}, {478, 2193}, {1015, 23090}, {1084, 65102}, {1086, 57081}, {1214, 78}, {1249, 2287}, {3160, 1812}, {4988, 34591}, {5139, 8641}, {5190, 1021}, {5521, 21789}, {6523, 4183}, {6609, 1437}, {7952, 56182}, {8054, 57134}, {15267, 228}, {15526, 68108}, {17113, 1444}, {20620, 58329}, {26932, 68151}, {34591, 57057}, {36103, 2328}, {36901, 15416}, {36908, 3}, {39053, 643}, {39060, 645}, {40586, 1802}, {40590, 219}, {40593, 332}, {40603, 1265}, {40611, 212}, {40617, 23189}, {40619, 15411}, {40622, 521}, {40837, 21}, {47345, 9}, {48317, 58331}, {50330, 3270}, {53982, 58328}, {53983, 58335}, {55060, 1946}, {56325, 3694}, {56905, 3965}, {59608, 63}, {62565, 3719}, {62570, 345}, {62602, 333}, {62605, 1043}
X(68576) = cevapoint of X(i) and X(j) for these (i,j): {1426, 1427}, {7178, 18210}, {16732, 24006}
X(68576) = crosspoint of X(55346) and X(58993)
X(68576) = crosssum of X(212) and X(6056)
X(68576) = trilinear pole of line {2501, 7178}
X(68576) = crossdifference of every pair of points on line {57134, 58340}
X(68576) = barycentric product X(i)*X(j) for these {i,j}: {4, 1446}, {7, 40149}, {10, 1847}, {34, 349}, {56, 52575}, {57, 57809}, {65, 331}, {76, 1426}, {85, 225}, {92, 3668}, {158, 56382}, {226, 273}, {264, 1427}, {278, 1441}, {279, 41013}, {286, 6354}, {313, 1435}, {321, 1119}, {342, 8808}, {523, 13149}, {653, 4077}, {658, 24006}, {693, 52607}, {850, 32714}, {934, 14618}, {1020, 46107}, {1042, 1969}, {1088, 1826}, {1118, 1231}, {1254, 44129}, {1396, 34388}, {1398, 27801}, {1400, 57787}, {1410, 18027}, {1434, 56285}, {1439, 2052}, {1577, 36118}, {1824, 57792}, {1880, 6063}, {1893, 62946}, {1896, 20618}, {2501, 4569}, {3676, 65207}, {4017, 46404}, {4466, 24032}, {4566, 17924}, {4572, 55208}, {4573, 66297}, {6046, 31623}, {7103, 60197}, {7128, 21207}, {7147, 44130}, {7178, 18026}, {7282, 43682}, {8736, 57785}, {16732, 55346}, {17094, 54240}, {18210, 57538}, {20567, 57652}, {23062, 53008}, {24002, 61178}, {40701, 52384}, {40933, 52581}, {51664, 52938}, {52373, 57806}, {52937, 55206}, {53237, 60229}, {55110, 57810}, {65296, 66299}
X(68576) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2327}, {2, 1792}, {4, 2287}, {7, 1812}, {10, 3692}, {12, 3694}, {19, 2328}, {27, 1098}, {28, 7054}, {34, 284}, {37, 1260}, {42, 1802}, {56, 2193}, {57, 283}, {65, 219}, {73, 2289}, {77, 6514}, {85, 332}, {92, 1043}, {108, 5546}, {158, 2322}, {181, 52370}, {223, 1819}, {225, 9}, {226, 78}, {227, 55111}, {269, 1790}, {273, 333}, {278, 21}, {279, 1444}, {281, 56182}, {286, 7058}, {307, 3719}, {313, 52406}, {321, 1265}, {331, 314}, {342, 27398}, {349, 3718}, {393, 4183}, {429, 3965}, {512, 65102}, {513, 23090}, {514, 57081}, {523, 57055}, {525, 68108}, {608, 2194}, {647, 58340}, {649, 57134}, {650, 58338}, {653, 643}, {656, 57057}, {658, 4592}, {661, 57108}, {693, 15411}, {850, 15416}, {905, 68151}, {934, 4558}, {1014, 65568}, {1020, 1331}, {1042, 48}, {1088, 17206}, {1096, 2332}, {1118, 1172}, {1119, 81}, {1172, 6061}, {1214, 1259}, {1231, 1264}, {1254, 71}, {1365, 53560}, {1395, 57657}, {1396, 60}, {1398, 1333}, {1400, 212}, {1402, 52425}, {1407, 1437}, {1409, 6056}, {1410, 577}, {1425, 3990}, {1426, 6}, {1427, 3}, {1435, 58}, {1439, 394}, {1441, 345}, {1446, 69}, {1461, 4575}, {1565, 16731}, {1824, 220}, {1825, 52405}, {1826, 200}, {1829, 46889}, {1835, 2323}, {1841, 8021}, {1847, 86}, {1848, 46877}, {1865, 64171}, {1867, 3713}, {1874, 3684}, {1880, 55}, {1893, 37658}, {1897, 7259}, {1990, 58337}, {2006, 1793}, {2171, 2318}, {2189, 23609}, {2333, 1253}, {2358, 2192}, {2489, 8641}, {2501, 3900}, {3064, 58329}, {3120, 34591}, {3125, 3270}, {3668, 63}, {3669, 23189}, {3701, 30681}, {4017, 652}, {4077, 6332}, {4466, 24031}, {4551, 4587}, {4552, 4571}, {4566, 1332}, {4569, 4563}, {4572, 55207}, {6046, 1214}, {6335, 7256}, {6354, 72}, {6356, 3998}, {6358, 3710}, {6591, 21789}, {7053, 18604}, {7103, 2303}, {7117, 66898}, {7128, 4570}, {7140, 4515}, {7143, 1409}, {7147, 73}, {7178, 521}, {7180, 1946}, {7216, 1459}, {7250, 22383}, {7282, 56440}, {7337, 2204}, {7649, 1021}, {8736, 210}, {8747, 2326}, {8754, 36197}, {8808, 271}, {8898, 7085}, {10376, 2286}, {13149, 99}, {13853, 52389}, {14273, 58331}, {14618, 4397}, {16732, 2968}, {17898, 57045}, {17924, 7253}, {17925, 65575}, {18026, 645}, {18210, 35072}, {20618, 52385}, {23984, 5379}, {24006, 3239}, {30572, 14418}, {32674, 65375}, {32714, 110}, {36118, 662}, {36127, 65201}, {36197, 65752}, {37755, 3682}, {40149, 8}, {40663, 52978}, {40837, 13614}, {40933, 15905}, {40961, 7124}, {41013, 346}, {43923, 7252}, {43932, 7254}, {46404, 7257}, {46406, 55202}, {51664, 57241}, {52373, 255}, {52374, 1789}, {52384, 268}, {52575, 3596}, {52577, 4319}, {52607, 100}, {52937, 55205}, {53008, 728}, {53237, 16713}, {53321, 906}, {53540, 7117}, {53545, 7004}, {54240, 36797}, {55110, 285}, {55206, 4105}, {55208, 663}, {55346, 4567}, {56285, 2321}, {56382, 326}, {56848, 1800}, {57652, 41}, {57787, 28660}, {57809, 312}, {57810, 55112}, {58757, 65103}, {59941, 15419}, {61178, 644}, {61400, 1806}, {61401, 1805}, {61411, 5324}, {62192, 603}, {65207, 3699}, {65232, 4636}, {66287, 8611}, {66297, 3700}
X(68576) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {273, 342, 4}, {273, 1847, 1119}, {278, 40837, 37388}, {1441, 57809, 41013}

X(68577) = X(4)X(1043)∩X(8)X(225)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^3 - a^2*b + a*b^2 + 3*b^3 - a^2*c + 2*a*b*c + b^2*c - a*c^2 - b*c^2 + c^3)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c + 2*a*b*c - b^2*c + a*c^2 + b*c^2 + 3*c^3) : :

X(68577) lies on the Huygens hyperbola and these lines: {4, 1043}, {8, 225}, {69, 407}, {280, 5177}, {341, 41013}, {346, 1826}, {388, 51565}, {393, 966}, {847, 44143}, {1792, 37194}, {1861, 56205}, {7046, 65822}, {26027, 56099}

X(68577) = polar conjugate of X(37642)
X(68577) = isotomic conjugate of the anticomplement of X(5816)
X(68577) = polar conjugate of the isotomic conjugate of X(60254)
X(68577) = X(i)-cross conjugate of X(j) for these (i,j): {5816, 2}, {28161, 1897}
X(68577) = X(i)-isoconjugate of X(j) for these (i,j): {48, 37642}, {184, 44735}, {603, 3486}
X(68577) = X(i)-Dao conjugate of X(j) for these (i,j): {1249, 37642}, {7952, 3486}, {62605, 44735}
X(68577) = cevapoint of X(10) and X(1867)
X(68577) = trilinear pole of line {2501, 3239}
X(68577) = barycentric product X(4)*X(60254)
X(68577) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 37642}, {92, 44735}, {281, 3486}, {60254, 69}

X(68578) = X(4)X(81)∩X(264)X(274)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^3 + a^2*b + a*b^2 + b^3 + a^2*c + 2*a*b*c + b^2*c - a*c^2 - b*c^2 - c^3)*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c + 2*a*b*c - b^2*c + a*c^2 + b*c^2 + c^3) : :

X(68578) lies on the Huygens hyperbola and these lines: {1, 1826}, {2, 1068}, {4, 81}, {28, 393}, {57, 225}, {105, 59083}, {254, 37117}, {264, 274}, {386, 56231}, {387, 2982}, {406, 37870}, {847, 66954}, {959, 34266}, {1217, 37305}, {1255, 38295}, {1300, 59130}, {1714, 39947}, {1870, 56041}, {2282, 5230}, {2990, 65303}, {3421, 40399}, {4200, 39747}, {5125, 55939}, {6344, 66922}, {6526, 37372}, {7498, 63157}, {8884, 37395}, {10529, 35058}, {14016, 37642}, {17905, 39954}, {23710, 25430}, {32085, 52376}, {39948, 40950}

X(68578) = polar conjugate of X(5739)
X(68578) = polar conjugate of the isotomic conjugate of X(60156)
X(68578) = polar conjugate of the isogonal conjugate of X(46010)
X(68578) = X(i)-cross conjugate of X(j) for these (i,j): {4185, 4}, {46010, 60156}
X(68578) = X(i)-isoconjugate of X(j) for these (i,j): {3, 12514}, {48, 5739}, {63, 36744}, {71, 27174}, {78, 64020}, {219, 45126}, {255, 406}, {326, 44086}, {1259, 1452}
X(68578) = X(i)-Dao conjugate of X(j) for these (i,j): {1249, 5739}, {3162, 36744}, {6523, 406}, {15259, 44086}, {36103, 12514}
X(68578) = cevapoint of X(4) and X(7521)
X(68578) = trilinear pole of line {513, 2501}
X(68578) = barycentric product X(i)*X(j) for these {i,j}: {4, 60156}, {264, 46010}, {273, 56225}, {393, 57832}, {693, 59083}, {2052, 57667}, {14618, 59130}, {17924, 65303}
X(68578) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 5739}, {19, 12514}, {25, 36744}, {28, 27174}, {34, 45126}, {393, 406}, {608, 64020}, {2207, 44086}, {41013, 42707}, {46010, 3}, {56225, 78}, {57667, 394}, {57832, 3926}, {59083, 100}, {59130, 4558}, {60156, 69}, {65303, 1332}
X(68578) = {X(225),X(5292)}-harmonic conjugate of X(14018)

X(68579) = X(4)X(75)∩X(264)X(561)

Barycentrics    b*c*(b + c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 + 2*a*b + b^2 + c^2)*(a^2 + b^2 + 2*a*c + c^2) : :

X(68579) lies on the Huygens hyperbola and these lines: {4, 75}, {92, 393}, {225, 1441}, {254, 57998}, {264, 561}, {313, 41013}, {321, 1826}, {429, 20336}, {847, 20571}, {1039, 2997}, {1093, 57806}, {1105, 57955}, {1300, 1310}, {1821, 2281}, {2995, 17863}, {3112, 32085}, {6344, 63759}, {8884, 40440}, {17983, 46277}, {18697, 39579}, {35142, 54982}, {44140, 64989}, {51686, 56875}, {57923, 58013}

X(68579) = polar conjugate of X(2303)
X(68579) = polar conjugate of the isotomic conjugate of X(60197)
X(68579) = polar conjugate of the isogonal conjugate of X(56219)
X(68579) = X(i)-cross conjugate of X(j) for these (i,j): {1867, 40149}, {56219, 60197}
X(68579) = X(i)-isoconjugate of X(j) for these (i,j): {3, 44119}, {48, 2303}, {58, 7085}, {163, 2522}, {184, 1010}, {212, 5323}, {255, 4206}, {283, 1460}, {284, 2286}, {612, 1437}, {1038, 2194}, {1333, 5227}, {1576, 23874}, {1790, 54416}, {2193, 2285}, {2206, 54433}, {2484, 4558}, {4575, 8678}, {4592, 8646}, {6590, 32661}, {14575, 44154}, {32656, 47844}, {51644, 65375}, {56367, 57657}
X(68579) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 7085}, {37, 5227}, {115, 2522}, {136, 8678}, {1214, 1038}, {1249, 2303}, {4858, 23874}, {5139, 8646}, {6523, 4206}, {18589, 19459}, {36103, 44119}, {40590, 2286}, {40603, 54433}, {40622, 51644}, {40837, 5323}, {47345, 2285}, {62570, 56367}, {62605, 1010}
X(68579) = trilinear pole of line {1577, 2501}
X(68579) = barycentric product X(i)*X(j) for these {i,j}: {4, 60197}, {225, 64989}, {264, 56219}, {349, 1039}, {523, 65341}, {850, 36099}, {1036, 52575}, {1245, 1969}, {1310, 14618}, {1826, 57923}, {2052, 66948}, {2281, 18022}, {2339, 57809}, {2501, 54982}, {20948, 32691}, {24006, 37215}, {27801, 51686}, {30479, 40149}, {44129, 68558}
X(68579) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 2303}, {10, 5227}, {19, 44119}, {37, 7085}, {65, 2286}, {92, 1010}, {225, 2285}, {226, 1038}, {278, 5323}, {313, 19799}, {321, 54433}, {393, 4206}, {523, 2522}, {1036, 2193}, {1039, 284}, {1089, 3610}, {1245, 48}, {1310, 4558}, {1441, 56367}, {1577, 23874}, {1824, 54416}, {1826, 612}, {1867, 34261}, {1880, 1460}, {1969, 44154}, {2221, 1437}, {2281, 184}, {2339, 283}, {2489, 8646}, {2501, 8678}, {7178, 51644}, {14618, 2517}, {16583, 19459}, {16732, 26933}, {17924, 47844}, {24006, 6590}, {30479, 1812}, {32691, 163}, {36099, 110}, {37215, 4592}, {40149, 388}, {41013, 2345}, {51686, 1333}, {53510, 7386}, {54982, 4563}, {56219, 3}, {56328, 1790}, {56841, 4267}, {57923, 17206}, {60197, 69}, {64989, 332}, {65207, 14594}, {65298, 4575}, {65341, 99}, {66948, 394}, {68558, 71}

X(68580) = X(4)X(653)∩X(264)X(21666)

Barycentrics    (a + b - c)*(a - b + c)*(b + c)*(a^2 - 2*a*b + b^2 + a*c + b*c - 2*c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2 + a*b - 2*b^2 - 2*a*c + b*c + c^2) : :

X(68580) lies on the Huygens hyperbola and these lines: {4, 653}, {225, 52607}, {254, 38295}, {264, 21666}, {393, 36127}, {1300, 14733}, {1826, 61178}, {17983, 17985}, {35142, 35157}, {37139, 52891}, {41013, 65207}

X(68580) = X(65340)-Ceva conjugate of X(62764)
X(68580) = X(i)-isoconjugate of X(j) for these (i,j): {3, 62756}, {110, 14414}, {255, 52891}, {283, 1155}, {284, 6510}, {527, 2193}, {1055, 1812}, {1437, 6745}, {1790, 6603}, {2327, 6610}, {4558, 65680}, {4575, 6366}, {4592, 6139}, {18604, 60431}, {23090, 23890}, {23346, 57081}, {56543, 57134}
X(68580) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 6366}, {244, 14414}, {5139, 6139}, {6523, 52891}, {36103, 62756}, {40590, 6510}, {47345, 527}
X(68580) = trilinear pole of line {225, 2501}
X(68580) = barycentric product X(i)*X(j) for these {i,j}: {92, 62764}, {225, 1121}, {226, 65340}, {523, 65335}, {1156, 40149}, {1826, 62723}, {2291, 57809}, {2501, 35157}, {14618, 14733}, {24006, 37139}, {34056, 41013}, {34068, 52575}, {35348, 65207}, {52607, 63748}, {60479, 61178}, {65304, 66299}
X(68580) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 62756}, {65, 6510}, {225, 527}, {393, 52891}, {661, 14414}, {1121, 332}, {1156, 1812}, {1426, 6610}, {1824, 6603}, {1826, 6745}, {1874, 24685}, {1880, 1155}, {2291, 283}, {2489, 6139}, {2501, 6366}, {4845, 2327}, {14733, 4558}, {23351, 23090}, {23893, 57081}, {32728, 32661}, {34056, 1444}, {34068, 2193}, {35157, 4563}, {36141, 4575}, {37139, 4592}, {40149, 30806}, {41798, 1792}, {52607, 56543}, {55206, 14392}, {55208, 14413}, {57652, 1055}, {60047, 6514}, {62723, 17206}, {62764, 63}, {63748, 15411}, {65335, 99}, {65340, 333}

X(68581) = X(1)X(264)∩X(4)X(31)

Barycentrics    (a+b)*(a+c)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^2-b*a+b^2-c^2)*(a^2-c*a-b^2+c^2) : :

X(68581) lies on the Huygens hyperbola and these lines: {1, 264}, {4, 31}, {42, 41013}, {213, 1826}, {225, 1402}, {393, 1973}, {923, 17983}, {1093, 1096}, {1300, 15440}, {2258, 39585}, {8884, 62268}, {32085, 46289}, {34208, 38252}, {35142, 36051}

X(68581) = polar conjugate of the isotomic conjugate of X(60088)
X(68581) = X(i)-isoconjugate of X(j) for these (i,j): {63, 4269}, {69, 4215}, {283, 37591}, {1444, 26893}
X(68581) = X(3162)-Dao conjugate of X(4269)
X(68581) = trilinear pole of line {798, 2501}
X(68581) = barycentric product X(i)*X(j) for these {i,j}: {4, 60088}, {14618, 15440}
X(68581) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 4269}, {1880, 37591}, {1973, 4215}, {2333, 26893}, {15440, 4558}, {60088, 69}

X(68582) = X(4)X(3945)∩X(391)X(461)

Barycentrics    -((a - b - c)*(3*a + b + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2 - (b - c)^2 + 2*a*(b + c))) : :

See David Nguyen, euclid 8509.

X(68582) lies on the Hatzipolakis-Lozada hyperbola and these lines: {4, 3945}, {6, 4207}, {33, 1839}, {185, 3332}, {193, 14004}, {391, 461}, {1827, 1829}, {1863, 40950}, {2905, 62997}, {2907, 4194}, {4196, 4648}, {4213, 37681}, {4307, 66248}, {55115, 58906}

X(68582) = orthic-isogonal conjugate of X(4196)
X(68582) = barycentric product X(i)*X(j) for these {i,j}: {391, 4196}, {461, 4648}
X(68582) = barycentric quotient X(i)/X(j) for these {i,j}: {461, 32022}, {4196, 57826}, {4648, 57873}, {5021, 57701}
X(68582) = trilinear product X(i)*X(j) for these {i,j}: {461, 62819}, {4196, 4512}
X(68582) = trilinear quotient X(i)/X(j) for these {i,j}: {57701, 62819}

X(68583) = X(1)X(5927)∩X(5)X(10)

Barycentrics    a*(a^5*(b + c) - a^4*(b^2 + 4*b*c + c^2) - (b^2 - c^2)^2*(b^2 + 4*b*c + c^2) + a*(b - c)^2*(b^3 - 5*b^2*c - 5*b*c^2 + c^3) - 2*a^3*(b^3 - 3*b^2*c - 3*b*c^2 + c^3) + 2*a^2*(b^4 + 4*b^3*c - 6*b^2*c^2 + 4*b*c^3 + c^4)) : :
X(68583) = X[1]+X[5927], X[4]+2*X[58679], 2*X[946]+X[960], X[3878]+2*X[5806], X[4301]+2*X[5044], X[65]-7*X[68034], X[72]+5*X[11522], X[354]+X[67998], 2*X[355]+X[66256], X[392]+X[1699], X[5919]+X[59387], X[962]+5*X[25917], X[962]+2*X[58637], 5*X[25917]-2*X[58637], X[1071]-7*X[9624], 2*X[1125]+X[9856], 4*X[1125]-X[9943], 2*X[1125]-X[11227], 2*X[9856]+X[9943], X[9856]+X[11227], X[9943]-2*X[11227], X[1320]+2*X[58687], X[1482]+2*X[58631], X[3057]+5*X[3091], X[3244]+2*X[9947], 5*X[3616]-X[11220], 5*X[3616]+X[12688], 5*X[3616]-2*X[58567], X[11220]+X[12688], X[11220]-2*X[58567], X[12688]+2*X[58567], 7*X[3622]-X[12680], 4*X[3634]-X[31798], 2*X[3636]+X[31871], 5*X[3697]+X[11531], X[3742]-2*X[5886], 3*X[3742]-2*X[10202], 3*X[5886]-X[10202], X[3753]-3*X[7988], 2*X[3812]-5*X[8227], 2*X[3812]+X[12672], 5*X[8227]+X[12672], 7*X[3832]+5*X[3890], X[3874]+2*X[31821], X[3877]+3*X[9779], X[3884]+2*X[12571], 2*X[4662]+X[7982], 2*X[5045]+X[31803], X[5657]-2*X[58451], X[5777]+2*X[13464], 2*X[5777]+X[34791], 4*X[13464]-X[34791], X[5887]+2*X[13374], X[5887]+5*X[18493], 2*X[13374]-5*X[18493], 4*X[5901]-X[12675], 2*X[5901]+X[31937], X[12675]+2*X[31937], X[5918]-3*X[54445], X[7970]+2*X[58682], X[7983]+2*X[58681], X[7984]+2*X[58680], 7*X[7989]-X[10914], X[9957]+2*X[19925], 2*X[10156]-3*X[19883], 2*X[10165]-X[10178], X[10167]-3*X[25055], 5*X[10595]+X[14872], 5*X[10595]-2*X[58609], X[14872]+2*X[58609], X[10695]+2*X[58686], X[10697]+2*X[58684], X[10698]+2*X[58683], X[10703]+2*X[58685], X[12528]+5*X[17609], X[16200]+X[18908], X[17638]+2*X[58613], 2*X[18483]+X[31786], 5*X[19862]-2*X[31787], X[22793]+2*X[31838], X[23340]+5*X[61261], X[31162]+X[64107], 2*X[33575]-X[50808], X[34339]-4*X[61272], X[37562]-7*X[61268], X[39898]+2*X[58694], X[40263]+5*X[61276], X[43166]+2*X[58634], 2*X[58591]+X[67988], 2*X[58595]+X[66044], X[59417]-3*X[61686], 3*X[61275]+X[61705]

See David Nguyen, euclid 8513.

X(68583) lies on these lines: {1, 5927}, {4, 58679}, {5, 10}, {12, 66216}, {65, 5704}, {72, 11522}, {165, 474}, {354, 1858}, {355, 66256}, {377, 9812}, {392, 1699}, {496, 12617}, {497, 5252}, {515, 10179}, {518, 5603}, {519, 10157}, {551, 971}, {912, 51709}, {952, 67875}, {962, 25917}, {997, 15587}, {999, 54370}, {1001, 1012}, {1064, 15569}, {1071, 9624}, {1125, 9856}, {1319, 6912}, {1320, 58687}, {1387, 2801}, {1482, 58631}, {1519, 3838}, {1538, 3822}, {2771, 38044}, {2800, 45310}, {3057, 3091}, {3244, 9947}, {3428, 15254}, {3616, 11220}, {3622, 12680}, {3634, 31798}, {3636, 31871}, {3697, 11531}, {3698, 5056}, {3742, 5886}, {3753, 7988}, {3812, 8227}, {3832, 3890}, {3848, 6833}, {3874, 31821}, {3877, 9779}, {3880, 5587}, {3884, 12571}, {4423, 64150}, {4428, 52026}, {4640, 22753}, {4662, 7982}, {4682, 64449}, {4711, 28234}, {4915, 11224}, {5045, 31803}, {5068, 14923}, {5219, 66226}, {5220, 68032}, {5302, 22770}, {5400, 64175}, {5657, 58451}, {5703, 9848}, {5777, 13464}, {5789, 5887}, {5901, 12675}, {5902, 23708}, {5918, 54445}, {6261, 10246}, {6326, 65466}, {6835, 12701}, {6854, 35514}, {6873, 7704}, {6915, 37568}, {6939, 64322}, {6946, 13528}, {6964, 37828}, {6984, 14110}, {7288, 17634}, {7951, 17618}, {7958, 25973}, {7966, 18529}, {7970, 58682}, {7983, 58681}, {7984, 58680}, {7989, 10914}, {8167, 30503}, {8226, 30384}, {8583, 64074}, {9549, 58655}, {9578, 17622}, {9614, 10827}, {9623, 18227}, {9858, 12446}, {9957, 19925}, {9961, 46934}, {10156, 19883}, {10165, 10178}, {10167, 25055}, {10391, 15950}, {10595, 14872}, {10695, 58686}, {10697, 58684}, {10698, 58683}, {10703, 58685}, {10863, 31397}, {11281, 18238}, {11496, 59691}, {12528, 17609}, {12629, 12635}, {12705, 21164}, {12709, 50443}, {12710, 37737}, {16112, 63430}, {17638, 58613}, {18446, 42819}, {18483, 31786}, {18528, 64735}, {19862, 31787}, {22793, 31838}, {23340, 61261}, {25466, 63989}, {28150, 31775}, {28236, 63999}, {31162, 38150}, {32537, 59388}, {33575, 50808}, {34339, 61272}, {37561, 63266}, {37562, 61268}, {37704, 65465}, {37735, 67970}, {38021, 44663}, {38757, 51782}, {39898, 58694}, {40263, 61276}, {40726, 52027}, {43166, 58634}, {44675, 63994}, {47357, 54051}, {51723, 54227}, {52264, 58441}, {58591, 67988}, {58595, 66044}, {59417, 61686}, {61275, 61705}, {63970, 63993}, {64131, 64160}

X(68583) = midpoint of X(i) and X(j) for these {i,j}: {1, 5927}, {354, 67998}, {392, 1699}, {5919, 59387}, {9856, 11227}, {11220, 12688}, {16200, 18908}, {31162, 64107}
X(68583 = reflection of X(i) in X(j) for these (i,j): {3742, 5886}, {5657, 58451}, {9943, 11227}, {10178, 10165}, {11220, 58567}, {11227, 1125}, {50808, 33575}
X(68583) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 45776, 5836}, {962, 25917, 58637}, {1001, 63992, 65404}, {1125, 9856, 9943}, {3616, 12688, 58567}, {5603, 5817, 34625}, {5777, 13464, 34791}, {5887, 18493, 13374}, {5901, 31937, 12675}, {8227, 12672, 3812}, {10595, 14872, 58609}, {12705, 25524, 64128}

X(68584) = X(1)X(5779)∩X(3)X(6)

Barycentrics    a^2*(3*a^4 + 2*a^3*(b + c) + a^2*(-8*b^2 + 2*b*c - 8*c^2) + (b - c)^2*(5*b^2 + 8*b*c + 5*c^2) - 2*a*(b^3 + b^2*c + b*c^2 + c^3)) : :

See David Nguyen, euclid 8509.

X(68584) lies on these lines: {1, 5779}, {3, 6}, {4, 62997}, {5, 3945}, {35, 38293}, {37, 51516}, {42, 67711}, {81, 19541}, {140, 37681}, {269, 5708}, {355, 4349}, {381, 63054}, {382, 3332}, {405, 63088}, {940, 5400}, {942, 1419}, {952, 4344}, {971, 1449}, {990, 16666}, {1100, 60884}, {1159, 2263}, {1260, 54444}, {1279, 37624}, {1353, 36474}, {1385, 16469}, {1442, 5729}, {1458, 67261}, {1656, 4648}, {1743, 59381}, {1993, 13615}, {1994, 20835}, {2293, 37698}, {3019, 62008}, {3247, 64198}, {3295, 67264}, {3517, 44100}, {3526, 37650}, {3527, 57701}, {3664, 38107}, {3672, 5843}, {3843, 5733}, {3851, 63323}, {4000, 59380}, {4644, 60922}, {4667, 5805}, {5055, 17392}, {5070, 17245}, {5222, 31657}, {5308, 61511}, {5644, 16409}, {5790, 64174}, {5927, 62808}, {6767, 23071}, {6846, 65460}, {7290, 10246}, {7411, 63095}, {7580, 37685}, {8148, 61086}, {8727, 63007}, {9777, 37269}, {10247, 20430}, {10601, 16411}, {11108, 37659}, {11898, 36530}, {12631, 67012}, {13243, 17013}, {13632, 50962}, {13727, 37677}, {14547, 22117}, {14912, 49131}, {15178, 16487}, {16667, 66660}, {16670, 31658}, {16853, 25878}, {16857, 50317}, {17014, 36996}, {17337, 46219}, {17365, 51514}, {20090, 36652}, {21002, 37621}, {34545, 37309}, {34748, 50130}, {36002, 63039}, {36682, 62999}, {36706, 51170}, {36707, 39899}, {36721, 63052}, {37260, 54349}, {37826, 66683}, {45942, 61953}, {46922, 48878}, {66549, 66661}

X(68584) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 62183, 3}, {3311, 3312, 4251}, {5050, 48908, 3}, {11485, 11486, 4262}

X(68585) = X(2)X(6)∩X(37)X(651)

Barycentrics    a*(a^4 - b*(b - c)^2*c - a^3*(b + c) - a^2*(b^2 - 3*b*c + c^2) + a*(b^3 + b^2*c + b*c^2 + c^3)) : :

See David Nguyen, euclid 8509.

X(68585) lies on these lines: {1, 60964}, {2, 6}, {8, 63341}, {10, 63309}, {37, 651}, {71, 7175}, {219, 4644}, {269, 18593}, {284, 67984}, {347, 2256}, {442, 64421}, {613, 38053}, {644, 17351}, {851, 64414}, {894, 1332}, {991, 6906}, {1014, 2245}, {1419, 18675}, {1633, 49537}, {2293, 64710}, {2323, 3664}, {2475, 64423}, {2886, 63324}, {3085, 56293}, {3142, 64408}, {3193, 26131}, {3332, 6850}, {3560, 5453}, {3908, 21865}, {3909, 4239}, {4223, 37516}, {4271, 11349}, {4349, 63370}, {4415, 37798}, {4422, 23617}, {4653, 28461}, {4675, 60988}, {4861, 49478}, {5323, 22076}, {5733, 6937}, {6842, 63323}, {6893, 63318}, {6897, 63297}, {6940, 13329}, {7277, 17796}, {7290, 63292}, {7330, 63450}, {8609, 60969}, {8766, 63446}, {10039, 63319}, {11038, 12595}, {12594, 39587}, {15501, 31397}, {15843, 63325}, {17262, 25256}, {17365, 62799}, {21811, 51653}, {22136, 49743}, {22759, 63295}, {24936, 64394}, {25255, 63782}, {28968, 63398}, {37157, 63360}, {37261, 44085}, {37438, 63374}, {40863, 63443}, {48828, 62828}, {50070, 62848}, {58786, 64912}, {63332, 67264}, {63388, 63396}

X(68585) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 25878, 37650}, {6, 63401, 81}, {86, 4585, 15988}, {86, 28980, 2}, {323, 37635, 81}, {2256, 6180, 4419}, {3945, 63088, 6}, {22136, 49743, 62843}

X(68586) = X(5)X(6)∩X(30)X(991)

Barycentrics    2*a^6 - (b - c)^4*(b + c)^2 + 2*a*(b - c)^2*(b + c)^3 - 3*a^4*(b^2 + c^2) + 2*a^2*(b - c)^2*(b^2 + b*c + c^2) - 2*a^3*(b^3 + b^2*c + b*c^2 + c^3) : :
X(68586) = X[991]-3*X[17392], X[10446]+X[49132], X[17378]+X[36722], 3*X[17378]+X[48878], 3*X[36722]-X[48878]

See David Nguyen, euclid 8509.

X(68586) lies on these lines: {1, 5805}, {3, 3332}, {4, 3945}, {5, 6}, {7, 64750}, {30, 991}, {37, 5762}, {44, 61511}, {45, 64065}, {81, 8226}, {140, 3019}, {182, 19512}, {193, 36660}, {226, 15252}, {269, 57282}, {381, 63054}, {442, 37659}, {495, 60691}, {516, 15569}, {517, 64174}, {524, 48888}, {546, 45942}, {550, 50677}, {580, 50205}, {581, 65460}, {916, 942}, {940, 8727}, {946, 4349}, {952, 49478}, {971, 3664}, {975, 5763}, {990, 4675}, {1060, 2263}, {1086, 61509}, {1100, 53599}, {1203, 7958}, {1279, 5901}, {1351, 36526}, {1386, 15251}, {1418, 24470}, {1419, 9612}, {1449, 38150}, {1456, 12047}, {1458, 18990}, {1471, 15325}, {1479, 67264}, {1503, 24220}, {1536, 14828}, {1656, 37650}, {1743, 38108}, {2293, 15171}, {2476, 63088}, {3008, 61595}, {3090, 37681}, {3091, 62997}, {3247, 5735}, {3542, 44100}, {3628, 17337}, {3672, 59386}, {4000, 38107}, {4312, 59215}, {4344, 5603}, {4419, 60922}, {4644, 5779}, {4667, 63970}, {4670, 12618}, {4888, 66661}, {5308, 5759}, {5658, 41825}, {5706, 8728}, {5712, 19541}, {5717, 5806}, {5757, 30810}, {5843, 17365}, {5880, 53996}, {5886, 7290}, {5921, 36662}, {6147, 37729}, {6354, 59611}, {6675, 37530}, {6678, 9306}, {6826, 48847}, {6839, 64167}, {6841, 45931}, {6842, 63323}, {6881, 45923}, {6894, 64377}, {6920, 63297}, {7522, 14826}, {7956, 26098}, {8227, 16469}, {9624, 16487}, {9840, 31774}, {9955, 29287}, {10446, 49132}, {10739, 44858}, {10883, 14996}, {11018, 40960}, {11245, 47513}, {13226, 37520}, {13408, 31789}, {13598, 49557}, {13727, 17300}, {15973, 28350}, {16884, 38137}, {17278, 38171}, {17378, 36722}, {17379, 36652}, {17390, 29016}, {18440, 36659}, {19624, 61533}, {21620, 67026}, {21850, 36661}, {22791, 61086}, {29331, 49776}, {29571, 31658}, {31672, 66660}, {31775, 48903}, {31799, 59305}, {35227, 61276}, {36002, 37635}, {36477, 48876}, {36663, 39884}, {36721, 63110}, {37355, 45298}, {37363, 63068}, {37364, 37674}, {37374, 37633}, {45728, 64443}, {50294, 51709}, {51755, 66683}, {52260, 54972}, {52682, 64168}, {52969, 61563}, {58906, 67930}

X(68586) = midpoint of X(i) and X(j) for these {i,j}: {10446, 49132}, {17378, 36722}
X(68586) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 3945, 62183}, {990, 4675, 31657}, {3332, 4648, 3}, {13329, 17245, 140}

X(68587) = X(1)X(5696)∩X(6)X(31)

Barycentrics    -(a^2*(a^3 + a^2*(b + c) + 3*(b - c)^2*(b + c) - a*(5*b^2 + 2*b*c + 5*c^2))) : :

See David Nguyen, euclid 8509.

X(68587) lies on these lines: {1, 5696}, {6, 31}, {37, 42014}, {56, 64751}, {198, 21746}, {269, 5173}, {516, 66683}, {517, 8147}, {528, 4307}, {991, 3428}, {2099, 2263}, {2191, 10579}, {2256, 4343}, {2334, 54421}, {2886, 4648}, {3332, 5842}, {3419, 64174}, {3434, 3945}, {3553, 14100}, {4000, 8255}, {4644, 38454}, {5573, 11018}, {6690, 37650}, {7023, 64206}, {7290, 24929}, {7322, 64171}, {16465, 62833}, {16469, 59337}, {17245, 31245}, {17365, 36971}, {17392, 31140}, {20075, 62997}, {47387, 64739}

X(68587) = barycentric product X(i)*X(j) for these {i,j}: {55, 30275}
X(68587) = barycentric quotient X(i)/X(j) for these {i,j}: {30275, 6063}
X(68587) = trilinear product X(i)*X(j) for these {i,j}: {41, 30275}
X(68587) = trilinear quotient X(i)/X(j) for these {i,j}: {85, 30275}
X(68587) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 2293, 21002}

X(68588) = X(1)X(6)∩X(40)X(991)

Barycentrics    a*(a^2 - (b - c)^2 + 4*a*(b + c)) : :
X(68588) = 2*X[4349]-3*X[63054], 2*X[50284]-3*X[66637], X[4307]-2*X[4667], 2*X[4356]-X[4419]

See David Nguyen, euclid 8509.

X(68588) lies on these lines: {1, 6}, {2, 4684}, {7, 3755}, {8, 3879}, {10, 4648}, {31, 10389}, {39, 61326}, {40, 991}, {42, 57}, {43, 5437}, {55, 62812}, {58, 21059}, {63, 17018}, {65, 269}, {75, 49495}, {77, 7672}, {81, 3870}, {84, 37529}, {86, 49450}, {145, 894}, {165, 50677}, {171, 3158}, {193, 3883}, {200, 940}, {210, 17022}, {223, 5173}, {228, 16878}, {354, 2999}, {386, 1066}, {387, 21620}, {515, 3332}, {516, 4644}, {517, 8147}, {519, 4349}, {527, 64168}, {537, 50281}, {581, 68036}, {612, 62821}, {614, 44841}, {678, 55935}, {740, 4659}, {750, 46917}, {756, 25430}, {938, 27384}, {968, 3929}, {997, 62844}, {1045, 2136}, {1064, 68032}, {1086, 59372}, {1103, 67930}, {1125, 37650}, {1126, 2191}, {1215, 39594}, {1253, 1468}, {1280, 39958}, {1390, 39948}, {1418, 3339}, {1419, 2263}, {1420, 1471}, {1456, 2099}, {1462, 4327}, {1469, 52020}, {1697, 2293}, {1698, 17245}, {1706, 50581}, {1707, 3750}, {1738, 6173}, {1834, 5290}, {2093, 64175}, {2177, 35445}, {2274, 42079}, {2279, 39258}, {2308, 67209}, {2550, 3664}, {2801, 66661}, {2809, 11529}, {3008, 38053}, {3057, 67264}, {3220, 37580}, {3240, 3306}, {3241, 3685}, {3244, 3923}, {3295, 21002}, {3296, 24171}, {3304, 15287}, {3305, 29814}, {3338, 5312}, {3361, 4255}, {3474, 62240}, {3475, 40940}, {3576, 13329}, {3616, 37681}, {3622, 17260}, {3623, 17350}, {3624, 17337}, {3632, 24342}, {3666, 62823}, {3677, 3873}, {3679, 17392}, {3680, 66646}, {3681, 5287}, {3686, 39581}, {3696, 25590}, {3714, 35629}, {3717, 17316}, {3720, 7308}, {3722, 62846}, {3729, 49470}, {3736, 18164}, {3742, 23511}, {3744, 62842}, {3745, 41711}, {3748, 62875}, {3749, 3979}, {3752, 10980}, {3779, 64006}, {3786, 64377}, {3789, 26102}, {3811, 37554}, {3823, 17313}, {3875, 24349}, {3912, 59406}, {3914, 4654}, {3920, 62808}, {3928, 17594}, {3931, 54422}, {3932, 29573}, {3935, 14996}, {3938, 36483}, {3946, 4310}, {3951, 62831}, {3957, 37685}, {4000, 5542}, {4026, 17272}, {4038, 5268}, {4090, 59599}, {4307, 4667}, {4312, 17365}, {4318, 11526}, {4321, 5228}, {4328, 8581}, {4334, 60955}, {4340, 63146}, {4356, 4419}, {4360, 49446}, {4363, 28581}, {4383, 4883}, {4413, 21870}, {4414, 67334}, {4421, 39980}, {4429, 17298}, {4430, 17011}, {4454, 28557}, {4512, 4641}, {4640, 62820}, {4661, 17019}, {4666, 32911}, {4675, 38052}, {4716, 31178}, {4722, 62849}, {4847, 5712}, {4849, 8580}, {4851, 49524}, {4859, 25557}, {4860, 62695}, {4862, 66071}, {4878, 56809}, {4888, 5880}, {4899, 29574}, {4901, 49529}, {4907, 10394}, {4966, 17284}, {4974, 50283}, {5014, 42045}, {5219, 11269}, {5222, 11038}, {5231, 5718}, {5262, 62861}, {5263, 49451}, {5285, 54312}, {5295, 56087}, {5308, 5686}, {5313, 51816}, {5438, 37607}, {5534, 5707}, {5625, 49449}, {5695, 49475}, {5698, 63977}, {5709, 37698}, {5711, 6765}, {5733, 5881}, {5772, 29616}, {5847, 36479}, {6180, 12560}, {6610, 18421}, {6738, 67026}, {6769, 35658}, {7074, 10383}, {7191, 62815}, {7226, 62816}, {7277, 64016}, {7613, 60980}, {7675, 62797}, {7719, 54407}, {7982, 12717}, {8236, 60960}, {8679, 64751}, {9041, 24358}, {9340, 17782}, {9345, 21805}, {9347, 62236}, {9580, 41011}, {9623, 66687}, {10529, 27254}, {10578, 37666}, {11019, 63089}, {11363, 44100}, {11518, 54418}, {11520, 17016}, {12526, 37548}, {12625, 67969}, {13405, 37642}, {14839, 66638}, {16569, 28600}, {16574, 19767}, {16688, 37507}, {16825, 49685}, {16834, 32922}, {17017, 62850}, {17118, 49468}, {17127, 62856}, {17151, 49483}, {17156, 32771}, {17306, 49511}, {17318, 28582}, {17319, 31302}, {17378, 32850}, {17449, 67211}, {17721, 31146}, {17723, 51463}, {17738, 24841}, {17768, 66673}, {18443, 44414}, {18789, 25426}, {19624, 59337}, {19765, 62824}, {19860, 37659}, {19868, 63055}, {19998, 26627}, {20009, 41247}, {20011, 63131}, {20090, 50289}, {20978, 37556}, {21371, 62832}, {21806, 36263}, {24159, 41870}, {24210, 28609}, {24248, 60933}, {24325, 49497}, {24392, 26098}, {25525, 33137}, {25568, 39595}, {25878, 64673}, {26015, 63008}, {28350, 59311}, {28503, 50126}, {29571, 38057}, {29597, 50075}, {29817, 63074}, {29828, 32919}, {29835, 31034}, {29839, 56519}, {29843, 62998}, {29866, 56521}, {29868, 56522}, {30331, 64017}, {31164, 33134}, {31249, 37663}, {31266, 33142}, {32004, 56696}, {32847, 49776}, {32857, 50080}, {32921, 49491}, {32935, 49471}, {33682, 49458}, {33771, 35242}, {34033, 62207}, {34255, 53663}, {34379, 50295}, {34471, 38293}, {35258, 62795}, {35613, 44417}, {36277, 61155}, {36740, 40910}, {36754, 64668}, {36845, 63007}, {37520, 64112}, {37537, 64679}, {37593, 62818}, {37633, 67097}, {37699, 67880}, {38036, 53599}, {38200, 49772}, {39586, 49457}, {42289, 60937}, {46904, 54352}, {48830, 50290}, {48851, 50308}, {48854, 50293}, {49455, 49535}, {49462, 55998}, {49466, 51192}, {49479, 49488}, {49676, 50287}, {49704, 63052}, {49707, 50286}, {49721, 50778}, {49768, 59408}, {50114, 51099}, {50128, 62392}, {50296, 50952}, {50301, 51102}, {51766, 60953}, {53114, 55923}, {56152, 63335}, {56255, 64845}, {59301, 62858}, {63430, 63982}

X(68588) = reflection of X(i) in X(j) for these (i,j): {4307, 4667}, {4419, 4356}, {7174, 1}
X(68588) = barycentric product X(i)*X(j) for these {i,j}: {1, 5308}, {57, 5686}, {100, 28878}, {190, 7659}
X(68588) = barycentric quotient X(i)/X(j) for these {i,j}: {692, 28879}, {5308, 75}, {5686, 312}, {7659, 514}, {28878, 693}
X(68588) = trilinear product X(i)*X(j) for these {i,j}: {6, 5308}, {56, 5686}, {100, 7659}, {101, 28878}
X(68588) = trilinear quotient X(i)/X(j) for these {i,j}: {2, 5308}, {8, 5686}, {101, 28879}, {513, 7659}, {514, 28878}
X(68588) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6, 7290}, {1, 238, 38316}, {1, 984, 3247}, {1, 1743, 1001}, {1, 3731, 15569}, {1, 3751, 9}, {1, 4649, 1449}, {1, 4663, 15601}, {1, 5223, 37}, {1, 5247, 5436}, {1, 5588, 30556}, {1, 5589, 30557}, {1, 5728, 18216}, {1, 7290, 35227}, {1, 16469, 1279}, {1, 16667, 1386}, {1, 42871, 15600}, {1, 49490, 3243}, {1, 49498, 16496}, {1, 49712, 16676}, {1, 60846, 42819}, {6, 1279, 16469}, {6, 49478, 1}, {8, 3945, 64174}, {37, 64070, 5223}, {42, 62819, 57}, {63, 17018, 37553}, {81, 3870, 5269}, {145, 894, 3886}, {145, 62997, 4344}, {354, 2999, 5573}, {614, 62867, 44841}, {968, 32912, 3929}, {1001, 1743, 15601}, {1001, 4663, 1743}, {1100, 3242, 1}, {1279, 16469, 7290}, {1386, 42871, 1}, {1419, 3340, 2263}, {1449, 3243, 1}, {3640, 3641, 11523}, {3681, 5287, 7322}, {3873, 5256, 3677}, {3957, 37685, 62834}, {3979, 62841, 3749}, {4360, 49499, 49446}, {4383, 4883, 10582}, {4430, 17011, 62833}, {4649, 49490, 1}, {4849, 37674, 8580}, {4864, 16666, 38315}, {4864, 38315, 1}, {4966, 38047, 17284}, {5220, 15569, 3731}, {6769, 36746, 35658}, {16670, 38316, 238}, {17594, 32913, 3928}, {32911, 62866, 4666}, {32913, 42042, 17594}, {49483, 49486, 17151}, {61358, 62867, 614}

X(68589) = X(1)X(142)∩X(6)X(8)

Barycentrics    3*a^3 - a^2*b + 3*a*b^2 - b^3 - a^2*c + 2*a*b*c + b^2*c + 3*a*c^2 + b*c^2 - c^3 : :
X(68589) = X[4419]-2*X[7174], 2*X[4307]-X[4644], 4*X[4349]-3*X[63054], 3*X[48856]-X[64168], X[3632]-2*X[4923], 5*X[4748]-8*X[36480], 5*X[4748]-4*X[50295], 2*X[36480]-X[50295], X[36479]-2*X[50302], 3*X[48802]-2*X[50308], 3*X[48830]-4*X[50293], 3*X[48854]-2*X[50290]

See David Nguyen, euclid 8509.

X(68589) lies on these lines: {1, 142}, {2, 1279}, {4, 23050}, {6, 8}, {7, 3242}, {9, 63969}, {10, 7290}, {19, 1697}, {37, 390}, {38, 3474}, {44, 5686}, {45, 52653}, {55, 4224}, {65, 28079}, {69, 50289}, {75, 145}, {144, 49515}, {171, 24477}, {175, 5604}, {176, 5605}, {193, 49450}, {200, 63089}, {238, 38057}, {269, 10106}, {321, 20020}, {329, 63979}, {344, 20179}, {346, 49484}, {388, 2263}, {474, 15287}, {497, 612}, {516, 4419}, {517, 3332}, {518, 4307}, {519, 4349}, {528, 48856}, {614, 26040}, {651, 12594}, {673, 5308}, {940, 36845}, {944, 991}, {948, 4318}, {952, 62183}, {958, 4339}, {966, 3883}, {968, 10385}, {975, 1058}, {976, 3485}, {984, 5698}, {990, 35514}, {993, 66680}, {1086, 59412}, {1125, 35227}, {1418, 3600}, {1419, 37709}, {1456, 5252}, {1458, 3476}, {1471, 1788}, {1698, 16487}, {1743, 24393}, {1757, 50303}, {1861, 31397}, {1890, 10624}, {2293, 3486}, {3008, 38200}, {3052, 5273}, {3241, 17392}, {3243, 3664}, {3247, 63977}, {3295, 7535}, {3434, 3920}, {3475, 3938}, {3488, 30116}, {3555, 4340}, {3616, 4429}, {3617, 17289}, {3621, 62997}, {3622, 16706}, {3632, 4923}, {3666, 17784}, {3679, 16469}, {3685, 50286}, {3689, 17723}, {3696, 4371}, {3717, 54389}, {3729, 49527}, {3745, 4863}, {3758, 49698}, {3826, 16020}, {3870, 5712}, {3879, 49451}, {3886, 17314}, {3932, 48805}, {3935, 63008}, {3961, 25568}, {4298, 23051}, {4308, 42314}, {4310, 5880}, {4359, 19993}, {4361, 51147}, {4363, 9053}, {4415, 9812}, {4452, 49463}, {4454, 28582}, {4643, 28566}, {4645, 36534}, {4656, 9580}, {4673, 20009}, {4675, 4864}, {4676, 27549}, {4696, 46738}, {4748, 17766}, {4847, 5269}, {4851, 49467}, {4899, 50127}, {4901, 17355}, {4916, 49460}, {5218, 29639}, {5222, 38315}, {5255, 21059}, {5266, 19843}, {5268, 26105}, {5275, 14942}, {5603, 30115}, {5657, 13329}, {5717, 6765}, {5718, 63168}, {5725, 34619}, {5731, 50677}, {5744, 37540}, {5817, 64013}, {6555, 59596}, {6666, 60846}, {7172, 44417}, {7222, 24349}, {7226, 44447}, {7229, 49690}, {7322, 40998}, {7365, 8270}, {9041, 35578}, {9776, 17597}, {9780, 17337}, {10327, 24552}, {10388, 40960}, {10436, 49466}, {10578, 17056}, {10580, 37674}, {10950, 67264}, {11024, 17054}, {12588, 21293}, {16475, 49772}, {16496, 50307}, {16517, 41325}, {16670, 59414}, {16696, 56984}, {16777, 62693}, {17022, 64162}, {17126, 64153}, {17140, 30614}, {17278, 40333}, {17279, 39570}, {17316, 20172}, {17334, 63975}, {17370, 46934}, {17371, 46933}, {17390, 20181}, {17446, 37598}, {17448, 61326}, {17580, 52541}, {17594, 34607}, {17599, 34612}, {17602, 31140}, {17716, 33137}, {17740, 29832}, {17765, 36479}, {17802, 45476}, {17805, 45477}, {19785, 29815}, {19822, 33090}, {19860, 25966}, {20015, 63007}, {20050, 63401}, {20075, 28606}, {21342, 21454}, {21949, 62208}, {24239, 59572},{24280, 49447}, {24331, 49696}, {24392, 39595}, {24695, 49448}, {25006, 62834}, {25524, 64442}, {25878, 37542}, {26228, 33108}, {26582, 26626}, {28570, 64015}, {28635, 49506}, {29571, 38316}, {30478, 37552}, {31091, 32779}, {32087, 49679}, {32846, 50316}, {32926, 42047}, {32932, 42049}, {32941, 50288}, {32945, 33088}, {33072, 33171}, {33109, 33144}, {37157, 63360}, {37520, 64151}, {37539, 64081}, {37548, 56936}, {37662, 64083}, {37665, 44798}, {38149, 53599}, {41242, 53661}, {41354, 56928}, {46916, 54390}, {47357, 50291}, {47359, 49694}, {48802, 50308}, {48830, 50293}, {48854, 50290}, {49675, 50301}, {49691, 50299}, {49693, 50300}, {49727, 50790}, {50101, 62392}, {50114, 51102}, {53529, 60909}, {54418, 64671}, {63126, 67097}

X(68589) = reflection of X(i) in X(j) for these (i,j): {3632, 4923}, {4419, 7174}, {4644, 4307}, {36479, 50302}, {50295, 36480}
X(68589) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2550, 4000}, {1, 64174, 4648}, {8, 4344, 6}, {8, 5263, 2345}, {8, 5749, 49524}, {8, 5772, 59407}, {8, 51192, 5839}, {10, 7290, 37650}, {145, 3945, 49478}, {390, 39587, 37}, {3886, 49476, 17314}, {3961, 26098, 25568}, {4675, 4864, 11038}, {4847, 5269, 37642}, {5880, 49465, 4310}, {17369, 59407, 5772}, {29815, 33110, 19785}, {49515, 64016, 144}

X(68590) = X(4)X(392)∩X(5)X(10)

Barycentrics    a*(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6 + a^5*c - 4*a^4*b*c + 4*a^3*b^2*c + 6*a^2*b^3*c - 5*a*b^4*c - 2*b^5*c - a^4*c^2 + 4*a^3*b*c^2 - 8*a^2*b^2*c^2 + 4*a*b^3*c^2 + b^4*c^2 - 2*a^3*c^3 + 6*a^2*b*c^3 + 4*a*b^2*c^3 + 4*b^3*c^3 + 2*a^2*c^4 - 5*a*b*c^4 + b^2*c^4 + a*c^5 - 2*b*c^5 - c^6) : : (Peter Moses, May 26, 2025)
Barycentrics    Sin[A]^3*(Cos[A] + Cos[B - C])^2*(-2 + Cos[B - C] + Cos[A]*(2 + Cos[B - C]) - 2*(Cos[A] - 2)*Cos[(B - C)/2]*Sin[A/2]) : : (Peter Moses, May 26, 2025)
X(68590) = X[1]+X[5777], 3*X[1]+X[14872], 3*X[5777]-X[14872], 3*X[2]+X[12672], 3*X[2]-X[31788], X[12672]+X[31788], X[3]+X[9856], X[4]+3*X[392], X[4]+X[31786], 3*X[392]-X[31786], X[946]+X[960], 3*X[3817]+X[3878], X[8]+X[13600], X[40]-5*X[25917], X[65]-5*X[8227], X[72]+3*X[5603], 2*X[140]-X[31787], X[145]+3*X[18908], 3*X[210]+X[7982], 3*X[354]+X[5693], 3*X[354]-7*X[9624], X[5693]+7*X[9624], X[355]+X[9957], X[355]-3*X[10157], X[9957]+3*X[10157], X[13464]+X[20117], 3*X[551]-X[12675], 3*X[551]+X[31803], X[12675]+X[31803], X[5045]-2*X[5901], X[5045]+X[31821], 2*X[5901]+X[31821], X[942]-3*X[5886], X[942]+X[5887], 3*X[5886]+X[5887], X[944]+3*X[5927], X[9947]+X[31792], X[962]+3*X[64107], 3*X[1385]+X[31828], X[1385]+X[31937], X[31828]-3*X[31937], X[11496]+X[18251], X[1071]-5*X[3616], X[1071]-3*X[66599], X[1071]+3*X[67998], 5*X[3616]-3*X[66599], 5*X[3616]+3*X[67998], X[66599]+X[67998], 2*X[1125]-X[9940], X[1482]+X[34790], 5*X[1656]-X[37562], 3*X[1699]+X[14110], X[3057]+3*X[5587], 7*X[3090]-3*X[3753], 5*X[3091]+3*X[3877], X[3244]+3*X[15064], X[3555]-5*X[10595], 3*X[3576]+X[12688], 7*X[3622]+X[12528], 3*X[3681]+5*X[5734], 5*X[3697]-X[12245], 5*X[3698]-9*X[54447], 3*X[3742]-X[5884], X[3754]-3*X[10171], X[3869]+7*X[68034], X[3884]+X[19925], 5*X[3890]+3*X[59387], X[3898]+X[67875], 7*X[3983]-3*X[63143], X[4662]-2*X[64693], 3*X[5049]-5*X[61276], 9*X[5055]-X[25413], 5*X[18493]-X[24474], 5*X[5439]-X[64021], 3*X[5692]+5*X[11522], X[5694]+3*X[51709], X[5697]+7*X[7989], X[5752]+X[31779], 3*X[5790]+X[23340], 5*X[5818]-X[10914], X[5881]+3*X[5919], X[5882]-3*X[10179], X[5903]-9*X[7988], 3*X[5918]-7*X[67706], X[6797]-3*X[23513], X[12699]+X[31793], X[50193]-7*X[61268], 3*X[11230]-X[34339], 2*X[13624]-X[31805], 2*X[31663]-3*X[33575], 3*X[26446]-X[31798], X[7957]+3*X[31162], X[9943]-3*X[10165], X[9961]-9*X[54445], 3*X[10156]-2*X[40296], 3*X[10202]+X[40266], 3*X[10246]+X[40263], 2*X[12571]-X[16616], X[12680]+3*X[61705], X[12680]-5*X[64953], 3*X[61705]+5*X[64953], X[13369]-3*X[38028], X[31794]-4*X[61272], X[15071]-9*X[25055], 3*X[17525]+X[67989], 5*X[17609]-9*X[61275], X[17653]+3*X[28461], X[17654]-5*X[31272], X[18254]+X[64192], 2*X[18483]-X[31822], 5*X[19862]-X[66019], X[24475]-2*X[50192], 7*X[30389]-3*X[63432], 3*X[31165]+X[37625], X[31165]+3*X[38021], X[37625]-9*X[38021], X[31775]+X[66992], X[31789]-3*X[40998], X[31797]-2*X[61524], X[33179]+X[56762], 3*X[34123]+X[67988], X[34791]+X[63967], 3*X[38032]+X[66044], 3*X[38036]+X[64723], 3*X[38038]+X[64139], X[40257]+X[68005], X[47745]+X[66256], X[51489]-3*X[66515], 3*X[52653]+X[67995], 3*X[61269]-X[61541], X[64742]+X[66049]}

See David Nguyen, euclid 8513.

X(68590) lies on these lines: {1, 1864}, {2, 12672}, {3, 4512}, {4, 392}, {5, 10}, {8, 6939}, {9, 22770}, {30, 31838}, {40, 4413}, {65, 8227}, {72, 5603}, {140, 31787}, {142, 54198}, {145, 18908}, {200, 5780}, {210, 7982}, {354, 5693}, {355, 497}, {388, 37822}, {405, 63986}, {442, 1519}, {495, 67855}, {496, 51755}, {515, 58679}, {516, 37281}, {518, 13464}, {519, 58631}, {551, 12675}, {758, 13374}, {908, 63257}, {912, 5045}, {936, 10306}, {942, 3086}, {944, 5927}, {952, 9947}, {962, 6864}, {971, 1001}, {975, 64449}, {997, 11496}, {999, 7330}, {1012, 19861}, {1056, 5811}, {1064, 6051}, {1071, 3616}, {1125, 6001}, {1158, 25524}, {1387, 16215}, {1482, 4853}, {1532, 24987}, {1538, 6842}, {1621, 33597}, {1656, 37562}, {1699, 14110}, {1737, 13601}, {1858, 15950}, {1898, 34471}, {2095, 12526}, {2550, 12700}, {2551, 64322}, {2646, 66248}, {2771, 11281}, {2800, 3812}, {2801, 3636}, {2818, 64817}, {3057, 5587}, {3085, 66226}, {3090, 3753}, {3091, 3877}, {3149, 5250}, {3244, 15064}, {3295, 5720}, {3303, 17857}, {3359, 16408}, {3419, 10531}, {3555, 10595}, {3576, 12688}, {3579, 6911}, {3622, 12528}, {3626, 58696}, {3646, 30503}, {3660, 67970}, {3681, 5734}, {3683, 11012}, {3697, 12245}, {3698, 54447}, {3742, 5884}, {3754, 10171}, {3816, 12616}, {3838, 64762}, {3869, 5775}, {3884, 19925}, {3890, 59387}, {3898, 67875}, {3983, 63143}, {4662, 28234}, {4861, 17615}, {5049, 61276}, {5055, 25413}, {5231, 18493}, {5248, 37837}, {5261, 39779}, {5436, 12664}, {5437, 54156}, {5439, 64021}, {5443, 13750}, {5534, 6767}, {5570, 37735}, {5572, 10283}, {5657, 6964}, {5692, 11522}, {5694, 34647}, {5697, 7989}, {5719, 16201}, {5730, 64171}, {5752, 31779}, {5771, 18249}, {5779, 7373}, {5790, 23340}, {5794, 26333}, {5818, 10598}, {5881, 5919}, {5882, 10179}, {5885, 10199}, {5903, 7988}, {5918, 67706}, {6265, 45230}, {6326, 37080}, {6797, 23513}, {6826, 12699}, {6831, 41012}, {6859, 50193}, {6862, 10200}, {6892, 11227}, {6906, 17614}, {6907, 63989}, {6909, 63266}, {6914, 13624}, {6916, 67999}, {6917, 22793}, {6924, 31663}, {6929, 18480}, {6930, 18481}, {6940, 17613}, {6941, 17618}, {6944, 26446}, {6956, 26129}, {6959, 11231}, {6973, 61261}, {6975, 17619}, {7082, 26437}, {7743, 26470}, {7957, 31162}, {7995, 37526}, {8273, 50528}, {8581, 61762}, {8666, 60911}, {9612, 64106}, {9709, 49163}, {9943, 10165}, {9961, 54445}, {10156, 40296}, {10202, 40266}, {10222, 12635}, {10246, 40263}, {10267, 64804}, {10269, 18237}, {10284, 11235}, {10532, 58798}, {10571, 52384}, {10589, 67937}, {10866, 31393}, {11018, 37737}, {11035, 51788}, {11249, 31445}, {11373, 12915}, {11375, 50195}, {11376, 50196}, {12047, 57285}, {12514, 22753}, {12571, 16616}, {12608, 25466}, {12611, 31936}, {12617, 63980}, {12680, 61705}, {13369, 38028}, {13462, 30290}, {14988, 31794}, {15071, 25055}, {15178, 51715}, {15733, 22836}, {15803, 17634}, {16007, 31871}, {17112, 23058}, {17525, 67989}, {17605, 64721}, {17609, 61275}, {17653, 28461}, {17654, 31272}, {17814, 37594}, {18254, 64192}, {18260, 58585}, {18483, 31822}, {19862, 66019}, {22758, 24928}, {23708, 64045}, {24390, 41389}, {24475, 50192}, {24703, 26332}, {24927, 26321}, {24929, 45770}, {28160, 37290}, {28194, 58637}, {28204, 49736}, {30294, 53053}, {30389, 63432}, {31165, 37625}, {31397, 66216}, {31658, 35239}, {31775, 66992}, {31789, 40998}, {31797, 61524}, {31870, 44663}, {32141, 51787}, {32613, 40262}, {33179, 56762}, {33596, 64792}, {34123, 67988}, {34791, 63967}, {37544, 39542}, {37548, 37732}, {37699, 66694}, {38032, 66044}, {38036, 64723}, {38038, 64139}, {40257, 68005}, {44547, 64160}, {44675, 58576}, {47745, 66256}, {49169, 64335}, {50190, 61274}, {50443, 67949}, {51489, 66515}, {52653, 59345}, {61269, 61541}, {61653, 64963}, {64673, 68001}, {64742, 66049}

X(68590) = midpoint of X(i) and X(j) for these {i,j}: {1, 5777}, {3, 9856}, {4, 31786}, {8, 13600}, {10, 45776}, {355, 9957}, {942, 5887}, {946, 960}, {1385, 31937}, {1482, 34790}, {3878, 7686}, {3884, 19925}, {3898, 67875}, {4301, 63976}, {5045, 31821}, {5752, 31779}, {9947, 31792}, {11496, 18251}, {12672, 31788}, {12675, 31803}, {12699, 31793}, {13464, 20117}, {18254, 64192}, {22791, 31837}, {31775, 66992}, {33179, 56762}, {34791, 63967}, {40257, 68005}, {47745, 66256}, {64742, 66049}, {66599, 67998}
X(68590) = reflection of X(i) in X(j) for these (i,j): {4662, 64693}, {5045, 5901}, {5806, 9955}, {9940, 1125}, {16616, 12571}, {24475, 50192}, {31787, 140}, {31797, 61524}, {31805, 13624}, {31822, 18483}, {58643, 5044}
X(68590) = complement of X(31788)
X(68590) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 12672, 31788}, {4, 392, 31786}, {10, 45776, 517}, {551, 31803, 12675}, {946, 960, 517}, {1001, 6261, 1385}, {1071, 3616, 66599}, {1385, 31937, 971}, {1858, 15950, 16193}, {3086, 12709, 942}, {3295, 5720, 64116}, {3616, 67998, 1071}, {3817, 3878, 7686}, {3878, 7686, 517}, {4301, 10176, 63976}, {4301, 63976, 517}, {5045, 31821, 912}, {5693, 9624, 354}, {5886, 5887, 942}, {8583, 12705, 3}, {9947, 31792, 952}, {9957, 10157, 355}, {11375, 64042, 50195}, {11376, 64041, 50196}, {12514, 22753, 37623}, {13464, 20117, 518}, {22791, 31837, 517}, {31435, 63992, 3}, {44675, 66250, 58576}, {61705, 64953, 12680}

X(68591) = X(1)X(6)∩X(36)X(59)

Barycentrics    a^2*(a^4 + 2*a*b*c*(b + c) + (b - c)^2*(b^2 + b*c + c^2) - a^2*(2*b^2 + b*c + 2*c^2)) : :
X(68591) = X[36]-2*X[59], 2*X[692]-X[3220], 3*X[3582]-4*X[33562], 5*X[31263]-4*X[46100]

See David Nguyen, euclid 8509.

X(68591) lies on these lines: {1, 6}, {8, 63088}, {10, 3562}, {35, 255}, {36, 59}, {40, 1419}, {46, 269}, {47, 21059}, {55, 2003}, {56, 38293}, {57, 61397}, {58, 61399}, {73, 59320}, {77, 67962}, {80, 8759}, {81, 13405}, {100, 22128}, {109, 5537}, {155, 5534}, {165, 222}, {171, 23131}, {200, 394}, {210, 65702}, {212, 15931}, {221, 7991}, {223, 41338}, {323, 3935}, {354, 52423}, {386, 1496}, {484, 6610}, {498, 4648}, {499, 37650}, {511, 40910}, {516, 651}, {517, 1456}, {521, 1734}, {580, 1066}, {595, 20978}, {603, 59326}, {614, 61357}, {677, 1815}, {692, 3220}, {916, 58326}, {942, 55102}, {971, 41339}, {1126, 13404}, {1210, 67026}, {1282, 20752}, {1331, 58328}, {1332, 3717}, {1406, 5128}, {1407, 53056}, {1418, 3336}, {1421, 18839}, {1428, 20958}, {1455, 5538}, {1462, 24231}, {1463, 5091}, {1465, 5536}, {1471, 5563}, {1478, 3332}, {1498, 63981}, {1621, 54444}, {1697, 64020}, {1698, 25878}, {1699, 34048}, {1735, 4880}, {1738, 64155}, {1768, 9371}, {1818, 3939}, {1936, 4551}, {1993, 3870}, {1994, 3957}, {2002, 5902}, {2077, 52407}, {2078, 2361}, {2114, 6211}, {2191, 52186}, {2195, 60025}, {2263, 5903}, {2293, 3746}, {2328, 56001}, {2330, 64006}, {2801, 3100}, {2810, 7193}, {3085, 3945}, {3086, 37681}, {3173, 7070}, {3218, 24025}, {3245, 6126}, {3295, 67264}, {3333, 36754}, {3361, 36745}, {3550, 20745}, {3579, 23070}, {3582, 33562}, {3584, 17392}, {3678, 66593}, {3781, 43146}, {3874, 66610}, {4000, 60924}, {4306, 59323}, {4312, 6180}, {4349, 31397}, {4512, 55400}, {4585, 32850}, {4644, 60923}, {4666, 5422}, {5010, 50677}, {5045, 37509}, {5228, 59372}, {5290, 5706}, {5399, 10902}, {5531, 51361}, {5587, 60691}, {5691, 9370}, {5697, 41733}, {5711, 51784}, {5732, 23144}, {5733, 37719}, {5850, 62799}, {5852, 59458}, {6149, 19624}, {6198, 63967}, {6745, 63068}, {6763, 17102}, {6765, 67012}, {6769, 37498}, {7292, 18240}, {7293, 23155}, {7957, 64055}, {7987, 34046}, {7994, 34033}, {8270, 15104}, {8580, 17811}, {8614, 37568}, {8757, 41869}, {9037, 20872}, {9441, 20744}, {9819, 64449}, {10025, 24014}, {10039, 63319}, {10056, 63054}, {10164, 17074}, {10310, 23072}, {10389, 61398}, {10578, 37685}, {10580, 63074}, {10582, 10601}, {10980, 52424}, {10982, 64669}, {11019, 32911}, {11399, 44100}, {11508, 21002}, {11529, 44414}, {11531, 34040}, {14872, 58906}, {15066, 67097}, {15071, 54295}, {15430, 64197}, {15934, 39523}, {17814, 67881}, {18451, 18528}, {19349, 59340}, {19369, 21746}, {20588, 64082}, {20780, 53298}, {20806, 56179}, {20999, 23202}, {22136, 34790}, {22161, 37619}, {24980, 49676}, {26884, 51377}, {26885, 61640}, {29817, 34545}, {31263, 46100}, {31508, 62207}, {32912, 62811}, {36752, 64668}, {37672, 66469}, {37800, 60895}, {41694, 64134}, {54408, 56418}, {55399, 62823}, {59645, 62798}, {61395, 62819}, {62805, 62997}, {64005, 64057}, {64679, 66608}, {66029, 66062}

X(68591) = reflection of X(i) in X(j) for these (i,j): {36, 59}, {3220, 692}
X(68591) = barycentric product X(i)*X(j) for these {i,j}: {190, 53300}, {329, 39558}
X(68591) = barycentric quotient X(i)/X(j) for these {i,j}: {198, 48357}, {39558, 189}, {53300, 514}
X(68591) = trilinear product X(i)*X(j) for these {i,j}: {40, 39558}, {100, 53300}, {32622, 32623}
X(68591) = trilinear quotient X(i)/X(j) for these {i,j}: {40, 48357}, {84, 39558}, {513, 53300}, {32622, 39145}, {32623, 39144}
X(68591) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1743, 15299}, {1, 1757, 1736}, {1, 3074, 5259}, {1, 3751, 18412}, {1, 54301, 1203}, {1, 56535, 54301}, {40, 3157, 34043}, {222, 7074, 165}, {991, 1253, 35}, {1124, 1335, 220}, {1458, 13329, 36}, {1736, 1757, 41700}, {1936, 4551, 44425}, {3299, 3301, 17745}, {4649, 9440, 1}, {4663, 30621, 5728}, {5353, 5357, 5526}, {5399, 52408, 10902}, {5728, 30621, 1}, {7078, 64069, 1}, {9370, 66249, 5691}, {13329, 44858, 1458}, {67012, 67963, 6765}

X(68592) = X(1)X(6)∩X(36)X(991)

Barycentrics    a^2*(a^4 - 2*a*b*c*(b + c) + a^2*(-2*b^2 + b*c - 2*c^2) + (b - c)^2*(b^2 + 3*b*c + c^2)) : :

See David Nguyen, euclid 8509.

X(68592) lies on these lines: {1, 6}, {3, 67264}, {35, 2293}, {36, 991}, {55, 52423}, {56, 62183}, {57, 61398}, {58, 13404}, {81, 11019}, {165, 52424}, {182, 40910}, {200, 10601}, {222, 10980}, {269, 3338}, {354, 1421}, {394, 10582}, {498, 37650}, {499, 4648}, {595, 61399}, {614, 61356}, {651, 5542}, {938, 62805}, {942, 1456}, {1125, 37659}, {1181, 64669}, {1210, 4349}, {1253, 1497}, {1418, 3337}, {1419, 3333}, {1428, 21746}, {1451, 59320}, {1458, 5563}, {1462, 50307}, {1479, 3332}, {1699, 37543}, {1737, 64174}, {1993, 4666}, {1994, 29817}, {2191, 57709}, {2263, 5902}, {3085, 37681}, {3086, 3945}, {3303, 38293}, {3361, 36746}, {3562, 6744}, {3582, 17392}, {3616, 63088}, {3624, 25878}, {3745, 64157}, {3870, 5422}, {3873, 54444}, {3935, 15018}, {3957, 34545}, {4000, 60923}, {4300, 59323}, {4312, 5228}, {4318, 30329}, {4344, 18391}, {4355, 64057}, {4512, 55399}, {4644, 60924}, {5045, 36750}, {5256, 62839}, {5262, 67944}, {5322, 20961}, {5537, 52428}, {5706, 66682}, {5710, 67942}, {5711, 67931}, {5733, 37720}, {5903, 61086}, {6180, 59372}, {6738, 57280}, {6767, 39523}, {6769, 37514}, {7190, 54370}, {7191, 62852}, {7280, 50677}, {8540, 64006}, {8580, 17825}, {9355, 41694}, {10072, 63054}, {10389, 61397}, {10578, 63074}, {10579, 56343}, {10580, 37685}, {11398, 44100}, {11402, 64685}, {11507, 21002}, {11518, 64020}, {12161, 64670}, {12564, 66610}, {12651, 66608}, {13405, 32911}, {14547, 15931}, {14986, 62997}, {17012, 24025}, {17017, 62811}, {18421, 64449}, {18527, 45923}, {20116, 62387}, {22128, 64149}, {23070, 50192}, {31393, 44414}, {36747, 64668}, {37587, 42314}, {37602, 44858}, {38035, 41004}, {40998, 62798}, {41339, 63972}, {42289, 64013}, {51090, 62799}, {55400, 62823}, {57281, 58469}, {61396, 62819}, {62797, 64017}, {63094, 64667}

X(68592) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1743, 15298}, {1, 16472, 54301}, {6, 16469, 1203}, {991, 1471, 36}, {1386, 5728, 1}, {2293, 13329, 35}, {14547, 55086, 15931}

X(68593) = X(1)X(5759)∩X(4)X(6)

Barycentrics    5*a^6 - 2*a^5*(b + c) - (b - c)^4*(b + c)^2 + 2*a*(b - c)^2*(b + c)^3 + 3*a^2*(b^2 - c^2)^2 - a^4*(7*b^2 + 2*b*c + 7*c^2) : :

See David Nguyen, euclid 8509.

X(68593) lies on these lines: {1, 5759}, {3, 1014}, {4, 6}, {5, 37681}, {12, 38293}, {20, 62183}, {37, 21168}, {40, 4349}, {57, 40945}, {98, 45098}, {182, 7397}, {184, 7490}, {185, 51223}, {193, 13727}, {269, 64347}, {376, 991}, {377, 63088}, {443, 37659}, {461, 11433}, {516, 1449}, {517, 4344}, {580, 16845}, {631, 4648}, {946, 16469}, {990, 4644}, {1253, 3072}, {1279, 10595}, {1352, 7402}, {1419, 4292}, {1456, 4295}, {1743, 5817}, {1754, 5712}, {1870, 2263}, {1992, 48878}, {2293, 37529}, {2323, 2550}, {2982, 19354}, {3019, 3090}, {3089, 44100}, {3333, 67026}, {3474, 45126}, {3524, 17392}, {3525, 17245}, {3528, 50677}, {3618, 36682}, {3664, 21151}, {3672, 5762}, {3946, 5735}, {4000, 59386}, {4196, 11402}, {4207, 11245}, {4213, 18950}, {4219, 44094}, {4253, 33536}, {4254, 36012}, {4294, 67264}, {4307, 35514}, {4312, 43035}, {4340, 37537}, {4346, 60922}, {4667, 5732}, {5050, 36670}, {5067, 17337}, {5222, 5805}, {5308, 31658}, {5603, 7290}, {5657, 64174}, {5707, 6865}, {5803, 30809}, {5921, 7377}, {6601, 45728}, {6817, 63174}, {6827, 45923}, {6849, 37509}, {6864, 36754}, {6892, 63307}, {6897, 63297}, {7580, 63007}, {7967, 30273}, {8727, 37666}, {10299, 45942}, {10431, 37685}, {10446, 25406}, {10883, 63067}, {11179, 48902}, {11372, 64017}, {11427, 57534}, {12848, 64750}, {13464, 16487}, {14004, 63031}, {15287, 45977}, {16670, 63970}, {17349, 36660}, {17379, 36706}, {17582, 25878}, {18440, 36671}, {18446, 66683}, {18925, 37379}, {21454, 65745}, {34631, 50130}, {36489, 63428}, {36652, 51171}, {36654, 53091}, {36722, 63086}, {37108, 49743}, {37441, 37538}, {37654, 48888}, {43065, 53014}, {43672, 54690}, {44547, 58906}, {44858, 63416}, {45097, 56144}, {54587, 54883}, {54790, 60112}

X(68593) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 3332, 4}, {20, 62997, 62183}, {990, 4644, 36996}, {4648, 13329, 631}, {5733, 13329, 4648}

X(68594) = X(394)X(2420)∩X(2407)X(3926)

Barycentrics    a^2*(a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 + a^8*c^2 + 3*a^6*b^2*c^2 - 8*a^4*b^4*c^2 + 3*a^2*b^6*c^2 + b^8*c^2 - 4*a^6*c^4 + 4*a^4*b^2*c^4 + 4*a^2*b^4*c^4 - 4*b^6*c^4 + 2*a^4*c^6 - 3*a^2*b^2*c^6 + 2*b^4*c^6 - a^2*c^8 - b^2*c^8 + c^10)*(a^10 + a^8*b^2 - 4*a^6*b^4 + 2*a^4*b^6 - a^2*b^8 + b^10 - 3*a^8*c^2 + 3*a^6*b^2*c^2 + 4*a^4*b^4*c^2 - 3*a^2*b^6*c^2 - b^8*c^2 + 2*a^6*c^4 - 8*a^4*b^2*c^4 + 4*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 3*a^2*b^2*c^6 - 4*b^4*c^6 - 3*a^2*c^8 + b^2*c^8 + c^10) : :

See Antreas Hatzipolakis and Peter Moses, euclid 8511.

X(68594) lies on these lines: {3, 23347}, {6, 52600}, {112, 14919}, {127, 67192}, {394, 2420}, {2407, 3926}, {2416, 5664}, {2781, 32738}, {5663, 17974}, {6720, 41392}, {9530, 18317}, {10766, 65839}, {13310, 14941}, {14685, 56961}, {15421, 51965}, {34897, 64923}, {35906, 40856}, {35911, 48453}

X(68594) = midpoint of X(112) and X(14919)
X(68594) = reflection of X(62583) in X(6720)
X(68594) = isogonal conjugate of X(6794)
X(68594) = trilinear pole of line {520, 1495}
X(68594) = barycentric quotient X(6)/X(6794)

X(68595) = X(63)X(1023)∩X(101)X(1797)

Barycentrics    a^2*(a^4 - 2*a^3*b + a^2*b^2 - a*b^3 + b^4 + 2*a^2*b*c - a*b^2*c - b^3*c - 2*a^2*c^2 + 2*a*b*c^2 + b^2*c^2 - 2*b*c^3 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^3*c + 2*a^2*b*c + 2*a*b^2*c - 2*b^3*c + a^2*c^2 - a*b*c^2 + b^2*c^2 - a*c^3 - b*c^3 + c^4) : :

See Antreas Hatzipolakis and Peter Moses, euclid 8511.

X(68595) lies on these lines: {3, 23157}, {63, 1023}, {101, 1797}, {116, 34234}, {222, 61210}, {2842, 17972}, {14377, 42754}

X(68595) = midpoint of X(101) and X(1797)
X(68595) = isogonal conjugate of X(61730)
X(68595) = isogonal conjugate of the complement of X(38941)
X(68595) = trilinear pole of line {902, 1459}
X(68595) = barycentric quotient X(6)/X(61730)

X(68596) = X(32)X(2452)∩X(184)X(5118)

Barycentrics    (2*a^6*b^2 - 3*a^4*b^4 + 2*a^2*b^6 - a^6*c^2 - b^6*c^2 + a^4*c^4 - a^2*b^2*c^4 + b^4*c^4)*(a^6*b^2 - a^4*b^4 - 2*a^6*c^2 + a^2*b^4*c^2 + 3*a^4*c^4 - b^4*c^4 - 2*a^2*c^6 + b^2*c^6) : :

See Antreas Hatzipolakis and Peter Moses, euclid 8511.

X(68596) lies on these lines: {3, 23342}, {32, 2452}, {184, 5118}, {669, 64479}, {878, 36822}, {1316, 14609}, {14908, 40866}, {17970, 53735}, {33813, 42065}

X(68596) = isogonal conjugate of X(6787) X(68596) = isogonal conjugate of the anticomplement of X(3111)
X(68596) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6787}, {75, 11332}
X(68596) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 6787}, {206, 11332}
X(68596) = cevapoint of X(i) and X(j) for these (i,j): {669, 46461}, {51492, 51493}
X(68596) = trilinear pole of line {3049, 3231}
X(68596) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6787}, {32, 11332}

X(68597) = X(6530)X(7473)∩X(8429)X(8430)

Barycentrics    a^2*(a^14 - 3*a^12*b^2 + 3*a^10*b^4 - a^8*b^6 - a^6*b^8 + 3*a^4*b^10 - 3*a^2*b^12 + b^14 - 3*a^12*c^2 + 6*a^10*b^2*c^2 - 4*a^8*b^4*c^2 + 2*a^6*b^6*c^2 - 4*a^4*b^8*c^2 + 6*a^2*b^10*c^2 - 3*b^12*c^2 + 5*a^10*c^4 - 5*a^8*b^2*c^4 - 5*a^2*b^8*c^4 + 5*b^10*c^4 - 6*a^8*c^6 + 4*a^6*b^2*c^6 + 3*a^4*b^4*c^6 + 4*a^2*b^6*c^6 - 6*b^8*c^6 + 4*a^6*c^8 - 2*a^4*b^2*c^8 - 2*a^2*b^4*c^8 + 4*b^6*c^8 - 2*a^4*c^10 - 2*a^2*b^2*c^10 - 2*b^4*c^10 + 2*a^2*c^12 + 2*b^2*c^12 - c^14)*(a^14 - 3*a^12*b^2 + 5*a^10*b^4 - 6*a^8*b^6 + 4*a^6*b^8 - 2*a^4*b^10 + 2*a^2*b^12 - b^14 - 3*a^12*c^2 + 6*a^10*b^2*c^2 - 5*a^8*b^4*c^2 + 4*a^6*b^6*c^2 - 2*a^4*b^8*c^2 - 2*a^2*b^10*c^2 + 2*b^12*c^2 + 3*a^10*c^4 - 4*a^8*b^2*c^4 + 3*a^4*b^6*c^4 - 2*a^2*b^8*c^4 - 2*b^10*c^4 - a^8*c^6 + 2*a^6*b^2*c^6 + 4*a^2*b^6*c^6 + 4*b^8*c^6 - a^6*c^8 - 4*a^4*b^2*c^8 - 5*a^2*b^4*c^8 - 6*b^6*c^8 + 3*a^4*c^10 + 6*a^2*b^2*c^10 + 5*b^4*c^10 - 3*a^2*c^12 - 3*b^2*c^12 + c^14) : :

See Antreas Hatzipolakis and Peter Moses, euclid 8511.

X(68597) lies on these lines: {3, 23350}, {325, 36166}, {842, 43754}, {1316, 14356}, {5968, 37930}, {6530, 7473}, {7575, 51543}, {8429, 8430}, {14687, 35908}, {32112, 64634}, {32662, 34370}

X(68597) = midpoint of X(842) and X(43754)
X(68597) = isogonal conjugate of X(11005)
X(68597) = isogonal conjugate of the anticomplement of X(53725)
X(68597) = barycentric quotient X(6)/X(11005)

X(68598) = X(1)X(5927)∩X(8)X(210)

Barycentrics    -(a*(a - b - c)*(2*a^3*b*c + 6*a*b*(b - c)^2*c + a^4*(b + c) - 2*a^2*(b^3 - 2*b^2*c - 2*b*c^2 + c^3) + (b - c)^2*(b^3 + 5*b^2*c + 5*b*c^2 + c^3))) : :
X(68598) = 3*X[1837]+X[3057], 3*X[496]-X[942], X[12053]+X[64131], 3*X[1898]+5*X[17609], 3*X[10072]-X[64132], 3*X[11238]+X[64042]

See David Nguyen, euclid 8513.

X(68598) lies on these lines: {1, 5927}, {2, 9848}, {8, 210}, {11, 3812}, {56, 8544}, {65, 5274}, {72, 51785}, {390, 25917}, {392, 66682}, {496, 942}, {518, 10392}, {950, 49736}, {952, 9947}, {971, 67042}, {1001, 3601}, {1071, 37704}, {1125, 10855}, {1682, 58655}, {1697, 3740}, {1864, 34647}, {1898, 3485}, {2136, 18236}, {2801, 16215}, {2886, 8582}, {3062, 7091}, {3086, 9943}, {3241, 17632}, {3486, 10179}, {3616, 10861}, {3649, 58563}, {3697, 9819}, {3742, 12711}, {4301, 64157}, {4342, 34790}, {4853, 18247}, {4915, 66197}, {5044, 12575}, {5225, 64106}, {5265, 5918}, {5439, 50444}, {5687, 8580}, {5722, 45776}, {5727, 17622}, {5728, 11522}, {5777, 63993}, {5836, 9581}, {5886, 12710}, {5903, 9614}, {6261, 30283}, {7288, 10178}, {7686, 9669}, {9951, 66065}, {9957, 47745}, {10072, 64132}, {10391, 11376}, {10393, 42819}, {10569, 30290}, {10624, 58637}, {10860, 15803}, {10864, 61762}, {10959, 18839}, {11035, 64110}, {11238, 44663}, {11281, 16120}, {11373, 12675}, {12688, 14986}, {12740, 45230}, {12915, 31803}, {15071, 17626}, {15558, 58683}, {17603, 28628}, {18492, 39779}, {20789, 28236}, {25524, 66239}, {30384, 44547}, {31793, 51783}, {31795, 31838}, {34471, 51715}, {35656, 35886}, {37080, 53055}, {37618, 65404}, {37722, 65465}, {42884, 63988}, {44675, 58567}, {60910, 62874}, 58576}, {47745, 66256}, {49169, 64335}, {50190, 61274}, {50443, 67949}, {51489, 66515}, {52653, 59345}, {61269, 61541}, {61653, 64963}, {64673, 68001}, {64742, 66049}

X(68598) = midpoint of X(i) and X(j) for these {i,j}: {1837, 66216}, {12053, 64131}
X(68598) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1125, 12446, 10855}, {1837, 66216, 3880}, {5727, 17622, 66256}, {9581, 66226, 5836}, {10394, 18220, 17609}, {10866, 17604, 8}, {12053, 64131, 518}, {12688, 14986, 63994}, {12711, 50443, 3742}, {37722, 66250, 65465}, {44675, 66248, 58567}

X(68599) = X(2)X(40)∩X(7)X(21)

Barycentrics    (a - b - c)*(a^3 + 3*a^2*(b + c) + (b - c)^2*(b + c) + a*(3*b^2 - 2*b*c + 3*c^2)) : :
X(68599) = X[6904]-2*X[8583]

See David Nguyen, euclid 8513.

X(68599) lies on these lines: {1, 329}, {2, 40}, {3, 54348}, {4, 392}, {7, 21}, {8, 210}, {9, 12053}, {10, 6919}, {20, 19861}, {29, 24552}, {46, 62773}, {55, 27383}, {63, 14986}, {65, 26105}, {72, 1058}, {78, 390}, {144, 62874}, {145, 5815}, {191, 10072}, {200, 12575}, {283, 17127}, {376, 17614}, {377, 9812}, {388, 24703}, {391, 40963}, {404, 9778}, {405, 5603}, {443, 12699}, {474, 6361}, {516, 6904}, {517, 5084}, {550, 35272}, {936, 10624}, {938, 3869}, {944, 5811}, {950, 15829}, {982, 8421}, {997, 4294}, {1000, 64087}, {1005, 56889}, {1056, 58798}, {1125, 4295}, {1145, 47744}, {1155, 5180}, {1193, 41828}, {1385, 11111}, {1479, 5175}, {1519, 6908}, {1616, 4415}, {1621, 5703}, {1697, 3452}, {1699, 5177}, {1706, 5316}, {1788, 3816}, {1848, 4194}, {2093, 9843}, {2094, 3338}, {2183, 5296}, {2476, 9779}, {2550, 12701}, {2886, 4193}, {3058, 3189}, {3085, 5748}, {3086, 5744}, {3091, 24987}, {3218, 10586}, {3219, 10529}, {3241, 5330}, {3295, 63168}, {3303, 25568}, {3333, 9965}, {3421, 9957}, {3436, 3890}, {3474, 25524}, {3476, 57288}, {3486, 5289}, {3487, 13615}, {3488, 5730}, {3496, 40127}, {3522, 35262}, {3523, 10270}, {3576, 17576}, {3579, 17567}, {3600, 64002}, {3601, 52457}, {3622, 5905}, {3623, 26792}, {3671, 10582}, {3681, 6764}, {3683, 11376}, {3753, 17559}, {3868, 10580}, {3871, 64083}, {3875, 28616}, {3878, 18391}, {3889, 5572}, {3897, 20323}, {3940, 15172}, {4187, 5657}, {4189, 37561}, {4208, 24564}, {4297, 64130}, {4301, 64673}, {4305, 30144}, {4310, 28011}, {4313, 4511}, {4329, 27402}, {4342, 4853}, {4345, 4861}, {4357, 37054}, {4420, 64146}, {4423, 28629}, {4640, 7288}, {4646, 63126}, {4647, 41915}, {4652, 5265}, {4654, 51723}, {4666, 11036}, {4847, 51785}, {4866, 64368}, {5044, 5082}, {5046, 48482}, {5087, 10588}, {5128, 6692}, {5129, 19860}, {5218, 25681}, {5225, 5794}, {5231, 18249}, {5253, 44447}, {5273, 10527}, {5274, 6734}, {5281, 27385}, {5328, 5552}, {5435, 56288}, {5436, 64160}, {5493, 64112}, {5554, 31806}, {5584, 25893}, {5686, 24389}, {5731, 6223}, {5745, 50443}, {5759, 37244}, {5768, 5887}, {5791, 7743}, {5795, 7962}, {5809, 64131}, {5818, 17556}, {5837, 9581}, {5886, 6857}, {5901, 16418}, {6172, 11240}, {6675, 18493}, {6700, 61763}, {6735, 8165}, {6736, 9819}, {6737, 66682}, {6745, 53053}, {6850, 31838}, {6856, 9955}, {6865, 12672}, {6921, 64108}, {6931, 19877}, {6983, 48363}, {6987, 63986}, {7091, 60961}, {7320, 30513}, {7520, 63968}, {7987, 50836}, {7991, 8582}, {7995, 64705}, {8726, 54198}, {9580, 57284}, {9654, 64109}, {9799, 67998}, {9800, 9856}, {9802, 13996}, {9874, 27065}, {10198, 11813}, {10246, 50241}, {10283, 50243}, {10385, 56176}, {10430, 12688}, {10528, 27131}, {10587, 31053}, {10589, 26066}, {11019, 12526}, {11038, 64674}, {11108, 22791}, {11114, 64000}, {11220, 54228}, {11235, 27870}, {11238, 21677}, {11373, 31445}, {11522, 60959}, {11523, 64162}, {12437, 41864}, {12447, 51783}, {12702, 17527}, {12709, 64747}, {13745, 48941}, {14022, 24390}, {15680, 63267}, {15933, 34195}, {16116, 17525}, {16127, 51705}, {16408, 28174}, {17257, 26116}, {17528, 40273}, {17558, 24541}, {17561, 51709}, {17571, 38028}, {17691, 26658}, {17781, 62832}, {18220, 60926}, {18237, 54052}, {19535, 34123}, {19843, 30384}, {20196, 63990}, {21075, 31393}, {21214, 24248}, {21625, 62823}, {21735, 35271}, {24477, 37722}, {24556, 37402}, {24954, 37568}, {24982, 59417}, {25904, 37024}, {26015, 54398}, {26363, 58449}, {27378, 32942}, {28146, 57000}, {28212, 51559}, {28370, 33100}, {28376, 37620}, {29814, 54356}, {31141, 45081}, {31142, 37556}, {31424, 44675}, {31627, 56929}, {31730, 37267}, {34625, 41229}, {35661, 35886}, {37037, 41827}, {37080, 47357}, {37282, 59418}, {37313, 59320}, {37403, 52148}, {37421, 63989}, {37423, 64150}, {37435, 41869}, {37462, 59412}, {37548, 63089}, {37601, 64154}, {38316, 61010}, {38693, 66060}, {40270, 41863}, {43161, 63988}, {43749, 56277}, {45776, 64111}, {51090, 62824}, {51423, 54392}, {54179, 66020}, {54227, 64679}, {54290, 64124}, {54422, 64151}, {56440, 56945}, {56887, 64211}, {57279, 63993}

X(68599) = reflection of X(6904) in X(8583)
X(68599) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 40998, 452}, {2, 40, 26062}, {2, 41012, 26129}, {8, 8055, 3701}, {9, 12053, 64081}, {72, 1058, 36845}, {145, 31018, 5815}, {200, 12575, 56936}, {210, 64068, 8}, {329, 452, 66940}, {497, 960, 8}, {936, 10624, 17784}, {946, 31435, 2}, {1001, 3485, 3616}, {1125, 4295, 9776}, {1265, 4514, 8}, {1697, 3452, 7080}, {2478, 3877, 8}, {2551, 3057, 8}, {3057, 4679, 2551}, {3085, 21616, 5748}, {3086, 12514, 5744}, {3522, 67043, 64696}, {3616, 11415, 7}, {3616, 52653, 21}, {3622, 5905, 11037}, {3683, 11376, 30478}, {4342, 18250, 4853}, {4673, 14555, 8}, {5250, 41012, 2}, {6684, 25522, 2}, {9785, 18228, 8}, {12701, 25917, 2550}, {17576, 24558, 3576}, {24703, 58679, 388}, {24954, 37568, 59572}

X(68600) = X(1)X(329)∩X(2)X(65)

Barycentrics    a^4 - 2*a^2*(b - c)^2 - 4*a^3*(b + c) + (b^2 - c^2)^2 + 4*a*(b^3 + c^3) : :
X(68600) = 3*X[2]-2*X[1788], 3*X[2]-4*X[25681], X[1788]-2*X[25681], 2*X[7080]-X[63133]

See David Nguyen, euclid 8513.

X(68600) lies on these lines: {1, 329}, {2, 65}, {4, 5730}, {7, 19861}, {8, 908}, {9, 64160}, {10, 5748}, {20, 224}, {40, 27383}, {56, 9965}, {63, 3333}, {72, 5603}, {78, 962}, {92, 3702}, {100, 20070}, {144, 1001}, {145, 497}, {149, 20013}, {200, 4301}, {221, 63068}, {226, 15829}, {279, 20347}, {346, 21801}, {347, 10571}, {388, 5289}, {390, 34772}, {392, 3487}, {443, 39542}, {498, 3899}, {517, 6848}, {519, 9614}, {527, 1420}, {551, 62824}, {758, 3086}, {938, 41012}, {944, 58798}, {956, 6920}, {997, 4295}, {1125, 5744}, {1159, 17527}, {1201, 4310}, {1210, 26129}, {1219, 17165}, {1319, 20214}, {1388, 34610}, {1479, 4867}, {1482, 3421}, {1697, 63168}, {1699, 5175}, {1836, 37435}, {1944, 24565}, {2093, 6700}, {2094, 3361}, {2099, 2551}, {2478, 62830}, {2646, 5698}, {2886, 3614}, {3057, 25568}, {3085, 3878}, {3146, 5057}, {3189, 12701}, {3218, 5265}, {3241, 63999}, {3339, 62773}, {3340, 3452}, {3434, 6894}, {3474, 37267}, {3475, 58679}, {3486, 24703}, {3522, 44447}, {3523, 56288}, {3529, 10609}, {3600, 5905}, {3621, 5176}, {3656, 34790}, {3671, 8583}, {3672, 20245}, {3746, 55160}, {3811, 30305}, {3832, 5086}, {3868, 14986}, {3870, 9785}, {3872, 5734}, {3901, 10072}, {3925, 45085}, {3927, 5901}, {3935, 12632}, {3940, 5082}, {3962, 11376}, {3999, 64442}, {4067, 45700}, {4084, 10200}, {4189, 22768}, {4293, 30144}, {4294, 22836}, {4308, 64143}, {4323, 18228}, {4345, 36846}, {4652, 54445}, {4673, 20921}, {4847, 11522}, {4848, 30827}, {4853, 21060}, {4855, 9778}, {4860, 11281}, {4869, 18589}, {4930, 37730}, {5080, 48482}, {5087, 54361}, {5177, 12047}, {5211, 11851}, {5219, 5837}, {5225, 44669}, {5226, 24987}, {5250, 5703}, {5253, 21454}, {5261, 31053}, {5273, 24541}, {5274, 12649}, {5327, 11115}, {5328, 24982}, {5440, 6361}, {5550, 59491}, {5552, 6960}, {5554, 8165}, {5572, 62854}, {5692, 19843}, {5697, 34619}, {5731, 56387}, {5741, 5799}, {5758, 63986}, {5795, 31142}, {5887, 6847}, {5904, 34625}, {6051, 62857}, {6282, 54198}, {6734, 68034}, {6736, 11531}, {6745, 7991}, {6857, 37737}, {6872, 45230}, {6884, 10527}, {6891, 14988}, {6909, 18237}, {6919, 18391}, {6926, 64021}, {6941, 12245}, {6944, 64044}, {6987, 21740}, {6990, 24390}, {8543, 37228}, {8582, 18421}, {9578, 66465}, {9580, 12437}, {9780, 30852}, {9812, 57287}, {9856, 12529}, {10106, 28609}, {10165, 54290}, {10176, 19855}, {10453, 30971}, {10529, 18220}, {10580, 11520}, {10582, 12563}, {10591, 11813}, {11224, 66205}, {11235, 31145}, {11523, 12053}, {12531, 20054}, {12609, 37436}, {12702, 59591}, {13464, 57279}, {14110, 37421}, {14557, 31779}, {15507, 28376}, {15950, 30478}, {16466, 60751}, {17014, 24612}, {17484, 20076}, {17567, 36279}, {17781, 38314}, {17950, 25905}, {18232, 62858}, {18393, 31418}, {20015, 64068}, {20036, 20557}, {20111, 56555}, {20220, 30807}, {20292, 56999}, {20535, 30694}, {21296, 41010}, {22299, 28830}, {24231, 56630}, {24316, 64015}, {24391, 50443}, {24470, 35272}, {27379, 36100}, {27522, 58894}, {28271, 38286}, {29958, 39550}, {30389, 60905}, {31162, 63146}, {31888, 48698}, {32850, 44722}, {34749, 49736}, {35659, 35886}, {37313, 59317}, {37434, 67998}, {37567, 59572}, {37614, 63089}, {38053, 64723}, {38316, 61003}, {38460, 67262}, {41828, 42289}, {41863, 63993}, {44675, 54422}, {44785, 50693}, {45770, 50701}, {50371, 64190}, {50696, 63988}, {52653, 60979}, {54199, 63985}, {54228, 63984}, {56545, 62874}, {56937, 57015}, {60966, 62832}, {63130, 64083}, {63144, 64108}, {67035, 67036}

X(68600) = reflection of X(i) in X(j) for these (i,j): {1788, 25681}, {63133, 7080}
X(68600) = anticomplement of X(1788)
X(68600) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {72, 5603, 64081}, {78, 962, 17784}, {78, 51423, 962}, {144, 3622, 2975}, {497, 12635, 145}, {908, 11682, 8}, {960, 3485, 2}, {960, 34647, 3485}, {997, 4295, 6904}, {1125, 12526, 5744}, {1699, 6737, 5175}, {1788, 25681, 2}, {2093, 6700, 26062}, {2646, 5698, 17576}, {3436, 62826, 145}, {3474, 59691, 37267}, {3617, 46873, 11681}, {3671, 8583, 9776}, {3811, 30305, 56936}, {3868, 14986, 64151}, {3940, 22791, 5082}, {3962, 11376, 24477}, {4323, 18228, 19860}, {4511, 11415, 20}, {5554, 27131, 8165}, {5730, 51409, 4}, {5734, 5815, 3872}, {7982, 21075, 8}, {9965, 24558, 56}, {11523, 12053, 36845}, {11813, 49168, 10591}, {18391, 21616, 6919}, {25917, 28629, 2}, {31142, 64964, 5795}, {31435, 64110, 3616}, {56387, 64002, 5731}, {63391, 63962, 20}

X(68601) = X(5)X(10)∩X(11)X(938)

Barycentrics    -(a^4*(b + c)^3) + 2*a^2*(b - c)^2*(b + c)^3 - (b - c)^4*(b + c)^3 + a^5*(b^2 + c^2) + a*(b^2 - c^2)^2*(b^2 + 4*b*c + c^2) - 2*a^3*(b^4 + 2*b^3*c - 2*b^2*c^2 + 2*b*c^3 + c^4) : :
X(68601) = 3*X[2]-X[5584], 5*X[3091]+X[3303], 7*X[3851]+X[12000], 3*X[381]+X[16202], 3*X[1699]+X[59340], 3*X[5886]-X[37615]

See David Nguyen, euclid 8513.

X(68601) lies on these lines: {1, 8226}, {2, 5584}, {4, 1001}, {5, 10}, {11, 938}, {12, 497}, {20, 7965}, {40, 3826}, {55, 6835}, {56, 6837}, {119, 3851}, {142, 9943}, {165, 17529}, {226, 64131}, {235, 1848}, {381, 16202}, {442, 1699}, {443, 64074}, {452, 68053}, {495, 19925}, {496, 6744}, {515, 51715}, {516, 8728}, {529, 10532}, {602, 67849}, {942, 12617}, {958, 6846}, {962, 3925}, {971, 51706}, {1071, 25557}, {1125, 8727}, {1376, 6864}, {1482, 64335}, {1621, 6253}, {2475, 54348}, {2476, 9779}, {2829, 37234}, {2883, 21239}, {3035, 6918}, {3149, 6690}, {3428, 6832}, {3545, 10598}, {3576, 37447}, {3614, 6945}, {3616, 10883}, {3624, 37374}, {3649, 67998}, {3673, 41003}, {3683, 64003}, {3742, 6245}, {3813, 5603}, {3816, 6831}, {3822, 12571}, {3829, 34647}, {3838, 63989}, {3847, 6830}, {3874, 20330}, {3877, 64754}, {3927, 60895}, {3931, 53599}, {3947, 9842}, {4187, 7988}, {4197, 9812}, {4423, 6836}, {4640, 64001}, {4882, 7989}, {4999, 6824}, {5068, 11681}, {5173, 10395}, {5204, 6974}, {5248, 20420}, {5249, 12688}, {5284, 6895}, {5432, 6915}, {5433, 6888}, {5493, 38151}, {5572, 21620}, {5587, 6765}, {5691, 41858}, {5709, 18253}, {5715, 24703}, {5787, 5886}, {5805, 12514}, {5840, 64473}, {5842, 44229}, {5880, 12705}, {5883, 33899}, {5927, 13407}, {6001, 55108}, {6067, 54398}, {6147, 31803}, {6173, 7992}, {6284, 6839}, {6668, 6834}, {6691, 6833}, {6706, 6823}, {6826, 11496}, {6829, 15908}, {6847, 25524}, {6849, 11500}, {6854, 10310}, {6859, 52795}, {6865, 8167}, {6867, 10893}, {6881, 12699}, {6882, 61268}, {6884, 24953}, {6893, 10894}, {6896, 64123}, {6901, 11826}, {6907, 18483}, {6912, 7354}, {6913, 26332}, {6920, 11827}, {6929, 65949}, {6957, 10895}, {7173, 64963}, {7741, 67931}, {7951, 9614}, {8255, 12710}, {8273, 10431}, {8583, 37363}, {9589, 41859}, {9654, 38757}, {9799, 38053}, {9856, 12609}, {9961, 27186}, {10157, 21077}, {10171, 17527}, {10198, 19541}, {10306, 49732}, {10523, 17618}, {10592, 67046}, {10942, 38140}, {10943, 51709}, {11224, 64200}, {11230, 37356}, {11495, 37407}, {11522, 24390}, {11698, 38161}, {12053, 64737}, {12436, 64128}, {12565, 41867}, {12599, 18247}, {13374, 51755}, {14872, 66009}, {15254, 64004}, {15845, 26481}, {15849, 23058}, {15888, 59387}, {15950, 45230}, {16160, 38028}, {16418, 64075}, {16617, 26286}, {17530, 30308}, {18249, 65452}, {18493, 26470}, {19860, 52254}, {19862, 37364}, {22793, 37438}, {24391, 64443}, {24541, 37358}, {25365, 34848}, {26066, 67880}, {28146, 44222}, {31787, 61595}, {34746, 64173}, {35663, 35886}, {36662, 41785}, {37424, 51118}, {37434, 63991}, {38036, 54422}, {38052, 67886}, {38454, 55104}, {40998, 47510}, {44847, 61924}, {54370, 57282}, {59389, 63981}

X(68601) = complement of X(5584)
X(68601) = Euler-isogonal conjugate of X(12858)
X(68601) = complement of isogonal conjugate of X(10429)
X(68601) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 946, 2886}, {5, 7680, 1329}, {5, 7956, 25639}, {5, 9955, 7681}, {142, 21628, 9943}, {946, 31806, 22791}, {962, 6991, 3925}, {1001, 31936, 25466}, {1125, 12558, 8727}, {1621, 6894, 6253}, {5587, 63257, 12607}, {5787, 5886, 64675}, {5886, 6261, 11281}, {5886, 6841, 63980}, {6824, 22753, 4999}, {6828, 68034, 11}, {6831, 8227, 3816}, {6913, 26332, 57288}, {8227, 63992, 28628}, {25466, 42356, 4}

X(68602) = X(1)X(6)∩X(8)X(908)

Barycentrics    -(a*(a^3 + 3*b^3 - a*(b - c)^2 + b^2*c + b*c^2 + 3*c^3 - 3*a^2*(b + c))) : :
X(68602) = X[57]-2*X[997], X[200]-2*X[3940], 2*X[200]-X[63137], 4*X[3940]-X[63137], 3*X[5720]-2*X[18491], X[3421]-2*X[21060], X[5727]-3*X[31142], 2*X[37611]-X[63430], 2*X[999]-X[62823], 2*X[1376]-X[2093], 2*X[3452]-X[18391], X[3586]-2*X[24703], 5*X[3616]-3*X[64151], 2*X[6282]-X[10860], 5*X[25917]-3*X[61660], X[30304]-2*X[63991]

See David Nguyen, euclid 8513.

X(68602) lies on these lines: {1, 6}, {2, 5775}, {3, 12526}, {4, 6737}, {8, 908}, {10, 3340}, {20, 54228}, {36, 3928}, {40, 78}, {46, 5438}, {55, 31165}, {56, 3962}, {57, 758}, {63, 3576}, {65, 936}, {69, 41010}, {80, 3632}, {84, 5693}, {90, 6596}, {144, 5731}, {145, 5815}, {165, 5440}, {191, 3612}, {200, 517}, {210, 2099}, {214, 67334}, {224, 16132}, {329, 515}, {388, 67850}, {442, 5665}, {443, 3671}, {474, 3339}, {495, 64734}, {497, 519}, {527, 4293}, {551, 44841}, {614, 49454}, {728, 4752}, {912, 37611}, {942, 8583}, {944, 12527}, {957, 39959}, {962, 20007}, {965, 54424}, {993, 3929}, {995, 3677}, {999, 62823}, {1038, 66693}, {1060, 30674}, {1064, 3190}, {1125, 11518}, {1145, 5660}, {1149, 62850}, {1210, 25522}, {1259, 11012}, {1260, 3428}, {1308, 2750}, {1320, 55931}, {1376, 2093}, {1385, 3927}, {1394, 61225}, {1420, 4067}, {1457, 2318}, {1478, 28609}, {1479, 12625}, {1482, 4853}, {1490, 14110}, {1512, 6736}, {1697, 3811}, {1698, 6668}, {1699, 3419}, {1706, 5903}, {1709, 5538}, {1737, 30827}, {1768, 51636}, {1788, 6700}, {2098, 12629}, {2136, 5697}, {2327, 2360}, {2478, 41575}, {2551, 64163}, {2646, 31424}, {2650, 9345}, {2771, 7171}, {2801, 36973}, {2886, 3679}, {2932, 12767}, {2975, 3951}, {3057, 6765}, {3059, 43166}, {3085, 5837}, {3086, 24391}, {3158, 3899}, {3189, 10624}, {3218, 35262}, {3333, 3868}, {3338, 3901}, {3359, 14988}, {3361, 17614}, {3434, 18406}, {3436, 5881}, {3452, 18391}, {3486, 12572}, {3488, 40998}, {3586, 24703}, {3600, 41572}, {3601, 12514}, {3616, 11520}, {3622, 62861}, {3624, 11281}, {3626, 63916}, {3646, 34195}, {3678, 64964}, {3681, 3872}, {3682, 10571}, {3687, 9549}, {3696, 27471}, {3698, 64963}, {3699, 51284}, {3702, 52345}, {3753, 8580}, {3812, 61158}, {3870, 3877}, {3876, 19860}, {3884, 37556}, {3894, 51816}, {3895, 3935}, {3898, 51779}, {3916, 7987}, {3976, 56630}, {3992, 59599}, {4005, 11011}, {4007, 21074}, {4127, 8666}, {4187, 67931}, {4294, 12437}, {4295, 57284}, {4301, 5082}, {4302, 34701}, {4304, 5698}, {4308, 41563}, {4312, 11112}, {4315, 5850}, {4323, 60943}, {4384, 27489}, {4420, 63130}, {4423, 44840}, {4512, 24929}, {4533, 4866}, {4536, 64849}, {4640, 30282}, {4652, 11684}, {4661, 38460}, {4666, 63159}, {4668, 11280}, {4673, 20927}, {4677, 11235}, {4678, 5828}, {4847, 5603}, {4848, 44848}, {4855, 35242}, {4857, 41709}, {4861, 63135}, {4870, 31245}, {4881, 67335}, {4882, 10914}, {4915, 11224}, {4930, 9708}, {5044, 64673}, {5084, 6738}, {5086, 18492}, {5128, 25440}, {5175, 18483}, {5231, 5886}, {5250, 34772}, {5268, 66640}, {5269, 30115}, {5330, 36846}, {5425, 51780}, {5427, 54302}, {5437, 5902}, {5562, 60018}, {5573, 49997}, {5657, 6745}, {5686, 11526}, {5687, 7991}, {5690, 37713}, {5691, 58798}, {5694, 7330}, {5705, 11375}, {5709, 45770}, {5732, 64723}, {5744, 10165}, {5748, 10175}, {5791, 37737}, {5794, 9612}, {5839, 40963}, {5851, 10609}, {5853, 30305}, {5884, 37526}, {5887, 12705}, {5919, 41711}, {6001, 6282}, {6173, 11551}, {6264, 46685}, {6508, 18446}, {6667, 25681}, {6684, 27383}, {6731, 11528}, {6734, 8227}, {6735, 63143}, {6763, 37618}, {6769, 12672}, {6790, 67570}, {6857, 18249}, {7070, 45272}, {7080, 11362}, {7280, 35204}, {7308, 10176}, {7322, 30116}, {7992, 37022}, {7995, 64074}, {9370, 34039}, {9578, 21077}, {9579, 17647}, {9581, 21616}, {9624, 10527}, {9709, 50193}, {9785, 12630}, {9819, 66469}, {9841, 15071}, {9856, 12651}, {9954, 64897}, {10032, 50820}, {10268, 33597}, {10310, 18237}, {10388, 66226}, {10582, 15934}, {10590, 66465}, {10698, 14740}, {10864, 12528}, {10916, 50443}, {10980, 24473}, {11111, 51090}, {11372, 41228}, {11415, 41869}, {11517, 59320}, {11522, 24390}, {11530, 17057}, {11680, 38021}, {12059, 12691}, {12520, 37551}, {12532, 66059}, {12565, 31793}, {12649, 41012}, {12650, 14872}, {13464, 64081}, {15803, 59691}, {16370, 53054}, {16371, 53056}, {16408, 31794}, {17022, 66687}, {17272, 41003}, {17296, 18589}, {17781, 50811}, {17784, 28194}, {18163, 18417}, {18206, 18465}, {18254, 64137}, {18467, 37787}, {19535, 51576}, {19843, 64160}, {20760, 37620}, {21031, 41687}, {21578, 60977}, {21617, 40333}, {21740, 55104}, {22275, 24806}, {23151, 51304}, {24349, 27492}, {24392, 30384}, {24467, 38602}, {24477, 44675}, {24914, 31235}, {24953, 31446}, {25568, 31397}, {25917, 61660}, {26015, 37704}, {27385, 31423}, {28629, 45085}, {29351, 32726}, {29960, 59557}, {30196, 67725}, {30284, 60949}, {30294, 66252}, {30304, 63991}, {30318, 50573}, {30503, 64107}, {30513, 56038}, {30852, 54447}, {31838, 64668}, {33099, 66672}, {33538, 51290}, {34606, 37740}, {34744, 59572}, {35665, 35886}, {36279, 64112}, {36845, 63993}, {37533, 42012}, {37554, 54421}, {37560, 64021}, {37567, 61152}, {37568, 61154}, {37625, 67880}, {37714, 52850}, {38314, 62815}, {40940, 60751}, {43161, 61003}, {43174, 59591}, {45391, 45393}, {46917, 54286}, {47623, 62865}, {49492, 56082}, {49736, 51093}, {50371, 52027}, {51281, 53388}, {51362, 59503}, {54348, 54391}, {56176, 61153}, {56821, 64708}, {56880, 64278}, {59319, 63915}, {59417, 64083}, {61156, 64047}, {61275, 64153}, {61762, 62874}, {62827, 64954}, {62860, 64675}, {63136, 64135}, {63976, 64328}, {63986, 68036}, {63988, 68057}, {64145, 66061}, {66197, 66231}

X(68602) = reflection of X(i) in X(j) for these (i,j): {1, 5289}, {57, 997}, {200, 3940}, {2093, 1376}, {3421, 21060}, {3586, 24703}, {10860, 6282}, {18391, 3452}, {30304, 63991}, {36845, 63993}, {62823, 999}, {63137, 200}, {63430, 37611}
X(68602) = barycentric quotient X(692)/X(26715)
X(68602) = trilinear quotient X(101)/X(26715)
X(68602) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 72, 57279}, {1, 238, 16485}, {1, 960, 31435}, {1, 5223, 956}, {1, 5692, 9}, {1, 5904, 6762}, {1, 11523, 41863}, {1, 54386, 1453}, {3, 12526, 54290}, {8, 908, 5587}, {8, 11682, 7982}, {40, 6326, 52026}, {56, 3962, 54422}, {63, 4511, 3576}, {72, 956, 5223}, {72, 5730, 1}, {72, 41389, 5692}, {78, 3869, 40}, {210, 2099, 9623}, {474, 4018, 3339}, {946, 64335, 5587}, {956, 5223, 57279}, {960, 12635, 1}, {962, 20007, 63146}, {1125, 12559, 11518}, {1385, 3927, 62824}, {1482, 34790, 4853}, {2975, 56387, 64953}, {3419, 51409, 1699}, {3434, 51423, 31162}, {3632, 30323, 3680}, {3671, 12447, 443}, {3681, 62826, 3872}, {3811, 3878, 1697}, {3868, 19861, 3333}, {3870, 3877, 31393}, {3872, 62826, 16200}, {3876, 62830, 19860}, {3929, 13384, 993}, {3951, 56387, 2975}, {3984, 11682, 8}, {4067, 30144, 62858}, {4301, 6743, 5082}, {4640, 56177, 30282}, {4677, 64896, 64203}, {4855, 56288, 35242}, {4864, 16486, 1}, {4867, 5692, 1}, {4882, 11531, 10914}, {4930, 9708, 50194}, {5223, 60885, 9}, {5587, 7982, 3577}, {5587, 36922, 8}, {5687, 7991, 63138}, {5693, 63391, 84}, {5887, 37531, 12705}, {6261, 31806, 40}, {8580, 18421, 3753}, {10176, 54318, 7308}, {10176, 62822, 54318}, {10179, 42871, 1}, {11375, 21677, 5705}, {11415, 57287, 41869}, {11523, 15829, 1}, {12514, 22836, 3601}, {21616, 49168, 9581}, {30144, 62858, 1420}, {54330, 57015, 9}

X(68603) = X(1)X(5809)∩X(7)X(21)

Barycentrics    -((a + b - c)*(a - b + c)*(3*a^4 - 8*a^3*(b + c) + 12*a*b*c*(b + c) - (b^2 - c^2)^2 + a^2*(6*b^2 + 4*b*c + 6*c^2))) : :

See David Nguyen, euclid 8513.

X(68603) lies on these lines: {1, 5809}, {7, 21}, {8, 60943}, {9, 64160}, {65, 62775}, {145, 24389}, {226, 38316}, {388, 42819}, {390, 946}, {497, 3748}, {516, 30275}, {518, 60995}, {551, 4321}, {954, 5603}, {960, 4323}, {1125, 8732}, {1445, 31435}, {2099, 38057}, {2346, 9785}, {2550, 11375}, {2886, 7679}, {3340, 6666}, {3486, 42356}, {3487, 42884}, {3600, 41857}, {3622, 64674}, {3947, 43179}, {4295, 52769}, {4308, 51715}, {4318, 5308}, {4870, 47357}, {5175, 66215}, {5219, 5853}, {5261, 18492}, {5265, 60938}, {5290, 51724}, {5542, 60934}, {5550, 61019}, {5686, 11526}, {5825, 18412}, {6172, 34647}, {6261, 30284}, {7675, 63992}, {7676, 64077}, {7688, 59418}, {8545, 11038}, {9612, 43175}, {9614, 30331}, {9776, 15804}, {9780, 61017}, {9856, 12706}, {11235, 50839}, {11522, 66210}, {12047, 43161}, {12709, 58564}, {12730, 66065}, {12848, 66515}, {13411, 43166}, {16020, 42289}, {18623, 29814}, {24393, 64964}, {28628, 60996}, {30340, 60936}, {30478, 64723}, {35668, 35886}, {38031, 39542}, {38108, 50194}, {38314, 61027}, {51099, 60909}, {59372, 60956}, {59412, 61008}

X(68603) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 38037, 5809}, {1001, 3485, 7}, {1125, 12560, 8732}, {4323, 18230, 7672}, {11526, 61015, 5686}

X(68604) = X(1)X(6)∩X(57)X(9943)

Barycentrics    a*(-(a^5*(b + c)) - a*(b - c)^2*(b + c)^3 + a^4*(b^2 + c^2) + (b^2 - c^2)^2*(b^2 + c^2) + 2*a^3*(b^3 + b^2*c + b*c^2 + c^3) - 2*a^2*(b^4 - 6*b^2*c^2 + c^4)) : :
X(68604) = 3*X[354]+X[1858], 3*X[354]-X[66250], X[1858]+X[66250], X[12432]+X[12575], X[3057]+3*X[61663], 7*X[3622]+X[64715], X[4292]+X[66248], 3*X[5603]+X[67919], X[15556]+3*X[64162], 5*X[17609]+3*X[61722], 5*X[18398]-X[67970]

See David Nguyen, euclid 8513.

X(68604) lies on these lines: {1, 6}, {3, 12710}, {4, 67966}, {7, 10429}, {8, 18928}, {10, 64157}, {11, 58613}, {20, 14100}, {55, 58637}, {56, 10391}, {57, 9943}, {65, 497}, {142, 18251}, {226, 64131}, {241, 4300}, {354, 1858}, {388, 1864}, {390, 7957}, {443, 15587}, {495, 58631}, {496, 942}, {498, 58451}, {499, 3848}, {516, 37544}, {517, 6738}, {758, 6744}, {774, 3666}, {912, 5045}, {971, 4298}, {999, 6261}, {1042, 21346}, {1044, 1418}, {1056, 14872}, {1070, 5721}, {1071, 3333}, {1125, 11018}, {1210, 2886}, {1319, 45230}, {1385, 13346}, {1445, 5584}, {1466, 64128}, {1496, 4641}, {1617, 10393}, {1660, 51695}, {1697, 41539}, {1788, 61660}, {1829, 44079}, {1848, 5244}, {1859, 1895}, {1898, 10404}, {2099, 66216}, {2646, 62873}, {2771, 18240}, {2800, 17706}, {2801, 12577}, {3057, 6992}, {3085, 3740}, {3086, 3742}, {3091, 17604}, {3295, 63976}, {3303, 41538}, {3338, 64132}, {3340, 66226}, {3361, 10167}, {3421, 18247}, {3486, 64106}, {3488, 14110}, {3600, 10394}, {3616, 11020}, {3622, 63092}, {3697, 51784}, {3745, 66593}, {3753, 67931}, {3868, 10580}, {3874, 12915}, {3876, 10578}, {3878, 51724}, {3880, 64163}, {3881, 16215}, {3947, 10157}, {4292, 15726}, {4301, 30329}, {4313, 7671}, {4314, 31793}, {4319, 37537}, {4326, 37551}, {4662, 31397}, {4719, 17102}, {4860, 64704}, {5044, 13405}, {5082, 5836}, {5173, 12053}, {5290, 5927}, {5542, 31803}, {5570, 67946}, {5603, 67919}, {5691, 9844}, {5703, 25917}, {5719, 16216}, {5722, 7686}, {5777, 21620}, {5881, 39779}, {5887, 15934}, {5902, 9614}, {6147, 31937}, {7288, 17603}, {7672, 9785}, {7675, 8273}, {7958, 21617}, {8581, 11037}, {8729, 12715}, {8734, 12716}, {9578, 61718}, {9654, 67875}, {9776, 12529}, {9940, 64124}, {9942, 22753}, {9947, 51782}, {9957, 31806}, {9961, 21454}, {10056, 58629}, {10072, 58560}, {10122, 11281}, {10178, 15803}, {10384, 12651}, {10395, 64737}, {10914, 67942}, {10956, 58687}, {10980, 15071}, {11235, 67937}, {11365, 40660}, {11518, 12709}, {11529, 12672}, {12005, 58576}, {12246, 66020}, {12262, 46373}, {12563, 20116}, {12573, 63998}, {12677, 26332}, {12736, 66065}, {15556, 64162}, {15733, 57284}, {15933, 31165}, {16127, 57282}, {16465, 19861}, {17170, 24471}, {17609, 61722}, {17622, 64964}, {17626, 18398}, {17753, 44735}, {18227, 21075}, {18237, 18238}, {18389, 50196}, {19520, 42012}, {20007, 30628}, {22476, 22970}, {25722, 56999}, {28150, 66254}, {29957, 37613}, {30290, 59372}, {31792, 61597}, {34371, 42450}, {34753, 40296}, {34772, 54348}, {34855, 53597}, {35672, 35886}, {37592, 62811}, {40636, 64208}, {44662, 58469}, {44663, 49736}, {52424, 54295}, {53021, 67964}, {58577, 67051}, {58649, 59722}, {60883, 64003}, {62805, 65702}, {64074, 66239}

X(68604) = midpoint of X(i) and X(j) for these {i,j}: {1, 44547}, {1858, 66250}, {4292, 66248}, {12432, 12575}
X(68604) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 7078, 30621}, {1, 10396, 958}, {1, 10398, 57279}, {1, 15299, 405}, {1, 18412, 3555}, {1, 44547, 518}, {1, 66235, 49465}, {56, 10391, 58567}, {57, 12711, 9943}, {65, 9848, 962}, {72, 31435, 960}, {354, 1858, 66250}, {496, 942, 13374}, {942, 9856, 3671}, {960, 5572, 1}, {960, 34791, 12635}, {960, 51715, 58679}, {1071, 3333, 63994}, {1125, 12564, 11018}, {1210, 50195, 3812}, {3086, 67930, 3742}, {3295, 67981, 63976}, {3600, 10394, 12680}, {3874, 21625, 12915}, {4292, 66248, 15726}, {5044, 16201, 13405}, {10122, 44675, 16193}, {11019, 67944, 942}, {11037, 12528, 8581}, {12432, 12575, 517}, {12635, 66009, 34791}, {14986, 62864, 354}, {31793, 63972, 4314}

X(68605) = X(1)X(6)∩X(946)X(3741)

Barycentrics    a*(b*c*(b + c)^2*(b^2 + c^2) + a*(b + c)^3*(b^2 + c^2) - a^4*(b^2 + b*c + c^2) - a^3*(b^3 + b^2*c + b*c^2 + c^3) + a^2*(b^4 + 2*b^3*c + 2*b*c^3 + c^4)) : :

See David Nguyen, euclid 8513.

X(68605) lies on these lines: {1, 6}, {2, 10822}, {56, 10461}, {443, 64007}, {497, 10449}, {946, 3741}, {1125, 35612}, {1764, 64077}, {2886, 10479}, {3085, 9564}, {3485, 10473}, {3616, 5208}, {3688, 54433}, {3757, 3876}, {3869, 24552}, {3913, 22325}, {4201, 64709}, {4300, 56509}, {4643, 42450}, {4894, 35634}, {5710, 22275}, {6261, 37620}, {8227, 30986}, {9856, 12544}, {9943, 10856}, {10381, 26098}, {10444, 12688}, {10463, 63991}, {10468, 41003}, {10472, 34830}, {10476, 63992}, {10882, 65404}, {11281, 35637}, {13725, 21746}, {17185, 23151}, {18417, 37357}, {19766, 52020}, {19863, 28628}, {22076, 33171}, {33082, 50617}, {35620, 64110}, {35636, 66065}, {35640, 35886}, {35645, 50608}, {63999, 67976}

X(68605) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1125, 35632, 35612}, {10856, 12548, 9943}

X(68606) = X(1)X(4)∩X(165)X(474)

Barycentrics    a*(a^6 - a^4*(b - c)^2 - 2*a^5*(b + c) + (b^2 - c^2)^2*(b^2 + 6*b*c + c^2) - a^2*(b - c)^2*(b^2 + 10*b*c + c^2) - 2*a*(b - c)^2*(b^3 - 3*b^2*c - 3*b*c^2 + c^3) + 4*a^3*(b^3 - 2*b^2*c - 2*b*c^2 + c^3)) : :

See David Nguyen, euclid 8513.

X(68606) lies on these lines: {1, 4}, {40, 4413}, {56, 11372}, {57, 7995}, {78, 43166}, {165, 474}, {200, 4301}, {210, 6766}, {392, 68057}, {516, 6904}, {517, 5780}, {551, 64679}, {936, 962}, {960, 1706}, {997, 12651}, {1001, 2951}, {1125, 10857}, {1210, 41824}, {1467, 8727}, {1697, 19541}, {1698, 6964}, {1709, 3361}, {1721, 21214}, {2093, 12672}, {2551, 10863}, {2886, 7989}, {3062, 7091}, {3086, 21628}, {3091, 9623}, {3149, 61763}, {3304, 10864}, {3333, 12688}, {3338, 7992}, {3340, 5806}, {3421, 9842}, {3555, 68000}, {3616, 5732}, {3624, 6926}, {3646, 5584}, {3649, 38036}, {3656, 5534}, {3680, 11224}, {3817, 6919}, {3832, 3872}, {3893, 7982}, {4187, 7988}, {4292, 67999}, {4298, 64130}, {4321, 63973}, {4668, 64335}, {4853, 19925}, {4882, 10914}, {4915, 37714}, {5231, 12617}, {5400, 5799}, {5563, 18237}, {5572, 30343}, {5720, 22791}, {5731, 51724}, {5735, 11415}, {5777, 68032}, {5881, 18529}, {5886, 8726}, {5901, 41854}, {5927, 6762}, {6264, 59390}, {6282, 12699}, {6765, 12541}, {6769, 12700}, {6848, 31434}, {6922, 8227}, {6940, 16192}, {6946, 63469}, {6967, 34595}, {6983, 64850}, {7271, 41010}, {7686, 68001}, {7956, 9581}, {7965, 11376}, {7993, 66065}, {8158, 10157}, {9580, 20420}, {9624, 50528}, {9779, 19860}, {9800, 64705}, {9812, 19861}, {9943, 11407}, {10222, 18528}, {10389, 64804}, {10398, 64131}, {10582, 12520}, {10624, 50700}, {10680, 18540}, {10860, 25524}, {10980, 15071}, {11037, 41561}, {11112, 50865}, {11281, 16143}, {11496, 30282}, {11519, 37712}, {12560, 65452}, {12629, 59387}, {12705, 15803}, {13865, 61275}, {15179, 62180}, {15911, 45770}, {17556, 30308}, {17857, 64754}, {18443, 18493}, {22793, 37611}, {28628, 37374}, {30326, 57279}, {30389, 41860}, {30392, 51715}, {31775, 41869}, {31803, 62823}, {34628, 49736}, {34647, 66469}, {35673, 35886}, {37240, 59340}, {37244, 59320}, {37403, 58221}, {37434, 44675}, {37447, 64684}, {38460, 50689}, {44425, 53053}, {51709, 64668}, {54370, 62824}, {54422, 67998}, {55922, 67701}, {59391, 64267}, {62874, 64197}, {63135, 64368}, {63257, 63966}, {63970, 64081}, {64150, 68034}

X(68606) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 946, 9614}, {4, 10106, 5691}, {57, 9856, 7995}, {474, 67886, 165}, {936, 962, 7994}, {946, 3485, 11522}, {946, 6261, 64669}, {946, 63992, 1}, {1125, 12565, 10857}, {1490, 5603, 1}, {3333, 12688, 30304}, {6261, 64669, 1}, {7987, 11379, 24644}, {10382, 64160, 1}, {11522, 63988, 1}, {12672, 67880, 2093}, {12705, 22753, 15803}, {31435, 64077, 165}, {63992, 64669, 6261}

X(68607) = X(1)X(1587)∩X(7)X(21)

Barycentrics    (a + b - c)*(a - b + c)*(a^4 - 2*a^3*(b + c) + 2*a*(b + c)*(b*c - S) - (b + c)^2*S + a^2*(b^2 + c^2 + S)) : :

See David Nguyen, euclid 8513.

X(68607) lies on these lines: {1, 1587}, {7, 21}, {497, 30334}, {946, 30381}, {960, 30413}, {1125, 30277}, {2886, 30314}, {4295, 31546}, {6204, 31435}, {6261, 18460}, {9614, 30432}, {30297, 64077}, {30342, 64110}, {30401, 63992}

X(68608) = X(1)X(4199)∩X(7)X(21)

Barycentrics    -(a*(b^2 + c^2 + a*(b + c))*(a^4 - 4*a*b*c*(b + c) - b*c*(b + c)^2 - a^2*(b^2 + b*c + c^2))) : :

See David Nguyen, euclid 8513.

X(68608) lies on these lines: {1, 4199}, {7, 21}, {350, 21405}, {497, 8240}, {846, 988}, {946, 4425}, {960, 1193}, {1125, 8731}, {1457, 58679}, {2886, 5051}, {3702, 49512}, {4220, 64077}, {4364, 23361}, {5260, 37759}, {6261, 30285}, {8235, 63992}, {9614, 30366}, {9856, 12713}, {11043, 64110}, {12635, 19767}, {12746, 66065}, {19259, 63997}, {19533, 24248}, {19786, 28265}, {20834, 51721}, {28258, 58386}, {28628, 37225}, {32776, 47521}, {35675, 35886}, {49735, 49736}

X(68608) = barycentric product X(3666)*X(16824)
X(68608) = barycentric quotient X(16824)/X(30710)
X(68608) = trilinear product X(1193)*X(16824)
X(68608) = trilinear quotient X(1220)/X(16824)
X(68608) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1125, 12567, 8731}

X(68609) = X(1)X(5927)∩X(11)X(65)

Barycentrics    a*(-(a^5*(b + c)) + a^4*(b + c)^2 + (b^2 - c^2)^2*(b^2 + 4*b*c + c^2) - 2*a^2*(b - c)^2*(b^2 + 5*b*c + c^2) - a*(b - c)^2*(b^3 - 3*b^2*c - 3*b*c^2 + c^3) + 2*a^3*(b^3 - 2*b^2*c - 2*b*c^2 + c^3)) : :

See David Nguyen, euclid 8513.

X(68609) lies on these lines: {1, 5927}, {5, 66226}, {11, 65}, {72, 3813}, {80, 32537}, {355, 497}, {392, 1479}, {496, 12709}, {517, 10591}, {960, 3679}, {1001, 3612}, {1071, 11376}, {1125, 17612}, {1376, 31435}, {1387, 40263}, {1388, 6261}, {1420, 12114}, {1470, 63266}, {1616, 36985}, {1698, 30294}, {1709, 3361}, {1864, 13464}, {2098, 18908}, {2886, 4002}, {3086, 9856}, {3485, 5045}, {3555, 34647}, {3586, 58679}, {3616, 17616}, {3660, 10785}, {3678, 17658}, {3753, 7741}, {4018, 45035}, {5044, 30305}, {5225, 31786}, {5439, 23708}, {5563, 51768}, {5587, 66216}, {5603, 64131}, {5886, 12711}, {5887, 7743}, {6001, 50443}, {7962, 58631}, {9581, 45776}, {9624, 10391}, {9668, 31838}, {9848, 13411}, {10043, 17648}, {10106, 34697}, {10523, 17618}, {10572, 49736}, {10589, 31788}, {10593, 37562}, {10624, 25917}, {10866, 31397}, {10893, 16616}, {10895, 39779}, {10944, 63999}, {11281, 17653}, {11522, 44547}, {12019, 23340}, {12528, 18220}, {12688, 44675}, {12701, 64107}, {15558, 38156}, {16483, 65128}, {17604, 64163}, {17609, 41562}, {17613, 58887}, {17624, 64110}, {17626, 18398}, {17634, 64124}, {17642, 20117}, {17652, 66065}, {18237, 67987}, {18483, 64106}, {22791, 41539}, {24926, 51715}, {26492, 40296}, {34717, 66256}, {35678, 35886}, {37704, 66250}, {37709, 67875}, {46677, 64767}, {50195, 68034}, {50196, 67998}, {64669, 64964}

X(68609) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11, 12672, 67937}, {960, 11235, 10914}, {1125, 17646, 17612}, {9614, 10826, 11235}, {11373, 31937, 17625}

X(68610) = X(1)X(1864)∩X(11)X(65)

Barycentrics    -(a*(a - b - c)*(4*a*b*(b - c)^2*c + a^4*(b + c) - 2*a^2*(b - c)^2*(b + c) + (b - c)^2*(b + c)^3)) : :
X(68610) = X[65]-2*X[67937], 2*X[3086]-X[37566]

See David Nguyen, euclid 8513.

X(68610) lies on these lines: {1, 1864}, {4, 64106}, {8, 210}, {10, 66226}, {11, 65}, {33, 1191}, {55, 936}, {56, 84}, {57, 7995}, {72, 12053}, {90, 22767}, {224, 1001}, {354, 1858}, {392, 950}, {496, 5887}, {517, 9581}, {519, 17622}, {614, 1854}, {912, 11373}, {942, 18493}, {971, 1420}, {978, 9371}, {999, 31937}, {1071, 44675}, {1108, 1903}, {1122, 41010}, {1125, 12711}, {1155, 63985}, {1156, 1476}, {1201, 2310}, {1319, 1898}, {1466, 12705}, {1479, 14110}, {1617, 63988}, {1697, 5044}, {1728, 22770}, {1836, 67999}, {2057, 3689}, {2078, 64804}, {2098, 12629}, {2099, 64669}, {2136, 51380}, {2886, 3698}, {3086, 6001}, {3295, 5780}, {3304, 8581}, {3340, 64157}, {3486, 58679}, {3576, 66248}, {3586, 31786}, {3616, 10391}, {3622, 10394}, {3660, 15071}, {3666, 24849}, {3678, 4342}, {3683, 26357}, {3812, 10589}, {3869, 5274}, {3873, 18220}, {3915, 51361}, {3962, 26015}, {4301, 41539}, {4315, 31871}, {5048, 12635}, {5173, 11522}, {5204, 5918}, {5265, 9961}, {5603, 44547}, {5692, 51785}, {5693, 37704}, {5710, 9817}, {5727, 9957}, {5728, 64160}, {5836, 54361}, {5886, 67930}, {5902, 50444}, {5919, 10950}, {5927, 10106}, {6734, 15845}, {6736, 18236}, {7004, 52541}, {7082, 10966}, {7288, 9943}, {7290, 58906}, {7686, 10591}, {7743, 24474}, {7957, 12701}, {7962, 34790}, {7991, 30294}, {8168, 58657}, {8227, 50195}, {8583, 18251}, {9355, 9363}, {9578, 10157}, {9580, 31793}, {9624, 16193}, {9844, 51724}, {9850, 20323}, {9947, 37709}, {10072, 67970}, {10085, 41426}, {10176, 12575}, {10395, 24390}, {10624, 64107}, {10896, 64721}, {10939, 38336}, {11019, 12709}, {11235, 25414}, {11238, 31165}, {11240, 34647}, {11281, 17637}, {11502, 37568}, {12047, 67966}, {12640, 46694}, {12664, 57278}, {12740, 38669}, {12773, 24928}, {12915, 31821}, {13601, 67931}, {13750, 23708}, {15829, 64171}, {17609, 64110}, {17625, 31803}, {17632, 37740}, {17636, 66065}, {17658, 21627}, {17661, 41554}, {18178, 64409}, {18391, 45776}, {19914, 23340}, {20117, 63993}, {20418, 65998}, {22791, 67981}, {23536, 38357}, {24005, 55120}, {30305, 63976}, {31231, 31787}, {32537, 58683}, {34772, 53055}, {34862, 64372}, {35679, 35886}, {37605, 65404}, {37618, 63432}, {37720, 64045}, {37722, 64041}, {40269, 62854}, {40944, 40963}, {42448, 61671}, {45230, 51715}, {47743, 64021}, {51768, 66020}, {52007, 60599}, {54348, 59691}, {55931, 56038}, {56176, 66199}, {57285, 63989}, {58649, 63137}, {61705, 61762}, {61718, 64964}, {63967, 64703}, {64368, 66197}, {65991, 66009}

X(68610) = reflection of X(i) in X(j) for these (i,j): {65, 67937}, {37566, 3086}
X(68610) = barycentric product X(9)*X(24213)
X(68610) = barycentric quotient X(24213)/X(85)
X(68610) = trilinear product X(55)*X(24213)
X(68610) = trilinear quotient X(7)/X(24213)
X(68610) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 64131, 1864}, {8, 66216, 3057}, {11, 64042, 65}, {56, 12688, 63995}, {57, 9856, 17634}, {72, 12053, 17642}, {210, 10866, 3057}, {496, 5887, 67949}, {497, 960, 3057}, {1125, 12711, 17603}, {1210, 12672, 65}, {1319, 1898, 12680}, {1858, 11376, 354}, {3057, 17604, 1837}, {5693, 37704, 50196}, {9848, 25917, 55}, {14986, 66250, 354}, {14986, 67998, 66250}

X(68611) = X(1)X(7593)∩X(236)X(960)

Barycentrics    -a^4 + 2*a^3*(b + c) - 2*a*(b - c)^2*(b + c) - (b^2 - c^2)^2 + 2*a^2*(b^2 + c^2) - (a^4 - 4*a*b*c*(b + c) - 2*a^2*(b + c)^2 + (b^2 - c^2)^2)*Sin[A/2] : :

See David Nguyen, euclid 8513.

X(68611) lies on these lines: {1, 7593}, {173, 31435}, {174, 3485}, {236, 960}, {497, 11924}, {946, 8351}, {1001, 7587}, {1125, 8729}, {2886, 8382}, {3616, 11890}, {3671, 8734}, {6261, 18454}, {7028, 28628}, {7589, 64077}, {7590, 63992}, {8092, 64110}, {9614, 30411}, {9856, 12716}, {11018, 12715}, {11281, 16151}, {12748, 66065}, {35681, 35886}

X(68611) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1125, 12570, 8729}

X(68612) = X(4)X(3427)∩X(355)X(960)

Barycentrics    -a^10 + 3*a^9*(b + c) - 12*a^5*b*(b - c)^2*c*(b + c) + (b - c)^6*(b + c)^4 - a^8*(b^2 + 4*b*c + c^2) - a*(b - c)^4*(b + c)^3*(3*b^2 + 2*b*c + 3*c^2) + 2*a^6*(b - c)^2*(4*b^2 + 9*b*c + 4*c^2) + a^7*(-6*b^3 + 2*b^2*c + 2*b*c^2 - 6*c^3) + a^2*(b^2 - c^2)^2*(b^4 - 2*b^3*c + 10*b^2*c^2 - 2*b*c^3 + c^4) - 2*a^4*(b - c)^2*(4*b^4 + 5*b^3*c - 2*b^2*c^2 + 5*b*c^3 + 4*c^4) + 2*a^3*(b - c)^2*(3*b^5 + 9*b^4*c + 4*b^3*c^2 + 4*b^2*c^3 + 9*b*c^4 + 3*c^5) : :
X(68612) = X[4]+X[3427], 2*X[5]-X[64328], X[48482]+X[64335], 3*X[381]-X[64326], 2*X[1125]-X[64310], 3*X[5587]-X[64319], 2*X[63980]-X[64334], 2*X[12616]-X[64311], X[64261]+X[64316]

See David Nguyen, euclid 8513.

X(68612) lies on these lines: {4, 3427}, {5, 64328}, {11, 63992}, {12, 64327}, {84, 16141}, {355, 960}, {381, 64326}, {392, 64317}, {497, 64322}, {515, 1001}, {946, 5722}, {1125, 64310}, {1837, 3577}, {2829, 31672}, {2886, 64733}, {3485, 64324}, {5252, 7966}, {5587, 64319}, {5787, 5886}, {5794, 64315}, {5805, 6001}, {5812, 68005}, {6245, 18237}, {7686, 12858}, {7702, 12688}, {9799, 58588}, {11374, 64266}, {12330, 21628}, {12599, 63999}, {12616, 64077}, {12677, 26332}, {16125, 64119}, {28234, 64767}, {28628, 63963}, {31435, 64261}, {52837, 54304}, {62354, 64138}, {64110, 64323}

X(68612) = midpoint of X(i) and X(j) for these {i,j}: {4, 3427}, {48482, 64335}, {64261, 64316}
X(68612) = reflection of X(i) in X(j) for these (i,j): {64310, 1125}, {64311, 12616}, {64328, 5}, {64334, 63980}
X(68612) = Euler-isogonal conjugate of X(2886)

X(68613) = X(1)X(5927)∩X(65)X(497)

Barycentrics    a*(-(a^5*(b + c)) - a*(b - c)^2*(b + c)^3 + a^4*(b^2 - 4*b*c + c^2) + (b^2 - c^2)^2*(b^2 + 4*b*c + c^2) + 2*a^3*(b^3 + b^2*c + b*c^2 + c^3) - 2*a^2*(b^4 - 14*b^2*c^2 + c^4)) : :
X(68613) = X[1]+X[9844], 3*X[1]-X[17644], 3*X[9844]+X[17644], X[65]-3*X[938], X[65]+3*X[9848], X[938]+X[9848], 2*X[5045]-3*X[21625], 2*X[1125]-X[9858], 5*X[3616]-X[9859], 5*X[3697]-3*X[4882], X[5697]+3*X[67942]

See David Nguyen, euclid 8513.

X(68613) lies on these lines: {1, 5927}, {65, 497}, {390, 58637}, {496, 3742}, {518, 1058}, {519, 960}, {936, 1001}, {946, 971}, {1071, 65465}, {1125, 9858}, {1420, 64679}, {2886, 9843}, {2951, 3361}, {3295, 3740}, {3333, 15726}, {3485, 9850}, {3488, 58679}, {3616, 9859}, {3697, 4882}, {3812, 11235}, {3817, 16201}, {3848, 47743}, {3947, 9842}, {4662, 31393}, {5044, 30331}, {5697, 67942}, {5722, 5836}, {5728, 51785}, {5777, 40270}, {5887, 18530}, {6744, 9856}, {6765, 18227}, {6767, 58631}, {7686, 18527}, {9614, 18398}, {9845, 63992}, {9943, 11019}, {10178, 64124}, {10384, 64074}, {10391, 37722}, {10580, 12688}, {11373, 41871}, {12128, 64110}, {12433, 45776}, {12575, 64157}, {14100, 14986}, {15172, 63976}, {16216, 38034}, {16616, 48482}, {17632, 37740}, {19843, 58608}, {34647, 58609}, {34791, 63993}, {37544, 51783}, {37723, 66226}, {63994, 66248}, {64128, 66239}, {64131, 64162}

X(68613) = midpoint of X(i) and X(j) for these {i,j}: {1, 9844}, {938, 9848}
X(68613) = reflection of X(9858) in X(1125) for these (i,j): {9858, 1125}
X(68613) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {496, 12710, 3742}, {14100, 14986, 58567}

X(68614) = X(11)X(65)∩X(90)X(104)

Barycentrics    -(a*(a^8*(b + c) - 2*a^7*(b + c)^2 - (b - c)^4*(b + c)^3*(b^2 + b*c + c^2) + a^4*b*c*(-13*b^3 + 11*b^2*c + 11*b*c^2 - 13*c^3) + a^6*(-2*b^3 + 5*b^2*c + 5*b*c^2 - 2*c^3) + 2*a*(b^4 - b^3*c + b*c^3 - c^4)^2 + 2*a^5*(3*b^4 + 2*b^3*c - 8*b^2*c^2 + 2*b*c^3 + 3*c^4) - 2*a^3*(b - c)^2*(3*b^4 + 4*b^3*c - 3*b^2*c^2 + 4*b*c^3 + 3*c^4) + a^2*(b - c)^2*(2*b^5 + 11*b^4*c + b^3*c^2 + b^2*c^3 + 11*b*c^4 + 2*c^5))) : :
X(68614) = X[1]+X[12691], 5*X[104]-X[64358], X[12247]+X[12758], 5*X[18254]-4*X[31835], 2*X[1125]-X[9946], X[1768]+X[67988], X[15528]-2*X[20418], 2*X[12619]-X[64745], 5*X[3616]-X[9964], X[6154]-3*X[64107], X[6264]+X[46685], X[12665]+X[38669], X[12690]+X[14110], X[13243]+3*X[67998], X[17660]-3*X[66627], X[49176]+X[64139]

See David Nguyen, euclid 8513.

X(68614) lies on these lines: {1, 12691}, {11, 65}, {80, 48482}, {90, 104}, {392, 5882}, {497, 10051}, {517, 66065}, {952, 960}, {1001, 2801}, {1125, 9946}, {1768, 63992}, {2771, 11281}, {2802, 19914}, {2886, 12619}, {3485, 11570}, {3616, 9964}, {3678, 12645}, {5083, 64110}, {5289, 63967}, {5884, 5886}, {6001, 13226}, {6154, 64107}, {6264, 46685}, {6326, 31435}, {12005, 15950}, {12515, 64077}, {12635, 12737}, {12665, 38669}, {12675, 51700}, {12690, 14110}, {12751, 30513}, {13243, 67998}, {13253, 64669}, {15558, 63999}, {17660, 66627}, {18237, 22775}, {19907, 51715}, {22753, 37545}, {23708, 64021}, {28628, 57298}, {31803, 51714}, {33598, 51636}, {38602, 65404}, {48695, 64372}, {49176, 64139}, {54441, 63988}

X(68614) = midpoint of X(i) and X(j) for these {i,j}: {1, 12691}, {1768, 67988}, {6264, 46685}, {12247, 12758}, {12665, 38669}, {12690, 14110}, {49176, 64139}
X(68614) = reflection of X(i) in X(j) for these (i,j): {9946, 1125}, {15528, 20418}, {64745, 12619}

X(68615) = X(1)X(329)∩X(3)X(142)

Barycentrics    -2*a^4 - a^3*(b + c) + a*(b + c)^3 + (b^2 - c^2)^2 + a^2*(b^2 + 10*b*c + c^2) : :
X(68615) = 3*X[1]+X[12527], X[1]+X[12572], X[1]+3*X[40998], X[12527]-3*X[12572], X[12527]-9*X[40998], X[12572]-3*X[40998], 3*X[2]+X[10624], 2*X[1125]-X[12436], X[10]+X[12575], X[72]+3*X[64162], 3*X[392]+X[950], X[960]+3*X[49736], X[960]+X[63999], X[5795]+X[9957], 3*X[15170]+X[34790], 3*X[49736]-X[63999], 3*X[549]-X[16004], 3*X[551]-X[4298], X[1770]-9*X[25055], 3*X[3058]+5*X[25917], 3*X[3058]+X[63146], 5*X[25917]-X[63146], 3*X[3576]+X[66992], 5*X[3616]-X[4292], 7*X[3622]+X[64002], 2*X[3636]-X[12577], 3*X[3877]+X[64163], X[3878]+X[6738], 5*X[3889]+3*X[17781], X[5044]+X[15172], 3*X[5603]+X[64004], X[66210]+3*X[66515], X[6743]-3*X[10176], X[9856]+X[64706], X[10106]+3*X[11113], 3*X[10179]+X[57288], 3*X[10179]-X[66230], X[57288]+X[66230], X[15171]+X[57284]

See David Nguyen, euclid 8513.

X(68615) lies on these lines: {1, 329}, {2, 9614}, {3, 142}, {5, 22801}, {8, 64368}, {9, 1058}, {10, 497}, {21, 44675}, {35, 54348}, {40, 9843}, {55, 6700}, {72, 64162}, {386, 63977}, {390, 936}, {392, 950}, {405, 12053}, {443, 9580}, {496, 5745}, {515, 58679}, {517, 11695}, {518, 40270}, {519, 960}, {522, 32195}, {527, 5045}, {549, 16004}, {551, 1420}, {595, 39595}, {596, 17132}, {758, 6744}, {944, 68000}, {958, 63993}, {975, 63969}, {997, 4314}, {1005, 12047}, {1210, 5250}, {1279, 34937}, {1467, 3671}, {1490, 43175}, {1621, 13411}, {1699, 37421}, {1706, 17559}, {1770, 25055}, {1848, 4222}, {2078, 11813}, {2478, 31397}, {2550, 3646}, {2551, 31393}, {2886, 3634}, {3058, 25917}, {3086, 4512}, {3295, 3452}, {3296, 60933}, {3303, 4679}, {3333, 5698}, {3361, 50836}, {3421, 37556}, {3487, 38316}, {3488, 15829}, {3576, 66992}, {3579, 6692}, {3616, 4292}, {3622, 64002}, {3626, 12640}, {3635, 12635}, {3636, 12577}, {3683, 37722}, {3702, 52369}, {3746, 6745}, {3754, 28228}, {3811, 30331}, {3812, 28194}, {3813, 15254}, {3816, 6684}, {3817, 6848}, {3822, 12571}, {3824, 40273}, {3828, 11235}, {3847, 10172}, {3848, 28232}, {3877, 64163}, {3878, 6738}, {3881, 5572}, {3889, 17781}, {4266, 40963}, {4294, 8583}, {4295, 10582}, {4297, 63992}, {4301, 54318}, {4304, 19861}, {4311, 6872}, {4353, 30148}, {4414, 28018}, {4423, 12701}, {4425, 28029}, {4428, 25681}, {4640, 64124}, {4652, 10586}, {4666, 11415}, {4701, 64768}, {4887, 41874}, {5044, 5853}, {5119, 8582}, {5126, 11281}, {5129, 9623}, {5218, 25522}, {5234, 34625}, {5259, 30384}, {5274, 5705}, {5316, 5687}, {5436, 5603}, {5437, 6361}, {5542, 64674}, {5722, 5837}, {5732, 67999}, {5766, 19843}, {5791, 24386}, {6261, 6930}, {6666, 31419}, {6675, 7743}, {6690, 40259}, {6702, 66065}, {6735, 37162}, {6743, 10176}, {6765, 18228}, {6857, 50443}, {6893, 19925}, {6919, 31434}, {6927, 8227}, {6944, 10171}, {8715, 20103}, {9708, 21627}, {9856, 64706}, {9955, 58463}, {10106, 11113}, {10164, 10200}, {10179, 57288}, {10580, 54422}, {10596, 55104}, {10916, 18249}, {11019, 12514}, {11023, 38399}, {11373, 16418}, {12705, 64705}, {12731, 21631}, {13405, 21616}, {14986, 31424}, {15171, 57284}, {16202, 67855}, {17531, 63145}, {17567, 19862}, {17580, 30332}, {17706, 44663}, {18443, 54198}, {18480, 64109}, {18483, 25466}, {19878, 52264}, {20196, 59591}, {21620, 24703}, {21625, 51090}, {23340, 31806}, {24171, 24248}, {24178, 33095}, {24564, 52367}, {24705, 35620}, {24982, 26127}, {25011, 63136}, {27385, 61155}, {28234, 66257}, {28629, 31162}, {29311, 58469}, {30305, 64673}, {30478, 37704}, {34471, 66229}, {34647, 51103}, {43174, 49163}, {43177, 63962}, {47742, 51787}, {49768, 56949}, {49987, 64071}, {51409, 63274}, {51723, 57282}, {52684, 64699}, {59576, 63147}, {64130, 64679}, {64734, 66251}

X(68615) = midpoint of X(i) and X(j) for these {i,j}: {1, 12572}, {10, 12575}, {960, 63999}, {3878, 6738}, {5044, 15172}, {5795, 9957}, {9856, 64706}, {15171, 57284}, {57288, 66230}
X(68615) = reflection of X(i) in X(j) for these (i,j): {12436, 1125}, {12577, 3636}
X(68615) = complement of X(31788)
X(68615) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 40998, 12572}, {2, 61763, 59675}, {40, 26105, 9843}, {55, 24954, 59587}, {390, 936, 64117}, {497, 31435, 10}, {946, 1001, 1125}, {960, 49736, 63999}, {960, 63999, 519}, {1621, 41012, 13411}, {1697, 5084, 10}, {3058, 25917, 63146}, {3295, 3452, 59722}, {3303, 4679, 21075}, {5044, 15172, 5853}, {5082, 7308, 10}, {5129, 9785, 9623}, {5795, 9957, 519}, {9623, 9785, 64767}, {10179, 57288, 66230}, {14986, 52653, 31424}, {17527, 63990, 3634}, {21625, 51090, 62858}, {24954, 59587, 6700}, {51715, 64110, 3636}, {51785, 66515, 19843}

X(68616) = X(1)X(442)∩X(4)X(518)

Barycentrics    (a - b - c)*(a^3 - 2*a*b*c + (b - c)^2*(b + c)) : :
X(68616) = X[1]-2*X[3813], X[1]+X[12625], X[1]-3*X[24392], 2*X[3813]+X[12625], 2*X[3813]-3*X[24392], X[12625]+3*X[24392], 3*X[2]-X[3189], 3*X[2]-2*X[56176], X[3189]-2*X[56176], X[3]-2*X[10916], 2*X[5]-X[3811], 2*X[10]-X[3913], X[1001]-2*X[24389], X[20]-3*X[24477], X[3632]+X[3680], X[145]-2*X[33895], 2*X[355]-X[32049], 3*X[381]-2*X[21077], 2*X[946]-3*X[11235], 2*X[946]-X[12635], 4*X[946]-3*X[34647], 2*X[1482]-3*X[34640], X[10912]-2*X[21627], 3*X[11235]-X[12635], 2*X[11235]-X[34647], 3*X[11236]-4*X[19925], 2*X[12635]-3*X[34647], 3*X[34700]-2*X[47745], 3*X[34706]-2*X[51118], X[5691]+X[6762], 2*X[10525]-X[64119], X[944]-2*X[11260], X[944]-3*X[34625], 2*X[11260]-3*X[34625], 2*X[1125]-X[12437], 2*X[1125]-3*X[24386], 4*X[1125]-3*X[56177], X[12437]-3*X[24386], 2*X[12437]-3*X[56177], 2*X[24386]-X[56177], 2*X[1385]-3*X[45700], 5*X[1656]-4*X[59719], 5*X[1698]-3*X[3158], 5*X[1698]-4*X[64123], 3*X[3158]-4*X[64123], 3*X[1699]-X[11523], X[2136]-3*X[3679], 5*X[3091]-3*X[25568], X[3174]-2*X[3826], X[3244]-2*X[64205], 5*X[3616]-X[12536], 5*X[3617]-X[12632], 2*X[3626]-X[12640], 4*X[3634]-3*X[59584], 6*X[3829]-5*X[8227], 3*X[3928]-X[64005], 2*X[4297]-3*X[11194], 3*X[4421]-4*X[6684], 3*X[4421]-2*X[64117], 2*X[6684]-X[64117], X[7674]-3*X[38057], X[5534]-2*X[18242], 3*X[5587]-X[6765], 3*X[5587]-2*X[12607], X[6765]-2*X[12607], 3*X[5770]-2*X[64118], 3*X[5790]-2*X[10915], 5*X[5818]-3*X[34619], X[7982]-2*X[13463], X[5881]+X[12629], 3*X[5886]-2*X[22836], 3*X[5886]-4*X[24387], X[22836]-2*X[24387], 2*X[6245]-X[64074], X[6764]+3*X[59387], 5*X[7987]-3*X[34701], 7*X[7989]-3*X[66469], 2*X[8666]-X[18481], 2*X[8715]-3*X[26446], 7*X[9780]-3*X[64146], 4*X[9956]-3*X[45701], 7*X[10005]-2*X[66681]

See David Nguyen, euclid 8513.

X(68616) lies on these lines: {1, 442}, {2, 3189}, {3, 10916}, {4, 518}, {5, 3811}, {8, 210}, {9, 66682}, {10, 1001}, {11, 78}, {12, 3870}, {20, 24477}, {30, 62858}, {37, 26036}, {38, 50065}, {40, 528}, {55, 6734}, {56, 1004}, {58, 66639}, {63, 6284}, {65, 3434}, {72, 1479}, {75, 41874}, {79, 3894}, {80, 3632}, {100, 20118}, {142, 6744}, {145, 3485}, {149, 3869}, {200, 1329}, {218, 21073}, {306, 41506}, {319, 20943}, {321, 36500}, {329, 5225}, {354, 377}, {355, 381}, {387, 1386}, {388, 5175}, {404, 17728}, {430, 5130}, {443, 3742}, {452, 5302}, {495, 31936}, {496, 997}, {499, 5440}, {515, 12513}, {516, 5787}, {517, 48482}, {521, 6238}, {524, 48937}, {527, 34706}, {529, 5691}, {536, 24316}, {674, 10441}, {740, 42440}, {758, 12699}, {908, 10896}, {912, 10525}, {936, 3816}, {938, 2550}, {942, 5880}, {944, 11260}, {950, 958}, {952, 6261}, {956, 10572}, {962, 44663}, {976, 17720}, {993, 16761}, {999, 17647}, {1043, 3705}, {1046, 64016}, {1056, 58609}, {1058, 58679}, {1104, 33137}, {1125, 12437}, {1145, 10073}, {1183, 44842}, {1193, 17721}, {1210, 1376}, {1259, 62333}, {1260, 10395}, {1319, 10529}, {1331, 7299}, {1385, 45700}, {1478, 3555}, {1532, 17857}, {1621, 67960}, {1656, 59719}, {1698, 3158}, {1699, 11523}, {1706, 66252}, {1714, 56178}, {1737, 5687}, {1738, 17054}, {1785, 66235}, {1788, 17784}, {1836, 3868}, {1848, 5090}, {1877, 9370}, {1891, 37385}, {1998, 15844}, {2000, 64349}, {2082, 40997}, {2099, 41575}, {2136, 3679}, {2280, 21029}, {2321, 40963}, {2475, 3873}, {2476, 17718}, {2646, 10527}, {2650, 33104}, {2800, 12700}, {2801, 6259}, {2802, 19914}, {2894, 5832}, {2975, 35989}, {3017, 48824}, {3056, 18178}, {3058, 5250}, {3086, 59691}, {3091, 25568}, {3169, 17275}, {3174, 3826}, {3187, 27052}, {3242, 13161}, {3243, 5290}, {3244, 64110}, {3303, 24987}, {3338, 11112}, {3416, 5015}, {3445, 53618}, {3452, 6743}, {3475, 5177}, {3486, 64081}, {3487, 3838}, {3488, 19843}, {3583, 5904}, {3586, 57279}, {3601, 4999}, {3616, 12536}, {3617, 12632}, {3621, 5176}, {3626, 12640}, {3633, 37710}, {3634, 59584}, {3649, 11520}, {3654, 34745}, {3673, 47595}, {3681, 5046}, {3684, 46835}, {3687, 9555}, {3689, 5552}, {3704, 3886}, {3710, 4387}, {3740, 5084}, {3744, 5230}, {3752, 36574}, {3756, 11512}, {3829, 8227}, {3847, 30827}, {3848, 17582}, {3872, 10950}, {3874, 13159}, {3875, 41003}, {3884, 64734}, {3900, 48090}, {3914, 37549}, {3916, 4302}, {3924, 33136}, {3925, 54392}, {3927, 9668}, {3928, 64005}, {3931, 30903}, {3935, 11681}, {3938, 21935}, {3940, 9669}, {3962, 11415}, {3991, 56746}, {4005, 31018}, {4007, 7323}, {4190, 32636}, {4193, 4420}, {4195, 33121}, {4204, 31330}, {4208, 38053}, {4255, 24239}, {4294, 4640}, {4297, 11194}, {4313, 30478}, {4314, 5745}, {4339, 37642}, {4361, 18589}, {4385, 49688}, {4421, 6684}, {4511, 11376}, {4512, 18253}, {4643, 24424}, {4652, 15338}, {4851, 34830}, {4853, 5727}, {4855, 5433}, {4857, 5692}, {4861, 37740}, {4865, 67976}, {4866, 59414}, {4894, 35634}, {4900, 34918}, {5014, 17751}, {5016, 10371}, {5044, 18527}, {5082, 5836}, {5087, 10591}, {5123, 7080}, {5129, 7674}, {5142, 56316}, {5217, 59491}, {5220, 12572}, {5248, 5791}, {5254, 16973}, {5266, 5292}, {5274, 20007}, {5289, 6737}, {5327, 56018}, {5434, 62832}, {5438, 6691}, {5442, 63752}, {5534, 18242}, {5562, 34372}, {5570, 18223}, {5587, 6765}, {5698, 54398}, {5704, 59572}, {5705, 6690}, {5706, 45728}, {5709, 5842}, {5720, 7681}, {5730, 30384}, {5768, 9943}, {5770, 64118}, {5777, 26333}, {5790, 10915}, {5799, 5846}, {5814, 35104}, {5818, 34619}, {5831, 55100}, {5837, 12575}, {5839, 54008}, {5840, 24467}, {5850, 31672}, {5855, 7982}, {5881, 12629}, {5886, 22836}, {6067, 7675}, {6075, 67439}, {6245, 64074}, {6735, 10965}, {6736, 8168}, {6763, 65134}, {6764, 59387}, {6826, 13374}, {6827, 63976}, {6831, 37569}, {6836, 7957}, {6850, 12675}, {6865, 58637}, {6872, 64153}, {6893, 58631}, {6916, 58567}, {6921, 61649}, {6925, 12680}, {6933, 61648}, {7173, 30852}, {7270, 10453}, {7280, 51636}, {7354, 51463}, {7368, 60444}, {7483, 59337}, {7987, 34701}, {7989, 66469}, {8256, 63137}, {8666, 12773}, {8715, 26446}, {8728, 64443}, {8818, 53426}, {9052, 15488}, {9579, 62823}, {9580, 12526}, {9612, 41863}, {9623, 66257}, {9640, 33178}, {9709, 52804}, {9710, 37723}, {9780, 64146}, {9799, 15726}, {9858, 58623}, {9956, 45701}, {10005, 66681}, {10043, 17648}, {10057, 25416}, {10072, 17614}, {10175, 59722}, {10248, 64143}, {10265, 13205}, {10306, 12616}, {10366, 68352}, {10448, 29690}, {10573, 10914}, {10588, 63168}, {10589, 27383}, {10609, 12750}, {10785, 50371}, {10826, 17757}, {10895, 41711}, {10944, 36846}, {10947, 64042}, {11019, 25524}, {11038, 37161}, {11113, 41229}, {11224, 64291}, {11238, 41012}, {11240, 20323}, {11269, 37539}, {11319, 33114}, {11373, 30144}, {11374, 25639}, {11375, 11680}, {11495, 64706}, {11496, 51755}, {11510, 51432}, {11519, 37712}, {11604, 43741}, {11679, 30847}, {11682, 64367}, {11826, 63399}, {12116, 14110}, {12164, 64875}, {12329, 37415}, {12433, 31419}, {12444, 36867}, {12512, 34626}, {12514, 15171}, {12528, 12679}, {12559, 39542}, {12609, 15934}, {12619, 25438}, {12641, 43731}, {12678, 37437}, {12704, 37468}, {12764, 46685}, {12953, 64002}, {13274, 64139}, {13407, 17532}, {13731, 15624}, {13740, 38047}, {14507, 67343}, {14923, 41687}, {15680, 62827}, {15829, 51785}, {15908, 18446}, {15954, 34036}, {16062, 19852}, {16465, 64086}, {16610, 28074}, {17151, 41010}, {17158, 56928}, {17276, 24851}, {17299, 17444}, {17528, 51706}, {17559, 58451}, {17597, 23536}, {17626, 58585}, {17676, 46909}, {17677, 47358}, {17697, 33118}, {17723, 19767}, {17768, 28646}, {18236, 58657}, {18250, 24393}, {18393, 41696}, {18395, 48696}, {18483, 67850}, {19589, 29674}, {19765, 29639}, {19785, 36579}, {19861, 37722}, {20018, 33071}, {20070, 34744}, {20075, 37568}, {21285, 30617}, {21620, 42871}, {21674, 62849}, {22837, 37727}, {23675, 49989}, {24174, 53619}, {24247, 40133}, {24468, 54302}, {24474, 37820}, {24789, 28082}, {24929, 26363}, {24953, 62829}, {24982, 61717}, {25005, 54348}, {25359, 41312}, {26286, 48713}, {26470, 37533}, {27368, 27553}, {28044, 46878}, {28234, 64767}, {29664, 64415}, {30513, 43745}, {31136, 50046}, {31394, 64753}, {31445, 31795}, {31777, 64129}, {32537, 59388}, {32777, 36568}, {33120, 54331}, {33133, 36565}, {33142, 62802}, {33838, 38186}, {34379, 48938}, {34606, 63135}, {34699, 45081}, {34719, 37563}, {34720, 63142}, {34742, 63984}, {34860, 58371}, {34862, 64076}, {35258, 63273}, {36974, 50625}, {36999, 64003}, {37108, 65426}, {37162, 63961}, {37359, 56278}, {37423, 65405}, {37435, 64151}, {37531, 63980}, {37552, 37646}, {37592, 48837}, {37717, 50581}, {37721, 64200}, {37738, 38460}, {37822, 63967}, {37826, 68366}, {38200, 64674}, {38211, 66200}, {40663, 63130}, {42819, 66215}, {43734, 56089}, {43740, 46354}, {48827, 61661}, {49169, 64335}, {49329, 49592}, {49330, 49593}, {49745, 62819}, {50736, 51099}, {51638, 67974}, {52362, 66724}, {53616, 64056}, {54286, 67980}, {54421, 63979}, {56090, 67330}, {56814, 64069}, {59416, 64199}, {59592, 59779}, {59598, 62297}, {60691, 67963}, {63143, 64202}, {64261, 68036}

X(68616) = midpoint of X(i) and X(j) for these {i,j}: {1, 12625}, {8, 64068}, {3632, 3680}, {5691, 6762}, {5881, 12629}, {12541, 67959}, {21627, 66251}, {41869, 54422}, {64261, 68036}
X(68616) = reflection of X(i) in X(j) for these (i,j): {1, 3813}, {3, 10916}, {145, 33895}, {944, 11260}, {1001, 24389}, {1482, 49600}, {3174, 3826}, {3189, 56176}, {3244, 64205}, {3811, 5}, {3913, 10}, {5534, 18242}, {6765, 12607}, {7982, 13463}, {10306, 12616}, {10912, 21627}, {12437, 1125}, {12635, 946}, {12640, 3626}, {13205, 10265}, {18481, 8666}, {22836, 24387}, {25438, 12619}, {28646, 54422}, {32049, 355}, {34647, 11235}, {37531, 63980}, {37727, 22837}, {56177, 24386}, {64074, 6245}, {64076, 34862}, {64117, 6684}, {64119, 10525}, {64744, 8}, {66215, 42819}, {66240, 47745}, {67850, 18483} X(68616) = anticomplement of X(56176)
X(68616) = complement of X(3189)
X(68616) = barycentric product X(i)*X(j) for these {i,j}: {8, 24789}, {312, 28082}
X(68616) = barycentric quotient X(i)/X(j) for these {i,j}: {24789, 7}, {28082, 57}
X(68616) = trilinear product X(i)*X(j) for these {i,j}: {8, 28082}, {9, 24789}
X(68616) = trilinear quotient X(i)/X(j) for these {i,j}: {56, 28082}, {57, 24789}
X(68616) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2886, 28628}, {1, 3419, 5794}, {1, 12625, 44669}, {1, 24392, 3813}, {2, 3189, 56176}, {8, 497, 960}, {8, 2478, 210}, {8, 2551, 4662}, {8, 3701, 30615}, {8, 12541, 67959}, {8, 36926, 44720}, {8, 64068, 3880}, {10, 63999, 1001}, {11, 78, 25681}, {55, 6734, 26066}, {72, 1479, 24703}, {80, 3632, 64087}, {145, 5086, 5252}, {149, 3869, 12701}, {200, 9581, 1329}, {388, 36845, 34791}, {938, 2550, 3812}, {944, 34625, 11260}, {946, 12635, 34647}, {950, 4847, 958}, {1125, 12437, 56177}, {1210, 63146, 1376}, {1482, 49600, 34640}, {1698, 3158, 64123}, {1737, 5687, 37828}, {1837, 4863, 8}, {2475, 3873, 10404}, {2894, 62864, 5832}, {3058, 21677, 5250}, {3242, 66104, 13161}, {3434, 12649, 65}, {3487, 31418, 3838}, {3583, 5904, 58798}, {3586, 57279, 57288}, {3601, 5231, 4999}, {3632, 3680, 5854}, {3689, 17606, 5552}, {3868, 52367, 1836}, {3940, 9669, 21616}, {5015, 10449, 3416}, {5016, 17135, 10371}, {5082, 18391, 5836}, {5175, 36845, 388}, {5587, 6765, 12607}, {5691, 6762, 529}, {5881, 12629, 38455}, {6598, 12625, 3419}, {6684, 64117, 4421}, {6737, 12053, 5289}, {7080, 54361, 5123}, {7354, 51463, 62874}, {11019, 57284, 25524}, {11235, 12635, 946}, {11680, 34772, 11375}, {12433, 31419, 54318}, {12437, 24386, 1125}, {12541, 67959, 3880}, {12625, 24392, 1}, {17647, 49627, 999}, {21627, 66251, 519}, {22836, 24387, 5886}, {26015, 57287, 56}, {34700, 66240, 47745}, {41869, 54422, 17768}, {64068, 67959, 12541}

X(68617) = X(56)X(84)∩X(496)X(942)

Barycentrics    a*(-(a^8*(b + c)) + 2*a^6*(b - c)^2*(b + c) + 8*a^4*b*(b - c)^2*c*(b + c) + (b - c)^4*(b + c)^5 + 2*a^7*(b^2 + b*c + c^2) - 2*a^5*(b - c)^2*(3*b^2 + 7*b*c + 3*c^2) - 2*a*(b^2 - c^2)^2*(b^4 - b^3*c + 4*b^2*c^2 - b*c^3 + c^4) + 2*a^3*(b - c)^2*(3*b^4 + 5*b^3*c + 5*b*c^3 + 3*c^4) - 2*a^2*(b - c)^2*(b^5 + 5*b^4*c + 2*b^3*c^2 + 2*b^2*c^3 + 5*b*c^4 + c^5)) : :
X(68617) = X[1]+X[12664], X[40]-2*X[58660], X[84]+X[12688], X[942]-2*X[63980], 3*X[5777]-2*X[32159], X[5777]-2*X[68005], X[32159]-3*X[68005], X[1071]-2*X[58588], 2*X[1125]-X[9942], 2*X[12616]-X[31788], 3*X[3576]-X[12671], 5*X[3616]-X[9960], 3*X[3742]-2*X[40249], 2*X[6705]-X[9943], 2*X[5044]-X[11500], X[5787]+X[5887], 3*X[5927]-X[12667], X[9799]+3*X[67998], 3*X[10157]-2*X[18242], X[10864]+X[18239], X[10864]+3*X[61705], X[18239]-3*X[61705], X[12650]+X[14872], X[14110]+X[64261], 5*X[25917]-3*X[52026], X[52359]-2*X[64818]

See David Nguyen, euclid 8513.

X(68617) lies on these lines: {1, 12664}, {3, 18251}, {4, 3427}, {11, 67996}, {19, 46022}, {40, 58660}, {56, 84}, {204, 16466}, {388, 12677}, {496, 942}, {497, 12672}, {515, 960}, {517, 48482}, {971, 1001}, {999, 49170}, {1012, 66248}, {1071, 3485}, {1125, 9942}, {1158, 37623}, {1490, 13615}, {1709, 8071}, {2800, 66065}, {2886, 12616}, {3304, 12687}, {3338, 7992}, {3576, 12671}, {3616, 9960}, {3660, 10785}, {3742, 40249}, {4999, 6705}, {5044, 11500}, {5173, 67919}, {5450, 65404}, {5572, 13464}, {5715, 67966}, {5768, 12709}, {5787, 5887}, {5842, 31793}, {5909, 44662}, {5927, 12667}, {6260, 25466}, {6831, 50195}, {6847, 12711}, {7686, 64157}, {7742, 63988}, {7971, 64320}, {9614, 64045}, {9799, 67998}, {9940, 28628}, {9947, 64335}, {10157, 18242}, {10864, 18239}, {11011, 64281}, {11108, 64328}, {11247, 30199}, {11281, 18238}, {12136, 40985}, {12330, 12705}, {12448, 28234}, {12650, 14872}, {12675, 64110}, {12676, 64130}, {12680, 34471}, {12684, 16203}, {13373, 65998}, {14110, 64171}, {15733, 37531}, {17646, 64129}, {18519, 61147}, {19904, 22654}, {21370, 63435}, {25917, 52026}, {26286, 34862}, {31806, 34790}, {31837, 40659}, {40257, 51715}, {45776, 63999}, {51706, 54227}, {52359, 64818}, {61663, 64754}, {63210, 64288}, {64265, 64721}, {67986, 67987}

X(68617) = midpoint of X(i) and X(j) for these {i,j}: {1, 12664}, {84, 12688}, {5787, 5887}, {10864, 18239}, {12650, 14872}, {14110, 64261}
X(68617) = reflection of X(i) in X(j) for these (i,j): {40, 58660}, {942, 63980}, {1071, 58588}, {5777, 68005}, {9942, 1125}, {9943, 6705}, {11500, 5044}, {31788, 12616}, {52359, 64818}
X(68617) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {84, 63992, 18237}, {10864, 61705, 18239}

X(68618 = X(1)X(12692)∩X(210)X(7160)

Barycentrics    -(a*(a^8*(b + c) - (b - c)^4*(b + c)^5 - 2*a^7*(b^2 + b*c + c^2) + 2*a*(b - c)^2*(b + c)^4*(b^2 + 5*b*c + c^2) - 2*a^6*(b^3 + 7*b^2*c + 7*b*c^2 + c^3) + 8*a^4*b*c*(3*b^3 + 7*b^2*c + 7*b*c^2 + 3*c^3) + 2*a^2*(b + c)^3*(b^4 - 8*b^3*c - 2*b^2*c^2 - 8*b*c^3 + c^4) + 2*a^5*(3*b^4 + 9*b^3*c + 8*b^2*c^2 + 9*b*c^3 + 3*c^4) - 2*a^3*(3*b^6 + 15*b^5*c + 17*b^4*c^2 - 6*b^3*c^3 + 17*b^2*c^4 + 15*b*c^5 + 3*c^6))) : :
X(68618) = X[1]+X[12692], X[72]+X[12777], 3*X[210]-X[7160], X[12599]-2*X[58631], 2*X[5044]-X[12260], 2*X[1125]-X[12439], X[12120]+X[14872], 5*X[3616]-X[12537], 3*X[3681]+X[9874], X[49110]-2*X[58630]

See David Nguyen, euclid 8513.

X(68618) lies on these lines: {1, 12692}, {72, 12777}, {200, 12333}, {210, 7160}, {497, 5920}, {517, 22801}, {518, 12864}, {946, 10157}, {960, 18255}, {1001, 3811}, {1125, 12439}, {1490, 3428}, {2886, 12620}, {3059, 61122}, {3485, 12854}, {3616, 12537}, {3681, 9874}, {11281, 18241}, {12329, 12411}, {12516, 64077}, {12658, 31435}, {12842, 63992}, {12853, 64110}, {12857, 17658}, {12859, 41538}, {49110, 58630}

X(68618) = reflection of X(i) in X(j) for these (i,j): {12260, 5044}, {12439, 1125}, {12599, 58631}, {49110, 58630}

X(68619) = X(1)X(329)∩X(5)X(10)

Barycentrics    3*a^3*(b + c) - (b^2 - c^2)^2 + a^2*(b^2 - 6*b*c + c^2) - a*(3*b^3 + b^2*c + b*c^2 + 3*c^3) : :
X(68619) = X[1]+X[329], 3*X[2]-X[2093], X[10]-2*X[3452], 3*X[10]-4*X[3820], 3*X[3452]-2*X[3820], 3*X[5603]-X[68032], X[57]-2*X[1125], X[72]+X[17642], X[200]+X[30305], X[43182]-2*X[54178], 3*X[392]-X[64106], X[3421]+X[7962], X[3421]-3*X[31142], X[4342]+X[21060], X[7962]+3*X[31142], X[5289]+X[24703], 3*X[551]-2*X[999], X[962]+X[7994], X[2094]-3*X[25055], X[2095]-3*X[5886], X[2096]-3*X[3576], X[3062]+X[54179], X[3244]-2*X[64897], 2*X[3359]-3*X[10164], 5*X[3616]-X[9965], 7*X[3622]+X[20214], 7*X[3624]-5*X[62773], 4*X[3634]-5*X[20196], 2*X[3678]-X[17658], X[3874]-2*X[12915], X[4297]-2*X[37611], X[5493]-2*X[6244], 4*X[6692]-5*X[19862], 2*X[6692]-X[36279], 5*X[19862]-2*X[36279], 2*X[8257]-3*X[38059], 3*X[11230]-2*X[61535], X[12848]-3*X[66515], X[60905]+X[60956], 4*X[18516]-3*X[34648], 2*X[20103]-X[54286], 4*X[35238]-3*X[50808], 3*X[38037]-X[54159], 3*X[38054]-2*X[61022], X[54135]-2*X[64699], X[64111]+X[68001]

See David Nguyen, euclid 8513.

X(68619) lies on these lines: {1, 329}, {2, 2093}, {3, 54198}, {4, 15829}, {5, 10}, {8, 9614}, {9, 5603}, {11, 31165}, {36, 54348}, {40, 6700}, {57, 1125}, {63, 44675}, {65, 9843}, {72, 12053}, {78, 10624}, {142, 39542}, {200, 30305}, {214, 38759}, {226, 392}, {381, 64734}, {405, 64160}, {495, 66465}, {496, 24391}, {497, 519}, {515, 5289}, {516, 997}, {518, 63993}, {527, 551}, {758, 11019}, {908, 3877}, {936, 962}, {950, 5730}, {958, 13464}, {995, 3663}, {1012, 61002}, {1056, 28609}, {1058, 11523}, {1191, 34937}, {1210, 3869}, {1376, 28194}, {1387, 60942}, {1457, 64708}, {1479, 6737}, {1482, 5795}, {1537, 64107}, {1697, 59722}, {1737, 3899}, {1788, 25522}, {2094, 11551}, {2095, 5745}, {2096, 3576}, {2099, 4679}, {2478, 11682}, {2550, 31162}, {2551, 7982}, {2810, 49505}, {2823, 21629}, {2835, 18589}, {3057, 21075}, {3062, 54179}, {3086, 12526}, {3244, 12635}, {3340, 5084}, {3359, 6954}, {3528, 45036}, {3577, 6939}, {3616, 9965}, {3622, 20214}, {3624, 62773}, {3634, 20196}, {3646, 28629}, {3647, 11281}, {3656, 9708}, {3678, 17658}, {3683, 15950}, {3753, 5316}, {3811, 10388}, {3816, 44663}, {3872, 31018}, {3874, 12915}, {3884, 21077}, {3892, 5572}, {3927, 11373}, {3940, 5853}, {3951, 10529}, {3962, 37722}, {4004, 17575}, {4067, 49627}, {4134, 18254}, {4187, 4848}, {4292, 11415}, {4295, 8583}, {4297, 6261}, {4304, 4511}, {4311, 20067}, {4314, 22836}, {4323, 5129}, {4533, 64200}, {4640, 10165}, {4662, 13463}, {4669, 11235}, {4847, 5692}, {4862, 46943}, {4896, 41874}, {5048, 34606}, {5119, 6745}, {5128, 17567}, {5223, 5825}, {5250, 13411}, {5267, 10269}, {5325, 51709}, {5328, 59417}, {5438, 6361}, {5493, 6244}, {5657, 30827}, {5697, 6736}, {5705, 68034}, {5731, 41561}, {5748, 31434}, {5785, 54206}, {5791, 18493}, {5794, 18483}, {5811, 12650}, {5815, 12629}, {5822, 40963}, {5850, 36973}, {5882, 57288}, {5887, 6245}, {5901, 31445}, {5903, 8582}, {5924, 6987}, {6001, 64705}, {6147, 51723}, {6260, 31786}, {6684, 25681}, {6692, 19862}, {6735, 27131}, {6744, 12559}, {6765, 9785}, {6865, 7971}, {6872, 56387}, {6907, 64315}, {6921, 63144}, {6926, 54156}, {7290, 60751}, {7688, 50908}, {7743, 24386}, {8257, 38059}, {9623, 18228}, {9624, 30478}, {9819, 34619}, {10106, 58798}, {11111, 13384}, {11230, 61535}, {11522, 19843}, {11529, 26105}, {12437, 15171}, {12513, 64703}, {12563, 64675}, {12699, 57284}, {12701, 63146}, {12702, 63990}, {12848, 66515}, {13462, 60905}, {14110, 63989}, {14986, 54422}, {15507, 24705}, {15863, 34641}, {17527, 50193}, {17647, 51118}, {17781, 54391}, {18237, 63983}, {18249, 26363}, {18250, 64669}, {18446, 43175}, {18516, 34648}, {18530, 36867}, {20103, 28228}, {21214, 24171}, {21620, 58679}, {21627, 34790}, {24177, 49997}, {24477, 37704}, {24954, 37567}, {25568, 31393}, {26364, 43174}, {26792, 38460}, {27383, 61763}, {30115, 63969}, {30323, 66205}, {30946, 67267}, {31730, 59691}, {33099, 47623}, {34371, 49511}, {35238, 50808}, {35262, 44447}, {35645, 50608}, {37568, 59587}, {38037, 54159}, {38054, 61022}, {38155, 64335}, {41010, 53598}, {41570, 47357}, {42884, 61003}, {46694, 64138}, {49736, 51071}, {50243, 51111}, {50836, 53054}, {51616, 59645}, {53020, 64017}, {53618, 62865}, {54135, 64699}, {60979, 62873}, {63391, 66992}, {63986, 64004}, {64045, 64658}, {64111, 68001}, {64192, 68298}

X(68619) = midpoint of X(i) and X(j) for these {i,j}: {1, 329}, {72, 17642}, {200, 30305}, {962, 7994}, {3062, 54179}, {3421, 7962}, {4342, 21060}, {5289, 24703}, {60905, 60956}, {64111, 68001}
X(68621) = reflection of X(i) in X(j) for these (i,j): {10, 3452}, {57, 1125}, {3244, 64897}, {3874, 12915}, {4297, 37611}, {5493, 6244}, {17658, 3678}, {36279, 6692}, {43182, 54178}, {54135, 64699}, {54286, 20103}
X(68619) = complement of X(2093)
X(68619) = Wasat-isogonal conjugate of X(64333)
X(68619) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 5837, 10}, {10, 11813, 3817}, {40, 6700, 59675}, {78, 10624, 64117}, {392, 51409, 226}, {551, 51090, 993}, {908, 3877, 31397}, {946, 960, 10}, {1001, 34647, 64110}, {1001, 64110, 551}, {1329, 11362, 10}, {2478, 11682, 64163}, {3421, 7962, 519}, {3485, 31435, 1125}, {3869, 41012, 1210}, {3878, 21616, 10}, {4295, 8583, 12436}, {4342, 21060, 519}, {5123, 38127, 10}, {5289, 24703, 515}, {5692, 30384, 4847}, {7962, 31142, 3421}, {11415, 19861, 4292}, {12635, 63999, 3244}

X(68620) = X(1)X(12695)∩X(21)X(60)

Barycentrics    -(a*(a - b - c)*(a^8 - a^7*(b + c) + b*(b - c)^4*c*(b + c)^2 - 3*a^6*(b^2 + c^2) + a^5*(3*b^3 - 2*b^2*c - 2*b*c^2 + 3*c^3) + a^4*(3*b^4 + b^3*c + 3*b^2*c^2 + b*c^3 + 3*c^4) + a^3*(-3*b^5 + 7*b^4*c + 7*b*c^4 - 3*c^5) + a*(b - c)^2*(b^5 - 2*b^4*c - 4*b^3*c^2 - 4*b^2*c^3 - 2*b*c^4 + c^5) - a^2*(b^6 + 2*b^5*c - b^4*c^2 - 6*b^3*c^3 - b^2*c^4 + 2*b*c^5 + c^6))) : :
X(68620) = X[1]+X[12695], 2*X[1125]-X[12444], 5*X[3616]-X[12540]

See David Nguyen, euclid 8513.

X(68620) lies on these lines: {1, 12695}, {21, 60}, {497, 12877}, {946, 12600}, {993, 16761}, {1001, 12267}, {1125, 12444}, {2886, 12623}, {3485, 12913}, {3616, 12540}, {5258, 44669}, {6597, 15446}, {6906, 54145}, {10572, 37286}, {11281, 18244}, {12519, 64077}, {12660, 31435}, {12845, 63992}, {12909, 64110}, {18237, 22782}, {28453, 34647}, {46816, 57002}

X(68620) = midpoint of X(1) and X(12695)

X(68621) = X(11)X(938)∩X(153)X(497)

Barycentrics    2*a^7 - 6*a^6*(b + c) + (b - c)^4*(b + c)^3 + 3*a*(b^2 - c^2)^2*(b^2 + c^2) - a^5*(b^2 - 12*b*c + c^2) - 8*a^2*(b - c)^2*(b^3 + c^3) + a^4*(13*b^3 - 9*b^2*c - 9*b*c^2 + 13*c^3) - 4*a^3*(b^4 + 3*b^3*c - 6*b^2*c^2 + 3*b*c^3 + c^4) : :
X(68621) = X[1]+X[13257], 2*X[10]-3*X[38758], 3*X[11]-X[9803], 5*X[11]-7*X[68034], X[40]-2*X[35023], 5*X[100]-X[20070], 2*X[119]-X[3036], 5*X[119]-3*X[5790], 3*X[119]-X[19914], X[119]+X[48667], 5*X[3036]-6*X[5790], 3*X[3036]-2*X[19914], X[3036]+2*X[48667], 9*X[5790]-5*X[19914], 3*X[5790]+5*X[48667], X[19914]+3*X[48667], X[153]+X[1317], 2*X[214]-X[38759], X[1537]+X[6326], 3*X[1537]-X[14217], 5*X[1537]-3*X[31162], X[1537]-3*X[50908], 3*X[6326]+X[14217], 5*X[6326]+3*X[31162], X[6326]+3*X[50908], 5*X[14217]-9*X[31162], X[14217]-9*X[50908], X[31162]-5*X[50908], 2*X[946]-X[66065], 2*X[12611]-X[65948], 2*X[21635]-X[38757], 5*X[3035]-4*X[6684], 3*X[3035]-2*X[64193], 3*X[3035]-4*X[68278], 6*X[6684]-5*X[64193], 3*X[6684]-5*X[68278], X[64193]-2*X[68278], X[962]+X[6154], 2*X[1125]-X[13226], X[1145]-3*X[5660], X[1145]+X[13253], 3*X[5660]+X[13253], 3*X[1699]-X[12690], X[1768]-3*X[34123], 2*X[11729]-X[20418], 3*X[6265]+X[16128], 5*X[6265]-X[18481], 5*X[16128]+3*X[18481], 5*X[3616]-X[13243], X[10698]+X[37725], 3*X[10698]+X[66008], 3*X[37725]-X[66008], 3*X[6174]-X[64189], X[6224]+X[52836], 2*X[6702]-X[9952], 3*X[9812]+X[9963], X[9964]+3*X[67998], 2*X[10265]-3*X[45310], X[10609]+X[34789], X[12019]-2*X[67876], X[12528]+X[27778], X[12736]-2*X[58613], X[12738]+X[64138], X[12767]-5*X[64012], X[12767]-3*X[66628], 5*X[64012]-3*X[66628], 5*X[15017]-3*X[34122], 5*X[19907]-3*X[50824], 3*X[38038]-X[49176], 3*X[50843]-X[64145], X[66059]-3*X[66627]

See David Nguyen, euclid 8513.

X(68621) lies on these lines: {1, 13257}, {10, 38758}, {11, 938}, {40, 35023}, {100, 20070}, {104, 1001}, {119, 2886}, {153, 497}, {214, 38759}, {516, 9945}, {517, 66051}, {519, 1538}, {528, 1537}, {546, 946}, {960, 2800}, {962, 6154}, {1125, 13226}, {1145, 5660}, {1387, 2801}, {1519, 44669}, {1532, 5855}, {1699, 12690}, {1768, 31435}, {1848, 12138}, {2771, 11281}, {2829, 6259}, {3616, 13243}, {3817, 14563}, {4999, 5887}, {5603, 42356}, {5854, 10698}, {5886, 17051}, {5901, 31803}, {6001, 9946}, {6174, 64189}, {6224, 52836}, {6264, 64669}, {6667, 28628}, {6691, 64021}, {6702, 9952}, {6953, 64963}, {7956, 62822}, {7971, 25681}, {7972, 9614}, {9812, 9963}, {9964, 15950}, {10265, 45310}, {10609, 34789}, {10703, 52659}, {10711, 11235}, {10742, 48482}, {11715, 51715}, {11734, 36949}, {12019, 67876}, {12528, 27778}, {12679, 56387}, {12735, 63999}, {12736, 58613}, {12738, 12858}, {12739, 68296}, {12740, 12831}, {12767, 64012}, {13463, 17857}, {15017, 30315}, {15507, 53292}, {18237, 48695}, {19907, 49736}, {34586, 59458}, {35272, 60896}, {38038, 49176}, {40257, 57288}, {50843, 64145}, {54198, 59691}, {61275, 64197}, {66059, 66627}

X(68621) = midpoint of X(i) and X(j) for these {i,j}: {1, 13257}, {119, 48667}, {153, 1317}, {962, 6154}, {1145, 13253}, {1537, 6326}, {6224, 52836}, {10609, 34789}, {10698, 37725}, {12528, 27778}, {12738, 64138}
X(68621) = reflection of X(i) in X(j) for these (i,j): {40, 35023}, {3036, 119}, {9952, 6702}, {12019, 67876}, {12736, 58613}, {13226, 1125}, {20418, 11729}, {38757, 21635}, {38759, 214}, {64193, 68278}, {65948, 12611}, {66065, 946}
X(68621) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1537, 6326, 528}, {5660, 13253, 1145}, {6326, 50908, 1537}, {10698, 37725, 5854}, {12767, 64012, 66628}, {64193, 68278, 3035}

X(68622) = X(9)X(6574)∩X(1743)X(63992)

Barycentrics    -(a*(a^8*(b + c) - 2*a^7*(b^2 + c^2) - (b - c)^2*(b + c)^5*(b^2 - b*c + c^2) + a^4*b*c*(-19*b^3 + 43*b^2*c + 43*b*c^2 - 19*c^3) + a^6*(-2*b^3 + 11*b^2*c + 11*b*c^2 - 2*c^3) + 2*a*(b^2 - c^2)^2*(b^4 - 4*b^3*c - 2*b^2*c^2 - 4*b*c^3 + c^4) + a^5*(6*b^4 - 44*b^2*c^2 + 6*c^4) - 2*a^3*(3*b^6 - 4*b^5*c + 13*b^4*c^2 + 8*b^3*c^3 + 13*b^2*c^4 - 4*b*c^5 + 3*c^6) + a^2*(2*b^7 + 9*b^6*c - 7*b^5*c^2 + 12*b^4*c^3 + 12*b^3*c^4 - 7*b^2*c^5 + 9*b*c^6 + 2*c^7))) : :

See David Nguyen, euclid 8513.

X(68622) lies on these lines: {9, 6574}, {1743, 63992}

X(68623) = X(2)X(11)∩X(960)X(2809)

Barycentrics    2*a^5 - 6*a^4*(b + c) + (b - c)^2*(b + c)^3 + 5*a^3*(b^2 + c^2) + a^2*(-3*b^3 + 7*b^2*c + 7*b*c^2 - 3*c^3) + a*(b^4 - 6*b^3*c + 2*b^2*c^2 - 6*b*c^3 + c^4) : :
X(68623) = 2*X[1125]-X[52826]

See David Nguyen, euclid 8513.

X(68623) lies on these lines: {2, 11}, {529, 45765}, {946, 28915}, {960, 2809}, {1125, 52826}, {1292, 64077}, {1358, 3485}, {2795, 11281}, {3716, 6084}, {4009, 50744}, {5540, 31435}, {8055, 25568}, {8834, 10699}, {10743, 48482}, {11716, 51715}, {24392, 60846}, {25525, 62221}, {34647, 50913}, {35652, 59732}

X(68623) = midpoint of X(3021) and X(20344)
X(68623) = reflection of X(52826) in X(1125)
X(68623) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3021, 20344, 528}, {8299, 62674, 3035}

X(68624) = X(4)X(512)∩X(264)X(523)

Barycentrics    b^2*(b^2 - c^2)*c^2*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^4 + b^4 - a^2*c^2 - b^2*c^2)*(-a^4 + a^2*b^2 + b^2*c^2 - c^4) : :

X(68624) lies on the Lemoine asymtotic hyperbola, the Huygens hyperbola, and these lines: {4, 512}, {98, 1300}, {107, 53327}, {225, 66928}, {264, 523}, {290, 35142}, {324, 62663}, {393, 2395}, {685, 10412}, {691, 22456}, {847, 34347}, {850, 55972}, {878, 8884}, {882, 16230}, {1093, 58757}, {1105, 65403}, {1217, 53173}, {1826, 4079}, {2052, 8029}, {2422, 6531}, {2501, 52631}, {4230, 53266}, {4705, 41013}, {5466, 46106}, {6344, 15475}, {8371, 52147}, {8723, 37124}, {8741, 58869}, {8742, 58870}, {9178, 16081}, {10278, 15466}, {18105, 23290}, {18829, 65272}, {18848, 46005}, {18850, 64788}, {32696, 41173}, {36120, 68571}, {41079, 45031}, {41515, 58825}, {41516, 58827}, {43187, 60054}, {46001, 68566}, {53245, 55121}, {60028, 60199}

X(68624) = polar conjugate of X(2421)
X(68624) = isotomic conjugate of the isogonal conjugate of X(53149)
X(68624) = polar conjugate of the isotomic conjugate of X(43665)
X(68624) = polar conjugate of the isogonal conjugate of X(2395)
X(68624) = X(22456)-Ceva conjugate of X(16081)
X(68624) = X(i)-cross conjugate of X(j) for these (i,j): {2395, 43665}, {17994, 2501}, {51441, 6531}
X(68624) = X(i)-isoconjugate of X(j) for these (i,j): {3, 23997}, {48, 2421}, {63, 14966}, {163, 36212}, {237, 4592}, {255, 4230}, {293, 68178}, {511, 4575}, {577, 62720}, {662, 3289}, {684, 1101}, {877, 52430}, {906, 17209}, {1102, 34859}, {1755, 4558}, {1959, 32661}, {2396, 9247}, {2491, 62719}, {4556, 42702}, {4563, 9417}, {6333, 23995}, {6507, 58070}, {9418, 55202}, {17932, 42075}, {23996, 43754}, {24041, 39469}, {32656, 51369}, {32676, 51386}, {36084, 65748}, {36134, 44716}
X(68624) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 36212}, {132, 68178}, {136, 511}, {137, 44716}, {523, 684}, {1084, 3289}, {1249, 2421}, {3005, 39469}, {3162, 14966}, {5139, 237}, {5190, 17209}, {6523, 4230}, {15526, 51386}, {18314, 6333}, {36103, 23997}, {36899, 4558}, {36901, 6393}, {38970, 36790}, {38987, 65748}, {39058, 4563}, {42426, 42743}, {48317, 9155}, {55152, 47406}, {56788, 52144}, {62562, 3}, {62576, 2396}, {62595, 15631}
X(68624) = cevapoint of X(i) and X(j) for these (i,j): {2395, 53149}, {2501, 17994}
X(68624) = crosspoint of X(16081) and X(22456)
X(68624) = crosssum of X(i) and X(j) for these (i,j): {684, 44716}, {3289, 39469}
X(68624) = trilinear pole of line {2501, 2970}
X(68624) = crossdifference of every pair of points on line {3289, 46094}
X(68624) = barycentric product X(i)*X(j) for these {i,j}: {4, 43665}, {76, 53149}, {98, 14618}, {115, 22456}, {264, 2395}, {275, 61196}, {287, 66299}, {290, 2501}, {338, 685}, {339, 20031}, {512, 60199}, {523, 16081}, {850, 6531}, {878, 18027}, {879, 2052}, {1093, 53173}, {1577, 36120}, {1821, 24006}, {2422, 18022}, {2489, 18024}, {2966, 2970}, {3124, 65272}, {6331, 51441}, {6528, 51404}, {8029, 41174}, {8754, 43187}, {14223, 52491}, {14265, 60338}, {16230, 34536}, {17994, 57541}, {18808, 60869}, {18817, 60777}, {23105, 60179}, {23962, 32696}, {23994, 36104}, {43673, 52641}, {44173, 57260}, {46105, 52076}, {46111, 52038}, {53245, 66300}, {57799, 58757}, {63746, 68572}
X(68624) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 2421}, {19, 23997}, {25, 14966}, {98, 4558}, {115, 684}, {158, 62720}, {232, 68178}, {264, 2396}, {290, 4563}, {297, 15631}, {338, 6333}, {393, 4230}, {512, 3289}, {523, 36212}, {525, 51386}, {685, 249}, {850, 6393}, {878, 577}, {879, 394}, {1821, 4592}, {1910, 4575}, {1976, 32661}, {2052, 877}, {2395, 3}, {2422, 184}, {2489, 237}, {2501, 511}, {2715, 47390}, {2970, 2799}, {2971, 2491}, {3124, 39469}, {3569, 65748}, {4705, 42702}, {6103, 42743}, {6344, 66075}, {6524, 58070}, {6531, 110}, {7649, 17209}, {8029, 41172}, {8754, 3569}, {9154, 65321}, {12077, 44716}, {14273, 9155}, {14618, 325}, {15422, 19189}, {15630, 3049}, {16081, 99}, {16229, 56437}, {16230, 36790}, {17924, 51369}, {17994, 11672}, {18024, 52608}, {18808, 35910}, {20031, 250}, {22456, 4590}, {23290, 60524}, {24006, 1959}, {32545, 62523}, {32696, 23357}, {34536, 17932}, {36036, 62719}, {36104, 1101}, {36120, 662}, {36897, 65327}, {41013, 42717}, {41174, 31614}, {41932, 43754}, {43187, 47389}, {43665, 69}, {43920, 7254}, {44427, 51383}, {46106, 66074}, {46107, 51370}, {46273, 55202}, {51404, 520}, {51441, 647}, {51820, 56389}, {52038, 3292}, {52076, 22151}, {52439, 34859}, {52491, 14999}, {52641, 34211}, {53149, 6}, {53173, 3964}, {55122, 47406}, {55208, 51651}, {57065, 51439}, {57071, 59707}, {57204, 9418}, {57260, 1576}, {58756, 41270}, {58757, 232}, {60179, 59152}, {60199, 670}, {60338, 52091}, {60517, 23181}, {60568, 65568}, {60777, 22115}, {61196, 343}, {62519, 40804}, {65272, 34537}, {66299, 297}, {66881, 6514}, {67102, 51440}, {68572, 63741}
X(68624) = {X(879),X(61196)}-harmonic conjugate of X(43665)

X(68625) = X(4)X(522)∩X(225)X(523)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - a^3*c + a^2*b*c + a*b^2*c - b^3*c + a^2*c^2 - 2*a*b*c^2 + b^2*c^2 + a*c^3 + b*c^3 - 2*c^4)*(-a^4 + a^3*b - a^2*b^2 - a*b^3 + 2*b^4 - a^2*b*c + 2*a*b^2*c - b^3*c + 2*a^2*c^2 - a*b*c^2 - b^2*c^2 + b*c^3 - c^4) : :

X(68625) lies on the Huygens hyperbola and these lines: {4, 522}, {102, 1300}, {225, 523}, {264, 35519}, {393, 3064}, {1826, 3700}, {2432, 45745}, {2689, 36067}, {4077, 68576}, {4086, 41013}, {7649, 68574}, {16230, 18013}, {17983, 52780}, {23289, 68573}, {34393, 35142}, {35154, 65295}, {36121, 68571}

X(68625) = polar conjugate of the isogonal conjugate of X(55255)
X(68625) = X(42759)-cross conjugate of X(225)
X(68625) = X(i)-isoconjugate of X(j) for these (i,j): {110, 46974}, {255, 7452}, {515, 4575}, {1812, 2425}, {2182, 4558}, {2193, 2406}, {9247, 55254}, {32661, 64194}, {46391, 52378}
X(68625) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 515}, {244, 46974}, {6523, 7452}, {47345, 2406}, {62566, 39471}, {62576, 55254}
X(68625) = trilinear pole of line {2501, 21044}
X(68625) = barycentric product X(i)*X(j) for these {i,j}: {10, 60584}, {102, 14618}, {225, 2399}, {226, 53152}, {264, 55255}, {523, 52780}, {1577, 36121}, {2432, 57809}, {2501, 34393}, {15633, 52607}, {21044, 65295}, {24006, 36100}
X(68625) = barycentric quotient X(i)/X(j) for these {i,j}: {102, 4558}, {225, 2406}, {264, 55254}, {393, 7452}, {661, 46974}, {2399, 332}, {2432, 283}, {2501, 515}, {4516, 46391}, {14618, 35516}, {15633, 15411}, {21044, 39471}, {24006, 64194}, {32677, 4575}, {34393, 4563}, {36067, 52378}, {36100, 4592}, {36121, 662}, {41013, 42718}, {52780, 99}, {53152, 333}, {55206, 51361}, {55208, 1455}, {55242, 68251}, {55255, 3}, {57652, 2425}, {58757, 8755}, {60584, 86}, {65295, 4620}, {66287, 51368}

X(68626) = X(4)X(514)∩X(225)X(7178)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^3 - a^2*b - a*b^2 + b^3 + a*c^2 + b*c^2 - 2*c^3)*(-a^3 - a*b^2 + 2*b^3 + a^2*c - b^2*c + a*c^2 - c^3) : :

X(68626) lies on the Huygens hyperbola and these lines: {4, 514}, {103, 1300}, {225, 7178}, {264, 3261}, {393, 7649}, {523, 1826}, {1577, 41013}, {2400, 68569}, {2424, 68574}, {2690, 40116}, {4049, 68563}, {10566, 32085}, {16230, 18014}, {17983, 52781}, {18025, 35142}, {18808, 66276}, {24006, 68576}, {36122, 68571}, {43927, 68567}, {65218, 65238}

X(68626) = polar conjugate of the isogonal conjugate of X(55257)
X(68626) = X(i)-isoconjugate of X(j) for these (i,j): {163, 26006}, {255, 4241}, {516, 4575}, {906, 14953}, {910, 4558}, {1437, 2398}, {1444, 2426}, {4565, 51376}, {9247, 55256}, {18604, 41321}, {30807, 32661}
X(68626) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 26006}, {136, 516}, {4988, 39470}, {5190, 14953}, {6523, 4241}, {55064, 51376}, {55065, 51366}, {62576, 55256}
X(68626) = trilinear pole of line {2501, 3120}
X(68626) = barycentric product X(i)*X(j) for these {i,j}: {10, 53150}, {103, 14618}, {226, 60583}, {264, 55257}, {523, 52781}, {648, 66276}, {1577, 36122}, {1815, 66299}, {1826, 2400}, {2501, 18025}, {16732, 65218}, {21207, 40116}, {24006, 36101}, {53008, 60581}
X(68626) = barycentric quotient X(i)/X(j) for these {i,j}: {103, 4558}, {264, 55256}, {393, 4241}, {523, 26006}, {911, 4575}, {1826, 2398}, {2333, 2426}, {2400, 17206}, {2424, 1790}, {2501, 516}, {3120, 39470}, {4024, 51366}, {4041, 51376}, {7649, 14953}, {14618, 35517}, {15634, 15419}, {18025, 4563}, {24006, 30807}, {36101, 4592}, {36122, 662}, {40116, 4570}, {41013, 42719}, {52781, 99}, {53150, 86}, {55206, 41339}, {55208, 1456}, {55257, 3}, {57996, 55202}, {58757, 1886}, {60583, 333}, {65218, 4567}, {66276, 525}, {68576, 24015}

X(68627) = X(4)X(2457)∩X(225)X(55244)

Barycentrics    (a + b - 2*c)*(b - c)*(a - 2*b + c)*(b + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

X(68627) lies on the Huygens hyperbola and these lines: {4, 2457}, {106, 1300}, {225, 55244}, {264, 46107}, {393, 55263}, {523, 68562}, {903, 35142}, {1022, 68578}, {1826, 2501}, {1869, 18011}, {4404, 24006}, {6336, 17983}, {6531, 8752}, {6548, 68569}, {16230, 18005}, {18808, 66288}, {23345, 68574}, {36125, 68571}, {54239, 68573}

X(68627) = polar conjugate of the isotomic conjugate of X(4049)
X(68627) = polar conjugate of the isogonal conjugate of X(55263)
X(68627) = X(55263)-cross conjugate of X(4049)
X(68627) = X(i)-isoconjugate of X(j) for these (i,j): {44, 4558}, {99, 23202}, {110, 5440}, {163, 3977}, {184, 55243}, {255, 46541}, {283, 23703}, {519, 4575}, {662, 22356}, {902, 4592}, {906, 16704}, {1023, 1790}, {1101, 14429}, {1331, 52680}, {1332, 3285}, {1437, 17780}, {1444, 23344}, {1812, 61210}, {2193, 62669}, {2251, 4563}, {4358, 32661}, {4565, 52978}, {4567, 22086}, {4570, 53532}, {4574, 30576}, {4622, 22371}, {9247, 55262}, {9459, 55202}, {14407, 62719}, {14418, 52378}, {30939, 32656}, {36058, 68149}, {61171, 65568}
X(68627) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 3977}, {136, 519}, {244, 5440}, {523, 14429}, {1084, 22356}, {5139, 902}, {5190, 16704}, {5521, 52680}, {6523, 46541}, {9460, 4563}, {20619, 68149}, {38986, 23202}, {40594, 4592}, {40595, 4558}, {40627, 22086}, {47345, 62669}, {50330, 53532}, {55064, 52978}, {62567, 39472}, {62576, 55262}, {62605, 55243}
X(68627) = trilinear pole of line {2501, 21950}
X(68627) = barycentric product X(i)*X(j) for these {i,j}: {4, 4049}, {27, 66285}, {88, 24006}, {92, 55244}, {106, 14618}, {225, 60480}, {264, 55263}, {273, 61179}, {514, 68563}, {523, 6336}, {648, 66288}, {850, 8752}, {903, 2501}, {1022, 41013}, {1577, 36125}, {1797, 66299}, {1826, 6548}, {2052, 66924}, {2403, 68562}, {2489, 57995}, {2970, 4591}, {3120, 65336}, {4013, 17925}, {4080, 7649}, {4615, 8754}, {4674, 17924}, {18808, 52753}, {23838, 40149}, {52031, 68561}, {60578, 61178}
X(68627) = barycentric quotient X(i)/X(j) for these {i,j}: {88, 4592}, {92, 55243}, {106, 4558}, {115, 14429}, {225, 62669}, {264, 55262}, {393, 46541}, {512, 22356}, {523, 3977}, {661, 5440}, {798, 23202}, {903, 4563}, {1022, 1444}, {1824, 1023}, {1826, 17780}, {1880, 23703}, {2333, 23344}, {2489, 902}, {2501, 519}, {2971, 14407}, {3122, 22086}, {3125, 53532}, {4013, 52609}, {4041, 52978}, {4049, 69}, {4080, 4561}, {4120, 65742}, {4516, 14418}, {4615, 47389}, {4622, 62719}, {4674, 1332}, {6336, 99}, {6548, 17206}, {6549, 15419}, {6591, 52680}, {7140, 4169}, {7649, 16704}, {8752, 110}, {8754, 4120}, {8756, 68149}, {9456, 4575}, {14407, 22371}, {14618, 3264}, {17924, 30939}, {20568, 55202}, {21950, 39472}, {23345, 1790}, {23838, 1812}, {24006, 4358}, {36125, 662}, {41013, 24004}, {43922, 7254}, {53008, 30731}, {55206, 3689}, {55208, 1319}, {55244, 63}, {55263, 3}, {57200, 30576}, {57204, 9459}, {57652, 61210}, {57995, 52608}, {58757, 8756}, {60480, 332}, {61179, 78}, {65336, 4600}, {66285, 306}, {66288, 525}, {66299, 46109}, {66924, 394}, {68562, 2415}, {68563, 190}

X(68628) = X(4)X(65)∩X(225)X(1254)

Barycentrics    b*(-a + b - c)*(a + b - c)*c*(b + c)*(-a^2 + b^2 - c^2)^2*(a^2 + b^2 - c^2)^2 : :

X(68628) lies on the Huygens hyperbola and these lines: {4, 65}, {12, 41013}, {19, 47345}, {21, 1940}, {34, 68574}, {53, 56908}, {92, 3485}, {107, 67753}, {108, 1300}, {225, 1254}, {243, 411}, {254, 1068}, {264, 1441}, {273, 68569}, {278, 68578}, {318, 68577}, {393, 1880}, {653, 3559}, {961, 54340}, {1096, 68573}, {1214, 1217}, {1231, 55972}, {1402, 8884}, {1409, 40402}, {1411, 8747}, {1780, 65180}, {1784, 10572}, {1826, 2171}, {1895, 3486}, {1896, 17097}, {1935, 56822}, {2052, 60321}, {6046, 57285}, {6867, 18855}, {6869, 18850}, {7233, 52938}, {7548, 14860}, {12047, 39574}, {17983, 54240}, {18026, 35142}, {18808, 66297}, {18848, 59355}, {24914, 37805}, {34401, 52385}, {41515, 61392}, {41516, 61393}, {41538, 61178}, {56300, 62691}, {57652, 68581}, {57809, 68579}, {61180, 64715}, {66275, 66299}

X(68628) = polar conjugate of X(1812)
polar conjugate of the isotomic conjugate of X(40149)
X(68628) = polar conjugate of the isogonal conjugate of X(1880)
X(68628) = X(158)-Ceva conjugate of X(225)
X(68628) = X(i)-cross conjugate of X(j) for these (i,j): {431, 4}, {1880, 40149}, {4516, 2501}
X(68628) = X(i)-isoconjugate of X(j) for these (i,j): {3, 283}, {6, 6514}, {9, 18604}, {21, 255}, {29, 1092}, {48, 1812}, {58, 1259}, {60, 3682}, {63, 2193}, {71, 65568}, {78, 1437}, {81, 2289}, {86, 6056}, {109, 68151}, {110, 57241}, {184, 332}, {212, 1444}, {219, 1790}, {222, 2327}, {261, 4055}, {284, 394}, {310, 62257}, {314, 52430}, {326, 2194}, {333, 577}, {520, 4636}, {521, 4575}, {603, 1792}, {643, 23224}, {652, 4558}, {662, 36054}, {822, 4612}, {1043, 7335}, {1069, 1800}, {1098, 22341}, {1102, 2204}, {1172, 6507}, {1264, 2206}, {1331, 23189}, {1333, 3719}, {1364, 4570}, {1414, 58340}, {1433, 1819}, {1576, 52616}, {1789, 52408}, {1793, 52407}, {1804, 2328}, {1808, 7193}, {1813, 23090}, {1946, 4592}, {2149, 16731}, {2150, 3998}, {2185, 3990}, {2287, 7125}, {2299, 3964}, {3193, 60794}, {3926, 57657}, {4091, 5546}, {4100, 31623}, {4131, 65375}, {4565, 57057}, {4587, 7254}, {4600, 61054}, {4620, 39687}, {5562, 35196}, {6332, 32661}, {6516, 57134}, {6517, 21789}, {7045, 66898}, {7054, 40152}, {14585, 28660}, {15411, 32660}, {17206, 52425}, {22361, 57668}, {23606, 44130}, {35072, 52378}, {35602, 52158}, {36059, 57081}, {52921, 68150}
X(68628) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 6514}, {10, 1259}, {11, 68151}, {37, 3719}, {136, 521}, {226, 3964}, {244, 57241}, {478, 18604}, {650, 16731}, {1084, 36054}, {1214, 326}, {1249, 1812}, {3162, 2193}, {4858, 52616}, {5139, 1946}, {5521, 23189}, {6523, 21}, {6741, 68108}, {7952, 1792}, {15259, 2194}, {15267, 22341}, {17115, 66898}, {20620, 57081}, {36103, 283}, {36908, 1804}, {38966, 58338}, {39053, 4592}, {39060, 4563}, {40586, 2289}, {40590, 394}, {40600, 6056}, {40603, 1264}, {40608, 58340}, {40611, 255}, {40622, 4131}, {40837, 1444}, {47345, 63}, {50330, 1364}, {50497, 61054}, {55060, 23224}, {55064, 57057}, {56325, 3998}, {59608, 7183}, {62565, 1102}, {62566, 24031}, {62570, 3926}, {62602, 17206}, {62605, 332}
X(68628) = cevapoint of X(2501) and X(4516)
X(68628) = crosspoint of X(158) and X(1093)
X(68628) = crosssum of X(i) and X(j) for these (i,j): {255, 1092}, {2289, 6056}
X(68628) = trilinear pole of line {2501, 57185}
X(68628) = barycentric product X(i)*X(j) for these {i,j}: {4, 40149}, {19, 57809}, {25, 52575}, {27, 56285}, {65, 2052}, {73, 6521}, {92, 225}, {108, 14618}, {158, 226}, {264, 1880}, {273, 1826}, {278, 41013}, {281, 68576}, {286, 8736}, {307, 6520}, {321, 1118}, {331, 1824}, {349, 1096}, {393, 1441}, {523, 54240}, {648, 66297}, {651, 66299}, {653, 24006}, {661, 52938}, {823, 66287}, {1093, 1214}, {1231, 6524}, {1396, 7141}, {1400, 57806}, {1402, 18027}, {1426, 7017}, {1446, 1857}, {1577, 36127}, {1847, 53008}, {1896, 6354}, {1969, 57652}, {2333, 57787}, {2501, 18026}, {4516, 57538}, {4554, 58757}, {5317, 34388}, {6358, 8747}, {6528, 57185}, {7337, 27801}, {7649, 65207}, {8808, 47372}, {17924, 61178}, {21044, 24032}, {36126, 57243}, {37790, 68563}, {37805, 68580}, {44426, 52607}, {57973, 66928}
X(68628) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 6514}, {4, 1812}, {10, 3719}, {11, 16731}, {12, 3998}, {19, 283}, {25, 2193}, {28, 65568}, {33, 2327}, {34, 1790}, {37, 1259}, {42, 2289}, {56, 18604}, {65, 394}, {73, 6507}, {92, 332}, {107, 4612}, {108, 4558}, {158, 333}, {181, 3990}, {213, 6056}, {225, 63}, {226, 326}, {273, 17206}, {278, 1444}, {281, 1792}, {307, 1102}, {321, 1264}, {393, 21}, {512, 36054}, {608, 1437}, {650, 68151}, {653, 4592}, {661, 57241}, {1020, 6517}, {1042, 7125}, {1093, 31623}, {1096, 284}, {1118, 81}, {1214, 3964}, {1231, 4176}, {1254, 40152}, {1400, 255}, {1402, 577}, {1409, 1092}, {1426, 222}, {1427, 1804}, {1441, 3926}, {1446, 7055}, {1577, 52616}, {1824, 219}, {1826, 78}, {1835, 22128}, {1857, 2287}, {1874, 20769}, {1880, 3}, {1893, 23151}, {1896, 7058}, {2052, 314}, {2171, 3682}, {2205, 62257}, {2207, 2194}, {2331, 1819}, {2333, 212}, {2358, 1433}, {2489, 1946}, {2501, 521}, {3064, 57081}, {3121, 61054}, {3125, 1364}, {3668, 7183}, {3700, 68108}, {3709, 58340}, {4017, 4091}, {4041, 57057}, {4077, 30805}, {4516, 35072}, {5317, 60}, {6354, 52385}, {6358, 52396}, {6520, 29}, {6521, 44130}, {6524, 1172}, {6528, 4631}, {6529, 52914}, {6591, 23189}, {7140, 3694}, {7178, 4131}, {7180, 23224}, {7337, 1333}, {8736, 72}, {8747, 2185}, {8748, 1098}, {8754, 53560}, {14618, 35518}, {14936, 66898}, {18026, 4563}, {18027, 40072}, {18344, 23090}, {20613, 1801}, {21044, 24031}, {24006, 6332}, {24019, 4636}, {24032, 4620}, {24033, 52378}, {30456, 35602}, {32674, 4575}, {36127, 662}, {37384, 23602}, {40149, 69}, {41013, 345}, {43923, 7254}, {44092, 22074}, {44426, 15411}, {46404, 55202}, {47372, 27398}, {52033, 1800}, {52439, 2204}, {52575, 305}, {52577, 1040}, {52607, 6516}, {52938, 799}, {53008, 3692}, {53861, 55466}, {54239, 57213}, {54240, 99}, {55197, 57109}, {55206, 57108}, {55208, 1459}, {56285, 306}, {57185, 520}, {57652, 48}, {57806, 28660}, {57809, 304}, {58757, 650}, {61058, 16730}, {61178, 1332}, {64834, 1789}, {64835, 1793}, {65103, 58338}, {65207, 4561}, {66287, 24018}, {66297, 525}, {66299, 4391}, {66928, 822}, {68327, 14395}, {68576, 348}
X(68628) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {158, 47372, 1857}, {653, 3559, 7098}

X(68629) = X(4)X(1499)∩X(225)X(23894)

Barycentrics    (b^2 - c^2)*(a^2 + b^2 - 2*c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-a^2 + 2*b^2 - c^2) : :

X(68629) lies on the Huygens hyperbola and these lines: {4, 1499}, {93, 55251}, {111, 1300}, {225, 23894}, {264, 8430}, {393, 9178}, {523, 68564}, {671, 35142}, {2408, 14273}, {2444, 44705}, {2489, 33885}, {3545, 55267}, {4235, 50941}, {6531, 8753}, {6563, 32984}, {8884, 15422}, {16230, 18007}, {18808, 64258}, {18850, 66124}, {23968, 53155}, {33228, 62645}, {34208, 59932}, {36128, 68571}, {43448, 65610}, {55972, 62629}, {62626, 68569}, {66945, 68574}

X(68629) = polar conjugate of X(5468)
X(68629) = polar conjugate of the isotomic conjugate of X(5466)
X(68629) = polar conjugate of the isogonal conjugate of X(9178)
X(68629) = X(65350)-Ceva conjugate of X(17983)
X(68629) = X(i)-cross conjugate of X(j) for these (i,j): {9178, 5466}, {14273, 2501}
X(68629) = X(i)-isoconjugate of X(j) for these (i,j): {3, 23889}, {48, 5468}, {63, 5467}, {163, 6390}, {184, 24039}, {187, 4592}, {255, 4235}, {326, 61207}, {351, 62719}, {524, 4575}, {662, 3292}, {799, 23200}, {896, 4558}, {906, 6629}, {922, 4563}, {1101, 14417}, {1331, 16702}, {14210, 32661}, {14567, 55202}, {16741, 32656}, {23995, 45807}, {36060, 68152}, {36142, 65747}, {42081, 65321}
X(68629) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 6390}, {136, 524}, {523, 14417}, {1084, 3292}, {1249, 5468}, {1560, 68152}, {3162, 5467}, {5139, 187}, {5190, 6629}, {5521, 16702}, {6523, 4235}, {15259, 61207}, {15477, 32661}, {15899, 4558}, {18314, 45807}, {23992, 65747}, {36103, 23889}, {38970, 50567}, {38996, 23200}, {39061, 4563}, {48317, 2482}, {53983, 7813}, {53992, 27088}, {62605, 24039}
X(68629) = cevapoint of X(i) and X(j) for these (i,j): {523, 9134}, {2501, 14273}
X(68629) = crosspoint of X(17983) and X(65350)
X(68629) = trilinear pole of line {2501, 6791}
X(68629) = crossdifference of every pair of points on line {3292, 23200}
X(68629) = barycentric product X(i)*X(j) for these {i,j}: {4, 5466}, {25, 52632}, {92, 23894}, {107, 51258}, {111, 14618}, {115, 65350}, {264, 9178}, {393, 14977}, {512, 46111}, {523, 17983}, {648, 64258}, {671, 2501}, {691, 2970}, {850, 8753}, {892, 8754}, {895, 66299}, {897, 24006}, {935, 10555}, {1577, 36128}, {1826, 62626}, {2052, 10097}, {2408, 68564}, {2489, 18023}, {2971, 53080}, {3124, 59762}, {6344, 9213}, {6531, 62629}, {7141, 43926}, {8430, 16081}, {9154, 16230}, {9214, 18808}, {10561, 46105}, {14246, 66943}, {14273, 57539}, {23288, 68566}, {30786, 58757}, {52450, 60338}, {60133, 65609}
X(68629) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 5468}, {19, 23889}, {25, 5467}, {92, 24039}, {111, 4558}, {115, 14417}, {338, 45807}, {393, 4235}, {468, 68152}, {512, 3292}, {523, 6390}, {669, 23200}, {671, 4563}, {690, 65747}, {892, 47389}, {897, 4592}, {923, 4575}, {2207, 61207}, {2489, 187}, {2501, 524}, {2970, 35522}, {2971, 351}, {5466, 69}, {6591, 16702}, {7649, 6629}, {8430, 36212}, {8753, 110}, {8754, 690}, {9134, 52881}, {9154, 17932}, {9178, 3}, {9213, 52437}, {10097, 394}, {10561, 22151}, {10630, 65321}, {14273, 2482}, {14618, 3266}, {14977, 3926}, {15475, 66125}, {16230, 50567}, {17924, 16741}, {17983, 99}, {17994, 9155}, {18023, 52608}, {18384, 14559}, {18808, 36890}, {23894, 63}, {24006, 14210}, {32729, 47390}, {32740, 32661}, {35235, 45808}, {36085, 62719}, {36128, 662}, {41013, 42721}, {46111, 670}, {46277, 55202}, {51258, 3265}, {51428, 39474}, {51513, 41586}, {52477, 66963}, {52632, 305}, {53149, 5967}, {55208, 51653}, {57071, 32459}, {57204, 14567}, {58757, 468}, {58780, 8030}, {59762, 34537}, {60040, 53784}, {60428, 68087}, {62626, 17206}, {62629, 6393}, {64258, 525}, {64619, 61198}, {65350, 4590}, {65472, 64724}, {65609, 62382}, {66299, 44146}, {66945, 1790}, {68327, 5642}, {68564, 2418}

X(68630) = X(4)X(18022)∩X(225)X(18833)

Barycentrics    b^4*(a^2 + b^2)*c^4*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 + c^2) : :

X(68630) lies on the Huygens hyperbola and these lines: {4, 18022}, {5, 18024}, {76, 68572}, {83, 6531}, {225, 18833}, {264, 44161}, {305, 66596}, {308, 393}, {689, 1300}, {1235, 68575}, {1502, 8801}, {1799, 8884}, {3112, 68581}, {6331, 37125}, {16044, 23962}, {16230, 18008}, {17984, 32085}, {32960, 46328}, {32968, 40822}, {35142, 42371}

X(68630) = isotomic conjugate of X(20775)
X(68630) = polar conjugate of X(3051)
X(68630) = isotomic conjugate of the isogonal conjugate of X(46104)
X(68630) = polar conjugate of the isotomic conjugate of X(40016)
X(68630) = polar conjugate of the isogonal conjugate of X(308)
X(68630) = X(i)-cross conjugate of X(j) for these (i,j): {264, 46104}, {308, 40016}, {17984, 60199}, {42394, 264}, {59635, 76}
X(68630) = X(i)-isoconjugate of X(j) for these (i,j): {3, 1923}, {31, 20775}, {32, 4020}, {38, 14575}, {39, 9247}, {48, 3051}, {63, 41331}, {184, 1964}, {255, 27369}, {560, 3917}, {688, 4575}, {822, 61218}, {1437, 41267}, {1843, 52430}, {1917, 3933}, {1930, 40373}, {2084, 32661}, {2169, 27374}, {4592, 9494}, {14585, 17442}, {20883, 61361}, {32656, 50521}, {40972, 52411}, {56915, 66942}
X(68630) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 20775}, {136, 688}, {308, 23174}, {1249, 3051}, {3162, 41331}, {5139, 9494}, {6374, 3917}, {6376, 4020}, {6523, 27369}, {14363, 27374}, {36103, 1923}, {40938, 59994}, {41884, 184}, {53983, 2531}, {62452, 32661}, {62576, 39}, {62605, 1964}
X(68630) = cevapoint of X(i) and X(j) for these (i,j): {76, 7752}, {264, 18022}, {308, 46104}
X(68630) = barycentric product X(i)*X(j) for these {i,j}: {4, 40016}, {76, 46104}, {83, 18022}, {92, 18833}, {251, 44161}, {264, 308}, {324, 41488}, {331, 62539}, {512, 42395}, {689, 14618}, {1502, 32085}, {1799, 18027}, {1969, 3112}, {2501, 42371}, {6331, 52618}, {17500, 57790}, {18070, 57968}, {20022, 60199}, {24006, 37204}, {31622, 42394}, {39287, 62274}, {40359, 61383}, {42396, 44173}, {44129, 56251}, {56186, 57796}
X(68630) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 20775}, {4, 3051}, {19, 1923}, {25, 41331}, {53, 27374}, {75, 4020}, {76, 3917}, {82, 9247}, {83, 184}, {92, 1964}, {107, 61218}, {251, 14575}, {264, 39}, {276, 16030}, {308, 3}, {318, 40972}, {331, 1401}, {393, 27369}, {419, 56915}, {427, 59994}, {689, 4558}, {1176, 14585}, {1235, 8041}, {1502, 3933}, {1799, 577}, {1826, 41267}, {1969, 38}, {2052, 1843}, {2489, 9494}, {2501, 688}, {3112, 48}, {3934, 23210}, {3963, 22367}, {4577, 32661}, {4580, 39201}, {4593, 4575}, {6331, 1634}, {6528, 35325}, {7017, 3688}, {10547, 61361}, {10549, 40951}, {14618, 3005}, {14970, 17970}, {16081, 51869}, {16890, 4173}, {17500, 217}, {17907, 23208}, {17924, 50521}, {17983, 41272}, {17984, 8623}, {18022, 141}, {18027, 427}, {18070, 810}, {18082, 2200}, {18833, 63}, {20022, 3289}, {24006, 2084}, {28724, 23606}, {32085, 32}, {34055, 52430}, {34294, 65751}, {36794, 3203}, {37204, 4592}, {39182, 58308}, {39287, 14533}, {39998, 22078}, {40016, 69}, {40826, 65006}, {41013, 21814}, {41488, 97}, {42371, 4563}, {42395, 670}, {42396, 1576}, {44129, 17187}, {44142, 11205}, {44144, 14096}, {44161, 8024}, {44173, 2525}, {46104, 6}, {46107, 21123}, {46111, 46154}, {46288, 40373}, {46404, 46153}, {51906, 23216}, {52395, 10547}, {52568, 4175}, {52570, 22352}, {52618, 647}, {52898, 23200}, {54412, 3787}, {56186, 228}, {56251, 71}, {57796, 16696}, {57806, 17442}, {58784, 3049}, {59762, 36827}, {60199, 20021}, {61383, 9233}, {61404, 22096}, {62275, 27371}, {62539, 219}
X(68630) = {X(18022),X(46104)}-harmonic conjugate of X(40016)

X(68631) = X(4)X(4444)∩X(225)X(876)

Barycentrics    (b - c)*(b^2 - a*c)*(a*b - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2) : :

X(68631) lies on the Huygens hyperbola and these lines: {4, 4444}, {225, 876}, {264, 24006}, {393, 3572}, {741, 1300}, {875, 68581}, {1826, 7649}, {16230, 18009}, {17924, 21108}, {17983, 65352}, {18827, 35142}, {32085, 57200}, {54247, 68573}, {65106, 68565}, {65336, 65338}, {66286, 68579}, {66937, 68574}

X(68631) = polar conjugate of X(3570)
X(68631) = polar conjugate of the isotomic conjugate of X(4444)
X(68631) = polar conjugate of the isogonal conjugate of X(3572)
X(68631) = X(3572)-cross conjugate of X(4444)
X(68631) = X(i)-isoconjugate of X(j) for these (i,j): {3, 3573}, {48, 3570}, {100, 7193}, {101, 20769}, {184, 874}, {238, 1331}, {239, 906}, {350, 32656}, {740, 4575}, {765, 22384}, {1332, 1914}, {1428, 4571}, {1429, 4587}, {1813, 3684}, {2210, 4561}, {2238, 4558}, {3685, 36059}, {3747, 4592}, {3948, 32661}, {3975, 32660}, {4435, 44717}, {4563, 41333}, {4570, 53556}, {9247, 27853}, {20778, 36086}, {43754, 50440}, {46390, 62719}
X(68631) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 740}, {513, 22384}, {1015, 20769}, {1249, 3570}, {4988, 24459}, {5139, 3747}, {5190, 239}, {5521, 238}, {8054, 7193}, {9470, 1331}, {16592, 12215}, {20620, 3685}, {36103, 3573}, {36906, 1332}, {38966, 58327}, {38989, 20778}, {50330, 53556}, {53985, 4432}, {62557, 4561}, {62576, 27853}, {62605, 874}
X(68631) = crosssum of X(7193) and X(22384)
X(68631) = trilinear pole of line {2501, 2969}
X(68631) = crossdifference of every pair of points on line {7193, 20778}
X(68631) = barycentric product X(i)*X(j) for these {i,j}: {4, 4444}, {19, 66286}, {27, 35352}, {92, 876}, {264, 3572}, {278, 60577}, {291, 17924}, {292, 46107}, {334, 6591}, {335, 7649}, {523, 65352}, {741, 14618}, {813, 2973}, {875, 1969}, {1086, 65338}, {1916, 54229}, {2052, 66938}, {2501, 18827}, {2969, 4562}, {3064, 7233}, {4369, 68575}, {8754, 65258}, {17925, 43534}, {24006, 37128}, {40098, 65106}, {43923, 66882}, {53559, 65351}
X(68631) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 3570}, {19, 3573}, {92, 874}, {264, 27853}, {291, 1332}, {292, 1331}, {335, 4561}, {513, 20769}, {649, 7193}, {665, 20778}, {741, 4558}, {875, 48}, {876, 63}, {1015, 22384}, {1911, 906}, {1922, 32656}, {2489, 3747}, {2501, 740}, {2969, 812}, {2971, 46390}, {2973, 65101}, {3064, 3685}, {3120, 24459}, {3125, 53556}, {3572, 3}, {4369, 12215}, {4444, 69}, {4876, 4571}, {6373, 20750}, {6591, 238}, {7077, 4587}, {7233, 65164}, {7649, 239}, {8735, 3716}, {14618, 35544}, {17924, 350}, {17925, 33295}, {18268, 4575}, {18344, 3684}, {18827, 4563}, {22093, 58354}, {24006, 3948}, {30671, 3781}, {35352, 306}, {36066, 62719}, {37128, 4592}, {39534, 51381}, {40017, 55202}, {42067, 8632}, {42069, 4148}, {43534, 52609}, {43923, 1429}, {43925, 5009}, {44426, 3975}, {46107, 1921}, {46110, 4087}, {53150, 65955}, {53559, 24284}, {54229, 385}, {55206, 4433}, {55208, 1284}, {60577, 345}, {65103, 58327}, {65106, 4366}, {65258, 47389}, {65338, 1016}, {65352, 99}, {66286, 304}, {66937, 1790}, {66938, 394}, {68575, 27805}

X(68632) = X(4)X(690)∩X(264)X(23350)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*c^2 - b^4*c^2 + 2*a^2*c^4 + 2*b^2*c^4 - 2*c^6)*(-a^6 + a^4*b^2 - 2*a^2*b^4 + 2*b^6 + a^4*c^2 - 2*b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6) : :

X(68632) lies on the Huygens hyperbola and these lines: {2, 65763}, {4, 690}, {99, 65754}, {107, 53691}, {264, 23350}, {393, 14273}, {523, 17983}, {648, 14559}, {842, 1300}, {1217, 35911}, {2501, 6531}, {5466, 65731}, {5641, 35142}, {6344, 14618}, {8754, 18808}, {9529, 18850}, {22105, 32085}, {23287, 68566}, {34765, 55972}, {41254, 51480}, {50942, 68564}, {52475, 58757}, {54395, 65714}, {65100, 68571}

X(68632) = polar conjugate of X(14999)
X(68632) = polar conjugate of the isotomic conjugate of X(14223)
X(68632) = polar conjugate of the isogonal conjugate of X(14998)
X(68632) = X(14998)-cross conjugate of X(14223)
X(68632) = X(i)-isoconjugate of X(j) for these (i,j): {48, 14999}, {163, 65722}, {255, 7473}, {293, 42743}, {542, 4575}, {1101, 65723}, {2247, 4558}, {4592, 5191}, {6041, 62719}, {6507, 35907}, {35200, 64607}
X(68632) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 65722}, {132, 42743}, {133, 64607}, {136, 542}, {523, 65723}, {1249, 14999}, {1649, 39474}, {5139, 5191}, {6523, 7473}, {42426, 68158}, {48317, 45662}, {65728, 65750}
X(68632) = cevapoint of X(523) and X(65610)
X(68632) = trilinear pole of line {1648, 2501}
X(68632) = barycentric product X(i)*X(j) for these {i,j}: {4, 14223}, {107, 65727}, {264, 14998}, {671, 53156}, {842, 14618}, {1093, 35911}, {2052, 35909}, {2501, 5641}, {2970, 5649}, {6035, 8754}, {6531, 34765}, {16081, 23350}, {17983, 50942}, {18808, 51228}, {34174, 60338}, {43665, 52492}, {44427, 54554}, {46105, 53177}, {65308, 66299}, {65350, 67082}
X(68632) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 14999}, {115, 65723}, {232, 42743}, {393, 7473}, {523, 65722}, {842, 4558}, {1640, 65750}, {1648, 39474}, {1990, 64607}, {2489, 5191}, {2501, 542}, {2970, 18312}, {2971, 6041}, {5466, 51405}, {5641, 4563}, {6035, 47389}, {6103, 68158}, {6524, 35907}, {6531, 34761}, {8029, 65724}, {8749, 51262}, {8754, 1640}, {8791, 53232}, {14223, 69}, {14273, 45662}, {14998, 3}, {17983, 50941}, {17994, 66354}, {18384, 23968}, {18808, 51227}, {23350, 36212}, {34765, 6393}, {35142, 67101}, {35909, 394}, {35911, 3964}, {50942, 6390}, {52492, 2421}, {53149, 34369}, {53156, 524}, {53177, 22151}, {54554, 60053}, {58757, 6103}, {65610, 65730}, {65727, 3265}, {66299, 60502}, {66943, 67199}, {67082, 14417}, {68572, 36885}

X(68633) = X(4)X(970)∩X(92)X(225)

Barycentrics    1/(a*(a^2 + a*b + a*c + 2*b*c)*(a^2 - b^2 - c^2)) :
Barycentrics    b*c*(a*b + b^2 + 2*a*c + b*c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(2*a*b + a*c + b*c + c^2) : ::

X(68633) lies on the Huygens hyperbola and these lines: {4, 970}, {25, 59482}, {29, 3192}, {53, 56914}, {92, 225}, {264, 429}, {286, 37384}, {318, 1826}, {331, 54314}, {393, 941}, {406, 37870}, {931, 1300}, {959, 16082}, {1105, 37194}, {2052, 60321}, {2258, 39585}, {4186, 32085}, {7017, 41013}, {14004, 68573}, {32693, 39429}, {34265, 37390}, {35142, 65280}, {44733, 64988}

X(68633) = polar conjugate of X(940)
X(68633) = polar conjugate of the isotomic conjugate of X(34258)
X(68633) = polar conjugate of the isogonal conjugate of X(941)
X(68633) = X(i)-cross conjugate of X(j) for these (i,j): {941, 34258}, {1904, 4}, {60321, 31359}
X(68633) = X(i)-isoconjugate of X(j) for these (i,j): {3, 1468}, {48, 940}, {63, 5019}, {73, 54417}, {184, 10436}, {222, 2268}, {255, 4185}, {577, 5307}, {603, 958}, {906, 48144}, {1437, 59305}, {3713, 7099}, {4575, 8672}, {4592, 8639}, {7335, 54396}, {9247, 34284}, {11679, 52411}, {17418, 36059}, {22383, 65168}, {23880, 32660}, {32656, 43067}, {32661, 50457}
X(68633) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 8672}, {1249, 940}, {3162, 5019}, {5139, 8639}, {5190, 48144}, {6523, 4185}, {7952, 958}, {20620, 17418}, {36103, 1468}, {38966, 58332}, {62576, 34284}, {62605, 10436}
X(68633) = cevapoint of X(i) and X(j) for these (i,j): {4, 406}, {281, 7102}, {3192, 44103}, {4385, 46937}
X(68633) = trilinear pole of line {2501, 44426}
X(68633) = barycentric product X(i)*X(j) for these {i,j}: {4, 34258}, {25, 40828}, {92, 31359}, {264, 941}, {281, 58008}, {318, 44733}, {469, 34265}, {931, 14618}, {959, 7017}, {1969, 2258}, {2052, 34259}, {2501, 65280}, {24006, 65230}, {31623, 60321}, {32038, 44426}, {37870, 41013}, {46110, 65225}, {57806, 66920}
X(68633) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 940}, {19, 1468}, {25, 5019}, {33, 2268}, {92, 10436}, {158, 5307}, {264, 34284}, {281, 958}, {318, 11679}, {393, 4185}, {931, 4558}, {941, 3}, {959, 222}, {1172, 54417}, {1826, 59305}, {1896, 44734}, {1897, 65168}, {2258, 48}, {2489, 8639}, {2501, 8672}, {2969, 53543}, {3064, 17418}, {5331, 1790}, {7046, 3713}, {7102, 34261}, {7649, 48144}, {17924, 43067}, {24006, 50457}, {31359, 63}, {32038, 6516}, {32693, 36059}, {34258, 69}, {34259, 394}, {34263, 56414}, {34265, 57876}, {37870, 1444}, {40828, 305}, {41013, 31993}, {42069, 53561}, {43742, 34279}, {44103, 34281}, {44426, 23880}, {44733, 77}, {52931, 52610}, {56914, 22076}, {58008, 348}, {60321, 1214}, {65103, 58332}, {65160, 65190}, {65225, 1813}, {65230, 4592}, {65280, 4563}, {66920, 255}

X(68634) = X(4)X(67)∩X(264)X(5169)

Barycentrics    b^2*c^2*(-a^2 + b^2 - c^2)^2*(a^2 + b^2 - c^2)^2*(a^4 - a^2*b^2 + b^4 - c^4)*(-a^4 + b^4 + a^2*c^2 - c^4) : :

X(68634) lies on the Huygens hyperbola and these lines: {4, 67}, {94, 66950}, {107, 9076}, {264, 5169}, {393, 2970}, {935, 1300}, {1105, 7527}, {1179, 44732}, {1217, 34897}, {1297, 65753}, {2052, 10511}, {3455, 8884}, {6344, 6530}, {6531, 52418}, {7565, 14860}, {10002, 52487}, {10415, 17983}, {11061, 60502}, {16263, 43976}, {18808, 66943}, {18848, 67339}, {18850, 49669}, {35142, 44138}, {37981, 67086}, {44135, 55972}, {57496, 68564}

X(68634) = isogonal conjugate of X(58357)
X(68634) = polar conjugate of X(22151)
X(68634) = polar conjugate of the isotomic conjugate of X(46105)
X(68634) = polar conjugate of the isogonal conjugate of X(8791)
X(68634) = X(i)-cross conjugate of X(j) for these (i,j): {8791, 46105}, {37981, 4}, {47298, 76}, {60428, 2052}
X(68634) = X(i)-isoconjugate of X(j) for these (i,j): {1, 58357}, {23, 255}, {48, 22151}, {63, 10317}, {316, 52430}, {326, 18374}, {577, 16568}, {822, 52630}, {4100, 37765}, {4575, 9517}, {4592, 42659}, {6507, 8744}, {9247, 37804}, {14585, 20944}, {16165, 35200}
X(68634) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 58357}, {133, 16165}, {136, 9517}, {1249, 22151}, {3162, 10317}, {5139, 42659}, {6523, 23}, {15259, 18374}, {15900, 394}, {62576, 37804}
X(68634) = trilinear pole of line {2501, 23105}
X(68634) = barycentric product X(i)*X(j) for these {i,j}: {4, 46105}, {67, 2052}, {264, 8791}, {338, 66950}, {393, 18019}, {523, 65356}, {648, 66943}, {935, 14618}, {1093, 34897}, {2157, 57806}, {2501, 65269}, {3455, 18027}, {10415, 37778}, {11605, 43678}, {17708, 66299}, {17983, 57496}, {39269, 60133}
X(68634) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 22151}, {6, 58357}, {25, 10317}, {67, 394}, {107, 52630}, {158, 16568}, {264, 37804}, {393, 23}, {935, 4558}, {1093, 37765}, {1990, 16165}, {2052, 316}, {2157, 255}, {2207, 18374}, {2489, 42659}, {2501, 9517}, {2970, 62563}, {3455, 577}, {6524, 8744}, {6528, 55226}, {6529, 52916}, {8791, 3}, {9076, 28724}, {11605, 20806}, {13854, 54060}, {17983, 57481}, {18019, 3926}, {18027, 40074}, {27376, 9019}, {34897, 3964}, {36820, 58354}, {37778, 7664}, {39269, 62382}, {46105, 69}, {57496, 6390}, {57806, 20944}, {58757, 2492}, {60428, 6593}, {60496, 51394}, {65269, 4563}, {65356, 99}, {66299, 9979}, {66943, 525}, {66950, 249}

X(68635) = X(4)X(2575)∩X(264)X(1346)

Barycentrics    b^2*c^2*(b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4 - a^2*(a^2 - b^2 - c^2)*J) : :

X(68635) lies on the Huygens hyperbola and these lines: {4, 2575}, {107, 1114}, {225, 2589}, {264, 1346}, {393, 8106}, {403, 523}, {1105, 14709}, {1113, 1300}, {1217, 46811}, {1312, 2970}, {1313, 34334}, {2586, 68571}, {2592, 16230}, {6528, 57544}, {8884, 42667}, {10287, 16868}, {15164, 35142}, {17983, 46815}, {18808, 39241}

X(68635) = polar-circle-inverse of X(43395)
X(68635) = polar conjugate of X(8116)
X(68635) = polar conjugate of the isotomic conjugate of X(2593)
X(68635) = polar conjugate of the isogonal conjugate of X(8106)
X(68635) = X(i)-Ceva conjugate of X(j) for these (i,j): {6528, 2592}, {46812, 393}
X(68635) = X(i)-cross conjugate of X(j) for these (i,j): {115, 2592}, {523, 53154}, {1312, 4}, {8106, 2593}, {39240, 66299}
X(68635) = X(i)-isoconjugate of X(j) for these (i,j): {3, 1823}, {48, 8116}, {63, 57025}, {110, 2584}, {163, 46814}, {255, 1114}, {326, 44124}, {394, 2577}, {577, 2581}, {822, 39299}, {1092, 2587}, {1822, 53385}, {2574, 4575}, {2578, 4558}, {2582, 32661}, {2585, 15460}, {4100, 46812}, {4592, 42668}, {9247, 46810}, {15165, 52430}
X(68635) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 46814}, {136, 2574}, {244, 2584}, {1249, 8116}, {1312, 3}, {1313, 53385}, {3162, 57025}, {5139, 42668}, {6523, 1114}, {8106, 46811}, {15167, 394}, {15259, 44124}, {36103, 1823}, {62576, 46810}, {62592, 3964}, {66877, 52613}
X(68635) = cevapoint of X(523) and X(39240)
X(68635) = crosspoint of X(264) and X(46812)
X(68635) = trilinear pole of line {2501, 39241}
X(68635) = barycentric product X(i)*X(j) for these {i,j}: {4, 2593}, {92, 2589}, {158, 2583}, {264, 8106}, {393, 22340}, {523, 46815}, {648, 39241}, {1093, 46811}, {1113, 14618}, {1312, 46812}, {1577, 2586}, {2052, 2575}, {2501, 15164}, {2579, 57806}, {2580, 24006}, {2585, 6521}, {2592, 53154}, {2970, 39298}, {6528, 66876}, {8115, 66299}, {18027, 42667}, {46813, 58757}
X(68635) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 8116}, {19, 1823}, {25, 57025}, {107, 39299}, {158, 2581}, {264, 46810}, {393, 1114}, {523, 46814}, {661, 2584}, {1093, 46812}, {1096, 2577}, {1113, 4558}, {1312, 46811}, {2052, 15165}, {2207, 44124}, {2489, 42668}, {2501, 2574}, {2575, 394}, {2576, 4575}, {2579, 255}, {2580, 4592}, {2583, 326}, {2585, 6507}, {2586, 662}, {2589, 63}, {2593, 69}, {6520, 2587}, {8105, 53385}, {8106, 3}, {8754, 66877}, {14618, 22339}, {15164, 4563}, {22340, 3926}, {24006, 2582}, {27376, 46167}, {39240, 62593}, {39241, 525}, {42667, 577}, {44123, 32661}, {46811, 3964}, {46815, 99}, {52418, 44068}, {53154, 8115}, {58757, 8105}, {66299, 2592}, {66357, 51394}, {66876, 520}, {68327, 66358}

X(68636) = X(4)X(2574)∩X(264)X(1347)

Barycentrics    b^2*c^2*(b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4 + a^2*(a^2 - b^2 - c^2)*J) : :

X(68636) lies on the Huygens hyperbola and these lines: {4, 2574}, {107, 1113}, {225, 2588}, {264, 1347}, {393, 8105}, {403, 523}, {1105, 14710}, {1114, 1300}, {1217, 46814}, {1312, 34334}, {1313, 2970}, {2587, 68571}, {2593, 16230}, {6528, 57543}, {8884, 42668}, {10288, 16868}, {15165, 35142}, {17983, 46812}, {18808, 39240}

X(68636) = polar-circle-inverse of X(43396)
X(68636) = polar conjugate of X(8115)
X(68636) = polar conjugate of the isotomic conjugate of X(2592)
X(68636) = polar conjugate of the isogonal conjugate of X(8105)
X(68636) = X(i)-Ceva conjugate of X(j) for these (i,j): {6528, 2593}, {46815, 393}
X(68636) = X(i)-cross conjugate of X(j) for these (i,j): {115, 2593}, {523, 53153}, {1313, 4}, {8105, 2592}, {39241, 66299}
X(68636) = X(i)-isoconjugate of X(j) for these (i,j): {3, 1822}, {48, 8115}, {63, 57026}, {110, 2585}, {163, 46811}, {255, 1113}, {326, 44123}, {394, 2576}, {577, 2580}, {822, 39298}, {1092, 2586}, {1823, 53384}, {2575, 4575}, {2579, 4558}, {2583, 32661}, {2584, 15461}, {4100, 46815}, {4592, 42667}, {9247, 46813}, {15164, 52430}
X(68636) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 46811}, {136, 2575}, {244, 2585}, {1249, 8115}, {1312, 53384}, {1313, 3}, {3162, 57026}, {5139, 42667}, {6523, 1113}, {8105, 46814}, {15166, 394}, {15259, 44123}, {36103, 1822}, {62576, 46813}, {62593, 3964}, {66876, 52613}
X(68636) = cevapoint of X(523) and X(39241)
X(68636) = crosspoint of X(264) and X(46815)
X(68636) = trilinear pole of line {2501, 39240}
X(68636) = barycentric product X(i)*X(j) for these {i,j}: {4, 2592}, {92, 2588}, {158, 2582}, {264, 8105}, {393, 22339}, {523, 46812}, {648, 39240}, {1093, 46814}, {1114, 14618}, {1313, 46815}, {1577, 2587}, {2052, 2574}, {2501, 15165}, {2578, 57806}, {2581, 24006}, {2584, 6521}, {2593, 53153}, {2970, 39299}, {6528, 66877}, {8116, 66299}, {18027, 42668}, {46810, 58757}
X(68636) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 8115}, {19, 1822}, {25, 57026}, {107, 39298}, {158, 2580}, {264, 46813}, {393, 1113}, {523, 46811}, {661, 2585}, {1093, 46815}, {1096, 2576}, {1114, 4558}, {1313, 46814}, {2052, 15164}, {2207, 44123}, {2489, 42667}, {2501, 2575}, {2574, 394}, {2577, 4575}, {2578, 255}, {2581, 4592}, {2582, 326}, {2584, 6507}, {2587, 662}, {2588, 63}, {2592, 69}, {6520, 2586}, {8105, 3}, {8106, 53384}, {8754, 66876}, {14618, 22340}, {15165, 4563}, {22339, 3926}, {24006, 2583}, {27376, 46166}, {39240, 525}, {39241, 62592}, {42668, 577}, {44124, 32661}, {46812, 99}, {46814, 3964}, {52418, 44067}, {53153, 8116}, {58757, 8106}, {66299, 2593}, {66358, 51394}, {66877, 520}, {68327, 66357}

X(68637) = X(4)X(66)∩X(264)X(5133)

Barycentrics    b^2*c^2*(-a^2 + b^2 - c^2)^2*(a^2 + b^2 - c^2)^2*(-a^4 + b^4 - c^4)*(a^4 + b^4 - c^4) : :

X(68637) lies on the Huygens hyperbola and these lines: {4, 66}, {107, 26284}, {254, 44145}, {264, 5133}, {311, 55972}, {393, 13854}, {458, 40404}, {847, 6530}, {1105, 7503}, {1179, 33971}, {1217, 7404}, {1289, 1300}, {2052, 16277}, {2353, 8884}, {2970, 52439}, {5596, 60516}, {6531, 8745}, {7566, 14249}, {10002, 18855}, {12225, 18848}, {17907, 37801}, {17983, 21447}, {34208, 37778}, {35142, 65266}, {40402, 60495}, {44766, 56017}, {46151, 64023}, {52487, 52661}

X(68637) = polar conjugate of X(20806)
X(68637) = polar conjugate of the isotomic conjugate of X(43678)
X(68637) = polar conjugate of the isogonal conjugate of X(13854)
X(68637) = X(i)-cross conjugate of X(j) for these (i,j): {338, 66299}, {2207, 2052}, {13854, 43678}
X(68637) = X(i)-isoconjugate of X(j) for these (i,j): {22, 255}, {48, 20806}, {63, 10316}, {163, 58359}, {206, 326}, {304, 22075}, {315, 52430}, {394, 2172}, {577, 1760}, {822, 4611}, {1101, 47413}, {1102, 17409}, {3719, 7251}, {3926, 17453}, {3998, 17186}, {4100, 17907}, {4123, 7335}, {4456, 18604}, {4548, 7183}, {4575, 8673}, {6056, 7210}, {6507, 8743}, {9247, 34254}, {14585, 20641}, {23995, 62573}
X(68637) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 58359}, {136, 8673}, {523, 47413}, {1249, 20806}, {3162, 10316}, {6523, 22}, {15259, 206}, {18314, 62573}, {62576, 34254}
X(68637) = cevapoint of X(i) and X(j) for these (i,j): {115, 65472}, {2970, 58757}
X(68637) = barycentric product X(i)*X(j) for these {i,j}: {4, 43678}, {66, 2052}, {264, 13854}, {393, 18018}, {1093, 14376}, {1096, 46244}, {1289, 14618}, {2156, 57806}, {2207, 40421}, {2353, 18027}, {2501, 65266}, {2970, 44183}, {8794, 41168}, {8884, 60515}, {44766, 66299}, {52583, 58075}
X(68637) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 20806}, {25, 10316}, {66, 394}, {107, 4611}, {115, 47413}, {158, 1760}, {264, 34254}, {338, 62573}, {393, 22}, {523, 58359}, {1093, 17907}, {1096, 2172}, {1289, 4558}, {1974, 22075}, {2052, 315}, {2156, 255}, {2207, 206}, {2353, 577}, {2501, 8673}, {2970, 127}, {6059, 4548}, {6524, 8743}, {6526, 67118}, {6528, 55225}, {6529, 52915}, {7337, 7251}, {8754, 38356}, {13854, 3}, {14376, 3964}, {14618, 57069}, {15388, 47390}, {16277, 28724}, {17407, 23115}, {18018, 3926}, {18027, 40073}, {27376, 3313}, {34138, 51386}, {36417, 20968}, {40144, 39172}, {40146, 14585}, {43678, 69}, {52439, 17409}, {57806, 20641}, {58075, 28419}, {58757, 2485}, {60495, 1092}, {60515, 52347}, {65266, 4563}, {66299, 33294}, {68327, 14396}

X(68638) = X(4)X(195)∩X(264)X(1272)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 3*a^4*c^2 + a^2*b^2*c^2 - 3*b^4*c^2 + 3*a^2*c^4 + 3*b^2*c^4 - c^6)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - a^4*c^2 + a^2*b^2*c^2 + 3*b^4*c^2 - a^2*c^4 - 3*b^2*c^4 + c^6) : :

X(68638) lies on the Huygens hyperbola and these lines: {4, 195}, {93, 2970}, {107, 43657}, {186, 14367}, {225, 51804}, {254, 21844}, {264, 1272}, {393, 14579}, {847, 14940}, {1217, 35475}, {1291, 1300}, {3043, 8754}, {3518, 30526}, {3575, 32639}, {6188, 52546}, {6344, 14993}, {8737, 10633}, {8738, 10632}, {8741, 56515}, {8742, 56514}, {14978, 35482}, {18808, 64935}, {35142, 65279}

X(68638) = isogonal conjugate of X(50461)
X(68638) = polar conjugate of the isotomic conjugate of X(13582)
X(68638) = polar conjugate of the isogonal conjugate of X(14579)
X(68638) = X(i)-cross conjugate of X(j) for these (i,j): {186, 4}, {14579, 13582}
X(68638) = X(i)-isoconjugate of X(j) for these (i,j): {1, 50461}, {3, 1749}, {48, 37779}, {63, 11063}, {255, 37943}, {265, 51802}, {343, 19306}, {656, 47053}, {1157, 44706}, {4575, 45147}, {4592, 6140}, {8562, 36061}, {10272, 35200}, {63760, 68468}
X(68638) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 50461}, {133, 10272}, {136, 45147}, {1249, 37779}, {3162, 11063}, {5139, 6140}, {6523, 37943}, {16221, 8562}, {36103, 1749}, {40596, 47053}
X(68638) = cevapoint of X(i) and X(j) for these (i,j): {8754, 47230}, {11062, 53386}
X(68638) = crosspoint of X(1138) and X(3459)
X(68638) = crosssum of X(195) and X(399)
X(68638) = trilinear pole of line {2501, 6748}
X(68638) = barycentric product X(i)*X(j) for these {i,j}: {4, 13582}, {92, 51804}, {107, 64938}, {264, 14579}, {275, 1263}, {340, 11071}, {470, 46072}, {471, 46076}, {648, 64935}, {1291, 14618}, {2052, 43704}, {2501, 65279}, {3471, 16080}, {6143, 30526}, {6331, 64937}, {9381, 38539}, {14165, 15392}, {16081, 64936}, {40684, 43657}
X(68638) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 37779}, {6, 50461}, {19, 1749}, {25, 11063}, {112, 47053}, {186, 40604}, {393, 37943}, {1263, 343}, {1291, 4558}, {1990, 10272}, {2489, 6140}, {2501, 45147}, {3471, 11064}, {8737, 51267}, {8738, 51274}, {8739, 5616}, {8740, 5612}, {8749, 3470}, {8754, 10413}, {8882, 1157}, {11071, 265}, {13582, 69}, {14579, 3}, {16080, 46751}, {18384, 56404}, {43657, 31626}, {43704, 394}, {44427, 45790}, {46072, 40709}, {46076, 40710}, {47230, 8562}, {51804, 63}, {52166, 15766}, {52418, 2914}, {62268, 19306}, {64935, 525}, {64936, 36212}, {64937, 647}, {64938, 3265}, {65279, 4563}
X(68638) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1263, 3471, 43704}, {1263, 38539, 59492}, {1263, 43704, 13582}

X(68639) = X(4)X(520)∩X(264)X(3265)

Barycentrics    b^2*(b - c)*c^2*(b + c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^8 + 2*a^6*b^2 - 6*a^4*b^4 + 2*a^2*b^6 + b^8 - 3*a^6*c^2 + 3*a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 3*b^6*c^2 + 3*a^4*c^4 - 4*a^2*b^2*c^4 + 3*b^4*c^4 - a^2*c^6 - b^2*c^6)*(-a^8 + 3*a^6*b^2 - 3*a^4*b^4 + a^2*b^6 - 2*a^6*c^2 - 3*a^4*b^2*c^2 + 4*a^2*b^4*c^2 + b^6*c^2 + 6*a^4*c^4 - 3*a^2*b^2*c^4 - 3*b^4*c^4 - 2*a^2*c^6 + 3*b^2*c^6 - c^8) : :

X(68639) lies on the Huygens hyperbola and these lines: {4, 520}, {264, 3265}, {393, 647}, {523, 1093}, {1217, 2416}, {1294, 1300}, {1304, 58071}, {2430, 40402}, {6344, 43083}, {6526, 66299}, {8884, 23286}, {14249, 62350}, {14380, 65488}, {18808, 58261}, {32085, 58353}, {35142, 54988}, {41013, 57109}

X(68639) = X(3134)-cross conjugate of X(4)
X(68639) = X(i)-isoconjugate of X(j) for these (i,j): {163, 44436}, {255, 46587}, {2404, 4100}, {2442, 6507}, {4575, 6000}, {32676, 62347}, {36034, 40948}
X(68639) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 44436}, {136, 6000}, {3258, 40948}, {6523, 46587}, {15526, 62347}
X(68639) = trilinear pole of line {2501, 3269}
X(68639) = barycentric product X(i)*X(j) for these {i,j}: {339, 32646}, {1093, 2416}, {1294, 14618}, {2052, 43701}, {2394, 58085}, {2501, 54988}, {20902, 36043}, {43665, 56605}
X(68639) = barycentric quotient X(i)/X(j) for these {i,j}: {393, 46587}, {523, 44436}, {525, 62347}, {1093, 2404}, {1294, 4558}, {1637, 40948}, {2416, 3964}, {2430, 1092}, {2433, 39174}, {2501, 6000}, {6524, 2442}, {18808, 57488}, {32646, 250}, {43665, 36893}, {43701, 394}, {54988, 4563}, {56605, 2421}, {58085, 2407}, {66299, 51358}, {68327, 47433}

X(68640) = X(4)X(525)∩X(264)X(2419)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 + a^2*c^4 + b^2*c^4 - 2*c^6)*(-a^6 - a^2*b^4 + 2*b^6 + a^4*c^2 - b^4*c^2 + a^2*c^4 - c^6) : :

X(68640) lies on the orthic-asymptotic hyperbola, the Huygens hyperbola, and these lines: {4, 525}, {225, 57243}, {264, 2419}, {393, 523}, {879, 6531}, {935, 32687}, {1093, 14618}, {1297, 1300}, {1316, 51937}, {1826, 4064}, {2409, 2966}, {4580, 32085}, {5489, 6526}, {6330, 14977}, {6344, 14592}, {6524, 55273}, {6587, 60341}, {6619, 23616}, {8057, 56013}, {8767, 68571}, {8884, 15412}, {10002, 65612}, {10723, 14944}, {16263, 64788}, {18850, 66077}, {35140, 35142}, {36092, 60056}, {44770, 60053}

X(68640) = polar conjugate of X(34211)
X(68640) = polar conjugate of the isotomic conjugate of X(43673)
X(68640) = polar conjugate of the isogonal conjugate of X(34212)
X(68640) = X(65265)-Ceva conjugate of X(43717)
X(68640) = X(i)-cross conjugate of X(j) for these (i,j): {868, 4}, {2395, 60338}, {34212, 43673}, {66164, 16080}
X(68640) = X(i)-isoconjugate of X(j) for these (i,j): {48, 34211}, {110, 8766}, {163, 441}, {255, 2409}, {326, 2445}, {662, 8779}, {1092, 24024}, {1503, 4575}, {2312, 4558}, {4592, 42671}, {6507, 23977}, {23997, 34156}
X(68640) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 441}, {136, 1503}, {244, 8766}, {647, 39473}, {1084, 8779}, {1249, 34211}, {5139, 42671}, {6523, 2409}, {15259, 2445}, {16178, 53568}, {38970, 15595}, {48317, 35282}, {50938, 68166}, {61505, 36212}, {62562, 34156}
X(68640) = cevapoint of X(523) and X(16230)
X(68640) = trilinear pole of line {125, 2501}
X(68640) = barycentric product X(i)*X(j) for these {i,j}: {4, 43673}, {125, 65265}, {264, 34212}, {338, 44770}, {339, 32687}, {393, 2419}, {459, 61189}, {523, 6330}, {850, 43717}, {1297, 14618}, {1577, 8767}, {2052, 2435}, {2394, 52485}, {2501, 35140}, {5466, 56601}, {6529, 66964}, {9476, 16230}, {14223, 47105}, {14944, 58759}, {15459, 65759}, {20902, 36092}, {23962, 32649}, {23994, 36046}, {39265, 43665}, {56687, 60338}, {64975, 66299}
X(68640) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 34211}, {125, 39473}, {393, 2409}, {512, 8779}, {523, 441}, {661, 8766}, {1297, 4558}, {2207, 2445}, {2395, 34156}, {2419, 3926}, {2435, 394}, {2489, 42671}, {2501, 1503}, {2970, 66161}, {5466, 36894}, {6330, 99}, {6520, 24024}, {6524, 23977}, {6530, 66076}, {6531, 60506}, {8767, 662}, {9476, 17932}, {14273, 35282}, {14618, 30737}, {14944, 36841}, {14998, 40080}, {16230, 15595}, {16318, 68166}, {17994, 9475}, {18808, 63856}, {32649, 23357}, {32687, 250}, {34212, 3}, {35140, 4563}, {36046, 1101}, {39265, 2421}, {43673, 69}, {43717, 110}, {44770, 249}, {47105, 14999}, {47236, 53568}, {51513, 51363}, {51822, 14966}, {52485, 2407}, {53149, 51963}, {55208, 51647}, {56601, 5468}, {58757, 16318}, {58759, 16096}, {60338, 56572}, {61189, 37669}, {62519, 51960}, {65265, 18020}, {65472, 51434}, {65759, 41077}, {66299, 60516}, {66964, 4143}, {67185, 43754}, {68327, 6793}
X(68640) = {X(43673),X(61189)}-harmonic conjugate of X(2435)

X(68641) = X(4)X(4846)∩X(264)X(403)

Barycentrics    b^2*c^2*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^4 + 4*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 + 4*a^2*c^2 - 2*b^2*c^2 + c^4) : :

X(68641) lies on the Huygens hyperbola and these lines: {4, 4846}, {24, 1105}, {25, 1300}, {93, 44958}, {235, 847}, {254, 3089}, {264, 403}, {378, 52147}, {393, 33885}, {1179, 1598}, {1217, 3542}, {1594, 66596}, {2052, 60119}, {5523, 68572}, {6240, 18848}, {6353, 18852}, {6526, 13450}, {6622, 18853}, {6623, 52487}, {7487, 51471}, {7576, 16263}, {8884, 10594}, {14618, 18808}, {14860, 35488}, {17983, 43976}, {18533, 18850}, {18560, 64844}, {34484, 59278}, {35142, 58782}, {44145, 68564}, {44146, 55972}, {52552, 60696}

X(68641) = polar conjugate of X(15066)
X(68641) = polar conjugate of the isotomic conjugate of X(34289)
X(68641) = polar conjugate of the isogonal conjugate of X(34288)
X(68641) = X(i)-cross conjugate of X(j) for these (i,j): {1596, 4}, {34288, 34289}
X(68641) = X(i)-isoconjugate of X(j) for these (i,j): {48, 15066}, {63, 5063}, {255, 378}, {304, 52438}, {326, 44080}, {2169, 5891}, {4575, 8675}, {4592, 42660}, {9247, 32833}, {10564, 35200}, {44134, 52430}
X(68641) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 10564}, {136, 8675}, {1249, 15066}, {3162, 5063}, {5139, 42660}, {6523, 378}, {14363, 5891}, {15259, 44080}, {62576, 32833}
X(68641) = cevapoint of X(4) and X(62961)
X(68641) = barycentric product X(i)*X(j) for these {i,j}: {4, 34289}, {264, 34288}, {393, 57819}, {1302, 14618}, {2052, 4846}, {2501, 65284}, {16081, 56925}, {39263, 56270}, {46106, 60119}, {65323, 66299}
X(68641) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 15066}, {25, 5063}, {53, 5891}, {264, 32833}, {393, 378}, {1302, 4558}, {1974, 52438}, {1990, 10564}, {2052, 44134}, {2207, 44080}, {2489, 42660}, {2501, 8675}, {4846, 394}, {6531, 11653}, {14618, 30474}, {32738, 32661}, {34288, 3}, {34289, 69}, {36149, 4575}, {41013, 42704}, {52165, 47391}, {52661, 62628}, {56925, 36212}, {57819, 3926}, {60119, 14919}, {65284, 4563}

X(68642) = X(4)X(74)∩X(264)X(339)

Barycentrics    (a^2 + b^2 - c^2)^2*(a^2 - b^2 + c^2)^2*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4) : :

X(68642) lies on the Huygens hyperbola and these lines: {2, 18850}, {3, 18848}, {4, 74}, {5, 1105}, {20, 18846}, {25, 16263}, {53, 51544}, {98, 57608}, {113, 648}, {115, 393}, {186, 14989}, {225, 36119}, {235, 8884}, {250, 16934}, {254, 56686}, {264, 339}, {265, 11251}, {275, 46090}, {376, 18847}, {403, 1300}, {546, 14860}, {631, 18849}, {847, 14249}, {1093, 2970}, {1217, 3091}, {1294, 1650}, {1514, 51358}, {1552, 10151}, {1593, 64844}, {1596, 16264}, {1826, 21046}, {2052, 60119}, {3081, 54512}, {3090, 18851}, {3545, 18852}, {3832, 18855}, {3839, 46808}, {3854, 62730}, {3855, 18853}, {4240, 10733}, {5627, 6344}, {6330, 62563}, {6530, 9139}, {6531, 20031}, {6624, 18918}, {6748, 18877}, {6761, 51403}, {7507, 46147}, {9717, 41372}, {10002, 68564}, {11060, 52418}, {11185, 55972}, {13450, 15424}, {13611, 64505}, {14380, 65488}, {14385, 16868}, {15291, 40065}, {15468, 44990}, {16077, 35142}, {16229, 32112}, {17702, 52913}, {18507, 20127}, {18808, 68327}, {18854, 61964}, {20774, 67478}, {23582, 65729}, {25641, 30716}, {30786, 42308}, {32111, 51939}, {33971, 68566}, {34208, 36875}, {34297, 66899}, {34329, 44872}, {36162, 44985}, {36164, 44992}, {37765, 47332}, {37943, 57471}, {39375, 46456}, {39464, 46065}, {41368, 48451}, {43817, 68536}, {44958, 59278}, {46927, 67201}, {51342, 68009}, {52172, 61462}, {52236, 57632}, {52448, 52487}, {52475, 58757}, {57587, 64510}, {65980, 68549}

X(68642) = isogonal conjugate of X(51394)
X(68642) = polar conjugate of X(11064)
X(68642) = isogonal conjugate of the complement of X(50435)
X(68642) = polar conjugate of the isotomic conjugate of X(16080)
X(68642) = polar conjugate of the isogonal conjugate of X(8749)
X(68642) = X(42308)-Ceva conjugate of X(15459)
X(68642) = X(i)-cross conjugate of X(j) for these (i,j): {8749, 16080}, {10151, 4}, {12079, 18808}, {35235, 66299}, {47236, 648}, {51385, 1093}, {52646, 68546}, {68327, 6529}
X(68642) = X(i)-isoconjugate of X(j) for these (i,j): {1, 51394}, {30, 255}, {48, 11064}, {63, 3284}, {163, 41077}, {326, 1495}, {394, 2173}, {577, 14206}, {662, 1636}, {822, 2407}, {1092, 1784}, {1101, 1650}, {1102, 14581}, {1259, 51654}, {1568, 2169}, {1813, 14395}, {1990, 6507}, {2289, 6357}, {2420, 24018}, {2631, 4558}, {3260, 52430}, {3682, 51420}, {3926, 9406}, {3990, 18653}, {4100, 46106}, {4575, 9033}, {4592, 9409}, {6149, 51254}, {7125, 7359}, {14585, 46234}, {15394, 52948}, {16163, 35200}, {22341, 51382}, {24001, 32320}, {36034, 68167}, {40152, 52949}, {52613, 56829}
X(68642) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 51394}, {115, 41077}, {133, 16163}, {136, 9033}, {523, 1650}, {1084, 1636}, {1249, 11064}, {3162, 3284}, {3258, 68167}, {5139, 9409}, {6523, 30}, {9410, 3926}, {14363, 1568}, {14993, 51254}, {15259, 1495}, {36896, 394}, {50937, 40948}, {62606, 3964}, {67191, 62569}
X(68642) = cevapoint of X(i) and X(j) for these (i,j): {4, 403}, {25, 52418}, {115, 68327}, {12079, 18808}, {51385, 52646}
X(68642) = crosspoint of X(15459) and X(42308)
X(68642) = trilinear pole of line {393, 2433}
X(68642) = barycentric product X(i)*X(j) for these {i,j}: {4, 16080}, {74, 2052}, {92, 36119}, {107, 2394}, {115, 42308}, {158, 2349}, {264, 8749}, {393, 1494}, {459, 10152}, {523, 15459}, {648, 18808}, {850, 32695}, {1093, 14919}, {1096, 33805}, {1304, 14618}, {2159, 57806}, {2433, 6528}, {2501, 16077}, {5627, 14165}, {6344, 57487}, {6521, 35200}, {6529, 34767}, {8794, 44715}, {8884, 62722}, {9139, 37778}, {12079, 23582}, {14380, 15352}, {16081, 35908}, {16263, 46808}, {18022, 40354}, {18027, 40352}, {20031, 65973}, {24006, 65263}, {40351, 44161}, {40384, 52661}, {40388, 44138}, {44769, 66299}, {46106, 68546}, {51385, 66764}, {52475, 65350}, {52493, 65359}
X(68642) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 11064}, {6, 51394}, {25, 3284}, {53, 1568}, {74, 394}, {107, 2407}, {115, 1650}, {158, 14206}, {393, 30}, {403, 62569}, {512, 1636}, {523, 41077}, {1093, 46106}, {1096, 2173}, {1118, 6357}, {1304, 4558}, {1494, 3926}, {1637, 68167}, {1857, 7359}, {1989, 51254}, {1990, 16163}, {2052, 3260}, {2159, 255}, {2207, 1495}, {2349, 326}, {2394, 3265}, {2433, 520}, {2489, 9409}, {2501, 9033}, {2970, 65753}, {5317, 51420}, {6344, 57482}, {6520, 1784}, {6524, 1990}, {6529, 4240}, {6530, 51389}, {6531, 35912}, {8744, 16165}, {8745, 51393}, {8747, 18653}, {8748, 51382}, {8749, 3}, {8794, 43752}, {8884, 43768}, {10152, 37669}, {10412, 18557}, {12079, 15526}, {14165, 6148}, {14380, 52613}, {14398, 58345}, {14569, 52945}, {14618, 66073}, {14919, 3964}, {15262, 59495}, {15291, 35602}, {15459, 99}, {15475, 18558}, {15627, 1259}, {16077, 4563}, {16080, 69}, {16263, 46809}, {17986, 65722}, {18344, 14395}, {18384, 56399}, {18808, 525}, {18877, 1092}, {20031, 65776}, {22455, 56266}, {27376, 51360}, {32695, 110}, {32713, 2420}, {32715, 32661}, {33971, 51372}, {34767, 4143}, {35200, 6507}, {35907, 64607}, {35908, 36212}, {35910, 51386}, {36119, 63}, {36126, 24001}, {36131, 4575}, {36417, 9407}, {40351, 14575}, {40352, 577}, {40354, 184}, {40355, 50433}, {40388, 5504}, {41489, 11589}, {42308, 4590}, {44084, 47405}, {44693, 3719}, {44705, 14345}, {51385, 62583}, {51513, 14391}, {51544, 63425}, {52418, 1511}, {52439, 14581}, {52475, 14417}, {52646, 44436}, {52661, 36789}, {57487, 52437}, {57488, 62347}, {57806, 46234}, {58757, 1637}, {60428, 5642}, {62722, 52347}, {65263, 4592}, {65478, 57290}, {66299, 41079}, {68327, 14401}, {68546, 14919}
X(68642) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 74, 10152}, {4, 5667, 13202}, {133, 7687, 4}, {403, 34150, 1304}, {1304, 34150, 10421}, {1552, 12079, 68546}, {6530, 17986, 15459}, {10151, 12079, 1552}, {10152, 16080, 74}, {13202, 47204, 5667}

X(68643) = X(4)X(3414)∩X(230)X(231)

Barycentrics    (b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4 - (a^2 - b^2 - c^2)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]) : :

X(68643) lies on the Feuerbach circumhyperbola of the orthic triangle, the Huygens hyperbola, and these lines: {4, 3414}, {230, 231}, {393, 5639}, {648, 6189}, {1300, 1379}, {3413, 44427}, {6190, 35142}, {6531, 66187}, {8754, 39022}, {13636, 40138}, {13722, 18808}, {17983, 57014}, {35913, 65563}, {41881, 63535}, {52723, 68564}

X(68643) = polar-circle-inverse of X(31863)
X(68643) = polar conjugate of X(6189)
X(68643) = polar conjugate of the isotomic conjugate of X(3414)
X(68643) = polar conjugate of the isogonal conjugate of X(5639)
X(68643) = X(57013)-Ceva conjugate of X(4)
X(68643) = X(5639)-cross conjugate of X(3414)
X(68643) = X(i)-isoconjugate of X(j) for these (i,j): {48, 6189}, {63, 1380}, {255, 57013}, {3413, 4575}, {4592, 5638}, {35200, 67680}, {36060, 66626}, {62719, 66885}
X(68643) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 67680}, {136, 3413}, {1249, 6189}, {1560, 66626}, {3162, 1380}, {5139, 5638}, {6523, 57013}, {13636, 525}, {39022, 69}, {39067, 3}, {48317, 52722}, {62560, 4563}
X(68643) = cevapoint of X(66187) and X(66884)
X(68643) = crosspoint of X(4) and X(57013)
X(68643) = trilinear pole of line {2501, 13722}
X(68643) = barycentric product X(i)*X(j) for these {i,j}: {4, 3414}, {264, 5639}, {523, 57014}, {648, 13722}, {1379, 14618}, {2501, 6190}, {6331, 66884}, {17983, 52723}, {18020, 66187}, {18808, 67691}, {39022, 57013}, {46463, 65350}
X(68643) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 6189}, {25, 1380}, {393, 57013}, {468, 66626}, {1379, 4558}, {1990, 67680}, {2489, 5638}, {2501, 3413}, {2971, 66885}, {3414, 69}, {5639, 3}, {6190, 4563}, {8754, 13636}, {13722, 525}, {14273, 52722}, {46463, 14417}, {52723, 6390}, {57013, 57575}, {57014, 99}, {66187, 125}, {66884, 647}

X(68644) = X(4)X(3413)∩X(230)X(231)

Barycentrics    (b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4 + (a^2 - b^2 - c^2)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]) : :

X(68644) lies on the Feuerbach circumhyperbola of the orthic triangle, the Huygens hyperbola, and these lines: {4, 3413}, {230, 231}, {393, 5638}, {648, 6190}, {1249, 40991}, {1300, 1380}, {3183, 40993}, {3414, 44427}, {6189, 35142}, {6531, 66186}, {8754, 39023}, {13636, 18808}, {13722, 40138}, {17983, 57013}, {35914, 65563}, {41880, 63535}, {52722, 68564}

X(68644) = polar-circle-inverse of X(31862)
X(68644) = polar conjugate of X(6190)
X(68644) = polar conjugate of the isotomic conjugate of X(3413)
X(68644) = polar conjugate of the isogonal conjugate of X(5638)
X(68644) = X(57014)-Ceva conjugate of X(4)
X(68644) = X(5638)-cross conjugate of X(3413)
X(68644) = X(i)-isoconjugate of X(j) for these (i,j): {48, 6190}, {63, 1379}, {255, 57014}, {3414, 4575}, {4592, 5639}, {35200, 67691}, {36060, 66625}, {62719, 66884}
X(68644) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 67691}, {136, 3414}, {1249, 6190}, {1560, 66625}, {3162, 1379}, {5139, 5639}, {6523, 57014}, {13722, 525}, {39023, 69}, {39068, 3}, {48317, 52723}, {62561, 4563}
X(68644) = cevapoint of X(66186) and X(66885)
X(68644) = crosspoint of X(4) and X(57014)
X(68644) = trilinear pole of line {2501, 13636}
X(68644) = barycentric product X(i)*X(j) for these {i,j}: {4, 3413}, {264, 5638}, {523, 57013}, {648, 13636}, {1380, 14618}, {2501, 6189}, {6331, 66885}, {17983, 52722}, {18020, 66186}, {18808, 67680}, {39023, 57014}, {46462, 65350}
X(68644) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 6190}, {25, 1379}, {393, 57014}, {468, 66625}, {1380, 4558}, {1990, 67691}, {2489, 5639}, {2501, 3414}, {2971, 66884}, {3413, 69}, {5638, 3}, {6189, 4563}, {8754, 13722}, {13636, 525}, {14273, 52723}, {46462, 14417}, {52722, 6390}, {57013, 99}, {57014, 57576}, {66186, 125}, {66885, 647}

X(68645) = X(4)X(80)∩X(264)X(20566)

Barycentrics    b*(-a + b - c)*(a + b - c)*c*(b + c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(-a^2 + b^2 + a*c - c^2) : :

X(68645) lies on the Huygens hyperbola and these lines: {4, 80}, {93, 2599}, {158, 8884}, {225, 52383}, {254, 56417}, {264, 20566}, {393, 64835}, {1300, 2222}, {1411, 68574}, {1835, 66289}, {2006, 68578}, {2970, 53982}, {15065, 41013}, {17983, 65329}, {18808, 66272}, {18815, 62771}, {34535, 68571}, {35142, 35174}, {52409, 68577}

X(68645) = polar conjugate of the isotomic conjugate of X(60091)
X(68645) = X(i)-isoconjugate of X(j) for these (i,j): {21, 52407}, {36, 283}, {63, 4282}, {215, 57985}, {255, 17515}, {284, 22128}, {332, 52434}, {643, 22379}, {654, 4558}, {1437, 4511}, {1444, 2361}, {1789, 6149}, {1790, 2323}, {1792, 52440}, {1812, 7113}, {2193, 3218}, {2245, 65568}, {3615, 22115}, {3738, 4575}, {3904, 32661}, {4592, 8648}, {4996, 57736}, {6369, 15958}, {6514, 52413}, {17206, 52426}
X(68645) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 3738}, {3162, 4282}, {5139, 8648}, {6523, 17515}, {14993, 1789}, {15898, 283}, {36909, 1792}, {39060, 55237}, {40590, 22128}, {40611, 52407}, {47345, 3218}, {53982, 4996}, {55060, 22379}
X(68645) = trilinear pole of line {2501, 8736}
X(68645) = barycentric product X(i)*X(j) for these {i,j}: {4, 60091}, {80, 40149}, {92, 52383}, {94, 1825}, {225, 18359}, {278, 15065}, {331, 34857}, {523, 65329}, {648, 66272}, {655, 24006}, {860, 34535}, {1441, 64835}, {1826, 18815}, {1880, 20566}, {2006, 41013}, {2052, 52391}, {2161, 57809}, {2222, 14618}, {2501, 35174}, {2599, 65360}, {6187, 52575}, {6344, 16577}, {6358, 68571}, {8736, 14616}, {14628, 68563}, {18026, 55238}, {18817, 21741}, {24624, 56285}, {36910, 68576}, {46102, 66289}, {47318, 66297}, {52356, 52607}, {60074, 61178}, {62735, 66300}, {65207, 66284}, {65299, 66299}
X(68645) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 4282}, {65, 22128}, {80, 1812}, {225, 3218}, {393, 17515}, {655, 4592}, {759, 65568}, {1400, 52407}, {1411, 1790}, {1807, 6514}, {1824, 2323}, {1825, 323}, {1826, 4511}, {1874, 27950}, {1877, 17191}, {1880, 36}, {1989, 1789}, {2006, 1444}, {2161, 283}, {2222, 4558}, {2333, 2361}, {2489, 8648}, {2501, 3738}, {6187, 2193}, {7180, 22379}, {8736, 758}, {15065, 345}, {16577, 52437}, {18026, 55237}, {18359, 332}, {18815, 17206}, {21741, 22115}, {24006, 3904}, {32675, 4575}, {34535, 57985}, {34857, 219}, {35174, 4563}, {36910, 1792}, {40149, 320}, {41013, 32851}, {44113, 34544}, {46405, 55202}, {51513, 2600}, {52356, 15411}, {52371, 2327}, {52383, 63}, {52391, 394}, {52575, 40075}, {55206, 53285}, {55208, 53314}, {55238, 521}, {56285, 3936}, {57652, 7113}, {57809, 20924}, {58756, 62734}, {58757, 65104}, {60091, 69}, {61178, 4585}, {64835, 21}, {65329, 99}, {66272, 525}, {66289, 26932}, {66297, 4707}, {68571, 2185}, {68576, 17078}

X(68646) = X(4)X(661)∩X(264)X(1577)

Barycentrics    (b - c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-(a^2*b^2) + b^4 - a^3*c + a*b^2*c + 2*a^2*c^2 - b^2*c^2 - a*c^3)*(a^3*b - 2*a^2*b^2 + a*b^3 + a^2*c^2 - a*b*c^2 + b^2*c^2 - c^4) : :

X(68646) lies on the Huygens hyperbola and these lines: {4, 661}, {225, 770}, {264, 1577}, {847, 55250}, {1021, 1105}, {1300, 2249}, {1826, 4705}, {1981, 41207}, {4024, 41013}, {17924, 42462}, {17983, 23894}, {23893, 65335}, {32085, 55240}, {35142, 35145}, {57094, 68581}, {59041, 59088}

X(68646) = X(i)-isoconjugate of X(j) for these (i,j): {109, 6518}, {184, 15418}, {249, 9391}, {255, 1981}, {394, 23353}, {851, 4558}, {906, 5088}, {1332, 26884}, {1813, 1936}, {1944, 36059}, {1951, 6516}, {2202, 6517}, {4563, 44112}, {4575, 8680}, {4592, 42669}, {23971, 65433}, {32661, 44150}
X(68646) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 6518}, {136, 8680}, {5139, 42669}, {5190, 5088}, {6523, 1981}, {20620, 1944}, {38966, 58325}, {62605, 15418}
X(68646) = trilinear pole of line {2501, 2643}
X(68646) = barycentric product X(i)*X(j) for these {i,j}: {1109, 59041}, {1937, 44426}, {1945, 46110}, {1952, 3064}, {2052, 52222}, {2249, 14618}, {2501, 35145}, {21044, 41207}, {24006, 37142}, {42069, 53211}, {46107, 61427}
X(68646) = barycentric quotient X(i)/X(j) for these {i,j}: {92, 15418}, {296, 6517}, {393, 1981}, {650, 6518}, {1096, 23353}, {1937, 6516}, {1945, 1813}, {1952, 65164}, {2249, 4558}, {2489, 42669}, {2501, 8680}, {2643, 9391}, {3064, 1944}, {7649, 5088}, {18344, 1936}, {24006, 44150}, {24010, 65433}, {35145, 4563}, {37142, 4592}, {41207, 4620}, {52222, 394}, {55208, 51645}, {57980, 55202}, {59041, 24041}, {61427, 1331}, {65103, 58325}

X(68647) = X(3)X(49)∩X(99)X(110)

Barycentrics    a^4*(a^2 - b^2)*(a^2 - c^2)*(a^2 - b^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4) : :

X(68647) lies on these lines: {3, 49}, {22, 67352}, {99, 110}, {237, 23098}, {511, 21525}, {868, 47049}, {1303, 59039}, {1316, 9306}, {1993, 31850}, {2421, 4230}, {3150, 11064}, {4558, 65310}, {5651, 7697}, {7422, 43574}, {9155, 40083}, {10420, 67807}, {14966, 68178}, {15920, 35922}, {34986, 67349}, {36213, 62590}, {43754, 65327}, {44077, 54096}, {51373, 54332}, {51389, 52128}

X(68647) = isotomic conjugate of the polar conjugate of X(14966)
X(68647) = isogonal conjugate of the polar conjugate of X(2421)
X(68647) = X(i)-Ceva conjugate of X(j) for these (i,j): {2421, 14966}, {17932, 4558}, {47390, 65748}, {57763, 62590}
X(68647) = X(i)-cross conjugate of X(j) for these (i,j): {38354, 3}, {39469, 3289}, {65748, 47390}
X(68647) = X(i)-isoconjugate of X(j) for these (i,j): {19, 43665}, {75, 53149}, {92, 2395}, {98, 24006}, {158, 879}, {293, 66299}, {336, 58757}, {338, 36104}, {523, 36120}, {661, 16081}, {685, 1109}, {798, 60199}, {811, 51441}, {823, 51404}, {878, 57806}, {1577, 6531}, {1821, 2501}, {1910, 14618}, {1969, 2422}, {2190, 61196}, {2489, 46273}, {2643, 22456}, {2970, 36084}, {6520, 53173}, {8754, 36036}, {15630, 57968}, {20031, 20902}, {20948, 57260}, {23994, 32696}, {56285, 60568}
X(68647) = X(i)-Dao conjugate of X(j) for these (i,j): {5, 61196}, {6, 43665}, {132, 66299}, {206, 53149}, {511, 16230}, {1147, 879}, {2679, 8754}, {11672, 14618}, {17423, 51441}, {22391, 2395}, {31998, 60199}, {36830, 16081}, {37867, 53173}, {38987, 2970}, {39000, 338}, {40601, 2501}, {41167, 23105}, {46094, 523}, {52878, 51513}, {55071, 35235}, {57294, 44114}, {62590, 850}
X(68647) = cevapoint of X(i) and X(j) for these (i,j): {684, 44716}, {3289, 39469}
X(68647) = crosspoint of X(4558) and X(17932)
X(68647) = crosssum of X(i) and X(j) for these (i,j): {2395, 53149}, {2501, 17994}
X(68647) = trilinear pole of line {3289, 46094}
X(68647) = crossdifference of every pair of points on line {2501, 2970}
X(68647) = barycentric product X(i)*X(j) for these {i,j}: {3, 2421}, {63, 23997}, {69, 14966}, {99, 3289}, {110, 36212}, {112, 51386}, {184, 2396}, {237, 4563}, {248, 15631}, {249, 684}, {255, 62720}, {287, 68178}, {325, 32661}, {394, 4230}, {511, 4558}, {577, 877}, {906, 51369}, {1331, 17209}, {1437, 42717}, {1576, 6393}, {1755, 4592}, {1959, 4575}, {2491, 47389}, {2799, 47390}, {2966, 65748}, {3964, 58070}, {4176, 34859}, {4590, 39469}, {6333, 23357}, {9155, 65321}, {9417, 55202}, {9418, 52608}, {10425, 47406}, {11672, 17932}, {15958, 60524}, {18315, 44716}, {18877, 66074}, {22115, 66075}, {32656, 51370}, {32662, 51383}, {36213, 65327}, {36790, 43754}, {40804, 62523}, {41172, 59152}, {42702, 52935}, {42743, 65308}, {52091, 56389}, {59707, 65311}
X(68647) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 43665}, {32, 53149}, {99, 60199}, {110, 16081}, {163, 36120}, {184, 2395}, {216, 61196}, {232, 66299}, {237, 2501}, {249, 22456}, {511, 14618}, {577, 879}, {684, 338}, {877, 18027}, {1092, 53173}, {1576, 6531}, {1755, 24006}, {2211, 58757}, {2396, 18022}, {2421, 264}, {2491, 8754}, {3049, 51441}, {3289, 523}, {3569, 2970}, {4230, 2052}, {4558, 290}, {4563, 18024}, {4575, 1821}, {4590, 65272}, {4592, 46273}, {6333, 23962}, {6393, 44173}, {9418, 2489}, {9419, 17994}, {10317, 52076}, {11672, 16230}, {14574, 57260}, {14575, 2422}, {14585, 878}, {14966, 4}, {15631, 44132}, {17209, 46107}, {17932, 57541}, {23098, 68089}, {23181, 53245}, {23200, 52038}, {23357, 685}, {23963, 32696}, {23995, 36104}, {23997, 92}, {32661, 98}, {34157, 60338}, {34859, 6524}, {36212, 850}, {39201, 51404}, {39469, 115}, {41172, 23105}, {41270, 66300}, {42702, 4036}, {42743, 60502}, {43754, 34536}, {44716, 18314}, {47390, 2966}, {51386, 3267}, {51394, 65778}, {52967, 51513}, {56389, 14265}, {57500, 62519}, {57655, 20031}, {58070, 1093}, {58306, 15422}, {59152, 41174}, {62720, 57806}, {65748, 2799}, {66075, 18817}, {68178, 297}

X(68648) = X(3)X(69)∩X(99)X(670)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a^2 - b^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4) : :

X(68648) lies on these lines: on lines {3, 69}, {99, 670}, {110, 35575}, {237, 36892}, {325, 7418}, {511, 34157}, {877, 2396}, {878, 17932}, {1624, 57216}, {1975, 44155}, {2421, 14966}, {2855, 2858}, {4558, 8552}, {4561, 23161}, {4563, 23181}, {4576, 50947}, {5467, 10411}, {5468, 15329}, {9146, 36829}, {15915, 37668}, {18020, 46587}, {32485, 32540}, {32815, 56957}, {32817, 37991}, {36212, 41172}, {40083, 50567}, {51374, 56574}, {52727, 59548}, {55227, 65182}, {62173, 68152}, {62431, 65975}

X(68648) = isogonal conjugate of X(53149)
X(68648) = isotomic conjugate of the polar conjugate of X(2421)
X(68648) = isogonal conjugate of the polar conjugate of X(2396)
X(68648) = X(2396)-Ceva conjugate of X(2421)
X(68648) = X(684)-cross conjugate of X(36212)
X(68648) = X(i)-isoconjugate of X(j) for these (i,j): {1, 53149}, {19, 2395}, {92, 2422}, {115, 36104}, {158, 878}, {162, 51441}, {293, 58757}, {512, 36120}, {661, 6531}, {685, 2643}, {798, 16081}, {811, 15630}, {879, 1096}, {1109, 32696}, {1577, 57260}, {1821, 2489}, {1910, 2501}, {1924, 60199}, {1973, 43665}, {1976, 24006}, {2971, 36036}, {3708, 20031}, {4117, 65272}, {8754, 36084}, {15628, 55208}, {24019, 51404}, {36128, 52038}, {46273, 57204}, {61196, 62268}
X(68648) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 53149}, {6, 2395}, {125, 51441}, {132, 58757}, {511, 17994}, {1147, 878}, {2679, 2971}, {5976, 14618}, {6337, 43665}, {6503, 879}, {9428, 60199}, {11672, 2501}, {17423, 15630}, {22391, 2422}, {31998, 16081}, {35071, 51404}, {35088, 2970}, {36830, 6531}, {38987, 8754}, {39000, 115}, {39040, 24006}, {39054, 36120}, {40601, 2489}, {41167, 8029}, {46094, 512}, {52032, 61196}, {57294, 58260}, {60596, 23290}, {62590, 523}, {62595, 66299}
X(68648) = cevapoint of X(684) and X(36212)
X(68648) = crosspoint of X(i) and X(j) for these (i,j): {99, 10425}, {4558, 65327}
X(68648) = crosssum of X(512) and X(55122)
X(68648) = trilinear pole of line {3289, 36212}
X(68648) = crossdifference of every pair of points on line {1084, 2489}
X(68648) = barycentric product X(i)*X(j) for these {i,j}: {3, 2396}, {69, 2421}, {99, 36212}, {110, 6393}, {237, 52608}, {249, 6333}, {287, 15631}, {304, 23997}, {305, 14966}, {325, 4558}, {326, 62720}, {394, 877}, {511, 4563}, {648, 51386}, {670, 3289}, {684, 4590}, {1331, 51370}, {1332, 51369}, {1444, 42717}, {1755, 55202}, {1959, 4592}, {3569, 47389}, {3926, 4230}, {4176, 58070}, {4561, 17209}, {4575, 46238}, {4623, 42702}, {5976, 65327}, {10425, 62590}, {14919, 66074}, {17932, 36790}, {31614, 41172}, {32458, 43754}, {34537, 39469}, {43187, 65748}, {47406, 65277}, {50567, 65321}, {51371, 65307}, {51373, 65310}, {51374, 65311}, {51383, 60053}, {51397, 65328}, {51439, 65309}, {51651, 55207}, {52437, 66075}, {57799, 68178}
X(68648) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 2395}, {6, 53149}, {69, 43665}, {99, 16081}, {110, 6531}, {184, 2422}, {232, 58757}, {237, 2489}, {249, 685}, {250, 20031}, {297, 66299}, {325, 14618}, {343, 61196}, {394, 879}, {511, 2501}, {520, 51404}, {577, 878}, {647, 51441}, {662, 36120}, {670, 60199}, {684, 115}, {877, 2052}, {1101, 36104}, {1576, 57260}, {1959, 24006}, {2396, 264}, {2421, 4}, {2491, 2971}, {2799, 2970}, {3049, 15630}, {3289, 512}, {3292, 52038}, {3569, 8754}, {3964, 53173}, {4230, 393}, {4558, 98}, {4563, 290}, {4575, 1910}, {4590, 22456}, {4592, 1821}, {6333, 338}, {6393, 850}, {6514, 66881}, {7254, 43920}, {9155, 14273}, {9418, 57204}, {11672, 17994}, {14966, 25}, {14999, 52491}, {15631, 297}, {17209, 7649}, {17932, 34536}, {19189, 15422}, {22115, 60777}, {22151, 52076}, {23181, 60517}, {23357, 32696}, {23997, 19}, {31614, 41174}, {32661, 1976}, {34211, 52641}, {34537, 65272}, {34859, 52439}, {35910, 18808}, {36212, 523}, {36790, 16230}, {39469, 3124}, {40804, 62519}, {41172, 8029}, {41270, 58756}, {42702, 4705}, {42717, 41013}, {42743, 6103}, {43754, 41932}, {44716, 12077}, {47389, 43187}, {47390, 2715}, {47406, 55122}, {51369, 17924}, {51370, 46107}, {51383, 44427}, {51386, 525}, {51439, 57065}, {51440, 67102}, {51651, 55208}, {52091, 60338}, {52608, 18024}, {55202, 46273}, {56389, 51820}, {56437, 16229}, {58070, 6524}, {59152, 60179}, {59707, 57071}, {60524, 23290}, {62523, 32545}, {62719, 36036}, {62720, 158}, {63741, 68572}, {65321, 9154}, {65327, 36897}, {65568, 60568}, {65748, 3569}, {66074, 46106}, {66075, 6344}, {68178, 232}

X(68649) = X(3)X(49)∩X(21)X(60)

Barycentrics    a^3*(a + b)*(a - b - c)*(a + c)*(a^2 - b^2 - c^2)^2 : :

X(68649) lies on these lines: {3, 49}, {21, 60}, {29, 56261}, {46, 62756}, {58, 22766}, {65, 3193}, {69, 5135}, {81, 37277}, {110, 1295}, {255, 820}, {268, 2327}, {343, 7483}, {377, 37669}, {442, 11064}, {453, 1155}, {511, 20832}, {662, 1816}, {1014, 62402}, {1043, 56099}, {1259, 6056}, {1444, 52385}, {1568, 37230}, {1792, 1809}, {1804, 7335}, {2193, 22074}, {2245, 22133}, {2328, 17104}, {4185, 9306}, {4259, 20806}, {4575, 36055}, {5794, 11103}, {6831, 68018}, {7414, 43574}, {10441, 37238}, {10974, 23130}, {11337, 44087}, {13323, 44117}, {13346, 37194}, {19607, 27378}, {35193, 37741}, {37298, 64062}, {57648, 57671}

X(68649) = isotomic conjugate of the polar conjugate of X(2193)
X(68649) = isogonal conjugate of the polar conjugate of X(1812)
X(68649) = X(1812)-Ceva conjugate of X(2193)
X(68649) = X(i)-cross conjugate of X(j) for these (i,j): {255, 283}, {2289, 6514}
X(68649) = X(i)-isoconjugate of X(j) for these (i,j): {4, 225}, {10, 1118}, {12, 8747}, {19, 40149}, {25, 57809}, {27, 8736}, {28, 56285}, {33, 68576}, {34, 41013}, {65, 158}, {73, 1093}, {92, 1880}, {107, 66287}, {108, 24006}, {109, 66299}, {162, 66297}, {226, 393}, {264, 57652}, {273, 1824}, {278, 1826}, {307, 6524}, {313, 7337}, {318, 1426}, {331, 2333}, {349, 2207}, {512, 52938}, {523, 36127}, {653, 2501}, {661, 54240}, {664, 58757}, {823, 57185}, {1096, 1441}, {1119, 53008}, {1214, 6520}, {1254, 1896}, {1400, 2052}, {1402, 57806}, {1409, 6521}, {1857, 3668}, {1865, 40573}, {1877, 68563}, {1973, 52575}, {2358, 64211}, {2489, 46404}, {3064, 52607}, {4516, 24032}, {5236, 68565}, {5317, 6358}, {5930, 6526}, {6335, 55208}, {6354, 8748}, {6528, 66928}, {6529, 57243}, {6591, 65207}, {7649, 61178}, {13149, 55206}, {14618, 32674}, {15352, 55234}, {18097, 27376}, {21044, 23984}, {23706, 68561}, {23710, 68580}, {36434, 52565}, {47372, 52384}, {52919, 55197}, {53009, 55110}, {56827, 68574}
X(68649) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 40149}, {11, 66299}, {125, 66297}, {1147, 65}, {6337, 52575}, {6503, 1441}, {6505, 57809}, {11517, 41013}, {22391, 1880}, {35072, 14618}, {36033, 225}, {36830, 54240}, {37867, 1214}, {38983, 24006}, {38985, 66287}, {39025, 58757}, {39054, 52938}, {40582, 2052}, {40591, 56285}, {40602, 158}, {40605, 57806}
X(68649) = cevapoint of X(i) and X(j) for these (i,j): {255, 1092}, {2289, 6056}
X(68649) = crosssum of X(2501) and X(4516)
X(68649) = crossdifference of every pair of points on line {2501, 57185}
X(68649) = barycentric product X(i)*X(j) for these {i,j}: {1, 6514}, {3, 1812}, {8, 18604}, {21, 394}, {29, 6507}, {48, 332}, {58, 3719}, {59, 16731}, {60, 3998}, {63, 283}, {69, 2193}, {72, 65568}, {77, 2327}, {78, 1790}, {81, 1259}, {86, 2289}, {99, 36054}, {163, 52616}, {212, 17206}, {219, 1444}, {222, 1792}, {255, 333}, {261, 3990}, {274, 6056}, {284, 326}, {314, 577}, {345, 1437}, {520, 4612}, {521, 4558}, {643, 4091}, {645, 23224}, {651, 68151}, {652, 4592}, {662, 57241}, {1021, 6517}, {1043, 7125}, {1092, 31623}, {1098, 40152}, {1102, 2299}, {1172, 3964}, {1264, 1333}, {1275, 66898}, {1332, 23189}, {1364, 4567}, {1414, 57057}, {1793, 22128}, {1800, 6513}, {1804, 2287}, {1808, 20769}, {1813, 57081}, {1819, 41081}, {1946, 4563}, {2150, 52396}, {2185, 3682}, {2194, 3926}, {2204, 4176}, {2328, 7183}, {2638, 4620}, {3193, 6512}, {4055, 52379}, {4100, 44130}, {4131, 5546}, {4565, 68108}, {4571, 7254}, {4573, 58340}, {4575, 6332}, {4601, 61054}, {4631, 39201}, {4636, 24018}, {6385, 62257}, {6516, 23090}, {7054, 52385}, {7058, 22341}, {14585, 40072}, {15411, 36059}, {22074, 57853}, {24031, 52378}, {28660, 52430}, {30805, 65375}, {31631, 60794}, {32661, 35518}, {52613, 52914}, {57134, 65164}, {58338, 65296}
X(68649) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 40149}, {21, 2052}, {29, 6521}, {48, 225}, {63, 57809}, {69, 52575}, {71, 56285}, {110, 54240}, {163, 36127}, {184, 1880}, {212, 1826}, {219, 41013}, {222, 68576}, {228, 8736}, {255, 226}, {283, 92}, {284, 158}, {314, 18027}, {326, 349}, {332, 1969}, {333, 57806}, {394, 1441}, {521, 14618}, {577, 65}, {647, 66297}, {650, 66299}, {652, 24006}, {662, 52938}, {822, 66287}, {906, 61178}, {1092, 1214}, {1172, 1093}, {1259, 321}, {1264, 27801}, {1331, 65207}, {1333, 1118}, {1364, 16732}, {1437, 278}, {1444, 331}, {1790, 273}, {1792, 7017}, {1802, 53008}, {1804, 1446}, {1812, 264}, {1819, 64211}, {1946, 2501}, {2150, 8747}, {2193, 4}, {2194, 393}, {2204, 6524}, {2289, 10}, {2299, 6520}, {2327, 318}, {2638, 21044}, {3063, 58757}, {3682, 6358}, {3694, 7141}, {3719, 313}, {3964, 1231}, {3990, 12}, {3998, 34388}, {4055, 2171}, {4091, 4077}, {4100, 73}, {4558, 18026}, {4575, 653}, {4592, 46404}, {4612, 6528}, {4636, 823}, {6056, 37}, {6507, 307}, {6514, 75}, {7054, 1896}, {7125, 3668}, {7335, 1427}, {9247, 57652}, {14585, 1402}, {16730, 1367}, {16731, 34387}, {17206, 57787}, {18604, 7}, {22074, 429}, {22134, 56827}, {22341, 6354}, {23090, 44426}, {23189, 17924}, {23207, 1865}, {23224, 7178}, {23606, 1409}, {23609, 36421}, {32661, 108}, {36054, 523}, {36059, 52607}, {39201, 57185}, {39687, 4516}, {52370, 7140}, {52378, 24032}, {52411, 1426}, {52425, 1824}, {52430, 1400}, {52616, 20948}, {52914, 15352}, {52949, 52661}, {53560, 2970}, {57057, 4086}, {57081, 46110}, {57134, 3064}, {57241, 1577}, {57657, 1096}, {58340, 3700}, {60794, 60249}, {61054, 3125}, {62257, 213}, {65568, 286}, {66898, 1146}, {68151, 4391}
X(68649) = {X(283),X(1800)}-harmonic conjugate of X(3)

X(68650) = X(3)X(69)∩X(21)X(261)

Barycentrics    a*(a + b)*(a - b - c)*(a + c)*(a^2 - b^2 - c^2)^2 : :

X(68650) lies on these lines: {3, 69}, {21, 261}, {99, 1295}, {255, 326}, {268, 44189}, {305, 37670}, {325, 4231}, {333, 53994}, {1014, 34399}, {1043, 4305}, {1259, 1264}, {1804, 7055}, {1812, 2193}, {2092, 15988}, {2289, 3719}, {3998, 18604}, {4592, 36055}, {5019, 56834}, {13425, 60848}, {13458, 60847}, {26357, 30479}, {34254, 59353}, {34409, 37741}, {44130, 56261}, {57785, 63185}, {57800, 57837}

X(68650) = isotomic conjugate of the polar conjugate of X(1812)
X(68650) = X(i)-cross conjugate of X(j) for these (i,j): {326, 332}, {1259, 6514}
X(68650) = X(i)-isoconjugate of X(j) for these (i,j): {4, 57652}, {10, 7337}, {19, 1880}, {25, 225}, {33, 1426}, {34, 1824}, {42, 1118}, {65, 1096}, {73, 6524}, {107, 66928}, {109, 58757}, {158, 1402}, {181, 8747}, {226, 2207}, {278, 2333}, {307, 52439}, {349, 36417}, {393, 1400}, {512, 36127}, {608, 1826}, {653, 2489}, {669, 52938}, {798, 54240}, {1042, 1857}, {1395, 41013}, {1398, 53008}, {1409, 6520}, {1474, 8736}, {1783, 55208}, {1973, 40149}, {1974, 57809}, {2171, 5317}, {2203, 56285}, {2212, 68576}, {2331, 2358}, {2501, 32674}, {3668, 6059}, {4516, 24033}, {5930, 61349}, {6528, 66975}, {6529, 55234}, {21044, 23985}, {24019, 57185}, {32676, 66297}, {32713, 66287}, {32714, 55206}, {36434, 40152}, {46404, 57204}
X(68650) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 1880}, {11, 58757}, {521, 4516}, {1147, 1402}, {6337, 40149}, {6338, 1441}, {6503, 65}, {6505, 225}, {11517, 1824}, {15526, 66297}, {31998, 54240}, {35071, 57185}, {35072, 2501}, {36033, 57652}, {37867, 1409}, {38985, 66928}, {39006, 55208}, {39054, 36127}, {40582, 393}, {40592, 1118}, {40602, 1096}, {40605, 158}, {40624, 66299}, {40626, 24006}, {51574, 8736}, {62564, 56285}, {62584, 41013}, {62604, 52575}, {62647, 1826}
X(68650) = cevapoint of X(i) and X(j) for these (i,j): {326, 3964}, {1259, 3719}, {16731, 68151}
X(68650) = barycentric product X(i)*X(j) for these {i,j}: {21, 3926}, {29, 1102}, {63, 332}, {69, 1812}, {75, 6514}, {78, 17206}, {81, 1264}, {86, 3719}, {255, 28660}, {261, 3998}, {274, 1259}, {283, 304}, {305, 2193}, {310, 2289}, {314, 394}, {326, 333}, {345, 1444}, {348, 1792}, {520, 4631}, {521, 4563}, {577, 40072}, {643, 30805}, {645, 4131}, {652, 55202}, {662, 52616}, {670, 36054}, {799, 57241}, {1043, 7183}, {1098, 52565}, {1172, 4176}, {1364, 4601}, {1437, 57919}, {1459, 55207}, {1790, 3718}, {1946, 52608}, {2185, 52396}, {2287, 7055}, {2327, 7182}, {3265, 4612}, {3596, 18604}, {3682, 52379}, {3964, 31623}, {3990, 18021}, {4091, 7257}, {4143, 52914}, {4554, 68151}, {4558, 35518}, {4571, 15419}, {4573, 68108}, {4592, 6332}, {4620, 24031}, {4625, 57057}, {4998, 16731}, {6056, 6385}, {6507, 44130}, {6516, 15411}, {7058, 52385}, {20336, 65568}, {23224, 62534}, {47389, 53560}, {55196, 57109}, {55205, 57108}, {57081, 65164}
X(68650) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 1880}, {21, 393}, {29, 6520}, {48, 57652}, {60, 5317}, {63, 225}, {69, 40149}, {72, 8736}, {78, 1826}, {81, 1118}, {99, 54240}, {212, 2333}, {219, 1824}, {222, 1426}, {255, 1400}, {283, 19}, {284, 1096}, {304, 57809}, {305, 52575}, {306, 56285}, {314, 2052}, {326, 226}, {332, 92}, {333, 158}, {345, 41013}, {348, 68576}, {394, 65}, {520, 57185}, {521, 2501}, {525, 66297}, {577, 1402}, {650, 58757}, {662, 36127}, {799, 52938}, {822, 66928}, {1040, 52577}, {1092, 1409}, {1098, 8748}, {1102, 307}, {1172, 6524}, {1259, 37}, {1264, 321}, {1332, 61178}, {1333, 7337}, {1364, 3125}, {1433, 2358}, {1437, 608}, {1444, 278}, {1459, 55208}, {1789, 64834}, {1790, 34}, {1792, 281}, {1793, 64835}, {1800, 52033}, {1801, 20613}, {1804, 1427}, {1812, 4}, {1819, 2331}, {1946, 2489}, {2185, 8747}, {2193, 25}, {2194, 2207}, {2204, 52439}, {2287, 1857}, {2289, 42}, {2327, 33}, {3682, 2171}, {3692, 53008}, {3694, 7140}, {3719, 10}, {3926, 1441}, {3964, 1214}, {3990, 181}, {3998, 12}, {4091, 4017}, {4131, 7178}, {4176, 1231}, {4391, 66299}, {4558, 108}, {4561, 65207}, {4563, 18026}, {4575, 32674}, {4592, 653}, {4612, 107}, {4620, 24032}, {4631, 6528}, {4636, 24019}, {6056, 213}, {6332, 24006}, {6507, 73}, {6514, 1}, {6516, 52607}, {6517, 1020}, {7055, 1446}, {7058, 1896}, {7125, 1042}, {7183, 3668}, {7254, 43923}, {14395, 68327}, {15411, 44426}, {16730, 61058}, {16731, 11}, {17206, 273}, {18604, 56}, {20769, 1874}, {22074, 44092}, {22128, 1835}, {23090, 18344}, {23151, 1893}, {23189, 6591}, {23224, 7180}, {23602, 37384}, {24018, 66287}, {24031, 21044}, {27398, 47372}, {28660, 57806}, {30805, 4077}, {31623, 1093}, {35072, 4516}, {35518, 14618}, {35602, 30456}, {36054, 512}, {40072, 18027}, {40152, 1254}, {44130, 6521}, {52378, 24033}, {52385, 6354}, {52396, 6358}, {52616, 1577}, {52914, 6529}, {53560, 8754}, {55202, 46404}, {55466, 53861}, {57057, 4041}, {57081, 3064}, {57108, 55206}, {57109, 55197}, {57213, 54239}, {57241, 661}, {58338, 65103}, {58340, 3709}, {61054, 3121}, {62257, 2205}, {65568, 28}, {66898, 14936}, {68108, 3700}, {68151, 650}
X(68650) = {X(1444),X(1792)}-harmonic conjugate of X(332)

X(68651) = X(3)X(49)∩X(110)X(384)

Barycentrics    a^6*(a^2 - b^2 - c^2)*(b^2 + c^2) : :

X(68651) lies on these lines: {3, 49}, {6, 41277}, {32, 3202}, {39, 3203}, {54, 1976}, {110, 384}, {157, 15648}, {182, 11285}, {187, 40643}, {206, 5017}, {217, 682}, {578, 13860}, {1176, 60702}, {1614, 11676}, {1899, 28407}, {1970, 46272}, {2056, 3499}, {3044, 5152}, {3051, 14820}, {3289, 4173}, {3552, 9544}, {5012, 7824}, {5999, 34148}, {6467, 14965}, {7470, 43574}, {7751, 54332}, {7770, 9306}, {7807, 36213}, {8356, 14133}, {10282, 37123}, {10539, 35930}, {10547, 17970}, {11003, 33004}, {11360, 47638}, {11634, 14135}, {13366, 43843}, {14575, 14585}, {20021, 37125}, {20968, 52438}, {22401, 58355}, {38654, 57011}, {40111, 44224}, {41334, 63556}, {50659, 64028}, {50672, 68513}, {50685, 64052}, {51887, 58306}

X(68651) = isogonal conjugate of the isotomic conjugate of X(20775)
X(68651) = isotomic conjugate of the polar conjugate of X(41331)
X(68651) = isogonal conjugate of the polar conjugate of X(3051)
X(68651) = X(i)-Ceva conjugate of X(j) for these (i,j): {184, 20775}, {3051, 41331}
X(68651) = X(i)-isoconjugate of X(j) for these (i,j): {4, 18833}, {19, 40016}, {75, 46104}, {82, 18022}, {83, 1969}, {92, 308}, {264, 3112}, {273, 62539}, {286, 56251}, {561, 32085}, {689, 24006}, {798, 42395}, {811, 52618}, {1799, 57806}, {2501, 37204}, {3405, 60199}, {4580, 57973}, {4593, 14618}, {6331, 18070}, {18027, 34055}, {18082, 57796}, {20948, 42396}, {39287, 62273}, {44129, 56186}, {44161, 46289}, {57968, 58784}
X(68651) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 40016}, {39, 44161}, {141, 18022}, {206, 46104}, {17423, 52618}, {22391, 308}, {31998, 42395}, {34452, 264}, {36033, 18833}, {40368, 32085}, {52042, 1235}, {55050, 14618}
X(68651) = crosspoint of X(i) and X(j) for these (i,j): {184, 14575}, {3051, 20775}
X(68651) = crosssum of X(i) and X(j) for these (i,j): {76, 7752}, {264, 18022}, {308, 46104}
X(68651) = barycentric product X(i)*X(j) for these {i,j}: {3, 3051}, {6, 20775}, {31, 4020}, {32, 3917}, {38, 9247}, {39, 184}, {48, 1964}, {63, 1923}, {69, 41331}, {97, 27374}, {141, 14575}, {217, 16030}, {394, 27369}, {427, 14585}, {520, 61218}, {577, 1843}, {603, 40972}, {688, 4558}, {906, 50521}, {1176, 59994}, {1235, 61361}, {1401, 52425}, {1437, 21814}, {1501, 3933}, {1634, 3049}, {1790, 41267}, {2084, 4575}, {2200, 17187}, {2525, 14574}, {2531, 65307}, {3005, 32661}, {3289, 51869}, {3292, 41272}, {3688, 52411}, {3787, 40319}, {4563, 9494}, {8024, 40373}, {8041, 10547}, {8623, 17970}, {17442, 52430}, {19606, 20794}, {21123, 32656}, {23200, 46154}, {23208, 60495}, {23210, 42346}, {23225, 46163}, {23606, 27376}, {27371, 62256}, {35319, 58308}, {35325, 39201}, {36214, 56915}, {37894, 57503}, {41676, 58310}
X(68651) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 40016}, {32, 46104}, {39, 18022}, {48, 18833}, {99, 42395}, {141, 44161}, {184, 308}, {688, 14618}, {1501, 32085}, {1843, 18027}, {1923, 92}, {1964, 1969}, {2200, 56251}, {3049, 52618}, {3051, 264}, {3917, 1502}, {3933, 40362}, {4020, 561}, {4558, 42371}, {4575, 37204}, {9247, 3112}, {9494, 2501}, {14533, 41488}, {14574, 42396}, {14575, 83}, {14585, 1799}, {16030, 57790}, {20775, 76}, {23209, 16890}, {23216, 34294}, {27369, 2052}, {27374, 324}, {32661, 689}, {40373, 251}, {41272, 46111}, {41331, 4}, {42548, 42394}, {51869, 60199}, {52425, 62539}, {56915, 17984}, {57503, 37892}, {58310, 4580}, {59994, 1235}, {61218, 6528}, {61361, 1176}, {62270, 39287}
X(68651) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 3202, 9418}, {217, 682, 65751}, {3051, 27369, 27374}, {14575, 14585, 40373}

X(68652) = X(3)X(49)∩X(6)X(1408)

Barycentrics    a^3*(a^2 + a*b + a*c + 2*b*c)*(a^2 - b^2 - c^2) : :

X(68652) lies on these lines: {1, 5197}, {2, 13323}, {3, 49}, {6, 1408}, {21, 9306}, {25, 36746}, {32, 45785}, {48, 577}, {51, 36742}, {54, 6942}, {56, 20986}, {58, 5320}, {60, 5138}, {69, 1798}, {73, 7335}, {78, 3955}, {110, 4189}, {125, 34120}, {154, 37501}, {182, 404}, {197, 16980}, {206, 4265}, {222, 1425}, {228, 255}, {377, 37527}, {386, 44104}, {405, 5651}, {408, 23606}, {411, 13346}, {474, 43650}, {500, 11334}, {511, 11337}, {569, 6924}, {578, 6905}, {580, 4191}, {581, 37259}, {692, 5217}, {851, 37530}, {936, 26890}, {940, 4185}, {944, 56863}, {970, 1993}, {980, 37237}, {991, 3145}, {1012, 26883}, {1038, 64040}, {1193, 1397}, {1376, 55098}, {1398, 34042}, {1399, 3185}, {1410, 7125}, {1495, 13730}, {1614, 6950}, {1724, 28238}, {1894, 49745}, {1974, 36740}, {2194, 4252}, {2203, 37245}, {2308, 28270}, {2360, 28348}, {2594, 2933}, {2646, 14529}, {2915, 37482}, {3098, 59354}, {3142, 13478}, {3149, 11424}, {3286, 44120}, {3651, 37480}, {3781, 54337}, {3916, 42463}, {4188, 5012}, {4255, 44085}, {4257, 17104}, {4340, 37384}, {4420, 43146}, {4652, 7193}, {5019, 34281}, {5061, 5230}, {5096, 64028}, {5396, 20842}, {5398, 16453}, {5707, 37241}, {6056, 22361}, {6759, 6906}, {6876, 43574}, {6914, 10539}, {6940, 37515}, {7508, 61753}, {7535, 61643}, {7536, 57876}, {7561, 61644}, {8192, 34046}, {9544, 17548}, {10391, 11363}, {10441, 16049}, {11003, 37307}, {11340, 35203}, {11344, 37474}, {11402, 36745}, {13366, 36754}, {13743, 46261}, {15004, 36750}, {15066, 59359}, {15489, 34986}, {15803, 26889}, {16187, 17536}, {16408, 22112}, {17811, 37246}, {17977, 54433}, {19548, 61220}, {20831, 44082}, {20833, 35268}, {20846, 48893}, {22129, 42461}, {22344, 23201}, {23085, 23095}, {24265, 25591}, {25526, 37056}, {26885, 31424}, {26892, 57281}, {26934, 41393}, {27622, 37522}, {30675, 63399}, {33597, 47371}, {34417, 51340}, {35980, 46623}, {37116, 55303}, {37227, 50317}, {37231, 37521}, {37285, 48929}, {37306, 64393}, {37399, 63068}, {40985, 67930}

X(68652) = isotomic conjugate of the polar conjugate of X(5019)
X(68652) = isogonal conjugate of the polar conjugate of X(940)
X(68652) = X(i)-Ceva conjugate of X(j) for these (i,j): {940, 5019}, {54417, 1468}
X(68652) = X(i)-isoconjugate of X(j) for these (i,j): {4, 31359}, {19, 34258}, {29, 60321}, {33, 58008}, {92, 941}, {158, 34259}, {264, 2258}, {281, 44733}, {318, 959}, {931, 24006}, {1826, 37870}, {1973, 40828}, {2052, 66920}, {2501, 65230}, {3064, 32038}, {5331, 41013}, {32693, 46110}, {44426, 65225}, {50040, 54396}
X(68652) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 34258}, {1147, 34259}, {6337, 40828}, {17417, 46110}, {22391, 941}, {34261, 7017}, {34281, 17555}, {36033, 31359}
X(68652) = crosspoint of X(3) and X(57667)
X(68652) = crosssum of X(i) and X(j) for these (i,j): {4, 406}, {281, 7102}, {3192, 44103}, {4385, 46937}
X(68652) = crossdifference of every pair of points on line {2501, 44426}
X(68652) = barycentric product X(i)*X(j) for these {i,j}: {3, 940}, {48, 10436}, {63, 1468}, {69, 5019}, {77, 2268}, {184, 34284}, {222, 958}, {255, 5307}, {394, 4185}, {603, 11679}, {906, 43067}, {1214, 54417}, {1331, 48144}, {1437, 31993}, {1459, 65168}, {1790, 59305}, {1813, 17418}, {1867, 18604}, {3713, 7053}, {4558, 8672}, {4563, 8639}, {4575, 50457}, {7125, 54396}, {22341, 44734}, {23880, 36059}, {34279, 56414}, {34281, 57876}, {58332, 65296}
X(68652) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 34258}, {48, 31359}, {69, 40828}, {184, 941}, {222, 58008}, {577, 34259}, {603, 44733}, {940, 264}, {958, 7017}, {1409, 60321}, {1437, 37870}, {1468, 92}, {2268, 318}, {4185, 2052}, {4558, 65280}, {4575, 65230}, {5019, 4}, {5307, 57806}, {8639, 2501}, {8672, 14618}, {9247, 2258}, {10436, 1969}, {17418, 46110}, {32660, 65225}, {32661, 931}, {34281, 469}, {34284, 18022}, {36059, 32038}, {48144, 46107}, {52411, 959}, {52430, 66920}, {53543, 2973}, {53561, 21666}, {54417, 31623}, {57704, 34265}
X(68652) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 394, 22076}, {3, 1437, 184}, {48, 603, 22345}, {222, 56414, 1425}, {2360, 37469, 28348}, {5707, 37241, 58889}, {36742, 37034, 51}

X(68653) = X(3)X(69)∩X(6)X(980)

Barycentrics    a*(a^2 + a*b + a*c + 2*b*c)*(a^2 - b^2 - c^2) : :
X(68653) = 3 X[16436] - 2 X[36744]

X(68653) lies on these lines: {1, 54344}, {2, 1014}, {3, 69}, {6, 980}, {7, 2975}, {9, 7175}, {20, 54303}, {21, 3945}, {25, 45962}, {36, 17272}, {48, 23151}, {55, 3879}, {56, 4357}, {63, 77}, {72, 326}, {75, 956}, {76, 37415}, {81, 16368}, {86, 405}, {99, 34282}, {100, 32099}, {141, 21477}, {183, 16434}, {193, 4254}, {198, 4416}, {261, 8033}, {269, 62824}, {286, 7497}, {298, 21476}, {299, 21475}, {302, 21481}, {303, 21480}, {305, 57853}, {307, 1804}, {314, 1975}, {319, 5687}, {325, 19544}, {333, 11347}, {348, 1791}, {379, 16713}, {385, 21485}, {391, 11349}, {404, 5232}, {474, 5224}, {491, 16432}, {492, 16433}, {517, 55392}, {524, 16436}, {597, 21515}, {599, 5124}, {604, 56509}, {859, 56433}, {940, 5019}, {958, 10436}, {966, 16412}, {969, 66687}, {993, 3664}, {999, 17321}, {1011, 30941}, {1012, 8822}, {1030, 21518}, {1270, 16441}, {1271, 16440}, {1330, 37320}, {1376, 17270}, {1385, 55391}, {1424, 27626}, {1434, 19520}, {1439, 7183}, {1442, 3869}, {1443, 62827}, {1654, 11329}, {1696, 50093}, {1761, 18161}, {1817, 14552}, {1992, 21509}, {2178, 4643}, {2256, 62817}, {2287, 37064}, {2322, 37378}, {2329, 44421}, {2893, 7580}, {2895, 11340}, {2897, 11413}, {3314, 56773}, {3428, 64700}, {3560, 17139}, {3589, 21496}, {3593, 21553}, {3595, 21492}, {3618, 21514}, {3619, 21526}, {3620, 21495}, {3629, 21510}, {3630, 21523}, {3631, 21524}, {3663, 8666}, {3672, 54391}, {3713, 65168}, {3763, 21519}, {3875, 12513}, {3882, 37499}, {3912, 54322}, {3916, 54404}, {3927, 17976}, {3942, 20742}, {4021, 62825}, {4185, 34284}, {4189, 62999}, {4220, 37668}, {4221, 32817}, {4644, 38871}, {4851, 54285}, {5227, 25083}, {5228, 29747}, {5258, 25590}, {5278, 16438}, {5279, 24635}, {5288, 17151}, {5730, 44179}, {5738, 11344}, {5739, 11350}, {5933, 37541}, {6144, 21517}, {6172, 38869}, {6359, 40702}, {6527, 37404}, {7023, 62786}, {7412, 32001}, {7776, 21287}, {7837, 21505}, {8021, 51893}, {9534, 37273}, {10444, 12114}, {10449, 37062}, {10461, 36746}, {10477, 37474}, {11108, 63014}, {11160, 35276}, {11194, 17274}, {11249, 64122}, {11320, 17178}, {11358, 30966}, {11509, 58800}, {13615, 14548}, {13723, 37492}, {14555, 37269}, {14829, 16435}, {15066, 21494}, {15413, 23187}, {15533, 21498}, {15589, 19649}, {16058, 30962}, {16367, 17300}, {16370, 17378}, {16371, 17271}, {16418, 63110}, {16439, 18139}, {16700, 19728}, {16738, 19281}, {16887, 19762}, {16986, 21504}, {17077, 24612}, {17103, 19533}, {17253, 21773}, {17277, 37272}, {17343, 19308}, {17398, 21986}, {18133, 29477}, {18134, 21483}, {18747, 28755}, {19529, 33947}, {20080, 21508}, {20582, 21541}, {20760, 60729}, {21273, 68344}, {21286, 40999}, {21356, 21539}, {21358, 21533}, {21371, 55432}, {21487, 37671}, {21488, 31017}, {21497, 22165}, {21503, 64062}, {21516, 51171}, {21520, 47355}, {21521, 47352}, {21527, 34573}, {21528, 51126}, {21529, 63119}, {21542, 63121}, {21543, 51128}, {21545, 32812}, {21546, 32807}, {21547, 32805}, {21548, 32806}, {21550, 32813}, {21558, 32810}, {21559, 32808}, {21560, 32809}, {21561, 32811}, {21566, 32814}, {21982, 36740}, {22370, 60701}, {22758, 64126}, {22770, 64694}, {24540, 25875}, {24581, 27507}, {25000, 25931}, {25728, 59221}, {26041, 33828}, {26045, 33035}, {26818, 37076}, {27317, 50200}, {28753, 30810}, {29472, 37674}, {32000, 37305}, {32830, 37399}, {34046, 64365}, {34120, 57832}, {34259, 57701}, {37250, 54429}, {37329, 44094}, {37344, 63070}, {37412, 48878}, {37592, 56328}, {44140, 56960}, {51368, 56367}, {54358, 62853}, {56772, 63046}, {57667, 66948}

X(68653) = isotomic conjugate of the polar conjugate of X(940)
X(68653) = isogonal conjugate of the polar conjugate of X(34284)
X(68653) = X(34284)-Ceva conjugate of X(940)
X(68653) = X(i)-isoconjugate of X(j) for these (i,j): {4, 2258}, {19, 941}, {25, 31359}, {33, 959}, {393, 66920}, {607, 44733}, {1096, 34259}, {1824, 5331}, {1973, 34258}, {2212, 58008}, {2299, 60321}, {2333, 37870}, {2489, 65230}, {3064, 32693}, {18344, 65225}
X(68653) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 941}, {226, 60321}, {940, 406}, {958, 7102}, {5743, 1904}, {6337, 34258}, {6503, 34259}, {6505, 31359}, {17417, 3064}, {34261, 281}, {34281, 3192}, {36033, 2258}, {62604, 40828}
X(68653) = crosspoint of X(69) and X(57832)
X(68653) = crosssum of X(25) and X(44086)
X(68653) = crossdifference of every pair of points on line {2489, 18344}
X(68653) = barycentric product X(i)*X(j) for these {i,j}: {3, 34284}, {63, 10436}, {69, 940}, {77, 11679}, {304, 1468}, {305, 5019}, {326, 5307}, {348, 958}, {1231, 54417}, {1332, 43067}, {1444, 31993}, {2268, 7182}, {3713, 7056}, {3926, 4185}, {4025, 65168}, {4561, 48144}, {4563, 8672}, {4592, 50457}, {6516, 23880}, {7053, 61414}, {7183, 54396}, {8639, 52608}, {17206, 59305}, {17418, 65164}, {44734, 52385}
X(68653) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 941}, {48, 2258}, {63, 31359}, {69, 34258}, {77, 44733}, {222, 959}, {255, 66920}, {305, 40828}, {348, 58008}, {394, 34259}, {940, 4}, {958, 281}, {1214, 60321}, {1444, 37870}, {1468, 19}, {1790, 5331}, {1813, 65225}, {2268, 33}, {3713, 7046}, {4185, 393}, {4558, 931}, {4563, 65280}, {4592, 65230}, {5019, 25}, {5307, 158}, {6516, 32038}, {8639, 2489}, {8672, 2501}, {10436, 92}, {11679, 318}, {17418, 3064}, {22076, 56914}, {23880, 44426}, {31993, 41013}, {34261, 7102}, {34279, 43742}, {34281, 44103}, {34284, 264}, {36059, 32693}, {43067, 17924}, {44734, 1896}, {48144, 7649}, {50457, 24006}, {52610, 52931}, {53543, 2969}, {53561, 42069}, {54417, 1172}, {56414, 34263}, {57876, 34265}, {58332, 65103}, {59305, 1826}, {65168, 1897}, {65190, 65160}
X(68653) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 22152, 23079}, {69, 1444, 3}, {141, 36743, 21477}, {193, 21511, 4254}, {219, 222, 23124}, {18747, 28755, 30808}

X(68654) = X(3)X(69)∩X(23)X(316)

Barycentrics    a^2*(a^2 - b^2 - c^2)^2*(a^4 - b^4 + b^2*c^2 - c^4) : :

X(68654) lies on these lines: {3, 69}, {22, 64018}, {23, 316}, {24, 32816}, {25, 32827}, {26, 32006}, {32, 28710}, {76, 14118}, {99, 1236}, {183, 35921}, {186, 325}, {187, 249}, {305, 7771}, {315, 7488}, {378, 32815}, {384, 28417}, {1007, 6644}, {1078, 28706}, {1593, 32826}, {1658, 7776}, {1975, 3520}, {2072, 65518}, {2079, 44377}, {2080, 44716}, {2854, 39659}, {3053, 64195}, {3265, 39201}, {3269, 58354}, {3518, 7773}, {3552, 28441}, {4558, 14961}, {4611, 52058}, {5504, 43705}, {6636, 14907}, {7395, 32838}, {7485, 62299}, {7496, 66767}, {7503, 32828}, {7512, 7750}, {7514, 34229}, {7527, 11185}, {7550, 37688}, {7752, 44802}, {7763, 22467}, {7793, 28728}, {7796, 38448}, {7799, 37941}, {7809, 37940}, {7850, 21395}, {9609, 32986}, {10298, 37668}, {10317, 22151}, {14865, 32819}, {15078, 32837}, {15270, 33801}, {15574, 44837}, {16041, 44524}, {17506, 32821}, {17928, 32829}, {17932, 40079}, {21844, 32818}, {32459, 62381}, {32534, 32825}, {32817, 35473}, {32822, 35475}, {32823, 44879}, {32824, 35477}, {32839, 66607}, {32883, 64585}, {32972, 44527}, {34990, 41336}, {35500, 59635}, {35936, 47286}, {37460, 55551}, {37645, 52275}, {37948, 59634}, {39854, 54103}, {43459, 45308}, {43574, 51439}, {46951, 54994}, {51386, 51394}, {54075, 56473}, {66717, 67536}

X(68654) = isotomic conjugate of the isogonal conjugate of X(58357)
X(68654) = isotomic conjugate of the polar conjugate of X(22151)
X(68654) = isogonal conjugate of the polar conjugate of X(37804)
X(68654) = X(37804)-Ceva conjugate of X(22151)
X(68654) = X(58357)-cross conjugate of X(22151)
X(68654) = X(i)-isoconjugate of X(j) for these (i,j): {19, 8791}, {67, 1096}, {158, 3455}, {393, 2157}, {798, 65356}, {1973, 46105}, {2643, 66950}, {32676, 66943}
X(68654) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 8791}, {187, 60428}, {1147, 3455}, {5099, 58757}, {6337, 46105}, {6338, 18019}, {6503, 67}, {15526, 66943}, {18311, 2970}, {31998, 65356}, {39169, 8753}, {40583, 393}, {52881, 57496}, {55048, 2501}, {62376, 37981}, {62597, 66299}, {65918, 27376}
X(68654) = crosssum of X(27376) and X(60428)
X(68654) = crossdifference of every pair of points on line {2489, 8029}
X(68654) = barycentric product X(i)*X(j) for these {i,j}: {3, 37804}, {23, 3926}, {69, 22151}, {76, 58357}, {255, 20944}, {305, 10317}, {316, 394}, {326, 16568}, {520, 55226}, {577, 40074}, {1259, 17088}, {3265, 52630}, {3964, 37765}, {4143, 52916}, {4176, 8744}, {4563, 9517}, {6390, 57481}, {34254, 54060}, {42659, 52608}
X(68654) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 8791}, {23, 393}, {69, 46105}, {99, 65356}, {249, 66950}, {255, 2157}, {316, 2052}, {394, 67}, {525, 66943}, {577, 3455}, {2492, 58757}, {3926, 18019}, {3964, 34897}, {4558, 935}, {4563, 65269}, {6390, 57496}, {6593, 60428}, {7664, 37778}, {8744, 6524}, {9019, 27376}, {9517, 2501}, {9979, 66299}, {10317, 25}, {16165, 1990}, {16568, 158}, {18374, 2207}, {20806, 11605}, {20944, 57806}, {22151, 4}, {28724, 9076}, {37765, 1093}, {37804, 264}, {40074, 18027}, {42659, 2489}, {51394, 60496}, {52630, 107}, {52916, 6529}, {54060, 13854}, {55226, 6528}, {57481, 17983}, {58354, 36820}, {58357, 6}, {62382, 39269}, {62563, 2970}
X(68654) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6390, 5866}, {5866, 52437, 6390}, {34883, 44180, 3}

X(68655) = X(3)X(49)∩X(110)X(1114)

Barycentrics    a^4*(a^2 - b^2)*(a^2 - c^2)*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4 - c^2*(a^2 + b^2 - c^2)*J)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4 - b^2*(a^2 - b^2 + c^2)*J) : :

X(68655) lies on these lines: {3, 49}, {30, 32615}, {110, 1114}, {323, 25407}, {511, 44123}, {520, 53385}, {539, 57323}, {912, 52481}, {1113, 43574}, {1312, 11064}, {1313, 44665}, {1344, 13352}, {1345, 9306}, {1347, 41171}, {1568, 10751}, {1823, 36059}, {2071, 32617}, {2574, 5504}, {3289, 57026}, {3564, 13415}, {10540, 15155}, {10750, 14499}, {14157, 15156}, {14374, 57648}, {14709, 34148}, {15035, 44067}, {15154, 37477}, {15165, 18831}, {17932, 46810}, {17974, 46814}, {20479, 61753}, {32661, 57025}, {34382, 44126}, {35232, 40111}, {38709, 43572}, {42065, 42668}

X(68655) = isotomic conjugate of the polar conjugate of X(57025)
X(68655) = isogonal conjugate of the polar conjugate of X(8116)
X(68655) = X(i)-Ceva conjugate of X(j) for these (i,j): {110, 53385}, {249, 57026}, {8116, 57025}, {15460, 3}
X(68655) = X(39201)-cross conjugate of X(57026)
X(68655) = X(i)-isoconjugate of X(j) for these (i,j): {4, 2589}, {19, 2593}, {92, 8106}, {158, 2575}, {162, 39241}, {393, 2583}, {523, 2586}, {661, 46815}, {823, 66876}, {1093, 2585}, {1096, 22340}, {1113, 24006}, {1312, 2587}, {1822, 66299}, {2052, 2579}, {2501, 2580}, {2576, 14618}, {2588, 53154}, {6520, 46811}, {42667, 57806}
X(68655) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 2593}, {125, 39241}, {1147, 2575}, {1313, 66299}, {2574, 39240}, {6503, 22340}, {15166, 14618}, {22391, 8106}, {36033, 2589}, {36830, 46815}, {37867, 46811}, {62581, 264}, {62593, 850}, {66876, 338}
X(68655) = crosssum of X(523) and X(39240)
X(68655) = crossdifference of every pair of points on line {2501, 39241}
X(68655) = barycentric product X(i)*X(j) for these {i,j}: {3, 8116}, {63, 1823}, {69, 57025}, {110, 46814}, {184, 46810}, {255, 2581}, {326, 2577}, {394, 1114}, {520, 39299}, {577, 15165}, {662, 2584}, {1092, 46812}, {2574, 4558}, {2578, 4592}, {2582, 4575}, {2587, 6507}, {3926, 44124}, {4563, 42668}, {8115, 53385}, {15460, 46811}, {22339, 32661}, {28724, 46167}
X(68655) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 2593}, {48, 2589}, {110, 46815}, {163, 2586}, {184, 8106}, {255, 2583}, {394, 22340}, {577, 2575}, {647, 39241}, {1092, 46811}, {1114, 2052}, {1823, 92}, {2574, 14618}, {2577, 158}, {2578, 24006}, {2581, 57806}, {2584, 1577}, {2587, 6521}, {4100, 2585}, {4558, 15164}, {4575, 2580}, {8105, 66299}, {8116, 264}, {14585, 42667}, {15165, 18027}, {15166, 39240}, {15460, 46812}, {32661, 1113}, {39201, 66876}, {39299, 6528}, {42668, 2501}, {44068, 14165}, {44124, 393}, {46810, 18022}, {46814, 850}, {47390, 39298}, {52430, 2579}, {53385, 2592}, {57025, 4}, {57026, 53154}, {66877, 2970}

X(68656) = X(3)X(69)∩X(99)X(1114)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4 - c^2*(a^2 + b^2 - c^2)*J)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4 - b^2*(a^2 - b^2 + c^2)*J) : :

X(68656) lies on these lines: {3, 69}, {99, 1114}, {316, 15157}, {325, 1113}, {1007, 1344}, {1345, 32815}, {1975, 14710}, {2482, 62580}, {2574, 43705}, {3265, 53385}, {4558, 8116}, {7763, 14709}, {7799, 38708}, {8115, 15167}, {10750, 65518}, {22339, 57829}, {38709, 59634}, {53384, 68152}

X(68656) = isotomic conjugate of the polar conjugate of X(8116)
X(68656) = isogonal conjugate of the polar conjugate of X(46810)
X(68656) = X(i)-Ceva conjugate of X(j) for these (i,j): {4590, 8115}, {46810, 8116}
X(68656) = X(i)-cross conjugate of X(j) for these (i,j): {520, 8115}, {46811, 3926}, {53385, 4558}
X(68656) = X(i)-isoconjugate of X(j) for these (i,j): {19, 8106}, {25, 2589}, {158, 42667}, {393, 2579}, {512, 2586}, {798, 46815}, {1096, 2575}, {1822, 58757}, {1973, 2593}, {2207, 2583}, {2489, 2580}, {2501, 2576}, {2585, 6524}, {2587, 44125}, {24006, 44123}, {24019, 66876}, {32676, 39241}
X(68656) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 8106}, {1147, 42667}, {1313, 58757}, {6337, 2593}, {6338, 22340}, {6503, 2575}, {6505, 2589}, {15166, 2501}, {15526, 39241}, {31998, 46815}, {35071, 66876}, {39054, 2586}, {46814, 39240}, {62580, 53154}, {62581, 4}, {62592, 1312}, {62593, 523}, {66876, 115}
X(68656) = cevapoint of X(3) and X(46811)
X(68656) = trilinear pole of line {394, 46814}
X(68656) = barycentric product X(i)*X(j) for these {i,j}: {3, 46810}, {69, 8116}, {99, 46814}, {304, 1823}, {305, 57025}, {326, 2581}, {394, 15165}, {799, 2584}, {1102, 2587}, {1114, 3926}, {2574, 4563}, {2578, 55202}, {2582, 4592}, {3265, 39299}, {3964, 46812}, {4558, 22339}, {42668, 52608}, {46813, 53385}, {47389, 66877}
X(68656) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 8106}, {63, 2589}, {69, 2593}, {99, 46815}, {255, 2579}, {326, 2583}, {394, 2575}, {520, 66876}, {525, 39241}, {577, 42667}, {662, 2586}, {1114, 393}, {1823, 19}, {2574, 2501}, {2577, 1096}, {2581, 158}, {2582, 24006}, {2584, 661}, {2587, 6520}, {2592, 66299}, {3926, 22340}, {3964, 46811}, {4558, 1113}, {4563, 15164}, {4575, 2576}, {4592, 2580}, {6507, 2585}, {8105, 58757}, {8115, 53154}, {8116, 4}, {15165, 2052}, {22339, 14618}, {32661, 44123}, {39299, 107}, {42668, 2489}, {44068, 52418}, {44124, 2207}, {46167, 27376}, {46810, 264}, {46811, 1312}, {46812, 1093}, {46814, 523}, {51394, 66357}, {53385, 8105}, {57025, 25}, {62593, 39240}, {66358, 68327}, {66877, 8754}

X(68657) = X(3)X(49)∩X(110)X(1113)

Barycentrics    a^4*(a^2 - b^2)*(a^2 - c^2)*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4 + c^2*(a^2 + b^2 - c^2)*J)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4 + b^2*(a^2 - b^2 + c^2)*J) : :

X(68657) lies on these lines: {3, 49}, {30, 32614}, {110, 1113}, {323, 25408}, {511, 44124}, {520, 53384}, {539, 57322}, {912, 52482}, {1114, 43574}, {1312, 44665}, {1313, 11064}, {1344, 9306}, {1345, 13352}, {1346, 41171}, {1568, 10750}, {1822, 36059}, {2071, 32616}, {2575, 5504}, {3289, 57025}, {3564, 13414}, {10540, 15154}, {10751, 14500}, {14157, 15157}, {14375, 57648}, {14710, 34148}, {15035, 44068}, {15155, 37477}, {15164, 18831}, {17932, 46813}, {17974, 46811}, {20478, 61753}, {32661, 57026}, {34382, 44125}, {35231, 40111}, {38708, 43572}, {42065, 42667}

X(68657) = isotomic conjugate of the polar conjugate of X(57026)
X(68657) = isogonal conjugate of the polar conjugate of X(8115)
X(68657) = X(i)-Ceva conjugate of X(j) for these (i,j): {110, 53384}, {249, 57025}, {8115, 57026}, {15461, 3}
X(68657) = X(39201)-cross conjugate of X(57025)
X(68657) = X(i)-isoconjugate of X(j) for these (i,j): {4, 2588}, {19, 2592}, {92, 8105}, {158, 2574}, {162, 39240}, {393, 2582}, {523, 2587}, {661, 46812}, {823, 66877}, {1093, 2584}, {1096, 22339}, {1114, 24006}, {1313, 2586}, {1823, 66299}, {2052, 2578}, {2501, 2581}, {2577, 14618}, {2589, 53153}, {6520, 46814}, {42668, 57806}
X(68657) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 2592}, {125, 39240}, {1147, 2574}, {1312, 66299}, {2575, 39241}, {6503, 22339}, {15167, 14618}, {22391, 8105}, {36033, 2588}, {36830, 46812}, {37867, 46814}, {62580, 264}, {62592, 850}, {66877, 338}
X(68657) = crosssum of X(523) and X(39241)
X(68657) = crossdifference of every pair of points on line {2501, 39240}
X(68657) = barycentric product X(i)*X(j) for these {i,j}: {3, 8115}, {63, 1822}, {69, 57026}, {110, 46811}, {184, 46813}, {255, 2580}, {326, 2576}, {394, 1113}, {520, 39298}, {577, 15164}, {662, 2585}, {1092, 46815}, {2575, 4558}, {2579, 4592}, {2583, 4575}, {2586, 6507}, {3926, 44123}, {4563, 42667}, {8116, 53384}, {15461, 46814}, {22340, 32661}, {28724, 46166}
X(68657) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 2592}, {48, 2588}, {110, 46812}, {163, 2587}, {184, 8105}, {255, 2582}, {394, 22339}, {577, 2574}, {647, 39240}, {1092, 46814}, {1113, 2052}, {1822, 92}, {2575, 14618}, {2576, 158}, {2579, 24006}, {2580, 57806}, {2585, 1577}, {2586, 6521}, {4100, 2584}, {4558, 15165}, {4575, 2581}, {8106, 66299}, {8115, 264}, {14585, 42668}, {15164, 18027}, {15167, 39241}, {15461, 46815}, {32661, 1114}, {39201, 66877}, {39298, 6528}, {42667, 2501}, {44067, 14165}, {44123, 393}, {46811, 850}, {46813, 18022}, {47390, 39299}, {52430, 2578}, {53384, 2593}, {57025, 53153}, {57026, 4}, {66876, 2970}

X(68658) = X(3)X(69)∩X(99)X(1113)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4 + c^2*(a^2 + b^2 - c^2)*J)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4 + b^2*(a^2 - b^2 + c^2)*J) : :

X(68658) lies on these lines: {3, 69}, {99, 1113}, {316, 15156}, {325, 1114}, {1007, 1345}, {1344, 32815}, {1975, 14709}, {2482, 62581}, {2575, 43705}, {3265, 53384}, {4558, 8115}, {7763, 14710}, {7799, 38709}, {8116, 15166}, {10751, 65518}, {22340, 57829}, {38708, 59634}, {53385, 68152}

X(68658) = isotomic conjugate of the polar conjugate of X(8115)
X(68658) = isogonal conjugate of the polar conjugate of X(46813)
X(68658) = X(i)-Ceva conjugate of X(j) for these (i,j): {4590, 8116}, {46813, 8115}
X(68658) = X(i)-cross conjugate of X(j) for these (i,j): {520, 8116}, {46814, 3926}, {53384, 4558}
X(68658) = X(i)-isoconjugate of X(j) for these (i,j): {19, 8105}, {25, 2588}, {158, 42668}, {393, 2578}, {512, 2587}, {798, 46812}, {1096, 2574}, {1823, 58757}, {1973, 2592}, {2207, 2582}, {2489, 2581}, {2501, 2577}, {2584, 6524}, {2586, 44126}, {24006, 44124}, {24019, 66877}, {32676, 39240}
X(68658) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 8105}, {1147, 42668}, {1312, 58757}, {6337, 2592}, {6338, 22339}, {6503, 2574}, {6505, 2588}, {15167, 2501}, {15526, 39240}, {31998, 46812}, {35071, 66877}, {39054, 2587}, {46811, 39241}, {62580, 4}, {62581, 53153}, {62592, 523}, {62593, 1313}, {66877, 115}
X(68658) = cevapoint of X(3) and X(46814)
X(68658) = trilinear pole of line {394, 46811}
X(68658) = barycentric product X(i)*X(j) for these {i,j}: {3, 46813}, {69, 8115}, {99, 46811}, {304, 1822}, {305, 57026}, {326, 2580}, {394, 15164}, {799, 2585}, {1102, 2586}, {1113, 3926}, {2575, 4563}, {2579, 55202}, {2583, 4592}, {3265, 39298}, {3964, 46815}, {4558, 22340}, {42667, 52608}, {46810, 53384}, {47389, 66876}
X(68658) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 8105}, {63, 2588}, {69, 2592}, {99, 46812}, {255, 2578}, {326, 2582}, {394, 2574}, {520, 66877}, {525, 39240}, {577, 42668}, {662, 2587}, {1113, 393}, {1822, 19}, {2575, 2501}, {2576, 1096}, {2580, 158}, {2583, 24006}, {2585, 661}, {2586, 6520}, {2593, 66299}, {3926, 22339}, {3964, 46814}, {4558, 1114}, {4563, 15165}, {4575, 2577}, {4592, 2581}, {6507, 2584}, {8106, 58757}, {8115, 4}, {8116, 53153}, {15164, 2052}, {22340, 14618}, {32661, 44124}, {39298, 107}, {42667, 2489}, {44067, 52418}, {44123, 2207}, {46166, 27376}, {46811, 523}, {46813, 264}, {46814, 1313}, {46815, 1093}, {51394, 66358}, {53384, 8106}, {57026, 25}, {62592, 39241}, {66357, 68327}, {66876, 8754}

X(68659) = X(3)X(49)∩X(6)X(5892)

Barycentrics    a^4*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 4*b^2*c^2 + c^4) : :
X(68659) = 3 X[3] - X[10605], 3 X[3] + X[58891], 3 X[394] + X[10605], 3 X[394] - X[58891], X[37483] + 2 X[43586], 2 X[37480] + X[46261], 5 X[631] - X[6515], 3 X[6090] - X[18451], 3 X[6090] + X[21312], X[12828] - 3 X[38793], X[18534] - 3 X[35259], 2 X[44233] - 3 X[61507]

X(68659) lies on these lines: {2, 13352}, {3, 49}, {4, 43614}, {5, 13346}, {6, 5892}, {20, 10539}, {22, 51393}, {24, 10625}, {25, 37483}, {26, 15644}, {30, 9306}, {52, 17928}, {54, 3523}, {68, 3546}, {69, 5504}, {74, 46202}, {110, 376}, {113, 44440}, {140, 578}, {141, 51739}, {154, 35243}, {156, 548}, {182, 524}, {186, 2979}, {206, 1511}, {249, 67639}, {323, 5890}, {343, 10257}, {378, 4550}, {381, 5651}, {389, 16266}, {403, 54040}, {417, 61355}, {511, 6644}, {539, 1899}, {547, 16187}, {550, 6759}, {567, 5054}, {569, 631}, {576, 5946}, {577, 18877}, {692, 35238}, {858, 18474}, {1154, 11438}, {1173, 46221}, {1196, 44415}, {1209, 37119}, {1350, 14070}, {1352, 44441}, {1368, 44665}, {1370, 44407}, {1495, 12083}, {1498, 14641}, {1578, 8909}, {1614, 3522}, {1656, 11424}, {1657, 18350}, {1658, 10627}, {1660, 6000}, {1974, 33878}, {1993, 9730}, {1994, 15045}, {2070, 13340}, {2071, 11459}, {2477, 63756}, {2888, 23294}, {2931, 18438}, {2972, 13496}, {3043, 15051}, {3146, 43598}, {3200, 10645}, {3201, 10646}, {3311, 9686}, {3357, 5876}, {3410, 44450}, {3431, 41462}, {3515, 37486}, {3518, 64050}, {3520, 11444}, {3524, 5012}, {3525, 13434}, {3526, 37472}, {3528, 52525}, {3530, 32046}, {3534, 10540}, {3537, 64177}, {3543, 43576}, {3547, 44516}, {3548, 5449}, {3549, 43839}, {3581, 54048}, {3819, 7514}, {4846, 61113}, {5020, 44413}, {5023, 9603}, {5063, 52438}, {5092, 64028}, {5171, 44221}, {5320, 51340}, {5446, 6642}, {5462, 9777}, {5480, 10127}, {5654, 37669}, {5899, 44082}, {5907, 12084}, {5943, 39522}, {5961, 20819}, {5972, 10201}, {6056, 52407}, {6090, 14915}, {6101, 37814}, {6193, 10116}, {6200, 9687}, {6243, 43809}, {6247, 31831}, {6396, 9676}, {6636, 11464}, {6638, 6760}, {6643, 12118}, {6776, 63649}, {6823, 9820}, {7335, 52408}, {7386, 66735}, {7393, 11425}, {7464, 15305}, {7484, 37506}, {7485, 37513}, {7492, 15034}, {7506, 44106}, {7512, 11449}, {7525, 32171}, {7526, 11793}, {7575, 52987}, {7691, 21844}, {7706, 66614}, {7811, 10411}, {7815, 36952}, {7818, 35088}, {7998, 35921}, {7999, 14118}, {8538, 61724}, {8588, 9696}, {8703, 40111}, {8718, 50693}, {9143, 43578}, {9544, 10304}, {9545, 15717}, {9586, 58221}, {9604, 53095}, {9653, 52793}, {9705, 21734}, {9706, 61138}, {9729, 12161}, {9818, 10170}, {9833, 34944}, {9927, 11585}, {9936, 18909}, {9967, 19141}, {10112, 18952}, {10192, 16618}, {10282, 13348}, {10298, 15035}, {10303, 43651}, {10519, 19131}, {10546, 52294}, {10574, 56292}, {10575, 11441}, {10602, 34382}, {10606, 58871}, {11003, 15692}, {11064, 15760}, {11134, 11481}, {11137, 11480}, {11178, 44287}, {11204, 34152}, {11250, 11591}, {11403, 46852}, {11412, 22467}, {11413, 12162}, {11426, 15805}, {11454, 37948}, {11455, 15052}, {11456, 14855}, {11470, 37933}, {11472, 54992}, {11511, 14984}, {11653, 32833}, {11693, 66368}, {11750, 47528}, {11935, 15700}, {12082, 35264}, {12085, 17814}, {12086, 15058}, {12106, 13391}, {12121, 18564}, {12228, 48378}, {12250, 43813}, {12358, 12901}, {12359, 16196}, {12605, 63631}, {12828, 15463}, {12893, 41673}, {12900, 15472}, {13339, 15693}, {13347, 15712}, {13353, 15720}, {13369, 42463}, {13472, 67302}, {13490, 48901}, {13596, 54434}, {13598, 13861}, {14379, 19210}, {14385, 51821}, {14531, 37490}, {14790, 45286}, {14791, 18400}, {14805, 54006}, {14806, 42445}, {14852, 30771}, {14865, 15056}, {15030, 66717}, {15032, 20791}, {15036, 41398}, {15053, 23061}, {15067, 18570}, {15078, 32110}, {15080, 44832}, {15087, 40280}, {15107, 47485}, {15111, 36188}, {15122, 23329}, {15303, 15462}, {15694, 22112}, {15801, 67924}, {16051, 36253}, {16063, 30714}, {16163, 66721}, {16252, 46374}, {16534, 67890}, {16836, 34986}, {16867, 34483}, {16976, 44683}, {17702, 18531}, {17704, 64026}, {17714, 63414}, {18281, 21243}, {18388, 50008}, {18390, 62378}, {18418, 47336}, {18435, 18859}, {18534, 35259}, {18537, 64096}, {18580, 40107}, {18876, 54032}, {18911, 61713}, {18925, 43617}, {19127, 54169}, {19128, 62174}, {19137, 21850}, {19139, 52520}, {20376, 21230}, {20806, 37511}, {21639, 67920}, {22401, 23128}, {22660, 31829}, {23293, 65085}, {23325, 37938}, {23335, 64035}, {24206, 60763}, {25738, 63652}, {25739, 31101}, {26884, 37584}, {26937, 52104}, {31074, 41171}, {31723, 51360}, {31834, 32138}, {31837, 47371}, {31861, 67891}, {32063, 35237}, {32139, 46850}, {32269, 44211}, {32284, 44503}, {32534, 43898}, {32609, 35268}, {32903, 40276}, {33522, 66589}, {33533, 44324}, {33543, 51933}, {33556, 34116}, {33813, 57011}, {34397, 55576}, {34417, 37496}, {34513, 55649}, {34565, 36749}, {34782, 44679}, {35254, 66718}, {35472, 52416}, {36753, 44111}, {37118, 37636}, {37347, 61743}, {37452, 67903}, {37471, 61811}, {37475, 37672}, {37484, 45735}, {38723, 54073}, {40112, 68500}, {40441, 68082}, {41716, 45170}, {43595, 64038}, {43894, 52398}, {44078, 54047}, {44233, 61507}, {44263, 51391}, {44279, 68426}, {44452, 61646}, {44470, 47549}, {44489, 64067}, {44752, 67913}, {44802, 64051}, {46029, 46114}, {46849, 47527}, {47066, 48366}, {47068, 48365}, {48906, 52016}, {54132, 66705}, {54183, 63180}, {55582, 56918}, {62082, 64491}, {63428, 66730}, {64099, 67067}

X(68659) = midpoint of X(i) and X(j) for these {i,j}: {3, 394}, {25, 37483}, {9306, 37480}, {9833, 34944}, {10605, 58891}, {10625, 65654}, {18451, 21312}, {54183, 63180}
X(68659) = reflection of X(i) in X(j) for these {i,j}: {5, 53415}, {25, 43586}, {13567, 140}, {41619, 1147}, {46261, 9306}, {64095, 6644}
X(68659) = isotomic conjugate of the polar conjugate of X(5063)
X(68659) = isogonal conjugate of the polar conjugate of X(15066)
X(68659) = X(i)-Ceva conjugate of X(j) for these (i,j): {15066, 5063}, {56266, 577}
X(68659) = X(i)-isoconjugate of X(j) for these (i,j): {19, 34289}, {92, 34288}, {158, 4846}, {1096, 57819}, {1302, 24006}, {1784, 60119}, {14618, 36149}, {36120, 56925}
X(68659) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 34289}, {1147, 4846}, {6503, 57819}, {22391, 34288}, {46094, 56925}, {53993, 66299}
X(68659) = crosssum of X(4) and X(62961)
X(68659) = barycentric product X(i)*X(j) for these {i,j}: {3, 15066}, {69, 5063}, {97, 5891}, {184, 32833}, {305, 52438}, {378, 394}, {577, 44134}, {1437, 42704}, {3926, 44080}, {4550, 56266}, {4558, 8675}, {4563, 42660}, {10564, 14919}, {11653, 36212}, {30474, 32661}
X(68659) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 34289}, {184, 34288}, {378, 2052}, {394, 57819}, {577, 4846}, {3289, 56925}, {4558, 65284}, {5063, 4}, {5891, 324}, {8675, 14618}, {10564, 46106}, {11653, 16081}, {15066, 264}, {18877, 60119}, {32661, 1302}, {32833, 18022}, {42660, 2501}, {44080, 393}, {44134, 18027}, {52438, 25}
X(68659) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 43574, 13352}, {3, 49, 10984}, {3, 155, 40647}, {3, 1092, 1147}, {3, 1147, 64049}, {3, 5562, 7689}, {3, 12163, 43604}, {3, 18436, 1204}, {3, 18445, 64100}, {3, 22115, 184}, {3, 23039, 63425}, {3, 35602, 12038}, {3, 47391, 18475}, {3, 58891, 10605}, {54, 3523, 13336}, {184, 1092, 22115}, {184, 22115, 1147}, {186, 2979, 37478}, {378, 5891, 4550}, {378, 15066, 5891}, {394, 10605, 58891}, {549, 48876, 44201}, {550, 61753, 6759}, {567, 5054, 43650}, {631, 34148, 569}, {1092, 43652, 3}, {1495, 36987, 12083}, {1511, 7502, 11202}, {1511, 54042, 7502}, {1656, 37495, 11424}, {1657, 18350, 26883}, {1658, 10627, 46728}, {3098, 11202, 7502}, {3292, 64100, 18445}, {3530, 32046, 37515}, {3547, 64181, 44516}, {3819, 11430, 7514}, {3917, 51394, 3}, {5447, 12038, 3}, {5891, 10564, 378}, {6090, 21312, 18451}, {6101, 37814, 46730}, {6642, 37498, 5446}, {7502, 54042, 3098}, {8703, 40111, 61752}, {9545, 15717, 61134}, {9818, 17811, 10170}, {10564, 15066, 4550}, {11464, 54041, 6636}, {11585, 68018, 9927}, {12358, 59495, 12901}, {13596, 54434, 66756}, {15052, 37944, 11455}, {15122, 67926, 23329}, {17811, 37497, 9818}, {23329, 34507, 67926}, {36747, 66607, 5462}

X(68660) = X(3)X(69)∩X(22)X(1634)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 4*b^2*c^2 + c^4) : :

X(68660) lies on these lines: {3, 69}, {22, 1634}, {23, 37668}, {24, 340}, {99, 43660}, {183, 3266}, {264, 35502}, {305, 57829}, {311, 63664}, {315, 12082}, {317, 10594}, {325, 1995}, {378, 32833}, {394, 4558}, {524, 52275}, {599, 50660}, {1007, 11284}, {1092, 62347}, {1273, 17928}, {1583, 32810}, {1584, 32811}, {1593, 52710}, {1599, 32808}, {1600, 32809}, {1609, 20080}, {1975, 3260}, {1992, 37344}, {2407, 48871}, {3284, 20806}, {3518, 32001}, {3630, 8553}, {3746, 55392}, {4121, 52153}, {5063, 15066}, {5158, 36212}, {5198, 63155}, {5563, 55391}, {6642, 32825}, {6644, 52149}, {7464, 32817}, {7485, 37671}, {7492, 10513}, {7496, 15589}, {7527, 32830}, {7530, 7776}, {7768, 10323}, {8573, 11008}, {9145, 51383}, {9737, 61667}, {9818, 32836}, {11063, 40341}, {11160, 35296}, {11413, 32820}, {12085, 32824}, {14264, 15919}, {14865, 32000}, {14907, 41463}, {16042, 63098}, {18354, 68346}, {20563, 57800}, {31861, 44135}, {35302, 50992}, {35479, 55551}, {36889, 54992}, {37183, 63180}, {37248, 63110}, {43705, 55977}, {44149, 52712}, {45808, 46616}, {59257, 65310}

X(68660) = isotomic conjugate of the polar conjugate of X(15066)
X(68660) = isogonal conjugate of the polar conjugate of X(32833)
X(68660) = X(32833)-Ceva conjugate of X(15066)
X(68660) = X(i)-isoconjugate of X(j) for these (i,j): {19, 34288}, {1096, 4846}, {1973, 34289}, {2501, 36149}, {24006, 32738}, {36083, 68327}
X(68660) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 34288}, {6337, 34289}, {6338, 57819}, {6503, 4846}, {15066, 62961}, {37648, 1596}, {53993, 58757}, {62590, 56925}, {62606, 60119}
X(68660) = crossdifference of every pair of points on line {2489, 68327}
X(68660) = barycentric product X(i)*X(j) for these {i,j}: {3, 32833}, {69, 15066}, {305, 5063}, {378, 3926}, {394, 44134}, {1444, 42704}, {4558, 30474}, {4563, 8675}, {5891, 34386}, {6393, 11653}, {40050, 52438}, {42660, 52608}
X(68660) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 34288}, {69, 34289}, {378, 393}, {394, 4846}, {3926, 57819}, {4558, 1302}, {4563, 65284}, {4575, 36149}, {5063, 25}, {5891, 53}, {8675, 2501}, {10564, 1990}, {11653, 6531}, {14919, 60119}, {15066, 4}, {30474, 14618}, {32661, 32738}, {32833, 264}, {36212, 56925}, {42660, 2489}, {42704, 41013}, {44080, 2207}, {44134, 2052}, {47391, 52165}, {52438, 1974}, {62628, 52661}
X(68660) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 3964, 52437}, {3, 52437, 9723}, {69, 3926, 62338}, {69, 3964, 9723}, {69, 45795, 52347}, {69, 52437, 3}

X(68661) = X(3)X(49)∩X(21)X(15446)

Barycentrics    a^3*(a + b)*(a - b - c)*(a + c)*(a^2 - b^2 - c^2)*(a^2 - b^2 + b*c - c^2) : :

X(68661) lies on these lines: {3, 49}, {21, 15446}, {36, 22128}, {60, 54356}, {81, 30274}, {110, 2716}, {255, 2169}, {484, 62756}, {3193, 5903}, {4282, 34544}, {7335, 56553}, {14799, 35193}, {34471, 54417}, {37227, 46920}, {43574, 61220}

X(68661) = isotomic conjugate of the polar conjugate of X(4282)
X(68661) = X(i)-isoconjugate of X(j) for these (i,j): {4, 52383}, {12, 68571}, {19, 60091}, {34, 15065}, {80, 225}, {158, 52391}, {162, 66272}, {226, 64835}, {273, 34857}, {653, 55238}, {655, 2501}, {661, 65329}, {759, 56285}, {860, 63750}, {1411, 41013}, {1426, 52409}, {1824, 18815}, {1825, 2166}, {1826, 2006}, {1880, 18359}, {2161, 40149}, {2222, 24006}, {2489, 46405}, {2594, 6344}, {6187, 57809}, {7012, 66289}, {8736, 24624}, {14584, 68563}, {14618, 32675}, {18384, 40999}, {20566, 57652}, {23290, 36078}, {36804, 55208}, {44113, 57645}, {52371, 68576}, {61178, 66284}
X(68661) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 60091}, {125, 66272}, {1147, 52391}, {6149, 860}, {11517, 15065}, {11597, 1825}, {13999, 66299}, {34586, 56285}, {35128, 14618}, {35204, 41013}, {36033, 52383}, {36830, 65329}, {38984, 24006}, {40584, 40149}, {40612, 57809}
X(68661) = crossdifference of every pair of points on line {2501, 8736}
X(68661) = barycentric product X(i)*X(j) for these {i,j}: {21, 22128}, {36, 1812}, {69, 4282}, {283, 3218}, {320, 2193}, {323, 1789}, {332, 7113}, {333, 52407}, {394, 17515}, {645, 22379}, {654, 4592}, {758, 65568}, {1437, 32851}, {1443, 2327}, {1444, 2323}, {1790, 4511}, {1808, 27950}, {1870, 6514}, {1946, 55237}, {2361, 17206}, {3738, 4558}, {3904, 4575}, {4563, 8648}, {4585, 23189}, {5081, 18604}, {34544, 57985}
X(68661) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 60091}, {36, 40149}, {48, 52383}, {50, 1825}, {110, 65329}, {219, 15065}, {283, 18359}, {320, 52575}, {577, 52391}, {647, 66272}, {654, 24006}, {1437, 2006}, {1789, 94}, {1790, 18815}, {1812, 20566}, {1946, 55238}, {1983, 61178}, {2150, 68571}, {2193, 80}, {2194, 64835}, {2245, 56285}, {2323, 41013}, {2327, 52409}, {2361, 1826}, {2600, 23290}, {3218, 57809}, {3724, 8736}, {3738, 14618}, {4282, 4}, {4558, 35174}, {4575, 655}, {4592, 46405}, {7113, 225}, {7117, 66289}, {8648, 2501}, {17515, 2052}, {18604, 52392}, {21828, 66297}, {22115, 16577}, {22128, 1441}, {22379, 7178}, {23090, 52356}, {23181, 62735}, {23189, 60074}, {32661, 2222}, {34544, 860}, {52059, 1835}, {52407, 226}, {52425, 34857}, {52426, 1824}, {52434, 1880}, {57736, 34535}, {62734, 66300}, {65104, 66299}, {65568, 14616}

X(68662) = EULER LINE INTERCEPT OF X(5023)X(7607)

Barycentrics    7*a^8 - 19*a^6*b^2 + 9*a^4*b^4 + 7*a^2*b^6 - 4*b^8 - 19*a^6*c^2 + 4*a^4*b^2*c^2 - 7*a^2*b^4*c^2 + 18*b^6*c^2 + 9*a^4*c^4 - 7*a^2*b^2*c^4 - 28*b^4*c^4 + 7*a^2*c^6 + 18*b^2*c^6 - 4*c^8 : : (Peter Moses)
X(68662) = 2*X[10992]-3*X[35705], 2*X[62203]-X[63424]

As a point on the Euler line, X(68662) has Shinagawa coefficients: {-3 S^2, (e + f)^2 + 12 S^2}

See Kadir Altintas and David Nguyen, euclid 8524.

X(68662) lies on these lines: {2, 3}, {5023, 7607}, {7608, 15815}, {8596, 50962}, {9830, 11477}, {10488, 10753}, {10992, 35705}, {11456, 38339}, {62203, 63424}

X(68662) = reflection of X(63424) in X(62203)

X(68663) = X(1)X(376)∩X(4)X(11)

Barycentrics    -5*a^4+4*a^2*b^2+b^4-8*a^2*b*c+4*a^2*c^2-2*b^2*c^2+c^4 : :
X(68663) = 6*X[11]-5*X[47744],X[14986]-2*X[67261],3*X[3361]-X[67931]

See David Nguyen and Ercole Suppa, euclid 8525.

X(68663) lies on these lines: {1, 376}, {2, 9654}, {3, 1056}, {4, 11},{5, 5265}, {7, 1385}, {8, 9352}, {10, 34610}, {12, 3525}, {20, 999}, {30, 14986}, {35, 21735}, {36, 388}, {40, 1000}, {46, 3476}, {55, 3528}, {57, 944}, {65, 7967}, {140, 5261}, {145, 10031}, {165, 66230}, {221, 65114}, {329, 17614}, {350, 32822}, {354, 4305}, {355, 5435}, {382, 5274}, {390, 550}, {404, 3421}, {443, 2975}, {495, 3523}, {496, 3146}, {497, 3529}, {499, 3545}, {515, 3361}, {516, 61762}, {517, 4308}, {535, 10200}, {548, 6767}, {551, 43733}, {614, 7714}, {920,7284}, {938,18481}, {942,5731}, {943,8273}, {946, 13462}, {952, 37545}, {956, 6904}, {957, 42448}, {958, 17582}, {962, 24928}, {993, 16845}, {1106, 3072}, {1125, 5714}, {1250, 52080}, {1285, 16502}, {1319, 4295}, {1388, 11246}, {1398, 7487}, {1420, 4292}, {1425, 18925}, {1450, 1745}, {1466, 11491}, {1469, 14912}, {1470, 6942}, {1476, 6948}, {1477, 53898}, {1478, 3090}, {1479, 4325}, {1617, 6906}, {1788, 45287}, {1870, 4320}, {2067, 7581}, {2093, 63987}, {2094, 4018}, {2096, 63986}, {2192, 12250}, {2242, 7738}, {2478, 20067}, {2550, 8666}, {3058, 62130}, {3085, 3524}, {3091, 9655}, {3183, 7049}, {3241, 50193}, {3244, 34607}, {3295, 3522}, {3297, 9541}, {3303, 62092}, {3304, 4294}, {3332, 42314}, {3333, 3488}, {3338, 3486}, {3339, 5882}, {3434, 57000}, {3436, 17567}, {3475, 3612}, {3485, 37618}, {3487, 3576}, {3518, 18954}

X(68663) = reflection of X(14986) in X(67261)
X(68663) = perspector of the circumconic through X(65331)and X(68185)
X(68663) = pole of line {4778, 53522} with respect to incircle
X(68663) = pole of line {6001, 10595} with respect to Feuerbach hyperbola
X(68663) = pole of line {11, 123} with respect to orthoptic circle of Feuerbach hyperbola
X(68663) = pole of line {4654, 34050} with respect to dual conic of Yff parabola
X(68663) = X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,3600,1056},{20,999,1058},{36,388,631},{36,4317,388},{46,3476,12245},{56,4293,4},{56,7354,3086},{57,4311,944},{388,631,8164},{404,20076,3421},{497,4299,3529},{499,5229,3545},{550,7373,390},{1319,4295,10595},{1420,4292,5603},{1478,7288,3090},{1788,45287,59388},{3085,5204,3524},{3086,4293,7354},{3086,7354,4},{3304,15326,4294},{3333,4297,3488},{3338,21578,3486},{3339,5882,11041},{3576,4298,3487},{4294,15326,17538},{4299,5563,497},{4325,37587,1479},{5204,5434,3085},{9655,15325,3091},{10591,12943,4},{22753,64120,4}

X(68664) = X(4)X(12)∩X(20)X(999)

Barycentrics    7*a^4 - 4*a^2*(b + c)^2 - 3*(b^2 - c^2)^2 : :
X(68664) = 2*X[390]-X[1056], 3*X[390]-2*X[6767], 3*X[1056]-4*X[6767], 3*X[1000]-4*X[9819], 3*X[3488]-2*X[11529], 4*X[6913]-3*X[38149], 2*X[6987]-X[35514], 3*X[11038]-2*X[18541], 6*X[16857]-5*X[40333]

See David Nguyen, euclid 8525.

X(68664) lies on these lines: {1, 3529}, {2, 9668}, {3, 5274}, {4, 12}, {7, 28146}, {11, 3524}, {20, 999}, {30, 390}, {35, 3090}, {36, 376}, {56, 17538}, {145, 50244}, {186, 10833}, {202, 42120}, {203, 42119}, {221, 64187}, {381, 5281}, {382, 10386}, {388, 4309}, {443, 5284}, {495, 3543}, {496, 3522}, {498, 3855}, {499, 10299}, {515, 1000}, {516, 3488}, {517, 10394}, {548, 5265}, {550, 14986}, {631, 1479}, {938, 31795}, {944, 7962}, {946, 53054}, {950, 2093}, {962, 50194}, {1159, 28216}, {1210, 63207}, {1478, 10385}, {1657, 3600}, {1870, 4319}, {2066, 23267}, {2478, 20066}, {2886, 50739}, {3056, 39874}, {3058, 4293}, {3086, 3528}, {3091, 64951}, {3146, 3295}, {3296, 4292}, {3297, 43407}, {3298, 43408}, {3303, 11541}, {3304, 62146}, {3421, 11114}, {3434, 11111}, {3486, 25415}, {3487, 4314}, {3523, 9669}, {3525, 5217}, {3533, 7741}, {3545, 3583}, {3576, 51783}, {3582, 15710}, {3584, 61980}, {3585, 62021}, {3586, 5657}, {3616, 57000}, {3627, 5261}, {3655, 4345}, {3746, 5229}, {3790, 48798}, {3816, 34626}, {3820, 34707}, {3839, 31479}, {4295, 44840}, {4299, 37602}, {4304, 5603}, {4305, 10595}, {4313, 12699}, {4316, 46333}, {4324, 37587}, {4342, 50811}, {4366, 32986}, {4845, 63416}, {4857, 7288}, {4972, 51673}, {4995, 41106}, {5046, 59591}, {5048, 7967}, {5059, 18990}, {5067, 5326}, {5071, 5432}, {5082, 6872}, {5119, 59388}, {5183, 18391}, {5204, 62092}, {5298, 62086}, {5310, 8889}, {5332, 9598}, {5414, 23273}, {5433, 61138}, {5434, 62161}, {5435, 18527}, {5563, 62133}, {5691, 53052}, {5703, 22793}, {5704, 31663}, {5714, 51118}, {5722, 9778}, {5727, 50810}, {5731, 25405}, {5768, 14646}, {5818, 61763}, {5840, 6916}, {6690, 34706}, {6827, 35000}, {6857, 52367}, {6865, 35238}, {6883, 7676}, {6905, 8166}, {6913, 38149}, {6947, 13199}, {6949, 40272}, {6954, 10738}, {6987, 35514}, {6988, 10525}, {7173, 60781}, {7280, 62084}, {7354, 8162}, {7373, 15704}, {7491, 44455}, {7735, 9664}, {7743, 54445}, {7951, 41099}, {8167, 17582}, {8168, 57288}, {8236, 28154}, {8540, 14912}, {9581, 63214}, {9654, 17578}, {9655, 49135}, {9659, 35475}, {9671, 52793}, {9673, 47485}, {9785, 18481}, {9812, 24929}, {10056, 62017}, {10175, 31508}, {10303, 10593}, {10304, 15325}, {10483, 62171}, {10572, 12245}, {10588, 61964}, {10592, 50689}, {10831, 14865}, {10980, 63999}, {11038, 18541}, {11041, 28194}, {11106, 31419}, {11113, 17784}, {11237, 62029}, {11238, 19708}, {12082, 16541}, {12103, 67261}, {12244, 46687}, {12317, 12896}, {12374, 20125}, {12572, 62218}, {12943, 62042}, {14039, 26590}, {15170, 15683}, {15326, 62130}, {15709, 65140}, {15933, 28202}, {15934, 28178}, {16857, 40333}, {17561, 33108}, {17576, 24390}, {18421, 28232}, {18513, 62011}, {18514, 31452}, {20070, 37730}, {24210, 66680}, {24552, 51665}, {26629, 33285}, {28158, 30331}, {28164, 31393}, {28172, 66686}, {28897, 64168}, {31156, 33110}, {31673, 53053}, {31730, 53056}, {31777, 37423}, {33095, 60751}, {34610, 34649}, {34631, 37740}, {35986, 41345}, {37720, 62066}, {37722, 62113}, {43509, 44623}, {43510, 44624}, {44858, 64500}, {50865, 64110}, {53058, 63993}, {54255, 64787}, {54361, 59316}, {57002, 64081}, {59325, 61807}, {61814, 63756}, {61881, 65142}, {62083, 64894}, {64133, 67536}, {64144, 66992}

X(68664) = reflection of X(i) in X(j) for these (i,j): {1056, 390}, {35514, 6987}
X(68664) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 55, 8164}, {20, 15171, 1058}, {35, 5225, 3090}, {55, 65632, 10590}, {388, 65134, 33703}, {497, 4302, 376}, {1479, 5010, 10589}, {1657, 15172, 3600}, {3085, 4294, 63273}, {3085, 12953, 4}, {3086, 15338, 3528}, {3583, 5218, 3545}, {4294, 6284, 4}, {4304, 9580, 5603}, {4305, 12701, 10595}, {4309, 65134, 388}, {4314, 41869, 3487}, {5010, 10589, 631}, {5217, 10591, 3525}, {6284, 63273, 12953}, {9670, 15338, 3086}, {10590, 65632, 4}, {11114, 20075, 3421}, {12953, 63273, 3085}

X(68665) = X(2)X(12)∩X(7)X(551)

Barycentrics    (a+b-c)*(a-b+c)*(11*a^2+b^2+2*b*c+c^2) : :
X(68665) = X[14986]-4*X[67261], 2*X[3361]+X[4308]

See David Nguyen, euclid 8525.

X(68665) lies on these lines: {1, 10304}, {2, 12}, {7, 551}, {11, 61985}, {20, 5563}, {30, 14986}, {35, 62059}, {36, 5281}, {55, 62063}, {57, 3241}, {145, 9352}, {376, 390}, {495, 15702}, {496, 15682}, {497, 15683}, {498, 61844}, {499, 61924}, {519, 3361}, {549, 1056}, {553, 1420}, {613, 51028}, {938, 50811}, {993, 61027}, {1014, 4234}, {1058, 3534}, {1210, 50864}, {1319, 21454}, {1398, 7714}, {1400, 63086}, {1428, 63127}, {1445, 50835}, {1447, 7268}, {1469, 5032}, {1476, 15998}, {1478, 61936}, {1479, 15640}, {1617, 17549}, {2067, 63059}, {2192, 68027}, {3058, 62120}, {3085, 15708}, {3086, 3839}, {3091, 3582}, {3146, 11238}, {3295, 19708}, {3303, 21734}, {3304, 3522}, {3333, 15933}, {3338, 18221}, {3339, 6049}, {3476, 31145}, {3487, 3653}, {3523, 10056}, {3543, 3583}, {3545, 18990}, {3576, 11038}, {3584, 15721}, {3585, 61966}, {3598, 17079}, {3616, 4654}, {3622, 44447}, {3623, 5221}, {3671, 51105}, {3679, 4315}, {3746, 62067}, {3828, 64114}, {3845, 47743}, {3929, 7091}, {3945, 37617}, {4292, 18220}, {4294, 15697}, {4298, 25055}, {4299, 62160}, {4302, 62122}, {4309, 62110}, {4321, 6172}, {4322, 42042}, {4323, 51103}, {4325, 49135}, {4345, 53058}, {4355, 51110}, {4428, 51773}, {4857, 49140}, {4995, 15717}, {5059, 37722}, {5068, 9657}, {5071, 15325}, {5126, 30340}, {5204, 15705}, {5217, 62056}, {5218, 61806}, {5225, 62032}, {5229, 61954}, {5270, 7486}, {5290, 19883}, {5432, 61825}, {5703, 50828}, {5704, 50796}, {5708, 50824}, {6284, 62148}, {6502, 63058}, {6767, 34200}, {7051, 63079}, {7280, 61781}, {7320, 63469}, {7354, 50687}, {7373, 8703}, {7677, 16418}, {7741, 61958}, {8164, 15694}, {8666, 17580}, {8732, 38092}, {9654, 61899}, {9655, 41099}, {9668, 62161}, {9669, 62017}, {9670, 62152}, {10106, 53620}, {10386, 15695}, {10404, 46934}, {10483, 62037}, {10589, 61944}, {10590, 61912}, {10591, 61989}, {10592, 61889}, {10593, 61967}, {10895, 61930}, {10896, 61992}, {11019, 34628}, {11036, 37618}, {12573, 59374}, {12632, 37267}, {12943, 62005}, {12953, 62051}, {13370, 34619}, {15171, 62130}, {15172, 15689}, {15326, 62129}, {15710, 64951}, {15888, 61820}, {17798, 66305}, {18419, 64106}, {19373, 63080}, {19876, 31188}, {20070, 20323}, {21625, 50815}, {24928, 67262}, {26437, 36004}, {27649, 55362}, {28194, 61762}, {30332, 34638}, {30652, 52440}, {31165, 63994}, {31410, 46936}, {31452, 61798}, {31475, 43883}, {31479, 61859}, {31480, 61807}, {32065, 64059}, {34631, 36279}, {34647, 60984}, {34753, 50798}, {37582, 50810}, {37666, 54310}, {37709, 51072}, {37719, 61856}, {37720, 50688}, {37817, 62782}, {42314, 63054}, {49736, 50738}, {50303, 62787}, {50829, 51784}, {50840, 60934}, {51082, 67942}, {51099, 60939}, {58184, 59325}, {58204, 65134}, {61780, 64894}, {62002, 65631}, {62007, 65140}, {62054, 63756}, {62060, 64950}, {62102, 63273}, {64108, 66228}, {66372, 66724}

X(68665) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {56, 3600, 5265}, {388, 5298, 2}, {553, 1420, 38314}, {3333, 51705, 15933}, {3543, 10072, 5274}, {3600, 5265, 5261}, {4293, 10072, 3543}, {7288, 11237, 2}, {34610, 40726, 2}, {37267, 62837, 12632}

X(68666) = X(1)X(2)∩X(30)X(390)

Barycentrics    a^4 + (b^2 - c^2)^2 - 2*a^2*(b^2 + 12*b*c + c^2) : :
X(68666) = X[7]+2*X[31393], X[390]+2*X[1056], X[390]-4*X[6767], X[1056]+2*X[6767], X[1000]+2*X[15934], 2*X[5542]+X[9819], 2*X[30331]+X[66686]

See David Nguyen and Ercole Suppa, euclid 8525.

X(68666) lies on these lines: {1, 2}, {4, 15170}, {7, 28194}, {11, 61924}, {12, 61936}, {20, 3303}, {30, 390}, {35, 62063}, {36, 15705}, {40, 65384}, {55, 10304}, {56, 15692}, {226, 51779}, {346, 48806}, {350, 32874}, {354, 59417}, {376, 3295}, {381, 1058}, {388, 3058}, {443, 12632}, {495, 3545}, {496, 5071}, {497, 3839}, {515, 8236}, {517, 11038}, {529, 47357}, {547, 47743}, {549, 5265}, {553, 1697}, {611, 5032}, {613, 63127}, {942, 50810}, {950, 50864}, {956, 17561}, {962, 4654}, {999, 3524}, {1000, 8732}, {1124, 63058}, {1335, 63059}, {1469, 54174}, {1478, 50687}, {1479, 61985}, {1909, 32869}, {2136, 11024}, {3056, 51028}, {3057, 11036}, {3091, 11238}, {3296, 12702}, {3297, 19053}, {3298, 19054}, {3304, 3523}, {3475, 5919}, {3476, 3748}, {3487, 3656}, {3488, 8232}, {3522, 3746}, {3525, 31480}, {3583, 61992}, {3585, 62005}, {3653, 51788}, {3654, 5045}, {3671, 30337}, {3672, 48819}, {3830, 15172}, {3880, 38053}, {3895, 9776}, {3913, 17580}, {3945, 48823}, {3947, 30308}, {4208, 64068}, {4293, 62120}, {4294, 15683}, {4299, 62129}, {4302, 62148}, {4307, 28854}, {4308, 51705}, {4309, 5059}, {4310, 66674}, {4313, 50811}, {4314, 34628}, {4317, 50693}, {4325, 62124}, {4330, 62152}, {4344, 48828}, {4345, 64110}, {4428, 34610}, {4857, 31410}, {5010, 62056}, {5049, 5657}, {5055, 8164}, {5056, 37722}, {5068, 37719}, {5082, 44217}, {5119, 21454}, {5204, 61781}, {5217, 62059}, {5218, 5298}, {5225, 61989}, {5226, 38021}, {5229, 62007}, {5270, 17578}, {5325, 6762}, {5432, 61844}, {5433, 61846}, {5542, 9819}, {5563, 15717}, {5603, 67889}, {5722, 38074}, {5726, 38076}, {5728, 50835}, {5731, 10389}, {5734, 37421}, {5882, 37434}, {6284, 15640}, {6392, 32095}, {6824, 61286}, {6846, 37727}, {6886, 37724}, {6908, 10222}, {6926, 15178}, {6944, 61278}, {6964, 61276}, {7278, 43983}, {7280, 61778}, {7288, 15721}, {7320, 7982}, {7354, 62160}, {7741, 61927}, {7951, 61930}, {8148, 44284}, {8965, 63689}, {9581, 66241}, {9654, 41099}, {9655, 62042}, {9657, 49135}, {9668, 62017}, {9669, 41106}, {9670, 50688}, {9785, 21620}, {9846, 63972}, {9957, 37427}, {10179, 25568}, {10386, 15681}, {10387, 54170}, {10483, 62168}, {10588, 61912}, {10589, 61906}, {10590, 61954}, {10591, 61944}, {10592, 61932}, {10593, 61926}, {10595, 63282}, {10629, 62969}, {10895, 61966}, {10896, 61958}, {11001, 18990}, {11374, 18220}, {11997, 51064}, {12000, 28458}, {12100, 67261}, {12245, 18221}, {12433, 50798}, {12513, 17558}, {12575, 50865}, {12577, 50808}, {12915, 64664}, {12943, 62037}, {14563, 61019}, {14930, 16784}, {15022, 37720}, {15171, 15682}, {15325, 15709}, {15326, 62112}, {15338, 62122}, {15677, 20076}, {16486, 63089}, {16777, 27508}, {16781, 63024}, {16785, 63005}, {16857, 42884}, {17469, 55912}, {17699, 23958}, {19708, 64951}, {25303, 32830}, {28228, 59372}, {28236, 38037}, {28629, 66256}, {28849, 40892}, {30331, 66686}, {31419, 50727}, {31452, 61820}, {31479, 61899}, {32836, 64133}, {33925, 37106}, {34648, 66682}, {34791, 54398}, {37462, 64199}, {37542, 37631}, {37576, 66305}, {37580, 66301}, {37602, 61812}, {38068, 64114}, {39873, 51215}, {40091, 48870}, {40270, 50796}, {44663, 51099}, {48818, 64168}, {49719, 56936}, {50802, 51785}, {50828, 61762}, {51816, 64142}, {54435, 63079}, {54436, 63080}, {54437, 63102}, {54438, 63103}, {54712, 60229}, {55908, 62833}, {55913, 62834}, {56028, 64329}, {59325, 62054}, {59387, 64162}, {61777, 64894}, {62051, 65134}, {62067, 64950}, {62828, 62997}, {62848, 63007}, {67993, 67999}

X(68666) = reflection of X(15933) in X(1)
X(68666) = pole of line {30, 35514} with respect to orthoptic circle of Kiepert parabola
X(68666) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 31397, 10580}, {1, 51784, 21625}, {388, 3058, 3543}, {497, 11237, 3839}, {551, 34619, 2}, {553, 1697, 34632}, {1056, 6767, 390}, {2136, 51723, 11024}, {3085, 10072, 2}, {3086, 3584, 2}, {3303, 5434, 10385}, {3623, 10587, 64081}, {4428, 34610, 50742}, {4857, 31410, 50689}, {5218, 5298, 15708}, {5434, 10385, 20}, {10389, 66228, 5731}, {10590, 65140, 61954}, {11037, 34632, 553}, {11239, 38314, 2}, {21625, 51784, 5704}, {36442, 36460, 499}

X(68667) = X(1)X(376)∩X(2)X(65)

Barycentrics    -((a + b - c)*(a - b + c)*(a^2 - 6*a*(b + c) - (b + c)^2)) : :
X(68667) = 2*X[1]-X[10385], X[3486]+2*X[4295], 2*X[4428]-3*X[38314], X[388]+2*X[3340], X[388]-4*X[3671], X[388]-2*X[4654], 5*X[388]-2*X[37709], X[3340]+2*X[3671], X[3340]+X[4654], 5*X[3340]+X[37709], 2*X[3671]-X[4654], 10*X[3671]-X[37709], 5*X[4654]-X[37709], X[5082]+2*X[12559], X[1697]-4*X[12563], 2*X[5325]-X[12526], 2*X[12514]-3*X[17561]

See David Nguyen, euclid 8525.

X(68667) lies on these lines: {1, 376}, {2, 65}, {4, 17098}, {7, 528}, {8, 3649}, {12, 53620}, {30, 3486}, {46, 3524}, {55, 34632}, {56, 4323}, {57, 551}, {73, 42042}, {145, 10404}, {196, 60681}, {226, 3679}, {279, 4955}, {381, 1159}, {388, 519}, {390, 44840}, {495, 34718}, {497, 11529}, {517, 3475}, {518, 60967}, {527, 12560}, {534, 2263}, {537, 7201}, {549, 36279}, {938, 11238}, {942, 3656}, {948, 50282}, {950, 50865}, {962, 3058}, {999, 38602}, {1056, 11551}, {1155, 15692}, {1210, 38021}, {1319, 21454}, {1420, 51103}, {1445, 38025}, {1452, 62979}, {1464, 17018}, {1466, 40726}, {1467, 64667}, {1469, 50999}, {1478, 9897}, {1482, 28458}, {1697, 12563}, {1737, 5071}, {1770, 11001}, {1836, 3543}, {1837, 3839}, {2093, 5218}, {2094, 11194}, {2098, 11037}, {2362, 19053}, {2646, 10304}, {2800, 5603}, {3057, 11036}, {3085, 3654}, {3086, 31794}, {3146, 37724}, {3185, 27654}, {3189, 34195}, {3303, 7411}, {3304, 5734}, {3338, 10595}, {3339, 7288}, {3361, 51105}, {3487, 5903}, {3488, 5425}, {3529, 4338}, {3534, 4305}, {3545, 12047}, {3577, 67993}, {3584, 5657}, {3586, 14563}, {3600, 11011}, {3601, 50808}, {3612, 19708}, {3616, 5221}, {3622, 32636}, {3653, 37582}, {3655, 4293}, {3674, 24797}, {3828, 5219}, {3830, 37730}, {3872, 60952}, {3877, 38053}, {3913, 35990}, {3947, 4745}, {3982, 34747}, {4018, 19843}, {4031, 13462}, {4292, 50811}, {4294, 28198}, {4298, 51071}, {4301, 11518}, {4307, 50070}, {4308, 36005}, {4312, 34628}, {4329, 15936}, {4333, 46333}, {4342, 44841}, {4355, 63987}, {4644, 49487}, {4669, 9578}, {4677, 5290}, {4757, 26363}, {4848, 10588}, {4995, 5703}, {5054, 37737}, {5055, 67980}, {5083, 50891}, {5173, 24473}, {5183, 5281}, {5225, 6738}, {5226, 40663}, {5229, 64163}, {5252, 31145}, {5261, 41687}, {5289, 9776}, {5323, 42028}, {5325, 12526}, {5435, 15950}, {5542, 7962}, {5543, 7198}, {5563, 6950}, {5665, 28609}, {5698, 31156}, {5714, 10573}, {5727, 34648}, {5731, 11246}, {5761, 35004}, {5775, 31245}, {5919, 11038}, {5933, 17271}, {6604, 50310}, {6916, 7982}, {6935, 13464}, {6948, 10222}, {7026, 30327}, {7043, 30328}, {7080, 10107}, {7098, 50739}, {7672, 61027}, {7991, 63274}, {8162, 30295}, {8543, 16861}, {9581, 50802}, {9612, 50796}, {9614, 17706}, {9657, 67972}, {9779, 61717}, {9799, 64754}, {10106, 51093}, {10389, 28228}, {10474, 48858}, {10572, 15682}, {10826, 41106}, {10950, 50864}, {10956, 64746}, {11028, 50905}, {11114, 17097}, {11224, 59372}, {11239, 34711}, {11374, 50821}, {11415, 66099}, {11509, 13587}, {11520, 64068}, {11544, 18525}, {12016, 50901}, {12245, 13407}, {12514, 17561}, {12575, 51120}, {12635, 49732}, {12702, 16137}, {12736, 50908}, {12832, 59377}, {13601, 34619}, {15170, 15934}, {15672, 41697}, {15677, 44447}, {15698, 58887}, {15803, 50828}, {15956, 17220}, {16232, 19054}, {16371, 37541}, {17164, 50043}, {17530, 64127}, {17605, 61936}, {17606, 61924}, {17718, 59417}, {17781, 19860}, {18541, 37728}, {20070, 37080}, {21161, 59317}, {24465, 50843}, {24470, 50824}, {24471, 47358}, {24472, 50881}, {24806, 42289}, {26105, 51423}, {28204, 57282}, {28208, 37739}, {30308, 67931}, {32808, 57267}, {32809, 57266}, {34200, 37606}, {34231, 52167}, {34606, 64143}, {34641, 51782}, {34716, 60963}, {34753, 38022}, {35176, 65288}, {36999, 64321}, {37558, 48855}, {37566, 58560}, {37600, 62063}, {37614, 37631}, {37692, 61899}, {38024, 60992}, {39782, 64142}, {39897, 51001}, {41712, 61023}, {45287, 50818}, {45701, 50841}, {47357, 60932}, {49492, 50045}, {49739, 64168}, {50080, 56821}, {50809, 59316}, {50836, 52819}, {50878, 59818}, {50886, 59815}, {50898, 59813}, {50918, 59816}, {50921, 59817}, {51077, 66230}, {51104, 64849}, {52374, 63354}, {54382, 63006}, {59387, 61716}, {61663, 67998}

X(68667) = midpoint of X(3340) and X(4654)
X(68667) = reflection of X(i) in X(j) for these (i,j): {388, 4654}, {4654, 3671}, {10385, 1}, {12526, 5325}
X(68667) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 2099, 3476}, {7, 3241, 5434}, {65, 3485, 1788}, {962, 15933, 3058}, {1159, 39542, 18391}, {2093, 64110, 5218}, {2099, 5434, 3241}, {3241, 5434, 3476}, {3339, 64160, 7288}, {3340, 3671, 388}, {3340, 4654, 519}, {3487, 50810, 10056}, {3649, 64963, 8}, {4323, 65384, 38314}, {5903, 10056, 50810}, {11224, 59372, 66228}, {11551, 25415, 1056}, {38314, 65384, 56}

X(68668) = X(1)X(3)∩X(7)X(944)

Barycentrics    -(a*(a^3 - 2*a^2*(b + c) + 2*(b - c)^2*(b + c) - a*(b^2 + c^2))) : :
X(68668) = X[8]-2*X[31419], X[3486]+X[4295], X[145]+X[5082], X[388]-2*X[6147], 4*X[3671]-X[9655], 2*X[3671]+X[37739], 2*X[3671]-X[57282], X[9655]+2*X[37739], X[9655]-2*X[57282], X[37739]+X[57282], X[5794]-2*X[12609], 2*X[5794]-3*X[17528], 2*X[12563]-X[21620], 4*X[12609]-3*X[17528], 2*X[958]-X[3927], X[958]-2*X[30147], X[3927]-4*X[30147], 2*X[1125]-X[5837], 5*X[3623]-X[56936], 2*X[18517]-X[18525], 2*X[3626]-3*X[61031], 3*X[4654]-X[9613], X[4853]+X[41863]

See David Nguyen, euclid 8525.

X(68668) lies on these lines: {1, 3}, {2, 4930}, {4, 15911}, {5, 3485}, {6, 1731}, {7, 944}, {8, 442}, {10, 3940}, {11, 18493}, {12, 5790}, {21, 64047}, {29, 1148}, {30, 3486}, {34, 1871}, {42, 47522}, {63, 4018}, {72, 9708}, {73, 37698}, {78, 3753}, {79, 12943}, {80, 10895}, {81, 37227}, {140, 1788}, {142, 66244}, {145, 377}, {150, 33949}, {158, 7524}, {218, 17451}, {219, 2294}, {221, 36742}, {224, 39779}, {226, 355}, {296, 64840}, {381, 1837}, {382, 1836}, {386, 34586}, {388, 952}, {392, 11682}, {405, 3869}, {407, 1068}, {474, 4511}, {496, 938}, {497, 12433}, {498, 40663}, {499, 15950}, {515, 3671}, {518, 12559}, {519, 5794}, {550, 3474}, {551, 64124}, {553, 3655}, {595, 36013}, {653, 7531}, {758, 958}, {851, 17018}, {912, 12709}, {916, 7352}, {946, 5722}, {950, 9668}, {954, 7672}, {956, 3868}, {959, 13731}, {960, 11108}, {962, 3488}, {971, 12560}, {993, 4084}, {994, 19763}, {995, 17054}, {997, 3812}, {1000, 37112}, {1001, 3878}, {1004, 3885}, {1012, 62864}, {1058, 6836}, {1118, 7510}, {1125, 5289}, {1145, 10528}, {1191, 30117}, {1210, 5886}, {1317, 10044}, {1320, 3296}, {1376, 3754}, {1387, 6833}, {1392, 15179}, {1411, 36750}, {1445, 38031}, {1449, 2182}, {1452, 3517}, {1468, 52407}, {1476, 14497}, {1478, 3649}, {1483, 3476}, {1490, 3577}, {1497, 3215}, {1537, 10531}, {1597, 67965}, {1598, 1905}, {1656, 1737}, {1657, 1770}, {1698, 4867}, {1732, 40937}, {1776, 3560}, {1845, 38573}, {1854, 6000}, {1857, 44225}, {1858, 37234}, {1864, 31937}, {1870, 4185}, {1953, 19350}, {2050, 44733}, {2217, 53114}, {2245, 16777}, {2263, 62183}, {2271, 3959}, {2278, 16884}, {2320, 5303}, {2362, 3312}, {2478, 51409}, {2550, 41548}, {2647, 8757}, {2650, 3157}, {2654, 67901}, {2771, 18761}, {2800, 11496}, {2886, 49168}, {3085, 5690}, {3086, 5901}, {3091, 12019}, {3149, 21740}, {3189, 39783}, {3218, 3897}, {3241, 11037}, {3242, 4259}, {3243, 5784}, {3244, 4780}, {3297, 35642}, {3298, 35641}, {3306, 17614}, {3311, 16232}, {3419, 41575}, {3475, 5844}, {3526, 24914}, {3555, 3872}, {3556, 20831}, {3584, 38066}, {3585, 61716}, {3586, 22793}, {3600, 6934}, {3616, 7483}, {3617, 8164}, {3622, 5330}, {3626, 61031}, {3632, 41711}, {3633, 41870}, {3635, 12577}, {3656, 12053}, {3679, 41696}, {3683, 16866}, {3697, 3984}, {3811, 5836}, {3827, 37492}, {3843, 37721}, {3851, 10826}, {3870, 10914}, {3871, 35979}, {3873, 4861}, {3874, 12513}, {3876, 51572}, {3881, 22837}, {3889, 38460}, {3894, 5288}, {3899, 5259}, {3901, 5258}, {3919, 25440}, {3924, 16466}, {3962, 41229}, {3970, 4513}, {4002, 67097}, {4004, 5440}, {4067, 5220}, {4253, 34522}, {4258, 5011}, {4292, 18481}, {4293, 24470}, {4294, 28174}, {4298, 5882}, {4299, 11246}, {4301, 63999}, {4302, 10543}, {4313, 6361}, {4314, 28194}, {4315, 13607}, {4317, 52783}, {4333, 15681}, {4338, 17800}, {4342, 40270}, {4393, 37233}, {4417, 5827}, {4640, 17571}, {4654, 9613}, {4658, 54417}, {4744, 5267}, {4848, 13411}, {4853, 41863}, {4870, 5055}, {4879, 34958}, {5044, 64673}, {5083, 12737}, {5086, 17532}, {5135, 38315}, {5218, 61524}, {5219, 5780}, {5225, 40273}, {5226, 5818}, {5252, 12645}, {5261, 6984}, {5270, 37706}, {5290, 5881}, {5396, 10571}, {5398, 55101}, {5399, 52544}, {5414, 38235}, {5434, 37734}, {5437, 64263}, {5439, 19861}, {5492, 24430}, {5530, 9567}, {5554, 17757}, {5555, 33898}, {5557, 21398}, {5587, 66700}, {5644, 54386}, {5657, 5703}, {5687, 34772}, {5693, 5779}, {5694, 18397}, {5698, 50241}, {5714, 59387}, {5718, 5754}, {5727, 9612}, {5728, 12672}, {5729, 6920}, {5734, 10580}, {5738, 41007}, {5747, 21933}, {5758, 31799}, {5763, 64111}, {5795, 67850}, {5804, 7956}, {5806, 63992}, {5855, 25466}, {5883, 25524}, {5884, 12114}, {5887, 6913}, {5904, 16126}, {6198, 37194}, {6261, 7686}, {6265, 12736}, {6284, 48661}, {6326, 6797}, {6734, 31493}, {6744, 63993}, {6842, 64127}, {6847, 33899}, {6860, 47743}, {6889, 12245}, {6897, 11038}, {6900, 45043}, {6911, 61541}, {6918, 45770}, {6985, 45230}, {7052, 54402}, {7078, 44414}, {7190, 62402}, {7288, 34753}, {7354, 18541}, {7465, 29815}, {7590, 11899}, {7741, 61717}, {7743, 11522}, {7971, 9856}, {8082, 11535}, {8094, 18454}, {8227, 67931}, {8256, 45701}, {8728, 28629}, {9352, 19537}, {9579, 28160}, {9580, 31795}, {9581, 9955}, {9614, 18527}, {9623, 11523}, {9642, 38336}, {9785, 15170}, {9895, 37697}, {9928, 66760}, {10039, 17718}, {10056, 34718}, {10073, 51517}, {10106, 37727}, {10107, 54286}, {10198, 11281}, {10384, 15008}, {10386, 28212}, {10390, 56038}, {10393, 37411}, {10404, 11551}, {10459, 28388}, {10483, 11552}, {10569, 17624}, {10585, 38058}, {10588, 38042}, {10589, 61272}, {10590, 18357}, {10591, 38034}, {10593, 68034}, {10742, 66206}, {10893, 64762}, {10896, 18393}, {10944, 67933}, {10956, 64140}, {10974, 30116}, {11019, 11373}, {11043, 37225}, {11101, 46441}, {11113, 11415}, {11194, 51111}, {11237, 37710}, {11362, 13405}, {11491, 18467}, {11499, 37733}, {11501, 12331}, {11502, 37251}, {11570, 12773}, {11826, 60923}, {12115, 52683}, {12432, 31806}, {12445, 18456}, {12514, 16418}, {12526, 31445}, {12607, 33559}, {12647, 15888}, {12649, 24390}, {12650, 12671}, {12675, 30283}, {12684, 15071}, {12717, 21848}, {12722, 61086}, {12743, 48680}, {12747, 13273}, {12832, 57298}, {12876, 63969}, {12898, 46683}, {12908, 55173}, {13733, 57280}, {13743, 16141}, {14151, 30340}, {14257, 34231}, {15172, 15935}, {15254, 16860}, {15346, 61030}, {15347, 41540}, {15570, 58563}, {15823, 54422}, {15844, 26470}, {15937, 63386}, {15952, 64420}, {16049, 64377}, {16117, 33857}, {16133, 40269}, {16370, 56288}, {16417, 59691}, {16483, 28082}, {16496, 66698}, {16844, 31359}, {16853, 25917}, {16857, 31165}, {17015, 37241}, {17024, 37449}, {17164, 49492}, {17619, 30852}, {17644, 30318}, {18240, 25485}, {18241, 64334}, {18389, 22758}, {18395, 37701}, {18413, 38572}, {18519, 67970}, {19349, 34040}, {19526, 62838}, {19854, 21677}, {19869, 56735}, {20050, 64201}, {20057, 62863}, {20122, 67011}, {20292, 50239}, {20818, 54405}, {21180, 53295}, {21616, 34647}, {21625, 64703}, {21767, 46882}, {21863, 54285}, {22457, 31880}, {22753, 31870}, {22759, 62859}, {23708, 39782}, {24473, 62874}, {27003, 51577}, {27383, 47742}, {27622, 54315}, {28109, 36558}, {28443, 41697}, {29814, 30944}, {30435, 54382}, {31162, 66682}, {31397, 63274}, {31410, 61249}, {31474, 35775}, {31837, 41539}, {32047, 54292}, {32850, 50624}, {33655, 54403}, {33895, 58609}, {34954, 48292}, {35768, 44635}, {35769, 44636}, {36011, 62843}, {36845, 37363}, {36846, 62861}, {37298, 38314}, {37414, 63965}, {37482, 67968}, {37703, 45081}, {37719, 41684}, {37828, 59719}, {38030, 60992}, {39523, 54301}, {40430, 51290}, {41558, 62354}, {41712, 59381}, {42884, 67934}, {43166, 63972}, {43175, 60945}, {43213, 64347}, {44447, 57002}, {44661, 59285}, {44675, 61276}, {45636, 64282}, {46681, 64137}, {47745, 51782}, {48333, 50336}, {48664, 64000}, {48907, 49745}, {48908, 50307}, {48909, 49743}, {48944, 66669}, {49626, 64744}, {49682, 62805}, {50443, 51709}, {50597, 50637}, {50806, 65140}, {51410, 63008}, {51783, 68035}, {51784, 63143}, {53115, 54354}, {54296, 62831}, {54424, 59681}, {56027, 56030}, {56029, 56040}, {56425, 58739}, {58679, 64675}, {59372, 61291}, {61287, 63987}, {61296, 66686}, {61650, 62215}, {62333, 64792}, {63976, 64733}, {64055, 66693}, {64074, 66019}, {64345, 64766}, {64669, 68001}

X(68668) = midpoint of X(i) and X(j) for these {i,j}: {1, 3340}, {145, 5082}, {3486, 4295}, {4853, 41863}, {37739, 57282}
X(68668) = reflection of X(i) in X(j) for these (i,j): {8, 31419}, {388, 6147}, {958, 30147}, {3295, 1}, {3927, 958}, {5794, 12609}, {5837, 1125}, {9655, 57282}, {12526, 31445}, {12702, 35239}, {18525, 18517}, {21620, 12563}, {57282, 3671}
X(68668) = barycentric product X(1)*X(31266)
X(68668) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 56062}, {6, 56027}, {19, 64843}, {48, 56336}, {31266, 75}
X(68668) = trilinear product X(6)*X(31266)
X(68668) = trilinear quotient X(i)/X(j) for these {i,j}: {2, 31266}, {2, 56062}, {3, 56336}, {4, 64843}
X(68668) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 36, 34471}, {1, 40, 24929}, {1, 46, 2646}, {1, 56, 10246}, {1, 57, 1385}, {1, 65, 3}, {1, 354, 7373}, {1, 484, 37571}, {1, 942, 999}, {1, 1159, 36279}, {1, 1319, 37624}, {1, 1420, 15178}, {1, 1467, 64668}, {1, 1482, 64897}, {1, 2093, 3601}, {1, 2099, 1482}, {1, 3057, 6767}, {1, 3333, 24928}, {1, 3336, 37525}, {1, 3337, 21842}, {1, 3338, 1319}, {1, 3339, 3576}, {1, 3340, 517}, {1, 3361, 64953}, {1, 4424, 19765}, {1, 5119, 37080}, {1, 5173, 22770}, {1, 5563, 1388}, {1, 5697, 3303}, {1, 5902, 56}, {1, 5903, 55}, {1, 7962, 31792}, {1, 7982, 9957}, {1, 10980, 61762}, {1, 11009, 2098}, {1, 11011, 10247}, {1, 11518, 5045}, {1, 11529, 942}, {1, 11531, 31393}, {1, 13462, 64952}, {1, 13601, 10306}, {1, 15803, 13384}, {1, 18398, 3304}, {1, 18421, 40}, {1, 18838, 16203}, {1, 25415, 3057}, {1, 30323, 5919}, {1, 36279, 37606}, {1, 37550, 24299}, {1, 37558, 50317}, {1, 50193, 64951}, {1, 51816, 20323}, {1, 53615, 8071}, {1, 61762, 25405}, {1, 64721, 16202}, {1, 64963, 12702}, {1, 64964, 10222}, {1, 66640, 5711}, {1, 67932, 37620}, {1, 67971, 5563}, {1, 67977, 35}, {2, 62830, 5730}, {3, 65, 36279}, {3, 1159, 65}, {7, 944, 18990}, {8, 3487, 495}, {10, 11374, 31479}, {10, 12635, 3940}, {10, 62822, 12635}, {10, 64110, 11374}, {12, 10573, 5790}, {35, 67977, 37567}, {36, 5221, 37545}, {40, 18421, 50193}, {40, 24929, 64951}, {46, 2646, 3}, {55, 56, 36152}, {55, 5903, 12702}, {55, 59317, 3}, {55, 64963, 5903}, {56, 5902, 5708}, {56, 10267, 41345}, {56, 11507, 3}, {65, 354, 13750}, {65, 1319, 1454}, {65, 2646, 46}, {65, 20323, 17700}, {65, 67930, 34339}, {72, 19860, 9708}, {78, 3753, 9709}, {145, 11036, 1056}, {158, 60681, 7524}, {226, 355, 9654}, {226, 64163, 355}, {354, 11011, 1}, {484, 37571, 5217}, {495, 16137, 3487}, {938, 4323, 5603}, {938, 5603, 496}, {942, 999, 67301}, {942, 5049, 58576}, {942, 24928, 3333}, {942, 50194, 1}, {946, 5722, 9669}, {946, 6738, 5722}, {946, 14563, 6738}, {950, 12699, 9668}, {960, 54318, 11108}, {962, 3488, 15171}, {997, 3812, 16408}, {1155, 3612, 3}, {1159, 10247, 37541}, {1210, 64160, 5886}, {1319, 3338, 67261}, {1385, 31794, 57}, {1385, 59318, 3}, {1388, 4860, 5563}, {1478, 10950, 18525}, {1482, 15934, 1}, {1482, 63210, 64893}, {1482, 64897, 67300}, {1697, 21842, 10267}, {1737, 11375, 1656}, {1836, 10572, 382}, {1836, 37724, 10572}, {1837, 12047, 381}, {2093, 3601, 3579}, {2098, 2099, 11009}, {2098, 11009, 1482}, {2099, 5425, 15934}, {2099, 15934, 64897}, {3057, 25415, 8148}, {3057, 44840, 1}, {3244, 5542, 66230}, {3295, 41345, 10267}, {3304, 11509, 22766}, {3306, 56387, 17614}, {3333, 24928, 999}, {3336, 37525, 5204}, {3337, 21842, 56}, {3339, 3576, 37582}, {3361, 64953, 5126}, {3474, 4305, 550}, {3485, 18391, 5}, {3486, 4295, 30}, {3487, 11041, 8}, {3649, 10950, 1478}, {3671, 37739, 9655}, {3754, 22836, 1376}, {3872, 11520, 3555}, {3878, 30143, 1001}, {4848, 13411, 26446}, {4870, 17606, 37692}, {5044, 64732, 64673}, {5045, 10222, 1}, {5045, 34339, 67930}, {5048, 17609, 1}, {5122, 7987, 64894}, {5128, 30282, 31663}, {5221, 34471, 36}, {5226, 5818, 10592}, {5563, 67971, 4860}, {5690, 5719, 3085}, {5708, 10246, 56}, {5727, 9612, 18480}, {5730, 62830, 4930}, {5794, 12609, 17528}, {5883, 30144, 25524}, {5885, 46920, 10269}, {5902, 21842, 3337}, {6261, 7686, 19541}, {6767, 8148, 3057}, {7373, 10247, 1}, {8071, 26437, 22765}, {9614, 37723, 18527}, {9623, 11523, 34790}, {10039, 41687, 59503}, {10107, 56176, 54286}, {10269, 40245, 3}, {10404, 37740, 45287}, {10592, 11545, 5818}, {10595, 14986, 1387}, {10826, 17605, 3851}, {10912, 42871, 3244}, {10980, 25405, 999}, {11019, 13464, 11373}, {11278, 31792, 7962}, {11509, 22766, 3}, {11518, 64964, 1}, {11529, 50194, 999}, {11551, 45287, 10404}, {11682, 54392, 392}, {12433, 22791, 497}, {12739, 67945, 12331}, {13384, 15803, 13624}, {13464, 17706, 11019}, {13601, 50195, 37562}, {17164, 49492, 50044}, {17606, 37692, 5055}, {17718, 41687, 10039}, {18393, 37702, 10896}, {18990, 37728, 944}, {24470, 34773, 4293}, {24474, 61146, 22770}, {24929, 50193, 40}, {25415, 44840, 6767}, {25524, 30144, 35272}, {26395, 26419, 44840}, {31788, 37531, 6244}, {31870, 40257, 22753}, {33179, 50192, 51788}, {33179, 51788, 1}, {34753, 38028, 7288}, {34879, 59321, 3}, {37468, 64283, 944}, {37533, 37562, 10306}, {37600, 58887, 3}, {37624, 67261, 1319}, {37730, 39542, 4}, {37737, 67980, 2}, {37739, 57282, 515}, {37740, 45287, 18526}, {45711, 45712, 67966}, {50371, 59333, 3}, {67261, 67710, 56}

X(68669) = X(3)X(758)∩X(56)X(58)

Barycentrics    a^2*(-a^5 + a*b*c*(b^2 + c^2) + a^3*(b^2 - b*c + c^2) + (b - c)^2*(b^3 + c^3) - a^2*(b^3 - 2*b^2*c - 2*b*c^2 + c^3)) : :

See David Nguyen, euclid 8525.

X(68669) lies on these lines: {1, 1283}, {3, 758}, {36, 1046}, {56, 58}, {65, 1324}, {72, 34868}, {172, 21744}, {191, 52273}, {411, 54181}, {511, 11249}, {540, 11194}, {956, 36974}, {958, 3454}, {993, 56949}, {999, 11365}, {1330, 2975}, {1473, 22766}, {2217, 57282}, {2392, 22765}, {2792, 5450}, {2825, 19162}, {2842, 22586}, {2933, 36279}, {3086, 28098}, {3149, 54136}, {3304, 20840}, {3428, 3430}, {3647, 20849}, {3649, 37227}, {3754, 38903}, {5083, 51628}, {5221, 20842}, {5429, 5563}, {5440, 67033}, {5692, 37247}, {5901, 53302}, {5902, 37259}, {6693, 25524}, {7683, 22753}, {8666, 38456}, {9798, 35650}, {10544, 10966}, {10974, 59317}, {11281, 36011}, {14127, 16116}, {14803, 23205}, {15952, 17768}, {16064, 37571}, {18954, 26308}, {19548, 49168}, {20834, 35016}, {22758, 37823}, {23846, 41345}, {26866, 40293}, {26889, 54427}, {34195, 37311}, {37308, 53280}, {37702, 52242}, {37730, 53279}, {51340, 53254}, {51716, 55010}, {54125, 65877}

X(68670) = X(4)X(8)∩X(24)X(55)

Barycentrics    a*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(b^5 - a^3*b*c - b^3*c^2 - b^2*c^3 + c^5 + a^4*(b + c) + a*b*c*(b^2 + c^2) - a^2*(2*b^3 + b^2*c + b*c^2 + 2*c^3)) : :
X(68670) = 2*X[3]-X[20243], X[4]-2*X[1824], 2*X[51755]-3*X[61720]

See David Nguyen, euclid 8525.

X(68670) lies on these lines: {1, 1825}, {3, 11401}, {4, 8}, {19, 101}, {24, 55}, {25, 10679}, {28, 37533}, {33, 5119}, {34, 25415}, {40, 7414}, {65, 1068}, {186, 32613}, {225, 5903}, {378, 3428}, {403, 7680}, {407, 64044}, {429, 5690}, {468, 61533}, {528, 7576}, {674, 6403}, {942, 38295}, {1172, 4280}, {1426, 50193}, {1482, 4185}, {1594, 2886}, {1598, 44455}, {1614, 10537}, {1831, 65128}, {1835, 67977}, {1844, 59337}, {1869, 37625}, {1870, 2099}, {1894, 21664}, {2915, 8141}, {3193, 9928}, {3542, 40635}, {3877, 5136}, {3878, 54396}, {4219, 37584}, {5692, 53008}, {5697, 40950}, {5842, 6240}, {5902, 23710}, {6690, 10018}, {6827, 66741}, {6832, 9895}, {6850, 64039}, {6889, 41340}, {6897, 37613}, {7487, 20075}, {7501, 24929}, {7511, 12135}, {7551, 17927}, {7713, 12703}, {8069, 15500}, {10222, 40985}, {10786, 52359}, {11248, 26377}, {11362, 39579}, {11491, 31384}, {11849, 20832}, {12115, 44662}, {12173, 18499}, {12528, 13754}, {12702, 37194}, {14016, 41723}, {14018, 24474}, {14257, 64721}, {16465, 37379}, {18446, 44661}, {18533, 30273}, {19128, 47373}, {20837, 37621}, {21670, 22276}, {22791, 37368}, {23839, 38461}, {26371, 48461}, {26372, 48460}, {31245, 52296}, {35480, 36999}, {37305, 61146}, {37310, 64750}, {37414, 37562}, {41338, 57276}, {50195, 63965}, {51755, 61720}, {54428, 65144}

X(68670) = reflection of X(i) in X(j) for these (i,j): {4, 1824}, {20243, 3}
X(68670) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1829, 1872, 4}, {1871, 1902, 4}, {6197, 6198, 24}, {10525, 11391, 4}

X(68671) = X(235)X(1824)∩X(517)X(546)

Barycentrics    a*(a^8*(b + c) - a^4*b*c*(b + c)^3 - 2*a^3*b*c*(b^2 - c^2)^2 + a^5*b*c*(b^2 + c^2) + a*b*c*(b^2 - c^2)^2*(b^2 + c^2) + 2*a^2*(b - c)^2*(b + c)^3*(b^2 + b*c + c^2) - (b - c)^4*(b + c)^3*(b^2 + 3*b*c + c^2) - 2*a^6*(b^3 + b^2*c + b*c^2 + c^3)) : :
X(68671) = X[1824]+X[7680]

See David Nguyen, euclid 8525.

X(68671) lies on these lines: {55, 10594}, {235, 1824}, {517, 546}, {674, 63688}, {1829, 65949}, {1871, 18242}, {2099, 63669}, {2161, 3072}, {2875, 10110}, {2886, 7403}, {3428, 63664}, {3434, 63666}, {5842, 6756}, {6690, 13383}, {10679, 63665}, {18407, 63672}, {20243, 63657}, {32613, 37440}, {44670, 64471}, {47373, 63663}, {58490, 63697}

X(68671) = midpoint of X(1824) and X(7680)
X(68671) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {63675, 63676, 10594}

X(68672) = X(2)X(1824)∩X(5)X(10)

Barycentrics    a*(-2*a^3*b*c + a^4*(b + c) + 2*a^2*b*c*(b + c) + 2*a*b*c*(b^2 + c^2) - (b - c)^2*(b^3 + 5*b^2*c + 5*b*c^2 + c^3)) : :
X(68672) = 3*X[2]+X[1824], 9*X[2]-X[20243], 3*X[1824]+X[20243], X[5905]+3*X[61662], X[17441]-5*X[31266], X[17441]+3*X[61720], 5*X[31266]+3*X[61720]

See David Nguyen, euclid 8525.

X(68672) lies on these lines: {2, 1824}, {5, 10}, {19, 19544}, {21, 1900}, {33, 25514}, {55, 5020}, {65, 17064}, {169, 20310}, {171, 40974}, {182, 10537}, {226, 34381}, {442, 37613}, {518, 59730}, {528, 10128}, {674, 9822}, {916, 58491}, {942, 3772}, {971, 13478}, {975, 7535}, {1155, 1719}, {1214, 47522}, {1766, 42316}, {1828, 6871}, {1829, 2476}, {1861, 37360}, {1871, 6825}, {1872, 6824}, {1878, 17577}, {1902, 6828}, {2099, 19372}, {2182, 37527}, {2355, 35996}, {3419, 54433}, {3428, 11479}, {3434, 4358}, {3812, 58462}, {3822, 44662}, {3827, 3838}, {4192, 40937}, {4698, 6677}, {5045, 17061}, {5173, 33137}, {5393, 67952}, {5405, 67951}, {5777, 5788}, {5784, 37521}, {5791, 67887}, {5816, 10157}, {5842, 9825}, {5905, 61662}, {6642, 32613}, {6678, 11018}, {6874, 41722}, {7401, 37820}, {7522, 10477}, {8769, 40809}, {10129, 41717}, {10679, 11484}, {13754, 58497}, {15509, 46475}, {15733, 29649}, {17070, 31794}, {17441, 31266}, {17605, 41581}, {17720, 43214}, {18252, 32916}, {18407, 18420}, {19137, 47373}, {20368, 24341}, {20718, 58651}, {21621, 25365}, {21848, 31187}, {23512, 60699}, {23841, 58631}, {26098, 40962}, {28600, 29635}, {29069, 59638}, {29311, 58699}, {30444, 60427}, {31053, 61669}, {32118, 40656}, {37697, 50194}, {44661, 58463}, {53861, 54320}, {58504, 58683}, {59520, 64780}

X(68672) = midpoint of X(2886) and X(40635)
X(68672) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2886, 40635, 517}, {9816, 9817, 5020}, {31266, 61720, 17441}, {40530, 58402, 6677}

X(68673) = X(1)X(3)∩X(30)X(993)

Barycentrics    -(a^2*(a^5 - a^4*(b + c) - 2*a^3*(b^2 + c^2) - (b - c)^2*(b^3 + c^3) + a^2*(2*b^3 - b^2*c - b*c^2 + 2*c^3) + a*(b^4 + 4*b^2*c^2 + c^4))) : :
X(68673) = 3*X[3]-X[55], X[20]+X[37820], 2*X[2886]-X[18407], 2*X[140]-X[7680], 3*X[376]+X[3434], 3*X[381]-5*X[31245], 3*X[549]-2*X[6690], X[1657]+X[36999], X[3419]+X[18481], 5*X[3522]-X[37000], 2*X[3530]-X[61533], 3*X[3534]+X[18499], X[3534]+X[31140], X[18499]-3*X[31140], 2*X[5092]-X[47373], X[5732]+X[54203], 3*X[5892]-2*X[58490], X[7580]+X[22758], 9*X[10304]-X[20075], 3*X[21165]+X[64150], X[54133]-3*X[59380]

See David Nguyen, euclid 8525.

X(68673) lies on these lines: {1, 3}, {4, 35250}, {8, 6876}, {11, 28459}, {20, 37820}, {21, 12699}, {30, 993}, {63, 2771}, {100, 3654}, {104, 7411}, {140, 7680}, {191, 40266}, {212, 34586}, {355, 411}, {376, 3434}, {378, 1824}, {381, 5251}, {392, 37300}, {405, 9955}, {500, 1468}, {516, 6914}, {528, 8703}, {548, 63983}, {549, 6690}, {550, 5450}, {573, 2174}, {582, 1193}, {674, 3098}, {912, 65404}, {956, 28204}, {958, 6985}, {962, 6875}, {1001, 28466}, {1006, 5284}, {1012, 28146}, {1064, 2308}, {1376, 50821}, {1480, 3052}, {1621, 3656}, {1657, 36999}, {1698, 37251}, {1699, 7489}, {2071, 20243}, {2807, 38599}, {2808, 34928}, {2975, 3419}, {3058, 5427}, {3149, 9956}, {3185, 54081}, {3522, 37000}, {3530, 61533}, {3534, 18499}, {3560, 22793}, {3655, 54391}, {3679, 18524}, {3681, 12738}, {3841, 64473}, {3868, 33858}, {3869, 16139}, {3877, 27086}, {3899, 35204}, {3925, 28452}, {4189, 6361}, {4192, 24892}, {4220, 29664}, {4225, 35193}, {4297, 32153}, {4299, 64086}, {4324, 15446}, {4996, 12515}, {4999, 37356}, {5047, 61268}, {5092, 47373}, {5130, 31384}, {5248, 5428}, {5250, 37308}, {5258, 18525}, {5259, 18493}, {5260, 61261}, {5267, 31730}, {5274, 6987}, {5288, 18526}, {5303, 37403}, {5396, 39523}, {5587, 62359}, {5603, 37106}, {5690, 6796}, {5694, 6261}, {5732, 54203}, {5790, 44425}, {5841, 6907}, {5843, 54175}, {5855, 8715}, {5892, 58490}, {6000, 10537}, {6097, 48919}, {6186, 14636}, {6210, 24436}, {6684, 6924}, {6713, 37364}, {6763, 16132}, {6825, 10526}, {6827, 10589}, {6838, 37821}, {6841, 24953}, {6842, 11827}, {6851, 30478}, {6865, 26492}, {6868, 10525}, {6869, 18517}, {6883, 8167}, {6889, 64079}, {6905, 26446}, {6911, 11231}, {6917, 64075}, {6932, 68367}, {6950, 9778}, {6954, 64111}, {6988, 8164}, {7171, 42012}, {7330, 31828}, {7354, 37401}, {7430, 26893}, {7491, 15908}, {7508, 28174}, {7514, 40635}, {7580, 22758}, {8143, 62871}, {8666, 34773}, {9708, 18491}, {9856, 37302}, {10304, 20075}, {10448, 48903}, {10483, 47032}, {10571, 52408}, {10884, 26201}, {11362, 32141}, {11699, 22583}, {12331, 63143}, {12512, 16004}, {12514, 22937}, {12520, 24467}, {12611, 24703}, {12619, 60782}, {12672, 52270}, {12773, 50811}, {13146, 66059}, {13743, 41869}, {13754, 35203}, {15326, 28458}, {16117, 26321}, {16370, 28198}, {17009, 21630}, {17614, 37301}, {18242, 64477}, {18519, 28208}, {18527, 57278}, {18761, 33697}, {19525, 35258}, {19541, 38140}, {19543, 24880}, {19649, 29681}, {19854, 44229}, {20330, 38028}, {20818, 62246}, {21165, 64150}, {22775, 22935}, {22936, 31424}, {23839, 34865}, {24390, 44238}, {25440, 61524}, {28202, 28444}, {28443, 31162}, {28453, 50865}, {28628, 33592}, {29243, 63968}, {29690, 49127}, {31423, 45976}, {31445, 31937}, {31493, 45630}, {31799, 52265}, {31806, 51717}, {31837, 37837}, {33142, 37400}, {33856, 64372}, {34718, 48696}, {37258, 39529}, {37309, 52148}, {37406, 57288}, {37429, 38761}, {37737, 54430}, {37960, 40592}, {38713, 51626}, {40262, 58643}, {41229, 56762}, {41854, 62824}, {45770, 55104}, {46441, 64720}, {46816, 48680}, {49107, 57282}, {52026, 62218}, {54133, 59380}, {59366, 64804}, {62805, 63307}, {62827, 64358}, {62844, 63393}

X(68673) = midpoint of X(i) and X(j) for these {i,j}: {1, 37584}, {3, 3428}, {20, 37820}, {40, 61146}, {1657, 36999}, {2099, 12702}, {3419, 18481}, {3534, 31140}, {5732, 54203}, {7580, 22758}, {37533, 41338}
X(68673) = reflection of X(i) in X(j) for these (i,j): {7680, 140}, {18407, 2886}, {24929, 13624}, {32613, 3}, {47373, 5092}, {50195, 40296}, {61533, 3530}
X(68673) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 37584, 517}, {3, 40, 26285}, {3, 56, 13624}, {3, 1482, 10902}, {3, 3428, 517}, {3, 5584, 31663}, {3, 10246, 15931}, {3, 10269, 17502}, {3, 10310, 26086}, {3, 11012, 26286}, {3, 11248, 33862}, {3, 11249, 1385}, {3, 11849, 59331}, {3, 12702, 35}, {3, 16203, 8273}, {3, 22765, 3576}, {3, 22770, 10267}, {3, 26286, 32612}, {3, 35000, 5010}, {3, 35239, 3579}, {3, 35251, 63756}, {3, 35252, 56}, {3, 35448, 5217}, {3, 35457, 14799}, {3, 35459, 34879}, {3, 35461, 35445}, {3, 37535, 7987}, {36, 7688, 3}, {36, 59320, 7688}, {40, 3576, 5538}, {40, 5010, 35000}, {40, 61146, 517}, {56, 40292, 24929}, {958, 6985, 18480}, {1381, 1382, 35459}, {1385, 23961, 60743}, {2099, 12702, 517}, {2975, 3651, 18481}, {3560, 64077, 22793}, {3576, 41338, 37533}, {5010, 25415, 55}, {5010, 35000, 26285}, {5045, 13624, 1385}, {5119, 7280, 5172}, {5428, 22791, 5248}, {5450, 12511, 550}, {6261, 26921, 5694}, {6869, 19843, 18517}, {6883, 22753, 11230}, {7280, 35242, 3}, {7580, 22758, 28160}, {7688, 11012, 36}, {7742, 10966, 24928}, {7991, 59331, 11849}, {10267, 22770, 10222}, {11012, 35242, 34880}, {11012, 59320, 3}, {11012, 65143, 35252}, {12704, 37615, 6583}, {13624, 40292, 32613}, {16192, 59332, 3}, {18761, 37411, 33697}, {22767, 37578, 5126}, {26357, 59317, 942}, {37533, 41338, 517}, {37587, 44841, 999}, {40294, 58219, 32613}

X(68674) = X(55)X(750)∩X(140)X(517)

Barycentrics    a*(a^2*b*c*(b + c) - 2*b*(b - c)^2*c*(b + c) + a^3*(b^2 + c^2) - a*(b - c)^2*(b^2 + b*c + c^2)) : :
X(68674) = 3*X[2]-X[22276], 3*X[354]+X[1824], X[5173]+X[40635], X[20243]-9*X[64149], X[26892]+3*X[61716], X[26893]-5*X[31245]

See David Nguyen, euclid 8525.

X(68674) lies on these lines: {1, 37836}, {2, 22276}, {5, 58493}, {51, 17605}, {55, 750}, {65, 37646}, {140, 517}, {141, 674}, {209, 24892}, {226, 8679}, {354, 1824}, {375, 908}, {511, 3838}, {518, 59730}, {942, 51716}, {960, 62689}, {1193, 2099}, {1953, 8608}, {2390, 39542}, {2646, 58889}, {2807, 58490}, {3060, 10129}, {3120, 67961}, {3434, 18141}, {3683, 61643}, {3742, 17045}, {3772, 10473}, {3911, 64550}, {3946, 58572}, {4259, 17064}, {4511, 61699}, {4847, 9049}, {4966, 35626}, {5087, 5943}, {5173, 15253}, {5249, 50362}, {5326, 67493}, {5707, 14529}, {5743, 30986}, {5745, 20718}, {5855, 59303}, {5880, 37521}, {6701, 65399}, {6703, 59701}, {6745, 22278}, {9038, 32853}, {9955, 13754}, {10404, 41682}, {10439, 25525}, {10441, 28628}, {10537, 37543}, {11018, 40646}, {11019, 58574}, {11263, 11573}, {11375, 19721}, {11553, 22345}, {12047, 18180}, {12609, 37536}, {13411, 22300}, {16608, 23304}, {17056, 21334}, {18165, 24210}, {18191, 41011}, {20243, 64149}, {22277, 33137}, {25415, 49997}, {25466, 35631}, {25639, 67975}, {26013, 44411}, {26725, 38474}, {26892, 61716}, {26893, 30970}, {29311, 58463}, {31760, 58476}, {34434, 64160}, {35633, 44669}, {38062, 62352}, {39793, 40974}, {44661, 58626}, {51377, 61648}, {58475, 67876}, {58617, 63980}

X(68674) = midpoint of X(5173) and X(40635) for these {i,j}: {5173, 40635}
X(68674) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11019, 64524, 58574}, {12047, 18180, 42450}

X(68675) = X(6)X(31)∩X(371)X(517)

Barycentrics    -(a^2*(a^3 - a^2*(b + c) - a*(b^2 + c^2 - 2*S) + (b + c)*(b^2 - 2*b*c + c^2 - 2*S))) : :
Barycentrics    a^2(a - r - R - s) : : (* Peter Moses, June 1, 2025)

See David Nguyen, euclid 8525.

X(68675) lies on these lines: {1, 44582}, {3, 19050}, {4, 44620}, {6, 31}, {35, 26464}, {56, 44645}, {371, 517}, {372, 32613}, {377, 13907}, {485, 37820}, {528, 32787}, {590, 2886}, {615, 6690}, {1124, 8069}, {1151, 3428}, {1386, 45583}, {1587, 37000}, {1702, 37569}, {1824, 5412}, {2067, 2099}, {2161, 60851}, {2174, 34125}, {2911, 60848}, {3068, 3434}, {3070, 5842}, {3071, 7680}, {3085, 44621}, {3295, 19049}, {3297, 44607}, {3299, 32760}, {3303, 44646}, {3311, 10679}, {3419, 13911}, {3746, 26458}, {5119, 18991}, {5172, 6502}, {5597, 26409}, {5598, 26385}, {6199, 44455}, {6284, 19026}, {6564, 18407}, {7362, 45436}, {7584, 61533}, {7585, 20075}, {7968, 24929}, {8186, 44583}, {8187, 44585}, {8253, 31245}, {8255, 60914}, {9616, 41338}, {10222, 45641}, {10533, 10537}, {11417, 20243}, {13665, 18499}, {13846, 31140}, {18992, 59337}, {18996, 44643}, {19013, 44644}, {19047, 31474}, {23251, 36999}, {26286, 45640}, {26359, 45368}, {26360, 45365}, {26465, 65139}, {31453, 49233}, {31472, 64086}, {33179, 35819}, {35768, 50194}, {35775, 37533}, {35882, 49240}, {36976, 60887}, {38454, 60913}, {44669, 49232}, {49234, 49248}, {63328, 63393}

X(68675) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2066, 5415, 6}, {19000, 19038, 6}, {44582, 44584, 1}

X(68676) = X(6)X(31)∩X(372)X(517)

Barycentrics    a^2*(a^3 - a^2*(b + c) - a*(b^2 + c^2 + 2*S) + (b + c)*(b^2 - 2*b*c + c^2 + 2*S)) : :
Barycentrics    a^2(a + r + R - s) : : (* Peter Moses, June 1, 2025)

See David Nguyen, euclid 8525.

X(68676) lies on these lines: {1, 44583}, {3, 19049}, {4, 44621}, {6, 31}, {35, 26458}, {56, 44646}, {371, 32613}, {372, 517}, {377, 13965}, {486, 37820}, {528, 32788}, {590, 6690}, {615, 2886}, {1152, 3428}, {1335, 8069}, {1386, 45582}, {1588, 37000}, {1703, 37569}, {1824, 5413}, {2067, 5172}, {2099, 6502}, {2161, 60852}, {2174, 34121}, {2911, 60847}, {3069, 3434}, {3070, 7680}, {3071, 5842}, {3085, 44620}, {3295, 19050}, {3298, 44606}, {3301, 32760}, {3303, 44645}, {3312, 10679}, {3419, 13973}, {3746, 26464}, {5119, 18992}, {5597, 26408}, {5598, 26384}, {6284, 19025}, {6395, 44455}, {6565, 18407}, {7353, 45437}, {7583, 61533}, {7586, 20075}, {7969, 24929}, {8186, 44582}, {8187, 44584}, {8252, 31245}, {8255, 60913}, {10222, 45640}, {10534, 10537}, {11418, 20243}, {13785, 18499}, {13847, 31140}, {18991, 59337}, {18995, 44644}, {19014, 44643}, {23261, 36999}, {26286, 45641}, {26359, 45367}, {26360, 45366}, {26459, 65138}, {33179, 35818}, {35769, 50194}, {35774, 37533}, {35883, 49241}, {38454, 60914}, {44622, 64086}, {44669, 49233}, {49235, 49249}, {63329, 63393}

X(68676) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5414, 5416, 6}, {18999, 19037, 6}, {44583, 44585, 1}

X(68677) = X(4)X(3871)∩X(19)X(25)

Barycentrics    -(a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^3 - a^2*(b + c) + (b + c)^3 - a*(b^2 + c^2))) : :

See David Nguyen, euclid 8525.

X(68677) lies on these lines: {3, 11401}, {4, 3871}, {8, 57530}, {19, 25}, {35, 26377}, {427, 3434}, {429, 5687}, {430, 6600}, {431, 3085}, {517, 1593}, {528, 5064}, {607, 14974}, {674, 12167}, {1260, 7140}, {1398, 2099}, {1597, 44455}, {1825, 37579}, {1829, 5119}, {1844, 8069}, {1900, 11398}, {1902, 37569}, {2177, 57652}, {2886, 5094}, {3295, 4185}, {3303, 40985}, {3419, 37318}, {3428, 3516}, {3515, 32613}, {3542, 61533}, {3575, 37000}, {3730, 26867}, {3913, 5130}, {4214, 56814}, {5597, 26372}, {5598, 26371}, {5842, 12173}, {6284, 11391}, {6690, 37453}, {7102, 56183}, {7414, 35448}, {7507, 37820}, {7680, 37197}, {8715, 39579}, {8750, 44086}, {10306, 37194}, {10537, 26864}, {10829, 66719}, {11363, 59337}, {11399, 32760}, {16202, 37117}, {18386, 18407}, {19118, 47373}, {19624, 44100}, {20832, 64951}, {24929, 37245}, {26358, 40950}, {26378, 65128}, {31140, 62980}, {31245, 52298}, {32929, 41013}, {36976, 60879}

X(68677) = barycentric product X(281)*X(11509)
X(68677) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 56050},{11509, 348}
X(68677) = trilinear product X(33)*X(11509)
X(68677) = trilinear quotient X(i)/X(j) for these {i,j}: {19, 56050},{77, 11509}
X(68677) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {33, 11383, 25}, {55, 1824, 25}, {7071, 11406, 25}

X(68678) = X(2)X(44619)∩X(6)X(31)

Barycentrics    a^2*(a - r + R - s) : :
Barycentrics    a^2*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c + 4*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3 + 2*(a - b - c)*S) : :

X(68678) lies on these lines: {2, 44619}, {3, 19047}, {6, 31}, {35, 26459}, {56, 44644}, {371, 1385}, {372, 26285}, {497, 44618}, {517, 45643}, {518, 45583}, {590, 3816}, {615, 3035}, {1124, 8071}, {1151, 44607}, {1388, 2067}, {1588, 10786}, {3058, 19024}, {3071, 18242}, {3295, 19048}, {3297, 44606}, {3303, 44643}, {3311, 16202}, {3612, 18992}, {3746, 26465}, {5432, 19023}, {6921, 13964}, {7113, 34125}, {7969, 9957}, {10222, 35817}, {10284, 35641}, {18996, 44646}, {19014, 44645}, {19050, 31474}, {25405, 35768}, {26458, 65139}, {31439, 40296}, {31453, 49232}, {31788, 49226}, {32787, 49736}, {35775, 61146}, {35789, 35882}

X(68678) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 2066, 68675}, {6, 44591, 68676}, {18999, 19038, 6}

X(68679) = X(2)X(44618)∩X(6)X(31)

Barycentrics    a^2*(a + r - R - s) : :
Barycentrics    a^2*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c + 4*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3 - 2*(a - b - c)*S) : :

X(68679) lies on these lines:{2, 44618}, {3, 19048}, {6, 31}, {35, 26465}, {56, 44643}, {371, 26285}, {372, 1385}, {497, 44619}, {517, 45642}, {518, 45582}, {590, 3035}, {615, 3816}, {1152, 44606}, {1335, 8071}, {1388, 6502}, {1587, 10786}, {3058, 19023}, {3070, 18242}, {3295, 19047}, {3298, 44607}, {3303, 44644}, {3312, 16202}, {3612, 18991}, {3746, 26459}, {5432, 19024}, {6921, 13906}, {7113, 34121}, {7968, 9957}, {10222, 35816}, {10284, 35642}, {18995, 44645}, {19013, 44646}, {19112, 63072}, {25405, 35769}, {26464, 65138}, {31788, 49227}, {32788, 49736}, {35774, 61146}, {35788, 35883}

X(68679) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 5414, 68676}, {6, 44590, 68675}, {19000, 19037, 6}

X(68680) = EULER LINE INTERCEPT OF X(1568)X(61713)

Barycentrics    a^10 - 2*(b^2 - c^2)^4*(b^2 + c^2) - 4*a^6*(b^4 + c^4) + 3*a^2*(b^2 - c^2)^2*(b^4 - b^2*c^2 + c^4) + a^4*(2*b^6 + 3*b^4*c^2 + 3*b^2*c^4 + 2*c^6) : :

As a point on the Euler line, X(68680) has Shinagawa coefficients: {1/3 (-((13 e)/4) + 2 (e + f)), -((7 e)/4) + 2 (e + f)}

See David Nguyen, euclid 8526.

X(68680) lies on these lines: {2, 3}, {1568, 61713}, {11178, 44668}, {13346, 43865}, {14644, 23039}, {15060, 23325}, {18390, 51391}, {18396, 40111}, {18418, 51548}, {20304, 63425}

X(68680) = midpoint of X(i) and X(j) for these {i,j}: {2, 18404}, {3830, 11413}
X(68680) = reflection of X(i) in X(j) for these (i,j): {235, 5066}, {8703, 16196}
X(68680) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 18404, 30}, {5, 18531, 7502}, {3830, 11413, 30}, {18586, 18587, 44802}

X(68681) = EULER LINE INTERCEPT OF X(49)X(15072)

Barycentrics    -(a^2*(a^8 + 8*a^4*b^2*c^2 - 2*a^6*(b^2 + c^2) - (b^2 - c^2)^2*(b^4 + 3*b^2*c^2 + c^4) + a^2*(2*b^6 - 5*b^4*c^2 - 5*b^2*c^4 + 2*c^6))) : :
X(68681) = 3*X[3]-X[24], 2*X[140]-X[235], 2*X[5092]-X[51730], 3*X[5892]-2*X[58482], X[12163]+X[44752], 2*X[13624]-X[51694], 5*X[15026]-4*X[58559], X[33563]-2*X[44158], 3*X[38028]-2*X[51702], 3*X[38110]-2*X[51734]

As a point on the Euler line, X(68681) has Shinagawa coefficients: {1/3 (-((33 e)/4) + 6 (e + f)), (13 e)/4 - 2 (e + f)}

See David Nguyen, euclid 8526.

X(68681) lies on these lines: {2, 3}, {49, 15072}, {64, 15068}, {68, 10264}, {74, 18436}, {110, 64030}, {155, 45957}, {156, 10575}, {185, 10564}, {511, 43604}, {567, 66606}, {568, 43601}, {578, 36153}, {1092, 5663}, {1147, 13491}, {1154, 1204}, {1216, 31978}, {1350, 8548}, {1495, 43898}, {1511, 6759}, {2935, 5878}, {2979, 11468}, {3098, 15578}, {3357, 5876}, {3564, 54217}, {3581, 64050}, {4549, 18431}, {5092, 51730}, {5206, 44468}, {5562, 32138}, {5651, 45958}, {5866, 7776}, {5889, 37477}, {5890, 37495}, {5892, 58482}, {6000, 61753}, {6101, 7689}, {6102, 13346}, {6241, 22115}, {6696, 67926}, {6776, 36966}, {7691, 16013}, {8185, 28190}, {8542, 55637}, {9703, 67879}, {10263, 11438}, {10282, 14641}, {10540, 12279}, {10574, 37472}, {10605, 16266}, {10625, 21663}, {10627, 32210}, {10938, 43617}, {11204, 54042}, {11270, 12307}, {11424, 12006}, {11440, 23039}, {11457, 32423}, {11464, 52093}, {11591, 43652}, {11750, 16163}, {12038, 46850}, {12042, 39831}, {12118, 45731}, {12121, 12289}, {12161, 37497}, {12163, 44752}, {12228, 18466}, {12290, 18350}, {13340, 15055}, {13352, 13630}, {13353, 20791}, {13358, 16270}, {13367, 14855}, {13381, 31381}, {13391, 63709}, {13434, 40280}, {13445, 18439}, {13470, 43589}, {13474, 43586}, {13496, 38621}, {13624, 51694}, {14156, 61749}, {14449, 37490}, {15026, 58559}, {15040, 52100}, {15114, 22109}, {15115, 37853}, {15579, 34507}, {15644, 45780}, {17821, 35237}, {18911, 43575}, {19357, 64098}, {19510, 35707}, {20417, 52104}, {20771, 65095}, {21230, 32345}, {22962, 43865}, {23329, 34826}, {25487, 44573}, {25564, 38726}, {31831, 61540}, {32046, 64100}, {32139, 35602}, {32600, 55655}, {33563, 44158}, {33813, 39860}, {34148, 43602}, {34469, 58891}, {34783, 43574}, {35228, 48898}, {37481, 43603}, {38028, 51702}, {38110, 51734}, {39812, 67268}, {39841, 51872}, {40647, 64026}, {41427, 47391}, {41725, 46374}, {43291, 44528}, {43394, 64049}, {43607, 54040}, {43619, 44523}, {43845, 61136}, {44883, 48876}, {50461, 64025}, {51491, 58885}, {55657, 61676}, {61757, 67832}

X(68681) = midpoint of X(i) and X(j) for these {i,j}: {3, 11413}, {20, 18404}, {12163, 44752}
X(68681) = reflection of X(i) in X(j) for these (i,j): {5, 16196}, {235, 140}, {33563, 44158}, {51694, 13624}, {51730, 5092}
X(68681) = complement of X(31725)
X(68681) = anticomplement of X(44235)
X(68681) = circumperp conjugate of X(31726) X(68681) = Ehrmann-cross-isogonal conjugate of X(37814)
X(68681) = Trinh-isogonal conjugate of X(3357)
X(68681) = circumperp conjugate of complement of X(34152)
X(68681) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 20, 1658}, {3, 22, 15331}, {3, 24, 43615}, {3, 26, 15646}, {3, 376, 7525}, {3, 378, 140}, {3, 382, 22467}, {3, 550, 7502}, {3, 1657, 186}, {3, 2071, 11250}, {3, 2937, 21844}, {3, 3516, 7514}, {3, 3534, 7488}, {3, 7464, 12106}, {3, 7485, 61792}, {3, 7503, 3530}, {3, 7509, 12100}, {3, 7514, 15712}, {3, 7516, 44682}, {3, 7517, 15078}, {3, 7526, 549}, {3, 11250, 18570}, {3, 11413, 30}, {3, 11414, 18324}, {3, 12082, 18571}, {3, 12083, 32534}, {3, 12084, 5}, {3, 12085, 6644}, {3, 13564, 10298}, {3, 14130, 631}, {3, 15688, 67321}, {3, 15696, 7512}, {3, 18859, 4}, {20, 7556, 47748}, {20, 18404, 30}, {24, 1593, 44226}, {24, 7393, 16238}, {140, 378, 63682}, {140, 13488, 5}, {186, 1657, 17714}, {548, 10226, 3}, {631, 14130, 49671}, {1216, 58871, 64027}, {2979, 11468, 63392}, {6101, 12041, 7689}, {6644, 12085, 3627}, {7689, 37480, 6101}, {10257, 15761, 60780}, {10298, 17538, 13564}, {10575, 51394, 156}, {10627, 32210, 63425}, {11424, 37470, 12006}, {12038, 46850, 61752}, {12103, 15331, 22}, {14784, 14785, 52403}, {15646, 15704, 26}, {18350, 64624, 12290}, {32139, 35602, 40111}

X(68682) = EULER LINE INTERCEPT OF X(1147)X(1539)

Barycentrics    3*a^10 - 4*a^8*(b^2 + c^2) - 2*(b^2 - c^2)^4*(b^2 + c^2) + a^2*(b^2 - c^2)^2*(b^4 - 9*b^2*c^2 + c^4) - 4*a^6*(b^4 - 4*b^2*c^2 + c^4) + a^4*(6*b^6 - 7*b^4*c^2 - 7*b^2*c^4 + 6*c^6) : :
X(68682) = 2*X[48895]-X[51730]

As a point on the Euler line, X(68682) has Shinagawa coefficients: {1/3 ((27 e)/4 - 6 (e + f)), -((47 e)/4) + 10 (e + f)}

See David Nguyen, euclid 8526.

X(68682) lies on these lines: {2, 3}, {1147, 1539}, {1204, 34584}, {3357, 10113}, {5878, 45731}, {5895, 45957}, {9820, 51998}, {10264, 20427}, {10721, 34783}, {10733, 18439}, {11455, 52863}, {12161, 61721}, {12370, 51491}, {13474, 45780}, {14677, 26937}, {20127, 26917}, {20424, 64096}, {34785, 51548}, {38790, 64025}, {44668, 48884}, {48895, 51730}

X(68682) = midpoint of X(i) and X(j) for these {i,j}: {3146, 18404}, {5073, 11413}
X(68682) = reflection of X(i) in X(j) for these (i,j): {235, 3853}, {15704, 16196}, {51730, 48895}
X(68682) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 3146, 18565}, {4, 6640, 546}, {4, 18565, 12106}, {546, 16976, 5}, {3146, 18404, 30}, {3853, 10224, 4}, {5073, 11413, 30}, {10736, 10737, 18403}

X(68683) = EULER LINE INTERCEPT OF X(113)X(156)

Barycentrics    a^10 - 2*a^6*(b^2 - c^2)^2 - a^8*(b^2 + c^2) - (b^2 - c^2)^4*(b^2 + c^2) + a^2*(b^2 - c^2)^2*(b^4 - 3*b^2*c^2 + c^4) + a^4*(2*b^6 - b^4*c^2 - b^2*c^4 + 2*c^6) : :
X(68683) = X[24]-3*X[381], 3*X[5946]-2*X[52003], 2*X[9955]-X[51694], X[12293]+X[44752], 2*X[19130]-X[51730], X[20771]-2*X[61574], 3*X[38034]-2*X[51702], 3*X[38136]-2*X[51734]

As a point on the Euler line, X(68683) has Shinagawa coefficients: {-(e/4), -((17 e)/4) + 4 (e + f)}

See David Nguyen, euclid 8526.

X(68683) lies on these lines: {2, 3}, {52, 1531}, {113, 156}, {125, 32138}, {184, 67869}, {265, 12111}, {399, 34799}, {578, 67861}, {1351, 54157}, {1352, 54148}, {1511, 63685}, {1539, 13491}, {1614, 12228}, {1899, 45957}, {3521, 10574}, {3583, 63669}, {3585, 37729}, {3818, 22804}, {5448, 13403}, {5449, 7687}, {5609, 61751}, {5663, 67903}, {5876, 9927}, {5890, 43821}, {5907, 45780}, {5925, 14677}, {5944, 61747}, {5946, 52003}, {6101, 63740}, {6102, 18390}, {6241, 7728}, {6288, 12280}, {7592, 43575}, {7689, 63839}, {7706, 15026}, {7730, 15800}, {9730, 63670}, {9955, 51694}, {10539, 30522}, {10540, 12289}, {10610, 63629}, {11424, 15807}, {11438, 34798}, {11440, 14644}, {11441, 32423}, {11449, 14643}, {11454, 11704}, {11464, 18504}, {11468, 15061}, {11550, 32137}, {11572, 16194}, {11750, 51403}, {11805, 17824}, {12041, 63695}, {12042, 63696}, {12118, 40111}, {12162, 13851}, {12278, 18350}, {12293, 15068}, {12370, 22660}, {13346, 51391}, {13630, 63738}, {14157, 65149}, {14531, 18555}, {14670, 22823}, {15087, 43835}, {15305, 18394}, {17702, 61753}, {18114, 18380}, {18379, 18474}, {18381, 22816}, {18382, 39884}, {18383, 34514}, {18396, 32139}, {18418, 34785}, {18435, 58922}, {18436, 50435}, {18439, 25739}, {18445, 45970}, {18513, 37696}, {18514, 37697}, {18952, 45956}, {19106, 63681}, {19107, 63680}, {19130, 51730}, {20304, 32210}, {20424, 39522}, {20771, 32171}, {21230, 64105}, {21400, 45736}, {22505, 39847}, {22515, 39818}, {22793, 63698}, {29012, 63663}, {32046, 43831}, {33563, 67926}, {33813, 63687}, {34114, 46686}, {34782, 46817}, {34786, 46261}, {35820, 63678}, {35821, 63677}, {38034, 51702}, {38136, 51734}, {38609, 63708}, {38610, 63715}, {39242, 58407}, {40647, 63697}, {43608, 45622}, {48901, 63688}, {48906, 63699}, {58789, 64032}, {63686, 66762}, {63721, 64729}

X(68683) = midpoint of X(i) and X(j) for these {i,j}: {4, 18404}, {382, 11413}, {12293, 44752}
X(68683) = reflection of X(i) in X(j) for these (i,j): {235, 546}, {550, 16196}, {20771, 61574}, {51694, 9955}, {51730, 19130}
X(68683) = anticomplement of X(43615)
X(68683) = Ehrmann-mid-isogonal conjugate of X(22660)
X(68683) = Ehrmann-vertex-isogonal conjugate of X(18381)
X(68683) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 4, 44279}, {3, 5, 60780}, {3, 381, 16868}, {4, 5, 44263}, {4, 20, 31726}, {4, 546, 63672}, {4, 3153, 382}, {4, 14790, 44276}, {4, 14791, 44267}, {4, 18377, 44288}, {4, 18403, 18377}, {4, 18404, 30}, {4, 18569, 3627}, {5, 550, 44452}, {24, 381, 44235}, {113, 21659, 156}, {265, 12111, 18356}, {381, 382, 13621}, {381, 3830, 67323}, {381, 7526, 5}, {381, 12173, 13861}, {381, 18561, 37943}, {382, 11413, 30}, {382, 13621, 18559}, {403, 18563, 1658}, {546, 13383, 37984}, {546, 13406, 35488}, {546, 18567, 4}, {548, 15330, 38438}, {1539, 13491, 22802}, {1885, 10297, 13371}, {2072, 18560, 11250}, {3520, 10255, 61736}, {3627, 18572, 18569}, {3843, 7517, 62974}, {3843, 7566, 546}, {5876, 10113, 9927}, {6102, 43865, 18390}, {10151, 12605, 15761}, {10750, 10751, 2071}, {11799, 12225, 17714}, {12173, 13861, 38322}, {12370, 58885, 22660}, {12605, 15761, 7502}, {13630, 63738, 67911}, {15058, 18392, 6288}, {18379, 45959, 18474}, {18396, 32139, 45731}

X(68684) = EULER LINE INTERCEPT OF X(18356)X(50434)

Barycentrics    -5*a^10 + 8*a^8*(b^2 + c^2) + 2*(b^2 - c^2)^4*(b^2 + c^2) + 4*a^6*(b^4 - 8*b^2*c^2 + c^4) + a^2*(b^2 - c^2)^2*(b^4 + 15*b^2*c^2 + c^4) + a^4*(-10*b^6 + 17*b^4*c^2 + 17*b^2*c^4 - 10*c^6) : :
X(68684) = 2*X[48891]-X[51730]}

As a point on the Euler line, X(68684) has Shinagawa coefficients: {1/3 (-((93 e)/4) + 18 (e + f)), (73 e)/4 - 14 (e + f)}

See David Nguyen, euclid 8526.

X(68684) lies on these lines: {2, 3}, {18356, 50434}, {20725, 63734}, {44668, 48879}, {48891, 51730}

X(68684) = midpoint of X(i) and X(j) for these {i,j}: {5059, 18404}, {11413, 17800}
X(68684) = reflection of X(i) in X(j) for these (i,j): {235, 12103}, {51730, 48891}
X(68684) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5059, 18404, 30}, {11413, 17800, 30}

X(68685) = EULER LINE INTERCEPT OF X(13)X(11409)

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (a^6-6 a^4 b^2+9 a^2 b^4-4 b^6-6 a^4 c^2-10 a^2 b^2 c^2+4 b^4 c^2+9 a^2 c^4+4 b^2 c^4-4 c^6) : :

As a point on the Euler line, X(68685) has Shinagawa coefficients: {5*F,5*F-3*E}

See David Nguyen and Ercole Suppa, euclid 8531.

X(68685) lies on these lines: {2, 3}, {13, 11409}, {14, 11408}, {542, 19118}, {599, 68023}, {1398, 3582}, {1829, 38021}, {1843, 38072}, {1902, 19875}, {1974, 47353}, {1989, 41489}, {3199, 18362}, {3584, 7071}, {5186, 23234}, {5309, 45141}, {5410, 35823}, {5411, 35822}, {5476, 12167}, {5655, 12165}, {7713, 30308}, {8780, 50435}, {9166, 12131}, {10606, 61691}, {11396, 51709}, {11398, 65140}, {11402, 61747}, {11470, 15533}, {12132, 14639}, {12135, 38074}, {12138, 59377}, {12294, 21358}, {12315, 26917}, {14831, 44084}, {18390, 26864}, {18405, 44082}, {18418, 32223}, {25561, 67882}, {26869, 67890}, {26958, 51403}, {32000, 32893}, {32062, 61735}, {32063, 61701}, {38073, 60879}, {38076, 49542}, {39871, 59373}, {40318, 45016}, {41584, 54132}, {51425, 63649}, {61139, 68009}, {61506, 67868}, {61680, 61744}, {64024, 67896}

X(68685) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(20),X(1989)}}, {{A,B,C,X(186),X(41489)}}, {{A,B,C,X(382),X(54658)}}, {{A,B,C,X(546),X(54550)}}, {{A,B,C,X(550),X(13380)}}, {{A,B,C,X(1368),X(60175)}}, {{A,B,C,X(3522),X(59496)}}, {{A,B,C,X(3529),X(54604)}}, {{A,B,C,X(3851),X(45300)}}, {{A,B,C,X(5020),X(60192)}}, {{A,B,C,X(5067),X(43891)}}, {{A,B,C,X(7396),X(54866)}}, {}A,B,C,X(7398),X(54521)}}, {{A,B,C,X(10302),X(41235)}}, {}A,B,C,X(15702),X(45972)}}, {{A,B,C,X(18368),X(61817)}}, {{A,B,C,X(34152),X(35372)}}, {{A,B,C,X(34609),X(54608)}}, {{A,B,C,X(47315),X(60323)}}, {{A,B,C,X(54643),X(66529)}}
X(68685) = pole of line {6103, 15905} with respect to Dao-Moses-Telv circle
X(68685) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 235, 62966}, {4, 5055, 62980}, {381, 10201, 54994}, {428, 468, 66373}, {428, 3545, 7507}, {468, 6623, 44438}, {3089, 3545, 428}, {3517, 14269, 18559}, {6353, 10151, 37196}, {7577, 18535, 62977}

X(68686) = X(1)X(381)∩X(30)X(40)

Barycentrics    -5*a^4+3*a^3*b+a^2*b^2-3*a*b^3+4*b^4+3*a^3*c-6*a^2*b*c+3*a*b^2*c+a^2*c^2+3*a*b*c^2-8*b^2*c^2-3*a*c^3+4*c^4 : :
X(68686) = X[1]-2*X[381], 5*X[1]-8*X[9955], X[1]-4*X[18480], 2*X[1]-5*X[18492], 7*X[1]-10*X[18493], X[1]+2*X[18525], 5*X[1]-2*X[18526], 3*X[1]-5*X[30308], 2*X[1]-3*X[38021], 3*X[1]-4*X[51709], 5*X[381]-4*X[9955], X[381]-2*X[18480], 4*X[381]-5*X[18492], 7*X[381]-5*X[18493], X[381]+X[18525], 5*X[381]-X[18526], 6*X[381]-5*X[30308], 4*X[381]-3*X[38021], 3*X[381]-2*X[51709], 2*X[9955]-5*X[18480], 4*X[9955]+5*X[18525], 4*X[9955]-X[18526], 6*X[9955]-5*X[51709], 8*X[18480]-5*X[18492], 2*X[18480]+X[18525], 10*X[18480]-X[18526], 8*X[18480]-3*X[38021], 3*X[18480]-X[51709], 7*X[18492]-4*X[18493], 5*X[18492]+4*X[18525], 3*X[18492]-2*X[30308], 5*X[18492]-3*X[38021], 5*X[18493]+7*X[18525], 6*X[18493]-7*X[30308], 5*X[18525]+X[18526], 6*X[18525]+5*X[30308], 4*X[18525]+3*X[38021], 3*X[18525]+2*X[51709], 6*X[18526]-25*X[30308], 4*X[18526]-15*X[38021], 3*X[18526]-10*X[51709], 10*X[30308]-9*X[38021], 5*X[30308]-4*X[51709], 6*X[32900]-11*X[51709], 9*X[38021]-8*X[51709], 4*X[2]-3*X[3576], 2*X[2]-3*X[5587], 5*X[2]-3*X[5731], 7*X[2]-6*X[10165], 5*X[2]-6*X[10175]

See David Nguyen and Ercole Suppa, euclid 8532.

X(68686) lies on the circunconic {{A,B,C,X(34393),X(50811)}} and these lines: {1, 381}, {2, 515}, {3, 19875}, {4, 519}, {5, 3655}, {8, 3543}, {10, 376}, {20, 53620}, {30, 40}, {35, 18518}, {36, 18491}, {57, 80}, {84, 17579}, {119, 64011}, {140, 61258}, {145, 18483}, {153, 3577}, {165, 3534}, {329, 36922}, {382, 7991}, {392, 67875}, {403, 47495}, {511, 50950}, {516, 4669}, {517, 3830}, {518, 47353}, {524, 39885}, {528, 11372}, {529, 12704}, {535, 3928}, {540, 54136}, {541, 13211}, {542, 3751}, {543, 9864}, {544, 50903}, {546, 11522}, {547, 3624}, {549, 1698}, {550, 9588}, {551, 944}, {553, 18391}, {671, 64749}, {942, 17632}, {946, 3241}, {952, 1699}, {962, 31145}, {1000, 51783}, {1001, 38075}, {1012, 4421}, {1125, 5071}, {1158, 40264}, {1385, 5055}, {1386, 38072}, {1387, 38077}, {1420, 3582}, {1478, 4654}, {1479, 37709}, {1482, 14269}, {1483, 23046}, {1490, 17532}, {1503, 47359}, {1532, 3829}, {1572, 14537}, {1656, 30389}, {1657, 63469}, {1697, 37710}, {1706, 37430}, {1750, 31140}, {1836, 62616}, {1837, 3333}, {2093, 12943}, {2136, 34629}, {2771, 67977}, {2782, 9875}, {2784, 12243}, {2800, 50890}, {2801, 60963}, {2802, 50907}, {2809, 50904}, {3058, 3586}, {3085, 66247}, {3090, 19883}, {3091, 5882}, {3099, 55007}, {3146, 11362}, {3149, 11194}, {3158, 50906}, {3244, 61980}, {3247, 32431}, {3338, 34698}, {3339, 9655}, {3340, 3585}, {3434, 11525}, {3476, 37704}, {3488, 51782}, {3522, 31425}, {3523, 31399}, {3524, 3828}, {3526, 30315}, {3529, 34638}

X(68686) = midpoint of X(i) and X(j) for these {i,j}: {2, 50864}, {4, 34627}, {8, 3543}, {381, 18525}, {382, 34718}, {962, 31145}, {3146, 34632}, {3679, 5691}, {3830, 50798}, {4669, 50862}, {4677, 50865}, {5881, 31162}, {15682, 50810}, {34697, 34746}, {50871, 51093}
X(68686) = reflection of X(i) in X(j) for these {i,j}: {1, 381}, {2, 50796}, {4, 34648}, {40, 3679}, {165, 5790}, {376, 10}, {381, 18480}, {392, 67875}, {549, 18357}, {551, 19925}, {944, 551}, {3241, 946}, {3529, 34638}, {3534, 50821}, {3543, 31673}, {3576, 5587}, {3655, 5}, {3656, 3845}, {3679, 355}, {4297, 3828}, {4669, 50801}, {4677, 50798}, {5587, 59387}, {5731, 10175}, {5881, 34627}, {7982, 31162}, {7991, 34718}, {12243, 50884}, {15682, 50862}, {16200, 1699}, {18481, 549}, {31145, 47745}, {31162, 4}, {34628, 3}, {34631, 4301}, {34632, 11362}, {34638, 43174}, {34747, 1482}, {34773, 547}, {41869, 3543}, {50810, 4669}, {50811, 2}, {50812, 51066}, {50817, 4677}, {50865, 3830}, {51066, 50797}, {51093, 3656}, {61296, 3241}, {63143, 59388}, {64011, 119}, {64749, 671}
X(68686) = anticomplement of X(51705)
X(68686) = reflection of X(i) in the line X(j)X(k) for these {i,j,k}: {47270, 381, 523}
X(68686) = X(8)-beth conjugate of X(376)
X(68686) = X(51705)-Dao conjugate of X(51705)
X(68686) = X(376)-of-outer-Garcia triangle
X(68686) = X(3655)-of-Johnson triangle
X(68686) = X(34627)-of-Euler triangle
X(68686) = X(34628)-of-ABC-X3 reflections triangle
X(68686) = X(34648)-of-anti-Euler triangle
X(68686) = X(51705)-of-anticomplementary triangle
X(68686) = X(51709)-of-anti-Ehrmann-mid triangle
X(68686) = center of circles {{ X(i), X(j), X(k)}} for these {i, j, k}: {2, 50145, 50864}, {8, 3543, 36154}
X(68686) = pole of line {28205, 48391} with respect to circumcircle
X(68686) = pole of line {4926, 39547} with respect to Conway circle
X(68686) = pole of line {4926, 39540} with respect to incircle
X(68686) = pole of line {4926, 44409} with respect to Suppa-Cucoanes circle
X(68686) = pole of line {5903, 64131} with respect to Feuerbach hyperbola
X(68686) = pole of line {523, 21385} with respect to orthoptic circle of Bevan circle
X(68686) = pole of line {4840, 4926} with respect to orthoptic circle of Conway circle
X(68686) = pole of line {522, 59912} with respect to orthoptic circle of orthoptic circle of the Steiner Inellipse
X(68686) = pole of line {1, 30} with respect to orthoptic circle of Kiepert parabola
X(68686) = pole of line {376, 516} with respect to orthoptic circle of Yff parabola
X(68686) = pole of line {1503, 31162} with respect to orthoptic circle of Moses HK-parabola
X(68686) = center of the orthopolar conic of X(34627)
X(68686) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 381, 38021}, {1, 18480, 18492}, {1, 30308, 51709}, {2, 5731, 50828}, {2, 50796, 5587}, {2, 50811, 3576}, {2, 50864, 515}, {2, 59387, 50796}, {4, 5881, 7982}, {4, 34627, 519}, {5, 3655, 25055}, {8, 3543, 28194}, {8, 31673, 41869}, {165, 3534, 50812}, {165, 51066, 50821}, {165, 61254, 5790}, {355, 5691, 40}, {376, 38074, 10}, {381, 18525, 28204}, {381, 51709, 30308}, {382, 34718, 28198}, {546, 37727, 11522}, {547, 3653, 3624}, {547, 34773, 3653}, {551, 3545, 8227}, {551, 19925, 3545}, {944, 3545, 551}, {944, 8227, 64952}, {944, 19925, 8227}, {1125, 38076, 5071}, {1478, 5727, 11529}, {1698, 18481, 67706}, {1699, 51093, 3656}, {1837, 9613, 3333}, {3091, 5882, 9624}, {3241, 3839, 946}, {3524, 3828, 31423}, {3524, 5818, 3828}, {3534, 5790, 50821}, {3534, 50797, 5790}, {3534, 50821, 165}, {3576, 5587, 54447}, {3585, 37711, 3340}, {3586, 5252, 31393}, {3624, 34773, 64954}, {3653, 61261, 547}, {3655, 25055, 64953}, {3656, 3845, 1699}, {3656, 51093, 16200}, {3679, 5691, 30}, {3828, 4297, 3524}, {3830, 37712, 50817}, {3830, 50798, 517}, {3845, 50871, 16200}, {4297, 5818, 31423}, {4669, 50801, 59388}, {4669, 50810, 63143}, {4669, 50862, 516}, {4677, 37712, 50798}, {4677, 50865, 517}, {5587, 50811, 2}, {5690, 64005, 40}, {5731, 54448, 10175}, {5731, 59387, 54448}, {5790, 50821, 51066}, {5881, 31162, 519}, {9900, 9901, 3751}, {9955, 18526, 1}, {10175, 50828, 2}, {10175, 54448, 5587}, {15682, 50801, 63143}, {15682, 50810, 516}, {15682, 59388, 50810}, {18357, 18481, 1698}, {18480, 18525, 1}, {18491, 18519, 36}, {18492, 38021, 381}, {18518, 18761, 35}, {19875, 34628, 3}, {30308, 51709, 38021}, {31145, 50687, 962}, {34627, 34648, 31162}, {34628, 37714, 19875}, {34638, 38098, 43174}, {34697, 34746, 30}, {34773, 61261, 3624}, {37712, 50865, 4677}, {41869, 61250, 8}, {50796, 50864, 50811}, {50798, 50865, 50817}, {50801, 50862, 50810}, {50810, 59388, 4669}, {50864, 59387, 2}, {50871, 51093, 952}

X(68687) = X(1)X(479)∩X(3)X(934)

Barycentrics    (a + b - c)*(a - b + c)*(a^5 + a^4 b - 2 a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 + a^4*c + 12*a^3*b*c - 6*a^2*b^2*c - 4*a*b^3*c - 3*b^4*c - 2*a^3*c^2 - 6*a^2*b*c^2 + 6*a*b^2*c^2 + 2*b^3*c^2 - 2*a^2*c^3 - 4*a*b*c^3 + 2*b^2*c^3 + a*c^4 - 3*b*c^4 + c^5) : :

See Keita Miyamoto and Francisco Javier García Capitán, euclid 8534

X(68687) lies on these lines: {1, 479}, {3, 934}, {4, 31527}, {5, 36620}, {7, 1058}, {8, 35312}, {20, 56309}, {40, 9533}, {279, 3333}, {347, 7177}, {381, 66467}, {938, 31526}, {946, 10004}, {962, 7056}, {1088, 11037}, {1323, 3361}, {1482, 66246}, {2551, 59601}, {3295, 3599}, {5558, 30494}, {5779, 15913}, {5815, 31627}, {6223, 50562}, {9312, 17580}, {9589, 10136}, {14986, 62786}, {17582, 31994}, {22153, 34028}, {26446, 32079}, {31788, 56870}, {34059, 50700}, {34060, 37434}, {36888, 63971}, {36991, 56310}, {37412, 57498}, {50561, 64696}

X(68688) = X(2)X(11)∩X(56)X(381)

Barycentrics    (a+2*b-2*c)*(a-b-c)*(a-2*b+2*c) : :
X(68688) = 4*X[499]-X[5204], X[499]+2*X[10593], 2*X[499]+X[10896], X[5204]+8*X[10593], X[5204]+2*X[10896], 4*X[10593]-X[10896], X[1388]+2*X[10826]

See David Nguyen and Ercole Suppa, euclid 8535.

X(68688) lies on these lines: {1, 5055}, {2, 11}, {3, 9671}, {4, 5298}, {5, 3304}, {12, 5071}, {30, 499}, {33, 62980}, {35, 15694}, {36, 3830}, {56, 381}, {57, 30308}, {65, 38021}, {115, 12351}, {140, 9670}, {202, 41121}, {203, 41122}, {354, 7988}, {376, 5433}, {388, 61936}, {495, 10109}, {496, 547}, {498, 15170}, {519, 11376}, {549, 1479}, {551, 1837}, {553, 3817}, {597, 12589}, {613, 11178}, {632, 4309}, {950, 19883}, {999, 19709}, {1015, 18362}, {1056, 61926}, {1058, 61895}, {1155, 50865}, {1387, 38084}, {1388, 10826}, {1428, 47353}, {1469, 38072}, {1478, 5066}, {1482, 15079}, {1656, 3303}, {1697, 19876}, {1699, 61649}, {1737, 3656}, {1836, 50802}, {1858, 64664}, {2098, 3679}, {2099, 23708}, {2482, 13183}, {2646, 50740}, {3023, 9166}, {3027, 23234}, {3056, 21358}, {3057, 19875}, {3085, 61899}, {3086, 3545}, {3090, 37722}, {3091, 9657}, {3241, 54361}, {3295, 15703}, {3524, 6284}, {3525, 63273}, {3526, 4857}, {3534, 3583}, {3543, 7288}, {3600, 61944}, {3614, 14986}, {3632, 65991}, {3653, 10572}, {3654, 30384}, {3711, 30827}, {3715, 5231}, {3746, 5070}, {3813, 6931}, {3828, 12053}, {3839, 7354}, {3845, 12943}, {3847, 10527}, {3850, 4317}, {3851, 5563}, {3928, 7082}, {4293, 41099}, {4294, 7294}, {4299, 15687}, {4302, 12100}, {4316, 62040}, {4324, 14093}, {4325, 61984}, {4330, 15720}, {4345, 51068}, {4370, 24837}, {4654, 4860}, {4677, 5048}, {4679, 5325}, {4870, 8227}, {4999, 31156}, {5010, 15701}, {5054, 5217}

X(68688) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(3035),X(65036)}, {{A,B,C,X(4995),X(32023)}}, {{A,B,C,X(6174),X(65065)}}, {{A,B,C,X(11238),X(56365)}}
X(68688) = pole of line {11, 1111} with respect to orthoptic circle of Feuerbach hyperbola
X(68688) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 11, 11238}, {2, 497, 4995}, {2, 3829, 31140}, {2, 5274, 10385}, {2, 10385, 5432}, {2, 10707, 4421}, {2, 11238, 55}, {5, 10072, 11237}, {11, 5432, 5274}, {381, 3582, 56}, {496, 547, 10056}, {499, 10593, 10896}, {499, 10896, 5204}, {1656, 37720, 3303}, {3086, 3545, 5434}, {3086, 7173, 10895}, {3526, 4857, 64950}, {3545, 5434, 10895}, {3582, 7741, 381}, {3817, 17728, 61716}, {3829, 45310, 2}, {3851, 5563, 9656}, {5433, 10591, 12953}, {5434, 7173, 3545}, {10072, 11237, 3304}, {15170, 15699, 498}, {17606, 50443, 2098}, {34612, 59376, 2}, {36441, 36459, 11238}

X(68689) = X(4)X(19172)∩X(13322)X(34782)

Barycentrics    2*a^24*b^4 - 17*a^22*b^6 + 63*a^20*b^8 - 132*a^18*b^10 + 168*a^16*b^12 - 126*a^14*b^14 + 42*a^12*b^16 + 12*a^10*b^18 - 18*a^8*b^20 + 7*a^6*b^22 - a^4*b^24 + 4*a^24*b^2*c^2 - 33*a^22*b^4*c^2 + 109*a^20*b^6*c^2 - 178*a^18*b^8*c^2 + 128*a^16*b^10*c^2 + 14*a^14*b^12*c^2 - 70*a^12*b^14*c^2 - 4*a^10*b^16*c^2 + 68*a^8*b^18*c^2 - 53*a^6*b^20*c^2 + 17*a^4*b^22*c^2 - 2*a^2*b^24*c^2 + 2*a^24*c^4 - 33*a^22*b^2*c^4 + 136*a^20*b^4*c^4 - 232*a^18*b^6*c^4 + 159*a^16*b^8*c^4 - 14*a^14*b^10*c^4 + 8*a^12*b^12*c^4 - 20*a^10*b^14*c^4 - 88*a^8*b^16*c^4 + 151*a^6*b^18*c^4 - 88*a^4*b^20*c^4 + 20*a^2*b^22*c^4 - b^24*c^4 - 17*a^22*c^6 + 109*a^20*b^2*c^6 - 232*a^18*b^4*c^6 + 202*a^16*b^6*c^6 - 66*a^14*b^8*c^6 - 2*a^12*b^10*c^6 + 28*a^10*b^12*c^6 + 44*a^8*b^14*c^6 - 205*a^6*b^16*c^6 + 213*a^4*b^18*c^6 - 84*a^2*b^20*c^6 + 10*b^22*c^6 + 63*a^20*c^8 - 178*a^18*b^2*c^8 + 159*a^16*b^4*c^8 - 66*a^14*b^6*c^8 + 44*a^12*b^8*c^8 - 16*a^10*b^10*c^8 - 22*a^8*b^12*c^8 + 134*a^6*b^14*c^8 - 263*a^4*b^16*c^8 + 190*a^2*b^18*c^8 - 45*b^20*c^8 - 132*a^18*c^10 + 128*a^16*b^2*c^10 - 14*a^14*b^4*c^10 - 2*a^12*b^6*c^10 - 16*a^10*b^8*c^10 + 32*a^8*b^10*c^10 - 34*a^6*b^12*c^10 + 154*a^4*b^14*c^10 - 236*a^2*b^16*c^10 + 120*b^18*c^10 + 168*a^16*c^12 + 14*a^14*b^2*c^12 + 8*a^12*b^4*c^12 + 28*a^10*b^6*c^12 - 22*a^8*b^8*c^12 - 34*a^6*b^10*c^12 - 64*a^4*b^12*c^12 + 112*a^2*b^14*c^12 - 210*b^16*c^12 - 126*a^14*c^14 - 70*a^12*b^2*c^14 - 20*a^10*b^4*c^14 + 44*a^8*b^6*c^14 + 134*a^6*b^8*c^14 + 154*a^4*b^10*c^14 + 112*a^2*b^12*c^14 + 252*b^14*c^14 + 42*a^12*c^16 - 4*a^10*b^2*c^16 - 88*a^8*b^4*c^16 - 205*a^6*b^6*c^16 - 263*a^4*b^8*c^16 - 236*a^2*b^10*c^16 - 210*b^12*c^16 + 12*a^10*c^18 + 68*a^8*b^2*c^18 + 151*a^6*b^4*c^18 + 213*a^4*b^6*c^18 + 190*a^2*b^8*c^18 + 120*b^10*c^18 - 18*a^8*c^20 - 53*a^6*b^2*c^20 - 88*a^4*b^4*c^20 - 84*a^2*b^6*c^20 - 45*b^8*c^20 + 7*a^6*c^22 + 17*a^4*b^2*c^22 + 20*a^2*b^4*c^22 + 10*b^6*c^22 - a^4*c^24 - 2*a^2*b^2*c^24 - b^4*c^24 : :

See Antreas Hatzipolakis and Peter Moses, euclid 8536.

X(68689) lies on these lines: {4, 19172}, {13322, 34782}

X(68690) = EULER LINE INTERCEPT OF X(159)X(38072)

Barycentrics    -(a^2*(3*a^8-6*a^6*b^2+6*a^2*b^6-3*b^8-6*a^6*c^2-8*a^4*b^2*c^2-14*a^2*b^4*c^2+28*b^6*c^2-14*a^2*b^2*c^4-50*b^4*c^4+6*a^2*c^6+28*b^2*c^6-3*c^8)) : :
X(68690) = 2*X[3]+7*X[5198], X[3]-7*X[7529], 4*X[3]-7*X[66607]

As a point on the Euler line, X(68690) has Shinagawa coefficients: {-E+3 F,-6 E-3 F}

See David Nguyen and Ercole Suppa, euclid 8539.

X(68690) lies on these lines: {2, 3}, {159, 38072}, {1181, 58470}, {1498, 16226}, {3426, 15053}, {5476, 19459}, {5640, 32063}, {8185, 30308}, {8192, 51709}, {9166, 9861}, {9777, 46261}, {9798, 38021}, {9876, 14639}, {9911, 19875}, {9913, 59377}, {10037, 65140}, {10110, 63094}, {12160, 21849}, {12168, 68317}, {12410, 38074}, {13175, 23234}, {14491, 38263}, {14831, 17810}, {15066, 58764}, {17814, 21969}, {19005, 35822}, {19006, 35823}, {31860, 63425}, {34469, 46849}, {35259, 67067}, {38073, 60897}, {38076, 49553}, {38077, 54065}, {39879, 59373}

X(68690) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(1885),X(54863)}, {{A,B,C,X(1989),X(3088)}}, {{A,B,C,X(5054),X(38263)}}, {{A,B,C,X(14491),X(38282)}}
X(68690) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 381, 54994}, {381, 7545, 51519}, {1995, 18535, 21312}, {3545, 9909, 7395}, {3545, 10594, 9909}, {5198, 7529, 66607}, {7545, 9818, 25}

X(68691) = EULER LINE INTERCEPT OF X(496)X(15654)

Barycentrics    a*(2*a^5*b + 2*a^4*b^2 - 2*a^3*b^3 - 2*a^2*b^4 + 2*a^5*c - 2*a^4*b*c + a^2*b^3*c - 2*a*b^4*c + b^5*c + 2*a^4*c^2 + 2*a*b^3*c^2 - 2*a^3*c^3 + a^2*b*c^3 + 2*a*b^2*c^3 - 2*b^3*c^3 - 2*a^2*c^4 - 2*a*b*c^4 + b*c^5) : :
X(68691) = X[851] + 3 X[855], X[851] - 3 X[859], 3 X[855] - X[13744], 3 X[859] + X[13744]

X(68691) lies on these lines: {2, 3}, {496, 15654}, {513, 1960}, {759, 53424}, {1420, 1464}, {3743, 9957}, {10058, 20989}, {12433, 22345}, {15171, 23361}, {15174, 64753}, {18990, 23383}, {20872, 51506}, {20991, 22758}, {22344, 34753}, {24928, 41682}, {48894, 64544}

X(68691) = midpoint of X(i) and X(j) for these {i,j}: {851, 13744}, {855, 859}
X(68691) = crossdifference of every pair of points on line {45, 647}
X(68691) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {851, 855, 13744}, {859, 13744, 851}, {7428, 13724, 140}

X(68692) = EULER LINE INTERCEPT OF X(39)X(2387)

Barycentrics    a^2*(2*a^6*b^4 - 2*a^4*b^6 + 2*a^6*b^2*c^2 - a^4*b^4*c^2 + b^8*c^2 + 2*a^6*c^4 - a^4*b^2*c^4 - b^6*c^4 - 2*a^4*c^6 - b^4*c^6 + b^2*c^8) : :
X(68692) = X[50706] - 3 X[66414]

X(68692) lies on these lines: {2, 3}, {39, 2387}, {187, 2679}, {230, 21444}, {512, 2021}, {1576, 1691}, {2882, 3003}, {3511, 47286}, {5167, 11672}, {7737, 46319}, {14601, 58312}, {36212, 55005}, {38650, 62366}, {51404, 51869}, {52460, 52967}

X(68692) = midpoint of X(237) and X(45900)
X(68692) = crossdifference of every pair of points on line {183, 647}
X(68692) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 21177, 237}

X(68693) = EULER LINE INTERCEPT OF X(1)X(18662)

Barycentrics    (a + b)*(a - b - c)*(a + c)*(2*a^4 - a^3*b - a^2*b^2 + a*b^3 - b^4 - a^3*c + 2*a^2*b*c - a*b^2*c - a^2*c^2 - a*b*c^2 + 2*b^2*c^2 + a*c^3 - c^4) : :

X(68693) lies on these lines: {1, 18662}, {2, 3}, {8, 283}, {58, 1771}, {78, 28950}, {81, 56943}, {86, 347}, {99, 53703}, {107, 67758}, {110, 2734}, {145, 3193}, {150, 51607}, {163, 45748}, {280, 285}, {281, 2193}, {476, 53870}, {519, 62756}, {522, 663}, {758, 39767}, {925, 953}, {944, 1437}, {1014, 55119}, {1259, 27410}, {1737, 52680}, {1790, 5731}, {1792, 7080}, {1793, 2222}, {2327, 27382}, {2328, 52365}, {2726, 3565}, {3007, 17195}, {3101, 17185}, {3465, 61185}, {3486, 54417}, {3877, 20243}, {4296, 37558}, {4305, 54323}, {4329, 17183}, {4653, 17188}, {5172, 26095}, {5842, 23541}, {6224, 68661}, {6360, 8025}, {6740, 16704}, {7283, 56187}, {9436, 17219}, {10420, 43655}, {10436, 17134}, {10454, 40456}, {14552, 23602}, {17139, 57985}, {17194, 54318}, {18659, 24549}, {22361, 34831}, {31631, 52352}, {34188, 68244}, {34851, 40950}, {40081, 63642}, {44316, 48390}, {46974, 64194}, {52954, 62757}, {53895, 53933}

X(68693) = midpoint of X(7424) and X(50403)
X(68693) = anticomplement of X(860)
X(68693) = deLongchamps-circle-inverse of X(36154)
X(68693) = anticomplement of the isogonal conjugate of X(57736)
X(68693) = anticomplement of the isotomic conjugate of X(57985)
X(68693) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {759, 4}, {1437, 6224}, {1793, 3436}, {1807, 1330}, {2341, 68335}, {9274, 110}, {14616, 11442}, {17104, 12383}, {23226, 14731}, {24624, 21270}, {32662, 1577}, {32671, 525}, {34079, 5905}, {36069, 7253}, {37140, 850}, {52351, 21287}, {52431, 2895}, {57736, 8}, {57985, 6327}, {65283, 21300}, {67166, 193}, {68571, 5906}
X(68693) = X(i)-Ceva conjugate of X(j) for these (i,j): {17139, 16704}, {57985, 2}
X(68693) = X(46391)-cross conjugate of X(2406)
X(68693) = X(i)-isoconjugate of X(j) for these (i,j): {65, 102}, {73, 36121}, {225, 36055}, {226, 32677}, {523, 36040}, {525, 32667}, {651, 55255}, {656, 36067}, {661, 65297}, {810, 65295}, {1020, 2432}, {1400, 36100}, {1402, 34393}, {1409, 52780}, {1427, 15629}, {1577, 32643}, {2250, 60000}, {36059, 68625}, {52383, 58741}
X(68693) = X(i)-Dao conjugate of X(j) for these (i,j): {515, 51421}, {10017, 523}, {20620, 68625}, {23986, 226}, {36830, 65297}, {36944, 38955}, {38991, 55255}, {39062, 65295}, {40582, 36100}, {40596, 36067}, {40602, 102}, {40605, 34393}, {46974, 758}, {51221, 225}
X(68693) = X(68693) = cevapoint of X(i) and X(j) for these (i,j): {1, 60018}, {515, 46974}
X(68693) = crosspoint of X(i) and X(j) for these (i,j): {14616, 31623}, {23582, 65283}
X(68693) = crosssum of X(i) and X(j) for these (i,j): {1409, 3724}, {3269, 42666}
X(68693) = crossdifference of every pair of points on line {647, 1400}
X(68693) = barycentric product X(i)*X(j) for these {i,j}: {21, 64194}, {274, 51361}, {284, 35516}, {314, 2182}, {332, 8755}, {333, 515}, {645, 53522}, {648, 39471}, {662, 14304}, {663, 55254}, {811, 46391}, {1043, 34050}, {2406, 7253}, {3737, 42718}, {6332, 7452}, {7058, 51421}, {15411, 23987}, {24035, 57081}, {31623, 46974}, {51368, 59482}
X(68693) = barycentric quotient X(i)/X(j) for these {i,j}: {21, 36100}, {29, 52780}, {110, 65297}, {112, 36067}, {163, 36040}, {284, 102}, {333, 34393}, {515, 226}, {648, 65295}, {663, 55255}, {859, 60000}, {1172, 36121}, {1455, 1427}, {1576, 32643}, {2182, 65}, {2193, 36055}, {2194, 32677}, {2328, 15629}, {2406, 4566}, {2425, 53321}, {3064, 68625}, {4282, 58741}, {7253, 2399}, {7452, 653}, {8755, 225}, {10017, 42761}, {11700, 18593}, {14304, 1577}, {17139, 56666}, {17926, 53152}, {21789, 2432}, {23986, 51421}, {23987, 52607}, {32676, 32667}, {34050, 3668}, {35516, 349}, {38554, 51368}, {39471, 525}, {46391, 656}, {46974, 1214}, {51361, 37}, {51368, 6356}, {51375, 64708}, {51414, 41003}, {51421, 6354}, {51424, 52023}, {53522, 7178}, {55254, 4572}, {59283, 60091}, {64194, 1441}, {68251, 52037}
X(68693) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 5136, 2}, {3, 27378, 27506}, {4, 27379, 27504}, {20, 27402, 27505}, {20, 36024, 22}, {21, 1325, 17515}, {21, 1816, 4225}, {21, 11103, 2}, {21, 17518, 37277}, {21, 17519, 4228}, {21, 35995, 3}, {448, 15149, 14953}, {859, 3109, 37168}, {4183, 4221, 21}, {7515, 37381, 2}, {7515, 37468, 24984}, {8021, 19259, 21}, {14955, 46582, 14953}

X(68694) = EULER LINE INTERCEPT OF X(50)X(340)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^2 - b^2 + c^2)*(2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6) : :

X(68694) lies on these lines: {2, 3}, {50, 340}, {67, 287}, {648, 45331}, {1990, 16237}, {2407, 56021}, {2420, 3580}, {3268, 5664}, {5649, 41253}, {6103, 60502}, {6749, 18122}, {7799, 57487}, {11064, 35910}, {16080, 39295}, {17986, 51262}, {18311, 62595}, {18384, 34365}, {32223, 58343}, {32313, 39905}, {37638, 67378}, {40879, 44134}, {41145, 49116}, {41254, 44529}, {43084, 44138}, {52469, 64607}

X(68694) = polar conjugate of X(54554)
X(68694) = X(i)-isoconjugate of X(j) for these (i,j): {48, 54554}, {293, 34370}, {647, 36096}, {656, 23969}, {14998, 36061}, {32678, 35909}
X(68694) = X(i)-Dao conjugate of X(j) for these (i,j): {132, 34370}, {1249, 54554}, {2493, 51847}, {5664, 65727}, {6103, 43090}, {14918, 51228}, {16221, 14998}, {18334, 35909}, {23967, 265}, {39052, 36096}, {40596, 23969}, {40604, 65308}, {42426, 1989}, {57464, 65723}, {65728, 14582}, {65732, 14592}
vcrossdifference of every pair of points on line {647, 52153}
X(68694) = barycentric product X(i)*X(j) for these {i,j}: {323, 60502}, {340, 542}, {3268, 7473}, {6103, 7799}, {6148, 17986}, {14165, 65722}, {14590, 18312}, {14920, 51227}, {14999, 44427}, {18020, 53132}, {35907, 45792}, {44146, 57470}, {45808, 53155}, {51383, 52491}
X(68694) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 54554}, {112, 23969}, {162, 36096}, {186, 842}, {232, 34370}, {323, 65308}, {340, 5641}, {526, 35909}, {542, 265}, {1640, 14582}, {3043, 52179}, {5191, 52153}, {6103, 1989}, {7473, 476}, {8552, 35911}, {14590, 5649}, {14920, 51228}, {14999, 60053}, {16188, 51847}, {17986, 5627}, {18312, 14592}, {38552, 43087}, {39176, 48453}, {42426, 43090}, {44427, 14223}, {45662, 66125}, {47230, 14998}, {48451, 11079}, {51456, 12028}, {52469, 65617}, {53132, 125}, {54380, 14356}, {57470, 895}, {60340, 65723}, {60502, 94}, {60505, 23968}, {62551, 65727}, {65723, 43083}
X(68694) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 401, 62563}, {2, 4235, 297}, {441, 62686, 2}, {470, 471, 35235}, {45662, 54380, 7473}

X(68695) = EULER LINE INTERCEPT OF X(52)X(40938)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - a^2*b^2 - a^2*c^2 - 2*b^2*c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4) : :
X(68695) = X[4] - 3 X[46511], 3 X[186] - X[7482]

X(68695) lies on these lines: {2, 3}, {52, 40938}, {53, 8266}, {74, 54998}, {98, 16083}, {107, 67728}, {112, 2080}, {114, 46094}, {132, 39569}, {160, 64711}, {182, 10311}, {183, 33971}, {232, 511}, {264, 22062}, {317, 9744}, {325, 19189}, {385, 41204}, {389, 1194}, {682, 36998}, {805, 3563}, {842, 53699}, {1235, 49111}, {1297, 14941}, {1350, 40801}, {1351, 45141}, {1503, 53174}, {1691, 57260}, {1799, 8884}, {1843, 13354}, {1968, 5171}, {1971, 17974}, {1974, 13355}, {1990, 5201}, {2456, 19128}, {2782, 44146}, {3001, 11062}, {3095, 39575}, {3199, 5188}, {3313, 14576}, {3329, 60693}, {3398, 10312}, {3569, 14696}, {5092, 10985}, {5976, 44132}, {5980, 67159}, {5981, 67158}, {6130, 65778}, {6331, 56442}, {6403, 56920}, {6530, 51862}, {6748, 41328}, {7710, 22655}, {8722, 34096}, {8744, 9301}, {9420, 23878}, {9729, 14133}, {9753, 17907}, {9756, 63421}, {10986, 26316}, {11257, 54412}, {11438, 40254}, {11574, 63634}, {12131, 44145}, {12188, 41377}, {13352, 52905}, {14978, 52787}, {15819, 39530}, {16308, 67529}, {16328, 46127}, {18860, 33874}, {20794, 32001}, {21163, 33843}, {22240, 30258}, {23347, 32217}, {30737, 32428}, {32515, 41676}, {35429, 39588}, {35908, 52692}, {38552, 62490}, {39604, 67854}, {40118, 53603}, {41363, 57262}, {42329, 62698}, {44524, 45030}, {46106, 47202}, {47151, 47584}, {52460, 61485}, {56437, 65748}, {59226, 59228}, {60516, 63736}

X(68695) = reflection of X(i) in X(j) for these {i,j}: {2967, 232}, {3289, 52128}, {65778, 6130}
X(68695) = isogonal conjugate of X(66879)
X(68695) = polar-circle-inverse of X(36183)
X(68695) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(57583)
X(68695) = X(40801)-Ceva conjugate of X(2967)
X(68695) = X(i)-isoconjugate of X(j) for these (i,j): {1, 66879}, {262, 293}, {263, 336}, {287, 2186}, {525, 36132}, {656, 6037}, {810, 53196}, {879, 65252}, {1821, 43718}, {1910, 42313}, {3402, 57799}, {14208, 32716}, {36120, 54032}
X(68695) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 66879}, {132, 262}, {11672, 42313}, {38997, 879}, {39009, 525}, {39062, 53196}, {40596, 6037}, {40601, 43718}, {46094, 54032}, {51580, 57799}, {62590, 59257}, {62595, 327}, {62596, 684}
X(68695) = crosspoint of X(i) and X(j) for these (i,j): {98, 3425}, {182, 39683}
X(68695) = crosssum of X(i) and X(j) for these (i,j): {262, 39682}, {511, 1352}
X(68695) = crossdifference of every pair of points on line {647, 879}
X(68695) = barycentric product X(i)*X(j) for these {i,j}: {25, 51373}, {182, 297}, {183, 232}, {237, 44144}, {240, 52134}, {250, 66192}, {325, 10311}, {458, 511}, {877, 3288}, {1959, 60685}, {2211, 20023}, {2967, 46806}, {3403, 57653}, {4230, 23878}, {6331, 9420}, {8842, 51324}, {19189, 59197}, {22456, 33569}, {33971, 36212}, {34396, 44132}, {35908, 51372}, {39683, 62595}
X(68695) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 66879}, {112, 6037}, {182, 287}, {183, 57799}, {232, 262}, {237, 43718}, {297, 327}, {458, 290}, {511, 42313}, {648, 53196}, {2211, 263}, {2967, 46807}, {3288, 879}, {3289, 54032}, {4230, 65271}, {6784, 51404}, {9420, 647}, {10311, 98}, {14966, 65310}, {17994, 66291}, {19189, 42300}, {32676, 36132}, {33569, 684}, {33971, 16081}, {34396, 248}, {34854, 68572}, {36212, 59257}, {39530, 53245}, {41270, 51444}, {44144, 18024}, {51373, 305}, {51542, 47388}, {52134, 336}, {57653, 2186}, {58070, 65349}, {59208, 53174}, {60685, 1821}, {61206, 32716}, {66192, 339}
X(68695) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 22, 418}, {4, 24, 27369}, {4, 631, 37337}, {232, 51324, 2211}, {237, 297, 15143}, {1350, 40805, 54032}, {1350, 59229, 40801}, {1513, 37123, 237}, {5000, 5001, 237}, {5004, 5005, 401}, {5112, 21525, 237}, {7418, 37123, 1513}, {16089, 17984, 60199}, {37114, 58883, 20885}, {42789, 42790, 11676}, {53138, 53139, 232}

X(68696) = EULER LINE INTERCEPT OF X(1)X(4463)

Barycentrics    a*(a^6 + a^5*b - a^2*b^4 - a*b^5 + a^5*c - 2*a^3*b^2*c - 2*a^2*b^3*c - 3*a*b^4*c - 2*b^5*c - 2*a^3*b*c^2 - 2*a^2*b^2*c^2 - 2*a*b^3*c^2 - 2*b^4*c^2 - 2*a^2*b*c^3 - 2*a*b^2*c^3 - a^2*c^4 - 3*a*b*c^4 - 2*b^2*c^4 - a*c^5 - 2*b*c^5) : :

X(68696) lies on these lines: {1, 4463}, {2, 3}, {6, 977}, {8, 41230}, {55, 5300}, {63, 1724}, {272, 1234}, {321, 19838}, {958, 4968}, {975, 62829}, {993, 23536}, {1441, 54394}, {1751, 6734}, {3869, 16471}, {4292, 48866}, {4972, 8193}, {5249, 43531}, {5250, 61086}, {5320, 10381}, {5324, 54429}, {11036, 19717}, {17595, 19728}, {18709, 27005}, {19734, 38904}, {19742, 54398}, {19752, 54433}, {19844, 33129}, {19845, 32779}, {26085, 36744}, {30811, 54371}, {31424, 62695}, {48863, 57287}, {54337, 56519}, {54392, 54405}

X(68696) = crossdifference of every pair of points on line {647, 832}
X(68696) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4189, 7523}, {2, 4198, 377}, {2, 5046, 5142}, {2, 6872, 37179}, {2, 7520, 404}, {2, 7557, 2476}, {2, 16865, 47512}, {2, 17521, 3}, {21, 4197, 37090}, {21, 37467, 37285}, {379, 964, 377}, {405, 4185, 964}, {405, 7535, 2}, {405, 11347, 37065}, {405, 16368, 21}, {405, 56960, 11319}, {4187, 7561, 2}, {4202, 48890, 377}, {5084, 7521, 2}, {5262, 5279, 3868}, {11102, 16062, 22}, {11108, 19285, 2}, {11347, 37065, 404}

X(68697) = EULER LINE INTERCEPT OF X(160)X(48837)

Barycentrics    a^2*(a + b)*(a + c)*(a^4*b^2 + a^3*b^3 - a^2*b^4 - a*b^5 + a^3*b^2*c - a^2*b^3*c - a*b^4*c + b^5*c + a^4*c^2 + a^3*b*c^2 - 2*a^2*b^2*c^2 + 2*a*b^3*c^2 + a^3*c^3 - a^2*b*c^3 + 2*a*b^2*c^3 - 2*b^3*c^3 - a^2*c^4 - a*b*c^4 - a*c^5 + b*c^5) : :

X(68697) lies on these lines: {2, 3}, {160, 48837}, {501, 23846}, {512, 1326}, {519, 1634}, {540, 5201}, {2617, 56884}, {5127, 23845}, {8266, 48835}, {8618, 53273}, {15792, 64753}, {17104, 23844}, {20775, 48847}, {35222, 48866}, {41328, 48843}, {48857, 66886}, {52604, 52954}

X(68697) = crosspoint of X(1126) and X(2372)
X(68697) = crosssum of X(1125) and X(2392)
X(68697) = crossdifference of every pair of points on line {647, 1213}

X(68698) = EULER LINE INTERCEPT OF X(8)X(3430)

Barycentrics    2*a^6 - a^5*b + 2*a^3*b^3 - 2*a^2*b^4 - a*b^5 - a^5*c + 2*a^4*b*c - a^3*b^2*c - a^2*b^3*c + 2*a*b^4*c - b^5*c - a^3*b*c^2 + 2*a^2*b^2*c^2 - a*b^3*c^2 + 2*a^3*c^3 - a^2*b*c^3 - a*b^2*c^3 + 2*b^3*c^3 - 2*a^2*c^4 + 2*a*b*c^4 - a*c^5 - b*c^5 : :

X(68698) lies on these lines: {2, 3}, {8, 3430}, {40, 17164}, {98, 901}, {111, 2737}, {165, 4418}, {226, 5764}, {355, 31079}, {511, 16704}, {515, 3006}, {516, 902}, {517, 20045}, {612, 12520}, {649, 3239}, {842, 53611}, {944, 29832}, {946, 26230}, {962, 26228}, {1150, 1350}, {1155, 4459}, {1297, 1309}, {1352, 31017}, {1385, 29823}, {1503, 3936}, {1699, 29855}, {1770, 66632}, {2373, 67777}, {2697, 53612}, {2716, 9070}, {2770, 67783}, {2784, 4062}, {2792, 17491}, {3007, 23710}, {3424, 60242}, {3564, 63071}, {3705, 64572}, {3952, 6211}, {4295, 5264}, {4297, 29639}, {4414, 24728}, {4427, 29057}, {5285, 52358}, {5297, 66106}, {5603, 29831}, {5691, 29857}, {5718, 44882}, {6684, 26251}, {6776, 31034}, {7081, 56288}, {7291, 7360}, {7987, 29826}, {8025, 37527}, {9956, 31098}, {14927, 30828}, {14963, 45748}, {17763, 18788}, {17977, 56555}, {19925, 30768}, {20070, 26245}, {20653, 35099}, {20872, 26095}, {20989, 26031}, {21368, 61185}, {24257, 64161}, {24597, 51212}, {25406, 63008}, {26244, 42316}, {27577, 64797}, {28164, 50752}, {28236, 50743}, {28796, 50861}, {29181, 35466}, {29641, 44039}, {30811, 36990}, {30834, 48905}, {31179, 43273}, {31187, 48872}, {31229, 48910}, {31303, 63428}, {31785, 64047}, {31884, 37660}, {33122, 64085}, {34168, 36067}, {37508, 64108}, {37798, 62314}, {44661, 65206}, {59417, 63400}

X(68698) = reflection of X(20045) in X(39572)
X(68698) = anticomplement of X(8229)
X(68698) = orthoptic-circle-of-the-Steiner-inellipse-inverse of X(36155)
X(68698) = orthoptic-circle-of-the-Steiner-circumellipse-inverse of X(36154)
X(68698) = crossdifference of every pair of points on line {647, 1201}
X(68698) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 35988, 36007}, {3, 15971, 11115}, {411, 37088, 50702}, {4220, 7413, 2}, {5000, 5001, 37168}, {5004, 5005, 859}, {7580, 19645, 50694}, {37370, 46549, 37009}, {37959, 50402, 23}

X(68699) = EULER LINE INTERCEPT OF X(1)X(19725)

Barycentrics    a*(a^5 - a*b^4 - 2*a^3*b*c + 2*a^2*b^2*c + 2*a*b^3*c - 2*b^4*c + 2*a^2*b*c^2 + 6*a*b^2*c^2 + 2*b^3*c^2 + 2*a*b*c^3 + 2*b^2*c^3 - a*c^4 - 2*b*c^4) : :

X(68699) lies on these lines: {1, 19725}, {2, 3}, {6, 354}, {41, 3720}, {81, 6090}, {142, 1473}, {169, 17441}, {198, 4423}, {228, 1001}, {373, 4383}, {584, 5275}, {612, 37080}, {748, 1400}, {940, 5320}, {946, 26935}, {1040, 1827}, {1211, 61506}, {1426, 19372}, {1486, 3925}, {1724, 3338}, {1730, 41338}, {1824, 9816}, {1876, 55875}, {1899, 25964}, {2355, 10319}, {3220, 41867}, {3624, 57281}, {3742, 51743}, {3826, 37577}, {3917, 25878}, {4387, 42713}, {4648, 5324}, {4847, 52015}, {5249, 24320}, {5268, 59337}, {5278, 64153}, {5310, 19760}, {6703, 61507}, {7085, 51687}, {7292, 67301}, {8012, 36808}, {9306, 44117}, {9776, 26866}, {10582, 16783}, {11365, 19854}, {12595, 64559}, {17017, 21808}, {17063, 36572}, {17123, 27659}, {19799, 19838}, {24309, 61029}, {26241, 29641}, {26885, 37543}, {34048, 45963}, {35259, 37527}, {37581, 54357}, {37674, 44098}, {48863, 62673}, {61657, 63009}

X(68699) = crossdifference of every pair of points on line {647, 3309}
X(68699) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 21, 16353}, {2, 405, 37060}, {2, 1995, 19544}, {2, 4223, 25}, {2, 4224, 7484}, {2, 4228, 3}, {2, 5020, 37366}, {2, 17522, 37090}, {2, 26118, 30739}, {2, 37254, 37261}, {2, 37449, 16419}, {2, 59358, 16849}, {28, 16845, 37246}, {405, 7522, 11323}, {405, 7535, 4185}, {405, 11347, 1011}, {405, 37063, 57523}, {1724, 5272, 19724}, {5004, 5005, 37105}, {5651, 61643, 940}, {6846, 37275, 1593}, {6878, 36009, 3}, {11108, 13737, 37225}, {11357, 56970, 405}, {16352, 50715, 2}, {16428, 17561, 4221}

X(68700) = EULER LINE INTERCEPT OF X(1)X(19735)

Barycentrics    a^2*(a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + a^4*c - a^3*b*c + a*b^3*c - b^4*c + a^3*c^2 + 4*a*b^2*c^2 + 5*b^3*c^2 - a^2*c^3 + a*b*c^3 + 5*b^2*c^3 - a*c^4 - b*c^4) : :

X(68700) lies on these lines: {1, 19735}, {2, 3}, {6, 1201}, {56, 748}, {58, 5651}, {228, 5436}, {373, 386}, {999, 28370}, {1451, 26885}, {1468, 28360}, {1724, 5563}, {1730, 7991}, {3303, 10459}, {3430, 34417}, {3616, 21319}, {3624, 15654}, {3746, 19763}, {3984, 10477}, {4423, 23361}, {5260, 23853}, {5437, 22344}, {12513, 19723}, {19721, 50034}, {19760, 64950}, {19811, 19838}, {19879, 37546}, {23085, 27003}, {23383, 64752}, {23858, 28353}, {24320, 28402}, {25524, 32944}, {28377, 41341}, {37502, 64415}, {62837, 63060}

X(68700) = crossdifference of every pair of points on line {647, 3667}
X(68700) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7419, 3}, {2, 28376, 47521}, {2, 28383, 28348}, {3, 25, 20851}, {3, 5047, 16373}, {3, 7419, 28348}, {3, 11108, 16296}, {3, 16421, 17572}, {3, 16855, 19249}, {3, 19253, 16862}, {3, 28383, 7419}, {25, 37244, 37247}, {405, 474, 11354}, {405, 4245, 13738}, {405, 11358, 11319}, {405, 13738, 1011}, {405, 37058, 57523}, {474, 19532, 11115}, {859, 16296, 3}, {4225, 16859, 16058}, {5020, 37248, 37259}, {5047, 19245, 3}, {5047, 19334, 16842}, {7428, 19265, 16408}, {13737, 37246, 199}, {16845, 19256, 37225}, {17697, 37442, 16405}, {19260, 37035, 19283}, {20851, 37247, 3}

X(68701) = EULER LINE INTERCEPT OF X(6)X(2501)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8 - 2*a^6*b^2 + a^4*b^4 - 2*a^6*c^2 + 3*a^4*b^2*c^2 - a^2*b^4*c^2 - b^6*c^2 + a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 - b^2*c^6) : :

X(68701) lies on these lines: {2, 3}, {6, 2501}, {107, 1112}, {110, 2970}, {136, 5972}, {184, 58261}, {264, 18020}, {338, 58943}, {476, 65586}, {685, 14355}, {847, 61753}, {1300, 1511}, {1304, 66790}, {1576, 47208}, {2052, 53176}, {2453, 47627}, {2790, 40352}, {2931, 60035}, {2986, 14984}, {3043, 6344}, {3233, 15928}, {5186, 64479}, {5642, 8754}, {5962, 63839}, {5967, 53149}, {6531, 44467}, {9717, 18808}, {10113, 44990}, {10605, 59291}, {12079, 16933}, {12228, 66953}, {12828, 47204}, {13414, 39241}, {13415, 39240}, {13558, 53577}, {14254, 38936}, {14385, 15111}, {14593, 59543}, {14685, 47217}, {14698, 44427}, {14920, 18384}, {15454, 47391}, {15462, 36789}, {15472, 34334}, {16080, 20774}, {16221, 22104}, {18121, 23292}, {18279, 18390}, {19504, 35360}, {22823, 33547}, {23306, 39118}, {26869, 52473}, {32227, 54395}, {32710, 38609}, {34397, 46106}, {36306, 56515}, {36309, 56514}, {36820, 66943}, {43530, 54554}, {44084, 61733}, {47146, 67531}, {52449, 59771}, {53319, 67529}, {58733, 61711}, {61743, 66939}

X(68701) = orthocentroidal-circle-inverse of X(35235)
X(68701) = polar-circle-inverse of X(47348)
X(68701) = polar conjugate of the isotomic conjugate of X(40879)
X(68701) = polar conjugate of the isogonal conjugate of X(32761)
X(68701) = X(32761)-cross conjugate of X(40879)
X(68701) = X(i)-isoconjugate of X(j) for these (i,j): {656, 9160}, {810, 53192}
X(68701) = X(i)-Dao conjugate of X(j) for these (i,j): {39062, 53192}, {40596, 9160}
X(68701) = crossdifference of every pair of points on line {647, 13754}
X(68701) = barycentric product X(i)*X(j) for these {i,j}: {4, 40879}, {264, 32761}, {340, 56396}, {648, 62489}, {14590, 68473}, {44427, 64221}
X(68701) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 9160}, {648, 53192}, {32761, 3}, {40879, 69}, {56396, 265}, {62489, 525}, {64221, 60053}, {68473, 14592}
X(68701) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 35235}, {2, 30512, 3}, {4, 4240, 25}, {421, 450, 468}, {460, 15144, 468}, {1316, 56962, 40856}, {3134, 36178, 3}, {57013, 57014, 186}

X(68702) = EULER LINE INTERCEPT OF X(107)X(53703)

Barycentrics    (a + b)*(a + c)*(2*a^4 - a^3*b + a^2*b^2 - 3*a*b^3 + b^4 - a^3*c - 2*a^2*b*c + 3*a*b^2*c + a^2*c^2 + 3*a*b*c^2 - 2*b^2*c^2 - 3*a*c^3 + c^4) : :

X(68702) lies on these lines: {2, 3}, {107, 53703}, {110, 2726}, {284, 40127}, {476, 53933}, {953, 1302}, {1150, 35259}, {1443, 1447}, {2222, 67755}, {2734, 9064}, {3006, 68149}, {3086, 5358}, {3936, 32269}, {5121, 52680}, {9060, 43655}, {15448, 35466}, {20769, 29824}, {20872, 26031}, {20989, 26095}, {24597, 35260}, {30738, 41869}, {40979, 46345}, {41610, 64151}, {47321, 67530}, {53870, 53944}, {63067, 64058}

X(68702) = X(656)-isoconjugate of X(67777)
X(68702) = X(40596)-Dao conjugate of X(67777)
X(68702) = crossdifference of every pair of points on line {647, 1334}
X(68702) = barycentric product X(86)*X(8074)
X(68702) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 67777}, {8074, 10}
X(68702) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 37254, 50404}, {468, 8229, 2}, {14956, 33325, 14953}

X(68703) = EULER LINE INTERCEPT OF X(1)X(21807)

Barycentrics    a*(a^6 - a^4*b^2 + a^3*b^3 - a*b^5 + a^4*b*c + a^2*b^3*c - 2*b^5*c - a^4*c^2 + 2*a^2*b^2*c^2 + a*b^3*c^2 + a^3*c^3 + a^2*b*c^3 + a*b^2*c^3 + 4*b^3*c^3 - a*c^5 - 2*b*c^5) : :

X(68703) lies on these lines: {1, 21807}, {2, 3}, {6, 1411}, {56, 60033}, {283, 15488}, {484, 1730}, {580, 58889}, {956, 17165}, {993, 25385}, {999, 33148}, {1324, 7951}, {1478, 20999}, {1626, 12943}, {1724, 3460}, {1790, 6176}, {1870, 21318}, {2217, 11375}, {2654, 11429}, {3465, 37525}, {3585, 23850}, {5434, 53302}, {9306, 68661}, {9708, 26867}, {10895, 23843}, {11681, 38903}, {15950, 50705}, {18393, 34300}, {23858, 45701}, {30438, 36280}

X(68703) = crossdifference of every pair of points on line {647, 3738}
X(68703) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 49128, 52273}, {4, 13733, 3145}, {5, 37227, 37259}, {28, 6920, 13724}, {381, 11334, 52242}, {405, 4185, 13738}, {405, 7535, 68700}, {405, 56960, 1011}, {405, 56970, 37061}, {4224, 6912, 855}, {6883, 37241, 4191}, X(68703) = {6906, 28349, 20843}, {11113, 36011, 47523}

X(68704) = EULER LINE INTERCEPT OF X(58)X(12241)

Barycentrics    (a^2 - b^2 - c^2)*(a^3*b + a^2*b^2 - a*b^3 - b^4 + a^3*c - 2*a^2*b*c + a*b^2*c + a^2*c^2 + a*b*c^2 + 2*b^2*c^2 - a*c^3 - c^4)*(2*a^4 - a^3*b - a^2*b^2 + a*b^3 - b^4 - a^3*c + 2*a^2*b*c - a*b^2*c - a^2*c^2 - a*b*c^2 + 2*b^2*c^2 + a*c^3 - c^4) : :

X(68704) lies on these lines: {2, 3}, {58, 12241}, {84, 5928}, {117, 515}, {386, 12233}, {517, 2968}, {946, 17102}, {1040, 63992}, {1062, 63986}, {1071, 1425}, {1512, 60427}, {1519, 60415}, {1528, 55058}, {1537, 35012}, {1809, 5080}, {5081, 60758}, {5179, 40616}, {5562, 41014}, {6253, 53850}, {6356, 64126}, {10446, 41005}, {10740, 56863}, {12114, 56414}, {15252, 45766}, {17747, 35072}, {18588, 63989}, {18643, 24220}, {42353, 48902}, {54083, 64501}, {56862, 64507}, {64194, 65584}

X(68704) = midpoint of X(10740) and X(56863)
X(68704) = reflection of X(i) in X(j) for these {i,j}: {5081, 60758}, {38554, 46974}, {45766, 15252}, {51421, 117}
X(68704) = complement of X(37420)
X(68704) = X(i)-isoconjugate of X(j) for these (i,j): {102, 40396}, {947, 36121}
X(68704) = X(i)-Dao conjugate of X(j) for these (i,j): {20262, 36100}, {40943, 52780}
X(68704) = crosssum of X(7412) and X(37305)
X(68704) = crossdifference of every pair of points on line {647, 2432}
X(68704) = barycentric product X(i)*X(j) for these {i,j}: {17102, 64194}, {22063, 35516}
X(68704) = barycentric quotient X(i)/X(j) for these {i,j}: {946, 52780}, {2182, 40396}, {2262, 36121}, {17102, 36100}, {22063, 102}, {34050, 63186}, {40945, 15629}, {46974, 55987}
X(68704) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20, 37410}, {1012, 3149, 25}, {6831, 37468, 407}, {6847, 50701, 7490}, {6905, 6906, 186}, {8727, 20420, 15762}

X(68705) = EULER LINE INTERCEPT OF X(6)X(9145)

Barycentrics    a^2*(a^6 - 5*a^4*b^2 + 5*a^2*b^4 - b^6 - 5*a^4*c^2 - 2*a^2*b^2*c^2 - 3*b^4*c^2 + 5*a^2*c^4 - 3*b^2*c^4 - c^6) : :

X(68705) lies on these lines: {2, 3}, {6, 9145}, {32, 3292}, {187, 5651}, {352, 1384}, {373, 574}, {575, 59707}, {576, 36212}, {2080, 15066}, {2482, 33929}, {3003, 8542}, {3284, 40799}, {5191, 35259}, {5201, 9516}, {5640, 54439}, {5650, 8722}, {5967, 64633}, {6800, 26316}, {7801, 41586}, {7820, 61644}, {9127, 37809}, {18860, 34417}, {21163, 22112}, {22330, 59559}, {22735, 33997}, {23235, 40814}, {32127, 58265}, {33705, 33975}, {34010, 47200}, {34236, 39498}, {36759, 61633}, {36760, 61632}, {37827, 44380}, {39099, 68660}, {46124, 47047}, {52771, 63128}

X(68705) = X(62912)-Ceva conjugate of X(6)
X(68705) = crossdifference of every pair of points on line {647, 2793}
X(68705) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 35298, 3}, {3, 25, 37916}, {3, 1995, 3148}, {3, 11328, 1995}, {3, 35298, 41275}, {3, 37465, 20897}, {3, 37914, 22}, {32, 61631, 3292}, {1344, 1345, 57618}, {1995, 52275, 3}, {5020, 35302, 37457}, {7496, 37184, 3}, {7550, 37114, 3}, {11328, 52275, 3148}, {44221, 49671, 3}

X(68706) = EULER LINE INTERCEPT OF X(110)X(33873)

Barycentrics    a^2*(a^6*b^2 - a^2*b^6 + a^6*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 + a^2*b^2*c^4 + 2*b^4*c^4 - a^2*c^6 - 2*b^2*c^6) : :

X(68706) lies on these lines: {2, 3}, {110, 33873}, {111, 694}, {230, 34294}, {323, 57258}, {325, 1634}, {385, 9512}, {511, 44420}, {524, 9149}, {669, 804}, {682, 7823}, {1194, 46305}, {1196, 46321}, {1302, 67728}, {1350, 34095}, {1495, 36213}, {1613, 33586}, {1915, 5012}, {2080, 46316}, {2373, 18024}, {2493, 9019}, {2770, 53603}, {2780, 11186}, {3051, 3060}, {3229, 40350}, {3291, 8623}, {3580, 20021}, {3978, 53765}, {5201, 22329}, {5422, 44508}, {5651, 52658}, {5968, 8705}, {5996, 56394}, {7752, 23208}, {7773, 15270}, {7777, 20775}, {7806, 40981}, {8266, 37688}, {8842, 17938}, {9463, 11173}, {10311, 60694}, {11163, 66886}, {11631, 17414}, {12220, 15355}, {15652, 52674}, {25046, 46818}, {25332, 56430}, {38998, 62578}, {40601, 64646}, {46127, 52692}, {53699, 53929}, {53708, 67760}, {56390, 64927}, {56393, 58849}, {56437, 61101}, {59226, 59229}

X(68706) = circumcircle-inverse of X(9832)
X(68706) = isogonal conjugate of the isotomic conjugate of X(39266)
X(68706) = X(60072)-Ceva conjugate of X(6)
X(68706) = X(2021)-Dao conjugate of X(15993)
X(68706) = crossdifference of every pair of points on line {647, 3117}
X(68706) = barycentric product X(6)*X(39266)
X(68706) = barycentric quotient X(39266)/X(76)
X(68706) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 23, 237}, {2, 14957, 858}, {2, 34098, 1995}, {23, 37183, 22}, {25, 6660, 23}, {468, 21531, 2}, {1113, 1114, 9832}, {5004, 5005, 11676}, {5201, 53264, 22329}, {5999, 15915, 47620}, {8352, 37927, 56957}, {14041, 37896, 46522}, {34008, 34009, 37916}, {46600, 46601, 37184}, {47620, 68695, 15915}

X(68707) = EULER LINE INTERCEPT OF X(6)X(924)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^6*c^2 - 4*a^4*b^2*c^2 + 4*a^2*b^4*c^2 - 3*b^6*c^2 - 2*a^4*c^4 + 4*a^2*b^2*c^4 + 2*b^4*c^4 + a^2*c^6 - 3*b^2*c^6) : :

X(68707) lies on these lines: {2, 3}, {6, 924}, {136, 51389}, {246, 1112}, {264, 65975}, {394, 31848}, {1624, 51431}, {1634, 8754}, {2088, 33843}, {2782, 2970}, {2971, 9155}, {3563, 54439}, {5186, 47202}, {6090, 6787}, {9723, 36891}, {11433, 18347}, {12828, 15357}, {17994, 46130}, {20775, 63549}, {23181, 23698}, {32761, 61207}, {34157, 59211}, {36822, 68624}, {47200, 53273}, {53793, 65586}, {57493, 58083}

X(68707) = crossdifference of every pair of points on line {647, 3564}
X(68707) = barycentric product X(6331)*X(56394)
X(68707) = barycentric quotient X(56394)/X(647)
X(68707) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 4230, 25}, {460, 15143, 44896}, {1316, 56961, 44889}, {1316, 56962, 56961}, {5112, 21177, 44890}, {56957, 56961, 1316}, {56957, 56962, 44889}

X(68708) = EULER LINE INTERCEPT OF X(32)X(30830)

Barycentrics    (a^2 - b*c)*(a^3 + a^2*b + a^2*c + a*b*c - b^2*c - b*c^2) : :

X(68708) lies on these lines: {2, 3}, {32, 30830}, {36, 25510}, {37, 41258}, {81, 1655}, {86, 53476}, {100, 26048}, {190, 65741}, {238, 239}, {274, 24271}, {312, 4426}, {333, 21024}, {385, 3948}, {595, 41232}, {894, 25371}, {904, 14823}, {1030, 27111}, {1078, 30819}, {1211, 33954}, {1333, 25660}, {1429, 39930}, {1580, 4368}, {1914, 3975}, {1975, 24621}, {1999, 5247}, {3912, 38456}, {4393, 16466}, {5337, 18140}, {6002, 24560}, {6645, 16826}, {6703, 23905}, {6707, 59631}, {7782, 31234}, {7783, 24598}, {7793, 30863}, {7816, 31198}, {7839, 31036}, {16800, 21080}, {17027, 18756}, {17128, 20913}, {17129, 31060}, {17277, 56953}, {17367, 66644}, {17397, 25496}, {17499, 60721}, {17789, 24358}, {23947, 35466}, {24514, 54419}, {24549, 27184}, {26042, 54285}, {26110, 63158}, {26147, 37796}, {26979, 56934}, {27064, 41239}, {27523, 37652}, {27911, 27944}, {27913, 27941}, {29456, 52680}, {30568, 54329}, {30939, 44352}, {33955, 50178}, {41839, 54416}, {49545, 59517}

X(68708) = X(1284)-Ceva conjugate of X(385)
X(68708) = X(i)-isoconjugate of X(j) for these (i,j): {875, 54986}, {876, 6010}, {18268, 43677}
X(68708) = X(i)-Dao conjugate of X(j) for these (i,j): {16613, 60577}, {35068, 43677}
X(68708) = crossdifference of every pair of points on line {647, 3572}
X(68708) = barycentric product X(i)*X(j) for these {i,j}: {239, 1999}, {350, 5247}, {1447, 56311}, {3570, 6002}, {33295, 63800}
X(68708) = barycentric quotient X(i)/X(j) for these {i,j}: {740, 43677}, {1999, 335}, {3570, 54986}, {5247, 291}, {6002, 4444}, {56311, 4518}, {63800, 43534}
X(68708) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 405, 19224}, {2, 3552, 11329}, {2, 6655, 37096}, {2, 11320, 384}, {2, 17685, 26601}, {2, 17692, 21511}, {2, 17693, 25946}, {2, 27907, 27933}, {2, 31015, 63795}, {2, 33824, 33736}, {3, 11353, 2}, {405, 19269, 11110}, {405, 37415, 16048}, {405, 41236, 2}, {16046, 25946, 17693}, {16287, 25520, 7824}, {33821, 37086, 2}

X(68709) = EULER LINE INTERCEPT OF X(55)X(65543)

Barycentrics    a*(a*b + b^2 + a*c + c^2)*(a^4 - a^2*b^2 - a^2*b*c - 2*a*b^2*c - b^3*c - a^2*c^2 - 2*a*b*c^2 - 2*b^2*c^2 - b*c^3) : :

X(68709) lies on these lines: {2, 3}, {55, 65543}, {56, 6703}, {283, 68488}, {386, 35623}, {846, 978}, {958, 5230}, {960, 1193}, {992, 4261}, {993, 6693}, {1001, 41877}, {1211, 4267}, {1284, 4657}, {1621, 29984}, {1837, 8240}, {2234, 15254}, {3683, 28275}, {4276, 24931}, {4357, 22345}, {4425, 21616}, {4447, 4682}, {4512, 35657}, {4679, 28271}, {5087, 28268}, {5260, 36926}, {9791, 41828}, {10461, 40952}, {10826, 30366}, {11031, 44547}, {11043, 37737}, {12019, 12746}, {12567, 27628}, {12683, 68005}, {17185, 22076}, {17740, 56313}, {17751, 37730}, {20891, 49512}, {24987, 39566}, {27659, 31424}, {28246, 59207}, {30285, 45770}, {30362, 37692}, {35293, 35675}

X(68709) = crossdifference of every pair of points on line {647, 62749}
X(68709) = barycentric product X(i)*X(j) for these {i,j}: {1193, 19810}, {3687, 54339}
X(68709) = barycentric quotient X(i)/X(j) for these {i,j}: {19810, 1240}, {54339, 64984}
X(68709) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 37255}, {2, 21, 37225}, {2, 4189, 37030}, {3, 28258, 28260}, {3, 37360, 28381}, {21, 5051, 9840}, {21, 26117, 855}, {21, 56778, 8731}, {18235, 68608, 2292}, {19260, 52258, 13724}, {19519, 50199, 37462}

X(68710) = EULER LINE INTERCEPT OF X(171)X(172)

Barycentrics    a*(a^2 + b*c)*(a^3*b^2 - a*b^4 - a^2*b^2*c - 2*a*b^3*c - b^4*c + a^3*c^2 - a^2*b*c^2 - 2*a*b^2*c^2 - b^3*c^2 - 2*a*b*c^3 - b^2*c^3 - a*c^4 - b*c^4) : :

X(68710) lies on these lines: {2, 3}, {171, 172}, {187, 56010}, {256, 1740}, {846, 3229}, {980, 35623}, {1284, 50302}, {2178, 8424}, {2198, 53129}, {2292, 3009}, {2311, 15984}, {4459, 27697}, {5106, 45705}, {8844, 41269}, {18758, 43223}, {28369, 40731}, {40937, 56679}

X(68710) = crossdifference of every pair of points on line {647, 25537}
X(68710) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 33745, 1009}, {21, 37237, 13723}, {11358, 37327, 4220}

X(68711) = EULER LINE INTERCEPT OF X(48)X(63)

Barycentrics    a*(a + b)*(a + c)*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 4*a^2*b*c - 2*a^2*c^2 - 2*b^2*c^2 + c^4) : :

X(68711) lies on these lines: {2, 3}, {48, 63}, {58, 54369}, {81, 1214}, {86, 6349}, {283, 64082}, {284, 1708}, {306, 1792}, {333, 6350}, {394, 23602}, {1014, 7011}, {1451, 5256}, {1778, 18591}, {1805, 13389}, {1806, 13388}, {3736, 22057}, {4288, 10319}, {4640, 10536}, {5279, 42700}, {5323, 22341}, {5333, 17073}, {6360, 56014}, {8822, 31631}, {16577, 62700}, {17011, 37565}, {17167, 18589}, {17185, 30675}, {22119, 61409}, {23171, 62800}, {40571, 55873}, {40592, 62777}, {41340, 41723}, {41610, 46885}, {54323, 62810}

X(68711) = isotomic conjugate of the polar conjugate of X(62843)
X(68711) = X(i)-isoconjugate of X(j) for these (i,j): {37, 55105}, {213, 55106}, {228, 55107}, {656, 58965}, {2501, 58992}
X(68711) = X(i)-Dao conjugate of X(j) for these (i,j): {6626, 55106}, {31653, 523}, {40589, 55105}, {40596, 58965}, {49183, 1826}
X(68711) = barycentric product X(i)*X(j) for these {i,j}: {69, 62843}, {81, 26872}, {86, 55104}, {314, 19349}, {326, 37383}, {332, 37550}, {662, 60494}, {1444, 3085}, {3553, 17206}
X(68711) = barycentric quotient X(i)/X(j) for these {i,j}: {27, 55107}, {58, 55105}, {86, 55106}, {112, 58965}, {3085, 41013}, {3553, 1826}, {4575, 58992}, {18909, 17869}, {19349, 65}, {26872, 321}, {37383, 158}, {37550, 225}, {55104, 10}, {60494, 1577}, {62843, 4}
X(68711) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20, 37181}, {3, 21482, 464}, {21, 1817, 27}, {21, 17518, 37228}, {1006, 37263, 2}, {1214, 2193, 81}, {8021, 36017, 4233}, {11347, 37227, 14014}

X(68712) = EULER LINE INTERCEPT OF X(1)X(24632)

Barycentrics    (a + b)*(a + c)*(a^3 - a^2*b - a*b^2 - b^3 - a^2*c - 2*a*b*c - b^2*c - a*c^2 - b*c^2 - c^3) : :

X(68712) lies on these lines: {1, 24632}, {2, 3}, {45, 30906}, {58, 17023}, {81, 2221}, {86, 1333}, {239, 257}, {284, 4357}, {306, 37573}, {332, 38814}, {894, 8822}, {1043, 3661}, {1211, 18755}, {1778, 3618}, {1801, 26006}, {2271, 5739}, {2287, 17257}, {2303, 17321}, {2352, 23407}, {3219, 41249}, {3285, 17325}, {3672, 56019}, {3912, 4653}, {3948, 26244}, {4026, 17798}, {4261, 17277}, {4273, 4643}, {4281, 5256}, {4393, 56018}, {4877, 17353}, {5235, 17740}, {5327, 24723}, {5337, 25499}, {6646, 56020}, {6703, 59625}, {16704, 17014}, {16826, 37539}, {16834, 64072}, {16887, 60721}, {17189, 17304}, {17254, 66450}, {17260, 62707}, {17292, 52352}, {17316, 64415}, {20172, 27164}, {23151, 40571}, {24271, 52538}, {25507, 29609}, {25526, 29603}, {29598, 52680}, {50129, 66212}, {55968, 56332}

X(68712) = crossdifference of every pair of points on line {647, 7234}
X(68712) = barycentric product X(86)*X(50295)
X(68712) = barycentric quotient X(50295)/X(10)
X(68712) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 21, 16050}, {2, 4201, 37096}, {2, 11320, 13740}, {2, 14953, 26643}, {2, 16865, 11342}, {2, 17676, 33736}, {2, 21997, 16054}, {2, 22267, 11329}, {2, 26117, 26601}, {2, 26830, 37095}, {2, 56769, 16349}, {27, 16054, 37233}, {379, 16342, 2}, {1333, 4657, 86}, {1817, 37442, 11329}, {11110, 16054, 2}, {11110, 35916, 1010}, {13725, 24609, 2}, {16350, 16368, 2}, {16927, 24610, 2}, {19270, 37086, 2}, {24580, 37314, 2}

X(68713) = EULER LINE INTERCEPT OF X(9)X(81)

Barycentrics    a*(a + b)*(a + c)*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 2*a*b*c - 7*b^2*c - a*c^2 - 7*b*c^2 - c^3) : :

X(68713) lies on these lines: {2, 3}, {6, 44306}, {9, 81}, {57, 4877}, {58, 17022}, {72, 17019}, {86, 329}, {226, 1014}, {284, 7308}, {306, 5260}, {333, 344}, {940, 1778}, {1255, 22021}, {1444, 25507}, {1621, 3886}, {1728, 64420}, {1790, 24557}, {2287, 3305}, {2303, 44307}, {2328, 66515}, {2360, 3646}, {2893, 41809}, {2999, 4653}, {3488, 4720}, {4656, 17189}, {4776, 57246}, {5235, 33157}, {5256, 5436}, {5259, 40940}, {5358, 29604}, {5777, 64393}, {8025, 56020}, {8822, 9776}, {12555, 17185}, {12572, 25526}, {17825, 46889}, {19753, 19767}, {27065, 40571}, {27164, 41260}, {28619, 67850}, {41839, 56019}, {47787, 65575}, {47790, 57093}, {50292, 66212}, {55104, 68031}, {55432, 56001}, {64004, 64400}

X(68713) = crossdifference of every pair of points on line {647, 4822}
X(68713) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 21, 1817}, {2, 16865, 16368}, {2, 31049, 469}, {2, 37312, 17531}, {21, 14005, 37402}, {28, 37322, 21}, {405, 47512, 21}, {11108, 21483, 2}, {11108, 37322, 28}, {16413, 16842, 2}, {16845, 25516, 17557}, {37276, 52891, 27}

X(68714) = EULER LINE INTERCEPT OF X(81)X(88)

Barycentrics    a*(a + b)*(a + c)*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 2*a*b*c + 3*b^2*c - a*c^2 + 3*b*c^2 - c^3) : :

X(68714) lies on these lines: {2, 3}, {57, 40571}, {81, 88}, {229, 1792}, {284, 3306}, {1014, 16704}, {1333, 16610}, {1444, 5235}, {1778, 37680}, {2287, 3218}, {2303, 4850}, {3187, 62837}, {3687, 52375}, {4292, 27412}, {4658, 5256}, {5165, 32911}, {5253, 29833}, {5271, 8666}, {5323, 62620}, {5333, 17173}, {5905, 27398}, {8822, 31018}, {16586, 62700}, {17013, 64377}, {17021, 54387}, {17384, 63158}, {17484, 58786}, {17495, 56019}, {31623, 55995}, {40592, 64425}, {52680, 54390}

X(68714) = X(38962)-Dao conjugate of X(523)
X(68714) = crossdifference of every pair of points on line {647, 4730}
X(68714) = barycentric product X(86)*X(54286)
X(68714) = barycentric quotient X(54286)/X(10)
X(68714) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1817, 27174}, {2, 4188, 21488}, {2, 26783, 30841}, {474, 52012, 21}, {851, 19329, 4239}, {4188, 21488, 37312}, {4273, 37520, 81}, {15771, 15772, 16049}, {33325, 35983, 21}, {35991, 37442, 16049}

X(68715) = EULER LINE INTERCEPT OF X(1)X(18750)

Barycentrics    (a + b)*(a - b - c)^2*(a + c)*(a^2 + b^2 - 2*b*c + c^2) : :

X(68715) lies on these lines: {1, 18750}, {2, 3}, {58, 11019}, {81, 10580}, {86, 269}, {200, 341}, {283, 1067}, {284, 40869}, {333, 643}, {497, 5324}, {614, 3673}, {1040, 40987}, {1944, 10391}, {3732, 17441}, {4313, 27410}, {4357, 4872}, {4423, 25521}, {4512, 5263}, {4640, 60705}, {4653, 13405}, {4666, 54356}, {5208, 10025}, {5327, 18165}, {6554, 30706}, {6740, 51564}, {6745, 52352}, {7675, 27413}, {8822, 9436}, {9535, 17810}, {10382, 27411}, {10383, 27384}, {10394, 28950}, {10453, 23151}, {10578, 64415}, {11220, 26651}, {18228, 27394}, {18651, 41012}, {24552, 52653}, {31146, 41629}, {31623, 56098}, {36845, 56018}, {36850, 68599}, {37658, 46889}, {44698, 65583}, {51351, 58786}, {54107, 62811}, {59646, 65671}, {65670, 65675}

X(68715) = X(799)-Ceva conjugate of X(1021)
X(68715) = X(i)-isoconjugate of X(j) for these (i,j): {42, 56359}, {65, 1037}, {73, 1041}, {213, 30705}, {512, 8269}, {647, 66952}, {656, 59128}, {1042, 56179}, {1334, 63178}, {1400, 7131}, {1402, 8817}, {1407, 56260}, {1427, 7123}, {2281, 8816}, {3668, 7084}, {7250, 52778}, {37755, 57386}, {53551, 59133}, {56243, 62192}
X(68715) = X(i)-Dao conjugate of X(j) for these (i,j): {1040, 41539}, {4000, 10}, {6554, 3668}, {6626, 30705}, {14936, 661}, {15487, 1427}, {16583, 6356}, {18589, 1254}, {24771, 56260}, {39052, 66952}, {39054, 8269}, {40582, 7131}, {40592, 56359}, {40596, 59128}, {40602, 1037}, {40605, 8817}, {59619, 1441}
X(68715) = cevapoint of X(i) and X(j) for these (i,j): {497, 1040}, {4319, 6554}
X(68715) = crosspoint of X(86) and X(1043)
X(68715) = crosssum of X(42) and X(1042)
X(68715) = barycentric product X(i)*X(j) for these {i,j}: {29, 27509}, {86, 6554}, {200, 16750}, {274, 4319}, {310, 30706}, {312, 5324}, {314, 2082}, {333, 497}, {799, 17115}, {1040, 31623}, {1043, 4000}, {1098, 53510}, {1434, 4012}, {1863, 17206}, {2287, 3673}, {2322, 17170}, {2326, 20235}, {3732, 7253}, {3914, 7058}, {4211, 52406}, {7083, 28660}, {7124, 44130}, {7256, 48398}, {16713, 64438}, {18589, 59482}, {28070, 57785}, {40965, 52379}, {41629, 62543}
X(68715) = barycentric quotient X(i)/X(j) for these {i,j}: {21, 7131}, {81, 56359}, {86, 30705}, {112, 59128}, {162, 66952}, {200, 56260}, {284, 1037}, {333, 8817}, {497, 226}, {614, 1427}, {662, 8269}, {1010, 8816}, {1014, 63178}, {1040, 1214}, {1043, 30701}, {1098, 40403}, {1172, 1041}, {1473, 52373}, {1633, 1020}, {1863, 1826}, {2082, 65}, {2287, 56179}, {2328, 7123}, {3673, 1446}, {3732, 4566}, {3914, 6354}, {4000, 3668}, {4012, 2321}, {4211, 1435}, {4319, 37}, {5324, 57}, {6554, 10}, {7083, 1400}, {7124, 73}, {7253, 48070}, {7258, 54967}, {7259, 52778}, {7289, 1439}, {16502, 1042}, {16583, 1254}, {16750, 1088}, {17115, 661}, {17170, 56382}, {17441, 37755}, {18589, 6356}, {23620, 1425}, {27509, 307}, {28070, 210}, {30706, 42}, {40961, 7147}, {40965, 2171}, {40987, 1880}, {41629, 62538}, {56182, 56243}, {59482, 40411}, {62543, 4052}, {64438, 60229}
X(68715) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4195, 37059}, {21, 29, 1010}, {21, 17584, 29}, {21, 17586, 11103}, {21, 68693, 4234}, {25, 30943, 23512}, {440, 33302, 37443}, {452, 27378, 13740}, {452, 37054, 13725}, {2478, 27505, 16062}, {4228, 14956, 27}, {6817, 68699, 17682}, {11103, 17584, 17586}, {11103, 17586, 29}, {11344, 26091, 19270}, {13746, 17585, 29}, {17188, 17194, 86}, {20834, 37370, 7413}, {25494, 37193, 37086}

X(68716) = EULER LINE INTERCEPT OF X(35)X(1819)

Barycentrics    a^2*(a + b)*(a - b - c)*(a + c)*(a^4*b - 2*a^2*b^3 + b^5 + a^4*c + a^3*b*c - a^2*b^2*c - a*b^3*c - a^2*b*c^2 - 2*a*b^2*c^2 - b^3*c^2 - 2*a^2*c^3 - a*b*c^3 - b^2*c^3 + c^5) : :

X(68716) lies on these lines: {2, 3}, {35, 1819}, {57, 54356}, {58, 40958}, {60, 283}, {81, 51496}, {255, 56001}, {285, 1436}, {603, 54411}, {1259, 2287}, {1466, 3286}, {2360, 10902}, {4278, 15803}, {5248, 17188}, {5709, 41723}, {7331, 56934}, {8822, 52673}, {11507, 62843}, {17167, 64001}, {18180, 37623}, {37504, 46889}, {37564, 40980}, {40214, 68649}, {54320, 54407}

X(68716) = X(i)-isoconjugate of X(j) for these (i,j): {65, 54972}, {226, 2219}, {661, 68206}, {1402, 57911}
X(68716) = X(i)-Dao conjugate of X(j) for these (i,j): {36830, 68206}, {40602, 54972}, {40605, 57911}
X(68716) = crossdifference of every pair of points on line {647, 66287}
X(68716) = barycentric product X(i)*X(j) for these {i,j}: {21, 62857}, {86, 15830}, {333, 581}
X(68716) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 68206}, {284, 54972}, {333, 57911}, {581, 226}, {2194, 2219}, {15830, 10}, {62857, 1441}
X(68716) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 28, 4225}, {3, 1011, 37297}, {3, 8021, 21}, {3, 11347, 16451}, {3, 37052, 4216}, {3, 37264, 4210}, {21, 1816, 2}, {21, 1817, 37277}, {21, 35995, 1010}, {4215, 37259, 27652}

X(68717) = EULER LINE INTERCEPT OF X(8)X(16752)

Barycentrics    (a + b)*(a + c)*(a^2*b^2 + b^4 + a^2*b*c - 2*a*b^2*c + b^3*c + a^2*c^2 - 2*a*b*c^2 + b*c^3 + c^4) : :

X(68717) lies on these lines: {2, 3}, {8, 16752}, {10, 274}, {86, 4429}, {310, 4385}, {314, 1738}, {350, 23537}, {387, 30962}, {899, 30984}, {978, 30953}, {986, 4476}, {1043, 24378}, {1193, 30969}, {1330, 2238}, {1575, 16716}, {1582, 50302}, {1655, 50177}, {1834, 24366}, {2239, 27660}, {3662, 3786}, {3695, 17759}, {3736, 3836}, {3741, 24178}, {3823, 16696}, {3826, 27164}, {3932, 56023}, {4026, 25508}, {4368, 24851}, {4645, 27644}, {4658, 33955}, {4972, 5333}, {5235, 24988}, {5295, 31027}, {5300, 63818}, {6327, 27643}, {8747, 40411}, {10449, 30945}, {10455, 38052}, {10458, 25961}, {13161, 31008}, {18600, 39570}, {19767, 31006}, {20929, 40973}, {24169, 35623}, {24514, 57282}, {25492, 30959}, {25507, 32773}, {25526, 29633}, {28620, 48822}, {29679, 30599}, {30941, 56018}, {40721, 49743}, {40908, 50153}, {43531, 56167}

X(68717) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 377, 1008}, {2, 4201, 16850}, {2, 52245, 13740}, {8728, 37148, 2}, {14005, 26643, 1010}

X(68718) = EULER LINE INTERCEPT OF X(6)X(2482)

Barycentrics    11*a^4 - 5*a^2*b^2 + 2*b^4 - 5*a^2*c^2 + 4*b^2*c^2 + 2*c^4 : :
X(68718) = X[2] - 3 X[33191], 4 X[2] - 3 X[33240], 5 X[2] - 3 X[33285], 4 X[33191] - X[33240], 5 X[33191] - X[33285], 5 X[33240] - 4 X[33285]

X(68718) lies on these lines: {2, 3}, {6, 2482}, {32, 12151}, {39, 51185}, {141, 8182}, {183, 26613}, {187, 599}, {230, 40727}, {524, 1384}, {574, 47352}, {590, 54627}, {597, 5024}, {598, 64019}, {615, 54628}, {620, 11184}, {671, 11164}, {1975, 11054}, {1992, 6390}, {3053, 7801}, {3618, 11147}, {3734, 7610}, {3763, 8588}, {3785, 50994}, {3793, 11160}, {3849, 7778}, {3926, 63064}, {3933, 50992}, {3972, 11163}, {5023, 7810}, {5026, 19911}, {5182, 11842}, {5206, 51186}, {5210, 7820}, {5215, 37637}, {5304, 9741}, {5306, 51122}, {5309, 15300}, {5461, 66587}, {5475, 9167}, {5485, 37689}, {5569, 15271}, {5642, 14653}, {6090, 52231}, {6337, 43136}, {6680, 34504}, {7615, 44401}, {7617, 58448}, {7620, 43291}, {7622, 7804}, {7735, 52229}, {7736, 12040}, {7737, 22110}, {7758, 63115}, {7767, 50990}, {7776, 7870}, {7789, 22165}, {7794, 51189}, {7795, 50991}, {7800, 51143}, {7806, 8591}, {7816, 34505}, {7835, 51224}, {7844, 32479}, {7863, 22331}, {7868, 55164}, {7880, 66455}, {7881, 9939}, {7891, 34604}, {7908, 63942}, {7913, 44541}, {8176, 22247}, {8584, 30435}, {8589, 47355}, {8593, 11156}, {8716, 36521}, {8724, 18800}, {9155, 37811}, {9169, 47077}, {9605, 63124}, {9740, 46453}, {9755, 64090}, {9770, 18907}, {10168, 52771}, {11148, 63097}, {11149, 55801}, {11168, 21843}, {11173, 41146}, {11178, 47113}, {12154, 42975}, {12155, 42974}, {14666, 47200}, {14830, 18440}, {15048, 53142}, {16279, 46987}, {16509, 62992}, {18860, 54131}, {19662, 47353}, {20112, 44381}, {22246, 63127}, {22253, 63065}, {22486, 32447}, {23055, 64093}, {23334, 37690}, {23967, 34360}, {29573, 37589}, {31489, 50571}, {31859, 52695}, {32456, 66616}, {32532, 63536}, {32815, 63107}, {34897, 40995}, {35007, 51188}, {36213, 50672}, {37512, 51584}, {37746, 61506}, {41139, 43620}, {44377, 66466}, {46958, 61507}, {50280, 65630}, {51010, 52021}, {51013, 52022}, {51123, 63006}, {59780, 63029}, {62993, 63647}, {67585, 67717}

X(68718) = reflection of X(i) in X(j) for these {i,j}: {381, 37071}, {1384, 37809}
X(68718) = orthocentroidal-circle-inverse of X(8355)
X(68718) = crossdifference of every pair of points on line {647, 6088}
X(68718) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 8355}, {2, 1003, 11159}, {2, 3363, 5055}, {2, 3552, 9855}, {2, 8352, 11318}, {2, 8598, 5077}, {2, 9855, 7841}, {2, 11001, 66394}, {2, 11159, 381}, {2, 13586, 35955}, {2, 14033, 3363}, {2, 19708, 8358}, {2, 27088, 3}, {2, 32985, 27088}, {2, 33007, 8352}, {2, 33187, 52942}, {2, 33255, 35954}, {2, 35287, 47061}, {2, 35954, 11286}, {2, 35955, 11287}, {2, 40246, 5025}, {2, 47061, 8359}, {2, 52942, 33228}, {2, 52943, 33251}, {2, 66391, 3830}, {2, 66407, 33291}, {2, 66409, 19709}, {3, 8369, 33237}, {6, 2482, 11165}, {20, 33197, 8360}, {32, 39785, 15534}, {439, 7819, 3}, {550, 8365, 33190}, {597, 7618, 5024}, {597, 32459, 7618}, {1003, 11288, 381}, {1003, 33191, 37071}, {1003, 33246, 11288}, {1657, 32954, 33241}, {1992, 19661, 21309}, {3053, 7801, 63950}, {3552, 32954, 1657}, {3972, 41134, 11163}, {5077, 8598, 3534}, {6390, 19661, 1992}, {6656, 68528, 62100}, {7622, 7804, 42849}, {7807, 8352, 2}, {7807, 33007, 11318}, {7807, 68513, 382}, {7833, 8366, 7866}, {7833, 33225, 8366}, {7866, 33235, 15696}, {7901, 66421, 7841}, {8359, 8369, 14001}, {8359, 35287, 3}, {8361, 33239, 5073}, {8366, 33235, 7833}, {8369, 27088, 2}, {8369, 32985, 3}, {8369, 33237, 33242}, {11149, 60855, 55801}, {11159, 11288, 2}, {11286, 35297, 5054}, {11287, 13586, 15688}, {11291, 11292, 15022}, {11295, 35303, 60660}, {11296, 35304, 60661}, {11318, 33007, 382}, {11318, 68513, 33007}, {13586, 33220, 11287}, {14001, 32985, 35287}, {14001, 35287, 8359}, {14001, 47061, 2}, {14063, 33007, 66425}, {15686, 33213, 33210}, {16043, 33227, 3}, {16368, 16436, 21509}, {16431, 16436, 11340}, {16925, 68527, 1656}, {21509, 21539, 37269}, {32952, 50693, 66347}, {32970, 68177, 3851}, {32973, 32985, 8369}, {32985, 33215, 439}, {33181, 33190, 8365}, {33184, 35927, 15681}, {33211, 44245, 33025}, {33216, 66415, 15694}, {33220, 35955, 2}, {33224, 35927, 33184}, {33225, 33235, 7866}, {33255, 35297, 11286}, {35297, 35954, 2}, {35303, 35304, 376}, {35305, 35306, 8370}, {37172, 37173, 3146}, {37340, 37341, 3090}, {44649, 46066, 2}, {47597, 56967, 381}

X(68719) = EULER LINE INTERCEPT OF X(6)X(1239)

Barycentrics    a^6 + 2*a^4*b^2 + a^2*b^4 + 2*a^4*c^2 + 2*a^2*b^2*c^2 + 2*b^4*c^2 + a^2*c^4 + 2*b^2*c^4 : :

X(68719) lies on these lines: {2, 3}, {6, 1239}, {32, 8891}, {76, 5359}, {83, 305}, {183, 1627}, {394, 46900}, {1180, 1975}, {1184, 39998}, {1194, 3734}, {1235, 3162}, {1369, 7879}, {1460, 26969}, {1611, 26235}, {1613, 24273}, {1799, 3972}, {3096, 16275}, {3108, 9462}, {3314, 8878}, {3329, 8264}, {4074, 42534}, {6563, 26225}, {7083, 27030}, {7754, 34482}, {7772, 19568}, {7822, 21248}, {7839, 40904}, {8877, 52756}, {11174, 18092}, {18018, 36879}, {20172, 29667}, {24256, 42295}, {24686, 25364}, {26687, 29679}, {30435, 31078}, {31088, 39951}, {40179, 52713}, {42052, 63938}, {57518, 60855}

X(68719) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 384, 22}, {2, 7391, 6656}, {2, 7409, 33180}, {2, 15246, 11285}, {2, 16920, 37325}, {2, 16924, 37990}, {2, 16932, 25}, {2, 16949, 3}, {2, 16950, 1995}, {2, 16951, 7485}, {2, 16952, 23}, {2, 16953, 6636}, {2, 26209, 26226}, {2, 32971, 6997}, {2, 37353, 7887}, {2, 63797, 7866}, {2, 68525, 16950}, {25, 11286, 16932}, {427, 7819, 2}, {5064, 7866, 63797}, {6636, 16953, 1003}, {7386, 16045, 2}, {7539, 32954, 2}, {7770, 11324, 2}, {19568, 37875, 7772}

X(68720) = EULER LINE INTERCEPT OF X(33)X(52793)

Barycentrics    (a^2+b^2-c^2)*(a^2-b^2+c^2)*(4*a^6-9*a^4*b^2+6*a^2*b^4-b^6-9*a^4*c^2+4*a^2*b^2*c^2+b^4*c^2+6*a^2*c^4+b^2*c^4-c^6) : :

As a point on the Euler line, X(68720) has Shinagawa coefficients: {5 f,-e-3 f}

See David Nguyen and Ercole Suppa, euclid 8548.

X(68720) lies on these lines: {2, 3}, {33, 52793}, {125, 34782}, {154, 26937}, {182, 41584}, {185, 10192}, {371, 13937}, {372, 13884}, {389, 10182}, {1112, 10625}, {1192, 61680}, {1204, 16252}, {1353, 9545}, {1495, 6247}, {1511, 63734}, {1620, 64024}, {1829, 10165}, {1862, 38760}, {1899, 17821}, {1902, 10164}, {2207, 21843}, {2883, 21663}, {2931, 23307}, {3043, 22251}, {3532, 64714}, {3580, 11449}, {3819, 68020}, {5085, 39871}, {5090, 31423}, {5185, 38772}, {5186, 38748}, {5206, 60428}, {5410, 13935}, {5411, 9540}, {5433, 52427}, {5642, 13148}, {5656, 34469}, {5889, 59553}, {5894, 51403}, {6000, 43903}, {6101, 52000}, {6146, 11202}, {6403, 15028}, {6684, 11363}, {6696, 15448}, {6746, 64854}, {7689, 51425}, {7749, 27376}, {8550, 64724}, {8739, 16772}, {8740, 16773}, {8901, 19185}, {8981, 10881}, {9707, 18914}, {9729, 13394}, {9940, 41609}, {10110, 46265}, {10193, 13474}, {10282, 44673}, {10312, 31406}, {10519, 19118}, {10539, 44158}, {10632, 42121}, {10633, 42124}, {10880, 13966}, {11064, 46730}, {11206, 58378}, {11245, 19357}, {11264, 32171}, {11381, 23328}, {11397, 51412}, {11425, 61506}, {11426, 61657}, {11430, 40240}, {11464, 26879}, {11468, 32111}, {11695, 47328}, {11745, 61743}, {12038, 41587}, {12131, 38737}, {12133, 38727}, {12135, 26446}, {12137, 38133}, {12138, 21154}, {12140, 34128}, {12143, 15819}, {12174, 18931}, {12241, 61645}, {12294, 21167}, {12359, 51393}, {13346, 32269}, {13367, 13567}, {13399, 44762}, {13568, 58434}, {14157, 43607}, {15059, 41482}, {15576, 51611}, {15644, 44084}, {15897, 67559}, {16621, 44082}

X(68720) = X(16624)-cross conjugate of X(4)
X(68720) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(5),X(45857)}}, {{A,B,C,X(20),X(45838)}}, {{A,B,C,X(382),X(46729)}}, {{A,B,C,X(3091),X(46952)}}, {{A,B,C,X(3543),X(43695)}}, {{A,B,C,X(3545),X(66597)}}, {{A,B,C,X(3832),X(14457)}}, {{A,B,C,X(3839),X(51032)}}, {{A,B,C,X(5056),X(5486)}}, {{A,B,C,X(6662),X(31283)}}, {{A,B,C,X(6676),X(53104)}}, {{A,B,C,X(7392),X(40102)}}, {{A,B,C,X(9909),X(60175)}}, {{A,B,C,X(10293),X(62036)}}, {{A,B,C,X(10565),X(60102)}}, {{A,B,C,X(11413),X(43713)}}, {{A,B,C,X(11585),X(34483)}}, {{A,B,C,X(12084),X(32902)}}, {{A,B,C,X(12085),X(44763)}}}, {{{A,B,C,X(15687),X(54895)}}, {{A,B,C,X(34484),X(57387)}}, {{A,B,C,X(44452),X(65088)}}
X(68720) = pole of line {5650, 21659} with respect to Thomson-Gibert-Moses hyperbola
X(68720) = pole of line {523, 13473} with respect to orthoptic circle of MacBeath inconic
X(68720) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3515, 3575}, {3, 468, 235}, {3, 3147, 468}, {3, 3542, 1885}, {4, 3526, 62958}, {4, 15717, 11410}, {4, 15750, 37931}, {5, 6240, 63662}, {5, 10018, 52297}, {5, 15331, 44249}, {24, 140, 427}, {140, 6756, 37119}, {186, 10018, 5}, {186, 14940, 6240}, {468, 1885, 3542}, {632, 1594, 52293}, {1595, 3518, 10301}, {1595, 14869, 37118}, {1596, 15712, 3520}, {1656, 18533, 23047}, {1658, 11585, 44239}, {1885, 3542, 235}, {3089, 3516, 62962}, {3089, 3524, 3516}, {3517, 3541, 428}, {3517, 5054, 3541}, {3522, 6622, 44438}, {3523, 6353, 1593}, {3525, 7487, 5094}, {3546, 9715, 7667}, {3580, 11449, 66762}, {6240, 10018, 14940}, {6240, 14940, 5}, {6644, 7542, 7399}, {6696, 15448, 26883}, {10020, 16531, 37814}, {10282, 44673, 67902}, {10295, 16868, 3627}, {11464, 26879, 31804}, {12100, 13488, 35477}, {15646, 15761, 44240}, {17506, 18560, 8703}

X(68721) = EULER LINE INTERCEPT OF X(1)X(3644)

Barycentrics    5*a^4 + 3*a^3*b + a^2*b^2 + 3*a*b^3 + 3*a^3*c + 5*a^2*b*c + 5*a*b^2*c + 3*b^3*c + a^2*c^2 + 5*a*b*c^2 + 6*b^2*c^2 + 3*a*c^3 + 3*b*c^3 : :

X(68721) lies on these lines: {1, 3644}, {2, 3}, {1104, 4739}, {1220, 8715}, {3626, 5247}, {4681, 7283}, {4686, 50054}, {5258, 5263}, {43531, 52352}

X(68721) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4195, 56992}, {20, 51670, 37036}, {405, 1010, 56991}, {405, 16394, 51674}, {405, 51674, 1010}, {405, 56991, 37035}, {964, 4234, 19270}, {1010, 4195, 13735}, {1010, 13735, 37035}, {1010, 37035, 51666}, {1010, X(68721) = 51595, 16458}, {4195, 16394, 1010}, {4195, 51674, 405}, {11319, 51669, 56766}, {13735, 56991, 405}, {16458, 56989, 51595}, {50242, 51590, 37164}, {51668, 56986, 33833}

X(68722) = EULER LINE INTERCEPT OF X(1)X(31238)

Barycentrics    a^4 + 4*a^3*b + 7*a^2*b^2 + 4*a*b^3 + 4*a^3*c + 18*a^2*b*c + 18*a*b^2*c + 4*b^3*c + 7*a^2*c^2 + 18*a*b*c^2 + 8*b^2*c^2 + 4*a*c^3 + 4*b*c^3 : :

X(68722) lies on these lines: {1, 31238}, {2, 3}, {10, 2334}, {956, 16828}, {1698, 37674}, {3624, 4255}, {3634, 37660}, {4687, 50044}, {5247, 64850}, {5290, 57663}, {5439, 19859}, {12513, 19871}, {15668, 28619}, {17259, 25526}, {19744, 37522}

X(68722) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 404, 16457}, {2, 443, 17514}, {2, 1010, 56993}, {2, 2049, 16842}, {2, 14005, 11108}, {2, 16454, 16844}, {2, 16458, 405}, {2, 17698, 50795}, {2, 19270, 19272}, {2, 37037, 17590}, {2, 37462, 50409}, {2, 51666, 51676}, {2, 51671, 51599}, {2, 56766, 16343}, {2, 56767, 474}, {2, 56985, 56996}, {2, 56988, 37035}, {2, 56991, 16458}, {405, 16458, 51602}, {443, 17514, 51677}, {1010, 37035, 56989}, {1010, 56989, 16394}, {1010, 56990, 56992}, {1010, 56993, 405}, {4195, 51676, 405}, {11108, 17524, 405}, {11110, 19290, 19535}, {11115, 11357, 19538}, {13725, 51671, 56997}, {16343, 56766, 16371}, {16394, 16458, 56988}, {16394, 37035, 405}, {16408, 19282, 16453}, {16453, 19282, 405}, {16454, 16844, 16370}, {16454, 17588, 19274}, {16458, 56993, 1010}, {16844, 19274, 17588}, {16862, 19534, 474}, {17588, 19274, 16370}, {17590, 51667, 37037}, {19286, 19309, 20833}, {19331, 56769, 19704}, {37035, 56988, 16394}, {51599, 56997, 13725}, {51601, 56996, 56985}, {56988, 56989, 1010}, {56990, 56992, 405}, {56992, 56993, 56990}

X(68723) = EULER LINE INTERCEPT OF X(1)X(4759)

Barycentrics    6*a^4 + a^3*b - 4*a^2*b^2 + a*b^3 + a^3*c - 7*a^2*b*c - 7*a*b^2*c + b^3*c - 4*a^2*c^2 - 7*a*b*c^2 + 2*b^2*c^2 + a*c^3 + b*c^3 : :

X(68723) lies on these lines: {1, 4759}, {2, 3}, {536, 19851}, {1104, 4664}, {1453, 29584}, {3241, 5247}, {3616, 32856}, {4304, 17338}, {4740, 7283}, {5302, 50075}, {5436, 50127}, {16485, 17261}, {16817, 50049}, {16824, 50126}, {50064, 51488}, {51055, 51715}

X(68723) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 13735, 4195}, {2, 16393, 56768}, {2, 50430, 51594}, {2, 51606, 51678}, {2, 51674, 51602}, {2, 56989, 13735}, {21, 19292, 4189}, {405, 4195, 56990}, {405, 13735, 2}, {405, 16394, 51595}, {405, 56989, 4195}, {405, 56992, 37035}, {1010, 51676, 2}, {4195, 56990, 56988}, {4234, 16857, 2}, {5047, 16393, 2}, {11346, 16858, 2}, {13735, 51595, 16394}, {13742, 51665, 2}, {16394, 51595, 2}, {16418, 33309, 2}, {16865, 17697, 56769}, {17547, 19336, 2}, {37035, 51602, 2}, {37035, 56992, 51674}, {37036, 51596, 2}, {48816, 50202, 2}, {50430, 51673, 2}, {51597, 51603, 2}, {51605, 51679, 2}, {51674, 56992, 4195}

X(68724) = EULER LINE INTERCEPT OF X(1)X(4740)

Barycentrics    6*a^4 + 5*a^3*b + 4*a^2*b^2 + 5*a*b^3 + 5*a^3*c + 13*a^2*b*c + 13*a*b^2*c + 5*b^3*c + 4*a^2*c^2 + 13*a*b*c^2 + 10*b^2*c^2 + 5*a*c^3 + 5*b*c^3 : :

X(68724) lies on these lines: {1, 4740}, {2, 3}, {8, 62846}, {75, 50064}, {192, 50049}, {1278, 50072}, {4664, 50054}, {4688, 19851}, {5247, 53620}, {19797, 50070}, {19808, 50046}, {20018, 46922}, {33954, 63110}

X(68724) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 11111, 51594}, {2, 16394, 4195}, {2, 16397, 19278}, {2, 50055, 37164}, {2, 51674, 16394}, {2, 51678, 51681}, {2, 56989, 51595}, {405, 51666, 2}, {1010, 4195, 56988}, {1010, 13735, 51602}, {1010, 16394, 2}, {1010, 51674, 4195}, {4195, 56988, 56990}, {4234, 19277, 2}, {11112, 51605, 2}, {13735, 51602, 2}, {13745, 51604, 2}, {16394, 51602, 13735}, {16458, 51595, 2}, {17678, 51590, 2}, {19332, 33309, 2}, {48816, 50059, 2}, {51597, 51601, 2}

X(68725) = EULER LINE INTERCEPT OF X(1)X(49513)

Barycentrics    7*a^4 + 2*a^3*b - 3*a^2*b^2 + 2*a*b^3 + 2*a^3*c - 4*a^2*b*c - 4*a*b^2*c + 2*b^3*c - 3*a^2*c^2 - 4*a*b*c^2 + 4*b^2*c^2 + 2*a*c^3 + 2*b*c^3 : :

X(68725) lies on these lines: {1, 49513}, {2, 3}, {1104, 4718}, {3633, 5247}, {4726, 50044}, {4764, 7283}, {9655, 24542}, {20041, 41453}, {24956, 65630}

X(68725) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {405, 1010, 51676}, {405, 4195, 16458}, {405, 13735, 56992}, {405, 16394, 56993}, {405, 51602, 56990}, {405, 56992, 16394}, {4195, 16458, 16394}, {4234, 16842, 19331}, {11106, 51673, 13728}, {11354, 16865, 16343}, {13735, 56989, 405}, {13740, 19526, 16351}, {13742, 51606, 57002}, {16408, 17539, 16401}, {16458, 56992, 4195}, {17526, 50241, 50056}

X(68726) = EULER LINE INTERCEPT OF X(1)X(25269)

Barycentrics    8*a^4 + 3*a^3*b - 2*a^2*b^2 + 3*a*b^3 + 3*a^3*c - a^2*b*c - a*b^2*c + 3*b^3*c - 2*a^2*c^2 - a*b*c^2 + 6*b^2*c^2 + 3*a*c^3 + 3*b*c^3 : :

X(68726) lies on these lines: {1, 25269}, {2, 3}, {145, 2308}, {902, 3617}, {1043, 63050}, {1104, 1278}, {3621, 5247}, {4772, 50054}, {4788, 7283}, {4821, 19851}, {11523, 17350}, {12513, 36635}, {17232, 64159}, {24549, 45789}

X(68726) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {405, 4195, 51674}, {405, 16394, 56991}, {405, 51674, 2}, {405, 56991, 56990}, {4195, 13735, 56989}, {4195, 56989, 2}, {4195, 56990, 16394}, {13735, 56992, 4195}, {37036, 57003, 51681}, {51674, 56989, 405}

X(68727) = EULER LINE INTERCEPT OF X(1453)X(29578)

Barycentrics    2*a^4 - 3*a^3*b - 8*a^2*b^2 - 3*a*b^3 - 3*a^3*c - 19*a^2*b*c - 19*a*b^2*c - 3*b^3*c - 8*a^2*c^2 - 19*a*b*c^2 - 6*b^2*c^2 - 3*a*c^3 - 3*b*c^3 : :

X(68727) lies on these lines: {2, 3}, {1453, 29578}, {2177, 9780}, {3616, 61358}, {4687, 19851}, {4699, 54287}, {5247, 5550}, {14997, 46934}, {16817, 27268}, {17259, 20018}, {17358, 19857}, {28619, 63108}

X(68727) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 405, 56988}, {2, 17544, 17589}, {2, 37035, 56990}, {2, 51674, 56991}, {2, 56989, 16458}, {2, 56990, 4195}, {405, 56988, 4195}, {405, 56991, 51674}, {1010, 37035, 51676}, {16458, 51595, 56989}, {17534, 19334, 2}, {17590, 37039, 2}, {37035, 56993, 2}, {51674, 56991, 56988}, {56988, 56990, 405}

X(68728) = EULER LINE INTERCEPT OF X(4751)X(19851)

Barycentrics    2*a^4 + 5*a^3*b + 8*a^2*b^2 + 5*a*b^3 + 5*a^3*c + 21*a^2*b*c + 21*a*b^2*c + 5*b^3*c + 8*a^2*c^2 + 21*a*b*c^2 + 10*b^2*c^2 + 5*a*c^3 + 5*b*c^3 : :

X(68728) lies on these lines: {2, 3}, {4751, 19851}, {5247, 17124}, {5288, 19853}, {5372, 46932}, {9534, 28619}, {9780, 37684}, {15668, 20018}, {17349, 25526}

X(68728) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1010, 56990}, {2, 16454, 56769}, {2, 16458, 56988}, {2, 17589, 17697}, {2, 51674, 37035}, {2, 56988, 4195}, {2, 56989, 56993}, {405, 16458, 51666}, {1010, 37035, 56992}, {1010, 56990, 4195}, {1010, 56992, 51674}, {1010, 56993, 56989}, {14007, 56767, 2}, {16456, 56766, 2}, {16458, 56991, 2}, {37035, 51602, 51674}, {51602, 56992, 1010}, {56988, 56990, 1010}, {56989, 56993, 56990}

X(68729) = BROCARD AXIS INTERCEPT OF X(9)X(22071)

Barycentrics    a^2*(a - b - c)*(a^3*b^2 + a^2*b^3 - a*b^4 - b^5 - a^2*b^2*c + b^4*c + a^3*c^2 - a^2*b*c^2 + a^2*c^3 - a*c^4 + b*c^4 - c^5) : :

X(68729) lies on these lines: {3, 6}, {9, 22071}, {37, 1939}, {101, 56911}, {517, 8607}, {522, 650}, {1415, 22123}, {1731, 14936}, {2077, 47434}, {2183, 13006}, {2288, 54427}, {2323, 7117}, {3208, 7105}, {3882, 36212}, {3959, 17053}, {5277, 62700}, {25059, 63078}, {26932, 45270}, {27379, 27395}, {27505, 27522}, {35069, 52949}, {47432, 58326}

X(68729) = crosssum of X(6) and X(855)
X(68729) = crossdifference of every pair of points on line {56, 523}
X(68729) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2183, 22059, 13006}, {2245, 3003, 3002}

X(68730) = BROCARD AXIS INTERCEPT OF X(8)X(29)

Barycentrics    a^2*(a + b)*(a - b - c)*(a + c)*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3) : :

X(68730) lies on these lines: {3, 6}, {8, 29}, {9, 16699}, {19, 22134}, {21, 55432}, {27, 34048}, {28, 7078}, {30, 3330}, {59, 7115}, {81, 5435}, {112, 2745}, {212, 8021}, {218, 62691}, {222, 1817}, {391, 26091}, {517, 14571}, {521, 650}, {651, 14953}, {859, 2183}, {965, 7532}, {966, 25490}, {1062, 66745}, {1449, 37523}, {1465, 62402}, {1474, 22132}, {1708, 18603}, {1743, 40979}, {1816, 56000}, {1958, 23124}, {2173, 61197}, {2194, 61397}, {2206, 61357}, {2238, 37370}, {2267, 17524}, {2269, 21798}, {2300, 41015}, {2303, 41344}, {2328, 7074}, {2332, 22131}, {2341, 17796}, {2360, 64069}, {2427, 21801}, {2999, 18163}, {3008, 17197}, {3157, 52012}, {3240, 35981}, {3293, 21860}, {3686, 34831}, {3990, 46884}, {4559, 16548}, {5778, 7524}, {7436, 63436}, {7677, 54358}, {11436, 66743}, {15149, 36949}, {16704, 34234}, {17139, 51987}, {20857, 44112}, {21965, 60691}, {22059, 67510}, {22074, 40937}, {22122, 41502}, {27174, 55400}, {30943, 37657}, {36086, 37142}, {40571, 55399}, {44151, 56958}, {52663, 68693}, {52890, 67494}, {53238, 54425}

X(68730) = X(6740)-Ceva conjugate of X(55)
X(68730) = X(46393)-cross conjugate of X(2427)
X(68730) = X(i)-isoconjugate of X(j) for these (i,j): {7, 2250}, {10, 34051}, {57, 38955}, {65, 34234}, {73, 16082}, {104, 226}, {225, 65302}, {349, 34858}, {523, 37136}, {525, 36110}, {604, 57984}, {656, 65331}, {661, 54953}, {662, 66275}, {664, 55259}, {850, 32669}, {909, 1441}, {1020, 43728}, {1042, 36795}, {1214, 36123}, {1309, 51664}, {1400, 18816}, {1427, 51565}, {1446, 2342}, {1577, 2720}, {1795, 40149}, {1813, 68561}, {2401, 4551}, {3668, 52663}, {4017, 13136}, {4077, 32641}, {4566, 61238}, {4674, 40218}, {7178, 36037}, {8611, 65537}, {8808, 15501}, {14208, 32702}, {14578, 57809}, {18210, 39294}, {18593, 40437}, {36819, 66941}, {37628, 52607}, {41804, 67178}, {43933, 65233}, {47317, 57099}, {51662, 64824}, {52378, 66294}, {53527, 53811}
X(68730) = X(i)-Dao conjugate of X(j) for these (i,j): {1084, 66275}, {1145, 321}, {2245, 41804}, {3161, 57984}, {3259, 7178}, {5452, 38955}, {16586, 349}, {23980, 1441}, {25640, 40149}, {34961, 13136}, {36830, 54953}, {38981, 1577}, {39004, 525}, {39025, 55259}, {40582, 18816}, {40596, 65331}, {40602, 34234}, {40613, 226}, {45247, 4080}, {55153, 850}
X(68730) = crosspoint of X(284) and X(2341)
X(68730) = crosssum of X(i) and X(j) for these (i,j): {37, 40663}, {226, 18593}, {3960, 24237}, {21044, 30572}
X(68730) = trilinear pole of line {52307, 53549}
X(68730) = crossdifference of every pair of points on line {65, 523}
X(68730) = barycentric product X(i)*X(j) for these {i,j}: {8, 859}, {21, 517}, {28, 51379}, {29, 22350}, {55, 17139}, {58, 6735}, {59, 14010}, {60, 17757}, {99, 53549}, {110, 2804}, {261, 51377}, {283, 1785}, {284, 908}, {333, 2183}, {521, 4246}, {643, 1769}, {645, 3310}, {648, 52307}, {650, 64828}, {662, 46393}, {759, 64139}, {1014, 51380}, {1021, 24029}, {1043, 1457}, {1169, 51407}, {1465, 2287}, {1792, 1875}, {1793, 1845}, {1812, 14571}, {2185, 21801}, {2189, 51367}, {2194, 3262}, {2311, 51381}, {2328, 22464}, {2341, 16586}, {2397, 7252}, {2427, 4560}, {3063, 55258}, {3939, 23788}, {4183, 62402}, {4570, 35015}, {5379, 35014}, {5546, 10015}, {6064, 42752}, {6740, 34586}, {7253, 23981}, {8677, 36797}, {14224, 42746}, {15507, 56154}, {23189, 53151}, {23706, 57081}, {36038, 65375}, {43728, 68147}, {47318, 53046}
X(68730) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 57984}, {21, 18816}, {41, 2250}, {55, 38955}, {110, 54953}, {112, 65331}, {163, 37136}, {284, 34234}, {512, 66275}, {517, 1441}, {859, 7}, {908, 349}, {1172, 16082}, {1333, 34051}, {1457, 3668}, {1465, 1446}, {1576, 2720}, {1769, 4077}, {1785, 57809}, {1875, 68576}, {2183, 226}, {2193, 65302}, {2194, 104}, {2287, 36795}, {2299, 36123}, {2328, 51565}, {2427, 4552}, {2804, 850}, {3063, 55259}, {3285, 40218}, {3310, 7178}, {4246, 18026}, {4516, 66294}, {5546, 13136}, {6735, 313}, {7252, 2401}, {8677, 17094}, {14010, 34387}, {14571, 40149}, {17139, 6063}, {17757, 34388}, {18344, 68561}, {21789, 43728}, {21801, 6358}, {22350, 307}, {23788, 52621}, {23981, 4566}, {32676, 36110}, {34586, 41804}, {35015, 21207}, {42752, 1365}, {46393, 1577}, {47434, 51365}, {51377, 12}, {51379, 20336}, {51380, 3701}, {51407, 1228}, {51987, 66941}, {52143, 61492}, {52307, 525}, {53046, 4707}, {53549, 523}, {57134, 37628}, {57657, 909}, {61206, 32702}, {64139, 35550}, {64828, 4554}, {65201, 65223}, {65375, 36037}
X(68730) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 284, 46882}, {284, 4266, 4267}, {284, 4282, 3285}, {1817, 56001, 222}

X(68731) = BROCARD AXIS INTERCEPT OF X(41)X(408)

Barycentrics    a^3*(b + c)*(a^2 - b^2 - c^2)*(a^4 - a^2*b^2 + a^2*b*c - b^3*c - a^2*c^2 + 2*b^2*c^2 - b*c^3) : :

X(68731) lies on these lines: {3, 6}, {41, 408}, {213, 22341}, {810, 822}, {851, 1430}, {856, 2238}, {906, 20741}, {1409, 1410}, {1415, 39690}, {2176, 20764}, {2205, 22057}, {2242, 18592}, {3230, 62736}, {16969, 38284}, {20963, 40946}, {22119, 62420}, {44093, 52411}

X(68731) = isotomic conjugate of the polar conjugate of X(44112)
X(68731) = isogonal conjugate of the polar conjugate of X(851)
X(68731) = X(i)-complementary conjugate of X(j) for these (i,j): {810, 15612}, {929, 21259}
X(68731) = X(i)-Ceva conjugate of X(j) for these (i,j): {851, 44112}, {1951, 42669}
X(68731) = X(i)-isoconjugate of X(j) for these (i,j): {4, 35145}, {19, 57980}, {29, 1952}, {92, 37142}, {99, 68646}, {264, 2249}, {522, 41207}, {1577, 59041}, {1896, 40843}, {1937, 31623}, {1945, 44130}, {6528, 52222}, {8062, 65357}, {8748, 57801}, {17926, 53211}, {41206, 44426}, {44129, 61427}
X(68731) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 57980}, {22391, 37142}, {35075, 1969}, {36033, 35145}, {38986, 68646}, {39032, 44130}, {39037, 31623}
X(68731) = crossdifference of every pair of points on line {92, 523}
X(68731) = barycentric product X(i)*X(j) for these {i,j}: {3, 851}, {48, 8680}, {63, 42669}, {69, 44112}, {72, 26884}, {73, 1936}, {110, 9391}, {184, 44150}, {219, 51645}, {228, 5088}, {243, 22341}, {520, 23353}, {822, 1981}, {1214, 1951}, {1400, 6518}, {1409, 1944}, {1410, 7360}, {1430, 3682}, {2202, 40152}, {3049, 15418}, {7138, 15146}, {51726, 52385}, {52373, 58325}
X(68731) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 57980}, {48, 35145}, {184, 37142}, {798, 68646}, {851, 264}, {1409, 1952}, {1415, 41207}, {1576, 59041}, {1936, 44130}, {1951, 31623}, {1981, 57973}, {5088, 57796}, {6518, 28660}, {8680, 1969}, {9247, 2249}, {9391, 850}, {22341, 57801}, {23353, 6528}, {26884, 286}, {32660, 41206}, {42669, 92}, {44112, 4}, {44150, 18022}, {51645, 331}, {51726, 1896}
X(68731) = {X(856),X(2238)}-harmonic conjugate of X(16573)

X(68732) = BROCARD AXIS INTERCEPT OF X(1)X(3255)

Barycentrics    a^2*(a^3*b - a^2*b^2 - a*b^3 + b^4 + a^3*c + 2*a^2*b*c - b^3*c - a^2*c^2 - a*c^3 - b*c^3 + c^4) : :

X(68732) lies on these lines: {1, 3255}, {3, 6}, {21, 68585}, {42, 9340}, {44, 1818}, {81, 35989}, {269, 34489}, {513, 663}, {741, 59088}, {851, 18191}, {896, 64710}, {971, 8609}, {1104, 4303}, {1108, 63395}, {1145, 62325}, {1397, 1626}, {1404, 20780}, {1408, 3145}, {1418, 37566}, {1469, 3941}, {1471, 34880}, {1756, 7290}, {1834, 48897}, {1914, 35505}, {2191, 52372}, {2223, 8679}, {2274, 4749}, {2293, 2650}, {2352, 26892}, {2478, 4648}, {3019, 48903}, {3052, 62207}, {3220, 7113}, {3271, 20470}, {3332, 6938}, {3554, 5732}, {3724, 53542}, {3752, 22053}, {3945, 6872}, {4186, 28014}, {4187, 17245}, {4322, 45219}, {4436, 35104}, {5061, 53279}, {5119, 68588}, {5495, 63307}, {5820, 36474}, {6149, 19624}, {6834, 68529}, {6921, 37650}, {7193, 17455}, {7202, 44661}, {8053, 64006}, {8557, 66660}, {8614, 11510}, {10966, 67264}, {11113, 17392}, {13724, 28350}, {13747, 17337}, {16064, 44085}, {16885, 56809}, {17056, 17194}, {17611, 53035}, {18178, 37425}, {20978, 37605}, {22435, 28272}, {23691, 59769}, {25875, 25878}, {28238, 57666}, {35466, 61220}, {37290, 68586}, {40956, 67961}, {41426, 42314}, {49745, 54356}, {63054, 64912}, {64070, 64739}

X(68732) = reflection of X(2245) in X(3286)
X(68732) = crosspoint of X(1) and X(28471)
X(68732) = crosssum of X(1) and X(17768)
X(68732) = crossdifference of every pair of points on line {9, 523}
X(68732) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 37516, 4271}, {58, 500, 52544}, {4259, 37507, 583}, {22053, 40958, 3752}, {36740, 37474, 2278}, {37507, 48908, 4259}

X(68733) = BROCARD AXIS INTERCEPT OF X(1)X(16608)

Barycentrics    a^2*(b + c)*(a^2 - b^2 - c^2)*(a^4 - b^4 - a^2*b*c + b^3*c + b*c^3 - c^4) : :

X(68733) lies on these lines: {1, 16608}, {2, 57820}, {3, 6}, {42, 1214}, {55, 22119}, {73, 1439}, {408, 45963}, {440, 1040}, {442, 54293}, {521, 656}, {851, 1465}, {857, 3100}, {971, 3330}, {990, 1901}, {1038, 7181}, {1062, 1834}, {1375, 54407}, {1801, 23130}, {2331, 30265}, {2594, 22341}, {3192, 7580}, {3194, 30267}, {3216, 7515}, {3220, 39690}, {3240, 6350}, {3682, 3694}, {3755, 18589}, {3914, 18588}, {3936, 34846}, {4035, 20309}, {4055, 22069}, {4551, 51368}, {4646, 41340}, {5399, 20764}, {5728, 17102}, {5738, 19767}, {6056, 22130}, {6349, 17018}, {6547, 18455}, {15526, 35093}, {17441, 22348}, {18210, 22349}, {22063, 63395}, {22124, 63434}, {23112, 44707}, {25015, 66610}, {25909, 64415}, {30674, 37553}, {33305, 61226}, {35071, 65910}, {40952, 52424}, {44244, 48897}, {46837, 56809}

X(68733) = isotomic conjugate of the polar conjugate of X(39690)
X(68733) = isogonal conjugate of the polar conjugate of X(857)
X(68733) = X(i)-complementary conjugate of X(j) for these (i,j): {73, 120}, {105, 34831}, {603, 8299}, {810, 1566}, {927, 21259}, {1214, 20540}, {1409, 16593}, {1410, 50441}, {1416, 942}, {1438, 6708}, {1462, 34830}, {1814, 21246}, {10099, 124}, {18785, 41883}, {32658, 5745}, {32735, 8062}, {36057, 960}, {36146, 30476}, {52373, 17060}, {56853, 20262}, {64216, 40942}, {65301, 42327}, {66941, 20305}
X(68733) = X(i)-Ceva conjugate of X(j) for these (i,j): {677, 520}, {857, 39690}, {3100, 44661}, {51568, 525}, {53160, 53300}, {64975, 219}, {67761, 3556}
X(68733) = X(i)-isoconjugate of X(j) for these (i,j): {4, 26702}, {19, 37202}, {92, 57735}, {850, 32673}, {1577, 36071}, {2867, 54247}
X(68733) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 37202}, {22391, 57735}, {36033, 26702}, {65887, 92}
X(68733) = crosssum of X(6) and X(37908)
X(68733) = crossdifference of every pair of points on line {19, 523}
X(68733) = barycentric product X(i)*X(j) for these {i,j}: {3, 857}, {63, 44661}, {69, 39690}, {71, 4872}, {72, 7291}, {73, 37774}, {228, 7112}, {306, 3220}, {1214, 3100}, {56382, 58326}
X(68733) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 37202}, {48, 26702}, {184, 57735}, {857, 264}, {1576, 36071}, {3100, 31623}, {3220, 27}, {4872, 44129}, {7112, 57796}, {7291, 286}, {37774, 44130}, {39690, 4}, {44661, 92}, {58326, 2322}
X(68733) = {X(42),X(22057)}-harmonic conjugate of X(1214)

X(68734) = BROCARD AXIS INTERCEPT OF X(2)X(1236)

Barycentrics    a^2*(a^6*b^2 - a^4*b^4 - a^2*b^6 + b^8 + a^6*c^2 + a^2*b^4*c^2 - a^4*c^4 + a^2*b^2*c^4 - 2*b^4*c^4 - a^2*c^6 + c^8) : :

X(68734) lies on these lines: {2, 1236}, {3, 6}, {115, 232}, {230, 44452}, {237, 55384}, {248, 38534}, {647, 826}, {1084, 65920}, {1180, 66378}, {1194, 5355}, {1196, 37453}, {1500, 9627}, {1648, 66133}, {2493, 43291}, {2548, 26216}, {2549, 22240}, {3199, 37197}, {3291, 64646}, {3455, 39840}, {3580, 7813}, {3767, 7505}, {5254, 15761}, {5305, 10020}, {5306, 34477}, {5309, 10201}, {5938, 18374}, {6128, 47167}, {6390, 34990}, {6781, 16386}, {7737, 35481}, {7745, 52070}, {7747, 18560}, {7753, 52069}, {7755, 10018}, {7756, 52071}, {7758, 28710}, {8779, 19627}, {8791, 14580}, {9418, 14917}, {9699, 10311}, {11672, 39021}, {12236, 44221}, {12825, 46301}, {15355, 43620}, {15525, 65923}, {16308, 44468}, {18334, 23976}, {19626, 61450}, {23967, 39013}, {23992, 61067}, {25337, 63633}, {33843, 44438}, {34397, 42671}, {35282, 47195}, {37784, 68654}, {39832, 44090}, {39857, 44089}, {44127, 67544}, {47421, 47427}, {54075, 62375}

X(68734) = complement of X(1236)
X(68734) = complement of the isotomic conjugate of X(1177)
X(68734) = isogonal conjugate of the isotomic conjugate of X(62376)
X(68734) = isogonal conjugate of the polar conjugate of X(37981)
X(68734) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 15116}, {560, 1560}, {1177, 2887}, {1917, 61067}, {2373, 21235}, {10422, 21256}, {10423, 21259}, {37220, 40379}, {65306, 42327}
X(68734) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 15116}, {935, 512}
X(68734) = X(i)-Dao conjugate of X(j) for these (i,j): {15116, 2}, {62376, 37804}
X(68734) = crosspoint of X(i) and X(j) for these (i,j): {2, 1177}, {6, 8791}, {37981, 62376}
X(68734) = crosssum of X(i) and X(j) for these (i,j): {2, 22151}, {6, 858}
X(68734) = crossdifference of every pair of points on line {22, 523}
X(68734) = barycentric product X(i)*X(j) for these {i,j}: {3, 37981}, {6, 62376}, {31, 18694}, {67, 40949}, {1177, 15116}, {5486, 35370}
X(68734) = barycentric quotient X(i)/X(j) for these {i,j}: {15116, 1236}, {18694, 561}, {35370, 11185}, {37981, 264}, {40949, 316}, {62376, 76}
X(68734) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 41336, 187}, {39, 187, 14961}, {39, 216, 574}, {574, 7772, 5065}, {574, 46432, 32}, {3003, 5421, 58267}, {3003, 14961, 187}, {15166, 15167, 32}

X(68735) = BROCARD AXIS INTERCEPT OF X(1)X(513)

Barycentrics    a^2*(a^3*b - 2*a^2*b^2 - a*b^3 + 2*b^4 + a^3*c + 2*a^2*b*c - b^3*c - 2*a^2*c^2 - a*c^3 - b*c^3 + 2*c^4) : :
X(68735) = 2 X[2245] - 3 X[3286]

X(68735) lies on these lines: {1, 513}, {2, 16500}, {3, 6}, {8, 16506}, {21, 4585}, {35, 23344}, {42, 23832}, {56, 59234}, {81, 13589}, {390, 2098}, {940, 52242}, {942, 53546}, {978, 16495}, {995, 16492}, {1125, 16494}, {1193, 16493}, {1201, 16501}, {1283, 53324}, {1284, 1388}, {1334, 2284}, {1482, 53792}, {1756, 21842}, {1834, 48921}, {2223, 9037}, {3271, 54333}, {3303, 38508}, {3811, 16504}, {4193, 17234}, {5046, 17300}, {5697, 20718}, {7186, 52139}, {7976, 38499}, {10391, 35014}, {11114, 17139}, {12635, 20077}, {14196, 54277}, {16610, 22067}, {16686, 36942}, {16917, 20144}, {17313, 17556}, {17352, 17566}, {18178, 48897}, {18191, 61220}, {19515, 37662}, {20086, 41711}, {20122, 56410}, {24433, 67428}, {24436, 35327}, {25557, 56850}, {37299, 63052}, {38511, 68668}, {48909, 64159}, {50259, 57006}, {53280, 53542}

X(68735) = crossdifference of every pair of points on line {44, 523}
X(68735) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {991, 37516, 5132}, {53542, 64710, 53280}

X(68736) = BROCARD AXIS INTERCEPT OF X(30)X(17197)

Barycentrics    a^2*(2*a^3*b - a^2*b^2 - 2*a*b^3 + b^4 + 2*a^3*c + 4*a^2*b*c - 2*b^3*c - a^2*c^2 - 2*a*c^3 - 2*b*c^3 + c^4) : :
X(68736) = X[2245] - 3 X[3286]

X(68736) lies on these lines: {3, 6}, {30, 17197}, {36, 3271}, {214, 49710}, {513, 1960}, {1284, 1420}, {1362, 2078}, {1412, 20834}, {2223, 2810}, {4667, 24929}, {4850, 22068}, {5542, 5625}, {9957, 20718}, {11111, 17139}, {15447, 18191}, {17758, 49738}, {18178, 48919}, {19256, 44151}, {22053, 40649}, {23634, 54310}, {28376, 28397}, {30117, 53543}, {48894, 64159}, {50264, 64912}, {56176, 64073}, {62882, 62921}

X(68736) = crossdifference of every pair of points on line {45, 523}
X(68736) = {X(991),X(37507)}-harmonic conjugate of X(4260)

X(68737) = BROCARD AXIS INTERCEPT OF X(30)X(1625)

Barycentrics    a^2*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^6*c^2 - 4*a^4*b^2*c^2 + 2*a^2*b^4*c^2 + b^6*c^2 - 2*a^4*c^4 + 2*a^2*b^2*c^4 - 2*b^4*c^4 + a^2*c^6 + b^2*c^6) : :
X(68737) = 2 X[6] - 3 X[14965], 3 X[1625] - 2 X[3331], 3 X[3289] - X[3331]

X(68737) lies on these lines: {3, 6}, {30, 1625}, {115, 45935}, {160, 35704}, {217, 550}, {276, 59157}, {316, 2421}, {323, 401}, {376, 60106}, {381, 40805}, {394, 51372}, {399, 57012}, {458, 15066}, {549, 59208}, {1154, 3269}, {1495, 60522}, {1511, 39000}, {1657, 32445}, {1971, 22115}, {1993, 31859}, {2387, 53273}, {3051, 15048}, {3231, 21531}, {3620, 37337}, {3631, 34850}, {5059, 41367}, {5073, 38297}, {5167, 56957}, {5189, 15340}, {5523, 35325}, {5891, 33843}, {5938, 52162}, {6101, 22416}, {8744, 35474}, {9463, 37190}, {10311, 68659}, {10313, 32661}, {10601, 37871}, {10985, 43586}, {11004, 51350}, {11641, 56923}, {15018, 60700}, {15107, 27867}, {16266, 39643}, {18371, 53505}, {28438, 60518}, {32515, 48452}, {36952, 37125}, {37126, 60589}, {37672, 51122}, {37971, 59558}, {39575, 64050}, {39849, 50461}, {45938, 53419}, {46841, 62334}, {48262, 63548}, {62260, 68084}, {64099, 66919}

X(68737) = reflection of X(1625) in X(3289)
X(68737) = isogonal conjugate of X(54547)
X(68737) = circumcircle-inverse of X(67381)
X(68737) = Brocard-circle-inverse of X(50678)
X(68737) = X(54973)-Ceva conjugate of X(3)
X(68737) = X(1)-isoconjugate of X(54547)
X(68737) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 54547}, {852, 64781}
X(68737) = crosssum of X(6) and X(44890)
X(68737) = crossdifference of every pair of points on line {51, 523}
X(68737) = barycentric quotient X(6)/X(54547)
X(68737) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 50678}, {6, 50678, 41334}, {39, 10625, 41480}, {1379, 1380, 67381}, {1971, 22115, 35324}, {10313, 43574, 32661}

X(68738) = BROCARD AXIS INTERCEPT OF X(30)X(250)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^8 - 2*a^6*b^2 + a^4*b^4 - 2*a^6*c^2 + a^4*b^2*c^2 + b^6*c^2 + a^4*c^4 - 2*b^4*c^4 + b^2*c^6) : :
X(68738) = X[3] + 2 X[3284], 4 X[140] - X[340], X[401] + 2 X[67405], 5 X[15694] - 2 X[45312]

X(68738) lies on these lines: {2, 23606}, {3, 6}, {4, 61532}, {5, 14152}, {20, 41371}, {30, 250}, {54, 43975}, {95, 140}, {97, 61355}, {110, 852}, {125, 51458}, {129, 1971}, {155, 23163}, {160, 64061}, {184, 6638}, {237, 19128}, {287, 441}, {323, 2972}, {401, 32428}, {417, 34148}, {418, 5012}, {426, 1993}, {436, 59532}, {520, 6760}, {549, 66449}, {550, 42873}, {631, 43980}, {933, 51939}, {1092, 40800}, {1147, 20794}, {1199, 42441}, {1216, 22138}, {1316, 44145}, {1353, 41005}, {1503, 1576}, {1513, 44089}, {1650, 52603}, {1657, 15274}, {1899, 52435}, {1942, 5504}, {1994, 13409}, {2072, 13557}, {2782, 15013}, {2967, 52058}, {3148, 39588}, {3167, 6617}, {3289, 14941}, {3292, 44436}, {3515, 59228}, {3526, 43998}, {3548, 41770}, {3580, 44888}, {5422, 6641}, {5462, 37081}, {5562, 63532}, {5622, 23200}, {5965, 15526}, {6146, 36245}, {6389, 63722}, {6509, 34986}, {6748, 42350}, {6776, 14575}, {7464, 40948}, {7574, 18402}, {7709, 35952}, {8550, 42353}, {8721, 20968}, {8779, 52170}, {9306, 38283}, {9475, 41363}, {10104, 28407}, {10313, 68695}, {10359, 37186}, {10539, 38281}, {10540, 62334}, {11245, 54034}, {11674, 62341}, {11898, 20208}, {12007, 34828}, {13366, 46832}, {13434, 26897}, {13754, 15781}, {14379, 36608}, {14912, 34396}, {15018, 54375}, {15694, 45312}, {18831, 57274}, {19124, 32444}, {22143, 34382}, {22151, 44716}, {23181, 58357}, {23583, 39569}, {26874, 61394}, {31388, 37126}, {32545, 44137}, {33753, 67872}, {34003, 39243}, {34117, 63419}, {34397, 44886}, {34545, 61378}, {34751, 35225}, {35311, 62308}, {35442, 41724}, {36153, 46025}, {37124, 56290}, {37182, 60694}, {39231, 51733}, {40647, 45842}, {40947, 67904}, {41202, 46106}, {44252, 44704}, {44715, 63720}, {45200, 59553}, {46841, 57011}, {49124, 66135}, {52090, 54080}, {61748, 63421}

X(68738) = midpoint of X(i) and X(j) for these {i,j}: {401, 41204}, {44252, 44704}
X(68738) = reflection of X(i) in X(j) for these {i,j}: {39569, 23583}, {41204, 67405}
X(68738) = Brocard-circle-inverse of X(30258)
X(68738) = isotomic conjugate of the polar conjugate of X(1971)
X(68738) = isogonal conjugate of the polar conjugate of X(401)
X(68738) = X(i)-Ceva conjugate of X(j) for these (i,j): {401, 1971}, {17974, 3}, {60179, 32661}
X(68738) = X(52128)-cross conjugate of X(3)
X(68738) = X(i)-isoconjugate of X(j) for these (i,j): {4, 1956}, {19, 1972}, {92, 1987}, {158, 14941}, {162, 60036}, {656, 65358}, {661, 53205}, {662, 62519}, {1577, 53708}, {8767, 51960}, {24006, 65305}, {36120, 40804}, {40703, 67190}, {52177, 57806}
X(68738) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 1972}, {125, 60036}, {1084, 62519}, {1147, 14941}, {6130, 868}, {14382, 60199}, {22391, 1987}, {36033, 1956}, {36830, 53205}, {38974, 14618}, {39038, 92}, {39045, 4}, {39071, 51960}, {39081, 264}, {40596, 65358}, {46094, 40804}, {52128, 5}, {65909, 324}
X(68738) = crosspoint of X(i) and X(j) for these (i,j): {95, 287}, {250, 43754}, {4558, 57991}
X(68738) = crosssum of X(i) and X(j) for these (i,j): {51, 232}, {125, 16230}, {2501, 44114}
X(68738) = crossdifference of every pair of points on line {53, 523}
X(68738) = barycentric product X(i)*X(j) for these {i,j}: {3, 401}, {63, 1955}, {69, 1971}, {97, 32428}, {184, 44137}, {287, 52128}, {394, 41204}, {523, 62523}, {577, 16089}, {2313, 62277}, {3926, 58311}, {4558, 6130}, {14919, 67405}, {15407, 66903}, {17974, 62595}, {32545, 36212}, {38974, 57991}
X(68738) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 1972}, {48, 1956}, {110, 53205}, {112, 65358}, {184, 1987}, {401, 264}, {512, 62519}, {577, 14941}, {647, 60036}, {933, 41210}, {1576, 53708}, {1955, 92}, {1971, 4}, {3289, 40804}, {6130, 14618}, {8779, 51960}, {14533, 1298}, {14585, 52177}, {14600, 67190}, {16089, 18027}, {18315, 41208}, {32428, 324}, {32545, 16081}, {32661, 65305}, {38974, 868}, {41204, 2052}, {44137, 18022}, {52128, 297}, {58311, 393}, {62523, 99}, {64227, 53245}, {67405, 46106}, {67409, 41203}
X(68738) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 30258}, {3, 10317, 2080}, {3, 22120, 3095}, {3, 38292, 1351}, {3, 44456, 34815}, {182, 577, 3}, {5092, 63433, 3}, {13335, 22401, 3}, {40349, 47113, 3}, {45410, 45411, 36752}

X(68739) = BROCARD AXIS INTERCEPT OF X(112)X(35474)

Barycentrics    a^4*(a^4 - a^2*b^2 - a^2*c^2 - 2*b^2*c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4) : :

X(68739) lies on these lines: {3, 6}, {112, 35474}, {183, 458}, {230, 8623}, {232, 51862}, {233, 6292}, {237, 3289}, {248, 290}, {276, 308}, {1968, 37200}, {1971, 3511}, {1974, 46312}, {2799, 55974}, {3087, 3785}, {3231, 66350}, {3329, 46807}, {4558, 39099}, {6660, 57261}, {7735, 37190}, {7766, 51350}, {7767, 34850}, {8779, 60528}, {9468, 32654}, {10312, 37124}, {10314, 15271}, {11174, 37067}, {11328, 40805}, {14096, 59208}, {14614, 35941}, {14957, 60517}, {35934, 54991}, {36212, 41270}, {37184, 43718}, {39469, 42660}, {40799, 46319}, {42671, 60514}, {44894, 46841}, {51869, 52144}

X(68739) = isogonal conjugate of the polar conjugate of X(68695)
X(68739) = X(i)-Ceva conjugate of X(j) for these (i,j): {3407, 36213}, {40799, 11672}, {46806, 182}
X(68739) = X(i)-isoconjugate of X(j) for these (i,j): {92, 66879}, {262, 1821}, {263, 46273}, {290, 2186}, {327, 1910}, {336, 68572}, {661, 53196}, {850, 36132}, {1577, 6037}, {3402, 18024}, {20948, 32716}, {36036, 66291}, {36120, 42313}, {43665, 65252}
X(68739) = X(i)-Dao conjugate of X(j) for these (i,j): {511, 46807}, {2679, 66291}, {11672, 327}, {22391, 66879}, {36830, 53196}, {38997, 43665}, {39009, 850}, {40601, 262}, {46094, 42313}, {51580, 18024}, {52878, 66919}, {62596, 2799}
X(68739) = crosspoint of X(182) and X(46806)
X(68739) = crosssum of X(i) and X(j) for these (i,j): {6, 37123}, {262, 51543}, {3569, 66459}
X(68739) = trilinear pole of line {9420, 33569}
X(68739) = crossdifference of every pair of points on line {262, 523}
X(68739) = barycentric product X(i)*X(j) for these {i,j}: {3, 68695}, {32, 51373}, {99, 9420}, {182, 511}, {183, 237}, {325, 34396}, {458, 3289}, {1755, 52134}, {2421, 3288}, {2966, 33569}, {3403, 9417}, {9418, 20023}, {10311, 36212}, {11672, 46806}, {14096, 51862}, {14966, 23878}, {17209, 60726}, {23357, 66192}, {36790, 51542}, {39683, 52128}, {41270, 59197}, {67172, 68178}
X(68739) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 53196}, {182, 290}, {183, 18024}, {184, 66879}, {237, 262}, {458, 60199}, {511, 327}, {1576, 6037}, {2211, 68572}, {2491, 66291}, {3288, 43665}, {3289, 42313}, {9417, 2186}, {9418, 263}, {9419, 51543}, {9420, 523}, {10311, 16081}, {11672, 46807}, {14574, 32716}, {14966, 65271}, {33569, 2799}, {34396, 98}, {39009, 66192}, {41270, 42300}, {46806, 57541}, {51373, 1502}, {51542, 34536}, {52134, 46273}, {52967, 66919}, {58262, 67173}, {59208, 53245}, {60497, 60520}, {66192, 23962}, {68695, 264}
X(68739) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 571, 41331}, {6, 8266, 216}, {237, 3289, 11672}, {248, 51327, 41932}, {385, 401, 290}, {37184, 60106, 43718}

X(68740) = BROCARD AXIS INTERCEPT OF X(30)X(3269)

Barycentrics    a^2*(a^4*b^4 - 2*a^2*b^6 + b^8 + a^2*b^4*c^2 - b^6*c^2 + a^4*c^4 + a^2*b^2*c^4 - 2*a^2*c^6 - b^2*c^6 + c^8) : :

X(68740) lies on these lines: {3, 6}, {5, 22416}, {23, 13509}, {24, 23128}, {26, 39643}, {30, 3269}, {115, 63735}, {186, 32661}, {217, 6102}, {230, 47421}, {232, 1625}, {237, 63712}, {265, 1987}, {287, 54076}, {297, 525}, {546, 62260}, {1154, 3289}, {1658, 14585}, {1968, 7689}, {1971, 2070}, {2207, 12163}, {2493, 45938}, {3124, 43291}, {3199, 12162}, {3331, 5663}, {3448, 15340}, {3567, 26216}, {5167, 44114}, {5254, 41587}, {5889, 39575}, {5890, 22240}, {5946, 59208}, {6390, 36790}, {7738, 64048}, {8791, 60039}, {9418, 39846}, {10311, 64095}, {10540, 39849}, {11062, 25711}, {11459, 15355}, {12359, 27376}, {14917, 60514}, {15048, 20859}, {15305, 33885}, {15341, 32269}, {16003, 52945}, {16194, 33842}, {16328, 47405}, {18439, 38297}, {18451, 59229}, {20966, 66743}, {20977, 44468}, {22241, 40802}, {23039, 40805}, {23639, 66745}, {26166, 36952}, {31406, 31802}, {32445, 34783}, {35324, 51393}, {37638, 52251}, {39000, 64645}, {41367, 64025}, {44221, 47406}, {44228, 51334}, {44683, 65809}, {53419, 53494}, {61213, 67544}

X(68740) = reflection of X(1625) in X(232)
X(68740) = crosssum of X(6) and X(44886)
X(68740) = crossdifference of every pair of points on line {184, 523}
X(68740) = barycentric product X(5)*X(19167)
X(68740) = barycentric quotient X(19167)/X(95)
X(68740) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 54082, 10317}, {216, 9730, 50678}, {2070, 22146, 1971}, {3581, 10317, 54082}, {14962, 45283, 3001}, {14962, 45284, 45283}

X(68741) = BROCARD AXIS INTERCEPT OF X(2)X(54832)

Barycentrics    a^2*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8 + a^8*c^2 - 2*a^6*b^2*c^2 + 2*a^2*b^6*c^2 - b^8*c^2 - 3*a^6*c^4 - 2*a^2*b^4*c^4 + b^6*c^4 + 3*a^4*c^6 + 2*a^2*b^2*c^6 + b^4*c^6 - a^2*c^8 - b^2*c^8) : :
X(68741) = 2 X[3] + X[5201], 4 X[3003] - X[67529], 2 X[237] + X[53246], X[1634] - 4 X[44221]

X(68741) lies on these lines: {2, 54832}, {3, 6}, {4, 34845}, {5, 35222}, {22, 45030}, {23, 8902}, {24, 157}, {25, 9756}, {26, 33801}, {98, 34133}, {125, 44886}, {160, 6776}, {185, 44088}, {186, 523}, {230, 68695}, {237, 1503}, {251, 5481}, {338, 32428}, {378, 39656}, {418, 13567}, {468, 1624}, {852, 47296}, {1199, 51255}, {1495, 44890}, {1576, 19128}, {1634, 3564}, {1899, 3135}, {2070, 30715}, {2351, 21213}, {2794, 21177}, {3050, 39215}, {3133, 12359}, {3168, 55354}, {3260, 17928}, {3515, 15653}, {3520, 60693}, {3580, 23181}, {3589, 54004}, {3613, 37121}, {4230, 66167}, {5480, 40981}, {5622, 39231}, {5961, 46155}, {5999, 51862}, {6530, 52604}, {6638, 26958}, {6644, 45847}, {7418, 38227}, {7482, 47000}, {7793, 40807}, {8550, 20775}, {10192, 61374}, {10516, 11328}, {10548, 37126}, {11245, 23195}, {11433, 26874}, {12006, 34292}, {13366, 16030}, {13367, 16035}, {13558, 21284}, {13568, 26897}, {14165, 65182}, {14561, 35934}, {14575, 64061}, {14703, 14729}, {14912, 66886}, {15576, 15750}, {15577, 40947}, {15648, 19357}, {16303, 59661}, {20885, 64711}, {20960, 59363}, {20975, 44668}, {21525, 53267}, {21663, 34950}, {22467, 56290}, {23200, 51733}, {23217, 41586}, {25406, 37184}, {29181, 47620}, {31952, 48910}, {32444, 53023}, {34397, 51458}, {34990, 44716}, {35296, 68178}, {35360, 47153}, {37457, 46124}, {37814, 63833}, {37920, 47213}, {37954, 54077}, {37991, 66170}, {38605, 62598}, {39058, 54086}, {41724, 50947}, {41760, 42329}, {44889, 61691}, {44896, 53568}, {45735, 45848}, {48906, 52274}, {51730, 61748}, {52276, 63419}

X(68741) = reflection of X(44716) in X(34990)
X(68741) = isogonal conjugate of X(35098)
X(68741) = circumcircle-inverse of X(31850)
X(68741) = isogonal conjugate of the anticomplement of X(46094)
X(68741) = isogonal conjugate of the polar conjugate of X(44893)
X(68741) = X(16081)-Ceva conjugate of X(6)
X(68741) = X(i)-isoconjugate of X(j) for these (i,j): {1, 35098}, {39452, 44706}
X(68741) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 35098}, {3289, 36212}
X(68741) = crosspoint of X(i) and X(j) for these (i,j): {54, 98}, {249, 22456}
X(68741) = crosssum of X(i) and X(j) for these (i,j): {5, 511}, {6, 60522}, {115, 39469}, {520, 3150}, {34983, 41172}
X(68741) = crossdifference of every pair of points on line {216, 523}
X(68741) = X(i)-line conjugate of X(j) for these (i,j): {3, 216}, {186, 523}
X(68741) = barycentric product X(i)*X(j) for these {i,j}: {3, 44893}, {16081, 46094}
X(68741) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 35098}, {8882, 39452}, {44893, 264}, {46094, 36212}
X(68741) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 182, 41328}, {3, 24729, 47619}, {98, 37123, 60514}, {186, 41204, 19189}, {800, 63433, 50649}, {1379, 1380, 31850}, {2351, 21213, 35225}, {6776, 37114, 160}, {34133, 34134, 37123}, {40981, 54003, 5480}

X(68742) = BROCARD AXIS INTERCEPT OF X(30)X(45930)

Barycentrics    a^2*(a + b)*(a - b - c)*(a + c)*(2*a^2 - a*b - b^2 - a*c + 2*b*c - c^2) : :

X(68742) lies on these lines: {3, 6}, {21, 34522}, {30, 45930}, {45, 62700}, {81, 65003}, {112, 53911}, {163, 859}, {220, 283}, {524, 51608}, {652, 663}, {857, 26145}, {1146, 68693}, {1172, 55918}, {1262, 17966}, {1437, 3207}, {1755, 53290}, {1790, 62207}, {1865, 46468}, {2327, 62245}, {5074, 17197}, {5718, 24581}, {5723, 14953}, {5829, 37113}, {5845, 51607}, {6149, 61197}, {6603, 62756}, {9247, 23361}, {24597, 31015}, {30808, 31187}, {36039, 57736}, {36075, 51654}, {42669, 53324}, {43065, 52680}, {45926, 45927}

X(68742) = isogonal conjugate of the polar conjugate of X(52891)
X(68742) = X(i)-isoconjugate of X(j) for these (i,j): {2, 62764}, {10, 34056}, {37, 62723}, {63, 68580}, {65, 1121}, {226, 1156}, {349, 34068}, {523, 37139}, {656, 65335}, {661, 35157}, {850, 36141}, {1020, 63748}, {1214, 65340}, {1441, 2291}, {1446, 4845}, {1577, 14733}, {3668, 41798}, {4041, 60487}, {4171, 65553}, {4551, 60479}, {4552, 35348}, {4566, 23893}, {20948, 32728}, {24006, 65304}, {39130, 61493}, {40149, 60047}
X(68742) = X(i)-Dao conjugate of X(j) for these (i,j): {3162, 68580}, {6594, 321}, {32664, 62764}, {35091, 850}, {35110, 349}, {36830, 35157}, {40589, 62723}, {40596, 65335}, {40602, 1121}, {52879, 1446}, {52880, 1231}
X(68742) = crosssum of X(i) and X(j) for these (i,j): {1214, 64888}, {21044, 30574}
X(68742) = crossdifference of every pair of points on line {226, 523}
X(68742) = barycentric product X(i)*X(j) for these {i,j}: {1, 62756}, {3, 52891}, {21, 1155}, {58, 6745}, {81, 6603}, {99, 6139}, {110, 6366}, {162, 14414}, {283, 23710}, {284, 527}, {333, 1055}, {643, 14413}, {662, 65680}, {1021, 23890}, {1172, 6510}, {1323, 2328}, {1414, 14392}, {1638, 5546}, {1790, 60431}, {1817, 56763}, {2193, 37805}, {2194, 30806}, {2287, 6610}, {2311, 24685}, {4636, 30574}, {7253, 23346}, {21789, 56543}, {33573, 52378}
X(68742) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 68580}, {31, 62764}, {58, 62723}, {110, 35157}, {112, 65335}, {163, 37139}, {284, 1121}, {527, 349}, {1055, 226}, {1155, 1441}, {1333, 34056}, {1576, 14733}, {2194, 1156}, {2299, 65340}, {4565, 60487}, {6139, 523}, {6366, 850}, {6510, 1231}, {6603, 321}, {6610, 1446}, {6745, 313}, {7252, 60479}, {14392, 4086}, {14413, 4077}, {14414, 14208}, {14574, 32728}, {21789, 63748}, {23346, 4566}, {23710, 57809}, {32661, 65304}, {37805, 52575}, {52891, 264}, {57657, 2291}, {62756, 75}, {65680, 1577}
X(68742) = {X({}),X(1)}-harmonic conjugate of X({}[[1]][[3]])

X(68743) = BROCARD AXIS INTERCEPT OF X(1)X(61326)

Barycentrics    a^2*(a*b - b^2 + a*c - c^2)*(a*b - b^2 + a*c + 2*b*c - c^2) : :

X(68743) lies on these lines: {1, 61326}, {2, 42290}, {3, 6}, {9, 4334}, {36, 51633}, {37, 5542}, {55, 23653}, {57, 16588}, {142, 1212}, {181, 23636}, {220, 42314}, {241, 39063}, {269, 27626}, {354, 21795}, {518, 6184}, {672, 1362}, {1015, 1279}, {1107, 64174}, {1155, 14936}, {1401, 5364}, {1456, 43039}, {1471, 22070}, {1475, 2293}, {1500, 49478}, {1536, 43672}, {1617, 16283}, {1738, 43065}, {2114, 43034}, {2195, 17798}, {2223, 20455}, {2225, 3937}, {2275, 7290}, {2276, 68588}, {2350, 40952}, {3218, 23988}, {3271, 20459}, {3662, 26690}, {3693, 4684}, {3755, 40133}, {3778, 20978}, {3917, 36808}, {4648, 5283}, {5257, 25914}, {5701, 17768}, {6007, 24727}, {8012, 61376}, {8607, 52962}, {9454, 20662}, {10581, 21127}, {11019, 21856}, {11038, 40779}, {13006, 68591}, {14827, 37578}, {14838, 14991}, {16502, 21002}, {16589, 17245}, {16970, 17053}, {16975, 68589}, {17435, 39077}, {20229, 22053}, {20470, 51436}, {20683, 42079}, {20974, 51377}, {21808, 39790}, {24036, 49676}, {24578, 35104}, {25066, 49511}, {25878, 68653}, {35227, 63493}, {35310, 51463}, {43059, 56379}, {60108, 62921}

X(68743) = isogonal conjugate of the isotomic conjugate of X(51384)
X(68743) = X(i)-Ceva conjugate of X(j) for these (i,j): {37128, 18164}, {41353, 926}, {53241, 354}, {54325, 665}
X(68743) = X(i)-isoconjugate of X(j) for these (i,j): {105, 32008}, {294, 21453}, {666, 58322}, {673, 2346}, {885, 65222}, {927, 62747}, {1024, 6606}, {1170, 14942}, {1174, 2481}, {1416, 63239}, {1438, 57815}, {1462, 56118}, {2195, 31618}, {6559, 61373}, {6605, 56783}, {10482, 34018}, {10509, 28071}, {36086, 56322}, {36146, 62725}, {47487, 54235}, {55261, 55281}, {59193, 63236}
X(68743) = X(i)-Dao conjugate of X(j) for these (i,j): {142, 36796}, {1212, 18031}, {6184, 57815}, {38989, 56322}, {39014, 62725}, {39046, 32008}, {39063, 31618}, {40606, 2481}, {40609, 63239}, {65525, 28132}
X(68743) = crosspoint of X(i) and X(j) for these (i,j): {241, 672}, {354, 53241}, {3252, 37128}, {17758, 43672}
X(68743) = crosssum of X(i) and X(j) for these (i,j): {2, 63087}, {6, 11349}, {294, 673}, {2238, 6654}, {4251, 13329}
X(68743) = crossdifference of every pair of points on line {523, 885}
X(68743) = barycentric product X(i)*X(j) for these {i,j}: {6, 51384}, {142, 672}, {241, 1212}, {354, 518}, {665, 65195}, {883, 2488}, {918, 35326}, {926, 35312}, {1025, 21127}, {1026, 48151}, {1229, 52635}, {1233, 9454}, {1418, 3693}, {1458, 4847}, {1475, 3912}, {1861, 22053}, {2223, 20880}, {2254, 35338}, {2283, 6362}, {2284, 21104}, {2293, 9436}, {2340, 10481}, {3059, 34855}, {3286, 3925}, {3717, 61376}, {3930, 18164}, {6184, 53241}, {6608, 41353}, {8012, 62786}, {15185, 57469}, {16708, 39258}, {17169, 20683}, {18206, 21808}, {20229, 40704}, {30941, 52020}, {34230, 51463}, {35341, 53544}, {52614, 61241}, {53539, 65198}, {54353, 55282}
X(68743) = barycentric quotient X(i)/X(j) for these {i,j}: {142, 18031}, {241, 31618}, {354, 2481}, {518, 57815}, {665, 56322}, {672, 32008}, {926, 62725}, {1212, 36796}, {1418, 34018}, {1458, 21453}, {1475, 673}, {2223, 2346}, {2283, 6606}, {2293, 14942}, {2340, 56118}, {2488, 885}, {3693, 63239}, {3930, 56127}, {8012, 6559}, {9454, 1174}, {10581, 28132}, {20229, 294}, {20683, 56157}, {22053, 31637}, {34855, 42311}, {35312, 46135}, {35326, 666}, {35338, 51560}, {39258, 56255}, {40983, 36124}, {46388, 62747}, {51384, 76}, {52020, 13576}, {52635, 1170}, {53241, 57537}, {53539, 65552}, {54353, 55281}, {59217, 63229}, {61241, 65847}, {61376, 56783}, {63203, 34085}, {65195, 36803}
X(68743) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {354, 40606, 21795}, {991, 4253, 6}, {1155, 43046, 14936}, {3002, 5030, 62371}

X(68744) = BROCARD AXIS INTERCEPT OF X(30)X(39569)

Barycentrics    a^2*(a^2 - b^2 - c^2)^2*(2*a^6 - a^4*b^2 - b^6 - a^4*c^2 + b^4*c^2 + b^2*c^4 - c^6) : :
X(68744) = 2 X[3] + X[3284], X[340] - 7 X[3523], 4 X[549] - X[45312], 3 X[3163] - 2 X[59661], 2 X[23583] + X[44252]

X(68744) lies on these lines: {3, 6}, {20, 10002}, {30, 39569}, {97, 5481}, {110, 34147}, {122, 11064}, {125, 44888}, {140, 37081}, {154, 6617}, {184, 426}, {185, 14575}, {206, 63419}, {250, 2071}, {325, 34841}, {340, 3523}, {394, 53852}, {417, 13367}, {418, 22352}, {441, 1503}, {458, 20792}, {520, 4091}, {549, 45312}, {852, 1495}, {1092, 53844}, {1196, 45030}, {1297, 52058}, {1576, 34146}, {2972, 3292}, {3148, 19124}, {3163, 59661}, {3564, 15526}, {3796, 26880}, {3917, 23606}, {3964, 14379}, {5012, 46832}, {5159, 13611}, {5447, 19210}, {6000, 15781}, {6389, 6776}, {6394, 12215}, {6527, 15258}, {6530, 23583}, {6641, 43650}, {7193, 35072}, {8550, 41005}, {8721, 28696}, {8798, 11441}, {9306, 46831}, {9475, 51437}, {9530, 44704}, {10192, 45200}, {10257, 12095}, {11793, 14152}, {12203, 28723}, {13198, 47195}, {13366, 13409}, {13557, 64689}, {13558, 60774}, {14767, 37124}, {14981, 54075}, {15069, 20208}, {15407, 36212}, {15512, 19360}, {15578, 61748}, {18592, 37527}, {19128, 52279}, {21663, 23200}, {22078, 31388}, {25406, 37188}, {28406, 59363}, {29012, 44231}, {29181, 44248}, {33582, 67888}, {34109, 40948}, {34156, 58258}, {36177, 45847}, {36794, 44924}, {36988, 42459}, {37072, 59767}, {39020, 63440}, {39071, 65749}, {41204, 64781}, {42353, 48906}, {44716, 47413}, {44891, 61691}, {46372, 52566}, {46841, 52128}, {52734, 58070}, {52772, 65753}, {52950, 53795}, {65735, 68158}

X(68744) = midpoint of X(i) and X(j) for these {i,j}: {250, 2071}, {6530, 44252}
X(68744) = reflection of X(6530) in X(23583)
X(68744) = isotomic conjugate of the polar conjugate of X(8779)
X(68744) = isogonal conjugate of the polar conjugate of X(441)
X(68744) = X(810)-complementary conjugate of X(38970)
X(68744) = X(i)-Ceva conjugate of X(j) for these (i,j): {441, 8779}, {43754, 520}
X(68744) = X(i)-isoconjugate of X(j) for these (i,j): {4, 8767}, {19, 6330}, {92, 43717}, {158, 1297}, {162, 68640}, {523, 36092}, {661, 65265}, {823, 34212}, {1096, 35140}, {1577, 32687}, {2435, 36126}, {6520, 64975}, {14618, 36046}, {24006, 44770}, {24019, 43673}, {24022, 66964}, {36119, 52485}, {36120, 39265}, {36128, 56601}
X(68744) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 6330}, {125, 68640}, {1147, 1297}, {1511, 52485}, {6503, 35140}, {15595, 264}, {22391, 43717}, {23976, 2052}, {33504, 14618}, {35071, 43673}, {36033, 8767}, {36830, 65265}, {37867, 64975}, {39071, 4}, {39073, 6530}, {39473, 58258}, {45248, 14944}, {46093, 2435}, {46094, 39265}, {50938, 1093}, {57296, 66161}, {65726, 16081}
X(68744) = crosspoint of X(i) and X(j) for these (i,j): {3, 17974}, {17932, 23582}
X(68744) = crosssum of X(i) and X(j) for these (i,j): {4, 6530}, {3269, 17994}
X(68744) = crossdifference of every pair of points on line {393, 523}
X(68744) = barycentric product X(i)*X(j) for these {i,j}: {3, 441}, {63, 8766}, {69, 8779}, {110, 39473}, {326, 2312}, {394, 1503}, {520, 34211}, {577, 30737}, {1092, 60516}, {1259, 43045}, {2409, 52613}, {2445, 4143}, {3292, 36894}, {3719, 51647}, {3926, 42671}, {3964, 16318}, {4176, 51437}, {6394, 9475}, {15595, 17974}, {15905, 16096}, {23357, 58258}, {34156, 36212}, {40080, 65722}, {51386, 51963}, {51394, 63856}, {64975, 65749}
X(68744) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 6330}, {48, 8767}, {110, 65265}, {163, 36092}, {184, 43717}, {394, 35140}, {441, 264}, {520, 43673}, {577, 1297}, {647, 68640}, {1092, 64975}, {1503, 2052}, {1576, 32687}, {2312, 158}, {2409, 15352}, {2445, 6529}, {3284, 52485}, {3289, 39265}, {3292, 56601}, {6793, 52661}, {8766, 92}, {8779, 4}, {9475, 6530}, {15905, 14944}, {16096, 52581}, {16318, 1093}, {17974, 9476}, {30737, 18027}, {32320, 2435}, {32661, 44770}, {34156, 16081}, {34211, 6528}, {35282, 37778}, {36894, 46111}, {39201, 34212}, {39473, 850}, {42671, 393}, {51363, 13450}, {51437, 6524}, {52613, 2419}, {58258, 23962}, {58796, 61189}, {60341, 66161}, {65749, 60516}
X(68744) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 182, 216}, {3, 577, 63433}, {3, 2055, 15644}, {3, 10316, 5188}, {3, 13346, 52543}, {3, 14961, 18860}, {3, 15905, 1350}, {3, 23115, 30270}, {3, 37893, 52520}, {3, 38292, 34815}, {110, 44436, 34147}, {184, 426, 6509}, {34815, 38292, 11477}, {36748, 53094, 3}, {45552, 45553, 37515}

X(68745) = BROCARD AXIS INTERCEPT OF X(20)X(160)

Barycentrics    a^2*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8 + a^8*c^2 - 2*a^6*b^2*c^2 + 4*a^4*b^4*c^2 - 2*a^2*b^6*c^2 - b^8*c^2 - 3*a^6*c^4 + 4*a^4*b^2*c^4 - 2*a^2*b^4*c^4 + b^6*c^4 + 3*a^4*c^6 - 2*a^2*b^2*c^6 + b^4*c^6 - a^2*c^8 - b^2*c^8) : :
X(68745) = 4 X[3] - X[5201], X[1634] + 2 X[47620]

X(68745) lies on these lines: {3, 6}, {20, 160}, {69, 63421}, {141, 54003}, {237, 29181}, {378, 44145}, {446, 5103}, {523, 2071}, {858, 23181}, {1503, 1634}, {1624, 11064}, {2781, 44716}, {3260, 9723}, {3564, 53246}, {3964, 63420}, {5480, 35222}, {5999, 60514}, {7526, 45847}, {7667, 23195}, {8719, 21312}, {10516, 32444}, {11328, 53023}, {12058, 52032}, {14096, 21167}, {15466, 55354}, {15577, 33801}, {15874, 34808}, {16030, 22352}, {18859, 62490}, {19189, 35474}, {20775, 44882}, {20794, 64080}, {23161, 41327}, {23217, 44886}, {25406, 66886}, {31952, 36990}, {34146, 36212}, {34968, 43574}, {37183, 68166}, {37465, 51538}, {40049, 67614}, {44704, 52604}, {48874, 52274}, {51255, 67321}, {53267, 66078}, {53273, 54996}

X(68745) = circumcircle-inverse of X(67352)
X(68745) = crossdifference of every pair of points on line {523, 800}
X(68745) = X(i)-line conjugate of X(j) for these (i,j): {3, 800}, {2071, 523}
X(68745) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1350, 8266}, {1379, 1380, 67352}, {9723, 11413, 63419}, {23217, 51360, 44886}, {30262, 50645, 37473}

X(68746) = BROCARD AXIS INTERCEPT OF X(1)X(29)

Barycentrics    a^2*(a + b)*(a + c)*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*b*c - 2*a^2*b^3*c + b^5*c + a^4*c^2 - b^4*c^2 - 2*a^2*b*c^3 - 2*b^3*c^3 - 2*a^2*c^4 - b^2*c^4 + b*c^5 + c^6) : :

X(68746) lies on these lines: {1, 29}, {3, 6}, {21, 3074}, {27, 1745}, {28, 73}, {35, 4055}, {42, 35981}, {81, 1816}, {223, 46883}, {255, 56001}, {943, 40591}, {1465, 18180}, {1817, 4303}, {1860, 12047}, {2194, 2594}, {2287, 3682}, {2360, 9798}, {2635, 31902}, {3194, 14547}, {3216, 7572}, {3362, 44698}, {4658, 41344}, {5327, 37529}, {7078, 8021}, {7538, 19767}, {7567, 37732}, {8555, 30117}, {8885, 22063}, {10458, 26091}, {10571, 54394}, {16609, 37523}, {18603, 44547}, {19366, 53819}, {25516, 37694}, {30268, 48897}, {32590, 32593}, {32591, 32592}, {34831, 45206}, {37142, 37573}, {37565, 66745}, {40602, 41608}, {54407, 64347}

X(68746) = isogonal conjugate of X(56227)
X(68746) = X(i)-isoconjugate of X(j) for these (i,j): {1, 56227}, {520, 681}, {1214, 53817}, {39201, 57930}
X(68746) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 56227}, {38969, 656}
X(68746) = crossdifference of every pair of points on line {523, 822}
X(68746) = barycentric product X(i)*X(j) for these {i,j}: {29, 53819}, {284, 53821}, {333, 19366}, {680, 823}, {2299, 53818}, {18180, 26941}, {24019, 35521}
X(68746) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 56227}, {680, 24018}, {823, 57930}, {2299, 53817}, {19366, 226}, {24019, 681}, {26941, 56189}, {53819, 307}, {53821, 349}
X(68746) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1047, 67847}, {81, 1816, 3075}, {56001, 68716, 255}

X(68747) = BROCARD AXIS INTERCEPT OF X(2)X(7054)

Barycentrics    a^3*(a^2 - b^2 - c^2)*(a^3 - a*b^2 - 2*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2) : :

X(68747) lies on these lines: {2, 7054}, {3, 6}, {9, 3465}, {19, 13733}, {34, 1950}, {37, 54430}, {48, 184}, {53, 31789}, {71, 52425}, {73, 22054}, {219, 1794}, {233, 6863}, {255, 14597}, {393, 6987}, {405, 56831}, {440, 1751}, {604, 22341}, {966, 27407}, {1006, 1172}, {1011, 2299}, {1062, 63848}, {1100, 1214}, {1104, 5301}, {1213, 7515}, {1409, 3215}, {1474, 13738}, {1841, 54368}, {1914, 10319}, {1968, 5317}, {1974, 33718}, {2189, 13726}, {2204, 37246}, {2241, 41340}, {2268, 40946}, {2269, 53850}, {2911, 14827}, {3087, 6908}, {3145, 44103}, {4220, 10311}, {5475, 30445}, {5839, 6350}, {6748, 6907}, {6928, 36412}, {7078, 22118}, {7536, 17056}, {10314, 19544}, {11429, 26908}, {14575, 22369}, {15830, 56911}, {15831, 16885}, {17277, 25950}, {17398, 18641}, {21566, 26912}, {23079, 52016}, {26880, 44087}, {37180, 63055}

X(68747) = Brocard-circle-inverse of X(18591)
X(68747) = isotomic conjugate of the polar conjugate of X(5320)
X(68747) = isogonal conjugate of the polar conjugate of X(405)
X(68747) = X(59097)-complementary conjugate of X(21259)
X(68747) = X(405)-Ceva conjugate of X(5320)
X(68747) = X(i)-isoconjugate of X(j) for these (i,j): {19, 57831}, {92, 51223}, {264, 2215}, {273, 2335}, {1577, 36077}, {7649, 54970}, {8747, 63235}, {17924, 65227}, {24006, 68202}, {24019, 63220}, {36080, 46107}
X(68747) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 57831}, {22391, 51223}, {35071, 63220}, {38967, 14618}
X(68747) = crosspoint of X(i) and X(j) for these (i,j): {3, 45127}, {1751, 54972}
X(68747) = crosssum of X(i) and X(j) for these (i,j): {6, 14017}, {579, 581}, {2335, 51875}
X(68747) = crossdifference of every pair of points on line {523, 17924}
X(68747) = barycentric product X(i)*X(j) for these {i,j}: {3, 405}, {48, 5271}, {69, 5320}, {78, 1451}, {184, 44140}, {219, 37543}, {255, 39585}, {662, 46382}, {906, 23882}, {1259, 54394}, {1331, 46385}, {1333, 42706}, {1437, 5295}, {1882, 68649}, {3682, 56831}, {36054, 65355}, {57241, 65180}
X(68747) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 57831}, {184, 51223}, {405, 264}, {520, 63220}, {906, 54970}, {1451, 273}, {1576, 36077}, {3990, 63235}, {5271, 1969}, {5320, 4}, {9247, 2215}, {32656, 65227}, {32661, 68202}, {37543, 331}, {39585, 57806}, {42706, 27801}, {44140, 18022}, {46382, 1577}, {46385, 46107}, {52425, 2335}, {65180, 52938}
X(68747) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 18591}, {3, 2193, 577}, {6, 41332, 32}, {48, 212, 3990}, {219, 11517, 52386}, {284, 580, 6}

X(68748) = BROCARD AXIS INTERCEPT OF X(44)X(9454)

Barycentrics    a^2*(a^4*b - a^2*b^3 + a^4*c + 2*a^3*b*c + a^2*b^2*c - a*b^3*c + a^2*b*c^2 - b^3*c^2 - a^2*c^3 - a*b*c^3 - b^2*c^3) : :

X(68748) lies on these lines: {3, 6}, {44, 9454}, {238, 7113}, {513, 1919}, {604, 1284}, {662, 20142}, {672, 64710}, {1100, 20718}, {1914, 1977}, {2223, 9016}, {7175, 52023}, {9456, 51333}, {16503, 17768}, {17139, 17379}, {34079, 37128}, {46922, 64912}

X(68748) = crossdifference of every pair of points on line {523, 984}
X(68748) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 3285, 41333}, {6, 3286, 2245}

X(68749) = X(2)-LINE CONJUGATE OF X(8)

Barycentrics    a^2*(a^2*b^2 + a*b^3 - 2*a*b^2*c - b^3*c + a^2*c^2 - 2*a*b*c^2 + 2*b^2*c^2 + a*c^3 - b*c^3) : :

X(68749) lies on these lines: {1, 2}, {35, 34876}, {221, 38286}, {373, 22172}, {649, 854}, {727, 901}, {750, 63504}, {902, 8054}, {1155, 3248}, {1279, 17477}, {2234, 4706}, {2239, 20464}, {2309, 46904}, {3121, 20459}, {3123, 67494}, {3218, 23579}, {4414, 22343}, {6377, 21760}, {9082, 33849}, {9364, 65231}, {9490, 21750}, {16405, 23493}, {17126, 23578}, {17187, 25059}, {46716, 56312}

X(68749) = X(65231)-Ceva conjugate of X(649)
X(68749) = crosspoint of X(1) and X(9432)
X(68749) = crosssum of X(1) and X(5205)
X(68749) = crossdifference of every pair of points on line {8, 649}
X(68749) = X(i)-line conjugate of X(j) for these (i,j): {1, 8}, {854, 649}
X(68749) = barycentric product X(1)*X(57037)
X(68749) = barycentric quotient X(57037)/X(75)

X(68750) = X(1)-LINE CONJUGATE OF X(10)

Barycentrics    a^2*(a^2*b^2 + a*b^3 - a*b^2*c - b^3*c + a^2*c^2 - a*b*c^2 + a*c^3 - b*c^3) : :

X(68750) lies on these lines: {1, 2}, {171, 40148}, {238, 8054}, {244, 50362}, {649, 834}, {896, 3248}, {902, 5143}, {1964, 46904}, {2229, 17475}, {3231, 6377}, {3666, 17187}, {3685, 14752}, {4093, 57024}, {4414, 7032}, {7109, 23533}, {8620, 21760}, {16373, 16969}, {17495, 18792}, {18170, 32917}, {20961, 66022}, {20965, 21827}, {21747, 23578}, {22024, 46720}, {43922, 61433}, {46153, 62739}, {57037, 67494}

X(68750) = isogonal conjugate of the isotomic conjugate of X(57029)
X(68750) = X(65239)-Ceva conjugate of X(649)
X(68750) = crosspoint of X(1) and X(17954)
X(68750) = crosssum of X(1) and X(17763)
X(68750) = crossdifference of every pair of points on line {10, 649}
X(68750) = X(i)-line conjugate of X(j) for these (i,j): {1, 10}, {834, 649}
X(68750) = barycentric product X(i)*X(j) for these {i,j}: {1, 57039}, {6, 57029}, {63, 52461}
X(68750) = barycentric quotient X(i)/X(j) for these {i,j}: {52461, 92}, {57029, 76}, {57039, 75}
X(68750) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 60684, 10459}, {8620, 21760, 46148}

X(68751) = X(1)-LINE CONJUGATE OF X(43)

Barycentrics    a*(a^2*b^2 - 2*a^2*b*c - a*b^2*c + a^2*c^2 - a*b*c^2 + 2*b^2*c^2) : :

X(68751) lies on these lines: {1, 2}, {38, 3727}, {75, 63504}, {87, 1278}, {192, 18194}, {194, 23457}, {330, 17155}, {350, 3226}, {518, 20356}, {536, 3248}, {649, 4083}, {660, 49675}, {672, 17475}, {726, 23579}, {748, 16969}, {894, 23532}, {1015, 20868}, {1100, 2235}, {1107, 3989}, {1266, 53541}, {1964, 4852}, {2170, 20590}, {2308, 62813}, {2309, 4360}, {3123, 9025}, {3729, 23524}, {3747, 14752}, {3875, 7032}, {3923, 23578}, {4365, 17144}, {4699, 24661}, {4772, 25528}, {8026, 32915}, {8621, 16971}, {9359, 23539}, {14839, 20456}, {16515, 59207}, {17187, 33296}, {17375, 25573}, {17793, 25298}, {20669, 20688}, {20963, 42548}, {20985, 54282}, {23427, 25264}, {23508, 49470}, {23633, 25048}, {24659, 27311}, {24672, 27343}, {24766, 53676}, {25284, 27011}, {25292, 34832}, {25505, 25618}, {25535, 25624}, {25726, 52161}, {32924, 34063}, {38247, 53678}, {40780, 49448}, {46901, 62803}, {64546, 64555}

X(68751) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2162, 20355}, {7121, 39354}, {34077, 21219}
X(68751) = X(20688)-cross conjugate of X(20530)
X(68751) = X(20530)-Dao conjugate of X(726)
X(68751) = crosspoint of X(1) and X(3226)
X(68751) = crosssum of X(1) and X(3009)
X(68751) = crossdifference of every pair of points on line {43, 649}
X(68751) = X(i)-line conjugate of X(j) for these (i,j): {1, 43}, {4083, 649}
X(68751) = barycentric product X(i)*X(j) for these {i,j}: {1, 20530}, {75, 20669}, {86, 20688}, {92, 20757}
X(68751) = barycentric quotient X(i)/X(j) for these {i,j}: {20530, 75}, {20669, 1}, {20688, 10}, {20757, 63}, {20787, 20785}
X(68751) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 239, 3009}, {1, 4393, 42}, {1, 16834, 869}, {1, 21352, 3720}, {1, 49477, 1193}, {1, 50023, 1149}, {192, 18194, 22343}, {239, 3009, 899}, {1278, 63527, 87}, {4360, 18170, 2309}, {46126, 62659, 899}

X(68752) = X(1)-LINE CONJUGATE OF X(145)

Barycentrics    a^2*(a^2*b^2 + a*b^3 - 4*a*b^2*c - b^3*c + a^2*c^2 - 4*a*b*c^2 + 6*b^2*c^2 + a*c^3 - b*c^3) : :

X(68752) lies on these lines: {1, 2}, {649, 6363}, {750, 23532}, {1054, 23579}, {3685, 16576}, {3689, 17477}, {4413, 63504}, {23524, 64112}, {23578, 56010}

X(68752) = crossdifference of every pair of points on line {145, 649}
X(68752) = X(i)-line conjugate of X(j) for these (i,j): {1, 145}, {6363, 649}

X(68753) = X(1)-LINE CONJUGATE OF X(200)

Barycentrics    a*(a^3*b + 2*a^2*b^2 + a*b^3 + a^3*c - 6*a^2*b*c - a*b^2*c - 2*b^3*c + 2*a^2*c^2 - a*b*c^2 + 4*b^2*c^2 + a*c^3 - 2*b*c^3) : :

X(68753) lies on these lines: {1, 2}, {649, 3669}, {2260, 64558}, {3000, 3248}, {6180, 38266}, {17474, 27499}, {21785, 28351}, {23579, 53602}, {23649, 37555}

X(68753) = crosspoint of X(i) and X(j) for these (i,j): {1, 51845}, {5382, 36146}
X(68753) = crosssum of X(i) and X(j) for these (i,j): {1, 56714}, {2340, 4936}
X(68753) = crossdifference of every pair of points on line {200, 649}
X(68753) = X(i)-line conjugate of X(j) for these (i,j): {1, 200}, {3669, 649}
X(68753) = barycentric product X(1)*X(57033)
X(68753) = barycentric quotient X(57033)/X(75)

X(68754) = X(1)-LINE CONJUGATE OF X(1125)

Barycentrics    a^2*(a^2*b^2 + a*b^3 + a*b^2*c - b^3*c + a^2*c^2 + a*b*c^2 - 4*b^2*c^2 + a*c^3 - b*c^3) : :

X(68725) lies on these lines: {1, 2}, {649, 4057}, {660, 741}, {756, 17187}, {1740, 64178}, {1757, 58292}, {2234, 3994}, {3231, 21830}, {3952, 18792}, {4557, 62740}, {5145, 9330}, {8054, 37680}, {32911, 40148}, {56878, 61433}

X(68754) = X(39441)-Ceva conjugate of X(42)
X(68754) = crossdifference of every pair of points on line {649, 1125}
X(68754) = X(i)-line conjugate of X(j) for these (i,j): {1, 1125}, {4057, 649}
X(68754) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2664, 17763, 899}, {3231, 21830, 46148}

X(68755) = X(1)-LINE CONJUGATE OF X(612)

Barycentrics    a*(a^3*b + 2*a^2*b^2 + a*b^3 + a^3*c - 2*a^2*b*c - a*b^2*c - 2*b^3*c + 2*a^2*c^2 - a*b*c^2 + a*c^3 - 2*b*c^3) : :

X(68755) lies on these lines: {1, 2}, {63, 23649}, {649, 905}, {902, 21495}, {1429, 33854}, {1457, 43053}, {1716, 20978}, {2183, 57039}, {2235, 28358}, {2269, 27633}, {2275, 56509}, {3666, 39244}, {3727, 3752}, {3915, 21477}, {4383, 9310}, {5109, 41311}, {8621, 45785}, {11512, 24590}, {14953, 18792}, {16483, 21526}, {16502, 25940}, {17187, 68712}, {17474, 37676}, {24403, 49757}, {28351, 28365}, {28367, 28391}, {39956, 44421}, {50995, 62214}

X(68755) = crossdifference of every pair of points on line {612, 649}
X(68755) = X(i)-line conjugate of X(j) for these (i,j): {1, 612}, {905, 649}
X(68755) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {239, 28254, 899}, {899, 3009, 56714}, {23511, 56630, 25930}

X(68756) = X(2)-LINE CONJUGATE OF X(2276)

Barycentrics    a^3*b + a^2*b^2 + a*b^3 + a^3*c - 2*a^2*b*c - a*b^2*c - b^3*c + a^2*c^2 - a*b*c^2 + a*c^3 - b*c^3 : :

X(68756) lies on these lines: {2, 37}, {190, 31348}, {213, 24166}, {244, 742}, {514, 659}, {524, 17449}, {596, 4721}, {614, 4376}, {712, 49997}, {726, 4465}, {899, 9055}, {982, 24690}, {1086, 4766}, {1149, 35101}, {1279, 4760}, {3006, 25357}, {3230, 57029}, {4363, 32944}, {4364, 46901}, {4392, 4643}, {4395, 26007}, {4396, 26273}, {4413, 17318}, {4694, 8682}, {4713, 17155}, {4725, 17145}, {7292, 24358}, {16604, 17489}, {21216, 63493}, {24165, 24330}, {24631, 46907}, {24699, 32844}, {28352, 59515}

X(68756) = midpoint of X(17495) and X(24403)
X(68756) = crossdifference of every pair of points on line {667, 2276}
X(68756) = X(i)-line conjugate of X(j) for these (i,j): {2, 2276}, {514, 667}
X(68756) = {X(26273),X(32922)}-harmonic conjugate of X(4396)

X(68757) = X(1)-LINE CONJUGATE OF X(1026)

Barycentrics    a*(b - c)^2*(a^4 + a^2*b^2 - 3*a^2*b*c - a*b^2*c + a^2*c^2 - a*b*c^2 + 2*b^2*c^2) : :

X(68757) lies on these lines: {1, 2}, {244, 649}, {660, 3315}, {1015, 4378}, {1086, 24719}, {3248, 4724}, {4063, 24196}, {7035, 25531}, {7192, 23824}, {14752, 68153}, {21210, 47691}

X(68757) = crossdifference of every pair of points on line {649, 1026}
X(68757) = X(i)-line conjugate of X(j) for these (i,j): {1, 1026}, {244, 649}
X(68757) = barycentric product X(513)*X(62324)
X(68757) = barycentric quotient X(62324)/X(668)

X(68758) = X(1)X(3)∩X(4)X(4306)

Barycentrics    a*(a^5*b - 2*a^3*b^3 + a*b^5 + a^5*c + a^3*b^2*c - a^2*b^3*c - 2*a*b^4*c + b^5*c + a^3*b*c^2 + 2*a^2*b^2*c^2 + a*b^3*c^2 - 2*a^3*c^3 - a^2*b*c^3 + a*b^2*c^3 - 2*b^3*c^3 - 2*a*b*c^4 + a*c^5 + b*c^5) : :

X(68758) lies on these lines: {1, 3}, {4, 4306}, {10, 1066}, {11, 1464}, {34, 34052}, {73, 1210}, {77, 17863}, {104, 59073}, {106, 53702}, {222, 57278}, {227, 67937}, {242, 514}, {405, 55406}, {495, 56191}, {500, 12433}, {515, 1458}, {519, 1818}, {580, 3562}, {581, 938}, {774, 5884}, {855, 3937}, {912, 1736}, {936, 58466}, {946, 1042}, {950, 4303}, {951, 7549}, {956, 17811}, {991, 3488}, {995, 37642}, {1012, 1407}, {1044, 41869}, {1056, 4648}, {1064, 11019}, {1068, 24159}, {1070, 12609}, {1106, 5450}, {1193, 64124}, {1254, 31870}, {1387, 43036}, {1406, 1777}, {1457, 32486}, {1647, 34913}, {1724, 3157}, {1725, 11570}, {1737, 4551}, {1745, 9581}, {1785, 64115}, {1877, 51649}, {2654, 4292}, {3000, 28150}, {3073, 34043}, {3086, 10571}, {3120, 51751}, {3191, 3868}, {3293, 5399}, {3465, 60062}, {3554, 40941}, {3682, 24391}, {3720, 64110}, {3739, 9623}, {3872, 4359}, {3911, 22350}, {3924, 59285}, {4300, 63999}, {4304, 22053}, {4322, 5882}, {4341, 64334}, {5262, 8555}, {5603, 7365}, {6180, 6913}, {8025, 51382}, {8608, 61237}, {8776, 61224}, {9312, 17866}, {9708, 25878}, {10459, 66230}, {11041, 50189}, {11573, 48883}, {12109, 48909}, {13724, 23154}, {14597, 40979}, {15325, 34586}, {17074, 37469}, {17495, 38460}, {19335, 64489}, {19684, 30807}, {21620, 59305}, {21669, 65114}, {22837, 24176}, {24470, 48903}, {37699, 67931}, {39544, 59611}, {42884, 64449}, {43744, 56273}, {45022, 46419}, {52407, 52680}, {52635, 58036}, {54391, 63068}, {56816, 60356}, {58587, 63346}

X(68758) = X(1)-Ceva conjugate of X(46419)
X(68758) = X(22350)-Dao conjugate of X(51379)
X(68758) = crossdifference of every pair of points on line {71, 650}
X(68758) = barycentric product X(i)*X(j) for these {i,j}: {514, 7451}, {34234, 45022}
X(68758) = barycentric quotient X(i)/X(j) for these {i,j}: {7451, 190}, {45022, 908}
X(68758) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 57, 63982}, {1, 3339, 37529}, {1, 60786, 37569}, {56, 41344, 37530}, {73, 1210, 37732}, {942, 37565, 3670}, {950, 4303, 48897}, {1319, 14115, 36}, {1406, 62333, 1777}, {1457, 44675, 32486}, {1877, 51649, 61227}, {2654, 4292, 52524}, {5399, 67980, 3293}, {8758, 18838, 1735}, {15325, 34586, 49997}, {15934, 50317, 1}, {15934, 52506, 11529}, {17074, 62873, 37469}

X(68759) = X(1)X(3)∩X(2)X(5826)

Barycentrics    a*(a^2*b^2 - b^4 - a*b^2*c + b^3*c + a^2*c^2 - a*b*c^2 + b*c^3 - c^4) : :

X(68759) lies on these lines: {1, 3}, {2, 5826}, {9, 18161}, {10, 30801}, {19, 53996}, {37, 4503}, {44, 7202}, {69, 21078}, {77, 1766}, {101, 7291}, {116, 48381}, {141, 41007}, {142, 1953}, {150, 6542}, {169, 25930}, {219, 16551}, {222, 21375}, {239, 24618}, {304, 20922}, {306, 21621}, {394, 1726}, {511, 1736}, {514, 661}, {527, 3942}, {572, 1442}, {664, 6996}, {674, 17463}, {760, 4447}, {1018, 25083}, {1020, 62402}, {1086, 17444}, {1111, 43040}, {1350, 64750}, {1375, 17044}, {1401, 21333}, {1441, 24220}, {1746, 1943}, {1765, 52385}, {1818, 44661}, {1830, 2356}, {1855, 17442}, {1930, 17864}, {2114, 6211}, {2170, 3008}, {2171, 3664}, {2182, 6510}, {2183, 16578}, {2224, 2990}, {2269, 25065}, {2310, 29353}, {2323, 16560}, {2324, 18725}, {2340, 2809}, {3009, 4475}, {3061, 17284}, {3191, 18732}, {3218, 41405}, {3430, 66593}, {3661, 17181}, {3663, 17452}, {3674, 65578}, {3688, 17447}, {3726, 67027}, {3730, 24635}, {3878, 56509}, {3917, 21318}, {3930, 49765}, {3970, 3995}, {4006, 17294}, {4051, 16833}, {4053, 17374}, {4312, 68033}, {4336, 24309}, {4359, 40687}, {4466, 63844}, {4552, 29069}, {4553, 35552}, {4559, 20744}, {4681, 29602}, {4847, 21328}, {4851, 18733}, {4862, 41777}, {4887, 53538}, {4888, 7201}, {5011, 11349}, {5195, 6999}, {5525, 18728}, {5723, 19512}, {5729, 11477}, {8555, 37431}, {9037, 24433}, {9317, 53591}, {13006, 43034}, {15299, 64084}, {16577, 22097}, {16783, 55871}, {17023, 24581}, {17052, 22073}, {17077, 21271}, {17234, 18041}, {17245, 17443}, {17296, 41010}, {17317, 18714}, {17451, 29571}, {17742, 18727}, {17868, 24199}, {17895, 24224}, {18139, 40677}, {18148, 18156}, {18184, 18206}, {20731, 45932}, {21253, 46534}, {21362, 34371}, {21368, 22128}, {21370, 64082}, {21446, 43166}, {21951, 31198}, {24209, 53526}, {24559, 25532}, {25935, 34847}, {25940, 30144}, {27248, 27249}, {28609, 29573}, {28978, 29497}, {29594, 33299}, {29598, 31190}, {29604, 39244}, {29960, 29961}, {30033, 30038}, {30059, 30087}, {31395, 40784}, {35338, 44670}, {40188, 56354}, {40965, 60785}, {41572, 64702}, {43044, 63203}, {44708, 64708}, {46468, 49542}

X(68759) = midpoint of X(3942) and X(21801)
X(68759) = reflection of X(i) in X(j) for these {i,j}: {2183, 16578}, {20367, 241}, {57022, 17463}
X(68759) = X(i)-isoconjugate of X(j) for these (i,j): {6, 1311}, {101, 60580}, {112, 66949}, {522, 32689}, {650, 36094}
X(68759) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 1311}, {1015, 60580}, {34591, 66949}
X(68759) = crossdifference of every pair of points on line {31, 650}
X(68759) = barycentric product X(i)*X(j) for these {i,j}: {1, 33864}, {75, 8679}, {7463, 14208}
X(68759) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1311}, {109, 36094}, {513, 60580}, {656, 66949}, {1415, 32689}, {7463, 162}, {8679, 1}, {33864, 75}
X(68759) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 18788, 40910}, {1959, 3912, 57015}, {37596, 68478, 3670}, {45267, 45268, 45266}, {45268, 45270, 45269}

X(68760) = X(1)X(3)∩X(2)X(15621)

Barycentrics    a^2*(a^3*b - a*b^3 + a^3*c - 2*a^2*b*c + 2*a*b^2*c - b^3*c + 2*a*b*c^2 - a*c^3 - b*c^3) : :

X(68760) lies on these lines: {1, 3}, {2, 15621}, {6, 23375}, {8, 19245}, {10, 19241}, {11, 19546}, {23, 385}, {100, 20470}, {145, 23361}, {238, 23404}, {333, 1222}, {497, 19647}, {516, 15626}, {518, 53280}, {519, 859}, {528, 851}, {529, 855}, {535, 13744}, {551, 16374}, {899, 54333}, {902, 3286}, {946, 15623}, {962, 15622}, {1001, 16373}, {1011, 4428}, {1125, 19249}, {1284, 17724}, {1376, 30942}, {1412, 51476}, {1486, 45738}, {1621, 29822}, {1698, 19253}, {1995, 26227}, {2177, 5132}, {2225, 2284}, {3058, 4192}, {3185, 3870}, {3218, 23845}, {3241, 4216}, {3616, 19292}, {3679, 4245}, {3722, 3724}, {3813, 27622}, {3828, 19265}, {3868, 23844}, {3911, 40451}, {3913, 13738}, {3915, 54300}, {3935, 4557}, {3996, 17751}, {4009, 33845}, {4184, 42028}, {4188, 15625}, {4191, 4421}, {4318, 53321}, {4413, 16421}, {4423, 29825}, {4669, 19293}, {4857, 19648}, {5047, 26115}, {5298, 19335}, {5399, 67967}, {5434, 37331}, {7292, 53307}, {7428, 8666}, {7496, 29823}, {8053, 17379}, {8229, 45920}, {8301, 15571}, {8715, 16453}, {9840, 15888}, {10072, 19550}, {10385, 37400}, {11238, 19540}, {11284, 29828}, {12329, 23397}, {12513, 28348}, {12607, 13724}, {12632, 27649}, {14942, 36002}, {16056, 34612}, {16059, 31137}, {16297, 50605}, {16405, 48805}, {17718, 31394}, {17735, 41395}, {19250, 19875}, {19251, 51093}, {19252, 51105}, {19254, 51071}, {19256, 34619}, {19261, 25055}, {19291, 53620}, {19513, 37722}, {19732, 68700}, {20041, 37683}, {20372, 61164}, {20605, 38347}, {20760, 41711}, {20872, 20999}, {20989, 51621}, {22345, 34791}, {23374, 64169}, {23393, 31999}, {23846, 34772}, {24488, 56878}, {27145, 34444}, {27621, 64068}, {28238, 64123}, {28353, 35466}, {28364, 37663}, {28365, 34445}, {28393, 51415}, {31140, 47522}, {34230, 67502}, {34607, 37262}, {35223, 36635}, {35224, 40916}, {37269, 52804}, {37425, 63273}, {37449, 53302}, {37631, 50423}, {37924, 62491}, {37959, 53246}, {39572, 53260}, {39914, 65163}, {46973, 57024}, {54391, 62401}, {55362, 62837}, {56181, 64199}, {62236, 65313}, {68147, 68148}

X(68760) = reflection of X(45916) in X(8758)
X(68760) = circumcircle-inverse of X(34583)
X(68760) = X(3227)-Ceva conjugate of X(6)
X(68760) = X(3230)-Dao conjugate of X(536)
X(68760) = crosspoint of X(i) and X(j) for these (i,j): {59, 898}, {251, 739}
X(68760) = crosssum of X(i) and X(j) for these (i,j): {11, 891}, {141, 536}
X(68760) = crossdifference of every pair of points on line {39, 650}
X(68760) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {55, 56, 37540}, {55, 23853, 16678}, {1381, 1382, 34583}, {1402, 3744, 16687}, {15621, 18613, 2}, {20475, 23398, 20875}

X(68761) = X(1)X(3)∩X(7)X(4392)

Barycentrics    a*(a + b - c)*(a - b + c)*(a*b^2 - b^3 + a*c^2 - c^3) : :

X(68761) lies on these lines: {1, 3}, {7, 4392}, {10, 1393}, {11, 1736}, {12, 67979}, {34, 62858}, {38, 226}, {63, 34036}, {73, 3874}, {109, 3218}, {201, 1125}, {222, 45728}, {223, 62823}, {225, 596}, {227, 3555}, {238, 1421}, {244, 3011}, {278, 24477}, {291, 2006}, {347, 64151}, {497, 62811}, {516, 7004}, {518, 1465}, {522, 693}, {553, 42038}, {595, 7098}, {603, 4347}, {614, 1708}, {651, 62235}, {758, 1457}, {774, 12053}, {899, 43048}, {946, 44706}, {984, 5219}, {1064, 18389}, {1072, 1210}, {1193, 15556}, {1254, 10106}, {1407, 12595}, {1427, 17625}, {1441, 46909}, {1456, 61225}, {1458, 5083}, {1463, 53540}, {1699, 24430}, {1725, 30384}, {1729, 22131}, {1739, 40663}, {1759, 56913}, {1776, 64013}, {1777, 24467}, {1788, 24046}, {1935, 6763}, {1937, 3254}, {2003, 32913}, {2292, 64160}, {2323, 3509}, {2635, 2801}, {3073, 54432}, {3149, 66235}, {3216, 41538}, {3315, 7677}, {3720, 16577}, {3741, 6358}, {3751, 56418}, {3752, 41539}, {3782, 64127}, {3817, 7069}, {3838, 24434}, {3868, 10571}, {3873, 17080}, {3987, 41687}, {4022, 4032}, {4303, 12005}, {4310, 54366}, {4492, 7204}, {4552, 29824}, {4848, 24443}, {4850, 7672}, {4858, 26013}, {4968, 6734}, {5087, 24433}, {5211, 17950}, {5226, 7226}, {5231, 49483}, {5262, 55101}, {5435, 26228}, {5904, 37694}, {5905, 34029}, {6745, 43068}, {7191, 55086}, {7292, 37787}, {8543, 62796}, {9335, 64114}, {9955, 35194}, {10198, 56444}, {10527, 56367}, {11019, 64708}, {12116, 26929}, {12649, 56819}, {14547, 62852}, {15253, 35466}, {16586, 39046}, {16823, 60705}, {17063, 31231}, {17140, 52358}, {17194, 58578}, {17605, 24431}, {17724, 43056}, {18183, 24471}, {19372, 41229}, {20470, 23067}, {21147, 62874}, {22465, 23710}, {24026, 64858}, {24028, 28234}, {24231, 64115}, {24539, 24541}, {24985, 24987}, {26363, 56366}, {28780, 33170}, {29676, 36482}, {29821, 52423}, {30493, 67967}, {32780, 56453}, {33139, 37771}, {33148, 37797}, {34589, 64194}, {37736, 49675}, {37800, 64153}, {42289, 46901}, {42753, 67494}, {43039, 57015}, {45126, 62819}, {51378, 61222}, {51706, 54346}, {54154, 56825}, {64176, 64736}

X(68761) = reflection of X(i) in X(j) for these {i,j}: {4551, 1465}, {45269, 1735}, {64194, 34589}
X(68761) = isotomic conjugate of the isogonal conjugate of X(51657)
X(68761) = X(29068)-anticomplementary conjugate of X(329)
X(68761) = X(674)-cross conjugate of X(57015)
X(68761) = X(i)-isoconjugate of X(j) for these (i,j): {9, 2224}, {11, 52941}, {41, 37130}, {55, 675}, {101, 60573}, {284, 60135}, {522, 32682}, {650, 36087}, {2175, 43093}, {4636, 66281}
X(68761) = X(i)-Dao conjugate of X(j) for these (i,j): {223, 675}, {478, 2224}, {1015, 60573}, {3160, 37130}, {38990, 650}, {40590, 60135}, {40593, 43093}, {43039, 29069}, {53980, 33}, {65919, 9}
X(68761) = cevapoint of X(674) and X(43039)
X(68761) = crossdifference of every pair of points on line {41, 650}
X(68761) = barycentric product X(i)*X(j) for these {i,j}: {7, 57015}, {57, 3006}, {75, 43039}, {76, 51657}, {85, 674}, {651, 23887}, {1441, 14964}, {2225, 6063}, {3676, 42723}, {4554, 65703}, {8618, 20567}
X(68761) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 37130}, {56, 2224}, {57, 675}, {65, 60135}, {85, 43093}, {109, 36087}, {513, 60573}, {674, 9}, {1415, 32682}, {2149, 52941}, {2225, 55}, {3006, 312}, {4249, 65201}, {8618, 41}, {14964, 21}, {23887, 4391}, {42723, 3699}, {43039, 1}, {51657, 6}, {57015, 8}, {57185, 66281}, {64611, 1320}, {65703, 650}
X(68761) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1758, 2078}, {1, 5536, 1936}, {241, 3999, 3660}, {1427, 21342, 17625}, {1458, 17449, 5083}, {3218, 4318, 109}, {5083, 18593, 1458}, {7191, 67120, 55086}, {8758, 18839, 1}, {9364, 18201, 57}, {18193, 60786, 57}

X(68762) = X(1)X(3)∩X(6)X(650)

Barycentrics    a*(a^5 - a^4*b - a^3*b^2 + a^2*b^3 - a^4*c + 3*a^3*b*c - a^2*b^2*c - 2*a*b^3*c + b^4*c - a^3*c^2 - a^2*b*c^2 + 4*a*b^2*c^2 - b^3*c^2 + a^2*c^3 - 2*a*b*c^3 - b^2*c^3 + b*c^4) : :

X(68762) lies on these lines: {1, 3}, {6, 650}, {7, 7045}, {11, 109}, {31, 1647}, {58, 3109}, {63, 24433}, {81, 3658}, {89, 37131}, {222, 11502}, {238, 61649}, {255, 24914}, {269, 15737}, {513, 52242}, {515, 52440}, {528, 35281}, {603, 1837}, {651, 45885}, {750, 4675}, {946, 18360}, {1054, 53743}, {1086, 45946}, {1100, 9356}, {1210, 1399}, {1331, 3035}, {1406, 3149}, {1411, 11700}, {1464, 6905}, {1478, 56416}, {1737, 52407}, {1768, 53524}, {1777, 10896}, {1795, 12832}, {1807, 11570}, {1836, 9316}, {1935, 17606}, {2006, 64155}, {2222, 3025}, {2361, 3911}, {2801, 52371}, {3700, 34361}, {3937, 53279}, {3939, 6174}, {4292, 51889}, {4369, 24279}, {4641, 61653}, {4644, 40629}, {5400, 61225}, {6126, 6127}, {6326, 53537}, {6911, 52005}, {7354, 18340}, {8614, 37732}, {8750, 23711}, {9440, 52638}, {10090, 34586}, {10950, 67476}, {11246, 61732}, {12740, 53530}, {13589, 67627}, {14127, 61731}, {14266, 14584}, {14812, 60698}, {15252, 24465}, {16173, 46819}, {17122, 61648}, {17126, 36086}, {17638, 64761}, {24457, 38530}, {24495, 35992}, {25005, 33118}, {36037, 51402}, {36280, 67213}, {38238, 57520}, {41701, 53531}, {51422, 66206}, {51476, 66485}, {53302, 61674}, {53389, 61221}, {64704, 67472}

X(68762) = crossdifference of every pair of points on line {517, 650}
X(68762) = X(i)-line conjugate of X(j) for these (i,j): {1, 517}, {6, 650}
X(68762) = barycentric product X(i)*X(j) for these {i,j}: {9, 67576}, {3218, 14204}
X(68762) = barycentric quotient X(i)/X(j) for these {i,j}: {14204, 18359}, {67576, 85}
X(68762) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 36, 60687}, {1, 57, 53525}, {1, 23703, 55}, {651, 60782, 45885}, {942, 33649, 1}, {1086, 51408, 45946}, {1936, 9364, 1155}, {11700, 12736, 1411}, {34345, 65702, 1}, {35015, 60718, 1836}

X(68763) = X(1)X(3)∩X(11)X(1785)

Barycentrics    a*(a^5*b + a^4*b^2 - 2*a^3*b^3 - 2*a^2*b^4 + a*b^5 + b^6 + a^5*c - 4*a^4*b*c + 2*a^3*b^2*c + 4*a^2*b^3*c - 3*a*b^4*c + a^4*c^2 + 2*a^3*b*c^2 - 4*a^2*b^2*c^2 + 2*a*b^3*c^2 - b^4*c^2 - 2*a^3*c^3 + 4*a^2*b*c^3 + 2*a*b^2*c^3 - 2*a^2*c^4 - 3*a*b*c^4 - b^2*c^4 + a*c^5 + c^6) : :

X(68763) lies on these lines: {1, 3}, {11, 1785}, {21, 16697}, {33, 22753}, {34, 12114}, {44, 52962}, {73, 12675}, {78, 66235}, {104, 1455}, {106, 2765}, {221, 63399}, {223, 63430}, {225, 63980}, {227, 944}, {475, 3086}, {515, 1465}, {518, 22350}, {519, 24025}, {522, 905}, {614, 57278}, {774, 1201}, {912, 34586}, {960, 44706}, {995, 62811}, {997, 25091}, {1012, 34036}, {1042, 64132}, {1064, 10391}, {1068, 10785}, {1071, 10571}, {1074, 2886}, {1103, 6762}, {1125, 59638}, {1158, 34040}, {1193, 44547}, {1210, 64172}, {1393, 7686}, {1427, 4293}, {1457, 6001}, {1468, 64722}, {1519, 38357}, {1736, 49997}, {1737, 16610}, {1741, 22124}, {1745, 12680}, {1777, 34862}, {1795, 52440}, {1854, 63986}, {1877, 2829}, {1878, 42753}, {2654, 13374}, {2975, 66610}, {3616, 24538}, {3693, 13006}, {3752, 18391}, {3880, 24028}, {3911, 51375}, {4297, 67022}, {4303, 58567}, {4305, 15852}, {4318, 6909}, {4347, 63983}, {5089, 7117}, {5253, 66593}, {5313, 18412}, {5731, 17080}, {5768, 56821}, {6681, 52537}, {6905, 51361}, {7078, 62858}, {7191, 62873}, {7354, 56814}, {8609, 23986}, {9119, 22063}, {10072, 50103}, {10085, 64057}, {11398, 22654}, {14872, 37694}, {14986, 19785}, {15252, 15325}, {16466, 62810}, {17916, 46830}, {19541, 36985}, {20122, 64662}, {20324, 26877}, {20418, 23710}, {22072, 63976}, {22758, 37697}, {22835, 35015}, {24026, 64930}, {24046, 67937}, {34046, 64347}, {36123, 61429}, {40943, 53994}, {44075, 51490}, {50604, 67944}, {51889, 67857}, {55399, 62874}

X(68763) = midpoint of X(i) and X(j) for these {i,j}: {1, 1735}, {1457, 7004}
X(68763) = reflection of X(9371) in X(60415)
X(68763) = incircle-inverse of X(31849)
X(68763) = X(i)-complementary conjugate of X(j) for these (i,j): {56, 117}, {102, 1329}, {1397, 23986}, {32643, 514}, {32677, 3452}, {36040, 513}, {36055, 34823}, {36067, 20316}, {36100, 21244}, {65297, 3835}
X(68763) = X(1309)-Ceva conjugate of X(513)
X(68763) = crosspoint of X(7) and X(16082)
X(68763) = crosssum of X(6) and X(51361)
X(68763) = crossdifference of every pair of points on line {198, 650}
X(68763) = barycentric product X(7)*X(62326)
X(68763) = barycentric quotient X(62326)/X(8)
X(68763) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 36, 46974}, {1, 982, 67949}, {1, 3338, 41344}, {1, 3953, 50196}, {1, 8069, 3744}, {1, 22766, 37539}, {1, 30274, 66687}, {1, 59335, 5710}, {104, 1870, 1455}, {2446, 2447, 31849}

X(68764) = X(1)X(3)∩X(11)X(68704)

Barycentrics    a^3*(a + b - c)*(a - b + c)*(a^2 - b^2 - c^2)*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3) : :

X(68764) lies on these lines: {1, 3}, {11, 68704}, {20, 54200}, {31, 26880}, {34, 28348}, {58, 19365}, {73, 22345}, {108, 2734}, {184, 603}, {208, 13738}, {216, 1400}, {222, 23206}, {225, 27622}, {227, 23361}, {386, 19366}, {388, 25876}, {404, 23661}, {408, 40952}, {418, 40956}, {441, 43053}, {577, 604}, {672, 35072}, {855, 1877}, {856, 3911}, {859, 1465}, {1106, 52218}, {1118, 27621}, {1359, 15326}, {1361, 1457}, {1404, 3284}, {1405, 5158}, {1455, 53321}, {1459, 1946}, {1532, 1846}, {1785, 64784}, {1795, 68240}, {1804, 54404}, {1809, 54391}, {1951, 32674}, {2171, 63848}, {2183, 47434}, {2361, 36040}, {2968, 40663}, {3937, 51649}, {4188, 53151}, {4216, 17080}, {4292, 51759}, {5396, 20122}, {5440, 23067}, {6509, 28274}, {8283, 11500}, {10571, 67011}, {13587, 44699}, {15654, 21147}, {15905, 67843}, {17408, 37257}, {19372, 28383}, {20803, 33597}, {20967, 46016}, {22350, 65743}, {22376, 57293}, {24027, 52440}, {28272, 46831}, {28386, 34823}, {36059, 52407}, {36743, 53853}, {37264, 44696}, {38292, 38296}, {40944, 63397}, {47410, 62371}, {47521, 54346}, {57672, 66921}

X(68764) = isogonal conjugate of the isotomic conjugate of X(62402)
X(68764) = isogonal conjugate of the polar conjugate of X(1465)
X(68764) = X(53702)-complementary conjugate of X(20316)
X(68764) = X(i)-Ceva conjugate of X(j) for these (i,j): {3, 56973}, {859, 1457}, {1295, 221}, {23981, 8677}, {36058, 603}, {53995, 23980}, {65297, 36054}
X(68764) = X(i)-isoconjugate of X(j) for these (i,j): {4, 51565}, {8, 36123}, {9, 16082}, {19, 36795}, {29, 38955}, {33, 18816}, {92, 52663}, {104, 318}, {158, 1809}, {264, 2342}, {281, 34234}, {522, 1309}, {643, 68561}, {650, 65223}, {909, 7017}, {1146, 39294}, {1897, 43728}, {2250, 31623}, {2299, 57984}, {2401, 65160}, {3064, 13136}, {3239, 65331}, {3699, 43933}, {4163, 65537}, {4397, 36110}, {5081, 40437}, {6335, 61238}, {7020, 15501}, {7101, 34051}, {14776, 35519}, {23104, 59103}, {32641, 46110}, {32702, 52622}, {36037, 44426}
X(68764) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 36795}, {226, 57984}, {478, 16082}, {1147, 1809}, {3259, 44426}, {8677, 35014}, {22391, 52663}, {23980, 7017}, {34467, 43728}, {36033, 51565}, {39004, 4397}, {40613, 318}, {55060, 68561}, {57293, 2804}, {60339, 23978}
X(68764) = crosspoint of X(i) and X(j) for these (i,j): {3, 36055}, {1465, 62402}, {24027, 36040}
X(68764) = crosssum of X(i) and X(j) for these (i,j): {318, 5081}, {14304, 24026}
X(68764) = crossdifference of every pair of points on line {281, 650}
X(68764) = barycentric product X(i)*X(j) for these {i,j}: {3, 1465}, {6, 62402}, {48, 22464}, {57, 22350}, {63, 1457}, {77, 2183}, {222, 517}, {394, 1875}, {603, 908}, {651, 8677}, {859, 1214}, {905, 23981}, {934, 52307}, {1262, 35014}, {1319, 57478}, {1361, 65302}, {1407, 51379}, {1408, 51367}, {1409, 17139}, {1415, 65868}, {1459, 24029}, {1769, 1813}, {1785, 7125}, {1797, 53530}, {1804, 14571}, {1814, 53548}, {3262, 52411}, {3310, 6516}, {4091, 23706}, {4554, 23220}, {6735, 7099}, {10015, 36059}, {32660, 36038}, {34051, 65743}, {36058, 52659}, {36100, 56973}, {42753, 44717}, {46974, 60000}, {47408, 63190}, {52212, 52407}, {53549, 65296}
X(68764) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 36795}, {48, 51565}, {56, 16082}, {109, 65223}, {184, 52663}, {222, 18816}, {517, 7017}, {577, 1809}, {603, 34234}, {604, 36123}, {859, 31623}, {1214, 57984}, {1409, 38955}, {1415, 1309}, {1457, 92}, {1465, 264}, {1769, 46110}, {1875, 2052}, {2183, 318}, {3310, 44426}, {7180, 68561}, {7335, 65302}, {8677, 4391}, {9247, 2342}, {22350, 312}, {22383, 43728}, {22464, 1969}, {23220, 650}, {23981, 6335}, {24027, 39294}, {32660, 36037}, {34346, 37768}, {35014, 23978}, {36059, 13136}, {51379, 59761}, {52307, 4397}, {52411, 104}, {53530, 46109}, {53548, 46108}, {56973, 64194}, {57181, 43933}, {61057, 14571}, {62258, 14578}, {62402, 76}
X(68764) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 17102, 40946}, {56, 64721, 10475}, {57, 63982, 65}, {603, 7114, 7335}, {1410, 22344, 603}, {1470, 37579, 10832}

X(68765) = X(1)X(3)∩X(10)X(6757)

Barycentrics    a*(a + b - c)*(a - b + c)*(b + c)^2*(a^2 - b^2 + b*c - c^2) : :
X(68765) = 4 X[18593] - X[53537]

X(68765) lies on these lines: {1, 3}, {10, 6757}, {12, 201}, {30, 1725}, {31, 66724}, {34, 7299}, {38, 5434}, {47, 32047}, {58, 41697}, {73, 59282}, {79, 5492}, {80, 62491}, {109, 5127}, {225, 41501}, {226, 60116}, {227, 4849}, {244, 5298}, {388, 7226}, {394, 1406}, {500, 67946}, {523, 656}, {664, 17731}, {758, 1464}, {774, 6284}, {860, 51462}, {896, 7286}, {920, 64053}, {984, 11237}, {1042, 45288}, {1046, 8614}, {1325, 52380}, {1393, 5433}, {1399, 4296}, {1411, 37791}, {1427, 64041}, {1441, 5224}, {1465, 32126}, {1478, 24431}, {1708, 57277}, {1737, 45926}, {1788, 24883}, {1835, 2245}, {1836, 45924}, {1870, 2361}, {2292, 3649}, {2310, 65632}, {2594, 15556}, {2650, 63295}, {3218, 52440}, {3485, 24936}, {3585, 35194}, {3743, 11553}, {3911, 24168}, {4053, 35069}, {4331, 64086}, {4351, 52407}, {4570, 35049}, {5018, 39137}, {5080, 24433}, {5620, 52383}, {6126, 56844}, {7004, 15326}, {7066, 7143}, {7100, 63339}, {7288, 9335}, {7354, 44706}, {8261, 54356}, {10122, 63356}, {11684, 52372}, {12943, 24430}, {15443, 20617}, {16139, 63388}, {16575, 60353}, {17637, 48897}, {17950, 44370}, {18360, 35193}, {18477, 66719}, {18524, 56812}, {18977, 66658}, {21677, 56839}, {21839, 61170}, {24443, 37646}, {24880, 24881}, {30358, 33097}, {35550, 41804}, {40149, 60112}, {41542, 52680}, {41547, 63292}, {41551, 63366}, {42440, 58889}, {49745, 67935}, {52378, 57649}, {54292, 67120}, {57722, 60321}

X(68765) = reflection of X(i) in X(j) for these {i,j}: {1464, 18593}, {53524, 1725}, {53537, 1464}
X(68765) = isogonal conjugate of X(52380)
X(68765) = X(i)-Ceva conjugate of X(j) for these (i,j): {65, 53537}, {18815, 226}
X(68765) = X(i)-isoconjugate of X(j) for these (i,j): {1, 52380}, {21, 759}, {28, 1793}, {29, 57736}, {36, 62713}, {58, 6740}, {60, 80}, {81, 2341}, {101, 60571}, {261, 6187}, {270, 1807}, {283, 68571}, {284, 24624}, {314, 67166}, {333, 34079}, {522, 36069}, {593, 36910}, {650, 37140}, {663, 65283}, {757, 52371}, {849, 52409}, {1098, 1411}, {2006, 7054}, {2150, 18359}, {2161, 2185}, {2189, 52351}, {2194, 14616}, {2222, 65575}, {2299, 57985}, {4391, 32671}, {4636, 66284}, {7252, 47318}, {9273, 21044}, {35192, 66922}, {35193, 68483}, {46103, 52431}, {52426, 57555}, {64835, 65568}
X(68765) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 52380}, {10, 6740}, {44, 30606}, {226, 57985}, {758, 4511}, {1015, 60571}, {1214, 14616}, {4075, 52409}, {15267, 1411}, {15898, 62713}, {34586, 21}, {35069, 333}, {35204, 1098}, {38982, 522}, {38984, 65575}, {40584, 2185}, {40586, 2341}, {40590, 24624}, {40591, 1793}, {40607, 52371}, {40611, 759}, {40612, 261}, {51583, 314}, {53982, 29}, {55065, 52356}, {56325, 18359}, {60342, 53524}
X(68765) = crosspoint of X(i) and X(j) for these (i,j): {226, 18815}, {758, 860}, {18593, 41804}
X(68765) = crosssum of X(i) and X(j) for these (i,j): {58, 51420}, {284, 2361}, {759, 57736}
X(68765) = trilinear pole of line {2610, 51663}
X(68765) = crossdifference of every pair of points on line {284, 650}
X(68765) = barycentric product X(i)*X(j) for these {i,j}: {7, 4053}, {10, 18593}, {12, 3218}, {36, 6358}, {37, 41804}, {56, 61410}, {65, 3936}, {181, 20924}, {190, 51663}, {201, 17923}, {226, 758}, {306, 1835}, {320, 2171}, {321, 1464}, {349, 3724}, {594, 1443}, {651, 6370}, {664, 2610}, {756, 17078}, {860, 1214}, {1231, 44113}, {1254, 32851}, {1400, 35550}, {1441, 2245}, {1870, 26942}, {2006, 4736}, {3028, 18359}, {3738, 4605}, {4080, 53537}, {4242, 57243}, {4453, 21859}, {4511, 6354}, {4551, 4707}, {4552, 53527}, {4554, 42666}, {4585, 66287}, {5081, 37755}, {7113, 34388}, {7265, 68430}, {18815, 35069}, {22128, 56285}, {28654, 52440}, {42701, 52382}, {52413, 57807}, {53546, 65958}
X(68765) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 52380}, {12, 18359}, {36, 2185}, {37, 6740}, {42, 2341}, {65, 24624}, {71, 1793}, {109, 37140}, {181, 2161}, {201, 52351}, {214, 30606}, {226, 14616}, {320, 52379}, {513, 60571}, {594, 52409}, {651, 65283}, {654, 65575}, {756, 36910}, {758, 333}, {860, 31623}, {1214, 57985}, {1254, 2006}, {1400, 759}, {1402, 34079}, {1409, 57736}, {1415, 36069}, {1443, 1509}, {1464, 81}, {1500, 52371}, {1835, 27}, {1870, 46103}, {1880, 68571}, {1983, 4636}, {2088, 53524}, {2161, 62713}, {2171, 80}, {2197, 1807}, {2245, 21}, {2323, 1098}, {2361, 7054}, {2610, 522}, {3028, 3218}, {3218, 261}, {3724, 284}, {3936, 314}, {4024, 52356}, {4053, 8}, {4511, 7058}, {4551, 47318}, {4605, 35174}, {4707, 18155}, {4736, 32851}, {6354, 18815}, {6358, 20566}, {6370, 4391}, {7113, 60}, {17078, 873}, {17923, 57779}, {18593, 86}, {18815, 57555}, {20924, 18021}, {21794, 56422}, {21828, 3737}, {21859, 51562}, {35069, 4511}, {35550, 28660}, {37755, 52392}, {41804, 274}, {42666, 650}, {44113, 1172}, {51663, 514}, {52382, 66922}, {52407, 65568}, {52413, 270}, {52427, 2326}, {52434, 2150}, {52440, 593}, {53525, 26856}, {53527, 4560}, {53537, 16704}, {53562, 1021}, {57185, 66284}, {61060, 7113}, {61410, 3596}, {66287, 60074}
X(68765) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {46, 1060, 5348}, {201, 1254, 12}, {4296, 7098, 1399}

X(68766) = X(1)X(3)∩X(8)X(27506)

Barycentrics    a*(a - b - c)*(a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + a^4*c - 2*a^3*b*c + 2*a*b^3*c - b^4*c + a^3*c^2 - 2*a*b^2*c^2 + b^3*c^2 - a^2*c^3 + 2*a*b*c^3 + b^2*c^3 - a*c^4 - b*c^4) : :

X(68766) lies on these lines: {1, 3}, {8, 27506}, {9, 22074}, {10, 22072}, {11, 49997}, {20, 10571}, {30, 34586}, {33, 997}, {43, 5727}, {73, 4297}, {78, 3701}, {109, 6909}, {212, 993}, {214, 38981}, {221, 37022}, {222, 63991}, {255, 5450}, {386, 3486}, {497, 995}, {515, 4551}, {516, 1457}, {519, 38955}, {522, 663}, {581, 4305}, {603, 63983}, {758, 7004}, {856, 55044}, {946, 1074}, {950, 1193}, {956, 7074}, {978, 9581}, {1064, 4304}, {1066, 4311}, {1103, 12650}, {1125, 2654}, {1201, 3914}, {1450, 11019}, {1464, 15326}, {1682, 8240}, {1724, 22760}, {1765, 22134}, {1837, 3216}, {1854, 5730}, {2238, 53561}, {2293, 50294}, {2361, 52680}, {2617, 7424}, {2635, 28164}, {3159, 3191}, {3293, 10950}, {3465, 6326}, {3872, 56860}, {4314, 50604}, {5080, 18340}, {5179, 61224}, {5218, 30116}, {5267, 22361}, {5399, 34773}, {5432, 56191}, {5440, 22306}, {5691, 37694}, {5692, 24430}, {5720, 36985}, {6001, 33810}, {6735, 61222}, {6836, 56819}, {6840, 38945}, {6890, 34030}, {6925, 34029}, {7069, 10176}, {7078, 12114}, {8804, 22063}, {9323, 54391}, {9841, 66693}, {10448, 54430}, {10572, 37732}, {10703, 62826}, {11813, 35015}, {11998, 45751}, {12047, 52524}, {13744, 56884}, {14942, 35333}, {14963, 47411}, {17064, 21214}, {17622, 45219}, {17647, 40950}, {17749, 54361}, {19861, 24537}, {23205, 53389}, {30384, 32486}, {30943, 46016}, {31806, 44706}, {34040, 64074}, {37374, 51421}, {37737, 48903}, {38357, 51409}, {39757, 47621}, {40091, 66199}, {40611, 50702}, {40945, 63389}, {40957, 41006}, {45770, 64054}, {48805, 59573}, {51558, 64577}, {52005, 68367}, {52428, 62828}, {54400, 64129}, {55086, 62873}, {62754, 67267}, {66957, 66977}

X(68766) = reflection of X(i) in X(j) for these {i,j}: {1735, 60415}, {4551, 22350}, {45269, 9371}
X(68766) = X(36795)-Ceva conjugate of X(9)
X(68766) = X(522)-isoconjugate of X(59073)
X(68766) = X(2183)-Dao conjugate of X(1465)
X(68766) = crosspoint of X(21) and X(51565)
X(68766) = crosssum of X(i) and X(j) for these (i,j): {65, 1457}, {35015, 66287}
X(68766) = crossdifference of every pair of points on line {650, 1400}
X(68766) = barycentric product X(i)*X(j) for these {i,j}: {662, 14310}, {6332, 7461}, {36795, 40613}
X(68766) = barycentric quotient X(i)/X(j) for these {i,j}: {1415, 59073}, {7461, 653}, {14310, 1577}, {40613, 1465}
X(68766) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3, 37558}, {1, 165, 24806}, {1, 6282, 8270}, {1, 7987, 37523}, {8, 27506, 56861}, {1193, 65670, 950}, {2646, 23207, 3576}, {3100, 4511, 45272}, {10572, 54427, 37732}

X(68767) = X(1)X(3)∩X(30)X(1875)

Barycentrics    a*(a + b - c)*(a - b + c)*(a^2 - b^2 - c^2)*(a^3*b - a^2*b^2 + a*b^3 - b^4 + a^3*c - a*b^2*c - a^2*c^2 - a*b*c^2 + 2*b^2*c^2 + a*c^3 - c^4) : :

X(68767) lies on these lines: {1, 3}, {5, 1887}, {30, 1875}, {34, 37241}, {108, 3100}, {109, 3827}, {123, 908}, {222, 17441}, {226, 34822}, {296, 64887}, {377, 5081}, {516, 59816}, {518, 2968}, {521, 4025}, {603, 18732}, {851, 1465}, {1012, 1905}, {1071, 67901}, {1118, 6851}, {1368, 18588}, {1452, 13730}, {1455, 44662}, {1829, 37397}, {1859, 8727}, {1888, 20420}, {1895, 6836}, {2262, 15905}, {2299, 37227}, {2823, 16870}, {3149, 67965}, {3869, 28921}, {4463, 28774}, {4848, 34823}, {6001, 68704}, {6350, 7672}, {7012, 37449}, {7125, 20277}, {7335, 66760}, {10391, 20122}, {11375, 34120}, {12671, 15498}, {13369, 30493}, {14961, 43039}, {15556, 34851}, {17080, 35980}, {17975, 22148}, {20243, 57477}, {21147, 52359}, {22053, 44708}, {22119, 34036}, {23115, 56913}, {34050, 44661}, {40663, 60427}, {41005, 54344}, {56549, 62811}, {61178, 64194}

X(68767) = X(i)-complementary conjugate of X(j) for these (i,j): {929, 20316}, {1459, 15612}
X(68767) = X(57801)-Ceva conjugate of X(1214)
X(68767) = X(i)-isoconjugate of X(j) for these (i,j): {33, 43363}, {607, 37214}
X(68767) = X(i)-Dao conjugate of X(j) for these (i,j): {851, 243}, {5179, 1948}, {15612, 44426}, {65935, 281}
X(68767) = crosspoint of X(77) and X(40843)
X(68767) = crosssum of X(33) and X(2202)
X(68767) = crossdifference of every pair of points on line {607, 650}
X(68767) = barycentric product X(i)*X(j) for these {i,j}: {77, 5179}, {348, 44670}, {1214, 14956}, {6516, 47137}
X(68767) = barycentric quotient X(i)/X(j) for these {i,j}: {77, 37214}, {222, 43363}, {5179, 318}, {14956, 31623}, {44670, 281}, {47137, 44426}
X(68767) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {603, 54360, 18732}, {10391, 46017, 20122}

X(68768) = X(1)X(3)∩X(100)X(101)

Barycentrics    a*(a - b)*(a - c)*(2*a^2 - a*b + b^2 - a*c - 2*b*c + c^2) : :

X(68768) lies on these lines: {1, 3}, {2, 67583}, {10, 37009}, {31, 16479}, {100, 101}, {103, 67754}, {108, 53888}, {109, 1292}, {200, 67385}, {238, 52902}, {244, 11716}, {513, 23344}, {514, 2398}, {516, 24980}, {519, 47043}, {572, 7676}, {677, 68267}, {692, 35338}, {901, 1022}, {902, 27846}, {910, 1642}, {934, 1293}, {1019, 4236}, {1020, 40576}, {1024, 36086}, {1025, 5377}, {1027, 54325}, {1083, 1376}, {1156, 12034}, {1253, 24309}, {1477, 63775}, {1633, 3939}, {2175, 60785}, {2222, 2742}, {2743, 14733}, {3008, 8647}, {3667, 62669}, {4557, 61232}, {5435, 67341}, {5528, 16554}, {5723, 28915}, {5853, 20780}, {6011, 53243}, {6014, 14074}, {6161, 11124}, {8683, 14723}, {9778, 67570}, {10196, 17780}, {11712, 35293}, {11714, 46684}, {13589, 14513}, {14942, 24618}, {16686, 53391}, {19240, 56191}, {21889, 53408}, {23343, 46973}, {23696, 35333}, {23838, 39154}, {23845, 53388}, {24029, 40577}, {28850, 45749}, {30236, 53622}, {32630, 58126}, {33952, 65166}, {35258, 46407}, {38371, 53337}, {50808, 67572}, {53279, 61222}, {63442, 68591}

X(68768) = isogonal conjugate of X(35355)
X(68768) = X(i)-Ceva conjugate of X(j) for these (i,j): {5377, 1}, {36146, 101}, {53337, 23704}
X(68768) = X(48032)-cross conjugate of X(1279)
X(68768) = X(i)-isoconjugate of X(j) for these (i,j): {1, 35355}, {9, 37626}, {56, 60576}, {513, 1280}, {522, 1477}, {649, 36807}, {650, 43760}, {663, 35160}, {885, 56643}, {1086, 6078}, {1810, 7649}, {3675, 39272}
X(68768) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 60576}, {3, 35355}, {478, 37626}, {1279, 47695}, {3823, 47705}, {5375, 36807}, {16593, 693}, {35111, 4391}, {39026, 1280}, {39048, 514}, {61074, 1111}
X(68768) = cevapoint of X(i) and X(j) for these (i,j): {1279, 48032}, {2254, 4864}
X(68768) = crosspoint of X(i) and X(j) for these (i,j): {100, 36086}, {190, 51568}, {6632, 39293}
X(68768) = crosssum of X(513) and X(2254)
X(68768) = trilinear pole of line {1279, 2348}
X(68768) = crossdifference of every pair of points on line {244, 650}
X(68768) = X(11716)-line conjugate of X(244)
X(68768) = barycentric product X(i)*X(j) for these {i,j}: {1, 53337}, {7, 23704}, {100, 3008}, {190, 1279}, {651, 5853}, {664, 2348}, {666, 53552}, {765, 6084}, {1016, 48032}, {1026, 52210}, {1110, 65869}, {1332, 54234}, {2976, 5382}, {3257, 53534}, {4554, 8647}, {4564, 53523}, {4567, 53558}, {6335, 20780}, {7035, 8659}, {16593, 36086}, {20662, 51560}, {36037, 51419}, {36146, 40609}, {39048, 51568}, {43290, 51839}, {54325, 56667}
X(68768) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 35355}, {9, 60576}, {56, 37626}, {100, 36807}, {101, 1280}, {109, 43760}, {651, 35160}, {906, 1810}, {1110, 6078}, {1279, 514}, {1415, 1477}, {2348, 522}, {3008, 693}, {5853, 4391}, {6084, 1111}, {8647, 650}, {8659, 244}, {20662, 2254}, {20680, 4088}, {20780, 905}, {23704, 8}, {39048, 47695}, {48032, 1086}, {51419, 36038}, {53337, 75}, {53523, 4858}, {53534, 3762}, {53552, 918}, {53558, 16732}, {54234, 17924}
X(68768) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 165, 67417}, {55, 5091, 1}, {100, 3573, 1026}, {100, 35280, 101}, {100, 54440, 1018}, {1026, 3573, 1023}, {1633, 3939, 21362}, {4236, 54353, 1019}, {9441, 40910, 20367}, {23703, 67434, 23890}, {38013, 38014, 41343}, {67434, 67445, 67446}

X(68769) = X(1)X(3)∩X(6)X(75)

Barycentrics    a*(a^3*b + a^2*b^2 + a^3*c + a^2*b*c + a*b^2*c + a^2*c^2 + a*b*c^2 + 2*b^2*c^2) : :

X(68769) lies on these lines: {1, 3}, {2, 1258}, {6, 75}, {7, 28369}, {8, 37676}, {31, 21352}, {37, 21371}, {43, 46032}, {63, 1107}, {76, 41232}, {81, 330}, {86, 21769}, {172, 52134}, {194, 32939}, {213, 4383}, {219, 21233}, {222, 40765}, {274, 40153}, {553, 24215}, {750, 3009}, {869, 1376}, {981, 2481}, {984, 20715}, {985, 1580}, {1001, 3747}, {1191, 16823}, {1407, 7176}, {1914, 54419}, {1999, 17144}, {2241, 60721}, {2298, 41527}, {2300, 10436}, {2303, 16516}, {2664, 4413}, {3008, 3997}, {3052, 23407}, {3218, 62803}, {3219, 19735}, {3230, 16831}, {3242, 62872}, {3474, 66657}, {3721, 19734}, {3739, 27623}, {3782, 5244}, {3948, 4713}, {3975, 24514}, {4306, 50627}, {4307, 34253}, {4386, 20769}, {4503, 17274}, {4641, 21384}, {4699, 27644}, {4850, 65695}, {5114, 65168}, {5156, 54410}, {5275, 16514}, {5276, 62799}, {5283, 19731}, {5712, 6604}, {5880, 23682}, {6383, 40409}, {7263, 63359}, {8624, 11343}, {10459, 56509}, {11329, 21008}, {14839, 41265}, {15668, 16685}, {15978, 49745}, {16367, 17735}, {16466, 16825}, {16476, 40749}, {16483, 24331}, {16515, 40750}, {16523, 17475}, {16524, 24512}, {16815, 37679}, {16816, 32911}, {16817, 19728}, {16819, 19732}, {16826, 16969}, {16827, 19804}, {16832, 54981}, {16834, 20963}, {16884, 18166}, {16975, 18206}, {17011, 19714}, {17014, 63066}, {17023, 17750}, {17050, 24789}, {17143, 30710}, {17234, 27272}, {17279, 29988}, {17379, 21785}, {17448, 62853}, {17739, 23151}, {17752, 20917}, {19271, 57280}, {19725, 55466}, {20131, 21788}, {20257, 40940}, {20259, 40942}, {20964, 64170}, {22275, 56542}, {24268, 56913}, {24308, 27626}, {24621, 34063}, {27950, 52127}, {29570, 37633}, {29576, 37673}, {29578, 36647}, {29586, 41849}, {29960, 32777}, {31036, 32933}, {44733, 56358}, {49477, 62805}, {50023, 62828}

X(68769) = crossdifference of every pair of points on line {650, 788}
X(68769) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 57, 37596}, {1, 171, 21010}, {1, 5264, 37590}, {1, 20367, 980}, {1, 37555, 3666}, {1, 64560, 17597}, {2, 62813, 2176}, {213, 4384, 4383}, {980, 20367, 17595}, {2300, 10436, 28365}, {3210, 4393, 33296}, {5228, 5710, 940}, {16969, 37674, 16826}, {17475, 63099, 16523}

X(68770) = X(1)X(3)∩X(6)X(11)

Barycentrics    a*(a - b - c)*(a^4 + a^3*b - a^2*b^2 - a*b^3 + a^3*c + 2*a^2*b*c + a*b^2*c - 2*b^3*c - a^2*c^2 + a*b*c^2 + 4*b^2*c^2 - a*c^3 - 2*b*c^3) : :

X(68770) lies on these lines: {1, 3}, {2, 61397}, {6, 11}, {7, 8759}, {8, 28920}, {12, 64069}, {33, 1430}, {42, 11502}, {58, 62333}, {81, 497}, {155, 26470}, {212, 3720}, {222, 1836}, {333, 3193}, {390, 14996}, {394, 2886}, {499, 36754}, {518, 2000}, {611, 17720}, {613, 17721}, {908, 45729}, {946, 64020}, {1001, 2361}, {1036, 18178}, {1181, 63980}, {1203, 50443}, {1253, 9345}, {1331, 42843}, {1332, 30741}, {1364, 21746}, {1398, 64522}, {1399, 11496}, {1406, 4295}, {1407, 4331}, {1411, 64731}, {1468, 2654}, {1479, 36742}, {1699, 2003}, {1737, 44414}, {1776, 62795}, {1856, 5155}, {1899, 23304}, {1993, 11680}, {2323, 5231}, {2550, 63068}, {2594, 3149}, {3025, 38530}, {3157, 12047}, {3240, 60782}, {3474, 17074}, {3485, 3562}, {3751, 9817}, {3816, 10601}, {3829, 63094}, {3873, 37782}, {3925, 17811}, {4042, 6734}, {4336, 7004}, {4383, 33140}, {4413, 25934}, {4641, 7082}, {4679, 55432}, {5094, 21252}, {5218, 37633}, {5219, 68591}, {5263, 26625}, {5274, 37685}, {5275, 29639}, {5326, 37682}, {5432, 7074}, {5433, 36745}, {5713, 26481}, {5727, 16474}, {6180, 61716}, {6284, 36746}, {7069, 32912}, {7071, 13476}, {7078, 11375}, {7083, 18191}, {7354, 34046}, {7681, 10982}, {7741, 16473}, {7986, 11570}, {8227, 54301}, {9346, 53561}, {9370, 10895}, {9668, 51340}, {9669, 36750}, {10529, 37683}, {10589, 32911}, {10739, 38964}, {10833, 36740}, {10916, 32853}, {10947, 63979}, {11365, 18180}, {11376, 16466}, {11393, 44105}, {12053, 62805}, {12595, 36488}, {13374, 64722}, {14547, 62821}, {14969, 67264}, {15066, 33108}, {15253, 20330}, {15338, 37501}, {15908, 37498}, {16472, 37720}, {17027, 28934}, {17126, 66199}, {17300, 27542}, {17365, 38357}, {17605, 34048}, {17728, 52424}, {17768, 22129}, {19732, 26363}, {19734, 21321}, {21328, 26934}, {22136, 31493}, {22300, 37257}, {24390, 67963}, {24430, 32913}, {24477, 62798}, {24703, 55400}, {24725, 35015}, {26015, 45728}, {26475, 37357}, {27518, 31631}, {29662, 61395}, {30223, 62812}, {36942, 47522}, {37415, 55098}, {37692, 56535}, {38945, 40271}, {49478, 51361}, {52371, 64165}, {54421, 64042}, {64737, 68586}, {64747, 68593}

X(68770) = X(44059)-Ceva conjugate of X(513)
X(68770) = crossdifference of every pair of points on line {650, 928}
X(68770) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 57, 8758}, {1, 1936, 55}, {1, 3072, 11510}, {1, 37530, 37579}, {81, 497, 61398}, {1468, 2654, 22760}, {3075, 37529, 11509}, {7074, 37674, 5432}

X(68771) = X(1)X(3)∩X(8)X(479)

Barycentrics    a*(a + b - c)*(a - b + c)*(a*b - b^2 + a*c - c^2)*(a^3 - a^2*b + a*b^2 - b^3 - a^2*c + 2*a*b*c + b^2*c + a*c^2 + b*c^2 - c^3) : :

X(68771) lies on these lines: {1, 3}, {8, 479}, {105, 4318}, {347, 40965}, {516, 11028}, {518, 1362}, {665, 1938}, {672, 1876}, {910, 3827}, {1002, 7672}, {1279, 1416}, {1282, 5018}, {1323, 2809}, {1439, 3779}, {1456, 2348}, {1458, 53552}, {1530, 1836}, {1788, 41785}, {2263, 40131}, {3056, 14524}, {3242, 52013}, {3509, 51376}, {3869, 26658}, {4712, 6168}, {6182, 47123}, {6610, 9004}, {14077, 43042}, {18413, 53617}, {24476, 59242}, {28850, 67654}

X(68771) = midpoint of X(18413) and X(53617)
X(68771) = reflection of X(1362) in X(34855)
X(68771) = X(1002)-Ceva conjugate of X(1362)
X(68771) = X(i)-isoconjugate of X(j) for these (i,j): {105, 56098}, {294, 39273}, {522, 58989}, {673, 949}, {1438, 58004}, {3423, 14942}, {23696, 65215}, {28071, 63150}
X(68771) = X(i)-Dao conjugate of X(j) for these (i,j): {6184, 58004}, {39046, 56098}, {55057, 885}
X(68771) = crosssum of X(3220) and X(7290)
X(68771) = crossdifference of every pair of points on line {650, 949}
X(68771) = barycentric product X(i)*X(j) for these {i,j}: {241, 2550}, {518, 948}, {1025, 47123}, {2263, 3912}, {9436, 40131}, {28043, 62786}, {37580, 40704}
X(68771) = barycentric quotient X(i)/X(j) for these {i,j}: {518, 58004}, {672, 56098}, {948, 2481}, {1415, 58989}, {1458, 39273}, {2223, 949}, {2263, 673}, {2550, 36796}, {6182, 28132}, {28043, 6559}, {37580, 294}, {40131, 14942}, {52635, 3423}
X(68771) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {57, 8270, 55}, {1282, 5018, 58320}, {5902, 52510, 65}

X(68772) = ORTHIC AXIS INTERCEPT OF X(2)X(2517)

Barycentrics    a*(b - c)*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 2*a*b*c - b^2*c - a*c^2 - b*c^2 - c^3) : :
X(68772) = X[7649] - 3 X[47800], 4 X[14838] + X[59972], X[693] - 3 X[48209], X[1577] - 3 X[48186], X[4036] - 3 X[48181], X[4086] - 3 X[47794], X[4391] - 3 X[48165], X[4397] - 5 X[31209], X[4397] - 3 X[48204], 5 X[31209] - 3 X[48204], X[4560] + 3 X[48173], X[7650] - 3 X[48173], X[47965] + 2 X[59837], X[4811] - 3 X[26144], 2 X[4885] - 3 X[48207], 3 X[48207] - X[50334], 2 X[6050] + X[48350], 3 X[14413] - X[48342], X[20294] + 3 X[47798], 3 X[48168] - X[50327], 4 X[31287] - 3 X[48205], X[42312] + 3 X[47828], 3 X[47828] - X[50338], 2 X[44316] - 3 X[47802], 3 X[44429] - X[44444], 3 X[47820] - X[47844], 3 X[47839] - X[50331], 3 X[47888] - X[50345]

X(68772) lies on these lines: {2, 2517}, {36, 238}, {37, 23282}, {56, 51644}, {230, 231}, {514, 51648}, {521, 2605}, {522, 8062}, {649, 50332}, {656, 663}, {659, 28374}, {693, 48209}, {832, 1960}, {834, 48136}, {866, 42752}, {1459, 9001}, {1491, 25537}, {1577, 48186}, {1734, 48307}, {1769, 17418}, {1946, 48383}, {2254, 48340}, {2509, 3709}, {3309, 48306}, {3646, 24457}, {3667, 38324}, {3669, 4977}, {3716, 28623}, {3900, 48302}, {3907, 20316}, {3960, 4778}, {4017, 46385}, {4036, 48181}, {4041, 48303}, {4064, 46380}, {4086, 47794}, {4132, 50501}, {4139, 50504}, {4391, 48165}, {4397, 31209}, {4467, 57054}, {4560, 7650}, {4724, 50354}, {4777, 8043}, {4794, 59753}, {4802, 21112}, {4811, 26144}, {4885, 48207}, {4985, 48321}, {6050, 48350}, {8672, 50507}, {8678, 47842}, {14077, 48292}, {14413, 48342}, {16751, 48080}, {16755, 50450}, {16757, 27648}, {20294, 47798}, {21120, 59975}, {22091, 48390}, {23585, 40560}, {23788, 43067}, {23880, 48168}, {23882, 30591}, {24562, 50347}, {24948, 47975}, {25430, 35348}, {28147, 48003}, {28175, 47921}, {28205, 31318}, {29051, 47843}, {30235, 39540}, {30913, 48223}, {31287, 48205}, {35057, 48294}, {38469, 48330}, {42312, 47828}, {43060, 47998}, {44316, 47802}, {44429, 44444}, {45316, 64905}, {47820, 47844}, {47827, 59829}, {47839, 50331}, {47888, 50345}, {48325, 59971}, {53322, 61226}

X(68772) = midpoint of X(i) and X(j) for these {i,j}: {649, 50332}, {650, 6129}, {656, 663}, {667, 50330}, {1459, 17420}, {1491, 50353}, {1734, 48307}, {1769, 17418}, {2254, 48340}, {3737, 21189}, {4017, 46385}, {4040, 23800}, {4041, 48303}, {4560, 7650}, {4724, 50354}, {4794, 59753}, {4985, 48321}, {42312, 50338}, {47827, 59829}, {48297, 53527}, {48302, 57099}, {48306, 50350}, {48325, 59971}
X(68772) = reflection of X(i) in X(j) for these {i,j}: {905, 31947}, {50334, 4885}
X(68772) = isogonal conjugate of X(65303)
X(68772) = complement of X(2517)
X(68772) = complement of the isotomic conjugate of X(1310)
X(68772) = isogonal conjugate of the anticomplement of X(17421)
X(68772) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 5517}, {560, 55046}, {1036, 124}, {1245, 125}, {1310, 2887}, {1472, 11}, {2221, 116}, {2281, 8287}, {32691, 5}, {36099, 20305}, {37215, 626}, {54982, 21235}, {56219, 21253}, {56328, 21252}, {65298, 141}
X(68772) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 5517}, {36099, 6}, {47844, 834}
X(68772) = X(i)-isoconjugate of X(j) for these (i,j): {1, 65303}, {10, 59130}, {63, 59083}, {101, 60156}, {190, 46010}, {651, 56225}, {1331, 68578}, {1897, 57667}, {8750, 57832}
X(68772) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 65303}, {1015, 60156}, {3162, 59083}, {5517, 2}, {5521, 68578}, {26932, 57832}, {34467, 57667}, {38991, 56225}, {55053, 46010}
X(68772) = crosspoint of X(i) and X(j) for these (i,j): {2, 1310}, {58, 32691}, {100, 56220}, {108, 959}, {651, 64979}, {37870, 52935}
X(68772) = crosssum of X(i) and X(j) for these (i,j): {6, 8678}, {10, 23874}, {521, 958}, {650, 61398}
X(68772) = crossdifference of every pair of points on line {3, 37}
X(68772) = X(23282)-line conjugate of X(37)
X(68772) = barycentric product X(i)*X(j) for these {i,j}: {406, 905}, {513, 5739}, {514, 12514}, {522, 45126}, {523, 27174}, {693, 36744}, {1310, 5517}, {1452, 6332}, {3733, 42707}, {4391, 64020}, {8678, 14258}, {15413, 44086}, {17421, 36099}
X(68772) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 65303}, {25, 59083}, {406, 6335}, {513, 60156}, {663, 56225}, {667, 46010}, {905, 57832}, {1333, 59130}, {1452, 653}, {5517, 2517}, {5739, 668}, {6591, 68578}, {12514, 190}, {14258, 54982}, {22383, 57667}, {27174, 99}, {36744, 100}, {42707, 27808}, {44086, 1783}, {45126, 664}, {64020, 651}
X(68772) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4397, 31209, 48204}, {4560, 48173, 7650}, {42312, 47828, 50338}, {48207, 50334, 4885}

X(68773) = ORTHIC AXIS INTERCEPT OF X(513)X(23740)

Barycentrics    a*(b - c)*(a^3*b - a*b^3 + a^3*c - a^2*b*c + a*b^2*c + b^3*c + a*b*c^2 + 2*b^2*c^2 - a*c^3 + b*c^3) : :

X(68773) lies on these lines: {230, 231}, {513, 23740}, {514, 25098}, {522, 7180}, {649, 4083}, {665, 4765}, {693, 7212}, {834, 4790}, {905, 21196}, {1459, 58773}, {2522, 29017}, {3005, 50502}, {3666, 43067}, {3669, 3910}, {3700, 64857}, {3900, 66928}, {4820, 21894}, {4885, 24622}, {4976, 43060}, {4988, 21118}, {20906, 26114}, {21105, 50522}, {21828, 48277}, {23655, 54271}, {23877, 57055}, {24900, 47655}, {25594, 31287}, {27648, 46915}, {27674, 47782}, {28374, 45746}, {28606, 47666}, {29146, 46380}, {43052, 64825}, {47878, 55210}, {48281, 57181}, {50519, 65703}

X(68773) = reflection of X(i) in X(j) for these {i,j}: {650, 6589}, {35519, 4885}
X(68773) = complement of the isotomic conjugate of X(43350)
X(68773) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 38961}, {43350, 2887}
X(68773) = X(2)-Ceva conjugate of X(38961)
X(68773) = X(38961)-Dao conjugate of X(2)
X(68773) = crosspoint of X(i) and X(j) for these (i,j): {1, 32038}, {2, 43350}
X(68773) = crossdifference of every pair of points on line {3, 43}
X(68773) = barycentric product X(i)*X(j) for these {i,j}: {523, 37442}, {38961, 43350}
X(68773) = barycentric quotient X(37442)/X(99)
X(68773) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {647, 45745, 650}, {3310, 6590, 650}

X(68774) = ORTHIC AXIS INTERCEPT OF X(1)X(513)

Barycentrics    a*(b - c)*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 2*a*b*c + 3*b^2*c - a*c^2 + 3*b*c^2 - c^3) : :
X(68774) = X[23838] + 3 X[48281], 3 X[48283] - X[53535], X[48327] + 2 X[55244], X[650] - 3 X[6129], 2 X[3960] - 3 X[51648], 3 X[656] - X[4814], X[659] - 3 X[59829], 3 X[2517] - 5 X[26985], 3 X[4017] + X[4895], X[4895] - 3 X[48303], X[4397] - 3 X[48209], X[4404] - 3 X[48186], X[4462] - 3 X[26144], X[4768] - 3 X[47795], X[4813] - 3 X[50332], 2 X[20317] - 3 X[48168], 3 X[21189] + X[50767], 3 X[48293] - X[50767], 3 X[50353] - X[50358], 3 X[59830] - X[67533]

X(68774) lies on these lines: {1, 513}, {225, 16228}, {230, 231}, {521, 48292}, {522, 3960}, {656, 4814}, {659, 59829}, {900, 3669}, {905, 4777}, {1068, 44426}, {1769, 4449}, {2323, 39521}, {2517, 26985}, {2804, 34958}, {3738, 48287}, {3900, 53527}, {3999, 50336}, {4017, 4895}, {4145, 50501}, {4397, 48209}, {4404, 48186}, {4462, 26144}, {4768, 47795}, {4802, 21111}, {4813, 50332}, {6615, 48342}, {7628, 60421}, {8678, 48350}, {8702, 57178}, {9002, 48346}, {10198, 48181}, {10527, 48246}, {11269, 47823}, {11809, 62492}, {14353, 50350}, {14838, 28169}, {20317, 48168}, {21102, 59975}, {21185, 55126}, {21189, 48293}, {26228, 47804}, {26357, 48390}, {26363, 48230}, {28151, 47965}, {28165, 31947}, {28179, 47921}, {28984, 64917}, {29639, 47802}, {29640, 48197}, {29675, 45666}, {33140, 48216}, {37565, 59973}, {37579, 48383}, {38295, 66512}, {42312, 50354}, {45700, 68101}, {48387, 53313}, {50353, 50358}, {59830, 67533}

X(68774) = midpoint of X(i) and X(j) for these {i,j}: {1769, 4449}, {4017, 48303}, {6615, 48342}, {21189, 48293}, {42312, 50354}
X(68774) = reflection of X(i) in X(j) for these {i,j}: {905, 59837}, {50350, 14353}
X(68774) = complement of the isotomic conjugate of X(46962)
X(68774) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 38962}, {46962, 2887}
X(68774) = X(2)-Ceva conjugate of X(38962)
X(68774) = X(38962)-Dao conjugate of X(2)
X(68774) = crosspoint of X(2) and X(46962)
X(68774) = crosssum of X(521) and X(5289)
X(68774) = crossdifference of every pair of points on line {3, 44}
X(68774) = barycentric product X(i)*X(j) for these {i,j}: {514, 54286}, {523, 68714}, {38962, 46962}
X(68774) = barycentric quotient X(i)/X(j) for these {i,j}: {54286, 190}, {68714, 99}

X(68775) = ORTHIC AXIS INTERCEPT OF X(513)X(1960)

Barycentrics    a*(b - c)*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 2*a*b*c - a*c^2 - c^3) : :
X(68775) = X[650] + 3 X[6129], 3 X[656] + X[4895], X[4895] - 3 X[48302], 3 X[1459] - X[53535], X[1491] + 3 X[59829], X[2517] - 3 X[48207], X[4036] - 3 X[48186], 3 X[4057] - X[50358], X[4086] - 3 X[48181], X[4391] - 3 X[48168], X[4397] - 3 X[48205], X[4814] + 3 X[48303], X[4814] - 3 X[57099], X[4879] + 3 X[28284], X[4979] + 3 X[50332], X[14288] - 3 X[47841], 3 X[21173] + X[23838], X[17496] + 3 X[26144], 5 X[26985] - 9 X[48209], 5 X[26985] - 3 X[50334], 3 X[48209] - X[50334], X[50328] + 3 X[50353], 3 X[45660] - 2 X[68119], 3 X[48173] - X[50327], 3 X[48292] - X[50767], 3 X[53304] + X[67533]

X(68775) lies on these lines: {11, 53826}, {37, 29370}, {230, 231}, {513, 1960}, {514, 59837}, {522, 1125}, {656, 4895}, {663, 53527}, {667, 48350}, {900, 905}, {1201, 1459}, {1491, 7292}, {1769, 55969}, {1946, 39478}, {2517, 48207}, {2605, 21189}, {3669, 28209}, {3777, 4491}, {4017, 48297}, {4036, 48186}, {4057, 8654}, {4086, 48181}, {4145, 50504}, {4391, 48168}, {4397, 48205}, {4526, 29078}, {4777, 14838}, {4814, 48303}, {4833, 16754}, {4871, 47830}, {4879, 28284}, {4977, 51648}, {4979, 50332}, {6370, 20517}, {8043, 28161}, {8674, 48294}, {9013, 48330}, {14288, 47841}, {15446, 21173}, {17322, 62415}, {17420, 48283}, {17496, 26144}, {19864, 48228}, {21180, 59831}, {21789, 53313}, {21828, 48177}, {22160, 39200}, {23383, 39199}, {23800, 48306}, {23809, 64905}, {26094, 48243}, {26230, 47798}, {26985, 48209}, {28151, 48003}, {28179, 47965}, {28221, 59972}, {28355, 28399}, {28374, 48248}, {30950, 47828}, {35057, 57178}, {42337, 59973}, {45660, 68119}, {48173, 50327}, {48292, 50767}, {48307, 50350}, {53248, 58336}, {53304, 67533}, {59895, 60414}

X(68775) = midpoint of X(i) and X(j) for these {i,j}: {656, 48302}, {663, 53527}, {667, 48350}, {1769, 55969}, {2605, 21189}, {3777, 4491}, {4017, 48297}, {17420, 48283}, {23800, 48306}, {39199, 59830}, {48303, 57099}, {48307, 50350}
X(68775) = complement of the isotomic conjugate of X(13396)
X(68775) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 38963}, {13396, 2887}
X(68775) = X(2)-Ceva conjugate of X(38963)
X(68775) = X(38963)-Dao conjugate of X(2)
X(68775) = crosspoint of X(2) and X(13396)
X(68775) = crossdifference of every pair of points on line {3, 45}
X(68775) = barycentric product X(13396)*X(38963)

X(68776) = ORTHIC AXIS INTERCEPT OF X(6)X(513)

Barycentrics    a*(b - c)*(a^3 - a^2*b + a*b^2 - b^3 - a^2*c + b^2*c + a*c^2 + b*c^2 - c^3) : :
X(68776) = X[6] - 4 X[54250], 3 X[1643] - X[3063]

X(68776) lies on these lines: {6, 513}, {53, 16228}, {111, 53943}, {112, 10101}, {216, 59973}, {230, 231}, {393, 44426}, {522, 2509}, {649, 50354}, {656, 45755}, {657, 1769}, {661, 48340}, {665, 51648}, {905, 4762}, {1015, 52946}, {1108, 66514}, {1249, 66512}, {1459, 17412}, {1609, 48383}, {1946, 53311}, {2402, 4000}, {2522, 4976}, {2775, 59920}, {2837, 30234}, {3008, 23798}, {3163, 65945}, {3553, 48307}, {3554, 48281}, {3669, 6084}, {3815, 47802}, {4148, 20316}, {4394, 7297}, {4435, 15313}, {4501, 24290}, {4765, 16612}, {4776, 63089}, {5222, 57167}, {5304, 47805}, {7661, 14330}, {7735, 47804}, {7736, 44429}, {8557, 21390}, {8678, 48025}, {9575, 48335}, {11672, 35084}, {13401, 22383}, {16757, 17494}, {21185, 55133}, {23972, 65889}, {23980, 35113}, {23990, 67434}, {28894, 47679}, {32644, 32703}, {34288, 55259}, {34958, 55137}, {37642, 47762}, {37646, 47761}, {37662, 47760}, {37665, 48164}, {37666, 47763}, {41015, 43052}, {46380, 48277}, {47322, 62492}

X(68776) = midpoint of X(i) and X(j) for these {i,j}: {2402, 24002}, {4501, 24290}
X(68776) = complement of the isotomic conjugate of X(1292)
X(68776) = isogonal conjugate of the isotomic conjugate of X(26546)
X(68776) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 5511}, {32, 40615}, {1292, 2887}, {2191, 21252}, {2428, 20540}, {32644, 20335}, {36041, 20544}, {37206, 626}, {54987, 21235}, {57656, 116}, {63906, 21262}
X(68776) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 5511}, {2402, 6084}, {4000, 1086}, {21185, 11934}, {24002, 513}, {32644, 3290}, {40576, 1486}, {67060, 21867}
X(68776) = X(i)-isoconjugate of X(j) for these (i,j): {63, 26706}, {100, 44178}, {101, 13577}, {190, 3433}, {664, 40141}, {692, 57773}, {765, 67014}, {1252, 26721}, {3912, 35185}, {3939, 64240}, {4468, 15402}, {25083, 36111}, {37206, 54236}
X(68776) = X(i)-Dao conjugate of X(j) for these (i,j): {55, 644}, {513, 67014}, {661, 26721}, {1015, 13577}, {1086, 57773}, {3162, 26706}, {5511, 2}, {8054, 44178}, {34847, 25266}, {39025, 40141}, {40617, 64240}, {55053, 3433}, {56796, 2414}
X(68776) = crosspoint of X(i) and X(j) for these (i,j): {2, 1292}, {108, 279}, {37800, 40576}
X(68776) = crosssum of X(i) and X(j) for these (i,j): {6, 3309}, {9, 1734}, {220, 521}, {44178, 67014}
X(68776) = crossdifference of every pair of points on line {3, 518}
X(68776) = barycentric product X(i)*X(j) for these {i,j}: {1, 21185}, {6, 26546}, {7, 11934}, {11, 40576}, {105, 55133}, {169, 514}, {244, 67060}, {513, 3434}, {522, 34036}, {523, 4228}, {649, 20927}, {650, 37800}, {693, 1486}, {905, 17905}, {1019, 21073}, {1111, 57250}, {1292, 5511}, {2826, 61491}, {3309, 14268}, {4391, 56913}, {4394, 27826}, {4581, 41581}, {5452, 24002}, {6591, 28420}, {7192, 21867}, {17924, 22131}, {41582, 58784}
X(68776) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 26706}, {169, 190}, {244, 26721}, {513, 13577}, {514, 57773}, {649, 44178}, {667, 3433}, {1015, 67014}, {1486, 100}, {3063, 40141}, {3434, 668}, {3669, 64240}, {4228, 99}, {5452, 644}, {8642, 54236}, {11934, 8}, {14268, 54987}, {17905, 6335}, {20927, 1978}, {21073, 4033}, {21185, 75}, {21867, 3952}, {22131, 1332}, {26546, 76}, {34036, 664}, {37800, 4554}, {40576, 4998}, {41581, 53332}, {41582, 4576}, {48398, 41788}, {55133, 3263}, {56913, 651}, {57250, 765}, {64216, 35185}, {67060, 7035}
X(68776) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {650, 6129, 6586}, {650, 6591, 6588}, {650, 47227, 40134}, {6591, 40134, 47227}, {40134, 47227, 6588}

X(68777) = ORTHIC AXIS INTERCEPT OF X(6)X(649)

Barycentrics    a^2*(b - c)*(a^2*b - b^3 + a^2*c - 3*a*b*c + 2*b^2*c + 2*b*c^2 - c^3) : :

X(688) lies on these lines: {6, 649}, {37, 4521}, {39, 28565}, {42, 65697}, {111, 53946}, {112, 67782}, {230, 231}, {514, 52595}, {665, 2516}, {1635, 43060}, {2487, 3669}, {2527, 54249}, {2999, 58324}, {3666, 4468}, {3676, 3752}, {3835, 37662}, {4255, 65659}, {4394, 7180}, {4646, 28292}, {4762, 24782}, {4850, 47676}, {5233, 26596}, {6084, 43051}, {6685, 59673}, {8034, 50513}, {8643, 17967}, {8664, 50503}, {8665, 50506}, {9000, 53521}, {11672, 35085}, {16614, 40621}, {17424, 20980}, {18134, 26571}, {20295, 63089}, {23972, 65922}, {23980, 35129}, {24622, 45659}, {25084, 44567}, {25924, 37679}, {25955, 37674}, {27013, 37642}, {27014, 35519}, {28374, 47784}, {31286, 37646}, {32645, 32705}, {45313, 61661}, {48281, 62748}

X(68777) = isogonal conjugate of X(55996)
X(68777) = complement of the isotomic conjugate of X(1293)
X(68777) = polar conjugate of the isotomic conjugate of X(32475)
X(68777) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 5510}, {32, 40617}, {560, 40621}, {692, 2885}, {1293, 2887}, {3445, 21252}, {5382, 21262}, {16945, 17059}, {27834, 626}, {32645, 3834}, {34080, 141}, {36042, 21241}, {38266, 116}, {38828, 17046}, {53647, 21235}, {56174, 53575}, {65173, 17047}
X(68777) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 5510}, {2403, 6085}, {32645, 8610}, {60482, 513}
X(68777) = X(i)-isoconjugate of X(j) for these (i,j): {1, 55996}, {63, 32704}, {3977, 36112}, {4358, 35186}, {4462, 15403}, {27834, 54237}
X(68777) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 55996}, {3057, 25268}, {3162, 32704}, {5510, 2}, {56795, 2415}
X(68777) = crosspoint of X(i) and X(j) for these (i,j): {2, 1293}, {57, 56194}
X(68777) = crosssum of X(i) and X(j) for these (i,j): {6, 3667}, {9, 21173}, {4370, 39472}
X(68777) = crossdifference of every pair of points on line {3, 519}
X(68777) = barycentric product X(i)*X(j) for these {i,j}: {4, 32475}, {106, 55134}, {513, 14923}, {523, 7419}, {1293, 5510}, {3667, 14261}
X(68777) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 55996}, {25, 32704}, {7419, 99}, {8643, 54237}, {14261, 53647}, {14923, 668}, {32475, 69}, {55134, 3264}
X(68777) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {650, 3310, 6589}, {650, 6589, 6586}, {650, 21348, 2490}

X(68778) = ORTHIC AXIS INTERCEPT OF X(6)X(512)

Barycentrics    a^2*(b^2 - c^2)*(a^4 - b^4 + 4*b^2*c^2 - c^4) : :
X(68778) = 3 X[2506] - 4 X[7651], 3 X[2506] - 2 X[59987], 2 X[647] - 3 X[2485], X[647] - 3 X[2489], 5 X[647] - 3 X[47133], 3 X[2485] - 4 X[2492], 5 X[2485] - 2 X[47133], 3 X[2489] - 2 X[2492], 5 X[2489] - X[47133], 10 X[2492] - 3 X[47133], 3 X[47125] - 4 X[59652], 3 X[2514] - X[8665], 3 X[3267] - 5 X[31072], X[3288] - 3 X[14398], 3 X[17994] - X[67534]

X(68778) lies on these lines: {6, 512}, {32, 65390}, {39, 8704}, {53, 16229}, {111, 9137}, {112, 10098}, {187, 19901}, {216, 18311}, {230, 231}, {393, 14618}, {520, 39232}, {566, 37742}, {570, 59896}, {574, 9175}, {888, 50550}, {924, 2451}, {1084, 61071}, {1499, 7652}, {1510, 39520}, {1989, 51479}, {2395, 34288}, {2408, 8599}, {2514, 8665}, {2524, 5421}, {2549, 21732}, {2780, 59928}, {2799, 7624}, {3050, 62176}, {3163, 65923}, {3267, 31072}, {3288, 14398}, {3569, 8675}, {3800, 52588}, {4108, 7735}, {4145, 54250}, {5063, 44823}, {5158, 33752}, {5661, 50149}, {5996, 7736}, {6088, 8644}, {8552, 64919}, {8651, 53272}, {9035, 54262}, {9220, 18313}, {10422, 10561}, {10562, 15899}, {11620, 23287}, {11672, 35087}, {14977, 15355}, {15328, 43718}, {17994, 67534}, {18309, 18424}, {18487, 23878}, {20403, 58267}, {30209, 53777}, {30476, 35522}, {32112, 51544}, {32648, 32709}, {33630, 66895}, {34958, 40941}, {39602, 65863}, {44568, 64925}, {46985, 60428}, {47322, 62489}, {62992, 63250}

X(68778) = midpoint of X(i) and X(j) for these {i,j}: {2408, 8599}, {2451, 55219}, {12077, 14273}
X(68778) = reflection of X(i) in X(j) for these {i,j}: {647, 2492}, {2485, 2489}, {3050, 62176}, {6131, 44451}, {8644, 46001}, {35522, 30476}, {47138, 2501}, {53272, 8651}, {59987, 7651}
X(68778) = isogonal conjugate of X(65324)
X(68778) = complement of the isotomic conjugate of X(1296)
X(68778) = polar conjugate of the isotomic conjugate of X(30209)
X(68778) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 5512}, {560, 35133}, {1296, 2887}, {21448, 21253}, {32648, 4892}, {35179, 21235}, {36045, 625}, {37216, 626}, {39238, 8287}, {55923, 53575}
X(68778) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 5512}, {1296, 52174}, {2408, 6088}, {8599, 512}, {32648, 3291}, {39382, 25}
X(68778) = X(i)-isoconjugate of X(j) for these (i,j): {1, 65324}, {3, 37217}, {63, 30247}, {662, 5486}, {4575, 60266}, {6390, 36115}, {13608, 37216}, {14207, 15406}, {14210, 35188}, {23889, 60317}
X(68778) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 65324}, {136, 60266}, {574, 9146}, {1084, 5486}, {3162, 30247}, {5512, 2}, {10354, 2418}, {15477, 35188}, {36103, 37217}
X(68778) = crosspoint of X(i) and X(j) for these (i,j): {2, 1296}, {112, 1383}, {18818, 34574}
X(68778) = crosssum of X(i) and X(j) for these (i,j): {6, 1499}, {523, 43620}, {525, 599}, {1649, 62657}
X(68778) = crossdifference of every pair of points on line {3, 524}
X(68778) = barycentric product X(i)*X(j) for these {i,j}: {4, 30209}, {19, 14209}, {74, 44203}, {111, 55135}, {512, 11185}, {523, 1995}, {842, 68326}, {850, 19136}, {1296, 5512}, {1499, 14262}, {2489, 66767}, {2501, 41614}, {2793, 34241}, {5466, 53777}, {6088, 34166}, {8542, 8599}, {10097, 37855}, {13493, 65870}, {14672, 39382}, {29959, 58784}
X(68778) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 65324}, {19, 37217}, {25, 30247}, {512, 5486}, {1995, 99}, {2501, 60266}, {6088, 39157}, {8542, 9146}, {8644, 13608}, {9135, 53764}, {9178, 60317}, {11185, 670}, {13493, 6082}, {14209, 304}, {14262, 35179}, {19136, 110}, {29959, 4576}, {30209, 69}, {32740, 35188}, {34241, 46144}, {41614, 4563}, {44203, 3260}, {52174, 1296}, {53777, 5468}, {55135, 3266}, {66767, 52608}
X(68778) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {647, 2489, 2492}, {647, 2492, 2485}, {2489, 14273, 6753}, {7651, 59987, 2506}

X(68779) = ORTHIC AXIS INTERCEPT OF X(1)X(19)

Barycentrics    a*(b + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + b*c - c^2)*(a^2 - b^2 + c^2) : :

X(68779) lies on these lines: {1, 19}, {4, 3743}, {6, 59282}, {9, 52033}, {12, 37}, {33, 1962}, {34, 2292}, {92, 58026}, {112, 12030}, {186, 39149}, {230, 231}, {250, 57649}, {278, 25080}, {281, 451}, {283, 1761}, {475, 4647}, {607, 53037}, {608, 4016}, {651, 64888}, {740, 1861}, {758, 1870}, {862, 54245}, {1735, 2252}, {1824, 37593}, {1835, 2245}, {1839, 1841}, {1869, 3931}, {1876, 20718}, {2006, 37799}, {2331, 8557}, {2333, 16600}, {2614, 54244}, {3192, 3725}, {3209, 37579}, {3330, 53560}, {3724, 52427}, {3747, 8750}, {3960, 44427}, {4053, 44113}, {4068, 7071}, {4200, 64071}, {5236, 8680}, {5307, 28606}, {6198, 58380}, {6675, 37565}, {7009, 58391}, {7079, 25088}, {7412, 58392}, {7438, 26242}, {8143, 46467}, {11683, 18721}, {16577, 40149}, {16578, 37805}, {16586, 17923}, {17913, 20883}, {21016, 28594}, {22059, 24025}, {23555, 54283}, {24394, 56183}, {25063, 61710}, {37908, 53323}, {40941, 56905}, {40978, 41320}, {46878, 58386}, {56319, 59727}

X(68779) = polar conjugate of X(14616)
X(68779) = complement of the isotomic conjugate of X(39435)
X(68779) = polar conjugate of the isotomic conjugate of X(758)
X(68779) = polar conjugate of the isogonal conjugate of X(3724)
X(68779) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 53982}, {39435, 2887}
X(68779) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 53982}, {92, 1845}, {1870, 44113}, {17923, 860}, {36119, 33}, {37799, 37982}, {39439, 25}, {65329, 24006}, {68563, 1824}
X(68779) = X(i)-cross conjugate of X(j) for these (i,j): {3724, 758}, {44113, 1835}
X(68779) = X(i)-isoconjugate of X(j) for these (i,j): {2, 57736}, {3, 24624}, {6, 57985}, {48, 14616}, {57, 1793}, {58, 52351}, {63, 759}, {69, 34079}, {77, 2341}, {80, 1790}, {81, 1807}, {86, 52431}, {125, 9273}, {216, 39277}, {222, 6740}, {265, 40214}, {283, 2006}, {284, 52392}, {304, 67166}, {394, 68571}, {525, 36069}, {647, 65283}, {655, 23189}, {656, 37140}, {1214, 52380}, {1411, 1812}, {1437, 18359}, {1444, 2161}, {1459, 47318}, {1789, 65228}, {1797, 56950}, {2185, 52391}, {2193, 18815}, {2605, 60053}, {3737, 65299}, {4467, 32662}, {4558, 66284}, {4575, 60074}, {6187, 17206}, {7254, 51562}, {9274, 20902}, {14208, 32671}, {14838, 36061}, {14919, 56645}, {22094, 39295}, {23067, 60571}, {23226, 32680}, {34016, 52153}, {34055, 46160}, {34535, 68661}, {52383, 65568}, {52408, 66922}
X(68779) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 57985}, {10, 52351}, {136, 60074}, {1249, 14616}, {3162, 759}, {5452, 1793}, {13999, 4560}, {16221, 14838}, {32664, 57736}, {34586, 63}, {35069, 69}, {35204, 1812}, {36103, 24624}, {38982, 525}, {39052, 65283}, {40584, 1444}, {40586, 1807}, {40590, 52392}, {40596, 37140}, {40600, 52431}, {40612, 17206}, {47345, 18815}, {51583, 304}, {53982, 2}, {57434, 15411}, {66508, 15419}
X(68779) = crosspoint of X(i) and X(j) for these (i,j): {2, 39435}, {1870, 17923}, {7012, 65329}
X(68779) = crosssum of X(1807) and X(52431)
X(68779) = trilinear pole of line {2610, 53562}
X(68779) = crossdifference of every pair of points on line {3, 656}
X(68779) = barycentric product X(i)*X(j) for these {i,j}: {1, 860}, {4, 758}, {8, 1835}, {10, 1870}, {12, 17515}, {19, 3936}, {25, 35550}, {27, 4053}, {33, 41804}, {36, 41013}, {37, 17923}, {65, 5081}, {75, 44113}, {92, 2245}, {162, 6370}, {186, 6757}, {214, 68563}, {225, 4511}, {264, 3724}, {281, 18593}, {318, 1464}, {320, 1824}, {321, 52413}, {451, 39149}, {523, 4242}, {648, 2610}, {654, 65207}, {661, 65162}, {811, 42666}, {1309, 42768}, {1441, 52427}, {1443, 53008}, {1474, 61410}, {1783, 4707}, {1825, 63642}, {1826, 3218}, {1832, 5239}, {1833, 5240}, {1845, 38955}, {1880, 32851}, {1897, 53527}, {1983, 14618}, {2323, 40149}, {2333, 20924}, {2361, 57809}, {2501, 4585}, {3738, 61178}, {4551, 44428}, {4552, 65104}, {4736, 68571}, {4881, 68562}, {4996, 68645}, {6335, 21828}, {6336, 40988}, {6739, 36119}, {8818, 52414}, {15455, 47230}, {18026, 53562}, {30250, 55149}, {31845, 39439}, {36797, 51663}, {39435, 53982}, {42701, 64834}, {52426, 52575}, {58328, 68576}
X(68779) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 57985}, {4, 14616}, {19, 24624}, {25, 759}, {31, 57736}, {33, 6740}, {36, 1444}, {37, 52351}, {42, 1807}, {55, 1793}, {65, 52392}, {112, 37140}, {162, 65283}, {181, 52391}, {186, 56934}, {213, 52431}, {215, 68661}, {225, 18815}, {607, 2341}, {758, 69}, {860, 75}, {862, 36815}, {1096, 68571}, {1464, 77}, {1783, 47318}, {1824, 80}, {1825, 63778}, {1826, 18359}, {1835, 7}, {1843, 46160}, {1845, 17139}, {1870, 86}, {1880, 2006}, {1973, 34079}, {1974, 67166}, {1983, 4558}, {2190, 39277}, {2245, 63}, {2299, 52380}, {2323, 1812}, {2333, 2161}, {2361, 283}, {2501, 60074}, {2610, 525}, {3218, 17206}, {3724, 3}, {3936, 304}, {3960, 15419}, {4053, 306}, {4242, 99}, {4282, 65568}, {4511, 332}, {4559, 65299}, {4585, 4563}, {4707, 15413}, {5081, 314}, {6370, 14208}, {6757, 328}, {7113, 1790}, {7140, 15065}, {8648, 23189}, {8736, 60091}, {8754, 66289}, {14270, 23226}, {17515, 261}, {17923, 274}, {18593, 348}, {21758, 7254}, {21828, 905}, {32676, 36069}, {34397, 17104}, {35235, 17886}, {35550, 305}, {39149, 57865}, {40988, 3977}, {41013, 20566}, {41804, 7182}, {42666, 656}, {42768, 65868}, {44097, 56405}, {44113, 1}, {44427, 18160}, {44428, 18155}, {47230, 14838}, {51663, 17094}, {52413, 81}, {52414, 34016}, {52426, 2193}, {52427, 21}, {52434, 1437}, {53008, 52409}, {53285, 57081}, {53525, 17219}, {53527, 4025}, {53562, 521}, {57652, 1411}, {57655, 9274}, {58313, 3737}, {58328, 1792}, {61178, 35174}, {61206, 32671}, {61410, 40071}, {64834, 66922}, {65104, 4560}, {65162, 799}, {65207, 46405}, {68563, 57788}, {68645, 57645}
X(68779) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1781, 62700}, {37, 1880, 1826}, {5089, 14571, 8756}, {5089, 47232, 468}, {8609, 14571, 8755}

X(68780) = ORTHIC AXIS INTERCEPT OF X(513)X(3004)

Barycentrics    (b - c)*(-a^3 + a^2*b + a*b^2 + b^3 + a^2*c + 2*a*b*c + b^2*c + a*c^2 + b*c^2 + c^3) : :
X(68780) = 3 X[650] - 2 X[2977], 2 X[676] - 3 X[48211], 4 X[2490] - 3 X[48219], 4 X[2977] - 3 X[48062], 2 X[4874] - 3 X[47800], X[6590] - 3 X[47800], X[7662] - 3 X[48211], 3 X[47766] - 2 X[48405], 3 X[48220] - X[48397], X[48007] + 2 X[50347], 3 X[48224] - X[48326], 2 X[3835] - 3 X[48555], X[4010] - 3 X[48177], X[693] - 3 X[47797], X[2526] - 3 X[47880], and many others

X(68780) lies on these lines: {2, 47690}, {10, 29192}, {230, 231}, {513, 3004}, {514, 659}, {522, 1491}, {525, 48099}, {649, 47701}, {663, 21124}, {693, 47797}, {784, 21185}, {812, 49295}, {824, 3716}, {826, 50507}, {900, 2526}, {905, 29142}, {918, 48029}, {1635, 47702}, {2254, 47886}, {2517, 24622}, {2533, 14837}, {2786, 48043}, {3239, 4122}, {3566, 50508}, {3667, 47877}, {3676, 21146}, {3700, 48179}, {3798, 4784}, {3800, 50501}, {3837, 47757}, {3910, 48136}, {4024, 47832}, {4083, 60492}, {4088, 4893}, {4129, 29062}, {4132, 50506}, {4379, 47703}, {4453, 48108}, {4467, 48080}, {4468, 62423}, {4477, 8058}, {4521, 48185}, {4522, 25666}, {4560, 47708}, {4724, 16892}, {4762, 23770}, {4765, 48349}, {4777, 47784}, {4778, 47968}, {4782, 48060}, {4802, 47890}, {4804, 48277}, {4806, 29078}, {4830, 28882}, {4885, 47799}, {4962, 48160}, {4977, 47960}, {4979, 47938}, {4988, 48142}, {6005, 21192}, {6332, 29017}, {6545, 48119}, {6546, 48118}, {7192, 47699}, {7265, 47838}, {7650, 35518}, {7658, 47823}, {7927, 50504}, {8678, 48402}, {9001, 53550}, {9508, 29144}, {11068, 28147}, {14838, 29021}, {17069, 50336}, {17161, 48172}, {17494, 47691}, {18004, 29370}, {21051, 29074}, {21115, 47933}, {21183, 48098}, {21186, 24782}, {21188, 50352}, {21212, 24720}, {21260, 29086}, {22388, 48383}, {23875, 48058}, {23880, 48400}, {23882, 48403}, {24719, 48552}, {25259, 47821}, {25380, 47882}, {26275, 28894}, {26277, 45746}, {27115, 48208}, {27486, 48158}, {28151, 47884}, {28155, 47885}, {28161, 47827}, {28169, 45684}, {28175, 48095}, {28183, 48193}, {28191, 48140}, {28213, 47919}, {28229, 47925}, {28423, 48209}, {28478, 48123}, {28481, 48092}, {28846, 48024}, {28851, 48001}, {28855, 47986}, {28863, 53580}, {28878, 47946}, {28898, 50326}, {29047, 48003}, {29066, 50453}, {29204, 48056}, {29288, 47965}, {29362, 48398}, {30520, 48055}, {30795, 44432}, {31150, 47692}, {31209, 47689}, {31287, 47807}, {44429, 47687}, {44433, 47697}, {44435, 46403}, {46919, 48235}, {47652, 48174}, {47653, 47696}, {47656, 47834}, {47657, 48237}, {47660, 47804}, {47662, 48250}, {47663, 47688}, {47672, 47887}, {47676, 47969}, {47677, 49275}, {47683, 49300}, {47685, 48159}, {47686, 48156}, {47693, 47771}, {47695, 47782}, {47698, 47775}, {47704, 47926}, {47707, 47793}, {47711, 47794}, {47715, 47795}, {47719, 47796}, {47762, 49283}, {47781, 47945}, {47783, 48030}, {47801, 48248}, {47810, 48077}, {47811, 48094}, {47813, 48275}, {47826, 48082}, {47878, 47934}, {47894, 53343}, {47902, 48104}, {47923, 48102}, {47924, 48101}, {47930, 48078}, {47931, 48105}, {47971, 48021}, {47973, 48032}, {47974, 48422}, {48028, 48038}, {48047, 64856}, {48089, 48192}, {48090, 48268}, {48120, 62552}, {48200, 53573}

X(68780) = midpoint of X(i) and X(j) for these {i,j}: {649, 47701}, {663, 21124}, {1491, 50340}, {2254, 47972}, {3004, 50347}, {4025, 48006}, {4467, 48080}, {4560, 47708}, {4724, 16892}, {4804, 48277}, {4979, 47938}, {4988, 48142}, {7192, 47699}, {17494, 47691}, {27486, 48158}, {45745, 47123}, {45746, 47694}, {47653, 47696}, {47663, 47688}, {47676, 47969}, {47677, 49275}, {47683, 49300}, {47692, 48408}, {47695, 47975}, {47702, 48106}, {47704, 47926}, {47782, 48223}, {47902, 48104}, {47923, 48102}, {47924, 48101}, {47930, 48078}, {47931, 48105}, {47968, 50358}, {47971, 48021}, {47973, 48032}, {47974, 49301}, {47979, 48013}, {48014, 48015}, {48024, 50342}
X(68780) = reflection of X(i) in X(j) for these {i,j}: {2533, 14837}, {4122, 3239}, {4522, 25666}, {4784, 3798}, {6590, 4874}, {7662, 676}, {21146, 3676}, {21183, 48212}, {24720, 21212}, {47787, 48195}, {47981, 47990}, {47982, 47999}, {47983, 47998}, {48007, 3004}, {48038, 48028}, {48039, 48030}, {48040, 48029}, {48060, 4782}, {48062, 650}, {48069, 9508}, {48103, 11068}, {48222, 14425}, {48235, 46919}, {48268, 48090}, {48269, 4806}, {48396, 4885}, {49285, 3837}, {49286, 3716}, {50336, 17069}, {50352, 21188}
X(68780) = complement of X(47690)
X(68780) = X(57726)-Ceva conjugate of X(11)
X(68780) = crossdifference of every pair of points on line {3, 172}
X(68780) = barycentric product X(i)*X(j) for these {i,j}: {514, 50295}, {523, 68712}
X(68780) = barycentric quotient X(i)/X(j) for these {i,j}: {50295, 190}, {68712, 99}
X(68780) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1635, 47702, 48106}, {4122, 47822, 3239}, {4467, 48161, 48080}, {6590, 47800, 4874}, {7662, 48211, 676}, {17494, 48203, 47691}, {21146, 48227, 3676}, {31150, 47692, 48408}, {31209, 47689, 47809}, {45746, 47798, 47694}, {47653, 47805, 47696}, {47688, 48240, 47663}, {47695, 47782, 47975}, {47757, 49285, 3837}, {47783, 48039, 48030}, {47785, 48069, 9508}, {47799, 48396, 4885}, {47886, 47972, 2254}, {47923, 48572, 48102}, {47969, 48241, 47676}, {47974, 48422, 49301}, {47975, 48223, 47695}, {48103, 48226, 11068}

X(68781) = ORTHIC AXIS INTERCEPT OF X(4)X(512)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-a^4 + a^2*b^2 + a^2*c^2 + 2*b^2*c^2) : :
X(68781) = X[4] - 3 X[14618], 2 X[4] - 3 X[16229], X[4] - 9 X[66895], X[14618] - 3 X[66895], X[16229] - 6 X[66895], 4 X[140] - 3 X[52584], 4 X[468] - 3 X[47221], 2 X[47214] - 3 X[47255], 7 X[3851] - 6 X[44918], 3 X[39201] - 2 X[65389]

X(68781) lies on these lines: {4, 512}, {140, 52584}, {230, 231}, {340, 520}, {458, 45321}, {525, 62438}, {648, 53199}, {684, 30476}, {826, 57065}, {924, 21646}, {1826, 4806}, {3800, 58757}, {3804, 65394}, {3851, 44918}, {3906, 44427}, {4108, 4232}, {4139, 44426}, {4784, 5307}, {5094, 53266}, {5466, 43530}, {5486, 15328}, {5996, 52284}, {6368, 33294}, {7607, 60338}, {7927, 51513}, {7950, 67102}, {8599, 18808}, {8673, 18314}, {9033, 62628}, {9409, 64788}, {9420, 23878}, {9979, 42651}, {10295, 62489}, {10311, 67172}, {11331, 41167}, {12073, 68327}, {14363, 60341}, {15451, 52585}, {16080, 32112}, {30209, 41079}, {33754, 39569}, {34291, 52292}, {37458, 65390}, {39201, 65389}, {39240, 44126}, {39241, 44125}, {40550, 52289}, {42658, 65403}, {44134, 53347}, {46812, 53153}, {46815, 53154}, {50351, 65100}, {50548, 65694}, {51428, 66939}, {53569, 57426}, {54260, 56370}, {58784, 66299}, {65723, 68089}

X(68781) = midpoint of X(850) and X(53345)
X(68781) = reflection of X(i) in X(j) for these {i,j}: {647, 6130}, {684, 30476}, {15451, 52585}, {16229, 14618}, {16230, 2501}, {59932, 51513}
X(68781) = isogonal conjugate of X(65310)
X(68781) = polar-circle-inverse of X(6785)
X(68781) = polar conjugate of X(65271)
X(68781) = polar conjugate of the isotomic conjugate of X(23878)
X(68781) = polar conjugate of the isogonal conjugate of X(3288)
X(68781) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {162, 9742}, {42298, 21294}, {47735, 21221}
X(68781) = X(i)-Ceva conjugate of X(j) for these (i,j): {33971, 66459}, {42374, 3269}, {65349, 4}, {67897, 115}
X(68781) = X(i)-cross conjugate of X(j) for these (i,j): {3288, 23878}, {6784, 10311}, {66459, 33971}
X(68781) = X(i)-isoconjugate of X(j) for these (i,j): {1, 65310}, {3, 65252}, {48, 65271}, {63, 26714}, {162, 54032}, {163, 42313}, {255, 65349}, {262, 4575}, {263, 4592}, {293, 63741}, {662, 43718}, {906, 60679}, {1813, 66936}, {2186, 4558}, {2617, 51444}, {3402, 4563}, {23997, 66879}, {32676, 59257}, {36061, 57268}, {36132, 36212}, {39681, 66942}, {46319, 55202}, {52631, 62719}, {52926, 62277}
X(68781) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 65310}, {115, 42313}, {125, 54032}, {132, 63741}, {136, 262}, {1084, 43718}, {1249, 65271}, {3162, 26714}, {5139, 263}, {5190, 60679}, {6523, 65349}, {15526, 59257}, {16221, 57268}, {36103, 65252}, {38970, 46807}, {38997, 3}, {39009, 36212}, {42426, 36885}, {51580, 4563}, {54262, 54272}, {55051, 3917}, {62562, 66879}, {62596, 65748}, {66459, 48876}
X(68781) = cevapoint of X(54262) and X(64919)
X(68781) = crosspoint of X(i) and X(j) for these (i,j): {4, 65349}, {6528, 64983}
X(68781) = crosssum of X(520) and X(63433)
X(68781) = crossdifference of every pair of points on line {3, 217}
X(68781) = barycentric product X(i)*X(j) for these {i,j}: {4, 23878}, {182, 14618}, {183, 2501}, {232, 63746}, {264, 3288}, {297, 67172}, {458, 523}, {512, 44144}, {525, 33971}, {648, 66459}, {656, 51315}, {685, 66192}, {850, 10311}, {1577, 60685}, {2489, 20023}, {6331, 6784}, {6591, 42711}, {7649, 60737}, {9420, 60199}, {15412, 39530}, {16230, 46806}, {17924, 60723}, {17984, 39680}, {18808, 51372}, {24006, 52134}, {43665, 68695}, {44427, 56401}, {46107, 60726}, {51373, 53149}, {59197, 66300}
X(68781) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 65271}, {6, 65310}, {19, 65252}, {25, 26714}, {182, 4558}, {183, 4563}, {232, 63741}, {393, 65349}, {419, 39681}, {458, 99}, {512, 43718}, {523, 42313}, {525, 59257}, {647, 54032}, {2395, 66879}, {2489, 263}, {2501, 262}, {2623, 51444}, {2971, 52631}, {3199, 52926}, {3288, 3}, {3403, 55202}, {6103, 36885}, {6531, 6037}, {6784, 647}, {7649, 60679}, {8754, 66291}, {9420, 3289}, {10311, 110}, {14618, 327}, {16081, 53196}, {16230, 46807}, {17994, 51543}, {18344, 66936}, {20023, 52608}, {23878, 69}, {33569, 65748}, {33971, 648}, {34396, 32661}, {39530, 14570}, {39680, 36214}, {44144, 670}, {45321, 65722}, {46806, 17932}, {47230, 57268}, {51315, 811}, {51513, 66919}, {51542, 43754}, {52134, 4592}, {54273, 51997}, {56401, 60053}, {57204, 46319}, {57260, 32716}, {58757, 68572}, {59208, 23181}, {60685, 662}, {60723, 1332}, {60726, 1331}, {60737, 4561}, {63746, 57799}, {66192, 6333}, {66300, 42300}, {66459, 525}, {67172, 287}, {68695, 2421}, {68739, 68647}

X(68782) = ORTHIC AXIS INTERCEPT OF X(2)X(65868)

Barycentrics    (a - b - c)*(b - c)*(2*a^4 - a^3*b - a^2*b^2 + a*b^3 - b^4 - a^3*c + 2*a^2*b*c - a*b^2*c - a^2*c^2 - a*b*c^2 + 2*b^2*c^2 + a*c^3 - c^4) : :
X(68782) = 3 X[1638] - X[23737]

X(68782) lies on these lines: {2, 65868}, {11, 1146}, {37, 52307}, {101, 24863}, {105, 28132}, {108, 40117}, {210, 65448}, {226, 39470}, {230, 231}, {521, 3239}, {522, 5583}, {652, 3700}, {654, 900}, {657, 47874}, {661, 8819}, {665, 60371}, {910, 60062}, {1635, 42462}, {1638, 23737}, {1864, 30692}, {1901, 2631}, {2006, 10015}, {2432, 14312}, {3004, 28834}, {4120, 65680}, {5540, 56423}, {6084, 40166}, {13405, 55123}, {14077, 14330}, {14308, 57108}, {14321, 46389}, {14344, 17924}, {15252, 65808}, {17728, 65483}, {17963, 18013}, {21120, 48300}, {23615, 65664}, {23741, 43049}, {23893, 64330}, {23986, 46391}, {48395, 58332}

X(68782) = isogonal conjugate of X(65297)
X(68782) = complement of X(65868)
X(68782) = polar conjugate of X(65295)
X(68782) = complement of the isogonal conjugate of X(14776)
X(68782) = complement of the isotomic conjugate of X(1309)
X(68782) = polar conjugate of the isotomic conjugate of X(39471)
X(68782) = tripolar centroid of X(63857)
X(68782) = X(i)-complementary conjugate of X(j) for these (i,j): {25, 57434}, {31, 10017}, {33, 63757}, {1110, 42769}, {1309, 2887}, {2212, 55153}, {2250, 127}, {2342, 123}, {2720, 34822}, {8750, 119}, {14776, 10}, {32641, 18589}, {32669, 17073}, {32676, 34586}, {32702, 142}, {34858, 2968}, {36037, 1368}, {36110, 2886}, {36123, 21252}, {37136, 18639}, {65223, 626}, {65331, 17046}
X(68782) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 10017}, {514, 42755}, {2006, 11}, {2399, 522}, {2406, 515}, {10015, 900}, {24035, 51421}, {32706, 42069}, {43737, 3900}, {52780, 38357}, {60074, 61041}
X(68782) = X(i)-isoconjugate of X(j) for these (i,j): {1, 65297}, {2, 36040}, {48, 65295}, {63, 36067}, {69, 32667}, {75, 32643}, {102, 651}, {109, 36100}, {223, 6081}, {653, 36055}, {655, 58741}, {664, 32677}, {934, 15629}, {1415, 34393}, {1813, 36121}, {2399, 24027}, {2432, 7045}, {17080, 35183}, {36037, 60000}, {36059, 52780}, {36088, 56550}, {36108, 56553}
X(68782) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 65297}, {11, 36100}, {206, 32643}, {515, 2406}, {522, 2399}, {860, 65162}, {1146, 34393}, {1249, 65295}, {3162, 36067}, {3259, 60000}, {10017, 2}, {14714, 15629}, {17115, 2432}, {20620, 52780}, {23986, 664}, {32664, 36040}, {36944, 13136}, {38991, 102}, {39025, 32677}, {46398, 56666}, {46974, 4585}, {51221, 653}, {53522, 3904}
X(68782) = crosspoint of X(i) and X(j) for these (i,j): {2, 1309}, {8, 655}, {104, 37141}, {107, 24624}, {514, 43728}, {515, 2406}, {522, 2399}, {23987, 34050}, {44426, 60074}, {52780, 65213}
X(68782) = crosssum of X(i) and X(j) for these (i,j): {6, 8677}, {56, 654}, {101, 23981}, {102, 2432}, {109, 2425}, {517, 14298}, {520, 2245}, {521, 62326}, {1983, 36059}, {2323, 61042}, {2804, 15849}
X(68782) = crossdifference of every pair of points on line {3, 102}
X(68782) = barycentric product X(i)*X(j) for these {i,j}: {1, 14304}, {4, 39471}, {8, 53522}, {92, 46391}, {280, 6087}, {515, 522}, {523, 68693}, {650, 64194}, {663, 35516}, {693, 51361}, {1146, 2406}, {1309, 10017}, {1455, 4397}, {2170, 42718}, {2182, 4391}, {2399, 23986}, {2425, 23978}, {2432, 59205}, {2804, 56638}, {2968, 23987}, {3239, 34050}, {3738, 59283}, {5081, 61041}, {6332, 8755}, {6366, 63857}, {7253, 51421}, {10570, 55128}, {11700, 52356}, {15633, 66957}, {17926, 51368}, {24035, 34591}, {38554, 53152}, {42755, 51565}, {44426, 46974}, {51408, 63748}, {51424, 62725}, {51562, 57446}, {54243, 55124}, {57291, 65213}
X(68782) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 65295}, {6, 65297}, {25, 36067}, {31, 36040}, {32, 32643}, {515, 664}, {522, 34393}, {650, 36100}, {657, 15629}, {663, 102}, {1146, 2399}, {1455, 934}, {1946, 36055}, {1973, 32667}, {2182, 651}, {2192, 6081}, {2399, 57551}, {2406, 1275}, {2425, 1262}, {3063, 32677}, {3064, 52780}, {3310, 60000}, {6087, 347}, {8648, 58741}, {8735, 60584}, {8755, 653}, {10015, 56666}, {10017, 65868}, {14304, 75}, {14936, 2432}, {18344, 36121}, {23615, 15633}, {23986, 2406}, {23987, 55346}, {34050, 658}, {35516, 4572}, {39471, 69}, {42069, 53152}, {42718, 67038}, {42755, 22464}, {46391, 63}, {46974, 6516}, {51361, 100}, {51408, 56543}, {51421, 4566}, {51424, 35312}, {53522, 7}, {56638, 54953}, {57446, 4453}, {59283, 35174}, {61041, 52392}, {63857, 35157}, {64194, 4554}, {68693, 99}
X(68782) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {650, 47227, 65104}, {3239, 14331, 14298}, {40134, 65103, 650}

X(68783) = ORTHIC AXIS INTERCEPT OF X(100)X(108)

Barycentrics    (b - c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-(a*b) + b^2 - a*c + c^2) : :

X(68783) lies on these lines: {1, 58313}, {4, 2826}, {11, 2969}, {24, 44805}, {25, 659}, {27, 24623}, {33, 53523}, {34, 30725}, {100, 108}, {105, 15344}, {120, 20621}, {230, 231}, {427, 3837}, {428, 64913}, {513, 57044}, {514, 18344}, {522, 43923}, {693, 17094}, {891, 1829}, {900, 1830}, {918, 53550}, {926, 44881}, {935, 59107}, {1360, 55145}, {1861, 4925}, {1876, 52305}, {1878, 6550}, {1890, 6009}, {1902, 2821}, {1946, 14344}, {1960, 11363}, {3700, 57173}, {3900, 57230}, {4088, 17464}, {4449, 29288}, {4804, 55208}, {4977, 54244}, {5064, 48167}, {5094, 30795}, {6336, 65340}, {6362, 49285}, {7178, 57092}, {7713, 21385}, {11028, 55125}, {11396, 21343}, {15904, 55121}, {16228, 48089}, {17924, 47690}, {17994, 65873}, {21108, 21118}, {21146, 23742}, {23741, 48398}, {24006, 48396}, {29162, 54247}, {36122, 36124}, {39471, 51643}, {45314, 62978}, {47660, 56322}, {47691, 59915}, {47693, 65100}, {47694, 66512}, {47934, 55206}, {54229, 63812}, {55124, 59816}, {60581, 60583}

X(68783) = polar-circle-inverse of X(10773)
X(68783) = polar conjugate of X(666)
X(68783) = polar conjugate of the isotomic conjugate of X(918)
X(68783) = polar conjugate of the isogonal conjugate of X(665)
X(68783) = X(15344)-anticomplementary conjugate of X(33650)
X(68783) = X(i)-Ceva conjugate of X(j) for these (i,j): {108, 20621}, {15344, 5521}, {36122, 11}, {53150, 3064}, {54235, 8735}, {60585, 513}, {65333, 4}
X(68783) = X(i)-cross conjugate of X(j) for these (i,j): {665, 918}, {3675, 1876}, {53551, 2254}
X(68783) = X(i)-isoconjugate of X(j) for these (i,j): {3, 36086}, {41, 65301}, {48, 666}, {59, 23696}, {63, 919}, {69, 32666}, {77, 52927}, {78, 32735}, {100, 36057}, {101, 1814}, {105, 1331}, {184, 51560}, {190, 32658}, {212, 927}, {219, 36146}, {255, 65333}, {294, 1813}, {603, 36802}, {673, 906}, {692, 31637}, {1024, 44717}, {1040, 59133}, {1176, 35333}, {1332, 1438}, {1416, 4571}, {1459, 5377}, {1462, 4587}, {1946, 39293}, {2195, 6516}, {2481, 32656}, {4558, 18785}, {4561, 64216}, {4570, 10099}, {4575, 13576}, {4592, 56853}, {7004, 59101}, {9247, 36803}, {14942, 36059}, {20780, 39272}, {23151, 36138}, {32660, 36796}, {34055, 46163}, {34085, 52425}
X(68783) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 13576}, {1015, 1814}, {1086, 31637}, {1249, 666}, {3126, 521}, {3160, 65301}, {3162, 919}, {3675, 34381}, {5139, 56853}, {5190, 673}, {5521, 105}, {6184, 1332}, {6523, 65333}, {6615, 23696}, {7952, 36802}, {8054, 36057}, {14936, 23601}, {17435, 25083}, {17755, 4561}, {20620, 14942}, {20621, 100}, {35094, 69}, {35509, 26932}, {36103, 36086}, {36905, 65164}, {38966, 28071}, {38980, 63}, {38989, 3}, {39012, 23151}, {39014, 219}, {39046, 1331}, {39053, 39293}, {39063, 6516}, {40609, 4571}, {40837, 927}, {50330, 10099}, {55053, 32658}, {62576, 36803}, {62602, 34085}, {62605, 51560}
X(68783) = cevapoint of X(3675) and X(52305)
X(68783) = crosspoint of X(i) and X(j) for these (i,j): {4, 65333}, {7649, 68631}, {18026, 54235}, {44426, 60583}
X(68783) = crosssum of X(i) and X(j) for these (i,j): {3, 53550}, {1459, 20780}, {1946, 20752}
X(68783) = crossdifference of every pair of points on line {3, 906}
X(68783) = barycentric product X(i)*X(j) for these {i,j}: {4, 918}, {27, 4088}, {92, 2254}, {241, 44426}, {264, 665}, {278, 50333}, {281, 43042}, {286, 24290}, {318, 53544}, {331, 926}, {513, 46108}, {514, 1861}, {518, 17924}, {522, 5236}, {523, 15149}, {672, 46107}, {693, 5089}, {883, 8735}, {1309, 42770}, {1458, 46110}, {1577, 54407}, {1783, 62429}, {1826, 23829}, {1876, 4391}, {2052, 53550}, {2284, 2973}, {2356, 3261}, {2501, 30941}, {2969, 42720}, {3064, 9436}, {3126, 54235}, {3263, 6591}, {3286, 14618}, {3675, 6335}, {3912, 7649}, {3932, 17925}, {4238, 16732}, {7017, 53539}, {8751, 62430}, {16082, 42758}, {17435, 18026}, {17755, 68631}, {18027, 23225}, {18206, 24006}, {18344, 40704}, {23770, 57499}, {31623, 53551}, {34337, 62635}, {35094, 65333}, {36124, 53583}, {39063, 60583}, {39534, 56753}, {40217, 65106}, {43933, 51390}, {46102, 52305}, {46388, 57787}, {50441, 53150}, {68161, 68565}
X(68783) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 666}, {7, 65301}, {19, 36086}, {25, 919}, {34, 36146}, {92, 51560}, {241, 6516}, {264, 36803}, {273, 34085}, {278, 927}, {281, 36802}, {331, 46135}, {393, 65333}, {513, 1814}, {514, 31637}, {518, 1332}, {607, 52927}, {608, 32735}, {649, 36057}, {653, 39293}, {665, 3}, {667, 32658}, {672, 1331}, {918, 69}, {926, 219}, {1458, 1813}, {1783, 5377}, {1843, 46163}, {1861, 190}, {1876, 651}, {1973, 32666}, {2170, 23696}, {2223, 906}, {2254, 63}, {2283, 44717}, {2340, 4587}, {2356, 101}, {2489, 56853}, {2501, 13576}, {2969, 62635}, {3064, 14942}, {3125, 10099}, {3126, 25083}, {3286, 4558}, {3675, 905}, {3693, 4571}, {3912, 4561}, {3932, 52609}, {4088, 306}, {4238, 4567}, {5089, 100}, {5236, 664}, {6591, 105}, {7115, 59101}, {7649, 673}, {8638, 52425}, {8735, 885}, {8754, 66282}, {9436, 65164}, {9454, 32656}, {15149, 99}, {17115, 23601}, {17435, 521}, {17442, 35333}, {17924, 2481}, {18157, 55202}, {18206, 4592}, {18344, 294}, {20683, 4574}, {23225, 577}, {23829, 17206}, {24290, 72}, {30941, 4563}, {34337, 42720}, {34855, 65296}, {35505, 53550}, {37908, 5546}, {42067, 43929}, {42069, 28132}, {42071, 2284}, {42770, 65868}, {42771, 52307}, {43042, 348}, {43923, 1462}, {43933, 55943}, {44426, 36796}, {46107, 18031}, {46108, 668}, {46388, 212}, {50333, 345}, {52305, 26932}, {52614, 1260}, {52635, 36059}, {53539, 222}, {53544, 77}, {53550, 394}, {53551, 1214}, {53555, 22128}, {54407, 662}, {55133, 28420}, {57468, 65302}, {57499, 35574}, {58757, 68565}, {62429, 15413}, {65103, 28071}, {65106, 6654}, {65333, 57536}, {68631, 52209}
X(68783) = {X(58838),X(58840)}-harmonic conjugate of X(65103)

X(68784) = ORTHIC AXIS INTERCEPT OF X(9)X(522)

Barycentrics    (b - c)*(-a + b + c)^2*(a^2 + b^2 - 2*b*c + c^2) : :
X(68784) = 2 X[14330] + X[42462], 3 X[14427] - 2 X[57064]

X(68784) lies on these lines: {6, 59976}, {9, 522}, {75, 48070}, {230, 231}, {514, 7216}, {649, 14331}, {661, 17898}, {1447, 11068}, {1459, 3554}, {1639, 7628}, {3008, 28590}, {3239, 4171}, {3667, 65680}, {3676, 23748}, {3700, 40137}, {3737, 17412}, {4130, 42337}, {4435, 65102}, {4468, 20906}, {4501, 57180}, {14282, 28161}, {14298, 48269}, {14300, 49293}, {14400, 47768}, {14427, 57064}, {20504, 21119}, {21120, 30520}, {26258, 47798}, {27486, 55868}, {28143, 47806}, {31231, 46919}, {34805, 40577}, {46389, 49284}, {46393, 47765}, {47785, 59491}, {48398, 51400}, {60479, 60946}

X(68784) = midpoint of X(657) and X(42462)
X(68784) = reflection of X(i) in X(j) for these {i,j}: {657, 14330}, {4171, 3239}, {23748, 3676}
X(68784) = X(i)-Ceva conjugate of X(j) for these (i,j): {75, 2310}, {277, 11}, {514, 48398}, {3732, 497}, {7046, 1146}
X(68784) = X(i)-isoconjugate of X(j) for these (i,j): {3, 66952}, {6, 8269}, {63, 59128}, {101, 56359}, {109, 7131}, {241, 59133}, {651, 1037}, {658, 7084}, {692, 30705}, {934, 7123}, {1041, 1813}, {1407, 52778}, {1415, 8817}, {1461, 56179}, {3939, 63178}, {4619, 68559}, {6614, 56243}, {24027, 48070}, {34080, 62538}, {40403, 53321}, {52410, 54967}
X(68784) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 8269}, {11, 7131}, {522, 48070}, {1015, 56359}, {1086, 30705}, {1146, 8817}, {1565, 7056}, {2968, 30701}, {3162, 59128}, {4000, 190}, {5515, 8816}, {6554, 658}, {14714, 7123}, {14936, 1}, {15487, 934}, {17060, 1025}, {18589, 1020}, {24771, 52778}, {35508, 56179}, {36103, 66952}, {38991, 1037}, {40194, 6574}, {40617, 63178}, {40621, 62538}, {55068, 40403}, {59619, 4554}
X(68784) = crosspoint of X(i) and X(j) for these (i,j): {497, 3732}, {514, 3239}
X(68784) = crosssum of X(101) and X(1461)
X(68784) = crossdifference of every pair of points on line {3, 1037}
X(68784) = barycentric product X(i)*X(j) for these {i,j}: {75, 17115}, {346, 48398}, {497, 522}, {514, 6554}, {523, 68715}, {614, 4397}, {693, 4319}, {1021, 53510}, {1040, 44426}, {1043, 48403}, {1119, 58776}, {1146, 3732}, {1633, 24026}, {1863, 4025}, {2082, 4391}, {2322, 21107}, {3064, 27509}, {3239, 4000}, {3261, 30706}, {3667, 62543}, {3673, 3900}, {3676, 4012}, {3914, 7253}, {4081, 65188}, {4086, 5324}, {4163, 7195}, {4171, 16750}, {6362, 64438}, {7083, 35519}, {7124, 46110}, {15411, 52577}, {16502, 52622}, {17926, 18589}, {18155, 40965}, {24002, 28070}, {28132, 51400}, {35518, 40987}, {57064, 62544}
X(68784) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 8269}, {19, 66952}, {25, 59128}, {200, 52778}, {341, 54967}, {497, 664}, {513, 56359}, {514, 30705}, {522, 8817}, {614, 934}, {650, 7131}, {657, 7123}, {663, 1037}, {1021, 40403}, {1040, 6516}, {1146, 48070}, {1633, 7045}, {1851, 36118}, {1863, 1897}, {2082, 651}, {2195, 59133}, {3239, 30701}, {3667, 62538}, {3669, 63178}, {3673, 4569}, {3732, 1275}, {3900, 56179}, {3914, 4566}, {4000, 658}, {4012, 3699}, {4130, 56243}, {4171, 56260}, {4319, 100}, {4397, 57925}, {5324, 1414}, {6554, 190}, {6590, 8816}, {7083, 109}, {7101, 42384}, {7124, 1813}, {7195, 4626}, {7289, 65296}, {8641, 7084}, {16502, 1461}, {16583, 1020}, {16750, 4635}, {17115, 1}, {17926, 40411}, {18344, 1041}, {21107, 56382}, {23620, 52610}, {27509, 65164}, {28017, 4617}, {28070, 644}, {30706, 101}, {40175, 6574}, {40934, 53321}, {40965, 4551}, {40987, 108}, {48398, 279}, {48403, 3668}, {50490, 1042}, {52577, 52607}, {58776, 1265}, {62543, 53647}, {64438, 6606}, {65188, 59457}, {68715, 99}
X(68784) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {650, 3064, 6590}, {58838, 58840, 47123}

X(68785) = ORTHIC AXIS INTERCEPT OF X(25)X(512)

Barycentrics    a^2*(b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 4*b^2*c^2 + c^4) : :

X(68785) lies on these lines: {2, 14618}, {4, 5996}, {24, 65390}, {25, 512}, {111, 32710}, {112, 7480}, {230, 231}, {427, 16229}, {686, 924}, {1194, 2506}, {1560, 25641}, {1636, 2451}, {1640, 59987}, {3569, 14696}, {4108, 6353}, {6997, 44918}, {7630, 30476}, {7631, 45688}, {8430, 14687}, {9064, 32681}, {10278, 59896}, {14401, 40938}, {17414, 17994}, {21828, 55206}, {23301, 65394}, {30211, 32320}, {30451, 65656}, {30474, 46229}, {38282, 66895}, {40353, 40388}, {53958, 58959}, {55208, 55210}, {57260, 60777}

X(68785) = reflection of X(i) in X(j) for these {i,j}: {647, 46425}, {47236, 2501}, {66073, 30476}
X(68785) = isogonal conjugate of X(65323)
X(68785) = polar conjugate of X(65284)
X(68785) = complement of the isotomic conjugate of X(53958)
X(68785) = polar conjugate of the isotomic conjugate of X(8675)
X(68785) = polar conjugate of the isogonal conjugate of X(42660)
X(68785) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 53993}, {53958, 2887}
X(68785) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 53993}, {9064, 25}
X(68785) = X(42660)-cross conjugate of X(8675)
X(68785) = X(i)-isoconjugate of X(j) for these (i,j): {1, 65323}, {48, 65284}, {63, 1302}, {69, 36149}, {163, 57819}, {304, 32738}, {662, 4846}, {4575, 34289}, {4592, 34288}, {11064, 36083}
X(68785) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 65323}, {115, 57819}, {136, 34289}, {1084, 4846}, {1249, 65284}, {3162, 1302}, {5139, 34288}, {53993, 2}
X(68785) = crosspoint of X(i) and X(j) for these (i,j): {2, 53958}, {40388, 58959}
X(68785) = crosssum of X(i) and X(j) for these (i,j): {525, 37638}, {647, 64100}, {1514, 55265}
X(68785) = crossdifference of every pair of points on line {3, 4549}
X(68785) = barycentric product X(i)*X(j) for these {i,j}: {4, 8675}, {25, 30474}, {264, 42660}, {378, 523}, {512, 44134}, {850, 44080}, {924, 51833}, {2433, 62628}, {2489, 32833}, {2501, 15066}, {5063, 14618}, {5891, 66300}, {6591, 42704}, {8749, 46229}, {9064, 53832}, {10564, 18808}, {11653, 16230}, {53958, 53993}, {58757, 68660}, {66299, 68659}
X(68785) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 65284}, {6, 65323}, {25, 1302}, {378, 99}, {512, 4846}, {523, 57819}, {1973, 36149}, {1974, 32738}, {2489, 34288}, {2501, 34289}, {5063, 4558}, {8675, 69}, {11653, 17932}, {15066, 4563}, {17994, 56925}, {30474, 305}, {32833, 52608}, {40354, 32681}, {42660, 3}, {44080, 110}, {44134, 670}, {47649, 65322}, {51833, 46134}, {52438, 32661}, {58757, 68641}
X(68785) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {647, 1637, 2485}, {647, 2489, 47230}, {647, 2501, 6753}, {2489, 47230, 6753}, {2501, 47230, 2489}

X(68786) = ORTHIC AXIS INTERCEPT OF X(51)X(512)

Barycentrics    (b^2 - c^2)*(-a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - 4*b^2*c^2 + 2*c^4) : :
X(68786) = X[647] - 4 X[2501], 5 X[647] - 8 X[6587], 3 X[647] - 4 X[9209], X[647] + 2 X[12077], 7 X[647] - 4 X[47122], 5 X[647] - 2 X[55280], 7 X[647] - 16 X[59652], 5 X[1637] - 4 X[6587], 3 X[1637] - 2 X[9209], 7 X[1637] - 2 X[47122], 5 X[1637] - X[55280], 7 X[1637] - 8 X[59652], 5 X[2501] - 2 X[6587], 3 X[2501] - X[9209], and many others

X(68786) lies on these lines: {51, 512}, {115, 6070}, {125, 65755}, {230, 231}, {262, 5466}, {520, 14391}, {525, 58351}, {686, 55219}, {1383, 2395}, {2028, 13636}, {2029, 13722}, {2052, 2394}, {2081, 18578}, {2524, 14773}, {2799, 30474}, {3268, 30476}, {3567, 65425}, {3569, 3906}, {4108, 62663}, {6563, 31277}, {7950, 9210}, {8371, 55267}, {8644, 55122}, {9979, 13306}, {10567, 65610}, {12079, 65613}, {14417, 64919}, {16229, 42400}, {18487, 32225}, {20577, 63175}, {20977, 33601}, {23357, 60603}, {27550, 47862}, {27551, 47861}, {35361, 65762}, {36830, 60604}, {44554, 44560}, {52742, 58903}, {55197, 58299}, {55273, 55276}, {64788, 66463}

X(68786) = midpoint of X(i) and X(j) for these {i,j}: {1637, 12077}, {4108, 62663}, {20578, 20579}
X(68786) = reflection of X(i) in X(j) for these {i,j}: {647, 1637}, {1637, 2501}, {3268, 30476}, {62568, 8371}
X(68786) = complement of the isotomic conjugate of X(53693)
X(68786) = polar conjugate of the isogonal conjugate of X(67534)
X(68786) = X(i)-complementary conjugate of X(j) for these (i,j): {52154, 21253}, {53693, 2887}
X(68786) = X(i)-Ceva conjugate of X(j) for these (i,j): {16813, 16265}, {34288, 115}, {52487, 8754}, {56270, 125}, {61116, 41221}
X(68786) = X(i)-isoconjugate of X(j) for these (i,j): {63, 58994}, {162, 56266}, {163, 57822}, {662, 3431}, {799, 58941}, {4575, 43530}, {6149, 54959}, {36034, 46809}, {58940, 65262}
X(68786) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 57822}, {125, 56266}, {136, 43530}, {1084, 3431}, {3162, 58994}, {3258, 46809}, {4550, 4558}, {14993, 54959}, {15295, 58983}, {38996, 58941}
X(68786) = crosspoint of X(2) and X(53693)
X(68786) = crossdifference of every pair of points on line {3, 323}
X(68786) = barycentric product X(i)*X(j) for these {i,j}: {30, 67183}, {264, 67534}, {381, 523}, {512, 44135}, {850, 34417}, {1531, 18808}, {1637, 46808}, {2394, 18487}, {2501, 37638}, {3581, 10412}, {4993, 12077}, {5158, 14618}, {5466, 32225}, {6344, 14314}, {6368, 58785}, {15475, 52149}, {18477, 24006}, {34416, 44173}, {41079, 51544}, {55121, 58942}, {58261, 65316}, {63425, 66299}
X(68786) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 58994}, {381, 99}, {512, 3431}, {523, 57822}, {647, 56266}, {669, 58941}, {1637, 46809}, {1989, 54959}, {2501, 43530}, {3581, 10411}, {5158, 4558}, {11060, 58983}, {14314, 52437}, {14398, 51545}, {15475, 18316}, {18477, 4592}, {18487, 2407}, {21731, 58940}, {21970, 57216}, {32225, 5468}, {34416, 1576}, {34417, 110}, {37638, 4563}, {44135, 670}, {51544, 44769}, {58308, 46091}, {58757, 16263}, {58785, 18831}, {58942, 18878}, {67183, 1494}, {67534, 3}
X(68786) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2501, 12077, 647}, {2501, 47122, 59652}, {2501, 47236, 2489}, {6587, 55280, 647}, {14582, 55265, 58900}

X(68787) = ORTHIC AXIS INTERCEPT OF X(2)X(881)

Barycentrics    (b^2 - c^2)*(a^4 + b^2*c^2) : :
X(68787) = 3 X[2] + X[58784], 2 X[3005] - 3 X[45333], 3 X[45333] + 2 X[58784], 2 X[647] - 3 X[11176], 3 X[1637] - X[47126], X[6131] - 4 X[6134], 3 X[11176] - 4 X[44451], 3 X[11176] + 2 X[47128], 2 X[44451] + X[47128], 3 X[45687] - X[55280], 3 X[351] - X[31296], 2 X[23301] - 3 X[45689], 4 X[30476] - 3 X[45689], X[669] - 3 X[4108], and many others

X(68787) lies on these lines: {2, 881}, {25, 65394}, {125, 46669}, {141, 9012}, {230, 231}, {351, 31296}, {512, 625}, {513, 24353}, {525, 50550}, {661, 35352}, {669, 804}, {670, 62410}, {689, 59026}, {826, 4142}, {1215, 29512}, {1510, 53567}, {1576, 9514}, {2514, 55190}, {2525, 9479}, {2528, 14420}, {3800, 14341}, {3804, 25423}, {4139, 47831}, {4369, 27799}, {4806, 50495}, {5466, 43528}, {5996, 8665}, {6704, 12073}, {7927, 44564}, {8034, 26983}, {8651, 23878}, {8664, 9148}, {8704, 44813}, {9035, 60372}, {9147, 66893}, {9185, 31950}, {9494, 52618}, {10189, 50543}, {10278, 12075}, {11123, 41298}, {14428, 60475}, {16536, 16537}, {17989, 58361}, {18105, 23285}, {21051, 50494}, {24325, 29402}, {24533, 24622}, {24924, 40471}, {27045, 58289}, {28729, 39201}, {31299, 53365}, {33294, 50552}, {42660, 44818}, {44821, 66122}, {46778, 53266}, {50496, 65152}

X(68787) = midpoint of X(i) and X(j) for these {i,j}: {647, 47128}, {669, 850}, {2525, 50548}, {3005, 58784}, {5113, 54263}, {8664, 44445}, {9494, 52618}, {12077, 50553}, {18105, 23285}, {33294, 50552}, {47173, 47175}
X(68787) = reflection of X(i) in X(j) for these {i,j}: {647, 44451}, {23301, 30476}, {42660, 44818}, {45333, 2}, {50548, 59740}
X(68787) = complement of X(3005)
X(68787) = complement of the isogonal conjugate of X(4577)
X(68787) = complement of the isotomic conjugate of X(689)
X(68787) = medial-isogonal conjugate of X(15449)
X(68787) = X(8601)-anticomplementary conjugate of X(21221)
X(68787) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 15449}, {31, 35971}, {75, 46654}, {82, 115}, {83, 8287}, {99, 21249}, {110, 16587}, {251, 16592}, {308, 21253}, {560, 55050}, {561, 55070}, {662, 6292}, {689, 2887}, {799, 21248}, {827, 37}, {1101, 52591}, {1176, 16573}, {1414, 17055}, {1580, 39079}, {1799, 34846}, {3112, 125}, {3405, 35088}, {4577, 10}, {4593, 141}, {4599, 2}, {4628, 16589}, {4630, 16584}, {18098, 6627}, {18833, 53575}, {24041, 3005}, {28724, 16595}, {33515, 65096}, {34055, 15526}, {34072, 39}, {36084, 8623}, {37204, 626}, {39179, 6547}, {42371, 21235}, {42396, 226}, {46289, 1084}, {52376, 1086}, {52394, 11}, {52936, 1215}, {53657, 16580}, {55240, 23991}, {56971, 35078}, {56982, 61063}, {57545, 8060}, {58784, 24040}, {62531, 63618}, {65307, 1214}, {67149, 41178}
X(68787) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 35971}, {882, 804}
X(68787) = X(35971)-cross conjugate of X(4074)
X(68787) = X(i)-isoconjugate of X(j) for these (i,j): {99, 9288}, {110, 9285}, {163, 9229}, {662, 695}, {670, 9236}, {783, 2236}, {799, 51948}, {1576, 9239}, {1740, 59028}, {4575, 37892}, {4599, 67165}, {37204, 57503}, {51982, 56982}
X(68787) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 9229}, {136, 37892}, {244, 9285}, {1084, 695}, {3124, 67165}, {4858, 9239}, {35078, 54129}, {35971, 2}, {37895, 99}, {38986, 9288}, {38996, 51948}, {46669, 3505}
X(68787) = crosspoint of X(i) and X(j) for these (i,j): {2, 689}, {99, 45857}, {107, 60125}, {22456, 36897}
X(68787) = crosssum of X(i) and X(j) for these (i,j): {6, 688}, {512, 5943}, {520, 11574}, {826, 21243}, {36213, 39469}
X(68787) = crossdifference of every pair of points on line {3, 695}
X(68787) = barycentric product X(i)*X(j) for these {i,j}: {384, 523}, {512, 9230}, {661, 1965}, {689, 35971}, {733, 35558}, {782, 14970}, {798, 1925}, {804, 54130}, {850, 1915}, {1577, 1582}, {1932, 20948}, {2501, 37894}, {3267, 11380}, {4074, 58784}, {4580, 12143}, {14618, 37893}, {16985, 66267}, {51320, 56981}
X(68787) = barycentric quotient X(i)/X(j) for these {i,j}: {384, 99}, {512, 695}, {523, 9229}, {661, 9285}, {669, 51948}, {733, 783}, {782, 732}, {798, 9288}, {804, 54129}, {881, 14946}, {882, 51982}, {1577, 9239}, {1582, 662}, {1915, 110}, {1924, 9236}, {1925, 4602}, {1932, 163}, {1965, 799}, {2501, 37892}, {3005, 67165}, {3224, 59028}, {3800, 3866}, {4074, 4576}, {9230, 670}, {9494, 57503}, {11380, 112}, {12143, 41676}, {14970, 18828}, {16985, 17941}, {35558, 35540}, {35971, 3005}, {37893, 4558}, {37894, 4563}, {51320, 56980}, {51904, 56982}, {54130, 18829}, {66267, 40847}
X(68787) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 58784, 3005}, {647, 2501, 54267}, {647, 44451, 11176}, {850, 4108, 669}, {8664, 9148, 44445}, {8665, 31279, 5996}, {23301, 30476, 45689}, {31072, 44445, 9148}

X(68788) = ORTHIC AXIS INTERCEPT OF X(1)X(7252)

Barycentrics    a*(b^2 - c^2)*(a^3 - a*b^2 + a*b*c + b^2*c - a*c^2 + b*c^2) : :
X(68788) = 3 X[1962] + X[40471], 3 X[36900] + X[50557], 5 X[26985] - X[66893], 4 X[30476] - 5 X[31250], 4 X[41300] + X[48125]

X(68788) lies on these lines: {1, 7252}, {37, 661}, {63, 18199}, {230, 231}, {312, 28758}, {321, 26983}, {512, 4162}, {513, 42664}, {514, 29717}, {522, 43060}, {525, 3669}, {649, 3726}, {652, 4449}, {665, 4151}, {693, 31296}, {850, 4885}, {905, 23879}, {1021, 48293}, {1214, 4077}, {1962, 40471}, {2352, 23864}, {2522, 51648}, {2611, 38347}, {2623, 57174}, {3050, 22383}, {3666, 4369}, {3700, 7180}, {3708, 11998}, {3709, 4841}, {3752, 24924}, {3931, 4761}, {4017, 8611}, {4024, 21828}, {4139, 66524}, {4145, 4394}, {4359, 27167}, {4762, 36900}, {4820, 64868}, {4988, 55210}, {6563, 28984}, {7192, 28606}, {8672, 48026}, {14589, 68154}, {16751, 17161}, {17147, 26822}, {20906, 25511}, {20949, 27293}, {20950, 26854}, {20952, 26114}, {21104, 25080}, {21225, 29771}, {23818, 64864}, {23878, 43051}, {24782, 47788}, {24900, 47792}, {24948, 46915}, {24960, 48207}, {24961, 26080}, {25084, 47784}, {25091, 26017}, {25098, 43067}, {25258, 25667}, {25666, 44307}, {26985, 66893}, {27648, 47657}, {28374, 28894}, {30476, 31250}, {37593, 63461}, {41300, 48125}, {46383, 53314}, {46389, 52310}, {48276, 52326}, {48281, 57042}, {48292, 65097}, {51662, 55234}, {52591, 52601}, {53527, 55232}, {55197, 56325}, {57148, 62851}, {57232, 58286}

X(68788) = midpoint of X(693) and X(31296)
X(68788) = reflection of X(i) in X(j) for these {i,j}: {650, 647}, {850, 4885}
X(68788) = isogonal conjugate of X(68205)
X(68788) = complement of the isotomic conjugate of X(34594)
X(68788) = X(i)-complementary conjugate of X(j) for these (i,j): {596, 53575}, {1576, 4075}, {2206, 8054}, {32739, 62588}, {34594, 2887}, {37205, 626}, {39798, 21253}, {39949, 21252}, {40148, 125}, {40519, 3454}, {65202, 21245}, {65286, 21235}
X(68788) = X(i)-Ceva conjugate of X(j) for these (i,j): {62748, 661}, {62918, 125}, {65207, 65}
X(68788) = X(i)-isoconjugate of X(j) for these (i,j): {1, 68205}, {29, 40518}, {58, 56248}, {163, 57830}, {662, 57666}, {4558, 60816}, {4565, 44040}
X(68788) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 68205}, {10, 56248}, {115, 57830}, {1084, 57666}, {4415, 21580}, {55064, 44040}
X(68788) = crosspoint of X(i) and X(j) for these (i,j): {1, 4552}, {2, 34594}, {47796, 48281}
X(68788) = crosssum of X(i) and X(j) for these (i,j): {1, 7252}, {6, 4132}
X(68788) = crossdifference of every pair of points on line {3, 1724}
X(68788) = barycentric product X(i)*X(j) for these {i,j}: {10, 48281}, {37, 47796}, {65, 20293}, {81, 21721}, {404, 523}, {512, 44139}, {513, 56318}, {661, 32939}, {850, 44085}, {905, 56319}, {1441, 48387}, {4132, 58073}, {4551, 44311}, {6358, 57212}, {6591, 42705}, {39006, 65207}, {40149, 57042}, {57103, 57809}
X(68788) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 68205}, {37, 56248}, {404, 99}, {512, 57666}, {523, 57830}, {1409, 40518}, {4041, 44040}, {20293, 314}, {21721, 321}, {32939, 799}, {44085, 110}, {44139, 670}, {44311, 18155}, {47796, 274}, {48281, 86}, {48387, 21}, {56318, 668}, {56319, 6335}, {57042, 1812}, {57103, 283}, {57212, 2185}, {58073, 65286}, {58315, 2194}
X(68788) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3700, 7180, 21894}, {6586, 45745, 650}, {6589, 6590, 650}, {26080, 48209, 24961}

X(68789) = ORTHIC AXIS INTERCEPT OF X(512)X(661)

Barycentrics    a*(b^2 - c^2)*(a^3 - a*b^2 - 2*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2) : :
X(68789) = 3 X[1637] - 2 X[47124], 3 X[4893] - X[42664], 4 X[4885] - 5 X[31277], 5 X[26777] - X[31296], X[26824] - 5 X[31072]

X(68789) lies on these lines: {2, 50557}, {230, 231}, {449, 525}, {512, 661}, {649, 8672}, {665, 48275}, {669, 50538}, {693, 30476}, {850, 17494}, {1021, 50346}, {1109, 14936}, {1577, 27588}, {2451, 17418}, {2522, 48069}, {3239, 4151}, {3700, 24089}, {3709, 4024}, {3737, 65097}, {4139, 4893}, {4762, 31174}, {4789, 24948}, {4802, 43060}, {4838, 55210}, {4885, 31277}, {4976, 64868}, {4988, 7180}, {7252, 50349}, {14618, 17926}, {21828, 47669}, {22222, 54256}, {23878, 31150}, {23879, 48003}, {24782, 24900}, {24960, 48204}, {24961, 48205}, {25667, 30864}, {26777, 31296}, {26824, 31072}, {27648, 47656}, {28423, 55190}, {48277, 52326}

X(68789) = midpoint of X(i) and X(j) for these {i,j}: {669, 50538}, {850, 17494}
X(68789) = reflection of X(i) in X(j) for these {i,j}: {647, 650}, {693, 30476}, {47174, 47252}
X(68789) = isogonal conjugate of X(68202)
X(68789) = complement of X(50557)
X(68789) = complement of the isotomic conjugate of X(43356)
X(68789) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 38967}, {39708, 53575}, {39983, 21253}, {43356, 2887}
X(68789) = X(2)-Ceva conjugate of X(38967)
X(68789) = X(i)-isoconjugate of X(j) for these (i,j): {1, 68202}, {58, 54970}, {63, 36077}, {81, 65227}, {86, 36080}, {99, 2215}, {163, 57831}, {662, 51223}, {1414, 2335}
X(68789) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 68202}, {10, 54970}, {115, 57831}, {647, 63220}, {1084, 51223}, {3162, 36077}, {23882, 15417}, {38967, 2}, {38986, 2215}, {40586, 65227}, {40600, 36080}, {40608, 2335}
X(68789) = crosspoint of X(i) and X(j) for these (i,j): {2, 43356}, {23882, 46385}
X(68789) = crosssum of X(i) and X(j) for these (i,j): {525, 32782}, {1019, 62812}, {4560, 25515}, {36080, 65227}
X(68789) = crossdifference of every pair of points on line {3, 81}
X(68789) = barycentric product X(i)*X(j) for these {i,j}: {10, 46385}, {37, 23882}, {92, 46382}, {405, 523}, {512, 44140}, {513, 5295}, {521, 1882}, {656, 39585}, {661, 5271}, {850, 5320}, {1451, 4086}, {1500, 15417}, {3700, 37543}, {4064, 56831}, {4151, 14549}, {6591, 42706}, {14618, 68747}, {38967, 43356}, {52355, 54394}, {53560, 65355}
X(68789) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 68202}, {25, 36077}, {37, 54970}, {42, 65227}, {125, 63220}, {213, 36080}, {405, 99}, {512, 51223}, {523, 57831}, {798, 2215}, {1451, 1414}, {1882, 18026}, {3709, 2335}, {5271, 799}, {5295, 668}, {5320, 110}, {14549, 53649}, {23882, 274}, {37543, 4573}, {38967, 50557}, {39585, 811}, {44140, 670}, {46382, 63}, {46385, 86}, {55232, 63235}, {68747, 4558}
X(68789) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {650, 3064, 33525}, {650, 45745, 3310}, {650, 48397, 6586}, {661, 4041, 55232}, {661, 21832, 50492}, {2501, 57094, 2489}, {4789, 24948, 25084}, {24900, 47782, 24782}

X(68790) = ORTHIC AXIS INTERCEPT OF X(4)X(32473)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(b^2 + c^2) : :
X(68790) = 3 X[1637] - 2 X[2485], 3 X[2489] - 2 X[57071], 4 X[2501] - X[14273], 3 X[2501] - X[57071], X[12077] + 2 X[47138], 3 X[14273] - 4 X[57071], X[47133] - 4 X[68436], 3 X[2525] - 2 X[57222], 3 X[23285] - X[57222], 3 X[9134] - 2 X[56739]

X(68790) lies on these lines: {4, 32473}, {24, 39214}, {25, 18105}, {112, 1287}, {115, 53987}, {230, 231}, {233, 55267}, {648, 65278}, {688, 1843}, {826, 21108}, {1510, 45801}, {2395, 8882}, {2451, 9033}, {2514, 12075}, {2525, 23285}, {2799, 3267}, {3005, 65472}, {3050, 64877}, {3569, 6368}, {6563, 52598}, {7927, 57204}, {9134, 56739}, {14977, 40413}, {17994, 39799}, {23290, 62384}, {23878, 57065}, {30476, 57069}, {34952, 55122}, {35325, 46155}, {46151, 61218}, {50548, 65394}, {55195, 55206}, {55197, 55208}, {55219, 55275}, {60101, 60338}, {66291, 66299}

X(68790) = reflection of X(i) in X(j) for these {i,j}: {647, 47125}, {2489, 2501}, {2514, 12075}, {2525, 23285}, {6563, 52598}, {14273, 2489}, {17994, 51513}, {57069, 30476}
X(68790) = isogonal conjugate of X(65307)
X(68790) = polar conjugate of X(4577)
X(68790) = complement of the isotomic conjugate of X(53949)
X(68790) = isotomic conjugate of the polar conjugate of X(65472)
X(68790) = polar conjugate of the isotomic conjugate of X(826)
X(68790) = polar conjugate of the isogonal conjugate of X(3005)
X(68790) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 53983}, {53949, 2887}
X(68790) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 53983}, {25, 115}, {264, 8754}, {648, 46026}, {27376, 39691}, {35325, 27371}, {41676, 427}, {42396, 4}, {43678, 125}, {46151, 1843}, {47847, 5139}
X(68790) = X(i)-cross conjugate of X(j) for these (i,j): {3005, 826}, {39691, 27376}, {66184, 68572}
X(68790) = X(i)-isoconjugate of X(j) for these (i,j): {1, 65307}, {3, 4599}, {48, 4577}, {63, 827}, {69, 34072}, {82, 4558}, {83, 4575}, {110, 34055}, {162, 28724}, {163, 1799}, {184, 4593}, {251, 4592}, {255, 42396}, {304, 4630}, {662, 1176}, {689, 9247}, {799, 10547}, {906, 52394}, {1101, 4580}, {1331, 52376}, {1444, 4628}, {3112, 32661}, {3405, 43754}, {4020, 52936}, {4563, 46289}, {14575, 37204}, {18070, 47390}, {18105, 62719}, {46288, 55202}, {56971, 65327}
X(68790) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 65307}, {39, 4563}, {115, 1799}, {125, 28724}, {136, 83}, {141, 4558}, {244, 34055}, {339, 305}, {427, 4611}, {523, 4580}, {826, 2525}, {1084, 1176}, {1249, 4577}, {3124, 3}, {3162, 827}, {5139, 251}, {5190, 52394}, {5521, 52376}, {6523, 42396}, {15449, 69}, {34452, 32661}, {36103, 4599}, {38970, 20022}, {38996, 10547}, {39691, 7767}, {40585, 4592}, {40938, 99}, {46654, 57480}, {47413, 20806}, {48317, 52898}, {53981, 17941}, {53983, 2}, {55043, 63}, {55050, 184}, {55070, 10316}, {62576, 689}, {62605, 4593}
X(68790) = crosspoint of X(i) and X(j) for these (i,j): {2, 53949}, {4, 42396}, {107, 54703}, {427, 41676}, {2501, 14618}
X(68790) = crosssum of X(i) and X(j) for these (i,j): {647, 22352}, {4558, 32661}
X(68790) = crossdifference of every pair of points on line {3, 1176}
X(68790) = barycentric product X(i)*X(j) for these {i,j}: {4, 826}, {10, 21108}, {19, 62418}, {25, 23285}, {38, 24006}, {39, 14618}, {69, 65472}, {92, 8061}, {115, 41676}, {125, 46151}, {141, 2501}, {225, 48278}, {264, 3005}, {338, 35325}, {393, 2525}, {427, 523}, {428, 31067}, {512, 1235}, {514, 21016}, {525, 27376}, {648, 39691}, {661, 20883}, {688, 18022}, {850, 1843}, {1577, 17442}, {1634, 2970}, {1824, 48084}, {1826, 16892}, {1969, 2084}, {2489, 8024}, {2528, 32085}, {2530, 41013}, {2531, 68630}, {3566, 47730}, {3806, 8801}, {3917, 66299}, {3933, 58757}, {3954, 17924}, {4024, 17171}, {4576, 8754}, {4705, 16747}, {5466, 64724}, {6368, 19174}, {7649, 15523}, {7813, 68629}, {9019, 66943}, {9494, 44161}, {13854, 23881}, {14273, 31125}, {14424, 17983}, {15412, 27371}, {15449, 42396}, {16030, 23290}, {16230, 20021}, {18808, 51360}, {21035, 46107}, {23962, 61218}, {27369, 44173}, {31065, 46026}, {35235, 46155}, {39240, 46166}, {39241, 46167}, {43673, 51434}, {46104, 57132}, {51371, 53149}, {52568, 57204}, {53949, 53983}, {58335, 68576}
X(68790) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 4577}, {6, 65307}, {19, 4599}, {25, 827}, {38, 4592}, {39, 4558}, {92, 4593}, {115, 4580}, {141, 4563}, {264, 689}, {393, 42396}, {427, 99}, {512, 1176}, {523, 1799}, {647, 28724}, {661, 34055}, {669, 10547}, {688, 184}, {826, 69}, {1235, 670}, {1843, 110}, {1930, 55202}, {1964, 4575}, {1969, 37204}, {1973, 34072}, {1974, 4630}, {2084, 48}, {2333, 4628}, {2474, 19459}, {2489, 251}, {2501, 83}, {2525, 3926}, {2528, 3933}, {2530, 1444}, {2531, 20775}, {2970, 52618}, {2971, 18105}, {3005, 3}, {3051, 32661}, {3186, 62531}, {3806, 3785}, {3954, 1332}, {4576, 47389}, {6591, 52376}, {7649, 52394}, {8024, 52608}, {8061, 63}, {8754, 58784}, {9494, 14575}, {13854, 53657}, {14273, 52898}, {14424, 6390}, {14618, 308}, {15449, 2525}, {15523, 4561}, {16230, 20022}, {16747, 4623}, {16892, 17206}, {17171, 4610}, {17442, 662}, {17994, 51862}, {18022, 42371}, {19174, 18831}, {20021, 17932}, {20883, 799}, {20975, 58353}, {21016, 190}, {21035, 1331}, {21108, 86}, {21123, 1790}, {21814, 906}, {23285, 305}, {23881, 34254}, {24006, 3112}, {27367, 32734}, {27369, 1576}, {27371, 14570}, {27373, 52915}, {27376, 648}, {28666, 61219}, {30489, 65328}, {31067, 57852}, {32085, 52936}, {35325, 249}, {39691, 525}, {40938, 4611}, {41267, 32656}, {41584, 57216}, {41676, 4590}, {42396, 57545}, {46026, 10330}, {46151, 18020}, {46154, 65321}, {47730, 35136}, {48278, 332}, {50521, 1437}, {51434, 34211}, {51513, 17500}, {51869, 43754}, {55206, 56245}, {56978, 65327}, {57132, 3917}, {57204, 46288}, {58335, 1792}, {58757, 32085}, {61218, 23357}, {62418, 304}, {64724, 5468}, {65472, 4}, {66299, 46104}, {66300, 39287}, {68575, 41209}

X(68791) = ORTHIC AXIS INTERCEPT OF X(122)X(125)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 + c^2)*(-2*a^6 + a^4*b^2 + b^6 + a^4*c^2 - b^4*c^2 - b^2*c^4 + c^6) : :
X(68791) = X[53345] + 3 X[53383], 2 X[684] - 3 X[14417], X[684] - 3 X[65723], 3 X[1637] - 4 X[6130], 3 X[1637] - 2 X[16230], 9 X[1637] - 8 X[45259], 3 X[6130] - 2 X[45259], 8 X[6132] - 9 X[9125], 3 X[16230] - 4 X[45259]

X(68791) lies on these lines: {2, 65871}, {3, 41077}, {4, 2848}, {67, 9003}, {74, 67763}, {98, 1297}, {107, 1301}, {122, 125}, {132, 50938}, {230, 231}, {248, 879}, {378, 52737}, {512, 62365}, {520, 2525}, {525, 42658}, {526, 54376}, {690, 5489}, {826, 39201}, {935, 32640}, {1649, 42736}, {1946, 48300}, {2881, 12145}, {3265, 8057}, {6334, 9517}, {6793, 9475}, {7669, 10117}, {7927, 15451}, {8673, 60597}, {12073, 67534}, {14223, 43537}, {15000, 65623}, {18556, 46229}, {31953, 55122}, {34156, 40080}, {34211, 60506}, {39473, 65749}, {43083, 52153}, {43754, 53379}, {44202, 46990}, {44203, 46996}, {44564, 65714}, {47082, 53783}, {47327, 55141}, {52585, 59932}, {58342, 58759}

X(68791) = midpoint of X(5489) and X(9409)
X(68791) = reflection of X(i) in X(j) for these {i,j}: {647, 47194}, {3265, 54260}, {14417, 65723}, {16230, 6130}, {41077, 3}, {59932, 52585}, {65714, 44564}
X(68791) = isogonal conjugate of X(44770)
X(68791) = complement of X(65871)
X(68791) = Dao-Moses-Telv-circle-inverse of X(6587)
X(68791) = polar conjugate of X(65265)
X(68791) = complement of the isotomic conjugate of X(2867)
X(68791) = isogonal conjugate of the polar conjugate of X(66161)
X(68791) = polar conjugate of the isotomic conjugate of X(39473)
X(68791) = tripolar centroid for these (i,j): {42287, 63856}
X(68791) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2155, 39359}, {19614, 14721}, {36084, 6225}, {36104, 14361}
X(68791) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 33504}, {2867, 2887}, {39297, 21259}
X(68791) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 33504}, {98, 125}, {1301, 50938}, {2409, 1503}, {2419, 525}, {2966, 51963}, {6330, 1562}, {20031, 1899}, {34211, 8779}, {43945, 23616}, {53173, 647}, {60506, 65749}, {60527, 3269}
X(68791) = X(i)-isoconjugate of X(j) for these (i,j): {1, 44770}, {2, 36046}, {3, 36092}, {48, 65265}, {63, 32687}, {75, 32649}, {110, 8767}, {162, 1297}, {163, 6330}, {662, 43717}, {1101, 68640}, {2435, 24000}, {24019, 64975}, {32676, 35140}, {36034, 52485}, {36036, 51822}, {36084, 39265}, {36142, 56601}, {51937, 65263}, {62720, 67185}
X(68791) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 44770}, {115, 6330}, {122, 14944}, {125, 1297}, {206, 32649}, {244, 8767}, {441, 877}, {523, 68640}, {525, 2419}, {647, 43673}, {1084, 43717}, {1249, 65265}, {1503, 2409}, {2679, 51822}, {3162, 32687}, {3258, 52485}, {14401, 66077}, {15526, 35140}, {15595, 99}, {23976, 648}, {23992, 56601}, {32664, 36046}, {33504, 2}, {35071, 64975}, {36103, 36092}, {38987, 39265}, {39071, 110}, {39073, 4230}, {50938, 107}, {57296, 441}, {60341, 8057}, {65726, 2966}, {65728, 47105}
X(68791) = crosspoint of X(i) and X(j) for these (i,j): {2, 2867}, {69, 2966}, {98, 60506}, {290, 15352}, {523, 879}, {525, 2419}, {1503, 2409}, {6330, 53639}, {30737, 34211}
X(68791) = crosssum of X(i) and X(j) for these (i,j): {4, 53345}, {6, 2881}, {25, 3569}, {110, 4230}, {112, 2445}, {237, 32320}, {297, 33294}, {1297, 2435}, {8779, 42658}
X(68791) = crossdifference of every pair of points on line {3, 112}
X(68791) = X(i)-line conjugate of X(j) for these (i,j): {98, 1297}, {6793, 9475}, {41077, 3}
X(68791) = barycentric product X(i)*X(j) for these {i,j}: {3, 66161}, {4, 39473}, {112, 58258}, {125, 34211}, {132, 53173}, {441, 523}, {520, 60516}, {525, 1503}, {647, 30737}, {684, 57490}, {690, 36894}, {850, 8779}, {879, 15595}, {1577, 8766}, {2312, 14208}, {2409, 15526}, {2419, 23976}, {2445, 36793}, {2525, 21458}, {2799, 34156}, {2867, 33504}, {3265, 16318}, {3267, 42671}, {6330, 60341}, {6333, 51963}, {6394, 55275}, {6587, 16096}, {6793, 34767}, {8552, 43089}, {9033, 63856}, {14376, 55129}, {14618, 68744}, {14977, 35282}, {15421, 53568}, {15639, 66964}, {17708, 57426}, {17932, 57430}, {18312, 40080}, {39469, 51257}, {43045, 52355}, {43673, 65749}, {51363, 62428}, {51437, 52617}, {53639, 57296}
X(68791) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 65265}, {6, 44770}, {19, 36092}, {25, 32687}, {31, 36046}, {32, 32649}, {115, 68640}, {125, 43673}, {441, 99}, {512, 43717}, {520, 64975}, {523, 6330}, {525, 35140}, {647, 1297}, {661, 8767}, {690, 56601}, {878, 67185}, {879, 9476}, {1503, 648}, {1562, 61189}, {1637, 52485}, {1640, 47105}, {1650, 66077}, {2312, 162}, {2409, 23582}, {2419, 57549}, {2445, 23964}, {2491, 51822}, {2525, 64974}, {3269, 2435}, {3569, 39265}, {6394, 55274}, {6587, 14944}, {6793, 4240}, {8766, 662}, {8779, 110}, {9409, 51937}, {9475, 4230}, {15526, 2419}, {15595, 877}, {16096, 44326}, {16318, 107}, {20975, 34212}, {21458, 42396}, {23616, 66964}, {23976, 2409}, {23977, 32230}, {28343, 52916}, {30737, 6331}, {33504, 65871}, {34156, 2966}, {34211, 18020}, {35282, 4235}, {36894, 892}, {39473, 69}, {40080, 5649}, {42671, 112}, {43089, 46456}, {51257, 65272}, {51363, 35360}, {51434, 46151}, {51437, 32713}, {51647, 65232}, {51960, 53205}, {51963, 685}, {53173, 57761}, {53568, 16237}, {55129, 17907}, {55275, 6530}, {56572, 65354}, {57296, 8057}, {57426, 9979}, {57430, 16230}, {57490, 22456}, {58258, 3267}, {60341, 441}, {60495, 46967}, {60506, 60179}, {60516, 6528}, {63856, 16077}, {65749, 34211}, {66161, 264}, {68744, 4558}
X(68791) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6103, 47138, 1637}, {6130, 16230, 1637}, {24007, 24008, 6587}

X(68792) = ORTHIC AXIS INTERCEPT OF X(37)X(4841)

Barycentrics    a*(b^2 - c^2)*(a^3 - a*b^2 + 2*a*b*c + 2*b^2*c - a*c^2 + 2*b*c^2) : :
X(68792) = 3 X[647] - 2 X[650], 3 X[850] - 5 X[26985], 3 X[1962] - 2 X[8653], X[4813] - 3 X[42664], 4 X[4885] - 3 X[31174], 3 X[17414] - X[50538], X[17494] - 3 X[36900], X[26824] + 3 X[31296], X[26824] - 3 X[50557], 5 X[31209] - 6 X[44560], 6 X[41300] - X[47664]

X(68792) lies on these lines: {37, 4841}, {230, 231}, {512, 4895}, {649, 4139}, {665, 48277}, {693, 23878}, {850, 26985}, {1962, 8653}, {2523, 51648}, {3288, 4449}, {3709, 4988}, {3960, 23879}, {4017, 55232}, {4024, 7180}, {4502, 4526}, {4777, 43060}, {4789, 24782}, {4838, 21828}, {4885, 31174}, {7252, 48292}, {7950, 50539}, {17414, 50538}, {17458, 50511}, {17494, 36900}, {21834, 50495}, {24960, 48209}, {25084, 47782}, {26824, 31296}, {31209, 44560}, {41300, 47664}, {46383, 48281}, {47669, 55210}, {48275, 52326}, {48293, 65097}, {48328, 57129}, {50492, 57234}

X(68792) = midpoint of X(31296) and X(50557)
X(68792) = reflection of X(12077) in X(47124)
X(68792) = X(39711)-complementary conjugate of X(53575)
X(68792) = X(i)-isoconjugate of X(j) for these (i,j): {163, 57877}, {662, 57705}
X(68792) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 57877}, {1084, 57705}
X(68792) = crosssum of X(525) and X(33172)
X(68792) = crossdifference of every pair of points on line {3, 16948}
X(68792) = barycentric product X(i)*X(j) for these {i,j}: {10, 48342}, {474, 523}, {512, 44147}, {850, 44104}
X(68792) = barycentric quotient X(i)/X(j) for these {i,j}: {474, 99}, {512, 57705}, {523, 57877}, {44104, 110}, {44147, 670}, {48342, 86}
X(68792) = {X(37),X(4841)}-harmonic conjugate of X(58299)

X(68793) = ORTHIC AXIS INTERCEPT OF X(187)X(690)

Barycentrics    (b^2 - c^2)*(-2*a^2 + b^2 + c^2)*(-2*a^6 + 2*a^4*b^2 - a^2*b^4 + b^6 + 2*a^4*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6) : :
X(68793) = 3 X[8859] - X[9979], 3 X[21445] - X[66463]

X(68793) lies on these lines: {2, 65872}, {187, 690}, {230, 231}, {351, 33919}, {385, 3268}, {524, 14417}, {525, 27088}, {1513, 2793}, {1640, 6041}, {1648, 1649}, {2021, 3906}, {2394, 2966}, {2799, 22329}, {3154, 44398}, {5912, 36166}, {6784, 55143}, {8859, 9979}, {9003, 15993}, {11063, 53263}, {14424, 33906}, {15544, 64461}, {16092, 34366}, {18334, 35078}, {21445, 66463}, {23878, 64943}, {34369, 48451}, {44564, 62629}, {51428, 57465}, {55141, 66266}, {58351, 66166}, {61070, 65908}

X(68793) = midpoint of X(385) and X(3268)
X(68793) = reflection of X(i) in X(j) for these {i,j}: {1637, 230}, {62629, 44564}
X(68793) = complement of X(65872)
X(68793) = complement of the isotomic conjugate of X(20404)
X(68793) = polar conjugate of the isotomic conjugate of X(39474)
X(68793) = tripolar centroid for these (i,j): {5967, 63858}
X(68793) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 35582}, {20404, 2887}
X(68793) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 35582}, {16092, 51428}, {18312, 32313}, {50941, 542}, {50942, 690}, {53232, 5191}
X(68793) = X(i)-isoconjugate of X(j) for these (i,j): {842, 36085}, {897, 5649}, {923, 6035}, {5641, 36142}
X(68793) = X(i)-Dao conjugate of X(j) for these (i,j): {542, 50941}, {690, 50942}, {1648, 52094}, {1649, 14223}, {2482, 6035}, {6593, 5649}, {21905, 14998}, {23967, 892}, {23992, 5641}, {35582, 2}, {38988, 842}, {42426, 65350}, {65728, 671}, {65731, 14977}, {65732, 18023}
X(68793) = crosspoint of X(i) and X(j) for these (i,j): {2, 20404}, {542, 50941}, {690, 50942}, {14999, 16092}
X(68793) = crosssum of X(i) and X(j) for these (i,j): {6, 20403}, {9979, 54395}
X(68793) = crossdifference of every pair of points on line {3, 691}
X(68793) = barycentric product X(i)*X(j) for these {i,j}: {4, 39474}, {67, 32313}, {187, 18312}, {468, 65723}, {523, 45662}, {524, 1640}, {542, 690}, {1648, 14999}, {1649, 16092}, {3266, 6041}, {4235, 65724}, {5099, 53232}, {5191, 35522}, {5468, 51428}, {6103, 14417}, {14273, 65722}, {14357, 55142}, {14559, 53132}, {20404, 35582}, {23967, 50942}, {23992, 50941}, {33921, 63858}, {34761, 51429}, {43087, 44814}, {48451, 66122}, {51405, 58780}, {51474, 55131}, {53156, 65750}, {66958, 66959}, {67082, 68158}
X(68793) = barycentric quotient X(i)/X(j) for these {i,j}: {187, 5649}, {351, 842}, {524, 6035}, {542, 892}, {690, 5641}, {1640, 671}, {1648, 14223}, {1649, 52094}, {2247, 36085}, {5191, 691}, {6041, 111}, {6103, 65350}, {11183, 57452}, {14443, 67082}, {14999, 52940}, {18312, 18023}, {21906, 14998}, {23967, 50941}, {23992, 50942}, {32313, 316}, {35582, 65872}, {39474, 69}, {45662, 99}, {46048, 66958}, {46049, 66959}, {50941, 57552}, {50942, 57547}, {51428, 5466}, {51429, 34765}, {55142, 52551}, {59175, 64775}, {60502, 59762}, {65723, 30786}, {65724, 14977}
X(68793) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {647, 3003, 2491}, {23992, 66123, 51429}

X(68794) = ORTHIC AXIS INTERCEPT OF X(2)X(3004)

Barycentrics    (b - c)*(2*a^2 - a*b + b^2 - a*c + 2*b*c + c^2) : :
X(68794) = 9 X[2] - X[47653], 3 X[2] + X[47660], 3 X[3004] - X[47653], X[47653] + 3 X[47660], 2 X[650] - 3 X[14425], 3 X[650] - X[45745], X[650] - 3 X[47766], X[650] + 3 X[47881], 5 X[650] - 3 X[47883], 3 X[650] + X[48397], X[676] + 2 X[48405], 2 X[2490] + X[6590], 4 X[2490] - 3 X[14425], 6 X[2490] - X[45745], 2 X[2490] - 3 X[47766], and many others

X(68794) lies on these lines: {2, 3004}, {6, 57164}, {37, 52326}, {230, 231}, {512, 4990}, {513, 2529}, {514, 4521}, {522, 4394}, {649, 900}, {659, 48231}, {661, 1639}, {665, 63812}, {667, 29278}, {693, 6084}, {824, 17069}, {918, 4369}, {1021, 22108}, {1491, 47807}, {1499, 49288}, {1577, 29162}, {1635, 4024}, {1638, 16892}, {2254, 48232}, {2483, 62749}, {2487, 4025}, {2516, 4765}, {2526, 47806}, {2533, 48299}, {2976, 48063}, {3261, 30024}, {3676, 30520}, {3738, 65450}, {3776, 47779}, {3798, 28898}, {3835, 47879}, {3910, 8045}, {4088, 47813}, {4106, 47787}, {4120, 4979}, {4374, 21611}, {4379, 21104}, {4380, 47790}, {4453, 49273}, {4467, 27013}, {4468, 43067}, {4477, 23865}, {4498, 48280}, {4500, 48008}, {4581, 57158}, {4728, 23729}, {4762, 11068}, {4763, 21196}, {4773, 4931}, {4776, 47988}, {4784, 50326}, {4789, 17494}, {4790, 4944}, {4791, 29126}, {4818, 47830}, {4820, 28221}, {4841, 4893}, {4843, 50501}, {4897, 25259}, {4927, 26985}, {4932, 48270}, {4940, 59751}, {4949, 6006}, {4988, 6544}, {6009, 49289}, {6362, 66514}, {6366, 48290}, {6545, 48130}, {6546, 47672}, {7178, 48300}, {7192, 28902}, {7265, 48566}, {9001, 14298}, {9029, 65448}, {9040, 50516}, {9362, 57030}, {10015, 47682}, {10196, 48000}, {14315, 55214}, {14331, 40137}, {14350, 28225}, {17989, 58364}, {20295, 48567}, {20950, 29488}, {20952, 24622}, {21183, 49299}, {21212, 28863}, {21786, 46383}, {21960, 48022}, {23220, 48387}, {23770, 47833}, {24099, 52745}, {26275, 50340}, {26777, 47661}, {26798, 49297}, {26824, 47892}, {27115, 47659}, {27138, 48550}, {27486, 47665}, {28042, 47921}, {28116, 50457}, {28161, 31182}, {28187, 47873}, {28209, 47765}, {28213, 47777}, {28882, 59522}, {28894, 31287}, {29047, 34958}, {29058, 58139}, {29212, 39545}, {29232, 50512}, {29266, 58145}, {29288, 52601}, {30061, 35519}, {30795, 47968}, {30835, 47756}, {31147, 48104}, {31148, 48082}, {31150, 47656}, {31207, 47886}, {31209, 45746}, {31250, 47757}, {44429, 47696}, {44435, 47662}, {44449, 47763}, {45320, 48095}, {45661, 48049}, {45663, 59749}, {45755, 48301}, {46403, 48250}, {46915, 47658}, {47664, 51317}, {47666, 47791}, {47676, 47891}, {47687, 47805}, {47689, 47798}, {47690, 47804}, {47691, 48236}, {47693, 47797}, {47694, 47809}, {47695, 48208}, {47697, 47808}, {47701, 48179}, {47703, 47811}, {47707, 47820}, {47711, 47818}, {47715, 47817}, {47719, 47815}, {47755, 49272}, {47760, 47995}, {47769, 48107}, {47778, 48404}, {47786, 48067}, {47794, 48402}, {47802, 48007}, {47821, 49283}, {47822, 47998}, {47823, 50348}, {47824, 49275}, {47832, 48106}, {47834, 48408}, {47872, 48400}, {47875, 48403}, {47885, 48120}, {47887, 48118}, {47932, 48416}, {47950, 48554}, {47961, 48555}, {48024, 48166}, {48030, 48199}, {48047, 48185}, {48056, 54265}, {48076, 48577}, {48077, 48578}, {48078, 48579}, {48083, 48253}, {48087, 48563}, {48097, 48221}, {48182, 50328}, {48217, 50335}, {48233, 58375}, {48247, 50358}, {48249, 50359}, {48252, 53343}, {48559, 60492}, {48576, 49284}, {49286, 50336}, {59630, 64862}

X(68794) = midpoint of X(i) and X(j) for these {i,j}: {649, 3700}, {650, 6590}, {659, 48396}, {661, 48276}, {667, 48395}, {693, 47890}, {2533, 48299}, {3004, 47660}, {4024, 4976}, {4025, 48271}, {4106, 48060}, {4468, 43067}, {4498, 48280}, {4500, 48008}, {4581, 57158}, {4773, 4931}, {4784, 50326}, {4789, 47884}, {4790, 48269}, {4841, 48275}, {4874, 48405}, {4897, 25259}, {4927, 47773}, {4932, 48270}, {4944, 47768}, {7178, 48300}, {7192, 48046}, {7662, 48062}, {10015, 47682}, {17494, 48274}, {21104, 48094}, {21146, 48055}, {21348, 47129}, {23729, 48101}, {23770, 48103}, {45745, 48397}, {47690, 50347}, {47694, 50333}, {47766, 47881}, {47767, 47874}, {47770, 47789}, {47771, 47788}, {47891, 48557}, {47988, 49282}, {47995, 49281}, {48026, 49293}, {48056, 54265}, {48087, 49296}, {48095, 48398}, {49275, 50357}, {49286, 50336}
X(68794) = reflection of X(i) in X(j) for these {i,j}: {649, 2527}, {650, 2490}, {676, 4874}, {1491, 53573}, {2976, 48063}, {4025, 2487}, {4394, 43061}, {4765, 2516}, {4940, 59751}, {14321, 3239}, {14425, 47766}, {17069, 31286}, {48269, 59589}
X(68794) = complement of X(3004)
X(68794) = complement of the isogonal conjugate of X(32736)
X(68794) = complement of the isotomic conjugate of X(8707)
X(68794) = polar conjugate of the isotomic conjugate of X(57156)
X(68794) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 15611}, {101, 51571}, {560, 39015}, {961, 17059}, {1110, 50330}, {1169, 17761}, {1220, 21252}, {2298, 116}, {2363, 53564}, {6648, 17046}, {8687, 142}, {8707, 2887}, {14624, 21253}, {32736, 10}, {35334, 21248}, {36098, 2886}, {36147, 141}, {46289, 55054}, {52928, 11019}, {59159, 40608}, {65229, 626}, {65282, 21235}
X(68794) = X(2)-Ceva conjugate of X(15611)
X(68794) = X(i)-Dao conjugate of X(j) for these (i,j): {15611, 2}, {17355, 21272}
X(68794) = crosspoint of X(i) and X(j) for these (i,j): {2, 8707}, {514, 56323}
X(68794) = crosssum of X(i) and X(j) for these (i,j): {6, 6371}, {101, 23845}, {3910, 41883}
X(68794) = crossdifference of every pair of points on line {3, 595}
X(68794) = barycentric product X(i)*X(j) for these {i,j}: {4, 57156}, {75, 48322}, {513, 4696}, {514, 17355}, {522, 10106}, {523, 11115}, {8707, 15611}
X(68794) = barycentric quotient X(i)/X(j) for these {i,j}: {4696, 668}, {10106, 664}, {11115, 99}, {15611, 3004}, {17355, 190}, {48322, 1}, {57156, 69}
X(68794) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 47660, 3004}, {649, 47767, 2527}, {649, 47874, 3700}, {650, 2490, 14425}, {650, 21348, 6589}, {650, 47766, 2490}, {650, 47881, 6590}, {650, 48397, 45745}, {693, 47771, 47890}, {1491, 47807, 53573}, {1635, 4024, 4976}, {1639, 48276, 661}, {3700, 47767, 649}, {4025, 47761, 2487}, {4379, 48094, 21104}, {4468, 47789, 43067}, {4728, 48101, 23729}, {4776, 49282, 47988}, {4789, 17494, 48274}, {4790, 4944, 48269}, {4893, 48275, 4841}, {4944, 48269, 59589}, {6590, 45745, 48397}, {6590, 47766, 650}, {7192, 30565, 48046}, {7662, 48219, 48062}, {16892, 24924, 1638}, {25259, 47762, 4897}, {26777, 47792, 47661}, {26985, 47652, 4927}, {26985, 47773, 47652}, {27013, 47870, 4467}, {27115, 47659, 47782}, {30795, 47968, 48178}, {30835, 47958, 47756}, {31209, 45746, 47784}, {31250, 47960, 47757}, {43067, 47770, 4468}, {45320, 48095, 48398}, {47690, 47804, 50347}, {47694, 47809, 50333}, {47760, 49281, 47995}, {47761, 48271, 4025}, {47765, 49293, 48026}, {47768, 48269, 4790}, {47787, 48060, 4106}, {47788, 47890, 693}, {47824, 49275, 50357}, {47833, 48103, 23770}, {47884, 48274, 17494}, {48087, 48563, 49296}, {48231, 48396, 659}

X(68795) = ORTHIC AXIS INTERCEPT OF X(1)X(39)

Barycentrics    a^2*(a^3*b^2 - a^2*b^3 - a*b^4 + b^5 - a^2*b^2*c - b^4*c + a^3*c^2 - a^2*b*c^2 + 4*a*b^2*c^2 - a^2*c^3 - a*c^4 - b*c^4 + c^5) : :

X(68795) lies on these lines: {1, 39}, {6, 33925}, {37, 17724}, {190, 24501}, {230, 231}, {800, 62214}, {1415, 3053}, {1575, 26015}, {1914, 2078}, {2260, 23622}, {2911, 65208}, {3002, 3230}, {3271, 24488}, {3726, 59734}, {3752, 16586}, {5206, 36152}, {6377, 17053}, {7117, 9259}, {7123, 9456}, {8557, 23980}, {9465, 26228}, {15815, 26357}, {16588, 21827}, {16969, 22070}, {17444, 45916}, {24499, 26273}, {24555, 24654}, {27918, 43063}, {34460, 37726}, {38865, 47431}, {43039, 62370}

X(68795) = complement of the isotomic conjugate of X(840)
X(68795) = polar conjugate of the isotomic conjugate of X(64889)
X(68795) = X(i)-complementary conjugate of X(j) for these (i,j): {560, 35113}, {840, 2887}, {18821, 21235}, {37131, 626}
X(68795) = crosspoint of X(2) and X(840)
X(68795) = crosssum of X(6) and X(528)
X(68795) = crossdifference of every pair of points on line {3, 659}
X(68795) = barycentric product X(4)*X(64889)
X(68795) = barycentric quotient X(64889)/X(69)

X(68796) = ORTHIC AXIS INTERCEPT OF X(1)X(21)

Barycentrics    a*(b + c)*(a^4 - 2*a^2*b^2 + b^4 + 3*a^2*b*c - b^3*c - 2*a^2*c^2 - b*c^3 + c^4) : :

X(68796) lies on these lines: {1, 21}, {2, 17874}, {42, 16577}, {55, 53035}, {56, 42440}, {225, 431}, {230, 231}, {499, 23555}, {740, 12080}, {851, 4516}, {899, 16578}, {1068, 17902}, {1108, 61647}, {1214, 3914}, {1284, 18210}, {1402, 21318}, {1711, 6505}, {1758, 9358}, {1776, 64888}, {1937, 43746}, {2006, 66289}, {2078, 2611}, {3120, 18593}, {3720, 16579}, {4028, 42700}, {4068, 10934}, {4133, 42704}, {4646, 40663}, {4647, 45700}, {5231, 21020}, {5292, 24907}, {6758, 62305}, {8286, 51462}, {10589, 53041}, {11240, 64071}, {11401, 11406}, {15076, 35980}, {15325, 37565}, {16586, 33140}, {16778, 20243}, {18839, 20718}, {25065, 46904}, {25078, 33156}, {25081, 53034}, {29639, 30778}, {34486, 58392}, {34977, 35466}, {37782, 68154}, {40950, 46366}, {53043, 60926}, {53524, 68732}

X(68796) = midpoint of X(i) and X(j) for these {i,j}: {2611, 3724}, {6758, 62305}
X(68796) = polar conjugate of the isotomic conjugate of X(64888)
X(68796) = X(i)-Ceva conjugate of X(j) for these (i,j): {37139, 661}, {68580, 65}
X(68796) = X(110)-isoconjugate of X(66277)
X(68796) = X(244)-Dao conjugate of X(66277)
X(68796) = crosspoint of X(1) and X(62764)
X(68796) = crosssum of X(1) and X(62756)
X(68796) = crossdifference of every pair of points on line {3, 661}
X(68796) = barycentric product X(i)*X(j) for these {i,j}: {4, 64888}, {19, 51608}, {226, 1776}, {52880, 68580}
X(68796) = barycentric quotient X(i)/X(j) for these {i,j}: {661, 66277}, {1776, 333}, {51608, 304}, {64888, 69}
X(68796) = {X(8609),X(8758)}-harmonic conjugate of X(3011)

X(68797) = ORTHIC AXIS INTERCEPT OF X(2)X(7)

Barycentrics    2*a^4 - a^3*b - a^2*b^2 - a*b^3 + b^4 - a^3*c + 2*a^2*b*c + a*b^2*c - a^2*c^2 + a*b*c^2 - 2*b^2*c^2 - a*c^3 + c^4 : :

X(68797) lies on these lines: {2, 7}, {6, 17728}, {10, 4390}, {11, 910}, {25, 1856}, {36, 5179}, {37, 5432}, {41, 1210}, {44, 60414}, {48, 24005}, {56, 46835}, {73, 52530}, {101, 1737}, {105, 61436}, {111, 2689}, {115, 36195}, {140, 16601}, {169, 499}, {198, 37366}, {220, 24914}, {230, 231}, {515, 1055}, {519, 17439}, {604, 20262}, {614, 3554}, {861, 2223}, {950, 36007}, {1111, 51775}, {1125, 17451}, {1146, 1319}, {1155, 17747}, {1212, 5433}, {1311, 32689}, {1323, 21139}, {1334, 6684}, {1415, 1877}, {1420, 23058}, {1429, 26001}, {1475, 64124}, {1566, 60060}, {1696, 17303}, {1738, 17737}, {1759, 21616}, {1770, 24045}, {1826, 2178}, {1837, 3207}, {1855, 37579}, {2082, 3086}, {2170, 8074}, {2202, 51359}, {2238, 20752}, {2246, 53579}, {2280, 11019}, {2325, 37762}, {2329, 24982}, {2348, 51406}, {2646, 21049}, {3008, 9502}, {3035, 3693}, {3263, 20881}, {3496, 41012}, {3582, 5540}, {3684, 26015}, {3723, 63287}, {3767, 23536}, {3879, 17001}, {3930, 6745}, {3970, 59719}, {3977, 3985}, {4070, 62621}, {4071, 26250}, {4119, 49991}, {4138, 30746}, {4292, 37050}, {4386, 36488}, {4513, 37828}, {4771, 50758}, {4919, 51433}, {4987, 26280}, {5011, 30384}, {5060, 51382}, {5121, 33854}, {5253, 27068}, {5265, 27541}, {5275, 29639}, {5276, 24239}, {5310, 8606}, {5513, 15608}, {5659, 37675}, {5819, 10589}, {6049, 63593}, {6506, 60431}, {6554, 7288}, {6603, 40663}, {6675, 25086}, {6681, 24036}, {6700, 33299}, {6735, 56530}, {7181, 44664}, {8776, 61224}, {9085, 67295}, {9367, 22070}, {10164, 41423}, {11415, 36643}, {13411, 21808}, {13747, 25066}, {15325, 43065}, {16502, 28018}, {16549, 58405}, {16609, 26006}, {16611, 50759}, {16780, 28074}, {17044, 43037}, {17048, 31284}, {17081, 30694}, {17278, 31203}, {17566, 25082}, {17732, 58887}, {17736, 21077}, {17742, 26364}, {17761, 67644}, {21029, 57284}, {21073, 25440}, {21096, 59587}, {21840, 39595}, {24247, 35262}, {24268, 35290}, {25068, 52264}, {25994, 26686}, {26232, 63134}, {26242, 66632}, {30807, 43054}, {34852, 43053}, {37519, 54008}, {37959, 68257}, {40968, 59644}, {40997, 59691}, {51364, 58320}, {63208, 63592}, {63595, 63987}

X(68797) = midpoint of X(1055) and X(21044)
X(68797) = isogonal conjugate of X(60025)
X(68797) = complement of X(33864)
X(68797) = complement of the isotomic conjugate of X(1311)
X(68797) = polar conjugate of the isotomic conjugate of X(64875)
X(68797) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 53839}, {1311, 2887}, {32689, 4885}, {36094, 17072}, {60580, 21252}
X(68797) = X(2)-Ceva conjugate of X(53839)
X(68797) = X(1)-isoconjugate of X(60025)
X(68797) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 60025}, {53839, 2}
X(68797) = crosspoint of X(2) and X(1311)
X(68797) = crosssum of X(6) and X(8679)
X(68797) = crossdifference of every pair of points on line {3, 663}
X(68797) = barycentric product X(i)*X(j) for these {i,j}: {4, 64875}, {318, 51661}, {523, 7462}, {1311, 53839}
X(68797) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 60025}, {7462, 99}, {51661, 77}, {53839, 33864}, {64875, 69}
X(68797) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7, 30742}, {2, 144, 30740}, {2, 329, 30757}, {2, 1447, 51400}, {2, 5905, 30782}, {2, 6646, 30798}, {2, 26229, 142}, {2, 26258, 9}, {2, 26265, 3452}, {2, 26267, 226}, {2, 26279, 4357}, {2, 40127, 40131}, {230, 3290, 3011}, {3911, 40869, 672}, {8074, 44675, 2170}, {15325, 65808, 43065}

X(68798) = ORTHIC AXIS INTERCEPT OF X(2)X(11)

Barycentrics    2*a^5 - 2*a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + b^5 - 2*a^4*c + a^2*b^2*c - b^4*c + a^3*c^2 + a^2*b*c^2 + 2*a*b^2*c^2 - a^2*c^3 - a*c^4 - b*c^4 + c^5 : :

X(68798) lies on these lines: {2, 11}, {12, 4223}, {25, 5521}, {108, 6353}, {140, 15251}, {226, 51435}, {230, 231}, {614, 3756}, {851, 20875}, {1083, 51390}, {1086, 24346}, {1155, 51400}, {1214, 6676}, {1329, 16048}, {1375, 2223}, {1421, 5272}, {1447, 37757}, {2006, 36815}, {2175, 16608}, {2330, 25964}, {2770, 59107}, {2829, 7427}, {3263, 3712}, {3271, 36949}, {3589, 63522}, {3688, 58457}, {3920, 63287}, {4124, 5723}, {4904, 5091}, {4998, 57754}, {4999, 16823}, {5121, 60058}, {5266, 52259}, {5433, 16020}, {5840, 57605}, {5852, 56513}, {6677, 15252}, {6679, 40533}, {7295, 25365}, {7354, 37254}, {10198, 19309}, {11028, 13405}, {14019, 25440}, {17044, 67428}, {17522, 57288}, {17602, 26242}, {17718, 40131}, {18343, 35280}, {18635, 19133}, {19310, 25466}, {19721, 68699}, {20359, 53415}, {24953, 39581}, {25129, 59695}, {26540, 39873}, {29681, 60359}, {30810, 37586}, {38282, 38300}, {46555, 53310}, {57600, 64503}, {58443, 58467}, {60756, 61567}

X(68798) = orthoptic-circle-of-the-Steiner-inellipse-inverse of X(10773)
X(68798) = complement of the isotomic conjugate of X(2862)
X(68798) = X(2862)-complementary conjugate of X(2887)
X(68798) = crosspoint of X(2) and X(2862)
X(68798) = crosssum of X(6) and X(2876)
X(68798) = crossdifference of every pair of points on line {3, 665}
X(68798) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 100, 120}, {2, 105, 11}, {2, 149, 30787}, {2, 3434, 30755}, {2, 26231, 3035}, {2, 26241, 2886}, {2, 60354, 105}, {3035, 6714, 2}

X(68799) = ORTHIC AXIS INTERCEPT OF X(6)X(514)

Barycentrics    (b - c)*(-a^4 + 2*a^3*b - 2*a^2*b^2 + b^4 + 2*a^3*c - 2*a^2*b*c - 2*a^2*c^2 - 2*b^2*c^2 + c^4) : :

X(68799) lies on these lines: {6, 514}, {37, 21185}, {230, 231}, {594, 44448}, {657, 21102}, {1086, 30181}, {1609, 48387}, {3239, 64917}, {3553, 4040}, {3554, 48282}, {3798, 53276}, {3815, 47757}, {4762, 10015}, {4786, 30184}, {4794, 62210}, {5304, 47773}, {6084, 43052}, {7735, 47771}, {7736, 44435}, {8553, 48386}, {8557, 49300}, {16672, 54261}, {21178, 21225}, {28151, 54250}, {31489, 44432}, {34288, 55257}, {37642, 47791}, {37646, 47789}, {37662, 47783}, {37665, 48156}, {47322, 62494}, {47781, 63089}, {60677, 66284}

X(68799) = reflection of X(53276) in X(3798)
X(68799) = complement of the isotomic conjugate of X(44876)
X(68799) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 25642}, {44876, 2887}
X(68799) = X(2)-Ceva conjugate of X(25642)
X(68799) = X(35190)-isoconjugate of X(57015)
X(68799) = X(25642)-Dao conjugate of X(2)
X(68799) = crosspoint of X(2) and X(44876)
X(68799) = crossdifference of every pair of points on line {3, 674}
X(68799) = barycentric product X(i)*X(j) for these {i,j}: {514, 56746}, {523, 7474}, {25642, 44876}
X(68799) = barycentric quotient X(i)/X(j) for these {i,j}: {7474, 99}, {56746, 190}

X(68800) = ORTHIC AXIS INTERCEPT OF X(19)X(27)

Barycentrics    (b + c)*(-a^2 + b*c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2) : :

X(68800) lies on these lines: {4, 59261}, {10, 52577}, {19, 27}, {25, 23398}, {112, 53920}, {169, 23555}, {210, 430}, {230, 231}, {242, 740}, {281, 4213}, {321, 8020}, {423, 39718}, {862, 4037}, {1840, 2333}, {1851, 21020}, {1863, 42446}, {1874, 2238}, {1880, 16606}, {1973, 56138}, {2294, 3475}, {2388, 5185}, {3186, 25124}, {3975, 35544}, {5819, 53041}, {14024, 56828}, {17442, 17911}, {17869, 23620}, {17874, 40131}, {17920, 49598}, {17927, 17982}, {40975, 56875}, {42440, 46835}, {59520, 59671}

X(68800) = polar conjugate of X(18827)
X(68800) = polar conjugate of the isotomic conjugate of X(740)
X(68800) = polar conjugate of the isogonal conjugate of X(3747)
X(68800) = X(i)-Ceva conjugate of X(j) for these (i,j): {242, 862}, {17982, 430}, {36120, 33}, {65352, 4}, {68565, 1826}
X(68800) = X(i)-cross conjugate of X(j) for these (i,j): {862, 1874}, {3747, 740}
X(68800) = X(i)-isoconjugate of X(j) for these (i,j): {3, 37128}, {6, 57738}, {32, 57987}, {48, 18827}, {57, 1808}, {63, 741}, {69, 18268}, {77, 2311}, {81, 295}, {86, 2196}, {184, 40017}, {222, 56154}, {255, 65352}, {291, 1790}, {292, 1444}, {335, 1437}, {337, 1333}, {603, 36800}, {647, 36066}, {660, 7254}, {662, 66938}, {810, 65258}, {875, 4563}, {876, 4558}, {1332, 66937}, {1459, 4584}, {1911, 17206}, {2193, 7233}, {3049, 65285}, {3572, 4592}, {3917, 39276}, {4367, 65327}, {4444, 4575}, {4589, 22383}, {8033, 17970}, {15419, 34067}, {17103, 66942}, {22093, 37134}, {22373, 39292}, {32661, 66286}, {34055, 46159}
X(68800) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 57738}, {37, 337}, {136, 4444}, {1084, 66938}, {1249, 18827}, {3162, 741}, {5139, 3572}, {5452, 1808}, {6376, 57987}, {6523, 65352}, {6651, 17206}, {7952, 36800}, {8299, 63}, {16591, 348}, {19557, 1444}, {35068, 69}, {35119, 15419}, {36103, 37128}, {38978, 647}, {39029, 1790}, {39052, 36066}, {39062, 65258}, {40586, 295}, {40600, 2196}, {47345, 7233}, {62553, 304}, {62605, 40017}
X(68800) = crosspoint of X(4) and X(65352)
X(68800) = crosssum of X(295) and X(2196)
X(68800) = crossdifference of every pair of points on line {3, 810}
X(68800) = barycentric product X(i)*X(j) for these {i,j}: {4, 740}, {8, 1874}, {10, 242}, {12, 14024}, {19, 3948}, {25, 35544}, {27, 4037}, {29, 7235}, {42, 40717}, {75, 862}, {92, 2238}, {225, 3685}, {238, 41013}, {239, 1826}, {264, 3747}, {273, 4433}, {278, 3985}, {281, 16609}, {313, 57654}, {318, 1284}, {321, 2201}, {350, 1824}, {594, 31905}, {811, 4155}, {860, 36815}, {1309, 42767}, {1447, 53008}, {1840, 17493}, {1880, 3975}, {1897, 4010}, {1921, 2333}, {1969, 41333}, {2489, 27853}, {2501, 3570}, {3573, 24006}, {3684, 40149}, {3695, 34856}, {3716, 61178}, {3952, 65106}, {4087, 57652}, {4093, 46104}, {4148, 52607}, {4154, 68575}, {4432, 68563}, {4435, 65207}, {4783, 36125}, {5009, 7141}, {6331, 46390}, {6335, 21832}, {7140, 33295}, {7212, 65160}, {11599, 52468}, {17755, 68565}, {17984, 66971}, {35068, 65352}, {36120, 50440}, {44129, 66878}, {58327, 68576}, {68153, 68631}
X(68800) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 57738}, {4, 18827}, {10, 337}, {19, 37128}, {25, 741}, {33, 56154}, {42, 295}, {55, 1808}, {75, 57987}, {92, 40017}, {162, 36066}, {213, 2196}, {225, 7233}, {238, 1444}, {239, 17206}, {242, 86}, {281, 36800}, {393, 65352}, {419, 17103}, {512, 66938}, {607, 2311}, {648, 65258}, {740, 69}, {811, 65285}, {812, 15419}, {862, 1}, {874, 55202}, {1284, 77}, {1783, 4584}, {1824, 291}, {1826, 335}, {1840, 30669}, {1843, 46159}, {1874, 7}, {1897, 4589}, {1914, 1790}, {1973, 18268}, {2201, 81}, {2210, 1437}, {2238, 63}, {2333, 292}, {2489, 3572}, {2501, 4444}, {3570, 4563}, {3573, 4592}, {3684, 1812}, {3685, 332}, {3747, 3}, {3948, 304}, {3985, 345}, {4010, 4025}, {4037, 306}, {4093, 3917}, {4124, 17219}, {4148, 15411}, {4154, 12215}, {4155, 656}, {4213, 52207}, {4433, 78}, {4455, 1459}, {4829, 4101}, {5027, 22093}, {6335, 4639}, {7140, 43534}, {7235, 307}, {8632, 7254}, {14024, 261}, {16369, 60701}, {16609, 348}, {21832, 905}, {24006, 66286}, {24459, 30805}, {27853, 52608}, {31905, 1509}, {35544, 305}, {36815, 57985}, {39786, 3942}, {40717, 310}, {40729, 66942}, {41013, 334}, {41333, 48}, {42767, 65868}, {46390, 647}, {52468, 17731}, {52651, 66933}, {53008, 4518}, {53556, 4131}, {57654, 58}, {58327, 1792}, {58757, 68631}, {65106, 7192}, {65352, 57554}, {66878, 71}, {66971, 36214}, {68565, 52209}
X(68800) = {X(2333),X(41013)}-harmonic conjugate of X(1840)

X(68801) = ORTHIC AXIS INTERCEPT OF X(1)X(29)

Barycentrics    (b + c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-a^4 + a^2*b^2 - a^2*b*c + b^3*c + a^2*c^2 - 2*b^2*c^2 + b*c^3) : :

X(68801) lies on these lines: {1, 29}, {4, 2650}, {42, 40149}, {65, 225}, {73, 68628}, {108, 3724}, {196, 4331}, {230, 231}, {240, 61180}, {243, 1430}, {278, 11269}, {318, 49598}, {393, 2294}, {612, 17874}, {651, 56822}, {653, 1758}, {740, 1897}, {758, 1785}, {896, 52891}, {899, 37805}, {1046, 3559}, {1068, 1148}, {1118, 18673}, {1249, 40977}, {1857, 53036}, {1859, 1860}, {1874, 44113}, {1940, 60682}, {1962, 63965}, {2202, 42669}, {2292, 7952}, {5307, 62819}, {6198, 31880}, {7046, 21020}, {12081, 15500}, {17719, 37770}, {17902, 33127}, {17918, 32775}, {17923, 33140}, {23688, 56905}, {24014, 42078}, {29640, 52412}, {37782, 62305}, {40950, 42385}, {41013, 59305}, {44661, 59816}, {56814, 63354}

X(68801) = polar conjugate of X(35145)
X(68801) = polar conjugate of the isotomic conjugate of X(8680)
X(68801) = polar conjugate of the isogonal conjugate of X(42669)
X(68801) = X(i)-Ceva conjugate of X(j) for these (i,j): {243, 851}, {821, 41500}
X(68801) = X(42669)-cross conjugate of X(8680)
X(68801) = X(i)-isoconjugate of X(j) for these (i,j): {3, 37142}, {21, 296}, {48, 35145}, {63, 2249}, {184, 57980}, {283, 1937}, {284, 40843}, {333, 1949}, {520, 59041}, {652, 41206}, {662, 52222}, {1444, 61427}, {1812, 1945}, {1952, 2193}, {2194, 57801}, {23090, 65214}, {36054, 41207}, {53211, 57134}, {57557, 68731}
X(68801) = X(i)-Dao conjugate of X(j) for these (i,j): {1084, 52222}, {1214, 57801}, {1249, 35145}, {3162, 2249}, {35075, 69}, {36103, 37142}, {39032, 1812}, {39033, 333}, {39035, 332}, {39036, 314}, {39037, 283}, {40590, 40843}, {40611, 296}, {47345, 1952}, {62605, 57980}
X(68801) = crosspoint of X(65) and X(57676)
X(68801) = crosssum of X(21) and X(416)
X(68801) = crossdifference of every pair of points on line {3, 822}
X(68801) = barycentric product X(i)*X(j) for these {i,j}: {4, 8680}, {19, 44150}, {65, 1948}, {92, 851}, {225, 1944}, {226, 243}, {264, 42669}, {318, 51645}, {321, 1430}, {349, 51726}, {523, 1981}, {823, 9391}, {1400, 57812}, {1441, 2202}, {1577, 23353}, {1826, 5088}, {1936, 40149}, {1951, 57809}, {1969, 44112}, {6354, 15146}, {6518, 68628}, {11608, 41499}, {39036, 57676}, {57806, 68731}, {58325, 68576}
X(68801) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 35145}, {19, 37142}, {25, 2249}, {65, 40843}, {92, 57980}, {108, 41206}, {225, 1952}, {226, 57801}, {243, 333}, {512, 52222}, {851, 63}, {1400, 296}, {1402, 1949}, {1430, 81}, {1880, 1937}, {1936, 1812}, {1944, 332}, {1948, 314}, {1951, 283}, {1981, 99}, {2202, 21}, {2333, 61427}, {5088, 17206}, {6518, 68650}, {8680, 69}, {9391, 24018}, {15146, 7058}, {15418, 55202}, {23353, 662}, {24019, 59041}, {26884, 1790}, {36127, 41207}, {41499, 40882}, {42669, 3}, {44112, 48}, {44150, 304}, {51645, 77}, {51726, 284}, {52607, 53211}, {57652, 1945}, {57812, 28660}, {58325, 1792}, {58757, 68646}, {68731, 255}
X(68801) = {X(8755),X(23710)}-harmonic conjugate of X(3011)

X(68802) = ORTHIC AXIS INTERCEPT OF X(6)X(22)

Barycentrics    a^2*(a^6*b^2 - a^4*b^4 - a^2*b^6 + b^8 + a^6*c^2 + 2*a^4*b^2*c^2 - b^6*c^2 - a^4*c^4 - a^2*c^6 - b^2*c^6 + c^8) : :

X(68802) lies on these lines: {2, 44136}, {6, 22}, {23, 55384}, {30, 68734}, {32, 1658}, {39, 550}, {50, 37978}, {111, 16166}, {112, 37970}, {115, 11563}, {186, 41336}, {187, 15646}, {216, 1368}, {230, 231}, {325, 34990}, {566, 7736}, {570, 7667}, {800, 5306}, {1576, 44090}, {1609, 8792}, {1691, 13195}, {1692, 65517}, {1971, 15647}, {2871, 9418}, {2967, 68741}, {3053, 32534}, {3199, 44960}, {3289, 41673}, {3425, 64061}, {5305, 64472}, {7737, 66721}, {7745, 12605}, {9722, 14577}, {11063, 30717}, {13337, 14930}, {13351, 37665}, {15355, 37637}, {18573, 31489}, {19128, 19165}, {20975, 60514}, {26216, 65630}, {28710, 63932}, {36212, 50771}, {37123, 44668}, {37969, 68070}, {40583, 63846}, {44267, 44468}, {51363, 53493}, {55415, 59229}

X(68802) = complement of the isotomic conjugate of X(29011)
X(68802) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 44953}, {29011, 2887}
X(68802) = X(2)-Ceva conjugate of X(44953)
X(68802) = X(63)-isoconjugate of X(67794)
X(68802) = X(i)-Dao conjugate of X(j) for these (i,j): {3162, 67794}, {44953, 2}
X(68802) = crosspoint of X(2) and X(29011)
X(68802) = crosssum of X(6) and X(29012)
X(68802) = crossdifference of every pair of points on line {3, 826}
X(68802) = barycentric product X(29011)*X(44953)
X(68802) = barycentric quotient X(25)/X(67794)
X(68802) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {230, 232, 2493}, {230, 16308, 232}, {232, 3003, 230}, {3003, 16303, 47228}, {3003, 16308, 2493}, {16308, 16328, 47169}

X(68803) = ORTHIC AXIS INTERCEPT OF X(649)X(693)

Barycentrics    (b - c)*(a^4 + a^2*b^2 + a^2*b*c + 2*a*b^2*c + b^3*c + a^2*c^2 + 2*a*b*c^2 + 2*b^2*c^2 + b*c^3) : :

X(68803) lies on these lines: {230, 231}, {649, 693}, {661, 3716}, {665, 52601}, {1635, 47834}, {2483, 30591}, {2484, 7650}, {2533, 4435}, {4024, 47695}, {4763, 4789}, {4765, 31286}, {4790, 48089}, {4813, 47697}, {4815, 21389}, {4893, 48237}, {4913, 47690}, {4976, 9508}, {4979, 46403}, {4988, 6546}, {8632, 29051}, {10566, 29771}, {17494, 26248}, {18154, 29443}, {21116, 47652}, {23770, 48276}, {24290, 48305}, {43067, 48398}, {45666, 47876}, {47669, 47693}, {47672, 62635}, {47685, 50525}, {47689, 50482}, {47691, 48275}, {47696, 48001}, {47844, 48022}, {47874, 64860}, {47878, 48562}, {48563, 48576}, {49285, 50336}, {50352, 66513}

X(68803) = crossdifference of every pair of points on line {3, 869}
X(68803) = {X(650),X(7662)}-harmonic conjugate of X(6590)

X(68804) = ORTHIC AXIS INTERCEPT OF X(2)X(670)

Barycentrics    a^2*(a^4*b^4 + a^2*b^6 - 4*a^4*b^2*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 + a^4*c^4 + a^2*b^2*c^4 + 2*b^4*c^4 + a^2*c^6 - 2*b^2*c^6) : :

X(68804) lies on these lines: {2, 670}, {6, 6786}, {39, 620}, {99, 14700}, {111, 62411}, {115, 9828}, {141, 48444}, {183, 9468}, {187, 37927}, {230, 231}, {385, 32526}, {694, 6784}, {729, 6787}, {1194, 34990}, {2086, 34383}, {3053, 11325}, {3229, 22329}, {5106, 9149}, {6375, 6719}, {6722, 51906}, {7664, 7806}, {7665, 23356}, {7735, 11672}, {7778, 63572}, {7875, 15302}, {9998, 46303}, {10311, 32654}, {17004, 39576}, {19118, 35325}, {30736, 48439}, {37667, 46948}, {37690, 63562}, {38880, 44127}, {44453, 47642}, {59994, 63019}

X(68804) = complement of the isotomic conjugate of X(5970)
X(68804) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 9152}, {560, 35077}, {5970, 2887}, {14606, 21253}, {35146, 21235}
X(68804) = X(2)-Ceva conjugate of X(9152)
X(68804) = X(9152)-Dao conjugate of X(2)
X(68804) = crosspoint of X(2) and X(5970)
X(68804) = crosssum of X(6) and X(5969)
X(68804) = crossdifference of every pair of points on line {3, 887}
X(68804) = barycentric product X(5970)*X(9152)
X(68804) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {385, 32526, 52961}, {2492, 11176, 2491}

X(68805) = ORTHIC AXIS INTERCEPT OF X(2)X(800)

Barycentrics    a^2*(a^6*b^2 - a^4*b^4 - a^2*b^6 + b^8 + a^6*c^2 + 8*a^4*b^2*c^2 - 3*a^2*b^4*c^2 + 2*b^6*c^2 - a^4*c^4 - 3*a^2*b^2*c^4 - 6*b^4*c^4 - a^2*c^6 + 2*b^2*c^6 + c^8) : :

X(68805) lies on these lines: {2, 800}, {6, 3917}, {32, 7503}, {39, 3523}, {115, 47096}, {187, 2071}, {216, 1194}, {230, 231}, {385, 36212}, {570, 5306}, {1180, 63097}, {1196, 22240}, {1495, 53500}, {1609, 10311}, {1611, 45141}, {1691, 8779}, {2092, 40129}, {2276, 22071}, {3053, 3516}, {3199, 9747}, {3767, 59349}, {3787, 60106}, {5065, 7485}, {6467, 20885}, {7667, 53420}, {8963, 13758}, {13341, 63005}, {14930, 15302}, {14961, 16976}, {16434, 50653}, {20975, 51412}, {34990, 50774}, {40135, 52058}, {44089, 52144}, {44468, 47337}, {51363, 53475}

X(68805) = complement of the isotomic conjugate of X(29180)
X(68805) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 44955}, {29180, 2887}
X(68805) = X(2)-Ceva conjugate of X(44955)
X(68805) = X(44955)-Dao conjugate of X(2)
X(68805) = crosspoint of X(2) and X(29180)
X(68805) = crosssum of X(6) and X(29181)
X(68805) = crossdifference of every pair of points on line {3, 3800}
X(68805) = barycentric product X(29180)*X(44955)
X(68805) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {216, 7735, 1194}, {230, 232, 3291}, {230, 3003, 232}, {16328, 47184, 47181}, {22240, 37689, 1196}

X(68806) = ORTHIC AXIS INTERCEPT OF X(9)X(46383)

Barycentrics    a^2*(b - c)*(a^2*b - b^3 + a^2*c + 2*a*b*c - 3*b^2*c - 3*b*c^2 - c^3) : :

X(68806) lies on these lines: {9, 46383}, {37, 4976}, {230, 231}, {513, 58299}, {514, 27674}, {649, 3709}, {657, 22383}, {661, 665}, {667, 50494}, {693, 25084}, {788, 58303}, {798, 50511}, {802, 30023}, {905, 4468}, {926, 58286}, {1635, 52326}, {1960, 57096}, {2509, 2523}, {3239, 8714}, {3250, 50492}, {3804, 50506}, {4893, 7180}, {6371, 27675}, {7252, 22108}, {9404, 22086}, {11068, 28374}, {16751, 30565}, {17990, 50510}, {21127, 55212}, {21225, 24622}, {24782, 31209}, {24948, 47660}, {25098, 47785}, {25594, 26114}, {27648, 47771}, {31947, 48056}, {46393, 52306}

X(68806) = midpoint of X(i) and X(j) for these {i,j}: {649, 58298}, {50510, 58291}
X(68806) = complement of the isotomic conjugate of X(8701)
X(68806) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 46660}, {560, 35076}, {872, 65787}, {1126, 21252}, {4629, 21240}, {6540, 21235}, {8701, 2887}, {28615, 116}, {37212, 626}, {52555, 21253}
X(68806) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 46660}, {40396, 3270}
X(68806) = X(46660)-Dao conjugate of X(2)
X(68806) = crosspoint of X(2) and X(8701)
X(68806) = crosssum of X(i) and X(j) for these (i,j): {6, 4977}, {525, 41809}
X(68806) = crossdifference of every pair of points on line {3, 962}
X(68806) = barycentric product X(i)*X(j) for these {i,j}: {1, 50338}, {513, 34790}, {521, 1887}, {523, 17524}, {693, 54327}, {8701, 46660}
X(68806) = barycentric quotient X(i)/X(j) for these {i,j}: {1887, 18026}, {17524, 99}, {34790, 668}, {50338, 75}, {54327, 100}
X(68806) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {647, 650, 3310}, {650, 6586, 647}, {650, 21348, 45745}, {1635, 55210, 52326}

X(68807) = X(100)X(101) INTERCEPT OF X(1)X(48408)

Barycentrics    a*(b - c)*(a^5 - 2*a^4*b + 2*a^2*b^3 - a*b^4 - 2*a^4*c - 3*a^3*b*c + a^2*b^2*c + a*b^3*c - b^4*c + a^2*b*c^2 + 4*a*b^2*c^2 - b^3*c^2 + 2*a^2*c^3 + a*b*c^3 - b^2*c^3 - a*c^4 - b*c^4) : :

X(68807) lies on these lines: {1, 48408}, {100, 101}, {242, 514}, {2977, 3811}, {3309, 4057}, {4040, 47804}, {4794, 43061}, {15344, 67810}, {23770, 64675}, {29015, 40101}, {29240, 54318}, {42325, 57155}, {47680, 54392}

X(68807) = crossdifference of every pair of points on line {71, 244}

X(68808) = X(100)X(101) INTERCEPT OF X(230)X(231)

Barycentrics    a*(b - c)*(a^4 - 2*a^3*b + 2*a*b^3 - b^4 - 2*a^3*c - a^2*b*c + 2*a*b^2*c - b^3*c + 2*a*b*c^2 + 2*a*c^3 - b*c^3 - c^4) : :

X(68808) lies on these lines: {100, 101}, {120, 5513}, {230, 231}, {661, 21390}, {665, 29102}, {3960, 21115}, {5190, 5521}, {6546, 43050}, {9085, 15344}, {11068, 26641}, {14838, 30911}, {23988, 65808}, {24562, 48398}, {47768, 66514}, {48003, 50457}

X(68808) = crosspoint of X(662) and X(39439)
X(68808) = crossdifference of every pair of points on line {3, 244}

X(68809) = X(100)X(101) INTERCEPT OF X(1)X(13256)

Barycentrics    a*(b - c)*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5 + a^4*c - 2*a^3*b*c + 2*a*b^3*c - b^4*c - 2*a^3*c^2 + 2*a*b*c^3 + 2*a*c^4 - b*c^4 - c^5) : :

X(68809) lies on these lines: {1, 13256}, {100, 101}, {240, 522}, {514, 66527}, {654, 2806}, {1714, 65106}, {2254, 4707}, {4041, 29066}, {4063, 6003}, {6608, 48339}, {8676, 63827}, {13258, 21385}, {15313, 50501}, {23800, 64917}

X(68809) = crossdifference of every pair of points on line {48, 244}

X(68810) = X(100)X(101) INTERCEPT OF X(23)X(385)

Barycentrics    a*(b - c)*(a^4 - 2*a^3*b + a^2*b^2 - 2*a^3*c + a^2*c^2 + b^2*c^2) : :

X(68810) lies on these lines: {2, 53284}, {23, 385}, {55, 47776}, {100, 101}, {105, 675}, {411, 38324}, {667, 29188}, {693, 8642}, {812, 1621}, {1001, 21297}, {1734, 30913}, {1960, 27950}, {2254, 3737}, {2820, 7411}, {3870, 53396}, {4380, 8641}, {4453, 45695}, {4455, 6652}, {4728, 5284}, {6986, 62432}, {7466, 65104}, {8299, 68156}, {13245, 48032}, {13266, 64359}, {17126, 21786}, {21789, 31291}, {32666, 36087}, {46403, 53309}, {52133, 63222}

X(68810) = reflection of X(1621) in X(8645)
X(68810) = crosspoint of X(662) and X(2368)
X(68810) = crosssum of X(661) and X(2388)
X(68810) = crossdifference of every pair of points on line {39, 244}
X(68810) = {X(53284),X(53287)}-harmonic conjugate of X(2)

X(68811) = X(100)X(101) INTERCEPT OF X(798)X(812)

Barycentrics    a*(b - c)*(a^4*b - 2*a^3*b^2 + a^2*b^3 + a^4*c - 2*a^3*b*c + a^2*b^2*c - 2*a^3*c^2 + a^2*b*c^2 - 2*a*b^2*c^2 + b^3*c^2 + a^2*c^3 + b^2*c^3) : :

X(68811) lies on these lines: {100, 101}, {798, 812}, {1734, 21901}, {2254, 16549}, {3294, 3716}, {3730, 53343}, {3762, 16552}, {4253, 21222}, {16783, 53308}, {17742, 53395}, {17761, 27009}, {29433, 65101}

X(68811) = crosspoint of X(i) and X(j) for these (i,j): {82, 36087}, {100, 37130}
X(68811) = crosssum of X(513) and X(2225)
X(68811) = crossdifference of every pair of points on line {244, 1964}
X(68811) = barycentric product X(1)*X(65660)
X(68811) = barycentric quotient X(65660)/X(75)
X(68811) = {X(3762),X(46388)}-harmonic conjugate of X(16552)

X(68812) = X(100)X(101) INTERCEPT OF X(2)X(6)

Barycentrics    a*(a - b)*(a - c)*(a^2*b - a*b^2 + a^2*c + 2*a*b*c - b^2*c - a*c^2 - b*c^2) : :

X(68812) lies on these lines: {2, 6}, {100, 101}, {651, 37209}, {799, 65256}, {813, 53685}, {2284, 17780}, {3952, 35326}, {8693, 9059}, {9067, 59022}, {23343, 43928}, {30729, 61197}

X(68812) = X(4607)-Ceva conjugate of X(100)
X(68812) = X(i)-isoconjugate of X(j) for these (i,j): {56, 60575}, {1086, 59071}
X(68812) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 60575}, {899, 4728}
X(68812) = crosspoint of X(i) and X(j) for these (i,j): {662, 898}, {5381, 57731}
X(68812) = crosssum of X(i) and X(j) for these (i,j): {661, 891}, {764, 1646}, {1084, 33917}
X(68812) = trilinear pole of line {44671, 45751}
X(68812) = crossdifference of every pair of points on line {244, 512}
X(68812) = barycentric product X(i)*X(j) for these {i,j}: {99, 44671}, {100, 29824}, {190, 45751}, {4607, 40614}
X(68812) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 60575}, {1110, 59071}, {29824, 693}, {40614, 4728}, {44671, 523}, {45751, 514}

X(68813) = X(100)X(101) INTERCEPT OF X(514)X(661)

Barycentrics    a*(a - b - c)*(b - c)*(a*b - b^2 + a*c - c^2) : :
X(68813) = 2 X[9] - 3 X[14427], 3 X[14427] - X[65680]

X(68813) lies on these lines: {1, 13259}, {2, 46793}, {6, 53532}, {8, 4148}, {9, 3738}, {11, 1146}, {37, 1769}, {100, 101}, {108, 24863}, {109, 64616}, {117, 5513}, {198, 53277}, {294, 2338}, {513, 4130}, {514, 661}, {521, 657}, {522, 4171}, {526, 3958}, {649, 3309}, {650, 663}, {654, 14418}, {656, 6586}, {665, 1642}, {672, 57468}, {812, 20533}, {885, 3716}, {918, 16593}, {926, 68115}, {1021, 62747}, {1024, 28071}, {1145, 6544}, {1320, 23893}, {1459, 2509}, {1643, 23057}, {2292, 3569}, {2294, 6370}, {2321, 4768}, {2371, 15731}, {2488, 58335}, {2533, 48299}, {2804, 42462}, {2826, 4120}, {3063, 66704}, {3161, 24118}, {3572, 23466}, {3667, 57064}, {3700, 6362}, {3709, 17420}, {3800, 42664}, {4017, 21348}, {4025, 25900}, {4051, 24143}, {4079, 48022}, {4088, 17464}, {4147, 4521}, {4369, 66516}, {4394, 48329}, {4406, 63231}, {4490, 21343}, {4529, 57091}, {4712, 34905}, {4813, 48116}, {4877, 35055}, {4893, 14077}, {4979, 42325}, {5029, 45695}, {5120, 23087}, {6003, 21390}, {6139, 58331}, {6161, 8659}, {8058, 68784}, {8674, 22108}, {14298, 40628}, {14321, 48265}, {14413, 44304}, {14825, 38329}, {14943, 28132}, {16550, 48032}, {17115, 28073}, {17304, 25602}, {20520, 29571}, {20901, 21429}, {20940, 21602}, {21123, 48033}, {21580, 33946}, {21921, 65707}, {21957, 47134}, {22116, 59489}, {22379, 36743}, {24002, 46399}, {28292, 47766}, {28843, 29624}, {30016, 30017}, {30805, 48070}, {30857, 30858}, {31058, 53362}, {34586, 40614}, {36101, 68267}, {38379, 46388}, {38955, 52222}, {40551, 47203}, {47906, 48026}, {47913, 48616}, {48025, 50332}, {48151, 54249}, {48269, 57049}, {48303, 68776}, {48322, 58339}, {50523, 66523}, {51058, 64860}, {53305, 54322}, {53343, 65970}, {55232, 68806}, {57057, 68136}

X(68813) = midpoint of X(4171) and X(21127)
X(68813) = reflection of X(i) in X(j) for these {i,j}: {649, 66514}, {693, 54264}, {885, 3716}, {2254, 3126}, {14330, 4521}, {24002, 46399}, {30804, 3835}, {45755, 650}, {65680, 9}, {66516, 4369}
X(68813) = isogonal conjugate of X(36146)
X(68813) = isotomic conjugate of X(34085)
X(68813) = complement of X(53357)
X(68813) = isotomic conjugate of the isogonal conjugate of X(46388)
X(68813) = tripolar centroid of X(60668)
X(68813) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {9442, 150}, {14943, 33650}, {52001, 37781}
X(68813) = X(i)-complementary conjugate of X(j) for these (i,j): {43672, 21252}, {59067, 3739}
X(68813) = X(i)-Ceva conjugate of X(j) for these (i,j): {190, 4712}, {918, 2254}, {1025, 518}, {1026, 2340}, {2284, 3930}, {2338, 34591}, {3570, 44694}, {3693, 17435}, {4562, 8}, {4876, 2170}, {14942, 2310}, {14943, 3119}, {18025, 24010}, {27805, 17755}, {36807, 244}, {42719, 1736}, {48070, 53583}, {60577, 4041}, {62747, 38379}
X(68813) = X(i)-cross conjugate of X(j) for these (i,j): {926, 2254}, {17435, 3693}
X(68813) = X(i)-isoconjugate of X(j) for these (i,j): {1, 36146}, {2, 32735}, {6, 927}, {7, 919}, {25, 65301}, {31, 34085}, {32, 46135}, {56, 666}, {57, 36086}, {59, 62635}, {85, 32666}, {100, 1462}, {101, 56783}, {103, 56786}, {105, 651}, {108, 1814}, {109, 673}, {110, 66941}, {190, 1416}, {222, 65333}, {279, 52927}, {294, 934}, {604, 51560}, {649, 39293}, {653, 36057}, {658, 2195}, {664, 1438}, {692, 34018}, {883, 41934}, {884, 1275}, {885, 1262}, {948, 58989}, {1024, 7045}, {1025, 51838}, {1027, 4564}, {1086, 59101}, {1252, 43930}, {1397, 36803}, {1407, 36802}, {1414, 18785}, {1415, 2481}, {1445, 36041}, {1461, 14942}, {1509, 66930}, {1813, 36124}, {2175, 65847}, {2283, 6185}, {3669, 5377}, {4000, 59133}, {4554, 64216}, {4565, 13576}, {4573, 56853}, {4617, 28071}, {4998, 43929}, {5545, 14625}, {5723, 59021}, {6180, 65371}, {6516, 8751}, {6559, 6614}, {6604, 32644}, {7128, 23696}, {7339, 28132}, {16588, 65562}, {18026, 32658}, {23981, 55943}, {24016, 56900}, {31615, 43921}, {31637, 32674}, {32724, 60720}, {35185, 37800}, {36059, 54235}, {36138, 40719}, {37136, 54364}, {51987, 54953}, {53241, 53243}, {53539, 57536}, {53607, 56896}
X(68813) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 666}, {2, 34085}, {3, 36146}, {9, 927}, {11, 673}, {241, 24015}, {244, 66941}, {518, 1025}, {661, 43930}, {665, 43041}, {926, 46388}, {1015, 56783}, {1086, 34018}, {1146, 2481}, {2968, 36796}, {3126, 514}, {3161, 51560}, {3716, 812}, {4904, 31638}, {5375, 39293}, {5452, 36086}, {5519, 1445}, {6184, 664}, {6376, 46135}, {6505, 65301}, {6608, 28132}, {6615, 62635}, {8054, 1462}, {14714, 294}, {17060, 3732}, {17115, 1024}, {17435, 9436}, {17755, 4554}, {20620, 54235}, {20621, 653}, {24771, 36802}, {27918, 10030}, {32664, 32735}, {33570, 66987}, {35072, 31637}, {35094, 85}, {35508, 14942}, {35509, 1111}, {36905, 4569}, {38980, 7}, {38983, 1814}, {38989, 57}, {38991, 105}, {39012, 40719}, {39014, 1}, {39025, 1438}, {39046, 651}, {39063, 658}, {40593, 65847}, {40608, 18785}, {40609, 190}, {40624, 18031}, {48315, 9312}, {55053, 1416}, {55064, 13576}, {62585, 36803}, {62587, 4572}
X(68813) = cevapoint of X(926) and X(52614)
X(68813) = crosspoint of X(i) and X(j) for these (i,j): {92, 65218}, {100, 36101}, {190, 14942}, {312, 36801}, {514, 35355}, {518, 1025}, {643, 36800}, {662, 26702}, {918, 50333}, {1026, 3912}, {4562, 22116}, {56359, 65245}
X(68813) = crosssum of X(i) and X(j) for these (i,j): {6, 53308}, {57, 53544}, {101, 68768}, {105, 1024}, {294, 23696}, {513, 910}, {649, 1458}, {661, 44661}, {919, 32735}, {1027, 1438}, {4319, 46392}, {62635, 66941}
X(68813) = crossdifference of every pair of points on line {31, 57}
X(68813) = X(64616)-line conjugate of X(109)
X(68813) = barycentric product X(i)*X(j) for these {i,j}: {1, 50333}, {8, 2254}, {9, 918}, {11, 1026}, {21, 4088}, {75, 926}, {76, 46388}, {78, 68783}, {85, 52614}, {190, 17435}, {200, 43042}, {210, 23829}, {241, 3239}, {294, 53583}, {312, 665}, {318, 53550}, {333, 24290}, {341, 53539}, {346, 53544}, {513, 3717}, {514, 3693}, {518, 522}, {521, 1861}, {561, 8638}, {650, 3912}, {652, 46108}, {657, 40704}, {663, 3263}, {672, 4391}, {693, 2340}, {765, 52305}, {883, 2310}, {885, 4712}, {1024, 4437}, {1025, 1146}, {1043, 53551}, {1458, 4397}, {1818, 44426}, {2170, 42720}, {2195, 62430}, {2223, 35519}, {2283, 24026}, {2284, 4858}, {2356, 35518}, {2414, 38375}, {2804, 36819}, {3064, 25083}, {3126, 14942}, {3286, 4086}, {3675, 3699}, {3680, 4925}, {3700, 18206}, {3709, 18157}, {3716, 22116}, {3737, 3932}, {3900, 9436}, {3930, 4560}, {3939, 62429}, {4041, 30941}, {4081, 41353}, {4130, 62786}, {4163, 34855}, {4435, 40217}, {4451, 53553}, {4768, 34230}, {4811, 14626}, {4866, 50357}, {4876, 62552}, {5089, 6332}, {5236, 57055}, {6735, 57468}, {8299, 60577}, {8611, 15149}, {14208, 37908}, {14430, 64612}, {14439, 60480}, {18155, 20683}, {20504, 56111}, {20752, 46110}, {23696, 34337}, {28071, 66967}, {28143, 46793}, {34387, 54325}, {35293, 63748}, {35355, 40609}, {36801, 38989}, {42341, 65952}, {42719, 56787}, {42758, 51565}, {44448, 57469}, {45755, 62622}, {46392, 56668}, {46393, 56753}, {51384, 62747}, {51390, 61238}, {52344, 53554}, {52355, 54407}, {52409, 53555}, {52622, 52635}, {53552, 60576}
X(68813) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 927}, {2, 34085}, {6, 36146}, {8, 51560}, {9, 666}, {31, 32735}, {33, 65333}, {41, 919}, {55, 36086}, {63, 65301}, {75, 46135}, {85, 65847}, {100, 39293}, {200, 36802}, {241, 658}, {244, 43930}, {312, 36803}, {513, 56783}, {514, 34018}, {518, 664}, {521, 31637}, {522, 2481}, {649, 1462}, {650, 673}, {652, 1814}, {657, 294}, {661, 66941}, {663, 105}, {665, 57}, {667, 1416}, {672, 651}, {872, 66930}, {884, 51838}, {910, 56786}, {918, 85}, {926, 1}, {1024, 6185}, {1025, 1275}, {1026, 4998}, {1110, 59101}, {1253, 52927}, {1362, 41353}, {1458, 934}, {1818, 6516}, {1861, 18026}, {1876, 36118}, {1946, 36057}, {2170, 62635}, {2175, 32666}, {2223, 109}, {2254, 7}, {2283, 7045}, {2284, 4564}, {2310, 885}, {2340, 100}, {2356, 108}, {3063, 1438}, {3064, 54235}, {3119, 28132}, {3126, 9436}, {3239, 36796}, {3263, 4572}, {3270, 23696}, {3271, 1027}, {3286, 1414}, {3675, 3676}, {3688, 35333}, {3693, 190}, {3709, 18785}, {3717, 668}, {3900, 14942}, {3912, 4554}, {3930, 4552}, {3939, 5377}, {4041, 13576}, {4088, 1441}, {4105, 28071}, {4130, 6559}, {4391, 18031}, {4435, 6654}, {4447, 6649}, {4526, 36816}, {4712, 883}, {4925, 39126}, {5089, 653}, {5236, 13149}, {6184, 1025}, {7084, 59133}, {8638, 31}, {8641, 2195}, {9436, 4569}, {9439, 65371}, {9454, 1415}, {14411, 2246}, {14439, 62669}, {14936, 1024}, {17435, 514}, {18206, 4573}, {18344, 36124}, {20683, 4551}, {20752, 1813}, {21127, 53241}, {23225, 603}, {23612, 66978}, {23829, 57785}, {24290, 226}, {25083, 65164}, {28143, 46792}, {30941, 4625}, {34855, 4626}, {35293, 56543}, {35505, 53544}, {36819, 54953}, {37908, 162}, {38375, 2402}, {38379, 63087}, {38989, 43041}, {39014, 46388}, {39063, 24015}, {39258, 4559}, {39959, 41075}, {40704, 46406}, {40781, 65237}, {40972, 46163}, {41353, 59457}, {42079, 2283}, {42341, 9312}, {42720, 67038}, {42758, 22464}, {42771, 1769}, {43042, 1088}, {45755, 63236}, {46108, 46404}, {46388, 6}, {46392, 56900}, {50333, 75}, {52213, 65245}, {52305, 1111}, {52614, 9}, {52635, 1461}, {52969, 52227}, {53539, 269}, {53544, 279}, {53547, 23973}, {53549, 54364}, {53550, 77}, {53551, 3668}, {53553, 7176}, {53554, 1442}, {53555, 1443}, {53583, 40704}, {54325, 59}, {55206, 68565}, {56718, 53640}, {60001, 65218}, {60577, 67197}, {60668, 53227}, {61238, 55943}, {62429, 52621}, {62552, 10030}, {62786, 36838}, {63188, 65562}, {63461, 56853}, {65664, 56639}, {65952, 14727}, {66982, 2223}, {68115, 2340}, {68743, 63203}, {68783, 273}
X(68813) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {661, 3250, 48131}, {665, 24290, 2254}, {14427, 65680, 9}

X(68814) = X(100)X(101) INTERCEPT OF X(1)X(665)

Barycentrics    a*(b - c)*(a^3 - 2*a^2*b + a*b^2 - 2*a^2*c - a*b*c + b^2*c + a*c^2 + b*c^2) : :
X(68814) = 3 X[47776] + X[53357]

X(68814) lies on these lines: {1, 665}, {9, 900}, {19, 44428}, {57, 43042}, {100, 101}, {104, 2291}, {239, 514}, {294, 1027}, {522, 21390}, {650, 1734}, {652, 14330}, {654, 1768}, {657, 3667}, {659, 8659}, {661, 42325}, {667, 3900}, {673, 812}, {798, 57155}, {885, 18785}, {918, 53396}, {1120, 23892}, {1449, 53314}, {2246, 13266}, {2516, 48331}, {2590, 3308}, {2591, 3307}, {2827, 65680}, {3063, 21173}, {3125, 55067}, {3126, 9508}, {3239, 60366}, {3250, 48337}, {3662, 27012}, {3731, 4526}, {3737, 21007}, {3762, 4148}, {3766, 4384}, {4010, 40551}, {4380, 24601}, {4501, 21348}, {4724, 60343}, {4827, 17410}, {5257, 26144}, {5296, 27545}, {5299, 22155}, {5541, 68401}, {5750, 26078}, {6003, 21127}, {6366, 66068}, {6544, 68137}, {6586, 48307}, {6646, 26851}, {14077, 66524}, {16550, 53395}, {17338, 27074}, {17754, 48244}, {20373, 21391}, {20520, 62635}, {21362, 57018}, {23800, 68776}, {24285, 62552}, {28073, 59751}, {28292, 48363}, {29485, 29492}, {31182, 68135}, {31286, 54264}, {36141, 37136}, {38379, 53343}, {40131, 44433}, {46393, 64446}, {47329, 51194}, {47890, 50556}, {48282, 54249}, {48305, 68794}

X(68814) = midpoint of X(i) and X(j) for these {i,j}: {649, 45755}, {885, 50343}, {4380, 30804}, {4827, 17410}, {17494, 66516}
X(68814) = reflection of X(i) in X(j) for these {i,j}: {1, 45695}, {9, 22108}, {3126, 9508}, {4010, 40551}, {35355, 2254}, {54264, 31286}, {60343, 4724}, {66514, 4394}
X(68814) = X(i)-Ceva conjugate of X(j) for these (i,j): {666, 1}, {6078, 9}
X(68814) = X(105)-isoconjugate of X(40526)
X(68814) = X(i)-Dao conjugate of X(j) for these (i,j): {2254, 918}, {39046, 40526}
X(68814) = crosspoint of X(i) and X(j) for these (i,j): {81, 36086}, {100, 673}, {190, 1280}, {662, 53707}, {21453, 34085}
X(68814) = crosssum of X(i) and X(j) for these (i,j): {37, 2254}, {513, 672}, {649, 1279}, {661, 20718}, {2293, 46388}, {23704, 53388}
X(68814) = crossdifference of every pair of points on line {42, 244}
X(68814) = barycentric product X(i)*X(j) for these {i,j}: {1, 53343}, {75, 53287}, {514, 63087}, {522, 7677}, {666, 38980}, {34018, 38379}
X(68814) = barycentric quotient X(i)/X(j) for these {i,j}: {672, 40526}, {7677, 664}, {38379, 3693}, {38980, 918}, {53287, 1}, {53343, 75}, {63087, 190}
X(68814) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {650, 66513, 4040}, {665, 4435, 1}

X(68815) = X(100)X(101) INTERCEPT OF X(320)X(350)

Barycentrics    a*(b - c)*(a^3*b - 2*a^2*b^2 + a*b^3 + a^3*c - a^2*b*c - a*b^2*c + b^3*c - 2*a^2*c^2 - a*b*c^2 + b^2*c^2 + a*c^3 + b*c^3) : :
X(68815) = 3 X[3681] - 4 X[30700], 3 X[30565] - 2 X[30700], 4 X[1639] - 3 X[63961], 4 X[2488] - X[4467], X[3868] + 2 X[49276], 2 X[4453] - 3 X[64149], X[12528] - 4 X[62434], X[17494] + 2 X[50518], X[25259] + 2 X[65697], 5 X[26985] - 2 X[44319], X[47666] + 2 X[50519]

X(68815) lies on these lines: {100, 101}, {312, 65660}, {320, 350}, {354, 48571}, {512, 48567}, {518, 47772}, {654, 62838}, {918, 3873}, {926, 3681}, {1639, 63961}, {2488, 4467}, {2832, 38478}, {3309, 47762}, {3738, 10707}, {3868, 49276}, {3900, 31150}, {4120, 37998}, {4448, 8674}, {4453, 64149}, {4928, 30993}, {6608, 35057}, {9029, 47769}, {10129, 46397}, {12528, 62434}, {15313, 47804}, {17494, 50518}, {20525, 32915}, {25259, 65697}, {26985, 44319}, {28143, 68388}, {28606, 65703}, {31148, 42325}, {47666, 50519}, {48383, 53278}, {68102, 68115}

X(68815) = reflection of X(i) in X(j) for these {i,j}: {3681, 30565}, {48571, 354}
X(68815) = crossdifference of every pair of points on line {213, 244}

X(68816) = X(100)X(101) INTERCEPT OF X(1)X(659)

Barycentrics    a*(b - c)*(3*a^2 - a*b - a*c - b*c) : :
X(68816) = X[1] + 2 X[659], X[1] - 4 X[1960], 5 X[1] - 2 X[21343], 2 X[1] + X[21385], 7 X[1] - 4 X[48296], X[659] + 2 X[1960], 5 X[659] + X[21343], 4 X[659] - X[21385], 7 X[659] + 2 X[48296], 10 X[1960] - X[21343], 8 X[1960] + X[21385], 7 X[1960] - X[48296], 4 X[21343] + 5 X[21385], X[21343] - 5 X[25569], 7 X[21343] - 10 X[48296], and many others

X(68816) lies on these lines: {1, 659}, {36, 238}, {40, 44805}, {100, 101}, {105, 106}, {165, 2821}, {191, 53403}, {214, 13266}, {514, 8643}, {649, 4794}, {650, 8657}, {663, 4063}, {676, 47680}, {830, 47810}, {900, 15015}, {978, 27675}, {1125, 46403}, {1281, 41190}, {1420, 30725}, {1697, 58369}, {1734, 6050}, {2108, 48244}, {2224, 34068}, {2787, 4448}, {2820, 38690}, {2826, 3576}, {2827, 38693}, {3251, 8298}, {3361, 53539}, {3601, 53523}, {3624, 3837}, {3669, 47977}, {3679, 45314}, {3762, 53580}, {3768, 23892}, {3907, 47817}, {3960, 48032}, {4041, 48345}, {4083, 58155}, {4129, 31291}, {4160, 47811}, {4367, 47970}, {4378, 58151}, {4449, 58153}, {4498, 48294}, {4707, 5592}, {4724, 8656}, {4763, 28521}, {4775, 4782}, {4784, 58139}, {4800, 29340}, {4809, 29102}, {4874, 47724}, {4879, 58156}, {4925, 5438}, {4928, 25531}, {4962, 57091}, {5029, 52745}, {6005, 58140}, {6009, 38316}, {6089, 28114}, {6161, 9508}, {6326, 19916}, {6366, 53401}, {6372, 58149}, {7713, 58313}, {8689, 48325}, {14407, 37998}, {14431, 45666}, {14838, 48150}, {16831, 24623}, {17030, 27075}, {18793, 23355}, {19903, 58034}, {23597, 52133}, {23887, 44433}, {25055, 64913}, {25574, 51093}, {26275, 29240}, {27015, 27255}, {27666, 27667}, {28183, 48305}, {28470, 47794}, {29033, 47832}, {29051, 47818}, {29066, 47804}, {29160, 48223}, {29182, 47872}, {29186, 47820}, {29192, 47771}, {29274, 47875}, {30795, 34595}, {31149, 48197}, {32665, 36086}, {38349, 47776}, {43928, 62763}, {47682, 50347}, {47683, 47694}, {47723, 68794}, {47726, 50340}, {47727, 47890}, {47929, 48343}, {47947, 48058}, {47948, 50507}, {47959, 50517}, {47976, 50508}, {48003, 48322}, {48008, 48339}, {48011, 48338}, {48045, 50526}, {48063, 48321}, {48064, 48367}, {48065, 48144}, {48248, 48288}, {48282, 48330}, {48286, 48408}, {48336, 50512}, {48341, 48623}, {50346, 50353}, {53285, 53395}, {53314, 53392}, {53571, 64850}, {66513, 66524}

X(68816) = midpoint of X(659) and X(25569)
X(68816) = reflection of X(i) in X(j) for these {i,j}: {1, 25569}, {1022, 14413}, {14431, 45666}, {25569, 1960}, {31149, 48197}
X(68816) = X(898)-Ceva conjugate of X(1)
X(68816) = crosspoint of X(i) and X(j) for these (i,j): {58, 59071}, {86, 4607}, {100, 37129}
X(68816) = crosssum of X(i) and X(j) for these (i,j): {42, 3768}, {513, 899}, {661, 44671}
X(68816) = trilinear pole of line {16507, 38349}
X(68816) = crossdifference of every pair of points on line {37, 244}
X(68816) = barycentric product X(i)*X(j) for these {i,j}: {1, 47776}, {190, 16507}, {1019, 19998}, {3227, 38349}
X(68816) = barycentric quotient X(i)/X(j) for these {i,j}: {16507, 514}, {19998, 4033}, {38349, 536}, {47776, 75}
X(68816) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 659, 21385}, {649, 4794, 48352}, {659, 1960, 1}, {663, 4063, 48337}, {663, 4401, 4063}, {667, 4040, 1019}, {667, 48331, 4040}, {3737, 4057, 57155}, {4449, 58153, 65428}, {4491, 53287, 54333}, {4498, 58154, 48294}, {5592, 13246, 4707}, {6050, 48329, 1734}, {47694, 48284, 47683}, {48058, 50523, 47947}, {48367, 58138, 48064}

X(68817) = X(100)X(101) INTERCEPT OF X(659)X(3004)

Barycentrics    a*(b - c)*(a^4 - 2*a^3*b + a^2*b^2 - 2*a^3*c - 4*a^2*b*c + 2*a*b^2*c + a^2*c^2 + 2*a*b*c^2 + b^2*c^2) : :

X(68817) lies on these lines: {100, 101}, {659, 3004}, {667, 31150}, {812, 5284}, {1621, 47776}, {2254, 24948}, {3873, 53396}, {4057, 48242}, {4401, 47785}, {4763, 9342}, {8702, 16158}, {45695, 47892}

X(68817) = Stevanovic-circle-inverse of X(792)
X(68817) = crossdifference of every pair of points on line {244, 1500}
X(68817) = {X(47776),X(53287)}-harmonic conjugate of X(1621)

X(68818) = X(100)X(101) INTERCEPT OF X(523)X(661)

Barycentrics    a*(b^2 - c^2)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2 + c^3) : :

X(68818) lies on these lines: {37, 2610}, {71, 2624}, {100, 101}, {523, 661}, {594, 1639}, {926, 3690}, {1500, 3310}, {2605, 46381}, {3969, 30565}, {21044, 21046}, {21090, 21092}, {21797, 21828}, {22037, 32679}, {32849, 65669}, {47971, 57106}, {55210, 55232}, {58294, 58842}

X(68818) = X(51562)-Ceva conjugate of X(756)
X(68818) = X(i)-isoconjugate of X(j) for these (i,j): {58, 65238}, {81, 1290}, {110, 21907}, {593, 66280}, {1333, 35156}, {4556, 5620}, {4565, 11604}
X(68818) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 65238}, {37, 35156}, {244, 21907}, {2610, 4453}, {35090, 86}, {40586, 1290}, {53988, 27}, {55064, 11604}, {55065, 68363}
X(68818) = crosssum of X(i) and X(j) for these (i,j): {81, 3960}, {284, 35055}, {513, 15586}
X(68818) = crossdifference of every pair of points on line {58, 229}
X(68818) = barycentric product X(i)*X(j) for these {i,j}: {10, 8674}, {306, 47235}, {313, 42670}, {656, 56877}, {661, 32849}, {756, 65669}, {1089, 42741}, {1577, 17796}, {2074, 4064}, {2677, 36037}, {3701, 51646}, {4024, 37783}, {4036, 5127}, {4086, 5172}, {8611, 37799}, {19622, 52623}, {38982, 51562}
X(68818) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 35156}, {37, 65238}, {42, 1290}, {661, 21907}, {756, 66280}, {2677, 36038}, {4024, 68363}, {4041, 11604}, {4705, 5620}, {5127, 52935}, {5172, 1414}, {8674, 86}, {17796, 662}, {19622, 4556}, {32849, 799}, {37783, 4610}, {38982, 4453}, {42670, 58}, {42741, 757}, {47235, 27}, {51646, 1014}, {56877, 811}, {65669, 873}
X(68818) = {X(661),X(4171)}-harmonic conjugate of X(4931)

X(68819) = X(100)X(101) INTERCEPT OF X(10)X(37)

Barycentrics    a*(a - b)*(a - c)*(b + c)^2*(a^2 - b^2 + b*c - c^2) : :

X(68819) lies on these lines: {10, 37}, {100, 101}, {2397, 60074}, {3570, 48003}, {4033, 18740}, {4103, 21859}, {4115, 61161}, {14838, 42717}, {21711, 24036}, {29044, 65370}, {61167, 61168}

X(68819) = X(36804)-Ceva conjugate of X(3952)
X(68819) = X(i)-cross conjugate of X(j) for these (i,j): {2610, 4053}, {42666, 758}
X(68819) = X(i)-isoconjugate of X(j) for these (i,j): {56, 60571}, {244, 37140}, {593, 66284}, {759, 1019}, {849, 60074}, {1015, 65283}, {1086, 36069}, {1111, 32671}, {2341, 7203}, {3669, 52380}, {3733, 24624}, {7192, 34079}, {7199, 67166}, {7254, 68571}, {14616, 57129}, {17925, 57736}, {39179, 46160}, {43925, 57985}
X(68819) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 60571}, {758, 3960}, {4075, 60074}, {34586, 1019}, {35069, 7192}, {38982, 1086}, {51583, 7199}, {53982, 17925}, {57434, 26856}, {66508, 61403}
X(68819) = cevapoint of X(2610) and X(4053)
X(68819) = crosspoint of X(3952) and X(36804)
X(68819) = crosssum of X(3733) and X(21758)
X(68819) = crossdifference of every pair of points on line {244, 3733}
X(68819) = barycentric product X(i)*X(j) for these {i,j}: {101, 61410}, {190, 4053}, {320, 40521}, {594, 4585}, {758, 3952}, {762, 55237}, {765, 6370}, {1016, 2610}, {1018, 3936}, {1983, 28654}, {2245, 4033}, {3218, 4103}, {3695, 4242}, {3699, 68765}, {3724, 27808}, {3738, 65958}, {3949, 65162}, {3960, 61402}, {4069, 41804}, {4076, 51663}, {4557, 35550}, {4736, 51562}, {7035, 42666}, {15065, 68154}, {18593, 30730}, {21859, 32851}, {35069, 36804}, {52609, 68779}
X(68819) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 60571}, {594, 60074}, {756, 66284}, {758, 7192}, {762, 55238}, {765, 65283}, {1018, 24624}, {1110, 36069}, {1252, 37140}, {1464, 7203}, {1983, 593}, {2245, 1019}, {2610, 1086}, {3724, 3733}, {3936, 7199}, {3939, 52380}, {3952, 14616}, {3960, 61403}, {4053, 514}, {4069, 6740}, {4103, 18359}, {4557, 759}, {4585, 1509}, {4707, 16727}, {4736, 4453}, {6057, 52356}, {6058, 66272}, {6370, 1111}, {18593, 17096}, {21828, 16726}, {21859, 2006}, {23990, 32671}, {35069, 3960}, {35550, 52619}, {36804, 57555}, {40521, 80}, {42666, 244}, {42701, 16755}, {44113, 57200}, {51663, 1358}, {53527, 17205}, {53562, 18191}, {55237, 57949}, {56193, 68483}, {58328, 65575}, {61402, 36804}, {61410, 3261}, {65958, 35174}, {68765, 3676}, {68779, 17925}

X(68820) = X(100)X(101) INTERCEPT OF X(513)X(3716)

Barycentrics    a*(b - c)*(a^3*b - 2*a^2*b^2 + a*b^3 + a^3*c + a^2*b*c - 2*a*b^2*c + b^3*c - 2*a^2*c^2 - 2*a*b*c^2 + b^2*c^2 + a*c^3 + b*c^3) : :
X(68820) = X[21115] - 3 X[64149]

X(68820) lies on these lines: {100, 101}, {354, 28890}, {513, 3716}, {918, 59842}, {926, 10196}, {1639, 37998}, {3309, 45313}, {3738, 4448}, {6003, 47804}, {8674, 45314}, {9001, 45673}, {21115, 64149}, {35057, 48226}, {42325, 47762}

X(68820) = crossdifference of every pair of points on line {244, 2176}

X(68821) = X(100)X(101) INTERCEPT OF X(190)X(670)

Barycentrics    a*(a - b)*(a - c)*(a*b^2 + a*b*c + b^2*c + a*c^2 + b*c^2) : :

X(68821) lies on these lines: {31, 7075}, {43, 38853}, {100, 101}, {190, 670}, {244, 24578}, {333, 40586}, {643, 32739}, {649, 65166}, {672, 32919}, {750, 3501}, {813, 835}, {1150, 3730}, {1334, 32917}, {2177, 3208}, {2225, 3685}, {2229, 21788}, {3294, 5235}, {3508, 3994}, {3882, 61160}, {4436, 35326}, {4513, 64752}, {4553, 35310}, {4712, 38479}, {4781, 25577}, {5208, 40463}, {5364, 32915}, {7239, 65314}, {9345, 17754}, {10453, 36808}, {16549, 37633}, {17277, 29797}, {17763, 39258}, {24036, 38484}, {25531, 40614}, {32930, 40972}, {32936, 53129}, {38871, 41423}, {43077, 53627}, {53355, 65195}

X(68821) = X(2978)-cross conjugate of X(10458)
X(68821) = X(i)-isoconjugate of X(j) for these (i,j): {649, 2296}, {667, 1218}, {785, 1086}, {1015, 57959}, {3121, 59093}
X(68821) = X(i)-Dao conjugate of X(j) for these (i,j): {5375, 2296}, {6631, 1218}, {10472, 514}, {31330, 47666}
X(68821) = crosssum of X(649) and X(48144)
X(68821) = X(24578)-line conjugate of X(244)
X(68821) = barycentric product X(i)*X(j) for these {i,j}: {100, 31330}, {190, 5283}, {765, 784}, {1018, 27164}, {1110, 35559}, {1185, 1978}, {2978, 7035}, {3699, 10473}, {3952, 10458}, {4557, 10471}
X(68821) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 2296}, {190, 1218}, {765, 57959}, {784, 1111}, {1110, 785}, {1185, 649}, {2978, 244}, {4600, 59093}, {5283, 514}, {10458, 7192}, {10471, 52619}, {10473, 3676}, {27164, 7199}, {31330, 693}
X(68821) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1018, 61234, 100}, {46148, 61163, 190}

X(68822) = X(100)X(101) INTERCEPT OF X(1)X(3310)

Barycentrics    a*(a - b - c)*(b - c)*(a^3 - a*b^2 + 3*a*b*c - b^2*c - a*c^2 - b*c^2) : :
X(68822) = 2 X[9511] - 3 X[64112]

X(68822) lies on these lines: {1, 3310}, {9, 654}, {46, 49276}, {57, 918}, {63, 30565}, {100, 101}, {200, 926}, {521, 650}, {612, 65703}, {649, 3239}, {652, 4521}, {663, 57121}, {667, 58339}, {900, 16561}, {908, 46401}, {1019, 4391}, {1024, 14942}, {1158, 62434}, {1311, 29068}, {1638, 5437}, {1743, 22086}, {2316, 3738}, {2488, 4477}, {2977, 53400}, {3218, 47772}, {3306, 4453}, {3762, 24618}, {3904, 21385}, {4057, 4394}, {4063, 6332}, {4130, 66524}, {4435, 4919}, {4468, 68259}, {4512, 6139}, {4885, 58324}, {5219, 46397}, {5223, 30700}, {5268, 24462}, {7110, 62746}, {7308, 45326}, {7658, 25955}, {9511, 64112}, {10980, 30704}, {14425, 22108}, {18004, 53403}, {20525, 29649}, {21044, 55067}, {21189, 40134}, {21786, 21894}, {27003, 48571}, {30199, 66514}, {31182, 57057}, {33811, 58038}, {42341, 62823}, {46396, 57140}, {47127, 62749}, {47884, 53396}, {48387, 58329}, {50333, 53395}, {50518, 57067}, {68135, 68142}

X(68822) = X(i)-Ceva conjugate of X(j) for these (i,j): {5548, 9}, {6079, 200}, {13136, 1}
X(68822) = X(i)-isoconjugate of X(j) for these (i,j): {56, 50039}, {109, 14554}, {649, 65014}
X(68822) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 50039}, {11, 14554}, {1769, 10015}, {5375, 65014}, {34590, 226}
X(68822) = crosspoint of X(i) and X(j) for these (i,j): {29, 65223}, {100, 34234}, {190, 1320}, {1476, 37136}
X(68822) = crosssum of X(i) and X(j) for these (i,j): {513, 2183}, {649, 1319}, {1400, 53528}, {3057, 46393}
X(68822) = crossdifference of every pair of points on line {65, 244}
X(68822) = barycentric product X(i)*X(j) for these {i,j}: {9, 21222}, {75, 53286}, {312, 21786}, {318, 23087}, {333, 21894}, {522, 54391}, {4391, 5053}, {4567, 52341}
X(68822) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 50039}, {100, 65014}, {650, 14554}, {5053, 651}, {21222, 85}, {21786, 57}, {21894, 226}, {23087, 77}, {52341, 16732}, {53286, 1}, {54391, 664}
X(68822) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 47766, 21390}, {654, 1639, 9}

X(68823) = X(100)X(101) INTERCEPT OF X(512)X(661)

Barycentrics    a*(-b + c)*(b + c)*(a^2 + a*b - b^2 + a*c - b*c - c^2) : :
X(68823) = 3 X[1635] - 2 X[8632], 3 X[4728] - 4 X[21261]

X(68823) lies on these lines: {37, 2642}, {100, 101}, {512, 661}, {594, 900}, {649, 50355}, {656, 21834}, {665, 1500}, {812, 6653}, {830, 4979}, {876, 2254}, {1734, 3250}, {2295, 4435}, {2511, 21858}, {2533, 4024}, {2610, 21888}, {3125, 21824}, {3661, 24287}, {3700, 59740}, {3766, 60736}, {4010, 21053}, {4079, 57099}, {4083, 50454}, {4120, 21013}, {4132, 8061}, {4147, 48269}, {4391, 48266}, {4490, 4813}, {4674, 23894}, {4707, 4838}, {4728, 21261}, {4826, 47842}, {4958, 14430}, {5029, 9508}, {17285, 29485}, {17303, 24506}, {17458, 50350}, {18785, 35347}, {20981, 35057}, {21055, 50329}, {21385, 47918}, {29106, 48264}, {48244, 60724}, {50343, 60577}, {53686, 67764}

X(68823) = reflection of X(38348) in X(9508)
X(68823) = X(18757)-anticomplementary conjugate of X(39362)
X(68823) = X(i)-Ceva conjugate of X(j) for these (i,j): {660, 756}, {668, 20710}, {2786, 18004}, {4562, 40794}, {9508, 17990}, {40521, 20687}
X(68823) = X(i)-isoconjugate of X(j) for these (i,j): {2, 17940}, {6, 17930}, {58, 35148}, {81, 37135}, {86, 2702}, {99, 17962}, {110, 6650}, {163, 18032}, {249, 18014}, {593, 66283}, {648, 17972}, {662, 1929}, {2054, 4610}, {4556, 11599}, {4558, 17982}, {4584, 40767}, {4590, 18001}, {9278, 52935}, {39921, 53628}, {51332, 53655}
X(68823) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 17930}, {10, 35148}, {115, 18032}, {244, 6650}, {1084, 1929}, {9508, 812}, {27929, 7199}, {32664, 17940}, {35080, 274}, {38986, 17962}, {39041, 99}, {39042, 4610}, {40586, 37135}, {40600, 2702}, {41841, 799}, {55066, 17972}
X(68823) = crosspoint of X(i) and X(j) for these (i,j): {10, 4562}, {876, 47947}, {2786, 9508}
X(68823) = crosssum of X(i) and X(j) for these (i,j): {58, 8632}, {2702, 37135}, {3573, 35342}
X(68823) = crossdifference of every pair of points on line {81, 244}
X(68823) = barycentric product X(i)*X(j) for these {i,j}: {1, 18004}, {10, 9508}, {37, 2786}, {75, 17990}, {321, 5029}, {423, 55232}, {512, 20947}, {513, 6541}, {514, 20693}, {523, 1757}, {656, 17927}, {661, 6542}, {693, 58287}, {850, 18266}, {1109, 17943}, {1326, 4036}, {1577, 17735}, {1931, 4024}, {2643, 17934}, {4010, 40794}, {4079, 52137}, {4674, 28602}, {4705, 17731}, {8298, 35352}, {9278, 68140}, {17976, 24006}, {37212, 57461}, {38348, 43534}, {52623, 64215}
X(68823) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17930}, {31, 17940}, {37, 35148}, {42, 37135}, {213, 2702}, {423, 55231}, {512, 1929}, {523, 18032}, {661, 6650}, {756, 66283}, {798, 17962}, {810, 17972}, {1326, 52935}, {1757, 99}, {1931, 4610}, {2643, 18014}, {2786, 274}, {4079, 9278}, {4455, 40767}, {4705, 11599}, {5029, 81}, {6541, 668}, {6542, 799}, {9508, 86}, {17731, 4623}, {17735, 662}, {17927, 811}, {17934, 24037}, {17943, 24041}, {17976, 4592}, {17990, 1}, {18004, 75}, {18266, 110}, {20693, 190}, {20947, 670}, {21832, 40725}, {27929, 30940}, {28602, 30939}, {38348, 33295}, {40794, 4589}, {50487, 2054}, {52137, 52612}, {55232, 57848}, {57461, 4978}, {58287, 100}, {58301, 53688}, {64215, 4556}, {68140, 52137}
X(68823) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2533, 4024, 54256}, {4730, 24290, 21832}, {21832, 24290, 661}

X(68824) = X(100)X(101) INTERCEPT OF X(1)X(513)

Barycentrics    a*(b - c)*(5*a^2 - 5*a*b + 2*b^2 - 5*a*c + b*c + 2*c^2) : :
X(68824) = 5 X[1] - 2 X[764], X[1] + 2 X[6161], 5 X[1] - 4 X[9269], 3 X[1] - 2 X[14421], 4 X[764] - 5 X[1022], X[764] - 5 X[3251], X[764] + 5 X[6161], 3 X[764] - 5 X[14421], X[1022] - 4 X[3251], X[1022] + 4 X[6161], 5 X[1022] - 8 X[9269], 3 X[1022] - 4 X[14421], 5 X[3251] - 2 X[9269], 3 X[3251] - X[14421], and many others

X(68824) lies on these lines: {1, 513}, {100, 101}, {145, 63246}, {244, 14835}, {514, 3241}, {522, 62634}, {663, 995}, {667, 5010}, {840, 2718}, {900, 30580}, {1019, 4653}, {1027, 4160}, {1643, 47777}, {1960, 48244}, {2320, 35355}, {2820, 15735}, {2832, 10699}, {3126, 30234}, {3240, 4794}, {3309, 3576}, {3622, 23814}, {3667, 5731}, {3679, 4448}, {3900, 53401}, {4040, 4490}, {4063, 5119}, {4162, 7962}, {4677, 30583}, {4724, 60347}, {4895, 21385}, {4928, 28521}, {4962, 8834}, {5091, 35348}, {5592, 28161}, {6006, 47357}, {9260, 34747}, {14077, 66232}, {14422, 24286}, {14437, 24482}, {14839, 64237}, {16785, 43929}, {17461, 29350}, {19875, 45666}, {24496, 64612}, {25055, 36848}, {28183, 50351}, {30947, 47779}, {35445, 66524}, {47724, 48189}, {47805, 50286}, {47821, 50287}, {47947, 48336}, {48018, 58154}, {48086, 58160}, {48150, 48337}, {50328, 58158}, {50335, 58157}, {67417, 68260}

X(68824) = midpoint of X(i) and X(j) for these {i,j}: {145, 63246}, {3251, 6161}
X(68824) = reflection of X(i) in X(j) for these {i,j}: {1, 3251}, {764, 9269}, {1022, 1}, {3679, 4448}, {4677, 30583}, {4893, 4794}, {47724, 48189}, {48244, 1960}, {60347, 4724}
X(68824) = reflection of X(1022) in the OI line
X(68824) = crossdifference of every pair of points on line {44, 244}
X(68824) = X(14835)-line conjugate of X(244)
X(68824) = barycentric product X(i)*X(j) for these {i,j}: {1, 31992}, {513, 41138}
X(68824) = barycentric quotient X(i)/X(j) for these {i,j}: {31992, 75}, {41138, 668}

X(68825) = X(100)X(101) INTERCEPT OF X(6)X(31)

Barycentrics    a^2*(a - b)*(a - c)*(a*b + a*c - 2*b*c) : :

X(68825) lies on these lines: {6, 31}, {37, 67430}, {100, 101}, {109, 6016}, {649, 2284}, {750, 5091}, {813, 901}, {883, 47971}, {899, 52902}, {1646, 3230}, {3570, 6632}, {3722, 23622}, {4427, 61164}, {4588, 43077}, {6014, 8693}, {8683, 35326}, {9259, 33846}, {14437, 23343}, {20672, 23858}, {23845, 46148}, {28210, 28841}, {28868, 29363}, {48266, 56881}, {56530, 56758}, {61160, 61222}

X(68825) = isogonal conjugate of X(62619)
X(68825) = isogonal conjugate of the isotomic conjugate of X(23891)
X(68825) = X(i)-Ceva conjugate of X(j) for these (i,j): {34075, 101}, {59071, 4557}
X(68825) = X(i)-cross conjugate of X(j) for these (i,j): {890, 62740}, {3768, 3230}
X(68825) = X(i)-isoconjugate of X(j) for these (i,j): {1, 62619}, {2, 43928}, {75, 23892}, {76, 23349}, {81, 35353}, {244, 4607}, {513, 3227}, {514, 37129}, {649, 31002}, {693, 739}, {764, 5381}, {889, 1015}, {891, 57542}, {898, 1086}, {1019, 41683}, {1022, 36872}, {1111, 34075}, {1977, 57994}, {2401, 63852}, {3669, 36798}, {3733, 60288}, {7199, 62763}, {23989, 32718}, {33917, 57572}, {52768, 64237}, {62635, 64612}
X(68825) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 62619}, {206, 23892}, {5375, 31002}, {13466, 3261}, {14434, 6545}, {32664, 43928}, {39011, 1111}, {39026, 3227}, {40586, 35353}, {40614, 693}, {52875, 1577}, {52882, 40495}
X(68825) = cevapoint of X(i) and X(j) for these (i,j): {3230, 3768}, {4526, 52959}
X(68825) = crosspoint of X(101) and X(34075)
X(68825) = crosssum of X(i) and X(j) for these (i,j): {514, 4728}, {19945, 21143}
X(68825) = crossdifference of every pair of points on line {244, 514}
X(68825) = barycentric product X(i)*X(j) for these {i,j}: {1, 23343}, {6, 23891}, {31, 41314}, {59, 14430}, {100, 899}, {101, 536}, {109, 4009}, {110, 3994}, {190, 3230}, {644, 52896}, {662, 52959}, {692, 6381}, {739, 68131}, {765, 891}, {813, 4465}, {890, 7035}, {898, 42083}, {1016, 3768}, {1018, 52897}, {1023, 52900}, {1026, 52902}, {1252, 4728}, {1646, 6632}, {2284, 36816}, {3699, 62739}, {3939, 43037}, {3952, 62740}, {4526, 4564}, {4557, 62755}, {4570, 14431}, {4588, 4937}, {4600, 14404}, {4607, 59797}, {4706, 8694}, {4752, 52901}, {5376, 14437}, {5381, 68134}, {9268, 30583}, {13466, 34075}, {19945, 57731}, {23344, 52755}, {23845, 62760}, {32739, 35543}, {36037, 61672}, {52626, 59149}, {53280, 62761}
X(68825) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 62619}, {31, 43928}, {32, 23892}, {42, 35353}, {100, 31002}, {101, 3227}, {536, 3261}, {560, 23349}, {692, 37129}, {765, 889}, {890, 244}, {891, 1111}, {899, 693}, {1018, 60288}, {1110, 898}, {1252, 4607}, {1646, 6545}, {3230, 514}, {3768, 1086}, {3939, 36798}, {3994, 850}, {4009, 35519}, {4465, 65101}, {4526, 4858}, {4557, 41683}, {4728, 23989}, {6381, 40495}, {7035, 57994}, {14404, 3120}, {14430, 34387}, {14431, 21207}, {23343, 75}, {23344, 36872}, {23891, 76}, {23990, 34075}, {32739, 739}, {34075, 57542}, {41314, 561}, {43037, 52621}, {52626, 23100}, {52896, 24002}, {52897, 7199}, {52959, 1577}, {54325, 64612}, {59149, 5381}, {59797, 4728}, {61672, 36038}, {62739, 3676}, {62740, 7192}, {62755, 52619}, {68131, 35543}, {68134, 52626}
X(68825) = {X(2284),X(23832)}-harmonic conjugate of X(649)

X(68826) = X(100)X(101) INTERCEPT OF X(6)X(650)

Barycentrics    a*(b - c)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + 7*a^2*b*c - 5*a*b^2*c + b^3*c - a^2*c^2 - 5*a*b*c^2 + 2*b^2*c^2 + a*c^3 + b*c^3) : :

X(68826) lies on these lines: {6, 650}, {9, 649}, {57, 6546}, {63, 10196}, {100, 101}, {198, 4394}, {513, 4413}, {514, 3306}, {654, 14425}, {661, 23617}, {667, 64752}, {750, 1027}, {996, 23892}, {1019, 5235}, {1146, 56761}, {3218, 31992}, {4040, 56010}, {4893, 17754}, {5437, 6545}, {6590, 42462}, {14442, 16561}, {14624, 62749}, {25955, 48398}, {31207, 58324}, {32212, 62874}, {37222, 46781}, {37998, 67097}, {40131, 47766}, {42316, 59239}, {45193, 57049}, {47045, 61730}, {47778, 56510}, {52148, 66514}, {68137, 68142}

X(68826) = crosspoint of X(100) and X(37222)
X(68826) = crossdifference of every pair of points on line {244, 517}
X(68826) = {X(649),X(6544)}-harmonic conjugate of X(9)

X(68827) = X(100)X(101) INTERCEPT OF X(1)X(649)

Barycentrics    a*(b - c)*(3*a^3 - 2*a^2*b + a*b^2 - 2*a^2*c - 3*a*b*c + b^2*c + a*c^2 + b*c^2) : :
X(68827) = X[4435] + 2 X[8659]

X(68827) lies on these lines: {1, 649}, {6, 513}, {9, 14437}, {43, 4040}, {45, 57051}, {55, 667}, {81, 1019}, {100, 101}, {239, 62324}, {514, 16834}, {650, 6161}, {659, 8658}, {665, 25569}, {764, 4790}, {891, 4435}, {894, 53376}, {1449, 21143}, {2384, 2718}, {3207, 4394}, {4063, 50458}, {4375, 4384}, {4765, 5592}, {4773, 30580}, {4976, 50351}, {14421, 14969}, {14433, 16833}, {14442, 16554}, {16666, 23345}, {17027, 47780}, {17244, 27013}, {17367, 20295}, {18154, 34023}, {23876, 62634}, {25381, 29603}, {39771, 54377}, {45755, 48572}, {46782, 52226}, {46922, 62619}, {52227, 66526}

X(68827) = crosssum of X(513) and X(20331)
X(68827) = crossdifference of every pair of points on line {244, 518}

X(68828) = X(100)X(101) INTERCEPT OF X(1)X(19)

Barycentrics    a^2*(a - b)*(a - c)*(a^3 + b^3 - a*b*c - b^2*c - b*c^2 + c^3) : :

X(68828) lies on these lines: {1, 19}, {100, 101}, {112, 58986}, {163, 53290}, {649, 1983}, {673, 2224}, {813, 29103}, {919, 32682}, {2087, 7113}, {4251, 54315}, {4429, 16788}, {14887, 56742}, {23981, 32665}, {28841, 29030}, {49998, 54316}, {53282, 65375}

X(68828) = X(i)-isoconjugate of X(j) for these (i,j): {525, 39439}, {39166, 60074}
X(68828) = X(i)-Dao conjugate of X(j) for these (i,j): {31845, 1577}, {62652, 693}
X(68828) = crosspoint of X(662) and X(36069)
X(68828) = crosssum of X(i) and X(j) for these (i,j): {523, 1639}, {661, 6370}, {758, 24036}
X(68828) = crossdifference of every pair of points on line {244, 656}
X(68828) = barycentric product X(i)*X(j) for these {i,j}: {1, 13589}, {100, 30117}, {101, 33129}, {163, 62305}, {651, 1731}, {1252, 47680}, {1290, 5497}, {1331, 5146}, {15906, 36037}, {31845, 36069}
X(68828) = barycentric quotient X(i)/X(j) for these {i,j}: {1731, 4391}, {5146, 46107}, {13589, 75}, {15906, 36038}, {30117, 693}, {32676, 39439}, {33129, 3261}, {47680, 23989}, {62305, 20948}
X(68828) = {X(53290),X(61197)}-harmonic conjugate of X(163)

X(68829) = X(100)X(101) INTERCEPT OF X(1)X(661)

Barycentrics    a*(b - c)*(a^3 + a^2*b - 2*a*b^2 + b^3 + a^2*c - 3*a*b*c + b^2*c - 2*a*c^2 + b*c^2 + c^3) : :

X(68829) lies on these lines: {1, 661}, {2, 40459}, {37, 513}, {42, 663}, {100, 101}, {514, 29574}, {523, 50113}, {649, 1334}, {650, 20691}, {764, 4813}, {846, 48150}, {1022, 1255}, {1083, 46407}, {2267, 20981}, {2642, 53289}, {2787, 4120}, {3169, 35057}, {3661, 27929}, {3738, 14408}, {3943, 62323}, {4024, 50351}, {4369, 17244}, {4444, 16826}, {4931, 62634}, {4944, 29236}, {5029, 14419}, {5592, 6590}, {7192, 29569}, {8674, 14407}, {9269, 48136}, {9810, 53339}, {14421, 48544}, {14438, 25569}, {17367, 25666}, {17369, 68160}, {21802, 57133}, {23650, 53532}, {25748, 25822}, {29066, 47874}, {38348, 47827}, {47678, 48300}, {48266, 66995}, {48329, 63211}

X(68829) = crosspoint of X(662) and X(59827)
X(68829) = crossdifference of every pair of points on line {238, 244}

X(68830) = X(100)X(101) INTERCEPT OF X(43)X(661)

Barycentrics    a^2*(b - c)*(a^3*b + a^3*c - a^2*b*c - 2*a*b^2*c + b^3*c - 2*a*b*c^2 + b^2*c^2 + b*c^3) : :

X(68830) lies on these lines: {42, 649}, {43, 661}, {100, 101}, {513, 21904}, {650, 21877}, {798, 2276}, {1024, 56856}, {2280, 20981}, {4369, 17027}, {7252, 21792}, {8631, 50501}, {14436, 22108}, {16834, 31148}, {21834, 26242}, {24533, 25837}, {25813, 25822}, {29574, 45313}, {40214, 57129}

X(68830) = crosspoint of X(662) and X(12031)
X(68830) = crossdifference of every pair of points on line {239, 244}
X(68830) = {X(42),X(649)}-harmonic conjugate of X(3572)

X(68831) = X(100)X(101) INTERCEPT OF X(55)X(650)

Barycentrics    a*(a - b - c)*(b - c)*(a^4 + a^3*b - 3*a^2*b^2 + a*b^3 + a^3*c - a^2*b*c + a*b^2*c + b^3*c - 3*a^2*c^2 + a*b*c^2 - 2*b^2*c^2 + a*c^3 + b*c^3) : :

X(68831) lies on these lines: {9, 53401}, {40, 14825}, {55, 650}, {100, 101}, {165, 649}, {513, 42316}, {659, 52614}, {665, 9511}, {1021, 56181}, {1027, 36258}, {1615, 4394}, {2717, 43079}, {3239, 4375}, {4521, 41795}, {4893, 52155}, {8578, 9593}, {9366, 17425}, {9746, 47766}, {24615, 59779}, {48572, 59216}

X(68831) = crossdifference of every pair of points on line {241, 244}
X(68831) = {X(55),X(650)}-harmonic conjugate of X(1024)

X(68832) = X(100)X(101) INTERCEPT OF X(1)X(3904)

Barycentrics    a*(a - b - c)*(b - c)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + a^2*b*c - a*b^2*c + b^3*c - a^2*c^2 - a*b*c^2 + a*c^3 + b*c^3) : :

X(68832) lies on these lines: {1, 3904}, {8, 66993}, {63, 928}, {78, 50333}, {100, 101}, {102, 26703}, {522, 663}, {676, 54392}, {1281, 2785}, {3239, 4794}, {3309, 66527}, {3872, 6366}, {4040, 4391}, {4057, 15313}, {4652, 53300}, {4855, 52739}, {5250, 53549}, {6003, 48150}, {10015, 19860}, {14942, 23696}, {21173, 44550}, {24203, 36038}, {28292, 47805}, {32667, 36093}, {44448, 57108}, {48303, 64917}, {48306, 57158}, {52614, 55337}, {53550, 62874}

X(68832) = X(29241)-Ceva conjugate of X(78)
X(68832) = X(1415)-isoconjugate of X(54739)
X(68832) = X(1146)-Dao conjugate of X(54739)
X(68832) = crosssum of X(649) and X(51657)
X(68832) = crossdifference of every pair of points on line {244, 1400}
X(68832) = barycentric product X(6332)*X(62971)
X(68832) = barycentric quotient X(i)/X(j) for these {i,j}: {522, 54739}, {62971, 653}
X(68832) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4040, 58339, 4391}, {50333, 53285, 78}

X(68833) = X(100)X(101) INTERCEPT OF X(1)X(514)

Barycentrics    a*(b - c)*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3 - 3*a^3*c + a^2*b*c + a*b^2*c - b^3*c + 3*a^2*c^2 + a*b*c^2 - a*c^3 - b*c^3) : :

X(68833) lies on these lines: {1, 514}, {3, 667}, {9, 37998}, {21, 1019}, {100, 101}, {200, 10196}, {224, 6332}, {513, 1001}, {649, 35258}, {764, 48136}, {840, 2725}, {876, 48336}, {1960, 24286}, {2726, 12032}, {2826, 6265}, {2827, 64765}, {3250, 48352}, {3251, 14077}, {3423, 57468}, {3667, 5732}, {3736, 3737}, {3870, 6546}, {3900, 3913}, {3935, 31992}, {4369, 58322}, {4666, 6545}, {4785, 50836}, {5218, 61230}, {6544, 67097}, {6765, 32212}, {7220, 8275}, {9746, 28292}, {10582, 21204}, {14432, 34600}, {16788, 48324}, {23892, 64612}, {30199, 64312}, {31286, 57511}, {48299, 48305}, {48320, 60343}, {66513, 66514}

X(68833) = midpoint of X(i) and X(j) for these {i,j}: {3126, 6161}, {38371, 66995}, {48320, 60343}
X(68833) = reflection of X(45695) in X(1960)
X(68833) = crossdifference of every pair of points on line {244, 672}
X(68833) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 4040, 1027}, {3573, 7437, 68768}

X(68834) = X(100)X(101) INTERCEPT OF X(6)X(661)

Barycentrics    a*(b - c)*(2*a^4 - a^2*b^2 + a*b^3 - 3*a*b^2*c + b^3*c - a^2*c^2 - 3*a*b*c^2 + 2*b^2*c^2 + a*c^3 + b*c^3) : :

X(68834) lies on these lines: {6, 661}, {37, 649}, {100, 101}, {244, 24286}, {572, 57129}, {650, 21858}, {940, 8042}, {1019, 37633}, {1100, 55263}, {2298, 4979}, {4271, 58288}, {4367, 17450}, {4893, 21904}, {33682, 35353}, {36509, 66704}, {38348, 46904}

X(68834) = crosspoint of X(662) and X(2718)
X(68834) = crosssum of X(i) and X(j) for these (i,j): {661, 2802}, {35069, 65854}
X(68834) = crossdifference of every pair of points on line {244, 758}
X(68834) = X(24286)-lineconjugate of X(244)
X(68834) = barycentric product X(1)*X(62323)
X(68834) = barycentric quotient X(62323)/X(75)

X(68835) = X(100)X(101) INTERCEPT OF X(1)X(650)

Barycentrics    a*(b - c)*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3 - 4*a^2*c + 3*a*b*c - b^2*c + 5*a*c^2 - b*c^2 - 2*c^3) : :
X(68835) = 4 X[3126] - X[35355]

X(68835) lies on these lines: {1, 650}, {2, 514}, {9, 513}, {44, 14812}, {45, 23838}, {100, 101}, {165, 3309}, {649, 3730}, {661, 46196}, {663, 41276}, {876, 48162}, {1027, 37675}, {1174, 61238}, {1639, 2826}, {2725, 43079}, {2827, 14427}, {3158, 3251}, {3239, 56937}, {4394, 6161}, {4521, 6554}, {4873, 68101}, {4962, 38371}, {5574, 59979}, {5592, 18231}, {5657, 28292}, {5659, 28473}, {6366, 14425}, {14330, 33994}, {16670, 53535}, {16676, 24457}, {17244, 17494}, {17284, 62552}, {17367, 27115}, {17756, 47828}, {21611, 33933}, {26777, 29569}, {27929, 56697}, {28161, 31325}, {29571, 62635}, {30199, 52026}, {31182, 68146}, {36014, 48387}, {36835, 47777}, {37131, 52228}, {45322, 45666}, {45755, 64343}, {48281, 62216}, {50351, 68794}

X(68835) = reflection of X(45322) in X(45666)
X(68835) = X(1110)-complementary conjugate of X(52873)
X(68835) = crosspoint of X(100) and X(37131)
X(68835) = crosssum of X(513) and X(2246)
X(68835) = crossdifference of every pair of points on line {244, 902}
X(68835) = barycentric product X(522)*X(14151)
X(68835) = barycentric quotient X(14151)/X(664)

X(68836) = X(100)X(101) INTERCEPT OF X(512)X(650)

Barycentrics    a*(b - c)*(a^3 + a^2*b + a*b^2 - b^3 + a^2*c + a*b*c - 2*b^2*c + a*c^2 - 2*b*c^2 - c^3) : :
X(68836) = 3 X[1635] - X[8632], X[21303] + 3 X[47776]

X(68836) lies on these lines: {100, 101}, {512, 650}, {522, 47127}, {649, 830}, {656, 21389}, {659, 24290}, {661, 4063}, {665, 9508}, {812, 21261}, {1919, 35057}, {2483, 57099}, {3250, 14838}, {3700, 29106}, {4151, 68803}, {4435, 4730}, {4763, 26629}, {4813, 48004}, {4976, 29086}, {6590, 23879}, {14288, 17303}, {21303, 47776}, {48269, 59672}, {50336, 60350}, {50355, 66513}, {50512, 66523}

X(68836) = X(i)-complementary conjugate of X(j) for these (i,j): {835, 20542}, {1911, 5515}, {14598, 39016}, {34067, 52782}
X(68836) = crossdifference of every pair of points on line {244, 940}

X(68837) = X(100)X(101) INTERCEPT OF X(522)X(649)

Barycentrics    a*(b - c)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c - a^2*b*c - a*b^2*c + b^3*c - a^2*c^2 - a*b*c^2 + 2*b^2*c^2 + a*c^3 + b*c^3) : :

X(68837) lies on these lines: {1, 21828}, {6, 21894}, {9, 4120}, {57, 4750}, {63, 2786}, {98, 2249}, {100, 101}, {228, 53258}, {522, 649}, {650, 15313}, {652, 48269}, {654, 900}, {657, 47765}, {661, 1021}, {663, 7654}, {798, 64905}, {812, 36038}, {968, 5075}, {1019, 31148}, {1024, 13576}, {1639, 22108}, {2341, 2610}, {2785, 21832}, {3218, 53333}, {3219, 53339}, {3305, 45661}, {3306, 45674}, {3310, 4435}, {3572, 56154}, {4010, 47203}, {4063, 23876}, {4088, 53400}, {4382, 58324}, {4384, 59736}, {4394, 56956}, {4729, 57121}, {4730, 23858}, {5709, 62436}, {6546, 53396}, {7180, 65097}, {9404, 14321}, {16561, 46457}, {19642, 68486}, {21385, 23884}, {21390, 47874}, {22060, 53269}, {22382, 57074}, {23865, 50518}, {24624, 60074}, {24900, 57148}, {28094, 48398}, {28292, 45755}, {33811, 58036}, {37593, 53763}, {47227, 53527}, {47971, 68259}, {53553, 62819}

X(68837) = X(47318)-Ceva conjugate of X(1)
X(68837) = X(53527)-Dao conjugate of X(4707)
X(68837) = crosspoint of X(i) and X(j) for these (i,j): {100, 24624}, {104, 662}, {655, 17097}, {1309, 40395}, {2363, 37140}, {32680, 57710}
X(68837) = crosssum of X(i) and X(j) for these (i,j): {500, 2624}, {513, 2245}, {517, 661}, {654, 2646}, {2292, 2610}, {8677, 18591}
X(68837) = crossdifference of every pair of points on line {244, 942}

X(68838) = X(100)X(101) INTERCEPT OF X(9)X(661)

Barycentrics    a*(b - c)*(a^4 - 2*a^3*b + 2*a*b^3 - b^4 - 2*a^3*c + a^2*b*c + a*b^2*c - b^3*c + a*b*c^2 + 2*a*c^3 - b*c^3 - c^4) : :

X(68838) lies on these lines: {9, 661}, {37, 650}, {71, 649}, {100, 101}, {514, 54357}, {647, 17053}, {1021, 56948}, {2177, 4041}, {2259, 61238}, {4850, 14838}, {4893, 59207}, {9168, 16598}, {16814, 61179}, {23775, 50351}, {26641, 48094}, {27754, 64934}, {32917, 60577}, {48266, 57055}

X(68838) = X(63755)-complementary conjugate of X(21252)
X(68838) = crosspoint of X(662) and X(12030)
X(68838) = crossdifference of every pair of points on line {36, 244}
X(68838) = barycentric product X(522)*X(12739)
X(68838) = barycentric quotient X(12739)/X(664)

X(68839) = X(1)X(30) INTERCEPT OF X(2)X(67598)

Barycentrics    2*a^6 - a^3*b^3 - 2*a^2*b^4 + a*b^5 + a^3*b^2*c - b^5*c + a^3*b*c^2 + 2*a^2*b^2*c^2 - a*b^3*c^2 - a^3*c^3 - a*b^2*c^3 + 2*b^3*c^3 - 2*a^2*c^4 + a*c^5 - b*c^5 : :
X(68839) = X[5189] - 3 X[68378], 3 X[7426] - 2 X[16305], 3 X[7426] - X[50144], X[16304] - 3 X[37904], X[16322] + 3 X[47312], 3 X[16323] - 4 X[47316], 3 X[37909] - X[50145]

X(68839) lies on these lines: {1, 30}, {2, 67598}, {23, 385}, {468, 29828}, {858, 16332}, {1495, 14985}, {3109, 51693}, {5189, 68378}, {6741, 32223}, {6742, 15107}, {7426, 16305}, {10989, 29823}, {16304, 37904}, {16309, 37897}, {16322, 47312}, {16323, 47316}, {29301, 65524}, {29826, 47097}, {37909, 50145}, {62490, 68282}

X(68839) = midpoint of X(6742) and X(15107)
X(68839) = reflection of X(i) in X(j) for these {i,j}: {858, 16332}, {3109, 51693}, {6741, 32223}, {14985, 1495}, {16309, 37897}, {50144, 16305}, {67596, 16272}, {67601, 468}
X(68839) = anticomplement of X(67598)
X(68839) = reflection of X(67596) in the Orthic axis
X(68839) = incircle-inverse of X(63979)
X(68839) = crossdifference of every pair of points on line {39, 9404}
X(68839) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7426, 50144, 16305}, {51873, 51874, 63979}

X(68840) = X(1)X(30) INTERCEPT OF X(514)X(661)

Barycentrics    a^4*b + a^3*b^2 - a*b^4 - b^5 + a^4*c - 2*a^3*b*c - a^2*b^2*c + a*b^3*c + b^4*c + a^3*c^2 - a^2*b*c^2 + a*b*c^3 - a*c^4 + b*c^4 - c^5 : :

X(68840) lies on these lines: {1, 30}, {150, 20016}, {226, 7278}, {514, 661}, {1375, 5437}, {1565, 20367}, {1790, 8025}, {3007, 29069}, {3674, 55090}, {4364, 41007}, {5697, 36482}, {6999, 38941}, {7202, 16581}, {7208, 53590}, {11349, 67625}, {17181, 29576}, {19867, 37613}, {24618, 29590}, {29659, 37165}, {37508, 41808}, {41003, 67984}, {47727, 62494}

X(68840) = crossdifference of every pair of points on line {31, 9404}

X(68841) = X(1)X(30) INTERCEPT OF X(44)X(513)

Barycentrics    a*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4 + a^4*c + 2*a^3*b*c - a*b^3*c - 2*b^4*c - a^3*c^2 + 2*b^3*c^2 - a^2*c^3 - a*b*c^3 + 2*b^2*c^3 + a*c^4 - 2*b*c^4) : :
X(68841) = 2 X[1] - 3 X[1464]

X(68841) lies on these lines: {1, 30}, {44, 513}, {846, 41695}, {855, 37605}, {859, 4278}, {991, 61716}, {1044, 1837}, {1406, 37411}, {1443, 4872}, {1474, 52890}, {1742, 17718}, {1756, 15447}, {1770, 2594}, {2947, 41706}, {3120, 68732}, {3240, 3474}, {3245, 61638}, {3579, 64539}, {4003, 11031}, {4306, 12953}, {4316, 34586}, {4338, 37698}, {4420, 33066}, {8544, 46553}, {8758, 15726}, {10123, 63366}, {17605, 22053}, {17768, 61220}, {18593, 53524}, {18607, 41871}, {19546, 34583}, {20470, 38389}, {29349, 68760}, {36002, 68762}, {37558, 65631}, {37567, 41329}, {41341, 64013}, {56198, 63206}, {65632, 68758}

X(68841) = reflection of X(53524) in X(18593)
X(68841) = crosspoint of X(7) and X(54497)
X(68841) = crossdifference of every pair of points on line {1, 9404}
X(68841) = X(i)-line conjugate of X(j) for these (i,j): {30, 1}, {44, 9404}
X(68841) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {79, 500, 11553}, {1155, 2635, 45885}, {2635, 3000, 1155}, {11544, 48927, 1}

X(68842) = X(1)X(30) INTERCEPT OF X(320)X(350)

Barycentrics    a^5*b + a^4*b^2 - a^2*b^4 - a*b^5 + a^5*c - a^3*b^2*c + a*b^4*c - b^5*c + a^4*c^2 - a^3*b*c^2 + 2*b^3*c^3 - a^2*c^4 + a*b*c^4 - a*c^5 - b*c^5 : :
X(68842) = 3 X[859] - 4 X[1125]

X(68842) lies on these lines: {1, 30}, {8, 64580}, {11, 1756}, {256, 17721}, {320, 350}, {516, 15626}, {851, 5880}, {855, 68608}, {859, 1125}, {1848, 52890}, {3120, 62740}, {4679, 24697}, {4683, 24703}, {4703, 50605}, {5928, 10381}, {7359, 39690}, {9554, 17595}, {11376, 30362}, {15314, 56077}, {17751, 33066}, {20292, 29822}, {22097, 30970}, {26013, 64904}, {29069, 45916}, {29301, 53524}, {48902, 61716}

X(68842) = crossdifference of every pair of points on line {213, 9404}

X(68843) = X(1)X(30) INTERCEPT OF X(43)X(46)

Barycentrics    a*(a^5*b - 2*a^3*b^3 + a*b^5 + a^5*c + 2*a^4*b*c + a^3*b^2*c - a^2*b^3*c - 2*a*b^4*c - b^5*c + a^3*b*c^2 + a*b^3*c^2 - 2*a^3*c^3 - a^2*b*c^3 + a*b^2*c^3 + 2*b^3*c^3 - 2*a*b*c^4 + a*c^5 - b*c^5) : :

X(68843) lies on these lines: {1, 30}, {36, 238}, {43, 46}, {73, 1770}, {226, 4337}, {256, 7284}, {484, 4551}, {517, 53537}, {758, 61220}, {855, 30362}, {997, 4683}, {1042, 10572}, {1079, 54295}, {1406, 6985}, {1457, 21578}, {1458, 30384}, {1478, 30116}, {1479, 4306}, {1725, 18593}, {1727, 1758}, {1737, 2635}, {1742, 59337}, {1777, 36152}, {2617, 10081}, {2771, 68765}, {3000, 22350}, {3074, 59321}, {3216, 4650}, {3336, 37732}, {3465, 5018}, {3583, 68758}, {3585, 37558}, {3720, 4303}, {4292, 63366}, {4295, 17018}, {4299, 10571}, {4300, 13407}, {4316, 68766}, {4334, 51816}, {4338, 37529}, {4351, 45272}, {4511, 17491}, {5131, 6127}, {5396, 11246}, {6149, 61225}, {6180, 40292}, {7163, 49553}, {9363, 65133}, {11263, 54356}, {11551, 14547}, {13744, 21842}, {15326, 34586}, {17194, 26725}, {23821, 44675}, {25502, 37370}, {26102, 46521}, {37694, 58887}, {38945, 38952}, {47402, 53406}, {48293, 62492}, {50317, 61716}, {56824, 60786}, {59317, 64057}, {62359, 68762}

X(68843) = reflection of X(i) in X(j) for these {i,j}: {1, 1464}, {1725, 18593}
X(68843) = X(74)-Ceva conjugate of X(1)
X(68843) = X(14206)-Dao conjugate of X(3260)
X(68843) = crosspoint of X(86) and X(2349)
X(68843) = crosssum of X(42) and X(2173)
X(68843) = crossdifference of every pair of points on line {37, 9404}
X(68843) = barycentric product X(1)*X(18668)
X(68843) = barycentric quotient X(18668)/X(75)
X(68843) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 16118, 52524}, {500, 3649, 1}, {1044, 1745, 46}, {5453, 11544, 11553}, {5453, 11553, 1}, {7100, 38336, 1}, {48903, 63295, 1}, {50528, 56848, 1}, {63295, 67973, 48903}

X(68844) = X(1)X(30) INTERCEPT OF X(11)X(244)

Barycentrics    (a - b - c)*(b - c)^2*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2 - c^3) : :

X(68844) lies on these lines: {1, 30}, {5, 18302}, {11, 244}, {12, 5492}, {55, 13589}, {56, 14127}, {65, 51889}, {115, 67293}, {513, 42759}, {846, 2607}, {1109, 14101}, {1365, 3024}, {1718, 16128}, {1770, 33649}, {2166, 10264}, {3025, 3326}, {3925, 24433}, {4415, 4995}, {6174, 26611}, {6354, 60718}, {8143, 14526}, {8614, 16116}, {10017, 13141}, {10950, 18340}, {11238, 33146}, {11246, 61732}, {11813, 52537}, {12079, 66284}, {12515, 56417}, {12679, 63669}, {15325, 46820}, {15950, 60687}, {17637, 36250}, {21105, 23763}, {24715, 52371}, {30384, 56421}, {31522, 65856}, {40663, 56416}, {52639, 52680}, {58322, 66484}, {62223, 68770}, {64935, 66289}

X(68844) = reflection of X(867) in X(45260)
X(68844) = X(i)-Ceva conjugate of X(j) for these (i,j): {1443, 10015}, {2166, 523}, {17484, 68388}, {21907, 650}
X(68844) = X(i)-isoconjugate of X(j) for these (i,j): {59, 3065}, {100, 34921}, {101, 68387}, {2149, 21739}, {4564, 19302}
X(68844) = X(i)-Dao conjugate of X(j) for these (i,j): {650, 21739}, {1015, 68387}, {1577, 40716}, {6615, 3065}, {8054, 34921}, {59837, 40663}, {68388, 323}
X(68844) = crosspoint of X(i) and X(j) for these (i,j): {7, 60074}, {94, 693}, {484, 35055}, {522, 11604}, {7372, 24624}
X(68844) = crosssum of X(i) and X(j) for these (i,j): {50, 692}, {55, 1983}, {100, 48698}, {109, 5172}, {2245, 21784}
X(68844) = crossdifference of every pair of points on line {101, 9404}
X(68844) = barycentric product X(i)*X(j) for these {i,j}: {11, 17484}, {484, 4858}, {693, 68388}, {1577, 35055}, {2170, 17791}, {3261, 42657}, {4391, 59837}, {19297, 34387}, {21044, 56935}, {21739, 31522}
X(68844) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 21739}, {484, 4564}, {513, 68387}, {649, 34921}, {2170, 3065}, {3271, 19302}, {4858, 40716}, {14158, 35049}, {17484, 4998}, {17791, 67038}, {19297, 59}, {21132, 60486}, {21864, 65573}, {23071, 44717}, {31522, 17484}, {35055, 662}, {42657, 101}, {56935, 4620}, {59837, 651}, {68388, 100}
X(68844) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3120, 53524, 11}, {35015, 53525, 11}

X(68845) = X(1)X(30) INTERCEPT OF X(11)X(6023)

Barycentrics    (b + c)*(-2*a^5 - a^4*b + 2*a*b^4 + b^5 - a^4*c + 2*a^3*b*c + a^2*b^2*c - a*b^3*c - b^4*c + a^2*b*c^2 - 2*a*b^2*c^2 - a*b*c^3 + 2*a*c^4 - b*c^4 + c^5) : :
X(68845) = X[41869] - 3 X[51883], 2 X[10] - 3 X[30447], 3 X[1325] - 5 X[3616], 7 X[3624] - 6 X[44898], 4 X[5087] - 3 X[5520], X[5196] - 3 X[68378], X[6361] - 3 X[36001]

X(68845) lies on these lines: {1, 30}, {10, 30447}, {11, 6023}, {19, 468}, {23, 1621}, {55, 37959}, {56, 37960}, {149, 4442}, {517, 16003}, {523, 661}, {858, 2886}, {1001, 7469}, {1155, 4934}, {1325, 3616}, {2651, 4831}, {3027, 31522}, {3624, 44898}, {3932, 27562}, {5087, 5520}, {5196, 25557}, {6057, 21081}, {6361, 36001}, {9639, 34666}, {13369, 68386}, {16597, 60423}, {33077, 50144}, {42422, 67293}, {51409, 62500}

X(68845) = incircle-inverse of X(4854)
X(68845) = crossdifference of every pair of points on line {58, 9404}
X(68845) = {X(51873),X(51874)}-harmonic conjugate of X(4854)

X(68846) = X(1)X(30) INTERCEPT OF X(10)X(37)

Barycentrics    (b + c)*(4*a^3 + 5*a^2*b + 2*a*b^2 + b^3 + 5*a^2*c + 2*a*b*c - b^2*c + 2*a*c^2 - b*c^2 + c^3) : :
X(68846) = 2 X[10] - 3 X[4205], X[145] + 3 X[26117], X[145] - 3 X[41813], 3 X[1010] - 5 X[3616]

X(68846) lies on these lines: {1, 30}, {5, 17592}, {10, 37}, {81, 3648}, {145, 2895}, {442, 1962}, {581, 18243}, {942, 4890}, {1010, 3616}, {1284, 3295}, {1479, 20182}, {1770, 37595}, {2292, 64167}, {3017, 18253}, {3178, 16052}, {3624, 24789}, {3635, 38456}, {3647, 61661}, {3672, 41804}, {3712, 25441}, {3841, 56221}, {3874, 17246}, {4021, 29040}, {4038, 24470}, {4187, 46904}, {4415, 59301}, {4425, 41014}, {4658, 17768}, {4857, 17726}, {5015, 17319}, {5045, 39793}, {5051, 27558}, {5442, 37634}, {5698, 54358}, {5711, 6361}, {5725, 18492}, {6051, 24564}, {6675, 33135}, {6693, 59592}, {6701, 17056}, {7359, 66101}, {8143, 22798}, {8728, 21926}, {9791, 56018}, {9955, 24210}, {10386, 17716}, {13728, 32915}, {13745, 27368}, {15569, 23537}, {17301, 64675}, {17514, 21020}, {17593, 34753}, {17673, 31308}, {24697, 49718}, {30143, 48846}, {33100, 63285}, {33132, 50205}, {33152, 63282}, {33697, 67969}, {35466, 63286}, {48848, 49488}, {49452, 50042}, {52367, 62851}, {56949, 58399}, {62831, 64172}, {63366, 63401}

X(68846) = midpoint of X(i) and X(j) for these {i,j}: {1, 66644}, {26117, 41813}
X(68846) = crosspoint of X(7) and X(60203)
X(68846) = crossdifference of every pair of points on line {3733, 9404}
X(68846) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 79, 37631}, {1, 4854, 63997}, {1, 10404, 48823}, {1, 24851, 49743}, {1, 33097, 65460}, {1, 33154, 6147}, {1, 41864, 48824}, {1, 50066, 10404}, {1, 66642, 64158}, {3649, 52382, 55010}, {11553, 52382, 3649}, {36250, 58380, 17056}

X(68847) = X(1)X(30) INTERCEPT OF X(10)X(82)

Barycentrics    3*a^4 + 2*a^3*b + a^2*b^2 + a*b^3 - b^4 + 2*a^3*c - a^2*b*c + 2*a*b^2*c + a^2*c^2 + 2*a*b*c^2 + 2*b^2*c^2 + a*c^3 - c^4 : :
X(68847) = 2 X[10] - 3 X[13740], 5 X[3616] - 3 X[4201], 7 X[3624] - 6 X[56734]

X(68847) lies on these lines: {1, 30}, {10, 82}, {35, 17722}, {38, 3648}, {145, 17165}, {149, 62847}, {986, 6361}, {987, 33120}, {1330, 49473}, {1479, 17716}, {1621, 27577}, {1770, 17598}, {3434, 16478}, {3616, 4201}, {3624, 37552}, {3813, 5429}, {4234, 49613}, {4307, 66666}, {4309, 17592}, {4894, 32780}, {5266, 9955}, {5716, 66674}, {5903, 66645}, {6701, 33130}, {17469, 52367}, {17721, 37603}, {17726, 63273}, {18492, 37716}, {22793, 33152}, {24161, 33104}, {24850, 29840}, {28646, 64016}, {50301, 64675}, {50635, 67964}, {66643, 66646}

X(68847) = crossdifference of every pair of points on line {9404, 21123}
X(68847) = {X(12699),X(48824)}-harmonic conjugate of X(1)

X(68848) = X(1)X(2) INTERCEPT OF X(6)X(17781)

Barycentrics    4*a^3 + 5*a^2*b + 2*a*b^2 + b^3 + 5*a^2*c + 2*a*b*c - b^2*c + 2*a*c^2 - b*c^2 + c^3 : :

X(68848) lies on these lines: {1, 2}, {6, 17781}, {72, 48861}, {81, 553}, {142, 62801}, {226, 54929}, {376, 62809}, {377, 48828}, {597, 3175}, {1051, 33152}, {1100, 5249}, {1104, 49739}, {1255, 6666}, {1386, 3058}, {1449, 4654}, {1453, 31156}, {2895, 4856}, {3219, 4021}, {3303, 16439}, {3304, 16438}, {3555, 48820}, {3578, 4357}, {3618, 42032}, {3663, 37685}, {3664, 33150}, {3666, 16585}, {3745, 49732}, {3755, 49719}, {3782, 16666}, {3875, 50043}, {3879, 32774}, {3969, 4464}, {4000, 62808}, {4001, 17302}, {4102, 17289}, {4349, 33131}, {4356, 17127}, {4360, 5294}, {4425, 4991}, {4641, 17395}, {4656, 63074}, {4667, 33146}, {4719, 5298}, {4852, 50052}, {4989, 5284}, {5224, 42030}, {5295, 50323}, {5325, 28606}, {5717, 6175}, {5814, 51593}, {5853, 62855}, {5905, 16667}, {6173, 15474}, {8025, 24199}, {10385, 62834}, {14996, 24177}, {16884, 24789}, {17045, 49730}, {17126, 50808}, {17184, 50256}, {17366, 37595}, {17380, 54311}, {17396, 37652}, {17525, 64166}, {17592, 61647}, {19723, 41312}, {19738, 50102}, {19796, 46922}, {20086, 53598}, {20182, 54357}, {20963, 50178}, {21454, 62781}, {23536, 48825}, {23537, 49744}, {23681, 66549}, {24208, 30690}, {25417, 27186}, {28194, 57280}, {30606, 37756}, {32776, 51196}, {32911, 52405}, {32915, 38049}, {32925, 59408}, {33100, 64017}, {33107, 50802}, {33129, 63343}, {34611, 50294}, {39980, 56050}, {41311, 49724}, {42044, 50115}, {42051, 50112}, {48842, 50070}, {48870, 50066}, {50048, 50120}, {50053, 64184}, {50063, 50124}, {50071, 50127}, {50093, 63060}, {50106, 50109}, {51171, 56082}, {54358, 60932}, {56037, 56228}, {60938, 65028}, {62240, 63039}, {62818, 63067}

X(68848) = X(i)-complementary conjugate of X(j) for these (i,j): {6186, 62586}, {34819, 3647}, {58954, 513}
X(68848) = crosspoint of X(1434) and X(30598)
X(68848) = crosssum of X(1334) and X(61358)
X(68848) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 16834, 50306}, {6, 50068, 17781}, {553, 52374, 62779}, {1051, 33152, 61652}, {29833, 45222, 3687}, {48842, 50070, 57287}, {48861, 50069, 72}, {48870, 50066, 64002}

X(68849) = X(6)X(30) INTERCEPT OF X(1)X(75)

Barycentrics    (a + b)*(a + c)*(a^4 + 4*a^2*b^2 + b^4 + 6*a^2*b*c + 4*a^2*c^2 - 2*b^2*c^2 + c^4) : :

X(68849) lies on these lines: {1, 75}, {6, 30}, {58, 3755}, {81, 49719}, {238, 66644}, {333, 4085}, {3286, 35239}, {3946, 4653}, {4205, 17259}, {4229, 59243}, {4483, 4780}, {4649, 57287}, {4658, 5853}, {5327, 29040}, {7474, 33128}, {16704, 63140}, {17349, 26117}, {18653, 44119}, {38456, 49685}, {41610, 50282}

X(68849) = crossdifference of every pair of points on line {798, 8675}

X(68850) = X(6)X(30) INTERCEPT OF X(44)X(513)

Barycentrics    a*(b + c)*(a^6 - 2*a^4*b^2 + a^2*b^4 + 5*a^4*b*c - 4*a^2*b^3*c - b^5*c - 2*a^4*c^2 + 4*a^2*b^2*c^2 - 4*a^2*b*c^3 + 2*b^3*c^3 + a^2*c^4 - b*c^5) : :

X(68850) lies on these lines: {6, 30}, {37, 1464}, {44, 513}, {855, 4268}, {859, 36743}, {1211, 55911}, {8818, 40401}, {14956, 63066}, {16581, 51645}, {24512, 46521}

X(68850) = X(53907)-complementary conjugate of X(3741)
X(68850) = X(43660)-Ceva conjugate of X(55)
X(68850) = crossdifference of every pair of points on line {1, 8675}
X(68850) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2238, 3013, 45886}, {2238, 3330, 3013}, {2245, 3013, 2238}, {2245, 3330, 45886}

X(68851) = X(6)X(30) INTERCEPT OF X(187)X(237)

Barycentrics    a^2*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^6*c^2 + 6*a^4*b^2*c^2 - 3*a^2*b^4*c^2 - 2*b^6*c^2 - 2*a^4*c^4 - 3*a^2*b^2*c^4 + 4*b^4*c^4 + a^2*c^6 - 2*b^2*c^6) : :

X(68851) lies on these lines: {6, 30}, {15, 23023}, {16, 23017}, {23, 67356}, {115, 1533}, {187, 237}, {323, 14712}, {394, 47102}, {574, 47620}, {1501, 19220}, {1971, 9934}, {1974, 5033}, {2387, 6195}, {2421, 8598}, {3003, 14915}, {3289, 6781}, {5063, 8717}, {5210, 44221}, {9862, 12112}, {10295, 61209}, {11456, 36998}, {14537, 20965}, {14567, 32761}, {14907, 15066}, {15544, 20977}, {21531, 31415}, {21843, 44215}, {32456, 56437}, {32681, 46233}, {43620, 44227}, {50678, 53418}, {51735, 56957}, {59208, 62203}, {67630, 68706}

X(68851) = reflection of X(56957) in X(51735)
X(68851) = crossdifference of every pair of points on line {2, 8675}
X(68851) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {187, 3016, 3231}, {187, 3331, 45938}, {3016, 3231, 45938}, {3231, 3331, 3016}

X(68852) = X(1)X(2) INTERCEPT OF X(3)X(61647)

Barycentrics    2*a^4 + 5*a^3*b + 3*a^2*b^2 + a*b^3 + b^4 + 5*a^3*c + 2*a^2*b*c - a*b^2*c + 3*a^2*c^2 - a*b*c^2 - 2*b^2*c^2 + a*c^3 + c^4 : :

X(68852) lies on these lines: {1, 2}, {3, 61647}, {6, 10404}, {31, 31730}, {58, 18653}, {65, 17366}, {79, 1203}, {81, 24178}, {113, 5515}, {238, 66644}, {377, 16475}, {443, 62845}, {946, 33128}, {950, 4989}, {964, 38049}, {988, 24597}, {1072, 36754}, {1104, 10543}, {1471, 5930}, {1738, 57280}, {1829, 1835}, {2082, 2260}, {2292, 3946}, {2308, 4292}, {3003, 41015}, {3296, 62819}, {3666, 18253}, {3710, 32921}, {3755, 3915}, {3782, 28645}, {3914, 12699}, {4000, 54421}, {4101, 26128}, {4202, 5847}, {4719, 35466}, {5300, 49684}, {5319, 61651}, {7290, 41864}, {8192, 20470}, {11512, 63078}, {13161, 32911}, {13407, 61652}, {13624, 63449}, {16468, 64002}, {16478, 57287}, {16948, 66692}, {17380, 31359}, {17469, 63146}, {17679, 51005}, {18480, 64172}, {19785, 54386}, {21620, 61358}, {32636, 61661}, {33135, 41012}, {33143, 67850}, {33178, 65673}, {56311, 63051}, {57284, 62847}

X(68852) = X(54586)-complementary conjugate of X(21245)
X(68852) = crosspoint of X(86) and X(65028)
X(68852) = {X(1203),X(23537)}-harmonic conjugate of X(41011)

X(68853) = X(6)X(30) INTERCEPT OF X(10)X(37)

Barycentrics    (b + c)*(2*a^4 + 5*a^3*b + 3*a^2*b^2 + a*b^3 + b^4 + 5*a^3*c + 2*a^2*b*c - a*b^2*c + 3*a^2*c^2 - a*b*c^2 - 2*b^2*c^2 + a*c^3 + c^4) : :

X(68853) lies on these lines: {6, 30}, {9, 66644}, {10, 37}, {391, 26117}, {1010, 19766}, {1100, 17647}, {1211, 3875}, {1901, 20970}, {3017, 37508}, {3247, 32865}, {3936, 19785}, {3946, 17056}, {4270, 5254}, {4272, 8818}, {4483, 6703}, {4854, 21810}, {4856, 38456}, {16052, 34528}, {16777, 31419}, {20966, 23668}, {32921, 41014}, {40952, 40965}, {49734, 56903}, {63051, 68708}

X(68853) = X(i)-complementary conjugate of X(j) for these (i,j): {1402, 51572}, {61375, 960}, {65028, 21240}, {68185, 42327}
X(68853) = crosspoint of X(10) and X(54586)
X(68853) = crossdifference of every pair of points on line {3733, 8675}

X(68854) = X(6)X(30) INTERCEPT OF X(2)X(37)

Barycentrics    a^5 + a^4*b + 4*a^3*b^2 + 4*a^2*b^3 + a*b^4 + b^5 + a^4*c + 6*a^3*b*c + 4*a^2*b^2*c + b^4*c + 4*a^3*c^2 + 4*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 + 4*a^2*c^3 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5 : :

X(68854) lies on these lines: {2, 37}, {4, 54693}, {6, 30}, {9, 50066}, {44, 50065}, {45, 50067}, {376, 1333}, {381, 53417}, {386, 66106}, {387, 5165}, {524, 50057}, {528, 54358}, {579, 3017}, {594, 50041}, {597, 50060}, {1100, 50070}, {1213, 51593}, {1714, 18253}, {1723, 50080}, {2092, 5309}, {2220, 63006}, {4263, 39593}, {4277, 5286}, {4286, 5292}, {5035, 7738}, {5227, 48818}, {5306, 36744}, {5319, 54409}, {5747, 66051}, {5749, 50045}, {5750, 50053}, {10056, 56926}, {16777, 50069}, {17275, 50051}, {17330, 50056}, {17369, 50044}, {17398, 51590}, {19766, 50054}, {32833, 56023}, {37631, 52025}, {37654, 50055}, {40138, 52955}, {50042, 61321}, {50046, 50082}, {50072, 50113}, {51591, 63055}

X(68854) = reflection of X(50060) in X(597)
X(68854) = crossdifference of every pair of points on line {667, 8675}
X(68854) = {X({}),X(1)}-harmonic conjugate of X({}[[1]][[3]])

X(68855) = X(2)X(37) INTERCEPT OF X(1)X(7739)

Barycentrics    a^4 + 4*a^2*b^2 + b^4 + 6*a^2*b*c + 4*a^2*c^2 - 2*b^2*c^2 + c^4 : :
X(68855) = X[9598] + 2 X[54416]

X(68855) lies on these lines: {1, 7739}, {2, 37}, {6, 3058}, {30, 9598}, {39, 10072}, {42, 34288}, {45, 4854}, {55, 5306}, {165, 61688}, {172, 376}, {213, 48857}, {218, 11113}, {220, 48842}, {381, 9596}, {390, 5332}, {497, 63024}, {549, 31448}, {1250, 61318}, {1478, 11648}, {1479, 7753}, {1500, 5309}, {1656, 31462}, {1914, 10385}, {2325, 25453}, {2548, 65140}, {2549, 16785}, {3017, 3730}, {3027, 6034}, {3304, 9607}, {3545, 31402}, {3584, 3767}, {3746, 5319}, {3815, 68688}, {3943, 33171}, {4029, 29642}, {4294, 7296}, {4309, 5007}, {4387, 17369}, {4426, 31156}, {4873, 32783}, {4995, 31477}, {5013, 5298}, {5054, 31461}, {5055, 31460}, {5254, 11237}, {5275, 49732}, {5276, 49719}, {5304, 10987}, {5434, 9597}, {5657, 61741}, {6057, 61321}, {7755, 31452}, {7788, 26590}, {9300, 9599}, {9650, 39563}, {9664, 14537}, {10638, 61317}, {10895, 63543}, {11354, 48830}, {15170, 16502}, {17732, 48870}, {17742, 50066}, {18362, 31476}, {25264, 32833}, {28194, 54382}, {31459, 32787}, {41325, 60697}, {49749, 50168}, {62210, 66719}

X(68855) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1500, 5309, 10056}, {9300, 11238, 9599}, {10385, 63006, 1914}

X(68856) = X(2)X(7) INTERCEPT OF X(11)X(44)

Barycentrics    2*a^4 - 5*a^3*b + 3*a^2*b^2 - a*b^3 + b^4 - 5*a^3*c + 2*a^2*b*c + a*b^2*c + 3*a^2*c^2 + a*b*c^2 - 2*b^2*c^2 - a*c^3 + c^4 : :

X(68856) lies on these lines: {2, 7}, {11, 44}, {37, 37703}, {41, 4304}, {45, 17718}, {55, 61651}, {80, 294}, {101, 21578}, {218, 5722}, {220, 5252}, {516, 2246}, {650, 900}, {1146, 36920}, {1155, 51406}, {1212, 15950}, {1317, 6603}, {1323, 9502}, {1387, 43065}, {1826, 5101}, {1830, 8756}, {1846, 60431}, {2082, 30305}, {2183, 61160}, {2280, 30331}, {2325, 3952}, {2348, 17747}, {3217, 8804}, {3259, 67496}, {3711, 17281}, {3977, 4070}, {4021, 21840}, {4059, 58458}, {4144, 49756}, {4315, 9310}, {4319, 4878}, {4465, 49775}, {4530, 28234}, {4867, 10699}, {4873, 53661}, {5519, 35111}, {5719, 16601}, {6326, 18461}, {6745, 14439}, {16572, 37704}, {17369, 61686}, {17464, 34587}, {21241, 63978}, {21801, 38375}, {31397, 61706}, {43038, 44664}, {54377, 64747}, {56744, 59316}

X(68856) = crossdifference of every pair of points on line {663, 999}
X(68856) = {X(672),X(40869)}-harmonic conjugate of X(68797)

X(68857) = X(1)X(99) INTERCEPT OF X(23)X(385)

Barycentrics    (b + c)*(a^6 - a^4*b^2 - 2*a^3*b^3 + a^4*b*c + 2*a^3*b^2*c - a^2*b^3*c - a^4*c^2 + 2*a^3*b*c^2 + a^2*b^2*c^2 - 2*a^3*c^3 - a^2*b*c^3 + b^3*c^3) : :

X(68857) lies on these lines: {1, 99}, {23, 385}, {111, 26227}, {545, 49749}, {740, 58863}, {3570, 5147}, {3747, 42046}, {9263, 23861}, {18099, 51906}, {20331, 68153}, {29823, 31128}

X(68857) = reflection of X(3570) in X(5147)
X(68857) = crossdifference of every pair of points on line {39, 46390}

X(68858) = X(1)X(99) INTERCEPT OF X(514)X(661)

Barycentrics    a^4*b^2 + a^3*b^3 - a^2*b^4 - a*b^5 - a^3*b^2*c + a^4*c^2 - a^3*b*c^2 - 2*a^2*b^2*c^2 + a*b^3*c^2 + b^4*c^2 + a^3*c^3 + a*b^2*c^3 - a^2*c^4 + b^2*c^4 - a*c^5 : :

X(68858) lies on these lines: {1, 99}, {304, 20951}, {514, 661}, {1930, 17886}, {2611, 4576}, {3061, 25682}, {4563, 21381}, {4934, 50567}, {17023, 24636}, {17284, 30856}, {29633, 40533}, {29960, 30015}, {30038, 30088}

X(68858) = crossdifference of every pair of points on line {31, 46390}

X(68859) = X(1)X(99) INTERCEPT OF X(75)X(2611)

Barycentrics    (b + c)*(-(a^4*b^2) - a^3*b^3 + a^2*b^4 + a*b^5 + a^4*b*c + a^3*b^2*c - a^2*b^3*c - a*b^4*c - a^4*c^2 + a^3*b*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - a^3*c^3 - a^2*b*c^3 + b^3*c^3 + a^2*c^4 - a*b*c^4 - b^2*c^4 + a*c^5) : :

X(68859) lies on these lines: {1, 99}, {75, 2611}, {325, 523}, {545, 37631}, {1494, 54986}, {3708, 33946}, {4736, 61187}, {7664, 26230}, {14360, 29832}, {29857, 30786}

X(68859) = isotomic conjugate of X(53920)
X(68859) = X(31)-isoconjugate of X(53920)
X(68859) = X(2)-Dao conjugate of X(53920)
X(68859) = crossdifference of every pair of points on line {32, 46390}
X(68859) = barycentric quotient X(2)/X(53920)
X(68859) = {X(22339),X(22340)}-harmonic conjugate of X(35544)

X(68860) = X(1)X(99) INTERCEPT OF X(44)X(513)

Barycentrics    a*(-(a^3*b^3) + 2*a^4*b*c + a^3*b^2*c - 2*a^2*b^3*c - a*b^4*c + a^3*b*c^2 + a*b^3*c^2 - a^3*c^3 - 2*a^2*b*c^3 + a*b^2*c^3 + 2*b^3*c^3 - a*b*c^4) : :

X(68860) lies on these lines: {1, 99}, {44, 513}, {3571, 37128}, {4760, 53541}, {6015, 35445}, {7207, 46369}, {9421, 65250}, {9780, 26072}, {16741, 57040}, {24714, 24722}, {27627, 27669}, {30950, 30996}

X(68860) = crossdifference of every pair of points on line {1, 46390}
X(68860) = X(i)-line conjugate of X(j) for these (i,j): {44, 46390}, {99, 1}
X(68860) = {X(99),X(5539)}-harmonic conjugate of X(4128)

X(68861) = X(1)X(99) INTERCEPT OF X(187)X(237)

Barycentrics    a^2*(b + c)*(-(a^3*b^3) + a^4*b*c + a^3*b^2*c - a^2*b^3*c + a^3*b*c^2 - a^3*c^3 - a^2*b*c^3 + b^3*c^3) : :

X(68861) lies on these lines: {1, 99}, {81, 62550}, {100, 4117}, {187, 237}, {213, 2107}, {729, 28841}, {1911, 4094}, {2176, 9431}, {3099, 4093}, {3511, 37590}, {4154, 18794}, {5255, 8298}, {9468, 68871}, {16369, 52894}, {16971, 52067}, {20668, 45216}, {23861, 52127}, {39337, 62530}

X(68861) = crossdifference of every pair of points on line {2, 46390}
X(68861) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5539, 18827}, {18773, 18774, 3747}

X(68862) = X(1)X(99) INTERCEPT OF X(239)X(514)

Barycentrics    (a + b)*(a + c)*(a^3*b - a^2*b^2 - a*b^3 + a^3*c - a^2*c^2 + 2*b^2*c^2 - a*c^3) : :

X(68862) lies on these lines: {1, 99}, {86, 4432}, {194, 20371}, {239, 514}, {274, 1111}, {759, 55945}, {799, 1054}, {846, 873}, {1018, 30669}, {1125, 4368}, {1434, 39775}, {1757, 2669}, {2108, 40017}, {4384, 24617}, {4418, 61407}, {4427, 61403}, {4495, 30938}, {5209, 40874}, {7170, 35040}, {8033, 17596}, {16741, 32845}, {16826, 31059}, {17210, 19856}, {17731, 50016}, {17738, 37128}, {21220, 21383}, {30941, 49764}, {32857, 51370}, {39367, 68450}, {39925, 39950}

X(68862) = X(213)-isoconjugate of X(35173)
X(68862) = X(i)-Dao conjugate of X(j) for these (i,j): {6626, 35173}, {35127, 10}
X(68862) = crossdifference of every pair of points on line {42, 46390}
X(68862) = barycentric quotient X(86)/X(35173)
X(68862) = {X(99),X(18827)}-harmonic conjugate of X(1)

X(68863) = X(1)X(99) INTERCEPT OF X(86)X(244)

Barycentrics    (a + b)*(a + c)*(a^3*b^2 - a*b^4 - a^2*b^2*c + a^3*c^2 - a^2*b*c^2 + b^3*c^2 + b^2*c^3 - a*c^4) : :

X(68863) lies on these lines: {1, 99}, {75, 4576}, {86, 244}, {261, 57060}, {314, 38484}, {319, 61174}, {320, 350}, {561, 670}, {2802, 55243}, {3666, 4760}, {4459, 21334}, {4594, 27922}, {17140, 21295}, {17731, 17763}, {17793, 30966}, {18205, 53681}, {21221, 32863}, {24413, 46159}, {30942, 30992}

X(68863) = X(65941)-Dao conjugate of X(37)
X(68863) = crossdifference of every pair of points on line {213, 46390}
X(68863) = barycentric product X(i)*X(j) for these {i,j}: {320, 46800}, {1921, 68235}
X(68863) = barycentric quotient X(i)/X(j) for these {i,j}: {46800, 80}, {68235, 292}

X(68864) = X(1)X(99) INTERCEPT OF X(36)X(238)

Barycentrics    a*(a + b)*(a + c)*(-(a^2*b^3) + 2*a^3*b*c - a*b^3*c + b^3*c^2 - a^2*c^3 - a*b*c^3 + b^2*c^3) : :

X(68864) lies on these lines: {1, 99}, {36, 238}, {58, 2665}, {291, 3110}, {662, 9359}, {757, 1045}, {849, 51903}, {978, 27665}, {1018, 68873}, {1414, 52161}, {1931, 2664}, {2667, 30593}, {3216, 9509}, {3624, 25530}, {3783, 6629}, {17962, 39949}, {18268, 24578}, {18794, 37128}

X(68864) = X(12031)-Ceva conjugate of X(1)
X(68864) = X(42)-isoconjugate of X(35166)
X(68864) = X(i)-Dao conjugate of X(j) for these (i,j): {35118, 321}, {40592, 35166}
X(68864) = crossdifference of every pair of points on line {37, 46390}
X(68864) = barycentric quotient X(81)/X(35166)
X(68864) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {99, 741, 1}, {4128, 11711, 1}

X(68865) = X(1)X(99) INTERCEPT OF X(523)X(661)

Barycentrics    (b + c)*(-(a^5*b) - a^4*b^2 + 2*a^2*b^4 + a*b^5 - a^5*c + 2*a^3*b^2*c - a^4*c^2 + 2*a^3*b*c^2 - 2*a*b^3*c^2 - b^4*c^2 - 2*a*b^2*c^3 + 2*a^2*c^4 - b^2*c^4 + a*c^5) : :

X(68865) lies on these lines: {1, 99}, {2, 2643}, {192, 6758}, {244, 17045}, {523, 661}, {1962, 4760}, {17286, 27805}, {20653, 21709}, {21085, 65191}, {27569, 61174}

X(68865) = crossdifference of every pair of points on line {58, 46390}

X(68866) = X(1)X(99) INTERCEPT OF X(10)X(37)

Barycentrics    (b + c)*(a^4 + 4*a^3*b + 3*a^2*b^2 + a*b^3 + 4*a^3*c + 6*a^2*b*c + a*b^2*c + 3*a^2*c^2 + a*b*c^2 - b^2*c^2 + a*c^3) : :
X(68866) = 5 X[1698] - 3 X[59261]

X(68866) lies on these lines: {1, 99}, {10, 37}, {519, 50273}, {730, 37548}, {1125, 24384}, {1698, 59261}, {1962, 16826}, {2796, 50262}, {3247, 23944}, {4368, 6155}, {4647, 33939}, {4854, 20337}, {6542, 27804}, {9791, 46707}, {17163, 60710}, {17319, 23928}, {17320, 57040}, {17395, 21254}, {17759, 25263}, {24342, 32014}, {24437, 50281}, {25457, 28612}, {31730, 58389}, {49753, 50016}

X(68866) = crossdifference of every pair of points on line {3733, 46390}
X(68866) = barycentric product X(321)*X(16480)
X(68866) = barycentric quotient X(16480)/X(81)
X(68866) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 13174, 1509}, {10, 55343, 3842}

X(68867) = X(1)X(99) INTERCEPT OF X(2)X(37)

Barycentrics    a^3*b + 2*a^2*b^2 + a*b^3 + a^3*c + 3*a^2*b*c + a*b^2*c + 2*a^2*c^2 + a*b*c^2 - b^2*c^2 + a*c^3 : :

X(68867) lies on these lines: {1, 99}, {2, 37}, {38, 3571}, {86, 1962}, {190, 21840}, {274, 3743}, {314, 58391}, {319, 3896}, {325, 4854}, {334, 1500}, {335, 60724}, {668, 4868}, {740, 30966}, {846, 33295}, {1045, 4093}, {1909, 3931}, {2292, 33296}, {3027, 7179}, {3228, 35181}, {3726, 17319}, {3873, 17393}, {3875, 32010}, {4062, 4389}, {4065, 16887}, {4115, 46913}, {4357, 4970}, {4485, 30964}, {4646, 25280}, {4981, 5564}, {5224, 8013}, {6155, 17499}, {6626, 27368}, {6646, 20536}, {9281, 54117}, {10026, 17246}, {14210, 16712}, {16600, 32026}, {16703, 20932}, {16705, 17762}, {17160, 46912}, {17206, 41813}, {17261, 46907}, {17394, 62840}, {17497, 50184}, {17592, 24259}, {18146, 40090}, {18600, 41875}, {21334, 23482}, {22184, 43263}, {24241, 40878}, {24338, 35147}, {25303, 37548}, {25599, 33939}, {27804, 30941}, {30138, 31451}, {31008, 35544}, {31997, 62831}, {37159, 50067}, {37664, 66071}, {37678, 46904}, {39252, 62817}, {39717, 40775}, {49474, 59261}

X(68867) = X(40776)-anticomplementary conjugate of X(1330)
X(68867) = crossdifference of every pair of points on line {667, 46390}
X(68867) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 192, 4037}, {16705, 64071, 17762}, {17592, 49518, 37632}

X(68868) = X(1)X(99) INTERCEPT OF X(10)X(82)

Barycentrics    a^5 + 3*a^4*b + a^3*b^2 + a^2*b^3 + 3*a^4*c + 2*a^3*b*c + a^2*b^2*c + 2*a*b^3*c + a^3*c^2 + a^2*b*c^2 + 3*a*b^2*c^2 + b^3*c^2 + a^2*c^3 + 2*a*b*c^3 + b^2*c^3 : :

X(68868) lies on these lines: {1, 99}, {8, 24275}, {10, 82}, {1621, 2223}, {2108, 55085}, {3923, 7760}, {4065, 4658}, {4301, 48925}, {5264, 54291}, {6625, 43671}, {20337, 63979}, {20963, 57280}, {32014, 50302}, {49470, 62805}

X(68868) = crossdifference of every pair of points on line {21123, 46390}

X(68869) = X(1)X(99) INTERCEPT OF X(10)X(190)

Barycentrics    a^5 + 3*a^4*b - a^3*b^2 - 3*a^2*b^3 - 2*a*b^4 + 3*a^4*c - 3*a^2*b^2*c - a^3*c^2 - 3*a^2*b*c^2 + 5*a*b^2*c^2 + 3*b^3*c^2 - 3*a^2*c^3 + 3*b^2*c^3 - 2*a*c^4 : :
X(68869) = 2 X[1] - 3 X[99], 4 X[1] - 3 X[7983], 5 X[1] - 6 X[11711], X[1] - 3 X[13174], 5 X[99] - 4 X[11711], 5 X[7983] - 8 X[11711], X[7983] - 4 X[13174], 2 X[11711] - 5 X[13174], 4 X[8] - 3 X[50885], 4 X[10] - 3 X[671], 2 X[10] - 3 X[9881], 3 X[98] - 4 X[3579], 6 X[115] - 7 X[9780], X[145] - 3 X[8591], 2 X[145], and others

X(68869) lies on these lines: {1, 99}, {8, 543}, {10, 190}, {40, 38664}, {98, 3579}, {115, 9780}, {145, 8591}, {148, 3617}, {517, 23235}, {542, 6361}, {620, 5550}, {690, 49274}, {944, 10992}, {962, 14981}, {1125, 41134}, {1482, 51524}, {1698, 9166}, {2482, 3616}, {2782, 12702}, {2795, 11684}, {3027, 5221}, {3241, 15300}, {3244, 50888}, {3339, 51795}, {3508, 41322}, {3621, 20094}, {3622, 52695}, {3624, 12258}, {3626, 13178}, {3634, 11599}, {3649, 12350}, {4663, 10754}, {4668, 9875}, {4678, 8596}, {4691, 50884}, {5204, 22514}, {5217, 13173}, {5461, 19877}, {5708, 52822}, {5818, 38734}, {5847, 45018}, {5969, 64070}, {6054, 12699}, {6321, 18357}, {6781, 50247}, {7970, 8148}, {8593, 67964}, {8598, 50776}, {8724, 22791}, {9778, 10991}, {9812, 38745}, {9864, 10723}, {9955, 23234}, {10069, 37524}, {10769, 12019}, {11623, 68545}, {11632, 61524}, {11710, 67706}, {11725, 46934}, {12117, 18481}, {12184, 67973}, {12357, 26015}, {13624, 21166}, {14360, 66236}, {14639, 61261}, {14928, 51192}, {15561, 61272}, {18525, 50880}, {19862, 38220}, {20399, 68034}, {28174, 52090}, {28530, 53426}, {29615, 64010}, {34473, 35242}, {36521, 38314}, {41135, 46933}, {43671, 65250}, {50250, 51224}, {61268, 64089}

X(68869) = reflection of X(i) in X(j) for these {i,j}: {99, 13174}, {671, 9881}, {944, 10992}, {962, 14981}, {1482, 51524}, {3241, 15300}, {7970, 13188}, {7983, 99}, {9884, 8591}, {10723, 9864}, {38664, 40}, {50247, 6781}, {51192, 14928}

X(68870) = X(30)X(511) INTERCEPT OF X(37)X(86)

Barycentrics    a^3*b - a^2*b^2 - a*b^3 + a^3*c - a^2*c^2 + 2*b^2*c^2 - a*c^3 : :

X(68870) lies on these lines: {2, 31310}, {6, 17738}, {30, 511}, {37, 86}, {38, 24330}, {44, 20142}, {69, 41842}, {75, 1654}, {141, 53600}, {192, 4644}, {244, 4465}, {274, 21879}, {291, 20716}, {312, 24691}, {319, 6653}, {320, 3797}, {321, 24690}, {599, 27474}, {668, 21888}, {673, 17348}, {903, 4688}, {982, 4713}, {984, 4363}, {1086, 1213}, {1100, 4366}, {1125, 59627}, {1215, 25349}, {1278, 64015}, {1761, 3512}, {1962, 49749}, {1992, 27480}, {2292, 4754}, {3120, 25383}, {3218, 4396}, {3248, 20356}, {3663, 49481}, {3670, 4721}, {3696, 4690}, {3721, 56024}, {3729, 49509}, {3758, 36409}, {3782, 25345}, {3799, 7245}, {3842, 4472}, {3959, 56025}, {3993, 4667}, {4037, 30941}, {4115, 17205}, {4364, 24325}, {4368, 9507}, {4370, 4755}, {4376, 32933}, {4389, 31317}, {4400, 56288}, {4409, 4726}, {4419, 9791}, {4422, 4698}, {4427, 4760}, {4432, 5625}, {4437, 17229}, {4454, 31302}, {4473, 4687}, {4553, 20694}, {4641, 19791}, {4659, 49448}, {4664, 4795}, {4665, 49457}, {4681, 4796}, {4699, 4748}, {4704, 4747}, {4732, 64712}, {4799, 33098}, {4851, 20533}, {4852, 32029}, {5839, 41845}, {7200, 53332}, {7781, 22836}, {8298, 13174}, {8716, 56177}, {10022, 50094}, {10180, 50180}, {11997, 24840}, {15624, 24820}, {16720, 56318}, {17154, 24403}, {17163, 50278}, {17165, 24326}, {17175, 21816}, {17239, 26582}, {17262, 51058}, {17279, 52157}, {17318, 49490}, {17326, 27191}, {17334, 49516}, {17759, 20693}, {17790, 52049}, {18157, 42713}, {20430, 24844}, {20500, 20548}, {20681, 53541}, {20722, 21080}, {21020, 49717}, {21093, 24318}, {21839, 62755}, {24215, 59512}, {24248, 49531}, {24441, 31178}, {24697, 49483}, {24699, 32857}, {24705, 42027}, {24813, 30271}, {24817, 30273}, {24828, 67858}, {24833, 64088}, {24841, 49478}, {25048, 36294}, {25350, 59511}, {25358, 64946}, {26273, 48443}, {27475, 41313}, {27478, 50093}, {27481, 50128}, {27798, 50158}, {27804, 50257}, {31306, 41311}, {31342, 50125}, {32935, 49519}, {36237, 66067}, {36494, 49748}, {40607, 58691}, {40857, 57039}, {44353, 57037}, {46458, 52068}, {49470, 64016}, {50086, 66454}, {52897, 66878}, {58485, 58553}, {58571, 58618}, {61522, 61558}, {64546, 67024}

X(68870) = isotomic conjugate of X(35173)
X(68870) = isotomic conjugate of the anticomplement of X(35127)
X(68870) = crossdifference of every pair of points on line {6, 4455}
X(68870) = X(17738)-line conjugate of X(6)
X(68870) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {38, 24330, 25368}, {86, 190, 6651}, {190, 335, 37}, {274, 21879, 59633}, {903, 31349, 4688}, {984, 4363, 25384}, {1086, 1213, 62675}, {1086, 17755, 3739}, {1654, 4440, 6650}, {4419, 24349, 24357}, {4440, 33888, 75}, {6646, 41844, 4440}

X(68871) = X(37)X(86) INTERCEPT OF X(514)X(661)

Barycentrics    a^3*b^2 - a*b^4 - a^2*b^2*c + a^3*c^2 - a^2*b*c^2 + b^3*c^2 + b^2*c^3 - a*c^4 : :

X(68871) lies on these lines: {2, 3125}, {37, 86}, {244, 1125}, {257, 4997}, {312, 65169}, {321, 4568}, {392, 16823}, {514, 661}, {869, 49526}, {1150, 5692}, {1655, 21220}, {1914, 30908}, {1978, 18359}, {3264, 17444}, {3747, 59720}, {3799, 4518}, {4016, 26979}, {4053, 30939}, {4154, 20356}, {4359, 17761}, {4427, 9978}, {4475, 17793}, {4516, 53338}, {4858, 18697}, {7202, 65161}, {16704, 21839}, {16706, 59626}, {16729, 24069}, {16738, 21810}, {16815, 59633}, {17174, 61410}, {17197, 65191}, {17202, 27697}, {17244, 18055}, {17738, 24279}, {17788, 18137}, {18148, 18149}, {18714, 25660}, {20716, 67428}, {21873, 29767}, {24285, 53331}, {27261, 27270}

X(68871) = X(35151)-Ceva conjugate of X(8)
X(68871) = X(65941)-Dao conjugate of X(1)
X(68871) = crossdifference of every pair of points on line {31, 4455}
X(68871) = barycentric product X(i)*X(j) for these {i,j}: {3264, 61433}, {3936, 46800}, {35544, 68235}
X(68871) = barycentric quotient X(i)/X(j) for these {i,j}: {46800, 24624}, {61433, 106}, {68235, 741}

X(68872) = X(2)X(799) INTERCEPT OF X(239)X(514)

Barycentrics    (a + b)*(a + c)*(-(a^2*b^3) + 2*a^3*b*c - a*b^3*c + b^3*c^2 - a^2*c^3 - a*b*c^3 + b^2*c^3) : :

X(68872) lies on these lines: {2, 799}, {81, 1977}, {86, 4465}, {99, 9263}, {239, 514}, {274, 4403}, {593, 4027}, {741, 17794}, {1509, 1655}, {1929, 39747}, {1931, 56431}, {3720, 4368}, {3797, 16741}, {3952, 18787}, {3995, 22033}, {4094, 4427}, {4576, 33888}, {6758, 13174}, {7257, 31298}, {17103, 21226}, {17126, 40731}, {17731, 17759}, {18827, 30667}, {26860, 60680}, {27189, 27348}, {31059, 31061}, {33908, 55243}, {39360, 55245}, {40908, 52379}

X(68872) = X(12031)-anticomplementary conjugate of X(69)
X(68872) = X(213)-isoconjugate of X(35166)
X(68872) = X(i)-Dao conjugate of X(j) for these (i,j): {6626, 35166}, {35118, 10}
X(68872) = crosssum of X(213) and X(5147)
X(68872) = barycentric quotient X(86)/X(35166)
X(68872) = {X(799),X(37128)}-harmonic conjugate of X(2)

X(68873) = X(10)X(274) INTERCEPT OF X(30)X(511)

Barycentrics    a^2*b^3 - 2*a^3*b*c + a*b^3*c - b^3*c^2 + a^2*c^3 + a*b*c^3 - b^2*c^3 : :

X(68873) lies on these lines: {1, 1655}, {6, 24294}, {8, 31298}, {10, 274}, {30, 511}, {76, 52044}, {99, 8298}, {190, 21100}, {551, 3227}, {596, 9278}, {984, 64133}, {1015, 1107}, {1054, 56801}, {1111, 19962}, {1573, 24325}, {1757, 40859}, {3122, 24517}, {3634, 25109}, {3664, 59565}, {3679, 39360}, {3770, 24437}, {3795, 7757}, {3807, 43262}, {3828, 13466}, {3879, 21080}, {3912, 59735}, {3952, 31061}, {3971, 29574}, {3975, 46843}, {4039, 18206}, {4075, 20529}, {4103, 59724}, {4115, 46460}, {4416, 42027}, {4738, 20010}, {4986, 20000}, {9359, 24485}, {10027, 62222}, {12263, 17448}, {12782, 24524}, {16709, 21699}, {17031, 45751}, {17155, 29617}, {17157, 17363}, {17389, 32925}, {18159, 19950}, {18792, 52894}, {19862, 27195}, {19878, 40479}, {19958, 20568}, {20456, 52043}, {20538, 24068}, {20606, 62858}, {20888, 56660}, {24165, 50095}, {24231, 44353}, {24261, 49489}, {24293, 49531}, {24575, 44139}, {25092, 59454}, {25264, 54101}, {30109, 40857}, {32913, 41232}, {36524, 51069}, {41683, 60725}, {53338, 53541}, {53601, 57038}

X(68873) = isotomic conjugate of X(35166)
X(68873) = isotomic conjugate of the anticomplement of X(35118)
X(68873) = X(24294)-line conjugate of X(6)
X(68873) = barycentric quotient X(23823)/X(5185)
X(68873) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {274, 668, 39028}, {291, 668, 10}, {1015, 17793, 1125}, {1655, 9263, 39925}, {9263, 17794, 1}, {27076, 40533, 3634}

X(68874) = X(1)X(2) INTERCEPT OF X(81)X(4610)

Barycentrics    a^4 + 4*a^3*b + 3*a^2*b^2 + a*b^3 + 4*a^3*c + 6*a^2*b*c + a*b^2*c + 3*a^2*c^2 + a*b*c^2 - b^2*c^2 + a*c^3 : :

X(68874) lies on these lines: {1, 2}, {81, 4610}, {1051, 17738}, {1100, 51356}, {1962, 20142}, {4037, 4360}, {4359, 51314}, {4716, 59261}, {5333, 27483}, {6650, 64164}, {6651, 27804}, {17277, 59218}, {17302, 20536}, {31323, 62851}, {32911, 66878}, {34016, 52548}

X(68874) = barycentric product X(75)*X(16480)
X(68874) = barycentric quotient X(16480)/X(1)
X(68874) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {17023, 21085, 2}

X(68875) = X(1)X(2) INTERCEPT OF X(81)X(4600)

Barycentrics    a^4 - a^2*b^2 + a*b^3 - 2*a^2*b*c + a*b^2*c - a^2*c^2 + a*b*c^2 - b^2*c^2 + a*c^3 : :

X(68875) lies on these lines: {1, 2}, {37, 3570}, {81, 4600}, {100, 335}, {171, 40217}, {190, 4760}, {192, 9318}, {321, 874}, {385, 3930}, {756, 27926}, {872, 27927}, {894, 42720}, {1621, 6652}, {3120, 6653}, {3711, 20154}, {3722, 4366}, {3750, 27916}, {3797, 32927}, {3807, 24358}, {3879, 24318}, {3952, 6651}, {3993, 24428}, {3995, 4024}, {4360, 27918}, {4437, 26629}, {6654, 46798}, {8298, 43534}, {9317, 27295}, {16997, 51058}, {17261, 53337}, {17319, 24403}, {17724, 26582}, {20142, 21805}, {20693, 33295}, {21021, 33954}, {24491, 27065}, {24502, 26223}, {27495, 32917}, {27804, 40725}, {35292, 62222}

X(68875) = isotomic conjugate of X(19975)
X(68875) = X(31)-isoconjugate of X(19975)
X(68875) = X(2)-Dao conjugate of X(19975)
X(68875) = barycentric product X(190)*X(40459)
X(68875) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 19975}, {40459, 514}
X(68875) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 23891, 66151}, {3935, 26247, 239}, {17319, 27912, 24403}

X(68876) = X(8)X(238) INTERCEPT OF X(37)X(86)

Barycentrics    a^4 + 2*a^3*b - a^2*b^2 + a*b^3 + 2*a^3*c - 2*a^2*b*c + b^3*c - a^2*c^2 + b^2*c^2 + a*c^3 + b*c^3 : :

X(68876) lies on these lines: {2, 41423}, {6, 17242}, {8, 238}, {9, 27495}, {29, 60699}, {37, 86}, {344, 14621}, {1001, 17368}, {1334, 33816}, {2321, 20142}, {3161, 17379}, {3501, 33817}, {3797, 3875}, {4366, 17353}, {4422, 20179}, {4967, 20138}, {5263, 15254}, {6541, 49685}, {6661, 40534}, {7321, 15668}, {15485, 48851}, {16484, 63053}, {16777, 27481}, {17000, 17355}, {17259, 17289}, {17277, 17359}, {17290, 17322}, {20147, 50115}, {20154, 48628}, {20159, 41313}, {20181, 29628}, {22016, 30940}, {25101, 49482}, {27268, 50302}, {27949, 49521}, {29585, 37677}

X(68877) = X(2)X(661) INTERCEPT OF X(10)X(75)

Barycentrics    a^3*b^3 + a^2*b^4 - a^2*b^3*c - a*b^4*c - 2*a^2*b^2*c^2 + a^3*c^3 - a^2*b*c^3 + 2*b^3*c^3 + a^2*c^4 - a*b*c^4 : :

X(68877) lies on these lines: {2, 661}, {10, 75}, {874, 24429}, {982, 24197}, {4708, 52626}, {17259, 29460}, {17318, 23891}, {23894, 52756}, {35956, 36218}, {52755, 55244}

X(68877) = crossdifference of every pair of points on line {1919, 3747}
X(68877) = {X(10),X(23822)}-harmonic conjugate of X(75)

X(68878) = ANTI-ORTHIC AXIS INTERCEPT OF X(6)X(48276)

Barycentrics    a*(b - c)*(a^4 - a^2*b^2 - 3*a^2*b*c - b^3*c - a^2*c^2 - 2*b^2*c^2 - b*c^3) : :

X(68878) lies on these lines: {6, 48276}, {44, 513}, {419, 2501}, {647, 21390}, {2527, 21786}, {3063, 6590}, {3700, 21007}, {3737, 68806}, {4024, 4435}, {4040, 58299}, {7252, 68794}, {20980, 49293}, {21389, 50492}, {23090, 57156}, {28960, 30061}, {30565, 57148}, {47768, 57181}, {48269, 66513}, {48336, 58298}, {48352, 58297}, {50353, 57096}

X(68878) = crossdifference of every pair of points on line {1, 3917}
X(68878) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 3287, 46383}, {3063, 6590, 65097}

X(68879) = ANTI-ORTHIC AXIS INTERCEPT OF X(10)X(82)

Barycentrics    a*(a^3*b + a^3*c - 2*a^2*b*c + a*b^2*c - b^3*c + a*b*c^2 - b*c^3) : :

X(68879) lies on these lines: {10, 82}, {42, 57024}, {43, 3764}, {44, 513}, {190, 59735}, {210, 28242}, {320, 37678}, {518, 872}, {597, 20985}, {1279, 10459}, {1757, 3216}, {2209, 17279}, {3248, 4447}, {3264, 39044}, {3507, 25048}, {3589, 20964}, {3747, 4422}, {3823, 28375}, {3836, 27042}, {4645, 26030}, {4798, 17124}, {6687, 30970}, {16690, 17338}, {16706, 51902}, {19863, 31289}, {20108, 49676}, {24517, 29504}, {49675, 50604}

X(68879) =crossdifference of every pair of points on line {1, 21123}
X(68879) ={X(899),X(2183)}-harmonic conjugate of X(2228)

X(68880) = ANTI-ORTHIC AXIS INTERCEPT OF X(514)X(1919)

Barycentrics    a*(b - c)*(a^3 - a*b*c - b^2*c - b*c^2) : :

X(68880) lies on these lines: {44, 513}, {514, 1919}, {523, 8632}, {663, 17458}, {693, 8060}, {3250, 48297}, {3737, 21123}, {4040, 4079}, {4057, 57234}, {4107, 20949}, {4374, 24354}, {4375, 20954}, {4448, 21960}, {4826, 48336}, {4977, 20981}, {4988, 57096}, {14407, 21390}, {17398, 40086}, {21007, 21832}, {21143, 21173}, {21225, 38367}, {21348, 48331}, {21834, 48306}, {23885, 48094}, {24623, 42327}, {28894, 50458}, {29051, 47127}, {50522, 57129}

X(68880) = reflection of X(i) in X(j) for these {i,j}: {693, 8060}, {8061, 650}
X(68880) = X(2)-isoconjugate of X(29071)
X(68880) = X(32664)-Dao conjugate of X(29071)
X(68880) = crosspoint of X(4610) and X(39949)
X(68880) = crosssum of X(3293) and X(4079)
X(68880) = crossdifference of every pair of points on line {1, 4283}
X(68880) = barycentric product X(i)*X(j) for these {i,j}: {1, 29070}, {513, 32914}, {3261, 5371}
X(68880) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 29071}, {5371, 101}, {29070, 75}, {32914, 668}
X(68880) = {X(4040),X(21389)}-harmonic conjugate of X(4079)

X(68881) = ANTI-ORTHIC AXIS INTERCEPT OF X(512)X(21123)

Barycentrics    a^2*(b - c)*(a*b + a*c + 2*b*c) : :
X(68881) = 4 X[649] - X[3768], 3 X[649] - X[20979], 3 X[798] - 2 X[20979], 3 X[3768] - 4 X[20979], X[23794] - 3 X[47762]

X(68881) lies on these lines: {44, 513}, {512, 21123}, {665, 4079}, {786, 17159}, {788, 4834}, {802, 4406}, {812, 1019}, {900, 57234}, {1919, 66513}, {3250, 4826}, {3572, 50344}, {3733, 8632}, {3766, 52602}, {4063, 28840}, {4367, 9400}, {4369, 20954}, {4378, 4501}, {4455, 50512}, {4502, 6586}, {4762, 48144}, {4785, 21191}, {4832, 6363}, {4977, 21832}, {5029, 48306}, {6371, 21143}, {6373, 58179}, {9010, 58178}, {17217, 26816}, {17458, 54249}, {20295, 27159}, {20981, 21007}, {21225, 64861}, {23655, 50500}, {23751, 50492}, {23794, 47762}, {27673, 48079}, {29450, 29457}, {40418, 62619}, {50509, 54251}

X(68881) = midpoint of X(i) and X(j) for these {i,j}: {17217, 26853}, {50509, 54251}
X(68881) = reflection of X(i) in X(j) for these {i,j}: {798, 649}, {3766, 52602}, {3768, 798}, {4079, 665}, {4455, 50512}, {4502, 6586}, {4826, 3250}, {17458, 54249}, {20295, 42327}, {20954, 4369}
X(68881) = isogonal conjugate of the isotomic conjugate of X(47672)
X(68881) = X(i)-Ceva conjugate of X(j) for these (i,j): {42, 1015}, {86, 3248}, {4436, 2667}, {39950, 244}
X(68881) = X(50497)-cross conjugate of X(6372)
X(68881) = X(i)-isoconjugate of X(j) for these (i,j): {2, 8708}, {100, 32009}, {190, 40433}, {668, 57397}, {1016, 50520}, {1018, 40439}, {3952, 40408}, {40521, 59147}
X(68881) = X(i)-Dao conjugate of X(j) for these (i,j): {2486, 4043}, {3121, 10}, {3739, 4033}, {6372, 47672}, {8054, 32009}, {16589, 1978}, {17205, 310}, {32664, 8708}, {55053, 40433}, {62646, 668}
X(68881) = crosspoint of X(i) and X(j) for these (i,j): {649, 1019}, {4436, 18166}
X(68881) = crosssum of X(190) and X(1018)
X(68881) = crossdifference of every pair of points on line {1, 872}
X(68881) = barycentric product X(i)*X(j) for these {i,j}: {1, 6372}, {6, 47672}, {56, 48264}, {58, 48393}, {86, 50497}, {244, 4436}, {512, 17175}, {513, 3720}, {514, 20963}, {649, 3739}, {661, 18166}, {663, 4059}, {667, 20888}, {757, 50538}, {798, 16748}, {905, 40975}, {1019, 16589}, {2667, 7192}, {3248, 53363}, {3669, 3691}, {3706, 43924}, {3733, 21020}, {3737, 39793}, {4111, 7203}, {7199, 21753}, {7649, 22060}, {16726, 61163}, {18089, 21123}, {40433, 68124}, {53478, 57129}
X(68881) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 8708}, {649, 32009}, {667, 40433}, {1919, 57397}, {2667, 3952}, {3248, 50520}, {3691, 646}, {3720, 668}, {3733, 40439}, {3739, 1978}, {4059, 4572}, {4436, 7035}, {6372, 75}, {16589, 4033}, {16748, 4602}, {17175, 670}, {18166, 799}, {20888, 6386}, {20963, 190}, {21020, 27808}, {21753, 1018}, {21820, 4103}, {22060, 4561}, {40975, 6335}, {47672, 76}, {48264, 3596}, {48393, 313}, {50497, 10}, {50538, 1089}, {57129, 40408}, {68124, 20888}
X(68881) = {X(2254),X(48022)}-harmonic conjugate of X(8061)

X(68882) = ANTI-ORTHIC AXIS INTERCEPT OF X(514)X(50011)

Barycentrics    a*(b - c)*(a*b^2 - b^3 + a*c^2 - c^3) : :
X(68882) = 4 X[2492] - 3 X[53037]

X(68882) lies on these lines: {6, 3738}, {37, 1769}, {44, 513}, {142, 20520}, {514, 50011}, {518, 13259}, {521, 68776}, {522, 68799}, {523, 50555}, {594, 4768}, {665, 53527}, {758, 39232}, {900, 24290}, {906, 35338}, {918, 1086}, {1100, 53532}, {1633, 64616}, {2492, 53037}, {2874, 3779}, {3250, 48350}, {3569, 4016}, {3735, 23884}, {3862, 60577}, {4000, 53357}, {4148, 17275}, {4435, 8674}, {4526, 24457}, {4809, 5098}, {4976, 55232}, {6003, 21007}, {6184, 23980}, {6369, 42445}, {6586, 21189}, {14427, 16814}, {15413, 26546}, {17323, 25602}, {21112, 55195}, {23687, 65697}, {23758, 29204}, {23887, 62556}, {25259, 50450}, {30520, 43052}, {35505, 66508}, {36743, 53305}, {36744, 53277}, {42662, 53285}, {49509, 64860}, {57185, 68794}

X(68882) = reflection of X(4016) in X(3569)
X(68882) = isogonal conjugate of X(36087)
X(68882) = X(i)-Ceva conjugate of X(j) for these (i,j): {29068, 38345}, {36087, 1}, {37130, 17463}, {42723, 57015}, {54739, 11}
X(68882) = X(i)-isoconjugate of X(j) for these (i,j): {1, 36087}, {2, 32682}, {59, 60573}, {100, 2224}, {101, 675}, {110, 60135}, {249, 66281}, {514, 52941}, {692, 37130}, {3730, 65554}, {4628, 46158}, {32739, 43093}, {35190, 56746}
X(68882) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 36087}, {244, 60135}, {1015, 675}, {1086, 37130}, {6615, 60573}, {8054, 2224}, {32664, 32682}, {38990, 1}, {40619, 43093}, {53980, 1783}, {65703, 68811}, {65919, 100}
X(68882) = crosspoint of X(i) and X(j) for these (i,j): {1, 36087}, {42723, 57015}
X(68882) = crosssum of X(60135) and X(60573)
X(68882) = crossdifference of every pair of points on line {1, 692}
X(68882) = barycentric product X(i)*X(j) for these {i,j}: {1, 23887}, {75, 65703}, {513, 3006}, {514, 57015}, {522, 68761}, {674, 693}, {1086, 42723}, {1577, 14964}, {2224, 62556}, {2225, 3261}, {3762, 64611}, {4391, 43039}, {8618, 40495}, {35519, 51657}
X(68882) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 36087}, {31, 32682}, {513, 675}, {514, 37130}, {649, 2224}, {661, 60135}, {674, 100}, {692, 52941}, {693, 43093}, {2170, 60573}, {2225, 101}, {2530, 46158}, {2643, 66281}, {3006, 668}, {4249, 5379}, {8618, 692}, {14964, 662}, {23887, 75}, {38990, 68811}, {42723, 1016}, {43039, 651}, {51657, 109}, {57015, 190}, {64611, 3257}, {65703, 1}, {68761, 664}
X(68882) = {X(1769),X(68813)}-harmonic conjugate of X(37)

X(68883) = ANTI-ORTHIC AXIS INTERCEPT OF X(10)X(37)

Barycentrics    a*(b + c)*(a^2*b + a*b^2 + a^2*c - 2*a*b*c - b^2*c + a*c^2 - b*c^2) : :

X(68883) lies on these lines: {2, 4277}, {6, 474}, {10, 37}, {39, 17330}, {44, 513}, {45, 24429}, {58, 5956}, {75, 26772}, {115, 35129}, {141, 64556}, {210, 4735}, {239, 57039}, {314, 27111}, {391, 5069}, {404, 5035}, {519, 8610}, {524, 16726}, {536, 3264}, {573, 19543}, {860, 14571}, {872, 22279}, {966, 4261}, {980, 17251}, {992, 4271}, {1015, 4969}, {1084, 35123}, {1100, 1193}, {1211, 3752}, {1230, 42051}, {1400, 56198}, {1574, 17369}, {1654, 16696}, {2276, 24517}, {2277, 17275}, {2609, 9423}, {2664, 4553}, {2667, 22174}, {2895, 16700}, {3006, 3290}, {3122, 44671}, {3125, 4053}, {3136, 21949}, {3285, 35342}, {3666, 4708}, {3723, 10459}, {3739, 27042}, {3759, 27641}, {3770, 40908}, {3778, 22271}, {3780, 28249}, {3783, 57024}, {3834, 3936}, {3882, 52897}, {3963, 59715}, {3965, 40941}, {4000, 27039}, {4016, 21033}, {4263, 17398}, {4266, 17749}, {4285, 24512}, {4286, 16552}, {4359, 27041}, {4360, 29399}, {4433, 39688}, {4446, 64581}, {4686, 53478}, {4690, 37596}, {4695, 21801}, {4849, 40952}, {4850, 17250}, {4852, 28366}, {5165, 37657}, {5275, 40126}, {5283, 51677}, {5291, 19297}, {5839, 63499}, {9350, 40401}, {14554, 51415}, {14973, 21936}, {16569, 20973}, {16590, 21838}, {16602, 17056}, {16604, 16666}, {16606, 28658}, {16702, 19308}, {16736, 17778}, {16753, 63071}, {17053, 17362}, {17147, 62588}, {17160, 57023}, {17227, 31037}, {17281, 25610}, {17299, 62214}, {17303, 26030}, {17346, 24598}, {17348, 27633}, {17392, 31198}, {17448, 50082}, {17490, 27792}, {17495, 26844}, {17790, 26048}, {18143, 26756}, {18172, 24897}, {18591, 63622}, {18601, 43990}, {20340, 64869}, {20683, 22323}, {21035, 40607}, {21864, 21888}, {22425, 25066}, {24478, 56537}, {25003, 25091}, {25004, 25067}, {25058, 41817}, {25457, 26110}, {25629, 52538}, {26242, 31079}, {26771, 27793}, {27095, 29484}, {30473, 40598}, {31144, 40773}, {33854, 33882}, {35068, 39011}, {35069, 35092}, {36744, 37246}, {37062, 37499}, {39956, 39984}, {41014, 52541}, {46826, 53486}, {50131, 63493}, {52529, 64185}, {56174, 58889}, {56185, 56249}, {62636, 65161}

X(68883) = midpoint of X(62636) and X(65161)
X(68883) = reflection of X(3264) in X(59738)
X(68883) = complement of X(30939)
X(68883) = complement of the isotomic conjugate of X(4674)
X(68883) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 34587}, {32, 51583}, {42, 121}, {88, 21240}, {106, 3741}, {213, 16594}, {512, 3259}, {667, 34590}, {669, 35092}, {798, 66508}, {901, 512}, {1402, 1145}, {1417, 3742}, {1918, 4370}, {2316, 21246}, {3257, 42327}, {4080, 626}, {4555, 23301}, {4591, 52602}, {4674, 2887}, {8752, 34830}, {9456, 3739}, {23345, 53564}, {32645, 2487}, {32665, 4369}, {32719, 523}, {41935, 4395}, {55244, 21252}, {55263, 116}, {66285, 53575}, {68563, 21243}
X(68883) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 34587}, {16704, 758}, {24004, 4145}, {31855, 21805}, {50039, 523}, {53905, 55}
X(68883) = X(i)-isoconjugate of X(j) for these (i,j): {2, 59072}, {58, 39698}, {1019, 53685}, {1333, 40039}
X(68883) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 39698}, {37, 40039}, {17495, 18145}, {32664, 59072}, {34587, 2}
X(68883) = crosspoint of X(i) and X(j) for these (i,j): {2, 4674}, {10, 14554}, {37, 39982}, {17495, 49997}
X(68883) = crosssum of X(i) and X(j) for these (i,j): {6, 52680}, {58, 5053}, {81, 37680}
X(68883) = crossdifference of every pair of points on line {1, 3733}
X(68883) = barycentric product X(i)*X(j) for these {i,j}: {10, 49997}, {37, 17495}, {42, 39995}, {4674, 34587}, {23169, 41013}
X(68883) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 40039}, {31, 59072}, {37, 39698}, {4557, 53685}, {17495, 274}, {23169, 1444}, {34587, 30939}, {39995, 310}, {49997, 86}
X(68883) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 46838, 39798}, {37, 21857, 21858}, {594, 21796, 37}, {1213, 2092, 37}, {1654, 24530, 16696}, {2238, 2245, 44}, {5257, 56926, 37}, {21857, 21892, 37}

X(68884) = ANTI-ORTHIC AXIS INTERCEPT OF X(1)X(87)

Barycentrics    a*(a^2*b^2 - 2*a^2*b*c + a*b^2*c + a^2*c^2 + a*b*c^2 - 2*b^2*c^2) : :
X(68884) = 3 X[24487] - X[39995]

X(68884) lies on these lines: {1, 87}, {42, 3758}, {44, 513}, {75, 22343}, {190, 3009}, {239, 9359}, {291, 23633}, {320, 24722}, {536, 3248}, {651, 56413}, {740, 23579}, {869, 50127}, {889, 3226}, {1045, 17120}, {1125, 26976}, {1149, 4432}, {1156, 43748}, {1193, 4672}, {1201, 4676}, {1266, 27846}, {1278, 18194}, {1740, 17350}, {1964, 17351}, {2210, 65168}, {3264, 25382}, {3720, 4670}, {3783, 20072}, {3875, 23524}, {3912, 53541}, {4360, 23532}, {4363, 21352}, {4440, 56805}, {4489, 63049}, {4641, 59308}, {4704, 24661}, {4753, 49984}, {4759, 49997}, {4788, 63527}, {4892, 29978}, {6007, 20456}, {7184, 17280}, {7240, 17300}, {7262, 28248}, {8026, 59182}, {8610, 60725}, {9780, 25624}, {15254, 28352}, {15966, 62574}, {16571, 17349}, {16672, 55919}, {17143, 23427}, {17144, 23457}, {17373, 25572}, {17449, 57024}, {17756, 36406}, {17759, 20464}, {20091, 25311}, {20864, 22413}, {23578, 32921}, {24487, 30950}, {25113, 27073}, {25140, 27136}, {25284, 26797}, {25528, 27268}, {25570, 26685}, {25618, 26042}, {25621, 33115}, {27627, 27641}, {30473, 36856}, {32845, 68749}, {35143, 53195}, {36646, 53676}

X(68884) = reflection of X(68751) in X(3248)
X(68884) = crossdifference of every pair of points on line {1, 20979}
X(68884) = X(i)-line conjugate of X(j) for these (i,j): {44, 20979}, {87, 1}
X(68884) = barycentric product X(i)*X(j) for these {i,j}: {192, 52899}, {330, 52895}
X(68884) = barycentric quotient X(i)/X(j) for these {i,j}: {52895, 192}, {52899, 330}
X(68884) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {44, 2234, 899}, {87, 192, 63504}, {2230, 20331, 899}, {3758, 24696, 42}, {7155, 41834, 17157}, {26069, 26077, 25624}

X(68885) = ANTI-ORTHIC AXIS INTERCEPT OF X(99)X(101)

Barycentrics    a*(b - c)*(a^3 + a*b*c - b^2*c - b*c^2) : :

X(68885) lies on these lines: {6, 21832}, {9, 14407}, {37, 5029}, {44, 513}, {45, 53289}, {75, 4107}, {99, 101}, {292, 2161}, {522, 1919}, {523, 20981}, {573, 44812}, {604, 30572}, {824, 47682}, {897, 37129}, {900, 8632}, {909, 1910}, {1100, 4145}, {1449, 53411}, {1459, 17458}, {1960, 4526}, {2170, 2643}, {2171, 51646}, {2291, 6015}, {2605, 21834}, {2640, 5539}, {3196, 9509}, {3226, 35143}, {3250, 55969}, {3733, 57234}, {3737, 4079}, {3738, 68836}, {3766, 24354}, {3835, 30183}, {4010, 24506}, {4024, 57129}, {4363, 53276}, {4465, 30996}, {4502, 48297}, {5040, 17989}, {8060, 50451}, {8674, 68823}, {9278, 61708}, {14437, 68816}, {14442, 19557}, {14750, 17439}, {16554, 24578}, {16777, 53315}, {17303, 21053}, {17455, 17475}, {17739, 24105}, {21123, 21173}, {21131, 21180}, {26072, 26074}, {27669, 28283}, {28898, 50458}, {42081, 42083}, {48278, 57047}, {53271, 53553}

X(68885) = reflection of X(50451) in X(8060)
X(68885) = isogonal conjugate of X(65239)
X(68885) = X(65239)-Ceva conjugate of X(1)
X(68885) = X(17989)-cross conjugate of X(2787)
X(68885) = X(i)-isoconjugate of X(j) for these (i,j): {1, 65239}, {2, 2703}, {6, 35147}, {37, 17929}, {59, 60484}, {100, 17946}, {110, 11611}, {112, 57847}, {190, 17954}, {321, 17939}, {648, 57680}, {651, 11609}, {668, 17961}, {1332, 17981}, {4567, 18015}, {4601, 18002}, {5040, 57559}, {6335, 17971}, {53332, 53689}
X(68885) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 65239}, {9, 35147}, {244, 11611}, {6615, 60484}, {8054, 17946}, {32664, 2703}, {34591, 57847}, {35079, 75}, {38991, 11609}, {40589, 17929}, {40627, 18015}, {55053, 17954}, {55066, 57680}
X(68885) = crosspoint of X(1) and X(65239)
X(68885) = crosssum of X(i) and X(j) for these (i,j): {649, 68750}, {11611, 60484}, {65864, 65941}
X(68885) = crossdifference of every pair of points on line {1, 3122}
X(68885) = barycentric product X(i)*X(j) for these {i,j}: {1, 2787}, {58, 18003}, {75, 5040}, {86, 17989}, {422, 656}, {512, 5209}, {513, 17763}, {514, 5291}, {522, 5061}, {649, 17790}, {661, 19623}, {1459, 17987}, {1577, 5006}, {3120, 17944}, {3122, 17935}, {4444, 67404}, {7649, 17977}, {35079, 65239}
X(68885) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 35147}, {6, 65239}, {31, 2703}, {58, 17929}, {422, 811}, {649, 17946}, {656, 57847}, {661, 11611}, {663, 11609}, {667, 17954}, {810, 57680}, {1919, 17961}, {2170, 60484}, {2206, 17939}, {2787, 75}, {3122, 18015}, {5006, 662}, {5040, 1}, {5061, 664}, {5209, 670}, {5291, 190}, {17763, 668}, {17790, 1978}, {17944, 4600}, {17977, 4561}, {17989, 10}, {18003, 313}, {19623, 799}, {57462, 21124}, {65239, 57559}, {67404, 3570}
X(68885) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2590, 2591, 798}, {21173, 21389, 21123}

X(68886) = ANTI-ORTHIC AXIS INTERCEPT OF X(190)X(670)

Barycentrics    a^2*(b - c)*(a^2*b^2 + a^2*b*c - a*b^2*c + a^2*c^2 - a*b*c^2 - b^2*c^2) : :

X(68886) lies on these lines: {44, 513}, {190, 670}, {292, 5029}, {512, 21763}, {663, 1912}, {669, 20981}, {875, 2107}, {890, 9299}, {894, 24356}, {1019, 53581}, {2176, 23394}, {2978, 4502}, {3248, 4117}, {3249, 68816}, {3835, 27140}, {4107, 30667}, {4375, 19579}, {4449, 57050}, {6373, 23466}, {7192, 25258}, {8632, 23569}, {8640, 20980}, {9359, 39337}, {17754, 25836}, {21832, 24578}, {21834, 50524}, {22224, 47836}, {37129, 37132}, {46403, 62558}, {57059, 57078}

X(68886) = reflection of X(649) in X(46386)
X(68886) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {7168, 150}, {51919, 149}
X(68886) = X(i)-Ceva conjugate of X(j) for these (i,j): {190, 39916}, {874, 3009}, {875, 649}, {4455, 5029}, {4589, 42}
X(68886) = X(i)-isoconjugate of X(j) for these (i,j): {2, 53624}, {6, 53216}, {99, 54980}, {100, 39925}, {110, 43685}, {190, 2665}, {668, 51333}, {799, 2107}, {874, 63874}, {3570, 63892}, {4562, 40769}, {36086, 64239}, {37207, 40798}
X(68886) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 53216}, {244, 43685}, {8054, 39925}, {27854, 514}, {32664, 53624}, {38986, 54980}, {38989, 64239}, {38996, 2107}, {39056, 668}, {39057, 670}, {46390, 4010}, {55053, 2665}
X(68886) = crosspoint of X(i) and X(j) for these (i,j): {6, 4584}, {190, 292}
X(68886) = crosssum of X(i) and X(j) for these (i,j): {2, 21832}, {239, 649}
X(68886) = crossdifference of every pair of points on line {1, 1655}
X(68886) = X(i)-line conjugate of X(j) for these (i,j): {875, 2107}, {4107, 30667}, {4375, 19579}
X(68886) = barycentric product X(i)*X(j) for these {i,j}: {292, 27854}, {512, 2669}, {513, 2664}, {514, 21788}, {523, 56837}, {649, 17759}, {656, 15148}, {659, 40796}, {661, 2106}, {667, 52049}, {669, 41535}, {798, 40874}, {875, 39028}, {1019, 21897}, {1577, 56388}, {2254, 56856}, {3572, 39916}, {4444, 51331}, {4589, 38978}, {7649, 20796}, {30665, 40772}, {57129, 58367}
X(68886) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 53216}, {31, 53624}, {649, 39925}, {661, 43685}, {665, 64239}, {667, 2665}, {669, 2107}, {798, 54980}, {875, 63892}, {1919, 51333}, {2106, 799}, {2664, 668}, {2669, 670}, {15148, 811}, {17759, 1978}, {20796, 4561}, {21788, 190}, {21897, 4033}, {27854, 1921}, {38978, 4010}, {39916, 27853}, {40772, 41072}, {40796, 4583}, {40874, 4602}, {41535, 4609}, {51331, 3570}, {52049, 6386}, {56388, 662}, {56837, 99}, {56856, 51560}, {58864, 40798}
X(68886) = {X(798),X(4784)}-harmonic conjugate of X(649)

X(68887) = ANTI-ORTHIC AXIS INTERCEPT OF X(10)X(190)

Barycentrics    a*(a^3*b - 2*a*b^3 + a^3*c - 2*a^2*b*c + a*b^2*c + b^3*c + a*b*c^2 - 2*a*c^3 + b*c^3) : :

X(68887) lies on these lines: {1, 3122}, {10, 190}, {44, 513}, {58, 662}, {191, 2640}, {292, 34916}, {320, 18201}, {524, 20984}, {894, 24350}, {1018, 60725}, {1193, 3248}, {2108, 24482}, {2292, 2643}, {3216, 9359}, {3792, 45763}, {4277, 67211}, {4368, 24517}, {4465, 4708}, {4643, 36263}, {8301, 36267}, {26030, 26076}, {26064, 26081}, {26273, 29639}, {64751, 67207}

X(68887) = X(i)-isoconjugate of X(j) for these (i,j): {2, 35107}, {6, 35155}
X(68887) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 35155}, {32664, 35107}, {35089, 75}
X(68887) = X(3122)-line conjugate of X(1)
X(68887) = barycentric product X(i)*X(j) for these {i,j}: {1, 35103}, {75, 5163}, {896, 46799}
X(68887) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 35155}, {31, 35107}, {5163, 1}, {35103, 75}, {46799, 46277}

X(68888) = X(30)X(511)∩X(650)X(21958)

Barycentrics    (b - c)*(-a^4 + a^2*b^2 + 3*a^2*b*c + b^3*c + a^2*c^2 + 2*b^2*c^2 + b*c^3) : :

X(68888) lies on these lines: {30, 511}, {650, 21958}, {656, 21302}, {663, 2517}, {693, 48303}, {1459, 47729}, {1577, 48307}, {2533, 50353}, {3716, 4036}, {4040, 4086}, {4391, 48340}, {4397, 46385}, {4560, 50338}, {4768, 50346}, {4775, 50331}, {4815, 47724}, {4978, 48293}, {6129, 47843}, {6133, 48331}, {7650, 42312}, {8632, 47127}, {17072, 68772}, {18070, 24006}, {21052, 48165}, {21301, 50332}, {23752, 47695}, {24720, 51648}, {25380, 31947}, {40086, 65482}, {43041, 48152}, {44444, 48131}, {46110, 54238}, {47844, 48322}, {48288, 50345}, {48302, 50334}

X(68888) = crossdifference of every pair of points on line {6, 4020}
X(68888) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {663, 2517, 8062}, {4036, 48306, 3716}

X(68889) = X(1)X(2)∩X(5)X(4514)

Barycentrics    a^4 - a^2*b^2 - 3*a^2*b*c - b^3*c - a^2*c^2 - 2*b^2*c^2 - b*c^3 : :

X(68889) lies on these lines: {1, 2}, {5, 4514}, {12, 4030}, {21, 4696}, {35, 4692}, {46, 24349}, {55, 4385}, {75, 5687}, {100, 4968}, {183, 39731}, {191, 62222}, {242, 4222}, {312, 3295}, {318, 11398}, {321, 3871}, {333, 34790}, {341, 405}, {442, 32850}, {495, 7270}, {595, 27064}, {596, 62300}, {894, 5264}, {958, 4737}, {986, 32920}, {993, 9369}, {1001, 46937}, {1058, 28808}, {1089, 3685}, {1215, 5255}, {1219, 3523}, {1220, 5266}, {1279, 13741}, {1330, 63134}, {1403, 11501}, {1621, 3701}, {1909, 5088}, {1914, 21021}, {2292, 32927}, {2476, 5014}, {2886, 5100}, {3496, 21101}, {3555, 14829}, {3579, 32939}, {3697, 17277}, {3699, 5044}, {3729, 61763}, {3744, 13740}, {3750, 63800}, {3883, 21075}, {3915, 32931}, {3931, 32926}, {3992, 5259}, {3996, 5295}, {4095, 41239}, {4388, 21077}, {4400, 24326}, {4434, 37607}, {4645, 13407}, {4647, 48696}, {4680, 37719}, {4723, 5260}, {4894, 7951}, {5011, 22011}, {5047, 52353}, {5192, 62806}, {5248, 56311}, {5795, 21290}, {5839, 31402}, {7155, 7162}, {7247, 7767}, {8715, 32932}, {9708, 44720}, {9709, 19804}, {11365, 26264}, {11491, 37619}, {12514, 32937}, {15254, 59577}, {17164, 63136}, {17165, 56288}, {24443, 32923}, {26066, 49688}, {37568, 63996}, {38000, 67979}, {46897, 57280}, {49553, 51630}, {62297, 68615}

X(68889) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 3757, 16817}, {10, 50607, 8}, {12, 4030, 5015}, {55, 4385, 7283}, {1089, 3746, 3685}, {16817, 68245, 10}

X(68890) = X(1)X(16720)∩X(30)X(511)

Barycentrics    a^3*b + a^3*c - 2*a^2*b*c + a*b^2*c - b^3*c + a*b*c^2 - b*c^3 : :

X(68890) lies on these lines: {1, 16720}, {10, 25368}, {30, 511}, {37, 30109}, {75, 24254}, {83, 213}, {192, 3735}, {1278, 24282}, {1500, 3666}, {1572, 3729}, {1573, 49516}, {1739, 68756}, {2176, 33937}, {2238, 4986}, {2896, 6542}, {3008, 6704}, {3230, 3263}, {3875, 9620}, {3884, 59515}, {3923, 10800}, {3934, 4095}, {3954, 17152}, {3992, 4465}, {3997, 49481}, {4362, 24264}, {4376, 37610}, {4424, 12783}, {4692, 24330}, {4696, 4721}, {4805, 5014}, {5091, 17763}, {6308, 37619}, {7764, 49613}, {7976, 32117}, {8290, 62609}, {9903, 50016}, {9997, 49455}, {10027, 20924}, {11606, 11611}, {12122, 63444}, {12264, 50023}, {13078, 42397}, {16969, 33942}, {17266, 31268}, {17310, 31168}, {17750, 39731}, {17752, 33940}, {17755, 52963}, {18061, 33889}, {20016, 20088}, {20432, 40886}, {20893, 49777}, {21802, 26965}, {21888, 57029}, {24281, 50010}, {24357, 30116}, {24358, 40091}, {36230, 49781}, {41623, 50026}, {49758, 49774}, {49773, 50011}, {50014, 57038}, {54933, 60181}

X(68890) = crossdifference of every pair of points on line {6, 50521}
X(68890) = barycentric product X(29669)*X(49375)
X(68890) = barycentric quotient X(21544)/X(16687)
X(68890) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {75, 40859, 50025}, {10027, 20924, 36226}, {17152, 28598, 3954}, {20432, 40886, 63443}

X(68891) = X(1)X(6)∩X(2)X(18209)

Barycentrics    a*(a^4 - 2*a^2*b*c - b^3*c - b^2*c^2 - b*c^3) : :

X(68891) lies on these lines: {1, 6}, {2, 18209}, {82, 21803}, {560, 17368}, {594, 8300}, {1030, 2108}, {1580, 3589}, {1582, 17369}, {2210, 17289}, {3770, 24294}, {5009, 24295}, {7246, 45782}, {17123, 18162}, {18082, 21094}, {24340, 27697}

X(68892) = X(1)X(6)∩X(63)X(18209)

Barycentrics    a*(b^4 + b^3*c + b^2*c^2 + b*c^3 + c^4) : :

X(68892) lies on these lines: {1, 6}, {63, 18209}, {75, 7237}, {141, 18168}, {190, 57946}, {313, 561}, {599, 18207}, {668, 57942}, {2643, 48628}, {3661, 4118}, {3721, 4446}, {3763, 18208}, {4016, 12782}, {4475, 17228}, {4665, 17891}, {4687, 20703}, {5224, 20590}, {17233, 17470}, {17292, 20274}, {17446, 33299}, {29667, 31090}, {49519, 56023}

X(68892) = isotomic conjugate of X(14623)
X(68892) = isotomic conjugate of the isogonal conjugate of X(14620)
X(68892) = X(i)-isoconjugate of X(j) for these (i,j): {31, 14623}, {514, 59001}, {1980, 9065}
X(68892) = X(2)-Dao conjugate of X(14623)
X(68892) = barycentric product X(i)*X(j) for these {i,j}: {1, 30149}, {72, 46506}, {76, 14620}, {100, 63814}
X(68892) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14623}, {692, 59001}, {1978, 9065}, {9008, 1919}, {14620, 6}, {30149, 75}, {46506, 286}, {63814, 693}
X(68892) = {X(141),X(51836)}-harmonic conjugate of X(18168)

X(68893) = X(1)X(6)∩X(2)X(2241)

Barycentrics    a*(a^3 - 3*a*b*c - b^2*c - b*c^2) : :

X(68898) lies on these lines: {1, 6}, {2, 2241}, {21, 1015}, {32, 3616}, {35, 16604}, {39, 1621}, {172, 551}, {187, 5253}, {274, 4366}, {593, 763}, {595, 24512}, {668, 16918}, {993, 63493}, {1086, 33870}, {1125, 1914}, {1333, 28619}, {1500, 33854}, {1572, 54392}, {1573, 5047}, {1574, 3871}, {1575, 3746}, {2242, 3622}, {2275, 5248}, {2295, 40091}, {2476, 9665}, {3496, 29820}, {3624, 4386}, {3691, 50028}, {3727, 30117}, {3735, 28082}, {3915, 17750}, {4428, 31448}, {4657, 17192}, {5035, 61302}, {5284, 16589}, {5337, 17397}, {6155, 29821}, {6537, 21341}, {7031, 25055}, {7824, 27195}, {7839, 32026}, {9599, 10198}, {9619, 62829}, {9651, 11114}, {10585, 31415}, {10587, 31409}, {10987, 25440}, {15172, 21956}, {16865, 31456}, {16915, 53680}, {16916, 64133}, {17000, 17143}, {17302, 33955}, {17962, 39949}, {18755, 49997}, {20179, 31996}, {21793, 37522}, {21888, 37563}, {22036, 32923}, {22065, 23530}, {30148, 41269}, {31451, 61155}, {32942, 52538}, {34460, 37621}, {54382, 64675}, {57017, 64536}

X(68893) = crossdifference of every pair of points on line {513, 58289}
X(68893) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {405, 16781, 16975}, {1001, 16502, 5283}, {1125, 1914, 5277}, {5259, 16784, 1107}

X(68894) = X(30)X(511)∩X(650)X(23815)

Barycentrics    (b - c)*(-a^3 + 3*a*b*c + b^2*c + b*c^2) : :

X(68894) lies on these lines: {30, 511}, {650, 23815}, {659, 4978}, {667, 4801}, {693, 47815}, {764, 4560}, {1635, 48569}, {2530, 17494}, {2533, 21385}, {3733, 68123}, {3837, 48003}, {4010, 47970}, {4040, 48279}, {4041, 48115}, {4063, 21146}, {4106, 47966}, {4382, 47929}, {4498, 48119}, {4705, 46403}, {4724, 48273}, {4728, 48553}, {4804, 47936}, {4806, 48004}, {4808, 47687}, {4822, 47933}, {4824, 48086}, {4830, 50512}, {4834, 48108}, {4983, 47969}, {4992, 48058}, {7265, 48083}, {8043, 40086}, {9508, 23789}, {14838, 19947}, {20295, 47949}, {21192, 58375}, {21260, 47965}, {23765, 48321}, {24719, 47959}, {24720, 50504}, {26824, 48393}, {27855, 52619}, {30592, 47840}, {31150, 47819}, {45320, 48561}, {47650, 47708}, {47663, 47719}, {47664, 48410}, {47679, 47968}, {47715, 48103}, {47716, 50340}, {47793, 48170}, {47794, 48184}, {47795, 48226}, {47796, 48240}, {47811, 47839}, {47812, 47837}, {47816, 48167}, {47817, 47833}, {47827, 48556}, {47904, 48597}, {47906, 48114}, {47926, 48122}, {47927, 48121}, {47932, 48151}, {47934, 48116}, {47935, 48148}, {47946, 48085}, {47962, 48092}, {47963, 48091}, {47974, 48351}, {47993, 48051}, {47994, 48049}, {48000, 48059}, {48001, 48053}, {48002, 48052}, {48005, 48050}, {48032, 48305}, {48090, 59672}, {48111, 48301}, {48150, 48291}, {48196, 48198}, {48214, 48218}, {48253, 48566}, {48280, 49290}, {48288, 48334}, {48295, 48331}, {48407, 50328}, {48570, 58144}

X(68894) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {659, 4978, 52601}, {659, 47889, 47818}, {693, 47815, 47875}, {3837, 48003, 65449}, {4382, 47929, 48267}, {4498, 48119, 50352}, {4978, 47818, 47889}, {14838, 48406, 19947}, {31150, 47819, 47888}, {47818, 47889, 52601}, {47965, 48089, 21260}

X(68895) = X(1)X(596)∩X(30)X(511)

Barycentrics    (b + c)*(-(a^2*b) - a*b^2 - a^2*c + 2*a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(68895) lies on these lines: {1, 596}, {2, 46895}, {8, 24068}, {10, 321}, {30, 511}, {31, 49683}, {36, 32845}, {58, 63996}, {65, 2901}, {75, 42285}, {145, 64429}, {191, 27368}, {192, 30116}, {210, 50083}, {244, 4975}, {350, 21208}, {354, 50122}, {386, 4734}, {392, 42051}, {442, 4918}, {484, 17763}, {551, 1962}, {846, 54335}, {942, 4891}, {960, 64185}, {986, 50605}, {993, 32934}, {995, 3210}, {1125, 3666}, {1149, 14752}, {1213, 24067}, {1215, 4868}, {1266, 20893}, {1278, 17461}, {1500, 22011}, {1698, 64436}, {1739, 4358}, {1764, 17733}, {1837, 44040}, {1854, 56146}, {2051, 17748}, {2238, 4115}, {2294, 3950}, {2321, 4016}, {2650, 3244}, {2667, 49479}, {3125, 4037}, {3175, 3753}, {3178, 11263}, {3187, 49500}, {3216, 25253}, {3293, 56318}, {3454, 3704}, {3616, 64431}, {3624, 64432}, {3626, 64428}, {3634, 17070}, {3635, 63354}, {3636, 58380}, {3663, 18697}, {3670, 3702}, {3678, 22325}, {3679, 17163}, {3685, 30117}, {3721, 21070}, {3728, 4793}, {3741, 4717}, {3747, 50023}, {3754, 63800}, {3797, 30109}, {3822, 48643}, {3828, 27798}, {3869, 64184}, {3872, 25254}, {3877, 50106}, {3891, 37610}, {3892, 42055}, {3918, 66043}, {3923, 48866}, {3930, 22035}, {3952, 31855}, {3956, 4096}, {3977, 50759}, {3985, 16611}, {3995, 56191}, {4009, 59669}, {4013, 17757}, {4052, 53036}, {4067, 59302}, {4084, 67976}, {4099, 21808}, {4103, 52959}, {4134, 4685}, {4276, 11688}, {4365, 22024}, {4427, 39766}, {4511, 44311}, {4568, 17759}, {4658, 41813}, {4674, 62227}, {4678, 64427}, {4680, 33094}, {4693, 24225}, {4694, 4742}, {4709, 21080}, {4745, 59718}, {4871, 24168}, {4880, 32919}, {5180, 32842}, {5251, 32936}, {5315, 32924}, {5603, 42049}, {5620, 34895}, {5657, 42047}, {5692, 32860}, {5695, 48863}, {5835, 50067}, {5902, 32915}, {6381, 35544}, {6735, 62305}, {6742, 14158}, {6765, 67848}, {7230, 21025}, {8669, 37619}, {8715, 64753}, {8720, 37620}, {9623, 25255}, {9708, 17262}, {10915, 23555}, {11552, 32949}, {11599, 11611}, {11684, 64072}, {14210, 17205}, {15953, 17760}, {16086, 62392}, {16474, 32940}, {16887, 17762}, {17157, 49469}, {17495, 34587}, {17734, 37759}, {17749, 19582}, {17874, 49626}, {18393, 29849}, {18673, 64117}, {18698, 53594}, {19862, 64433}, {19875, 27812}, {19877, 64435}, {19883, 53034}, {20681, 52894}, {20691, 21067}, {21024, 24044}, {21081, 56949}, {22836, 63982}, {24058, 27042}, {24077, 26772}, {24080, 53675}, {24170, 33939}, {24248, 48835}, {24850, 63292}, {24851, 36974}, {24880, 56313}, {25055, 27811}, {25081, 59585}, {25248, 29433}, {25295, 49532}, {25439, 32920}, {30115, 32932}, {32921, 62828}, {32922, 40091}, {32927, 48696}, {33936, 49518}, {37614, 50044}, {40977, 59579}, {42027, 53114}, {42440, 49600}, {43972, 49743}, {43993, 62804}, {46932, 64437}, {48843, 66071}, {49452, 66640}, {49462, 66687}, {49477, 54282}, {49483, 66694}, {49493, 66674}, {49560, 68478}, {50115, 53037}, {51108, 58381}, {51578, 62609}, {53332, 62755}, {53601, 57040}, {58393, 58565}, {59261, 60276}, {59722, 67847}

X(68895) = isogonal conjugate of X(59072)
X(68895) = crossdifference of every pair of points on line {6, 57129}
X(68895) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 64071, 4065}, {10, 3159, 4075}, {10, 4135, 4125}, {321, 4424, 10}, {350, 57029, 21208}, {756, 4714, 10}, {1089, 4642, 10}, {1125, 3743, 58387}, {1125, 24176, 6532}, {1125, 67983, 24176}, {1739, 4358, 49993}, {2292, 4647, 10}, {3701, 3987, 10}, {3704, 63997, 3454}, {3743, 49598, 1125}, {3992, 4695, 10}, {3994, 4695, 3992}, {4427, 39766, 52680}, {17164, 64071, 1}, {20691, 22036, 21067}

X(68896) = X(1)X(21222)∩X(30)X(511)

Barycentrics    (b - c)*(-(a^2*b) + a*b^2 - a^2*c - 2*a*b*c + b^2*c + a*c^2 + b*c^2) : :

X(68896) lies on these lines: {1, 21222}, {10, 2254}, {30, 511}, {244, 1111}, {551, 14413}, {596, 50342}, {665, 1639}, {672, 68811}, {764, 4010}, {876, 43534}, {905, 59672}, {946, 68329}, {1026, 3952}, {1125, 3716}, {1577, 23789}, {1638, 45338}, {1734, 4462}, {2530, 4129}, {3159, 22037}, {3244, 4895}, {3634, 25380}, {3669, 59590}, {3679, 53364}, {3777, 48267}, {3828, 45328}, {3837, 23814}, {3904, 49276}, {4040, 17496}, {4075, 18004}, {4088, 66995}, {4120, 52745}, {4147, 48018}, {4170, 48334}, {4368, 14432}, {4391, 4905}, {4448, 14419}, {4458, 21201}, {4504, 48345}, {4560, 47970}, {4674, 53356}, {4707, 21132}, {4712, 4738}, {4724, 48284}, {4728, 35353}, {4750, 14433}, {4791, 23796}, {4794, 48325}, {4922, 6161}, {4944, 54249}, {4958, 22035}, {4978, 23738}, {4984, 21832}, {4985, 50354}, {5248, 53308}, {5267, 8648}, {8666, 53286}, {8715, 53278}, {10015, 50357}, {14315, 23808}, {14431, 36848}, {14838, 48562}, {17072, 48075}, {18160, 23790}, {19957, 21198}, {20888, 65101}, {21385, 50343}, {23057, 51071}, {23765, 48273}, {23815, 59714}, {23828, 62415}, {25259, 49278}, {30565, 35293}, {42285, 46781}, {44902, 58467}, {45671, 47811}, {47661, 60529}, {47676, 49300}, {47680, 49301}, {47682, 49275}, {47683, 47969}, {47694, 48320}, {47725, 49302}, {47726, 49273}, {47872, 48569}, {47893, 48553}, {47906, 50449}, {47918, 48409}, {47959, 48410}, {48032, 53536}, {48080, 48335}, {48083, 50351}, {48144, 48578}, {48270, 59721}, {48286, 53523}, {48298, 48352}, {48305, 48323}, {48339, 50761}, {48571, 53382}, {50348, 50453}, {52872, 59724}, {53389, 58036}, {53395, 62858}, {59753, 59971}

X(68896) = isogonal conjugate of X(59071)
X(68896) = crossdifference of every pair of points on line {6, 23404}
X(68896) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 23795, 2254}, {1577, 48151, 23789}, {2254, 3762, 10}, {2530, 48265, 4129}, {3716, 3960, 1125}, {3766, 23829, 20520}, {4391, 4905, 50337}, {4724, 48321, 48284}, {10015, 50357, 62435}, {21222, 53343, 1}, {23738, 48264, 4978}, {23814, 59737, 3837}

X(68897) = X(1)X(21021)∩X(10)X(37)

Barycentrics    (b + c)*(-a^3 - a*b*c + b^2*c + b*c^2) : :

X(68897) lies on these lines: {1, 21021}, {8, 762}, {10, 37}, {44, 57017}, {65, 22036}, {80, 4876}, {115, 4071}, {187, 4434}, {190, 66152}, {213, 3701}, {239, 1016}, {312, 41232}, {512, 3700}, {519, 4103}, {536, 57029}, {668, 8682}, {742, 6381}, {1018, 4037}, {1055, 62659}, {1089, 2295}, {2238, 3992}, {2242, 29649}, {2298, 41261}, {3263, 50025}, {3509, 68482}, {3726, 49999}, {3952, 21839}, {3954, 17751}, {3985, 52963}, {3997, 4125}, {4075, 21879}, {4385, 17750}, {4692, 24512}, {4696, 20963}, {4723, 50012}, {4865, 5475}, {5291, 17763}, {5722, 17299}, {5724, 49476}, {7794, 24211}, {7813, 24318}, {14210, 36226}, {17023, 30818}, {17033, 33932}, {17034, 18137}, {17131, 24333}, {17752, 33939}, {20891, 33941}, {20947, 40859}, {21802, 27040}, {24254, 30114}, {29671, 31476}, {33889, 49753}, {34542, 37716}, {39774, 44418}, {56530, 61164}

X(68897) = reflection of X(20693) in X(4103)
X(68897) = X(i)-isoconjugate of X(j) for these (i,j): {27, 17971}, {58, 17946}, {81, 17954}, {86, 17961}, {514, 17939}, {649, 17929}, {849, 11611}, {1019, 2703}, {1412, 11609}, {1790, 17981}, {3733, 65239}, {4556, 18015}, {4610, 18002}, {35147, 57129}, {53689, 54308}
X(68897) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 17946}, {4075, 11611}, {5375, 17929}, {6741, 60484}, {35079, 7192}, {40586, 17954}, {40599, 11609}, {40600, 17961}
X(68897) = crosspoint of X(17763) and X(17790)
X(68897) = crosssum of X(i) and X(j) for these (i,j): {3666, 57039}, {17954, 17961}
X(68897) = crossdifference of every pair of points on line {3733, 8027}
X(68897) = barycentric product X(i)*X(j) for these {i,j}: {10, 17763}, {37, 17790}, {72, 17987}, {100, 18003}, {321, 5291}, {422, 3695}, {594, 19623}, {668, 17989}, {756, 5209}, {2787, 3952}, {3701, 5061}, {4036, 17944}, {4705, 17935}, {5006, 28654}, {5040, 27808}, {17977, 41013}
X(68897) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 17946}, {42, 17954}, {100, 17929}, {210, 11609}, {213, 17961}, {228, 17971}, {594, 11611}, {692, 17939}, {1018, 65239}, {1824, 17981}, {2680, 42753}, {2787, 7192}, {3690, 57680}, {3695, 57847}, {3700, 60484}, {3952, 35147}, {4557, 2703}, {4705, 18015}, {5006, 593}, {5040, 3733}, {5061, 1014}, {5209, 873}, {5291, 81}, {17763, 86}, {17790, 274}, {17935, 4623}, {17944, 52935}, {17977, 1444}, {17987, 286}, {17989, 513}, {18003, 693}, {19623, 1509}, {50487, 18002}
X(68897) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4095, 63800, 1500}, {30114, 33931, 24254}

X(68898) = X(1)X(6)∩X(36)X(2108)

Barycentrics    a*(a^4 + a*b^2*c - b^3*c + a*b*c^2 - b^2*c^2 - b*c^3) : :

X(68898) lies on these lines: {1, 6}, {36, 2108}, {100, 8626}, {101, 3783}, {312, 4112}, {350, 24294}, {519, 8300}, {560, 17233}, {667, 1734}, {730, 18047}, {740, 8625}, {753, 898}, {761, 813}, {899, 5168}, {1016, 1110}, {1580, 3912}, {1582, 2321}, {1739, 1929}, {2210, 6542}, {2239, 3573}, {2243, 5184}, {2251, 8298}, {3507, 40910}, {3802, 8624}, {4475, 56513}, {5144, 58863}, {7122, 17280}, {8299, 11364}, {8622, 40790}, {10789, 37610}, {10791, 54291}, {14665, 28883}, {17231, 18209}, {17799, 68759}, {29824, 41252}, {30113, 50775}

X(68898) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {238, 56530, 1}, {1083, 5291, 238}

X(68899) = X(1)X(4486)∩X(30)X(511)

Barycentrics    (b - c)*(-a^4 - a*b^2*c + b^3*c - a*b*c^2 + b^2*c^2 + b*c^3) : :

X(68899) lies on these lines: {1, 4486}, {2, 14438}, {30, 511}, {101, 668}, {116, 1015}, {118, 44951}, {150, 9263}, {291, 23596}, {693, 4508}, {764, 24721}, {1960, 27929}, {3227, 10708}, {3762, 4375}, {3960, 25381}, {4107, 27919}, {4435, 24287}, {4922, 62552}, {6710, 27076}, {10725, 44941}, {11712, 17793}, {13466, 65899}, {20096, 31298}, {25356, 59837}, {27195, 31273}, {30565, 68829}, {40479, 58418}

X(68899) = barycentric quotient X(39323)/X(42703)

X(68900) = X(1)X(4455)∩X(30)X(511)

Barycentrics    a*(b - c)*(a^2*b^2 + a^2*b*c - a*b^2*c + a^2*c^2 - a*b*c^2 - b^2*c^2) : :

X(68900) lies on these lines: {1, 4455}, {2, 14404}, {6, 4164}, {30, 511}, {81, 5040}, {291, 9508}, {650, 24534}, {661, 50524}, {668, 670}, {693, 20983}, {694, 46149}, {798, 4367}, {940, 53281}, {1015, 1084}, {1469, 7212}, {1635, 38238}, {2533, 7199}, {2978, 47666}, {3227, 3228}, {3572, 21832}, {3768, 9267}, {4010, 17794}, {4147, 21191}, {4369, 50491}, {4394, 43931}, {4449, 20979}, {4481, 4705}, {4498, 54251}, {4507, 4932}, {4649, 50456}, {4885, 25142}, {4922, 9263}, {4928, 14426}, {4979, 50485}, {5752, 62437}, {7192, 50487}, {7234, 18197}, {8027, 47776}, {13466, 35073}, {14296, 52044}, {16482, 24508}, {17494, 50521}, {17592, 22216}, {17946, 60045}, {19581, 27855}, {20954, 48279}, {21051, 42327}, {24747, 24768}, {25050, 25332}, {25316, 25319}, {25323, 25327}, {25511, 25636}, {27076, 36950}, {27195, 31639}, {31290, 50497}, {39360, 39361}, {39548, 50449}, {40478, 40479}, {48000, 50510}, {48008, 50514}, {48141, 50481}, {48147, 50483}, {50493, 52601}

X(68900) = isogonal conjugate of X(53624)
X(68900) = isotomic conjugate of X(53216)
X(68900) = crossdifference of every pair of points on line {6, 1045}
X(68900) = X(i)-line conjugate of X(j) for these (i,j): {3572, 54980}, {4164, 6}
X(68900) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 53271, 4164}, {798, 4367, 54254}

X(68901) = X(1)X(4107)∩X(514)X(661)

Barycentrics    (b - c)*(-(a^2*b^2) - a^2*b*c + a*b^2*c - a^2*c^2 + a*b*c^2 + b^2*c^2) : :
X(68901) = X[192] + 3 X[53368], X[3766] - 3 X[4728], 5 X[4687] - 3 X[14407], 3 X[4927] - X[20507], 2 X[20520] - 3 X[21204]

X(68901) lies on these lines: {1, 4107}, {2, 21832}, {192, 53368}, {244, 17761}, {335, 2786}, {512, 4369}, {514, 661}, {522, 21191}, {523, 42327}, {665, 812}, {784, 4500}, {802, 21348}, {876, 4010}, {891, 4928}, {900, 21211}, {1912, 50516}, {1978, 4568}, {2978, 4932}, {3005, 4151}, {3261, 17458}, {3661, 21053}, {3739, 4145}, {3837, 21261}, {4041, 30765}, {4079, 7199}, {4083, 4885}, {4106, 54249}, {4132, 52602}, {4139, 17066}, {4374, 21834}, {4411, 64867}, {4444, 24290}, {4458, 18009}, {4687, 14407}, {4762, 45658}, {4927, 20507}, {4940, 29198}, {5029, 16826}, {6372, 48049}, {8632, 24623}, {8714, 23803}, {9508, 40549}, {16777, 53276}, {17217, 57234}, {18061, 18149}, {18080, 21113}, {18197, 53581}, {20520, 21204}, {20907, 21206}, {20954, 21123}, {21146, 60349}, {21180, 21200}, {21297, 52745}, {21836, 52619}, {22044, 22046}, {25667, 40627}, {27647, 50557}, {28225, 60348}, {29350, 47779}, {30023, 45745}, {30094, 48268}, {53523, 65490}

X(68901) = midpoint of X(i) and X(j) for these {i,j}: {693, 3250}, {876, 4010}, {3261, 17458}, {4079, 7199}, {4106, 54249}, {4374, 21834}, {17217, 57234}, {18080, 21113}, {20954, 21123}, {21146, 60349}, {21297, 52745}, {21836, 52619}
X(68901) = reflection of X(i) in X(j) for these {i,j}: {9508, 40549}, {20907, 21206}
X(68901) = complement of X(21832)
X(68901) = complement of the isogonal conjugate of X(4584)
X(68901) = complement of the isotomic conjugate of X(4639)
X(68901) = X(i)-complementary conjugate of X(j) for these (i,j): {58, 35119}, {81, 38989}, {99, 20333}, {100, 46842}, {101, 35068}, {110, 17755}, {172, 35078}, {190, 45162}, {291, 8287}, {292, 115}, {295, 15526}, {334, 21253}, {335, 125}, {337, 127}, {651, 50440}, {660, 1211}, {662, 17793}, {741, 1086}, {799, 20542}, {805, 4357}, {813, 1213}, {894, 2679}, {1808, 16596}, {1911, 16592}, {1922, 1084}, {2196, 16573}, {2311, 1146}, {4562, 3454}, {4570, 27929}, {4583, 21245}, {4584, 10}, {4589, 141}, {4600, 27854}, {4639, 2887}, {5378, 4129}, {9506, 57461}, {18268, 1015}, {18827, 116}, {18895, 53575}, {34067, 16589}, {36066, 3739}, {36800, 124}, {37128, 11}, {37134, 3846}, {39276, 64523}, {40017, 21252}, {56154, 26932}, {65258, 3741}, {65285, 21240}, {66937, 6547}
X(68901) = X(i)-Ceva conjugate of X(j) for these (i,j): {876, 514}, {4010, 2786}
X(68901) = X(i)-isoconjugate of X(j) for these (i,j): {6, 53624}, {32, 53216}, {100, 51333}, {101, 2665}, {110, 54980}, {662, 2107}, {692, 39925}, {813, 40769}, {1576, 43685}, {3573, 63874}, {32666, 64239}
X(68901) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 53624}, {244, 54980}, {350, 874}, {1015, 2665}, {1084, 2107}, {1086, 39925}, {4858, 43685}, {6376, 53216}, {8054, 51333}, {27854, 649}, {35094, 64239}, {39056, 100}, {39057, 99}, {40623, 40769}, {46390, 4455}
X(68901) = crosspoint of X(i) and X(j) for these (i,j): {2, 4639}, {335, 1978}
X(68901) = crosssum of X(1914) and X(1919)
X(68901) = crossdifference of every pair of points on line {31, 1979}
X(68901) = barycentric product X(i)*X(j) for these {i,j}: {335, 27854}, {512, 41535}, {513, 52049}, {514, 17759}, {523, 2669}, {661, 40874}, {693, 2664}, {850, 56837}, {876, 39028}, {1019, 58367}, {1577, 2106}, {2254, 64238}, {3261, 21788}, {3766, 40796}, {4444, 39916}, {4486, 40742}, {7199, 21897}, {14208, 15148}, {20796, 46107}, {20948, 56388}
X(68901) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 53624}, {75, 53216}, {512, 2107}, {513, 2665}, {514, 39925}, {649, 51333}, {659, 40769}, {661, 54980}, {876, 63892}, {918, 64239}, {1577, 43685}, {2106, 662}, {2664, 100}, {2669, 99}, {3572, 63874}, {15148, 162}, {17759, 190}, {20796, 1331}, {21788, 101}, {21897, 1018}, {27854, 239}, {30665, 40798}, {38978, 4455}, {39028, 874}, {39916, 3570}, {40742, 37207}, {40772, 30664}, {40796, 660}, {40874, 799}, {41535, 670}, {52049, 668}, {56388, 163}, {56837, 110}, {56856, 36086}, {58367, 4033}, {64238, 51560}

X(68902) = X(2)X(17989)∩X(37)X(804)

Barycentrics    (b^2 - c^2)*(-(a^2*b^2) - a^2*b*c + a*b^2*c - a^2*c^2 + a*b*c^2 + b^2*c^2) : :
X(68902) = 3 X[21052] + X[21834]

X(68902) lies on these lines: {2, 17989}, {10, 4155}, {37, 804}, {321, 9148}, {523, 1577}, {661, 23949}, {693, 58289}, {756, 4728}, {812, 3842}, {814, 6586}, {824, 21260}, {850, 58360}, {900, 24462}, {918, 3837}, {1089, 20908}, {1215, 4928}, {1824, 47206}, {2486, 3125}, {2533, 4079}, {3004, 58364}, {3005, 58361}, {3700, 23301}, {3766, 21349}, {3805, 42327}, {3995, 53365}, {4024, 65152}, {4505, 6386}, {4557, 40865}, {17069, 25126}, {21052, 21834}, {21053, 21959}, {21350, 59713}, {21721, 23948}, {23815, 28851}, {23818, 48270}, {24085, 46390}, {30968, 47874}, {31993, 45689}, {48273, 58304}, {60028, 60288}, {66267, 66282}

X(68902) = midpoint of X(i) and X(j) for these {i,j}: {10, 22043}, {2533, 4079}, {3766, 21349}
X(68902) = reflection of X(17990) in X(3842)
X(68902) = X(i)-isoconjugate of X(j) for these (i,j): {58, 53624}, {110, 2665}, {163, 39925}, {662, 51333}, {2107, 52935}, {2206, 53216}, {4556, 54980}, {8937, 53628}
X(68902) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 53624}, {115, 39925}, {244, 2665}, {1084, 51333}, {27854, 3733}, {39056, 662}, {39057, 4610}, {40603, 53216}, {46390, 8632}
X(68902) = crosspoint of X(i) and X(j) for these (i,j): {10, 4583}, {27808, 43534}
X(68902) = crossdifference of every pair of points on line {1333, 51328}
X(68902) = barycentric product X(i)*X(j) for these {i,j}: {513, 58367}, {523, 17759}, {661, 52049}, {693, 21897}, {850, 21788}, {1577, 2664}, {2106, 4036}, {2669, 4024}, {4079, 41535}, {4705, 40874}, {14618, 20796}, {24290, 64238}, {27854, 43534}, {35352, 39916}, {52623, 56837}
X(68902) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 53624}, {321, 53216}, {512, 51333}, {523, 39925}, {661, 2665}, {2106, 52935}, {2664, 662}, {2669, 4610}, {4036, 43685}, {4079, 2107}, {4088, 64239}, {4705, 54980}, {17759, 99}, {20796, 4558}, {21788, 110}, {21832, 40769}, {21897, 100}, {27854, 33295}, {38978, 8632}, {40796, 4584}, {40874, 4623}, {41535, 52612}, {52049, 799}, {56837, 4556}, {58367, 668}
X(68902) = {X(3837),X(58363)}-harmonic conjugate of X(18004)

X(68903) = X(1)X(3)∩X(11)X(145)

Barycentrics    a*(a-b-c)*(3*a^2-a*b-4*b^2-a*c+8*b*c-4*c^2) : :
X(68903) = 2*X[1]-X[1388], 4*X[1]-X[5204], 3*X[1]-X[37618], 5*X[1]-X[58887], 2*X[1388]-X[5204], 3*X[1388]-2*X[37618], 5*X[1388]-2*X[58887]

See David Nguyen and Ercole Suppa, euclid 8555.

X(68903) lies on these lines: {1, 3}, {2, 34710}, {4, 1317}, {8, 17774}, {11, 145}, {12, 10595}, {33, 33963}, {78, 33895}, {119, 54134}, {381, 37707}, {388, 10947}, {496, 61597}, {497, 3623}, {498, 10283}, {499, 5844}, {519, 11376}, {550, 11046}, {764, 4162}, {938, 67972}, {944, 12953}, {946, 37738}, {950, 51071}, {952, 10896}, {958, 5330}, {1001, 61026}, {1058, 10953}, {1124, 35810}, {1279, 38293}, {1320, 1392}, {1335, 35811}, {1387, 10573}, {1389, 11501}, {1479, 1483}, {1537, 37001}, {1616, 61397}, {1836, 63987}, {1837, 3244}, {1858, 17622}, {3036, 6931}, {3056, 67538}, {3058, 34605}, {3241, 10950}, {3243, 60910}, {3326, 67476}, {3474, 6049}, {3476, 5734}, {3486, 4345}, {3582, 50805}, {3583, 18526}, {3586, 61288}, {3621, 10589}, {3622, 5432}, {3632, 17606}, {3633, 50443}, {3635, 12053}, {3656, 45287}, {3680, 3689}, {3711, 4853}, {3715, 15829}, {3872, 4662}, {3874, 15558}, {3876, 4861}, {3880, 56387}, {3885, 56177}, {4308, 11246}, {4428, 51683}, {4511, 10912}, {4863, 64205}, {5252, 13464}, {5298, 34631}, {5326, 46934}, {5433, 12245}, {5434, 34629}, {5445, 34718}, {5533, 6971}, {5552, 5854}, {5603, 10893}, {5730, 22837}, {5790, 37735}, {5855, 10529}, {5882, 12701}, {5901, 12647}, {6174, 63133}, {6284, 7967}, {6762, 7082}, {6788, 65153}, {6924, 10087}, {7680, 10949}, {7681, 10956}, {7741, 12645}, {7743, 37711}, {7972, 12611}, {8192, 9673}, {8236, 60919}, {9581, 51093}, {9614, 61291}, {9656, 18393}, {9669, 37706}, {9670, 61286}, {9671, 30384}

X(68903) = reflection of X(i) in X(j) for these {i,j}: {1388, 1}, {5204, 1388}
X(68903) = X(4902)-reciprocal conjugate of X(85)
X(68903) = X(1)-crosssum of X(63208)
X(68903) = X(1388)-of-5th mixtilinear triangle
X(68903) = X(5204)-of-Mandart-incircle triangle
X(68903) = X(32534)-of-Hutson intouch triangle
X(68903) = pole of line {44426, 68118} with respect to polar circle
X(68903) = pole of line {1, 4004} with respect to Feuerbach hyperbola
X(68903) = pole of line {56, 32537} with respect to dual conic of Moses-Feuerbach circumconic
X(68903) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(1),X(4902)}}, {{A,B,C,X(3),X(33963)}}, {{A,B,C,X(4),X(25405)}}, {{A,B,C,X(9),X(64849)}}, {{A,B,C,X(939),X(37535)}}
X(68903) = barycentric product X(9)*X(4902)
X(68903) = barycentric quotient X(4902)/X(85)
X(68903) = trilinear product X(55)*X(4902)
X(68903) = trilinear quotient X(7)/X(4902)
X(68903) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 35, 37624}, {1, 46, 25405}, {1, 165, 64849}, {1, 1482, 56}, {1, 2098, 55}, {1, 2099, 3304}, {1, 3057, 34471}, {1, 3340, 20323}, {1, 5048, 2098}, {1, 5119, 15178}, {1, 5697, 10246}, {1, 7962, 2646}, {1, 7982, 1319}, {1, 10222, 2099}, {1, 11009, 999}, {1, 11224, 1420}, {1, 11531, 63208}, {1, 16189, 57}, {1, 16191, 3361}, {1, 16200, 65}, {1, 16204, 37550}, {1, 25415, 24928}, {1, 30323, 1385}, {1, 63210, 3}, {1, 64896, 21842}, {1, 64897, 3303}, {1, 64953, 64848}, {1, 64964, 354}, {1, 67300, 5217}, {56, 3303, 11508}, {56, 10965, 55}, {145, 5154, 12531}, {145, 11681, 66240}, {497, 3623, 37734}, {999, 11009, 64963}, {1317, 64192, 12763}, {1319, 7982, 37567}, {1482, 36279, 11280}, {2098, 34471, 3057}, {2646, 3057, 53053}, {3057, 34471, 55}, {3295, 37564, 55}, {3303, 10966, 55}, {3340, 20323, 4860}, {3623, 67262, 497}, {3635, 12053, 37740}, {4345, 20057, 3486}, {5048, 33176, 1}, {5603, 10944, 10895}, {5697, 10246, 5217}, {7962, 53053, 3057}, {7991, 37605, 63212}, {10222, 13600, 16200}, {10222, 46920, 1482}, {10246, 67300, 5697}, {11278, 25405, 46}, {11531, 63208, 1155}, {13601, 64964, 2099}, {21842, 64896, 12702}, {22767, 37622, 14882}, {24928, 25415, 5221}, {26351, 26352, 10388}, {37568, 64848, 64953}

X(68904) = X(1)X(3)∩X(7)X(5554)

Barycentrics    -(a*(a+b-c)*(a-b+c)*(a^3*b-a^2*b^2-a*b^3+b^4+a^3*c-a^2*b*c+2*a*b^2*c-a^2*c^2+2*a*b*c^2-2*b^2*c^2-a*c^3+c^4)) : :
X(68904) = 2*X[65]+X[1420], X[65]+X[37566], X[1420]-2*X[37566], X[9581]-2*X[67937]

See David Nguyen and Ercole Suppa, euclid 8556.

X(68904) lies on these lines: {1, 3}, {4, 12736}, {7, 5554}, {8, 18419}, {10, 64115}, {11, 33899}, {80, 18961}, {109, 3924}, {119, 18397}, {145, 5083}, {226, 11681}, {388, 3754}, {497, 66019}, {553, 34605}, {758, 1788}, {960, 31231}, {1071, 5727}, {1193, 26742}, {1210, 1519}, {1412, 41723}, {1421, 17054}, {1457, 24046}, {1698, 64041}, {1737, 5693}, {1739, 37694}, {1768, 22760}, {1835, 14257}, {1837, 6259}, {1845, 54200}, {1858, 61718}, {2003, 54418}, {2006, 34030}, {2057, 11523}, {2362, 26465}, {2771, 45631}, {2800, 3086}, {2950, 57278}, {3474, 64076}, {3476, 67051}, {3485, 5883}, {3555, 64736}, {3585, 7702}, {3671, 33815}, {3753, 9578}, {3812, 5219}, {3868, 4848}, {3869, 3911}, {3874, 49169}, {3878, 7288}, {3901, 41538}, {3913, 37736}, {3919, 4298}, {3922, 8581}, {3959, 52635}, {4018, 41389}, {4188, 18467}, {4293, 15528}, {4295, 5804}, {4551, 24440}, {4654, 11236}, {5435, 64047}, {5552, 15556}, {5587, 67970}, {5691, 64704}, {5692, 24914}, {5775, 40661}, {5806, 17634}, {5836, 17625}, {5884, 6256}, {5904, 26482}, {6001, 9581}, {6797, 18525}, {6921, 64139}, {7195, 59813}, {7354, 24465}, {7686, 9579}, {10107, 63994}, {10396, 12686}, {10571, 24443}, {10573, 11570}, {10598, 67988}, {10826, 61705}, {10895, 30290}, {10940, 39772}, {11220, 66247}, {11571, 12832}, {11729, 34753}, {12115, 18389}, {12432, 45701}, {12672, 50443}, {12711, 37723}, {14450, 41824}, {14923, 63987}, {16232, 26459}, {16616, 51790}, {17165, 56173}, {17451, 56546}, {17626, 66216}, {20320, 38955}, {22836, 66630}

X(68904) = midpoint of X(65) and X(37566)
X(68904) = reflection of X(i) in X(j) for these {i,j}: {1420, 37566}, {9581, 67937}
X(68904) = Zosma transform of X(52860)
X(68904) = X(25005)-beth conjugate of X(25005)
X(68904) = X(25005)-reciprocal conjugate of X(312)
X(68904) = X(7)-crosspoint of X(65002)
X(68904) = X(55)-crosssum of X(34524)
X(68904) = X(30552)-of-inverse-in-incircle triangle
X(68904) = X(37197)-of-intouch triangle
X(68904) = pole of line {1756, 65119} with respect to Evans 1st circle
X(68904) = pole of line {513, 46004} with respect to incircle
X(68904) = pole of line {513, 46004} with respect to de Longchamps ellipse
X(68904) = pole of line {1, 22792} with respect to Feuerbach hyperbola
X(68904) = pole of line {56, 4193} with respect to dual conic of Moses-Feuerbach circumconic
X(68904) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(1),X(25005)}}, {{A,B,C,X(3),X(46435)}}, {{A,B,C,X(4),X(2077)}}, {{A,B,C,X(55),X(34918)}}, {{A,B,C,X(58),X(32612)}}, {{A,B,C,X(80),X(65119)}}, {{A,B,C,X(959),X(5563)}}, {{A,B,C,X(961),X(5903)}}, {{A,B,C,X(994),X(1482)}, }{A,B,C,X(1168),X(35460)}}, {{A,B,C,X(3255),X(22768)}}, {{A,B,C,X(3577),X(49163)}, {{A,B,C,X(5048),X(34434)}}, {{A,B,C,X(5553),X(37561)}}, {{A,B,C,X(8069),X(52186)}}, {{A,B,C,X(12641),X(26358)}}, {{A,B,C,X(13476),X(20323)}}, {{A,B,C,X(54197),X(63391)}}
X(68904) = barycentric product X(57)*X(25005)
X(68904) = barycentric quotient X(25005)/X(312)
X(68904) = trilinear product X(56)*X(25005)
X(68904) = trilinear quotient X(8)/X(25005)
X(68904) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 46, 2077}, {1, 484, 65119}, {1, 2093, 49163}, {1, 10270, 3601}, {40, 34489, 2078}, {46, 53615, 37625}, {56, 65, 5903}, {56, 59334, 37583}, {65, 354, 13601}, {65, 942, 3340}, {65, 18838, 1}, {65, 37566, 517}, {65, 64106, 50193}, {65, 64721, 67977}, {354, 13601, 64964}, {942, 7373, 18398}, {942, 37562, 1}, {1858, 67931, 61718}, {3057, 3660, 63208}, {3057, 31787, 35445}, {3361, 67977, 64721}, {3753, 66250, 9578}, {3812, 12709, 5219}, {3874, 64745, 49169}, {5425, 14803, 1}, {5836, 17625, 37709}, {5903, 30274, 46920}, {11529, 59333, 1}, {17054, 34040, 1421}, {24914, 45288, 5692}, {24927, 50194, 1}, {31788, 67949, 1697}

X(68905) = X(4)X(653)∩X(8)X(496)

Barycentrics    -a^4+2*a^3*b-a^2*b^2-2*a*b^3+2*b^4+2*a^3*c+2*a*b^2*c-a^2*c^2+2*a*b*c^2-4*b^2*c^2-2*a*c^3+2*c^4 : :
X(68905) = 3*X[9581]-X[9614], 2*X[9581]-X[9669], 2*X[9614]-3*X[9669]

See David Nguyen and Ercole Suppa, euclid 8557.

X(68905) lies on the circumconic {{A,B,C,X(43731),X(65340)}} and these lines: {1, 1656}, {2, 37730}, {3, 1737}, {4, 653}, {5, 3485}, {8, 496}, {10, 1001}, {11, 1482}, {12, 15934}, {30, 1788}, {40, 9668}, {46, 382}, {55, 18395}, {56, 80}, {57, 9655}, {65, 381}, {78, 17619}, {140, 3486}, {145, 1387}, {218, 21044}, {226, 61261}, {354, 10827}, {355, 999}, {388, 18357}, {474, 5086}, {484, 12953}, {495, 938}, {497, 5690}, {499, 10246}, {517, 9581}, {519, 11373}, {546, 4295}, {549, 4305}, {912, 67937}, {942, 5290}, {944, 5704}, {950, 26446}, {952, 3086}, {986, 35194}, {1000, 4678}, {1058, 3617}, {1062, 1722}, {1125, 37739}, {1148, 7541}, {1155, 1657}, {1159, 3851}, {1319, 18526}, {1329, 3940}, {1385, 5727}, {1388, 3582}, {1420, 28204}, {1452, 18494}, {1466, 18761}, {1467, 18528}, {1470, 26321}, {1478, 5708}, {1479, 12702}, {1537, 10598}, {1538, 7971}, {1617, 18518}, {1697, 18527}, {1698, 24929}, {1699, 50193}, {1728, 59318}, {1768, 37001}, {1770, 3830}, {1807, 3924}, {1836, 3843}, {1854, 20299}, {1864, 34339}, {2093, 22793}, {2095, 10526}, {2098, 37720}, {2099, 7741}, {2362, 13665}, {2646, 3526}, {2771, 45631}, {2800, 10893}, {3036, 49169}, {3057, 58630}, {3058, 38066}, {3085, 12433}, {3090, 37737}, {3091, 39542}, {3297, 35788}, {3298, 35789}, {3303, 18530}, {3304, 37710}, {3333, 37714}, {3336, 12943}, {3337, 9657}, {3339, 18492}, {3340, 9955}, {3419, 9709}, {3474, 3627}, {3476, 37705}, {3487, 10592}, {3488, 9780}, {3534, 58887}, {3579, 3586}

X(68905) = reflection of X(9669) and X(9581)
X(68905) = pole of line {5697, 5881} with respect to Feuerbach hyperbola
X(68905) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 9956, 31479}, {1, 17606, 1656}, {4, 67980, 36279}, {8, 496, 64897}, {10, 5722, 3295}, {11, 10573, 1482}, {56, 80, 18525}, {57, 18480, 9655}, {65, 10826, 381}, {80, 20118, 12773}, {140, 3486, 37606}, {145, 47743, 1387}, {355, 1210, 999}, {496, 11545, 8}, {499, 10950, 10246}, {938, 5818, 495}, {942, 5587, 9654}, {944, 5704, 15325}, {950, 26446, 64951}, {1159, 3851, 12047}, {1319, 37711, 18526}, {1329, 49168, 3940}, {1479, 40663, 12702}, {1737, 1837, 3}, {1737, 10572, 24914}, {1837, 24914, 10572}, {2099, 7741, 18493}, {3419, 24982, 9709}, {3582, 37706, 1388}, {5587, 67931, 942}, {5729, 45043, 52682}, {10572, 24914, 3}, {12019, 67980, 4}, {12433, 38042, 3085}, {18391, 54361, 5}, {18395, 37702, 55}, {37720, 41684, 2098}

X(68906) = EULER LINE INTERCEPT OF X(33)X(3614)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^4*(b^2 + c^2) + 3*(b^2 - c^2)^2*(b^2 + c^2) + a^2*(-6*b^4 + 4*b^2*c^2 - 6*c^4)) : :

As a point on the Euler line, X(68906) has Shinagawa coefficients: {3 f, -4 e + 3 (e + f)}

See Gabi Cuc Cucoanes and David Nguyen, euclid 8558.

X(68906) lies on these lines: {2, 3}, {33, 3614}, {34, 7173}, {52, 43823}, {113, 12359}, {125, 2883}, {154, 68009}, {185, 67868}, {232, 63534}, {1112, 5562}, {1204, 5893}, {1249, 51316}, {1352, 46444}, {1398, 10589}, {1495, 41362}, {1514, 3357}, {1829, 3817}, {1870, 10593}, {1899, 64024}, {1902, 10175}, {1974, 67865}, {1990, 9722}, {2207, 43620}, {3070, 13937}, {3071, 13884}, {3092, 42277}, {3093, 42274}, {3532, 5895}, {3574, 15873}, {5090, 7989}, {5186, 36519}, {5410, 42561}, {5411, 31412}, {5412, 42270}, {5413, 42273}, {5448, 41587}, {5480, 41584}, {5587, 12135}, {5806, 41609}, {5876, 52000}, {5907, 44084}, {5972, 63631}, {6102, 58546}, {6146, 61747}, {6198, 10592}, {6247, 51403}, {6403, 38136}, {7071, 10588}, {7687, 10282}, {7699, 67883}, {7746, 60428}, {8550, 21637}, {8739, 42166}, {8740, 42163}, {8743, 43291}, {8778, 62992}, {8901, 38808}, {9707, 61606}, {9927, 51425}, {10192, 21659}, {10575, 12133}, {10632, 42135}, {10633, 42138}, {10641, 42107}, {10642, 42110}, {11363, 19925}, {11381, 23332}, {11396, 68034}, {11408, 42139}, {11409, 42142}, {11441, 61544}, {11473, 42582}, {11474, 42583}, {12131, 23514}, {12137, 38161}, {12138, 23513}, {12174, 23291}, {12241, 61690}, {12293, 51933}, {13434, 22750}, {13567, 43831}, {13568, 61645}, {13851, 34782}, {13881, 16318}, {14249, 61381}, {14561, 39871}, {14644, 34224}, {15010, 16879}, {15311, 43903}, {15738, 41725}, {16252, 67903}, {16655, 23325}, {18388, 65093}, {18874, 66604}, {18914, 61701}, {18918, 64717}, {18945, 26864}, {19128, 39884}, {19132, 64080}, {20302, 30714}, {20303, 45185}, {21663, 51491}, {22660, 63735}, {23294, 32111}, {23324, 61139}, {27371, 39601}, {27376, 39565}, {32204, 39533}, {34469, 58378}, {38034, 41722}, {38150, 60879}, {40316, 64067}, {40330, 68023}, {41588, 66727}, {41602, 68025}, {43719, 48672}, {44829, 46682}, {44870, 45303}, {47730, 56390}, {50435, 66762}, {61749, 67902}, {67891, 68020}

X(68906) = orthic-isogonal conjugate of X(16879)
X(68906) = complement of circumperp conjugate of X(57584)
X(68906) = isotomic conjugate of isogonal conjugate of X(15010)
X(68906) = barycentric product X(i)*X(j) for these {i,j}:{76, 15010}, {2052, 45187}, {16879, 66732}
X(68906) = barycentric quotient X(i)/X(j) for these {i,j}:{16879, 37672}, {45187, 394}
X(68906) = trilinear product X(i)*X(j) for these {i,j}:{75, 15010}, {158, 45187}
X(68906) = trilinear quotient X(i)/X(j) for these {i,j}:{31, 15010}, {255, 45187}
X(68906) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k):{2, 4, 3516}, {4, 403, 44960}, {4, 3516, 1885}, {4, 3517, 3575}, {4, 3542, 3517}, {4, 5056, 5094}, {4, 7505, 32534}, {4, 10018, 550}, {4, 10295, 62036}, {4, 10299, 49670}, {4, 16868, 35487}, {5, 235, 427}, {5, 403, 235}, {5, 1595, 7577}, {5, 1596, 1594}, {5, 10024, 7399}, {5, 11563, 13371}, {5, 13160, 37439}, {5, 13406, 15760}, {5, 15761, 11585}, {24, 546, 66725}, {25, 3091, 23047}, {235, 403, 45004}, {235, 427, 1906}, {235, 1907, 1596}, {381, 3517, 4}, {381, 3542, 3575}, {381, 3575, 63662}, {382, 3147, 37931}, {403, 1594, 44958}, {403, 7577, 44957}, {403, 16868, 5}, {468, 10019, 4}, {547, 13488, 37119}, {1312, 1313, 13473}, {1593, 3090, 62958}, {1594, 1596, 1907}, {1594, 1907, 427}, {3089, 3545, 7507}, {3089, 7507, 428}, {3090, 6623, 1593}, {3091, 6622, 25}, {3091, 7503, 44920}, {3542, 3575, 62978}, {3832, 6353, 12173}, {3854, 4232, 4}, {3855, 7487, 18386}, {5066, 6756, 7547}, {5893, 47296, 1204}, {10255, 11799, 23335}, {13383, 18404, 44239}, {14813, 14814, 10257}, {14940, 18560, 549}, {23291, 68024, 12174}, {58378, 66752, 34469}

X(68907) = EULER LINE INTERCEPT OF X(165)X(12135)

Barycentrics    ((a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(8*a^6 - 15*a^4*(b^2 + c^2) + (b^2 - c^2)^2*(b^2 + c^2) + 6*a^2*(b^2 + c^2)^2)) : :

As a point on the Euler line, X(68907) has Shinagawa coefficients: {7 f, 8 e - 9 (e + f)}

See Gabi Cuc Cucoanes and David Nguyen, euclid 8558.

X(68907) lies on these lines: {2, 3}, {165, 12135}, {1192, 11245}, {1350, 46444}, {1495, 5894}, {1503, 43903}, {1620, 1899}, {5023, 16318}, {5090, 16192}, {5410, 42637}, {5411, 42638}, {7689, 32145}, {8567, 31383}, {11204, 16655}, {11206, 34469}, {11363, 12512}, {13568, 61690}, {13884, 42259}, {13937, 42258}, {15513, 27376}, {17704, 47328}, {18931, 64717}, {19128, 48874}, {19467, 37487}, {21663, 34782}, {23328, 61139}, {31805, 41609}, {41584, 44882}, {43601, 48906}, {44078, 46374}, {64196, 64724}

X(68907) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k):{20, 15750, 468}, {186, 550, 235}, {376, 3515, 1885}, {1657, 3147, 10151}, {1885, 3515, 62978}, {3523, 12173, 62958}, {6240, 17506, 549}, {10018, 13619, 3627}, {15332, 18571, 5}

X(68908) = SODDY LINE INTERCEPT OF X(2)X(31897)

Barycentrics    a*(a^5 - a^3*b^2 - a^2*b^3 + b^5 + a^3*b*c + a^2*b^2*c - a*b^3*c - b^4*c - a^3*c^2 + a^2*b*c^2 + 2*a*b^2*c^2 - a^2*c^3 - a*b*c^3 - b*c^4 + c^5) : :
X(68908) = 3 X[38941] - X[62314], X[41327] - 3 X[67625]

X(68908) lies on these lines: {1, 7}, {2, 31897}, {11, 6357}, {33, 56848}, {34, 18328}, {57, 20277}, {58, 18734}, {86, 7112}, {101, 34381}, {103, 34855}, {105, 35184}, {106, 2728}, {109, 8758}, {150, 29219}, {222, 62811}, {241, 13329}, {242, 514}, {284, 501}, {386, 51775}, {518, 6510}, {580, 66760}, {581, 5803}, {651, 1736}, {664, 29016}, {741, 53182}, {758, 6518}, {759, 2727}, {774, 34043}, {927, 29015}, {971, 6610}, {975, 5821}, {999, 2097}, {1038, 67652}, {1040, 67632}, {1060, 5820}, {1062, 67721}, {1088, 56144}, {1210, 3468}, {1308, 67810}, {1354, 3328}, {1412, 40959}, {1427, 65702}, {1428, 3675}, {1449, 67648}, {1456, 43044}, {1461, 1876}, {1464, 45272}, {1503, 1565}, {1735, 24027}, {1776, 61225}, {1936, 18593}, {2328, 18607}, {2360, 18732}, {2361, 65524}, {2717, 3660}, {2725, 58967}, {2911, 3157}, {2999, 67643}, {3190, 6505}, {3218, 62756}, {3220, 3942}, {3323, 34228}, {3333, 5144}, {3487, 5074}, {3583, 56844}, {3670, 64421}, {3772, 59606}, {4038, 58626}, {4415, 59611}, {4649, 30329}, {4989, 64124}, {5127, 52407}, {5179, 5747}, {5181, 34586}, {5453, 15939}, {5760, 50317}, {6180, 64750}, {6198, 31851}, {7175, 32118}, {9364, 24025}, {9629, 68841}, {10980, 17017}, {11028, 58320}, {14547, 47057}, {15524, 68763}, {16475, 39273}, {16578, 23693}, {16972, 41323}, {17044, 51366}, {18161, 51687}, {18210, 26884}, {21346, 68592}, {22350, 67647}, {23710, 64115}, {28125, 38052}, {30116, 49653}, {31852, 66610}, {34036, 68379}, {37374, 43036}, {37523, 67654}, {37697, 61730}, {37729, 66106}, {37782, 61220}, {41700, 42082}, {43035, 53599}, {45126, 67651}, {52023, 68586}, {52305, 53314}, {52946, 62211}, {53996, 56809}, {54411, 63388}, {56139, 56359}, {59005, 65876}

X(68908) = midpoint of X(i) and X(j) for these {i,j}: {1, 5018}, {1323, 51617}
X(68908) = reflection of X(i) in X(j) for these {i,j}: {51366, 17044}, {62343, 51775}
X(68908) = anticomplement of X(31897)
X(68908) = incircle-inverse of X(4292)
X(68908) = crosspoint of X(81) and X(43736)
X(68908) = crosssum of X(i) and X(j) for these (i,j): {35, 68591}, {37, 41339}
X(68908) = crossdifference of every pair of points on line {71, 657}
X(68908) = barycentric product X(i)*X(j) for these {i,j}: {7, 8558}, {56, 45798}, {514, 53160}
X(68908) = barycentric quotient X(i)/X(j) for these {i,j}: {8558, 8}, {45798, 3596}, {53160, 190}
X(68908) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 77, 991}, {1, 269, 990}, {1, 1323, 47621}, {1, 4312, 4336}, {1, 68448, 1770}, {942, 7100, 8555}, {1870, 68758, 30117}, {1876, 51661, 1461}, {39152, 39153, 5011}

X(68909) = SODDY LINE INTERCEPT OF X(2)X(32)

Barycentrics    3*a^4 + a^3*b + a*b^3 - b^4 + a^3*c + a^2*b*c + a*b^2*c + b^3*c + a*b*c^2 + a*c^3 + b*c^3 - c^4 : :

X(68909) lies on these lines: {1, 7}, {2, 32}, {69, 4195}, {86, 4201}, {144, 3954}, {145, 17141}, {150, 5264}, {194, 20090}, {384, 17300}, {940, 6999}, {987, 7261}, {1014, 37328}, {1654, 7893}, {1975, 17378}, {3172, 37448}, {3661, 14552}, {3673, 66639}, {3744, 7247}, {3933, 4234}, {4644, 25242}, {4645, 24549}, {4648, 17691}, {4869, 33198}, {4872, 37539}, {4911, 5266}, {5232, 56986}, {5255, 56928}, {5712, 37416}, {7179, 37552}, {7379, 9863}, {7738, 63054}, {7762, 16061}, {7766, 33818}, {7767, 13740}, {7791, 17379}, {7826, 24275}, {7839, 63052}, {7876, 63053}, {14828, 49745}, {14929, 17698}, {15668, 63938}, {16898, 17232}, {16992, 26051}, {16995, 33822}, {17126, 21285}, {17137, 20101}, {17681, 18907}, {18600, 37256}, {20056, 33941}, {20102, 26759}, {20533, 54416}, {24342, 49562}, {26244, 63928}, {30435, 33838}, {32830, 62999}, {33298, 37540}, {39731, 49704}, {55082, 63979}, {63401, 63548}

X(68909) = crossdifference of every pair of points on line {657, 3005}
X(68909) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {20, 3945, 4352}, {86, 7750, 4201}, {7893, 17688, 1654}

X(68910) = SODDY LINE INTERCEPT OF X(23)X(385)

Barycentrics    2*a^5 - a^4*b + a^3*b^2 - 3*a^2*b^3 + a*b^4 - a^4*c + 2*a^2*b^2*c - b^4*c + a^3*c^2 + 2*a^2*b*c^2 - 2*a*b^2*c^2 + b^3*c^2 - 3*a^2*c^3 + b^2*c^3 + a*c^4 - b*c^4 : :

X(68910) lies on these lines: {1, 7}, {23, 385}, {2183, 2398}, {4232, 8756}, {4552, 40910}, {5773, 57022}, {8618, 58939}, {8758, 68698}, {12329, 25243}, {14953, 44670}, {16063, 29823}, {29353, 63782}

X(68910) = reflection of X(3007) in X(3012)
X(68910) = crossdifference of every pair of points on line {39, 657}

X(68911) = SODDY LINE INTERCEPT OF X(573)X(60720)

Barycentrics    a^5*b + a^4*b^2 - a^3*b^3 - a^2*b^4 + a^5*c - a*b^4*c + a^4*c^2 + a*b^3*c^2 - 2*b^4*c^2 - a^3*c^3 + a*b^2*c^3 + 4*b^3*c^3 - a^2*c^4 - a*b*c^4 - 2*b^2*c^4 : :

X(68911) lies on these lines: {1, 7}, {573, 60720}, {812, 1019}, {1111, 1756}, {1759, 1760}, {1764, 6063}, {3882, 20448}, {7243, 24310}, {10030, 20367}, {17030, 27170}, {18045, 29474}, {18142, 29418}, {21609, 54035}, {23512, 33765}, {26801, 26827}, {29069, 40704}, {29377, 29382}, {29742, 29747}

X(68911) = crossdifference of every pair of points on line {657, 872}

X(68912) = SODDY LINE INTERCEPT OF X(2)X(2183)

Barycentrics    a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + a^4*c - 2*a^3*b*c + 2*a*b^3*c - b^4*c + a^3*c^2 - 2*a*b^2*c^2 + b^3*c^2 - a^2*c^3 + 2*a*b*c^3 + b^2*c^3 - a*c^4 - b*c^4 : :

X(68912) lies on these lines: {1, 7}, {2, 2183}, {4, 11573}, {19, 26651}, {30, 41801}, {69, 313}, {73, 50702}, {75, 14923}, {86, 4225}, {150, 66846}, {222, 19645}, {307, 63397}, {320, 350}, {329, 4001}, {346, 20348}, {391, 41828}, {497, 67961}, {514, 52341}, {519, 41830}, {527, 22031}, {535, 61435}, {573, 17077}, {651, 6996}, {857, 26932}, {942, 48941}, {1229, 43216}, {1246, 30712}, {1266, 62401}, {1370, 68336}, {1441, 64126}, {1447, 24192}, {1565, 41804}, {1763, 28951}, {1764, 52358}, {1766, 28968}, {1848, 17184}, {1936, 68698}, {1944, 7291}, {2260, 26818}, {2262, 20905}, {2269, 30097}, {2323, 5773}, {2654, 50419}, {2979, 20242}, {3262, 21272}, {3562, 37088}, {3888, 20556}, {3936, 51414}, {3942, 8680}, {4357, 17183}, {4452, 20041}, {4552, 29069}, {4566, 62402}, {4651, 25288}, {5232, 8165}, {5435, 9535}, {5812, 37536}, {5813, 52457}, {5905, 63057}, {5928, 32859}, {6327, 36844}, {6840, 36918}, {7518, 54234}, {8049, 39704}, {10436, 26115}, {11521, 64573}, {12699, 64538}, {15571, 17768}, {15983, 20891}, {16609, 53526}, {16713, 28287}, {17074, 23512}, {17164, 41600}, {17209, 26856}, {17272, 50605}, {17378, 34605}, {17863, 24471}, {17895, 43037}, {18607, 53043}, {18725, 45738}, {18726, 25255}, {18732, 67848}, {20243, 68345}, {20244, 42697}, {20248, 20927}, {20367, 24237}, {21228, 41010}, {21239, 25000}, {21246, 27039}, {21296, 68334}, {21363, 40687}, {21370, 28950}, {22370, 27514}, {24705, 56509}, {26871, 37185}, {27108, 29965}, {29822, 35980}, {30807, 34371}, {30942, 30946}, {31778, 56927}, {36589, 53380}, {44670, 68350}, {54452, 66862}

X(68912) = reflection of X(i) in X(j) for these {i,j}: {4552, 68759}, {20367, 24237}
X(68912) = anticomplement of X(2183)
X(68912) = anticomplement of the isogonal conjugate of X(34234)
X(68912) = isotomic conjugate of the anticomplement of X(40613)
X(68912) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2, 153}, {7, 36918}, {104, 2}, {909, 192}, {1309, 4391}, {1795, 6360}, {1809, 56943}, {2250, 1654}, {2342, 3177}, {2401, 149}, {2423, 9263}, {2720, 17496}, {10428, 17495}, {13136, 513}, {14578, 3164}, {15501, 20211}, {15635, 54102}, {16082, 4}, {18816, 69}, {32641, 17494}, {34051, 145}, {34234, 8}, {34858, 194}, {36037, 514}, {36123, 5905}, {36795, 3436}, {36819, 20533}, {36944, 30578}, {37136, 522}, {38955, 2895}, {39294, 61185}, {40218, 64743}, {40437, 17484}, {43728, 37781}, {45145, 39360}, {47317, 35057}, {51565, 329}, {52663, 144}, {53811, 3738}, {54953, 693}, {55259, 148}, {55943, 518}, {56753, 20344}, {57468, 39353}, {57495, 34550}, {57753, 3262}, {57984, 21287}, {59196, 517}, {61238, 39351}, {65223, 20293}, {65302, 20}, {65331, 521}, {65537, 17896}, {67178, 20072}
X(68912) = X(40613)-cross conjugate of X(2)
X(68912) = X(3900)-isoconjugate of X(59073)
X(68912) = crosspoint of X(i) and X(j) for these (i,j): {86, 18816}, {668, 65014}, {54953, 57757}
X(68912) = crosssum of X(3270) and X(23220)
X(68912) = crossdifference of every pair of points on line {213, 657}
X(68912) = barycentric product X(i)*X(j) for these {i,j}: {85, 68766}, {4573, 14310}, {7461, 15413}
X(68912) = barycentric quotient X(i)/X(j) for these {i,j}: {1461, 59073}, {7461, 1783}, {14310, 3700}, {40613, 2183}, {68766, 9}
X(68912) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 10446, 17220}, {69, 21279, 21270}, {77, 10444, 17134}, {269, 10442, 18655}, {320, 17139, 20347}, {1944, 7291, 14543}, {22370, 30035, 27514}, {22464, 67267, 3007}

X(68913) = SODDY LINE INTERCEPT OF X(36)X(238)

Barycentrics    a*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4 + a^4*c + 2*a^3*b*c - 2*a*b^3*c - b^4*c - a^3*c^2 + 2*a*b^2*c^2 + b^3*c^2 - a^2*c^3 - 2*a*b*c^3 + b^2*c^3 + a*c^4 - b*c^4) : :
X(68913) = X[1] + 2 X[3000]

X(68913) lies on these lines: {1, 7}, {3, 6180}, {11, 68841}, {30, 68758}, {35, 9440}, {36, 238}, {40, 11573}, {42, 62240}, {43, 5756}, {46, 3293}, {57, 5751}, {78, 40905}, {100, 50003}, {144, 56809}, {165, 22097}, {200, 4001}, {222, 1754}, {226, 22053}, {241, 971}, {484, 51766}, {500, 24470}, {511, 20367}, {518, 35338}, {527, 1818}, {553, 14547}, {579, 610}, {651, 13329}, {851, 3937}, {942, 15937}, {948, 21151}, {954, 50677}, {978, 27624}, {1004, 22129}, {1026, 62222}, {1040, 56848}, {1066, 31730}, {1086, 68732}, {1155, 4551}, {1193, 64017}, {1214, 63995}, {1284, 4014}, {1331, 36003}, {1407, 7580}, {1418, 5728}, {1427, 10167}, {1463, 2223}, {1464, 15326}, {1465, 61227}, {1474, 39949}, {1478, 56191}, {1496, 12511}, {1724, 64057}, {1730, 26892}, {1757, 39341}, {1758, 1768}, {1764, 3784}, {1777, 7742}, {1838, 53238}, {2340, 5850}, {2361, 61225}, {2635, 3911}, {2947, 30304}, {3190, 9965}, {3218, 61220}, {3624, 25513}, {3651, 65114}, {3670, 64132}, {3942, 44661}, {3953, 14523}, {4191, 21361}, {4674, 61435}, {5126, 32486}, {5228, 62183}, {5247, 59323}, {5249, 17194}, {5753, 37582}, {5903, 49490}, {6211, 67417}, {7004, 18593}, {7354, 37558}, {7677, 64013}, {8229, 44313}, {8732, 68529}, {9026, 53397}, {9364, 44425}, {9436, 58035}, {9441, 20744}, {9501, 35333}, {9579, 37523}, {9612, 15669}, {10394, 17092}, {10398, 51302}, {10861, 24635}, {11019, 61376}, {11220, 62811}, {11575, 16610}, {12109, 48921}, {12433, 48916}, {13226, 43056}, {15252, 43036}, {15299, 64741}, {17077, 48888}, {17616, 18607}, {18541, 50317}, {19546, 64489}, {20078, 56813}, {20605, 24484}, {22068, 30944}, {22128, 62756}, {22350, 62320}, {24177, 40958}, {24708, 66515}, {25557, 55340}, {26884, 61221}, {30117, 31647}, {30295, 44858}, {30379, 53599}, {34043, 37570}, {36706, 63152}, {37270, 55406}, {37575, 49537}, {38054, 59217}, {39796, 41342}, {40910, 63442}, {42314, 42884}, {46882, 52949}, {48926, 64535}, {50584, 50630}, {59215, 66661}, {61228, 68761}

X(68913) = midpoint of X(1458) and X(3000)
X(68913) = reflection of X(i) in X(j) for these {i,j}: {1, 1458}, {1736, 241}
X(68913) = X(103)-Ceva conjugate of X(1)
X(68913) = X(10)-isoconjugate of X(59074)
X(68913) = X(30807)-Dao conjugate of X(35517)
X(68913) = crosspoint of X(i) and X(j) for these (i,j): {86, 36101}, {24011, 65245}
X(68913) = crosssum of X(i) and X(j) for these (i,j): {42, 910}, {24012, 46392}
X(68913) = crossdifference of every pair of points on line {37, 657}
X(68913) = barycentric quotient X(1333)/X(59074)
X(68913) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 991, 1}, {20, 4306, 1}, {77, 990, 1}, {77, 8544, 990}, {269, 5732, 1}, {1042, 4297, 1}, {1448, 10884, 1}, {1464, 15326, 68766}, {1742, 4334, 1}, {1745, 15803, 3216}, {2293, 5542, 1}, {2635, 3911, 5400}, {3668, 63395, 1}, {3751, 60785, 3293}, {4292, 4303, 1}, {4298, 4300, 1}, {4301, 4322, 1}, {4320, 12520, 1}, {4341, 30265, 1}, {38459, 47621, 1}, {41353, 56381, 1}

X(68914) = SODDY LINE INTERCEPT OF X(11)X(244)

Barycentrics    (a - b - c)*(b - c)^2*(2*a^2 - a*b - b^2 - a*c + 2*b*c - c^2) : :

X(68914) lies on these lines: {1, 7}, {2, 24411}, {11, 244}, {55, 24405}, {57, 61732}, {513, 42754}, {522, 60491}, {885, 23838}, {903, 14942}, {1040, 33098}, {1253, 17276}, {3270, 4014}, {3328, 30573}, {3675, 43909}, {3826, 24433}, {4389, 36219}, {4414, 4419}, {4530, 14393}, {4542, 52304}, {4845, 34578}, {5432, 49742}, {5723, 5851}, {5727, 18340}, {6603, 44785}, {8275, 24864}, {9355, 37771}, {9371, 32856}, {10391, 33145}, {10394, 33149}, {10427, 35293}, {15845, 42040}, {20119, 24715}, {21035, 21698}, {21105, 23730}, {23757, 66508}, {24341, 33108}, {24980, 62669}, {26932, 52335}, {27529, 27547}, {28160, 34232}, {30311, 64134}, {33573, 35091}, {48629, 65952}, {60479, 60579}, {66276, 66284}

X(68914) = reflection of X(62669) in X(24980)
X(68914) = X(i)-Ceva conjugate of X(j) for these (i,j): {11, 3328}, {527, 65680}, {1111, 40629}, {1155, 30574}, {1323, 1638}, {1638, 52334}, {6745, 6366}, {23710, 14413}, {34578, 650}, {43762, 657}, {60479, 42462}, {60579, 11}, {62723, 514}, {68241, 513}
X(68914) = X(i)-cross conjugate of X(j) for these (i,j): {3328, 11}, {52333, 42462}, {52334, 1638}
X(68914) = X(i)-isoconjugate of X(j) for these (i,j): {59, 1156}, {100, 14733}, {101, 37139}, {190, 36141}, {668, 32728}, {692, 35157}, {906, 65335}, {1110, 62723}, {1121, 2149}, {1252, 34056}, {1262, 41798}, {1275, 18889}, {1783, 65304}, {2291, 4564}, {3900, 59105}, {4570, 62764}, {4619, 23893}, {4845, 7045}, {4998, 34068}, {7012, 60047}, {65646, 67434}
X(68914) = X(i)-Dao conjugate of X(j) for these (i,j): {514, 62723}, {650, 1121}, {661, 34056}, {1015, 37139}, {1086, 35157}, {5190, 65335}, {6366, 6745}, {6594, 765}, {6615, 1156}, {8054, 14733}, {17115, 4845}, {35091, 190}, {35110, 4998}, {39006, 65304}, {40615, 60487}, {40629, 664}, {50330, 62764}, {52870, 1275}, {52879, 7045}, {55053, 36141}, {62579, 8}, {64440, 60579}
X(68914) = crosspoint of X(i) and X(j) for these (i,j): {1, 61230}, {7, 60479}, {11, 60579}, {514, 62723}, {522, 3254}, {1323, 1638}, {6366, 6745}
X(68914) = crosssum of X(109) and X(2078)
X(68914) = trilinear pole of line {3328, 52334}
X(68914) = crossdifference of every pair of points on line {59, 101}
X(68914) = barycentric product X(i)*X(j) for these {i,j}: {7, 33573}, {11, 527}, {142, 66484}, {514, 6366}, {522, 1638}, {664, 52334}, {693, 65680}, {1055, 34387}, {1086, 6745}, {1111, 6603}, {1121, 3328}, {1146, 1323}, {1155, 4858}, {1565, 60431}, {2170, 30806}, {2310, 37780}, {3254, 40629}, {3261, 6139}, {4391, 14413}, {4466, 52891}, {4530, 36887}, {4560, 30574}, {6174, 60578}, {6610, 24026}, {7004, 37805}, {14392, 24002}, {14414, 17924}, {16732, 62756}, {21207, 68742}, {23710, 26932}, {23890, 42455}, {30573, 60480}, {34591, 38461}, {35091, 62723}, {35110, 60579}, {42462, 56543}, {42762, 43728}, {58817, 65448}, {60479, 62579}
X(68914) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 1121}, {244, 34056}, {513, 37139}, {514, 35157}, {527, 4998}, {649, 14733}, {667, 36141}, {1055, 59}, {1086, 62723}, {1155, 4564}, {1323, 1275}, {1459, 65304}, {1461, 59105}, {1638, 664}, {1919, 32728}, {2170, 1156}, {2310, 41798}, {3125, 62764}, {3271, 2291}, {3328, 527}, {3676, 60487}, {4530, 52746}, {6139, 101}, {6366, 190}, {6603, 765}, {6610, 7045}, {6745, 1016}, {7117, 60047}, {7649, 65335}, {8735, 65340}, {14392, 644}, {14413, 651}, {14414, 1332}, {14936, 4845}, {21132, 60479}, {23346, 4619}, {23710, 46102}, {30573, 62669}, {30574, 4552}, {30806, 67038}, {33573, 8}, {35091, 6745}, {42462, 63748}, {52333, 33573}, {52334, 522}, {58817, 65553}, {60431, 15742}, {60579, 57565}, {62723, 57563}, {62756, 4567}, {64445, 60579}, {65448, 6558}, {65680, 100}, {66484, 32008}, {68742, 4570}
X(68914) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14393, 52338, 42462}, {17435, 52946, 4530}, {35015, 53525, 1647}, {53525, 68844, 35015}

X(68915) = SODDY LINE INTERCEPT OF X(523)X(661)

Barycentrics    (b + c)*(-2*a^4 - a^2*b^2 + 2*a*b^3 + b^4 + 4*a^2*b*c - 2*a*b^2*c - 2*b^3*c - a^2*c^2 - 2*a*b*c^2 + 2*b^2*c^2 + 2*a*c^3 - 2*b*c^3 + c^4) : :

X(68915) lies on these lines: {1, 7}, {523, 661}, {857, 52335}, {2938, 41808}, {3698, 27685}, {10727, 44967}, {16581, 21945}, {18589, 21931}, {20653, 21673}, {24931, 24949}, {41003, 42446}

X(68915) = reflection of X(52335) in X(857)
X(68915) = crossdifference of every pair of points on line {58, 657}

X(68916) = SODDY LINE INTERCEPT OF X(348)X(30228)

Barycentrics    (b - c)^2*(a^4 - 2*a^2*b^2 + b^4 - a^2*b*c + 2*a*b^2*c - b^3*c - 2*a^2*c^2 + 2*a*b*c^2 - b*c^3 + c^4) : :

X(68916) lies on these lines: {1, 7}, {348, 30228}, {918, 1086}, {1565, 14205}, {3676, 35015}, {7658, 33573}, {24002, 58259}, {24778, 56445}, {34789, 65188}, {42386, 65702}, {53525, 66486}

X(68916) = incircle-inverse of X(14116)
X(68916) = X(52156)-Ceva conjugate of X(514)
X(68916) = crosspoint of X(i) and X(j) for these (i,j): {7, 2400}, {24002, 67128}
X(68916) = crosssum of X(55) and X(2426)
X(68916) = crossdifference of every pair of points on line {657, 692}
X(68916) = barycentric product X(i)*X(j) for these {i,j}: {3261, 53300}, {23989, 68591}
X(68916) = barycentric quotient X(i)/X(j) for these {i,j}: {53300, 101}, {68591, 1252}
X(68916) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 45276, 1}, {22106, 22107, 1146}

X(68917) = SODDY LINE INTERCEPT OF X(10)X(35026)

Barycentrics    (b - c)^2*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c - a^2*b*c + a*b^2*c + b^3*c - a^2*c^2 + a*b*c^2 - 2*b^2*c^2 + a*c^3 + b*c^3) : :

X(68917) lies on these lines: {1, 7}, {10, 35026}, {142, 5701}, {514, 17435}, {522, 62429}, {527, 2284}, {812, 1015}, {1111, 2310}, {1358, 2820}, {3323, 44043}, {3676, 53525}, {6549, 23838}, {14505, 52946}, {15634, 62635}, {16580, 40690}, {21201, 21202}, {23813, 56893}, {24715, 61480}, {25101, 28742}, {34018, 43672}, {35015, 66486}, {38357, 40615}, {48888, 60720}, {56783, 64013}

X(68917) = X(i)-Ceva conjugate of X(j) for these (i,j): {34018, 514}, {43736, 3676}
X(68917) = X(i)-isoconjugate of X(j) for these (i,j): {1018, 59067}, {1110, 43672}
X(68917) = X(i)-Dao conjugate of X(j) for these (i,j): {514, 43672}, {68813, 3693}
X(68917) = crosspoint of X(i) and X(j) for these (i,j): {673, 56322}, {1088, 60581}
X(68917) = crosssum of X(i) and X(j) for these (i,j): {55, 54325}, {672, 35326}
X(68917) = crossdifference of every pair of points on line {657, 4557}
X(68917) = barycentric product X(i)*X(j) for these {i,j}: {514, 53357}, {1111, 62799}, {1565, 26003}, {3261, 53308}, {13329, 23989}
X(68917) = barycentric quotient X(i)/X(j) for these {i,j}: {1086, 43672}, {3733, 59067}, {13329, 1252}, {26003, 15742}, {53308, 101}, {53357, 190}, {62799, 765}
X(68917) = {X(481),X(482)}-harmonic conjugate of X(67577)

X(68918) = SODDY LINE INTERCEPT OF X(513)X(3716)

Barycentrics    3*a^3*b^2 - 2*a^2*b^3 - a*b^4 - 4*a^3*b*c + a^2*b^2*c + 4*a*b^3*c - b^4*c + 3*a^3*c^2 + a^2*b*c^2 - 6*a*b^2*c^2 + b^3*c^2 - 2*a^2*c^3 + 4*a*b*c^3 + b^2*c^3 - a*c^4 - b*c^4 : :

X(68918) lies on these lines: {1, 7}, {513, 3716}, {527, 4447}, {1944, 53394}, {2183, 17754}, {3741, 3784}, {4459, 43040}, {4795, 64904}, {5205, 56555}, {8256, 8679}, {18788, 40862}, {20258, 56861}, {21060, 64015}, {24237, 29353}, {26929, 32946}, {30097, 49537}

X(68918) = crossdifference of every pair of points on line {657, 2176}

X(68919) = SODDY LINE INTERCEPT OF X(10)X(190)

Barycentrics    4*a^3 - 3*a^2*b - 2*a*b^2 - 3*b^3 - 3*a^2*c + 3*b^2*c - 2*a*c^2 + 3*b*c^2 - 3*c^3 : :
X(68919) = 2 X[1] - 3 X[3663], 5 X[1] - 6 X[4353], X[1] - 3 X[24248], 4 X[1] - 3 X[63969], 5 X[3663] - 4 X[4353], 2 X[4353] - 5 X[24248], 8 X[4353] - 5 X[63969], 4 X[24248] - X[63969], X[30332] - 3 X[64695], X[8] - 3 X[64299], 2 X[10] - 3 X[49630], 4 X[10] - 3 X[50118], 5 X[3617] - 3 X[3729], 2 X[3626] - 3 X[4660], 4 X[3634] - 3 X[3923], 3 X[3755] - 2 X[4663], 6 X[3821] - 5 X[19862], 2 X[4527] - 3 X[50781], 2 X[4672] - 3 X[50091], 2 X[5695] - 3 X[29594], 7 X[9780] - 6 X[17355], 7 X[9780] - 3 X[24280], 3 X[12722] - 4 X[50192], 7 X[15808] - 6 X[49482], 15 X[17304] - 13 X[46934], X[20050] - 3 X[49446], X[24695] - 3 X[50080], 3 X[50114] - 4 X[66071], 4 X[31794] - 3 X[32118], 2 X[49484] - 3 X[50092], 3 X[49543] - 2 X[67964]

X(68919) lies on these lines: {1, 7}, {8, 17132}, {10, 190}, {226, 4689}, {497, 51615}, {519, 50992}, {527, 64070}, {551, 66692}, {726, 3625}, {752, 50108}, {896, 3914}, {946, 37599}, {982, 51783}, {986, 51118}, {1001, 63589}, {1125, 11147}, {1284, 41430}, {1738, 51090}, {1743, 63975}, {1766, 5128}, {1836, 9554}, {2321, 28530}, {3008, 5698}, {3057, 4014}, {3244, 28562}, {3416, 28557}, {3474, 39595}, {3617, 3729}, {3626, 4660}, {3634, 3923}, {3666, 65698}, {3685, 21255}, {3705, 66444}, {3731, 59412}, {3755, 4663}, {3817, 17596}, {3821, 17400}, {3828, 66691}, {3883, 53594}, {3886, 53598}, {3944, 10164}, {3946, 64016}, {3950, 4645}, {3986, 9791}, {4008, 24208}, {4052, 7081}, {4054, 44006}, {4061, 4683}, {4082, 32948}, {4085, 28546}, {4114, 4883}, {4133, 50304}, {4416, 62392}, {4429, 59579}, {4440, 49466}, {4527, 50781}, {4640, 17070}, {4655, 28580}, {4656, 5297}, {4672, 50091}, {4676, 31191}, {4743, 28558}, {4780, 17770}, {4847, 33094}, {4859, 52653}, {4923, 17344}, {4924, 5850}, {5204, 63968}, {5217, 24309}, {5325, 21949}, {5493, 13161}, {5524, 21060}, {5550, 31312}, {5695, 29594}, {5853, 17276}, {5880, 29571}, {5988, 49631}, {7292, 24177}, {7613, 66515}, {9780, 17355}, {11019, 18201}, {12053, 24237}, {12722, 50192}, {15254, 53600}, {15808, 49482}, {16020, 50836}, {16484, 38054}, {16815, 41842}, {17304, 46934}, {17334, 24393}, {17738, 29596}, {17764, 49511}, {17767, 49529}, {20050, 49446}, {24175, 40998}, {24695, 50080}, {24703, 45204}, {25072, 38052}, {28198, 66675}, {28494, 49684}, {28508, 51196}, {28534, 50114}, {29057, 31673}, {31730, 37589}, {31794, 32118}, {33098, 67207}, {33151, 63145}, {36277, 40940}, {41011, 67211}, {45763, 68619}, {49478, 60962}, {49484, 50092}, {49543, 67964}, {50808, 66632}

X(68919) = reflection of X(i) in X(j) for these {i,j}: {3663, 24248}, {3886, 53598}, {4133, 50304}, {24280, 17355}, {50118, 49630}, {63969, 3663}, {64016, 3946}, {64073, 4743}
X(68919) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 30424, 4896}, {7, 66673, 63977}, {4312, 64168, 3664}, {4346, 30332, 1}

X(68920) = X(2)X(85)∩X(21)X(36)

Barycentrics    a^5 - a^3*b^2 - a^2*b^3 + b^5 + a^3*b*c + a^2*b^2*c - a*b^3*c - b^4*c - a^3*c^2 + a^2*b*c^2 + 2*a*b^2*c^2 - a^2*c^3 - a*b*c^3 - b*c^4 + c^5 : :

X(68920) lies on these lines: {2, 85}, {7, 17073}, {21, 36}, {77, 18634}, {81, 18652}, {86, 7054}, {239, 25593}, {307, 37659}, {379, 17181}, {664, 48381}, {693, 905}, {857, 5088}, {908, 67647}, {934, 5236}, {936, 32782}, {946, 37048}, {1146, 43066}, {1375, 1565}, {1442, 16608}, {1443, 18644}, {1817, 18651}, {1944, 41804}, {3188, 37448}, {3218, 11064}, {3306, 29598}, {3616, 36706}, {3662, 24540}, {3664, 24780}, {3811, 6505}, {3936, 27399}, {4357, 24557}, {4657, 62778}, {4872, 14953}, {5011, 68840}, {5222, 20269}, {5703, 37180}, {6349, 28606}, {6610, 37781}, {6708, 18625}, {7269, 17043}, {9310, 36482}, {9317, 26012}, {9436, 26006}, {9776, 24609}, {11349, 51775}, {11413, 64077}, {11681, 30782}, {14986, 19785}, {15474, 40836}, {16706, 27161}, {17080, 20266}, {17086, 20905}, {17092, 56445}, {17170, 24580}, {17184, 24556}, {18726, 24884}, {18732, 24882}, {20292, 48900}, {21617, 58412}, {22128, 37783}, {22464, 44356}, {25066, 33157}, {25083, 28757}, {25584, 26059}, {26001, 43035}, {26011, 37798}, {26669, 28739}, {27065, 58458}, {27396, 28753}, {28982, 51390}, {30379, 37141}, {31191, 31222}, {32779, 56457}, {36101, 62786}, {36949, 37787}, {37782, 46488}, {40937, 41808}, {40942, 62779}, {43054, 51400}, {52156, 59195}, {56227, 57722}, {58457, 61024}, {67625, 68759}

X(68920) = crosspoint of X(86) and X(52156)
X(68920) = crossdifference of every pair of points on line {228, 8641}
X(68920) = barycentric product X(i)*X(j) for these {i,j}: {57, 45798}, {85, 8558}, {693, 53160}
X(68920) = barycentric quotient X(i)/X(j) for these {i,j}: {8558, 9}, {45798, 312}, {53160, 100}
X(68920) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 348, 24635}, {77, 18634, 26540}, {1375, 1565, 7291}, {9436, 26006, 62799}, {43035, 62388, 26001}

X(68921) = X(2)X(85)∩X(20)X(145)

Barycentrics    a^4*b - a^3*b^2 - a^2*b^3 + a*b^4 + a^4*c + 2*a^3*b*c - 2*a*b^3*c - b^4*c - a^3*c^2 + 2*a*b^2*c^2 + b^3*c^2 - a^2*c^3 - 2*a*b*c^3 + b^2*c^3 + a*c^4 - b*c^4 : :
X(68921) = 3 X[2] - 4 X[241], 9 X[2] - 8 X[34852], 15 X[2] - 16 X[58466], 3 X[241] - 2 X[34852], 5 X[241] - 4 X[58466], 3 X[30807] - 4 X[34852], 5 X[30807] - 8 X[58466], 5 X[34852] - 6 X[58466]

X(68921) lies on these lines: {2, 85}, {7, 18726}, {20, 145}, {28, 39747}, {38, 66668}, {63, 3188}, {69, 40905}, {75, 40903}, {144, 25243}, {192, 20059}, {193, 20082}, {239, 514}, {269, 26651}, {320, 65205}, {347, 18734}, {516, 66225}, {527, 4552}, {664, 62799}, {672, 43040}, {857, 1565}, {1323, 26006}, {1418, 20905}, {1443, 1944}, {1448, 19860}, {1959, 20347}, {2323, 63782}, {2398, 9441}, {3000, 68350}, {3661, 25244}, {3664, 25255}, {3732, 11349}, {3912, 65195}, {3942, 8680}, {3952, 56714}, {3970, 3995}, {4001, 6737}, {4303, 67848}, {4329, 18733}, {4359, 4875}, {4427, 58327}, {4480, 25268}, {4513, 32933}, {4936, 25734}, {6542, 25257}, {7289, 17134}, {8025, 51382}, {9436, 48381}, {9801, 67063}, {10481, 25935}, {14570, 56935}, {16609, 21139}, {16728, 20448}, {16826, 25261}, {17075, 56445}, {17136, 20769}, {17181, 31014}, {17220, 18161}, {17221, 18162}, {17244, 31053}, {17364, 25241}, {17389, 17479}, {17863, 26818}, {17950, 37781}, {18186, 24219}, {18651, 31045}, {18655, 18725}, {18656, 18727}, {18657, 18728}, {18658, 18729}, {18659, 18730}, {18660, 18731}, {18661, 18735}, {20078, 20111}, {20110, 53997}, {20211, 63009}, {20247, 62853}, {20248, 21371}, {20930, 27514}, {21296, 25252}, {22097, 53043}, {22128, 39767}, {24599, 24604}, {25083, 30806}, {25242, 29616}, {25245, 31300}, {25718, 28610}, {25930, 30625}, {26538, 26836}, {26932, 41804}, {28950, 56848}, {29621, 31035}, {32859, 56187}, {35058, 40188}, {35596, 39357}, {36850, 50697}, {39698, 68239}, {43035, 65174}, {46793, 54118}, {49496, 56185}, {52980, 61241}, {59195, 65294}

X(68921) = reflection of X(i) in X(j) for these {i,j}: {30807, 241}, {68350, 3000}
X(68921) = anticomplement of X(30807)
X(68921) = anticomplement of the isogonal conjugate of X(911)
X(68921) = anticomplement of the isotomic conjugate of X(36101)
X(68921) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 152}, {103, 69}, {677, 20295}, {911, 8}, {1815, 1370}, {2338, 3436}, {2424, 150}, {9503, 20556}, {15380, 35517}, {18025, 315}, {24016, 46402}, {32642, 514}, {32657, 20}, {32668, 3900}, {32684, 9000}, {32701, 64878}, {34181, 57768}, {35184, 3261}, {36039, 513}, {36056, 4329}, {36101, 6327}, {36122, 21270}, {40116, 20293}, {43736, 21285}, {45144, 21290}, {52156, 21280}, {52781, 11442}, {55257, 3448}, {57928, 21304}, {57996, 21275}
X(68921) = X(36101)-Ceva conjugate of X(2)
X(68921) = X(37)-isoconjugate of X(59074)
X(68921) = X(40589)-Dao conjugate of X(59074)
X(68921) = crosspoint of X(i) and X(j) for these (i,j): {274, 18025}, {57581, 65294}
X(68921) = crossdifference of every pair of points on line {42, 8641}
X(68921) = barycentric quotient X(58)/X(59074)
X(68921) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {85, 24635, 2}, {241, 30807, 2}, {269, 45738, 26651}, {5088, 7291, 14953}, {6360, 9965, 17147}, {17316, 25237, 3995}

X(68922) = X(1)X(2)∩X(244)X(516)

Barycentrics    2*a^3 - a^2*b + 4*a*b^2 - b^3 - a^2*c - 8*a*b*c + b^2*c + 4*a*c^2 + b*c^2 - c^3 : :
X(68922) = 3 X[4871] - X[49994], X[50003] - 9 X[64149]

X(68922) lies on these lines: {1, 2}, {11, 16870}, {88, 63145}, {142, 17721}, {238, 24216}, {244, 516}, {390, 62695}, {497, 5573}, {513, 676}, {527, 3999}, {901, 15637}, {908, 3315}, {950, 52541}, {982, 40998}, {1279, 3756}, {1386, 17051}, {1416, 9364}, {1420, 64442}, {1421, 34050}, {1435, 2137}, {1462, 67653}, {1479, 24171}, {1616, 4848}, {1743, 64151}, {2078, 43068}, {2177, 43179}, {3242, 5316}, {3243, 63126}, {3306, 63969}, {3434, 24175}, {3445, 63987}, {3452, 17597}, {3664, 50003}, {3668, 64747}, {3670, 68615}, {3677, 4656}, {3717, 25531}, {3744, 6692}, {3752, 64162}, {3816, 4906}, {3953, 12572}, {3976, 12527}, {4031, 64016}, {4850, 63977}, {4864, 51415}, {4887, 5057}, {4899, 58371}, {5218, 35227}, {5274, 23681}, {5435, 62875}, {5537, 37815}, {5744, 60846}, {5850, 17449}, {5853, 16610}, {8056, 17784}, {8074, 16784}, {9812, 63583}, {10171, 33127}, {10624, 24046}, {10980, 62240}, {12053, 17054}, {12575, 24443}, {12577, 46190}, {14956, 17205}, {17626, 65687}, {17765, 58467}, {19925, 23675}, {21060, 62850}, {24386, 24789}, {24725, 43180}, {25405, 55317}, {26139, 62297}, {30829, 49527}, {31197, 46916}, {31788, 55287}, {44841, 63089}, {46909, 63978}, {62815, 63090}

X(68922) = midpoint of X(i) and X(j) for these {i,j}: {899, 49989}, {29824, 49987}, {49986, 50001}
X(68922) = complement of X(49991)
X(68922) = X(68239)-complementary conjugate of X(141)
X(68922) = crossdifference of every pair of points on line {220, 649}
X(68922) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5121, 6745}, {497, 5573, 24177}, {614, 11019, 40940}, {1279, 3756, 3911}, {3008, 51615, 26015}, {3677, 26105, 4656}, {7292, 26015, 3008}, {28011, 28074, 10}, {28016, 28080, 1}, {28018, 28082, 1125}, {28027, 28096, 3634}

X(68923) = X(1)X(6)∩X(910)X(1015)

Barycentrics    a*(2*a^3 - a^2*b + 4*a*b^2 - b^3 - a^2*c - 8*a*b*c + b^2*c + 4*a*c^2 + b*c^2 - c^3) : :

X(68923) lies on these lines: {1, 6}, {649, 3669}, {910, 1015}, {919, 8686}, {1149, 2348}, {1308, 17222}, {1462, 43064}, {2082, 52541}, {3010, 3248}, {3207, 15854}, {3732, 57033}, {3756, 8074}, {23649, 37568}, {26015, 57019}, {29627, 63126}, {51415, 62398}

X(68923) = crosspoint of X(1) and X(68239)
X(68923) = crosssum of X(1) and X(41391)
X(68923) = crossdifference of every pair of points on line {200, 513}
X(68923) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 62370, 6603}, {16502, 40133, 1104}, {16784, 43065, 1279}

X(68924) = X(1)X(38503)∩X(57)X(279)

Barycentrics    (b - c)^2*(-2*a^2 + a*b + b^2 + a*c - 2*b*c + c^2) : :
X(68924) = 4 X[4089] - X[4530], 2 X[4089] + X[23766], X[4530] + 2 X[23766]

X(68924) lies on these lines: {1, 38503}, {7, 60692}, {57, 279}, {514, 4089}, {527, 36887}, {764, 1647}, {1022, 6549}, {1055, 1323}, {1086, 1358}, {1565, 21044}, {2099, 60718}, {2246, 43057}, {3120, 4403}, {3241, 60963}, {3328, 30573}, {4927, 54270}, {5845, 43038}, {9317, 17089}, {9318, 38941}, {17213, 21138}, {17316, 31164}, {17483, 29588}, {20089, 29572}, {27541, 63620}, {30694, 30852}, {33573, 40629}

X(68924) = reflection of X(2246) in X(43057)
X(68924) = X(i)-Ceva conjugate of X(j) for these (i,j): {527, 1638}, {1323, 14413}, {30806, 30574}, {36887, 30573}
X(68924) = X(i)-isoconjugate of X(j) for these (i,j): {59, 41798}, {644, 14733}, {646, 32728}, {765, 2291}, {1016, 34068}, {1110, 1121}, {1156, 1252}, {3699, 36141}, {3939, 37139}, {4130, 59105}, {4564, 4845}, {4998, 18889}, {6065, 34056}, {23351, 31615}, {35348, 59149}, {52985, 65646}, {56183, 65304}
X(68924) = X(i)-Dao conjugate of X(j) for these (i,j): {513, 2291}, {514, 1121}, {661, 1156}, {1638, 17264}, {6544, 52746}, {6615, 41798}, {35091, 3699}, {35110, 1016}, {40615, 35157}, {40617, 37139}, {40629, 190}, {52870, 4998}, {52879, 4564}, {62579, 346}
X(68924) = crosspoint of X(i) and X(j) for these (i,j): {514, 34578}, {527, 1638}
X(68924) = crosssum of X(101) and X(5526)
X(68924) = crossdifference of every pair of points on line {1252, 3939}
X(68924) = barycentric product X(i)*X(j) for these {i,j}: {11, 1323}, {244, 30806}, {279, 33573}, {514, 1638}, {527, 1086}, {658, 52334}, {693, 14413}, {1055, 23989}, {1111, 1155}, {1358, 6745}, {1565, 23710}, {1647, 36887}, {2170, 37780}, {2401, 42762}, {3321, 60579}, {3328, 62723}, {3676, 6366}, {3942, 37805}, {4858, 6610}, {6139, 52621}, {6174, 6549}, {6548, 30573}, {7004, 38461}, {7192, 30574}, {10481, 66484}, {14392, 59941}, {21132, 56543}, {23890, 40166}, {24002, 65680}, {34578, 40629}
X(68924) = barycentric quotient X(i)/X(j) for these {i,j}: {244, 1156}, {527, 1016}, {764, 35348}, {1015, 2291}, {1055, 1252}, {1086, 1121}, {1155, 765}, {1323, 4998}, {1358, 62723}, {1638, 190}, {1647, 52746}, {2170, 41798}, {2969, 65340}, {3248, 34068}, {3271, 4845}, {3328, 6745}, {3669, 37139}, {3676, 35157}, {3937, 60047}, {6139, 3939}, {6366, 3699}, {6545, 60479}, {6610, 4564}, {6614, 59105}, {6745, 4076}, {7336, 60579}, {14392, 4578}, {14413, 100}, {14414, 4571}, {21132, 63748}, {23710, 15742}, {23890, 31615}, {30573, 17780}, {30574, 3952}, {30806, 7035}, {33573, 346}, {36887, 62536}, {37780, 67038}, {40629, 17264}, {42762, 2397}, {43924, 14733}, {52334, 3239}, {53538, 34056}, {53540, 62764}, {57181, 36141}, {58817, 60487}, {65680, 644}, {66484, 56118}
X(68924) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1565, 21139, 21044}, {4089, 23766, 4530}

X(68925) = X(6)X(34619)∩X(10)X(37)

Barycentrics    (a - b - c)*(b + c)*(2*a^2 - a*b - b^2 - a*c + 2*b*c - c^2) : :

X(68925) lies on these lines: {6, 34619}, {8, 34522}, {10, 37}, {200, 52818}, {220, 7080}, {325, 4595}, {391, 3713}, {728, 46835}, {1018, 17747}, {1055, 6174}, {1145, 57015}, {1146, 3693}, {1212, 6736}, {1329, 3208}, {1334, 21031}, {2325, 5199}, {2329, 64123}, {3035, 56530}, {3161, 6554}, {3207, 59591}, {3230, 51415}, {3421, 42316}, {3496, 32157}, {3501, 12607}, {3554, 6765}, {3700, 4041}, {3813, 4050}, {3912, 58466}, {3930, 21013}, {3933, 29697}, {3936, 51367}, {4034, 4882}, {4390, 5432}, {4482, 6390}, {4513, 5552}, {4898, 67931}, {5179, 51362}, {5233, 28830}, {5657, 50995}, {6547, 62398}, {6603, 6745}, {6706, 24199}, {9327, 52264}, {10915, 25066}, {17044, 40872}, {17234, 21258}, {17362, 62216}, {17768, 41322}, {21075, 21872}, {21859, 51421}, {31397, 44798}, {31406, 50637}, {34606, 41423}, {34852, 62297}, {39244, 45081}, {41391, 51406}

X(68925) = X(i)-isoconjugate of X(j) for these (i,j): {58, 34056}, {593, 62764}, {1014, 2291}, {1019, 14733}, {1121, 1408}, {1156, 1412}, {1333, 62723}, {1396, 60047}, {1434, 34068}, {3733, 37139}, {4565, 35348}, {4637, 23351}, {7192, 36141}, {7199, 32728}, {35157, 57129}, {57200, 65304}
X(68925) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 34056}, {37, 62723}, {6594, 81}, {6741, 60479}, {35091, 7192}, {35110, 1434}, {40599, 1156}, {40629, 17096}, {55064, 35348}, {59577, 1121}, {62579, 17197}
X(68925) = crossdifference of every pair of points on line {1412, 3733}
X(68925) = barycentric product X(i)*X(j) for these {i,j}: {10, 6745}, {210, 30806}, {306, 60431}, {321, 6603}, {527, 2321}, {1055, 30713}, {1089, 62756}, {1155, 3701}, {1323, 4082}, {1638, 30730}, {3694, 37805}, {3695, 52891}, {3699, 30574}, {3710, 23710}, {3952, 6366}, {4033, 65680}, {4515, 37780}, {4566, 65448}, {6139, 27808}, {28654, 68742}, {56157, 61035}
X(68925) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 62723}, {37, 34056}, {210, 1156}, {527, 1434}, {756, 62764}, {1018, 37139}, {1055, 1412}, {1155, 1014}, {1334, 2291}, {1638, 17096}, {2318, 60047}, {2321, 1121}, {3700, 60479}, {3952, 35157}, {4041, 35348}, {4171, 23893}, {4515, 41798}, {4524, 23351}, {4552, 60487}, {4557, 14733}, {4566, 65553}, {4574, 65304}, {6139, 3733}, {6366, 7192}, {6603, 81}, {6745, 86}, {7140, 68580}, {14392, 3737}, {14413, 7203}, {21871, 61493}, {23890, 4637}, {30574, 3676}, {30806, 57785}, {33573, 17197}, {52335, 60579}, {53008, 65340}, {56543, 4616}, {60431, 27}, {61035, 17169}, {62756, 757}, {65448, 7253}, {65680, 1019}, {68742, 593}
X(68925) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 3991, 21049}, {1018, 17757, 17747}, {1334, 21031, 38930}, {3693, 6735, 1146}, {3930, 21013, 40663}

X(68926) = X(2)X(37)∩X(7)X(59572)

Barycentrics    a^4 - 2*a^2*b^2 + b^4 - a^2*b*c + 2*a*b^2*c - b^3*c - 2*a^2*c^2 + 2*a*b*c^2 - b*c^3 + c^4 : :

X(68926) lies on these lines: {2, 37}, {7, 59572}, {8, 17095}, {69, 64083}, {78, 33298}, {85, 5552}, {86, 13405}, {100, 4872}, {150, 5440}, {190, 40869}, {200, 319}, {241, 27526}, {279, 27525}, {320, 765}, {322, 31627}, {325, 32850}, {341, 3926}, {348, 7080}, {404, 7247}, {461, 55394}, {497, 30740}, {664, 6735}, {894, 25355}, {910, 56555}, {1088, 20930}, {1376, 7179}, {1447, 3035}, {3085, 25583}, {3086, 17158}, {3212, 37828}, {3617, 27187}, {3664, 59593}, {3673, 26364}, {3674, 63990}, {3684, 24318}, {3685, 62674}, {3717, 57548}, {3870, 44179}, {3912, 62388}, {4130, 31287}, {4357, 20103}, {4360, 11019}, {4554, 35517}, {4847, 5564}, {4911, 25440}, {5088, 17757}, {5176, 17136}, {5224, 8580}, {5687, 17181}, {5836, 17084}, {6604, 27383}, {7176, 12607}, {7196, 27517}, {7321, 40719}, {9780, 25581}, {10578, 17394}, {10580, 17393}, {14548, 63168}, {17044, 40872}, {17078, 30806}, {17143, 54443}, {17170, 59591}, {17315, 31038}, {17377, 66469}, {17791, 37780}, {20880, 27529}, {21453, 40424}, {24241, 60714}, {25083, 37774}, {25242, 46835}, {25244, 27068}, {25723, 66205}, {26590, 30798}, {27006, 27096}, {27385, 55082}, {28795, 31225}, {30827, 64695}, {32049, 59537}, {34254, 57925}, {35518, 57244}, {42697, 62710}, {46749, 59181}, {53597, 59722}, {55393, 57534}, {56928, 59691}, {59584, 64702}

X(68926) = isotomic conjugate of the isogonal conjugate of X(68591)
X(68926) = X(2208)-isoconjugate of X(48357)
X(68926) = crosspoint of X(4998) and X(57928)
X(68926) = barycentric product X(i)*X(j) for these {i,j}: {76, 68591}, {1978, 53300}
X(68926) = barycentric quotient X(i)/X(j) for these {i,j}: {329, 48357}, {39558, 1436}, {53300, 649}, {68591, 6}
X(68926) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 44798, 17289}, {100, 33864, 4872}, {348, 7080, 16284}, {1267, 5391, 346}, {32791, 32792, 17263}

X(68927) = X(7)X(2804)∩X(100)X(658)

Barycentrics    (b - c)*(a^4 - 2*a^2*b^2 + b^4 - a^2*b*c + 2*a*b^2*c - b^3*c - 2*a^2*c^2 + 2*a*b*c^2 - b*c^3 + c^4) : :
X(68927) = 3 X[4453] - 4 X[43042], 3 X[4453] - 2 X[53357], 3 X[30565] - 4 X[68813], 4 X[14330] - 5 X[31209]

X(68927) lies on these lines: {7, 2804}, {100, 658}, {514, 661}, {523, 46402}, {885, 31637}, {918, 20533}, {1295, 43363}, {2826, 9809}, {3676, 14392}, {3900, 4131}, {3945, 53522}, {4397, 30805}, {4648, 47137}, {4851, 63813}, {6362, 46400}, {7192, 65674}, {8713, 48107}, {10436, 23678}, {14330, 31209}, {17880, 20901}, {18025, 50333}, {20520, 35293}, {20940, 65867}, {21105, 43041}, {21202, 53532}, {24582, 51381}, {28473, 66516}, {30857, 53362}, {36101, 39470}, {39775, 47695}, {57196, 57241}, {66270, 66286}

X(68927) = reflection of X(53357) in X(43042)
X(68927) = isotomic conjugate of the isogonal conjugate of X(53300)
X(68927) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {103, 37781}, {651, 152}, {677, 329}, {911, 39351}, {1815, 34188}, {24016, 7}, {32642, 3177}, {32668, 145}, {36039, 144}, {36101, 33650}, {40116, 5942}, {43736, 150}, {52156, 21293}, {57928, 21286}, {65218, 68335}, {65245, 3434}, {65294, 21285}, {65538, 32003}
X(68927) = X(32652)-isoconjugate of X(48357)
X(68927) = X(16596)-Dao conjugate of X(48357)
X(68927) = crosspoint of X(i) and X(j) for these (i,j): {85, 65294}, {664, 18025}
X(68927) = crossdifference of every pair of points on line {31, 14936}
X(68927) = barycentric product X(i)*X(j) for these {i,j}: {76, 53300}, {3261, 68591}
X(68927) = barycentric quotient X(i)/X(j) for these {i,j}: {14837, 48357}, {39558, 36049}, {53300, 6}, {68591, 101}
X(68927) = {X(43042),X(53357)}-harmonic conjugate of X(4453)

X(68928) = X(2)X(6)∩X(7)X(480)

Barycentrics    a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c - a^2*b*c + a*b^2*c + b^3*c - a^2*c^2 + a*b*c^2 - 2*b^2*c^2 + a*c^3 + b*c^3 : :

X(68928) lies on these lines: {1, 16284}, {2, 6}, {7, 480}, {8, 5543}, {55, 30946}, {56, 36854}, {75, 200}, {76, 1043}, {77, 31627}, {78, 85}, {100, 20347}, {144, 42316}, {150, 17757}, {190, 3693}, {192, 24352}, {241, 27399}, {264, 14004}, {315, 36652}, {319, 4847}, {320, 765}, {322, 3870}, {326, 1088}, {332, 44139}, {348, 27383}, {350, 14942}, {404, 1434}, {461, 32000}, {518, 1447}, {519, 24203}, {664, 4511}, {666, 2338}, {673, 3684}, {894, 44798}, {903, 51567}, {908, 4872}, {1078, 17206}, {1222, 24524}, {1260, 33765}, {1330, 7767}, {2340, 56783}, {2893, 8226}, {3158, 64695}, {3212, 12635}, {3262, 3935}, {3263, 3699}, {3452, 64702}, {3509, 24685}, {3663, 60714}, {3664, 17122}, {3673, 3811}, {3713, 26125}, {3726, 26273}, {3732, 57015}, {3759, 24600}, {3785, 36706}, {3873, 26229}, {3875, 66469}, {3879, 11019}, {3930, 9318}, {3964, 37309}, {3965, 41246}, {3996, 4441}, {4184, 40007}, {4251, 17681}, {4255, 4352}, {4258, 17691}, {4357, 13405}, {4389, 63168}, {4419, 31477}, {4420, 20880}, {4561, 20924}, {4645, 50441}, {4666, 55392}, {4851, 31038}, {4864, 57033}, {4911, 21077}, {4966, 62674}, {5088, 5440}, {5195, 51409}, {5231, 17360}, {5552, 33298}, {5687, 17753}, {5845, 56555}, {6603, 40872}, {6604, 7080}, {6700, 53597}, {6765, 17158}, {7176, 59691}, {7179, 47595}, {7190, 63151}, {7196, 27391}, {7411, 8822}, {8580, 10436}, {9778, 59253}, {10446, 19541}, {10578, 17321}, {10582, 17394}, {11681, 21285}, {12607, 56928}, {13741, 33953}, {14208, 18155}, {14615, 18147}, {16061, 17499}, {17095, 27385}, {17139, 36002}, {17169, 17531}, {17201, 17536}, {17241, 30813}, {17246, 59238}, {17272, 32916}, {17336, 59216}, {17365, 25355}, {17377, 36845}, {17747, 20533}, {18133, 68715}, {18140, 18738}, {18162, 60726}, {18822, 53647}, {20173, 20928}, {20448, 34018}, {20449, 34063}, {20935, 30963}, {21272, 62826}, {21320, 51928}, {21581, 33954}, {21596, 33296}, {24283, 49518}, {24499, 59734}, {25006, 32025}, {25082, 32024}, {25532, 45751}, {25940, 41245}, {26563, 34772}, {28742, 32008}, {30997, 63236}, {31146, 50132}, {32001, 57534}, {32029, 33891}, {32828, 36660}, {33101, 33869}, {33949, 41826}, {36003, 56935}, {36027, 44150}, {36485, 41711}, {36722, 64093}, {39959, 49499}, {52156, 56110}, {53598, 59679}, {56999, 57826}

X(68928) = isotomic conjugate of X(43672)
X(68928) = isotomic conjugate of the complement of X(58035)
X(68928) = isotomic conjugate of the isogonal conjugate of X(13329)
X(68928) = isotomic conjugate of the polar conjugate of X(26003)
X(68928) = X(13329)-cross conjugate of X(26003)
X(68928) = X(i)-isoconjugate of X(j) for these (i,j): {31, 43672}, {661, 59067}
X(68928) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 43672}, {36830, 59067}, {68813, 17435}
X(68928) = cevapoint of X(2) and X(58035)
X(68928) = crosspoint of X(4998) and X(51560)
X(68928) = crossdifference of every pair of points on line {512, 40978}
X(68928) = barycentric product X(i)*X(j) for these {i,j}: {69, 26003}, {75, 62799}, {76, 13329}, {190, 53357}, {1978, 53308}
X(68928) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 43672}, {110, 59067}, {13329, 6}, {26003, 4}, {53308, 649}, {53357, 514}, {62799, 1}
X(68928) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 14828, 86}, {2, 37658, 17277}, {69, 183, 14829}, {69, 63014, 54303}, {200, 40719, 75}, {298, 299, 17297}, {322, 55391, 4360}, {350, 33677, 35517}, {491, 492, 17234}, {3684, 20335, 673}, {3693, 10025, 190}, {3870, 62697, 4360}, {4511, 30806, 664}

X(68929) = X(2)X(6)∩X(513)X(676)

Barycentrics    2*a^4 + a^2*b^2 + 2*a*b^3 - b^4 - 4*a^2*b*c - 2*a*b^2*c + 2*b^3*c + a^2*c^2 - 2*a*b*c^2 - 2*b^2*c^2 + 2*a*c^3 + 2*b*c^3 - c^4 : :

X(68929) lies on these lines: {2, 6}, {30, 17205}, {274, 49734}, {513, 676}, {1086, 4872}, {1104, 53597}, {1279, 9436}, {1434, 64159}, {1447, 3756}, {1565, 30117}, {1616, 6604}, {3263, 9053}, {3290, 5845}, {3663, 4906}, {3664, 3742}, {3665, 28082}, {3752, 64702}, {3879, 4849}, {4328, 39688}, {4904, 16784}, {5272, 47595}, {5543, 17084}, {5572, 6007}, {7181, 17213}, {7195, 28080}, {9507, 17768}, {16502, 21258}, {16887, 49728}, {17054, 17170}, {17061, 24241}, {17169, 49745}, {17724, 33864}, {18600, 64158}, {24215, 57288}, {28011, 30617}, {33953, 65543}, {49466, 59509}

X(68929) = crossdifference of every pair of points on line {220, 512}
X(68929) = {X(28014),X(28078)}-harmonic conjugate of X(141)

X(68930) = X(1)X(2)∩X(661)X(6587)

Barycentrics    (b + c)*(-2*a^4 - a^2*b^2 - 2*a*b^3 + b^4 + 4*a^2*b*c + 2*a*b^2*c - 2*b^3*c - a^2*c^2 + 2*a*b*c^2 + 2*b^2*c^2 - 2*a*c^3 - 2*b*c^3 + c^4) : :

X(68930) lies on these lines: {1, 2}, {661, 6587}, {857, 3120}, {4414, 37169}, {16609, 21950}, {17054, 21671}, {17056, 21921}, {17067, 24086}, {17278, 21675}, {18635, 40977}, {21076, 53665}, {21094, 27191}, {21931, 40934}, {21945, 39688}, {24161, 30839}, {24443, 30810}

X(68930) = crossdifference of every pair of points on line {649, 2328}

X(68931) = X(6)X(57)∩X(37)X(100)

Barycentrics    a*(4*a^3 - 3*a^2*b - 2*a*b^2 - 3*b^3 - 3*a^2*c + 3*b^2*c - 2*a*c^2 + 3*b*c^2 - 3*c^3) : :

X(68931) lies on these lines: {6, 57}, {37, 100}, {169, 5024}, {187, 16583}, {574, 1212}, {1104, 1384}, {1376, 3731}, {1575, 15492}, {2329, 56174}, {3290, 10987}, {3509, 4849}, {3684, 21342}, {3723, 4386}, {3772, 37689}, {4000, 63097}, {4646, 16785}, {5210, 16968}, {8301, 49465}, {8589, 49758}, {16784, 52541}, {16814, 44798}, {25269, 40883}, {43448, 46835}, {62693, 62992}

X(68931) = crosssum of X(9) and X(64070)
X(68931) = crossdifference of every pair of points on line {3900, 14419}

X(68932) = X(2)X(85)∩X(9)X(25727)

Barycentrics    b*c*(-4*a^3 + 3*a^2*b + 2*a*b^2 + 3*b^3 + 3*a^2*c - 3*b^2*c + 2*a*c^2 - 3*b*c^2 + 3*c^3) : :

X(68932) lies on these lines: {2, 85}, {9, 25727}, {92, 62955}, {321, 668}, {524, 53510}, {599, 1229}, {1441, 50093}, {1992, 17863}, {3262, 49748}, {3436, 64143}, {3452, 21139}, {3765, 4980}, {3928, 24612}, {3975, 21432}, {4696, 42029}, {11160, 20171}, {17132, 20895}, {17677, 52345}, {17789, 52353}, {17895, 54280}, {20927, 21356}, {20956, 42724}, {24993, 35578}, {26660, 43066}, {29573, 30806}, {29577, 52043}

X(68933) = X(2)X(523)∩X(513)X(676)

Barycentrics    (b - c)*(-4*a^3 + 3*a^2*b + 2*a*b^2 + 3*b^3 + 3*a^2*c - 3*b^2*c + 2*a*c^2 - 3*b*c^2 + 3*c^3) : :
X(68933) = 3 X[47797] + X[47834], 3 X[47799] - X[47827], 4 X[48206] - 9 X[59943], 2 X[2490] + X[47691], 2 X[4458] + X[14321], X[4879] + 2 X[65494], 2 X[7657] + X[65685], X[47131] + 2 X[53573], X[47826] + 3 X[47887], X[47826] - 3 X[48179]

X(68933) lies on these lines: {2, 523}, {513, 676}, {522, 45677}, {918, 48547}, {1499, 21181}, {2490, 47691}, {3004, 48237}, {3837, 28221}, {4160, 34958}, {4458, 14321}, {4874, 28175}, {4879, 65494}, {4885, 28161}, {4927, 47798}, {6084, 47800}, {7657, 65685}, {14419, 48403}, {23770, 48226}, {28147, 68794}, {28183, 48198}, {29162, 30234}, {45318, 47803}, {47123, 48193}, {47131, 53573}, {47826, 47887}, {47891, 48161}, {48166, 58372}, {48170, 50347}

X(68933) = midpoint of X(i) and X(j) for these {i,j}: {3004, 48237}, {4927, 47798}, {14419, 48403}, {23770, 48226}, {47123, 48193}, {47788, 48203}, {47887, 48179}, {47891, 48161}, {48166, 58372}, {48170, 50347}
X(68933) = reflection of X(47803) in X(45318)
X(68933) = crossdifference of every pair of points on line {187, 220}

X(68934) = X(10)X(37) INTERCEPT OF X(2)X(32)

Barycentrics    (b + c)*(2*a^3 + 2*a^2*b + 2*a*b^2 + b^3 + 2*a^2*c + 2*a*b*c + b^2*c + 2*a*c^2 + b*c^2 + c^3) : :

X(68934) lies on these lines: {2, 32}, {10, 37}, {39, 13728}, {115, 1281}, {239, 41809}, {964, 7747}, {1211, 17023}, {2238, 5280}, {3125, 27714}, {3936, 29614}, {3948, 33941}, {4204, 5310}, {4687, 30173}, {5224, 17034}, {5254, 50058}, {5259, 8299}, {5309, 51593}, {6155, 20653}, {6536, 21816}, {6781, 11115}, {7737, 37037}, {7745, 50318}, {7748, 50056}, {7756, 17676}, {7765, 26035}, {7794, 25499}, {7813, 16705}, {8040, 21718}, {11108, 36744}, {14537, 50323}, {16846, 19763}, {17385, 53486}, {17588, 24956}, {17589, 26079}, {17744, 36478}, {17750, 32784}, {18755, 24931}, {19863, 31488}, {19866, 31416}, {24275, 26117}, {29610, 33157}, {41258, 46828}, {50055, 65633}

X(68934) = X(i)-complementary conjugate of X(j) for these (i,j): {512, 53835}, {831, 512}, {57975, 23301}, {58956, 8060}
X(68934) = crossdifference of every pair of points on line {3005, 3733}
X(68934) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1213, 4205, 16589}, {1213, 53423, 10}, {5051, 52538, 115}

X(68935) = X(10)X(37) INTERCEPT OF X(230)X(231)

Barycentrics    a*(b + c)*(a^3 + b^3 - a*b*c - b^2*c - b*c^2 + c^3) : :

X(68935) lies on these lines: {1, 4273}, {2, 20896}, {6, 2294}, {9, 4016}, {10, 37}, {19, 2204}, {44, 758}, {45, 2292}, {55, 40973}, {142, 63398}, {187, 15586}, {230, 231}, {292, 66274}, {518, 15990}, {524, 49760}, {748, 4137}, {976, 16777}, {1015, 56531}, {1084, 65944}, {1086, 8680}, {1279, 44661}, {1333, 1781}, {1400, 56908}, {1731, 30117}, {1841, 1842}, {1880, 66918}, {1914, 7297}, {1953, 16685}, {1959, 52897}, {1962, 16672}, {2160, 2305}, {2161, 56911}, {2173, 3285}, {2178, 23843}, {2220, 16547}, {2238, 4053}, {2245, 3125}, {2300, 17443}, {2325, 68895}, {2643, 3747}, {3230, 17444}, {3666, 4395}, {3724, 6187}, {3752, 25080}, {3958, 16885}, {4000, 25255}, {4068, 23668}, {4274, 61704}, {4286, 24443}, {4384, 30903}, {4429, 49512}, {4466, 68930}, {4516, 39688}, {4647, 17281}, {4771, 50746}, {4850, 31205}, {4969, 50014}, {4999, 40937}, {5165, 5902}, {5540, 33882}, {7113, 42669}, {7124, 18674}, {8053, 17872}, {8772, 35327}, {9020, 20590}, {9022, 30059}, {11221, 58390}, {12081, 62239}, {14735, 61668}, {16050, 24335}, {16585, 16700}, {16599, 57037}, {16666, 63354}, {16684, 17446}, {16752, 65205}, {17054, 56839}, {17160, 28606}, {17164, 54389}, {17278, 18698}, {17279, 18697}, {17303, 19846}, {17369, 49598}, {17760, 25136}, {18179, 28287}, {18714, 27644}, {20271, 40978}, {21020, 61321}, {21864, 52963}, {25065, 64556}, {25078, 46838}, {25090, 40986}, {25370, 26601}, {26227, 26242}, {31880, 62210}, {33129, 62305}, {39232, 65680}, {58401, 65561}

X(68935) = midpoint of X(2643) and X(3747)
X(68935) = complement of X(35550)
X(68935) = complement of the isogonal conjugate of X(67166)
X(68935) = complement of the isotomic conjugate of X(759)
X(68935) = isogonal conjugate of the isotomic conjugate of X(62305)
X(68935) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 31845}, {560, 35069}, {759, 2887}, {1397, 6739}, {2161, 21245}, {2206, 214}, {2341, 21244}, {6187, 3454}, {9274, 21254}, {14616, 21235}, {24624, 626}, {32671, 4369}, {34079, 141}, {36069, 512}, {37140, 42327}, {47318, 21262}, {57736, 1368}, {65283, 23301}, {66284, 53575}, {67166, 10}, {68571, 21243}
X(68935) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 31845}, {66280, 512}
X(68935) = X(47417)-cross conjugate of X(6)
X(68935) = X(i)-isoconjugate of X(j) for these (i,j): {63, 39439}, {24624, 39166}
X(68935) = X(i)-Dao conjugate of X(j) for these (i,j): {3162, 39439}, {31845, 2}, {62652, 86}
X(68935) = crosspoint of X(i) and X(j) for these (i,j): {2, 759}, {30117, 33129}
X(68935) = crosssum of X(i) and X(j) for these (i,j): {6, 758}, {81, 37783}
X(68935) = crossdifference of every pair of points on line {3, 3733}
X(68935) = barycentric product X(i)*X(j) for these {i,j}: {6, 62305}, {10, 30117}, {37, 33129}, {72, 5146}, {226, 1731}, {523, 13589}, {758, 38938}, {759, 31845}, {1018, 47680}, {1577, 68828}, {5497, 5620}, {15906, 38955}
X(68935) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 39439}, {1731, 333}, {3724, 39166}, {5146, 286}, {13589, 99}, {15906, 17139}, {30117, 86}, {31845, 35550}, {33129, 274}, {38938, 14616}, {47680, 7199}, {62305, 76}, {68828, 662}
X(68935) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 21858, 59733}, {2294, 40977, 6}, {3011, 47209, 230}, {3290, 8609, 8610}, {3290, 47231, 3291}, {3924, 54324, 6}, {5089, 47232, 14580}, {8609, 14571, 8608}

X(68936) = X(10)X(37) INTERCEPT OF X(2)X(60227)

Barycentrics    (b + c)*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4 + a^4*c + 2*a^3*b*c + b^4*c - a^3*c^2 - 2*a*b^2*c^2 - b^3*c^2 - a^2*c^3 - b^2*c^3 + a*c^4 + b*c^4) : :

X(68936) lies on these lines: {2, 60227}, {4, 579}, {5, 4260}, {6, 48888}, {8, 27039}, {10, 37}, {12, 52020}, {25, 1751}, {43, 41797}, {115, 65892}, {142, 442}, {209, 22000}, {226, 3136}, {240, 522}, {242, 1731}, {284, 6998}, {387, 4270}, {406, 1714}, {407, 1893}, {429, 1827}, {511, 17197}, {516, 2245}, {518, 20544}, {851, 3911}, {938, 25521}, {946, 10974}, {950, 37225}, {984, 55076}, {1125, 52544}, {1211, 4847}, {1230, 63147}, {1536, 43672}, {1716, 56861}, {1865, 65950}, {1901, 63970}, {2238, 40869}, {2350, 60617}, {2486, 20718}, {3008, 68733}, {3220, 46501}, {3454, 10916}, {3717, 3948}, {3763, 50591}, {3778, 21935}, {3914, 20966}, {3936, 4684}, {4129, 37998}, {4199, 5745}, {4255, 16844}, {4259, 24220}, {4357, 5051}, {4516, 7235}, {4848, 58889}, {5138, 36477}, {5746, 36695}, {5747, 36672}, {5800, 5816}, {5802, 7390}, {5822, 6776}, {8609, 29016}, {9581, 27626}, {9840, 64582}, {10381, 24391}, {10477, 52257}, {11019, 17056}, {12053, 22076}, {12618, 53417}, {13329, 36027}, {15970, 40979}, {16052, 50092}, {16609, 44661}, {17606, 61034}, {17768, 23821}, {21075, 22312}, {21807, 56326}, {24209, 64858}, {24982, 25001}, {25005, 25243}, {25006, 41809}, {26000, 56317}, {30444, 51755}, {31397, 64167}, {33140, 34589}, {33305, 35466}, {37500, 49130}, {37507, 48938}, {38057, 46196}, {62305, 65196}

X(68936) = midpoint of X(4516) and X(7235)
X(68936) = X(i)-complementary conjugate of X(j) for these (i,j): {56, 8299}, {65, 120}, {105, 960}, {226, 20540}, {512, 1566}, {669, 39014}, {673, 21246}, {884, 34591}, {927, 512}, {1027, 34589}, {1042, 50441}, {1400, 16593}, {1402, 6184}, {1416, 1125}, {1427, 17060}, {1438, 5745}, {1462, 3739}, {2195, 59646}, {4559, 62552}, {7180, 35094}, {8034, 35509}, {8751, 6708}, {10099, 123}, {13576, 1329}, {18785, 3452}, {32735, 523}, {34018, 21240}, {34085, 42327}, {36057, 34851}, {36124, 34831}, {36146, 4369}, {43929, 4858}, {43930, 53564}, {46135, 23301}, {53321, 3126}, {55261, 26932}, {56783, 3741}, {56853, 9}, {64216, 40937}, {65301, 52598}, {66930, 661}, {66941, 141}, {68565, 41883}
X(68936) = crosspoint of X(i) and X(j) for these (i,j): {10, 43672}, {75, 65264}
X(68936) = crosssum of X(58) and X(13329)
X(68936) = crossdifference of every pair of points on line {48, 3733}
X(68936) = barycentric product X(1577)*X(7437)
X(68936) = barycentric quotient X(7437)/X(662)
X(68936) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1834, 2092, 3755}, {1834, 53425, 2092}, {3136, 40952, 226}

X(68937) = X(10)X(37) INTERCEPT OF X(732)X(3589)

Barycentrics    (b + c)*(-a^2 + b*c)*(a^2*b - a*b^2 + a^2*c + 2*a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(68937) lies on these lines: {10, 37}, {732, 3589}, {798, 812}, {874, 29375}, {2238, 4974}, {2240, 4434}, {2350, 40013}, {3948, 17755}, {4253, 18144}, {14991, 23789}, {16552, 18133}, {17277, 18046}, {18143, 29447}, {22011, 22279}, {24211, 25373}, {26965, 26971}, {27026, 27032}, {27040, 33159}, {62588, 62646}

X(68937) = X(i)-complementary conjugate of X(j) for these (i,j): {18082, 20542}, {18098, 20333}, {18105, 38989}, {34067, 3005}, {36081, 512}, {67149, 6682}
X(68937) = X(i)-Ceva conjugate of X(j) for these (i,j): {17277, 17755}, {18140, 17793}, {30940, 740}
X(68937) = X(18268)-isoconjugate of X(27807)
X(68937) = X(35068)-Dao conjugate of X(27807)
X(68937) = crossdifference of every pair of points on line {1964, 3733}
X(68937) = barycentric product X(i)*X(j) for these {i,j}: {239, 22011}, {350, 22279}, {740, 17140}, {874, 40471}, {2238, 18143}, {14991, 27853}, {16679, 35544}
X(68937) = barycentric quotient X(i)/X(j) for these {i,j}: {740, 27807}, {14991, 3572}, {16679, 741}, {17140, 18827}, {18143, 40017}, {22011, 335}, {22279, 291}, {40471, 876}
X(68937) = {X(1213),X(16589)}-harmonic conjugate of X(3842)

X(68938) = X(10)X(37) INTERCEPT OF X(1)X(27040)

Barycentrics    (b + c)*(-(a^2*b) + a*b^2 - a^2*c - 2*a*b*c + b^2*c + a*c^2 + b*c^2) : :

X(68938) lies on these lines: {1, 27040}, {2, 60276}, {8, 46196}, {10, 37}, {56, 62426}, {76, 17234}, {115, 35123}, {142, 53478}, {238, 57017}, {350, 17761}, {391, 5129}, {442, 51972}, {514, 661}, {519, 2238}, {538, 17205}, {672, 49999}, {758, 3985}, {1001, 34812}, {1089, 21808}, {1107, 52539}, {1125, 52538}, {1211, 29594}, {1575, 49993}, {1655, 16887}, {1855, 44143}, {2140, 3760}, {2170, 4975}, {2229, 4871}, {2240, 49994}, {2245, 2325}, {2300, 4856}, {3125, 4037}, {3159, 3721}, {3161, 3730}, {3244, 20970}, {3294, 17751}, {3661, 17497}, {3685, 5011}, {3701, 3970}, {3726, 59717}, {3797, 57029}, {3831, 25092}, {3840, 21838}, {3930, 3992}, {3954, 4075}, {4006, 52353}, {4016, 24067}, {4044, 4054}, {4071, 21090}, {4082, 40952}, {4099, 4642}, {4109, 24065}, {4125, 21101}, {4253, 27523}, {4568, 20947}, {4645, 5134}, {4647, 21921}, {4653, 26244}, {4898, 50581}, {5233, 30830}, {5275, 48863}, {5283, 50605}, {6376, 17240}, {6651, 66152}, {6703, 50162}, {6707, 50163}, {8074, 33845}, {8620, 20340}, {8818, 52569}, {14964, 61234}, {17034, 63051}, {17056, 29600}, {17135, 62646}, {17200, 33816}, {17235, 21240}, {17238, 27269}, {17242, 26772}, {17499, 20090}, {17729, 24602}, {20659, 21044}, {20888, 20891}, {20963, 64536}, {21255, 53476}, {21753, 42057}, {22214, 23444}, {24049, 64071}, {24170, 25264}, {24189, 31348}, {24275, 40750}, {26959, 27033}, {27109, 29438}, {27255, 27261}, {29571, 30818}, {29577, 31037}, {29824, 40614}, {34587, 35068}, {38930, 41006}, {39011, 65940}, {40515, 60244}, {42285, 60676}, {48866, 63099}

X(68938) = midpoint of X(3125) and X(4037)
X(68938) = reflection of X(4115) in X(3985)
X(68938) = complement of X(62755)
X(68938) = complement of the isogonal conjugate of X(62763)
X(68938) = complement of the isotomic conjugate of X(41683)
X(68938) = X(i)-complementary conjugate of X(j) for these (i,j): {213, 13466}, {669, 39011}, {739, 3739}, {889, 23301}, {898, 512}, {3227, 21240}, {4557, 14434}, {4607, 42327}, {23349, 244}, {23892, 17761}, {32718, 523}, {34075, 4369}, {35353, 21252}, {37129, 3741}, {41683, 2887}, {43928, 53564}, {60288, 626}, {62763, 10}
X(68938) = X(i)-Ceva conjugate of X(j) for these (i,j): {29824, 44671}, {60288, 10}
X(68938) = X(1019)-isoconjugate of X(59071)
X(68938) = X(i)-Dao conjugate of X(j) for these (i,j): {899, 52897}, {6741, 60575}
X(68938) = crosspoint of X(2) and X(41683)
vcrosssum of X(6) and X(62740)
X(68938) = crossdifference of every pair of points on line {31, 3733}
X(68938) = barycentric product X(i)*X(j) for these {i,j}: {10, 29824}, {75, 44671}, {321, 45751}, {1577, 68812}, {3952, 68896}, {40614, 60288}
X(68938) = barycentric quotient X(i)/X(j) for these {i,j}: {3700, 60575}, {4557, 59071}, {29824, 86}, {40614, 52897}, {44671, 1}, {45751, 81}, {68812, 662}, {68896, 7192}
X(68938) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 21071, 21070}, {350, 30109, 17761}, {1089, 21808, 22011}, {1500, 21025, 10}, {3701, 3970, 21067}, {3760, 29966, 2140}, {3930, 3992, 4103}, {16589, 21024, 10}, {20947, 49753, 4568}

X(68939) = X(10)X(37) INTERCEPT OF X(325)X(523)

Barycentrics    (b + c)*(-(a^2*b^2) + a*b^3 - a*b^2*c + b^3*c - a^2*c^2 - a*b*c^2 + a*c^3 + b*c^3) : :

X(68939) lies on these lines: {10, 37}, {141, 3728}, {321, 58365}, {325, 523}, {344, 4068}, {674, 30059}, {714, 1086}, {2388, 30109}, {2486, 3948}, {2667, 17243}, {3216, 24746}, {3703, 4665}, {3717, 20718}, {3747, 4422}, {3912, 44671}, {4395, 30970}, {4439, 68895}, {4516, 42713}, {4647, 9710}, {9053, 10459}, {17163, 32862}, {17234, 25295}, {17245, 25124}, {18134, 25294}, {21926, 53478}, {22008, 22271}, {25123, 37662}, {27812, 33089}, {29639, 30748}, {31079, 31087}, {34337, 68779}, {40005, 40088}

X(68939) = reflection of X(3747) in X(4422)
X(68939) = isotomic conjugate of X(2368)
X(68939) = isotomic conjugate of the isogonal conjugate of X(2388)
X(68939) = X(31)-isoconjugate of X(2368)
X(68939) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 2368}, {30109, 56431}
X(68939) = crossdifference of every pair of points on line {32, 3733}
X(68939) = barycentric product X(i)*X(j) for these {i,j}: {10, 30109}, {76, 2388}, {321, 57024}
X(68939) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2368}, {2388, 6}, {30109, 86}, {57024, 81}

X(68940) = X(10)X(37) INTERCEPT OF X(187)X(237)

Barycentrics    a^2*(b + c)*(a^2*b^2 + a*b^3 - a*b^2*c - b^3*c + a^2*c^2 - a*b*c^2 + a*c^3 - b*c^3) : :

X(68940) lies on these lines: {10, 37}, {39, 2292}, {187, 237}, {595, 1333}, {758, 1015}, {960, 52535}, {1574, 4647}, {1575, 68895}, {2241, 64753}, {2300, 23525}, {2388, 4093}, {3121, 21839}, {3571, 34996}, {4016, 17053}, {5291, 17961}, {21685, 52529}, {21816, 52539}, {27076, 35544}, {57029, 57039}

X(68940) = reflection of X(35544) in X(27076)
X(68940) = X(i)-Ceva conjugate of X(j) for these (i,j): {5291, 5164}, {17961, 2092}
X(68940) = crosspoint of X(57039) and X(68750)
X(68940) = crosssum of X(81) and X(19623)
X(68940) = crossdifference of every pair of points on line {2, 3733}
X(68940) = barycentric product X(i)*X(j) for these {i,j}: {10, 68750}, {37, 57039}, {42, 57029}, {72, 52461}
X(68940) = barycentric quotient X(i)/X(j) for these {i,j}: {52461, 286}, {57029, 310}, {57039, 274}, {68750, 86}
X(68940) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2292, 40986, 39}, {3230, 5163, 187}

X(68941) = X(10)X(37) INTERCEPT OF X(239)X(514)

Barycentrics    (b + c)*(-2*a^3 - a^2*b - a*b^2 - a^2*c + 4*a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(68941) lies on these lines: {10, 37}, {239, 514}, {519, 3125}, {596, 3780}, {758, 4771}, {1125, 6155}, {2238, 4115}, {3214, 21067}, {3293, 22011}, {4103, 31855}, {4386, 49683}, {4706, 43065}, {4714, 21840}, {5134, 62392}, {6533, 39247}, {8682, 17205}, {9278, 39697}, {11611, 60094}, {14210, 17023}, {16971, 49477}, {20693, 22035}, {20963, 24176}, {24044, 27040}, {24185, 30941}, {35342, 39766}, {41015, 64185}, {46196, 64071}

X(68941) = midpoint of X(17497) and X(62755)
X(68941) = reflection of X(i) in X(j) for these {i,j}: {4115, 2238}, {30941, 24185}
X(68941) = crossdifference of every pair of points on line {42, 3733}
X(68941) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {239, 5011, 57017}, {4065, 16589, 24051}

X(68942) = X(10)X(37) INTERCEPT OF X(2)X(2234)

Barycentrics    (b + c)*(a^2*b^2 - 3*a^2*b*c + a^2*c^2 + b^2*c^2) : :
X(68942) = 3 X[24517] + X[30939], 7 X[3624] - 3 X[18792]

X(689412) lies on these lines: {2, 2234}, {10, 37}, {75, 22174}, {190, 20984}, {244, 39995}, {256, 25660}, {313, 22172}, {513, 3716}, {536, 20340}, {714, 3122}, {730, 8610}, {751, 17250}, {1045, 27111}, {1086, 46843}, {2228, 4358}, {2245, 4368}, {2667, 26772}, {3248, 27166}, {3264, 64874}, {3616, 26107}, {3624, 18792}, {3741, 4708}, {3770, 63520}, {3840, 17237}, {3971, 4735}, {3994, 20711}, {4039, 39688}, {4443, 30830}, {6376, 22189}, {8287, 21256}, {14210, 68877}, {16597, 19563}, {16609, 21254}, {17227, 30957}, {17229, 25121}, {17231, 34832}, {17242, 31337}, {17793, 57024}, {18133, 21330}, {20923, 45197}, {22167, 56249}, {22190, 56250}, {22214, 56253}, {23634, 46898}, {24327, 30819}, {24351, 30863}, {24429, 60861}, {25351, 49993}, {25624, 48630}, {27042, 58396}, {30982, 64554}, {31036, 46905}, {52529, 58380}

X(68942) = midpoint of X(i) and X(j) for these {i,j}: {3122, 3948}, {57034, 68883}
X(68942) = complement of X(2234)
X(68942) = complement of the isogonal conjugate of X(37132)
X(68942) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 35073}, {99, 62611}, {512, 9151}, {669, 39010}, {729, 2}, {886, 23301}, {3228, 141}, {9150, 512}, {14608, 126}, {32717, 523}, {34087, 626}, {35366, 46654}, {36133, 4369}, {37132, 10}, {41309, 2482}, {46156, 6292}, {51510, 39080}, {52752, 113}, {52762, 44956}, {52765, 114}, {57459, 44949}, {57540, 59765}, {60028, 125}, {63749, 115}, {66278, 53575}
X(68942) = crossdifference of every pair of points on line {2176, 3733}
X(68942) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 21238, 28593}, {37, 21257, 21238}

X(68943) = X(10)X(37) INTERCEPT OF X(513)X(663)

Barycentrics    a*(b + c)*(a^3*b + 2*a^2*b^2 + a*b^3 + a^3*c - 4*a^2*b*c - 2*a*b^2*c - b^3*c + 2*a^2*c^2 - 2*a*b*c^2 + 2*b^2*c^2 + a*c^3 - b*c^3) : :

X(68943) lies on these lines: {10, 37}, {513, 663}, {516, 8610}, {1001, 18792}, {2234, 28352}, {3122, 20718}, {3685, 57039}, {3752, 4854}, {3923, 39798}, {3936, 58401}, {4442, 16610}, {4657, 26094}, {4890, 49478}, {5695, 46838}, {9791, 16696}, {16700, 33100}, {16726, 17768}, {16753, 44006}, {17321, 30939}, {17384, 19847}, {25881, 25887}, {28018, 28022}, {28366, 49484}, {32922, 57023}, {49740, 66696}, {52541, 63997}, {64556, 66071}

X(68943) = midpoint of X(18792) and X(24429)
X(68943) = crossdifference of every pair of points on line {9, 3733}
X(68943) = {X(1284),X(39688)}-harmonic conjugate of X(1279)

X(68944) = X(10)X(37) INTERCEPT OF X(8)X(16974)

Barycentrics    (b + c)*(a^3 + a*b^2 + b^3 + a*b*c + b^2*c + a*c^2 + b*c^2 + c^3) : :

X(68944) lies on these lines: {8, 16974}, {10, 37}, {75, 17673}, {172, 32779}, {333, 3661}, {429, 68800}, {1211, 21879}, {1761, 3501}, {2176, 32778}, {2238, 20653}, {2295, 15523}, {2345, 26051}, {3844, 53475}, {3912, 6703}, {4037, 5051}, {4204, 4433}, {8258, 50252}, {14007, 17303}, {16519, 32783}, {17280, 17685}, {17281, 17677}, {17698, 49560}, {17762, 31090}, {17776, 26044}, {18755, 33160}, {20970, 21081}, {24275, 36974}, {25345, 33935}, {29576, 41817}, {29594, 61661}, {29674, 40750}, {39597, 52964}, {42707, 59212}

X(68944) = barycentric product X(i)*X(j) for these {i,j}: {10, 32783}, {321, 16519}
X(68944) = barycentric quotient X(i)/X(j) for these {i,j}: {16519, 81}, {32783, 86}

X(68945) = X(1)X(2)∩X(31)X(4657)

Barycentrics    2*a^3 + 2*a^2*b + 2*a*b^2 + b^3 + 2*a^2*c + 2*a*b*c + b^2*c + 2*a*c^2 + b*c^2 + c^3 : :

X(68945) lies on these lines: {1, 2}, {31, 4657}, {56, 37061}, {86, 2206}, {199, 20470}, {238, 6536}, {244, 6703}, {756, 3589}, {1001, 5347}, {1100, 33081}, {1962, 17045}, {2260, 5282}, {2308, 4357}, {2916, 20988}, {3120, 14012}, {3683, 41311}, {3715, 47352}, {3745, 17384}, {3989, 5294}, {3995, 24295}, {4026, 17469}, {4418, 17302}, {4423, 57522}, {4974, 41809}, {5322, 37033}, {6535, 17289}, {8025, 49676}, {8040, 17322}, {9347, 33174}, {10180, 24542}, {13728, 62847}, {16437, 24789}, {16555, 56532}, {17184, 33682}, {17305, 33067}, {17306, 33080}, {17320, 32936}, {17368, 32925}, {17370, 25961}, {17379, 33069}, {17380, 32860}, {17381, 32771}, {17383, 33125}, {17495, 59628}, {17600, 32779}, {17602, 31264}, {18139, 50293}, {19684, 26128}, {19717, 33064}, {19808, 32924}, {19812, 25760}, {19832, 33071}, {20182, 33156}, {20985, 25499}, {21027, 33132}, {24342, 33150}, {25539, 33172}, {27184, 61707}, {27186, 43997}, {32774, 50302}, {32784, 62807}, {33073, 48650}, {33076, 62855}, {33087, 62801}, {33155, 48642}, {33158, 62851}, {39979, 44307}

X(68945) = crosssum of X(42) and X(5280)
X(68945) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2, 15523}, {1, 29647, 29685}, {1, 29648, 29686}, {2, 5311, 29687}, {2, 17150, 10}, {2, 29644, 29682}, {2, 29645, 29683}, {2, 29646, 29684}, {612, 29598, 29663}, {1125, 29654, 2}, {3624, 5287, 29677}, {3745, 17384, 32781}, {17011, 32783, 4062}, {17184, 33682, 64164}, {17289, 32928, 6535}, {17306, 62845, 33080}, {19786, 32772, 3120}, {21085, 45222, 49985}, {24542, 41820, 10180}, {26230, 43223, 29689}, {26626, 33171, 67208}

X(68946) = X(1)X(442)∩X(2)X(4256)

Barycentrics    (b + c)*(a^3 + b^3 - a*b*c - b^2*c - b*c^2 + c^3) : :
X(68946) = X[1] - 4 X[17070], 2 X[10] + X[4442], X[24222] + 2 X[33136], 5 X[1698] - 2 X[3712]

X(68946) lies on these lines: {1, 442}, {2, 4256}, {4, 580}, {5, 3216}, {6, 17532}, {8, 3454}, {9, 53417}, {10, 321}, {11, 49997}, {12, 3293}, {21, 24880}, {30, 35466}, {35, 37225}, {36, 851}, {42, 3822}, {43, 3136}, {46, 407}, {53, 1713}, {58, 2475}, {79, 1046}, {80, 3465}, {81, 3017}, {100, 17734}, {101, 17737}, {115, 2238}, {149, 40091}, {191, 24851}, {225, 23604}, {230, 35342}, {238, 3583}, {240, 522}, {316, 33295}, {325, 62755}, {333, 17677}, {355, 30449}, {377, 5292}, {381, 4383}, {386, 2476}, {387, 5177}, {403, 61226}, {404, 45939}, {405, 66104}, {429, 1717}, {440, 3586}, {484, 2161}, {499, 37154}, {500, 5499}, {515, 50759}, {519, 3936}, {540, 16704}, {581, 6937}, {595, 52367}, {673, 46497}, {740, 20657}, {758, 3120}, {759, 37311}, {857, 3008}, {899, 3814}, {940, 17528}, {964, 20083}, {978, 3142}, {993, 24892}, {995, 11680}, {1010, 25441}, {1018, 21956}, {1040, 9581}, {1043, 25645}, {1054, 36195}, {1074, 1210}, {1111, 44150}, {1150, 48835}, {1193, 25639}, {1201, 24387}, {1211, 3679}, {1213, 3731}, {1228, 33937}, {1230, 33117}, {1330, 64072}, {1478, 33137}, {1532, 5400}, {1647, 52537}, {1654, 19570}, {1698, 3712}, {1718, 37982}, {1723, 1865}, {1730, 1894}, {1731, 5146}, {1743, 1901}, {1757, 53501}, {1771, 67933}, {1836, 49500}, {1884, 61221}, {2092, 17057}, {2170, 14963}, {2177, 10197}, {2650, 11263}, {2893, 17189}, {2901, 57808}, {3434, 37610}, {3545, 63126}, {3570, 14568}, {3582, 6739}, {3584, 60714}, {3585, 5247}, {3632, 41014}, {3670, 6734}, {3722, 50749}, {3743, 21674}, {3753, 21949}, {3792, 38474}, {3825, 27627}, {3836, 49999}, {3839, 37681}, {3841, 59305}, {3894, 33103}, {3925, 37715}, {3944, 5692}, {3953, 10916}, {3959, 23639}, {4000, 17052}, {4085, 27042}, {4193, 17749}, {4199, 5251}, {4202, 50605}, {4234, 41806}, {4252, 50239}, {4257, 17579}, {4272, 5949}, {4340, 37161}, {4361, 21245}, {4362, 4680}, {4384, 26601}, {4658, 26131}, {4674, 40663}, {4677, 66551}, {4692, 29673}, {4694, 26015}, {4720, 30831}, {4737, 27792}, {4859, 18635}, {4880, 32857}, {5080, 33139}, {5123, 68883}, {5222, 31043}, {5224, 18145}, {5230, 5264}, {5252, 49494}, {5254, 16552}, {5313, 17717}, {5315, 31159}, {5497, 30117}, {5540, 39690}, {5587, 30444}, {5697, 22076}, {5712, 50741}, {5713, 6984}, {5718, 48847}, {5721, 6907}, {5722, 24789}, {5737, 50056}, {5791, 50065}, {5853, 50745}, {5902, 17889}, {5903, 10974}, {5904, 10381}, {6149, 56790}, {6656, 29433}, {6675, 24902}, {6693, 11115}, {6701, 63354}, {6703, 50169}, {6840, 13329}, {6841, 52524}, {6842, 37732}, {6917, 37530}, {6951, 37469}, {7683, 15971}, {7797, 60149}, {8025, 50226}, {8143, 9956}, {8185, 23847}, {8286, 41684}, {10479, 16062}, {11112, 37646}, {11235, 16483}, {11359, 37660}, {11545, 56416}, {12649, 24159}, {13161, 67979}, {13576, 60135}, {13589, 38938}, {13745, 62689}, {14041, 20142}, {14829, 17678}, {14996, 48868}, {15079, 27685}, {15680, 24898}, {16086, 37759}, {16137, 63415}, {16173, 47623}, {16370, 31187}, {16572, 53422}, {16589, 23903}, {16600, 21029}, {16610, 60415}, {16611, 21017}, {17034, 33841}, {17046, 24790}, {17327, 51593}, {17514, 64850}, {17530, 37662}, {17533, 51415}, {17556, 37679}, {17577, 32911}, {17670, 29438}, {17680, 29473}, {17757, 31855}, {18395, 24440}, {18641, 63621}, {19732, 54367}, {19867, 44417}, {19925, 35099}, {20337, 50016}, {20483, 68897}, {20888, 24995}, {20966, 33131}, {21214, 37720}, {21677, 63997}, {21838, 63605}, {22461, 30436}, {23675, 49627}, {23897, 36478}, {23942, 27040}, {24160, 34772}, {24217, 25055}, {24439, 24780}, {24864, 25040}, {24988, 49993}, {25446, 26117}, {25460, 68708}, {25526, 26051}, {25669, 65543}, {26558, 29773}, {26561, 29742}, {26723, 27052}, {26727, 36909}, {27022, 41785}, {27081, 53620}, {27368, 36974}, {27577, 58380}, {27687, 37702}, {27690, 64071}, {29455, 33840}, {29460, 33034}, {29561, 50319}, {30115, 33133}, {30116, 33108}, {30447, 37718}, {31254, 64415}, {33076, 53486}, {33079, 51285}, {33096, 61703}, {33104, 62828}, {33132, 37346}, {33165, 53478}, {37374, 62320}, {37375, 37680}, {37401, 48897}, {37666, 50736}, {37674, 44217}, {37682, 57005}, {37719, 50581}, {37989, 54407}, {38456, 50755}, {41329, 41723}, {44909, 45802}, {45923, 50461}, {46870, 62843}, {48857, 63008}, {48870, 63067}, {49448, 53476}, {50066, 62818}, {51409, 62221}, {54354, 65134}, {56402, 63317}, {56417, 56812}, {56843, 61261}, {58658, 63396}, {63443, 68840}

X(68946) = reflection of X(i) in X(j) for these {i,j}: {3722, 50749}, {52680, 35466}
X(68946) = X(i)-complementary conjugate of X(j) for these (i,j): {56, 34587}, {65, 121}, {88, 21246}, {106, 960}, {604, 51583}, {1042, 1145}, {1400, 16594}, {1402, 4370}, {1417, 1125}, {4017, 3259}, {4080, 21244}, {4674, 1329}, {7180, 66508}, {8752, 6708}, {9456, 5745}, {23345, 34589}, {36058, 34851}, {36125, 34831}, {43924, 34590}, {51641, 35092}, {55244, 124}, {55263, 26932}, {56049, 3741}, {66924, 123}
X(68946) = X(i)-Ceva conjugate of X(j) for these (i,j): {51562, 523}, {68571, 1}
X(68946) = X(i)-isoconjugate of X(j) for these (i,j): {3, 39439}, {759, 39166}
X(68946) = X(i)-Dao conjugate of X(j) for these (i,j): {30117, 68244}, {31845, 1}, {34586, 39166}, {36103, 39439}, {62652, 81}
X(68946) = crosspoint of X(i) and X(j) for these (i,j): {10, 5620}, {75, 24624}
X(68946) = crosssum of X(i) and X(j) for these (i,j): {31, 2245}, {58, 5127}
X(68946) = crossdifference of every pair of points on line {48, 57129}
X(68946) = barycentric product X(i)*X(j) for these {i,j}: {1, 62305}, {10, 33129}, {306, 5146}, {321, 30117}, {850, 68828}, {1441, 1731}, {1577, 13589}, {3936, 38938}, {3952, 47680}, {5497, 68363}, {24624, 31845}
X(68946) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 39439}, {1731, 21}, {2245, 39166}, {5146, 27}, {5497, 37783}, {13589, 662}, {30117, 81}, {31845, 3936}, {33129, 86}, {38938, 24624}, {47680, 7192}, {62305, 75}, {62652, 68244}, {68828, 110}
X(68946) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1714, 1724}, {10, 3914, 4424}, {10, 36250, 2292}, {58, 2475, 66658}, {377, 5292, 37522}, {386, 2476, 37693}, {442, 1834, 1}, {442, 64167, 17056}, {1737, 1738, 1739}, {1737, 1785, 1736}, {1834, 17056, 64167}, {2475, 24883, 58}, {2886, 64172, 1}, {3017, 6175, 49744}, {3419, 3772, 1}, {3925, 37715, 56191}, {5315, 31159, 33106}, {6734, 23537, 3670}, {10916, 23536, 3953}, {10974, 58889, 5903}, {17056, 64167, 1}, {23897, 53423, 52538}, {32865, 37716, 3679}, {37401, 63318, 48897}

X(68947) = X(1)X(2)∩X(7)X(2260)

Barycentrics    a^4*b - a^3*b^2 - a^2*b^3 + a*b^4 + a^4*c + 2*a^3*b*c + b^4*c - a^3*c^2 - 2*a*b^2*c^2 - b^3*c^2 - a^2*c^3 - b^2*c^3 + a*c^4 + b*c^4 : :

X(68947) lies on these lines: {1, 2}, {7, 2260}, {11, 857}, {27, 19850}, {31, 48900}, {35, 31016}, {36, 14953}, {56, 379}, {81, 20744}, {169, 26267}, {226, 1475}, {241, 53241}, {272, 1474}, {278, 58074}, {312, 27109}, {321, 25066}, {496, 30810}, {497, 14021}, {579, 17220}, {673, 7677}, {693, 905}, {910, 14625}, {993, 37076}, {1086, 41804}, {1108, 1441}, {1111, 68921}, {1375, 15325}, {1429, 5773}, {1447, 7291}, {1468, 27142}, {1479, 31015}, {1713, 52673}, {1723, 24179}, {1731, 14543}, {1754, 59362}, {2170, 16609}, {2223, 13576}, {2275, 3772}, {2898, 3188}, {2975, 37086}, {3120, 53590}, {3361, 27171}, {3434, 37280}, {3664, 26818}, {3673, 24635}, {3686, 27039}, {3875, 27514}, {3945, 25521}, {4000, 17077}, {4044, 26770}, {4357, 16713}, {4552, 8609}, {5249, 17169}, {5253, 16054}, {5265, 24604}, {5284, 16053}, {5303, 35935}, {5433, 24581}, {5736, 54358}, {5740, 16608}, {6349, 26747}, {7288, 24580}, {7318, 67181}, {7741, 31014}, {10436, 27161}, {10446, 27624}, {10591, 31042}, {11025, 27475}, {11680, 37445}, {16412, 24596}, {16610, 58466}, {16705, 19786}, {16732, 56531}, {17183, 28287}, {17197, 68912}, {17257, 27317}, {17304, 27170}, {17352, 28748}, {17761, 20367}, {17863, 25255}, {17885, 40903}, {17895, 64780}, {18206, 20347}, {18650, 40963}, {19645, 64077}, {19788, 62857}, {20173, 25082}, {20978, 45305}, {21384, 30961}, {23536, 26961}, {23947, 27071}, {24203, 62799}, {24390, 37326}, {24554, 44735}, {25078, 45744}, {25639, 31043}, {26074, 56883}, {26978, 37596}, {27254, 51171}, {27334, 31995}, {27339, 62208}, {30807, 43065}, {30809, 47743}, {30985, 62853}, {33108, 37097}, {35270, 66670}, {36022, 40980}, {37089, 54313}, {37111, 37722}, {46792, 66941}, {55082, 62797}

X(68947) = X(i)-complementary conjugate of X(j) for these (i,j): {43672, 21245}, {59067, 513}
X(68947) = crosspoint of X(86) and X(34018)
X(68947) = crosssum of X(17435) and X(55232)
X(68947) = crossdifference of every pair of points on line {228, 649}
X(68947) = barycentric product X(693)*X(7437)
X(68947) = barycentric quotient X(7437)/X(100)
X(68947) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 17316, 28742}, {2, 26964, 17023}, {2, 27146, 29603}, {36, 53591, 14953}, {239, 24559, 4511}, {1193, 40940, 5222}, {2260, 34830, 7}, {3008, 44675, 26006}, {3912, 29769, 29824}, {17863, 40937, 25255}, {33129, 68920, 24781}

X(68948) = X(1)X(2)∩X(32)X(24166)

Barycentrics    2*a^4 + a^2*b^2 + a*b^3 - 2*a^2*b*c - a*b^2*c - b^3*c + a^2*c^2 - a*b*c^2 + a*c^3 - b*c^3 : :

X(68948) lies on these lines: {1, 2}, {32, 24166}, {83, 22011}, {384, 596}, {385, 21208}, {514, 1919}, {754, 1086}, {1100, 8682}, {1914, 57029}, {2170, 18715}, {2251, 68756}, {4075, 16918}, {4366, 68895}, {4568, 33854}, {6532, 16917}, {16787, 33945}, {16915, 64431}, {16916, 24068}, {16920, 64429}, {18089, 32923}, {24631, 49480}, {31317, 48866}, {35092, 61064}

X(68948) = crossdifference of every pair of points on line {649, 21035}
X(68948) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {239, 30117, 30109}

X(68949) = X(1)X(2)∩X(667)X(693)

Barycentrics    (a^2 - b*c)*(a^2*b - a*b^2 + a^2*c + 2*a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(68949) lies on these lines: {1, 2}, {667, 693}, {894, 24340}, {4972, 6292}, {8053, 29764}, {12264, 17469}, {16679, 18143}, {16684, 18046}, {17140, 22011}, {17176, 17200}, {20985, 30982}, {20990, 29484}, {22289, 34017}, {26768, 38302}, {29453, 64524}, {34443, 63230}, {37686, 62872}, {41681, 56537}

X(68949) = X(1911)-isoconjugate of X(27807)
X(68949) = X(6651)-Dao conjugate of X(27807)
X(68949) = crossdifference of every pair of points on line {649, 21814}
X(68949) = barycentric product X(i)*X(j) for these {i,j}: {238, 18143}, {239, 17140}, {1921, 16679}, {3570, 23789}, {22011, 33295}, {22279, 30940}
X(68949) = barycentric quotient X(i)/X(j) for these {i,j}: {239, 27807}, {16679, 292}, {17140, 335}, {18143, 334}, {22011, 43534}, {23789, 4444}, {29447, 40094}
X(68949) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1125, 3720, 16826}, {1125, 50023, 49997}, {4039, 27846, 239}, {29824, 49997, 56811}

X(68950) = X(1)X(39)∩X(2)X(2350)

Barycentrics    a*(a^2*b - a*b^2 + a^2*c + 2*a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(68950) lies on these lines: {1, 39}, {2, 2350}, {3, 16783}, {5, 58036}, {6, 474}, {9, 583}, {10, 1475}, {21, 5030}, {35, 16503}, {36, 41239}, {37, 3953}, {40, 6205}, {56, 16788}, {57, 1759}, {58, 33854}, {63, 24583}, {75, 29742}, {76, 29438}, {83, 1019}, {85, 63203}, {101, 5253}, {141, 18164}, {169, 3306}, {171, 5299}, {213, 16604}, {218, 25524}, {244, 16600}, {274, 29433}, {321, 29769}, {330, 30114}, {350, 29750}, {354, 3970}, {386, 63066}, {404, 4251}, {405, 5022}, {411, 572}, {518, 25068}, {519, 17474}, {551, 1334}, {573, 3523}, {579, 6857}, {597, 53391}, {609, 37608}, {644, 9327}, {649, 23814}, {668, 29375}, {672, 1125}, {750, 61235}, {756, 25089}, {758, 39244}, {894, 20372}, {940, 9605}, {942, 57015}, {992, 1743}, {1025, 55082}, {1107, 56191}, {1111, 17048}, {1201, 3997}, {1212, 5439}, {1434, 17681}, {1449, 5153}, {1509, 55085}, {1574, 3780}, {1575, 3293}, {1621, 24047}, {1698, 21384}, {1724, 5021}, {1730, 24580}, {1739, 41015}, {1756, 56533}, {1764, 63013}, {1765, 6837}, {1930, 24631}, {2170, 3754}, {2245, 37298}, {2260, 5750}, {2280, 25440}, {2303, 4284}, {2329, 5563}, {3061, 5902}, {3169, 66549}, {3218, 29614}, {3219, 29609}, {3248, 23629}, {3333, 17698}, {3336, 3496}, {3337, 3509}, {3485, 56546}, {3555, 4006}, {3616, 3730}, {3634, 3691}, {3693, 5045}, {3720, 25092}, {3742, 16601}, {3753, 40133}, {3758, 21362}, {3812, 43065}, {3815, 37693}, {3833, 21921}, {3848, 25086}, {3874, 33299}, {3881, 3930}, {3882, 46922}, {3934, 4754}, {3960, 68811}, {3991, 17609}, {4050, 51093}, {4188, 4262}, {4258, 16371}, {4340, 37665}, {4390, 62825}, {4441, 29748}, {4513, 7373}, {4568, 17141}, {4672, 24491}, {4689, 31430}, {4721, 20530}, {4868, 39247}, {5024, 19765}, {5035, 51303}, {5044, 17746}, {5120, 16293}, {5255, 16784}, {5264, 16502}, {5280, 37607}, {5305, 37634}, {5437, 16572}, {5718, 31406}, {5883, 17451}, {6168, 58816}, {6626, 46194}, {7031, 16779}, {7278, 21232}, {7745, 66658}, {7765, 23903}, {7772, 63099}, {7786, 37632}, {7824, 20147}, {8359, 50260}, {8568, 21620}, {8583, 54330}, {9300, 49744}, {10582, 37319}, {10583, 63053}, {10601, 25931}, {11019, 21073}, {11109, 61236}, {13571, 17207}, {14520, 58035}, {16061, 33953}, {16408, 37658}, {16574, 17381}, {16679, 22279}, {16706, 20605}, {16709, 41681}, {16781, 37610}, {16831, 56507}, {16862, 56527}, {16971, 20691}, {17023, 20367}, {17026, 32092}, {17034, 62755}, {17140, 22011}, {17205, 26978}, {17277, 29459}, {17298, 29964}, {17308, 62853}, {17370, 29749}, {17379, 67984}, {17384, 29812}, {17397, 62817}, {17439, 51714}, {17489, 24166}, {17531, 63087}, {17575, 38930}, {17683, 60075}, {17735, 68893}, {17749, 37657}, {17761, 26964}, {18140, 29440}, {18143, 29447}, {19718, 21483}, {19812, 29788}, {19862, 59207}, {20269, 38186}, {20371, 51860}, {20459, 33682}, {20913, 29561}, {21070, 29824}, {21214, 54981}, {21372, 27003}, {21495, 60721}, {21808, 24036}, {21868, 62325}, {23632, 59315}, {23649, 59305}, {24051, 61163}, {24170, 26965}, {24220, 27161}, {24342, 24727}, {24524, 29691}, {24629, 33945}, {25082, 35341}, {25280, 29720}, {25303, 29699}, {25510, 27064}, {25583, 59405}, {26004, 37649}, {26035, 50605}, {27162, 39797}, {28369, 64713}, {29381, 64133}, {29455, 34284}, {29552, 63119}, {29773, 60706}, {31429, 54287}, {32911, 51311}, {33863, 52680}, {34460, 37732}, {36743, 37284}, {37149, 43531}, {37558, 43039}, {37563, 41322}, {37633, 61234}, {39248, 49500}, {44424, 58567}, {44562, 49749}, {48883, 66313}, {50190, 51058}, {50637, 63499}, {56985, 57279}, {63548, 66659}

X(68950) = isogonal conjugate of the isotomic conjugate of X(18143)
X(68950) = X(1509)-Ceva conjugate of X(1)
X(68950) = X(22279)-cross conjugate of X(17140)
X(68950) = X(6)-isoconjugate of X(27807)
X(68950) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 27807}, {756, 594}, {14991, 21208}, {23789, 59746}, {61404, 18070}
X(68950) = crosspoint of X(662) and X(63918)
X(68950) = crosssum of X(i) and X(j) for these (i,j): {661, 64523}, {798, 8054}
X(68950) = crossdifference of every pair of points on line {659, 4093}
X(68950) = barycentric product X(i)*X(j) for these {i,j}: {1, 17140}, {6, 18143}, {75, 16679}, {81, 22011}, {86, 22279}, {99, 40471}, {100, 23789}, {799, 14991}, {2350, 29447}
X(68950) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 27807}, {14991, 661}, {16679, 1}, {17140, 75}, {18143, 76}, {22011, 321}, {22279, 10}, {23789, 693}, {29447, 18152}, {40471, 523}
X(68950) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 16549, 1018}, {1, 17754, 16549}, {2, 4253, 16552}, {2, 16552, 46196}, {2, 17169, 17758}, {9, 3338, 17736}, {10, 1475, 45751}, {39, 24512, 1}, {213, 16604, 49997}, {274, 37686, 29433}, {354, 25066, 3970}, {404, 4251, 35342}, {583, 17398, 9}, {672, 1125, 3294}, {1015, 2295, 1}, {1574, 3780, 31855}, {1575, 20963, 3293}, {2260, 5750, 21061}, {2275, 17750, 1}, {3336, 56532, 3496}, {3555, 44798, 4006}, {16779, 37603, 7031}, {24036, 58565, 21808}

X(68951) = X(1)X(2)∩X(693)X(784)

Barycentrics    a^2*b^3 - a*b^4 + a*b^3*c - b^4*c + a^2*c^3 + a*b*c^3 - a*c^4 - b*c^4 : :

X(68951) lies on these lines: {1, 2}, {11, 3948}, {76, 11680}, {325, 64223}, {693, 784}, {2486, 39995}, {2886, 20913}, {3662, 4022}, {4040, 26775}, {12746, 46501}, {16574, 17138}, {17153, 21061}, {17177, 21415}, {17234, 58571}, {17447, 20234}, {17463, 35550}, {17499, 33107}, {20453, 52922}, {20470, 32850}, {20544, 52043}, {20912, 23772}, {21746, 30034}, {29981, 35892}

X(68951) = crossdifference of every pair of points on line {649, 2205}
X(68951) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {}

X(68952) = X(1)X(6)∩X(514)X(1921)

Barycentrics    a*(a^2*b^3 - a*b^4 + a*b^3*c - b^4*c + a^2*c^3 + a*b*c^3 - a*c^4 - b*c^4) : :

X(68952) lies on these lines: {1, 6}, {257, 26965}, {514, 1921}, {537, 20706}, {726, 20593}, {740, 2170}, {982, 23632}, {1959, 17755}, {2887, 23636}, {3006, 7239}, {3739, 18726}, {3842, 39244}, {3936, 20974}, {4051, 49459}, {4645, 24484}, {6376, 17241}, {6682, 22230}, {16708, 62677}, {16728, 20367}, {17451, 24325}, {17793, 38980}, {18061, 52049}, {18176, 21240}, {20435, 20892}, {22285, 23370}, {24036, 39258}, {29965, 29966}, {33299, 49457}, {50362, 61234}

X(68952) = reflection of X(39258) in X(24036)
X(68952) = crossdifference of every pair of points on line {513, 1918}

X(68953) = X(1)X(4491)∩X(30)X(511)

Barycentrics    a*(b - c)*(a^2*b + a*b^2 + a^2*c - 2*a*b*c - b^2*c + a*c^2 - b*c^2) : :

X(68953) lies on these lines: {1, 4491}, {10, 53565}, {30, 511}, {37, 3768}, {44, 23650}, {65, 53528}, {244, 659}, {319, 21303}, {764, 53527}, {1027, 46149}, {1100, 8632}, {1635, 39982}, {1769, 55244}, {2254, 22313}, {3762, 14288}, {3837, 14426}, {3909, 3952}, {4057, 48281}, {4528, 14287}, {4670, 24354}, {4674, 21385}, {4690, 24721}, {4708, 25356}, {4833, 47970}, {8683, 53388}, {14404, 36848}, {14407, 52745}, {17154, 25048}, {17239, 21261}, {17460, 21343}, {17929, 17930}, {21143, 21832}, {21191, 59747}, {21297, 39996}, {21606, 53368}, {22271, 22319}, {23789, 50493}, {24097, 34434}, {24128, 64580}, {24471, 43041}, {39200, 41343}, {41179, 41180}, {48032, 53532}, {48334, 48350}, {53305, 53389}, {53315, 68816}

X(68953) = isogonal conjugate of X(53685)
X(68953) = crossdifference of every pair of points on line {6, 1018}
X(68953) = {X(1),X(53392)}-harmonic conjugate of X(4491)

X(68954) = X(10)X(321)∩X(44)X(513)

Barycentrics    a*(b + c)*(a^2*b^2 + a*b^3 - a*b^2*c - b^3*c + a^2*c^2 - a*b*c^2 + a*c^3 - b*c^3) : :

X(68954) lies on these lines: {10, 321}, {38, 1211}, {44, 513}, {58, 404}, {81, 24911}, {244, 3936}, {982, 31037}, {984, 27081}, {1193, 17614}, {1215, 27041}, {2092, 46904}, {3122, 4062}, {3454, 24443}, {3882, 62740}, {3909, 18792}, {4201, 26064}, {4283, 37656}, {4446, 33065}, {4553, 68754}, {14815, 20653}, {15523, 21936}, {17155, 27792}, {17165, 26771}, {17763, 17954}, {19513, 28273}, {21805, 52784}, {21830, 35309}, {22024, 40603}, {22174, 53034}, {22200, 40463}, {23639, 33299}, {24165, 27793}, {24478, 32775}, {24923, 37635}, {25624, 64178}, {26772, 32931}, {29398, 32926}, {31247, 35623}, {37247, 54371}, {57039, 68750}, {59712, 64873}, {60090, 60097}

X(68954) = X(17954)-Ceva conjugate of X(2292)
X(68954) = X(57039)-Dao conjugate of X(5209)
X(68954) = crosspoint of X(57029) and X(57039)
X(68954) = crossdifference of every pair of points on line {1, 57129}
X(68954) = barycentric product X(i)*X(j) for these {i,j}: {10, 57039}, {37, 57029}, {306, 52461}, {321, 68750}
X(68954) = barycentric quotient X(i)/X(j) for these {i,j}: {52461, 27}, {57029, 274}, {57039, 86}, {68750, 81}
X(68954) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {899, 68887, 896}, {1211, 20966, 38}

X(68955) = X(30)X(511)∩X(313)X(321)

Barycentrics    (b + c)*(-(a^2*b^2) - a*b^3 + a*b^2*c + b^3*c - a^2*c^2 + a*b*c^2 - a*c^3 + b*c^3) : :

X(68955) lies on these lines: {1, 24335}, {10, 25370}, {30, 511}, {75, 17202}, {81, 4360}, {141, 4016}, {313, 321}, {597, 53037}, {1086, 35550}, {2292, 4364}, {2294, 17243}, {3125, 42713}, {3666, 6703}, {3703, 4137}, {3782, 20896}, {3963, 40085}, {4026, 4424}, {4363, 17164}, {4377, 22036}, {4472, 49598}, {4647, 4665}, {5044, 49558}, {5137, 32845}, {5710, 17318}, {7228, 63398}, {7263, 18698}, {11611, 44396}, {17262, 25255}, {17790, 17946}, {23928, 24326}, {24317, 49609}, {24704, 36974}, {25080, 59583}, {25358, 58386}, {40937, 59640}, {44378, 62609}, {52897, 53332}, {57029, 57039}, {65191, 68883}

X(68955) = crossdifference of every pair of points on line {6, 50493}
X(68955) = {X(4016),X(18697)}-harmonic conjugate of X(141)

X(68956) = X(75)X(53565)∩X(325)X(523)

Barycentrics    (b - c)*(-(a^2*b^2) - a*b^3 + a*b^2*c + b^3*c - a^2*c^2 + a*b*c^2 - a*c^3 + b*c^3) : :

X(68956) lies on these lines: {75, 53565}, {325, 523}, {594, 21261}, {812, 1015}, {1213, 25356}, {3768, 4364}, {4106, 68774}, {4132, 23790}, {4360, 21303}, {4374, 40086}, {4411, 23815}, {4491, 17321}, {8632, 17045}, {10566, 26968}, {17332, 23650}, {17395, 24721}, {17398, 24354}, {20949, 31946}, {26809, 26855}, {31118, 31129}, {39714, 55244}, {41003, 43041}, {46403, 53276}

X(68956) = midpoint of X(4360) and X(21303)
X(68956) = reflection of X(i) in X(j) for these {i,j}: {594, 21261}, {8632, 17045}
X(68956) = crossdifference of every pair of points on line {32, 4557}
X(68956) = barycentric product X(i)*X(j) for these {i,j}: {514, 57029}, {693, 57039}, {3261, 68750}, {15413, 52461}
X(68956) = barycentric quotient X(i)/X(j) for these {i,j}: {52461, 1783}, {57029, 190}, {57039, 100}, {68750, 101}

X(68957) = X(76)X(321)∩X(239)X(514)

Barycentrics    a^2*b^3 + a*b^4 - a*b^3*c - b^4*c + a^2*c^3 - a*b*c^3 + a*c^4 - b*c^4 : :

X(68957) lies on these lines: {8, 20068}, {76, 321}, {239, 514}, {335, 20448}, {696, 2228}, {698, 62553}, {714, 21142}, {1086, 20912}, {3662, 18179}, {3705, 18203}, {3752, 17367}, {3948, 21138}, {4266, 30931}, {4568, 27044}, {5262, 52564}, {17178, 24219}, {17292, 31026}, {21208, 27166}, {29613, 30818}, {29617, 42051}, {50582, 50633}

X(68957) = crossdifference of every pair of points on line {42, 1980}

X(68958) = X(1)X(2)∩X(238)X(4427)

Barycentrics    2*a^3 + a^2*b + a*b^2 + a^2*c - 4*a*b*c - b^2*c + a*c^2 - b*c^2 : :

X(68958) lies on these lines: {1, 2}, {238, 4427}, {244, 4974}, {659, 3004}, {748, 17147}, {982, 19742}, {985, 39706}, {1001, 64161}, {1086, 17491}, {1386, 24589}, {1738, 21282}, {1757, 17154}, {3218, 53393}, {3246, 4706}, {3315, 17145}, {3751, 17146}, {3759, 64149}, {3891, 37679}, {3952, 32922}, {3995, 17123}, {4383, 17165}, {4392, 17349}, {4395, 4442}, {4679, 50102}, {4722, 42053}, {5057, 37756}, {5196, 24378}, {5284, 27804}, {5846, 24988}, {8054, 18792}, {9335, 37683}, {14997, 24349}, {16475, 26627}, {16478, 19284}, {17063, 37639}, {17125, 31035}, {17126, 24620}, {17127, 17490}, {17140, 32911}, {17163, 32942}, {17278, 33070}, {17348, 46909}, {17352, 33089}, {17356, 48647}, {17450, 49489}, {17489, 31088}, {17960, 24200}, {19740, 40328}, {26724, 33071}, {31025, 32944}, {31037, 33123}, {31289, 32848}, {32926, 37687}, {32937, 63096}, {33148, 63002}, {33170, 63051}, {41002, 59477}, {41241, 49483}, {41629, 65112}, {49676, 63071}

X(68958) = crossdifference of every pair of points on line {649, 1500}
X(68958) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {238, 17495, 4427}, {239, 7292, 29824}, {239, 29824, 17162}, {244, 4974, 16704}, {899, 20045, 17780}, {899, 50023, 20045}, {3006, 49987, 62667}, {3008, 49987, 3006}, {5211, 29590, 33139}, {16823, 17012, 29822}, {17123, 32924, 3995}, {17125, 32921, 31035}, {32922, 37680, 3952}, {50017, 60423, 50000}

X(68959) = X(30)X(511)∩X(659)X(1125)

Barycentrics    (b - c)*(-2*a^3 - a^2*b - a*b^2 - a^2*c + 4*a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(68959) lies on these lines: {10, 21385}, {30, 511}, {551, 68816}, {659, 1125}, {1960, 3636}, {3008, 24623}, {3634, 3837}, {3635, 21343}, {3828, 48167}, {3960, 4830}, {4063, 23789}, {4089, 17205}, {4170, 47936}, {4298, 30725}, {4448, 30592}, {4498, 50337}, {4761, 48115}, {4978, 47813}, {6545, 21181}, {9508, 23814}, {12575, 53523}, {13266, 21630}, {16819, 26855}, {22037, 48083}, {23795, 50343}, {23796, 50336}, {23815, 48216}, {25569, 51103}, {30795, 31253}, {47650, 49300}, {47663, 49278}, {48061, 49288}, {48116, 48407}, {48284, 48335}, {48324, 50761}

X(68959) = barycentric quotient X(17311)/X(65501)
X(68959) = {X(21385),X(46403)}-harmonic conjugate of X(10)

X(68960) = X(1)X(6)∩X(83)X(29383)

Barycentrics    a*(2*a^3 + a^2*b + a*b^2 + a^2*c - 4*a*b*c - b^2*c + a*c^2 - b*c^2) : :

X(68960) lies on these lines: {1, 6}, {83, 29383}, {551, 21764}, {1015, 52680}, {1018, 33854}, {1019, 3960}, {1914, 35342}, {2241, 3216}, {3290, 21372}, {3915, 16549}, {4366, 62755}, {5011, 7292}, {6629, 26860}, {7031, 21214}, {14964, 38346}, {17200, 26807}, {17736, 28011}, {24602, 29603}, {24902, 31488}, {29824, 57017}

X(68960) = crossdifference of every pair of points on line {513, 756}
X(68960) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {238, 16784, 45751}, {1914, 49997, 35342}, {33854, 40091, 1018}

X(68961) = X(10)X(762)∩X(274)X(514)

Barycentrics    (b + c)*(a^2*b - a*b^2 + a^2*c + 6*a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(68961) lies on these lines: {10, 762}, {274, 514}, {551, 6155}, {596, 1573}, {1015, 6532}, {1107, 24176}, {2170, 6533}, {3968, 4095}, {4006, 22017}, {4115, 17164}, {4165, 41859}, {4647, 21921}, {4714, 21808}, {4771, 63354}, {5011, 16817}, {16589, 68895}, {16611, 49598}, {16819, 57029}, {16887, 24185}, {16975, 64431}, {17175, 17497}, {17451, 28611}, {22036, 25614}, {24051, 64071}, {27040, 46895}, {27812, 61165}, {29460, 53332}, {33944, 59746}, {35101, 36812}, {35103, 59633}

X(68961) = crosspoint of X(10) and X(58279)
X(68961) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 22011, 4103}, {10, 22035, 762}, {17164, 46196, 4115}

X(68962) = X(10)X(514)∩X(522)X(68123)

Barycentrics    (b - c)*(a^2*b - a*b^2 + a^2*c + 6*a*b*c + b^2*c - a*c^2 + b*c^2) : :
X(68962) = 9 X[10] - 8 X[4147], 7 X[10] - 8 X[17072], 5 X[10] - 8 X[24720], 3 X[10] - 4 X[50337], 5 X[3777] - X[47928], 7 X[4147] - 9 X[17072], 4 X[4147] - 9 X[23789], 5 X[4147] - 9 X[24720], 2 X[4147] - 3 X[50337], 4 X[17072] - 7 X[23789], 5 X[17072] - 7 X[24720], 6 X[17072] - 7 X[50337], 5 X[23789] - 4 X[24720], 3 X[23789] - 2 X[50337], and many others

X(68962) lies on these lines: {10, 514}, {522, 68123}, {551, 4040}, {1125, 47970}, {1734, 23796}, {3244, 48282}, {3309, 50761}, {3625, 21302}, {3669, 48284}, {3835, 68124}, {3840, 47780}, {4129, 48406}, {4151, 23795}, {4406, 53594}, {4449, 51071}, {4724, 15808}, {4801, 8714}, {4978, 23738}, {4992, 6372}, {10481, 57247}, {19862, 47796}, {19883, 47929}, {21185, 21625}, {21189, 68122}, {29130, 49299}, {45667, 51106}, {47773, 62673}, {47793, 51073}, {48058, 65482}, {48623, 51109}, {49300, 49627}

X(68962) = midpoint of X(i) and X(j) for these {i,j}: {4978, 23738}, {21189, 68122}
X(68962) = reflection of X(i) in X(j) for these {i,j}: {10, 23789}, {1734, 23796}, {3244, 48282}, {3625, 21302}, {4129, 48406}, {4705, 23814}, {23795, 48151}, {47970, 1125}, {48058, 65482}, {48284, 3669}

X(68963) = X(2)X(37)∩X(244)X(30941)

Barycentrics    a^3*b + 2*a^2*b^2 + a*b^3 + a^3*c - 2*a^2*b*c - 2*a*b^2*c - b^3*c + 2*a^2*c^2 - 2*a*b*c^2 + a*c^3 - b*c^3 : :

X(68963) lies on these lines: {2, 37}, {244, 30941}, {659, 3004}, {1015, 17497}, {1086, 31062}, {1111, 16711}, {1574, 28598}, {3216, 17141}, {3509, 53391}, {5211, 20553}, {6532, 17175}, {9335, 30962}, {17137, 24046}, {17140, 37678}, {17152, 24443}, {17480, 25278}, {18600, 33944}, {21021, 26779}, {21208, 62755}, {26094, 33935}, {27162, 33945}, {39044, 62636}, {49997, 53332}

X(68963) = crossdifference of every pair of points on line {667, 1500}
X(68963) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3216, 24166, 17141}, {49997, 57029, 53332}

X(68964) = X(1)X(2)∩X(6)X(29429)

Barycentrics    a*(a^3*b + 2*a^2*b^2 + a*b^3 + a^3*c - 2*a^2*b*c - 2*a*b^2*c - b^3*c + 2*a^2*c^2 - 2*a*b*c^2 + a*c^3 - b*c^3) : :

X(68964) lies on these lines: {1, 2}, {6, 29429}, {1015, 18206}, {1019, 3960}, {1616, 21477}, {2087, 49760}, {2275, 62817}, {3882, 57039}, {4360, 29388}, {4366, 18792}, {4426, 29418}, {6996, 32486}, {8572, 16436}, {9336, 62853}, {16604, 54282}, {16752, 17761}, {16784, 20769}, {17045, 54308}, {17798, 23404}, {20228, 28366}, {21000, 21524}, {21495, 40091}, {27918, 63443}, {29529, 39248}, {29531, 54406}

X(68964) = crossdifference of every pair of points on line {649, 756}
X(68964) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {239, 27166, 29456}, {1149, 68755, 3912}

X(68965) = X(10)X(321)∩X(239)X(514)

Barycentrics    (b + c)*(-(a^3*b) - 2*a^2*b^2 - a*b^3 - a^3*c + 2*a^2*b*c + 2*a*b^2*c + b^3*c - 2*a^2*c^2 + 2*a*b*c^2 - a*c^3 + b*c^3) : :

X(68965) lies on these lines: {10, 321}, {239, 514}, {538, 42051}, {1266, 16732}, {3125, 3912}, {3687, 18202}, {3752, 6703}, {4016, 24090}, {4393, 24166}, {4686, 49521}, {9278, 39714}, {10914, 49476}, {11611, 14554}, {20691, 22048}, {20715, 22313}, {21858, 22012}, {24195, 30939}, {25298, 59717}, {40085, 59715}, {41327, 62392}, {65191, 68883}

X(68965) = reflection of X(i) in X(j) for these {i,j}: {30939, 24195}, {65191, 68883}
X(68965) = crossdifference of every pair of points on line {42, 57129}

X(68966) = X(1)X(4753)∩X(2)X(6)

Barycentrics    3*a^2 - a*b - a*c - b*c : :
X(68966) = X[1] + 2 X[4753], X[86] - 4 X[20142], X[17297] + 2 X[63049], 4 X[44] - X[190], 2 X[44] + X[239], 8 X[44] + X[17160], X[190] + 2 X[239], 2 X[190] + X[17160], 4 X[239] - X[17160], 2 X[17264] - 3 X[41138], X[320] - 4 X[3008], 2 X[320] - 5 X[27191], 8 X[3008] - 5 X[27191], X[903] - 4 X[41140], X[1757] + 2 X[4974], and many others

X(68966) lies on these lines: {1, 4753}, {2, 6}, {8, 17354}, {9, 3759}, {10, 16477}, {30, 37510}, {37, 17121}, {39, 21937}, {44, 190}, {45, 4393}, {61, 21869}, {62, 21898}, {75, 1743}, {83, 60276}, {89, 24594}, {100, 62296}, {144, 4398}, {182, 13634}, {187, 22355}, {192, 16885}, {213, 16829}, {238, 519}, {319, 17285}, {320, 3008}, {344, 17377}, {511, 13635}, {527, 666}, {537, 1757}, {551, 4649}, {574, 22351}, {575, 6998}, {576, 21554}, {577, 22359}, {598, 60079}, {648, 26003}, {662, 5053}, {664, 37787}, {671, 60094}, {672, 41142}, {798, 1019}, {894, 4688}, {899, 4607}, {1001, 3241}, {1043, 1724}, {1051, 10180}, {1086, 20072}, {1100, 4755}, {1268, 5750}, {1350, 66302}, {1351, 66308}, {1386, 51034}, {1405, 41245}, {1449, 4687}, {1740, 36634}, {1778, 16046}, {1918, 4685}, {2234, 9359}, {2239, 64911}, {2322, 36794}, {2325, 49770}, {2664, 3248}, {3052, 59295}, {3098, 66304}, {3227, 45751}, {3284, 21940}, {3286, 13587}, {3524, 37474}, {3644, 25728}, {3679, 5263}, {3686, 17289}, {3707, 17023}, {3731, 17393}, {3739, 16671}, {3751, 51055}, {3758, 4384}, {3791, 42056}, {3828, 33682}, {3834, 29607}, {3875, 3973}, {3879, 17263}, {3912, 4700}, {3923, 50086}, {3943, 4473}, {3946, 17258}, {3996, 17127}, {4000, 17347}, {4034, 48630}, {4102, 40394}, {4234, 4279}, {4361, 4740}, {4363, 16816}, {4366, 4370}, {4389, 5222}, {4395, 4440}, {4416, 16706}, {4421, 20992}, {4422, 4969}, {4432, 50016}, {4437, 50030}, {4445, 17358}, {4464, 59585}, {4465, 17029}, {4604, 37222}, {4643, 17305}, {4657, 17331}, {4663, 16823}, {4669, 49482}, {4670, 16815}, {4672, 50096}, {4675, 29628}, {4676, 50126}, {4677, 32941}, {4690, 17292}, {4693, 4759}, {4698, 16668}, {4708, 29614}, {4725, 17310}, {4741, 17290}, {4851, 17338}, {4852, 15492}, {4886, 5294}, {4887, 62424}, {4908, 28329}, {4937, 50756}, {4954, 19998}, {4980, 20174}, {5007, 16061}, {5050, 66307}, {5085, 66301}, {5092, 66303}, {5093, 66310}, {5097, 66312}, {5132, 17549}, {5327, 17577}, {5540, 16568}, {5564, 17355}, {5686, 38048}, {5723, 17950}, {5822, 18747}, {5839, 17233}, {6007, 24482}, {6172, 49748}, {6419, 21992}, {6420, 21909}, {6626, 46913}, {6646, 17366}, {6666, 17317}, {6687, 17266}, {6749, 54372}, {7263, 31300}, {7277, 26806}, {7290, 49450}, {7385, 8550}, {7474, 15019}, {7757, 47039}, {7760, 17681}, {7772, 16060}, {7827, 17677}, {7878, 13740}, {8692, 49680}, {9041, 49706}, {9534, 16394}, {11286, 48869}, {11354, 48850}, {11477, 66314}, {12017, 66315}, {13741, 64072}, {14621, 55955}, {15485, 49497}, {15601, 49495}, {16226, 21162}, {16367, 37503}, {16370, 37502}, {16371, 37507}, {16397, 16948}, {16475, 48854}, {16482, 44671}, {16484, 49685}, {16494, 62236}, {16495, 21805}, {16503, 29574}, {16505, 17780}, {16552, 33296}, {16666, 16826}, {16667, 17394}, {16779, 29573}, {16814, 17319}, {16825, 31178}, {16832, 41847}, {16857, 48858}, {16884, 27268}, {17117, 17351}, {17132, 53602}, {17237, 29630}, {17243, 31333}, {17250, 29598}, {17252, 17384}, {17253, 17383}, {17254, 17382}, {17257, 17380}, {17267, 17373}, {17268, 17372}, {17269, 20055}, {17270, 17371}, {17272, 17370}, {17275, 17368}, {17278, 17364}, {17279, 17295}, {17280, 17362}, {17281, 29617}, {17282, 17361}, {17284, 17360}, {17287, 17357}, {17288, 17356}, {17291, 17344}, {17294, 17342}, {17296, 17341}, {17299, 17339}, {17301, 17333}, {17302, 17332}, {17303, 32089}, {17304, 17329}, {17306, 17328}, {17315, 25101}, {17320, 50093}, {17322, 63978}, {17359, 29615}, {17389, 41313}, {17399, 66451}, {17487, 28297}, {17686, 50155}, {17781, 19796}, {18145, 30940}, {18206, 53391}, {19279, 50598}, {19323, 37492}, {19325, 36740}, {19326, 36741}, {19875, 50302}, {19876, 43997}, {20018, 68723}, {20162, 66702}, {20176, 27065}, {20179, 50095}, {20180, 50113}, {20683, 25048}, {21384, 34063}, {22377, 41335}, {24295, 42334}, {24378, 56935}, {24599, 42697}, {24625, 57039}, {24715, 49710}, {24723, 50091}, {24841, 49712}, {25055, 41848}, {25531, 32919}, {25534, 26982}, {26723, 33066}, {27036, 27166}, {27064, 55095}, {29484, 44139}, {29570, 62212}, {29575, 50125}, {29591, 64712}, {29592, 61302}, {29593, 61344}, {29609, 52706}, {30593, 33766}, {31136, 32864}, {31137, 32853}, {31161, 32914}, {31323, 36409}, {31884, 66306}, {32040, 53210}, {32921, 51035}, {33761, 45222}, {33878, 66316}, {33908, 40859}, {36647, 54098}, {36807, 49783}, {37128, 39982}, {38088, 47356}, {39252, 42046}, {39561, 66311}, {39952, 39960}, {39971, 39974}, {40433, 45223}, {40861, 64462}, {40954, 58470}, {42030, 56224}, {42033, 50306}, {47359, 50310}, {48814, 48857}, {48816, 48870}, {48849, 59406}, {48852, 48867}, {48853, 59408}, {48878, 68539}, {49477, 50777}, {49486, 51054}, {49489, 50111}, {49543, 50110}, {49707, 62390}, {49709, 49772}, {49714, 49771}, {49720, 50303}, {49726, 50098}, {49742, 50112}, {49746, 50282}, {50019, 52885}, {50084, 68876}, {50088, 50107}, {50090, 50109}, {50099, 50118}, {50100, 59579}, {50121, 50129}, {50287, 50296}, {50291, 51005}, {53091, 66309}, {53093, 66313}, {53094, 66305}, {54333, 54391}, {54676, 54686}, {56983, 66212}, {60954, 68344}, {60971, 62403}, {60984, 62383}, {62872, 64523}

X(68966) = midpoint of X(2) and X(63049)
X(68966) = reflection of X(i) in X(j) for these {i,j}: {903, 37756}, {17297, 2}, {17310, 41310}, {37756, 41140}
X(68966) = complement of the isotomic conjugate of X(54795)
X(68966) = X(54795)-complementary conjugate of X(2887)
X(68966) = X(5381)-Ceva conjugate of X(190)
X(68966) = X(i)-Dao conjugate of X(j) for these (i,j): {4728, 52626}, {52959, 3994}
X(68966) = cevapoint of X(899) and X(45751)
X(68966) = crosspoint of X(i) and X(j) for these (i,j): {2, 54795}, {1016, 4607}, {37129, 40433}
X(68966) = crosssum of X(i) and X(j) for these (i,j): {899, 3720}, {1015, 3768}
X(68966) = trilinear pole of line {38349, 47776}
X(68966) = crossdifference of every pair of points on line {512, 2667}
X(68966) = barycentric product X(i)*X(j) for these {i,j}: {86, 19998}, {190, 47776}, {668, 68816}, {889, 38349}, {4954, 39704}, {7035, 16507}
X(68966) = barycentric quotient X(i)/X(j) for these {i,j}: {4954, 3679}, {16507, 244}, {19998, 10}, {38349, 891}, {47776, 514}, {52875, 3994}, {67401, 68895}, {68816, 513}
X(68966) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6, 46922}, {2, 1992, 17378}, {2, 5032, 63054}, {2, 17330, 31144}, {2, 17346, 17271}, {2, 37654, 17346}, {2, 46922, 86}, {2, 50074, 599}, {2, 50133, 17313}, {2, 63052, 17392}, {2, 63086, 1992}, {6, 15668, 37677}, {6, 17259, 17379}, {6, 17277, 86}, {6, 17349, 17277}, {9, 3759, 4360}, {9, 16834, 4664}, {37, 50124, 29584}, {44, 239, 190}, {69, 17352, 17283}, {69, 37681, 17352}, {190, 239, 17160}, {193, 37650, 17234}, {319, 17353, 17285}, {320, 3008, 27191}, {391, 3618, 5224}, {597, 17330, 2}, {966, 51171, 17381}, {1100, 4755, 29580}, {1213, 6329, 63053}, {1654, 3589, 17307}, {1743, 16833, 50127}, {1757, 4974, 32922}, {3629, 17337, 17300}, {3679, 16468, 50300}, {3679, 50300, 5263}, {3686, 17289, 32025}, {3707, 17023, 17256}, {3739, 16671, 17120}, {3759, 4664, 16834}, {3875, 3973, 17336}, {3912, 4700, 62231}, {4361, 49721, 4740}, {4383, 19750, 37652}, {4383, 37652, 14829}, {4384, 16670, 3758}, {4416, 16706, 17273}, {4422, 4969, 6542}, {4473, 20016, 3943}, {4643, 17367, 17305}, {4664, 16834, 4360}, {4740, 17350, 49721}, {4759, 50018, 4693}, {4852, 15492, 17261}, {5222, 54280, 4389}, {5839, 26685, 17233}, {6144, 17265, 17375}, {6172, 50101, 49748}, {6687, 17374, 17266}, {8584, 17000, 46922}, {8584, 17392, 63052}, {15534, 17313, 50133}, {16669, 17348, 894}, {16833, 50127, 75}, {17121, 29584, 50124}, {17245, 32455, 20090}, {17251, 47352, 2}, {17260, 29580, 4755}, {17277, 46922, 2}, {17279, 17363, 17295}, {17279, 50076, 29577}, {17342, 50077, 17294}, {17349, 63050, 6}, {17359, 50082, 29615}, {17363, 29577, 50076}, {19742, 32911, 333}, {20072, 29590, 1086}, {27644, 29767, 86}, {29577, 50076, 17295}, {37681, 62985, 69}, {41313, 50131, 17389}, {49712, 50023, 24841}, {50093, 50114, 17320}, {62989, 63051, 141}

X(68967) = X(1)X(2)∩X(106)X(673)

Barycentrics    (b - c)^2*(-3*a^2 + a*b + a*c + b*c) : :
X(68967) = 3 X[2087] + X[52626], 3 X[27918] - X[52626]

X(68967) lies on these lines: {1, 2}, {75, 46914}, {106, 673}, {335, 39697}, {514, 2087}, {527, 52900}, {812, 1015}, {1022, 6549}, {2140, 63493}, {2170, 21208}, {3227, 30997}, {3248, 23821}, {3730, 24737}, {4000, 67625}, {4089, 31647}, {4473, 27348}, {5701, 57023}, {7200, 59746}, {17214, 26847}, {17755, 34587}, {18061, 22035}, {19945, 62619}, {21630, 53600}, {27929, 35092}, {37756, 52759}, {49751, 60692}

X(68967) = midpoint of X(2087) and X(27918)
X(68967) = reflection of X(52908) in X(1125)
X(68967) = complement of X(23891)
X(68967) = complement of the isogonal conjugate of X(23892)
X(68967) = complement of the isotomic conjugate of X(62619)
X(68967) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 14434}, {667, 13466}, {739, 513}, {898, 27076}, {1977, 39011}, {3227, 21260}, {23349, 2}, {23892, 10}, {31002, 21262}, {32718, 4422}, {34075, 24003}, {35353, 21245}, {37129, 3835}, {43928, 141}, {62619, 2887}, {62763, 4129}
X(68967) = X(i)-Ceva conjugate of X(j) for these (i,j): {3227, 514}, {62755, 68896}
X(68967) = X(4728)-Dao conjugate of X(536)
X(68967) = crosspoint of X(i) and X(j) for these (i,j): {2, 62619}, {7199, 31002}
X(68967) = crosssum of X(i) and X(j) for these (i,j): {6, 68825}, {1018, 45751}
X(68967) = crossdifference of every pair of points on line {649, 4557}
X(68967) = barycentric product X(i)*X(j) for these {i,j}: {75, 16507}, {514, 47776}, {693, 68816}, {17205, 19998}
X(68967) = barycentric quotient X(i)/X(j) for these {i,j}: {16507, 1}, {38349, 23343}, {47776, 190}, {68816, 100}
X(68967) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 57038, 10}, {1015, 17761, 17205}

X(68968) = X(1)X(50343)∩X(30)X(511)

Barycentrics    (b^2 - c^2)*(-3*a^2 + a*b + a*c + b*c) : :

X(68968) lies on these lines: {1, 50343}, {10, 4010}, {30, 511}, {244, 17761}, {551, 14419}, {649, 48339}, {665, 4773}, {764, 23795}, {798, 22042}, {876, 39697}, {1018, 3952}, {1125, 9508}, {1577, 4729}, {1734, 44429}, {2254, 23814}, {3244, 4922}, {3671, 53551}, {3679, 30709}, {3746, 16158}, {3766, 4986}, {3828, 45342}, {4040, 48240}, {4041, 4129}, {4049, 4674}, {4063, 47805}, {4088, 22037}, {4378, 50761}, {4380, 48324}, {4467, 47727}, {4560, 48337}, {4707, 53558}, {4761, 4804}, {4765, 42664}, {4770, 4806}, {4775, 48284}, {4784, 48291}, {4794, 48008}, {4822, 48407}, {4834, 48301}, {4839, 24290}, {4927, 20520}, {4984, 52745}, {5466, 60094}, {5996, 44432}, {7253, 66518}, {8299, 34587}, {8666, 53270}, {8715, 53257}, {9147, 14752}, {9168, 55162}, {12514, 53400}, {13246, 42666}, {13277, 21630}, {14349, 48175}, {14435, 27811}, {14618, 39532}, {17072, 59714}, {17154, 23829}, {17494, 48352}, {21385, 53343}, {23770, 62435}, {23789, 48279}, {30476, 59752}, {30592, 36848}, {32104, 53370}, {38349, 47776}, {44445, 49287}, {46457, 46912}, {47711, 57068}, {47766, 68811}, {47803, 50501}, {48062, 49288}, {48099, 48210}, {48142, 58170}, {48184, 48273}, {48214, 50504}, {48220, 50499}, {48251, 48305}, {48288, 50339}, {48295, 50336}, {48304, 48320}, {48335, 50356}, {48408, 49276}, {53334, 62634}, {53582, 68153}

X(68968) = crossdifference of every pair of points on line {6, 33846}
X(68968) = barycentric quotient X(7040)/X(15869)
X(68968) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 4010, 59737}, {1577, 4729, 4807}, {4010, 4730, 10}, {4041, 4170, 4129}, {48273, 50355, 50337}

X(68969) = X(1)X(75)∩X(6)X(145)

Barycentrics    a^3 - 2*a^2*b + a*b^2 - 2*a^2*c - a*b*c + b^2*c + a*c^2 + b*c^2 : :
X(68969) = 4 X[1] - X[17160], X[145] + 2 X[3943], X[190] - 4 X[4702], 3 X[190] - 2 X[62222], 3 X[3685] - X[62222], 6 X[4702] - X[62222], 2 X[3717] - 3 X[17264], 3 X[17264] - X[49698], 2 X[4645] - 3 X[17297], 4 X[4966] - 3 X[17297], 2 X[4693] + X[24841], 3 X[903] - 4 X[24231], 4 X[1738] - 5 X[27191], 5 X[27191] - 8 X[49768], 4 X[2325] - X[49714], 7 X[3622] - 4 X[4395], 4 X[3823] - 5 X[17266], X[6542] + 2 X[53534], X[49709] + 2 X[49763]

X(68969) lies on these lines: {1, 75}, {2, 3996}, {3, 29746}, {6, 145}, {8, 344}, {9, 49450}, {10, 16484}, {21, 29767}, {31, 41629}, {37, 49467}, {42, 32942}, {55, 10453}, {65, 35617}, {69, 390}, {78, 20946}, {100, 20470}, {110, 30606}, {144, 4779}, {149, 3936}, {171, 42057}, {190, 518}, {192, 3242}, {200, 18743}, {214, 47626}, {238, 519}, {239, 1279}, {306, 4514}, {312, 3870}, {319, 3883}, {320, 516}, {321, 3957}, {333, 1621}, {341, 6765}, {345, 36845}, {350, 14942}, {354, 32932}, {497, 4417}, {523, 4833}, {528, 4645}, {536, 4864}, {595, 56018}, {664, 4318}, {673, 3912}, {726, 4693}, {894, 49478}, {899, 25531}, {902, 32919}, {903, 24231}, {908, 50744}, {960, 58365}, {984, 49458}, {996, 55932}, {1086, 62392}, {1120, 23579}, {1150, 61155}, {1191, 20018}, {1215, 3979}, {1220, 40433}, {1229, 34772}, {1319, 38475}, {1330, 15171}, {1386, 49475}, {1616, 20036}, {1738, 27191}, {1757, 4432}, {1918, 37588}, {1997, 64083}, {1999, 3744}, {2110, 4433}, {2177, 30942}, {2321, 16503}, {2325, 4899}, {2481, 20448}, {2550, 17234}, {2809, 49753}, {2886, 29839}, {2891, 49716}, {2975, 8053}, {3052, 37683}, {3058, 4388}, {3158, 30567}, {3187, 62806}, {3210, 17597}, {3218, 65166}, {3241, 28503}, {3243, 3729}, {3244, 4649}, {3286, 54391}, {3293, 13741}, {3295, 5774}, {3315, 17495}, {3416, 17295}, {3434, 18134}, {3555, 7283}, {3616, 17380}, {3617, 17259}, {3621, 10005}, {3622, 4395}, {3623, 4461}, {3632, 15485}, {3633, 16468}, {3635, 33682}, {3644, 49446}, {3687, 64162}, {3689, 5205}, {3696, 16823}, {3699, 3935}, {3706, 3748}, {3712, 51463}, {3720, 32945}, {3722, 17763}, {3741, 3750}, {3749, 3769}, {3751, 4676}, {3755, 16706}, {3758, 68588}, {3759, 7290}, {3771, 33141}, {3790, 49688}, {3797, 32029}, {3823, 17266}, {3840, 60714}, {3871, 5132}, {3873, 32929}, {3874, 63996}, {3877, 49687}, {3879, 63969}, {3893, 63522}, {3896, 7191}, {3914, 33124}, {3920, 34064}, {3923, 49490}, {3938, 32915}, {3950, 49527}, {3952, 62236}, {3969, 33090}, {3999, 62300}, {4026, 17307}, {4028, 33071}, {4062, 32844}, {4085, 29637}, {4184, 29766}, {4225, 23374}, {4307, 17378}, {4310, 4398}, {4356, 17320}, {4357, 63977}, {4359, 29817}, {4362, 17715}, {4365, 32923}, {4366, 4437}, {4383, 20012}, {4384, 38316}, {4387, 32937}, {4389, 64168}, {4393, 38315}, {4413, 30947}, {4418, 62867}, {4423, 59296}, {4427, 17145}, {4429, 17283}, {4430, 32933}, {4440, 28530}, {4441, 14828}, {4442, 33148}, {4450, 32863}, {4511, 4742}, {4562, 52030}, {4648, 20181}, {4651, 5284}, {4660, 33087}, {4664, 7174}, {4666, 19804}, {4685, 17123}, {4689, 24627}, {4716, 50023}, {4764, 15600}, {4847, 33116}, {4851, 50289}, {4861, 28974}, {4863, 29641}, {4869, 62383}, {4871, 56009}, {4903, 59597}, {4923, 50095}, {4924, 59579}, {4970, 17598}, {4972, 33173}, {4974, 50016}, {4997, 52925}, {5014, 32858}, {5156, 37610}, {5223, 17336}, {5248, 50625}, {5255, 35633}, {5271, 62856}, {5327, 62830}, {5484, 64158}, {5524, 24003}, {5542, 7321}, {5564, 43179}, {5687, 16297}, {5695, 24349}, {5698, 17347}, {5844, 37510}, {5847, 49709}, {6063, 21453}, {6327, 34611}, {6745, 37758}, {7175, 63987}, {7967, 37474}, {8167, 26038}, {8299, 37686}, {8616, 32853}, {8692, 20053}, {9453, 56851}, {9454, 56530}, {10385, 63140}, {10389, 11679}, {10459, 45223}, {10699, 33946}, {11038, 42697}, {11115, 18166}, {11281, 30543}, {11682, 45738}, {12513, 20992}, {13136, 51565}, {14621, 17389}, {14997, 20048}, {15172, 41014}, {15254, 60731}, {15569, 16830}, {15570, 49483}, {16477, 49685}, {16496, 49447}, {16824, 51715}, {16825, 49459}, {17018, 24552}, {17117, 49468}, {17121, 68876}, {17140, 62863}, {17142, 34195}, {17147, 62814}, {17155, 62869}, {17261, 49515}, {17262, 31302}, {17269, 59407}, {17271, 49746}, {17273, 24723}, {17274, 66673}, {17280, 20180}, {17315, 20179}, {17316, 20172}, {17317, 64174}, {17318, 67538}, {17322, 19868}, {17335, 66515}, {17346, 47357}, {17350, 64070}, {17354, 59406}, {17364, 64016}, {17374, 28566}, {17377, 51192}, {17449, 32845}, {17592, 29652}, {17681, 40006}, {17719, 50748}, {17724, 37759}, {17751, 64169}, {17764, 32857}, {17765, 32847}, {17766, 32846}, {17769, 49691}, {17772, 49700}, {17778, 63979}, {17784, 18141}, {17785, 29866}, {18139, 33110}, {19239, 37502}, {19998, 37680}, {20011, 32911}, {20014, 63050}, {20016, 20142}, {20040, 27644}, {20050, 49680}, {20051, 56983}, {20131, 29585}, {20132, 29588}, {20135, 29624}, {20170, 37614}, {20352, 38878}, {20718, 62872}, {20942, 66469}, {20955, 66669}, {21242, 29640}, {21283, 29830}, {21296, 30332}, {21896, 25965}, {23361, 23375}, {24210, 33126}, {24390, 25444}, {24393, 25101}, {24542, 33139}, {24715, 49676}, {24727, 39258}, {24813, 29327}, {24987, 65822}, {25001, 58366}, {25048, 35104}, {25496, 42042}, {26015, 32851}, {26139, 51415}, {27164, 64415}, {28808, 63168}, {28850, 49779}, {29584, 50778}, {29627, 59413}, {29632, 33136}, {29638, 33128}, {29642, 32865}, {29655, 33160}, {29656, 33135}, {29672, 33132}, {29673, 33158}, {29818, 32924}, {29835, 32779}, {29844, 32855}, {30829, 67097}, {30832, 33175}, {31136, 32917}, {31144, 49740}, {31330, 62849}, {32004, 68868}, {32771, 67209}, {32773, 33171}, {32784, 50311}, {32928, 67210}, {32931, 67207}, {32934, 62865}, {32935, 49498}, {32947, 33081}, {33064, 33095}, {33069, 33094}, {33076, 49560}, {33082, 50315}, {33120, 33156}, {33121, 59692}, {33122, 33134}, {33142, 41806}, {33944, 66666}, {34747, 50283}, {37540, 37684}, {37552, 39584}, {37573, 50608}, {37678, 54291}, {37679, 59295}, {38478, 50362}, {40954, 58535}, {41263, 64163}, {42033, 63147}, {44669, 60452}, {48696, 49999}, {49452, 49455}, {49461, 49463}, {49462, 49465}, {49488, 49678}, {49722, 51099}, {49748, 50999}, {49765, 53602}, {50126, 51055}, {50300, 51093}, {51384, 52164}, {52653, 54280}, {55082, 60720}, {68148, 68153}

X(68969) = midpoint of X(i) and X(j) for these {i,j}: {4693, 49675}, {6542, 49704}
X(68969) = reflection of X(i) in X(j) for these {i,j}: {8, 3932}, {190, 3685}, {239, 1279}, {320, 4684}, {1738, 49768}, {1757, 4432}, {3685, 4702}, {4645, 4966}, {4716, 50023}, {4899, 2325}, {17160, 32922}, {24715, 49676}, {24841, 49675}, {32846, 49764}, {32850, 3912}, {32922, 1}, {49698, 3717}, {49704, 53534}, {49714, 4899}, {50016, 4974}, {62392, 1086}
X(68969) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1280, 1330}, {1477, 2475}, {1810, 52364}, {35355, 3448}, {36807, 21287}, {43760, 2893}
X(68969) = X(40526)-isoconjugate of X(43929)
X(68969) = X(2254)-Dao conjugate of X(3675)
X(68969) = crosspoint of X(1016) and X(51560)
X(68969) = crosssum of X(i) and X(j) for these (i,j): {3248, 8659}, {8647, 21748}
X(68969) = trilinear pole of line {38379, 53343}
X(68969) = crossdifference of every pair of points on line {798, 6363}
X(68969) = barycentric product X(i)*X(j) for these {i,j}: {75, 63087}, {190, 53343}, {312, 7677}, {668, 68814}, {1978, 53287}, {38379, 46135}
X(68969) = barycentric quotient X(i)/X(j) for these {i,j}: {1026, 40526}, {7677, 57}, {38379, 926}, {38980, 3675}, {53287, 649}, {53343, 514}, {63087, 1}, {68814, 513}
X(68969) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3886, 75}, {1, 5263, 86}, {1, 32941, 5263}, {1, 49469, 32921}, {1, 49470, 4360}, {8, 1001, 17277}, {9, 49451, 49450}, {42, 32943, 32942}, {55, 10453, 14829}, {902, 50001, 32919}, {1001, 49460, 8}, {1621, 17135, 333}, {2886, 29839, 41878}, {3241, 48805, 46922}, {3243, 3729, 49499}, {3244, 49482, 4649}, {3633, 16468, 49497}, {3689, 5205, 43290}, {3696, 42819, 16823}, {3706, 3748, 3757}, {3706, 3757, 55095}, {3749, 39594, 3769}, {3873, 32929, 32939}, {3935, 4358, 3699}, {3938, 32915, 32926}, {4387, 41711, 32937}, {4427, 17145, 62235}, {4645, 4966, 17297}, {4666, 63131, 19804}, {5695, 42871, 24349}, {7290, 49495, 3759}, {15570, 49485, 49483}, {17264, 49698, 3717}, {17388, 51147, 145}, {21283, 29830, 33108}, {24723, 49511, 17273}, {49471, 49473, 1}, {49478, 49484, 894}, {49746, 50316, 17271}, {62863, 64010, 17140}

X(68970) = X(1)X(2)∩X(11)X(116)

Barycentrics    (b - c)^2*(a^3 - 2*a^2*b + a*b^2 - 2*a^2*c - a*b*c + b^2*c + a*c^2 + b*c^2) : :
X(68970) = 9 X[2] - X[20038], 3 X[1026] - X[20038]

X(68970) lies on these lines: {1, 2}, {9, 24409}, {11, 116}, {36, 7437}, {57, 67577}, {105, 24618}, {226, 36008}, {244, 1111}, {514, 3675}, {516, 56850}, {885, 35355}, {958, 16444}, {993, 16375}, {1015, 1146}, {1086, 23821}, {2283, 3911}, {2293, 24778}, {2401, 35348}, {2424, 60580}, {3271, 24237}, {3663, 24338}, {3772, 67214}, {3816, 24250}, {3942, 4965}, {4357, 24517}, {4530, 68835}, {4778, 53546}, {4858, 24225}, {5745, 16443}, {17063, 19950}, {17197, 53564}, {17337, 55076}, {21201, 53525}, {23352, 60578}, {24231, 36814}, {27854, 38989}, {32919, 68812}, {35094, 35119}, {43672, 61480}, {45305, 63589}, {46660, 53835}, {53600, 56855}, {53602, 60857}, {58259, 68917}, {64705, 67574}, {67576, 67658}

X(68970) = midpoint of X(3675) and X(4124)
X(68970) = complement of X(1026)
X(68970) = complement of the isogonal conjugate of X(1027)
X(68970) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 62552}, {56, 3126}, {105, 513}, {294, 20317}, {513, 120}, {514, 20540}, {649, 16593}, {666, 27076}, {667, 6184}, {673, 3835}, {884, 9}, {885, 1329}, {919, 4422}, {1015, 35094}, {1024, 3452}, {1027, 10}, {1416, 522}, {1438, 514}, {1462, 4885}, {2195, 4521}, {2481, 21260}, {3271, 1566}, {3669, 17060}, {3733, 8299}, {6654, 27854}, {7336, 35967}, {10099, 21530}, {13576, 31946}, {14942, 59971}, {15382, 2977}, {18031, 21262}, {18785, 4129}, {23355, 22116}, {23696, 34823}, {32644, 40534}, {32666, 24036}, {32735, 3035}, {36057, 20315}, {36086, 24003}, {36124, 20316}, {36146, 21232}, {36802, 3038}, {41934, 918}, {43921, 11}, {43924, 50441}, {43929, 2}, {43930, 2886}, {51838, 3716}, {51845, 4147}, {51866, 812}, {52030, 3837}, {52209, 21261}, {52902, 14434}, {52927, 3039}, {55261, 1211}, {56783, 17072}, {56853, 661}, {62635, 141}, {64216, 650}
X(68970) = X(i)-Ceva conjugate of X(j) for these (i,j): {1280, 522}, {2481, 514}, {30941, 812}
X(68970) = X(919)-isoconjugate of X(40526)
X(68970) = X(i)-Dao conjugate of X(j) for these (i,j): {2254, 518}, {38980, 40526}
X(68970) = crosspoint of X(673) and X(7192)
X(68970) = crosssum of X(i) and X(j) for these (i,j): {6, 54325}, {672, 4557}
X(68970) = crossdifference of every pair of points on line {649, 23845}
X(68970) = barycentric product X(i)*X(j) for these {i,j}: {514, 53343}, {693, 68814}, {1111, 63087}, {2481, 38980}, {3261, 53287}, {4858, 7677}
X(68970) = barycentric quotient X(i)/X(j) for these {i,j}: {2254, 40526}, {7677, 4564}, {38980, 518}, {53287, 101}, {53343, 190}, {63087, 765}, {68814, 100}
X(68970) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3756, 34589, 40451}, {53564, 64523, 17197}

X(68971) = X(2)X(37)∩X(8)X(3294)

Barycentrics    (b + c)*(a^3 - 2*a^2*b + a*b^2 - 2*a^2*c - a*b*c + b^2*c + a*c^2 + b*c^2) : :

X(68971) lies on these lines: {1, 26770}, {2, 37}, {8, 3294}, {9, 17135}, {42, 3950}, {72, 56937}, {145, 213}, {172, 17539}, {190, 30941}, {304, 25237}, {325, 31058}, {349, 60229}, {672, 2325}, {1334, 17751}, {1500, 27040}, {1655, 26759}, {2176, 20040}, {2238, 3943}, {2321, 4651}, {2348, 4702}, {3161, 10453}, {3187, 16970}, {3239, 4024}, {3509, 4427}, {3663, 30821}, {3685, 20605}, {3701, 3991}, {3702, 16601}, {3710, 21096}, {3720, 17355}, {3726, 17154}, {3731, 31330}, {3741, 59585}, {3760, 28742}, {3912, 20347}, {3930, 3952}, {3932, 13576}, {3936, 17747}, {3970, 56318}, {3986, 59306}, {3992, 30730}, {4029, 29822}, {4072, 4685}, {4098, 43223}, {4099, 16600}, {4109, 27558}, {4368, 6541}, {4515, 52353}, {4742, 43065}, {4975, 24036}, {5749, 29814}, {6376, 26757}, {7283, 56984}, {8055, 22020}, {11319, 54416}, {14210, 65195}, {16369, 20016}, {16705, 32026}, {17027, 17339}, {17164, 21808}, {17165, 51058}, {17242, 24514}, {17243, 24330}, {17261, 31027}, {17262, 30945}, {17268, 31004}, {17314, 20011}, {17332, 50278}, {17340, 24512}, {17390, 50257}, {18135, 27096}, {18140, 27025}, {18600, 25264}, {20012, 61036}, {20244, 29966}, {20335, 22031}, {20911, 25261}, {20954, 26824}, {21029, 27690}, {21073, 57808}, {21840, 27804}, {21935, 63607}, {22008, 30946}, {22010, 30985}, {22013, 29839}, {24067, 33173}, {24592, 25101}, {26964, 27109}, {30822, 55998}, {32929, 40131}, {32937, 40463}, {42057, 59579}, {49687, 54330}, {49753, 53332}, {52241, 56538}, {54389, 63066}, {68128, 68132}

X(68971) = X(i)-Ceva conjugate of X(j) for these (i,j): {18031, 4651}, {36807, 10}
X(68971) = barycentric product X(i)*X(j) for these {i,j}: {321, 63087}, {3701, 7677}, {3952, 53343}, {4033, 68814}, {27808, 53287}
X(68971) = barycentric quotient X(i)/X(j) for these {i,j}: {7677, 1014}, {53287, 3733}, {53343, 7192}, {63087, 81}, {68814, 1019}
X(68971) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {344, 4441, 2}, {1334, 21071, 17751}, {2321, 59207, 4651}, {3294, 21070, 8}, {3930, 3985, 3952}, {4099, 16600, 64071}, {17264, 42722, 32849}, {17314, 37657, 20011}, {25264, 27097, 18600}

X(68972) = X(115)X(124)∩X(514)X(661)

Barycentrics    (b^2 - c^2)*(a^3 - 2*a^2*b + a*b^2 - 2*a^2*c - a*b*c + b^2*c + a*c^2 + b*c^2) : :
X(68972) = X[3766] + 3 X[30565], 3 X[45661] - X[68901], X[665] - 3 X[1639], 3 X[4120] + X[21832], X[20507] - 3 X[45338]

X(68972) lies on these lines: {2, 23829}, {10, 24290}, {115, 124}, {512, 14321}, {514, 661}, {523, 22042}, {650, 8714}, {665, 1639}, {798, 3667}, {918, 20520}, {1018, 3952}, {2786, 17755}, {3008, 24285}, {3686, 8674}, {3700, 4151}, {3887, 4148}, {4013, 61179}, {4025, 27045}, {4171, 4404}, {4384, 53335}, {4521, 52592}, {5257, 53527}, {6184, 35068}, {6372, 68794}, {15413, 40474}, {20507, 45338}, {20623, 31845}, {29512, 48270}, {38379, 53343}, {40619, 59736}, {48269, 58288}

X(68972) = complement of X(23829)
X(68972) = X(i)-complementary conjugate of X(j) for these (i,j): {105, 53564}, {213, 35094}, {666, 21240}, {692, 8299}, {919, 3739}, {1018, 20540}, {1438, 17761}, {2195, 34589}, {4516, 35967}, {4557, 120}, {4559, 17060}, {5377, 512}, {13576, 21252}, {18785, 116}, {32666, 1125}, {32724, 15569}, {32735, 3742}, {36086, 3741}, {36146, 17050}, {52927, 960}, {56853, 11}, {64216, 244}, {66930, 442}
X(68972) = X(42720)-Ceva conjugate of X(740)
X(68972) = crosssum of X(649) and X(20470)
X(68972) = crossdifference of every pair of points on line {31, 18613}
X(68972) = barycentric product X(i)*X(j) for these {i,j}: {10, 53343}, {313, 53287}, {321, 68814}, {1577, 63087}, {4086, 7677}
X(68972) = barycentric quotient X(i)/X(j) for these {i,j}: {3930, 40526}, {7677, 1414}, {53287, 58}, {53343, 86}, {63087, 662}, {68814, 81}

X(68973) = X(1)X(2)∩X(11)X(33228)

Barycentrics    a^2*b^2 - 3*a^2*b*c + a^2*c^2 + b^2*c^2 : :
X(68973) = 4 X[1125] - X[3783], X[42881] + 2 X[49764], X[350] + 2 X[1015], X[668] - 4 X[20530], 2 X[1575] - 5 X[27195], X[3227] + 2 X[41144], 2 X[6381] + X[9263], 4 X[40479] - X[52959]

X(68973) lies on these lines: {1, 2}, {11, 33228}, {35, 33273}, {36, 4366}, {56, 1003}, {76, 63493}, {172, 12150}, {330, 3760}, {335, 4694}, {350, 538}, {384, 5563}, {385, 16784}, {388, 32983}, {496, 26561}, {497, 32986}, {536, 9296}, {668, 20530}, {999, 11286}, {1019, 4785}, {1428, 5182}, {1449, 26107}, {1469, 22486}, {1478, 33016}, {1479, 33017}, {1500, 44562}, {1575, 27195}, {1870, 46511}, {1909, 9466}, {1966, 37756}, {2087, 49755}, {2275, 7757}, {3058, 8356}, {3227, 18145}, {3230, 37686}, {3303, 11285}, {3304, 7770}, {3329, 16785}, {3570, 50028}, {3726, 18061}, {3746, 7824}, {3761, 30998}, {3797, 4975}, {3813, 17670}, {3934, 25303}, {4299, 33193}, {4302, 33207}, {4309, 32965}, {4317, 14035}, {4325, 6658}, {4330, 33260}, {4358, 24625}, {4496, 51974}, {4857, 6655}, {5025, 37720}, {5048, 28798}, {5204, 68519}, {5258, 16918}, {5270, 16044}, {5298, 35297}, {5299, 63038}, {5434, 8370}, {6284, 8353}, {6381, 9263}, {6656, 37722}, {7187, 7264}, {7288, 33216}, {7354, 66408}, {7771, 10987}, {7841, 11238}, {8354, 15171}, {8358, 15172}, {8359, 15170}, {8368, 26686}, {8666, 16916}, {8667, 16781}, {8924, 28194}, {9651, 63957}, {9665, 63956}, {9670, 33234}, {10385, 33215}, {10483, 66405}, {11237, 44543}, {12953, 66396}, {14041, 65140}, {14210, 33891}, {14614, 16502}, {15325, 26629}, {15888, 32992}, {16604, 17143}, {16884, 25505}, {16921, 37719}, {16971, 37678}, {16975, 30963}, {17178, 17324}, {17270, 25535}, {17272, 26143}, {17319, 26963}, {17374, 25534}, {17394, 25538}, {17448, 18140}, {17474, 17499}, {17541, 62837}, {17731, 25530}, {18135, 63499}, {18152, 63508}, {18153, 63528}, {18990, 66409}, {20176, 39974}, {20688, 62319}, {20691, 24739}, {20924, 27918}, {23478, 42040}, {23660, 46922}, {23892, 62638}, {24387, 33841}, {24519, 25661}, {27285, 62866}, {27318, 32104}, {31276, 31999}, {31348, 68895}, {31410, 32991}, {31452, 33001}, {34914, 40738}, {40479, 52959}, {41142, 59454}, {41310, 65040}, {43266, 63242}, {50093, 56533}, {65134, 66406}

X(68973) = midpoint of X(3227) and X(18145)
X(68973) = reflection of X(i) in X(j) for these {i,j}: {18145, 41144}, {65040, 41310}
X(68973) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 26959, 27020}, {1, 29438, 17752}, {1, 29750, 41240}, {239, 27166, 25510}, {1125, 26801, 16819}, {3616, 17030, 31996}, {3760, 9336, 330}, {26821, 27166, 239}

X(68974) = X(1)X(6)∩X(36)X(5201)

Barycentrics     a*(a^2*b^2 - 3*a^2*b*c + a^2*c^2 + b^2*c^2) : :

X(68974) lies on these lines: {1, 6}, {36, 5201}, {75, 23524}, {86, 23532}, {87, 4361}, {190, 68751}, {239, 2234}, {320, 27846}, {385, 7292}, {524, 56805}, {536, 9359}, {597, 40790}, {609, 34097}, {614, 14614}, {674, 63526}, {889, 3226}, {899, 9362}, {1266, 24722}, {1914, 33875}, {1964, 17121}, {2228, 24625}, {2230, 17029}, {3329, 5297}, {3733, 4782}, {3736, 4991}, {3759, 7032}, {3783, 4969}, {4360, 22343}, {5035, 12194}, {5063, 10801}, {5263, 23578}, {5272, 8667}, {5370, 51862}, {7113, 8300}, {7184, 17366}, {7191, 63038}, {7202, 18208}, {7232, 25572}, {7240, 7263}, {8054, 32919}, {8264, 32005}, {8266, 59319}, {10802, 33872}, {11364, 33882}, {15953, 51005}, {16694, 20475}, {16834, 24696}, {17049, 46189}, {17120, 17445}, {17259, 24661}, {17277, 63504}, {17311, 25573}, {17349, 63527}, {20148, 29578}, {20456, 25048}, {21352, 46922}, {21790, 23566}, {23427, 33296}, {23457, 34063}, {23473, 24519}, {23579, 32922}, {30950, 37678}, {34916, 40763}, {37756, 53541}, {41328, 59325}

X(68974) = crosssum of X(37) and X(52894)
X(68974) = crossdifference of every pair of points on line {513, 20691}
X(68974) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 18194, 18170}, {15571, 16493, 238}, {17160, 37129, 68884}, {23539, 68884, 37129}

X(68975) = X(2)X(37)∩X(190)X(26982)

Barycentrics   a^3*b^2 + a^2*b^3 - 2*a^3*b*c - 2*a^2*b^2*c - 2*a*b^3*c + a^3*c^2 - 2*a^2*b*c^2 + 4*a*b^2*c^2 + b^3*c^2 + a^2*c^3 - 2*a*b*c^3 + b^2*c^3 : :

X(68975) lies on these lines: {2, 37}, {190, 26982}, {320, 26821}, {903, 36799}, {1019, 4785}, {1086, 27166}, {3123, 64909}, {3663, 26963}, {3875, 27095}, {3943, 27113}, {3946, 26772}, {4440, 26975}, {4852, 26756}, {6542, 26142}, {16738, 17324}, {17023, 26976}, {17160, 25534}, {17178, 17235}, {17304, 27145}, {26113, 27159}, {26768, 62231}, {27036, 29590}

X(68975) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6542, 26142, 27106}, {17160, 25534, 27044}

X(68976) = X(2)X(37)∩X(7)X(9335)

Barycentrics   a^3*b + 2*a^2*b^2 + a*b^3 + a^3*c - 4*a^2*b*c - 2*a*b^2*c - b^3*c + 2*a^2*c^2 - 2*a*b*c^2 + 2*b^2*c^2 + a*c^3 - b*c^3 : :

X(68976) lies on these lines: {2, 37}, {7, 9335}, {39, 25261}, {244, 20347}, {614, 26229}, {978, 20247}, {1149, 21272}, {1443, 1447}, {3212, 28370}, {3315, 68928}, {5121, 24191}, {6586, 27115}, {16583, 26964}, {16756, 26845}, {20244, 24174}, {21208, 49997}, {21273, 24173}, {21285, 28074}, {24166, 56318}, {24172, 27627}, {26094, 33945}, {26273, 33854}, {26563, 52541}, {26805, 41015}

X(68976) = crossdifference of every pair of points on line {667, 1334}

X(68977) = X(1)X(2)∩X(6)X(29740)

Barycentrics   a*(a^3*b + 2*a^2*b^2 + a*b^3 + a^3*c - 4*a^2*b*c - 2*a*b^2*c - b^3*c + 2*a^2*c^2 - 2*a*b*c^2 + 2*b^2*c^2 + a*c^3 - b*c^3) : :

X(68977) lies on these lines: {1, 2}, {6, 29740}, {57, 9336}, {106, 11349}, {1015, 20367}, {1019, 1429}, {1086, 68840}, {1100, 4503}, {1743, 29696}, {3445, 37272}, {4360, 29705}, {8610, 53391}, {11320, 39748}, {16779, 29698}, {21362, 57037}, {28358, 67984}, {62370, 68759}

X(68977) = crossdifference of every pair of points on line {210, 649}
X(68977) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {239, 26821, 29769}, {1149, 68753, 3008}, {28254, 49770, 31855}

X(68978) = X(10)X(321)∩X(239)X(21208)

Barycentrics   (b + c)*(-(a^3*b) - 2*a^2*b^2 - a*b^3 - a^3*c + 4*a^2*b*c + 2*a*b^2*c + b^3*c - 2*a^2*c^2 + 2*a*b*c^2 - 2*b^2*c^2 - a*c^3 + b*c^3) : :

X(68978) lies on these lines: {10, 321}, {239, 21208}, {241, 514}, {524, 24195}, {538, 49730}, {596, 3765}, {712, 25125}, {1738, 31897}, {1739, 30807}, {3686, 24219}, {3975, 57029}, {4384, 62636}, {4568, 26048}, {4771, 50019}, {17497, 29456}, {17749, 56882}, {21921, 29571}, {29604, 35101}

X(68978) = crossdifference of every pair of points on line {55, 57129}
X(68978) = {X(16609),X(16611)}-harmonic conjugate of X(3008)

X(68979) = X(30)X(511)∩X(659)X(21124)

Barycentrics   (b - c)*(a^3 + a*b^2 + b^3 + a*b*c + b^2*c + a*c^2 + b*c^2 + c^3) : :

X(68979) lies on these lines: {30, 511}, {659, 21124}, {1491, 48300}, {2522, 54249}, {2530, 47682}, {2533, 47660}, {3004, 48299}, {3777, 47973}, {3801, 47694}, {3837, 8045}, {4041, 48103}, {4122, 21301}, {4142, 48248}, {4367, 8635}, {4435, 4988}, {4449, 47923}, {4468, 47967}, {4490, 48094}, {4705, 48056}, {4729, 48146}, {4822, 47944}, {4983, 47990}, {6332, 48007}, {6545, 47889}, {7255, 57149}, {14349, 47999}, {17072, 48405}, {17166, 48326}, {21192, 50512}, {21302, 47693}, {21791, 22157}, {44435, 47841}, {47652, 48279}, {47680, 48393}, {47684, 48410}, {47691, 48301}, {47701, 48336}, {47712, 48305}, {47716, 48291}, {47771, 47835}, {47814, 48185}, {47820, 48227}, {47840, 48552}, {47913, 48078}, {47918, 48083}, {47921, 48096}, {47922, 48614}, {47924, 48338}, {47931, 48334}, {47951, 48128}, {47957, 48040}, {47958, 48123}, {47959, 48048}, {47960, 48136}, {47961, 50508}, {47968, 48131}, {47995, 48093}, {48087, 48607}, {48092, 49280}, {48106, 50355}, {48129, 48621}, {48150, 50340}, {48215, 48564}, {48265, 49275}, {48278, 50328}, {48331, 68780}, {48409, 50351}, {50342, 50523}

X(68979) = {X(6332),X(48007)}-harmonic conjugate of X(48100)

X(68980) = X(30)X(511)∩X(86)X(310)

Barycentrics   a^3*b^2 + a^2*b^2*c + a^3*c^2 + a^2*b*c^2 - 2*a*b^2*c^2 - b^3*c^2 - b^2*c^3 : :

X(68980) lies on these lines: {6, 24259}, {30, 511}, {39, 24688}, {76, 24327}, {86, 310}, {190, 58287}, {194, 4443}, {256, 56023}, {1045, 3770}, {1213, 1575}, {1654, 4651}, {1655, 24450}, {2234, 3948}, {2901, 63366}, {3122, 62636}, {3912, 23823}, {4039, 4436}, {4368, 52897}, {4416, 22316}, {4446, 21299}, {6707, 20530}, {7228, 64545}, {9791, 24338}, {16726, 57034}, {17142, 20090}, {17346, 32860}, {17378, 32915}, {17392, 23812}, {17790, 24342}, {20081, 24351}, {21022, 26764}, {21254, 63443}, {25354, 57039}, {30939, 53541}, {31144, 41142}, {40085, 52208}, {51575, 53478}, {63401, 67024}

X(68980) = isogonal conjugate of X(39441)
X(68980) = crossdifference of every pair of points on line {6, 53581}
X(68980) = X(24259)-line conjugate of X(6)
X(68980) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {39, 24688, 25347}, {76, 24696, 24327}, {194, 24717, 4443}

X(68981) = X(2)X(39)∩X(523)X(661)

Barycentrics   (b + c)*(a^3*b^2 + a^2*b^2*c + a^3*c^2 + a^2*b*c^2 - 2*a*b^2*c^2 - b^3*c^2 - b^2*c^3) : :

X(68981) lies on these lines: {2, 39}, {115, 3006}, {523, 661}, {726, 3124}, {899, 35068}, {2653, 56318}, {3685, 21341}, {4358, 16592}, {4427, 20666}, {4970, 61324}, {8013, 21685}, {9280, 29822}, {21220, 52137}, {22038, 24070}, {23903, 33120}, {23917, 29690}, {29673, 63604}, {30970, 61342}, {32925, 52651}, {33117, 63605}, {46714, 64224}

X(68981) = X(81)-isoconjugate of X(39441)
X(68981) = X(40586)-Dao conjugate of X(39441)
X(68981) = crossdifference of every pair of points on line {58, 669}
X(68981) = barycentric quotient X(42)/X(39441)
X(68981) = {X(194),X(27570)}-harmonic conjugate of X(27698)

X(68982) = X(1)X(75) INTERCEPT OF X(3)X(28014)

Barycentrics   a*(a + b)*(a + c)*(a^2*b + b^3 + a^2*c - 2*a*b*c - b^2*c - b*c^2 + c^3) : :

X(68982) = {1, 75}, {3, 28014}, {7, 37819}, {21, 28011}, {55, 16700}, {57, 38832}, {58, 2191}, {81, 614}, {100, 16753}, {238, 18206}, {239, 27169}, {242, 514}, {244, 62740}, {333, 5272}, {354, 40153}, {386, 4648}, {516, 17205}, {518, 52897}, {612, 5333}, {741, 927}, {752, 17179}, {942, 2300}, {975, 28620}, {982, 17185}, {995, 38053}, {1001, 16696}, {1014, 2263}, {1279, 3286}, {1284, 43034}, {1352, 28078}, {1376, 16736}, {1386, 18166}, {1412, 34036}, {1449, 2303}, {1474, 46883}, {1503, 68929}, {1621, 18601}, {1714, 28753}, {1722, 56018}, {1738, 16752}, {1756, 3122}, {3216, 17234}, {3246, 18198}, {3290, 34381}, {3677, 35623}, {3685, 62636}, {3749, 13588}, {3751, 27644}, {3752, 18185}, {3786, 16496}, {3976, 10461}, {4229, 12652}, {4260, 28350}, {4267, 52541}, {4278, 16688}, {4307, 17169}, {4310, 17183}, {4653, 38316}, {4666, 10458}, {5009, 51838}, {5262, 17497}, {5268, 25507}, {5573, 18163}, {5847, 30941}, {6776, 28089}, {7032, 28082}, {7191, 8025}, {7292, 16704}, {10449, 26106}, {10457, 46190}, {10477, 28365}, {16020, 16713}, {16705, 50290}, {16711, 28580}, {16716, 16781}, {16738, 16823}, {16782, 20605}, {16876, 28026}, {16887, 50295}, {17054, 18178}, {17139, 24231}, {17171, 24159}, {17174, 33148}, {17182, 33144}, {17210, 50298}, {17749, 53665}, {17768, 39688}, {18169, 29820}, {18186, 60846}, {18600, 64168}, {21214, 46877}, {25508, 39586}, {27660, 62874}, {28368, 40952}, {37042, 44139}, {40934, 50307}, {46972, 52680}, {52564, 55340}, {54418, 64377}, {61409, 62819}, {63055, 68938}

X(68982) = midpoint of X(5018) and X(61434)
X(68982) = X(3290)-cross conjugate of X(16752)
X(68982) = X(i)-isoconjugate of X(j) for these (i,j): {37, 2991}, {65, 56111}, {72, 15344}, {512, 35574}, {3932, 15382}, {13576, 34159}, {20683, 57754}
X(68982) = X(i)-Dao conjugate of X(j) for these (i,j): {120, 10}, {3675, 4088}, {39054, 35574}, {40589, 2991}, {40602, 56111}, {65924, 306}
X(68982) = crosssum of X(i) and X(j) for these (i,j): {37, 4433}, {42, 3930}
X(68982) = crossdifference of every pair of points on line {71, 798}
X(68982) = barycentric product X(i)*X(j) for these {i,j}: {1, 16752}, {27, 34381}, {81, 1738}, {86, 3290}, {514, 4236}, {662, 23770}, {757, 21956}, {1019, 53358}, {14267, 18206}
X(68982) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 2991}, {284, 56111}, {662, 35574}, {1474, 15344}, {1738, 321}, {3290, 10}, {4236, 190}, {16752, 75}, {17464, 3932}, {20455, 3930}, {21956, 1089}, {23770, 1577}, {34381, 306}, {53358, 4033}, {54407, 57499}
X(68982) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1279, 16726, 3286}, {7290, 18164, 58}

X(68983) = X(1)X(75) INTERCEPT OF X(81)X(39713)

Barycentrics   (a + b)*(a + c)*(a^3*b + a*b^3 + a^3*c - 2*a^2*b*c - b^3*c + a*c^3 - b*c^3) : :

X(68983) lies on these lines: {1, 75}, {81, 39713}, {105, 33295}, {514, 1919}, {528, 16711}, {594, 26759}, {1438, 34081}, {1909, 18082}, {4026, 16705}, {4366, 62636}, {5846, 30941}, {7191, 16707}, {9055, 52897}, {16752, 26582}, {16887, 33076}, {17045, 26807}, {17176, 32923}, {17205, 17766}, {17381, 26035}, {18827, 52030}, {35172, 53649}, {53219, 65275}

X(68983) = crosssum of X(39258) and X(41267)
X(68983) = crossdifference of every pair of points on line {798, 21035}

X(68984) = X(1)X(75) INTERCEPT OF X(2)X(32)

Barycentrics   (a + b)*(a + c)*(a^2 + b^2 + b*c + c^2) : :

X(68984) lies on these lines: {1, 75}, {2, 32}, {10, 17200}, {21, 3415}, {28, 63158}, {37, 16735}, {58, 30966}, {69, 37037}, {76, 964}, {81, 3661}, {99, 5992}, {172, 27274}, {316, 5051}, {333, 17308}, {384, 25499}, {385, 52538}, {443, 63014}, {612, 33932}, {668, 1220}, {894, 3954}, {1509, 30941}, {1655, 24275}, {1724, 5224}, {1975, 16394}, {2049, 16992}, {3920, 33941}, {3933, 50059}, {4352, 51674}, {4389, 33868}, {4426, 27164}, {4594, 17946}, {4596, 13576}, {5235, 29608}, {5259, 11102}, {5280, 17289}, {5283, 17688}, {5291, 16738}, {5300, 14005}, {5333, 17397}, {6542, 8025}, {6626, 17210}, {6651, 21816}, {7261, 52935}, {7750, 13728}, {7760, 26035}, {7767, 50318}, {7782, 16393}, {7788, 51590}, {7802, 17676}, {7859, 17672}, {11057, 50321}, {11321, 15668}, {13588, 37586}, {13740, 18140}, {15149, 46103}, {16054, 25500}, {16589, 33816}, {16704, 29591}, {16712, 51669}, {16752, 29586}, {16783, 17381}, {16818, 20179}, {16887, 17103}, {16928, 16994}, {16930, 26978}, {17056, 25665}, {17095, 37583}, {17139, 33865}, {17167, 37089}, {17183, 33867}, {17294, 42028}, {17307, 29562}, {17378, 51605}, {17389, 42025}, {17398, 17670}, {17686, 29479}, {17698, 37664}, {18171, 37128}, {18206, 36483}, {18754, 25496}, {19284, 27162}, {26110, 56903}, {26828, 27189}, {30142, 33931}, {32819, 50391}, {32833, 51591}, {37632, 40024}, {37671, 50323}, {49560, 51356}, {57919, 64133}

X(68984) = anticomplement of X(68934)
X(68984) = X(35137)-Ceva conjugate of X(7192)
X(68984) = X(i)-isoconjugate of X(j) for these (i,j): {512, 831}, {669, 57975}, {8061, 58956}
X(68984) = X(i)-Dao conjugate of X(j) for these (i,j): {17289, 4425}, {39054, 831}, {55054, 3005}
X(68984) = cevapoint of X(3920) and X(17289)
X(68984) = trilinear pole of line {2483, 23885}
X(68984) = crossdifference of every pair of points on line {798, 3005}
X(68984) = barycentric product X(i)*X(j) for these {i,j}: {81, 33941}, {86, 17289}, {99, 47660}, {274, 3920}, {310, 5280}, {333, 7247}, {670, 2483}, {799, 830}, {873, 28594}, {4577, 23885}, {4596, 27610}, {4602, 8635}, {4610, 47711}, {5314, 44129}, {50496, 52612}
X(68984) = barycentric quotient X(i)/X(j) for these {i,j}: {662, 831}, {799, 57975}, {827, 58956}, {830, 661}, {2483, 512}, {3920, 37}, {5280, 42}, {5314, 71}, {7247, 226}, {7859, 4972}, {8635, 798}, {17289, 10}, {17672, 3925}, {23885, 826}, {27610, 30591}, {28594, 756}, {33941, 321}, {47660, 523}, {47711, 4024}, {50496, 4079}, {54308, 17108}
X(68984) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 17200, 33295}, {58, 30966, 34016}, {86, 274, 33955}, {86, 1010, 274}, {86, 5263, 30940}, {86, 33954, 1}, {11115, 16705, 99}, {17210, 52680, 6626}, {25526, 33953, 86}

X(68985) = X(1)X(75) INTERCEPT OF X(230)X(231)

Barycentrics   (b + c)*(a^4 + a^2*b^2 - 3*a^2*b*c + b^3*c + a^2*c^2 - 2*b^2*c^2 + b*c^3) : :

X(68985) lies on these lines: {1, 75}, {230, 231}, {614, 17874}, {714, 21254}, {1441, 40934}, {1716, 20930}, {1918, 4032}, {2228, 21232}, {3120, 68915}, {3122, 16609}, {3747, 8680}, {3778, 21231}, {4000, 23668}, {4022, 21233}, {4451, 41876}, {6007, 63443}, {10180, 29675}, {16732, 39688}, {18040, 59667}, {18589, 23663}, {23682, 44670}, {23688, 40941}, {25132, 66971}, {26228, 26274}, {27798, 29676}, {29639, 30748}, {52335, 68930}

X(68985) = midpoint of X(3747) and X(53559)
X(68985) = crossdifference of every pair of points on line {3, 798}

X(68986) = X(1)X(75) INTERCEPT OF X(23)X(385)

Barycentrics   (b + c)*(-a^4 - a^2*b^2 + 2*a^2*b*c - a^2*c^2 + b^2*c^2) : :
X(68986) = 3 X[46912] - 4 X[68935]

X(68986) lies on these lines: {1, 75}, {2, 68939}, {6, 25295}, {23, 385}, {37, 58365}, {190, 714}, {192, 4068}, {239, 44671}, {256, 29509}, {313, 40934}, {674, 40886}, {744, 53559}, {758, 24841}, {1201, 46898}, {1897, 68779}, {1918, 42027}, {1962, 32926}, {2388, 40859}, {3122, 4039}, {3303, 49453}, {3728, 17277}, {3891, 17318}, {3932, 26115}, {3943, 29822}, {3948, 39688}, {3963, 18082}, {4150, 23663}, {4383, 25294}, {4395, 29824}, {4436, 62636}, {8053, 17148}, {9053, 15985}, {9362, 17763}, {16672, 27811}, {17119, 17163}, {17164, 37542}, {17475, 68153}, {17962, 66283}, {18785, 41683}, {21295, 68955}, {21330, 29437}, {22174, 29399}, {24437, 29767}, {24575, 41681}, {26227, 26242}, {28309, 50180}, {29453, 63497}, {29558, 63515}, {29559, 63520}, {29823, 31077}, {41261, 58393}, {46910, 60737}, {52897, 53338}, {56185, 64169}

X(68986) = reflection of X(190) in X(3747)
X(68986) = anticomplement of X(68939)
X(68986) = anticomplement of the isotomic conjugate of X(2368)
X(68986) = X(2368)-anticomplementary conjugate of X(6327)
X(68986) = X(2368)-Ceva conjugate of X(2)
X(68986) = crossdifference of every pair of points on line {39, 798}
X(68986) = barycentric product X(304)*X(46573)
X(68986) = barycentric quotient X(46573)/X(19)

X(68987) = X(1)X(75) INTERCEPT OF X(187)X(237)

Barycentrics   a^2*(b + c)*(a^2*b^2 - a^2*b*c + a^2*c^2 - b^2*c^2) : :

X(68987) lies on these lines: {1, 75}, {2, 4116}, {10, 23629}, {31, 36873}, {42, 52959}, {187, 237}, {213, 6378}, {214, 38978}, {668, 3510}, {718, 35544}, {758, 4093}, {904, 5277}, {1015, 2388}, {1125, 4161}, {1911, 5291}, {2176, 20990}, {2230, 6381}, {2238, 52894}, {2295, 41267}, {2664, 4434}, {3216, 63506}, {3248, 45751}, {3725, 5269}, {3728, 7032}, {6196, 18140}, {9259, 23398}, {14963, 23651}, {16613, 51464}, {16829, 18170}, {16969, 38301}, {16971, 63504}, {20727, 23626}, {20963, 23532}, {27880, 52538}, {28594, 40936}, {29580, 62550}, {38986, 49997}, {40986, 66971}

X(68987) = midpoint of X(4093) and X(4128)
X(68987) = isogonal conjugate of X(18826)
X(68987) = isogonal conjugate of the anticomplement of X(65940)
X(68987) = isogonal conjugate of the isotomic conjugate of X(714)
X(68987) = X(i)-Ceva conjugate of X(j) for these (i,j): {715, 6}, {4607, 798}
X(68987) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18826}, {75, 715}
X(68987) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 18826}, {206, 715}, {714, 35532}, {65940, 76}
X(68987) = crosspoint of X(i) and X(j) for these (i,j): {1, 62763}, {6, 715}
X(68987) = crosssum of X(i) and X(j) for these (i,j): {1, 62755}, {2, 714}
X(68987) = crossdifference of every pair of points on line {2, 798}
X(68987) = X(4116)-line conjugate of X(2)
X(68987) = barycentric product X(i)*X(j) for these {i,j}: {1, 2229}, {6, 714}, {32, 35532}, {213, 62234}, {715, 65940}, {798, 53366}, {2238, 36817}
X(68987) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 18826}, {32, 715}, {714, 76}, {2229, 75}, {35532, 1502}, {36817, 40017}, {53366, 4602}, {62234, 6385}, {65940, 35532}
X(68987) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 40935, 23629}, {3747, 42669, 8625}, {20727, 23664, 23626}

X(68988) = X(1)X(75) INTERCEPT OF X(6)X(16710)

Barycentrics   (a + b)*(a + c)*(a^2*b + a*b^2 + a^2*c - 2*a*b*c - b^2*c + a*c^2 - b*c^2) : :

X(68988) lies on these lines: {1, 75}, {6, 16710}, {81, 39706}, {88, 16704}, {190, 52897}, {192, 29393}, {239, 16726}, {319, 24170}, {320, 17205}, {333, 18601}, {659, 3004}, {660, 18827}, {903, 16711}, {3759, 18164}, {4361, 17178}, {4389, 18600}, {4398, 17183}, {5224, 27162}, {7035, 68754}, {14829, 16700}, {16696, 17277}, {16714, 17273}, {16752, 27191}, {16887, 17271}, {17169, 17378}, {17197, 37756}, {17377, 20051}, {18133, 29454}, {18206, 53391}, {18822, 53649}, {19998, 30941}, {24530, 29509}, {25508, 27261}, {26860, 62212}, {26963, 34017}, {27164, 27311}, {29379, 46838}, {32911, 39747}, {35175, 65275}, {39995, 49997}

X(68988) = X(i)-Ceva conjugate of X(j) for these (i,j): {1509, 26844}, {4555, 7192}
X(68988) = X(26844)-cross conjugate of X(1509)
X(68988) = X(i)-isoconjugate of X(j) for these (i,j): {213, 39698}, {512, 53685}, {756, 59072}, {1918, 40039}
X(68988) = X(i)-Dao conjugate of X(j) for these (i,j): {4358, 3992}, {6626, 39698}, {34021, 40039}, {34587, 756}, {39054, 53685}, {68883, 3943}
X(68988) = cevapoint of X(17495) and X(49997)
X(68988) = crosssum of X(i) and X(j) for these (i,j): {42, 58292}, {872, 52963}
X(68988) = crossdifference of every pair of points on line {798, 1500}
X(68988) = barycentric product X(i)*X(j) for these {i,j}: {81, 39995}, {86, 17495}, {274, 49997}, {799, 68953}, {873, 68883}, {1509, 68895}, {23169, 44129}
X(68988) = barycentric quotient X(i)/X(j) for these {i,j}: {86, 39698}, {274, 40039}, {593, 59072}, {662, 53685}, {17495, 10}, {23169, 71}, {26844, 68895}, {34587, 3943}, {39995, 321}, {49997, 37}, {62571, 3992}, {68883, 756}, {68895, 594}, {68953, 661}
X(68988) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {86, 17160, 30939}, {16709, 54308, 86}, {30939, 62755, 17160}, {52897, 62636, 190}

X(68989) = X(1)X(75) INTERCEPT OF X(100)X(190)

Barycentrics   (a - b)*(a - c)*(a^2*b - a*b^2 + a^2*c + 2*a*b*c - b^2*c - a*c^2 - b*c^2) : :

X(68989) lies on these lines: {1, 75}, {8, 16506}, {42, 25382}, {100, 190}, {239, 16507}, {670, 53649}, {846, 42056}, {1026, 24004}, {2284, 30731}, {2805, 68871}, {3571, 9458}, {4033, 35338}, {4256, 6789}, {4418, 24345}, {4582, 60574}, {6558, 61197}, {14829, 34583}, {16482, 17277}, {18830, 53651}, {19945, 52908}, {23891, 62619}, {24346, 32929}, {24482, 56801}, {29423, 60785}, {32932, 67430}, {36872, 68966}, {38348, 68153}

X(68989) = X(889)-Ceva conjugate of X(190)
X(68989) = X(68896)-cross conjugate of X(29824)
X(68989) = X(i)-isoconjugate of X(j) for these (i,j): {244, 59071}, {604, 60575}
X(68989) = X(i)-Dao conjugate of X(j) for these (i,j): {899, 891}, {3161, 60575}, {29824, 68968}, {68938, 4728}
X(68989) = cevapoint of X(i) and X(j) for these (i,j): {899, 68968}, {29824, 68896}
X(68989) = crosspoint of X(99) and X(4607)
X(68989) = crosssum of X(i) and X(j) for these (i,j): {512, 3768}, {899, 14752}
X(68989) = trilinear pole of line {29824, 40614}
X(68989) = crossdifference of every pair of points on line {798, 1015}
X(68989) = barycentric product X(i)*X(j) for these {i,j}: {75, 68812}, {99, 68938}, {190, 29824}, {668, 45751}, {799, 44671}, {889, 40614}, {1016, 68896}
X(68989) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 60575}, {1252, 59071}, {29824, 514}, {40614, 891}, {44671, 661}, {45751, 513}, {68812, 1}, {68896, 1086}, {68938, 523}
X(68989) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4436, 53338, 190}, {4557, 61183, 190}, {17780, 53340, 23343}, {23343, 53340, 190}

X(68990) = X(1)X(75) INTERCEPT OF X(11)X(244)

Barycentrics   (b - c)^2*(-a^3 - a*b*c + b^2*c + b*c^2) : :

X(68990) lies on these lines: {1, 75}, {11, 244}, {31, 24346}, {58, 2607}, {81, 24345}, {320, 19955}, {321, 25382}, {518, 19961}, {741, 14616}, {876, 66284}, {942, 19938}, {1227, 17793}, {2643, 17197}, {3248, 4858}, {3836, 19964}, {3942, 21142}, {5883, 6788}, {9359, 18151}, {16704, 53341}, {16726, 53559}, {16732, 53541}, {17467, 18645}, {21105, 21135}, {21131, 21132}, {21140, 53546}, {21200, 21201}, {21210, 24237}, {23823, 24224}, {25005, 31337}, {39949, 42005}, {62619, 62626}

X(68990) = X(17763)-Ceva conjugate of X(2787)
X(68990) = X(i)-isoconjugate of X(j) for these (i,j): {59, 11609}, {100, 2703}, {101, 65239}, {692, 35147}, {765, 17954}, {1016, 17961}, {1252, 17946}, {3952, 17939}, {4557, 17929}, {5379, 57680}, {15742, 17971}
X(68990) = X(i)-Dao conjugate of X(j) for these (i,j): {513, 17954}, {661, 17946}, {1015, 65239}, {1086, 35147}, {2787, 17763}, {4988, 11611}, {6615, 11609}, {8054, 2703}, {35079, 190}
X(68990) = crosspoint of X(2787) and X(17763)
X(68990) = crosssum of X(2703) and X(17954)
X(68990) = crossdifference of every pair of points on line {101, 798}
X(68990) = barycentric product X(i)*X(j) for these {i,j}: {244, 17790}, {422, 4466}, {514, 2787}, {693, 68885}, {1019, 18003}, {1086, 17763}, {1111, 5291}, {3120, 19623}, {3125, 5209}, {3261, 5040}, {3942, 17987}, {4858, 5061}, {5006, 21207}, {7199, 17989}, {17205, 68897}
X(68990) = barycentric quotient X(i)/X(j) for these {i,j}: {244, 17946}, {513, 65239}, {514, 35147}, {649, 2703}, {1015, 17954}, {1019, 17929}, {2170, 11609}, {2680, 21801}, {2787, 190}, {3120, 11611}, {3248, 17961}, {4466, 57847}, {5006, 4570}, {5040, 101}, {5061, 4564}, {5209, 4601}, {5291, 765}, {17763, 1016}, {17790, 7035}, {17989, 1018}, {18003, 4033}, {19623, 4600}, {21132, 60484}, {35079, 17763}, {57129, 17939}, {57462, 2292}, {68885, 100}
X(68990) = {X(3942),X(23676)}-harmonic conjugate of X(21142)

X(68991) = X(1)X(75) INTERCEPT OF X(10)X(27705)

Barycentrics   b*c*(b + c)*(-2*a^3 - a^2*b + a*b^2 + b^3 - a^2*c + a*c^2 + c^3) : :
X(68991) = X[4647] - 4 X[35550]

X(68991) lies on these lines: {1, 75}, {10, 27705}, {35, 17797}, {238, 21254}, {350, 20437}, {523, 1577}, {726, 53559}, {758, 60452}, {846, 20929}, {1089, 3178}, {1109, 4358}, {1125, 21442}, {1962, 20896}, {2611, 3006}, {2643, 3836}, {3263, 49544}, {3821, 23928}, {3993, 20234}, {4062, 61410}, {4359, 62548}, {4645, 21295}, {4710, 20336}, {4934, 51417}, {6758, 32849}, {10026, 20685}, {16598, 32851}, {16741, 20903}, {17874, 29671}, {20236, 25650}, {20339, 62553}, {20360, 49676}, {20432, 28522}, {20488, 20546}, {20924, 32857}, {21020, 29655}, {25441, 25457}, {25645, 25660}, {30171, 68939}, {33064, 42066}, {33940, 68866}, {51370, 68858}

X(68991) = midpoint of X(4645) and X(21295)
X(68991) = reflection of X(i) in X(j) for these {i,j}: {238, 21254}, {2643, 3836}, {20360, 49676}
X(68991) = X(20947)-Ceva conjugate of X(3948)
X(68991) = X(i)-isoconjugate of X(j) for these (i,j): {58, 28482}, {110, 60050}, {1576, 60042}, {2206, 35162}
X(68991) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 28482}, {244, 60050}, {4858, 60042}, {10026, 1931}, {35114, 81}, {40603, 35162}, {41180, 1019}, {51578, 1}
X(68991) = crosssum of X(31) and X(64215)
X(68991) = crossdifference of every pair of points on line {798, 1333}
X(68991) = barycentric product X(i)*X(j) for these {i,j}: {75, 10026}, {310, 20685}, {321, 17770}, {561, 20666}, {1577, 62644}, {1969, 20754}, {4601, 65704}, {4647, 31064}
X(68991) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 28482}, {321, 35162}, {661, 60050}, {1577, 60042}, {10026, 1}, {17770, 81}, {20666, 31}, {20685, 42}, {20754, 48}, {31064, 40438}, {51578, 1931}, {62644, 662}, {65704, 3125}

X(68992) = X(1)X(75) INTERCEPT OF X(513)X(23822)

Barycentrics   (a + b)*(b - c)^2*(a + c)*(a^2 - b*c) : :

X(68992) lies on these lines: {1, 75}, {513, 23822}, {732, 3589}, {741, 65636}, {812, 1015}, {1111, 3248}, {1215, 16707}, {2284, 52897}, {4375, 27918}, {4472, 40548}, {4972, 17193}, {5701, 16696}, {7192, 47070}, {8042, 46051}, {8054, 40619}, {12264, 17200}, {16727, 63222}, {18191, 23824}, {19947, 23816}, {21208, 64523}, {26805, 26850}, {26813, 26964}, {27846, 27855}, {27922, 33295}, {40148, 40216}, {44312, 64643}

X(68992) = X(i)-complementary conjugate of X(j) for these (i,j): {82, 27854}, {733, 21051}, {875, 6292}, {876, 21248}, {3572, 21249}, {10566, 20542}, {18105, 46842}, {18108, 20333}, {18268, 3005}, {21755, 39079}, {36081, 27076}, {39276, 512}, {46289, 27929}, {55240, 45162}, {56242, 61063}, {67149, 25666}
X(68992) = X(i)-Ceva conugate of X(j) for these (i,j): {18827, 7192}, {30940, 812}, {32014, 27929}, {40834, 29402}, {61403, 35119}
X(68992) = X(35119)-cross conjugate of X(61403)
X(68992) = X(i)-isoconjugate of X(j) for these (i,j): {42, 5378}, {512, 65363}, {660, 4557}, {813, 1018}, {1110, 43534}, {3952, 34067}, {7077, 65573}, {8684, 62753}, {18268, 61402}, {41333, 57566}
X(68992) = X(i)-Dao conjugate of X(j) for these (i,j): {514, 43534}, {665, 3930}, {812, 740}, {16591, 65958}, {35068, 61402}, {35119, 3952}, {39054, 65363}, {40592, 5378}, {40620, 4562}, {40623, 1018}, {40625, 36801}, {62552, 10}, {62558, 37}
X(68992) = cevapoint of X(27846) and X(27918)
X(68992) = crosspoint of X(7192) and X(18827)
X(68992) = crosssum of X(i) and X(j) for these (i,j): {3747, 4557}, {3930, 35309}
X(68992) = crossdifference of every pair of points on line {798, 4557}
X(68992) = barycentric product X(i)*X(j) for these {i,j}: {86, 27918}, {238, 16727}, {239, 17205}, {244, 30940}, {274, 27846}, {350, 16726}, {659, 7199}, {693, 50456}, {740, 61403}, {812, 7192}, {873, 39786}, {874, 8042}, {1019, 3766}, {1086, 33295}, {1434, 4124}, {1447, 17197}, {1565, 31905}, {3716, 17096}, {3733, 65101}, {4444, 68156}, {4481, 63222}, {4560, 43041}, {4601, 24193}, {5009, 23989}, {8632, 52619}, {10030, 18191}, {15419, 65106}, {18827, 35119}, {23824, 39914}
X(68992) = barycentric quotient X(i)/X(j) for these {i,j}: {81, 5378}, {659, 1018}, {662, 65363}, {740, 61402}, {812, 3952}, {1019, 660}, {1086, 43534}, {1429, 65573}, {3716, 30730}, {3733, 813}, {3766, 4033}, {3808, 7239}, {4010, 4103}, {4124, 2321}, {4164, 61164}, {4435, 4069}, {4448, 4169}, {4481, 65210}, {4560, 36801}, {5009, 1252}, {6545, 35352}, {7192, 4562}, {7199, 4583}, {8042, 876}, {8632, 4557}, {16609, 65958}, {16726, 291}, {16727, 334}, {16742, 41531}, {17197, 4518}, {17205, 335}, {17925, 65338}, {18191, 4876}, {18827, 57566}, {21832, 40521}, {22384, 4574}, {23597, 4613}, {23824, 40848}, {24193, 3125}, {27846, 37}, {27918, 10}, {30940, 7035}, {31905, 15742}, {33295, 1016}, {35119, 740}, {38989, 3930}, {39179, 36081}, {39786, 756}, {43041, 4552}, {50456, 100}, {57129, 34067}, {61403, 18827}, {65101, 27808}, {68156, 3570}
X(68992) = {X(16707),X(17176)}-harmonic conjugate of X(1215)

X(68993) = X(1)X(75) INTERCEPT OF X(6)X(25106)

Barycentrics   a^3*b^2 - 2*a^3*b*c - a^2*b^2*c + a*b^3*c + a^3*c^2 - a^2*b*c^2 - 2*a*b^2*c^2 + b^3*c^2 + a*b*c^3 + b^2*c^3 : :

X(68993) lies on these lines: {1, 75}, {6, 25106}, {36, 4432}, {37, 24659}, {44, 24003}, {69, 25120}, {81, 25123}, {87, 20923}, {192, 24672}, {320, 17793}, {385, 5205}, {513, 3716}, {714, 16726}, {1215, 4670}, {2228, 40533}, {3122, 64909}, {3123, 27166}, {3271, 46843}, {3454, 34832}, {3728, 17178}, {3758, 37604}, {3844, 15985}, {3879, 59562}, {3943, 25382}, {3948, 53541}, {4358, 68884}, {4672, 25079}, {4675, 30982}, {4676, 37608}, {8424, 20849}, {9025, 20340}, {12263, 50116}, {17233, 59668}, {17392, 24327}, {18194, 30090}, {20532, 35079}, {20892, 63504}, {21257, 49537}, {22343, 29982}, {24653, 26106}, {25492, 25535}, {26093, 26143}, {26131, 26135}, {30947, 30998}, {30984, 30989}

X(68993) = midpoint of X(i) and X(j) for these {i,j}: {2234, 30939}, {3948, 53541}
X(68993) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 35070}, {35105, 2}, {35143, 141}
X(68993) = crossdifference of every pair of points on line {798, 2176}
X(68993) = {X(86),X(51575)}-harmonic conjugate of X(25124)

X(68994) = X(1)X(75) INTERCEPT OF X(76)X(63506)

Barycentrics   a*(a^3*b^3 - a^3*b^2*c - a^2*b^3*c - a^3*b*c^2 + a^3*c^3 - a^2*b*c^3 + 2*b^3*c^3) : :

X(68994) lies on these lines: {1, 75}, {76, 63506}, {194, 63518}, {330, 63527}, {350, 38986}, {649, 4083}, {893, 17599}, {1107, 23485}, {2227, 16742}, {17147, 23489}, {18135, 63480}, {24660, 27312}, {24662, 27318}, {25506, 25619}, {25864, 68944}

X(68994) = reflection of X(2227) in X(16742)
X(68994) = crosspoint of X(1) and X(18826)
X(68994) = crossdifference of every pair of points on line {43, 798}

X(68995) = X(1)X(75) INTERCEPT OF X(241)X(514)

Barycentrics   a^3*b + a^2*b^2 + a^3*c - 2*a^2*b*c - a*b^2*c + b^3*c + a^2*c^2 - a*b*c^2 - 2*b^2*c^2 + b*c^3 : :
X(68995) = 3 X[16711] + X[53332]

X(68995) lies on these lines: {1, 75}, {39, 25073}, {72, 24215}, {76, 25079}, {85, 978}, {238, 5088}, {241, 514}, {277, 1432}, {392, 3663}, {538, 3985}, {664, 60353}, {758, 17205}, {899, 30806}, {960, 24214}, {1015, 50025}, {1046, 1434}, {1111, 49997}, {1122, 10481}, {1201, 20880}, {1575, 21232}, {1722, 9312}, {1738, 67267}, {1959, 16752}, {2238, 7200}, {2292, 18600}, {2650, 17169}, {3212, 24174}, {3290, 46180}, {3293, 7278}, {3673, 21214}, {3674, 24178}, {3725, 16708}, {3729, 35274}, {3756, 24240}, {3761, 59511}, {4383, 7223}, {4695, 21272}, {4771, 8682}, {4920, 23536}, {5195, 24715}, {5222, 17497}, {5247, 7176}, {5291, 6647}, {5527, 61434}, {5529, 68928}, {6048, 16284}, {6381, 24003}, {7187, 16827}, {7208, 45751}, {8680, 52897}, {9317, 33854}, {9441, 62385}, {14949, 16912}, {16604, 17048}, {16705, 58386}, {16711, 53332}, {17084, 24161}, {17739, 33828}, {17946, 34578}, {24172, 52541}, {24203, 47623}, {24241, 64172}, {24628, 66152}, {25570, 59620}, {26563, 27627}, {26689, 56024}, {26978, 39244}, {27304, 27343}, {28253, 59181}, {29571, 30818}, {31183, 31205}, {35079, 35110}, {36226, 52959}, {44669, 68929}, {50114, 68941}

X(68995) = midpoint of X(i) and X(j) for these {i,j}: {2238, 7200}, {14210, 62755}
X(68995) = X(i)-complementary conjugate of X(j) for these (i,j): {32, 65929}, {6015, 141}
X(68995) = X(7)-Ceva conjugate of X(5579)
X(68995) = X(5579)-cross conjugate of X(7)
X(68995) = X(41)-isoconjugate of X(35176)
X(68995) = X(i)-Dao conjugate of X(j) for these (i,j): {3160, 35176}, {35130, 8}
X(68995) = crossdifference of every pair of points on line {55, 798}
X(68995) = barycentric product X(5579)*X(35176)
X(68995) = barycentric quotient X(7)/X(35176)
X(68995) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {274, 41805, 59509}, {274, 59509, 49598}, {1575, 49777, 21232}

X(68996) = X(1)X(75) INTERCEPT OF X(6)X(20236)

Barycentrics   b*c*(2*a^3 - a*b^2 + b^3 - b^2*c - a*c^2 - b*c^2 + c^3) : :

X(68996) lies on these lines: {1, 75}, {6, 20236}, {7, 2475}, {9, 20171}, {37, 25589}, {69, 17861}, {80, 21277}, {192, 25601}, {239, 1944}, {312, 41806}, {320, 1111}, {350, 33140}, {448, 19623}, {519, 3262}, {522, 693}, {524, 16732}, {536, 8609}, {758, 17139}, {903, 35156}, {1227, 39995}, {1229, 17353}, {1278, 27334}, {1441, 3879}, {1743, 20927}, {1959, 17197}, {1998, 40719}, {1999, 6358}, {2006, 37759}, {2321, 26665}, {2481, 2648}, {2995, 2997}, {3008, 37788}, {3011, 3263}, {3187, 14213}, {3210, 27339}, {3254, 7261}, {3260, 21207}, {3264, 4986}, {3663, 10916}, {3666, 20882}, {3672, 10527}, {3673, 17274}, {3729, 8557}, {3882, 16609}, {3936, 63334}, {3946, 26538}, {3950, 28974}, {4357, 5051}, {4389, 7264}, {4406, 49300}, {4416, 53510}, {4441, 11269}, {4452, 10529}, {4459, 6007}, {4957, 4969}, {4967, 24987}, {5231, 62697}, {5292, 44140}, {5709, 10444}, {5847, 23690}, {6703, 42708}, {7018, 29676}, {7235, 35104}, {8680, 18206}, {8755, 46108}, {8822, 54302}, {10446, 37625}, {14054, 54344}, {14206, 16704}, {14616, 65283}, {15149, 68779}, {15314, 21279}, {15936, 20880}, {16585, 25254}, {16586, 17495}, {16713, 25255}, {17135, 17884}, {17147, 20879}, {17156, 17871}, {17162, 17897}, {17321, 25598}, {17882, 44356}, {17885, 20930}, {17895, 30806}, {18151, 68966}, {18166, 63398}, {18805, 24425}, {18816, 35169}, {19788, 25527}, {20086, 30690}, {21413, 60691}, {22021, 29967}, {23521, 26563}, {23689, 49511}, {24179, 55391}, {24208, 53598}, {24329, 60722}, {24424, 62817}, {24474, 46704}, {24541, 24547}, {24773, 28738}, {24779, 28753}, {24780, 28755}, {24781, 37756}, {25940, 30885}, {26228, 31130}, {26234, 29639}, {28612, 37153}, {29640, 60706}, {29658, 33931}, {30699, 56367}, {38468, 62789}, {39126, 62780}, {39766, 68693}, {40950, 54314}, {43683, 67125}, {45700, 50101}, {48482, 64694}, {49627, 53594}, {56019, 62700}

X(68996) = midpoint of X(17139) and X(39765)
X(68996) = reflection of X(i) in X(j) for these {i,j}: {1959, 17197}, {3262, 24209}, {3882, 16609}, {44694, 68936}
X(68996) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {759, 329}, {1411, 2895}, {1412, 6224}, {2006, 1330}, {6740, 54113}, {14616, 21286}, {18815, 21287}, {24624, 3436}, {32675, 31290}, {34079, 144}, {52377, 65161}, {52380, 18750}, {57736, 56943}, {67166, 3177}, {68571, 68335}
X(68996) = X(38370)-cross conjugate of X(99)
X(68996) = X(i)-isoconjugate of X(j) for these (i,j): {32, 60251}, {163, 35354}, {512, 6083}, {1576, 66279}
X(68996) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 35354}, {4858, 66279}, {6376, 60251}, {35466, 758}, {39054, 6083}, {62694, 2341}, {65927, 9}
X(68996) = cevapoint of X(35466) and X(44669)
X(68996) = crosspoint of X(i) and X(j) for these (i,j): {75, 14616}, {7035, 35174}
X(68996) = crosssum of X(i) and X(j) for these (i,j): {31, 3724}, {3248, 8648}
X(68996) = crossdifference of every pair of points on line {41, 798}
X(68996) = barycentric product X(i)*X(j) for these {i,j}: {75, 35466}, {85, 44669}, {304, 1884}, {799, 6089}, {7018, 27970}
X(68996) = barycentric quotient X(i)/X(j) for these {i,j}: {75, 60251}, {523, 35354}, {662, 6083}, {1577, 66279}, {1884, 19}, {6089, 661}, {18593, 56648}, {27970, 171}, {35466, 1}, {44669, 9}
X(68996) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {75, 86, 18698}, {75, 314, 18697}, {75, 30939, 35550}, {75, 44735, 10436}, {16704, 62305, 14206}, {30939, 35550, 14210}

X(68997) = X(1)X(75) INTERCEPT OF X(8)X(24672)

Barycentrics   a^2*(a^2*b^2 + a*b^3 - a*b^2*c + a^2*c^2 - a*b*c^2 - 2*b^2*c^2 + a*c^3) : :
X(68997) = 2 X[1] + X[2234], 5 X[3616] - 2 X[68942]

X(68997) lies on these lines: {1, 75}, {8, 24672}, {10, 24659}, {42, 4682}, {44, 23579}, {56, 2305}, {81, 3725}, {87, 4676}, {106, 2703}, {238, 3248}, {244, 50362}, {292, 21788}, {511, 3122}, {513, 663}, {518, 3009}, {741, 36066}, {872, 4649}, {922, 5009}, {978, 3759}, {1001, 7032}, {1100, 1193}, {1125, 27042}, {1201, 1386}, {1431, 3445}, {1911, 9470}, {1918, 37609}, {1962, 17187}, {2076, 17962}, {2235, 20363}, {2275, 16525}, {2292, 16696}, {2309, 15569}, {3121, 3231}, {3216, 29559}, {3286, 3747}, {3616, 26107}, {3720, 37869}, {3724, 62740}, {3743, 52564}, {3783, 41851}, {3786, 24437}, {3915, 3941}, {4259, 63497}, {4974, 49997}, {6631, 60353}, {7184, 24723}, {7292, 17477}, {8610, 8679}, {9047, 23633}, {9454, 16782}, {10180, 18169}, {15254, 22343}, {15991, 49511}, {16610, 68749}, {16710, 17164}, {16726, 20718}, {16823, 18170}, {17053, 64006}, {17348, 24739}, {17469, 40148}, {17768, 53541}, {18192, 59624}, {18194, 21214}, {19861, 25895}, {21805, 68754}, {24429, 35016}, {24656, 28639}, {24663, 56805}, {28011, 28023}, {28082, 28087}, {28370, 63527}, {39046, 40623}, {46190, 64555}, {49490, 56800}, {54407, 57653}

X(68997) = midpoint of X(1) and X(18792)
X(68997) = reflection of X(2234) in X(18792)
X(68997) = X(i)-Ceva conjugate of X(j) for these (i,j): {16609, 1755}, {65258, 798}
X(68997) = X(i)-isoconjugate of X(j) for these (i,j): {8, 35108}, {55, 35159}, {512, 65635}
X(68997) = X(i)-Dao conjugate of X(j) for these (i,j): {223, 35159}, {35095, 312}, {39054, 65635}, {46842, 75}
X(68997) = crosspoint of X(1) and X(741)
X(68997) = crosssum of X(i) and X(j) for these (i,j): {1, 740}, {4155, 16613}
X(68997) = crossdifference of every pair of points on line {9, 798}
X(68997) = barycentric product X(i)*X(j) for these {i,j}: {57, 35104}, {662, 65873}, {741, 46842}
X(68997) = barycentric quotient X(i)/X(j) for these {i,j}: {57, 35159}, {604, 35108}, {662, 65635}, {35104, 312}, {46842, 35544}, {65873, 1577}
X(68997) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1740, 49470}, {1, 3736, 2667}, {1, 24325, 17445}, {292, 21788, 39258}, {1201, 63504, 1386}

X(68998) = X(1)X(75) INTERCEPT OF X(99)X(101)

Barycentrics   (a - b)*(a + b)*(a - c)*(a + c)*(a*b - b^2 + a*c - c^2) : :

X(68998) lies on these lines: {1, 75}, {99, 101}, {100, 4576}, {333, 9319}, {537, 68862}, {643, 4563}, {664, 670}, {668, 53649}, {673, 36800}, {799, 3699}, {883, 2283}, {1026, 55260}, {2284, 42720}, {3589, 24384}, {3935, 16741}, {4436, 16680}, {4467, 62643}, {4557, 61219}, {4573, 7256}, {4600, 55237}, {4997, 30992}, {6629, 49712}, {7035, 55262}, {7192, 17780}, {16751, 42723}, {17136, 53338}, {17245, 23897}, {18155, 55258}, {18827, 24841}, {18830, 65275}, {24505, 68450}, {27805, 29441}, {27949, 31059}, {30790, 31001}, {30941, 34230}, {32029, 37128}, {32850, 51370}, {36803, 55256}, {43290, 62530}, {53651, 65286}, {62644, 68153}

X(68998) = isotomic conjugate of the isogonal conjugate of X(54353)
X(68998) = X(4639)-Ceva conjugate of X(190)
X(68998) = X(i)-cross conjugate of X(j) for these (i,j): {4088, 3912}, {23829, 30941}, {42720, 55260}, {53553, 518}
X(68998) = X(i)-isoconjugate of X(j) for these (i,j): {6, 55261}, {25, 10099}, {37, 43929}, {42, 1027}, {65, 884}, {105, 512}, {213, 62635}, {294, 7180}, {513, 56853}, {523, 64216}, {647, 8751}, {649, 18785}, {661, 1438}, {666, 3121}, {667, 13576}, {669, 2481}, {673, 798}, {810, 36124}, {885, 1402}, {919, 3125}, {1024, 1400}, {1333, 66282}, {1416, 4041}, {1462, 3709}, {1576, 66290}, {1814, 2489}, {1924, 18031}, {2195, 4017}, {2501, 32658}, {3049, 54235}, {3063, 66941}, {3120, 32666}, {3122, 36086}, {4455, 52030}, {4516, 32735}, {4557, 43921}, {5377, 8034}, {7250, 28071}, {14942, 51641}, {18105, 46149}, {18191, 66930}, {21832, 51866}, {22383, 68565}, {23696, 57652}, {24290, 41934}, {29956, 40747}, {51987, 55259}, {52927, 53540}, {56783, 63461}
X(68998) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 55261}, {37, 66282}, {2238, 21832}, {3008, 53558}, {3912, 4010}, {4858, 66290}, {5375, 18785}, {6184, 661}, {6505, 10099}, {6626, 62635}, {6631, 13576}, {9428, 18031}, {10001, 66941}, {17755, 523}, {31998, 673}, {34961, 2195}, {35094, 3120}, {36830, 1438}, {36905, 7178}, {38980, 3125}, {38989, 3122}, {39026, 56853}, {39046, 512}, {39052, 8751}, {39054, 105}, {39062, 36124}, {39063, 4017}, {40582, 1024}, {40589, 43929}, {40592, 1027}, {40602, 884}, {40605, 885}, {40609, 4041}, {45250, 55257}, {62587, 1577}
X(68998) = cevapoint of X(i) and X(j) for these (i,j): {239, 53343}, {918, 4966}, {1026, 42720}, {3912, 4088}, {23829, 30941}
X(68998) = crosspoint of X(99) and X(4589)
X(68998) = crosssum of X(512) and X(4455)
X(68998) = trilinear pole of line {672, 3912}
X(68998) = crossdifference of every pair of points on line {798, 3122}
X(68998) = barycentric product X(i)*X(j) for these {i,j}: {1, 55260}, {76, 54353}, {86, 42720}, {99, 3912}, {100, 18157}, {190, 30941}, {241, 7257}, {274, 1026}, {304, 4238}, {310, 2284}, {314, 1025}, {333, 883}, {518, 799}, {643, 40704}, {645, 9436}, {662, 3263}, {668, 18206}, {670, 672}, {811, 25083}, {918, 4600}, {1016, 23829}, {1458, 62534}, {1818, 6331}, {1861, 4563}, {1876, 55207}, {1978, 3286}, {2223, 4602}, {2254, 4601}, {2283, 28660}, {2356, 52608}, {3693, 4625}, {3717, 4573}, {3930, 4623}, {3932, 4610}, {4088, 4590}, {4447, 7260}, {4561, 15149}, {4584, 64223}, {4589, 17755}, {4592, 46108}, {4609, 9454}, {4620, 50333}, {4632, 4966}, {4633, 4684}, {4634, 14439}, {4639, 8299}, {5089, 55202}, {6385, 54325}, {7256, 62786}, {7258, 34855}, {16728, 51560}, {18031, 68148}, {20683, 52612}, {20752, 57968}, {24037, 24290}, {34230, 55262}, {36819, 55258}, {51384, 55281}, {60735, 63743}
X(68998) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 55261}, {10, 66282}, {21, 1024}, {58, 43929}, {63, 10099}, {81, 1027}, {86, 62635}, {99, 673}, {100, 18785}, {101, 56853}, {110, 1438}, {162, 8751}, {163, 64216}, {190, 13576}, {241, 4017}, {284, 884}, {333, 885}, {518, 661}, {643, 294}, {645, 14942}, {648, 36124}, {662, 105}, {664, 66941}, {665, 3122}, {670, 18031}, {672, 512}, {799, 2481}, {811, 54235}, {883, 226}, {918, 3120}, {1019, 43921}, {1025, 65}, {1026, 37}, {1043, 28132}, {1414, 1462}, {1434, 43930}, {1458, 7180}, {1577, 66290}, {1812, 23696}, {1818, 647}, {1861, 2501}, {1876, 55208}, {1897, 68565}, {2223, 798}, {2254, 3125}, {2283, 1400}, {2284, 42}, {2340, 3709}, {2356, 2489}, {3263, 1577}, {3286, 649}, {3693, 4041}, {3717, 3700}, {3736, 29956}, {3912, 523}, {3930, 4705}, {3932, 4024}, {4088, 115}, {4238, 19}, {4437, 4088}, {4447, 57234}, {4558, 36057}, {4563, 31637}, {4565, 1416}, {4567, 36086}, {4570, 919}, {4573, 56783}, {4575, 32658}, {4584, 52030}, {4589, 52209}, {4592, 1814}, {4600, 666}, {4601, 51560}, {4620, 927}, {4625, 34018}, {4639, 67197}, {4684, 4841}, {4712, 24290}, {4899, 14321}, {4925, 21950}, {4966, 4988}, {5546, 2195}, {7256, 6559}, {7257, 36796}, {7259, 28071}, {8299, 21832}, {9436, 7178}, {9454, 669}, {9455, 1924}, {14439, 4730}, {15149, 7649}, {16593, 53558}, {16728, 2254}, {17755, 4010}, {18157, 693}, {18206, 513}, {20683, 4079}, {20752, 810}, {23829, 1086}, {24290, 2643}, {25083, 656}, {30941, 514}, {34230, 55263}, {34855, 7216}, {36819, 55259}, {39258, 50487}, {39775, 7212}, {40217, 35352}, {40704, 4077}, {40883, 21052}, {41353, 1427}, {42720, 10}, {43042, 53545}, {46108, 24006}, {50333, 21044}, {51384, 55282}, {51400, 48403}, {52378, 32735}, {52635, 51641}, {53544, 53540}, {53553, 16592}, {53554, 20982}, {54325, 213}, {54353, 6}, {54407, 6591}, {55194, 39293}, {55260, 75}, {60735, 63221}, {63743, 60677}, {64828, 54364}, {65166, 14625}, {68086, 2356}, {68106, 3930}, {68128, 20683}, {68148, 672}, {68813, 4516}

X(68999) = X(1)X(75) INTERCEPT OF X(10)X(190)

Barycentrics   a^3 + a*b^2 + 3*a*b*c + 3*b^2*c + a*c^2 + 3*b*c^2 : :
X(68999) = 2 X[1] - 3 X[86], 5 X[1] - 6 X[5625], X[1] - 3 X[24342], 5 X[86] - 4 X[5625], 2 X[5625] - 5 X[24342], 4 X[10] - 3 X[31144], 2 X[24697] - 3 X[31144], 6 X[1213] - 7 X[9780], 7 X[9780] - 3 X[9791], 3 X[1654] - 5 X[3617], 5 X[3617] - 6 X[4733], 4 X[3634] - 3 X[25354], 16 X[3634] - 15 X[31248], 4 X[25354] - 5 X[31248], 4 X[3579] - 3 X[63402], X[3621] + 3 X[20090], 4 X[3626] - 3 X[42334], 11 X[5550] - 12 X[6707], X[20050] - 6 X[63401]

X(68999) lies on these lines: {1, 75}, {2, 3712}, {8, 524}, {10, 190}, {21, 4436}, {45, 1213}, {81, 17162}, {100, 31025}, {105, 31077}, {144, 1654}, {171, 55095}, {191, 16568}, {310, 18075}, {319, 50307}, {321, 5297}, {333, 896}, {516, 4967}, {518, 17116}, {536, 16830}, {594, 4645}, {596, 55930}, {612, 42029}, {673, 6651}, {846, 25999}, {894, 3696}, {966, 24280}, {1001, 4699}, {1125, 4693}, {1215, 5524}, {1220, 34916}, {1222, 7312}, {1266, 19868}, {1268, 50298}, {1279, 4739}, {1281, 26244}, {1376, 11688}, {1386, 17117}, {1738, 3634}, {1757, 4732}, {1821, 37138}, {1962, 25507}, {2292, 24438}, {2783, 6998}, {3120, 30832}, {3416, 48628}, {3434, 19825}, {3579, 63402}, {3616, 17395}, {3619, 7613}, {3621, 20090}, {3625, 4924}, {3626, 17770}, {3661, 5880}, {3679, 28558}, {3685, 3739}, {3704, 26051}, {3706, 38473}, {3741, 18201}, {3743, 14007}, {3775, 17273}, {3786, 20718}, {3821, 17307}, {3826, 17280}, {3836, 17285}, {3914, 19808}, {3920, 4980}, {3923, 17277}, {3936, 46918}, {3980, 14829}, {3996, 32771}, {4000, 5550}, {4026, 28604}, {4046, 17778}, {4133, 17315}, {4234, 54335}, {4307, 42696}, {4312, 17270}, {4331, 33298}, {4359, 7292}, {4365, 34064}, {4384, 4676}, {4388, 65698}, {4427, 5235}, {4431, 64174}, {4495, 20888}, {4514, 19833}, {4527, 50299}, {4649, 4709}, {4655, 17271}, {4659, 49447}, {4664, 39586}, {4670, 49468}, {4672, 50096}, {4683, 8013}, {4688, 16823}, {4689, 31993}, {4697, 41629}, {4716, 33682}, {4740, 49453}, {4835, 56174}, {4886, 41011}, {4966, 26806}, {5189, 33110}, {5224, 24248}, {5241, 17777}, {5253, 15571}, {5268, 42034}, {5271, 36277}, {5333, 27804}, {5564, 5847}, {5699, 21898}, {5700, 21869}, {5793, 24451}, {5988, 30761}, {5992, 31090}, {6536, 41817}, {6650, 26582}, {7229, 59406}, {7321, 49511}, {11246, 37653}, {12702, 46704}, {13740, 28612}, {13741, 28611}, {14005, 64071}, {16477, 20179}, {16706, 19862}, {16816, 20142}, {16824, 50054}, {16826, 49462}, {17118, 24349}, {17164, 53338}, {17258, 28526}, {17286, 38052}, {17297, 49560}, {17305, 33149}, {17319, 28484}, {17346, 24695}, {17351, 60731}, {17371, 19872}, {17379, 49486}, {17491, 31143}, {17731, 59261}, {19822, 32773}, {19853, 50044}, {20050, 63401}, {20292, 56810}, {20884, 23555}, {21027, 33115}, {21085, 33097}, {21949, 50052}, {22184, 23577}, {24589, 25531}, {24693, 29674}, {24841, 49483}, {27081, 44006}, {27191, 29637}, {27785, 56991}, {28546, 51297}, {28605, 32926}, {28634, 64016}, {28639, 49461}, {28653, 50290}, {29595, 31308}, {29596, 62675}, {29617, 67964}, {29641, 50048}, {30818, 48443}, {30861, 61158}, {30970, 32845}, {31178, 49458}, {31330, 32939}, {31359, 55923}, {32025, 33082}, {32087, 51192}, {32860, 67211}, {32917, 65166}, {32934, 59312}, {32936, 59306}, {33100, 41809}, {33135, 59628}, {33160, 41878}, {36480, 49493}, {36531, 49456}, {39581, 49746}, {39708, 55927}, {39766, 51669}, {40875, 68980}, {41814, 63366}, {42289, 55096}, {46922, 49488}, {47272, 67570}, {47687, 62635}, {48802, 49722}, {49451, 51055}, {49515, 68870}, {49721, 53620}, {50015, 62225}, {50995, 59412}, {55955, 64908}

X(68999) = reflection of X(i) in X(j) for these {i,j}: {86, 24342}, {1654, 4733}, {9791, 1213}, {24697, 10}
X(68999) = barycentric product X(i)*X(j) for these {i,j}: {75, 37675}, {190, 47834}
X(68999) = barycentric quotient X(i)/X(j) for these {i,j}: {37675, 1}, {47834, 514}
X(68999) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 24697, 31144}, {75, 1966, 60735}, {75, 5263, 32922}, {75, 50314, 5263}, {171, 62226, 55095}, {3775, 32857, 17273}, {4418, 21020, 333}, {4427, 27812, 5235}, {4683, 8013, 41816}, {4688, 49484, 16823}, {28604, 62392, 4026}, {33082, 50312, 32025}, {49474, 50302, 4360}

X(69000) = X(1)X(24484)∩X(4)X(9)

Barycentrics   a*(a - b)*(a - c)*(a^2*b + b^3 + a^2*c - 2*a*b*c - b^2*c - b*c^2 + c^3) : :

X(69000) lies on these lines: {1, 24484}, {3, 56381}, {4, 9}, {100, 28847}, {101, 1292}, {190, 646}, {513, 2284}, {517, 17435}, {644, 3888}, {650, 23832}, {661, 68825}, {813, 927}, {883, 28846}, {1023, 54328}, {1024, 36086}, {1252, 13589}, {1738, 14267}, {1756, 39258}, {2082, 38503}, {2176, 24289}, {3294, 24723}, {3662, 27093}, {4169, 32212}, {4419, 68917}, {4427, 61173}, {4676, 16549}, {6011, 59063}, {6184, 15507}, {6603, 63390}, {6646, 26754}, {7239, 61223}, {8750, 61236}, {16283, 49127}, {18142, 29506}, {18785, 24715}, {19515, 46101}, {20482, 47104}, {21856, 37619}, {23703, 41405}, {26705, 29014}, {29377, 29382}, {29434, 29460}, {29693, 29698}, {35310, 53280}, {38531, 42316}, {45751, 49709}, {48269, 56881}, {68127, 68141}, {68128, 68144}

X(69000) = reflection of X(56381) in X(3)
X(69000) = X(i)-isoconjugate of X(j) for these (i,j): {513, 2991}, {665, 57754}, {905, 15344}, {918, 15382}, {1015, 35574}, {3669, 56111}, {34159, 62635}
X(69000) = X(i)-Dao conjugate of X(j) for these (i,j): {120, 514}, {39026, 2991}, {65924, 4025}
X(69000) = crosspoint of X(i) and X(j) for these (i,j): {190, 36086}, {673, 46972}
X(69000) = crosssum of X(i) and X(j) for these (i,j): {513, 4435}, {649, 2254}, {672, 3722}
X(69000) = trilinear pole of line {3290, 17464}
X(69000) = crossdifference of every pair of points on line {1459, 3248}
X(69000) = barycentric product X(i)*X(j) for these {i,j}: {1, 53358}, {10, 4236}, {100, 1738}, {120, 36086}, {190, 3290}, {662, 21956}, {666, 17464}, {765, 23770}, {919, 20431}, {1018, 16752}, {1026, 14267}, {1897, 34381}, {5377, 20504}, {20455, 51560}
X(69000) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 2991}, {765, 35574}, {1738, 693}, {3290, 514}, {3939, 56111}, {4236, 86}, {8750, 15344}, {16752, 7199}, {17464, 918}, {20455, 2254}, {20504, 62429}, {20702, 4088}, {21956, 1577}, {23770, 1111}, {32666, 15382}, {34381, 4025}, {36086, 57754}, {53358, 75}, {54325, 34159}

X(69001) = X(75)X(29479)∩X(83)X(28598)

Barycentrics   (a - b)*(a - c)*(a^2 + b^2 + b*c + c^2) : :

X(68001) lies on these lines: {75, 29479}, {83, 28598}, {99, 42720}, {101, 3807}, {190, 646}, {644, 33948}, {666, 6540}, {1016, 53332}, {1783, 65204}, {3570, 4103}, {3573, 3952}, {3729, 33934}, {3954, 17741}, {4115, 32094}, {4482, 33946}, {4568, 18047}, {4737, 17336}, {5291, 33889}, {5525, 20947}, {8707, 29143}, {17742, 33932}, {17744, 33938}, {21604, 29420}, {23343, 23861}, {25468, 28604}, {35177, 58129}

X(69001) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {765, 4B}, {1110, 51860}
X(69001) = X(i)-cross conjugate of X(j) for these (i,j): {2483, 3920}, {47660, 17289}
X(69001) = X(i)-isoconjugate of X(j) for these (i,j): {831, 1015}, {1977, 57975}
X(69001) = X(i)-Dao conjugate of X(j) for these (i,j): {17384, 47916}, {28594, 7927}
X(69001) = cevapoint of X(i) and X(j) for these (i,j): {830, 28594}, {2483, 3920}, {7927, 68934}, {17289, 47660}
X(69001) = crosspoint of X(799) and X(35137)
X(69001) = crosssum of X(798) and X(8664)
X(69001) = trilinear pole of line {3920, 17289}
X(69001) = barycentric product X(i)*X(j) for these {i,j}: {100, 33941}, {190, 17289}, {668, 3920}, {799, 28594}, {830, 7035}, {1016, 47660}, {1978, 5280}, {2483, 31625}, {3699, 7247}, {4538, 4625}, {4600, 47711}
X(69001) = barycentric quotient X(i)/X(j) for these {i,j}: {765, 831}, {830, 244}, {2483, 1015}, {3882, 17108}, {3920, 513}, {4538, 4041}, {5280, 649}, {5314, 1459}, {7035, 57975}, {7247, 3676}, {7859, 47652}, {8635, 3248}, {17289, 514}, {17672, 21104}, {28594, 661}, {33941, 693}, {47660, 1086}, {47711, 3120}, {50496, 3122}
X(69001) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {190, 668, 33951}, {190, 4595, 33952}

X(69002) = X(8)X(7878)∩X(76)X(2345)

Barycentrics   (a^2 + b^2 + b*c + c^2)^2 : :

X(69002) lies on these lines: {8, 7878}, {76, 2345}, {83, 594}, {595, 17354}, {1015, 39722}, {1086, 10159}, {1500, 17280}, {3096, 17293}, {3661, 7768}, {3743, 17264}, {4385, 7827}, {4445, 7877}, {5222, 43527}, {5749, 7894}, {7760, 17369}, {7762, 48636}, {7770, 61321}, {7858, 30179}, {7859, 17289}, {32027, 48635}, {52570, 56186}

X(69002) = X(830)-Dao conjugate of X(1015)
X(69002) = barycentric product X(i)*X(j) for these {i,j}: {668, 68113}, {3920, 33941}, {17289, 17289}
X(69002) = barycentric quotient X(68113)/X(513)
X(69002) = {X(17289),X(33941)}-harmonic conjugate of X(7859)

X(69003) = X(1)X(24333)∩X(6)X(49777)

Barycentrics   a^4 + a^2*b^2 - 3*a^2*b*c + b^3*c + a^2*c^2 - 2*b^2*c^2 + b*c^3 : :

X(69003) lies on these lines: {1, 24333}, {6, 49777}, {7, 1572}, {32, 7176}, {36, 7208}, {81, 239}, {85, 16502}, {86, 49755}, {238, 35102}, {242, 514}, {278, 28039}, {519, 4349}, {538, 3685}, {894, 10027}, {1015, 1447}, {1056, 4644}, {1111, 16784}, {1146, 68929}, {1149, 9318}, {1279, 44664}, {1914, 5088}, {2345, 49773}, {3008, 5437}, {3230, 10025}, {3290, 3732}, {3496, 24215}, {3673, 16781}, {3912, 5712}, {3945, 50027}, {5011, 17205}, {5280, 7278}, {7754, 18156}, {9312, 16780}, {10436, 49774}, {16486, 24352}, {16720, 68889}, {16779, 24249}, {20228, 41246}, {24203, 62370}, {24699, 25432}, {28089, 53994}, {30806, 33854}, {36226, 40872}, {36854, 54406}, {43040, 45126}, {51384, 57019}, {52963, 60960}

X(69003) = crossdifference of every pair of points on line {71, 50487}
X(69003) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1914, 7200, 5088}, {50025, 50028, 239}

X(69004) = X(2)X(4491)∩X(86)X(8632)

Barycentrics   (b - c)*(-a^4 - a^2*b^2 + 2*a^2*b*c - a^2*c^2 + b^2*c^2) : :
X(69004) = 2 X[23650] - 3 X[68966]

X(69004) lies on these lines: {2, 4491}, {86, 8632}, {141, 21303}, {190, 4598}, {239, 68953}, {513, 24601}, {649, 4406}, {659, 27927}, {673, 1027}, {812, 1019}, {900, 4784}, {903, 18825}, {1086, 1977}, {2827, 6996}, {3716, 29489}, {3768, 17277}, {4010, 29809}, {4107, 21143}, {4374, 10566}, {4384, 53392}, {6084, 17496}, {14621, 59488}, {17302, 68956}, {17305, 24721}, {17307, 21261}, {21211, 68885}, {21606, 62324}, {23650, 68966}, {29546, 48032}, {41245, 53528}

X(69004) = crosspoint of X(799) and X(2368)
X(69004) = crosssum of X(798) and X(2388)
X(69004) = crossdifference of every pair of points on line {872, 38986}

X(69005) = X(2)X(6)∩X(75)X(17052)

Barycentrics   (b + c)*(-(a^2*b^2) + b^4 + a^2*b*c - b^3*c - a^2*c^2 + b^2*c^2 - b*c^3 + c^4) : :

X(69005) lies on these lines: {2, 6}, {75, 17052}, {190, 857}, {307, 4150}, {313, 20305}, {579, 18744}, {714, 21256}, {824, 1577}, {1631, 21275}, {2887, 46910}, {3454, 17227}, {3662, 21245}, {3778, 21235}, {3948, 8287}, {4389, 26601}, {4398, 53417}, {4466, 35550}, {5051, 17305}, {6631, 16086}, {8626, 33911}, {8680, 21094}, {16607, 40071}, {19308, 25469}, {20891, 21236}, {21237, 52043}, {21252, 68951}, {21287, 36022}, {27052, 32939}, {27487, 51417}, {31043, 42697}, {35552, 63819}, {37756, 68946}, {44150, 57435}

X(69005) = isotomic conjugate of X(60134)
X(69005) = isotomic conjugate of the isogonal conjugate of X(14963)
X(69005) = X(43362)-complementary conjugate of X(4369)
X(69005) = X(i)-isoconjugate of X(j) for these (i,j): {31, 60134}, {560, 37219}
X(69005) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 60134}, {6374, 37219}, {14963, 46513}
X(69005) = crossdifference of every pair of points on line {512, 560}
X(69005) = barycentric product X(76)*X(14963)
X(69005) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 60134}, {76, 37219}, {14963, 6}
X(69005) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 69, 30882}, {2, 193, 30902}, {2, 1654, 30906}, {2, 2895, 30905}, {2, 31034, 30937}

X(69006) = X(39)X(23656)∩X(649)X(891)

Barycentrics   a^2*(b - c)*(a^2*b^2 - a^2*b*c + a^2*c^2 - b^2*c^2) : :

X(69006) lies on these lines: {39, 23656}, {213, 23569}, {291, 27810}, {649, 891}, {659, 3249}, {667, 6373}, {668, 4598}, {812, 1019}, {1015, 1977}, {1960, 68886}, {3227, 53641}, {3768, 23892}, {3960, 46387}, {4607, 57572}, {8634, 22093}, {20963, 23458}, {23866, 33863}

X(69006) = X(23349)-Ceva conjugate of X(649)
X(69006) = X(i)-isoconjugate of X(j) for these (i,j): {715, 4033}, {1018, 18826}
X(69006) = X(65940)-Dao conjugate of X(27808)
X(69006) = crosspoint of X(i) and X(j) for these (i,j): {6, 4607}, {1019, 23892}, {4598, 37129}
X(69006) = crosssum of X(i) and X(j) for these (i,j): {2, 3768}, {899, 20979}, {1018, 23891}
X(69006) = crossdifference of every pair of points on line {192, 872}
X(69006) = barycentric product X(i)*X(j) for these {i,j}: {659, 36817}, {667, 62234}, {714, 3733}, {1019, 2229}, {3248, 53366}, {23349, 52882}
X(69006) = barycentric quotient X(i)/X(j) for these {i,j}: {714, 27808}, {2229, 4033}, {3733, 18826}, {36817, 4583}, {62234, 6386}
X(69006) = {X(667),X(23572)}-harmonic conjugate of X(21763)

X(69007) = X(1)X(24517)∩X(2)X(6)

Barycentrics   (a - b)*(a - c)*(a^2*b + a*b^2 + a^2*c - 2*a*b*c - b^2*c + a*c^2 - b*c^2) : :

X(69007) lies on these lines: {1, 24517}, {2, 6}, {100, 53637}, {190, 646}, {662, 37205}, {894, 59715}, {1043, 13744}, {1227, 51381}, {2397, 46779}, {3699, 3909}, {4360, 57023}, {4598, 4604}, {4638, 62536}, {16552, 24493}, {17160, 52900}, {17499, 36275}, {17780, 23345}, {18150, 29483}, {23343, 23354}, {25272, 52609}, {35147, 35148}, {43290, 61220}, {54986, 65275}, {62231, 68967}, {68873, 68887}

X(69007) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1018, 66862}, {1110, 30579}, {5376, 17135}, {6551, 7192}, {6635, 17217}, {9268, 75}, {32665, 17154}, {42372, 53363}, {62536, 17137}
X(69007) = X(3257)-Ceva conjugate of X(190)
X(69007) = X(i)-isoconjugate of X(j) for these (i,j): {661, 59072}, {667, 39698}, {1015, 53685}, {1919, 40039}, {4491, 60810}
X(69007) = X(i)-Dao conjugate of X(j) for these (i,j): {4358, 3762}, {6631, 39698}, {9296, 40039}, {17495, 59737}, {34587, 661}, {36830, 59072}, {49997, 21894}, {68883, 900}
X(69007) = cevapoint of X(68883) and X(68953)
X(69007) = crosspoint of X(i) and X(j) for these (i,j): {799, 4555}, {4590, 6635}
X(69007) = crosssum of X(i) and X(j) for these (i,j): {798, 1960}, {3124, 8661}
X(69007) = trilinear pole of line {17495, 34587}
X(69007) = crossdifference of every pair of points on line {512, 3248}
X(69007) = barycentric product X(i)*X(j) for these {i,j}: {99, 68895}, {100, 39995}, {190, 17495}, {668, 49997}, {799, 68883}, {3257, 62571}, {4555, 34587}, {7035, 68953}
X(69007) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 59072}, {190, 39698}, {668, 40039}, {765, 53685}, {17495, 514}, {23169, 1459}, {34587, 900}, {39995, 693}, {49997, 513}, {62571, 3762}, {67401, 68968}, {68883, 661}, {68895, 523}, {68953, 244}
X(69007) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 29695, 29379}, {69, 29395, 29437}, {69, 29711, 29395}, {86, 29714, 29399}, {193, 29507, 29453}, {1654, 29509, 17277}, {2895, 29508, 14829}, {3882, 65161, 190}, {4033, 21362, 190}, {18150, 53391, 29483}

X(69008) = X(2)X(6)∩X(75)X(18164)

Barycentrics   (a + b)*(a + c)*(a^2*b - a*b^2 + a^2*c + 2*a*b*c - b^2*c - a*c^2 - b*c^2) : :

X(69008) lies on these lines: {2, 6}, {75, 18164}, {190, 18206}, {239, 16726}, {320, 17197}, {812, 1019}, {3227, 62755}, {3257, 65264}, {3286, 54391}, {3875, 18186}, {4360, 16696}, {4361, 16710}, {4562, 37128}, {16709, 17207}, {16723, 17310}, {16887, 17305}, {17143, 39950}, {17160, 18198}, {17183, 17347}, {17202, 17273}, {17205, 37756}, {18143, 34017}, {18150, 29561}, {18171, 33296}, {18172, 34063}, {18792, 31855}, {24517, 38302}, {29456, 65161}, {29769, 39995}, {52564, 56018}

X(69008) = X(4607)-Ceva conjugate of X(1019)
X(69008) = X(661)-isoconjugate of X(59071)
X(69008) = X(i)-Dao conjugate of X(j) for these (i,j): {899, 52959}, {36830, 59071}, {40625, 60575}, {68938, 3994}
X(69008) = cevapoint of X(29824) and X(45751)
X(69008) = crossdifference of every pair of points on line {512, 872}
X(69008) = barycentric product X(i)*X(j) for these {i,j}: {86, 29824}, {99, 68896}, {274, 45751}, {873, 44671}, {1509, 68938}, {7199, 68812}
X(69008) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 59071}, {4560, 60575}, {29824, 10}, {40614, 52959}, {44671, 756}, {45751, 37}, {68812, 1018}, {68896, 523}, {68938, 594}
X(69008) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 17178, 86}, {6, 29437, 17277}, {6, 29746, 29437}, {69, 29453, 29379}, {69, 29763, 29453}, {81, 86, 46922}, {81, 29766, 14829}, {86, 29767, 17277}, {86, 68966, 52897}, {193, 29558, 29395}, {1654, 29559, 29399}, {14829, 46922, 17277}, {16704, 52897, 68966}, {16738, 18166, 86}, {18206, 30939, 190}

X(69009) = X(2)X(6)∩X(11)X(244)

Barycentrics   (b - c)^2*(-a^2 - a*b + b^2 - a*c + b*c + c^2) : :

X(69009) lies on these lines: {2, 6}, {10, 19895}, {11, 244}, {37, 24318}, {106, 10708}, {115, 17205}, {116, 1015}, {121, 13466}, {150, 9259}, {274, 23897}, {291, 20531}, {310, 23918}, {320, 27912}, {514, 23816}, {528, 58863}, {544, 8649}, {1506, 17758}, {1565, 21138}, {1737, 49777}, {1834, 37165}, {2087, 45213}, {2886, 24396}, {3124, 17198}, {3664, 25342}, {3726, 33864}, {3943, 42720}, {4023, 9458}, {4089, 52626}, {4364, 40878}, {4675, 17717}, {4688, 21242}, {4966, 52908}, {5121, 50011}, {5205, 49752}, {5988, 9507}, {6543, 18827}, {6545, 21131}, {6546, 21135}, {6547, 68967}, {6548, 62626}, {6627, 8287}, {6745, 49776}, {6788, 24281}, {7113, 27943}, {7200, 17213}, {7238, 27922}, {7261, 17962}, {7813, 68938}, {9318, 17365}, {9651, 62425}, {9665, 14377}, {13589, 15447}, {16604, 17046}, {16613, 16742}, {16705, 23905}, {16748, 23917}, {16786, 31210}, {16969, 33298}, {17055, 29655}, {17058, 17197}, {17175, 62680}, {17181, 20271}, {17237, 25377}, {17246, 24403}, {17276, 24398}, {17301, 24217}, {17374, 62630}, {17390, 68875}, {17669, 33947}, {17768, 24428}, {18184, 20982}, {18600, 23903}, {19974, 20255}, {20490, 20548}, {21200, 21202}, {21208, 58898}, {23947, 62636}, {24384, 33954}, {24918, 35342}, {24982, 59524}, {25350, 26590}, {25381, 39786}, {25383, 62675}, {27920, 41851}, {27929, 41180}, {28602, 35080}, {30790, 32029}, {31001, 37128}, {33297, 68450}, {33816, 59627}, {35094, 35119}, {35131, 35134}, {36230, 46914}, {37009, 64159}, {46526, 53417}, {53426, 62755}

X(69009) = complement of X(3570)
X(69009) = complement of the isogonal conjugate of X(3572)
X(69009) = complement of the isotomic conjugate of X(4444)
X(69009) = medial-isogonal conjugate of X(27854)
X(69009) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 27854}, {31, 27929}, {291, 3835}, {292, 513}, {334, 21262}, {335, 21260}, {512, 46842}, {513, 20333}, {514, 20542}, {649, 17793}, {660, 27076}, {661, 45162}, {667, 17755}, {694, 21051}, {741, 4369}, {798, 35068}, {805, 40546}, {813, 24003}, {875, 2}, {876, 141}, {882, 46826}, {1015, 38989}, {1911, 514}, {1922, 650}, {1967, 25666}, {2196, 20315}, {3248, 35119}, {3572, 10}, {4367, 39080}, {4444, 2887}, {4876, 59971}, {7077, 20317}, {7180, 50440}, {9468, 3709}, {14598, 6586}, {16592, 2679}, {18268, 523}, {18827, 42327}, {18897, 52589}, {20981, 19563}, {21755, 35078}, {30663, 21261}, {34067, 4422}, {35352, 21245}, {36806, 3037}, {37128, 512}, {39276, 68787}, {40017, 23301}, {40730, 62552}, {51856, 665}, {51858, 4521}, {51866, 3716}, {52205, 3837}, {56242, 5976}, {57738, 52598}, {60577, 21244}, {63874, 68900}, {65285, 36950}, {65352, 21259}, {66286, 626}, {66937, 3739}, {66938, 18589}, {68631, 20305}
X(69009) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 27929}, {239, 918}, {310, 20511}, {6542, 2786}, {6650, 514}, {7261, 513}, {18827, 523}, {20947, 18004}, {32014, 68168}, {55025, 512}
X(69009) = X(41180)-cross conjugate of X(3120)
X(69009) = X(i)-isoconjugate of X(j) for these (i,j): {100, 2702}, {101, 37135}, {163, 66283}, {692, 35148}, {765, 17962}, {1018, 17940}, {1101, 6543}, {1110, 6650}, {1252, 1929}, {2054, 4567}, {4570, 9278}, {5379, 57681}, {18032, 23990}, {18266, 57560}
X(69009) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 66283}, {513, 17962}, {514, 6650}, {523, 6543}, {661, 1929}, {1015, 37135}, {1086, 35148}, {2786, 6542}, {4988, 11599}, {8054, 2702}, {9508, 2238}, {27929, 2}, {35080, 190}, {39041, 765}, {39042, 4567}, {40620, 17930}, {40627, 2054}, {41841, 1016}, {50330, 9278}, {62552, 40725}, {62558, 40767}
X(69009) = crosspoint of X(i) and X(j) for these (i,j): {2, 4444}, {514, 6650}, {693, 40017}, {2786, 6542}
X(69009) = crosssum of X(i) and X(j) for these (i,j): {101, 17735}, {692, 41333}, {2702, 17962}
X(69009) = crossdifference of every pair of points on line {101, 512}
X(69009) = barycentric product X(i)*X(j) for these {i,j}: {244, 20947}, {423, 4466}, {514, 2786}, {693, 9508}, {1086, 6542}, {1111, 1757}, {1326, 21207}, {1565, 17927}, {1931, 16732}, {2973, 17976}, {3120, 17731}, {3125, 52137}, {3261, 5029}, {4444, 27929}, {6541, 17205}, {6548, 28602}, {6650, 35080}, {7192, 18004}, {7199, 68823}, {15634, 28346}, {16727, 20693}, {17735, 23989}, {17990, 52619}, {18014, 68168}, {32014, 57461}, {38348, 66286}, {60669, 66191}
X(69009) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 6543}, {244, 1929}, {513, 37135}, {514, 35148}, {523, 66283}, {649, 2702}, {1015, 17962}, {1086, 6650}, {1111, 18032}, {1326, 4570}, {1757, 765}, {1931, 4567}, {2681, 17747}, {2786, 190}, {2969, 17982}, {3120, 11599}, {3122, 2054}, {3125, 9278}, {3733, 17940}, {3937, 17972}, {4466, 57848}, {5029, 101}, {6542, 1016}, {6650, 57560}, {7192, 17930}, {8034, 18001}, {9508, 100}, {17731, 4600}, {17735, 1252}, {17927, 15742}, {17990, 4557}, {18004, 3952}, {18266, 1110}, {20947, 7035}, {21140, 64236}, {27846, 40767}, {27918, 40725}, {27929, 3570}, {28602, 17780}, {35080, 6542}, {38348, 3573}, {40794, 5378}, {41180, 10026}, {46668, 41324}, {52137, 4601}, {57461, 1213}, {66191, 60710}, {68168, 17934}, {68823, 1018}
X(69009) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1086, 66189, 3120}, {3756, 61673, 1086}, {6788, 67625, 24281}, {17213, 21044, 7200}, {25745, 30945, 141}

X(69010) = X(1)X(41843)∩X(2)X(38)

Barycentrics   (a^2 + a*b - b^2 + a*c - b*c - c^2)^2 : :

X(69010) lies on these lines: {1, 41843}, {2, 38}, {6, 4473}, {10, 6650}, {115, 6540}, {190, 594}, {346, 20055}, {1086, 1268}, {1500, 40776}, {1757, 6541}, {3219, 3512}, {3799, 20670}, {4366, 20016}, {4370, 28337}, {4422, 63053}, {4437, 27949}, {4440, 5224}, {5220, 20533}, {5686, 41845}, {17251, 17487}, {18037, 52662}, {20090, 24358}, {20679, 29593}, {21879, 39722}, {24504, 35309}, {24821, 41844}, {27495, 31310}, {28604, 68870}, {29591, 53600}, {60710, 62675}

X(69010) = midpoint of X(190) and X(32025)
X(69010) = X(6540)-Ceva conjugate of X(18004)
X(69010) = X(35080)-cross conjugate of X(68140)
X(69010) = X(i)-isoconjugate of X(j) for these (i,j): {1929, 17962}, {3248, 57560}, {9506, 40767}
X(69010) = X(i)-Dao conjugate of X(j) for these (i,j): {239, 40725}, {2786, 1086}, {39041, 1929}, {41841, 6650}, {57461, 4977}
X(69010) = cevapoint of X(35080) and X(68140)
X(69010) = barycentric product X(i)*X(j) for these {i,j}: {190, 68140}, {1016, 35080}, {1757, 20947}, {3952, 68168}, {6541, 17731}, {6542, 6542}, {17934, 18004}, {20693, 52137}
X(69010) = barycentric quotient X(i)/X(j) for these {i,j}: {1016, 57560}, {1757, 1929}, {6541, 11599}, {6542, 6650}, {6651, 40725}, {8298, 40767}, {17735, 17962}, {17927, 17982}, {17934, 17930}, {17943, 17940}, {17976, 17972}, {17990, 18001}, {18004, 18014}, {20693, 9278}, {20947, 18032}, {35080, 1086}, {40794, 9505}, {58287, 2054}, {59724, 64236}, {68140, 514}, {68168, 7192}
X(69010) = {X(6542),X(41841)}-harmonic conjugate of X(6651)

X(69011) = X(2)X(18004)∩X(11)X(244)

Barycentrics   (b - c)*(-2*a^3 - a^2*b + a*b^2 + b^3 - a^2*c + a*c^2 + c^3) : :
X(69011) = 3 X[2] + X[50342], 3 X[1638] - X[3837], X[2254] + 3 X[4809], X[4010] + 3 X[4750], X[4458] + 3 X[45674], X[9508] - 3 X[45674], X[649] + 3 X[48227], X[659] + 3 X[4453], 3 X[659] + X[49301], 9 X[4453] - X[49301], 3 X[4453] - X[58375], X[49301] - 3 X[58375], 3 X[1635] + X[48326], X[3700] - 3 X[48206], X[3835] - 3 X[48215], and many others

X(69011) lies on these lines: {2, 18004}, {11, 244}, {56, 18006}, {88, 60043}, {100, 5606}, {513, 13246}, {522, 59749}, {523, 2487}, {649, 48227}, {659, 3004}, {690, 1125}, {814, 21188}, {918, 27929}, {1054, 2606}, {1635, 2527}, {1649, 51583}, {1960, 62435}, {3670, 42666}, {3676, 29362}, {3700, 48206}, {3720, 53563}, {3762, 6533}, {3776, 4782}, {3798, 29328}, {3835, 48215}, {3887, 58565}, {3907, 59743}, {3960, 24099}, {4025, 4874}, {4088, 28602}, {4122, 24924}, {4467, 47833}, {4468, 48214}, {4500, 48221}, {4522, 48216}, {4707, 14419}, {4763, 48056}, {4784, 31095}, {4786, 49295}, {4790, 48192}, {4806, 4897}, {4885, 29078}, {4922, 30574}, {4979, 48552}, {5957, 16158}, {6532, 68896}, {6682, 25380}, {7659, 48211}, {8674, 58591}, {14837, 29324}, {18005, 62732}, {19862, 22037}, {20470, 53263}, {21051, 41800}, {21192, 52601}, {21222, 24093}, {23770, 24623}, {23875, 31288}, {24382, 57099}, {24920, 25469}, {26275, 50357}, {27013, 48103}, {27486, 48120}, {28179, 46915}, {28183, 47132}, {28209, 47999}, {28217, 48050}, {29266, 59714}, {31207, 48185}, {31286, 62423}, {39386, 44433}, {44551, 64914}, {45314, 48055}, {45323, 48039}, {47676, 48226}, {47755, 48024}, {47758, 68780}, {47761, 48405}, {47763, 47944}, {47784, 48002}, {47798, 50359}, {47822, 47971}, {47824, 50340}, {47829, 48047}, {47882, 48030}, {47983, 48574}, {47989, 48032}, {48013, 48555}, {48043, 48195}, {48046, 48180}, {48048, 48562}, {48083, 48571}, {48106, 48224}, {48183, 50326}, {48197, 48270}, {48229, 50333}, {48238, 48277}, {48245, 50347}, {48248, 50348}

X(69011) = midpoint of X(i) and X(j) for these {i,j}: {659, 58375}, {1960, 62435}, {3776, 4782}, {4025, 4874}, {4458, 9508}, {4806, 4897}, {18004, 50342}, {21192, 52601}, {21196, 54265}, {21222, 24093}, {48248, 50348}
X(69011) = reflection of X(24099) in X(3960)
X(69011) = complement of X(18004)
X(69011) = complement of the isogonal conjugate of X(17940)
X(69011) = complement of the isotomic conjugate of X(17930)
X(69011) = tripolar centroid of X(59267)
X(69011) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 41180}, {81, 46668}, {163, 6651}, {1333, 35080}, {1929, 125}, {2702, 1211}, {4556, 20529}, {4610, 20548}, {6650, 21253}, {17930, 2887}, {17940, 10}, {17962, 8287}, {17972, 34846}, {18001, 24040}, {18032, 53575}, {35148, 21245}, {37135, 3454}, {52935, 20339}
X(69011) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 41180}, {18014, 4977}, {60042, 514}, {62644, 17770}
X(69011) = X(i)-isoconjugate of X(j) for these (i,j): {100, 28482}, {692, 35162}, {765, 60050}, {1110, 60042}
X(69011) = X(i)-Dao conjugate of X(j) for these (i,j): {513, 60050}, {514, 60042}, {1086, 35162}, {8054, 28482}, {17770, 62644}, {35114, 190}, {41180, 2}, {51578, 3952}
X(69011) = crosspoint of X(i) and X(j) for these (i,j): {2, 17930}, {514, 60042}, {17770, 62644}
X(69011) = crosssum of X(i) and X(j) for these (i,j): {6, 17990}, {28482, 60050}
X(69011) = crossdifference of every pair of points on line {101, 1500}
X(69011) = barycentric product X(i)*X(j) for these {i,j}: {514, 17770}, {1086, 62644}, {4610, 65704}, {4977, 31064}, {7192, 10026}, {17930, 41180}, {20666, 52619}, {35114, 60042}
X(69011) = barycentric quotient X(i)/X(j) for these {i,j}: {514, 35162}, {649, 28482}, {1015, 60050}, {1086, 60042}, {10026, 3952}, {17770, 190}, {20666, 4557}, {20685, 40521}, {20754, 4574}, {31064, 6540}, {35114, 62644}, {41180, 18004}, {62644, 1016}, {65704, 4024}
X(69011) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 50342, 18004}, {659, 4453, 58375}, {4458, 45674, 9508}, {4897, 47799, 4806}, {27013, 48241, 48103}

X(69012) = X(10)X(37)∩X(190)X(646)

Barycentrics   (a - b)*(a - c)*(b + c)^2*(a^2 - b*c) : :

X(69012) lies on these lines: {10, 37}, {190, 646}, {192, 68877}, {536, 23822}, {813, 65635}, {1016, 17934}, {1966, 29699}, {3572, 23354}, {4103, 4155}, {4369, 42721}, {4444, 42720}, {4472, 40548}, {26759, 26764}, {27097, 27102}, {29383, 29388}, {30730, 56257}, {30731, 53341}, {35310, 61175}, {40614, 62571}, {61163, 61174}, {61165, 61172}

X(69012) = X(i)-Ceva conjugate of X(j) for these (i,j): {4562, 3952}, {61402, 35068}
X(69012) = X(i)-cross conjugate of X(j) for these (i,j): {4155, 740}, {35068, 61402}
vX(i)-isoconjugate of X(j) for these (i,j): {81, 66937}, {593, 876}, {741, 1019}, {757, 3572}, {849, 4444}, {875, 1509}, {1015, 36066}, {1977, 65285}, {2311, 7203}, {3248, 65258}, {3733, 37128}, {7192, 18268}, {18827, 57129}, {34067, 61403}, {36806, 61048}, {39179, 46159}, {43925, 57738}
X(69012) = X(i)-Dao conjugate of X(j) for these (i,j): {740, 812}, {4075, 4444}, {8299, 1019}, {16591, 17096}, {35068, 7192}, {35119, 61403}, {38978, 1015}, {40586, 66937}, {40607, 3572}, {62553, 7199}
X(69012) = crosspoint of X(3952) and X(4562)
X(69012) = crosssum of X(3733) and X(8632)
X(69012) = trilinear pole of line {4037, 4829}
X(69012) = crossdifference of every pair of points on line {3248, 3733}
X(69012) = barycentric product X(i)*X(j) for these {i,j}: {190, 4037}, {239, 4103}, {335, 68132}, {350, 40521}, {594, 3570}, {740, 3952}, {756, 874}, {812, 61402}, {1018, 3948}, {1089, 3573}, {1500, 27853}, {1978, 66878}, {2238, 4033}, {3027, 36801}, {3699, 7235}, {3716, 65958}, {3747, 27808}, {3975, 21859}, {3985, 4552}, {4094, 4583}, {4155, 7035}, {4557, 35544}, {4562, 35068}, {4829, 53658}, {16609, 30730}, {31625, 46390}, {43534, 68153}, {52609, 68800}
X(69012) = barycentric quotient X(i)/X(j) for these {i,j}: {42, 66937}, {594, 4444}, {740, 7192}, {756, 876}, {765, 36066}, {812, 61403}, {862, 57200}, {872, 875}, {874, 873}, {1016, 65258}, {1018, 37128}, {1089, 66286}, {1284, 7203}, {1500, 3572}, {2238, 1019}, {3027, 43041}, {3570, 1509}, {3573, 757}, {3690, 66938}, {3747, 3733}, {3948, 7199}, {3952, 18827}, {3985, 4560}, {4010, 17205}, {4033, 40017}, {4037, 514}, {4039, 17212}, {4069, 56154}, {4094, 659}, {4103, 335}, {4148, 26856}, {4155, 244}, {4433, 3737}, {4557, 741}, {4562, 57554}, {4771, 48580}, {4829, 4778}, {6057, 60577}, {6535, 35352}, {7035, 65285}, {7140, 68631}, {7235, 3676}, {16609, 17096}, {21832, 16726}, {30730, 36800}, {35068, 812}, {35544, 52619}, {39786, 8042}, {40521, 291}, {41333, 57129}, {46390, 1015}, {58327, 65575}, {61402, 4562}, {66878, 649}, {68132, 239}, {68153, 33295}, {68800, 17925}

X(69013) = X(2)X(39)∩X(32)X(11339)

Barycentrics   a^3*b^3 - a^3*b^2*c - a^2*b^3*c - a^3*b*c^2 + a^3*c^3 - a^2*b*c^3 + 2*b^3*c^3 : :

X(69013) lies on these lines: {2, 39}, {32, 11339}, {256, 17306}, {350, 45214}, {513, 3716}, {698, 23824}, {899, 27076}, {1015, 62234}, {1920, 21827}, {1921, 6377}, {1965, 23533}, {1978, 20688}, {3712, 40548}, {3734, 16405}, {3739, 53039}, {3741, 52547}, {4358, 40562}, {4970, 59570}, {7816, 11322}, {8620, 64870}, {17070, 21264}, {17149, 22199}, {19579, 23552}, {21224, 40089}, {21345, 59505}, {22184, 51863}, {23470, 39914}, {24656, 43223}, {35466, 40546}, {36950, 41144}

X(69013) = complement of X(2229)
X(69013) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 65940}, {58, 52882}, {715, 2}, {18826, 141}
X(69013) = X(53366)-Ceva conjugate of X(891)
X(69013) = crossdifference of every pair of points on line {669, 2176}
X(69013) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 30955, 3934}, {2, 30964, 39}, {2, 31000, 76}, {2, 31008, 21838}, {2, 62709, 16589}

X(69014) = X(38)X(75)∩X(39)X(25114)

Barycentrics   a^3*b^4 - a^3*b^3*c - a^2*b^4*c - a*b^4*c^2 - a^3*b*c^3 + 2*a*b^3*c^3 + b^4*c^3 + a^3*c^4 - a^2*b*c^4 - a*b^2*c^4 + b^3*c^4 : :

X(69014) lies on these lines: {38, 75}, {39, 25114}, {76, 25122}, {194, 25141}, {513, 3716}, {1213, 34832}, {3741, 17235}, {3840, 48632}, {18144, 45782}, {24214, 52547}, {24660, 26108}, {25121, 50158}, {25349, 28593}, {25508, 25528}, {26110, 26135}, {50092, 64909}

X(69014) = crossdifference of every pair of points on line {1924, 2176}

X(69015) = X(38)X(75)∩X(244)X(350)

Barycentrics   a^2*b^2 + a*b^3 - a*b^2*c + a^2*c^2 - a*b*c^2 - 2*b^2*c^2 + a*c^3 : :
X(69015) = 3 X[16711] - X[53332]

X(69015) lies on these lines: {1, 55945}, {2, 3985}, {7, 3210}, {38, 75}, {69, 32860}, {76, 24443}, {86, 1962}, {194, 17451}, {244, 350}, {257, 40908}, {274, 2292}, {291, 18034}, {319, 32948}, {320, 4938}, {325, 3120}, {335, 3930}, {522, 693}, {536, 3726}, {538, 3125}, {596, 39712}, {668, 4695}, {726, 3263}, {740, 30941}, {756, 60706}, {758, 62755}, {894, 21840}, {896, 33295}, {903, 31177}, {982, 4441}, {986, 34284}, {1089, 24170}, {1111, 57029}, {1447, 62300}, {1655, 21921}, {1739, 6381}, {1909, 4642}, {1959, 62636}, {1975, 3924}, {1999, 60717}, {2234, 4093}, {2238, 68870}, {2294, 56023}, {2650, 33296}, {3663, 4425}, {3664, 4970}, {3670, 20888}, {3729, 26242}, {3743, 17175}, {3760, 24046}, {3873, 3875}, {3879, 3896}, {3994, 20947}, {4087, 62234}, {4357, 4359}, {4360, 62867}, {4363, 41269}, {4373, 7249}, {4414, 16992}, {4446, 4493}, {4479, 42040}, {4647, 16887}, {4967, 4981}, {4986, 59717}, {5224, 33125}, {5333, 10436}, {6650, 7779}, {6707, 59218}, {7200, 35101}, {7264, 24166}, {8264, 23484}, {10455, 58391}, {14210, 17205}, {16583, 56024}, {16703, 18697}, {16705, 49598}, {16711, 53332}, {17148, 17868}, {17151, 62865}, {17158, 34860}, {17164, 18600}, {17169, 64071}, {17206, 27368}, {17274, 31143}, {17351, 46907}, {17490, 30946}, {17495, 20347}, {17497, 35102}, {17596, 37670}, {17738, 33854}, {17762, 33947}, {18135, 24174}, {18827, 65285}, {20553, 24715}, {20911, 24214}, {21044, 47286}, {21808, 25264}, {21816, 36812}, {23537, 24995}, {24185, 68938}, {24248, 45962}, {24291, 54315}, {26042, 27285}, {27633, 30004}, {28530, 68929}, {30758, 32925}, {30962, 32915}, {31348, 33891}, {32864, 60729}, {33937, 64429}, {37632, 46904}, {49514, 59207}

X(69015) = reflection of X(i) in X(j) for these {i,j}: {14210, 17205}, {68938, 24185}
X(69015) = anticomplement of X(3985)
X(69015) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {604, 39367}, {741, 329}, {1014, 20345}, {1408, 33888}, {1412, 17794}, {1434, 20554}, {7233, 21287}, {16947, 30667}, {18268, 144}, {18827, 21286}, {37128, 3436}, {56154, 54113}, {66937, 37781}
X(69015) = X(i)-isoconjugate of X(j) for these (i,j): {55, 35108}, {669, 65635}, {2175, 35159}
X(69015) = X(i)-Dao conjugate of X(j) for these (i,j): {223, 35108}, {35095, 9}, {40593, 35159}, {46842, 1}
X(69015) = crosspoint of X(75) and X(18827)
X(69015) = crosssum of X(31) and X(3747)
X(69015) = crossdifference of every pair of points on line {41, 1924}
X(69015) = barycentric product X(i)*X(j) for these {i,j}: {85, 35104}, {799, 65873}, {18827, 46842}
X(69015) = barycentric quotient X(i)/X(j) for these {i,j}: {57, 35108}, {85, 35159}, {799, 65635}, {35104, 9}, {46842, 740}, {65873, 661}
X(69015) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {75, 30966, 21020}, {86, 68867, 1962}, {335, 17759, 3930}, {3663, 24165, 26234}, {17164, 18600, 59509}, {24214, 67983, 20911}

X(69016) = X(1)X(4433)∩X(2)X(39)

Barycentrics   a*(a^2*b^2 + a*b^3 - a*b^2*c + a^2*c^2 - a*b*c^2 - 2*b^2*c^2 + a*c^3) : :
X(69016) = 3 X[2] + X[62636]

X(69016) lies on these lines: {1, 4433}, {2, 39}, {6, 16412}, {9, 28252}, {10, 37596}, {11, 45270}, {32, 11329}, {37, 4472}, {57, 978}, {58, 19329}, {63, 28251}, {75, 17053}, {81, 5277}, {86, 2092}, {88, 17946}, {99, 68708}, {115, 26019}, {141, 46838}, {142, 27633}, {187, 19308}, {232, 15149}, {239, 1015}, {241, 514}, {291, 2664}, {320, 53543}, {333, 23447}, {335, 21830}, {350, 25510}, {386, 474}, {404, 5337}, {511, 18792}, {524, 16726}, {536, 8610}, {573, 28365}, {574, 16367}, {579, 27623}, {620, 5977}, {626, 37096}, {668, 26048}, {730, 20340}, {894, 21796}, {899, 20456}, {992, 16574}, {995, 68769}, {1010, 50622}, {1045, 4890}, {1054, 5184}, {1086, 57039}, {1107, 24603}, {1125, 3666}, {1201, 54282}, {1211, 16700}, {1213, 16696}, {1266, 57037}, {1432, 8056}, {1500, 16826}, {1573, 29576}, {1574, 3661}, {1575, 3912}, {1739, 68759}, {1740, 17065}, {1764, 15489}, {1929, 17799}, {1959, 3125}, {1964, 17049}, {1975, 11353}, {2140, 40688}, {2234, 3122}, {2238, 18206}, {2245, 52897}, {2275, 4384}, {2276, 16831}, {2277, 10436}, {2893, 28091}, {2915, 4278}, {3009, 14839}, {3110, 25825}, {3216, 37676}, {3218, 27646}, {3230, 28371}, {3286, 20857}, {3290, 25083}, {3589, 39798}, {3662, 27641}, {3663, 28366}, {3688, 4446}, {3729, 62214}, {3736, 24923}, {3752, 6703}, {3753, 15953}, {3770, 27111}, {3778, 64007}, {3879, 21857}, {3936, 16753}, {3963, 27102}, {4111, 24437}, {4253, 4383}, {4261, 15668}, {4263, 17379}, {4272, 18166}, {4357, 28244}, {4359, 26747}, {4416, 21892}, {4436, 39688}, {4850, 17397}, {4859, 40784}, {5069, 17259}, {5121, 37370}, {5247, 60715}, {5272, 8731}, {5294, 58452}, {5308, 17756}, {5956, 41014}, {6542, 52959}, {6675, 37597}, {6996, 62371}, {7146, 24174}, {7816, 11320}, {7853, 63794}, {11349, 33854}, {11350, 19724}, {12263, 24165}, {13588, 40984}, {16050, 53387}, {16602, 62689}, {16709, 27042}, {16710, 26772}, {16736, 17056}, {16834, 63493}, {16849, 19758}, {17030, 19804}, {17058, 37796}, {17121, 46189}, {17197, 68729}, {17261, 21826}, {17367, 41805}, {17390, 21858}, {17448, 50095}, {17495, 27166}, {17759, 26113}, {18601, 41809}, {18755, 60721}, {18904, 62675}, {19728, 19762}, {19808, 27274}, {20137, 39971}, {20142, 37128}, {20269, 24776}, {20367, 49997}, {20544, 23682}, {20691, 29574}, {20917, 27091}, {21214, 37555}, {21352, 46908}, {24003, 59735}, {24199, 28358}, {24220, 50650}, {24443, 68478}, {24625, 29590}, {25068, 25089}, {25298, 33908}, {25457, 27164}, {25593, 33129}, {25660, 56023}, {25860, 63822}, {27076, 52043}, {27233, 27239}, {27627, 56509}, {28395, 48627}, {28530, 68943}, {28606, 29612}, {28639, 56926}, {29456, 62755}, {30110, 32777}, {34063, 41232}, {37165, 68946}, {37595, 59301}, {37681, 39956}, {42027, 59562}, {46548, 52680}, {52541, 65543}, {56518, 68709}, {65168, 65741}, {68871, 68940}, {68942, 68980}

X(69016) = midpoint of X(i) and X(j) for these {i,j}: {2234, 3122}, {3948, 62636}, {16726, 68883}
X(69016) = complement of X(3948)
X(69016) = complement of the isogonal conjugate of X(18268)
X(69016) = complement of the isotomic conjugate of X(37128)
X(69016) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 45162}, {31, 46842}, {32, 35068}, {58, 20333}, {81, 20542}, {110, 27854}, {291, 21245}, {292, 3454}, {604, 50440}, {741, 141}, {813, 31946}, {875, 8287}, {876, 21253}, {1333, 17793}, {1576, 27929}, {1911, 1211}, {1922, 1213}, {1967, 46826}, {2196, 21530}, {2206, 17755}, {2311, 1329}, {3572, 125}, {4444, 53575}, {4584, 21260}, {4589, 21262}, {9506, 20546}, {14598, 16589}, {17938, 25666}, {18263, 10026}, {18265, 38930}, {18268, 10}, {18827, 626}, {18897, 21838}, {20981, 2679}, {34067, 4129}, {36066, 42327}, {37128, 2887}, {39276, 21238}, {40017, 21235}, {46159, 21248}, {56154, 21244}, {57129, 38989}, {65258, 23301}, {65285, 21263}, {65352, 21243}, {66937, 116}, {66938, 127}
X(69016) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 46842}, {65285, 512}
X(69016) = X(i)-isoconjugate of X(j) for these (i,j): {9, 35108}, {41, 35159}, {798, 65635}
X(69016) = X(i)-Dao conjugate of X(j) for these (i,j): {478, 35108}, {3160, 35159}, {31998, 65635}, {35095, 8}, {46842, 2}
X(69016) = crosspoint of X(i) and X(j) for these (i,j): {2, 37128}, {37137, 39292}
X(69016) = crosssum of X(i) and X(j) for these (i,j): {6, 2238}, {2086, 3287}
X(69016) = crossdifference of every pair of points on line {55, 669}
X(69016) = barycentric product X(i)*X(j) for these {i,j}: {7, 35104}, {99, 65873}, {37128, 46842}
X(69016) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 35159}, {56, 35108}, {99, 65635}, {35104, 8}, {46842, 3948}, {65873, 523}
X(69016) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 76, 30819}, {2, 194, 30830}, {2, 20081, 30863}, {2, 20913, 3934}, {2, 24598, 39}, {2, 24621, 76}, {2, 40773, 16589}, {2, 62636, 3948}, {39, 31198, 2}, {76, 31234, 2}, {81, 25946, 5277}, {86, 24530, 2092}, {291, 2664, 20683}, {1045, 63520, 4890}, {1740, 17065, 21746}, {3752, 16604, 17023}, {3963, 27102, 44418}, {16609, 43034, 43059}, {24621, 31234, 30819}, {29576, 62803, 1573}, {39798, 64556, 3589}

X(69017) = X(2)X(31)∩X(320)X(350)

Barycentrics   a^4*b - a*b^4 + a^4*c + 2*a^3*b*c - b^4*c - a*c^4 - b*c^4 : :

X(69017) lies on these lines: {2, 31}, {6, 17138}, {42, 17766}, {226, 26237}, {291, 32844}, {320, 350}, {321, 518}, {672, 3006}, {730, 6542}, {894, 17153}, {1001, 31006}, {1279, 37869}, {1281, 3218}, {1707, 30752}, {1757, 26223}, {2276, 33070}, {3112, 54112}, {3120, 17031}, {3720, 4425}, {3741, 4001}, {3783, 32843}, {3936, 8299}, {4184, 40605}, {4358, 20716}, {4651, 4886}, {5905, 10453}, {11355, 49492}, {13576, 21282}, {16801, 19740}, {17018, 49704}, {17027, 33134}, {17300, 45223}, {17484, 17794}, {17721, 24691}, {17726, 25349}, {17759, 32842}, {17765, 20011}, {18206, 68951}, {20992, 29981}, {24552, 49706}, {24587, 60722}, {24690, 25368}, {26238, 30985}, {29822, 49709}, {29829, 63066}, {30052, 36635}, {30751, 56520}, {30965, 32942}, {30970, 49710}, {32928, 49675}, {33106, 60090}, {43223, 49705}, {52256, 57280}

X(69017) = anticomplement of X(2239)
X(69017) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {743, 2}, {57944, 69}
X(69017) = crosspoint of X(86) and X(57944)
X(69017) = crosssum of X(42) and X(8624)
X(69017) = crossdifference of every pair of points on line {213, 3250}
X(69017) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {31, 30953, 2}, {238, 30969, 2}

X(69018) = X(1)X(3) INTERCEPT OF X(100)X(10027)

Barycentrics   a^3*(a^2*b^2 - a*b^3 + b^3*c + a^2*c^2 - 2*b^2*c^2 - a*c^3 + b*c^3) : :

X(69018) lies on these lines: {1, 3}, {100, 10027}, {184, 2210}, {667, 788}, {1015, 8618}, {1424, 32462}, {1500, 22367}, {1575, 53268}, {1716, 20794}, {2653, 20456}, {3747, 20777}, {3778, 22389}, {4020, 4531}, {17448, 23851}, {18266, 41268}, {20544, 27181}, {20775, 40934}, {21384, 23863}, {26959, 33821}, {27020, 33828}, {34067, 51973}, {51928, 56530}

X(69018) = X(i)-isoconjugate of X(j) for these (i,j): {2, 60014}, {9, 34084}, {76, 59020}, {85, 30627}
X(69018) = X(i)-Dao conjugate of X(j) for these (i,j): {478, 34084}, {32664, 60014}, {65942, 561}
X(69018) = crossdifference of every pair of points on line {75, 650}
X(69018) = barycentric product X(31)*X(46180)
X(69018) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 60014}, {56, 34084}, {560, 59020}, {2175, 30627}, {46180, 561}

X(69019) = X(1)X(3) INTERCEPT OF X(100)X(17145)

Barycentrics   a^2*(a^3*b - a*b^3 + a^3*c + 2*a^2*b*c - 3*b^3*c + 4*b^2*c^2 - a*c^3 - 3*b*c^3) : :

X(69019) lies on these lines: {1, 3}, {100, 17145}, {244, 3286}, {659, 3004}, {859, 4973}, {896, 54333}, {3218, 20470}, {3715, 16409}, {3742, 22060}, {4184, 65112}, {4392, 20990}, {4436, 29824}, {4557, 62235}, {6763, 16414}, {7292, 53310}, {8053, 64149}, {9047, 22067}, {9352, 15621}, {13731, 52783}, {14956, 53564}, {15571, 32845}, {16708, 33944}, {17524, 58565}, {19998, 30577}, {24237, 68842}, {25557, 30944}, {27003, 52139}, {33846, 68754}, {36002, 53296}, {37449, 53312}

X(69019) = crossdifference of every pair of points on line {650, 1500}
X(69019) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1155, 68760, 23832}, {3218, 20470, 53280}

X(69020) = X(1)X(3) INTERCEPT OF X(523)X(661)

Barycentrics   a*(b + c)*(a^3*b + a^2*b^2 - a*b^3 - b^4 + a^3*c - 2*a^2*b*c + b^3*c + a^2*c^2 - a*c^3 + b*c^3 - c^4) : :

X(69020) lies on these lines: {1, 3}, {11, 7235}, {511, 53524}, {523, 661}, {896, 7202}, {1710, 22136}, {1959, 3712}, {2245, 68796}, {2783, 68842}, {2941, 47057}, {3011, 17444}, {3430, 38336}, {4125, 21075}, {4220, 7073}, {7359, 16309}, {17452, 17602}, {18210, 20718}, {20539, 32842}, {20653, 21031}, {21318, 22276}, {21333, 59174}, {22076, 42440}, {22080, 53035}, {24433, 56878}, {24931, 24932}, {61716, 68033}

X(69020) = crossdifference of every pair of points on line {58, 650}
X(69020) = barycentric product X(190)*X(65665)
X(69020) = barycentric quotient X(65665)/X(514)

X(69021) = X(1)X(3) INTERCEPT OF X(523)X(1577)

Barycentrics   a*(b + c)*(a^3*b^2 + a^2*b^3 - a*b^4 - b^5 - a^2*b^2*c + b^4*c + a^3*c^2 - a^2*b*c^2 + a^2*c^3 - a*c^4 + b*c^4 - c^5) : :

X(69021) lies on these lines: {1, 3}, {10, 21318}, {191, 22139}, {442, 42440}, {511, 1725}, {523, 1577}, {758, 18210}, {774, 67967}, {859, 34977}, {1737, 7235}, {3142, 23555}, {3178, 3971}, {3822, 21807}, {4516, 68946}, {5080, 38514}, {10056, 31395}, {10742, 20957}, {11813, 42753}, {14963, 53560}, {17637, 48907}, {18175, 18417}, {18669, 59734}, {22345, 59729}, {25441, 25442}, {25645, 25646}, {41697, 64419}, {44410, 66704}, {48909, 67946}, {57590, 68244}

X(69021) = reflection of X(859) in X(34977)
X(69021) = crossdifference of every pair of points on line {650, 1333}
X(69021) = barycentric product X(1441)*X(68729)
X(69021) = barycentric quotient X(68729)/X(21)

X(69022) = X(1)X(3) INTERCEPT OF X(11)X(25642)

Barycentrics   a*(b - c)^2*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + a^2*b*c - a*b^2*c + b^3*c - a^2*c^2 - a*b*c^2 + a*c^3 + b*c^3) : :

X(69022) lies on these lines: {1, 3}, {11, 25642}, {37, 9321}, {104, 1462}, {150, 34460}, {244, 45695}, {812, 1015}, {952, 43063}, {1111, 11998}, {2401, 62635}, {3669, 4089}, {4530, 47965}, {4670, 5701}, {4904, 7117}, {14714, 68559}, {26803, 26805}, {26961, 26964}, {34253, 34586}, {35094, 35128}

X(69022) = X(i)-complementary conjugate of X(j) for these (i,j): {9085, 20316}, {29241, 27076}, {35365, 141}, {60049, 3835}
X(69022) = X(675)-Ceva conjugate of X(513)
X(69022) = X(1110)-isoconjugate of X(54739)
X(69022) = X(i)-Dao conjugate of X(j) for these (i,j): {514, 54739}, {68882, 3006}
X(69022) = crosspoint of X(1019) and X(2224)
X(69022) = crosssum of X(1018) and X(57015)
X(69022) = crossdifference of every pair of points on line {650, 4557}
X(69022) = barycentric product X(i)*X(j) for these {i,j}: {1565, 62971}, {3676, 68832}
X(69022) = barycentric quotient X(i)/X(j) for these {i,j}: {1086, 54739}, {62971, 15742}, {68832, 3699}

X(69023) = X(1)X(3) INTERCEPT OF X(513)X(3716)

Barycentrics   a*(a^3*b^2 - a*b^4 - 2*a^3*b*c + a^2*b^2*c + 2*a*b^3*c - b^4*c + a^3*c^2 + a^2*b*c^2 - 4*a*b^2*c^2 + b^3*c^2 + 2*a*b*c^3 + b^2*c^3 - a*c^4 - b*c^4) : :
X(69023) = 3 X[3] + X[18330], X[1155] - 3 X[34583], 3 X[5048] - X[67459], 3 X[5131] + X[38474], 3 X[34583] + X[50362], 3 X[35059] - X[67418], X[3218] + 3 X[33852], X[5057] + 3 X[67627], X[67499] - 3 X[67627], X[26015] - 3 X[61674], X[56878] - 3 X[67493]

X(69023) lies on these lines: {1, 3}, {2, 3784}, {11, 15310}, {29, 1878}, {73, 19514}, {140, 11573}, {222, 16434}, {295, 660}, {511, 3911}, {513, 3716}, {518, 4434}, {603, 13732}, {908, 3937}, {916, 13226}, {1125, 67968}, {1357, 24231}, {1401, 66632}, {1463, 17719}, {1465, 51651}, {1745, 19549}, {1829, 37304}, {2392, 6681}, {2635, 19546}, {2810, 6745}, {3025, 59806}, {3035, 8679}, {3218, 33852}, {3271, 5121}, {3781, 5744}, {3814, 14058}, {3819, 5745}, {3917, 59491}, {3955, 17074}, {4192, 22053}, {4303, 19513}, {4655, 30986}, {5057, 30943}, {5123, 34831}, {5650, 54357}, {5907, 6705}, {5943, 6692}, {6666, 15082}, {6676, 58460}, {6691, 42450}, {6700, 29958}, {6712, 22102}, {7193, 63068}, {7248, 33144}, {9037, 38472}, {9955, 64539}, {10624, 53002}, {12607, 41682}, {13747, 67893}, {16610, 18191}, {17717, 49537}, {18141, 26929}, {19335, 22350}, {19522, 20731}, {22066, 63496}, {22067, 61220}, {22148, 23693}, {23154, 27385}, {23440, 30037}, {24320, 25934}, {24685, 34371}, {25075, 40972}, {25490, 25492}, {25526, 64544}, {25528, 63522}, {26015, 61674}, {26091, 26093}, {26884, 37449}, {26910, 31053}, {32932, 35626}, {34753, 67975}, {37634, 67961}, {37684, 67978}, {40649, 40940}, {46850, 67041}, {52264, 58497}, {56878, 67493}, {64528, 64538}

X(69023) = midpoint of X(i) and X(j) for these {i,j}: {908, 3937}, {1155, 50362}, {5057, 67499}, {31849, 50371}
X(69023) = reflection of X(3911) in X(64489)
X(69023) = complement of X(67494)
X(69023) = circumcircle-inverse of X(23853)
X(69023) = incircle-inverse of X(982)
X(69023) = Conway-circle-inverse of X(35621)
X(69023) = X(59131)-complementary conjugate of X(2)
X(69023) = X(43362)-Ceva conjugate of X(513)
X(69023) = crossdifference of every pair of points on line {650, 2176}
X(69023) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1155, 1319, 5143}, {1381, 1382, 23853}, {2446, 2447, 982}, {5057, 67627, 67499}, {17074, 19649, 3955}, {34583, 50362, 1155}

X(69024) = X(1)X(3) INTERCEPT OF X(2)X(1914)

Barycentrics   a*(a^4 + a^3*b + a^3*c - a*b^2*c + b^3*c - a*b*c^2 + b*c^3) : :

X(69024) lies on these lines: {1, 3}, {2, 1914}, {6, 22370}, {31, 8299}, {32, 3912}, {37, 1760}, {43, 8298}, {63, 17735}, {81, 2276}, {100, 63818}, {141, 5301}, {172, 17316}, {312, 385}, {321, 4376}, {333, 3661}, {345, 6542}, {384, 20917}, {609, 29573}, {614, 8301}, {902, 56509}, {1107, 16367}, {1279, 26241}, {1333, 4851}, {1958, 28350}, {2176, 20769}, {2185, 56441}, {2220, 17279}, {2241, 17023}, {2242, 29574}, {2268, 28369}, {2275, 21495}, {2295, 54419}, {3052, 49706}, {3684, 4383}, {3693, 3719}, {3703, 32853}, {3761, 24271}, {3782, 33869}, {3879, 5019}, {3915, 25940}, {4044, 7751}, {4366, 52138}, {5275, 44307}, {5277, 16831}, {5287, 40750}, {5291, 17294}, {5332, 32911}, {6376, 68708}, {7031, 17284}, {10315, 37642}, {10987, 37633}, {11320, 52043}, {14974, 23151}, {16502, 21477}, {16946, 17353}, {16968, 49760}, {16992, 31993}, {17019, 41269}, {17308, 19732}, {17750, 60721}, {20284, 62421}, {21001, 21775}, {25440, 69016}, {27623, 54316}, {29603, 68893}, {32942, 52133}, {33106, 36481}, {33863, 62853}, {36475, 56010}, {36528, 56518}, {46178, 62806}, {49560, 59692}

X(69024) = barycentric product X(190)*X(50458)
X(69024) = barycentric quotient X(50458)/X(514)
X(69024) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {55, 940, 3666}, {3550, 3749, 55}

X(69025) = X(1)X(3) INTERCEPT OF X(10)X(190)

Barycentrics   a*(a^3 + a^2*b - 2*a*b^2 - 2*b^3 + a^2*c - 3*a*b*c + b^2*c - 2*a*c^2 + b*c^2 - 2*c^3) : : X(69025) =
X[1] - 3 X[17596], 2 X[1] - 3 X[37617], 3 X[17596] - 2 X[37599], 4 X[37599] - 3 X[37617]

X(69025) lies on these lines: {1, 3}, {6, 41319}, {8, 32845}, {10, 190}, {37, 41322}, {38, 63136}, {44, 3496}, {45, 3501}, {63, 64176}, {72, 5524}, {191, 3987}, {392, 1054}, {495, 32857}, {516, 37717}, {519, 66692}, {758, 60714}, {846, 3753}, {896, 4642}, {902, 54315}, {950, 53614}, {984, 54286}, {1046, 4646}, {1247, 5302}, {1571, 3061}, {1706, 24341}, {1722, 15601}, {1737, 33095}, {1739, 17123}, {2292, 5297}, {3058, 53619}, {3125, 60711}, {3175, 68482}, {3214, 11684}, {3219, 4695}, {3617, 33083}, {3679, 66672}, {3868, 67207}, {3878, 45763}, {3919, 4653}, {3927, 59294}, {3944, 26446}, {3979, 24473}, {4084, 33771}, {4304, 66643}, {4640, 60353}, {4649, 4868}, {4674, 5251}, {5015, 49609}, {5250, 24174}, {5529, 31165}, {5657, 24248}, {5692, 56009}, {5774, 49474}, {5883, 16484}, {7286, 18360}, {7292, 24443}, {9593, 16670}, {10056, 33103}, {10544, 53002}, {11238, 31520}, {12514, 24440}, {12575, 51615}, {12782, 49712}, {13161, 43174}, {17757, 33099}, {19875, 66691}, {22166, 50444}, {23844, 35206}, {24239, 28194}, {25073, 27000}, {25439, 49675}, {28174, 33106}, {31433, 51058}, {32913, 64175}, {33101, 45701}, {36277, 54418}, {41012, 60414}, {42054, 68245}, {48915, 50618}, {50044, 59313}, {50581, 64070}, {61524, 63997}

X(69025) = reflection of X(i) in X(j) for these {i,j}: {1, 37599}, {37617, 17596}
X(69025) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 4689, 37573}, {1, 17596, 37599}, {1, 37599, 37617}, {40, 986, 5255}, {46, 37598, 37607}, {65, 4689, 1}, {484, 4424, 171}, {988, 7991, 66650}, {2093, 17594, 66640}, {3670, 11010, 37588}, {4642, 56288, 5247}, {5657, 24248, 37716}

X(69026) = X(2)X(7)∩X(6)X(17073)

Barycentrics   a^4*b - 2*a^2*b^3 + b^5 + a^4*c - 2*a^3*b*c + 2*a^2*b^2*c - b^4*c + 2*a^2*b*c^2 - 2*a^2*c^3 - b*c^4 + c^5 : :
X(69026) = 3 X[2] + X[17950]

X(69026) lies on these lines: {2, 7}, {6, 17073}, {37, 16608}, {44, 18644}, {46, 406}, {69, 53996}, {101, 9028}, {116, 29069}, {169, 24316}, {198, 41004}, {240, 522}, {241, 26932}, {347, 53994}, {371, 31534}, {372, 31535}, {461, 3474}, {475, 1728}, {516, 36027}, {518, 51366}, {524, 6510}, {534, 5011}, {573, 18589}, {580, 1125}, {651, 68920}, {674, 25379}, {914, 3580}, {997, 49511}, {1020, 5236}, {1086, 65943}, {1100, 17043}, {1146, 64780}, {1155, 33305}, {1158, 3089}, {1214, 13567}, {1375, 2182}, {1427, 41883}, {1441, 25000}, {1443, 37781}, {1465, 26005}, {1536, 15726}, {1565, 34371}, {1730, 1848}, {1758, 50366}, {1937, 26013}, {1948, 53211}, {1952, 67654}, {2003, 18652}, {2052, 60249}, {2092, 18592}, {2183, 4466}, {2225, 25344}, {2262, 41007}, {2270, 41010}, {2310, 45281}, {2323, 26006}, {2340, 21914}, {3002, 44360}, {3542, 63437}, {3664, 58412}, {3668, 20262}, {3755, 18391}, {3812, 4205}, {3821, 12618}, {3834, 58466}, {3912, 16578}, {4260, 67981}, {4270, 54369}, {4329, 54420}, {4364, 21258}, {4511, 4684}, {4552, 48381}, {4667, 53597}, {4708, 6706}, {4858, 22464}, {5053, 43054}, {5120, 55118}, {5125, 10395}, {5179, 8680}, {5845, 44355}, {5928, 11347}, {6349, 11433}, {6350, 37643}, {6354, 6708}, {6356, 9119}, {6505, 6515}, {10393, 37180}, {15733, 50441}, {15836, 18909}, {16581, 67501}, {16596, 62326}, {16844, 28628}, {17046, 24336}, {17050, 24319}, {17861, 24005}, {17903, 56300}, {18206, 62328}, {18641, 44547}, {18650, 36016}, {20268, 26723}, {21062, 24310}, {21239, 64126}, {23693, 49676}, {23980, 35094}, {24607, 62691}, {24880, 58462}, {25001, 40999}, {25091, 26942}, {25964, 40937}, {28629, 57007}, {28849, 50290}, {30686, 41342}, {30807, 41804}, {31600, 34492}, {34822, 62811}, {35075, 35086}, {35091, 65901}, {38454, 62674}, {40535, 44334}, {44661, 62343}, {52020, 61663}, {52381, 54444}, {56848, 68335}, {61034, 61653}, {64875, 68908}

X(69026) = midpoint of X(i) and X(j) for these {i,j}: {1944, 17950}, {1948, 53211}, {9436, 40880}
X(69026) = reflection of X(i) in X(j) for these {i,j}: {6510, 17044}, {44356, 62388}
X(69026) = isotomic conjugate of X(8777)
X(69026) = complement of X(1944)
X(69026) = polar conjugate of X(43764)
X(69026) = complement of the isogonal conjugate of X(1945)
X(69026) = complement of the isotomic conjugate of X(1952)
X(69026) = isotomic conjugate of the isogonal conjugate of X(8776)
X(69026) = X(i)-complementary conjugate of X(j) for these (i,j): {296, 18589}, {1402, 35075}, {1937, 141}, {1945, 10}, {1949, 3}, {1952, 2887}, {2249, 960}, {37142, 21246}, {40843, 1368}, {41206, 512}, {41207, 21259}, {52222, 123}, {61427, 3452}, {65214, 17072}
X(69026) = X(53211)-Ceva conjugate of X(522)
X(69026) = X(i)-isoconjugate of X(j) for these (i,j): {3, 20624}, {6, 8759}, {31, 8777}, {48, 43764}
X(69026) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 8777}, {9, 8759}, {1249, 43764}, {8758, 5179}, {20623, 1}, {36103, 20624}, {39070, 6}
X(69026) = crosspoint of X(i) and X(j) for these (i,j): {2, 1952}, {75, 37214}
X(69026) = crosssum of X(6) and X(1951)
X(69026) = crossdifference of every pair of points on line {48, 663}
X(69026) = barycentric product X(i)*X(j) for these {i,j}: {75, 8758}, {76, 8776}, {92, 64887}, {20623, 37214}
X(69026) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 8759}, {2, 8777}, {4, 43764}, {19, 20624}, {8758, 1}, {8776, 6}, {20623, 5179}, {56904, 1936}, {64887, 63}
X(69026) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 17950, 1944}, {2, 60705, 5745}, {307, 25019, 9}, {1214, 13567, 45206}, {6349, 11433, 45126}, {16578, 63844, 3912}, {22464, 26001, 4858}, {27508, 68349, 61010}

X(69027) = X(2)X(7)∩X(110)X(415)

Barycentrics   a^4*b^2 - 2*a^2*b^4 + b^6 - a^3*b^2*c + a^2*b^3*c + a*b^4*c - b^5*c + a^4*c^2 - a^3*b*c^2 + 2*a^2*b^2*c^2 - a*b^3*c^2 - b^4*c^2 + a^2*b*c^3 - a*b^2*c^3 + 2*b^3*c^3 - 2*a^2*c^4 + a*b*c^4 - b^2*c^4 - b*c^5 + c^6 : :

X(69027) lies on these lines: {1, 5906}, {2, 7}, {4, 66757}, {33, 68336}, {72, 24984}, {110, 415}, {124, 68761}, {141, 26591}, {244, 26010}, {278, 68335}, {297, 525}, {320, 63068}, {321, 343}, {394, 32859}, {518, 23541}, {651, 17923}, {756, 25970}, {860, 912}, {914, 4552}, {942, 24983}, {1086, 26005}, {1262, 68920}, {1861, 61185}, {2988, 52780}, {2989, 36107}, {3006, 44694}, {3120, 26013}, {3187, 6515}, {3729, 53816}, {3782, 13567}, {3868, 17555}, {4091, 25924}, {4194, 55109}, {4318, 33650}, {4358, 26611}, {4389, 26635}, {4703, 25885}, {5044, 24985}, {5081, 36121}, {5125, 12528}, {5136, 37826}, {5709, 27504}, {5758, 27505}, {5762, 33305}, {5812, 27378}, {7232, 25934}, {8679, 45917}, {8822, 26645}, {10601, 32774}, {11105, 24474}, {11433, 19785}, {14544, 23710}, {17220, 30687}, {17768, 25968}, {18662, 45206}, {20905, 37648}, {21361, 40677}, {23542, 44547}, {24473, 25016}, {24537, 57282}, {24539, 31445}, {24987, 67848}, {25243, 31017}, {25941, 33064}, {26540, 33151}, {26575, 33736}, {26579, 26665}, {26871, 57477}, {26932, 64194}, {27380, 37623}, {29085, 46549}, {29307, 61221}, {29369, 46484}, {33124, 54348}, {33146, 54284}, {34822, 68345}, {35194, 63850}, {37781, 37798}, {44706, 63840}

X(69027) = reflection of X(14544) in X(23710)
X(69027) = isotomic conjugate of X(2988)
X(69027) = polar conjugate of X(32706)
X(69027) = anticomplement of the isotomic conjugate of X(52780)
X(69027) = isotomic conjugate of the isogonal conjugate of X(8607)
X(69027) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {19, 151}, {102, 4329}, {2432, 34188}, {15629, 52366}, {32643, 66520}, {32667, 522}, {32677, 20}, {32700, 57091}, {34393, 68347}, {36055, 6527}, {36067, 693}, {36100, 1370}, {36108, 35519}, {36121, 69}, {52780, 6327}, {60584, 21293}, {68625, 21294}
X(69027) = X(i)-Ceva conjugate of X(j) for these (i,j): {34393, 54242}, {52780, 2}
X(69027) = X(i)-isoconjugate of X(j) for these (i,j): {31, 2988}, {48, 32706}, {521, 32707}, {650, 35187}, {652, 36113}, {692, 60567}, {2182, 15379}, {32677, 54243}
X(69027) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 2988}, {117, 6}, {1086, 60567}, {1249, 32706}, {8607, 515}, {23986, 54243}
X(69027) = crosspoint of X(i) and X(j) for these (i,j): {76, 34393}, {1275, 65295}
X(69027) = trilinear pole of line {117, 55124}
X(69027) = crossdifference of every pair of points on line {184, 663}
X(69027) = barycentric product X(i)*X(j) for these {i,j}: {75, 1735}, {76, 8607}, {117, 34393}, {664, 55124}, {850, 7450}, {35516, 54242}
X(69027) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2988}, {4, 32706}, {102, 15379}, {108, 36113}, {109, 35187}, {117, 515}, {514, 60567}, {515, 54243}, {1735, 1}, {7450, 110}, {8607, 6}, {32674, 32707}, {34393, 57751}, {54242, 102}, {55124, 522}
X(69027) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3580, 48380, 48381}, {3782, 13567, 17862}, {17257, 27184, 26580}

X(69028) = X(2)X(3230)∩X(7)X(8)

Barycentrics   a^3*b - a*b^3 + a^3*c - 2*a^2*b*c + 2*a*b^2*c - b^3*c + 2*a*b*c^2 - a*c^3 - b*c^3 : :

X(69028) lies on these lines: {1, 24170}, {2, 3230}, {7, 8}, {10, 17152}, {76, 20244}, {150, 20553}, {304, 14923}, {316, 512}, {517, 3263}, {519, 16711}, {668, 20347}, {672, 49774}, {742, 17497}, {758, 4986}, {1018, 30109}, {1222, 1434}, {1228, 58889}, {1334, 25101}, {1739, 68963}, {2140, 29381}, {2295, 3589}, {2802, 14210}, {3125, 68890}, {3208, 29966}, {3241, 30962}, {3264, 17139}, {3501, 27109}, {3596, 17220}, {3629, 3780}, {3721, 28598}, {3727, 25263}, {3735, 31087}, {3753, 26234}, {3763, 27116}, {3877, 30758}, {3885, 18156}, {3959, 17489}, {3977, 30059}, {4019, 17868}, {4039, 24194}, {4360, 54315}, {4390, 24586}, {4561, 62826}, {4651, 54112}, {4674, 32847}, {4850, 17316}, {5697, 33942}, {5903, 33937}, {6542, 17495}, {10459, 16705}, {15983, 17116}, {15991, 50171}, {16752, 40886}, {17143, 17751}, {17164, 33941}, {17169, 25303}, {17752, 26978}, {17758, 29699}, {20028, 28660}, {20255, 27097}, {20345, 21290}, {20549, 26778}, {20924, 21272}, {21240, 26759}, {21760, 26825}, {21788, 27169}, {21839, 31025}, {24254, 31077}, {24282, 31130}, {24602, 56530}, {25253, 33932}, {25278, 56799}, {27036, 66878}, {29824, 31002}, {33938, 56318}, {42720, 57015}, {45751, 57038}, {45913, 63018}, {45962, 48849}, {49481, 65695}, {49755, 65195}

X(69028) = reflection of X(53332) in X(3263)
X(69028) = anticomplement of X(3230)
X(69028) = anticomplement of the isogonal conjugate of X(3227)
X(69028) = isotomic conjugate of the isogonal conjugate of X(68760)
X(69028) = anticomplementary isogonal conjugate of X(39360)
X(69028) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 39360}, {190, 44008}, {739, 192}, {889, 20295}, {898, 514}, {3112, 25333}, {3227, 8}, {4607, 513}, {5381, 3952}, {23349, 21224}, {23892, 9263}, {31002, 69}, {32718, 21225}, {34075, 17494}, {35353, 21221}, {36798, 329}, {36872, 30578}, {37129, 2}, {41683, 2895}, {43928, 4440}, {57542, 29824}, {57994, 21304}, {60288, 1330}, {62619, 149}, {62763, 1655}, {64612, 20533}
X(69028) = crosspoint of X(i) and X(j) for these (i,j): {83, 3227}, {889, 4998}
X(69028) = crosssum of X(i) and X(j) for these (i,j): {39, 3230}, {890, 3271}
X(69028) = crossdifference of every pair of points on line {3051, 3063}
X(69028) = barycentric product X(76)*X(68760)
X(69028) = barycentric quotient X(68760)/X(6)
X(69028) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 21281, 17137}, {3501, 30036, 27109}

X(69029) = X(1)X(19238)∩X(2)X(7)

Barycentrics   a*(a^3*b - a*b^3 + a^3*c - 2*a^2*b*c + 2*a*b^2*c - b^3*c + 2*a*b*c^2 - a*c^3 - b*c^3) : :

X(69029) lies on these lines: {1, 19238}, {2, 7}, {37, 54282}, {44, 16726}, {45, 62817}, {71, 25101}, {72, 19241}, {78, 19245}, {190, 20367}, {239, 53391}, {312, 1730}, {314, 32017}, {320, 21362}, {344, 573}, {517, 33845}, {518, 54333}, {674, 1026}, {798, 812}, {859, 5440}, {942, 19244}, {995, 3751}, {1014, 23617}, {1018, 17264}, {1052, 1757}, {1756, 3836}, {1764, 18743}, {1766, 41316}, {1959, 16578}, {2183, 3882}, {2245, 4422}, {2250, 24630}, {2347, 3879}, {3122, 68879}, {3227, 45751}, {3271, 4447}, {3294, 33792}, {3685, 63136}, {3886, 63137}, {3916, 19249}, {3927, 19253}, {3940, 4245}, {4266, 17316}, {4271, 17243}, {4557, 57024}, {4652, 19292}, {5044, 10461}, {5782, 16412}, {8610, 68964}, {10446, 28778}, {11349, 65168}, {16549, 17354}, {16552, 17335}, {16560, 20602}, {16561, 17738}, {16670, 62853}, {17120, 18164}, {17185, 44307}, {17277, 21061}, {17317, 67984}, {17469, 20964}, {18040, 29395}, {18134, 21361}, {18785, 46798}, {19255, 24929}, {21363, 33116}, {22047, 61168}, {24220, 28748}, {24310, 30568}, {24727, 40859}, {25917, 64365}, {29069, 37788}, {29353, 63852}, {29504, 65161}, {30109, 32094}, {35892, 41711}, {37510, 56529}, {39979, 57039}, {41310, 67501}, {50126, 54286}, {52923, 62872}, {62826, 65573}

X(69029) = X(31002)-Ceva conjugate of X(1)
X(69029) = X(3230)-Dao conjugate of X(899)
X(69029) = crosspoint of X(i) and X(j) for these (i,j): {82, 37129}, {4564, 4607}
X(69029) = crosssum of X(i) and X(j) for these (i,j): {38, 899}, {2170, 3768}
X(69029) = crossdifference of every pair of points on line {663, 1964}
X(69029) = barycentric product X(75)*X(68760)
X(69029) = barycentric quotient X(68760)/X(1)
X(69029) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 29497, 29698}, {7, 29552, 29749}, {9, 57, 50127}, {9, 21371, 16574}, {894, 17260, 27078}, {2183, 3912, 3882}, {16574, 29380, 9}, {29382, 29439, 142}, {29417, 29472, 3452}, {29418, 29474, 226}, {29429, 29492, 4357}, {29497, 29552, 7}, {29696, 29747, 144}, {29698, 29749, 7}, {29740, 29812, 6646}

X(69030) = X(2)X(7)∩X(187)X(237)

Barycentrics   a^2*(a^2*b^2 - a*b^3 + b^3*c + a^2*c^2 - 2*b^2*c^2 - a*c^3 + b*c^3) : :

X(69030) lies on these lines: {2, 7}, {48, 1613}, {187, 237}, {244, 2225}, {614, 20665}, {726, 61234}, {813, 9082}, {899, 46148}, {919, 59019}, {1155, 39258}, {1201, 23544}, {1334, 4414}, {1357, 68743}, {1401, 16588}, {1403, 51949}, {1424, 3177}, {1755, 3290}, {2178, 17735}, {2238, 20785}, {2294, 3726}, {2311, 37791}, {3051, 9449}, {3210, 7075}, {3508, 5205}, {3721, 22065}, {3752, 40972}, {3840, 21369}, {3917, 20684}, {3924, 20460}, {4020, 16583}, {4871, 20372}, {6377, 21760}, {9315, 62875}, {9454, 35326}, {10453, 21387}, {14964, 18173}, {15082, 25100}, {16975, 17449}, {17082, 21218}, {19554, 26884}, {19591, 26274}, {20593, 50362}, {23443, 28082}, {23531, 30117}, {23535, 28011}, {23622, 62739}, {23863, 56931}, {24578, 62300}, {28360, 30646}, {32845, 68821}

X(69030) = isogonal conjugate of X(60014)
X(69030) = isogonal conjugate of the anticomplement of X(65942)
X(69030) = isogonal conjugate of the isotomic conjugate of X(46180)
X(69030) = X(59020)-Ceva conjugate of X(6)
X(69030) = X(i)-isoconjugate of X(j) for these (i,j): {1, 60014}, {7, 30627}, {55, 34084}, {75, 59020}
X(69030) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 60014}, {206, 59020}, {223, 34084}, {65942, 76}
X(69030) = crosspoint of X(6) and X(59020)
X(69030) = crosssum of X(2) and X(46180)
X(69030) = crossdifference of every pair of points on line {2, 663}
X(69030) = X(i)-line conjugate of X(j) for these (i,j): {7, 2}, {187, 663}
X(69030) = barycentric product X(i)*X(j) for these {i,j}: {6, 46180}, {59020, 65942}
X(69030) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 60014}, {32, 59020}, {41, 30627}, {57, 34084}, {46180, 76}
X(69030) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {244, 2225, 20459}, {1401, 16588, 23636}, {3231, 8620, 3009}, {4020, 16583, 23640}, {6377, 21760, 68749}

X(69031) = X(3)X(238)∩X(7)X(8)

Barycentrics   a*(a^2*b^2 - a*b^3 + b^3*c + a^2*c^2 - 2*b^2*c^2 - a*c^3 + b*c^3) : :
X(69031) = 4 X[6687] - 3 X[16482], X[25048] - 3 X[37756]

X(69031) lies on these lines: {3, 238}, {6, 49537}, {7, 8}, {44, 513}, {46, 1757}, {72, 4655}, {142, 21746}, {192, 25279}, {209, 11246}, {210, 4643}, {239, 3888}, {244, 23633}, {312, 24717}, {354, 4675}, {442, 3831}, {511, 1738}, {517, 3792}, {527, 4014}, {536, 4553}, {660, 40848}, {674, 1086}, {750, 61687}, {752, 11112}, {758, 24692}, {766, 50025}, {959, 56999}, {960, 4201}, {1004, 19591}, {1201, 1279}, {1266, 14839}, {1284, 1818}, {1362, 62789}, {1401, 4847}, {1423, 34247}, {1581, 2652}, {1654, 58655}, {1788, 26050}, {1836, 6817}, {1864, 26052}, {2175, 24309}, {2223, 35338}, {2278, 52086}, {2323, 5091}, {2340, 21320}, {2810, 49772}, {2876, 50011}, {2979, 33131}, {3008, 3271}, {3009, 3123}, {3022, 45275}, {3056, 4000}, {3210, 25308}, {3246, 36289}, {3474, 37109}, {3551, 53676}, {3662, 64709}, {3663, 3688}, {3664, 52020}, {3675, 57022}, {3681, 4741}, {3683, 37329}, {3740, 17256}, {3747, 22399}, {3752, 3764}, {3753, 24693}, {3755, 64006}, {3772, 20359}, {3778, 22064}, {3781, 24248}, {3784, 33137}, {3812, 26051}, {3819, 24210}, {3834, 21264}, {3912, 6007}, {3914, 3917}, {3925, 67961}, {3952, 25288}, {3959, 44453}, {4260, 50307}, {4292, 10822}, {4300, 28265}, {4357, 64007}, {4384, 60929}, {4395, 9024}, {4413, 5782}, {4419, 4517}, {4443, 37596}, {4516, 68759}, {4640, 37467}, {4864, 17445}, {5044, 24697}, {5211, 33891}, {5249, 67892}, {5308, 66677}, {5918, 37419}, {6173, 64560}, {6687, 16482}, {6999, 15726}, {7146, 24341}, {7184, 24575}, {7186, 33132}, {7238, 9054}, {7248, 24477}, {7483, 31289}, {7998, 33134}, {8539, 62797}, {8609, 67428}, {9052, 24231}, {9309, 37681}, {9365, 9442}, {10391, 37107}, {10427, 38989}, {10609, 49700}, {10974, 17770}, {11509, 23693}, {11677, 12589}, {12609, 49676}, {12723, 17668}, {13576, 68912}, {15346, 19584}, {16514, 24289}, {17049, 24199}, {17064, 37521}, {17082, 20935}, {17114, 24391}, {17235, 56537}, {17258, 58693}, {17263, 25108}, {17274, 56542}, {17278, 63522}, {17289, 25144}, {17298, 35892}, {17300, 64546}, {17317, 58620}, {17344, 22271}, {17345, 64581}, {17365, 22277}, {17374, 44671}, {17490, 25306}, {17528, 31151}, {17635, 21871}, {17647, 17766}, {17796, 38530}, {18726, 21804}, {20456, 53541}, {20715, 24699}, {20777, 23363}, {20892, 21278}, {20917, 24351}, {20923, 21299}, {21387, 52657}, {21620, 50583}, {21922, 24993}, {21926, 39780}, {22370, 64727}, {23772, 43040}, {23774, 30807}, {24239, 40649}, {24463, 40790}, {24484, 43065}, {24914, 26027}, {25048, 37756}, {25120, 34832}, {25137, 33116}, {25903, 37228}, {26029, 26041}, {26806, 58583}, {27622, 28256}, {28389, 59691}, {28600, 41847}, {31136, 31138}, {31604, 40593}, {36279, 46032}, {36574, 67930}, {40875, 56802}, {43921, 53241}, {45782, 47522}, {49675, 68668}, {49699, 62401}, {49777, 68985}, {52043, 53338}, {52562, 52563}, {61672, 62660}

X(69031) = midpoint of X(i) and X(j) for these {i,j}: {239, 3888}, {3792, 24715}, {4014, 20683}
X(69031) = reflection of X(i) in X(j) for these {i,j}: {3271, 3008}, {20358, 1086}, {23772, 43040}, {57024, 3834}
X(69031) = X(i)-isoconjugate of X(j) for these (i,j): {2, 59020}, {6, 60014}, {41, 34084}, {57, 30627}
X(69031) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 60014}, {3160, 34084}, {5452, 30627}, {32664, 59020}, {65942, 75}
X(69031) = crossdifference of every pair of points on line {1, 3063}
X(69031) = barycentric product X(1)*X(46180)
X(69031) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 60014}, {7, 34084}, {31, 59020}, {55, 30627}, {46180, 75}
X(69031) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1740, 41886, 27633}, {1742, 27626, 20992}, {2228, 2234, 1575}, {2340, 52896, 21320}, {3009, 3123, 57037}, {3914, 3917, 21334}, {4259, 5880, 65}, {4675, 64751, 354}, {49537, 61034, 6}

X(69032) = X(38)X(3663)∩X(69)X(350)

Barycentrics   b*c*(a^2*b^2 - a*b^3 + b^3*c + a^2*c^2 - 2*b^2*c^2 - a*c^3 + b*c^3) : :

X(69032) lies on these lines: {38, 3663}, {55, 18043}, {69, 350}, {305, 561}, {312, 59508}, {354, 18045}, {514, 661}, {799, 2651}, {1111, 21241}, {1150, 17026}, {1233, 2886}, {3006, 20435}, {3717, 21404}, {3933, 21422}, {4009, 21580}, {4554, 5205}, {17605, 18142}, {18066, 62234}, {18153, 32023}, {20880, 33108}, {20930, 20945}, {25074, 27038}, {25760, 26563}, {27436, 27458}, {27523, 59619}, {28071, 51560}, {30545, 32937}, {30741, 60720}, {33129, 62803}

X(69032) = X(i)-isoconjugate of X(j) for these (i,j): {6, 59020}, {32, 60014}, {604, 30627}, {9447, 34084}
X(69032) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 59020}, {3161, 30627}, {6376, 60014}, {65942, 1}
X(69032) = crossdifference of every pair of points on line {31, 57171}
X(69032) = barycentric product X(75)*X(46180)
X(69032) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 59020}, {8, 30627}, {75, 60014}, {6063, 34084}, {46180, 1}
X(69032) = {X(3006),X(23989)}-harmonic conjugate of X(20435)

X(69033) = X(2)X(7)∩X(320)X(3882)

Barycentrics   a*(a^3*b - a*b^3 + a^3*c + 2*a^2*b*c - 3*b^3*c + 4*b^2*c^2 - a*c^3 - 3*b*c^3) : :

X(69033) lies on these lines: {2, 7}, {320, 3882}, {583, 48631}, {812, 1019}, {1018, 17297}, {1086, 18206}, {2245, 7238}, {3894, 49498}, {4675, 62817}, {7321, 21061}, {10461, 24470}, {16549, 17227}, {16726, 68964}, {17045, 17207}, {17139, 24237}, {17302, 18164}, {17305, 68950}, {18150, 29401}, {20072, 53391}, {29456, 39995}, {35338, 62872}, {37756, 45751}, {49764, 53410}, {52783, 64365}

X(69033) = crossdifference of every pair of points on line {663, 872}
X(69033) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 16574, 29382}, {7, 29747, 16574}, {9, 29749, 29439}, {57, 29788, 29472}, {63, 29790, 29474}, {144, 29552, 29380}, {320, 20367, 3882}, {894, 29812, 29492}, {21371, 60933, 29698}

X(69034) = X(1)X(24482)∩X(2)X(7)

Barycentrics   a*(a - b)*(a - c)*(a*b + b^2 + a*c - 4*b*c + c^2) : :

X(69034) lies on these lines: {1, 24482}, {2, 7}, {6, 57023}, {44, 52900}, {72, 13744}, {100, 6014}, {101, 46962}, {109, 9104}, {190, 646}, {513, 1026}, {514, 2397}, {522, 53358}, {651, 23704}, {666, 1024}, {765, 68768}, {813, 9089}, {1022, 1023}, {1441, 25730}, {2183, 4480}, {3570, 6634}, {3573, 6163}, {3758, 24491}, {3888, 4069}, {3899, 49448}, {4266, 20073}, {4440, 53391}, {4499, 35338}, {6546, 42720}, {16482, 24405}, {16610, 52140}, {18150, 29541}, {21093, 53393}, {21129, 61186}, {23703, 23831}, {23705, 23832}, {25737, 65233}, {36222, 56801}, {36278, 67385}, {36814, 49997}, {37206, 65226}, {54280, 57038}, {57151, 61220}, {68128, 68143}

X(69034) = reflection of X(1026) in X(23343)
X(69034) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {50039, 21293}, {65014, 21285}
X(69034) = X(i)-Ceva conjugate of X(j) for these (i,j): {61186, 23705}, {62536, 1}
X(69034) = X(61176)-cross conjugate of X(61186)
X(69034) = X(i)-isoconjugate of X(j) for these (i,j): {6, 23836}, {9, 37627}, {513, 40400}, {649, 1120}, {650, 8686}, {663, 65241}, {667, 36805}, {1015, 6079}, {1811, 6591}, {23345, 52556}
X(69034) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 23836}, {478, 37627}, {1266, 21433}, {2087, 1647}, {2325, 4768}, {5375, 1120}, {6631, 36805}, {16594, 514}, {16610, 3762}, {39026, 40400}
X(69034) = crosspoint of X(190) and X(3257)
X(69034) = crosssum of X(i) and X(j) for these (i,j): {513, 21343}, {649, 1635}
X(69034) = trilinear pole of line {1149, 3880}
X(69034) = crossdifference of every pair of points on line {663, 3248}
X(69034) = barycentric product X(i)*X(j) for these {i,j}: {1, 61186}, {7, 23705}, {75, 23832}, {86, 61176}, {99, 4695}, {100, 1266}, {190, 16610}, {651, 62297}, {664, 3880}, {668, 1149}, {765, 4927}, {901, 20900}, {1018, 16711}, {1023, 52574}, {1878, 4561}, {3257, 16594}, {4555, 17460}, {4618, 62666}, {4622, 21041}, {5376, 21129}, {6085, 7035}, {24004, 52206}, {52140, 62669}
X(69034) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 23836}, {56, 37627}, {100, 1120}, {101, 40400}, {109, 8686}, {190, 36805}, {651, 65241}, {765, 6079}, {1023, 52556}, {1149, 513}, {1266, 693}, {1331, 1811}, {1878, 7649}, {3880, 522}, {4695, 523}, {4927, 1111}, {6085, 244}, {8660, 3248}, {16594, 3762}, {16610, 514}, {16711, 7199}, {17109, 23345}, {17460, 900}, {20900, 65867}, {20972, 1635}, {22082, 53532}, {23205, 1459}, {23703, 56642}, {23705, 8}, {23832, 1}, {45247, 23838}, {52140, 60480}, {52206, 1022}, {52871, 4768}, {61176, 10}, {61186, 75}, {62297, 4391}, {62666, 68101}, {67445, 61483}
X(69034) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 29380, 29439}, {7, 29696, 29380}, {9, 6646, 29492}, {9, 29698, 29382}, {144, 29497, 16574}, {190, 21362, 3882}, {190, 69007, 24004}, {329, 29529, 29472}, {894, 29740, 29429}, {3257, 46779, 1022}, {4499, 52923, 35338}, {5905, 29531, 29474}, {23832, 61176, 23705}, {24004, 69007, 23891}, {24029, 62669, 1025}

X(69035) = X(7)X(21)∩X(514)X(661)

Barycentrics   a^3*b + a^2*b^2 - a*b^3 - b^4 + a^3*c - 2*a^2*b*c + b^3*c + a^2*c^2 - a*c^3 + b*c^3 - c^4 : :

X(69035) lies on these lines: {7, 21}, {30, 17136}, {100, 5195}, {150, 62826}, {322, 3260}, {325, 53332}, {404, 33867}, {496, 20247}, {514, 661}, {517, 33864}, {644, 56555}, {664, 5080}, {934, 16091}, {946, 20880}, {1111, 11813}, {1201, 4920}, {1320, 7261}, {1565, 20347}, {3061, 33839}, {3120, 68995}, {3212, 4193}, {3306, 29603}, {3674, 41012}, {3869, 17181}, {3877, 7179}, {3933, 25253}, {4056, 30144}, {4427, 6390}, {4442, 62755}, {4495, 30019}, {4511, 4872}, {4561, 20553}, {5011, 24582}, {5051, 59509}, {5057, 5088}, {5087, 43037}, {5250, 25581}, {5253, 33865}, {5328, 28827}, {5330, 56928}, {5730, 21285}, {5741, 33936}, {6710, 21372}, {7906, 25270}, {9436, 51423}, {12514, 27187}, {17095, 56288}, {17672, 39244}, {17747, 65195}, {17757, 21272}, {18600, 63997}, {18697, 20895}, {20244, 22791}, {20245, 41007}, {21616, 26563}, {25082, 27129}, {27270, 27276}, {28986, 59615}, {29597, 31164}, {29624, 37635}, {29965, 29985}, {30635, 31909}, {33946, 41324}, {49997, 65116}, {51567, 60458}, {57808, 59504}, {63136, 68926}

X(69035) = crossdifference of every pair of points on line {31, 3709}
X(69035) = barycentric product X(799)*X(65665)
X(69035) = barycentric quotient X(65665)/X(661)
X(69035) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {908, 67267, 30806}, {1565, 51409, 20347}

X(69036) = X(2)X(7)∩X(523)X(1577)

Barycentrics   (b + c)*(-(a^3*b) - a^2*b^2 + a*b^3 + b^4 - a^3*c + 2*a^2*b*c - b^3*c - a^2*c^2 + a*c^3 - b*c^3 + c^4) : :

X(69036) lies on these lines: {2, 7}, {523, 1577}, {4053, 8287}, {4552, 21066}, {4647, 51557}, {5603, 54335}, {6510, 61170}, {7110, 11683}, {16552, 24317}, {17052, 21033}, {17132, 24070}, {18698, 50036}, {20305, 21078}, {21065, 45744}, {24435, 32431}, {25359, 46196}, {26019, 68996}, {27697, 46826}

X(69036) = crossdifference of every pair of points on line {663, 1333}
X(69036) = barycentric product X(668)*X(65665)
X(69036) = barycentric quotient X(65665)/X(513)
X(69036) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {27557, 27689, 7}, {27559, 27691, 9}

X(69037) = X(30)X(511)∩X(57)X(77)

Barycentrics   a*(a^3*b + a^2*b^2 - a*b^3 - b^4 + a^3*c - 2*a^2*b*c + b^3*c + a^2*c^2 - a*c^3 + b*c^3 - c^4) : :

X(69037) lies on these lines: {2, 61695}, {30, 511}, {57, 77}, {65, 4667}, {142, 2262}, {329, 2893}, {374, 60986}, {573, 18161}, {651, 16548}, {910, 6510}, {957, 34914}, {999, 21009}, {1211, 3452}, {1332, 20602}, {1439, 65415}, {1482, 24328}, {1781, 68585}, {1959, 3882}, {2003, 3101}, {2183, 16578}, {2245, 7202}, {2269, 18726}, {2270, 53996}, {2294, 63387}, {2316, 3512}, {2321, 43216}, {2323, 7291}, {3060, 30437}, {3678, 4690}, {3727, 4503}, {3754, 4670}, {3812, 4758}, {3873, 66637}, {3878, 4643}, {3884, 4364}, {3898, 41312}, {3918, 4472}, {3939, 18788}, {3942, 20367}, {3946, 24471}, {4015, 64712}, {4266, 7146}, {4271, 25065}, {4396, 38484}, {4419, 5697}, {4466, 68840}, {4585, 16568}, {4644, 5903}, {4659, 14923}, {4858, 68912}, {5884, 24683}, {5902, 63054}, {6282, 63400}, {6610, 59813}, {6692, 6703}, {7013, 34492}, {7015, 29652}, {7277, 21863}, {9965, 18668}, {10176, 17251}, {13156, 42549}, {14953, 63782}, {15556, 64039}, {15983, 18697}, {16547, 37659}, {16579, 22097}, {16609, 17197}, {16611, 52897}, {17392, 61704}, {17868, 29382}, {18178, 35650}, {18261, 64827}, {18417, 54245}, {18725, 60974}, {20117, 24682}, {20196, 31247}, {21362, 21801}, {21809, 29740}, {21871, 60942}, {23839, 60982}, {24315, 31870}, {24316, 31806}, {24318, 51377}, {24685, 50362}, {24712, 56878}, {31142, 31143}, {32118, 37516}, {43034, 68729}, {43035, 62402}, {44352, 66152}, {53042, 62189}, {58607, 64524}, {61535, 61554}

X(69037) = crossdifference of every pair of points on line {6, 4041}
X(69037) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2183, 68759, 16578}, {2294, 67984, 63387}

X(69038) = X(10)X(75)∩X(99)X(515)

Barycentrics   a^3*b - a^2*b^2 - a*b^3 + b^4 + a^3*c - 2*a^2*b*c + a*b^2*c - a^2*c^2 + a*b*c^2 - a*c^3 + c^4 : :

X(69038) lies on these lines: {1, 7763}, {2, 9620}, {8, 3926}, {10, 75}, {40, 315}, {46, 55416}, {69, 5657}, {99, 515}, {145, 32831}, {165, 14907}, {183, 26446}, {190, 5179}, {274, 24987}, {304, 33298}, {316, 516}, {325, 517}, {332, 947}, {345, 60705}, {350, 1737}, {355, 1975}, {491, 35774}, {492, 35775}, {518, 6393}, {519, 7799}, {668, 6735}, {760, 32458}, {944, 6337}, {946, 7752}, {952, 6390}, {962, 32816}, {1007, 5603}, {1016, 1429}, {1018, 63817}, {1078, 6684}, {1125, 7769}, {1565, 59513}, {1571, 7791}, {1572, 7774}, {1698, 32832}, {1909, 10039}, {2784, 5152}, {3241, 32837}, {3262, 44150}, {3501, 56547}, {3579, 7750}, {3616, 32829}, {3617, 32830}, {3621, 32841}, {3622, 32835}, {3654, 7788}, {3661, 17740}, {3679, 32833}, {3710, 33932}, {3753, 37664}, {3760, 18395}, {3785, 68545}, {3933, 5690}, {3948, 48380}, {4163, 29037}, {4297, 7782}, {4301, 7814}, {4437, 68925}, {4561, 68926}, {4595, 49773}, {4642, 24995}, {4678, 32840}, {5493, 7860}, {5550, 32839}, {5587, 11185}, {6361, 32006}, {6734, 17143}, {7179, 24282}, {7767, 61524}, {7768, 43174}, {7771, 10164}, {7773, 12699}, {7776, 12702}, {7786, 31396}, {7796, 11362}, {7802, 31730}, {7803, 9593}, {7809, 28194}, {7983, 8781}, {8356, 31443}, {9436, 20924}, {9778, 64018}, {9780, 32828}, {9812, 32827}, {9956, 59635}, {10915, 24524}, {10916, 17144}, {11057, 50808}, {11230, 37647}, {11231, 37688}, {12245, 32818}, {12514, 55469}, {14839, 51373}, {18135, 25005}, {18140, 24982}, {18480, 32819}, {19877, 32838}, {20053, 32876}, {20541, 21888}, {20553, 63136}, {24996, 31008}, {24997, 34020}, {26526, 27109}, {28204, 59634}, {29311, 51370}, {31397, 64133}, {31422, 32965}, {31444, 33258}, {32815, 59387}, {32817, 59388}, {32834, 46933}, {32836, 53620}, {32870, 46931}, {32896, 51072}, {32897, 46930}, {33864, 53332}, {36796, 60444}, {37668, 59417}, {37671, 50821}, {38042, 64093}, {41229, 55417}, {50440, 55004}, {53127, 54447}, {55418, 56288}, {55470, 62858}, {61187, 67267}

X(69038) = X(i)-isoconjugate of X(j) for these (i,j): {667, 929}, {1980, 58000}
X(69038) = X(i)-Dao conjugate of X(j) for these (i,j): {6631, 929}, {39017, 649}
X(69038) = barycentric product X(928)*X(1978)
X(69038) = barycentric quotient X(i)/X(j) for these {i,j}: {190, 929}, {928, 649}, {1978, 58000}, {35519, 65852}

X(69039) = X(2)X(37)∩X(9)X(326)

Barycentrics   a*(a^3*b - a^2*b^2 - a*b^3 + b^4 + a^3*c - 2*a^2*b*c + a*b^2*c - a^2*c^2 + a*b*c^2 - a*c^3 + c^4) : :

X(69039) lies on these lines: {1, 41263}, {2, 37}, {6, 44179}, {9, 326}, {59, 518}, {63, 28920}, {78, 60682}, {100, 44670}, {190, 1944}, {198, 1760}, {239, 8609}, {241, 320}, {319, 3965}, {329, 27472}, {662, 2182}, {740, 1737}, {908, 8680}, {910, 16568}, {984, 997}, {1108, 3759}, {1212, 17335}, {1214, 4417}, {1445, 55391}, {1792, 66593}, {1812, 55987}, {1818, 44694}, {1959, 2183}, {2262, 18041}, {2270, 18713}, {3262, 4552}, {3687, 16577}, {3694, 17233}, {3882, 68759}, {3912, 16578}, {4130, 28898}, {4150, 51574}, {4203, 59705}, {4357, 25065}, {4389, 37597}, {4557, 35552}, {4585, 6510}, {4967, 59727}, {5279, 36016}, {6350, 28807}, {6826, 67853}, {6827, 30273}, {6830, 64088}, {6839, 67858}, {6880, 63427}, {6881, 61522}, {6882, 29010}, {6883, 54410}, {6911, 20430}, {8257, 51058}, {9025, 67428}, {11502, 64727}, {14555, 62857}, {14829, 60705}, {16560, 20769}, {16574, 21078}, {17073, 28738}, {17277, 40937}, {17353, 25078}, {17354, 25066}, {18151, 34852}, {18391, 49470}, {18607, 33066}, {20149, 20250}, {20927, 25252}, {21094, 61161}, {23585, 36212}, {24635, 54280}, {25076, 62398}, {25241, 53510}, {27471, 30852}, {27473, 27492}, {27475, 60987}, {28058, 67060}, {28965, 53996}, {30807, 65205}, {34247, 41346}, {35069, 44396}, {43065, 68966}, {44355, 62390}, {50701, 51063}, {51052, 52457}, {54344, 67120}, {60782, 66067}, {61663, 64546}, {67978, 67981}

X(69039) = X(i)-isoconjugate of X(j) for these (i,j): {649, 929}, {1919, 58000}
X(69039) = X(i)-Dao conjugate of X(j) for these (i,j): {5375, 929}, {9296, 58000}, {39017, 513}, {40624, 65852}, {68731, 26884}
X(69039) = barycentric product X(668)*X(928)
X(69039) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 929}, {668, 58000}, {928, 513}, {4391, 65852}
X(69039) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25243, 27514, 28974}, {26669, 27396, 344}

X(69040) = X(2)X(47137)∩X(37)X(63813)

Barycentrics   (b - c)*(a^3*b - a^2*b^2 - a*b^3 + b^4 + a^3*c - 2*a^2*b*c + a*b^2*c - a^2*c^2 + a*b*c^2 - a*c^3 + c^4) : :

X(69090) lies on these lines: {2, 47137}, {37, 63813}, {69, 68927}, {75, 2804}, {86, 53522}, {190, 644}, {325, 523}, {337, 66273}, {522, 30805}, {676, 21205}, {2501, 25511}, {2799, 5977}, {3766, 55126}, {6129, 21178}, {6586, 26641}, {18025, 34393}, {18160, 57158}, {21348, 28984}, {23827, 62435}

X(69040) = isotomic conjugate of X(929)
X(69040) = anticomplement of X(47137)
X(69040) = isotomic conjugate of the anticomplement of X(15612)
X(69040) = isotomic conjugate of the isogonal conjugate of X(928)
X(69040) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {37214, 21293}, {43363, 150}
X(69040) = X(15418)-Ceva conjugate of X(37796)
X(69040) = X(15612)-cross conjugate of X(2)
X(69040) = X(i)-isoconjugate of X(j) for these (i,j): {31, 929}, {560, 58000}
X(69040) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 929}, {6374, 58000}, {39017, 6}
X(69040) = crosspoint of X(99) and X(57980)
X(69040) = crosssum of X(512) and X(44112)
X(69040) = crossdifference of every pair of points on line {32, 3271}
X(69040) = barycentric product X(76)*X(928)
X(69040) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 929}, {76, 58000}, {928, 6}, {15612, 47137}, {34387, 65852}, {52331, 3271}

X(69041) = X(100)X(109)∩X(514)X(661)

Barycentrics   a*(b - c)*(a^3*b - a^2*b^2 - a*b^3 + b^4 + a^3*c - 2*a^2*b*c + a*b^2*c - a^2*c^2 + a*b*c^2 - a*c^3 + c^4) : :

X(69041) lies on these lines: {2, 46393}, {100, 109}, {226, 36038}, {514, 661}, {650, 4131}, {652, 26641}, {654, 9034}, {812, 43991}, {918, 23737}, {1444, 45756}, {1491, 9001}, {1621, 65664}, {1769, 9978}, {2294, 9979}, {2774, 68809}, {2812, 68832}, {3119, 40618}, {3219, 14418}, {4010, 55126}, {4453, 53544}, {6563, 52310}, {8611, 25259}, {13256, 53554}, {14298, 24562}, {14404, 47329}, {17494, 46400}, {20940, 65101}, {21127, 31150}, {25900, 31209}, {25954, 60025}, {26114, 26545}, {26546, 46396}, {28984, 46389}, {36100, 36101}, {40166, 46397}, {46402, 47775}

X(69041) = midpoint of X(43991) and X(46401)
X(69041) = reflection of X(i) in X(j) for these {i,j}: {36038, 226}, {40166, 46397}
X(69041) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1937, 150}, {1945, 149}, {1952, 21293}, {41206, 17135}, {41207, 20242}, {53211, 21285}, {59041, 68338}, {61427, 37781}, {65214, 3434}
X(69041) = X(i)-isoconjugate of X(j) for these (i,j): {6, 929}, {32, 58000}
X(69041) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 929}, {1577, 65852}, {6376, 58000}, {39017, 1}
X(69041) = crosspoint of X(i) and X(j) for these (i,j): {85, 53211}, {190, 8777}, {662, 35145}, {664, 37214}
X(69041) = crosssum of X(i) and X(j) for these (i,j): {649, 8776}, {661, 42669}
X(69041) = crossdifference of every pair of points on line {31, 1572}
X(69041) = barycentric product X(i)*X(j) for these {i,j}: {75, 928}, {52331, 67038}
X(69041) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 929}, {75, 58000}, {928, 1}, {4858, 65852}, {52331, 2170}

X(69042) = X(7)X(21)∩X(325)X(523)

Barycentrics   a^3*b^2 + a^2*b^3 - a*b^4 - b^5 - a^2*b^2*c + b^4*c + a^3*c^2 - a^2*b*c^2 + a^2*c^3 - a*c^4 + b*c^4 - c^5 : :

X(69042) lies on these lines: {7, 21}, {75, 17181}, {320, 1565}, {325, 523}, {326, 41010}, {664, 21277}, {857, 65205}, {1375, 16568}, {1760, 17073}, {1959, 4466}, {2178, 17481}, {3882, 68840}, {4934, 6007}, {5748, 41878}, {5988, 68985}, {6376, 18749}, {6511, 18664}, {6703, 27003}, {7249, 58028}, {11683, 28755}, {16596, 37774}, {16608, 18041}, {16701, 43034}, {16732, 26019}, {17043, 18042}, {17056, 17775}, {17482, 19308}, {17757, 17791}, {18049, 18636}, {18634, 18713}, {18635, 18714}, {18637, 18715}, {18638, 18716}, {18639, 18717}, {18640, 18718}, {18641, 18719}, {18642, 18720}, {18643, 18721}, {18644, 18722}, {21376, 52381}, {30740, 30756}, {36100, 65238}, {37135, 37214}, {41004, 44179}

X(69042) = crossdifference of every pair of points on line {32, 3709}
X(69042) = barycentric product X(6063)*X(68729)
X(69042) = barycentric quotient X(68729)/X(55)
X(69042) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1959, 4466, 37796}, {3007, 33864, 3262}, {17084, 41874, 86}

X(69043) = X(57)X(77)∩X(514)X(661)

Barycentrics   a*(a^3*b^2 + a^2*b^3 - a*b^4 - b^5 - a^2*b^2*c + b^4*c + a^3*c^2 - a^2*b*c^2 + a^2*c^3 - a*c^4 + b*c^4 - c^5) : :

X(69043) lies on these lines: {1, 52017}, {2, 18161}, {22, 20277}, {38, 23155}, {57, 77}, {73, 64039}, {102, 1290}, {192, 3151}, {445, 30687}, {511, 18210}, {514, 661}, {651, 21368}, {1432, 67929}, {1725, 2392}, {1756, 68796}, {1763, 6505}, {1953, 31019}, {1993, 26934}, {2170, 33129}, {2173, 40612}, {2294, 37635}, {2650, 5903}, {2702, 43363}, {2915, 7100}, {3100, 53557}, {3101, 37755}, {3218, 3942}, {3430, 52362}, {3468, 37231}, {3580, 4466}, {3909, 44694}, {5497, 17460}, {7146, 63008}, {7202, 35466}, {7966, 7982}, {8679, 45916}, {11680, 21328}, {14544, 68698}, {15988, 30675}, {17209, 18609}, {17452, 33151}, {17484, 21801}, {18651, 45206}, {20237, 20896}, {20254, 26892}, {20892, 30067}, {20928, 20929}, {21033, 31143}, {24474, 48909}, {25941, 41717}, {29984, 30006}, {30267, 37531}, {30690, 53036}, {30827, 30831}, {33946, 42709}, {42045, 64700}, {43058, 62402}, {44661, 61220}

X(69043) = crossdifference of every pair of points on line {31, 4041}
X(69043) = barycentric product X(85)*X(68729)
X(69043) = barycentric quotient X(68729)/X(9)

X(69044) = X(1)X(75)∩X(150)X(325)

Barycentrics   a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + a^2*b*c - a*b^2*c + b^3*c - a^2*c^2 - a*b*c^2 + a*c^3 + b*c^3 : :

X(69044) lies on these lines: {1, 75}, {3, 21281}, {8, 37670}, {21, 17152}, {29, 1969}, {41, 30036}, {69, 956}, {76, 55082}, {101, 30109}, {106, 35574}, {150, 325}, {190, 57015}, {345, 37543}, {350, 24203}, {664, 20924}, {668, 68928}, {675, 3006}, {789, 12032}, {997, 30758}, {999, 30962}, {1016, 1429}, {1055, 24602}, {1102, 62874}, {1125, 54443}, {1170, 30701}, {1240, 60041}, {1268, 19867}, {1975, 17753}, {2099, 24282}, {2176, 33821}, {2295, 16061}, {2329, 29960}, {2751, 13396}, {2975, 17137}, {3263, 4511}, {3684, 49774}, {3718, 55391}, {3732, 49755}, {3926, 6604}, {4184, 54112}, {4441, 49492}, {6563, 17161}, {7763, 33298}, {9310, 29966}, {9619, 25918}, {14828, 64133}, {14974, 17696}, {16050, 62813}, {16600, 30128}, {16788, 17277}, {17141, 34195}, {17316, 17740}, {17322, 19869}, {18659, 68340}, {20255, 21008}, {20267, 30120}, {20769, 30059}, {20925, 40719}, {22836, 33937}, {26639, 48380}, {30038, 41239}, {30144, 33942}, {30147, 33945}, {30941, 54391}, {31002, 51567}, {31637, 51560}, {37030, 51612}, {38459, 40704}, {53332, 62826}

X(69044) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {35365, 21221}, {60049, 2895}
X(69044) = X(32)-isoconjugate of X(54739)
X(69044) = X(6376)-Dao conjugate of X(54739)
X(69044) = crosspoint of X(7035) and X(54979)
X(69044) = barycentric product X(i)*X(j) for these {i,j}: {304, 62971}, {4554, 68832}
X(69044) = barycentric quotient X(i)/X(j) for these {i,j}: {75, 54739}, {62971, 19}, {68832, 650}
X(69044) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2975, 17137, 17206}, {3263, 4511, 4561}, {20255, 21008, 33828}

X(69045) = X(2)X(6)∩X(100)X(674)

Barycentrics   a*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + a^2*b*c - a*b^2*c + b^3*c - a^2*c^2 - a*b*c^2 + a*c^3 + b*c^3) : :
X(69045) = 3 X[13587] - 2 X[33844]

X(69045) lies on these lines: {2, 6}, {8, 45728}, {9, 55391}, {21, 5135}, {48, 21371}, {59, 518}, {63, 2267}, {75, 28916}, {78, 1451}, {100, 674}, {145, 12595}, {182, 1006}, {190, 62799}, {219, 344}, {239, 3262}, {264, 1172}, {294, 46798}, {312, 62798}, {320, 651}, {322, 3759}, {326, 1445}, {345, 55399}, {350, 28955}, {404, 4259}, {511, 6905}, {572, 1444}, {576, 6946}, {580, 1792}, {613, 18391}, {644, 17264}, {662, 11349}, {666, 1814}, {692, 57024}, {748, 16793}, {758, 5150}, {908, 9028}, {980, 5114}, {997, 3751}, {1203, 19869}, {1232, 3770}, {1332, 2323}, {1351, 6911}, {1352, 6830}, {1405, 25940}, {1449, 55392}, {1503, 6840}, {1621, 47373}, {1737, 5847}, {1843, 4231}, {1959, 16560}, {2183, 20769}, {2224, 15397}, {2245, 21495}, {2278, 21511}, {2651, 35992}, {2911, 26685}, {2982, 30710}, {2991, 9456}, {3187, 20928}, {3218, 34377}, {3564, 6882}, {3681, 16799}, {3687, 52423}, {3713, 42696}, {3758, 62797}, {3883, 68592}, {3964, 5120}, {4001, 54444}, {4253, 40403}, {4274, 5337}, {4358, 28952}, {4422, 17796}, {4579, 62872}, {4586, 53210}, {4684, 68591}, {5035, 56834}, {5050, 6883}, {5053, 18206}, {5085, 37106}, {5091, 6007}, {5092, 21161}, {5096, 27086}, {5124, 44180}, {5138, 37306}, {5228, 42697}, {5284, 16792}, {5480, 6839}, {6776, 6827}, {6826, 14853}, {6829, 14561}, {6859, 40330}, {6880, 63428}, {6881, 18583}, {6947, 14912}, {6954, 10519}, {6963, 63722}, {6987, 25406}, {6996, 17139}, {8257, 51194}, {8540, 9025}, {8609, 26639}, {9020, 62235}, {9054, 51157}, {9723, 36743}, {11681, 12587}, {12017, 28466}, {13587, 33844}, {13740, 62843}, {13742, 16471}, {14868, 16453}, {16347, 64422}, {16475, 54318}, {17335, 31269}, {17950, 34253}, {19270, 64420}, {20367, 65168}, {21850, 28452}, {23151, 54280}, {25101, 52405}, {26034, 61356}, {28459, 48906}, {31438, 55423}, {33171, 61395}, {34259, 37246}, {34283, 44149}, {36741, 37300}, {37086, 62691}, {37509, 41014}, {47595, 61008}, {50701, 51212}, {51190, 52457}, {51841, 55453}, {51842, 55452}, {59405, 60987}, {61398, 63140}

X(69045) = reflection of X(10755) in X(8540)
X(69045) = isotomic conjugate of X(54739)
X(69045) = isotomic conjugate of the polar conjugate of X(62971)
X(69045) = X(31)-isoconjugate of X(54739)
X(69045) = X(2)-Dao conjugate of X(54739)
X(69045) = barycentric product X(i)*X(j) for these {i,j}: {69, 62971}, {664, 68832}
X(69045) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 54739}, {62971, 4}, {68832, 522}
X(69045) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 141, 15988}, {572, 16574, 1444}, {2323, 3912, 1332}, {4869, 63088, 28965}, {23151, 55432, 54280}

X(69046) = X(2)X(7)∩X(244)X(1111)

Barycentrics   (b - c)^2*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + a^2*b*c - a*b^2*c + b^3*c - a^2*c^2 - a*b*c^2 + a*c^3 + b*c^3) : :

X(69046) lies on these lines: {2, 7}, {244, 1111}, {693, 68917}, {1565, 61674}, {1647, 24192}, {3912, 42723}, {4025, 53525}, {4106, 56893}, {4292, 37040}, {4468, 17435}, {16507, 66483}, {16586, 39775}, {20901, 44311}, {23989, 34589}, {24602, 68812}, {34234, 56783}, {42753, 47995}

X(69046) = X(37130)-Ceva conjugate of X(514)
X(69046) = X(23990)-isoconjugate of X(54739)
X(69046) = X(68882)-Dao conjugate of X(57015)
X(69046) = crosspoint of X(675) and X(7192)
X(69046) = crosssum of X(674) and X(4557)
X(69046) = barycentric product X(24002)*X(68832)
X(69046) = barycentric quotient X(i)/X(j) for these {i,j}: {1111, 54739}, {68832, 644}
X(69046) = {X(53525),X(62429)}-harmonic conjugate of X(4025)

X(69047) = X(190)X(644)∩X(320)X(350)

Barycentrics   (b - c)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + a^2*b*c - a*b^2*c + b^3*c - a^2*c^2 - a*b*c^2 + a*c^3 + b*c^3) : :
X(69047) = 2 X[13259] - 3 X[59406]

X(69047) lies on these lines: {2, 68882}, {69, 3738}, {75, 53357}, {190, 644}, {320, 350}, {344, 68813}, {885, 63229}, {900, 53276}, {927, 13136}, {1156, 37214}, {1444, 53305}, {1769, 17321}, {2412, 23887}, {2481, 18816}, {2509, 26641}, {3261, 57091}, {4406, 66516}, {4768, 42696}, {6370, 53331}, {10436, 20520}, {13259, 59406}, {15419, 21173}, {23820, 23821}, {23884, 24282}, {28846, 48320}, {30805, 31605}, {34496, 57242}, {54280, 65680}, {62430, 68103}

X(69047) = anticomplement of X(68882)
X(69047) = anticomplement of the isogonal conjugate of X(36087)
X(69047) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {675, 150}, {2224, 149}, {32682, 2}, {36087, 8}, {37130, 21293}, {52941, 514}, {60135, 3448}, {65554, 17753}
X(69047) = X(32739)-isoconjugate of X(54739)
X(69047) = X(40619)-Dao conjugate of X(54739)
X(69047) = crosspoint of X(4554) and X(43093)
X(69047) = crosssum of X(i) and X(j) for these (i,j): {42, 65703}, {3063, 8618}
X(69047) = crossdifference of every pair of points on line {213, 3271}
X(69047) = barycentric product X(i)*X(j) for these {i,j}: {85, 68832}, {15413, 62971}
X(69047) = barycentric quotient X(i)/X(j) for these {i,j}: {693, 54739}, {62971, 1783}, {68832, 9}
X(69047) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21173, 23785, 15419}, {57091, 57167, 3261}

X(69048) = X(2)X(7)∩X(75)X(18161)

Barycentrics   a^3*b^2 - a*b^4 - 2*a^3*b*c + a^2*b^2*c + 2*a*b^3*c - b^4*c + a^3*c^2 + a^2*b*c^2 - 4*a*b^2*c^2 + b^3*c^2 + 2*a*b*c^3 + b^2*c^3 - a*c^4 - b*c^4 : :

X(69048) lies on these lines: {2, 7}, {75, 18161}, {514, 4374}, {1149, 68982}, {1266, 17197}, {1401, 24997}, {2796, 30384}, {2810, 6735}, {3000, 20556}, {3262, 3942}, {3888, 68918}, {3912, 24237}, {4014, 20544}, {4562, 18816}, {6007, 26015}, {7321, 34830}, {10538, 62314}, {18207, 23688}, {20343, 66508}, {20881, 68759}, {22464, 43034}, {31623, 31917}, {34371, 51381}, {34852, 41772}, {37331, 63439}

X(69048) = crossdifference of every pair of points on line {663, 2209}
X(69048) = {X(7),X(30035)}-harmonic conjugate of X(29967)

X(69049) = X(76)X(85)∩X(514)X(4374)

Barycentrics   a^2*b^2 - a*b^3 - 2*a^2*b*c + 3*b^3*c + a^2*c^2 - 4*b^2*c^2 - a*c^3 + 3*b*c^3 : :
X(69049) = 3 X[3912] - 2 X[49753], 3 X[20924] - X[49753], 4 X[20893] - X[41773], 3 X[41140] - 4 X[50025]

X(69049) lies on these lines: {76, 85}, {239, 62812}, {514, 4374}, {519, 4740}, {712, 4403}, {1266, 35101}, {3008, 27318}, {3263, 21139}, {3729, 40872}, {4089, 63817}, {4431, 33936}, {5429, 50023}, {8682, 49770}, {17755, 44664}, {20089, 56025}, {20237, 20432}, {20335, 33946}, {24215, 33890}, {24631, 41140}, {30038, 33930}, {30054, 30082}, {30063, 33943}, {49771, 66668}, {50010, 69003}

X(69049) = reflection of X(i) in X(j) for these {i,j}: {3912, 20924}, {41773, 49774}, {49774, 20893}
X(69049) = {X(17789),X(43040)}-harmonic conjugate of X(3912)

X(69050) = X(7)X(8)∩X(513)X(3716)

Barycentrics   a*(a^2*b^2 - a*b^3 - 2*a^2*b*c + 3*b^3*c + a^2*c^2 - 4*b^2*c^2 - a*c^3 + 3*b*c^3) : :
X(69050) = X[4499] - 5 X[17266], 3 X[16482] - 5 X[31243], 3 X[31138] - X[57024]

X(69050) lies on these lines: {7, 8}, {9, 25108}, {44, 17754}, {57, 25135}, {63, 25137}, {210, 4741}, {238, 4252}, {513, 3716}, {517, 24692}, {674, 7238}, {894, 25144}, {946, 15310}, {960, 4655}, {1086, 9025}, {1279, 24661}, {1329, 3836}, {3000, 8299}, {3662, 49537}, {3740, 4643}, {3742, 4675}, {3792, 44663}, {3823, 25121}, {3880, 24715}, {3888, 20358}, {3912, 4014}, {3999, 23633}, {4447, 52896}, {4499, 17266}, {4887, 14839}, {4912, 40521}, {6168, 43946}, {6646, 58693}, {7184, 28358}, {10912, 49675}, {11236, 31151}, {12436, 17770}, {16482, 31243}, {17049, 60980}, {17256, 58451}, {17300, 58620}, {17344, 58655}, {17364, 61034}, {17376, 64546}, {17766, 66230}, {20072, 41836}, {20544, 24237}, {23774, 37788}, {24655, 26125}, {24669, 26111}, {24699, 34371}, {24723, 58679}, {27429, 53677}, {27637, 28252}, {29349, 49768}, {29632, 67499}, {30823, 34583}, {30948, 30998}, {30960, 30969}, {31137, 31138}, {35552, 53546}, {41772, 56555}, {53598, 64007}, {60929, 64560}, {63442, 64074}

X(69050) = midpoint of X(i) and X(j) for these {i,j}: {1463, 4645}, {3888, 20358}, {3912, 4014}
X(69050) = crossdifference of every pair of points on line {2176, 3063}

X(69051) = X(2)X(7)∩X(65)X(4202)

Barycentrics   (a + b - c)*(a - b + c)*(a*b^2 - b^3 + a*c^2 - c^3) : :

X(69051) lies on these lines: {2, 7}, {65, 4202}, {69, 28916}, {77, 17075}, {85, 17227}, {141, 1441}, {239, 37771}, {241, 3834}, {320, 651}, {335, 18815}, {347, 4869}, {448, 27418}, {524, 5723}, {664, 17297}, {860, 1876}, {918, 3261}, {997, 2263}, {1150, 37695}, {1214, 18139}, {1442, 17086}, {1457, 1818}, {1458, 49676}, {1465, 3936}, {1737, 24231}, {1758, 29632}, {1764, 40677}, {1892, 5136}, {1943, 32863}, {2099, 48829}, {3006, 68761}, {3007, 68759}, {3668, 21255}, {3674, 17192}, {3739, 40999}, {3821, 42289}, {3912, 4552}, {4000, 56927}, {4298, 19869}, {4310, 18391}, {4327, 54318}, {4359, 26942}, {4429, 7672}, {4851, 68344}, {4858, 48381}, {4872, 28955}, {4972, 5173}, {5228, 17290}, {5252, 47358}, {5764, 13411}, {5773, 9028}, {6180, 7232}, {6327, 34036}, {6610, 31138}, {7190, 17304}, {7269, 17302}, {7678, 63597}, {8543, 24723}, {10394, 36652}, {11375, 16342}, {12610, 17220}, {15950, 49740}, {16608, 20905}, {17059, 57022}, {17080, 18134}, {17284, 62780}, {17305, 55082}, {17307, 55096}, {17310, 41803}, {17923, 63068}, {18141, 57477}, {20367, 65233}, {20891, 34388}, {21296, 54425}, {21931, 59621}, {24593, 43043}, {24630, 52392}, {24692, 60718}, {26012, 53526}, {26932, 30807}, {27521, 41338}, {31034, 56418}, {32774, 37543}, {32859, 34048}, {35516, 40624}, {37591, 57808}, {39126, 48629}, {40013, 40149}, {43035, 63782}, {43040, 53545}, {43048, 62620}, {44717, 57750}, {45126, 63056}, {45281, 68350}, {45917, 50362}, {46909, 64737}, {48632, 52023}, {48646, 64127}, {50011, 66941}, {50441, 61035}

X(69051) = isotomic conjugate of the isogonal conjugate of X(43039)
X(69051) = X(57015)-cross conjugate of X(3006)
X(69051) = X(i)-isoconjugate of X(j) for these (i,j): {41, 675}, {55, 2224}, {650, 32682}, {663, 36087}, {692, 60573}, {2170, 52941}, {2175, 37130}, {2194, 60135}, {9447, 43093}
X(69051) = X(i)-Dao conjugate of X(j) for these (i,j): {223, 2224}, {1086, 60573}, {1214, 60135}, {3160, 675}, {38990, 663}, {40593, 37130}, {53980, 607}, {65919, 55}
X(69051) = cevapoint of X(57015) and X(68761)
X(69051) = crossdifference of every pair of points on line {663, 2175}
X(69051) = barycentric product X(i)*X(j) for these {i,j}: {7, 3006}, {75, 68761}, {76, 43039}, {85, 57015}, {349, 14964}, {561, 51657}, {664, 23887}, {674, 6063}, {2225, 20567}, {4554, 68882}, {4572, 65703}, {8618, 41283}, {24002, 42723}
X(69051) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 675}, {57, 2224}, {59, 52941}, {85, 37130}, {109, 32682}, {226, 60135}, {514, 60573}, {651, 36087}, {674, 55}, {2225, 41}, {3006, 8}, {3665, 46158}, {6063, 43093}, {8618, 2175}, {14964, 284}, {23887, 522}, {42723, 644}, {43039, 6}, {51657, 31}, {57015, 9}, {64611, 2316}, {65703, 663}, {66287, 66281}, {68761, 1}, {68882, 650}
X(69051) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 17950, 37787}, {7, 28739, 28968}, {7, 28780, 894}, {142, 307, 17077}, {894, 3662, 26573}, {3912, 22464, 4552}, {4858, 63844, 48381}, {17086, 17300, 1442}, {36589, 37787, 17950}

X(69052) = X(7)X(8)∩X(76)X(239)

Barycentrics   b*c*(a^4 + a^3*b + a^3*c - a*b^2*c + b^3*c - a*b*c^2 + b*c^3) : :

X(69052) lies on these lines: {1, 64223}, {7, 8}, {76, 239}, {274, 3661}, {304, 6542}, {305, 1999}, {321, 20641}, {2082, 17738}, {2991, 14621}, {3187, 8024}, {3673, 20345}, {3975, 5222}, {3978, 28659}, {4657, 24735}, {4851, 18157}, {17033, 17743}, {17144, 50015}, {17302, 41838}, {17367, 18140}, {17752, 30092}, {18052, 24789}, {18067, 33132}, {20081, 21216}, {20247, 20352}, {21415, 32852}, {21416, 32946}, {21590, 62998}, {24656, 25384}, {25298, 26563}, {25303, 36534}, {32847, 33937}, {33135, 59510}, {33137, 51861}, {33890, 40844}, {40071, 41233}, {54120, 64989}

X(69052) = barycentric product X(1978)*X(50458)
X(69052) = barycentric quotient X(50458)/X(649)
X(69052) = {X(8),X(34284)}-harmonic conjugate of X(20911)

X(69053) = X(2)X(6)∩X(513)X(663)

Barycentrics   a*(a^3*b + a^2*b^2 + a^3*c - 2*a^2*b*c - a*b^2*c + b^3*c + a^2*c^2 - a*b*c^2 - 2*b^2*c^2 + b*c^3) : :

X(69053) lies on these lines: {1, 6007}, {2, 6}, {7, 21769}, {142, 20228}, {213, 4667}, {320, 24663}, {511, 68982}, {513, 663}, {527, 3230}, {651, 65741}, {851, 62740}, {991, 37331}, {1014, 2305}, {1086, 68912}, {1122, 28022}, {1125, 4503}, {1418, 59173}, {1431, 2191}, {1616, 3600}, {1944, 69003}, {1966, 10027}, {2092, 67984}, {2176, 4644}, {2234, 4433}, {2245, 16726}, {2295, 4670}, {2300, 3664}, {3009, 21320}, {3290, 34371}, {3726, 34377}, {3747, 53541}, {3882, 69016}, {4000, 21785}, {4039, 68993}, {4419, 16969}, {4459, 68985}, {4465, 62234}, {4643, 24652}, {4708, 24668}, {4888, 41418}, {5429, 7290}, {7200, 8680}, {7202, 68935}, {9055, 25302}, {13329, 19335}, {16469, 38302}, {16483, 50303}, {16486, 47357}, {16685, 17365}, {17183, 53476}, {17954, 67146}, {20367, 53543}, {21826, 29740}, {24215, 24705}, {25570, 64727}, {40934, 49537}, {44671, 49478}, {50116, 54282}, {63443, 68996}

X(69053) = reflection of X(2238) in X(52897)
X(69053) = X(55)-isoconjugate of X(35176)
X(69053) = X(i)-Dao conjugate of X(j) for these (i,j): {223, 35176}, {35130, 312}
X(69053) = crosspoint of X(1) and X(6015)
X(69053) = crosssum of X(1) and X(6007)
X(69053) = crossdifference of every pair of points on line {9, 512}
X(69053) = barycentric product X(1)*X(68995)
X(69053) = barycentric quotient X(i)/X(j) for these {i,j}: {57, 35176}, {68995, 75}
X(69053) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 28365, 992}, {86, 29509, 26979}, {1149, 52896, 57037}

X(69054) = X(2)X(39)∩X(522)X(693)

Barycentrics   b*c*(a^3*b + a^2*b^2 + a^3*c - 2*a^2*b*c - a*b^2*c + b^3*c + a^2*c^2 - a*b*c^2 - 2*b^2*c^2 + b*c^3) : :

X(69054) lies on these lines: {2, 39}, {522, 693}, {726, 21404}, {1233, 3752}, {1975, 11345}, {1999, 7196}, {2092, 53236}, {2481, 5211}, {3210, 6063}, {3936, 16727}, {4554, 37759}, {10030, 62300}, {16708, 17056}, {17495, 23989}, {17778, 57785}, {18045, 40688}, {18142, 37663}, {18206, 46574}, {20880, 29639}, {21405, 24214}, {21795, 25249}, {30699, 61413}, {34018, 65740}, {35104, 53355}, {44150, 62755}, {52563, 59181}, {62234, 64223}

X(69054) = X(6015)-anticomplementary conjugate of X(329)
X(69054) = X(2175)-isoconjugate of X(35176)
X(69054) = X(i)-Dao conjugate of X(j) for these (i,j): {35130, 9}, {40593, 35176}, {68995, 6007}
X(69054) = crossdifference of every pair of points on line {41, 669}
X(69054) = barycentric product X(75)*X(68995)
X(69054) = barycentric quotient X(i)/X(j) for these {i,j}: {85, 35176}, {68995, 1}

X(69055) = X(6)X(75)∩X(241)X(514)

Barycentrics   a^4*b + a^3*b^2 + a^4*c - 2*a^3*b*c - a*b^3*c + b^4*c + a^3*c^2 - b^3*c^2 - a*b*c^3 - b^2*c^3 + b*c^4 : :

X(69055) lies on these lines: {1, 36486}, {2, 21332}, {6, 75}, {241, 514}, {312, 10027}, {519, 1215}, {716, 17755}, {940, 69003}, {1404, 27970}, {1575, 51381}, {3666, 46180}, {3912, 5718}, {5396, 29331}, {5725, 32847}, {6542, 63008}, {16584, 49758}, {16606, 50014}, {17292, 59524}, {20171, 21785}, {20228, 20236}, {20927, 21769}, {24597, 24620}, {24631, 41140}, {24806, 57277}, {27317, 27343}, {29607, 31187}, {29988, 41877}, {31993, 49774}, {35119, 65899}

X(69055) = crossdifference of every pair of points on line {55, 788}
X(69055) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3008, 49759, 49760}, {5723, 57019, 35466}

X(69056) = X(2)X(31)∩X(513)X(663)

Barycentrics   a*(a^4*b + a^3*b^2 + a^4*c - 2*a^3*b*c - a*b^3*c + b^4*c + a^3*c^2 - b^3*c^2 - a*b*c^3 - b^2*c^3 + b*c^4) : :

X(69056) lies on these lines: {1, 23634}, {2, 31}, {44, 2295}, {244, 67502}, {388, 3915}, {513, 663}, {535, 40091}, {730, 3685}, {902, 40109}, {1064, 15310}, {2230, 8299}, {3009, 15507}, {3834, 24668}, {6687, 24751}, {7032, 31394}, {14956, 62740}, {16796, 36267}, {20039, 49691}, {29046, 64013}

X(69056) = reflection of X(2239) in X(238)
X(69056) = crossdifference of every pair of points on line {9, 3250}
X(69056) = {X(238),X(4645)}-harmonic conjugate of X(28256)

X(69057) = X(2)X(7)∩X(72)X(17677)

Barycentrics   a^3 + a^2*b - 2*a*b^2 - 2*b^3 + a^2*c - 3*a*b*c + b^2*c - 2*a*c^2 + b*c^2 - 2*c^3 : :
X(69057) = X[1999] - 4 X[4415], X[1999] + 2 X[33066], 2 X[4415] + X[33066], X[32932] + 2 X[33099]

X(69057) lies on these lines: {2, 7}, {72, 17677}, {92, 52282}, {171, 28558}, {190, 50104}, {200, 64299}, {239, 33151}, {312, 599}, {321, 668}, {497, 50999}, {519, 4388}, {524, 1999}, {597, 19786}, {756, 31177}, {982, 50533}, {1211, 53501}, {1654, 4054}, {1836, 49720}, {2796, 32932}, {3120, 60731}, {3487, 50430}, {3663, 63002}, {3679, 32937}, {3685, 33065}, {3687, 17132}, {3757, 4703}, {3782, 37756}, {3876, 17679}, {3936, 17261}, {3961, 28562}, {3995, 22047}, {4052, 54119}, {4358, 17288}, {4416, 37759}, {4417, 49748}, {4514, 9041}, {4651, 53372}, {4655, 5205}, {4656, 17778}, {4671, 17287}, {4683, 7081}, {4721, 17292}, {4756, 48647}, {4759, 29860}, {4862, 24620}, {4886, 50098}, {4912, 32939}, {4933, 32936}, {4956, 17135}, {5233, 17276}, {5241, 7321}, {5297, 17491}, {5718, 17258}, {5739, 29617}, {5743, 49727}, {7232, 30829}, {16823, 32856}, {16826, 50179}, {16830, 24725}, {17056, 49737}, {17117, 37656}, {17121, 33155}, {17247, 63008}, {17268, 31017}, {17273, 30818}, {17310, 49753}, {17312, 31035}, {17319, 31034}, {17329, 37660}, {17334, 32851}, {17336, 30811}, {17347, 17720}, {17351, 30832}, {17777, 49511}, {17794, 31136}, {18134, 41313}, {21060, 49630}, {21093, 33082}, {21356, 56084}, {22034, 50084}, {24703, 47358}, {25378, 54352}, {25760, 62222}, {26117, 67850}, {27777, 32916}, {28538, 32926}, {28606, 31179}, {29573, 41839}, {29630, 41241}, {31140, 50075}, {31144, 31993}, {31151, 42056}, {32773, 47359}, {32942, 51003}, {33097, 50299}, {34064, 50125}, {34772, 50165}, {42334, 48641}, {48627, 63003}, {48643, 50309}, {50103, 68966}, {51678, 64002}, {53193, 53208}, {53598, 62297}, {56311, 56949}, {58820, 66637}, {60235, 64424}

X(69057) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5905, 50128}, {226, 50093, 2}, {321, 31143, 29615}, {329, 27184, 27064}, {908, 6646, 24627}, {3452, 26840, 27002}, {4415, 33066, 1999}, {4703, 33101, 3757}, {5233, 17276, 62300}, {17274, 31142, 2}, {17484, 26580, 894}

X(69058) = EULER LINE INTERCEPT OF X(5892)X(61690)

Barycentrics    4*a^10 - 9*a^8*(b^2 + c^2) + (b^2 - c^2)^4*(b^2 + c^2) + 2*a^6*(b^4 + 14*b^2*c^2 + c^4) - 2*a^2*(b^2 - c^2)^2*(3*b^4 - 2*b^2*c^2 + 3*c^4) + 8*a^4*(b^6 - 4*b^4*c^2 - 4*b^2*c^4 + c^6) : :

As a point on the Euler line, X(69058) has Shinagawa coefficients: {1/3 (-8 e + 5 (e + f)), -f}

See Gabi Cuc Cucoanes and David Nguyen, euclid 8571.

X(69058) lies on these lines: {2, 3}, {5892, 61690}, {10821, 39562}, {11202, 64730}, {11245, 63649}, {12099, 38793}, {15045, 59553}, {16226, 65094}, {23329, 35283}, {50979, 66730}, {61507, 64100}

X(69058) = midpoint of X(2) and X(17928)
X(69058) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3524, 7395}, {2, 6815, 5055}, {2, 10304, 6804}, {2, 13160, 15699}, {2, 15078, 67263}, {2, 17928, 30}, {140, 15330, 549}, {547, 549, 44218}, {631, 5020, 47090}

X(69059) = EULER LINE INTERCEPT OF X(373)X(67868)

Barycentrics    3*a^8*(b^2 + c^2) + 16*a^4*b^2*c^2*(b^2 + c^2) - 3*(b^2 - c^2)^4*(b^2 + c^2) + a^6*(-6*b^4 + 4*b^2*c^2 - 6*c^4) + 2*a^2*(b^2 - c^2)^2*(3*b^4 - 10*b^2*c^2 + 3*c^4) : :

As a point on the Euler line, X(69059) has Shinagawa coefficients: {1/3 (-5 e + 3 (e + f)), -e + f}

See Gabi Cuc Cucoanes and David Nguyen, euclid 8571.

X(69059) lies on these lines: {2, 3}, {373, 67868}, {12099, 36518}, {15045, 54039}, {15873, 21969}, {18358, 40318}, {47352, 67870}, {61507, 61744}

X(69059) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 381, 62962}, {5, 403, 37439}, {381, 5055, 44441}, {381, 5071, 47097}, {3091, 5020, 10151}, {5066, 10127, 381}

X(69060) = EULER LINE INTERCEPT OF X(567)X(59543)

Barycentrics    -a^10 + 3*a^2*(b^2 - c^2)^4 + 3*a^8*(b^2 + c^2) - (b^2 - c^2)^4*(b^2 + c^2) - 2*a^6*(b^4 + 3*b^2*c^2 + c^4) - 2*a^4*(b^6 - 6*b^4*c^2 - 6*b^2*c^4 + c^6) : :

As a point on the Euler line, X(69060) has Shinagawa coefficients: {1/3 (-((13 e)/4) + 2 (e + f)), -(e/4)}

See Gabi Cuc Cucoanes and David Nguyen, euclid 8571.

X(69060) lies on these lines: {2, 3}, {567, 59543}, {569, 61681}, {974, 5655}, {3167, 45967}, {3618, 39562}, {5065, 6128}, {9306, 61713}, {10170, 61645}, {11433, 50461}, {14561, 40670}, {14643, 68292}, {15037, 18928}, {15038, 37645}, {15087, 63084}, {18445, 37648}, {18952, 43598}, {23039, 61506}, {29959, 38317}, {31267, 38064}, {32140, 43614}, {36753, 59659}, {37506, 59648}, {38079, 63612}, {40280, 67890}, {55039, 64177}, {59373, 63703}

X(69060) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3091, 65085}, {2, 3545, 18281}, {2, 5055, 14787}, {2, 5071, 60763}, {5, 140, 63664}, {5, 3628, 52296}, {5, 10019, 5072}, {381, 5054, 54992}, {547, 6677, 67263}, {1656, 5020, 2072}, {6642, 16072, 38321}, {14782, 14783, 12086}, {15765, 18585, 11413}, {18586, 18587, 50143}

X(69061) = EULER LINE INTERCEPT OF X(6467)X(51737)

Barycentrics    -8*a^10 + 15*a^8*(b^2 + c^2) + (b^2 - c^2)^4*(b^2 + c^2) + 6*a^2*(b^4 - c^4)^2 + 2*a^6*(b^4 - 30*b^2*c^2 + c^4) - 16*a^4*(b^6 - 3*b^4*c^2 - 3*b^2*c^4 + c^6) : :

As a point on the Euler line, X(69061) has Shinagawa coefficients: {1/3 (-11 e + 7 (e + f)), 4 e - 3 (e + f)}

See Gabi Cuc Cucoanes and David Nguyen, euclid 8571.

X(69061) lies on these lines: {2, 3}, {6467, 51737}, {19161, 51132}, {23327, 55673}

X(69062) = BROCARD AXIS INTERCEPT OF X(230)X(1625)

Barycentrics    a^2*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^6*c^2 + 4*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + a^2*c^6 + b^2*c^6) : :

X(69062) lies on these lines: {3, 6}, {230, 1625}, {403, 61204}, {419, 2501}, {729, 53701}, {1495, 34982}, {3231, 52261}, {3331, 43291}, {5305, 48262}, {5915, 13509}, {10540, 53500}, {10601, 51372}, {11360, 63554}, {13754, 68805}, {18907, 59208}, {20960, 63555}, {21177, 65751}, {45935, 46301}, {46453, 60106}, {47421, 68802}, {53419, 68851}, {56957, 67540}

X(69062) = Schoutte-circle-inverse of X(68745)
X(69062) = crossdifference of every pair of points on line {523, 3917}
X(69062) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 187, 68737}, {15, 16, 68745}, {3003, 50387, 68740}, {41336, 58312, 2420}

X(69063) = BROCARD AXIS INTERCEPT OF X(242)X(514)

Barycentrics    a^2*(a + b)*(a + c)*(a^3*b^2 - a^2*b^3 - a*b^4 + b^5 + a^3*c^2 + 2*a*b^2*c^2 - b^3*c^2 - a^2*c^3 - b^2*c^3 - a*c^4 + c^5) : :

X(69063) lies on these lines: {1, 25255}, {3, 6}, {28, 4306}, {59, 51625}, {77, 44709}, {242, 514}, {759, 56910}, {916, 8608}, {954, 4653}, {995, 7677}, {1465, 18191}, {2287, 56809}, {2309, 64017}, {2999, 17187}, {3008, 18792}, {3190, 40571}, {3937, 46513}, {5736, 44150}, {8747, 58074}, {10571, 40980}, {14953, 68913}, {17010, 22350}, {17197, 53599}, {17205, 62786}, {18603, 62811}

X(69063) = X(i)-isoconjugate of X(j) for these (i,j): {37, 2989}, {65, 56110}, {72, 917}, {228, 57997}, {525, 36107}, {662, 66270}, {1577, 35182}, {4551, 60569}, {14208, 32699}
X(69063) = X(i)-Dao conjugate of X(j) for these (i,j): {118, 10}, {1084, 66270}, {39003, 525}, {40589, 2989}, {40602, 56110}
X(69063) = crosssum of X(42) and X(17747)
X(69063) = crossdifference of every pair of points on line {71, 523}
X(69063) = barycentric product X(i)*X(j) for these {i,j}: {27, 916}, {58, 48381}, {81, 1736}, {86, 8608}, {110, 55125}, {286, 2253}, {514, 4243}, {7192, 56742}, {14953, 54232}
X(69063) = barycentric quotient X(i)/X(j) for these {i,j}: {27, 57997}, {58, 2989}, {284, 56110}, {512, 66270}, {916, 306}, {1474, 917}, {1576, 35182}, {1736, 321}, {2253, 72}, {4243, 190}, {7252, 60569}, {8608, 10}, {32676, 36107}, {47407, 51366}, {48381, 313}, {55125, 850}, {56742, 3952}, {61206, 32699}
X(69063) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {58, 68746, 386}, {284, 54411, 991}, {3286, 68730, 13329}

X(69064) = BROCARD AXIS INTERCEPT OF X(240)X(522)

Barycentrics    a^2*(b + c)*(a^4*b^2 - 2*a^2*b^4 + b^6 - a^4*b*c + 2*a^2*b^3*c - b^5*c + a^4*c^2 - a^2*b^2*c^2 + 2*a^2*b*c^3 - 2*a^2*c^4 - b*c^5 + c^6) : :

X(69064) lies on these lines: {1, 55351}, {3, 6}, {109, 44113}, {226, 22069}, {240, 522}, {851, 22094}, {916, 8607}, {1708, 20966}, {1779, 3192}, {1834, 67980}, {2238, 8558}, {3755, 3778}, {4055, 63807}, {4357, 18690}, {22084, 68913}, {24984, 25000}, {40952, 56549}

X(69064) = X(59016)-complementary conjugate of X(960)
X(69064) = crossdifference of every pair of points on line {48, 523}

X(69065) = BROCARD AXIS INTERCEPT OF X(320)X(350)

Barycentrics    a^2*(a + b)*(a + c)*(a*b^3 - b^4 + a*c^3 - c^4) : :

X(69065) lies on these lines: {3, 6}, {81, 64751}, {320, 350}, {674, 18206}, {692, 16876}, {766, 35552}, {1985, 30945}, {3056, 16696}, {3271, 52897}, {3739, 22412}, {3764, 17187}, {3770, 24523}, {3784, 4675}, {3792, 52680}, {3868, 20718}, {4643, 26892}, {4749, 40153}, {4852, 63516}, {5880, 18180}, {7184, 18195}, {7186, 18169}, {7976, 56023}, {16700, 63513}, {16736, 63511}, {16752, 64523}, {17197, 29353}, {17444, 55004}, {17768, 35637}, {18166, 21746}, {18184, 57022}, {18191, 69031}, {18610, 23124}, {21252, 46515}, {25048, 62636}, {25505, 26094}, {29767, 64709}, {39551, 64016}, {41582, 47595}

X(69065) = X(i)-isoconjugate of X(j) for these (i,j): {42, 767}, {1918, 57951}, {4551, 60572}
X(69065) = X(i)-Dao conjugate of X(j) for these (i,j): {34021, 57951}, {40592, 767}
X(69065) = crossdifference of every pair of points on line {213, 523}
X(69065) = barycentric product X(i)*X(j) for these {i,j}: {21, 45267}, {58, 63817}, {81, 35552}, {110, 63813}, {274, 766}, {6385, 8629}
X(69065) = barycentric quotient X(i)/X(j) for these {i,j}: {81, 767}, {274, 57951}, {766, 37}, {7252, 60572}, {8629, 213}, {35552, 321}, {45267, 1441}, {63813, 850}, {63817, 313}

X(69066) = BROCARD AXIS INTERCEPT OF X(513)X(3716)

Barycentrics    a^2*(a^3*b^3 - a*b^5 - a^3*b^2*c + a^2*b^3*c + a*b^4*c - b^5*c - a^3*b*c^2 + a^3*c^3 + a^2*b*c^3 + a*b*c^4 - a*c^5 - b*c^5) : :

X(69066) lies on these lines: {2, 44151}, {3, 6}, {75, 22413}, {320, 3937}, {513, 3716}, {1284, 37523}, {3271, 69016}, {3784, 8731}, {8609, 55004}, {9017, 25083}, {16876, 44112}, {17139, 30943}, {18156, 23420}, {20718, 64546}, {22412, 28287}, {23456, 68974}, {23526, 44179}, {25490, 25504}, {26091, 26106}, {34831, 52547}, {35466, 69031}, {37142, 57738}, {41877, 49537}

X(69066) = complement of X(44151)
X(69066) = X(59040)-complementary conjugate of X(2)
X(69066) = crosssum of X(6) and X(863)
X(69066) = crossdifference of every pair of points on line {523, 2176}
X(69066) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {22413, 23440, 75}

X(69067) = BROCARD AXIS INTERCEPT OF X(2)X(39952)

Barycentrics    a^2*(-(a*b^3) + 2*a^2*b*c + a*b^2*c - b^3*c + a*b*c^2 - a*c^3 - b*c^3) : :

X(69067) lies on these lines: {2, 39952}, {3, 6}, {31, 22199}, {37, 5625}, {38, 23533}, {63, 23543}, {81, 21838}, {86, 16589}, {87, 21384}, {190, 20688}, {238, 1015}, {292, 1757}, {538, 30940}, {667, 6373}, {672, 21760}, {799, 69013}, {896, 3121}, {1001, 31456}, {1107, 33682}, {1475, 14758}, {1500, 4649}, {1573, 50302}, {1575, 49650}, {1911, 20683}, {2229, 16704}, {2238, 18792}, {2241, 20992}, {2260, 59481}, {2275, 16468}, {2308, 23632}, {2309, 20963}, {2350, 20965}, {3218, 6377}, {3219, 21827}, {3229, 18206}, {3550, 24528}, {3774, 4663}, {4418, 22184}, {4434, 21893}, {4641, 16584}, {4661, 30651}, {4722, 21814}, {4753, 21897}, {5283, 17379}, {6683, 20148}, {8616, 63509}, {9359, 24727}, {15485, 63493}, {16502, 23433}, {16503, 46189}, {16592, 35466}, {16738, 52538}, {17178, 27040}, {17187, 21753}, {17448, 49482}, {17768, 39786}, {17770, 18904}, {18166, 52539}, {20142, 37128}, {20154, 31198}, {20158, 24598}, {20456, 20861}, {20691, 49685}, {21788, 52963}, {36598, 51974}, {36812, 51314}, {45913, 63822}, {46922, 50264}, {47080, 62313}, {62478, 62481}, {66878, 68997}

X(69067) = Moses-circle-inverse of X(3110)
X(69067) = X(36066)-Ceva conjugate of X(512)
X(69067) = crosspoint of X(6) and X(37128)
X(69067) = crosssum of X(i) and X(j) for these (i,j): {2, 2238}, {6, 19308}, {2086, 24533}
X(69067) = crossdifference of every pair of points on line {192, 523}
X(69067) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 3286, 41333}, {6, 3736, 20970}, {6, 33863, 4279}, {6, 37507, 32}, {6, 62692, 2092}, {292, 1757, 21830}, {672, 23579, 21760}, {1475, 22343, 23660}, {2028, 2029, 3110}, {3286, 41333, 187}, {20142, 37128, 69016}

X(69068) = BROCARD AXIS INTERCEPT OF X(81)X(172)

Barycentrics    a^3*(a + b)*(a + c)*(a*b + a*c - b*c) : :

X(69068) lies on these lines: {3, 6}, {21, 1107}, {81, 172}, {112, 15323}, {163, 34077}, {333, 3661}, {560, 1932}, {849, 59192}, {859, 16502}, {904, 2194}, {906, 5247}, {910, 16716}, {1010, 4386}, {1408, 1415}, {1437, 64215}, {1438, 52150}, {1778, 1792}, {2176, 20760}, {2204, 8852}, {2206, 7121}, {2238, 27660}, {2240, 30984}, {2241, 4653}, {2242, 4658}, {2275, 4225}, {2280, 10457}, {5277, 25526}, {5291, 64072}, {5301, 54354}, {5331, 40746}, {5546, 56012}, {7031, 21384}, {9310, 62740}, {10461, 16973}, {13588, 16606}, {16519, 35623}, {16604, 37442}, {20691, 51319}, {21879, 28631}, {23640, 52426}, {27164, 27274}, {27632, 30945}, {33296, 33890}, {39248, 46877}

X(69068) = isogonal conjugate of X(60244)
X(69068) = isogonal conjugate of the complement of X(36857)
X(69068) = isogonal conjugate of the isotomic conjugate of X(27644)
X(69068) = X(i)-Ceva conjugate of X(j) for these (i,j): {2206, 1333}, {4600, 692}, {38813, 2194}
X(69068) = X(i)-cross conjugate of X(j) for these (i,j): {2209, 38832}, {21835, 8640}, {38832, 1333}, {56806, 33296}
X(69068) = X(i)-isoconjugate of X(j) for these (i,j): {1, 60244}, {2, 42027}, {10, 330}, {37, 6384}, {42, 6383}, {65, 27424}, {75, 16606}, {76, 23493}, {87, 321}, {210, 7209}, {226, 7155}, {274, 7148}, {310, 6378}, {313, 2162}, {349, 2053}, {523, 4598}, {561, 21759}, {661, 18830}, {693, 65167}, {850, 34071}, {932, 1577}, {1215, 27447}, {1237, 51974}, {1240, 45218}, {1441, 2319}, {1969, 22381}, {2998, 63618}, {3120, 5383}, {3261, 65163}, {3701, 7153}, {3967, 27498}, {3971, 53677}, {4024, 56053}, {4033, 43931}, {7121, 27801}, {18082, 62541}, {20691, 53679}, {21051, 32039}, {27432, 34860}, {27438, 39742}, {27496, 56174}, {27809, 67196}, {30710, 45197}, {34475, 40720}, {39914, 43534}, {40718, 51837}
X(69068) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 60244}, {206, 16606}, {798, 3120}, {3835, 16732}, {6377, 20948}, {32664, 42027}, {36830, 18830}, {40368, 21759}, {40589, 6384}, {40592, 6383}, {40598, 27801}, {40602, 27424}, {40610, 850}
X(69068) = cevapoint of X(i) and X(j) for these (i,j): {32, 56836}, {2209, 62420}, {8640, 21835}
X(69068) = crosspoint of X(163) and X(4567)
X(69068) = crosssum of X(i) and X(j) for these (i,j): {10, 21071}, {1577, 3125}
X(69068) = trilinear pole of line {8640, 57074}
X(69068) = crossdifference of every pair of points on line {523, 20910}
X(69068) = barycentric product X(i)*X(j) for these {i,j}: {1, 38832}, {6, 27644}, {21, 1403}, {28, 20760}, {31, 33296}, {32, 31008}, {43, 58}, {56, 56181}, {81, 2176}, {86, 2209}, {99, 8640}, {100, 16695}, {101, 18197}, {110, 4083}, {112, 25098}, {162, 22090}, {163, 3835}, {190, 57074}, {192, 1333}, {213, 7304}, {274, 62420}, {284, 1423}, {333, 41526}, {593, 20691}, {662, 20979}, {667, 62530}, {692, 17217}, {849, 3971}, {906, 17921}, {1110, 23824}, {1178, 51902}, {1252, 16742}, {1408, 27538}, {1412, 3208}, {1415, 27527}, {1474, 22370}, {1576, 20906}, {1783, 23092}, {1919, 36860}, {2194, 3212}, {2206, 6376}, {2328, 62791}, {3123, 4570}, {3733, 52923}, {4110, 16947}, {4556, 21834}, {4567, 6377}, {4590, 21835}, {4591, 14408}, {4595, 57129}, {4600, 38986}, {4601, 21762}, {5009, 41531}, {5546, 43051}, {13588, 57505}, {27891, 66931}, {30545, 57657}, {34476, 40780}, {38813, 41886}, {40415, 56806}, {40432, 51319}, {50491, 52935}
X(69068) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 60244}, {31, 42027}, {32, 16606}, {43, 313}, {58, 6384}, {81, 6383}, {110, 18830}, {163, 4598}, {192, 27801}, {284, 27424}, {560, 23493}, {1333, 330}, {1403, 1441}, {1412, 7209}, {1423, 349}, {1501, 21759}, {1576, 932}, {1918, 7148}, {2176, 321}, {2194, 7155}, {2205, 6378}, {2206, 87}, {2209, 10}, {3123, 21207}, {3208, 30713}, {3835, 20948}, {4083, 850}, {6377, 16732}, {7304, 6385}, {8640, 523}, {14575, 22381}, {16695, 693}, {16742, 23989}, {16946, 27432}, {16947, 7153}, {17217, 40495}, {18197, 3261}, {20284, 20234}, {20691, 28654}, {20760, 20336}, {20906, 44173}, {20979, 1577}, {21762, 3125}, {21834, 52623}, {21835, 115}, {22090, 14208}, {22370, 40071}, {22386, 18210}, {23092, 15413}, {25098, 3267}, {27644, 76}, {31008, 1502}, {32739, 65167}, {33296, 561}, {33628, 27496}, {38832, 75}, {38986, 3120}, {41526, 226}, {50491, 4036}, {51319, 3963}, {51902, 1237}, {52923, 27808}, {56181, 3596}, {56806, 2887}, {56836, 63618}, {57074, 514}, {57657, 2319}, {62420, 37}, {62530, 6386}
X(69068) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 58, 1333}, {1333, 33882, 4273}

X(69069) = BROCARD AXIS INTERCEPT OF X(99)X(101)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a*b^2 - b^2*c + a*c^2 - b*c^2) : :

X(69069) lies on these lines: {3, 6}, {21, 24491}, {99, 101}, {110, 43077}, {112, 28469}, {163, 34071}, {692, 8671}, {805, 2702}, {1293, 26714}, {2284, 3733}, {2715, 67801}, {3737, 69000}, {7252, 23832}, {25424, 30554}, {26716, 39639}, {40499, 53273}, {53891, 59067}, {57129, 68825}

X(69069) = X(i)-isoconjugate of X(j) for these (i,j): {37, 62638}, {75, 66947}, {321, 23355}, {512, 32020}, {513, 27809}, {514, 18793}, {523, 20332}, {661, 3226}, {727, 1577}, {850, 34077}, {3121, 54985}, {3125, 8709}, {4017, 36799}, {7178, 8851}, {18098, 35367}, {55244, 60865}
X(69069) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 66947}, {17793, 1577}, {20532, 850}, {34961, 36799}, {36830, 3226}, {39026, 27809}, {39054, 32020}, {40589, 62638}
X(69069) = crosssum of X(i) and X(j) for these (i,j): {523, 21053}, {661, 4010}, {4024, 21722}
X(69069) = trilinear pole of line {3009, 20785}
X(69069) = crossdifference of every pair of points on line {523, 3122}
X(69069) = barycentric product X(i)*X(j) for these {i,j}: {58, 23354}, {99, 3009}, {100, 18792}, {101, 62636}, {110, 726}, {163, 52043}, {249, 21053}, {643, 1463}, {648, 20785}, {662, 1575}, {799, 21760}, {811, 20777}, {1576, 35538}, {3837, 4570}, {4584, 17475}, {4589, 20663}, {4600, 6373}, {4610, 21830}, {5546, 43040}, {17940, 59724}, {17943, 64236}, {20681, 36066}, {36860, 51864}
X(69069) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 66947}, {58, 62638}, {101, 27809}, {110, 3226}, {163, 20332}, {662, 32020}, {692, 18793}, {726, 850}, {1463, 4077}, {1575, 1577}, {1576, 727}, {2206, 23355}, {3009, 523}, {3837, 21207}, {4570, 8709}, {4600, 54985}, {5546, 36799}, {6373, 3120}, {17187, 35367}, {18792, 693}, {20663, 4010}, {20671, 21053}, {20750, 24459}, {20777, 656}, {20785, 525}, {21053, 338}, {21760, 661}, {21830, 4024}, {22092, 4466}, {23354, 313}, {35538, 44173}, {38367, 39786}, {40155, 35352}, {52043, 20948}, {62530, 64226}, {62636, 3261}, {65375, 8851}

X(69070) = BROCARD AXIS INTERCEPT OF X(30)X(34978)

Barycentrics    a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^6*c^2 + 4*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + a^2*c^6 + b^2*c^6 : :
X(69070) = 2 X[237] + X[25051], 4 X[3003] - X[14570], 2 X[3003] + X[51481], X[14570] + 2 X[51481], X[5201] + 2 X[7668], 2 X[5201] + X[14957], 4 X[7668] - X[14957], X[20975] + 2 X[63736], X[25046] - 4 X[51735]

X(69070) lies on these lines: {2, 6}, {30, 34978}, {237, 25051}, {311, 800}, {576, 37121}, {598, 54843}, {648, 44893}, {671, 54547}, {2489, 4580}, {3003, 14570}, {3564, 25314}, {5201, 7668}, {6179, 37125}, {20975, 63736}, {22087, 32515}, {25046, 51735}, {34288, 42354}, {37943, 41205}, {41678, 44138}, {51833, 62961}, {53245, 64781}

X(69070) = crossdifference of every pair of points on line {512, 20775}
X(69070) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3003, 51481, 14570}, {5201, 7668, 14957}, {37784, 44375, 2407}

X(69071) = X(1)X(21)∩X(2)X(16699)

Barycentrics    a*(a + b)*(a + c)*(a^3*b^2 - a^2*b^3 - a*b^4 + b^5 + a^3*c^2 + 2*a*b^2*c^2 - b^3*c^2 - a^2*c^3 - b^2*c^3 - a*c^4 + c^5) : :

X(69071) lies on these lines: {1, 21}, {2, 16699}, {27, 3188}, {77, 40979}, {86, 16743}, {216, 5740}, {241, 14953}, {348, 16697}, {693, 905}, {800, 25000}, {857, 3002}, {859, 7291}, {1333, 26215}, {1442, 46882}, {2287, 26669}, {3100, 3286}, {3672, 16696}, {4850, 26647}, {7190, 18164}, {8608, 48381}, {10394, 54411}, {14964, 68759}, {16050, 26690}, {16700, 62208}, {16738, 24547}, {17197, 22464}, {17863, 18606}, {24609, 63078}, {24624, 65214}, {37169, 63008}, {37787, 68730}, {46889, 60970}

X(69071) = X(i)-isoconjugate of X(j) for these (i,j): {42, 2989}, {71, 917}, {110, 66270}, {523, 35182}, {525, 32699}, {656, 36107}, {1400, 56110}, {2200, 57997}, {4559, 60569}, {15380, 17747}, {51436, 57752}
X(69071) = X(i)-Dao conjugate of X(j) for these (i,j): {118, 37}, {244, 66270}, {39003, 656}, {40582, 56110}, {40592, 2989}, {40596, 36107}, {55067, 60569}
X(69071) = cevapoint of X(1736) and X(8608)
X(69071) = crossdifference of every pair of points on line {228, 661}
X(69071) = barycentric product X(i)*X(j) for these {i,j}: {81, 48381}, {86, 1736}, {274, 8608}, {286, 916}, {662, 55125}, {693, 4243}, {2253, 44129}, {7199, 56742}
X(69071) = barycentric quotient X(i)/X(j) for these {i,j}: {21, 56110}, {28, 917}, {81, 2989}, {112, 36107}, {163, 35182}, {286, 57997}, {661, 66270}, {916, 72}, {1736, 10}, {2253, 71}, {3737, 60569}, {4243, 100}, {8608, 37}, {32676, 32699}, {48381, 321}, {55125, 1577}, {56742, 1018}
X(69071) = {X(40773),X(64415)}-harmonic conjugate of X(28606)

X(69072) = X(1)X(21)∩X(667)X(693)

Barycentrics    a*(a + b)*(a + c)*(a^3*b^2 - a^2*b^3 + a^3*c^2 + 2*a*b^2*c^2 - b^3*c^2 - a^2*c^3 - b^2*c^3) : :

X(69072) lies on these lines: {1, 21}, {160, 29763}, {172, 18098}, {667, 693}, {1634, 69008}, {2862, 53683}, {3226, 33295}, {3286, 4366}, {4228, 26236}, {8266, 29453}, {11349, 29561}, {14665, 34594}, {16704, 20044}, {20352, 20475}, {29437, 35222}, {30664, 53707}

X(69072) = X(523)-isoconjugate of X(20696)
X(69072) = cevapoint of X(20372) and X(20475)
X(69072) = crossdifference of every pair of points on line {661, 21814}
X(69072) = barycentric product X(i)*X(j) for these {i,j}: {58, 20453}, {81, 20352}, {86, 20372}, {274, 20475}, {286, 20747}, {662, 20525}, {757, 20501}, {1509, 20723}
X(69072) = barycentric quotient X(i)/X(j) for these {i,j}: {163, 20696}, {20352, 321}, {20372, 10}, {20453, 313}, {20475, 37}, {20501, 1089}, {20525, 1577}, {20723, 594}, {20747, 72}

X(69073) = X(2)X(6)∩X(274)X(20247)

Barycentrics    (a + b)*(a + c)*(a^2*b^2 - a*b^3 + a*b^2*c - b^3*c + a^2*c^2 + a*b*c^2 - a*c^3 - b*c^3) : :

X(69073) lies on these lines: {2, 6}, {274, 20247}, {354, 16707}, {1621, 54112}, {3315, 68983}, {3766, 6372}, {3840, 17176}, {4576, 18157}, {8033, 39734}, {16703, 18165}, {16741, 18191}, {17198, 51370}, {17208, 18169}, {18138, 33798}, {21281, 61155}, {29824, 30940}

X(69073) = X(872)-isoconjugate of X(2368)
X(69073) = cevapoint of X(30109) and X(57024)
X(69073) = crossdifference of every pair of points on line {512, 7109}
X(69073) = barycentric product X(i)*X(j) for these {i,j}: {86, 30109}, {274, 57024}, {1509, 68939}
X(69073) = barycentric quotient X(i)/X(j) for these {i,j}: {1509, 2368}, {2388, 1500}, {30109, 10}, {57024, 37}, {68939, 594}
X(69073) = {X(17208),X(18169)}-harmonic conjugate of X(61407)

X(69074) = X(6)X(31)∩X(239)X(514)

Barycentrics    a^2*(a^2*b^2 - a*b^3 + a*b^2*c - b^3*c + a^2*c^2 + a*b*c^2 - a*c^3 - b*c^3) : :
X(69074) = 4 X[43046] - 3 X[46125]

X(69074) lies on these lines: {6, 31}, {171, 2350}, {238, 38346}, {239, 514}, {511, 20974}, {518, 2225}, {899, 20459}, {1015, 3231}, {1150, 24586}, {1197, 17187}, {1621, 40586}, {2368, 56431}, {2702, 53696}, {3051, 22199}, {3121, 68750}, {3509, 16547}, {3685, 61163}, {3691, 40955}, {3722, 39258}, {3730, 61155}, {3757, 21061}, {3917, 23653}, {3938, 5364}, {3952, 20372}, {4253, 17126}, {8622, 20456}, {9052, 23988}, {14751, 17011}, {16738, 17208}, {16975, 17449}, {17155, 21387}, {17165, 21369}, {17474, 22065}, {17763, 24727}, {20665, 32912}, {20718, 38347}, {29824, 61234}, {36213, 38990}, {68821, 68969}

X(69074) = reflection of X(46148) in X(2225)
X(69074) = isogonal conjugate of the isotomic conjugate of X(30109)
X(69074) = X(2388)-cross conjugate of X(57024)
X(69074) = X(37)-isoconjugate of X(2368)
X(69074) = X(i)-Dao conjugate of X(j) for these (i,j): {40589, 2368}, {57024, 49753}
X(69074) = crosssum of X(2) and X(40859)
X(69074) = crossdifference of every pair of points on line {42, 514}
X(69074) = X(i)-line conjugate of X(j) for these (i,j): {6, 42}, {239, 514}
X(69074) = barycentric product X(i)*X(j) for these {i,j}: {1, 57024}, {6, 30109}, {58, 68939}, {86, 2388}
X(69074) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 2368}, {2388, 10}, {30109, 76}, {57024, 75}, {68939, 313}
X(69074) = {X(1197),X(23632)}-harmonic conjugate of X(17187)

X(69075) = X(6)X(69047)∩X(30)X(511)

Barycentrics    (b - c)*(-(a^2*b^2) + a*b^3 - a*b^2*c + b^3*c - a^2*c^2 - a*b*c^2 + a*c^3 + b*c^3) : :

X(69075) lies on these lines: {6, 69047}, {30, 511}, {141, 68882}, {190, 46725}, {1086, 16727}, {1769, 4364}, {3739, 20520}, {4361, 53357}, {4406, 47676}, {4422, 23988}, {4444, 53565}, {4665, 4768}, {6546, 14407}, {7192, 69004}, {13259, 49524}, {17243, 68813}, {23810, 68972}, {24354, 62323}, {53527, 62552}

X(69075) = crossdifference of every pair of points on line {6, 23392}
X(69075) = X(69047)-line conjugate of X(6)

X(69076) = X(2)X(6)∩X(38)X(17203)

Barycentrics    (a + b)*(a + c)*(a*b^3 - b^4 + a*c^3 - c^4) : :

X(69076) lies on these lines: {2, 6}, {38, 17203}, {274, 33108}, {310, 11680}, {626, 18171}, {693, 784}, {982, 16891}, {2886, 16748}, {2887, 17208}, {3006, 17198}, {3705, 16703}, {4766, 18206}, {16887, 25760}, {17205, 21241}, {33136, 62755}

X(69076) = X(767)-isoconjugate of X(1918)
X(69076) = X(i)-Dao conjugate of X(j) for these (i,j): {34021, 767}, {40625, 60572}
X(69076) = cevapoint of X(35552) and X(63817)
X(69076) = crossdifference of every pair of points on line {512, 2205}
X(69076) = barycentric product X(i)*X(j) for these {i,j}: {86, 63817}, {99, 63813}, {274, 35552}, {314, 45267}, {766, 6385}
X(69076) = barycentric quotient X(i)/X(j) for these {i,j}: {274, 767}, {766, 213}, {4560, 60572}, {6385, 57951}, {8629, 2205}, {35552, 37}, {45267, 65}, {63813, 523}, {63817, 10}
X(69076) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3006, 17198, 18157}, {3705, 17177, 16703}

X(69077) = X(1)X(21)∩X(514)X(1921)

Barycentrics    a*(a + b)*(a + c)*(a*b^3 - b^4 + a*c^3 - c^4) : :

X(69077) lies on these lines: {1, 21}, {39, 18189}, {86, 30105}, {325, 14963}, {333, 30108}, {514, 1921}, {517, 16728}, {760, 3286}, {766, 35552}, {2170, 62755}, {2275, 18167}, {3061, 16887}, {3721, 18171}, {3735, 16696}, {7202, 8682}, {14964, 51369}, {16609, 30107}, {16726, 21331}, {16735, 52564}, {17175, 17451}, {17197, 46180}, {17203, 23636}, {17210, 39244}, {17671, 18134}, {18161, 33936}, {18184, 30941}, {18204, 56805}, {30939, 49753}, {49758, 52897}

X(69077) = X(i)-isoconjugate of X(j) for these (i,j): {213, 767}, {2205, 57951}, {4559, 60572}
X(69077) = X(i)-Dao conjugate of X(j) for these (i,j): {6626, 767}, {55067, 60572}
X(69077) = crossdifference of every pair of points on line {661, 1918}
X(69077) = barycentric product X(i)*X(j) for these {i,j}: {81, 63817}, {86, 35552}, {310, 766}, {333, 45267}, {662, 63813}
X(69077) = barycentric quotient X(i)/X(j) for these {i,j}: {86, 767}, {310, 57951}, {766, 42}, {3737, 60572}, {8629, 1918}, {35552, 10}, {45267, 226}, {63813, 1577}, {63817, 321}
X(69077) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3061, 18176, 16887}, {18184, 57015, 30941}

X(69078) = X(2)X(39)∩X(100)X(68983)

Barycentrics    (a + b)*(a + c)*(a^2*b^2 + a*b^3 - a*b^2*c - b^3*c + a^2*c^2 - a*b*c^2 + a*c^3 - b*c^3) : :

X(69078) lies on these lines: {2, 39}, {100, 68983}, {659, 3004}, {3752, 16707}, {4576, 18792}, {8033, 39747}, {15523, 24170}, {16700, 16703}, {16726, 16741}, {16739, 18601}, {16753, 18157}, {17150, 34063}, {17176, 24165}, {17495, 30940}, {17763, 62755}, {29823, 68984}, {30941, 32842}, {33296, 35983}, {40087, 46720}, {57029, 68750}

X(69078) = X(57039)-Dao conjugate of X(68897)
X(69078) = cevapoint of X(57029) and X(57039)
X(69078) = crossdifference of every pair of points on line {669, 1500}
X(69078) = barycentric product X(i)*X(j) for these {i,j}: {86, 57029}, {99, 68956}, {274, 57039}, {310, 68750}, {873, 68954}, {1509, 68955}
X(69078) = barycentric quotient X(i)/X(j) for these {i,j}: {52461, 1824}, {57029, 10}, {57039, 37}, {68750, 42}, {68940, 1500}, {68954, 756}, {68955, 594}, {68956, 523}
X(69078) = {X(16739),X(18601)}-harmonic conjugate of X(61407)

X(69079) = X(76)X(321)∩X(320)X(350)

Barycentrics    b*c*(-(a^2*b^2) - a*b^3 + a*b^2*c + b^3*c - a^2*c^2 + a*b*c^2 - a*c^3 + b*c^3) : :

X(69079) lies on these lines: {76, 321}, {305, 33146}, {320, 350}, {1086, 3266}, {1920, 40013}, {1978, 62304}, {3782, 8024}, {3995, 18052}, {4389, 26234}, {4415, 39998}, {4442, 64223}, {16703, 17184}, {16705, 18601}, {16707, 19786}, {16727, 40075}, {17152, 33075}, {17155, 59510}, {17165, 18057}, {18066, 52049}, {18067, 32925}, {18140, 41242}, {32845, 42721}, {40087, 40603}, {50102, 69052}, {57029, 68954}

X(69079) = isotomic conjugate of the isogonal conjugate of X(57039)
X(69079) = X(68955)-cross conjugate of X(57029)
X(69079) = X(57039)-Dao conjugate of X(5291)
X(69079) = crossdifference of every pair of points on line {213, 1980}
X(69079) = barycentric product X(i)*X(j) for these {i,j}: {75, 57029}, {76, 57039}, {274, 68955}, {305, 52461}, {310, 68954}, {561, 68750}, {668, 68956}, {6385, 68940}
X(69079) = barycentric quotient X(i)/X(j) for these {i,j}: {52461, 25}, {57029, 1}, {57039, 6}, {68750, 31}, {68940, 213}, {68954, 42}, {68955, 37}, {68956, 513}

X(69080) = X(10)X(75)∩X(190)X(316)

Barycentrics    a^3*b - a^2*b^2 - a*b^3 + b^4 + a^3*c - a^2*b*c - a*b^2*c + b^3*c - a^2*c^2 - a*b*c^2 + b^2*c^2 - a*c^3 + b*c^3 + c^4 : :

X(69080) lies on these lines: {10, 75}, {99, 16086}, {190, 316}, {325, 33948}, {671, 54979}, {1016, 1252}, {1265, 7763}, {3685, 19987}, {3695, 33297}, {3710, 33939}, {4115, 41324}, {4561, 7799}, {21711, 40859}, {29615, 50105}, {32014, 64985}, {46136, 58128}

X(69080) = isotomic conjugate of the isogonal conjugate of X(56808)
X(69080) = X(i)-isoconjugate of X(j) for these (i,j): {667, 2690}, {2203, 38535}, {57129, 66274}
X(69080) = X(i)-Dao conjugate of X(j) for these (i,j): {6631, 2690}, {62564, 38535}
X(69080) = crossdifference of every pair of points on line {1919, 8034}
X(69080) = barycentric product X(i)*X(j) for these {i,j}: {76, 56808}, {305, 56747}, {1978, 2774}, {2073, 40071}, {27808, 42744}
X(69080) = barycentric quotient X(i)/X(j) for these {i,j}: {190, 2690}, {306, 38535}, {2073, 1474}, {2774, 649}, {3952, 66274}, {17233, 39993}, {42744, 3733}, {56747, 25}, {56808, 6}

X(69081) = X(58)X(86)∩X(325)X(523)

Barycentrics    a^3*b^2 + a^2*b^3 - a*b^4 - b^5 + a^3*c^2 + a^2*c^3 - a*c^4 - c^5 : :

X(69081) lies on these lines: {58, 86}, {325, 523}, {1959, 69005}, {4360, 24211}, {5988, 68986}, {6703, 24593}, {15586, 24636}, {17791, 56249}, {19867, 59509}, {20531, 58365}, {21094, 65205}, {21287, 41808}, {27691, 29509}, {27725, 29508}, {29823, 31117}, {30759, 30761}, {65161, 69036}

X(69081) = isotomic conjugate of X(2372)
X(69081) = isotomic conjugate of the isogonal conjugate of X(2392)
X(69081) = X(31)-isoconjugate of X(2372)
X(69081) = X(2)-Dao conjugate of X(2372)
X(69081) = crossdifference of every pair of points on line {32, 4079}
X(69081) = barycentric product X(76)*X(2392)
X(69081) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2372}, {2392, 6}
X(69081) = {X(325),X(69042)}-harmonic conjugate of X(35550)

X(69082) = X(1)X(21)∩X(239)X(244)

Barycentrics    a*(-(a*b^3) + 2*a^2*b*c + a*b^2*c - b^3*c + a*b*c^2 - a*c^3 - b*c^3) : :

X(69082) lies on these lines: {1, 21}, {6, 21330}, {37, 64561}, {42, 4447}, {69, 63497}, {86, 3728}, {192, 67024}, {193, 63515}, {194, 32915}, {213, 4722}, {239, 244}, {256, 20090}, {274, 18059}, {291, 6542}, {292, 3930}, {320, 3123}, {330, 10453}, {354, 17448}, {518, 3009}, {524, 3122}, {649, 4083}, {661, 9422}, {714, 30939}, {740, 62636}, {748, 21384}, {749, 17240}, {756, 16826}, {894, 22167}, {980, 46904}, {982, 4393}, {984, 5625}, {1100, 4022}, {1107, 3720}, {1654, 22174}, {1739, 50018}, {1757, 22220}, {2176, 32912}, {2223, 3722}, {2227, 30941}, {2228, 17374}, {2234, 16726}, {2309, 64546}, {2664, 21805}, {2667, 16696}, {2895, 63519}, {3121, 3229}, {3227, 53195}, {3248, 57024}, {3726, 20590}, {3778, 3879}, {3882, 20984}, {3912, 20456}, {3914, 24215}, {3938, 21010}, {3948, 68873}, {3953, 49477}, {4416, 22172}, {4436, 18198}, {4443, 17378}, {4446, 17377}, {4694, 50023}, {4695, 50016}, {4735, 50125}, {4741, 24456}, {4938, 68954}, {5282, 16524}, {6007, 53541}, {6707, 21699}, {7032, 35892}, {7238, 19945}, {8298, 19308}, {8610, 9038}, {9025, 23633}, {9507, 49760}, {12782, 17389}, {13476, 17445}, {16738, 25124}, {16816, 17063}, {16823, 46190}, {16827, 32864}, {16834, 42040}, {16973, 21346}, {16975, 17450}, {17065, 17363}, {17144, 17155}, {17178, 25295}, {17187, 18172}, {17300, 24575}, {17375, 41886}, {17390, 21035}, {17499, 22189}, {17793, 27166}, {18170, 64555}, {18827, 65285}, {20055, 46032}, {20340, 25298}, {23682, 33136}, {24168, 50021}, {24214, 33145}, {24429, 28558}, {24443, 49488}, {24621, 32860}, {24653, 27303}, {24923, 42334}, {25504, 25611}, {26061, 27248}, {27044, 40533}, {27272, 33115}, {28269, 29960}, {28288, 33087}, {29150, 48114}, {29580, 42039}, {29584, 42038}, {29597, 42041}, {31330, 31997}, {36812, 59219}, {39925, 40858}, {40783, 63527}, {41815, 67979}, {52655, 63509}, {53338, 68993}, {53559, 68996}, {65161, 68942}, {68986, 69008}

X(69082) = reflection of X(i) in X(j) for these {i,j}: {2234, 16726}, {25298, 20340}, {53338, 68993}, {65161, 68942}
X(69082) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {7121, 39367}, {18268, 21219}
X(69082) = X(65258)-Ceva conjugate of X(661)
X(69082) = crosspoint of X(1) and X(18827)
X(69082) = crosssum of X(1) and X(3747)
X(69082) = crossdifference of every pair of points on line {43, 661}
X(69082) = X(9422)-line conjugate of X(661)
X(69082) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 18206, 3747}, {1, 20985, 17469}, {1, 32913, 62813}, {1, 40773, 1962}, {1, 62853, 31}, {86, 24437, 3728}, {354, 17448, 21352}, {1654, 63520, 22174}, {3747, 18206, 896}, {17178, 25295, 51575}, {17449, 68751, 20358}

X(69083) = X(2)X(6)∩X(7)X(11680)

Barycentrics    a^3*b - a^2*b^2 - a*b^3 + b^4 + a^3*c + 2*a^2*b*c - b^3*c - a^2*c^2 - a*c^3 - b*c^3 + c^4 : :

X(69083) lies on these lines: {1, 25581}, {2, 6}, {7, 11680}, {8, 25583}, {10, 7278}, {11, 20347}, {38, 24241}, {56, 21285}, {116, 45751}, {145, 25592}, {150, 54391}, {218, 28734}, {239, 25593}, {348, 12649}, {442, 17169}, {518, 33864}, {522, 693}, {758, 69035}, {857, 18206}, {1014, 2893}, {1210, 26563}, {1280, 51614}, {1434, 2475}, {1442, 67126}, {1444, 35989}, {1475, 17046}, {1737, 30806}, {1834, 18600}, {3207, 20071}, {3218, 4872}, {3509, 24712}, {3663, 42038}, {3664, 33105}, {3665, 20247}, {3684, 24582}, {3756, 68976}, {3813, 20244}, {3868, 17181}, {3873, 7179}, {3879, 40999}, {3930, 24318}, {3935, 68926}, {4253, 33839}, {4425, 10868}, {4460, 36845}, {4847, 17874}, {4875, 26532}, {4938, 6745}, {5021, 26099}, {5051, 16887}, {5086, 7176}, {6604, 10529}, {6629, 51382}, {6675, 17201}, {6734, 53597}, {7181, 17136}, {10543, 59602}, {10916, 20880}, {11681, 36854}, {14963, 18184}, {16284, 25005}, {16572, 27132}, {17052, 18164}, {17062, 17474}, {17084, 34195}, {17095, 34772}, {17177, 40952}, {17205, 68946}, {17213, 68995}, {17272, 25960}, {17539, 59538}, {17672, 68950}, {17675, 56527}, {17728, 26229}, {17747, 31058}, {18738, 26541}, {20292, 60717}, {21044, 35102}, {21272, 40663}, {21935, 24215}, {26258, 51190}, {26526, 40133}, {26593, 44798}, {30742, 51194}, {31071, 51150}, {31080, 50995}, {33953, 56778}, {39765, 69042}, {56928, 62837}, {57002, 62400}, {59491, 64702}

X(69083) = reflection of X(17136) in X(7181)
X(69083) = isotomic conjugate of the isogonal conjugate of X(68732)
X(69083) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {28471, 329}, {35141, 21286}, {65261, 3436}
X(69083) = crosspoint of X(75) and X(35141)
X(69083) = crossdifference of every pair of points on line {41, 512}
X(69083) = barycentric product X(76)*X(68732)
X(69083) = barycentric quotient X(68732)/X(6)
X(69083) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 5740, 24986}, {325, 30941, 3936}, {9436, 51364, 37780}, {17728, 47595, 26229}

X(69084) = X(1)X(21)∩X(3)X(63450)

Barycentrics    a*(a^3*b - a^2*b^2 - a*b^3 + b^4 + a^3*c + 2*a^2*b*c - b^3*c - a^2*c^2 - a*c^3 - b*c^3 + c^4) : :

X(69084) lies on these lines: {1, 21}, {3, 63450}, {6, 25065}, {9, 68585}, {36, 7291}, {37, 4667}, {44, 6510}, {69, 25078}, {77, 1723}, {86, 25081}, {214, 20769}, {238, 68908}, {241, 514}, {277, 52374}, {279, 18625}, {323, 5526}, {519, 25083}, {527, 8609}, {579, 18161}, {604, 16551}, {672, 68759}, {740, 16728}, {894, 59727}, {1014, 1781}, {1015, 35115}, {1086, 56531}, {1108, 3663}, {1111, 68921}, {1212, 3452}, {1412, 1762}, {1418, 63589}, {1743, 60947}, {2170, 20367}, {2173, 53408}, {2223, 2809}, {2245, 7202}, {2260, 18726}, {2294, 18164}, {2340, 14740}, {3286, 44661}, {3554, 60974}, {3664, 40937}, {3693, 49765}, {3731, 63384}, {3879, 59733}, {3912, 24036}, {3973, 26669}, {4253, 7146}, {4436, 24394}, {4641, 16577}, {4859, 17092}, {4888, 24554}, {5053, 16560}, {5088, 53591}, {5184, 9441}, {5308, 31018}, {5453, 31837}, {5540, 11349}, {5733, 24695}, {6184, 35080}, {8680, 17197}, {10481, 55010}, {11362, 63356}, {16572, 47057}, {16600, 37596}, {16601, 37631}, {16702, 52949}, {16713, 18698}, {16726, 68935}, {17075, 24773}, {17078, 24781}, {17213, 68930}, {17284, 26690}, {17757, 19931}, {18176, 40978}, {18607, 40940}, {20683, 67428}, {21811, 67984}, {21907, 34578}, {23151, 30144}, {24209, 64780}, {24703, 63324}, {24780, 41808}, {25066, 29594}, {25077, 60986}, {25082, 29573}, {25255, 26818}, {27338, 27340}, {29624, 41819}, {31183, 31195}, {32118, 37507}, {34522, 63382}, {35090, 35110}, {37128, 60055}, {37597, 40133}, {39273, 52769}, {40965, 41430}, {51302, 62793}, {53590, 65116}, {53600, 68743}, {59720, 68873}, {65205, 68996}

X(69084) = midpoint of X(i) and X(j) for these {i,j}: {1959, 18206}, {2245, 7202}, {65205, 68996}
X(69084) = complement of the isotomic conjugate of X(65261)
X(69084) = X(i)-complementary conjugate of X(j) for these (i,j): {32, 35066}, {28471, 141}, {35141, 626}, {35347, 21253}, {65261, 2887}
X(69084) = crosspoint of X(2) and X(65261)
X(69084) = crossdifference of every pair of points on line {55, 661}
X(69084) = barycentric product X(75)*X(68732)
X(69084) = barycentric quotient X(68732)/X(1)
X(69084) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 63401, 63387}, {81, 16585, 25080}, {241, 43044, 1323}, {241, 43065, 3008}, {241, 43066, 18593}, {37597, 40133, 50114}, {49760, 69016, 16611}, {68921, 68947, 1111}

X(69085) = X(100)X(101)∩X(109)X(813)

Barycentrics    a^2*(a - b)*(a - c)*(a*b + a*c - b*c) : :

X(69085) lies on these lines: {1, 23622}, {6, 23470}, {10, 25633}, {37, 22225}, {100, 101}, {109, 813}, {110, 43077}, {190, 61164}, {572, 41423}, {579, 68795}, {646, 3570}, {692, 34071}, {799, 4482}, {1252, 32739}, {1293, 8693}, {1334, 54388}, {2176, 6377}, {2177, 4251}, {2284, 23845}, {2702, 29363}, {3550, 51949}, {3699, 61165}, {3730, 23988}, {4551, 69000}, {4595, 62530}, {4919, 38484}, {5143, 39258}, {6016, 28162}, {8694, 28841}, {9310, 56010}, {9327, 37633}, {9351, 62712}, {14408, 52923}, {16680, 40499}, {17735, 68748}, {18047, 65185}, {23832, 25577}, {29271, 59012}, {38832, 51319}, {41322, 42669}, {41323, 51377}, {46407, 65739}, {53145, 62420}, {60711, 60726}

X(69085) = isogonal conjugate of the isotomic conjugate of X(4595)
X(69085) = X(i)-Ceva conjugate of X(j) for these (i,j): {692, 101}, {1252, 62420}, {59102, 4557}
X(69085) = X(i)-cross conjugate of X(j) for these (i,j): {8640, 38832}, {20979, 2176}, {52923, 101}
X(69085) = X(i)-isoconjugate of X(j) for these (i,j): {2, 43931}, {87, 514}, {244, 4598}, {330, 513}, {522, 7153}, {649, 6384}, {663, 7209}, {667, 6383}, {693, 2162}, {764, 5383}, {876, 39914}, {932, 1086}, {1015, 18830}, {1019, 42027}, {1111, 34071}, {2053, 24002}, {2319, 3676}, {3123, 32039}, {3125, 56053}, {3261, 7121}, {3669, 7155}, {3733, 60244}, {3835, 53678}, {4083, 53677}, {4367, 27447}, {4374, 51974}, {4444, 34252}, {4449, 27498}, {4581, 27455}, {4817, 45782}, {7192, 16606}, {7199, 23493}, {15373, 46107}, {16727, 65163}, {17205, 65167}, {17924, 23086}, {18108, 62541}, {20906, 53146}, {20979, 53679}, {21759, 52619}, {27424, 43924}, {27499, 56323}, {40881, 62638}, {51321, 66286}, {52195, 60482}, {52621, 57264}
X(69085) = X(i)-Dao conjugate of X(j) for these (i,j): {75, 40495}, {798, 764}, {3835, 6545}, {5375, 6384}, {6377, 23989}, {6631, 6383}, {32664, 43931}, {39026, 330}, {40598, 3261}, {40610, 1111}, {55062, 4858}
X(69085) = cevapoint of X(i) and X(j) for these (i,j): {649, 63509}, {2176, 20979}, {4083, 61034}
X(69085) = crosspoint of X(1252) and X(6632)
X(69085) = crosssum of X(i) and X(j) for these (i,j): {514, 59522}, {1086, 21143}
X(69085) = trilinear pole of line {2176, 2209}
X(69085) = crossdifference of every pair of points on line {244, 4124}
X(69085) = barycentric product X(i)*X(j) for these {i,j}: {1, 52923}, {6, 4595}, {31, 36863}, {41, 66991}, {42, 62530}, {43, 100}, {59, 4147}, {101, 192}, {109, 27538}, {110, 3971}, {190, 2176}, {213, 36860}, {644, 1423}, {646, 41526}, {651, 3208}, {662, 20691}, {668, 2209}, {692, 6376}, {765, 4083}, {932, 53676}, {1016, 20979}, {1018, 27644}, {1110, 20906}, {1252, 3835}, {1403, 3699}, {1415, 4110}, {1783, 22370}, {1897, 20760}, {1978, 62420}, {3123, 57731}, {3212, 3939}, {3257, 52964}, {3570, 51973}, {3573, 41531}, {3903, 51902}, {3952, 38832}, {4551, 56181}, {4557, 33296}, {4567, 21834}, {4570, 21051}, {4578, 62791}, {4598, 53145}, {4600, 50491}, {4621, 20284}, {4734, 8694}, {4970, 8701}, {5376, 14408}, {5383, 57050}, {6377, 6632}, {6382, 32739}, {7035, 8640}, {15742, 22090}, {21138, 59149}, {25312, 57400}, {27805, 51319}, {34071, 53675}, {38986, 57950}, {46148, 62537}, {53268, 63232}, {59102, 59565}, {61234, 63238}
X(69085) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 43931}, {43, 693}, {100, 6384}, {101, 330}, {190, 6383}, {192, 3261}, {644, 27424}, {651, 7209}, {692, 87}, {765, 18830}, {932, 53679}, {1018, 60244}, {1110, 932}, {1252, 4598}, {1403, 3676}, {1415, 7153}, {1423, 24002}, {2176, 514}, {2209, 513}, {3208, 4391}, {3212, 52621}, {3835, 23989}, {3939, 7155}, {3971, 850}, {4083, 1111}, {4147, 34387}, {4557, 42027}, {4570, 56053}, {4595, 76}, {6376, 40495}, {6377, 6545}, {8640, 244}, {16695, 17205}, {18197, 16727}, {20284, 3776}, {20691, 1577}, {20760, 4025}, {20971, 48406}, {20979, 1086}, {21051, 21207}, {21138, 23100}, {21762, 21143}, {21834, 16732}, {22090, 1565}, {22370, 15413}, {23990, 34071}, {27538, 35519}, {27644, 7199}, {32656, 23086}, {32739, 2162}, {33296, 52619}, {34071, 53677}, {36860, 6385}, {36863, 561}, {38832, 7192}, {38986, 764}, {41526, 3669}, {41531, 66286}, {45216, 63812}, {46148, 62541}, {50491, 3120}, {51319, 4369}, {51902, 4374}, {51973, 4444}, {52923, 75}, {52964, 3762}, {53145, 3835}, {53676, 20906}, {56181, 18155}, {56806, 3777}, {57050, 21138}, {57074, 16726}, {57177, 21139}, {57192, 27496}, {59149, 5383}, {61234, 61417}, {62420, 649}, {62530, 310}, {62791, 59941}, {65498, 52633}, {66991, 20567}
X(69085) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {100, 644, 61234}, {23832, 35326, 25577}

X(69086) = X(2)X(6)∩X(100)X(1634)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a*b^2 - b^2*c + a*c^2 - b*c^2) : :

X(69086) lies on these lines: {2, 6}, {99, 4482}, {100, 1634}, {190, 670}, {274, 46894}, {662, 4598}, {1019, 23891}, {1043, 37019}, {3807, 4576}, {4560, 61186}, {4589, 68998}, {7192, 42720}, {17135, 24447}, {18829, 35148}, {20331, 68872}, {20785, 35538}, {24281, 33296}, {46143, 53199}, {53647, 65271}, {61234, 65161}

X(69086) = isogonal conjugate of X(66947)
X(69086) = X(4584)-Ceva conjugate of X(190)
X(69086) = X(21053)-cross conjugate of X(726)
X(69086) = X(i)-isoconjugate of X(j) for these (i,j): {1, 66947}, {37, 23355}, {213, 62638}, {512, 20332}, {523, 34077}, {649, 18793}, {661, 727}, {667, 27809}, {669, 32020}, {798, 3226}, {3121, 8709}, {7180, 8851}, {21832, 63881}, {36799, 51641}
X(69086) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 66947}, {726, 21053}, {1575, 4010}, {5375, 18793}, {6626, 62638}, {6631, 27809}, {17793, 661}, {20532, 523}, {27846, 39786}, {31998, 3226}, {36830, 727}, {39054, 20332}, {40589, 23355}
X(69086) = cevapoint of X(726) and X(21053)
X(69086) = crosspoint of X(799) and X(4589)
X(69086) = crosssum of X(798) and X(4455)
X(69086) = trilinear pole of line {726, 3009}
X(69086) = barycentric product X(i)*X(j) for these {i,j}: {86, 23354}, {99, 726}, {110, 35538}, {190, 62636}, {645, 43040}, {662, 52043}, {668, 18792}, {670, 3009}, {799, 1575}, {1463, 7257}, {3837, 4600}, {4567, 20908}, {4584, 62553}, {4589, 17793}, {4590, 21053}, {4602, 21760}, {4639, 17475}, {6331, 20785}, {17930, 59724}, {17934, 64236}, {20681, 65285}, {20777, 57968}, {21830, 52612}, {27044, 37205}, {36814, 55243}, {36860, 40881}, {62530, 67196}
X(69086) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 66947}, {58, 23355}, {86, 62638}, {99, 3226}, {100, 18793}, {110, 727}, {163, 34077}, {190, 27809}, {643, 8851}, {645, 36799}, {662, 20332}, {726, 523}, {799, 32020}, {1463, 4017}, {1575, 661}, {3009, 512}, {3837, 3120}, {4600, 8709}, {6373, 3122}, {16887, 35367}, {17475, 21832}, {17793, 4010}, {18792, 513}, {20532, 21053}, {20663, 4455}, {20681, 4155}, {20777, 810}, {20785, 647}, {20908, 16732}, {21053, 115}, {21760, 798}, {21830, 4079}, {23354, 10}, {24816, 30572}, {27044, 4129}, {35538, 850}, {36814, 55244}, {36860, 40844}, {42766, 42759}, {43040, 7178}, {52043, 1577}, {52633, 8034}, {59724, 18004}, {62558, 39786}, {62636, 514}, {64236, 18014}
X(69086) = {X(1019),X(23891)}-harmonic conjugate of X(55243)

X(69087) = X(1)X(21)∩X(99)X(100)

Barycentrics    a*(a^2 - b^2)*(a^2 - c^2)*(a*b^2 - b^2*c + a*c^2 - b*c^2) : :

X(69087) lies on these lines: {1, 21}, {8, 37019}, {86, 24405}, {99, 100}, {101, 34594}, {110, 932}, {190, 1634}, {662, 52923}, {931, 58117}, {1019, 1026}, {2397, 47844}, {3570, 46597}, {3733, 23343}, {3737, 69034}, {4243, 53337}, {4367, 62644}, {4427, 23390}, {4560, 53358}, {4584, 5378}, {6163, 64828}, {8299, 31059}, {8690, 29227}, {11364, 30576}, {18792, 36814}, {20777, 52043}, {23363, 53338}, {27644, 36267}, {27834, 65252}, {28828, 30993}, {36860, 57150}, {37128, 38878}, {37134, 37135}, {43076, 53627}, {43350, 43359}, {68148, 68156}

X(69087) = X(i)-isoconjugate of X(j) for these (i,j): {2, 66947}, {10, 23355}, {42, 62638}, {512, 3226}, {513, 18793}, {523, 727}, {649, 27809}, {661, 20332}, {798, 32020}, {1577, 34077}, {3122, 8709}, {4010, 63881}, {4017, 8851}, {7180, 36799}, {55263, 60865}
X(69087) = X(i)-Dao conjugate of X(j) for these (i,j): {5375, 27809}, {17793, 523}, {20532, 1577}, {22116, 35352}, {31998, 32020}, {32664, 66947}, {34961, 8851}, {36830, 20332}, {39026, 18793}, {39054, 3226}, {40592, 62638}
X(69087) = crosspoint of X(99) and X(4584)
X(69087) = crosssum of X(512) and X(21832)
X(69087) = trilinear pole of line {1575, 18792}
X(69087) = crossdifference of every pair of points on line {661, 3121}
X(69087) = barycentric product X(i)*X(j) for these {i,j}: {81, 23354}, {99, 1575}, {100, 62636}, {110, 52043}, {163, 35538}, {190, 18792}, {643, 43040}, {645, 1463}, {662, 726}, {670, 21760}, {799, 3009}, {811, 20785}, {3837, 4567}, {4570, 20908}, {4584, 17793}, {4589, 17475}, {4601, 6373}, {4623, 21830}, {4639, 20663}, {6331, 20777}, {20681, 65258}, {21053, 24041}, {27044, 34594}, {40881, 62530}
X(69087) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 66947}, {81, 62638}, {99, 32020}, {100, 27809}, {101, 18793}, {110, 20332}, {163, 727}, {643, 36799}, {662, 3226}, {726, 1577}, {1333, 23355}, {1463, 7178}, {1575, 523}, {1576, 34077}, {3009, 661}, {3837, 16732}, {4567, 8709}, {4601, 54985}, {5546, 8851}, {6373, 3125}, {8850, 7212}, {16696, 35367}, {17475, 4010}, {18792, 514}, {20663, 21832}, {20750, 53556}, {20777, 647}, {20785, 656}, {20908, 21207}, {21053, 1109}, {21760, 512}, {21830, 4705}, {22092, 18210}, {23354, 321}, {35538, 20948}, {36814, 4049}, {36860, 64226}, {43040, 4077}, {52043, 850}, {52656, 35352}, {62530, 40844}, {62636, 693}, {65498, 21835}

X(69088) = X(1)X(76)∩X(239)X(514)

Barycentrics    a^4*b^2 + a^3*b^3 + a^4*c^2 - a*b^3*c^2 + a^3*c^3 - a*b^2*c^3 - 2*b^3*c^3 : :

X(69088) lies on these lines: {1, 76}, {239, 514}, {538, 17475}, {716, 2235}, {1921, 18792}, {3009, 6381}, {3875, 49532}, {4384, 24598}, {14210, 20651}, {16823, 68976}, {16825, 68963}, {16826, 31026}, {17141, 49488}, {17175, 52564}, {17486, 21369}, {19565, 20372}, {23543, 29557}, {30667, 40859}, {39044, 49997}, {57039, 68877}

X(69088) = X(57933)-Ceva conjugate of X(1)
X(69088) = crossdifference of every pair of points on line {42, 46386}

X(69089) = X(8)X(210)∩X(9)X(6057)

Barycentrics    a^3 - (b + c)^3 : :

X(69089) lies on these lines: {1, 50318}, {2, 48648}, {8, 210}, {9, 6057}, {31, 6535}, {42, 17299}, {46, 50042}, {55, 2321}, {171, 50048}, {181, 41687}, {200, 4007}, {306, 17718}, {319, 32937}, {321, 1836}, {333, 3790}, {346, 3683}, {354, 34255}, {390, 56086}, {519, 25496}, {536, 26034}, {594, 612}, {756, 17275}, {968, 3943}, {1089, 5814}, {1278, 33068}, {1859, 3059}, {2345, 3745}, {3052, 53664}, {3175, 50295}, {3187, 38047}, {3474, 4461}, {3475, 29616}, {3661, 32926}, {3679, 7322}, {3686, 3715}, {3689, 7172}, {3696, 10327}, {3703, 11679}, {3711, 4060}, {3717, 4042}, {3757, 17233}, {3772, 15523}, {3773, 4362}, {3844, 19785}, {3883, 4387}, {3886, 4030}, {3923, 48644}, {3932, 5271}, {3967, 5739}, {3969, 26227}, {3971, 50308}, {3996, 4102}, {4066, 58798}, {4078, 19732}, {4135, 4703}, {4365, 33074}, {4385, 10371}, {4388, 42034}, {4512, 4873}, {4643, 32925}, {4645, 42029}, {4657, 32928}, {4659, 11246}, {4671, 24703}, {4682, 19822}, {4851, 32771}, {4942, 17781}, {5295, 43214}, {5311, 17303}, {5564, 30963}, {5695, 63134}, {5880, 28605}, {14973, 26893}, {17135, 49688}, {17156, 49524}, {17229, 33171}, {17230, 33124}, {17276, 33080}, {17278, 29687}, {17279, 32914}, {17301, 32781}, {17314, 37593}, {17369, 62845}, {17720, 32778}, {17721, 32866}, {17723, 33088}, {17769, 29652}, {20017, 46897}, {20075, 49485}, {24552, 49681}, {24789, 29674}, {26037, 28634}, {26040, 32087}, {26061, 50756}, {26223, 67964}, {26446, 51285}, {28503, 62833}, {29641, 55095}, {29669, 49471}, {30568, 41002}, {30623, 59200}, {31134, 48642}, {31161, 50076}, {32784, 50068}, {32920, 49560}, {33079, 49474}, {33083, 42044}, {33085, 49493}, {33086, 50106}, {35613, 51463}, {37653, 49447}, {37674, 49990}, {49453, 54311}, {49995, 62821}, {50105, 59536}, {50107, 63140}, {53673, 58629}, {59506, 63003}

X(69089) = barycentric product X(i)*X(j) for these {i,j}: {8, 17303}, {210, 30599}, {312, 5311}, {346, 10404}, {644, 50334}, {2321, 25526}, {3699, 48275}
X(69089) = barycentric quotient X(i)/X(j) for these {i,j}: {646, 54957}, {5311, 57}, {10404, 279}, {17303, 7}, {25526, 1434}, {30599, 57785}, {48275, 3676}, {50334, 24002}
X(69089) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 312, 3966}, {8, 497, 4914}, {8, 3706, 4863}, {8, 3714, 1837}, {8, 3974, 210}, {8, 27538, 4886}, {31, 6535, 17281}, {200, 4007, 4046}, {312, 3966, 4679}, {321, 3416, 1836}, {3686, 4082, 3715}, {3773, 4362, 32777}, X(69089) = {4519, 4914, 497}, {4671, 33075, 24703}, {28605, 33078, 5880}, {33088, 44417, 17723}

X(69090) = X(2)X(308)∩X(32)X(6697)

Barycentrics    a^8 + (b^4 - c^4)^2 : :
X(69090) = 3 X[2] + X[33785]

X(69090) lies on these lines: {2, 308}, {32, 6697}, {66, 9233}, {315, 40380}, {5596, 52958}, {7800, 26190}, {14003, 41331}, {16985, 33797}, {28419, 52961}

X(69090) = X(i)-complementary conjugate of X(j) for these (i,j): {2156, 52532}, {2353, 21247}, {40146, 16582}
X(69090) = {X(9233),X(15449)}-harmonic conjugate of X(66)





leftri   Additive associates, X(69091) - X(69098)  rightri

Contributed by Clark Kimberling and Peter Moses, June 18-30, 2025.

Suppose that X is a triangle center, given by barycentrics

f(a,b,c) : f(b,c,a) : f(c,a,b),

where f(a,b,c) is a sum of n terms t(i) each of the form

k a^p b^q c^r,

where k is a nonzero constant, n>=1, and p,q,r are real numbers. Referring to the representation

(*)        f(a,b,c) = + k(1) a^p(1) b^q(1) c^r(1) + . . . + k(n) a^p(n) b^q(n) c^r(n),

note that f(a,b,c) is a polynomial in a,b,c such that, in accord with the definition of triangle center, f(a,b,c) = f(b,c,a) = f(c,a,b) and f(a,b,c) = f(a,c,b), so that X is, in the literature, a polynomial triangle center, abbreviated as PTC.

In (*), the symbol + occurs n times, for which we write the n-tuple (+,+,...+). By varying the symbols so that the first + remains fixed, and the others can be +, 0, or - , or equivalently, 1, 0, or -1, we obtain 3^(n-1) n-tuples, and a corresponding set of 3^(n-1) points by varying the additive symbols in (*), by which we mean coefficient 1, 0, or -1. The resulting family of n-tuples, or points, are here named primary additive associates (AAs) of X. These points are of two kinds: triangle centers and bicentric pairs.

Let S(X) denote the set of AA's of X. If X' and X'' are distinct PTC's points in S(X), then S(X) possibly also contains a pair of (X',X'')-harmonic conjugates, specifically, the sum and difference of X' and X'', which we denote by X' + X'' and X' - X''. Either these are PTC's, or else they are a bicentric pair. See Example 3 below.

Example 1. Represent the b^2 + c^2 + b c as (1,1,1). The AA for (1,1,-1) is the PTC b^2 - c^2 + b c : : . On the other hand, (1,-1,1) and (1,-1,-1) represent the bicentric pair of points, b^2 - c^2 + b c : : and b^2 - c^2 - b c.

In most of the rest of this preamble, AA's are understood to represent triangle centers, not bicentric points.

Example 2. Taking f(a,b,c) = a^6 - a^4 b^2 - a^4 c^2 + a^2 b^2 c^2, so that X = X(110), the family of AA's of X has ten PTC's, as follows:

Additive associates of X(110)
X(110) (1,1,1,1)
X(5012) (1,1,1,-1)
X(251) (1,-1,-1,1)
X(1627) (1,-1,-1,-1)
X(184) (1,1,1,0)
X(1915) (1,0,0,1)
X(1501) (1,0,0,0)
X(2) (0,0,0,1)
X(3051) (0,1,1,0)
X(1613) (0,1,1,1)

In the following tables, the AA consisting of all 1's has terms in the order given by the Mathematica command Expand, as in Examples 1 and 2.

Additive associates of X(3)
X(3) (1,1,1)
X(6) (1,-1,-1)
Additive associates of X(4)
X(4) (1,1,1,1)
X(69) (1,1,-1,1)
X(7795) (1,-1,1,-1)
X(3767) (1,-1,-1,-11)
X(315) (1,1,0,1)
X(3734) (1,0,1,0)
X(32) (1,0,0,0)
X(76) (0,0,1,0)
X(626) (0,1,0,1)
X(115) (0,1,1,1)
Additive associates of X(5)
X(5) (1,1,1,1,1)
X(3933) (1,1,1,-1,1)
X(141) (1,-1,1,1,-1)
X(5254) (1,-1,1,-1,-1)
X(325) (1,1,1,0,1)
X(3934) (1,0,1,1,0)
X(39) (1,0,1,0,0)
X(76) (0,0,0,1,0)
X(626) (0,1,0,0,1)
X(115) (0,1,0,1,1)
Additive associates of X(7)
X(7) (1,1,1,1)
X(8) (1,1,-1,1)
X(2345) (1,-1,1,-1)
X(4000) (1,-1,-1,-1)
X(69) (1,1,0,1)
X(4363) (1,0,1,0)
X(6) (1,0,0,0)
X(75) (0,0,1,0)
X(141) (0,1,0,1)
X(1086) (0,1,1,1)
Additive associates of X(8)
X(8) (1,1,1)
X(2) (1,-1,1)
X(1) (1,0,0)
X(10) (0,1,1)
Additive associates of X(9)
X(9) (1,1,1)
X(1) (1,-1,-1)
X(6) (1,0,0)
X(37) (0,1,1)
Additive associates of X(10)
X(10) (1,1)
X(514) (1,-1)
Additive associates of X(11)
X(11) (1,1,1,1,1,1,1)
X(69134) (1,1,1,0,1,0,1)
X(69091) (1,1,1,-1,1,-1,1)
X(2886) (1,1,0,1,1,1,1)
X(918) (1,1,0,1,-1,-1,-1)
X(3006) (1,1,0,0,1,0,1)
X(25259) (1,1,0,0,-1,0,-1)
X(3703) ()
X(3703) (1,1,0,-1,1,-1,1)
X(3700) (1,1,0,-1,-1,1,-1)
X(3925) (1,1,-1,1,1,1,1)
X(69250) (1,1,-1,0,1,0,1)
X(3932) (1,1,-1,-1,1,-1,1)
X(3840) (1,0,1,1,1,1,0)
X(244) (1,0,1,0,1,0,0)
X(24165) (1,0,1,-1,1,-1,0)
X(3741) (1,0,0,1,1,1,0)
X(514) (1,0,0,1,-1,-1,0)
X(38) (1,0,0,0,1,0,0)
X(661) (1,0,0,0,-1,0,0)
X(726) (1,0,0,-1,1,-1,00)
X(3835) (1,0,0,-1,-1,1,0)
X(10) (1,0,-1,1,1,1,0)
X(756) (1,0,-1,0,1,0,0)
X(3971) (1,0,-1,-1,1,-1,0)
X(69092) (1,-1,1,1,1,1,-1)
X(69251) (1,-1,1,0,1,0,-1)
X(1086) (1,-1,1,-1,1,-1,-1)
X(141) (1,-1,0,1,1,1,-1)
X(523) (1,-1,0,1,-1,-1,1)
X(17184) (1,-1,0,0,1,0,-1)
X(45746) (1,-1,0,0,-1,0,1)
X(3782) (1,-1,0,-1,1,-1,-1)
X(3004) (1,-1,0,-1,-1,1,1)
X(1211) (1,-1,-1,1,1,1,-1)
X(26580) (1,-1,-1,0,1,0,-1)
X(4415) (1,-1,-1,-1,1,-1,-1)
X69173) (0,1,1,1,0,1,1)
X(3846) (0,1,1,0,0,0,1)
X(69252) (0,1,1,-1,0,-1,1)
X(3120) (0,1,0,1,0,1,1)
X(16892) (0,1,0,1,0,-1,-1)
X(2887) (0,1,0,0,0,0,1)
X(824) (0,1,0,0,0,0,-1)
X(4024) (0,1,0,-1,0,1,-1)
X(15523) (0,1,0,-1,0,-1,1)
X(69253) (0,1,-1,1,0,1,1)
X(3836) (0,1,-1,0,0,0,1)
X(29687) (0,1,-1,-1,0,-1,1)
X(4358) (0,0,1,1,0,1,0)
X(2) (0,0,1,0,0,0,0)
X(4359) (0,0,1,-1,0,-1,0)
X(321) (0,0,0,1,0,1,0)
X(693) (0,0,0,1,0,-1,0)
Example 3. Continuing the above discussion of harmonic conjugates, There are C(53,2) = 1378 pairs of points X', X'' that can be chosen from the 53 additive associates of X(11). One such choice is X' = X(141) and X''=X(3703), with respect to which X(38) and X(15523) are harmonic conjugates. To verify this, it is expedient to use the equals symbol, =, between pairs of 7-tuples that represent the same point. Then

X' = X(141) = (1,-1,0,1,1,1,-1)
X'' = X(3703) = (1,1,0,-1,1,-1,1)
X' + X'' = (2,0,0,0,2,0,0) = (1,0,0,0,1,0,0) = X(38)
X' - X'' = (0,-2,0,2,0,2,-2) = (0,1,0 -1,0,-1,1) = X(15523)

To illustrate the possiblity that for other choices of X' and X'', the harmonic conjugates are a bicentric pair, we have

X' = X(321) = (0,0,0,1,0,1,0)
X'' = X(693) = (0,0,0,1,0,-1,0)
X' + X'' = (0,0,0,2,0,0,0) = (0,0,0,1,0,0,0), which represents b^2 c : c^2 a : a^2 b
X' - X'' = (0,0,0,0,0,2,0) = (0,0,0,0,0,1,0), which represents b c^2 : c a^2 : a b^2
Note that the bicentric pair, (X' + X'', X' - X'') = (a/b : : , a/c : : ).

For some choices of X' and X'', at least one of the harrnonic conjugates is a primary additive associate, while the other is a more general additive associate (not formally defined above). For example,

X'= X(11) = (1,1,1,1,1,1,1)
X'' = X(523) = (1,-1,0,1,-1,-1,1)
X' + X'' = (2,1,1,1,2,1,1) = 2 a^b^2 - b^3 - 2 a b c + b^2 c + 2 a c^2 + b c^2 - c^3 : :
X' - X'' = (0,1,1,1,0,1,1) = X(69173).

The above discussion of AA's is based on the Mathematica order in which the terms of the first barycentric of a point are arranged. The set of these AA's include not only polynomial triangle centers, but also bicentric pairs. For each degree n of a PTC, there is a natural way to represent all PTC's of that degree, without bicentric pairs. We begin with collections of elements, denoted by E(n), and defined inductively as follows:

E(0) = {1}
E(1) = {b + c}
E(n) = {b^n + c^2}∪ b c E(n-2), for n>=2.

E(2) = {b^2 + c^2, b c}
E(3) = {b^3 + c^3, b c (b + c)}
E(4) = {b^4 + c^4, b c (b^2 + c^2), b^2 c^2)}.

It is easy to find that the cardinality of E(n) is [(n+2)/(2)], where [ ] = floor.

In general, each even PTC of degree n has a representation

h(1) a^n + h(2) a^(n-1) p(1,b,c) + h(3) a^(n-2) p(2,b,c) + . . . ,

where each p(i,b,c) is a linear combination of the elements in E(n-2 i), and h(i) is a constant. The trivial linear combination, given by h(i)=0 for all i, does not represent a triangle center. The odd PTC's, as defined elsewhere in the literature, are simply those of the form (b - c)*e(a,b,c), where e(a,b,c) is an even PTC. Thus, every odd PTC of degree n+1 has a representation

(b - c)(h(1) a^n + h(2) a^(n-1) p(1,b,c) + h(3) a^(n-2) p(2,b,c) + . . . ).

While this manner of expressing PTC's is conceptually natural, it nevertheless leads to a conclusion that various sets of barycentric products and isotomic conjugates, etc., can be highly elaborate and difficult to enumerate.

For a further discussion of additive associates, see the preamble just before X(69302).

underbar



X(69091) = X(11)X(321)∩X(12)X(4968)

Barycentrics    a*b^2 - b^3 - 2*a*b*c - b^2*c + a*c^2 - b*c^2 - c^3 : :

X(69091) lies on these lines: {1, 3704}, {2, 1390}, {8, 56}, {10, 3752}, {11, 321}, {12, 4968}, {31, 59574}, {38, 1211}, {43, 33169}, {55, 17740}, {57, 3416}, {63, 3966}, {75, 325}, {81, 32842}, {100, 4030}, {141, 982}, {149, 64010}, {171, 5846}, {181, 518}, {197, 956}, {200, 49688}, {210, 63147}, {226, 49483}, {238, 33167}, {239, 33121}, {244, 15523}, {306, 354}, {312, 3816}, {333, 8301}, {341, 9711}, {344, 8167}, {345, 1001}, {346, 26105}, {442, 30171}, {496, 50042}, {497, 5695}, {524, 32861}, {528, 4514}, {536, 24210}, {545, 33099}, {594, 1575}, {596, 3454}, {614, 32777}, {726, 3846}, {740, 29655}, {748, 33161}, {750, 32854}, {756, 5241}, {846, 59583}, {894, 33071}, {899, 33162}, {910, 3686}, {940, 33088}, {968, 49740}, {984, 4884}, {1054, 33079}, {1086, 2887}, {1089, 4187}, {1125, 3695}, {1155, 4914}, {1210, 3714}, {1214, 2968}, {1215, 37662}, {1279, 59692}, {1329, 4385}, {1479, 50044}, {1621, 3712}, {1647, 6535}, {1699, 4659}, {2321, 11019}, {2611, 62548}, {2895, 62235}, {2999, 38047}, {3004, 20898}, {3006, 3925}, {3035, 7081}, {3058, 32929}, {3210, 32773}, {3218, 33075}, {3219, 41002}, {3315, 33173}, {3452, 3967}, {3550, 49506}, {3589, 29821}, {3666, 4026}, {3681, 4023}, {3683, 3977}, {3685, 49736}, {3702, 37722}, {3706, 26015}, {3710, 25917}, {3715, 63003}, {3717, 3740}, {3720, 32848}, {3729, 24703}, {3742, 3912}, {3756, 3773}, {3757, 6690}, {3782, 17155}, {3790, 18743}, {3791, 61661}, {3794, 25048}, {3819, 4553}, {3825, 4066}, {3826, 19804}, {3829, 42029}, {3844, 39597}, {3873, 33077}, {3874, 41014}, {3881, 21081}, {3883, 4640}, {3896, 29835}, {3914, 42051}, {3933, 33945}, {3936, 17140}, {3944, 49493}, {3961, 9053}, {3969, 29824}, {3980, 4865}, {3996, 36528}, {4003, 54311}, {4007, 35613}, {4011, 17340}, {4028, 49478}, {4035, 5542}, {4038, 17390}, {4042, 64153}, {4046, 17135}, {4054, 17605}, {4062, 62867}, {4082, 5316}, {4104, 49515}, {4126, 63961}, {4358, 6057}, {4360, 29837}, {4361, 33137}, {4362, 37646}, {4363, 26098}, {4383, 33163}, {4386, 17362}, {4388, 17768}, {4392, 32782}, {4395, 29861}, {4399, 24374}, {4413, 10327}, {4417, 24349}, {4418, 32844}, {4422, 17123}, {4423, 17776}, {4425, 17246}, {4429, 17490}, {4431, 40883}, {4438, 16825}, {4439, 59517}, {4461, 5274}, {4512, 59536}, {4647, 24390}, {4656, 49523}, {4665, 29676}, {4679, 56082}, {4680, 11112}, {4682, 49476}, {4692, 17757}, {4696, 21031}, {4702, 64162}, {4703, 17334}, {4733, 24643}, {4759, 59664}, {4819, 20011}, {4849, 49529}, {4850, 29667}, {4854, 17147}, {4863, 63131}, {4901, 8580}, {4942, 56084}, {4972, 17495}, {5014, 34612}, {5016, 7354}, {5208, 66027}, {5211, 32942}, {5220, 14555}, {5231, 65684}, {5233, 32937}, {5248, 59592}, {5263, 29840}, {5269, 49681}, {5271, 24446}, {5272, 17279}, {5284, 32849}, {5294, 49987}, {5295, 10916}, {5432, 26227}, {5573, 17284}, {5718, 29849}, {5737, 19758}, {5741, 17165}, {5814, 62858}, {5852, 33066}, {6327, 11246}, {6533, 17529}, {6679, 50023}, {6763, 49716}, {7172, 59572}, {7186, 9024}, {7191, 32779}, {7227, 17722}, {7228, 33097}, {7263, 17889}, {7292, 33157}, {8616, 59580}, {8728, 30172}, {8731, 16684}, {9342, 60459}, {10072, 50041}, {10371, 62874}, {10980, 17296}, {11238, 50043}, {11680, 28605}, {15254, 56078}, {15985, 24478}, {16569, 33165}, {16602, 62673}, {16706, 59477}, {16823, 33116}, {17045, 17600}, {17049, 18165}, {17051, 17233}, {17056, 24325}, {17063, 29674}, {17069, 24382}, {17122, 32847}, {17142, 29985}, {17154, 31037}, {17229, 24216}, {17243, 26102}, {17245, 29653}, {17299, 39594}, {17365, 32946}, {17366, 25453}, {17369, 25496}, {17398, 29644}, {17449, 33081}, {17591, 32784}, {17595, 26034}, {17596, 33076}, {17597, 33171}, {17598, 32783}, {17670, 30149}, {17694, 30167}, {17716, 51147}, {17721, 50048}, {17724, 29846}, {17726, 32772}, {17763, 37634}, {17769, 58443}, {18134, 25557}, {18201, 33085}, {18990, 36974}, {20340, 21025}, {20486, 20913}, {20544, 20888}, {20883, 37362}, {21020, 29690}, {21242, 62226}, {21342, 49511}, {22279, 58572}, {24169, 28595}, {24239, 44417}, {24387, 42031}, {24434, 44694}, {24789, 29857}, {25441, 43993}, {25524, 54433}, {25957, 40688}, {25958, 33146}, {25960, 32925}, {26582, 30179}, {26724, 29873}, {26942, 68761}, {27003, 33078}, {27798, 68939}, {28530, 33095}, {28612, 31419}, {29631, 32924}, {29635, 32921}, {29639, 31993}, {29645, 49472}, {29685, 46904}, {29820, 33158}, {29843, 49470}, {29844, 32941}, {29845, 32928}, {29872, 33129}, {30615, 67097}, {30831, 33148}, {31137, 50097}, {32843, 32940}, {32845, 32947}, {32850, 49732}, {32858, 64149}, {32860, 33120}, {32911, 33170}, {32914, 33119}, {32931, 37663}, {33064, 42055}, {33068, 62300}, {33084, 62865}, {33093, 37633}, {33101, 49532}, {33134, 50106}, {33141, 49474}, {33166, 37680}, {33175, 62814}, {36498, 40980}, {36845, 49460}, {37326, 58624}, {42049, 64168}, {42053, 49676}, {45713, 56385}, {45714, 56386}, {46909, 56810}, {47356, 62842}, {48643, 50117}, {50038, 52353}, {51415, 59511}, {60446, 68999}, {60736, 68951}, {62812, 67964}, {64123, 68889}

X(69091) = midpoint of X(i) and X(j) for these {i,j}: {171, 32866}, {4388, 32939}, {4514, 32932}, {32861, 32913}
X(69091) = reflection of X(4415) in X(3846)
X(69091) = complement of X(32926)
X(69091) = X(i)-complementary conjugate of X(j) for these (i,j): {667, 55053}, {54458, 21260}
X(69091) = crosspoint of X(6063) and X(40013)
X(69091) = crosssum of X(i) and X(j) for these (i,j): {2175, 2220}, {2483, 3271}
X(69091) = barycentric product X(75)*X(39244)
X(69091) = barycentric quotient X(39244)/X(1)
X(69091) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3703, 3932}, {2, 3891, 17602}, {2, 32922, 17061}, {2, 33089, 3703}, {43, 33169, 49524}, {75, 3705, 2886}, {100, 33090, 4030}, {238, 33167, 44416}, {306, 354, 4966}, {982, 32778, 141}, {1155, 4914, 63134}, {1621, 33168, 3712}, {2887, 24165, 1086}, {3006, 4359, 3925}, {3210, 32773, 66071}, {3757, 32851, 6690}, {4046, 51463, 17135}, {4082, 5316, 59506}, {4418, 32844, 63979}, {4884, 5743, 984}, {17123, 33164, 4422}, {17155, 25760, 3782}, {17596, 33076, 44419}, {19804, 29641, 3826}, {24325, 29671, 17056}, {26102, 33092, 17243}, {29821, 32780, 3589}, {29846, 32923, 17724}, {29849, 32771, 5718}, {32914, 33119, 35466}

X(69092) = X(11)X(2887)∩X(12)X(3831)

Barycentrics    a*b^2 + b^3 - 2*a*b*c + b^2*c + a*c^2 + b*c^2 + c^3 : :
X(69092) = 3 X[2] + X[32863]

X(69092) lies on these lines: {1, 4030}, {2, 6}, {5, 37482}, {8, 17054}, {10, 354}, {11, 2887}, {12, 3831}, {31, 29677}, {37, 54311}, {38, 3932}, {42, 4966}, {43, 33087}, {44, 4001}, {57, 7198}, {63, 17279}, {66, 21489}, {75, 40688}, {76, 40012}, {100, 33173}, {140, 25645}, {142, 31993}, {171, 29637}, {190, 26840}, {210, 20455}, {226, 1122}, {238, 33085}, {244, 15523}, {306, 3752}, {312, 3662}, {320, 27064}, {321, 1086}, {335, 40033}, {345, 17595}, {404, 5347}, {442, 50597}, {474, 37538}, {528, 32943}, {553, 17355}, {594, 3726}, {614, 3416}, {726, 6057}, {740, 24169}, {748, 33080}, {750, 24943}, {899, 33081}, {980, 8362}, {982, 3703}, {986, 4918}, {1001, 26034}, {1054, 33160}, {1107, 29988}, {1125, 3745}, {1155, 59692}, {1193, 25914}, {1215, 49676}, {1230, 40013}, {1279, 63134}, {1330, 13741}, {1352, 16434}, {1376, 33171}, {1407, 28739}, {1500, 3666}, {1503, 19649}, {1621, 33086}, {1647, 48650}, {1722, 10371}, {1738, 3706}, {1746, 19512}, {1834, 4202}, {1943, 5723}, {1999, 16706}, {2321, 24177}, {2345, 9776}, {2886, 25957}, {2999, 17296}, {3035, 29846}, {3058, 4660}, {3175, 3663}, {3187, 17366}, {3210, 17233}, {3216, 41014}, {3218, 33157}, {3219, 4422}, {3242, 10327}, {3305, 4643}, {3315, 33090}, {3452, 16594}, {3454, 4187}, {3661, 19804}, {3670, 3695}, {3685, 33068}, {3687, 16610}, {3704, 24443}, {3712, 17596}, {3714, 23536}, {3720, 4026}, {3740, 60423}, {3741, 3836}, {3742, 3844}, {3748, 49768}, {3756, 48647}, {3771, 5432}, {3773, 24165}, {3816, 25760}, {3821, 4854}, {3823, 25006}, {3826, 25961}, {3834, 5249}, {3846, 4871}, {3873, 29679}, {3911, 20106}, {3923, 11246}, {3934, 20337}, {3943, 17147}, {3948, 18136}, {3966, 5272}, {3969, 17495}, {3974, 4310}, {3995, 17246}, {4000, 34255}, {4005, 59685}, {4011, 4655}, {4023, 16569}, {4038, 29633}, {4046, 49560}, {4085, 42057}, {4104, 61686}, {4126, 49448}, {4138, 17605}, {4201, 64158}, {4203, 15447}, {4228, 35283}, {4252, 17526}, {4265, 37325}, {4267, 50717}, {4357, 44307}, {4358, 4415}, {4387, 24248}, {4389, 41839}, {4392, 4884}, {4423, 50295}, {4429, 10453}, {4434, 29656}, {4447, 16687}, {4535, 24200}, {4641, 17353}, {4645, 32942}, {4651, 24988}, {4656, 50092}, {4657, 5287}, {4659, 63583}, {4671, 33146}, {4685, 50315}, {4697, 24295}, {4850, 32858}, {4851, 5256}, {4852, 50292}, {4865, 29668}, {4886, 17287}, {4904, 24175}, {4972, 29824}, {4995, 59679}, {5047, 49728}, {5133, 33852}, {5205, 33126}, {5226, 30824}, {5228, 56460}, {5247, 25992}, {5271, 17278}, {5273, 37111}, {5284, 33083}, {5292, 56780}, {5294, 17357}, {5337, 7819}, {5437, 16608}, {5542, 53663}, {5711, 19836}, {5846, 7191}, {5852, 32938}, {5905, 7232}, {5918, 21629}, {6247, 6926}, {6354, 69051}, {6656, 23947}, {6682, 29653}, {6690, 29632}, {6692, 63844}, {6964, 15873}, {6967, 67902}, {7018, 18149}, {7081, 17724}, {7179, 21471}, {7228, 26842}, {7238, 17483}, {7263, 28605}, {7292, 33075}, {7308, 17272}, {7536, 18642}, {7767, 33953}, {7789, 21495}, {7795, 21477}, {7800, 11343}, {8728, 10479}, {9053, 33091}, {9345, 29647}, {9347, 29648}, {9965, 54389}, {10159, 14534}, {10167, 12618}, {10319, 18639}, {10449, 33833}, {10516, 26118}, {11112, 48863}, {11113, 48835}, {11319, 64159}, {11679, 17282}, {11680, 25959}, {13740, 49745}, {14376, 21499}, {16413, 62772}, {16496, 30615}, {16593, 30821}, {16703, 39998}, {17011, 17390}, {17019, 17045}, {17022, 17306}, {17023, 37595}, {17024, 51147}, {17061, 17763}, {17063, 32778}, {17074, 28780}, {17120, 62230}, {17122, 32783}, {17123, 33082}, {17185, 29492}, {17227, 18743}, {17230, 17490}, {17235, 35652}, {17243, 28606}, {17266, 38000}, {17267, 17776}, {17268, 42033}, {17274, 30568}, {17276, 56082}, {17280, 32939}, {17286, 50048}, {17288, 33066}, {17290, 19785}, {17291, 19786}, {17292, 19808}, {17293, 19822}, {17302, 34064}, {17304, 50068}, {17316, 20182}, {17332, 27065}, {17340, 32933}, {17345, 17781}, {17356, 26723}, {17365, 26223}, {17372, 50306}, {17380, 58820}, {17449, 33162}, {17450, 29685}, {17514, 52782}, {17536, 26064}, {17591, 33092}, {17598, 32847}, {17602, 26128}, {17698, 37522}, {17716, 29660}, {17720, 25527}, {17726, 33073}, {17758, 60084}, {17768, 32930}, {18137, 18739}, {18150, 27792}, {18201, 33167}, {18214, 56518}, {18229, 41867}, {18840, 60076}, {19529, 40980}, {19765, 56737}, {19792, 37096}, {19825, 61321}, {19881, 37559}, {20020, 67538}, {20892, 30713}, {20913, 21024}, {20917, 26561}, {20942, 48637}, {21025, 52043}, {21026, 29690}, {21258, 29611}, {21342, 63147}, {21487, 46264}, {21537, 59545}, {21581, 24994}, {21596, 24548}, {24206, 37360}, {24250, 34583}, {24586, 69024}, {24589, 48635}, {24627, 33116}, {24691, 56507}, {25079, 56949}, {25501, 50298}, {25526, 50318}, {25557, 32771}, {25882, 25941}, {25973, 26013}, {26102, 32784}, {26132, 28808}, {26582, 31027}, {26590, 31028}, {26686, 30175}, {26724, 40480}, {27002, 43055}, {27003, 32779}, {27191, 55095}, {27248, 68769}, {28530, 33102}, {28595, 29655}, {28654, 62304}, {29596, 37520}, {29642, 32916}, {29662, 31237}, {29663, 62821}, {29666, 62807}, {29667, 64149}, {29670, 37703}, {29673, 51463}, {29820, 33076}, {29821, 32846}, {29827, 33111}, {29850, 32919}, {29851, 32917}, {29960, 37596}, {30614, 49690}, {30827, 41883}, {30829, 48633}, {31137, 33141}, {31151, 33109}, {31884, 50698}, {32849, 59583}, {32913, 33159}, {32915, 33125}, {32924, 59477}, {32931, 33069}, {32941, 34612}, {32944, 32949}, {32945, 49732}, {32947, 49736}, {33064, 59511}, {33165, 62865}, {33166, 62235}, {33828, 54365}, {34829, 64528}, {36990, 50699}, {37366, 43653}, {37456, 67865}, {37543, 56444}, {37549, 54433}, {37715, 49999}, {38047, 62819}, {42029, 48627}, {42034, 48629}, {48644, 50117}, {48646, 62221}, {48843, 64167}, {49688, 62850}, {50043, 53664}, {50104, 51390}, {50115, 62240}, {50320, 53417}, {56771, 59695}, {57024, 67892}, {57663, 62771}

X(69092) = midpoint of X(i) and X(j) for these {i,j}: {7191, 33078}, {32863, 32911}, {32930, 33067}, {32943, 32948}, {33091, 62814}
X(69092) = reflection of X(32924) in X(59477)
X(69092) = complement of X(32911)
X(69092) = complement of the isogonal conjugate of X(39798)
X(69092) = complement of the isotomic conjugate of X(40013)
X(69092) = isotomic conjugate of the polar conjugate of X(1883)
X(69092) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 4075}, {42, 62588}, {596, 141}, {649, 8054}, {8050, 3835}, {20615, 142}, {34594, 4369}, {37205, 512}, {39747, 3741}, {39798, 10}, {39949, 3739}, {40013, 2887}, {40085, 3454}, {40086, 116}, {40148, 2}, {40519, 514}, {57915, 626}, {59014, 4977}, {65202, 513}, {65286, 42327}
X(69092) = X(i)-Ceva conjugate of X(j) for these (i,j): {65286, 523}, {68205, 525}
X(69092) = crosspoint of X(i) and X(j) for these (i,j): {2, 40013}, {76, 57830}
X(69092) = crosssum of X(i) and X(j) for these (i,j): {6, 2220}, {32, 44085}
X(69092) = barycentric product X(i)*X(j) for these {i,j}: {69, 1883}, {141, 27005}
X(69092) = barycentric quotient X(i)/X(j) for these {i,j}: {1883, 4}, {27005, 83}
X(69092) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 33079, 4030}, {2, 69, 4383}, {2, 81, 3589}, {2, 141, 1211}, {2, 343, 26005}, {2, 1211, 5241}, {2, 2895, 37680}, {2, 3620, 14555}, {2, 3936, 37662}, {2, 4417, 37663}, {2, 4648, 19701}, {2, 4869, 5712}, {2, 5278, 17337}, {2, 5739, 37679}, {2, 5741, 51415}, {2, 14552, 37650}, {2, 14829, 35466}, {2, 17232, 18134}, {2, 18134, 5718}, {2, 18139, 17056}, {2, 18141, 940}, {2, 31017, 5741}, {2, 32782, 5743}, {2, 32863, 32911}, {2, 33172, 141}, {2, 37633, 6703}, {2, 37652, 17352}, {2, 37653, 17277}, {2, 37656, 37687}, {2, 63057, 3618}, {2, 63068, 23292}, {38, 29687, 3932}, {57, 17284, 32777}, {141, 5743, 32782}, {141, 30945, 51384}, {312, 3662, 3782}, {599, 37679, 5739}, {982, 29674, 3703}, {1621, 33086, 44419}, {2321, 24177, 42051}, {2887, 3840, 11}, {3218, 33157, 44416}, {3720, 32781, 4026}, {3741, 3836, 3925}, {3752, 17231, 306}, {3763, 37674, 2}, {3834, 44417, 5249}, {3873, 29679, 49524}, {4358, 17184, 4415}, {4392, 32862, 4884}, {4415, 48632, 17184}, {4645, 32942, 63979}, {5737, 17265, 2}, {5743, 32782, 1211}, {6703, 34573, 2}, {7081, 33124, 17724}, {11679, 17282, 24789}, {14552, 37650, 19723}, {14829, 17283, 2}, {16569, 33084, 4023}, {17123, 33082, 41002}, {17227, 18743, 27184}, {17277, 37653, 49724}, {17596, 33158, 3712}, {17763, 33123, 17061}, {25527, 30567, 17720}, {25760, 30957, 3816}, {25957, 30942, 2886}, {25961, 31330, 3826}, {26128, 29649, 17602}, {29632, 32918, 6690}, {32915, 33125, 66071}, {49511, 62673, 210}

X(69093) = X(11)X(325)∩X(12)X(76)

Barycentrics    (a*b - b^2 + a*c - c^2)*(a*b + b^2 + a*c + c^2) : :

X(69093) lies on these lines: {1, 3933}, {2, 17224}, {5, 3760}, {7, 345}, {11, 325}, {12, 76}, {35, 7767}, {36, 6390}, {55, 69}, {56, 3926}, {75, 3703}, {99, 15326}, {116, 68938}, {120, 3263}, {141, 2276}, {172, 7789}, {183, 5432}, {192, 3314}, {194, 26561}, {209, 52396}, {230, 4396}, {241, 3693}, {304, 3665}, {312, 7179}, {315, 6284}, {316, 65632}, {319, 4030}, {320, 3712}, {385, 26629}, {388, 32830}, {390, 10513}, {442, 20888}, {491, 19030}, {492, 19029}, {495, 3761}, {497, 37668}, {524, 1914}, {528, 20553}, {536, 20541}, {594, 24326}, {599, 31477}, {609, 8369}, {664, 60452}, {1001, 45962}, {1015, 7813}, {1040, 41005}, {1078, 52793}, {1086, 69015}, {1211, 2092}, {1233, 1441}, {1329, 18135}, {1358, 20924}, {1403, 51612}, {1447, 32851}, {1479, 7776}, {1500, 7794}, {1565, 14210}, {1655, 26558}, {1834, 24995}, {1909, 15888}, {1930, 3695}, {1975, 7354}, {1997, 30740}, {2241, 7855}, {2242, 7801}, {2886, 4441}, {2887, 3663}, {3004, 4509}, {3027, 32458}, {3058, 7788}, {3086, 32818}, {3286, 4966}, {3600, 32840}, {3614, 59635}, {3629, 5332}, {3630, 10987}, {3664, 4697}, {3672, 32773}, {3685, 4872}, {3704, 20911}, {3705, 62697}, {3744, 3879}, {3782, 49518}, {3785, 5217}, {3930, 4437}, {3934, 31460}, {3936, 20347}, {3964, 10832}, {3995, 31117}, {4293, 32817}, {4358, 33864}, {4364, 41269}, {4366, 7779}, {4376, 17365}, {4417, 30946}, {4534, 49755}, {4766, 17747}, {4904, 30109}, {4911, 7283}, {4995, 37671}, {5204, 6337}, {5205, 68926}, {5218, 15589}, {5224, 59296}, {5265, 32841}, {5280, 7819}, {5298, 7799}, {5305, 30104}, {5326, 37688}, {5433, 7763}, {5434, 32833}, {5724, 24291}, {6057, 33931}, {6376, 21031}, {6381, 17757}, {6656, 25264}, {6690, 37670}, {7173, 7752}, {7185, 21605}, {7264, 30171}, {7288, 32831}, {7294, 7769}, {7750, 15338}, {7758, 16502}, {7768, 63273}, {7777, 30998}, {7784, 9598}, {7795, 54416}, {7796, 37722}, {7800, 31448}, {7818, 9664}, {7836, 26686}, {7854, 31451}, {7864, 32107}, {7877, 53680}, {7903, 9665}, {7951, 64093}, {8728, 32092}, {9466, 31476}, {9599, 9766}, {9650, 17130}, {10436, 32777}, {10588, 32834}, {10589, 63098}, {10590, 52713}, {10591, 32823}, {10896, 32816}, {11237, 32836}, {12185, 54103}, {12943, 32815}, {12953, 32006}, {14828, 29839}, {15271, 31497}, {17046, 21071}, {17151, 32865}, {17211, 63997}, {17272, 17594}, {17302, 31090}, {17321, 17599}, {17759, 26582}, {18146, 44847}, {20880, 57808}, {21258, 29966}, {21264, 37661}, {24211, 63800}, {25466, 34284}, {25599, 56734}, {25914, 27162}, {26932, 59734}, {29824, 69083}, {30966, 39915}, {31130, 32862}, {31419, 32104}, {32819, 65631}, {32934, 33869}, {33865, 63996}, {35517, 69032}, {40663, 69038}, {40959, 54344}, {42703, 50747}, {42758, 62430}, {50029, 57019}, {50104, 50116}, {52347, 55391}, {53598, 59547}, {65116, 68895}

X(69093) = reflection of X(21956) in X(20541)
X(69093) = X(55260)-Ceva conjugate of X(918)
X(69093) = X(i)-isoconjugate of X(j) for these (i,j): {884, 36098}, {919, 62749}, {961, 2195}, {1024, 8687}, {1027, 32736}, {1169, 18785}, {1220, 64216}, {1438, 2298}, {2359, 8751}, {2363, 56853}, {4581, 32666}, {36147, 43929}
X(69093) = X(i)-Dao conjugate of X(j) for these (i,j): {960, 56853}, {1211, 105}, {2092, 294}, {3125, 55261}, {3666, 13576}, {6184, 2298}, {17419, 1024}, {17755, 1220}, {35094, 4581}, {36905, 64984}, {38980, 62749}, {38992, 884}, {39015, 43929}, {39063, 961}, {52087, 1438}, {56905, 68565}, {59509, 673}, {62587, 30710}
X(69093) = crosspoint of X(3263) and X(30941)
X(69093) = crosssum of X(56853) and X(64216)
X(69093) = barycentric product X(i)*X(j) for these {i,j}: {518, 20911}, {883, 3910}, {918, 53332}, {960, 40704}, {1026, 4509}, {1211, 30941}, {1228, 3286}, {2292, 18157}, {3004, 42720}, {3263, 3666}, {3674, 3717}, {3687, 9436}, {3912, 4357}, {3930, 16739}, {3932, 16705}, {18206, 18697}, {21124, 68998}, {23829, 65191}, {25083, 54314}, {50330, 55260}
X(69093) = barycentric quotient X(i)/X(j) for these {i,j}: {241, 961}, {429, 68565}, {518, 2298}, {883, 6648}, {918, 4581}, {960, 294}, {1025, 36098}, {1026, 36147}, {1193, 1438}, {1211, 13576}, {1818, 2359}, {1829, 8751}, {1848, 36124}, {2092, 56853}, {2254, 62749}, {2269, 2195}, {2283, 8687}, {2284, 32736}, {2292, 18785}, {2300, 64216}, {3004, 62635}, {3263, 30710}, {3286, 1169}, {3666, 105}, {3674, 56783}, {3687, 14942}, {3882, 36086}, {3910, 885}, {3912, 1220}, {3932, 14624}, {3965, 28071}, {4357, 673}, {6371, 43929}, {9436, 64984}, {17420, 1024}, {18206, 2363}, {20911, 2481}, {22097, 36057}, {22345, 32658}, {24290, 57162}, {24471, 1462}, {25083, 1791}, {30941, 14534}, {40704, 31643}, {41003, 66941}, {42720, 8707}, {43042, 66488}, {48131, 1027}, {50330, 55261}, {52326, 884}, {53280, 919}, {53332, 666}, {54314, 54235}, {57158, 28132}, {61412, 1416}
X(69093) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {75, 37664, 3925}, {192, 3314, 26590}, {325, 350, 11}

X(69094) = X(11)X(76)∩X(12)X(325)

Barycentrics    (a*b - b^2 - a*c - c^2)*(a*b + b^2 - a*c + c^2) : :

X(69094) lies on these lines: {1, 3933}, {5, 3761}, {8, 348}, {10, 68995}, {11, 76}, {12, 325}, {35, 6390}, {36, 7767}, {55, 3926}, {56, 69}, {75, 3665}, {85, 3705}, {99, 15338}, {141, 2275}, {172, 524}, {183, 5433}, {194, 26590}, {230, 4400}, {241, 3687}, {274, 3925}, {304, 3703}, {315, 7354}, {316, 65631}, {319, 7181}, {320, 7198}, {330, 3314}, {337, 17762}, {349, 20436}, {350, 37722}, {385, 26686}, {388, 37668}, {390, 32840}, {491, 19028}, {492, 19027}, {496, 3760}, {497, 32830}, {594, 16720}, {596, 65116}, {668, 21031}, {958, 45962}, {1015, 7794}, {1038, 41005}, {1211, 37596}, {1358, 33930}, {1434, 4645}, {1478, 7776}, {1500, 7813}, {1565, 1930}, {1914, 7789}, {1920, 20487}, {1943, 43045}, {1975, 6284}, {2241, 7801}, {2242, 7855}, {2886, 34284}, {2887, 24215}, {3023, 32458}, {3058, 32833}, {3085, 32818}, {3600, 10513}, {3614, 7752}, {3629, 7296}, {3661, 25918}, {3695, 14210}, {3710, 59504}, {3785, 5204}, {3813, 4441}, {3816, 18135}, {3879, 37539}, {3964, 10831}, {4026, 16705}, {4095, 24318}, {4109, 35102}, {4187, 6381}, {4294, 32817}, {4352, 32773}, {4372, 17362}, {4385, 17181}, {4416, 51436}, {4417, 36854}, {4437, 33299}, {4696, 33864}, {4851, 54317}, {4972, 18600}, {4995, 7799}, {4999, 37670}, {5015, 5088}, {5195, 63996}, {5217, 6337}, {5218, 32831}, {5281, 32841}, {5298, 37671}, {5299, 7819}, {5305, 30103}, {5326, 7769}, {5432, 7763}, {5434, 7788}, {6057, 33939}, {6645, 7779}, {7031, 8369}, {7081, 17095}, {7173, 59635}, {7176, 7270}, {7187, 30179}, {7200, 16886}, {7288, 15589}, {7294, 37688}, {7741, 64093}, {7750, 15326}, {7758, 54416}, {7764, 31460}, {7784, 9597}, {7795, 16502}, {7796, 15888}, {7818, 9651}, {7836, 26629}, {7864, 32005}, {7903, 9650}, {8728, 52716}, {9596, 9766}, {9665, 17130}, {10588, 63098}, {10589, 32834}, {10590, 32823}, {10591, 52713}, {10895, 32816}, {11238, 32836}, {12184, 54103}, {12943, 32006}, {12953, 32815}, {14829, 43053}, {16593, 27109}, {16992, 24953}, {17081, 32099}, {17206, 17798}, {17747, 56024}, {17751, 69083}, {20888, 24390}, {21025, 69009}, {21226, 26558}, {21258, 30036}, {21422, 35517}, {21927, 60719}, {24349, 33949}, {26227, 27187}, {26942, 43034}, {29960, 51384}, {30058, 45226}, {31419, 32092}, {31448, 34511}, {31997, 37664}, {32819, 65632}, {32820, 63273}, {32939, 33867}, {37609, 41014}, {46951, 68688}, {52347, 55392}, {56318, 69035}

X(69094) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {325, 1909, 12}, {330, 3314, 26561}

X(69095) = X(11)X(3934)∩X(12)X(626)

Barycentrics    a^2*b^2 + b^4 + 2*a^2*b*c + a^2*c^2 + 2*b^2*c^2 + c^4 : :

X(69095) lies on these lines: {1, 141}, {2, 16502}, {10, 116}, {11, 3934}, {12, 626}, {35, 7789}, {55, 7795}, {56, 7800}, {69, 54416}, {76, 26590}, {172, 7767}, {230, 30103}, {274, 26582}, {304, 3661}, {325, 27020}, {330, 7876}, {442, 20541}, {498, 7778}, {499, 15271}, {524, 5280}, {594, 1930}, {609, 63928}, {620, 52793}, {625, 3614}, {668, 26558}, {1015, 6292}, {1058, 18840}, {1078, 26686}, {1086, 17192}, {1211, 3912}, {1213, 16782}, {1255, 17316}, {1478, 7784}, {1500, 7794}, {1565, 16720}, {1909, 6656}, {1914, 7819}, {2241, 7822}, {2242, 7854}, {2275, 8362}, {2276, 3933}, {2329, 26932}, {2345, 17170}, {2896, 6645}, {3096, 26561}, {3589, 5299}, {3631, 16785}, {3662, 39731}, {3734, 6284}, {3761, 5254}, {3763, 16781}, {3788, 5432}, {3926, 31448}, {3954, 4437}, {4095, 24211}, {4366, 46226}, {4400, 5305}, {4422, 7313}, {4475, 21336}, {4643, 17742}, {4721, 17747}, {4920, 21101}, {4972, 26978}, {4995, 7880}, {5010, 59545}, {5218, 53033}, {5224, 27248}, {5241, 29596}, {5433, 7815}, {5434, 7865}, {5743, 17284}, {7146, 26942}, {7354, 7761}, {7776, 9596}, {7796, 31462}, {7801, 31451}, {7816, 15338}, {7818, 9650}, {7821, 31476}, {7830, 15326}, {7832, 26629}, {7842, 65631}, {7849, 15888}, {7888, 31501}, {7935, 9651}, {9597, 11287}, {9646, 45473}, {9664, 17130}, {14210, 48635}, {15668, 19784}, {15934, 36479}, {15985, 41239}, {16519, 49752}, {16784, 34573}, {16818, 17245}, {17023, 37595}, {17048, 29655}, {17228, 18156}, {17239, 59633}, {17257, 30701}, {17327, 19836}, {17332, 17744}, {17670, 31997}, {17686, 20553}, {17757, 25102}, {17760, 51390}, {18642, 37613}, {20255, 29659}, {20888, 21956}, {21240, 51384}, {21264, 24390}, {21965, 63817}, {22011, 65116}, {26034, 37580}, {26601, 52043}, {26626, 33172}, {27255, 37664}, {31402, 37668}, {32459, 59325}, {40910, 44419}

X(69095) = crossdifference of every pair of points on line {2483, 16874}
X(69095) = barycentric product X(141)*X(27066)
X(69095) = barycentric quotient X(27066)/X(83)
X(69095) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 34847, 30748}, {325, 27020, 31460}, {3096, 64133, 26561}

X(69096) = X(11)X(39)∩X(12)X(115)

Barycentrics    (b + c)^2*(a^2 + b^2 - 2*b*c + c^2) : :

X(69096) lies on these lines: {1, 5254}, {2, 31448}, {3, 9598}, {4, 54416}, {5, 2276}, {6, 1479}, {10, 3985}, {11, 39}, {12, 115}, {21, 17737}, {30, 172}, {32, 6284}, {33, 27376}, {35, 230}, {36, 63548}, {37, 442}, {41, 8735}, {42, 430}, {53, 3553}, {55, 3767}, {56, 2549}, {58, 5134}, {76, 26590}, {99, 26686}, {141, 3760}, {148, 6645}, {187, 15338}, {192, 5025}, {205, 37226}, {213, 1834}, {220, 66104}, {321, 1228}, {325, 25264}, {350, 6656}, {381, 9596}, {386, 24045}, {388, 43448}, {485, 31459}, {496, 2275}, {497, 5286}, {498, 13881}, {499, 5013}, {574, 5433}, {594, 762}, {609, 65134}, {740, 4109}, {999, 9597}, {1015, 7765}, {1086, 7264}, {1107, 24390}, {1211, 4044}, {1334, 21935}, {1425, 1562}, {1478, 44518}, {1504, 19030}, {1505, 19029}, {1506, 7173}, {1571, 24914}, {1572, 12701}, {1575, 4187}, {1656, 31461}, {1698, 31426}, {1824, 22283}, {1837, 9620}, {1855, 20310}, {1865, 39579}, {1909, 47286}, {1914, 5305}, {2066, 13081}, {2082, 61160}, {2171, 8736}, {2176, 64172}, {2241, 3058}, {2242, 7354}, {2292, 21029}, {2295, 23903}, {2345, 52258}, {2548, 10896}, {2886, 5283}, {2887, 21071}, {2901, 4153}, {3053, 4302}, {3054, 65142}, {3055, 65141}, {3086, 7738}, {3091, 31402}, {3094, 10079}, {3120, 21808}, {3125, 21049}, {3208, 37716}, {3269, 26955}, {3290, 23537}, {3294, 68946}, {3454, 21070}, {3496, 33095}, {3509, 24851}, {3583, 5280}, {3585, 16785}, {3614, 31476}, {3663, 17046}, {3673, 17060}, {3691, 33136}, {3695, 4037}, {3721, 63997}, {3735, 40997}, {3761, 63923}, {3772, 30810}, {3813, 16975}, {3815, 7741}, {3914, 16583}, {3925, 16589}, {3954, 4415}, {4000, 17671}, {4024, 21134}, {4026, 52538}, {4056, 17365}, {4071, 63800}, {4193, 17756}, {4294, 7735}, {4299, 44526}, {4366, 7797}, {4396, 7767}, {4424, 21965}, {4426, 11113}, {4441, 17550}, {4642, 21044}, {4857, 5299}, {4894, 17362}, {4972, 27040}, {5021, 11269}, {5028, 39873}, {5080, 63537}, {5179, 41015}, {5230, 14974}, {5231, 31429}, {5276, 52367}, {5291, 57288}, {5292, 17732}, {5306, 7031}, {5319, 9670}, {5414, 13082}, {5432, 7746}, {5434, 9651}, {5523, 6198}, {6034, 10070}, {6057, 7230}, {6155, 66655}, {6354, 20618}, {6421, 44624}, {6422, 44623}, {6781, 9341}, {7109, 63604}, {7296, 18907}, {7736, 10591}, {7737, 12953}, {7739, 11238}, {7747, 65632}, {7749, 52793}, {7755, 63273}, {7756, 15326}, {7772, 9665}, {7789, 30103}, {7790, 26561}, {7828, 26629}, {7856, 53680}, {7876, 30998}, {7906, 32107}, {7951, 63534}, {9300, 65140}, {9331, 37719}, {9575, 9614}, {9581, 9593}, {9592, 50443}, {9599, 9605}, {9607, 37720}, {9619, 11376}, {9660, 12963}, {9668, 30435}, {10086, 44534}, {10386, 10987}, {10527, 31449}, {10589, 31400}, {10590, 63533}, {10593, 31406}, {10799, 13183}, {10895, 31409}, {10959, 11998}, {11680, 31466}, {12428, 23128}, {12699, 54382}, {16049, 32758}, {16455, 54285}, {16600, 21090}, {16780, 66682}, {17048, 53600}, {17143, 26558}, {17280, 33834}, {17302, 33835}, {17606, 31398}, {17669, 17759}, {17670, 30963}, {17757, 20691}, {18140, 26582}, {18514, 53418}, {18600, 31058}, {18965, 62206}, {18966, 62205}, {19028, 31471}, {20236, 65117}, {20337, 53509}, {21031, 52959}, {21384, 33141}, {21843, 63756}, {24047, 45939}, {24054, 54426}, {24703, 54406}, {25092, 25639}, {27020, 59635}, {28146, 61688}, {28174, 61741}, {31140, 31416}, {31231, 31421}, {33034, 60706}, {33296, 41324}, {35802, 45512}, {35803, 45513}, {40965, 52577}, {41269, 50067}, {49736, 68893}, {50036, 56926}, {53416, 62210}

X(69096) = isotomic conjugate of the isogonal conjugate of X(21813)
X(69096) = polar conjugate of the isotomic conjugate of X(21015)
X(69096) = X(i)-isoconjugate of X(j) for these (i,j): {58, 40403}, {60, 7131}, {249, 68559}, {593, 56179}, {757, 7123}, {849, 30701}, {1037, 2185}, {1041, 65568}, {1437, 40411}, {1444, 57386}, {1509, 7084}, {2150, 8817}, {6061, 63178}, {7054, 56359}, {7341, 56243}
X(69096) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 40403}, {4000, 7058}, {4075, 30701}, {6554, 1509}, {14936, 65575}, {15487, 757}, {16583, 17206}, {17463, 16751}, {18589, 81}, {40607, 7123}, {55065, 48070}, {56325, 8817}, {59619, 52379}
X(69096) = crosspoint of X(i) and X(j) for these (i,j): {321, 1826}, {594, 6354}, {3914, 53510}
X(69096) = crosssum of X(i) and X(j) for these (i,j): {593, 7054}, {1333, 1790}
X(69096) = barycentric product X(i)*X(j) for these {i,j}: {4, 21015}, {10, 3914}, {12, 497}, {37, 53510}, {76, 21813}, {306, 52577}, {313, 40934}, {321, 16583}, {594, 4000}, {614, 1089}, {756, 3673}, {762, 16750}, {1040, 56285}, {1441, 40965}, {1473, 7141}, {1577, 61160}, {1633, 4036}, {1824, 20235}, {1826, 18589}, {1851, 3695}, {1863, 6356}, {2082, 6358}, {3701, 40961}, {3732, 4024}, {3952, 48403}, {4012, 6046}, {4086, 62752}, {4103, 48398}, {4605, 68784}, {6057, 7195}, {6354, 6554}, {7083, 34388}, {7140, 17170}, {8020, 40071}, {8736, 27509}, {16502, 28654}, {17441, 41013}, {18084, 21016}, {21750, 27801}, {27808, 50490}, {40987, 57807}
X(69096) = barycentric quotient X(i)/X(j) for these {i,j}: {12, 8817}, {37, 40403}, {181, 1037}, {497, 261}, {594, 30701}, {614, 757}, {756, 56179}, {762, 56260}, {872, 7084}, {1089, 57925}, {1254, 56359}, {1500, 7123}, {1633, 52935}, {1826, 40411}, {1863, 59482}, {2082, 2185}, {2171, 7131}, {2333, 57386}, {2643, 68559}, {3673, 873}, {3732, 4610}, {3914, 86}, {4000, 1509}, {4024, 48070}, {4319, 1098}, {6354, 30705}, {6554, 7058}, {7083, 60}, {7124, 65568}, {7147, 63178}, {7195, 552}, {8020, 1474}, {16502, 593}, {16583, 81}, {16750, 57949}, {17115, 65575}, {17441, 1444}, {18589, 17206}, {21015, 69}, {21107, 15419}, {21750, 1333}, {21813, 6}, {22363, 1437}, {23620, 1790}, {30706, 7054}, {40521, 52778}, {40934, 58}, {40961, 1014}, {40965, 21}, {40987, 270}, {48403, 7192}, {50490, 3733}, {52577, 27}, {53510, 274}, {61160, 662}, {62752, 1414}, {66930, 59133}
X(69096) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 2276, 31460}, {32, 9664, 6284}, {115, 1500, 12}, {496, 15048, 2275}, {497, 5286, 16502}, {1656, 31461, 31497}, {1834, 17747, 213}, {2242, 7748, 7354}, {2276, 31460, 31462}, {2295, 23903, 37715}, {3583, 5280, 7745}, {4037, 16886, 3695}, {5305, 15171, 1914}, {7746, 31451, 5432}, {9605, 9669, 9599}, {13881, 31477, 498}, {21090, 36250, 16600}, {25092, 25639, 37661}, {31476, 39565, 3614}

X(69097) = X(11)X(626)∩X(12)X(3934)

Barycentrics    a^2*b^2 + b^4 - 2*a^2*b*c + a^2*c^2 + 2*b^2*c^2 + c^4 : :

X(69097) lies on these lines: {1, 141}, {2, 31402}, {3, 8299}, {10, 17048}, {11, 626}, {12, 3934}, {36, 7789}, {55, 7800}, {56, 7795}, {69, 16502}, {75, 17670}, {76, 26561}, {88, 17740}, {142, 49598}, {172, 7819}, {192, 7876}, {230, 30104}, {304, 3662}, {321, 50319}, {325, 26959}, {350, 6656}, {442, 21264}, {498, 15271}, {499, 7778}, {524, 5299}, {599, 16781}, {625, 7173}, {1015, 7794}, {1056, 18840}, {1078, 26629}, {1086, 1930}, {1211, 17023}, {1213, 16818}, {1333, 29789}, {1429, 26942}, {1479, 7784}, {1500, 3666}, {1914, 7767}, {1973, 25940}, {2241, 7854}, {2242, 7822}, {2275, 3933}, {2276, 8362}, {2896, 4366}, {3058, 7865}, {3061, 18730}, {3096, 26590}, {3589, 5280}, {3631, 16784}, {3661, 39731}, {3734, 7354}, {3739, 17529}, {3760, 5254}, {3788, 5433}, {3831, 17062}, {3836, 17050}, {3840, 17046}, {3936, 27067}, {4000, 33833}, {4136, 24172}, {4187, 20530}, {4301, 12610}, {4396, 5305}, {4422, 17744}, {4426, 17540}, {4441, 33840}, {4904, 20255}, {5025, 30998}, {5298, 7880}, {5432, 7815}, {5437, 17284}, {5692, 59703}, {5743, 29598}, {6284, 7761}, {6645, 46226}, {6707, 19881}, {7031, 63928}, {7237, 21336}, {7280, 59545}, {7288, 53033}, {7776, 9599}, {7811, 53680}, {7816, 15326}, {7818, 9665}, {7830, 15338}, {7832, 26686}, {7842, 65632}, {7849, 37722}, {7935, 9664}, {9598, 11287}, {9651, 17130}, {9661, 45473}, {14210, 17192}, {15668, 19836}, {16043, 31448}, {16785, 34573}, {16886, 27918}, {17030, 37664}, {17143, 26582}, {17227, 18156}, {17234, 27248}, {17279, 17742}, {17316, 33172}, {17327, 19784}, {17514, 25498}, {18140, 26558}, {18636, 62772}, {19786, 41251}, {20541, 24390}, {20549, 30038}, {20860, 33299}, {21031, 27076}, {25500, 29642}, {25532, 25645}, {26626, 32782}, {26639, 37636}, {27633, 29960}, {27950, 34055}, {30965, 68712}, {31239, 31476}, {32459, 59319}, {33171, 37580}, {37539, 51384}

X(69097) = X(i)-complementary conjugate of X(j) for these (i,j): {649, 55053}, {54458, 3835}
X(69097) = barycentric product X(141)*X(27004)
X(69097) = barycentric quotient X(27004)/X(83)
X(69097) = {X(141),X(4657)}-harmonic conjugate of X(51571)

X(69098) = X(11)X(115)∩X(12)X(39)

Barycentrics    (b - c)^2*(a^2 + b^2 + 2*b*c + c^2) : :

X(69098) lies on these lines: {1, 5254}, {2, 31449}, {3, 9597}, {4, 16502}, {5, 2275}, {6, 1478}, {11, 115}, {12, 39}, {30, 1914}, {32, 7354}, {34, 27376}, {35, 63548}, {36, 230}, {53, 3554}, {55, 2549}, {56, 3767}, {76, 26561}, {99, 26629}, {116, 17205}, {119, 34460}, {141, 3761}, {148, 4366}, {172, 5305}, {187, 15326}, {244, 21044}, {274, 26558}, {330, 5025}, {350, 47286}, {381, 9599}, {388, 5286}, {442, 1107}, {495, 2276}, {496, 63493}, {497, 43448}, {498, 5013}, {499, 13881}, {514, 65116}, {519, 21956}, {529, 5291}, {540, 57017}, {574, 5432}, {594, 4692}, {604, 8736}, {609, 5306}, {668, 26582}, {918, 1086}, {1018, 24222}, {1100, 12690}, {1104, 53422}, {1145, 21888}, {1358, 4403}, {1475, 21935}, {1479, 16781}, {1500, 7765}, {1504, 19028}, {1505, 19027}, {1506, 3614}, {1562, 3270}, {1565, 7200}, {1572, 1836}, {1573, 3925}, {1574, 21031}, {1575, 17757}, {1826, 20227}, {1834, 20963}, {1870, 5523}, {1901, 2300}, {1909, 6656}, {2067, 18989}, {2087, 62221}, {2170, 2969}, {2221, 60156}, {2241, 6284}, {2242, 5309}, {2476, 31466}, {2548, 10895}, {2886, 16975}, {3053, 4299}, {3058, 9664}, {3085, 7738}, {3094, 10063}, {3122, 23686}, {3136, 23632}, {3230, 17747}, {3269, 26956}, {3290, 5179}, {3295, 9598}, {3419, 16973}, {3583, 16784}, {3585, 5299}, {3662, 20925}, {3670, 21965}, {3703, 34542}, {3708, 53559}, {3727, 63997}, {3735, 3782}, {3760, 63923}, {3765, 37096}, {3815, 7951}, {3822, 37661}, {4165, 17155}, {4167, 67983}, {4187, 16604}, {4293, 7735}, {4302, 44526}, {4386, 11112}, {4400, 7767}, {4534, 29102}, {4680, 17362}, {4872, 69003}, {5021, 5230}, {5024, 31479}, {5028, 39897}, {5030, 17734}, {5080, 33854}, {5134, 40091}, {5177, 31405}, {5190, 53825}, {5219, 9592}, {5252, 9620}, {5261, 31402}, {5270, 5280}, {5283, 25466}, {5319, 9657}, {5332, 18907}, {5433, 7746}, {5514, 15611}, {5515, 5517}, {5518, 53823}, {5691, 16780}, {6034, 10054}, {6376, 17670}, {6421, 44622}, {6422, 31472}, {6502, 18988}, {6547, 47915}, {6645, 7797}, {7031, 10483}, {7173, 39565}, {7272, 17365}, {7736, 10590}, {7737, 12943}, {7739, 11237}, {7741, 63534}, {7747, 65631}, {7756, 15338}, {7772, 9650}, {7789, 30104}, {7790, 26590}, {7828, 26686}, {7906, 32005}, {9336, 37720}, {9456, 38950}, {9561, 10406}, {9574, 31434}, {9575, 9612}, {9578, 9593}, {9596, 9605}, {9607, 31462}, {9619, 11375}, {9647, 12963}, {9655, 30435}, {10056, 31477}, {10089, 44534}, {10588, 31400}, {10591, 63533}, {10592, 31406}, {10956, 21859}, {12723, 23663}, {12835, 13182}, {13901, 62206}, {13958, 62205}, {15325, 43291}, {16519, 63360}, {16583, 23536}, {16609, 53590}, {17046, 24215}, {17062, 24214}, {17448, 24390}, {17737, 54391}, {17754, 37716}, {18513, 53418}, {18961, 56913}, {18970, 23128}, {19854, 31490}, {20455, 22308}, {21226, 33841}, {21232, 53600}, {23537, 41015}, {24070, 34587}, {24247, 33144}, {24512, 37715}, {24630, 35466}, {24953, 31456}, {26073, 27546}, {26564, 26565}, {26845, 27009}, {26959, 59635}, {30384, 62370}, {31426, 51784}, {31501, 53096}, {35076, 61073}, {35092, 53167}, {35800, 45512}, {35801, 45513}, {37512, 52793}, {38962, 53838}, {39248, 58798}, {41269, 66675}, {45751, 68946}, {47515, 62692}, {50319, 52043}, {51421, 52635}, {53416, 62211}, {54382, 57282}, {63543, 65140}

X(69098) = polar conjugate of the isotomic conjugate of X(26933)
X(69098) = X(i)-complementary conjugate of X(j) for these (i,j): {649, 10472}, {941, 3835}, {959, 17072}, {2258, 513}, {3122, 53829}, {5331, 512}, {31359, 21260}, {32693, 21232}, {34258, 21262}, {34260, 48044}, {37870, 42327}
X(69098) = X(i)-Ceva conjugate of X(j) for these (i,j): {388, 8678}, {2345, 6590}, {4385, 48395}, {34284, 784}, {56044, 514}, {60156, 513}, {68567, 649}
X(69098) = X(i)-isoconjugate of X(j) for these (i,j): {59, 2339}, {100, 65298}, {101, 1310}, {249, 68558}, {692, 37215}, {765, 2221}, {1016, 1472}, {1036, 4564}, {1039, 44717}, {1245, 4567}, {1252, 56328}, {1331, 36099}, {1332, 32691}, {2149, 30479}, {2281, 4600}, {4570, 56219}, {23990, 57923}, {32656, 65341}, {32739, 54982}
X(69098) = X(i)-Dao conjugate of X(j) for these (i,j): {513, 2221}, {650, 30479}, {661, 56328}, {1015, 1310}, {1086, 37215}, {1577, 64989}, {5515, 190}, {5521, 36099}, {6590, 5224}, {6615, 2339}, {8054, 65298}, {8678, 54416}, {17421, 1332}, {40181, 765}, {40619, 54982}, {40627, 1245}, {47842, 28606}, {50330, 56219}, {50497, 2281}, {55046, 100}, {68772, 5739}
X(69098) = crosspoint of X(i) and X(j) for these (i,j): {513, 46331}, {514, 43531}, {2345, 6590}, {17924, 68578}
X(69098) = crosssum of X(i) and X(j) for these (i,j): {101, 386}, {2221, 65298}
X(69098) = crossdifference of every pair of points on line {692, 1331}
X(69098) = barycentric product X(i)*X(j) for these {i,j}: {4, 26933}, {11, 388}, {244, 4385}, {513, 2517}, {514, 6590}, {523, 47844}, {612, 1111}, {693, 8678}, {1010, 3120}, {1086, 2345}, {1146, 7365}, {1358, 3974}, {1460, 34387}, {1565, 7102}, {2285, 4858}, {2303, 16732}, {2484, 3261}, {2522, 17924}, {2968, 7103}, {2969, 54433}, {2973, 7085}, {3122, 44154}, {4081, 7197}, {4320, 24026}, {5515, 43531}, {5517, 60156}, {7192, 48395}, {7649, 23874}, {8646, 40495}, {8735, 56367}, {14594, 21132}, {17421, 68578}, {21207, 44119}, {23989, 54416}, {44426, 51644}, {50494, 52619}
X(69098) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 30479}, {244, 56328}, {388, 4998}, {513, 1310}, {514, 37215}, {612, 765}, {649, 65298}, {693, 54982}, {1010, 4600}, {1015, 2221}, {1111, 57923}, {1460, 59}, {2170, 2339}, {2285, 4564}, {2286, 44717}, {2303, 4567}, {2345, 1016}, {2484, 101}, {2517, 668}, {2522, 1332}, {2643, 68558}, {3121, 2281}, {3122, 1245}, {3125, 56219}, {3248, 1472}, {3271, 1036}, {3974, 4076}, {4206, 5379}, {4320, 7045}, {4385, 7035}, {4858, 64989}, {5515, 5224}, {5517, 5739}, {6590, 190}, {6591, 36099}, {7102, 15742}, {7103, 55346}, {7197, 59457}, {7365, 1275}, {8646, 692}, {8678, 100}, {16732, 60197}, {17924, 65341}, {18210, 66948}, {23874, 4561}, {26933, 69}, {42067, 51686}, {44119, 4570}, {47844, 99}, {48395, 3952}, {50494, 4557}, {51644, 6516}, {54416, 1252}, {55046, 54416}
X(69098) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {12, 39, 31460}, {32, 9651, 7354}, {115, 1015, 11}, {388, 5286, 54416}, {495, 15048, 2276}, {1086, 1146, 3125}, {2241, 7748, 6284}, {3085, 7738, 31448}, {3585, 5299, 7745}, {5024, 31479, 31497}, {5305, 18990, 172}, {7790, 64133, 26590}, {9605, 9654, 9596}, {9607, 37719, 31462}, {16781, 44518, 1479}, {26564, 26572, 26565}

X(69099) = X(11)-LINE CONJUGATE Of X(2)

Barycentrics    a^3*(a^2*b^2 - a*b^3 - a*b^2*c + a^2*c^2 - a*b*c^2 + 2*b^2*c^2 - a*c^3) : :

X(69099) lies on these lines: {2, 11}, {31, 1911}, {42, 20457}, {109, 61055}, {187, 237}, {739, 8693}, {869, 30647}, {1086, 24447}, {1149, 69018}, {1201, 18758}, {1402, 51329}, {1613, 1979}, {2112, 18265}, {2177, 40732}, {2179, 5369}, {2195, 61050}, {2238, 20475}, {2295, 16683}, {2876, 68795}, {4433, 20352}, {8616, 18794}, {10987, 15624}, {12338, 18135}, {13077, 26541}, {14096, 37586}, {16691, 33863}, {16693, 17735}, {16704, 20044}, {17002, 64170}, {17126, 21010}, {17127, 20332}, {20670, 20974}, {20678, 20999}, {20683, 69074}, {21000, 21001}, {24501, 25050}, {30646, 51985}, {33854, 51928}, {40910, 56805}, {68872, 69087}

X(69099) = isogonal conjugate of X(53219)
X(69099) = isogonal conjugate of the anticomplement of X(65888)
X(69099) = isogonal conjugate of the isotomic conjugate of X(14839)
X(69099) = X(14665)-Ceva conjugate of X(6)
X(69099) = X(i)-isoconjugate of X(j) for these (i,j): {1, 53219}, {75, 14665}, {673, 46802}, {2254, 65636}
X(69099) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 53219}, {206, 14665}, {65888, 76}
X(69099) = crosspoint of X(6) and X(14665)
X(69099) = crosssum of X(2) and X(14839)
X(69099) = crossdifference of every pair of points on line {2, 665}
X(69099) = X(i)-line conjugate of X(j) for these (i,j): {11, 2}, {100,2}, {187, 665}
X(69099) = barycentric product X(i)*X(j) for these {i,j}: {6, 14839}, {55, 43063}, {919, 65874}, {2223, 46798}, {14665, 65888}
X(69099) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 53219}, {32, 14665}, {919, 65636}, {2223, 46802}, {14839, 76}, {43063, 6063}
X(69099) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {31, 1911, 1977}, {55, 2110, 100}, {55, 23404, 56853}, {100, 1621, 4366}, {100, 38878, 2}, {902, 3009, 8622}, {3009, 8622, 3231}

X(69100) = X(98)-LINE CONJUGATE Of X(2)

Barycentrics    a^2*(a^6*b^2 - a^4*b^4 + a^6*c^2 - 2*a^4*b^2*c^2 + a^2*b^4*c^2 - b^6*c^2 - a^4*c^4 + a^2*b^2*c^4 + 2*b^4*c^4 - b^2*c^6) : :
X(69100) = 3 X[44420] - X[57257]

X(69100) lies on these lines: {2, 98}, {3, 54998}, {5, 68651}, {6, 6784}, {23, 11673}, {25, 694}, {31, 61053}, {51, 1196}, {99, 34341}, {111, 263}, {154, 20885}, {187, 237}, {217, 63556}, {230, 9418}, {251, 46306}, {323, 33873}, {385, 33758}, {394, 57258}, {420, 50188}, {511, 44420}, {543, 5118}, {1003, 35399}, {1316, 22735}, {1506, 3203}, {1560, 44080}, {1576, 44127}, {1625, 65751}, {1634, 6786}, {1915, 51318}, {2393, 18371}, {2421, 9149}, {2493, 2871}, {3098, 34095}, {3117, 3148}, {3202, 7746}, {3917, 7467}, {3978, 56430}, {4563, 25332}, {5108, 43765}, {5650, 9155}, {6072, 51455}, {7749, 40643}, {7824, 14133}, {9129, 34010}, {9157, 9998}, {9512, 63464}, {9696, 39834}, {9698, 42444}, {10192, 60527}, {10539, 37466}, {11064, 21531}, {11159, 50672}, {11183, 67595}, {11328, 35259}, {11672, 51869}, {13366, 20965}, {14811, 15080}, {14957, 51360}, {15004, 39024}, {15066, 52658}, {18024, 46247}, {21650, 38551}, {26864, 46276}, {26885, 56558}, {32217, 46998}, {34396, 45901}, {35266, 44215}, {35268, 37184}, {35901, 46253}, {36425, 61194}, {37988, 61743}, {39087, 51983}, {39689, 44109}, {39846, 47421}, {42742, 62516}, {44089, 53500}, {45317, 46302}, {51431, 57615}

X(69100) = midpoint of X(2421) and X(9149)
X(69100) = reflection of X(44114) in X(2493)
X(69100) = isogonal conjugate of X(46142)
X(69100) = Parry-circle-inverse of X(237)
X(69100) = Parry-isodynamic-circle-inverse of X(3288)
X(69100) = isogonal conjugate of the anticomplement of X(61070)
X(69100) = isogonal conjugate of the isotomic conjugate of X(2782)
X(69100) = psi-transform of X(37991)
X(69100) = X(2698)-Ceva conjugate of X(6)
X(69100) = X(i)-isoconjugate of X(j) for these (i,j): {1, 46142}, {75, 2698}, {662, 46040}, {1581, 16069}, {1821, 51229}, {8773, 46039}
X(69100) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 46142}, {206, 2698}, {1084, 46040}, {19576, 16069}, {39072, 46039}, {40601, 51229}, {61070, 76}
X(69100) = crosspoint of X(6) and X(2698)
X(69100) = crosssum of X(2) and X(2782)
X(69100) = crossdifference of every pair of points on line {2, 3569}
X(69100) = X(i)-line conjugate of X(j) for these (i,j): {98, 2}, {110, 2}{187, 3569}
X(69100) = barycentric product X(i)*X(j) for these {i,j}: {6, 2782}, {230, 51455}, {385, 16068}, {511, 48452}, {1691, 67078}, {2698, 61070}, {2966, 55143}, {4590, 6071}, {6072, 41932}
X(69100) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 46142}, {32, 2698}, {237, 51229}, {512, 46040}, {1691, 16069}, {1692, 46039}, {2715, 64779}, {2782, 76}, {6071, 115}, {6072, 32458}, {16068, 1916}, {48452, 290}, {51455, 8781}, {55143, 2799}, {67078, 18896}
X(69100) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 110, 36213}, {2, 20021, 125}, {6, 53264, 6784}, {110, 1976, 184}, {154, 21001, 20885}, {237, 3231, 47638}, {263, 34098, 34417}, {1495, 47638, 237}, {1495, 52144, 67541}, {1495, 67550, 56393}, {1613, 20998, 694}, {2502, 3231, 5106}, {2502, 5191, 1495}, {3124, 51335, 51}, {3229, 42671, 237}, {3506, 9306, 110}, {5638, 5639, 237}, {9463, 34098, 263}, {18773, 18774, 67551}, {20998, 52162, 25}, {42667, 42668, 56393}, {42671, 67543, 1495}, {61053, 61059, 31}

X(69101) = X(514)-LINE CONJUGATE Of X(2)

Barycentrics    a^2*(b - c)*(a*b - 2*b^2 + a*c - b*c - 2*c^2) : :
X(69101) = X[4893] + 2 X[52745], X[649] - 4 X[665], X[649] + 2 X[3250], 2 X[665] + X[3250], X[661] + 2 X[54249], 2 X[876] + X[4724], 2 X[3766] - 5 X[30835], X[4382] - 4 X[68901], 4 X[6586] - X[20979], 2 X[6586] + X[21123], X[20979] + 2 X[21123], 2 X[21191] + X[21225], X[31147] - 4 X[45658]

X(69101) lies on these lines: {2, 514}, {187, 237}, {263, 55257}, {513, 14408}, {650, 29226}, {657, 47329}, {661, 3777}, {784, 47873}, {876, 4724}, {1459, 9011}, {1613, 21791}, {1635, 4083}, {1919, 53315}, {3063, 23562}, {3709, 6085}, {3766, 30835}, {4145, 17458}, {4382, 68901}, {4449, 21352}, {4813, 48086}, {4988, 47720}, {6004, 68829}, {6372, 48544}, {6586, 9002}, {9029, 19586}, {21191, 21225}, {21389, 53390}, {21828, 40627}, {21832, 56805}, {24577, 45323}, {25258, 30094}, {30665, 47828}, {30671, 33570}, {31147, 45658}, {47874, 63812}, {50454, 68813}, {50525, 57133}, {52660, 66947}

X(69101) = midpoint of X(i) and X(j) for these {i,j}: {14407, 21123}, {21225, 53368}
X(69101) = reflection of X(i) in X(j) for these {i,j}: {14407, 6586}, {20979, 14407}, {53368, 21191}
X(69101) = isogonal conjugate of the isotomic conjugate of X(30519)
X(69101) = X(30554)-Ceva conjugate of X(6)
X(69101) = X(i)-isoconjugate of X(j) for these (i,j): {75, 30554}, {100, 60873}, {662, 60624}
X(69101) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 30554}, {1084, 60624}, {8054, 60873}
X(69101) = crosspoint of X(6) and X(30554)
X(69101) = crosssum of X(i) and X(j) for these (i,j): {2, 30519}, {514, 17367}
X(69101) = crossdifference of every pair of points on line {2, 902}
X(69101) = X(i)-line conjugate of X(j) for these (i,j): {187, 902}, {514, 2}
X(69101) = barycentric product X(i)*X(j) for these {i,j}: {1, 50335}, {6, 30519}, {513, 49448}, {647, 31916}, {649, 17230}, {903, 9461}
X(69101) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 30554}, {512, 60624}, {649, 60873}, {9461, 519}, {17230, 1978}, {30519, 76}, {31916, 6331}, {49448, 668}, {50335, 75}
X(69101) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {665, 3250, 649}, {6586, 21123, 20979}

X(69102) = X(513)-LINE CONJUGATE Of X(2)

Barycentrics    a^2*(b - c)*(a*b^2 - a*b*c + b^2*c + a*c^2 + b*c^2) : :
X(69102) = 3 X[14474] - 2 X[47761], 3 X[43928] - X[47763], 2 X[649] + X[2978], X[649] + 2 X[50510], X[2978] - 4 X[50510], 4 X[650] - X[20983], 2 X[650] + X[50521], X[20983] + 2 X[50521], X[661] + 2 X[50514], 4 X[4394] - X[50487], 2 X[4790] + X[50497], X[7192] - 4 X[43931], X[17494] + 2 X[50516], 4 X[25142] - 7 X[27115], X[39548] + 2 X[48011], 2 X[48008] + X[50524]

X(69102) lies on these lines: {2, 513}, {187, 237}, {263, 55259}, {650, 9010}, {659, 9002}, {661, 50514}, {788, 1635}, {1491, 47330}, {1613, 21007}, {3051, 3063}, {3805, 47771}, {3808, 44435}, {4083, 47776}, {4378, 21352}, {4394, 50487}, {4724, 6085}, {4790, 50497}, {4893, 6373}, {6363, 48572}, {6372, 48577}, {7192, 16737}, {9040, 47884}, {9980, 11673}, {14407, 46386}, {17494, 29226}, {20965, 20980}, {25142, 27115}, {27486, 30665}, {30094, 31003}, {31150, 68900}, {39548, 48011}, {40790, 48324}, {48008, 50524}, {48335, 56805}, {50489, 57162}

X(69102) = midpoint of X(14404) and X(50521)
X(69102) = reflection of X(i) in X(j) for these {i,j}: {14404, 650}, {20983, 14404}, {47762, 38238}
X(69102) = X(i)-Ceva conjugate of X(j) for these (i,j): {36873, 3248}, {55919, 1015}
X(69102) = X(i)-isoconjugate of X(j) for these (i,j): {100, 56166}, {101, 56129}, {190, 60871}
X(69102) = X(i)-Dao conjugate of X(j) for these (i,j): {1015, 56129}, {8054, 56166}, {55053, 60871}
X(69102) = crossdifference of every pair of points on line {2, 3230}
X(69102) = X(i)-line conjugate of X(j) for these (i,j): {187, 3230}, {513, 2}
X(69102) = barycentric product X(i)*X(j) for these {i,j}: {513, 16975}, {649, 30942}
X(69102) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 56129}, {649, 56166}, {667, 60871}, {16975, 668}, {30942, 1978}
X(69102) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 663, 890}, {649, 50510, 2978}, {650, 50521, 20983}

X(69103) = X(650)-LINE CONJUGATE Of X(2)

Barycentrics    a^2*(b - c)*(a^2*b^2 - a*b^3 + a^2*b*c - a*b^2*c - b^3*c + a^2*c^2 - a*b*c^2 - a*c^3 - b*c^3) : :

X(69103) lies on these lines: {2, 650}, {187, 237}, {659, 6586}, {2276, 50335}, {2530, 52589}, {3051, 22383}, {3709, 4724}, {4790, 50520}, {6004, 21837}, {6377, 68757}, {7192, 54249}, {21007, 53309}, {21123, 46386}, {21348, 47694}, {43931, 52655}, {48008, 68901}, {50521, 68112}

X(69103) = crossdifference of every pair of points on line {2, 2223}
X(69103) = X(i)-line conjugate of X(j) for these (i,j): {187, 2223}, {650, 2}
X(69103) = barycentric product X(513)*X(56542)
X(69103) = barycentric quotient X(56542)/X(668)
X(69103) = {X(649),X(3250)}-harmonic conjugate of X(2978)

X(69104) = X(661)-LINE CONJUGATE Of X(2)

Barycentrics    a^2*(b - c)*(-(a*b^3) + a^2*b*c - a*b^2*c - b^3*c - a*b*c^2 - b^2*c^2 - a*c^3 - b*c^3) : :

X(69104) lies on these lines: {2, 661}, {187, 237}, {650, 24534}, {659, 21123}, {1491, 23656}, {1613, 7252}, {2084, 14838}, {3709, 68886}, {4079, 4784}, {4107, 24626}, {4367, 40627}, {6586, 46386}, {20979, 27675}, {21763, 52592}, {21832, 50524}, {21834, 50343}, {23657, 47888}, {63461, 68830}

X(69104) = crossdifference of every pair of points on line {2, 3747}
X(69104) = X(i)-line conjugate of X(j) for these (i,j): {187, 3747}, {661, 2}
X(69104) = barycentric product X(649)*X(31027)
X(69104) = barycentric quotient X(31027)/X(1978)
X(69104) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 5029, 669}, {665, 50510, 649}

X(69105) = X(38)-LINE CONJUGATE Of X(2)

Barycentrics    a^2*(-(a*b^4) + a^2*b^2*c + b^4*c + a^2*b*c^2 - 2*a*b^2*c^2 - a*c^4 + b*c^4) : :

X(69105) lies on these lines: {2, 38}, {42, 6377}, {187, 237}, {190, 24413}, {1613, 35326}, {2228, 68756}, {3116, 24513}, {3123, 24403}, {3862, 25817}, {17449, 20456}, {20284, 42079}, {20331, 52633}, {20663, 46148}, {32925, 34020}, {36817, 42046}, {38986, 62753}

X(69105) = crossdifference of every pair of points on line {2, 8632}
X(69105) = X(i)-line conjugate of X(j) for these (i,j): {38, 2}, {187, 8632}
X(69105) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3009, 69030, 8622}, {3116, 26242, 24513}

X(69106) = (1,1,1,1,-1,1,1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 - 4*b^2*c^2 - 2*c^4 + 2*Sqrt[3]*a^2*S : :
Barycentrics    (a^2 - b^2 - c^2)*(a^2 + 2*b^2 + 2*c^2) + 2*Sqrt[3]*a^2*S : :

For an introduction to additive associates, see the preamble just before X(69091).

X(69106) lies on these lines: {6, 7794}, {13, 76}, {14, 7776}, {15, 7767}, {16, 3926}, {18, 325}, {61, 69}, {62, 3933}, {99, 42433}, {202, 69094}, {302, 7871}, {315, 16964}, {622, 3146}, {628, 5980}, {634, 41020}, {1007, 42489}, {1975, 42158}, {3180, 46226}, {3411, 7796}, {3785, 5238}, {3818, 5865}, {5237, 6390}, {5351, 6337}, {5352, 59541}, {7006, 69093}, {7750, 36967}, {7752, 42580}, {7763, 16242}, {7769, 33416}, {7773, 16809}, {7789, 41406}, {7797, 34541}, {7800, 63201}, {7802, 42099}, {7810, 36775}, {7874, 62199}, {7888, 62197}, {10513, 42999}, {10653, 32830}, {11132, 22914}, {14023, 41407}, {14539, 59363}, {14541, 48881}, {14929, 42147}, {15589, 42152}, {16962, 37671}, {16966, 32832}, {18581, 32823}, {18582, 32834}, {19106, 32819}, {22845, 44777}, {30471, 43459}, {32006, 36970}, {32815, 42431}, {32817, 42151}, {32818, 42149}, {32822, 42086}, {32828, 37832}, {32833, 41100}, {32835, 42089}, {32836, 41107}, {32869, 41112}, {32874, 41119}, {34229, 42936}, {37668, 40694}, {41121, 46951}, {42162, 52713}, {43632, 64018}, {47005, 66445}, {53428, 66347}

X(69107) = (1,1,1,1,-1,1,-1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 - 4*b^2*c^2 - 2*c^4 - 2*Sqrt[3]*a^2*S : :
Barycentrics    (a^2 - b^2 - c^2)*(a^2 + 2*b^2 + 2*c^2) - 2*Sqrt[3]*a^2*S : :

X(69107) lies on these lines: {6, 7794}, {13, 7776}, {14, 76}, {15, 3926}, {16, 7767}, {17, 325}, {61, 3933}, {62, 69}, {99, 42434}, {203, 69094}, {303, 7871}, {315, 16965}, {621, 3146}, {627, 5981}, {633, 41021}, {1007, 42488}, {1975, 42157}, {3181, 46226}, {3412, 7796}, {3785, 5237}, {3818, 5864}, {5238, 6390}, {5351, 59542}, {5352, 6337}, {7005, 69093}, {7750, 36968}, {7752, 42581}, {7763, 16241}, {7769, 33417}, {7773, 16808}, {7788, 61719}, {7789, 41407}, {7797, 34540}, {7800, 63200}, {7802, 42100}, {7874, 62200}, {7888, 62198}, {10513, 42998}, {10654, 32830}, {11133, 22869}, {14023, 41406}, {14538, 59363}, {14540, 48881}, {14929, 42148}, {15589, 42149}, {16963, 37671}, {16967, 32832}, {18581, 32834}, {18582, 32823}, {19107, 32819}, {22844, 44776}, {30472, 43459}, {32006, 36969}, {32815, 42432}, {32817, 42150}, {32818, 42152}, {32822, 42085}, {32828, 37835}, {32833, 41101}, {32835, 42092}, {32836, 41108}, {32869, 41113}, {32874, 41120}, {34229, 42937}, {37668, 40693}, {41122, 46951}, {42159, 52713}, {43633, 64018}, {47005, 66446}, {53440, 66347}

X(69108) = (1, 1, -1, 1, 1, -1, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 + 4*b^2*c^2 + 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69108) lies on these lines: {6, 7794}, {13, 626}, {15, 7789}, {16, 7800}, {17, 7778}, {18, 3934}, {61, 7795}, {62, 141}, {83, 298}, {202, 69097}, {303, 7909}, {624, 3090}, {635, 22687}, {3104, 23024}, {3314, 9112}, {3734, 16964}, {3926, 63201}, {5149, 6777}, {5352, 59545}, {5864, 19130}, {7006, 69095}, {7761, 42158}, {7767, 41406}, {7784, 16965}, {7785, 34540}, {7801, 36775}, {7815, 16242}, {7816, 36967}, {7830, 42433}, {7842, 19106}, {7849, 42990}, {7862, 16966}, {7865, 41100}, {7870, 62600}, {7874, 62198}, {7880, 16962}, {7883, 12155}, {8359, 59542}, {8362, 63200}, {14001, 41407}, {18840, 37641}, {22712, 37463}, {42152, 53033}, {42488, 44377}, {42937, 58446}, {51200, 54298}, {54116, 56055}

X(69109) = (1, 1, -1, 1, 1, -1, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 + 4*b^2*c^2 + 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69109) lies on these lines: {6, 7794}, {14, 626}, {15, 7800}, {16, 7789}, {17, 3934}, {18, 7778}, {61, 141}, {62, 7795}, {83, 299}, {203, 69097}, {302, 7909}, {623, 3090}, {636, 22689}, {3105, 23018}, {3314, 9113}, {3734, 16965}, {3926, 63200}, {5149, 6778}, {5351, 59545}, {5865, 19130}, {7005, 69095}, {7761, 42157}, {7767, 41407}, {7784, 16964}, {7785, 34541}, {7815, 16241}, {7816, 36968}, {7830, 42434}, {7842, 19107}, {7849, 42991}, {7862, 16967}, {7865, 41101}, {7870, 62601}, {7874, 62197}, {7880, 16963}, {7883, 12154}, {8359, 59541}, {8362, 63201}, {14001, 41406}, {18840, 37640}, {22712, 37464}, {42149, 53033}, {42489, 44377}, {42936, 58446}, {51203, 54297}, {54115, 56056}

X(69110) = (1, 1, -1, 1, -1, -1, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 - 4*b^2*c^2 + 2*c^4 + 2*Sqrt[3]*a^2*S : :
X(69110) = X[13] + 2 X[42975], 2 X[115] + X[9113]

X(69110) lies on these lines: {2, 22574}, {3, 36772}, {4, 54569}, {5, 36771}, {6, 13}, {15, 230}, {16, 2549}, {17, 13881}, {18, 39}, {30, 41406}, {32, 16964}, {61, 3767}, {62, 5254}, {187, 36967}, {202, 69098}, {299, 7934}, {302, 7757}, {303, 14061}, {393, 35714}, {395, 5463}, {396, 22489}, {398, 5305}, {530, 37641}, {574, 16242}, {616, 63033}, {618, 11489}, {671, 12155}, {1384, 42154}, {1506, 42580}, {2023, 3107}, {3053, 42157}, {3068, 6306}, {3069, 6302}, {3087, 31687}, {3364, 36762}, {3365, 49221}, {3411, 7765}, {3815, 37835}, {5023, 42434}, {5024, 16645}, {5210, 42529}, {5237, 63548}, {5286, 6782}, {5304, 5334}, {5306, 41108}, {5318, 59393}, {5321, 18907}, {5335, 5478}, {5339, 30435}, {5344, 63536}, {5353, 10078}, {5357, 10062}, {5459, 13705}, {5460, 9762}, {5461, 9763}, {5523, 8739}, {5611, 38224}, {5617, 11543}, {6108, 7735}, {6114, 36776}, {6115, 7736}, {6303, 13653}, {6307, 13773}, {6669, 11488}, {6770, 47863}, {6771, 11485}, {6776, 41045}, {6781, 19780}, {6792, 30468}, {7006, 69096}, {7685, 9749}, {7737, 36970}, {7738, 42149}, {7739, 16268}, {7745, 42814}, {7748, 42158}, {7756, 42433}, {8742, 57382}, {8836, 25155}, {9115, 35751}, {9166, 37786}, {9300, 41122}, {9605, 42153}, {10188, 12815}, {10645, 21843}, {10653, 31710}, {11296, 31696}, {11306, 51388}, {11409, 12142}, {11537, 34315}, {11542, 59401}, {11648, 41100}, {11725, 66238}, {12817, 14537}, {13711, 13929}, {13834, 13876}, {14136, 18582}, {14482, 43543}, {14639, 41044}, {14904, 62983}, {15655, 42626}, {15905, 40682}, {16241, 37637}, {16962, 62198}, {16963, 63198}, {16965, 44518}, {16966, 36763}, {16967, 31489}, {18362, 49907}, {19099, 25185}, {19100, 25186}, {20425, 44498}, {22580, 31694}, {22846, 22856}, {22847, 23302}, {22892, 43029}, {23004, 36759}, {23303, 36770}, {30439, 61675}, {31401, 42489}, {31406, 42599}, {33416, 36782}, {34321, 36300}, {34541, 46708}, {35749, 63111}, {35750, 63113}, {35752, 41745}, {36761, 41034}, {36767, 49906}, {36768, 49861}, {36769, 49812}, {36968, 44526}, {36969, 53419}, {37351, 51010}, {37665, 43404}, {37832, 43620}, {39593, 42507}, {40671, 59373}, {41039, 67863}, {41113, 63006}, {41120, 63024}, {41408, 42085}, {41409, 42117}, {41620, 51482}, {41751, 53441}, {42119, 46453}, {42162, 63533}, {42910, 62993}, {42973, 63543}, {43454, 51207}, {44534, 60069}, {47361, 67690}, {47362, 67679}, {47363, 52051}, {47364, 52052}, {47857, 59378}, {47861, 59394}, {47865, 63102}, {48913, 66445}, {51206, 54140}, {51728, 62201}, {56055, 60252}, {61576, 63732}

X(69110) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 22574, 42035}, {6, 13, 9112}, {6, 115, 13}, {6, 42975, 9113}, {13, 14, 41042}, {62, 23005, 23006}, {115, 5471, 11646}, {395, 6772, 5463}, {395, 15048, 63200}, {574, 62197, 16242}, {5321, 22513, 36961}, {5355, 5471, 6}, {5471, 11646, 6777}, {6115, 18581, 36765}, {7735, 10654, 41407}, {37637, 63199, 16241}, {41745, 53435, 35752}, {43229, 53435, 41745}, {43620, 61332, 37832}, {51482, 63080, 41620}, {59378, 63079, 47857}

X(69111) = (1, 1, -1, 1, -1, -1, -1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 - 4*b^2*c^2 + 2*c^4 - 2*Sqrt[3]*a^2*S : :
X(69111) = X[14] + 2 X[42974], 2 X[115] + X[9112]

X(69111) lies on these lines: {2, 22573}, {4, 54570}, {6, 13}, {15, 2549}, {16, 230}, {17, 39}, {18, 13881}, {30, 41407}, {32, 16965}, {61, 5254}, {62, 3767}, {114, 36771}, {187, 36968}, {203, 69098}, {298, 7934}, {302, 14061}, {303, 7757}, {393, 35715}, {395, 22490}, {396, 5464}, {397, 5305}, {531, 37640}, {574, 16241}, {617, 63032}, {619, 11488}, {671, 12154}, {1384, 42155}, {1506, 42581}, {2023, 3106}, {2482, 36764}, {3053, 42158}, {3068, 6307}, {3069, 6303}, {3087, 31688}, {3389, 49220}, {3390, 49221}, {3412, 7765}, {3815, 37832}, {5023, 42433}, {5024, 16644}, {5210, 42528}, {5238, 63548}, {5286, 6783}, {5304, 5335}, {5306, 41107}, {5318, 18907}, {5321, 59395}, {5334, 5479}, {5340, 30435}, {5343, 63536}, {5353, 10061}, {5357, 10077}, {5459, 9760}, {5460, 13703}, {5461, 9761}, {5523, 8740}, {5613, 11542}, {5615, 38224}, {6109, 7735}, {6114, 7736}, {6115, 61634}, {6302, 13653}, {6306, 13773}, {6670, 11489}, {6773, 47864}, {6774, 11486}, {6776, 41044}, {6781, 19781}, {6792, 30465}, {7005, 69096}, {7684, 9750}, {7737, 36969}, {7738, 42152}, {7739, 16267}, {7745, 42813}, {7748, 42157}, {7756, 42434}, {8741, 57383}, {8838, 25165}, {9117, 36329}, {9166, 37785}, {9300, 41121}, {9605, 42156}, {10187, 12815}, {10646, 21843}, {10654, 31709}, {11295, 31695}, {11305, 51387}, {11408, 12141}, {11543, 59402}, {11549, 34316}, {11648, 41101}, {11725, 66237}, {12816, 14537}, {13711, 13928}, {13834, 13875}, {14137, 18581}, {14482, 43542}, {14639, 41045}, {14905, 62984}, {15655, 42625}, {15905, 40683}, {16242, 37637}, {16962, 63199}, {16963, 62197}, {16964, 44518}, {16966, 31489}, {18362, 49908}, {19099, 25189}, {19100, 25190}, {20426, 44497}, {22579, 31693}, {22848, 43028}, {22891, 22900}, {22893, 23303}, {23005, 36760}, {23698, 36772}, {30440, 61675}, {31401, 42488}, {31406, 42598}, {34322, 36301}, {34540, 46709}, {36327, 63112}, {36330, 41746}, {36331, 63114}, {36766, 44534}, {36967, 44526}, {36970, 53419}, {37352, 51013}, {37665, 43403}, {37835, 43620}, {39593, 42506}, {40672, 59373}, {41035, 41458}, {41038, 67863}, {41112, 63006}, {41119, 63024}, {41408, 42118}, {41409, 42086}, {41621, 51483}, {41753, 53429}, {42120, 46453}, {42159, 63533}, {42911, 62993}, {42972, 63543}, {43455, 51206}, {47361, 52051}, {47362, 52052}, {47363, 67690}, {47364, 67679}, {47858, 59379}, {47862, 59396}, {47866, 63103}, {47867, 49813}, {48913, 66446}, {51207, 54141}, {56056, 60253}, {61576, 63731}

X(69111) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 22573, 42036}, {6, 14, 9113}, {6, 115, 14}, {6, 42974, 9112}, {13, 14, 41043}, {14, 61719, 51203}, {61, 23004, 23013}, {115, 5472, 11646}, {396, 6775, 5464}, {396, 15048, 63201}, {574, 62198, 16241}, {5318, 22512, 36962}, {5355, 5472, 6}, {5472, 11646, 6778}, {7735, 10653, 41406}, {37637, 63198, 16242}, {41746, 53447, 36330}, {43228, 53447, 41746}, {43620, 61331, 37835}, {51483, 63079, 41621}, {59379, 63080, 47858}

X(69112) = (1, -1, 1, -1, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 - a^2*b^2 - 2*b^4 - a^2*c^2 + 4*b^2*c^2 - 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69112) lies on these lines: {3, 62198}, {4, 6}, {13, 39}, {15, 7756}, {16, 7746}, {17, 574}, {18, 39565}, {32, 16965}, {61, 7748}, {62, 115}, {187, 42158}, {203, 9651}, {230, 42148}, {298, 7885}, {299, 20081}, {302, 32966}, {303, 7783}, {395, 63534}, {396, 63548}, {621, 15514}, {634, 53458}, {1506, 63200}, {1656, 63198}, {2548, 42162}, {2549, 40693}, {3053, 42155}, {3054, 42944}, {3091, 61331}, {3105, 20425}, {3767, 10653}, {3815, 42166}, {5013, 42156}, {5023, 43193}, {5038, 36251}, {5111, 20428}, {5116, 53428}, {5206, 36968}, {5237, 7749}, {5309, 41107}, {5471, 42814}, {5472, 7765}, {5475, 42813}, {6034, 25154}, {6114, 16627}, {6775, 34509}, {6781, 43633}, {7005, 9664}, {7737, 42161}, {7738, 61332}, {7739, 41112}, {7747, 36969}, {7753, 42973}, {7755, 41406}, {7817, 12155}, {9761, 33006}, {10668, 42256}, {10672, 42254}, {11289, 53452}, {11303, 44453}, {11304, 51159}, {11481, 44535}, {11646, 36252}, {11648, 61719}, {13881, 22238}, {15513, 42433}, {15515, 16241}, {15815, 16644}, {16631, 46053}, {16770, 39691}, {16963, 18362}, {18582, 31401}, {19780, 42118}, {19781, 42086}, {22236, 44526}, {22862, 36756}, {23005, 54297}, {31400, 43403}, {31415, 42921}, {31455, 37832}, {32448, 63732}, {32967, 62601}, {32993, 62983}, {33517, 37825}, {36757, 65417}, {36836, 44519}, {36843, 37637}, {37641, 63533}, {39601, 42580}, {41094, 52643}, {41407, 42431}, {42149, 43620}, {42150, 43619}, {42157, 65633}, {42494, 62993}, {42815, 43451}, {42924, 43291}, {42943, 62232}, {42988, 63199}, {42992, 63201}, {43028, 47520}, {43229, 63543}, {43238, 53095}, {44377, 59542}, {63080, 63536}

X(69112) = crosspoint of X(4) and X(54116)
X(69112) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {397, 5254, 6}, {13881, 22238, 62197}, {42645, 42646, 41039}, {53458, 53463, 634}

X(69113) = (1, -1, 1, -1, 1, 1, -1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 - a^2*b^2 - 2*b^4 - a^2*c^2 + 4*b^2*c^2 - 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69113) lies on these lines: {3, 62197}, {4, 6}, {14, 39}, {15, 7746}, {16, 7756}, {17, 39565}, {18, 574}, {32, 16964}, {61, 115}, {62, 7748}, {187, 42157}, {202, 9651}, {230, 42147}, {298, 20081}, {299, 7885}, {302, 7783}, {303, 32966}, {395, 63548}, {396, 63534}, {622, 15514}, {633, 53452}, {1506, 63201}, {1656, 63199}, {2548, 42159}, {2549, 40694}, {3053, 42154}, {3054, 42945}, {3091, 61332}, {3104, 20426}, {3767, 10654}, {3815, 42163}, {5013, 42153}, {5023, 43194}, {5038, 36252}, {5111, 20429}, {5116, 53440}, {5206, 36967}, {5238, 7749}, {5309, 41108}, {5471, 7765}, {5472, 42813}, {5475, 42814}, {6034, 25164}, {6115, 16626}, {6772, 34508}, {6781, 43632}, {7006, 9664}, {7737, 42160}, {7738, 61331}, {7739, 41113}, {7747, 36970}, {7753, 42972}, {7755, 41407}, {7817, 12154}, {9763, 33006}, {10667, 42257}, {10671, 42255}, {11290, 53463}, {11303, 51160}, {11304, 44453}, {11480, 44535}, {11646, 36251}, {13881, 22236}, {15513, 42434}, {15515, 16242}, {15815, 16645}, {16630, 46054}, {16771, 39691}, {16962, 18362}, {18581, 31401}, {19780, 42085}, {19781, 42117}, {22238, 44526}, {22906, 36755}, {23004, 54298}, {31400, 43404}, {31415, 42920}, {31455, 37835}, {32448, 63731}, {32967, 62600}, {32993, 62984}, {33518, 37824}, {36758, 65417}, {36836, 37637}, {36843, 44519}, {37640, 63533}, {39563, 61719}, {39601, 42581}, {41098, 52642}, {41406, 42432}, {42151, 43619}, {42152, 43620}, {42158, 65633}, {42495, 62993}, {42816, 43452}, {42925, 43291}, {42942, 62233}, {42989, 63198}, {42993, 63200}, {43029, 47518}, {43228, 63543}, {43239, 53095}, {44377, 59541}, {63079, 63536}

X(69113) = crosspoint of X(4) and X(54115)
X(69113) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {398, 5254, 6}, {13881, 22236, 62198}, {42645, 42646, 41038}, {53452, 53469, 633}

X(69114) = (1, -1, 1, -1, -1, 1, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 - a^2*b^2 - 2*b^4 - a^2*c^2 - 4*b^2*c^2 - 2*c^4 + 2*Sqrt[3]*a^2*S : :
Barycentrics    (a^2 - 2*b^2 - 2*c^2)*(a^2 + b^2 + c^2) + 2*Sqrt[3]*a^2*S : :
X(69114) = 2 X[43229] - 3 X[47352]

X(69114) lies on these lines: {2, 6}, {13, 9466}, {14, 7818}, {15, 7810}, {16, 7801}, {18, 7821}, {61, 7854}, {62, 7794}, {511, 37332}, {616, 51160}, {619, 6582}, {621, 51162}, {624, 51016}, {2482, 10646}, {3094, 6581}, {3098, 5464}, {3642, 23018}, {3818, 50858}, {5092, 5463}, {5237, 7863}, {5969, 11129}, {6299, 51013}, {7813, 63200}, {7820, 41406}, {7873, 16964}, {7947, 62601}, {8369, 19780}, {11178, 21360}, {12154, 66455}, {14994, 33482}, {15810, 36775}, {16809, 31173}, {16965, 17130}, {18358, 52649}, {18470, 34897}, {19130, 50855}, {20425, 25157}, {22911, 51020}, {30471, 33273}, {37352, 59410}, {42118, 59780}

X(69114) = reflection of X(66446) in X(597)
X(69114) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 3618, 40900}, {69, 34541, 141}, {141, 23302, 3763}

X(69115) = (1, -1, 1, -1, -1, 1, -1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 - a^2*b^2 - 2*b^4 - a^2*c^2 - 4*b^2*c^2 - 2*c^4 - 2*Sqrt[3]*a^2*S : :
Barycentrics    (a^2 - 2*b^2 - 2*c^2)*(a^2 + b^2 + c^2) - 2*Sqrt[3]*a^2*S : :
X(69115) = 2 X[43228] - 3 X[47352]

X(69115) lies on these lines: {2, 6}, {13, 7818}, {14, 9466}, {15, 7801}, {16, 7810}, {17, 7821}, {61, 7794}, {62, 7854}, {511, 37333}, {617, 51159}, {618, 6295}, {622, 51161}, {623, 51018}, {2482, 10645}, {3094, 6294}, {3098, 5463}, {3643, 23024}, {3818, 50855}, {5092, 5464}, {5238, 7863}, {5969, 11128}, {6298, 51010}, {7813, 63201}, {7820, 41407}, {7873, 16965}, {7947, 62600}, {8369, 19781}, {11178, 21359}, {12155, 66455}, {14994, 33483}, {16808, 31173}, {16964, 17130}, {18358, 44289}, {18468, 34897}, {19130, 50858}, {20426, 25167}, {22866, 51021}, {30472, 33273}, {42117, 59780}

X(69115) = reflection of X(66445) in X(597)
X(69115) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 3618, 40901}, {69, 34540, 141}, {141, 23303, 3763}

X(69116) = (1, -1, -1, -1, 1, -1, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 + 4*b^2*c^2 + 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69116) lies on these lines: {6, 3933}, {13, 7821}, {14, 17130}, {15, 7863}, {16, 7854}, {17, 7888}, {18, 9466}, {61, 7801}, {62, 7794}, {298, 384}, {302, 31276}, {303, 7947}, {599, 22238}, {633, 35917}, {2482, 5352}, {3181, 19689}, {3788, 62198}, {5237, 7810}, {5858, 6661}, {5864, 44230}, {5868, 44251}, {5872, 44224}, {6292, 63200}, {6655, 34540}, {7751, 62199}, {7818, 16965}, {7826, 41406}, {7873, 42158}, {8556, 43239}, {11178, 37332}, {11304, 42163}, {14069, 61317}, {19692, 40900}, {19780, 63928}, {19781, 32973}, {22110, 42598}, {30472, 33004}, {32818, 61332}, {32954, 62200}, {66322, 66446}

X(69117) = (1, -1, -1, -1, 1, -1, -1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 + 4*b^2*c^2 + 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69117) lies on these lines: {6, 3933}, {13, 17130}, {14, 7821}, {15, 7854}, {16, 7863}, {17, 9466}, {18, 7888}, {61, 7794}, {62, 7801}, {299, 384}, {302, 7947}, {303, 31276}, {599, 22236}, {634, 35918}, {2482, 5351}, {3180, 19689}, {3788, 62197}, {5238, 7810}, {5859, 6661}, {5865, 44230}, {5869, 44251}, {5873, 44224}, {6292, 63201}, {6655, 34541}, {7751, 62200}, {7818, 16964}, {7826, 41407}, {7873, 42157}, {8556, 43238}, {11178, 37333}, {11303, 42166}, {14069, 61318}, {19692, 40901}, {19780, 32973}, {19781, 63928}, {22110, 42599}, {30471, 33004}, {32818, 61331}, {32954, 62199}, {66322, 66445}

X(69118) = (1, -1, -1, -1, -1, -1, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - 4*b^2*c^2 + 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69118) lies on these lines: {2, 53463}, {3, 62197}, {4, 53442}, {5, 6}, {13, 39565}, {14, 32}, {15, 7749}, {16, 7748}, {18, 39}, {61, 7746}, {62, 115}, {187, 16964}, {194, 302}, {230, 398}, {298, 17129}, {299, 7912}, {303, 32967}, {384, 53440}, {395, 616}, {397, 63534}, {571, 8174}, {624, 5111}, {628, 23302}, {1506, 37835}, {1691, 7685}, {2549, 42149}, {3053, 5339}, {3054, 16772}, {3090, 61332}, {3094, 23018}, {3105, 46053}, {3107, 52643}, {3124, 8836}, {3180, 22893}, {3526, 63199}, {3815, 42599}, {5013, 16645}, {5023, 42154}, {5206, 42157}, {5210, 43194}, {5237, 7756}, {5286, 61331}, {5304, 22237}, {5309, 16268}, {5321, 19780}, {5334, 19781}, {5460, 6034}, {5471, 7755}, {6114, 44534}, {6781, 42432}, {6782, 37825}, {7603, 42580}, {7737, 42159}, {7738, 11489}, {7745, 42163}, {7747, 41406}, {7753, 41122}, {7765, 63200}, {7783, 62601}, {8259, 11488}, {8588, 42434}, {9112, 39601}, {10616, 67072}, {10654, 62233}, {11307, 23303}, {11481, 44519}, {11614, 42596}, {11648, 16963}, {12155, 47617}, {15513, 36967}, {15815, 40921}, {16242, 37512}, {16773, 63548}, {16809, 39590}, {18362, 61719}, {18424, 42813}, {20425, 22891}, {21843, 42150}, {22236, 37637}, {22238, 44518}, {22511, 54297}, {31455, 42489}, {31467, 42129}, {31694, 49948}, {36836, 44535}, {36843, 44526}, {36968, 65633}, {36998, 41038}, {37334, 53465}, {39563, 41100}, {39593, 49904}, {42148, 53419}, {42491, 53095}, {42975, 62200}, {43404, 61318}, {43455, 59384}, {53428, 62983}

X(69118) = X(1973)-complementary conjugate of X(62601)
X(69118) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {61, 7746, 62198}, {485, 486, 63731}, {3767, 40694, 6}, {41406, 42814, 7747}, {42489, 63201, 31455}

X(69119) = (1, -1, -1, -1, -1, -1, -1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - 4*b^2*c^2 + 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69119) lies on these lines: {2, 53452}, {3, 62198}, {4, 53430}, {5, 6}, {13, 32}, {14, 39565}, {15, 7748}, {16, 7749}, {17, 39}, {61, 115}, {62, 7746}, {187, 16965}, {194, 303}, {230, 397}, {298, 7912}, {299, 17129}, {302, 32967}, {384, 53428}, {396, 617}, {398, 63534}, {571, 8175}, {623, 5111}, {627, 23303}, {1506, 37832}, {1691, 7684}, {2549, 42152}, {3053, 5340}, {3054, 16773}, {3090, 61331}, {3094, 23024}, {3104, 46054}, {3106, 52642}, {3124, 8838}, {3181, 22847}, {3526, 63198}, {3815, 42598}, {5013, 16644}, {5023, 42155}, {5206, 42158}, {5210, 43193}, {5238, 7756}, {5286, 61332}, {5304, 22235}, {5309, 16267}, {5318, 19781}, {5335, 19780}, {5459, 6034}, {5472, 7755}, {6115, 44534}, {6781, 42431}, {6783, 37824}, {7603, 42581}, {7737, 42162}, {7738, 11488}, {7745, 42166}, {7747, 41407}, {7753, 41121}, {7765, 63201}, {7783, 62600}, {8260, 11489}, {8588, 42433}, {9113, 39601}, {10617, 67071}, {10653, 62232}, {11308, 23302}, {11480, 44519}, {11614, 42597}, {11648, 16962}, {12154, 47617}, {15513, 36968}, {15815, 40922}, {16241, 37512}, {16772, 63548}, {16808, 39590}, {18424, 42814}, {20426, 22846}, {21843, 42151}, {22236, 44518}, {22238, 37637}, {22510, 54298}, {31455, 42488}, {31467, 42132}, {31693, 49947}, {36836, 44526}, {36843, 44535}, {36967, 65633}, {36998, 41039}, {37334, 53454}, {39563, 41101}, {39593, 49903}, {42147, 53419}, {42490, 53095}, {42974, 62199}, {43403, 61317}, {43454, 59383}, {53440, 62984}

X(69119) = X(1973)-complementary conjugate of X(62600)
X(69119) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {62, 7746, 62197}, {485, 486, 63732}, {3767, 40693, 6}, {41407, 42813, 7747}, {42488, 63200, 31455}

X(69120) = (1, 1, 1, 1, -1, 1, 1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 - 3*a^2*b^2 + 2*b^4 - 3*a^2*c^2 + 4*b^2*c^2 + 2*c^4 - 2*Sqrt[3]*a^2*S : :
Barycentrics    (a^2 - 2*b^2 - 2*c^2)*(a^2 - b^2 - c^2) - 2*Sqrt[3]*a^2*S : :

X(69120) lies on these lines: {6, 7801}, {14, 325}, {15, 69}, {16, 6390}, {17, 76}, {61, 3933}, {62, 3926}, {99, 299}, {141, 63201}, {183, 16241}, {203, 69093}, {302, 16961}, {315, 42157}, {316, 19107}, {524, 41407}, {599, 36775}, {624, 43276}, {634, 3522}, {1007, 37835}, {1503, 14539}, {1975, 16965}, {3785, 5352}, {5237, 6337}, {5238, 7767}, {5335, 34509}, {5995, 53186}, {6775, 42036}, {7005, 69094}, {7750, 42434}, {7771, 30471}, {7776, 16964}, {7788, 41101}, {7796, 42991}, {7799, 16963}, {7840, 12154}, {8550, 52194}, {10653, 32817}, {10654, 37668}, {11128, 44361}, {11133, 22683}, {11185, 16808}, {14907, 42529}, {14929, 42942}, {16267, 32836}, {17131, 62198}, {18019, 40710}, {18581, 63098}, {18582, 52713}, {31274, 62233}, {32006, 42432}, {32815, 36969}, {32816, 42814}, {32818, 40694}, {32822, 42161}, {32823, 42159}, {32824, 41974}, {32825, 42993}, {32828, 42488}, {32829, 42937}, {32830, 40693}, {32831, 42149}, {32833, 61719}, {32837, 41944}, {32840, 42998}, {32876, 42801}, {33417, 37688}, {34511, 63200}, {36776, 51388}, {37832, 64093}, {42581, 59635}, {51277, 62431}, {51387, 54141}, {52629, 57123}

X(69120) = {X(7767),X(59541)}-harmonic conjugate of X(5238)

X(69121) = (1, 1, 1, 1, -1, 1, -1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 - 3*a^2*b^2 + 2*b^4 - 3*a^2*c^2 + 4*b^2*c^2 + 2*c^4 + 2*Sqrt[3]*a^2*S : :
Barycentrics    (a^2 - 2*b^2 - 2*c^2)*(a^2 - b^2 - c^2) + 2*Sqrt[3]*a^2*S : :

X(69121) lies on these lines: {6, 7801}, {13, 325}, {15, 6390}, {16, 69}, {18, 76}, {61, 3926}, {62, 3933}, {99, 298}, {141, 63200}, {183, 16242}, {202, 69093}, {303, 16960}, {315, 42158}, {316, 19106}, {524, 41406}, {599, 63198}, {623, 43277}, {633, 3522}, {1007, 37832}, {1503, 14538}, {1975, 16964}, {3785, 5351}, {5237, 7767}, {5238, 6337}, {5334, 34508}, {5994, 53186}, {6772, 42035}, {7006, 69094}, {7750, 42433}, {7771, 30472}, {7776, 16965}, {7788, 41100}, {7796, 42990}, {7799, 16962}, {7840, 12155}, {8550, 52193}, {10653, 37668}, {10654, 32817}, {11129, 44362}, {11132, 22685}, {11185, 16809}, {14907, 42528}, {14929, 42943}, {16268, 32836}, {17131, 62197}, {18019, 40709}, {18581, 52713}, {18582, 63098}, {31274, 62232}, {32006, 42431}, {32815, 36970}, {32816, 42813}, {32818, 40693}, {32822, 42160}, {32823, 42162}, {32824, 41973}, {32825, 42992}, {32828, 42489}, {32829, 42936}, {32830, 40694}, {32831, 42152}, {32837, 41943}, {32840, 42999}, {32876, 42802}, {33416, 37688}, {34511, 63201}, {36775, 39785}, {37835, 64093}, {42580, 59635}, {51270, 62431}, {51387, 61634}, {51388, 54140}, {52629, 57122}

X(69121) = {X(7767),X(59542)}-harmonic conjugate of X(5237)

X(69122) = (1, 1, -1, 1, 1, -1, 1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 - 3*a^2*b^2 - 2*b^4 - 3*a^2*c^2 - 4*b^2*c^2 - 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69122) lies on these lines: {6, 6292}, {13, 7784}, {14, 3934}, {15, 7795}, {17, 626}, {18, 15271}, {61, 141}, {62, 7800}, {98, 635}, {203, 69095}, {298, 7786}, {299, 32027}, {303, 7922}, {625, 42581}, {629, 3525}, {3105, 51010}, {3412, 7849}, {3734, 42157}, {3785, 41406}, {3788, 16241}, {3933, 63201}, {5238, 7789}, {7005, 69097}, {7761, 16965}, {7816, 42434}, {7819, 41407}, {7825, 16808}, {7830, 36968}, {7852, 62200}, {7865, 61719}, {7867, 62198}, {7909, 62600}, {10645, 59545}, {16043, 63200}, {22997, 44383}, {37340, 51013}, {37641, 55732}, {38412, 52644}, {41753, 53464}, {42489, 58446}, {42936, 44377}

X(69123) = (1, 1, -1, 1, 1, -1, -1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 - 3*a^2*b^2 - 2*b^4 - 3*a^2*c^2 - 4*b^2*c^2 - 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69123) lies on these lines: {6, 6292}, {13, 3934}, {14, 7784}, {16, 7795}, {17, 15271}, {18, 626}, {61, 7800}, {62, 141}, {98, 636}, {202, 69095}, {298, 32027}, {299, 7786}, {302, 7922}, {625, 42580}, {630, 3525}, {3104, 51013}, {3411, 7849}, {3734, 42158}, {3785, 41407}, {3788, 16242}, {3933, 63200}, {5237, 7789}, {7006, 69097}, {7761, 16964}, {7816, 42433}, {7819, 41406}, {7825, 16809}, {7830, 36967}, {7852, 62199}, {7867, 62197}, {7909, 62601}, {8359, 36775}, {10646, 59545}, {16043, 63201}, {22998, 36770}, {37341, 51010}, {37640, 55732}, {41751, 53453}, {42488, 58446}, {42937, 44377}

X(69124) = (1, 1, -1, 1, -1, -1, 1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 - 3*a^2*b^2 - 2*b^4 - 3*a^2*c^2 + 4*b^2*c^2 - 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69124) lies on these lines: {3, 36772}, {4, 16941}, {5, 63201}, {6, 382}, {13, 44518}, {14, 39}, {15, 3767}, {16, 22890}, {17, 115}, {18, 5013}, {20, 41406}, {32, 42157}, {61, 5254}, {62, 2549}, {187, 42434}, {202, 9597}, {203, 69096}, {230, 5238}, {299, 7911}, {397, 52838}, {398, 9113}, {1285, 43770}, {1384, 43194}, {2548, 42814}, {3053, 36967}, {3104, 54141}, {3105, 46855}, {3411, 63198}, {5024, 42153}, {5111, 51208}, {5206, 42529}, {5286, 10654}, {5305, 41407}, {5309, 41101}, {5339, 9605}, {5340, 9112}, {5343, 37665}, {7005, 69098}, {7006, 9598}, {7735, 42150}, {7736, 42159}, {7737, 42432}, {7738, 40694}, {7739, 41108}, {7745, 36970}, {7746, 16241}, {7756, 36968}, {7827, 12154}, {7940, 30471}, {8361, 59541}, {9300, 42972}, {10722, 47863}, {11318, 36775}, {11648, 61719}, {13881, 63199}, {14061, 62600}, {15815, 16242}, {16630, 52642}, {16772, 43291}, {16967, 31455}, {18581, 31400}, {18582, 63533}, {18907, 42164}, {23303, 33387}, {30435, 42154}, {31401, 37835}, {31404, 42920}, {31406, 42163}, {31489, 42580}, {37512, 62197}, {37832, 63534}, {39565, 42581}, {40693, 43448}, {41121, 63543}, {41408, 42087}, {41409, 42119}, {42158, 44526}, {42433, 44519}, {42488, 43620}, {42813, 53419}, {42993, 61331}, {43403, 63536}, {43619, 43633}, {49220, 51728}

X(69124) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 7748, 16965}, {5305, 42147, 41407}, {7738, 40694, 63200}

X(69125) = (1, 1, -1, 1, -1, -1, -1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 - 3*a^2*b^2 - 2*b^4 - 3*a^2*c^2 + 4*b^2*c^2 - 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69125) lies on these lines: {4, 16940}, {5, 63200}, {6, 382}, {13, 39}, {14, 44518}, {15, 22843}, {16, 3767}, {17, 5013}, {18, 115}, {20, 41407}, {32, 42158}, {61, 2549}, {62, 5254}, {187, 42433}, {202, 69096}, {203, 9597}, {230, 5237}, {298, 7911}, {397, 9112}, {398, 52839}, {1285, 43769}, {1384, 43193}, {2548, 42813}, {3053, 36968}, {3104, 46854}, {3105, 54140}, {3412, 63199}, {5024, 42156}, {5111, 51209}, {5206, 42528}, {5286, 10653}, {5305, 41406}, {5309, 41100}, {5339, 9113}, {5340, 9605}, {5344, 37665}, {7005, 9598}, {7006, 69098}, {7735, 42151}, {7736, 42162}, {7737, 42431}, {7738, 40693}, {7739, 41107}, {7745, 36969}, {7746, 16242}, {7756, 36967}, {7827, 12155}, {7940, 30472}, {8361, 59542}, {9300, 42973}, {10722, 47864}, {13881, 63198}, {14061, 62601}, {15815, 16241}, {16631, 52643}, {16773, 43291}, {16966, 31455}, {18581, 63533}, {18582, 31400}, {18907, 42165}, {23302, 33386}, {30435, 42155}, {31401, 37832}, {31404, 42921}, {31406, 42166}, {31489, 42581}, {37512, 62198}, {37835, 63534}, {39565, 42580}, {40694, 43448}, {41122, 63543}, {41408, 42120}, {41409, 42088}, {42157, 44526}, {42434, 44519}, {42489, 43620}, {42814, 53419}, {42992, 61332}, {43404, 63536}, {43619, 43632}

X(69125) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 7748, 16964}, {5305, 42148, 41406}, {7738, 40693, 63201}

X(69126) = (1, -1, 1, -1, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 + 3*a^2*b^2 + 2*b^4 + 3*a^2*c^2 - 4*b^2*c^2 + 2*c^4 - 2*Sqrt[3]*a^2*S : :
X(69126) = 2 X[397] + X[5339]

X(69126) lies on these lines: {2, 5488}, {3, 43454}, {4, 6}, {13, 7772}, {16, 7755}, {17, 39}, {20, 61317}, {32, 42158}, {61, 7765}, {62, 5309}, {230, 42944}, {298, 7933}, {396, 9607}, {550, 19781}, {631, 62232}, {633, 53440}, {2548, 42921}, {2549, 42150}, {3767, 42149}, {5007, 16965}, {5013, 43238}, {5305, 42924}, {5306, 42148}, {5319, 10653}, {5346, 41406}, {5355, 43775}, {5858, 33251}, {6034, 36251}, {6694, 42674}, {7736, 42494}, {7739, 40693}, {7746, 42937}, {7748, 42432}, {7753, 42813}, {7756, 41407}, {9300, 42166}, {9606, 42598}, {9698, 37832}, {11134, 30391}, {11290, 53428}, {11648, 16964}, {14930, 43556}, {15815, 42773}, {16241, 31652}, {16644, 22332}, {16773, 62233}, {18362, 42580}, {19780, 42151}, {22236, 44461}, {22238, 62199}, {22331, 43193}, {30472, 33245}, {33654, 61741}, {35007, 36968}, {36252, 37824}, {37171, 49948}, {39593, 61719}, {42776, 63533}, {43495, 63097}

X(69127) = (1, -1, 1, -1, 1, 1, -1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 + 3*a^2*b^2 + 2*b^4 + 3*a^2*c^2 - 4*b^2*c^2 + 2*c^4 + 2*Sqrt[3]*a^2*S : :
X(69127) = 2 X[398] + X[5340]

X(69127) lies on these lines: {2, 5487}, {3, 43455}, {4, 6}, {14, 7772}, {15, 7755}, {18, 39}, {20, 61318}, {32, 42157}, {61, 5309}, {62, 7765}, {230, 42945}, {299, 7933}, {395, 9607}, {550, 19780}, {631, 62233}, {634, 53428}, {2306, 61741}, {2548, 42920}, {2549, 42151}, {3767, 42152}, {5007, 16964}, {5013, 43239}, {5305, 42925}, {5306, 42147}, {5319, 10654}, {5346, 41407}, {5355, 43776}, {5859, 33251}, {6034, 36252}, {6695, 42675}, {7736, 42495}, {7739, 40694}, {7746, 42936}, {7748, 42431}, {7753, 42814}, {7756, 41406}, {9300, 42163}, {9606, 42599}, {9698, 37835}, {11137, 30390}, {11289, 53440}, {11648, 16965}, {14930, 43557}, {15815, 42774}, {16242, 31652}, {16645, 22332}, {16772, 62232}, {18362, 42581}, {19781, 42150}, {22236, 62200}, {22238, 44465}, {22331, 43194}, {30471, 33245}, {35007, 36967}, {36251, 37825}, {37170, 49947}, {42775, 63533}, {43496, 63097}

X(69128) = (1, -1, 1, -1, -1, 1, 1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 + 3*a^2*b^2 + 2*b^4 + 3*a^2*c^2 + 4*b^2*c^2 + 2*c^4 - 2*Sqrt[3]*a^2*S : :
(a^2 + b^2 + c^2)*(a^2 + 2*b^2 + 2*c^2) - 2*Sqrt[3]*a^2*S : : Barycentrics    X(69128) lies on these lines: {2, 6}, {14, 7853}, {16, 7820}, {17, 31239}, {18, 7867}, {61, 6292}, {62, 7822}, {511, 47519}, {622, 24273}, {636, 51018}, {5026, 14904}, {7801, 63200}, {7810, 41407}, {7935, 16964}, {7945, 62601}, {10007, 11132}, {11299, 51160}, {11303, 51162}, {16967, 31275}, {22512, 22689}, {24206, 40335}, {33479, 35439}, {38317, 40334}, {44465, 48905}, {51265, 51848}

X(69128) = {X(141),X(3589)}-harmonic conjugate of X(302)

X(69129) = (1, -1, 1, -1, -1, 1, -1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 + 3*a^2*b^2 + 2*b^4 + 3*a^2*c^2 + 4*b^2*c^2 + 2*c^4 + 2*Sqrt[3]*a^2*S : :
Barycentrics    (a^2 + b^2 + c^2)*(a^2 + 2*b^2 + 2*c^2) + 2*Sqrt[3]*a^2*S ::

X(69129) lies on these lines: {2, 6}, {13, 7853}, {15, 7820}, {17, 7867}, {18, 31239}, {61, 7822}, {62, 6292}, {511, 47517}, {621, 24273}, {635, 51016}, {5026, 14905}, {7801, 63201}, {7810, 41406}, {7935, 16965}, {7945, 62600}, {10007, 11133}, {11300, 51159}, {11304, 51161}, {16966, 31275}, {22513, 22687}, {24206, 40334}, {33478, 35439}, {38317, 40335}, {44461, 48905}, {51272, 51848}

X(69129) = {X(141),X(3589)}-harmonic conjugate of X(303)

X(69130) = (1, -1, -1, -1, 1, -1, 1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 + 3*a^2*b^2 - 2*b^4 + 3*a^2*c^2 - 4*b^2*c^2 - 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69130) lies on these lines: {6, 3933}, {13, 7903}, {16, 7826}, {18, 17131}, {61, 7813}, {62, 7855}, {194, 298}, {302, 17129}, {397, 50771}, {3788, 62200}, {5858, 32833}, {7751, 62197}, {7845, 16965}, {7854, 63200}, {7863, 41407}, {19781, 59545}, {22238, 40341}, {53033, 61317}, {59542, 63928}

X(69131) = (1, -1, -1, -1, 1, -1, -1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 + 3*a^2*b^2 - 2*b^4 + 3*a^2*c^2 - 4*b^2*c^2 - 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69131) lies on these lines: {6, 3933}, {14, 7903}, {15, 7826}, {17, 17131}, {61, 7855}, {62, 7813}, {194, 299}, {303, 17129}, {398, 50771}, {3788, 62199}, {5859, 32833}, {7751, 62198}, {7845, 16964}, {7854, 63201}, {7863, 41406}, {19780, 59545}, {22236, 40341}, {53033, 61318}, {59541, 63928}

X(69132) = (1, -1, -1, -1, -1, -1, 1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 + 3*a^2*b^2 - 2*b^4 + 3*a^2*c^2 + 4*b^2*c^2 - 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69132) lies on these lines: {2, 34534}, {4, 53443}, {5, 6}, {14, 39}, {15, 31455}, {16, 7747}, {17, 7603}, {18, 32}, {61, 1506}, {62, 5475}, {202, 9650}, {230, 42599}, {298, 31276}, {299, 7941}, {302, 384}, {382, 63198}, {395, 7745}, {398, 3815}, {574, 16964}, {623, 1691}, {1656, 62198}, {2549, 42159}, {3053, 16645}, {3055, 16772}, {3105, 20426}, {3364, 31481}, {3552, 62601}, {5013, 5339}, {5023, 43239}, {5111, 7684}, {5206, 16242}, {5210, 42491}, {5254, 42163}, {5286, 43404}, {5309, 41122}, {5321, 63548}, {5334, 31400}, {5351, 6781}, {5477, 20415}, {6114, 37824}, {7005, 31476}, {7006, 9665}, {7737, 42149}, {7739, 41120}, {7746, 37835}, {7748, 42814}, {7749, 41407}, {7753, 16268}, {7756, 36970}, {8019, 21461}, {8260, 63033}, {8370, 9761}, {8589, 42434}, {8838, 20976}, {9698, 63201}, {10653, 52647}, {10654, 31401}, {11308, 19781}, {11489, 19780}, {13330, 34508}, {14537, 16963}, {15484, 42989}, {15515, 36967}, {15815, 42154}, {16044, 62983}, {16965, 39590}, {17005, 62600}, {22236, 31489}, {22237, 37665}, {22238, 65630}, {30435, 62199}, {31404, 42999}, {31467, 63199}, {31693, 49948}, {33013, 37785}, {37334, 53466}, {37512, 42157}, {42139, 63533}, {42148, 53418}, {42158, 62203}, {42813, 43457}, {42910, 62232}, {43194, 53095}, {43543, 61318}, {54115, 60105}

X(69132) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {18, 32, 62197}, {1506, 5471, 61}, {2548, 40694, 6}, {31404, 42999, 61332}, {41407, 42489, 7749}, {42814, 63200, 7748}

X(69133) = (1, -1, -1, -1, -1, -1, -1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 + 3*a^2*b^2 - 2*b^4 + 3*a^2*c^2 + 4*b^2*c^2 - 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69133) lies on these lines: {2, 34533}, {4, 53431}, {5, 6}, {13, 39}, {15, 7747}, {16, 31455}, {17, 32}, {18, 7603}, {61, 5475}, {62, 1506}, {203, 9650}, {230, 42598}, {298, 7941}, {299, 31276}, {303, 384}, {382, 63199}, {396, 7745}, {397, 3815}, {574, 16965}, {624, 1691}, {1656, 62197}, {2549, 42162}, {3053, 16644}, {3055, 16773}, {3104, 20425}, {3389, 31481}, {3552, 62600}, {5013, 5340}, {5023, 43238}, {5111, 7685}, {5206, 16241}, {5210, 42490}, {5254, 42166}, {5286, 43403}, {5309, 41121}, {5318, 63548}, {5335, 31400}, {5352, 6781}, {5477, 20416}, {6115, 37825}, {7005, 9665}, {7006, 31476}, {7737, 42152}, {7739, 41119}, {7746, 37832}, {7748, 42813}, {7749, 41406}, {7753, 16267}, {7756, 36969}, {8018, 21462}, {8259, 63032}, {8370, 9763}, {8589, 42433}, {8836, 20976}, {9698, 63200}, {10653, 31401}, {10654, 52648}, {11307, 19780}, {11488, 19781}, {13330, 34509}, {14537, 16962}, {15484, 42988}, {15515, 36968}, {15815, 42155}, {16044, 62984}, {16964, 39590}, {17005, 62601}, {22235, 37665}, {22236, 65630}, {22238, 31489}, {30435, 62200}, {31404, 42998}, {31467, 63198}, {31694, 49947}, {33013, 37786}, {37334, 53455}, {37512, 42158}, {42142, 63533}, {42147, 53418}, {42157, 62203}, {42814, 43457}, {42911, 62233}, {43193, 53095}, {43542, 61317}, {54116, 60105}

X(69133) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {17, 32, 62198}, {1506, 5472, 62}, {2548, 40693, 6}, {31404, 42998, 61331}, {41406, 42488, 7749}, {42813, 63201, 7748}

X(69134) = (1, 1, 1, 0, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(11)

Barycentrics    a*b^2 - b^3 - 2*a*b*c + a*c^2 - c^3 : :

X(69134) lies on these lines: {1, 2}, {5, 4968}, {11, 321}, {38, 3846}, {56, 5016}, {57, 6327}, {69, 26229}, {75, 11680}, {81, 33071}, {100, 4514}, {104, 4220}, {110, 333}, {141, 46910}, {149, 32932}, {171, 32844}, {226, 17140}, {238, 33119}, {244, 2887}, {312, 33089}, {325, 26234}, {354, 3936}, {391, 26258}, {404, 5015}, {427, 38462}, {474, 5300}, {496, 3702}, {497, 17740}, {518, 5741}, {748, 4438}, {750, 4865}, {894, 33107}, {908, 17165}, {940, 33070}, {946, 17164}, {956, 37366}, {982, 17184}, {984, 25960}, {988, 17676}, {1001, 33113}, {1054, 32948}, {1086, 48646}, {1089, 3825}, {1150, 3966}, {1155, 4450}, {1211, 46909}, {1233, 40619}, {1281, 56512}, {1329, 4696}, {1376, 5014}, {1400, 17153}, {1475, 4109}, {1621, 32851}, {1654, 26279}, {2530, 44435}, {2757, 53942}, {2886, 4359}, {2975, 35996}, {2979, 25306}, {3035, 4030}, {3120, 24165}, {3210, 33134}, {3218, 4388}, {3315, 30831}, {3416, 17728}, {3452, 3952}, {3454, 3953}, {3578, 26280}, {3662, 25958}, {3663, 31117}, {3681, 5233}, {3685, 33168}, {3686, 68797}, {3701, 4187}, {3703, 3816}, {3704, 37722}, {3712, 49736}, {3729, 31080}, {3742, 18139}, {3751, 63010}, {3752, 4972}, {3756, 48647}, {3814, 4692}, {3817, 4054}, {3820, 4723}, {3829, 4980}, {3841, 6533}, {3873, 4417}, {3883, 59491}, {3891, 17720}, {3911, 63134}, {3914, 17495}, {3925, 24589}, {3944, 17155}, {3967, 30566}, {3977, 40998}, {3980, 33104}, {4011, 33161}, {4136, 39244}, {4153, 68950}, {4193, 4385}, {4228, 26261}, {4383, 33114}, {4387, 50105}, {4392, 27184}, {4418, 33106}, {4425, 46901}, {4442, 42051}, {4544, 17439}, {4640, 51583}, {4645, 27003}, {4647, 24387}, {4703, 36263}, {4766, 24631}, {4850, 32773}, {4892, 42053}, {4894, 25440}, {4901, 20196}, {4914, 61649}, {4981, 5743}, {5051, 37592}, {5057, 32939}, {5081, 35973}, {5253, 7270}, {5284, 33116}, {5563, 36974}, {5695, 11238}, {5722, 49492}, {5739, 24477}, {5846, 37634}, {5847, 37639}, {5887, 8229}, {6636, 34758}, {6703, 17726}, {7018, 65209}, {7485, 38901}, {7788, 26240}, {7998, 25308}, {8050, 40451}, {9059, 38452}, {9284, 23632}, {9335, 25959}, {9669, 50044}, {10584, 28808}, {10707, 64010}, {14009, 30599}, {14555, 26265}, {14829, 26250}, {15246, 17100}, {16067, 19835}, {16393, 66639}, {16602, 24988}, {16604, 16886}, {16703, 17198}, {16739, 69076}, {17001, 17363}, {17063, 25957}, {17117, 30798}, {17122, 33072}, {17123, 33115}, {17147, 24210}, {17163, 24386}, {17211, 24166}, {17449, 33064}, {17490, 33131}, {17591, 32776}, {17595, 32950}, {17596, 32947}, {17597, 33122}, {17598, 32775}, {17605, 49483}, {17717, 32771}, {17719, 32923}, {17721, 24552}, {17722, 32772}, {17723, 19684}, {17774, 64361}, {17776, 26105}, {18134, 64149}, {18201, 33067}, {18743, 32862}, {19804, 33108}, {20544, 20913}, {21020, 21242}, {21241, 31075}, {21283, 24392}, {21616, 56318}, {23921, 63514}, {24199, 31071}, {24216, 31017}, {24217, 32855}, {24325, 33105}, {24349, 31053}, {24627, 33083}, {24703, 32933}, {25253, 41012}, {25345, 68756}, {25385, 31115}, {26239, 37671}, {26563, 69094}, {26758, 49505}, {26792, 62222}, {27064, 33170}, {27131, 32937}, {27512, 56461}, {28774, 34036}, {28968, 34029}, {30740, 32087}, {31034, 62819}, {31037, 49511}, {32779, 32942}, {32780, 32944}, {32843, 32913}, {32845, 33095}, {32856, 42055}, {32860, 33141}, {32861, 32919}, {32911, 33121}, {32918, 33076}, {32922, 33133}, {32924, 33135}, {32930, 33167}, {32931, 33169}, {32940, 33096}, {32943, 33160}, {33065, 62865}, {33066, 62235}, {33073, 37633}, {33102, 62300}, {33118, 37680}, {33126, 62814}, {33162, 59511}, {37162, 56311}, {37662, 46897}, {37663, 49524}, {38456, 54310}, {39998, 51861}, {48164, 66517}, {48642, 50117}, {48645, 62221}, {50698, 64120}, {51192, 63078}, {59406, 63090}

X(69134) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3705, 3006}, {2, 5211, 7191}, {2, 7191, 26230}, {2, 17024, 29634}, {2, 20045, 66632}, {2, 29667, 26251}, {2, 29832, 612}, {2, 29840, 3920}, {2, 33090, 7081}, {2, 33091, 5205}, {11, 69091, 321}, {38, 3846, 26580}, {42, 29655, 29835}, {238, 33119, 56520}, {306, 11019, 29824}, {497, 17740, 32929}, {982, 25760, 17184}, {1125, 30171, 57808}, {1647, 15523, 3840}, {3315, 30831, 33124}, {3452, 63147, 3952}, {3687, 26015, 17135}, {3703, 3816, 4358}, {5272, 29857, 2}, {7292, 29872, 2}, {17017, 29635, 29833}, {24217, 32855, 32915}, {24392, 63131, 21283}, {27757, 29817, 29839}, {29649, 32854, 50000}

X(69135) = (1, 1, 1, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(12)

Barycentrics    a^2*b^2 - b^4 + 2*a^2*b*c + a^2*c^2 - c^4 : :

X(69135) lies on these lines: {1, 325}, {2, 31402}, {5, 350}, {7, 51612}, {10, 37664}, {11, 7752}, {12, 76}, {32, 26629}, {35, 7750}, {39, 26561}, {42, 24995}, {55, 315}, {56, 7763}, {65, 69038}, {69, 611}, {75, 442}, {99, 7354}, {141, 27020}, {150, 25650}, {172, 7807}, {183, 498}, {192, 5025}, {194, 69098}, {274, 25466}, {304, 7179}, {316, 6284}, {317, 11398}, {319, 68889}, {330, 7906}, {344, 17671}, {345, 37445}, {388, 3926}, {491, 1335}, {492, 1124}, {495, 1909}, {497, 32816}, {594, 30149}, {607, 27516}, {612, 45201}, {626, 1500}, {668, 12607}, {1007, 3086}, {1015, 7764}, {1056, 32818}, {1058, 32823}, {1078, 5432}, {1329, 18140}, {1334, 4766}, {1454, 55416}, {1469, 6393}, {1478, 1975}, {1479, 7773}, {1575, 17670}, {1914, 7762}, {2241, 7759}, {2242, 3788}, {2276, 6656}, {2476, 4441}, {2886, 17143}, {3058, 7809}, {3178, 24211}, {3263, 40071}, {3295, 7776}, {3314, 69095}, {3584, 37671}, {3585, 32819}, {3600, 32831}, {3665, 20924}, {3695, 33931}, {3703, 33941}, {3704, 33935}, {3705, 39731}, {3734, 9650}, {3760, 7951}, {3761, 37719}, {3771, 24549}, {3782, 20337}, {3785, 5218}, {3815, 26959}, {3822, 20888}, {3871, 20553}, {3912, 16603}, {3932, 33932}, {3934, 31476}, {3936, 17137}, {3970, 63817}, {4109, 49516}, {4187, 30963}, {4293, 6337}, {4294, 32006}, {4357, 5530}, {4364, 44396}, {4366, 7785}, {4424, 17211}, {4479, 17530}, {4995, 7811}, {5217, 14907}, {5225, 32827}, {5229, 32815}, {5254, 25264}, {5261, 32830}, {5265, 32835}, {5270, 32820}, {5280, 7792}, {5283, 26558}, {5299, 41624}, {5433, 7769}, {5434, 7799}, {6376, 17757}, {6390, 18990}, {6604, 30828}, {6645, 7836}, {7098, 55418}, {7247, 32851}, {7288, 32829}, {7761, 31451}, {7770, 9596}, {7771, 52793}, {7774, 16502}, {7775, 9665}, {7781, 9651}, {7782, 15326}, {7784, 31477}, {7788, 10056}, {7791, 31448}, {7795, 31409}, {7796, 15888}, {7802, 15338}, {7812, 53680}, {7814, 37722}, {7815, 31501}, {7825, 9664}, {7841, 9598}, {7849, 31478}, {7860, 63273}, {7876, 31462}, {8728, 60706}, {9597, 31859}, {9766, 16781}, {10198, 16992}, {10588, 32828}, {10592, 64093}, {10895, 11185}, {11237, 32833}, {11285, 31497}, {11287, 31461}, {11681, 18135}, {13077, 39266}, {13733, 38906}, {14986, 63098}, {16785, 30103}, {16886, 24326}, {16921, 30998}, {17030, 37661}, {17045, 30134}, {17062, 29960}, {17084, 60452}, {17144, 24390}, {17181, 18156}, {17206, 37232}, {17280, 33835}, {17302, 33834}, {17321, 52258}, {17390, 30141}, {17398, 30176}, {17756, 33840}, {17759, 33841}, {18134, 26942}, {20541, 20691}, {20561, 30034}, {24282, 33949}, {26601, 28606}, {26801, 31466}, {30172, 33937}, {30810, 33116}, {33219, 68855}, {33296, 64172}, {37159, 50067}, {40997, 49753}, {45198, 55391}, {51370, 64006}, {56928, 69044}

X(69135) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {12, 69093, 76}, {192, 5025, 69096}, {495, 3933, 1909}, {626, 1500, 26590}, {2242, 3788, 26686}, {3760, 7951, 59635}, {5280, 30104, 7792}, {7796, 64133, 69094}, {15888, 69094, 64133}, {31460, 69097, 2}

X(69136) = (1, 0, 1, 1, 1, 0)-ADDITIVE ASSOCIATE OF X(12)

Barycentrics    a^2*b^2 + 2*a^2*b*c + a^2*c^2 + 2*b^2*c^2 : :

X(69136) lies on these lines: {1, 3934}, {2, 668}, {10, 141}, {12, 626}, {35, 7816}, {37, 716}, {39, 1909}, {55, 3734}, {56, 7815}, {69, 31409}, {75, 52959}, {76, 192}, {115, 26590}, {172, 7780}, {183, 2242}, {274, 1574}, {315, 9650}, {325, 31476}, {330, 7786}, {350, 9466}, {388, 7800}, {498, 3788}, {519, 21264}, {538, 2276}, {551, 20530}, {620, 3023}, {625, 7951}, {712, 21101}, {996, 55161}, {999, 15271}, {1018, 24330}, {1078, 6645}, {1111, 24326}, {1125, 24656}, {1215, 21232}, {1334, 4721}, {1478, 7761}, {1698, 36812}, {1914, 7804}, {1930, 21021}, {1975, 31451}, {2241, 7770}, {2275, 6683}, {2295, 29381}, {3085, 7795}, {3584, 7880}, {3585, 7842}, {3634, 25107}, {3661, 3936}, {3780, 29433}, {3822, 20541}, {3912, 5718}, {3920, 8891}, {3948, 31035}, {4035, 29594}, {4045, 69098}, {4396, 16785}, {4400, 5280}, {4686, 20691}, {4687, 6376}, {4698, 52708}, {4754, 16549}, {4772, 26817}, {4851, 5725}, {5010, 32456}, {5149, 10053}, {5291, 37670}, {5297, 30749}, {6238, 59556}, {6292, 26561}, {7354, 7830}, {7749, 26686}, {7751, 54416}, {7758, 31402}, {7759, 9596}, {7763, 31501}, {7764, 31460}, {7778, 31479}, {7781, 31448}, {7784, 9654}, {7791, 9651}, {7793, 9341}, {7808, 16502}, {7820, 26629}, {7825, 10895}, {7849, 37719}, {7865, 11237}, {7886, 30103}, {7915, 30104}, {9664, 11185}, {9665, 16924}, {10436, 44418}, {12782, 49532}, {14767, 37697}, {15325, 58446}, {15888, 69097}, {16819, 25280}, {16826, 30819}, {17030, 24524}, {17033, 20970}, {17034, 34017}, {17175, 29375}, {17205, 25350}, {17243, 68938}, {17270, 50271}, {17303, 50163}, {17369, 40548}, {17495, 20913}, {17499, 29509}, {17541, 68893}, {17750, 28369}, {19862, 25130}, {19977, 33159}, {21024, 40006}, {21827, 27285}, {24222, 29659}, {24249, 29670}, {24514, 52963}, {24654, 25492}, {25109, 51073}, {25303, 26959}, {25499, 26115}, {26035, 27096}, {27091, 31997}, {29593, 31017}, {29822, 30955}, {30114, 37632}, {31052, 53332}, {32107, 46846}, {32777, 50162}, {37678, 40859}, {50097, 60276}, {53680, 68525}, {59212, 62304}, {63924, 69096}

X(69136) = midpoint of X(2276) and X(3761)
X(69136) = complement of X(16975)
X(69136) = complement of the isogonal conjugate of X(60871)
X(69136) = complement of the isotomic conjugate of X(56129)
X(69136) = X(i)-complementary conjugate of X(j) for these (i,j): {56129, 2887}, {56166, 141}, {60871, 10}
X(69136) = crosspoint of X(2) and X(56129)
X(69136) = crossdifference of every pair of points on line {890, 21007}
X(69136) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 668, 1573}, {2, 64133, 1015}, {10, 17758, 20255}, {12, 69095, 626}, {274, 26752, 1574}, {1215, 21232, 24254}, {1909, 27020, 39}, {4400, 5280, 7805}, {6376, 27255, 16589}, {31460, 69094, 7764}

X(69137) = (1, 0, 1, 1, 1, 0)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 - 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69137) lies on these lines: {2, 69123}, {3, 51388}, {5, 51016}, {6, 626}, {13, 76}, {14, 7773}, {15, 7750}, {16, 7763}, {17, 183}, {18, 7752}, {20, 622}, {61, 315}, {62, 325}, {69, 635}, {99, 42158}, {194, 69125}, {298, 7917}, {302, 3411}, {303, 636}, {316, 16964}, {396, 7767}, {397, 3933}, {491, 3389}, {492, 3390}, {538, 69112}, {550, 59541}, {621, 51753}, {624, 5207}, {625, 69118}, {634, 5980}, {1007, 42149}, {1975, 16965}, {2896, 62984}, {3180, 7797}, {3314, 9112}, {3643, 37172}, {3734, 69117}, {3785, 42152}, {3926, 10653}, {3934, 69114}, {5025, 69110}, {5238, 14907}, {5309, 5859}, {5335, 32830}, {5472, 7794}, {5873, 14880}, {5989, 6778}, {6337, 42151}, {6390, 42148}, {7751, 69119}, {7754, 69111}, {7769, 16242}, {7770, 69109}, {7771, 62600}, {7775, 69132}, {7780, 62198}, {7782, 42433}, {7788, 61719}, {7791, 63201}, {7796, 42990}, {7798, 69126}, {7799, 41100}, {7800, 61332}, {7801, 12155}, {7802, 36967}, {7807, 41406}, {7808, 69128}, {7810, 9763}, {7811, 16962}, {7825, 69113}, {7833, 36775}, {7841, 69124}, {7849, 69129}, {7861, 69127}, {7862, 62197}, {7879, 69122}, {7883, 37786}, {7884, 66445}, {7886, 62199}, {7895, 69116}, {7896, 69115}, {7916, 69130}, {8357, 53428}, {10008, 33382}, {10654, 32006}, {11133, 37008}, {11185, 42813}, {16267, 37671}, {18582, 32828}, {20065, 41407}, {24273, 33468}, {30471, 42528}, {32815, 42161}, {32816, 40694}, {32819, 36969}, {32820, 41974}, {32823, 37641}, {32827, 42159}, {32832, 37832}, {32833, 41107}, {32834, 43403}, {32836, 41112}, {32837, 42510}, {32838, 42911}, {32839, 42089}, {32869, 49825}, {32874, 49874}, {33465, 51272}, {34541, 46226}, {37647, 42937}, {37668, 42998}, {37688, 42488}, {37825, 41035}, {41119, 46951}, {42123, 59539}, {42150, 64018}, {42166, 64093}, {42924, 59542}, {49111, 63732}

X(69137) = midpoint of X(69112) and X(69131)
X(69137) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 69106, 76}, {76, 299, 69106}, {16965, 69120, 1975}, {69114, 69133, 3934}

X(69138) = (1, 1, 0, 1, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 + a^2*b^2 + a^2*c^2 + 4*b^2*c^2 + 2*Sqrt[3]*a^2*S : :

X(69138) lies on these lines: {2, 53452}, {3, 22687}, {4, 51161}, {6, 76}, {13, 626}, {15, 7816}, {16, 7815}, {17, 3788}, {61, 3734}, {62, 3934}, {69, 53429}, {141, 397}, {298, 7785}, {302, 16921}, {303, 7836}, {315, 69115}, {325, 69116}, {396, 7789}, {622, 5103}, {623, 37825}, {624, 1656}, {3926, 61332}, {4045, 69125}, {5340, 7784}, {5352, 32456}, {5472, 7794}, {5611, 42674}, {5858, 7753}, {5864, 14881}, {5869, 18440}, {6581, 9605}, {6656, 69112}, {6683, 63200}, {7759, 69107}, {7761, 16965}, {7764, 69121}, {7774, 69130}, {7778, 42156}, {7780, 41406}, {7781, 63201}, {7793, 19780}, {7795, 40693}, {7800, 10653}, {7801, 9763}, {7803, 69126}, {7807, 62198}, {7808, 33483}, {7810, 12155}, {7825, 42813}, {7830, 42158}, {7834, 69111}, {7842, 36969}, {7857, 62232}, {7862, 37832}, {7865, 41107}, {7869, 42992}, {7880, 16267}, {7891, 62600}, {9112, 69106}, {9989, 42155}, {11185, 69113}, {15271, 22238}, {16627, 33411}, {16772, 59545}, {16773, 58446}, {16924, 69132}, {25157, 54298}, {32459, 42945}, {32832, 62197}, {32968, 61331}, {33020, 62983}, {34509, 42974}, {41039, 54393}, {42598, 44377}, {42777, 44382}, {42990, 69123}, {43104, 44383}, {47286, 69127}, {59635, 69118}, {61719, 69109}, {63924, 69110}

X(69138) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 69108, 626}, {16965, 69122, 7761}, {69112, 69129, 6656}, {69116, 69133, 325}

X(69139) = (1, 1, 0, 1, 1, 0, 0)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 + a^2*b^2 + a^2*c^2 + 4*b^2*c^2 : :

X(69139) lies on these lines: {2, 1975}, {3, 3734}, {4, 141}, {5, 7778}, {6, 76}, {10, 20181}, {13, 69109}, {14, 69108}, {22, 10130}, {25, 8891}, {30, 7800}, {32, 8667}, {39, 17130}, {69, 7745}, {75, 26687}, {99, 11285}, {115, 7822}, {140, 52771}, {148, 7876}, {183, 384}, {187, 68527}, {194, 11174}, {218, 4721}, {220, 4713}, {230, 14001}, {311, 26164}, {315, 599}, {316, 7879}, {325, 16924}, {381, 626}, {382, 6287}, {385, 68525}, {394, 41231}, {458, 4074}, {538, 7808}, {574, 31239}, {598, 60210}, {620, 3526}, {625, 3851}, {631, 58446}, {639, 6251}, {640, 6250}, {671, 7918}, {958, 21264}, {964, 26100}, {1003, 1078}, {1007, 32987}, {1008, 19768}, {1184, 39998}, {1235, 2207}, {1351, 10358}, {1384, 7780}, {1478, 69097}, {1479, 69095}, {1498, 59530}, {1506, 7801}, {1611, 11324}, {1656, 3788}, {1657, 7830}, {1909, 16781}, {2076, 39266}, {2453, 36165}, {2548, 3933}, {2549, 8362}, {2896, 11361}, {3054, 32838}, {3055, 32829}, {3090, 39663}, {3096, 7841}, {3114, 36615}, {3314, 7773}, {3329, 20081}, {3523, 32459}, {3529, 55732}, {3549, 54075}, {3552, 5210}, {3589, 5286}, {3618, 6392}, {3619, 32974}, {3620, 32006}, {3673, 4363}, {3760, 54416}, {3761, 16502}, {3763, 6656}, {3767, 7819}, {3785, 14033}, {3796, 10328}, {3815, 3926}, {3830, 7842}, {3843, 7825}, {3913, 25102}, {3972, 22331}, {4048, 5085}, {4252, 37100}, {4361, 4385}, {4445, 5015}, {4657, 13161}, {4911, 7232}, {5007, 17131}, {5021, 29455}, {5024, 6683}, {5025, 7868}, {5026, 10541}, {5041, 14711}, {5055, 7862}, {5056, 37690}, {5064, 21248}, {5077, 65633}, {5103, 43453}, {5108, 46512}, {5149, 12188}, {5192, 26978}, {5201, 10790}, {5206, 68513}, {5275, 17686}, {5305, 63955}, {5306, 46951}, {5309, 7889}, {5337, 50060}, {5395, 20080}, {5475, 7776}, {5485, 48310}, {5585, 43459}, {5737, 37086}, {6144, 53489}, {6292, 7748}, {6376, 20172}, {6389, 6823}, {6390, 31401}, {6421, 13930}, {6422, 13877}, {6645, 30998}, {6655, 16986}, {6658, 7904}, {6680, 33237}, {7383, 59702}, {7388, 42262}, {7389, 42265}, {7400, 34828}, {7470, 55646}, {7603, 7888}, {7610, 8369}, {7615, 8360}, {7620, 33230}, {7735, 32834}, {7736, 32830}, {7737, 7767}, {7746, 7820}, {7747, 7854}, {7749, 11288}, {7750, 14035}, {7751, 7804}, {7752, 7881}, {7753, 7855}, {7759, 15484}, {7762, 40341}, {7763, 31489}, {7771, 33235}, {7772, 22253}, {7774, 33269}, {7775, 7895}, {7777, 32821}, {7785, 7788}, {7786, 22332}, {7787, 14614}, {7790, 59236}, {7791, 32819}, {7792, 16898}, {7797, 16895}, {7798, 14535}, {7803, 47286}, {7805, 43136}, {7806, 19689}, {7807, 32832}, {7810, 11159}, {7812, 15533}, {7817, 40727}, {7818, 39590}, {7823, 63044}, {7824, 53095}, {7828, 33217}, {7831, 33234}, {7832, 7887}, {7834, 63924}, {7835, 33233}, {7836, 16921}, {7843, 7896}, {7844, 7915}, {7846, 14568}, {7848, 63931}, {7857, 33220}, {7860, 32027}, {7861, 7914}, {7863, 31455}, {7867, 11318}, {7872, 63922}, {7873, 62203}, {7883, 11317}, {7885, 33018}, {7898, 14042}, {7902, 32457}, {7906, 11163}, {7910, 31168}, {7912, 33013}, {7920, 19570}, {7925, 33002}, {7928, 33019}, {7930, 14061}, {7931, 32966}, {7932, 19694}, {7934, 15031}, {7937, 10159}, {7938, 14041}, {7944, 33219}, {7945, 32967}, {8352, 51186}, {8356, 44519}, {8361, 43620}, {8365, 16509}, {8367, 31406}, {8588, 68528}, {8623, 11335}, {9300, 32836}, {9317, 25591}, {9596, 69093}, {9599, 69094}, {9606, 63041}, {9709, 27076}, {10302, 53107}, {10356, 35456}, {10359, 13196}, {10601, 51481}, {11168, 32985}, {11284, 30749}, {11311, 36251}, {11312, 36252}, {11321, 18140}, {11338, 21001}, {11477, 12251}, {11479, 14767}, {12164, 59556}, {12203, 53094}, {13468, 14039}, {13740, 15668}, {14023, 18907}, {14034, 14712}, {14037, 17008}, {14064, 63534}, {14376, 15760}, {14532, 52854}, {14907, 19687}, {14930, 32882}, {15483, 51524}, {15491, 31400}, {15589, 63928}, {15597, 52718}, {15681, 40344}, {16043, 32815}, {16062, 17327}, {16405, 30955}, {16412, 30819}, {16853, 36812}, {16913, 16999}, {16916, 16992}, {16920, 37670}, {16925, 37688}, {16932, 52898}, {16964, 69122}, {16965, 69123}, {17004, 33225}, {17030, 31490}, {17042, 42359}, {17118, 33940}, {17119, 33941}, {17259, 17681}, {17265, 33838}, {17541, 34284}, {17691, 26244}, {17814, 59698}, {17825, 40814}, {18501, 18806}, {18584, 32962}, {18841, 60636}, {18843, 60143}, {18845, 60285}, {19568, 39951}, {19690, 60728}, {19692, 63047}, {20065, 37671}, {20179, 20943}, {20255, 62383}, {20269, 30754}, {20530, 25524}, {20582, 33190}, {21167, 46034}, {21477, 24271}, {23235, 61132}, {25878, 26678}, {26166, 26226}, {27020, 31477}, {28723, 36748}, {31404, 32818}, {32816, 32983}, {32822, 32960}, {32826, 32986}, {32831, 62993}, {32833, 66416}, {32867, 32977}, {32869, 63024}, {32870, 33203}, {32872, 37689}, {32874, 63006}, {32885, 33224}, {32894, 63005}, {32956, 34573}, {32973, 34229}, {32999, 37647}, {33004, 45017}, {33023, 67536}, {33180, 63533}, {33181, 62992}, {33185, 43291}, {33189, 44381}, {33197, 44401}, {33215, 66616}, {33223, 63543}, {33249, 53127}, {33261, 63083}, {33586, 62949}, {34138, 62368}, {34227, 38675}, {34506, 68718}, {34816, 41328}, {35930, 49111}, {36697, 59625}, {36751, 37186}, {37174, 41244}, {37344, 59197}, {37674, 41236}, {38526, 47284}, {39142, 61870}, {40332, 50659}, {40405, 56067}, {41235, 59563}, {43527, 60250}, {43676, 60644}, {44562, 51122}, {52288, 53415}, {52289, 59767}, {53017, 54393}, {54858, 60099}, {55164, 66395}, {55676, 61102}, {60128, 60151}, {60187, 60619}, {61689, 63556}, {69112, 69128}, {69113, 69129}, {69116, 69132}, {69117, 69133}

X(69139) = midpoint of X(32869) and X(63024)
X(69139) = reflection of X(9605) in X(7808)
X(69139) = complement of X(7738)
X(69139) = X(82)-anticomplementary conjugate of X(41927)
X(69139) = crossdifference of every pair of points on line {688, 8651}
X(69139) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1975, 5013}, {2, 17128, 1975}, {2, 59635, 13881}, {3, 3934, 15271}, {4, 141, 7784}, {5, 7795, 7778}, {6, 76, 63933}, {69, 7745, 63932}, {69, 32971, 7745}, {76, 83, 7754}, {76, 7770, 6}, {76, 60855, 7760}, {83, 7754, 6}, {99, 11285, 15815}, {115, 7822, 7866}, {183, 384, 3053}, {194, 68522, 11174}, {315, 8370, 65630}, {384, 31276, 183}, {599, 65630, 315}, {1003, 1078, 5023}, {1975, 5013, 8716}, {2548, 3933, 9766}, {3090, 53033, 44377}, {3314, 16044, 7773}, {3589, 63923, 5286}, {3620, 32979, 32006}, {3734, 3934, 3}, {3734, 7815, 7816}, {3763, 44518, 6656}, {3926, 32968, 3815}, {3933, 66415, 2548}, {3934, 7816, 7815}, {5023, 8556, 1078}, {5025, 46226, 7868}, {5286, 16045, 3589}, {5286, 52713, 63923}, {5475, 7794, 7776}, {6292, 7748, 11287}, {6656, 11185, 44518}, {6683, 7781, 5024}, {7737, 7767, 63938}, {7746, 7820, 32954}, {7751, 7804, 30435}, {7754, 7770, 83}, {7763, 32992, 31489}, {7786, 31859, 22332}, {7787, 17129, 14614}, {7791, 32819, 44526}, {7807, 32832, 37637}, {7815, 7816, 3}, {7819, 64093, 3767}, {7867, 39565, 11318}, {7881, 44543, 7752}, {7914, 18546, 7861}, {7930, 14061, 33218}, {8367, 34511, 42849}, {8367, 59780, 34511}, {9466, 11286, 8667}, {11324, 40022, 1611}, {11338, 60707, 21001}, {14001, 32828, 230}, {14035, 16990, 7750}, {15491, 59546, 31400}, {16043, 32815, 63548}, {16045, 52713, 5286}, {16925, 37688, 44535}, {17686, 18135, 5275}, {24273, 44530, 7770}, {31400, 32817, 59546}, {31400, 32957, 15491}, {32006, 32979, 53418}, {32817, 32957, 31400}, {32829, 32975, 3055}, {32834, 33198, 7735}, {32838, 32970, 3054}, {39998, 68719, 1184}, {43136, 63954, 7805}, {43459, 68516, 5585}, {58446, 59545, 631}

X(69140) = (1, 0, 1, 0, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 - 2*b^4 + 4*b^2*c^2 - 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69140) lies on these lines: {4, 54860}, {5, 33517}, {6, 3843}, {13, 39}, {16, 16629}, {17, 37512}, {18, 39601}, {32, 5340}, {62, 39565}, {115, 397}, {187, 16965}, {396, 7756}, {546, 5471}, {574, 42156}, {635, 53458}, {1506, 42166}, {3412, 63196}, {3767, 5335}, {5007, 69111}, {5041, 69126}, {5206, 42155}, {5254, 5472}, {5309, 41112}, {5318, 7747}, {5344, 7737}, {5475, 42162}, {6781, 42165}, {7739, 49825}, {7745, 43416}, {7746, 10653}, {7748, 40693}, {7749, 42148}, {8588, 43193}, {9112, 69127}, {11542, 63548}, {11646, 44512}, {14537, 42973}, {14711, 69106}, {15513, 42158}, {15515, 16644}, {16808, 69132}, {18424, 40694}, {18582, 31455}, {22236, 65633}, {31401, 43403}, {31703, 55716}, {31704, 50664}, {37007, 69124}, {39563, 61719}, {42921, 61331}, {42974, 44518}, {42988, 44526}, {42990, 69118}

X(69140) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 42813, 39590}, {13, 69112, 39}, {13, 69125, 69133}, {16965, 69119, 187}, {42158, 62198, 15513}, {69112, 69133, 69125}, {69125, 69133, 39}

X(69141) = (1, 0, 1, 0, 1, 1, 0)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 - 2*b^4 + 4*b^2*c^2 - 2*c^4 : :
X(69141) = X[5206] - 3 X[18362], X[5206] - 4 X[63534], 2 X[7746] - 3 X[18362], 3 X[18362] - 4 X[63534], X[7763] - 3 X[33006]

X(69141) lies on these lines: {2, 7756}, {3, 39565}, {4, 32}, {5, 574}, {6, 3843}, {11, 9651}, {12, 9664}, {13, 69113}, {14, 69112}, {15, 16631}, {16, 16630}, {20, 7749}, {30, 5206}, {39, 381}, {53, 44226}, {76, 7818}, {83, 7902}, {99, 7862}, {111, 31857}, {148, 7752}, {172, 18513}, {183, 7842}, {187, 382}, {194, 671}, {230, 3627}, {232, 35488}, {315, 17131}, {316, 7751}, {384, 7844}, {385, 14044}, {427, 34481}, {485, 62241}, {486, 62242}, {538, 7773}, {543, 7763}, {546, 5254}, {548, 3054}, {571, 18353}, {576, 11646}, {577, 1879}, {578, 9697}, {620, 32961}, {625, 1975}, {626, 11185}, {631, 43619}, {1003, 7886}, {1015, 10896}, {1078, 33019}, {1194, 63538}, {1196, 5064}, {1384, 62008}, {1500, 10895}, {1503, 39764}, {1504, 6564}, {1505, 6565}, {1506, 2549}, {1570, 18440}, {1571, 7989}, {1593, 44527}, {1656, 37512}, {1657, 15513}, {1691, 48884}, {1692, 36990}, {1914, 18514}, {1971, 34786}, {1990, 46257}, {2021, 52854}, {2076, 48904}, {2088, 23325}, {2241, 3583}, {2242, 3585}, {2482, 32984}, {2548, 3832}, {2909, 58907}, {2996, 7758}, {3053, 3830}, {3095, 38732}, {3098, 31703}, {3146, 6781}, {3199, 37197}, {3291, 31133}, {3520, 50718}, {3522, 12815}, {3526, 8589}, {3529, 21843}, {3534, 44535}, {3545, 31401}, {3552, 14061}, {3614, 31501}, {3734, 5025}, {3785, 54097}, {3788, 32819}, {3815, 3850}, {3818, 5028}, {3839, 5286}, {3845, 5309}, {3851, 5013}, {3853, 43291}, {3854, 31404}, {3855, 7738}, {3856, 9607}, {3857, 31406}, {3858, 15048}, {3859, 9606}, {3861, 5305}, {3934, 7841}, {3972, 14042}, {4045, 16924}, {5007, 61984}, {5008, 61990}, {5017, 48895}, {5019, 53421}, {5023, 5073}, {5024, 61953}, {5033, 29012}, {5034, 19130}, {5041, 15484}, {5052, 53023}, {5055, 15815}, {5058, 23261}, {5062, 23251}, {5063, 9220}, {5065, 36412}, {5070, 53095}, {5072, 31489}, {5076, 35007}, {5094, 40350}, {5107, 15069}, {5158, 53416}, {5210, 17800}, {5283, 17577}, {5306, 14893}, {5319, 14075}, {5346, 18907}, {5355, 50689}, {5368, 61985}, {5461, 33007}, {5471, 42159}, {5472, 42162}, {5569, 66424}, {5585, 62121}, {5868, 16942}, {5869, 16943}, {5872, 33517}, {5873, 33518}, {6033, 62356}, {6128, 36430}, {6248, 32452}, {6292, 32974}, {6310, 41262}, {6321, 9737}, {6392, 7890}, {6655, 7815}, {6658, 7857}, {6680, 14035}, {6683, 44543}, {6704, 33269}, {6722, 16925}, {7503, 9700}, {7506, 44528}, {7507, 33843}, {7615, 7810}, {7617, 7833}, {7618, 32839}, {7620, 32830}, {7689, 8571}, {7697, 46283}, {7736, 61964}, {7739, 41099}, {7750, 8352}, {7754, 7843}, {7759, 47286}, {7761, 33229}, {7762, 41748}, {7766, 62427}, {7770, 7861}, {7771, 33256}, {7776, 31173}, {7782, 32967}, {7783, 53105}, {7784, 9466}, {7785, 7798}, {7786, 33013}, {7790, 7808}, {7795, 16041}, {7800, 32982}, {7801, 37350}, {7803, 33016}, {7804, 7851}, {7806, 14066}, {7809, 7916}, {7812, 41135}, {7813, 32816}, {7816, 7887}, {7817, 11317}, {7820, 14064}, {7822, 33184}, {7823, 14568}, {7826, 32006}, {7828, 11361}, {7830, 32832}, {7834, 8370}, {7838, 36523}, {7845, 63933}, {7847, 15482}, {7852, 11286}, {7854, 64093}, {7855, 63923}, {7859, 66413}, {7860, 17129}, {7863, 32815}, {7865, 7911}, {7869, 7934}, {7874, 11318}, {7877, 19570}, {7889, 32971}, {7908, 7912}, {7914, 7933}, {7915, 33219}, {7918, 68522}, {7919, 68525}, {7923, 60855}, {7930, 14046}, {7931, 33289}, {7941, 48913}, {7951, 31451}, {8359, 20112}, {8541, 41221}, {8860, 66397}, {9166, 66419}, {9300, 23046}, {9465, 63539}, {9540, 9684}, {9598, 31476}, {9605, 61970}, {9620, 18492}, {9635, 66610}, {9650, 69096}, {9665, 69098}, {9674, 10576}, {9675, 35821}, {9696, 10539}, {9699, 10594}, {9855, 17503}, {10113, 14901}, {10151, 27376}, {10301, 47298}, {10316, 18403}, {10989, 39576}, {11178, 44453}, {11287, 31239}, {11572, 60501}, {11606, 18548}, {11614, 55863}, {11742, 62085}, {13192, 38397}, {13474, 50387}, {13509, 18394}, {13585, 63762}, {13851, 39643}, {14130, 44523}, {14269, 14537}, {14561, 65417}, {14832, 68417}, {14907, 33279}, {14971, 32985}, {15602, 61911}, {15603, 58207}, {15655, 49134}, {15720, 44541}, {15820, 62702}, {15980, 30270}, {16589, 17532}, {16628, 51207}, {16629, 51206}, {16808, 69124}, {16809, 69125}, {16964, 69119}, {16965, 69118}, {17006, 33267}, {17578, 43618}, {18584, 31467}, {19220, 44263}, {19695, 37688}, {20970, 66104}, {22331, 62004}, {22332, 61955}, {22615, 62201}, {22644, 62202}, {22682, 46305}, {23004, 47068}, {23005, 47066}, {23514, 37466}, {25639, 31456}, {26613, 66422}, {31274, 32969}, {31421, 61264}, {31422, 54447}, {31481, 42273}, {32456, 33233}, {32826, 32972}, {32838, 33272}, {32873, 53141}, {32965, 53127}, {33192, 34506}, {33235, 58448}, {33264, 43459}, {33703, 62992}, {34417, 39691}, {34507, 53505}, {34508, 53435}, {34509, 53447}, {34571, 61981}, {34864, 44521}, {34866, 45735}, {37004, 39266}, {37348, 37479}, {39593, 61974}, {39913, 67903}, {42157, 62198}, {42158, 62197}, {42262, 62205}, {42265, 62206}, {42283, 49220}, {42284, 49221}, {42920, 61331}, {42921, 61332}, {46893, 66396}, {48901, 53475}, {52625, 63561}, {54005, 54991}, {59768, 62937}, {61755, 67672}, {61945, 62993}, {61973, 63024}, {61983, 63006}, {62982, 63550}, {63493, 65140}

X(69141) = reflection of X(i) in X(j) for these {i,j}: {5206, 7746}, {7746, 63534}, {7903, 7773}
X(69141) = isogonal conjugate of X(63811)
X(69141) = X(1)-isoconjugate of X(63811)
X(69141) = X(3)-Dao conjugate of X(63811)
X(69141) = barycentric quotient X(6)/X(63811)
X(69141) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7756, 15515}, {4, 32, 62203}, {4, 115, 32}, {4, 3767, 7747}, {4, 59363, 39838}, {4, 63533, 3767}, {5, 7748, 574}, {5, 53419, 7748}, {5, 63548, 31455}, {6, 3843, 39590}, {20, 7749, 8588}, {20, 43620, 7749}, {39, 39563, 44518}, {39, 44518, 11648}, {76, 7825, 7818}, {76, 7885, 7896}, {76, 14041, 7825}, {99, 32966, 7862}, {115, 7747, 3767}, {148, 7752, 7781}, {148, 32993, 7752}, {315, 63924, 17131}, {381, 39563, 11648}, {381, 44518, 39}, {382, 13881, 187}, {546, 5254, 5475}, {625, 1975, 7888}, {625, 63922, 1975}, {626, 11185, 17130}, {1506, 2549, 53096}, {1656, 44526, 37512}, {1657, 37637, 15513}, {2548, 3832, 43457}, {2548, 43448, 7765}, {2549, 3091, 1506}, {2996, 32827, 7758}, {3526, 44519, 8589}, {3734, 5025, 7867}, {3767, 7747, 32}, {3767, 63533, 115}, {3788, 63957, 32819}, {3832, 43448, 2548}, {3839, 63536, 5286}, {3845, 63543, 5309}, {3851, 5013, 7603}, {3855, 7738, 31415}, {3861, 5305, 53418}, {3934, 7841, 7935}, {5063, 9220, 61327}, {5064, 63541, 1196}, {5206, 18362, 7746}, {5254, 5475, 7772}, {7737, 7755, 32}, {7738, 31415, 9698}, {7746, 63534, 18362}, {7748, 18424, 5}, {7748, 31455, 63548}, {7765, 43457, 2548}, {7770, 7861, 7913}, {7790, 16044, 7808}, {7809, 20081, 7916}, {7825, 7896, 7885}, {7825, 18546, 76}, {7843, 32457, 7754}, {7847, 16921, 15482}, {7885, 7896, 7818}, {7911, 31276, 7865}, {7934, 17128, 7869}, {11185, 14063, 626}, {14041, 18546, 7818}, {14045, 17128, 7934}, {16808, 69124, 69133}, {16809, 69125, 69132}, {17577, 63537, 5283}, {18424, 53419, 574}, {31455, 63548, 574}, {31467, 61946, 18584}, {31703, 31704, 3098}, {32006, 63955, 7826}, {32819, 33228, 3788}, {32832, 33017, 7830}, {33229, 59635, 7761}, {35786, 35787, 48889}, {37512, 39601, 1656}, {62702, 62977, 15820}

X(69142) = (1, 0, 1, 0, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 - 2*b^4 - 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69142) lies on these lines: {2, 47067}, {6, 7818}, {13, 9466}, {14, 31173}, {61, 7873}, {62, 7821}, {141, 5472}, {299, 538}, {303, 14904}, {396, 7810}, {397, 7794}, {530, 619}, {599, 22495}, {624, 53442}, {3180, 7827}, {5340, 17130}, {5859, 39593}, {7801, 10653}, {7831, 62984}, {7854, 40693}, {7863, 42148}, {7888, 22238}, {9112, 69129}, {9763, 15810}, {11645, 48656}, {13349, 38750}, {15819, 63732}, {16965, 69117}, {22110, 42913}, {31239, 69123}, {31275, 62197}, {33215, 63105}, {40706, 43535}, {42850, 43542}, {42990, 69116}, {61719, 69115}, {69106, 69112}, {69125, 69131}

X(69142) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 69114, 9466}, {69123, 69133, 31239}

X(69143) = (1, 0, 0, 0, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^4 + 4*b^2*c^2 + 2*Sqrt[3]*a^2*S : :

X(69143) lies on these lines: {6, 17130}, {13, 7821}, {62, 9466}, {298, 7843}, {396, 7863}, {397, 7794}, {2482, 16772}, {3933, 5472}, {5340, 7818}, {7801, 40693}, {7810, 42148}, {7845, 69107}, {7852, 69111}, {7853, 69108}, {7854, 10653}, {7873, 16965}, {7874, 69119}, {7888, 42156}, {9112, 69131}, {31173, 42813}, {42990, 69114}, {61719, 69117}, {69121, 69133}, {69125, 69129}

X(69143) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 69116, 7821}, {16965, 69115, 7873}, {69108, 69112, 7853}

X(69144) = (1, 0, 0, 0, 0, 0, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^2*(a^2 + 2*Sqrt[3]*S) : :

X(69144) lies on these lines: {3, 6}, {4, 54861}, {5, 5472}, {13, 39565}, {14, 39590}, {17, 62197}, {18, 7603}, {51, 34394}, {115, 397}, {232, 64468}, {298, 7882}, {299, 7895}, {395, 1506}, {396, 7749}, {398, 7747}, {532, 53452}, {629, 23302}, {635, 53453}, {1500, 7127}, {2241, 54402}, {2242, 54403}, {2482, 59541}, {2548, 37641}, {3180, 7905}, {3199, 8739}, {3457, 44109}, {3458, 44107}, {3767, 42998}, {3815, 61319}, {5254, 31710}, {5339, 62203}, {5346, 61318}, {5471, 7745}, {5475, 40694}, {6114, 12830}, {6781, 42147}, {7737, 42999}, {7746, 40693}, {7748, 10653}, {7753, 31693}, {7756, 42148}, {7769, 62984}, {7805, 33482}, {7858, 62983}, {8604, 34321}, {9115, 37340}, {9697, 11134}, {11304, 41745}, {11542, 61515}, {12829, 22692}, {13366, 21461}, {13881, 42974}, {16806, 57384}, {16965, 69113}, {18424, 42162}, {31404, 63033}, {31406, 42634}, {31455, 42149}, {33526, 40696}, {37332, 51200}, {37637, 42988}, {37786, 62601}, {39563, 41107}, {40350, 54363}, {42155, 65633}, {42163, 43457}, {42562, 49238}, {42563, 49239}, {42924, 63548}, {42975, 65630}, {42990, 69110}, {45212, 64609}, {49208, 51852}, {49209, 51854}, {51753, 53442}, {61719, 69119}, {69106, 69116}, {69108, 69114}, {69121, 69131}, {69123, 69129}, {69125, 69127}

X(69144) = isogonal conjugate of the isotomic conjugate of X(23302)
X(69144) = X(16806)-Ceva conjugate of X(512)
X(69144) = X(629)-Dao conjugate of X(76)
X(69144) = crosspoint of X(6) and X(21461)
X(69144) = crosssum of X(2) and X(302)
X(69144) = crossdifference of every pair of points on line {523, 44361}
X(69144) = barycentric product X(i)*X(j) for these {i,j}: {6, 23302}, {61, 8018}, {629, 21461}, {11139, 34327}
X(69144) = barycentric quotient X(i)/X(j) for these {i,j}: {8018, 34389}, {23302, 76}
X(69144) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 62, 39}, {6, 13330, 44511}, {13, 69118, 39565}, {18, 9112, 69133}, {18, 69133, 7603}, {61, 35230, 21401}, {62, 3398, 10614}, {371, 372, 9736}, {36843, 63199, 15515}, {42149, 61332, 31455}, {42990, 69110, 69112}

X(69145) = (1, 1, 1, 1, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(14)

Barycentrics    a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 - 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69145) lies on these lines: {2, 69122}, {3, 51387}, {5, 51018}, {6, 626}, {13, 7773}, {14, 76}, {15, 7763}, {16, 7750}, {17, 7752}, {18, 183}, {20, 621}, {61, 325}, {62, 315}, {69, 636}, {99, 42157}, {194, 69124}, {299, 7917}, {302, 635}, {303, 3412}, {316, 16965}, {395, 7767}, {398, 3933}, {491, 3364}, {492, 3365}, {538, 69113}, {550, 59542}, {622, 51754}, {623, 5207}, {625, 69119}, {633, 5981}, {1007, 42152}, {1975, 16964}, {2896, 62983}, {3181, 7797}, {3314, 9113}, {3642, 37173}, {3734, 69116}, {3785, 42149}, {3926, 10654}, {3934, 69115}, {5025, 69111}, {5237, 14907}, {5309, 5858}, {5334, 32830}, {5471, 7794}, {5872, 14880}, {5989, 6777}, {6337, 42150}, {6390, 42147}, {7751, 69118}, {7754, 69110}, {7769, 16241}, {7770, 69108}, {7771, 62601}, {7775, 69133}, {7780, 62197}, {7782, 42434}, {7788, 69106}, {7791, 63200}, {7796, 42991}, {7798, 69127}, {7799, 41101}, {7800, 61331}, {7801, 12154}, {7802, 36968}, {7807, 41407}, {7808, 69129}, {7809, 61719}, {7810, 9761}, {7811, 16963}, {7825, 69112}, {7841, 69125}, {7849, 69128}, {7861, 69126}, {7862, 62198}, {7879, 69123}, {7883, 37785}, {7884, 66446}, {7886, 62200}, {7895, 69117}, {7896, 69114}, {7916, 69131}, {8357, 53440}, {10008, 33383}, {10653, 32006}, {11132, 37007}, {11185, 42814}, {16268, 37671}, {18581, 32828}, {20065, 41406}, {24273, 33469}, {30472, 42529}, {32815, 42160}, {32816, 40693}, {32819, 36970}, {32820, 41973}, {32823, 37640}, {32827, 42162}, {32832, 37835}, {32833, 41108}, {32834, 43404}, {32836, 41113}, {32837, 42511}, {32838, 42910}, {32839, 42092}, {32869, 49824}, {32874, 49873}, {33464, 51265}, {34540, 46226}, {37647, 42936}, {37668, 42999}, {37688, 42489}, {37824, 41034}, {41120, 46951}, {42122, 59540}, {42151, 64018}, {42163, 64093}, {42925, 59541}, {49111, 63731}

X(69145) = midpoint of X(69113) and X(69130)
X(69145) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 69107, 76}, {76, 298, 69107}, {16964, 69121, 1975}, {69115, 69132, 3934}

X(69146) = (1, 1, 0, 1, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(14)

Barycentrics    a^4 + a^2*b^2 + a^2*c^2 + 4*b^2*c^2 - 2*Sqrt[3]*a^2*S : :

X(69146) lies on these lines: {2, 53463}, {3, 22689}, {4, 51162}, {6, 76}, {14, 626}, {15, 7815}, {16, 7816}, {18, 3788}, {61, 3934}, {62, 3734}, {69, 53441}, {141, 398}, {299, 7785}, {302, 7836}, {303, 16921}, {315, 69114}, {325, 69117}, {395, 7789}, {621, 5103}, {623, 1656}, {624, 37824}, {3926, 61331}, {4045, 69124}, {5339, 7784}, {5351, 32456}, {5471, 7794}, {5615, 42675}, {5859, 7753}, {5865, 14881}, {5868, 18440}, {6294, 9605}, {6656, 69113}, {6683, 63201}, {7759, 69106}, {7761, 16964}, {7764, 69120}, {7774, 69131}, {7778, 42153}, {7780, 41407}, {7781, 63200}, {7793, 19781}, {7795, 40694}, {7800, 10654}, {7801, 9761}, {7803, 69127}, {7807, 62197}, {7808, 33482}, {7810, 12154}, {7825, 42814}, {7830, 42157}, {7834, 69110}, {7842, 36970}, {7857, 62233}, {7862, 37835}, {7865, 41108}, {7869, 42993}, {7880, 16268}, {7891, 62601}, {9113, 69107}, {9988, 42154}, {11185, 69112}, {15271, 22236}, {16626, 33410}, {16772, 58446}, {16773, 59545}, {16924, 69133}, {25167, 54297}, {32459, 42944}, {32832, 62198}, {32968, 61332}, {33020, 62984}, {34508, 42975}, {41038, 54393}, {42599, 44377}, {42778, 44383}, {42991, 69122}, {43101, 44382}, {47286, 69126}, {59635, 69119}, {63924, 69111}

X(69146) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 69109, 626}, {16964, 69123, 7761}, {69113, 69128, 6656}, {69117, 69132, 325}

X(69147) = (1, 0, 1, 0, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(14)

Barycentrics    a^4 - 2*b^4 + 4*b^2*c^2 - 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69147) lies on these lines: {4, 54861}, {5, 33518}, {6, 3843}, {14, 39}, {15, 16628}, {17, 39601}, {18, 37512}, {32, 5339}, {61, 39565}, {115, 398}, {187, 16964}, {395, 7756}, {546, 5472}, {574, 42153}, {636, 53469}, {1506, 42163}, {3411, 63197}, {3767, 5334}, {5007, 69110}, {5041, 69127}, {5206, 42154}, {5254, 5471}, {5309, 41113}, {5321, 7747}, {5343, 7737}, {5475, 42159}, {6781, 42164}, {7739, 49824}, {7745, 43417}, {7746, 10654}, {7748, 40694}, {7749, 42147}, {8588, 43194}, {9113, 69126}, {11543, 63548}, {11646, 44511}, {14537, 42972}, {14711, 69107}, {15513, 42157}, {15515, 16645}, {16809, 69133}, {18424, 40693}, {18581, 31455}, {22238, 65633}, {31401, 43404}, {31703, 50664}, {31704, 55716}, {37008, 69125}, {39563, 69112}, {42920, 61332}, {42975, 44518}, {42989, 44526}, {42991, 69119}

X(69147) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 42814, 39590}, {14, 69113, 39}, {14, 69124, 69132}, {16964, 69118, 187}, {42157, 62197, 15513}, {69113, 69132, 69124}, {69124, 69132, 39}

X(69148) = (1, 0, 1, 0, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(14)

Barycentrics    a^4 - 2*b^4 - 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69148) lies on these lines: {2, 47069}, {6, 7818}, {13, 31173}, {14, 9466}, {61, 7821}, {62, 7873}, {141, 5471}, {298, 538}, {302, 14905}, {395, 7810}, {398, 7794}, {531, 618}, {599, 22496}, {623, 53430}, {3181, 7827}, {5339, 17130}, {5858, 39593}, {7801, 10654}, {7831, 62983}, {7854, 40694}, {7863, 42147}, {7888, 22236}, {9113, 69128}, {9761, 15810}, {11645, 48655}, {13350, 38750}, {15819, 63731}, {16964, 69116}, {22110, 42912}, {31239, 69122}, {31275, 62198}, {33215, 63106}, {40707, 43535}, {42850, 43543}, {42991, 69117}, {69107, 69113}, {69124, 69130}

X(69148) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 69115, 9466}, {69122, 69132, 31239}

X(69149) = (1, 0, 0, 0, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(14)

Barycentrics    a^4 + 4*b^2*c^2 - 2*Sqrt[3]*a^2*S : :

X(69149) lies on these lines: {6, 17130}, {14, 7821}, {61, 9466}, {299, 7843}, {395, 7863}, {398, 7794}, {2482, 16773}, {3933, 5471}, {5339, 7818}, {7801, 40694}, {7810, 42147}, {7845, 69106}, {7852, 69110}, {7853, 69109}, {7854, 10654}, {7873, 16964}, {7874, 69118}, {7888, 42153}, {9113, 69130}, {31173, 42814}, {42991, 69115}, {69120, 69132}, {69124, 69128}

X(69149) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 69117, 7821}, {16964, 69114, 7873}, {69109, 69113, 7853}

X(69150) = (1, 0, 0, 0, 0, 0, 1)-ADDITIVE ASSOCIATE OF X(14)

Barycentrics    a^2*(a^2 - 2*Sqrt[3]*S) : :

X(69150) lies on these lines: {3, 6}, {4, 54860}, {5, 5471}, {13, 39590}, {14, 39565}, {17, 7603}, {18, 62198}, {51, 34395}, {115, 398}, {232, 64469}, {298, 7895}, {299, 7882}, {395, 7749}, {396, 1506}, {397, 7747}, {533, 53463}, {630, 23303}, {636, 53464}, {1015, 2307}, {2241, 54403}, {2242, 54402}, {2482, 59542}, {2548, 37640}, {3181, 7905}, {3199, 8740}, {3457, 44107}, {3458, 44109}, {3767, 42999}, {3815, 61320}, {5254, 31709}, {5340, 62203}, {5346, 61317}, {5472, 7745}, {5475, 40693}, {6115, 12830}, {6781, 42148}, {7737, 42998}, {7746, 40694}, {7748, 10654}, {7753, 31694}, {7756, 42147}, {7769, 62983}, {7805, 33483}, {7858, 62984}, {8603, 34322}, {9117, 37341}, {9697, 11137}, {11303, 41746}, {11543, 61516}, {12829, 22691}, {13366, 21462}, {13881, 42975}, {14537, 61719}, {16807, 57385}, {16964, 69112}, {18424, 42159}, {31404, 63032}, {31406, 42633}, {31455, 42152}, {33527, 40695}, {37333, 51203}, {37637, 42989}, {37785, 62600}, {39563, 41108}, {40350, 54362}, {42154, 65633}, {42166, 43457}, {42564, 49236}, {42565, 49237}, {42925, 63548}, {42974, 65630}, {42991, 69111}, {45212, 64608}, {49210, 51853}, {49211, 51855}, {51754, 53430}, {69107, 69117}, {69109, 69115}, {69120, 69130}, {69122, 69128}, {69124, 69126}

X(69150) = isogonal conjugate of the isotomic conjugate of X(23303)
X(69150) = X(16807)-Ceva conjugate of X(512)
X(69150) = X(630)-Dao conjugate of X(76)
X(69150) = crosspoint of X(6) and X(21462)
X(69150) = crosssum of X(2) and X(303)
X(69150) = crossdifference of every pair of points on line {523, 44362}
X(69150) = barycentric product X(i)*X(j) for these {i,j}: {6, 23303}, {62, 8019}, {630, 21462}, {11138, 34328}
X(69150) = barycentric quotient X(i)/X(j) for these {i,j}: {8019, 34390}, {23303, 76}
X(69150) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 61, 39}, {6, 13330, 44512}, {14, 69119, 39565}, {17, 9113, 69132}, {17, 69132, 7603}, {61, 3398, 10613}, {62, 35229, 21402}, {371, 372, 9735}, {36836, 63198, 15515}, {42152, 61331, 31455}, {42991, 69111, 69113}

X(69151) = X(2)X(313)∩X(48)X(141)

Barycentrics    a^5 + b^5 + b^3*c^2 + b^2*c^3 + c^5 : :

X(69151) lies on these lines: {2, 313}, {48, 141}, {1631, 2887}, {2178, 30811}, {3771, 20990}, {4112, 21235}, {16578, 17073}, {17265, 25524}, {17356, 19846}, {21249, 37586}, {26222, 69005}, {30746, 44412}

X(69152) = (1,-1,1,1,-1)-ADDITIVE ASSOCIATE OF X(69151)

Barycentrics    a^5 - b^5 + b^3*c^2 + b^2*c^3 - c^5 : :
X(69152) = 5 X[31265] - 3 X[35267]

X(69152) lies on these lines: {4, 15320}, {19, 21045}, {48, 44412}, {66, 11391}, {313, 21275}, {674, 21270}, {1631, 20305}, {1836, 53421}, {4112, 21235}, {4497, 26012}, {12586, 21279}, {21301, 55035}, {21931, 61725}, {31265, 35267}

X(69152) = reflection of X(i) in X(j) for these {i,j}: {48, 44412}, {1631, 20305}
X(69152) = crosspoint of X(4) and X(7357)
X(69152) = crosssum of X(3) and X(1631)
X(69152) = barycentric product X(92)*X(20279)
X(69152) = barycentric quotient X(20279)/X(63)

X(69153) = (1,1,-1,-1,1)-ADDITIVE ASSOCIATE OF X(69151)

Barycentrics    a^5 + b^5 - b^3*c^2 - b^2*c^3 + c^5 : :

X(69153) lies on these lines: {2, 313}, {6, 20305}, {31, 44412}, {857, 5301}, {3771, 17267}, {21270, 30902}, {26176, 30882}

X(69153) = X(i)-complementary conjugate of X(j) for these (i,j): {1973, 32664}, {7087, 18589}, {7096, 1368}, {7213, 18639}, {40145, 3}
X(69153) = crosssum of X(6) and X(20739)

X(69154) = (1,-1,-1,-1,-1)-ADDITIVE ASSOCIATE OF X(69151)

Barycentrics    a^5 - b^5 - b^3*c^2 - b^2*c^3 - c^5 : :
X(69154) = 4 X[3589] - 5 X[31237], 5 X[3620] - X[20064], 5 X[3763] - 4 X[6679], 3 X[21356] - X[42058]

X(69154) lies on these lines: {6, 2887}, {31, 141}, {69, 674}, {313, 21275}, {518, 4680}, {524, 31134}, {599, 752}, {734, 3094}, {744, 49509}, {758, 3416}, {1503, 30269}, {2273, 21249}, {3589, 31237}, {3620, 20064}, {3763, 6679}, {3936, 47373}, {4112, 33911}, {4769, 16732}, {5846, 49454}, {5847, 49683}, {9020, 32859}, {9022, 32852}, {9055, 37003}, {16792, 18134}, {16798, 25527}, {17137, 46158}, {21356, 42058}, {24315, 57032}, {26232, 69005}, {29181, 52847}

X(69154) = midpoint of X(69) and X(6327)
X(69154) = reflection of X(i) in X(j) for these {i,j}: {6, 2887}, {31, 141}

X(69155) = (1,0,0,1)-ADDITIVE ASSOCIATE OF X(15)

Barycentrics    a^2*(Sqrt[3]*a^2 - 2*S) : :
X(69155) = X[16] + 3 X[61], X[16] - 3 X[36760], 3 X[14136] - X[33517]

X(69155) lies on these lines: {2, 47069}, {3, 6}, {4, 54847}, {5, 31706}, {13, 14537}, {14, 62200}, {51, 3457}, {111, 39411}, {115, 5321}, {172, 5357}, {217, 21647}, {230, 5471}, {232, 10632}, {251, 2981}, {298, 7880}, {299, 63939}, {302, 58448}, {384, 69143}, {395, 533}, {396, 624}, {397, 44667}, {398, 7685}, {530, 43228}, {531, 53440}, {622, 37640}, {633, 11489}, {635, 7749}, {754, 69142}, {1015, 7051}, {1196, 54362}, {1495, 3458}, {1500, 10638}, {1506, 6694}, {1914, 5353}, {1989, 11582}, {2004, 58470}, {2005, 34534}, {2207, 11408}, {2241, 54435}, {2242, 54436}, {2381, 5994}, {2482, 59540}, {2548, 11488}, {2549, 42119}, {2715, 58921}, {3055, 43103}, {3170, 34986}, {3181, 7799}, {3199, 3490}, {3291, 37775}, {3441, 8740}, {3480, 15610}, {3767, 5334}, {3815, 42124}, {5026, 41640}, {5215, 9761}, {5254, 42117}, {5277, 5367}, {5306, 6108}, {5309, 10654}, {5318, 7747}, {5335, 7737}, {5339, 41037}, {5340, 36994}, {5472, 14136}, {5475, 18582}, {5978, 41746}, {6151, 11145}, {6781, 42088}, {6782, 51872}, {7603, 16966}, {7745, 11542}, {7746, 18581}, {7748, 42085}, {7751, 69149}, {7756, 42087}, {7765, 42147}, {7768, 34541}, {7805, 33466}, {7832, 34540}, {7836, 40900}, {7874, 69145}, {8015, 61370}, {8259, 33465}, {8584, 52021}, {9112, 42896}, {9113, 16961}, {9300, 42912}, {9698, 16772}, {10409, 14658}, {11086, 19627}, {11135, 58478}, {11475, 33843}, {11648, 42154}, {13881, 42125}, {14137, 44223}, {14585, 21648}, {15442, 18487}, {15484, 42817}, {16268, 62233}, {16529, 22850}, {16589, 54379}, {16644, 40335}, {16808, 39590}, {16809, 39565}, {16960, 69133}, {16967, 69132}, {18424, 42103}, {19106, 69112}, {19107, 69111}, {20429, 40693}, {21461, 34565}, {22511, 22856}, {22689, 41753}, {22861, 37825}, {22893, 33561}, {23715, 38932}, {30471, 66446}, {31455, 42092}, {32552, 41754}, {34394, 44109}, {36970, 39563}, {36995, 42998}, {37007, 43031}, {37637, 42129}, {37689, 42983}, {37835, 62232}, {37852, 61656}, {39593, 41101}, {39601, 42918}, {40694, 59404}, {42089, 61331}, {42093, 69141}, {42094, 62203}, {42096, 65633}, {42099, 69125}, {42110, 43457}, {42122, 63548}, {42126, 44518}, {42128, 65630}, {42130, 44526}, {42135, 63534}, {42136, 53419}, {42138, 53418}, {42139, 43620}, {42141, 43618}, {42157, 69126}, {42671, 44083}, {42999, 59398}, {43229, 45880}, {43448, 43466}, {43463, 62993}, {49947, 50858}, {51485, 63103}, {51546, 57413}, {51547, 57385}, {63079, 67071}

X(69155) = midpoint of X(61) and X(36760)
X(69155) = isogonal conjugate of X(40706)
X(69155) = circumcircle-inverse of X(19780)
X(69155) = Schoutte-circle-inverse of X(36756)
X(69155) = isogonal conjugate of the anticomplement of X(22848)
X(69155) = isogonal conjugate of the isotomic conjugate of X(395)
X(69155) = isogonal conjugate of the polar conjugate of X(462)
X(69155) = X(i)-Ceva conjugate of X(j) for these (i,j): {5994, 512}, {11081, 1495}, {11088, 51}
X(69155) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40706}, {63, 38427}, {75, 6151}, {304, 51447}, {662, 62632}, {1577, 10410}, {11118, 65569}, {11120, 65570}
X(69155) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 40706}, {206, 6151}, {619, 76}, {1084, 62632}, {3162, 38427}, {33527, 7799}
X(69155) = crosspoint of X(i) and X(j) for these (i,j): {6, 3458}, {395, 462}
X(69155) = crosssum of X(i) and X(j) for these (i,j): {2, 299}, {6, 34009}, {62551, 68172}
X(69155) = crossdifference of every pair of points on line {298, 523}
X(69155) = X(7880)-line conjugate of X(298)
X(69155) = barycentric product X(i)*X(j) for these {i,j}: {3, 462}, {6, 395}, {15, 61370}, {16, 8015}, {25, 52193}, {32, 41001}, {62, 36305}, {111, 9117}, {512, 35315}, {523, 35330}, {533, 3457}, {619, 3458}, {1976, 51387}, {1989, 19295}, {2381, 30459}, {3129, 3480}, {3130, 38932}, {3441, 15769}, {3490, 15778}, {5994, 35444}, {5995, 14447}, {6672, 21462}, {8740, 59210}, {9202, 13305}, {11060, 14921}, {11139, 52971}, {11142, 40668}, {23715, 36296}, {34395, 43086}, {35344, 50344}
X(69155) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40706}, {25, 38427}, {32, 6151}, {395, 76}, {462, 264}, {512, 62632}, {1576, 10410}, {1974, 51447}, {3130, 46757}, {3457, 11118}, {3458, 11120}, {8015, 301}, {9117, 3266}, {15778, 46756}, {19295, 7799}, {34395, 38404}, {35315, 670}, {35330, 99}, {36305, 34390}, {41001, 1502}, {52193, 305}, {61370, 300}
X(69155) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 36758, 10614}, {6, 15, 39}, {6, 3053, 11486}, {6, 13330, 44498}, {6, 19780, 62}, {6, 19781, 16}, {6, 41407, 187}, {6, 51207, 1570}, {15, 16, 36756}, {15, 1691, 187}, {15, 54297, 5092}, {16, 19781, 187}, {16, 41407, 19781}, {32, 69150, 69144}, {62, 39555, 13349}, {62, 41409, 19780}, {371, 372, 47068}, {395, 10616, 6672}, {396, 53441, 624}, {398, 53454, 7685}, {1379, 1380, 19780}, {1384, 59245, 19780}, {3458, 34395, 1495}, {5058, 5062, 69150}, {5994, 11088, 54473}, {10654, 61317, 5309}, {13350, 44511, 3106}, {19780, 39555, 187}, {19780, 41409, 35007}, {47069, 62876, 2}

X(69156) = (1,0,0,1)-ADDITIVE ASSOCIATE OF X(16)

Barycentrics    a^2*(Sqrt[3]*a^2 + 2*S) : :
X(69156) = X[15] + 3 X[62], X[15] - 3 X[36759], 3 X[14137] - X[33518]

X(69156) lies on these lines: {2, 47067}, {3, 6}, {4, 54848}, {5, 31705}, {13, 62199}, {14, 14537}, {51, 3458}, {111, 39410}, {115, 5318}, {172, 5353}, {217, 21648}, {230, 5472}, {232, 10633}, {251, 6151}, {298, 63939}, {299, 7880}, {303, 58448}, {384, 69149}, {395, 623}, {396, 532}, {397, 7684}, {398, 44666}, {530, 53428}, {531, 43229}, {621, 37641}, {634, 11488}, {636, 7749}, {754, 69148}, {1015, 19373}, {1196, 54363}, {1250, 1500}, {1495, 3457}, {1506, 6695}, {1914, 5357}, {1989, 11581}, {2004, 34533}, {2005, 58470}, {2207, 11409}, {2241, 54436}, {2242, 54435}, {2307, 9341}, {2380, 5995}, {2482, 59539}, {2548, 11489}, {2549, 42120}, {2715, 58920}, {2981, 11146}, {3055, 43102}, {3171, 34986}, {3180, 7799}, {3199, 3489}, {3291, 37776}, {3440, 8739}, {3479, 15609}, {3767, 5335}, {3815, 42121}, {5026, 41630}, {5215, 9763}, {5254, 42118}, {5277, 5362}, {5306, 6109}, {5309, 10653}, {5321, 7747}, {5334, 7737}, {5339, 36992}, {5340, 41036}, {5471, 14137}, {5475, 18581}, {5979, 41745}, {6781, 42087}, {6783, 51872}, {7603, 16967}, {7745, 11543}, {7746, 18582}, {7748, 42086}, {7751, 69143}, {7756, 42088}, {7765, 42148}, {7768, 34540}, {7805, 33467}, {7832, 34541}, {7836, 40901}, {7874, 69137}, {8014, 61371}, {8260, 33464}, {8584, 52022}, {9112, 16960}, {9113, 42897}, {9300, 42913}, {9698, 16773}, {10410, 14658}, {11081, 19627}, {11136, 58477}, {11476, 33843}, {11648, 42155}, {13881, 42128}, {14136, 52650}, {14585, 21647}, {15441, 18487}, {15484, 42818}, {16267, 62232}, {16530, 22894}, {16589, 54378}, {16645, 40334}, {16808, 39565}, {16809, 39590}, {16961, 69132}, {16966, 69133}, {18424, 42106}, {19106, 69110}, {19107, 69113}, {20428, 40694}, {21462, 34565}, {22510, 22900}, {22687, 41751}, {22847, 33560}, {22907, 37824}, {23714, 38931}, {30472, 66445}, {31455, 42089}, {32553, 41752}, {34395, 44109}, {36969, 39563}, {36993, 42999}, {37008, 43030}, {37637, 42132}, {37689, 42982}, {37832, 62233}, {37851, 61656}, {39593, 41100}, {39601, 42919}, {40693, 59403}, {42092, 61332}, {42093, 62203}, {42094, 69141}, {42097, 65633}, {42100, 69124}, {42107, 43457}, {42123, 63548}, {42125, 65630}, {42127, 44518}, {42131, 44526}, {42135, 53418}, {42137, 53419}, {42138, 63534}, {42140, 43618}, {42142, 43620}, {42158, 69127}, {42671, 44122}, {42998, 59397}, {43228, 45879}, {43448, 43465}, {43464, 62993}, {49948, 50855}, {51484, 63102}, {51546, 57384}, {51547, 57412}, {61719, 62200}, {63080, 67072}

X(69156) = midpoint of X(62) and X(36759)
X(69156) = isogonal conjugate of X(40707)
X(69156) = circumcircle-inverse of X(19781)
X(69156) = Schoutte-circle-inverse of X(36755)
X(69156) = isogonal conjugate of the anticomplement of X(22892)
X(69156) = isogonal conjugate of the isotomic conjugate of X(396)
X(69156) = isogonal conjugate of the polar conjugate of X(463)
X(69156) = X(i)-Ceva conjugate of X(j) for these (i,j): {5995, 512}, {11083, 51}, {11086, 1495}
X(69156) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40707}, {63, 38428}, {75, 2981}, {304, 51446}, {662, 62631}, {1577, 10409}, {11117, 65570}, {11119, 65569}
X(69156) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 40707}, {206, 2981}, {618, 76}, {1084, 62631}, {3162, 38428}, {33526, 7799}
X(69156) = crosspoint of X(i) and X(j) for these (i,j): {6, 3457}, {396, 463}
X(69156) = crosssum of X(i) and X(j) for these (i,j): {2, 298}, {6, 34008}, {62551, 68171}
X(69156) = crossdifference of every pair of points on line {299, 523}
X(69156) = X(7880)-line conjugate of X(299)
X(69156) = barycentric product X(i)*X(j) for these {i,j}: {3, 463}, {6, 396}, {15, 8014}, {16, 61371}, {25, 52194}, {32, 41000}, {61, 36304}, {111, 9115}, {512, 35314}, {523, 35329}, {532, 3458}, {618, 3457}, {1976, 51388}, {1989, 19294}, {2380, 30462}, {3129, 38931}, {3130, 3479}, {3440, 15768}, {3489, 15802}, {5994, 14446}, {5995, 35443}, {6671, 21461}, {8739, 59209}, {9203, 13304}, {11060, 14922}, {11138, 52972}, {11141, 40667}, {23714, 36297}, {34394, 43085}, {35343, 50344}
X(69156) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40707}, {25, 38428}, {32, 2981}, {396, 76}, {463, 264}, {512, 62631}, {1576, 10409}, {1974, 51446}, {3129, 46758}, {3457, 11119}, {3458, 11117}, {8014, 300}, {9115, 3266}, {15802, 46755}, {19294, 7799}, {34394, 38403}, {35314, 670}, {35329, 99}, {36304, 34389}, {41000, 1502}, {52194, 305}, {61371, 301}
X(69156) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 36757, 10613}, {6, 16, 39}, {6, 3053, 11485}, {6, 13330, 44497}, {6, 19780, 15}, {6, 19781, 61}, {6, 41406, 187}, {6, 51206, 1570}, {15, 16, 36755}, {15, 19780, 187}, {15, 41406, 19780}, {16, 1691, 187}, {16, 54298, 5092}, {32, 69144, 69150}, {61, 39554, 13350}, {61, 41408, 19781}, {371, 372, 47066}, {395, 53429, 623}, {396, 10617, 6671}, {397, 53465, 7684}, {1379, 1380, 19781}, {1384, 59244, 19781}, {3457, 34394, 1495}, {5058, 5062, 69144}, {5995, 11083, 54472}, {10653, 61318, 5309}, {13349, 44512, 3107}, {19781, 39554, 187}, {19781, 41408, 35007}, {47067, 62877, 2}

X(69157) = (1,1,1,1,0,1,1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 - 3*a^2*b^2 + 2*b^4 - 3*a^2*c^2 + 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69157) lies on these lines: {2, 69109}, {3, 51388}, {6, 3788}, {13, 1975}, {14, 7752}, {15, 315}, {17, 76}, {30, 59541}, {61, 325}, {62, 7763}, {69, 6671}, {99, 16965}, {147, 628}, {183, 69106}, {194, 69111}, {203, 69135}, {298, 7871}, {299, 618}, {302, 6694}, {316, 42157}, {396, 3933}, {397, 6390}, {491, 3365}, {492, 3364}, {538, 69119}, {619, 6337}, {625, 69113}, {634, 3523}, {1007, 40694}, {1506, 69146}, {3105, 5976}, {3314, 69122}, {3412, 7796}, {3734, 69133}, {3926, 40693}, {3934, 69117}, {5025, 69124}, {5152, 6778}, {5238, 7750}, {5352, 14907}, {5472, 7863}, {5873, 12042}, {6656, 63201}, {6683, 69128}, {7603, 69149}, {7751, 62198}, {7762, 41407}, {7767, 16772}, {7773, 16964}, {7775, 12154}, {7776, 22236}, {7781, 69112}, {7782, 30471}, {7784, 63199}, {7788, 16962}, {7795, 61332}, {7799, 61719}, {7801, 9763}, {7802, 42434}, {7805, 62200}, {7809, 41101}, {7814, 42991}, {7815, 69114}, {7836, 62984}, {7841, 36775}, {7844, 69127}, {7862, 69118}, {7869, 69129}, {7870, 37786}, {7881, 69108}, {7887, 69110}, {7895, 69115}, {7908, 69116}, {8359, 53452}, {9113, 63021}, {10654, 32816}, {11285, 69123}, {16267, 32833}, {16925, 41406}, {16963, 62601}, {25167, 52642}, {31859, 69125}, {32006, 42150}, {32450, 69126}, {32815, 42162}, {32818, 37640}, {32819, 42813}, {32820, 42992}, {32821, 69121}, {32827, 42160}, {32828, 33411}, {32829, 42149}, {32831, 42998}, {32832, 42488}, {32896, 49811}, {34540, 46710}, {37512, 69142}, {37647, 42489}, {37671, 41943}, {37688, 42936}, {37824, 52193}, {37832, 59635}, {41107, 59634}, {42036, 54116}, {42118, 59539}, {42598, 64093}, {42999, 63098}, {44512, 51397}, {48906, 51018}

X(69157) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {17, 69120, 76}, {61, 325, 69145}, {76, 303, 17}, {299, 62600, 1078}, {1078, 62600, 16241}, {62198, 69131, 7751}

X(69158) = (1,1,1,1,0,1,0)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 - 3*a^2*b^2 + 2*b^4 - 3*a^2*c^2 + 2*c^4 : :
X(69158) = X[6337] - 3 X[32837], X[32816] + 3 X[32837], X[2996] - 3 X[32984]

X(69158) lies on these lines: {2, 3933}, {3, 315}, {4, 6390}, {5, 1007}, {6, 3788}, {17, 69121}, {18, 69120}, {20, 32823}, {26, 9723}, {30, 6337}, {32, 9766}, {39, 7778}, {52, 51386}, {69, 140}, {75, 31493}, {76, 1656}, {83, 7870}, {99, 382}, {141, 31401}, {183, 3526}, {187, 7903}, {193, 32970}, {194, 7887}, {230, 7758}, {298, 62600}, {299, 62601}, {302, 42989}, {303, 42988}, {316, 1657}, {317, 3517}, {381, 1975}, {384, 15484}, {385, 33233}, {394, 64665}, {491, 3312}, {492, 3311}, {498, 69094}, {499, 69093}, {538, 7862}, {546, 32815}, {547, 32836}, {548, 64018}, {549, 3785}, {550, 32006}, {574, 7784}, {576, 51397}, {599, 5038}, {620, 3053}, {625, 7781}, {626, 5013}, {631, 7767}, {632, 32839}, {641, 19145}, {642, 19146}, {754, 5023}, {999, 69135}, {1003, 7785}, {1078, 5054}, {1235, 3266}, {1285, 33205}, {1351, 6393}, {1384, 7762}, {1482, 69038}, {1506, 7801}, {1909, 31479}, {2023, 6309}, {2548, 7789}, {2549, 59546}, {2782, 8781}, {2996, 32984}, {3090, 32830}, {3091, 32817}, {3095, 37071}, {3146, 32881}, {3314, 11285}, {3329, 7945}, {3525, 15589}, {3530, 14929}, {3533, 32871}, {3534, 7782}, {3546, 41005}, {3552, 7941}, {3618, 33185}, {3620, 32978}, {3627, 32827}, {3628, 32828}, {3763, 6683}, {3767, 22253}, {3815, 7795}, {3830, 59634}, {3832, 32822}, {3843, 32819}, {3845, 32826}, {3850, 32824}, {3851, 11185}, {3853, 32891}, {3934, 7908}, {3964, 6642}, {4045, 22332}, {4121, 10601}, {5020, 34254}, {5024, 6656}, {5025, 31859}, {5055, 32833}, {5056, 32840}, {5067, 32834}, {5070, 32832}, {5093, 20576}, {5206, 7845}, {5210, 63935}, {5254, 11318}, {5286, 8361}, {5304, 33189}, {5475, 7863}, {5864, 51388}, {5865, 51387}, {5866, 11250}, {5976, 48673}, {6033, 62348}, {6144, 58448}, {6243, 51439}, {6392, 32969}, {6421, 45473}, {6422, 45472}, {7395, 22241}, {7483, 45962}, {7499, 40123}, {7525, 44180}, {7539, 8024}, {7603, 17130}, {7664, 62981}, {7715, 63155}, {7736, 7819}, {7737, 59545}, {7738, 33184}, {7745, 68527}, {7746, 7813}, {7748, 8716}, {7749, 7855}, {7751, 37637}, {7757, 7851}, {7760, 7940}, {7761, 15815}, {7766, 33245}, {7768, 15720}, {7770, 7777}, {7771, 7917}, {7772, 7874}, {7774, 7807}, {7775, 7816}, {7779, 7907}, {7780, 7916}, {7783, 7841}, {7786, 7868}, {7787, 33220}, {7790, 33241}, {7793, 7840}, {7794, 15271}, {7797, 33218}, {7798, 7886}, {7802, 15696}, {7806, 13571}, {7808, 7880}, {7811, 15693}, {7812, 68718}, {7818, 37512}, {7822, 9698}, {7823, 33235}, {7824, 7879}, {7825, 44526}, {7830, 53095}, {7832, 11174}, {7835, 7858}, {7842, 44519}, {7844, 32450}, {7849, 15482}, {7850, 43459}, {7853, 53096}, {7857, 7905}, {7860, 62100}, {7864, 33219}, {7873, 15515}, {7876, 31470}, {7885, 33234}, {7890, 31274}, {7892, 63018}, {7893, 33259}, {7900, 13586}, {7914, 44562}, {7915, 47355}, {7921, 33225}, {7929, 33273}, {7930, 55085}, {7935, 31652}, {7939, 33004}, {8362, 31400}, {8365, 59373}, {8366, 10583}, {8367, 63025}, {8369, 9770}, {8573, 42406}, {9732, 51401}, {9733, 51395}, {9855, 45017}, {9917, 11360}, {9974, 35685}, {9975, 35684}, {10008, 34380}, {10104, 11898}, {10109, 32896}, {10303, 10513}, {10352, 38905}, {10653, 59542}, {10654, 59541}, {11057, 14093}, {11064, 44141}, {11477, 20399}, {12040, 33215}, {12042, 54103}, {12106, 52437}, {12215, 18440}, {12812, 32875}, {12962, 42009}, {12969, 42060}, {13925, 32806}, {13993, 32805}, {14001, 62988}, {14023, 50771}, {14037, 53489}, {14043, 62994}, {14062, 20094}, {14064, 15048}, {14067, 63020}, {14069, 37665}, {14482, 32953}, {14535, 16898}, {14712, 68516}, {14869, 32887}, {14930, 33183}, {15022, 32879}, {15031, 61946}, {15533, 34506}, {15561, 32458}, {15694, 37671}, {15699, 46951}, {15712, 32889}, {15717, 32895}, {15905, 28697}, {16041, 51123}, {16051, 59766}, {16408, 37664}, {16419, 45201}, {17005, 31276}, {17128, 44543}, {17169, 30834}, {17181, 32851}, {17211, 17595}, {18281, 62338}, {18377, 65518}, {18907, 32973}, {19547, 68653}, {20065, 35297}, {20081, 32967}, {20088, 33246}, {20181, 31488}, {20208, 28407}, {20478, 68656}, {20479, 68658}, {21843, 63928}, {22236, 69145}, {22238, 69137}, {23335, 40697}, {25581, 69134}, {25583, 30741}, {26341, 45508}, {26348, 45509}, {26558, 31468}, {26590, 31461}, {27162, 56780}, {30270, 64711}, {30771, 41009}, {30786, 31856}, {31173, 65633}, {31245, 32092}, {31404, 66415}, {31497, 69095}, {31670, 59548}, {32456, 63931}, {32459, 68528}, {32838, 55856}, {32867, 48154}, {32868, 61894}, {32869, 61899}, {32870, 60781}, {32872, 46935}, {32874, 61895}, {32877, 61907}, {32883, 55861}, {32884, 55859}, {32885, 61885}, {32892, 61898}, {32893, 61889}, {32897, 61881}, {32898, 61867}, {32959, 37689}, {32961, 47286}, {32968, 63077}, {32977, 37667}, {32992, 63083}, {33000, 63046}, {33015, 63044}, {33186, 63633}, {33203, 63091}, {33216, 63940}, {33228, 51122}, {33274, 41136}, {34505, 39565}, {34780, 57275}, {35287, 63945}, {36212, 52251}, {37474, 51417}, {37688, 46219}, {37814, 68654}, {38335, 48913}, {39142, 46453}, {39560, 39603}, {40107, 44508}, {41624, 43136}, {42085, 59539}, {42086, 59540}, {43620, 63923}, {44456, 51374}, {46236, 51872}, {46264, 59552}, {47090, 68355}, {48657, 53765}, {48875, 51370}, {50571, 63942}, {51396, 53097}, {51438, 55724}, {52695, 66395}, {53127, 61905}, {59211, 60524}, {62197, 69131}, {62198, 69130}, {63119, 66344}

X(69158) = midpoint of X(6337) and X(32816)
X(69158) = reflection of X(13881) in X(7862)
X(69158) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7906, 7754}, {2, 7947, 7881}, {2, 32818, 3933}, {3, 325, 7776}, {4, 32831, 6390}, {6, 3788, 32954}, {39, 7778, 7866}, {39, 7888, 7778}, {69, 32829, 140}, {99, 7773, 382}, {99, 7814, 7773}, {183, 7769, 3526}, {187, 7903, 63932}, {194, 7925, 7887}, {325, 7763, 3}, {574, 7821, 7784}, {620, 7759, 3053}, {625, 7781, 44518}, {626, 5013, 11287}, {631, 37668, 7767}, {1007, 3926, 5}, {1078, 7871, 7788}, {1506, 7801, 69139}, {1975, 7752, 381}, {2548, 7789, 11286}, {3090, 32830, 64093}, {3091, 32841, 32817}, {3329, 7945, 33217}, {3788, 7764, 6}, {5056, 32840, 52713}, {5206, 7845, 63938}, {5286, 37690, 8361}, {6392, 32969, 43291}, {6683, 7869, 3763}, {7736, 53033, 7819}, {7737, 59545, 68513}, {7746, 7813, 63933}, {7749, 7855, 8667}, {7752, 7799, 1975}, {7757, 7899, 7851}, {7762, 16925, 1384}, {7769, 7796, 183}, {7774, 7807, 30435}, {7775, 7816, 65630}, {7777, 7836, 7770}, {7780, 7916, 40341}, {7783, 7912, 7841}, {7785, 7891, 1003}, {7786, 7909, 7868}, {7794, 31455, 15271}, {7815, 7895, 599}, {7816, 65630, 11159}, {7824, 7897, 7879}, {7851, 7899, 33240}, {7857, 7905, 14614}, {7870, 11163, 33237}, {7917, 62362, 7771}, {10513, 32873, 10303}, {11184, 69139, 1506}, {22110, 34511, 11318}, {32816, 32837, 6337}, {32825, 32829, 69}, {32828, 34803, 3628}, {32831, 63098, 4}, {32832, 37647, 5070}, {32835, 37668, 631}, {32839, 34229, 632}, {40341, 44535, 7780}

X(69159) = (1,1,0,1,1,0,1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 - 3*a^2*b^2 - 3*a^2*c^2 - 4*b^2*c^2 - 2*Sqrt[3]*a^2*S : :

X(69159) lies on these lines: {2, 6}, {3, 22687}, {13, 7761}, {15, 3734}, {17, 626}, {61, 3934}, {62, 7815}, {203, 69136}, {315, 69133}, {383, 51161}, {538, 63201}, {576, 33478}, {620, 16241}, {622, 53431}, {623, 5613}, {625, 37832}, {1506, 69145}, {3131, 8266}, {3412, 69109}, {3642, 12188}, {3643, 9301}, {3788, 69108}, {3972, 19781}, {4045, 69111}, {5026, 5981}, {5238, 7816}, {5472, 7810}, {5865, 32521}, {6115, 51018}, {6656, 69119}, {7603, 69148}, {7763, 69116}, {7764, 69107}, {7784, 42156}, {7789, 16772}, {7791, 69112}, {7795, 42152}, {7800, 40693}, {7804, 41407}, {7830, 16965}, {7836, 62600}, {7842, 42813}, {7854, 69137}, {7862, 42488}, {7865, 16267}, {7880, 41943}, {10645, 32456}, {11303, 53430}, {11305, 46054}, {11477, 44460}, {11480, 35917}, {15482, 63200}, {15822, 54363}, {21462, 34816}, {22236, 69139}, {22253, 41630}, {25183, 63199}, {31239, 69150}, {32224, 32460}, {32832, 69118}, {32992, 69132}, {33015, 62601}, {33878, 36364}, {36345, 55582}, {36384, 44456}, {36766, 51388}, {37348, 41038}, {37512, 69143}, {40344, 41107}, {42675, 59244}, {42945, 59545}, {59635, 69113}, {63924, 69124}

X(69159) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {17, 69122, 626}, {61, 3934, 69146}, {299, 396, 9763}, {62198, 69129, 2}, {62984, 63044, 299}

X(69160) = (1,1,0,1,0,0,1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^2*(a^2 - 3*b^2 - 3*c^2 - 2*Sqrt[3]*S) : :

X(69160) lies on these lines: {2, 53463}, {3, 6}, {4, 53431}, {5, 69113}, {13, 7748}, {14, 1506}, {17, 115}, {18, 31455}, {69, 69131}, {140, 62197}, {141, 69117}, {203, 1500}, {230, 16772}, {298, 7906}, {299, 2896}, {303, 5025}, {396, 617}, {397, 16941}, {398, 3815}, {630, 43029}, {1015, 7005}, {1975, 69138}, {2023, 5981}, {2276, 2307}, {2548, 10654}, {2549, 40693}, {2981, 3124}, {3055, 42599}, {3071, 41018}, {3131, 34394}, {3201, 9697}, {3205, 9696}, {3411, 31457}, {3412, 7765}, {3439, 8603}, {3642, 6300}, {3767, 42152}, {3926, 69116}, {3933, 69115}, {3981, 52349}, {5305, 42912}, {5309, 16962}, {5334, 31404}, {5339, 41040}, {5340, 22513}, {5464, 6034}, {5471, 9698}, {5472, 7756}, {5475, 16964}, {5859, 7811}, {5869, 9873}, {5873, 32151}, {6115, 37824}, {6292, 69109}, {6655, 53428}, {6772, 34509}, {6781, 42434}, {7006, 31451}, {7603, 69147}, {7737, 42150}, {7738, 37640}, {7745, 42147}, {7746, 69110}, {7747, 42157}, {7749, 16241}, {7753, 41101}, {7761, 69137}, {7764, 69145}, {7789, 59541}, {7794, 69120}, {7795, 69129}, {7797, 53440}, {7800, 69114}, {7801, 36775}, {7813, 69107}, {7833, 37786}, {7841, 9763}, {7854, 69106}, {7881, 11129}, {7891, 30471}, {8259, 53453}, {8362, 69128}, {9112, 42158}, {9225, 44719}, {9603, 11137}, {9981, 18503}, {10329, 34395}, {11296, 49947}, {11309, 43028}, {11542, 22907}, {11648, 16267}, {12155, 34504}, {14169, 20976}, {14567, 34009}, {15048, 69126}, {16626, 33518}, {16629, 23005}, {20977, 38432}, {22113, 53458}, {22691, 44531}, {22715, 44530}, {31239, 69149}, {31400, 42999}, {31401, 40694}, {31415, 42159}, {31448, 54403}, {31467, 42975}, {31477, 54438}, {31489, 42153}, {34321, 51890}, {34534, 39951}, {34540, 44029}, {35731, 49209}, {35932, 43228}, {36251, 42156}, {36969, 65633}, {36970, 39590}, {37637, 43238}, {37832, 39565}, {39563, 41121}, {39575, 56514}, {39593, 42976}, {41021, 64092}, {41034, 53505}, {42154, 65630}, {42155, 44519}, {42161, 43619}, {42164, 53418}, {42166, 53419}, {42432, 62203}, {42490, 44535}, {42598, 63534}, {42992, 69140}, {44361, 46710}, {52193, 53475}, {52643, 59384}, {52688, 53442}

X(69160) = isogonal conjugate of the isotomic conjugate of X(34540)
X(69160) = X(30252)-Ceva conjugate of X(512)
X(69160) = X(i)-isoconjugate of X(j) for these (i,j): {75, 34533}, {18813, 65571}
X(69160) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 34533}, {44029, 76}
X(69160) = crosspoint of X(40157) and X(53032)
X(69160) = barycentric product X(i)*X(j) for these {i,j}: {6, 34540}, {21461, 44776}
X(69160) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 34533}, {21461, 18813}, {34540, 76}
X(69160) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 61332, 69133}, {6, 15, 19781}, {6, 11480, 19780}, {6, 15815, 22238}, {6, 36836, 3053}, {15, 3098, 11480}, {15, 3105, 3}, {17, 69124, 115}, {39, 61, 6}, {61, 63201, 39}, {396, 5254, 69119}, {398, 3815, 69132}, {1504, 3365, 6}, {1505, 3364, 6}, {2549, 40693, 69112}, {3389, 3390, 39554}, {3767, 42152, 62198}, {5028, 36757, 6}, {5352, 41406, 5206}, {5472, 7756, 16965}, {14630, 14631, 69144}, {31400, 42999, 61331}, {37512, 69144, 16}, {62198, 69127, 3767}, {69120, 69122, 7794}

X(69161) = (1,0,1,0,1,1,1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 + 2*b^4 - 4*b^2*c^2 + 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69161) lies on these lines: {2, 69143}, {4, 54847}, {5, 14137}, {6, 3851}, {13, 5007}, {17, 39}, {32, 5340}, {61, 13102}, {115, 398}, {187, 42158}, {230, 42924}, {299, 63925}, {303, 32450}, {396, 7765}, {397, 7684}, {574, 43238}, {2548, 42494}, {3767, 42998}, {5025, 69148}, {5041, 69133}, {5237, 62232}, {5305, 5472}, {5309, 37824}, {5350, 7747}, {5355, 43773}, {5366, 7737}, {5471, 63534}, {5475, 42921}, {6034, 44511}, {6670, 8260}, {6694, 53428}, {7746, 42149}, {7748, 42150}, {7749, 42944}, {7751, 69142}, {7753, 42166}, {7772, 42156}, {7852, 69138}, {9698, 42598}, {11648, 22236}, {12815, 23303}, {13881, 42989}, {14537, 42813}, {14711, 69117}, {15515, 42773}, {16267, 39593}, {16644, 53096}, {16964, 39563}, {16965, 35007}, {18362, 42153}, {19780, 41974}, {19781, 42431}, {20252, 47855}, {31274, 59542}, {36251, 61537}, {37512, 62198}, {39601, 69132}, {42162, 61317}, {42990, 62199}, {63924, 69149}

X(69161) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {17, 69111, 69126}, {17, 69126, 39}, {115, 69150, 69147}, {16965, 62200, 35007}, {62198, 69125, 37512}, {69111, 69119, 39}, {69119, 69126, 17}

X(69162) = (1,0,1,0,1,1,0)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 + 2*b^4 - 4*b^2*c^2 + 2*c^4 : :
X(69162) = 3 X[7887] - X[32821], 3 X[7888] - 2 X[32821]

X(69162) lies on these lines: {2, 7765}, {3, 11648}, {4, 32}, {5, 5309}, {6, 3851}, {17, 69127}, {18, 69126}, {39, 1656}, {61, 16628}, {62, 16629}, {76, 7844}, {140, 574}, {148, 7857}, {183, 7861}, {187, 1657}, {194, 7862}, {230, 550}, {231, 577}, {315, 33290}, {381, 5007}, {382, 35007}, {384, 18546}, {385, 7825}, {403, 41366}, {468, 34481}, {538, 7887}, {543, 16925}, {546, 5306}, {547, 9606}, {576, 6034}, {599, 33241}, {625, 7754}, {626, 17131}, {632, 31457}, {671, 3552}, {754, 14063}, {1003, 63922}, {1078, 7872}, {1180, 7570}, {1184, 62977}, {1196, 5094}, {1384, 62023}, {1504, 8960}, {1505, 58866}, {1506, 5056}, {1692, 64080}, {1975, 7886}, {1990, 44960}, {2070, 44537}, {2165, 5065}, {2241, 4857}, {2242, 5270}, {2482, 32970}, {2548, 5068}, {2549, 3523}, {3053, 5073}, {3087, 40136}, {3090, 7739}, {3091, 5319}, {3515, 44527}, {3522, 7756}, {3526, 31652}, {3533, 7738}, {3544, 31417}, {3627, 63543}, {3628, 9607}, {3734, 7828}, {3788, 32820}, {3815, 35018}, {3830, 22331}, {3843, 14537}, {3850, 5305}, {3854, 5368}, {3855, 63006}, {3858, 5346}, {3934, 7851}, {4045, 32832}, {5008, 61975}, {5013, 46219}, {5023, 62131}, {5024, 55860}, {5025, 7751}, {5028, 34507}, {5034, 25555}, {5041, 39601}, {5055, 39593}, {5059, 6781}, {5070, 22332}, {5072, 41940}, {5158, 10024}, {5210, 62107}, {5237, 62233}, {5238, 62232}, {5339, 41037}, {5340, 41036}, {5461, 7764}, {6102, 61675}, {6179, 14041}, {6292, 32828}, {6337, 31274}, {6392, 7813}, {6658, 41135}, {6680, 11185}, {6722, 7763}, {6794, 10413}, {7603, 9605}, {7615, 32971}, {7617, 7827}, {7620, 33201}, {7622, 16923}, {7736, 61921}, {7752, 7798}, {7757, 32967}, {7758, 32972}, {7759, 33228}, {7760, 7775}, {7770, 7817}, {7773, 7805}, {7780, 7841}, {7787, 15031}, {7788, 63925}, {7790, 7815}, {7794, 14064}, {7796, 19570}, {7797, 7808}, {7801, 8361}, {7802, 63047}, {7806, 14034}, {7810, 32974}, {7812, 32993}, {7821, 11318}, {7822, 64093}, {7829, 16924}, {7830, 17008}, {7834, 59635}, {7843, 14614}, {7847, 17004}, {7849, 33219}, {7852, 69139}, {7854, 33184}, {7856, 16044}, {7864, 15482}, {7865, 7933}, {7866, 9466}, {7873, 8667}, {7878, 33013}, {7880, 33218}, {7884, 43527}, {7885, 33293}, {7890, 32816}, {7896, 7934}, {7899, 7908}, {7912, 7916}, {7914, 7919}, {7922, 14046}, {7942, 17128}, {8182, 33247}, {8357, 13468}, {8550, 39764}, {8589, 44535}, {8859, 33256}, {9167, 32959}, {9220, 33872}, {9341, 12943}, {9593, 30315}, {9664, 63273}, {9737, 38224}, {10299, 62992}, {10415, 51819}, {10611, 16626}, {10612, 16627}, {12150, 33018}, {13342, 36412}, {13345, 18353}, {13711, 62241}, {13834, 62242}, {13854, 61303}, {14023, 16041}, {14581, 37197}, {14711, 33240}, {14971, 32969}, {15048, 31455}, {15513, 44526}, {15544, 34783}, {15712, 63548}, {15720, 37512}, {15815, 61832}, {15820, 40126}, {16964, 62200}, {16965, 62199}, {17734, 41326}, {17737, 56880}, {18325, 40320}, {18907, 61976}, {19687, 63957}, {19695, 47101}, {19780, 42431}, {19781, 42432}, {21735, 21843}, {22329, 33229}, {22510, 47068}, {22511, 47066}, {22847, 35688}, {22893, 35689}, {23055, 33226}, {23251, 62220}, {23261, 62219}, {26456, 43411}, {26463, 43412}, {26613, 33268}, {30435, 39590}, {31173, 63932}, {31400, 46935}, {31401, 61886}, {31406, 61907}, {31407, 61924}, {31412, 61328}, {31465, 42602}, {31470, 61887}, {31478, 68855}, {31483, 43879}, {31492, 55857}, {32833, 33248}, {32954, 34505}, {32965, 34506}, {33002, 55085}, {33007, 36523}, {33212, 59780}, {33232, 42850}, {33239, 63107}, {33412, 37641}, {33413, 37640}, {34288, 61314}, {34809, 55571}, {37689, 49135}, {39646, 67872}, {40326, 47629}, {40350, 62981}, {41154, 52942}, {42159, 61317}, {42162, 61318}, {42561, 61329}, {42978, 63200}, {42979, 63201}, {43618, 50690}, {43619, 62127}, {44519, 62082}, {44533, 45735}, {44534, 62356}, {44904, 63633}, {46951, 60202}, {47298, 52293}, {48884, 59232}, {50691, 63536}, {52292, 62702}, {53095, 61815}, {53419, 62036}, {55738, 66324}, {56548, 56820}, {62197, 69125}, {62198, 69124}, {62322, 63099}, {62968, 63541}, {69110, 69119}, {69111, 69118}

X(69162) = reflection of X(7888) in X(7887)
X(69162) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7765, 53096}, {2, 63924, 17130}, {4, 3767, 7755}, {4, 7755, 32}, {5, 5309, 7772}, {32, 115, 69141}, {32, 69141, 62203}, {76, 7844, 7867}, {76, 7901, 7869}, {115, 3767, 32}, {115, 7755, 4}, {183, 7861, 7935}, {187, 44518, 65633}, {194, 14061, 7862}, {230, 7748, 5206}, {385, 7860, 63937}, {385, 14045, 7860}, {625, 7754, 7903}, {2549, 7749, 15515}, {3090, 7739, 9698}, {3091, 5319, 7753}, {3544, 63024, 31417}, {3934, 7851, 7913}, {5025, 7751, 7818}, {5025, 14568, 7751}, {5254, 7746, 574}, {5254, 43291, 7746}, {5286, 43620, 1506}, {5305, 63534, 5475}, {5346, 18424, 7745}, {5461, 7764, 32961}, {6179, 14041, 63931}, {7735, 7747, 32}, {7760, 9166, 32966}, {7760, 32966, 7775}, {7772, 18362, 5}, {7825, 63937, 7860}, {7844, 7869, 7901}, {7860, 14045, 7825}, {7869, 7901, 7867}, {7886, 32457, 1975}, {7899, 20081, 7908}, {7919, 31276, 7914}, {7934, 17129, 7896}, {8361, 63923, 7801}, {11318, 63933, 7821}, {14064, 63955, 7794}, {22329, 33229, 63935}, {35007, 39563, 382}

X(69163) = (1,0,1,0,0,1,1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 + 2*b^4 + 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69163) lies on these lines: {2, 69144}, {6, 7821}, {17, 31239}, {18, 31275}, {32, 69142}, {61, 7853}, {62, 7874}, {115, 69149}, {299, 7805}, {303, 6683}, {396, 6292}, {397, 7820}, {626, 69148}, {3734, 69140}, {3763, 42988}, {5007, 69137}, {5025, 69147}, {5472, 7819}, {7795, 69143}, {7822, 40693}, {7859, 62984}, {7935, 22236}, {9466, 69109}, {16773, 31274}, {39565, 69146}, {40335, 43150}, {62198, 69123}, {69111, 69117}, {69120, 69126}

X(69163) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {17, 69128, 31239}, {626, 69150, 69148}, {69109, 69119, 9466}

X(69164) = (1,0,0,0,1,0,1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    a^4 - 4*b^2*c^2 - 2*Sqrt[3]*a^2*S : :

X(69164) lies on these lines: {6, 9466}, {13, 7845}, {17, 69130}, {32, 69143}, {69, 69142}, {76, 69149}, {115, 69148}, {298, 625}, {315, 69140}, {396, 7813}, {397, 7826}, {524, 5472}, {5007, 69138}, {5026, 25183}, {5471, 64093}, {7751, 69144}, {7821, 69107}, {7852, 69108}, {7853, 69111}, {7855, 40693}, {7873, 69112}, {7874, 69116}, {7903, 42156}, {11542, 50771}, {20425, 36765}, {25167, 44497}, {39565, 69145}, {40341, 42974}, {62198, 69121}, {63924, 69147}, {69122, 69126}

X(69164) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 69150, 69149}, {69107, 69119, 7821}, {69111, 69115, 7853}

X(69165) = (1,1,1,1,0,1,1)-ADDITIVE ASSOCIATE OF X(18)

Barycentrics    a^4 - 3*a^2*b^2 + 2*b^4 - 3*a^2*c^2 + 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69165) lies on these lines: {2, 69108}, {3, 51387}, {6, 3788}, {13, 7752}, {14, 1975}, {16, 315}, {18, 76}, {30, 59542}, {61, 7763}, {62, 325}, {69, 6672}, {99, 16964}, {147, 627}, {183, 69107}, {194, 69110}, {202, 69135}, {298, 619}, {299, 7871}, {303, 6695}, {316, 42158}, {395, 3933}, {398, 6390}, {491, 3390}, {492, 3389}, {538, 69118}, {618, 6337}, {625, 69112}, {633, 3523}, {1007, 40693}, {1506, 69138}, {3104, 5976}, {3314, 69123}, {3411, 7796}, {3734, 69132}, {3926, 40694}, {3934, 69116}, {5025, 69125}, {5152, 6777}, {5237, 7750}, {5351, 14907}, {5471, 7863}, {5872, 12042}, {6656, 63200}, {6683, 69129}, {7603, 69143}, {7751, 62197}, {7762, 41406}, {7767, 16773}, {7773, 16965}, {7775, 12155}, {7776, 22238}, {7781, 69113}, {7782, 30472}, {7784, 63198}, {7788, 16963}, {7795, 61331}, {7801, 9761}, {7802, 42433}, {7805, 62199}, {7809, 41100}, {7814, 42990}, {7815, 69115}, {7836, 62983}, {7844, 69126}, {7862, 69119}, {7869, 69128}, {7870, 37785}, {7881, 69109}, {7887, 69111}, {7895, 69114}, {7908, 69117}, {8359, 53463}, {9112, 63021}, {10653, 32816}, {11285, 69122}, {16268, 32833}, {16925, 41407}, {16962, 62600}, {25157, 52643}, {31859, 69124}, {32006, 42151}, {32450, 69127}, {32815, 42159}, {32818, 37641}, {32819, 42814}, {32820, 42993}, {32821, 69120}, {32827, 42161}, {32828, 33410}, {32829, 42152}, {32831, 42999}, {32832, 42489}, {32896, 49810}, {34541, 46711}, {37512, 69148}, {37647, 42488}, {37671, 41944}, {37688, 42937}, {37825, 52194}, {37835, 59635}, {41108, 59634}, {42035, 54115}, {42117, 59540}, {42599, 64093}, {42998, 63098}, {44511, 51397}, {48906, 51016}

X(69165) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {18, 69121, 76}, {62, 325, 69137}, {76, 302, 18}, {298, 62601, 1078}, {1078, 62601, 16242}, {62197, 69130, 7751}

X(69166) = (1,1,0,1,1,0,1)-ADDITIVE ASSOCIATE OF X(18)

Barycentrics    a^4 - 3*a^2*b^2 - 3*a^2*c^2 - 4*b^2*c^2 + 2*Sqrt[3]*a^2*S : :

X(69166) lies on these lines: {2, 6}, {3, 22689}, {14, 7761}, {16, 3734}, {18, 626}, {61, 7815}, {62, 3934}, {202, 69136}, {315, 69132}, {538, 63200}, {576, 33479}, {620, 16242}, {621, 53443}, {624, 5617}, {625, 37835}, {1080, 51162}, {1506, 69137}, {3132, 8266}, {3411, 69108}, {3642, 9301}, {3643, 12188}, {3788, 69109}, {3972, 19780}, {4045, 69110}, {5026, 5980}, {5237, 7816}, {5471, 7810}, {5864, 32521}, {6114, 51016}, {6656, 69118}, {7603, 69142}, {7763, 69117}, {7764, 69106}, {7784, 42153}, {7789, 16773}, {7791, 69113}, {7795, 42149}, {7800, 40694}, {7804, 41406}, {7830, 16964}, {7836, 62601}, {7842, 42814}, {7854, 69145}, {7862, 42489}, {7865, 16268}, {7880, 41944}, {10646, 32456}, {11304, 53442}, {11306, 46053}, {11477, 44464}, {11481, 35918}, {15482, 63201}, {15822, 54362}, {21461, 34816}, {22238, 69139}, {22253, 41640}, {25187, 63198}, {31239, 69144}, {32224, 32461}, {32832, 69119}, {32992, 69133}, {33015, 62600}, {33878, 36365}, {36347, 55582}, {36385, 44456}, {37348, 41039}, {37512, 69149}, {40344, 41108}, {42674, 59245}, {42944, 59545}, {51387, 60069}, {59635, 69112}, {63924, 69125}

X(69166) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {18, 69123, 626}, {62, 3934, 69138}, {298, 395, 9761}, {62197, 69128, 2}, {62983, 63044, 298}

X(69167) = (1,1,0,1,1,0,1)-ADDITIVE ASSOCIATE OF X(18)

Barycentrics    a^2*(a^2 - 3*b^2 - 3*c^2 + 2*Sqrt[3]*S) : :

X(69167) lies on these lines: {2, 53452}, {3, 6}, {4, 53443}, {5, 69112}, {13, 1506}, {14, 7748}, {17, 31455}, {18, 115}, {69, 69130}, {140, 62198}, {141, 69116}, {202, 1500}, {230, 16773}, {298, 2896}, {299, 7906}, {302, 5025}, {395, 616}, {397, 3815}, {398, 16940}, {629, 43028}, {1015, 7006}, {1975, 69146}, {2023, 5980}, {2275, 7127}, {2548, 10653}, {2549, 40694}, {3055, 42598}, {3124, 6151}, {3132, 34395}, {3200, 9697}, {3206, 9696}, {3411, 7765}, {3412, 31457}, {3438, 8604}, {3643, 6301}, {3767, 42149}, {3926, 69117}, {3933, 69114}, {3981, 52348}, {5305, 42913}, {5309, 16963}, {5335, 31404}, {5339, 22512}, {5340, 41041}, {5463, 6034}, {5471, 7756}, {5472, 9698}, {5475, 16965}, {5858, 7811}, {5868, 9873}, {5872, 32151}, {6114, 37825}, {6292, 69108}, {6655, 53440}, {6775, 34508}, {6781, 42433}, {7005, 31451}, {7603, 69140}, {7737, 42151}, {7738, 37641}, {7745, 42148}, {7746, 69111}, {7747, 42158}, {7749, 16242}, {7753, 41100}, {7761, 69145}, {7764, 69137}, {7789, 59542}, {7794, 69121}, {7795, 69128}, {7797, 53428}, {7800, 69115}, {7801, 69109}, {7813, 69106}, {7833, 37785}, {7841, 9761}, {7854, 69107}, {7881, 11128}, {7891, 30472}, {8260, 53464}, {8362, 69129}, {9113, 42157}, {9225, 44718}, {9603, 11134}, {9982, 18503}, {10329, 34394}, {11295, 49948}, {11310, 43029}, {11543, 22861}, {11648, 16268}, {12154, 34504}, {14170, 20976}, {14567, 34008}, {15048, 69127}, {16627, 33517}, {16628, 23004}, {20977, 38431}, {22114, 53469}, {22692, 44531}, {22714, 44530}, {31239, 69143}, {31400, 42998}, {31401, 40693}, {31415, 42162}, {31448, 54402}, {31467, 42974}, {31477, 54437}, {31489, 42156}, {34322, 51891}, {34533, 39951}, {34541, 44031}, {35931, 43229}, {36252, 42153}, {36969, 39590}, {36970, 65633}, {37637, 43239}, {37835, 39565}, {39563, 41122}, {39575, 56515}, {39593, 42977}, {40350, 59710}, {41020, 64092}, {41035, 53505}, {42154, 44519}, {42155, 65630}, {42160, 43619}, {42163, 53419}, {42165, 53418}, {42431, 62203}, {42491, 44535}, {42599, 63534}, {42993, 69147}, {44362, 46711}, {52194, 53475}, {52642, 59383}, {52689, 53430}

X(69167) = isogonal conjugate of the isotomic conjugate of X(34541)
X(69167) = X(30253)-Ceva conjugate of X(512)
X(69167) = X(i)-isoconjugate of X(j) for these (i,j): {75, 34534}, {18814, 65572}
X(69167) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 34534}, {44031, 76}
X(69167) = crosspoint of X(40156) and X(53031)
X(69167) = barycentric product X(i)*X(j) for these {i,j}: {6, 34541}, {21462, 44777}
X(69167) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 34534}, {21462, 18814}, {34541, 76}
X(69167) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 61331, 69132}, {6, 16, 19780}, {6, 11481, 19781}, {6, 15815, 22236}, {6, 36843, 3053}, {16, 3098, 11481}, {16, 3104, 3}, {18, 69125, 115}, {39, 62, 6}, {62, 63200, 39}, {395, 5254, 69118}, {397, 3815, 69133}, {1504, 3390, 6}, {1505, 3389, 6}, {2549, 40694, 69113}, {3364, 3365, 39555}, {3767, 42149, 62197}, {5028, 36758, 6}, {5351, 41407, 5206}, {5471, 7756, 16964}, {14630, 14631, 69150}, {31400, 42998, 61332}, {37512, 69150, 15}, {62197, 69126, 3767}, {69121, 69123, 7794}

X(69168) = (1,0,1,0,1,1,1)-ADDITIVE ASSOCIATE OF X(18)

Barycentrics    a^4 + 2*b^4 - 4*b^2*c^2 + 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69168) lies on these lines: {2, 69149}, {4, 54848}, {5, 14136}, {6, 3851}, {14, 5007}, {18, 39}, {32, 5339}, {62, 13103}, {115, 397}, {187, 42157}, {230, 42925}, {298, 63925}, {302, 32450}, {395, 7765}, {398, 7685}, {574, 43239}, {2548, 42495}, {3767, 42999}, {5025, 69142}, {5041, 69132}, {5238, 62233}, {5305, 5471}, {5309, 37825}, {5349, 7747}, {5355, 43774}, {5365, 7737}, {5472, 63534}, {5475, 42920}, {6034, 44512}, {6669, 8259}, {6695, 53440}, {7746, 42152}, {7748, 42151}, {7749, 42945}, {7751, 69148}, {7753, 42163}, {7772, 42153}, {7852, 69146}, {9698, 42599}, {11648, 22238}, {12815, 23302}, {13881, 42988}, {14537, 42814}, {14711, 69116}, {15515, 42774}, {16268, 39593}, {16645, 53096}, {16964, 35007}, {16965, 39563}, {18362, 42156}, {19780, 42432}, {19781, 41973}, {20253, 47856}, {31274, 59541}, {36252, 61538}, {37512, 62197}, {39601, 69133}, {42159, 61318}, {42991, 62200}, {63924, 69143}

X(69168) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {18, 69110, 69127}, {18, 69127, 39}, {115, 69144, 69140}, {16964, 62199, 35007}, {62197, 69124, 37512}, {69110, 69118, 39}, {69118, 69127, 18}

X(69169) = (1,0,1,0,0,1,1)-ADDITIVE ASSOCIATE OF X(18)

Barycentrics    a^4 + 2*b^4 + 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69169) lies on these lines: {2, 69150}, {6, 7821}, {17, 31275}, {18, 31239}, {32, 69148}, {61, 7874}, {62, 7853}, {115, 69143}, {298, 7805}, {302, 6683}, {395, 6292}, {398, 7820}, {626, 69142}, {3734, 69147}, {3763, 42989}, {5007, 69145}, {5025, 69140}, {5471, 7819}, {7795, 69149}, {7822, 40694}, {7859, 62983}, {7935, 22238}, {9466, 69108}, {16772, 31274}, {39565, 69138}, {40334, 43150}, {62197, 69122}, {69110, 69116}, {69121, 69127}

X(69169) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {18, 69129, 31239}, {626, 69144, 69142}, {69108, 69118, 9466}

X(69170) = (1,0,0,0,1,0,1)-ADDITIVE ASSOCIATE OF X(18)

Barycentrics    a^4 - 4*b^2*c^2 + 2*Sqrt[3]*a^2*S : :

X(69170) lies on these lines: {6, 9466}, {14, 7845}, {18, 69131}, {32, 69149}, {69, 69148}, {76, 69143}, {115, 69142}, {299, 625}, {315, 69147}, {395, 7813}, {398, 7826}, {524, 5471}, {5007, 69146}, {5026, 25187}, {5472, 64093}, {7751, 69150}, {7821, 69106}, {7852, 69109}, {7853, 69110}, {7855, 40694}, {7873, 69113}, {7874, 69117}, {7903, 42153}, {11543, 50771}, {25157, 44498}, {39565, 69137}, {40341, 42975}, {62197, 69120}, {63924, 69140}, {69123, 69127}

X(69170) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 69144, 69143}, {69106, 69118, 7821}, {69110, 69114, 7853}

X(69171) = (1,1,0,1,1,0)-ADDITIVE ASSOCIATE OF X(20)

Barycentrics    3*a^4 - 2*a^2*b^2 - 2*a^2*c^2 + 2*b^2*c^2 : :
X(69171) = X[7825] - 4 X[59545], X[3767] - 3 X[32985], 7 X[3767] - 9 X[63107], 7 X[32985] - 3 X[63107], X[3926] + 3 X[35927], X[7916] + 6 X[35927], 2 X[7886] - 3 X[11288], 3 X[11288] - X[44518]

X(69171) lies on these lines: {2, 7756}, {3, 3734}, {4, 620}, {5, 9734}, {6, 68513}, {20, 626}, {22, 30749}, {30, 3788}, {32, 99}, {39, 1003}, {76, 5206}, {83, 45017}, {98, 61126}, {115, 16925}, {141, 548}, {148, 7857}, {182, 32516}, {183, 15513}, {187, 1975}, {211, 58212}, {315, 6781}, {316, 7888}, {325, 33250}, {376, 7795}, {382, 625}, {384, 574}, {439, 32815}, {538, 3053}, {543, 3767}, {550, 7761}, {576, 5026}, {698, 41412}, {754, 3926}, {980, 16046}, {1078, 8588}, {1384, 7805}, {1506, 14035}, {1657, 7778}, {1968, 4235}, {2482, 7747}, {2548, 32981}, {2549, 6680}, {3096, 33260}, {3098, 4048}, {3199, 37199}, {3314, 33268}, {3329, 68524}, {3522, 7800}, {3528, 55732}, {3534, 7784}, {3627, 44377}, {3793, 63934}, {3815, 68177}, {3818, 59695}, {3849, 7776}, {3933, 63935}, {3972, 7772}, {4045, 14001}, {4576, 44116}, {5007, 31859}, {5013, 7804}, {5023, 7780}, {5025, 65633}, {5031, 48884}, {5033, 18906}, {5103, 48904}, {5149, 9737}, {5171, 32521}, {5189, 30747}, {5248, 25130}, {5461, 63533}, {5475, 19687}, {6292, 32965}, {6337, 7737}, {6390, 7759}, {6655, 7835}, {6658, 7752}, {6683, 11286}, {6704, 33198}, {6722, 32970}, {7603, 51581}, {7617, 11164}, {7618, 31400}, {7619, 11147}, {7622, 8370}, {7738, 7829}, {7745, 66391}, {7746, 18546}, {7748, 7807}, {7749, 11185}, {7750, 7801}, {7753, 33187}, {7754, 35007}, {7758, 14148}, {7762, 59634}, {7767, 47101}, {7769, 11361}, {7770, 15482}, {7771, 17128}, {7773, 66387}, {7787, 68520}, {7790, 33225}, {7791, 7820}, {7793, 17131}, {7794, 14907}, {7796, 14712}, {7799, 7823}, {7802, 7818}, {7803, 33255}, {7810, 33208}, {7812, 52695}, {7813, 20065}, {7817, 68718}, {7822, 8356}, {7826, 32833}, {7828, 11648}, {7831, 33275}, {7832, 7833}, {7834, 8369}, {7838, 34511}, {7841, 7874}, {7845, 32821}, {7847, 7892}, {7849, 15696}, {7852, 33220}, {7853, 33234}, {7855, 32820}, {7859, 14036}, {7860, 7947}, {7861, 32954}, {7866, 44519}, {7870, 7885}, {7873, 7881}, {7882, 63938}, {7886, 11288}, {7889, 14037}, {7893, 51224}, {7898, 7909}, {7899, 33019}, {7910, 7931}, {7911, 7945}, {7915, 11287}, {7918, 14043}, {7924, 7930}, {7925, 19696}, {7934, 33256}, {7939, 11057}, {7940, 14041}, {8290, 18548}, {8589, 11285}, {8716, 30435}, {9167, 33006}, {9466, 68519}, {9651, 26629}, {9664, 26686}, {10684, 52773}, {11160, 32875}, {11174, 31652}, {11202, 59530}, {11257, 18806}, {11318, 32479}, {11676, 30270}, {12110, 35951}, {13108, 38225}, {13196, 37517}, {13881, 58448}, {14023, 32817}, {14033, 31401}, {14064, 43619}, {14068, 43457}, {14981, 36998}, {15520, 61625}, {15688, 40344}, {15712, 58446}, {16393, 25499}, {16395, 69013}, {16397, 26100}, {16418, 36812}, {16953, 38862}, {17508, 24256}, {17539, 25497}, {18424, 33249}, {18840, 54891}, {18907, 59546}, {20081, 68521}, {21508, 24271}, {22247, 32984}, {22253, 22331}, {22267, 24275}, {22676, 40253}, {22802, 59706}, {23698, 37466}, {25107, 25440}, {28695, 40349}, {31088, 35929}, {31173, 66395}, {31274, 32961}, {31276, 43459}, {32816, 43618}, {32822, 63955}, {32823, 44678}, {32824, 63927}, {32826, 32989}, {32828, 34506}, {32829, 51579}, {33205, 43448}, {33206, 53127}, {33227, 64093}, {33233, 39565}, {33237, 66616}, {33703, 37690}, {35925, 37479}, {36156, 51999}, {38747, 59363}, {40341, 63930}, {40842, 49112}, {41133, 66425}, {41134, 66419}, {42266, 45472}, {42267, 45473}, {42528, 69123}, {42529, 69122}, {44240, 54075}, {47391, 59556}, {48863, 59625}, {52886, 63021}, {63534, 63957}

X(69171) = reflection of X(i) in X(j) for these {i,j}: {3788, 59545}, {7825, 3788}, {7916, 3926}, {44518, 7886}
X(69171) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7756, 7872}, {3, 3734, 7815}, {3, 7816, 3734}, {4, 620, 7862}, {32, 99, 7781}, {32, 7781, 7798}, {76, 13586, 5206}, {99, 3552, 32}, {183, 68516, 15513}, {187, 1975, 7751}, {315, 7863, 7908}, {315, 33244, 6781}, {316, 7891, 7888}, {376, 7795, 7830}, {384, 574, 7808}, {384, 7782, 574}, {550, 7789, 7761}, {1078, 33014, 8588}, {1657, 7778, 7842}, {1975, 33235, 187}, {2482, 7747, 7763}, {2482, 33007, 7775}, {2549, 6680, 7902}, {2549, 32973, 6680}, {3972, 7783, 7772}, {5013, 68527, 7804}, {6337, 7737, 7764}, {6337, 33239, 7737}, {6655, 7835, 7867}, {6658, 7752, 62203}, {6781, 7863, 315}, {7746, 32819, 18546}, {7747, 7763, 7775}, {7748, 7807, 7844}, {7750, 7801, 7896}, {7750, 7896, 66455}, {7761, 7789, 7869}, {7763, 33007, 7747}, {7764, 35022, 6337}, {7770, 37512, 15482}, {7791, 7820, 7914}, {7795, 7830, 7865}, {7799, 7823, 7903}, {7802, 7836, 7818}, {7816, 32456, 3}, {7832, 7833, 7935}, {7834, 34504, 63548}, {7836, 33265, 7802}, {7847, 7892, 7913}, {7891, 33257, 316}, {7931, 33267, 7910}, {7945, 33264, 7911}, {8369, 63548, 7834}, {8588, 17130, 1078}, {8716, 30435, 32450}, {11185, 32964, 7749}, {11286, 15815, 6683}, {11288, 44518, 7886}, {17128, 33276, 7771}, {32819, 35297, 7746}, {32826, 32989, 43620}, {32954, 44526, 7861}, {58448, 63922, 13881}

X(69172) = (1,0,0,0,1,0)-ADDITIVE ASSOCIATE OF X(20)

Barycentrics    3*a^4 + 2*b^2*c^2 : :
X(69172) = X[7795] - 3 X[14039], X[7896] - 6 X[14039], X[7784] - 3 X[33237], 2 X[7915] - 3 X[33237], X[32822] + 3 X[63006]

X(69172) lies on these lines: {2, 5206}, {3, 6683}, {4, 6680}, {5, 44381}, {6, 7781}, {20, 4045}, {30, 7834}, {32, 76}, {39, 1003}, {69, 63937}, {83, 574}, {99, 7772}, {115, 14035}, {141, 19697}, {148, 7856}, {183, 35007}, {187, 7770}, {194, 12150}, {251, 9464}, {315, 7820}, {316, 7867}, {376, 14492}, {381, 7886}, {382, 7861}, {538, 30435}, {543, 5286}, {550, 3589}, {576, 32134}, {597, 34504}, {598, 7940}, {620, 2548}, {625, 32954}, {626, 7737}, {754, 7795}, {1078, 68525}, {1285, 14023}, {1384, 7780}, {1506, 16925}, {1627, 16952}, {1656, 13449}, {1968, 44142}, {1975, 5007}, {2241, 25303}, {2549, 7829}, {2896, 19692}, {3053, 3934}, {3091, 6722}, {3096, 14712}, {3098, 42534}, {3266, 16949}, {3314, 14038}, {3329, 7782}, {3528, 9751}, {3530, 15491}, {3618, 33239}, {3767, 14033}, {3788, 7745}, {3849, 7784}, {3926, 7838}, {4048, 5039}, {5008, 7754}, {5013, 32456}, {5025, 62203}, {5041, 31859}, {5149, 13356}, {5188, 39656}, {5254, 68177}, {5277, 16920}, {5306, 66321}, {5309, 32819}, {5319, 32815}, {5346, 47286}, {5475, 7807}, {5569, 8367}, {6292, 14907}, {6645, 53680}, {6655, 7846}, {6658, 7790}, {6661, 7750}, {6704, 16043}, {6781, 7791}, {7603, 33233}, {7617, 7746}, {7622, 31401}, {7735, 63924}, {7748, 7792}, {7749, 16924}, {7752, 33225}, {7753, 7763}, {7755, 11185}, {7756, 7803}, {7758, 32840}, {7759, 7789}, {7760, 20105}, {7761, 7819}, {7762, 7801}, {7764, 32831}, {7765, 16989}, {7767, 66318}, {7768, 14040}, {7769, 33246}, {7771, 68522}, {7773, 7874}, {7774, 7863}, {7776, 7880}, {7778, 7843}, {7783, 7878}, {7785, 7835}, {7786, 13586}, {7794, 20065}, {7796, 20088}, {7797, 11648}, {7799, 7921}, {7800, 33198}, {7806, 14034}, {7809, 7945}, {7811, 46226}, {7812, 7836}, {7817, 11159}, {7818, 7823}, {7824, 8588}, {7828, 11361}, {7831, 16895}, {7833, 7859}, {7841, 7852}, {7842, 7866}, {7845, 7881}, {7847, 7875}, {7848, 63938}, {7853, 33217}, {7857, 16044}, {7858, 7891}, {7860, 7931}, {7868, 7873}, {7870, 7941}, {7877, 34604}, {7879, 66455}, {7884, 66328}, {7885, 7930}, {7887, 39590}, {7890, 32833}, {7894, 14075}, {7895, 63932}, {7898, 7944}, {7900, 7909}, {7904, 51224}, {7910, 7948}, {7918, 33256}, {7919, 33019}, {7923, 19696}, {7924, 7943}, {7926, 7947}, {7928, 11057}, {7932, 66419}, {7934, 14043}, {7937, 19694}, {7942, 14041}, {8361, 53418}, {8366, 31173}, {8368, 63956}, {8589, 68516}, {9650, 26629}, {9665, 26686}, {9734, 11272}, {9737, 10796}, {9939, 32027}, {9996, 44237}, {9997, 51710}, {10007, 55649}, {10359, 35951}, {11059, 16951}, {11174, 33235}, {11178, 32151}, {11285, 15513}, {11676, 37479}, {12110, 30270}, {12191, 51932}, {13335, 35930}, {13862, 36997}, {14030, 14568}, {14042, 16984}, {14061, 33018}, {15031, 18362}, {15815, 44562}, {16932, 26235}, {18501, 35002}, {20081, 41748}, {20179, 31456}, {21309, 63933}, {21843, 32968}, {22234, 51524}, {24256, 41412}, {31404, 32871}, {31406, 32459}, {31415, 32970}, {31417, 34803}, {31455, 35297}, {32448, 39561}, {32822, 63006}, {32824, 51170}, {32838, 32971}, {32884, 32989}, {32886, 37667}, {32961, 43457}, {32974, 43618}, {32979, 43620}, {33197, 66466}, {33226, 63119}, {33843, 37199}, {34506, 37809}, {35022, 37665}, {44116, 46900}, {47005, 66327}, {48884, 51848}, {53095, 68528}, {55085, 68520}, {55774, 62092}, {59545, 66393}, {59780, 63953}, {63534, 66409}, {63548, 66391}

X(69172) = reflection of X(i) in X(j) for these {i,j}: {7784, 7915}, {7872, 7834}, {7896, 7795}
X(69172) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7747, 7825}, {2, 7802, 7935}, {3, 7804, 7808}, {3, 7808, 15482}, {3, 18502, 58851}, {4, 6680, 7844}, {6, 7816, 7781}, {6, 68527, 7816}, {32, 384, 3734}, {32, 3734, 7751}, {32, 17130, 385}, {32, 17131, 6179}, {83, 3552, 574}, {99, 7787, 7772}, {187, 7770, 7815}, {315, 7820, 7869}, {315, 14037, 7820}, {316, 7892, 7867}, {384, 3972, 32}, {626, 7737, 63931}, {1975, 5007, 7798}, {2548, 32973, 620}, {3053, 11286, 3934}, {3329, 7782, 53096}, {3788, 7745, 7775}, {5013, 68513, 32456}, {5395, 32829, 2548}, {5395, 32973, 32829}, {5475, 7807, 7862}, {6179, 17128, 17131}, {6655, 7846, 7913}, {6658, 7790, 65633}, {6658, 10583, 7790}, {6661, 7750, 7822}, {6781, 7889, 7791}, {7737, 14001, 626}, {7745, 8369, 3788}, {7748, 7792, 7902}, {7750, 7822, 7865}, {7759, 7789, 7908}, {7761, 7819, 7914}, {7762, 7801, 7916}, {7773, 33220, 7874}, {7784, 33237, 7915}, {7785, 7835, 7888}, {7786, 13586, 15515}, {7787, 68517, 99}, {7789, 18907, 7759}, {7792, 19687, 7748}, {7803, 33007, 7756}, {7812, 7836, 7903}, {7823, 7832, 7818}, {7823, 14036, 7832}, {7874, 14537, 7773}, {7875, 33257, 7847}, {11174, 33235, 37512}, {12110, 35925, 30270}, {14712, 19689, 3096}, {14907, 16898, 6292}, {32954, 65630, 625}

X(69173) = (0, 1, 1, 1, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(11)

Barycentrics    b^3 + 2*a*b*c - b^2*c - b*c^2 + c^3 : :

X(69173) lies on these lines: {1, 5046}, {2, 846}, {5, 2292}, {9, 17737}, {10, 25253}, {11, 38}, {31, 17720}, {37, 17605}, {42, 908}, {43, 27131}, {45, 31245}, {55, 35015}, {57, 33098}, {63, 29662}, {75, 25960}, {81, 33096}, {100, 33095}, {149, 3961}, {171, 5057}, {190, 33119}, {191, 45939}, {192, 29849}, {210, 33136}, {226, 1458}, {238, 33133}, {244, 3782}, {312, 15523}, {321, 3846}, {329, 11269}, {354, 32856}, {404, 24851}, {497, 3938}, {546, 63360}, {612, 1699}, {614, 33143}, {726, 69134}, {740, 5741}, {748, 3772}, {750, 1836}, {756, 2886}, {894, 29845}, {896, 37646}, {899, 3452}, {902, 66632}, {940, 24725}, {946, 10459}, {960, 21935}, {968, 5219}, {976, 1479}, {982, 1647}, {984, 11680}, {986, 4193}, {1001, 29689}, {1125, 11319}, {1150, 4703}, {1193, 21616}, {1201, 13161}, {1215, 29685}, {1316, 46410}, {1329, 4642}, {1376, 33094}, {1402, 40109}, {1468, 58798}, {1621, 17719}, {1757, 26792}, {1961, 33112}, {1962, 5718}, {1999, 32843}, {2476, 21674}, {2478, 3924}, {2887, 4358}, {3006, 3971}, {3011, 40998}, {3057, 67021}, {3058, 3722}, {3159, 30171}, {3175, 32848}, {3216, 36250}, {3218, 33099}, {3219, 33140}, {3242, 11238}, {3266, 21416}, {3305, 17064}, {3583, 30115}, {3662, 30957}, {3663, 33864}, {3666, 5087}, {3670, 3825}, {3681, 33141}, {3685, 29846}, {3687, 4365}, {3703, 3994}, {3705, 32925}, {3724, 4192}, {3735, 21044}, {3741, 26580}, {3743, 37693}, {3752, 33145}, {3756, 42040}, {3814, 4424}, {3817, 3989}, {3820, 4695}, {3829, 42039}, {3836, 48646}, {3838, 44307}, {3840, 17184}, {3873, 24217}, {3877, 37716}, {3898, 24222}, {3912, 31023}, {3920, 33106}, {3925, 62221}, {3952, 29673}, {3967, 33162}, {3995, 29671}, {4026, 31264}, {4062, 4417}, {4080, 17140}, {4082, 4937}, {4187, 24443}, {4202, 25079}, {4300, 12608}, {4357, 31241}, {4359, 48643}, {4383, 33128}, {4387, 33156}, {4388, 17763}, {4419, 10589}, {4428, 17783}, {4434, 4450}, {4465, 25345}, {4514, 32927}, {4653, 7424}, {4671, 6535}, {4683, 14829}, {4850, 33154}, {4854, 37662}, {4862, 31249}, {4892, 18139}, {4972, 30566}, {4980, 48641}, {4981, 21242}, {5014, 49996}, {5047, 24161}, {5205, 32948}, {5233, 32860}, {5249, 30950}, {5254, 39244}, {5259, 24160}, {5284, 33130}, {5293, 52367}, {5294, 29863}, {5297, 33109}, {5306, 39251}, {5311, 26098}, {5396, 12081}, {5722, 49454}, {5743, 21020}, {5748, 64168}, {5880, 17124}, {6327, 29649}, {6545, 24111}, {6690, 37691}, {7018, 18152}, {7081, 32947}, {7191, 33152}, {7226, 29676}, {7292, 33147}, {7988, 62818}, {8616, 29665}, {9284, 21327}, {10129, 33111}, {10176, 68946}, {10448, 11375}, {10453, 33065}, {11019, 17449}, {11031, 37358}, {11681, 37598}, {12047, 59305}, {12579, 16347}, {13097, 19546}, {14008, 35623}, {15485, 29681}, {16062, 25591}, {16569, 33131}, {16891, 31008}, {17056, 33329}, {17063, 33146}, {17122, 20292}, {17123, 33129}, {17125, 24789}, {17127, 29658}, {17165, 21093}, {17174, 18169}, {17182, 17187}, {17247, 24230}, {17276, 17728}, {17279, 31237}, {17353, 29867}, {17469, 17602}, {17470, 53510}, {17484, 32913}, {17594, 30852}, {17717, 28606}, {17718, 62849}, {17724, 49736}, {17725, 62806}, {17748, 64071}, {17768, 37634}, {17770, 37639}, {17862, 26010}, {18393, 30116}, {18743, 25957}, {19742, 50755}, {19786, 29684}, {20347, 24241}, {21027, 26037}, {21241, 59517}, {21330, 53476}, {21334, 67494}, {21635, 64710}, {21949, 61686}, {23536, 28352}, {23537, 27627}, {24239, 46901}, {24250, 68478}, {24255, 68871}, {24387, 67979}, {24589, 48645}, {24695, 63078}, {24709, 33123}, {24723, 32918}, {25378, 32771}, {25496, 68945}, {25502, 27186}, {25527, 29677}, {25568, 67207}, {25681, 50065}, {25958, 29674}, {25959, 46938}, {25961, 30829}, {26034, 28808}, {26102, 31019}, {26223, 29635}, {26229, 33869}, {27003, 32857}, {27064, 29631}, {27065, 33138}, {27184, 30942}, {27538, 33117}, {27577, 27785}, {27714, 52258}, {28027, 68615}, {28377, 37620}, {28609, 62819}, {29641, 64178}, {29643, 41839}, {29653, 31035}, {29661, 31266}, {29686, 32775}, {29820, 33148}, {29821, 33155}, {29824, 33064}, {29857, 30568}, {29861, 30578}, {29872, 33164}, {30818, 32781}, {30831, 33158}, {30867, 32932}, {30992, 32010}, {31018, 33137}, {31037, 49560}, {32773, 32931}, {32780, 41242}, {32844, 32926}, {32851, 32936}, {32852, 49995}, {32855, 42044}, {32865, 63961}, {32911, 33135}, {32914, 37759}, {32919, 33066}, {32928, 33071}, {32937, 33120}, {32938, 33121}, {32943, 33126}, {33068, 37758}, {33097, 37633}, {33103, 64149}, {33132, 37680}, {33161, 56082}, {33163, 56084}, {33299, 69096}, {35262, 66672}, {37375, 37717}, {37527, 59787}, {37663, 66071}, {37674, 61716}, {37692, 62871}, {37715, 51409}, {39595, 41011}, {49488, 63010}, {49608, 50165}, {59593, 63145}, {63008, 67208}, {63089, 67211}

X(69173) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3944, 3120}, {2, 17777, 32930}, {2, 33100, 17596}, {2, 33102, 1054}, {11, 4415, 38}, {31, 17720, 29683}, {37, 17605, 33105}, {37, 33105, 29682}, {81, 33096, 61707}, {312, 25760, 15523}, {329, 11269, 32912}, {612, 1699, 33104}, {908, 24210, 42}, {940, 24725, 64164}, {968, 5219, 29678}, {984, 11680, 29690}, {1001, 33127, 29689}, {2887, 4358, 29687}, {3452, 3914, 899}, {3670, 3825, 28096}, {3772, 4679, 748}, {3782, 3816, 244}, {3817, 4656, 29639}, {4187, 63997, 24443}, {4417, 32915, 4062}, {4656, 29639, 3989}, {4671, 32778, 6535}, {4854, 37662, 46904}, {4972, 30566, 59511}, {11814, 24169, 2}, {13161, 41012, 1201}, {17717, 28606, 29688}, {17720, 24703, 31}, {19786, 32944, 29684}, {21093, 29655, 17165}, {24217, 33101, 3873}, {25958, 29674, 48650}, {26792, 33142, 1757}, {27131, 33134, 43}, {32775, 32942, 29686}

X(69174) = (0, 1, 1, 0, 0, 1)-ADDITIVE ASSOCIATE OF X(12)

Barycentrics    b^4 - 2*a^2*b*c + c^4 : :

X(69174) lies on these lines: {1, 626}, {2, 2242}, {10, 141}, {11, 625}, {12, 3934}, {35, 7830}, {36, 620}, {39, 26561}, {55, 7761}, {56, 3788}, {75, 34542}, {115, 350}, {127, 18455}, {172, 6680}, {187, 26629}, {192, 7790}, {239, 3936}, {315, 2241}, {316, 4366}, {320, 66152}, {325, 1015}, {330, 7796}, {388, 7795}, {498, 7815}, {499, 7862}, {519, 20541}, {524, 57017}, {538, 69093}, {754, 1914}, {999, 7778}, {1086, 57029}, {1478, 3734}, {1479, 7825}, {1500, 6656}, {1506, 26959}, {1573, 37664}, {1574, 17670}, {1757, 19977}, {1909, 7794}, {1975, 9651}, {2275, 7764}, {2276, 4045}, {2280, 4805}, {3085, 7800}, {3230, 4766}, {3295, 7784}, {3314, 64133}, {3600, 53033}, {3726, 63817}, {3727, 17211}, {3760, 63924}, {3814, 20530}, {3822, 21264}, {3912, 68897}, {3920, 21248}, {3963, 33941}, {4071, 68890}, {4121, 63503}, {4299, 69171}, {4376, 7272}, {4657, 5725}, {4885, 8678}, {4986, 20483}, {4995, 40344}, {5011, 24699}, {5149, 12184}, {5280, 7829}, {5299, 7838}, {5434, 7880}, {5718, 17023}, {6284, 7842}, {6292, 27020}, {6645, 7832}, {6683, 31460}, {7354, 7816}, {7759, 16502}, {7765, 25264}, {7770, 9650}, {7773, 9665}, {7775, 9599}, {7776, 16781}, {7781, 9597}, {7789, 18990}, {7791, 31451}, {7808, 9596}, {7823, 53680}, {7834, 54416}, {7841, 9664}, {7849, 15888}, {7853, 26590}, {7861, 69096}, {7865, 10056}, {7872, 9598}, {7874, 26686}, {7876, 31478}, {7895, 69094}, {9654, 69139}, {11285, 31501}, {11287, 31477}, {15271, 31479}, {15325, 44377}, {15326, 32456}, {15482, 31497}, {16589, 26558}, {16611, 25357}, {17045, 21245}, {17143, 33841}, {17322, 46826}, {17528, 20181}, {17757, 27076}, {18040, 30173}, {18134, 41232}, {19786, 46828}, {20963, 24995}, {21057, 27918}, {21760, 29990}, {24222, 32847}, {26582, 52959}, {26639, 60524}, {26801, 31488}, {27042, 68934}, {27838, 30969}, {28369, 30077}, {29830, 30954}, {30837, 56530}, {31041, 69028}, {33888, 69080}, {35101, 65116}, {35768, 45473}, {35769, 45472}, {36230, 57015}, {40690, 59512}, {44396, 57039}, {46671, 50362}

X(69174) = midpoint of X(69093) and X(69098)
X(69174) = complement of X(5291)
X(69174) = complement of the isogonal conjugate of X(17946)
X(69174) = X(i)-complementary conjugate of X(j) for these (i,j): {58, 62609}, {514, 46671}, {649, 35079}, {661, 41179}, {2703, 514}, {11609, 3452}, {11611, 3454}, {17929, 4369}, {17939, 14838}, {17946, 10}, {17954, 2}, {17961, 37}, {17971, 1214}, {17981, 226}, {18002, 16592}, {18015, 8287}, {35147, 3835}, {53689, 5750}, {57680, 440}, {60484, 124}, {65239, 513}
X(69174) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {12, 69097, 3934}, {141, 495, 69136}, {172, 30104, 6680}, {26561, 69135, 39}

X(69175) = (0, 1, 1, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(12)

Barycentrics    b^4 - 2*a^2*b*c - 2*b^2*c^2 + c^4 : :

X(69175) lies on these lines: {1, 115}, {2, 31456}, {3, 9651}, {4, 2241}, {5, 1015}, {6, 9650}, {11, 39565}, {12, 39}, {32, 1478}, {34, 27371}, {35, 7756}, {36, 7749}, {55, 7748}, {56, 7746}, {172, 5270}, {187, 7354}, {202, 69118}, {203, 69119}, {230, 18990}, {330, 7752}, {350, 63924}, {381, 9665}, {388, 2242}, {442, 1573}, {495, 1500}, {496, 63534}, {498, 574}, {538, 69135}, {626, 1909}, {668, 33841}, {999, 13881}, {1058, 63533}, {1107, 3822}, {1479, 69141}, {1504, 31472}, {1505, 44622}, {1506, 2275}, {1570, 39897}, {1571, 31434}, {1572, 9612}, {1574, 17757}, {1914, 3585}, {2085, 21725}, {2276, 7765}, {2330, 65417}, {2476, 16975}, {2548, 10590}, {2549, 3085}, {3027, 62356}, {3053, 9655}, {3058, 39563}, {3086, 43620}, {3136, 22199}, {3295, 9664}, {3614, 7603}, {3665, 4403}, {3761, 7794}, {3780, 68946}, {3814, 16604}, {3815, 10592}, {3934, 26561}, {4045, 27020}, {4299, 5206}, {4302, 65633}, {4385, 34542}, {4400, 7826}, {4692, 16886}, {5013, 31479}, {5025, 64133}, {5028, 12588}, {5046, 68893}, {5177, 31416}, {5219, 9619}, {5229, 7737}, {5261, 5286}, {5280, 5355}, {5291, 20060}, {5292, 9346}, {5297, 59768}, {5299, 7753}, {5309, 11237}, {5319, 31410}, {5368, 7296}, {5432, 37512}, {5475, 10895}, {5726, 9593}, {6537, 31339}, {6645, 7828}, {6656, 69136}, {6781, 10483}, {7005, 69113}, {7006, 69112}, {7051, 18972}, {7173, 39601}, {7738, 8164}, {7739, 31402}, {7741, 63493}, {7772, 9596}, {7816, 26629}, {7820, 30104}, {7821, 69094}, {7853, 69095}, {7861, 26590}, {7886, 26686}, {8589, 52793}, {8818, 16685}, {9466, 69097}, {9578, 9620}, {9598, 10056}, {9646, 62206}, {9955, 62370}, {10063, 32452}, {10072, 18362}, {10588, 31401}, {10896, 18424}, {10987, 65134}, {11361, 53680}, {12607, 52959}, {12903, 14901}, {13905, 62241}, {13963, 62242}, {15170, 63543}, {15171, 53419}, {15326, 15513}, {15888, 69096}, {16589, 25466}, {16784, 43457}, {16969, 24045}, {17053, 50036}, {17448, 25639}, {17670, 27076}, {17734, 33863}, {17750, 37716}, {18055, 36230}, {18973, 19373}, {20963, 21935}, {20970, 64172}, {21057, 30171}, {21243, 63525}, {31433, 51784}, {31497, 53096}, {36707, 38865}, {39764, 39901}, {40690, 49777}, {41345, 44517}, {44526, 64951}, {62992, 68663}

X(69175) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 9654, 9650}, {12, 39, 31476}, {12, 69098, 39}, {381, 16781, 9665}, {388, 3767, 2242}, {495, 5254, 1500}, {498, 9597, 574}, {1914, 3585, 7747}, {2275, 7951, 1506}, {2476, 16975, 31488}, {2549, 3085, 31451}, {3295, 44518, 9664}, {5013, 31479, 31501}, {5261, 5286, 31409}, {7765, 37719, 31478}, {10895, 16502, 5475}

X(69176) = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(74)

Barycentrics    a^2*(a^8 - a^6*b^2 - 3*a^4*b^4 + 5*a^2*b^6 - 2*b^8 - a^6*c^2 + 7*a^4*b^2*c^2 - 5*a^2*b^4*c^2 - b^6*c^2 - 3*a^4*c^4 - 5*a^2*b^2*c^4 - 6*b^4*c^4 + 5*a^2*c^6 - b^2*c^6 - 2*c^8) : :

X(69176) lies on these lines: {69, 11438}, {74, 76}, {325, 43584}, {3581, 7767}, {3785, 37478}, {3926, 37470}, {4550, 32828}, {7750, 15107}, {7773, 10545}, {7788, 15053}, {7796, 43597}, {7860, 38848}, {16836, 52437}, {32006, 34417}, {37475, 68660}, {37513, 68654}

X(69177) = (1, 1, 1, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(74)

Barycentrics    a^2*(a^8 - a^6*b^2 - 3*a^4*b^4 + 5*a^2*b^6 - 2*b^8 - a^6*c^2 - 7*a^4*b^2*c^2 + 5*a^2*b^4*c^2 - b^6*c^2 - 3*a^4*c^4 + 5*a^2*b^2*c^4 + 6*b^4*c^4 + 5*a^2*c^6 - b^2*c^6 - 2*c^8) : :

X(69177) lies on these lines: {6, 7509}, {74, 5210}, {574, 8623}, {1384, 33533}, {1614, 45769}, {3098, 43448}, {5206, 11440}, {8553, 11459}, {15107, 43620}, {37478, 62992}

X(69178) = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(74)

Barycentrics    a^2*(a^8 - a^6*b^2 - 3*a^4*b^4 + 5*a^2*b^6 - 2*b^8 - a^6*c^2 + 7*a^4*b^2*c^2 - 5*a^2*b^4*c^2 - b^6*c^2 - 3*a^4*c^4 - 5*a^2*b^2*c^4 + 5*a^2*c^6 - b^2*c^6 - 2*c^8) : :

X(69178) lies on these lines: {3, 51383}, {74, 76}, {315, 11438}, {1078, 61188}, {3581, 7750}, {4550, 32832}, {4563, 41462}, {7752, 43584}, {7763, 37470}, {7796, 43601}, {7802, 15107}, {7809, 15053}, {7814, 43597}, {14907, 37478}, {32819, 64624}

X(69179) = (1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(74)

Barycentrics    a^2*(a^8 - a^6*b^2 - 3*a^4*b^4 + 5*a^2*b^6 - 2*b^8 - a^6*c^2 - b^6*c^2 - 3*a^4*c^4 + 6*b^4*c^4 + 5*a^2*c^6 - b^2*c^6 - 2*c^8) : :

X(69179) lies on these lines: {6, 5889}, {74, 5210}, {230, 63425}, {574, 33533}, {2979, 34866}, {3054, 11438}, {3581, 43620}, {4550, 53418}, {5023, 11440}, {5206, 32138}, {7691, 44518}, {9699, 15060}, {11464, 45769}, {37478, 53419}, {46730, 63534}

X(69180) = (0, 0, 0, 0, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    2*b^2*c^2 + Sqrt[3]*a^2*S : :

X(69180) lies on these lines: {6, 76}, {13, 7776}, {17, 69121}, {32, 69143}, {69, 397}, {99, 36836}, {183, 22238}, {298, 7773}, {303, 32821}, {315, 5340}, {325, 42156}, {350, 54437}, {381, 69145}, {394, 41000}, {395, 32828}, {396, 3926}, {621, 5869}, {631, 59542}, {1007, 42598}, {1078, 36843}, {1656, 69165}, {1909, 54438}, {1975, 22236}, {3760, 54402}, {3761, 54403}, {3785, 42148}, {3933, 40693}, {5013, 69159}, {5318, 32006}, {5339, 11185}, {5472, 7855}, {6337, 16772}, {6390, 42152}, {7750, 42155}, {7763, 16644}, {7767, 10653}, {7778, 69116}, {7781, 63199}, {7784, 69112}, {7799, 49905}, {7815, 63198}, {7818, 69140}, {7866, 69108}, {7867, 69161}, {9763, 53453}, {9766, 69130}, {10601, 41001}, {11287, 69122}, {11488, 32831}, {12155, 63950}, {14907, 43193}, {15271, 69167}, {16645, 32832}, {16773, 34229}, {17130, 69150}, {17131, 69144}, {23302, 32829}, {23303, 32838}, {32815, 42147}, {32816, 42166}, {32817, 59541}, {32819, 42154}, {32823, 43403}, {32826, 42164}, {32830, 37640}, {32833, 49947}, {32834, 37641}, {32836, 43228}, {32840, 63032}, {32869, 63103}, {32872, 63033}, {32874, 63102}, {32893, 49812}, {32984, 60222}, {34505, 69113}, {37688, 43239}, {40694, 64093}, {41407, 68527}, {42153, 59635}, {42165, 64018}, {42974, 69137}, {42988, 69157}, {42999, 52713}, {43101, 63106}, {43229, 46951}, {61719, 69106}, {69126, 69129}, {69141, 69148}, {69162, 69169}

X(69180) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 69107, 7776}, {17, 69121, 69158}, {69108, 69111, 7866}, {69112, 69115, 7784}, {69116, 69119, 7778}, {69122, 69125, 11287}, {69130, 69133, 9766}, {69143, 69164, 32}

X(69181) = (0, 0, 1, 0, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    b^4 + c^4 - Sqrt[3]*a^2*S : :

X(69181) lies on these lines: {6, 626}, {13, 69109}, {17, 15271}, {32, 69142}, {61, 7784}, {62, 7778}, {141, 11311}, {299, 7754}, {303, 11285}, {381, 69146}, {396, 7800}, {397, 7795}, {620, 36843}, {624, 18440}, {625, 42153}, {636, 3526}, {1656, 69166}, {3180, 7920}, {3734, 5340}, {3788, 22238}, {3934, 42156}, {5013, 69157}, {5339, 7825}, {5472, 7822}, {7761, 22236}, {7763, 63198}, {7789, 10653}, {7791, 63199}, {7815, 16644}, {7816, 42155}, {7818, 69150}, {7830, 36836}, {7842, 42154}, {7862, 16645}, {7865, 49947}, {7867, 69144}, {7876, 62984}, {8362, 61332}, {11287, 69160}, {11308, 53033}, {11318, 69118}, {17130, 69140}, {17131, 69161}, {22253, 69126}, {34509, 42974}, {41407, 63938}, {42149, 44377}, {42151, 59545}, {42988, 69159}, {43193, 69171}, {61719, 69108}, {63933, 69106}, {69112, 69117}, {69114, 69119}, {69120, 69125}, {69128, 69133}, {69141, 69149}, {69158, 69167}, {69162, 69170}

X(69181) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 69109, 69139}, {17, 69123, 15271}, {69106, 69111, 63933}, {69126, 69131, 22253}, {69142, 69163, 32}

X(69182) = (0, 0, 1, 0, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    b^4 - 2*b^2*c^2 + c^4 - Sqrt[3]*a^2*S : :

X(69182) lies on these lines: {2, 63198}, {3, 62198}, {4, 16943}, {5, 61331}, {6, 13}, {15, 44526}, {16, 37637}, {17, 5013}, {32, 5340}, {39, 42156}, {61, 44518}, {62, 13881}, {187, 42155}, {230, 10653}, {303, 31859}, {395, 22492}, {396, 2549}, {397, 3767}, {574, 16644}, {1080, 5335}, {1384, 62200}, {1656, 69167}, {1692, 41036}, {2548, 42166}, {3053, 16965}, {3055, 42911}, {3815, 18582}, {3851, 69132}, {5023, 42158}, {5052, 22693}, {5094, 30468}, {5107, 50855}, {5206, 43193}, {5210, 36968}, {5237, 44535}, {5238, 44519}, {5254, 40693}, {5306, 41112}, {5318, 7737}, {5339, 69141}, {5585, 42528}, {6115, 37071}, {6772, 11159}, {6775, 9763}, {6792, 52267}, {7603, 42098}, {7612, 43954}, {7736, 43403}, {7745, 42162}, {7746, 22238}, {7748, 22236}, {7749, 36843}, {7756, 36836}, {7818, 69164}, {7866, 69138}, {7867, 69143}, {8588, 42625}, {9300, 41119}, {9605, 69126}, {10645, 44541}, {10654, 53419}, {11165, 42036}, {11287, 69159}, {11295, 53435}, {11300, 11488}, {11302, 23302}, {11311, 53452}, {11485, 43454}, {11486, 62197}, {11542, 15048}, {11648, 49947}, {12154, 66587}, {15515, 42490}, {15993, 20425}, {16241, 53095}, {16267, 63201}, {16645, 50858}, {17130, 69163}, {17131, 69142}, {18362, 49948}, {18907, 43416}, {19781, 42127}, {21843, 42943}, {22512, 41039}, {26869, 30465}, {31401, 42598}, {31404, 42494}, {31489, 37832}, {32519, 43539}, {36449, 49261}, {36468, 49262}, {36969, 41407}, {37512, 43238}, {37640, 43448}, {39565, 42153}, {39601, 42095}, {40694, 63534}, {41038, 41045}, {41041, 53443}, {41107, 41406}, {41895, 54618}, {42094, 62203}, {42152, 63548}, {42813, 65630}, {42941, 43618}, {42942, 43619}, {42988, 69160}, {42999, 63533}, {43194, 65633}, {43277, 54141}, {49825, 63006}, {49874, 63024}, {52649, 61318}, {54570, 54670}, {63933, 69137}, {69144, 69162}

X(69182) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 69111, 6}, {17, 69125, 5013}, {32, 69140, 5340}, {396, 2549, 63199}, {5309, 5472, 6}, {11542, 15048, 61332}, {37832, 63200, 31489}, {61719, 69110, 6}, {69112, 69119, 3}, {69126, 69133, 9605}, {69140, 69161, 32}, {69141, 69150, 5339}

X(69183) = (0, 1, 0, 1, 0, 0, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^2*(b^2 + c^2 + 2*Sqrt[3]*S) : :

X(69183) lies on these lines: {2, 46710}, {3, 6}, {4, 5472}, {13, 69113}, {14, 69133}, {17, 69118}, {115, 22509}, {298, 7916}, {299, 7896}, {303, 7862}, {381, 69147}, {395, 31455}, {396, 7746}, {397, 7748}, {398, 5475}, {1015, 54402}, {1500, 54403}, {1506, 40694}, {1968, 56514}, {2242, 2307}, {2548, 5471}, {2549, 42998}, {2981, 11004}, {3171, 20998}, {3180, 7877}, {3412, 62198}, {3458, 20976}, {3642, 53453}, {3767, 37640}, {5286, 41746}, {5309, 31693}, {5339, 39590}, {5357, 63493}, {5368, 61317}, {5872, 6115}, {6781, 42150}, {7127, 31451}, {7603, 42153}, {7737, 61319}, {7739, 63103}, {7747, 10654}, {7749, 42152}, {7752, 62984}, {7756, 10653}, {7775, 37786}, {7776, 69148}, {7778, 69169}, {7784, 69142}, {7818, 69137}, {7866, 69163}, {7888, 69157}, {7903, 69145}, {9112, 16964}, {9115, 37172}, {9698, 61331}, {10311, 64469}, {11542, 63534}, {11648, 61719}, {13881, 42988}, {16267, 18362}, {16626, 18582}, {16965, 65633}, {17130, 69138}, {18424, 42166}, {20977, 34395}, {22114, 63032}, {22496, 49947}, {22892, 63106}, {31400, 63080}, {31401, 37641}, {31489, 42989}, {37333, 51200}, {37665, 61320}, {39565, 42156}, {42159, 43457}, {42974, 44518}, {42982, 63536}, {47860, 62356}, {47863, 51753}, {63933, 69164}, {69106, 69115}, {69107, 69131}, {69108, 69117}, {69109, 69129}, {69110, 69119}, {69111, 69127}, {69114, 69122}, {69116, 69120}, {69139, 69149}

X(69183) = Brocard-circle-inverse of X(69144)
X(69183) = crosssum of X(2) and X(11488)
X(69183) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 69144}, {6, 61, 32}, {6, 11485, 69156}, {6, 36757, 39764}, {6, 69160, 62}, {13, 69113, 69141}, {62, 69160, 574}, {371, 372, 13350}, {2548, 42999, 5471}, {15513, 63196, 36836}, {22238, 63199, 37512}, {40694, 61332, 1506}, {42974, 44518, 69140}, {61719, 69124, 69112}, {63201, 69167, 53096}, {69110, 69119, 69162}, {69112, 69124, 11648}

X(69184) = (0, 1, 0, 1, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^2*b^2 + a^2*c^2 + 4*b^2*c^2 + 2*Sqrt[3]*a^2*S : :

X(69184) lies on these lines: {2, 6151}, {3, 69143}, {6, 9466}, {13, 7818}, {17, 7888}, {32, 69138}, {61, 17130}, {69, 5472}, {182, 25157}, {298, 7775}, {303, 7908}, {381, 69148}, {396, 7801}, {397, 7854}, {574, 69159}, {576, 25167}, {599, 22495}, {5340, 7873}, {6295, 22687}, {7622, 30472}, {7784, 69140}, {7794, 40693}, {7810, 10653}, {7813, 61332}, {7821, 42156}, {7863, 42152}, {7866, 69161}, {7867, 69108}, {7903, 69107}, {7913, 69111}, {7935, 69112}, {8556, 11486}, {8667, 69156}, {11168, 42913}, {11286, 69155}, {13881, 69169}, {42633, 59780}, {48656, 50855}, {61719, 69114}, {69139, 69150}

X(69184) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 69115, 7818}, {17, 69116, 7888}, {599, 42974, 69142}, {69107, 69133, 7903}, {69108, 69119, 7867}, {69111, 69129, 7913}, {69112, 69122, 7935}

X(69185) = (0, 1, 1, 1, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^2*b^2 - 2*b^4 + a^2*c^2 - 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69185) lies on these lines: {3, 69142}, {6, 7821}, {13, 17130}, {17, 69114}, {32, 69137}, {61, 7818}, {62, 7888}, {299, 7751}, {303, 7815}, {381, 69149}, {396, 7854}, {397, 7801}, {574, 69157}, {599, 42988}, {2482, 42151}, {3096, 62984}, {3180, 7856}, {5339, 31173}, {5471, 32816}, {5472, 7795}, {6292, 61332}, {7776, 69150}, {7778, 69144}, {7794, 40693}, {7810, 42152}, {7863, 10653}, {7873, 22236}, {7935, 69160}, {9466, 42156}, {11318, 69168}, {13881, 69170}, {16043, 63105}, {17131, 69106}, {32954, 69156}, {33458, 66326}, {37243, 47068}, {42974, 69143}, {61719, 69116}, {63932, 69155}, {63933, 69161}, {69109, 69133}, {69111, 69131}, {69112, 69120}

X(69185) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 69117, 17130}, {69106, 69119, 17131}

X(69186) = (0, 0, 0, 0, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(14)

Barycentrics    2*b^2*c^2 - Sqrt[3]*a^2*S : :

X(69186) lies on these lines: {6, 76}, {14, 7776}, {18, 69120}, {32, 69149}, {69, 398}, {99, 36843}, {183, 22236}, {299, 7773}, {302, 32821}, {315, 5339}, {325, 42153}, {350, 54438}, {381, 69137}, {394, 41001}, {395, 3926}, {396, 32828}, {622, 5868}, {631, 59541}, {1007, 42599}, {1078, 36836}, {1656, 69157}, {1909, 54437}, {1975, 22238}, {3760, 54403}, {3761, 54402}, {3785, 42147}, {3933, 40694}, {5013, 69166}, {5321, 32006}, {5340, 11185}, {5471, 7855}, {6337, 16773}, {6390, 42149}, {7750, 42154}, {7763, 16645}, {7767, 10654}, {7778, 69117}, {7781, 63198}, {7784, 69113}, {7799, 49906}, {7815, 63199}, {7818, 69147}, {7866, 69109}, {7867, 69168}, {9761, 53464}, {9766, 69131}, {10601, 41000}, {11287, 69123}, {11489, 32831}, {12154, 63950}, {14907, 43194}, {15271, 69160}, {16644, 32832}, {16772, 34229}, {17130, 69144}, {17131, 69150}, {23302, 32838}, {23303, 32829}, {32815, 42148}, {32816, 42163}, {32817, 59542}, {32819, 42155}, {32823, 43404}, {32826, 42165}, {32830, 37641}, {32833, 49948}, {32834, 37640}, {32836, 43229}, {32840, 63033}, {32869, 63102}, {32872, 63032}, {32874, 63103}, {32893, 49813}, {34505, 69112}, {37688, 43238}, {40693, 64093}, {41406, 68527}, {42156, 59635}, {42164, 64018}, {42975, 69145}, {42989, 69165}, {42998, 52713}, {43104, 63105}, {43228, 46951}, {69127, 69128}, {69141, 69142}, {69162, 69163}

X(69186) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 69106, 7776}, {18, 69120, 69158}, {69109, 69110, 7866}, {69113, 69114, 7784}, {69117, 69118, 7778}, {69123, 69124, 11287}, {69131, 69132, 9766}, {69149, 69170, 32}

X(69187) = (0, 0, 1, 0, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(14)

Barycentrics    b^4 + c^4 + Sqrt[3]*a^2*S : :

X(69187) lies on these lines: {6, 626}, {14, 69108}, {18, 15271}, {32, 69148}, {61, 7778}, {62, 7784}, {141, 11312}, {298, 7754}, {302, 11285}, {381, 69138}, {395, 7800}, {398, 7795}, {620, 36836}, {623, 18440}, {625, 42156}, {635, 3526}, {1656, 69159}, {3181, 7920}, {3734, 5339}, {3788, 22236}, {3934, 42153}, {5013, 69165}, {5340, 7825}, {5471, 7822}, {7761, 22238}, {7763, 63199}, {7789, 10654}, {7791, 63198}, {7815, 16645}, {7816, 42154}, {7818, 69144}, {7830, 36843}, {7842, 42155}, {7862, 16644}, {7865, 49948}, {7867, 69150}, {7876, 62983}, {8362, 61331}, {11287, 69167}, {11307, 53033}, {11318, 69119}, {17130, 69147}, {17131, 69168}, {22253, 69127}, {34508, 42975}, {41406, 63938}, {42150, 59545}, {42152, 44377}, {42989, 69166}, {43194, 69171}, {63933, 69107}, {69113, 69116}, {69115, 69118}, {69121, 69124}, {69129, 69132}, {69141, 69143}, {69158, 69160}, {69162, 69164}

X(69187) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 69108, 69139}, {18, 69122, 15271}, {69107, 69110, 63933}, {69127, 69130, 22253}, {69148, 69169, 32}

X(69188) = (0, 0, 1, 0, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(14)

Barycentrics    b^4 - 2*b^2*c^2 + c^4 + Sqrt[3]*a^2*S : :

X(69188) lies on these lines: {2, 63199}, {3, 62197}, {4, 16942}, {5, 61332}, {6, 13}, {15, 37637}, {16, 44526}, {18, 5013}, {32, 5339}, {39, 42153}, {61, 13881}, {62, 44518}, {187, 42154}, {230, 10654}, {302, 31859}, {383, 5334}, {395, 2549}, {396, 22491}, {398, 3767}, {574, 16645}, {1384, 62199}, {1656, 69160}, {1692, 41037}, {2548, 42163}, {3053, 16964}, {3055, 42910}, {3815, 18581}, {3851, 69133}, {5023, 42157}, {5052, 22694}, {5094, 30465}, {5107, 50858}, {5206, 43194}, {5210, 36967}, {5237, 44519}, {5238, 44535}, {5254, 40694}, {5306, 41113}, {5321, 7737}, {5340, 69141}, {5585, 42529}, {6114, 37071}, {6772, 9761}, {6775, 11159}, {6792, 52268}, {7603, 42095}, {7612, 43953}, {7736, 43404}, {7745, 42159}, {7746, 22236}, {7748, 22238}, {7749, 36836}, {7756, 36843}, {7818, 69170}, {7866, 69146}, {7867, 69149}, {8588, 42626}, {9300, 41120}, {9605, 69127}, {10646, 44541}, {10653, 53419}, {11165, 42035}, {11287, 69166}, {11296, 53447}, {11299, 11489}, {11301, 23303}, {11312, 53463}, {11485, 62198}, {11486, 43455}, {11543, 15048}, {11648, 49948}, {12155, 66587}, {15515, 42491}, {15993, 20426}, {16242, 53095}, {16268, 63200}, {16644, 50855}, {17130, 69169}, {17131, 69148}, {18362, 49947}, {18907, 43417}, {19780, 42126}, {21843, 42942}, {22513, 41038}, {26869, 30468}, {31401, 42599}, {31404, 42495}, {31489, 37835}, {32519, 43538}, {36450, 49262}, {36467, 49261}, {36970, 41406}, {37512, 43239}, {37641, 43448}, {39565, 42156}, {39601, 42098}, {40693, 63534}, {41039, 41044}, {41040, 53431}, {41108, 41407}, {41895, 54617}, {42093, 62203}, {42149, 63548}, {42814, 65630}, {42940, 43618}, {42943, 43619}, {42989, 69167}, {42998, 63533}, {43193, 65633}, {43276, 54140}, {44289, 61317}, {49824, 63006}, {49873, 63024}, {54569, 54669}, {63933, 69145}, {69150, 69162}

X(69188) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 69110, 6}, {18, 69124, 5013}, {32, 69147, 5339}, {395, 2549, 63198}, {5309, 5471, 6}, {11543, 15048, 61331}, {37835, 63201, 31489}, {69113, 69118, 3}, {69127, 69132, 9605}, {69141, 69144, 5340}, {69147, 69168, 32}

X(69189) = (0, 1, 0, 1, 0, 0, 1)-ADDITIVE ASSOCIATE OF X(14)

Barycentrics    a^2*(b^2 + c^2 - 2*Sqrt[3]*S) : :

X(69189) lies on these lines: {2, 46711}, {3, 6}, {4, 5471}, {13, 69132}, {14, 69112}, {18, 69119}, {115, 22507}, {298, 7896}, {299, 7916}, {302, 7862}, {381, 69140}, {395, 7746}, {396, 31455}, {397, 5475}, {398, 7748}, {1015, 54403}, {1500, 54402}, {1506, 40693}, {1968, 56515}, {2241, 7127}, {2548, 5472}, {2549, 42999}, {3170, 20998}, {3181, 7877}, {3411, 62197}, {3457, 20976}, {3643, 53464}, {3767, 37641}, {5286, 41745}, {5309, 31694}, {5340, 39590}, {5353, 63493}, {5368, 61318}, {5873, 6114}, {6151, 11004}, {6781, 42151}, {7603, 42156}, {7737, 61320}, {7739, 63102}, {7747, 10653}, {7749, 42149}, {7752, 62983}, {7756, 10654}, {7775, 37785}, {7776, 69142}, {7778, 69163}, {7784, 69148}, {7818, 69145}, {7866, 69169}, {7888, 69165}, {7903, 69137}, {9113, 16965}, {9117, 37173}, {9698, 61332}, {10311, 64468}, {11543, 63534}, {11648, 69113}, {13881, 42989}, {16268, 18362}, {16627, 18581}, {16964, 65633}, {17130, 69146}, {18424, 42163}, {20977, 34394}, {22113, 63033}, {22495, 49948}, {22848, 63105}, {31400, 63079}, {31401, 37640}, {31489, 42988}, {37332, 51203}, {37665, 61319}, {39565, 42153}, {42162, 43457}, {42975, 44518}, {42983, 63536}, {47859, 62356}, {47864, 51754}, {61719, 69133}, {63933, 69170}, {69106, 69130}, {69107, 69114}, {69108, 69128}, {69109, 69116}, {69110, 69126}, {69111, 69118}, {69115, 69123}, {69117, 69121}, {69139, 69143}

X(69189) = Brocard-circle-inverse of X(69150)
X(69189) = crosssum of X(2) and X(11489)
X(69189) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 69150}, {6, 62, 32}, {6, 11486, 69155}, {6, 36758, 39764}, {6, 69167, 61}, {14, 69112, 69141}, {61, 69167, 574}, {371, 372, 13349}, {2548, 42998, 5472}, {15513, 63197, 36843}, {22236, 63198, 37512}, {40693, 61331, 1506}, {42975, 44518, 69147}, {63200, 69160, 53096}, {69111, 69118, 69162}, {69113, 69125, 11648}

X(69190) = (0, 1, 0, 1, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(14)

Barycentrics    a^2*b^2 + a^2*c^2 + 4*b^2*c^2 - 2*Sqrt[3]*a^2*S : :

X(69190) lies on these lines: {2, 2981}, {3, 69149}, {6, 9466}, {14, 7818}, {18, 7888}, {32, 69146}, {62, 17130}, {69, 5471}, {182, 25167}, {299, 7775}, {302, 7908}, {381, 69142}, {395, 7801}, {398, 7854}, {574, 69166}, {576, 25157}, {599, 22496}, {5339, 7873}, {6582, 22689}, {7622, 30471}, {7784, 69147}, {7794, 40694}, {7810, 10654}, {7813, 61331}, {7821, 42153}, {7863, 42149}, {7866, 69168}, {7867, 69109}, {7903, 69106}, {7913, 69110}, {7935, 69113}, {8556, 11485}, {8667, 69155}, {11168, 42912}, {11286, 69156}, {13881, 69163}, {42634, 59780}, {48655, 50858}, {69139, 69144}

X(69190) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 69114, 7818}, {18, 69117, 7888}, {599, 42975, 69148}, {69106, 69132, 7903}, {69109, 69118, 7867}, {69110, 69128, 7913}, {69113, 69123, 7935}

X(69191) = (0, 1, 1, 1, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(14)

Barycentrics    a^2*b^2 - 2*b^4 + a^2*c^2 - 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69191) lies on these lines: {3, 69148}, {6, 7821}, {14, 17130}, {18, 69115}, {32, 69145}, {61, 7888}, {62, 7818}, {298, 7751}, {302, 7815}, {381, 69143}, {395, 7854}, {398, 7801}, {574, 69165}, {599, 42989}, {2482, 42150}, {3096, 62983}, {3181, 7856}, {5340, 31173}, {5471, 7795}, {5472, 32816}, {6292, 61331}, {7776, 69144}, {7778, 69150}, {7794, 40694}, {7810, 42149}, {7863, 10654}, {7873, 22238}, {7935, 69167}, {9466, 42153}, {11318, 69161}, {13881, 69164}, {16043, 63106}, {17131, 69107}, {32954, 69155}, {33459, 66326}, {37243, 47066}, {42975, 69149}, {63932, 69156}, {63933, 69168}, {69108, 69132}, {69110, 69130}, {69113, 69121}

X(69191) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 69116, 17130}, {69107, 69118, 17131}

X(69192) = (0, 1, 1, 1, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(13)

Barycentrics    a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69192) lies on these lines: {3, 69140}, {5, 69189}, {6, 3851}, {13, 32}, {15, 65633}, {17, 574}, {39, 42156}, {61, 69141}, {76, 69185}, {115, 22509}, {187, 5340}, {303, 7781}, {381, 69150}, {396, 7748}, {397, 7746}, {398, 18424}, {625, 69191}, {626, 69184}, {1506, 18582}, {2548, 43403}, {3054, 42924}, {3091, 5471}, {3767, 5472}, {5206, 16965}, {5254, 11542}, {5286, 22235}, {5335, 22531}, {5366, 43618}, {5475, 42166}, {6781, 42161}, {7738, 43542}, {7747, 42162}, {7749, 10653}, {7753, 41119}, {7756, 42152}, {7765, 61332}, {7772, 69111}, {7776, 69164}, {7778, 69143}, {7821, 69180}, {7867, 69138}, {7935, 69159}, {8588, 42158}, {8589, 43238}, {9466, 69181}, {11318, 69169}, {11648, 16267}, {13881, 42974}, {15513, 42155}, {16644, 37512}, {17131, 69137}, {18362, 61719}, {31415, 42494}, {31455, 42598}, {34509, 53463}, {37832, 69167}, {39563, 49947}, {39601, 42153}, {42128, 65630}, {42813, 62203}, {42815, 69156}, {42921, 43457}, {42988, 44518}, {42990, 62197}, {42998, 43620}, {43239, 63197}, {43418, 62232}, {46054, 47066}, {53096, 69125}, {59635, 69190}, {69139, 69163}

X(69192) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 69119, 32}, {17, 69112, 574}, {115, 40693, 69183}, {13881, 42974, 69144}, {16965, 62198, 5206}, {42156, 69182, 39}, {69111, 69133, 7772}

X(69193) = (0, 1, 1, 1, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(14)

Barycentrics    a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69193) lies on these lines: {3, 69147}, {5, 69183}, {6, 3851}, {14, 32}, {16, 65633}, {18, 574}, {39, 42153}, {62, 69141}, {76, 69191}, {115, 22507}, {187, 5339}, {302, 7781}, {381, 69144}, {395, 7748}, {397, 18424}, {398, 7746}, {625, 69185}, {626, 69190}, {1506, 18581}, {2548, 43404}, {3054, 42925}, {3091, 5472}, {3767, 5471}, {5206, 16964}, {5254, 11543}, {5286, 22237}, {5334, 22532}, {5365, 43618}, {5475, 42163}, {6781, 42160}, {7738, 43543}, {7747, 42159}, {7749, 10654}, {7753, 41120}, {7756, 42149}, {7765, 61331}, {7772, 69110}, {7776, 69170}, {7778, 69149}, {7821, 69186}, {7867, 69146}, {7935, 69166}, {8588, 42157}, {8589, 43239}, {9466, 69187}, {11318, 69163}, {11648, 16268}, {13881, 42975}, {15513, 42154}, {16645, 37512}, {17131, 69145}, {18362, 69119}, {31415, 42495}, {31455, 42599}, {34508, 53452}, {37835, 69160}, {39563, 49948}, {39601, 42156}, {42125, 65630}, {42814, 62203}, {42816, 69155}, {42920, 43457}, {42989, 44518}, {42991, 62198}, {42999, 43620}, {43238, 63196}, {43419, 62233}, {46053, 47068}, {53096, 69124}, {59635, 69184}, {69139, 69169}

X(69193) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 69118, 32}, {18, 69113, 574}, {115, 40694, 69189}, {13881, 42975, 69150}, {16964, 62197, 5206}, {42153, 69188, 39}, {69110, 69132, 7772}

X(69194) = (0, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(15)

Barycentrics    a^2*(Sqrt[3]*(b^2 + c^2) + 2*S) : :

X(69194) lies on these lines: {2, 2981}, {3, 6}, {13, 11648}, {17, 69127}, {115, 5613}, {230, 42124}, {299, 7865}, {303, 7844}, {396, 3642}, {398, 53455}, {538, 69184}, {617, 6775}, {622, 6772}, {628, 3767}, {1015, 54436}, {1250, 31451}, {1500, 54435}, {1506, 18581}, {1656, 69168}, {2241, 10638}, {2242, 7051}, {2275, 5357}, {2276, 5353}, {2548, 5334}, {2549, 5335}, {2896, 40901}, {3054, 43103}, {3096, 34541}, {3180, 7811}, {3457, 20859}, {3815, 11543}, {5254, 11542}, {5283, 5362}, {5286, 63032}, {5306, 42912}, {5318, 7748}, {5321, 5475}, {5471, 7736}, {6151, 63040}, {6656, 69185}, {6771, 43455}, {6778, 9981}, {6781, 42090}, {7603, 42095}, {7737, 42119}, {7745, 42117}, {7746, 23302}, {7747, 42085}, {7749, 42092}, {7753, 10654}, {7755, 42152}, {7756, 42086}, {7764, 69191}, {7765, 22907}, {7790, 62984}, {7796, 34540}, {7867, 69157}, {7935, 69137}, {8259, 42672}, {9698, 40694}, {9766, 69148}, {9886, 59373}, {10311, 10632}, {10616, 13083}, {11287, 69142}, {11466, 13509}, {11489, 31401}, {13084, 43229}, {13571, 40900}, {13881, 42132}, {14537, 42154}, {14609, 66873}, {15271, 69170}, {16241, 62199}, {16529, 69111}, {16773, 31457}, {16808, 69124}, {16809, 69113}, {16960, 69119}, {16962, 62200}, {16966, 36763}, {16967, 69118}, {17131, 69159}, {18362, 37832}, {18424, 42110}, {19106, 65633}, {19107, 62203}, {21647, 39643}, {22253, 69164}, {22532, 42998}, {22745, 22855}, {23303, 31455}, {31239, 69186}, {31400, 63033}, {31415, 42139}, {31467, 42818}, {31489, 42129}, {31704, 46855}, {33223, 63105}, {33416, 62197}, {33458, 66335}, {34509, 53428}, {36251, 63732}, {39565, 42098}, {39590, 42093}, {39593, 49947}, {39913, 67909}, {41745, 67071}, {41943, 62232}, {42103, 43457}, {42118, 63548}, {42125, 69147}, {42126, 65630}, {42127, 44526}, {42128, 44518}, {42131, 44519}, {42136, 53418}, {42138, 53419}, {42141, 43619}, {42146, 63534}, {42627, 43291}, {42815, 69140}, {42988, 69161}, {43463, 62992}, {69120, 69129}, {69122, 69131}, {69158, 69169}

X(69194) = isogonal conjugate of X(62877)
X(69194) = Brocard-circle-inverse of X(69156)
X(69194) = X(1)-isoconjugate of X(62877)
X(69194) = X(3)-Dao conjugate of X(62877)
X(69194) = crosssum of X(2) and X(37640)
X(69194) = barycentric quotient X(6)/X(62877)
X(69194) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 69156}, {6, 15, 32}, {6, 5013, 11486}, {6, 11485, 69155}, {6, 11486, 69144}, {6, 63199, 187}, {6, 63201, 574}, {6, 69160, 15}, {15, 3094, 574}, {15, 3105, 3098}, {17, 69127, 69162}, {39, 69183, 69189}, {61, 3107, 182}, {1504, 1505, 69183}, {10645, 19780, 5206}, {69124, 69133, 69141}

X(69195) = (0, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(16)

Barycentrics    a^2*(Sqrt[3]*(b^2 + c^2) - 2*S) : :

X(69195) lies on these lines: {2, 6151}, {3, 6}, {14, 11648}, {18, 69126}, {115, 5617}, {230, 42121}, {298, 7865}, {302, 7844}, {395, 3643}, {397, 53466}, {538, 69190}, {616, 6772}, {621, 6775}, {627, 3767}, {1015, 54435}, {1250, 2241}, {1500, 54436}, {1506, 18582}, {1656, 69161}, {2242, 19373}, {2275, 5353}, {2276, 5357}, {2548, 5335}, {2549, 5334}, {2896, 40900}, {2981, 63040}, {3054, 43102}, {3096, 34540}, {3181, 7811}, {3458, 20859}, {3815, 11542}, {5254, 11543}, {5283, 5367}, {5286, 63033}, {5306, 42913}, {5318, 5475}, {5321, 7748}, {5472, 7736}, {6656, 69191}, {6774, 43454}, {6777, 9982}, {6781, 42091}, {7603, 42098}, {7737, 42120}, {7745, 42118}, {7746, 23303}, {7747, 42086}, {7749, 42089}, {7753, 10653}, {7755, 42149}, {7756, 42085}, {7764, 69185}, {7765, 22861}, {7790, 62983}, {7796, 34541}, {7867, 69165}, {7935, 69145}, {8260, 42673}, {9698, 40693}, {9766, 69142}, {9885, 59373}, {10311, 10633}, {10617, 13084}, {10638, 31451}, {11287, 69148}, {11467, 13509}, {11488, 31401}, {13083, 43228}, {13571, 40901}, {13881, 42129}, {14537, 42155}, {14609, 66872}, {15271, 69164}, {16242, 62200}, {16530, 69110}, {16772, 31457}, {16808, 69112}, {16809, 69125}, {16961, 69118}, {16963, 62199}, {16966, 69119}, {16967, 69111}, {17131, 69166}, {18362, 37835}, {18424, 42107}, {19106, 62203}, {19107, 65633}, {21648, 39643}, {22253, 69170}, {22531, 42999}, {22746, 22901}, {23302, 31455}, {31239, 69180}, {31400, 63032}, {31415, 42142}, {31467, 42817}, {31489, 42132}, {31703, 46854}, {33223, 63106}, {33417, 62198}, {33459, 66335}, {34508, 53440}, {36252, 63731}, {39565, 42095}, {39590, 42094}, {39593, 49948}, {39913, 67910}, {41746, 67072}, {41944, 62233}, {42106, 43457}, {42117, 63548}, {42125, 44518}, {42126, 44526}, {42127, 65630}, {42128, 69140}, {42130, 44519}, {42135, 53419}, {42137, 53418}, {42140, 43619}, {42143, 63534}, {42628, 43291}, {42816, 69147}, {42989, 69168}, {43464, 62992}, {69121, 69128}, {69123, 69130}, {69158, 69163}

X(69195) = isogonal conjugate of X(62876)
X(69195) = Brocard-circle-inverse of X(69155)
X(69195) = X(1)-isoconjugate of X(62876)
X(69195) = X(3)-Dao conjugate of X(62876)
X(69195) = crosssum of X(2) and X(37641)
X(69195) = barycentric quotient X(6)/X(62876)
X(69195) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 69155}, {6, 16, 32}, {6, 5013, 11485}, {6, 11485, 69150}, {6, 11486, 69156}, {6, 63198, 187}, {6, 63200, 574}, {6, 69167, 16}, {16, 3094, 574}, {16, 3104, 3098}, {18, 69126, 69162}, {39, 69189, 69183}, {62, 3106, 182}, {1504, 1505, 69189}, {10646, 19781, 5206}, {69125, 69132, 69141}

X(69196) = (0, 1, 0, 1, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    3*a^2*b^2 + 3*a^2*c^2 + 4*b^2*c^2 + 2*Sqrt[3]*a^2*S : :

X(69196) lies on these lines: {2, 46711}, {6, 24861}, {13, 7935}, {17, 7867}, {32, 69159}, {39, 69180}, {141, 69185}, {303, 7869}, {396, 7822}, {574, 69138}, {1506, 69191}, {1656, 69169}, {3763, 42988}, {3934, 69183}, {5013, 69143}, {5471, 32968}, {5472, 7800}, {6292, 40693}, {7603, 69187}, {7794, 61332}, {7818, 69122}, {7820, 42152}, {7853, 42156}, {7874, 16644}, {7888, 69108}, {7903, 69115}, {7913, 69119}, {9605, 69164}, {11287, 69140}, {15271, 69144}, {17130, 69160}, {31457, 59542}

X(69196) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {17, 69129, 7867}, {3763, 42988, 69163}, {3934, 69183, 69190}, {69122, 69133, 7818}

X(69197) = (0, 1, 1, 1, 0, 1, 0)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    3*a^2*b^2 - 2*b^4 + 3*a^2*c^2 - 2*c^4 : :
X(69197) = 2 X[7906] + X[39565]

X(69197) lies on these lines: {2, 5041}, {3, 7845}, {5, 7813}, {6, 7874}, {17, 69130}, {18, 69131}, {32, 9766}, {39, 325}, {69, 31455}, {76, 7603}, {83, 7880}, {99, 7843}, {140, 7826}, {141, 9698}, {183, 7916}, {187, 7759}, {194, 625}, {211, 6786}, {230, 7890}, {315, 33008}, {316, 19691}, {524, 7749}, {538, 7752}, {574, 7776}, {599, 31467}, {620, 7762}, {754, 15513}, {1007, 7746}, {1078, 7840}, {1506, 3933}, {1656, 17131}, {1975, 7775}, {2023, 41756}, {2548, 7801}, {2549, 32823}, {2896, 15810}, {3054, 63926}, {3096, 44562}, {3266, 15820}, {3314, 6683}, {3329, 7909}, {3526, 39561}, {3552, 7926}, {3734, 32821}, {3767, 31275}, {3788, 5007}, {3815, 7794}, {3849, 7782}, {3926, 5475}, {3934, 7777}, {5008, 7807}, {5013, 7818}, {5024, 7935}, {5025, 32450}, {5031, 41622}, {5052, 51371}, {5206, 63932}, {5305, 22110}, {5319, 37690}, {5355, 8361}, {5650, 52906}, {6179, 58448}, {6292, 31406}, {6390, 7747}, {6680, 41624}, {6704, 63101}, {7736, 7822}, {7737, 32831}, {7745, 7863}, {7748, 31173}, {7750, 8589}, {7753, 7789}, {7754, 7862}, {7755, 44377}, {7756, 59546}, {7757, 7861}, {7760, 7886}, {7761, 31652}, {7766, 7940}, {7769, 7779}, {7770, 7908}, {7771, 7946}, {7772, 7778}, {7773, 7781}, {7783, 7809}, {7784, 53096}, {7785, 7799}, {7786, 7849}, {7787, 7870}, {7788, 7815}, {7792, 41940}, {7793, 7949}, {7795, 32825}, {7798, 7887}, {7804, 7836}, {7808, 7881}, {7812, 7891}, {7817, 7839}, {7823, 32456}, {7824, 7848}, {7825, 31859}, {7828, 13571}, {7832, 63018}, {7835, 7921}, {7837, 7857}, {7846, 63028}, {7850, 33004}, {7851, 39593}, {7854, 31401}, {7867, 9605}, {7868, 39784}, {7869, 11174}, {7877, 7907}, {7878, 7945}, {7879, 15482}, {7889, 9300}, {7896, 11285}, {7929, 40344}, {7930, 62994}, {7931, 55085}, {8588, 63938}, {8716, 65633}, {11008, 32977}, {11165, 44519}, {13330, 51397}, {13586, 51581}, {14023, 32829}, {14148, 32819}, {14711, 59635}, {14981, 52854}, {15704, 51587}, {16923, 63929}, {17008, 63934}, {19569, 45017}, {20080, 32839}, {21843, 32835}, {22253, 69162}, {31415, 32830}, {31476, 69094}, {32447, 37004}, {32457, 32966}, {32458, 46321}, {32826, 66466}, {33222, 62995}, {33250, 35022}, {33274, 63942}, {34504, 52943}, {34571, 63017}, {39601, 63924}, {44535, 63936}, {46313, 51373}, {57518, 59560}, {62362, 63930}, {69120, 69132}, {69121, 69133}, {69142, 69167}, {69144, 69165}, {69148, 69160}, {69150, 69157}

X(69197) = midpoint of X(i) and X(j) for these {i,j}: {7752, 7906}, {7782, 7900}, {7793, 7949}
X(69197) = reflection of X(39565) in X(7752)
X(69197) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7871, 7895}, {2, 7905, 7805}, {3, 7903, 7845}, {6, 7888, 7874}, {17, 69130, 69164}, {18, 69131, 69170}, {39, 325, 7821}, {39, 7821, 7853}, {83, 7947, 7880}, {99, 7941, 7843}, {140, 50771, 7826}, {194, 7814, 625}, {325, 7764, 39}, {574, 7776, 7873}, {620, 7762, 35007}, {1007, 7758, 7746}, {1078, 7840, 7882}, {1506, 3933, 9466}, {1975, 7775, 39590}, {2548, 32818, 7801}, {3329, 7909, 7915}, {3788, 7774, 5007}, {3815, 7794, 31239}, {7748, 32816, 31173}, {7757, 7912, 7861}, {7759, 7763, 187}, {7760, 7925, 7886}, {7769, 7779, 7780}, {7772, 7778, 7852}, {7777, 7796, 3934}, {7783, 7809, 7842}, {7785, 7799, 7816}, {7785, 7816, 14537}, {7786, 7897, 7849}, {7807, 7838, 5008}, {7824, 7917, 7848}, {7836, 7858, 7804}, {7839, 7899, 7817}, {7881, 11163, 7808}, {9766, 69158, 32}, {9770, 32818, 2548}, {31401, 37668, 7854}, {32816, 34511, 7748}, {32825, 62988, 7795}, {39590, 39785, 1975}, {69120, 69132, 69149}, {69121, 69133, 69143}

X(69198) = (0, 1, 1, 1, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    3*a^2*b^2 - 2*b^4 + 3*a^2*c^2 - 2*c^4 : :3*a^2*b^2 - 2*b^4 + 3*a^2*c^2 - 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69198) lies on these lines: {6, 7874}, {17, 17131}, {32, 69157}, {39, 69181}, {61, 7903}, {299, 7815}, {303, 7751}, {325, 69183}, {396, 7855}, {574, 69137}, {1506, 69190}, {1656, 69170}, {3107, 37004}, {3180, 7857}, {3926, 5472}, {3933, 69184}, {5013, 69142}, {6033, 47068}, {7603, 69186}, {7764, 69189}, {7794, 61332}, {7796, 62984}, {7813, 40693}, {7818, 69160}, {7826, 42152}, {7845, 22236}, {7873, 63199}, {7935, 63201}, {9605, 69163}, {9766, 69150}, {17130, 69120}, {22253, 69161}, {42988, 69164}, {69144, 69158}

X(69198) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {17, 69131, 17131}, {325, 69183, 69191}, {69120, 69133, 17130}

X(69199) = (0, 1, 1, 1, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(17)

Barycentrics    3*a^2*b^2 - 2*b^4 + 3*a^2*c^2 + 4*b^2*c^2 - 2*c^4 + 2*Sqrt[3]*a^2*S : :

X(69199) lies on these lines: {2, 5472}, {5, 69183}, {6, 5055}, {13, 574}, {15, 62203}, {16, 59383}, {17, 32}, {39, 42156}, {61, 41094}, {115, 5613}, {187, 16644}, {303, 3734}, {325, 69184}, {396, 5475}, {397, 31455}, {1506, 40693}, {1656, 69144}, {2549, 43403}, {3094, 22688}, {3643, 23302}, {3815, 11542}, {3851, 69147}, {3934, 69185}, {5013, 69140}, {5340, 37512}, {5471, 31415}, {6115, 63732}, {6775, 59378}, {7617, 37786}, {7736, 43542}, {7737, 11488}, {7746, 42598}, {7747, 42152}, {7748, 42166}, {7752, 69191}, {7756, 42162}, {7772, 69119}, {7818, 69159}, {7888, 69138}, {8018, 15004}, {8588, 16241}, {8589, 42155}, {9112, 16530}, {9605, 69161}, {9736, 47859}, {9766, 69164}, {10654, 43457}, {11648, 41121}, {13083, 53434}, {14537, 49905}, {15271, 69142}, {15484, 42817}, {15513, 43238}, {15515, 16965}, {17130, 69157}, {18362, 49907}, {18584, 42975}, {22235, 31400}, {22236, 39590}, {22512, 59397}, {27088, 33475}, {31239, 69181}, {31489, 42974}, {37637, 42132}, {39601, 42098}, {42127, 44541}, {42128, 44526}, {42154, 63196}, {42581, 69118}, {42813, 65633}, {42912, 53418}, {42986, 61320}, {42988, 69150}, {42992, 69167}, {46053, 64092}, {47066, 52642}, {53096, 69112}, {62600, 69171}, {69141, 69160}, {69143, 69158}

X(69199) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {17, 69133, 32}, {1506, 40693, 69189}, {9112, 16966, 62197}, {18582, 61332, 115}, {31415, 37640, 5471}, {42098, 69188, 39601}

X(69200) = (0, 1, 0, 1, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(18)

Barycentrics    3*a^2*b^2 + 3*a^2*c^2 + 4*b^2*c^2 - 2*Sqrt[3]*a^2*S : :

X(69200) lies on these lines: {2, 46710}, {6, 24861}, {14, 7935}, {18, 7867}, {32, 69166}, {39, 69186}, {141, 69191}, {302, 7869}, {395, 7822}, {574, 69146}, {1506, 69185}, {1656, 69163}, {3763, 42989}, {3934, 69184}, {5013, 69149}, {5471, 7800}, {5472, 32968}, {6292, 40694}, {7603, 69181}, {7794, 61331}, {7818, 69123}, {7820, 42149}, {7853, 42153}, {7874, 16645}, {7888, 69109}, {7903, 69114}, {7913, 69118}, {9605, 69170}, {11287, 69147}, {15271, 69150}, {17130, 69167}, {31457, 59541}

X(69200) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {18, 69128, 7867}, {3763, 42989, 69169}, {3934, 69189, 69184}, {69123, 69132, 7818}

X(69201) = (0, 1, 1, 1, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(18)

Barycentrics    3*a^2*b^2 - 2*b^4 + 3*a^2*c^2 - 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69201) lies on these lines: {6, 7874}, {18, 17131}, {32, 69165}, {39, 69187}, {62, 7903}, {298, 7815}, {302, 7751}, {325, 69185}, {395, 7855}, {574, 69145}, {1506, 69184}, {1656, 69164}, {3106, 37004}, {3181, 7857}, {3926, 5471}, {3933, 69190}, {5013, 69148}, {6033, 47066}, {7603, 69180}, {7764, 69183}, {7794, 61331}, {7796, 62983}, {7813, 40694}, {7818, 69167}, {7826, 42149}, {7845, 22238}, {7873, 63198}, {7935, 63200}, {9605, 69169}, {9766, 69144}, {17130, 69121}, {22253, 69168}, {42989, 69170}, {69150, 69158}

X(69201) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {18, 69130, 17131}, {325, 69189, 69185}, {69121, 69132, 17130}

X(69202) = (0, 1, 1, 1, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(18)

Barycentrics    3*a^2*b^2 - 2*b^4 + 3*a^2*c^2 + 4*b^2*c^2 - 2*c^4 - 2*Sqrt[3]*a^2*S : :

X(69202) lies on these lines: {2, 5471}, {5, 69189}, {6, 5055}, {14, 574}, {15, 59384}, {16, 62203}, {18, 32}, {39, 42153}, {62, 41098}, {115, 5617}, {187, 16645}, {302, 3734}, {325, 69190}, {395, 5475}, {398, 31455}, {1506, 40694}, {1656, 69150}, {2549, 43404}, {3094, 22690}, {3642, 23303}, {3815, 11543}, {3851, 69140}, {3934, 69191}, {5013, 69147}, {5339, 37512}, {5472, 31415}, {6114, 63731}, {6772, 59379}, {7617, 37785}, {7736, 43543}, {7737, 11489}, {7746, 42599}, {7747, 42149}, {7748, 42163}, {7752, 69185}, {7756, 42159}, {7772, 69118}, {7818, 69166}, {7888, 69146}, {8019, 15004}, {8588, 16242}, {8589, 42154}, {9113, 16529}, {9605, 69168}, {9735, 47860}, {9766, 69170}, {10653, 43457}, {11648, 41122}, {13084, 53446}, {14537, 49906}, {15271, 69148}, {15484, 42818}, {15513, 43239}, {15515, 16964}, {17130, 69165}, {18362, 49908}, {18584, 42974}, {22237, 31400}, {22238, 39590}, {22513, 59398}, {27088, 33474}, {31239, 69187}, {31489, 42975}, {37637, 42129}, {39601, 42095}, {42125, 44526}, {42126, 44541}, {42155, 63197}, {42580, 69119}, {42814, 65633}, {42913, 53418}, {42987, 61319}, {42989, 69144}, {42993, 69160}, {46054, 64092}, {47068, 52643}, {53096, 69113}, {62601, 69171}, {69141, 69167}, {69149, 69158}

X(69202) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {18, 69132, 32}, {1506, 40694, 69183}, {9113, 16967, 62198}, {18581, 61331, 115}, {31415, 37641, 5472}, {42095, 69182, 39601}

X(69203) = (1, -1, 1, -1)-ADDITIVE ASSOCIATE OF X(19)

Barycentrics    a*(a^4 + b^4 + 2*b^2*c^2 + c^4) : :

X(69203) lies on these lines: {1, 75}, {19, 4118}, {38, 48}, {63, 560}, {976, 1818}, {1496, 8766}, {1582, 1760}, {1910, 17467}, {1958, 17446}, {3938, 46395}, {17457, 62833}, {18596, 21336}, {24207, 30885}

X(69203) = X(2)-isoconjugate of X(56344)
X(69203) = X(32664)-Dao conjugate of X(56344)
X(69203) = barycentric product X(i)*X(j) for these {i,j}: {1, 7795}, {662, 50552}
X(69203) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 56344}, {7795, 75}, {50552, 1577}
X(69203) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 326, 1964}, {1582, 51836, 1760}

X(69204) = (1, -1, -1, -1)-ADDITIVE ASSOCIATE OF X(19)

Barycentrics    a*(a^4 + b^4 - 2*b^2*c^2 + c^4) : :

X(69204) lies on these lines: {1, 75}, {19, 560}, {31, 1820}, {42, 21675}, {48, 8772}, {57, 20274}, {63, 4118}, {610, 922}, {614, 30742}, {872, 3553}, {1096, 6520}, {1580, 1760}, {1582, 17891}, {1707, 17472}, {1918, 5336}, {2273, 21804}, {2654, 3924}, {3248, 3554}, {3959, 23868}, {9620, 21035}, {16968, 20593}, {17446, 52134}, {17901, 21593}, {18041, 33760}, {21045, 69153}, {22343, 62216}, {23688, 24315}, {36120, 57806}

X(69204) = isogonal conjugate of the isotomic conjugate of X(17871)
X(69204) = polar conjugate of the isotomic conjugate of X(2083)
X(69204) = X(i)-Ceva conjugate of X(j) for these (i,j): {823, 798}, {17871, 2083}
X(69204) = X(i)-isoconjugate of X(j) for these (i,j): {2, 56004}, {3, 34405}, {6, 42407}, {69, 56307}, {512, 42297}, {1092, 57851}, {3926, 56364}, {3964, 57684}
X(69204) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 42407}, {6389, 75}, {14713, 1}, {32664, 56004}, {36103, 34405}, {39054, 42297}, {52532, 38}, {53848, 1102}
X(69204) = crosspoint of X(1) and X(1096)
X(69204) = crosssum of X(1) and X(326)
X(69204) = crossdifference of every pair of points on line {798, 63827}
X(69204) = barycentric product X(i)*X(j) for these {i,j}: {1, 3767}, {4, 2083}, {6, 17871}, {19, 1899}, {31, 41760}, {63, 41762}, {75, 42295}, {92, 40947}, {158, 39643}, {426, 6520}, {610, 62545}, {661, 1632}, {1096, 6389}, {1910, 2450}, {1973, 41009}, {2156, 41761}, {2168, 27362}, {57806, 61360}
X(69204) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 42407}, {19, 34405}, {31, 56004}, {426, 1102}, {662, 42297}, {1632, 799}, {1899, 304}, {1973, 56307}, {2083, 69}, {2450, 46238}, {3767, 75}, {6520, 57851}, {17871, 76}, {39643, 326}, {40947, 63}, {41009, 40364}, {41760, 561}, {41761, 20641}, {41762, 92}, {42295, 1}, {61360, 255}, {62545, 57921}
X(69204) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1740, 44179}, {1, 33781, 326}, {326, 33781, 2234}, {560, 2643, 19}, {3924, 4336, 40934}, {8772, 17872, 48}

X(69205) = (1, 0, 1, 0)-ADDITIVE ASSOCIATE OF X(19)

Barycentrics    a*(a^4 + 2*b^2*c^2) : :

X(69205) lies on these lines: {1, 662}, {6, 7241}, {19, 4118}, {31, 2234}, {38, 2173}, {48, 17445}, {63, 2244}, {75, 560}, {82, 1740}, {922, 52134}, {1958, 1964}, {2210, 4363}, {2278, 21352}, {3264, 4112}, {3758, 8300}, {3764, 8301}, {4361, 7122}, {4381, 20432}, {4412, 21442}, {9459, 24264}, {12263, 30882}, {16545, 21336}, {16568, 51836}, {16571, 33760}, {18209, 48627}, {18837, 37204}, {20274, 27059}, {24688, 26232}, {36289, 60686}

X(69205) = X(i)-isoconjugate of X(j) for these (i,j): {2, 44557}, {523, 65836}
X(69205) = X(32664)-Dao conjugate of X(44557)
X(69205) = barycentric product X(1)*X(3734)
X(69205) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 44557}, {163, 65836}, {3734, 75}
X(69205) = {X(75),X(1582)}-harmonic conjugate of X(560)

X(69206) = (1, 1, -1, 1, 1, -1)-ADDITIVE ASSOCIATE OF X(20)

Barycentrics    3*a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4 : :
X(69206) = 2 X[1384] - 3 X[37809], X[1384] - 3 X[68718], X[7735] - 3 X[33191], X[32817] + 3 X[33191]

X(69206) lies on these lines: {2, 99}, {3, 66}, {4, 625}, {6, 6390}, {20, 626}, {30, 7778}, {32, 193}, {39, 3618}, {69, 187}, {76, 2021}, {83, 14037}, {114, 7694}, {140, 52771}, {183, 21843}, {194, 5319}, {230, 11288}, {315, 3552}, {316, 7870}, {325, 1003}, {343, 35302}, {376, 7761}, {381, 44377}, {384, 2548}, {385, 32833}, {439, 3785}, {441, 34360}, {491, 35306}, {492, 35305}, {524, 1384}, {538, 7735}, {549, 15271}, {550, 7784}, {591, 61389}, {597, 11165}, {599, 5210}, {621, 37172}, {622, 37173}, {631, 3934}, {754, 7908}, {1007, 5475}, {1078, 32964}, {1211, 16436}, {1506, 32829}, {1609, 22241}, {1899, 2936}, {1975, 3767}, {1991, 61388}, {1992, 5008}, {2715, 53186}, {2896, 33014}, {2996, 69162}, {3053, 3793}, {3091, 7862}, {3096, 32965}, {3146, 7825}, {3229, 34341}, {3314, 13586}, {3329, 14036}, {3363, 18584}, {3455, 11442}, {3522, 7830}, {3523, 7815}, {3528, 7849}, {3529, 7842}, {3580, 52275}, {3589, 5024}, {3619, 8589}, {3620, 7810}, {3629, 21309}, {3630, 63950}, {3631, 15655}, {3763, 8359}, {3815, 11286}, {3972, 7774}, {4159, 35926}, {4293, 69174}, {4657, 37599}, {4851, 37589}, {5007, 62995}, {5013, 7819}, {5021, 59538}, {5023, 7767}, {5026, 8724}, {5032, 14075}, {5054, 58446}, {5067, 39142}, {5108, 14685}, {5149, 9744}, {5204, 69097}, {5215, 23055}, {5217, 69095}, {5218, 69136}, {5224, 21937}, {5237, 69109}, {5238, 69108}, {5241, 11343}, {5254, 32954}, {5286, 6680}, {5304, 7798}, {5306, 22253}, {5309, 33224}, {5351, 69123}, {5352, 69122}, {5486, 9145}, {5569, 42850}, {5743, 21509}, {5921, 10991}, {6250, 37342}, {6251, 37343}, {6292, 15515}, {6309, 13357}, {6329, 9605}, {6392, 7755}, {6394, 67008}, {6560, 45473}, {6561, 45472}, {6655, 7945}, {6658, 7912}, {6661, 11174}, {6683, 16045}, {6776, 14981}, {6781, 7818}, {7493, 30749}, {7603, 32983}, {7610, 59780}, {7697, 38750}, {7736, 7804}, {7738, 7834}, {7739, 7792}, {7745, 68527}, {7746, 32970}, {7747, 7888}, {7748, 7874}, {7749, 17130}, {7750, 7881}, {7751, 32830}, {7752, 14035}, {7753, 32837}, {7754, 32820}, {7756, 7867}, {7757, 16989}, {7759, 32818}, {7762, 32821}, {7764, 32831}, {7769, 16924}, {7770, 31401}, {7771, 16990}, {7772, 63123}, {7773, 19687}, {7775, 63098}, {7776, 68513}, {7782, 7791}, {7783, 7803}, {7785, 68517}, {7786, 16898}, {7796, 20065}, {7802, 7909}, {7808, 31400}, {7809, 33187}, {7811, 33266}, {7817, 15301}, {7821, 32006}, {7822, 16043}, {7823, 7947}, {7831, 33008}, {7833, 7931}, {7838, 32841}, {7841, 43619}, {7843, 32823}, {7847, 7930}, {7848, 47101}, {7850, 51224}, {7853, 32986}, {7854, 15513}, {7855, 35007}, {7861, 32951}, {7864, 14043}, {7865, 10304}, {7866, 63548}, {7868, 8356}, {7872, 33180}, {7879, 68516}, {7883, 33208}, {7885, 33257}, {7886, 32822}, {7887, 32819}, {7889, 53096}, {7895, 63935}, {7898, 33265}, {7899, 14063}, {7900, 68520}, {7902, 33183}, {7903, 32825}, {7904, 33276}, {7907, 17006}, {7910, 33253}, {7911, 32997}, {7914, 33202}, {7915, 32956}, {7922, 33254}, {7923, 14067}, {7925, 11361}, {7928, 33275}, {7929, 68518}, {7934, 33017}, {7935, 33023}, {7938, 33260}, {7940, 32961}, {7941, 68524}, {8352, 11164}, {8354, 44541}, {8355, 66587}, {8357, 44519}, {8361, 44518}, {8362, 15815}, {8368, 8716}, {8370, 31415}, {8722, 10519}, {9306, 10836}, {9466, 33216}, {9597, 30104}, {9598, 30103}, {9734, 24206}, {9737, 19130}, {9766, 18907}, {10299, 18840}, {10328, 37457}, {10334, 39652}, {11147, 15810}, {11157, 13789}, {11158, 13669}, {11159, 22110}, {11163, 35954}, {11171, 24256}, {11291, 26362}, {11292, 26361}, {11315, 39660}, {11316, 39661}, {11317, 41133}, {11318, 53419}, {11328, 59765}, {11336, 24855}, {11842, 13196}, {12040, 42849}, {12150, 63017}, {13085, 62367}, {14357, 34161}, {14482, 66447}, {15031, 32963}, {15534, 19661}, {15561, 37348}, {15692, 34624}, {16041, 67536}, {16431, 69092}, {16845, 36812}, {16909, 27162}, {16951, 34254}, {16986, 33273}, {17005, 66413}, {17044, 59580}, {17131, 32836}, {17539, 26099}, {18424, 31275}, {18581, 37340}, {18582, 37341}, {18880, 18881}, {18911, 34013}, {19697, 31406}, {19708, 40344}, {20423, 44380}, {22401, 28696}, {22491, 44383}, {22492, 44382}, {26978, 56781}, {27076, 59572}, {30435, 32455}, {30747, 31099}, {31239, 32978}, {31276, 33259}, {31404, 32835}, {31455, 32968}, {31489, 66415}, {31670, 35002}, {32457, 33231}, {32782, 35276}, {32826, 32972}, {32827, 62203}, {32839, 32987}, {32955, 63533}, {32969, 39565}, {32982, 65633}, {33004, 46226}, {33184, 44526}, {33190, 34504}, {33203, 63924}, {33233, 59635}, {33269, 62362}, {33813, 37242}, {34505, 43291}, {34507, 47113}, {35282, 61631}, {36163, 47326}, {36212, 37645}, {36890, 48451}, {37188, 44436}, {37344, 37648}, {37450, 63424}, {37459, 64653}, {37461, 54173}, {37637, 64093}, {37647, 44543}, {38907, 46323}, {40727, 44401}, {41275, 43653}, {41406, 69120}, {41407, 69121}, {41760, 58846}, {42147, 69187}, {42148, 69181}, {42149, 69146}, {42152, 69138}, {44395, 50955}, {44678, 66391}, {46127, 51999}, {47426, 59543}, {50774, 63954}, {51123, 63633}, {52438, 58354}, {52713, 58448}, {55732, 61138}, {55823, 60143}, {56967, 67396}, {57630, 67681}, {57631, 67692}, {59706, 67890}, {61304, 66458}, {61506, 68705}, {65630, 68177}, {66455, 66699}

X(69206) = midpoint of X(7735) and X(32817)
X(69206) = reflection of X(i) in X(j) for these {i,j}: {37668, 7908}, {37809, 68718}, {43448, 7844}
X(69206) = complement of X(43448)
X(69206) = anticomplement of X(7844)
X(69206) = complement of the isotomic conjugate of X(63179)
X(69206) = X(i)-complementary conjugate of X(j) for these (i,j): {10603, 20305}, {63179, 2887}, {63181, 226}
X(69206) = X(60317)-Ceva conjugate of X(524)
X(69206) = crosspoint of X(i) and X(j) for these (i,j): {2, 63179}, {4590, 65324}
X(69206) = crosssum of X(i) and X(j) for these (i,j): {6, 62702}, {3124, 68778}
X(69206) = crossdifference of every pair of points on line {351, 2485}
X(69206) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 99, 2549}, {2, 2482, 7618}, {2, 7620, 5461}, {2, 11185, 43620}, {2, 32815, 115}, {2, 43448, 7844}, {3, 7789, 7795}, {3, 7795, 7800}, {3, 54075, 6389}, {4, 37690, 625}, {6, 6390, 34511}, {20, 53033, 626}, {32, 3926, 7758}, {32, 7813, 193}, {32, 7863, 3926}, {69, 32985, 187}, {99, 7835, 2}, {141, 32459, 3}, {141, 59545, 32459}, {183, 35297, 21843}, {187, 7801, 69}, {193, 3926, 7813}, {193, 7813, 7758}, {316, 33007, 43618}, {325, 1003, 7737}, {384, 7763, 2548}, {384, 7891, 7763}, {439, 3785, 5206}, {574, 7820, 2}, {599, 27088, 8182}, {620, 2482, 15483}, {620, 3734, 2}, {625, 3788, 37690}, {626, 69171, 20}, {1007, 14033, 5475}, {1975, 7807, 3767}, {2482, 7820, 574}, {3053, 3933, 14023}, {3053, 40341, 3793}, {3314, 13586, 14907}, {3552, 7836, 315}, {3552, 7897, 14712}, {3763, 53095, 8359}, {3788, 7816, 4}, {3793, 3933, 40341}, {3793, 40341, 14023}, {3926, 32973, 32}, {3972, 7799, 7774}, {5024, 33237, 3589}, {5206, 7794, 3785}, {5286, 33181, 6680}, {6292, 15515, 32990}, {6337, 14001, 39}, {6390, 8369, 6}, {6680, 7781, 5286}, {6781, 7818, 64018}, {7617, 22247, 2}, {7736, 14039, 7804}, {7738, 14069, 7834}, {7747, 7888, 32816}, {7748, 7874, 14064}, {7749, 17130, 32828}, {7756, 7867, 32974}, {7761, 32456, 376}, {7774, 33255, 3972}, {7782, 7832, 7791}, {7783, 7892, 7803}, {7789, 32459, 141}, {7789, 59545, 3}, {7789, 59702, 54075}, {7792, 31859, 7739}, {7792, 59634, 31859}, {7822, 37512, 16043}, {7836, 14712, 7897}, {7863, 32973, 7758}, {7880, 32456, 7761}, {7881, 33235, 7750}, {7897, 14712, 315}, {7907, 17128, 32832}, {9885, 9886, 53142}, {11159, 22110, 66466}, {11185, 43620, 7615}, {18424, 31275, 32984}, {31400, 33198, 7808}, {31859, 33220, 7792}, {32816, 32981, 7747}, {32817, 33191, 7735}, {32826, 32972, 69141}, {32828, 32989, 7749}, {32829, 32971, 1506}, {32836, 37667, 17131}, {32983, 34803, 7603}, {33220, 59634, 7739}, {35927, 64018, 6781}, {68527, 69158, 7745}

X(69207) = (1, 1, -1, 1, -1, -1)-ADDITIVE ASSOCIATE OF X(20)

Barycentrics    3*a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4 : :
X(69207) = X[6337] - 3 X[33216], X[2996] + 3 X[35287]

X(69207) lies on these lines: {2, 32}, {3, 230}, {4, 187}, {5, 3053}, {6, 140}, {17, 41406}, {18, 41407}, {20, 115}, {22, 44527}, {26, 2165}, {30, 5023}, {39, 631}, {50, 18281}, {53, 3517}, {66, 45838}, {68, 32661}, {69, 1692}, {76, 2021}, {98, 8721}, {99, 32964}, {112, 7505}, {141, 32954}, {148, 33014}, {172, 498}, {183, 7795}, {193, 7764}, {194, 33259}, {217, 3231}, {232, 3147}, {252, 40633}, {262, 46321}, {316, 32961}, {325, 14023}, {371, 62202}, {372, 62201}, {376, 7748}, {382, 63534}, {384, 17004}, {385, 7758}, {439, 32815}, {468, 2207}, {485, 12968}, {486, 12963}, {499, 1914}, {524, 69158}, {538, 6337}, {543, 2996}, {548, 44526}, {549, 5013}, {550, 5210}, {574, 3523}, {575, 63043}, {577, 3546}, {590, 6423}, {609, 9596}, {615, 6424}, {620, 3926}, {625, 32006}, {632, 31489}, {633, 11489}, {634, 11488}, {637, 50374}, {638, 50375}, {671, 33208}, {1003, 59635}, {1007, 7759}, {1015, 7288}, {1125, 1572}, {1151, 49221}, {1152, 49220}, {1153, 59373}, {1184, 7499}, {1196, 7494}, {1285, 5067}, {1352, 1691}, {1384, 1656}, {1500, 5218}, {1504, 9540}, {1505, 13935}, {1513, 59363}, {1569, 20081}, {1570, 63428}, {1571, 10164}, {1573, 30478}, {1574, 59572}, {1587, 33372}, {1588, 9675}, {1609, 6642}, {1611, 6676}, {1657, 15655}, {1658, 2079}, {1899, 14585}, {1968, 3542}, {1970, 39571}, {1971, 14216}, {1975, 35297}, {1992, 45796}, {2023, 9821}, {2028, 52095}, {2029, 52096}, {2030, 34507}, {2076, 31670}, {2241, 3086}, {2242, 3085}, {2271, 37646}, {2305, 5747}, {2387, 63555}, {2794, 39647}, {3003, 46262}, {3016, 41367}, {3055, 21309}, {3068, 5062}, {3069, 5058}, {3090, 5475}, {3091, 7747}, {3094, 43456}, {3146, 6781}, {3172, 37453}, {3199, 6353}, {3291, 7493}, {3314, 33245}, {3329, 33015}, {3515, 27376}, {3522, 7756}, {3524, 5309}, {3525, 5007}, {3526, 3815}, {3529, 63533}, {3530, 15048}, {3533, 5008}, {3541, 10311}, {3543, 18362}, {3545, 39590}, {3548, 10316}, {3552, 11185}, {3618, 5052}, {3619, 7915}, {3620, 7869}, {3628, 18907}, {3734, 32828}, {3763, 33185}, {3787, 11427}, {3793, 63932}, {3832, 62203}, {3849, 32984}, {3851, 53418}, {3855, 39601}, {3933, 8667}, {3934, 14001}, {3972, 16924}, {4045, 32990}, {4262, 45939}, {4293, 69175}, {4386, 26363}, {4426, 26364}, {5010, 9598}, {5017, 14561}, {5024, 15720}, {5025, 14907}, {5028, 10519}, {5032, 7619}, {5033, 6776}, {5038, 38064}, {5039, 58445}, {5041, 15702}, {5054, 5306}, {5068, 12815}, {5070, 15484}, {5071, 14537}, {5171, 6036}, {5204, 69098}, {5215, 7801}, {5217, 69096}, {5237, 69111}, {5238, 69110}, {5275, 7483}, {5280, 31497}, {5283, 6910}, {5291, 5552}, {5292, 18755}, {5304, 7772}, {5334, 22532}, {5335, 22531}, {5351, 69125}, {5352, 69124}, {5355, 53096}, {5368, 61834}, {5432, 54416}, {5433, 16502}, {5477, 20399}, {5523, 32534}, {5562, 50387}, {5569, 7817}, {5585, 33923}, {5976, 31981}, {6103, 35486}, {6179, 7769}, {6194, 32452}, {6375, 61305}, {6389, 28697}, {6390, 63933}, {6392, 7781}, {6636, 9700}, {6640, 10317}, {6684, 9620}, {6722, 7825}, {6784, 35704}, {6857, 16589}, {6914, 44517}, {6924, 44542}, {6961, 62371}, {7031, 9599}, {7280, 9597}, {7388, 39654}, {7389, 39655}, {7404, 10314}, {7487, 27371}, {7502, 44523}, {7525, 44525}, {7610, 8369}, {7612, 60619}, {7615, 26613}, {7618, 7783}, {7691, 69177}, {7694, 9754}, {7750, 7887}, {7754, 22329}, {7760, 33206}, {7761, 7886}, {7765, 15515}, {7767, 7778}, {7768, 7940}, {7770, 37688}, {7771, 7791}, {7773, 33249}, {7776, 44377}, {7777, 16923}, {7782, 14568}, {7784, 8361}, {7786, 16989}, {7789, 11288}, {7790, 32965}, {7792, 11285}, {7794, 15589}, {7796, 33262}, {7797, 33004}, {7802, 14061}, {7803, 7806}, {7804, 32968}, {7805, 63034}, {7813, 32831}, {7816, 23055}, {7819, 15271}, {7820, 33181}, {7821, 32959}, {7822, 14069}, {7823, 32967}, {7826, 7888}, {7829, 15482}, {7830, 7844}, {7831, 7942}, {7832, 16990}, {7834, 16043}, {7838, 32839}, {7841, 8182}, {7842, 16041}, {7843, 32976}, {7845, 32823}, {7847, 33008}, {7849, 33222}, {7851, 8356}, {7852, 32956}, {7853, 32951}, {7854, 7874}, {7855, 32818}, {7856, 33012}, {7858, 63083}, {7861, 32986}, {7863, 17131}, {7864, 33273}, {7872, 33023}, {7873, 32955}, {7876, 16984}, {7878, 33003}, {7881, 37671}, {7882, 63952}, {7890, 32835}, {7891, 17129}, {7893, 7925}, {7894, 63065}, {7901, 7904}, {7903, 63098}, {7905, 63093}, {7910, 33251}, {7911, 33283}, {7913, 33202}, {7916, 20080}, {7921, 17005}, {7923, 66414}, {7928, 14065}, {7932, 33021}, {7934, 33248}, {7935, 33180}, {7945, 63044}, {8365, 21358}, {8368, 8556}, {8370, 8860}, {8589, 10299}, {8591, 45017}, {8703, 44519}, {8743, 10018}, {8770, 10154}, {8778, 60428}, {8782, 62355}, {8972, 61335}, {9166, 33192}, {9167, 9740}, {9300, 15694}, {9341, 9650}, {9466, 33191}, {9606, 55863}, {9607, 61811}, {9619, 10165}, {9665, 10589}, {9697, 11003}, {9698, 37665}, {9699, 44802}, {9722, 23335}, {9752, 67854}, {9753, 37334}, {9890, 53765}, {10257, 23115}, {10304, 11648}, {10312, 37119}, {10441, 50361}, {10446, 20666}, {10565, 34481}, {10576, 41411}, {10577, 41410}, {10616, 53463}, {10617, 53452}, {10653, 62232}, {10654, 62233}, {11001, 39563}, {11063, 12106}, {11160, 22247}, {11168, 33237}, {11179, 39560}, {11272, 13330}, {11318, 44401}, {11360, 15270}, {11442, 19627}, {11580, 52300}, {11614, 61873}, {11646, 40278}, {12042, 43449}, {12105, 47275}, {12108, 22332}, {12829, 15561}, {12962, 19102}, {12969, 19105}, {13087, 13843}, {13088, 13720}, {13172, 62356}, {13340, 61675}, {13357, 15819}, {13638, 39387}, {13711, 42261}, {13754, 15575}, {13758, 39388}, {13834, 42260}, {13846, 44647}, {13847, 44648}, {13880, 32419}, {13882, 13910}, {13921, 32421}, {13934, 13972}, {13941, 61336}, {14043, 16986}, {14068, 15031}, {14581, 38282}, {14712, 32966}, {14791, 44529}, {14826, 62194}, {14930, 61848}, {14971, 66699}, {15325, 16781}, {15603, 62131}, {15681, 63543}, {15709, 63024}, {15712, 53095}, {15719, 39593}, {15761, 49123}, {15810, 33230}, {15880, 37454}, {15993, 63722}, {16044, 53127}, {16045, 31239}, {16241, 61318}, {16242, 61317}, {16306, 44214}, {16310, 34351}, {16318, 68720}, {16780, 31231}, {16921, 17006}, {16966, 41408}, {16967, 41409}, {16992, 17694}, {17128, 33246}, {17130, 32834}, {17566, 33854}, {17966, 34030}, {18282, 38463}, {18373, 69060}, {18546, 32826}, {18553, 38010}, {18581, 19781}, {18582, 19780}, {18584, 35018}, {18840, 33197}, {18906, 32189}, {18913, 39913}, {19145, 49104}, {19146, 49103}, {19547, 36744}, {19761, 37047}, {20885, 23208}, {20970, 37642}, {21001, 52261}, {21073, 39255}, {21448, 62981}, {22110, 63950}, {22253, 59546}, {24206, 41412}, {24855, 52292}, {26937, 39643}, {27040, 56781}, {27088, 34505}, {27385, 54406}, {30270, 38737}, {31152, 47298}, {31275, 32958}, {31276, 33225}, {31377, 35071}, {31396, 58441}, {31398, 31423}, {31402, 31501}, {31448, 52793}, {31459, 31499}, {31481, 32785}, {31492, 61837}, {31652, 61814}, {32445, 45938}, {32459, 63923}, {32786, 62219}, {32819, 33235}, {32827, 32988}, {32838, 32971}, {32867, 32987}, {32952, 55732}, {33006, 51224}, {33043, 36812}, {33188, 55085}, {33204, 63017}, {33223, 40344}, {33228, 47102}, {34782, 53496}, {34866, 68681}, {35022, 51579}, {35812, 45515}, {35813, 45514}, {37114, 56688}, {37482, 67522}, {37640, 69144}, {37641, 69150}, {38230, 44530}, {38297, 68851}, {38317, 41413}, {39095, 43183}, {40693, 62198}, {40694, 62197}, {42147, 69188}, {42148, 69182}, {42150, 69113}, {42151, 69112}, {42295, 43653}, {42459, 44277}, {42944, 63198}, {42945, 63199}, {44116, 54013}, {44141, 63464}, {44210, 62702}, {44499, 50977}, {44532, 67268}, {44541, 46853}, {46264, 53475}, {47638, 63556}, {50570, 68518}, {50771, 63936}, {52959, 59591}, {52987, 68315}, {53497, 61097}, {53498, 61096}, {54752, 60128}, {54816, 60184}, {54833, 62880}, {54916, 60263}, {56891, 63734}, {61754, 65726}, {61842, 63005}, {62199, 69160}, {62200, 69167}, {63392, 69179}, {63541, 66381}, {63922, 67536}

X(69207) = midpoint of X(i) and X(j) for these {i,j}: {5023, 13881}, {8981, 13966}
X(69207) = reflection of X(32816) in X(7862)
X(69207) = complement of X(32816)
X(69207) = anticomplement of X(7862)
X(69207) = complement of the isotomic conjugate of X(34285)
X(69207) = isogonal conjugate of the polar conjugate of X(61381)
X(69207) = X(34285)-complementary conjugate of X(2887)
X(69207) = crosspoint of X(i) and X(j) for these (i,j): {2, 34285}, {925, 4590}
X(69207) = crosssum of X(i) and X(j) for these (i,j): {6, 33586}, {924, 3124}
X(69207) = crossdifference of every pair of points on line {3005, 46953}
X(69207) = barycentric product X(3)*X(61381)
X(69207) = barycentric quotient X(61381)/X(264)
X(69207) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 32, 2548}, {2, 1078, 7800}, {2, 3785, 626}, {2, 7793, 315}, {2, 20065, 7752}, {2, 32816, 7862}, {3, 230, 3767}, {3, 3767, 2549}, {4, 7746, 43620}, {4, 62992, 7746}, {5, 3053, 7737}, {6, 140, 31401}, {6, 44535, 140}, {32, 7749, 2}, {39, 7735, 5319}, {69, 32970, 3788}, {115, 5206, 20}, {172, 498, 31409}, {183, 7807, 7795}, {187, 7746, 4}, {187, 43620, 43618}, {187, 62992, 43620}, {193, 32829, 7764}, {230, 21843, 2549}, {384, 17004, 32832}, {385, 7763, 7758}, {385, 7907, 7763}, {439, 32815, 69171}, {549, 5305, 5013}, {550, 43291, 44518}, {550, 44518, 43619}, {574, 7755, 5286}, {590, 6423, 31411}, {620, 7751, 3926}, {625, 63935, 32006}, {631, 5319, 31450}, {631, 7735, 39}, {1078, 7857, 2}, {1384, 1656, 7745}, {1384, 3054, 31415}, {1656, 7745, 31415}, {3053, 37637, 5}, {3054, 7745, 1656}, {3522, 43448, 7756}, {3523, 5286, 574}, {3523, 37689, 5286}, {3524, 7738, 37512}, {3525, 7736, 31455}, {3526, 30435, 3815}, {3530, 15048, 15815}, {3618, 32978, 6683}, {3767, 21843, 3}, {3788, 7780, 69}, {3788, 58448, 32970}, {3926, 32989, 620}, {3926, 37667, 7751}, {4386, 26363, 31416}, {5007, 31455, 7736}, {5013, 5305, 7739}, {5210, 43291, 43619}, {5210, 44518, 550}, {5280, 65142, 31497}, {5286, 37689, 7755}, {5304, 10303, 31400}, {5304, 31400, 7772}, {5309, 37512, 7738}, {5569, 7817, 33215}, {6179, 7769, 7774}, {6680, 7815, 2}, {6680, 34506, 7815}, {6722, 7825, 32972}, {6781, 69141, 3146}, {7748, 15513, 376}, {7753, 31404, 2548}, {7756, 8588, 3522}, {7756, 69162, 43448}, {7761, 7886, 14064}, {7771, 7828, 7791}, {7774, 33000, 7769}, {7780, 58448, 3788}, {7790, 43459, 32965}, {7802, 14061, 14063}, {7806, 7824, 7803}, {7826, 7888, 37668}, {7826, 31274, 7888}, {7830, 7844, 32974}, {7863, 17131, 32830}, {7891, 17129, 32833}, {7916, 63927, 20080}, {8588, 69162, 7756}, {9540, 44595, 1504}, {9754, 21445, 7694}, {9754, 36998, 37446}, {10104, 14693, 37466}, {10104, 37466, 1352}, {12042, 44534, 43449}, {13935, 44596, 1505}, {14001, 34229, 3934}, {14064, 63104, 7886}, {15589, 33203, 53033}, {15589, 53033, 7794}, {15694, 43136, 31467}, {16925, 17008, 76}, {16989, 33001, 7786}, {20080, 32825, 7916}, {21445, 37446, 36998}, {31467, 43136, 9300}, {32006, 32969, 625}, {32828, 32973, 3734}, {32972, 64018, 7825}, {32989, 37667, 3926}, {33215, 63107, 7817}, {33259, 63047, 194}, {36998, 37446, 7694}, {41975, 41976, 48876}, {44377, 63928, 7776}, {63924, 69171, 32815}

X(69208) = (1, -1, 1, -1, 1, 1)-ADDITIVE ASSOCIATE OF X(20)

Barycentrics    3*a^4 + 2*a^2*b^2 - b^4 + 2*a^2*c^2 + 2*b^2*c^2 - c^4 : :
X(69208) = 3 X[2] - 4 X[7808], X[7738] - 3 X[63024], 2 X[9605] - 3 X[63024]

X(69208) lies on these lines: {2, 32}, {3, 7736}, {4, 6}, {5, 7735}, {8, 1572}, {20, 39}, {30, 7738}, {51, 6620}, {55, 31402}, {66, 8801}, {69, 7762}, {76, 193}, {99, 32981}, {112, 3541}, {115, 3832}, {140, 1384}, {141, 16045}, {148, 14068}, {165, 31396}, {172, 3086}, {183, 32968}, {186, 9608}, {187, 3523}, {194, 14035}, {226, 16780}, {230, 3090}, {232, 7487}, {237, 10790}, {238, 26036}, {262, 13357}, {297, 11427}, {316, 1692}, {324, 40814}, {325, 14001}, {371, 31403}, {372, 21737}, {376, 5013}, {377, 33854}, {381, 5305}, {382, 15048}, {384, 3926}, {385, 16924}, {388, 16502}, {390, 1500}, {427, 3172}, {452, 5283}, {458, 11433}, {460, 9777}, {485, 26469}, {486, 26468}, {497, 54416}, {498, 7031}, {499, 609}, {515, 9575}, {516, 9593}, {524, 69139}, {546, 63533}, {549, 31467}, {550, 5024}, {571, 7383}, {572, 50425}, {574, 3522}, {577, 7400}, {590, 7376}, {595, 56746}, {597, 33190}, {598, 2996}, {599, 18840}, {615, 7375}, {620, 32835}, {625, 33199}, {631, 1285}, {671, 18845}, {847, 47735}, {958, 31405}, {962, 9620}, {966, 13740}, {1003, 6337}, {1007, 7807}, {1015, 3600}, {1056, 16781}, {1151, 36701}, {1152, 36703}, {1180, 7500}, {1184, 7392}, {1194, 3199}, {1196, 7398}, {1352, 5039}, {1449, 13161}, {1478, 5299}, {1479, 5280}, {1569, 20094}, {1571, 9778}, {1593, 39662}, {1609, 7509}, {1656, 21309}, {1691, 10359}, {1914, 3085}, {1915, 14585}, {1968, 3088}, {1975, 14033}, {1992, 7754}, {2021, 14712}, {2023, 9862}, {2031, 38227}, {2241, 31409}, {2242, 14986}, {2271, 6996}, {2275, 4293}, {2276, 4294}, {2345, 5015}, {2475, 63075}, {2478, 5276}, {2546, 12051}, {2547, 12050}, {2549, 3146}, {3051, 14826}, {3054, 61886}, {3055, 3533}, {3068, 6424}, {3069, 6423}, {3089, 10311}, {3091, 3767}, {3102, 49038}, {3103, 49039}, {3314, 16898}, {3329, 7791}, {3424, 22682}, {3509, 36574}, {3524, 5023}, {3525, 22331}, {3528, 9606}, {3529, 63548}, {3542, 10312}, {3543, 5041}, {3545, 5306}, {3547, 10316}, {3549, 10317}, {3552, 63018}, {3575, 45141}, {3589, 7784}, {3613, 34285}, {3618, 6656}, {3619, 7879}, {3620, 7768}, {3627, 63633}, {3629, 52713}, {3673, 4644}, {3734, 7758}, {3788, 33181}, {3839, 5309}, {3851, 43291}, {3854, 5368}, {3855, 63534}, {3933, 11286}, {3934, 14023}, {3972, 7763}, {3981, 62260}, {4000, 4911}, {4045, 33025}, {4048, 35701}, {4251, 36670}, {4254, 37415}, {4258, 7397}, {4297, 9592}, {4307, 17750}, {4340, 24512}, {4385, 5839}, {4426, 19843}, {4648, 17681}, {5008, 5056}, {5017, 10519}, {5021, 13727}, {5022, 64159}, {5025, 16989}, {5034, 12203}, {5037, 5747}, {5046, 63004}, {5058, 7585}, {5059, 7756}, {5062, 7586}, {5065, 52404}, {5067, 37637}, {5068, 7755}, {5073, 22246}, {5084, 5275}, {5111, 43453}, {5116, 33750}, {5129, 16589}, {5140, 48445}, {5206, 9698}, {5210, 10299}, {5218, 31460}, {5225, 69096}, {5229, 69098}, {5230, 21764}, {5261, 9650}, {5274, 9665}, {5291, 64081}, {5292, 36671}, {5332, 10590}, {5354, 62937}, {5355, 50689}, {5359, 6997}, {5418, 41410}, {5420, 41411}, {5477, 38664}, {5485, 18843}, {5585, 61787}, {5712, 37086}, {5731, 9619}, {5819, 16583}, {6179, 32832}, {6194, 46321}, {6248, 44500}, {6389, 28717}, {6390, 68527}, {6392, 7760}, {6409, 36717}, {6410, 36702}, {6419, 39660}, {6420, 39661}, {6421, 6460}, {6422, 6459}, {6515, 41231}, {6525, 27373}, {6531, 18855}, {6560, 45513}, {6561, 45512}, {6655, 62994}, {6661, 7881}, {6683, 63935}, {6781, 50693}, {6792, 46512}, {6823, 15905}, {6835, 40129}, {6857, 37661}, {6885, 34460}, {7296, 10591}, {7377, 37642}, {7378, 14581}, {7394, 34482}, {7395, 8573}, {7402, 37646}, {7406, 20970}, {7470, 50659}, {7486, 7603}, {7507, 16318}, {7610, 52718}, {7667, 39951}, {7694, 9748}, {7750, 11174}, {7751, 32834}, {7759, 7795}, {7761, 33202}, {7764, 32831}, {7765, 17578}, {7766, 16044}, {7769, 32989}, {7773, 7792}, {7776, 7819}, {7777, 16925}, {7778, 14069}, {7779, 68525}, {7782, 35927}, {7783, 33007}, {7786, 14907}, {7789, 9766}, {7790, 32982}, {7794, 10513}, {7797, 14063}, {7802, 33023}, {7805, 63955}, {7806, 32961}, {7813, 32840}, {7816, 34511}, {7817, 66466}, {7820, 7903}, {7822, 7845}, {7825, 7829}, {7827, 63127}, {7828, 32972}, {7832, 7926}, {7834, 7843}, {7836, 14037}, {7837, 17128}, {7839, 11361}, {7841, 23334}, {7842, 33210}, {7847, 33272}, {7851, 16041}, {7852, 33182}, {7855, 41750}, {7856, 32980}, {7859, 7860}, {7861, 63956}, {7863, 32841}, {7864, 33017}, {7874, 33183}, {7875, 7885}, {7877, 20080}, {7890, 17130}, {7891, 32837}, {7892, 7941}, {7893, 16990}, {7897, 19689}, {7905, 32833}, {7907, 32839}, {7920, 14041}, {7923, 33251}, {7932, 33283}, {7933, 63020}, {7939, 16895}, {7946, 46226}, {7947, 14036}, {8352, 66763}, {8367, 42850}, {8369, 9770}, {8584, 34505}, {8588, 61791}, {8589, 62067}, {8972, 31481}, {9112, 36251}, {9113, 36252}, {9463, 54013}, {9540, 12963}, {9574, 31730}, {9600, 42638}, {9607, 14482}, {9609, 12088}, {9700, 37913}, {9744, 12110}, {9752, 22521}, {9833, 68572}, {10159, 60145}, {10164, 31428}, {10303, 21843}, {10304, 37512}, {10313, 59349}, {10323, 46952}, {10530, 65203}, {10788, 34870}, {10796, 43450}, {11001, 44519}, {11163, 32985}, {11285, 63041}, {11289, 11489}, {11290, 11488}, {11303, 37641}, {11304, 37640}, {11317, 63022}, {11648, 50687}, {12212, 40330}, {12251, 13330}, {12252, 13331}, {12512, 31421}, {12572, 16517}, {12829, 14651}, {12968, 13935}, {13567, 52288}, {13571, 14031}, {13595, 44527}, {13596, 44528}, {13910, 53492}, {13941, 62220}, {13972, 53491}, {14061, 52250}, {14246, 52450}, {14535, 14929}, {14542, 43717}, {14555, 41236}, {14561, 54393}, {14614, 32983}, {14683, 46301}, {14789, 46262}, {15043, 50387}, {15045, 15575}, {15271, 32957}, {15513, 15692}, {15515, 21734}, {15603, 61794}, {15655, 15712}, {15820, 30769}, {16062, 63055}, {16317, 30734}, {16787, 33144}, {16921, 17008}, {16922, 32883}, {16923, 32884}, {16984, 33248}, {17000, 33028}, {17004, 32867}, {17005, 33000}, {17129, 46951}, {17500, 52580}, {17538, 22332}, {17686, 45962}, {18362, 61944}, {18373, 44441}, {18391, 54382}, {18533, 39575}, {18584, 61921}, {18755, 36697}, {18841, 47355}, {18844, 60219}, {18928, 41235}, {18993, 45510}, {18994, 45511}, {19053, 44648}, {19054, 44647}, {19102, 35823}, {19103, 35771}, {19104, 35770}, {19105, 35822}, {19569, 33263}, {19687, 31859}, {19766, 56994}, {20154, 32022}, {20423, 44499}, {20965, 37190}, {21735, 53095}, {22110, 33197}, {22401, 61113}, {23292, 52283}, {23311, 45487}, {23312, 45486}, {25066, 66639}, {26098, 41239}, {26619, 32497}, {26620, 32494}, {27318, 33058}, {28723, 40680}, {30478, 31466}, {31276, 33269}, {31398, 68545}, {31412, 49220}, {31457, 61788}, {31470, 46853}, {31492, 61138}, {31636, 57260}, {31652, 62097}, {32134, 37466}, {32452, 44434}, {32455, 63923}, {32480, 66398}, {32954, 37690}, {32960, 63938}, {32962, 63048}, {32966, 63019}, {32975, 37688}, {32977, 37647}, {32992, 34229}, {33002, 63047}, {33013, 63065}, {33016, 63038}, {33106, 54329}, {33189, 44377}, {33215, 63101}, {33225, 63021}, {33229, 63011}, {33230, 47352}, {33233, 34803}, {33249, 63104}, {33838, 37650}, {33863, 36489}, {33871, 34621}, {34484, 44524}, {34571, 61982}, {36181, 38526}, {36434, 52448}, {36664, 42265}, {36665, 42262}, {36744, 37431}, {36997, 53016}, {37174, 63030}, {37334, 39647}, {37643, 52289}, {37953, 47169}, {38224, 41675}, {38259, 53109}, {39563, 61989}, {39593, 62007}, {40050, 44152}, {40123, 68719}, {40126, 67238}, {40823, 55972}, {41237, 63085}, {41238, 63084}, {41254, 66914}, {41406, 42149}, {41407, 42152}, {41408, 42089}, {41409, 42092}, {41760, 44142}, {41891, 46432}, {41895, 53107}, {41940, 50688}, {42150, 63201}, {42151, 63200}, {42159, 69110}, {42160, 69124}, {42161, 69125}, {42162, 69111}, {42561, 49221}, {43403, 61317}, {43404, 61318}, {43511, 62205}, {43512, 62206}, {43527, 60647}, {43619, 49135}, {43681, 60146}, {44116, 64058}, {44523, 47485}, {44541, 62113}, {44543, 63034}, {44562, 47102}, {44839, 63043}, {46264, 64713}, {46767, 52583}, {47286, 62995}, {47846, 62303}, {51481, 63012}, {52187, 59430}, {52483, 59423}, {53101, 53105}, {53102, 60285}, {54639, 60278}, {54800, 60118}, {56344, 60501}, {56395, 59428}, {58212, 58556}, {59363, 67859}, {60133, 62917}, {60250, 60650}, {60282, 60639}, {60284, 60636}, {60601, 65005}, {60644, 60648}, {60779, 63899}, {61328, 63015}, {61329, 63016}, {61980, 63543}, {62949, 62991}, {63032, 69155}, {63033, 69156}, {63079, 69150}, {63080, 69144}, {63616, 64621}, {63954, 66412}

X(69208) = reflection of X(i) in X(j) for these {i,j}: {7738, 9605}, {7800, 7808}
X(69208) = anticomplement of X(7800)
X(69208) = orthosymmedial circle inverse of X(5286)
X(69208) = polar conjugate of the isotomic conjugate of X(7494)
X(69208) = polar conjugate of the isogonal conjugate of X(19125)
X(69208) = X(19125)-cross conjugate of X(7494)
X(69208) = X(7494)-Dao conjugate of X(3619)
X(69208) = crosspoint of X(i) and X(j) for these (i,j): {4, 18841}, {4590, 68220}
X(69208) = crosssum of X(3) and X(9605)
X(69208) = crossdifference of every pair of points on line {520, 3005}
X(69208) = barycentric product X(i)*X(j) for these {i,j}: {4, 7494}, {264, 19125}
X(69208) = barycentric quotient X(i)/X(j) for these {i,j}: {7494, 69}, {19125, 3}
X(69208) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2548, 31404}, {2, 7785, 32816}, {2, 20065, 3785}, {2, 20088, 20065}, {3, 7736, 31400}, {4, 6, 5286}, {4, 1249, 27376}, {4, 5286, 43448}, {4, 8743, 393}, {4, 14912, 39646}, {4, 39646, 46034}, {5, 30435, 7735}, {6, 7745, 4}, {6, 65630, 5254}, {20, 37665, 39}, {32, 2548, 2}, {32, 7753, 2548}, {39, 7737, 20}, {83, 315, 2}, {83, 7812, 315}, {172, 9599, 3086}, {187, 31401, 3523}, {193, 5395, 32971}, {193, 32971, 76}, {194, 14035, 32815}, {316, 7803, 32974}, {316, 7878, 7803}, {325, 14001, 53033}, {381, 43136, 5305}, {384, 7774, 3926}, {384, 7921, 7774}, {385, 16924, 32828}, {598, 5032, 7620}, {631, 1285, 3053}, {1587, 1588, 14853}, {1914, 9596, 3085}, {2549, 7747, 3146}, {3053, 3815, 631}, {3055, 44535, 3533}, {3070, 3071, 53023}, {3091, 5304, 3767}, {3329, 7823, 7791}, {3589, 7784, 32956}, {3618, 32006, 6656}, {3734, 7758, 32830}, {3734, 7838, 7758}, {3767, 5007, 5304}, {3767, 5475, 3091}, {3934, 14023, 15589}, {3972, 7763, 32973}, {3972, 7858, 7763}, {5007, 5475, 3767}, {5007, 39565, 5346}, {5008, 31415, 37689}, {5041, 7748, 7739}, {5041, 14537, 7748}, {5056, 37689, 7746}, {5058, 31411, 7585}, {5254, 7745, 65630}, {5254, 65630, 4}, {5305, 43136, 63006}, {5346, 5475, 39565}, {5346, 39565, 3767}, {5368, 43457, 69162}, {6179, 32832, 37667}, {6392, 32979, 11185}, {6392, 51170, 7760}, {7738, 63024, 9605}, {7739, 14537, 3543}, {7746, 31415, 5056}, {7747, 7772, 2549}, {7750, 11174, 16043}, {7756, 43618, 5059}, {7758, 7838, 63091}, {7759, 7795, 37668}, {7759, 7804, 7795}, {7760, 11185, 6392}, {7762, 7770, 69}, {7762, 53489, 7770}, {7763, 7858, 62988}, {7773, 7792, 14064}, {7777, 16925, 32829}, {7785, 7787, 2}, {7785, 10583, 7912}, {7786, 14907, 32990}, {7787, 7912, 10583}, {7789, 9766, 32818}, {7791, 7823, 64018}, {7795, 7804, 33198}, {7800, 7808, 2}, {7803, 7878, 51171}, {7893, 68522, 16990}, {7907, 63083, 32839}, {7912, 10583, 2}, {8367, 63950, 42850}, {12963, 31463, 9540}, {13571, 14031, 32824}, {14035, 63017, 194}, {14039, 32818, 7789}, {14063, 63045, 7797}, {14069, 32823, 7778}, {14075, 69162, 5368}, {15484, 30435, 5}, {16921, 17008, 32838}, {17004, 32999, 32867}, {21843, 31455, 10303}, {31455, 35007, 21843}, {32830, 63091, 7758}, {32834, 63042, 7751}, {32835, 33205, 620}, {32973, 62988, 7763}, {32974, 51171, 7803}, {32979, 51170, 6392}, {32987, 37667, 32832}, {32989, 63077, 7769}, {33181, 63098, 3788}, {33198, 37668, 7795}, {33269, 63046, 31276}, {44518, 53418, 4}, {62950, 63031, 40814}, {63093, 66413, 46951}

X(69209) = (1, -1, -1, -1, 1, -1)-ADDITIVE ASSOCIATE OF X(20)

Barycentrics    3*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 + c^4 : :

X(69209) lies on these lines: {2, 32}, {3, 3589}, {4, 7804}, {5, 7694}, {6, 3933}, {20, 4045}, {39, 3618}, {66, 15257}, {69, 5007}, {76, 5319}, {99, 14037}, {115, 32971}, {141, 14023}, {182, 8721}, {187, 16043}, {193, 7794}, {194, 19689}, {216, 28696}, {262, 631}, {263, 41262}, {316, 7943}, {325, 33217}, {384, 2549}, {385, 16895}, {439, 15515}, {441, 17825}, {485, 39679}, {486, 39648}, {524, 43136}, {574, 32973}, {597, 7789}, {620, 31400}, {621, 37178}, {622, 37177}, {625, 32951}, {1007, 7874}, {1285, 63935}, {1352, 3398}, {1384, 51126}, {1656, 14535}, {1975, 6661}, {1992, 7855}, {3053, 8362}, {3090, 7886}, {3091, 7844}, {3095, 31958}, {3146, 7872}, {3314, 19694}, {3329, 7763}, {3523, 15482}, {3524, 55774}, {3525, 58448}, {3526, 15491}, {3619, 5008}, {3620, 7826}, {3631, 63936}, {3734, 5286}, {3763, 7767}, {3767, 7770}, {3788, 7736}, {3815, 32954}, {3849, 33230}, {3926, 7772}, {3934, 7735}, {3972, 7791}, {5013, 7618}, {5023, 8359}, {5024, 33242}, {5041, 7801}, {5056, 6722}, {5070, 44381}, {5171, 58445}, {5206, 32990}, {5254, 11286}, {5266, 17279}, {5275, 17540}, {5304, 7751}, {5305, 63955}, {5346, 9466}, {5355, 6392}, {5368, 17131}, {5395, 32827}, {5475, 7852}, {6179, 16990}, {6309, 64713}, {6459, 11291}, {6460, 11292}, {6655, 43618}, {6656, 7737}, {6703, 21514}, {6720, 53767}, {6781, 33023}, {7392, 42671}, {7603, 32969}, {7615, 7817}, {7738, 7816}, {7745, 7866}, {7746, 32968}, {7747, 7913}, {7748, 14033}, {7755, 32828}, {7756, 32981}, {7759, 7915}, {7760, 63045}, {7761, 32956}, {7762, 7868}, {7764, 37665}, {7765, 32815}, {7766, 46226}, {7771, 43527}, {7773, 8363}, {7774, 7832}, {7777, 14043}, {7778, 33185}, {7782, 33255}, {7783, 14036}, {7784, 8364}, {7786, 16925}, {7788, 66325}, {7790, 14035}, {7796, 63017}, {7797, 11185}, {7798, 32830}, {7805, 63006}, {7806, 32832}, {7807, 11174}, {7813, 63123}, {7823, 7948}, {7825, 33180}, {7828, 16924}, {7830, 33202}, {7835, 55085}, {7836, 62994}, {7837, 66323}, {7838, 7869}, {7839, 32833}, {7842, 33190}, {7843, 33194}, {7847, 33007}, {7851, 8370}, {7853, 32006}, {7858, 7930}, {7876, 14907}, {7880, 32818}, {7881, 41624}, {7887, 31415}, {7888, 62988}, {7890, 51170}, {7898, 66345}, {7902, 43448}, {7904, 16897}, {7908, 14930}, {7916, 63091}, {7918, 33017}, {7919, 14063}, {7920, 17128}, {7921, 7931}, {7923, 11361}, {7925, 14067}, {7932, 16044}, {7935, 64018}, {7940, 63083}, {7942, 32961}, {7945, 63018}, {7947, 63028}, {8182, 48310}, {8360, 66466}, {8365, 11184}, {8366, 63101}, {8368, 31406}, {8781, 31274}, {9300, 69158}, {9596, 30104}, {9599, 30103}, {9698, 32829}, {9737, 25555}, {9744, 10359}, {9821, 10007}, {9822, 15594}, {9825, 20204}, {10601, 52350}, {10653, 37340}, {10654, 37341}, {11285, 21843}, {11324, 62310}, {11328, 23208}, {11648, 32826}, {12054, 38064}, {12251, 35437}, {13335, 38317}, {13881, 66415}, {14003, 54012}, {14061, 32962}, {14075, 20080}, {14537, 33223}, {14853, 30270}, {15048, 19697}, {15270, 31267}, {15513, 33215}, {16041, 39590}, {16060, 17381}, {16061, 17352}, {16896, 16986}, {16909, 27109}, {16921, 16984}, {17259, 17698}, {19569, 66336}, {19687, 43619}, {20083, 36674}, {20179, 31416}, {20582, 63950}, {20960, 35222}, {21309, 34573}, {24256, 31981}, {31239, 34229}, {31276, 63019}, {31417, 33218}, {31450, 33225}, {31455, 32970}, {31652, 51581}, {32450, 32817}, {32952, 37690}, {32957, 62992}, {32975, 63104}, {32979, 69141}, {32980, 43457}, {32982, 62203}, {32983, 39565}, {32985, 37512}, {33020, 53127}, {33184, 65630}, {33189, 62993}, {33191, 44562}, {33196, 63956}, {33201, 69171}, {33222, 34803}, {33258, 43459}, {34571, 62995}, {35007, 39784}, {36212, 63085}, {36474, 48866}, {37176, 37650}, {37344, 37649}, {38905, 42421}, {39143, 39601}, {41940, 63011}, {42442, 50666}, {44152, 59560}, {44230, 46264}, {44519, 66391}, {44526, 68177}, {46544, 67599}, {47005, 63093}, {48674, 51848}, {60234, 60239}, {63548, 68527}, {63932, 66344}

X(69209) = anticomplement of X(7914)
X(69209) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 32, 7800}, {2, 83, 2548}, {2, 3785, 6292}, {2, 7787, 315}, {2, 20065, 3096}, {2, 20088, 7938}, {2, 31404, 7862}, {2, 32816, 7867}, {6, 7795, 7758}, {6, 7819, 7795}, {32, 6292, 3785}, {32, 7815, 6308}, {32, 7889, 2}, {76, 16989, 5319}, {83, 7846, 2}, {141, 30435, 14023}, {384, 7803, 2549}, {384, 7875, 7803}, {597, 7789, 9605}, {597, 33237, 34511}, {2548, 7800, 31982}, {3053, 47355, 8362}, {3096, 12150, 20065}, {3329, 7892, 7763}, {3618, 14001, 39}, {3734, 7829, 5286}, {3785, 6292, 7800}, {3926, 51171, 7772}, {3972, 7859, 7791}, {4045, 69172, 20}, {5007, 7822, 69}, {5024, 33242, 59545}, {5286, 33198, 3734}, {5305, 69139, 63955}, {5355, 17130, 6392}, {5475, 7852, 14064}, {6680, 7808, 2}, {6704, 7815, 2}, {7735, 16045, 3934}, {7736, 14069, 3788}, {7738, 14039, 7816}, {7747, 7913, 32974}, {7753, 7867, 32816}, {7762, 66342, 7868}, {7770, 7792, 3767}, {7772, 7820, 3926}, {7787, 7938, 20088}, {7789, 9605, 34511}, {7797, 68525, 11185}, {7804, 7834, 4}, {7806, 68522, 32832}, {7807, 11174, 31401}, {7828, 16924, 43620}, {7828, 60855, 16924}, {7832, 7878, 7774}, {7838, 7869, 37668}, {7938, 20088, 315}, {8363, 53489, 7773}, {8364, 18907, 7784}, {9605, 33237, 7789}, {16898, 16989, 76}, {19689, 63020, 194}, {31400, 33181, 620}, {32006, 33221, 7853}, {32970, 63041, 31455}, {37665, 53033, 7764}

X(69210) = (1, -1, 1, -1, -1, -1)-ADDITIVE ASSOCIATE OF X(21)

Barycentrics    a*(a^3 + a*b^2 - a*b*c + b^2*c + a*c^2 + b*c^2) : :

X(69210) lies on these lines: {1, 41}, {2, 16502}, {6, 8}, {9, 3915}, {10, 5299}, {21, 1107}, {31, 21384}, {32, 2975}, {37, 33882}, {38, 3496}, {39, 100}, {58, 45751}, {78, 9575}, {81, 239}, {83, 668}, {85, 1462}, {86, 26965}, {145, 54416}, {149, 69096}, {171, 1475}, {172, 17448}, {187, 5303}, {190, 28598}, {213, 62804}, {220, 37542}, {238, 3691}, {257, 7104}, {330, 16915}, {384, 21226}, {385, 26801}, {388, 56913}, {404, 2275}, {514, 40398}, {519, 5280}, {595, 16552}, {607, 16066}, {609, 8666}, {651, 56928}, {672, 5255}, {673, 26978}, {940, 5222}, {956, 30435}, {976, 3061}, {993, 7031}, {999, 63499}, {1015, 5253}, {1100, 2303}, {1104, 4875}, {1125, 16784}, {1172, 1829}, {1191, 37658}, {1193, 3684}, {1212, 3744}, {1334, 37588}, {1449, 54418}, {1572, 3869}, {1573, 5260}, {1621, 2241}, {1655, 4366}, {1870, 56832}, {1909, 17686}, {1930, 30133}, {2176, 36534}, {2220, 38871}, {2221, 14552}, {2242, 62837}, {2276, 3871}, {2287, 2300}, {2475, 69098}, {2548, 11681}, {2991, 14621}, {3244, 16785}, {3294, 40091}, {3329, 26752}, {3434, 5286}, {3583, 63537}, {3616, 5275}, {3617, 63075}, {3661, 32911}, {3666, 7291}, {3670, 5011}, {3681, 54406}, {3686, 16470}, {3727, 16519}, {3730, 37610}, {3746, 25092}, {3767, 11680}, {3815, 27529}, {3868, 16973}, {3876, 39248}, {3924, 16787}, {3954, 18098}, {3959, 54315}, {3961, 33299}, {3997, 17745}, {4000, 41826}, {4051, 49487}, {4109, 32844}, {4189, 31449}, {4193, 9599}, {4253, 5264}, {4254, 8192}, {4357, 65115}, {4360, 17489}, {4383, 29611}, {4390, 54329}, {4393, 21216}, {4400, 21264}, {4426, 5332}, {4441, 7754}, {4855, 9592}, {5007, 5291}, {5021, 17126}, {5022, 37540}, {5041, 52959}, {5080, 7745}, {5247, 21764}, {5250, 16517}, {5254, 52367}, {5257, 16488}, {5266, 43065}, {5269, 28043}, {5284, 16589}, {5293, 39244}, {5304, 64081}, {5305, 17737}, {5552, 7736}, {5687, 9605}, {5711, 63066}, {6376, 17541}, {6645, 9263}, {6656, 20553}, {6734, 40129}, {7080, 37665}, {7191, 16583}, {7735, 10527}, {7739, 49719}, {7760, 17143}, {7839, 17759}, {9593, 63130}, {9597, 17579}, {9620, 14923}, {10459, 41239}, {10528, 31402}, {10800, 56542}, {11349, 37596}, {13588, 23632}, {14210, 30130}, {16370, 31468}, {16466, 37657}, {16474, 50017}, {16503, 59305}, {16604, 17531}, {16691, 20990}, {16779, 59311}, {16780, 19860}, {16782, 16830}, {16783, 30116}, {16916, 41838}, {17030, 37670}, {17141, 32029}, {17292, 37680}, {17367, 37633}, {17474, 37607}, {17497, 68983}, {17499, 64223}, {17721, 46835}, {17752, 23660}, {19649, 20606}, {19825, 37685}, {20172, 34284}, {20964, 24727}, {20965, 56802}, {21071, 32943}, {21219, 68525}, {21511, 62803}, {24211, 24712}, {24549, 30036}, {25542, 52708}, {25598, 30121}, {27040, 32942}, {28795, 63089}, {28813, 37662}, {29613, 37687}, {30109, 33953}, {31416, 33108}, {31429, 35258}, {31442, 62838}, {33090, 34482}, {34253, 40765}, {35978, 62692}, {37539, 40133}, {40400, 40401}, {40406, 60082}, {46196, 68960}, {51194, 54421}, {53675, 62994}, {54282, 62799}, {56882, 62798}, {57017, 64072}, {62797, 68769}, {66655, 67726}

X(69210) = anticomplement of X(69095)
X(69210) = X(i)-Dao conjugate of X(j) for these (i,j): {18235, 27697}, {27040, 26563}
X(69210) = crosspoint of X(i) and X(j) for these (i,j): {81, 23617}, {765, 4603}
X(69210) = crosssum of X(i) and X(j) for these (i,j): {37, 3752}, {244, 57234}, {30618, 56176}
X(69210) = crossdifference of every pair of points on line {2254, 6371}
X(69210) = barycentric product X(i)*X(j) for these {i,j}: {1, 32942}, {81, 27040}, {101, 18071}
X(69210) = barycentric quotient X(i)/X(j) for these {i,j}: {18071, 3261}, {27040, 321}, {32942, 75}, {57048, 16892}
X(69210) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 169, 26242}, {1, 5540, 16600}, {10, 5299, 33854}, {32, 16975, 2975}, {145, 63004, 54416}, {172, 17448, 54391}, {1015, 5277, 5253}, {1100, 41015, 5262}, {1107, 1914, 21}, {1909, 20179, 17686}, {2241, 5283, 1621}, {2275, 4386, 404}, {5275, 16781, 3616}, {5305, 24390, 17737}, {5687, 9605, 17756}, {16589, 68893, 5284}, {16973, 54382, 3868}, {54329, 59310, 4390}, {62804, 63087, 213}

X(69211) = (1, -1, -1, 1, -1, 1)-ADDITIVE ASSOCIATE OF X(21)

Barycentrics    a*(a^3 + a*b^2 + a*b*c - b^2*c + a*c^2 - b*c^2) : :

X(69211) lies on these lines: {1, 1390}, {2, 31402}, {3, 17756}, {4, 218}, {5, 17737}, {6, 8}, {9, 2292}, {10, 5276}, {21, 2276}, {31, 3501}, {32, 100}, {37, 5047}, {39, 2975}, {41, 43}, {42, 41239}, {44, 33950}, {58, 16549}, {63, 9593}, {75, 17686}, {78, 40129}, {81, 3661}, {83, 213}, {101, 3216}, {145, 16502}, {172, 404}, {190, 17489}, {192, 16916}, {205, 33849}, {220, 4383}, {230, 27529}, {238, 1334}, {346, 17697}, {350, 17541}, {384, 17759}, {385, 26752}, {386, 16788}, {519, 5299}, {574, 5303}, {595, 1018}, {609, 25440}, {644, 2176}, {651, 3212}, {668, 7760}, {672, 5247}, {728, 7290}, {894, 20911}, {940, 29611}, {956, 9605}, {978, 9310}, {985, 46032}, {986, 5282}, {1015, 62837}, {1104, 3693}, {1125, 16785}, {1172, 5090}, {1191, 4513}, {1193, 2329}, {1201, 56530}, {1203, 3997}, {1278, 68525}, {1332, 41316}, {1333, 35978}, {1462, 6604}, {1468, 17754}, {1500, 1621}, {1572, 14923}, {1574, 5277}, {1698, 37675}, {1714, 56746}, {1722, 40131}, {1724, 3730}, {1743, 1766}, {1914, 3871}, {1922, 40848}, {2220, 21858}, {2238, 5051}, {2242, 5253}, {2271, 3240}, {2273, 2287}, {2275, 54391}, {2280, 50581}, {2303, 14005}, {2321, 16470}, {2476, 9596}, {2548, 11680}, {3051, 56802}, {3061, 49487}, {3208, 3915}, {3214, 3684}, {3241, 16781}, {3244, 16784}, {3293, 4251}, {3329, 26801}, {3436, 5286}, {3496, 4642}, {3509, 24443}, {3617, 63004}, {3721, 54315}, {3744, 4515}, {3759, 46738}, {3767, 11681}, {3869, 9620}, {3870, 16780}, {3872, 9575}, {3875, 65115}, {3877, 39248}, {3938, 16787}, {3970, 30117}, {3987, 5011}, {4189, 31448}, {4203, 21877}, {4261, 38871}, {4360, 28598}, {4386, 7296}, {4400, 25102}, {4429, 26085}, {4441, 7770}, {4641, 7291}, {4644, 41826}, {4652, 9574}, {4734, 21775}, {4850, 21516}, {4986, 30133}, {5007, 52959}, {5016, 63087}, {5046, 69096}, {5080, 5254}, {5120, 8192}, {5251, 25092}, {5255, 21764}, {5260, 5283}, {5275, 9780}, {5304, 7080}, {5305, 17757}, {5315, 50017}, {5336, 27396}, {5354, 60459}, {5359, 10327}, {5369, 7077}, {5375, 54230}, {5552, 7735}, {5687, 30435}, {6163, 36287}, {6653, 20088}, {7031, 8715}, {7718, 45786}, {7736, 10527}, {7745, 21956}, {7762, 20553}, {7766, 53675}, {7772, 16975}, {7839, 21226}, {8370, 37857}, {8743, 56876}, {9598, 11114}, {9665, 10707}, {11342, 17776}, {14974, 17127}, {14997, 30578}, {16370, 31461}, {16600, 17744}, {16704, 26843}, {16926, 28604}, {16968, 25082}, {16970, 55337}, {17033, 17743}, {17034, 64223}, {17277, 26965}, {17280, 33816}, {17292, 37633}, {17350, 21216}, {17367, 37680}, {17451, 60353}, {17736, 24046}, {17742, 26242}, {17745, 49772}, {17752, 21760}, {17863, 27420}, {18047, 34063}, {18135, 26687}, {18343, 40754}, {20060, 69098}, {20331, 33863}, {21029, 37717}, {21796, 38869}, {24047, 52680}, {24170, 29473}, {24883, 26074}, {26030, 26244}, {26042, 55094}, {27020, 37670}, {27525, 63097}, {28082, 51058}, {28794, 63126}, {28795, 37642}, {28813, 37646}, {29630, 37687}, {30706, 50698}, {31396, 59491}, {31426, 35258}, {33091, 34482}, {33953, 40006}, {37037, 63066}, {37539, 44798}, {37549, 50995}, {37665, 64081}, {37676, 62797}, {38859, 43063}, {41423, 54354}, {50015, 62848}, {52136, 56011}, {52405, 68852}, {52530, 60444}, {53417, 56534}, {54361, 62372}, {56883, 62798}, {56983, 68971}, {57397, 60082}, {66099, 68855}

X(69211) = anticomplement of X(69097)
X(69211) = X(i)-Ceva conjugate of X(j) for these (i,j): {31625, 100}, {52395, 1621}
X(69211) = X(i)-cross conjugate of X(j) for these (i,j): {55053, 21005}, {57097, 100}, {64643, 21389}
X(69211) = X(649)-isoconjugate of X(54458)
X(69211) = X(i)-Dao conjugate of X(j) for these (i,j): {667, 1015}, {5375, 54458}, {64643, 48084}
X(69211) = cevapoint of X(i) and X(j) for these (i,j): {21005, 55053}, {21389, 64643}
X(69211) = crosssum of X(i) and X(j) for these (i,j): {1015, 2530}, {3121, 50330}
X(69211) = trilinear pole of line {21005, 21389}
X(69211) = crossdifference of every pair of points on line {6371, 50521}
X(69211) = barycentric product X(i)*X(j) for these {i,j}: {1, 32926}, {100, 21301}, {101, 20952}, {190, 21389}, {662, 21099}, {668, 21005}, {765, 21210}, {1016, 64643}, {1783, 28423}, {1978, 57047}, {6335, 22157}, {6386, 57097}, {31625, 55053}
X(69211) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 54458}, {20952, 3261}, {21005, 513}, {21099, 1577}, {21210, 1111}, {21301, 693}, {21389, 514}, {22157, 905}, {28423, 15413}, {32926, 75}, {55053, 1015}, {57047, 649}, {57097, 667}, {64643, 1086}
X(69211) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 2295, 57280}, {6, 2345, 2298}, {10, 5280, 5276}, {39, 5291, 2975}, {43, 54329, 41}, {44, 41015, 33950}, {145, 63075, 16502}, {172, 1575, 404}, {1914, 20691, 3871}, {2276, 4426, 21}, {7745, 21956, 52367}, {9620, 54406, 3869}

X(69212) = (1, 0, 1, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(21)

Barycentrics    a*(a^3 - a*b*c - b^2*c - b*c^2) : :

X(69212) lies on these lines: {1, 6}, {2, 32}, {3, 34460}, {4, 60081}, {8, 2241}, {10, 1914}, {21, 39}, {31, 17750}, {35, 1575}, {36, 16604}, {41, 748}, {58, 16850}, {63, 19729}, {76, 16916}, {86, 25499}, {99, 17692}, {100, 1574}, {112, 52252}, {115, 5046}, {141, 17540}, {172, 1125}, {187, 404}, {194, 16914}, {230, 4187}, {274, 384}, {284, 992}, {316, 33841}, {377, 7737}, {385, 16918}, {406, 10311}, {427, 2204}, {442, 7745}, {451, 10312}, {452, 5286}, {474, 3053}, {475, 1968}, {572, 9840}, {573, 13732}, {574, 4189}, {580, 15972}, {595, 2295}, {598, 7621}, {609, 3624}, {612, 57522}, {614, 37061}, {672, 40955}, {762, 3961}, {966, 13742}, {993, 2275}, {1011, 5310}, {1015, 2975}, {1030, 46838}, {1055, 28352}, {1086, 33868}, {1194, 37325}, {1213, 2220}, {1285, 17582}, {1333, 13728}, {1384, 16408}, {1415, 5433}, {1500, 1621}, {1571, 35258}, {1572, 19860}, {1573, 5260}, {1655, 7760}, {1698, 4386}, {1759, 20271}, {1930, 24358}, {1931, 46194}, {1935, 52635}, {2110, 40638}, {2207, 62972}, {2223, 57519}, {2238, 4251}, {2242, 3616}, {2251, 17123}, {2271, 4383}, {2276, 5248}, {2475, 7747}, {2476, 5475}, {2478, 3767}, {2549, 6872}, {2999, 19737}, {3125, 3496}, {3199, 62971}, {3216, 18755}, {3329, 33047}, {3501, 8616}, {3552, 27318}, {3661, 5278}, {3721, 30117}, {3734, 16920}, {3735, 3924}, {3746, 20691}, {3815, 7483}, {3871, 52959}, {3934, 17541}, {3972, 16915}, {4188, 5206}, {4193, 7746}, {4252, 16851}, {4258, 37679}, {4262, 17749}, {4284, 4877}, {4366, 17143}, {4372, 33942}, {4376, 33945}, {4400, 6381}, {4417, 25665}, {4512, 9593}, {5007, 5047}, {5008, 17536}, {5011, 21951}, {5013, 16370}, {5016, 34542}, {5019, 13725}, {5023, 16371}, {5024, 17571}, {5035, 13745}, {5041, 16858}, {5042, 13736}, {5052, 15988}, {5062, 63072}, {5080, 69175}, {5084, 7735}, {5091, 25806}, {5124, 20833}, {5129, 5304}, {5187, 43620}, {5210, 19537}, {5250, 9620}, {5254, 11113}, {5264, 21793}, {5275, 11108}, {5282, 28082}, {5301, 17303}, {5309, 66099}, {5362, 69155}, {5367, 69156}, {5395, 33037}, {6175, 14537}, {6179, 16997}, {6196, 30649}, {6389, 28722}, {6423, 31473}, {6542, 19742}, {6675, 37661}, {6690, 31460}, {6781, 37256}, {6857, 7736}, {6906, 62371}, {6910, 31401}, {6921, 21843}, {6933, 31415}, {7504, 7603}, {7738, 11111}, {7739, 31156}, {7748, 11114}, {7750, 17670}, {7751, 17002}, {7755, 37162}, {7756, 15680}, {7761, 33840}, {7762, 37664}, {7766, 27269}, {7770, 16992}, {7772, 16865}, {7786, 17684}, {7790, 33824}, {7795, 45962}, {7797, 17685}, {7802, 33823}, {7804, 17686}, {7806, 33046}, {7828, 17669}, {7834, 17550}, {8588, 37307}, {8624, 16823}, {8666, 63493}, {8715, 10987}, {8728, 18907}, {9300, 15670}, {9596, 10198}, {9599, 26363}, {9605, 16418}, {9607, 57003}, {9665, 11680}, {11010, 21888}, {11319, 24275}, {11320, 24271}, {11346, 48864}, {11349, 31198}, {11354, 48851}, {13587, 15513}, {13730, 36743}, {13740, 52538}, {13741, 26244}, {13881, 17556}, {14712, 17565}, {15048, 50241}, {15171, 21956}, {15515, 17548}, {15815, 19535}, {15820, 30770}, {16368, 19724}, {16549, 17735}, {16686, 20994}, {16705, 16931}, {16819, 20179}, {16853, 21309}, {16857, 43136}, {16859, 63004}, {16862, 22331}, {16919, 69172}, {16989, 33029}, {16996, 31276}, {17034, 33295}, {17050, 53602}, {17053, 38871}, {17129, 18145}, {17277, 33821}, {17294, 19723}, {17308, 19732}, {17330, 33882}, {17389, 63060}, {17397, 19684}, {17531, 35007}, {17532, 65630}, {17543, 41940}, {17549, 37512}, {17558, 37665}, {17561, 63024}, {17574, 31652}, {17577, 39590}, {17680, 25468}, {17754, 54354}, {17756, 31451}, {17798, 20988}, {17962, 39748}, {19237, 40773}, {19701, 25527}, {19717, 29586}, {19750, 29605}, {19865, 56903}, {20331, 24047}, {20947, 30167}, {20970, 32911}, {21008, 49997}, {21341, 37656}, {21511, 69016}, {21764, 59305}, {21826, 38869}, {21868, 48696}, {21921, 39251}, {22065, 23531}, {22398, 30904}, {24482, 36287}, {24891, 29564}, {24953, 31466}, {26243, 30819}, {27040, 56983}, {30646, 40986}, {31295, 43618}, {33033, 53489}, {33040, 51171}, {33762, 33834}, {33863, 52680}, {33950, 49758}, {37179, 68747}, {37316, 54426}, {37375, 39565}, {37676, 60721}, {37817, 54317}, {40718, 40746}, {40984, 47511}, {41656, 59517}, {43619, 50244}, {44519, 57006}, {44526, 50242}, {46922, 50179}, {48822, 48867}, {49560, 51331}, {50243, 63633}, {54318, 54382}, {54409, 68883}, {57002, 63548}, {57288, 69098}

X(69212) = isotomic conjugate of the isogonal conjugate of X(5371)
X(69212) = X(514)-isoconjugate of X(29071)
X(69212) = crosspoint of X(4590) and X(34594)
X(69212) = crosssum of X(i) and X(j) for these (i,j): {6, 5347}, {3124, 4132}
X(69212) = crossdifference of every pair of points on line {513, 3005}
X(69212) = barycentric product X(i)*X(j) for these {i,j}: {1, 32914}, {76, 5371}, {100, 29070}, {190, 68880}
X(69212) = barycentric quotient X(i)/X(j) for these {i,j}: {692, 29071}, {5371, 6}, {29070, 693}, {32914, 75}, {68880, 514}
X(69212) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 9, 3954}, {1, 4426, 5291}, {2, 32, 5277}, {6, 405, 5283}, {21, 33854, 39}, {238, 41239, 213}, {384, 17000, 274}, {385, 16918, 18140}, {958, 16502, 16975}, {1698, 7031, 4386}, {1724, 16783, 6}, {5007, 16589, 5276}, {5047, 5276, 16589}, {5247, 16503, 20963}, {5251, 5299, 1107}, {5258, 16784, 17448}, {5259, 5280, 37}, {5291, 68893, 1}, {11108, 30435, 5275}, {11319, 26035, 24275}, {16916, 16998, 76}, {16920, 34284, 3734}, {17002, 18135, 7751}, {17541, 37670, 3934}, {52680, 68950, 33863}

X(69213) = (1, -1, -1, -1)-ADDITIVE ASSOCIATE OF X(23)

Barycentrics    a^2*(a^4 + b^4 - b^2*c^2 + c^4) : :

X(69213) lies on these lines: {2, 6}, {3, 39524}, {4, 14715}, {22, 40825}, {23, 1501}, {25, 41363}, {32, 3060}, {51, 251}, {110, 1196}, {112, 65654}, {182, 1180}, {184, 9465}, {324, 6531}, {376, 44415}, {384, 33798}, {511, 1627}, {1194, 1692}, {1199, 37446}, {1691, 6636}, {1899, 34137}, {1915, 3124}, {1974, 40366}, {2001, 5640}, {2030, 22352}, {2056, 20976}, {2422, 62663}, {2450, 5305}, {2979, 5028}, {3094, 15246}, {3148, 9777}, {3167, 40126}, {3410, 53475}, {3767, 11442}, {3787, 23061}, {3917, 44499}, {3978, 8039}, {5007, 15019}, {5017, 62187}, {5025, 33796}, {5038, 11205}, {5039, 15004}, {5052, 53863}, {5092, 38862}, {6525, 56364}, {6620, 8743}, {7496, 8041}, {7755, 41724}, {8267, 12215}, {9998, 57258}, {10312, 47328}, {10329, 35006}, {14912, 40179}, {15038, 37345}, {16949, 18906}, {19128, 40938}, {19136, 40146}, {21969, 41413}, {23291, 60495}, {25047, 35929}, {26881, 62194}, {33873, 35388}, {34481, 44116}, {34565, 44500}, {34986, 40130}, {35264, 62702}, {36990, 63538}, {38905, 46546}, {41334, 52580}, {41472, 69155}, {41473, 69156}, {51437, 64820}, {56344, 60501}, {57481, 61384}

X(69213) = isogonal conjugate of the isotomic conjugate of X(7828)
X(69213) = crosssum of X(i) and X(j) for these (i,j): {2, 7836}, {6, 50669}, {39, 16893}
X(69213) = barycentric product X(6)*X(7828)
X(69213) = barycentric quotient X(7828)/X(76)
X(69213) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6, 34945}, {6, 1184, 1993}, {6, 3051, 1994}, {6, 5359, 62991}, {6, 42295, 2}, {251, 39024, 51}, {1184, 1993, 9463}, {1194, 1692, 5012}, {1501, 3981, 23}, {1691, 20859, 6636}, {1915, 3124, 13595}, {1994, 5354, 3051}, {34482, 34545, 6}

X(69214) = (1, 1, 1, -1, 1, 1, 1, 1, 1, -1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 + a^2*b - a*b^2 + b^3 + a^2*c - 2*a*b*c + b^2*c - a*c^2 + b*c^2 + c^3) : :

X(69214) lies on these lines: {1, 39}, {3, 39255}, {6, 3692}, {8, 16968}, {9, 983}, {19, 25}, {31, 3930}, {32, 17742}, {38, 41423}, {40, 3721}, {42, 16972}, {45, 39254}, {57, 3726}, {63, 17735}, {72, 14974}, {78, 2176}, {100, 26242}, {171, 51058}, {172, 37552}, {200, 2238}, {213, 3811}, {345, 1999}, {346, 5304}, {594, 5271}, {595, 54406}, {609, 5525}, {614, 1575}, {672, 3938}, {893, 56279}, {902, 5282}, {976, 1334}, {984, 60711}, {997, 3230}, {1104, 4515}, {1279, 44798}, {1376, 3290}, {1449, 3979}, {1571, 3670}, {1572, 37610}, {1697, 3727}, {1706, 21951}, {1724, 4006}, {1743, 5332}, {1961, 3247}, {1965, 17786}, {2000, 8609}, {2082, 20692}, {2177, 21840}, {2178, 37577}, {2280, 3722}, {2285, 41346}, {2321, 4362}, {2345, 3757}, {3052, 50995}, {3061, 37588}, {3125, 54286}, {3158, 46907}, {3242, 42316}, {3306, 3666}, {3509, 3550}, {3555, 5021}, {3677, 9574}, {3703, 17156}, {3723, 17599}, {3729, 4376}, {3731, 10987}, {3735, 5119}, {3751, 60697}, {3767, 21073}, {3771, 4071}, {3772, 21956}, {3780, 6765}, {3815, 17721}, {3862, 19591}, {3913, 41015}, {3915, 33299}, {3923, 21101}, {3935, 37657}, {3954, 12514}, {3957, 63066}, {3959, 63130}, {3970, 5264}, {3991, 5266}, {4030, 17275}, {4119, 4438}, {4424, 31433}, {4441, 26247}, {4561, 35274}, {4689, 16672}, {4851, 69093}, {4855, 21008}, {5248, 28594}, {5255, 54382}, {5256, 56510}, {5269, 63099}, {5291, 37817}, {5687, 16583}, {5750, 29651}, {7031, 17744}, {7191, 17756}, {8557, 10315}, {8715, 16600}, {9259, 35262}, {9598, 13161}, {10319, 36572}, {10436, 24326}, {11529, 65695}, {14001, 30701}, {16061, 39731}, {16502, 25066}, {16503, 17715}, {16604, 28011}, {16969, 19861}, {16974, 20691}, {17595, 31443}, {17737, 29665}, {17784, 62693}, {21853, 40959}, {21859, 57277}, {24239, 31497}, {24403, 26229}, {24514, 68875}, {25092, 30145}, {25264, 30141}, {25355, 68929}, {31442, 67979}, {31448, 37592}, {33854, 62806}, {33863, 62874}, {37673, 67097}, {38874, 40217}, {50087, 50104}, {52963, 54330}

X(69214) = X(4025)-isoconjugate of X(58977)
X(69214) = crossdifference of every pair of points on line {659, 905}
X(69214) = barycentric product X(i)*X(j) for these {i,j}: {37, 16050}, {100, 48062}
X(69214) = barycentric quotient X(i)/X(j) for these {i,j}: {16050, 274}, {48062, 693}
X(69214) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1018, 9620}, {9, 3749, 1914}, {37, 4386, 40131}, {200, 16970, 2238}, {672, 3938, 16973}, {3247, 17594, 41269}, {3693, 3744, 6}, {3915, 33299, 39248}, {3991, 5266, 54416}, {16777, 31477, 3666}, {16974, 20691, 54418}, {17735, 49509, 63}, {37610, 57015, 1572}

X(69215) = (1, 1, 1, -1, 1, 1, -1, 1, -1, -1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 + a^2*b - a*b^2 + b^3 + a^2*c - 2*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3) : :

X(69215) lies on these lines: {1, 6}, {3, 3290}, {21, 26242}, {25, 2223}, {31, 21808}, {32, 40131}, {38, 31442}, {39, 614}, {40, 3125}, {46, 17735}, {55, 16583}, {65, 14974}, {75, 33821}, {115, 27553}, {169, 1914}, {172, 37817}, {239, 17776}, {274, 16050}, {321, 11342}, {346, 19851}, {354, 5021}, {500, 15976}, {595, 54382}, {612, 4204}, {672, 28082}, {748, 33299}, {975, 16850}, {976, 59207}, {980, 37323}, {986, 60711}, {1015, 28011}, {1125, 54317}, {1201, 9619}, {1334, 3924}, {1400, 4332}, {1500, 54418}, {1571, 24443}, {1572, 3915}, {1766, 13732}, {2079, 2915}, {2082, 2241}, {2178, 13730}, {2191, 68743}, {2207, 5089}, {2212, 17442}, {2218, 18785}, {2238, 3811}, {2271, 37080}, {2295, 54318}, {2345, 13742}, {2549, 23536}, {3011, 3767}, {3177, 69003}, {3208, 60353}, {3295, 41015}, {3338, 33863}, {3496, 8616}, {3509, 54354}, {3612, 21008}, {3677, 31429}, {3679, 68944}, {3691, 3938}, {3721, 12514}, {3726, 62858}, {3730, 30117}, {3735, 5250}, {3752, 31448}, {3771, 4109}, {3772, 30810}, {3959, 5119}, {3997, 30143}, {4205, 39586}, {4294, 62693}, {4362, 21071}, {4384, 32777}, {4512, 46902}, {4642, 31433}, {5019, 54385}, {5248, 16600}, {5266, 5275}, {5277, 37552}, {5364, 40955}, {5687, 16605}, {6155, 37553}, {8624, 13733}, {8715, 16611}, {9259, 37618}, {9598, 23537}, {9840, 62214}, {10436, 25497}, {15981, 28631}, {16716, 17524}, {16823, 17526}, {16826, 27184}, {16830, 37314}, {16831, 25499}, {17054, 42316}, {17279, 17540}, {17696, 26274}, {17721, 31466}, {17732, 24159}, {17736, 52680}, {17750, 54392}, {18135, 26247}, {18755, 59337}, {20227, 54322}, {20861, 27626}, {21951, 54286}, {22036, 56082}, {22200, 27659}, {23407, 37325}, {24045, 24160}, {24046, 24047}, {24512, 64675}, {24549, 49516}, {25082, 33854}, {25354, 49561}, {25440, 39255}, {26227, 27040}, {26234, 33819}, {27802, 37609}, {29597, 37631}, {31421, 62695}, {37225, 53387}, {37246, 37575}, {40941, 54285}, {41269, 62871}, {48824, 48854}, {56525, 62817}

X(69215) = crossdifference of every pair of points on line {513, 24562}
X(69215) = barycentric product X(9)*X(28081)
X(69215) = barycentric quotient X(28081)/X(85)
X(69215) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 16552, 16973}, {1, 16970, 213}, {37, 1104, 54416}, {37, 4426, 17742}, {37, 16974, 1}, {1212, 1279, 16502}, {1334, 3924, 9620}, {2241, 49758, 2082}, {3915, 17451, 1572}, {17735, 20271, 46}, {24443, 41423, 1571}

X(69216) = (1, 1, 1, -1, 1, -1, -1, 1, -1, -1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 + a^2*b - a*b^2 + b^3 + a^2*c + 2*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3) : :

X(69216) lies on these lines: {1, 6}, {3, 41015}, {10, 54317}, {19, 5019}, {31, 1572}, {32, 1951}, {33, 53561}, {39, 54418}, {46, 3959}, {56, 16583}, {57, 3125}, {58, 54382}, {63, 3735}, {65, 5021}, {77, 43059}, {169, 172}, {239, 17740}, {292, 998}, {474, 16605}, {604, 40977}, {609, 5540}, {612, 1573}, {614, 1015}, {672, 9620}, {997, 2238}, {999, 3290}, {1018, 49494}, {1146, 37646}, {1193, 9619}, {1201, 8776}, {1398, 1880}, {1411, 2279}, {1420, 17966}, {1468, 17451}, {1475, 3924}, {1571, 4642}, {1914, 37817}, {2242, 40131}, {2271, 2646}, {2275, 3002}, {2285, 5042}, {2292, 31442}, {2305, 54420}, {2345, 16821}, {2549, 3914}, {2901, 62426}, {2999, 9592}, {3010, 20978}, {3057, 14974}, {3338, 20271}, {3612, 18755}, {3666, 31449}, {3721, 62858}, {3727, 12514}, {3730, 15955}, {3772, 69098}, {3780, 3811}, {4000, 5088}, {4051, 5255}, {4165, 33119}, {4257, 5011}, {4293, 62693}, {4384, 35466}, {4386, 50014}, {4393, 63067}, {4511, 37657}, {4646, 31448}, {4674, 6205}, {4875, 37539}, {5115, 17443}, {5119, 17735}, {5716, 31405}, {5718, 16831}, {5725, 37661}, {7176, 37800}, {8666, 16600}, {8715, 39255}, {9346, 62819}, {9597, 23537}, {16826, 63008}, {16832, 31187}, {16834, 56523}, {17026, 24291}, {17474, 28082}, {17750, 19860}, {17754, 60353}, {18156, 33821}, {20282, 34591}, {21008, 37618}, {21044, 29662}, {21332, 63099}, {24247, 33137}, {24268, 40940}, {24512, 54318}, {26242, 54391}, {29857, 34542}, {31433, 41423}, {31477, 64175}, {39253, 62805}, {43039, 57277}, {49496, 69044}, {60711, 66674}

X(69216) = crosssum of X(1) and X(54330)
X(69216) = crossdifference of every pair of points on line {513, 69041}
X(69216) = barycentric product X(1)*X(11269)
X(69216) = barycentric quotient X(11269)/X(75)
X(69216) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 16970, 3230}, {1, 45751, 16973}, {31, 2170, 1572}, {213, 53165, 1}, {672, 49487, 9620}, {1104, 40133, 16502}, {2242, 49758, 40131}, {3061, 5247, 54406}, {3959, 33863, 46}, {16974, 17448, 1}

X(69217) = (1, 1, -1, 1, 1, -1, 1, -1, 1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 + a^2*b + a*b^2 - b^3 + a^2*c + 2*a*b*c + b^2*c + a*c^2 + b*c^2 - c^3) : :

X(69217) lies on these lines: {1, 32}, {2, 54406}, {6, 169}, {7, 5286}, {9, 1046}, {10, 31409}, {31, 21808}, {37, 5711}, {39, 57}, {40, 1500}, {41, 2650}, {42, 15496}, {46, 1571}, {56, 9619}, {58, 16968}, {63, 5283}, {65, 9620}, {72, 5275}, {78, 5277}, {115, 9612}, {165, 31451}, {187, 3601}, {213, 40131}, {226, 3767}, {230, 11374}, {354, 16502}, {387, 62693}, {553, 7739}, {574, 15803}, {612, 3954}, {750, 33299}, {910, 2271}, {950, 7737}, {975, 40750}, {1015, 3333}, {1107, 62858}, {1125, 39248}, {1155, 31422}, {1210, 2548}, {1212, 5021}, {1395, 17442}, {1468, 17451}, {1475, 69030}, {1504, 51841}, {1505, 51842}, {1573, 57279}, {1698, 31476}, {1702, 31471}, {1706, 52959}, {1737, 9596}, {1745, 2653}, {1770, 9598}, {1788, 31398}, {1836, 69096}, {1858, 62372}, {1892, 27376}, {1944, 4644}, {2082, 20229}, {2171, 2199}, {2270, 4263}, {2275, 3338}, {2294, 5336}, {2295, 17742}, {2300, 54385}, {2329, 66640}, {2549, 4292}, {3053, 24929}, {3061, 37607}, {3125, 54418}, {3290, 16466}, {3294, 49500}, {3339, 9593}, {3361, 9592}, {3487, 7735}, {3579, 31477}, {3586, 7747}, {3691, 32912}, {3811, 4386}, {3868, 5276}, {3874, 16973}, {3876, 37675}, {3911, 31401}, {3923, 21071}, {3928, 31429}, {3970, 5264}, {4109, 32946}, {4426, 54318}, {4654, 5309}, {4666, 68893}, {4847, 31416}, {5007, 11518}, {5013, 37582}, {5024, 37545}, {5045, 16781}, {5122, 15815}, {5128, 31426}, {5206, 30282}, {5219, 7746}, {5231, 31488}, {5254, 57282}, {5255, 51058}, {5280, 5902}, {5282, 59305}, {5290, 69175}, {5291, 19860}, {5299, 18398}, {5304, 11036}, {5305, 6147}, {5435, 31400}, {5475, 9581}, {5587, 9650}, {5704, 31404}, {5708, 9605}, {5722, 7745}, {5725, 21965}, {5808, 21049}, {5903, 16785}, {7373, 62370}, {7748, 9579}, {9331, 11010}, {9588, 31478}, {9664, 41869}, {10404, 69098}, {10436, 21240}, {12433, 18907}, {13006, 17700}, {15048, 24470}, {15934, 30435}, {16517, 54422}, {16600, 16972}, {16784, 50190}, {16975, 62874}, {17200, 30105}, {17594, 36643}, {17799, 37604}, {20665, 40955}, {20691, 54286}, {21384, 32913}, {21764, 28082}, {23621, 51949}, {24477, 31405}, {24914, 31441}, {26223, 27040}, {26242, 57280}, {27255, 66152}, {31231, 31455}, {31406, 34753}, {31421, 53056}, {31423, 31501}, {31437, 31459}, {31443, 31461}, {31444, 31462}, {31456, 62824}, {33950, 63066}, {36744, 37547}, {37522, 54317}, {37534, 62371}, {40997, 49745}, {41883, 46835}, {46902, 62845}, {49758, 62812}, {51816, 63493}, {56530, 66646}

X(69217) = crossdifference of every pair of points on line {1491, 15313}
X(69217) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 54382, 1572}, {40, 1500, 31433}, {46, 2276, 1571}, {63, 5283, 31442}, {65, 54416, 9620}, {1155, 31448, 31422}, {1788, 31402, 31398}, {2082, 62819, 20963}, {3333, 9575, 1015}, {3721, 63099, 1}, {16600, 62805, 16972}, {24914, 31460, 31441}, {37522, 57015, 54317}, {40131, 54421, 213}

X(69218) = (1, 1, -1, 1, 1, -1, -1, -1, -1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 + a^2*b + a*b^2 - b^3 + a^2*c + 2*a*b*c - b^2*c + a*c^2 - b*c^2 - c^3) : :

X(69218) lies on these lines: {1, 6}, {7, 50011}, {8, 54382}, {10, 31409}, {19, 1843}, {31, 3930}, {32, 3811}, {39, 62858}, {40, 20691}, {41, 22061}, {42, 5282}, {43, 3509}, {57, 1575}, {63, 2276}, {69, 894}, {78, 172}, {141, 4472}, {169, 23841}, {190, 49496}, {193, 346}, {198, 17798}, {200, 4386}, {210, 5275}, {292, 62853}, {511, 1766}, {519, 1572}, {524, 17281}, {527, 33869}, {594, 3416}, {604, 53531}, {612, 63099}, {614, 3726}, {672, 32912}, {742, 3729}, {758, 9620}, {896, 41423}, {910, 4849}, {966, 59406}, {968, 60724}, {992, 54385}, {997, 2242}, {1018, 49500}, {1045, 1761}, {1046, 3501}, {1213, 38047}, {1445, 34253}, {1468, 33299}, {1469, 2285}, {1500, 12514}, {1706, 21868}, {1707, 17735}, {1722, 20271}, {1740, 3862}, {1759, 3293}, {1778, 41610}, {1836, 21956}, {1914, 3870}, {1966, 17786}, {1992, 17389}, {2082, 3780}, {2093, 21888}, {2112, 3217}, {2178, 36741}, {2238, 40131}, {2275, 62874}, {2277, 21371}, {2295, 54421}, {2303, 3786}, {2321, 3923}, {2325, 64073}, {2548, 10916}, {3053, 56176}, {3056, 42447}, {3216, 17736}, {3218, 17756}, {3290, 4383}, {3333, 16604}, {3496, 50581}, {3589, 29603}, {3618, 17260}, {3619, 29608}, {3620, 29591}, {3629, 29605}, {3681, 5276}, {3685, 17314}, {3686, 49529}, {3693, 3719}, {3721, 54418}, {3749, 21793}, {3759, 32029}, {3767, 21077}, {3827, 21853}, {3873, 33854}, {3875, 9055}, {3886, 5846}, {3916, 31448}, {3928, 9574}, {3938, 21764}, {3943, 67964}, {3949, 20964}, {3950, 51196}, {3986, 59408}, {3991, 14974}, {4006, 5264}, {4037, 56082}, {4071, 32946}, {4119, 4865}, {4260, 54405}, {4261, 16574}, {4360, 49502}, {4362, 21101}, {4384, 49481}, {4430, 63075}, {4437, 4851}, {4640, 31477}, {4661, 63004}, {4908, 15534}, {5021, 25066}, {5089, 44105}, {5256, 41269}, {5268, 40750}, {5279, 54383}, {5750, 49511}, {6651, 17242}, {6734, 9596}, {7262, 60711}, {7289, 44421}, {7735, 25568}, {7736, 24477}, {7745, 68616}, {8666, 9619}, {9041, 50131}, {9593, 54422}, {9598, 64002}, {9599, 26015}, {12329, 36744}, {17151, 49533}, {17275, 49524}, {17278, 51150}, {17330, 47359}, {17337, 38186}, {17355, 34379}, {17362, 49688}, {17365, 47595}, {17388, 49681}, {17737, 31053}, {17754, 32913}, {17781, 68855}, {18206, 62692}, {20785, 56556}, {21840, 61358}, {22035, 49683}, {22769, 36743}, {24342, 59772}, {24512, 62819}, {25092, 31442}, {26242, 32911}, {26685, 49775}, {28538, 50087}, {28594, 62805}, {29380, 68964}, {29586, 51171}, {29588, 51170}, {31426, 54290}, {31497, 59491}, {34377, 68478}, {36483, 62812}, {36740, 54285}, {37650, 59405}, {37654, 50310}, {37657, 63818}, {37675, 63961}, {39254, 67207}, {40746, 56179}, {41325, 51190}, {47356, 50113}, {48876, 59680}, {49764, 59579}, {50015, 50026}, {50115, 50311}, {50123, 51000}, {50996, 63054}, {52959, 54286}, {58798, 69096}, {62832, 63493}

X(69218) = reflection of X(16973) in X(6)
X(69218) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 54406, 39248}, {6, 9, 36404}, {6, 37, 16972}, {6, 3242, 1100}, {6, 16777, 1386}, {6, 49509, 1}, {6, 50995, 37}, {9, 3751, 6}, {1468, 33299, 54317}, {1743, 51194, 6}, {4386, 20693, 200}, {6762, 9575, 17448}

X(69219) = (1, -1, 1, 1, -1, 1, 1, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 - a^2*b - a*b^2 - b^3 - a^2*c - 2*a*b*c + b^2*c - a*c^2 + b*c^2 - c^3) : :

X(69219) lies on these lines: {1, 1929}, {3, 41015}, {6, 46}, {9, 6048}, {19, 2092}, {32, 54418}, {35, 16968}, {37, 5687}, {39, 2082}, {40, 213}, {41, 4642}, {42, 15496}, {43, 3496}, {55, 16583}, {57, 20963}, {65, 2271}, {78, 3735}, {169, 2276}, {200, 3954}, {386, 5011}, {405, 16605}, {614, 2241}, {672, 1571}, {893, 20606}, {910, 4646}, {968, 16589}, {986, 3684}, {988, 16975}, {997, 3727}, {1155, 5021}, {1193, 1572}, {1197, 54373}, {1212, 31448}, {1334, 31433}, {1500, 40131}, {1697, 3230}, {1739, 16783}, {1759, 3293}, {2083, 3764}, {2170, 9619}, {2176, 5119}, {2177, 21808}, {2238, 12514}, {2273, 56911}, {2277, 68729}, {2280, 24443}, {2285, 4263}, {2295, 54286}, {2329, 64176}, {3085, 62693}, {3158, 46902}, {3214, 5282}, {3216, 39248}, {3269, 62321}, {3290, 3295}, {3333, 16971}, {3419, 21965}, {3509, 50581}, {3576, 53165}, {3670, 16973}, {3691, 4414}, {3721, 3811}, {3751, 36643}, {3752, 16502}, {3767, 3914}, {3780, 62858}, {3871, 26242}, {3931, 5275}, {3987, 16788}, {4051, 37617}, {4109, 4660}, {4875, 31449}, {5013, 43065}, {5179, 9598}, {5248, 16611}, {5264, 16972}, {5272, 68893}, {5283, 17594}, {5336, 36744}, {5819, 31402}, {8715, 16600}, {9574, 16572}, {11010, 54981}, {12526, 21839}, {14974, 37568}, {16503, 24174}, {16549, 36404}, {16601, 31477}, {16970, 61763}, {17596, 21384}, {17735, 59316}, {17742, 20691}, {17756, 33950}, {18156, 33828}, {20970, 54421}, {21796, 54359}, {21951, 54318}, {24440, 41239}, {25440, 54317}, {27040, 32929}, {29639, 31416}, {31451, 49758}, {33863, 58887}, {37657, 56288}, {37675, 62831}, {46835, 69096}, {50314, 52538}

X(69219) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {41, 4642, 9620}, {43, 3496, 54406}, {386, 5011, 54382}, {910, 4646, 54416}, {3691, 4414, 31442}, {3959, 18755, 1}

X(69220) = (1, -1, 1, 1, -1, 1, -1, 1, -1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 - a^2*b - a*b^2 - b^3 - a^2*c - 2*a*b*c - b^2*c - a*c^2 - b*c^2 - c^3) : :

X(69220) lies on these lines: {1, 32}, {2, 62693}, {3, 54317}, {6, 63}, {9, 43}, {19, 25}, {21, 16968}, {31, 16972}, {35, 39255}, {38, 2280}, {40, 2295}, {41, 2292}, {42, 5282}, {45, 3693}, {46, 17750}, {48, 20282}, {57, 24512}, {58, 39253}, {71, 5364}, {72, 2271}, {78, 18755}, {169, 5283}, {193, 60729}, {213, 12514}, {230, 17720}, {284, 35623}, {304, 16060}, {312, 26244}, {345, 966}, {350, 52652}, {386, 54406}, {405, 16583}, {594, 63131}, {604, 22099}, {672, 4414}, {941, 5279}, {958, 41015}, {975, 5277}, {980, 16782}, {982, 16503}, {984, 3684}, {986, 41239}, {988, 2275}, {1001, 3290}, {1018, 31433}, {1040, 68747}, {1100, 17599}, {1107, 2082}, {1193, 39248}, {1213, 32777}, {1247, 60676}, {1282, 7174}, {1403, 2285}, {1449, 5332}, {1500, 17742}, {1571, 16549}, {1621, 26242}, {1698, 68934}, {1707, 60697}, {1766, 37619}, {1837, 21965}, {1973, 68558}, {2093, 65695}, {2176, 5250}, {2177, 3930}, {2268, 2312}, {2269, 53129}, {2277, 30646}, {2288, 20752}, {2303, 25058}, {2329, 37598}, {2345, 32932}, {3053, 37539}, {3125, 54318}, {3218, 63066}, {3219, 37657}, {3224, 21379}, {3247, 3749}, {3305, 37673}, {3403, 41250}, {3553, 10315}, {3663, 24333}, {3670, 16783}, {3672, 5304}, {3703, 17275}, {3719, 15984}, {3729, 24326}, {3744, 16777}, {3750, 51058}, {3752, 56518}, {3780, 57279}, {3811, 3954}, {3870, 49509}, {3916, 5021}, {3931, 54416}, {3959, 19860}, {3980, 5750}, {4030, 17299}, {4071, 4660}, {4254, 20760}, {4262, 30115}, {4376, 10436}, {4419, 10025}, {4424, 9620}, {4426, 54418}, {4512, 16970}, {4643, 69093}, {4652, 33863}, {4760, 16831}, {4850, 33854}, {4875, 31490}, {5011, 30116}, {5248, 16600}, {5254, 50065}, {5257, 59692}, {5276, 28606}, {5287, 40750}, {5305, 50067}, {5306, 50068}, {5309, 50066}, {5525, 9331}, {5530, 9596}, {8715, 28594}, {9574, 20331}, {9599, 24239}, {10319, 18592}, {10393, 60586}, {10448, 17451}, {11517, 16601}, {16502, 37592}, {16517, 62818}, {16524, 37596}, {16525, 34251}, {16552, 31442}, {16572, 31429}, {16589, 54287}, {16779, 17591}, {17017, 21764}, {17596, 17754}, {17735, 35258}, {17737, 33134}, {19861, 21008}, {20271, 54392}, {20963, 62858}, {21101, 29670}, {21793, 62834}, {21951, 64673}, {22060, 36743}, {25066, 31448}, {26085, 57808}, {31449, 43065}, {37553, 60724}, {37577, 54285}, {38316, 46178}, {40997, 64158}, {46835, 54396}, {56512, 63075}, {56513, 63004}, {56530, 66674}, {59547, 63978}, {62796, 63087}

X(69220) = X(68712)-Ceva conjugate of X(50295)
X(69220) = crossdifference of every pair of points on line {905, 1491}
X(69220) = barycentric product X(i)*X(j) for these {i,j}: {1, 50295}, {37, 68712}, {100, 68780}
X(69220) = barycentric quotient X(i)/X(j) for these {i,j}: {50295, 75}, {68712, 274}, {68780, 693}
X(69220) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3496, 54382}, {9, 17594, 2276}, {31, 21840, 16972}, {37, 910, 5275}, {37, 4386, 612}, {38, 2280, 16973}, {45, 31477, 3693}, {169, 62871, 5283}, {968, 40131, 37}, {1914, 41269, 1}, {3693, 4689, 31477}, {4424, 16788, 9620}

X(69221) = (1, -1, 1, 1, -1, -1, 1, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 - a^2*b - a*b^2 - b^3 - a^2*c + 2*a*b*c + b^2*c - a*c^2 + b*c^2 - c^3) : :

X(69221) lies on these lines: {1, 1929}, {3, 3290}, {6, 3338}, {9, 9367}, {19, 17053}, {32, 614}, {36, 16968}, {37, 474}, {39, 40131}, {40, 3230}, {41, 244}, {45, 25068}, {46, 2176}, {56, 16583}, {57, 213}, {63, 28251}, {75, 33828}, {101, 24046}, {115, 27685}, {169, 2275}, {198, 20227}, {354, 2271}, {404, 26242}, {762, 8580}, {851, 53387}, {859, 16716}, {910, 16502}, {936, 3954}, {956, 16605}, {975, 41269}, {978, 3509}, {988, 5283}, {995, 54382}, {997, 3721}, {999, 41015}, {1015, 2082}, {1054, 3501}, {1055, 3924}, {1086, 20269}, {1155, 14974}, {1196, 28259}, {1201, 1572}, {1334, 1571}, {1375, 3772}, {1420, 53165}, {1575, 17742}, {1722, 5291}, {1759, 39248}, {1766, 19514}, {2092, 54385}, {2170, 32577}, {2178, 5336}, {2238, 62858}, {2241, 28011}, {2242, 54418}, {2251, 5573}, {2270, 20228}, {2277, 28258}, {2285, 21796}, {2329, 24174}, {2352, 20857}, {3086, 62693}, {3207, 17054}, {3216, 17736}, {3306, 17750}, {3333, 20963}, {3336, 54981}, {3496, 21214}, {3666, 16412}, {3673, 26273}, {3684, 3976}, {3726, 3811}, {3735, 19861}, {3752, 37272}, {3767, 23536}, {3953, 16973}, {4372, 68756}, {4419, 25583}, {5013, 16601}, {5021, 32636}, {5119, 16969}, {5179, 9597}, {5275, 37592}, {5282, 27627}, {5286, 40127}, {5319, 40128}, {7131, 43063}, {8572, 34522}, {8666, 16611}, {9310, 9620}, {9593, 62695}, {9619, 17451}, {15803, 16970}, {16600, 54317}, {16972, 37522}, {17063, 41239}, {17735, 58887}, {17966, 34489}, {19522, 20284}, {20970, 62819}, {21744, 41264}, {21769, 54420}, {21809, 33589}, {21816, 62818}, {21839, 54422}, {24166, 30108}, {24440, 56530}, {24598, 56517}, {25593, 33146}, {25946, 28606}, {26085, 69134}, {26229, 26978}, {26234, 33830}, {26279, 34284}, {28017, 51436}, {31198, 56518}, {31422, 41423}, {31442, 59207}, {33825, 33891}, {36404, 68950}, {37673, 41229}, {37817, 46548}, {41526, 42669}, {46835, 69098}

X(69221) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {910, 52541, 16502}, {978, 3509, 54406}, {1759, 49997, 39248}, {2178, 40941, 5336}, {3959, 9259, 1}, {9310, 24443, 9620}, {20271, 21008, 1}, {23536, 68797, 3767}

X(69222) = (1, -1, 1, 1, -1, -1, -1, 1, -1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 - a^2*b - a*b^2 - b^3 - a^2*c + 2*a*b*c - b^2*c - a*c^2 - b*c^2 - c^3) : :

X(69222) lies on these lines: {1, 32}, {3, 39255}, {6, 62874}, {9, 1050}, {19, 20471}, {37, 56}, {38, 9310}, {39, 17742}, {41, 16973}, {55, 39653}, {57, 2295}, {63, 2176}, {78, 21008}, {169, 16975}, {190, 25918}, {213, 62858}, {348, 4419}, {612, 17798}, {614, 4426}, {728, 9574}, {956, 16583}, {958, 3290}, {968, 21010}, {976, 1055}, {982, 2329}, {986, 56530}, {988, 2276}, {995, 54406}, {997, 3954}, {1018, 1571}, {1107, 40131}, {1201, 5282}, {1449, 7296}, {1468, 16972}, {1475, 36404}, {2053, 40881}, {2082, 17448}, {2238, 57279}, {2271, 3555}, {2277, 5227}, {2549, 21073}, {2975, 16968}, {3053, 3744}, {3207, 3242}, {3208, 17596}, {3230, 12514}, {3294, 31442}, {3333, 24512}, {3338, 17750}, {3339, 65695}, {3665, 17276}, {3670, 9620}, {3693, 5013}, {3729, 16720}, {3780, 6762}, {3870, 18755}, {3872, 3959}, {3875, 4372}, {3916, 14974}, {3953, 16788}, {3976, 41239}, {3991, 31448}, {4016, 64349}, {4390, 24443}, {4447, 17594}, {4513, 17595}, {4643, 69094}, {4652, 17735}, {4656, 43054}, {4675, 7198}, {4694, 16783}, {5011, 50637}, {5250, 16969}, {5252, 21965}, {6763, 54981}, {7079, 9367}, {7745, 17721}, {8545, 28391}, {8666, 16600}, {9259, 19861}, {9336, 56532}, {9596, 24239}, {9619, 57015}, {9623, 21951}, {12513, 41015}, {16043, 30701}, {16060, 39731}, {16519, 62833}, {16601, 31449}, {16777, 37539}, {16970, 62824}, {19860, 20271}, {22061, 41526}, {23222, 54359}, {24172, 24249}, {24215, 24333}, {24358, 24652}, {32577, 39244}, {33890, 62650}, {37592, 54416}, {37609, 62871}, {46902, 53165}, {56517, 62803}, {62693, 64081}

X(69222) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1759, 1572}, {1, 3509, 54382}, {37, 56, 54317}, {1201, 5282, 39248}, {1468, 16972, 39253}, {2975, 26242, 16968}, {3991, 37599, 31448}, {21008, 49509, 78}

X(69223) = (1, -1, -1, 1, -1, 1, -1, -1, -1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 - a^2*b + a*b^2 - b^3 - a^2*c - 2*a*b*c - b^2*c + a*c^2 - b*c^2 - c^3) : :

X(69223) lies on these lines: {1, 6}, {2, 169}, {10, 2082}, {19, 475}, {21, 26690}, {32, 54317}, {40, 16549}, {41, 997}, {46, 3496}, {55, 25066}, {57, 1759}, {63, 4253}, {78, 4251}, {85, 17681}, {90, 2344}, {101, 19861}, {120, 1698}, {190, 39731}, {241, 21514}, {294, 56775}, {304, 17277}, {391, 54433}, {443, 5819}, {474, 910}, {497, 21073}, {572, 2172}, {614, 16600}, {644, 3890}, {672, 12514}, {728, 31393}, {894, 17753}, {962, 1766}, {968, 25092}, {975, 5276}, {1018, 1697}, {1024, 66995}, {1125, 40131}, {1211, 7308}, {1334, 36479}, {1376, 25068}, {1438, 5248}, {1445, 3674}, {1475, 5282}, {1571, 20331}, {1572, 2295}, {1593, 2297}, {1621, 25082}, {1752, 19854}, {1760, 17381}, {1829, 7719}, {1930, 4384}, {1973, 2267}, {2264, 5783}, {2270, 18641}, {2280, 3811}, {2285, 3671}, {2345, 5082}, {2348, 25917}, {2478, 5179}, {2999, 39951}, {3002, 68709}, {3008, 56518}, {3198, 37059}, {3207, 17614}, {3219, 26626}, {3295, 3693}, {3303, 3991}, {3305, 3912}, {3333, 17736}, {3336, 36643}, {3338, 3509}, {3501, 5119}, {3646, 16550}, {3666, 9605}, {3683, 37580}, {3727, 9620}, {3730, 5250}, {3876, 63087}, {3916, 5022}, {3923, 66670}, {4006, 6765}, {4187, 46835}, {4193, 27068}, {4205, 39690}, {4258, 5440}, {4262, 4855}, {4424, 9593}, {4512, 7085}, {4513, 9957}, {4643, 69097}, {4652, 5030}, {4689, 31461}, {4872, 33838}, {4875, 9708}, {4876, 7162}, {5044, 37658}, {5084, 6554}, {5088, 17691}, {5128, 6205}, {5195, 17368}, {5257, 19836}, {5262, 63075}, {5271, 42715}, {5279, 11036}, {5305, 17720}, {5437, 21372}, {5438, 35342}, {5687, 44798}, {5722, 40997}, {6684, 8568}, {7031, 37552}, {7079, 41006}, {7187, 33827}, {7289, 17306}, {7330, 58036}, {7739, 50066}, {8074, 8582}, {8167, 25086}, {8193, 54322}, {8270, 56913}, {8728, 24701}, {9843, 46345}, {9856, 64121}, {10200, 68797}, {10436, 16818}, {10572, 24247}, {12575, 17355}, {15048, 50065}, {15288, 16293}, {15503, 26890}, {16060, 25918}, {16545, 16547}, {16699, 19259}, {17095, 68537}, {17107, 63578}, {17192, 17282}, {17260, 27248}, {17279, 69095}, {17316, 27065}, {17335, 18156}, {17367, 56511}, {17397, 56517}, {17451, 54318}, {17527, 65808}, {17687, 31269}, {17750, 54382}, {17795, 21387}, {18636, 25964}, {19869, 54324}, {20009, 62985}, {20602, 29603}, {21096, 64162}, {21390, 48335}, {21808, 64675}, {24047, 35258}, {24204, 53510}, {24388, 41789}, {24987, 56746}, {26793, 37162}, {29579, 35595}, {30107, 39712}, {30435, 37539}, {36483, 37555}, {37317, 44081}, {37652, 41251}, {39254, 39255}, {40571, 60721}, {49466, 55337}, {50067, 63633}, {51972, 63999}, {54405, 63055}, {60958, 64702}

X(69223) = complement of X(41826)
X(69223) = X(3618)-Ceva conjugate of X(2999)
X(69223) = X(40181)-cross conjugate of X(40226)
X(69223) = X(i)-isoconjugate of X(j) for these (i,j): {2297, 40225}, {40193, 56179}
X(69223) = X(612)-Dao conjugate of X(2345)
X(69223) = crosspoint of X(765) and X(37215)
X(69223) = crosssum of X(244) and X(2484)
X(69223) = barycentric product X(3672)*X(40226)
X(69223) = barycentric quotient X(i)/X(j) for these {i,j}: {1191, 40225}, {16502, 40193}, {40181, 2345}, {40226, 1219}
X(69223) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 9, 17742}, {1, 1743, 5280}, {2, 33950, 169}, {6, 1212, 54305}, {9, 1449, 5227}, {9, 2257, 21061}, {9, 16572, 16552}, {9, 21384, 41229}, {9, 31435, 3294}, {37, 16502, 1}, {41, 39244, 997}, {218, 960, 54330}, {1475, 5282, 62858}, {1759, 68950, 57}, {2280, 33299, 3811}, {3061, 41239, 1}, {3496, 17754, 46}, {5283, 16782, 1}, {16783, 57015, 1}, {17691, 27340, 5088}, {36404, 39248, 213}

X(69224) = (1, -1, -1, -1, -1, 1, -1, -1, -1, -1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 - a^2*b + a*b^2 + b^3 - a^2*c - 2*a*b*c - b^2*c + a*c^2 - b*c^2 + c^3) : :

X(69224) lies on these lines: {1, 39}, {6, 169}, {10, 16973}, {32, 57}, {34, 52635}, {35, 31422}, {37, 64675}, {40, 2241}, {41, 244}, {46, 1914}, {55, 1571}, {63, 19729}, {65, 1572}, {115, 9581}, {171, 16787}, {172, 3338}, {187, 15803}, {200, 1574}, {213, 614}, {218, 3290}, {226, 2548}, {354, 54416}, {405, 31442}, {498, 31441}, {517, 16781}, {574, 3601}, {604, 1254}, {607, 57656}, {609, 3337}, {672, 28082}, {758, 39248}, {938, 5286}, {950, 2549}, {975, 16519}, {982, 41239}, {986, 16503}, {997, 16604}, {1100, 4646}, {1104, 5021}, {1107, 54318}, {1210, 3767}, {1279, 14974}, {1384, 37545}, {1449, 34543}, {1475, 3924}, {1482, 62370}, {1506, 5219}, {1573, 64673}, {1575, 3811}, {1699, 9665}, {1722, 51194}, {1834, 5808}, {1837, 69098}, {1876, 2207}, {2066, 31437}, {2082, 3125}, {2242, 3333}, {2260, 5336}, {2271, 3752}, {2273, 54385}, {2280, 24443}, {2329, 3976}, {2999, 20970}, {3053, 37582}, {3085, 31398}, {3212, 69003}, {3230, 28011}, {3295, 31433}, {3306, 5277}, {3336, 7031}, {3475, 31402}, {3487, 7736}, {3488, 7738}, {3496, 16779}, {3553, 17053}, {3586, 7748}, {3670, 16783}, {3684, 24174}, {3726, 17742}, {3815, 11374}, {3868, 33854}, {3953, 16788}, {3997, 30148}, {4251, 24046}, {4253, 16968}, {4292, 7737}, {4384, 21240}, {4426, 62858}, {4515, 4864}, {4654, 7753}, {4861, 63499}, {5013, 24929}, {5023, 5122}, {5058, 51841}, {5062, 51842}, {5226, 31404}, {5250, 68893}, {5254, 5722}, {5256, 19714}, {5262, 63066}, {5275, 5439}, {5280, 18398}, {5283, 54392}, {5290, 9650}, {5291, 62874}, {5299, 5902}, {5436, 31429}, {5475, 9612}, {5587, 69175}, {5691, 9651}, {5703, 31400}, {5708, 30435}, {5719, 31406}, {5903, 16784}, {6765, 52959}, {7079, 17435}, {7745, 57282}, {7747, 9579}, {7749, 31231}, {7765, 37723}, {7772, 11518}, {8610, 62215}, {9312, 24281}, {9574, 31451}, {9575, 11529}, {9596, 13407}, {9597, 10572}, {9599, 12047}, {9605, 15934}, {9664, 66682}, {10389, 31426}, {10987, 59316}, {11036, 37665}, {12433, 15048}, {13405, 31396}, {13411, 31401}, {16517, 16589}, {16572, 49758}, {16600, 36404}, {16785, 50190}, {16975, 19860}, {17306, 68934}, {17474, 49487}, {17718, 31460}, {18838, 56913}, {18907, 24470}, {20963, 54418}, {21620, 31409}, {24172, 24333}, {24215, 24249}, {24549, 24631}, {25979, 27546}, {28628, 31466}, {28629, 31405}, {30282, 37512}, {31428, 31501}, {31443, 64951}, {31444, 31452}, {31448, 37080}, {31497, 63259}, {33863, 37817}, {34460, 37700}, {36743, 37547}, {37531, 62371}, {39244, 49454}, {40955, 56556}

X(69224) = X(i)-complementary conjugate of X(j) for these (i,j): {1395, 6600}, {2191, 1368}, {17107, 18639}, {57656, 18589}, {65339, 21262}
X(69224) = crosssum of X(11517) and X(22131)
X(69224) = crossdifference of every pair of points on line {659, 15313}
X(69224) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2275, 9619}, {1, 9593, 1500}, {1, 68950, 54317}, {6, 17054, 16583}, {6, 20271, 169}, {57, 16780, 32}, {65, 16502, 1572}, {3868, 33854, 54406}, {4253, 30117, 16968}, {5299, 5902, 54382}

X(69225) = (1, -1, -1, -1, -1, -1, 1, -1, 1, -1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 - a^2*b + a*b^2 + b^3 - a^2*c + 2*a*b*c + b^2*c + a*c^2 + b*c^2 + c^3) : :

X(69225) lies on these lines: {1, 6}, {10, 3767}, {32, 12514}, {33, 862}, {38, 9310}, {39, 997}, {40, 4386}, {41, 2292}, {46, 5277}, {63, 172}, {65, 5275}, {78, 2276}, {169, 3735}, {191, 609}, {200, 20691}, {230, 26066}, {239, 14555}, {607, 41609}, {612, 2295}, {672, 54317}, {869, 2318}, {908, 9596}, {936, 1575}, {966, 16824}, {968, 5320}, {975, 17750}, {976, 1334}, {988, 21008}, {993, 31442}, {1265, 2345}, {1385, 31449}, {1500, 3811}, {1571, 25440}, {1572, 3878}, {1706, 21888}, {1722, 37673}, {1812, 40773}, {1914, 5250}, {1975, 49514}, {2082, 3727}, {2238, 54418}, {2242, 62858}, {2271, 3931}, {2275, 19861}, {2548, 21616}, {2549, 17647}, {3053, 4640}, {3208, 3961}, {3290, 37549}, {3419, 69096}, {3501, 5293}, {3556, 36744}, {3576, 31429}, {3589, 59703}, {3601, 20672}, {3684, 37598}, {3691, 49487}, {3721, 40131}, {3730, 30115}, {3744, 4520}, {3772, 4384}, {3815, 25681}, {3869, 5276}, {3905, 49516}, {3924, 59207}, {3997, 30142}, {4357, 16822}, {4393, 63037}, {4559, 64349}, {4670, 59554}, {4919, 59310}, {5013, 59691}, {5254, 5794}, {5266, 14974}, {5438, 9574}, {5440, 31448}, {5603, 31405}, {5886, 31466}, {6700, 31396}, {6703, 16831}, {7745, 24703}, {8583, 16604}, {8715, 31433}, {9598, 57287}, {9599, 41012}, {9619, 30144}, {10246, 31468}, {10436, 59557}, {11375, 37661}, {16589, 54318}, {16825, 34937}, {16826, 63013}, {17016, 37657}, {17144, 20173}, {17185, 69068}, {17594, 18755}, {17735, 37552}, {17966, 54320}, {18231, 37689}, {21077, 31409}, {21096, 49458}, {21711, 39586}, {21808, 49454}, {21965, 46835}, {22061, 51949}, {22065, 56556}, {22836, 25092}, {25091, 37596}, {26030, 30729}, {26364, 31398}, {27385, 31497}, {31036, 56187}, {31469, 61276}, {31477, 56176}, {37614, 37658}, {39255, 41423}, {39946, 60665}, {54421, 63099}, {56524, 62817}, {60697, 62809}, {62372, 64043}

X(69225) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 9, 16968}, {1, 213, 16972}, {1, 16517, 1107}, {1, 16970, 16974}, {1, 54330, 213}, {6, 960, 39248}, {936, 9593, 1575}, {2176, 16519, 1}, {2273, 21810, 9}, {3869, 5276, 54382}, {4426, 21879, 9}, {5280, 5692, 54406}, {37614, 37658, 41015}

X(69226) = (1, 1, 1, 1, 1, 1, 0, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 2*a*b*c - a*c^2 - c^3) : :
X(69226) = X[1] - 3 X[35258], 3 X[63] + X[3895], 7 X[63] + X[51786], X[3895] - 3 X[5119], 7 X[3895] - 3 X[51786], 7 X[5119] - X[51786], 3 X[165] - X[64150], 2 X[226] - 3 X[10197], X[1320] - 3 X[10058], X[2099] - 3 X[16370], X[5905] - 3 X[10056], 3 X[11239] + X[20078], X[20059] - 3 X[60923], 3 X[59417] + X[64078], 3 X[60925] + X[60957]

X(69226) lies on these lines: {1, 89}, {2, 484}, {3, 214}, {4, 9}, {6, 4868}, {8, 191}, {15, 5239}, {16, 5240}, {21, 5903}, {31, 4424}, {35, 3869}, {36, 3877}, {37, 36283}, {38, 37610}, {42, 49500}, {46, 1125}, {55, 758}, {56, 3884}, {57, 551}, {58, 37598}, {63, 519}, {65, 5248}, {72, 3689}, {78, 59316}, {80, 11114}, {100, 5692}, {144, 34619}, {145, 6763}, {165, 997}, {192, 66152}, {200, 4134}, {212, 45269}, {226, 10197}, {238, 69025}, {329, 45701}, {392, 1155}, {404, 37572}, {405, 3754}, {495, 17768}, {498, 11415}, {499, 33709}, {517, 993}, {518, 25439}, {535, 5252}, {595, 986}, {672, 50287}, {748, 1739}, {759, 51290}, {846, 30116}, {920, 64163}, {936, 63469}, {942, 42819}, {944, 66059}, {946, 6862}, {956, 2802}, {958, 3647}, {960, 3579}, {962, 6888}, {982, 40091}, {984, 5184}, {995, 17596}, {999, 3898}, {1000, 34610}, {1001, 5883}, {1005, 18397}, {1018, 5282}, {1100, 2241}, {1145, 34606}, {1158, 4297}, {1320, 2975}, {1329, 61524}, {1334, 1759}, {1376, 10176}, {1454, 64160}, {1478, 44447}, {1482, 51111}, {1571, 39248}, {1621, 5902}, {1697, 3244}, {1698, 5141}, {1709, 28164}, {1724, 4642}, {1749, 15677}, {1768, 5731}, {1770, 24987}, {1776, 5727}, {1836, 3822}, {2082, 50022}, {2093, 3919}, {2099, 16370}, {2160, 58386}, {2161, 60079}, {2292, 5264}, {2305, 3743}, {2328, 17515}, {2329, 12497}, {2651, 9275}, {2886, 28174}, {3036, 5690}, {3052, 49480}, {3057, 3916}, {3061, 24047}, {3158, 4525}, {3189, 12695}, {3219, 3679}, {3241, 67335}, {3245, 5251}, {3295, 3874}, {3303, 3881}, {3305, 3828}, {3336, 3616}, {3337, 3622}, {3338, 3636}, {3339, 60948}, {3340, 7098}, {3359, 6954}, {3419, 54288}, {3488, 34744}, {3550, 30115}, {3556, 39582}, {3584, 31053}, {3587, 50808}, {3612, 11682}, {3625, 57279}, {3626, 41229}, {3634, 6933}, {3635, 62874}, {3648, 20060}, {3652, 18525}, {3654, 35460}, {3663, 54404}, {3666, 62828}, {3670, 3915}, {3671, 37550}, {3678, 3711}, {3681, 48696}, {3683, 3753}, {3688, 67546}, {3715, 3956}, {3735, 17735}, {3746, 3868}, {3747, 24464}, {3748, 24473}, {3749, 49686}, {3811, 4067}, {3814, 6980}, {3817, 6859}, {3825, 24914}, {3833, 4423}, {3871, 5904}, {3873, 4880}, {3885, 5288}, {3890, 5563}, {3892, 6767}, {3894, 3957}, {3897, 11009}, {3899, 4511}, {3911, 10199}, {3913, 3927}, {3928, 31393}, {3929, 4669}, {3940, 4421}, {3944, 17734}, {4018, 37080}, {4125, 56082}, {4184, 18417}, {4193, 5445}, {4256, 17601}, {4266, 49710}, {4294, 49168}, {4295, 10198}, {4301, 5709}, {4309, 12649}, {4311, 7284}, {4313, 54302}, {4342, 54408}, {4362, 68895}, {4428, 15934}, {4450, 4680}, {4641, 64175}, {4646, 16669}, {4650, 66674}, {4653, 66640}, {4671, 51285}, {4692, 32933}, {4701, 63135}, {4707, 42657}, {4714, 5278}, {4717, 11679}, {4744, 11529}, {4746, 63142}, {4850, 5315}, {4900, 64202}, {4999, 22791}, {5036, 54367}, {5046, 18395}, {5057, 7951}, {5086, 65134}, {5087, 11231}, {5123, 50821}, {5127, 17512}, {5128, 19862}, {5217, 5730}, {5221, 58565}, {5230, 36250}, {5253, 37524}, {5255, 9941}, {5258, 14923}, {5272, 24168}, {5273, 34632}, {5303, 5330}, {5328, 6960}, {5432, 51409}, {5437, 19883}, {5440, 31165}, {5506, 19877}, {5535, 5603}, {5693, 11491}, {5694, 32141}, {5695, 5774}, {5705, 9589}, {5722, 34741}, {5725, 64016}, {5726, 29007}, {5744, 21630}, {5745, 28194}, {5775, 30332}, {5836, 31445}, {5837, 17647}, {5882, 24467}, {5884, 10267}, {5887, 6796}, {5905, 10056}, {6224, 37299}, {6284, 12690}, {6684, 6863}, {6690, 39542}, {6700, 6962}, {6702, 17556}, {6872, 10573}, {6928, 64763}, {6932, 34789}, {6974, 28228}, {7262, 64176}, {7289, 49505}, {7675, 53053}, {7991, 31424}, {8424, 31395}, {8616, 30117}, {9623, 63468}, {9957, 62825}, {10039, 64002}, {10200, 68599}, {10266, 31888}, {10268, 12520}, {10624, 10916}, {10860, 59420}, {10884, 16208}, {10902, 64021}, {10915, 12527}, {10950, 57002}, {11113, 40663}, {11194, 64897}, {11235, 51113}, {11237, 16140}, {11239, 20078}, {11248, 31806}, {11362, 26921}, {11375, 58404}, {11499, 20117}, {11500, 31803}, {11531, 51576}, {11552, 31019}, {11571, 65739}, {12005, 16202}, {12331, 47320}, {12512, 59340}, {12575, 49627}, {12635, 64951}, {12699, 25639}, {12701, 24387}, {12732, 34720}, {12747, 15863}, {12758, 65120}, {13464, 37532}, {13528, 64107}, {14794, 45392}, {14974, 16600}, {14988, 32613}, {15228, 17579}, {15556, 37284}, {15829, 35242}, {15950, 37298}, {16160, 18253}, {16371, 63212}, {16474, 62795}, {16483, 17595}, {16568, 49720}, {17057, 17577}, {17548, 37616}, {17549, 37525}, {17574, 51683}, {17576, 54432}, {17692, 30136}, {17696, 30128}, {17742, 49766}, {18231, 31418}, {18233, 22793}, {18389, 20835}, {18421, 67120}, {18517, 67873}, {18540, 34648}, {19535, 34471}, {19618, 34431}, {19843, 20070}, {19854, 58449}, {19861, 58887}, {19869, 35263}, {19875, 27065}, {19919, 37705}, {20059, 60923}, {20104, 37692}, {20367, 24331}, {20832, 49553}, {21384, 50018}, {21620, 60962}, {21740, 59331}, {22267, 30132}, {22370, 50304}, {23958, 38314}, {24036, 42316}, {24325, 47042}, {24468, 30478}, {24590, 31211}, {24929, 44663}, {25055, 27003}, {25092, 54382}, {25270, 30167}, {25728, 51284}, {25917, 63206}, {26065, 48831}, {26877, 64953}, {28534, 61004}, {30172, 56313}, {30331, 60974}, {30384, 59491}, {31453, 35611}, {31663, 59691}, {31786, 63983}, {31794, 51715}, {31871, 58636}, {33094, 68946}, {33144, 50749}, {33815, 54392}, {34587, 56758}, {34637, 66686}, {34639, 61005}, {34649, 60970}, {34701, 36922}, {35016, 68668}, {35445, 68602}, {36480, 62817}, {37295, 60712}, {37555, 50023}, {37559, 62831}, {37571, 62830}, {37582, 58679}, {37602, 62835}, {37614, 63292}, {37623, 45776}, {37787, 50836}, {37817, 49682}, {40265, 68366}, {41012, 59330}, {41348, 51073}, {41423, 57015}, {41697, 64963}, {43677, 43680}, {44425, 67998}, {45751, 49488}, {47043, 49458}, {47357, 60989}, {48924, 49728}, {49487, 52680}, {50241, 66257}, {50286, 56513}, {50291, 56517}, {50305, 56511}, {50310, 56512}, {50311, 56509}, {50761, 68259}, {50889, 51768}, {51103, 51816}, {51506, 64189}, {51706, 60980}, {53340, 67343}, {54421, 59301}, {56551, 60905}, {59317, 66219}, {59417, 64078}, {60925, 60957}, {62245, 67963}, {62844, 66694}, {64108, 68277}

X(69226) = midpoint of X(i) and X(j) for these {i,j}: {8, 4302}, {63, 5119}, {1478, 44447}
X(69226) = reflection of X(i) in X(j) for these {i,j}: {993, 4640}, {1836, 3822}, {3419, 54288}, {39542, 6690}, {62822, 24929}
X(69226) = X(i)-Dao conjugate of X(j) for these (i,j): {37520, 50116}, {38963, 514}
X(69226) = crossdifference of every pair of points on line {1459, 4893}
X(69226) = barycentric product X(190)*X(68775)
X(69226) = barycentric quotient X(68775)/X(514)
X(69226) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5180, 18393}, {3, 3878, 30144}, {3, 5289, 214}, {9, 40, 54286}, {9, 41319, 5011}, {9, 54286, 10}, {21, 5903, 30147}, {35, 3869, 22836}, {40, 5587, 48363}, {40, 12514, 10}, {40, 55104, 43174}, {46, 5250, 1125}, {65, 5248, 30143}, {72, 37568, 8715}, {191, 11010, 8}, {214, 3878, 5289}, {214, 5289, 30144}, {405, 37567, 3754}, {960, 3579, 25440}, {1001, 36279, 5883}, {1276, 1277, 573}, {1697, 54290, 62858}, {1697, 62858, 3244}, {2093, 4512, 54318}, {2093, 54318, 3919}, {2292, 5264, 30142}, {2975, 5697, 22837}, {3057, 3916, 8666}, {3065, 5541, 20085}, {3219, 63136, 3679}, {3670, 3915, 30148}, {3683, 5183, 3753}, {3811, 12526, 4067}, {3871, 11684, 5904}, {3885, 62827, 5288}, {3898, 4973, 999}, {3899, 5010, 4511}, {4018, 37080, 62860}, {4257, 17461, 1}, {4295, 10198, 11263}, {5250, 63144, 46}, {5303, 5330, 21842}, {5744, 30305, 45700}, {5837, 31730, 17647}, {6684, 64762, 6863}, {6763, 37563, 145}, {10268, 54156, 12520}, {12514, 54286, 9}, {12526, 61763, 3811}, {12572, 43174, 10}, {12699, 26066, 25639}, {17549, 62826, 37525}, {24703, 26446, 3814}, {30305, 45700, 21630}, {31165, 63211, 5440}, {31786, 64118, 63983}, {41229, 63130, 3626}, {59340, 63985, 12512}

X(69227) = (1, 1, 1, 1, 1, 0, 0, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - a*c^2 - c^3) : :
X(69227) = 3 X[3] - X[12635], 5 X[3] - 3 X[56177], 2 X[12635] - 3 X[22836], 5 X[12635] - 9 X[56177], 5 X[22836] - 6 X[56177], 5 X[40] - X[2136], X[40] + 3 X[3928], 3 X[40] + X[6762], X[944] + 3 X[34744], X[2136] + 15 X[3928], 3 X[2136] + 5 X[6762], X[2136] + 5 X[62858], 9 X[3928] - X[6762], 3 X[3928] - X[62858], X[6762] - 3 X[62858], and many others

X(69227) lies on these lines: {1, 89}, {2, 191}, {3, 758}, {4, 5535}, {5, 17768}, {7, 10198}, {8, 484}, {9, 3634}, {10, 46}, {20, 1768}, {21, 5902}, {31, 3670}, {35, 3868}, {36, 3869}, {38, 5264}, {40, 376}, {55, 3874}, {56, 3878}, {57, 1125}, {58, 986}, {65, 993}, {69, 21081}, {72, 1155}, {78, 4067}, {79, 2476}, {84, 28164}, {100, 5904}, {109, 4347}, {145, 11010}, {165, 3811}, {169, 36643}, {171, 30142}, {194, 66152}, {214, 5204}, {226, 1454}, {238, 24046}, {255, 1735}, {329, 26364}, {333, 28612}, {355, 535}, {381, 3652}, {386, 1046}, {392, 32636}, {404, 5692}, {405, 3647}, {411, 15071}, {442, 11246}, {474, 10176}, {498, 5905}, {499, 11415}, {515, 24467}, {516, 1158}, {517, 5450}, {518, 3098}, {522, 68263}, {524, 48924}, {527, 6684}, {529, 5690}, {540, 46617}, {550, 44669}, {551, 3338}, {553, 51706}, {573, 17748}, {579, 1761}, {595, 982}, {614, 24167}, {654, 62435}, {672, 1759}, {896, 1724}, {912, 6796}, {920, 1210}, {936, 53056}, {940, 3743}, {942, 4640}, {946, 37532}, {956, 37567}, {958, 3754}, {960, 37582}, {962, 5536}, {988, 50604}, {997, 12526}, {999, 3884}, {1001, 5708}, {1005, 10399}, {1006, 15016}, {1013, 1844}, {1054, 17749}, {1089, 32933}, {1150, 4647}, {1193, 49500}, {1203, 4850}, {1214, 59729}, {1255, 31320}, {1276, 49591}, {1277, 49590}, {1323, 7183}, {1334, 17736}, {1376, 3678}, {1385, 44663}, {1403, 19762}, {1406, 18593}, {1445, 9843}, {1468, 4424}, {1479, 1727}, {1482, 11194}, {1617, 67051}, {1621, 18398}, {1697, 3635}, {1698, 3219}, {1706, 4691}, {1708, 12572}, {1709, 51118}, {1710, 21367}, {1714, 21376}, {1715, 43160}, {1722, 16570}, {1737, 55873}, {1738, 1760}, {1741, 59646}, {1748, 1838}, {1749, 3648}, {1754, 56839}, {1758, 4306}, {1764, 17733}, {1766, 28526}, {1770, 6734}, {1771, 44706}, {1776, 9581}, {1782, 24310}, {1828, 67509}, {1836, 25639}, {2093, 62824}, {2095, 11496}, {2099, 51111}, {2245, 3454}, {2292, 37522}, {2392, 5752}, {2646, 4018}, {2651, 5197}, {2795, 49129}, {2796, 16560}, {2800, 11249}, {2801, 11500}, {2802, 11256}, {2901, 32934}, {2949, 6908}, {2975, 5903}, {3057, 62825}, {3061, 5030}, {3085, 9965}, {3149, 31803}, {3159, 29649}, {3178, 35468}, {3241, 37563}, {3244, 5119}, {3245, 5288}, {3293, 32912}, {3295, 3881}, {3303, 3892}, {3304, 3898}, {3305, 51073}, {3306, 19862}, {3333, 3636}, {3337, 3616}, {3339, 31424}, {3359, 10915}, {3428, 66019}, {3452, 58405}, {3496, 4253}, {3509, 3730}, {3543, 64740}, {3555, 25439}, {3560, 31870}, {3576, 26877}, {3601, 12559}, {3621, 5541}, {3624, 27003}, {3625, 63130}, {3626, 5128}, {3632, 63136}, {3649, 7483}, {3650, 4187}, {3654, 32049}, {3664, 54404}, {3666, 62805}, {3683, 5439}, {3729, 4066}, {3735, 33863}, {3746, 3873}, {3751, 50587}, {3784, 65399}, {3812, 31445}, {3813, 28174}, {3814, 7702}, {3816, 34753}, {3822, 26066}, {3825, 24703}, {3828, 3929}, {3829, 40273}, {3830, 16138}, {3833, 11108}, {3841, 5791}, {3870, 59316}, {3871, 62235}, {3876, 9352}, {3877, 5563}, {3901, 5010}, {3911, 21616}, {3915, 3953}, {3918, 9708}, {3919, 19860}, {3923, 16574}, {3924, 52680}, {3940, 4127}, {3944, 45939}, {3951, 4134}, {3962, 5440}, {3970, 41423}, {3976, 40091}, {3984, 4525}, {4015, 5220}, {4184, 35637}, {4188, 5131}, {4252, 63292}, {4295, 5744}, {4298, 37550}, {4301, 12704}, {4302, 12649}, {4312, 5705}, {4340, 63370}, {4354, 37782}, {4418, 10479}, {4450, 4894}, {4511, 7280}, {4512, 64675}, {4643, 5955}, {4645, 30172}, {4658, 17592}, {4669, 63135}, {4697, 43531}, {4701, 41348}, {4757, 68668}, {4867, 59319}, {4996, 11571}, {4999, 39542}, {5011, 21384}, {5016, 67502}, {5057, 7741}, {5080, 18395}, {5083, 11510}, {5086, 10483}, {5122, 59691}, {5125, 52414}, {5141, 61703}, {5183, 10914}, {5187, 59419}, {5219, 20104}, {5223, 8544}, {5225, 64372}, {5258, 62827}, {5259, 62838}, {5278, 28611}, {5279, 59682}, {5282, 16549}, {5292, 16566}, {5303, 37525}, {5433, 51409}, {5435, 10200}, {5437, 19878}, {5442, 17566}, {5445, 11681}, {5493, 41338}, {5496, 50317}, {5542, 60968}, {5552, 20078}, {5693, 6905}, {5694, 6924}, {5696, 30295}, {5697, 54391}, {5698, 60989}, {5704, 63975}, {5745, 12609}, {5771, 5857}, {5789, 52682}, {5794, 54288}, {5805, 12558}, {5850, 37560}, {5852, 64123}, {5855, 34773}, {5881, 48363}, {6001, 37623}, {6051, 37520}, {6147, 6690}, {6191, 49588}, {6192, 49589}, {6210, 28508}, {6326, 6942}, {6327, 30171}, {6361, 24468}, {6681, 25681}, {6700, 59336}, {6738, 62810}, {6744, 62839}, {6765, 63469}, {6834, 21635}, {6845, 49177}, {6876, 16132}, {6885, 64335}, {6906, 37625}, {6911, 20117}, {6928, 10265}, {7078, 24025}, {7262, 24174}, {7289, 34379}, {7308, 31253}, {7330, 19925}, {7504, 63285}, {7705, 31160}, {7751, 68870}, {7807, 30146}, {7957, 17613}, {8141, 64780}, {8669, 20368}, {8728, 18253}, {8822, 17861}, {9052, 53002}, {9340, 62847}, {9940, 52769}, {9943, 12511}, {10122, 20835}, {10129, 31262}, {10164, 55104}, {10165, 37612}, {10197, 13407}, {10199, 41012}, {10267, 12005}, {10310, 46684}, {10319, 59547}, {10461, 68508}, {10526, 64763}, {10529, 21630}, {10624, 49627}, {11012, 64021}, {11112, 21677}, {11114, 37702}, {11235, 48661}, {11374, 58404}, {11499, 63967}, {11507, 18389}, {11509, 15556}, {11520, 59337}, {11523, 35242}, {11570, 36152}, {11573, 22276}, {11679, 42031}, {11682, 37618}, {11826, 34695}, {11827, 34742}, {12047, 59491}, {12329, 34378}, {12432, 37541}, {12436, 18249}, {12528, 44425}, {12571, 54370}, {12575, 54408}, {12607, 61524}, {12617, 64001}, {12629, 63468}, {12645, 34740}, {12647, 20076}, {12667, 66058}, {12699, 24387}, {13405, 59335}, {13411, 17700}, {13463, 28212}, {13465, 37251}, {14110, 63983}, {14829, 63996}, {14988, 26286}, {15015, 37307}, {15079, 37375}, {15296, 60962}, {15654, 23085}, {15792, 56439}, {16118, 52126}, {16126, 17548}, {16140, 38062}, {16141, 50242}, {16370, 35016}, {16453, 53280}, {16466, 17595}, {16763, 37291}, {16767, 37299}, {16825, 20367}, {16948, 54315}, {17080, 34043}, {17206, 33936}, {17437, 44675}, {17484, 27529}, {17532, 51113}, {17536, 41872}, {17549, 34195}, {17570, 56203}, {17579, 47033}, {17594, 59301}, {17601, 33771}, {17614, 31165}, {17692, 30139}, {17696, 30131}, {17698, 59574}, {17732, 21090}, {17738, 29455}, {17889, 24880}, {18206, 35915}, {18221, 50742}, {18233, 61716}, {18391, 54432}, {19541, 31871}, {19765, 63354}, {19846, 56520}, {19854, 55868}, {19858, 38000}, {19872, 35595}, {20070, 34625}, {20107, 31231}, {20320, 54107}, {20369, 25446}, {20718, 37536}, {20846, 67946}, {21060, 59675}, {21371, 46827}, {21385, 24128}, {21842, 62826}, {22099, 23621}, {22267, 30127}, {22793, 28534}, {23169, 23361}, {23887, 53300}, {24391, 31730}, {24470, 25466}, {24473, 37080}, {24475, 32613}, {24725, 37693}, {24850, 48863}, {24883, 33102}, {24929, 62860}, {25078, 37500}, {25441, 32776}, {25512, 26627}, {25524, 37545}, {26878, 28609}, {26893, 31737}, {27065, 64850}, {27247, 56880}, {27784, 37674}, {27785, 37633}, {28158, 68057}, {28194, 49600}, {28228, 68036}, {28606, 37559}, {28610, 45701}, {29747, 49676}, {30115, 37603}, {30117, 54354}, {30150, 33820}, {30165, 33831}, {31446, 38052}, {31663, 56176}, {31793, 64128}, {32845, 64184}, {32860, 64072}, {33830, 46899}, {34641, 63142}, {34718, 66240}, {34997, 56313}, {35010, 54445}, {35445, 41863}, {36263, 67979}, {37285, 39772}, {37286, 47319}, {37294, 37783}, {37548, 62844}, {37549, 49480}, {37555, 49477}, {37581, 49553}, {37592, 62828}, {37639, 64071}, {37646, 63997}, {37787, 60905}, {37816, 51290}, {37822, 67046}, {37826, 67856}, {37960, 64720}, {38054, 60938}, {38059, 60985}, {38456, 49609}, {38472, 56885}, {39580, 56518}, {40592, 46441}, {41696, 59325}, {41733, 62315}, {42819, 50192}, {44229, 67873}, {48931, 64912}, {49169, 59417}, {51071, 62832}, {52407, 59285}, {55405, 67963}, {57287, 59324}, {57288, 67980}, {59420, 63141}, {59639, 62673}, {60948, 66515}, {61763, 62823}, {63278, 66099}, {63437, 64004}, {64068, 66068}, {64887, 66761}, {67420, 68261}, {67436, 67518}

X(69227) = midpoint of X(i) and X(j) for these {i,j}: {20, 49168}, {40, 62858}, {1158, 5709}, {3811, 54422}, {12513, 12702}, {24391, 31730}, {24467, 59318}, {28610, 45701}, {60896, 60950}, {60990, 67962}
X(69227) = reflection of X(i) in X(j) for these {i,j}: {8715, 3579}, {10526, 64763}, {10915, 43174}, {12607, 61524}, {12699, 24387}, {21077, 6684}, {22836, 3}, {22837, 8666}, {40257, 26286}, {56176, 31663}, {60911, 60994}, {67850, 59719}
X(69227) = X(4641)-Dao conjugate of X(3879)
X(69227) = barycentric product X(1)*X(32859)
X(69227) = barycentric quotient X(32859)/X(75)
X(69227) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 4652, 5267}, {8, 67335, 6763}, {21, 5902, 30143}, {35, 4880, 3868}, {36, 3869, 30144}, {38, 5264, 30145}, {40, 3928, 62858}, {40, 63399, 4297}, {40, 64129, 12512}, {46, 63, 10}, {57, 12514, 1125}, {57, 54290, 12514}, {65, 993, 30147}, {65, 3916, 993}, {72, 1155, 25440}, {109, 37591, 4347}, {165, 54422, 3811}, {191, 3336, 2}, {404, 11684, 5692}, {405, 5221, 5883}, {484, 6763, 8}, {499, 11415, 11813}, {595, 982, 30148}, {896, 24443, 1724}, {942, 4640, 5248}, {958, 36279, 3754}, {986, 4650, 58}, {1001, 5708, 58565}, {1046, 17596, 386}, {1376, 3927, 3678}, {2646, 4018, 62822}, {3218, 56288, 1}, {3245, 5288, 14923}, {3338, 5250, 551}, {3339, 31424, 54318}, {3339, 54318, 33815}, {3555, 37568, 25439}, {3616, 23958, 3337}, {3647, 5883, 405}, {3821, 8258, 20083}, {3878, 4973, 56}, {3901, 5010, 34772}, {4084, 5267, 1}, {4295, 5744, 26363}, {5119, 62874, 3244}, {5128, 57279, 54286}, {5128, 67334, 57279}, {5204, 5730, 214}, {5220, 9709, 4015}, {5292, 24248, 36250}, {5303, 62830, 37525}, {5692, 37524, 404}, {5694, 41347, 6924}, {5791, 5880, 3841}, {5904, 37572, 100}, {6684, 26921, 60912}, {10164, 67850, 59719}, {12526, 15803, 997}, {17549, 34195, 37571}, {24914, 58798, 3814}, {26066, 57282, 3822}, {41338, 63985, 5493}, {54286, 57279, 3626}, {54370, 67880, 12571}, {55104, 59333, 10164}, {62874, 63144, 5119}

X(69228) = (1, 1, 1, 0, 1, 1, 1, 1, 1, 0)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 + a^2*b - a*b^2 + a^2*c - 2*a*b*c + b^2*c - a*c^2 + b*c^2) : :

X(69228) lies on these lines: {1, 574}, {6, 3871}, {8, 17735}, {31, 20691}, {32, 1018}, {37, 37568}, {39, 37610}, {40, 3721}, {46, 3726}, {55, 2295}, {100, 2176}, {145, 33863}, {172, 3208}, {213, 8715}, {404, 16969}, {902, 4426}, {1107, 41423}, {1334, 4386}, {1500, 5264}, {1575, 3915}, {1697, 54317}, {1724, 52959}, {1914, 3501}, {1931, 20055}, {2238, 5687}, {2241, 16549}, {2243, 17742}, {2275, 37588}, {2276, 5255}, {2305, 17314}, {2802, 53165}, {3053, 4513}, {3230, 25440}, {3295, 24512}, {3552, 18047}, {3689, 21874}, {3727, 5119}, {3729, 4400}, {3734, 29381}, {3735, 11010}, {3746, 17750}, {3780, 3913}, {3923, 21021}, {3924, 21888}, {3959, 63136}, {4188, 9259}, {4203, 53145}, {4642, 16974}, {4950, 69038}, {5013, 37542}, {5710, 31477}, {5711, 60724}, {9310, 52964}, {9651, 24222}, {10987, 41239}, {16502, 20331}, {16968, 63130}, {16975, 24047}, {17475, 37590}, {19767, 59238}, {20963, 25439}, {21951, 54286}, {32468, 51319}, {33074, 68944}, {49509, 56288}, {50581, 60697}

X(69228) = barycentric product X(100)*X(47835)
X(69228) = barycentric quotient X(47835)/X(693)
X(69228) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1500, 5264, 63099}, {3208, 3550, 172}, {5687, 14974, 2238}

X(69229) = (1, 1, 1, 0, 1, 1, 0, 1, 0, 0)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a^2*(a^2 + a*b - b^2 + a*c - 2*b*c - c^2) : :

X(69229) lies on these lines: {1, 17735}, {3, 3230}, {6, 3746}, {9, 762}, {31, 1500}, {32, 902}, {35, 2176}, {36, 16969}, {37, 1759}, {39, 3915}, {40, 3125}, {41, 52963}, {55, 213}, {101, 28502}, {187, 9310}, {191, 49509}, {220, 2251}, {484, 20271}, {574, 1201}, {595, 2276}, {612, 21816}, {614, 1571}, {672, 2241}, {728, 21821}, {748, 1574}, {997, 39255}, {1018, 4426}, {1055, 5206}, {1107, 37610}, {1191, 31448}, {1193, 31451}, {1621, 17750}, {1724, 20691}, {1914, 3730}, {1931, 17389}, {2177, 20970}, {2179, 39258}, {2238, 8715}, {2273, 5301}, {2275, 24047}, {2295, 5248}, {2300, 54285}, {2305, 3247}, {3052, 54416}, {3057, 53165}, {3208, 5291}, {3290, 3579}, {3294, 4386}, {3295, 20963}, {3303, 5021}, {3501, 8616}, {3550, 5277}, {3747, 37586}, {3780, 25439}, {3811, 21839}, {3954, 12514}, {3959, 11010}, {4251, 10987}, {4264, 33635}, {4424, 16974}, {5006, 35193}, {5010, 21008}, {5013, 16483}, {5030, 63493}, {5119, 16968}, {5255, 5283}, {5280, 21793}, {5312, 59238}, {6155, 16972}, {6651, 33932}, {7280, 9259}, {7290, 31426}, {7299, 21859}, {8053, 40728}, {8649, 15513}, {9593, 62875}, {9664, 21935}, {11510, 52635}, {15624, 66878}, {16367, 54282}, {16466, 31477}, {16502, 42316}, {16583, 37568}, {16777, 37559}, {16970, 61763}, {16975, 37588}, {17299, 64072}, {17754, 68893}, {18755, 54981}, {20228, 54322}, {20992, 21760}, {22080, 53387}, {30143, 65695}, {31433, 54418}, {31449, 37542}, {36647, 59325}, {36744, 61036}, {37508, 62214}, {60724, 62805}, {62817, 69024}

X(69229) = isogonal conjugate of the isotomic conjugate of X(17299)
X(69229) = crosssum of X(1086) and X(4897)
X(69229) = crossdifference of every pair of points on line {26248, 28195}
X(69229) = barycentric product X(i)*X(j) for these {i,j}: {6, 17299}, {42, 64072}, {55, 24914}, {100, 50504}, {101, 48266}
X(69229) = barycentric quotient X(i)/X(j) for these {i,j}: {17299, 76}, {24914, 6063}, {48266, 3261}, {50504, 693}, {64072, 310}
X(69229) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {55, 14974, 213}, {902, 1334, 32}, {3208, 54354, 5291}, {3303, 5021, 16971}, {3915, 41423, 39}, {5206, 9351, 1055}, {5255, 60711, 5283}, {24047, 40091, 2275}

X(69230) = (1, 1, 1, 0, 1, 0, 1, 1, 1, 0)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 + a^2*b - a*b^2 + a^2*c + b^2*c - a*c^2 + b*c^2) : :

X(69230) lies on these lines: {1, 574}, {2, 17735}, {3, 2295}, {6, 100}, {8, 33863}, {9, 2243}, {31, 1575}, {32, 16549}, {35, 17750}, {37, 750}, {39, 5264}, {40, 3727}, {43, 60697}, {45, 37675}, {46, 3721}, {55, 16693}, {57, 3726}, {99, 30114}, {171, 2276}, {172, 3501}, {183, 24330}, {187, 16788}, {190, 16997}, {213, 25440}, {404, 2176}, {474, 14974}, {484, 3735}, {594, 1150}, {672, 4386}, {799, 30473}, {813, 52656}, {940, 31477}, {985, 2108}, {1015, 37610}, {1018, 2242}, {1100, 2177}, {1376, 2238}, {1468, 20691}, {1500, 37522}, {1574, 1724}, {1580, 19584}, {1914, 3550}, {1931, 29593}, {2241, 68950}, {2275, 5255}, {2305, 2345}, {3208, 37608}, {3218, 49509}, {3570, 17350}, {3666, 31443}, {3723, 4003}, {3729, 4396}, {3730, 5277}, {3780, 5021}, {3815, 63979}, {3915, 16604}, {4037, 29649}, {4188, 21008}, {4257, 5291}, {4363, 37670}, {4438, 20483}, {5013, 5710}, {5030, 16975}, {5078, 5124}, {5115, 21858}, {5253, 16969}, {5269, 9574}, {5275, 42316}, {5283, 24047}, {5711, 31448}, {8715, 20963}, {9352, 26242}, {9598, 23903}, {9650, 66658}, {10987, 16503}, {15447, 15989}, {16777, 17595}, {16968, 21951}, {16970, 64112}, {16971, 25439}, {16974, 24443}, {17018, 59238}, {17122, 60711}, {17279, 24602}, {17299, 32919}, {17303, 32917}, {17314, 37684}, {17475, 21010}, {17596, 41269}, {20255, 33819}, {20693, 32912}, {20990, 25804}, {21495, 68769}, {21793, 33854}, {21888, 49487}, {21956, 37646}, {24170, 25497}, {26098, 31497}, {31426, 37554}, {31501, 37693}, {36743, 64752}, {37588, 63493}, {40127, 56758}, {40401, 67501}, {67425, 68825}

X(69230) = crossdifference of every pair of points on line {891, 48226}
X(69230) = barycentric product X(i)*X(j) for these {i,j}: {1, 32935}, {100, 47823}
X(69230) = barycentric quotient X(i)/X(j) for these {i,j}: {32935, 75}, {47823, 693}
X(69230) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 54317, 3727}, {171, 2276, 63099}, {750, 41423, 37}, {940, 31477, 60724}, {3501, 37603, 172}, {3550, 17754, 1914}, {5021, 5687, 3780}, {17126, 17756, 6}

X(69231) = (1, 1, 1, 0, 1, 0, 0, 1, 0, 0)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a^2*(a^2 + a*b - b^2 + a*c - c^2) : :

X(69231) lies on these lines: {1, 17735}, {3, 213}, {6, 35}, {9, 2305}, {21, 17750}, {31, 39}, {32, 672}, {36, 2176}, {37, 3916}, {41, 187}, {42, 31451}, {46, 3125}, {47, 13006}, {55, 5021}, {56, 3230}, {58, 2276}, {63, 3954}, {71, 5019}, {78, 21839}, {99, 17033}, {101, 28488}, {171, 5283}, {172, 3730}, {183, 4721}, {218, 2251}, {386, 60697}, {484, 3959}, {517, 53165}, {574, 1193}, {579, 20861}, {583, 5301}, {595, 2275}, {602, 62371}, {612, 31442}, {750, 16589}, {762, 41229}, {896, 33299}, {902, 1475}, {975, 21816}, {980, 60701}, {993, 2295}, {1015, 3915}, {1078, 24514}, {1107, 5264}, {1155, 16583}, {1333, 2273}, {1334, 2242}, {1453, 9574}, {1468, 1500}, {1509, 17032}, {1571, 54418}, {1575, 1724}, {1707, 54406}, {1914, 4253}, {1931, 3661}, {2178, 61036}, {2209, 69067}, {2238, 25440}, {2245, 23639}, {2267, 4274}, {2269, 5042}, {2271, 5217}, {2300, 36743}, {2549, 5230}, {2999, 31421}, {3052, 5022}, {3286, 40728}, {3290, 37582}, {3295, 16971}, {3336, 20271}, {3501, 5291}, {3550, 21384}, {3579, 41015}, {3670, 16974}, {3735, 56288}, {3780, 8715}, {3788, 4766}, {3811, 39255}, {3997, 5267}, {4020, 39258}, {4252, 42316}, {4277, 54351}, {4386, 16552}, {4426, 16549}, {4652, 21802}, {4805, 7750}, {5006, 56840}, {5010, 18755}, {5013, 16466}, {5115, 56926}, {5120, 20228}, {5124, 39578}, {5134, 45939}, {5248, 24512}, {5255, 16975}, {5269, 31429}, {5292, 9598}, {5299, 21793}, {5440, 21874}, {5563, 16969}, {5710, 31449}, {6155, 17594}, {6186, 65027}, {6629, 40006}, {6763, 49509}, {7280, 21008}, {7748, 21935}, {7751, 56024}, {7761, 24995}, {7772, 21764}, {7793, 17350}, {7874, 30816}, {8557, 51281}, {8616, 68893}, {8671, 20457}, {9310, 52963}, {10459, 31456}, {11679, 52579}, {12514, 54317}, {15803, 16970}, {16782, 37576}, {17103, 27255}, {17279, 29473}, {17448, 37610}, {17754, 54354}, {18206, 69024}, {20669, 36635}, {20992, 23660}, {21059, 68743}, {21240, 33819}, {21760, 37507}, {21837, 22093}, {22036, 32933}, {24587, 56520}, {24632, 32777}, {24691, 33953}, {25092, 63099}, {27631, 28245}, {30147, 65695}, {31466, 63979}, {31488, 33104}, {32916, 52538}, {33830, 58452}, {37579, 52635}, {37607, 60711}, {37646, 69096}, {40091, 63493}, {53850, 68731}, {56836, 59454}, {63548, 64172}

X(69231) = isogonal conjugate of the isotomic conjugate of X(4851)
X(69231) = crosssum of X(2) and X(5839)
X(69231) = crossdifference of every pair of points on line {4802, 7662}
X(69231) = barycentric product X(i)*X(j) for these {i,j}: {1, 32912}, {6, 4851}, {31, 33942}, {101, 47971}
X(69231) = barycentric quotient X(i)/X(j) for these {i,j}: {4851, 76}, {32912, 75}, {33942, 561}, {47971, 3261}
X(69231) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9, 37603, 5277}, {46, 16968, 3125}, {55, 5021, 20963}, {56, 14974, 3230}, {58, 24047, 2276}, {218, 3053, 2251}, {595, 5030, 2275}, {902, 1475, 2241}, {1468, 41423, 1500}, {3052, 5022, 16502}, {3730, 4257, 172}, {4252, 42316, 54416}, {7280, 54981, 21008}, {16549, 52680, 4426}, {17735, 33863, 1}

X(69232) = (1, 1, 0, 1, 1, 1, 1, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 + a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3) : :

X(69232) lies on these lines: {1, 187}, {6, 12702}, {8, 7737}, {10, 5475}, {32, 517}, {39, 40}, {46, 1015}, {65, 2241}, {115, 12699}, {165, 9619}, {169, 52963}, {172, 5697}, {230, 22791}, {355, 7747}, {484, 2275}, {516, 7748}, {574, 3579}, {595, 3959}, {946, 7746}, {962, 3767}, {1384, 8148}, {1385, 5206}, {1386, 5033}, {1482, 3053}, {1500, 5119}, {1504, 49226}, {1505, 49227}, {1506, 26446}, {1570, 64084}, {1573, 12514}, {1574, 39248}, {1698, 7603}, {1699, 39565}, {1737, 9665}, {1770, 9651}, {1836, 69175}, {1914, 5903}, {1968, 41722}, {2176, 5011}, {2242, 3057}, {2276, 11010}, {2548, 5657}, {2549, 6361}, {2959, 66638}, {3054, 61272}, {3125, 3915}, {3245, 5299}, {3336, 63493}, {3576, 15513}, {3616, 21843}, {3654, 7753}, {3679, 14537}, {3721, 37610}, {3727, 5264}, {3735, 5255}, {3780, 49500}, {3815, 61524}, {3877, 5277}, {3878, 4386}, {3905, 35103}, {4640, 31456}, {5007, 7991}, {5023, 10246}, {5041, 9593}, {5058, 35774}, {5062, 35775}, {5250, 16589}, {5254, 28174}, {5286, 20070}, {5291, 14923}, {5305, 28212}, {5309, 28194}, {5346, 28228}, {5477, 67964}, {5587, 39590}, {5690, 7745}, {5790, 65630}, {5886, 7749}, {6684, 31455}, {6781, 18481}, {7713, 33842}, {7739, 34632}, {7759, 69038}, {7969, 9675}, {7982, 35007}, {8588, 13624}, {8589, 35242}, {9341, 30323}, {9588, 31441}, {9592, 31422}, {9650, 10039}, {9780, 31415}, {11173, 64070}, {11648, 28198}, {12778, 46301}, {12782, 46313}, {12963, 35641}, {12968, 35642}, {14377, 49777}, {14839, 46321}, {14974, 49758}, {15515, 31663}, {16502, 37567}, {16781, 36279}, {16783, 65695}, {16946, 21853}, {16975, 56288}, {17144, 66152}, {18357, 53418}, {18424, 18483}, {18480, 62203}, {18493, 37637}, {20271, 40091}, {20666, 31394}, {21796, 54420}, {22793, 69141}, {26066, 31488}, {28146, 65633}, {31398, 43174}, {31401, 68545}, {31439, 62241}, {31443, 53096}, {31451, 37568}, {36643, 66647}, {37582, 62370}, {39563, 50865}, {40273, 63534}, {43457, 61261}, {44517, 64792}, {44518, 48661}, {52959, 54406}, {64532, 67522}

X(69232) = midpoint of X(35610) and X(35611)
X(69232) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 1572, 39}, {40, 9575, 1571}, {165, 9619, 37512}, {1571, 1572, 9575}, {1571, 9575, 39}, {5119, 54382, 1500}, {9592, 31422, 31652}, {9592, 63469, 31422}, {39248, 54286, 1574}, {54406, 63130, 52959}

X(69233) = (1, 1, 0, 1, 1, 1, 0, 0, 0, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 + a^2*b - b^3 + a^2*c - 2*a*b*c - c^3) : :

X(69233) lies on these lines: {1, 21793}, {6, 37598}, {10, 5475}, {31, 3727}, {32, 3878}, {37, 5250}, {40, 1575}, {44, 2082}, {46, 16604}, {63, 17448}, {71, 23544}, {172, 3877}, {187, 30144}, {191, 16975}, {214, 5206}, {238, 3959}, {517, 4426}, {595, 3735}, {748, 21951}, {758, 2241}, {960, 4386}, {978, 41319}, {1100, 54421}, {1107, 1572}, {1759, 3230}, {1761, 21769}, {1914, 3869}, {2176, 3496}, {2242, 3884}, {2243, 9310}, {2275, 56288}, {2278, 23623}, {3053, 5289}, {3061, 17735}, {3218, 63493}, {3305, 25614}, {3509, 16969}, {3647, 31456}, {3721, 3915}, {3899, 7031}, {3913, 20693}, {3954, 37610}, {4011, 21025}, {4051, 7262}, {4372, 53332}, {4376, 17152}, {4713, 17739}, {5119, 20691}, {5291, 5697}, {5902, 68893}, {7746, 11813}, {9597, 44447}, {10987, 34772}, {12526, 16973}, {12702, 21888}, {16602, 24590}, {17278, 27000}, {17742, 52964}, {20963, 49500}, {21281, 24358}, {21868, 63130}, {21892, 54420}, {21904, 54386}, {24586, 59512}, {28358, 54404}, {37588, 49509}, {40256, 62371}, {41269, 62804}, {52680, 53165}, {63499, 67335}

X(69233) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 39248, 1575}, {595, 3735, 16974}, {1572, 12514, 1107}, {5119, 54406, 20691}, {5250, 54382, 37}

X(69234) = (1, 1, 0, 1, 1, 0, 1, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 + a^2*b - b^3 + a^2*c + b^2*c + b*c^2 - c^3) : :

X(69234) lies on these lines: {1, 187}, {6, 5011}, {10, 9650}, {31, 3125}, {32, 65}, {37, 36283}, {39, 46}, {40, 1500}, {57, 1015}, {58, 3959}, {63, 1573}, {75, 66152}, {115, 1836}, {171, 3735}, {172, 5903}, {230, 39542}, {484, 2276}, {516, 9664}, {517, 2242}, {574, 1155}, {595, 20271}, {609, 67977}, {758, 4386}, {942, 2241}, {1159, 1384}, {1210, 9665}, {1452, 3199}, {1506, 24914}, {1571, 5128}, {1574, 54406}, {1724, 21951}, {1737, 5475}, {1759, 2295}, {1770, 7748}, {1788, 2548}, {1837, 7747}, {1914, 5902}, {2093, 9620}, {2238, 49500}, {2243, 16788}, {2262, 5042}, {2275, 3336}, {2362, 5058}, {2549, 3474}, {2646, 5206}, {3053, 68668}, {3218, 16975}, {3245, 16785}, {3337, 63493}, {3340, 9341}, {3496, 17750}, {3579, 31451}, {3612, 15513}, {3721, 5264}, {3726, 37610}, {3727, 37522}, {3754, 4426}, {3767, 4295}, {3869, 5277}, {3916, 31456}, {4263, 54420}, {4292, 9651}, {4424, 63099}, {4645, 34542}, {5062, 16232}, {5165, 7297}, {5210, 37606}, {5221, 16502}, {5283, 56288}, {5348, 38345}, {5657, 31409}, {5708, 16781}, {6205, 20331}, {6684, 31501}, {7737, 18391}, {7745, 67980}, {7746, 12047}, {7749, 11375}, {8588, 37600}, {9592, 53056}, {9619, 15803}, {10826, 39590}, {11246, 69098}, {12019, 53418}, {12514, 16589}, {20970, 54421}, {20982, 61366}, {21746, 67546}, {21793, 30117}, {24254, 24586}, {24464, 40747}, {25497, 26562}, {26446, 31476}, {31426, 41348}, {31442, 54290}, {31471, 49226}, {37512, 58887}, {37567, 54416}, {40131, 52963}, {52959, 54286}, {57282, 69175}, {60717, 69003}, {64016, 67726}, {65630, 68905}

X(69234) = barycentric product X(1)*X(24725)
X(69234) = barycentric quotient X(24725)/X(75)
X(69234) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {46, 54382, 39}, {57, 1572, 1015}, {2243, 65695, 16788}

X(69235) = (1, 1, 0, 1, 1, 0, 0, 0, 0, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 + a^2*b - b^3 + a^2*c - c^3) : :

X(69235) lies on these lines: {1, 21793}, {6, 986}, {10, 9650}, {31, 3721}, {32, 758}, {37, 5711}, {40, 20691}, {41, 2243}, {44, 169}, {46, 1575}, {57, 16604}, {58, 3735}, {63, 1107}, {65, 4426}, {72, 4386}, {76, 66152}, {172, 3869}, {187, 22836}, {191, 5283}, {213, 1759}, {238, 20271}, {518, 5017}, {579, 18904}, {752, 4136}, {896, 17451}, {910, 21874}, {1468, 3727}, {1572, 17448}, {1707, 16968}, {1724, 3125}, {1757, 46032}, {1914, 3868}, {2176, 3509}, {2241, 3874}, {2242, 3878}, {2275, 3218}, {2276, 56288}, {2292, 63099}, {2295, 5282}, {2911, 21767}, {3053, 12635}, {3061, 4650}, {3230, 17736}, {3726, 3915}, {3780, 32912}, {3827, 21861}, {3901, 7031}, {3923, 21024}, {3928, 9575}, {3954, 5264}, {3959, 5247}, {4016, 4275}, {4376, 17137}, {4641, 41015}, {4797, 24549}, {4799, 24995}, {4880, 5299}, {5023, 56177}, {5254, 17768}, {5255, 49509}, {5275, 21879}, {5276, 11684}, {5277, 5692}, {5291, 5903}, {5301, 35632}, {5329, 21771}, {5687, 20693}, {6327, 16886}, {6763, 16975}, {7737, 49168}, {7754, 68870}, {9598, 44447}, {14377, 50025}, {16600, 36283}, {16606, 27425}, {16973, 54422}, {17733, 50252}, {18398, 68893}, {21868, 54286}, {21888, 37567}, {22099, 23623}, {33035, 59633}, {36643, 54386}, {41269, 57280}, {41319, 50581}

X(69235) = crossdifference of every pair of points on line {38469, 48030}
X(69235) = barycentric product X(1)*X(32946)
X(69235) = barycentric quotient X(32946)/X(75)
X(69235) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {31, 3721, 16974}, {46, 54406, 1575}, {57, 39248, 16604}, {63, 54382, 1107}, {1046, 3496, 6}, {1572, 62858, 17448}, {1759, 49500, 213}, {3061, 4650, 33863}

X(69236) = (1, 0, 1, 1, 0, 1, 1, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 - a*b^2 - b^3 - 2*a*b*c + b^2*c - a*c^2 + b*c^2 - c^3) : :

X(69236) lies on these lines: {3, 53165}, {6, 484}, {32, 4642}, {35, 3959}, {40, 213}, {43, 41319}, {46, 20963}, {55, 3125}, {57, 16971}, {100, 3735}, {115, 33094}, {187, 49487}, {574, 2170}, {1571, 2082}, {1573, 4414}, {1759, 20691}, {2087, 67417}, {2176, 11010}, {2241, 24443}, {2251, 9620}, {2271, 37567}, {2276, 5011}, {3230, 5119}, {3579, 41015}, {3684, 69025}, {3721, 8715}, {3726, 25439}, {3727, 25440}, {3746, 20271}, {3930, 36283}, {3954, 5687}, {3987, 4426}, {4386, 4424}, {4805, 69038}, {4868, 63099}, {5248, 21951}, {5277, 37598}, {5282, 52959}, {5291, 64176}, {5697, 21008}, {5711, 6155}, {5903, 18755}, {9664, 21044}, {10987, 30117}, {16583, 37568}, {16788, 21888}, {16968, 59316}, {16969, 37563}, {16975, 17596}, {17451, 31451}, {21904, 49500}, {24174, 68893}, {24296, 51381}, {31433, 40131}, {31443, 43065}, {32948, 34542}, {33863, 37572}, {41423, 49758}, {48696, 49509}

X(69237) = (1, 0, 1, 1, 0, 1, 0, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 - a*b^2 - b^3 - 2*a*b*c - a*c^2 - c^3) : :

X(69237) lies on these lines: {1, 187}, {3, 3727}, {6, 56288}, {21, 3959}, {32, 4424}, {35, 3735}, {37, 37568}, {40, 2295}, {46, 24512}, {55, 3721}, {63, 3780}, {165, 54317}, {172, 37598}, {405, 21951}, {484, 17750}, {902, 16974}, {986, 1914}, {1107, 4414}, {1500, 1759}, {1571, 20331}, {1621, 20271}, {2238, 12514}, {2241, 3670}, {2243, 54416}, {2275, 17596}, {2276, 3496}, {2292, 4386}, {3053, 37614}, {3061, 17601}, {3125, 5248}, {3295, 3726}, {3683, 16605}, {3869, 18755}, {3871, 49509}, {3877, 21008}, {3890, 9259}, {3931, 63099}, {3954, 8715}, {3970, 36283}, {4400, 49518}, {4426, 4642}, {4640, 41015}, {4660, 16886}, {4760, 24549}, {4799, 69135}, {5011, 5283}, {5255, 41269}, {5262, 21793}, {5267, 53165}, {5282, 20691}, {6155, 62805}, {6284, 21965}, {9607, 57019}, {14974, 46907}, {16780, 39254}, {16781, 17595}, {16968, 35258}, {17594, 54382}, {17742, 31433}, {20970, 49500}, {21024, 32929}, {24046, 68893}, {30111, 53680}, {31451, 57015}, {33830, 59512}, {35445, 39255}, {37567, 65695}, {40750, 62831}, {41239, 69025}

X(69237) = barycentric product X(1)*X(4703)
X(69237) = barycentric quotient X(4703)/X(75)

X(69238) = (1, 0, 1, 1, 0, 0, 1, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 - a*b^2 - b^3 + b^2*c - a*c^2 + b*c^2 - c^3) : :

X(69238) lies on these lines: {3, 3125}, {6, 3336}, {32, 24443}, {35, 20271}, {36, 3959}, {40, 3230}, {46, 213}, {57, 20963}, {187, 3924}, {191, 37673}, {244, 2241}, {404, 3735}, {484, 2176}, {574, 17451}, {762, 9709}, {851, 23639}, {940, 6155}, {986, 5277}, {993, 21951}, {1054, 3496}, {1155, 16583}, {1376, 3954}, {1571, 40131}, {1574, 5282}, {1575, 1759}, {1729, 1939}, {1739, 4426}, {1914, 24046}, {2160, 4261}, {2242, 4642}, {2271, 5221}, {2275, 5011}, {3120, 7746}, {3290, 3579}, {3338, 16971}, {3670, 4386}, {3721, 25440}, {3726, 8715}, {3916, 16605}, {3980, 52538}, {4372, 57029}, {4376, 24170}, {4414, 16589}, {4987, 17748}, {5006, 37405}, {5283, 17596}, {5291, 24440}, {5697, 9259}, {5902, 18755}, {5903, 21008}, {7748, 21044}, {7751, 69015}, {9665, 28096}, {9941, 58863}, {11010, 16969}, {11112, 21965}, {11329, 68478}, {16601, 31443}, {16968, 58887}, {17063, 68893}, {17735, 37572}, {17736, 20691}, {17966, 68904}, {20228, 54420}, {21214, 41319}, {21808, 31451}, {23627, 61366}, {24254, 33830}, {33299, 36283}, {33863, 37524}, {36643, 54406}, {37582, 41015}

X(69238) = barycentric product X(1)*X(33098)
X(69238) = barycentric quotient X(33098)/X(75)
X(69238) = {X(36),X(3959)}-harmonic conjugate of X(53165)

X(69239) = (1, 0, 1, 1, 0, 0, 0, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 - a*b^2 - b^3 - a*c^2 - c^3) : :

X(69239) lies on these lines: {1, 187}, {3, 3721}, {6, 2243}, {9, 1054}, {21, 20271}, {32, 3670}, {36, 3735}, {37, 750}, {38, 4386}, {39, 1759}, {46, 2295}, {55, 3726}, {56, 3727}, {57, 24512}, {63, 2238}, {71, 69030}, {89, 16522}, {100, 49509}, {171, 41269}, {172, 986}, {230, 3782}, {325, 4799}, {574, 36283}, {958, 21951}, {980, 40747}, {982, 1914}, {988, 36643}, {993, 3125}, {1100, 4003}, {1500, 17736}, {1571, 17742}, {1575, 5282}, {1631, 8628}, {1761, 2277}, {2176, 56288}, {2241, 3953}, {2242, 4424}, {2275, 3496}, {2276, 3509}, {2344, 18208}, {2975, 3959}, {3053, 37549}, {3054, 37691}, {3116, 66149}, {3187, 50252}, {3219, 37673}, {3231, 16520}, {3290, 4640}, {3336, 17750}, {3666, 63099}, {3693, 31443}, {3724, 4016}, {3780, 62858}, {3788, 17211}, {3868, 18755}, {3869, 21008}, {3877, 9259}, {3916, 16583}, {3954, 25440}, {3970, 31451}, {3972, 30111}, {4037, 32934}, {4346, 37689}, {4390, 21888}, {4396, 49518}, {4415, 26267}, {4426, 24443}, {4650, 60697}, {4652, 16968}, {4713, 26279}, {4950, 7750}, {4987, 29671}, {5011, 16975}, {5078, 21773}, {5744, 62693}, {7191, 21793}, {7354, 21965}, {7761, 63817}, {7771, 30113}, {7824, 18055}, {8624, 24464}, {9318, 17276}, {9347, 16777}, {10026, 32859}, {11337, 21771}, {15803, 54317}, {16503, 18201}, {17398, 26627}, {17475, 37555}, {17594, 60724}, {17601, 51058}, {17735, 26242}, {17737, 33102}, {19557, 51836}, {20684, 20729}, {21101, 59679}, {21332, 31449}, {21904, 32912}, {23903, 50065}, {23958, 63066}, {26244, 32939}, {28350, 54404}, {28606, 40750}, {35242, 39255}, {36075, 66724}, {36279, 65695}, {37657, 67335}, {37675, 62796}, {40997, 63548}, {56530, 69025}

X(69239) = crossdifference of every pair of points on line {25569, 47827}
X(69239) = barycentric product X(i)*X(j) for these {i,j}: {1, 4655}, {100, 48227}
X(69239) = barycentric quotient X(i)/X(j) for these {i,j}: {4655, 75}, {48227, 693}
X(69239) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {574, 36283, 57015}, {988, 36643, 54382}, {3509, 17596, 2276}

X(69240) = (1, 0, 0, 1, 0, 1, 1, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 - b^3 - 2*a*b*c + b^2*c + b*c^2 - c^3) : :

X(69240) lies on these lines: {1, 1055}, {3, 2170}, {6, 4642}, {8, 3496}, {9, 3617}, {19, 2269}, {31, 41015}, {32, 49487}, {40, 672}, {41, 517}, {42, 54382}, {46, 1475}, {48, 64046}, {55, 17451}, {57, 3160}, {65, 2280}, {100, 3061}, {101, 5697}, {145, 3509}, {169, 1334}, {187, 53165}, {198, 17452}, {218, 12702}, {220, 2246}, {244, 16781}, {284, 41723}, {484, 4253}, {519, 1759}, {528, 40997}, {573, 40968}, {609, 15955}, {649, 21105}, {748, 16605}, {899, 39248}, {902, 16968}, {910, 3057}, {940, 39247}, {976, 3735}, {1001, 21921}, {1107, 4414}, {1121, 55965}, {1146, 6284}, {1155, 40133}, {1193, 1572}, {1200, 41338}, {1212, 37568}, {1276, 5335}, {1277, 5334}, {1376, 39244}, {1400, 54420}, {1479, 21044}, {1697, 39587}, {1729, 4304}, {1761, 5839}, {1766, 2347}, {1914, 3924}, {1953, 36744}, {2098, 3207}, {2099, 4258}, {2160, 16884}, {2171, 4254}, {2183, 38008}, {2241, 3125}, {2262, 2268}, {2270, 54359}, {2301, 37625}, {2329, 14923}, {2348, 21872}, {2975, 4051}, {3058, 21049}, {3101, 54373}, {3169, 5279}, {3214, 54406}, {3217, 21871}, {3244, 17736}, {3245, 17745}, {3295, 21808}, {3434, 21029}, {3501, 33950}, {3512, 41845}, {3579, 43065}, {3684, 3869}, {3691, 12514}, {3721, 3938}, {3727, 4386}, {3730, 5540}, {3754, 16783}, {3780, 32912}, {3885, 56530}, {3913, 3930}, {3915, 16583}, {3970, 25439}, {4136, 5014}, {4165, 5015}, {4167, 5016}, {4251, 5903}, {4266, 16548}, {4302, 4530}, {4336, 23868}, {4372, 35101}, {4390, 10914}, {4441, 17739}, {4534, 15338}, {4640, 4875}, {4950, 62467}, {5030, 37572}, {5060, 54313}, {5184, 16476}, {5217, 34522}, {5250, 59207}, {5254, 33094}, {5276, 37598}, {5541, 17744}, {5687, 33299}, {5710, 21840}, {6658, 54120}, {7291, 37555}, {7297, 54324}, {8074, 10624}, {8558, 63592}, {8715, 57015}, {9580, 23058}, {9785, 40127}, {10441, 23623}, {12053, 68797}, {12701, 46835}, {14936, 22070}, {15624, 20593}, {16058, 22173}, {16502, 24443}, {16600, 37610}, {16716, 62740}, {16784, 24046}, {16787, 54315}, {17017, 53128}, {17440, 37504}, {18156, 24602}, {20244, 24333}, {20606, 23544}, {21384, 41319}, {21764, 54418}, {22370, 26998}, {24440, 33854}, {24712, 33298}, {26242, 37588}, {27000, 30949}, {28125, 37586}, {28174, 61706}, {28228, 61651}, {30144, 35342}, {32577, 62370}, {33937, 33952}, {36643, 62874}, {40256, 58036}, {45219, 68931}

X(69240) = barycentric product X(1)*X(24703)
X(69240) = barycentric quotient X(24703)/X(75)
X(69240) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 3496, 5282}, {40, 2082, 672}, {169, 5119, 1334}, {910, 3057, 9310}, {1212, 37568, 41423}, {1914, 3959, 3924}, {2098, 3207, 17439}, {2241, 3125, 28082}, {5540, 11010, 3730}, {21384, 41319, 56288}, {33950, 63136, 3501}

X(69241) = (1, 0, 0, 1, 0, 1, 0, 0, 0, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 - b^3 - 2*a*b*c - c^3) : :

X(69241) lies on these lines: {1, 32}, {2, 5011}, {6, 4424}, {9, 80}, {19, 5136}, {21, 1729}, {35, 3061}, {37, 37610}, {40, 16549}, {41, 3878}, {46, 68950}, {55, 57015}, {57, 7181}, {63, 16834}, {65, 16783}, {75, 33952}, {87, 5539}, {101, 3877}, {163, 17512}, {169, 3294}, {191, 21384}, {257, 7096}, {284, 18417}, {392, 910}, {484, 6205}, {517, 16788}, {519, 5282}, {614, 68960}, {672, 50287}, {748, 16611}, {758, 2280}, {982, 16784}, {986, 5299}, {993, 2170}, {997, 35342}, {1021, 55936}, {1100, 24473}, {1449, 1761}, {1697, 17742}, {1723, 68853}, {1724, 41015}, {1764, 2339}, {2082, 12514}, {2329, 5697}, {2802, 4390}, {3187, 57017}, {3208, 17744}, {3216, 39248}, {3219, 7349}, {3293, 54406}, {3295, 3970}, {3338, 36643}, {3501, 11010}, {3512, 57725}, {3670, 16502}, {3684, 5692}, {3730, 33950}, {3760, 17739}, {3761, 17738}, {3869, 4251}, {3884, 9310}, {3913, 4006}, {3915, 16600}, {3916, 40133}, {3930, 25439}, {3953, 16781}, {4051, 5258}, {4056, 17062}, {4136, 4894}, {4253, 56288}, {4254, 21078}, {4258, 5730}, {4262, 4511}, {4302, 24247}, {4366, 20373}, {4512, 28121}, {4640, 43065}, {4799, 69174}, {4875, 31445}, {5248, 17451}, {5280, 37598}, {5291, 49494}, {5724, 18907}, {5749, 34632}, {5902, 16503}, {5903, 41239}, {7054, 7094}, {8074, 40998}, {8558, 11111}, {8715, 33299}, {10624, 21073}, {11343, 68759}, {11545, 65683}, {14210, 24586}, {15048, 57019}, {15171, 40997}, {16370, 34522}, {16782, 24464}, {16785, 66674}, {17027, 66152}, {18156, 29473}, {20271, 68893}, {21372, 40131}, {21840, 62828}, {24036, 41423}, {24045, 27068}, {24047, 26690}, {24296, 69055}, {24496, 67417}, {25066, 37568}, {25440, 39244}, {26242, 40091}, {27958, 38477}, {30108, 53332}, {30435, 37614}, {33790, 53680}, {35258, 46407}, {48809, 59207}, {53018, 54370}, {54440, 62838}

X(69241) = X(2)-isoconjugate of X(7236)
X(69241) = X(i)-Dao conjugate of X(j) for these (i,j): {750, 4363}, {32664, 7236}
X(69241) = crossdifference of every pair of points on line {1491, 53314}
X(69241) = barycentric product X(75)*X(4471)
X(69241) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 7236}, {4471, 1}
X(69241) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1759, 17736}, {1, 3496, 1759}, {9, 5119, 1018}, {9, 5540, 21373}, {32, 3727, 1}, {169, 5250, 3294}, {484, 17754, 6205}, {484, 56532, 17754}, {1914, 3735, 1}, {2082, 12514, 16552}, {2241, 3721, 1}, {17744, 37563, 3208}, {17754, 41319, 484}, {41319, 56532, 6205}

X(69242) = (1, 0, 0, 1, 0, 0, 1, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 - b^3 + b^2*c + b*c^2 - c^3) : :

X(69242) lies on these lines: {1, 1055}, {2, 3496}, {3, 17451}, {4, 21044}, {6, 1406}, {8, 3509}, {9, 5128}, {10, 1759}, {19, 208}, {28, 57657}, {31, 2355}, {32, 3125}, {37, 37568}, {40, 1334}, {41, 65}, {42, 15496}, {46, 169}, {48, 54360}, {55, 21808}, {56, 2170}, {57, 279}, {63, 3691}, {101, 5903}, {116, 4056}, {172, 3959}, {198, 2171}, {218, 2246}, {220, 37567}, {244, 16502}, {257, 16915}, {267, 16562}, {304, 24602}, {379, 16609}, {404, 3061}, {405, 21921}, {474, 39244}, {484, 3730}, {517, 9310}, {519, 17736}, {573, 1781}, {579, 16547}, {604, 2262}, {607, 1254}, {610, 21748}, {649, 21132}, {899, 54406}, {942, 2280}, {946, 68797}, {962, 40127}, {966, 1761}, {976, 3721}, {986, 5276}, {1042, 8776}, {1046, 37657}, {1111, 14377}, {1146, 7354}, {1155, 1212}, {1193, 54382}, {1201, 1572}, {1204, 53560}, {1376, 33299}, {1393, 56913}, {1405, 2182}, {1447, 27000}, {1454, 2272}, {1468, 41015}, {1482, 17439}, {1652, 63032}, {1653, 63033}, {1696, 21809}, {1724, 16611}, {1729, 4292}, {1730, 20665}, {1744, 5816}, {1770, 5179}, {1788, 5819}, {1836, 21912}, {1839, 24005}, {1880, 2199}, {1914, 20271}, {1953, 2178}, {1959, 11329}, {2083, 40975}, {2097, 53538}, {2099, 3207}, {2183, 9596}, {2243, 4426}, {2245, 54324}, {2269, 54405}, {2270, 2285}, {2271, 2650}, {2278, 15586}, {2292, 5275}, {2294, 36744}, {2312, 4185}, {3119, 64152}, {3120, 3767}, {3188, 67654}, {3204, 21863}, {3208, 63136}, {3212, 4209}, {3218, 16816}, {3290, 3915}, {3332, 62340}, {3336, 4253}, {3338, 17474}, {3474, 6554}, {3579, 16601}, {3684, 3868}, {3735, 5277}, {3754, 16788}, {3871, 51058}, {3923, 27040}, {3930, 5687}, {3937, 23630}, {3954, 36283}, {3970, 8715}, {3980, 26035}, {4051, 54391}, {4063, 39797}, {4109, 6327}, {4136, 5300}, {4165, 7270}, {4190, 24247}, {4251, 5902}, {4293, 4530}, {4332, 52425}, {4376, 20255}, {4390, 5836}, {4414, 5283}, {4642, 54416}, {5030, 37524}, {5053, 61695}, {5060, 11101}, {5183, 21872}, {5204, 34522}, {5255, 26242}, {5264, 16600}, {5299, 24046}, {5711, 21840}, {5834, 43053}, {5883, 16783}, {6155, 67208}, {6284, 21049}, {7031, 30117}, {7146, 11349}, {7208, 62425}, {7614, 56547}, {7737, 21950}, {7754, 69015}, {8558, 9579}, {9318, 17753}, {9352, 26690}, {9599, 28096}, {9860, 18784}, {10473, 23623}, {11246, 61706}, {11358, 22230}, {11383, 17442}, {12514, 59207}, {13738, 45208}, {14439, 56536}, {14923, 56530}, {14953, 24268}, {16549, 21372}, {16589, 22080}, {16919, 24291}, {17680, 19555}, {17738, 18135}, {17739, 34284}, {17742, 54286}, {17754, 33950}, {17798, 28125}, {18785, 56639}, {20267, 65116}, {20292, 27068}, {20459, 28110}, {20606, 53129}, {20706, 34247}, {20911, 24586}, {21071, 32929}, {21371, 27059}, {21741, 21767}, {21811, 37499}, {22836, 35342}, {23619, 61366}, {24047, 37572}, {24174, 33854}, {24549, 26562}, {25440, 57015}, {27627, 39248}, {29473, 33936}, {29690, 31416}, {32636, 40133}, {33094, 69096}, {33830, 59509}, {33866, 40690}, {33942, 33952}, {36118, 63187}, {37274, 56882}, {37582, 43065}, {41882, 60245}, {46345, 67880}, {56382, 67651}

X(69242) = isogonal conjugate of X(55965)
X(69242) = polar conjugate of the isotomic conjugate of X(20277)
X(69242) = X(i)-Ceva conjugate of X(j) for these (i,j): {1836, 4336}, {36118, 663}
X(69242) = X(i)-isoconjugate of X(j) for these (i,j): {1, 55965}, {2, 37741}, {6, 34409}, {8, 56005}, {78, 63187}, {219, 34398}, {36054, 54968}
X(69242) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 55965}, {9, 34409}, {4640, 4416}, {17073, 312}, {32664, 37741}, {46835, 304}, {53846, 326}
X(69242) = crosspoint of X(19) and X(57)
X(69242) = crosssum of X(i) and X(j) for these (i,j): {9, 63}, {3647, 17746}
X(69242) = crossdifference of every pair of points on line {4105, 35057}
X(69242) = barycentric product X(i)*X(j) for these {i,j}: {1, 1836}, {4, 20277}, {7, 4336}, {19, 17073}, {28, 21912}, {56, 17860}, {57, 46835}, {65, 17188}, {158, 53847}, {664, 2520}, {759, 51462}, {1783, 23727}, {4394, 27833}
X(69242) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 34409}, {6, 55965}, {31, 37741}, {34, 34398}, {604, 56005}, {608, 63187}, {1836, 75}, {2520, 522}, {4336, 8}, {17073, 304}, {17188, 314}, {17860, 3596}, {20277, 69}, {21912, 20336}, {23727, 15413}, {36127, 54968}, {46835, 312}, {51462, 35550}, {53847, 326}
X(69242) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 1759, 5282}, {19, 1400, 40968}, {32, 3125, 3924}, {40, 40131, 1334}, {46, 169, 672}, {57, 2082, 1475}, {65, 910, 41}, {172, 3959, 49487}, {1042, 40957, 8776}, {1914, 20271, 28082}, {2243, 21951, 4426}, {2270, 2285, 2347}, {3212, 4209, 9317}, {3336, 5540, 4253}, {3579, 16601, 41423}, {3721, 4386, 976}, {15586, 61704, 2278}, {54405, 54420, 2269}

X(69243) = (1, 0, 0, 0, 0, 1, 1, 0, 1, 0)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^3 - 2*a*b*c + b^2*c + b*c^2) : :

X(69243) lies on these lines: {1, 1929}, {3, 17448}, {6, 979}, {8, 1914}, {10, 2241}, {21, 10987}, {31, 3780}, {32, 519}, {35, 16975}, {37, 169}, {39, 8715}, {40, 16973}, {44, 4515}, {55, 1107}, {100, 2275}, {145, 172}, {187, 8666}, {200, 39248}, {213, 37610}, {220, 52964}, {230, 3813}, {239, 69024}, {315, 62467}, {384, 24524}, {385, 17144}, {404, 63493}, {517, 21861}, {518, 5017}, {528, 5254}, {536, 7754}, {609, 3633}, {976, 3727}, {1015, 25440}, {1043, 69068}, {1100, 4646}, {1279, 16605}, {1333, 56018}, {1376, 16604}, {1415, 37738}, {1500, 25439}, {1572, 3811}, {1573, 5248}, {1575, 5687}, {1698, 68893}, {2082, 20692}, {2176, 3684}, {2220, 17299}, {2238, 3915}, {2242, 3244}, {2276, 3871}, {2280, 2295}, {2321, 16946}, {2548, 45701}, {3053, 12513}, {3158, 9575}, {3189, 10315}, {3303, 5275}, {3496, 49509}, {3550, 33863}, {3632, 5291}, {3680, 17967}, {3704, 5301}, {3721, 3938}, {3722, 17451}, {3744, 16974}, {3746, 5283}, {3749, 16968}, {3814, 9665}, {3815, 64123}, {3870, 54382}, {3905, 35101}, {4050, 54329}, {4136, 17765}, {4188, 63499}, {4203, 24528}, {4253, 50028}, {4262, 50637}, {4366, 6376}, {4400, 4441}, {4421, 5013}, {4479, 17129}, {4595, 17743}, {4856, 5042}, {5014, 16886}, {5023, 11194}, {5035, 50131}, {5247, 21793}, {5250, 21879}, {5264, 20963}, {5299, 48696}, {5304, 12632}, {5337, 16834}, {5552, 9599}, {7735, 64068}, {7738, 34607}, {7745, 12607}, {7746, 24387}, {7770, 25102}, {7816, 33908}, {8624, 49458}, {9596, 10528}, {9598, 20075}, {10198, 31416}, {10311, 68004}, {10965, 62372}, {11108, 25614}, {11235, 13881}, {11236, 65630}, {11320, 25298}, {11321, 24656}, {11353, 25125}, {13588, 63509}, {13741, 25610}, {16283, 41006}, {16466, 21904}, {16517, 53053}, {16519, 37598}, {16780, 63137}, {16787, 64176}, {16915, 25303}, {16916, 25280}, {16920, 25278}, {16955, 25286}, {16957, 25287}, {16971, 37522}, {16995, 32095}, {17275, 68944}, {17281, 33882}, {17735, 21384}, {18055, 68875}, {20065, 62463}, {20363, 34247}, {20693, 54406}, {20970, 62828}, {21024, 32941}, {21049, 53534}, {21341, 33104}, {21769, 54316}, {21888, 63130}, {21951, 28082}, {24249, 59524}, {24652, 33828}, {24735, 33821}, {25130, 33035}, {31449, 64951}, {31999, 33062}, {34626, 44519}, {35977, 63528}, {40133, 52804}, {59691, 62370}

X(69243) = barycentric product X(1)*X(4011)
X(69243) = barycentric quotient X(4011)/X(75)
X(69243) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 3913, 20691}, {8, 1914, 4426}, {1376, 16781, 16604}, {3632, 7031, 5291}, {3684, 37588, 2176}, {3744, 41015, 16974}, {5687, 16502, 1575}

X(69244) = (0, 0, 0, 1, 0, 1, 1, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(b^3 + 2*a*b*c - b^2*c - b*c^2 + c^3) : :

X(69244) lies on these lines: {1, 1929}, {3, 53165}, {6, 5903}, {8, 3735}, {10, 3727}, {32, 49487}, {37, 10914}, {38, 22184}, {39, 2170}, {65, 20963}, {148, 54120}, {172, 5011}, {213, 517}, {239, 68478}, {257, 17143}, {392, 16605}, {484, 33863}, {519, 3721}, {758, 3780}, {762, 3679}, {942, 16971}, {986, 4051}, {1015, 24443}, {1107, 4424}, {1125, 21951}, {1146, 69096}, {1334, 49758}, {1500, 17451}, {1572, 54418}, {1573, 2292}, {1574, 4695}, {1575, 3987}, {1739, 16604}, {1759, 49494}, {1834, 23639}, {1930, 35101}, {1953, 2092}, {2082, 9620}, {2087, 2275}, {2099, 2271}, {2171, 4263}, {2176, 5697}, {2238, 3878}, {2241, 3924}, {2802, 16600}, {3057, 3230}, {3061, 64176}, {3120, 69175}, {3187, 18202}, {3244, 3726}, {3290, 9957}, {3419, 60586}, {3496, 5291}, {3632, 49509}, {3670, 17448}, {3673, 52626}, {3752, 68759}, {3754, 24512}, {3755, 20861}, {3764, 4516}, {3774, 20593}, {3869, 21839}, {3880, 46902}, {3884, 16611}, {3885, 26242}, {3968, 25089}, {3970, 21331}, {4016, 17362}, {4286, 56531}, {4414, 31456}, {4674, 68950}, {4696, 22036}, {4852, 18179}, {5021, 37567}, {5119, 16968}, {5283, 37598}, {7264, 21138}, {7748, 33094}, {8682, 17137}, {10950, 38345}, {11010, 17735}, {14210, 20255}, {14923, 21802}, {16549, 21888}, {16552, 50014}, {16974, 37610}, {17152, 17497}, {17275, 21810}, {17443, 56926}, {17452, 21796}, {17489, 68890}, {18061, 26752}, {18167, 33296}, {18189, 62755}, {20691, 57015}, {21226, 35957}, {21272, 26978}, {21332, 25092}, {21746, 23668}, {21965, 24390}, {22171, 32915}, {24046, 63493}, {24254, 26965}, {24790, 49777}, {25264, 49755}, {31448, 34522}, {33299, 52959}, {49760, 62817}, {54286, 54317}

X(69244) = reflection of X(213) in X(41015)
X(69244) = barycentric product X(1)*X(69173)
X(69244) = barycentric quotient X(69173)/X(75)
X(69244) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3959, 3125}, {8, 3735, 3954}, {986, 4051, 16975}, {2170, 4642, 39}, {3057, 16583, 3230}, {4695, 39244, 1574}, {5011, 15955, 172}

X(69245) = (0, 1, 0, 0, 1, 1, 1, 0, 1, 0)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^2*b + a^2*c - 2*a*b*c + b^2*c + b*c^2) : :
X(69245) = 2 X[41015] - 3 X[46907]

X(69245) lies on these lines: {1, 39}, {2, 16969}, {6, 145}, {8, 2176}, {9, 59310}, {10, 3230}, {32, 37610}, {37, 1953}, {42, 4161}, {43, 4050}, {55, 237}, {65, 3726}, {75, 28350}, {81, 29588}, {83, 1016}, {100, 21008}, {141, 17152}, {172, 5255}, {190, 21226}, {192, 698}, {213, 519}, {239, 60730}, {321, 40886}, {350, 17752}, {384, 18047}, {404, 9259}, {517, 3721}, {524, 20109}, {594, 992}, {595, 5291}, {668, 29706}, {672, 17448}, {728, 9575}, {762, 10176}, {869, 4433}, {894, 25303}, {899, 21868}, {940, 29585}, {942, 65695}, {956, 14974}, {1086, 20244}, {1107, 1334}, {1149, 16604}, {1185, 19994}, {1193, 20691}, {1201, 1575}, {1572, 17742}, {1573, 3294}, {1574, 49997}, {1909, 10027}, {1914, 2329}, {1930, 68890}, {2235, 49484}, {2241, 16788}, {2242, 5264}, {2300, 2321}, {2345, 21769}, {2802, 16600}, {2975, 17735}, {3051, 20044}, {3216, 52959}, {3244, 3997}, {3263, 59512}, {3290, 5836}, {3576, 39255}, {3617, 36647}, {3621, 37657}, {3623, 63066}, {3632, 54981}, {3635, 16971}, {3686, 61036}, {3730, 16975}, {3735, 5697}, {3869, 49509}, {3871, 18755}, {3872, 16968}, {3878, 3954}, {3880, 41015}, {3884, 28594}, {3903, 59480}, {3912, 54282}, {3915, 4390}, {3934, 29381}, {3959, 14923}, {3996, 21779}, {4168, 17765}, {4363, 69053}, {4386, 9310}, {4465, 6376}, {4559, 10944}, {4595, 26752}, {4665, 52897}, {4754, 64133}, {4853, 16970}, {4879, 45240}, {5263, 21788}, {5275, 36007}, {5332, 54329}, {5710, 63099}, {5749, 21785}, {6542, 37676}, {7109, 17135}, {8610, 28249}, {10914, 16583}, {12194, 68898}, {14621, 56353}, {15989, 64158}, {16499, 31456}, {16552, 52963}, {16601, 21332}, {16777, 37614}, {16782, 49466}, {16974, 49487}, {17033, 17144}, {17143, 40859}, {17267, 29986}, {17279, 30036}, {17316, 68769}, {17355, 20228}, {17489, 35101}, {17745, 50028}, {17759, 34063}, {18140, 29547}, {20011, 21753}, {20016, 32911}, {20255, 27097}, {20880, 49777}, {21281, 30945}, {21760, 49482}, {21888, 24443}, {22837, 53165}, {23354, 25285}, {23416, 24494}, {24222, 69175}, {24326, 59509}, {24514, 24524}, {25414, 38345}, {26563, 59524}, {26765, 37678}, {27076, 29691}, {27271, 53665}, {27623, 42696}, {28368, 28605}, {28598, 53332}, {29511, 30819}, {29624, 37674}, {29699, 69136}, {30577, 37633}, {31087, 59515}, {32941, 40728}, {33863, 54391}, {34064, 45897}, {36409, 39247}, {37598, 41269}, {40747, 49476}, {44798, 45219}, {49457, 66878}, {49704, 51921}, {50015, 62848}, {51058, 66646}, {52134, 69024}, {52635, 63987}

X(69245) = reflection of X(3780) in X(213)
X(69245) = X(694)-Ceva conjugate of X(2238)
X(69245) = X(59511)-Dao conjugate of X(3662)
X(69245) = crosspoint of X(i) and X(j) for these (i,j): {1, 17743}, {1016, 3903}
X(69245) = crosssum of X(i) and X(j) for these (i,j): {1, 2275}, {1015, 4367}
X(69245) = crossdifference of every pair of points on line {659, 6363}
X(69245) = barycentric product X(i)*X(j) for these {i,j}: {1, 59511}, {1018, 29402}
X(69245) = barycentric quotient X(i)/X(j) for these {i,j}: {29402, 7199}, {59511, 75}
X(69245) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1018, 39}, {1, 2295, 24512}, {1, 3208, 2276}, {1, 3501, 2275}, {1, 16549, 1015}, {1, 17754, 63493}, {8, 2176, 2238}, {37, 3057, 3727}, {594, 16685, 992}, {1107, 52964, 1334}, {2275, 3501, 20331}, {2329, 37588, 1914}, {3244, 3997, 20963}, {3290, 5836, 21951}, {3730, 50637, 16975}, {3915, 4390, 4426}, {4513, 37542, 6}, {5255, 56530, 172}, {6542, 62813, 37676}, {14923, 26242, 3959}, {17152, 26759, 141}, {27097, 69028, 20255}

X(69246) = (0, 1, 1, 1, 1, 0, 0, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^2*b - a*b^2 - b^3 + a^2*c - a*c^2 - c^3) : :

X(69246) lies on these lines: {1, 17735}, {2, 21879}, {6, 986}, {8, 9597}, {9, 20271}, {10, 69175}, {37, 579}, {38, 2295}, {39, 758}, {40, 16973}, {44, 16583}, {46, 4386}, {63, 4426}, {65, 1107}, {72, 1575}, {76, 68870}, {141, 68944}, {171, 16519}, {172, 3218}, {213, 3670}, {304, 24691}, {517, 17448}, {518, 3094}, {537, 4095}, {574, 22836}, {583, 4016}, {672, 3721}, {960, 16604}, {982, 2176}, {992, 3958}, {1015, 3878}, {1100, 2241}, {1159, 31468}, {1212, 21331}, {1334, 3726}, {1475, 3727}, {1500, 3874}, {1571, 3811}, {1573, 3754}, {1574, 3678}, {1739, 17746}, {1914, 56288}, {2238, 24443}, {2275, 3869}, {2276, 3868}, {2292, 24512}, {2329, 18208}, {2549, 49168}, {3125, 16552}, {3208, 62865}, {3216, 21839}, {3230, 3953}, {3336, 5277}, {3339, 16517}, {3501, 49509}, {3649, 37661}, {3673, 17276}, {3691, 21951}, {3735, 4253}, {3739, 24190}, {3752, 21874}, {3780, 4642}, {3821, 53423}, {3877, 63493}, {3954, 16549}, {3959, 21384}, {3976, 16969}, {3999, 4520}, {4047, 20227}, {4053, 4286}, {4424, 20963}, {4880, 5280}, {5013, 12635}, {5221, 5275}, {5262, 60697}, {5283, 5902}, {5291, 6763}, {5694, 34460}, {5883, 16589}, {5903, 16975}, {5904, 20693}, {5905, 9596}, {6646, 33944}, {6650, 60149}, {7745, 17768}, {7760, 66152}, {9560, 10026}, {9574, 11523}, {9593, 54422}, {9598, 12649}, {9599, 11415}, {9620, 62858}, {10459, 65695}, {11529, 31429}, {11684, 33854}, {12526, 39248}, {15815, 56177}, {16780, 54290}, {16787, 21793}, {16887, 24254}, {16974, 37549}, {17141, 24326}, {17165, 21021}, {17398, 58386}, {17596, 18755}, {17755, 20255}, {18189, 18206}, {20331, 33299}, {20698, 24530}, {20911, 24690}, {21077, 31398}, {21342, 21872}, {21868, 34790}, {21873, 46838}, {21897, 64581}, {22061, 23622}, {24068, 68897}, {24174, 37673}, {24215, 49777}, {24652, 35274}, {25248, 30941}, {30147, 31456}, {30173, 69080}, {31396, 67850}, {31426, 41863}, {31442, 54318}, {31443, 56176}, {31449, 68668}, {31466, 39542}, {31806, 62371}, {33888, 33938}, {43039, 45288}, {44669, 63548}

X(69246) = X(33064)-Dao conjugate of X(4851)
X(69246) = crossdifference of every pair of points on line {38469, 48297}
X(69246) = barycentric product X(1)*X(33064)
X(69246) = barycentric quotient X(33064)/X(75)
X(69246) = {X(3959),X(21384)}-harmonic conjugate of X(50014)

X(69247) = (0, 1, 1, 1, 1, 0, 1, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(a^2*b - a*b^2 - b^3 + a^2*c + b^2*c - a*c^2 + b*c^2 - c^3) : :
X(69247) = 3 X[1015] - 2 X[62370]

X(69247) lies on these lines: {1, 574}, {6, 5011}, {7, 31409}, {9, 36283}, {10, 69175}, {32, 46}, {37, 5883}, {39, 65}, {40, 2241}, {44, 16611}, {57, 2242}, {72, 1574}, {115, 1737}, {172, 3336}, {187, 1155}, {213, 24443}, {226, 31398}, {244, 3230}, {291, 760}, {355, 9651}, {484, 1914}, {517, 1015}, {518, 52959}, {519, 21888}, {672, 3125}, {726, 68897}, {742, 57029}, {758, 1575}, {899, 21839}, {942, 1500}, {986, 17750}, {1018, 3726}, {1100, 4868}, {1107, 3754}, {1159, 5024}, {1504, 2362}, {1505, 16232}, {1506, 12047}, {1572, 2093}, {1573, 3753}, {1739, 2238}, {1770, 7747}, {1788, 3767}, {1836, 5475}, {1837, 7748}, {1905, 33843}, {2176, 24046}, {2275, 5903}, {2276, 5902}, {2295, 3670}, {2548, 4295}, {2549, 18391}, {2646, 37512}, {3199, 67965}, {3218, 5291}, {3245, 16784}, {3290, 52963}, {3339, 9593}, {3340, 9619}, {3474, 7737}, {3485, 31401}, {3601, 31422}, {3612, 15515}, {3634, 21879}, {3649, 31460}, {3671, 31396}, {3721, 16549}, {3727, 68950}, {3730, 20271}, {3735, 17754}, {3780, 3987}, {3812, 16589}, {3815, 39542}, {3874, 20691}, {3878, 16604}, {3959, 4253}, {4037, 49999}, {4115, 49993}, {4403, 43037}, {4424, 24512}, {4642, 20963}, {4674, 45751}, {5013, 68668}, {5074, 69009}, {5088, 24281}, {5128, 16780}, {5134, 6788}, {5164, 16592}, {5206, 58887}, {5213, 10026}, {5219, 31441}, {5221, 54416}, {5254, 67980}, {5530, 9560}, {5697, 63493}, {5722, 9664}, {6381, 68870}, {6603, 8649}, {7603, 17605}, {7746, 24914}, {7756, 10572}, {7772, 54382}, {8589, 37600}, {8610, 21864}, {8624, 20367}, {9574, 11529}, {9592, 18421}, {9597, 10573}, {9650, 57282}, {9665, 12699}, {10826, 69141}, {11374, 31501}, {11375, 31455}, {11518, 31426}, {12019, 53419}, {12702, 16781}, {13006, 53615}, {14974, 17054}, {14988, 34460}, {15934, 31477}, {16502, 37567}, {16503, 69025}, {16552, 21951}, {16779, 41319}, {16973, 54286}, {17053, 21853}, {17205, 49777}, {17606, 39565}, {17735, 30117}, {18201, 56530}, {19860, 31456}, {19950, 32857}, {20227, 21866}, {20331, 57015}, {20718, 68940}, {21331, 24036}, {23622, 42669}, {24691, 33936}, {24929, 31443}, {25092, 33815}, {31442, 64673}, {37606, 53095}, {37730, 63548}, {40663, 69098}, {44518, 68905}, {69038, 69174}

X(69247) = barycentric product X(1)*X(32856)
X(69247) = barycentric quotient X(32856)/X(75)
X(69247) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1571, 31451}, {57, 9620, 2242}, {226, 31398, 31476}, {672, 3125, 49758}

X(69248) = (0, 1, 1, 1, 1, 1, 0, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(b + c)*(a^2 - a*b - b^2 - a*c + b*c - c^2) : :

X(69248) lies on these lines: {1, 17735}, {6, 37598}, {8, 9598}, {9, 3959}, {10, 115}, {37, 65}, {39, 3878}, {40, 4386}, {44, 33950}, {72, 20691}, {85, 17276}, {100, 25435}, {172, 56288}, {190, 257}, {191, 5291}, {210, 21868}, {213, 4424}, {214, 37512}, {392, 16604}, {484, 5277}, {517, 1107}, {574, 30144}, {672, 3727}, {758, 1500}, {960, 1575}, {982, 16969}, {986, 2176}, {997, 1571}, {1015, 3884}, {1018, 3954}, {1100, 5035}, {1482, 31449}, {1506, 11813}, {1574, 10176}, {1580, 4640}, {1697, 16973}, {1909, 68870}, {2238, 4642}, {2275, 3877}, {2276, 3869}, {2650, 60724}, {3057, 11998}, {3125, 3294}, {3159, 68897}, {3208, 49509}, {3212, 4419}, {3219, 49760}, {3230, 3670}, {3290, 4520}, {3293, 21839}, {3666, 51319}, {3678, 52959}, {3698, 25614}, {3730, 3735}, {3754, 16589}, {3811, 31433}, {3890, 63493}, {3901, 9331}, {3985, 21025}, {4037, 17751}, {4050, 49448}, {4253, 17461}, {4415, 16609}, {4426, 9620}, {4646, 21874}, {4674, 46196}, {4868, 20970}, {5013, 5289}, {5239, 69160}, {5240, 69167}, {5255, 16519}, {5275, 37567}, {5283, 5903}, {5293, 41322}, {5697, 16975}, {5730, 31448}, {6646, 20955}, {7982, 31429}, {7991, 16517}, {8148, 31468}, {9574, 15829}, {9593, 39248}, {9596, 11415}, {12635, 31477}, {14974, 16974}, {16552, 50014}, {16600, 52963}, {16601, 21331}, {16720, 53332}, {16814, 56244}, {17016, 60697}, {17152, 25248}, {17303, 31359}, {17452, 63496}, {17596, 21008}, {17747, 21965}, {18156, 24691}, {18905, 40966}, {20331, 39244}, {20461, 23905}, {21021, 56318}, {21035, 66971}, {21231, 53476}, {21616, 31398}, {21677, 21956}, {21816, 56191}, {21858, 21873}, {21883, 22275}, {21887, 63886}, {21889, 22297}, {21902, 22325}, {21951, 59207}, {22791, 31466}, {22836, 31451}, {24214, 49777}, {24440, 37673}, {25270, 33931}, {25349, 59509}, {31396, 68619}, {31426, 68602}, {31443, 59691}, {31460, 51409}, {35128, 65922}, {59633, 60706}, {62817, 68478}

X(69248) = X(52651)-Ceva conjugate of X(37)
X(69248) = X(58)-isoconjugate of X(54120)
X(69248) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 54120}, {894, 8033}
X(69248) = crosspoint of X(i) and X(j) for these (i,j): {1, 60149}, {6646, 17596}
X(69248) = crosssum of X(1) and X(33863)
X(69248) = barycentric product X(i)*X(j) for these {i,j}: {10, 17596}, {37, 6646}, {42, 20955}, {321, 21008}, {1018, 21212}, {22161, 41013}, {52651, 62650}
X(69248) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 54120}, {6646, 274}, {17596, 86}, {20955, 310}, {21008, 81}, {21212, 7199}, {22161, 1444}, {62650, 8033}
X(69248) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {72, 20691, 20693}, {1334, 3721, 37}, {2292, 2295, 37}, {4646, 21874, 21904}, {9620, 12514, 4426}, {21879, 21888, 10}

X(69249) = (0, 1, 1, 1, 1, 1, 1, 1, 1, 1)-ADDITIVE ASSOCIATE OF X(40)

Barycentrics    a*(b + c)*(a^2 - a*b - b^2 - a*c + 2*b*c - c^2) : :
X(69249) = 3 X[40] - X[12497]

X(69249) lies on these lines: {1, 574}, {6, 12702}, {8, 2549}, {10, 115}, {32, 40}, {37, 3754}, {39, 517}, {46, 2242}, {65, 1500}, {72, 52959}, {165, 5206}, {172, 484}, {187, 3579}, {213, 4642}, {230, 61524}, {355, 7748}, {512, 4531}, {515, 7756}, {516, 7747}, {626, 69038}, {758, 20691}, {760, 12782}, {946, 1506}, {952, 63548}, {960, 1574}, {962, 2548}, {1015, 3057}, {1018, 3721}, {1334, 3125}, {1385, 31443}, {1482, 5013}, {1504, 35774}, {1505, 35775}, {1572, 7772}, {1573, 5836}, {1575, 3878}, {1829, 33843}, {1836, 9650}, {1837, 9664}, {1902, 3199}, {1914, 11010}, {2087, 23649}, {2092, 21853}, {2099, 31448}, {2238, 3987}, {2241, 5119}, {2275, 5697}, {2276, 5903}, {2295, 4424}, {2329, 69025}, {2362, 31471}, {2802, 17448}, {3055, 61272}, {3214, 21839}, {3230, 24443}, {3245, 5280}, {3294, 4674}, {3340, 31426}, {3501, 3735}, {3576, 15515}, {3617, 43448}, {3654, 5309}, {3678, 21868}, {3679, 11648}, {3727, 16549}, {3730, 3959}, {3753, 16589}, {3767, 5657}, {3774, 20718}, {3815, 22791}, {3884, 16604}, {3968, 25614}, {4067, 20693}, {4095, 22036}, {4286, 17444}, {4295, 31409}, {4301, 9698}, {4646, 20970}, {4663, 5107}, {5024, 8148}, {5058, 49226}, {5062, 49227}, {5252, 9651}, {5254, 5690}, {5286, 59417}, {5291, 56288}, {5475, 12699}, {5587, 69141}, {5603, 31401}, {5691, 65633}, {5734, 31450}, {5790, 44518}, {5886, 31455}, {6361, 7737}, {6684, 7749}, {6781, 31730}, {7187, 61187}, {7603, 9955}, {7738, 12245}, {7739, 50810}, {7745, 28174}, {7746, 26446}, {7753, 28194}, {7755, 43174}, {7765, 11362}, {7968, 62205}, {7969, 62206}, {7982, 9574}, {8227, 31441}, {8588, 35242}, {8589, 13624}, {9341, 63206}, {9592, 11531}, {9597, 12647}, {9598, 10573}, {9600, 44635}, {9623, 31442}, {9624, 31444}, {9665, 12701}, {9699, 49553}, {9780, 43620}, {9956, 39565}, {10039, 69175}, {10222, 31430}, {10246, 15815}, {11375, 31501}, {11522, 31428}, {12047, 31476}, {12778, 14901}, {14537, 28198}, {14923, 16975}, {15513, 31663}, {15955, 33863}, {16583, 21872}, {16969, 24046}, {17742, 36283}, {17750, 37598}, {17760, 35103}, {18357, 53419}, {18362, 19875}, {18424, 61261}, {18483, 43457}, {18493, 31489}, {18525, 44526}, {18991, 31437}, {18992, 62242}, {21796, 21871}, {21859, 45288}, {21863, 56926}, {22793, 39590}, {24170, 59512}, {24215, 36226}, {24918, 26660}, {25248, 69028}, {25270, 33932}, {31457, 61276}, {31459, 38235}, {31477, 68668}, {31490, 40587}, {37567, 54416}, {38042, 63534}, {39913, 64040}, {40663, 69096}, {41319, 54329}, {41869, 62203}, {48661, 65630}, {59415, 63537}

X(69249) = reflection of X(22036) in X(4095)
X(69249) = barycentric product X(37)*X(17276)
X(69249) = barycentric quotient X(17276)/X(274)
X(69249) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1571, 574}, {40, 9620, 32}, {946, 31398, 1506}, {1385, 31443, 37512}, {1572, 9593, 7772}, {3294, 4674, 21951}, {3576, 31422, 15515}, {3730, 3959, 49758}, {7982, 9574, 9619}, {7991, 9593, 1572}, {9574, 9619, 53096}, {16583, 21872, 52963}, {18991, 31437, 62241}

X(69250) = (1,1,-1,0,1,0,1)-ADDITIVE ASSOCIATE OF X(11)

Barycentrics    (b + c)*(-(a*b) + b^2 - a*c - b*c + c^2) : :

X(69250) lies on these lines: {1, 2}, {9, 6327}, {12, 52353}, {33, 27521}, {37, 4972}, {38, 3836}, {75, 32862}, {81, 33118}, {100, 4228}, {120, 40603}, {141, 4981}, {142, 17140}, {171, 33115}, {190, 20292}, {192, 33131}, {210, 3936}, {226, 3952}, {238, 33072}, {312, 33108}, {313, 1233}, {321, 3925}, {333, 33078}, {344, 3434}, {405, 5300}, {427, 56875}, {442, 1230}, {495, 4723}, {518, 18139}, {594, 48648}, {748, 4865}, {750, 4438}, {756, 2887}, {846, 32948}, {894, 33166}, {940, 33114}, {982, 25961}, {984, 17184}, {1001, 5014}, {1089, 3841}, {1211, 48647}, {1213, 46907}, {1376, 33113}, {1621, 32850}, {1738, 17147}, {1757, 32949}, {1826, 7378}, {1869, 6995}, {1962, 4085}, {2321, 17163}, {2475, 56311}, {2476, 46937}, {2533, 47771}, {2550, 17776}, {2886, 4358}, {2895, 60731}, {2899, 6871}, {2979, 25279}, {3060, 25308}, {3120, 3971}, {3175, 4442}, {3219, 4645}, {3266, 51861}, {3290, 21858}, {3295, 50715}, {3314, 21699}, {3416, 5278}, {3662, 7226}, {3666, 3823}, {3671, 52354}, {3681, 18134}, {3683, 4450}, {3685, 33110}, {3696, 3969}, {3699, 41878}, {3702, 31419}, {3703, 3826}, {3710, 17164}, {3712, 49732}, {3717, 5249}, {3740, 5741}, {3744, 24542}, {3751, 63056}, {3752, 24988}, {3755, 27804}, {3773, 21020}, {3790, 28605}, {3821, 3989}, {3822, 3992}, {3838, 4009}, {3842, 28595}, {3869, 26911}, {3873, 17234}, {3883, 28599}, {3891, 24789}, {3914, 3995}, {3944, 64178}, {3980, 33161}, {3994, 48643}, {4011, 33104}, {4036, 47808}, {4054, 4082}, {4071, 59207}, {4096, 4892}, {4104, 31037}, {4135, 48642}, {4136, 21921}, {4153, 46196}, {4197, 4385}, {4233, 5174}, {4312, 25734}, {4365, 6541}, {4383, 33070}, {4388, 27065}, {4415, 48646}, {4416, 20290}, {4417, 63961}, {4418, 33164}, {4422, 63979}, {4427, 56078}, {4429, 28606}, {4472, 50265}, {4514, 5284}, {4518, 30690}, {4663, 42045}, {4670, 50261}, {4690, 50274}, {4696, 25466}, {4703, 31134}, {4705, 44435}, {4884, 40688}, {4901, 41867}, {4968, 8728}, {5015, 5047}, {5086, 7474}, {5133, 44411}, {5220, 32859}, {5260, 7270}, {5263, 33157}, {5294, 64174}, {5640, 25306}, {5687, 68699}, {5739, 38057}, {5847, 19742}, {5880, 32933}, {5905, 27549}, {5996, 8034}, {6535, 21027}, {7179, 30636}, {7206, 42031}, {7413, 24808}, {7485, 52139}, {8270, 28776}, {9330, 25958}, {9335, 56192}, {9708, 16353}, {9709, 16352}, {11346, 66639}, {11680, 18743}, {11681, 20921}, {12609, 56318}, {17056, 46897}, {17122, 33119}, {17123, 32844}, {17127, 50289}, {17261, 33100}, {17265, 17597}, {17277, 33075}, {17279, 24552}, {17282, 62833}, {17469, 50288}, {17483, 62222}, {17491, 17781}, {17598, 31252}, {17605, 30566}, {17674, 37592}, {17740, 26040}, {17792, 30056}, {17889, 32925}, {18082, 39668}, {18141, 64153}, {19684, 38047}, {19743, 59408}, {19804, 33089}, {20068, 24231}, {20486, 59212}, {20905, 25973}, {20930, 30758}, {21241, 59517}, {21282, 25101}, {21666, 25985}, {21711, 65695}, {21714, 48182}, {21926, 22016}, {21946, 40599}, {22276, 22294}, {22278, 61172}, {23682, 31036}, {24169, 46901}, {24210, 31035}, {24325, 33162}, {24349, 27186}, {24589, 69091}, {24715, 32936}, {24723, 33761}, {25959, 27184}, {26061, 50302}, {26242, 56926}, {26262, 35996}, {26724, 32922}, {27064, 33112}, {27538, 31053}, {28741, 34036}, {28996, 34029}, {31019, 32937}, {31151, 33067}, {32771, 33165}, {32772, 33159}, {32846, 32864}, {32849, 32932}, {32856, 42054}, {32860, 33092}, {32865, 32915}, {32911, 33073}, {32917, 33079}, {32926, 33129}, {32927, 33130}, {32928, 33132}, {32930, 33109}, {32931, 33111}, {32938, 33097}, {32945, 33158}, {33068, 62796}, {33069, 49448}, {33071, 37680}, {33081, 49457}, {33086, 38000}, {33105, 59511}, {33121, 37633}, {33134, 41839}, {33145, 49456}, {33146, 49447}, {37353, 39583}, {37693, 59666}, {38200, 63131}, {42033, 64010}, {46909, 69092}, {47798, 57099}, {48652, 50312}, {48800, 50714}, {48806, 50793}, {50256, 64073}, {56246, 62265}, {61155, 63139}, {63060, 67964}

X(69250) = isotomic conjugate of the isogonal conjugate of X(22277)
X(69250) = X(i)-Ceva conjugate of X(j) for these (i,j): {6063, 321}, {17234, 3970}, {32019, 37}, {55082, 22012}
X(69250) = X(1333)-isoconjugate of X(60075)
X(69250) = X(i)-Dao conjugate of X(j) for these (i,j): {37, 60075}, {210, 55}, {17059, 21789}, {52594, 17197}
X(69250) = crosspoint of X(17234) and X(33933)
X(69250) = barycentric product X(i)*X(j) for these {i,j}: {10, 17234}, {37, 33933}, {75, 3970}, {76, 22277}, {313, 4253}, {321, 3873}, {349, 64739}, {1441, 25082}, {1502, 61038}, {3701, 17092}, {3941, 27801}, {3952, 47676}, {4033, 4905}, {4998, 21946}, {6063, 40599}, {27827, 52353}
X(69250) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 60075}, {3873, 81}, {3941, 1333}, {3970, 1}, {4253, 58}, {4905, 1019}, {17059, 17197}, {17092, 1014}, {17234, 86}, {21946, 11}, {22277, 6}, {25082, 21}, {33933, 274}, {40599, 55}, {47676, 7192}, {52594, 21789}, {61038, 32}, {64739, 284}, {65697, 7252}
X(69250) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3006, 69134}, {2, 3920, 26230}, {2, 10327, 26227}, {2, 19993, 16020}, {2, 29641, 3006}, {2, 29679, 26251}, {2, 29832, 614}, {2, 29838, 29871}, {2, 29840, 7292}, {2, 33090, 16823}, {2, 33091, 3757}, {2, 39570, 10327}, {2, 60459, 7081}, {10, 306, 4651}, {10, 3178, 3214}, {10, 15523, 56810}, {10, 29653, 42}, {142, 63147, 17140}, {171, 33115, 56520}, {756, 2887, 26580}, {756, 21026, 2887}, {984, 25957, 17184}, {2550, 17776, 32929}, {3175, 21949, 4442}, {3703, 3826, 4359}, {3717, 5249, 17165}, {3720, 29673, 29835}, {3741, 49769, 29687}, {3912, 25006, 17135}, {3914, 4078, 3995}, {3920, 29871, 29838}, {3925, 3932, 321}, {4514, 17263, 5284}, {5268, 29857, 2}, {5297, 29873, 2}, {5311, 25453, 29833}, {6535, 21027, 62226}, {16569, 29849, 62620}, {17605, 59506, 30566}, {21241, 59517, 69173}, {24239, 60423, 2}, {26723, 49476, 17150}, {29639, 62673, 2}, {29838, 29871, 26230}, {29854, 33117, 1}

X(69251) = (1,-1,1,0,1,0,-1)-ADDITIVE ASSOCIATE OF X(11)

Barycentrics    a*b^2 + b^3 - 2*a*b*c + a*c^2 + c^3 : :

X(69251) lies on these lines: {1, 32948}, {2, 7}, {10, 17140}, {11, 48646}, {38, 3836}, {42, 24169}, {43, 33069}, {72, 17674}, {75, 33172}, {76, 65049}, {81, 16706}, {88, 30831}, {100, 33124}, {141, 4359}, {171, 26230}, {210, 24988}, {238, 33067}, {239, 24790}, {244, 2887}, {306, 17495}, {312, 33146}, {313, 40013}, {320, 32911}, {321, 1086}, {333, 26724}, {354, 4972}, {379, 26634}, {614, 6327}, {726, 29687}, {748, 4655}, {750, 26128}, {846, 29851}, {899, 33064}, {902, 29672}, {940, 17290}, {942, 4202}, {982, 3006}, {984, 25961}, {1001, 32950}, {1054, 29846}, {1125, 16347}, {1150, 24789}, {1211, 24589}, {1233, 16727}, {1255, 17320}, {1279, 4450}, {1376, 33122}, {1621, 33068}, {1738, 17135}, {1943, 37771}, {1999, 33150}, {2895, 17288}, {2999, 31034}, {3008, 4001}, {3120, 3840}, {3187, 4000}, {3210, 17232}, {3315, 4514}, {3419, 17679}, {3550, 29638}, {3578, 17348}, {3619, 19822}, {3649, 25914}, {3661, 24190}, {3663, 3995}, {3664, 19717}, {3666, 3834}, {3670, 57808}, {3677, 29832}, {3685, 33102}, {3687, 31017}, {3705, 25959}, {3717, 20068}, {3720, 3821}, {3721, 8041}, {3729, 63584}, {3752, 3936}, {3757, 33086}, {3782, 4358}, {3826, 4981}, {3868, 33833}, {3873, 4429}, {3879, 45222}, {3891, 50000}, {3896, 4966}, {3912, 17147}, {3914, 29824}, {3923, 29677}, {3925, 46909}, {3944, 30957}, {3952, 62673}, {3969, 17231}, {3980, 24943}, {4011, 33098}, {4085, 62867}, {4292, 11319}, {4298, 25904}, {4310, 10327}, {4383, 7232}, {4388, 7292}, {4392, 29641}, {4398, 42044}, {4414, 29642}, {4417, 62620}, {4418, 29637}, {4425, 30950}, {4640, 24542}, {4641, 17356}, {4645, 7191}, {4651, 49511}, {4667, 19743}, {4675, 19684}, {4683, 17123}, {4684, 20011}, {4703, 17125}, {4850, 18134}, {4859, 5271}, {4862, 56082}, {4871, 69173}, {4980, 7263}, {4987, 21764}, {5014, 17597}, {5016, 17054}, {5051, 5439}, {5192, 57282}, {5205, 33153}, {5256, 17298}, {5269, 29831}, {5278, 17278}, {5284, 24723}, {5287, 17304}, {5708, 56780}, {5741, 16610}, {5880, 24552}, {6384, 30632}, {6535, 50117}, {6536, 25501}, {7081, 33148}, {7238, 41241}, {7290, 20064}, {8025, 17023}, {8040, 19862}, {8616, 29853}, {9335, 25958}, {10453, 33131}, {11220, 36652}, {11263, 19864}, {12047, 26094}, {12436, 19284}, {12527, 25967}, {13407, 26030}, {14829, 33129}, {15523, 24165}, {15803, 56781}, {16569, 33065}, {16602, 24183}, {16704, 26723}, {16823, 33083}, {16825, 33080}, {16891, 17198}, {17011, 17300}, {17012, 17778}, {17019, 17302}, {17020, 62998}, {17024, 50289}, {17063, 25760}, {17121, 20086}, {17122, 32775}, {17154, 63147}, {17155, 29674}, {17165, 24231}, {17205, 39747}, {17227, 19804}, {17233, 50106}, {17234, 28606}, {17235, 44307}, {17237, 41809}, {17263, 33761}, {17272, 63100}, {17279, 32933}, {17283, 32939}, {17284, 63583}, {17296, 20017}, {17313, 20182}, {17317, 62851}, {17364, 63074}, {17367, 37685}, {17376, 42045}, {17380, 62801}, {17382, 37595}, {17449, 29673}, {17490, 33077}, {17591, 29643}, {17594, 29830}, {17595, 33113}, {17596, 29632}, {17598, 31151}, {17676, 54392}, {17690, 57287}, {17751, 23536}, {17763, 33147}, {17766, 29818}, {17776, 53665}, {17863, 50320}, {17889, 30942}, {18141, 19785}, {18193, 29857}, {18201, 33119}, {18743, 33151}, {19786, 37633}, {19789, 34255}, {19825, 29611}, {20255, 52043}, {20290, 68958}, {20292, 32942}, {20880, 50319}, {20892, 28654}, {20913, 21240}, {21026, 42038}, {21296, 63009}, {24167, 30171}, {24175, 31037}, {24214, 29988}, {24325, 32781}, {24349, 29679}, {24592, 62675}, {24593, 37646}, {24690, 25357}, {24715, 32943}, {25498, 41818}, {25536, 40592}, {26085, 69224}, {26102, 32776}, {26115, 51706}, {26150, 29648}, {26251, 32771}, {26526, 26527}, {26567, 64967}, {26571, 47676}, {26582, 26593}, {26978, 41233}, {27145, 34830}, {27521, 54408}, {27627, 56949}, {27633, 29984}, {28595, 42053}, {28605, 48627}, {28639, 41820}, {29615, 41821}, {29636, 37604}, {29649, 33143}, {29653, 46901}, {29668, 33104}, {29684, 33682}, {29820, 32947}, {29821, 32949}, {29848, 56010}, {29850, 32913}, {29852, 62841}, {29960, 62636}, {29982, 53476}, {30631, 62234}, {30713, 62304}, {30821, 53600}, {31025, 63589}, {31191, 62240}, {32773, 64149}, {32845, 33158}, {32846, 32924}, {32850, 62814}, {32856, 59511}, {32857, 32930}, {32860, 33087}, {32914, 33085}, {32915, 33149}, {32918, 33130}, {32919, 33132}, {32922, 33078}, {32923, 33079}, {32931, 33103}, {32932, 33173}, {32940, 33159}, {32944, 33097}, {33066, 37680}, {33117, 62865}, {33118, 62235}, {33162, 42055}, {33168, 62300}, {33826, 40744}, {36505, 37603}, {37545, 56779}, {37582, 56778}, {37639, 40940}, {48381, 54284}, {48647, 69091}, {56983, 64002}

X(69251) = isotomic conjugate of X(55990)
X(69251) = X(31)-isoconjugate of X(55990)
X(69251) = X(2)-Dao conjugate of X(55990)
X(69251) = crosspoint of X(85) and X(40010)
X(69251) = barycentric product X(75)*X(3953)
X(69251) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 55990}, {3953, 1}
X(69251) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7, 26223}, {2, 329, 26688}, {2, 3218, 56520}, {2, 3662, 17184}, {2, 6646, 27065}, {2, 9776, 26627}, {2, 17184, 26580}, {2, 17483, 27064}, {2, 20078, 26685}, {2, 26840, 3219}, {2, 26842, 894}, {57, 69051, 56559}, {63, 17282, 2}, {141, 4359, 56810}, {141, 40688, 4359}, {142, 54311, 2}, {171, 33123, 26230}, {244, 2887, 69134}, {306, 24177, 17495}, {312, 48629, 33146}, {333, 27191, 26724}, {354, 4972, 29835}, {940, 17290, 32774}, {940, 32774, 29833}, {982, 25957, 3006}, {1086, 69092, 321}, {3008, 4001, 19742}, {3210, 17232, 32858}, {3306, 25527, 2}, {3666, 3834, 18139}, {4383, 7232, 32859}, {5256, 17298, 63056}, {17227, 19804, 32782}, {17231, 42051, 3969}, {17283, 32939, 33157}, {17598, 31151, 33072}, {21255, 24177, 306}, {24169, 49676, 42}, {24214, 29988, 31036}, {26561, 26562, 26526}, {32771, 33174, 26251}

X(69252) = (0,1,1,-1,0,-1,1)-ADDITIVE ASSOCIATE OF X(11)

Barycentrics    b^3 + 2*a*b*c + b^2*c + b*c^2 + c^3 : :

X(69252) lies on these lines: {1, 2}, {9, 33161}, {11, 594}, {31, 3966}, {37, 32848}, {38, 1211}, {48, 17275}, {57, 33080}, {75, 3120}, {81, 32861}, {100, 33076}, {141, 244}, {149, 46918}, {171, 33075}, {199, 23361}, {210, 33162}, {238, 32779}, {274, 30984}, {312, 6535}, {319, 32919}, {321, 3846}, {333, 2206}, {354, 33081}, {599, 4860}, {726, 26580}, {748, 32777}, {750, 3416}, {756, 3703}, {846, 33168}, {894, 32843}, {902, 3883}, {940, 32852}, {982, 32782}, {984, 33089}, {1001, 33156}, {1054, 33086}, {1150, 50308}, {1215, 5741}, {1376, 5078}, {1468, 5814}, {1491, 29144}, {1621, 33160}, {1654, 16556}, {1757, 33170}, {1930, 3266}, {1985, 21029}, {2183, 5282}, {2228, 3116}, {2229, 9284}, {2246, 17330}, {2886, 21020}, {2887, 4359}, {2895, 32913}, {3210, 32776}, {3218, 33082}, {3219, 33167}, {3264, 33931}, {3681, 33169}, {3696, 33136}, {3711, 59407}, {3752, 32781}, {3756, 48635}, {3773, 4358}, {3775, 46909}, {3790, 64178}, {3821, 17495}, {3826, 21026}, {3836, 24589}, {3844, 16610}, {3873, 33084}, {3932, 5241}, {3936, 24325}, {3944, 28605}, {3980, 6327}, {4003, 17237}, {4023, 21805}, {4026, 46904}, {4104, 63147}, {4109, 26035}, {4141, 50093}, {4144, 17369}, {4357, 46901}, {4361, 33128}, {4363, 24725}, {4365, 24210}, {4383, 26061}, {4388, 4418}, {4414, 17740}, {4417, 32771}, {4425, 17147}, {4438, 5278}, {4514, 32945}, {4643, 36263}, {4679, 17281}, {4683, 32939}, {4703, 32933}, {4710, 59712}, {4850, 32784}, {4851, 9345}, {4884, 42039}, {4886, 32864}, {4933, 49740}, {4966, 17450}, {4980, 48643}, {5233, 32931}, {5263, 32844}, {5284, 33158}, {5739, 32912}, {5880, 31134}, {6536, 28606}, {6541, 31035}, {7018, 40087}, {10009, 30632}, {14555, 33163}, {15254, 50104}, {16887, 69078}, {17063, 33172}, {17118, 61716}, {17122, 33078}, {17123, 33157}, {17125, 17279}, {17140, 31037}, {17155, 27184}, {17184, 24165}, {17277, 33115}, {17285, 25531}, {17289, 32944}, {17303, 17723}, {17449, 49511}, {17490, 33125}, {17596, 33083}, {17889, 25958}, {19786, 32924}, {19804, 25957}, {19808, 32772}, {19822, 26098}, {20333, 30955}, {21027, 33108}, {21140, 33934}, {21242, 50312}, {21911, 31245}, {24342, 33112}, {24349, 33065}, {24697, 62796}, {24703, 50048}, {24723, 32845}, {24789, 31237}, {25385, 31025}, {26010, 26591}, {27003, 33085}, {27065, 33164}, {30831, 33130}, {30832, 32775}, {30969, 60706}, {31017, 49676}, {31143, 62235}, {31177, 49727}, {31264, 37662}, {31993, 33105}, {32773, 32860}, {32780, 32911}, {32846, 37633}, {32851, 32917}, {32856, 49483}, {32923, 33126}, {32932, 32947}, {32940, 33066}, {32946, 64164}, {33070, 50302}, {33087, 64149}, {33095, 64010}, {33104, 50314}, {33134, 49474}, {33145, 42051}, {33151, 49493}, {33154, 50106}, {33159, 37680}, {33165, 63961}, {33299, 37315}, {41002, 44416}, {43534, 60097}, {50296, 62838}, {57040, 68939}, {62846, 67964}, {67598, 68845}

X(69252) = X(513)-isoconjugate of X(29313)
X(69252) = X(39026)-Dao conjugate of X(29313)
X(69252) = barycentric product X(190)*X(29312)
X(69252) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 29313}, {29312, 514}
X(69252) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 33077, 4062}, {2, 8, 17763}, {2, 4362, 29683}, {2, 15523, 29687}, {2, 17017, 68945}, {2, 17025, 29646}, {2, 17150, 29645}, {2, 29671, 29682}, {2, 29821, 29684}, {2, 29849, 29688}, {2, 32778, 15523}, {2, 32842, 1}, {2, 33088, 5311}, {2, 33093, 1961}, {10, 29639, 30970}, {75, 25760, 3120}, {321, 3846, 69173}, {894, 32843, 61707}, {1211, 69091, 38}, {3703, 5743, 756}, {3705, 31330, 29690}, {3944, 28605, 48642}, {4023, 49524, 21805}, {4886, 33121, 32864}, {7191, 32783, 29686}, {8013, 29690, 31330}, {17140, 31037, 33064}, {17740, 50295, 4414}, {19808, 33071, 32772}, {21085, 29655, 17135}, {24589, 48647, 3836}, {30832, 32922, 32775}, {33170, 37656, 1757}, {56810, 69134, 3741}

X(69253) = (0,1,-1,1,0,1,1)-ADDITIVE ASSOCIATE OF X(11)

Barycentrics     b^3 - 2*a*b*c - b^2*c - b*c^2 + c^3 : :

X(69253) lies on these lines: {1, 27186}, {2, 846}, {6, 64164}, {7, 32912}, {8, 33069}, {9, 33098}, {10, 17165}, {31, 5880}, {37, 33145}, {38, 1086}, {42, 1738}, {43, 31019}, {46, 27661}, {55, 29689}, {57, 24892}, {75, 15523}, {81, 33132}, {100, 33130}, {141, 21020}, {142, 3720}, {149, 29820}, {171, 33129}, {192, 29854}, {210, 32856}, {226, 899}, {238, 20292}, {239, 32949}, {244, 2886}, {274, 16891}, {310, 30631}, {312, 25961}, {320, 32864}, {321, 3836}, {333, 33067}, {334, 40087}, {354, 21949}, {377, 3924}, {404, 24161}, {442, 20966}, {443, 60751}, {495, 4695}, {612, 23681}, {614, 4859}, {740, 18139}, {748, 1836}, {750, 3772}, {756, 3782}, {894, 29850}, {896, 11246}, {940, 33128}, {946, 28352}, {968, 41867}, {976, 24159}, {982, 29690}, {984, 33146}, {986, 4197}, {1001, 33094}, {1125, 19284}, {1193, 12609}, {1201, 24178}, {1215, 25351}, {1376, 17783}, {1621, 24715}, {1647, 11680}, {1739, 3822}, {1757, 17483}, {1961, 33155}, {1962, 66071}, {2210, 24588}, {2292, 8728}, {2308, 26723}, {2476, 24174}, {2550, 3938}, {2887, 4359}, {3006, 24165}, {3008, 41011}, {3125, 20859}, {3210, 29643}, {3214, 13407}, {3216, 11263}, {3218, 33138}, {3219, 32857}, {3306, 17064}, {3336, 24880}, {3454, 28611}, {3475, 67207}, {3550, 29681}, {3662, 21027}, {3663, 3989}, {3666, 29682}, {3670, 3841}, {3681, 33103}, {3685, 29851}, {3696, 33081}, {3703, 7263}, {3706, 3834}, {3722, 34612}, {3724, 16056}, {3752, 33105}, {3757, 32948}, {3773, 4980}, {3812, 21935}, {3813, 46190}, {3838, 16610}, {3846, 24589}, {3873, 32865}, {3912, 4365}, {3920, 33147}, {3961, 33148}, {3966, 31134}, {4000, 17017}, {4042, 7232}, {4054, 62673}, {4062, 18134}, {4137, 18698}, {4202, 49598}, {4256, 26725}, {4357, 59306}, {4358, 48643}, {4361, 32852}, {4363, 26061}, {4383, 24725}, {4387, 17265}, {4429, 32771}, {4642, 25466}, {4645, 32914}, {4651, 33064}, {4655, 5278}, {4656, 38204}, {4675, 62821}, {4682, 50103}, {4683, 17277}, {4722, 17365}, {4847, 17449}, {4850, 29688}, {4854, 17245}, {4863, 62869}, {4892, 5741}, {4972, 24325}, {5047, 24851}, {5057, 17123}, {5254, 21921}, {5263, 29686}, {5271, 33080}, {5284, 33095}, {5293, 26060}, {5297, 33152}, {5311, 19785}, {5712, 67211}, {5883, 68946}, {6173, 62819}, {6175, 37717}, {6327, 16825}, {6358, 17888}, {6535, 28605}, {6701, 37693}, {7191, 33109}, {7292, 33106}, {7321, 32940}, {8013, 32782}, {9316, 37695}, {9776, 11269}, {9782, 24883}, {10171, 60414}, {10436, 29647}, {10459, 23536}, {12047, 27627}, {16062, 27714}, {16569, 31053}, {16602, 17605}, {16706, 29684}, {16823, 32947}, {17050, 21352}, {17056, 46904}, {17070, 37634}, {17122, 33133}, {17124, 17720}, {17125, 24703}, {17135, 49676}, {17140, 29673}, {17147, 29653}, {17155, 29641}, {17156, 17298}, {17163, 49560}, {17173, 18792}, {17234, 32915}, {17282, 29677}, {17490, 29849}, {17495, 29671}, {17529, 63997}, {17591, 29664}, {17594, 29661}, {17715, 49719}, {17724, 49732}, {17726, 59477}, {17770, 19742}, {17861, 21717}, {17862, 25970}, {17871, 21931}, {19717, 23812}, {19796, 32928}, {19804, 25760}, {20068, 53601}, {20257, 68751}, {20271, 21029}, {20590, 20880}, {20653, 28612}, {21085, 31017}, {21241, 31075}, {21330, 21926}, {21416, 39998}, {21616, 28257}, {23537, 59305}, {24176, 30171}, {24177, 29639}, {24210, 30950}, {24231, 25006}, {24254, 25468}, {24330, 25357}, {24349, 33117}, {24430, 68914}, {24693, 26128}, {24943, 50314}, {24988, 59511}, {25348, 28283}, {25525, 29678}, {25557, 62867}, {25639, 28096}, {25959, 32778}, {26037, 27184}, {26102, 33134}, {26627, 29635}, {26727, 59416}, {26842, 32913}, {27003, 33140}, {27065, 33099}, {27131, 62711}, {27191, 32942}, {27458, 56663}, {27577, 41862}, {27690, 49609}, {28074, 31418}, {28606, 33149}, {29632, 32932}, {29642, 32929}, {29665, 56010}, {29816, 64174}, {29821, 33112}, {29862, 33168}, {29869, 59692}, {29873, 33167}, {30379, 61376}, {30970, 54311}, {31151, 33078}, {31993, 32781}, {32774, 50302}, {32845, 33116}, {32848, 42051}, {32850, 32923}, {32858, 49474}, {32862, 49493}, {32911, 33097}, {32917, 33068}, {32920, 49996}, {32922, 33072}, {32924, 33073}, {32939, 33115}, {32945, 33124}, {33065, 59296}, {33092, 50106}, {33096, 37680}, {33101, 63961}, {33135, 37633}, {33141, 64149}, {33158, 64010}, {33162, 49483}, {33163, 42697}, {37445, 68930}, {37639, 50755}, {37664, 69015}, {37679, 61716}, {42045, 49489}, {49488, 63056}, {49991, 59730}, {50238, 63360}, {50301, 62807}, {50759, 54310}, {57808, 67983}

X(69253) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3120, 69173}, {2, 17889, 3120}, {2, 32776, 6536}, {2, 33102, 846}, {75, 25957, 15523}, {142, 3914, 3720}, {321, 3836, 29687}, {354, 21949, 33136}, {612, 23681, 33143}, {750, 3772, 29683}, {982, 33108, 29690}, {986, 4197, 21674}, {1086, 3925, 38}, {1738, 5249, 42}, {1836, 17278, 748}, {2886, 40688, 244}, {3306, 17064, 29662}, {3782, 3826, 756}, {4850, 33111, 29688}, {4972, 24325, 29685}, {5263, 33123, 29686}, {5880, 24789, 31}, {7321, 33118, 32940}, {11680, 17063, 1647}, {16706, 32772, 29684}, {17050, 23682, 21352}, {18134, 32860, 4062}, {20292, 26724, 238}, {23681, 38052, 612}, {24589, 48646, 3846}, {25959, 32778, 48650}, {26723, 50307, 2308}, {26842, 33139, 32913}, {27186, 33131, 1}, {28605, 29674, 6535}, {29641, 48627, 17155}, {32774, 50302, 68945}, {32911, 33097, 61707}, {33073, 37756, 32924}

X(69254) = (1, 1, -1, 1, 0, 1)-ADDITIVE ASSOCIATE OF X(12)

Barycentrics     a^2*b^2 - b^4 - 2*a^2*b*c + a^2*c^2 - c^4 : :

X(69254) lies on these lines: {1, 325}, {2, 16502}, {5, 1909}, {8, 4561}, {11, 76}, {12, 7752}, {32, 26686}, {36, 7750}, {39, 26590}, {55, 7763}, {56, 315}, {69, 613}, {75, 17181}, {99, 6284}, {141, 26959}, {172, 7762}, {183, 499}, {192, 7906}, {194, 69096}, {202, 69137}, {203, 69145}, {274, 2886}, {304, 3705}, {316, 7354}, {317, 11399}, {330, 5025}, {333, 23150}, {348, 65188}, {350, 496}, {388, 32816}, {390, 32831}, {404, 20553}, {442, 31997}, {491, 1124}, {492, 1335}, {495, 25303}, {497, 3926}, {614, 45201}, {625, 69175}, {626, 1015}, {668, 1329}, {999, 7776}, {1007, 3085}, {1056, 32823}, {1058, 32818}, {1078, 5433}, {1125, 37664}, {1201, 24995}, {1475, 4766}, {1478, 7773}, {1479, 1975}, {1500, 7764}, {1506, 69136}, {1565, 33930}, {1776, 55419}, {1914, 7807}, {2241, 3788}, {2242, 7759}, {2275, 6656}, {3056, 6393}, {3057, 69038}, {3058, 7799}, {3295, 69158}, {3314, 69097}, {3582, 37671}, {3583, 32819}, {3665, 33940}, {3703, 33939}, {3734, 9665}, {3760, 37720}, {3761, 7741}, {3765, 26019}, {3785, 7288}, {3813, 17143}, {3815, 27020}, {3816, 18140}, {3820, 25280}, {3825, 6381}, {3953, 17211}, {4187, 6376}, {4293, 32006}, {4294, 6337}, {4366, 7836}, {4386, 17694}, {4388, 17206}, {4514, 17095}, {4857, 32820}, {4972, 27162}, {5204, 14907}, {5218, 32829}, {5225, 32815}, {5229, 32827}, {5274, 32830}, {5280, 41624}, {5281, 32835}, {5298, 7811}, {5299, 7792}, {5432, 7769}, {5434, 7809}, {6390, 15171}, {6645, 7785}, {7005, 69157}, {7006, 69165}, {7018, 20256}, {7081, 54443}, {7082, 55417}, {7179, 39731}, {7767, 15325}, {7770, 9599}, {7774, 54416}, {7775, 9650}, {7777, 31460}, {7778, 16781}, {7781, 9664}, {7782, 15338}, {7788, 10072}, {7796, 37722}, {7802, 15326}, {7814, 15888}, {7821, 69174}, {7825, 9651}, {7835, 53680}, {7841, 9597}, {9598, 31859}, {10385, 32837}, {10527, 45962}, {10589, 32828}, {10593, 64093}, {10896, 11185}, {11238, 32833}, {11680, 34284}, {14210, 30171}, {14986, 37668}, {16067, 40071}, {16604, 17670}, {16705, 69076}, {16784, 30104}, {16893, 63505}, {16975, 26558}, {16992, 26363}, {17045, 30137}, {17046, 30038}, {17137, 69083}, {17362, 30167}, {17390, 30138}, {17669, 21226}, {17757, 24524}, {18982, 39266}, {19974, 20255}, {20487, 59518}, {20888, 24387}, {20911, 69134}, {21219, 33061}, {21746, 51370}, {24166, 65116}, {24241, 49613}, {25598, 30148}, {26601, 62803}, {26770, 31058}, {27255, 37661}, {28773, 56913}, {31402, 62988}, {31419, 60706}, {33046, 41838}, {33935, 69091}, {34063, 64172}, {45198, 55392}, {51369, 67967}

X(69254) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 325, 69135}, {11, 69094, 76}, {330, 5025, 69098}, {496, 3933, 350}, {626, 1015, 26561}, {2241, 3788, 26629}, {3761, 7741, 59635}, {5299, 30103, 7792}, {7752, 64133, 12}, {16604, 20541, 17670}

X(69255) = (1, 0, 1, 1, -1, 0)-ADDITIVE ASSOCIATE OF X(12)

Barycentrics     a^2*b^2 + 2*a^2*b*c + a^2*c^2 - 2*b^2*c^2 : :

X(69255) lies on these lines: {1, 538}, {2, 32026}, {10, 536}, {11, 7764}, {12, 63924}, {35, 4396}, {37, 20888}, {39, 350}, {42, 4721}, {55, 7751}, {56, 7781}, {75, 16589}, {76, 192}, {115, 69135}, {141, 21070}, {145, 33908}, {172, 7816}, {183, 31451}, {190, 17034}, {194, 1015}, {315, 9664}, {386, 4713}, {495, 63923}, {497, 7758}, {524, 15171}, {527, 35633}, {543, 7354}, {626, 69093}, {716, 49462}, {736, 13077}, {742, 2901}, {754, 6284}, {1089, 24326}, {1111, 4099}, {1266, 29968}, {1278, 27269}, {1479, 7759}, {1573, 1655}, {1574, 17759}, {1914, 7805}, {1930, 4037}, {1975, 2242}, {2241, 7754}, {2275, 32450}, {2276, 3760}, {3027, 47025}, {3085, 63955}, {3086, 34511}, {3175, 3782}, {3210, 30830}, {3216, 4465}, {3295, 63933}, {3501, 55998}, {3552, 9341}, {3583, 7843}, {3585, 63922}, {3644, 6376}, {3661, 42044}, {3663, 21071}, {3666, 4044}, {3729, 17750}, {3730, 17262}, {3734, 54416}, {3746, 4400}, {3797, 33940}, {3849, 65134}, {3874, 68870}, {3943, 40006}, {3948, 17147}, {3995, 20913}, {4045, 69097}, {4260, 64871}, {4294, 14023}, {4302, 63935}, {4309, 63934}, {4330, 63930}, {4360, 17499}, {4366, 7760}, {4398, 24190}, {4419, 10449}, {4441, 5283}, {4479, 17030}, {4664, 27255}, {4718, 6381}, {4726, 52708}, {4890, 17157}, {5254, 69174}, {5280, 7804}, {6646, 33297}, {7191, 19568}, {7230, 33931}, {7755, 26629}, {7757, 32107}, {7761, 9598}, {7765, 26561}, {7766, 53680}, {7774, 9665}, {7775, 10896}, {7786, 30998}, {7794, 26590}, {7798, 16502}, {7815, 31448}, {7817, 30104}, {7863, 26686}, {7880, 30103}, {8149, 10079}, {8667, 64951}, {9055, 24068}, {9466, 27020}, {9650, 11185}, {9654, 34505}, {9668, 63932}, {9669, 9766}, {10386, 63926}, {10895, 18546}, {11055, 32005}, {12782, 49445}, {12953, 63931}, {13466, 53675}, {15271, 31461}, {15325, 59546}, {15488, 29069}, {16781, 22253}, {16818, 48860}, {17033, 52963}, {17053, 56023}, {17243, 17758}, {17246, 21024}, {17301, 30107}, {17489, 64223}, {18135, 27076}, {18145, 26752}, {19858, 50174}, {20081, 64133}, {20170, 34283}, {20255, 68938}, {20331, 29438}, {20683, 32925}, {20688, 32453}, {20963, 56024}, {20970, 24514}, {21264, 25092}, {24603, 42051}, {25345, 36250}, {25349, 50605}, {26978, 68971}, {27248, 48840}, {27299, 50101}, {29571, 35652}, {29576, 50106}, {29983, 56185}, {31476, 59635}, {31501, 32832}, {32092, 36812}, {34506, 52793}, {41312, 50163}, {46893, 59325}, {47286, 69175}, {49763, 50122}, {50068, 50162}, {51122, 67261}, {63273, 63927}, {64054, 64781}

X(69255) = crosspoint of X(31625) and X(54118)
X(69255) = crosssum of X(1977) and X(21007)
X(69255) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {35, 4396, 7780}, {76, 192, 1500}, {76, 1500, 69136}, {350, 25264, 39}, {1655, 17143, 1573}, {2276, 3760, 3934}, {3663, 21071, 21240}, {17759, 18140, 1574}, {69093, 69096, 626}

X(69256) = (1, 0, -1, 1, 1, 0)-ADDITIVE ASSOCIATE OF X(12)

Barycentrics     a^2*b^2 - 2*a^2*b*c + a^2*c^2 + 2*b^2*c^2 : :

X(69256) lies on these lines: {1, 3934}, {2, 1500}, {5, 69174}, {6, 29748}, {8, 27076}, {10, 20530}, {11, 626}, {36, 7816}, {39, 350}, {55, 7815}, {56, 3734}, {75, 20688}, {76, 330}, {115, 26561}, {141, 496}, {172, 7804}, {183, 2241}, {192, 7786}, {202, 69138}, {203, 69146}, {239, 30819}, {315, 9665}, {497, 7800}, {499, 3788}, {538, 2275}, {620, 3027}, {625, 7741}, {712, 24172}, {999, 69139}, {1078, 4366}, {1107, 41144}, {1125, 21264}, {1475, 4721}, {1479, 7761}, {1506, 69135}, {1573, 18140}, {1909, 9466}, {1914, 7780}, {1930, 27918}, {2092, 25505}, {2176, 29455}, {2238, 29742}, {2242, 7770}, {2276, 6683}, {2295, 29438}, {3086, 7795}, {3244, 25102}, {3295, 15271}, {3582, 7880}, {3583, 7842}, {3624, 30571}, {3626, 25107}, {3636, 24656}, {3739, 3993}, {3761, 63493}, {3840, 21240}, {3875, 44418}, {3972, 9341}, {4000, 25492}, {4045, 69096}, {4253, 4713}, {4361, 17749}, {4396, 5299}, {4400, 16784}, {4405, 59738}, {4441, 40479}, {4465, 16552}, {4871, 17050}, {4975, 24786}, {5149, 10069}, {5204, 69171}, {5291, 17541}, {6284, 7830}, {6292, 26590}, {6381, 17448}, {7005, 69159}, {7006, 69166}, {7191, 8891}, {7264, 16720}, {7280, 32456}, {7288, 69206}, {7292, 30749}, {7352, 59556}, {7749, 26629}, {7751, 16502}, {7759, 9599}, {7764, 69093}, {7784, 9669}, {7789, 15325}, {7791, 9664}, {7793, 53680}, {7808, 54416}, {7820, 26686}, {7825, 10896}, {7849, 37720}, {7865, 10877}, {7886, 30104}, {7915, 30103}, {9650, 16924}, {9651, 11185}, {11285, 31451}, {14767, 37696}, {15482, 31448}, {15808, 25130}, {16408, 20181}, {16589, 17030}, {16604, 20888}, {16975, 18135}, {17027, 20970}, {17048, 24254}, {17144, 27091}, {17306, 50616}, {17490, 31234}, {17761, 20255}, {18145, 21226}, {18146, 41838}, {18152, 22199}, {20257, 46827}, {20333, 49560}, {20363, 21443}, {20541, 24387}, {20549, 21257}, {20913, 27166}, {21796, 26107}, {24330, 68950}, {24512, 29750}, {25089, 25384}, {25510, 31198}, {25530, 52379}, {26035, 27146}, {26639, 59197}, {26770, 26980}, {26821, 31026}, {26964, 27040}, {26979, 29764}, {27020, 31239}, {27432, 30038}, {29768, 37657}, {29769, 37676}, {29824, 30955}, {30942, 69013}, {31276, 64133}, {31409, 32968}, {31476, 32992}, {31488, 37664}, {33891, 33939}, {34283, 46189}, {36230, 40997}, {37670, 68893}, {37722, 69095}, {39698, 61404}, {59635, 69175}, {63924, 69098}

X(69256) = midpoint of X(2275) and X(3760)
X(69256) = X(63480)-Dao conjugate of X(16549)
X(69256) = barycentric product X(6384)*X(63480)
X(69256) = barycentric quotient X(63480)/X(43)
X(69256) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3934, 69136}, {2, 17143, 1574}, {11, 69097, 626}, {350, 26959, 39}, {4396, 5299, 7805}, {17030, 30963, 16589}, {17144, 27091, 52959}, {18140, 26801, 1573}, {18152, 26815, 22199}

X(69257) = (1, 0, -1, 1, -1, 0)-ADDITIVE ASSOCIATE OF X(12)

Barycentrics     a^2*b^2 - 2*a^2*b*c + a^2*c^2 - 2*b^2*c^2 : :
X(69257) = 5 X[1698] - 9 X[36871]

X(69257) lies on these lines: {1, 538}, {8, 33908}, {10, 25350}, {11, 63924}, {12, 7764}, {36, 4400}, {39, 1909}, {55, 7781}, {56, 7751}, {76, 330}, {172, 7805}, {194, 1500}, {274, 1573}, {310, 22199}, {315, 9651}, {325, 69175}, {388, 7758}, {496, 63923}, {514, 596}, {524, 18990}, {536, 3244}, {543, 6284}, {626, 69094}, {668, 1574}, {716, 49483}, {736, 18982}, {754, 7354}, {999, 63933}, {1089, 7208}, {1201, 4721}, {1478, 7759}, {1698, 36871}, {1914, 7816}, {1930, 7200}, {1975, 2241}, {2242, 7754}, {2275, 3761}, {2276, 32450}, {3085, 34511}, {3086, 63955}, {3230, 56024}, {3583, 63922}, {3585, 7843}, {3734, 16502}, {3760, 63493}, {3765, 69016}, {3770, 17053}, {3780, 62755}, {3813, 23880}, {3849, 10483}, {3878, 68870}, {3920, 19568}, {3933, 69174}, {3975, 31198}, {4045, 69095}, {4293, 14023}, {4299, 63935}, {4317, 63934}, {4325, 63930}, {4396, 5563}, {4403, 33930}, {4692, 16720}, {5299, 7804}, {6179, 9341}, {6381, 16604}, {6645, 7760}, {7187, 33941}, {7755, 26686}, {7757, 32005}, {7761, 9597}, {7765, 26590}, {7774, 9650}, {7775, 10895}, {7794, 26561}, {7798, 54416}, {7813, 69135}, {7817, 30103}, {7863, 26629}, {7880, 30104}, {7976, 24349}, {8149, 10063}, {8716, 64951}, {9263, 17143}, {9466, 26959}, {9654, 9766}, {9655, 63932}, {9665, 11185}, {9669, 34505}, {10896, 18546}, {10987, 15301}, {11055, 32107}, {12943, 63931}, {14711, 68973}, {15172, 52229}, {16589, 31997}, {16818, 48844}, {16975, 34284}, {16992, 31456}, {17205, 20255}, {17448, 20888}, {17499, 34063}, {19858, 50163}, {20331, 29381}, {21138, 24166}, {21219, 27318}, {21240, 24215}, {21796, 34283}, {24068, 59515}, {24524, 52959}, {24656, 25092}, {25264, 25303}, {27248, 48864}, {27437, 30038}, {31451, 31859}, {36812, 52716}, {42051, 49770}, {46893, 59319}, {63954, 67261}, {64053, 64781}, {69015, 69244}

X(69257) = reflection of X(24068) in X(59515)
X(69257) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {36, 4400, 7780}, {39, 1909, 69136}, {76, 330, 1015}, {194, 64133, 1500}, {274, 21226, 1573}, {2275, 3761, 3934}, {9263, 40908, 17143}, {69094, 69098, 626}

X(69258) = (0, 1, 1, 0, -1, 1)-ADDITIVE ASSOCIATE OF X(12)

Barycentrics     b^4 - 2*a^2*b*c + 2*b^2*c^2 + c^4 : :

X(69258) lies on these lines: {1, 7794}, {11, 7821}, {12, 9466}, {35, 7810}, {36, 7863}, {39, 69093}, {55, 7854}, {56, 7801}, {69, 2241}, {75, 30173}, {76, 69174}, {115, 3760}, {141, 1500}, {172, 7820}, {192, 3096}, {202, 69116}, {203, 69117}, {350, 626}, {499, 7888}, {538, 26561}, {599, 3295}, {1015, 3933}, {1062, 15526}, {1086, 3695}, {1478, 17130}, {1479, 7818}, {1914, 7826}, {2242, 7795}, {2275, 7813}, {2276, 6292}, {2482, 7280}, {3673, 34542}, {3825, 41144}, {3931, 17237}, {3934, 31476}, {3991, 17231}, {4045, 25264}, {4366, 7768}, {4396, 7755}, {4657, 68934}, {5280, 7889}, {5299, 7890}, {6284, 7873}, {6547, 30036}, {7005, 69115}, {7006, 69114}, {7264, 16886}, {7752, 30998}, {7764, 26959}, {7776, 9665}, {7780, 26629}, {7784, 9664}, {7800, 31451}, {7822, 54416}, {7849, 26590}, {7853, 69096}, {7855, 16502}, {7880, 26686}, {7893, 53680}, {7903, 9599}, {7935, 9598}, {8369, 9341}, {9650, 69139}, {15271, 31501}, {27020, 31478}, {27918, 30171}, {31239, 31460}, {35094, 51989}

X(69258) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 69174, 69175}, {3934, 69135, 31476}, {4396, 30104, 7755}, {69093, 69097, 39}

X(69259) = (0, 1, -1, 0, 1, 1)-ADDITIVE ASSOCIATE OF X(12)

Barycentrics     b^4 + 2*a^2*b*c - 2*b^2*c^2 + c^4 : :

X(69259) lies on these lines: {1, 115}, {2, 31451}, {3, 9664}, {4, 2242}, {5, 1500}, {6, 9665}, {11, 39}, {12, 39565}, {32, 1479}, {33, 27371}, {35, 7749}, {36, 7756}, {37, 25639}, {55, 7746}, {56, 7748}, {172, 3583}, {187, 6284}, {192, 7752}, {202, 69112}, {203, 69113}, {230, 15171}, {350, 626}, {381, 9650}, {485, 31471}, {495, 63534}, {496, 1015}, {497, 2241}, {499, 574}, {625, 69135}, {999, 9651}, {1056, 63533}, {1107, 24387}, {1210, 69247}, {1250, 22910}, {1329, 52959}, {1428, 65417}, {1478, 69141}, {1504, 44623}, {1505, 44624}, {1506, 2276}, {1570, 39873}, {1572, 9614}, {1573, 24390}, {1574, 4187}, {1575, 3825}, {1656, 31477}, {1698, 31433}, {1699, 69217}, {1737, 69249}, {1909, 63924}, {1914, 4857}, {2082, 64445}, {2275, 7765}, {2548, 10591}, {2549, 3086}, {2886, 16589}, {3023, 62356}, {3053, 9668}, {3085, 43620}, {3091, 31409}, {3096, 30998}, {3230, 21935}, {3295, 13881}, {3614, 39601}, {3703, 7230}, {3760, 7794}, {3814, 20691}, {3815, 10593}, {3829, 31466}, {3934, 26590}, {3954, 69173}, {4037, 30171}, {4045, 26959}, {4294, 69207}, {4299, 65633}, {4302, 5206}, {4366, 7828}, {4396, 7826}, {5025, 69174}, {5028, 12589}, {5046, 5291}, {5134, 33863}, {5225, 7737}, {5231, 31442}, {5274, 5286}, {5277, 52367}, {5280, 7753}, {5283, 11680}, {5299, 5355}, {5309, 11238}, {5332, 5368}, {5433, 37512}, {5434, 39563}, {5475, 10896}, {6537, 31330}, {6781, 65134}, {7005, 69119}, {7006, 69118}, {7173, 7603}, {7292, 59768}, {7738, 47743}, {7764, 25264}, {7772, 9599}, {7806, 53680}, {7816, 26686}, {7820, 30103}, {7821, 69093}, {7853, 69097}, {7861, 26561}, {7886, 26629}, {9466, 69095}, {9581, 9620}, {9592, 50444}, {9597, 10072}, {9619, 50443}, {9661, 62206}, {10056, 18362}, {10079, 32452}, {10527, 31456}, {10589, 31401}, {10638, 22865}, {10707, 69210}, {10895, 18424}, {12699, 69234}, {12701, 69232}, {12904, 14901}, {13904, 62241}, {13962, 62242}, {14986, 43448}, {15172, 43291}, {15325, 63548}, {15338, 15513}, {16785, 43457}, {17034, 41324}, {17053, 53417}, {17143, 17669}, {17735, 45939}, {17737, 69212}, {18990, 53419}, {21044, 69244}, {26100, 31031}, {26475, 53561}, {26978, 31058}, {29662, 69231}, {31231, 31422}, {31402, 31415}, {31426, 31441}, {31448, 31455}, {31459, 31481}, {31461, 31489}, {33094, 69238}, {37637, 64951}, {37693, 60724}, {37722, 69098}, {39583, 56926}, {39764, 39900}, {44541, 64894}, {52579, 69252}, {54391, 63537}, {54437, 69182}, {54438, 69188}, {59635, 69136}

X(69259) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 115, 69175}, {5, 1500, 31476}, {6, 9669, 9665}, {11, 69096, 39}, {172, 3583, 7747}, {496, 5254, 1015}, {497, 3767, 2241}, {499, 9598, 574}, {999, 44518, 9651}, {1500, 31476, 31478}, {1656, 31477, 31501}, {2276, 7741, 1506}, {4187, 21956, 1574}, {5283, 11680, 31488}, {7173, 31460, 7603}, {10896, 54416, 5475}

X(69260) = (0, 1, -1, 0, 0, 1)-ADDITIVE ASSOCIATE OF X(12)

Barycentrics     b^4 + 2*a^2*b*c + c^4 : :

X(69260) lies on these lines: {1, 626}, {2, 2241}, {5, 69136}, {10, 20529}, {11, 3934}, {12, 625}, {35, 620}, {36, 7830}, {39, 26590}, {55, 3788}, {56, 7761}, {115, 1909}, {127, 18447}, {141, 496}, {172, 754}, {187, 26686}, {192, 7796}, {315, 2242}, {316, 6645}, {319, 46826}, {325, 1500}, {330, 7790}, {350, 7794}, {390, 53033}, {491, 31471}, {497, 7795}, {498, 7862}, {499, 7815}, {538, 69094}, {668, 17669}, {999, 7784}, {1015, 6656}, {1125, 20541}, {1478, 7825}, {1479, 3734}, {1506, 27020}, {1914, 6680}, {1975, 9664}, {2275, 4045}, {2276, 7764}, {3058, 7880}, {3086, 7800}, {3230, 24995}, {3295, 7778}, {3726, 17211}, {3727, 63817}, {3761, 63924}, {3814, 25102}, {3822, 24656}, {3925, 36812}, {4109, 8682}, {4187, 27076}, {4294, 69206}, {4302, 69171}, {4366, 7832}, {4372, 4894}, {4766, 20963}, {4799, 17736}, {4805, 9310}, {5025, 64133}, {5149, 12185}, {5277, 20553}, {5280, 7838}, {5298, 40344}, {5299, 7829}, {6284, 7816}, {6292, 26959}, {7191, 21248}, {7354, 7842}, {7752, 31476}, {7759, 54416}, {7763, 31451}, {7770, 9665}, {7773, 9650}, {7775, 9596}, {7781, 9598}, {7789, 15171}, {7808, 9599}, {7813, 25264}, {7814, 31478}, {7821, 69135}, {7834, 16502}, {7841, 9651}, {7849, 37722}, {7853, 26561}, {7861, 69098}, {7865, 10072}, {7866, 16781}, {7872, 9597}, {7874, 26629}, {7892, 53680}, {7895, 69093}, {9669, 69139}, {14210, 16886}, {15338, 32456}, {16975, 17550}, {17192, 27918}, {17390, 21245}, {17499, 41324}, {18156, 34542}, {21264, 24387}, {24170, 69009}, {24211, 68890}, {26035, 31031}, {27040, 31058}, {27274, 68934}, {29829, 30954}, {30106, 53423}, {30125, 41269}, {30179, 33939}, {31409, 32816}, {31477, 69158}, {35808, 45473}, {35809, 45472}, {40690, 49481}, {54437, 69181}, {54438, 69187}

X(69260) = midpoint of X(69094) and X(69096)
X(69260) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 626, 69174}, {11, 69095, 3934}, {1914, 30103, 6680}, {5025, 64133, 69175}

X(69261) = (0, 1, -1, 0, -1, 1)-ADDITIVE ASSOCIATE OF X(12)

Barycentrics     b^4 + 2*a^2*b*c + 2*b^2*c^2 + c^4 : :

X(69261) lies on these lines: {1, 7794}, {10, 49777}, {11, 9466}, {12, 7821}, {35, 7863}, {36, 7810}, {39, 69094}, {55, 7801}, {56, 7854}, {69, 2242}, {115, 3761}, {141, 1015}, {172, 7826}, {202, 69114}, {203, 69115}, {325, 31476}, {330, 3096}, {498, 7888}, {538, 26590}, {594, 1565}, {599, 999}, {626, 1909}, {1060, 15526}, {1478, 7818}, {1479, 17130}, {1500, 3933}, {1914, 7820}, {2241, 7795}, {2275, 6292}, {2276, 7813}, {2482, 5010}, {3314, 64133}, {3675, 69252}, {3679, 35094}, {3820, 13466}, {3926, 31451}, {4400, 7755}, {5280, 7890}, {5299, 7889}, {6645, 7768}, {7005, 69117}, {7006, 69116}, {7354, 7873}, {7764, 27020}, {7776, 9650}, {7780, 26686}, {7784, 9651}, {7796, 31478}, {7822, 16502}, {7849, 26561}, {7853, 69098}, {7855, 54416}, {7880, 26629}, {7895, 69135}, {7903, 9596}, {7935, 9597}, {9341, 63928}, {9665, 69139}, {20924, 30179}, {24241, 68897}, {29673, 50025}, {30115, 49752}, {30173, 41875}, {31409, 37668}, {31460, 69197}, {31501, 69158}, {32773, 48840}

X(69261) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {325, 69136, 31476}, {626, 1909, 69175}, {3314, 64133, 69174}, {4400, 30103, 7755}, {69094, 69095, 39}

X(69262) = X(2)X(6)∩X(31)X(350)

Barycentrics     a^4 + a^3*b + a^3*c + a^2*b*c - b^2*c^2 : :

X(69262) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (1, 1, 0, 1, 0, 0, 0, 0, -1).

X(69262) lies on these lines: {2, 6}, {3, 33296}, {31, 350}, {32, 17034}, {55, 4360}, {56, 664}, {58, 76}, {75, 171}, {83, 29455}, {85, 54339}, {99, 4257}, {172, 17033}, {192, 17735}, {194, 33863}, {238, 30963}, {239, 4386}, {274, 37522}, {290, 54951}, {308, 52394}, {315, 5292}, {319, 32778}, {320, 29658}, {386, 1078}, {387, 3785}, {750, 60706}, {757, 8033}, {870, 20985}, {904, 63518}, {985, 17031}, {1043, 19761}, {1386, 52133}, {1468, 1909}, {1724, 18140}, {1760, 3509}, {1799, 54426}, {1834, 7750}, {1914, 17027}, {1966, 4485}, {1975, 4252}, {1999, 69024}, {2242, 40859}, {2248, 2998}, {3017, 7811}, {3099, 49526}, {3550, 3875}, {3684, 3759}, {3718, 20947}, {3735, 66152}, {3750, 17393}, {3770, 5115}, {3879, 29671}, {3954, 30138}, {4038, 17394}, {4042, 32025}, {4251, 6179}, {4253, 7760}, {4256, 7771}, {4259, 56154}, {4357, 29645}, {4366, 21793}, {4396, 24514}, {4414, 68867}, {4426, 41240}, {4441, 17126}, {4650, 49518}, {4664, 60711}, {5021, 7754}, {5030, 7757}, {5156, 30940}, {5247, 6376}, {5255, 17144}, {5264, 17143}, {5291, 30114}, {5429, 24291}, {6384, 34252}, {7751, 17499}, {7752, 45939}, {7780, 20970}, {7793, 18755}, {8266, 16678}, {8298, 49488}, {10030, 60716}, {10436, 37604}, {10449, 33745}, {10479, 68984}, {12188, 36730}, {14621, 21264}, {14907, 48837}, {16476, 39044}, {16706, 24586}, {16825, 20955}, {17017, 18170}, {17026, 20179}, {17103, 19271}, {17160, 37540}, {17200, 50605}, {17377, 33088}, {17688, 21024}, {17733, 17762}, {17759, 69230}, {17763, 33931}, {18021, 33769}, {18194, 29821}, {19270, 41849}, {20553, 33142}, {23903, 33824}, {24241, 49711}, {24366, 33829}, {24378, 37274}, {24549, 29674}, {25264, 69231}, {28660, 34281}, {32819, 64159}, {32820, 59538}, {35154, 35165}, {37596, 52136}, {39915, 56934}, {40415, 44153}, {49683, 57029}, {49755, 69216}, {50591, 60702}

X(69262) = crosssum of X(4386) and X(4426)
X(69262) = barycentric product X(i)*X(j) for these {i,j}: {86, 41233}, {664, 28960}
X(69262) = barycentric quotient X(i)/X(j) for these {i,j}: {28960, 522}, {41233, 10}
X(69262) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 63099, 20159}, {6, 183, 37678}, {69, 37683, 17731}, {81, 37670, 37632}, {940, 16992, 86}, {4396, 60697, 24514}, {16999, 20142, 37673}, {17001, 37657, 3570}

X(69263) = X(2)X(6)∩X(4)X(40411)

Barycentrics     a^4 - a^3*b + a^2*b^2 - a*b^3 - a^3*c + a^2*b*c + a*b^2*c - b^3*c + a^2*c^2 + a*b*c^2 - a*c^3 - b*c^3 : :

X(69263) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (1, -1, 1, 1, -1, 1, 0, -1, 0).

X(69263) lies on these lines: {2, 6}, {4, 40411}, {9, 24586}, {32, 33821}, {39, 33828}, {41, 29966}, {55, 344}, {75, 40131}, {76, 17682}, {83, 13741}, {100, 20875}, {101, 30109}, {105, 68969}, {120, 4645}, {169, 304}, {171, 17353}, {190, 3263}, {194, 33825}, {220, 21281}, {239, 3290}, {312, 9816}, {315, 17671}, {320, 51400}, {350, 673}, {379, 28809}, {404, 27109}, {612, 4687}, {614, 3759}, {644, 69028}, {664, 49755}, {672, 24602}, {894, 30748}, {1043, 4223}, {1194, 24530}, {1438, 3684}, {1572, 35274}, {1655, 33826}, {1760, 15487}, {1975, 4209}, {2082, 18156}, {2329, 30030}, {3006, 57750}, {3053, 17696}, {3059, 7081}, {3509, 17755}, {3705, 30620}, {3726, 32029}, {3732, 20924}, {3757, 64546}, {3932, 8301}, {3948, 50200}, {3972, 13735}, {3985, 17738}, {3996, 10327}, {4042, 51463}, {4044, 24588}, {4071, 27916}, {4360, 26242}, {4361, 26274}, {4386, 17279}, {4422, 17735}, {4434, 5268}, {4441, 24596}, {4518, 9470}, {4561, 57015}, {4966, 6714}, {4986, 5525}, {5089, 69039}, {5205, 9362}, {5254, 17680}, {5272, 32853}, {5277, 16061}, {5283, 16060}, {5540, 14210}, {7776, 17675}, {7785, 33837}, {7787, 33817}, {7790, 17678}, {7797, 17673}, {7803, 33833}, {7823, 33822}, {7864, 33829}, {9310, 30036}, {9534, 19309}, {13742, 19761}, {14942, 32850}, {16552, 17206}, {16600, 30130}, {16823, 17609}, {17350, 30791}, {17522, 52352}, {17550, 26085}, {17681, 18140}, {17683, 34284}, {17686, 27040}, {18145, 29479}, {20179, 32942}, {20646, 36796}, {20769, 29988}, {21372, 33952}, {23398, 68939}, {25101, 60711}, {26007, 69093}, {26273, 33891}, {27097, 69210}, {27269, 33827}, {27299, 54416}, {28420, 62372}, {29455, 60075}, {29968, 41239}, {30790, 33864}, {30830, 37086}, {32008, 52133}, {32022, 60165}, {32939, 56517}, {46738, 57925}, {49774, 56530}, {50029, 68995}

X(69263) = X(5377)-Ceva conjugate of X(190)
X(69263) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5276, 86}, {101, 30109, 69044}, {4209, 27523, 1975}, {10327, 26241, 3996}, {16552, 29473, 17206}

X(69264) = X(2)X(6)∩X(7)X(2886)

Barycentrics     a^3*b - a^2*b^2 - a*b^3 + b^4 + a^3*c + a^2*b*c - a*b^2*c - b^3*c - a^2*c^2 - a*b*c^2 - a*c^3 - b*c^3 + c^4 : :

X(69264) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (0, 1, -1, 1, -1, -1, 1, -1, 0).

X(69264) lies on these lines: {2, 6}, {7, 2886}, {8, 348}, {10, 16284}, {11, 30946}, {12, 36854}, {37, 31038}, {63, 4872}, {72, 17181}, {75, 1088}, {76, 18738}, {78, 17095}, {85, 6734}, {144, 17747}, {150, 956}, {200, 319}, {253, 57873}, {307, 31627}, {309, 56944}, {315, 13727}, {317, 14004}, {320, 5231}, {322, 25006}, {377, 1434}, {461, 32001}, {518, 7179}, {594, 25355}, {984, 24241}, {1043, 3926}, {1098, 4592}, {1330, 7776}, {1440, 56026}, {1444, 7411}, {1447, 47595}, {1536, 48878}, {1834, 4352}, {2893, 7580}, {2975, 21285}, {3177, 40997}, {3419, 5088}, {3509, 24694}, {3661, 44798}, {3663, 33141}, {3664, 33111}, {3673, 10916}, {3674, 24391}, {3681, 33864}, {3686, 62388}, {3693, 17233}, {3769, 3879}, {3846, 17272}, {3868, 33949}, {3870, 17377}, {3933, 10449}, {3964, 20835}, {4056, 6763}, {4197, 17169}, {4252, 68909}, {4253, 33838}, {4357, 11019}, {4360, 36845}, {4389, 26015}, {4416, 40869}, {4666, 55391}, {4712, 33931}, {4875, 26531}, {4880, 33866}, {4911, 62858}, {5220, 20531}, {5282, 24712}, {5745, 64702}, {5794, 7176}, {5880, 60717}, {6604, 64081}, {6646, 24352}, {6745, 17360}, {7210, 54289}, {7247, 62874}, {7796, 33297}, {7881, 54365}, {8568, 17228}, {8580, 17270}, {8727, 10446}, {8822, 10431}, {10025, 17347}, {10436, 33121}, {10453, 69093}, {10527, 55082}, {10580, 17321}, {10582, 17322}, {10883, 17139}, {11679, 52089}, {11680, 20347}, {12513, 56928}, {12635, 17084}, {16062, 16887}, {16552, 17671}, {16699, 24555}, {16706, 24600}, {17046, 21384}, {17210, 37039}, {17263, 30813}, {17264, 59216}, {17320, 31146}, {17753, 24390}, {18133, 18153}, {18206, 37445}, {18827, 41501}, {20533, 42316}, {20541, 24691}, {21049, 27288}, {21258, 27304}, {24203, 45700}, {24392, 64695}, {25353, 51058}, {26059, 52818}, {27187, 34772}, {27484, 61673}, {28043, 63150}, {28740, 32008}, {30758, 60668}, {32000, 57534}, {32099, 40999}, {32816, 36660}, {33677, 60706}, {34282, 51612}, {35140, 54951}, {35150, 35154}, {35171, 53210}, {42697, 51351}, {50995, 56555}, {60227, 62946}

X(69264) = isotomic conjugate of X(56144)
X(69264) = isotomic conjugate of the isogonal conjugate of X(991)
X(69264) = isotomic conjugate of the polar conjugate of X(37448)
X(69264) = X(991)-cross conjugate of X(37448)
X(69264) = X(i)-isoconjugate of X(j) for these (i,j): {31, 56144}, {41, 63149}, {560, 58024}, {63461, 68232}
X(69264) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 56144}, {3160, 63149}, {6374, 58024}, {24635, 4312}
X(69264) = cevapoint of X(24635) and X(41228)
X(69264) = barycentric product X(i)*X(j) for these {i,j}: {69, 37448}, {75, 24635}, {76, 991}, {85, 41228}
X(69264) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 56144}, {7, 63149}, {76, 58024}, {991, 6}, {4573, 68232}, {24635, 1}, {37448, 4}, {41228, 9}
X(69264) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 69, 68928}, {2, 14548, 86}, {2, 51384, 17234}, {69, 325, 4417}, {75, 20935, 35517}, {86, 17731, 56439}, {298, 299, 17346}, {307, 51364, 31627}, {319, 68926, 200}, {491, 492, 17277}, {1270, 1271, 63001}, {4847, 9436, X(69264) = 75}, {10513, 37655, 69}, {30962, 37664, 17234}

X(69265) = X(2)X(6)∩X(8)X(291)

Barycentrics     a^3*b - a^2*b^2 - a*b^3 + a^3*c + a^2*b*c - a*b^2*c - b^3*c - a^2*c^2 - a*b*c^2 - b^2*c^2 - a*c^3 - b*c^3 : :

X(69265) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (0, 1, -1, 1, -1, -1, 0, -1, -1).

X(69265) lies on these lines: {2, 6}, {8, 291}, {9, 31028}, {37, 41851}, {38, 192}, {39, 33297}, {43, 17363}, {75, 24691}, {194, 10449}, {310, 9230}, {319, 1575}, {320, 21264}, {350, 6646}, {384, 17206}, {672, 17280}, {894, 3741}, {1008, 20077}, {1011, 20794}, {1043, 7783}, {1330, 7785}, {2108, 49560}, {2227, 24575}, {2276, 6542}, {2975, 8299}, {3219, 20785}, {3661, 17754}, {3662, 17026}, {3720, 25420}, {3840, 4416}, {4184, 20775}, {4210, 22062}, {4360, 25349}, {4440, 4441}, {4479, 17276}, {4643, 30963}, {4661, 31111}, {4713, 17347}, {4741, 24712}, {5021, 17688}, {6625, 60110}, {7836, 54365}, {9534, 27318}, {10477, 56838}, {16058, 22152}, {16887, 17034}, {17027, 17302}, {17028, 30949}, {17135, 17148}, {17137, 26801}, {17248, 26102}, {17288, 20335}, {17319, 42057}, {17333, 31137}, {17338, 30822}, {17344, 20530}, {17362, 25350}, {17499, 50605}, {17680, 40017}, {17751, 21226}, {17762, 69246}, {17787, 61417}, {20072, 24514}, {21904, 62231}, {24318, 27920}, {24330, 31300}, {25311, 41836}, {27484, 30791}, {28604, 31330}, {29588, 60724}, {29824, 60725}, {30967, 59207}, {30989, 33080}, {33863, 33954}, {33888, 33931}, {33889, 49509}, {34016, 69212}, {37042, 49716}, {37148, 56018}, {56210, 62921}

X(69265) = anticomplement of X(37678)
X(69265) = anticomplement of the isotomic conjugate of X(60090)
X(69265) = X(60090)-anticomplementary conjugate of X(6327)
X(69265) = X(60090)-Ceva conjugate of X(2)
X(69265) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20090, 37632}, {2, 30941, 17300}, {2, 62989, 2238}, {2, 63066, 63053}, {141, 37686, 2}, {350, 24690, 6646}, {672, 31027, 17280}, {4741, 30998, 30946}, {24512, 30966, 2}

X(69266) = X(2)X(6)∩X(9)X(124)

Barycentrics     a^3*b - a^2*b^2 + a*b^3 + b^4 + a^3*c + b^3*c - a^2*c^2 + a*c^3 + b*c^3 + c^4 : :

X(69266) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (0, 1, -1, 0, 1, 0, 1, 1, 0).

X(69266) lies on these lines: {2, 6}, {3, 26085}, {5, 27040}, {9, 124}, {10, 17451}, {32, 56778}, {37, 3006}, {39, 4202}, {194, 16906}, {281, 427}, {315, 33819}, {319, 26247}, {442, 26035}, {573, 8229}, {594, 31079}, {626, 33839}, {672, 2887}, {1100, 26230}, {1449, 29855}, {1655, 33834}, {1975, 16910}, {2276, 4972}, {2280, 3771}, {2321, 33136}, {2345, 33108}, {2548, 5192}, {3011, 3686}, {3053, 56781}, {3263, 20234}, {3290, 69134}, {3454, 16552}, {3509, 33119}, {3684, 29846}, {3705, 21796}, {3846, 30751}, {3930, 29673}, {3933, 26978}, {3985, 69173}, {4251, 25645}, {4429, 17756}, {4438, 5282}, {4450, 17735}, {4660, 41423}, {4766, 17353}, {5013, 56782}, {5014, 69214}, {5016, 16968}, {5051, 5283}, {5254, 26770}, {5257, 29639}, {5296, 30741}, {5750, 30768}, {5839, 26228}, {6656, 27109}, {6679, 21764}, {7745, 11319}, {7759, 25497}, {7763, 33830}, {7776, 26099}, {7783, 33831}, {7785, 33816}, {7787, 16905}, {7836, 33826}, {7839, 16908}, {7885, 33820}, {7905, 33955}, {7912, 33836}, {9605, 56780}, {11112, 26079}, {14963, 37050}, {16503, 29632}, {16600, 30171}, {16777, 29832}, {16779, 29858}, {16787, 36505}, {16830, 19942}, {16884, 29831}, {16972, 33070}, {16973, 33122}, {17243, 31093}, {17260, 30763}, {17275, 26227}, {17314, 31091}, {17355, 21241}, {17362, 20045}, {17526, 69208}, {17537, 53418}, {17740, 62693}, {17754, 25957}, {20444, 30758}, {21101, 33162}, {21840, 29671}, {22021, 55076}, {24995, 29960}, {25960, 30752}, {26030, 31460}, {26965, 69135}, {27097, 69254}, {28742, 69095}, {30435, 56779}, {30748, 33864}, {32947, 60711}, {33113, 69220}, {33114, 69218}, {33120, 51058}, {40585, 41797}, {47809, 60577}, {50752, 63978}, {51583, 69239}

X(69266) = crossdifference of every pair of points on line {512, 66522}
X(69266) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1654, 37670}, {2, 7766, 17003}, {2, 17007, 183}, {1213, 37661, 2}

X(69267) = X(1)X(2)∩X(31)X(46)

Barycentrics     a*(a^3 + a^2*b + a*b^2 + b^3 + a^2*c - b^2*c + a*c^2 - b*c^2 + c^3) : :

X(69267) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (1, 1, 1, 0, 1, -1, 0, 0, 0).

X(69267) lies on these lines: {1, 2}, {3, 1104}, {4, 990}, {5, 3772}, {6, 169}, {9, 16600}, {19, 8743}, {21, 4850}, {28, 34}, {29, 19788}, {31, 46}, {33, 5142}, {36, 11337}, {37, 11108}, {38, 41229}, {39, 16968}, {40, 595}, {44, 3927}, {56, 998}, {63, 1724}, {65, 16466}, {72, 4383}, {73, 34489}, {75, 13740}, {77, 44709}, {86, 57748}, {106, 61762}, {141, 5814}, {142, 5717}, {171, 16478}, {172, 69221}, {222, 37566}, {223, 1467}, {225, 52489}, {226, 19372}, {227, 1617}, {238, 986}, {244, 1468}, {269, 937}, {273, 8747}, {312, 13741}, {321, 5192}, {345, 13742}, {346, 37024}, {392, 37614}, {404, 62802}, {405, 3666}, {442, 24789}, {443, 5716}, {474, 16610}, {497, 54295}, {517, 1191}, {553, 48870}, {579, 4456}, {580, 5709}, {581, 18443}, {582, 37584}, {601, 59333}, {613, 67963}, {748, 2292}, {750, 62847}, {758, 54386}, {774, 15299}, {902, 59316}, {910, 30435}, {940, 5439}, {950, 1040}, {958, 37592}, {964, 4359}, {968, 5259}, {977, 56220}, {979, 7194}, {982, 5247}, {988, 993}, {991, 8726}, {999, 52541}, {1001, 3931}, {1010, 19804}, {1038, 3911}, {1046, 16468}, {1054, 37603}, {1060, 7561}, {1062, 1834}, {1086, 57282}, {1096, 1838}, {1100, 16605}, {1126, 2191}, {1158, 3073}, {1169, 52375}, {1178, 60245}, {1203, 5902}, {1212, 9605}, {1228, 3760}, {1245, 12609}, {1254, 1471}, {1279, 3295}, {1329, 17061}, {1330, 3662}, {1333, 52012}, {1376, 5266}, {1385, 19547}, {1386, 3812}, {1407, 64055}, {1421, 3340}, {1449, 2303}, {1473, 1828}, {1478, 23536}, {1479, 3914}, {1571, 17735}, {1572, 3959}, {1575, 16974}, {1612, 6986}, {1616, 9957}, {1697, 40091}, {1707, 69227}, {1708, 37591}, {1716, 3821}, {1721, 51118}, {1723, 40977}, {1728, 44706}, {1739, 5264}, {1743, 5279}, {1766, 37415}, {1788, 8270}, {1791, 8666}, {1829, 21370}, {1837, 64172}, {1854, 64131}, {1870, 7521}, {1914, 69219}, {2049, 3739}, {2082, 5299}, {2176, 9620}, {2221, 4185}, {2257, 2331}, {2260, 2333}, {2263, 3339}, {2275, 3002}, {2276, 69215}, {2277, 5336}, {2300, 10441}, {2352, 16453}, {2363, 11116}, {2476, 33129}, {2478, 19785}, {2802, 66647}, {2901, 3875}, {3052, 3579}, {3057, 16483}, {3075, 37253}, {3091, 62208}, {3125, 54382}, {3159, 30568}, {3195, 67965}, {3210, 7283}, {3242, 34790}, {3247, 28594}, {3290, 54416}, {3303, 64175}, {3306, 37522}, {3315, 3889}, {3333, 5573}, {3361, 4320}, {3445, 51788}, {3452, 34937}, {3454, 25527}, {3487, 26728}, {3553, 56746}, {3554, 8555}, {3555, 17597}, {3586, 52364}, {3601, 4256}, {3660, 34046}, {3663, 12572}, {3672, 5129}, {3673, 20914}, {3677, 57279}, {3684, 16787}, {3695, 17279}, {3701, 3891}, {3703, 25992}, {3721, 54406}, {3730, 9593}, {3735, 39248}, {3737, 50453}, {3744, 5687}, {3749, 8715}, {3751, 3874}, {3753, 5710}, {3754, 62828}, {3755, 63999}, {3759, 56018}, {3782, 58798}, {3797, 33817}, {3868, 32911}, {3869, 54315}, {3871, 62806}, {3876, 37680}, {3915, 4642}, {3916, 17595}, {3923, 41247}, {3946, 37819}, {3953, 62874}, {3987, 37610}, {4187, 17720}, {4193, 33133}, {4195, 17490}, {4197, 26724}, {4198, 4292}, {4202, 5016}, {4205, 4657}, {4251, 16780}, {4252, 37582}, {4255, 24929}, {4257, 7520}, {4259, 67975}, {4260, 12109}, {4261, 16290}, {4267, 19264}, {4296, 5435}, {4298, 24171}, {4303, 40958}, {4319, 66682}, {4327, 5290}, {4340, 9776}, {4344, 11024}, {4347, 60786}, {4353, 18250}, {4361, 5295}, {4385, 32922}, {4423, 6051}, {4424, 5250}, {4429, 5015}, {4652, 17521}, {4653, 5436}, {4662, 49465}, {4675, 49743}, {4676, 63996}, {4694, 62832}, {4749, 5880}, {4751, 14007}, {4858, 54396}, {4906, 34791}, {5020, 27802}, {5044, 37679}, {5046, 33150}, {5047, 28606}, {5051, 32774}, {5126, 8572}, {5179, 5286}, {5219, 24160}, {5248, 17594}, {5255, 24440}, {5280, 40131}, {5284, 62831}, {5291, 69222}, {5305, 46835}, {5310, 37557}, {5315, 5903}, {5322, 8185}, {5359, 15487}, {5396, 37615}, {5398, 37532}, {5429, 37608}, {5437, 37554}, {5493, 12652}, {5586, 52332}, {5713, 55108}, {5725, 25466}, {5808, 21258}, {5883, 16475}, {6583, 39523}, {7078, 67949}, {7270, 33833}, {7288, 54292}, {7354, 34653}, {7373, 15663}, {7483, 64348}, {7557, 9612}, {7741, 37983}, {7754, 25994}, {7772, 49758}, {7986, 31937}, {8074, 68908}, {8225, 45501}, {8728, 17278}, {9370, 17625}, {9574, 24047}, {9630, 61717}, {9799, 68529}, {9840, 61325}, {9940, 36746}, {10202, 36742}, {10396, 62811}, {10436, 33945}, {10826, 21935}, {10914, 37542}, {10974, 39598}, {11109, 54284}, {11113, 50065}, {11227, 37501}, {11246, 63669}, {11319, 17495}, {11354, 50054}, {11359, 50050}, {11374, 37662}, {11512, 63292}, {11573, 37516}, {12233, 23982}, {12433, 48847}, {12436, 24175}, {12578, 52808}, {12705, 64013}, {14923, 62848}, {15844, 37695}, {16062, 16706}, {16408, 16602}, {16414, 64556}, {16454, 24589}, {16456, 31238}, {16471, 55399}, {16474, 50190}, {16486, 31792}, {16487, 53053}, {16492, 67418}, {16498, 56009}, {16502, 41015}, {16519, 37673}, {16609, 37523}, {16842, 44307}, {16863, 31197}, {16972, 17750}, {17063, 37607}, {17064, 25639}, {17102, 57278}, {17127, 56288}, {17147, 56983}, {17189, 45196}, {17281, 50042}, {17301, 50067}, {17303, 50318}, {17382, 54367}, {17383, 37164}, {17526, 17740}, {17556, 50103}, {17596, 54354}, {17717, 24161}, {17721, 24390}, {17742, 26242}, {17861, 39585}, {17863, 20235}, {18147, 40071}, {18178, 68730}, {18180, 40153}, {18193, 24167}, {18398, 62819}, {18446, 37732}, {18785, 21742}, {18838, 64020}, {19239, 56538}, {19523, 19765}, {19673, 51710}, {19724, 56219}, {19786, 52258}, {19808, 37036}, {19834, 26601}, {19850, 35996}, {20267, 30742}, {20966, 27661}, {21077, 33144}, {21309, 68931}, {21764, 69242}, {23664, 63497}, {24068, 49446}, {24176, 48866}, {24178, 67969}, {24474, 36754}, {24597, 54289}, {24703, 63997}, {24851, 33149}, {24914, 64349}, {25440, 37552}, {25496, 49598}, {25542, 27785}, {25939, 37244}, {26131, 27186}, {26685, 59639}, {28358, 46475}, {28612, 50314}, {31266, 37693}, {31546, 45500}, {31788, 64449}, {32047, 34753}, {32921, 63800}, {32926, 46937}, {33131, 52367}, {33155, 37162}, {33854, 69223}, {33934, 33955}, {33950, 63075}, {34048, 66250}, {36279, 64558}, {36741, 37547}, {37469, 37534}, {37550, 55086}, {37559, 62845}, {37588, 64176}, {37594, 37674}, {37681, 54398}, {38832, 54373}, {39688, 68846}, {40688, 49745}, {41344, 64722}, {42051, 50044}, {45219, 64897}, {46878, 56445}, {48824, 49732}, {48863, 64185}, {50066, 66099}, {50196, 64069}, {51724, 63977}, {52542, 64706}, {54343, 62810}, {54400, 68904}, {56535, 61357}, {57288, 59477}, {58393, 64554}, {58469, 63511}, {58624, 67974}, {59285, 64124}, {61397, 64046}, {61661, 63388}, {61763, 62875}, {62693, 69208}, {62833, 67979}, {63513, 67967}, {64054, 66104}, {64057, 64132}, {64174, 64671}, {64347, 68758}, {65687, 67930}, {69212, 69220}

X(69267) = reflection of X(1) in X(30148)
X(69267) = isogonal conjugate of X(56220)
X(69267) = complement of X(54433)
X(69267) = complement of the isogonal conjugate of X(51686)
X(69267) = X(i)-complementary conjugate of X(j) for these (i,j): {1036, 34823}, {1039, 1329}, {1245, 21530}, {1395, 34261}, {1472, 3}, {2221, 18589}, {2281, 440}, {32691, 513}, {36099, 3835}, {51686, 10}, {56328, 1368}, {65341, 21262}
X(69267) = X(i)-Ceva conjugate of X(j) for these (i,j): {977, 1}, {65303, 513}
X(69267) = X(i)-isoconjugate of X(j) for these (i,j): {1, 56220}, {37, 56045}, {513, 65372}
X(69267) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 56220}, {26933, 23874}, {39026, 65372}, {40589, 56045}
X(69267) = crosspoint of X(i) and X(j) for these (i,j): {39267, 68578}, {46740, 57923}
X(69267) = crosssum of X(6) and X(7085)
X(69267) = crossdifference of every pair of points on line {649, 8611}
X(69267) = barycentric product X(i)*X(j) for these {i,j}: {1, 19785}, {27, 41340}, {57, 2478}
X(69267) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 56220}, {58, 56045}, {101, 65372}, {2478, 312}, {19785, 75}, {19852, 19787}, {41340, 306}
X(69267) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2, 975}, {1, 43, 3811}, {1, 936, 30115}, {1, 978, 997}, {1, 1698, 612}, {1, 1722, 10}, {1, 2999, 386}, {1, 3216, 78}, {1, 3293, 3870}, {1, 4853, 50637}, {1, 5268, 30142}, {1, 5272, 1125}, {1, 6048, 3961}, {1, 16569, 5293}, {1, 23511, 936}, {1, 49494, 36846}, {1, 49997, 19861}, {1, 64673, 30116}, {2, 5262, 1}, {3, 1104, 37817}, {4, 4000, 23537}, {6, 16583, 169}, {6, 17054, 942}, {6, 20271, 69217}, {6, 40941, 54405}, {8, 7191, 1}, {31, 24443, 46}, {34, 57, 1448}, {40, 7290, 595}, {42, 28082, 1}, {56, 57277, 21147}, {57, 1453, 58}, {58, 24046, 57}, {223, 1467, 4306}, {238, 986, 12514}, {244, 1468, 3338}, {386, 30117, 1}, {405, 3666, 62871}, {614, 54418, 1}, {936, 23511, 17749}, {938, 5222, 387}, {938, 41785, 10}, {938, 66610, 1}, {942, 7535, 54405}, {982, 5247, 62858}, {1104, 3752, 3}, {1125, 17748, 3771}, {1193, 3924, 1}, {1201, 49487, 1}, {1203, 5902, 54421}, {1210, 40940, 5292}, {1279, 4646, 3295}, {1386, 3812, 5711}, {1393, 1451, 57}, {1724, 3670, 63}, {3210, 17697, 7283}, {3306, 62809, 37522}, {3616, 17016, 1}, {3622, 17015, 1}, {3634, 30142, 5268}, {3915, 4642, 5119}, {3987, 37610, 63130}, {4383, 37549, 72}, {5047, 28606, 54287}, {5255, 24440, 54286}, {5256, 54392, 1}, {6734, 26723, 1714}, {7292, 17016, 3616}, {9581, 33178, 33}, {9593, 16970, 3730}, {10449, 27299, 10}, {16478, 24174, 171}, {16610, 37539, 474}, {17017, 59305, 1}, {17749, 30115, 936}, {19846, 30171, 29857}, {25440, 49480, 37552}, {26242, 69211, 17742}, {30144, 49682, 1}, {30147, 50604, 1}, {33137, 36574, 10916}, {34489, 56418, 73}, {36103, 54293, 4}, {37582, 64166, 4252}

X(69268) = X(1)X(2)∩X(6)X(18398)

Barycentrics     a*(a^3 + a^2*b + a*b^2 + b^3 + a^2*c - a*b*c - b^2*c + a*c^2 - b*c^2 + c^3) : :

X(69268) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (1, 1, 1, -1, 1, -1, 0, 0, 0).

X(69268) lies on these lines: {1, 2}, {6, 18398}, {11, 33178}, {28, 58624}, {31, 3336}, {34, 2163}, {35, 3752}, {36, 1104}, {37, 25542}, {46, 7290}, {56, 8185}, {57, 1399}, {58, 229}, {65, 1421}, {79, 1086}, {81, 58565}, {191, 238}, {404, 49480}, {484, 595}, {580, 5536}, {582, 24468}, {593, 52375}, {609, 69221}, {651, 67051}, {942, 1203}, {982, 1724}, {988, 37317}, {1010, 6533}, {1040, 66682}, {1089, 13741}, {1126, 62867}, {1191, 5903}, {1224, 50318}, {1279, 3746}, {1386, 5439}, {1393, 15932}, {1420, 57277}, {1453, 3338}, {1455, 13370}, {1479, 4000}, {1717, 65140}, {1718, 5563}, {1739, 5255}, {1768, 3073}, {1770, 24177}, {1781, 16470}, {1870, 64124}, {1929, 52376}, {2901, 32924}, {3052, 37572}, {3218, 24167}, {3290, 5280}, {3315, 3881}, {3333, 37697}, {3339, 34036}, {3454, 33123}, {3555, 4906}, {3583, 23537}, {3585, 23536}, {3612, 16485}, {3666, 5259}, {3677, 41229}, {3678, 37680}, {3697, 49465}, {3743, 5284}, {3754, 62804}, {3772, 7741}, {3825, 33133}, {3841, 26724}, {3874, 32911}, {3878, 54315}, {3901, 54386}, {3914, 4857}, {3915, 11010}, {3953, 5247}, {3987, 5541}, {4003, 31445}, {4040, 8578}, {4187, 17061}, {4347, 5435}, {4383, 5904}, {4418, 24176}, {4423, 27785}, {4429, 4894}, {4642, 37563}, {4647, 32942}, {4680, 33833}, {4850, 5248}, {4973, 16948}, {4974, 64072}, {5045, 16474}, {5251, 37592}, {5253, 63292}, {5263, 28611}, {5264, 24174}, {5266, 16610}, {5290, 19372}, {5299, 5540}, {5506, 17123}, {5525, 69211}, {5570, 54301}, {5692, 37549}, {5697, 16483}, {5883, 57280}, {5902, 16466}, {6583, 37509}, {7194, 39748}, {7280, 37817}, {8167, 31318}, {8715, 62806}, {9957, 16489}, {10591, 62208}, {11102, 52680}, {11108, 17599}, {11263, 33107}, {11509, 26742}, {13462, 21147}, {15852, 35202}, {16297, 20990}, {16302, 16684}, {16414, 16687}, {16478, 17063}, {16487, 61763}, {16600, 33854}, {16611, 69210}, {16784, 41015}, {17127, 69227}, {17278, 41859}, {17527, 17602}, {17598, 67979}, {17744, 26242}, {18180, 64523}, {21793, 69238}, {21935, 37718}, {24440, 37610}, {24552, 28612}, {25639, 33129}, {26127, 33155}, {26842, 63366}, {27644, 35637}, {31231, 64349}, {32636, 64166}, {32772, 41812}, {32857, 64289}, {32927, 59666}, {33130, 37693}, {33150, 36250}, {33944, 33955}, {34043, 37566}, {35010, 37469}, {37662, 37731}, {37702, 64172}, {43993, 63800}, {48866, 64431}, {50196, 68591}, {51785, 54295}, {54293, 65128}, {54421, 67971}, {56343, 62812}, {58887, 62695}, {59316, 62875}, {61086, 63469}, {61649, 64339}, {67930, 68592}

X(69268) = crosspoint of X(86) and X(40143)
X(69268) = crosssum of X(42) and X(21873)
X(69268) = crossdifference of every pair of points on line {649, 68818}
X(69268) = barycentric product X(i)*X(j) for these {i,j}: {1, 33150}, {81, 36250}
X(69268) = barycentric quotient X(i)/X(j) for these {i,j}: {33150, 75}, {36250, 321}
X(69268) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1722, 3679}, {1, 2999, 5312}, {1, 5272, 3624}, {1, 34595, 975}, {1, 64850, 612}, {8, 30148, 1}, {10, 7191, 1}, {31, 24046, 3336}, {58, 244, 3337}, {238, 3670, 191}, {386, 28082, 1}, {551, 17016, 1}, {595, 24443, 484}, {982, 1724, 6763}, {995, 3924, 1}, {1125, 5262, 1}, {1149, 15955, 1}, {1193, 30117, 1}, {1386, 5439, 37559}, {1393, 55086, 15932}, {1453, 5573, 3338}, {3636, 17015, 1}, {3987, 37588, 5541}, {4642, 40091, 37563}, {5256, 64675, 1}, {5262, 7292, 1125}, {5299, 16583, 5540}, {9780, 17024, 30145}, {11019, 66610, 1}, {13741, 32922, 1089}, {16466, 17054, 5902}, {16470, 40941, 1781}, {16478, 17063, 37522}, {17024, 30145, 1}

X(69269) = X(1)X(2)∩X(9)X(7766)

Barycentrics     a^4 + a^3*b + a*b^3 + a^3*c + a*b^2*c + a*b*c^2 - b^2*c^2 + a*c^3 : :

X(69269) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (1, 1, 0, 0, 1, 1, 0, 0, -1).

X(69269) lies on these lines: {1, 2}, {9, 7766}, {37, 385}, {44, 63038}, {45, 14614}, {81, 4621}, {171, 335}, {183, 16777}, {190, 4797}, {192, 69214}, {210, 20142}, {226, 8857}, {312, 52138}, {319, 31090}, {321, 1966}, {325, 17390}, {330, 54317}, {333, 27495}, {344, 16989}, {384, 5266}, {894, 21101}, {988, 33004}, {1100, 3329}, {1194, 59454}, {1255, 26243}, {1281, 3993}, {1447, 17319}, {1449, 62994}, {1621, 63237}, {1799, 62537}, {1965, 52043}, {3219, 3508}, {3290, 16999}, {3314, 4851}, {3403, 28605}, {3552, 37552}, {3570, 46907}, {3701, 33816}, {3744, 4366}, {3745, 20132}, {3797, 32926}, {3879, 7779}, {3914, 6653}, {3971, 6651}, {3985, 68876}, {4339, 14035}, {4357, 63044}, {4359, 60683}, {4360, 33891}, {4364, 37671}, {4385, 17688}, {4416, 50248}, {4518, 4649}, {4657, 16986}, {5015, 33834}, {5300, 16906}, {5990, 67428}, {5992, 50307}, {6194, 46475}, {6376, 16974}, {6645, 37539}, {6655, 13161}, {7146, 56358}, {7179, 17391}, {7792, 17243}, {7824, 37592}, {7839, 25066}, {7840, 50125}, {7868, 17311}, {7875, 17279}, {8356, 66675}, {8667, 16672}, {11174, 16884}, {11361, 66639}, {13586, 37589}, {14621, 17716}, {16968, 41838}, {16987, 17357}, {16988, 17384}, {16990, 17321}, {16995, 40131}, {16997, 26242}, {17061, 26582}, {17200, 21067}, {17257, 63046}, {17332, 50251}, {17353, 63020}, {17565, 23536}, {17602, 26590}, {18152, 63242}, {18758, 19308}, {19701, 24679}, {20090, 56555}, {20145, 62845}, {20165, 64169}, {25497, 33932}, {26685, 63045}, {27064, 39914}, {27065, 56533}, {27269, 69215}, {31317, 32920}, {33007, 66680}, {33264, 66672}, {33273, 37599}, {33817, 46937}, {37187, 63965}, {44367, 50093}, {48819, 66414}, {48820, 66417}, {48824, 66413}, {54280, 63093}

X(69269) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6542, 30179}, {2, 29588, 29840}, {3661, 29634, 2}, {3920, 26247, 2}, {4393, 29569, 25943}, {7081, 16826, 2}, {30138, 30141, 1}

X(69270) = X(1)X(2)∩X(3)X(3797)

Barycentrics     a^4 + a^2*b*c - b^3*c - b^2*c^2 - b*c^3 : :

X(69270) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (1, 0, 0, 1, 0, 0, 0, -1, -1).

X(69270) lies on these lines: {1, 2}, {3, 3797}, {32, 33939}, {75, 16917}, {85, 17129}, {172, 33931}, {192, 22267}, {304, 385}, {312, 384}, {321, 16915}, {345, 16925}, {346, 32973}, {350, 4372}, {1104, 33817}, {2242, 33941}, {2303, 56696}, {3497, 51837}, {3552, 7283}, {3695, 7807}, {3703, 26686}, {3772, 17673}, {4358, 16916}, {4385, 6645}, {4386, 17762}, {4396, 33930}, {4403, 63925}, {4426, 20947}, {4671, 16919}, {4687, 16929}, {4872, 7893}, {5016, 17669}, {5025, 7270}, {5088, 20081}, {5132, 20164}, {5277, 33935}, {5291, 33932}, {6651, 54354}, {7187, 7754}, {7230, 7816}, {7751, 20924}, {7777, 54443}, {7779, 17181}, {7828, 34542}, {7839, 25918}, {7906, 17095}, {7907, 32851}, {8782, 18596}, {16905, 33157}, {16906, 33133}, {16918, 18743}, {16924, 28808}, {16931, 31035}, {16997, 20911}, {17170, 63046}, {17680, 37759}, {17688, 37539}, {17689, 62871}, {17720, 33834}, {17742, 33889}, {23537, 33829}, {25270, 56288}, {31317, 37607}, {33246, 42033}, {33255, 42032}, {33816, 62802}, {33888, 62858}, {41771, 50029}

X(69270) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {16822, 29649, 41240}

X(69271) = X(1)X(2)∩X(21)X(3797)

Barycentrics     a^4 - a*b^2*c - b^3*c - a*b*c^2 - b^2*c^2 - b*c^3 : :

X(69271) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (1, 0, 0, 0, 0, -1, 0, -1, -1).

X(69271) lies on these lines: {1, 2}, {21, 3797}, {32, 33935}, {37, 16927}, {75, 172}, {304, 16998}, {312, 16916}, {321, 384}, {385, 20911}, {1003, 50044}, {1055, 17117}, {1104, 33816}, {1468, 31317}, {1914, 17762}, {3210, 22267}, {3702, 4366}, {3704, 26629}, {3772, 16906}, {3995, 16931}, {4358, 16918}, {4359, 16917}, {4361, 21008}, {4396, 33944}, {4400, 20955}, {4426, 33931}, {4461, 33201}, {4671, 16920}, {4968, 6645}, {5016, 5025}, {5088, 40908}, {5291, 33941}, {7270, 33841}, {7283, 17692}, {7751, 33934}, {7906, 27187}, {8369, 50042}, {8624, 17143}, {12514, 25270}, {16519, 30966}, {16905, 32777}, {16919, 28605}, {16925, 17740}, {16926, 31993}, {16930, 31025}, {16951, 19835}, {17129, 26563}, {17673, 33129}, {17688, 62802}, {23537, 33831}, {26686, 69091}, {33133, 33834}, {33187, 50045}, {33220, 50041}, {33255, 50043}, {33939, 69212}, {38810, 52379}, {41826, 63046}

X(69271) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {239, 27954, 1193}, {4362, 16822, 17033}, {30130, 54335, 16819}

X(69272) = X(1)X(2)∩X(35)X(38)

Barycentrics     a*(a^3 + b^3 - a*b*c + b^2*c + b*c^2 + c^3) : :
X(69272) = 2 X[3454] - 3 X[33126], X[5015] - 3 X[33126], 4 X[6693] - 3 X[33121], 3 X[33152] - 2 X[36250]

X(69272) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (1, 0, 0, -1, 1, 1, 0, 0, 0).

X(69272) lies on these lines: {1, 2}, {3, 3242}, {21, 67979}, {31, 5904}, {32, 49509}, {35, 38}, {37, 1731}, {55, 1782}, {58, 518}, {65, 56149}, {72, 595}, {79, 32856}, {100, 3670}, {106, 17614}, {171, 3874}, {191, 902}, {201, 2078}, {238, 3678}, {242, 6198}, {404, 3953}, {442, 17724}, {474, 17597}, {500, 542}, {517, 3430}, {528, 63997}, {537, 24850}, {674, 41329}, {750, 18398}, {756, 5259}, {758, 5255}, {952, 63318}, {960, 40091}, {982, 25440}, {984, 5248}, {986, 8715}, {990, 6769}, {996, 40436}, {1054, 24167}, {1066, 68908}, {1089, 32927}, {1104, 34790}, {1191, 3940}, {1203, 17469}, {1279, 5044}, {1283, 2292}, {1376, 24046}, {1469, 50599}, {1490, 61086}, {1497, 18397}, {1500, 16519}, {1697, 17461}, {1724, 3681}, {1807, 9957}, {1914, 3954}, {2223, 4278}, {2271, 16777}, {2550, 24159}, {2647, 51782}, {2886, 24160}, {2901, 32926}, {3052, 3927}, {3056, 50593}, {3057, 45272}, {3073, 63967}, {3159, 3685}, {3189, 48837}, {3243, 37554}, {3263, 33953}, {3315, 17531}, {3337, 17449}, {3338, 62850}, {3445, 35272}, {3454, 5015}, {3465, 10624}, {3487, 68589}, {3550, 69227}, {3555, 37539}, {3646, 35227}, {3663, 64117}, {3666, 33771}, {3684, 16600}, {3699, 13741}, {3721, 5011}, {3726, 5277}, {3730, 69214}, {3735, 69243}, {3743, 3750}, {3745, 4658}, {3748, 6051}, {3749, 12514}, {3756, 52264}, {3841, 33130}, {3868, 5264}, {3869, 37610}, {3871, 4424}, {3873, 37522}, {3876, 62806}, {3878, 37588}, {3881, 37607}, {3891, 64184}, {3897, 16499}, {3905, 33936}, {3915, 5692}, {3930, 5280}, {3949, 16470}, {3970, 5276}, {3987, 54315}, {4006, 69211}, {4253, 16973}, {4255, 67538}, {4256, 37592}, {4257, 16496}, {4306, 8270}, {4332, 5290}, {4385, 48863}, {4415, 15171}, {4437, 7819}, {4642, 48696}, {4647, 32945}, {4653, 37080}, {4692, 54331}, {4694, 5253}, {4857, 69173}, {4864, 5045}, {4894, 25760}, {5178, 68946}, {5247, 49480}, {5269, 41863}, {5282, 7031}, {5299, 33299}, {5300, 33122}, {5399, 8555}, {5687, 37549}, {5730, 37542}, {5777, 64013}, {5846, 41014}, {5853, 34937}, {5903, 49454}, {6555, 37024}, {6693, 33121}, {7174, 62871}, {7186, 23156}, {7283, 24068}, {7794, 49752}, {9052, 10974}, {9053, 65543}, {9364, 67051}, {9709, 17054}, {11263, 33109}, {12594, 36742}, {12595, 36754}, {12699, 66106}, {13329, 63976}, {15172, 53534}, {15888, 63360}, {16061, 32029}, {16484, 27784}, {16784, 39244}, {17056, 63282}, {17122, 58565}, {17211, 20553}, {17698, 49524}, {17716, 62805}, {17719, 25639}, {17766, 56949}, {18446, 20280}, {19762, 37590}, {19947, 48283}, {20964, 38485}, {21342, 37582}, {23537, 63146}, {23814, 48281}, {24281, 59524}, {24549, 33937}, {25439, 37598}, {27785, 62849}, {28594, 41239}, {30435, 50995}, {31445, 49515}, {32922, 64185}, {33096, 68847}, {33105, 37731}, {33152, 36250}, {33153, 52367}, {33941, 33954}, {34707, 49747}, {37594, 49478}, {37603, 62865}, {37817, 57279}, {41538, 55086}, {44315, 48287}, {49448, 49530}, {50190, 62869}, {56179, 56220}, {57015, 69210}, {58405, 59593}, {60751, 64068}, {62811, 66235}, {62822, 66646}, {62860, 66640}, {63969, 67850}

X(69272) = reflection of X(i) in X(j) for these {i,j}: {58, 5266}, {5015, 3454}, {15955, 1}
X(69272) = X(513)-isoconjugate of X(43348)
X(69272) = X(39026)-Dao conjugate of X(43348)
X(69272) = barycentric product X(1)*X(33157)
X(69272) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 43348}, {33157, 75}
X(69272) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 10, 30117}, {1, 78, 995}, {1, 976, 30115}, {1, 978, 30148}, {1, 1698, 28082}, {1, 3216, 7191}, {1, 3293, 5262}, {1, 3632, 49487}, {1, 3679, 3924}, {1, 3811, 386}, {1, 3961, 10}, {1, 5268, 64675}, {1, 5293, 1125}, {1, 5312, 17017}, {1, 19861, 56804}, {1, 59311, 30143}, {8, 26228, 1714}, {8, 36565, 1}, {10, 49686, 1}, {72, 3744, 595}, {404, 62814, 3953}, {976, 3938, 1}, {1193, 67210, 1}, {2292, 3722, 3746}, {3699, 13741, 59666}, {3935, 5262, 3293}, {3961, 49686, 30117}, {4420, 7191, 3216}, {5015, 33126, 3454}, {10916, 66632, 45939}, {16496, 37552, 62858}, {17733, 49458, 50625}, {19767, 29815, 1}, {33117, 36505, 19846}, {37552, 62858, 4257}, {37592, 56176, 4256}, {37607, 49675, 3881}, {49465, 56176, 37592}

X(69273) = X(1)X(2)∩X(32)X(18061)

Barycentrics     a^4 - a^2*b^2 + a*b^3 + a^2*b*c - a^2*c^2 - b^2*c^2 + a*c^3 : :

X(69273) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (1, 0, -1, 1, 1, 0, 0, 0, -1).

X(69273) lies on these lines: {1, 2}, {32, 18061}, {56, 335}, {172, 18055}, {183, 257}, {385, 3061}, {668, 53165}, {846, 33063}, {894, 54317}, {986, 7824}, {1078, 3735}, {1447, 33890}, {2292, 17684}, {3120, 33823}, {3496, 7793}, {3552, 17738}, {3905, 33891}, {3944, 6655}, {6651, 17696}, {7755, 36230}, {7783, 49518}, {7833, 24851}, {8356, 63997}, {14621, 37539}, {16043, 60751}, {16921, 37717}, {16997, 17451}, {16998, 39244}, {17084, 17300}, {17565, 17889}, {17596, 33004}, {21965, 37688}, {24248, 32965}, {24291, 31276}, {25079, 33817}, {25591, 33816}, {27980, 48330}, {33824, 69173}, {60128, 60245}

X(69273) = crosssum of X(4426) and X(69243)
X(69273) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 29438, 30139}, {17696, 19582, 6651}, {30138, 30140, 1}

X(69274) = X(1)X(2)∩X(36)X(72)

Barycentrics     a*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c + a*b*c + b^2*c - a*c^2 + b*c^2 + c^3) : :
X(69274) = X[1] + 2 X[4420], 5 X[1698] - 4 X[24982]

X(69274) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (1, -1, -1, 1, 1, 1, 0, 0, 0).

X(69274) lies on these lines: {1, 2}, {3, 191}, {4, 5538}, {5, 47033}, {9, 2174}, {21, 3467}, {35, 960}, {36, 72}, {40, 3899}, {46, 5438}, {56, 3940}, {57, 3901}, {58, 37783}, {63, 7280}, {65, 4867}, {79, 11112}, {80, 1329}, {100, 3878}, {101, 17744}, {104, 63967}, {119, 61551}, {140, 21677}, {141, 24780}, {165, 6261}, {210, 1385}, {214, 2975}, {224, 18232}, {326, 17272}, {329, 4299}, {376, 16143}, {392, 3746}, {404, 758}, {405, 5426}, {442, 37701}, {474, 5902}, {484, 3869}, {500, 49723}, {515, 6903}, {517, 37251}, {518, 5563}, {549, 33858}, {572, 21033}, {609, 54406}, {908, 3585}, {912, 37561}, {940, 63310}, {942, 41696}, {944, 5531}, {946, 6900}, {952, 21031}, {956, 21842}, {958, 37525}, {993, 3876}, {1006, 40661}, {1054, 67033}, {1203, 37539}, {1259, 14793}, {1260, 8071}, {1319, 5288}, {1376, 5730}, {1388, 3711}, {1490, 52050}, {1699, 10525}, {1706, 25415}, {1818, 33082}, {1930, 4561}, {2077, 5887}, {2098, 12653}, {2099, 9709}, {2163, 56220}, {2169, 44687}, {2287, 25078}, {2292, 4256}, {2318, 33167}, {2475, 61703}, {2646, 5044}, {2802, 5330}, {2886, 5443}, {3035, 5445}, {3057, 48696}, {3061, 5540}, {3191, 42708}, {3218, 4067}, {3219, 5267}, {3304, 35272}, {3306, 12559}, {3337, 3868}, {3338, 3894}, {3419, 7741}, {3430, 37959}, {3452, 10572}, {3465, 56862}, {3496, 35342}, {3534, 63267}, {3576, 32153}, {3579, 31165}, {3583, 21616}, {3601, 30223}, {3647, 17549}, {3648, 36004}, {3681, 8666}, {3689, 9957}, {3697, 24926}, {3698, 50194}, {3702, 23580}, {3754, 62830}, {3812, 5425}, {3813, 16173}, {3814, 5086}, {3820, 10950}, {3833, 17535}, {3871, 3884}, {3874, 5253}, {3877, 8715}, {3890, 25439}, {3893, 41702}, {3897, 63961}, {3916, 59319}, {3925, 37737}, {3927, 5204}, {3929, 67706}, {3954, 21008}, {3962, 4880}, {3984, 35262}, {3987, 56009}, {4005, 37605}, {4006, 56530}, {4015, 51111}, {4040, 66995}, {4127, 4973}, {4134, 4881}, {4187, 37702}, {4188, 5131}, {4189, 12786}, {4190, 64289}, {4193, 37718}, {4251, 39244}, {4305, 18228}, {4309, 68599}, {4311, 21060}, {4316, 64002}, {4371, 18261}, {4413, 68668}, {4640, 59325}, {4855, 5010}, {4857, 41012}, {4863, 11373}, {4866, 30392}, {4930, 64963}, {4999, 5444}, {5047, 35016}, {5110, 21810}, {5119, 15829}, {5124, 21873}, {5178, 24387}, {5221, 16417}, {5223, 60994}, {5259, 24929}, {5266, 5315}, {5270, 21077}, {5289, 5541}, {5433, 12739}, {5441, 11113}, {5453, 49724}, {5496, 21020}, {5525, 9310}, {5526, 25066}, {5535, 6924}, {5537, 12672}, {5657, 40257}, {5660, 18242}, {5690, 6265}, {5691, 5720}, {5696, 36868}, {5718, 63319}, {5722, 24954}, {5741, 36974}, {5777, 50371}, {5784, 60885}, {5790, 46920}, {5794, 7951}, {5836, 11009}, {5837, 59587}, {5883, 17531}, {5884, 6940}, {6001, 59326}, {6174, 61524}, {6196, 34996}, {6264, 12645}, {6282, 6869}, {6597, 15175}, {6684, 21740}, {6833, 64335}, {6857, 45085}, {6873, 7989}, {6905, 31806}, {6906, 20117}, {6909, 31803}, {6911, 37625}, {6941, 15017}, {6946, 31870}, {6949, 68278}, {7031, 39248}, {7343, 56280}, {7373, 41711}, {7982, 46917}, {7987, 18446}, {7991, 48363}, {8227, 37533}, {8703, 63278}, {8728, 26725}, {9589, 63992}, {9708, 34471}, {9945, 15338}, {9955, 31159}, {10393, 53054}, {10399, 37244}, {10483, 58798}, {10609, 57288}, {10826, 30827}, {10884, 58221}, {10912, 11524}, {10914, 63210}, {11012, 31837}, {11235, 45035}, {11246, 17563}, {11279, 32635}, {11280, 62826}, {11281, 17529}, {11522, 37569}, {11551, 12436}, {11567, 38176}, {11571, 66630}, {11682, 54286}, {11684, 13587}, {11813, 52367}, {11826, 34789}, {12047, 57284}, {12520, 16192}, {12526, 58887}, {12527, 21578}, {12528, 63983}, {12532, 34758}, {12625, 25522}, {12680, 64659}, {12738, 34606}, {12740, 64056}, {13146, 15680}, {13370, 17625}, {13729, 54193}, {14110, 44425}, {14432, 44827}, {14804, 37308}, {14923, 64896}, {15104, 22770}, {15931, 33597}, {15950, 31419}, {16061, 46899}, {16118, 17579}, {16371, 37524}, {16374, 64753}, {16418, 41872}, {16454, 41812}, {17270, 44179}, {17606, 31263}, {17757, 37710}, {17857, 37611}, {18243, 37429}, {18253, 37298}, {18397, 22766}, {18398, 25524}, {18465, 64072}, {18480, 31160}, {18493, 31140}, {18525, 31141}, {19649, 54180}, {19765, 27785}, {19907, 38763}, {21075, 45287}, {21635, 37437}, {21839, 33863}, {22072, 45272}, {22791, 34612}, {24046, 49454}, {24390, 37735}, {24467, 59332}, {24703, 65134}, {25466, 37731}, {25590, 55391}, {25591, 48863}, {26066, 65142}, {26321, 56762}, {26770, 30729}, {27784, 64415}, {28452, 49177}, {28458, 49178}, {28628, 41859}, {30196, 62320}, {30323, 63137}, {30384, 63146}, {30556, 65147}, {30557, 65148}, {31246, 68905}, {31259, 64680}, {31435, 59337}, {31445, 37600}, {31838, 34486}, {32911, 63292}, {33096, 66658}, {33857, 63286}, {34200, 63276}, {34720, 50891}, {34791, 37602}, {35202, 45120}, {35637, 37442}, {36005, 63280}, {37300, 54302}, {37587, 62874}, {37603, 49500}, {37617, 67979}, {37618, 57279}, {37633, 63354}, {37656, 64710}, {37659, 63446}, {37662, 63360}, {37707, 64087}, {37720, 68616}, {37837, 59320}, {39778, 68277}, {40663, 47742}, {41389, 59327}, {41698, 67855}, {41863, 51816}, {45047, 66551}, {45825, 56179}, {46877, 52680}, {48307, 68101}, {50082, 62211}, {51714, 62837}, {51975, 56422}, {52026, 59340}, {58565, 63159}, {61705, 64740}, {63068, 63339}, {63469, 64150}

X(69274) = reflection of X(i) in X(j) for these {i,j}: {3336, 404}, {4857, 41012}, {5563, 17614}, {37702, 4187}, {62837, 51714}
X(69274) = X(513)-isoconjugate of X(26711)
X(69274) = X(i)-Dao conjugate of X(j) for these (i,j): {18447, 3585}, {39026, 26711}
X(69274) = crossdifference of every pair of points on line {649, 47227}
X(69274) = barycentric product X(1)*X(33168)
X(69274) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 26711}, {33168, 75}
X(69274) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 200, 3632}, {1, 936, 1698}, {1, 4668, 3872}, {1, 5529, 3216}, {1, 6765, 51093}, {1, 8583, 25055}, {1, 19875, 19860}, {1, 34595, 54392}, {1, 59294, 49494}, {1, 64850, 54318}, {2, 22836, 1}, {3, 5692, 191}, {3, 5693, 1768}, {8, 26364, 18395}, {8, 30144, 1}, {10, 4511, 1}, {36, 72, 6763}, {56, 3940, 5904}, {72, 59691, 36}, {78, 997, 1}, {78, 19861, 3811}, {100, 3878, 11010}, {101, 33299, 17744}, {191, 15015, 3}, {210, 1385, 5258}, {214, 3678, 2975}, {392, 56176, 3746}, {405, 37571, 5426}, {405, 56177, 37571}, {474, 12635, 5902}, {908, 17647, 3585}, {960, 5440, 35}, {976, 995, 1}, {997, 3811, 19861}, {1125, 34772, 1}, {1193, 30115, 1}, {1319, 34790, 5288}, {1376, 5730, 5903}, {2646, 5044, 5251}, {3306, 12559, 67971}, {3338, 11523, 3894}, {3419, 25681, 7741}, {3636, 3957, 1}, {3811, 19861, 1}, {3869, 25440, 484}, {3877, 8715, 37563}, {3920, 50604, 1}, {3962, 37582, 4880}, {3984, 35262, 62858}, {4188, 69227, 5131}, {4251, 39244, 56532}, {4855, 12514, 5010}, {5289, 5687, 5697}, {5426, 5506, 405}, {5438, 68602, 46}, {5687, 5697, 5541}, {5690, 6265, 11014}, {5720, 63391, 5691}, {5902, 12635, 16126}, {6282, 63988, 64005}, {6326, 15015, 34600}, {6700, 6737, 1737}, {17531, 34195, 5883}, {18395, 26364, 1698}, {20117, 54192, 6906}, {21616, 57287, 3583}, {22837, 26364, 64673}, {22935, 35204, 15015}, {24929, 25917, 5259}, {24987, 59719, 3584}, {30148, 36565, 1}, {35204, 44782, 191}, {37837, 64107, 59320}

X(69275) = X(1)X(2)∩X(35)X(72)

Barycentrics     a*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2 + c^3) : :
X(69275) = X[1] + 2 X[3935], 3 X[1] - 2 X[38460], 5 X[1698] - 4 X[1737], 5 X[1698] - 8 X[6745], 6 X[3582] - 7 X[3624], 3 X[3582] - 2 X[26015], 7 X[3624] - 4 X[26015], 3 X[3679] - 4 X[6735], 3 X[3679] - 2 X[41684], 3 X[3935] + X[38460], 3 X[4511] - X[38460], 9 X[25055] - 8 X[44675], 3 X[1727] - 4 X[4640], 2 X[36] - 3 X[15015], 5 X[36] - 6 X[35271], 4 X[5440] - 3 X[15015], 5 X[5440] - 3 X[35271], 5 X[15015] - 4 X[35271], 2 X[214] + X[62236], 2 X[3689] + X[4867], 4 X[3689] - X[5541], 2 X[4867] + X[5541], 2 X[1532] - 3 X[5660], 3 X[1749] - 4 X[59670], 2 X[3218] - 3 X[5131], 4 X[3814] - 3 X[37718], 2 X[4973] - 3 X[13587], 3 X[13587] - X[62235], 4 X[5087] - 3 X[65140], 3 X[35204] - 2 X[41542], 5 X[15017] - 4 X[67857], 4 X[15325] - 5 X[64012], 2 X[51463] - 5 X[64012]

X(69275) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (1, -1, -1, -1, 1, 1, 0, 0, 0).

X(69275) lies on these lines: {1, 2}, {3, 5904}, {9, 584}, {12, 47033}, {21, 3678}, {30, 12738}, {35, 72}, {36, 518}, {38, 4256}, {40, 14988}, {41, 17744}, {46, 3901}, {55, 3940}, {57, 3894}, {58, 4722}, {63, 5010}, {65, 16126}, {79, 66107}, {80, 6596}, {100, 484}, {101, 2752}, {104, 54192}, {119, 54154}, {149, 11813}, {153, 54193}, {165, 18446}, {210, 5251}, {214, 54391}, {224, 54422}, {284, 3949}, {329, 4302}, {404, 3337}, {442, 37731}, {474, 18398}, {480, 18412}, {515, 5531}, {517, 3689}, {522, 3465}, {528, 51409}, {529, 10609}, {609, 69218}, {756, 4653}, {765, 1110}, {859, 4557}, {908, 3583}, {912, 1768}, {943, 15910}, {956, 37525}, {958, 37571}, {960, 3746}, {993, 3681}, {999, 41711}, {1043, 1089}, {1046, 67033}, {1071, 59326}, {1111, 68928}, {1145, 5855}, {1155, 4880}, {1168, 52925}, {1203, 5266}, {1259, 41686}, {1260, 8069}, {1319, 37736}, {1320, 13143}, {1324, 2948}, {1329, 37702}, {1331, 6149}, {1376, 5902}, {1385, 5288}, {1420, 63914}, {1478, 25568}, {1479, 3189}, {1490, 16127}, {1532, 5660}, {1621, 10176}, {1699, 5720}, {1739, 56009}, {1743, 5037}, {1749, 59670}, {1770, 64289}, {1776, 10393}, {1780, 52387}, {1781, 22021}, {1784, 1897}, {2136, 30323}, {2163, 56179}, {2280, 56532}, {2287, 59733}, {2292, 33771}, {2329, 4006}, {2646, 5258}, {2771, 35000}, {2801, 6909}, {2802, 45764}, {2886, 37701}, {2900, 3586}, {2975, 37616}, {3065, 47320}, {3158, 3899}, {3174, 52050}, {3218, 5131}, {3219, 4134}, {3243, 51816}, {3245, 44663}, {3256, 64041}, {3336, 3868}, {3338, 5438}, {3419, 7951}, {3428, 15104}, {3434, 18393}, {3509, 35342}, {3510, 34996}, {3550, 49500}, {3555, 5563}, {3570, 49753}, {3579, 3962}, {3585, 21077}, {3601, 41229}, {3612, 57279}, {3647, 3988}, {3680, 21398}, {3684, 5540}, {3693, 5526}, {3699, 3992}, {3711, 9708}, {3715, 16418}, {3722, 40091}, {3744, 5315}, {3753, 5425}, {3754, 34195}, {3813, 37735}, {3814, 37718}, {3833, 9342}, {3869, 8715}, {3871, 3878}, {3876, 5248}, {3877, 25439}, {3880, 12653}, {3881, 5253}, {3893, 10222}, {3913, 5697}, {3916, 59325}, {3925, 5719}, {3927, 5217}, {3943, 7359}, {3954, 18755}, {3984, 12514}, {4005, 31445}, {4015, 5260}, {4053, 16548}, {4067, 56288}, {4088, 44827}, {4127, 11684}, {4149, 49469}, {4251, 33299}, {4257, 32912}, {4262, 5282}, {4304, 21060}, {4305, 5815}, {4312, 61011}, {4324, 64002}, {4413, 15934}, {4417, 4680}, {4424, 60714}, {4533, 5302}, {4539, 15481}, {4547, 32635}, {4561, 14210}, {4641, 37589}, {4694, 49675}, {4727, 62239}, {4731, 64732}, {4855, 7280}, {4857, 21616}, {4860, 16417}, {4863, 5886}, {4866, 56027}, {4873, 62215}, {4900, 14497}, {4973, 13587}, {4975, 68969}, {4986, 69044}, {4995, 33857}, {5044, 5259}, {5048, 41702}, {5087, 65140}, {5127, 37783}, {5172, 33667}, {5176, 9897}, {5178, 25639}, {5180, 20095}, {5220, 16370}, {5223, 30282}, {5270, 17647}, {5291, 20693}, {5396, 32861}, {5441, 57288}, {5443, 24390}, {5497, 32849}, {5528, 28534}, {5534, 63391}, {5536, 6905}, {5537, 6001}, {5587, 37533}, {5687, 5903}, {5690, 37733}, {5691, 10526}, {5693, 11248}, {5694, 11849}, {5763, 6253}, {5771, 21155}, {5794, 37719}, {5844, 6265}, {5853, 30384}, {5883, 63159}, {6154, 28174}, {6198, 53008}, {6261, 7991}, {6583, 45976}, {6600, 40292}, {6740, 15065}, {6762, 37618}, {6769, 9589}, {6882, 49176}, {6906, 63967}, {6912, 15064}, {6920, 64693}, {6940, 12005}, {7031, 54406}, {7741, 68616}, {7957, 64804}, {7972, 38455}, {7982, 45770}, {9020, 33844}, {9581, 41709}, {9591, 57281}, {9945, 15326}, {10164, 18444}, {10310, 15071}, {10572, 12437}, {10826, 12625}, {10884, 16192}, {10902, 31837}, {10914, 11009}, {11280, 14923}, {11362, 21740}, {11491, 31806}, {11499, 37625}, {11517, 36152}, {11520, 67971}, {11529, 46917}, {11531, 63986}, {12047, 63146}, {12245, 40257}, {12432, 57283}, {12513, 21842}, {12520, 63469}, {12526, 59316}, {12607, 37710}, {12645, 46920}, {12739, 40663}, {12740, 26726}, {13146, 17484}, {13407, 57284}, {13465, 67712}, {13743, 56762}, {14110, 64116}, {15017, 67857}, {15228, 17768}, {15325, 51463}, {15556, 15932}, {15733, 51768}, {15931, 64107}, {16143, 31730}, {16200, 64203}, {17057, 31479}, {17151, 55391}, {17531, 58565}, {17543, 51573}, {17563, 52783}, {17602, 48847}, {17605, 31159}, {17614, 34791}, {17719, 68946}, {17735, 21839}, {17745, 25066}, {18254, 63281}, {18417, 56181}, {21031, 37730}, {21033, 55100}, {22765, 22935}, {23071, 33649}, {23708, 24392}, {24036, 63087}, {24299, 58630}, {24391, 59587}, {25415, 63137}, {25416, 32426}, {25524, 50190}, {25542, 51715}, {25681, 37720}, {26921, 59331}, {28204, 35459}, {29311, 38483}, {30305, 64146}, {30729, 68971}, {31018, 54342}, {31053, 61703}, {31423, 37615}, {31493, 64677}, {32049, 37707}, {32760, 51379}, {32775, 48843}, {32911, 49480}, {32931, 48863}, {33065, 48835}, {33597, 59320}, {33858, 61524}, {34612, 39542}, {34784, 52769}, {34997, 37311}, {35200, 36037}, {35258, 51817}, {35262, 37587}, {35338, 68843}, {36006, 65112}, {36909, 52371}, {37106, 60912}, {37559, 63310}, {37583, 41538}, {37706, 64087}, {37799, 56877}, {40263, 64740}, {41338, 52026}, {41389, 65144}, {41701, 64011}, {46974, 68591}, {47270, 50747}, {50082, 62238}, {52362, 63802}, {54286, 64135}, {54981, 69214}, {56220, 56343}, {58798, 65134}, {59329, 67970}, {61030, 64154}, {61035, 64155}, {61146, 63143}, {63468, 64150}

X(69275) = midpoint of X(i) and X(j) for these {i,j}: {153, 54193}, {3935, 4511}, {4867, 48696}, {5180, 20095}, {5531, 5538}, {54391, 62236}
X(69275) = reflection of X(i) in X(j) for these {i,j}: {1, 4511}, {36, 5440}, {80, 17757}, {104, 54192}, {149, 11813}, {484, 100}, {1737, 6745}, {1757, 765}, {1768, 2077}, {3065, 48698}, {3583, 908}, {4880, 1155}, {5536, 6905}, {5541, 48696}, {9897, 5176}, {12653, 63210}, {15326, 9945}, {22765, 22935}, {36975, 10609}, {41684, 6735}, {41702, 5048}, {48696, 3689}, {49176, 6882}, {51463, 15325}, {51768, 60885}, {54154, 119}, {54391, 214}, {62235, 4973}, {64155, 61035}, {64896, 62826}
X(69275) = X(56121)-anticomplementary conjugate of X(3436)
X(69275) = X(i)-Ceva conjugate of X(j) for these (i,j): {18359, 9}, {37783, 17796}
X(69275) = X(5497)-cross conjugate of X(1)
X(69275) = X(i)-isoconjugate of X(j) for these (i,j): {6, 21907}, {56, 11604}, {58, 5620}, {513, 1290}, {649, 65238}, {667, 35156}, {1333, 68363}, {3733, 66280}, {5172, 55012}, {34442, 58076}, {39267, 61453}
X(69275) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 11604}, {9, 21907}, {10, 5620}, {37, 68363}, {2323, 3218}, {5375, 65238}, {6631, 35156}, {18455, 3583}, {35090, 514}, {39026, 1290}, {53988, 7649}
X(69275) = cevapoint of X(1) and X(13146)
X(69275) = crosspoint of X(765) and X(51562)
X(69275) = crosssum of X(244) and X(53314)
X(69275) = trilinear pole of line {8674, 68818}
X(69275) = crossdifference of every pair of points on line {649, 2260}
X(69275) = barycentric product X(i)*X(j) for these {i,j}: {1, 32849}, {10, 37783}, {63, 56877}, {75, 17796}, {78, 37799}, {99, 68818}, {190, 8674}, {306, 2074}, {312, 5172}, {313, 19622}, {321, 5127}, {646, 51646}, {1018, 65669}, {1978, 42670}, {4033, 42741}, {4102, 41542}, {4561, 47235}, {4997, 41541}, {18359, 35204}, {20920, 51470}
X(69275) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 21907}, {9, 11604}, {10, 68363}, {37, 5620}, {100, 65238}, {101, 1290}, {190, 35156}, {1018, 66280}, {2074, 27}, {5127, 81}, {5172, 57}, {5497, 33129}, {8674, 514}, {15910, 68358}, {16164, 18653}, {16548, 58076}, {17796, 1}, {19622, 58}, {32849, 75}, {35204, 3218}, {37783, 86}, {37799, 273}, {41541, 3911}, {41542, 553}, {42670, 649}, {42741, 1019}, {47235, 7649}, {51646, 3669}, {56877, 92}, {65669, 7199}, {66016, 64115}, {68818, 523}
X(69275) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 200, 3679}, {1, 936, 3624}, {1, 4677, 3872}, {1, 5529, 49997}, {1, 6765, 3633}, {1, 19875, 54318}, {1, 31855, 60353}, {1, 34595, 64675}, {1, 64850, 54392}, {3, 5904, 6763}, {8, 22836, 1}, {10, 34772, 1}, {35, 72, 191}, {36, 5440, 15015}, {42, 30115, 1}, {46, 11523, 3901}, {55, 3940, 5692}, {65, 41696, 16126}, {72, 56176, 35}, {78, 3811, 1}, {78, 3870, 997}, {100, 41689, 34600}, {101, 3930, 5525}, {145, 30144, 1}, {210, 24929, 5251}, {386, 976, 1}, {404, 3874, 3337}, {551, 3957, 1}, {956, 56177, 37525}, {995, 3938, 1}, {997, 3811, 3870}, {997, 3870, 1}, {2646, 34790, 5258}, {3158, 68602, 5119}, {3555, 59691, 5563}, {3684, 57015, 5540}, {3689, 4867, 5541}, {3868, 25440, 3336}, {3869, 8715, 11010}, {3871, 3878, 37563}, {3913, 5730, 5697}, {3925, 5719, 26725}, {4015, 35016, 5260}, {4420, 34772, 10}, {4855, 62858, 7280}, {5044, 5259, 5506}, {5044, 37080, 5259}, {5119, 68602, 3899}, {5251, 24929, 5426}, {5438, 41863, 3338}, {5524, 60353, 31855}, {5552, 20013, 49168}, {5552, 49168, 18395}, {5687, 12635, 5903}, {5720, 37569, 1699}, {6735, 38460, 9623}, {6735, 41684, 3679}, {6737, 59722, 10039}, {6769, 63988, 9589}, {6940, 12005, 35010}, {7191, 49686, 1}, {12437, 21075, 10572}, {13587, 62235, 4973}, {17857, 37531, 5691}, {19767, 30142, 1}, {21077, 57287, 3585}, {22021, 54316, 1781}, {33597, 63976, 59320}, {41686, 59334, 54432}, {54318, 67097, 19875}

X(69276) = X(1)X(2)∩X(9)X(7774)

Barycentrics     a^3*b + a^2*b^2 + a*b^3 - b^4 + a^3*c + a*b^2*c + a^2*c^2 + a*b*c^2 + a*c^3 - c^4 : :

X(69276) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (0, 1, 1, 0, 1, 1, -1, 0, 0).

X(69276) lies on these lines: {1, 2}, {9, 7774}, {35, 56772}, {36, 56771}, {37, 325}, {38, 4766}, {44, 41624}, {45, 9766}, {55, 21485}, {56, 56773}, {142, 33891}, {183, 4851}, {192, 7179}, {226, 335}, {230, 17390}, {319, 26244}, {321, 46238}, {344, 7736}, {385, 3879}, {908, 31120}, {988, 7791}, {1003, 66639}, {1100, 7792}, {1255, 30760}, {1281, 50307}, {1447, 17300}, {1449, 16989}, {1743, 63017}, {1785, 62954}, {1848, 46507}, {1959, 20684}, {3290, 37664}, {3314, 4357}, {3329, 17353}, {3666, 26590}, {3674, 33890}, {3744, 26629}, {3752, 26582}, {3815, 17243}, {3835, 30671}, {3948, 51861}, {3950, 33889}, {3971, 63234}, {3993, 5988}, {3995, 22032}, {4078, 4518}, {4104, 27495}, {4110, 30758}, {4136, 59509}, {4144, 24690}, {4153, 16887}, {4339, 32973}, {4416, 7779}, {4422, 9300}, {4437, 37662}, {4643, 7788}, {4657, 7868}, {4865, 24586}, {5015, 16060}, {5025, 13161}, {5249, 19600}, {5257, 31090}, {5266, 7807}, {5300, 33830}, {5847, 52133}, {6645, 67969}, {6651, 56078}, {6656, 37592}, {7778, 16777}, {7833, 66692}, {7840, 50093}, {8356, 37599}, {9865, 17760}, {11163, 41313}, {11174, 17279}, {15271, 17311}, {16667, 63045}, {16925, 37552}, {16990, 17296}, {16997, 68797}, {17147, 31031}, {17184, 31023}, {17257, 37668}, {17261, 56555}, {17332, 50771}, {17374, 37671}, {17598, 30837}, {17738, 33106}, {17747, 49514}, {18743, 52662}, {18905, 37596}, {20496, 59505}, {20533, 61018}, {21073, 25264}, {21993, 37607}, {22329, 50125}, {23536, 33841}, {24177, 24221}, {24602, 33072}, {25101, 63018}, {26685, 37665}, {26686, 37539}, {27474, 32855}, {28773, 64349}, {30077, 50618}, {30701, 31402}, {30710, 57859}, {31041, 69251}, {31087, 33864}, {32985, 66680}, {33006, 66691}, {33017, 66672}, {33184, 66675}, {33219, 48819}, {33220, 48824}, {33251, 48818}, {33255, 48827}, {35297, 37589}, {37554, 56733}, {37573, 56561}, {37594, 56731}, {40099, 52135}, {41310, 63101}, {41318, 66882}, {41323, 56517}, {49563, 49676}, {57518, 59518}, {63065, 66637}

X(69276) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6542, 7081}, {2, 29570, 29634}, {2, 29840, 239}, {2, 30179, 10}, {3912, 24239, 2}, {30122, 30125, 1}

X(69277) = X(1)X(2)∩X(6)X(274)

Barycentrics     a^3*b + a^3*c - a*b^2*c - a*b*c^2 - b^2*c^2 : :

X(69277) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (0, 1, 0, 0, 0, -1, 0, 0, -1).

X(69277) lies on these lines: {1, 2}, {6, 274}, {9, 25264}, {32, 33295}, {45, 32026}, {58, 16915}, {75, 213}, {76, 2238}, {192, 3294}, {194, 16552}, {292, 4372}, {304, 16782}, {314, 27623}, {330, 39797}, {333, 980}, {335, 5904}, {384, 1724}, {391, 4352}, {405, 20154}, {579, 8866}, {668, 29486}, {673, 29471}, {742, 33933}, {894, 32092}, {940, 33035}, {992, 30022}, {1043, 33821}, {1100, 20150}, {1107, 17348}, {1150, 33830}, {1203, 14621}, {1330, 17680}, {1730, 4209}, {2176, 4361}, {2271, 16992}, {3210, 62817}, {3230, 17144}, {3454, 16906}, {3552, 52680}, {3570, 7751}, {3662, 24790}, {3702, 26689}, {3730, 17759}, {3747, 32860}, {3759, 20963}, {3780, 64133}, {3797, 64184}, {3952, 24080}, {3980, 40749}, {4083, 29458}, {4251, 16998}, {4256, 17684}, {4257, 17693}, {4270, 26110}, {4295, 39721}, {4340, 33039}, {4383, 7770}, {4416, 24214}, {4647, 46899}, {4673, 35274}, {4974, 5156}, {5025, 68946}, {5115, 59631}, {5224, 25499}, {5278, 40773}, {5283, 17277}, {5712, 33026}, {5718, 33033}, {7754, 37658}, {7785, 60149}, {7793, 35342}, {7807, 35466}, {10471, 27644}, {14829, 33828}, {16466, 20172}, {16574, 21384}, {16672, 32090}, {16675, 32102}, {16690, 22316}, {16710, 39950}, {16777, 32009}, {16783, 17000}, {16911, 20132}, {16912, 20138}, {16913, 20158}, {16917, 37522}, {16926, 43531}, {16975, 34063}, {17117, 32104}, {17121, 52716}, {17137, 24190}, {17175, 17379}, {17448, 64556}, {17490, 20367}, {17499, 34284}, {17541, 37680}, {17686, 32911}, {17694, 37646}, {17750, 60706}, {18140, 37673}, {18144, 29757}, {18147, 25661}, {18171, 29767}, {19742, 62636}, {19765, 33036}, {20888, 24514}, {20970, 37632}, {24522, 29552}, {24535, 31286}, {25270, 68895}, {25497, 33954}, {26558, 64172}, {27269, 46196}, {27318, 68950}, {28350, 34282}, {29473, 33825}, {29544, 65169}, {30945, 33297}, {32922, 56542}, {32968, 63126}, {32992, 37663}, {33045, 37693}, {33198, 37681}, {33816, 48863}, {33831, 48835}, {33888, 64429}, {33930, 50025}, {50577, 67984}, {63518, 68987}, {66152, 69242}

X(69277) = X(21877)-Dao conjugate of X(21080)
X(69277) = crosssum of X(i) and X(j) for these (i,j): {661, 23500}, {3248, 23503}
X(69277) = barycentric product X(i)*X(j) for these {i,j}: {75, 16690}, {86, 22316}
X(69277) = barycentric quotient X(i)/X(j) for these {i,j}: {16690, 1}, {22316, 10}
X(69277) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 4384, 16819}, {10, 17033, 30114}, {194, 17349, 16552}, {239, 16826, 3187}, {239, 16827, 1}, {384, 20142, 1724}, {978, 17026, 26959}, {1193, 17030, 30112}, {1193, 24592, 17030}, {2176, 4361, 17143}, {3008, 59303, 29960}, {3216, 29433, 2}, {3759, 31997, 20963}, {16552, 62755, 194}, {16569, 29557, 2}, {17277, 33296, 5283}, {17749, 29455, 2}, {20036, 24599, 27304}, {24621, 37652, 18206}, {29381, 31855, 53675}, {34284, 37657, 17499}

X(69278) = X(1)X(2)∩X(12)X(335)

Barycentrics     a^2*b^2 + a*b^3 - b^4 + a^2*b*c + a^2*c^2 + b^2*c^2 + a*c^3 - c^4 : :

X(69278) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (0, 0, 1, 1, 1, 0, -1, 0, 1).

X(69278) lies on these lines: {1, 2}, {12, 335}, {41, 28000}, {75, 23897}, {192, 27691}, {257, 325}, {384, 37717}, {423, 2907}, {846, 17685}, {986, 5025}, {1054, 17565}, {1506, 18061}, {1655, 21044}, {1916, 7179}, {2292, 17669}, {3061, 7777}, {3496, 7785}, {3662, 21057}, {3684, 27967}, {3735, 7752}, {3903, 30544}, {3944, 32966}, {4414, 33824}, {4865, 27963}, {5724, 26686}, {6376, 27733}, {6651, 56313}, {6655, 17596}, {7779, 17739}, {7836, 24291}, {9698, 36230}, {13571, 50029}, {14041, 24851}, {14063, 24248}, {16044, 17738}, {17062, 33891}, {17247, 27704}, {17248, 27688}, {17363, 27958}, {17694, 63360}, {17759, 21029}, {21018, 56023}, {23903, 68867}, {24443, 33841}, {26098, 27997}, {27481, 30436}, {27697, 48628}, {27976, 32861}, {27982, 50289}, {32969, 60751}, {33228, 63997}, {41324, 69248}, {41875, 69009}

X(69278) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 8, 27954}, {10, 30170, 40859}, {10, 49544, 8}, {325, 21965, 257}, {18395, 30122, 30114}

X(69279) = X(1)X(2)∩X(3)X(7270)

Barycentrics     a^2*b^2 - b^4 + a^2*b*c - b^3*c + a^2*c^2 - b*c^3 - c^4 : :

X(69279) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (0, 0, 1, 1, 0, 0, -1, -1, 0).

X(69279) lies on these lines: {1, 2}, {3, 7270}, {4, 345}, {5, 312}, {9, 4109}, {12, 3703}, {21, 5016}, {28, 1792}, {35, 4680}, {37, 52258}, {39, 34542}, {46, 4645}, {55, 5015}, {63, 1330}, {72, 4417}, {75, 442}, {76, 17866}, {85, 3933}, {100, 5300}, {147, 18596}, {190, 58798}, {264, 1969}, {304, 325}, {321, 2476}, {322, 41005}, {333, 5791}, {341, 17757}, {344, 5084}, {346, 3091}, {348, 32818}, {350, 52257}, {377, 17740}, {381, 42033}, {405, 33116}, {406, 28420}, {427, 19799}, {469, 42706}, {573, 1759}, {908, 3710}, {942, 18134}, {964, 32779}, {970, 4158}, {986, 2887}, {993, 36974}, {1043, 3419}, {1046, 32946}, {1062, 4123}, {1089, 3790}, {1220, 5725}, {1229, 6991}, {1259, 5081}, {1329, 3932}, {1453, 56519}, {1468, 33119}, {1479, 3685}, {1565, 21605}, {1791, 3417}, {1836, 63996}, {1838, 25252}, {1930, 7179}, {2049, 19808}, {2276, 16886}, {2292, 25760}, {2345, 31402}, {2475, 33168}, {2478, 17776}, {2886, 3704}, {2968, 52346}, {3090, 28808}, {3210, 23537}, {3263, 20235}, {3295, 4514}, {3416, 5135}, {3454, 27184}, {3487, 30828}, {3501, 4071}, {3545, 42032}, {3614, 6057}, {3662, 3670}, {3666, 16062}, {3673, 69093}, {3693, 36652}, {3701, 11681}, {3702, 11680}, {3712, 6284}, {3717, 21075}, {3719, 5709}, {3729, 9612}, {3730, 4153}, {3746, 4894}, {3752, 33833}, {3797, 5025}, {3836, 24174}, {3841, 28612}, {3868, 3936}, {3871, 5014}, {3876, 5741}, {3913, 5100}, {3915, 32844}, {3926, 5088}, {3927, 33066}, {3931, 32773}, {3974, 10588}, {3977, 64002}, {3998, 5125}, {4035, 24391}, {4150, 41507}, {4165, 17451}, {4185, 19845}, {4187, 18743}, {4193, 4358}, {4197, 4359}, {4202, 4850}, {4273, 17275}, {4276, 17521}, {4296, 28774}, {4387, 10896}, {4388, 12514}, {4438, 5247}, {4671, 5141}, {4673, 24390}, {4692, 37719}, {4737, 12607}, {4865, 5255}, {4872, 7776}, {4968, 33089}, {4975, 37720}, {5044, 5233}, {5046, 32849}, {5051, 28606}, {5086, 49492}, {5179, 27523}, {5192, 33157}, {5264, 50289}, {5439, 17234}, {5687, 32850}, {5711, 33073}, {5716, 37176}, {5717, 39559}, {5758, 9535}, {5827, 9708}, {6327, 56288}, {6679, 16478}, {7162, 43749}, {7187, 7906}, {7230, 39565}, {7385, 17742}, {7513, 40445}, {7522, 19838}, {7752, 33939}, {7796, 20924}, {8258, 62841}, {8715, 63139}, {8728, 19804}, {11491, 37431}, {11501, 41346}, {12572, 56078}, {12618, 27544}, {13407, 24349}, {13740, 32777}, {13741, 17279}, {14020, 27754}, {16052, 50067}, {16466, 33071}, {16706, 56780}, {16842, 17263}, {17038, 34920}, {17095, 69158}, {17170, 37668}, {17264, 17556}, {17303, 19280}, {17363, 64072}, {17527, 30829}, {17530, 42034}, {17532, 50044}, {17533, 20942}, {17577, 50105}, {17677, 50065}, {17889, 67983}, {18156, 69254}, {19582, 21616}, {19763, 68696}, {19810, 37056}, {19827, 56985}, {20254, 21530}, {20946, 50206}, {21077, 32937}, {21281, 69038}, {21857, 40941}, {21935, 32848}, {24443, 25957}, {24851, 32934}, {25248, 31023}, {25306, 67967}, {25466, 69091}, {25962, 54284}, {26117, 62871}, {27339, 54346}, {27542, 54295}, {28611, 41859}, {30811, 37549}, {31053, 56318}, {31424, 59779}, {32929, 52367}, {32939, 57282}, {33070, 57280}, {33092, 63800}, {33111, 49598}, {33134, 64071}, {33745, 41258}, {33746, 44115}, {34188, 52364}, {35203, 45048}, {37038, 50050}, {37151, 37527}, {38408, 54405}, {41230, 67960}, {42029, 50042}, {44694, 44706}, {47712, 48012}, {53821, 57807}, {54429, 55868}, {56564, 64969}, {56729, 62338}, {56778, 62802}, {60108, 60197}, {60448, 62330}, {64184, 68946}

X(69279) = isotomic conjugate of the isogonal conjugate of X(26893)
X(69279) = X(i)-isoconjugate of X(j) for these (i,j): {513, 15440}, {1333, 60088}, {1437, 68581}
X(69279) = X(i)-Dao conjugate of X(j) for these (i,j): {37, 60088}, {5509, 667}, {39026, 15440}
X(69279) = barycentric product X(i)*X(j) for these {i,j}: {76, 26893}, {312, 37591}, {313, 4269}, {4215, 27801}
X(69279) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 60088}, {101, 15440}, {1826, 68581}, {4215, 1333}, {4269, 58}, {26893, 6}, {37591, 57}
X(69279) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 25441, 29841}, {1, 30171, 3705}, {4, 345, 7283}, {5, 3695, 312}, {8, 3085, 68889}, {8, 19843, 16821}, {8, 30741, 19843}, {8, 34772, 50624}, {10, 3687, 9534}, {10, 17748, 43}, {10, 29671, 1}, {10, 30172, 29641}, {12, 3703, 4385}, {304, 325, 17181}, {306, 6734, 10449}, {1329, 3932, 46937}, {4388, 56313, 12514}, {5016, 33113, 21}, {5791, 5814, 333}, {7270, 32851, 3}, {10528, 31091, 8}, {11681, 32862, 3701}, {21674, 69252, 31339}, {27020, 30149, 3661}, {30134, 30176, 17397}

X(69280) = X(1)X(2)∩X(3)X(5016)

Barycentrics     (a - b - c)*(a*b^2 + b^3 + a*c^2 + c^3) : :

X(69280) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (0, 0, 1, 0, 0, -1, -1, -1, 0).

X(69280) lies on these lines: {1, 2}, {3, 5016}, {4, 17740}, {5, 321}, {11, 3702}, {12, 4968}, {21, 32851}, {28, 5081}, {34, 28774}, {36, 36974}, {46, 6327}, {60, 333}, {72, 5741}, {75, 2476}, {100, 5015}, {242, 28101}, {312, 4193}, {318, 5142}, {325, 20911}, {345, 2478}, {346, 6919}, {381, 50044}, {404, 7270}, {405, 33113}, {442, 4359}, {672, 4109}, {908, 56318}, {942, 3936}, {956, 5827}, {986, 25760}, {1046, 32843}, {1089, 3814}, {1104, 56778}, {1150, 5814}, {1228, 21422}, {1259, 68696}, {1329, 3701}, {1330, 3218}, {1376, 5300}, {1393, 69051}, {1428, 3416}, {1441, 15844}, {1479, 32929}, {1491, 47708}, {1575, 16886}, {1724, 56520}, {1837, 49492}, {1930, 33864}, {2170, 4167}, {2183, 5279}, {2260, 21076}, {2269, 38408}, {2292, 3846}, {2308, 8258}, {2887, 24443}, {2968, 21530}, {3061, 4165}, {3100, 27534}, {3120, 67983}, {3142, 20891}, {3264, 20336}, {3452, 3710}, {3454, 3670}, {3545, 50043}, {3579, 4450}, {3666, 5051}, {3686, 21014}, {3695, 4187}, {3714, 17606}, {3752, 4202}, {3797, 17669}, {3813, 3902}, {3820, 52353}, {3839, 50045}, {3841, 28611}, {3868, 4417}, {3871, 4514}, {3876, 5233}, {3909, 11573}, {3916, 51583}, {3933, 26563}, {3965, 40941}, {3966, 26066}, {3977, 12572}, {4030, 64123}, {4101, 24391}, {4136, 33299}, {4153, 16549}, {4197, 19804}, {4385, 11681}, {4388, 56288}, {4391, 56283}, {4451, 52344}, {4461, 5068}, {4647, 25639}, {4671, 5154}, {4680, 25440}, {4696, 17757}, {4723, 21031}, {4742, 37722}, {4850, 16062}, {4886, 64401}, {4894, 8715}, {4980, 17530}, {5014, 5687}, {5046, 7283}, {5047, 33116}, {5055, 50041}, {5057, 63996}, {5084, 17776}, {5133, 19835}, {5141, 28605}, {5179, 26770}, {5192, 32777}, {5247, 33119}, {5255, 32844}, {5278, 5791}, {5294, 39559}, {5439, 18139}, {5695, 10896}, {5711, 33070}, {5744, 54429}, {6684, 63134}, {6931, 28808}, {7752, 33935}, {7796, 33934}, {8728, 24589}, {9369, 56880}, {10538, 52364}, {11337, 23361}, {12047, 17164}, {13407, 17140}, {13740, 32779}, {13741, 33157}, {16610, 17674}, {17054, 30811}, {17077, 54346}, {17152, 69038}, {17165, 21077}, {17211, 57029}, {17263, 17534}, {17495, 23537}, {17556, 50105}, {18253, 41002}, {19372, 28776}, {20237, 56564}, {21075, 63147}, {21616, 25253}, {23640, 23921}, {24046, 69251}, {24174, 25957}, {24210, 64071}, {24593, 34753}, {24851, 32845}, {25017, 25939}, {26064, 38000}, {26234, 69135}, {27187, 69158}, {27566, 46826}, {28606, 52258}, {30806, 69094}, {32848, 63800}, {32849, 37162}, {32862, 46937}, {32932, 52367}, {32933, 58798}, {33071, 57280}, {33105, 49598}, {36624, 56349}, {37668, 41826}, {37717, 54331}, {37817, 56781}, {37983, 55089}, {39566, 51558}, {45744, 53599}, {45964, 60197}, {64185, 68946}

X(69280) = X(27808)-Ceva conjugate of X(4391)
X(69280) = X(i)-isoconjugate of X(j) for these (i,j): {65, 3453}, {604, 40394}
X(69280) = X(i)-Dao conjugate of X(j) for these (i,j): {3161, 40394}, {3454, 56}, {18191, 3733}, {40602, 3453}
X(69280) = crosspoint of X(333) and X(3596)
X(69280) = crosssum of X(1397) and X(1400)
X(69280) = barycentric product X(i)*X(j) for these {i,j}: {8, 17184}, {21, 20896}, {261, 20654}, {312, 3670}, {314, 4016}, {333, 3454}, {645, 21121}, {3701, 18601}, {3909, 4391}, {7017, 11573}, {18155, 61167}, {20966, 28660}, {22073, 44130}, {30713, 52564}, {40072, 40986}, {59761, X(69280) = 68372}
X(69280) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 40394}, {284, 3453}, {3454, 226}, {3670, 57}, {3909, 651}, {4016, 65}, {11573, 222}, {17184, 7}, {18601, 1014}, {20654, 12}, {20896, 1441}, {20966, 1400}, {21121, 7178}, {22073, 73}, {23197, 52411}, {30713, 59138}, {40986, 1402}, {52564, 1412}, {61167, 4551}, {68372, 1407}
X(69280) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 27529, 7081}, {10, 20108, 19867}, {10, 30171, 3006}, {11, 3704, 3702}, {12, 69091, 4968}, {1329, 3703, 3701}, {1698, 30172, 69250}, {3454, 3670, 17184}, {3687, 6734, 8}, {3695, 4187, 4358}, {5046, 33168, 7283}, {11681, 33089, 4385}, {27714, 29688, 1125}, {56385, 56386, 3187}

X(69281) = X(1)X(6)∩X(38)X(101)

Barycentrics     a*(a^3 - a^2*b + a*b^2 + b^3 - a^2*c + a*b*c + b^2*c + a*c^2 + b*c^2 + c^3) : :

X(69281) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (1, -1, 1, 1, 1, 1, 0, 0, 0).

X(69281) lies on these lines: {1, 6}, {2, 30729}, {32, 191}, {38, 101}, {63, 609}, {86, 46899}, {172, 6763}, {239, 26580}, {484, 4386}, {574, 15015}, {672, 30115}, {758, 5276}, {846, 5371}, {976, 3730}, {1018, 3961}, {1572, 3899}, {2292, 4251}, {3336, 5277}, {3612, 31429}, {3624, 69224}, {3678, 69211}, {3679, 5309}, {3684, 4424}, {3735, 5540}, {3870, 9331}, {3878, 69210}, {3901, 69217}, {3920, 3997}, {3958, 4264}, {4016, 16547}, {4115, 32930}, {4262, 4414}, {4384, 33129}, {5254, 47033}, {5275, 5902}, {5293, 16549}, {5305, 21677}, {5443, 31466}, {5883, 37675}, {6205, 56010}, {7031, 12514}, {10176, 33854}, {11010, 69248}, {16611, 54315}, {16705, 30127}, {16833, 50103}, {17137, 30130}, {17152, 30133}, {17596, 35342}, {18785, 56149}, {20367, 36572}, {25092, 34772}, {27269, 30139}, {30117, 59207}, {30136, 41838}, {31422, 63752}, {31449, 37525}, {31468, 34471}, {37563, 69243}, {37661, 37701}, {37783, 40773}

X(69281) = barycentric product X(1)*X(33175)
X(69281) = barycentric quotient X(33175)/X(75)
X(69281) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 54330, 54981}, {213, 16519, 1}, {5277, 69246, 3336}

X(69282) = X(1)X(6)∩X(39)X(101)

Barycentrics     a^2*(a^2 - a*b + b^2 - a*c + b*c + c^2) : :

X(69282) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (1, -1, 1, 1, 0, 0, 0, 0, 0).

X(69282) lies on these lines: {1, 6}, {31, 4517}, {32, 3730}, {35, 2251}, {39, 101}, {41, 2276}, {42, 47523}, {55, 2210}, {81, 16047}, {110, 25433}, {169, 3959}, {172, 672}, {187, 24047}, {190, 384}, {198, 21771}, {404, 20331}, {595, 5007}, {607, 2201}, {609, 69231}, {644, 69210}, {894, 33943}, {940, 17244}, {995, 7772}, {1010, 17369}, {1017, 4256}, {1026, 21792}, {1086, 17682}, {1265, 4195}, {1334, 1914}, {1500, 4251}, {1783, 11105}, {1922, 40794}, {2162, 7123}, {2174, 5110}, {2238, 5051}, {2242, 4253}, {2243, 56288}, {2246, 4642}, {2275, 9259}, {2295, 5276}, {2348, 41015}, {3053, 42316}, {3204, 4261}, {3207, 5013}, {3208, 69243}, {3244, 50028}, {3285, 17524}, {3494, 34249}, {3496, 69248}, {3501, 4386}, {3509, 69246}, {3556, 56911}, {3570, 26752}, {3573, 52651}, {3684, 20691}, {3688, 12212}, {3727, 33950}, {3758, 41875}, {3767, 56746}, {3780, 63087}, {3782, 50200}, {3915, 5332}, {3952, 18099}, {4258, 31477}, {4262, 31451}, {4284, 17053}, {4286, 16453}, {4363, 11321}, {4370, 13735}, {4383, 17367}, {4389, 33827}, {4415, 37086}, {4419, 17691}, {4422, 33821}, {4465, 17541}, {4628, 14885}, {4643, 25364}, {4713, 7770}, {5011, 69249}, {5091, 38472}, {5134, 7747}, {5145, 17976}, {5262, 46907}, {5277, 16549}, {5452, 63479}, {5475, 24045}, {5540, 69244}, {5710, 50286}, {6161, 21007}, {6180, 7185}, {6376, 17743}, {6586, 8578}, {6646, 39724}, {7031, 69229}, {7116, 40972}, {7252, 68829}, {7737, 17732}, {7745, 17747}, {7755, 17734}, {7760, 40859}, {7785, 41324}, {7839, 34063}, {8258, 17750}, {8301, 12782}, {10311, 41320}, {10312, 56747}, {11343, 56524}, {14370, 20994}, {14887, 23990}, {14974, 21793}, {15990, 49728}, {16048, 37549}, {16060, 25349}, {17277, 27296}, {17279, 24549}, {17280, 33954}, {17354, 17688}, {17686, 24330}, {17741, 33938}, {18047, 21226}, {18278, 25800}, {19237, 33761}, {20271, 40131}, {20602, 68478}, {21772, 68828}, {21791, 57053}, {22116, 66973}, {23631, 24488}, {23851, 51928}, {23861, 66971}, {24320, 50591}, {25350, 33828}, {26039, 56988}, {26082, 27958}, {27064, 41258}, {30133, 68890}, {33955, 52897}, {37111, 62797}, {37588, 52964}, {37674, 56519}, {41265, 60722}, {47352, 50073}, {50127, 59557}, {50287, 54367}, {55337, 69214}, {56989, 62706}, {67417, 67507}

X(69282) = isogonal conjugate of X(39724)
X(69282) = isogonal conjugate of the isotomic conjugate of X(17280)
X(69282) = X(i)-Ceva conjugate of X(j) for these (i,j): {256, 55}, {56547, 41346}
X(69282) = X(3961)-cross conjugate of X(34249)
X(69282) = X(i)-isoconjugate of X(j) for these (i,j): {1, 39724}, {2, 7194}, {6, 40038}, {57, 43749}, {330, 3502}, {514, 65364}, {55014, 56547}
X(69282) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 39724}, {9, 40038}, {2329, 1909}, {5452, 43749}, {32664, 7194}
X(69282) = crosspoint of X(3961) and X(56547)
X(69282) = crosssum of X(1086) and X(4369)
X(69282) = crossdifference of every pair of points on line {513, 3776}
X(69282) = barycentric product X(i)*X(j) for these {i,j}: {1, 3961}, {6, 17280}, {8, 41346}, {9, 56547}, {31, 33938}, {42, 33954}, {43, 3494}, {55, 56928}, {192, 34249}, {350, 66997}, {893, 17741}
X(69282) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40038}, {6, 39724}, {31, 7194}, {55, 43749}, {692, 65364}, {2209, 3502}, {3494, 6384}, {3961, 75}, {17280, 76}, {17741, 1920}, {33938, 561}, {33954, 310}, {34249, 330}, {41346, 7}, {56547, 85}, {56928, 6063}, {66997, 291}
X(69282) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 45, 65741}, {6, 220, 2176}, {6, 2256, 21785}, {6, 16969, 16502}, {9, 2273, 6}, {9, 54329, 4426}, {32, 3730, 17735}, {39, 101, 21008}, {41, 2276, 18755}, {169, 9620, 3959}, {172, 672, 33863}, {213, 5280, 6}, {213, 16514, 2176}, {218, 54416, 6}, {644, 69210, 69245}, {2275, 9310, 9259}, {5007, 52963, 595}, {5280, 5526, 213}, {14974, 30435, 21793}, {16785, 17745, 20963}, {17745, 20963, 6}

X(69283) = X(1)X(6)∩X(32)X(63)

Barycentrics     a*(a^3 - a^2*b + a*b^2 + b^3 - a^2*c + b^2*c + a*c^2 + b*c^2 + c^3) : :

X(69283) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (1, -1, 1, 0, 1, 1, 0, 0, 0).

X(69283) lies on these lines: {1, 6}, {2, 69224}, {8, 5286}, {21, 31442}, {32, 63}, {38, 41}, {39, 78}, {46, 4386}, {57, 5277}, {100, 1571}, {101, 69222}, {169, 3721}, {172, 62858}, {187, 4652}, {191, 7031}, {200, 9593}, {239, 5739}, {292, 56220}, {329, 69208}, {391, 19851}, {519, 68855}, {573, 21744}, {574, 4855}, {609, 6763}, {612, 17750}, {672, 976}, {758, 54382}, {908, 2548}, {942, 5275}, {965, 20227}, {966, 16817}, {968, 60722}, {975, 24512}, {982, 69221}, {986, 3684}, {997, 2275}, {1015, 19861}, {1038, 52635}, {1211, 4384}, {1334, 3938}, {1500, 3870}, {1506, 30852}, {1572, 3869}, {1573, 19860}, {1574, 67097}, {1763, 37555}, {1914, 12514}, {2082, 3735}, {2198, 36808}, {2223, 7085}, {2241, 5250}, {2242, 62874}, {2269, 23620}, {2271, 3666}, {2276, 3811}, {2280, 2292}, {2304, 64040}, {2549, 57287}, {2646, 31449}, {3053, 3916}, {3158, 31426}, {3419, 5254}, {3485, 31405}, {3501, 3961}, {3601, 31429}, {3681, 69211}, {3691, 3924}, {3730, 69214}, {3744, 14974}, {3749, 69229}, {3767, 6734}, {3868, 5276}, {3871, 31433}, {3876, 33854}, {3927, 30435}, {3940, 9605}, {3951, 5007}, {3984, 7772}, {3997, 30145}, {4251, 69220}, {4253, 30115}, {4254, 42461}, {4372, 24690}, {4393, 63009}, {4420, 17756}, {4511, 9619}, {4561, 25918}, {4920, 24694}, {5013, 5440}, {5021, 37539}, {5119, 69243}, {5175, 43448}, {5256, 20970}, {5262, 37657}, {5293, 17754}, {5304, 54398}, {5552, 31398}, {5748, 31404}, {5794, 69098}, {6745, 31396}, {7737, 64002}, {7745, 58798}, {9454, 69203}, {9596, 21077}, {9597, 17647}, {9599, 21616}, {10544, 23630}, {11374, 37661}, {11375, 31466}, {16583, 37549}, {16589, 54392}, {16834, 48848}, {16887, 30108}, {17184, 26085}, {17451, 49454}, {17475, 50594}, {17755, 24549}, {18206, 56525}, {19684, 41249}, {24047, 39255}, {24291, 56025}, {25426, 57748}, {25568, 31402}, {26036, 33144}, {26094, 30729}, {27383, 31400}, {27385, 31401}, {27529, 31441}, {28082, 59207}, {31448, 56176}, {31497, 59719}, {32029, 39731}, {36278, 36287}, {37552, 69231}, {40571, 40773}, {41609, 45786}, {56913, 64041}, {59491, 69207}, {62372, 64046}, {63130, 69249}, {68616, 69096}

X(69283) = X(25453)-Dao conjugate of X(4812)
X(69283) = barycentric product X(1)*X(33171)
X(69283) = barycentric quotient X(33171)/X(75)
X(69283) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 9, 69215}, {1, 16517, 5283}, {1, 16552, 16968}, {1, 21384, 69216}, {1, 51194, 20963}, {1, 54330, 2176}, {6, 72, 54406}, {6, 16519, 1}, {9, 16780, 69212}, {986, 3684, 69219}, {1100, 21874, 16466}, {3868, 5276, 69217}, {3869, 69210, 1572}, {4253, 30115, 54317}, {4386, 69246, 46}, {5280, 5904, 69218}, {5299, 5692, 39248}, {16973, 69225, 1}, {37549, 37658, 16583}, {69243, 69248, 5119}

X(69284) = X(1)X(6)∩X(2)X(17048)

Barycentrics     a*(a*b^2 - b^3 + a*b*c + a*c^2 - c^3) : :

X(69284) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (0, 0, 1, 1, -1, 0, 0, 0, 0).

X(69284) lies on these lines: {1, 6}, {2, 17048}, {3, 3509}, {4, 1840}, {7, 25242}, {8, 3930}, {10, 60108}, {12, 40997}, {21, 5282}, {35, 1759}, {36, 17736}, {39, 982}, {41, 17522}, {43, 16583}, {55, 3496}, {57, 21477}, {63, 16367}, {65, 3501}, {75, 17050}, {76, 85}, {78, 19310}, {100, 69242}, {101, 22836}, {145, 2170}, {165, 36643}, {169, 3684}, {171, 69217}, {192, 20706}, {194, 335}, {198, 20836}, {228, 33714}, {284, 11102}, {329, 1655}, {337, 30962}, {346, 2171}, {350, 18055}, {386, 16600}, {388, 24247}, {442, 19584}, {517, 3208}, {519, 4051}, {522, 14825}, {573, 59733}, {594, 9710}, {644, 62830}, {661, 49303}, {672, 3868}, {728, 3340}, {758, 3730}, {869, 20703}, {894, 17688}, {910, 56176}, {936, 16852}, {942, 17754}, {943, 2344}, {946, 21096}, {950, 49476}, {966, 3949}, {976, 5276}, {978, 3290}, {986, 2276}, {1018, 5903}, {1146, 12607}, {1193, 26242}, {1260, 37580}, {1329, 21049}, {1334, 3869}, {1400, 27396}, {1429, 1708}, {1432, 30701}, {1475, 3873}, {1482, 4919}, {1500, 3735}, {1572, 37588}, {1575, 20271}, {1742, 12689}, {1761, 54285}, {1766, 12520}, {1953, 17314}, {2082, 3870}, {2099, 4513}, {2241, 17715}, {2262, 3169}, {2268, 5279}, {2275, 3726}, {2280, 33950}, {2294, 2345}, {2295, 66640}, {2476, 21029}, {2810, 23630}, {2900, 19589}, {3116, 23414}, {3119, 27541}, {3125, 24440}, {3185, 40910}, {3189, 5819}, {3207, 56177}, {3212, 21232}, {3219, 54419}, {3252, 52085}, {3263, 29966}, {3419, 32847}, {3502, 39967}, {3601, 21982}, {3616, 39244}, {3662, 7876}, {3665, 51384}, {3673, 20335}, {3679, 4006}, {3681, 3691}, {3727, 66674}, {3729, 7201}, {3746, 69241}, {3767, 17719}, {3812, 44798}, {3871, 69240}, {3874, 4253}, {3876, 59207}, {3889, 17474}, {3913, 20692}, {3924, 69211}, {3934, 30869}, {3938, 69210}, {3944, 69096}, {3950, 4301}, {3959, 20691}, {4032, 25252}, {4050, 10914}, {4073, 21746}, {4109, 4417}, {4168, 50624}, {4335, 18252}, {4385, 21101}, {4447, 54474}, {4485, 18050}, {4511, 9310}, {4515, 5836}, {4520, 31165}, {4568, 33942}, {4645, 32117}, {4650, 69231}, {4812, 29964}, {4851, 18161}, {5011, 8715}, {5021, 32913}, {5044, 16846}, {5060, 56946}, {5179, 21077}, {5208, 5364}, {5255, 54382}, {5257, 37039}, {5275, 5293}, {5286, 33144}, {5296, 21033}, {5311, 47511}, {5439, 25068}, {5750, 37036}, {5776, 19782}, {5902, 16549}, {6007, 24341}, {6048, 16605}, {6167, 62792}, {6356, 18639}, {6554, 25568}, {6706, 17284}, {7179, 17046}, {7308, 21986}, {7413, 29649}, {7580, 18788}, {7735, 36573}, {7736, 36574}, {8074, 59722}, {8256, 68925}, {9596, 37717}, {9598, 24851}, {9780, 21921}, {9785, 17452}, {11517, 19557}, {11681, 21044}, {12047, 21073}, {12514, 60711}, {12563, 17355}, {16720, 30945}, {17023, 31269}, {17141, 27109}, {17143, 49507}, {17158, 63587}, {17165, 26770}, {17170, 61010}, {17233, 17762}, {17234, 33943}, {17243, 59515}, {17296, 18726}, {17298, 41777}, {17299, 17443}, {17315, 18041}, {17449, 23649}, {17464, 17794}, {17596, 31448}, {17601, 31451}, {17735, 69235}, {17756, 24443}, {17786, 21608}, {18156, 52049}, {18176, 30941}, {18398, 68950}, {18730, 41004}, {19767, 21840}, {20018, 22197}, {20037, 63501}, {20089, 29583}, {20171, 29967}, {20258, 44735}, {20347, 25237}, {20533, 33867}, {20593, 49470}, {20769, 56517}, {20861, 41886}, {20880, 30949}, {21090, 24045}, {21872, 44663}, {21933, 38406}, {22035, 24068}, {22173, 59295}, {22230, 59297}, {22285, 64169}, {24047, 69227}, {24266, 41245}, {24445, 24696}, {24524, 49755}, {25253, 68971}, {25264, 49518}, {26035, 32771}, {26793, 61706}, {27040, 32931}, {27248, 41876}, {27383, 40127}, {27385, 68797}, {27474, 33935}, {27523, 32937}, {28082, 33854}, {28594, 30116}, {28609, 29573}, {28830, 54366}, {29574, 31169}, {29643, 37330}, {29676, 31466}, {29968, 30758}, {29986, 31130}, {30038, 39731}, {30109, 33937}, {30546, 67850}, {31359, 60668}, {32915, 40463}, {33936, 40006}, {34064, 41260}, {36565, 63004}, {37567, 41322}, {37568, 41319}, {37573, 69220}, {37607, 54317}, {37617, 69222}, {41015, 50581}, {41423, 56288}, {41796, 46835}, {44421, 54344}, {45039, 54433}, {45240, 52135}, {49168, 56746}, {60714, 69219}, {61003, 64702}, {66650, 69245}

X(69284) = midpoint of X(3177) and X(36854)
X(69284) = reflection of X(i) in X(j) for these {i,j}: {3208, 3991}, {21384, 1212}
X(69284) = isotomic conjugate of the isogonal conjugate of X(5364)
X(69284) = X(5208)-Ceva conjugate of X(3779)
X(69284) = X(i)-isoconjugate of X(j) for these (i,j): {514, 58947}, {3063, 34083}
X(69284) = X(10001)-Dao conjugate of X(34083)
X(69284) = crossdifference of every pair of points on line {513, 22384}
X(69284) = barycentric product X(i)*X(j) for these {i,j}: {1, 29641}, {10, 5208}, {75, 3779}, {76, 5364}, {100, 23877}
X(69284) = barycentric quotient X(i)/X(j) for these {i,j}: {664, 34083}, {692, 58947}, {3779, 1}, {5208, 86}, {5364, 6}, {23877, 693}, {29641, 75}
X(69284) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6, 16787}, {1, 9, 41239}, {1, 1743, 16780}, {1, 3970, 51058}, {1, 17742, 2329}, {1, 17744, 16788}, {1, 57015, 3061}, {1, 69223, 16503}, {6, 16974, 16478}, {65, 3693, 3501}, {72, 16601, 9}, {169, 3811, 3684}, {218, 56536, 9}, {942, 25066, 17754}, {1500, 3735, 37598}, {1575, 20271, 24174}, {2275, 3726, 3976}, {2276, 3721, 986}, {3061, 51058, 1}, {3868, 25082, 672}, {3873, 26690, 1475}, {3874, 24036, 4253}, {3912, 17760, 304}, {3930, 17451, 8}, {3954, 5283, 984}, {3959, 20691, 64176}, {3970, 57015, 1}, {16572, 41863, 51194}, {16968, 69218, 5247}, {20691, 21331, 3959}, {21808, 33299, 2}, {30038, 49528, 39731}, {54382, 69214, 5255}, {54406, 69215, 238}

X(69285) = X(1)X(6)∩X(2)X(3721)

Barycentrics     a*(b^3 + a*b*c + b^2*c + b*c^2 + c^3) : :

X(69285) is obtained from the general form for 4th degree even PTC's,

k1 a^4 + k2 a^3(b+c) + k3 a^2 (b^2+c^2) + k4 a^2 b c + k5 a (b^3+c^3) + k6 a b c (b+c) + k7 (b^4+c^4) + k8 b c (b^2+c^2) + k9 b^2 c^2

by putting (k1, k2, k3, k4, k5, k6, k7, k8, k9) = (0, 0, 0, 1, 1, 1, 0, 0, 0).

X(69285) lies on these lines: {1, 6}, {2, 3721}, {3, 21771}, {8, 3727}, {10, 3735}, {32, 30115}, {38, 2275}, {63, 33863}, {78, 18755}, {100, 69237}, {141, 304}, {171, 69235}, {172, 5282}, {191, 69231}, {210, 41015}, {257, 312}, {335, 31997}, {337, 25918}, {404, 69239}, {612, 20715}, {712, 33945}, {756, 4517}, {758, 17750}, {762, 3679}, {966, 1265}, {975, 40750}, {976, 1914}, {982, 16604}, {986, 1575}, {997, 21008}, {1193, 41269}, {1329, 21965}, {1573, 23841}, {1697, 20692}, {1698, 3125}, {1759, 5277}, {1840, 5130}, {1921, 33935}, {2238, 3876}, {2271, 3940}, {2276, 2292}, {2295, 3869}, {3094, 4357}, {3185, 20994}, {3290, 25917}, {3454, 34542}, {3496, 4386}, {3501, 69248}, {3556, 54285}, {3616, 3726}, {3662, 33943}, {3681, 3780}, {3718, 15985}, {3740, 16605}, {3846, 4136}, {3868, 24512}, {3877, 69245}, {3878, 28594}, {3927, 5021}, {3961, 69243}, {4016, 17303}, {4364, 59703}, {4415, 5254}, {4419, 7738}, {4469, 33296}, {4485, 22028}, {4518, 21711}, {4561, 16060}, {4656, 49757}, {4698, 59554}, {5013, 25083}, {5044, 16583}, {5248, 51328}, {5250, 69214}, {5255, 69233}, {5266, 21793}, {5312, 6155}, {5337, 56524}, {5739, 6542}, {5814, 63800}, {5883, 25089}, {7085, 17798}, {9055, 39731}, {9259, 19861}, {9780, 21951}, {10176, 16600}, {10974, 16589}, {12514, 17735}, {13881, 17308}, {16886, 25760}, {17030, 18061}, {17137, 25263}, {17152, 31087}, {17210, 18167}, {17294, 35652}, {17327, 18179}, {17456, 24943}, {17754, 69246}, {17789, 27184}, {17793, 20600}, {17962, 56220}, {18189, 30966}, {20255, 30758}, {20593, 22231}, {20672, 53560}, {20691, 37598}, {20693, 50581}, {21004, 54371}, {21025, 46937}, {21029, 69173}, {21071, 49560}, {21240, 33942}, {21868, 64176}, {22065, 53129}, {24358, 24549}, {24723, 32117}, {26035, 56318}, {27248, 59512}, {29588, 63009}, {29603, 56519}, {29608, 37691}, {30038, 49521}, {31004, 33930}, {32916, 59720}, {33890, 33944}, {35258, 39255}, {39967, 56219}, {50087, 50122}, {52085, 52656}, {56288, 69230}, {60724, 62831}, {64047, 65695}

X(69285) = complement of the isotomic conjugate of X(56238)
X(69285) = X(56238)-complementary conjugate of X(2887)
X(69285) = X(i)-isoconjugate of X(j) for these (i,j): {6, 56065}, {514, 29018}, {649, 62464}, {667, 62465}
X(69285) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 56065}, {5375, 62464}, {6631, 62465}
X(69285) = crosspoint of X(2) and X(56238)
X(69285) = crossdifference of every pair of points on line {513, 8633}
X(69285) = barycentric product X(i)*X(j) for these {i,j}: {1, 32778}, {10, 35623}, {100, 29017}
X(69285) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 56065}, {100, 62464}, {190, 62465}, {692, 29018}, {29017, 693}, {32778, 75}, {35623, 86}
X(69285) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 9, 4426}, {1, 3954, 49509}, {1, 54406, 6}, {2, 3721, 20271}, {10, 3735, 3959}, {37, 960, 2176}, {38, 39244, 2275}, {63, 54317, 33863}, {78, 69220, 18755}, {141, 59515, 304}, {762, 69244, 3679}, {975, 69217, 40750}, {984, 3061, 1107}, {2292, 33299, 2276}, {3496, 5293, 4386}, {5044, 16583, 37673}, {19861, 69222, 9259}

X(69286) = EULER LINE INTERCEPT OF X(51)X(53415)

Barycentrics    2 a^6-a^4 b^2-2 a^2 b^4+b^6-a^4 c^2+12 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6 : :

See Gabi Cuc Cucoanes and Francisco Javier García Capitán, euclid 8572.

X(69286) lies on these lines: {2, 3}, {51, 53415}, {110, 45298}, {125, 64605}, {141, 61645}, {154, 54012}, {184, 59699}, {251, 5913}, {373, 23292}, {974, 6053}, {1196, 5355}, {1993, 61657}, {3167, 63084}, {3589, 6467}, {3819, 32269}, {3917, 47582}, {4319, 5432}, {4320, 5433}, {5422, 59553}, {5651, 13567}, {5892, 51425}, {5943, 11064}, {5972, 6688}, {6090, 11433}, {6329, 61659}, {6696, 30443}, {6716, 67281}, {7917, 45201}, {9306, 11245}, {9777, 37669}, {9924, 23327}, {10192, 43650}, {10278, 47173}, {10601, 59543}, {11402, 18928}, {11451, 18583}, {11793, 15739}, {13857, 51130}, {14826, 26869}, {15018, 61655}, {15043, 61607}, {15066, 41588}, {15448, 22352}, {15873, 43652}, {16187, 61646}, {17811, 61506}, {18358, 23293}, {18914, 43598}, {21243, 35283}, {29181, 44106}, {34750, 58434}, {35264, 48906}, {37480, 44935}, {39562, 51732}, {40112, 53863}, {41670, 58480}, {43620, 47297}, {44082, 44882}, {47260, 66122}, {51185, 63650}, {51360, 66531}, {58447, 63632}, {59208, 59558}, {59659, 64854}, {61676, 62375}

X(69286) = midpoint of X(125) and X(64605)
X(69286) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5972, 6688, 37649}, {10601, 59543, 61690}

X(69287) = X(1)X(3)∩X(12)X(3742)

Barycentrics    a*(a + b - c)*(a - b + c)*(a^3*(b + c) - a*(b + c)^3 + (b^2 - c^2)^2 - a^2*(b^2 - 4*b*c + c^2)) : :
X(69287) = 3*X[354]+X[37605]

See Gabi Cuc Cucoanes, David Nguyen and Ercole Suppa, euclid 8575 and euclid 8577.

X(69287) lies on these lines: {1, 3}, {2, 58585}, {11, 12675}, {12, 3742}, {201, 17449}, {388, 64149}, {442, 41556}, {497, 7702}, {499, 37713}, {518, 5433}, {950, 18240}, {960, 31157}, {1071, 11376}, {1125, 5083}, {1279, 1399}, {1317, 5836}, {1357, 58597}, {1358, 58596}, {1359, 58599}, {1361, 58600}, {1362, 58592}, {1364, 58593}, {1393, 4322}, {1404, 61650}, {1457, 46190}, {1463, 58627}, {1469, 58562}, {1788, 3889}, {1858, 12005}, {1898, 50443}, {1935, 29820}, {3022, 58594}, {3023, 58589}, {3024, 58582}, {3027, 58590}, {3028, 58601}, {3296, 64747}, {3306, 11501}, {3320, 58603}, {3324, 58598}, {3325, 58602}, {3555, 24914}, {3616, 58578}, {3649, 58568}, {3697, 26364}, {3740, 7294}, {3753, 37738}, {3812, 10944}, {3873, 7288}, {3881, 3911}, {3892, 4848}, {3918, 41558}, {4067, 41389}, {5249, 10957}, {5252, 5439}, {5434, 58560}, {5552, 24477}, {5603, 64704}, {5777, 17660}, {5883, 63987}, {5901, 67970}, {6284, 58567}, {6285, 58579}, {6797, 32900}, {7354, 13374}, {7743, 26201}, {8581, 58564}, {9850, 11237}, {9956, 20118}, {10039, 12832}, {10106, 58565}, {10167, 12701}, {10391, 37722}, {10527, 41537}, {10528, 64151}, {10896, 12680}, {10956, 24982}, {11375, 17625}, {12573, 58607}, {12608, 41561}, {12672, 18260}, {12688, 58588}, {12736, 13607}, {12739, 17614}, {12953, 63432}, {13369, 30384}, {14151, 17531}, {15172, 24465}, {15950, 66250}, {17626, 18961}, {18247, 44847}, {18970, 58580}, {18976, 58587}, {18977, 58586}, {18982, 58584}, {18990, 58561}, {20418, 67919}, {22759, 54392}, {23708, 40263}, {24816, 58618}, {26481, 51706}, {26741, 50587}, {31937, 37735}, {33812, 64745}, {34790, 38411}, {34791, 40663}, {37740, 67937}, {38053, 60943}, {39897, 58581}, {45288, 58679}, {54377, 54385}, {58563, 60883}, {61663, 64124}, {64160, 67051}

X(69287) = pole of line {650,21103} with respect to Hofstadter inellipse
X(69287) = pole of line {56,5082} with respect to dual conic of Moses-Feuerbach circumconic
X(69287) = X(21)-beth conjugate of X(33596)
X(69287) = X(10018)-of-intouch triangle
X(69287) = X(37605)-of-2nd anti-circumperp-tangential triangle
X(69287) = intersection,other than A,B,C,of the circumconics: {{A,B,C,X(4),X(37622)}}, {{A,B,C,X(46),X(2191)}}, {{A,B,C,X(104),X(33596)}}, {{A,B,C,X(5553),X(10679)}} ,{{A,B,C,X(5708),X(13476)}}, {{A,B,C,X(10965),X(30513)}}, {{A,B,C,X(37579),X(56155)}
X(69287) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 36, 33596}, {1, 46, 37622}, {1, 3359, 10965}, {1, 37534, 55}, {1, 59333, 26358}, {354, 2646, 50196}, {942, 1319, 64721}, {1319, 13751, 942}, {1385, 5570, 64043}, {1420, 18398, 65}, {3660, 5045, 65}, {3873, 7288, 41538}, {12005, 44675, 1858}, {16193, 58576, 354}, {17609, 37566, 2099}, {34489, 51816, 26437}, {50196, 66599, 2646}

X(69288) = EULER LINE INTERCEPT OF X(141)X(1568)

Barycentrics    3*a^8*(b^2 + c^2) - 8*a^4*b^2*c^2*(b^2 + c^2) - 3*(b^2 - c^2)^4*(b^2 + c^2) + a^6*(-6*b^4 + 4*b^2*c^2 - 6*c^4) + 2*a^2*(b^2 - c^2)^2*(3*b^4 + 2*b^2*c^2 + 3*c^4) : :
X(69288) = X[343]+2*X[18388], 2*X[5480]+X[16789], X[5907]+2*X[58480], 2*X[7687]+X[16165], 5*X[8227]-2*X[51718], 4*X[9729]-X[67921], X[19127]+2*X[67865], 2*X[19925]+X[51692], 5*X[64854]-2*X[66713]

As a point on the Euler line, X(69288) has Shinagawa coefficients: {1/3 (-2 e + 3 (e + f)), f}

See Gabi Cuc Cucoanes and David Nguyen, euclid 8576.

X(69288) lies on these lines: {2, 3}, {141, 1568}, {343, 18388}, {1092, 66712}, {1209, 22660}, {2781, 36518}, {3574, 21969}, {3614, 66719}, {5480, 16789}, {5907, 58480}, {6000, 45303}, {6247, 64179}, {7173, 66724}, {7592, 61544}, {7687, 16165}, {7699, 48876}, {8227, 51718}, {9722, 14836}, {9729, 67921}, {10592, 66610}, {10593, 66593}, {11245, 14852}, {12233, 14831}, {13394, 18400}, {13565, 67869}, {13567, 16226}, {13599, 66732}, {14157, 39884}, {14389, 50435}, {14568, 66449}, {14644, 38110}, {15030, 41580}, {18390, 37649}, {19127, 67865}, {19925, 51692}, {20299, 66758}, {20300, 51737}, {21357, 66739}, {21849, 45089}, {23300, 43273}, {23332, 64100}, {25739, 48906}, {31166, 47353}, {31804, 58922}, {33547, 68316}, {34774, 47354}, {35264, 61606}, {37514, 66716}, {38042, 66741}, {38136, 66750}, {38443, 64037}, {43572, 59553}, {44665, 61690}, {45298, 61701}, {60121, 60241}, {64854, 66713}

X(69288) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 54994}, {2, 381, 34664}, {2, 3545, 16072}, {2, 15078, 140}, {4, 6676, 44239}, {4, 7569, 63679}, {5, 1368, 7577}, {5, 1596, 5133}, {5, 6823, 1594}, {5, 10024, 235}, {5, 13160, 7399}, {5, 15760, 427}, {5, 15761, 7403}, {22, 3091, 66728}, {378, 3090, 64852}, {381, 7540, 3845}, {381, 9909, 4}, {1312, 1313, 47339}, {2043, 2044, 7507}, {6676, 9909, 44210}, {7403, 15761, 1906}, {7547, 7558, 12362}, {7552, 7576, 10154}, {14788, 16868, 5}, {15765, 18585, 12605}

X(69289) = EULER LINE INTERCEPT OF X(14)X(10634)

Barycentrics    -((a^2 - b^2 - c^2)*(a^8 - 6*a^4*b^2*c^2 - (b^2 - c^2)^4 - 2*a^6*(b^2 + c^2) + 2*a^2*(b^2 - c^2)^2*(b^2 + c^2))) : :
X(69289) = 2*X[5]+X[22], 4*X[140]-X[378], X[52]-4*X[58480], X[265]+2*X[16165], 2*X[343]+X[18445], X[355]+2*X[51692], 2*X[1216]+X[54384], X[1351]+2*X[16789], X[1352]+2*X[19127], X[1993]-4*X[61619], X[11442]+2*X[61752], X[11456]+2*X[67926], X[11605]-4*X[61586], X[13352]-4*X[58447], X[14983]+2*X[38624], 2*X[18388]+X[37478], X[19149]+2*X[34177], 5*X[31267]+X[54146], 5*X[40686]+X[66723]

As a point on the Euler line, X(69289) has Shinagawa coefficients: {1/3 (-((5 e)/4) + 2 (e + f)), -(e/4)}

See Gabi Cuc Cucoanes and David Nguyen, euclid 8576 .

X(69289) lies on these lines:{2, 3}, {13, 10635}, {14, 10634}, {49, 63649}, {52, 58480}, {69, 50461}, {127, 7865}, {141, 51425}, {182, 63735}, {184, 539}, {265, 16165}, {343, 18445}, {355, 51692}, {498, 66719}, {499, 66724}, {542, 19131}, {1060, 3582}, {1062, 3584}, {1092, 44516}, {1147, 64064}, {1209, 6759}, {1216, 54384}, {1351, 16789}, {1352, 10540}, {1879, 10979}, {1993, 61619}, {2781, 14643}, {2980, 18437}, {3098, 51392}, {3519, 9936}, {3796, 14852}, {4550, 51403}, {5050, 45967}, {5449, 10984}, {5476, 9967}, {5654, 23039}, {5655, 7723}, {5891, 41580}, {5892, 61645}, {6193, 9704}, {6243, 12606}, {6288, 9833}, {6515, 15087}, {6689, 11424}, {7592, 63734}, {7603, 22052}, {7689, 64179}, {7736, 22121}, {7737, 18472}, {7749, 19220}, {7753, 10316}, {9019, 14561}, {9300, 22120}, {9730, 61646}, {10056, 18455}, {10072, 18447}, {10263, 12363}, {10653, 18470}, {10654, 18468}, {10897, 35823}, {10898, 35822}, {11178, 19126}, {11179, 19129}, {11433, 15037}, {11442, 61752}, {11456, 67926}, {11457, 34826}, {11515, 37835}, {11516, 37832}, {11605, 61586}, {12161, 41628}, {13340, 61711}, {13352, 58447}, {13353, 39571}, {13394, 44665}, {13557, 37893}, {13754, 61644}, {14389, 39522}, {14627, 64048}, {14845, 38317}, {14855, 23329}, {14983, 38624}, {14993, 43089}, {15038, 63085}, {15068, 37636}, {15080, 25739}, {15151, 20126}, {15362, 38064}, {16226, 32225}, {17508, 23515}, {18388, 37478}, {18390, 37513}, {18435, 67890}, {18438, 20423}, {18911, 63839}, {18951, 43845}, {18952, 61134}, {19149, 34177}, {24301, 28204}, {25043, 46025}, {25406, 38724}, {25738, 64049}, {25740, 50955}, {26881, 41171}, {26937, 67921}, {30522, 34513}, {31267, 54146}, {32140, 52525}, {32348, 61749}, {32885, 40680}, {35268, 44407}, {36753, 41587}, {37470, 44673}, {37495, 66712}, {37511, 50977}, {37515, 43817}, {38224, 57314}, {38796, 57357}, {40686, 66723}, {43608, 66715}, {43652, 43839}, {52520, 64689}, {57305, 57346}, {57307, 57380}, {57311, 57366}, {57344, 57365}, {57355, 57379}, {59648, 61680}, {61715, 62187}, {64036, 67878}, {66716, 67902}

X(69289) = complement of polar conjugate of X(62925)
X(69289) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 60763}, {2, 376, 18281}, {2, 381, 14787}, {2, 7552, 10201}, {2, 10304, 65085}, {3, 381, 67337}, {3, 1656, 37452}, {3, 3549, 6639}, {3, 6639, 6640}, {3, 10024, 18404}, {3, 10254, 18531}, {3, 10255, 6643}, {5, 22, 31723}, {5, 428, 381}, {5, 7525, 37444}, {5, 17714, 4}, {381, 3534, 34725}, {381, 7517, 428}, {381, 9909, 7540}, {403, 7495, 7514}, {427, 16618, 12083}, {465, 466, 37068}, {547, 10691, 11585}, {547, 16197, 10691}, {1598, 1656, 50137}, {1656, 12083, 427}, {2043, 2044, 18569}, {2937, 7517, 17714}, {3089, 14786, 3851}, {3090, 7391, 39504}, {3547, 3549, 3}, {3547, 7558, 34002}, {3548, 7400, 3}, {5654, 43653, 23039}, {6636, 7577, 14791}, {6676, 7502, 47525}, {6676, 15760, 3}, {6823, 7542, 3}, {7399, 13383, 7506}, {7493, 18420, 2070}, {7494, 18531, 3}, {7502, 17714, 22}, {7568, 15761, 7503}, {10024, 18404, 63671}, {10127, 13383, 62978}, {11585, 16197, 3}, {14784, 14785, 52295}, {15765, 18585, 7503}, {18586, 18587, 5576}

X(69290) = EULER LINE INTERCEPT OF X(1350)X(31166)

Barycentrics    -8*a^10 + 15*a^8*(b^2 + c^2) + (b^2 - c^2)^4*(b^2 + c^2) + 2*a^6*(b^2 + c^2)^2 + 2*a^2*(b^2 - c^2)^2*(3*b^4 + 2*b^2*c^2 + 3*c^4) - 8*a^4*(2*b^6 + b^4*c^2 + b^2*c^4 + 2*c^6) : :
X(69290) = X[20]+5*X[22], 2*X[6053]-5*X[16165]

As a point on the Euler line, X(69290) has Shinagawa coefficients: {1/3 (-4 e + 7 (e + f)), e - 3 (e + f)}

See Gabi Cuc Cucoanes and David Nguyen, euclid 8576 .

X(69290) lies on these lines: {2, 3}, {1350, 31166}, {6053, 16165}, {12220, 50979}, {13345, 14836}, {16226, 32191}, {31804, 41628}, {36987, 41580}, {37486, 63649}, {41464, 48906}, {43574, 48874}, {45186, 65094}, {46728, 64062}, {48881, 51394}, {50824, 64039}

X(69290) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {24, 376, 10691}, {376, 9909, 34664}, {550, 2937, 235}

X(69291) = X(513)X(3776)∩X(514)X(661)

Barycentrics    (b - c)*(a^2 + a*b + b^2 + a*c - b*c + c^2) : :
X(69291) = X[661] - 3 X[48550], 3 X[693] - X[48275], 2 X[3239] - 3 X[3835], 8 X[3239] - 9 X[45661], 5 X[3239] - 6 X[59751], 4 X[3835] - 3 X[45661], 5 X[3835] - 4 X[59751], X[3904] - 3 X[48131], X[4468] - 3 X[48554], 3 X[4728] - X[47660], 3 X[4728] + X[47916], 3 X[4776] + X[47651], 3 X[4776] - X[48094], 3 X[30565] - X[48130], 15 X[45661] - 16 X[59751], and many others

X(69291) lies on these lines: {2, 48101}, {513, 3776}, {514, 661}, {522, 4810}, {523, 47999}, {649, 21212}, {650, 28882}, {659, 48552}, {812, 3004}, {824, 4106}, {830, 48286}, {918, 48049}, {2254, 48159}, {2490, 45684}, {2527, 44902}, {2530, 29118}, {2786, 16892}, {2789, 48298}, {3667, 48015}, {3676, 4817}, {3700, 28863}, {3716, 4977}, {3798, 48016}, {3808, 50510}, {3810, 48616}, {4010, 47968}, {4024, 21297}, {4025, 4785}, {4071, 48152}, {4088, 47688}, {4120, 26798}, {4369, 6703}, {4375, 6545}, {4379, 47907}, {4380, 47886}, {4382, 45746}, {4394, 47882}, {4453, 4979}, {4467, 48114}, {4481, 23777}, {4500, 23813}, {4522, 4802}, {4724, 47686}, {4750, 26853}, {4761, 44314}, {4762, 48404}, {4778, 47983}, {4790, 47754}, {4813, 28855}, {4838, 47654}, {4893, 47663}, {4927, 48276}, {4928, 68794}, {4940, 30520}, {4988, 26824}, {4992, 68979}, {6006, 48224}, {6084, 48000}, {7658, 45313}, {7662, 47951}, {10196, 25666}, {11068, 47778}, {13246, 47797}, {17069, 45679}, {20949, 29739}, {21104, 28840}, {21115, 48019}, {21146, 47944}, {21183, 49293}, {22044, 24085}, {23755, 28115}, {23770, 47989}, {23877, 48092}, {24924, 48145}, {25259, 31147}, {25380, 48178}, {27138, 47773}, {28041, 48341}, {28225, 47887}, {28229, 48238}, {28487, 47708}, {28846, 48041}, {28851, 48026}, {28859, 43067}, {28867, 48426}, {28878, 47984}, {28890, 48046}, {28898, 48427}, {28906, 49284}, {29025, 48100}, {29082, 48129}, {29098, 48059}, {29120, 48137}, {29158, 48066}, {30519, 48269}, {30835, 47771}, {31148, 47900}, {31286, 47757}, {43061, 44432}, {44429, 48106}, {44449, 47930}, {45320, 48605}, {46403, 47701}, {47123, 47982}, {47650, 47781}, {47656, 53585}, {47659, 48416}, {47664, 47878}, {47671, 47869}, {47677, 48266}, {47685, 47972}, {47687, 47702}, {47690, 47924}, {47691, 48023}, {47692, 48077}, {47694, 47943}, {47695, 48020}, {47696, 47832}, {47697, 48585}, {47699, 48119}, {47703, 48170}, {47704, 47945}, {47712, 48086}, {47716, 47948}, {47720, 47912}, {47755, 50525}, {47758, 48067}, {47759, 48082}, {47760, 48095}, {47762, 48104}, {47769, 48117}, {47770, 48132}, {47780, 48414}, {47782, 47932}, {47792, 48418}, {47809, 48146}, {47810, 48408}, {47812, 47902}, {47821, 48102}, {47919, 48271}, {47931, 49275}, {47937, 48107}, {47938, 48108}, {47961, 48089}, {47971, 48079}, {47973, 48080}, {47981, 49296}, {48021, 49301}, {48032, 48161}, {48090, 48621}, {48098, 48611}, {48140, 48185}, {48177, 50358}, {48179, 53580}, {48184, 48599}, {48397, 48417}, {48425, 48571}, {50453, 50492}

X(69291) = midpoint of X(i) and X(j) for these {i,j}: {649, 49298}, {661, 47652}, {693, 47958}, {3004, 23729}, {4010, 47968}, {4024, 47653}, {4025, 49294}, {4088, 47688}, {4106, 47960}, {4382, 45746}, {4467, 48114}, {4724, 47686}, {4813, 47676}, {4838, 47654}, {4979, 49297}, {4988, 26824}, {7192, 23731}, {7662, 47951}, {16892, 20295}, {21104, 47988}, {21146, 47944}, {23770, 47989}, {25259, 47923}, {43067, 47950}, {44449, 47930}, {46403, 47701}, {47123, 47982}, {47650, 47926}, {47651, 48094}, {47660, 47916}, {47677, 48266}, {47685, 47972}, {47687, 47702}, {47690, 47924}, {47691, 48023}, {47692, 48077}, {47694, 47943}, {47695, 48020}, {47696, 48598}, {47697, 48585}, {47699, 48119}, {47704, 47945}, {47708, 48122}, {47712, 48086}, {47716, 47948}, {47720, 47912}, {47902, 49283}, {47907, 49282}, {47919, 48271}, {47931, 49275}, {47937, 48107}, {47938, 48108}, {47961, 48089}, {47971, 48079}, {47973, 48080}, {47981, 49296}, {47995, 48398}, {48007, 49295}, {48021, 49301}, {48026, 49299}, {48082, 49302}, {48090, 48621}, {48098, 48611}, {48349, 50328}, {48605, 49281}
X(69291) = reflection of X(i) in X(j) for these {i,j}: {649, 21212}, {4500, 23813}, {4761, 44314}, {4932, 3676}, {4979, 59630}, {6590, 59522}, {10196, 47756}, {21196, 3004}, {43067, 48415}, {45674, 44435}, {47890, 25666}, {48016, 3798}, {48060, 31286}, {48270, 4940}, {48397, 48417}, {48567, 59755}
X(69291) = complement of X(48101)
X(69291) = X(i)-complementary conjugate of X(j) for these (i,j): {37, 46665}, {213, 15527}, {3108, 11}, {7953, 3739}, {10159, 21252}, {35137, 21240}, {57421, 64523}
X(69291) = X(i)-Ceva conjugate of X(j) for these (i,j): {6625, 1086}, {8818, 59746}, {57949, 17205}
X(69291) = X(i)-isoconjugate of X(j) for these (i,j): {6, 65369}, {101, 39977}, {692, 39722}, {32739, 40033}
X(69291) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 65369}, {1015, 39977}, {1086, 39722}, {2643, 762}, {17289, 69001}, {40619, 40033}
X(69291) = crossdifference of every pair of points on line {31, 4517}
X(69291) = barycentric product X(i)*X(j) for these {i,j}: {513, 33944}, {514, 17302}, {693, 29821}, {4425, 7192}, {16726, 21604}, {16727, 21383}, {17205, 21295}
X(69291) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 65369}, {513, 39977}, {514, 39722}, {693, 40033}, {4425, 3952}, {17302, 190}, {23928, 40521}, {29821, 100}, {33944, 668}
X(69291) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 21212, 45674}, {649, 44435, 21212}, {4379, 47907, 49282}, {4453, 4979, 59630}, {4453, 49297, 4979}, {4728, 47916, 47660}, {4776, 47651, 48094}, {6545, 23731, 7192}, {20295, 48156, 16892}, {21297, 47653, 4024}, {24924, 48145, 48567}, {25666, 47890, 10196}, {26798, 49273, 4120}, {30835, 48138, 47771}, {31147, 47923, 25259}, {44435, 49298, 649}, {45320, 48605, 49281}, {47650, 47781, 47926}, {47652, 48550, 661}, {47676, 48543, 4813}, {47756, 47890, 25666}, {47757, 48060, 31286}, {47759, 49302, 48082}, {47812, 47902, 49283}, {47832, 48598, 47696}, {48079, 48422, 47971}, {48414, 50522, 47780}

X(69292) = X(513)X(3776)∩X(523)X(49291)

Barycentrics    (b - c)*(-a^2 - a*b + b^2 - a*c - b*c + c^2) : :
X(69292) = 3 X[2] - 4 X[59749], X[48082] - 4 X[59749], 4 X[3676] - 3 X[21204], 2 X[3835] - 3 X[21204], 3 X[649] - X[47663], X[649] - 3 X[47755], 4 X[3798] - 3 X[45679], 3 X[4025] - X[45745], 3 X[4750] - X[17494], X[4988] - 3 X[47894], 3 X[7192] + X[47653], X[7192] + 3 X[48571], 3 X[7192] - X[50522], 3 X[16892] - X[47653], X[16892] - 3 X[48571], 3 X[16892] + X[50522], and many others

X(69292) lies on these lines: {2, 48082}, {226, 3676}, {239, 514}, {513, 3776}, {522, 21146}, {523, 49291}, {647, 3960}, {650, 28851}, {661, 4453}, {664, 41405}, {693, 2786}, {812, 4897}, {824, 43067}, {900, 49289}, {918, 4369}, {1638, 25666}, {2487, 4763}, {2529, 30520}, {2785, 4378}, {3004, 28840}, {3239, 47779}, {3667, 46403}, {3669, 25098}, {3700, 47891}, {3762, 24622}, {4024, 47780}, {4049, 60258}, {4084, 29350}, {4088, 47824}, {4106, 28867}, {4120, 26985}, {4122, 48253}, {4142, 6372}, {4160, 62435}, {4379, 25259}, {4467, 47672}, {4468, 10196}, {4500, 28898}, {4521, 45684}, {4724, 13246}, {4728, 44449}, {4776, 48076}, {4778, 47968}, {4784, 48326}, {4785, 48013}, {4790, 28882}, {4813, 44435}, {4820, 48417}, {4885, 45661}, {4928, 14321}, {4979, 21115}, {5098, 20520}, {5270, 44314}, {5592, 8654}, {5882, 28292}, {6003, 53550}, {6006, 60962}, {6545, 17483}, {6546, 27013}, {6548, 26798}, {6590, 30519}, {7658, 47778}, {8045, 23875}, {11068, 45313}, {14475, 27138}, {17069, 48000}, {17161, 47671}, {17660, 37998}, {19882, 44315}, {21116, 26824}, {21183, 28906}, {21202, 64866}, {21297, 48414}, {22388, 39476}, {23093, 44408}, {23731, 48156}, {23813, 48413}, {24924, 30565}, {25380, 48047}, {28161, 47703}, {28225, 47943}, {28468, 68773}, {28487, 48151}, {28859, 47960}, {28863, 48276}, {28878, 47996}, {28886, 47754}, {28890, 47890}, {28894, 48427}, {28902, 47991}, {28910, 47882}, {29037, 50352}, {29078, 48098}, {29252, 52601}, {29304, 48343}, {30704, 42325}, {30835, 47769}, {31148, 47660}, {31603, 53544}, {31605, 57235}, {35505, 44312}, {44432, 59612}, {47651, 48104}, {47659, 48428}, {47666, 47886}, {47674, 50482}, {47675, 48277}, {47677, 48275}, {47690, 48579}, {47698, 47828}, {47700, 48252}, {47701, 48241}, {47704, 50343}, {47720, 50509}, {47757, 48038}, {47761, 48087}, {47762, 48094}, {47771, 48117}, {47781, 47908}, {47782, 47917}, {47792, 48429}, {47797, 48021}, {47800, 48036}, {47804, 48078}, {47813, 49275}, {47871, 48114}, {47874, 49272}, {47880, 47952}, {47887, 48080}, {47937, 48432}, {47938, 48174}, {47958, 48107}, {48019, 48550}, {48024, 48227}, {48034, 48554}, {48079, 48421}, {48113, 48250}, {48130, 48567}, {48271, 48563}, {48274, 64859}, {48281, 53556}, {48300, 48570}, {48349, 58372}, {49298, 50525}

X(69292) = midpoint of X(i) and X(j) for these {i,j}: {649, 47676}, {693, 47971}, {4025, 49296}, {4467, 47672}, {4707, 48320}, {4784, 48326}, {4790, 49299}, {4897, 21104}, {4979, 47652}, {7192, 16892}, {17161, 47671}, {17496, 23755}, {21116, 53333}, {21146, 50342}, {45746, 48141}, {47651, 48104}, {47653, 50522}, {47659, 48428}, {47660, 47930}, {47674, 50482}, {47675, 48277}, {47677, 48275}, {47704, 50343}, {47720, 50509}, {47923, 49282}, {47958, 48107}, {48013, 48398}, {48101, 49302}, {49298, 50525}
X(69292) = reflection of X(i) in X(j) for these {i,j}: {649, 59630}, {661, 21212}, {3835, 3676}, {4106, 48415}, {4468, 31286}, {4724, 13246}, {4765, 59550}, {4820, 48417}, {10196, 47758}, {21196, 4025}, {47769, 59755}, {47960, 48426}, {48000, 17069}, {48008, 3798}, {48046, 25666}, {48047, 25380}, {48269, 59522}, {48270, 4885}
X(69292) = complement of X(48082)
X(69292) = X(692)-isoconjugate of X(60149)
X(69292) = X(1086)-Dao conjugate of X(60149)
X(69292) = crosspoint of X(4610) and X(39734)
X(69292) = crosssum of X(4079) and X(64169)
X(69292) = crossdifference of every pair of points on line {42, 47523}
X(69292) = barycentric product X(i)*X(j) for these {i,j}: {513, 33943}, {514, 17300}, {693, 32913}, {3261, 33863}, {4025, 4212}, {7192, 29653}
X(69292) = barycentric quotient X(i)/X(j) for these {i,j}: {514, 60149}, {4212, 1897}, {17300, 190}, {29653, 3952}, {32913, 100}, {33863, 101}, {33943, 668}
X(69292) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 47755, 59630}, {661, 4453, 21212}, {1638, 48046, 25666}, {3676, 3835, 21204}, {3798, 48008, 45679}, {4468, 31286, 10196}, {4468, 47758, 31286}, {4885, 48270, 45661}, {4979, 21115, 47652}, {7192, 47653, 50522}, {7192, 48571, 16892}, {16892, 50522, 47653}, {21183, 48269, 59522}, {23731, 48425, 48156}, {24924, 48112, 30565}, {31148, 47930, 47660}, {47676, 47755, 649}, {47763, 49302, 48101}, {47923, 48577, 49282}, {48047, 48245, 25380}, {48107, 48422, 47958}

X(69293) = X(513)X(4522)∩X(523)X(3716)

Barycentrics    (b - c)*(a^2 - a*b + b^2 - a*c + b*c + c^2) : :
X(69293) = 3 X[2] + X[49273], 2 X[21212] + X[49273], X[661] - 3 X[30565], X[693] - 3 X[47874], X[693] + 3 X[48557], 3 X[1577] - X[47680], 4 X[3239] - 3 X[45661], 3 X[3239] - 2 X[59751], 2 X[3835] - 3 X[45661], 3 X[3835] - 4 X[59751], 3 X[4391] + X[47684], 3 X[4728] - X[47652], 3 X[4728] + X[48130], 3 X[4776] + X[47662], 3 X[4776] - X[47958], 3 X[4789] - X[47672], and many others

X(69293) lies on these lines: {2, 16892}, {75, 29404}, {190, 41405}, {312, 18071}, {513, 4522}, {514, 661}, {522, 659}, {523, 3716}, {649, 2786}, {650, 824}, {663, 47707}, {667, 29037}, {768, 6586}, {812, 3700}, {826, 4142}, {830, 50494}, {891, 49290}, {918, 4369}, {1215, 23805}, {1491, 48185}, {1635, 4467}, {1639, 3004}, {1734, 23954}, {1960, 29110}, {2254, 47809}, {2488, 68820}, {2490, 4763}, {2785, 49279}, {2789, 4474}, {2977, 4913}, {3125, 24194}, {3261, 21612}, {3667, 48016}, {3676, 47779}, {3776, 4885}, {3798, 43061}, {3801, 47872}, {3805, 50510}, {3907, 48299}, {3995, 4024}, {4010, 48103}, {4025, 26248}, {4040, 47711}, {4063, 7265}, {4088, 47694}, {4106, 4944}, {4120, 20295}, {4379, 47676}, {4380, 48266}, {4382, 47663}, {4394, 28898}, {4401, 29062}, {4448, 50340}, {4453, 24924}, {4458, 4874}, {4500, 4762}, {4521, 47778}, {4568, 7035}, {4681, 4777}, {4724, 47690}, {4750, 27013}, {4761, 49276}, {4778, 47984}, {4782, 29078}, {4785, 48060}, {4790, 28867}, {4794, 29192}, {4800, 48349}, {4804, 48408}, {4808, 48305}, {4813, 47769}, {4838, 47661}, {4893, 45746}, {4897, 47767}, {4931, 47892}, {4932, 28846}, {4951, 50358}, {4976, 47884}, {4977, 47991}, {4979, 44449}, {4988, 47659}, {5592, 29066}, {6084, 49289}, {6544, 27115}, {6545, 26985}, {7192, 28855}, {7662, 48088}, {8060, 21194}, {8714, 17989}, {9957, 14077}, {13246, 47804}, {14321, 48049}, {14432, 48298}, {14475, 48425}, {14837, 24793}, {14838, 68806}, {15584, 52305}, {17161, 26777}, {17496, 28005}, {18108, 32927}, {18743, 29427}, {18788, 28487}, {20517, 29358}, {20907, 21439}, {20908, 24622}, {20947, 21123}, {21051, 68979}, {21104, 28890}, {21124, 47793}, {21125, 32775}, {21146, 48083}, {21206, 30025}, {21438, 30061}, {23731, 47759}, {23879, 48003}, {23886, 47130}, {24089, 57068}, {25380, 47807}, {25381, 50454}, {26824, 48416}, {26853, 53339}, {27138, 48156}, {28147, 48162}, {28161, 47811}, {28225, 48036}, {28468, 49280}, {28840, 48046}, {28851, 43067}, {28859, 48026}, {28894, 48404}, {28906, 47768}, {29021, 59672}, {29051, 48395}, {29074, 48331}, {29090, 50512}, {29116, 48400}, {29118, 48267}, {29216, 48011}, {29350, 49288}, {29354, 52601}, {29545, 33889}, {30584, 63812}, {30665, 50491}, {30835, 44435}, {31147, 48138}, {31148, 48112}, {31150, 47665}, {31209, 47677}, {31250, 47754}, {31287, 45684}, {31290, 50522}, {44312, 64644}, {44429, 47973}, {45320, 48124}, {45344, 64914}, {46403, 48102}, {47656, 47873}, {47657, 47878}, {47658, 47669}, {47664, 48423}, {47671, 47792}, {47673, 47782}, {47685, 48105}, {47686, 48139}, {47687, 48032}, {47689, 47972}, {47691, 47832}, {47693, 47701}, {47695, 47700}, {47696, 48023}, {47697, 48077}, {47698, 48142}, {47699, 47826}, {47702, 48161}, {47703, 47969}, {47704, 47834}, {47715, 47970}, {47719, 47929}, {47760, 47960}, {47762, 47971}, {47764, 47981}, {47786, 49294}, {47789, 49296}, {47791, 48141}, {47806, 48015}, {47812, 48113}, {47833, 48326}, {47869, 48418}, {47907, 48543}, {47919, 48558}, {47931, 48159}, {47962, 48397}, {47998, 48166}, {48006, 48546}, {48021, 49283}, {48038, 49293}, {48055, 48396}, {48061, 49285}, {48067, 49284}, {48076, 48107}, {48078, 48108}, {48079, 48104}, {48080, 48106}, {48089, 48096}, {48090, 48097}, {48098, 48614}, {48125, 48417}, {48145, 49297}, {48172, 53558}, {48184, 48604}, {48201, 50335}, {48208, 53343}, {48219, 50336}, {48232, 50357}, {48235, 50359}, {48422, 59755}, {50347, 53580}

X(69293) = midpoint of X(i) and X(j) for these {i,j}: {649, 25259}, {650, 48271}, {659, 4122}, {661, 47660}, {663, 47707}, {693, 48094}, {2254, 49275}, {3700, 47890}, {3762, 47682}, {4010, 48103}, {4024, 17494}, {4040, 47711}, {4063, 7265}, {4088, 47694}, {4106, 48095}, {4120, 47773}, {4380, 48266}, {4382, 47663}, {4391, 48300}, {4468, 6590}, {4474, 47728}, {4724, 47690}, {4761, 49276}, {4804, 48408}, {4808, 48305}, {4813, 49282}, {4838, 47661}, {4931, 47892}, {4979, 44449}, {4988, 47659}, {6546, 47870}, {7192, 48082}, {7662, 48088}, {16892, 49273}, {17161, 48429}, {20295, 48101}, {21146, 48083}, {31290, 50522}, {43067, 48087}, {46403, 48102}, {47652, 48130}, {47656, 47926}, {47658, 47669}, {47662, 47958}, {47665, 48277}, {47666, 48275}, {47676, 48117}, {47685, 48105}, {47686, 48139}, {47687, 48032}, {47689, 47972}, {47691, 48118}, {47693, 47701}, {47695, 47700}, {47696, 48023}, {47697, 48077}, {47698, 48142}, {47703, 47969}, {47715, 47970}, {47719, 47929}, {47874, 48557}, {47962, 48397}, {47971, 49272}, {48021, 49283}, {48026, 49281}, {48038, 49293}, {48046, 48276}, {48055, 48396}, {48060, 48269}, {48061, 49285}, {48062, 49286}, {48067, 49284}, {48076, 48107}, {48078, 48108}, {48079, 48104}, {48080, 48106}, {48089, 48096}, {48090, 48097}, {48098, 48614}, {48113, 49301}, {48124, 49299}, {48138, 49298}, {48145, 49297}
X(69293) = reflection of X(i) in X(j) for these {i,j}: {3004, 25666}, {3776, 4885}, {3798, 43061}, {3835, 3239}, {4025, 31286}, {4369, 68794}, {4458, 4874}, {4913, 2977}, {10196, 47770}, {16892, 21212}, {17069, 2490}, {21194, 8060}, {21196, 650}, {21204, 47879}, {45674, 47766}, {47971, 59630}, {48008, 11068}, {48049, 14321}, {48125, 48417}, {48398, 59522}, {48422, 59755}, {49299, 48415}, {50347, 53580}, {50348, 25380}
X(69293) = complement of X(16892)
X(69293) = anticomplement of X(21212)
X(69293) = complement of the isogonal conjugate of X(4628)
X(69293) = X(54120)-anticomplementary conjugate of X(21293)
X(69293) = X(i)-complementary conjugate of X(j) for these (i,j): {37, 46654}, {82, 116}, {83, 21252}, {100, 21248}, {101, 21249}, {213, 15449}, {251, 11}, {321, 55070}, {692, 6292}, {827, 3739}, {1415, 17055}, {2205, 35971}, {4577, 21240}, {4599, 3741}, {4628, 10}, {4630, 3666}, {18082, 21253}, {18098, 125}, {32739, 16587}, {34072, 1125}, {36081, 20541}, {46288, 1015}, {46289, 1086}, {52376, 53564}, {56186, 53575}, {56245, 124}, {58113, 40959}, {59996, 64523}
X(69293) = X(190)-Ceva conjugate of X(17741)
X(69293) = X(i)-isoconjugate of X(j) for these (i,j): {6, 65364}, {101, 7194}, {692, 39724}, {1415, 43749}, {3502, 34071}, {32739, 40038}
X(69293) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 65364}, {1015, 7194}, {1086, 39724}, {1146, 43749}, {2329, 4579}, {8061, 2530}, {16706, 33951}, {40610, 3502}, {40619, 40038}
X(69293) = crosspoint of X(190) and X(257)
X(69293) = crosssum of X(i) and X(j) for these (i,j): {6, 21123}, {172, 649}, {1459, 26923}
X(69293) = crossdifference of every pair of points on line {31, 2275}
X(69293) = barycentric product X(i)*X(j) for these {i,j}: {37, 18077}, {513, 33938}, {514, 17280}, {522, 56928}, {523, 33954}, {693, 3961}, {3261, 69282}, {3494, 20906}, {4391, 56547}, {35519, 41346}
X(69293) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 65364}, {513, 7194}, {514, 39724}, {522, 43749}, {693, 40038}, {3494, 932}, {3961, 100}, {4083, 3502}, {17280, 190}, {17741, 18047}, {18077, 274}, {33938, 668}, {33954, 99}, {34249, 34071}, {41346, 109}, {55043, 2530}, {56547, 651}, {56928, 664}, {66997, 34067}, {69282, 101}
X(69293) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 16892, 21212}, {2, 49273, 16892}, {661, 4486, 3835}, {693, 48557, 48094}, {1639, 3004, 25666}, {2490, 17069, 4763}, {3239, 3835, 45661}, {3776, 4885, 21204}, {3776, 47879, 4885}, {3798, 43061, 45313}, {4024, 6546, 17494}, {4025, 31286, 45674}, {4025, 47766, 31286}, {4120, 48101, 20295}, {4379, 48117, 47676}, {4453, 24924, 59749}, {4728, 48130, 47652}, {4776, 47662, 47958}, {4944, 48095, 4106}, {7192, 47772, 48082}, {10196, 21196, 650}, {17161, 31992, 26777}, {17494, 47870, 4024}, {20295, 47773, 48101}, {24924, 47930, 4453}, {25259, 47771, 649}, {26985, 49302, 6545}, {30565, 47660, 661}, {30835, 47923, 44435}, {31147, 48138, 49298}, {31150, 47665, 48277}, {31209, 47677, 47886}, {44449, 48567, 4979}, {45320, 48124, 49299}, {45320, 49299, 48415}, {47659, 47775, 4988}, {47663, 47790, 4382}, {47693, 47821, 47701}, {47694, 48171, 4088}, {47762, 47971, 59630}, {47762, 49272, 47971}, {47769, 49282, 4813}, {47770, 48271, 650}, {47787, 48398, 59522}, {47807, 50348, 25380}, {47809, 49275, 2254}, {47812, 48113, 49301}, {47832, 48118, 47691}, {47873, 47926, 47656}, {47874, 48094, 693}, {47881, 48087, 43067}, {48080, 48236, 48106}

X(69294) = X(2)X(31)nX(10)X(321) Barycentrics (b + c)*(a^2 + a*b + b^2 + a*c + c^2) : : X(69294) lies on these lines: {1, 32782}, {2, 31}, {6, 29647}, {8, 32928}, {9, 26061}, {10, 321}, {11, 31241}, {12, 28387}, {35, 24931}, {37, 6536}, {38, 3778}, {42, 1211}, {65, 27714}, {69, 62821}, {71, 1213}, {75, 32776}, {79, 1224}, {81, 33082}, {86, 32949}, {100, 31247}, {141, 3720}, {213, 68934}, {244, 54311}, {306, 1962}, {333, 29631}, {350, 5224}, {354, 17237}, {427, 2354}, {474, 28268}, {484, 1698}, {518, 29685}, {594, 4365}, {612, 33074}, {614, 17306}, {740, 56810}, {846, 32779}, {894, 4683}, {899, 5743}, {908, 31264}, {940, 33080}, {958, 28377}, {968, 33156}, {982, 24164}, {984, 29667}, {1001, 24943}, {1125, 18139}, {1150, 29635}, {1193, 13728}, {1215, 26580}, {1245, 4205}, {1279, 29686}, {1281, 33115}, {1330, 19865}, {1386, 68945}, {1621, 32783}, {1654, 32864}, {1834, 59307}, {1836, 17303}, {1853, 3556}, {1961, 33078}, {2886, 30970}, {2895, 4649}, {3112, 16889}, {3120, 20716}, {3175, 6535}, {3187, 50308}, {3219, 24697}, {3416, 5311}, {3589, 41002}, {3661, 32915}, {3666, 69252}, {3681, 29659}, {3687, 46904}, {3696, 8013}, {3703, 3989}, {3706, 17239}, {3717, 42041}, {3739, 69253}, {3750, 33175}, {3757, 32775}, {3763, 4423}, {3769, 29847}, {3773, 3995}, {3775, 17135}, {3791, 29833}, {3821, 4359}, {3842, 28595}, {3844, 29687}, {3883, 17469}, {3896, 21085}, {3920, 33076}, {3931, 20653}, {3936, 43223}, {3966, 4657}, {3969, 3993}, {3980, 32950}, {4028, 21806}, {4038, 32863}, {4042, 17251}, {4062, 37593}, {4071, 5257}, {4085, 4651}, {4104, 21805}, {4202, 23682}, {4363, 33098}, {4383, 29663}, {4387, 17293}, {4389, 17155}, {4416, 4722}, {4418, 19808}, {4429, 26037}, {4643, 32912}, {4670, 64164}, {4687, 29854}, {4703, 26223}, {4708, 25368}, {4710, 40603}, {4981, 29673}, {5014, 36480}, {5235, 33138}, {5247, 26064}, {5260, 19879}, {5263, 32947}, {5271, 33128}, {5278, 25453}, {5284, 29637}, {5297, 33079}, {5361, 29864}, {5737, 24892}, {5739, 61358}, {5741, 6685}, {5750, 41011}, {5846, 29816}, {6541, 48648}, {6646, 32940}, {6682, 69134}, {7226, 33169}, {7295, 59358}, {8025, 20290}, {8040, 48650}, {9256, 30911}, {9780, 69025}, {9791, 32936}, {10176, 19867}, {10448, 13725}, {10453, 17238}, {11599, 60203}, {12609, 19857}, {14829, 29845}, {16062, 31339}, {16484, 33173}, {16589, 21813}, {16823, 33123}, {16825, 32774}, {16830, 33072}, {16874, 24921}, {16927, 30175}, {16991, 17248}, {17011, 32861}, {17018, 33084}, {17019, 32846}, {17156, 17270}, {17163, 50312}, {17184, 24325}, {17250, 33120}, {17256, 33118}, {17257, 33163}, {17272, 62819}, {17277, 29850}, {17289, 32930}, {17302, 32924}, {17307, 32942}, {17321, 33088}, {17322, 33073}, {17325, 17599}, {17326, 32844}, {17327, 33104}, {17384, 29684}, {17592, 33077}, {17600, 32842}, {19684, 32946}, {19786, 32914}, {19804, 33125}, {19812, 29636}, {19822, 24248}, {19854, 56456}, {19856, 33109}, {19863, 52782}, {20017, 50281}, {20292, 24342}, {20727, 39244}, {21027, 21949}, {23868, 37061}, {23928, 62548}, {24169, 24589}, {25354, 29653}, {25539, 29666}, {26102, 33172}, {26117, 54331}, {26251, 59511}, {26942, 42289}, {27065, 33159}, {27184, 32771}, {27186, 40328}, {27627, 56734}, {28599, 50288}, {28605, 33154}, {28606, 32778}, {29604, 40998}, {29633, 32911}, {29640, 30831}, {29644, 33070}, {29651, 33122}, {29655, 46909}, {29661, 30811}, {29662, 37660}, {29814, 33087}, {29815, 49506}, {29822, 31037}, {29829, 32853}, {29837, 32919}, {29846, 30832}, {29863, 35466}, {30768, 37330}, {30950, 69092}, {31025, 48643}, {33085, 37633}, {33094, 50314}, {33108, 59312}, {33119, 38000}, {33164, 33761}, {33167, 62796}, {33171, 62849}, {42039, 63147}, {44417, 69173}, {46901, 69091}, {48644, 62227}, {49511, 62867}, {50058, 64172}, {56520, 59624}, {57808, 58386}, {62846, 63013} X(69294) = X(4657)-Dao conjugate of X(1010) X(69294) = crosspoint of X(i) and X(j) for these (i,j): {75, 43531}, {4657, 33945} X(69294) = crosssum of X(i) and X(j) for these (i,j): {31, 386}, {58, 44119} X(69294) = crossdifference of every pair of points on line {3250, 57047} X(69294) = barycentric product X(i)*X(j) for these {i,j}: {10, 4657}, {37, 33945}, {226, 3966}, {321, 17017}, {523, 33952}, {3952, 47958}, {27808, 50455} X(69294) = barycentric quotient X(i)/X(j) for these {i,j}: {3966, 333}, {4657, 86}, {17017, 81}, {33945, 274}, {33952, 99}, {47958, 7192}, {50455, 3733} X(69294) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 32782, 33081}, {2, 4388, 32772}, {2, 6327, 50302}, {2, 25760, 33105}, {2, 25958, 33111}, {2, 26034, 750}, {2, 32784, 32781}, {2, 33083, 171}, {2, 33086, 17122}, {2, 50295, 31}, {10, 3914, 21020}, {10, 4425, 321}, {10, 5051, 21935}, {75, 32776, 33145}, {306, 50290, 1962}, {594, 4854, 4365}, {984, 29667, 33162}, {1211, 4026, 42}, {1213, 3925, 59306}, {2887, 50298, 2}, {3703, 4364, 3989}, {3763, 4423, 29677}, {3842, 28595, 69250}, {3844, 44307, 29687}, {3966, 4657, 17017}, {4972, 41809, 10}, {5224, 32773, 31330}, {6536, 15523, 37}, {19808, 24723, 4418}, {24697, 32780, 3219}, {27184, 32771, 32856}, {28606, 32778, 32848}, {29837, 37653, 32919}, {31330, 32773, 33136} X(69295) = X(1)X(26061)nX(2)X(3773) Barycentrics (b + c)*(-a^2 - a*b + b^2 - a*c + c^2) : : X(69295) lies on these lines: {1, 26061}, {2, 3773}, {8, 56990}, {9, 32852}, {10, 1962}, {31, 17776}, {37, 6536}, {38, 3912}, {42, 3932}, {75, 29854}, {81, 33164}, {141, 3989}, {171, 32849}, {190, 32949}, {192, 25957}, {210, 4062}, {226, 3994}, {238, 33093}, {306, 756}, {312, 29643}, {321, 6541}, {344, 748}, {345, 750}, {354, 21865}, {536, 69253}, {594, 59306}, {612, 33156}, {726, 18139}, {740, 69250}, {846, 33078}, {896, 56078}, {940, 33161}, {968, 33074}, {984, 32858}, {1001, 32854}, {1621, 32847}, {1698, 62816}, {1961, 32779}, {1999, 33115}, {2177, 10327}, {2321, 21020}, {2325, 41011}, {2340, 65684}, {2650, 3710}, {2887, 3995}, {3120, 3175}, {3178, 3701}, {3210, 25961}, {3219, 32846}, {3632, 62856}, {3666, 29687}, {3685, 33072}, {3694, 21717}, {3695, 59305}, {3703, 3720}, {3714, 21674}, {3750, 33091}, {3790, 32771}, {3799, 5208}, {3836, 17147}, {3842, 56810}, {3846, 31035}, {3879, 4722}, {3891, 29642}, {3914, 3950}, {3920, 33158}, {3925, 3943}, {3936, 3971}, {3966, 41313}, {3980, 50105}, {3993, 4972}, {4011, 33070}, {4028, 21805}, {4038, 33170}, {4042, 17309}, {4085, 27804}, {4358, 29671}, {4360, 29850}, {4387, 33104}, {4417, 64178}, {4418, 42033}, {4425, 48647}, {4439, 17165}, {4527, 17163}, {4535, 27798}, {4645, 32936}, {4649, 33166}, {4664, 32776}, {4671, 33111}, {4682, 50104}, {4683, 17261}, {4693, 33110}, {4851, 32912}, {4884, 17449}, {4970, 49769}, {4981, 49560}, {5284, 32866}, {5297, 33160}, {5311, 32777}, {5432, 62659}, {5741, 59517}, {5745, 49990}, {6057, 17056}, {6535, 31993}, {6542, 32864}, {7226, 33087}, {9340, 59544}, {10448, 54433}, {10458, 40790}, {15569, 29685}, {16484, 33090}, {16777, 29647}, {17011, 33159}, {17017, 17279}, {17018, 33165}, {17019, 32780}, {17061, 29869}, {17122, 33168}, {17123, 32842}, {17124, 17740}, {17155, 17234}, {17184, 49456}, {17233, 31330}, {17242, 29641}, {17262, 33098}, {17264, 32930}, {17267, 17599}, {17280, 32772}, {17300, 32940}, {17315, 33118}, {17316, 33163}, {17351, 64164}, {17357, 29684}, {17469, 49476}, {17592, 29679}, {17602, 29865}, {17725, 29866}, {17763, 33116}, {17772, 19742}, {17778, 32938}, {17889, 42044}, {18134, 32856}, {18743, 29849}, {20011, 49693}, {20182, 29663}, {21828, 21959}, {21912, 21920}, {21935, 57808}, {22034, 48642}, {24725, 56082}, {25760, 41839}, {25959, 33154}, {26065, 62846}, {26102, 33089}, {27065, 32861}, {27186, 49493}, {28606, 29674}, {29573, 62819}, {29583, 54352}, {29631, 34064}, {29632, 32926}, {29633, 62851}, {29649, 33113}, {29659, 62840}, {29682, 44417}, {29688, 30818}, {29814, 33169}, {29830, 32920}, {29839, 32927}, {29840, 56196}, {29851, 32922}, {29862, 33133}, {29873, 33135}, {30615, 67207}, {30950, 69091}, {30965, 40774}, {31025, 48644}, {31151, 33102}, {32935, 63056}, {33069, 49447}, {33082, 33761}, {33085, 62796}, {33117, 49470}, {33131, 49452}, {33146, 49445}, {33167, 37633}, {35309, 40463}, {35652, 69173}, {38047, 67208}, {42039, 49511}, {44307, 69252}, {46901, 69092}, {48643, 62227}, {48652, 50298}, {49688, 67209}, {49766, 63134}, {50743, 64162}, {60459, 60714}, {62867, 63147} X(69295) = crosspoint of X(4851) and X(33942) X(69295) = barycentric product X(i)*X(j) for these {i,j}: {10, 4851}, {37, 33942}, {313, 69231}, {321, 32912}, {3952, 47971} X(69295) = barycentric quotient X(i)/X(j) for these {i,j}: {4851, 86}, {32912, 81}, {33942, 274}, {47971, 7192}, {69231, 58} X(69295) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 32862, 33162}, {2, 33092, 32848}, {192, 25957, 33145}, {306, 4078, 756}, {312, 29643, 33105}, {344, 33088, 748}, {984, 32858, 33081}, {3703, 17243, 3720}, {3925, 3943, 4365}, {6541, 29653, 321}, {17242, 29641, 32915}, {17264, 33073, 32930}, {17267, 17599, 29677}, {17316, 33163, 62821}, {18134, 32925, 32856}, {28606, 29674, 32781}, {29641, 32915, 33136}, {57808, 63800, 21935} X(69296) = X(2)X(1390)nX(10)X(321) Barycentrics (b + c)*(a^2 + b^2 + b*c + c^2) : : X(69296) lies on these lines: {2, 1390}, {8, 13740}, {10, 321}, {42, 3773}, {75, 29679}, {100, 8852}, {141, 17165}, {190, 33083}, {210, 4463}, {226, 48647}, {281, 6605}, {306, 46897}, {312, 29667}, {333, 33166}, {594, 2238}, {726, 32781}, {740, 6535}, {894, 33078}, {908, 39597}, {1150, 33163}, {1211, 3952}, {1215, 3936}, {1621, 17280}, {1698, 32774}, {1738, 4980}, {1757, 3578}, {1824, 3697}, {1962, 6541}, {2229, 16587}, {2321, 3896}, {2345, 5276}, {2551, 3434}, {2886, 31079}, {3006, 44417}, {3120, 28595}, {3187, 38047}, {3416, 26223}, {3589, 17150}, {3661, 3681}, {3666, 26251}, {3679, 5014}, {3695, 26115}, {3715, 63100}, {3717, 4981}, {3729, 32950}, {3741, 33162}, {3743, 7206}, {3744, 17359}, {3745, 50000}, {3757, 24542}, {3790, 28606}, {3844, 17184}, {3846, 30566}, {3870, 17286}, {3920, 17289}, {3923, 4450}, {3925, 31025}, {3943, 27804}, {3967, 26580}, {3989, 4439}, {3995, 4026}, {4015, 21683}, {4046, 19998}, {4054, 48646}, {4085, 4365}, {4359, 24186}, {4362, 26061}, {4388, 41242}, {4418, 33079}, {4427, 44419}, {4429, 28605}, {4467, 23954}, {4671, 32773}, {4733, 6539}, {4854, 62227}, {4892, 48650}, {5263, 33091}, {5297, 19808}, {5741, 32778}, {5772, 34255}, {6685, 32848}, {7081, 32779}, {8013, 20679}, {8300, 32945}, {9564, 63961}, {9780, 19785}, {11679, 33114}, {11680, 20545}, {14829, 33170}, {17018, 17233}, {17127, 17354}, {17135, 49524}, {17140, 69092}, {17155, 33174}, {17242, 62840}, {17281, 32929}, {17285, 33173}, {17355, 63134}, {17368, 62807}, {17371, 29648}, {17469, 24295}, {17594, 50105}, {17757, 37346}, {17763, 32780}, {17769, 29819}, {17873, 21408}, {18139, 29674}, {19738, 50284}, {19825, 26040}, {19875, 50102}, {20056, 63020}, {20069, 63053}, {20553, 32025}, {21021, 68944}, {21085, 21805}, {21407, 40999}, {21730, 27700}, {21865, 22275}, {24325, 29687}, {24349, 33172}, {24358, 50274}, {24589, 62673}, {24943, 32920}, {25496, 32854}, {26034, 32933}, {26047, 41915}, {26227, 32777}, {27064, 33075}, {28599, 63979}, {29633, 32928}, {29637, 32923}, {29659, 32915}, {29663, 32921}, {29669, 62849}, {29670, 33156}, {29671, 31264}, {29684, 49472}, {29828, 33113}, {30699, 46933}, {30818, 69134}, {30942, 33169}, {31161, 33064}, {31330, 33165}, {31993, 69250}, {32772, 32847}, {32782, 32937}, {32783, 32927}, {32784, 32925}, {32846, 42045}, {32866, 32944}, {32912, 50313}, {32914, 33159}, {32916, 33161}, {32917, 33164}, {32918, 33167}, {32930, 33076}, {32935, 33080}, {32938, 33082}, {32939, 33086}, {32940, 33085}, {32942, 33090}, {33110, 37349}, {33125, 49493}, {33131, 42029}, {33134, 42034}, {33139, 55095}, {46909, 63147}, {49483, 69251}, {49994, 59628}, {58443, 62659}, {58644, 61172}, {59511, 69252} X(69296) = X(i)-Ceva conjugate of X(j) for these (i,j): {17289, 28594}, {40845, 3930} X(69296) = X(i)-isoconjugate of X(j) for these (i,j): {831, 3733}, {1169, 17108} X(69296) = X(28594)-Dao conjugate of X(3589) X(69296) = cevapoint of X(4538) and X(28594) X(69296) = crosspoint of X(i) and X(j) for these (i,j): {75, 10159}, {17289, 33941} X(69296) = crosssum of X(31) and X(5007) X(69296) = barycentric product X(i)*X(j) for these {i,j}: {10, 17289}, {37, 33941}, {75, 28594}, {85, 4538}, {190, 47711}, {313, 5280}, {321, 3920}, {523, 69001}, {594, 68984}, {830, 4033}, {1978, 50496}, {2321, 7247}, {2483, 27808}, {3952, 47660}, {14622, 28593}, {17672, 56157} X(69296) = barycentric quotient X(i)/X(j) for these {i,j}: {830, 1019}, {1018, 831}, {2292, 17108}, {2483, 3733}, {3920, 81}, {4033, 57975}, {4538, 9}, {5280, 58}, {5314, 1790}, {7247, 1434}, {7859, 33955}, {8635, 57129}, {17289, 86}, {17672, 17169}, {28594, 1}, {33941, 274}, {47660, 7192}, {47711, 514}, {50496, 649}, {68984, 1509}, {69001, 99}, {69002, 68984} X(69296) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 321, 4972}, {10, 756, 41809}, {10, 1089, 5051}, {42, 3773, 3969}, {306, 53663, 46897}, {321, 4972, 4442}, {1215, 15523, 3936}, {3757, 33157, 24542}, {3923, 33074, 4450}, {4026, 6057, 3995}, {4085, 48644, 4365}, {29674, 32771, 18139}, {32778, 32931, 5741}, {32918, 33167, 51583}, {38047, 69089, 3187}, {46897, 48648, 306} X(69297) = X(2)X(49455)nX(10)X(321) Barycentrics (b + c)*(a^2 - a*b + b^2 - a*c + b*c + c^2) : : X(69297) lies on these lines: {2, 49455}, {8, 4011}, {10, 321}, {42, 6541}, {43, 3790}, {141, 42054}, {171, 49994}, {190, 33079}, {210, 3773}, {306, 4090}, {312, 29673}, {344, 29651}, {519, 5315}, {537, 69092}, {594, 3985}, {726, 24169}, {740, 6057}, {940, 50313}, {1125, 32923}, {1211, 4096}, {1215, 3932}, {2238, 2321}, {2308, 50000}, {2796, 32948}, {2887, 3967}, {3175, 4085}, {3293, 7206}, {3626, 40998}, {3634, 26724}, {3666, 4439}, {3678, 22076}, {3679, 30568}, {3681, 49560}, {3696, 48644}, {3699, 33160}, {3703, 37663}, {3706, 49693}, {3715, 50308}, {3717, 3741}, {3750, 17264}, {3821, 29679}, {3840, 63147}, {3846, 4009}, {3920, 24295}, {3923, 10327}, {3950, 21840}, {3952, 15523}, {3961, 17280}, {3969, 21805}, {3974, 4362}, {3977, 59679}, {3989, 26251}, {4030, 4432}, {4046, 4535}, {4078, 43223}, {4109, 21082}, {4126, 49457}, {4358, 29655}, {4368, 4651}, {4415, 28595}, {4418, 60459}, {4434, 44416}, {4660, 56082}, {4671, 33117}, {4683, 4756}, {5205, 33167}, {5249, 49769}, {5276, 17355}, {5297, 59628}, {5743, 42056}, {6538, 56132}, {7018, 36863}, {7081, 33164}, {8616, 17339}, {11814, 69134}, {16587, 21902}, {17135, 49697}, {17155, 24200}, {17165, 29687}, {17242, 42042}, {17279, 29672}, {17354, 17716}, {17763, 33166}, {17766, 32930}, {17770, 32938}, {17776, 29670}, {17781, 50304}, {18139, 31161}, {18743, 33169}, {20942, 24217}, {21803, 59315}, {21949, 48641}, {22325, 40521}, {24003, 69091}, {24165, 24175}, {24821, 26840}, {25385, 29641}, {25591, 49613}, {26061, 29645}, {27064, 32847}, {27538, 32778}, {29649, 33163}, {29654, 32926}, {29656, 32927}, {29659, 41839}, {29667, 64178}, {29671, 32862}, {29674, 32937}, {29685, 31035}, {30615, 32941}, {31079, 69173}, {31204, 33115}, {32779, 59726}, {32865, 42034}, {33072, 41242}, {33085, 62222}, {33118, 50755}, {33158, 50748}, {33171, 53661}, {33174, 49447}, {33175, 53660}, {42033, 60714}, {42057, 49529}, {49520, 54311}, {49685, 50292}, {51857, 52662}, {53601, 69251}, {63800, 64167} X(69297) = midpoint of X(i) and X(j) for these {i,j}: {32930, 33091}, {32938, 33078} X(69297) = X(i)-isoconjugate of X(j) for these (i,j): {58, 7194}, {1333, 39724}, {1408, 43749}, {2206, 40038}, {3733, 65364} X(69297) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 7194}, {37, 39724}, {16706, 33955}, {40603, 40038}, {59577, 43749} X(69297) = crosspoint of X(17280) and X(33938) X(69297) = barycentric product X(i)*X(j) for these {i,j}: {10, 17280}, {37, 33938}, {313, 69282}, {321, 3961}, {594, 33954}, {2321, 56928}, {3701, 56547}, {18077, 40521}, {30713, 41346} X(69297) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 39724}, {37, 7194}, {321, 40038}, {1018, 65364}, {2321, 43749}, {3961, 81}, {17280, 86}, {17741, 17103}, {20691, 3502}, {33938, 274}, {33954, 1509}, {41346, 1412}, {56547, 1014}, {56928, 1434}, {66997, 18268}, {69282, 58} X(69297) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 3971, 4425}, {10, 4082, 3971}, {10, 4135, 3914}, {210, 3773, 21085}, {312, 33165, 29673}, {1215, 3932, 29653}, {2887, 3967, 21093}, {4078, 53663, 43223}, {4358, 33162, 29655}, {17165, 29687, 49676}, {17279, 32920, 29672}, {29674, 32937, 33064}, {29679, 32925, 3821}, {32862, 32931, 29671}, {32926, 33159, 29654}, {32927, 33157, 29656} X(69298) = X(2)X(17598)nX(10)X(321) Barycentrics (b + c)*(a^2 - a*b + b^2 - a*c + c^2) : : X(69298) lies on these lines: {2, 17598}, {8, 748}, {9, 33074}, {10, 321}, {31, 10327}, {38, 3717}, {42, 3932}, {43, 32848}, {100, 33164}, {141, 4126}, {171, 33166}, {190, 32948}, {200, 33156}, {210, 15523}, {226, 21026}, {238, 33091}, {244, 62673}, {306, 21805}, {312, 33117}, {329, 31134}, {344, 62849}, {518, 29687}, {537, 69251}, {594, 59207}, {612, 26061}, {614, 4901}, {750, 33163}, {899, 3703}, {984, 29679}, {1215, 69250}, {1376, 33161}, {1698, 26724}, {1757, 33078}, {1962, 4078}, {2177, 17776}, {2887, 3952}, {3006, 59511}, {3120, 3967}, {3214, 3695}, {3219, 33079}, {3240, 33092}, {3242, 29677}, {3305, 3679}, {3589, 29816}, {3610, 40977}, {3681, 29674}, {3696, 6535}, {3697, 20653}, {3699, 29846}, {3705, 27130}, {3720, 49524}, {3740, 69252}, {3744, 49996}, {3748, 41310}, {3773, 4651}, {3790, 32860}, {3791, 50000}, {3823, 69253}, {3836, 17165}, {3846, 31079}, {3896, 6541}, {3920, 33159}, {3935, 33158}, {3936, 4090}, {3938, 4952}, {3961, 33157}, {3969, 4685}, {3995, 4085}, {4009, 69173}, {4011, 5014}, {4030, 4422}, {4062, 4849}, {4080, 48649}, {4096, 26580}, {4113, 17229}, {4357, 42041}, {4358, 29673}, {4365, 6057}, {4383, 32854}, {4423, 59407}, {4429, 32925}, {4434, 56520}, {4437, 30821}, {4439, 17147}, {4457, 4535}, {4645, 32938}, {4661, 33087}, {4671, 32865}, {4756, 33099}, {4767, 30831}, {5205, 33119}, {5220, 33080}, {5249, 31161}, {5297, 32780}, {5311, 38047}, {5423, 31237}, {7081, 33115}, {7226, 33174}, {9053, 29818}, {9350, 17740}, {16569, 33089}, {17024, 49534}, {17122, 33170}, {17123, 33090}, {17135, 49693}, {17163, 48644}, {17184, 42054}, {17267, 41711}, {17280, 32945}, {17339, 63139}, {17340, 34612}, {17341, 29853}, {17353, 17469}, {17357, 29686}, {17483, 31151}, {17602, 29867}, {17719, 29873}, {17724, 29869}, {17763, 33118}, {18139, 49769}, {18141, 54352}, {18743, 33120}, {20684, 35309}, {21015, 21031}, {21085, 48648}, {21093, 48646}, {21813, 52959}, {21949, 48642}, {23636, 40585}, {24003, 69134}, {24165, 24988}, {24217, 46938}, {24349, 25961}, {24995, 33932}, {25760, 27538}, {25957, 32856}, {25959, 33101}, {26034, 27549}, {27064, 33072}, {27065, 33076}, {28387, 40663}, {29529, 67502}, {29641, 32931}, {29649, 33114}, {29653, 46897}, {29685, 44307}, {29690, 30818}, {29850, 32926}, {30171, 59666}, {30811, 59597}, {31037, 48651}, {32773, 64178}, {32778, 63961}, {32847, 32911}, {32849, 60714}, {32850, 32930}, {32863, 49712}, {32866, 37680}, {33067, 62222}, {33094, 56082}, {33109, 41242}, {33125, 49447}, {33144, 53661}, {33153, 53660}, {33168, 56009}, {33172, 49448}, {36568, 46937}, {37646, 62659}, {40172, 50104}, {42039, 54311}, {49529, 62867}, {49991, 59692}, {59406, 62821} X(69298) = X(3733)-isoconjugate of X(6012) X(69298) = crosspoint of X(17279) and X(33937) X(69298) = barycentric product X(i)*X(j) for these {i,j}: {10, 17279}, {37, 33937}, {190, 4808}, {226, 30615}, {306, 5101}, {321, 3938}, {594, 33953}, {2321, 30617}, {3952, 48094}, {4033, 6004}, {4052, 4952} X(69298) = barycentric quotient X(i)/X(j) for these {i,j}: {1018, 6012}, {3938, 81}, {4808, 514}, {4952, 41629}, {5101, 27}, {6004, 1019}, {8654, 57129}, {17279, 86}, {30615, 333}, {30617, 1434}, {33937, 274}, {33953, 1509}, {48094, 7192} X(69298) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 33165, 33162}, {10, 3701, 21935}, {10, 3710, 4642}, {10, 3971, 4972}, {10, 4082, 3914}, {10, 4125, 68946}, {43, 32862, 32848}, {312, 33117, 33136}, {984, 29679, 32781}, {3681, 29674, 33081}, {3914, 4082, 3994}, {4096, 28595, 26580}, {4429, 32925, 33145}, {17279, 30615, 3938}, {25957, 32937, 32856}, {29641, 32931, 33105}, {33166, 60459, 171}, {62673, 63147, 244} X(69299) = X(2)X(1757)nX(10)X(12) Barycentrics (b + c)*(-a^2 + a*b + b^2 + a*c - b*c + c^2) : : X(69299) = X[33066] + 2 X[59726], X[32932] + 3 X[69057], X[33099] - 3 X[69057] X(69299) lies on these lines: {1, 62998}, {2, 1757}, {6, 29645}, {8, 3944}, {9, 3771}, {10, 12}, {31, 49710}, {37, 10026}, {38, 5741}, {42, 4425}, {43, 3821}, {44, 6679}, {55, 4703}, {69, 24241}, {75, 33101}, {100, 4683}, {101, 40597}, {141, 59511}, {171, 17770}, {190, 33160}, {200, 4660}, {238, 29656}, {239, 33152}, {306, 3971}, {312, 33084}, {320, 17122}, {321, 1109}, {329, 3923}, {333, 17719}, {385, 4416}, {497, 49458}, {518, 3846}, {519, 4514}, {537, 69091}, {612, 32946}, {661, 22026}, {668, 7018}, {726, 3687}, {740, 4415}, {748, 29672}, {750, 32859}, {756, 3936}, {894, 59628}, {899, 17184}, {908, 3741}, {946, 45955}, {982, 5233}, {984, 4417}, {1054, 26840}, {1125, 5247}, {1329, 20256}, {1330, 5293}, {1376, 4655}, {1621, 50748}, {1961, 17778}, {2321, 4135}, {2796, 32932}, {2810, 3038}, {2886, 49457}, {2895, 17763}, {3120, 4651}, {3159, 21081}, {3219, 29846}, {3240, 32776}, {3305, 29642}, {3475, 24331}, {3662, 16569}, {3681, 25760}, {3696, 48643}, {3699, 33079}, {3703, 42054}, {3705, 49448}, {3715, 30811}, {3740, 3836}, {3744, 49705}, {3751, 29635}, {3773, 3967}, {3775, 44417}, {3782, 4023}, {3789, 30953}, {3791, 17602}, {3823, 58629}, {3834, 58451}, {3842, 17056}, {3844, 59596}, {3873, 25960}, {3912, 59517}, {3914, 4685}, {3920, 32843}, {3932, 4096}, {3935, 32947}, {3938, 49696}, {3952, 15523}, {3954, 4109}, {3961, 4388}, {3966, 32920}, {3969, 3994}, {3980, 5905}, {3993, 4028}, {3995, 4062}, {3996, 33095}, {4011, 31018}, {4035, 4078}, {4054, 62226}, {4061, 4709}, {4080, 17163}, {4085, 4849}, {4101, 67976}, {4357, 6685}, {4358, 33081}, {4359, 32856}, {4362, 5739}, {4383, 26128}, {4418, 17484}, {4438, 5220}, {4527, 22034}, {4643, 25353}, {4650, 17347}, {4661, 33120}, {4847, 49510}, {4903, 17230}, {4972, 21805}, {4974, 17061}, {4981, 33105}, {5057, 32945}, {5205, 33085}, {5263, 33096}, {5278, 33127}, {5297, 32949}, {5311, 31034}, {5743, 24325}, {5988, 7081}, {6536, 29822}, {6646, 17596}, {6682, 37662}, {6690, 17332}, {6745, 59679}, {7179, 17272}, {7226, 29849}, {8013, 31025}, {9330, 29854}, {9436, 20103}, {10164, 24685}, {11019, 49505}, {11813, 67213}, {12579, 37573}, {13161, 59303}, {14555, 16825}, {14997, 29852}, {16468, 29634}, {16475, 29842}, {16496, 29844}, {16704, 29683}, {17017, 63010}, {17127, 29848}, {17135, 69173}, {17165, 69252}, {17231, 59506}, {17236, 59298}, {17277, 33130}, {17364, 37604}, {17605, 21242}, {17720, 32853}, {17724, 41002}, {17771, 58443}, {17795, 49612}, {17889, 59296}, {18743, 33087}, {18905, 21830}, {19804, 33103}, {20653, 56318}, {21076, 56541}, {21090, 22015}, {21240, 59690}, {21241, 25006}, {21616, 50608}, {21629, 59687}, {21873, 65096}, {21949, 48649}, {22325, 61166}, {24003, 69092}, {24165, 53601}, {24260, 30946}, {24295, 27064}, {24318, 24690}, {24703, 32941}, {24723, 60714}, {25354, 43223}, {25385, 31053}, {25568, 29670}, {25957, 63961}, {25958, 33117}, {26037, 31019}, {26098, 36480}, {26364, 27339}, {26688, 29677}, {26792, 32930}, {27065, 29632}, {27130, 31242}, {27131, 30942}, {27538, 29674}, {28599, 49996}, {28609, 50314}, {29636, 63074}, {29644, 63008}, {29650, 63089}, {29654, 32775}, {29658, 37652}, {29676, 31056}, {29687, 31017}, {29821, 63002}, {29843, 49498}, {29847, 37685}, {29851, 35595}, {30115, 38456}, {30831, 33115}, {30832, 32780}, {31142, 50311}, {32778, 32937}, {32779, 32938}, {32782, 32931}, {32855, 49447}, {32858, 64178}, {32860, 33151}, {32864, 33133}, {32914, 33153}, {32925, 33077}, {32927, 33075}, {33068, 56009}, {33078, 49994}, {33121, 49712}, {33123, 37680}, {33138, 60731}, {33141, 49450}, {33167, 62222}, {34379, 39595}, {41014, 63800}, {42334, 55095}, {49509, 63479}, {59544, 60942} X(69299) = midpoint of X(i) and X(j) for these {i,j}: {171, 33066}, {3961, 4388}, {3996, 33095}, {32861, 32926}, {32932, 33099} X(69299) = reflection of X(i) in X(j) for these {i,j}: {171, 59726}, {29655, 3846} X(69299) = complement of X(32913) X(69299) = X(60149)-complementary conjugate of X(141) X(69299) = X(6646)-Ceva conjugate of X(69248) X(69299) = X(1333)-isoconjugate of X(54120) X(69299) = X(i)-Dao conjugate of X(j) for these (i,j): {37, 54120}, {894, 17103} X(69299) = crosspoint of X(6646) and X(20955) X(69299) = barycentric product X(i)*X(j) for these {i,j}: {10, 6646}, {37, 20955}, {75, 69248}, {313, 21008}, {321, 17596}, {3952, 21212} X(69299) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 54120}, {6646, 86}, {17596, 81}, {20955, 274}, {21008, 58}, {21212, 7192}, {22161, 1790}, {62650, 17103}, {69248, 1} X(69299) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 33064, 49676}, {2, 33065, 33064}, {10, 21060, 4090}, {42, 26580, 4425}, {43, 27184, 3821}, {210, 2887, 10}, {226, 4104, 10}, {238, 33126, 29656}, {306, 3971, 6541}, {312, 33084, 49560}, {748, 33122, 29672}, {756, 3936, 29653}, {899, 17184, 24169}, {984, 4417, 29671}, {1211, 1215, 10}, {3452, 3840, 11814}, {3452, 49511, 3840}, {3454, 3678, 10}, {3681, 25760, 29673}, {3681, 29673, 49697}, {3952, 31037, 15523}, {4028, 4656, 3993}, {4080, 17163, 48642}, {6690, 17332, 59624}, {14555, 33144, 16825}, {17123, 33124, 1125}, {21085, 21093, 321}, {25568, 50295, 29670}, {26792, 33175, 32930}, {27064, 32783, 24295}, {31018, 33171, 4011}, {31053, 31330, 25385}, {32775, 32911, 29654}, {32864, 33133, 50755}, {32932, 69057, 33099}, {33153, 37656, 32914} X(69300) = X(2)X(3775)nX(10)X(2650) Barycentrics (b + c)*(-a^2 + a*b + b^2 + a*c + c^2) : : X(69300) lies on these lines: {2, 3775}, {8, 21935}, {9, 33156}, {10, 2650}, {11, 31136}, {12, 59307}, {31, 5739}, {37, 4062}, {38, 3687}, {42, 1211}, {43, 32781}, {69, 750}, {72, 20653}, {75, 32856}, {100, 31143}, {141, 899}, {171, 2895}, {193, 62846}, {200, 33074}, {210, 15523}, {226, 21020}, {238, 33175}, {239, 32775}, {244, 49511}, {306, 756}, {319, 17763}, {321, 1109}, {333, 29846}, {518, 69252}, {599, 4413}, {612, 32852}, {740, 26580}, {748, 14555}, {896, 4416}, {976, 5814}, {984, 32848}, {1079, 54289}, {1155, 17344}, {1215, 56810}, {1376, 33080}, {1654, 32917}, {1757, 32779}, {1962, 4028}, {2177, 50295}, {2321, 3994}, {2550, 31134}, {2887, 4651}, {3006, 49457}, {3011, 3686}, {3120, 3696}, {3219, 33160}, {3240, 32784}, {3626, 21241}, {3661, 4766}, {3679, 3822}, {3681, 32778}, {3706, 69173}, {3712, 17332}, {3720, 5743}, {3722, 3883}, {3740, 29687}, {3741, 5741}, {3759, 29636}, {3771, 5278}, {3773, 3952}, {3778, 68954}, {3836, 31017}, {3846, 17135}, {3879, 4938}, {3896, 4425}, {3914, 4061}, {3920, 32861}, {3935, 33076}, {3938, 3966}, {3946, 49986}, {3961, 33075}, {3967, 6535}, {3969, 3971}, {3980, 32859}, {3996, 32947}, {4009, 17229}, {4042, 24892}, {4046, 4365}, {4080, 48641}, {4085, 19998}, {4357, 46904}, {4358, 49560}, {4359, 33064}, {4360, 14459}, {4361, 33143}, {4383, 24943}, {4388, 32945}, {4414, 4643}, {4417, 31330}, {4418, 33066}, {4442, 4709}, {4527, 62227}, {4657, 67211}, {4661, 33169}, {4683, 32932}, {4684, 17450}, {4685, 4972}, {4703, 32929}, {4706, 17235}, {4716, 33155}, {4732, 4892}, {4852, 49985}, {4886, 32914}, {4933, 50093}, {4937, 50097}, {4966, 5241}, {4974, 26230}, {4981, 29671}, {5051, 59302}, {5205, 17287}, {5220, 33161}, {5233, 30942}, {5235, 29640}, {5257, 53034}, {5263, 32843}, {5271, 33127}, {5297, 32846}, {5718, 30970}, {5737, 29678}, {5750, 61652}, {6536, 37593}, {6646, 32845}, {6679, 19742}, {6690, 49724}, {7081, 31089}, {7226, 32855}, {8013, 31993}, {10453, 25960}, {15481, 50104}, {16569, 33172}, {16825, 33122}, {17056, 59306}, {17122, 32863}, {17123, 33173}, {17125, 63003}, {17163, 48643}, {17231, 61686}, {17233, 64178}, {17270, 29828}, {17275, 17718}, {17277, 29632}, {17362, 17602}, {17372, 49995}, {17484, 46918}, {17719, 42334}, {17740, 36263}, {18134, 26037}, {19732, 29661}, {19804, 33069}, {20955, 40075}, {21242, 26758}, {21684, 22020}, {21806, 50290}, {22069, 24031}, {23619, 33299}, {24295, 41241}, {24589, 49676}, {24725, 50314}, {25496, 63010}, {25745, 25800}, {25957, 59296}, {25958, 32865}, {25961, 26038}, {26064, 37573}, {26227, 50308}, {27065, 33158}, {27081, 29822}, {27184, 32860}, {28605, 33101}, {29631, 30832}, {29637, 37680}, {29674, 63961}, {29677, 37679}, {29824, 50315}, {29827, 37651}, {29829, 49497}, {29833, 49489}, {30831, 33138}, {31025, 50312}, {31034, 50302}, {31079, 49693}, {31179, 48809}, {31241, 37662}, {32772, 62998}, {32783, 32911}, {32918, 37653}, {32944, 63002}, {33070, 36480}, {33083, 60714}, {33086, 56009}, {33089, 49448}, {33094, 63131}, {33099, 64010}, {33115, 60731}, {33120, 49450}, {33134, 49459}, {33151, 49474}, {33170, 49712}, {37639, 58443}, {41014, 59305}, {41809, 43223}, {50296, 61155}, {50753, 63978}, {63802, 69274} X(69300) = crosspoint of X(4643) and X(33936) X(69300) = barycentric product X(i)*X(j) for these {i,j}: {10, 4643}, {37, 33936}, {321, 4414}, {3952, 47886} X(69300) = barycentric quotient X(i)/X(j) for these {i,j}: {4414, 81}, {4643, 86}, {33936, 274}, {47886, 7192} X(69300) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 33084, 33081}, {8, 25760, 33136}, {10, 4101, 2650}, {43, 32782, 32781}, {75, 33065, 32856}, {100, 31143, 33082}, {141, 4023, 899}, {306, 4104, 756}, {321, 61174, 4710}, {984, 33077, 32848}, {3681, 32778, 33162}, {4046, 4415, 4365}, {4417, 31330, 33105}, {4651, 31037, 2887}, {4886, 33126, 32914}, {4966, 5241, 30950}, {14555, 33171, 748}, {17362, 17602, 50756}, {27081, 29822, 50298}, {27184, 32860, 33145}, {33175, 37656, 238} X(69301) = X(2)X(3790)nX(8)X(7162) Barycentrics (b + c)*(-(a*b) + b^2 - a*c + b*c + c^2) : : X(69301) lies on these lines: {2, 3790}, {8, 7162}, {10, 3995}, {19, 346}, {31, 50000}, {38, 4439}, {42, 6541}, {100, 42033}, {190, 33078}, {192, 29679}, {210, 3969}, {306, 3952}, {312, 3006}, {321, 3925}, {429, 3695}, {726, 29687}, {756, 3773}, {1089, 3822}, {1211, 48648}, {1376, 50105}, {1621, 17264}, {1999, 33166}, {2321, 4651}, {2325, 63134}, {2887, 3994}, {3120, 4135}, {3175, 4972}, {3434, 42032}, {3578, 15481}, {3681, 17233}, {3685, 33091}, {3703, 3816}, {3704, 52353}, {3710, 17751}, {3717, 17135}, {3775, 42041}, {3826, 4980}, {3891, 17279}, {3896, 3943}, {3912, 17165}, {3914, 62227}, {3920, 17280}, {3936, 3967}, {3950, 27804}, {3971, 15523}, {3974, 17776}, {4009, 5741}, {4011, 32854}, {4062, 4090}, {4080, 4138}, {4136, 21928}, {4387, 5014}, {4415, 48647}, {4429, 42044}, {4442, 22034}, {4671, 29641}, {4756, 33066}, {4873, 63131}, {4903, 27131}, {5205, 33168}, {5278, 69089}, {6327, 56082}, {7081, 32849}, {8026, 30632}, {17018, 17242}, {17127, 17339}, {17150, 17353}, {17184, 29674}, {17261, 33083}, {17262, 32950}, {17268, 33173}, {17354, 62807}, {17358, 29648}, {17495, 62673}, {17763, 33164}, {17781, 20290}, {18743, 33089}, {21020, 48644}, {21026, 48643}, {21075, 27558}, {22275, 40521}, {24210, 31079}, {24295, 29816}, {24988, 42051}, {25253, 51423}, {25961, 49493}, {26040, 50043}, {26061, 29833}, {26230, 32926}, {26251, 28606}, {27064, 33093}, {27538, 33077}, {29649, 33161}, {29667, 41839}, {29677, 49455}, {29822, 53663}, {29824, 63147}, {29835, 33162}, {29873, 37759}, {32778, 64178}, {32781, 49456}, {32846, 32938}, {32847, 32930}, {32848, 59511}, {32855, 62620}, {32858, 32937}, {32863, 62222}, {32915, 33165}, {32927, 33158}, {32928, 33159}, {32931, 33092}, {32932, 60459}, {32936, 33079}, {33073, 41242}, {33081, 42054}, {33108, 42034}, {33125, 49445}, {33172, 49447}, {37639, 49990}, {41011, 49766}, {49769, 69253}, {50313, 62821}, {59517, 69252} X(69301) = reflection of X(69251) in X(29687) X(69301) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {7123, 41821}, {56179, 2891} X(69301) = X(17233)-Ceva conjugate of X(4006) X(69301) = X(i)-isoconjugate of X(j) for these (i,j): {849, 15320}, {1333, 14377}, {8042, 31616}, {15378, 16726}, {43190, 57129} X(69301) = X(i)-Dao conjugate of X(j) for these (i,j): {37, 14377}, {116, 3733}, {4075, 15320}, {6586, 17205} X(69301) = crosspoint of X(i) and X(j) for these (i,j): {6540, 15742}, {17233, 33932} X(69301) = crosssum of X(3937) and X(50512) X(69301) = barycentric product X(i)*X(j) for these {i,j}: {10, 17233}, {37, 33932}, {75, 4006}, {313, 3730}, {321, 3681}, {594, 33297}, {1016, 21045}, {1734, 4033}, {1978, 58286}, {2321, 33298}, {3952, 25259}, {4103, 57214}, {4184, 28654}, {6386, 21837}, {6586, 27808}, {15624, 27801}, {17198, 61402}, {17916, 20336} X(69301) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 14377}, {116, 17205}, {594, 15320}, {1734, 1019}, {3681, 81}, {3730, 58}, {3952, 43190}, {4006, 1}, {4184, 593}, {6586, 3733}, {15624, 1333}, {17198, 61403}, {17233, 86}, {17463, 16726}, {17916, 28}, {20901, 16727}, {21045, 1086}, {21837, 667}, {25259, 7192}, {26911, 61409}, {27808, 31624}, {33297, 1509}, {33298, 1434}, {33932, 274}, {38358, 18191}, {56813, 1790}, {57054, 15419}, {58286, 649}, {64878, 7254} X(69301) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {306, 4082, 3952}, {312, 32862, 3006}, {321, 3932, 69250}, {346, 10327, 32929}, {756, 3773, 56810}, {3703, 4358, 69134}, {3932, 6057, 321}, {3971, 15523, 26580}, {3974, 17776, 26227}, {17763, 33164, 56520}, {29674, 32925, 17184}, {32926, 33157, 26230}

leftri   Additive associates, X(69302) - X(69376)  rightri

Contributed by Clark Kimberling and Peter Moses, July 4, 2025.

Continuing the discussion of odd polynomial triangle centers in the preamble just before X(69091), every odd polynomial center of degree 4 has a representation

(b-c)(k1*a^3 + k2*a^2 (b+c) + k3*a (b^2+c^2) + k4 a b c + k5 (b*3+c^3) + k6 b c (b+c)),

which we abbreviate as ODD<k1,k2,k3,k4,k5,k6>. Here, the use of angle brackets, < and >, distinguishes this type of representation from the additive associate notation introduced in the preamble just before X(69091), which uses parentheses and depend on Mathematica ordering of terms instead of elements E(n) as defined near X(69091). (Of course, the latter are additive associates, too, but are restricted to triangle centers, whereas in general, additive associates include bicentric pairs of points.)

Centers X(69302)-X(69376) are polynomial triangle centers of degree 4, in addition to centers listed below with indexes < 69302. The appearance of

ODD<k1,k2,k3,k4,k5,k6>, n

in the list means that X(n) is the center given by ODD<k1,k2,k3,k4,k5,k6>. If H and K are two such distinct centers, then H+K and H-K are also of this type, where + an - are the usual vector addition and subtraction. The centers, H, K, H+K, H-K comprise a harmonic range. In many cases, if all the coefficients in the representations for H and K are in the set {-1,0,1}, then the coefficients for H+K and H-K are also in the set {-1,0,1}.

ODD<1,1,1,1,1,1>, 514
ODD<1,1,1,1,0,0>, 513
ODD<1,1,1,0,1,1>, 514
ODD<1,1,1,0,0,-1>, 24719
ODD<1,1,1,-1,1,1>, 514
ODD<1,1,1,-1,1,0>, 47652
ODD<1,1,0,1,1,0>, 29029
ODD<1,1,0,1,0,1>, 4369
ODD<1,1,0,1,0,0>, 1019
ODD<1,1,0,1,0,-1>, 6002
ODD<1,1,0,1,-1,-1>, 50342
ODD<1,1,0,0,1,0>, 29025
ODD<1,1,0,0,0,0>, 649
ODD<1,1,0,0,0,-1>, 29013
ODD<1,1,0,-1,1,1>, 48103
ODD<1,1,0,-1,1,0>, 29098
ODD<1,1,0,-1,0,0>, 4063
ODD<1,1,0,-1,0,-1>, 812
ODD<1,1,-1,1,0,1>, 48569
ODD<1,1,-1,1,-1,0>, 4453
ODD<1,1,-1,0,0,1>, 47823
ODD<1,1,-1,0,-1,1>, 21188
ODD<1,1,-1,0,-1,-1>, 4025
ODD<1,1,-1,-1,0,1>, 47837
ODD<1,1,-1,-1,0,0>, 9508
ODD<1,1,-1,-1,-1,-1>, 21192
ODD<1,0,1,1,1,1>, 68979
ODD<1,0,1,1,0,1>, 47694
ODD<1,0,1,1,0,0>, 830
ODD<1,0,1,1,0,-1>, 21301
ODD<1,0,1,0,0,0>, 513
ODD<1,0,1,-1,0,-1>, 46403
ODD<1,0,0,1,1,1>, 47682
ODD<1,0,0,1,1,0>, 514
ODD<1,0,0,1,0,1>, 52601
ODD<1,0,0,1,0,0>, 4367
ODD<1,0,0,1,0,-1>, 2787
ODD<1,0,0,1,-1,0>, 4458
ODD<1,0,0,0,1,1>, 48300
ODD<1,0,0,0,1,0>, 514
ODD<1,0,0,0,0,1>, 4874
ODD<1,0,0,0,0,0>, 667
ODD<1,0,0,0,0,-1>, 814
ODD<1,0,0,0,-1,0>, 20517
ODD<1,0,0,-1,1,0>, 514
ODD<1,0,0,-1,1,-1>, 47680
ODD<1,0,0,-1,0,0>, 659
ODD<1,0,0,-1,0,-1>, 29070
ODD<1,0,0,-1,-1,0>, 4142
ODD<1,0,-1,1,0,1>, 47796
ODD<1,0,-1,1,0,0>, 3960
ODD<1,0,-1,1,0,-1>, 17496
ODD<1,0,-1,0,0,1>, 47795
ODD<1,0,-1,0,0,0>, 905
ODD<1,0,-1,0,0,-1>, 48321
ODD<1,0,-1,0,-1,0>, 48227
ODD<1,0,-1,-1,1,0>, 50351
ODD<1,0,-1,-1,0,1>, 2
ODD<1,0,-1,-1,0,0>, 14838
ODD<1,0,-1,-1,0,-1>, 4560
ODD<1,-1,1,1,0,1>, 48305
ODD<1,-1,1,1,-1,0>, 47695
ODD<1,-1,1,0,0,0>, 6004
ODD<1,-1,1,0,-1,1>, 21185
ODD<1,-1,1,0,-1,-1>, 522
ODD<1,-1,1,-1,0,0>, 513
ODD<1,-1,1,-1,-1,1>, 21201
ODD<1,-1,0,1,1,0>, 29094
ODD<1,-1,0,1,0,0>, 1
ODD<1,-1,0,1,0,-1>, 3907
ODD<1,-1,0,0,1,1>, 49279
ODD<1,-1,0,0,1,0>, 29082
ODD<1,-1,0,0,0,0>, 663
ODD<1,-1,0,0,0,-1>, 29066
ODD<1,-1,0,-1,1,0>, 29102
ODD<1,-1,0,-1,0,1>, 3716
ODD<1,-1,0,-1,0,0>, 4040
ODD<1,-1,0,-1,0,-1>, 29051
ODD<1,-1,0,-1,-1,-1>, 50340
ODD<1,-1,-1,1,1,0>, 3904
ODD<1,-1,-1,1,1,-1>, 514
ODD<1,-1,-1,0,1,1>, 6332
ODD<1,-1,-1,0,1,-1>, 514
ODD<1,-1,-1,0,0,1>, 47841
ODD<1,-1,-1,-1,1,-1>, 514
ODD<1,-1,-1,-1,0,1>, 47839
ODD<1,-1,-1,-1,0,-1>, 48288
ODD<1,-1,-1,-1,-1,0>, 47797
ODD<0,1,1,1,0,1>, 514
ODD<0,1,1,1,0,0>, 14349
ODD<0,1,1,1,0,-1>, 3835
ODD<0,1,1,0,0,1>, 514
ODD<0,1,1,0,0,0>, 48131
ODD<0,1,1,-1,0,1>, 514
ODD<0,1,1,-1,0,0>, 48335
ODD<0,1,0,1,1,1>, 29021
ODD<0,1,0,1,1,0>, 47708
ODD<0,1,0,1,0,1>, 50352
ODD<0,1,0,1,0,0>, 513
ODD<0,1,0,1,0,-1>, 48267
ODD<0,1,0,1,-1,-1>, 23875
ODD<0,1,0,0,1,1>, 523
ODD<0,1,0,0,1,0>, 47712
ODD<0,1,0,0,1,-1>, 48403
ODD<0,1,0,0,0,1>, 2533
ODD<0,1,0,0,0,0>, 512
ODD<0,1,0,0,0,-1>, 4010
ODD<0,1,0,0,-1,1>, 7178
ODD<0,1,0,0,-1,0>, 4707
ODD<0,1,0,0,-1,-1>, 525
ODD<0,1,0,-1,1,1>, 29047
ODD<0,1,0,-1,1,0>, 47691
ODD<0,1,0,-1,0,0>, 4083
ODD<0,1,0,-1,0,-1>, 48273
ODD<0,1,0,-1,-1,-1>, 23876
ODD<0,1,-1,1,0,1>, 24720
ODD<0,1,-1,1,0,0>, 4905
ODD<0,1,-1,0,0,1>, 50337
ODD<0,1,-1,0,0,0>, 2254
ODD<0,1,-1,0,0,-1>, 8714
ODD<0,1,-1,-1,0,1>, 17072
ODD<0,1,-1,-1,0,0>, 1734
ODD<0,1,-1,-1,0,-1>, 522
ODD<0,0,1,1,0,1>, 784
ODD<0,0,1,1,0,0>, 1491
ODD<0,0,1,1,0,-1>, 21260
ODD<0,0,1,1,-1,1>, 49300
ODD<0,0,1,1,-1,0>, 23877
ODD<0,0,1,1,-1,-1>, 48272
ODD<0,0,1,0,1,1>, 16892
ODD<0,0,1,0,0,1>, 63812
ODD<0,0,1,0,0,0>, 2530
ODD<0,0,1,0,0,-1>, 3837
ODD<0,0,1,0,-1,1>, 21118
ODD<0,0,1,0,-1,0>, 23887
ODD<0,0,1,0,-1,-1>, 48278
ODD<0,0,1,-1,1,0>, 3776
ODD<0,0,1,-1,0,0>, 3777
ODD<0,0,1,-1,0,-1>, 23815
ODD<0,0,1,-1,-1,0>, 3810
ODD<0,0,1,-1,-1,-1>, 49278
ODD<0,0,0,1,1,1>, 29017
ODD<0,0,0,1,0,1>, 693
ODD<0,0,0,1,0,0>, 514
ODD<0,0,0,1,0,-1>, 4391
ODD<0,0,0,1,-1,-1>, 62423
ODD<0,0,0,0,1,1>, 826
ODD<0,0,0,0,1,0>, 3801
ODD<0,0,0,0,0,1>, 1577

underbar



X(69302) = ODD<1,1,1,1,1,0> POINT

Barycentrics    (b - c)*(a^3 + a^2*b + a*b^2 + b^3 + a^2*c + a*b*c + a*c^2 + c^3) : :
X(69302) = 4 X[3835] - 3 X[57066], X[3904] - 4 X[69291], 4 X[4129] - 3 X[30565], 3 X[4728] - 2 X[8045], 3 X[4789] - 4 X[4823], 2 X[4978] - 3 X[47871], 2 X[14349] - 3 X[48550], 2 X[48300] - 3 X[57066], 2 X[667] - 3 X[47797], 2 X[905] - 3 X[44435], 2 X[1019] - 3 X[4453], 2 X[2530] - 3 X[48159], 4 X[3676] - 3 X[48570], 2 X[3803] - 3 X[47798], and many others

X(69302) lies on these lines: {513, 3801}, {514, 661}, {522, 47709}, {523, 21301}, {525, 20295}, {667, 47797}, {812, 21124}, {830, 47695}, {905, 44435}, {1019, 4453}, {1491, 29025}, {1734, 29158}, {2254, 29118}, {2530, 29029}, {3004, 4560}, {3267, 20949}, {3676, 48570}, {3776, 48144}, {3777, 29120}, {3800, 21302}, {3803, 47798}, {3810, 48122}, {3910, 23729}, {4010, 68979}, {4040, 48161}, {4063, 50453}, {4160, 47716}, {4458, 50523}, {4467, 29013}, {4498, 28882}, {4705, 29098}, {4707, 49297}, {4778, 21179}, {4802, 47707}, {4833, 4977}, {4905, 29132}, {4983, 29102}, {6002, 16892}, {6372, 49301}, {7178, 10566}, {7192, 7254}, {8678, 47691}, {10015, 68878}, {14837, 48060}, {15420, 21222}, {17072, 48106}, {17166, 23770}, {17494, 48402}, {17496, 29126}, {20317, 48095}, {21051, 48103}, {21052, 48146}, {21118, 47943}, {21120, 57164}, {21185, 47697}, {21188, 47762}, {21260, 47809}, {23731, 23755}, {23875, 44449}, {23877, 48023}, {23880, 47960}, {23882, 45746}, {23887, 48086}, {24601, 47782}, {24719, 29017}, {27013, 41800}, {28147, 47706}, {28175, 30709}, {28423, 47793}, {28478, 49294}, {28487, 48116}, {28579, 58168}, {28851, 47911}, {28855, 48582}, {29021, 47687}, {29051, 47701}, {29082, 48123}, {29114, 48321}, {29116, 48050}, {29122, 48100}, {29128, 31131}, {29130, 49278}, {29140, 48066}, {29142, 46403}, {29160, 48272}, {29184, 48059}, {29220, 49277}, {29288, 47688}, {29336, 48288}, {31291, 48203}, {47658, 47678}, {47661, 47679}, {47663, 47965}, {47693, 48395}, {47694, 48403}, {47698, 47956}, {47718, 49285}, {47719, 48089}, {47722, 47961}, {47725, 47948}, {47728, 48136}, {47814, 48062}, {47840, 48299}, {47892, 48003}, {47968, 63812}, {47989, 49303}, {48007, 48410}, {48091, 48543}, {48149, 69292}, {48177, 48331}, {48252, 50337}, {48267, 49275}, {49283, 50352}

X(69302) = midpoint of X(i) and X(j) for these {i,j}: {4462, 47651}, {21118, 47943}, {23731, 23755}, {47725, 47948}
X(69302) = reflection of X(i) in X(j) for these {i,j}: {3904, 48131}, {4063, 50453}, {4560, 3004}, {4801, 48398}, {17166, 23770}, {17494, 48402}, {47658, 47678}, {47660, 1577}, {47661, 47679}, {47663, 47965}, {47684, 6332}, {47693, 48395}, {47694, 48403}, {47695, 47712}, {47697, 21185}, {47698, 47956}, {47718, 49285}, {47719, 48089}, {47728, 48136}, {48060, 14837}, {48095, 20317}, {48103, 21051}, {48106, 17072}, {48131, 69291}, {48144, 3776}, {48149, 69292}, {48278, 48050}, {48300, 3835}, {48408, 4705}, {48410, 48007}, {49275, 48267}, {49283, 50352}, {50351, 48059}, {50523, 4458}
X(69302) = X(14534)-Ceva conjugate of X(1086)
X(69302) = crossdifference of every pair of points on line {31, 2273}
X(69302) = barycentric product X(i)*X(j) for these {i,j}: {514, 19786}, {693, 5262}, {3261, 16470}, {5051, 7192}, {18155, 35650}
X(69302) = barycentric quotient X(i)/X(j) for these {i,j}: {5051, 3952}, {5262, 100}, {16470, 101}, {19786, 190}, {35650, 4551}
X(69302) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3835, 48300, 57066}, {14837, 48060, 48565}

X(69303) = ODD<1, 1, 1, 1, 0, 1> POINT

Barycentrics    (b - c)*(a^3 + a^2*b + a*b^2 + a^2*c + a*b*c + b^2*c + a*c^2 + b*c^2) : :
X(69303) = 2 X[661] - 3 X[48553], X[663] - 3 X[48578], 2 X[1491] - 3 X[47837], 2 X[2530] - 3 X[48569], 4 X[4369] - 3 X[48569], 2 X[3835] - 3 X[47875], 2 X[4129] - 3 X[47872], 3 X[4379] - 2 X[23815], 3 X[4379] - X[48122], 3 X[4448] - 2 X[48058], 4 X[4874] - 3 X[47839], 2 X[14349] - 3 X[47839], 2 X[9508] - 3 X[48566], X[48409] - 3 X[48566],and many others

X(69303) lies on these lines: {2, 48059}, {512, 47694}, {513, 1577}, {514, 659}, {522, 4834}, {523, 4063}, {649, 784}, {661, 48553}, {663, 48578}, {812, 48393}, {826, 47660}, {830, 2533}, {834, 39547}, {891, 17166}, {1019, 63812}, {1491, 47837}, {1734, 64914}, {2530, 4369}, {2787, 50523}, {3716, 4983}, {3737, 4960}, {3766, 6372}, {3835, 47875}, {3837, 48086}, {4024, 29106}, {4040, 48248}, {4041, 48153}, {4083, 48291}, {4129, 47872}, {4379, 23815}, {4448, 48058}, {4498, 48142}, {4560, 50512}, {4581, 18003}, {4707, 68979}, {4784, 8714}, {4804, 47935}, {4806, 48085}, {4823, 24719}, {4824, 48003}, {4874, 14349}, {4885, 48092}, {4978, 54265}, {4979, 29150}, {6004, 47697}, {6366, 57159}, {6371, 47844}, {7662, 48273}, {7927, 47695}, {7950, 47693}, {8633, 47708}, {9508, 48409}, {15309, 48265}, {20317, 47956}, {21051, 47948}, {21052, 47905}, {21118, 29029}, {21188, 48007}, {21260, 48023}, {23755, 29102}, {23789, 48253}, {23880, 50515}, {28481, 48396}, {28840, 47949}, {29013, 48392}, {29025, 49300}, {29070, 50457}, {29098, 48101}, {29142, 48276}, {29168, 49283}, {29170, 48110}, {29182, 31291}, {29184, 49303}, {29186, 50358}, {29188, 48150}, {29246, 48111}, {29252, 49275}, {29298, 48322}, {29302, 48120}, {29328, 47976}, {29350, 48301}, {29366, 48324}, {31148, 48151}, {31286, 47888}, {31290, 47994}, {47666, 47815}, {47672, 68894}, {47719, 47791}, {47762, 48410}, {47793, 47945}, {47794, 48030}, {47795, 48100}, {47804, 50507}, {47810, 65449}, {47812, 48116}, {47813, 48131}, {47814, 47940}, {47817, 50449}, {47821, 48053}, {47822, 48054}, {47823, 48066}, {47832, 48121}, {47835, 48012}, {47838, 48093}, {47906, 48147}, {47929, 48141}, {47936, 48148}, {47946, 48004}, {47975, 48565}, {48024, 59672}, {48063, 48351}, {48123, 48234}, {48128, 48220}, {48251, 48336}, {48272, 48405}, {48298, 48328}, {48573, 50335}, {49276, 59629}, {50328, 50337}, {50343, 58179}

X(69303) = midpoint of X(i) and X(j) for these {i,j}: {4041, 48153}, {4498, 48142}, {4804, 47935}, {4960, 47970}, {4979, 48264}, {47708, 49282}, {47906, 48147}, {47929, 48141}, {47936, 48148}, {48101, 55282}
X(69303) = reflection of X(i) in X(j) for these {i,j}: {2530, 4369}, {4040, 48248}, {4560, 50512}, {4824, 48003}, {4978, 54265}, {4983, 3716}, {14349, 4874}, {24719, 4823}, {31290, 47994}, {47945, 48005}, {47946, 48004}, {47948, 21051}, {47956, 20317}, {47975, 50504}, {48007, 21188}, {48023, 21260}, {48024, 59672}, {48085, 4806}, {48086, 3837}, {48092, 4885}, {48122, 23815}, {48131, 52601}, {48272, 48405}, {48273, 7662}, {48288, 667}, {48298, 48328}, {48305, 47694}, {48351, 48063}, {48409, 9508}, {50328, 50337}, {50343, 58179}
X(69303) = anticomplement of X(48059)
X(69303) = X(40827)-Ceva conjugate of X(1015)
X(69303) = X(52538)-Dao conjugate of X(53332)
X(69303) = crosspoint of X(4581) and X(7192)
X(69303) = crosssum of X(4557) and X(53280)
X(69303) = crossdifference of every pair of points on line {2276, 5153}
X(69303) = barycentric product X(i)*X(j) for these {i,j}: {514, 32772}, {7192, 52538}
X(69303) = barycentric quotient X(i)/X(j) for these {i,j}: {32772, 190}, {52538, 3952}
X(69303) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2530, 4369, 48569}, {4379, 48122, 23815}, {4874, 14349, 47839}, {47793, 47945, 48005}, {47813, 48131, 52601}, {47975, 48565, 50504}, {48409, 48566, 9508}

X(69304) = ODD<1, 1, 1, 1, 0, -1> POINT

Barycentrics    (b - c)*(-a^3 - a^2*b - a*b^2 - a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2) : :
X(69304) = 3 X[20295] + X[21302], 3 X[21301] - X[21302], 3 X[48267] - 2 X[59590], 2 X[649] - 3 X[47837], 4 X[21260] - 3 X[47837], 2 X[659] - 3 X[48553], 4 X[4129] - 3 X[48553], X[663] - 3 X[31147], 2 X[667] - 3 X[47839], 4 X[3835] - 3 X[47839], 2 X[1019] - 3 X[48569], 4 X[3837] - 3 X[48569], 3 X[1635] - 4 X[65449], 2 X[1960] - 5 X[26798], and many others

X(69304) lies on these lines: {1, 4992}, {2, 50512}, {10, 58301}, {79, 18014}, {316, 512}, {513, 1577}, {514, 4122}, {523, 47948}, {649, 21260}, {659, 4129}, {661, 29070}, {663, 31147}, {667, 3835}, {784, 48023}, {812, 4705}, {814, 14349}, {830, 4010}, {1019, 3837}, {1491, 29013}, {1635, 65449}, {1734, 29328}, {1960, 26798}, {2254, 29150}, {2483, 55240}, {2530, 6002}, {2787, 48131}, {3004, 29232}, {3667, 4142}, {3777, 29148}, {4040, 4806}, {4041, 48114}, {4063, 21051}, {4088, 29098}, {4106, 8678}, {4151, 4810}, {4160, 48279}, {4379, 50526}, {4380, 47814}, {4382, 47912}, {4401, 47822}, {4467, 29266}, {4490, 29302}, {4560, 29340}, {4728, 50523}, {4762, 47956}, {4775, 28470}, {4776, 50507}, {4782, 47794}, {4784, 50337}, {4785, 4834}, {4804, 47905}, {4822, 29188}, {4824, 48613}, {4885, 50515}, {4905, 29170}, {4922, 48348}, {4940, 48099}, {4977, 32846}, {4983, 29051}, {6004, 48080}, {6008, 50501}, {6050, 47760}, {6371, 18003}, {6372, 46403}, {6373, 17217}, {7265, 68979}, {8714, 50328}, {9508, 47816}, {10404, 30724}, {15309, 21146}, {16892, 29090}, {17166, 21297}, {17494, 48005}, {17496, 29176}, {21052, 47935}, {21124, 29106}, {21385, 48401}, {21836, 23656}, {23301, 29807}, {23729, 29288}, {23789, 48167}, {23815, 48144}, {23880, 48092}, {23882, 48027}, {24561, 26596}, {26853, 47836}, {27013, 58145}, {27138, 58139}, {28306, 48178}, {28475, 48136}, {28481, 48400}, {28525, 48325}, {29025, 48272}, {29029, 48278}, {29033, 48054}, {29037, 69291}, {29058, 48550}, {29066, 48123}, {29082, 49277}, {29086, 47701}, {29120, 49278}, {29152, 48100}, {29162, 50351}, {29168, 47687}, {29178, 48066}, {29186, 48024}, {29236, 48129}, {29238, 48030}, {29246, 48081}, {29252, 44449}, {29268, 48298}, {29270, 48012}, {29272, 49274}, {29274, 48093}, {29324, 48335}, {29354, 47652}, {29362, 47959}, {30835, 31288}, {31207, 58143}, {31251, 31286}, {45313, 58146}, {45316, 58152}, {47707, 49298}, {47724, 48085}, {47759, 48053}, {47812, 48149}, {47823, 48064}, {47833, 59714}, {47835, 48011}, {47838, 48331}, {47872, 59737}, {47906, 48115}, {47911, 48119}, {47918, 68894}, {47946, 48612}, {47969, 47994}, {47970, 64913}, {48016, 58181}, {48019, 54256}, {48020, 48264}, {48043, 48351}, {48052, 64934}, {48086, 63812}, {48148, 48582}, {48320, 48406}, {48586, 64914}, {50358, 59672}, {50452, 50556}

X(69304) = midpoint of X(i) and X(j) for these {i,j}: {4041, 48114}, {4382, 47912}, {4804, 47905}, {20295, 21301}, {47707, 49298}, {47724, 48085}, {47906, 48115}, {47911, 48119}, {48020, 48264}, {48148, 48582}
X(69304) = reflection of X(i) in X(j) for these {i,j}: {1, 4992}, {649, 21260}, {659, 4129}, {667, 3835}, {1019, 3837}, {2530, 48050}, {4040, 4806}, {4063, 21051}, {4380, 50504}, {4560, 48059}, {4784, 50337}, {4824, 48613}, {4834, 17072}, {4922, 48348}, {4983, 48049}, {17494, 48005}, {21385, 48401}, {26853, 58179}, {31291, 1960}, {47946, 48612}, {47969, 47994}, {48099, 4940}, {48144, 23815}, {48273, 4106}, {48288, 14349}, {48291, 48273}, {48305, 4010}, {48320, 48406}, {48321, 48100}, {48351, 48043}, {50358, 59672}, {50515, 4885}, {50523, 52601}, {58179, 53571}
X(69304) = anticomplement of X(50512)
X(69304) = anticomplement of the isogonal conjugate of X(6540)
X(69304) = anticomplementary isogonal conjugate of X(39348)
X(69304) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 39348}, {190, 41821}, {668, 2891}, {765, 14779}, {1126, 9263}, {1255, 4440}, {1268, 149}, {4102, 37781}, {4596, 1}, {4629, 17147}, {4632, 75}, {6539, 21221}, {6540, 8}, {8701, 192}, {28615, 21224}, {32018, 150}, {32635, 39351}, {37212, 2}, {40438, 17154}, {47947, 54102}, {52555, 21220}, {57731, 24074}, {62535, 4360}
X(69304) = crosspoint of X(83) and X(6540)
X(69304) = crosssum of X(i) and X(j) for these (i,j): {39, 50512}, {669, 4272}
X(69304) = crossdifference of every pair of points on line {3051, 28643}
X(69304) = barycentric product X(514)*X(32928)
X(69304) = barycentric quotient X(32928)/X(190)
X(69304) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 21260, 47837}, {659, 4129, 48553}, {667, 3835, 47839}, {1019, 3837, 48569}, {4380, 47814, 50504}, {4728, 50523, 52601}, {4834, 31149, 17072}, {26798, 31291, 47840}, {26853, 47836, 58179}, {30835, 58140, 31288}, {31251, 58144, 31286}, {31291, 47840, 1960}, {53571, 58179, 47836}

X(69305) = ODD<1, 1, 1, 0, 0, 1> POINT

Barycentrics    (b - c)*(a^3 + a^2*b + a*b^2 + a^2*c + b^2*c + a*c^2 + b*c^2) : :
X(69305) = 4 X[20517] - 3 X[48224], 2 X[1491] - 3 X[47835], 2 X[2530] - 3 X[47823], 2 X[3835] - 3 X[47872], 3 X[4379] - 2 X[48406], 3 X[4448] - 2 X[48099], X[4449] - 3 X[48578], 4 X[4874] - 3 X[47841], 3 X[47841] - 2 X[48131], X[4879] - 3 X[48251], 2 X[4978] - 3 X[48238], 2 X[4992] - 3 X[47832], 3 X[8643] - 2 X[48289], 2 X[9508] - 3 X[48565],and many others

X(69305) lies on these lines: {2, 48100}, {513, 2517}, {514, 659}, {523, 4498}, {649, 63812}, {663, 48248}, {784, 4063}, {812, 48392}, {900, 50509}, {1491, 47835}, {1577, 24719}, {2530, 47823}, {3716, 48123}, {3777, 4369}, {3835, 47872}, {3837, 48122}, {4041, 64914}, {4083, 47694}, {4379, 48406}, {4448, 48099}, {4449, 48578}, {4560, 4782}, {4761, 6004}, {4774, 28470}, {4801, 54265}, {4806, 48121}, {4824, 47965}, {4834, 8714}, {4874, 47841}, {4879, 48251}, {4885, 48616}, {4922, 50517}, {4977, 7178}, {4978, 48238}, {4979, 29170}, {4983, 59672}, {4992, 47832}, {5040, 49282}, {7192, 16737}, {7662, 8712}, {8643, 48289}, {9508, 48410}, {14349, 47822}, {14430, 47905}, {14837, 48007}, {17072, 50328}, {17166, 29226}, {20317, 48027}, {20507, 48141}, {21051, 48023}, {21052, 48020}, {21118, 29025}, {21132, 29120}, {21145, 28195}, {21185, 48349}, {21260, 48086}, {23738, 31148}, {23755, 48102}, {23877, 48103}, {28478, 49286}, {28481, 48395}, {28840, 47913}, {29017, 47660}, {29051, 50358}, {29098, 49300}, {29146, 47693}, {29150, 47976}, {29174, 48146}, {29188, 48111}, {29200, 49275}, {29208, 47695}, {29236, 31291}, {29246, 48032}, {29280, 49273}, {29298, 48324}, {29302, 48393}, {29324, 50523}, {29328, 47935}, {29350, 48305}, {29362, 50457}, {29366, 48150}, {31286, 47893}, {31290, 47957}, {35352, 47959}, {47696, 68979}, {47793, 48030}, {47794, 48059}, {47796, 48137}, {47805, 48331}, {47813, 48334}, {47817, 50507}, {47821, 48093}, {47836, 50335}, {47837, 48066}, {47840, 48129}, {47912, 48401}, {47945, 47967}, {47946, 47966}, {48003, 48176}, {48054, 48553}, {48063, 48336}, {48188, 48272}, {48189, 48273}, {48225, 48409}, {48278, 48405}, {48298, 48330}, {48321, 50512}, {48335, 52601}, {50341, 50501}

X(69305) = midpoint of X(i) and X(j) for these {i,j}: {21118, 48101}, {23755, 48102}, {47935, 48264}
X(69305) = reflection of X(i) in X(j) for these {i,j}: {663, 48248}, {3777, 4369}, {4560, 4782}, {4801, 54265}, {4824, 47965}, {4922, 50517}, {4983, 59672}, {24719, 1577}, {31290, 47957}, {47912, 48401}, {47945, 47967}, {47946, 47966}, {48007, 14837}, {48023, 21051}, {48027, 20317}, {48086, 21260}, {48121, 4806}, {48122, 3837}, {48123, 3716}, {48131, 4874}, {48278, 48405}, {48279, 7662}, {48288, 4401}, {48298, 48330}, {48301, 47694}, {48321, 50512}, {48335, 52601}, {48336, 48063}, {48349, 21185}, {48409, 50504}, {48410, 9508}, {48616, 4885}, {50328, 17072}, {50341, 50501}
X(69305) = anticomplement of X(48100)
X(69305) = crosspoint of X(668) and X(62906)
X(69305) = crossdifference of every pair of points on line {2276, 2300}
X(69305) = barycentric product X(514)*X(25496)
X(69305) = barycentric quotient X(25496)/X(190)
X(69305) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4874, 48131, 47841}, {48409, 50504, 48225}, {48410, 48565, 9508}

X(69306) = ODD<1, 1, 1, 0, 0, 0> POINT

Barycentrics    a*(b - c)*(a^2 + a*b + b^2 + a*c + c^2) : :

X(69306) lies on these lines: {30, 511}, {649, 2473}, {650, 48059}, {659, 14349}, {667, 48131}, {764, 24286}, {905, 48616}, {1019, 3777}, {1491, 4063}, {1577, 24719}, {1635, 47888}, {1734, 48596}, {1960, 3803}, {2254, 4834}, {2520, 49294}, {2526, 50501}, {3669, 50515}, {3960, 48137}, {4040, 48123}, {4041, 48020}, {4142, 69291}, {4367, 48335}, {4369, 23815}, {4378, 48334}, {4380, 48410}, {4382, 48393}, {4448, 47838}, {4490, 21385}, {4498, 4705}, {4724, 4983}, {4728, 47875}, {4775, 48150}, {4776, 47815}, {4782, 14838}, {4784, 4905}, {4790, 8659}, {4806, 59672}, {4808, 48077}, {4813, 24290}, {4822, 48032}, {4879, 48324}, {4932, 24285}, {4960, 48143}, {4979, 22383}, {4992, 48248}, {5027, 9811}, {6161, 48338}, {9208, 9810}, {9508, 48011}, {14419, 58140}, {17072, 48042}, {20295, 48267}, {21052, 31149}, {21124, 47943}, {21185, 49295}, {21260, 48050}, {23729, 48403}, {23738, 48149}, {23765, 48320}, {30234, 58137}, {30592, 48578}, {31291, 48298}, {36848, 48573}, {41800, 48178}, {44429, 47837}, {46403, 50352}, {47694, 48273}, {47697, 48305}, {47708, 49298}, {47719, 49282}, {47760, 48561}, {47762, 47819}, {47804, 47839}, {47805, 47840}, {47816, 47835}, {47817, 47822}, {47818, 47841}, {47823, 48556}, {47836, 48164}, {47906, 48019}, {47913, 47947}, {47921, 47956}, {47926, 50489}, {47936, 48021}, {47942, 48595}, {47955, 48618}, {47957, 48612}, {47965, 48005}, {47966, 47994}, {47967, 48613}, {47970, 48024}, {47977, 48081}, {47982, 60492}, {47987, 48602}, {47989, 48402}, {48003, 48030}, {48004, 48028}, {48012, 48603}, {48029, 48053}, {48045, 48623}, {48058, 48093}, {48099, 48128}, {48101, 48278}, {48103, 48272}, {48107, 50556}, {48111, 48336}, {48114, 48264}, {48129, 48331}, {48250, 57066}, {48322, 48333}, {48327, 48347}, {48328, 48332}, {48329, 58160}, {48330, 48348}, {48341, 50526}, {50336, 58179}, {50525, 54253}

X(69306) = crossdifference of every pair of points on line {6, 3681}
X(69306) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 48122, 2530}, {650, 48092, 48059}, {659, 14349, 50507}, {1491, 4063, 50504}, {1734, 48596, 50328}, {2254, 47935, 4834}, {3803, 48136, 1960}, {4063, 48086, 1491}, {4498, 48023, 4705}, {4724, 48121, 4983}, {4776, 47815, 48553}, {4782, 48100, 14838}, {4813, 47929, 47949}, {4822, 48032, 48351}, {4905, 47976, 4784}, {21385, 47948, 4490}, {44429, 48565, 47837}, {47762, 47819, 48569}, {47935, 48116, 2254}, {47936, 48597, 48021}, {47965, 48027, 48005}, {47966, 48026, 47994}, {47970, 48085, 48024}, {48003, 48052, 48030}, {48004, 48051, 48028}, {48011, 48066, 9508}, {48029, 48091, 48053}, {48123, 50358, 4040}, {48332, 50517, 48328}, {48334, 50523, 4378}, {48556, 48566, 47823}

X(69307) = ODD<1, 1, 0, 1, 1, 1> POINT

Barycentrics    (b - c)*(a^3 + a^2*b + b^3 + a^2*c + a*b*c + b^2*c + b*c^2 + c^3) : :
X(69307) = 2 X[17072] - 3 X[48235], 2 X[47726] + X[50342], 2 X[3004] - 3 X[47893], X[4462] - 3 X[48236], 2 X[4806] - 3 X[57066], 2 X[20317] - 3 X[48219], 2 X[21051] - 3 X[47809], 2 X[21185] - 3 X[48234], X[21302] - 3 X[48254], 2 X[23770] - 3 X[47889], X[47709] - 3 X[47820], 3 X[47827] - 2 X[48402], 3 X[47833] - 2 X[48403], 3 X[47837] - 2 X[50453], 3 X[47872] - 2 X[48400], 3 X[47872] - 4 X[68794], 3 X[47885] - 2 X[47965]

X(69307) lies on these lines: {1, 7927}, {10, 514}, {512, 47682}, {513, 4064}, {523, 1325}, {525, 4784}, {649, 29017}, {659, 29142}, {663, 29144}, {667, 29021}, {693, 29025}, {814, 47690}, {826, 1019}, {1577, 29029}, {2254, 68979}, {2528, 7192}, {2787, 47711}, {3004, 47893}, {3669, 4802}, {3800, 4879}, {3801, 4369}, {4010, 8045}, {4024, 66523}, {4040, 29168}, {4063, 29312}, {4083, 48106}, {4122, 6002}, {4160, 4808}, {4170, 49290}, {4378, 29047}, {4379, 29863}, {4391, 29120}, {4449, 29208}, {4462, 48236}, {4468, 47913}, {4707, 29154}, {4761, 29094}, {4777, 50517}, {4806, 57066}, {4823, 29140}, {4834, 23876}, {4874, 29134}, {4977, 60465}, {4978, 29098}, {6005, 49279}, {6332, 48123}, {6367, 47681}, {6372, 48083}, {6590, 48392}, {7265, 29150}, {9508, 21124}, {12073, 48337}, {14349, 47944}, {16737, 66286}, {17496, 47693}, {17989, 47660}, {20317, 48219}, {20715, 48101}, {21051, 47809}, {21118, 29683}, {21185, 48234}, {21302, 48254}, {23738, 48130}, {23770, 47889}, {25259, 29170}, {29070, 47715}, {29074, 47689}, {29082, 47684}, {29086, 47714}, {29110, 47710}, {29126, 48395}, {29128, 47712}, {29132, 48267}, {29158, 48273}, {29162, 48396}, {29166, 50512}, {29174, 47691}, {29182, 47723}, {29184, 47680}, {29198, 48094}, {29256, 58179}, {29260, 48343}, {29280, 47971}, {29284, 50509}, {29288, 48323}, {29318, 48064}, {29324, 47707}, {29336, 47724}, {29354, 48320}, {29362, 47719}, {29366, 47728}, {47652, 48406}, {47709, 47820}, {47727, 48328}, {47827, 48402}, {47833, 48403}, {47837, 50453}, {47872, 48400}, {47885, 47965}, {47890, 53281}, {47906, 48048}, {47918, 48056}, {47938, 48093}, {47958, 48100}, {47972, 48331}, {48069, 50355}, {48118, 48341}, {48144, 62423}, {48146, 48334}, {48265, 69293}, {48275, 54249}, {48299, 48336}, {49274, 59629}, {54265, 55282}

X(69307) = midpoint of X(i) and X(j) for these {i,j}: {1019, 47726}, {17496, 47693}, {23738, 48130}, {23765, 48140}, {48118, 48341}, {48146, 48334}
X(69307) = reflection of X(i) in X(j) for these {i,j}: {3801, 4369}, {4010, 8045}, {4170, 49290}, {4391, 48405}, {4490, 48062}, {4879, 48290}, {21124, 9508}, {47652, 48406}, {47708, 4874}, {47712, 52601}, {47727, 48328}, {47906, 48048}, {47913, 4468}, {47918, 48056}, {47938, 48093}, {47944, 14349}, {47958, 48100}, {47968, 2530}, {47972, 48331}, {48123, 6332}, {48265, 69293}, {48336, 48299}, {48392, 6590}, {48400, 68794}, {50340, 667}, {50342, 1019}, {50355, 48069}, {55282, 54265}
X(69307) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {39722, 21294}, {39977, 3448}, {65369, 1330}
X(69307) = crosssum of X(512) and X(5280)
X(69307) = crossdifference of every pair of points on line {1914, 2092}
X(69307) = barycentric product X(i)*X(j) for these {i,j}: {514, 32780}, {40017, 50540}
X(69307) = barycentric quotient X(i)/X(j) for these {i,j}: {32780, 190}, {50540, 2238}
X(69307) = {X(48400),X(68794)}-harmonic conjugate of X(47872)

X(69308) = ODD<1, 1, 0, 0, 1, 1> POINT

Barycentrics    (b - c)*(a^3 + a^2*b + b^3 + a^2*c + b^2*c + b*c^2 + c^3) : :
X(69308) = 3 X[48235] - 2 X[50337], 2 X[48106] + X[49279], 2 X[3004] - 3 X[47888], 2 X[3776] - 3 X[48569], 2 X[4129] - 3 X[48185], X[4391] - 3 X[48236], 2 X[21260] - 3 X[47809], X[21301] - 3 X[48208], X[47708] - 3 X[47771], 5 X[31251] - 6 X[47807], 4 X[31288] - 3 X[47797], X[47651] - 3 X[47819], X[47688] - 3 X[47796], X[47692] - 3 X[47820], X[47709] - 3 X[47804], X[47713] - 3 X[47818], 3 X[47835] - 2 X[50453], 3 X[47875] - 2 X[48403], 3 X[47875] - 4 X[68794], 3 X[47885] - 2 X[48003]

X(69308) lies on these lines: {1, 29208}, {10, 514}, {512, 48106}, {513, 48272}, {523, 667}, {525, 4834}, {649, 826}, {659, 29021}, {663, 7927}, {665, 48275}, {690, 50509}, {693, 29098}, {784, 47660}, {814, 47711}, {905, 4802}, {1019, 62423}, {1499, 58171}, {1577, 29025}, {1734, 68979}, {2787, 47707}, {2977, 48402}, {3004, 47888}, {3566, 58173}, {3762, 29120}, {3776, 48569}, {3800, 4775}, {3801, 29160}, {3803, 4777}, {3906, 58179}, {4010, 29158}, {4040, 29144}, {4063, 29017}, {4083, 47682}, {4122, 29013}, {4129, 48185}, {4367, 29047}, {4378, 29288}, {4379, 29867}, {4380, 29106}, {4391, 29029}, {4401, 29164}, {4468, 47949}, {4498, 29312}, {4560, 47693}, {4707, 29332}, {4724, 29168}, {4761, 29082}, {4782, 29146}, {4784, 23875}, {4791, 29140}, {4808, 8678}, {4874, 29174}, {6372, 48094}, {6590, 48393}, {7265, 29328}, {7950, 50512}, {8045, 48273}, {12073, 48338}, {14419, 28147}, {21124, 50504}, {21260, 47809}, {21301, 48208}, {23815, 47652}, {25259, 29150}, {29070, 47690}, {29074, 47710}, {29086, 47689}, {29094, 47684}, {29110, 47706}, {29118, 48267}, {29128, 47708}, {29132, 48265}, {29142, 47890}, {29162, 48395}, {29198, 48097}, {29244, 47724}, {29274, 47723}, {29298, 47728}, {29318, 48011}, {29354, 48118}, {29358, 48064}, {29362, 47715}, {31208, 47784}, {31251, 47807}, {31288, 47797}, {32478, 58172}, {47651, 47819}, {47662, 48410}, {47663, 47719}, {47688, 47796}, {47691, 52601}, {47692, 47820}, {47700, 50523}, {47701, 50507}, {47709, 47804}, {47713, 47818}, {47727, 48330}, {47835, 50453}, {47875, 48403}, {47885, 48003}, {47938, 48053}, {47942, 48048}, {47944, 48054}, {47958, 48059}, {47959, 48056}, {48101, 48278}, {48122, 48138}, {48130, 48151}, {48131, 48146}, {48290, 48333}, {50515, 64856}

X(69308) = midpoint of X(i) and X(j) for these {i,j}: {3777, 48140}, {4063, 47726}, {4560, 47693}, {47662, 48410}, {47663, 47719}, {47700, 50523}, {48101, 48278}, {48106, 48300}, {48118, 48144}, {48122, 48138}, {48130, 48151}, {48131, 48146}
X(69308) = reflection of X(i) in X(j) for these {i,j}: {1577, 48405}, {4705, 48062}, {4775, 48299}, {21124, 50504}, {47652, 23815}, {47691, 52601}, {47701, 50507}, {47712, 4874}, {47727, 48330}, {47938, 48053}, {47942, 48048}, {47944, 48054}, {47949, 4468}, {47958, 48059}, {47959, 48056}, {47968, 48066}, {48267, 69293}, {48273, 8045}, {48333, 48290}, {48393, 6590}, {48402, 2977}, {48403, 68794}, {49279, 48300}, {50340, 4401}, {50342, 48064}
X(69308) = crossdifference of every pair of points on line {1914, 4261}
X(69308) = barycentric product X(514)*X(26061)
X(69308) = barycentric quotient X(26061)/X(190)
X(69308) = {X(48403),X(68794)}-harmonic conjugate of X(47875)

X(69309) = ODD<1, 1, 0, 0, 0, 1> POINT

Barycentrics    (b - c)*(a^3 + a^2*b + a^2*c + b^2*c + b*c^2) : :
X(69309) = X[1] - 3 X[47820], X[905] - 3 X[47761], X[3004] - 3 X[41800], X[7178] + 3 X[47767], X[43067] + 3 X[48559], X[47921] + 3 X[48563], X[47965] - 3 X[48559], X[649] - 3 X[48566], X[1577] + 3 X[48566], X[661] - 3 X[47794], X[663] - 3 X[47818], X[4761] + 3 X[47818], X[693] + 3 X[48565], X[4063] - 3 X[48565], X[1019] - 3 X[47762], and many others

X(69309) lies on these lines: {1, 47820}, {2, 14349}, {10, 8678}, {241, 514}, {512, 4874}, {513, 3814}, {523, 20517}, {525, 68794}, {649, 1577}, {659, 29186}, {661, 47794}, {663, 4761}, {667, 2533}, {676, 3800}, {693, 4063}, {784, 9508}, {812, 4823}, {814, 50512}, {824, 21192}, {826, 48405}, {830, 17072}, {1019, 4391}, {1125, 48136}, {1491, 47837}, {1499, 4990}, {1635, 50457}, {1698, 47814}, {1734, 47694}, {1960, 29366}, {2254, 48573}, {2527, 29162}, {2530, 47823}, {3566, 49288}, {3634, 48027}, {3700, 29216}, {3716, 6005}, {3762, 48144}, {3777, 48569}, {3801, 29160}, {3900, 4807}, {4010, 4834}, {4040, 47804}, {4041, 47813}, {4083, 48295}, {4106, 59714}, {4122, 29294}, {4142, 29021}, {4147, 4160}, {4151, 7662}, {4170, 47832}, {4379, 4498}, {4394, 23882}, {4401, 29051}, {4448, 48351}, {4458, 29047}, {4462, 48320}, {4560, 27013}, {4581, 21189}, {4705, 47835}, {4707, 29220}, {4724, 47817}, {4728, 47935}, {4729, 48339}, {4730, 48301}, {4776, 48085}, {4777, 31010}, {4778, 48004}, {4782, 29070}, {4784, 47872}, {4785, 45324}, {4790, 59737}, {4791, 6002}, {4801, 21385}, {4802, 21181}, {4813, 48551}, {4822, 47838}, {4844, 65428}, {4885, 29457}, {4893, 50449}, {4905, 47824}, {4932, 15309}, {4960, 47666}, {4983, 47822}, {4992, 48206}, {6004, 48248}, {6050, 48284}, {6133, 8672}, {6590, 23879}, {7192, 47793}, {7265, 47874}, {7649, 59932}, {7659, 59590}, {8045, 23876}, {8714, 50336}, {19947, 48137}, {20295, 47976}, {20317, 29402}, {21052, 50523}, {21185, 48069}, {21186, 60493}, {21302, 48324}, {23724, 68878}, {23875, 69293}, {24924, 47795}, {25380, 48066}, {25666, 48054}, {27322, 27323}, {28840, 47997}, {29062, 48395}, {29132, 48400}, {29158, 48403}, {29182, 58139}, {29188, 48331}, {29190, 48396}, {29196, 47711}, {29284, 49290}, {29298, 48285}, {29304, 48299}, {29328, 58179}, {29340, 58145}, {29487, 30024}, {29512, 48026}, {29717, 47955}, {30474, 47766}, {30835, 48121}, {31148, 47918}, {31250, 48128}, {42325, 48063}, {43927, 47842}, {44429, 48086}, {45313, 64934}, {47678, 48277}, {47679, 48275}, {47712, 48106}, {47716, 47887}, {47759, 48595}, {47760, 48091}, {47763, 48110}, {47789, 60492}, {47796, 48335}, {47802, 48092}, {47803, 48099}, {47805, 48111}, {47809, 48272}, {47815, 47970}, {47816, 48023}, {47821, 48081}, {47828, 48409}, {47833, 48273}, {47839, 48123}, {47911, 48577}, {47929, 48579}, {47947, 48107}, {48024, 48553}, {48029, 48561}, {48030, 65449}, {48065, 53580}, {48093, 48197}, {48098, 68894}, {48100, 48216}, {48122, 48556}, {48142, 48407}, {48164, 48596}, {48186, 50332}, {48220, 50499}, {48233, 48406}, {48234, 48305}, {49277, 57066}, {52623, 57234}

X(69309) = midpoint of X(i) and X(j) for these {i,j}: {649, 1577}, {659, 50352}, {663, 4761}, {667, 2533}, {693, 4063}, {1019, 4391}, {1734, 47694}, {3762, 48144}, {4010, 4834}, {4170, 50509}, {4462, 48320}, {4498, 4978}, {4581, 21189}, {4707, 48300}, {4729, 48339}, {4730, 48301}, {4784, 48267}, {4791, 48064}, {4801, 21385}, {4823, 48011}, {4960, 47666}, {7192, 47959}, {7659, 59590}, {7662, 50501}, {20295, 47976}, {21185, 48069}, {21186, 60493}, {21302, 48324}, {43067, 47965}, {43927, 47842}, {47678, 48277}, {47679, 48275}, {47712, 48106}, {47947, 48107}, {47970, 48108}, {48142, 48407}, {48276, 48402}, {48305, 50355}
X(69309) = reflection of X(i) in X(j) for these {i,j}: {4106, 59714}, {14838, 31286}, {48030, 65449}, {48054, 25666}, {48065, 53580}, {48066, 25380}, {48136, 1125}, {48137, 19947}, {48284, 6050}, {48285, 48330}, {48295, 52601}, {50453, 14837}
X(69309) = complement of X(14349)
X(69309) = complement of the isotomic conjugate of X(37218)
X(69309) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 5515}, {32, 39016}, {101, 52782}, {835, 141}, {2214, 11}, {32653, 34281}, {37218, 2887}, {43531, 116}, {56047, 53564}, {57704, 2968}, {57977, 626}, {58951, 1125}, {68197, 3741}
X(69309) = X(101)-isoconjugate of X(15315)
X(69309) = X(1015)-Dao conjugate of X(15315)
X(69309) = crosspoint of X(2) and X(37218)
X(69309) = crossdifference of every pair of points on line {55, 16685}
X(69309) = barycentric product X(i)*X(j) for these {i,j}: {514, 26223}, {693, 5264}
X(69309) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 15315}, {5264, 100}, {26223, 190}
X(69309) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {693, 48565, 4063}, {1577, 48566, 649}, {1698, 47948, 47814}, {3762, 48568, 48144}, {4379, 4498, 4978}, {4391, 47762, 1019}, {4462, 48570, 48320}, {4761, 47818, 663}, {4784, 47872, 48267}, {4834, 47875, 4010}, {7192, 47793, 47959}, {24924, 48131, 47795}, {43067, 48559, 47965}, {47694, 47836, 1734}, {47815, 48108, 47970}, {47832, 50509, 4170}, {48054, 48196, 25666}, {48136, 48564, 1125}, {48234, 50355, 48305}

X(69310) = ODD<1, 1, 0, -1, 0, 1> POINT

Barycentrics    (b - c)*(a^3 + a^2*b + a^2*c - a*b*c + b^2*c + b*c^2) : :
X(69310) = X[1] - 3 X[47818], X[650] - 3 X[48559], X[3669] - 3 X[47761], 3 X[4763] - 2 X[14838], X[21120] + 3 X[47767], 2 X[21212] - 3 X[41800], X[649] - 3 X[48565], X[4391] + 3 X[48565], X[661] - 3 X[47793], X[663] - 3 X[47804], X[764] - 3 X[48569], X[1019] - 3 X[48566], X[3762] + 3 X[48566], X[1491] - 3 X[47835], 3 X[1635] - X[4560], and many others

X(69310) lies on these lines: {1, 47818}, {2, 48131}, {8, 48322}, {10, 830}, {241, 514}, {512, 3716}, {513, 3823}, {522, 48395}, {523, 4142}, {525, 69293}, {649, 4391}, {659, 2533}, {661, 26049}, {663, 47804}, {667, 3907}, {693, 4498}, {764, 48569}, {784, 4913}, {798, 812}, {814, 4782}, {824, 47129}, {834, 8062}, {891, 52601}, {918, 52602}, {1019, 3762}, {1125, 48348}, {1268, 47947}, {1491, 47835}, {1635, 4560}, {1698, 47816}, {1960, 29298}, {2254, 47836}, {2530, 25380}, {2785, 48231}, {2787, 50512}, {2832, 23789}, {3239, 28478}, {3309, 48063}, {3667, 59590}, {3777, 47823}, {3801, 48103}, {3803, 28470}, {3887, 4807}, {3904, 28497}, {3910, 8045}, {4010, 47872}, {4040, 4761}, {4041, 47694}, {4057, 68888}, {4083, 4874}, {4129, 21261}, {4147, 8678}, {4379, 4801}, {4394, 23880}, {4401, 29066}, {4448, 48336}, {4449, 47820}, {4458, 29288}, {4462, 47762}, {4474, 58140}, {4581, 17420}, {4724, 47815}, {4730, 48305}, {4776, 48121}, {4778, 47966}, {4784, 48265}, {4785, 45664}, {4791, 29013}, {4808, 64860}, {4822, 47821}, {4823, 29302}, {4830, 29070}, {4834, 48267}, {4885, 8712}, {4905, 48573}, {4978, 21385}, {4983, 48553}, {6005, 59672}, {6332, 28468}, {6546, 23755}, {6590, 60492}, {7192, 47918}, {8643, 47729}, {9508, 63812}, {9780, 48020}, {14349, 25666}, {14430, 50523}, {14825, 27929}, {15309, 29717}, {17166, 47813}, {17494, 50457}, {17496, 27013}, {20295, 47935}, {20517, 29047}, {21052, 21301}, {21124, 47660}, {21260, 48050}, {21302, 28521}, {21832, 68803}, {23877, 48062}, {23882, 48008}, {24232, 43921}, {24924, 27014}, {25981, 26017}, {28493, 48269}, {28501, 57066}, {28519, 31291}, {28840, 47959}, {29017, 48405}, {29118, 48400}, {29148, 48064}, {29150, 58179}, {29176, 58145}, {29268, 58139}, {29366, 48331}, {30574, 48250}, {42661, 58386}, {44429, 48122}, {45315, 48054}, {45324, 59714}, {45332, 64913}, {47136, 60493}, {47708, 48106}, {47720, 47887}, {47759, 48597}, {47760, 48128}, {47763, 48149}, {47771, 48300}, {47788, 48280}, {47795, 48335}, {47802, 48616}, {47803, 48136}, {47809, 48278}, {47814, 48023}, {47822, 48123}, {47824, 48151}, {47828, 48410}, {47833, 48279}, {47845, 53532}, {47875, 48273}, {47911, 48107}, {47929, 48108}, {47991, 47997}, {47992, 48005}, {48059, 65449}, {48080, 50509}, {48085, 48551}, {48099, 48561}, {48116, 48164}, {48129, 48197}, {48137, 48216}, {48165, 50332}, {48234, 48301}, {48264, 50343}, {48285, 65428}, {48320, 48568}, {48332, 48564}, {48341, 48570}, {48408, 55282}, {48562, 50507}

X(69310) = midpoint of X(i) and X(j) for these {i,j}: {8, 48322}, {649, 4391}, {659, 2533}, {693, 4498}, {1019, 3762}, {1577, 4063}, {3801, 48103}, {4040, 4761}, {4041, 47694}, {4462, 48144}, {4581, 17420}, {4730, 48305}, {4784, 48265}, {4791, 48011}, {4834, 48267}, {4978, 21385}, {6590, 60492}, {7178, 47890}, {7192, 47918}, {17494, 50457}, {20295, 47935}, {21124, 47660}, {21302, 48150}, {30574, 48250}, {43067, 47921}, {47136, 60493}, {47708, 48106}, {47911, 48107}, {47929, 48108}, {48080, 50509}, {48264, 50343}, {48408, 55282}
X(69310) = reflection of X(i) in X(j) for these {i,j}: {905, 31286}, {2530, 25380}, {3776, 21188}, {4040, 53580}, {4913, 50504}, {8045, 68794}, {14349, 25666}, {42661, 58386}, {47991, 47997}, {47992, 48005}, {48000, 48003}, {48049, 4129}, {48050, 21260}, {48059, 65449}, {48285, 65428}, {48334, 65482}, {48348, 1125}, {49289, 4823}
X(69310) = complement of X(48131)
X(69310) = complement of the isogonal conjugate of X(36147)
X(69310) = complement of the isotomic conjugate of X(65229)
X(69310) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 15611}, {32, 39015}, {100, 51571}, {251, 55054}, {961, 4904}, {1169, 244}, {1220, 116}, {1252, 50330}, {2298, 11}, {2359, 2968}, {2363, 17761}, {6648, 2886}, {8687, 1}, {8707, 141}, {14534, 53564}, {14624, 125}, {30710, 21252}, {32736, 2}, {35334, 21249}, {36098, 142}, {36147, 10}, {52928, 4000}, {58982, 17045}, {58997, 20270}, {60264, 53575}, {64984, 17059}, {65229, 2887}, {65282, 626}
X(69310) = X(64978)-Ceva conjugate of X(11)
X(69310) = X(101)-isoconjugate of X(45989)
X(69310) = X(i)-Dao conjugate of X(j) for these (i,j): {1015, 45989}, {52326, 17420}
X(69310) = cevapoint of X(50353) and X(57164)
X(69310) = crosspoint of X(i) and X(j) for these (i,j): {2, 65229}, {82, 36098}
X(69310) = crosssum of X(38) and X(17420)
X(69310) = crossdifference of every pair of points on line {55, 1964}
X(69310) = barycentric product X(i)*X(j) for these {i,j}: {75, 50353}, {85, 57164}, {514, 27064}, {693, 5255}, {24994, 62748}
X(69310) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 45989}, {5255, 100}, {24994, 21580}, {27064, 190}, {38992, 17420}, {50353, 1}, {57164, 9}
X(69310) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1698, 48086, 47816}, {2530, 47837, 25380}, {3762, 48566, 1019}, {4040, 47817, 53580}, {4391, 48565, 649}, {4462, 47762, 48144}, {4761, 47817, 4040}, {14349, 47794, 25666}, {21302, 47805, 48150}, {24924, 48334, 47796}, {47796, 48334, 65482}

X(69311) = ODD<1, 1, -1, 1, 0, 0> POINT

Barycentrics    a*(b - c)*(a^2 + a*b - b^2 + a*c + b*c - c^2) : :
X(69311) = 3 X[667] - X[48111], 3 X[905] - X[48099], 3 X[1019] + X[48086], 3 X[2530] - X[48086], X[4040] - 3 X[14419], 3 X[4905] + X[48111], 3 X[3669] - X[48346], 3 X[3669] + X[50499], X[48346] + 3 X[50336], 3 X[50336] - X[50499], 3 X[3960] - X[48348], 3 X[9508] - 2 X[50504], 3 X[650] - X[48618], 3 X[659] - X[47936], X[47936] + 3 X[48151], and many others

X(69311) lies on these lines: {2, 48265}, {36, 238}, {65, 876}, {512, 3960}, {514, 9508}, {649, 3777}, {650, 29198}, {659, 47936}, {661, 47893}, {663, 50359}, {693, 16737}, {764, 3336}, {784, 54265}, {810, 53314}, {812, 48406}, {814, 24720}, {918, 59866}, {1385, 3309}, {1491, 47912}, {1577, 48569}, {1635, 23738}, {1638, 48400}, {1734, 4378}, {1960, 42325}, {2254, 4367}, {2486, 24234}, {2533, 17496}, {2787, 50337}, {3762, 47837}, {3776, 29025}, {3801, 4453}, {3808, 45902}, {3822, 21260}, {3835, 29170}, {3837, 6002}, {3887, 48328}, {3900, 48344}, {4010, 47796}, {4025, 29017}, {4041, 48244}, {4142, 69011}, {4147, 48575}, {4369, 63812}, {4379, 48392}, {4391, 47823}, {4449, 50355}, {4462, 47835}, {4490, 47828}, {4498, 23765}, {4560, 21146}, {4705, 48320}, {4729, 21343}, {4730, 48282}, {4777, 47715}, {4782, 37582}, {4784, 48131}, {4790, 48616}, {4804, 47889}, {4834, 48335}, {4879, 14413}, {4885, 24287}, {4893, 47913}, {4922, 21302}, {6004, 48075}, {6332, 29200}, {6372, 14838}, {7178, 48245}, {7659, 48136}, {8678, 50335}, {8714, 52601}, {9269, 50194}, {11263, 19947}, {14422, 58160}, {15309, 48059}, {17072, 29324}, {17166, 50341}, {17990, 28855}, {21051, 25380}, {21192, 29312}, {21222, 47836}, {21301, 36848}, {22090, 43924}, {23770, 30724}, {23789, 29070}, {23815, 29013}, {23882, 48098}, {24719, 47819}, {24924, 47872}, {28217, 59836}, {29094, 62435}, {29208, 48069}, {29226, 50501}, {29238, 48089}, {29246, 48073}, {29276, 49285}, {29366, 48325}, {31288, 45666}, {37605, 48329}, {38469, 50350}, {47707, 48235}, {47708, 48227}, {47709, 48224}, {47795, 48267}, {47827, 47918}, {47833, 48264}, {47841, 48080}, {47888, 47959}, {47905, 48160}, {47906, 48162}, {47922, 48213}, {47929, 48226}, {48018, 48343}, {48066, 48601}, {48150, 58374}, {48193, 48607}, {48194, 48609}, {48229, 48401}, {48253, 50457}, {48278, 50342}, {48279, 50343}, {48299, 50357}, {48301, 50356}, {48321, 50352}, {48410, 48570}, {48564, 59590}, {49291, 50544}, {50328, 50523}, {50348, 68979}, {50358, 58140}, {50454, 66523}

X(69311) = midpoint of X(i) and X(j) for these {i,j}: {649, 3777}, {659, 48151}, {663, 50359}, {667, 4905}, {764, 4063}, {1019, 2530}, {1491, 48144}, {1734, 4378}, {2254, 4367}, {2533, 17496}, {3669, 50336}, {4041, 48323}, {4449, 50355}, {4490, 48341}, {4498, 23765}, {4560, 21146}, {4705, 48320}, {4729, 21343}, {4730, 48282}, {4784, 48131}, {4790, 48616}, {4834, 48335}, {4922, 21302}, {7659, 48136}, {17166, 50341}, {48018, 48343}, {48150, 58374}, {48278, 50342}, {48279, 50343}, {48299, 50357}, {48301, 50356}, {48321, 50352}, {48346, 50499}, {50328, 50523} X(69311) = reflection of X(i) in X(j) for these {i,j}: {4142, 69011}, {21051, 25380}, {59672, 31288}
X(69311) = complement of X(48265)
X(69311) = X(101)-isoconjugate of X(60149)
X(69311) = X(1015)-Dao conjugate of X(60149)
X(69311) = crosspoint of X(39950) and X(52935)
X(69311) = crosssum of X(i) and X(j) for these (i,j): {513, 29820}, {3294, 4705}
X(69311) = crossdifference of every pair of points on line {37, 3684}
X(69311) = barycentric product X(i)*X(j) for these {i,j}: {1, 69292}, {513, 17300}, {514, 32913}, {649, 33943}, {693, 33863}, {905, 4212}, {1019, 29653}
X(69311) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 60149}, {4212, 6335}, {17300, 668}, {29653, 4033}, {32913, 190}, {33863, 100}, {33943, 1978}, {69292, 75}
X(69311) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3669, 50499, 48346}, {17496, 47824, 2533}, {31288, 59672, 45666}, {47828, 48341, 4490}, {48244, 48323, 4041}, {48346, 50336, 50499}

X(69312) = ODD<1, 1, -1, 1, 0, -1> POINT

Barycentrics    (b - c)*(-a^3 - a^2*b + a*b^2 - a^2*c - a*b*c + b^2*c + a*c^2 + b*c^2) : :
X(69312) = 2 X[10] - 3 X[48244], 3 X[17496] - X[48298], 3 X[667] - 2 X[48063], X[48142] - 3 X[48144], 4 X[905] - 3 X[47839], 3 X[47839] - 2 X[48267], 4 X[1125] - 3 X[4800], 2 X[1577] - 3 X[48569], 5 X[1698] - 6 X[48229], 3 X[2530] - 2 X[48050], 5 X[3616] - 6 X[14422], 7 X[3624] - 6 X[48183], 2 X[3716] - 3 X[14419], 2 X[4129] - 3 X[47893], and many others

X(69312) lies on these lines: {1, 900}, {10, 48244}, {239, 6550}, {274, 3766}, {512, 17496}, {513, 48288}, {514, 4784}, {522, 4378}, {523, 48320}, {659, 68896}, {665, 5283}, {667, 48063}, {690, 3904}, {764, 812}, {784, 48142}, {814, 4905}, {891, 21222}, {905, 47839}, {918, 50351}, {1019, 63812}, {1125, 4800}, {1491, 29148}, {1577, 48569}, {1698, 48229}, {1734, 29324}, {1960, 53343}, {2254, 2787}, {2530, 6002}, {3616, 14422}, {3624, 48183}, {3667, 4775}, {3669, 48273}, {3716, 14419}, {3762, 9508}, {3777, 29013}, {3887, 4922}, {3960, 4010}, {4129, 47893}, {4151, 48323}, {4160, 50341}, {4367, 8714}, {4391, 47837}, {4435, 20963}, {4462, 50504}, {4467, 29312}, {4486, 50179}, {4508, 4750}, {4560, 6372}, {4728, 19947}, {4770, 48242}, {4791, 47823}, {4809, 21201}, {4926, 48339}, {4977, 47683}, {4984, 23764}, {5088, 52305}, {9780, 28603}, {13245, 42670}, {14286, 14315}, {14349, 29170}, {14431, 25380}, {14838, 48265}, {16552, 22108}, {16823, 44433}, {16830, 31131}, {16892, 29029}, {19853, 26078}, {21146, 64934}, {21301, 29176}, {21302, 29268}, {21343, 68968}, {21384, 68814}, {23738, 68894}, {23765, 29302}, {23795, 58374}, {23814, 48167}, {23880, 50352}, {23882, 48126}, {23887, 50342}, {23888, 24097}, {24287, 25499}, {24719, 29178}, {25512, 48168}, {28217, 48289}, {29058, 47687}, {29066, 50359}, {29070, 48115}, {29078, 49278}, {29090, 48278}, {29126, 50348}, {29150, 48131}, {29224, 47930}, {29240, 50357}, {29328, 48335}, {29340, 46403}, {29344, 48075}, {30592, 65482}, {30709, 53571}, {30795, 59737}, {39586, 48182}, {44550, 48080}, {47680, 58375}, {47805, 58139}, {47940, 48410}, {48264, 52601}, {48301, 48343}, {48571, 49303}, {53578, 68480}

X(69312) = midpoint of X(i) and X(j) for these {i,j}: {2254, 53536}, {21222, 50343}
X(69312) = reflection of X(i) in X(j) for these {i,j}: {3762, 9508}, {4010, 3960}, {4462, 50504}, {4775, 48325}, {14286, 14315}, {47680, 58375}, {48264, 52601}, {48265, 14838}, {48267, 905}, {48273, 3669}, {48288, 48321}, {48291, 4378}, {48301, 48343}, {48305, 4367}, {48339, 48344}, {48352, 48289}, {53343, 1960}, {58374, 23795}
X(69312) = barycentric product X(514)*X(32919)
X(69312) = barycentric quotient X(32919)/X(190)
X(69312) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {905, 48267, 47839}, {14838, 48265, 48553}

X(69313) = ODD<1, 1, -1, 0, 1, 1> POINT

Barycentrics    (b - c)*(a^3 + a^2*b - a*b^2 + b^3 + a^2*c + b^2*c - a*c^2 + b*c^2 + c^3) : :
X(69313) = X[2533] - 3 X[48235], 2 X[676] - 3 X[48564], 4 X[2490] - 3 X[48561], 2 X[3676] - 3 X[48569], X[3801] - 3 X[47823], 2 X[21188] - 3 X[47823], 3 X[4379] - X[55282], X[4391] - 3 X[47809], 4 X[4521] - 3 X[48553], X[4707] - 3 X[48573], 3 X[6546] - X[47929], X[7178] - 3 X[48232], 3 X[14432] - X[48338], 2 X[14837] - 3 X[47837], and many others

X(69313) lies on these lines: {2, 47708}, {10, 514}, {512, 6332}, {513, 28591}, {522, 667}, {523, 905}, {525, 50336}, {649, 48278}, {650, 29142}, {665, 784}, {676, 48564}, {826, 4025}, {900, 3803}, {1019, 48272}, {1734, 47682}, {2254, 48300}, {2490, 48561}, {2522, 50539}, {2826, 48219}, {2933, 48387}, {2977, 47965}, {3239, 48267}, {3309, 48299}, {3566, 49280}, {3669, 29288}, {3676, 48569}, {3800, 48136}, {3801, 21188}, {3835, 29118}, {3837, 29025}, {3900, 48290}, {3910, 50501}, {3960, 29047}, {4063, 49278}, {4088, 48144}, {4129, 29132}, {4142, 31286}, {4369, 23877}, {4378, 4808}, {4379, 24892}, {4391, 47809}, {4449, 49487}, {4468, 6372}, {4521, 48553}, {4522, 6002}, {4560, 47690}, {4581, 20294}, {4707, 48573}, {4801, 48408}, {4834, 28478}, {4874, 21185}, {4885, 48403}, {6050, 50347}, {6362, 66514}, {6546, 47929}, {7178, 48232}, {8678, 50333}, {8714, 49286}, {9508, 29017}, {14419, 28161}, {14432, 48338}, {14837, 47837}, {14838, 29021}, {17494, 47719}, {17496, 47707}, {18004, 29170}, {20315, 50330}, {21051, 29120}, {21124, 47828}, {21192, 29318}, {21260, 29029}, {21301, 47808}, {21302, 47728}, {23815, 29098}, {23880, 48395}, {23882, 48396}, {25380, 29116}, {28183, 30234}, {29070, 49285}, {29126, 48200}, {29128, 47757}, {29130, 50453}, {29138, 53571}, {29150, 48269}, {29158, 49295}, {29166, 47785}, {29168, 48006}, {29198, 48056}, {29220, 62435}, {29312, 50504}, {31208, 46919}, {31288, 47800}, {31291, 48169}, {34958, 47131}, {44550, 47706}, {45671, 47714}, {45695, 48301}, {45902, 47130}, {47123, 52601}, {47652, 47819}, {47660, 48410}, {47691, 47796}, {47695, 47820}, {47709, 47797}, {47711, 48321}, {47712, 47795}, {47807, 48400}, {47841, 48349}, {47874, 48264}, {47979, 48053}, {47983, 48054}, {47995, 48059}, {48077, 50523}, {48080, 57066}, {48094, 48151}, {48101, 48122}, {48106, 48131}, {48185, 48265}, {48405, 63812}, {50335, 68979}

X(69313) = midpoint of X(i) and X(j) for these {i,j}: {649, 48278}, {1019, 48272}, {1734, 47682}, {2254, 48300}, {3777, 48103}, {4063, 49278}, {4088, 48144}, {4378, 4808}, {4560, 47690}, {4581, 20294}, {4801, 48408}, {6332, 48069}, {17494, 47719}, {17496, 47707}, {21302, 47728}, {47660, 48410}, {47711, 48321}, {48077, 50523}, {48094, 48151}, {48101, 48122}, {48106, 48131}, {50351, 50352}
X(69313) = reflection of X(i) in X(j) for these {i,j}: {3801, 21188}, {4142, 31286}, {21185, 4874}, {47123, 52601}, {47131, 34958}, {47965, 2977}, {47979, 48053}, {47983, 48054}, {47995, 48059}, {48006, 50507}, {48007, 48066}, {48267, 3239}, {48398, 23815}, {48403, 4885}, {50330, 20315}, {50347, 6050}, {60492, 50504}, {68780, 14838}
X(69313) = complement of X(47708)
X(69313) = crosssum of X(6) and X(50503)
X(69313) = crossdifference of every pair of points on line {1914, 2277}
X(69313) = barycentric product X(514)*X(33163)
X(69313) = barycentric quotient X(33163)/X(190)
X(69313) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3801, 47823, 21188}, {17496, 48208, 47707}

X(69314) = ODD<1, 1, -1, 0, 0, 0> POINT

Barycentrics    a*(b - c)*(a^2 + a*b - b^2 + a*c - c^2) : :
X(69314) = 3 X[905] - X[48136], X[48136] + 3 X[50336], 3 X[14838] - X[48058], 2 X[48058] - 3 X[50507], 3 X[48064] + X[48603], 3 X[48066] - X[48603], 3 X[649] + X[48122], 3 X[2530] - X[48122], 3 X[650] - X[47966], 3 X[659] - X[47977], 3 X[4905] + X[47977], X[661] - 3 X[47888], X[663] - 3 X[14419], 3 X[667] - X[48150], 3 X[2254] + X[48150], and many others

X(69314) lies on these lines: {1, 50355}, {2, 48267}, {10, 29324}, {512, 905}, {513, 4401}, {514, 9508}, {522, 52601}, {649, 2473}, {650, 6372}, {659, 4905}, {661, 47888}, {663, 14419}, {665, 52589}, {667, 2254}, {690, 6332}, {693, 48569}, {764, 4498}, {784, 4369}, {812, 23815}, {814, 50337}, {826, 4025}, {830, 50335}, {832, 3733}, {891, 3669}, {1019, 1491}, {1577, 47823}, {1635, 48151}, {1638, 48403}, {1734, 4367}, {1960, 3309}, {2487, 6362}, {2526, 50515}, {2533, 48321}, {2787, 17072}, {3126, 24286}, {3572, 23657}, {3716, 31288}, {3762, 47835}, {3776, 29098}, {3777, 4063}, {3798, 24285}, {3803, 58139}, {3835, 29150}, {3837, 29013}, {3887, 48330}, {3900, 48328}, {3960, 4083}, {4010, 47795}, {4040, 50359}, {4041, 4378}, {4129, 29170}, {4139, 51648}, {4142, 45674}, {4170, 47841}, {4379, 48393}, {4380, 47819}, {4391, 47837}, {4449, 4730}, {4490, 48320}, {4522, 29090}, {4560, 47824}, {4705, 47828}, {4729, 14413}, {4750, 48278}, {4784, 14349}, {4790, 48092}, {4834, 48131}, {4874, 8714}, {4893, 47949}, {6002, 21260}, {6161, 8643}, {6705, 40551}, {7659, 48099}, {7927, 48069}, {8651, 24562}, {14422, 48347}, {15309, 48030}, {17069, 29142}, {17166, 48242}, {17496, 47836}, {17990, 28846}, {20517, 69011}, {21051, 29148}, {21052, 53536}, {21192, 29017}, {21212, 29118}, {21385, 23765}, {23789, 29362}, {24719, 48556}, {24720, 29070}, {24924, 47875}, {27486, 47719}, {29082, 62435}, {29120, 50453}, {29168, 68780}, {29176, 53571}, {29198, 48003}, {29246, 48284}, {29298, 48325}, {29302, 48406}, {29354, 48062}, {30234, 48329}, {31209, 48553}, {41800, 48400}, {42325, 48331}, {43067, 50544}, {47711, 48235}, {47712, 48227}, {47713, 48224}, {47762, 48410}, {47794, 48265}, {47796, 48273}, {47803, 59590}, {47810, 48149}, {47820, 48305}, {47827, 47959}, {47830, 65449}, {47839, 48080}, {47889, 50339}, {47935, 58181}, {47942, 48162}, {47945, 48580}, {47956, 48193}, {47957, 48194}, {47967, 48213}, {47970, 48226}, {47975, 48570}, {48008, 68894}, {48111, 58374}, {48116, 58146}, {48160, 48586}, {48198, 59714}, {48225, 48407}, {48230, 50329}, {48232, 48395}, {48246, 50331}, {48272, 50342}, {48332, 50499}, {48409, 48568}, {48616, 58182}, {54249, 68836}

X(69314) = midpoint of X(i) and X(j) for these {i,j}: {1, 50355}, {649, 2530}, {659, 4905}, {667, 2254}, {764, 4498}, {905, 50336}, {1019, 1491}, {1734, 4367}, {2526, 50515}, {2533, 48321}, {3669, 50501}, {3733, 50350}, {3777, 4063}, {4040, 50359}, {4041, 4378}, {4401, 48075}, {4449, 4730}, {4490, 48320}, {4560, 50352}, {4705, 48144}, {4729, 48333}, {4784, 14349}, {4790, 48092}, {4834, 48131}, {7659, 48099}, {21385, 23765}, {48064, 48066}, {48111, 58374}, {48272, 50342}, {48273, 50343}, {48305, 50356}, {48332, 50499}
X(69314) = reflection of X(i) in X(j) for these {i,j}: {3716, 31288}, {3803, 58139}, {20517, 69011}, {21260, 25380}, {48329, 58150}, {50504, 9508}, {50507, 14838}
X(69314) = complement of X(48267)
X(69314) = crossdifference of every pair of points on line {3920, 5275}
X(69314) = barycentric product X(i)*X(j) for these {i,j}: {1, 47971}, {513, 4851}, {514, 32912}, {649, 33942}, {693, 69231}, {1019, 69295}
X(69314) = barycentric quotient X(i)/X(j) for these {i,j}: {4851, 668}, {32912, 190}, {33942, 1978}, {47971, 75}, {69231, 100}, {69295, 4033}
X(69314) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4367, 48244, 1734}, {4560, 47824, 50352}, {4729, 14413, 48333}, {4784, 47893, 14349}, {24924, 48264, 47875}, {30234, 48329, 58150}, {47796, 50343, 48273}, {47820, 50356, 48305}, {47828, 48144, 4705}, {48321, 48573, 2533}

X(69315) = ODD<1, 1, -1, 0, 0, -1> POINT

Barycentrics    (b - c)*(-a^3 - a^2*b + a*b^2 - a^2*c + b^2*c + a*c^2 + b*c^2) : :
X(69315) = 4 X[905] - 3 X[47841], 2 X[4010] - 3 X[47841], 2 X[1577] - 3 X[47823], 2 X[3835] - 3 X[47893], 2 X[4129] - 3 X[47888], 2 X[4391] - 3 X[47835], 4 X[9508] - 3 X[47835], 3 X[4448] - 4 X[6050], 3 X[4448] - 2 X[59590], 2 X[4705] - 3 X[48225], 3 X[4750] - X[21118], 2 X[4791] - 3 X[47837], 3 X[4809] - 2 X[21185], 2 X[4823] - 3 X[48569], 4 X[14838] - 3 X[47822], and many others

X(69315) lies on these lines: {512, 48321}, {513, 4560}, {514, 4784}, {522, 4367}, {523, 48144}, {649, 63812}, {650, 48265}, {661, 29170}, {663, 900}, {667, 8714}, {693, 16737}, {764, 29302}, {784, 1019}, {812, 3777}, {814, 2254}, {905, 4010}, {1491, 6002}, {1577, 47823}, {1734, 2787}, {2530, 24719}, {2533, 23880}, {3667, 48336}, {3669, 48279}, {3762, 50504}, {3801, 4025}, {3810, 4107}, {3835, 47893}, {3900, 4922}, {3904, 29284}, {3907, 50355}, {3960, 48273}, {4041, 29324}, {4083, 17496}, {4129, 47888}, {4142, 59748}, {4151, 4378}, {4369, 48392}, {4382, 48406}, {4391, 9508}, {4435, 22383}, {4448, 6050}, {4467, 29017}, {4490, 4913}, {4705, 29148}, {4750, 21118}, {4777, 17166}, {4791, 47837}, {4808, 29212}, {4809, 21185}, {4823, 48569}, {4874, 48264}, {4879, 48325}, {4905, 29070}, {4926, 48330}, {4962, 25569}, {5029, 48266}, {6005, 48288}, {6008, 48616}, {8631, 68885}, {8639, 64868}, {8678, 50341}, {14349, 29150}, {14838, 47822}, {16892, 29025}, {17069, 48400}, {17072, 48244}, {17494, 29198}, {20295, 48100}, {21051, 47828}, {21124, 29120}, {21146, 23882}, {21222, 29226}, {21301, 29152}, {21302, 29236}, {23738, 47932}, {23875, 50351}, {23877, 50342}, {24286, 66995}, {26732, 48396}, {28217, 48367}, {29033, 48075}, {29051, 50359}, {29078, 48278}, {29090, 48272}, {29106, 49278}, {29162, 50348}, {29178, 48066}, {29238, 46403}, {29276, 47687}, {29328, 48131}, {29344, 48018}, {29362, 48151}, {31251, 59737}, {31286, 47872}, {45671, 50507}, {47712, 48224}, {47775, 47957}, {47796, 48090}, {47825, 47967}, {47889, 48394}, {47911, 48002}, {47913, 48000}, {47959, 48176}, {48189, 52601}, {48227, 48403}, {48229, 59521}, {48235, 48395}, {48238, 48393}, {48248, 58140}, {48277, 54253}, {48284, 48351}, {48289, 48338}, {48291, 48343}, {48323, 50339}, {48328, 48339}, {48331, 53343}, {48333, 68968}, {48570, 54265}, {50352, 64934}, {50523, 64914}

X(69315) = midpoint of X(i) and X(j) for these {i,j}: {4041, 53536}, {17496, 50343}, {23738, 47932}, {48323, 50339}
X(69315) = reflection of X(i) in X(j) for these {i,j}: {2533, 50336}, {3762, 50504}, {3801, 4025}, {4010, 905}, {4142, 59748}, {4382, 48406}, {4391, 9508}, {4490, 4913}, {4879, 48325}, {20295, 48100}, {21301, 50335}, {24719, 2530}, {47911, 48002}, {47913, 48000}, {48264, 4874}, {48265, 650}, {48267, 14838}, {48273, 3960}, {48279, 3669}, {48291, 48343}, {48301, 4367}, {48338, 48289}, {48339, 48328}, {48351, 48284}, {48392, 4369}, {48400, 17069}, {53343, 48331}, {59590, 6050}
X(69315) = barycentric product X(i)*X(j) for these {i,j}: {274, 50544}, {514, 32853}
X(69315) = barycentric quotient X(i)/X(j) for these {i,j}: {32853, 190}, {50544, 37}
X(69315) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {905, 4010, 47841}, {4391, 9508, 47835}, {6050, 59590, 4448}, {14838, 48267, 47822}

X(69316) = ODD<1, 1, -1, -1, 1, 1> POINT

Barycentrics    (b - c)*(a^3 + a^2*b - a*b^2 + b^3 + a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2 + c^3) : :
X(69316) = 3 X[48235] - X[50352], X[1577] - 3 X[47809], X[3801] - 3 X[47837], X[4170] - 3 X[57066], X[4560] + 3 X[48208], X[47711] - 3 X[48208], X[4707] - 3 X[47836], 3 X[6546] - X[47970], 3 X[14432] - X[48337], X[21185] - 3 X[47766], 3 X[28602] - 2 X[65449], 3 X[31150] + X[47718], 5 X[31209] - X[47709], and many others

X(69316) lies on these lines: {2, 47712}, {10, 514}, {522, 1324}, {523, 8043}, {649, 48272}, {650, 29021}, {663, 30115}, {759, 2752}, {784, 48405}, {826, 9508}, {830, 50333}, {905, 29047}, {1019, 4088}, {1577, 47809}, {1734, 48300}, {2785, 4807}, {2977, 29142}, {3801, 47837}, {3835, 29158}, {3837, 29098}, {3887, 48299}, {3960, 29288}, {4025, 29358}, {4040, 5293}, {4041, 47682}, {4063, 48278}, {4129, 29118}, {4151, 8045}, {4170, 57066}, {4367, 4808}, {4379, 33138}, {4498, 49278}, {4522, 29013}, {4560, 47711}, {4707, 47836}, {4777, 6050}, {4905, 48094}, {4913, 23879}, {4978, 48408}, {6005, 48069}, {6332, 29350}, {6372, 48056}, {6546, 47970}, {7081, 47771}, {7265, 50343}, {8714, 17989}, {10196, 59726}, {11068, 53281}, {14349, 48106}, {14432, 48337}, {15309, 48047}, {16086, 21302}, {17166, 54335}, {17494, 47715}, {17734, 21118}, {18004, 29150}, {20517, 31286}, {21051, 29029}, {21124, 47726}, {21185, 47766}, {21260, 29025}, {23875, 50336}, {23876, 50501}, {24880, 55282}, {24931, 47701}, {28161, 59829}, {28591, 48068}, {28602, 29128}, {29017, 50504}, {29116, 50453}, {29144, 50507}, {29164, 68780}, {29184, 53571}, {29190, 48008}, {31150, 47718}, {31209, 47709}, {34868, 44408}, {36499, 47725}, {45671, 47710}, {47131, 48564}, {47652, 48556}, {47660, 48409}, {47679, 47825}, {47691, 47795}, {47695, 47818}, {47707, 48321}, {47708, 47794}, {47713, 47797}, {47716, 47796}, {47770, 59590}, {47807, 48403}, {47839, 48349}, {48086, 48101}, {48185, 48267}, {48236, 48410}, {48282, 60353}, {48287, 49682}, {48326, 48569}, {48395, 64934}, {49277, 50509}, {49279, 50355}, {49280, 50499}, {49283, 50449}, {59842, 68794}

X(69316) = midpoint of X(i) and X(j) for these {i,j}: {649, 48272}, {1019, 4088}, {1734, 48300}, {2530, 48103}, {2533, 50351}, {4041, 47682}, {4063, 48278}, {4367, 4808}, {4498, 49278}, {4560, 47711}, {4905, 48094}, {4978, 48408}, {7265, 50343}, {14349, 48106}, {17494, 47715}, {21124, 47726}, {28591, 48068}, {47660, 48409}, {47707, 48321}, {48086, 48101}, {49277, 50509}, {49279, 50355}, {49280, 50499}, {49283, 50449}
X(69316) = reflection of X(i) in X(j) for these {i,j}: {20517, 31286}, {21192, 9508}, {48003, 2977}
X(69316) = complement of X(47712)
X(69316) = X(i)-complementary conjugate of X(j) for these (i,j): {40398, 116}, {57420, 64523}
X(69316) = crosssum of X(6) and X(50486)
X(69316) = crossdifference of every pair of points on line {1030, 1914}
X(69316) = barycentric product X(514)*X(33166)
X(69316) = barycentric quotient X(33166)/X(190)
X(69316) = {X(4560),X(48208)}-harmonic conjugate of X(47711)

X(69317) = ODD<1, 1, -1, -1, 0, -1> POINT

Barycentrics    (b - c)*(-a^3 - a^2*b + a*b^2 - a^2*c + a*b*c + b^2*c + a*c^2 + b*c^2) : :
X(69317) = X[48288] + 2 X[50343], 4 X[650] - 3 X[48553], 2 X[48267] - 3 X[48553], 2 X[693] - 3 X[48569], 2 X[1577] - 3 X[47837], 4 X[9508] - 3 X[47837], 3 X[1635] - X[48264], 5 X[1698] - 4 X[59521], 4 X[2516] - 3 X[48561], 2 X[3835] - 3 X[47888], 2 X[4010] - 3 X[47839], 4 X[14838] - 3 X[47839], 2 X[4129] - 3 X[47827], and many others

X(69317) lies on these lines: {8, 29268}, {512, 4560}, {513, 48409}, {514, 4784}, {522, 667}, {523, 1019}, {525, 50351}, {649, 784}, {650, 48267}, {654, 56324}, {659, 8714}, {661, 29150}, {693, 48569}, {812, 2530}, {814, 1734}, {826, 4467}, {830, 50341}, {891, 17496}, {900, 4040}, {905, 48273}, {1491, 29013}, {1577, 9508}, {1635, 48264}, {1698, 59521}, {1924, 68885}, {2254, 29070}, {2516, 48561}, {2533, 64934}, {2787, 4041}, {3667, 48351}, {3733, 39547}, {3777, 29302}, {3801, 21192}, {3835, 47888}, {3907, 4730}, {3960, 48279}, {4010, 14838}, {4063, 63812}, {4083, 48321}, {4088, 29090}, {4129, 47827}, {4132, 39548}, {4151, 4367}, {4170, 45671}, {4369, 48393}, {4380, 48410}, {4382, 23815}, {4391, 50504}, {4458, 59748}, {4490, 29148}, {4705, 4913}, {4729, 29298}, {4750, 55282}, {4770, 29176}, {4774, 4807}, {4777, 47714}, {4791, 47835}, {4804, 52601}, {4808, 29037}, {4810, 47893}, {4823, 47823}, {4824, 15309}, {4843, 48290}, {4879, 68968}, {4905, 29362}, {4926, 5440}, {4976, 29142}, {6004, 50356}, {6008, 48092}, {6362, 68814}, {6372, 17494}, {8043, 50329}, {8632, 29106}, {9218, 65283}, {14349, 29328}, {16695, 64868}, {16892, 29098}, {17069, 48403}, {18155, 50544}, {20295, 48059}, {21124, 29029}, {21196, 29118}, {21260, 47828}, {21301, 29340}, {21302, 29182}, {21763, 21832}, {21836, 69104}, {23880, 50501}, {23882, 50336}, {24601, 27486}, {24719, 29270}, {26732, 48395}, {28221, 68816}, {29033, 48018}, {29066, 50355}, {29078, 48272}, {29170, 47959}, {29178, 48012}, {29186, 50359}, {29232, 50333}, {29238, 50335}, {29292, 47700}, {29354, 48408}, {30580, 65685}, {30795, 59714}, {31251, 47830}, {31286, 47875}, {31288, 47832}, {47694, 50512}, {47775, 47994}, {47795, 48090}, {47825, 48005}, {47932, 48151}, {47934, 48149}, {47947, 48002}, {47949, 48000}, {47997, 48176}, {48003, 48265}, {48080, 50507}, {48226, 59672}, {48244, 50337}, {48284, 48336}, {48289, 48337}, {48325, 48333}, {48330, 48339}, {48568, 54265}, {48578, 58143}

X(69317) = midpoint of X(i) and X(j) for these {i,j}: {4367, 50339}, {4380, 48410}, {4560, 50343}, {47932, 48151}, {47934, 48149}
X(69317) = reflection of X(i) in X(j) for these {i,j}: {1577, 9508}, {3801, 21192}, {4010, 14838}, {4382, 23815}, {4391, 50504}, {4458, 59748}, {4705, 4913}, {4774, 4807}, {4804, 52601}, {20295, 48059}, {24719, 48066}, {39547, 3733}, {47694, 50512}, {47947, 48002}, {47949, 48000}, {48080, 50507}, {48265, 48003}, {48267, 650}, {48273, 905}, {48279, 3960}, {48288, 4560}, {48291, 4367}, {48305, 667}, {48333, 48325}, {48336, 48284}, {48337, 48289}, {48339, 48330}, {48393, 4369}, {48403, 17069}, {50329, 8043}, {50352, 50336}
X(69317) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {749, 3448}, {30651, 21221}
X(69317) = crossdifference of every pair of points on line {2277, 4272}
X(69317) = barycentric product X(514)*X(32864)
X(69317) = barycentric quotient X(32864)/X(190)
X(69317) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {650, 48267, 48553}, {1577, 9508, 47837}, {4010, 14838, 47839}

X(69318) = ODD<1, 1, -1, -1, -1, 0> POINT

Barycentrics    (b - c)*(-a^3 - a^2*b + a*b^2 + b^3 - a^2*c + a*b*c + a*c^2 + c^3) : :
X(69318) = 4 X[41800] - X[57066], X[8] - 4 X[55285], X[4453] + 8 X[21199], X[44550] - 4 X[44551], 2 X[44550] + X[44553], 8 X[44551] + X[44553], X[656] + 2 X[21187], 2 X[656] + X[65099], 4 X[21187] - X[65099], X[693] - 4 X[21188], 4 X[905] - X[3904], X[1019] + 2 X[50453], 2 X[1577] + X[4467], X[1577] + 2 X[21192], and many others

X(69318) lies on these lines: {2, 525}, {8, 55285}, {244, 24234}, {512, 47797}, {514, 1635}, {523, 47836}, {648, 38340}, {656, 21187}, {690, 47839}, {693, 21188}, {826, 47809}, {905, 3904}, {918, 47793}, {1019, 50453}, {1577, 4467}, {1638, 3910}, {1734, 20517}, {2254, 4142}, {3309, 47798}, {3566, 47799}, {3616, 65685}, {3676, 4801}, {3776, 4498}, {3800, 48203}, {3801, 9508}, {3907, 30574}, {4025, 4391}, {4040, 62435}, {4041, 4458}, {4063, 47652}, {4083, 48227}, {4129, 44449}, {4151, 21181}, {4359, 17899}, {4367, 69011}, {4369, 21124}, {4560, 7178}, {4707, 14838}, {4750, 6002}, {4774, 59743}, {4789, 23879}, {4807, 47727}, {4850, 16754}, {4913, 55282}, {6004, 44433}, {6005, 48161}, {6332, 7658}, {7192, 48402}, {8045, 24924}, {8712, 47754}, {10015, 17496}, {11125, 64905}, {13246, 48150}, {14018, 59932}, {14419, 29094}, {14431, 29090}, {16451, 39201}, {16452, 22089}, {17896, 60494}, {21051, 50342}, {21052, 29037}, {21120, 21222}, {21185, 50356}, {21189, 25020}, {21196, 50457}, {21212, 48131}, {23678, 64857}, {23799, 41299}, {23875, 30565}, {23876, 47795}, {23877, 47828}, {23882, 27486}, {23883, 45324}, {24443, 27951}, {25380, 48278}, {25996, 57091}, {26732, 45669}, {28473, 53356}, {28478, 47757}, {28481, 48164}, {28493, 31147}, {28579, 58161}, {29017, 47823}, {29021, 48252}, {29142, 47824}, {29146, 48235}, {29200, 47822}, {29202, 48216}, {29208, 48224}, {29252, 48553}, {29280, 48185}, {29284, 47841}, {29288, 48241}, {29302, 47871}, {29312, 48569}, {30520, 48559}, {31286, 48300}, {31288, 49279}, {47676, 47965}, {47687, 50337}, {47691, 50501}, {47708, 50336}, {47709, 48069}, {47835, 62423}, {47918, 69292}, {47935, 69291}, {47976, 49297}, {48149, 59630}, {48403, 50343}, {48408, 50504}, {54392, 58329}

X(69318) = reflection of X(i) in X(j) for these {i,j}: {2, 41800}, {25020, 21189}, {30565, 47794}, {47796, 1638}, {47809, 47837}, {47840, 47799}, {47841, 48215}, {48252, 48573}, {48567, 48566}, {48570, 47758}, {57066, 2}, {57091, 25996}
X(69318) = cevapoint of X(14837) and X(21192)
X(69318) = crosspoint of X(75) and X(4573)
X(69318) = crosssum of X(31) and X(3709)
X(69318) = crossdifference of every pair of points on line {1495, 2177}
X(69318) = barycentric product X(i)*X(j) for these {i,j}: {514, 33066}, {693, 56288}, {34043, 35519}
X(69318) = barycentric quotient X(i)/X(j) for these {i,j}: {33066, 190}, {34043, 109}, {56288, 100}
X(69318) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {656, 21187, 65099}, {1577, 21192, 4467}, {1734, 20517, 47695}, {3676, 60492, 4801}, {4025, 14837, 4391}, {7178, 17069, 4560}

X(69319) = ODD<1, 0, 1, 1, -1, -1> POINT

Barycentrics    (b - c)*(-a^3 - a*b^2 + b^3 - a*b*c + b^2*c - a*c^2 + b*c^2 + c^3) : :

X(69319) lies on these lines: {2, 4809}, {30, 511}, {38, 2254}, {190, 692}, {659, 4088}, {661, 50340}, {667, 48272}, {903, 43097}, {1086, 21210}, {1215, 3716}, {1227, 62430}, {1491, 47886}, {1638, 48182}, {1639, 26275}, {2483, 57234}, {3251, 30605}, {3270, 24840}, {3700, 68803}, {3762, 4692}, {3801, 21301}, {3837, 4458}, {3904, 4922}, {4010, 47695}, {4025, 50335}, {4063, 4808}, {4064, 50353}, {4106, 47131}, {4120, 4800}, {4122, 47694}, {4142, 21051}, {4367, 48278}, {4370, 65899}, {4374, 48084}, {4378, 49278}, {4440, 21293}, {4448, 30565}, {4453, 31131}, {4467, 50341}, {4502, 48031}, {4522, 4874}, {4724, 48048}, {4750, 48244}, {4763, 28602}, {4775, 49277}, {4776, 48177}, {4782, 48062}, {4810, 53558}, {4817, 68133}, {4951, 47874}, {6161, 49276}, {6332, 48330}, {6545, 48167}, {6682, 25380}, {7265, 48305}, {9508, 50333}, {10196, 45314}, {14429, 28284}, {14432, 25569}, {16892, 50328}, {17165, 53343}, {17475, 23757}, {20295, 48349}, {20517, 21260}, {21146, 47687}, {21198, 28603}, {21222, 53533}, {24462, 53527}, {24623, 47690}, {24719, 47691}, {24813, 53291}, {24816, 53548}, {27486, 48225}, {30792, 44902}, {38324, 68333}, {39200, 53258}, {39714, 55244}, {44429, 48227}, {44435, 48224}, {45323, 47882}, {45340, 45668}, {45661, 48183}, {45674, 48229}, {45676, 47876}, {46403, 48326}, {47123, 48090}, {47700, 48103}, {47701, 47990}, {47702, 47944}, {47757, 48212}, {47759, 48158}, {47760, 48195}, {47761, 48200}, {47762, 48187}, {47763, 48254}, {47766, 48201}, {47771, 48188}, {47783, 48030}, {47785, 48213}, {47787, 48202}, {47790, 48189}, {47798, 47822}, {47800, 48197}, {47802, 48215}, {47803, 48199}, {47804, 48185}, {47805, 48171}, {47806, 48216}, {47808, 47823}, {47821, 48239}, {47824, 48169}, {47883, 48191}, {47887, 48184}, {47894, 48157}, {47925, 48585}, {47968, 48020}, {47971, 50359}, {47972, 48024}, {47982, 48621}, {47999, 48023}, {48006, 48028}, {48007, 48035}, {48014, 48040}, {48032, 48083}, {48047, 50347}, {48061, 48614}, {48094, 50358}, {48098, 49285}, {48105, 48604}, {48164, 48241}, {48203, 48552}, {48248, 69293}, {48324, 49279}, {48327, 49280}, {48396, 54265}, {50326, 53523}, {53314, 53553}, {53339, 53361}, {66284, 68882}

X(69319) = barycentric quotient X(47802)/X(45065)
X(69319) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {659, 4088, 48056}, {1639, 26275, 45666}, {4453, 31131, 36848}, {4776, 48223, 48177}, {4951, 48234, 47874}, {30565, 44433, 4448}, {47760, 48211, 48195}, {47761, 48200, 48217}, {47762, 48187, 48235}, {48039, 68780, 48030}, {48167, 58372, 6545}

X(69320) = ODD<1, 0, 1, 0, 0, 1> POINT

Barycentrics    (b - c)*(a^3 + a*b^2 + b^2*c + a*c^2 + b*c^2) : :
X(69320) = X[17166] - 3 X[47694], 2 X[21201] + X[47696], 2 X[650] - 3 X[47817], 3 X[47817] - X[48409], 4 X[48248] - X[48321], 2 X[905] - 3 X[47818], 2 X[1491] - 3 X[47794], 3 X[1491] - 4 X[65449], 9 X[47794] - 8 X[65449], 2 X[2254] - 3 X[48573], 2 X[2526] - 3 X[47816], 2 X[2530] - 3 X[47795], 4 X[4874] - 3 X[47795], X[4801] - 3 X[48237], and many others

X(69320) lies on these lines: {1, 514}, {2, 48066}, {513, 1577}, {522, 4063}, {649, 8714}, {650, 47817}, {659, 784}, {661, 59672}, {667, 48248}, {830, 4391}, {891, 48301}, {900, 4834}, {905, 47818}, {1491, 47794}, {2254, 48573}, {2526, 47816}, {2530, 4874}, {2533, 6004}, {2832, 4801}, {3309, 4761}, {3667, 4581}, {3700, 28481}, {3716, 14349}, {3762, 8678}, {3777, 48234}, {3800, 53523}, {3803, 23880}, {3810, 47682}, {3831, 50337}, {3835, 48086}, {3837, 47875}, {3840, 4379}, {3907, 48324}, {3960, 47820}, {4024, 29190}, {4083, 48305}, {4129, 48023}, {4151, 4498}, {4160, 4462}, {4367, 48251}, {4369, 4905}, {4401, 4560}, {4448, 50507}, {4705, 64914}, {4728, 48116}, {4776, 48052}, {4777, 47710}, {4778, 4960}, {4791, 21301}, {4802, 47713}, {4804, 29302}, {4823, 46403}, {4885, 48556}, {4893, 6685}, {4977, 48403}, {4978, 7662}, {5216, 7253}, {6005, 53343}, {6050, 45671}, {6133, 50345}, {6161, 29366}, {6545, 29668}, {6546, 29670}, {6590, 47715}, {7081, 47771}, {7265, 49286}, {8045, 28487}, {8689, 47683}, {14838, 47804}, {16892, 20517}, {20979, 55240}, {21188, 48015}, {21222, 48343}, {21260, 47872}, {23815, 47833}, {23875, 49275}, {23887, 48300}, {28840, 47942}, {29013, 48264}, {29021, 47660}, {29047, 47695}, {29051, 48111}, {29066, 48150}, {29070, 48392}, {29148, 50523}, {29158, 48101}, {29164, 47693}, {29186, 48032}, {29226, 48291}, {29344, 31291}, {29358, 49273}, {29362, 48393}, {30567, 47789}, {31149, 59521}, {31288, 47893}, {31290, 47987}, {32094, 53360}, {47131, 47717}, {47662, 47709}, {47666, 48004}, {47672, 47936}, {47724, 48072}, {47729, 48345}, {47775, 59297}, {47793, 48012}, {47813, 48151}, {47815, 47975}, {47821, 48054}, {47822, 48059}, {47824, 48075}, {47832, 48122}, {47836, 48018}, {47837, 50335}, {47839, 48100}, {47918, 48153}, {47940, 48613}, {47945, 47997}, {47965, 48407}, {48011, 50343}, {48027, 48551}, {48029, 50449}, {48030, 48553}, {48043, 48085}, {48050, 48596}, {48120, 68894}, {48144, 48578}, {48272, 69293}, {48288, 48331}, {48289, 58155}, {48295, 48334}, {48565, 50356}, {48566, 50336}, {50341, 50504}

X(69320) = midpoint of X(i) and X(j) for these {i,j}: {4391, 47697}, {47662, 47709}, {47672, 47936}, {47696, 47708}, {47918, 48153}, {47929, 48142}, {48032, 50457}, {48102, 55282}, {48392, 50358}
X(69320) = reflection of X(i) in X(j) for these {i,j}: {661, 59672}, {667, 48248}, {2530, 4874}, {3777, 52601}, {4040, 48063}, {4560, 4401}, {4905, 4369}, {4978, 7662}, {7265, 49286}, {14349, 3716}, {16892, 20517}, {21222, 48343}, {21301, 4791}, {31290, 47987}, {46403, 4823}, {47666, 48004}, {47708, 21201}, {47712, 21185}, {47715, 6590}, {47716, 47123}, {47717, 47131}, {47729, 48345}, {47940, 48613}, {47945, 47997}, {47969, 48623}, {47975, 48003}, {48015, 21188}, {48023, 4129}, {48085, 48043}, {48086, 3835}, {48272, 69293}, {48288, 48331}, {48298, 48294}, {48321, 667}, {48334, 48295}, {48339, 48305}, {48407, 47965}, {48409, 650}, {48410, 14838}, {48596, 48050}, {49278, 8045}, {50328, 21260}, {50341, 50504}, {50343, 48011}, {50345, 6133}, {50449, 48029}
X(69320) = anticomplement of X(48066)
X(69320) = crossdifference of every pair of points on line {672, 5069}
X(69320) = barycentric product X(514)*X(24552)
X(69320) = barycentric quotient X(24552)/X(190)
X(69320) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2530, 4874, 47795}, {3716, 14349, 47838}, {3777, 48234, 52601}, {4560, 47805, 4401}, {47804, 48410, 14838}, {47815, 47975, 48003}, {47817, 48409, 650}, {47872, 50328, 21260}

X(69321) = ODD<1, 0, 1, 0, 0, -1> POINT

Barycentrics    (b - c)*(-a^3 - a*b^2 + b^2*c - a*c^2 + b*c^2) : :
X(69321) = X[47724] + 2 X[48042], 2 X[649] - 3 X[48573], 3 X[48573] - 4 X[50337], 2 X[650] - 3 X[47816], 2 X[659] - 3 X[47794], 4 X[21260] - 3 X[47794], 2 X[667] - 3 X[47795], 4 X[3837] - 3 X[47795], 2 X[905] - 3 X[48556], 4 X[1125] - 3 X[8643], 2 X[1960] - 3 X[47841], 5 X[3616] - 4 X[65428], 2 X[3803] - 3 X[47818], 4 X[4885] - 3 X[47818], and many others

X(69321) lies on these lines: {1, 28470}, {2, 4401}, {4, 28591}, {10, 4498}, {226, 51652}, {388, 30719}, {512, 24719}, {513, 1577}, {514, 4088}, {522, 44444}, {649, 23791}, {650, 47816}, {659, 21260}, {661, 29186}, {667, 3837}, {693, 830}, {764, 29324}, {784, 50328}, {802, 24462}, {812, 1734}, {814, 2530}, {900, 48403}, {905, 48556}, {1019, 24720}, {1056, 28304}, {1125, 8643}, {1491, 29070}, {1960, 47841}, {2254, 29013}, {2526, 23882}, {2787, 3777}, {2820, 23819}, {2832, 4462}, {3309, 4077}, {3583, 3667}, {3616, 65428}, {3669, 28475}, {3716, 48111}, {3800, 23729}, {3803, 4885}, {3835, 4040}, {3907, 48335}, {3960, 47819}, {4010, 6004}, {4041, 29302}, {4057, 48186}, {4063, 17072}, {4129, 4724}, {4147, 21385}, {4151, 4382}, {4160, 4801}, {4162, 28585}, {4367, 23815}, {4378, 48406}, {4391, 47685}, {4490, 68894}, {4504, 28519}, {4560, 29033}, {4705, 29362}, {4728, 48150}, {4762, 48407}, {4775, 4992}, {4776, 48058}, {4777, 47713}, {4778, 47947}, {4782, 47837}, {4794, 47840}, {4802, 47710}, {4806, 48351}, {4823, 47694}, {4865, 23777}, {4905, 6002}, {4927, 34958}, {4943, 64068}, {4977, 48395}, {4978, 8678}, {4983, 29246}, {5216, 17217}, {5561, 23836}, {5711, 57238}, {6005, 20295}, {6050, 47802}, {7178, 28481}, {10436, 30183}, {14349, 29051}, {14838, 44429}, {15309, 48108}, {16892, 29062}, {17166, 48170}, {17494, 48012}, {17496, 29344}, {17738, 30187}, {21051, 31149}, {21124, 29190}, {21302, 29350}, {22160, 53284}, {23789, 48144}, {23877, 47680}, {25299, 48008}, {26148, 59522}, {27294, 27322}, {28533, 30725}, {28569, 43052}, {29021, 47687}, {29047, 47652}, {29066, 48131}, {29148, 48151}, {29150, 50359}, {29178, 48075}, {29188, 48123}, {29232, 50348}, {29236, 48137}, {29238, 50335}, {29244, 50351}, {29260, 47688}, {29270, 48018}, {29274, 48100}, {30724, 39545}, {30795, 31288}, {31147, 48367}, {31291, 47796}, {42325, 48080}, {44316, 48228}, {47651, 47706}, {47666, 48613}, {47672, 47905}, {47716, 48398}, {47729, 48348}, {47759, 48045}, {47812, 50523}, {47814, 48003}, {47821, 48065}, {47823, 50512}, {47824, 48064}, {47832, 59714}, {47835, 53571}, {47836, 48011}, {47839, 48331}, {47875, 48248}, {47918, 48115}, {47941, 48612}, {47969, 47997}, {47974, 48004}, {48020, 50457}, {48027, 50449}, {48029, 48551}, {48032, 59672}, {48049, 48081}, {48090, 48305}, {48184, 52601}, {48226, 65449}, {48273, 48339}, {48295, 48322}, {48393, 64914}, {48410, 64934}, {48568, 50515}, {48579, 50526}

X(69321) = midpoint of X(i) and X(j) for these {i,j}: {4391, 47685}, {21301, 46403}, {47651, 47706}, {47672, 47905}, {47686, 47707}, {47724, 48086}, {47912, 48119}, {47918, 48115}, {48020, 50457}
X(69321) = reflection of X(i) in X(j) for these {i,j}: {649, 50337}, {659, 21260}, {667, 3837}, {1019, 24720}, {3803, 4885}, {4040, 3835}, {4063, 17072}, {4170, 4106}, {4367, 23815}, {4378, 48406}, {4498, 10}, {4504, 65482}, {4560, 48066}, {4724, 4129}, {4775, 4992}, {4978, 48089}, {14349, 48050}, {17494, 48012}, {21385, 4147}, {47666, 48613}, {47694, 4823}, {47715, 49285}, {47716, 48398}, {47729, 48348}, {47941, 48612}, {47945, 48601}, {47969, 47997}, {47974, 48004}, {48032, 59672}, {48081, 48049}, {48086, 48042}, {48111, 3716}, {48144, 23789}, {48288, 48100}, {48305, 48090}, {48321, 2530}, {48322, 48295}, {48339, 48273}, {48351, 4806}, {48409, 2526}, {50343, 48018}, {50449, 48027}
X(69321) = anticomplement of X(4401)
X(69321) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {7241, 149}, {30636, 21293}, {65027, 4440}
X(69321) = crosspoint of X(7) and X(8050)
X(69321) = crosssum of X(i) and X(j) for these (i,j): {55, 4057}, {3953, 4905}
X(69321) = barycentric product X(i)*X(j) for these {i,j}: {513, 18044}, {514, 3891}, {3667, 27817}
X(69321) = barycentric quotient X(i)/X(j) for these {i,j}: {3891, 190}, {18044, 668}, {27817, 53647}
X(69321) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 50337, 48573}, {659, 21260, 47794}, {667, 3837, 47795}, {3803, 4885, 47818}, {3835, 4040, 47838}, {4367, 48167, 23815}, {4560, 48164, 48066}

X(69322) = ODD<1, 0, 1, -1, 1, 1> POINT

Barycentrics    (b - c)*(a^3 + a*b^2 + b^3 - a*b*c + b^2*c + a*c^2 + b*c^2 + c^3) : :

X(69322) lies on these lines: {30, 511}, {649, 48139}, {650, 48096}, {659, 16892}, {661, 47931}, {764, 47682}, {1491, 47973}, {1638, 48231}, {1639, 48178}, {2254, 48103}, {2509, 54249}, {2526, 48088}, {2605, 21126}, {3004, 48055}, {3250, 48031}, {3287, 21106}, {3766, 18160}, {3776, 4874}, {3777, 48300}, {3837, 69293}, {4010, 47652}, {4025, 4782}, {4088, 50328}, {4122, 46403}, {4369, 58375}, {4448, 47797}, {4453, 48250}, {4458, 48248}, {4468, 48007}, {4724, 47923}, {4784, 48101}, {4806, 69291}, {4809, 47805}, {4813, 48598}, {4893, 47877}, {5592, 48289}, {6161, 47727}, {6332, 48137}, {6545, 47833}, {6546, 47827}, {6590, 48098}, {7659, 48132}, {7662, 49299}, {8045, 48406}, {9508, 47890}, {10196, 47829}, {18004, 48050}, {21104, 54265}, {21115, 47813}, {21146, 47660}, {21183, 48221}, {21204, 48206}, {23729, 50326}, {24719, 25259}, {24720, 48405}, {30565, 48159}, {36848, 47809}, {44429, 48185}, {44435, 47822}, {45666, 47799}, {47651, 48080}, {47653, 47969}, {47662, 48108}, {47676, 47696}, {47688, 48349}, {47694, 48326}, {47716, 48305}, {47720, 48301}, {47754, 47803}, {47756, 48166}, {47757, 48197}, {47766, 48216}, {47767, 48245}, {47770, 47802}, {47771, 47823}, {47773, 47824}, {47798, 48224}, {47800, 48212}, {47804, 48227}, {47806, 48201}, {47808, 48188}, {47821, 48156}, {47828, 47885}, {47870, 48170}, {47874, 48184}, {47879, 48198}, {47882, 48214}, {47886, 48226}, {47887, 48234}, {47894, 48240}, {47901, 48112}, {47916, 47944}, {47919, 47961}, {47925, 47958}, {47930, 48626}, {47938, 48599}, {47943, 48082}, {47950, 48606}, {47951, 48026}, {47960, 48029}, {47983, 48036}, {47989, 48046}, {47995, 48028}, {48015, 48062}, {48023, 48117}, {48027, 48087}, {48032, 50340}, {48057, 68901}, {48061, 68780}, {48089, 48271}, {48090, 48398}, {48095, 48622}, {48105, 50358}, {48106, 48140}, {48126, 48397}, {48143, 48275}, {48152, 62430}, {48164, 48171}, {48174, 48177}, {48192, 48195}, {48217, 48219}, {48235, 48236}, {48251, 58372}, {48335, 49279}, {48408, 50341}, {48546, 48555}

X(69322) = barycentric quotient X(i)/X(j) for these {i,j}: {8422, 12197}, {39446, 28447}
X(69322) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {661, 47931, 47968}, {661, 47968, 47999}, {661, 48083, 48048}, {661, 48113, 48083}, {1491, 48094, 48056}, {1491, 48604, 48094}, {2254, 48130, 48103}, {2526, 48124, 48088}, {4468, 48007, 48030}, {16892, 48102, 659}, {25259, 47686, 24719}, {44429, 48557, 48185}, {46403, 49273, 4122}, {47652, 49275, 4010}, {47660, 49301, 21146}, {47688, 53343, 48349}, {47694, 49302, 48326}, {47754, 47803, 48215}, {47770, 47802, 48199}, {47804, 48422, 48227}, {47805, 48241, 4809}, {47821, 48156, 48552}, {47890, 50348, 9508}, {47916, 48021, 47944}, {47925, 48024, 47958}, {47925, 48078, 47990}, {47931, 48083, 47999}, {47931, 48113, 661}, {47958, 48024, 47990}, {47958, 48078, 48024}, {47968, 48083, 661}, {47968, 48113, 48048}, {47973, 48094, 1491}, {47973, 48604, 48056}, {47995, 48040, 48028}, {47999, 48048, 661}, {48015, 48062, 50335}, {48028, 48621, 47995}, {48030, 48614, 4468}, {48097, 50335, 48062}, {48140, 50359, 48106}, {48398, 49286, 48090}, {48622, 50336, 48095}

X(69323) = ODD<1, 0, 1, -1, 0, 1> POINT

Barycentrics    (b - c)*(a^3 + a*b^2 - a*b*c + b^2*c + a*c^2 + b*c^2) : :
X(69323) = 4 X[21201] - X[47688], 4 X[48063] - X[48298], 2 X[48102] + X[49303], 4 X[4391] - 3 X[30709], 2 X[21301] - 3 X[30709], 2 X[650] - 3 X[47815], 3 X[47815] - X[48410], 2 X[667] - 3 X[47805], X[17496] - 3 X[47805], 2 X[905] - 3 X[47804], 2 X[1491] - 3 X[47793], 2 X[2254] - 3 X[47836], 2 X[2526] - 3 X[47814], 4 X[20317] - 3 X[47814],and many others

X(69323) lies on these lines: {1, 514}, {2, 2530}, {350, 18081}, {512, 53343}, {513, 2517}, {522, 4498}, {523, 4491}, {525, 49275}, {650, 47815}, {659, 4560}, {667, 17496}, {764, 52601}, {784, 8640}, {812, 48264}, {814, 50358}, {826, 14318}, {830, 3762}, {832, 20293}, {891, 48305}, {905, 47804}, {1016, 53360}, {1019, 68896}, {1491, 26030}, {1577, 46403}, {2254, 47836}, {2526, 20317}, {2787, 31291}, {2826, 48250}, {2832, 4978}, {3667, 50509}, {3669, 47820}, {3716, 47840}, {3766, 6372}, {3777, 4874}, {3810, 48300}, {3835, 48122}, {3837, 47872}, {3904, 48299}, {3907, 48150}, {3960, 47818}, {4039, 29545}, {4063, 8714}, {4129, 48086}, {4142, 16892}, {4151, 21385}, {4367, 21222}, {4369, 48151}, {4379, 30957}, {4401, 48321}, {4462, 8678}, {4474, 28470}, {4477, 6362}, {4490, 64914}, {4761, 42325}, {4776, 48092}, {4777, 47706}, {4800, 4992}, {4801, 7662}, {4802, 47709}, {4823, 48170}, {4833, 4977}, {4885, 47819}, {4905, 47824}, {6004, 21302}, {6133, 48243}, {6161, 29298}, {6371, 7253}, {6590, 47719}, {7770, 18110}, {8643, 48325}, {14349, 47821}, {14837, 48015}, {14838, 47817}, {18344, 66512}, {18788, 28487}, {20295, 48267}, {20517, 48241}, {21051, 50328}, {21260, 48164}, {23738, 47813}, {23765, 48234}, {23815, 26985}, {23877, 48094}, {24601, 47763}, {26144, 48350}, {26824, 48393}, {26853, 29150}, {27040, 57048}, {27115, 47888}, {28569, 49280}, {28840, 47906}, {29021, 47693}, {29051, 48032}, {29066, 48111}, {29118, 48101}, {29142, 47660}, {29174, 48140}, {29186, 47977}, {29226, 48301}, {29288, 47695}, {29362, 48392}, {30061, 50505}, {30804, 43067}, {31209, 48561}, {31242, 47779}, {31290, 47949}, {43930, 52621}, {47127, 48033}, {47652, 48403}, {47666, 47966}, {47687, 48395}, {47722, 48068}, {47729, 48329}, {47774, 47994}, {47794, 48066}, {47822, 48100}, {47825, 48003}, {47833, 48406}, {47835, 50335}, {47841, 48137}, {47845, 53314}, {47936, 50457}, {47940, 47956}, {47945, 47959}, {47965, 47975}, {48004, 50449}, {48012, 48157}, {48043, 48121}, {48050, 48116}, {48052, 48551}, {48059, 48553}, {48075, 48573}, {48080, 59590}, {48171, 48272}, {48172, 48273}, {48242, 50504}, {48251, 48323}, {48341, 48578}, {48565, 50336}, {50356, 50501}, {50556, 53357}

X(69323) = midpoint of X(i) and X(j) for these {i,j}: {4462, 47697}, {21118, 48102}, {47936, 50457}
X(69323) = reflection of X(i) in X(j) for these {i,j}: {663, 48063}, {764, 52601}, {2526, 20317}, {3777, 4874}, {3904, 48299}, {4367, 48248}, {4560, 659}, {4801, 7662}, {14349, 59672}, {16892, 4142}, {17166, 47694}, {17496, 667}, {20295, 48267}, {21222, 4367}, {21301, 4391}, {26824, 48393}, {31290, 47949}, {44444, 4036}, {46403, 1577}, {47652, 48403}, {47666, 47966}, {47687, 48395}, {47688, 47712}, {47691, 21185}, {47712, 21201}, {47719, 6590}, {47720, 47123}, {47729, 48329}, {47940, 47956}, {47945, 47959}, {47969, 47970}, {47975, 47965}, {48015, 14837}, {48080, 59590}, {48086, 4129}, {48116, 48050}, {48121, 48043}, {48122, 3835}, {48131, 3716}, {48151, 4369}, {48278, 69293}, {48298, 663}, {48304, 48301}, {48321, 4401}, {48409, 48003}, {48410, 650}, {49303, 21118}, {50328, 21051}, {50343, 4063}, {50356, 50501}, {50449, 48004}
X(69323) = anticomplement of X(2530)
X(69323) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {42, 39346}, {82, 149}, {83, 150}, {100, 21289}, {101, 2896}, {190, 1369}, {251, 4440}, {689, 17138}, {692, 21217}, {827, 1}, {3112, 21293}, {4577, 17135}, {4593, 17137}, {4599, 75}, {4628, 2}, {4630, 17148}, {8750, 8878}, {18082, 3448}, {18098, 21221}, {32739, 52637}, {34072, 17147}, {36081, 4645}, {42396, 17220}, {46288, 21224}, {46289, 9263}, {52395, 25049}, {52936, 17142}, {53657, 18656}, {56186, 21294}, {56245, 37781}, {58956, 17302}, {65307, 17134}
X(69323) = X(27040)-Dao conjugate of X(21272)
X(69323) = crosspoint of X(i) and X(j) for these (i,j): {190, 32010}, {668, 10159}, {4577, 14534}, {7192, 56323}
X(69323) = crosssum of X(i) and X(j) for these (i,j): {649, 20964}, {667, 5007}, {2092, 3005}, {4557, 23845}
X(69323) = crossdifference of every pair of points on line {672, 2300}
X(69323) = barycentric product X(i)*X(j) for these {i,j}: {1, 18071}, {514, 32942}, {693, 69210}, {3112, 57048}, {7192, 27040}
X(69323) = barycentric quotient X(i)/X(j) for these {i,j}: {18071, 75}, {27040, 3952}, {32942, 190}, {57048, 38}, {69210, 100}
X(69323) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2526, 20317, 47814}, {3716, 48131, 47840}, {3777, 4874, 47796}, {4391, 21301, 30709}, {14349, 59672, 47821}, {17496, 47805, 667}, {23815, 47875, 26985}, {47815, 48410, 650}, {48003, 48409, 47825}

X(69324) = ODD<1, 0, 1, -1, 0, 0> POINT

Barycentrics    a*(b - c)*(a^2 + b^2 - b*c + c^2) : :

X(69324) lies on these lines: {1, 48150}, {2, 47817}, {30, 511}, {58, 1019}, {649, 4905}, {650, 48066}, {659, 2530}, {661, 16546}, {663, 48111}, {667, 3777}, {764, 4367}, {905, 4401}, {1046, 47935}, {1111, 18101}, {1491, 48003}, {1577, 46403}, {1734, 4498}, {2254, 4063}, {2526, 47965}, {3454, 4129}, {3600, 60482}, {3669, 3803}, {3762, 21301}, {3776, 20517}, {3835, 59672}, {4017, 57155}, {4040, 48032}, {4041, 21385}, {4086, 44444}, {4106, 59590}, {4170, 53343}, {4369, 23789}, {4378, 23765}, {4391, 47685}, {4448, 47839}, {4449, 48324}, {4482, 4499}, {4491, 48350}, {4705, 50328}, {4724, 14349}, {4794, 48136}, {4801, 47697}, {4813, 47942}, {4823, 48089}, {4834, 50359}, {4874, 23815}, {4879, 6161}, {4960, 48148}, {4978, 47694}, {6332, 48068}, {6615, 40091}, {6693, 23814}, {7178, 35650}, {7265, 49275}, {14422, 58149}, {17494, 48409}, {21108, 54244}, {21118, 47680}, {21125, 47652}, {21185, 48398}, {21192, 50348}, {21201, 48403}, {21222, 31291}, {21389, 48033}, {21763, 46387}, {23599, 43930}, {23738, 48320}, {24719, 48267}, {31647, 43921}, {36848, 47837}, {44429, 47794}, {45324, 47872}, {47651, 47709}, {47660, 47715}, {47662, 47718}, {47686, 47708}, {47687, 47711}, {47688, 47713}, {47693, 47714}, {47695, 47716}, {47696, 47719}, {47726, 48626}, {47793, 47816}, {47795, 47804}, {47796, 47805}, {47802, 48196}, {47803, 48218}, {47824, 48566}, {47875, 48184}, {47888, 48226}, {47889, 48251}, {47906, 47947}, {47912, 48586}, {47918, 47948}, {47929, 47959}, {47949, 48612}, {47956, 48601}, {47966, 47997}, {47969, 50449}, {47987, 48026}, {48008, 50505}, {48011, 48075}, {48018, 50501}, {48021, 48085}, {48024, 48051}, {48029, 48054}, {48045, 48091}, {48065, 48099}, {48072, 48295}, {48081, 48121}, {48094, 48272}, {48100, 50507}, {48102, 48278}, {48105, 48300}, {48115, 50457}, {48123, 48351}, {48137, 48331}, {48248, 48406}, {48279, 48305}, {48282, 48322}, {48287, 48327}, {48294, 48329}, {48343, 50517}, {48565, 48573}, {48591, 48602}, {50335, 50504}, {50454, 68836}, {50490, 69291}

X(69324) = crossdifference of every pair of points on line {6, 3930}
X(69324) = barycentric quotient X(i)/X(j) for these {i,j}: {29627, 55503}, {51912, 61080}
X(69324) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 48150, 48345}, {659, 2530, 14838}, {661, 47936, 47970}, {661, 47970, 48004}, {661, 48086, 48052}, {661, 48116, 48086}, {663, 48335, 48348}, {667, 3777, 3960}, {2526, 47965, 48012}, {3777, 50358, 667}, {4724, 14349, 48058}, {4724, 48122, 14349}, {14349, 47977, 4724}, {23738, 50523, 48320}, {44429, 47815, 47794}, {47793, 48164, 47816}, {47796, 47805, 47818}, {47802, 48561, 48196}, {47804, 47819, 47795}, {47817, 48556, 2}, {47918, 48020, 47948}, {47929, 48023, 47959}, {47929, 48596, 48613}, {47936, 48086, 48004}, {47936, 48116, 661}, {47959, 48023, 48613}, {47959, 48596, 48023}, {47966, 48027, 47997}, {47970, 48086, 661}, {47970, 48116, 48052}, {47977, 48122, 48058}, {47997, 48603, 48027}, {48004, 48052, 661}, {48011, 48075, 50336}, {48029, 48092, 48054}, {48032, 48131, 4040}, {48054, 48623, 48029}, {48111, 48335, 663}, {48150, 48334, 1}, {48248, 48406, 52601}, {48327, 48346, 48287}, {48329, 48332, 48294}

X(69325) = ODD<1, 0, 0, 1, -1, -1> POINT

Barycentrics    (b - c)*(-a^3 + b^3 - a*b*c + b^2*c + b*c^2 + c^3) : :
X(69325) = 3 X[1] - 2 X[65685], X[47727] + 2 X[50342], 4 X[1125] - 3 X[57066], 5 X[1698] - 6 X[41800], 3 X[1734] - 2 X[44448], 3 X[4025] - X[44448], 3 X[3679] - 4 X[55285], 2 X[4129] - 3 X[47797], 3 X[4453] - 2 X[50337], 2 X[4522] - 3 X[47795], 2 X[4823] - 3 X[47887], 4 X[13246] - 3 X[47817], 2 X[18004] - 3 X[47839], 4 X[21212] - 3 X[47816], and many others

X(69325) lies on these lines: {1, 525}, {46, 57121}, {512, 47727}, {514, 50017}, {522, 4905}, {523, 1019}, {596, 60574}, {649, 29047}, {659, 29354}, {663, 23875}, {667, 62423}, {690, 4879}, {693, 29062}, {812, 47716}, {814, 47680}, {826, 4367}, {830, 16892}, {905, 48272}, {918, 4040}, {1125, 57066}, {1577, 4458}, {1698, 41800}, {1734, 4025}, {2533, 29110}, {2786, 4170}, {2787, 3801}, {3004, 47948}, {3566, 48337}, {3669, 49278}, {3679, 55285}, {3700, 34958}, {3762, 4142}, {3800, 4897}, {3803, 30520}, {3906, 48328}, {3907, 4707}, {3910, 48282}, {3960, 48278}, {4010, 29090}, {4041, 21192}, {4063, 29288}, {4086, 21187}, {4088, 14838}, {4122, 29292}, {4129, 47797}, {4151, 4467}, {4160, 21124}, {4369, 47711}, {4378, 29017}, {4391, 20517}, {4401, 48094}, {4449, 23876}, {4453, 50337}, {4522, 47795}, {4775, 29200}, {4784, 7927}, {4801, 29190}, {4802, 50515}, {4808, 9508}, {4810, 29266}, {4823, 47887}, {4834, 29208}, {4922, 29094}, {4962, 28591}, {5307, 59932}, {6002, 47712}, {6005, 47971}, {6050, 48088}, {6372, 50340}, {8714, 47695}, {13246, 47817}, {15309, 47701}, {17166, 23879}, {17496, 23887}, {18004, 47839}, {21146, 29086}, {21189, 23874}, {21196, 48407}, {21212, 47816}, {21260, 48227}, {21301, 48241}, {21302, 62435}, {23770, 29232}, {23785, 24462}, {23789, 47687}, {23877, 48321}, {23880, 49300}, {28846, 48081}, {29013, 47691}, {29021, 48144}, {29058, 58372}, {29070, 48326}, {29074, 47723}, {29078, 48273}, {29106, 48279}, {29118, 47713}, {29132, 47709}, {29142, 48320}, {29148, 47708}, {29150, 48349}, {29158, 47692}, {29162, 47725}, {29186, 47676}, {29196, 47690}, {29202, 48344}, {29220, 47728}, {29252, 48336}, {29260, 48064}, {29278, 47724}, {29280, 48330}, {29284, 48333}, {29294, 48295}, {29302, 47720}, {29304, 47729}, {29312, 48323}, {29318, 48343}, {29358, 48300}, {31251, 48215}, {31288, 48185}, {39201, 39578}, {39545, 47726}, {47678, 49292}, {47689, 48570}, {47702, 48149}, {47706, 47762}, {47710, 48568}, {47798, 59672}, {47818, 69293}, {47837, 69011}, {47838, 48270}, {47886, 48012}, {47924, 50526}, {47930, 48150}, {47942, 48006}, {47947, 47998}, {47959, 68780}, {47970, 50347}, {47983, 48584}, {48007, 48586}, {48045, 48076}, {48058, 48082}, {48065, 48078}, {48066, 48077}, {48103, 50512}, {48118, 58140}, {48136, 49277}, {55282, 64934}

X(69325) = midpoint of X(i) and X(j) for these {i,j}: {47702, 48149}, {47924, 50526}, {47930, 48150}
X(69325) = reflection of X(i) in X(j) for these {i,j}: {1577, 4458}, {1734, 4025}, {3700, 34958}, {3762, 4142}, {4041, 21192}, {4086, 21187}, {4088, 14838}, {4122, 52601}, {4391, 20517}, {4808, 9508}, {21302, 62435}, {24462, 23785}, {47678, 49292}, {47682, 4367}, {47687, 23789}, {47711, 4369}, {47723, 50352}, {47942, 48006}, {47947, 47998}, {47948, 3004}, {47959, 68780}, {47970, 50347}, {48076, 48045}, {48077, 48066}, {48078, 48065}, {48082, 48058}, {48088, 6050}, {48094, 4401}, {48103, 50512}, {48106, 48064}, {48272, 905}, {48278, 3960}, {48407, 21196}, {48584, 47983}, {48586, 48007}, {49276, 663}, {49277, 48136}, {49278, 3669}, {49279, 48330}
X(69325) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {163, 17732}, {14377, 21294}, {31616, 65161}, {43190, 21287}
X(69325) = crosspoint of X(75) and X(6742)
X(69325) = crosssum of X(31) and X(2605)
X(69325) = crossdifference of every pair of points on line {4272, 39690}
X(69325) = barycentric product X(514)*X(33078)
X(69325) = barycentric quotient X(33078)/X(190)

X(69326) = ODD<1, 0, 0, 0, -1, -1> POINT

Barycentrics    (b - c)*(-a^3 + b^3 + b^2*c + b*c^2 + c^3) : :
X(69326) = X[47971] + 2 X[50340], X[47972] + 2 X[50342], 2 X[693] - 3 X[47887], 4 X[3676] - 3 X[47812], 3 X[4453] - 2 X[24720], 3 X[4453] - X[47687], 4 X[4458] - 3 X[47887], 3 X[47812] - 2 X[49285], 4 X[676] - 3 X[47832], 2 X[3700] - 3 X[47832], X[24719] - 3 X[48224], X[48078] - 4 X[50347], 2 X[1491] - 3 X[47886], 3 X[47886] - X[48077], and many others

X(69326) lies on these lines: {1, 23876}, {2, 4522}, {65, 3900}, {513, 16892}, {514, 50017}, {521, 53554}, {522, 693}, {523, 649}, {525, 663}, {596, 8714}, {650, 4088}, {656, 62418}, {659, 48094}, {661, 68780}, {667, 826}, {676, 3700}, {690, 4775}, {812, 47691}, {814, 3801}, {824, 4817}, {900, 24719}, {905, 37592}, {918, 4724}, {1019, 29021}, {1491, 47886}, {1499, 58166}, {1577, 20517}, {1635, 47700}, {1734, 3670}, {1936, 32845}, {1960, 3906}, {2487, 48232}, {2517, 21187}, {2522, 6129}, {2533, 29074}, {2785, 47729}, {2786, 48080}, {3004, 48023}, {3239, 47800}, {3309, 62436}, {3566, 48338}, {3667, 48015}, {3716, 25259}, {3762, 29212}, {3776, 46403}, {3798, 28161}, {3800, 50509}, {3810, 17496}, {3835, 47797}, {3837, 48227}, {3904, 48325}, {3910, 4449}, {3960, 49278}, {4010, 29078}, {4024, 7662}, {4040, 23875}, {4063, 29047}, {4064, 46380}, {4120, 48211}, {4122, 4809}, {4142, 4391}, {4170, 29216}, {4367, 29017}, {4369, 47690}, {4378, 29312}, {4379, 48396}, {4380, 47692}, {4382, 23770}, {4401, 29358}, {4468, 47811}, {4474, 10015}, {4498, 29288}, {4500, 47834}, {4560, 23877}, {4707, 29066}, {4750, 4777}, {4761, 29192}, {4762, 47704}, {4778, 47931}, {4782, 29204}, {4784, 29144}, {4786, 28169}, {4794, 49276}, {4802, 48101}, {4806, 48177}, {4808, 50504}, {4813, 47998}, {4818, 47894}, {4824, 47878}, {4830, 47663}, {4834, 7927}, {4841, 47909}, {4879, 29284}, {4893, 48047}, {4913, 27486}, {4932, 49283}, {4977, 47923}, {4978, 29190}, {4979, 47702}, {6002, 47708}, {6545, 48089}, {6546, 48088}, {6590, 47813}, {7178, 29278}, {7265, 29294}, {7658, 47806}, {7950, 50512}, {8045, 47820}, {8643, 48299}, {8678, 21124}, {12073, 58173}, {13246, 47804}, {14321, 48179}, {14432, 49280}, {14837, 21052}, {14838, 48272}, {17069, 47828}, {17420, 23874}, {18004, 47822}, {20295, 48203}, {21104, 48119}, {21115, 48115}, {21116, 48126}, {21118, 23880}, {21185, 48264}, {21189, 21193}, {21196, 47975}, {21212, 44429}, {23882, 55282}, {23887, 48321}, {24382, 35518}, {25380, 47808}, {27013, 48208}, {28147, 47661}, {28175, 48138}, {28183, 48238}, {28468, 48298}, {28481, 48122}, {28840, 47699}, {28846, 48006}, {28851, 47969}, {28855, 47941}, {28863, 47696}, {28867, 48158}, {28878, 47904}, {28882, 47688}, {28898, 49483}, {28906, 48037}, {29013, 47712}, {29033, 47680}, {29086, 50352}, {29090, 48267}, {29106, 48273}, {29118, 47709}, {29142, 48144}, {29158, 47713}, {29164, 48064}, {29194, 52601}, {29196, 47711}, {29200, 48336}, {29202, 48330}, {29232, 48403}, {29252, 48351}, {29256, 48328}, {29260, 48011}, {29280, 48331}, {29302, 47716}, {29318, 47682}, {29328, 48349}, {29350, 47727}, {29362, 48326}, {30519, 44433}, {30520, 48102}, {30795, 48215}, {30835, 47799}, {31207, 47807}, {31286, 47809}, {32478, 58165}, {43067, 47703}, {44435, 48050}, {44449, 48043}, {45674, 48187}, {45745, 47934}, {46381, 68775}, {47131, 53558}, {47656, 49292}, {47665, 48237}, {47671, 48134}, {47673, 48153}, {47677, 47697}, {47685, 48422}, {47689, 47762}, {47698, 48000}, {47705, 47932}, {47706, 48565}, {47710, 48566}, {47714, 48568}, {47718, 48570}, {47781, 47992}, {47782, 48010}, {47805, 49273}, {47810, 48039}, {47821, 48270}, {47823, 69011}, {47826, 48046}, {47890, 48118}, {47912, 48402}, {47930, 48032}, {47945, 48404}, {47983, 48019}, {48007, 48020}, {48008, 48408}, {48024, 48076}, {48029, 48082}, {48040, 48112}, {48042, 48159}, {48055, 48117}, {48056, 48226}, {48061, 48113}, {48105, 50358}, {48108, 69292}, {48114, 49295}, {48148, 49296}, {48170, 48415}, {48239, 53343}, {48286, 48339}, {48302, 53563}, {48388, 53284}, {49300, 64934}, {56125, 66284}, {58161, 59549}, {64868, 65703}

X(69326) = midpoint of X(i) and X(j) for these {i,j}: {4380, 47692}, {4467, 47695}, {4979, 47702}, {47673, 48153}, {47677, 47697}, {47705, 47932}, {47930, 48032}, {47971, 47972}, {50340, 50342}
X(69326) = reflection of X(i) in X(j) for these {i,j}: {661, 68780}, {693, 4458}, {1577, 20517}, {1734, 21192}, {2254, 4025}, {2517, 21187}, {3700, 676}, {3904, 48325}, {4024, 7662}, {4064, 68772}, {4088, 650}, {4120, 48211}, {4122, 4874}, {4382, 23770}, {4391, 4142}, {4474, 10015}, {4724, 50347}, {4804, 47123}, {4808, 50504}, {4813, 47998}, {23731, 47961}, {25259, 3716}, {44449, 48043}, {46403, 3776}, {47656, 49292}, {47663, 4830}, {47671, 48134}, {47687, 24720}, {47690, 4369}, {47698, 48000}, {47700, 48062}, {47703, 43067}, {47874, 4809}, {47909, 4841}, {47912, 48402}, {47934, 45745}, {47937, 47944}, {47938, 47701}, {47943, 47960}, {47945, 48404}, {47971, 50342}, {47972, 50340}, {47973, 16892}, {47975, 21196}, {48019, 47983}, {48020, 48007}, {48021, 48006}, {48023, 3004}, {48069, 3798}, {48076, 48024}, {48077, 1491}, {48078, 4724}, {48082, 48029}, {48094, 659}, {48103, 4782}, {48105, 50358}, {48106, 649}, {48108, 69292}, {48112, 48040}, {48113, 48061}, {48114, 49295}, {48117, 48055}, {48118, 47890}, {48119, 21104}, {48146, 48060}, {48148, 49296}, {48187, 45674}, {48264, 21185}, {48266, 4010}, {48272, 14838}, {48278, 905}, {48300, 667}, {48339, 48286}, {48408, 48008}, {48585, 47968}, {49275, 48063}, {49276, 4794}, {49278, 3960}, {49279, 1960}, {49283, 4932}, {49285, 3676}, {50333, 17069}, {53558, 47131}, {69293, 13246}
X(69326) = anticomplement of X(4522)
X(69326) = isotomic conjugate of the isogonal conjugate of X(50459)
X(69326) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {604, 39345}, {825, 329}, {985, 33650}, {1492, 3436}, {4586, 21286}, {34069, 144}, {40746, 37781}, {41280, 39347}
X(69326) = X(100)-isoconjugate of X(3415)
X(69326) = X(i)-Dao conjugate of X(j) for these (i,j): {8054, 3415}, {55061, 1}
X(69326) = crosspoint of X(75) and X(4586)
X(69326) = crosssum of X(i) and X(j) for these (i,j): {31, 3250}, {663, 21764}, {21407, 62415}
X(69326) = crossdifference of every pair of points on line {41, 386}
X(69326) = barycentric product X(i)*X(j) for these {i,j}: {76, 50459}, {514, 3416}, {523, 24632}, {693, 5282}, {1892, 6332}, {3261, 37586}, {4586, 55061}, {16892, 26270}
X(69326) = barycentric quotient X(i)/X(j) for these {i,j}: {649, 3415}, {1892, 653}, {3416, 190}, {5282, 100}, {24632, 99}, {37586, 101}, {50459, 6}, {55061, 824}
X(69326) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {676, 3700, 47832}, {693, 4458, 47887}, {1635, 47700, 48062}, {3676, 49285, 47812}, {4122, 4809, 4874}, {4122, 4874, 47874}, {4453, 47687, 24720}, {13246, 69293, 47804}, {17069, 50333, 47828}, {25259, 47798, 3716}, {44433, 49275, 48063}, {44449, 48161, 48043}, {46403, 48241, 3776}, {47886, 48077, 1491}, {48117, 48572, 48055}

X(69327) = ODD<1, 0, 0, -1, 1, 1> POINT

Barycentrics    (b - c)*(a^3 + b^3 - a*b*c + b^2*c + b*c^2 + c^3) : :
X(69327) = 2 X[48103] + X[49276], 2 X[4129] - 3 X[30565], X[4391] - 3 X[48557], 2 X[4823] - 3 X[47874], X[47652] - 3 X[57066], X[47680] - 4 X[69293], X[47682] + 2 X[48094], X[47727] + 2 X[48118], 3 X[1635] - 2 X[21192], 4 X[2490] - 3 X[41800], 2 X[3776] - 3 X[47795], 2 X[48097] + X[49279], 2 X[4142] - 3 X[47817], 2 X[4458] - 3 X[47818], and many others

X(69327) lies on these lines: {1, 29288}, {512, 48103}, {513, 48272}, {514, 661}, {522, 48111}, {523, 4040}, {525, 4063}, {649, 23875}, {659, 826}, {663, 29047}, {667, 62423}, {812, 7265}, {824, 68880}, {830, 4088}, {905, 30520}, {918, 1019}, {1635, 21192}, {1734, 48062}, {2259, 56320}, {2490, 41800}, {2533, 29102}, {3669, 48124}, {3716, 47712}, {3776, 47795}, {3777, 48604}, {3800, 48352}, {3801, 29224}, {3803, 64856}, {3910, 21385}, {4010, 29098}, {4083, 48097}, {4122, 29070}, {4142, 47817}, {4151, 48408}, {4367, 29354}, {4380, 29216}, {4401, 29358}, {4458, 47818}, {4498, 23876}, {4560, 49273}, {4705, 48056}, {4724, 29021}, {4775, 29208}, {4778, 48596}, {4782, 29280}, {4784, 29252}, {4794, 29260}, {4802, 48099}, {4822, 48146}, {4834, 29200}, {4960, 48276}, {4977, 48086}, {6005, 48106}, {6372, 48083}, {6546, 21124}, {7927, 48336}, {7950, 50340}, {8678, 48088}, {8712, 49280}, {8714, 49275}, {14344, 68259}, {14432, 48348}, {14838, 16892}, {15309, 48082}, {17494, 23879}, {18077, 21612}, {20517, 47804}, {20908, 55180}, {21188, 47766}, {21260, 48185}, {21301, 48171}, {23731, 48051}, {23789, 49301}, {23882, 48271}, {23885, 50456}, {25259, 29013}, {28195, 48092}, {28846, 48110}, {29025, 48267}, {29029, 48265}, {29051, 47711}, {29066, 47707}, {29142, 47726}, {29144, 48351}, {29158, 48080}, {29160, 47708}, {29164, 47972}, {29186, 47690}, {29192, 47706}, {29198, 48614}, {29204, 48331}, {29270, 48266}, {29302, 47663}, {31251, 48199}, {31288, 48227}, {43050, 57243}, {47679, 48000}, {47688, 47840}, {47700, 48150}, {47701, 48058}, {47718, 47974}, {47720, 48295}, {47724, 48395}, {47725, 48403}, {47793, 50453}, {47796, 49302}, {47809, 50337}, {47836, 62435}, {47885, 50504}, {47937, 48602}, {47938, 48045}, {47942, 48040}, {47943, 48052}, {47944, 48053}, {47947, 48046}, {47948, 48047}, {47949, 48048}, {47968, 48059}, {47971, 48064}, {47973, 48066}, {47976, 48060}, {47977, 48061}, {48038, 48584}, {48039, 48586}, {48095, 49277}, {48096, 49278}, {48102, 48278}, {48112, 48149}, {48113, 48151}, {48116, 48626}, {48117, 48144}, {48121, 48138}, {48122, 48139}, {48123, 48140}, {48128, 48132}, {48136, 48615}, {48145, 48597}, {48279, 49290}, {48282, 48290}, {48326, 52601}, {48405, 50352}, {48568, 69292}, {48569, 58375}, {48585, 48603}, {48616, 48622}, {50342, 50512}, {50351, 63812}, {50541, 68836}

X(69327) = midpoint of X(i) and X(j) for these {i,j}: {663, 48118}, {3669, 48124}, {3777, 48604}, {4462, 47684}, {4560, 49273}, {4822, 48146}, {47700, 48150}, {47718, 47974}, {47726, 47970}, {48094, 48300}, {48102, 48278}, {48112, 48149}, {48113, 48151}, {48116, 48626}, {48117, 48144}, {48121, 48138}, {48122, 48139}, {48123, 48140}, {48128, 48132}, {48130, 48131}, {48136, 48615}, {48145, 48597}, {48616, 48622}
X(69327) = reflection of X(i) in X(j) for these {i,j}: {1, 48299}, {1577, 69293}, {1734, 48062}, {4063, 47890}, {4705, 48056}, {4960, 48276}, {4978, 8045}, {16892, 14838}, {21124, 48003}, {23731, 48051}, {47679, 48000}, {47680, 1577}, {47682, 48300}, {47701, 48058}, {47708, 59672}, {47712, 3716}, {47720, 48295}, {47723, 47711}, {47724, 48395}, {47725, 48403}, {47727, 663}, {47937, 48602}, {47938, 48045}, {47942, 48040}, {47943, 48052}, {47944, 48053}, {47947, 48046}, {47948, 48047}, {47949, 48048}, {47958, 48054}, {47959, 4468}, {47968, 48059}, {47970, 48055}, {47971, 48064}, {47972, 48065}, {47973, 48066}, {47976, 48060}, {47977, 48061}, {48279, 49290}, {48282, 48290}, {48326, 52601}, {48335, 6332}, {48584, 48038}, {48585, 48603}, {48586, 48039}, {49301, 23789}, {50342, 50512}, {50352, 48405}, {68259, 14344}
X(69327) = X(40394)-anticomplementary conjugate of X(21293)
X(69327) = X(6)-isoconjugate of X(43348)
X(69327) = X(9)-Dao conjugate of X(43348)
X(69327) = crossdifference of every pair of points on line {31, 583}
X(69327) = barycentric product X(i)*X(j) for these {i,j}: {514, 33157}, {693, 69272}
X(69327) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 43348}, {33157, 190}, {69272, 100}
X(69327) = {X(6546),X(21124)}-harmonic conjugate of X(48003)

X(69328) = ODD<1, 0, 0, -1, 0, 1> POINT

Barycentrics    (b - c)*(a^3 - a*b*c + b^2*c + b*c^2) : :
X(69328) = X[650] - 3 X[48561], X[659] - 3 X[47817], X[659] + 3 X[47872], X[1577] + 3 X[47817], X[1577] - 3 X[47872], X[661] - 3 X[48553], X[667] - 3 X[47804], X[4391] + 3 X[47804], X[693] + 3 X[47815], X[693] - 3 X[47875], X[764] - 3 X[47796], X[905] - 3 X[47803], 2 X[31288] - 3 X[47803], X[1491] - 3 X[47794], 3 X[47794] - 2 X[65449],and many others

X(69328) lies on these lines: {2, 2530}, {512, 3716}, {513, 3814}, {514, 1125}, {522, 6133}, {523, 21179}, {649, 29150}, {650, 784}, {659, 1577}, {661, 48553}, {663, 10459}, {667, 2787}, {668, 9266}, {676, 29288}, {693, 47815}, {764, 47796}, {814, 4401}, {826, 4142}, {830, 21051}, {832, 20316}, {905, 31288}, {1019, 48265}, {1149, 4449}, {1491, 47794}, {1635, 48264}, {1734, 47835}, {1960, 3907}, {2254, 47837}, {2490, 6362}, {2533, 4040}, {2832, 48206}, {3669, 48564}, {3700, 29106}, {3709, 47127}, {3762, 4367}, {3766, 26248}, {3777, 19847}, {3801, 29224}, {3803, 45664}, {3910, 49290}, {4010, 4063}, {4036, 4057}, {4041, 48305}, {4086, 50353}, {4122, 29194}, {4160, 48401}, {4170, 4800}, {4369, 6372}, {4378, 4462}, {4379, 47929}, {4458, 29354}, {4474, 8643}, {4490, 48234}, {4498, 47832}, {4705, 26115}, {4724, 50352}, {4761, 48336}, {4776, 30836}, {4782, 29013}, {4784, 48566}, {4794, 29366}, {4808, 47695}, {4823, 29362}, {4834, 48080}, {4885, 23815}, {4905, 47823}, {4960, 47946}, {4977, 48004}, {4978, 47833}, {4983, 47821}, {4992, 48183}, {6002, 50512}, {6004, 17072}, {6050, 23880}, {6161, 21302}, {6371, 8062}, {6546, 29689}, {7178, 29102}, {7192, 47949}, {7662, 47965}, {8045, 29312}, {8640, 30061}, {8678, 20317}, {8714, 9508}, {9013, 53574}, {10015, 29094}, {13246, 29037}, {14349, 47822}, {14419, 17496}, {14430, 48322}, {14431, 21301}, {14838, 63812}, {17494, 48393}, {17541, 18110}, {18081, 30963}, {18107, 29546}, {20517, 62423}, {21052, 48150}, {21118, 50351}, {21120, 48290}, {21146, 47970}, {21185, 48062}, {21385, 48279}, {21389, 21960}, {24924, 48151}, {25666, 48059}, {26275, 29110}, {26277, 62415}, {28027, 47123}, {28840, 47994}, {29021, 48405}, {29029, 48231}, {29051, 53580}, {29066, 48331}, {29086, 48395}, {29098, 47890}, {29128, 47708}, {29142, 68794}, {29154, 47203}, {29170, 48064}, {29176, 58139}, {29226, 48295}, {29246, 48065}, {29268, 58150}, {29284, 49288}, {29302, 48090}, {29328, 48011}, {29402, 46843}, {30709, 31291}, {30795, 48556}, {30835, 48122}, {31148, 47906}, {31209, 47888}, {31251, 44429}, {41800, 50348}, {42455, 48387}, {43067, 47966}, {45314, 64934}, {45324, 64913}, {47697, 47814}, {47706, 48223}, {47707, 47798}, {47709, 48236}, {47711, 50340}, {47712, 48103}, {47729, 58155}, {47760, 48092}, {47811, 50457}, {47812, 47936}, {47813, 47918}, {47816, 50328}, {47827, 48409}, {47836, 53343}, {47838, 48123}, {47839, 48131}, {47841, 48335}, {47912, 48578}, {47921, 48220}, {48012, 64914}, {48066, 48196}, {48100, 48197}, {48162, 50449}, {48165, 50330}, {48168, 48350}, {48185, 48272}, {48204, 50345}, {48226, 48392}, {48559, 50501}, {48573, 50359}, {50336, 59590}, {50453, 68979}

X(69328) = midpoint of X(i) and X(j) for these {i,j}: {649, 48267}, {659, 1577}, {667, 4391}, {1019, 48265}, {2533, 4040}, {3762, 4367}, {4010, 4063}, {4036, 4057}, {4041, 48305}, {4086, 50353}, {4142, 69293}, {4378, 4462}, {4401, 4791}, {4498, 48273}, {4705, 47694}, {4724, 50352}, {4761, 48336}, {4808, 47695}, {4834, 48080}, {4960, 47946}, {6161, 21302}, {7192, 47949}, {7662, 47965}, {10015, 48299}, {14431, 47805}, {17072, 48063}, {17494, 48393}, {21051, 48248}, {21118, 50351}, {21120, 48290}, {21146, 47970}, {21185, 48062}, {21385, 48279}, {42455, 48387}, {43067, 47966}, {47711, 50340}, {47712, 48103}, {47815, 47875}, {47817, 47872}, {47890, 48403}, {48395, 50347}, {50336, 59590}
X(69328) = reflection of X(i) in X(j) for these {i,j}: {905, 31288}, {1491, 65449}, {3777, 19947}, {23815, 4885}, {48059, 25666}, {52601, 4874}
X(69328) = complement of X(2530)
X(69328) = X(i)-complementary conjugate of X(j) for these (i,j): {10, 46654}, {42, 15449}, {82, 11}, {83, 116}, {100, 21249}, {101, 6292}, {109, 17055}, {190, 21248}, {251, 1086}, {313, 55070}, {692, 16587}, {827, 1125}, {1176, 2968}, {1918, 35971}, {3112, 21252}, {4570, 3005}, {4577, 3741}, {4593, 21240}, {4599, 3739}, {4628, 2}, {18082, 125}, {18098, 8287}, {34072, 3666}, {36081, 3836}, {42396, 34830}, {46288, 6377}, {46289, 1015}, {52376, 17761}, {52394, 53564}, {52395, 44312}, {56186, 21253}, {56245, 26932}, {56251, 53575}, {58956, 17384}, {59996, 21208}
X(69328) = crosssum of X(6) and X(50521)
X(69328) = crossdifference of every pair of points on line {5124, 10329}
X(69328) = barycentric product X(514)*X(32930)
X(69328) = barycentric quotient X(32930)/X(190)
X(69328) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {659, 47872, 1577}, {905, 47803, 31288}, {1491, 47794, 65449}, {1577, 47817, 659}, {2533, 4448, 4040}, {3762, 47818, 4367}, {3777, 47795, 19947}, {4391, 47804, 667}, {4462, 47820, 4378}, {4498, 47832, 48273}, {24924, 48151, 48569}, {31209, 48410, 47888}, {47694, 47793, 4705}, {48080, 48565, 4834}

X(69329) = ODD<1, 0, 0, -1, -1, 1> POINT

Barycentrics    (b - c)*(-a^3 + b^3 + a*b*c - b^2*c - b*c^2 + c^3) : :
X(69329) = X[1] - 4 X[676], X[1] + 2 X[10015], 2 X[676] + X[10015], X[8] + 2 X[48286], 2 X[10] + X[47695], X[656] + 2 X[21179], X[1577] + 2 X[4142], X[1734] - 4 X[14837], X[1734] + 2 X[21185], X[1769] + 2 X[21180], 2 X[7649] + X[21189], 2 X[7661] + X[21186], 2 X[14837] + X[21185], 2 X[551] + X[44553], 2 X[650] + X[49300], 2 X[659] + X[47680], X[1019] + 2 X[48400],and many others

X(69329) lies on these lines: {1, 676}, {2, 23887}, {8, 48286}, {10, 47695}, {106, 1311}, {109, 26705}, {165, 9521}, {240, 522}, {244, 1111}, {514, 14413}, {551, 44553}, {650, 49300}, {659, 47680}, {667, 29156}, {690, 4800}, {826, 47872}, {900, 37718}, {928, 5902}, {1019, 48400}, {1089, 23684}, {1125, 3904}, {1638, 2826}, {1639, 17719}, {1698, 50333}, {1929, 18014}, {2254, 21201}, {2775, 11193}, {2787, 4809}, {2832, 6545}, {3245, 42763}, {3336, 53300}, {3679, 44566}, {3716, 4707}, {3737, 4960}, {3738, 11125}, {3762, 4458}, {3801, 29224}, {3810, 47795}, {3887, 30574}, {3960, 21132}, {4040, 7178}, {4049, 44433}, {4063, 48403}, {4391, 20517}, {4435, 65707}, {4448, 21145}, {4802, 21112}, {4874, 29172}, {4885, 49278}, {4905, 21188}, {4985, 21187}, {5903, 53549}, {6089, 28284}, {6362, 41800}, {6370, 48168}, {6615, 59750}, {7987, 44819}, {7989, 68332}, {13259, 14430}, {14315, 56419}, {14349, 48555}, {14400, 65104}, {14432, 23884}, {14838, 21118}, {18398, 53550}, {19947, 53533}, {21111, 31947}, {21120, 34958}, {21192, 48264}, {21385, 23770}, {23755, 48058}, {23786, 24622}, {23876, 47832}, {23877, 47794}, {24014, 24028}, {24795, 44902}, {25380, 66995}, {26275, 29240}, {28225, 53528}, {28487, 48556}, {29017, 47875}, {29066, 47798}, {29114, 58140}, {29118, 48566}, {29124, 58144}, {29132, 47762}, {29158, 48565}, {29160, 47771}, {29192, 48223}, {29312, 47833}, {29318, 47874}, {33148, 47772}, {37524, 52730}, {37571, 52739}, {45318, 45341}, {45664, 64856}, {47234, 68882}, {47694, 50453}, {47723, 50340}, {47724, 50347}, {47725, 47890}, {47726, 68794}, {47871, 68959}, {48003, 55282}, {53343, 62435}, {53395, 53407}, {53522, 64155}, {53573, 64850}, {54318, 68832}

X(69329) = midpoint of X(4448) and X(21145)
X(69329) = reflection of X(i) in X(j) for these {i,j}: {4453, 21181}, {5902, 30691}, {14349, 48555}, {14430, 21198}, {68816, 26275}
X(69329) = crosspoint of X(i) and X(j) for these (i,j): {75, 37143}, {653, 62723}, {7192, 60479}
X(69329) = crosssum of X(31) and X(22108)
X(69329) = crossdifference of every pair of points on line {48, 34879}
X(69329) = barycentric product X(i)*X(j) for these {i,j}: {514, 5057}, {693, 5011}, {1577, 33325}
X(69329) = barycentric quotient X(i)/X(j) for these {i,j}: {5011, 100}, {5057, 190}, {33325, 662}
X(69329) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {676, 10015, 1}, {3716, 4707, 49276}, {14837, 21185, 1734}, {21120, 34958, 48282}

X(69330) = ODD<1, 0, 0, -1, -1, -1> POINT

Barycentrics    (b - c)*(-a^3 + b^3 + a*b*c + b^2*c + b*c^2 + c^3) : :
X(69330) = X[49276] - 4 X[50347], 3 X[4453] - 2 X[23789], 2 X[4522] - 3 X[47794], 3 X[4809] - 2 X[52601], 2 X[8045] - 3 X[47818], 4 X[13246] - 3 X[47818], 2 X[18004] - 3 X[48553], 4 X[21212] - 3 X[48556], 2 X[23815] - 3 X[48227], X[47689] - 3 X[48565], X[47714] - 3 X[48566], X[47718] - 3 X[47762], 3 X[47817] - 2 X[69293], 3 X[47886] - 2 X[48066], 2 X[48299] - 3 X[68816], 3 X[48569] - 4 X[69011]

X(69330) lies on these lines: {1, 3910}, {240, 522}, {512, 50340}, {514, 50017}, {523, 4063}, {525, 4040}, {649, 29021}, {650, 48272}, {659, 826}, {663, 23876}, {667, 29017}, {690, 48336}, {693, 20517}, {812, 47712}, {830, 21124}, {905, 49278}, {918, 47970}, {1019, 29142}, {1960, 29256}, {2254, 21192}, {2533, 29086}, {2804, 57121}, {3004, 28481}, {3566, 48352}, {3716, 7265}, {3762, 29037}, {3766, 4467}, {3801, 29070}, {3810, 48321}, {4010, 29106}, {4024, 68836}, {4025, 4905}, {4083, 47727}, {4088, 48003}, {4122, 29194}, {4151, 47695}, {4367, 29312}, {4369, 47715}, {4380, 29158}, {4391, 29062}, {4401, 29318}, {4453, 23789}, {4458, 4978}, {4462, 29212}, {4498, 29047}, {4522, 47794}, {4560, 23887}, {4707, 29051}, {4724, 23875}, {4775, 29284}, {4777, 50501}, {4782, 29146}, {4784, 29168}, {4809, 52601}, {4834, 29144}, {4874, 29248}, {4976, 68814}, {6005, 47972}, {6372, 50342}, {7178, 47724}, {7950, 48103}, {8045, 13246}, {9237, 23597}, {10015, 29278}, {14349, 68780}, {14838, 48278}, {18004, 48553}, {21118, 64934}, {21188, 49285}, {21196, 48409}, {21201, 48264}, {21212, 48556}, {21301, 50453}, {21385, 29288}, {23815, 48227}, {23879, 47694}, {23882, 49300}, {25259, 29294}, {28161, 60493}, {28541, 48128}, {28846, 47942}, {28898, 59590}, {29013, 47708}, {29078, 48267}, {29090, 48265}, {29164, 48011}, {29166, 50512}, {29196, 47707}, {29200, 48351}, {29202, 48331}, {29216, 48080}, {29232, 48400}, {29302, 47691}, {29358, 48094}, {34958, 48280}, {47687, 50337}, {47689, 48565}, {47702, 47935}, {47714, 48566}, {47718, 47762}, {47817, 69293}, {47886, 48066}, {47930, 47936}, {47948, 48402}, {47965, 64856}, {47983, 48595}, {47987, 48076}, {47998, 48085}, {48004, 48082}, {48006, 48081}, {48007, 48596}, {48012, 48077}, {48078, 48623}, {48099, 49277}, {48299, 68816}, {48326, 68894}, {48569, 69011}

X(69330) = midpoint of X(i) and X(j) for these {i,j}: {4380, 47709}, {47702, 47935}, {47930, 47936}
X(69330) = reflection of X(i) in X(j) for these {i,j}: {693, 20517}, {1577, 4142}, {2254, 21192}, {4040, 50347}, {4088, 48003}, {4905, 4025}, {4978, 4458}, {7265, 3716}, {8045, 13246}, {14349, 68780}, {21301, 50453}, {25259, 59672}, {47680, 3801}, {47682, 667}, {47687, 50337}, {47715, 4369}, {47723, 2533}, {47724, 7178}, {47948, 48402}, {48076, 47987}, {48077, 48012}, {48078, 48623}, {48081, 48006}, {48082, 48004}, {48085, 47998}, {48086, 3004}, {48106, 48011}, {48264, 21201}, {48272, 650}, {48278, 14838}, {48280, 34958}, {48300, 4401}, {48409, 21196}, {48595, 47983}, {48596, 48007}, {49276, 4040}, {49277, 48099}, {49278, 905}, {49279, 48331}, {49285, 21188}
X(69330) = X(53835)-Dao conjugate of X(1)
X(69330) = crosssum of X(31) and X(2483)
X(69330) = crossdifference of every pair of points on line {48, 5153}
X(69330) = barycentric product X(514)*X(33075)
X(69330) = barycentric quotient X(i)/X(j) for these {i,j}: {33075, 190}, {53835, 47660}
X(69330) = {X(8045),X(13246)}-harmonic conjugate of X(47818)

X(69331) = ODD<1, 0, -1, 1, -1, 1> POINT

Barycentrics    (b - c)*(-a^3 + a*b^2 + b^3 - a*b*c - b^2*c + a*c^2 - b*c^2 + c^3) : :
X(69331) = 3 X[2] + X[48326], 5 X[676] - X[2976], X[650] - 3 X[48215], X[659] + 3 X[6545], X[693] + 3 X[48227], X[1491] + 3 X[47887], 3 X[1638] - X[9508], 3 X[1638] + X[23770], X[2977] - 3 X[44902], X[3801] + 3 X[47796], X[3837] - 3 X[21204], X[4458] + 3 X[21204], X[3904] + 3 X[21145], X[4010] + 3 X[4453], X[4088] - 9 X[14475], and many others

X(69331) lies on these lines: {2, 48056}, {513, 676}, {514, 31288}, {523, 21212}, {650, 48215}, {659, 6545}, {693, 48227}, {812, 69011}, {1125, 29102}, {1491, 47887}, {1638, 9508}, {2785, 44315}, {2977, 4802}, {3004, 54265}, {3716, 58375}, {3776, 4874}, {3801, 47796}, {3837, 4458}, {3904, 21145}, {4010, 4453}, {4025, 48090}, {4083, 21188}, {4088, 14475}, {4122, 26985}, {4142, 48406}, {4448, 49301}, {4468, 48197}, {4522, 48198}, {4728, 50342}, {4750, 4810}, {4777, 4925}, {4782, 48398}, {4806, 69292}, {4809, 6548}, {4885, 62423}, {4928, 18004}, {4977, 53580}, {6590, 48221}, {7192, 47990}, {7662, 47754}, {13246, 45668}, {14419, 47680}, {14425, 28199}, {14837, 29226}, {16892, 47833}, {19947, 23887}, {20517, 23815}, {20520, 30665}, {21104, 47799}, {21115, 48083}, {21124, 47889}, {21146, 47797}, {21183, 48098}, {21343, 30574}, {24924, 48103}, {28175, 45675}, {29078, 59522}, {29090, 59714}, {29362, 48415}, {30724, 48400}, {31148, 47944}, {31207, 47885}, {31250, 48088}, {36848, 47695}, {43051, 51648}, {43067, 48192}, {44435, 47999}, {45342, 50326}, {45666, 48055}, {45745, 48127}, {45746, 48238}, {47123, 50335}, {47676, 47822}, {47690, 48224}, {47691, 47823}, {47692, 48235}, {47701, 48253}, {47704, 47827}, {47712, 48569}, {47716, 47837}, {47720, 47835}, {47757, 48030}, {47758, 49295}, {47766, 48097}, {47779, 48405}, {47783, 47964}, {47803, 49299}, {47804, 48421}, {47812, 50340}, {47813, 47968}, {47824, 48349}, {47877, 48142}, {47880, 48134}, {47886, 48120}, {47891, 47998}, {47893, 55282}, {47961, 48563}, {47973, 48234}, {48028, 48555}, {48029, 48195}, {48062, 48216}, {48108, 48177}, {48202, 49286}, {48206, 69293}, {48244, 53558}, {48611, 49293}, {52601, 68979}

X(69331) = midpoint of X(i) and X(j) for these {i,j}: {3004, 54265}, {3716, 58375}, {3776, 4874}, {3837, 4458}, {4025, 48090}, {4142, 48406}, {4782, 48398}, {4806, 69292}, {7192, 47990}, {9508, 23770}, {20517, 23815}, {21183, 48212}, {45745, 48127}, {47123, 50335}, {47676, 48048}, {48028, 49296}, {48056, 48326}, {48098, 68780}, {48611, 49293}
X(69331) = complement of X(48056)
X(69331) = barycentric product X(514)*X(32857)
X(69331) = barycentric quotient X(32857)/X(190)
X(69331) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 48326, 48056}, {1638, 23770, 9508}, {4088, 14475, 30795}, {4458, 21204, 3837}, {7192, 48552, 47990}, {21183, 68780, 48098}, {26985, 48241, 4122}, {30795, 58372, 4088}, {31250, 48088, 48199}, {47676, 47822, 48048}, {48098, 48212, 68780}, {48555, 49296, 48028}

X(69332) = ODD<1, 0, -1, -1, -1, -1> POINT

Barycentrics    (b - c)*(-a^3 + a*b^2 + b^3 + a*b*c + b^2*c + a*c^2 + b*c^2 + c^3) : :
X(69332) = 3 X[3004] - X[47989], 3 X[4025] + X[48006], 2 X[47989] - 3 X[47999], X[48006] - 3 X[68780], 3 X[48007] - X[48589], X[48090] - 3 X[48212], 3 X[649] + X[47924], 3 X[650] - X[48088], 3 X[48056] - 2 X[48088], 3 X[659] - X[48102], 3 X[16892] + X[48102], X[693] - 3 X[48227], X[1491] - 3 X[47886], 3 X[1491] - X[48077], and many others

X(69332) lies on these lines: {2, 4122}, {10, 29110}, {512, 21192}, {513, 3004}, {514, 4782}, {522, 3837}, {523, 2487}, {647, 31947}, {649, 47924}, {650, 48056}, {656, 24382}, {659, 16892}, {661, 50342}, {662, 53379}, {667, 68979}, {693, 48227}, {784, 20517}, {824, 4874}, {826, 14838}, {900, 48050}, {905, 29017}, {918, 48048}, {1125, 49290}, {1491, 47886}, {1635, 48103}, {1638, 4777}, {2254, 50340}, {2529, 4394}, {2605, 53556}, {2785, 48289}, {2786, 4806}, {2787, 50453}, {3239, 48197}, {3676, 48098}, {3700, 47799}, {3776, 29362}, {3801, 4560}, {3835, 29078}, {3960, 29312}, {4010, 4467}, {4024, 47833}, {4036, 24622}, {4041, 24443}, {4088, 47827}, {4106, 48192}, {4129, 29090}, {4142, 63812}, {4151, 24176}, {4367, 21124}, {4380, 48174}, {4448, 49275}, {4453, 21146}, {4522, 29370}, {4707, 48288}, {4730, 47727}, {4750, 4784}, {4774, 30574}, {4790, 47961}, {4809, 47694}, {4818, 64914}, {4824, 47782}, {4885, 48215}, {4925, 28183}, {4926, 48178}, {4976, 23770}, {4977, 69292}, {4979, 47944}, {6332, 29202}, {7265, 47839}, {7658, 48216}, {11068, 48097}, {13246, 48248}, {14419, 47682}, {17072, 29074}, {17161, 47834}, {17494, 48241}, {18004, 25666}, {20295, 48552}, {21051, 29037}, {21260, 29062}, {23093, 48383}, {23282, 48207}, {23815, 29190}, {23875, 50507}, {23879, 52601}, {24623, 27486}, {24719, 44435}, {25098, 68772}, {25259, 47822}, {27013, 47693}, {27115, 48171}, {28161, 48233}, {28195, 49296}, {28478, 48129}, {28846, 48028}, {28855, 47993}, {28878, 47954}, {28898, 48195}, {29047, 50504}, {29086, 50337}, {29102, 48284}, {29118, 59748}, {29144, 50336}, {29188, 62435}, {29200, 48099}, {29204, 47785}, {29208, 50501}, {29226, 60492}, {29252, 48058}, {29284, 48136}, {29292, 65449}, {29354, 48003}, {31095, 48203}, {31209, 48185}, {31286, 48405}, {31287, 48199}, {36848, 47687}, {41800, 48395}, {45671, 50351}, {46919, 48201}, {47224, 62492}, {47656, 48238}, {47673, 47813}, {47677, 47804}, {47688, 47776}, {47689, 48235}, {47690, 47823}, {47695, 50341}, {47698, 48176}, {47699, 47755}, {47703, 48253}, {47704, 58372}, {47707, 47835}, {47711, 47837}, {47715, 48569}, {47754, 48089}, {47784, 48047}, {47800, 49286}, {47803, 48271}, {47811, 47930}, {47877, 48023}, {47878, 47928}, {47880, 48027}, {47885, 48118}, {47887, 48120}, {47888, 48272}, {47893, 48278}, {47901, 47968}, {47960, 48606}, {47969, 48571}, {47971, 48024}, {47972, 50359}, {47973, 50358}, {48080, 48177}, {48082, 48162}, {48094, 48226}, {48104, 48599}, {48179, 50326}, {48223, 50356}, {48240, 49302}, {48269, 48555}, {50339, 53558}, {51648, 68773}

X(69332) = midpoint of X(i) and X(j) for these {i,j}: {659, 16892}, {661, 50342}, {2254, 50340}, {3801, 4560}, {4010, 4467}, {4025, 68780}, {4367, 21124}, {4458, 21196}, {4707, 48288}, {4730, 47727}, {4784, 47701}, {4790, 47961}, {4809, 47894}, {4897, 47998}, {4976, 23770}, {4979, 47944}, {17494, 48326}, {27486, 48224}, {47695, 50341}, {47930, 48083}, {47971, 48024}, {47972, 50359}, {47973, 50358}, {47983, 48013}, {48104, 48599}, {48120, 48277}, {48349, 50343}, {50339, 53558}, {50347, 50348}
X(69332) = reflection of X(i) in X(j) for these {i,j}: {3837, 21212}, {4369, 69011}, {9508, 17069}, {18004, 25666}, {47999, 3004}, {48056, 650}, {48097, 11068}, {48098, 3676}, {48201, 46919}, {48248, 13246}, {48405, 31286}, {49290, 1125}
X(69332) = complement of X(4122)
X(69332) = X(i)-complementary conjugate of X(j) for these (i,j): {58, 55061}, {825, 1211}, {870, 53575}, {985, 125}, {1333, 61065}, {1492, 3454}, {2206, 53823}, {4586, 21245}, {5384, 31946}, {14621, 21253}, {34069, 1213}, {40746, 8287}, {58111, 59511}
X(69332) = crossdifference of every pair of points on line {2915, 18755}
X(69332) = barycentric product X(514)*X(33082)
X(69332) = barycentric quotient X(33082)/X(190)
X(69332) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4467, 47797, 4010}, {4750, 47701, 4784}, {17494, 48241, 48326}, {47811, 47930, 48083}, {47887, 48277, 48120}, {48203, 50343, 48349}

X(69333) = ODD<1, -1, 1, 1, 0, 0> POINT

Barycentrics    a*(b - c)*(a^2 - a*b + b^2 - a*c + b*c + c^2) : :

X(69333) lies on these lines: {1, 2530}, {30, 511}, {239, 18107}, {649, 50355}, {650, 8632}, {656, 28284}, {659, 4041}, {661, 4435}, {663, 1193}, {667, 1734}, {693, 21303}, {764, 48282}, {905, 48327}, {1016, 4553}, {1577, 48305}, {1960, 14838}, {2254, 4367}, {2526, 4162}, {2533, 17751}, {3271, 45213}, {3293, 4040}, {3669, 48344}, {3716, 21051}, {3737, 50345}, {3777, 4449}, {3801, 47695}, {3803, 4782}, {3831, 4874}, {3888, 6631}, {3960, 48328}, {4010, 21301}, {4057, 57099}, {4063, 4730}, {4147, 48063}, {4378, 4905}, {4394, 68836}, {4401, 50504}, {4448, 47793}, {4490, 4724}, {4498, 4814}, {4560, 50341}, {4770, 48003}, {4775, 14349}, {4784, 50523}, {4794, 48012}, {4822, 47905}, {4879, 4895}, {4885, 21261}, {4922, 17496}, {4959, 48122}, {4978, 48291}, {4983, 47948}, {4992, 48050}, {6542, 50521}, {6547, 64523}, {8027, 30613}, {8643, 47828}, {14296, 57110}, {17144, 18081}, {17166, 21146}, {18344, 65106}, {20051, 47969}, {21052, 47872}, {21124, 50340}, {21212, 58628}, {21304, 50451}, {21343, 48334}, {21958, 47127}, {22320, 57162}, {23506, 28374}, {23815, 48295}, {25048, 54102}, {25569, 47893}, {26078, 47845}, {28831, 47760}, {31149, 45342}, {31291, 50343}, {32094, 40521}, {36848, 47796}, {42661, 53563}, {44408, 55362}, {44429, 47841}, {44448, 48062}, {45316, 45323}, {45332, 48234}, {45666, 47794}, {46403, 48279}, {47724, 48393}, {47729, 48410}, {47757, 59980}, {47804, 47835}, {47812, 47889}, {47814, 47822}, {47816, 47839}, {47818, 47837}, {47820, 47823}, {47842, 48306}, {47888, 58155}, {47912, 48024}, {47922, 47966}, {47956, 48028}, {47957, 48607}, {47959, 48351}, {47967, 48029}, {47976, 58173}, {48005, 48058}, {48023, 48123}, {48027, 48093}, {48030, 48099}, {48052, 58163}, {48053, 48613}, {48059, 58160}, {48066, 48294}, {48075, 48343}, {48086, 48337}, {48092, 48129}, {48121, 58166}, {48137, 48332}, {48144, 50359}, {48151, 48323}, {48216, 48564}, {48265, 53343}, {48272, 49279}, {48273, 48339}, {48288, 48409}, {48299, 50333}, {48307, 50330}, {48333, 48335}, {48347, 48348}, {48400, 53523}, {48586, 58165}, {50336, 50517}, {50337, 52601}, {50352, 54265}, {50487, 68113}, {50505, 50516}, {55244, 56149}, {57048, 69245}, {59836, 60353}

X(69333) = isogonal conjugate of X(65364)
X(69333) = crossdifference of every pair of points on line {6, 982}
X(69333) = barycentric quotient X(i)/X(j) for these {i,j}: {10752, 15268}, {17575, 42445}, {25811, 33272}, {60883, 29326}
X(69333) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {650, 48329, 48331}, {667, 1734, 9508}, {905, 48327, 48330}, {1734, 48324, 667}, {2254, 48322, 4367}, {2526, 4162, 48136}, {2526, 48136, 48100}, {3803, 50501, 4782}, {4041, 48150, 659}, {4705, 6161, 4040}, {4794, 48012, 50507}, {4879, 50328, 48131}, {4895, 48131, 4879}, {14838, 48345, 1960}, {21302, 47694, 2533}, {47912, 48367, 48024}, {47948, 48352, 4983}, {48023, 48338, 48123}, {48027, 50508, 48093}, {48323, 58374, 48151}, {48330, 50335, 905}, {50505, 50519, 50516}

X(69334) = ODD<1, -1, 1, 1, 0, -1> POINT

Barycentrics    (b - c)*(-a^3 + a^2*b - a*b^2 + a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2) : :
X(69334) = 3 X[2] - 4 X[53571], X[20295] - 3 X[21301], X[20295] + 3 X[21302], X[21343] - 3 X[48167], 3 X[663] - 5 X[30835], 2 X[663] - 3 X[47839], 6 X[21260] - 5 X[30835], 4 X[21260] - 3 X[47839], 10 X[30835] - 9 X[47839], 3 X[667] - 4 X[31286], 2 X[667] - 3 X[47837], 3 X[17072] - 2 X[31286], 4 X[17072] - 3 X[47837], 8 X[31286] - 9 X[47837], and many others

X(69334) lies on these lines: {1, 3837}, {2, 1960}, {4, 2821}, {8, 891}, {10, 659}, {11, 58369}, {46, 53403}, {69, 9032}, {80, 900}, {145, 48296}, {150, 926}, {316, 512}, {334, 875}, {355, 2826}, {388, 53539}, {427, 58313}, {513, 3762}, {514, 4774}, {519, 21343}, {523, 47680}, {661, 29188}, {663, 15283}, {667, 17072}, {668, 3888}, {693, 48291}, {804, 24462}, {812, 4730}, {814, 1734}, {830, 2533}, {832, 2517}, {1125, 25569}, {1491, 29066}, {1577, 48305}, {1698, 68816}, {1837, 53523}, {2254, 2787}, {2530, 3907}, {2605, 44316}, {3251, 4928}, {3309, 48267}, {3626, 68959}, {3632, 25574}, {3661, 24623}, {3679, 21385}, {3716, 6161}, {3835, 4775}, {3887, 4010}, {3900, 23813}, {3904, 31131}, {3960, 4922}, {4040, 21051}, {4041, 29070}, {4088, 29102}, {4107, 21261}, {4129, 48336}, {4160, 21146}, {4367, 48569}, {4374, 21304}, {4378, 24720}, {4382, 4814}, {4391, 6004}, {4401, 47835}, {4444, 68899}, {4449, 23815}, {4458, 44314}, {4467, 29058}, {4486, 40459}, {4490, 29186}, {4491, 53574}, {4522, 49279}, {4528, 6084}, {4560, 29182}, {4645, 6165}, {4705, 29051}, {4728, 4895}, {4738, 68953}, {4770, 17494}, {4785, 58173}, {4794, 47822}, {4800, 59737}, {4806, 48352}, {4810, 68968}, {4823, 48301}, {4834, 48016}, {4874, 48324}, {4885, 48327}, {4905, 29324}, {4925, 5794}, {4992, 48337}, {5090, 68783}, {5176, 6550}, {5252, 30725}, {6085, 21290}, {6363, 20293}, {6371, 44444}, {6702, 41191}, {8632, 21053}, {8637, 28373}, {8643, 31288}, {8656, 31207}, {8674, 14288}, {8678, 43067}, {9014, 50765}, {9461, 32032}, {12619, 19916}, {13266, 59415}, {14077, 48089}, {14349, 29366}, {14413, 19947}, {14419, 25380}, {14421, 65482}, {14430, 48032}, {15313, 50331}, {16892, 29110}, {17023, 30865}, {17496, 29268}, {17752, 30095}, {18004, 49276}, {19875, 45314}, {21052, 48150}, {21055, 68880}, {21124, 29086}, {21297, 63216}, {21305, 52619}, {21389, 21958}, {21439, 33941}, {21612, 33938}, {23655, 23818}, {23789, 48323}, {24222, 53314}, {24618, 53286}, {24698, 64867}, {24719, 29350}, {25055, 45340}, {25299, 50510}, {26446, 44805}, {26752, 27015}, {26798, 58164}, {26801, 27140}, {26853, 58175}, {27013, 58139}, {27138, 58158}, {27675, 31339}, {27773, 30968}, {28294, 30580}, {28475, 50336}, {28585, 48329}, {29013, 50355}, {29082, 48272}, {29094, 48278}, {29148, 50359}, {29224, 47700}, {29236, 48321}, {29240, 50333}, {29246, 47959}, {29298, 48131}, {29312, 47687}, {29340, 50343}, {29344, 48018}, {30025, 41240}, {30709, 53343}, {31147, 58166}, {31251, 58155}, {31291, 47836}, {31946, 48306}, {39545, 48245}, {39547, 50334}, {42325, 48265}, {44429, 47729}, {45313, 58141}, {45316, 58157}, {47684, 48187}, {47711, 68979}, {47721, 47975}, {47728, 47808}, {47794, 48331}, {47795, 48330}, {47796, 48328}, {47814, 50507}, {47827, 48284}, {47840, 58160}, {47841, 48294}, {47908, 47912}, {47970, 48401}, {48090, 48339}, {48164, 48298}, {48170, 48304}, {48183, 68824}, {48184, 48295}, {48218, 65428}, {48282, 48406}, {48322, 52601}, {50330, 68888}, {50340, 50453}, {50341, 64934}, {50342, 62435}, {50556, 62415}, {50764, 64914}, {58374, 68896}

X(69334) = midpoint of X(i) and X(j) for these {i,j}: {8, 46403}, {4382, 4814}, {4774, 50328}, {21301, 21302}, {47721, 47975}
X(69334) = reflection of X(i) in X(j) for these {i,j}: {1, 3837}, {145, 48296}, {659, 10}, {663, 21260}, {667, 17072}, {1960, 53571}, {2605, 44316}, {3251, 4928}, {4040, 21051}, {4367, 50337}, {4378, 24720}, {4449, 23815}, {4458, 44314}, {4491, 53574}, {4775, 3835}, {4922, 3960}, {6161, 3716}, {17494, 4770}, {19916, 12619}, {26853, 58175}, {30580, 48182}, {31291, 50512}, {39547, 50334}, {41191, 6702}, {47970, 48401}, {48282, 48406}, {48288, 1491}, {48291, 693}, {48301, 4823}, {48305, 1577}, {48306, 31946}, {48321, 50335}, {48322, 52601}, {48323, 23789}, {48324, 4874}, {48327, 4885}, {48336, 4129}, {48337, 4992}, {48339, 48090}, {48352, 4806}, {49276, 18004}, {49279, 4522}, {50340, 50453}, {50342, 62435}, {50351, 50333}, {53314, 53565}, {68824, 48183}
X(69334) = anticomplement of X(1960)
X(69334) = anticomplement of the isogonal conjugate of X(4555)
X(69334) = isotomic conjugate of the anticomplement of X(55055)
X(69334) = anticomplementary isogonal conjugate of X(39349)
X(69334) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 39349}, {75, 66862}, {88, 4440}, {100, 17487}, {106, 9263}, {190, 30578}, {651, 30577}, {662, 30579}, {664, 64743}, {668, 21290}, {679, 20042}, {765, 44009}, {901, 192}, {903, 149}, {1016, 63246}, {1022, 54102}, {1320, 39351}, {3257, 2}, {4080, 21221}, {4555, 8}, {4564, 63252}, {4567, 62634}, {4582, 329}, {4591, 17147}, {4615, 75}, {4618, 519}, {4622, 1}, {4634, 17135}, {4638, 17495}, {4674, 148}, {4792, 39364}, {4997, 37781}, {5376, 514}, {5548, 3177}, {6551, 65195}, {6635, 3952}, {9268, 17494}, {9271, 62413}, {9456, 21224}, {20568, 150}, {23838, 17036}, {32665, 194}, {32719, 17486}, {46162, 21217}, {52925, 17488}, {53656, 9802}, {56049, 58371}, {57564, 21297}, {57995, 21293}, {62536, 513}, {65336, 5905}
X(69334) = X(55055)-cross conjugate of X(2)
X(69334) = crosspoint of X(i) and X(j) for these (i,j): {83, 4555}, {655, 56358}, {668, 32012}, {670, 24624}, {903, 4598}
X(69334) = crosssum of X(i) and X(j) for these (i,j): {39, 1960}, {649, 23633}, {654, 3056}, {669, 2245}, {902, 20979}
X(69334) = crossdifference of every pair of points on line {3051, 5332}
X(69334) = barycentric product X(514)*X(32927)
X(69334) = barycentric quotient X(i)/X(j) for these {i,j}: {32927, 190}, {55055, 1960}
X(69334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {663, 21260, 47839}, {667, 17072, 47837}, {1960, 53571, 2}, {4040, 21051, 48553}, {4367, 50337, 48569}, {4775, 31149, 3835}, {4922, 36848, 3960}, {6161, 14431, 3716}, {25569, 30795, 1125}, {31291, 47836, 50512}

X(69335) = ODD<1, -1, 1, 0, 1, 1> POINT

Barycentrics    (b - c)*(a^3 - a^2*b + a*b^2 + b^3 - a^2*c + b^2*c + a*c^2 + b*c^2 + c^3) : :

X(69335) lies on these lines: {30, 511}, {649, 50541}, {663, 16892}, {665, 2522}, {667, 4025}, {905, 48299}, {1577, 49286}, {1638, 48564}, {1734, 48062}, {2254, 48300}, {2530, 6332}, {3004, 48099}, {3239, 21260}, {3669, 48290}, {3676, 52601}, {3798, 50512}, {3801, 21185}, {4040, 68780}, {4041, 48094}, {4142, 48063}, {4170, 49295}, {4391, 49275}, {4401, 21192}, {4453, 47820}, {4468, 4705}, {4490, 48083}, {4498, 48102}, {4521, 65449}, {4724, 21124}, {4729, 48130}, {4750, 58140}, {4765, 8659}, {4786, 58144}, {4801, 49301}, {4808, 44448}, {4822, 47958}, {4834, 48060}, {4874, 21188}, {4897, 50515}, {4905, 47682}, {4983, 47995}, {5592, 24286}, {6050, 17069}, {6590, 24290}, {7626, 14838}, {7658, 31288}, {8045, 24720}, {11068, 50504}, {14349, 48007}, {17072, 69293}, {17166, 47676}, {17496, 47728}, {21301, 25259}, {21302, 47707}, {23747, 47123}, {23815, 49290}, {24285, 50507}, {30565, 47814}, {41800, 47803}, {44429, 57066}, {44435, 47840}, {45745, 66513}, {45746, 53335}, {47701, 48367}, {47708, 53343}, {47716, 48339}, {47720, 49302}, {47757, 47839}, {47766, 47837}, {47771, 47836}, {47838, 48555}, {47890, 50501}, {47912, 48082}, {47918, 48078}, {47923, 48338}, {47930, 48322}, {47943, 48121}, {47949, 48036}, {47956, 48046}, {47959, 48040}, {47960, 50508}, {47965, 48055}, {47967, 48048}, {47968, 48123}, {47971, 50523}, {47973, 48131}, {47983, 48081}, {47989, 48091}, {47999, 48093}, {48006, 48351}, {48029, 48402}, {48061, 60492}, {48086, 49277}, {48095, 50499}, {48101, 50509}, {48103, 50355}, {48138, 58172}, {48250, 48565}, {48271, 48395}, {48273, 48398}, {48301, 48326}, {48332, 65685}, {48400, 59590}, {48546, 48553}, {50453, 59672}, {51652, 57243}, {58139, 59550}

X(69335) = crossdifference of every pair of points on line {6, 63513}
X(69335) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2530, 49279, 6332}, {6332, 48015, 2530}, {21302, 49273, 47707}, {48299, 50348, 905}

X(69336) = ODD<1, -1, 1, 0, 0, 1> POINT

Barycentrics    (b - c)*(a^3 - a^2*b + a*b^2 - a^2*c + b^2*c + a*c^2 + b*c^2) : :
X(69336) = 2 X[693] - 3 X[48189], 3 X[4010] - 2 X[4106], 4 X[4106] - 3 X[24719], X[7192] - 3 X[47694], X[7192] + 3 X[53343], 4 X[7662] - 3 X[48238], 2 X[21146] - 3 X[48238], X[46403] - 3 X[48172], 3 X[47697] + X[48079], 3 X[47834] - 2 X[48098], X[48079] - 3 X[48080], 2 X[48090] - 3 X[48172], X[48108] - 3 X[48237], 3 X[48237] - 2 X[54265], and many others

X(69336) lies on these lines: {2, 50335}, {37, 69103}, {192, 4777}, {320, 350}, {514, 4775}, {522, 659}, {523, 4724}, {649, 900}, {650, 2276}, {661, 64914}, {663, 48289}, {667, 8714}, {669, 64868}, {676, 48227}, {764, 48295}, {784, 4040}, {812, 50358}, {814, 48150}, {824, 4375}, {830, 48267}, {832, 4985}, {885, 41527}, {891, 48339}, {1491, 3716}, {1577, 6004}, {1734, 47835}, {1960, 48321}, {2254, 4874}, {2309, 55969}, {2530, 47841}, {2533, 3309}, {2786, 4817}, {2787, 48324}, {2826, 48290}, {3004, 48177}, {3056, 9001}, {3251, 48285}, {3667, 4784}, {3777, 65482}, {3801, 21185}, {3835, 4800}, {3837, 47832}, {4025, 4809}, {4057, 23403}, {4358, 69102}, {4369, 48234}, {4378, 68896}, {4382, 64913}, {4408, 53370}, {4411, 53377}, {4467, 26277}, {4491, 68170}, {4553, 42722}, {4560, 48331}, {4705, 59672}, {4778, 39547}, {4782, 4926}, {4794, 48288}, {4802, 47969}, {4804, 29362}, {4806, 48023}, {4810, 48072}, {4824, 48029}, {4885, 36848}, {4905, 52601}, {4913, 48226}, {4922, 48327}, {4925, 47807}, {4932, 6006}, {4948, 45673}, {4963, 47986}, {4977, 47907}, {4992, 48122}, {6084, 68145}, {6133, 50338}, {6161, 29066}, {6362, 48299}, {7155, 23836}, {8678, 48265}, {9002, 50524}, {9508, 47804}, {16892, 48224}, {17069, 26275}, {17072, 47872}, {17166, 29198}, {17496, 48330}, {17787, 30061}, {18004, 48077}, {21104, 47132}, {21118, 29082}, {21189, 24793}, {21222, 48344}, {23729, 28209}, {23742, 47123}, {23880, 48329}, {23887, 49279}, {24623, 47790}, {24720, 47833}, {25569, 48325}, {26144, 27293}, {26985, 30998}, {27115, 48213}, {28147, 48009}, {28175, 47927}, {28183, 48572}, {28205, 48240}, {28217, 48578}, {28372, 45686}, {28623, 50353}, {29051, 48392}, {29070, 48111}, {29102, 49300}, {29144, 47660}, {29152, 31291}, {29170, 50523}, {29186, 48393}, {29204, 49273}, {29246, 50457}, {29324, 48322}, {30520, 47131}, {30795, 47831}, {30835, 48183}, {31095, 45342}, {31149, 59737}, {31207, 48229}, {31209, 45666}, {31286, 48244}, {42325, 50352}, {47653, 48158}, {47677, 48223}, {47695, 49275}, {47698, 48048}, {47705, 48113}, {47708, 68979}, {47817, 50504}, {47821, 48030}, {47826, 48002}, {47827, 48017}, {47837, 48018}, {47838, 48059}, {47839, 48066}, {47840, 48100}, {47875, 50337}, {47887, 58375}, {47903, 48021}, {47909, 47993}, {47914, 47946}, {47928, 48001}, {47945, 48028}, {47975, 48176}, {47977, 68894}, {47991, 48024}, {48007, 48552}, {48010, 48162}, {48012, 48553}, {48037, 48588}, {48073, 48253}, {48075, 48569}, {48102, 53558}, {48143, 49292}, {48160, 48547}, {48167, 59522}, {48185, 50333}, {48409, 50507}, {49278, 49290}

X(69336) = midpoint of X(i) and X(j) for these {i,j}: {4804, 48032}, {47694, 53343}, {47695, 49275}, {47697, 48080}, {47705, 48113}, {48021, 48153}, {48102, 53558}, {48150, 48264}
X(69336) = reflection of X(i) in X(j) for these {i,j}: {649, 48248}, {659, 48063}, {764, 48295}, {1491, 3716}, {2254, 4874}, {3801, 21185}, {4122, 49286}, {4560, 48331}, {4705, 59672}, {4824, 48029}, {4905, 52601}, {4913, 53580}, {4922, 48327}, {4948, 45673}, {4963, 47986}, {17496, 48330}, {21104, 47132}, {21146, 7662}, {21222, 48344}, {24719, 4010}, {46403, 48090}, {47698, 48048}, {47909, 47993}, {47928, 48001}, {47945, 48028}, {48008, 8689}, {48023, 4806}, {48077, 18004}, {48108, 54265}, {48122, 4992}, {48143, 49292}, {48160, 48547}, {48225, 4448}, {48265, 59590}, {48288, 4794}, {48301, 48305}, {48321, 1960}, {48326, 47123}, {48409, 50507}, {49278, 49290}, {50328, 3835}, {50338, 6133}, {50339, 48008}, {50341, 650}, {50343, 4782}, {50348, 676}, {50356, 9508}, {50359, 4369}, {58374, 24720}
X(69336) = anticomplement of X(50335)
X(69336) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {30554, 2}, {60624, 3448}, {60873, 150}
X(69336) = crosssum of X(i) and X(j) for these (i,j): {42, 69101}, {667, 5332}
X(69336) = crossdifference of every pair of points on line {213, 995}
X(69336) = barycentric product X(514)*X(32941)
X(69336) = barycentric quotient X(32941)/X(190)
X(69336) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {650, 50341, 48225}, {659, 50339, 48008}, {676, 50348, 48227}, {1491, 3716, 47822}, {2254, 4874, 47823}, {4448, 50341, 650}, {4800, 50328, 3835}, {4913, 53580, 48226}, {7662, 21146, 48238}, {8689, 48008, 659}, {46403, 48172, 48090}, {47804, 50356, 9508}, {47805, 50343, 4782}, {47833, 58374, 24720}, {48008, 48063, 8689}, {48108, 48237, 54265}, {48234, 50359, 4369}

X(69337) = ODD<1, -1, 0, 1, 1, 1> POINT

Barycentrics    (b - c)*(a^3 - a^2*b + b^3 - a^2*c + a*b*c + b^2*c + b*c^2 + c^3) : :
X(69337) = X[4983] - 3 X[30605], X[48083] - 4 X[49279], 5 X[4367] - 4 X[39545], 2 X[39545] - 5 X[48290], 8 X[39545] - 5 X[50342], 4 X[48290] - X[50342], 2 X[4147] - 3 X[48185], 3 X[4800] - 4 X[4990], 3 X[4800] - 2 X[48400], 2 X[7178] - 3 X[47833], 2 X[10015] - 3 X[47872], 3 X[14419] - 2 X[21192], 3 X[14432] - X[21124], 2 X[21051] - 3 X[57066], X[24097] + 2 X[49275], 3 X[30565] - 2 X[48401], 3 X[47807] - 2 X[55285], 3 X[47839] - 2 X[50453], 3 X[48226] - 2 X[60492], 3 X[48569] - 2 X[62435], 3 X[53356] - 4 X[59743]

X(69337) lies on these lines: {1, 826}, {512, 47682}, {514, 4010}, {523, 4833}, {525, 4367}, {649, 29284}, {659, 3910}, {663, 29017}, {667, 23876}, {690, 1019}, {693, 29082}, {814, 47728}, {900, 31291}, {918, 48323}, {1491, 6332}, {1577, 29094}, {1960, 29256}, {2533, 2785}, {2787, 7265}, {3566, 4784}, {3904, 63812}, {3906, 48328}, {3907, 4122}, {4040, 29312}, {4064, 38469}, {4083, 48103}, {4147, 48185}, {4151, 50351}, {4162, 4777}, {4170, 29029}, {4378, 23875}, {4382, 29244}, {4449, 62423}, {4560, 38348}, {4707, 52601}, {4774, 28473}, {4775, 29021}, {4800, 4990}, {4810, 29162}, {4843, 50339}, {4897, 59549}, {4922, 29037}, {4978, 29102}, {6004, 49278}, {6367, 47683}, {6372, 49276}, {7178, 47833}, {7192, 59629}, {7927, 47726}, {7950, 47727}, {8678, 49280}, {10015, 47872}, {14419, 21192}, {14432, 21124}, {17166, 49274}, {17762, 18077}, {21051, 57066}, {21343, 29288}, {23755, 54265}, {23877, 48301}, {23879, 48288}, {23887, 48305}, {24097, 49275}, {25259, 29324}, {26148, 47660}, {29025, 47684}, {29047, 48333}, {29074, 47729}, {29116, 48349}, {29120, 48080}, {29142, 48336}, {29144, 48338}, {29152, 48266}, {29154, 47712}, {29166, 58160}, {29168, 48352}, {29172, 47708}, {29188, 47715}, {29196, 48285}, {29200, 48144}, {29202, 48330}, {29220, 48295}, {29224, 47716}, {29226, 48094}, {29246, 47719}, {29252, 48320}, {29272, 47680}, {29280, 48344}, {29298, 47711}, {29304, 50352}, {29318, 48294}, {29332, 47691}, {29354, 48282}, {29358, 48287}, {29366, 47690}, {30520, 48346}, {30565, 48401}, {47807, 55285}, {47839, 50453}, {47958, 48129}, {47968, 48131}, {47973, 48137}, {48226, 60492}, {48569, 62435}, {51642, 57243}, {53356, 59743}, {68155, 68157}

X(69337) = midpoint of X(i) and X(j) for these {i,j}: {17166, 49274}, {47726, 48337}
X(69337) = reflection of X(i) in X(j) for these {i,j}: {659, 48299}, {1491, 6332}, {1577, 49290}, {2533, 8045}, {4367, 48290}, {4707, 52601}, {4774, 48395}, {4879, 65685}, {23755, 54265}, {47727, 48347}, {47944, 48123}, {47958, 48129}, {47968, 48131}, {47973, 48137}, {48103, 48300}, {48267, 49288}, {48400, 4990}, {50340, 663}, {50342, 4367}
X(69337) = X(54120)-anticomplementary conjugate of X(21294)
X(69337) = crosspoint of X(99) and X(257)
X(69337) = crosssum of X(172) and X(512)
X(69337) = crossdifference of every pair of points on line {20461, 60697}
X(69337) = barycentric product X(514)*X(33160)
X(69337) = barycentric quotient X(33160)/X(190)
X(69337) = {X(4990),X(48400)}-harmonic conjugate of X(4800)

X(69338) = ODD<1, -1, 0, 1, 1, -1> POINT

Barycentrics    (b - c)*(a^3 - a^2*b + b^3 - a^2*c + a*b*c - b^2*c - b*c^2 + c^3) : :
X(69338) = 3 X[36848] - 4 X[44314], 4 X[676] - 3 X[25569], 2 X[1125] - 3 X[4049], 4 X[1125] - 3 X[30580], 2 X[24093] - 3 X[60480], 3 X[1577] - 2 X[49290], 5 X[1698] - 3 X[62634], 4 X[2496] - 3 X[48329], 2 X[4458] - 3 X[21145], X[4922] - 3 X[21145], 3 X[4448] - 2 X[5592], 3 X[5587] - 2 X[68333], 3 X[6545] - X[21105], 3 X[8029] - 2 X[50574], and many others

X(69338) lies on these lines: {8, 523}, {10, 514}, {56, 4367}, {65, 513}, {79, 15475}, {257, 66267}, {512, 5903}, {522, 66251}, {649, 29156}, {650, 16605}, {659, 10015}, {661, 17451}, {676, 25569}, {885, 17097}, {891, 47680}, {918, 24835}, {1019, 3336}, {1022, 18011}, {1125, 4049}, {1220, 4581}, {1482, 4879}, {1577, 29094}, {1698, 62634}, {1909, 66286}, {2496, 48329}, {2605, 28082}, {2785, 4010}, {2787, 4707}, {2789, 4458}, {2826, 58374}, {3762, 29102}, {3801, 3907}, {3837, 3904}, {3959, 63462}, {4063, 29336}, {4124, 42753}, {4374, 33930}, {4378, 37609}, {4391, 29082}, {4448, 5592}, {4474, 62423}, {4498, 29244}, {4543, 28165}, {4761, 29029}, {4777, 49468}, {4784, 29126}, {4789, 17230}, {4791, 49279}, {4834, 29114}, {4874, 47728}, {5098, 20271}, {5466, 6625}, {5587, 68333}, {6161, 21201}, {6366, 21343}, {6545, 21105}, {7192, 17103}, {7199, 20955}, {8029, 50574}, {9508, 30574}, {9780, 28602}, {11236, 48047}, {11280, 48337}, {11681, 21051}, {14430, 48056}, {14450, 59629}, {14804, 39577}, {14888, 53578}, {16020, 47799}, {16816, 47782}, {16825, 48288}, {18004, 30709}, {18123, 35364}, {19931, 19947}, {21129, 28195}, {21222, 58375}, {23752, 38469}, {24719, 28468}, {28213, 63246}, {28537, 48332}, {29066, 50340}, {29122, 48106}, {29124, 50509}, {29154, 47711}, {29166, 47723}, {29172, 47690}, {29298, 47712}, {29304, 48267}, {29312, 47724}, {29332, 47707}, {29362, 47722}, {29366, 47708}, {29579, 47788}, {29674, 47682}, {35050, 50330}, {37559, 57076}, {41015, 55261}, {44553, 64913}, {47684, 48405}, {47797, 48289}, {47833, 48290}, {47836, 59743}, {47872, 48299}, {47887, 48344}, {48227, 48325}, {48291, 49458}, {50343, 53356}, {57066, 59521}

X(69338) = midpoint of X(8) and X(49303)
X(69338) = reflection of X(i) in X(j) for these {i,j}: {659, 10015}, {3904, 3837}, {4367, 7178}, {4879, 48403}, {4922, 4458}, {6161, 21201}, {21222, 58375}, {21343, 23770}, {24097, 764}, {30580, 4049}, {47684, 48405}, {47728, 4874}, {48083, 3762}, {48288, 50453}, {48336, 48400}, {49274, 18004}, {49279, 4791}, {50342, 4707}, {50351, 10}, {62323, 66285}
X(69338) = X(i)-Ceva conjugate of X(j) for these (i,j): {4998, 1086}, {39185, 17719}
X(69338) = X(7336)-Dao conjugate of X(11)
X(69338) = crosspoint of X(17719) and X(39185)
X(69338) = crossdifference of every pair of points on line {1914, 2323}
X(69338) = barycentric product X(i)*X(j) for these {i,j}: {514, 17719}, {1086, 39185}
X(69338) = barycentric quotient X(i)/X(j) for these {i,j}: {17719, 190}, {39185, 1016}
X(69338) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4922, 21145, 4458}, {30709, 49274, 18004}

X(69339) = ODD<1, -1, 0, 1, 0, 1> POINT

Barycentrics    (b - c)*(a^3 - a^2*b - a^2*c + a*b*c + b^2*c + b*c^2) : :
X(69339) = X[8] - 3 X[21052], X[3716] + 2 X[48295], 3 X[3716] - 2 X[59672], 3 X[48295] + X[59672], X[649] - 3 X[47820], X[656] - 3 X[48209], X[661] - 3 X[47840], X[17166] + 3 X[47840], 4 X[1125] - X[4913], X[1491] - 3 X[47841], 3 X[47841] + X[48301], X[1734] - 3 X[47795], 2 X[25380] - 3 X[47795], 2 X[25380] + X[48339], 3 X[47795] + X[48339], and many others

X(69339) lies on these lines: {1, 810}, {2, 4041}, {8, 21052}, {11, 46670}, {238, 21761}, {512, 4369}, {513, 4992}, {514, 3716}, {519, 45324}, {522, 905}, {523, 8045}, {525, 4458}, {551, 64934}, {649, 47820}, {656, 48209}, {659, 48279}, {661, 17166}, {663, 693}, {667, 812}, {676, 3910}, {811, 22456}, {814, 48090}, {826, 49290}, {830, 48050}, {918, 4990}, {1001, 21789}, {1019, 4170}, {1086, 24234}, {1125, 4151}, {1215, 29110}, {1459, 7650}, {1491, 47841}, {1734, 25380}, {1848, 54247}, {1960, 29070}, {2254, 47796}, {2517, 22090}, {2530, 48305}, {2533, 4879}, {2605, 30591}, {2785, 7178}, {2787, 4504}, {3250, 68803}, {3309, 24720}, {3485, 51664}, {3616, 4560}, {3700, 29037}, {3737, 4815}, {3741, 21259}, {3762, 48282}, {3777, 65482}, {3810, 21185}, {3835, 8678}, {3842, 58304}, {3887, 50337}, {3900, 4885}, {3904, 21118}, {3960, 8714}, {3976, 66938}, {4010, 4367}, {4017, 7253}, {4036, 48292}, {4040, 4978}, {4063, 47818}, {4077, 58329}, {4083, 4874}, {4086, 48293}, {4088, 57066}, {4106, 50517}, {4129, 4160}, {4147, 14077}, {4162, 45320}, {4378, 48267}, {4379, 48338}, {4380, 58140}, {4382, 8643}, {4391, 4449}, {4401, 4830}, {4490, 47822}, {4498, 47804}, {4529, 45902}, {4705, 25666}, {4724, 4801}, {4728, 21301}, {4729, 24924}, {4730, 47837}, {4761, 48337}, {4762, 45316}, {4763, 31288}, {4770, 65449}, {4775, 50352}, {4776, 47912}, {4785, 50515}, {4791, 48287}, {4794, 29186}, {4800, 48265}, {4811, 43924}, {4822, 7192}, {4823, 29066}, {4843, 17069}, {4895, 21302}, {4928, 21260}, {4931, 32771}, {4983, 28840}, {4985, 48281}, {5886, 39212}, {6004, 23815}, {6050, 48008}, {6182, 46399}, {6332, 23877}, {6362, 11281}, {7199, 57113}, {10026, 40466}, {10099, 12053}, {10180, 42653}, {10453, 21300}, {14413, 17496}, {14432, 55282}, {15283, 65445}, {15313, 47843}, {17420, 48173}, {17458, 47127}, {18154, 63461}, {18155, 51641}, {20295, 50523}, {20517, 23876}, {21124, 47797}, {21146, 47889}, {21297, 31291}, {21343, 47872}, {21385, 47817}, {22093, 37607}, {22318, 59302}, {23789, 42325}, {23794, 54251}, {23875, 49288}, {23880, 48325}, {23882, 48394}, {24325, 64859}, {24718, 50608}, {25055, 45671}, {28470, 48327}, {28521, 48184}, {28623, 51648}, {29033, 65428}, {29116, 47682}, {29148, 48343}, {29188, 58160}, {29208, 48405}, {29246, 48098}, {29288, 69293}, {29298, 48347}, {29324, 48344}, {29362, 48331}, {29512, 50493}, {30023, 50502}, {30584, 48395}, {30724, 50357}, {30835, 47814}, {31251, 45678}, {31286, 48564}, {37617, 42767}, {39547, 50330}, {43067, 50508}, {43223, 57131}, {44315, 50348}, {44448, 47806}, {45315, 48005}, {45328, 48018}, {45337, 50760}, {46402, 65664}, {46403, 48150}, {47691, 48300}, {47694, 48131}, {47695, 48278}, {47697, 48122}, {47707, 47874}, {47711, 47727}, {47713, 47726}, {47719, 47972}, {47720, 48094}, {47761, 50499}, {47762, 50509}, {47793, 48304}, {47800, 60492}, {47821, 47918}, {47823, 50355}, {47834, 50457}, {47838, 47959}, {47844, 50332}, {47875, 48333}, {47893, 50341}, {47991, 48053}, {47992, 48054}, {48000, 50507}, {48003, 48562}, {48079, 50526}, {48080, 48144}, {48089, 48329}, {48100, 64914}, {48108, 48367}, {48151, 53343}, {48183, 48401}, {48189, 48392}, {48207, 57099}, {48246, 50338}, {48272, 64860}, {48280, 50347}, {48283, 50327}, {48288, 48393}, {48302, 50334}, {48417, 59673}, {49285, 59980}, {53527, 64905}, {59837, 64868}, {64857, 68774}

X(69339) = midpoint of X(i) and X(j) for these {i,j}: {1, 1577}, {659, 48279}, {661, 17166}, {663, 693}, {667, 48273}, {1019, 4170}, {1459, 7650}, {1491, 48301}, {1734, 48339}, {2517, 48303}, {2530, 48305}, {2533, 4879}, {2605, 30591}, {3737, 4815}, {3762, 48282}, {3904, 21118}, {4010, 4367}, {4017, 7253}, {4036, 48292}, {4040, 4978}, {4077, 58329}, {4086, 48293}, {4106, 50517}, {4378, 48267}, {4391, 4449}, {4560, 4804}, {4705, 48291}, {4724, 4801}, {4761, 48337}, {4775, 50352}, {4791, 48287}, {4811, 43924}, {4822, 7192}, {4823, 48294}, {4895, 21302}, {4985, 48281}, {6332, 47123}, {7178, 65685}, {7662, 48136}, {14413, 48172}, {17496, 48264}, {18155, 51641}, {20295, 50523}, {21146, 48336}, {21301, 48322}, {23770, 48299}, {23794, 54251}, {39547, 50330}, {43067, 50508}, {46402, 65664}, {46403, 48150}, {47682, 47712}, {47691, 48300}, {47694, 48131}, {47695, 48278}, {47697, 48122}, {47711, 47727}, {47713, 47726}, {47719, 47972}, {47720, 48094}, {47844, 50332}, {48079, 50526}, {48080, 48144}, {48089, 48329}, {48090, 48330}, {48108, 48367}, {48151, 53343}, {48265, 48323}, {48280, 50347}, {48283, 50327}, {48288, 48393}, {48290, 48403}, {48302, 50334}
X(69339) = reflection of X(i) in X(j) for these {i,j}: {1734, 25380}, {3777, 65482}, {4142, 676}, {4369, 52601}, {4458, 34958}, {4504, 48328}, {4705, 25666}, {4770, 65449}, {4830, 4401}, {4913, 14838}, {14838, 1125}, {17072, 4885}, {47991, 48053}, {47992, 48054}, {48000, 50507}, {48001, 48058}, {48008, 6050}, {50501, 31286}, {50504, 31288}
X(69339) = complement of X(4041)
X(69339) = complement of the isogonal conjugate of X(1414)
X(69339) = complement of the isotomic conjugate of X(4625)
X(69339) = X(i)-complementary conjugate of X(j) for these (i,j): {7, 125}, {21, 5514}, {28, 6506}, {56, 115}, {57, 8287}, {58, 1146}, {59, 661}, {60, 34591}, {77, 34846}, {81, 26932}, {85, 21253}, {86, 124}, {99, 1329}, {101, 38930}, {107, 15849}, {108, 50036}, {109, 1213}, {110, 9}, {112, 46835}, {162, 20262}, {163, 1212}, {222, 15526}, {269, 8286}, {283, 40616}, {284, 13609}, {348, 127}, {552, 53564}, {593, 4858}, {603, 16573}, {604, 16592}, {608, 6388}, {648, 41883}, {651, 1211}, {658, 17052}, {662, 3452}, {664, 3454}, {757, 34589}, {799, 21244}, {859, 55153}, {934, 442}, {1014, 11}, {1106, 16613}, {1262, 1577}, {1397, 1084}, {1400, 6627}, {1401, 15449}, {1407, 17058}, {1408, 1015}, {1412, 1086}, {1414, 10}, {1415, 16589}, {1434, 116}, {1437, 35072}, {1444, 123}, {1461, 17056}, {1576, 16588}, {1790, 16596}, {1804, 122}, {1813, 440}, {2194, 35508}, {2360, 61075}, {2720, 2245}, {3286, 1566}, {3450, 20982}, {3649, 65787}, {3665, 46654}, {3733, 46101}, {4017, 24040}, {4554, 21245}, {4556, 5745}, {4559, 6537}, {4564, 4129}, {4565, 2}, {4566, 34829}, {4567, 20317}, {4570, 4521}, {4573, 141}, {4592, 34823}, {4600, 59971}, {4610, 21246}, {4616, 2886}, {4617, 18635}, {4620, 3835}, {4622, 5123}, {4625, 2887}, {4627, 18228}, {4635, 17046}, {4636, 59646}, {4637, 142}, {4998, 31946}, {5061, 41179}, {5323, 5517}, {5545, 1698}, {5546, 6554}, {6063, 53575}, {6516, 21530}, {6578, 18253}, {6614, 1834}, {7055, 55069}, {7125, 16595}, {7180, 23991}, {7181, 5099}, {7192, 46100}, {7198, 46665}, {7316, 1648}, {7335, 35071}, {7338, 13613}, {7339, 656}, {7340, 512}, {7341, 244}, {8059, 1901}, {8687, 52538}, {8822, 46663}, {16947, 6377}, {17081, 5139}, {17139, 63757}, {18191, 34530}, {26700, 5949}, {30493, 39019}, {32676, 20310}, {32735, 2238}, {36059, 18591}, {36069, 7359}, {36075, 51586}, {37137, 46826}, {43034, 35088}, {44717, 52599}, {52378, 514}, {52920, 9119}, {52935, 960}, {53243, 46196}, {54951, 56861}, {55194, 27076}, {55213, 21235}, {56053, 20545}, {57785, 21252}, {59065, 26066}, {59069, 17303}, {59075, 7110}, {59124, 17369}, {63190, 3139}, {65232, 226}, {65296, 18642}, {68232, 48888}
X(69339) = X(32932)-Ceva conjugate of X(23774)
X(69339) = X(i)-cross conjugate of X(j) for these (i,j): {21960, 24622}, {23774, 32932}
X(69339) = crosspoint of X(i) and X(j) for these (i,j): {2, 4625}, {7, 65332}
X(69339) = crosssum of X(6) and X(63461)
X(69339) = crossdifference of every pair of points on line {198, 1755}
X(69339) = barycentric product X(i)*X(j) for these {i,j}: {1, 24622}, {86, 21960}, {190, 23774}, {514, 32932}, {799, 65546}
X(69339) = barycentric quotient X(i)/X(j) for these {i,j}: {21960, 10}, {23774, 514}, {24622, 75}, {32932, 190}, {65546, 661}
X(69339) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1734, 47795, 25380}, {4449, 47832, 4391}, {4705, 47839, 25666}, {4728, 48322, 21301}, {4729, 24924, 47836}, {4800, 48323, 48265}, {4879, 47833, 2533}, {14413, 48264, 17496}, {17166, 47840, 661}, {17496, 48172, 48264}, {31288, 50504, 4763}, {47795, 48339, 1734}, {47839, 48291, 4705}, {47841, 48301, 1491}, {47889, 48336, 21146}, {48564, 50501, 31286}

X(69340) = ODD<1, -1, 0, 1, -1, 1 > POINT

Barycentrics    (b - c)*(-a^3 + a^2*b + b^3 + a^2*c - a*b*c - b^2*c - b*c^2 + c^3) : :
X(69340) = X[47782] - 3 X[47797], 2 X[47788] - 3 X[47833], X[47792] - 3 X[47834], X[47792] + 3 X[48203], X[47827] - 4 X[68933], X[659] - 4 X[676], X[659] + 2 X[23770], 2 X[676] + X[23770], 2 X[693] + X[50340], X[764] + 2 X[21201], 3 X[6548] + X[53361], X[4367] - 4 X[34958], X[4367] + 2 X[48403], 2 X[34958] + X[48403], 4 X[1125] - X[50351], and many others

X(69340) lies on these lines: {2, 523}, {105, 659}, {115, 23894}, {350, 66286}, {354, 513}, {512, 5902}, {514, 551}, {519, 4049}, {522, 21204}, {690, 50886}, {693, 26234}, {764, 21201}, {812, 4809}, {824, 48189}, {850, 4485}, {876, 3675}, {900, 903}, {918, 4800}, {999, 4367}, {1022, 6550}, {1125, 50351}, {1290, 13589}, {1491, 29639}, {1577, 4692}, {1638, 48244}, {1647, 42754}, {1960, 47680}, {2099, 4879}, {2785, 21145}, {2786, 4010}, {3004, 47132}, {3290, 55261}, {3570, 18014}, {3616, 49303}, {3676, 50359}, {3716, 28890}, {3737, 29820}, {3777, 21185}, {3837, 31126}, {3938, 48292}, {3961, 48293}, {4120, 45342}, {4132, 64550}, {4142, 48279}, {4369, 48349}, {4375, 28859}, {4379, 29144}, {4688, 4777}, {4762, 48211}, {4774, 36479}, {4784, 4860}, {4802, 6546}, {4874, 26230}, {4885, 30748}, {4893, 48195}, {4928, 64860}, {4951, 47787}, {4977, 48161}, {6544, 28151}, {7192, 51356}, {7662, 28894}, {8227, 68333}, {8643, 29244}, {9269, 30573}, {9508, 53558}, {10015, 21343}, {10196, 28147}, {10247, 28473}, {14015, 17925}, {14425, 45318}, {15475, 50148}, {16064, 48382}, {17069, 50339}, {17780, 66285}, {19947, 66995}, {20271, 63462}, {20508, 24768}, {20517, 48273}, {21130, 50760}, {21131, 36227}, {21132, 24097}, {21181, 68968}, {21188, 50355}, {21198, 30583}, {21212, 50341}, {21828, 47227}, {23598, 33920}, {23838, 65866}, {23877, 47841}, {24331, 48288}, {24403, 24405}, {25055, 62634}, {25381, 48394}, {25569, 29240}, {26985, 31077}, {27728, 53342}, {27929, 49292}, {28161, 59755}, {28179, 31992}, {28878, 48024}, {29025, 47820}, {29047, 47875}, {29098, 47818}, {29128, 47712}, {29142, 47889}, {29204, 47874}, {29288, 47872}, {29362, 47798}, {29370, 47790}, {29652, 39547}, {29659, 47727}, {29660, 47682}, {29823, 47694}, {30565, 48183}, {30787, 30790}, {36480, 48291}, {37165, 62494}, {44433, 47871}, {44435, 64914}, {45314, 47892}, {45665, 64859}, {45668, 45674}, {45677, 48182}, {45684, 45685}, {47652, 48248}, {47692, 48405}, {47701, 54265}, {47705, 48056}, {47779, 48235}, {47780, 48158}, {47800, 48226}, {47803, 47885}, {47808, 48198}, {47828, 48215}, {47831, 48185}, {47832, 62423}, {47877, 48192}, {47879, 48188}, {47882, 48225}, {47972, 48098}, {47999, 48153}, {48006, 48143}, {48120, 62552}, {48160, 48178}, {48162, 48179}, {48163, 50328}, {48170, 48239}, {48172, 48241}, {48174, 48237}, {48233, 48252}, {48323, 48400}, {48398, 50358}, {50343, 69011}, {52135, 66267}, {53343, 58375}, {53523, 58374}, {53533, 65482}, {59887, 64917}

X(69340) = midpoint of X(i) and X(j) for these {i,j}: {693, 48223}, {4800, 58372}, {21130, 50760}, {23770, 26275}, {31131, 47695}, {44433, 47871}, {47123, 47757}, {47131, 48200}, {47691, 47771}, {47694, 48156}, {47780, 48158}, {47834, 48203}, {48170, 48239}, {48172, 48241}, {48174, 48237}, {48189, 48224}
X(69340) = reflection of X(i) in X(j) for these {i,j}: {659, 26275}, {1491, 47757}, {4120, 45342}, {4893, 48195}, {4948, 47784}, {4951, 47787}, {6546, 45666}, {14425, 45318}, {26275, 676}, {30565, 48183}, {30573, 9269}, {30580, 551}, {30583, 21198}, {31131, 3837}, {36848, 21204}, {45674, 45668}, {47771, 4874}, {47799, 68933}, {47808, 48198}, {47809, 48206}, {47827, 47799}, {47828, 48215}, {47874, 48202}, {47877, 48192}, {47885, 47803}, {47886, 48212}, {47892, 45314}, {47968, 48156}, {48103, 47771}, {48160, 48178}, {48162, 48179}, {48167, 4927}, {48182, 45677}, {48185, 47831}, {48188, 47879}, {48200, 4885}, {48225, 47882}, {48226, 47800}, {48235, 47779}, {48244, 1638}, {48252, 48233}, {50328, 48163}, {50333, 30792}, {50340, 48223}, {53342, 27728}, {53372, 18005}
X(69340) = anticomplement of X(28602)
X(69340) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1929, 66862}, {2702, 30578}, {4591, 20538}, {4615, 20560}, {4622, 20351}, {9456, 39368}, {37135, 21290}
X(69340) = X(36236)-Ceva conjugate of X(24715)
X(69340) = crosspoint of X(i) and X(j) for these (i,j): {105, 1290}, {335, 4555}, {903, 35148}, {927, 2006}, {6548, 62635}, {24715, 36236}
X(69340) = crosssum of X(i) and X(j) for these (i,j): {518, 8674}, {902, 5029}, {926, 2323}, {1914, 1960}, {2284, 23344}
X(69340) = crossdifference of every pair of points on line {187, 1017}
X(69340) = barycentric product X(i)*X(j) for these {i,j}: {514, 24715}, {523, 24617}, {1086, 36236}, {4444, 24428}
X(69340) = barycentric quotient X(i)/X(j) for these {i,j}: {24428, 3570}, {24617, 99}, {24715, 190}, {36236, 1016}
X(69340) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {676, 23770, 659}, {3716, 48326, 48083}, {4010, 4458, 50342}, {4874, 47691, 48103}, {34958, 48403, 4367}

X(69341) = ODD<1, -1, 0, 1, -1, -1> POINT

Barycentrics    (b - c)*(-a^3 + a^2*b + b^3 + a^2*c - a*b*c + b^2*c + b*c^2 + c^3) : :
X(69341) = 2 X[47727] + X[50342], 4 X[676] - 3 X[47872], X[4462] - 3 X[48223], 2 X[4522] - 3 X[47841], 4 X[6050] - 3 X[47885], 3 X[8643] - X[48118], 2 X[17072] - 3 X[48227], 2 X[18004] - 3 X[47840], 2 X[20317] - 3 X[48211], 2 X[21051] - 3 X[47797], X[21302] - 3 X[48241], 3 X[25569] - 2 X[48299], 4 X[34958] - 3 X[47833], 3 X[47833] - 2 X[48395], X[47706] - 3 X[47820], 3 X[47820] - 2 X[48405], 3 X[47836] - 4 X[69011], 3 X[47889] - 2 X[48396], 3 X[47893] - 2 X[50333]

X(69341) lies on these lines: {1, 826}, {512, 47727}, {514, 4922}, {522, 3777}, {523, 1325}, {525, 4879}, {596, 876}, {649, 29208}, {659, 29288}, {663, 62423}, {667, 29047}, {676, 47872}, {690, 48337}, {693, 29074}, {814, 47691}, {824, 48301}, {830, 47968}, {918, 48336}, {1019, 7927}, {1577, 4692}, {2533, 4458}, {2787, 47712}, {3669, 4777}, {3800, 4784}, {3801, 3907}, {3906, 48347}, {3910, 21343}, {4010, 29037}, {4025, 50355}, {4040, 29354}, {4041, 24443}, {4170, 29090}, {4378, 29021}, {4382, 29276}, {4449, 29017}, {4462, 48223}, {4490, 68780}, {4504, 29116}, {4522, 47841}, {4707, 29298}, {4730, 21192}, {4774, 7178}, {4775, 23875}, {4802, 50517}, {4808, 14838}, {4810, 29232}, {4874, 47707}, {4978, 29086}, {4988, 66523}, {6002, 48349}, {6050, 47885}, {7265, 29292}, {7950, 47682}, {8643, 48118}, {17072, 48227}, {18004, 47840}, {20317, 48211}, {21051, 47797}, {21302, 48241}, {23770, 29278}, {23876, 48333}, {23879, 48291}, {23880, 47131}, {25569, 48299}, {29025, 47692}, {29029, 47713}, {29051, 48326}, {29062, 48273}, {29070, 47716}, {29082, 47729}, {29098, 47717}, {29120, 47709}, {29142, 48323}, {29144, 48144}, {29146, 48344}, {29164, 48343}, {29168, 48320}, {29182, 47680}, {29192, 50352}, {29196, 48295}, {29198, 47972}, {29200, 48338}, {29204, 48300}, {29212, 48267}, {29220, 48285}, {29246, 47676}, {29250, 47690}, {29252, 48352}, {29256, 48296}, {29312, 48282}, {29318, 48287}, {29324, 47708}, {29332, 47728}, {29336, 47725}, {29358, 48294}, {29362, 47720}, {30520, 48329}, {31291, 47688}, {34958, 47833}, {47123, 48392}, {47687, 48406}, {47695, 63812}, {47706, 47820}, {47711, 52601}, {47836, 69011}, {47889, 48396}, {47893, 50333}, {47905, 47999}, {47913, 48006}, {48077, 48100}, {48094, 48331}, {48136, 64856}, {48277, 54249}, {48286, 48305}, {48322, 68979}, {49276, 58160}

X(69341) = midpoint of X(31291) and X(47688)
X(69341) = reflection of X(i) in X(j) for these {i,j}: {2533, 4458}, {4490, 68780}, {4730, 21192}, {4774, 7178}, {4808, 14838}, {47682, 48328}, {47687, 48406}, {47706, 48405}, {47707, 4874}, {47711, 52601}, {47905, 47999}, {47913, 48006}, {48077, 48100}, {48083, 4040}, {48094, 48331}, {48103, 667}, {48300, 48330}, {48305, 48286}, {48392, 47123}, {48395, 34958}, {49276, 58160}, {49279, 48294}, {50355, 4025}
X(69341) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {4556, 20934}, {7194, 3448}, {39724, 21294}, {65364, 1330}
X(69341) = crosssum of X(512) and X(5299)
X(69341) = barycentric product X(514)*X(33079)
X(69341) = barycentric quotient X(33079)/X(190)
X(69341) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {34958, 48395, 47833}, {47706, 47820, 48405}

X(69342) = ODD<1, -1, 0, 0, 0, 1> POINT

Barycentrics    (b - c)*(a^3 - a^2*b - a^2*c + b^2*c + b*c^2) : :
X(69342) = 2 X[3716] + X[48295], 2 X[676] + X[49288], 2 X[4990] + X[20517], X[649] - 3 X[47818], X[4170] + 3 X[47818], X[656] - 3 X[48186], X[661] - 3 X[47838], X[663] + 3 X[47832], X[1577] - 3 X[47832], X[1019] - 3 X[47820], 3 X[47820] + X[48080], X[1491] - 3 X[47839], 3 X[47839] + X[48305], X[1769] + 3 X[45686], X[2254] - 3 X[47795], and many others

X(69342) lies on these lines: {1, 4391}, {2, 1734}, {10, 3900}, {37, 52589}, {86, 50450}, {350, 40495}, {512, 4874}, {513, 11813}, {514, 3716}, {519, 45664}, {522, 8062}, {523, 50507}, {525, 676}, {649, 4170}, {650, 4151}, {656, 48186}, {659, 29302}, {661, 47838}, {663, 1577}, {667, 4010}, {693, 4040}, {812, 4401}, {814, 1960}, {830, 3835}, {905, 1125}, {918, 34958}, {1001, 22160}, {1019, 47820}, {1215, 4944}, {1459, 4985}, {1491, 47839}, {1769, 45686}, {1946, 5248}, {2254, 47795}, {2517, 48307}, {2530, 47841}, {2533, 4775}, {2605, 50327}, {2787, 48330}, {2820, 46399}, {2887, 11193}, {2901, 21831}, {3063, 21960}, {3239, 6591}, {3309, 4885}, {3616, 17496}, {3624, 50356}, {3669, 59590}, {3700, 29062}, {3737, 7650}, {3741, 59754}, {3762, 4449}, {3800, 68794}, {3801, 29220}, {3803, 4106}, {3810, 21201}, {3837, 6004}, {3887, 17072}, {3907, 4791}, {4036, 48302}, {4041, 47794}, {4063, 47804}, {4086, 48303}, {4122, 29196}, {4129, 8678}, {4139, 6133}, {4142, 23876}, {4160, 48547}, {4367, 4800}, {4369, 6005}, {4378, 48265}, {4379, 48367}, {4448, 48279}, {4458, 23875}, {4462, 48282}, {4490, 48291}, {4498, 47817}, {4560, 48172}, {4705, 47822}, {4724, 4978}, {4728, 48150}, {4730, 47835}, {4761, 48338}, {4776, 47948}, {4794, 4823}, {4801, 47970}, {4808, 48185}, {4811, 21173}, {4815, 46385}, {4822, 47813}, {4879, 47872}, {4893, 48407}, {4895, 21052}, {4905, 47796}, {4992, 48248}, {6332, 21185}, {6554, 40465}, {7178, 29304}, {7192, 48081}, {7253, 21189}, {7265, 29294}, {7649, 57065}, {7927, 48405}, {8045, 29021}, {8642, 27731}, {9508, 31288}, {10015, 65685}, {14077, 20317}, {14349, 47694}, {14432, 21118}, {15309, 48043}, {17166, 47821}, {17924, 58329}, {19883, 44561}, {20316, 35057}, {21051, 48183}, {21146, 48351}, {21188, 62435}, {21301, 48324}, {21385, 47815}, {23800, 48209}, {23879, 68780}, {23882, 48284}, {24003, 36954}, {24325, 28898}, {24720, 42325}, {24924, 48573}, {25055, 44550}, {25380, 48018}, {25666, 48012}, {25961, 30613}, {28143, 57180}, {28470, 48345}, {28840, 48045}, {29017, 49290}, {29047, 69293}, {29070, 48090}, {29130, 47682}, {29160, 47712}, {29182, 58156}, {29188, 48202}, {29190, 50347}, {29192, 48395}, {29324, 48328}, {29328, 50512}, {29340, 58150}, {29344, 65428}, {29366, 58160}, {30144, 42455}, {30591, 48297}, {30835, 47816}, {30963, 57110}, {39540, 57156}, {39585, 59935}, {44409, 57158}, {45316, 64934}, {45322, 50311}, {46403, 48111}, {47695, 48272}, {47697, 48086}, {47707, 47727}, {47709, 47726}, {47711, 47874}, {47715, 47972}, {47716, 48094}, {47717, 48118}, {47803, 50501}, {47833, 48336}, {47837, 50355}, {47888, 50341}, {47912, 48551}, {48059, 64914}, {48121, 48578}, {48123, 48234}, {48142, 50449}, {48179, 48402}, {48181, 57099}, {48189, 48393}, {48197, 65449}, {48207, 50350}, {48220, 50508}, {48228, 50338}, {48264, 48321}, {48288, 48392}, {48306, 50334}, {48327, 59737}, {48329, 59714}, {48564, 50336}, {48566, 50509}, {48569, 50359}, {50331, 50353}, {55180, 64185}, {64868, 68775}

X(69342) = midpoint of X(i) and X(j) for these {i,j}: {1, 4391}, {649, 4170}, {659, 48273}, {663, 1577}, {667, 4010}, {676, 4990}, {693, 4040}, {1019, 48080}, {1459, 4985}, {1491, 48305}, {2517, 48307}, {2533, 4775}, {2605, 50327}, {3669, 59590}, {3737, 7650}, {3762, 4449}, {3801, 49279}, {3803, 4106}, {4036, 48302}, {4041, 48339}, {4086, 48303}, {4367, 48267}, {4378, 48265}, {4462, 48282}, {4490, 48291}, {4705, 48301}, {4724, 4978}, {4761, 48338}, {4791, 48294}, {4794, 4823}, {4801, 47970}, {4811, 21173}, {4815, 46385}, {4905, 53343}, {4992, 48248}, {6332, 21185}, {7192, 48081}, {7253, 21189}, {7662, 48099}, {10015, 65685}, {14349, 47694}, {17166, 47959}, {17924, 58329}, {20517, 49288}, {21146, 48351}, {21301, 48324}, {30591, 48297}, {44409, 57158}, {46403, 48111}, {47682, 47708}, {47695, 48272}, {47697, 48086}, {47707, 47727}, {47709, 47726}, {47712, 48300}, {47715, 47972}, {47716, 48094}, {47717, 48118}, {48090, 48331}, {48142, 50449}, {48264, 48321}, {48288, 48392}, {48290, 48400}, {48295, 59672}, {48299, 48403}, {48306, 50334}, {48336, 50352}, {50331, 50353}
X(69342) = reflection of X(i) in X(j) for these {i,j}: {905, 1125}, {9508, 31288}, {20517, 676}, {48012, 25666}, {48018, 25380}, {48285, 48294}, {49288, 4990}, {50337, 4885}, {59672, 3716}, {62435, 21188}
X(69342) = complement of X(1734)
X(69342) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 40618}, {692, 40606}, {14377, 116}, {15320, 125}, {15378, 514}, {26705, 5}, {31616, 4422}, {31624, 626}, {35184, 516}, {43190, 141}, {57750, 3835}, {65554, 674}
X(69342) = crosspoint of X(86) and X(1897)
X(69342) = crosssum of X(42) and X(1459)
X(69342) = crossdifference of every pair of points on line {1473, 2178}
X(69342) = barycentric product X(514)*X(32929)
X(69342) = barycentric quotient X(32929)/X(190)
X(69342) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {663, 47832, 1577}, {4170, 47818, 649}, {4367, 4800, 48267}, {4775, 47875, 2533}, {7253, 48173, 21189}, {17166, 47821, 47959}, {21831, 52623, 2901}, {47694, 47840, 14349}, {47695, 57066, 48272}, {47794, 48339, 4041}, {47796, 53343, 4905}, {47820, 48080, 1019}, {47822, 48301, 4705}, {47833, 48336, 50352}, {47839, 48305, 1491}, {48018, 48218, 25380}, {48291, 48553, 4490}

X(69343) = ODD<1, -1, 0, 0, -1, 0> POINT

Barycentrics    (b - c)*(-a^3 + a^2*b + b^3 + a^2*c + c^3) : :
X(69343) = X[650] - 3 X[48211], 4 X[676] - X[48405], 3 X[676] - X[68794], 2 X[2977] - 3 X[48214], 3 X[4874] - 2 X[68794], 3 X[7662] - X[48397], X[45745] + 3 X[47123], X[45745] - 3 X[68780], X[47131] + 3 X[48211], 3 X[47800] - X[48062], 3 X[48405] - 4 X[68794], 3 X[4458] - X[69292], 2 X[21212] - 3 X[48212], 3 X[48212] - X[50335], and many others

X(69343) lies on these lines: {230, 231}, {512, 20517}, {513, 3776}, {514, 1960}, {522, 3837}, {649, 4809}, {659, 8654}, {661, 48177}, {663, 3801}, {667, 29025}, {693, 26234}, {814, 48403}, {824, 25368}, {900, 4025}, {1491, 47695}, {1577, 29074}, {2254, 48227}, {2526, 48192}, {3004, 64914}, {3239, 48183}, {3700, 29370}, {3716, 62423}, {3739, 4500}, {3906, 49288}, {4010, 29078}, {4024, 48189}, {4083, 4142}, {4088, 47822}, {4122, 47832}, {4367, 29120}, {4369, 29144}, {4401, 29098}, {4448, 48094}, {4453, 50359}, {4707, 4775}, {4724, 48326}, {4778, 48611}, {4782, 13246}, {4791, 29110}, {4794, 29102}, {4800, 25259}, {4808, 47794}, {4823, 29086}, {4977, 47961}, {7178, 29366}, {7192, 48158}, {7658, 48229}, {8045, 29146}, {11068, 45314}, {16892, 48224}, {18004, 64856}, {21121, 48302}, {21124, 48301}, {21130, 50767}, {21146, 47887}, {21185, 63812}, {22388, 39478}, {23093, 39200}, {23770, 29362}, {24924, 48235}, {25380, 48215}, {25569, 47728}, {25666, 48195}, {26275, 47890}, {27486, 50339}, {28161, 48206}, {28183, 48198}, {28209, 47950}, {28487, 48137}, {28602, 31287}, {29021, 52601}, {29094, 48294}, {29142, 34958}, {29158, 50512}, {29172, 48290}, {29184, 58150}, {29204, 69293}, {29250, 48395}, {29272, 58156}, {29304, 58160}, {29312, 48295}, {29318, 49290}, {29324, 48400}, {29332, 48299}, {29354, 59672}, {30795, 47808}, {30865, 47790}, {31250, 48200}, {44433, 47652}, {44435, 50328}, {45668, 59749}, {46403, 48239}, {47653, 47694}, {47660, 48234}, {47676, 58372}, {47687, 48184}, {47688, 47805}, {47690, 47833}, {47692, 47804}, {47696, 48251}, {47697, 47968}, {47698, 48162}, {47700, 48185}, {47701, 50522}, {47702, 47813}, {47703, 48238}, {47705, 47811}, {47707, 47872}, {47709, 47820}, {47711, 47875}, {47713, 47818}, {47717, 47817}, {47719, 47889}, {47725, 68816}, {47799, 50333}, {47839, 48272}, {47841, 48278}, {47886, 50341}, {47924, 48578}, {48023, 48552}, {48024, 48161}, {48039, 48555}, {48047, 48179}, {48080, 50342}, {48140, 48250}, {48226, 48408}, {48241, 53343}, {48286, 50453}, {48288, 49300}, {48398, 64913}, {50336, 69011}, {50348, 53523}, {50508, 59629}

X(69343) = midpoint of X(i) and X(j) for these {i,j}: {649, 48349}, {650, 47131}, {659, 47691}, {663, 3801}, {667, 47712}, {693, 50340}, {1491, 47695}, {4367, 47708}, {4707, 4775}, {4724, 48326}, {21121, 48302}, {21124, 48301}, {21146, 47972}, {23770, 50347}, {47123, 68780}, {47652, 50358}, {47692, 48103}, {47697, 47968}, {48080, 50342}, {48286, 50453}, {48288, 49300}, {50348, 53523}
X(69343) = reflection of X(i) in X(j) for these {i,j}: {4782, 13246}, {4874, 676}, {48198, 68933}, {48405, 4874}, {50335, 21212}, {50336, 69011}
X(69343) = crossdifference of every pair of points on line {3, 69282}
X(69343) = barycentric product X(i)*X(j) for these {i,j}: {514, 4660}, {523, 21997}
X(69343) = barycentric quotient X(i)/X(j) for these {i,j}: {4660, 190}, {21997, 99}
X(69343) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {693, 48223, 50340}, {4809, 48349, 649}, {44433, 47652, 50358}, {47131, 48211, 650}, {47691, 47798, 659}, {47692, 47804, 48103}, {47695, 47797, 1491}, {47697, 48174, 47968}, {47887, 47972, 21146}, {48212, 50335, 21212}

X(69344) = ODD<1, -1, 0, 0, -1, -1> POINT

Barycentrics    (b - c)*(-a^3 + a^2*b + b^3 + a^2*c + b^2*c + b*c^2 + c^3) : :
X(69344) = 4 X[676] - 3 X[47875], 3 X[47875] - 2 X[48395], 2 X[4129] - 3 X[48177], X[4391] - 3 X[48223], 2 X[4522] - 3 X[47839], 2 X[18004] - 3 X[47838], 2 X[21260] - 3 X[47797], X[21301] - 3 X[48203], 5 X[31251] - 6 X[47799], 4 X[31288] - 3 X[47809], X[47689] - 3 X[47820], X[47706] - 3 X[47804], X[47707] - 3 X[47798], X[47710] - 3 X[47818], 3 X[47818] - 2 X[48405], 3 X[47888] - 2 X[50333], 3 X[48227] - 2 X[50337], 2 X[48299] - 3 X[58155], 3 X[48573] - 4 X[69011]

X(69344) lies on these lines: {1, 29017}, {514, 4922}, {522, 2530}, {523, 667}, {525, 4775}, {649, 7927}, {650, 4808}, {659, 29047}, {663, 826}, {665, 48277}, {676, 47875}, {690, 48338}, {693, 29086}, {784, 47695}, {814, 47712}, {824, 48305}, {905, 4777}, {918, 48351}, {1019, 29144}, {1499, 58167}, {1577, 29074}, {1960, 7950}, {2533, 20517}, {2787, 47708}, {3566, 58165}, {3800, 4834}, {3801, 29066}, {3803, 4802}, {3906, 58160}, {3910, 48333}, {4010, 29062}, {4040, 62423}, {4063, 29208}, {4083, 47727}, {4088, 50507}, {4122, 29196}, {4129, 48177}, {4170, 29078}, {4367, 29021}, {4378, 29142}, {4391, 29110}, {4401, 29260}, {4449, 29312}, {4458, 50352}, {4522, 47839}, {4705, 68780}, {4707, 29366}, {4724, 29354}, {4794, 29358}, {4874, 29250}, {4879, 23876}, {6004, 16892}, {6005, 50342}, {6372, 47972}, {7265, 29370}, {8642, 50545}, {12073, 50509}, {14419, 28161}, {18004, 47838}, {21192, 50355}, {21260, 47797}, {21301, 48203}, {23815, 47687}, {23875, 48336}, {23877, 48288}, {23879, 48286}, {23882, 47131}, {25259, 29292}, {29013, 48349}, {29025, 47713}, {29029, 47709}, {29037, 48267}, {29070, 47691}, {29090, 48080}, {29094, 47729}, {29098, 47692}, {29146, 47682}, {29154, 47728}, {29166, 48328}, {29168, 48144}, {29186, 48326}, {29190, 48279}, {29200, 48352}, {29204, 48331}, {29212, 48265}, {29244, 47725}, {29252, 48367}, {29256, 48347}, {29274, 47680}, {29278, 48403}, {29280, 49276}, {29284, 48337}, {29288, 50347}, {29318, 48294}, {29362, 47716}, {31208, 47788}, {31251, 47799}, {31288, 47809}, {32478, 58166}, {34958, 48396}, {47123, 48393}, {47689, 47820}, {47690, 52601}, {47702, 50523}, {47706, 47804}, {47707, 47798}, {47710, 47818}, {47720, 68894}, {47888, 50333}, {47949, 48006}, {47999, 48586}, {48059, 48077}, {48065, 48083}, {48099, 64856}, {48106, 50512}, {48227, 50337}, {48299, 58155}, {48324, 68979}, {48573, 69011}

X(69344) = midpoint of X(47702) and X(50523)
X(69344) = reflection of X(i) in X(j) for these {i,j}: {2533, 20517}, {4088, 50507}, {4705, 68780}, {4808, 650}, {47682, 48330}, {47687, 23815}, {47690, 52601}, {47710, 48405}, {47711, 4874}, {47949, 48006}, {48077, 48059}, {48083, 48065}, {48103, 4401}, {48106, 50512}, {48300, 1960}, {48301, 48286}, {48393, 47123}, {48395, 676}, {48396, 34958}, {48586, 47999}, {49279, 663}, {50352, 4458}, {50355, 21192}
X(69344) = barycentric product X(514)*X(33074)
X(69344) = barycentric quotient X(33074)/X(190)
X(69344) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {676, 48395, 47875}, {47710, 47818, 48405}

X(69345) = ODD<1, -1, 0, -1, 1, 1> POINT

Barycentrics    (b - c)*(a^3 - a^2*b + b^3 - a^2*c - a*b*c + b^2*c + b*c^2 + c^3) : :
X(69345) = X[48103] + 2 X[49276], X[48083] + 2 X[49279], 2 X[3776] - 3 X[47841], 2 X[3837] - 3 X[57066], 2 X[4142] - 3 X[4448], 3 X[4800] - 2 X[48403], 2 X[7178] - 3 X[47872], 2 X[17072] - 3 X[48185], 2 X[21051] - 3 X[30565], 2 X[21104] - 3 X[47889], X[21302] - 3 X[48171], 4 X[34958] - 3 X[58372], 3 X[47796] - 2 X[58375], 3 X[47837] - 2 X[62435], 3 X[47885] - 2 X[50501], 3 X[47893] - 2 X[50348], 3 X[48162] - 2 X[48402], 3 X[48553] - 2 X[50453]

X(69345) lies on these lines: {1, 29354}, {512, 48103}, {513, 4064}, {514, 4010}, {523, 48336}, {525, 659}, {649, 29200}, {650, 50541}, {661, 68979}, {663, 62423}, {667, 23875}, {690, 4063}, {814, 25259}, {826, 4040}, {918, 4367}, {1019, 29252}, {1577, 29102}, {2533, 69293}, {3566, 47890}, {3716, 3801}, {3762, 29094}, {3776, 47841}, {3777, 6332}, {3837, 57066}, {3887, 4808}, {3900, 48088}, {3910, 48055}, {4041, 48056}, {4083, 48094}, {4122, 29051}, {4142, 4448}, {4170, 29098}, {4391, 29082}, {4449, 48117}, {4468, 4490}, {4498, 29284}, {4724, 29017}, {4775, 29047}, {4794, 29358}, {4800, 48403}, {4802, 50508}, {4879, 29288}, {4978, 49290}, {4990, 23770}, {4992, 47652}, {5592, 29037}, {6004, 48272}, {6372, 47682}, {7178, 47872}, {7265, 29070}, {7927, 48352}, {8045, 21146}, {8712, 48096}, {8714, 50351}, {14349, 47968}, {17072, 48185}, {18004, 21301}, {21051, 30565}, {21104, 47889}, {21302, 48171}, {21343, 65685}, {28195, 48128}, {29021, 48351}, {29025, 48080}, {29120, 47684}, {29144, 48367}, {29146, 47972}, {29162, 50326}, {29168, 47726}, {29188, 47711}, {29198, 48078}, {29208, 48118}, {29220, 59672}, {29224, 47712}, {29226, 48614}, {29238, 48266}, {29246, 47690}, {29280, 48331}, {29312, 47970}, {29318, 48065}, {29324, 47728}, {29332, 47708}, {29366, 47707}, {30520, 48136}, {34958, 58372}, {47727, 58160}, {47796, 58375}, {47837, 62435}, {47885, 50501}, {47893, 50348}, {47918, 48048}, {47958, 48093}, {47973, 48100}, {48062, 50355}, {48113, 48334}, {48162, 48402}, {48290, 48323}, {48329, 64856}, {48406, 49301}, {48553, 50453}, {49275, 63812}

X(69345) = midpoint of X(i) and X(j) for these {i,j}: {4449, 48117}, {48113, 48334}, {48118, 48338}
X(69345) = reflection of X(i) in X(j) for these {i,j}: {2533, 69293}, {3777, 6332}, {3801, 3716}, {4041, 48056}, {4367, 48299}, {4490, 4468}, {4978, 49290}, {21146, 8045}, {21301, 18004}, {21343, 65685}, {23770, 4990}, {47652, 4992}, {47727, 58160}, {47913, 48040}, {47918, 48048}, {47944, 4983}, {47958, 48093}, {47968, 14349}, {47973, 48100}, {48273, 49288}, {48323, 48290}, {48392, 49286}, {49301, 48406}, {50340, 4040}, {50342, 667}, {50355, 48062}
X(69345) = crossdifference of every pair of points on line {16470, 60697}
X(69345) = barycentric product X(514)*X(33158)
X(69345) = barycentric quotient X(33158)/X(190)

X(69346) = ODD<1, -1, -1, 1, 0, 1> POINT

Barycentrics    (b - c)*(a^3 - a^2*b - a*b^2 - a^2*c + a*b*c + b^2*c - a*c^2 + b*c^2) : :
X(69346) = X[1] + 2 X[3837], X[8] + 2 X[48296], X[8] - 4 X[53571], X[48296] + 2 X[53571], 2 X[10] + X[21343], 2 X[10] - 5 X[30795], X[21343] + 5 X[30795], 3 X[47796] - X[47824], 2 X[47824] - 3 X[48569], 2 X[47822] - 3 X[47839], X[47822] - 3 X[47841], 4 X[47822] - 3 X[48553], 4 X[47841] - X[48553], 2 X[551] + X[48167], X[659] - 4 X[1125], and many others

X(69346) lies on these lines: {1, 3837}, {2, 891}, {8, 48296}, {10, 21343}, {512, 47796}, {514, 47822}, {522, 59887}, {551, 25569}, {659, 1125}, {663, 23815}, {690, 4453}, {693, 48288}, {764, 3716}, {812, 14419}, {900, 16173}, {905, 48273}, {1019, 4992}, {1022, 48183}, {1491, 48291}, {1499, 48245}, {1960, 3616}, {2254, 19947}, {2530, 48305}, {2533, 48348}, {2785, 21204}, {2787, 4728}, {2821, 5603}, {2826, 5886}, {2832, 4448}, {3251, 28521}, {3338, 53403}, {3485, 53539}, {3624, 21385}, {3669, 48267}, {3679, 25574}, {3776, 49279}, {3835, 4378}, {3887, 36848}, {3906, 48241}, {3912, 30865}, {3960, 4010}, {4040, 48406}, {4083, 47795}, {4129, 48323}, {4139, 48246}, {4147, 31251}, {4151, 47893}, {4449, 21260}, {4458, 44315}, {4498, 31288}, {4730, 25380}, {4770, 48304}, {4775, 24720}, {4800, 68896}, {4801, 50507}, {4806, 48320}, {4874, 48335}, {4879, 50337}, {4885, 48332}, {4927, 29240}, {4928, 14421}, {5443, 24099}, {6004, 47819}, {6085, 26144}, {6363, 48173}, {6366, 45677}, {6371, 48209}, {6372, 47840}, {6545, 14432}, {6548, 53334}, {8712, 48564}, {9508, 19949}, {11375, 30725}, {11376, 53523}, {12073, 48252}, {14077, 47802}, {14288, 59837}, {14422, 21297}, {14475, 30574}, {14838, 48279}, {16892, 49290}, {17072, 48333}, {17166, 48059}, {17397, 24623}, {19882, 49276}, {21051, 48282}, {21145, 23884}, {21201, 53533}, {21301, 48328}, {21302, 48347}, {21439, 33939}, {23765, 59672}, {23770, 50351}, {23789, 48336}, {23814, 58374}, {23876, 48227}, {24674, 48325}, {25055, 64913}, {26801, 27194}, {26854, 50510}, {26985, 48298}, {27167, 50485}, {27193, 50491}, {28603, 53364}, {29066, 48184}, {29166, 48203}, {29188, 47812}, {29198, 47838}, {29226, 47794}, {29312, 47797}, {29318, 48224}, {29350, 47823}, {29354, 57066}, {30583, 45678}, {31149, 45667}, {40086, 48306}, {44316, 48292}, {45323, 50760}, {47724, 48289}, {47813, 48131}, {47835, 48218}, {48066, 48301}, {48090, 48321}, {48136, 50352}, {48168, 68953}, {48244, 68968}, {48339, 50335}, {50331, 51648}, {62558, 69006}

X(69346) = midpoint of X(i) and X(j) for these {i,j}: {4728, 14413}, {6545, 14432}, {14419, 30592}, {14421, 14431}, {25569, 48167}, {47813, 48131}
X(69346) = reflection of X(i) in X(j) for these {i,j}: {14431, 4928}, {25569, 551}, {47813, 52601}, {47835, 48218}, {47837, 47795}, {47839, 47841}, {48553, 47839}, {48569, 47796}, {53364, 28603}
X(69346) = barycentric product X(514)*X(32845)
X(69346) = barycentric quotient X(32845)/X(190)
X(69346) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1491, 48295, 48291}, {3616, 46403, 1960}, {3716, 65482, 764}, {21343, 30795, 10}, {48296, 53571, 8}

X(69347) = ODD<1, -1, -1, 1, 0, 0> POINT

Barycentrics    a*(b - c)*(a^2 - a*b - b^2 - a*c + b*c - c^2) : :
X(69347) = 3 X[3669] + X[50508], X[4367] - 3 X[14413], 3 X[4367] - X[50523], 3 X[14413] + X[48131], 9 X[14413] - X[50523], 3 X[25569] - X[48150], 3 X[48131] + X[50523], 3 X[48136] - X[50508], 3 X[905] - X[50501], X[9508] + 2 X[48332], 3 X[9508] - 2 X[50501], 3 X[48332] + X[50501], 3 X[1022] + X[47970], X[2533] - 3 X[47796], and many others

X(69347) lies on these lines: {1, 2530}, {512, 3960}, {513, 663}, {514, 1125}, {650, 29226}, {659, 48334}, {661, 48323}, {667, 48335}, {693, 24674}, {764, 4040}, {784, 48295}, {814, 48325}, {830, 48328}, {891, 14838}, {905, 4083}, {1022, 47970}, {1491, 4449}, {1734, 48333}, {2254, 4879}, {2533, 26115}, {3676, 17992}, {3762, 47839}, {3776, 29082}, {3801, 3904}, {3835, 29324}, {3837, 3907}, {3887, 48347}, {3900, 50335}, {4010, 17496}, {4025, 29284}, {4041, 21343}, {4063, 14419}, {4106, 29152}, {4160, 48059}, {4378, 14349}, {4391, 47841}, {4462, 47822}, {4504, 48050}, {4560, 48279}, {4705, 14421}, {4724, 23765}, {4729, 48244}, {4761, 48569}, {4775, 4905}, {4782, 8712}, {4802, 47720}, {4885, 24665}, {4922, 21301}, {4928, 59521}, {4978, 48288}, {4983, 48320}, {4992, 6002}, {6004, 48294}, {6332, 62423}, {6545, 29689}, {8643, 50358}, {8678, 48100}, {9002, 68775}, {9269, 48066}, {9479, 21194}, {14422, 50512}, {14837, 48215}, {18081, 31997}, {19847, 47794}, {19947, 29298}, {20317, 48197}, {21052, 30795}, {21120, 47799}, {21222, 47840}, {21302, 36848}, {23506, 50505}, {23789, 29188}, {23815, 29066}, {23880, 48090}, {24720, 29366}, {24748, 31287}, {25666, 48401}, {25904, 48062}, {28374, 50516}, {29051, 48289}, {29124, 49295}, {29198, 48099}, {29244, 48398}, {29274, 48089}, {29280, 49280}, {30725, 48400}, {38469, 48283}, {42325, 58160}, {45667, 64914}, {47716, 50351}, {47729, 47819}, {47889, 50457}, {48024, 48341}, {48111, 58155}, {48273, 48321}, {48281, 50330}, {48284, 68894}, {48290, 68979}, {48291, 48409}, {48293, 50345}, {48301, 48410}, {48322, 50328}, {48338, 50359}, {49993, 65449}, {50348, 65685}, {59629, 69292}

X(69347) = midpoint of X(i) and X(j) for these {i,j}: {1, 2530}, {650, 48346}, {659, 48334}, {661, 48323}, {663, 3777}, {667, 48335}, {764, 4040}, {905, 48332}, {1491, 4449}, {1734, 48333}, {2254, 4879}, {2533, 48298}, {3669, 48136}, {3801, 3904}, {3960, 48348}, {4010, 17496}, {4041, 21343}, {4367, 48131}, {4378, 14349}, {4504, 48050}, {4560, 48279}, {4705, 48282}, {4724, 23765}, {4775, 4905}, {4922, 21301}, {4978, 48288}, {4983, 48320}, {21222, 48265}, {30725, 48400}, {47716, 50351}, {48024, 48341}, {48066, 48287}, {48100, 48344}, {48123, 48144}, {48137, 48330}, {48151, 48336}, {48273, 48321}, {48281, 50330}, {48289, 48406}, {48291, 48409}, {48293, 50345}, {48301, 48410}, {48322, 50328}, {48338, 50359}, {48350, 53314}, {48616, 50517}, {50348, 65685}
X(69347) = reflection of X(i) in X(j) for these {i,j}: {9508, 905}, {48401, 25666}, {48406, 65482}, {50337, 19947}
X(69347) = X(101)-isoconjugate of X(54120)
X(69347) = X(1015)-Dao conjugate of X(54120)
X(69347) = crosspoint of X(i) and X(j) for these (i,j): {1, 65364}, {934, 1432}
X(69347) = crosssum of X(i) and X(j) for these (i,j): {522, 29655}, {2329, 3900}
X(69347) = crossdifference of every pair of points on line {9, 3550}
X(69347) = barycentric product X(i)*X(j) for these {i,j}: {1, 21212}, {513, 6646}, {514, 17596}, {649, 20955}, {693, 21008}, {1019, 69299}, {7192, 69248}, {17924, 22161}
X(69347) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 54120}, {6646, 668}, {17596, 190}, {20955, 1978}, {21008, 100}, {21212, 75}, {22161, 1332}, {69248, 3952}, {69299, 4033}
X(69347) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4705, 14421, 48282}, {14413, 48131, 4367}, {21222, 47840, 48265}, {21343, 47893, 4041}, {47796, 48298, 2533}

X(69348) = ODD<1, -1, -1, 0, 0, 0> POINT

Barycentrics    a*(b - c)*(a^2 - a*b - b^2 - a*c - c^2) : :
X(69348) = 3 X[2] + X[48298], 3 X[905] - X[50336], 3 X[48136] + X[50336], X[649] - 3 X[14419], X[661] + 3 X[14413], X[4378] - 3 X[14413], 3 X[663] - X[6161], 3 X[2530] + X[6161], X[1577] - 3 X[47841], X[1734] - 3 X[47893], X[4879] + 3 X[47893], X[2533] - 3 X[47795], 5 X[3616] - X[47694], X[3762] - 3 X[47822], X[48616] + 2 X[58150], and many others

X(69348) lies on these lines: {1, 1491}, {2, 48298}, {512, 905}, {513, 1960}, {514, 1125}, {519, 45323}, {523, 48295}, {551, 64914}, {649, 14419}, {650, 891}, {659, 48335}, {661, 4378}, {663, 1201}, {667, 48131}, {690, 4025}, {693, 48288}, {764, 4724}, {824, 49290}, {826, 6332}, {830, 48100}, {832, 2605}, {834, 31947}, {1015, 38989}, {1019, 48123}, {1386, 9014}, {1459, 50330}, {1492, 13396}, {1577, 47841}, {1734, 4879}, {2254, 4775}, {2473, 8641}, {2523, 42664}, {2526, 48327}, {2533, 47795}, {2785, 21212}, {2787, 3835}, {3004, 48290}, {3011, 47757}, {3250, 27233}, {3309, 58160}, {3616, 47694}, {3669, 6372}, {3676, 51642}, {3762, 47822}, {3776, 29102}, {3777, 4040}, {3803, 48616}, {3831, 4147}, {3837, 29066}, {3887, 50335}, {3900, 48347}, {3904, 47797}, {3906, 49280}, {3907, 21260}, {4010, 48321}, {4041, 47888}, {4083, 14838}, {4106, 29340}, {4129, 29324}, {4160, 48030}, {4367, 14349}, {4382, 30592}, {4391, 47839}, {4449, 4705}, {4462, 48553}, {4474, 14431}, {4490, 48282}, {4522, 29110}, {4560, 48273}, {4707, 48227}, {4730, 47828}, {4761, 47823}, {4770, 14077}, {4774, 30795}, {4806, 29148}, {4871, 47778}, {4893, 14421}, {4905, 48336}, {4948, 50760}, {4979, 8657}, {4983, 48144}, {4992, 29013}, {6050, 8712}, {6371, 68772}, {8640, 28372}, {8643, 48122}, {8672, 51648}, {8678, 48059}, {9508, 29350}, {10015, 47799}, {12073, 48069}, {14432, 16892}, {15309, 48093}, {17072, 29298}, {17496, 47840}, {18004, 29212}, {19864, 47794}, {19869, 19948}, {19947, 24720}, {21052, 31251}, {21192, 29284}, {21222, 47821}, {21343, 47827}, {21385, 48226}, {23383, 44408}, {23765, 47970}, {23789, 29246}, {23815, 29051}, {23884, 48212}, {25055, 48234}, {25569, 48324}, {26094, 47793}, {26230, 30580}, {27138, 30709}, {27648, 50524}, {27674, 50491}, {28374, 50510}, {29186, 48406}, {29198, 48058}, {29226, 48003}, {29312, 68780}, {29362, 48284}, {29366, 50337}, {30234, 58139}, {30725, 48179}, {33920, 62621}, {38314, 48157}, {44429, 47729}, {44550, 48080}, {45316, 48063}, {45667, 48010}, {45902, 52589}, {47683, 48120}, {47684, 48174}, {47691, 50351}, {47724, 48184}, {47725, 62634}, {47796, 50352}, {47802, 53571}, {47825, 48304}, {47838, 48265}, {47842, 48283}, {47935, 58144}, {47949, 48341}, {47959, 48323}, {47965, 48346}, {47975, 48291}, {48012, 48287}, {48024, 48320}, {48054, 48343}, {48066, 48294}, {48090, 64934}, {48092, 50517}, {48116, 58152}, {48128, 50515}, {48137, 48331}, {48150, 58155}, {48151, 48351}, {48241, 49274}, {48301, 48409}, {48303, 50345}, {48305, 48410}, {48329, 58156}, {48337, 50355}, {48339, 50341}, {48350, 55969}, {48352, 50359}, {49277, 50342}, {49278, 50340}, {50358, 68816}

X(69348) = midpoint of X(i) and X(j) for these {i,j}: {1, 1491}, {650, 48332}, {659, 48335}, {661, 4378}, {663, 2530}, {667, 48131}, {693, 48288}, {764, 4724}, {905, 48136}, {1019, 48123}, {1459, 50330}, {1734, 4879}, {2254, 4775}, {2526, 48327}, {3004, 48290}, {3669, 48099}, {3777, 4040}, {3803, 48616}, {3835, 48325}, {3837, 48289}, {4010, 48321}, {4041, 48333}, {4367, 14349}, {4449, 4705}, {4490, 48282}, {4560, 48273}, {4770, 48296}, {4893, 14421}, {4905, 48336}, {4948, 50760}, {4983, 48144}, {14838, 48348}, {16892, 49279}, {17496, 48267}, {23765, 47970}, {30580, 44435}, {47683, 48120}, {47691, 50351}, {47842, 48283}, {47949, 48341}, {47959, 48323}, {47965, 48346}, {47975, 48291}, {48012, 48287}, {48024, 48320}, {48030, 48344}, {48054, 48343}, {48059, 48328}, {48066, 48294}, {48092, 50517}, {48100, 48330}, {48128, 50515}, {48137, 48331}, {48151, 48351}, {48301, 48409}, {48303, 50345}, {48305, 48410}, {48324, 50328}, {48337, 50355}, {48339, 50341}, {48350, 55969}, {48352, 50359}, {49277, 50342}, {49278, 50340}
X(69348) = reflection of X(i) in X(j) for these {i,j}: {3803, 58150}, {4147, 65449}, {4874, 1125}, {24720, 19947}, {48329, 58156}, {50504, 14838}
X(69348) = complement of the isotomic conjugate of X(35008)
X(69348) = X(i)-complementary conjugate of X(j) for these (i,j): {35008, 2887}, {35009, 141}
X(69348) = crosspoint of X(2) and X(35008)
X(69348) = crossdifference of every pair of points on line {45, 2243}
X(69348) = barycentric product X(i)*X(j) for these {i,j}: {1, 47886}, {513, 4643}, {514, 4414}, {649, 33936}, {1019, 69300}
X(69348) = barycentric quotient X(i)/X(j) for these {i,j}: {4414, 190}, {4643, 668}, {33936, 1978}, {47886, 75}, {69300, 4033}
X(69348) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {661, 14413, 4378}, {4474, 30835, 14431}, {4879, 47893, 1734}, {14432, 16892, 49279}, {17496, 47840, 48267}, {25569, 50328, 48324}, {47888, 48333, 4041}

X(69349) = ODD<1, -1, -1, 0, 0, -1> POINT

Barycentrics    (b - c)*(-a^3 + a^2*b + a*b^2 + a^2*c + b^2*c + a*c^2 + b*c^2) : :
X(69349) = 2 X[10] - 3 X[47888], 2 X[3801] - 3 X[48224], 4 X[905] - 3 X[47823], 2 X[2533] - 3 X[47823], 4 X[1125] - 3 X[47875], 2 X[1577] - 3 X[47841], 2 X[4041] - 3 X[48225], 2 X[4147] - 3 X[47827], 2 X[4391] - 3 X[47822], 2 X[4490] - 3 X[48176], X[4774] - 3 X[47893], 2 X[17072] - 3 X[47893], 2 X[4791] - 3 X[47839], 3 X[4893] - 2 X[48401], 2 X[7178] - 3 X[48227], 3 X[8643] - 2 X[48248], 3 X[14413] - X[50457], 3 X[48238] - 2 X[50457], 4 X[14838] - 3 X[47835], 5 X[30835] - 4 X[59521], 3 X[45671] - 2 X[50504], 2 X[47707] - 3 X[48188], X[47721] - 3 X[47819], 3 X[47775] - 2 X[47922], 3 X[48177] - 2 X[48400], 3 X[48189] - 2 X[48392]

X(69349) lies on these lines: {1, 784}, {10, 47888}, {512, 48321}, {513, 17496}, {514, 659}, {522, 4879}, {523, 1459}, {650, 24755}, {661, 29324}, {663, 48289}, {693, 24674}, {764, 29186}, {814, 24719}, {900, 48338}, {905, 2533}, {1125, 47875}, {1491, 3907}, {1577, 47841}, {1734, 29298}, {2254, 29366}, {2530, 29066}, {2787, 14349}, {2789, 47877}, {3669, 21146}, {3762, 50507}, {3777, 29051}, {3810, 50340}, {3900, 50341}, {3904, 29017}, {3960, 50352}, {4010, 23880}, {4041, 48225}, {4083, 4560}, {4122, 6332}, {4147, 47827}, {4151, 48333}, {4391, 47822}, {4467, 29284}, {4474, 21051}, {4490, 48176}, {4762, 48346}, {4774, 17072}, {4775, 8714}, {4791, 47839}, {4802, 47844}, {4822, 29170}, {4824, 15420}, {4844, 48018}, {4893, 48401}, {4905, 29188}, {4922, 8678}, {4977, 48341}, {4983, 29148}, {5040, 45746}, {6002, 48123}, {7178, 48227}, {7192, 25300}, {8643, 48248}, {14413, 48238}, {14838, 47835}, {16892, 29082}, {17166, 48344}, {17494, 29226}, {20295, 29152}, {21222, 29198}, {21301, 29236}, {21302, 50335}, {23815, 47724}, {23882, 48279}, {24381, 35518}, {25636, 27648}, {28470, 50328}, {28475, 48092}, {29047, 50351}, {29070, 48335}, {29074, 48278}, {29086, 49278}, {29090, 49277}, {29110, 48272}, {29120, 47701}, {29134, 47702}, {29244, 47652}, {29246, 48151}, {29268, 48059}, {29274, 46403}, {29280, 49274}, {29362, 48334}, {30835, 59521}, {45671, 50504}, {47683, 48282}, {47694, 48330}, {47707, 48188}, {47721, 47819}, {47728, 68979}, {47729, 48410}, {47775, 47922}, {47971, 59629}, {48099, 48265}, {48177, 48400}, {48189, 48392}, {48194, 53364}, {48273, 48348}, {48287, 48291}, {48294, 48305}, {48295, 48393}, {48322, 64914}, {48339, 48347}, {48351, 68896}

X(69349) = midpoint of X(i) and X(j) for these {i,j}: {4560, 48298}, {4822, 53536}, {47683, 48282}, {47729, 48410}
X(69349) = reflection of X(i) in X(j) for these {i,j}: {663, 48289}, {2533, 905}, {3762, 50507}, {4010, 48136}, {4122, 6332}, {4367, 48325}, {4474, 21051}, {4774, 17072}, {17166, 48344}, {20295, 48129}, {21146, 3669}, {21301, 48100}, {21302, 50335}, {24719, 48131}, {46403, 48137}, {47694, 48330}, {47724, 23815}, {48238, 14413}, {48265, 48099}, {48273, 48348}, {48279, 48332}, {48291, 48287}, {48301, 1}, {48305, 48294}, {48339, 48347}, {48393, 48295}, {50352, 3960}, {53364, 48194}
X(69349) = crossdifference of every pair of points on line {573, 2276}
X(69349) = barycentric product X(514)*X(32916)
X(69349) = barycentric quotient X(32916)/X(190)
X(69349) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {905, 2533, 47823}, {4774, 47893, 17072}

X(69350) = ODD<1, -1, -1, 0, -1, 1> POINT

Barycentrics    (b - c)*(-a^3 + a^2*b + a*b^2 + b^3 + a^2*c - b^2*c + a*c^2 - b*c^2 + c^3) : :
X(69350) = 3 X[2] + X[47691], X[676] - 3 X[68933], X[48090] + 3 X[48212], X[650] - 3 X[47799], X[23770] + 3 X[47799], X[659] - 3 X[47800], 3 X[47800] + X[48398], X[661] + 3 X[47887], X[661] - 3 X[48555], X[693] + 3 X[47797], 3 X[47797] - X[68780], X[1491] - 3 X[47757], X[47123] + 3 X[47757], 3 X[1638] - X[50336], 3 X[1639] - X[48088], and many others

X(69350) lies on these lines: {2, 47691}, {512, 21188}, {513, 676}, {514, 1125}, {522, 3837}, {523, 4885}, {649, 49295}, {650, 23770}, {659, 47800}, {661, 47887}, {663, 28082}, {693, 47797}, {905, 48403}, {1001, 48388}, {1491, 29639}, {1638, 50336}, {1639, 48088}, {2526, 48178}, {2530, 21185}, {2977, 31287}, {3004, 7662}, {3239, 62423}, {3616, 47728}, {3624, 47725}, {3669, 48400}, {3716, 3776}, {3777, 28018}, {3798, 29328}, {3801, 6332}, {3835, 4458}, {4010, 4025}, {4083, 14837}, {4088, 30835}, {4122, 30865}, {4379, 47701}, {4448, 48061}, {4453, 48080}, {4468, 47822}, {4521, 48056}, {4522, 4928}, {4724, 6545}, {4777, 45677}, {4778, 47990}, {4784, 47758}, {4785, 45668}, {4802, 68794}, {4804, 47886}, {4806, 28846}, {4808, 31251}, {4809, 24719}, {4824, 47783}, {4841, 48134}, {4893, 47704}, {4913, 47882}, {4927, 48089}, {4992, 28478}, {5592, 45316}, {6129, 28042}, {6588, 7180}, {6590, 47833}, {7178, 48136}, {7192, 47983}, {7649, 28116}, {7658, 9508}, {7659, 48245}, {8678, 34958}, {10006, 55126}, {10015, 48332}, {16892, 47832}, {18004, 59751}, {21104, 48029}, {21115, 48078}, {21116, 47927}, {21120, 48346}, {21146, 21183}, {21196, 48394}, {21204, 24720}, {23282, 24084}, {24924, 48106}, {25259, 48241}, {26985, 47690}, {28006, 47136}, {28147, 48194}, {28161, 48198}, {28199, 59943}, {28878, 48028}, {29062, 59714}, {29098, 31288}, {29212, 59737}, {30795, 47806}, {31148, 47938}, {31209, 48408}, {31250, 47807}, {41800, 50501}, {43067, 47998}, {44429, 47695}, {44433, 47685}, {44435, 47694}, {44928, 59903}, {45320, 48396}, {45745, 48120}, {45746, 47834}, {46403, 47798}, {47131, 47802}, {47650, 48240}, {47651, 48250}, {47652, 47804}, {47660, 48174}, {47676, 47821}, {47686, 47805}, {47687, 48223}, {47688, 47771}, {47692, 47809}, {47696, 48156}, {47697, 48159}, {47699, 47780}, {47708, 47796}, {47712, 47795}, {47716, 47794}, {47720, 47793}, {47754, 50348}, {47756, 48027}, {47760, 48047}, {47765, 58372}, {47766, 48103}, {47801, 50358}, {47803, 47890}, {47812, 47972}, {47813, 47958}, {47823, 48069}, {47828, 53558}, {47831, 69293}, {47943, 48578}, {47944, 49293}, {47960, 48220}, {47961, 48276}, {47968, 48234}, {47974, 48420}, {47989, 48558}, {47995, 48552}, {48024, 49296}, {48043, 69292}, {48055, 49299}, {48083, 48546}, {48087, 48166}, {48095, 48231}, {48108, 48161}, {48119, 48414}, {48184, 49285}, {48269, 50342}, {48279, 60492}, {48295, 50453}, {48330, 52596}, {48404, 49292}, {48421, 49301}, {48422, 49275}

X(69350) = midpoint of X(i) and X(j) for these {i,j}: {649, 49295}, {650, 23770}, {659, 48398}, {693, 68780}, {905, 48403}, {1491, 47123}, {2530, 21185}, {3004, 7662}, {3669, 48400}, {3716, 3776}, {3801, 6332}, {3835, 4458}, {4010, 4025}, {4468, 48326}, {4841, 48134}, {4927, 48211}, {7178, 48136}, {7192, 47983}, {10015, 48332}, {16892, 49286}, {21104, 48029}, {21120, 48346}, {21146, 48006}, {21183, 48177}, {21196, 48394}, {43067, 47998}, {45745, 48120}, {47131, 50333}, {47676, 48040}, {47691, 48062}, {47694, 48007}, {47765, 58372}, {47787, 48224}, {47887, 48555}, {47944, 49293}, {47961, 48276}, {48024, 49296}, {48043, 69292}, {48055, 49299}, {48069, 48349}, {48089, 50347}, {48269, 50342}, {48279, 60492}, {48295, 50453}, {48404, 49292}, {49285, 50340}
X(69350) = reflection of X(i) in X(j) for these {i,j}: {2977, 31287}, {3798, 69011}, {9508, 7658}, {18004, 59751}, {48056, 4521}, {48330, 52596}, {59903, 44928}
X(69350) = complement of X(48062)
X(69350) = X(58977)-complementary conjugate of X(226)
X(69350) = crossdifference of every pair of points on line {220, 3053}
X(69350) = barycentric product X(514)*X(24248)
X(69350) = barycentric quotient X(24248)/X(190)
X(69350) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 47691, 48062}, {693, 47797, 68780}, {3801, 47841, 6332}, {4010, 48227, 4025}, {4927, 50347, 48089}, {7662, 48192, 3004}, {9508, 48215, 7658}, {16892, 47832, 49286}, {21104, 48179, 48029}, {21146, 48177, 48006}, {21183, 48006, 21146}, {23770, 47799, 650}, {26985, 48203, 47690}, {44435, 47694, 48007}, {47123, 47757, 1491}, {47131, 47802, 50333}, {47676, 47821, 48040}, {47800, 48398, 659}, {47822, 48326, 4468}, {47823, 48349, 48069}, {48056, 48197, 4521}, {48089, 48211, 50347}, {48184, 50340, 49285}

X(69351) = ODD<1, -1, -1, 0, -1, -1> POINT

Barycentrics    (b - c)*(-a^3 + a^2*b + a*b^2 + b^3 + a^2*c + b^2*c + a*c^2 + b*c^2 + c^3) : :
X(69351) = X[3801] - 3 X[48224], X[2533] - 3 X[48227], 2 X[21188] - 3 X[48227], 2 X[3239] - 3 X[47839], X[4041] - 3 X[47886], X[4122] - 3 X[47841], 2 X[4129] - 3 X[48555], X[4391] - 3 X[47797], 3 X[4750] - X[50509], 3 X[4786] - 2 X[58179], X[4808] - 3 X[47888], 4 X[7658] - 3 X[47837], 3 X[8643] + X[47923], X[17496] + 3 X[48203], and many others

X(69351) lies on these lines: {2, 47707}, {512, 4025}, {514, 659}, {522, 2530}, {523, 905}, {525, 48136}, {650, 29288}, {663, 16892}, {665, 45745}, {784, 47123}, {826, 6332}, {830, 48007}, {891, 60492}, {918, 48099}, {1734, 47727}, {2533, 21188}, {3004, 8678}, {3126, 12864}, {3239, 47839}, {3309, 50348}, {3669, 29142}, {3676, 50352}, {3776, 29051}, {3777, 50340}, {3798, 4834}, {3800, 50336}, {3803, 4977}, {3835, 29037}, {3837, 29074}, {3910, 48332}, {3960, 29021}, {4041, 47886}, {4106, 29232}, {4122, 47841}, {4129, 29212}, {4391, 47797}, {4449, 21124}, {4468, 29354}, {4560, 47691}, {4750, 50509}, {4786, 58179}, {4808, 47888}, {4822, 47971}, {4885, 48395}, {4983, 28846}, {4992, 29078}, {6004, 48015}, {6050, 47890}, {6372, 48006}, {6590, 52601}, {7658, 47837}, {7662, 34958}, {7927, 48069}, {8643, 47923}, {9508, 29208}, {14419, 28147}, {14838, 29047}, {15309, 47983}, {17069, 50501}, {17072, 21212}, {17166, 45746}, {17494, 47720}, {17496, 47708}, {21185, 63812}, {21192, 29350}, {21196, 21197}, {21260, 29110}, {21301, 44435}, {23731, 50526}, {23770, 23882}, {23815, 29086}, {23874, 50330}, {23876, 48348}, {23879, 48295}, {23880, 48403}, {25259, 47840}, {28175, 30234}, {28481, 48616}, {29013, 49295}, {29070, 48398}, {29082, 48289}, {29090, 48269}, {29192, 50337}, {29246, 58375}, {31251, 44432}, {31288, 47766}, {31291, 48156}, {39545, 47961}, {44550, 47709}, {45671, 47717}, {47660, 47820}, {47687, 47819}, {47690, 47796}, {47695, 48410}, {47701, 48144}, {47706, 47809}, {47711, 47795}, {47712, 48321}, {47783, 48005}, {47785, 50504}, {47887, 50457}, {47938, 48149}, {47958, 50523}, {47960, 50517}, {47972, 48151}, {47973, 48150}, {48038, 48053}, {48039, 48059}, {48040, 48058}, {48060, 50512}, {48101, 58140}, {48123, 50342}, {48138, 58138}, {48177, 48265}, {48330, 68979}, {48564, 68794}, {48569, 65412}, {48570, 49283}, {58173, 59550}

X(69351) = midpoint of X(i) and X(j) for these {i,j}: {663, 16892}, {1734, 47727}, {3777, 50340}, {4449, 21124}, {4560, 47691}, {4822, 47971}, {17166, 45746}, {17494, 47720}, {17496, 47708}, {23731, 50526}, {47695, 48410}, {47701, 48144}, {47712, 48321}, {47938, 48149}, {47958, 50523}, {47960, 50517}, {47972, 48151}, {47973, 48150}, {48123, 50342}
X(69351) = reflection of X(i) in X(j) for these {i,j}: {2533, 21188}, {4468, 50507}, {4834, 3798}, {6590, 52601}, {7662, 34958}, {17072, 21212}, {47890, 6050}, {48038, 48053}, {48039, 48059}, {48040, 48058}, {48060, 50512}, {48062, 14838}, {48395, 4885}, {49285, 23815}, {50352, 3676}, {50501, 17069}
X(69351) = complement of X(47707)
X(69351) = crosssum of X(6) and X(50506)
X(69351) = crossdifference of every pair of points on line {2276, 36744}
X(69351) = barycentric product X(514)*X(26034)
X(69351) = barycentric quotient X(26034)/X(190)
X(69351) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2533, 48227, 21188}, {17496, 48203, 47708}

X(69352) = ODD<1, -1, -1, -1, 1, 1> POINT

Barycentrics    (b - c)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2 + c^3) : :
X(69352) = X[1] - 3 X[14432], X[4088] + 3 X[14432], 3 X[2] + X[49274], X[8] + 3 X[53334], X[1577] - 3 X[57066], X[3762] - 3 X[30565], X[3904] + 3 X[30565], X[4468] + 3 X[6332], 3 X[4728] - X[47680], 3 X[4776] + X[47684], 3 X[14349] - X[47958], 3 X[45661] - 2 X[59737], X[47958] + 3 X[48300], X[48130] + 3 X[48131], 3 X[14838] - 2 X[17069], and many others

X(69352) lies on these lines: {1, 4088}, {2, 4707}, {8, 53334}, {10, 2785}, {72, 2774}, {78, 44827}, {99, 101}, {386, 53556}, {514, 661}, {519, 45344}, {522, 4794}, {523, 49290}, {525, 14838}, {649, 49277}, {650, 23876}, {663, 48272}, {690, 9508}, {764, 48083}, {830, 48299}, {891, 48056}, {900, 9945}, {905, 23875}, {918, 3960}, {1121, 35141}, {1125, 4458}, {1491, 49279}, {1639, 10015}, {1698, 30574}, {2254, 49276}, {2349, 34234}, {2787, 18004}, {2832, 48055}, {3676, 45683}, {3700, 64934}, {3716, 21201}, {3737, 4064}, {3738, 18254}, {3801, 47839}, {3810, 59672}, {3837, 29102}, {3887, 50333}, {3910, 48003}, {3927, 53301}, {3940, 53249}, {4010, 50351}, {4024, 47683}, {4025, 46381}, {4040, 48278}, {4049, 4997}, {4120, 30578}, {4122, 48288}, {4160, 48047}, {4375, 62556}, {4467, 45671}, {4522, 29066}, {4560, 7265}, {4724, 49278}, {4730, 30605}, {4761, 47809}, {4806, 29029}, {4808, 4879}, {4858, 17761}, {4922, 30580}, {4988, 47681}, {4992, 29098}, {5692, 53562}, {5904, 53554}, {6003, 52355}, {6366, 12019}, {6370, 21180}, {6546, 21385}, {7308, 53407}, {9780, 53356}, {14321, 29126}, {14419, 50342}, {14837, 48196}, {16550, 16563}, {17072, 29304}, {17749, 24094}, {17882, 20881}, {17899, 55184}, {18061, 18151}, {19862, 21181}, {19875, 21952}, {19883, 45668}, {19947, 58375}, {20315, 59753}, {21051, 29094}, {21188, 48218}, {21222, 40612}, {21260, 29082}, {21630, 52356}, {24636, 45674}, {25259, 48321}, {25380, 62435}, {25666, 50453}, {27929, 62609}, {28478, 48011}, {28890, 65482}, {29017, 50507}, {29021, 48099}, {29047, 48136}, {29110, 48289}, {29132, 48043}, {29142, 48058}, {29148, 48270}, {29178, 48269}, {29212, 48325}, {29284, 50504}, {29288, 48348}, {29318, 68780}, {29350, 48062}, {30015, 30017}, {30115, 42662}, {30725, 45341}, {34586, 34587}, {43052, 45664}, {44550, 49272}, {47700, 47727}, {47701, 47726}, {47708, 47838}, {47712, 47840}, {47832, 49300}, {48059, 68979}, {48077, 48324}, {48082, 48320}, {48088, 48332}, {48171, 48298}, {48286, 64860}, {53343, 66995}, {53583, 68137}

X(69352) = midpoint of X(i) and X(j) for these {i,j}: {1, 4088}, {649, 49277}, {650, 49280}, {661, 47682}, {663, 48272}, {764, 48083}, {1491, 49279}, {2254, 49276}, {2582, 2583}, {3737, 4064}, {3762, 3904}, {4010, 50351}, {4024, 47683}, {4040, 48278}, {4120, 62634}, {4122, 48288}, {4560, 7265}, {4707, 49274}, {4724, 49278}, {4808, 4879}, {4988, 47681}, {5904, 53554}, {14349, 48300}, {25259, 48321}, {47700, 47727}, {47701, 47726}, {48047, 48290}, {48077, 48324}, {48082, 48320}, {48088, 48332}, {48094, 48335}, {53343, 66995}
X(69352) = reflection of X(i) in X(j) for these {i,j}: {4458, 1125}, {4791, 3239}, {21192, 14838}, {21198, 1639}, {21201, 3716}, {50453, 25666}, {58375, 19947}, {59753, 20315}, {62435, 25380}
X(69352) = isotomic conjugate of X(65238)
X(69352) = complement of X(4707)
X(69352) = complement of the isotomic conjugate of X(47318)
X(69352) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3065, 150}, {14147, 5080}, {19302, 149}, {21739, 21293}, {34921, 7}, {68387, 3434}
X(69352) = X(i)-complementary conjugate of X(j) for these (i,j): {80, 21253}, {101, 31845}, {163, 214}, {476, 21236}, {759, 116}, {1333, 51402}, {1415, 6739}, {1576, 16586}, {1807, 127}, {2161, 125}, {2174, 3258}, {2194, 46398}, {2222, 17052}, {2341, 124}, {6187, 8287}, {9273, 52601}, {9274, 21196}, {14560, 5249}, {18359, 53575}, {24624, 21252}, {32671, 1125}, {32675, 442}, {32678, 25639}, {32739, 35069}, {34079, 11}, {36069, 3739}, {37140, 3741}, {46073, 46651}, {46077, 46650}, {47318, 2887}, {51562, 21245}, {52431, 34846}, {57657, 35128}, {65283, 21240}, {67166, 1086}
X(69352) = X(65669)-Ceva conjugate of X(8674)
X(69352) = X(68818)-cross conjugate of X(8674)
X(69352) = X(i)-isoconjugate of X(j) for these (i,j): {6, 1290}, {31, 65238}, {32, 35156}, {163, 5620}, {692, 21907}, {1333, 66280}, {1415, 11604}, {1576, 68363}, {2836, 59107}
X(69352) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 65238}, {9, 1290}, {37, 66280}, {115, 5620}, {1086, 21907}, {1146, 11604}, {2610, 53527}, {4858, 68363}, {6376, 35156}, {35090, 1}, {47227, 21180}, {53988, 19}
X(69352) = crosspoint of X(i) and X(j) for these (i,j): {2, 47318}, {190, 18359}
X(69352) = crosssum of X(i) and X(j) for these (i,j): {6, 21828}, {41, 42657}, {649, 7113}
X(69352) = crossdifference of every pair of points on line {31, 3122}
X(69352) = barycentric product X(i)*X(j) for these {i,j}: {10, 65669}, {75, 8674}, {274, 68818}, {304, 47235}, {313, 42741}, {514, 32849}, {561, 42670}, {693, 69275}, {850, 5127}, {1577, 37783}, {2074, 14208}, {3261, 17796}, {3596, 51646}, {4025, 56877}, {5172, 35519}, {6332, 37799}, {19622, 20948}
X(69352) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1290}, {2, 65238}, {10, 66280}, {75, 35156}, {514, 21907}, {522, 11604}, {523, 5620}, {1577, 68363}, {2074, 162}, {5127, 110}, {5172, 109}, {5497, 13589}, {5520, 21180}, {8674, 1}, {17796, 101}, {19622, 163}, {21180, 58076}, {32849, 190}, {37783, 662}, {37799, 653}, {38982, 53527}, {41541, 23703}, {41542, 61225}, {42670, 31}, {42741, 58}, {47235, 19}, {51646, 56}, {56877, 1897}, {57447, 11125}, {65669, 86}, {66016, 61231}, {68164, 5127}, {68818, 37}, {69275, 100}
X(69352) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 49274, 4707}, {3762, 30566, 59737}, {3904, 30565, 3762}, {4088, 14432, 1}

X(69353) = ODD<1, -1, -1, -1, 1, 0> POINT

Barycentrics    (b - c)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - a*b*c - a*c^2 + c^3) : :
X(69353) = X[1019] - 3 X[62634], X[4467] + 2 X[49274], 2 X[1577] - 3 X[57066], 2 X[4391] - 3 X[30565], 3 X[4789] - 4 X[8045], 3 X[4789] - 2 X[50457], 4 X[14349] - 3 X[48550], 3 X[53334] - 2 X[65685], 4 X[905] - 3 X[4453], 4 X[2487] - 5 X[57088], 2 X[2533] - 3 X[47809], 5 X[3616] - 4 X[34958], 5 X[3617] - 6 X[44729], 2 X[3801] - 3 X[47797], 3 X[3873] - 4 X[39541], and many others

X(69353) lies on these lines: {2, 7178}, {8, 28473}, {63, 1019}, {92, 14618}, {100, 65882}, {329, 29126}, {448, 525}, {512, 3869}, {513, 20294}, {514, 661}, {523, 4833}, {645, 65236}, {647, 16754}, {663, 23877}, {784, 49279}, {826, 48288}, {905, 4453}, {918, 3287}, {1491, 29082}, {1734, 29304}, {2487, 57088}, {2530, 29102}, {2533, 47809}, {2785, 4041}, {2975, 4367}, {3265, 7192}, {3436, 48047}, {3566, 50343}, {3616, 34958}, {3617, 44729}, {3669, 24562}, {3716, 21118}, {3737, 65099}, {3777, 49301}, {3801, 47797}, {3810, 4724}, {3868, 44410}, {3873, 39541}, {3907, 4088}, {3910, 17494}, {4040, 23887}, {4083, 48408}, {4369, 23755}, {4380, 28478}, {4498, 28468}, {4705, 29094}, {4707, 14838}, {4784, 11684}, {4808, 29298}, {4822, 29118}, {4858, 45234}, {4983, 29029}, {4990, 48172}, {5086, 68333}, {5592, 48150}, {6002, 44449}, {6362, 53343}, {6370, 50349}, {7216, 25900}, {7265, 64934}, {8062, 23752}, {8678, 47728}, {8712, 47663}, {8714, 49276}, {10015, 47793}, {11681, 21051}, {11682, 48337}, {14432, 55282}, {14837, 31209}, {15420, 17498}, {17166, 48290}, {20295, 29162}, {20317, 43052}, {21124, 47782}, {21192, 45671}, {21301, 29240}, {21302, 50333}, {23617, 60482}, {23875, 48321}, {23879, 47683}, {23880, 25259}, {23882, 49280}, {23884, 48003}, {24719, 29244}, {26824, 48280}, {28292, 44448}, {28470, 48077}, {28487, 48032}, {28602, 59743}, {28851, 48341}, {29013, 49277}, {29025, 48123}, {29051, 47687}, {29066, 48272}, {29116, 47701}, {29120, 48024}, {29122, 48093}, {29132, 48081}, {29138, 48053}, {29186, 49278}, {29272, 48059}, {29288, 48298}, {30580, 48328}, {30719, 31605}, {31150, 60492}, {32008, 65552}, {35614, 39548}, {39351, 39368}, {42325, 66995}, {44433, 48331}, {47686, 48616}, {47691, 48136}, {47694, 48299}, {47708, 48099}, {47716, 48348}, {47720, 48332}, {47821, 48400}, {47840, 48403}, {48100, 48159}, {48121, 49297}, {48128, 49298}, {48393, 49290}, {49275, 63812}, {53356, 55285}, {54444, 57076}, {57247, 65869}, {58333, 66527}

X(69353) = midpoint of X(4560) and X(49274)
X(69353) = reflection of X(i) in X(j) for these {i,j}: {693, 6332}, {3868, 44410}, {4462, 4468}, {4467, 4560}, {4707, 14838}, {17166, 48290}, {21118, 3716}, {21302, 50333}, {23752, 8062}, {23755, 4369}, {26824, 48280}, {43052, 20317}, {47652, 48131}, {47660, 48300}, {47676, 3669}, {47686, 48616}, {47687, 48278}, {47691, 48136}, {47694, 48299}, {47695, 663}, {47708, 48099}, {47716, 48348}, {47720, 48332}, {48150, 5592}, {48393, 49290}, {49297, 48121}, {49298, 48128}, {49301, 3777}, {49303, 48403}, {50457, 8045}, {65099, 3737}
X(69353) = isotomic conjugate of X(65236)
X(69353) = anticomplement of X(7178)
X(69353) = anticomplement of the isogonal conjugate of X(5546)
X(69353) = anticomplement of the isotomic conjugate of X(645)
X(69353) = isotomic conjugate of the anticomplement of X(40622)
X(69353) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {8, 21294}, {9, 3448}, {21, 150}, {41, 148}, {55, 21221}, {78, 13219}, {99, 21285}, {100, 2893}, {101, 2475}, {110, 7}, {112, 12649}, {162, 56927}, {163, 145}, {212, 39352}, {284, 149}, {333, 21293}, {643, 69}, {644, 1330}, {645, 6327}, {648, 68336}, {662, 3434}, {692, 17778}, {799, 21280}, {827, 20247}, {906, 3152}, {1101, 4467}, {1331, 2897}, {1333, 58371}, {1414, 6604}, {1576, 3210}, {2150, 17154}, {2175, 21220}, {2194, 4440}, {2287, 33650}, {2289, 34186}, {2327, 34188}, {2328, 37781}, {3699, 21287}, {3939, 2895}, {4556, 3873}, {4558, 52365}, {4565, 36845}, {4567, 21302}, {4570, 693}, {4575, 347}, {4587, 52364}, {4592, 68351}, {4612, 17135}, {4629, 20292}, {4631, 17138}, {4636, 75}, {5379, 46400}, {5546, 8}, {6064, 17217}, {7256, 21286}, {7257, 315}, {7259, 3436}, {9447, 25054}, {18315, 68345}, {24000, 23683}, {24041, 4374}, {32656, 18667}, {32676, 30699}, {34072, 3891}, {36034, 41804}, {36069, 68996}, {36134, 68344}, {36142, 4442}, {36797, 21270}, {40972, 39346}, {46639, 68352}, {52378, 3900}, {52914, 17220}, {52935, 20244}, {56245, 25051}, {57657, 9263}, {59079, 5175}, {59149, 3909}, {62530, 20559}, {62534, 21275}, {63461, 54104}, {65201, 4}, {65375, 2}
X(69353) = X(66634)-complementary conjugate of X(21252)
X(69353) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 56946}, {645, 2}
X(69353) = X(i)-cross conjugate of X(j) for these (i,j): {6003, 31603}, {40622, 2}
X(69353) = X(i)-isoconjugate of X(j) for these (i,j): {6, 6011}, {31, 65236}, {112, 43708}, {163, 41501}, {692, 37887}, {1415, 6598}, {1576, 43683}
X(69353) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 65236}, {9, 6011}, {115, 41501}, {1086, 37887}, {1146, 6598}, {4858, 43683}, {6734, 61233}, {8286, 9}, {34591, 43708}, {35193, 5546}, {35583, 2245}
X(69353) = crosspoint of X(85) and X(99)
X(69353) = crosssum of X(i) and X(j) for these (i,j): {41, 512}, {649, 21748}, {669, 9449}
X(69353) = crossdifference of every pair of points on line {31, 21746}
X(69353) = barycentric product X(i)*X(j) for these {i,j}: {8, 31603}, {75, 6003}, {99, 8286}, {314, 57107}, {514, 33116}, {645, 40622}, {693, 34772}, {850, 56840}, {1577, 56439}, {3596, 57139}, {4025, 5174}, {4077, 56946}, {4391, 67120}, {4600, 23775}, {4610, 21961}, {7199, 59733}, {13739, 14208}, {15556, 18155}, {35519, 37583}
X(69353) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 6011}, {2, 65236}, {514, 37887}, {522, 6598}, {523, 41501}, {656, 43708}, {1577, 43683}, {5174, 1897}, {6003, 1}, {8286, 523}, {13739, 162}, {15556, 4551}, {21961, 4024}, {23775, 3120}, {31603, 7}, {33116, 190}, {34772, 100}, {37583, 109}, {39772, 61220}, {40622, 7178}, {41547, 61225}, {56316, 56183}, {56439, 662}, {56840, 110}, {56946, 643}, {56948, 5546}, {57107, 65}, {57139, 56}, {59733, 1018}, {67120, 651}
X(69353) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8045, 50457, 4789}, {47708, 48099, 48161}, {47840, 49303, 48403}

X(69354) = ODD<1, -1, -1, -1, 0, 0> POINT

Barycentrics    a*(b - c)*(a^2 - a*b - b^2 - a*c - b*c - c^2) : :
X(69354) = 2 X[2533] - 3 X[45332], X[1019] - 3 X[14419], X[4983] + 3 X[14419], 3 X[30234] + X[48091], 3 X[30234] - X[50515], X[48086] + 3 X[68816], 4 X[1125] - X[54265], 2 X[551] + X[45676], X[693] - 3 X[47841], X[1577] - 3 X[47839], 3 X[47839] + X[48288], X[1734] - 3 X[47888], X[4775] + 3 X[47888], X[2254] - 3 X[47893], 3 X[47893] + X[48336], and many others

X(69354) lies on these lines: {1, 4705}, {2, 2533}, {10, 29298}, {36, 238}, {110, 39054}, {214, 41185}, {512, 9508}, {514, 1125}, {517, 44824}, {523, 8045}, {551, 45676}, {649, 48123}, {650, 3250}, {659, 48131}, {661, 4367}, {662, 9218}, {663, 1193}, {669, 27647}, {672, 57176}, {690, 21192}, {693, 18077}, {764, 47970}, {810, 2605}, {812, 4992}, {814, 3835}, {830, 1960}, {891, 48003}, {960, 41167}, {1015, 9422}, {1201, 58302}, {1577, 47839}, {1734, 4775}, {2084, 68900}, {2175, 39496}, {2254, 47893}, {2260, 59833}, {2526, 48329}, {2664, 22320}, {2787, 4129}, {2978, 27648}, {3004, 48299}, {3309, 50335}, {3566, 17069}, {3616, 4824}, {3669, 29198}, {3709, 3805}, {3716, 63812}, {3720, 4449}, {3741, 4147}, {3762, 48553}, {3777, 4724}, {3801, 47797}, {3837, 29051}, {3887, 58160}, {3907, 21051}, {3960, 6372}, {4010, 4560}, {4025, 29200}, {4041, 4879}, {4106, 29238}, {4122, 57066}, {4132, 8043}, {4160, 48005}, {4162, 48193}, {4170, 45671}, {4369, 44451}, {4378, 47959}, {4391, 47822}, {4498, 48226}, {4504, 45315}, {4522, 29074}, {4730, 48337}, {4761, 47837}, {4770, 48347}, {4774, 21052}, {4782, 6050}, {4784, 4822}, {4794, 6004}, {4800, 48264}, {4802, 47716}, {4806, 6002}, {4808, 47727}, {5529, 57099}, {6332, 29017}, {6588, 47967}, {7178, 47799}, {8643, 48023}, {8678, 48030}, {8702, 69275}, {9260, 29824}, {10099, 34435}, {14077, 48194}, {14413, 47918}, {14422, 47994}, {14432, 21124}, {15309, 48053}, {16604, 63886}, {16751, 50497}, {16823, 35352}, {17072, 29366}, {17494, 48279}, {17496, 47821}, {18004, 29037}, {18081, 27855}, {19863, 47794}, {19947, 23789}, {20906, 21613}, {21146, 47796}, {21188, 48215}, {21260, 29066}, {23506, 50502}, {23765, 47929}, {23815, 29186}, {23879, 49290}, {23882, 48090}, {24720, 29246}, {24948, 50487}, {25512, 47795}, {25569, 47810}, {29070, 48284}, {29094, 50453}, {29156, 48555}, {29170, 48043}, {29188, 50337}, {29202, 49280}, {29208, 48062}, {29226, 47965}, {29236, 47760}, {29288, 48056}, {29324, 48325}, {29350, 50504}, {30203, 57131}, {31209, 47835}, {39548, 50488}, {39577, 42670}, {40501, 46369}, {42653, 52597}, {45316, 64914}, {45342, 64934}, {47672, 47889}, {47683, 48393}, {47706, 48188}, {47707, 48185}, {47708, 48177}, {47712, 50351}, {47729, 47814}, {47777, 48607}, {47784, 65685}, {47793, 48298}, {47811, 48334}, {47826, 47913}, {47828, 48338}, {47832, 48392}, {47833, 50457}, {47838, 48267}, {47921, 48346}, {47949, 48320}, {47975, 48301}, {47976, 58144}, {47997, 48343}, {48012, 48294}, {48024, 48144}, {48027, 50517}, {48052, 58150}, {48121, 58140}, {48122, 50358}, {48150, 50328}, {48179, 48400}, {48180, 48401}, {48278, 50340}, {48290, 48402}, {48291, 48407}, {48305, 48409}, {48307, 50345}, {48324, 58155}, {48345, 58156}, {48350, 50349}, {48367, 50359}, {48586, 58152}, {50336, 50508}, {59629, 69011}

X(69354) = midpoint of X(i) and X(j) for these {i,j}: {1, 4705}, {649, 48123}, {650, 48136}, {659, 48131}, {661, 4367}, {663, 1491}, {667, 14349}, {764, 47970}, {905, 48099}, {1019, 4983}, {1577, 48288}, {1734, 4775}, {1960, 48059}, {2254, 48336}, {2526, 48329}, {2530, 4040}, {2605, 47842}, {3004, 48299}, {3669, 48029}, {3737, 50330}, {3777, 4724}, {3803, 48092}, {3960, 48058}, {4010, 4560}, {4041, 4879}, {4378, 47959}, {4449, 4490}, {4455, 4481}, {4730, 48337}, {4770, 48347}, {4782, 48129}, {4784, 4822}, {4794, 48066}, {4808, 47727}, {4824, 17166}, {4833, 53527}, {4905, 48351}, {6332, 68780}, {14413, 48162}, {17494, 48279}, {17496, 48265}, {21051, 48289}, {23765, 47929}, {25569, 47810}, {39548, 50488}, {47683, 48393}, {47712, 50351}, {47913, 48341}, {47918, 48323}, {47921, 48346}, {47949, 48320}, {47965, 48332}, {47967, 48344}, {47975, 48301}, {47997, 48343}, {48003, 48348}, {48005, 48328}, {48012, 48294}, {48024, 48144}, {48027, 50517}, {48030, 48330}, {48091, 50515}, {48100, 48331}, {48122, 50358}, {48150, 50328}, {48267, 48321}, {48278, 50340}, {48290, 48402}, {48291, 48407}, {48305, 48409}, {48307, 50345}, {48338, 50355}, {48350, 50349}, {48367, 50359}, {50336, 50508}
X(69354) = reflection of X(i) in X(j) for these {i,j}: {10, 65449}, {4782, 6050}, {9508, 14838}, {21051, 25666}, {23789, 19947}, {45332, 2}, {48345, 58156}, {52601, 1125}, {54265, 52601}
X(69354) = complement of X(2533)
X(69354) = complement of the isotomic conjugate of X(4594)
X(69354) = isogonal conjugate of the isotomic conjugate of X(50451)
X(69354) = X(i)-complementary conjugate of X(j) for these (i,j): {58, 40608}, {110, 51575}, {163, 59509}, {238, 2679}, {256, 125}, {257, 21253}, {805, 3836}, {893, 8287}, {904, 115}, {1178, 11}, {1431, 8286}, {2210, 35078}, {3903, 3454}, {4594, 2887}, {4603, 141}, {7015, 34846}, {7018, 53575}, {7104, 16592}, {7116, 15526}, {7260, 626}, {7303, 53564}, {17938, 1575}, {27805, 21245}, {29055, 442}, {32010, 21252}, {37134, 20541}, {37137, 17052}, {40432, 116}, {40729, 6627}, {66931, 1084}, {66996, 17058}, {67144, 16613}
X(69354) = X(i)-Ceva conjugate of X(j) for these (i,j): {661, 513}, {4367, 4083}
X(69354) = X(i)-isoconjugate of X(j) for these (i,j): {10, 53628}, {37, 65257}, {42, 53655}, {100, 13610}, {101, 6625}, {110, 63885}, {190, 2248}, {648, 15377}, {662, 52208}, {668, 18757}, {692, 51865}, {1492, 40777}, {4557, 40164}
X(69354) = X(i)-Dao conjugate of X(j) for these (i,j): {86, 799}, {244, 63885}, {1015, 6625}, {1084, 52208}, {1086, 51865}, {6627, 75}, {8054, 13610}, {21196, 4036}, {38995, 40777}, {40589, 65257}, {40592, 53655}, {55053, 2248}, {55066, 15377}, {63486, 56241}
X(69354) = crosspoint of X(i) and X(j) for these (i,j): {1, 52935}, {2, 4594}, {58, 29038}, {100, 39977}
X(69354) = crosssum of X(i) and X(j) for these (i,j): {1, 4705}, {6, 7234}, {10, 29037}, {513, 29821}, {662, 65257}
X(69354) = crossdifference of every pair of points on line {37, 171}
X(69354) = barycentric product X(i)*X(j) for these {i,j}: {1, 21196}, {6, 50451}, {513, 1654}, {514, 846}, {523, 38814}, {649, 17762}, {650, 17084}, {656, 2905}, {661, 6626}, {667, 51857}, {693, 18755}, {798, 64224}, {824, 40751}, {905, 4213}, {1019, 21085}, {1491, 40722}, {3125, 57060}, {3733, 27569}, {3737, 27691}, {4010, 45783}, {4369, 63627}, {4978, 38836}, {6627, 52935}, {7192, 21879}, {8937, 68901}, {9508, 39921}, {14838, 14844}, {17924, 22139}, {21832, 52207}, {24381, 40432}
X(69354) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 65257}, {81, 53655}, {512, 52208}, {513, 6625}, {514, 51865}, {649, 13610}, {661, 63885}, {667, 2248}, {810, 15377}, {846, 190}, {1019, 40164}, {1333, 53628}, {1654, 668}, {1919, 18757}, {2905, 811}, {3250, 40777}, {4213, 6335}, {6626, 799}, {6627, 4036}, {14844, 15455}, {17084, 4554}, {17689, 65185}, {17762, 1978}, {18755, 100}, {21085, 4033}, {21196, 75}, {21879, 3952}, {22139, 1332}, {24381, 3963}, {27569, 27808}, {38348, 39922}, {38814, 99}, {38836, 37212}, {40722, 789}, {40751, 4586}, {45783, 4589}, {50451, 76}, {51332, 37135}, {51857, 6386}, {51867, 4584}, {52207, 4639}, {57060, 4601}, {63627, 27805}, {64224, 4602}
X(69354) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {661, 5029, 66523}, {4449, 4893, 4490}, {4560, 47840, 4010}, {4775, 47888, 1734}, {4879, 47827, 4041}, {4983, 14419, 1019}, {14413, 47918, 48323}, {17496, 47821, 48265}, {30234, 48091, 50515}, {47826, 48341, 47913}, {47828, 48338, 50355}, {47838, 48321, 48267}, {47839, 48288, 1577}, {47893, 48336, 2254}, {48162, 48323, 47918}

X(69355) = ODD<0, 1, 1, 1, 1, 1> POINT

Barycentrics    (b - c)*(a^2*b + a*b^2 + b^3 + a^2*c + a*b*c + b^2*c + a*c^2 + b*c^2 + c^3) : :
X(69355) = X[693] - 3 X[48174], 2 X[1491] - 3 X[47877], 3 X[1491] - 2 X[50333], 4 X[3004] - 3 X[47877], 3 X[3004] - X[50333], 2 X[3837] - 3 X[44435], 3 X[44429] - X[47689], 3 X[44435] - X[47690], 9 X[47877] - 4 X[50333], 3 X[48184] - 2 X[48396], 2 X[47960] + X[50340], 2 X[47961] + X[50342], 2 X[4458] - 3 X[48224], 4 X[650] - 3 X[47885], and many others

X(69355) lies on these lines: {2, 47693}, {325, 523}, {513, 16892}, {514, 659}, {522, 4810}, {525, 48123}, {649, 47924}, {650, 4802}, {661, 62423}, {663, 68979}, {676, 48234}, {784, 47712}, {824, 4010}, {826, 14349}, {918, 47998}, {1635, 48146}, {1734, 7927}, {2254, 29144}, {2526, 4777}, {2530, 29021}, {2977, 28179}, {3239, 48555}, {3676, 48253}, {3716, 28863}, {3776, 21146}, {3777, 29142}, {3800, 50355}, {3835, 4122}, {3904, 29172}, {3906, 49277}, {4024, 24085}, {4025, 4784}, {4036, 59713}, {4041, 29208}, {4083, 21124}, {4088, 29204}, {4369, 48227}, {4453, 49283}, {4467, 29328}, {4468, 48162}, {4490, 29288}, {4560, 29025}, {4705, 29047}, {4724, 47923}, {4776, 18004}, {4782, 48101}, {4800, 49286}, {4804, 47673}, {4806, 25259}, {4808, 29260}, {4813, 47990}, {4818, 50341}, {4822, 29200}, {4824, 48404}, {4834, 21192}, {4841, 47928}, {4874, 47660}, {4885, 48192}, {4893, 48056}, {4905, 29168}, {4951, 47756}, {4977, 47676}, {4979, 47902}, {4983, 23875}, {6545, 47703}, {6546, 48097}, {6590, 47833}, {7192, 48241}, {7662, 28894}, {7950, 48059}, {8045, 47841}, {9508, 47886}, {11068, 28191}, {14475, 28151}, {17494, 47688}, {17496, 29120}, {20295, 29078}, {21051, 47707}, {21104, 48143}, {21115, 48148}, {21116, 48135}, {21212, 47823}, {21260, 47711}, {21301, 29074}, {23815, 47715}, {23879, 48273}, {24924, 48215}, {25380, 48235}, {25666, 48185}, {28147, 47779}, {28161, 48160}, {28175, 47667}, {28195, 47919}, {28199, 48095}, {28846, 47983}, {28851, 47946}, {28890, 48001}, {29017, 48131}, {29029, 48321}, {29146, 48100}, {29160, 50351}, {29164, 48066}, {29166, 49278}, {29202, 48129}, {29252, 48081}, {29280, 48093}, {29312, 48335}, {29354, 47959}, {29358, 48054}, {29362, 47652}, {29370, 48550}, {30519, 48043}, {30520, 48029}, {30766, 30795}, {31209, 48236}, {31287, 48219}, {43051, 66287}, {46403, 48156}, {47653, 47694}, {47659, 47834}, {47662, 47804}, {47671, 48127}, {47677, 48080}, {47679, 47716}, {47683, 47725}, {47686, 64913}, {47687, 48159}, {47695, 64914}, {47696, 47798}, {47697, 48223}, {47698, 47781}, {47700, 47810}, {47705, 47934}, {47706, 47814}, {47708, 63812}, {47709, 48410}, {47710, 47816}, {47713, 48409}, {47714, 48556}, {47717, 48407}, {47718, 47819}, {47719, 48406}, {47728, 48289}, {47762, 69011}, {47782, 48408}, {47799, 68794}, {47811, 48130}, {47821, 49273}, {47822, 69293}, {47826, 48048}, {47842, 62418}, {47894, 50343}, {47930, 48021}, {47931, 48032}, {47969, 49302}, {47999, 48023}, {48015, 58374}, {48027, 64856}, {48028, 48082}, {48055, 48604}, {48069, 48244}, {48108, 48422}, {48139, 48572}, {48158, 53343}, {48161, 49275}, {48167, 49285}, {48392, 48403}, {50348, 50359}

X(69355) = midpoint of X(i) and X(j) for these {i,j}: {649, 47924}, {2254, 47702}, {4724, 47923}, {4804, 47673}, {4979, 47902}, {4988, 47704}, {16892, 47701}, {17494, 47688}, {45746, 47691}, {47653, 47694}, {47676, 47699}, {47677, 48080}, {47679, 47716}, {47683, 47725}, {47692, 47975}, {47705, 47934}, {47709, 48410}, {47713, 48409}, {47717, 48407}, {47925, 50358}, {47930, 48021}, {47931, 48032}, {47938, 47971}, {47944, 50342}, {47968, 50340}, {47969, 49302}, {47972, 47973}
X(69355) = reflection of X(i) in X(j) for these {i,j}: {659, 68780}, {1491, 3004}, {4024, 48090}, {4088, 48030}, {4122, 3835}, {4490, 48402}, {4784, 4025}, {4808, 48012}, {4810, 49295}, {4813, 47990}, {4824, 48404}, {4834, 21192}, {4951, 47756}, {21146, 3776}, {23731, 48611}, {24719, 69291}, {25259, 4806}, {47660, 4874}, {47671, 48127}, {47690, 3837}, {47693, 48405}, {47696, 48248}, {47698, 48002}, {47703, 48098}, {47707, 21051}, {47711, 21260}, {47715, 23815}, {47719, 48406}, {47728, 48289}, {47928, 4841}, {47943, 48621}, {47944, 47961}, {47968, 47960}, {48023, 47999}, {48024, 47998}, {48082, 48028}, {48083, 48029}, {48101, 4782}, {48103, 650}, {48106, 9508}, {48108, 58375}, {48117, 48048}, {48118, 48056}, {48120, 23770}, {48140, 47890}, {48143, 21104}, {48272, 48059}, {48275, 54265}, {48278, 48100}, {48392, 48403}, {48604, 48055}, {50328, 48007}, {50341, 4818}, {50358, 50347}, {50359, 50348}, {58374, 48015}
X(69355) = complement of X(47693)
X(69355) = anticomplement of X(48405)
X(69355) = crossdifference of every pair of points on line {32, 35}
X(69355) = barycentric product X(i)*X(j) for these {i,j}: {514, 32784}, {693, 41269}
X(69355) = barycentric quotient X(i)/X(j) for these {i,j}: {32784, 190}, {41269, 100}
X(69355) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 47693, 48405}, {650, 48103, 47885}, {1491, 3004, 47877}, {4122, 48552, 3835}, {4893, 48118, 48056}, {6545, 47703, 48098}, {44435, 47690, 3837}, {47653, 48203, 47694}, {47660, 47797, 4874}, {47696, 47798, 48248}, {47698, 47781, 48002}, {47826, 48117, 48048}, {47886, 48106, 9508}, {47887, 48275, 54265}, {48108, 48422, 58375}, {48140, 48226, 47890}

X(69356) = ODD<0, 1, 1, 1, -1, -1> POINT

Barycentrics    (b - c)*(-(a^2*b) - a*b^2 + b^3 - a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2 + c^3) : :
X(69356) = 4 X[3239] - 3 X[47872], 3 X[47877] - 4 X[48059], 2 X[4025] - 3 X[47893], 2 X[4063] - 3 X[47885], 3 X[4120] - X[21118], 2 X[4142] - 3 X[47822], 2 X[4458] - 3 X[47841], 3 X[4800] - 2 X[21185], 2 X[4874] - 3 X[57066], 3 X[4951] - 2 X[48395], 3 X[14432] - 2 X[48330], 2 X[20517] - 3 X[47839], 4 X[21188] - 5 X[30795], 2 X[21192] - 3 X[47888], 3 X[47819] - 2 X[58375]

X(69356) lies on these lines: {512, 48272}, {513, 4064}, {514, 4122}, {522, 48336}, {523, 48123}, {525, 1491}, {661, 29017}, {690, 1734}, {784, 7265}, {826, 14349}, {830, 49279}, {905, 50342}, {918, 3777}, {2254, 29200}, {2530, 23875}, {2533, 4522}, {3239, 47872}, {3566, 50333}, {3700, 48392}, {3801, 3835}, {3810, 48265}, {3904, 29324}, {3906, 47877}, {3910, 4490}, {4010, 23877}, {4025, 47893}, {4041, 29284}, {4063, 47885}, {4083, 4088}, {4120, 21118}, {4142, 47822}, {4367, 6332}, {4391, 18004}, {4458, 47841}, {4498, 48056}, {4560, 29078}, {4705, 23876}, {4707, 21260}, {4777, 50508}, {4800, 21185}, {4802, 48128}, {4806, 47708}, {4808, 29350}, {4822, 29144}, {4874, 57066}, {4905, 29252}, {4951, 48395}, {4977, 47719}, {4983, 29021}, {4992, 47691}, {6004, 49276}, {6362, 50326}, {6372, 49278}, {8678, 49280}, {8712, 48088}, {8714, 22037}, {14321, 48400}, {14432, 48330}, {16892, 29280}, {20295, 29025}, {20517, 47839}, {21124, 29202}, {21188, 30795}, {21192, 47888}, {21301, 29082}, {23738, 48112}, {23887, 48267}, {25259, 63812}, {28478, 48062}, {28481, 50358}, {29013, 50351}, {29062, 48288}, {29090, 48321}, {29116, 48049}, {29142, 48024}, {29146, 47701}, {29166, 48053}, {29168, 48081}, {29170, 44449}, {29198, 48082}, {29204, 48129}, {29208, 47700}, {29246, 47687}, {29256, 48005}, {29312, 47959}, {29318, 48054}, {29354, 48335}, {30520, 48616}, {47676, 48406}, {47726, 48085}, {47819, 58375}, {47913, 48046}, {47929, 48048}, {47944, 48091}, {47968, 48092}, {48023, 68979}, {48090, 55282}, {48099, 50340}, {48131, 62423}, {48136, 64856}, {48305, 49288}

X(69356) = midpoint of X(i) and X(j) for these {i,j}: {21301, 49274}, {23738, 48112}, {47726, 48085}, {48272, 49277}
X(69356) = reflection of X(i) in X(j) for these {i,j}: {2533, 4522}, {3801, 3835}, {4367, 6332}, {4391, 18004}, {4490, 48047}, {4498, 48056}, {4707, 21260}, {16892, 48100}, {21124, 48030}, {47676, 48406}, {47691, 4992}, {47701, 48093}, {47708, 4806}, {47913, 48046}, {47929, 48048}, {47944, 48091}, {47968, 48092}, {48265, 48270}, {48305, 49288}, {48392, 3700}, {48400, 14321}, {50340, 48099}, {50342, 905}, {50355, 50333}, {55282, 48090}
X(69356) = barycentric product X(514)*X(33092)
X(69356) = barycentric quotient X(33092)/X(190)

X(69357) = ODD<0, 1, 1, 0, 1, 1> POINT

Barycentrics    (b - c)*(a^2*b + a*b^2 + b^3 + a^2*c + b^2*c + a*c^2 + b*c^2 + c^3) : :
X(69357) = 2 X[20517] - 3 X[48224], 4 X[3798] - 3 X[58181], 2 X[4129] - 3 X[48552], X[4391] - 3 X[48174], 3 X[4750] - 2 X[58179], 4 X[21212] - 3 X[47837], 2 X[21260] - 3 X[44435], 3 X[44435] - X[47707], X[21301] - 3 X[48156], 5 X[31251] - 6 X[47757], 4 X[31288] - 3 X[47771], 3 X[44429] - X[47706], X[47662] - 3 X[47820], X[47689] - 3 X[47819], and many others

X(69357) lies on these lines: {1, 68979}, {512, 16892}, {514, 659}, {523, 2530}, {661, 29354}, {663, 47923}, {665, 4988}, {764, 29142}, {784, 47691}, {824, 48273}, {826, 48131}, {830, 47968}, {891, 21124}, {905, 4802}, {918, 4983}, {1491, 4808}, {1734, 29208}, {2254, 7927}, {3004, 4705}, {3776, 50352}, {3777, 29021}, {3798, 58181}, {3800, 50348}, {3803, 28195}, {3837, 47711}, {3904, 29154}, {4025, 4834}, {4088, 48059}, {4129, 48552}, {4391, 48174}, {4560, 29098}, {4750, 58179}, {4822, 29252}, {4905, 29144}, {4992, 7265}, {6004, 47973}, {6050, 48095}, {6367, 47673}, {6372, 47701}, {7950, 48278}, {8678, 47960}, {8714, 48349}, {14349, 62423}, {14419, 28175}, {14838, 48103}, {15309, 47944}, {17166, 47653}, {17496, 29029}, {20295, 29090}, {21212, 47837}, {21260, 44435}, {21301, 29110}, {23729, 29232}, {23770, 48393}, {23815, 47690}, {23875, 48123}, {23879, 48279}, {24719, 29062}, {29017, 48335}, {29025, 48321}, {29037, 69291}, {29070, 47652}, {29086, 46403}, {29094, 48298}, {29136, 53536}, {29146, 48137}, {29168, 47702}, {29204, 48100}, {29260, 48066}, {29266, 48114}, {29280, 48129}, {29312, 48334}, {30520, 48099}, {31208, 47789}, {31251, 47757}, {31288, 47771}, {44429, 47706}, {45746, 47720}, {47660, 52601}, {47662, 47820}, {47689, 47819}, {47692, 48410}, {47693, 47796}, {47710, 48556}, {47712, 63812}, {47795, 48405}, {47839, 69293}, {47840, 49273}, {47877, 48012}, {47886, 50504}, {47888, 48062}, {47902, 48149}, {47907, 50526}, {47916, 50523}, {47919, 50517}, {47924, 48144}, {47931, 48150}, {47947, 47990}, {47948, 47999}, {47949, 47998}, {48053, 48082}, {48058, 48083}, {48060, 58144}, {48092, 64856}, {48094, 50507}, {48101, 50512}, {48136, 49279}, {48138, 58140}, {48177, 59672}, {48566, 69011}

X(69357) = midpoint of X(i) and X(j) for these {i,j}: {663, 47923}, {4560, 47688}, {4822, 47930}, {17166, 47653}, {45746, 47720}, {47692, 48410}, {47702, 48151}, {47717, 48409}, {47902, 48149}, {47907, 50526}, {47916, 50523}, {47919, 50517}, {47924, 48144}, {47931, 48150}
X(69357) = reflection of X(i) in X(j) for these {i,j}: {4088, 48059}, {4705, 3004}, {4808, 1491}, {4834, 4025}, {7265, 4992}, {47660, 52601}, {47690, 23815}, {47707, 21260}, {47711, 3837}, {47715, 48406}, {47947, 47990}, {47948, 47999}, {47949, 47998}, {48082, 48053}, {48083, 48058}, {48094, 50507}, {48095, 6050}, {48101, 50512}, {48103, 14838}, {48272, 48100}, {48393, 23770}, {49277, 48129}, {49278, 48137}, {49279, 48136}, {50352, 3776}
X(69357) = crossdifference of every pair of points on line {2220, 2276}
X(69357) = barycentric product X(514)*X(32781)
X(69357) = barycentric quotient X(32781)/X(190)
X(69357) = {X(44435),X(47707)}-harmonic conjugate of X(21260)

X(69358) = ODD<0, 1, 1, 0, 1, 0> POINT

Barycentrics    (b - c)*(a^2*b + a*b^2 + b^3 + a^2*c + a*c^2 + c^3) : :
X(69358) = 3 X[2] + X[47688], X[693] + 3 X[48174], X[1491] - 3 X[44435], 3 X[4927] - X[48396], 3 X[44429] + X[47692], 3 X[44435] + X[47691], X[47690] - 3 X[48184], 3 X[47877] - X[47975], 3 X[48178] - X[50333], X[649] - 3 X[48227], 3 X[48227] - 2 X[69011], X[650] - 3 X[48192], X[659] - 3 X[47797], X[47652] + 3 X[47797], X[661] - 3 X[48552], and many others

X(69358) lies on these lines: {2, 47688}, {325, 523}, {512, 62435}, {513, 3776}, {514, 1125}, {525, 4992}, {649, 48227}, {650, 48192}, {659, 47652}, {661, 48326}, {676, 1459}, {824, 48090}, {891, 50453}, {900, 24719}, {905, 29025}, {918, 4806}, {2254, 48349}, {2526, 47131}, {2530, 47712}, {2977, 47829}, {3669, 29120}, {3777, 47708}, {3801, 48131}, {3810, 48137}, {3835, 18004}, {3960, 29029}, {4010, 16892}, {4025, 29328}, {4106, 29078}, {4122, 4728}, {4129, 29354}, {4379, 47924}, {4448, 48102}, {4453, 4784}, {4467, 4810}, {4468, 48555}, {4490, 47720}, {4522, 29204}, {4705, 47716}, {4777, 4818}, {4782, 28882}, {4789, 30865}, {4800, 49275}, {4802, 4885}, {4808, 47717}, {4824, 47704}, {4841, 4893}, {6332, 29332}, {6545, 21146}, {7192, 47944}, {7292, 59841}, {7662, 47960}, {7927, 50337}, {9508, 21212}, {11068, 48214}, {14838, 29098}, {17072, 29208}, {19947, 29128}, {20295, 48241}, {21051, 29288}, {21115, 48021}, {21119, 28006}, {21124, 48279}, {21260, 29047}, {23752, 28116}, {23789, 29168}, {23815, 29021}, {23877, 48100}, {24720, 29144}, {24924, 48146}, {25666, 48056}, {26985, 47693}, {27138, 48171}, {28147, 48198}, {28179, 45677}, {28213, 68933}, {28217, 50357}, {28602, 30768}, {28840, 47990}, {28851, 48028}, {28859, 48611}, {28890, 48048}, {29082, 48136}, {29094, 48348}, {29142, 48406}, {29156, 48325}, {29240, 48289}, {29362, 48398}, {29823, 47694}, {30795, 47809}, {30835, 48118}, {31148, 47902}, {31209, 47885}, {31250, 48219}, {31286, 48215}, {31946, 62418}, {43067, 47961}, {46403, 48203}, {47123, 48007}, {47651, 47804}, {47653, 47834}, {47660, 47833}, {47663, 48226}, {47676, 48024}, {47680, 48288}, {47685, 48223}, {47686, 47798}, {47687, 48167}, {47695, 48159}, {47696, 47925}, {47699, 48143}, {47702, 47812}, {47703, 48414}, {47705, 47810}, {47709, 47819}, {47713, 48556}, {47725, 50351}, {47754, 50336}, {47756, 48047}, {47760, 48088}, {47770, 48615}, {47771, 48140}, {47781, 47928}, {47799, 47890}, {47803, 48095}, {47813, 47916}, {47821, 48083}, {47822, 48094}, {47823, 48106}, {47827, 48408}, {47832, 47923}, {47841, 48300}, {47919, 48220}, {47983, 49296}, {48027, 48558}, {48029, 49299}, {48055, 48179}, {48080, 48422}, {48097, 48197}, {48098, 48415}, {48108, 48421}, {48123, 59629}, {48161, 49301}, {48253, 49283}, {48403, 63812}, {48550, 58372}, {48599, 49282}, {48617, 49281}, {50341, 53558}, {50347, 64913}

X(69358) = midpoint of X(i) and X(j) for these {i,j}: {659, 47652}, {661, 48326}, {1491, 47691}, {2254, 48349}, {2526, 47131}, {2530, 47712}, {3004, 23770}, {3777, 47708}, {3801, 48131}, {4010, 16892}, {4025, 49295}, {4458, 69291}, {4467, 4810}, {4490, 47720}, {4705, 47716}, {4808, 47717}, {4824, 47704}, {7192, 47944}, {7662, 47960}, {20295, 50342}, {21104, 47998}, {21124, 48279}, {21146, 47701}, {43067, 47961}, {45746, 48120}, {46403, 50340}, {47123, 48007}, {47676, 48024}, {47680, 48288}, {47686, 50358}, {47688, 48103}, {47694, 47968}, {47695, 50328}, {47696, 47925}, {47699, 48143}, {47725, 50351}, {47983, 49296}, {48029, 49299}, {48083, 49302}, {48398, 68780}, {48550, 58372}, {48599, 49282}, {48617, 49281}, {50341, 53558}
X(69358) = reflection of X(i) in X(j) for these {i,j}: {649, 69011}, {9508, 21212}, {18004, 3835}, {28602, 47757}, {48056, 25666}, {48098, 48415}, {48248, 676}, {48405, 4885}, {58375, 3776}
X(69358) = complement of X(48103)
X(69358) = X(870)-Ceva conjugate of X(1086)
X(69358) = X(4475)-Dao conjugate of X(984)
X(69358) = crosspoint of X(693) and X(4817)
X(69358) = crossdifference of every pair of points on line {32, 3730}
X(69358) = barycentric product X(514)*X(3821)
X(69358) = barycentric quotient X(i)/X(j) for these {i,j}: {3821, 190}, {23231, 906}
X(69358) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 47688, 48103}, {649, 48227, 69011}, {6545, 47701, 21146}, {20295, 48241, 50342}, {30835, 48118, 48185}, {44435, 47691, 1491}, {46403, 48203, 50340}, {47652, 47797, 659}, {47686, 47798, 50358}, {47694, 48156, 47968}, {47695, 48159, 50328}, {47717, 47816, 4808}, {47821, 49302, 48083}, {47925, 48234, 47696}, {48326, 48552, 661}

X(69359) = ODD<0, 1, 1, 0, 0, -1> POINT

Barycentrics    (b - c)*(-(a^2*b) - a*b^2 - a^2*c + b^2*c - a*c^2 + b*c^2) : :
X(69359) = 2 X[4992] + X[50337], X[1577] - 3 X[4728], 3 X[4728] + X[48131], 3 X[4728] - 2 X[59714], 3 X[4776] + X[4801], 3 X[4776] - X[47959], X[47652] + 3 X[57066], X[47918] - 3 X[48551], X[48131] + 2 X[59714], X[48335] + 2 X[59737], 3 X[31149] + X[48333], 3 X[551] - 2 X[48330], X[649] - 3 X[47795], X[659] - 3 X[47839], X[667] - 3 X[47841], and many others

X(69359) lies on these lines: {1, 21301}, {2, 4063}, {10, 4083}, {142, 6008}, {226, 3669}, {512, 3837}, {513, 11813}, {514, 661}, {519, 31149}, {522, 14288}, {523, 48059}, {551, 48330}, {649, 27167}, {650, 29302}, {659, 47839}, {667, 1125}, {676, 28481}, {764, 48265}, {784, 48090}, {812, 14838}, {824, 22007}, {830, 48050}, {891, 21051}, {905, 4106}, {918, 48057}, {946, 3309}, {1019, 17174}, {1022, 31053}, {1491, 4151}, {1519, 1769}, {1734, 44429}, {1848, 17924}, {2051, 4049}, {2254, 4170}, {2530, 4010}, {2786, 21194}, {2978, 23791}, {3004, 23879}, {3261, 57082}, {3452, 20317}, {3566, 62435}, {3616, 31291}, {3634, 31251}, {3661, 31040}, {3741, 30968}, {3776, 22037}, {3777, 48267}, {3907, 48348}, {3910, 50453}, {3960, 6002}, {3993, 64864}, {4025, 29216}, {4040, 46403}, {4041, 47816}, {4088, 47716}, {4132, 44316}, {4162, 12053}, {4375, 17192}, {4379, 48121}, {4444, 40515}, {4481, 20954}, {4498, 30835}, {4500, 31010}, {4522, 29047}, {4560, 21297}, {4705, 30592}, {4724, 47838}, {4778, 48045}, {4782, 19862}, {4784, 48569}, {4785, 21191}, {4804, 48409}, {4806, 6372}, {4807, 17072}, {4810, 47893}, {4822, 47812}, {4834, 47823}, {4885, 29457}, {4905, 12047}, {4960, 47780}, {4977, 48053}, {4979, 48568}, {4983, 21146}, {5249, 23825}, {5333, 57058}, {6005, 24720}, {6545, 22000}, {7192, 48085}, {7265, 16892}, {7662, 48092}, {8678, 48295}, {9010, 49511}, {10165, 39227}, {11263, 19947}, {12575, 58334}, {12609, 50336}, {15309, 48049}, {16783, 57171}, {17023, 24601}, {17030, 27168}, {17166, 47948}, {17217, 52615}, {17458, 21099}, {17758, 35367}, {17761, 40623}, {18004, 29354}, {18107, 30106}, {18197, 27293}, {20888, 23807}, {20908, 21834}, {21070, 21836}, {21077, 48346}, {21120, 23888}, {21188, 28478}, {21192, 21212}, {21196, 27609}, {21201, 23797}, {21302, 48337}, {21385, 27113}, {21613, 33939}, {21616, 59672}, {23731, 27587}, {23813, 23882}, {23829, 50452}, {23887, 48403}, {24924, 47935}, {25356, 64901}, {25511, 31286}, {25666, 48003}, {26114, 29487}, {26798, 48320}, {27046, 27255}, {27648, 30023}, {28225, 48594}, {28470, 48294}, {28759, 30103}, {28840, 48051}, {29066, 48136}, {29070, 48284}, {29126, 46397}, {29158, 49295}, {29182, 48289}, {29186, 48089}, {29190, 68780}, {29198, 68962}, {29220, 49280}, {29252, 58375}, {29270, 30094}, {29344, 48325}, {29362, 50507}, {29604, 30836}, {29667, 31096}, {29989, 29991}, {30038, 30060}, {30795, 47837}, {31147, 48144}, {31148, 48597}, {31946, 50493}, {41012, 47685}, {43067, 48091}, {44444, 48307}, {45320, 48128}, {47678, 48416}, {47691, 48272}, {47694, 48086}, {47697, 48596}, {47700, 47717}, {47701, 47715}, {47702, 47714}, {47708, 49278}, {47712, 48278}, {47756, 48280}, {47759, 47947}, {47760, 47965}, {47762, 47976}, {47802, 50501}, {47810, 48407}, {47821, 47970}, {47822, 68959}, {47832, 48122}, {48079, 48110}, {48081, 48108}, {48093, 48098}, {48107, 48595}, {48123, 48184}, {48167, 48336}, {48173, 57155}, {48305, 50328}, {48573, 50509}, {49290, 68979}

X(69359) = midpoint of X(i) and X(j) for these {i,j}: {1, 21301}, {661, 4978}, {667, 24719}, {693, 14349}, {764, 48265}, {905, 4106}, {1019, 20295}, {1491, 48273}, {1577, 48131}, {2254, 4170}, {2530, 4010}, {3762, 48334}, {3777, 48267}, {3837, 4992}, {4040, 46403}, {4088, 47716}, {4391, 48335}, {4481, 20954}, {4705, 48279}, {4801, 47959}, {4804, 48409}, {4806, 48406}, {4905, 48080}, {4983, 21146}, {7192, 48085}, {7265, 16892}, {7662, 48092}, {8045, 69291}, {14288, 48350}, {17166, 47948}, {20908, 21834}, {21302, 48337}, {43067, 48091}, {44444, 48307}, {47672, 50449}, {47685, 48111}, {47691, 48272}, {47694, 48086}, {47697, 48596}, {47700, 47717}, {47701, 47715}, {47702, 47714}, {47708, 49278}, {47712, 48278}, {48079, 48110}, {48081, 48108}, {48089, 48099}, {48090, 48100}, {48093, 48098}, {48107, 48595}, {48123, 50352}, {48280, 48402}, {48305, 50328}
X(69359) = reflection of X(i) in X(j) for these {i,j}: {10, 21260}, {667, 1125}, {1577, 59714}, {4129, 3835}, {4391, 59737}, {4782, 31288}, {4807, 17072}, {4823, 59522}, {4905, 23814}, {21192, 21212}, {23789, 23815}, {31010, 4500}, {48003, 25666}, {48011, 31286}, {50337, 3837}
X(69359) = complement of X(4063)
X(69359) = complement of the isogonal conjugate of X(65202)
X(69359) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 8054}, {101, 4075}, {596, 116}, {4557, 62588}, {8050, 141}, {20615, 4904}, {34594, 3739}, {37205, 3741}, {39747, 53564}, {39798, 11}, {39949, 17761}, {40013, 21252}, {40085, 125}, {40148, 1086}, {40519, 2}, {59014, 3634}, {65202, 10}, {65286, 21240}
X(69359) = X(i)-Ceva conjugate of X(j) for these (i,j): {1019, 514}, {20295, 513}, {47796, 905}
X(69359) = X(i)-isoconjugate of X(j) for these (i,j): {6, 53627}, {100, 39964}, {101, 39748}, {163, 42471}, {692, 35058}, {32739, 40010}
X(69359) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 53627}, {115, 42471}, {321, 4033}, {1015, 39748}, {1086, 35058}, {8054, 39964}, {31946, 649}, {39798, 8050}, {40619, 40010}, {58288, 4057}
X(69359) = crosspoint of X(i) and X(j) for these (i,j): {86, 1978}, {190, 55990}
X(69359) = crosssum of X(i) and X(j) for these (i,j): {6, 57096}, {42, 1919}
X(69359) = crossdifference of every pair of points on line {31, 20990}
X(69359) = barycentric product X(i)*X(j) for these {i,j}: {86, 31946}, {310, 50493}, {513, 18133}, {514, 17147}, {649, 40034}, {693, 3216}, {1019, 40603}, {1111, 57151}, {1509, 21720}, {3159, 7192}, {3261, 16685}, {7199, 21858}, {22458, 46107}
X(69359) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 53627}, {513, 39748}, {514, 35058}, {523, 42471}, {649, 39964}, {693, 40010}, {3159, 3952}, {3216, 100}, {16685, 101}, {17147, 190}, {18133, 668}, {21720, 594}, {21858, 1018}, {22458, 1331}, {31946, 10}, {40034, 1978}, {40086, 60790}, {40603, 4033}, {50493, 42}, {57151, 765}
X(69359) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {667, 47841, 1125}, {1577, 4728, 59714}, {1577, 52586, 4129}, {4170, 48556, 2254}, {4498, 30835, 47794}, {4705, 30592, 48279}, {4728, 48131, 1577}, {4776, 4801, 47959}, {4905, 47819, 23814}, {20295, 47796, 1019}, {24719, 47841, 667}, {24924, 47935, 48566}, {31251, 47835, 3634}, {46403, 47840, 4040}, {47756, 48280, 48402}, {47819, 48080, 4905}, {48011, 48218, 31286}, {48079, 48570, 48110}, {48123, 48184, 50352}

X(69360) = ODD<0, 1, 1, -1, 0, -1> POINT

Barycentrics    (b - c)*(-(a^2*b) - a*b^2 - a^2*c + a*b*c + b^2*c - a*c^2 + b*c^2) : :
X(69360) = 3 X[693] - X[50457], 3 X[3835] - 2 X[4129], X[4391] - 3 X[4728], 3 X[4728] + X[48334], 3 X[4776] - X[47918], 3 X[4978] + X[50449], 3 X[14349] - X[50449], X[48094] - 3 X[57066], 3 X[48131] + X[50457], X[48335] + 2 X[59522], X[2530] + 3 X[30592], 3 X[30592] - X[48273], 3 X[551] - 2 X[65428], X[649] - 3 X[47796], X[659] - 3 X[47841], and many others

X(69360) lies on these lines: {1, 28470}, {2, 4498}, {11, 24232}, {86, 30183}, {150, 54102}, {226, 30719}, {512, 23815}, {513, 4992}, {514, 661}, {522, 2530}, {523, 48100}, {525, 3776}, {551, 65428}, {649, 26854}, {659, 47841}, {663, 46403}, {764, 48267}, {784, 48394}, {812, 905}, {814, 48325}, {830, 48042}, {834, 47843}, {891, 4147}, {940, 57238}, {1019, 4785}, {1022, 39703}, {1125, 4401}, {1491, 48279}, {1734, 48556}, {2254, 47819}, {2525, 23879}, {2533, 48184}, {2832, 48547}, {3004, 48280}, {3261, 18081}, {3485, 51652}, {3616, 8643}, {3667, 4017}, {3669, 4106}, {3676, 28478}, {3777, 4010}, {3798, 65412}, {3810, 48403}, {3837, 4083}, {3907, 48332}, {3960, 23724}, {4041, 44429}, {4049, 60087}, {4063, 26114}, {4088, 47720}, {4151, 48017}, {4162, 28521}, {4366, 30187}, {4367, 24719}, {4375, 17170}, {4382, 4560}, {4449, 21301}, {4504, 28475}, {4522, 29288}, {4724, 47840}, {4762, 42327}, {4778, 4983}, {4804, 48410}, {4806, 29198}, {4822, 48108}, {4830, 6050}, {4834, 48569}, {4879, 48167}, {4885, 8712}, {4897, 28493}, {4927, 7178}, {4943, 21627}, {4977, 48093}, {4979, 48570}, {6005, 23789}, {6372, 48043}, {7192, 48121}, {7265, 22043}, {7658, 28025}, {7662, 48616}, {8678, 48050}, {14288, 64857}, {14837, 28116}, {14838, 28374}, {15309, 48041}, {16754, 29270}, {17166, 48023}, {17181, 19974}, {17458, 20950}, {17494, 30023}, {17496, 21297}, {18134, 23777}, {20295, 48144}, {20908, 23685}, {21051, 29226}, {21124, 44435}, {21129, 27131}, {21146, 48123}, {21185, 28487}, {21188, 21204}, {21385, 27346}, {23572, 62558}, {23755, 48414}, {23765, 48265}, {23770, 23877}, {23813, 23880}, {23814, 48075}, {23819, 53544}, {23825, 29178}, {23882, 49289}, {24924, 48565}, {25300, 40598}, {25380, 50501}, {25666, 47965}, {28161, 48409}, {28225, 48081}, {28481, 34958}, {28490, 30725}, {28497, 43052}, {28576, 53523}, {28840, 48091}, {29051, 48089}, {29066, 48348}, {29118, 49295}, {29162, 46396}, {29200, 58375}, {29274, 48289}, {29350, 50337}, {30520, 48057}, {30795, 47835}, {30835, 47793}, {31147, 48341}, {31250, 48559}, {36848, 50355}, {43067, 48128}, {44444, 48303}, {44448, 48545}, {45313, 48011}, {45316, 48331}, {45667, 48328}, {47685, 48150}, {47691, 48278}, {47694, 48122}, {47697, 48116}, {47701, 47719}, {47702, 47718}, {47712, 49278}, {47716, 48272}, {47757, 60492}, {47759, 47911}, {47760, 47921}, {47762, 47935}, {47778, 48003}, {47821, 47929}, {47824, 50509}, {47830, 50504}, {47838, 47970}, {47976, 48568}, {47980, 48045}, {47984, 48051}, {47985, 48052}, {47986, 48053}, {48009, 48058}, {48010, 48059}, {48016, 48064}, {48018, 68968}, {48065, 48625}, {48079, 48149}, {48080, 48151}, {48090, 48137}, {48098, 48129}, {48107, 48597}, {48301, 50328}, {48580, 50525}, {48588, 48602}, {48590, 48603}, {50507, 68894}

X(69360) = midpoint of X(i) and X(j) for these {i,j}: {661, 4801}, {663, 46403}, {693, 48131}, {764, 48267}, {1491, 48279}, {1577, 48335}, {2530, 48273}, {3004, 48280}, {3669, 4106}, {3777, 4010}, {4088, 47720}, {4170, 4905}, {4367, 24719}, {4382, 4560}, {4391, 48334}, {4449, 21301}, {4804, 48410}, {4822, 48108}, {4978, 14349}, {4992, 48406}, {6332, 48398}, {7192, 48121}, {7662, 48616}, {17166, 48023}, {17458, 20950}, {20295, 48144}, {21146, 48123}, {23765, 48265}, {23819, 53544}, {43067, 48128}, {44444, 48303}, {47652, 48300}, {47685, 48150}, {47691, 48278}, {47694, 48122}, {47697, 48116}, {47701, 47719}, {47702, 47718}, {47712, 49278}, {47716, 48272}, {48079, 48149}, {48080, 48151}, {48089, 48136}, {48090, 48137}, {48098, 48129}, {48107, 48597}, {48301, 50328}
X(69360) = reflection of X(i) in X(j) for these {i,j}: {1577, 59522}, {3669, 65482}, {3798, 65412}, {4063, 31286}, {4147, 21260}, {4401, 1125}, {4791, 59714}, {4830, 6050}, {17072, 3837}, {24720, 23815}, {47965, 25666}, {47980, 48045}, {47984, 48051}, {47985, 48052}, {47986, 48053}, {47996, 48054}, {48008, 14838}, {48009, 48058}, {48010, 48059}, {48016, 48064}, {48017, 48066}, {48073, 23789}, {48075, 23814}, {48588, 48602}, {48590, 48603}, {48625, 48065}, {50501, 25380}
X(69360) = complement of X(4498)
X(69360) = X(i)-complementary conjugate of X(j) for these (i,j): {101, 59577}, {8690, 3739}, {34860, 116}, {39956, 11}, {40012, 21252}, {42304, 17059}, {56123, 125}, {56155, 4904}, {56192, 8287}, {65059, 53564}
X(69360) = X(i)-Ceva conjugate of X(j) for these (i,j): {3669, 514}, {4106, 3667}, {60257, 1086}
X(69360) = X(i)-isoconjugate of X(j) for these (i,j): {6, 53625}, {101, 979}, {109, 56279}, {692, 39694}, {1415, 56276}, {32739, 58019}, {34080, 39701}
X(69360) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 53625}, {11, 56279}, {312, 646}, {1015, 979}, {1086, 39694}, {1146, 56276}, {16614, 8}, {40619, 58019}, {40621, 39701}, {42336, 57181}, {59971, 649}
X(69360) = crosspoint of X(i) and X(j) for these (i,j): {7, 1978}, {190, 55988}
X(69360) = crosssum of X(55) and X(1919)
X(69360) = crossdifference of every pair of points on line {31, 3217}
X(69360) = barycentric product X(i)*X(j) for these {i,j}: {7, 59971}, {514, 3210}, {693, 978}, {1978, 16614}, {3169, 24002}, {3261, 21769}, {3667, 27835}, {3669, 62585}, {3676, 19582}, {7199, 21857}, {20805, 46107}
X(69360) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 53625}, {513, 979}, {514, 39694}, {522, 56276}, {650, 56279}, {693, 58019}, {978, 100}, {3169, 644}, {3210, 190}, {3667, 39701}, {4397, 60814}, {16614, 649}, {19582, 3699}, {20805, 1331}, {21769, 101}, {21857, 1018}, {27835, 53647}, {59971, 8}, {62585, 646}
X(69360) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2530, 30592, 48273}, {3835, 68901, 54264}, {4063, 47795, 31286}, {4728, 48334, 4391}

X(69361) = ODD<0, 1, 0, 1, -1, 0> POINT

Barycentrics    (b - c)*(-(a^2*b) + b^3 - a^2*c - a*b*c + c^3) : :
X(69361) = 3 X[2] - 4 X[21188], 2 X[1019] - 3 X[47755], 2 X[4707] + X[47676], X[17496] - 3 X[48571], 4 X[21192] - 3 X[27486], X[47722] + 2 X[50342], 2 X[905] - 3 X[4453], 3 X[2457] - 2 X[20316], 4 X[3676] - 3 X[47796], 4 X[3676] - X[49274], 2 X[6332] - 3 X[47796], 3 X[47796] - X[49274], 2 X[4040] - 3 X[47798], 4 X[20517] - 3 X[47798], and many others

X(69361) lies on these lines: {1, 29304}, {2, 21188}, {8, 20504}, {92, 57224}, {189, 60480}, {239, 514}, {512, 47691}, {513, 3801}, {522, 55282}, {523, 50355}, {525, 693}, {663, 4458}, {664, 44717}, {667, 29102}, {690, 48273}, {814, 47722}, {824, 50457}, {826, 47690}, {905, 4453}, {918, 3261}, {1577, 23875}, {1734, 62435}, {2254, 23877}, {2457, 20316}, {2533, 47707}, {2785, 4449}, {3309, 47695}, {3566, 23770}, {3669, 3904}, {3676, 6332}, {3762, 26570}, {3776, 48131}, {3777, 58375}, {3800, 47692}, {3810, 48151}, {3868, 8676}, {3910, 4801}, {4010, 29200}, {4040, 20517}, {4064, 47843}, {4083, 47720}, {4088, 17072}, {4122, 29280}, {4129, 47769}, {4142, 4724}, {4147, 30574}, {4367, 8636}, {4369, 48300}, {4378, 29094}, {4379, 8045}, {4462, 10015}, {4467, 23882}, {4468, 14837}, {4705, 47698}, {4729, 47705}, {4761, 29047}, {4784, 29025}, {4809, 48331}, {4823, 7265}, {4834, 29098}, {4879, 58372}, {4885, 57066}, {4897, 29162}, {4905, 23887}, {4978, 23876}, {5592, 8643}, {6002, 47971}, {6005, 47712}, {6362, 50357}, {6590, 50541}, {8712, 49299}, {8714, 49300}, {14349, 44435}, {17096, 30805}, {17922, 19785}, {18623, 30719}, {20294, 23800}, {20317, 48087}, {20580, 65868}, {21115, 28468}, {21145, 48265}, {21146, 29017}, {21185, 53343}, {21187, 46385}, {22037, 59714}, {22443, 57243}, {23685, 65867}, {23752, 28623}, {23789, 49278}, {23879, 47656}, {24720, 48278}, {28478, 48398}, {28481, 47685}, {28493, 48114}, {28851, 47918}, {28855, 47911}, {28882, 47935}, {28886, 48582}, {28910, 47915}, {29013, 47680}, {29062, 47724}, {29142, 48108}, {29158, 47725}, {29196, 47723}, {29202, 48098}, {29220, 47682}, {29246, 50340}, {29252, 48267}, {29278, 47721}, {29284, 48279}, {29302, 47650}, {29318, 47715}, {29350, 47716}, {29358, 47711}, {30704, 39541}, {31209, 41800}, {31605, 57245}, {47666, 48402}, {47684, 48570}, {47781, 50449}, {47797, 48099}, {47808, 48272}, {47814, 48047}, {47815, 48055}, {47820, 48299}, {47835, 48056}, {47836, 48062}, {47890, 48565}, {47959, 50453}, {48075, 66995}, {48080, 48403}, {48085, 48543}, {48091, 48550}, {48092, 48159}, {48093, 48552}, {48121, 69291}, {48123, 59629}, {48281, 57081}, {48408, 50501}, {48410, 50348}, {49279, 52601}, {57167, 64886}, {63136, 66520}

X(69361) = midpoint of X(i) and X(j) for these {i,j}: {4729, 47705}, {16892, 23755}
X(69361) = reflection of X(i) in X(j) for these {i,j}: {663, 4458}, {1734, 62435}, {3777, 58375}, {3904, 3669}, {4040, 20517}, {4064, 47843}, {4088, 17072}, {4391, 7178}, {4462, 10015}, {4468, 14837}, {4560, 4025}, {4724, 4142}, {4801, 21104}, {6332, 3676}, {7265, 4823}, {20294, 23800}, {22037, 59714}, {25259, 1577}, {46385, 21187}, {47663, 4063}, {47666, 48402}, {47667, 47679}, {47690, 50352}, {47698, 4705}, {47707, 2533}, {47708, 3801}, {47719, 21146}, {47720, 48326}, {47728, 4367}, {47959, 50453}, {48080, 48403}, {48087, 20317}, {48121, 69291}, {48131, 3776}, {48144, 69292}, {48272, 50337}, {48278, 24720}, {48300, 4369}, {48408, 50501}, {48410, 50348}, {49274, 6332}, {49278, 23789}, {49279, 52601}, {53343, 21185}, {66995, 48075}
X(69361) = anticomplement of the isotomic conjugate of X(65247)
X(69361) = isotomic conjugate of the isogonal conjugate of X(43060)
X(69361) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {110, 56729}, {13397, 69}, {15474, 21293}, {23604, 21294}, {28787, 13219}, {39943, 33650}, {56269, 34188}, {59060, 72}, {65247, 6327}
X(69361) = X(i)-Ceva conjugate of X(j) for these (i,j): {18134, 65118}, {65164, 7}, {65247, 2}
X(69361) = X(65118)-cross conjugate of X(18134)
X(69361) = X(i)-isoconjugate of X(j) for these (i,j): {32, 51566}, {37, 58986}, {41, 1305}, {42, 65254}, {101, 2218}, {163, 41506}, {213, 65274}, {228, 68211}, {692, 1751}, {906, 68573}, {1415, 56146}, {1783, 66951}, {2149, 23289}, {2997, 32739}, {32676, 40161}
X(69361) = X(i)-Dao conjugate of X(j) for these (i,j): {72, 4574}, {115, 41506}, {650, 23289}, {1015, 2218}, {1086, 1751}, {1146, 56146}, {3160, 1305}, {5190, 68573}, {6376, 51566}, {6626, 65274}, {7649, 3064}, {15526, 40161}, {39006, 66951}, {40589, 58986}, {40592, 65254}, {40619, 2997}, {40620, 272}, {65118, 24789}
X(69361) = crosspoint of X(i) and X(j) for these (i,j): {286, 4554}, {331, 664}
X(69361) = crosssum of X(i) and X(j) for these (i,j): {228, 3063}, {663, 52425}
X(69361) = crossdifference of every pair of points on line {42, 2175}
X(69361) = barycentric product X(i)*X(j) for these {i,j}: {7, 20294}, {75, 23800}, {76, 43060}, {190, 65118}, {209, 52619}, {304, 57173}, {307, 57072}, {314, 51658}, {348, 57043}, {514, 18134}, {579, 3261}, {651, 17878}, {693, 3868}, {1088, 58333}, {2352, 40495}, {3190, 52621}, {4025, 5125}, {4306, 35519}, {4560, 56559}, {5190, 65164}, {6063, 8676}, {7182, 57092}, {7192, 57808}, {7199, 22021}, {23989, 57217}, {24002, 27396}, {34387, 65315}
X(69361) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 1305}, {11, 23289}, {27, 68211}, {58, 58986}, {75, 51566}, {81, 65254}, {86, 65274}, {209, 4557}, {513, 2218}, {514, 1751}, {522, 56146}, {523, 41506}, {525, 40161}, {579, 101}, {693, 2997}, {1459, 66951}, {2352, 692}, {3190, 3939}, {3261, 40011}, {3868, 100}, {4306, 109}, {5125, 1897}, {5190, 3064}, {7192, 272}, {7649, 68573}, {8676, 55}, {17094, 28786}, {17878, 4391}, {17925, 40574}, {18134, 190}, {20294, 8}, {22021, 1018}, {23800, 1}, {27396, 644}, {43060, 6}, {51574, 4574}, {51658, 65}, {52619, 57784}, {52621, 15467}, {56000, 5546}, {56559, 4552}, {57043, 281}, {57072, 29}, {57092, 33}, {57173, 19}, {57217, 1252}, {57808, 3952}, {58333, 200}, {65118, 514}, {65315, 59}, {66918, 4559}
X(69361) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3676, 6332, 47796}, {4040, 20517, 47798}, {4468, 14837, 47793}, {4823, 7265, 47790}, {47796, 49274, 6332}, {48272, 50337, 47808}

X(69362) = ODD<1, -1, 1, 0, 0, -1> POINT

Barycentrics    (b - c)*(-a^3 + a^2*b - a*b^2 + a^2*c + b^2*c - a*c^2 + b*c^2) : :
X(69362) = 2 X[659] - 3 X[47835], 4 X[17072] - 3 X[47835], 2 X[663] - 3 X[47841], 4 X[3837] - 3 X[47841], 2 X[667] - 3 X[47823], 3 X[47823] - 4 X[50337], 2 X[905] - 3 X[36848], 4 X[1125] - 3 X[58155], 2 X[1960] - 3 X[47795], 2 X[4040] - 3 X[47822], 4 X[21260] - 3 X[47822], 2 X[4129] - 3 X[31149], 3 X[31149] - X[48351], 2 X[4401] - 3 X[47837], and many others

X(69362) lies on these lines: {1, 23815}, {2, 48331}, {8, 29226}, {512, 24719}, {513, 2517}, {514, 4774}, {522, 3801}, {659, 17072}, {661, 29246}, {663, 3837}, {667, 47823}, {693, 21303}, {784, 47724}, {812, 50355}, {814, 2254}, {830, 50352}, {900, 7178}, {905, 36848}, {1125, 58155}, {1491, 29051}, {1577, 6004}, {1734, 29070}, {1960, 47795}, {2530, 29066}, {2787, 4905}, {3309, 4010}, {3669, 4922}, {3777, 3907}, {3835, 48336}, {3887, 48273}, {3900, 48089}, {4040, 21260}, {4041, 29362}, {4083, 21302}, {4129, 31149}, {4142, 44314}, {4367, 24720}, {4378, 23789}, {4382, 20507}, {4401, 47837}, {4449, 48406}, {4467, 29276}, {4498, 64913}, {4560, 29274}, {4705, 29186}, {4724, 21051}, {4730, 29302}, {4782, 47836}, {4794, 47839}, {4806, 48367}, {4809, 21188}, {4823, 48189}, {4874, 48150}, {4879, 48167}, {4885, 48329}, {4926, 21145}, {4977, 47912}, {4992, 48338}, {6002, 50359}, {7199, 21305}, {8640, 23791}, {8678, 21146}, {14349, 29188}, {14430, 47936}, {14431, 59672}, {16695, 28373}, {16892, 29074}, {17166, 48098}, {17496, 29236}, {20983, 22319}, {21052, 48032}, {21901, 69103}, {23506, 26854}, {23818, 24674}, {23882, 50341}, {24698, 64861}, {28209, 47911}, {28521, 48184}, {28585, 48327}, {29017, 47687}, {29033, 48018}, {29082, 48278}, {29094, 49278}, {29102, 48272}, {29182, 48321}, {29208, 47652}, {29238, 50343}, {29278, 50348}, {29298, 48335}, {29324, 48151}, {29344, 48075}, {29366, 48131}, {31288, 68816}, {31291, 47824}, {40086, 48283}, {42325, 48267}, {44316, 48297}, {47690, 68979}, {47721, 48410}, {47729, 47819}, {47794, 53571}, {47796, 48330}, {47812, 48322}, {47816, 50507}, {47843, 50353}, {47872, 48063}, {47888, 48284}, {47929, 48401}, {47946, 47956}, {47967, 47969}, {48012, 48176}, {48050, 48123}, {48065, 48553}, {48066, 48288}, {48100, 48164}, {48137, 48298}, {48324, 52601}, {48573, 50512}, {50457, 64914}, {53565, 55969}, {57114, 67196}

X(69362) = midpoint of X(i) and X(j) for these {i,j}: {21302, 46403}, {47721, 48410}
X(69362) = reflection of X(i) in X(j) for these {i,j}: {1, 23815}, {659, 17072}, {663, 3837}, {667, 50337}, {4040, 21260}, {4142, 44314}, {4367, 24720}, {4378, 23789}, {4449, 48406}, {4560, 50335}, {4724, 21051}, {4922, 3669}, {17166, 48098}, {47929, 48401}, {47946, 47956}, {47969, 47967}, {48123, 48050}, {48150, 4874}, {48279, 48089}, {48283, 40086}, {48288, 48066}, {48297, 44316}, {48298, 48137}, {48301, 693}, {48305, 4823}, {48324, 52601}, {48329, 4885}, {48336, 3835}, {48338, 4992}, {48351, 4129}, {48367, 4806}, {50353, 47843}, {55969, 53565}
X(69362) = anticomplement of X(48331)
X(69362) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {39742, 149}, {39966, 4440}, {60236, 150}
X(69362) = crosspoint of X(668) and X(60075)
X(69362) = crosssum of X(i) and X(j) for these (i,j): {43, 48307}, {667, 4253}
X(69362) = crossdifference of every pair of points on line {2300, 5037}
X(69362) = barycentric product X(514)*X(32920)
X(69362) = barycentric quotient X(32920)/X(190)
X(69362) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {659, 17072, 47835}, {663, 3837, 47841}, {667, 50337, 47823}, {4040, 21260, 47822}, {4823, 48305, 48189}, {31149, 48351, 4129}

X(69363) = ODD<0, 1, 0, -1, 0, 1> POINT

Barycentrics    (b - c)*(a^2*b + a^2*c - a*b*c + b^2*c + b*c^2) : :
X(69363) = 3 X[2533] - X[21146], 3 X[3777] - 4 X[23814], 3 X[4147] - X[48010], 3 X[4490] - X[47928], 3 X[4705] - 2 X[48010], 2 X[21146] - 3 X[50352], 2 X[23814] - 3 X[50337], 3 X[4391] - X[48080], 2 X[48080] - 3 X[48267], X[661] - 3 X[14430], 2 X[905] - 3 X[47837], 3 X[1577] - 2 X[48090], 4 X[48090] - 3 X[48273], 2 X[1960] - 3 X[47804], and many others

X(69363) lies on these lines: {1, 4874}, {2, 48298}, {8, 47694}, {10, 514}, {512, 4391}, {513, 3762}, {519, 48234}, {522, 4730}, {523, 10015}, {649, 2787}, {650, 48288}, {659, 4774}, {660, 668}, {661, 14430}, {663, 10459}, {667, 3907}, {690, 25259}, {693, 891}, {784, 4041}, {814, 4063}, {826, 47707}, {834, 4036}, {899, 4379}, {905, 47837}, {1019, 29324}, {1193, 4449}, {1499, 50326}, {1577, 4083}, {1734, 63812}, {1960, 47729}, {2517, 6371}, {2785, 49279}, {3006, 47728}, {3063, 47127}, {3216, 48282}, {3264, 4406}, {3416, 9014}, {3669, 48569}, {3679, 64914}, {3716, 4775}, {3737, 6133}, {3800, 48400}, {3801, 29047}, {3835, 14431}, {3837, 48335}, {3900, 48305}, {3904, 47809}, {3910, 48395}, {3960, 47823}, {4010, 4791}, {4040, 29366}, {4122, 23876}, {4129, 48123}, {4132, 50327}, {4139, 7650}, {4151, 48392}, {4369, 4378}, {4380, 29340}, {4448, 4794}, {4462, 6372}, {4498, 29070}, {4507, 18155}, {4522, 28468}, {4560, 50504}, {4581, 20293}, {4707, 62423}, {4724, 29188}, {4729, 48264}, {4770, 47975}, {4776, 28603}, {4782, 29236}, {4784, 29148}, {4807, 8714}, {4808, 23877}, {4817, 68899}, {4823, 48279}, {4834, 6002}, {4879, 47872}, {4885, 48332}, {4893, 30970}, {4978, 29226}, {4992, 59521}, {6004, 21302}, {6005, 48265}, {6050, 48559}, {6161, 48063}, {6366, 48290}, {7178, 29288}, {7192, 53364}, {7265, 29284}, {7662, 14077}, {7927, 47708}, {7950, 47706}, {9508, 48321}, {14349, 21051}, {14413, 24924}, {14419, 31286}, {14421, 47779}, {14838, 47835}, {16892, 30574}, {17496, 47836}, {17924, 68781}, {18003, 35519}, {18004, 49277}, {19863, 47794}, {19875, 45323}, {20295, 30709}, {20316, 50330}, {20317, 48099}, {21052, 21260}, {21120, 29142}, {21198, 48177}, {21222, 47824}, {21343, 47833}, {21385, 29362}, {23361, 48387}, {23815, 48334}, {23880, 50501}, {23884, 48188}, {25569, 48285}, {25574, 50767}, {26030, 47796}, {26251, 44435}, {26546, 47128}, {28294, 48247}, {28473, 48299}, {28537, 48219}, {29017, 47711}, {29029, 48106}, {29094, 48300}, {29102, 48094}, {29146, 47710}, {29150, 50509}, {29166, 47689}, {29172, 47726}, {29176, 58179}, {29192, 50340}, {29208, 47712}, {29212, 50342}, {29224, 48118}, {29240, 47890}, {29246, 47970}, {29268, 48565}, {29312, 47690}, {29344, 48011}, {29639, 30580}, {30061, 50510}, {30592, 59522}, {30725, 48232}, {31079, 47773}, {31149, 48050}, {33920, 62620}, {44429, 53571}, {47663, 47722}, {47682, 48405}, {47684, 48236}, {47793, 50507}, {47814, 48059}, {47816, 48100}, {47817, 48331}, {47818, 48330}, {47820, 48328}, {47834, 48304}, {47839, 48136}, {47841, 48348}, {47874, 49290}, {47875, 48333}, {47904, 47918}, {47922, 48610}, {47949, 47980}, {47959, 47993}, {47967, 50449}, {48093, 48551}, {48137, 48556}, {48157, 53620}, {48171, 49274}, {48226, 48284}, {48248, 48324}, {48254, 60480}, {48287, 50604}, {48336, 59672}, {49273, 53356}, {50359, 68896}, {50764, 64913}, {55182, 69280}

X(69363) = midpoint of X(i) and X(j) for these {i,j}: {8, 47694}, {649, 4474}, {659, 4774}, {3762, 4761}, {4581, 20293}, {4729, 48264}, {21385, 47724}, {47663, 47722}, {48254, 60480}
X(69363) = reflection of X(i) in X(j) for these {i,j}: {1, 4874}, {764, 24720}, {1491, 10}, {2530, 17072}, {3737, 6133}, {3777, 50337}, {4010, 4791}, {4378, 4369}, {4449, 52601}, {4560, 50504}, {4705, 4147}, {4775, 3716}, {4776, 28603}, {4992, 59521}, {6161, 48063}, {14349, 21051}, X(69363) = {14421, 47779}, {21343, 48295}, {23765, 23789}, {30580, 47766}, {47682, 48405}, {47729, 1960}, {47959, 48401}, {47975, 4770}, {48099, 20317}, {48123, 4129}, {48131, 21260}, {48177, 21198}, {48267, 4391}, {48273, 1577}, {48279, 4823}, {48288, 650}, {48290, 68794}, {48291, 7662}, {48321, 9508}, {48324, 48248}, {48325, 31286}, {48332, 4885}, {48334, 23815}, {48335, 3837}, {48336, 59672}, {49277, 18004}, {49279, 69293}, {50330, 20316}, {50331, 4036}, {50351, 48062}, {50352, 2533}, {50355, 4807}, {50449, 47967}
X(69363) = isotomic conjugate of X(35008)
X(69363) = complement of X(48298)
X(69363) = X(i)-isoconjugate of X(j) for these (i,j): {6, 35009}, {31, 35008}
X(69363) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 35008}, {9, 35009}
X(69363) = crossdifference of every pair of points on line {1914, 2278}
X(69363) = barycentric product X(514)*X(32931)
X(69363) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 35009}, {2, 35008}, {32931, 190}
X(69363) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {20317, 48099, 48553}, {21052, 48131, 21260}, {21343, 47833, 48295}, {31286, 48325, 14419}, {47729, 47804, 1960}

X(69364) = ODD<0, 1, 0, -1, -1, 0> POINT

Barycentrics    (b - c)*(-(a^2*b) + b^3 - a^2*c + a*b*c + c^3) : :
X(69364) = 3 X[2] - 4 X[14837], 5 X[2] - 4 X[45683], 5 X[6332] - 6 X[45683], 5 X[14837] - 3 X[45683], 2 X[4560] - 3 X[27486], 4 X[4707] - X[47676], 3 X[47755] - 2 X[48144], 4 X[10015] - X[25259], X[21302] - 3 X[53356], 2 X[663] - 3 X[47798], 4 X[4142] - 3 X[47798], 3 X[693] - 2 X[48280], 3 X[7178] - X[48280], 4 X[1577] - 3 X[47790], and many others

X(69364) lies on these lines: {1, 20517}, {2, 2399}, {10, 48272}, {239, 514}, {297, 525}, {321, 52622}, {512, 47708}, {521, 43923}, {522, 17950}, {653, 44765}, {656, 20294}, {659, 29082}, {663, 2785}, {664, 1262}, {667, 29094}, {676, 65685}, {690, 48267}, {693, 3910}, {826, 47707}, {834, 21121}, {891, 47720}, {905, 3904}, {918, 4462}, {1018, 25267}, {1459, 21187}, {1577, 2610}, {1734, 23887}, {1993, 57223}, {2254, 3810}, {2400, 39721}, {2457, 47843}, {2533, 29017}, {2994, 60480}, {3265, 25902}, {3566, 48080}, {3617, 4163}, {3622, 52596}, {3669, 4453}, {3676, 66986}, {3762, 23875}, {3776, 28024}, {3800, 47709}, {3801, 4083}, {3868, 64878}, {3900, 47695}, {4010, 29284}, {4040, 29304}, {4041, 23877}, {4049, 59714}, {4064, 20316}, {4088, 4147}, {4122, 29202}, {4129, 49277}, {4151, 49300}, {4373, 60479}, {4380, 29162}, {4449, 4458}, {4467, 23880}, {4474, 29037}, {4490, 47698}, {4522, 21052}, {4605, 56188}, {4761, 29021}, {4774, 29074}, {4784, 29120}, {4791, 7265}, {4809, 48330}, {4834, 29029}, {4850, 52595}, {4905, 62435}, {6362, 50356}, {6589, 16754}, {7253, 7649}, {8643, 13246}, {8712, 47652}, {8713, 48014}, {14298, 20296}, {14349, 50453}, {14838, 23884}, {17072, 30574}, {19860, 58339}, {20293, 23874}, {20295, 28478}, {20317, 30565}, {21102, 28623}, {21108, 21300}, {21130, 47959}, {21145, 48279}, {21186, 57091}, {21188, 47796}, {21189, 57111}, {23765, 58375}, {23984, 65295}, {28292, 48239}, {28468, 44435}, {28473, 47729}, {28490, 53536}, {28515, 48042}, {29070, 47722}, {29116, 48106}, {29118, 50509}, {29138, 58179}, {29190, 47724}, {29200, 48265}, {29226, 48326}, {29302, 47680}, {29312, 47719}, {29318, 47711}, {29324, 50342}, {29332, 48103}, {29350, 47712}, {29366, 50340}, {32714, 46640}, {33890, 53583}, {34772, 57108}, {44433, 48329}, {47656, 50457}, {47684, 48565}, {47696, 68979}, {47771, 48300}, {47781, 48402}, {47793, 49274}, {47797, 48136}, {47800, 53334}, {47804, 48299}, {47820, 48290}, {47875, 49290}, {48018, 66995}, {48024, 59629}, {48121, 48543}, {48128, 48550}, {48129, 48552}, {48159, 48616}, {48161, 50508}, {49276, 59672}, {49278, 50337}, {49280, 57066}, {50326, 59549}, {50333, 55285}, {50351, 50504}, {51664, 69041}, {56560, 57243}

X(69364) = reflection of X(i) in X(j) for these {i,j}: {1, 20517}, {663, 4142}, {693, 7178}, {1459, 21187}, {3904, 905}, {4064, 20316}, {4088, 4147}, {4391, 10015}, {4449, 4458}, {4462, 21120}, {4905, 62435}, {6332, 14837}, {7253, 7649}, {7265, 4791}, {14349, 50453}, {17494, 60492}, {17496, 4025}, {20294, 656}, {20296, 14298}, {23765, 58375}, {25259, 4391}, {45746, 21124}, {47656, 50457}, {47663, 4498}, {47690, 2533}, {47691, 3801}, {47698, 4490}, {47719, 50352}, {47728, 667}, {47808, 30574}, {48080, 48400}, {48272, 10}, {48278, 17072}, {48321, 21192}, {48334, 3776}, {48341, 69292}, {49276, 59672}, {49277, 4129}, {49278, 50337}, {50333, 55285}, {50351, 50504}, {53334, 47800}, {57091, 21186}, {65685, 676}, {66995, 48018}
X(69364) = isogonal conjugate of X(32653)
X(69364) = isotomic conjugate of X(44765)
X(69364) = anticomplement of X(6332)
X(69364) = polar conjugate of X(26704)
X(69364) = anticomplement of the isogonal conjugate of X(32674)
X(69364) = anticomplement of the isotomic conjugate of X(653)
X(69364) = isotomic conjugate of the anticomplement of X(40626)
X(69364) = isotomic conjugate of the isogonal conjugate of X(6589)
X(69364) = isotomic conjugate of the polar conjugate of X(59915)
X(69364) = polar conjugate of the isotomic conjugate of X(57242)
X(69364) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 34188}, {19, 33650}, {25, 37781}, {34, 150}, {65, 13219}, {101, 52366}, {108, 69}, {109, 4329}, {112, 3869}, {162, 20245}, {225, 21294}, {278, 21293}, {608, 149}, {651, 1370}, {653, 6327}, {664, 68347}, {692, 56943}, {934, 68351}, {1395, 4440}, {1402, 39352}, {1409, 34186}, {1414, 18659}, {1415, 20}, {1461, 52365}, {1783, 3436}, {1880, 3448}, {1897, 21286}, {1973, 39351}, {2149, 20294}, {2443, 34550}, {4559, 52364}, {4565, 20243}, {6516, 68355}, {7012, 20295}, {7115, 513}, {7128, 21302}, {8687, 64039}, {8750, 329}, {13149, 21280}, {14776, 64194}, {15385, 35518}, {18026, 315}, {23979, 66520}, {23985, 521}, {24033, 46400}, {32647, 6001}, {32667, 515}, {32669, 10538}, {32674, 8}, {32676, 63}, {32688, 3827}, {32702, 517}, {32713, 92}, {32714, 3434}, {32727, 64780}, {36059, 6527}, {36110, 68912}, {36113, 35516}, {36118, 21285}, {36127, 21270}, {40097, 8048}, {46102, 21301}, {46152, 1369}, {46404, 21275}, {52914, 54109}, {52920, 68338}, {53321, 2897}, {54240, 11442}, {56183, 54113}, {57193, 6225}, {57220, 64584}, {57652, 21221}, {58997, 67074}, {61178, 21287}, {65232, 17135}
X(69364) = X(i)-Ceva conjugate of X(j) for these (i,j): {653, 2}, {664, 10571}, {4565, 17184}, {18026, 56827}, {65297, 69027}
X(69364) = X(i)-cross conjugate of X(j) for these (i,j): {6589, 59915}, {38345, 3869}, {40626, 2}
X(69364) = X(i)-isoconjugate of X(j) for these (i,j): {1, 32653}, {6, 36050}, {31, 44765}, {37, 59005}, {42, 65253}, {48, 26704}, {101, 2217}, {163, 15232}, {213, 54951}, {604, 56112}, {650, 15386}, {692, 13478}, {906, 68574}, {1415, 10570}, {2182, 35183}, {2995, 32739}, {3063, 57757}, {32700, 46974}, {40160, 65375}
X(69364) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 44765}, {3, 32653}, {9, 36050}, {65, 4559}, {115, 15232}, {124, 6}, {1015, 2217}, {1086, 13478}, {1146, 10570}, {1249, 26704}, {3125, 42550}, {3161, 56112}, {5190, 68574}, {6589, 522}, {6626, 54951}, {10001, 57757}, {34588, 37}, {38345, 64042}, {40589, 59005}, {40592, 65253}, {40619, 2995}, {40622, 40160}, {40625, 19607}, {56973, 2425}, {57502, 32660}
X(69364) = crosspoint of X(i) and X(j) for these (i,j): {76, 664}, {190, 59759}, {274, 18026}, {648, 60235}, {57551, 65295}
X(69364) = crosssum of X(i) and X(j) for these (i,j): {32, 663}, {213, 1946}
X(69364) = trilinear pole of line {124, 34588}
X(69364) = crossdifference of every pair of points on line {42, 184}
X(69364) = barycentric product X(i)*X(j) for these {i,j}: {4, 57242}, {69, 59915}, {75, 21189}, {76, 6589}, {92, 57184}, {124, 664}, {273, 57111}, {321, 16754}, {514, 4417}, {573, 3261}, {653, 40626}, {693, 3869}, {850, 4225}, {3185, 40495}, {4025, 17555}, {4391, 17080}, {4554, 38345}, {7199, 21078}, {7649, 51612}, {10571, 35519}, {18026, 34588}, {22276, 52619}, {34393, 55128}, {44129, 52310}, {46110, 56553}, {46404, 47411}
X(69364) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 36050}, {2, 44765}, {4, 26704}, {6, 32653}, {8, 56112}, {58, 59005}, {81, 65253}, {86, 54951}, {102, 35183}, {109, 15386}, {124, 522}, {513, 2217}, {514, 13478}, {522, 10570}, {523, 15232}, {573, 101}, {664, 57757}, {693, 2995}, {3185, 692}, {3192, 8750}, {3261, 57906}, {3869, 100}, {3910, 19608}, {4225, 110}, {4417, 190}, {4560, 19607}, {6589, 6}, {7178, 40160}, {7649, 68574}, {10571, 109}, {14349, 53082}, {16754, 81}, {17080, 651}, {17555, 1897}, {19367, 65315}, {21078, 1018}, {21189, 1}, {22134, 906}, {22276, 4557}, {34242, 2222}, {34588, 521}, {36121, 36108}, {37836, 53324}, {38345, 650}, {39992, 2689}, {40590, 4559}, {40626, 6332}, {47411, 652}, {50330, 42550}, {51612, 4561}, {52310, 71}, {55128, 515}, {56553, 1813}, {56827, 61178}, {57111, 78}, {57184, 63}, {57220, 7115}, {57242, 69}, {59915, 4}
X(69364) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {663, 4142, 47798}, {6332, 14837, 2}, {17072, 48278, 47808}, {30574, 48278, 17072}

X(69365) = ODD<0, 1, -1, 1, 0, -1> POINT

Barycentrics    (b - c)*(-(a^2*b) + a*b^2 - a^2*c - a*b*c + b^2*c + a*c^2 + b*c^2) : :

X(69365) lies on these lines: {10, 48018}, {30, 511}, {145, 4959}, {656, 59971}, {661, 48410}, {663, 17496}, {665, 3239}, {667, 48063}, {693, 48151}, {764, 48273}, {905, 3716}, {1491, 48265}, {1577, 4905}, {1635, 47815}, {1734, 3762}, {2254, 4391}, {2530, 3835}, {2533, 50359}, {3250, 48269}, {3676, 18033}, {3700, 54249}, {3766, 4025}, {3776, 48403}, {3777, 4010}, {4017, 4811}, {4024, 47719}, {4040, 48321}, {4041, 4462}, {4120, 69101}, {4129, 48066}, {4163, 44720}, {4170, 48335}, {4378, 48305}, {4401, 8689}, {4403, 24225}, {4449, 21222}, {4458, 21185}, {4474, 21302}, {4490, 50341}, {4498, 50343}, {4504, 48327}, {4521, 52592}, {4560, 4724}, {4705, 48017}, {4728, 47819}, {4763, 48561}, {4791, 23795}, {4800, 47841}, {4801, 4804}, {4806, 48100}, {4818, 48402}, {4823, 23789}, {4824, 47913}, {4913, 47965}, {4978, 48394}, {4983, 48037}, {4985, 23800}, {4992, 48137}, {6050, 53580}, {6332, 19582}, {7178, 50357}, {7253, 43924}, {7265, 49278}, {7636, 50501}, {7650, 50354}, {8045, 49286}, {14349, 48043}, {14837, 19950}, {14838, 59672}, {16892, 47708}, {17166, 48341}, {17494, 47929}, {20295, 48122}, {20316, 50350}, {20517, 21201}, {21051, 50335}, {21146, 48392}, {21836, 57169}, {22319, 50513}, {23765, 48279}, {23814, 59714}, {23815, 59522}, {25242, 60490}, {25259, 48278}, {30719, 51641}, {30725, 65685}, {44550, 45316}, {44551, 45338}, {44561, 45337}, {45328, 45664}, {45671, 45673}, {47129, 68881}, {47666, 47906}, {47676, 55282}, {47694, 48144}, {47697, 50523}, {47720, 53558}, {47778, 47888}, {47779, 47875}, {47793, 47828}, {47794, 47830}, {47795, 47831}, {47796, 47832}, {47805, 58140}, {47813, 48570}, {47822, 47893}, {47823, 47872}, {47835, 48244}, {47837, 48575}, {47839, 48547}, {47843, 50327}, {47889, 48189}, {47911, 47945}, {47918, 47975}, {47942, 47986}, {47949, 47996}, {47955, 47992}, {47957, 48002}, {47959, 48010}, {47966, 48000}, {48049, 48092}, {48065, 48284}, {48073, 50352}, {48080, 48131}, {48090, 48406}, {48108, 50457}, {48150, 53536}, {48272, 66995}, {48282, 48339}, {48288, 48351}, {48298, 48338}, {48300, 49275}, {48301, 48323}, {48393, 48399}, {48400, 50348}, {56174, 60574}

X(69365) = isogonal conjugate of X(59113)
X(69365) = crossdifference of every pair of points on line {6, 18613}
X(69365) = barycentric product X(i)*X(j) for these {i,j}: {1369, 44934}, {33820, 44952}
X(69365) = barycentric quotient X(13435)/X(25159)
X(69365) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {663, 17496, 48325}, {905, 59590, 3716}, {1577, 4905, 24720}, {1734, 3762, 4147}, {2254, 4391, 17072}, {2530, 48267, 3835}, {4462, 50356, 4041}, {4791, 48075, 50337}, {4804, 23738, 4801}, {17496, 53343, 663}, {23795, 50337, 48075}, {47875, 48569, 47779}, {47888, 48553, 47778}, {47942, 50449, 47986}, {47959, 48409, 48010}, {48151, 48264, 693}

X(69366) = ODD<0, 1, -1, 1, -1, -1> POINT

Barycentrics    (b - c)*(-(a^2*b) + a*b^2 + b^3 - a^2*c - a*b*c + b^2*c + a*c^2 + b*c^2 + c^3) : :
X(69366) = 3 X[16892] - X[47701], 2 X[47951] - 3 X[47968], 3 X[47971] + X[48585], 3 X[47973] - X[48585], 3 X[649] - X[48139], 3 X[659] - 2 X[48061], 3 X[4025] - X[48061], 2 X[661] - 3 X[47877], 3 X[1491] - 2 X[48047], X[48047] - 3 X[50348], 3 X[1635] - X[48113], 3 X[2254] - X[47700], X[47700] + 3 X[47930], 4 X[2487] - 3 X[48231], and many others

X(69366) lies on these lines: {513, 16892}, {514, 4784}, {522, 48326}, {523, 47674}, {525, 3777}, {649, 48139}, {650, 48083}, {659, 4025}, {661, 47877}, {690, 48335}, {693, 58375}, {764, 23876}, {824, 21146}, {826, 4905}, {900, 47685}, {918, 1491}, {1635, 48113}, {1734, 29354}, {2254, 47700}, {2487, 48231}, {2530, 23875}, {2786, 24719}, {3004, 48024}, {3239, 30795}, {3667, 48349}, {3676, 47833}, {3700, 48184}, {3716, 48227}, {3776, 4010}, {3837, 25259}, {3906, 49278}, {3910, 23765}, {3960, 49279}, {4024, 48098}, {4088, 50335}, {4122, 24720}, {4394, 48096}, {4453, 4874}, {4467, 29362}, {4468, 47827}, {4522, 36848}, {4750, 4782}, {4777, 47704}, {4790, 4988}, {4802, 7659}, {4804, 21115}, {4806, 44435}, {4808, 48018}, {4809, 48063}, {4810, 48398}, {4813, 47999}, {4818, 4824}, {4841, 47910}, {4893, 48048}, {4897, 4977}, {4913, 28890}, {4926, 53558}, {4963, 28878}, {4979, 47931}, {6545, 48090}, {6546, 48614}, {6590, 48253}, {7265, 23815}, {9508, 47885}, {14321, 48178}, {14349, 29252}, {17069, 48055}, {17496, 29082}, {18004, 44429}, {21104, 48120}, {21116, 48127}, {21124, 29198}, {21188, 47872}, {21212, 47822}, {25380, 48185}, {28209, 47699}, {28213, 47667}, {28846, 48007}, {28871, 47992}, {28898, 48089}, {28910, 47953}, {29017, 48151}, {29078, 46403}, {29102, 48321}, {29156, 53536}, {29200, 48131}, {29280, 48278}, {29284, 48334}, {29288, 50355}, {29328, 47652}, {29358, 48075}, {29370, 47687}, {30520, 48103}, {44449, 48159}, {47123, 58372}, {47671, 48135}, {47673, 48148}, {47694, 48571}, {47696, 47755}, {47703, 48428}, {47781, 47993}, {47804, 69011}, {47810, 48112}, {47823, 69293}, {47824, 48405}, {47828, 48056}, {47886, 48078}, {47890, 48604}, {47894, 47969}, {47913, 48402}, {47946, 48404}, {48015, 48035}, {48030, 48082}, {48039, 48160}, {48040, 48162}, {48043, 48552}, {48050, 64862}, {48062, 48244}, {48080, 48422}, {48144, 68979}, {48174, 48433}, {48241, 53343}, {48245, 68794}, {48598, 50525}, {49302, 50343}, {50541, 54249}

X(69366) = midpoint of X(i) and X(j) for these {i,j}: {2254, 47930}, {4467, 49301}, {4979, 47931}, {47673, 48148}, {47677, 48108}, {47703, 48428}, {47971, 47973}, {48598, 50525}, {49302, 50343}
X(69366) = reflection of X(i) in X(j) for these {i,j}: {659, 4025}, {693, 58375}, {1491, 50348}, {4010, 3776}, {4024, 48098}, {4088, 50335}, {4122, 24720}, {4808, 48018}, {4810, 48398}, {4813, 47999}, {4824, 4818}, {7265, 23815}, {23731, 48621}, {25259, 3837}, {47671, 48135}, {47910, 4841}, {47913, 48402}, {47944, 47960}, {47946, 48404}, {48024, 3004}, {48055, 17069}, {48082, 48030}, {48083, 650}, {48094, 9508}, {48096, 4394}, {48102, 4782}, {48103, 50336}, {48117, 48056}, {48120, 21104}, {48604, 47890}, {49272, 18004}, {49273, 48405}, {49275, 4874}, {49279, 3960}, {49286, 3676}, {50328, 48015}, {50359, 50357}
X(69366) = barycentric product X(514)*X(33087)
X(69366) = barycentric quotient X(33087)/X(190)
X(69366) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3676, 49286, 47833}, {4453, 49275, 4874}, {4750, 48102, 4782}, {9508, 48094, 47885}, {17069, 48055, 48226}, {44429, 49272, 18004}, {47824, 49273, 48405}, {47828, 48117, 48056}

X(69367) = ODD<0, 1, -1, -1, 1, 1> POINT

Barycentrics    (b - c)*(a^2*b - a*b^2 + b^3 + a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2 + c^3) : :
X(69367) = X[693] - 3 X[48187], 3 X[1491] - 2 X[3004], 4 X[1491] - 3 X[47877], 8 X[3004] - 9 X[47877], X[3004] - 3 X[50333], 2 X[3837] - 3 X[47808], 2 X[23770] - 3 X[48184], 3 X[44429] - X[47692], X[47656] - 3 X[47690], X[47657] + 3 X[47689], X[47657] - 3 X[47975], X[47691] - 3 X[47808], 3 X[47877] - 8 X[50333], 3 X[48272] - X[49277], and many others

X(69367) lies on these lines: {37, 650}, {325, 523}, {512, 48272}, {513, 4088}, {514, 4774}, {522, 659}, {525, 50355}, {661, 29144}, {676, 47807}, {784, 47711}, {812, 68133}, {824, 24326}, {826, 1734}, {891, 49278}, {900, 4380}, {918, 50359}, {2254, 47700}, {2490, 26275}, {2526, 4802}, {2530, 29047}, {2533, 23877}, {2977, 28183}, {3239, 4800}, {3676, 58372}, {3700, 4951}, {3716, 48185}, {3776, 36848}, {3777, 29288}, {3800, 48123}, {3801, 17072}, {3835, 48349}, {3887, 49279}, {4010, 4522}, {4025, 48244}, {4041, 29017}, {4083, 48278}, {4142, 47835}, {4369, 48235}, {4458, 47823}, {4467, 29370}, {4490, 29142}, {4560, 29074}, {4705, 29021}, {4724, 48056}, {4729, 29284}, {4730, 23876}, {4770, 29166}, {4784, 48013}, {4809, 31286}, {4874, 47695}, {4879, 6332}, {4885, 30748}, {4905, 29354}, {4925, 50348}, {4948, 45745}, {4977, 47698}, {7192, 48254}, {7927, 14349}, {8045, 48301}, {16892, 29204}, {17494, 31087}, {18004, 48080}, {20517, 47837}, {21051, 47708}, {21124, 29146}, {21185, 47872}, {21186, 24793}, {21196, 48225}, {21212, 48224}, {21260, 47712}, {21301, 29025}, {21302, 29082}, {23815, 47716}, {24533, 64857}, {24720, 48326}, {24924, 48217}, {25380, 48227}, {25666, 48177}, {28147, 48007}, {28151, 47671}, {28161, 47827}, {28175, 47925}, {28179, 47674}, {28220, 48132}, {28602, 31209}, {29066, 50351}, {29078, 50343}, {29110, 48321}, {29164, 48012}, {29168, 47959}, {29192, 48288}, {29208, 48131}, {29260, 48066}, {29358, 48018}, {29362, 47687}, {30520, 53583}, {30572, 43051}, {30580, 48285}, {30795, 47806}, {31131, 47652}, {31287, 48211}, {46403, 48169}, {47123, 47833}, {47132, 47788}, {47653, 48157}, {47660, 64914}, {47663, 64913}, {47688, 48164}, {47694, 48208}, {47697, 48236}, {47699, 48002}, {47701, 48030}, {47702, 47810}, {47704, 48098}, {47705, 47812}, {47706, 48410}, {47707, 63812}, {47709, 47814}, {47710, 48409}, {47713, 47816}, {47714, 48407}, {47717, 48556}, {47720, 48406}, {47771, 48248}, {47799, 53573}, {47907, 48023}, {47924, 47999}, {47944, 48027}, {47981, 48039}, {47989, 48599}, {48006, 48162}, {48020, 48146}, {48024, 48047}, {48090, 53558}, {48167, 48398}, {48171, 53343}, {48234, 68794}, {48339, 49290}, {48392, 48395}, {50336, 50342}

X(69367) = midpoint of X(i) and X(j) for these {i,j}: {2254, 47700}, {47687, 48408}, {47689, 47975}, {47706, 48410}, {47710, 48409}, {47714, 48407}, {48020, 48146}, {48077, 48106}
X(69367) = reflection of X(i) in X(j) for these {i,j}: {659, 48062}, {1491, 50333}, {3801, 17072}, {4010, 4522}, {4724, 48056}, {4784, 48069}, {4879, 6332}, {16892, 50335}, {47131, 4885}, {47691, 3837}, {47694, 48405}, {47695, 4874}, {47699, 48002}, {47701, 48030}, {47704, 48098}, {47708, 21051}, {47712, 21260}, {47716, 23815}, {47720, 48406}, {47924, 47999}, {47944, 48027}, {47968, 2526}, {48024, 48047}, {48080, 18004}, {48083, 48088}, {48102, 48097}, {48120, 48396}, {48223, 28602}, {48301, 8045}, {48326, 24720}, {48339, 49290}, {48349, 3835}, {48392, 48395}, {48599, 47989}, {50340, 650}, {50342, 50336}, {50347, 2977}, {50348, 4925}, {50358, 47890}, {53558, 48090}
X(69367) = crossdifference of every pair of points on line {32, 36}
X(69367) = barycentric product X(514)*X(33165)
X(69367) = barycentric quotient X(33165)/X(190)
X(69367) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {659, 48062, 47885}, {2977, 50347, 48226}, {47131, 48200, 4885}, {47691, 47808, 3837}, {47694, 48208, 48405}, {47695, 47809, 4874}

X(69368) = ODD<0, 0, 1, 1, 1, 1> POINT

Barycentrics    (b - c)*(a*b^2 + b^3 + a*b*c + b^2*c + a*c^2 + b*c^2 + c^3) : :
X(69368) = X[4560] - 3 X[47894], 3 X[4750] - 2 X[48064], 4 X[3004] - X[49277], 4 X[3676] - X[47681], X[3762] + 2 X[48427], 3 X[47877] - 2 X[48059], X[47677] + 2 X[50453], X[4462] + 3 X[48434], 2 X[4522] - 3 X[47816], 2 X[4791] + X[48428], X[4801] - 3 X[48422], 2 X[8045] - 3 X[47795], 4 X[21212] - 3 X[47795], 2 X[14838] - 3 X[47886], and many others

X(69368) lies on these lines: {10, 47707}, {239, 514}, {522, 44444}, {523, 1734}, {525, 3004}, {661, 23875}, {667, 68979}, {690, 48123}, {693, 23685}, {784, 3801}, {824, 1577}, {826, 1491}, {905, 47682}, {918, 47959}, {2254, 29021}, {2530, 29017}, {3676, 47681}, {3762, 48427}, {3776, 4978}, {3777, 29312}, {3835, 7265}, {3900, 47727}, {3906, 47877}, {3910, 48335}, {4024, 4823}, {4040, 68780}, {4041, 29047}, {4088, 29358}, {4122, 21260}, {4129, 25259}, {4151, 47691}, {4391, 47677}, {4462, 48434}, {4467, 29013}, {4490, 29354}, {4522, 47816}, {4705, 62423}, {4730, 29208}, {4791, 48428}, {4801, 48422}, {4802, 50501}, {4808, 29204}, {4818, 23877}, {4897, 48110}, {4905, 29142}, {4983, 29200}, {6004, 50340}, {6005, 47701}, {6367, 48120}, {6590, 21188}, {7654, 23752}, {7927, 50355}, {8045, 21212}, {8714, 47708}, {14838, 30911}, {15309, 47971}, {17072, 47711}, {17166, 48241}, {20295, 29216}, {20517, 47694}, {20909, 52623}, {21120, 21130}, {21301, 29062}, {21302, 29192}, {23789, 47719}, {23876, 48131}, {23882, 47680}, {23883, 48266}, {23887, 48410}, {24719, 29106}, {24720, 47715}, {28481, 48596}, {28846, 47947}, {29082, 48288}, {29146, 50335}, {29158, 50343}, {29164, 48018}, {29168, 50359}, {29190, 46403}, {29202, 48100}, {29252, 48024}, {29280, 48030}, {29302, 47652}, {29318, 48066}, {29332, 50351}, {30520, 47965}, {41800, 68794}, {42325, 47972}, {47662, 48565}, {47673, 50457}, {47690, 50337}, {47693, 47836}, {47709, 50356}, {47790, 57068}, {47793, 49273}, {47794, 69293}, {47837, 48405}, {47840, 49288}, {47841, 49290}, {47916, 47935}, {47918, 47930}, {47924, 50509}, {47988, 48595}, {47995, 48085}, {47997, 48082}, {47998, 48081}, {48003, 48094}, {48004, 48078}, {48007, 48086}, {48076, 48612}, {48099, 49276}, {48103, 50504}, {48111, 50347}, {48185, 65449}, {48224, 48301}, {48227, 52601}, {48270, 48551}, {48275, 68836}, {48404, 50449}, {49275, 59672}, {55282, 60350}

X(69368) = midpoint of X(i) and X(j) for these {i,j}: {4391, 47677}, {4498, 47923}, {16892, 21124}, {47673, 50457}, {47709, 50356}, {47916, 47935}, {47918, 47930}, {47924, 50509}
X(69368) = reflection of X(i) in X(j) for these {i,j}: {649, 21192}, {1019, 4025}, {4024, 4823}, {4040, 68780}, {4088, 48012}, {4122, 21260}, {4391, 50453}, {4905, 50348}, {4978, 3776}, {6590, 21188}, {7265, 3835}, {8045, 21212}, {14349, 3004}, {21385, 60492}, {25259, 4129}, {47682, 905}, {47690, 50337}, {47694, 20517}, {47707, 10}, {47711, 17072}, {47715, 24720}, {47719, 23789}, {47959, 48402}, {48076, 48612}, {48078, 48004}, {48081, 47998}, {48082, 47997}, {48085, 47995}, {48086, 48007}, {48094, 48003}, {48101, 48011}, {48103, 50504}, {48110, 4897}, {48111, 50347}, {48272, 1491}, {48278, 48066}, {48300, 14838}, {48409, 4818}, {48595, 47988}, {49275, 59672}, {49276, 48099}, {49277, 14349}, {49278, 2530}, {49300, 3801}, {50449, 48404}, {57068, 59714}
X(69368) = X(i)-isoconjugate of X(j) for these (i,j): {32, 839}, {37, 59112}, {101, 54336}, {692, 60082}, {1501, 57979}, {32739, 66946}
X(69368) = X(i)-Dao conjugate of X(j) for these (i,j): {1015, 54336}, {1086, 60082}, {6376, 839}, {40589, 59112}, {40619, 66946}
X(69368) = crosssum of X(37) and X(46380)
X(69368) = crossdifference of every pair of points on line {42, 560}
X(69368) = barycentric product X(i)*X(j) for these {i,j}: {514, 32782}, {561, 838}, {1019, 56564}, {1111, 65314}, {3261, 4261}, {4025, 5142}, {7199, 56541}
X(69368) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 59112}, {75, 839}, {513, 54336}, {514, 60082}, {561, 57979}, {693, 66946}, {838, 31}, {4261, 101}, {5142, 1897}, {32782, 190}, {56541, 1018}, {56564, 4033}, {65314, 765}
X(69368) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8045, 21212, 47795}, {47886, 48300, 14838}, {57068, 59714, 47790}

X(69369) = ODD<0, 0, 1, 1, 1, 0> POINT

Barycentrics    (b - c)*(a*b^2 + b^3 + a*b*c + a*c^2 + c^3) : :
X(69369) = X[3669] - 3 X[47754], X[3776] + 2 X[50453], 2 X[14838] - 3 X[47882], 2 X[31286] - 3 X[41800], X[48095] - 3 X[48559], X[663] - 3 X[47797], X[48136] - 3 X[48192], X[3762] + 2 X[48426], X[4088] - 3 X[47814], X[4367] - 3 X[48227], 3 X[4453] - X[48144], X[4462] + 3 X[48422], X[4560] - 3 X[47886], 2 X[4791] + X[48427], X[4801] - 3 X[6545], and many others

X(69369) lies on these lines: {2, 48300}, {10, 29047}, {241, 514}, {513, 4142}, {522, 48403}, {523, 17072}, {525, 3835}, {663, 47797}, {693, 21124}, {784, 4818}, {824, 1577}, {826, 4522}, {830, 20517}, {918, 48044}, {1491, 3801}, {1734, 47712}, {2254, 47708}, {2530, 3810}, {2785, 48136}, {3762, 48426}, {3803, 13246}, {3837, 29017}, {4025, 6002}, {4041, 47691}, {4049, 60084}, {4063, 28882}, {4088, 47814}, {4129, 23875}, {4147, 29288}, {4367, 48227}, {4391, 16892}, {4444, 60197}, {4453, 48144}, {4458, 8678}, {4462, 48422}, {4490, 48326}, {4498, 47652}, {4500, 4823}, {4560, 47886}, {4707, 14349}, {4791, 48427}, {4801, 6545}, {4806, 29200}, {4874, 68979}, {4885, 8045}, {4927, 48280}, {4978, 48415}, {4992, 29284}, {6005, 62435}, {6332, 47757}, {7950, 53571}, {9508, 29025}, {17069, 29162}, {17166, 47887}, {17496, 28490}, {18004, 29280}, {20295, 28493}, {20317, 30520}, {21051, 62423}, {21052, 47707}, {21118, 48410}, {21145, 47877}, {21192, 29013}, {21196, 23882}, {21302, 48203}, {21789, 48388}, {23815, 29312}, {23887, 48066}, {24720, 29142}, {25380, 29116}, {28468, 44435}, {28481, 48042}, {28504, 47729}, {28579, 48338}, {28851, 47959}, {28855, 47955}, {28859, 68880}, {28886, 47947}, {29021, 50337}, {29051, 68780}, {29098, 50504}, {29102, 50507}, {29118, 50336}, {29198, 58375}, {29224, 65449}, {30574, 48174}, {30835, 57066}, {44429, 48278}, {45746, 50457}, {47676, 47918}, {47682, 47795}, {47719, 47812}, {47755, 48149}, {47793, 48094}, {47798, 48150}, {47799, 48299}, {47815, 48102}, {47816, 48272}, {47835, 48103}, {47836, 48106}, {47838, 49276}, {47839, 49279}, {47888, 50351}, {47929, 49301}, {47935, 49298}, {47975, 55282}, {48093, 59629}, {48101, 48565}, {48121, 48550}, {48122, 48159}, {48123, 48552}, {48128, 48558}, {48161, 48367}, {48177, 48336}, {48211, 48329}, {48212, 48330}, {48349, 50355}, {48398, 60492}, {48400, 50348}, {48409, 49300}, {48543, 48597}, {48556, 49278}

X(69369) = midpoint of X(i) and X(j) for these {i,j}: {693, 21124}, {1491, 3801}, {1734, 47712}, {2254, 47708}, {3004, 7178}, {4041, 47691}, {4391, 16892}, {4490, 48326}, {4498, 47652}, {4707, 14349}, {21118, 48410}, {21145, 47877}, {30574, 48174}, {45746, 50457}, {47676, 47918}, {47921, 49299}, {47929, 49301}, {47935, 49298}, {47975, 55282}, {48349, 50355}, {48398, 60492}, {48400, 50348}, {48409, 49300}
X(69369) = reflection of X(i) in X(j) for these {i,j}: {905, 21212}, {3803, 13246}, {4369, 21188}, {4500, 4823}, {4522, 21260}, {4978, 48415}, {8045, 4885}, {48270, 4129}
X(69369) = complement of X(48300)
X(69369) = X(i)-complementary conjugate of X(j) for these (i,j): {833, 141}, {977, 116}, {56342, 11}, {57976, 626}
X(69369) = X(34284)-Ceva conjugate of X(65116)
X(69369) = X(i)-isoconjugate of X(j) for these (i,j): {9, 59015}, {32, 65373}, {101, 987}, {692, 56046}, {1415, 56202}, {32739, 58021}
X(69369) = X(i)-Dao conjugate of X(j) for these (i,j): {478, 59015}, {1015, 987}, {1086, 56046}, {1146, 56202}, {2277, 65190}, {6376, 65373}, {40619, 58021}
X(69369) = crossdifference of every pair of points on line {55, 560}
X(69369) = barycentric product X(i)*X(j) for these {i,j}: {514, 27184}, {693, 986}, {2277, 3261}
X(69369) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 59015}, {75, 65373}, {513, 987}, {514, 56046}, {522, 56202}, {693, 58021}, {986, 100}, {2277, 101}, {27184, 190}

X(69370) = ODD<0, 0, 1, 0, 1, 0> POINT

Barycentrics    (b - c)*(a*b^2 + b^3 + a*c^2 + c^3) : :
X(69370) = X[905] - 3 X[47754], 2 X[3776] + X[50453], 3 X[41800] - X[47890], X[667] - 3 X[48227], X[1019] - 3 X[4453], X[3762] + 5 X[48425], X[4040] - 3 X[47797], X[4088] - 3 X[47816], X[4391] + 3 X[48422], X[4462] + 7 X[48432], 3 X[4728] - X[7265], X[4791] + 2 X[48426], X[4801] - 5 X[48421], X[4978] - 3 X[6545], 3 X[6545] + X[21124], and many others

X(69370) lies on these lines: {10, 29288}, {241, 514}, {512, 62435}, {513, 20517}, {523, 50337}, {667, 48227}, {690, 4992}, {693, 23685}, {812, 21192}, {824, 4823}, {826, 3837}, {830, 4458}, {918, 4129}, {1019, 4453}, {1125, 48299}, {1577, 16892}, {1734, 47691}, {2254, 47712}, {2530, 3801}, {2785, 48348}, {3261, 44173}, {3700, 59714}, {3762, 48425}, {3835, 23875}, {4025, 29013}, {4040, 47797}, {4041, 47716}, {4063, 47652}, {4088, 47816}, {4106, 29216}, {4151, 23770}, {4391, 48422}, {4462, 48432}, {4522, 29358}, {4560, 47680}, {4705, 48326}, {4707, 48131}, {4728, 7265}, {4791, 48426}, {4801, 48421}, {4806, 29252}, {4905, 47708}, {4977, 21181}, {4978, 6545}, {4983, 48552}, {6332, 29220}, {6372, 58375}, {8045, 21204}, {8714, 48403}, {9508, 29098}, {14349, 44435}, {15309, 69292}, {17072, 29047}, {19947, 29154}, {20908, 52623}, {21051, 29354}, {21115, 47918}, {21185, 48015}, {21260, 62423}, {21301, 48241}, {21302, 47727}, {22160, 48388}, {23789, 29142}, {23815, 29017}, {23877, 48066}, {23888, 48334}, {24720, 29021}, {28851, 47997}, {28855, 48612}, {28882, 48011}, {28886, 48600}, {29158, 50336}, {29186, 68780}, {29190, 48089}, {29302, 48398}, {29304, 48136}, {29312, 48406}, {31251, 48185}, {31288, 48215}, {44429, 48272}, {47651, 48565}, {47672, 47679}, {47673, 47678}, {47676, 47959}, {47682, 47796}, {47688, 47836}, {47704, 48407}, {47715, 47812}, {47755, 48110}, {47793, 49302}, {47794, 48094}, {47795, 48300}, {47798, 48111}, {47817, 48102}, {47819, 49278}, {47837, 48103}, {47840, 49276}, {47841, 49279}, {47893, 50351}, {47970, 49301}, {47976, 49298}, {48056, 65449}, {48082, 48551}, {48083, 48553}, {48085, 48550}, {48086, 48159}, {48091, 48558}, {48096, 48561}, {48099, 48192}, {48101, 48566}, {48106, 48573}, {48177, 48351}, {48212, 48331}, {48278, 48556}, {48400, 68896}, {48409, 55282}, {48410, 49300}, {48543, 48595}, {50512, 69011}, {52601, 68979}

X(69370) = midpoint of X(i) and X(j) for these {i,j}: {1577, 16892}, {1734, 47691}, {2254, 47712}, {2530, 3801}, {4041, 47716}, {4063, 47652}, {4560, 47680}, {4705, 48326}, {4707, 48131}, {4905, 47708}, {4978, 21124}, {21104, 48402}, {21185, 48015}, {21302, 47727}, {47672, 47679}, {47673, 47678}, {47676, 47959}, {47704, 48407}, {47965, 49299}, {47970, 49301}, {47976, 49298}, {48403, 50348}, {48409, 55282}, {48410, 49300}
X(69370) = reflection of X(i) in X(j) for these {i,j}: {3700, 59714}, {14838, 21212}, {48056, 65449}, {48299, 1125}, {50512, 69011}
X(69370) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 38958}, {43348, 141}
X(69370) = X(i)-Ceva conjugate of X(j) for these (i,j): {349, 1111}, {39747, 1086}
X(69370) = X(i)-isoconjugate of X(j) for these (i,j): {692, 40394}, {1018, 3453}
X(69370) = X(i)-Dao conjugate of X(j) for these (i,j): {1086, 40394}, {3454, 101}, {18191, 284}, {36901, 59138}
X(69370) = crosspoint of X(3261) and X(7192)
X(69370) = crosssum of X(4557) and X(32739)
X(69370) = barycentric product X(i)*X(j) for these {i,j}: {86, 21121}, {514, 17184}, {693, 3670}, {850, 52564}, {1019, 20896}, {1111, 3909}, {1577, 18601}, {3454, 7192}, {3676, 69280}, {4016, 7199}, {11573, 46107}, {16727, 61167}, {20966, 52619}, {35519, 68372}
X(69370) = barycentric quotient X(i)/X(j) for these {i,j}: {514, 40394}, {850, 59138}, {3454, 3952}, {3670, 100}, {3733, 3453}, {3909, 765}, {4016, 1018}, {11573, 1331}, {17184, 190}, {18601, 662}, {20654, 4103}, {20896, 4033}, {20966, 4557}, {21121, 10}, {22073, 4574}, {23197, 32656}, {52564, 110}, {68372, 109}, {69280, 3699}
X(69370) = {X(6545),X(21124)}-harmonic conjugate of X(4978)

X(69371) = ODD<0, 0, 1, 0, 1, -1> POINT

Barycentrics    (b-c)*((b^2+c^2)*a-b^2*c-b*c^2+c^3+b^3) : :
X(69371) = 3 X[6544] - 4 X[48196], 3 X[14475] - 2 X[47795], 2 X[10] + X[47716], X[649] - 4 X[21188], 4 X[676] - X[48150], 2 X[693] + X[21124], 2 X[1491] + X[55282], 2 X[1577] + X[16892], 2 X[1734] + X[53558], X[2254] + 2 X[48403], 2 X[2530] + X[21118], 2 X[3004] + X[50457], 4 X[3676] - X[48144], 2 X[3776] + X[4391], 2 X[3777] + X[21132], and many others

X(69371) lies on these lines: {2, 514}, {10, 47716}, {513, 11125}, {525, 4728}, {649, 21188}, {663, 28082}, {676, 48150}, {693, 21124}, {814, 48227}, {834, 2457}, {976, 4449}, {1491, 55282}, {1577, 16892}, {1635, 41800}, {1638, 29162}, {1734, 53558}, {2254, 48403}, {2530, 21118}, {3004, 50457}, {3145, 44408}, {3676, 28094}, {3776, 4391}, {3777, 21132}, {3801, 3837}, {3810, 47819}, {3910, 4927}, {4024, 4823}, {4041, 23770}, {4083, 30574}, {4088, 21260}, {4120, 23875}, {4129, 48082}, {4142, 46403}, {4147, 47720}, {4170, 62435}, {4453, 6002}, {4458, 21301}, {4462, 48421}, {4498, 14837}, {4560, 21212}, {4705, 47704}, {4750, 29013}, {4791, 48425}, {4801, 48415}, {4808, 53571}, {4885, 48300}, {4928, 57066}, {4978, 48414}, {4984, 29270}, {5293, 48282}, {7178, 28116}, {7265, 59714}, {8045, 26985}, {8678, 47887}, {10015, 48334}, {14349, 23755}, {14419, 29336}, {14431, 29354}, {14432, 29082}, {14838, 47680}, {17072, 36568}, {20317, 49299}, {21051, 48326}, {21052, 29288}, {21104, 47918}, {21174, 28041}, {21185, 36574}, {21202, 28091}, {21302, 44314}, {23729, 47935}, {23877, 44429}, {23879, 48416}, {23880, 47754}, {23882, 47886}, {23887, 48556}, {24720, 47708}, {28093, 28115}, {28882, 48565}, {29017, 48184}, {29025, 47823}, {29029, 48569}, {29037, 48241}, {29051, 47797}, {29074, 48224}, {29098, 47837}, {29102, 47839}, {29118, 47824}, {29142, 47812}, {29158, 48573}, {29174, 48235}, {29244, 48215}, {29246, 48177}, {29274, 48212}, {29332, 48198}, {34958, 48322}, {36499, 47725}, {36505, 47728}, {36565, 48287}, {47671, 47679}, {47672, 48402}, {47701, 50352}, {47712, 50337}, {47833, 68979}, {47911, 49296}, {48066, 49300}, {48151, 48400}, {48264, 50348}, {48265, 58375}, {49295, 50509}

X(69371) = reflection of X(i) in X(j) for these {i,j}: {1635, 41800}, {6546, 47794}, {14432, 47841}, {47796, 21204}, {57066, 4928}
X(69371) = isotomic conjugate of the isogonal conjugate of X(23751)
X(69371) = X(i)-Ceva conjugate of X(j) for these (i,j): {20565, 3120}, {57905, 1111}
X(69371) = X(i)-isoconjugate of X(j) for these (i,j): {101, 55991}, {644, 3450}, {692, 2985}
X(69371) = X(i)-Dao conjugate of X(j) for these (i,j): {1015, 55991}, {1086, 2985}, {1329, 101}, {17053, 3699}, {20982, 35}, {61051, 2269}
X(69371) = crosspoint of X(3261) and X(3676)
X(69371) = crosssum of X(3939) and X(32739)
X(69371) = crossdifference of every pair of points on line {902, 52405}
X(69371) = barycentric product X(i)*X(j) for these {i,j}: {7, 21119}, {76, 23751}, {514, 3782}, {693, 24443}, {1329, 3676}, {1577, 16700}, {3261, 17053}, {3669, 20237}, {4077, 18178}, {7178, 17182}, {17096, 21030}, {17114, 35519}, {17452, 24002}, {21936, 52619}, {23154, 46107}, {23638, 52621}
X(69371) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 55991}, {514, 2985}, {1329, 3699}, {3782, 190}, {16700, 662}, {17053, 101}, {17114, 109}, {17182, 645}, {17452, 644}, {18178, 643}, {20237, 646}, {21030, 30730}, {21119, 8}, {21936, 4557}, {22071, 4587}, {23154, 1331}, {23196, 32656}, {23638, 3939}, {23751, 6}, {24443, 100}, {43924, 3450}
X(69371) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3801, 3837, 48278}, {14837, 48398, 4498}

X(69372) = ODD<0, 0, 1, -1, 1, 1> POINT

Barycentrics    (b - c)*(a*b^2 + b^3 - a*b*c + b^2*c + a*c^2 + b*c^2 + c^3) : :
X(69372) = 3 X[4750] - 2 X[48011], 3 X[16892] - X[21124], X[3762] - 4 X[48426], 2 X[4129] - 3 X[44435], X[4391] - 3 X[48422], X[4462] - 5 X[48433], 5 X[48433] - 2 X[50453], 2 X[4522] - 3 X[48556], 2 X[4791] - 5 X[48425], 2 X[4823] - 3 X[6545], 4 X[7657] - 3 X[10015], 2 X[20517] - 3 X[48241], 3 X[21115] - X[50457], 4 X[21212] - 3 X[47794], 2 X[47930] + X[49277], 2 X[31010] - 3 X[51317], X[47662] - 3 X[48570], 3 X[47795] - 2 X[69293], 3 X[47796] - X[49273], 3 X[47797] - 2 X[59672], 3 X[47877] - 2 X[48005], 3 X[47886] - 2 X[48003], 3 X[47888] - 2 X[48056], X[48139] - 3 X[58140], 2 X[48405] - 3 X[48569]

X(69372) lies on these lines: {239, 514}, {522, 28591}, {523, 4905}, {525, 48335}, {764, 29017}, {784, 48326}, {824, 4978}, {826, 3777}, {830, 47973}, {905, 30520}, {918, 14349}, {1491, 29354}, {1577, 3776}, {1734, 29288}, {2254, 29047}, {2530, 48272}, {3004, 47959}, {3309, 47727}, {3669, 47682}, {3762, 48426}, {3800, 50357}, {3904, 29220}, {3960, 48300}, {4088, 48066}, {4122, 23815}, {4129, 44435}, {4151, 47720}, {4378, 68979}, {4391, 48422}, {4401, 48102}, {4462, 48433}, {4467, 29302}, {4522, 48556}, {4791, 48425}, {4801, 23879}, {4808, 50335}, {4818, 48407}, {4823, 6545}, {4897, 47976}, {6050, 48096}, {7208, 23776}, {7265, 22043}, {7657, 10015}, {7927, 50359}, {8714, 47691}, {14838, 48094}, {15309, 47958}, {20517, 48241}, {21115, 50457}, {21212, 47794}, {21301, 29212}, {23765, 29312}, {23789, 47690}, {23828, 47656}, {23875, 47930}, {23876, 48334}, {23880, 47680}, {23882, 49299}, {24719, 29090}, {24720, 47711}, {28195, 50515}, {28846, 48085}, {28851, 50449}, {29013, 47652}, {29021, 48151}, {29062, 46403}, {29114, 53536}, {29158, 47688}, {29186, 49301}, {29196, 47687}, {29252, 48123}, {29260, 48075}, {29280, 48137}, {29304, 48298}, {29358, 48278}, {30719, 57243}, {31010, 51317}, {35355, 43745}, {47662, 48570}, {47678, 48399}, {47707, 50337}, {47708, 68896}, {47795, 69293}, {47796, 49273}, {47797, 59672}, {47877, 48005}, {47886, 48003}, {47888, 48056}, {47916, 48149}, {47931, 50523}, {47942, 47998}, {47947, 47995}, {47948, 48007}, {47970, 68780}, {47977, 50347}, {47988, 48584}, {48051, 48076}, {48054, 48082}, {48058, 48078}, {48083, 50507}, {48136, 49276}, {48139, 58140}, {48405, 48569}, {48598, 50526}, {49300, 63812}, {50352, 58375}

X(69372) = midpoint of X(i) and X(j) for these {i,j}: {4560, 49302}, {4801, 47677}, {47916, 48149}, {47923, 48144}, {47930, 48131}, {47931, 50523}, {48598, 50526}
X(69372) = reflection of X(i) in X(j) for these {i,j}: {1577, 3776}, {1734, 50348}, {4063, 4025}, {4088, 48066}, {4122, 23815}, {4462, 50453}, {4498, 21192}, {4808, 50335}, {4960, 49296}, {47678, 48399}, {47682, 3669}, {47690, 23789}, {47707, 50337}, {47711, 24720}, {47942, 47998}, {47947, 47995}, {47948, 48007}, {47959, 3004}, {47970, 68780}, {47976, 4897}, {47977, 50347}, {48076, 48051}, {48078, 48058}, {48082, 48054}, {48083, 50507}, {48094, 14838}, {48096, 6050}, {48101, 48064}, {48102, 4401}, {48272, 2530}, {48300, 3960}, {48407, 4818}, {48584, 47988}, {49276, 48136}, {49277, 48131}, {49278, 3777}, {50352, 58375}
X(69372) = X(692)-isoconjugate of X(62923)
X(69372) = X(1086)-Dao conjugate of X(62923)
X(69372) = barycentric product X(i)*X(j) for these {i,j}: {514, 33172}, {3261, 5069}
X(69372) = barycentric quotient X(i)/X(j) for these {i,j}: {514, 62923}, {5069, 101}, {33172, 190}

X(69373) = ODD<0, 0, 1, -1, 0, 1> POINT

Barycentrics    (b - c)*(a*b^2 - a*b*c + b^2*c + a*c^2 + b*c^2) : :

X(69373) lies on these lines: {10, 50335}, {30, 511}, {39, 650}, {76, 693}, {194, 17494}, {659, 48321}, {665, 47766}, {667, 17496}, {668, 36238}, {905, 31288}, {1022, 48189}, {1491, 3762}, {1577, 3777}, {1960, 48063}, {2530, 4391}, {2533, 4905}, {3097, 48225}, {3250, 4120}, {3669, 48220}, {3766, 44435}, {3837, 4791}, {3904, 49275}, {3934, 4885}, {3960, 4874}, {4010, 48335}, {4024, 22036}, {4049, 60090}, {4122, 49278}, {4129, 48100}, {4367, 48251}, {4378, 21222}, {4411, 21443}, {4449, 48305}, {4462, 4705}, {4490, 48409}, {4560, 48240}, {4724, 48288}, {4730, 50356}, {4761, 50359}, {4770, 48017}, {4774, 58374}, {4775, 48298}, {4794, 48289}, {4801, 48393}, {4823, 48406}, {4922, 48324}, {4978, 23765}, {5145, 55969}, {5283, 69103}, {6161, 7976}, {6683, 31287}, {7757, 31150}, {7786, 31209}, {9466, 45320}, {9902, 47724}, {10015, 50348}, {12263, 48295}, {12782, 50341}, {14349, 48265}, {14419, 44550}, {14433, 47886}, {14838, 48214}, {15280, 42455}, {16892, 21132}, {17760, 48271}, {20081, 26824}, {20317, 65449}, {21051, 48066}, {21343, 48339}, {21894, 47881}, {23738, 50457}, {23746, 47703}, {26985, 31276}, {30709, 31149}, {30725, 48290}, {31239, 31250}, {33890, 47677}, {39547, 48342}, {44562, 44567}, {45324, 48198}, {45338, 47757}, {45658, 45661}, {45660, 59895}, {45664, 47802}, {45671, 48226}, {47793, 47888}, {47794, 47893}, {47795, 47872}, {47796, 47875}, {47874, 52745}, {47913, 50449}, {47965, 48210}, {48012, 48401}, {48094, 50351}, {48124, 60478}, {48131, 48267}, {48136, 59590}, {48151, 50352}, {48211, 49563}, {48264, 48273}, {48282, 48301}, {48326, 49300}, {49286, 49290}, {49302, 49303}

X(69373) = isotomic conjugate of the isogonal conjugate of X(69102)
X(69373) = crossdifference of every pair of points on line {6, 23375}
X(69373) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1577, 3777, 23815}, {2530, 4391, 21260}, {2530, 14431, 44429}, {3904, 49275, 49279}, {4122, 53533, 49278}, {4391, 44429, 14431}, {4462, 48410, 4705}, {14431, 44429, 21260}, {21222, 47694, 4378}, {23765, 48392, 4978}, {30709, 48164, 31149}, {44550, 47804, 14419}, {48063, 48325, 1960}, {48264, 48334, 48273}, {48298, 53343, 4775}

X(69374) = ODD<0, 0, 0, 1, 1, 0> POINT

Barycentrics    (b - c)*(b^3 + a*b*c + c^3) : :
X(69374) = 3 X[667] - 4 X[13246], 3 X[3801] - X[48326], 3 X[4142] - 2 X[13246], 3 X[48267] - 2 X[50326], 3 X[48400] - X[50326], 2 X[1960] - 3 X[47798], X[47728] - 3 X[47798], X[3904] - 3 X[47797], 2 X[3960] - 3 X[48227], 2 X[4522] - 3 X[14431], 3 X[4800] - 2 X[49288], 2 X[6332] - 3 X[47839], 2 X[8045] - 3 X[47875], 3 X[14430] - X[47700], 4 X[14837] - 3 X[47837], 3 X[21145] - X[21146], 4 X[21188] - 3 X[48569], X[21222] - 3 X[48241], X[23764] - 5 X[48425], X[47684] - 3 X[47804], X[47729] - 3 X[48223], 3 X[47808] - 4 X[53571], 3 X[47821] - X[49274], 3 X[47832] - 2 X[49290], 3 X[48203] - X[48298]

X(69374) lies on these lines: {512, 47708}, {513, 4707}, {514, 659}, {523, 10015}, {525, 48267}, {649, 29029}, {650, 16583}, {663, 29094}, {676, 48290}, {690, 48080}, {693, 18015}, {764, 3776}, {784, 21118}, {826, 4391}, {891, 47691}, {1019, 29120}, {1491, 23887}, {1577, 29017}, {1960, 47728}, {2530, 3810}, {2533, 29021}, {2785, 4775}, {2826, 50348}, {3696, 4777}, {3716, 49279}, {3762, 62423}, {3837, 49278}, {3904, 47797}, {3906, 25259}, {3910, 48273}, {3960, 48227}, {4010, 23876}, {4040, 29082}, {4063, 29025}, {4083, 47712}, {4122, 4791}, {4147, 4808}, {4170, 29284}, {4462, 29354}, {4474, 29110}, {4498, 29098}, {4522, 14431}, {4705, 23877}, {4724, 29102}, {4761, 29144}, {4774, 29192}, {4782, 29122}, {4784, 29132}, {4800, 49288}, {4806, 49277}, {4825, 28161}, {4834, 29118}, {4874, 29172}, {6332, 47839}, {6550, 48422}, {7178, 29142}, {7265, 29202}, {7927, 47709}, {7950, 47707}, {8045, 47875}, {9001, 24476}, {14077, 47131}, {14430, 47700}, {14837, 47837}, {16230, 17924}, {16892, 21132}, {17494, 49303}, {21051, 48272}, {21120, 29288}, {21125, 29224}, {21134, 23735}, {21145, 21146}, {21185, 48305}, {21188, 48569}, {21198, 48188}, {21222, 48241}, {21260, 48278}, {21385, 47725}, {23764, 48425}, {23775, 48277}, {23875, 48265}, {23879, 48392}, {23884, 48177}, {26546, 47126}, {28569, 48616}, {29066, 50340}, {29128, 48106}, {29138, 50512}, {29140, 48011}, {29146, 47711}, {29148, 50342}, {29154, 47203}, {29160, 48103}, {29166, 47690}, {29188, 47972}, {29208, 47713}, {29220, 59672}, {29226, 47716}, {29240, 50347}, {29304, 48336}, {29350, 48349}, {29362, 47680}, {30580, 48211}, {47123, 48291}, {47684, 47804}, {47726, 48405}, {47729, 48223}, {47808, 53571}, {47821, 49274}, {47832, 49290}, {47886, 50025}, {48081, 59629}, {48203, 48298}, {50335, 66995}, {50359, 62435}

X(69374) = midpoint of X(i) and X(j) for these {i,j}: {16892, 21132}, {17494, 49303}, {21118, 21124}, {21385, 47725}
X(69374) = reflection of X(i) in X(j) for these {i,j}: {667, 4142}, {764, 3776}, {1491, 50453}, {4122, 4791}, {4367, 20517}, {4378, 4458}, {4808, 4147}, {30580, 48211}, {47682, 4874}, {47726, 48405}, {47728, 1960}, {48188, 21198}, {48267, 48400}, {48272, 21051}, {48273, 48403}, {48278, 21260}, {48288, 68780}, {48290, 676}, {48291, 47123}, {48305, 21185}, {49277, 4806}, {49278, 3837}, {49279, 3716}, {50351, 650}, {50352, 7178}, {50359, 62435}, {66995, 50335}
X(69374) = crossdifference of every pair of points on line {2276, 2278}
X(69374) = barycentric product X(i)*X(j) for these {i,j}: {514, 25760}, {693, 3735}, {3261, 3764}, {3676, 4165}, {30647, 40495}
X(69374) = barycentric quotient X(i)/X(j) for these {i,j}: {3735, 100}, {3764, 101}, {4165, 3699}, {25760, 190}, {30647, 692}
X(69374) = {X(47728),X(47798)}-harmonic conjugate of X(1960)

X(69375) = ODD<0, 0, 0, 1, -1, 0> POINT

Barycentrics    (b - c)*(b^3 - a*b*c + c^3) : :
X(69375) = X[4367] - 3 X[58372], 2 X[4401] - 3 X[4809], 2 X[2977] - 3 X[41800], 4 X[3676] - 3 X[48569], 2 X[4468] - 3 X[48553], X[4560] - 3 X[48241], 2 X[5592] - 3 X[58155], 3 X[6545] - 2 X[23815], 3 X[6545] - X[48278], 2 X[14838] - 3 X[48227], 3 X[21115] - X[48151], 4 X[21188] - 3 X[47837], 3 X[47837] - 2 X[48062], 4 X[21212] - 3 X[47888], 3 X[47887] - X[48300], 3 X[47887] - 2 X[52601], 3 X[44435] - 2 X[48059], 3 X[47794] - 2 X[48056], 3 X[47797] - 2 X[50507], 3 X[47819] - 5 X[48421], 3 X[47875] - 2 X[69293], 2 X[48054] - 3 X[48552], 2 X[48058] - 3 X[48177], X[48410] - 3 X[48422]

X(69375) lies on these lines: {1, 29082}, {512, 47691}, {513, 47712}, {514, 659}, {523, 1734}, {525, 23770}, {649, 29098}, {663, 29102}, {693, 826}, {764, 3810}, {784, 16892}, {814, 47680}, {824, 48393}, {891, 47720}, {905, 50351}, {918, 48267}, {1019, 29025}, {1577, 62423}, {2530, 3776}, {2533, 29047}, {2785, 48333}, {2977, 41800}, {3309, 47131}, {3676, 48569}, {3777, 23887}, {3837, 48272}, {4010, 23875}, {4041, 47705}, {4083, 4707}, {4088, 21260}, {4122, 4823}, {4170, 29200}, {4382, 29106}, {4391, 29354}, {4449, 29094}, {4468, 48553}, {4490, 50453}, {4560, 48241}, {4761, 29208}, {4784, 29158}, {4801, 29312}, {4802, 21121}, {4808, 17072}, {4810, 29216}, {4879, 29304}, {4905, 58375}, {4977, 47977}, {4978, 29017}, {4992, 49277}, {5592, 58155}, {6004, 47695}, {6005, 48349}, {6372, 47676}, {6545, 23815}, {7178, 29288}, {7265, 29280}, {7927, 47692}, {7950, 47690}, {14838, 48227}, {17496, 49303}, {21104, 29142}, {21115, 48151}, {21124, 47704}, {21125, 47958}, {21146, 29021}, {21181, 47885}, {21188, 47837}, {21212, 47888}, {23747, 47123}, {23876, 48279}, {23879, 48120}, {28851, 47949}, {29013, 50342}, {29029, 48144}, {29074, 47724}, {29118, 69292}, {29120, 48320}, {29144, 47713}, {29146, 47715}, {29150, 47971}, {29166, 47719}, {29168, 47709}, {29182, 47722}, {29186, 50340}, {29204, 47711}, {29220, 48295}, {29224, 47887}, {29250, 47723}, {29252, 48080}, {29272, 47728}, {29332, 47682}, {29366, 47727}, {34958, 48299}, {40495, 66286}, {44435, 48059}, {47698, 48005}, {47794, 48056}, {47797, 50507}, {47819, 48421}, {47875, 69293}, {47930, 48264}, {48054, 48552}, {48058, 48177}, {48083, 59672}, {48406, 49278}, {48408, 50504}, {48410, 48422}, {49300, 63812}, {50355, 62435}

X(69375) = midpoint of X(i) and X(j) for these {i,j}: {1019, 47725}, {3801, 48326}, {4041, 47705}, {4707, 47716}, {4761, 47717}, {16892, 55282}, {17496, 49303}, {21124, 47704}, {47676, 47708}, {47709, 48108}, {47930, 48264}
X(69375) = reflection of X(i) in X(j) for these {i,j}: {659, 20517}, {667, 4458}, {2530, 3776}, {4088, 21260}, {4122, 4823}, {4490, 50453}, {4808, 17072}, {4905, 58375}, {7265, 48090}, {47698, 48005}, {47715, 48098}, {47728, 48328}, {47885, 21181}, {48062, 21188}, {48083, 59672}, {48267, 48403}, {48272, 3837}, {48273, 23770}, {48278, 23815}, {48299, 34958}, {48300, 52601}, {48305, 47123}, {48408, 50504}, {49277, 4992}, {49278, 48406}, {50351, 905}, {50355, 62435}
X(69375) = crossdifference of every pair of points on line {584, 2276}
X(69375) = barycentric product X(514)*X(25957)
X(69375) = barycentric quotient X(25957)/X(190)
X(69375) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6545, 48278, 23815}, {21188, 48062, 47837}, {47887, 48300, 52601}

X(69376) = ODD<0, 0, 0, 0, 1, -1> POINT

Barycentrics    (b - c)^3*(b + c) : :
X(69376) = 3 X[2] + X[49303], X[10] - 3 X[4049], X[4010] + 3 X[21145], X[4707] - 3 X[21145], X[764] - 3 X[6545], 3 X[764] - X[23764], 3 X[6545] + X[21132], 9 X[6545] - X[23764], 3 X[21132] + X[23764], 3 X[1577] - X[4122], 3 X[3801] + X[4122], 3 X[1022] - X[24097], 5 X[3616] - 3 X[30580], 7 X[3624] - 3 X[62634], 4 X[3634] - 3 X[28602], X[4088] - 3 X[14431], and many others

X(69376) lies on these lines: {2, 49303}, {5, 68333}, {10, 523}, {65, 512}, {76, 66286}, {513, 942}, {514, 1125}, {520, 3657}, {522, 44314}, {649, 40955}, {659, 47680}, {661, 21808}, {667, 29336}, {676, 1960}, {690, 4010}, {693, 18015}, {764, 1647}, {784, 4818}, {814, 20517}, {826, 1089}, {850, 27801}, {876, 19950}, {891, 10015}, {900, 62435}, {1019, 3337}, {1022, 24097}, {1254, 4017}, {1426, 7250}, {1491, 49300}, {1509, 7192}, {1649, 4988}, {1842, 7649}, {2501, 52577}, {2530, 21118}, {2533, 7927}, {2787, 4458}, {3125, 63462}, {3616, 30580}, {3624, 62634}, {3634, 28602}, {3700, 3906}, {3716, 29102}, {3733, 52375}, {3762, 48326}, {3810, 23815}, {3837, 23887}, {4024, 52579}, {4036, 52576}, {4088, 14431}, {4142, 29070}, {4170, 32478}, {4367, 5563}, {4369, 17048}, {4374, 33940}, {4378, 47887}, {4391, 29354}, {4401, 29244}, {4705, 21674}, {4730, 30574}, {4761, 12073}, {4774, 47727}, {4789, 17292}, {4791, 62423}, {4800, 49276}, {4802, 21198}, {4807, 59743}, {4808, 21052}, {4823, 29017}, {4843, 7657}, {4879, 11009}, {4977, 68160}, {4983, 23755}, {6002, 46976}, {6004, 21185}, {6089, 53527}, {6366, 48296}, {6367, 21124}, {6372, 48400}, {6788, 38938}, {7199, 33944}, {7212, 55122}, {7950, 48395}, {8045, 29154}, {8672, 43220}, {8678, 20516}, {10222, 28473}, {11607, 36236}, {14421, 21105}, {14430, 47705}, {14505, 23100}, {14837, 50504}, {15475, 52382}, {16583, 55261}, {16815, 47782}, {16823, 47797}, {18004, 59737}, {19947, 21204}, {21121, 30591}, {21131, 21134}, {21181, 69011}, {21260, 23877}, {21832, 65707}, {23752, 42768}, {23775, 42759}, {23876, 48090}, {23894, 23903}, {24928, 34958}, {27712, 47126}, {28147, 32212}, {28840, 50262}, {29124, 48064}, {29160, 48405}, {29162, 50512}, {29166, 48396}, {29168, 47708}, {29224, 69293}, {29252, 48267}, {29272, 48299}, {29591, 47792}, {29596, 47788}, {29637, 47682}, {31149, 48077}, {32014, 68168}, {36848, 66995}, {43052, 48332}, {45664, 48088}, {47721, 48223}, {47722, 47798}, {47724, 50340}, {47725, 48103}, {47832, 49279}, {47875, 48300}, {48184, 49278}, {48227, 48321}, {50333, 53571}, {52567, 66287}, {53400, 53407}, {56283, 68970}, {58375, 68896}

X(69376) = midpoint of X(i) and X(j) for these {i,j}: {659, 47680}, {764, 21132}, {1491, 49300}, {1577, 3801}, {2530, 21118}, {2533, 47712}, {3762, 48326}, {4010, 4707}, {4705, 55282}, {4730, 53558}, {4761, 48349}, {4774, 47727}, {4983, 23755}, {7178, 48403}, {10015, 23770}, {21121, 30591}, {21124, 48393}, {23752, 50330}, {43052, 48332}, {47708, 50352}, {47724, 50340}, {47725, 48103}, {49303, 50351}
X(69376) = reflection of X(i) in X(j) for these {i,j}: {1960, 676}, {4807, 59743}, {18004, 59737}, {48328, 34958}, {50333, 53571}, {50504, 14837}, {68333, 5}
X(69376) = complement of X(50351)
X(69376) = isotomic conjugate of the isogonal conjugate of X(8034)
X(69376) = X(i)-Ceva conjugate of X(j) for these (i,j): {244, 7336}, {523, 3120}, {850, 16732}, {1577, 115}, {4017, 1365}, {7178, 3125}, {7192, 1086}, {17925, 1015}, {55244, 42759}, {62635, 39786}
X(69376) = X(i)-isoconjugate of X(j) for these (i,j): {41, 55194}, {58, 57731}, {59, 643}, {81, 59149}, {99, 1110}, {100, 4570}, {101, 4567}, {110, 765}, {163, 1016}, {249, 1018}, {284, 31615}, {644, 52378}, {645, 2149}, {662, 1252}, {692, 4600}, {756, 59152}, {799, 23990}, {872, 31614}, {1101, 3952}, {1262, 7259}, {1331, 5379}, {1333, 6632}, {1414, 6065}, {1576, 7035}, {2206, 57950}, {2287, 4619}, {3949, 47443}, {4033, 23357}, {4557, 24041}, {4564, 5546}, {4575, 15742}, {4601, 32739}, {4625, 6066}, {4636, 65573}, {4998, 65375}, {5009, 65363}, {5377, 54353}, {6551, 52680}, {7256, 24027}, {7258, 23979}, {9273, 68819}, {23995, 27808}, {34072, 61406}, {44717, 65201}, {56182, 59151}
X(69376) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 57731}, {37, 6632}, {115, 1016}, {136, 15742}, {244, 765}, {513, 110}, {514, 99}, {522, 7256}, {523, 3952}, {647, 52609}, {650, 645}, {661, 662}, {900, 68149}, {1015, 4567}, {1084, 1252}, {1086, 4600}, {1577, 7257}, {3005, 4557}, {3160, 55194}, {4369, 4579}, {4858, 7035}, {4988, 190}, {5521, 5379}, {6615, 643}, {6741, 4076}, {8054, 4570}, {15449, 61406}, {18314, 27808}, {27929, 17934}, {36901, 31625}, {38986, 1110}, {38996, 23990}, {40586, 59149}, {40590, 31615}, {40603, 57950}, {40608, 6065}, {40615, 4620}, {40619, 4601}, {40620, 4590}, {40622, 4998}, {40625, 6064}, {40627, 101}, {42769, 3658}, {50330, 100}, {50497, 692}, {55060, 59}, {62566, 3699}, {62567, 43290}, {64440, 7253}
X(69376) = crosspoint of X(i) and X(j) for these (i,j): {244, 4017}, {523, 3120}, {850, 16732}, {1086, 7192}, {1111, 1577}, {2973, 17925}
X(69376) = crosssum of X(i) and X(j) for these (i,j): {110, 4570}, {163, 1110}, {643, 765}, {1252, 4557}, {52609, 61406}
X(69376) = crossdifference of every pair of points on line {1252, 3285}
X(69376) = barycentric product X(i)*X(j) for these {i,j}: {7, 55195}, {10, 6545}, {11, 7178}, {27, 21134}, {42, 23100}, {65, 40166}, {76, 8034}, {79, 21141}, {86, 21131}, {115, 7192}, {125, 17925}, {226, 21132}, {244, 1577}, {313, 21143}, {321, 764}, {338, 3733}, {339, 43925}, {512, 23989}, {513, 16732}, {514, 3120}, {522, 53545}, {523, 1086}, {525, 2969}, {593, 23105}, {647, 2973}, {649, 21207}, {661, 1111}, {693, 3125}, {826, 61404}, {850, 1015}, {1019, 1109}, {1020, 1090}, {1089, 8042}, {1254, 40213}, {1358, 3700}, {1365, 4560}, {1427, 42455}, {1509, 8029}, {1565, 2501}, {1647, 4049}, {1977, 44173}, {2170, 4077}, {2401, 42759}, {2643, 7199}, {2799, 43920}, {2970, 7254}, {3121, 40495}, {3122, 3261}, {3124, 52619}, {3248, 20948}, {3267, 42067}, {3668, 42462}, {3676, 21044}, {3937, 14618}, {3942, 24006}, {3960, 66289}, {4017, 4858}, {4024, 17205}, {4036, 16726}, {4052, 23764}, {4080, 6550}, {4086, 53538}, {4089, 55238}, {4092, 17096}, {4120, 6549}, {4122, 43266}, {4391, 53540}, {4466, 7649}, {4516, 24002}, {4552, 7336}, {4566, 64445}, {4705, 16727}, {4957, 55246}, {5489, 36419}, {6063, 63462}, {6354, 56283}, {7180, 34387}, {7216, 24026}, {7250, 23978}, {8027, 27801}, {8735, 17094}, {8754, 15419}, {15320, 21133}, {17197, 66287}, {17880, 55208}, {17924, 18210}, {18014, 69009}, {20902, 57200}, {23104, 62192}, {23775, 37887}, {23994, 57129}, {27918, 35352}, {30572, 60578}, {35353, 52626}, {36197, 59941}, {39786, 66286}, {42761, 43933}, {43926, 52628}, {52023, 56284}, {52305, 66941}, {52335, 58817}, {55257, 58259}, {55261, 62429}
X(69376) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 55194}, {10, 6632}, {11, 645}, {37, 57731}, {42, 59149}, {65, 31615}, {115, 3952}, {125, 52609}, {244, 662}, {321, 57950}, {338, 27808}, {512, 1252}, {513, 4567}, {514, 4600}, {523, 1016}, {593, 59152}, {649, 4570}, {661, 765}, {669, 23990}, {693, 4601}, {764, 81}, {798, 1110}, {826, 61406}, {850, 31625}, {1015, 110}, {1019, 24041}, {1042, 4619}, {1086, 99}, {1109, 4033}, {1111, 799}, {1146, 7256}, {1357, 4565}, {1358, 4573}, {1365, 4552}, {1509, 31614}, {1565, 4563}, {1577, 7035}, {1977, 1576}, {2170, 643}, {2310, 7259}, {2501, 15742}, {2643, 1018}, {2969, 648}, {2973, 6331}, {3120, 190}, {3121, 692}, {3122, 101}, {3124, 4557}, {3125, 100}, {3248, 163}, {3249, 2206}, {3271, 5546}, {3676, 4620}, {3700, 4076}, {3709, 6065}, {3733, 249}, {3937, 4558}, {3942, 4592}, {4017, 4564}, {4049, 62536}, {4077, 67038}, {4080, 6635}, {4089, 55237}, {4092, 30730}, {4122, 65040}, {4466, 4561}, {4516, 644}, {4560, 6064}, {4858, 7257}, {4957, 55245}, {6545, 86}, {6549, 4615}, {6550, 16704}, {6591, 5379}, {7178, 4998}, {7180, 59}, {7192, 4590}, {7199, 24037}, {7216, 7045}, {7250, 1262}, {7336, 4560}, {8027, 1333}, {8029, 594}, {8034, 6}, {8042, 757}, {8661, 3285}, {8735, 36797}, {14321, 44724}, {15419, 47389}, {16592, 4579}, {16726, 52935}, {16727, 4623}, {16732, 668}, {17096, 7340}, {17205, 4610}, {17880, 55207}, {17925, 18020}, {18191, 4612}, {18210, 1332}, {20975, 4574}, {21043, 4103}, {21044, 3699}, {21131, 10}, {21132, 333}, {21133, 33297}, {21134, 306}, {21138, 62530}, {21140, 69086}, {21141, 319}, {21143, 58}, {21207, 1978}, {21725, 61164}, {21833, 40521}, {21950, 43290}, {22096, 32661}, {22260, 1500}, {23099, 7109}, {23100, 310}, {23105, 28654}, {23759, 3879}, {23764, 41629}, {23775, 33116}, {23989, 670}, {24026, 7258}, {24193, 50456}, {34387, 62534}, {35092, 68149}, {35353, 5381}, {35505, 68148}, {36197, 4578}, {39786, 3573}, {40166, 314}, {40525, 59102}, {42067, 112}, {42462, 1043}, {42752, 2427}, {42753, 64828}, {42759, 2397}, {43920, 2966}, {43922, 4591}, {43924, 52378}, {43925, 250}, {43926, 66929}, {51641, 2149}, {52335, 6558}, {52619, 34537}, {52633, 69069}, {53538, 1414}, {53540, 651}, {53545, 664}, {53559, 18047}, {53560, 4571}, {55195, 8}, {55197, 65958}, {55208, 7012}, {55244, 5376}, {55246, 5385}, {55261, 5377}, {55263, 9268}, {56283, 7058}, {57129, 1101}, {57185, 65573}, {58259, 55256}, {61052, 4559}, {61404, 4577}, {62192, 59151}, {62429, 55260}, {63462, 55}, {64445, 7253}, {66289, 36804}, {69009, 17934}
X(69376) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 49303, 50351}, {4010, 21145, 4707}, {4049, 55244, 18011}, {6545, 21132, 764}, {30574, 53558, 4730}

X(69377) = X(2)X(3933)∩X(4)X(69)

Barycentrics    2*b^2*c^2 - SB*SC : :
X(69377) = 4 X[7808] - 3 X[63024]

X(69377) lies on these lines: {2, 3933}, {3, 15589}, {4, 69}, {5, 32823}, {6, 16045}, {7, 4385}, {8, 3673}, {20, 7767}, {30, 32869}, {32, 14039}, {39, 32960}, {83, 1992}, {85, 54433}, {99, 3528}, {115, 33292}, {140, 32831}, {141, 5286}, {148, 33238}, {183, 631}, {193, 7770}, {194, 16043}, {230, 33189}, {253, 52404}, {274, 17582}, {310, 6821}, {312, 17170}, {321, 41826}, {325, 3090}, {339, 6643}, {343, 14361}, {350, 1058}, {376, 1975}, {381, 32874}, {382, 14929}, {384, 1285}, {385, 14001}, {388, 3761}, {391, 17681}, {394, 52288}, {443, 34284}, {487, 36703}, {488, 36701}, {491, 7375}, {492, 7376}, {497, 3760}, {524, 69139}, {538, 7738}, {599, 5254}, {626, 33285}, {671, 60636}, {1007, 5067}, {1056, 1909}, {1078, 3524}, {1249, 56015}, {1270, 7388}, {1271, 7389}, {1369, 62964}, {1370, 40002}, {1384, 33201}, {1506, 9770}, {1656, 63098}, {1930, 3974}, {2481, 15998}, {2548, 7855}, {2549, 7854}, {2550, 20888}, {2551, 6381}, {2896, 32986}, {2996, 5485}, {3085, 69093}, {3086, 69094}, {3091, 7776}, {3096, 21356}, {3146, 32882}, {3314, 14064}, {3399, 60212}, {3522, 32880}, {3523, 6390}, {3525, 7763}, {3526, 32835}, {3529, 7750}, {3533, 32821}, {3538, 62698}, {3545, 7788}, {3618, 7760}, {3619, 7803}, {3620, 6392}, {3628, 32870}, {3630, 63932}, {3631, 7784}, {3714, 47595}, {3734, 14023}, {3767, 7794}, {3788, 32959}, {3793, 68527}, {3832, 32894}, {3855, 7773}, {3934, 7736}, {3945, 13740}, {3964, 7509}, {4176, 20023}, {4352, 56737}, {4417, 7402}, {4441, 5082}, {4869, 33838}, {4911, 21296}, {5015, 32099}, {5055, 32893}, {5056, 32872}, {5070, 32897}, {5071, 7752}, {5084, 18135}, {5205, 25583}, {5215, 7801}, {5232, 16062}, {5304, 7819}, {5319, 7822}, {6194, 15428}, {6292, 7739}, {6527, 11414}, {6680, 7735}, {6803, 26166}, {6822, 18152}, {6823, 40995}, {6857, 37670}, {6996, 37655}, {7383, 52347}, {7386, 8024}, {7392, 39998}, {7397, 14829}, {7400, 41005}, {7550, 68660}, {7737, 7826}, {7745, 40341}, {7746, 32958}, {7748, 14711}, {7762, 20080}, {7764, 62993}, {7766, 16898}, {7769, 61867}, {7771, 61138}, {7774, 31276}, {7777, 32975}, {7778, 32955}, {7779, 16924}, {7780, 69206}, {7782, 19708}, {7783, 9741}, {7785, 32983}, {7787, 63093}, {7789, 8667}, {7790, 33232}, {7791, 20081}, {7793, 32985}, {7797, 33221}, {7799, 15702}, {7802, 49138}, {7804, 63934}, {7805, 63006}, {7807, 37667}, {7808, 63024}, {7809, 41106}, {7811, 11001}, {7812, 50992}, {7813, 31401}, {7814, 53127}, {7815, 34511}, {7821, 43620}, {7828, 32953}, {7832, 32952}, {7834, 63925}, {7835, 33236}, {7836, 17008}, {7851, 33196}, {7857, 33231}, {7859, 63121}, {7863, 21843}, {7868, 33194}, {7870, 23055}, {7871, 61899}, {7878, 62995}, {7879, 32974}, {7883, 50990}, {7889, 41748}, {7891, 33216}, {7892, 63048}, {7893, 14035}, {7894, 59373}, {7896, 63533}, {7897, 32961}, {7900, 33016}, {7903, 31415}, {7904, 33226}, {7907, 39142}, {7912, 32984}, {7917, 61945}, {7918, 11054}, {7921, 33269}, {7925, 32976}, {7929, 33017}, {7938, 19570}, {7939, 14063}, {7941, 32962}, {7945, 33222}, {7999, 51386}, {8164, 69135}, {8362, 22253}, {8369, 9740}, {8370, 11160}, {8743, 56013}, {8889, 45201}, {9464, 46336}, {9766, 31404}, {10159, 60183}, {10299, 32820}, {10303, 32841}, {10327, 20880}, {10519, 39646}, {10583, 63065}, {11057, 62161}, {11488, 69157}, {11489, 69165}, {12088, 15574}, {13161, 17272}, {14033, 17128}, {14552, 37086}, {14651, 32458}, {14907, 17538}, {15048, 33202}, {15271, 31400}, {15533, 65630}, {15598, 59546}, {15682, 32826}, {15698, 59634}, {15709, 32837}, {15719, 32896}, {15905, 28717}, {16239, 32898}, {16589, 33027}, {16845, 16992}, {16895, 63045}, {16989, 46226}, {16999, 33043}, {17004, 32977}, {17181, 28808}, {17206, 36697}, {17559, 18140}, {17928, 22241}, {18841, 51171}, {18842, 60639}, {18843, 60628}, {18845, 60637}, {18907, 63936}, {19697, 21309}, {20094, 33253}, {20105, 33021}, {20208, 28425}, {21264, 31405}, {21735, 32824}, {22165, 34505}, {22329, 33197}, {26040, 32092}, {30270, 53015}, {30435, 33198}, {30567, 53597}, {30737, 52398}, {31859, 32990}, {32819, 32878}, {32825, 32838}, {32827, 32888}, {32839, 61870}, {32867, 37647}, {32871, 46219}, {32873, 61863}, {32875, 61807}, {32879, 61820}, {32881, 61848}, {32885, 61895}, {32886, 61921}, {32890, 62066}, {32954, 37689}, {32973, 46453}, {32992, 62988}, {32999, 63021}, {33199, 43291}, {33274, 55823}, {33651, 62979}, {33940, 42696}, {33941, 42697}, {34803, 60781}, {35513, 68355}, {36682, 69264}, {36851, 42554}, {37201, 51884}, {37431, 68653}, {37946, 67603}, {38664, 50567}, {38940, 46512}, {40693, 69106}, {40694, 69107}, {41236, 63057}, {42149, 69121}, {42152, 69120}, {42998, 69180}, {42999, 69186}, {43459, 61787}, {43678, 60114}, {43681, 60219}, {45807, 53345}, {47743, 69254}, {48817, 68909}, {48913, 61959}, {50248, 68525}, {51374, 61102}, {52422, 69279}, {53105, 60627}, {54616, 60210}, {54637, 60250}, {59780, 63950}, {60221, 62917}, {60278, 60629}, {60855, 62996}, {61331, 69130}, {61332, 69131}, {62959, 64062}, {63017, 68522}, {63927, 69172}

X(69377) = reflection of X(i) in X(j) for these {i,j}: {7738, 7800}, {69208, 69139}
X(69377) = anticomplement of X(9605)
X(69377) = anticomplement of the isogonal conjugate of X(18841)
X(69377) = isotomic conjugate of the isogonal conjugate of X(7484)
X(69377) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 41928}, {18841, 8}, {58102, 4560}
X(69377) = cevapoint of X(i) and X(j) for these (i,j): {69, 41927}, {51171, 59343}
X(69377) = crossdifference of every pair of points on line {3049, 3804}
X(69377) = barycentric product X(i)*X(j) for these {i,j}: {76, 7484}, {670, 47133}
X(69377) = barycentric quotient X(i)/X(j) for these {i,j}: {7484, 6}, {47133, 512}
X(69377) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3933, 32818}, {3, 32830, 32817}, {4, 76, 52713}, {5, 37668, 32823}, {69, 76, 4}, {69, 32006, 7768}, {69, 52710, 44133}, {76, 7768, 11185}, {141, 5286, 32956}, {141, 63933, 5286}, {183, 3926, 631}, {194, 16990, 16043}, {230, 53033, 33189}, {325, 32828, 3090}, {1007, 32832, 5067}, {1078, 6337, 3524}, {1078, 32833, 6337}, {1975, 3785, 376}, {1975, 37671, 3785}, {3091, 10513, 7776}, {3523, 32840, 6390}, {3620, 6392, 6656}, {3631, 63923, 7784}, {3785, 32836, 1975}, {3788, 62992, 32959}, {3934, 7736, 32957}, {3934, 7758, 7736}, {3974, 7195, 1930}, {7735, 7795, 14069}, {7746, 37690, 32958}, {7750, 32815, 3529}, {7751, 7795, 7735}, {7763, 34229, 3525}, {7768, 11185, 32006}, {7774, 31276, 32968}, {7776, 64093, 3091}, {7784, 63923, 43448}, {7788, 46951, 3545}, {7788, 59635, 32816}, {7794, 17131, 3767}, {7796, 32832, 1007}, {7802, 67536, 49138}, {7805, 69209, 63006}, {7826, 17130, 7737}, {7836, 17008, 32970}, {7855, 9466, 2548}, {7879, 47286, 32974}, {11185, 32006, 4}, {15589, 32830, 3}, {16043, 16990, 55732}, {17128, 20065, 14033}, {18135, 45962, 5084}, {20080, 32971, 7762}, {20081, 63044, 7791}, {30435, 63926, 63042}, {32816, 46951, 59635}, {32816, 59635, 3545}, {32819, 64018, 33703}, {32821, 37688, 32829}, {32829, 37688, 3533}, {32834, 37668, 5}, {32836, 37671, 376}, {32867, 37647, 61881}, {33198, 63042, 30435}, {39998, 40123, 7392}

X(69378) = X(2)X(1975)∩X(4)X(69)

Barycentrics    b^2*c^2 + SB*SC : :
X(69378) = X[3785] - 3 X[46951], X[32826] + 3 X[46951], 2 X[2548] - 3 X[32983], 4 X[7815] - 3 X[33215], 3 X[8556] - X[44519]

X(69378) lies on these lines: {2, 1975}, {3, 32815}, {4, 69}, {5, 1007}, {6, 6392}, {20, 183}, {30, 3785}, {32, 14033}, {39, 32968}, {65, 40028}, {75, 2551}, {83, 5485}, {99, 631}, {115, 7795}, {140, 32838}, {141, 32974}, {148, 5976}, {187, 33239}, {193, 7745}, {194, 7736}, {230, 32973}, {274, 5084}, {290, 15740}, {305, 7392}, {310, 6818}, {325, 3091}, {327, 40830}, {343, 37174}, {350, 388}, {376, 1078}, {377, 18135}, {381, 3933}, {382, 7767}, {384, 7735}, {385, 14035}, {394, 62950}, {397, 69186}, {398, 69180}, {443, 18140}, {452, 16992}, {458, 37669}, {460, 14826}, {491, 31412}, {492, 42561}, {497, 1909}, {524, 65630}, {538, 2548}, {543, 7815}, {546, 7776}, {549, 32885}, {550, 32886}, {574, 32978}, {598, 60636}, {620, 32977}, {626, 16041}, {632, 32883}, {668, 5082}, {671, 3096}, {683, 6524}, {1058, 64133}, {1230, 7382}, {1231, 1882}, {1285, 6179}, {1370, 39998}, {1384, 68177}, {1444, 37415}, {1478, 3760}, {1479, 3761}, {1506, 34511}, {1656, 6390}, {1799, 34608}, {1867, 64989}, {1992, 7754}, {2207, 41614}, {2478, 34284}, {2549, 3934}, {2550, 6376}, {2896, 33017}, {2899, 30758}, {3053, 32981}, {3090, 7763}, {3146, 7750}, {3314, 14063}, {3329, 33269}, {3421, 17143}, {3436, 4441}, {3522, 32872}, {3523, 37688}, {3524, 7782}, {3526, 32867}, {3528, 7771}, {3529, 14907}, {3543, 32874}, {3545, 7752}, {3552, 17008}, {3618, 5286}, {3619, 6656}, {3620, 7784}, {3627, 32888}, {3628, 32839}, {3642, 60253}, {3643, 60252}, {3673, 42697}, {3734, 3767}, {3763, 59548}, {3788, 32969}, {3815, 32987}, {3832, 7773}, {3839, 7788}, {3843, 32878}, {3845, 32892}, {3850, 32877}, {3851, 32825}, {3853, 14929}, {3854, 32880}, {3855, 7796}, {3964, 11479}, {4048, 44530}, {4385, 42696}, {4994, 34386}, {5023, 13468}, {5025, 63533}, {5046, 45962}, {5055, 32837}, {5056, 32820}, {5067, 7769}, {5068, 32821}, {5071, 7799}, {5072, 32875}, {5177, 37664}, {5305, 11286}, {5309, 69209}, {5319, 7804}, {5395, 43681}, {5475, 7758}, {5490, 35831}, {5491, 35830}, {5818, 69038}, {5866, 17928}, {6108, 37173}, {6109, 37172}, {6292, 11648}, {6393, 40330}, {6515, 33798}, {6526, 34403}, {6620, 37894}, {6655, 16990}, {6816, 26166}, {6817, 18152}, {6823, 40680}, {6872, 37670}, {6997, 8024}, {7375, 32813}, {7376, 32812}, {7378, 45201}, {7383, 18354}, {7386, 40022}, {7388, 32805}, {7389, 32806}, {7394, 40123}, {7395, 9723}, {7399, 40697}, {7401, 28706}, {7406, 14829}, {7486, 32835}, {7494, 16276}, {7509, 44180}, {7526, 68654}, {7529, 22241}, {7610, 35287}, {7615, 7801}, {7620, 7841}, {7714, 33651}, {7737, 7751}, {7739, 7808}, {7746, 32970}, {7747, 14023}, {7748, 7800}, {7749, 33216}, {7756, 33226}, {7760, 62995}, {7761, 33238}, {7762, 11008}, {7764, 31415}, {7774, 16044}, {7777, 32962}, {7778, 32972}, {7779, 33018}, {7781, 31401}, {7785, 33016}, {7786, 32957}, {7787, 19570}, {7790, 32956}, {7792, 33198}, {7793, 33007}, {7794, 69141}, {7797, 16898}, {7802, 33703}, {7803, 16045}, {7806, 14037}, {7807, 63104}, {7809, 41099}, {7810, 65633}, {7811, 15682}, {7812, 63064}, {7814, 61945}, {7816, 23055}, {7820, 69162}, {7821, 18424}, {7822, 33221}, {7823, 14068}, {7826, 62203}, {7827, 63109}, {7828, 14069}, {7830, 33247}, {7831, 55732}, {7832, 32951}, {7833, 42850}, {7834, 32457}, {7835, 33189}, {7836, 32961}, {7839, 63024}, {7842, 63957}, {7855, 14711}, {7857, 33191}, {7861, 33223}, {7863, 32976}, {7868, 33180}, {7871, 41106}, {7879, 33229}, {7881, 33228}, {7882, 63956}, {7883, 50994}, {7885, 32996}, {7886, 33222}, {7887, 39143}, {7893, 14042}, {7894, 11054}, {7895, 47617}, {7897, 32993}, {7898, 33279}, {7903, 43457}, {7904, 32997}, {7906, 9770}, {7912, 33006}, {7918, 33230}, {7919, 33194}, {7925, 32963}, {7930, 32953}, {7931, 33283}, {7934, 33292}, {7937, 33232}, {7938, 33251}, {7939, 14062}, {7940, 32958}, {7942, 32952}, {7944, 33196}, {7945, 33248}, {8352, 50990}, {8369, 40727}, {8556, 44519}, {8781, 23514}, {8797, 27356}, {9464, 62937}, {9605, 66415}, {9939, 52942}, {10159, 60219}, {10303, 32870}, {10304, 32893}, {10513, 32882}, {10522, 34387}, {10590, 69135}, {10591, 69254}, {10895, 69093}, {10896, 69094}, {10996, 62698}, {11057, 62042}, {11147, 33274}, {11257, 35438}, {11291, 39661}, {11292, 39660}, {11303, 43403}, {11304, 43404}, {11317, 50992}, {11318, 59780}, {11361, 17129}, {11413, 34883}, {11427, 41231}, {11433, 51481}, {11444, 51439}, {12243, 53765}, {13108, 37348}, {13161, 17321}, {13740, 63014}, {14031, 63048}, {14039, 14568}, {14061, 32955}, {14361, 59528}, {14382, 36874}, {14482, 55085}, {14639, 32458}, {14712, 33280}, {15022, 32841}, {15077, 54124}, {15271, 32990}, {15491, 22332}, {15810, 53143}, {15815, 58446}, {16589, 33026}, {16605, 39721}, {16644, 59541}, {16645, 59542}, {16921, 62993}, {16925, 62992}, {16989, 68525}, {16999, 33058}, {17004, 32964}, {17005, 33009}, {17006, 33206}, {17040, 56067}, {17170, 20925}, {17578, 32894}, {18581, 69165}, {18582, 69157}, {18841, 43676}, {18842, 60250}, {18843, 60216}, {18845, 60200}, {18928, 40814}, {19709, 32896}, {20080, 53418}, {20088, 63093}, {20094, 33004}, {20105, 63018}, {20477, 52404}, {21073, 30701}, {21735, 43459}, {21843, 69171}, {22331, 50774}, {22712, 60212}, {24467, 55448}, {24524, 64068}, {25264, 31402}, {25406, 39646}, {25994, 62693}, {26105, 31997}, {26164, 41760}, {26235, 46336}, {26870, 48716}, {26921, 55449}, {27269, 33028}, {30737, 37201}, {31400, 31859}, {31404, 44543}, {31406, 52229}, {31409, 69255}, {31467, 51122}, {31489, 59546}, {31994, 59513}, {32459, 44535}, {32532, 60210}, {32871, 46935}, {32876, 61919}, {32884, 55856}, {32887, 61900}, {32889, 61907}, {32890, 61953}, {32897, 55864}, {32954, 43291}, {32988, 44377}, {32989, 37637}, {32991, 62988}, {33002, 63083}, {33019, 63044}, {33023, 44526}, {33024, 63021}, {33200, 63536}, {33201, 37689}, {33296, 63089}, {34208, 40405}, {34504, 47061}, {35687, 61689}, {35925, 38907}, {36901, 46231}, {37242, 61550}, {37532, 55419}, {38259, 60285}, {38734, 50567}, {39127, 53420}, {40132, 66767}, {41370, 56015}, {41895, 60639}, {41916, 62964}, {42159, 69145}, {42162, 69137}, {42813, 69106}, {42814, 69107}, {43028, 59540}, {43029, 59539}, {43618, 63935}, {44440, 68355}, {44540, 59695}, {45198, 50572}, {46034, 60702}, {46927, 52283}, {47355, 59552}, {48913, 61967}, {50009, 51884}, {51580, 68522}, {52145, 56688}, {52289, 62708}, {52756, 59423}, {53105, 60143}, {53107, 60627}, {53489, 63073}, {54103, 67862}, {54122, 60151}, {54637, 60278}, {54703, 60221}, {54796, 60256}, {57819, 59429}, {60183, 60209}, {61881, 62362}, {63047, 68517}, {63954, 66409}, {66395, 66699}

X(69378) = midpoint of X(3785) and X(32826)
X(69378) = reflection of X(i) in X(j) for these {i,j}: {10983, 5}, {11257, 35438}
X(69378) = isotomic conjugate of X(17040)
X(69378) = anticomplement of X(5013)
X(69378) = anticomplement of the isogonal conjugate of X(5395)
X(69378) = anticomplement of the isotomic conjugate of X(56067)
X(69378) = isotomic conjugate of the isogonal conjugate of X(5020)
X(69378) = isotomic conjugate of the polar conjugate of X(43981)
X(69378) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {19, 8892}, {5395, 8}, {31506, 21217}, {56067, 6327}, {58100, 4560}
X(69378) = X(i)-Ceva conjugate of X(j) for these (i,j): {56067, 2}, {60114, 69}
X(69378) = X(5020)-cross conjugate of X(43981)
X(69378) = X(i)-isoconjugate of X(j) for these (i,j): {31, 17040}, {560, 59756}, {1973, 56339}
X(69378) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 17040}, {5020, 19459}, {6337, 56339}, {6374, 59756}, {40680, 11433}
X(69378) = cevapoint of X(10565) and X(51170)
X(69378) = crossdifference of every pair of points on line {3049, 8651}
X(69378) = barycentric product X(i)*X(j) for these {i,j}: {69, 43981}, {76, 5020}
X(69378) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 17040}, {69, 56339}, {76, 59756}, {5020, 6}, {43981, 4}
X(69378) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1975, 6337}, {2, 2996, 5254}, {3, 32828, 34229}, {3, 64093, 32828}, {4, 69, 32006}, {4, 76, 69}, {4, 52713, 76}, {5, 3926, 1007}, {6, 63923, 6392}, {20, 32819, 67536}, {20, 32834, 183}, {39, 32968, 63041}, {69, 44135, 52710}, {69, 51537, 5207}, {76, 11185, 4}, {99, 32832, 631}, {115, 7795, 14064}, {115, 17130, 7795}, {141, 44518, 32974}, {148, 31276, 7791}, {183, 32819, 20}, {193, 32979, 7745}, {194, 16924, 7736}, {381, 3933, 32816}, {382, 7767, 64018}, {546, 7776, 32827}, {631, 32822, 99}, {1656, 6390, 32829}, {1975, 59635, 2}, {2549, 3934, 16043}, {3090, 7763, 34803}, {3090, 32817, 7763}, {3091, 32830, 325}, {3146, 15589, 7750}, {3543, 32874, 37671}, {3545, 32818, 7752}, {3620, 32982, 7784}, {3734, 3767, 14001}, {3734, 63924, 3767}, {3788, 43620, 32969}, {3832, 37668, 7773}, {3839, 32869, 7788}, {5068, 32840, 63098}, {5254, 34505, 2996}, {5254, 69139, 2}, {5286, 7770, 3618}, {6392, 32971, 6}, {7486, 32835, 37647}, {7615, 7801, 32984}, {7745, 63933, 193}, {7746, 69206, 32970}, {7747, 17131, 14023}, {7748, 7800, 32986}, {7748, 9466, 7800}, {7752, 32833, 32818}, {7754, 8370, 69208}, {7754, 69208, 1992}, {7769, 53127, 5067}, {7770, 47286, 5286}, {7778, 63534, 32972}, {7784, 53419, 32982}, {7789, 13881, 2}, {7803, 16045, 63119}, {7816, 69207, 32985}, {7830, 43619, 33247}, {7836, 32961, 37690}, {11185, 52713, 69}, {11361, 17129, 20065}, {14033, 63955, 63034}, {14068, 63046, 7823}, {14711, 39590, 7855}, {15271, 63548, 32990}, {16044, 20081, 7774}, {18840, 33190, 3096}, {31859, 32992, 31400}, {32815, 32828, 3}, {32815, 64093, 34229}, {32816, 32836, 3933}, {32824, 32829, 6390}, {32826, 46951, 3785}, {32840, 63098, 32821}, {32882, 50689, 10513}, {32981, 37667, 3053}, {34505, 69139, 5254}, {37637, 59545, 32989}, {44780, 44781, 32000}

X(69379) = X(2)X(1975)∩X(20)X(76)

Barycentrics    2*b^2*c^2 - S^2 + 2*SB*SC : :
X(69379) = 3 X[2] - 4 X[69139]

X(69379) lies on these lines: {2, 1975}, {3, 32822}, {4, 3933}, {5, 32817}, {20, 76}, {30, 32869}, {69, 3146}, {99, 3523}, {115, 33199}, {140, 32870}, {141, 33025}, {148, 7938}, {183, 3522}, {193, 732}, {194, 32971}, {230, 33205}, {274, 5129}, {305, 7398}, {315, 3543}, {316, 50688}, {325, 3832}, {350, 3600}, {376, 32874}, {381, 32818}, {384, 5304}, {385, 32981}, {390, 1909}, {439, 17008}, {452, 34284}, {491, 1131}, {492, 1132}, {538, 69208}, {543, 7800}, {546, 32823}, {631, 64093}, {637, 12296}, {638, 12297}, {1007, 5068}, {1078, 10304}, {1285, 68177}, {1885, 32000}, {2549, 6292}, {2896, 33272}, {3090, 6390}, {3091, 3926}, {3212, 24280}, {3314, 32982}, {3524, 32893}, {3525, 32897}, {3529, 7767}, {3545, 69158}, {3552, 37667}, {3619, 59548}, {3620, 6655}, {3628, 32898}, {3734, 5286}, {3760, 4293}, {3761, 4294}, {3767, 33181}, {3839, 7871}, {3854, 32821}, {3964, 63664}, {5023, 63029}, {5024, 32957}, {5025, 63536}, {5054, 52718}, {5056, 7763}, {5059, 7750}, {5067, 32871}, {5073, 14929}, {5225, 69094}, {5229, 69093}, {5305, 14039}, {5343, 69145}, {5344, 69137}, {5395, 20105}, {5485, 8369}, {5976, 20094}, {6527, 37201}, {6658, 63046}, {6904, 18135}, {6995, 8024}, {7408, 40123}, {7486, 32829}, {7487, 44146}, {7615, 7862}, {7620, 7801}, {7735, 33201}, {7745, 63091}, {7748, 33210}, {7754, 14033}, {7766, 14031}, {7768, 50691}, {7769, 46936}, {7771, 62067}, {7773, 50689}, {7774, 32979}, {7777, 32991}, {7778, 63533}, {7779, 14068}, {7781, 31400}, {7782, 15692}, {7788, 50687}, {7793, 35927}, {7795, 7861}, {7796, 32827}, {7799, 61936}, {7802, 32878}, {7809, 61989}, {7811, 32892}, {7816, 63955}, {7823, 20080}, {7826, 43618}, {7828, 33183}, {7832, 33182}, {7836, 32972}, {7854, 43619}, {7863, 43620}, {7879, 33238}, {7881, 16041}, {7882, 44678}, {7885, 54097}, {7893, 33280}, {7895, 63957}, {7897, 32996}, {7906, 33016}, {7917, 32890}, {7925, 52250}, {7929, 33192}, {7939, 33279}, {7947, 33006}, {9605, 52229}, {9740, 17129}, {10303, 32832}, {10513, 17578}, {10565, 16276}, {10594, 22241}, {11057, 62166}, {11106, 16992}, {11160, 66419}, {11285, 47287}, {11286, 33685}, {11287, 18840}, {11488, 59541}, {11489, 59542}, {11606, 35369}, {12087, 15574}, {12173, 32001}, {12215, 51171}, {12632, 24524}, {14001, 47286}, {14826, 59527}, {15048, 16045}, {15056, 51386}, {15683, 37671}, {15717, 32872}, {15721, 32885}, {16044, 62988}, {16990, 33023}, {17576, 37670}, {17580, 18140}, {27269, 33039}, {30270, 46034}, {30758, 66681}, {31276, 32990}, {31404, 34511}, {31406, 51122}, {31467, 51123}, {31859, 32968}, {32837, 61924}, {32838, 55864}, {32839, 46935}, {32867, 61863}, {32868, 62097}, {32873, 37647}, {32884, 62362}, {32886, 61788}, {32888, 62110}, {32894, 50693}, {32896, 61966}, {32962, 63077}, {32973, 37689}, {32995, 63021}, {32997, 63044}, {33004, 46318}, {33189, 43291}, {33190, 59780}, {33200, 44518}, {33203, 63924}, {33215, 53141}, {33259, 51579}, {33283, 46236}, {33798, 63031}, {34803, 61914}, {37161, 37664}, {37512, 53142}, {37688, 61820}, {37690, 63534}, {41009, 61113}, {41916, 52397}, {42159, 69121}, {42160, 69107}, {42161, 69106}, {42162, 69120}, {43403, 69157}, {43404, 69165}, {43459, 58188}, {43681, 62930}, {46453, 68513}, {49135, 64018}, {59545, 62992}, {59546, 62993}, {59552, 63119}, {60184, 60200}, {63042, 63933}, {63048, 68517}

X(69379) = reflection of X(7738) in X(69139)
X(69379) = isotomic conjugate of X(45833)
X(69379) = anticomplement of X(7738)
X(69379) = X(31)-isoconjugate of X(45833)
X(69379) = X(2)-Dao conjugate of X(45833)
X(69379) = barycentric quotient X(2)/X(45833)
X(69379) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 52713, 32834}, {4, 32830, 37668}, {5, 32817, 32831}, {20, 76, 15589}, {69, 32819, 3146}, {76, 32815, 20}, {99, 32828, 3523}, {115, 53033, 33199}, {194, 32971, 37665}, {315, 32826, 3543}, {384, 6392, 5304}, {1007, 32820, 32841}, {1975, 59635, 6337}, {3090, 6390, 32835}, {3091, 3926, 63098}, {3734, 5286, 33198}, {3832, 32840, 325}, {3926, 11185, 3091}, {5068, 32841, 1007}, {6337, 59635, 2}, {7738, 69139, 2}, {7750, 67536, 5059}, {7795, 43448, 33180}, {10513, 17578, 32006}, {14035, 20081, 193}, {15717, 32872, 34229}, {17578, 32880, 10513}, {32822, 52713, 3}, {32826, 32836, 315}, {32839, 53127, 46935}

X(69380) = X(2)X(2418)∩X(3)X(76)

Barycentrics    2*b^2*c^2 - S^2 + SB*SC : :
X(69380) = X[15048] - 3 X[59780], 3 X[6] - 2 X[7798], 3 X[6] - 4 X[7804], 2 X[6] - 3 X[11286], 3 X[3734] - X[7798], 3 X[3734] - 2 X[7804], 4 X[3734] - 3 X[11286], 4 X[3734] - X[22253], 4 X[7798] - 9 X[11286], 4 X[7798] - 3 X[22253], 8 X[7804] - 9 X[11286], 8 X[7804] - 3 X[22253], 3 X[11286] - X[22253], 3 X[69] - 2 X[14929], X[69] - 3 X[32836], and many others

X(69380) lies on these lines: {2, 2418}, {3, 76}, {4, 3933}, {5, 1007}, {6, 538}, {13, 69120}, {14, 69121}, {20, 7767}, {22, 5938}, {25, 8024}, {30, 69}, {32, 63933}, {39, 17130}, {55, 3761}, {56, 3760}, {61, 69180}, {62, 69186}, {75, 9708}, {81, 50060}, {85, 7283}, {86, 11354}, {115, 7778}, {140, 6337}, {141, 2549}, {148, 3314}, {187, 8667}, {193, 14033}, {194, 3329}, {218, 56024}, {230, 11288}, {264, 1597}, {274, 11108}, {298, 11295}, {299, 11296}, {302, 11297}, {303, 11298}, {305, 5020}, {310, 16058}, {311, 3964}, {312, 5088}, {315, 382}, {316, 3830}, {317, 18494}, {325, 381}, {333, 56963}, {343, 54074}, {350, 999}, {376, 15589}, {384, 7754}, {385, 1003}, {394, 51372}, {405, 34284}, {428, 40123}, {474, 18135}, {476, 53186}, {491, 13665}, {492, 13785}, {524, 7737}, {525, 2395}, {543, 599}, {546, 32816}, {547, 32837}, {548, 32878}, {549, 34229}, {550, 3785}, {574, 8716}, {620, 37637}, {625, 7908}, {626, 44518}, {631, 32834}, {632, 32838}, {671, 7934}, {698, 11356}, {736, 5017}, {754, 40341}, {892, 53221}, {940, 24271}, {956, 4441}, {958, 20888}, {1228, 37034}, {1230, 11350}, {1232, 35243}, {1235, 1593}, {1236, 66717}, {1238, 11818}, {1285, 63042}, {1316, 5468}, {1350, 14532}, {1351, 18906}, {1376, 6381}, {1478, 69093}, {1479, 69094}, {1502, 16084}, {1573, 20181}, {1598, 54412}, {1655, 11321}, {1656, 7763}, {1657, 7750}, {1909, 3295}, {1992, 44839}, {1995, 5971}, {2049, 50177}, {2140, 62426}, {2407, 40856}, {2482, 7610}, {2794, 15069}, {2795, 5695}, {2797, 45807}, {2896, 33234}, {2899, 25583}, {2996, 14064}, {3051, 11335}, {3053, 7751}, {3090, 32831}, {3091, 32818}, {3146, 32880}, {3172, 56015}, {3212, 63996}, {3266, 11284}, {3363, 9770}, {3522, 32882}, {3524, 32874}, {3526, 32832}, {3530, 32868}, {3533, 32870}, {3534, 14907}, {3543, 10513}, {3545, 63098}, {3552, 17129}, {3589, 7739}, {3618, 63633}, {3620, 32986}, {3627, 32006}, {3628, 32829}, {3630, 43618}, {3631, 43619}, {3763, 4045}, {3767, 7789}, {3770, 4254}, {3788, 6722}, {3793, 66391}, {3815, 34511}, {3832, 32823}, {3843, 7773}, {3845, 32827}, {3849, 15533}, {3850, 32825}, {3851, 7752}, {3853, 32890}, {3879, 66639}, {3934, 5013}, {3945, 48817}, {3948, 16412}, {3972, 14614}, {4011, 68995}, {4048, 40825}, {4352, 13740}, {4383, 62755}, {4387, 7223}, {4417, 36731}, {4423, 52716}, {5008, 41748}, {5022, 29455}, {5023, 7780}, {5025, 7881}, {5050, 12215}, {5054, 37688}, {5055, 7799}, {5056, 32841}, {5066, 32896}, {5067, 32835}, {5068, 32879}, {5070, 7769}, {5073, 7768}, {5093, 10796}, {5108, 35606}, {5192, 18600}, {5210, 32456}, {5224, 11359}, {5232, 48813}, {5254, 7795}, {5278, 50154}, {5286, 7819}, {5304, 14039}, {5305, 6392}, {5309, 7820}, {5339, 69145}, {5340, 69137}, {5475, 7813}, {5564, 48804}, {5569, 36521}, {5585, 46893}, {5790, 69038}, {5891, 51386}, {5913, 11336}, {5941, 52628}, {6194, 54993}, {6248, 58851}, {6321, 32458}, {6333, 46229}, {6376, 9709}, {6393, 37242}, {6527, 35513}, {6642, 28706}, {6655, 7879}, {6658, 7893}, {6661, 16989}, {6683, 22332}, {6767, 64133}, {6786, 12525}, {6787, 34203}, {7395, 26166}, {7484, 39906}, {7514, 9723}, {7603, 11184}, {7615, 22110}, {7618, 11168}, {7620, 37350}, {7697, 51373}, {7735, 8369}, {7736, 66415}, {7738, 8362}, {7745, 7758}, {7746, 7863}, {7747, 7855}, {7748, 7784}, {7756, 7854}, {7757, 11174}, {7759, 65630}, {7760, 43136}, {7762, 14035}, {7765, 7822}, {7774, 8370}, {7777, 44543}, {7779, 11361}, {7783, 11285}, {7790, 7868}, {7792, 33237}, {7793, 33235}, {7797, 33217}, {7800, 63548}, {7802, 17800}, {7805, 69172}, {7806, 19570}, {7808, 32450}, {7809, 14269}, {7811, 15681}, {7814, 15031}, {7815, 15815}, {7821, 69141}, {7825, 7895}, {7826, 63938}, {7830, 44519}, {7832, 7851}, {7833, 20094}, {7835, 14568}, {7836, 7887}, {7839, 20105}, {7840, 11317}, {7842, 7896}, {7843, 7916}, {7844, 7880}, {7845, 62203}, {7849, 7872}, {7850, 15684}, {7853, 11648}, {7860, 62023}, {7861, 7869}, {7864, 46226}, {7870, 14061}, {7871, 61970}, {7873, 65633}, {7874, 69162}, {7876, 60728}, {7882, 63931}, {7888, 39565}, {7891, 33233}, {7897, 14041}, {7898, 66388}, {7900, 14042}, {7902, 7915}, {7903, 39590}, {7906, 16044}, {7909, 62427}, {7910, 32027}, {7917, 62008}, {7920, 19689}, {7929, 33256}, {7931, 33219}, {7939, 33019}, {7941, 33018}, {7945, 33218}, {7947, 32966}, {8267, 68719}, {8356, 16990}, {8361, 53033}, {8366, 16984}, {8367, 63041}, {8556, 15301}, {8591, 35955}, {8703, 32892}, {8719, 8722}, {8724, 40248}, {8780, 37894}, {8781, 61576}, {8860, 41134}, {9146, 15066}, {9182, 36207}, {9230, 22152}, {9308, 15014}, {9597, 69097}, {9598, 69095}, {9651, 69258}, {9654, 69135}, {9664, 69261}, {9669, 69254}, {9755, 35925}, {9756, 18860}, {9761, 40672}, {9763, 40671}, {9777, 33798}, {9832, 47293}, {9909, 16276}, {9939, 66395}, {10008, 61545}, {10303, 32872}, {10449, 49130}, {10516, 51371}, {11055, 22246}, {11057, 15685}, {11113, 45962}, {11160, 63945}, {11164, 51224}, {11178, 51397}, {11328, 20023}, {11329, 31060}, {11338, 40858}, {11352, 16704}, {11353, 24621}, {11355, 30941}, {11403, 44142}, {11539, 32885}, {11898, 44369}, {12079, 36194}, {12083, 15574}, {12812, 32876}, {13167, 33962}, {13468, 21843}, {13586, 15655}, {13637, 40286}, {13644, 62986}, {13757, 40287}, {13763, 62987}, {14034, 20088}, {14036, 63019}, {14148, 31489}, {14712, 66387}, {14869, 32886}, {14928, 43273}, {14981, 64711}, {15013, 15905}, {15300, 66616}, {15668, 24275}, {15702, 32893}, {15712, 32888}, {15717, 32894}, {15819, 52771}, {15928, 62431}, {16059, 18152}, {16239, 32867}, {16370, 37670}, {16408, 18140}, {16411, 18153}, {16417, 18145}, {16418, 16992}, {16419, 40022}, {16436, 26243}, {16747, 37318}, {16781, 69257}, {16916, 40908}, {16950, 40904}, {16964, 69107}, {16965, 69106}, {16983, 64023}, {17008, 35297}, {17030, 31468}, {17206, 49129}, {17259, 48864}, {17272, 66672}, {17277, 48869}, {17321, 66675}, {17528, 37664}, {17682, 27523}, {17814, 59527}, {18362, 31275}, {18420, 52347}, {18535, 58782}, {18570, 68654}, {18840, 33202}, {19118, 37912}, {19684, 50183}, {19686, 50248}, {19687, 20065}, {19701, 50178}, {19732, 50159}, {19744, 50162}, {20080, 63940}, {20208, 53481}, {20347, 49492}, {20477, 44149}, {20850, 33651}, {20911, 50044}, {21264, 31449}, {21312, 30737}, {22329, 68718}, {22331, 63925}, {22505, 54103}, {23039, 51439}, {23342, 44155}, {23698, 34507}, {24291, 49518}, {24296, 37660}, {25183, 63199}, {25187, 63198}, {25590, 48812}, {26541, 37248}, {26864, 35356}, {27020, 31461}, {27088, 63029}, {27269, 33035}, {28809, 37272}, {30109, 62383}, {30749, 62702}, {30806, 32929}, {30966, 56969}, {31173, 66587}, {31239, 53096}, {31401, 59546}, {31406, 32968}, {31467, 32992}, {31477, 69136}, {31861, 44135}, {32063, 57275}, {32087, 48806}, {32479, 66455}, {32519, 51580}, {32782, 50057}, {32809, 66435}, {32839, 55856}, {32871, 60781}, {32873, 46935}, {32883, 55859}, {32884, 55861}, {32897, 61867}, {32898, 61881}, {32981, 63926}, {32983, 62988}, {32985, 37667}, {33007, 63046}, {33013, 63021}, {33184, 43448}, {33191, 37689}, {33246, 63047}, {33255, 63048}, {33292, 63536}, {34227, 45012}, {34504, 44541}, {34506, 35022}, {34508, 47862}, {34509, 47861}, {34609, 45201}, {35001, 67606}, {35705, 48657}, {35954, 63065}, {36165, 47289}, {36721, 69264}, {36891, 46081}, {37344, 51481}, {37485, 42554}, {37638, 51389}, {38224, 62348}, {39113, 56965}, {39266, 47618}, {39809, 54393}, {40279, 61599}, {40879, 68152}, {41235, 59567}, {41676, 45141}, {42089, 59540}, {42092, 59539}, {42149, 59542}, {42152, 59541}, {42153, 69165}, {42156, 69157}, {42850, 53142}, {43620, 44377}, {44140, 56960}, {44141, 68018}, {44228, 55972}, {44461, 52193}, {44465, 52194}, {44558, 46154}, {45143, 52756}, {45672, 62191}, {46236, 67268}, {47288, 59227}, {48913, 61974}, {49671, 52437}, {50008, 62338}, {50170, 63056}, {50771, 53418}, {51356, 56968}, {51396, 54131}, {51440, 54048}, {51884, 52071}, {52629, 62489}, {52718, 55864}, {53489, 63017}, {54713, 60202}, {55629, 60702}, {55866, 62362}, {57521, 69263}, {57594, 68455}, {59197, 59211}, {59373, 66458}, {59545, 69207}, {60200, 62913}, {61923, 64809}, {63006, 66318}, {63018, 66413}, {63034, 66393}, {63093, 66319}, {66625, 67677}, {66626, 67688}, {69108, 69124}, {69109, 69125}, {69112, 69117}, {69113, 69116}, {69140, 69185}, {69143, 69183}, {69147, 69191}, {69149, 69189}

X(69380) = midpoint of X(i) and X(j) for these {i,j}: {69, 32815}, {64018, 67536}
X(69380) = reflection of X(i) in X(j) for these {i,j}: {2, 59780}, {3, 64653}, {6, 3734}, {193, 18907}, {1351, 35930}, {2549, 141}, {5077, 599}, {7798, 7804}, {14532, 1350}, {22253, 6}, {44526, 7761}, {45012, 34227}, {64018, 14929}, {64023, 16983}
X(69380) = anticomplement of X(15048)
X(69380) = isotomic conjugate of the isogonal conjugate of X(5651)
X(69380) = crossdifference of every pair of points on line {2491, 8644}
X(69380) = barycentric product X(i)*X(j) for these {i,j}: {76, 5651}, {99, 31174}
X(69380) = barycentric quotient X(i)/X(j) for these {i,j}: {5651, 6}, {31174, 523}
X(69380) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 31859, 5024}, {2, 32817, 6390}, {2, 52713, 64093}, {4, 3933, 7776}, {4, 32830, 3933}, {5, 3926, 69158}, {6, 3734, 11286}, {39, 17130, 69139}, {69, 64018, 14929}, {69, 67536, 64018}, {76, 99, 183}, {76, 1975, 3}, {99, 183, 3}, {115, 7778, 11318}, {115, 7801, 7778}, {141, 2549, 11287}, {148, 3314, 7841}, {183, 1975, 99}, {187, 14711, 17131}, {187, 17131, 8667}, {193, 14033, 18907}, {194, 7770, 9605}, {194, 17128, 7770}, {230, 69206, 11288}, {315, 32819, 382}, {325, 11185, 381}, {384, 7754, 30435}, {384, 20081, 7754}, {385, 1003, 1384}, {574, 9466, 15271}, {599, 44526, 7761}, {1384, 63954, 385}, {3053, 7816, 68513}, {3091, 32840, 32818}, {3329, 7770, 14535}, {3734, 7798, 7804}, {3767, 7789, 32954}, {3788, 63924, 13881}, {3934, 7781, 5013}, {3972, 14614, 21309}, {4387, 7223, 14210}, {5023, 69171, 68528}, {5024, 51122, 31859}, {5254, 7795, 7866}, {5475, 7813, 9766}, {6337, 32828, 140}, {6390, 64093, 2}, {6392, 14001, 5305}, {7747, 7855, 63932}, {7748, 7794, 7784}, {7751, 7816, 3053}, {7761, 44526, 5077}, {7763, 53127, 37647}, {7763, 59635, 1656}, {7774, 8370, 15484}, {7778, 34505, 115}, {7780, 69171, 5023}, {7783, 31276, 11285}, {7789, 63923, 3767}, {7798, 7804, 6}, {7801, 34505, 11318}, {7880, 32457, 7844}, {7895, 63922, 7825}, {7908, 18546, 625}, {8716, 15271, 574}, {9605, 14535, 3329}, {11185, 32833, 325}, {11286, 22253, 6}, {13468, 32459, 21843}, {20094, 63044, 7833}, {20105, 68525, 7839}, {22712, 23235, 63424}, {22712, 63424, 3}, {24275, 48840, 15668}, {32006, 32826, 3627}, {32815, 32836, 69}, {32815, 64018, 67536}, {32817, 52713, 2}, {32820, 59635, 7763}, {32824, 32828, 6337}, {37647, 53127, 1656}, {37647, 59635, 53127}, {63955, 69206, 230}, {67676, 67687, 50567}, {69112, 69117, 69181}, {69113, 69116, 69187}

X(69381) = X(2)X(3933)∩X(3)X(76)

Barycentrics    2*b^2*c^2 + S^2 - SB*SC : :
X(69381) = 3 X[3785] + X[32826], X[3785] + 3 X[46951], X[32826] - 9 X[46951], X[5013] - 3 X[8556], 2 X[7815] - 3 X[8556]

X(69381) lies on these lines: {2, 3933}, {3, 76}, {4, 7767}, {5, 69}, {6, 3934}, {8, 24203}, {17, 69106}, {18, 69107}, {20, 52713}, {25, 1235}, {30, 3785}, {32, 8667}, {39, 15271}, {55, 3760}, {56, 3761}, {61, 69186}, {62, 69180}, {75, 9709}, {83, 14614}, {115, 7784}, {140, 3926}, {141, 3767}, {148, 7904}, {187, 17130}, {193, 32968}, {194, 5024}, {230, 7795}, {264, 1598}, {274, 16408}, {298, 11306}, {299, 11305}, {300, 21310}, {301, 21311}, {302, 11312}, {303, 11311}, {305, 16419}, {310, 16059}, {311, 7387}, {315, 381}, {316, 3843}, {325, 1656}, {343, 44141}, {350, 3295}, {376, 32874}, {382, 7750}, {384, 1384}, {385, 7770}, {394, 59197}, {405, 18135}, {474, 34284}, {491, 8976}, {492, 11314}, {498, 69093}, {499, 69094}, {524, 2548}, {538, 5013}, {543, 44519}, {546, 14929}, {547, 32885}, {548, 32888}, {549, 6337}, {550, 32815}, {599, 626}, {620, 44535}, {625, 7896}, {631, 6390}, {632, 32829}, {633, 41041}, {634, 41040}, {637, 36655}, {638, 36656}, {671, 7910}, {754, 65630}, {940, 29456}, {958, 6381}, {999, 1909}, {1003, 7793}, {1007, 3628}, {1184, 8891}, {1232, 3964}, {1236, 7506}, {1376, 20888}, {1447, 4385}, {1506, 7855}, {1593, 44146}, {1597, 54412}, {1657, 14907}, {1799, 9909}, {1992, 8367}, {2549, 15598}, {2896, 7841}, {2996, 32986}, {3053, 3734}, {3089, 32000}, {3090, 37668}, {3091, 32872}, {3096, 7851}, {3314, 7887}, {3522, 32822}, {3523, 32817}, {3524, 32869}, {3525, 32831}, {3526, 7763}, {3533, 32835}, {3545, 32893}, {3547, 41005}, {3589, 5319}, {3619, 8364}, {3620, 14064}, {3627, 64018}, {3630, 31415}, {3631, 43620}, {3673, 7081}, {3763, 7834}, {3770, 5120}, {3788, 37637}, {3815, 7758}, {3830, 7811}, {3850, 32827}, {3851, 7768}, {4187, 45962}, {4396, 54416}, {4400, 16502}, {4413, 32092}, {4441, 5687}, {4513, 29381}, {4713, 14974}, {4968, 26229}, {5017, 18806}, {5020, 40022}, {5023, 7816}, {5025, 7879}, {5041, 41748}, {5054, 32833}, {5055, 7752}, {5056, 10513}, {5067, 32870}, {5070, 7796}, {5073, 7802}, {5077, 7748}, {5079, 53127}, {5188, 14532}, {5198, 44142}, {5206, 68528}, {5210, 69171}, {5224, 25446}, {5254, 7800}, {5286, 8362}, {5304, 16045}, {5306, 69209}, {5309, 6292}, {5346, 7889}, {5475, 7826}, {6130, 45807}, {6144, 7838}, {6179, 21309}, {6376, 9708}, {6392, 15048}, {6643, 41008}, {6656, 16990}, {6677, 41927}, {6683, 7798}, {6704, 47352}, {7373, 64133}, {7484, 8024}, {7516, 9723}, {7517, 15574}, {7530, 44135}, {7571, 40002}, {7603, 7903}, {7610, 7749}, {7697, 18502}, {7734, 19583}, {7735, 7819}, {7737, 63928}, {7738, 8359}, {7745, 14023}, {7746, 7778}, {7747, 63938}, {7753, 63951}, {7755, 7822}, {7759, 40341}, {7760, 11174}, {7761, 44518}, {7762, 15484}, {7764, 31489}, {7766, 14535}, {7769, 32821}, {7772, 31239}, {7774, 32992}, {7775, 7882}, {7779, 16921}, {7781, 15815}, {7783, 51122}, {7785, 44543}, {7789, 11288}, {7791, 47286}, {7797, 16986}, {7799, 15694}, {7806, 33217}, {7807, 17008}, {7809, 19709}, {7813, 31455}, {7814, 61905}, {7817, 7914}, {7818, 39565}, {7824, 20081}, {7825, 7848}, {7828, 7868}, {7829, 47355}, {7830, 44526}, {7833, 8596}, {7836, 17004}, {7842, 18546}, {7844, 7849}, {7850, 61953}, {7853, 69162}, {7860, 15031}, {7861, 7865}, {7862, 7895}, {7864, 19570}, {7869, 7886}, {7870, 8860}, {7871, 15703}, {7872, 32457}, {7873, 69141}, {7892, 63047}, {7893, 16044}, {7897, 32967}, {7900, 33013}, {7905, 11163}, {7917, 61919}, {7918, 31168}, {7921, 33020}, {7922, 14061}, {7929, 14041}, {7931, 33218}, {7934, 32027}, {7938, 33219}, {7939, 32966}, {7941, 33002}, {7943, 10159}, {8149, 50659}, {8150, 39560}, {8177, 40825}, {8355, 50990}, {8357, 43448}, {8360, 21356}, {8365, 63107}, {8366, 8859}, {8369, 63029}, {8370, 20065}, {8716, 14711}, {8781, 34127}, {9301, 39266}, {9464, 40916}, {9756, 30270}, {9939, 11317}, {10303, 32840}, {10449, 68928}, {10516, 67854}, {10983, 32515}, {11057, 15684}, {11108, 16992}, {11168, 34511}, {11184, 69197}, {11284, 26235}, {11321, 16997}, {11343, 26243}, {11414, 30737}, {11477, 67859}, {11482, 39099}, {11539, 32837}, {11623, 50567}, {12017, 12215}, {12100, 32892}, {12108, 32877}, {12251, 13860}, {12322, 36657}, {12323, 36658}, {14001, 37667}, {14069, 37689}, {14269, 22803}, {14530, 57275}, {14881, 44456}, {15482, 22332}, {15534, 63953}, {15655, 33235}, {15693, 59634}, {15704, 67536}, {15712, 32824}, {15713, 32896}, {15717, 32882}, {15720, 32820}, {15905, 28695}, {16058, 18152}, {16197, 40680}, {16198, 63155}, {16239, 32839}, {16367, 31060}, {16412, 20913}, {16418, 18145}, {16509, 32984}, {16619, 67603}, {16644, 69157}, {16645, 69165}, {16781, 69256}, {16857, 18146}, {16895, 63019}, {16898, 63048}, {16918, 16996}, {16922, 63021}, {16999, 33035}, {17001, 17686}, {17002, 17541}, {17170, 28808}, {18906, 32521}, {18907, 32971}, {20080, 32987}, {20088, 66413}, {20094, 33275}, {20208, 28405}, {20477, 64495}, {21443, 64170}, {21531, 62217}, {21615, 54410}, {21843, 59545}, {22241, 26166}, {22246, 55085}, {22329, 33237}, {22331, 69172}, {24172, 32920}, {25264, 31461}, {26182, 26190}, {26184, 26192}, {27269, 33036}, {29433, 37658}, {31401, 58446}, {31477, 69255}, {31479, 69135}, {32458, 38224}, {32480, 63651}, {32810, 42640}, {32811, 42639}, {32825, 32867}, {32841, 55864}, {32871, 61870}, {32875, 61837}, {32876, 61852}, {32879, 61842}, {32880, 61820}, {32883, 48154}, {32884, 55862}, {32890, 61824}, {32897, 61886}, {32898, 61873}, {32957, 37665}, {32975, 62988}, {32978, 55797}, {32983, 63940}, {32985, 59780}, {33197, 60143}, {33201, 46453}, {33202, 55732}, {33215, 52229}, {33269, 53489}, {33482, 69167}, {33483, 69160}, {34883, 68681}, {35930, 61550}, {36165, 47292}, {36662, 37655}, {36952, 40802}, {37439, 40123}, {37638, 62573}, {37647, 55857}, {37924, 67606}, {38739, 62348}, {42153, 69145}, {42156, 69137}, {42368, 44144}, {43183, 44772}, {44147, 68653}, {44149, 64585}, {45141, 56015}, {46236, 61560}, {47037, 69212}, {47285, 68506}, {47617, 66455}, {48913, 61948}, {49112, 55697}, {52422, 54433}, {52710, 64471}, {53033, 62992}, {54103, 61575}, {55610, 60702}, {61849, 62362}, {62197, 69116}, {62198, 69117}, {62697, 68889}, {63093, 66416}, {66415, 69208}, {69110, 69122}, {69111, 69123}, {69114, 69119}, {69115, 69118}, {69142, 69192}, {69144, 69184}, {69148, 69193}, {69150, 69190}, {69164, 69189}, {69170, 69183}

X(69381) = reflection of X(5013) in X(7815)
X(69381) = anticomplement of X(31406)
X(69381) = isotomic conjugate of the isogonal conjugate of X(43650)
X(69381) = crossdifference of every pair of points on line {2491, 3804}
X(69381) = barycentric product X(76)*X(43650)
X(69381) = barycentric quotient X(43650)/X(6)
X(69381) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3933, 69158}, {2, 7754, 9605}, {2, 17129, 7754}, {4, 15589, 7767}, {4, 32834, 64093}, {5, 69, 7776}, {32, 9466, 69139}, {32, 69139, 11286}, {39, 17131, 63933}, {39, 63933, 22253}, {69, 32828, 5}, {76, 183, 3}, {76, 1078, 1975}, {83, 14614, 43136}, {115, 7854, 7784}, {141, 3767, 7866}, {148, 7904, 33234}, {183, 1975, 1078}, {194, 11285, 5024}, {230, 7795, 32954}, {315, 59635, 381}, {325, 32832, 1656}, {385, 7770, 30435}, {385, 31276, 7770}, {546, 14929, 32006}, {599, 13881, 626}, {626, 13881, 11318}, {631, 32830, 6390}, {1007, 32838, 3628}, {1078, 1975, 3}, {1506, 7855, 9766}, {3053, 3734, 68527}, {3096, 14568, 7851}, {3734, 7780, 3053}, {3926, 34229, 140}, {3934, 7751, 6}, {3934, 7805, 7808}, {5013, 8556, 7815}, {5025, 63044, 7879}, {5056, 10513, 32823}, {5254, 7800, 11287}, {5475, 7826, 63932}, {6392, 16043, 15048}, {6683, 63925, 7798}, {7746, 7794, 7778}, {7750, 11185, 382}, {7751, 7808, 7805}, {7754, 17129, 63954}, {7761, 63924, 44518}, {7762, 16924, 15484}, {7762, 63046, 63936}, {7763, 37688, 3526}, {7767, 64093, 4}, {7789, 13468, 69207}, {7789, 69207, 11288}, {7793, 17128, 1003}, {7800, 63955, 5254}, {7805, 7808, 6}, {7806, 46226, 33217}, {7810, 34505, 5077}, {7824, 20081, 31859}, {7836, 17004, 33233}, {7838, 63934, 6144}, {8667, 9466, 11286}, {8667, 69139, 32}, {9605, 63954, 7754}, {10104, 64653, 3}, {14907, 32819, 1657}, {15271, 17131, 22253}, {15271, 63933, 39}, {15482, 32450, 22332}, {15484, 63936, 7762}, {15589, 32834, 4}, {16924, 63046, 7762}, {16992, 18140, 11108}, {18135, 37670, 405}, {22712, 39646, 3}, {32825, 32867, 34803}, {32867, 34803, 55856}, {32870, 63098, 5067}, {33020, 50248, 7921}, {37671, 59635, 315}, {42009, 42060, 599}, {69114, 69119, 69181}, {69115, 69118, 69187}

X(69382) = X(2)X(99)∩X(69)X(381)

Barycentrics    b^2*c^2 + S^2 + 2*SB*SC : :
X(69382) = 3 X[32885] - 2 X[34229], 2 X[44541] - 3 X[47061]

X(69382) lies on these lines: {2, 99}, {3, 32826}, {4, 183}, {5, 1007}, {6, 32983}, {20, 7771}, {30, 32885}, {32, 32979}, {39, 2996}, {69, 381}, {76, 3091}, {140, 32883}, {141, 16041}, {193, 5475}, {194, 31404}, {230, 14033}, {264, 6623}, {274, 6919}, {302, 37171}, {303, 37170}, {315, 3832}, {316, 3839}, {325, 3545}, {339, 44135}, {350, 10590}, {376, 37688}, {385, 33016}, {439, 7749}, {538, 31415}, {546, 14929}, {599, 20112}, {626, 32980}, {631, 32819}, {637, 26469}, {638, 26468}, {877, 66192}, {1003, 62992}, {1078, 3146}, {1285, 22329}, {1352, 67862}, {1384, 66409}, {1656, 6337}, {1799, 7408}, {1909, 10591}, {1975, 3090}, {1992, 3363}, {2548, 6392}, {2896, 32996}, {3054, 33216}, {3055, 8716}, {3096, 33200}, {3314, 33006}, {3329, 5286}, {3522, 32870}, {3528, 52718}, {3543, 14907}, {3544, 32818}, {3618, 14535}, {3619, 33184}, {3620, 9466}, {3628, 32884}, {3642, 16634}, {3643, 16635}, {3763, 33223}, {3767, 7804}, {3788, 32988}, {3815, 34505}, {3843, 7767}, {3850, 7776}, {3851, 3933}, {3855, 7773}, {3886, 64303}, {3934, 32974}, {3964, 8797}, {3972, 37689}, {4232, 11056}, {4357, 66691}, {5007, 5395}, {5013, 32975}, {5055, 6390}, {5056, 7763}, {5066, 32892}, {5067, 32822}, {5068, 7752}, {5071, 32817}, {5072, 32877}, {5177, 18140}, {5187, 34284}, {5254, 32968}, {5274, 64133}, {5304, 14568}, {5309, 51171}, {5355, 63123}, {5468, 54013}, {5485, 11163}, {5709, 55449}, {5921, 7694}, {5939, 14651}, {5971, 62937}, {6031, 7519}, {6033, 11180}, {6376, 31418}, {6462, 31481}, {6528, 10002}, {6656, 63533}, {6776, 12188}, {6871, 18135}, {7330, 55448}, {7378, 40022}, {7486, 7769}, {7603, 34511}, {7697, 10008}, {7735, 8370}, {7736, 44543}, {7737, 37667}, {7738, 32992}, {7739, 32457}, {7746, 32973}, {7748, 32990}, {7753, 51170}, {7761, 47617}, {7766, 16044}, {7774, 33013}, {7777, 33005}, {7778, 32984}, {7782, 10303}, {7783, 32999}, {7785, 32995}, {7787, 50570}, {7788, 41106}, {7789, 32969}, {7793, 14068}, {7795, 32972}, {7797, 33269}, {7799, 61924}, {7800, 32982}, {7801, 39601}, {7802, 17578}, {7809, 10513}, {7811, 32893}, {7812, 63042}, {7815, 33023}, {7816, 32989}, {7828, 33198}, {7831, 33210}, {7832, 33199}, {7836, 32963}, {7845, 11160}, {7850, 61966}, {7851, 16045}, {7857, 33201}, {7868, 33285}, {7891, 32998}, {7904, 33279}, {7917, 32894}, {7933, 60728}, {7938, 33290}, {7945, 33277}, {8352, 42850}, {8361, 39143}, {8369, 63104}, {8542, 63179}, {8667, 53418}, {8722, 60101}, {9752, 35930}, {9818, 68654}, {10519, 15980}, {11008, 63954}, {11057, 62007}, {11159, 16509}, {11168, 66587}, {11286, 43291}, {11317, 63029}, {11361, 17008}, {12150, 63097}, {12243, 35705}, {12812, 32889}, {13175, 37457}, {13881, 14001}, {14023, 39590}, {14041, 16990}, {14063, 31276}, {14064, 63534}, {14482, 63101}, {14853, 39099}, {14928, 38064}, {15022, 32831}, {15048, 63041}, {15271, 32986}, {15513, 55819}, {15717, 32897}, {15760, 40680}, {15810, 41895}, {16043, 44518}, {16589, 33037}, {16921, 31400}, {16986, 33251}, {16989, 66413}, {16999, 33031}, {17004, 33007}, {17128, 32961}, {17130, 52250}, {17131, 20080}, {18840, 33292}, {18907, 63034}, {19570, 63017}, {20023, 64621}, {20065, 33018}, {20081, 33024}, {21356, 37350}, {21843, 35927}, {23053, 27088}, {26235, 31099}, {27269, 33056}, {31652, 55797}, {31693, 63105}, {31694, 63106}, {31859, 62993}, {32022, 38930}, {32820, 61921}, {32821, 32875}, {32823, 32878}, {32833, 61936}, {32835, 61914}, {32869, 61944}, {32872, 50689}, {32887, 35018}, {32896, 61932}, {32898, 62362}, {32977, 59545}, {32978, 63548}, {32981, 69207}, {32985, 37637}, {33202, 63536}, {33215, 44526}, {33283, 46226}, {34883, 66717}, {37174, 59197}, {37182, 46034}, {37347, 40697}, {37375, 45962}, {37671, 41099}, {37984, 52710}, {38907, 68525}, {39482, 52629}, {39785, 68325}, {41748, 63061}, {42163, 69180}, {42166, 69186}, {42920, 69145}, {42921, 69137}, {43459, 50693}, {43619, 63957}, {44541, 47061}, {46236, 61576}, {47332, 67603}, {48913, 61958}, {50280, 63118}, {51372, 62950}, {51898, 53126}, {51899, 53125}, {52229, 63025}, {52674, 63045}, {52942, 66699}, {53489, 63006}, {54033, 60707}, {54814, 60259}, {59634, 61899}, {63155, 66728}, {69162, 69209}

X(69382) = reflection of X(62988) in X(31415)
X(69382) = isotomic conjugate of the isogonal conjugate of X(3066)
X(69382) = barycentric product X(76)*X(3066)
X(69382) = barycentric quotient X(i)/X(j) for these {i,j}: {3066, 6}, {39453, 1285}
X(69382) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 11185, 32815}, {4, 183, 64018}, {4, 32828, 3785}, {4, 59635, 32828}, {69, 381, 32827}, {69, 64093, 46951}, {76, 3091, 32816}, {99, 53127, 2}, {183, 64018, 3785}, {194, 32962, 31404}, {315, 15031, 3832}, {325, 52713, 32836}, {381, 64093, 69}, {1656, 6337, 32839}, {1975, 3090, 32829}, {2548, 63924, 6392}, {2996, 32987, 39}, {3363, 40727, 1992}, {3545, 52713, 325}, {3734, 43620, 2}, {3832, 32834, 315}, {3839, 15589, 316}, {5055, 6390, 34803}, {5068, 32830, 7752}, {5475, 63955, 193}, {5485, 11163, 66458}, {6392, 32991, 2548}, {7603, 34511, 63077}, {7752, 32830, 32825}, {7795, 39565, 32972}, {7800, 69141, 32982}, {11159, 16509, 23055}, {11185, 53127, 99}, {15271, 53419, 32986}, {17128, 32961, 53033}, {18906, 40330, 10008}, {32826, 32838, 3}, {32827, 46951, 69}, {32828, 64018, 183}, {32874, 61954, 7809}, {32893, 61985, 7811}, {44526, 58446, 33215}, {44543, 47286, 7736}, {63534, 69139, 14064}

X(69383) = X(2)X(1975)∩X(20)X(1078)

Barycentrics    2*b^2*c^2 + S^2 + 2*SB*SC : :

X(69383) lies on these lines: {2, 1975}, {4, 7767}, {5, 32818}, {20, 1078}, {30, 32893}, {69, 3832}, {75, 8165}, {76, 3091}, {99, 10303}, {115, 33180}, {140, 32822}, {141, 33200}, {148, 32990}, {183, 3146}, {193, 16044}, {194, 32987}, {230, 33201}, {315, 3839}, {316, 61982}, {325, 5068}, {350, 5261}, {381, 32874}, {384, 37689}, {385, 32979}, {439, 17004}, {538, 31404}, {549, 52718}, {626, 7615}, {631, 32870}, {1007, 15022}, {1235, 6623}, {1656, 32817}, {1909, 5274}, {3090, 32831}, {3314, 32980}, {3522, 32819}, {3523, 7782}, {3543, 3785}, {3545, 3933}, {3552, 46318}, {3620, 14063}, {3628, 32871}, {3734, 33181}, {3760, 10590}, {3761, 10591}, {3767, 33198}, {3851, 32823}, {3854, 7773}, {3855, 7776}, {3926, 5056}, {3934, 33202}, {4208, 18140}, {5032, 9731}, {5067, 6390}, {5071, 69158}, {5177, 18135}, {5286, 7808}, {5304, 7787}, {5305, 40727}, {5395, 7766}, {6381, 31418}, {6392, 7839}, {6919, 34284}, {6995, 26233}, {7378, 39998}, {7396, 40022}, {7486, 7763}, {7620, 7748}, {7736, 63923}, {7745, 63042}, {7746, 33203}, {7750, 17578}, {7752, 32836}, {7754, 32983}, {7769, 32824}, {7771, 62097}, {7774, 32991}, {7779, 32995}, {7788, 61954}, {7795, 33199}, {7799, 61912}, {7800, 18546}, {7802, 50691}, {7805, 63955}, {7809, 61958}, {7811, 62007}, {7836, 32988}, {7871, 61938}, {7881, 32984}, {7882, 66466}, {7900, 11160}, {7906, 33005}, {8357, 55732}, {8367, 51588}, {9740, 20065}, {11168, 44519}, {14031, 63047}, {14035, 37667}, {14069, 43291}, {14907, 49135}, {14929, 61984}, {15031, 32827}, {15048, 32957}, {15692, 32885}, {15717, 37688}, {16990, 32982}, {17008, 32981}, {17129, 33016}, {17130, 43620}, {18840, 33184}, {19690, 55735}, {20081, 32962}, {20094, 33012}, {21581, 44140}, {23047, 32001}, {27269, 33037}, {30962, 57826}, {31276, 32974}, {31467, 52229}, {31859, 32975}, {32000, 37197}, {32006, 50689}, {32820, 34803}, {32829, 46936}, {32833, 61924}, {32837, 61906}, {32841, 61914}, {32867, 61856}, {32886, 50688}, {32892, 61943}, {32898, 61886}, {32968, 47286}, {32996, 63044}, {33002, 63077}, {33014, 55819}, {33018, 63046}, {33025, 44518}, {33205, 62992}, {33206, 51579}, {33230, 55741}, {33269, 42534}, {33290, 60285}, {33798, 57008}, {37671, 61985}, {42920, 69107}, {42921, 69106}, {43136, 66412}, {43459, 62083}, {45833, 56067}, {46453, 68177}, {50693, 67536}, {54889, 60639}, {59548, 63121}, {59552, 63120}, {63091, 63933}

X(69383) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 32834, 15589}, {4, 64093, 32834}, {5, 32830, 63098}, {5, 52713, 32830}, {76, 3091, 37668}, {99, 32838, 10303}, {141, 63533, 33200}, {1078, 11185, 32826}, {1078, 32826, 20}, {1656, 32817, 32835}, {3146, 32872, 183}, {3854, 10513, 7773}, {3854, 32894, 10513}, {3934, 43448, 33202}, {6392, 16924, 37665}, {11185, 32828, 20}, {15022, 32840, 1007}, {17130, 43620, 53033}, {20081, 32962, 62988}, {32815, 32832, 3523}, {32819, 34229, 3522}, {32826, 32828, 1078}, {32829, 53127, 46936}

X(69384) = X(2)X(3933)∩X(4)X(183)

Barycentrics    2*b^2*c^2 + 2*S^2 - SB*SC : :

X(69384) lies on these lines: {2, 3933}, {3, 32822}, {4, 183}, {5, 9748}, {6, 32957}, {20, 32872}, {30, 32893}, {32, 63029}, {69, 576}, {76, 631}, {83, 63034}, {99, 10299}, {140, 32830}, {141, 10542}, {148, 33226}, {193, 32992}, {194, 32978}, {230, 14069}, {315, 3545}, {316, 61964}, {325, 5067}, {376, 1078}, {384, 46453}, {385, 32968}, {491, 3316}, {492, 3317}, {524, 31404}, {549, 32874}, {632, 32835}, {671, 55726}, {1007, 7871}, {1235, 6353}, {1285, 32971}, {1594, 32001}, {1656, 32870}, {1799, 7714}, {1975, 3524}, {2548, 41750}, {2896, 16041}, {2996, 8356}, {3091, 7767}, {3096, 33196}, {3314, 32969}, {3525, 3926}, {3526, 32831}, {3528, 32815}, {3529, 11185}, {3533, 7763}, {3542, 32000}, {3544, 7773}, {3618, 14994}, {3619, 7828}, {3620, 7887}, {3628, 32897}, {3760, 5218}, {3761, 7288}, {3767, 6292}, {3851, 14929}, {3855, 6249}, {3934, 7735}, {4396, 31402}, {4441, 59591}, {5013, 11168}, {5054, 32869}, {5056, 7776}, {5071, 32816}, {5084, 37670}, {5254, 8556}, {5286, 15271}, {5319, 31239}, {5485, 32480}, {6381, 30478}, {6390, 10303}, {6392, 11285}, {6656, 55732}, {6857, 18135}, {7486, 10513}, {7494, 39998}, {7558, 62338}, {7610, 7789}, {7615, 7842}, {7736, 7751}, {7738, 7815}, {7746, 32955}, {7758, 62993}, {7760, 63041}, {7761, 63533}, {7762, 32987}, {7768, 53127}, {7769, 61870}, {7770, 37667}, {7771, 21735}, {7774, 32975}, {7778, 32958}, {7779, 32999}, {7782, 15698}, {7784, 15598}, {7786, 14482}, {7788, 61899}, {7793, 14033}, {7794, 37690}, {7795, 33189}, {7796, 34803}, {7799, 61859}, {7800, 7861}, {7802, 62028}, {7805, 63024}, {7808, 63006}, {7809, 61926}, {7811, 41099}, {7819, 37689}, {7826, 31415}, {7832, 33195}, {7836, 32977}, {7851, 33230}, {7854, 43620}, {7855, 9770}, {7856, 63119}, {7857, 23055}, {7858, 11008}, {7864, 16043}, {7868, 32953}, {7870, 23053}, {7879, 32972}, {7893, 32962}, {7897, 32998}, {7900, 33005}, {7904, 33238}, {7921, 33261}, {7929, 33006}, {7934, 39143}, {7938, 14064}, {7939, 32963}, {7941, 33009}, {7949, 50992}, {8367, 43136}, {8667, 69208}, {8860, 60143}, {9466, 33191}, {11001, 32826}, {11056, 62960}, {11057, 62011}, {13468, 14039}, {13881, 33285}, {14001, 17008}, {14907, 33703}, {14912, 44508}, {15574, 34484}, {15702, 32836}, {15709, 32833}, {16239, 32871}, {16845, 18140}, {16898, 63047}, {16921, 63046}, {16924, 20088}, {16986, 33221}, {16989, 18841}, {16992, 17559}, {16999, 33026}, {17004, 32970}, {17128, 32985}, {17130, 21843}, {17131, 31401}, {17538, 32819}, {17567, 34284}, {20065, 32983}, {20081, 33001}, {20888, 59572}, {26235, 40132}, {27269, 33044}, {31400, 58446}, {32820, 61836}, {32821, 32839}, {32824, 32888}, {32825, 32883}, {32827, 61945}, {32829, 61867}, {32837, 61861}, {32840, 55864}, {32841, 61856}, {32867, 60781}, {32868, 61814}, {32880, 61842}, {32882, 61834}, {32892, 61833}, {32894, 61820}, {32898, 55858}, {32959, 37637}, {32961, 63044}, {32990, 47286}, {33180, 43291}, {35287, 55823}, {40995, 63667}, {43459, 62066}, {43463, 62600}, {43464, 62601}, {51371, 63121}, {52288, 59197}, {59634, 61822}, {62127, 67536}, {63048, 68522}, {63091, 63926}

X(69384) = anticomplement of X(31467)
X(69384) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 32834, 52713}, {3, 52713, 32822}, {20, 32872, 64093}, {69, 3090, 32823}, {69, 32832, 3090}, {76, 631, 32817}, {76, 34229, 631}, {183, 32828, 4}, {183, 59635, 3785}, {325, 32838, 5067}, {3619, 7828, 33194}, {3785, 32828, 59635}, {3785, 59635, 4}, {3926, 37688, 3525}, {3934, 7735, 16045}, {5286, 15271, 32960}, {7795, 62992, 33189}, {7815, 63955, 7738}, {7832, 63104, 33195}, {17008, 31276, 14001}, {32818, 52718, 2}, {32825, 32883, 37647}, {32870, 37668, 1656}, {32885, 37671, 5071}, {32897, 63098, 3628}, {37637, 53033, 32959}, {58446, 63933, 31400}

X(69385) = X(2)X(1975)∩X(5)X(69)

Barycentrics    b^2*c^2 + 2*S^2 + SB*SC : :

X(69385) lies on these lines: {2, 1975}, {3, 32826}, {4, 1078}, {5, 69}, {6, 32987}, {20, 32870}, {32, 32983}, {39, 32975}, {76, 1007}, {99, 3525}, {115, 16043}, {140, 32815}, {141, 32972}, {148, 33001}, {183, 3091}, {194, 32999}, {230, 32971}, {264, 6622}, {311, 50572}, {315, 3545}, {316, 3855}, {325, 5056}, {350, 10588}, {376, 52718}, {381, 3785}, {384, 62992}, {385, 32962}, {439, 44535}, {459, 59528}, {546, 64018}, {547, 32836}, {626, 7617}, {631, 7782}, {639, 5491}, {640, 5490}, {1506, 63955}, {1593, 34883}, {1656, 3926}, {1909, 10589}, {1992, 2548}, {2549, 32978}, {2896, 33006}, {3053, 32979}, {3054, 32989}, {3096, 33285}, {3314, 32963}, {3329, 33261}, {3523, 32819}, {3526, 32883}, {3529, 7771}, {3533, 32822}, {3618, 3767}, {3619, 3934}, {3620, 52250}, {3628, 32829}, {3734, 32970}, {3788, 32976}, {3815, 6392}, {3832, 7750}, {3851, 7767}, {3933, 5055}, {5067, 7763}, {5068, 7773}, {5070, 6390}, {5071, 7752}, {5079, 32886}, {5154, 45962}, {5187, 37670}, {5207, 67872}, {5210, 55819}, {5286, 32992}, {5305, 16509}, {5319, 63011}, {5461, 7914}, {5485, 62881}, {5866, 66607}, {6248, 58883}, {6292, 18362}, {6816, 65518}, {6856, 18140}, {6919, 16992}, {6931, 34284}, {6933, 18135}, {6997, 26233}, {7393, 36948}, {7486, 32830}, {7603, 7758}, {7612, 13335}, {7615, 7748}, {7735, 7787}, {7736, 7839}, {7745, 32991}, {7746, 14001}, {7749, 23053}, {7751, 11008}, {7754, 31404}, {7769, 32817}, {7770, 46318}, {7774, 33002}, {7775, 50992}, {7777, 33009}, {7778, 32988}, {7779, 33010}, {7784, 32980}, {7785, 33005}, {7788, 32893}, {7790, 32960}, {7791, 63533}, {7793, 33016}, {7795, 32969}, {7799, 61895}, {7800, 16041}, {7803, 32957}, {7806, 33269}, {7809, 61932}, {7811, 41106}, {7815, 32986}, {7816, 33216}, {7820, 33222}, {7823, 32995}, {7828, 16045}, {7832, 32955}, {7834, 63120}, {7835, 32959}, {7836, 32998}, {7838, 31417}, {7844, 33221}, {7854, 39601}, {7857, 14039}, {7866, 63121}, {7868, 33199}, {7871, 61915}, {7882, 8176}, {7904, 32996}, {7925, 33270}, {7928, 33290}, {7931, 33277}, {8367, 63109}, {8370, 23055}, {8889, 54412}, {9166, 33230}, {9722, 39127}, {9723, 19418}, {9753, 60212}, {10449, 36684}, {10576, 32806}, {10577, 32805}, {11056, 40132}, {11057, 61980}, {11285, 43448}, {11313, 32813}, {11314, 32812}, {12215, 43291}, {12811, 14929}, {13468, 65630}, {14033, 69207}, {14035, 17004}, {14061, 32951}, {14651, 53765}, {14880, 25406}, {15022, 32872}, {15271, 32974}, {15597, 35287}, {15699, 32837}, {16044, 17008}, {16589, 33041}, {16986, 33283}, {16989, 33020}, {16990, 32966}, {16999, 33056}, {17006, 32964}, {17538, 43459}, {18584, 20080}, {18840, 60095}, {20065, 33013}, {20081, 63083}, {21843, 33239}, {27269, 33052}, {31276, 32961}, {31400, 47286}, {31401, 63924}, {31467, 40727}, {31489, 63923}, {32820, 32835}, {32823, 61921}, {32824, 55856}, {32825, 32868}, {32831, 37647}, {32833, 61899}, {32869, 61906}, {32874, 61912}, {32877, 61903}, {32878, 61907}, {32884, 55857}, {32888, 61911}, {32892, 61908}, {32896, 61898}, {32967, 37690}, {32973, 37637}, {32977, 69206}, {32990, 44518}, {33011, 63044}, {33023, 53419}, {33196, 55741}, {33238, 69141}, {33248, 46226}, {33249, 53033}, {33292, 55732}, {34208, 56067}, {35438, 60619}, {37532, 55449}, {37671, 61936}, {37832, 63105}, {37835, 63106}, {41009, 44135}, {41927, 45201}, {42598, 69186}, {42599, 69180}, {42910, 69165}, {42911, 69157}, {44144, 62576}, {44543, 63034}, {47743, 64133}, {51538, 60702}, {52585, 57146}, {54753, 60128}, {57008, 59197}, {59741, 62642}, {61914, 63098}, {62988, 63933}

X(69385) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2996, 5013}, {4, 32832, 34229}, {5, 32828, 69}, {20, 32870, 37688}, {76, 3090, 1007}, {76, 53127, 3090}, {183, 3091, 32006}, {194, 32999, 62993}, {1656, 3926, 34803}, {1656, 64093, 3926}, {3619, 39143, 14064}, {3767, 32968, 3618}, {3851, 7767, 32827}, {3934, 14064, 3619}, {3934, 43620, 14064}, {5056, 32834, 325}, {5067, 52713, 7763}, {5068, 15589, 7773}, {5070, 6390, 32839}, {5286, 32992, 63041}, {7746, 14001, 63104}, {7800, 39565, 16041}, {14064, 43620, 39143}, {14907, 15031, 4}, {15022, 32872, 37668}, {15271, 63534, 32974}, {32815, 32867, 140}, {32817, 61886, 7769}, {32831, 46936, 37647}, {32893, 61924, 7788}, {32991, 37667, 7745}, {44518, 58446, 32990}

X(69386) = X(2)X(2418)∩X(4)X(183)

Barycentrics    2*b^2*c^2 + 2*S^2 + SB*SC : :

X(69386) lies on these lines: {2, 2418}, {4, 183}, {5, 32823}, {69, 1568}, {76, 1007}, {99, 631}, {115, 33190}, {140, 32870}, {141, 33285}, {148, 33215}, {193, 44543}, {194, 32975}, {230, 14039}, {315, 3855}, {316, 41099}, {325, 5071}, {350, 8164}, {376, 7771}, {381, 14929}, {385, 32983}, {403, 32000}, {538, 62993}, {547, 32869}, {1078, 3529}, {1235, 6622}, {1285, 8370}, {1656, 32830}, {1909, 47743}, {1916, 14064}, {1975, 3525}, {2996, 11285}, {3091, 32872}, {3314, 32984}, {3329, 32968}, {3363, 9740}, {3524, 32815}, {3526, 32897}, {3528, 32819}, {3533, 6337}, {3544, 32816}, {3618, 14568}, {3619, 33196}, {3620, 33228}, {3628, 32831}, {3734, 33191}, {3760, 10588}, {3761, 10589}, {3767, 7889}, {3832, 7767}, {3926, 5067}, {3933, 5056}, {3934, 32956}, {5055, 32874}, {5068, 7776}, {5070, 32835}, {5254, 32960}, {5286, 32957}, {5304, 66415}, {5305, 14535}, {6392, 32992}, {6722, 7795}, {6856, 18135}, {7486, 69158}, {7603, 9770}, {7612, 35925}, {7615, 7761}, {7620, 11168}, {7735, 7804}, {7736, 7798}, {7737, 63029}, {7738, 15482}, {7746, 33189}, {7752, 61921}, {7754, 32987}, {7762, 32991}, {7763, 61886}, {7766, 16924}, {7769, 61881}, {7773, 61945}, {7779, 33005}, {7782, 61807}, {7788, 61932}, {7789, 32959}, {7799, 61889}, {7800, 63533}, {7802, 62021}, {7811, 61980}, {7814, 64809}, {7836, 32976}, {7839, 33261}, {7850, 61951}, {7879, 32980}, {7881, 32988}, {7884, 63120}, {7890, 31417}, {7893, 32995}, {7906, 33009}, {7919, 63121}, {7934, 21356}, {7947, 33270}, {8357, 63536}, {8556, 53419}, {8598, 55823}, {8797, 44149}, {8889, 40022}, {9166, 50567}, {9466, 43620}, {9748, 60259}, {10484, 60637}, {10513, 61936}, {11056, 52290}, {11057, 62009}, {11286, 37689}, {13881, 32951}, {14033, 17008}, {14069, 69139}, {14482, 63041}, {14907, 15682}, {15271, 43448}, {15484, 63042}, {15574, 52294}, {16041, 16990}, {16051, 26235}, {16986, 33223}, {17004, 32985}, {17128, 32970}, {17129, 32962}, {17130, 31274}, {17131, 31415}, {17538, 32826}, {18584, 50771}, {18841, 68522}, {18842, 63065}, {18906, 22677}, {20081, 32999}, {21309, 66412}, {27269, 33041}, {31400, 63923}, {31404, 63933}, {32006, 61964}, {32820, 32839}, {32824, 32883}, {32825, 32888}, {32827, 37671}, {32829, 60781}, {32833, 34803}, {32836, 61899}, {32837, 61888}, {32840, 46936}, {32841, 46935}, {32867, 61867}, {32871, 55857}, {32892, 61904}, {32894, 61914}, {32898, 48154}, {32958, 53033}, {32974, 55732}, {33004, 35369}, {33006, 63044}, {33013, 63046}, {33197, 63104}, {33292, 63534}, {34505, 58446}, {37353, 41916}, {39663, 40330}, {40824, 60126}, {42494, 69137}, {42495, 69145}, {42910, 69121}, {42911, 69120}, {43459, 62084}, {45018, 51176}, {51171, 66416}, {51372, 62959}, {52283, 59197}, {55164, 55726}, {55819, 68516}, {59634, 61861}, {63048, 66413}, {63091, 63954}

X(69386) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 52713, 32817}, {2, 64093, 52713}, {76, 3090, 32818}, {76, 53127, 1007}, {631, 32832, 52718}, {1007, 53127, 3090}, {1975, 32838, 3525}, {3734, 62992, 33191}, {7771, 11185, 67536}, {7771, 67536, 376}, {8370, 37667, 1285}, {11185, 34229, 376}, {14033, 17008, 46453}, {14064, 31276, 18840}, {32815, 32885, 37688}, {32815, 37688, 3524}, {32822, 52718, 631}, {32828, 59635, 4}, {34229, 67536, 7771}

X(69387) = X(2)X(99)∩X(5)X(76)

Barycentrics    b^2*c^2 + 2*S^2 + 2*SB*SC : :

X(69387) lies on these lines: {2, 99}, {4, 1078}, {5, 76}, {6, 14568}, {11, 64133}, {20, 32838}, {30, 7771}, {32, 16044}, {39, 16921}, {69, 1568}, {75, 3814}, {83, 3767}, {98, 37348}, {114, 51520}, {140, 7782}, {141, 7934}, {182, 14651}, {183, 316}, {187, 11361}, {192, 31476}, {194, 1506}, {230, 3972}, {264, 403}, {274, 4193}, {298, 50858}, {299, 50855}, {302, 3643}, {303, 3642}, {305, 37990}, {315, 3091}, {339, 10254}, {350, 7951}, {384, 7746}, {385, 5475}, {427, 58782}, {491, 41484}, {492, 41483}, {523, 40826}, {524, 7926}, {538, 7603}, {546, 7750}, {547, 6390}, {598, 3363}, {621, 5613}, {622, 5617}, {623, 25157}, {624, 25167}, {625, 3314}, {626, 31276}, {668, 11680}, {754, 43457}, {868, 40877}, {877, 14967}, {1003, 37637}, {1007, 5071}, {1235, 16868}, {1236, 62947}, {1346, 15165}, {1347, 15164}, {1352, 10753}, {1594, 54412}, {1648, 34512}, {1656, 1975}, {1799, 7394}, {1909, 7741}, {1995, 11056}, {2070, 21395}, {2476, 18140}, {2548, 7760}, {2679, 6787}, {2782, 43461}, {2896, 7825}, {2980, 41296}, {2996, 31400}, {3054, 35297}, {3090, 7763}, {3096, 3934}, {3146, 32870}, {3329, 5309}, {3522, 32897}, {3523, 32826}, {3524, 58831}, {3529, 52718}, {3552, 7749}, {3589, 7884}, {3614, 69135}, {3619, 33285}, {3763, 33219}, {3785, 3832}, {3788, 17128}, {3815, 7757}, {3822, 30963}, {3825, 31997}, {3839, 32885}, {3845, 11057}, {3850, 7767}, {3851, 7768}, {3855, 6249}, {3926, 5056}, {5055, 7799}, {5066, 7850}, {5067, 6337}, {5068, 7917}, {5072, 7776}, {5079, 69158}, {5133, 40022}, {5141, 18135}, {5152, 38224}, {5154, 34284}, {5167, 61727}, {5169, 26235}, {5206, 6658}, {5210, 66387}, {5224, 17196}, {5254, 7786}, {5286, 32987}, {5305, 7878}, {5306, 53489}, {5355, 62994}, {5476, 39099}, {5485, 62895}, {5569, 9855}, {5651, 10411}, {5939, 49102}, {5976, 61576}, {6036, 35925}, {6071, 67630}, {6179, 7745}, {6248, 37446}, {6292, 7933}, {6376, 25639}, {6392, 31404}, {6655, 7815}, {6656, 63534}, {6680, 68525}, {6683, 7864}, {6781, 34506}, {6997, 33651}, {7173, 69254}, {7426, 64982}, {7486, 32829}, {7533, 26233}, {7577, 44146}, {7606, 51798}, {7610, 11317}, {7735, 12150}, {7737, 17008}, {7738, 32975}, {7747, 7793}, {7748, 7824}, {7751, 7785}, {7753, 7766}, {7754, 7858}, {7755, 7787}, {7756, 33004}, {7759, 17129}, {7761, 14041}, {7762, 15480}, {7764, 20081}, {7770, 7828}, {7772, 50570}, {7774, 31415}, {7775, 7779}, {7780, 7823}, {7783, 16922}, {7788, 19709}, {7789, 7940}, {7792, 43291}, {7794, 7912}, {7795, 7899}, {7797, 7808}, {7798, 19570}, {7800, 7911}, {7801, 7925}, {7803, 32968}, {7804, 7806}, {7805, 7921}, {7807, 44381}, {7810, 7898}, {7813, 32994}, {7816, 7907}, {7817, 7875}, {7818, 63044}, {7819, 7942}, {7822, 7901}, {7826, 7900}, {7827, 11174}, {7830, 33019}, {7831, 7841}, {7832, 7887}, {7833, 47617}, {7834, 16987}, {7836, 7862}, {7840, 8176}, {7842, 7904}, {7843, 7893}, {7847, 11285}, {7848, 31173}, {7851, 7859}, {7852, 16895}, {7854, 7885}, {7855, 7941}, {7861, 7876}, {7867, 46226}, {7868, 11318}, {7870, 44377}, {7872, 33021}, {7880, 31275}, {7883, 16990}, {7886, 7892}, {7889, 7932}, {7905, 63933}, {7909, 32963}, {7910, 33229}, {7915, 14065}, {7918, 8362}, {7928, 14045}, {7930, 8361}, {7937, 33184}, {7944, 14064}, {8182, 52942}, {8352, 11168}, {8356, 53419}, {8542, 31998}, {8588, 33265}, {8598, 15597}, {8860, 11159}, {9516, 40429}, {9607, 32477}, {9734, 13172}, {9744, 38664}, {9766, 18584}, {9873, 40279}, {9880, 57633}, {10024, 41009}, {10159, 33283}, {10168, 14928}, {10170, 51439}, {10175, 69038}, {10303, 32883}, {10449, 36677}, {11054, 11163}, {11059, 62299}, {11171, 67268}, {11179, 45018}, {11261, 50640}, {11284, 37803}, {11632, 35705}, {11681, 17143}, {12156, 63034}, {12215, 38317}, {12525, 61742}, {12833, 67215}, {13468, 53418}, {13586, 17006}, {13596, 34883}, {13862, 39266}, {14033, 62992}, {14035, 69207}, {14039, 63104}, {14295, 39509}, {14356, 35139}, {14485, 60212}, {14501, 53125}, {14502, 53126}, {14614, 15484}, {14639, 37242}, {14712, 62203}, {14929, 38071}, {14999, 33509}, {15022, 32830}, {15067, 51440}, {15513, 33257}, {15699, 59634}, {15760, 62698}, {15980, 22712}, {15993, 22486}, {16041, 31168}, {16043, 63533}, {16589, 33045}, {16992, 17556}, {17530, 18146}, {18906, 23514}, {20065, 32995}, {21843, 33007}, {22687, 46053}, {22689, 46054}, {22846, 51265}, {22891, 51272}, {23055, 46453}, {24250, 24282}, {24387, 24524}, {25303, 37720}, {26316, 61560}, {27269, 33060}, {29438, 62322}, {30736, 67131}, {30998, 69174}, {31268, 32956}, {31401, 32999}, {31488, 41838}, {31489, 31859}, {32451, 53484}, {32456, 33274}, {32525, 52625}, {32818, 61921}, {32820, 35018}, {32821, 61919}, {32822, 61886}, {32824, 32835}, {32831, 61914}, {32836, 61924}, {32837, 61912}, {32839, 46936}, {32869, 61927}, {32874, 61930}, {32893, 61944}, {32951, 39143}, {32988, 53033}, {32991, 69208}, {33008, 43619}, {33015, 37512}, {33196, 63121}, {33235, 44535}, {33246, 58448}, {33259, 69171}, {33260, 65633}, {33269, 69209}, {33273, 63957}, {34508, 44362}, {34509, 44361}, {34511, 63083}, {34624, 61102}, {35287, 51237}, {35520, 44135}, {35930, 38227}, {35954, 44401}, {35955, 66587}, {36165, 67616}, {36183, 46999}, {37353, 39998}, {37375, 37670}, {37439, 57518}, {37668, 46951}, {37988, 60707}, {38526, 67622}, {38735, 58445}, {39113, 64495}, {39563, 66414}, {39785, 66511}, {40332, 51848}, {41133, 59780}, {42488, 62600}, {42489, 62601}, {42580, 69165}, {42581, 69157}, {43453, 58851}, {44128, 54105}, {44142, 44958}, {44173, 59745}, {44185, 68630}, {44961, 67606}, {45198, 69288}, {46893, 66405}, {48901, 60702}, {50149, 52551}, {50280, 63942}, {51128, 66339}, {52247, 59197}, {54013, 57216}, {54507, 66428}, {54753, 62892}, {54826, 60217}, {54869, 60248}, {55795, 60219}, {57275, 61747}, {57583, 67614}, {61738, 67619}, {61920, 64809}, {62274, 62576}, {62983, 69202}, {62984, 69199}, {63101, 63633}, {63543, 66417}

X(69387) = reflection of X(i) in X(j) for these {i,j}: {7771, 37688}, {7777, 7603}
X(69387) = isotomic conjugate of the isogonal conjugate of X(5640)
X(69387) = barycentric product X(i)*X(j) for these {i,j}: {76, 5640}, {305, 33885}
X(69387) = barycentric quotient X(i)/X(j) for these {i,j}: {5640, 6}, {33885, 25}, {44204, 1637}
X(69387) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 115, 7790}, {2, 148, 574}, {2, 671, 52691}, {2, 3734, 7835}, {2, 7615, 671}, {2, 8591, 7622}, {2, 11185, 99}, {2, 43620, 14061}, {4, 1078, 7802}, {4, 32832, 1078}, {4, 34229, 14907}, {5, 76, 7752}, {5, 59635, 76}, {5, 64093, 325}, {76, 7752, 7796}, {76, 7814, 3933}, {83, 3767, 7856}, {140, 32819, 7782}, {141, 33228, 7934}, {183, 316, 7811}, {183, 381, 316}, {194, 33002, 1506}, {230, 8370, 3972}, {325, 59635, 64093}, {325, 64093, 76}, {384, 7746, 7857}, {385, 5475, 7812}, {385, 33013, 5475}, {547, 6390, 37647}, {574, 18546, 148}, {625, 9466, 3314}, {1007, 52713, 32833}, {1078, 15031, 4}, {1506, 63924, 194}, {1656, 1975, 7769}, {2896, 32993, 7825}, {3090, 32817, 34803}, {3091, 15589, 32827}, {3091, 32828, 315}, {3363, 16509, 22329}, {3363, 22329, 598}, {3767, 16924, 83}, {3815, 47286, 7757}, {3934, 5025, 3096}, {3934, 7853, 16986}, {3934, 39565, 5025}, {5025, 16986, 7853}, {5066, 37671, 48913}, {5068, 32834, 32816}, {5071, 52713, 1007}, {5254, 32992, 7786}, {6722, 7820, 2}, {7610, 11317, 51224}, {7615, 9890, 18546}, {7751, 7785, 7877}, {7761, 18424, 14041}, {7770, 7828, 7846}, {7770, 13881, 7828}, {7774, 33005, 31415}, {7775, 17131, 7779}, {7780, 39590, 7823}, {7783, 16922, 31455}, {7789, 33249, 7940}, {7792, 66415, 60855}, {7793, 33018, 7747}, {7795, 32961, 7899}, {7797, 33020, 7808}, {7800, 14063, 7911}, {7806, 66413, 7804}, {7808, 69162, 7797}, {7815, 69141, 6655}, {7841, 15271, 7831}, {7853, 16986, 3096}, {7861, 31239, 7876}, {7862, 17130, 7836}, {7887, 69139, 7832}, {7904, 14062, 7842}, {8352, 11168, 55164}, {8860, 11159, 26613}, {9466, 39601, 625}, {11054, 11163, 66703}, {11163, 40727, 11054}, {11168, 20112, 8352}, {11185, 53127, 2}, {11285, 44518, 7847}, {11361, 17004, 187}, {14907, 32832, 34229}, {14907, 34229, 1078}, {15031, 32832, 7802}, {15589, 32827, 315}, {17008, 33016, 7737}, {17128, 32967, 3788}, {19570, 63018, 7798}, {25157, 41098, 623}, {25167, 41094, 624}, {31276, 32966, 626}, {31415, 63955, 7774}, {31489, 34505, 31859}, {32817, 34803, 7763}, {32826, 32867, 3523}, {32827, 32828, 15589}, {43291, 66415, 7792}, {53419, 58446, 8356}

X(69388) = X(514)X(17215)∩X(693)X(25022)

Barycentrics    (b - c)*(b^2*c^2 + SB*SC) : :

X(69288) lies on these lines: {514, 17215}, {693, 25022}, {2517, 48268}, {3261, 4025}, {4468, 47134}, {20906, 47123}, {20907, 21185}, {21102, 30805}, {26114, 45745}, {47127, 47789}, {57054, 64917}

X(69388) = X(692)-isoconjugate of X(17040)
X(69388) = X(i)-Dao conjugate of X(j) for these (i,j): {1086, 17040}, {40618, 56339}
X(69388) = barycentric product X(i)*X(j) for these {i,j}: {3261, 5020}, {4025, 43981}
X(69388) = barycentric quotient X(i)/X(j) for these {i,j}: {514, 17040}, {3261, 59756}, {4025, 56339}, {5020, 101}, {43981, 1897}
X(69388) = {X(3261),X(7649)}-harmonic conjugate of X(4025)

X(69389) = X(2)X(68799)∩X(86)X(514)

Barycentrics    (b - c)*(b^2*c^2 + 2*SB*SC) : :

X(69389) lies on these lines: {2, 68799}, {75, 21185}, {86, 514}, {3261, 4025}, {3667, 30184}, {4389, 30181}, {17160, 54261}, {17215, 21102}, {20907, 21179}, {20954, 21186}, {27191, 30188}, {55135, 68778}

X(69389) = X(i)-isoconjugate of X(j) for these (i,j): {213, 65324}, {228, 30247}, {692, 5486}, {2200, 37217}, {21839, 35188}
X(69389) = X(i)-Dao conjugate of X(j) for these (i,j): {1086, 5486}, {5512, 42}, {6626, 65324}
X(69389) = barycentric product X(i)*X(j) for these {i,j}: {286, 14209}, {310, 68778}, {514, 11185}, {1995, 3261}, {7649, 66767}, {30209, 44129}, {41614, 46107}
X(69389) = barycentric quotient X(i)/X(j) for these {i,j}: {27, 30247}, {86, 65324}, {286, 37217}, {514, 5486}, {1995, 101}, {4786, 13608}, {11185, 190}, {14209, 72}, {19136, 32739}, {21109, 57466}, {29959, 46148}, {30209, 71}, {41614, 1331}, {46107, 60266}, {55135, 4062}, {62626, 60317}, {66767, 4561}, {68778, 42}
X(69389) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3261, 7649, 21178}, {3261, 21205, 4025}, {4025, 7649, 21205}, {4025, 21205, 21178}

X(69390) = X(6)X(45659)∩X(86)X(1459)

Barycentrics    (b - c)*(-2*b^2*c^2 + S^2 - SB*SC) : :

X(69390) lies on these lines: {6, 45659}, {86, 1459}, {522, 25590}, {657, 17259}, {663, 4411}, {2345, 54264}, {3739, 45755}, {3875, 20907}, {4025, 11125}, {4374, 48340}, {4408, 43924}, {5232, 20316}, {8062, 24002}, {8643, 53276}, {17218, 48342}, {20293, 62999}, {21102, 30805}, {26277, 53368}, {46399, 53665}

X(69390) = barycentric product X(i)*X(j) for these {i,j}: {86, 31174}, {3261, 5651}
X(69390) = barycentric quotient X(i)/X(j) for these {i,j}: {5651, 101}, {31174, 10}
X(69390) = {X(3261),X(17215)}-harmonic conjugate of X(1459)

X(69391) = X(86)X(1459)∩X(239)X(514)

Barycentrics    (b - c)*(b^2*c^2 + S^2 - SB*SC) : :

X(69391) = 3 X[47828] - 2 X[60577]

X(69391) lies on these lines: {86, 1459}, {239, 514}, {522, 3875}, {657, 17349}, {663, 3766}, {693, 3907}, {786, 2484}, {1919, 21113}, {2400, 60873}, {2605, 4408}, {2789, 44435}, {3004, 69350}, {3239, 25943}, {3250, 4508}, {3288, 23878}, {3810, 4467}, {4106, 4879}, {4373, 20091}, {4374, 43924}, {4378, 58862}, {4393, 68145}, {4411, 53314}, {4474, 62415}, {4724, 63222}, {5232, 20293}, {5936, 48243}, {6586, 17259}, {7199, 48342}, {8643, 26277}, {9002, 53276}, {15668, 45659}, {17316, 54264}, {17393, 20954}, {17418, 20906}, {17896, 29051}, {20295, 28470}, {20907, 21173}, {20949, 46385}, {21053, 30764}, {21102, 21178}, {23794, 42312}, {24533, 43067}, {25258, 25924}, {28565, 53333}, {29066, 36038}, {29116, 47657}, {29186, 55186}, {30804, 47729}, {46402, 62999}, {47828, 60577}, {48109, 68888}, {50556, 69373}, {53335, 69365}, {64857, 69047}

X(69391) = reflection of X(i) in X(j) for these {i,j}: {649, 4107}, {693, 43041}
X(69391) = X(i)-isoconjugate of X(j) for these (i,j): {37, 26714}, {42, 65252}, {100, 263}, {101, 2186}, {190, 3402}, {213, 65271}, {228, 65349}, {262, 692}, {668, 46319}, {906, 68572}, {1783, 43718}, {1824, 65310}, {4553, 42288}, {4559, 66936}, {4567, 52631}, {5360, 6037}, {52926, 56254}
X(69391) = X(i)-Dao conjugate of X(j) for these (i,j): {1015, 2186}, {1086, 262}, {4988, 66291}, {5190, 68572}, {6626, 65271}, {8054, 263}, {38997, 42}, {39006, 43718}, {40589, 26714}, {40592, 65252}, {40618, 42313}, {40620, 60679}, {40627, 52631}, {51580, 190}, {55051, 21035}, {55053, 3402}, {55067, 66936}
X(69391) = crosspoint of X(664) and X(870)
X(69391) = crosssum of X(663) and X(869)
X(69391) = crossdifference of every pair of points on line {42, 263}
X(69391) = barycentric product X(i)*X(j) for these {i,j}: {86, 23878}, {182, 3261}, {183, 514}, {310, 3288}, {458, 4025}, {513, 3403}, {649, 20023}, {693, 52134}, {1019, 42711}, {1459, 44144}, {4107, 8842}, {4131, 51315}, {4391, 60716}, {4610, 66459}, {6784, 52612}, {7192, 60737}, {7199, 60723}, {10566, 14994}, {15413, 60685}, {17206, 68781}, {17209, 63746}, {30805, 33971}, {51370, 67172}, {52619, 60726}
X(69391) = barycentric quotient X(i)/X(j) for these {i,j}: {27, 65349}, {58, 26714}, {81, 65252}, {86, 65271}, {182, 101}, {183, 190}, {458, 1897}, {513, 2186}, {514, 262}, {649, 263}, {667, 3402}, {1459, 43718}, {1790, 65310}, {1919, 46319}, {3120, 66291}, {3122, 52631}, {3261, 327}, {3288, 42}, {3403, 668}, {3737, 66936}, {4025, 42313}, {4091, 54032}, {6784, 4079}, {7192, 60679}, {7649, 68572}, {10311, 8750}, {10566, 42299}, {14096, 46148}, {14994, 4568}, {17209, 63741}, {20023, 1978}, {21102, 66919}, {23878, 10}, {30805, 59257}, {34396, 32739}, {42711, 4033}, {52134, 100}, {53521, 51543}, {60685, 1783}, {60716, 651}, {60723, 1018}, {60726, 4557}, {60737, 3952}, {66459, 4024}, {68781, 1826}
X(69391) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1459, 3261, 17215}, {17496, 53357, 4025}

X(69392) = X(86)X(1459)∩X(522)X(17151)

Barycentrics    (b - c)*(2*b^2*c^2 + S^2 - SB*SC) : :

X(69392) lies on these lines: {86, 1459}, {522, 17151}, {649, 21113}, {663, 4408}, {2484, 4508}, {2517, 43041}, {3766, 48340}, {4025, 21102}, {4411, 43924}, {4474, 48084}, {21103, 30805}, {28623, 53357}, {30804, 68888}, {57167, 64857}, {64856, 66987}

X(69392) = barycentric product X(3261)*X(43650)
X(69392) = barycentric quotient X(43650)/X(101)

X(69393) = X(1459)X(4025)∩X(1919)X(4750)

Barycentrics    (b - c)*(-2*b^2*c^2 + S^2 + SB*SC) : :

X(69393) lies on these lines: {1459, 4025}, {1919, 4750}, {3261, 21102}, {4064, 15413}, {4408, 21132}, {4411, 66287}, {11125, 17215}, {14429, 57054}, {16892, 21123}, {21202, 57091}, {23748, 69075}, {48084, 48278}, {64856, 66967}

X(69393) = isotomic conjugate of the polar conjugate of X(16892)
X(69393) = X(1790)-Ceva conjugate of X(17216)
X(69393) = X(i)-isoconjugate of X(j) for these (i,j): {19, 4628}, {82, 8750}, {112, 18098}, {213, 42396}, {251, 1783}, {668, 61383}, {692, 32085}, {827, 1824}, {1826, 34072}, {1897, 46289}, {2333, 4599}, {4630, 41013}, {5379, 18105}, {6335, 46288}, {18082, 32676}, {32674, 56245}, {36081, 57654}, {56186, 61206}
X(69393) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 4628}, {39, 1897}, {141, 8750}, {1086, 32085}, {3124, 2333}, {6626, 42396}, {15449, 1826}, {15526, 18082}, {26932, 82}, {34467, 46289}, {34591, 18098}, {35072, 56245}, {39006, 251}, {40585, 1783}, {40618, 83}, {55043, 1824}
X(69393) = barycentric product X(i)*X(j) for these {i,j}: {38, 15413}, {63, 48084}, {69, 16892}, {86, 2525}, {141, 4025}, {304, 2530}, {305, 21123}, {348, 48278}, {427, 30805}, {514, 3933}, {525, 16887}, {656, 16703}, {826, 17206}, {905, 1930}, {1235, 4091}, {1444, 62418}, {1459, 8024}, {1565, 4568}, {1790, 23285}, {3261, 3917}, {3265, 17171}, {3267, 17187}, {3665, 6332}, {3926, 21108}, {4020, 40495}, {4064, 61407}, {4131, 20883}, {4175, 10566}, {4466, 4576}, {14208, 16696}, {15419, 15523}, {16747, 24018}, {17216, 41676}, {18210, 55239}, {21126, 57852}, {40364, 50521}
X(69393) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 4628}, {38, 1783}, {39, 8750}, {86, 42396}, {141, 1897}, {514, 32085}, {521, 56245}, {525, 18082}, {656, 18098}, {826, 1826}, {905, 82}, {1401, 32674}, {1437, 34072}, {1444, 4599}, {1459, 251}, {1565, 10566}, {1790, 827}, {1919, 61383}, {1930, 6335}, {2525, 10}, {2528, 21016}, {2530, 19}, {3005, 2333}, {3261, 46104}, {3267, 56251}, {3665, 653}, {3703, 65160}, {3917, 101}, {3933, 190}, {3942, 18108}, {4020, 692}, {4025, 83}, {4064, 61405}, {4091, 1176}, {4131, 34055}, {4175, 4568}, {4466, 58784}, {4568, 15742}, {8061, 1824}, {14208, 56186}, {15413, 3112}, {15419, 52394}, {16696, 162}, {16703, 811}, {16747, 823}, {16887, 648}, {16892, 4}, {17094, 18097}, {17171, 107}, {17187, 112}, {17206, 4577}, {17216, 4580}, {18210, 55240}, {20775, 32739}, {21108, 393}, {21109, 21459}, {21123, 25}, {21126, 428}, {21134, 34294}, {22383, 46289}, {30805, 1799}, {33299, 56183}, {46153, 7115}, {46387, 57654}, {48084, 92}, {48278, 281}, {50521, 1973}, {58335, 7079}, {62418, 41013}
X(69393) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4025, 30805, 1459}, {17215, 21178, 11125}

X(69394) = X(86)X(11125)∩X(239)X(514)

Barycentrics    (b - c)*(-(b^2*c^2) + S^2 + SB*SC) : :

X(69394) = 2 X[4107] - 3 X[4750], 3 X[4453] - 2 X[43041]

X(69394) lies on these lines: {86, 11125}, {239, 514}, {693, 3810}, {918, 23781}, {1111, 3120}, {1459, 21178}, {1769, 69040}, {2400, 6650}, {2785, 53335}, {2786, 41842}, {2789, 53333}, {2799, 3569}, {3261, 21102}, {3265, 25008}, {3766, 21132}, {3907, 4467}, {3945, 53352}, {4024, 9237}, {4064, 18160}, {4088, 30639}, {4374, 66287}, {4453, 24126}, {5224, 14429}, {5232, 53342}, {7199, 23752}, {7649, 17215}, {10436, 21180}, {15419, 21187}, {15668, 47234}, {17896, 23877}, {20295, 28487}, {20316, 57054}, {20508, 23756}, {20906, 21119}, {21053, 23596}, {21108, 46107}, {21110, 21123}, {21131, 21136}, {21133, 69075}, {21202, 69047}, {28565, 44435}, {29094, 50556}, {39775, 47695}, {48278, 62415}, {63813, 68882}

X(69394) = reflection of X(i) in X(j) for these {i,j}: {4088, 60577}, {69047, 21202}
X(69394) = anticomplement of the isotomic conjugate of X(65237)
X(69394) = isotomic conjugate of the isogonal conjugate of X(53521)
X(69394) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3512, 33650}, {8852, 37781}, {51614, 21286}, {65237, 6327}, {65293, 315}
X(69394) = X(65237)-Ceva conjugate of X(2)
X(69394) = X(i)-isoconjugate of X(j) for these (i,j): {37, 2715}, {42, 36084}, {71, 36104}, {72, 32696}, {98, 692}, {100, 1976}, {101, 1910}, {213, 2966}, {228, 685}, {248, 1783}, {293, 8750}, {668, 14601}, {878, 5379}, {906, 6531}, {1332, 57260}, {1415, 15628}, {1821, 32739}, {1824, 43754}, {1918, 36036}, {2205, 43187}, {2330, 36065}, {2422, 4567}, {3404, 4628}, {3990, 20031}, {4705, 57742}, {5360, 41173}, {6335, 14600}, {32656, 36120}, {32716, 60723}, {36132, 60726}, {42717, 67167}, {50487, 57991}
X(69394) = X(i)-Dao conjugate of X(j) for these (i,j): {132, 8750}, {905, 66881}, {1015, 1910}, {1086, 98}, {1146, 15628}, {2679, 1918}, {4988, 2395}, {5190, 6531}, {5976, 190}, {6626, 2966}, {8054, 1976}, {11672, 101}, {26932, 293}, {34021, 36036}, {35088, 10}, {38970, 1826}, {38987, 42}, {39000, 71}, {39006, 248}, {39009, 60726}, {39039, 1783}, {39040, 100}, {40589, 2715}, {40592, 36084}, {40601, 32739}, {40616, 66880}, {40618, 287}, {40619, 1821}, {40627, 2422}, {41167, 55230}, {46094, 32656}, {50440, 3939}, {53521, 53258}, {55267, 4024}, {62590, 1331}, {62595, 1897}
X(69394) = crosspoint of X(i) and X(j) for these (i,j): {274, 65293}, {334, 664}
X(69394) = crosssum of X(663) and X(2210)
X(69394) = crossdifference of every pair of points on line {42, 1976}
X(69394) = barycentric product X(i)*X(j) for these {i,j}: {27, 6333}, {76, 53521}, {86, 2799}, {240, 15413}, {297, 4025}, {310, 3569}, {325, 514}, {511, 3261}, {513, 46238}, {523, 51370}, {684, 44129}, {693, 1959}, {850, 17209}, {868, 4610}, {877, 4466}, {905, 40703}, {1019, 42703}, {1111, 42717}, {1459, 44132}, {1577, 51369}, {1755, 40495}, {2396, 3120}, {2421, 21207}, {3405, 48084}, {4556, 62431}, {6393, 7649}, {6530, 30805}, {6629, 62629}, {10566, 51371}, {16230, 17206}, {16892, 20022}, {18653, 65973}, {21178, 34138}, {24002, 44694}, {35519, 43034}, {36212, 46107}, {41172, 55229}, {44114, 52612}, {50567, 62626}, {52621, 59734}
X(69394) = barycentric quotient X(i)/X(j) for these {i,j}: {27, 685}, {28, 36104}, {58, 2715}, {81, 36084}, {86, 2966}, {232, 8750}, {237, 32739}, {240, 1783}, {274, 36036}, {297, 1897}, {310, 43187}, {325, 190}, {511, 101}, {513, 1910}, {514, 98}, {522, 15628}, {649, 1976}, {684, 71}, {693, 1821}, {868, 4024}, {905, 293}, {1432, 36065}, {1459, 248}, {1474, 32696}, {1755, 692}, {1790, 43754}, {1919, 14601}, {1959, 100}, {2396, 4600}, {2421, 4570}, {2491, 1918}, {2530, 3404}, {2799, 10}, {3120, 2395}, {3122, 2422}, {3261, 290}, {3289, 32656}, {3569, 42}, {4025, 287}, {4091, 17974}, {4107, 40820}, {4466, 879}, {4556, 57742}, {4610, 57991}, {4750, 5967}, {6333, 306}, {6393, 4561}, {6545, 43920}, {7649, 6531}, {8747, 20031}, {11125, 35906}, {15413, 336}, {16230, 1826}, {16725, 23997}, {16892, 20021}, {17197, 60568}, {17206, 17932}, {17209, 110}, {17216, 53173}, {17924, 36120}, {17994, 2333}, {18653, 65776}, {21102, 60517}, {21109, 52672}, {21123, 51869}, {21131, 51441}, {21134, 51404}, {21172, 66880}, {21178, 31636}, {21207, 43665}, {26932, 66881}, {30805, 6394}, {36212, 1331}, {39469, 2200}, {40495, 46273}, {40703, 6335}, {41172, 55230}, {42703, 4033}, {42717, 765}, {42751, 2183}, {43034, 109}, {44114, 4079}, {44129, 22456}, {44694, 644}, {44728, 30720}, {46107, 16081}, {46238, 668}, {51369, 662}, {51370, 99}, {51371, 4568}, {51417, 4115}, {51651, 1415}, {51862, 4628}, {53521, 6}, {55229, 41174}, {58260, 53581}, {59734, 3939}, {60679, 6037}, {62431, 52623}, {62626, 9154}, {62720, 5379}
X(69394) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {86, 21205, 11125}, {7649, 30805, 17215}

X(69395) = X(86)X(21103)∩X(514)X(17215)

Barycentrics    (b - c)*(b^2*c^2 + S^2 + SB*SC) : :

X(69395) lies on these lines: {86, 21103}, {514, 17215}, {693, 21119}, {3261, 21102}, {3766, 66287}, {4374, 21132}, {4988, 26114}, {20906, 21118}, {20949, 23752}, {21113, 21126}

X(69395) = X(692)-isoconjugate of X(45857)
X(69395) = X(1086)-Dao conjugate of X(45857)
X(69395) = barycentric product X(i)*X(j) for these {i,j}: {514, 59635}, {693, 17868}, {3261, 5943}, {4025, 56022}
X(69395) = barycentric quotient X(i)/X(j) for these {i,j}: {514, 45857}, {5943, 101}, {17868, 100}, {56022, 1897}, {59635, 190}

X(69396) = X(242)X(514)∩X(513)X(65445)

Barycentrics    (b - c)*(S^2 + 2*SB*SC) : :

X(69396) = X[1459] - 3 X[7649], 5 X[1459] - 9 X[11125], X[1459] + 3 X[21102], 7 X[1459] - 3 X[21103], 2 X[1459] - 3 X[21172], 5 X[7649] - 3 X[11125], 7 X[7649] - X[21103], 3 X[11125] + 5 X[21102], 21 X[11125] - 5 X[21103], 6 X[11125] - 5 X[21172], 7 X[21102] + X[21103], 2 X[21102] + X[21172], 2 X[21103] - 7 X[21172]

X(69396) lies on these lines: {242, 514}, {513, 65445}, {522, 10015}, {523, 7661}, {676, 4802}, {1769, 28161}, {4778, 53522}, {6129, 28147}, {8058, 21185}, {14282, 68799}, {20316, 64917}, {21118, 47136}, {21119, 47123}, {28191, 52596}, {28225, 59976}, {43050, 51648}, {48303, 54261}

X(69396) = midpoint of X(i) and X(j) for these {i,j}: {7649, 21102}, {21118, 47136}, {21119, 47123}
X(69396) = reflection of X(i) in X(j) for these {i,j}: {21172, 7649}, {48303, 54261}
X(69396) = X(i)-isoconjugate of X(j) for these (i,j): {100, 14528}, {906, 56346}
X(69396) = X(i)-Dao conjugate of X(j) for these (i,j): {5190, 56346}, {8054, 14528}, {33537, 1331}
X(69396) = crossdifference of every pair of points on line {71, 14528}
X(69396) = barycentric product X(i)*X(j) for these {i,j}: {514, 3091}, {3261, 17810}, {19188, 21102}
X(69396) = barycentric quotient X(i)/X(j) for these {i,j}: {649, 14528}, {3091, 190}, {7649, 56346}, {17810, 101}, {33578, 32739}

X(69397) = X(86)X(1459)∩X(514)X(1919)

Barycentrics    (b - c)*(b^2*c^2 + 2*S^2 - 2*SB*SC) : :

X(69397) lies on these lines: {1, 20954}, {6, 21225}, {59, 664}, {75, 21173}, {86, 1459}, {514, 1919}, {522, 4360}, {657, 68966}, {693, 48283}, {786, 20981}, {1268, 48228}, {2605, 3766}, {3050, 31296}, {3226, 53219}, {3737, 20949}, {4107, 21123}, {4374, 53314}, {4406, 43924}, {4508, 42327}, {4828, 17218}, {5224, 20293}, {6586, 17277}, {7199, 48281}, {15419, 53357}, {17378, 46402}, {17496, 57214}, {18080, 21389}, {20906, 55969}, {21102, 21205}, {23355, 39693}, {23794, 48302}, {33296, 48321}, {34063, 48325}, {34064, 47790}, {47729, 48109}, {48144, 57059}

X(69397) = reflection of X(i) in X(j) for these {i,j}: {10566, 1919}, {21110, 21194}
X(69397) = X(4556)-Ceva conjugate of X(86)
X(69397) = X(i)-isoconjugate of X(j) for these (i,j): {100, 27375}, {213, 11794}, {692, 3613}, {4705, 27867}, {5360, 53701}
X(69397) = X(i)-Dao conjugate of X(j) for these (i,j): {850, 52623}, {1086, 3613}, {6626, 11794}, {7668, 21035}, {8054, 27375}, {40618, 36952}, {52591, 4024}
X(69397) = crosssum of X(4079) and X(41267)
X(69397) = crossdifference of every pair of points on line {21035, 27375}
X(69397) = barycentric product X(i)*X(j) for these {i,j}: {58, 57082}, {86, 31296}, {310, 3050}, {513, 33764}, {514, 1078}, {649, 33769}, {664, 27010}, {667, 33778}, {693, 18042}, {1629, 30805}, {3261, 5012}, {4025, 36794}, {4091, 54100}, {4556, 36901}, {4610, 7668}, {16892, 41296}, {38352, 55229}
X(69397) = barycentric quotient X(i)/X(j) for these {i,j}: {86, 11794}, {514, 3613}, {649, 27375}, {1078, 190}, {3050, 42}, {4025, 36952}, {4091, 42487}, {4556, 27867}, {5012, 101}, {7668, 4024}, {10312, 8750}, {10566, 30505}, {18042, 100}, {27010, 522}, {31296, 10}, {33764, 668}, {33769, 1978}, {33778, 6386}, {36794, 1897}, {36901, 52623}, {38352, 55230}, {41328, 46148}, {52591, 21035}, {57082, 313}
X(69397) = {X(1459),X(3261)}-harmonic conjugate of X(86)

X(69398) = X(86)X(7649)∩X(514)X(17215)

Barycentrics    (b - c)*(-(b^2*c^2) + 2*S^2 - SB*SC) : :

X(69398) lies on these lines: {69, 20315}, {77, 23465}, {86, 7649}, {514, 17215}, {1459, 4025}, {1790, 1919}, {3945, 20294}, {4874, 4977}, {9031, 57054}, {18648, 21191}, {18650, 29328}, {21123, 48060}, {21304, 48545}, {26114, 49293}, {59976, 69040}, {63110, 64917}

X(69398) = reflection of X(4025) in X(15419)
X(69398) = isotomic conjugate of the polar conjugate of X(3798)
X(69398) = X(514)-Ceva conjugate of X(4025)
X(69398) = X(i)-isoconjugate of X(j) for these (i,j): {100, 14248}, {692, 34208}, {1783, 8770}, {1824, 3565}, {1897, 38252}, {6335, 53059}, {8750, 8769}, {41013, 65178}
X(69398) = X(i)-Dao conjugate of X(j) for these (i,j): {69, 190}, {1086, 34208}, {6388, 10}, {8054, 14248}, {15525, 1826}, {26932, 8769}, {34467, 38252}, {39006, 8770}, {40618, 2996}, {51579, 1897}
X(69398) = crosspoint of X(514) and X(3798)
X(69398) = crossdifference of every pair of points on line {2333, 14248}
X(69398) = barycentric product X(i)*X(j) for these {i,j}: {69, 3798}, {193, 4025}, {514, 6337}, {905, 18156}, {1459, 57518}, {1707, 15413}, {3167, 3261}, {3566, 17206}, {4028, 15419}, {4091, 54412}, {4466, 57216}, {6332, 17081}, {6353, 30805}, {10607, 46107}
X(69398) = barycentric quotient X(i)/X(j) for these {i,j}: {193, 1897}, {514, 34208}, {649, 14248}, {905, 8769}, {1459, 8770}, {1707, 1783}, {1790, 3565}, {3053, 8750}, {3167, 101}, {3566, 1826}, {3798, 4}, {4025, 2996}, {4091, 6391}, {4750, 5203}, {4786, 52454}, {6337, 190}, {8651, 2333}, {10607, 1331}, {16892, 47730}, {17081, 653}, {17206, 35136}, {18156, 6335}, {22383, 38252}, {30805, 6340}
X(69398) = {X(1459),X(30805)}-harmonic conjugate of X(4025)

X(69399) = X(242)X(514)∩X(513)X(30725)

Barycentrics     (b - c)*(2*S^2 - SB*SC) : :

X(69399) = 5 X[1459] - 3 X[11125], 3 X[1459] - X[21102], 3 X[1459] - 2 X[21172], 5 X[7649] - 6 X[11125], 3 X[7649] - 2 X[21102], X[7649] + 2 X[21103], 3 X[7649] - 4 X[21172], 9 X[11125] - 5 X[21102], 3 X[11125] + 5 X[21103], 9 X[11125] - 10 X[21172], X[21102] + 3 X[21103], 3 X[21103] + 2 X[21172], 3 X[2457] - 4 X[65412]

X(69399) lies on these lines: {242, 514}, {513, 30725}, {522, 53532}, {523, 59976}, {650, 59975}, {676, 28213}, {1769, 4778}, {2457, 65412}, {3554, 17412}, {3904, 23874}, {4449, 21106}, {4802, 53522}, {4977, 6129}, {6589, 49293}, {7661, 28229}, {9031, 20294}, {17418, 21105}, {20293, 20315}, {28529, 44444}, {30719, 50354}, {47123, 48283}, {47136, 55969}, {48060, 53521}, {48275, 52587}, {49296, 53308}

X(69399) = midpoint of X(i) and X(j) for these {i,j}: {1459, 21103}, {4449, 21106}, {17418, 21105}
X(69399) = reflection of X(i) in X(j) for these {i,j}: {7649, 1459}, {20293, 20315}, {21102, 21172}, {47123, 48283}, {47136, 55969}, {50354, 30719}
X(69399) = X(i)-isoconjugate of X(j) for these (i,j): {72, 58950}, {100, 3527}, {101, 56033}, {692, 8797}, {906, 8796}, {1332, 34818}, {1783, 63154}
X(69399) = X(i)-Dao conjugate of X(j) for these (i,j): {1015, 56033}, {1086, 8797}, {5190, 8796}, {5522, 10}, {8054, 3527}, {39006, 63154}
X(69399) = crosspoint of X(36118) and X(65028)
X(69399) = crosssum of X(1459) and X(26928)
X(69399) = crossdifference of every pair of points on line {71, 3527}
X(69399) = barycentric product X(i)*X(j) for these {i,j}: {86, 47122}, {514, 631}, {649, 44149}, {3087, 4025}, {3261, 11402}, {4466, 65177}, {30805, 61348}, {36748, 46107}
X(69399) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 56033}, {514, 8797}, {631, 190}, {649, 3527}, {1459, 63154}, {1474, 58950}, {3087, 1897}, {7649, 8796}, {11402, 101}, {36748, 1331}, {44149, 1978}, {47122, 10}
X(69399) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1459, 21102, 21172}, {21102, 21172, 7649}

X(69400) = X(86)X(7649)∩X(514)X(1921)

Barycentrics   (b - c)*(-(b^2*c^2) + 2*S^2) : :

X(69400 lies on these lines: {69, 20294}, {86, 7649}, {513, 69040}, {514, 1921}, {924, 6563}, {1459, 21178}, {1919, 21135}, {3904, 15413}, {5224, 20315}, {6332, 18160}, {7192, 57242}, {15419, 69364}, {17215, 21102}, {17378, 64917}, {20293, 57054}, {21172, 21205}, {34805, 57750}

X(69400) = reflection of X(3261) in X(30805)
X(69400) = X(i)-isoconjugate of X(j) for these (i,j): {37, 32734}, {42, 36145}, {91, 32739}, {100, 60501}, {213, 925}, {228, 65176}, {692, 2165}, {906, 14593}, {1783, 2351}, {1820, 8750}, {1918, 65251}, {2205, 46134}, {21807, 32692}
X(69400) = X(i)-Dao conjugate of X(j) for these (i,j): {135, 2333}, {577, 32656}, {1086, 2165}, {5190, 14593}, {6626, 925}, {8054, 60501}, {26932, 1820}, {34021, 65251}, {34116, 32739}, {39006, 2351}, {39013, 42}, {40589, 32734}, {40592, 36145}, {40618, 68}, {40619, 91}, {52584, 4024}, {63827, 46389}
X(69400) = cevapoint of X(6563) and X(63827)
X(69400) = crosspoint of X(4610) and X(44129)
X(69400) = crosssum of X(2200) and X(4079)
X(69400) = crossdifference of every pair of points on line {1918, 60501}
X(69400) = barycentric product X(i)*X(j) for these {i,j}: {47, 40495}, {86, 6563}, {274, 63827}, {310, 924}, {317, 4025}, {514, 7763}, {561, 34948}, {693, 44179}, {1748, 15413}, {1993, 3261}, {4466, 55227}, {6385, 55216}, {7199, 42700}, {9723, 46107}, {11547, 30805}, {16732, 55249}, {17206, 57065}, {17881, 52935}, {18605, 20948}, {44129, 52584}, {47421, 52612}, {57796, 63832}
X(69400) = barycentric quotient X(i)/X(j) for these {i,j}: {24, 8750}, {27, 65176}, {47, 692}, {58, 32734}, {81, 36145}, {86, 925}, {274, 65251}, {310, 46134}, {317, 1897}, {514, 2165}, {571, 32739}, {649, 60501}, {693, 91}, {905, 1820}, {924, 42}, {1147, 32656}, {1459, 2351}, {1748, 1783}, {1993, 101}, {3261, 5392}, {3798, 56891}, {4025, 68}, {4091, 55549}, {6385, 55215}, {6563, 10}, {6753, 2333}, {7199, 66954}, {7649, 14593}, {7763, 190}, {9723, 1331}, {16732, 55250}, {17206, 65309}, {17881, 4036}, {18605, 163}, {30451, 2200}, {30805, 52350}, {34948, 31}, {34952, 1918}, {40495, 20571}, {42700, 1018}, {44129, 30450}, {44179, 100}, {46107, 847}, {47421, 4079}, {52584, 71}, {55216, 213}, {55249, 4567}, {57065, 1826}, {63827, 37}, {63829, 21011}, {63832, 228}

X(69401) = X(86)X(7649)∩X(514)X(1919)

Barycentrics   (b - c)*(-(b^2*c^2) + 2*S^2 + 2*SB*SC) : :

X(69401) = 3 X[21110] - X[21126]

X(69401) lies on these lines: {86, 7649}, {514, 1919}, {693, 21111}, {1459, 21205}, {3261, 21102}, {4406, 66287}, {5224, 20294}, {17271, 64917}, {20906, 21112}, {21123, 21136}, {21131, 21191}, {57214, 69364}

X(69401) = X(692)-isoconjugate of X(45838)
X(69401) = X(1086)-Dao conjugate of X(45838)
X(69401) = barycentric product X(i)*X(j) for these {i,j}: {514, 7752}, {693, 18041}, {3060, 3261}, {4610, 34981}
X(69401) = barycentric quotient X(i)/X(j) for these {i,j}: {514, 45838}, {3060, 101}, {7752, 190}, {18041, 100}, {34981, 4024}

X(69402) = EULER LINE INTERCEPT OF X(145)X(3817)

Barycentrics   5*S^2 + 8*SB*SC : :
X(69402) = 39 X[2] - 16 X[3], 15 X[2] + 8 X[4], 9 X[2] - 32 X[5], 27 X[2] - 4 X[20], 87 X[2] - 64 X[140], 31 X[2] - 8 X[376], 7 X[2] + 16 X[381], 99 X[2] + 16 X[382],, and many others

X(68402) lies on these lines: {2, 3}, {145, 3817}, {262, 20105}, {316, 32870}, {325, 32880}, {373, 11439}, {390, 3614}, {395, 42775}, {396, 42776}, {516, 46931}, {944, 68403}, {946, 4678}, {962, 31399}, {1131, 6442}, {1132, 6441}, {1151, 43520}, {1152, 43519}, {1352, 63061}, {1699, 46933}, {1975, 32881}, {2996, 60331}, {3241, 61252}, {3411, 5335}, {3412, 5334}, {3590, 3592}, {3591, 3594}, {3600, 7173}, {3617, 4301}, {3621, 5587}, {3622, 19925}, {3623, 5881}, {3634, 10248}, {3815, 63536}, {4772, 67858}, {4788, 67853}, {5226, 37723}, {5261, 37722}, {5274, 15888}, {5286, 31417}, {5339, 43877}, {5340, 43878}, {5343, 37832}, {5344, 37835}, {5395, 60336}, {5603, 20014}, {5735, 61006}, {5943, 64025}, {5984, 23514}, {6200, 43558}, {6396, 43559}, {6419, 43343}, {6420, 43342}, {6427, 14226}, {6428, 14241}, {6439, 9543}, {6440, 32786}, {6459, 43561}, {6460, 43560}, {6476, 9540}, {6477, 13935}, {6478, 10576}, {6479, 10577}, {6480, 12819}, {6481, 12818}, {6561, 9692}, {6564, 43431}, {6565, 43430}, {6688, 12279}, {7585, 42270}, {7586, 31414}, {7738, 18584}, {7752, 32840}, {7765, 31404}, {7768, 32893}, {7814, 32830}, {7850, 32828}, {7860, 32885}, {7988, 46934}, {7999, 44863}, {8165, 9710}, {9143, 15029}, {9588, 9812}, {9589, 9780}, {9607, 63533}, {9624, 13607}, {9657, 10589}, {9670, 10588}, {9680, 35787}, {9681, 23263}, {9698, 43448}, {9778, 46930}, {9842, 31019}, {9862, 68406}, {9955, 20052}, {10110, 16981}, {10141, 42642}, {10142, 42641}, {10172, 31425}, {10175, 20070}, {10194, 43525}, {10195, 43526}, {10302, 60328}, {10513, 59635}, {10516, 20080}, {10590, 37720}, {10591, 37719}, {10595, 61249}, {10653, 43023}, {10654, 43022}, {11002, 14531}, {11004, 17814}, {11185, 32841}, {11427, 68009}, {11431, 18388}, {11441, 63076}, {11451, 44870}, {11465, 16194}, {11482, 51215}, {11522, 31145}, {11542, 42690}, {11543, 42691}, {11695, 66747}, {12007, 63123}, {12046, 40280}, {12111, 27355}, {12245, 61262}, {13881, 63097}, {13886, 42604}, {13939, 42605}, {14484, 60639}, {14639, 35369}, {14683, 36518}, {14845, 15058}, {14853, 43150}, {14986, 31410}, {15025, 68317}, {15069, 51170}, {16772, 43466}, {16773, 43465}, {16808, 42935}, {16809, 42934}, {18424, 31400}, {18489, 18916}, {18493, 61255}, {18538, 43341}, {18581, 43012}, {18582, 43013}, {18762, 43340}, {18845, 60102}, {19053, 43376}, {19054, 43377}, {19876, 51076}, {20049, 38021}, {20054, 38155}, {20059, 38150}, {20085, 38161}, {20094, 36519}, {22236, 43541}, {22238, 43540}, {22615, 43513}, {22644, 43514}, {23241, 58431}, {23249, 35813}, {23259, 35812}, {23273, 31487}, {23302, 43365}, {23303, 43364}, {23332, 68015}, {23513, 64009}, {23515, 64102}, {25555, 50956}, {31407, 31415}, {31412, 63016}, {31450, 69141}, {31454, 42539}, {31673, 61265}, {32205, 61136}, {32879, 63098}, {32894, 37668}, {32897, 64018}, {32900, 38140}, {33606, 49874}, {33607, 49873}, {34627, 61282}, {35814, 42274}, {35815, 42277}, {37665, 63534}, {38259, 60333}, {39663, 50248}, {40693, 42918}, {40694, 42919}, {42101, 43869}, {42102, 43870}, {42103, 42488}, {42106, 42489}, {42107, 42472}, {42110, 42473}, {42111, 42813}, {42114, 42814}, {42121, 42689}, {42124, 42688}, {42139, 42156}, {42140, 42490}, {42141, 42491}, {42142, 42153}, {42150, 43483}, {42151, 43484}, {42154, 43479}, {42155, 43480}, {42163, 42494}, {42166, 42495}, {42283, 43339}, {42284, 43338}, {42474, 42945}, {42475, 42944}, {42561, 63015}, {42582, 43512}, {42583, 43511}, {42686, 43193}, {42687, 43194}, {42920, 42991}, {42921, 42990}, {42930, 43472}, {42931, 43471}, {42964, 43544}, {42965, 43545}, {43238, 43553}, {43239, 43552}, {43252, 63113}, {43253, 63114}, {43537, 60650}, {43605, 63040}, {43681, 54521}, {45184, 64756}, {45958, 67924}, {46852, 66606}, {46944, 60728}, {47354, 63000}, {47586, 54639}, {50796, 61288}, {50960, 63109}, {50964, 52987}, {51068, 58245}, {51171, 67865}, {51709, 61248}, {53099, 60625}, {53513, 63058}, {53516, 63059}, {54706, 60278}, {54866, 60145}, {60100, 60327}, {60118, 60200}, {60239, 60324}, {60291, 60300}, {60292, 60299}, {61716, 65383}, {63122, 63722}

X(69402) = midpoint of X(4) and X(61807)
X(69402) = reflection of X(i) in X(j) for these {i,j}: {46936, 61921}, {61788, 61867}, {61794, 41992}, {61807, 55860}, {61834, 46936}, {61867, 61911}, {61911, 5}, {61921, 61935}, {62061, 61850}, {62078, 61834}
X(69402) = complement of X(62078)
X(69402) = anticomplement of X(61834)
X(69402) = orthocentroidal-circle-inverse of X(50693)
X(69402) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 50693}, {2, 20, 61816}, {2, 381, 61992}, {2, 3091, 3854}, {2, 3146, 61791}, {2, 3543, 62056}, {2, 3832, 17578}, {2, 3839, 62048}, {2, 3854, 50689}, {2, 5059, 61804}, {2, 17578, 21734}, {2, 50687, 62081}, {2, 50689, 5059}, {2, 50690, 3}, {2, 50692, 15717}, {2, 61952, 61962}, {2, 61954, 61972}, {2, 61962, 62005}, {2, 61972, 62032}, {2, 61985, 62148}, {2, 61992, 62129}, {2, 62005, 62095}, {2, 62032, 61778}, and many others

X(69403) = EULER LINE INTERCEPT OF X(8)X(68403)

Barycentrics   6*S^2 + 7*SB*SC : :
X(69403) = 39 X[2] - 14 X[3], 18 X[2] + 7 X[4], 3 X[2] - 28 X[5], 57 X[2] - 7 X[20], 81 X[2] - 56 X[140], 32 X[2] - 7 X[376], 11 X[2] + 14 X[381], 111 X[2] + 14 X[382], 69 X[2] + 56 X[546], and many others

X(68403) lies on these lines: {2, 3}, {8, 68403}, {15, 43447}, {16, 43446}, {17, 42139}, {18, 42142}, {114, 62427}, {145, 61262}, {147, 68406}, {316, 52718}, {325, 32878}, {340, 8797}, {371, 43433}, {372, 43432}, {485, 60304}, {486, 60303}, {625, 18840}, {1056, 7173}, {1058, 3614}, {1249, 61327}, {1352, 63073}, {1587, 6442}, {1588, 6441}, {1698, 28232}, {1699, 68037}, {2979, 12002}, {3311, 3590}, {3312, 3591}, {3316, 6565}, {3317, 6564}, {3616, 61266}, {3625, 5603}, {3630, 14853}, {3633, 7989}, {3635, 5587}, {3817, 4691}, {4668, 5818}, {4726, 67853}, {4857, 10588}, {5270, 10589}, {5286, 18584}, {5339, 42806}, {5340, 42805}, {5343, 42107}, {5344, 42110}, {5346, 43620}, {5365, 23302}, {5366, 23303}, {5485, 7764}, {5493, 22266}, {5657, 30315}, {5817, 60962}, {5882, 7988}, {5965, 62996}, {6419, 14226}, {6420, 14241}, {6427, 43387}, {6428, 43386}, {6433, 43378}, {6434, 43379}, {6439, 23261}, {6440, 23251}, {6445, 43520}, {6446, 43519}, {6449, 43517}, {6450, 43518}, {6459, 6478}, {6460, 6479}, {6476, 32785}, {6477, 32786}, {6484, 12819}, {6485, 12818}, {6486, 43558}, {6487, 43559}, {6688, 12290}, {6761, 8888}, {7320, 7743}, {7581, 42274}, {7582, 42277}, {7603, 63533}, {7607, 18844}, {7612, 60146}, {7617, 63927}, {7737, 12815}, {7781, 68325}, {7844, 18841}, {7860, 53127}, {7914, 54890}, {7967, 61268}, {7998, 44863}, {8227, 28236}, {8960, 42561}, {9624, 38076}, {10187, 36969}, {10188, 36970}, {10516, 32455}, {10576, 23275}, {10577, 23269}, {10595, 61261}, {10653, 43025}, {10654, 43024}, {10982, 54434}, {10992, 52886}, {11002, 14128}, {11459, 27355}, {11465, 44870}, {11488, 42802}, {11489, 42801}, {11542, 22237}, {11543, 22235}, {11695, 16261}, {12112, 15805}, {12317, 36518}, {13382, 15024}, {14482, 31404}, {14494, 60209}, {14845, 15056}, {14862, 32064}, {14912, 18553}, {15003, 44324}, {15031, 34803}, {15045, 64029}, {15092, 52090}, {15105, 61735}, {16772, 42474}, {16773, 42475}, {16808, 42978}, {16809, 42979}, {16960, 42114}, {16961, 42111}, {16962, 33603}, {16963, 33602}, {18362, 31417}, {18392, 25712}, {18489, 66729}, {18510, 60292}, {18512, 60291}, {18525, 61267}, {18581, 42494}, {18582, 42495}, {19925, 61265}, {20053, 61263}, {20057, 61257}, {20791, 46852}, {24828, 62424}, {30308, 31399}, {31410, 68688}, {31412, 58866}, {31415, 69162}, {31467, 63536}, {32820, 32876}, {32821, 32875}, {32825, 32877}, {32888, 59635}, {33884, 67867}, {34089, 35255}, {34091, 35256}, {37832, 42435}, {37835, 42436}, {38072, 51179}, {38150, 61000}, {40273, 46932}, {42095, 42987}, {42098, 42986}, {42103, 42936}, {42106, 42937}, {42115, 43876}, {42116, 43875}, {42133, 43238}, {42134, 43239}, {42149, 42775}, {42150, 42915}, {42151, 42914}, {42152, 42776}, {42159, 42512}, {42162, 42513}, {42268, 43509}, {42269, 43510}, {42431, 52080}, {42432, 52079}, {42518, 49873}, {42519, 49874}, {42598, 43877}, {42599, 43878}, {42610, 42940}, {42611, 42941}, {42637, 43506}, {42638, 43505}, {42682, 42927}, {42683, 42926}, {42777, 43404}, {42778, 43403}, {43102, 43444}, {43103, 43445}, {43416, 43555}, {43417, 43554}, {43430, 43571}, {43431, 43570}, {43527, 60325}, {46931, 48661}, {48889, 63120}, {50964, 55606}, {50994, 55721}, {51176, 53093}, {53098, 53106}, {53107, 60123}, {54448, 61272}, {59386, 60977}, {60127, 60640}, {60250, 60330}, {60337, 60649}, {64809, 67536}, {66756, 67924}

X(69403) = midpoint of X(i) and X(j) for these {i,j}: {4, 61787}, {3091, 61914}, {3843, 61840}, {5076, 58192}, {17578, 58195}
X(69403) = reflection of X(i) in X(j) for these {i,j}: {631, 60781}, {3522, 61815}, {17538, 58188}, {58188, 61840}, {58195, 61793}, {60781, 61914}, {61787, 61856}, {61793, 632}, {61804, 55866}, {61856, 1656}, {61902, 5071}, {61914, 61923}, {61923, 5}, {62152, 58198}
X(69403) = complement of X(58188)
X(69403) = anticomplement of X(61840)
X(69403) = orthocentroidal-circle-inverse of X(21735)
X(69403) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 21735}, {2, 20, 12108}, {2, 381, 62029}, {2, 1657, 61817}, {2, 3091, 3843}, {2, 3543, 15706}, {2, 3545, 61951}, {2, 3627, 61138}, {2, 3832, 49140}, {2, 3839, 15686}, {2, 3843, 17538}, {2, 3850, 4}, {2, 5056, 61907}, {2, 5072, 61945}, {2, 14893, 62086}, {2, 15684, 3524}, and many others

X(69404) = EULER LINE INTERCEPT OF X(8)X(61264)

Barycentrics   7*S^2 + 8*SB*SC : :
X(69404) = 45 X[2] - 16 X[3], 21 X[2] + 8 X[4], 3 X[2] - 32 X[5], 33 X[2] - 4 X[20], 93 X[2] - 64 X[140], 37 X[2] - 8 X[376], 13 X[2] + 16 X[381], 129 X[2] + 16 X[382], 81 X[2] + 64 X[546], and many others

X(68404) lies on these lines: {2, 3}, {8, 61264}, {145, 7989}, {146, 38725}, {147, 38735}, {148, 38746}, {149, 38758}, {150, 38770}, {151, 38782}, {316, 32897}, {325, 32882}, {516, 46930}, {944, 61266}, {1131, 3594}, {1132, 3592}, {1352, 63122}, {1699, 46932}, {3068, 41948}, {3069, 41947}, {3070, 43884}, {3071, 43883}, {3411, 42953}, {3412, 42952}, {3448, 15029}, {3614, 5274}, {3617, 3817}, {3619, 55591}, {3620, 55722}, {3621, 16200}, {3622, 7988}, {3623, 5587}, {3626, 58241}, {3984, 46873}, {4678, 7982}, {4821, 67853}, {5041, 31415}, {5097, 63061}, {5102, 20080}, {5261, 7173}, {5286, 39601}, {5334, 42581}, {5335, 42580}, {5343, 42911}, {5344, 42910}, {5365, 42488}, {5366, 42489}, {5395, 54921}, {5603, 20052}, {5640, 45187}, {5645, 32605}, {5818, 11278}, {5921, 39561}, {5984, 20398}, {6425, 42539}, {6426, 42540}, {6427, 42604}, {6428, 42605}, {6429, 32785}, {6430, 32786}, {6431, 42561}, {6432, 31412}, {6437, 42582}, {6438, 42583}, {6447, 23275}, {6448, 23269}, {6480, 42268}, {6481, 42269}, {6482, 6561}, {6483, 6560}, {6486, 43314}, {6487, 43315}, {6688, 11439}, {6759, 46865}, {7581, 43316}, {7582, 43317}, {7585, 53516}, {7586, 53513}, {7814, 32869}, {7991, 9779}, {8176, 63925}, {8227, 54448}, {8972, 42270}, {9342, 44846}, {9543, 10141}, {9656, 68665}, {9780, 63468}, {9812, 46931}, {9842, 27186}, {10142, 23251}, {10171, 30389}, {10222, 20014}, {10248, 64850}, {10516, 51170}, {10541, 51537}, {10595, 61262}, {11180, 22234}, {11441, 63040}, {11444, 16981}, {11451, 15012}, {11480, 43474}, {11481, 43473}, {12571, 19877}, {13464, 20049}, {13881, 63005}, {13941, 42273}, {14360, 38802}, {15020, 68280}, {15025, 36518}, {15056, 16625}, {15069, 63000}, {15431, 63128}, {16189, 31145}, {16644, 42776}, {16645, 42775}, {16772, 43496}, {16773, 43495}, {16964, 41978}, {16965, 41977}, {17814, 62990}, {19883, 58229}, {19925, 30392}, {20094, 20399}, {20095, 20400}, {20096, 20401}, {20397, 64102}, {22235, 42153}, {22237, 42156}, {22332, 63533}, {25565, 55708}, {30315, 50802}, {31417, 41940}, {32205, 67925}, {32787, 43377}, {32788, 43376}, {32807, 51953}, {32815, 32873}, {32816, 32872}, {32826, 32871}, {32827, 32870}, {32894, 59635}, {33179, 61261}, {34573, 55622}, {34754, 42159}, {34755, 42162}, {35369, 38734}, {35770, 42274}, {35771, 42277}, {37517, 40330}, {37832, 43009}, {37835, 43008}, {38072, 51214}, {38076, 50871}, {38150, 61006}, {38757, 66063}, {41961, 43512}, {41962, 43511}, {42104, 43469}, {42105, 43470}, {42126, 42590}, {42127, 42591}, {42139, 42598}, {42142, 42599}, {42143, 42982}, {42146, 42983}, {42147, 42474}, {42148, 42475}, {42160, 42915}, {42161, 42914}, {42163, 42473}, {42164, 43365}, {42165, 43364}, {42166, 42472}, {42262, 63016}, {42265, 63015}, {42417, 43413}, {42418, 43414}, {42494, 63080}, {42495, 63079}, {42512, 42934}, {42513, 42935}, {42786, 55627}, {42803, 43554}, {42804, 43555}, {42813, 43200}, {42814, 43199}, {42890, 42936}, {42891, 42937}, {42906, 43463}, {42907, 43464}, {42960, 49874}, {42961, 49873}, {42964, 43497}, {42965, 43498}, {43101, 43556}, {43104, 43557}, {43252, 49812}, {43253, 49813}, {43621, 55645}, {43681, 54522}, {47586, 60648}, {51027, 63127}, {51538, 55607}, {53099, 60635}, {53620, 58245}, {53858, 63027}, {54706, 56059}, {55614, 63121}, {55684, 63120}, {55685, 67884}, {55699, 66755}, {55703, 67865}, {59387, 64952}, {60118, 60628}, {60238, 60324}, {60277, 60328}, {60291, 60623}, {60292, 60622}, {60327, 60644}, {61265, 64953}, {62993, 63536}, {63119, 64196}, {65434, 66893}

X(69404) = reflection of X(i) in X(j) for these {i,j}: {61798, 61870}, {61817, 61878}, {61842, 46935}, {61870, 61903}, {62060, 61842}
X(69404) = complement of X(62060)
X(69404) = anticomplement of X(61842)
X(69404) = orthocentroidal-circle-inverse of X(21734)
X(69404) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 21734}, {2, 381, 62032}, {2, 3091, 50689}, {2, 3146, 61804}, {2, 3545, 61952}, {2, 3832, 5059}, {2, 3839, 62129}, {2, 3854, 17578}, {2, 5059, 61816}, {2, 17578, 61791}, {2, 50687, 61778}, {2, 50689, 50693}, {2, 50690, 15717}, {2, 50692, 3523}, and many others

X(69405) = EULER LINE INTERCEPT OF X(325)X(32886)

Barycentrics   8*S^2 + 5*SB*SC : :
X(69405) = 39 X[2] - 10 X[3], 24 X[2] + 5 X[4], 9 X[2] + 20 X[5], 63 X[2] - 5 X[20], 69 X[2] - 40 X[140], 34 X[2] - 5 X[376], 19 X[2] + 10 X[381], 27 X[2] + 2 X[382], 21 X[2] + 8 X[546], 11 X[2] - 40 X[547], 243 X[2] - 40 X[548], 49 X[2] - 20 X[549], 33 X[2] - 4 X[550], 54 X[2] - 25 X[631], 21 X[2] - 50 X[1656], 213 X[2] - 10 X[1657], and many others

X(68405) lies on these lines: {2, 3}, {325, 32886}, {373, 67924}, {397, 42474}, {398, 42475}, {568, 12046}, {590, 43798}, {615, 43797}, {625, 55732}, {962, 61266}, {1151, 43517}, {1152, 43518}, {1327, 43524}, {1328, 43523}, {1698, 68035}, {1975, 32887}, {3244, 5818}, {3316, 6441}, {3317, 6442}, {3411, 42910}, {3412, 42911}, {3590, 6427}, {3591, 6428}, {3592, 14226}, {3594, 14241}, {3614, 31410}, {3616, 61258}, {3617, 61269}, {3618, 33749}, {3622, 61249}, {3626, 8227}, {3629, 40330}, {3632, 10175}, {3634, 61265}, {3636, 5881}, {4031, 5714}, {4301, 54447}, {4317, 65141}, {5274, 31480}, {5319, 7603}, {5418, 9693}, {5550, 61263}, {5587, 15808}, {5603, 31399}, {5734, 9956}, {5790, 20054}, {5886, 20050}, {6200, 12819}, {6329, 15069}, {6396, 12818}, {6419, 43387}, {6420, 43386}, {6439, 23259}, {6440, 23249}, {6459, 6476}, {6460, 6477}, {6478, 6565}, {6479, 6564}, {6480, 43558}, {6481, 43559}, {6688, 15058}, {7581, 42583}, {7582, 42582}, {7735, 31417}, {7751, 68325}, {7765, 62993}, {7773, 52718}, {7814, 53127}, {7844, 55774}, {7914, 55744}, {7967, 37714}, {7988, 11362}, {7999, 27355}, {8164, 37722}, {8165, 31494}, {8176, 63930}, {8252, 23269}, {8253, 23275}, {8797, 57897}, {9540, 34089}, {9541, 43505}, {9588, 10172}, {9656, 68663}, {9781, 15606}, {9812, 31447}, {10141, 43378}, {10142, 43379}, {10155, 60219}, {10588, 37720}, {10589, 37719}, {11008, 14561}, {11178, 63062}, {11465, 67891}, {11488, 42991}, {11489, 42990}, {11695, 67925}, {11704, 18950}, {12245, 61268}, {12317, 20396}, {13935, 34091}, {14494, 60636}, {15081, 24981}, {15605, 61715}, {15888, 47743}, {16966, 42939}, {16967, 42938}, {17825, 67879}, {18553, 63109}, {18583, 63026}, {18841, 60322}, {18843, 53103}, {18874, 62187}, {20057, 59388}, {20125, 23515}, {22236, 43447}, {22238, 43446}, {23273, 31454}, {25555, 50974}, {28174, 46930}, {31272, 66052}, {31412, 35813}, {31414, 42277}, {31425, 51073}, {31450, 39565}, {31457, 39601}, {31492, 43448}, {32785, 43792}, {32786, 43791}, {32822, 37647}, {34631, 38098}, {35812, 42561}, {37640, 42580}, {37641, 42581}, {37643, 44300}, {38034, 46932}, {38108, 60957}, {40693, 42914}, {40694, 42915}, {41112, 42978}, {41113, 42979}, {41119, 42636}, {41120, 42635}, {42090, 42596}, {42091, 42597}, {42107, 43488}, {42110, 43487}, {42111, 42488}, {42114, 42489}, {42119, 42901}, {42120, 42900}, {42133, 42490}, {42134, 42491}, {42147, 43463}, {42148, 43464}, {42149, 43020}, {42152, 43021}, {42154, 43445}, {42155, 43444}, {42270, 43509}, {42273, 43510}, {42433, 43195}, {42434, 43196}, {42472, 42813}, {42473, 42814}, {42590, 43479}, {42591, 43480}, {42775, 43485}, {42776, 43486}, {42786, 51212}, {42928, 43470}, {42929, 43469}, {42946, 43546}, {42947, 43547}, {42980, 43248}, {42981, 43249}, {42992, 49812}, {42993, 49813}, {42998, 43104}, {42999, 43101}, {43012, 43014}, {43013, 43015}, {43028, 43106}, {43029, 43105}, {43238, 43482}, {43239, 43481}, {43403, 43555}, {43404, 43554}, {46934, 61259}, {50818, 61248}, {50964, 55631}, {50994, 55718}, {51068, 58240}, {51176, 55701}, {52102, 68024}, {52519, 60183}, {53098, 60631}, {53100, 60616}, {54616, 60337}, {59386, 60983}, {60142, 60629}, {60143, 60330}, {60332, 60627}, {63011, 63722}, {63632, 66606}, {63922, 67292}, {64008, 66065}, {66801, 66816}, {68403, 68545}

X(69405) = reflection of X(i) in X(j) for these {i,j}: {46935, 61903}, {61798, 61855}, {61817, 61870}, {61842, 61878}, {61870, 46935}, {62060, 61831}
X(69405) = complement of X(61798)
X(69405) = anticomplement of X(61855)
X(69405) = orthocentroidal-circle-inverse of X(61814)
X(69405) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 67095}, {2, 4, 61814}, {2, 5, 3855}, {2, 20, 55863}, {2, 381, 15710}, {2, 382, 631}, {2, 546, 10299}, {2, 3091, 550}, {2, 3523, 61858}, {2, 3529, 61836}, {2, 3543, 61827}, {2, 3544, 4}, {2, 3545, 62042}, {2, 3832, 61788}, {2, 3839, 15700}, {2, 3851, 3529}, and many others

X(69406) = EULER LINE INTERCEPT OF X(8)X(61266)

Barycentrics   8*S^2 + 7*SB*SC : :
X(69406) = 45 X[2] - 14 X[3], 24 X[2] + 7 X[4], 3 X[2] + 28 X[5], 69 X[2] - 7 X[20], 87 X[2] - 56 X[140], 38 X[2] - 7 X[376], 17 X[2] + 14 X[381], 141 X[2] + 14 X[382], 99 X[2] + 56 X[546], 25 X[2] - 56 X[547], 39 X[2] - 8 X[548], 59 X[2] - 28 X[549], 183 X[2] - 28 X[550], 66 X[2] - 35 X[631], 39 X[2] - 70 X[1656], and many others

X(68406) lies on these lines: {2, 3}, {8, 61266}, {40, 22266}, {325, 32888}, {397, 42475}, {398, 42474}, {944, 61264}, {1181, 51959}, {1587, 41954}, {1588, 41953}, {1698, 68037}, {1975, 32889}, {3311, 43890}, {3312, 43889}, {3316, 3592}, {3317, 3594}, {3567, 40247}, {3614, 47743}, {3619, 55587}, {3622, 61262}, {3623, 61270}, {3625, 5818}, {3630, 5102}, {3633, 7988}, {3635, 8227}, {4114, 5714}, {4668, 5603}, {4691, 7982}, {5041, 43620}, {5097, 62996}, {5225, 51817}, {5418, 6482}, {5420, 6483}, {5550, 31662}, {5817, 61020}, {6425, 23275}, {6426, 23269}, {6429, 23259}, {6430, 23249}, {6431, 53516}, {6432, 53513}, {6437, 42270}, {6438, 42273}, {6480, 43374}, {6481, 43375}, {6484, 42268}, {6485, 42269}, {7173, 8164}, {7581, 43880}, {7582, 43879}, {7693, 31815}, {7738, 39601}, {7967, 7989}, {8252, 10142}, {8253, 10141}, {9780, 68403}, {10147, 43257}, {10148, 43256}, {10171, 64953}, {10172, 63469}, {10175, 11531}, {10222, 20053}, {11278, 68034}, {11465, 67925}, {11480, 43780}, {11481, 43779}, {12046, 34783}, {12317, 15088}, {12900, 15044}, {13886, 35771}, {13939, 35770}, {14094, 38792}, {14561, 63073}, {14843, 34567}, {14927, 55685}, {15012, 15058}, {15029, 23515}, {15034, 68280}, {15054, 38725}, {15178, 61263}, {15749, 58266}, {18581, 42896}, {18582, 42897}, {18844, 53103}, {19925, 64954}, {20125, 36253}, {20190, 51537}, {21356, 55721}, {22234, 25565}, {23235, 38746}, {23267, 42583}, {23273, 42582}, {23514, 62427}, {30315, 50810}, {31399, 58245}, {31414, 42603}, {31447, 50807}, {31487, 43377}, {32205, 64025}, {32818, 32877}, {32827, 52718}, {32878, 59635}, {33179, 59388}, {34089, 41959}, {34091, 41960}, {34573, 55618}, {34599, 63645}, {34754, 42139}, {34755, 42142}, {35814, 42570}, {35815, 42571}, {37832, 42495}, {37835, 42494}, {38127, 58248}, {38664, 38735}, {38665, 38758}, {38666, 38770}, {38667, 38782}, {38675, 38802}, {38751, 52886}, {39874, 55703}, {40273, 46931}, {41971, 42776}, {41972, 42775}, {42107, 43463}, {42110, 43464}, {42111, 42581}, {42114, 42580}, {42115, 42907}, {42116, 42906}, {42143, 42986}, {42146, 42987}, {42147, 43493}, {42148, 43494}, {42150, 42592}, {42151, 42593}, {42153, 43542}, {42156, 43543}, {42159, 42473}, {42162, 42472}, {42258, 43505}, {42259, 43506}, {42512, 43776}, {42513, 43775}, {42610, 43445}, {42611, 43444}, {42786, 51538}, {42805, 43556}, {42806, 43557}, {42936, 43245}, {42937, 43244}, {42946, 43499}, {42947, 43500}, {42960, 42990}, {42961, 42991}, {42998, 43101}, {42999, 43104}, {43102, 43364}, {43103, 43365}, {43403, 56613}, {43404, 56612}, {43521, 43565}, {43522, 43564}, {43536, 60304}, {46930, 48661}, {50818, 61258}, {51128, 55641}, {51214, 55718}, {52666, 60315}, {52667, 60316}, {53096, 63533}, {54597, 60303}, {54857, 60616}, {55606, 63121}, {55683, 67884}, {55687, 63120}, {55695, 63119}, {55698, 64014}, {55699, 67865}, {55704, 63109}, {60329, 60629}, {60976, 64198}, {63924, 68325}, {66801, 66818}, {67891, 67924}

X(69406) = reflection of X(i) in X(j) for these {i,j}: {61783, 61849}, {61795, 61863}, {61816, 61875}, {61836, 61881}, {61863, 61892}
X(69406) = complement of X(61783)
X(69406) = anticomplement of X(61849)
X(69406) = orthocentroidal-circle-inverse of X(61138)
X(69406) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 61138}, {2, 20, 61832}, {2, 381, 46333}, {2, 1657, 631}, {2, 3091, 3627}, {2, 3523, 45760}, {2, 3543, 41983}, {2, 3545, 61973}, {2, 3627, 61807}, {2, 3832, 62110}, {2, 3839, 14093}, {2, 3843, 21735}, {2, 3850, 33703}, {2, 5056, 61911}, {2, 5072, 61964}, {2, 12812, 3090}, and many others

X(69407) = X(2)X(99)∩X(5)X(69)

Barycentrics   2*b^2*c^2 + 2*S^2 - SA^2 + 2*SB*SC : :

X(69407) lies on these lines: {2, 99}, {3, 32867}, {4, 32838}, {5, 69}, {32, 32991}, {76, 5056}, {141, 32984}, {183, 3545}, {193, 31415}, {194, 33009}, {230, 32983}, {315, 5068}, {316, 3091}, {325, 5071}, {381, 34229}, {385, 33005}, {524, 18584}, {538, 63077}, {547, 32837}, {549, 67536}, {625, 3620}, {626, 52250}, {631, 32826}, {1007, 5055}, {1078, 3832}, {1352, 51520}, {1384, 3363}, {1506, 6392}, {1656, 6390}, {1975, 5067}, {1992, 16509}, {2548, 51170}, {2996, 31401}, {3054, 32985}, {3055, 34505}, {3090, 3926}, {3146, 15031}, {3525, 32819}, {3543, 7771}, {3544, 7773}, {3589, 13881}, {3618, 43291}, {3619, 11318}, {3628, 6337}, {3763, 14064}, {3767, 32987}, {3839, 14907}, {3851, 32006}, {3855, 7750}, {3933, 5079}, {3934, 32972}, {4232, 64982}, {5059, 43459}, {5072, 7767}, {5210, 15597}, {5254, 32975}, {5286, 16921}, {5305, 63011}, {5475, 37667}, {5490, 23312}, {5491, 23311}, {6776, 51523}, {7486, 7763}, {7603, 62988}, {7610, 53418}, {7735, 44543}, {7746, 32971}, {7749, 32981}, {7752, 15022}, {7769, 46936}, {7775, 20080}, {7779, 32994}, {7782, 55864}, {7788, 61926}, {7789, 32976}, {7793, 32995}, {7795, 31275}, {7797, 33261}, {7799, 61906}, {7800, 32980}, {7802, 50689}, {7805, 31417}, {7806, 16924}, {7809, 61930}, {7811, 61944}, {7814, 67096}, {7815, 32982}, {7830, 54097}, {7836, 33270}, {7839, 31407}, {7847, 63536}, {7850, 15589}, {7851, 32957}, {7866, 39143}, {7931, 32961}, {8176, 11160}, {8355, 51438}, {8360, 63121}, {8367, 63119}, {8370, 62992}, {10008, 24206}, {10513, 32893}, {10553, 14826}, {11057, 61966}, {11159, 23053}, {11285, 63533}, {11286, 63104}, {11623, 45018}, {11737, 14929}, {12812, 32878}, {14033, 37637}, {14568, 37665}, {14651, 35705}, {14928, 20398}, {15271, 16041}, {15484, 63034}, {16043, 63534}, {17004, 33016}, {17006, 33007}, {17008, 33013}, {17128, 32998}, {20065, 33024}, {28706, 64495}, {31276, 32963}, {31400, 32999}, {31404, 33002}, {31861, 34883}, {32817, 37647}, {32818, 69405}, {32821, 32877}, {32822, 60781}, {32823, 69406}, {32830, 61914}, {32833, 61912}, {32875, 61911}, {32876, 61907}, {32884, 61886}, {32886, 61921}, {32892, 61915}, {32896, 61910}, {32962, 63048}, {32967, 53033}, {32969, 69139}, {32973, 58448}, {32974, 39565}, {32978, 44518}, {32979, 69207}, {32986, 58446}, {33023, 69141}, {33215, 53419}, {33221, 51128}, {33224, 44381}, {33239, 44535}, {33277, 46226}, {34506, 43618}, {35018, 69158}, {37668, 61924}, {37671, 61932}, {40727, 63025}, {40826, 68564}, {40927, 51538}, {42914, 69121}, {42915, 69120}, {43619, 47617}, {47061, 66587}, {47286, 62993}, {48913, 61938}, {52284, 58782}, {59634, 61889}

X(69407) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 69382, 32815}, {2, 69387, 69382}, {5, 32828, 32816}, {5, 69385, 32828}, {183, 3545, 32827}, {325, 69386, 46951}, {381, 34229, 64018}, {547, 69380, 34803}, {1007, 64093, 32836}, {1656, 69378, 32829}, {1975, 5067, 32839}, {3090, 59635, 3926}, {3091, 32832, 3785}, {3544, 69384, 7773}, {3832, 32870, 1078}, {5055, 64093, 1007}, {5071, 69386, 325}, {7486, 69383, 7763}, {7603, 63955, 62988}, {15022, 32834, 7752}, {32817, 61899, 37647}, {32826, 32883, 631}, {32827, 32885, 183}, {32829, 69378, 32824}, {34803, 69380, 32837}, {40727, 63025, 66458}, {46936, 69379, 7769}, {53127, 69387, 2}

X(69408) = X(2)X(32)∩X(5)X(1007)

Barycentrics   2*S^2 - SA^2 + 2*SB*SC : :

X(69408) lies on these lines: {2, 32}, {3, 32827}, {4, 32829}, {5, 1007}, {6, 32969}, {20, 7769}, {39, 32972}, {69, 1656}, {76, 5056}, {99, 3832}, {115, 52250}, {140, 32006}, {141, 32975}, {183, 5067}, {193, 7746}, {194, 32963}, {230, 32976}, {302, 42998}, {303, 42999}, {316, 3523}, {325, 3090}, {381, 6337}, {385, 32998}, {439, 7747}, {546, 32887}, {547, 32885}, {574, 32982}, {620, 32981}, {625, 31401}, {631, 7773}, {1975, 3545}, {2549, 32980}, {2996, 34511}, {3053, 32977}, {3054, 63932}, {3055, 7784}, {3091, 7763}, {3311, 32490}, {3312, 32491}, {3314, 32999}, {3329, 31407}, {3522, 32871}, {3525, 7750}, {3543, 7782}, {3544, 32817}, {3549, 68354}, {3618, 8361}, {3620, 7821}, {3628, 7776}, {3734, 32991}, {3767, 32988}, {3788, 31415}, {3815, 14064}, {3843, 67536}, {3850, 32889}, {3851, 6390}, {3855, 32819}, {3933, 5055}, {3964, 11484}, {3972, 33203}, {5013, 16041}, {5025, 31400}, {5032, 5346}, {5068, 11185}, {5070, 7767}, {5071, 32818}, {5079, 64093}, {5254, 11184}, {5286, 7777}, {5304, 7858}, {5319, 6722}, {5368, 63122}, {5475, 32973}, {5576, 40697}, {6392, 7764}, {6642, 68654}, {6656, 62993}, {7486, 7814}, {7504, 45962}, {7517, 44180}, {7529, 9723}, {7571, 40123}, {7603, 7795}, {7618, 65633}, {7735, 33249}, {7736, 7887}, {7737, 32989}, {7738, 33228}, {7745, 32970}, {7754, 9770}, {7755, 51170}, {7756, 54097}, {7759, 37667}, {7762, 62992}, {7768, 46935}, {7770, 37690}, {7771, 55864}, {7774, 32967}, {7778, 32968}, {7783, 33006}, {7786, 33180}, {7788, 61899}, {7789, 32983}, {7791, 17005}, {7792, 32955}, {7796, 32834}, {7797, 33277}, {7799, 61936}, {7801, 68325}, {7802, 15717}, {7803, 33199}, {7804, 31417}, {7816, 8176}, {7823, 33000}, {7825, 33023}, {7828, 37665}, {7836, 32962}, {7843, 21843}, {7856, 14930}, {7859, 33182}, {7860, 61856}, {7866, 63041}, {7868, 32957}, {7871, 61912}, {7872, 31450}, {7885, 33001}, {7891, 33016}, {7895, 66511}, {7898, 33012}, {7903, 20080}, {7904, 33003}, {7917, 32897}, {7925, 16924}, {7934, 33202}, {7940, 33181}, {7941, 17008}, {7945, 33269}, {8182, 55812}, {8781, 36519}, {8797, 52347}, {9167, 18845}, {9771, 33215}, {9781, 51439}, {10008, 14853}, {10109, 32892}, {10303, 14907}, {10304, 48913}, {10513, 32870}, {10588, 69254}, {10589, 69135}, {11057, 15721}, {11059, 52284}, {11174, 32951}, {11318, 31406}, {11585, 40680}, {12812, 32888}, {14001, 44377}, {14712, 33206}, {14929, 48154}, {15022, 32830}, {15031, 69402}, {15513, 44678}, {15589, 46936}, {15815, 33238}, {16043, 31489}, {16922, 16990}, {17128, 33005}, {17578, 62362}, {18581, 69157}, {18582, 69165}, {18840, 60268}, {21841, 63155}, {22110, 69139}, {23053, 63950}, {23334, 33274}, {30435, 63104}, {30744, 68355}, {31275, 51171}, {31276, 33009}, {31455, 32990}, {31467, 33184}, {31859, 63533}, {32820, 32876}, {32821, 52713}, {32822, 61945}, {32833, 61924}, {32841, 69404}, {32868, 69386}, {32873, 50689}, {32875, 69406}, {32877, 61919}, {32886, 35018}, {32896, 61926}, {32966, 43448}, {32979, 69206}, {32985, 65630}, {33190, 55794}, {33194, 55762}, {33239, 53418}, {33247, 53095}, {33261, 46226}, {33272, 37512}, {33283, 39266}, {35287, 66466}, {37242, 42788}, {37671, 61895}, {37688, 61886}, {41106, 59634}, {43459, 61834}, {44144, 55079}, {46236, 61575}, {63955, 69197}, {68034, 69038}

X(69408) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7752, 32816}, {2, 32816, 3785}, {3, 34803, 32839}, {5, 1007, 3926}, {5, 3926, 69382}, {5, 69158, 69378}, {69, 1656, 32838}, {76, 63098, 32825}, {183, 5067, 32867}, {325, 3090, 32828}, {381, 6337, 32826}, {625, 31401, 32974}, {631, 7773, 64018}, {1007, 69378, 69158}, {2548, 7862, 2}, {3055, 7784, 32978}, {3091, 7763, 32815}, {3628, 7776, 34229}, {3628, 34229, 32883}, {3788, 31415, 32971}, {3832, 32835, 99}, {3933, 5055, 69385}, {3933, 69385, 46951}, {5025, 63083, 31400}, {5056, 63098, 76}, {5067, 32823, 183}, {5068, 32831, 11185}, {5071, 32818, 59635}, {7486, 37668, 32832}, {7603, 7795, 32987}, {7764, 43620, 6392}, {7773, 37647, 631}, {7777, 32961, 5286}, {7796, 53127, 32834}, {7814, 32832, 37668}, {7925, 16924, 53033}, {11185, 32831, 32824}, {15022, 32830, 69387}, {32818, 59635, 32836}, {32827, 32839, 3}, {32834, 61914, 53127}, {32972, 63077, 39}, {32988, 62988, 3767}, {34511, 39565, 2996}, {69158, 69378, 3926}

X(69409) = X(2)X(3108)∩X(3)X(69)

Barycentrics   2*b^2*c^2 + SA^2 - 2*SB*SC : :

X(69409) = 3 X[2] - 4 X[7869]

X(69409) lies on these lines: {2, 3108}, {3, 69}, {4, 7788}, {5, 46951}, {20, 7768}, {32, 20080}, {39, 3620}, {76, 3091}, {83, 63091}, {99, 32840}, {183, 3525}, {193, 5007}, {274, 37436}, {298, 37177}, {299, 37178}, {315, 3146}, {316, 50688}, {325, 3090}, {340, 7487}, {376, 32820}, {381, 32892}, {385, 53033}, {439, 7863}, {524, 14001}, {538, 32974}, {546, 7776}, {599, 16043}, {626, 6392}, {631, 32821}, {632, 32839}, {754, 32981}, {1007, 3628}, {1078, 32831}, {1595, 52710}, {1656, 32885}, {1975, 3529}, {1992, 7819}, {1995, 40123}, {2548, 7916}, {2549, 7896}, {3053, 3630}, {3088, 44134}, {3095, 40330}, {3303, 69093}, {3304, 69094}, {3314, 5286}, {3522, 7811}, {3523, 7799}, {3528, 59634}, {3543, 7860}, {3544, 32823}, {3619, 9605}, {3627, 32006}, {3631, 5013}, {3767, 7895}, {3788, 37667}, {3832, 7809}, {3934, 62988}, {5047, 45962}, {5056, 7814}, {5068, 32874}, {5072, 32888}, {5079, 32886}, {5304, 7832}, {5739, 21516}, {5921, 30270}, {6179, 33181}, {6919, 18145}, {7620, 32996}, {7735, 7881}, {7737, 7882}, {7738, 7879}, {7739, 7849}, {7750, 17538}, {7752, 15022}, {7754, 8363}, {7757, 33202}, {7759, 32971}, {7763, 10303}, {7769, 61863}, {7771, 61798}, {7773, 32878}, {7774, 68522}, {7775, 32991}, {7779, 68525}, {7780, 32989}, {7781, 33023}, {7782, 62083}, {7789, 22331}, {7800, 7813}, {7801, 11160}, {7802, 62152}, {7818, 32982}, {7821, 32972}, {7822, 41940}, {7826, 69206}, {7836, 63046}, {7837, 16898}, {7840, 16924}, {7846, 63005}, {7850, 49140}, {7854, 15810}, {7870, 9740}, {7871, 32832}, {7873, 33272}, {7880, 63934}, {7883, 33025}, {7889, 63123}, {7890, 51170}, {7892, 63093}, {7905, 37665}, {7906, 16990}, {7908, 69207}, {7912, 50570}, {7917, 11185}, {7922, 33180}, {7945, 63048}, {7946, 14035}, {7947, 17008}, {8024, 31099}, {8359, 50990}, {8361, 63954}, {8362, 21356}, {8369, 50992}, {8588, 51585}, {8591, 33209}, {8667, 32970}, {8716, 33226}, {8721, 62174}, {8781, 38740}, {9466, 32987}, {9766, 32968}, {9770, 32992}, {9939, 33244}, {10300, 19583}, {10983, 61545}, {11008, 30435}, {11163, 32957}, {11174, 18840}, {11291, 32809}, {11292, 32808}, {11541, 32822}, {12103, 14929}, {12322, 18511}, {12323, 18509}, {13468, 32977}, {14033, 63932}, {14042, 23334}, {14043, 63065}, {14064, 63933}, {14069, 14614}, {14555, 21496}, {14568, 33199}, {14907, 62097}, {15513, 51579}, {15515, 51587}, {15533, 32985}, {16041, 63923}, {16045, 41624}, {16051, 45201}, {17130, 32979}, {17271, 56737}, {17378, 37176}, {17698, 63110}, {18141, 21519}, {19570, 33283}, {20065, 68517}, {20081, 32452}, {22110, 32976}, {22165, 33215}, {22329, 33189}, {28419, 28436}, {31276, 31404}, {31457, 55729}, {32458, 38664}, {32819, 32890}, {32835, 61848}, {32841, 61804}, {32867, 60781}, {32872, 53127}, {32876, 61807}, {32879, 62078}, {32883, 34803}, {32884, 55858}, {32887, 61850}, {32889, 61840}, {32893, 61914}, {32954, 63034}, {32980, 63924}, {33002, 41136}, {33190, 66458}, {33217, 63006}, {33224, 63951}, {33233, 63029}, {33237, 63064}, {33239, 63938}, {33254, 66699}, {33260, 53142}, {35927, 63935}, {37188, 64062}, {37688, 61870}, {38745, 54103}, {40268, 51538}, {40691, 63129}, {42988, 63105}, {42989, 63106}, {43136, 62996}, {44230, 54132}, {46226, 63017}, {46236, 51523}, {47061, 50989}, {49136, 67536}, {50771, 69139}, {55626, 59548}, {59257, 63176}, {59349, 65711}, {60285, 62893}, {61834, 62362}, {63077, 69197}, {63940, 68527}, {69386, 69406}, {69387, 69404}

X(69409) = reflection of X(5319) in X(7869)
X(69409) = anticomplement of X(5319)
X(69409) = isotomic conjugate of the polar conjugate of X(3619)
X(69409) = X(1973)-isoconjugate of X(18841)
X(69409) = X(i)-Dao conjugate of X(j) for these (i,j): {3619, 7714}, {6337, 18841}, {47355, 62976}
X(69409) = crossdifference of every pair of points on line {2489, 8664}
X(69409) = barycentric product X(i)*X(j) for these {i,j}: {69, 3619}, {304, 62833}, {305, 9605}, {3926, 7378}
X(69409) = barycentric quotient X(i)/X(j) for these {i,j}: {69, 18841}, {3619, 4}, {4558, 58102}, {7378, 393}, {7716, 2207}, {9605, 25}, {62833, 19}
X(69409) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7796, 32825}, {20, 32833, 32824}, {69, 1238, 40680}, {69, 3926, 3785}, {69, 3933, 3926}, {69, 6337, 7767}, {76, 32816, 69382}, {76, 37668, 32816}, {183, 32818, 32829}, {315, 32830, 32815}, {325, 69377, 32828}, {631, 32821, 32837}, {1007, 69381, 32838}, {5319, 7869, 2}, {7758, 7794, 2}, {7768, 32833, 20}, {7776, 69378, 32827}, {7795, 7855, 193}, {7801, 14023, 32973}, {7821, 63955, 32972}, {7854, 34511, 32990}, {7871, 32832, 63098}, {7890, 69209, 51170}, {7906, 16990, 31400}, {10513, 32830, 315}, {11160, 32973, 14023}, {32006, 69380, 32826}, {32821, 37671, 631}, {32826, 32877, 69380}, {32954, 63926, 63034}, {34229, 69158, 32839}

X(69410) = X(2)X(3108)∩X(4)X(69)

Barycentrics   2*b^2*c^2 + S^2 + SA^2 - 2*SB*SC : :
X(69410) = 7 X[3523] - 6 X[37479], 2 X[8550] - 3 X[35423]

X(69410) lies on the cubic K917 and these lines: {2, 3108}, {3, 32820}, {4, 69}, {5, 7788}, {7, 33941}, {8, 33934}, {20, 7811}, {32, 14037}, {39, 16990}, {83, 193}, {99, 3522}, {115, 7896}, {140, 183}, {141, 7754}, {148, 7929}, {190, 56744}, {194, 7800}, {230, 7881}, {274, 37462}, {290, 42021}, {298, 11290}, {299, 11289}, {305, 46336}, {319, 3673}, {320, 4385}, {325, 1656}, {343, 11331}, {384, 14023}, {385, 7795}, {394, 52289}, {458, 64062}, {491, 8960}, {492, 58866}, {524, 7770}, {538, 7791}, {543, 32997}, {550, 1975}, {599, 6656}, {626, 17131}, {631, 7799}, {635, 43455}, {636, 43454}, {648, 56865}, {668, 56879}, {671, 32982}, {754, 14035}, {1003, 63928}, {1007, 7871}, {1078, 3523}, {1204, 69178}, {1494, 64252}, {1506, 7916}, {1657, 7750}, {1992, 7878}, {2478, 18145}, {2481, 43745}, {2548, 7779}, {2549, 2896}, {2996, 43676}, {3090, 7814}, {3091, 7809}, {3096, 3620}, {3146, 32869}, {3314, 3767}, {3519, 54124}, {3529, 11057}, {3533, 7769}, {3543, 32892}, {3618, 7894}, {3619, 7859}, {3630, 7745}, {3631, 5254}, {3734, 7826}, {3760, 4857}, {3761, 5270}, {3788, 17008}, {3832, 32874}, {3849, 33280}, {3850, 7773}, {3851, 7776}, {3854, 69382}, {3855, 48913}, {3934, 7774}, {4201, 48838}, {4232, 33651}, {4911, 17361}, {5007, 16898}, {5015, 17360}, {5025, 63955}, {5032, 10302}, {5056, 7752}, {5059, 7802}, {5068, 7917}, {5073, 32819}, {5084, 18146}, {5094, 45201}, {5304, 7846}, {5305, 7868}, {5306, 33217}, {5309, 7849}, {5346, 7915}, {5355, 7914}, {5395, 60640}, {5475, 7882}, {5485, 60209}, {5976, 52090}, {6179, 14001}, {6194, 8721}, {6249, 55720}, {6292, 7798}, {6337, 7771}, {6390, 15712}, {6392, 7790}, {6658, 9939}, {6661, 63951}, {6680, 63048}, {6770, 11129}, {6773, 11128}, {6823, 40996}, {7270, 20925}, {7375, 32811}, {7376, 32810}, {7388, 32808}, {7389, 32809}, {7470, 54173}, {7486, 32885}, {7495, 34254}, {7509, 68660}, {7615, 32993}, {7618, 33022}, {7620, 54097}, {7735, 7832}, {7736, 7905}, {7737, 7893}, {7738, 7831}, {7739, 7876}, {7746, 7895}, {7748, 7848}, {7749, 7908}, {7753, 33269}, {7757, 16043}, {7759, 9466}, {7762, 40341}, {7765, 7865}, {7766, 46226}, {7775, 32962}, {7780, 7801}, {7781, 7810}, {7782, 21735}, {7784, 47286}, {7787, 50248}, {7793, 69206}, {7805, 7822}, {7807, 8667}, {7808, 7890}, {7812, 11160}, {7813, 7815}, {7818, 14063}, {7819, 14614}, {7821, 32961}, {7824, 34511}, {7827, 21356}, {7836, 69207}, {7837, 68522}, {7839, 16986}, {7840, 16921}, {7841, 22165}, {7843, 33016}, {7857, 37667}, {7858, 32968}, {7862, 12815}, {7863, 32964}, {7866, 63954}, {7870, 32970}, {7873, 14711}, {7877, 20080}, {7883, 32974}, {7884, 33221}, {7889, 63045}, {7891, 21843}, {7906, 31401}, {7911, 43448}, {7912, 43620}, {7922, 14064}, {7933, 19570}, {7936, 32986}, {7939, 14045}, {7940, 62992}, {7941, 31415}, {7945, 63047}, {7946, 16044}, {7947, 17004}, {8024, 16063}, {8182, 33276}, {8352, 50989}, {8370, 15533}, {8550, 35423}, {9740, 33181}, {9744, 49111}, {9766, 32992}, {9770, 32975}, {10303, 32837}, {10327, 33933}, {10519, 12203}, {11008, 60855}, {11054, 33190}, {11286, 63936}, {11303, 41112}, {11304, 41113}, {11623, 32458}, {12150, 33198}, {13468, 33233}, {13740, 17378}, {14042, 44678}, {14068, 63931}, {14069, 63034}, {14148, 15515}, {14789, 52149}, {14929, 62036}, {14976, 68420}, {15022, 32893}, {15031, 32827}, {15118, 55225}, {15692, 32896}, {16062, 17271}, {16712, 56737}, {16895, 63038}, {17007, 26978}, {17170, 33939}, {17297, 33838}, {17346, 17681}, {17907, 41366}, {18135, 37162}, {18140, 45962}, {18546, 32996}, {18841, 60642}, {19687, 63938}, {19689, 44367}, {19768, 49716}, {20023, 52568}, {20062, 42052}, {20924, 54433}, {22329, 32954}, {22332, 66417}, {23055, 32959}, {28419, 53025}, {30435, 50251}, {31168, 33202}, {32822, 62147}, {32823, 61921}, {32826, 32878}, {32829, 61856}, {32831, 61834}, {32838, 46935}, {32840, 43459}, {32875, 61783}, {32877, 62110}, {32880, 62124}, {32882, 50690}, {32978, 42850}, {33007, 63935}, {33192, 66455}, {33195, 63107}, {33206, 34506}, {33226, 55164}, {33229, 34505}, {33230, 50994}, {33239, 51224}, {33254, 47101}, {33255, 35007}, {33257, 47102}, {33258, 53096}, {33279, 63922}, {33769, 40050}, {33935, 41826}, {34384, 40421}, {36998, 64653}, {37647, 55860}, {37688, 46219}, {39646, 48876}, {39978, 40022}, {39998, 62937}, {42992, 69137}, {42993, 69145}, {44519, 47287}, {47005, 63006}, {49135, 64018}, {50567, 53765}, {52695, 68514}, {53102, 60143}, {59780, 68177}, {60145, 60210}, {60146, 60639}, {60183, 62941}, {62171, 67536}, {63065, 63953}, {63083, 69197}, {63950, 68527}, {69130, 69166}, {69131, 69159}, {69386, 69403}

X(69410) = reflection of X(i) in X(j) for these {i,j}: {7791, 7854}, {14035, 17130}
X(69410) = isotomic conjugate of X(43726)
X(69410) = anticomplement of X(7772)
X(69410) = anticomplement of the isogonal conjugate of X(43527)
X(69410) = isotomic conjugate of the isogonal conjugate of X(7485)
X(69410) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 41917}, {7954, 4560}, {39955, 192}, {43527, 8}, {56034, 2}, {56072, 6360}, {65031, 21289}, {65032, 4329}
X(69410) = X(31)-isoconjugate of X(43726)
X(69410) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 43726}, {26880, 33578}
X(69410) = cevapoint of X(69) and X(41916)
X(69410) = crosssum of X(8665) and X(20975)
X(69410) = crossdifference of every pair of points on line {3049, 8664}
X(69410) = barycentric product X(i)*X(j) for these {i,j}: {76, 7485}, {1502, 64028}, {14259, 40022}, {32830, 52455}
X(69410) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 43726}, {7485, 6}, {14259, 39951}, {52455, 52223}, {64028, 32}
X(69410) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 69, 7768}, {4, 7768, 315}, {69, 76, 315}, {69, 1232, 317}, {69, 22468, 38437}, {69, 52713, 7850}, {69, 69377, 76}, {76, 315, 11185}, {76, 316, 69378}, {76, 7768, 4}, {140, 3933, 32821}, {140, 32821, 7763}, {141, 7754, 7803}, {183, 3933, 7763}, {183, 32821, 140}, {194, 63044, 7800}, {325, 69381, 32832}, {599, 63933, 6656}, {626, 69162, 33283}, {633, 634, 1352}, {637, 638, 3818}, {1232, 1235, 76}, {1975, 7767, 14907}, {1992, 16045, 7878}, {2896, 20081, 2549}, {3314, 17129, 3767}, {3522, 32824, 99}, {3522, 32830, 32824}, {3620, 5286, 3096}, {3631, 5254, 7879}, {3734, 7826, 20065}, {3785, 32824, 3522}, {3785, 32830, 99}, {3926, 15589, 1078}, {3934, 7855, 7774}, {5007, 63934, 63093}, {7751, 7794, 2}, {7751, 7869, 7755}, {7752, 32828, 53127}, {7755, 7794, 7869}, {7755, 7869, 2}, {7759, 9466, 16924}, {7766, 46226, 69209}, {7779, 31276, 2548}, {7780, 7801, 16925}, {7781, 7810, 32965}, {7805, 7822, 16989}, {7808, 7890, 63017}, {7818, 63924, 14063}, {7819, 63926, 14614}, {7849, 63925, 5309}, {7850, 32006, 315}, {7893, 17128, 7737}, {7917, 69387, 32816}, {7922, 14568, 14064}, {10303, 32837, 62362}, {10513, 32816, 7917}, {10513, 32834, 32816}, {16898, 63093, 5007}, {20080, 69208, 7877}, {32816, 32834, 69387}, {32818, 34229, 7769}, {32827, 32868, 69383}, {32827, 69383, 15031}, {32828, 37668, 7752}, {37667, 53033, 7857}, {40123, 41916, 40022}, {40341, 69139, 7762}

X(69411) = X(2)X(3108)∩X(5)X(69)

Barycentrics   2*b^2*c^2 + 2*S^2 + SA^2 - 2*SB*SC : :
X(69411) = 3 X[32990] - 5 X[55729]

X(69411) lies on these lines: {2, 3108}, {3, 32824}, {4, 37671}, {5, 69}, {20, 76}, {83, 63042}, {99, 21734}, {140, 32837}, {141, 33221}, {183, 631}, {193, 3934}, {194, 33258}, {230, 33222}, {315, 3832}, {316, 61982}, {325, 5067}, {376, 32892}, {382, 7767}, {385, 16898}, {452, 18145}, {524, 32968}, {538, 32990}, {548, 32878}, {549, 32896}, {599, 14064}, {754, 32979}, {1007, 5070}, {1078, 15717}, {1232, 41009}, {1235, 37122}, {1598, 52710}, {1975, 3528}, {1992, 63926}, {2548, 20080}, {2896, 43448}, {2996, 7761}, {3089, 44134}, {3090, 7788}, {3091, 7768}, {3146, 7811}, {3314, 33248}, {3522, 32869}, {3523, 32833}, {3524, 32820}, {3525, 32821}, {3526, 3933}, {3530, 6337}, {3598, 33941}, {3619, 5305}, {3620, 3767}, {3631, 13881}, {3760, 4309}, {3761, 4317}, {3839, 7860}, {3843, 32006}, {3855, 32827}, {3861, 14929}, {5013, 15598}, {5020, 41927}, {5032, 63953}, {5068, 7809}, {5129, 18146}, {5286, 7876}, {6179, 9740}, {6390, 32875}, {6392, 7765}, {6776, 49111}, {7172, 33940}, {7393, 68660}, {7486, 7814}, {7493, 41916}, {7516, 52437}, {7610, 32977}, {7620, 33019}, {7689, 69176}, {7735, 33217}, {7737, 63930}, {7739, 63925}, {7750, 32826}, {7752, 10513}, {7759, 11160}, {7763, 55864}, {7771, 61788}, {7773, 61945}, {7774, 31407}, {7779, 31404}, {7780, 32973}, {7782, 58188}, {7792, 18840}, {7795, 37667}, {7799, 10303}, {7801, 32989}, {7802, 32894}, {7803, 55738}, {7805, 51171}, {7807, 63029}, {7808, 51170}, {7810, 33023}, {7815, 31450}, {7818, 32980}, {7819, 63034}, {7821, 32988}, {7832, 37689}, {7836, 33262}, {7840, 32999}, {7846, 63097}, {7854, 32974}, {7855, 62988}, {7866, 21356}, {7881, 62992}, {7882, 31415}, {7883, 33200}, {7896, 43620}, {7917, 53127}, {7922, 33199}, {7933, 63044}, {7936, 33210}, {7946, 32962}, {8359, 66458}, {8360, 50994}, {8362, 63954}, {8367, 63064}, {8556, 31492}, {8667, 14001}, {9466, 14023}, {9606, 15271}, {9607, 16043}, {9766, 32975}, {9939, 14068}, {10008, 40107}, {10299, 59634}, {11057, 49135}, {11185, 17578}, {11285, 42850}, {11311, 63105}, {11312, 63106}, {11313, 32811}, {11314, 32810}, {11318, 50990}, {13468, 32970}, {14033, 63928}, {14069, 22329}, {14568, 33180}, {14614, 16045}, {14994, 63722}, {16239, 69158}, {16895, 63065}, {16896, 16989}, {16992, 17552}, {17008, 33245}, {17130, 32981}, {18546, 54097}, {18806, 20065}, {22165, 32984}, {28706, 44149}, {31276, 33269}, {31457, 34511}, {32817, 32877}, {32818, 32839}, {32819, 49138}, {32822, 62113}, {32823, 69405}, {32831, 61842}, {32840, 61816}, {32872, 69387}, {32876, 61837}, {32880, 43459}, {32882, 62102}, {32883, 61881}, {32890, 44682}, {32957, 41624}, {32982, 63924}, {32983, 63932}, {32986, 63923}, {33238, 34505}, {33257, 66699}, {33290, 50570}, {34803, 48154}, {37339, 48838}, {37647, 52718}, {39647, 64653}, {46226, 63048}, {59780, 68513}, {62155, 67536}, {63093, 68522}, {63936, 66415}

X(69411) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 32836, 32824}, {69, 32828, 32816}, {69, 69381, 32828}, {69, 69385, 7776}, {76, 3785, 32815}, {76, 14907, 69379}, {76, 15589, 3785}, {183, 69377, 3926}, {315, 32834, 69382}, {325, 69384, 32838}, {3933, 34229, 32829}, {7486, 37668, 7814}, {7750, 52713, 32826}, {7767, 69378, 64018}, {7800, 17131, 6392}, {7814, 32832, 7486}, {7854, 63955, 32974}, {9466, 14023, 32971}, {9740, 33198, 6179}, {11160, 32987, 7759}, {16990, 17129, 5286}, {31276, 63046, 69208}, {32818, 37688, 32839}, {32826, 32888, 52713}, {32827, 32886, 59635}, {32831, 61842, 62362}, {32868, 64018, 69378}

X(69412) = X(2)X(1975)∩X(5)X(183)

Barycentrics   2*b^2*c^2 + 2*S^2 - SA^2 + SB*SC : :

X(69412) lies on these lines: {2, 1975}, {3, 69387}, {4, 32838}, {5, 183}, {6, 16921}, {20, 32897}, {32, 44543}, {69, 5056}, {76, 1656}, {83, 11167}, {99, 3526}, {115, 11285}, {140, 11185}, {141, 32961}, {148, 15815}, {194, 16922}, {230, 16924}, {302, 69180}, {303, 69186}, {316, 3851}, {325, 3090}, {381, 1078}, {382, 7771}, {384, 37637}, {385, 33002}, {491, 23311}, {492, 23312}, {546, 14907}, {547, 3933}, {599, 7912}, {625, 7879}, {631, 32819}, {1003, 7749}, {1007, 7486}, {1506, 7754}, {1657, 43459}, {2548, 14614}, {2896, 8556}, {3053, 16044}, {3054, 16925}, {3055, 63923}, {3091, 7750}, {3096, 11318}, {3491, 61689}, {3524, 32826}, {3525, 32815}, {3544, 32827}, {3545, 3785}, {3552, 17006}, {3619, 33199}, {3628, 7763}, {3734, 33233}, {3761, 65141}, {3763, 7901}, {3767, 11174}, {3815, 32999}, {3843, 7802}, {3855, 64018}, {3926, 5067}, {3934, 7867}, {4193, 16992}, {5023, 11361}, {5025, 15271}, {5054, 7782}, {5055, 7752}, {5068, 32006}, {5070, 7769}, {5071, 32816}, {5079, 7776}, {5094, 54412}, {5154, 37670}, {5210, 6658}, {5286, 32975}, {5569, 66395}, {5585, 33268}, {5976, 7866}, {5989, 38224}, {6179, 15484}, {6292, 33219}, {6390, 55856}, {6392, 62993}, {6656, 43620}, {6680, 7746}, {6683, 69162}, {6722, 7822}, {6933, 37664}, {7504, 18135}, {7539, 40022}, {7571, 39998}, {7603, 7751}, {7610, 7793}, {7617, 7815}, {7735, 32987}, {7745, 17008}, {7757, 31467}, {7762, 31415}, {7768, 61919}, {7774, 33009}, {7777, 63933}, {7778, 31276}, {7784, 32966}, {7785, 8667}, {7791, 58446}, {7792, 32968}, {7795, 33249}, {7796, 61905}, {7799, 15703}, {7800, 33228}, {7803, 43291}, {7806, 33020}, {7809, 61920}, {7811, 19709}, {7814, 61911}, {7820, 12815}, {7823, 33024}, {7824, 44518}, {7825, 39601}, {7827, 51588}, {7830, 18424}, {7834, 51580}, {7844, 31239}, {7857, 11286}, {7860, 61937}, {7861, 18362}, {7862, 7881}, {7869, 31275}, {7871, 61908}, {7876, 44536}, {7885, 33011}, {7886, 33217}, {7900, 32994}, {7904, 32993}, {7905, 63954}, {7906, 11184}, {7918, 9166}, {7926, 63936}, {7932, 12215}, {7937, 33241}, {7941, 40341}, {7944, 33240}, {7999, 51440}, {8370, 8860}, {9605, 14568}, {9766, 17129}, {11056, 11284}, {11164, 33274}, {11168, 33006}, {11317, 34506}, {11623, 35705}, {12110, 60101}, {13468, 20065}, {14065, 46236}, {15022, 15589}, {15513, 66387}, {15515, 63922}, {15597, 33007}, {15699, 32833}, {15717, 67536}, {16509, 31406}, {16966, 69157}, {16967, 69165}, {16989, 33261}, {16990, 32963}, {16999, 33060}, {17005, 20081}, {18546, 37512}, {18584, 33010}, {19687, 21843}, {20112, 33192}, {22329, 69208}, {22332, 50570}, {30737, 63657}, {31246, 60706}, {31263, 32092}, {31401, 47286}, {31404, 41624}, {31455, 31859}, {32457, 53096}, {32459, 33206}, {32817, 32839}, {32818, 46951}, {32820, 32829}, {32822, 61867}, {32823, 69405}, {32824, 32884}, {32825, 32886}, {32830, 34803}, {32831, 46935}, {32836, 61895}, {32837, 61889}, {32869, 61897}, {32872, 63098}, {32893, 61906}, {32965, 53419}, {32971, 62992}, {32976, 53033}, {32978, 43448}, {32990, 63533}, {32995, 53418}, {32998, 44377}, {33000, 59545}, {33001, 63548}, {33004, 44526}, {33180, 39143}, {33182, 63121}, {33198, 63104}, {33234, 69141}, {33273, 44519}, {35955, 47617}, {37638, 57008}, {37668, 61914}, {40410, 44149}, {42580, 69145}, {42581, 69137}, {42786, 51371}, {43457, 63935}, {44146, 52296}, {46318, 68522}, {48913, 61933}, {49102, 53765}, {51127, 59552}, {51128, 59548}, {51520, 54393}, {53023, 60702}, {61878, 62362}, {62981, 64982}, {66416, 69209}

X(69412) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 13881, 7851}, {2, 59635, 1975}, {2, 69383, 6337}, {2, 69385, 59635}, {4, 32838, 37688}, {5, 183, 7773}, {5, 32832, 183}, {148, 33015, 15815}, {194, 16922, 31489}, {631, 69382, 32819}, {1506, 7754, 11163}, {3090, 32828, 325}, {3091, 32870, 34229}, {3091, 34229, 7750}, {3552, 17006, 44535}, {3628, 64093, 7763}, {3767, 32992, 11174}, {3926, 5067, 37647}, {3934, 7887, 7868}, {5055, 69381, 7752}, {5067, 69386, 3926}, {5070, 69380, 7769}, {5071, 32885, 37671}, {5071, 69384, 32816}, {6722, 7822, 33218}, {7486, 32834, 1007}, {7610, 65630, 7793}, {7617, 7815, 39565}, {7752, 69381, 7788}, {7771, 15031, 382}, {7793, 33013, 65630}, {7815, 39565, 7841}, {7862, 9466, 7881}, {16044, 17004, 3053}, {17008, 32962, 7745}, {31276, 32967, 7778}, {31455, 63924, 31859}, {32815, 32883, 3525}, {32816, 32885, 69384}, {32816, 69384, 37671}, {32817, 60781, 32839}, {32829, 52713, 32820}, {32830, 46936, 34803}, {32832, 53127, 5}, {32867, 69382, 631}, {52713, 61886, 32829}, {58446, 63534, 7791}, {59634, 69379, 1975}

X(69413) = X(2)X(6)∩X(3)X(316)

Barycentrics   2*S^2 - SA^2 + SB*SC : :
X(69413) = 4 X[3054] - 3 X[8860], 3 X[8860] - 2 X[17008]

X(69413) lies on these lines: {2, 6}, {3, 316}, {4, 32829}, {5, 1975}, {17, 69165}, {18, 69157}, {20, 53017}, {23, 44180}, {25, 32002}, {32, 33233}, {39, 7844}, {51, 51439}, {76, 1656}, {83, 7940}, {95, 64982}, {98, 39899}, {99, 381}, {110, 19156}, {114, 3818}, {115, 31859}, {140, 315}, {147, 9756}, {148, 8716}, {160, 68706}, {187, 7775}, {194, 13881}, {262, 5976}, {264, 2970}, {290, 41259}, {305, 7539}, {311, 3266}, {317, 468}, {327, 3978}, {340, 52292}, {376, 32827}, {382, 7782}, {383, 14145}, {458, 65722}, {498, 69254}, {499, 69135}, {543, 18424}, {547, 32833}, {549, 14907}, {574, 625}, {576, 15850}, {598, 64019}, {620, 1003}, {621, 37463}, {622, 37464}, {626, 11285}, {631, 7750}, {632, 7767}, {671, 11165}, {850, 34291}, {858, 20477}, {1078, 3526}, {1080, 14144}, {1232, 26235}, {1235, 52296}, {1351, 38227}, {1384, 7812}, {1506, 3788}, {1513, 31670}, {1799, 63173}, {1853, 57275}, {1916, 60233}, {1995, 7664}, {2482, 8176}, {2548, 7807}, {2549, 33228}, {2896, 33015}, {3053, 7785}, {3060, 51440}, {3090, 3926}, {3091, 6337}, {3106, 40335}, {3107, 40334}, {3260, 46127}, {3523, 32006}, {3524, 64018}, {3525, 3785}, {3533, 32884}, {3534, 48913}, {3544, 32822}, {3545, 32815}, {3552, 65630}, {3628, 3933}, {3705, 5564}, {3734, 7603}, {3760, 65141}, {3767, 33249}, {3832, 32873}, {3839, 67536}, {3849, 8588}, {3855, 32826}, {3934, 7881}, {3964, 11284}, {3972, 11288}, {4045, 33219}, {4232, 63155}, {5013, 5025}, {5023, 7823}, {5024, 7790}, {5031, 50659}, {5054, 7771}, {5055, 7799}, {5056, 32820}, {5067, 32818}, {5070, 7796}, {5071, 32817}, {5072, 15031}, {5085, 5207}, {5152, 38743}, {5159, 41005}, {5169, 11671}, {5206, 7843}, {5210, 14712}, {5254, 32961}, {5286, 32969}, {5309, 6722}, {5476, 50567}, {5503, 60211}, {5611, 11132}, {5615, 11133}, {5640, 51383}, {5866, 7527}, {5886, 69038}, {5939, 6054}, {5943, 51386}, {5999, 8350}, {6032, 65595}, {6036, 9755}, {6393, 14561}, {6527, 30769}, {6563, 65610}, {6655, 15815}, {6656, 31401}, {6683, 7867}, {6721, 32458}, {6781, 63956}, {6786, 61742}, {7179, 7321}, {7484, 15574}, {7486, 32830}, {7495, 44128}, {7504, 34284}, {7569, 28706}, {7570, 9464}, {7571, 8024}, {7607, 60198}, {7617, 39785}, {7618, 8352}, {7622, 8589}, {7694, 54996}, {7737, 35297}, {7738, 32972}, {7745, 16925}, {7746, 7754}, {7747, 33235}, {7749, 7759}, {7751, 69197}, {7753, 31274}, {7757, 14061}, {7762, 69207}, {7768, 46219}, {7772, 7886}, {7780, 7903}, {7781, 39565}, {7783, 32966}, {7784, 7824}, {7786, 7866}, {7787, 33245}, {7789, 16924}, {7793, 7941}, {7795, 32992}, {7803, 8361}, {7804, 33220}, {7808, 7874}, {7811, 15694}, {7815, 7821}, {7825, 33234}, {7828, 9605}, {7833, 53095}, {7834, 9698}, {7835, 11286}, {7836, 16921}, {7842, 15515}, {7853, 15482}, {7857, 7858}, {7860, 15720}, {7861, 53096}, {7864, 22332}, {7869, 31239}, {7871, 55857}, {7872, 31652}, {7885, 33004}, {7891, 16044}, {7898, 33273}, {7900, 63938}, {7906, 63933}, {7908, 9466}, {7913, 44562}, {7917, 55858}, {7918, 33241}, {7919, 33240}, {7933, 31492}, {7934, 11287}, {7942, 55085}, {7945, 68522}, {7947, 16922}, {7949, 63936}, {8370, 31415}, {8591, 66587}, {8597, 66616}, {8598, 43618}, {8786, 10811}, {8842, 40810}, {9114, 9760}, {9116, 9762}, {9149, 20794}, {9220, 62298}, {9596, 26686}, {9599, 26629}, {9606, 33248}, {9607, 33277}, {9734, 13449}, {9744, 48906}, {9753, 10011}, {9832, 67615}, {10155, 60212}, {10184, 57518}, {10303, 32898}, {10335, 44536}, {10352, 42535}, {10484, 10487}, {10486, 15069}, {10516, 12215}, {10567, 31277}, {10607, 17035}, {11057, 15693}, {11159, 41134}, {11171, 39266}, {11188, 12093}, {11331, 66354}, {11539, 14929}, {11669, 60101}, {11742, 66421}, {12040, 37350}, {12833, 41330}, {13862, 46236}, {14001, 31404}, {14035, 59545}, {14041, 44526}, {14063, 63548}, {14064, 31400}, {14494, 40824}, {14568, 22253}, {14853, 51374}, {14995, 35520}, {15022, 69379}, {15059, 52693}, {15301, 47617}, {15513, 63931}, {15561, 35930}, {15631, 67220}, {15655, 51224}, {15980, 63424}, {16042, 52437}, {16051, 40680}, {16760, 37930}, {16966, 69121}, {16967, 69120}, {17128, 33002}, {17377, 37764}, {18362, 32457}, {18546, 39601}, {18584, 33013}, {18896, 45146}, {20023, 54002}, {20065, 33000}, {20088, 22331}, {20423, 51438}, {20885, 45093}, {22236, 62600}, {22238, 62601}, {26276, 47596}, {26590, 31497}, {30737, 30744}, {30745, 44136}, {30771, 62698}, {31152, 46724}, {31245, 60706}, {31262, 32092}, {31407, 33222}, {31479, 64133}, {31501, 69260}, {32000, 62960}, {32001, 52290}, {32432, 45565}, {32435, 45564}, {32450, 69162}, {32456, 62203}, {32459, 33007}, {32824, 61921}, {32825, 32838}, {32834, 46936}, {32836, 61899}, {32841, 61914}, {32867, 60781}, {32874, 61897}, {32883, 61881}, {32885, 61889}, {32889, 69403}, {32895, 69404}, {32896, 61904}, {32963, 59546}, {32968, 53033}, {32970, 69208}, {32984, 43448}, {33006, 53419}, {33019, 44519}, {33237, 60855}, {33264, 44541}, {33270, 63923}, {33651, 38434}, {33878, 40248}, {33928, 61700}, {34254, 37439}, {34511, 43620}, {36519, 62348}, {36898, 63535}, {37067, 60524}, {37182, 48881}, {37446, 43453}, {37454, 52347}, {37459, 39656}, {37532, 55469}, {38317, 51371}, {39590, 69171}, {40332, 54189}, {40428, 40820}, {40697, 62310}, {41324, 42316}, {42107, 59539}, {42110, 59540}, {42163, 59541}, {42166, 59542}, {43529, 60098}, {44134, 52293}, {44149, 55081}, {44504, 63722}, {45103, 51589}, {46951, 61895}, {47211, 59569}, {47285, 53136}, {48895, 58851}, {50149, 67603}, {50977, 51396}, {51373, 52997}, {52081, 60872}, {52193, 63731}, {52194, 63732}, {52247, 57009}, {52250, 63533}, {52284, 63551}, {52288, 55972}, {52300, 68354}, {53099, 60262}, {53108, 60248}, {53843, 62299}, {54443, 69279}, {54645, 60202}, {55646, 60654}, {56064, 60093}, {58831, 64802}, {59229, 62954}, {59552, 67865}, {60096, 60213}, {60099, 62922}, {60129, 60231}, {60187, 62881}, {60201, 60333}, {61925, 64809}, {63557, 68804}

X(69413) = midpoint of X(491) and X(492)
X(69413) = reflection of X(i) in X(j) for these {i,j}: {8860, 2}, {17008, 3054}
X(69413) = isogonal conjugate of X(61379)
X(69413) = isotomic conjugate of X(7607)
X(69413) = complement of X(17008)
X(69413) = anticomplement of X(3054)
X(69413) = anticomplement of the isotomic conjugate of X(60198)
X(69413) = complement of the isotomic conjugate of X(60234)
X(69413) = isotomic conjugate of the anticomplement of X(15850)
X(69413) = isotomic conjugate of the isogonal conjugate of X(576)
X(69413) = isotomic conjugate of the polar conjugate of X(52282)
X(69413) = X(60198)-anticomplementary conjugate of X(6327)
X(69413) = X(60234)-complementary conjugate of X(2887)
X(69413) = X(60198)-Ceva conjugate of X(2)
X(69413) = X(i)-cross conjugate of X(j) for these (i,j): {576, 52282}, {15850, 2}
X(69413) = X(i)-isoconjugate of X(j) for these (i,j): {1, 61379}, {31, 7607}, {560, 57908}, {661, 59007}, {798, 35178}
X(69413) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 7607}, {3, 61379}, {576, 41275}, {6374, 57908}, {8786, 65001}, {31998, 35178}, {35132, 523}, {36830, 59007}
X(69413) = crosspoint of X(2) and X(60234)
X(69413) = barycentric product X(i)*X(j) for these {i,j}: {69, 52282}, {76, 576}, {15850, 60198}
X(69413) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 7607}, {6, 61379}, {76, 57908}, {99, 35178}, {110, 59007}, {576, 6}, {8827, 8587}, {8859, 65001}, {15850, 3054}, {52282, 4}
X(69413) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 69, 37688}, {2, 193, 62992}, {2, 325, 183}, {2, 385, 37637}, {2, 1007, 325}, {2, 3314, 15271}, {2, 3815, 11174}, {2, 5304, 63104}, {2, 7736, 7792}, {2, 7774, 230}, {2, 7777, 6}, {2, 7778, 7868}, {2, 7779, 17004}, {2, 7840, 7610}, {2, 7925, 7778}, {2, 7931, 3763}, {2, 9740, 23053}, {2, 9770, 22329}, {2, 11184, 11163}, {2, 16990, 58446}, {2, 17005, 31489}, {2, 17008, 3054}, {2, 34803, 37647}, {2, 37668, 34229}, {2, 62988, 7735}, {2, 62994, 16984}, {2, 63018, 7806}, {2, 63021, 385}, {2, 63025, 63101}, {2, 63065, 44401}, {2, 63077, 7736}, {2, 63083, 3815}, {2, 63098, 69}, {3, 7752, 7773}, {5, 6390, 11185}, {5, 7763, 1975}, {6, 7777, 11163}, {6, 11184, 7777}, {39, 7862, 7887}, {39, 7887, 7851}, {39, 31275, 7844}, {69, 1007, 63098}, {69, 37688, 183}, {69, 63098, 325}, {76, 69158, 32821}, {83, 7940, 32954}, {141, 3055, 2}, {141, 9771, 3055}, {183, 325, 7788}, {193, 62992, 22329}, {194, 32967, 13881}, {230, 3629, 63048}, {230, 7774, 14614}, {262, 64089, 37071}, {298, 299, 15533}, {302, 303, 6}, {325, 37647, 2}, {325, 37671, 37668}, {325, 37688, 69}, {385, 63021, 9766}, {547, 64093, 53127}, {574, 625, 7841}, {598, 64019, 68718}, {620, 5475, 1003}, {626, 31455, 11285}, {631, 32816, 7750}, {1007, 34803, 2}, {1007, 37647, 183}, {1078, 7814, 7776}, {1506, 3788, 7770}, {1656, 69158, 76}, {2482, 8176, 11317}, {2482, 11317, 11164}, {3054, 17008, 8860}, {3055, 22110, 141}, {3090, 3926, 59635}, {3091, 6337, 32819}, {3091, 32835, 6337}, {3525, 32823, 3785}, {3526, 7776, 1078}, {3593, 3595, 51170}, {3628, 3933, 32832}, {3629, 63048, 14614}, {3734, 7603, 44543}, {3763, 7778, 7931}, {3763, 7931, 7868}, {3815, 44377, 2}, {3934, 7888, 7881}, {5024, 11318, 7790}, {5055, 69380, 69387}, {5056, 32831, 69378}, {5067, 32818, 32828}, {5071, 32817, 69382}, {6390, 11185, 1975}, {7486, 32830, 69385}, {7622, 31173, 35955}, {7735, 62988, 41624}, {7746, 7764, 7754}, {7752, 7769, 3}, {7763, 11185, 6390}, {7774, 63048, 3629}, {7777, 7806, 63018}, {7778, 31489, 2}, {7779, 17004, 8667}, {7783, 32966, 44518}, {7785, 7907, 3053}, {7786, 7899, 7866}, {7793, 7941, 63932}, {7793, 16923, 44535}, {7799, 69387, 69380}, {7806, 63018, 6}, {7808, 7874, 33217}, {7815, 7821, 7879}, {7823, 33259, 5023}, {7824, 7912, 7784}, {7825, 37512, 33234}, {7836, 16921, 69139}, {7844, 7862, 31275}, {7844, 31275, 7887}, {7857, 7858, 30435}, {7866, 31467, 7786}, {7925, 17005, 2}, {7925, 31489, 7868}, {7941, 16923, 7793}, {7947, 16922, 31276}, {8361, 31406, 7803}, {8781, 64089, 5976}, {9766, 37637, 385}, {9770, 62992, 193}, {9771, 22110, 2}, {11288, 15484, 3972}, {13468, 50771, 63046}, {13860, 51580, 5989}, {14712, 33274, 5210}, {31415, 69206, 8370}, {32456, 62203, 66387}, {32459, 53418, 33007}, {32805, 32806, 59373}, {32816, 32839, 631}, {32825, 32838, 69377}, {32831, 69378, 32820}, {32833, 53127, 64093}, {32837, 69382, 32817}, {32841, 61914, 69383}, {34229, 37668, 37671}, {34229, 37671, 183}, {34511, 43620, 47286}, {37690, 62993, 2}, {44377, 63083, 11174}, {44535, 63932, 7793}, {56435, 63098, 39113}, {60781, 69384, 32867}, {61886, 69377, 32838}, {62995, 63107, 63097}, {63019, 63028, 6}, {63105, 63106, 63064}

X(69414) = X(2)X(9606)∩X(3)X(69)

Barycentrics   2*b^2*c^2 - S^2 + SA^2 - SB*SC : :
X(69414) = X[32824] - 3 X[32896], 2 X[22331] - 3 X[32973]

X(69414) lies on these lines: {2, 9606}, {3, 69}, {4, 7796}, {5, 32825}, {20, 7788}, {23, 40123}, {30, 32824}, {32, 11008}, {39, 3619}, {76, 1007}, {99, 17538}, {140, 32837}, {141, 22332}, {183, 10303}, {193, 7789}, {305, 6340}, {315, 3529}, {316, 32822}, {325, 3091}, {376, 7768}, {439, 11160}, {524, 22331}, {532, 37173}, {533, 37172}, {538, 14064}, {546, 32816}, {599, 32990}, {631, 7799}, {632, 32829}, {754, 33239}, {1078, 61814}, {1273, 6803}, {1285, 7877}, {1656, 46951}, {1975, 3146}, {1992, 5007}, {2549, 7895}, {3053, 20080}, {3088, 52710}, {3303, 69094}, {3304, 69093}, {3314, 7738}, {3522, 59634}, {3523, 37671}, {3525, 7763}, {3528, 7811}, {3544, 7752}, {3545, 7814}, {3618, 7772}, {3620, 5013}, {3627, 7776}, {3628, 32828}, {3630, 5023}, {3631, 15815}, {3767, 7908}, {3788, 63104}, {4175, 4176}, {5055, 32892}, {5056, 32869}, {5070, 32885}, {5072, 32877}, {5076, 32826}, {5079, 64093}, {5210, 51579}, {5286, 7881}, {5319, 7880}, {6179, 33191}, {6392, 7778}, {7486, 32874}, {7735, 7836}, {7736, 7906}, {7737, 7916}, {7739, 7869}, {7750, 10513}, {7751, 32970}, {7754, 53033}, {7755, 33222}, {7757, 32956}, {7759, 14033}, {7760, 14069}, {7764, 32968}, {7765, 33223}, {7769, 61870}, {7771, 61795}, {7773, 50689}, {7774, 68525}, {7777, 33261}, {7779, 68517}, {7780, 33216}, {7781, 32986}, {7782, 62084}, {7786, 18840}, {7794, 16043}, {7800, 31652}, {7802, 62146}, {7807, 63034}, {7810, 51587}, {7818, 33238}, {7819, 59373}, {7820, 63073}, {7821, 16041}, {7822, 63120}, {7827, 33194}, {7837, 14037}, {7840, 14035}, {7850, 62133}, {7854, 33215}, {7855, 35007}, {7856, 32952}, {7859, 14482}, {7860, 33703}, {7863, 14023}, {7870, 33189}, {7871, 11185}, {7873, 33247}, {7883, 9741}, {7888, 32969}, {7891, 63046}, {7892, 63006}, {7896, 14148}, {7907, 63029}, {7909, 32951}, {7912, 63533}, {7917, 11541}, {7922, 33190}, {7946, 33007}, {7947, 37690}, {8357, 51122}, {8359, 50994}, {8360, 66458}, {8369, 63064}, {8591, 33271}, {8667, 32989}, {8716, 33023}, {8781, 20398}, {8798, 34403}, {9466, 32975}, {9605, 63119}, {9766, 32971}, {9770, 16924}, {9939, 33254}, {10983, 40330}, {11055, 33196}, {11057, 62127}, {11288, 63926}, {11291, 32811}, {11292, 32810}, {11433, 40691}, {11511, 39127}, {12108, 32876}, {12609, 42697}, {12811, 32890}, {12812, 32878}, {13571, 16898}, {14555, 21516}, {14568, 32955}, {14614, 33181}, {14907, 62092}, {14927, 30270}, {14929, 44245}, {15022, 59635}, {15533, 35287}, {15589, 32841}, {15702, 62362}, {15704, 64018}, {16865, 45962}, {17129, 62992}, {17130, 32983}, {17271, 37339}, {18141, 21540}, {19570, 33248}, {19583, 45201}, {22110, 32988}, {22329, 33203}, {23055, 33233}, {23235, 32458}, {30435, 62996}, {31276, 62993}, {31407, 66416}, {32819, 50688}, {32827, 61984}, {32832, 60781}, {32834, 46936}, {32835, 37688}, {32838, 55857}, {32839, 55858}, {32867, 55861}, {32868, 61900}, {32880, 69383}, {32886, 61892}, {32888, 61903}, {32893, 46935}, {32961, 50570}, {32972, 63923}, {32980, 34505}, {32981, 63932}, {32984, 63924}, {32992, 63025}, {33001, 42850}, {33018, 41136}, {33198, 41624}, {33225, 63093}, {33237, 63022}, {35927, 63938}, {36890, 52130}, {37176, 63110}, {37177, 40694}, {37178, 40693}, {37201, 65711}, {37436, 37664}, {38664, 46236}, {40341, 59545}, {40824, 60619}, {41940, 63011}, {55614, 59548}, {55684, 59552}, {62988, 69139}, {63940, 68513}, {69387, 69406}

X(69414) = isotomic conjugate of the isogonal conjugate of X(62217)
X(69414) = isotomic conjugate of the polar conjugate of X(3620)
X(69414) = X(62217)-cross conjugate of X(3620)
X(69414) = X(1973)-isoconjugate of X(5395)
X(69414) = X(i)-Dao conjugate of X(j) for these (i,j): {3618, 6995}, {3620, 62979}, {6337, 5395}, {62604, 56067}
X(69414) = barycentric product X(i)*X(j) for these {i,j}: {69, 3620}, {76, 62217}, {305, 5013}, {3926, 8889}
X(69414) = barycentric quotient X(i)/X(j) for these {i,j}: {69, 5395}, {305, 56067}, {3620, 4}, {3917, 31506}, {4558, 58100}, {5013, 25}, {8889, 393}, {12167, 2207}, {62217, 6}
X(69414) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 3926, 6337}, {76, 1007, 69385}, {76, 32818, 1007}, {325, 32830, 69378}, {439, 11160, 63928}, {487, 488, 48906}, {599, 59546, 32990}, {1975, 32006, 67536}, {1975, 37668, 32006}, {3785, 3926, 6390}, {3926, 3933, 69}, {7739, 7869, 33221}, {7758, 7801, 14001}, {7758, 14001, 1992}, {7763, 69377, 34229}, {7788, 32820, 20}, {7794, 16043, 21356}, {7794, 34511, 16043}, {7796, 32833, 4}, {7863, 14023, 32985}, {7871, 11185, 32823}, {7888, 63955, 32969}, {13571, 16898, 63024}, {32825, 32836, 5}, {32828, 69158, 34803}, {32840, 37668, 1975}

X(69415) = X(5)X(76)∩X(20)X(64)

Barycentrics   2*b^2*c^2 + SA^2 - SB*SC : :
X(69415) = 5 X[384] - 3 X[34604], 3 X[6656] - 2 X[7765], 3 X[6656] - 4 X[7849], X[7765] - 3 X[7794], 3 X[7794] - 2 X[7849], 2 X[5007] - 3 X[6661], 3 X[7827] - 4 X[8364], 4 X[7829] - 5 X[66342], 3 X[7883] - 2 X[8357], 3 X[12150] - 4 X[19697], 5 X[19689] - 3 X[63038], 2 X[39593] - 3 X[66339]

X(69415) lies on these lines: {2, 9606}, {3, 32820}, {4, 7788}, {5, 76}, {6, 16898}, {20, 64}, {30, 7768}, {32, 50251}, {75, 9710}, {99, 548}, {115, 7895}, {140, 7799}, {141, 194}, {148, 7939}, {183, 631}, {230, 7836}, {264, 1907}, {274, 17529}, {298, 398}, {299, 397}, {305, 30739}, {315, 382}, {316, 3853}, {339, 37452}, {340, 3575}, {350, 37722}, {376, 32824}, {384, 524}, {385, 7789}, {401, 64062}, {428, 57852}, {538, 6656}, {542, 44251}, {543, 7873}, {546, 7809}, {550, 7811}, {597, 16895}, {599, 7791}, {626, 47286}, {754, 19687}, {858, 8024}, {1003, 14023}, {1007, 7486}, {1078, 3530}, {1216, 51383}, {1232, 1238}, {1235, 15559}, {1329, 20943}, {1565, 33939}, {1593, 44134}, {1906, 54412}, {1909, 15888}, {1992, 33198}, {2549, 7879}, {2896, 3631}, {3053, 63046}, {3090, 32825}, {3091, 32869}, {3096, 15048}, {3314, 5254}, {3411, 69121}, {3412, 69120}, {3524, 32896}, {3525, 32837}, {3526, 7763}, {3528, 3785}, {3545, 32892}, {3552, 63928}, {3589, 7839}, {3620, 7738}, {3627, 7860}, {3629, 7787}, {3630, 7893}, {3665, 33931}, {3695, 20924}, {3703, 33930}, {3704, 20955}, {3734, 7762}, {3760, 37720}, {3761, 37719}, {3767, 7881}, {3788, 17131}, {3790, 7185}, {3813, 4479}, {3815, 7906}, {3832, 7773}, {3843, 7776}, {3855, 32816}, {3858, 48913}, {3859, 15031}, {3861, 7917}, {3932, 33943}, {3934, 7813}, {3969, 25257}, {4089, 7206}, {4187, 18145}, {4195, 17378}, {4197, 34284}, {4201, 17271}, {4352, 5224}, {4396, 26686}, {4400, 26629}, {5007, 6661}, {5013, 16990}, {5025, 63923}, {5056, 32874}, {5067, 32818}, {5070, 32832}, {5286, 7868}, {5305, 7832}, {5306, 7892}, {5309, 7869}, {5319, 7754}, {5355, 7915}, {5475, 7916}, {5485, 33292}, {5976, 14981}, {6179, 8369}, {6292, 32450}, {6337, 15589}, {6376, 9711}, {6392, 7851}, {6393, 14994}, {6658, 63941}, {7503, 68660}, {7610, 33000}, {7745, 7779}, {7746, 7908}, {7747, 7882}, {7748, 7896}, {7751, 7801}, {7755, 7880}, {7756, 7848}, {7757, 8362}, {7758, 7770}, {7759, 8370}, {7760, 7819}, {7764, 9466}, {7769, 16239}, {7771, 44682}, {7774, 33269}, {7778, 33248}, {7780, 7863}, {7781, 7854}, {7782, 46853}, {7783, 63044}, {7785, 50771}, {7793, 59545}, {7798, 7822}, {7800, 31859}, {7802, 14929}, {7803, 22253}, {7804, 7890}, {7805, 7820}, {7812, 59780}, {7815, 31457}, {7816, 7826}, {7818, 33229}, {7821, 14711}, {7824, 59546}, {7827, 8364}, {7829, 66342}, {7833, 22165}, {7837, 68525}, {7838, 53489}, {7840, 16044}, {7847, 32027}, {7850, 62041}, {7856, 33185}, {7858, 66415}, {7859, 63633}, {7864, 20105}, {7877, 18907}, {7883, 8357}, {7885, 53419}, {7887, 63955}, {7888, 33249}, {7899, 43291}, {7900, 53418}, {7901, 19570}, {7907, 13468}, {7909, 8361}, {7912, 63534}, {7922, 33184}, {7946, 11361}, {7947, 44377}, {8352, 63922}, {8360, 11054}, {8556, 33001}, {8591, 33267}, {8667, 16925}, {8716, 32965}, {8860, 32977}, {9300, 13571}, {9464, 62310}, {9740, 33205}, {9766, 16924}, {9770, 32987}, {9939, 33257}, {9983, 15993}, {10513, 17578}, {11057, 15704}, {11160, 32981}, {11163, 31407}, {11168, 33015}, {11184, 32999}, {11257, 48876}, {11285, 31450}, {11293, 32809}, {11294, 32808}, {11898, 36998}, {12110, 34380}, {12150, 19697}, {13881, 33277}, {14001, 14614}, {14035, 63932}, {14037, 63093}, {14045, 63543}, {14063, 34505}, {14148, 37512}, {14907, 15696}, {15246, 40002}, {15271, 31492}, {15533, 33007}, {15606, 51439}, {16062, 48838}, {16275, 47095}, {16276, 37899}, {16711, 17674}, {16712, 56734}, {16897, 20582}, {16926, 50265}, {16930, 50261}, {16931, 50274}, {17008, 33262}, {17030, 31469}, {17143, 64200}, {17233, 25242}, {17330, 33827}, {17346, 17691}, {17392, 17688}, {17527, 18146}, {17575, 18140}, {18160, 56283}, {19686, 63944}, {19689, 63038}, {20065, 40341}, {20925, 69279}, {21356, 33202}, {22110, 32967}, {22331, 33255}, {25264, 69095}, {27020, 31462}, {30471, 42945}, {30472, 42944}, {31478, 69136}, {31829, 40996}, {32815, 32877}, {32822, 32890}, {32823, 32878}, {32826, 62021}, {32829, 61867}, {32831, 34229}, {32838, 61881}, {32841, 61842}, {32868, 69386}, {32875, 61138}, {32879, 61816}, {32882, 69383}, {32885, 61886}, {32893, 46936}, {32954, 63954}, {32989, 63029}, {33024, 41136}, {33181, 63034}, {33222, 53033}, {33230, 66458}, {33250, 63935}, {33289, 41135}, {33651, 37897}, {33838, 48869}, {33944, 69091}, {33947, 69092}, {34386, 52984}, {34501, 60706}, {35007, 63927}, {37126, 52437}, {39593, 66339}, {42990, 69106}, {42991, 69107}, {43459, 61790}, {43601, 69176}, {44148, 62338}, {44149, 52347}, {44166, 45921}, {50692, 67536}, {53096, 66417}, {53127, 61911}, {56986, 63110}, {59548, 60702}, {61914, 63098}, {62332, 67606}, {63931, 66408}, {63936, 68527}, {63939, 66319}, {63940, 68177}, {63950, 68513}, {66424, 66455}, {69130, 69146}, {69131, 69138}, {69255, 69261}, {69257, 69258}

X(69415) = reflection of X(i) in X(j) for these {i,j}: {6656, 7794}, {7760, 7819}, {7765, 7849}, {19695, 7873}
X(69415) = isotomic conjugate of X(57408)
X(69415) = isotomic conjugate of the isogonal conjugate of X(3819)
X(69415) = X(31)-isoconjugate of X(57408)
X(69415) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 57408}, {3819, 40981}
X(69415) = cevapoint of X(20) and X(51860)
X(69415) = barycentric product X(76)*X(3819)
X(69415) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 57408}, {3819, 6}
X(69415) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 32820, 59634}, {3, 32833, 32820}, {5, 3933, 7796}, {5, 7796, 325}, {69, 1975, 7750}, {69, 32830, 1975}, {76, 325, 59635}, {76, 3933, 325}, {76, 7752, 64093}, {76, 7796, 5}, {76, 7871, 69387}, {141, 9607, 7876}, {194, 7876, 9607}, {315, 69380, 32819}, {489, 490, 48905}, {1232, 1238, 45198}, {3314, 20081, 5254}, {3631, 63548, 2896}, {3734, 7855, 7762}, {3926, 69377, 183}, {5309, 7869, 8363}, {5319, 7795, 33217}, {5319, 33217, 7792}, {7751, 7801, 7807}, {7751, 7807, 22329}, {7754, 7795, 7792}, {7754, 33217, 5319}, {7758, 7770, 41624}, {7759, 17130, 8370}, {7763, 69381, 37688}, {7764, 9466, 32992}, {7765, 7794, 7849}, {7765, 7849, 6656}, {7779, 17128, 7745}, {7780, 7863, 35297}, {7781, 7854, 8356}, {7821, 14711, 63924}, {7821, 63924, 33228}, {7836, 17129, 230}, {7839, 16896, 51860}, {7839, 46226, 3589}, {7880, 63925, 7755}, {7888, 33249, 41133}, {7906, 31276, 3815}, {7909, 14568, 8361}, {8369, 63926, 6179}, {10513, 32880, 69379}, {10513, 69379, 32006}, {13571, 68522, 9300}, {15589, 32840, 6337}, {16896, 51860, 3589}, {32820, 37671, 3}, {32825, 46951, 3090}, {32832, 69158, 37647}, {32833, 37671, 59634}, {37668, 69378, 7773}, {46226, 51860, 16896}

X(69416) = X(2)X(9609)∩X(4)X(69)

Barycentrics   2*b^2*c^2 + S^2 + SA^2 - SB*SC : :

X(69416) lies on these lines: {2, 9606}, {3, 32824}, {4, 69}, {5, 46951}, {20, 32869}, {30, 32892}, {83, 62995}, {99, 21735}, {140, 3926}, {141, 6392}, {183, 3523}, {193, 69139}, {325, 5056}, {385, 14037}, {524, 32971}, {538, 16043}, {543, 33247}, {550, 3785}, {599, 32974}, {631, 32833}, {635, 60252}, {636, 60253}, {1007, 1656}, {1078, 10299}, {1204, 69176}, {1270, 43377}, {1271, 43376}, {1657, 7767}, {1975, 3522}, {1992, 7770}, {2551, 20943}, {2996, 7784}, {3090, 7796}, {3091, 7788}, {3314, 33283}, {3525, 7799}, {3526, 32837}, {3529, 7811}, {3533, 7763}, {3618, 7754}, {3619, 5286}, {3620, 5254}, {3628, 32885}, {3630, 65630}, {3631, 44518}, {3673, 42696}, {3734, 63927}, {3767, 7869}, {3850, 7776}, {3851, 32816}, {3854, 7773}, {3855, 7809}, {3858, 32827}, {3934, 63041}, {3974, 33930}, {4385, 42697}, {4479, 64068}, {5054, 32896}, {5059, 7750}, {5068, 37668}, {5071, 7814}, {5073, 64018}, {5084, 18145}, {5309, 33221}, {5319, 63925}, {5485, 7883}, {6179, 14039}, {6390, 15720}, {6656, 21356}, {7195, 33931}, {7388, 32810}, {7389, 32811}, {7391, 40002}, {7395, 68660}, {7486, 32893}, {7509, 52437}, {7735, 7892}, {7736, 13571}, {7737, 63937}, {7738, 10335}, {7745, 20080}, {7751, 14001}, {7752, 61921}, {7755, 7795}, {7757, 32960}, {7758, 9466}, {7759, 32983}, {7760, 16045}, {7764, 32975}, {7769, 61873}, {7771, 61787}, {7774, 33020}, {7780, 32985}, {7781, 33215}, {7782, 62061}, {7789, 37667}, {7791, 35700}, {7794, 14064}, {7801, 23055}, {7802, 62171}, {7803, 10159}, {7810, 33226}, {7819, 63954}, {7821, 32984}, {7824, 42850}, {7836, 62992}, {7837, 33269}, {7840, 32962}, {7841, 50990}, {7849, 33223}, {7854, 14711}, {7863, 33216}, {7870, 32959}, {7871, 53127}, {7878, 63022}, {7879, 43448}, {7880, 33222}, {7888, 12815}, {7895, 43620}, {7906, 62993}, {7909, 32955}, {7916, 31415}, {7922, 33285}, {7946, 33016}, {8024, 19583}, {8370, 50992}, {8556, 59546}, {8667, 32973}, {9740, 33201}, {9766, 32987}, {9770, 16921}, {9939, 33280}, {10150, 32969}, {11008, 69208}, {11054, 33230}, {11057, 49138}, {11160, 32979}, {11286, 63926}, {11289, 63105}, {11290, 63106}, {11303, 49825}, {11304, 49824}, {11623, 46236}, {13468, 32989}, {13740, 63110}, {14023, 14033}, {14031, 17128}, {14034, 20065}, {14045, 63533}, {14568, 32951}, {14614, 33198}, {14907, 32822}, {14929, 62026}, {15598, 15815}, {15709, 62362}, {15712, 32877}, {15717, 59634}, {16041, 63924}, {16898, 63006}, {16925, 63029}, {17559, 18146}, {18841, 60640}, {19689, 63065}, {20925, 54433}, {22329, 33181}, {32818, 32832}, {32819, 49135}, {32823, 69387}, {32826, 62036}, {32829, 46219}, {32831, 37688}, {32838, 55856}, {32839, 55859}, {32840, 61834}, {32867, 55860}, {32870, 37647}, {32872, 63098}, {32875, 61832}, {32880, 61791}, {32883, 61877}, {32886, 35018}, {32890, 61803}, {32954, 63107}, {32978, 34511}, {32981, 63928}, {32982, 34505}, {33010, 41136}, {33190, 50994}, {34284, 37462}, {34621, 36889}, {37162, 45962}, {37669, 52289}, {40123, 62937}, {42773, 59539}, {42774, 59540}, {42992, 69106}, {42993, 69107}, {43527, 60143}, {45201, 52284}, {48838, 56737}, {53033, 63104}, {56015, 56865}, {59780, 68527}, {60145, 60628}, {60146, 60637}, {60183, 60642}, {60209, 60636}, {60639, 60647}, {62950, 64062}, {63024, 68522}, {63093, 68525}, {63950, 68177}

X(69416) = anticomplement of the isogonal conjugate of X(60647)
X(69416) = isotomic conjugate of the isogonal conjugate of X(16419)
X(69416) = X(60647)-anticomplementary conjugate of X(8)
X(69416) = barycentric product X(76)*X(16419)
X(69416) = barycentric quotient X(16419)/X(6)
X(69416) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 76, 69378}, {69, 69378, 32006}, {76, 315, 52713}, {76, 69377, 69}, {183, 32820, 3523}, {183, 32830, 6337}, {325, 32834, 69385}, {599, 63923, 32974}, {1656, 3933, 32825}, {1656, 32825, 1007}, {3523, 32820, 6337}, {3523, 32830, 32820}, {3926, 69381, 34229}, {3933, 32828, 1007}, {7750, 69379, 67536}, {7751, 14001, 63034}, {7758, 9466, 32968}, {7760, 16045, 59373}, {7764, 32975, 63025}, {7794, 63955, 14064}, {7803, 18840, 63121}, {7860, 11185, 4}, {10513, 32894, 69383}, {10513, 69383, 7773}, {11160, 32979, 63932}, {14023, 17130, 14033}, {16990, 20081, 7738}, {32816, 32868, 64093}, {32818, 32832, 34803}, {32825, 32828, 1656}

X(69417) = X(2)X(9606)∩X(3)X(76)

Barycentrics   2*b^2*c^2 + 2*S^2 + SA^2 - SB*SC : :

X(69417) = 3 X[11285] - 2 X[53096]

X(69417) lies on these lines: {2, 9606}, {3, 76}, {4, 37671}, {5, 7788}, {6, 17129}, {20, 32874}, {69, 3091}, {115, 7879}, {140, 32833}, {141, 7851}, {194, 15271}, {235, 44134}, {264, 5198}, {274, 16862}, {298, 42153}, {299, 42156}, {315, 546}, {316, 61984}, {325, 3090}, {340, 7507}, {350, 3303}, {381, 7768}, {382, 7811}, {384, 8667}, {385, 68525}, {405, 18145}, {491, 43879}, {492, 43880}, {524, 16924}, {538, 11285}, {576, 14994}, {599, 5025}, {631, 32820}, {632, 7763}, {671, 7936}, {958, 20943}, {1003, 7780}, {1007, 46936}, {1232, 68660}, {1235, 10594}, {1236, 13861}, {1656, 7796}, {1909, 3304}, {1995, 39998}, {2896, 44518}, {3053, 17128}, {3146, 7750}, {3314, 13881}, {3523, 32869}, {3524, 32824}, {3525, 3926}, {3526, 7799}, {3529, 3785}, {3533, 32837}, {3544, 32816}, {3627, 7767}, {3628, 3933}, {3631, 63534}, {3734, 35007}, {3746, 3760}, {3761, 5563}, {3763, 7797}, {3767, 7868}, {3843, 7860}, {3851, 7809}, {3913, 4479}, {3934, 7754}, {3964, 52712}, {4968, 26240}, {5007, 7751}, {5013, 20081}, {5047, 16992}, {5055, 7814}, {5056, 32893}, {5067, 32825}, {5072, 7776}, {5073, 11057}, {5079, 7752}, {5210, 68526}, {5254, 16990}, {5306, 16898}, {6144, 7921}, {6179, 11286}, {6337, 61820}, {6390, 14869}, {6655, 34505}, {6656, 63955}, {7603, 7916}, {7610, 7907}, {7745, 63046}, {7746, 7881}, {7753, 63934}, {7755, 33217}, {7758, 11163}, {7759, 44543}, {7760, 63954}, {7769, 55858}, {7774, 33261}, {7781, 14711}, {7784, 63044}, {7785, 40341}, {7786, 22253}, {7789, 17008}, {7791, 63923}, {7794, 7887}, {7798, 31239}, {7800, 47286}, {7801, 8860}, {7802, 49136}, {7808, 41940}, {7810, 33234}, {7812, 63936}, {7815, 31652}, {7816, 68515}, {7824, 8556}, {7836, 37637}, {7837, 33020}, {7840, 33002}, {7841, 7854}, {7848, 69141}, {7849, 33219}, {7850, 15031}, {7866, 14568}, {7869, 33218}, {7871, 61903}, {7873, 18546}, {7876, 19570}, {7877, 15484}, {7883, 40727}, {7884, 10159}, {7891, 44535}, {7893, 65630}, {7896, 39565}, {7904, 44526}, {7906, 31489}, {7917, 61935}, {7920, 47355}, {7922, 11318}, {7935, 32457}, {7946, 15533}, {7948, 21358}, {7999, 51383}, {8024, 40916}, {8370, 14023}, {8716, 33004}, {9766, 16921}, {9939, 14042}, {10303, 32830}, {10513, 69404}, {10541, 12215}, {11108, 18146}, {11128, 59384}, {11129, 59383}, {11160, 32991}, {11164, 33244}, {11168, 33001}, {11184, 16922}, {11284, 40022}, {11317, 63931}, {11361, 63938}, {11403, 54412}, {11440, 69176}, {11541, 32826}, {12812, 53127}, {13468, 16925}, {14001, 22329}, {14035, 63928}, {14068, 63941}, {14537, 63937}, {14907, 15704}, {14929, 61988}, {15022, 32872}, {15598, 63548}, {15694, 62362}, {15709, 32896}, {16044, 63932}, {16373, 18152}, {16781, 30998}, {16842, 18140}, {16865, 37670}, {16916, 47037}, {17251, 33834}, {17313, 33835}, {17531, 34284}, {17538, 32815}, {18906, 53097}, {20398, 32458}, {21356, 33180}, {22110, 32998}, {22165, 33006}, {23055, 33203}, {30471, 42490}, {30472, 42491}, {30737, 33524}, {32006, 50689}, {32138, 69178}, {32488, 32808}, {32489, 32809}, {32817, 61814}, {32818, 32838}, {32822, 62092}, {32823, 69406}, {32829, 61870}, {32831, 61863}, {32839, 52718}, {32840, 61848}, {32870, 34803}, {32878, 61807}, {32882, 61804}, {32894, 50693}, {32952, 60143}, {32957, 63101}, {32968, 41624}, {32973, 63029}, {32976, 41133}, {32990, 42850}, {33183, 63107}, {33198, 63034}, {33269, 63093}, {35502, 44146}, {37353, 40002}, {37946, 67606}, {39099, 53858}, {41916, 45201}, {43136, 60855}, {48913, 61953}, {50251, 69208}, {55614, 60702}, {55857, 69158}, {61964, 69382}, {62028, 64018}, {62152, 67536}, {63657, 65711}, {63922, 66388}, {63926, 66415}, {63951, 66413}

X(69417) = anticomplement of X(9606)
X(69417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 32834, 59635}, {69, 59635, 7773}, {76, 183, 1975}, {76, 1078, 69380}, {76, 69381, 183}, {631, 32836, 32820}, {3785, 32868, 52713}, {3785, 52713, 32819}, {3926, 69384, 37688}, {3934, 7754, 11174}, {3934, 17131, 7754}, {3934, 63925, 7772}, {7751, 7770, 14614}, {7751, 9466, 7770}, {7758, 32992, 11163}, {7772, 17131, 63925}, {7772, 63925, 7754}, {7780, 17130, 1003}, {7849, 69162, 33219}, {7854, 63924, 7841}, {11168, 59546, 33001}, {15589, 69378, 7750}, {17129, 31276, 6}, {32816, 32886, 69386}, {32818, 32838, 37647}, {32825, 32885, 5067}, {32828, 69377, 325}, {32872, 37668, 69385}, {44776, 44777, 599}

X(69418) = X(3)X(315)∩X(5)X(99)

Barycentrics   2*S^2 - SA^2 - SB*SC : :

X(69418) = 5 X[7752] - 3 X[48913], 5 X[7782] + 3 X[48913], X[7793] - 3 X[33274], X[7906] + 3 X[33274]

X(69418) lies on these lines: {2, 1975}, {3, 315}, {4, 32829}, {5, 99}, {6, 16925}, {15, 69165}, {16, 69157}, {20, 1007}, {30, 7752}, {32, 35297}, {35, 69254}, {36, 69135}, {39, 620}, {52, 51440}, {69, 3523}, {75, 4999}, {76, 140}, {83, 5503}, {95, 52347}, {115, 33249}, {141, 5116}, {148, 32967}, {182, 6393}, {183, 631}, {187, 7762}, {194, 230}, {217, 2421}, {274, 7483}, {302, 398}, {303, 397}, {305, 7499}, {316, 550}, {317, 3515}, {343, 60700}, {350, 5433}, {371, 35684}, {372, 35685}, {376, 32816}, {384, 3815}, {385, 33259}, {389, 51439}, {395, 30471}, {396, 30472}, {404, 37664}, {439, 62988}, {468, 11059}, {491, 1152}, {492, 1151}, {524, 7793}, {538, 7749}, {543, 39565}, {548, 7802}, {549, 1078}, {566, 66143}, {570, 52636}, {574, 3788}, {575, 50567}, {576, 51438}, {597, 10583}, {625, 7756}, {626, 8356}, {632, 64093}, {754, 15513}, {877, 42873}, {970, 51369}, {1003, 2548}, {1235, 37118}, {1385, 69038}, {1506, 2482}, {1513, 9737}, {1656, 11185}, {1909, 5432}, {2275, 26629}, {2276, 26686}, {2549, 7887}, {2896, 7947}, {3035, 6376}, {3053, 7774}, {3054, 16923}, {3055, 16921}, {3090, 32815}, {3091, 34803}, {3095, 37459}, {3096, 7870}, {3146, 32873}, {3199, 35920}, {3266, 7495}, {3314, 33004}, {3329, 9606}, {3398, 38750}, {3432, 7488}, {3491, 6786}, {3522, 32006}, {3524, 3785}, {3525, 32817}, {3526, 32832}, {3528, 32823}, {3529, 32827}, {3530, 7767}, {3533, 32824}, {3545, 32826}, {3552, 7745}, {3589, 7892}, {3618, 33181}, {3628, 69387}, {3629, 13571}, {3734, 31455}, {3761, 65142}, {3767, 31859}, {3793, 7877}, {3832, 67536}, {3934, 7863}, {4045, 7874}, {4173, 35060}, {4201, 30761}, {4558, 66449}, {4563, 61690}, {4576, 14389}, {5023, 9766}, {5024, 7803}, {5025, 44377}, {5054, 32833}, {5056, 32871}, {5067, 32822}, {5070, 53127}, {5092, 51371}, {5152, 51872}, {5206, 7759}, {5207, 44882}, {5210, 63932}, {5237, 69137}, {5238, 69145}, {5283, 17694}, {5286, 32970}, {5305, 7757}, {5306, 7839}, {5475, 19687}, {5718, 17103}, {5743, 6626}, {5866, 14118}, {5939, 14981}, {5972, 59527}, {5989, 37334}, {6247, 57275}, {6292, 7880}, {6331, 9291}, {6392, 62992}, {6655, 7925}, {6658, 53418}, {6661, 7808}, {6676, 57518}, {6683, 7820}, {6690, 31997}, {6691, 30963}, {6703, 41849}, {6781, 7843}, {6910, 16992}, {7283, 54443}, {7484, 34254}, {7485, 45201}, {7486, 32898}, {7603, 35022}, {7618, 7841}, {7622, 7801}, {7709, 10256}, {7735, 32989}, {7736, 32973}, {7737, 33235}, {7746, 7781}, {7747, 32456}, {7748, 7862}, {7754, 22329}, {7755, 32450}, {7758, 21843}, {7761, 7888}, {7765, 7886}, {7768, 15712}, {7770, 31401}, {7775, 8598}, {7778, 7791}, {7779, 63928}, {7780, 7813}, {7784, 32965}, {7785, 13586}, {7786, 7819}, {7787, 9300}, {7790, 7940}, {7795, 11285}, {7797, 9607}, {7800, 7881}, {7804, 9698}, {7809, 8703}, {7810, 7895}, {7811, 7871}, {7812, 27088}, {7817, 9167}, {7821, 7830}, {7822, 15482}, {7823, 33014}, {7825, 19695}, {7827, 64019}, {7828, 15048}, {7831, 7909}, {7832, 8362}, {7833, 7912}, {7834, 53096}, {7838, 35007}, {7842, 8353}, {7846, 8368}, {7847, 7899}, {7850, 61792}, {7854, 7908}, {7856, 63633}, {7858, 18907}, {7859, 33185}, {7860, 33923}, {7868, 16043}, {7876, 7945}, {7882, 46893}, {7885, 33260}, {7889, 44562}, {7893, 50771}, {7897, 7904}, {7898, 33275}, {7900, 63941}, {7903, 8588}, {7910, 8354}, {7917, 14929}, {7918, 8360}, {7919, 33186}, {7930, 8364}, {7931, 33021}, {7934, 8357}, {7938, 66414}, {7941, 14712}, {7949, 63940}, {8352, 34504}, {8724, 53765}, {8781, 21166}, {9605, 11288}, {9723, 17928}, {9729, 51386}, {9730, 51383}, {9734, 54393}, {9753, 10983}, {9770, 35287}, {9771, 33013}, {10008, 14912}, {10018, 44146}, {10257, 41009}, {10299, 32825}, {10303, 32830}, {10352, 34870}, {10513, 32881}, {11057, 34200}, {11064, 57008}, {11132, 52194}, {11133, 52193}, {11163, 32985}, {11172, 18840}, {11174, 14001}, {11184, 33007}, {11257, 56370}, {11286, 31467}, {12506, 65595}, {13219, 35497}, {13334, 37450}, {13335, 32458}, {13394, 56430}, {13468, 17129}, {13747, 18140}, {14033, 31404}, {14035, 63083}, {14063, 44526}, {14537, 51581}, {14558, 15646}, {14568, 51123}, {14588, 18122}, {14614, 33216}, {14928, 18553}, {14961, 28697}, {15122, 67606}, {15271, 33001}, {15300, 47617}, {15484, 68513}, {15491, 68522}, {15589, 32841}, {15631, 67349}, {15702, 32836}, {15709, 46951}, {15717, 37668}, {15850, 38734}, {16044, 17005}, {16196, 62698}, {16276, 37439}, {16320, 36182}, {16898, 31492}, {16915, 37661}, {16924, 31489}, {16990, 33012}, {17004, 20081}, {17008, 33206}, {17566, 18135}, {17932, 58728}, {18114, 66074}, {18584, 32995}, {19694, 51126}, {19697, 60855}, {20190, 51397}, {20576, 32447}, {21167, 60702}, {21734, 32895}, {21735, 32889}, {21841, 58782}, {23302, 59540}, {23303, 59539}, {24467, 55469}, {24524, 64123}, {24953, 60706}, {26877, 55418}, {26878, 55419}, {26921, 55470}, {27162, 56778}, {28407, 54075}, {28438, 37125}, {30739, 37804}, {30786, 52293}, {31173, 66424}, {31276, 33015}, {31450, 33217}, {32834, 55864}, {32840, 61834}, {32867, 61867}, {32869, 61844}, {32870, 61863}, {32874, 61846}, {32876, 61817}, {32883, 61870}, {32884, 61886}, {32885, 61859}, {32886, 67095}, {32896, 61833}, {32961, 44518}, {32966, 53419}, {32969, 43448}, {32971, 62993}, {32974, 37690}, {32981, 51579}, {32984, 53142}, {32988, 63533}, {33000, 37637}, {33017, 44519}, {33189, 39142}, {33192, 66616}, {33198, 63041}, {33205, 37665}, {33220, 69209}, {33253, 44541}, {33296, 37646}, {34007, 65518}, {34383, 63556}, {34506, 39785}, {35937, 60524}, {37291, 37670}, {37612, 55416}, {37662, 59538}, {37803, 59766}, {39590, 66408}, {40405, 45857}, {40951, 51427}, {41259, 52261}, {41360, 63544}, {43118, 45509}, {43119, 45508}, {43141, 51395}, {43144, 51401}, {43238, 69180}, {43239, 69186}, {44369, 63722}, {44380, 44453}, {45017, 66419}, {46226, 46318}, {48886, 51370}, {50007, 52149}, {50571, 63065}, {51396, 55606}, {52289, 65722}, {52793, 69094}, {53489, 69172}, {54833, 60234}, {56297, 59528}, {58447, 59535}, {66395, 66466}

X(69418) = midpoint of X(i) and X(j) for these {i,j}: {7752, 7782}, {7793, 7906}, {15513, 69197}
X(69418) = isotomic conjugate of the isogonal conjugate of X(34986)
X(69418) = isotomic conjugate of the polar conjugate of X(27377)
X(69418) = X(i)-Ceva conjugate of X(j) for these (i,j): {42351, 183}, {60241, 69}
X(69418) = X(34986)-cross conjugate of X(27377)
X(69418) = X(41008)-Dao conjugate of X(23292)
X(69418) = barycentric product X(i)*X(j) for these {i,j}: {69, 27377}, {76, 34986}
X(69418) = barycentric quotient X(i)/X(j) for these {i,j}: {27377, 4}, {34986, 6}
X(69418) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1975, 59635}, {2, 6337, 1975}, {2, 7738, 7851}, {2, 7783, 5254}, {2, 7891, 7789}, {2, 69379, 69385}, {3, 325, 7750}, {3, 7763, 325}, {3, 7776, 14907}, {3, 69158, 315}, {5, 99, 32819}, {5, 7769, 37647}, {20, 1007, 7773}, {20, 32835, 1007}, {39, 620, 7807}, {39, 7807, 7792}, {69, 32831, 32821}, {76, 140, 37688}, {76, 6390, 32820}, {83, 31406, 63101}, {99, 7769, 5}, {99, 62362, 7769}, {140, 6390, 76}, {141, 59552, 12215}, {148, 32967, 63534}, {187, 7764, 7762}, {194, 7907, 230}, {230, 59546, 194}, {315, 7763, 69158}, {315, 69158, 325}, {549, 3933, 1078}, {549, 7799, 37671}, {574, 3788, 6656}, {620, 62348, 5976}, {625, 7756, 33229}, {626, 37512, 8356}, {631, 3926, 183}, {1078, 3933, 37671}, {1078, 7799, 3933}, {1506, 2482, 7816}, {1506, 7816, 8370}, {1975, 6337, 59634}, {3522, 63098, 32006}, {3523, 32831, 69}, {3524, 32818, 3785}, {3524, 32837, 7788}, {3525, 32817, 32828}, {3526, 69380, 32832}, {3528, 32823, 64018}, {3530, 7767, 7771}, {3533, 52713, 32838}, {3552, 7777, 7745}, {3589, 59548, 18906}, {3734, 31455, 32992}, {3785, 32818, 7788}, {3785, 32837, 32818}, {3815, 59545, 384}, {4045, 7874, 8363}, {5023, 9766, 20065}, {5024, 32954, 7803}, {5067, 32822, 69382}, {5475, 69171, 19687}, {7745, 32459, 3552}, {7746, 7781, 47286}, {7747, 32456, 33250}, {7748, 7862, 33228}, {7754, 69207, 22329}, {7757, 7857, 5305}, {7765, 31274, 7886}, {7771, 7796, 7767}, {7774, 32964, 3053}, {7778, 15815, 7791}, {7784, 53095, 32965}, {7786, 7835, 7819}, {7790, 7940, 8361}, {7821, 8589, 7830}, {7822, 31457, 15482}, {7824, 7836, 141}, {7847, 7899, 33184}, {7874, 31652, 4045}, {7888, 15515, 7761}, {7897, 33022, 7904}, {7903, 8588, 63935}, {7906, 33274, 7793}, {7941, 33276, 14712}, {7947, 33273, 2896}, {8369, 31406, 83}, {10303, 32830, 34229}, {12040, 41134, 63101}, {14001, 31400, 11174}, {16043, 53033, 7868}, {17008, 33206, 44535}, {31276, 33015, 58446}, {31401, 69206, 7770}, {31859, 33233, 3767}, {32450, 58448, 7755}, {32815, 32839, 3090}, {32819, 37647, 5}, {32820, 37688, 76}, {32824, 32838, 52713}, {32841, 61820, 15589}, {32881, 61804, 10513}, {33014, 63021, 7823}, {34511, 69207, 7754}, {44180, 45799, 39113}, {44377, 63548, 5025}, {44535, 63933, 17008}, {51579, 63077, 32981}, {59634, 59635, 1975}, {61867, 69386, 32867}

X(69419) = X(2)X(9607)∩X(20)X(64)

Barycentrics   2*b^2*c^2 - S^2 + SA^2 + SB*SC : :

X(69419) = 2 X[5319] - 3 X[14001], 4 X[7869] - 3 X[33223]

X(69419) lies on these lines: {2, 9607}, {3, 32824}, {4, 7796}, {5, 1007}, {20, 64}, {76, 631}, {99, 3528}, {140, 46951}, {183, 15717}, {194, 3618}, {274, 17552}, {305, 40132}, {315, 32822}, {316, 62021}, {325, 3832}, {381, 32825}, {382, 3933}, {384, 1992}, {439, 8667}, {491, 31414}, {524, 32981}, {538, 5319}, {543, 33238}, {548, 3785}, {549, 32892}, {599, 33023}, {632, 32885}, {671, 33292}, {1078, 61138}, {1238, 63155}, {1593, 52710}, {1656, 32837}, {2549, 7849}, {2996, 7778}, {3090, 7799}, {3091, 32821}, {3146, 7788}, {3522, 37671}, {3523, 32869}, {3526, 6390}, {3529, 7768}, {3530, 69381}, {3619, 7738}, {3620, 63548}, {3631, 44519}, {3761, 31452}, {3767, 33222}, {3843, 32816}, {3853, 7776}, {3855, 7814}, {3861, 32827}, {3934, 31450}, {4195, 63110}, {4352, 16714}, {5067, 7763}, {5070, 32829}, {5286, 33217}, {5485, 7870}, {6392, 7789}, {7484, 41927}, {7486, 32831}, {7493, 8024}, {7503, 52437}, {7519, 40123}, {7735, 20081}, {7736, 17128}, {7751, 32985}, {7752, 61945}, {7755, 33224}, {7757, 16045}, {7758, 14033}, {7760, 14039}, {7764, 31417}, {7765, 7795}, {7767, 15696}, {7769, 61881}, {7773, 61982}, {7781, 16043}, {7782, 62066}, {7783, 33258}, {7787, 63073}, {7791, 21356}, {7794, 32986}, {7801, 14064}, {7807, 63107}, {7811, 17538}, {7816, 63934}, {7833, 50990}, {7836, 33248}, {7837, 14031}, {7839, 63011}, {7840, 14068}, {7854, 33226}, {7860, 15682}, {7863, 32970}, {7866, 52229}, {7869, 33223}, {7879, 47287}, {7881, 43448}, {7882, 43618}, {7888, 32984}, {7891, 33262}, {7896, 43619}, {7909, 33285}, {7946, 33280}, {8362, 51122}, {8591, 33253}, {8716, 32990}, {9466, 31457}, {9606, 63041}, {9657, 69093}, {9670, 69094}, {9698, 17130}, {9714, 22241}, {9741, 32960}, {9766, 32979}, {9770, 16044}, {10303, 32874}, {10513, 50692}, {11054, 33197}, {11057, 62147}, {11160, 63938}, {11293, 32811}, {11294, 32810}, {14023, 33239}, {14037, 63006}, {14148, 31401}, {14568, 33189}, {14614, 33201}, {14711, 33216}, {14907, 62113}, {14929, 62151}, {15589, 21734}, {16239, 32838}, {16896, 63120}, {16989, 20105}, {17567, 18145}, {17578, 32819}, {17800, 64018}, {18840, 55738}, {18906, 35439}, {19583, 30739}, {20888, 31458}, {20943, 59572}, {22110, 52250}, {22329, 33205}, {23053, 33000}, {23055, 32989}, {25242, 29001}, {28629, 42697}, {31410, 69135}, {32832, 61867}, {32834, 55864}, {32839, 48154}, {32841, 61914}, {32867, 55866}, {32868, 55863}, {32876, 61911}, {32878, 61811}, {32879, 63098}, {32882, 61816}, {32883, 61876}, {32886, 61853}, {32888, 61837}, {32893, 61856}, {32964, 63029}, {32969, 63924}, {32972, 34505}, {32973, 63034}, {32987, 63025}, {33004, 42850}, {33007, 50992}, {33198, 59373}, {33218, 47286}, {33237, 66458}, {33245, 63104}, {33277, 37690}, {33955, 51673}, {35927, 63928}, {37176, 48838}, {37667, 59545}, {37688, 61842}, {42490, 59539}, {42491, 59540}, {44251, 64014}, {44367, 68520}, {50156, 51554}, {63024, 68525}, {63093, 68517}, {63926, 68513}, {63936, 66391}, {69387, 69405}

X(69419) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 6337, 34229}, {76, 32817, 6337}, {315, 32822, 67536}, {1975, 32830, 69}, {3855, 32818, 7814}, {3926, 69378, 1007}, {3926, 69380, 69378}, {3926, 69382, 69158}, {3933, 32815, 32006}, {7763, 52713, 69385}, {7765, 7795, 33221}, {7814, 11185, 3855}, {7863, 63955, 32970}, {17130, 34511, 32968}, {32824, 32836, 3}, {32831, 59635, 34803}, {32832, 62362, 61867}, {32840, 69379, 325}, {32973, 63933, 63034}

X(69420) = X(2)X(9607)∩X(3)X(76)

Barycentrics   2*b^2*c^2 + SA^2 + SB*SC : :

X(69420) = 3 X[7770] - 2 X[7772], X[7772] - 3 X[17130]

X(69420) lies on these lines: {2, 9607}, {3, 76}, {4, 7788}, {5, 32821}, {6, 17128}, {20, 32869}, {32, 63925}, {69, 3146}, {83, 22253}, {115, 7881}, {148, 7784}, {187, 68515}, {194, 11174}, {264, 11403}, {274, 16842}, {298, 5339}, {299, 5340}, {305, 11284}, {310, 16373}, {311, 63664}, {315, 3627}, {316, 5076}, {325, 3091}, {340, 12173}, {350, 3304}, {376, 32892}, {381, 7796}, {382, 7768}, {384, 14614}, {385, 22331}, {439, 63029}, {474, 18145}, {491, 53513}, {492, 53516}, {524, 14035}, {538, 7770}, {543, 7854}, {546, 3933}, {599, 6655}, {631, 32824}, {632, 6390}, {671, 7922}, {1003, 7751}, {1007, 15022}, {1232, 20477}, {1235, 35502}, {1376, 20943}, {1656, 7799}, {1657, 7811}, {1885, 44134}, {1909, 3303}, {1995, 8024}, {2896, 44526}, {3053, 17129}, {3090, 3926}, {3314, 44518}, {3329, 20105}, {3523, 32874}, {3525, 32817}, {3529, 7750}, {3533, 32885}, {3544, 32818}, {3545, 32825}, {3552, 8667}, {3628, 7763}, {3734, 5007}, {3746, 3761}, {3760, 5563}, {3763, 7864}, {3770, 37503}, {3785, 17538}, {3815, 33261}, {3830, 7860}, {3843, 7809}, {3851, 7814}, {3934, 31859}, {3964, 64495}, {4389, 5793}, {4479, 12513}, {5013, 31276}, {5023, 68518}, {5025, 34505}, {5047, 34284}, {5067, 32837}, {5071, 32896}, {5072, 7752}, {5077, 7936}, {5079, 69158}, {5198, 54412}, {5210, 68523}, {5254, 7868}, {5306, 14037}, {5309, 33217}, {5319, 6661}, {5461, 7801}, {5485, 32951}, {5695, 20955}, {6144, 20088}, {6179, 63954}, {6337, 10303}, {6392, 7792}, {6658, 63938}, {7283, 20925}, {7610, 33259}, {7748, 7879}, {7755, 33220}, {7758, 8370}, {7760, 11286}, {7764, 44543}, {7767, 15704}, {7769, 55857}, {7776, 61984}, {7779, 65630}, {7780, 33235}, {7781, 9466}, {7783, 15271}, {7794, 7841}, {7795, 7851}, {7798, 41940}, {7802, 49137}, {7807, 63955}, {7816, 17131}, {7818, 63922}, {7819, 59780}, {7821, 18546}, {7823, 40341}, {7824, 8716}, {7836, 13881}, {7840, 33018}, {7843, 11317}, {7848, 65633}, {7849, 11648}, {7850, 62024}, {7856, 11054}, {7863, 9167}, {7867, 32457}, {7869, 33219}, {7870, 40727}, {7871, 15031}, {7873, 66388}, {7880, 33218}, {7882, 62203}, {7891, 37637}, {7892, 19570}, {7895, 69141}, {7904, 20094}, {7905, 15484}, {7907, 8860}, {7908, 39565}, {7909, 11318}, {7916, 39590}, {7917, 61991}, {7946, 14042}, {8362, 52229}, {8556, 33004}, {8591, 33275}, {9300, 33269}, {9464, 16042}, {9766, 16044}, {9770, 32991}, {9939, 19696}, {10594, 44146}, {11055, 55085}, {11057, 17800}, {11128, 13102}, {11129, 13103}, {11159, 63936}, {11163, 16924}, {11164, 33250}, {11168, 33012}, {11184, 33002}, {11361, 63932}, {11444, 51383}, {11477, 18906}, {11541, 64018}, {12103, 14907}, {12215, 53093}, {13468, 32964}, {13571, 66413}, {13740, 48838}, {14023, 19687}, {14031, 63093}, {14032, 34604}, {14044, 66587}, {14148, 31455}, {14568, 32954}, {14929, 62044}, {14994, 52987}, {15301, 15515}, {15533, 66419}, {15589, 32882}, {15597, 33204}, {16408, 18146}, {16862, 18140}, {16865, 16992}, {16990, 63548}, {17008, 59545}, {17251, 33832}, {17313, 33818}, {17531, 18135}, {17681, 48869}, {17692, 47037}, {19686, 63951}, {19693, 44367}, {19768, 50156}, {20023, 37338}, {20181, 41838}, {21356, 33025}, {22110, 32963}, {22165, 33192}, {22235, 63105}, {22237, 63106}, {22329, 32973}, {30471, 43238}, {30472, 43239}, {31099, 45201}, {32006, 50688}, {32458, 38734}, {32816, 32877}, {32823, 32890}, {32826, 62028}, {32829, 60781}, {32831, 37647}, {32838, 61870}, {32841, 34803}, {32868, 61814}, {32872, 61848}, {32875, 69406}, {32880, 37668}, {32888, 61807}, {32893, 55864}, {32894, 61804}, {32971, 41624}, {32988, 41133}, {32992, 34511}, {33007, 63928}, {33201, 63034}, {33280, 63941}, {33290, 63543}, {33934, 50044}, {34229, 61820}, {34254, 37454}, {38740, 62348}, {39998, 40916}, {40246, 50989}, {46219, 62362}, {48913, 61970}, {49140, 67536}, {53127, 61900}, {55626, 60702}, {63098, 69404}, {63926, 68177}, {63935, 66387}, {66397, 66455}

X(69420) = reflection of X(i) in X(j) for these {i,j}: {7770, 17130}, {33234, 7854}
X(69420) = anticomplement of X(9607)
X(69420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 32833, 32821}, {69, 69379, 32819}, {76, 99, 69381}, {76, 1975, 183}, {76, 69380, 1975}, {194, 69139, 11174}, {384, 63933, 14614}, {385, 68517, 22331}, {631, 32824, 59634}, {3926, 52713, 59635}, {3933, 11185, 7773}, {6337, 32834, 37688}, {7781, 9466, 11285}, {7795, 47286, 7851}, {7801, 63924, 7887}, {7880, 69162, 33218}, {7904, 20094, 44519}, {13108, 64653, 39646}, {17128, 20081, 6}, {32815, 69377, 7750}, {32824, 46951, 631}, {32830, 69378, 325}, {32831, 69385, 37647}, {32840, 69383, 1007}, {63954, 68527, 6179}

X(69421) = X(2)X(9607)∩X(4)X(69)

Barycentrics   2*b^2*c^2 + S^2 + SA^2 + SB*SC : :

X(69421) lies on these lines: {2, 9607}, {3, 46951}, {4, 69}, {5, 32825}, {20, 32874}, {99, 10299}, {140, 6337}, {141, 2996}, {183, 3522}, {194, 63041}, {325, 5068}, {381, 32892}, {384, 63034}, {385, 14031}, {439, 13468}, {443, 18145}, {524, 32979}, {538, 32968}, {543, 33226}, {547, 32896}, {550, 32815}, {597, 60200}, {599, 32982}, {1007, 5056}, {1078, 21735}, {1656, 3926}, {1657, 3785}, {1975, 3523}, {1992, 32971}, {2550, 20943}, {3090, 32833}, {3091, 32869}, {3146, 37671}, {3314, 33290}, {3357, 69176}, {3526, 32885}, {3533, 32817}, {3544, 7814}, {3545, 7796}, {3552, 63029}, {3618, 6392}, {3619, 5254}, {3620, 44518}, {3628, 32837}, {3629, 5395}, {3812, 42697}, {3832, 7788}, {3850, 32816}, {3851, 3933}, {3854, 32882}, {3858, 7776}, {4441, 56879}, {4662, 42696}, {5059, 15589}, {5067, 7799}, {5073, 7767}, {5286, 63119}, {5485, 10159}, {6390, 32838}, {7386, 41927}, {7395, 52437}, {7533, 40123}, {7615, 7821}, {7620, 33229}, {7735, 14037}, {7736, 20081}, {7737, 63927}, {7738, 31276}, {7745, 11008}, {7750, 49135}, {7751, 14033}, {7752, 69403}, {7754, 62995}, {7755, 14001}, {7757, 32957}, {7758, 14711}, {7760, 63022}, {7763, 61886}, {7770, 59373}, {7771, 62061}, {7778, 39143}, {7780, 33239}, {7781, 32978}, {7782, 61787}, {7789, 63104}, {7790, 18840}, {7794, 16041}, {7795, 69162}, {7801, 32969}, {7809, 61964}, {7810, 33247}, {7811, 33703}, {7840, 32995}, {7841, 50994}, {7847, 55732}, {7854, 33238}, {7863, 32977}, {7869, 14064}, {7870, 32958}, {7922, 33292}, {8361, 40727}, {8367, 66458}, {8370, 63064}, {8667, 32981}, {9466, 16043}, {9766, 32991}, {9770, 32962}, {10303, 32893}, {11057, 11541}, {11147, 34506}, {11303, 49874}, {11304, 49873}, {11479, 68660}, {13571, 16924}, {14034, 17129}, {14069, 14568}, {14907, 62147}, {14929, 62013}, {15598, 44519}, {15720, 32886}, {16045, 63109}, {16898, 19570}, {16921, 63025}, {16925, 23055}, {17582, 18146}, {18135, 37462}, {18928, 51481}, {19583, 40022}, {20080, 65630}, {21356, 32974}, {22329, 33201}, {32818, 61921}, {32827, 61970}, {32829, 55856}, {32831, 46935}, {32839, 55860}, {32841, 37647}, {32867, 55859}, {32872, 37688}, {32875, 61907}, {32877, 61919}, {32880, 63098}, {32883, 61875}, {32890, 44904}, {32954, 59780}, {32965, 42850}, {32975, 34511}, {33181, 63107}, {33248, 50570}, {33269, 63024}, {33410, 47520}, {33411, 47518}, {35018, 69158}, {39998, 46336}, {42153, 63106}, {42156, 63105}, {43527, 60636}, {47286, 63121}, {47338, 67603}, {47730, 63195}, {50992, 63932}, {52718, 67095}, {53102, 60627}, {60143, 60209}, {60145, 63122}, {60219, 60640}, {61870, 62362}, {62036, 64018}, {62996, 69208}, {63006, 68525}, {63936, 66409}

X(69421) = anticomplement of X(22332)
X(69421) = anticomplement of the isogonal conjugate of X(60145)
X(69421) = X(60145)-anticomplementary conjugate of X(8)
X(69421) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 7768, 32006}, {4, 69377, 7768}, {76, 11185, 69377}, {76, 52713, 69378}, {76, 54412, 1232}, {76, 69378, 69}, {140, 32824, 6337}, {140, 69380, 32824}, {1975, 32834, 34229}, {3926, 64093, 69385}, {3926, 69385, 34803}, {5056, 32821, 1007}, {5056, 32830, 32821}, {6392, 69139, 3618}, {7768, 11185, 4}, {11185, 69377, 32006}, {12322, 12323, 3818}, {17130, 63955, 14001}, {32006, 69377, 69}, {32006, 69378, 11185}, {32815, 32868, 69381}, {32821, 59635, 5056}, {32824, 32828, 140}, {32828, 69380, 6337}, {32830, 59635, 1007}, {32971, 63933, 1992}

X(69422) = X(2)X(9607)∩X(5)X(76)

Barycentrics   2*b^2*c^2 + 2*S^2 + SA^2 + SB*SC : :

X(69422) = 2 X[9698] - 3 X[32992]

X(69422) lies on these lines: {2, 9607}, {4, 37671}, {5, 76}, {6, 33269}, {20, 183}, {69, 3832}, {75, 9711}, {99, 3530}, {115, 7849}, {140, 59634}, {141, 7933}, {194, 9606}, {230, 17128}, {264, 1906}, {274, 17575}, {298, 42163}, {299, 42166}, {311, 17703}, {315, 3843}, {316, 3861}, {340, 23047}, {350, 15888}, {382, 7750}, {384, 22329}, {442, 18145}, {524, 16044}, {538, 9698}, {546, 7768}, {548, 1078}, {599, 14063}, {631, 1975}, {637, 45439}, {638, 45438}, {668, 64200}, {671, 8357}, {858, 39998}, {1007, 61914}, {1656, 32833}, {1799, 37899}, {1907, 54412}, {1909, 37722}, {2886, 20943}, {2896, 53419}, {3054, 7891}, {3090, 32821}, {3091, 7788}, {3314, 63534}, {3523, 32893}, {3525, 32824}, {3526, 32832}, {3528, 32815}, {3552, 13468}, {3620, 63533}, {3627, 7811}, {3628, 7799}, {3630, 7900}, {3631, 7885}, {3760, 37719}, {3761, 37720}, {3767, 33217}, {3785, 33703}, {3815, 20081}, {3845, 7860}, {3850, 7809}, {3853, 7767}, {3855, 7773}, {3856, 15031}, {3857, 48913}, {3926, 5067}, {3934, 7765}, {4197, 18135}, {4479, 12607}, {5056, 32869}, {5070, 7763}, {5071, 32825}, {5169, 45201}, {5254, 7876}, {5306, 68525}, {5319, 7770}, {5485, 32960}, {6292, 32457}, {6337, 55864}, {6376, 9710}, {6390, 16239}, {6392, 11174}, {6656, 9466}, {6661, 7755}, {7396, 41927}, {7486, 32830}, {7610, 32964}, {7620, 33238}, {7745, 17129}, {7747, 63930}, {7751, 8370}, {7753, 63925}, {7758, 31417}, {7760, 66415}, {7762, 17131}, {7764, 14711}, {7769, 48154}, {7771, 46853}, {7772, 66416}, {7776, 61953}, {7778, 33277}, {7780, 19687}, {7781, 31457}, {7782, 44682}, {7783, 58446}, {7789, 33245}, {7790, 55738}, {7791, 34505}, {7792, 16898}, {7794, 33228}, {7795, 33218}, {7797, 16896}, {7801, 33249}, {7802, 62041}, {7805, 53489}, {7807, 17130}, {7810, 19695}, {7812, 63926}, {7817, 66342}, {7819, 14568}, {7827, 55767}, {7832, 43291}, {7840, 33024}, {7851, 33221}, {7854, 18546}, {7866, 40727}, {7870, 16509}, {7873, 8352}, {7881, 43620}, {7882, 43457}, {7893, 53418}, {7896, 18424}, {7901, 50570}, {7904, 15598}, {7923, 34573}, {7936, 66392}, {8363, 69162}, {8367, 11054}, {8556, 32965}, {8667, 14035}, {8716, 33001}, {8728, 18146}, {8860, 32989}, {9300, 33020}, {9766, 32962}, {9939, 14066}, {10159, 66326}, {11057, 62036}, {11128, 20253}, {11129, 20252}, {11163, 32987}, {11168, 33004}, {11184, 33009}, {11361, 63928}, {11793, 51383}, {13881, 33248}, {14031, 22331}, {14042, 63941}, {14068, 63938}, {14537, 63927}, {14614, 32971}, {14907, 17800}, {15062, 69176}, {15271, 33258}, {15480, 20088}, {15559, 44146}, {15589, 17578}, {15606, 51440}, {15717, 32872}, {16924, 41624}, {16990, 44518}, {17004, 59545}, {17529, 18140}, {19570, 51860}, {19768, 51554}, {20582, 66345}, {21356, 33200}, {23055, 33205}, {25264, 31462}, {26235, 62310}, {30739, 40022}, {31168, 66347}, {31450, 31859}, {31478, 69255}, {32006, 61982}, {32816, 32888}, {32817, 32838}, {32818, 69405}, {32822, 61138}, {32826, 49138}, {32829, 61881}, {32837, 61886}, {32840, 34803}, {32882, 63098}, {32894, 37668}, {32896, 61895}, {32967, 41133}, {32968, 63101}, {32981, 63029}, {33016, 63932}, {33023, 42850}, {33262, 37637}, {35007, 66319}, {37197, 44134}, {37512, 47287}, {40107, 51374}, {43459, 58190}, {53127, 61905}, {58212, 61689}, {62021, 64018}, {62344, 67606}, {63046, 65630}, {63935, 66408}

X(69422) = isotomic conjugate of the isogonal conjugate of X(6688)
X(69422) = barycentric product X(76)*X(6688)
X(69422) = barycentric quotient X(6688)/X(6)
X(69422) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 3933, 7814}, {76, 59635, 325}, {76, 64093, 59635}, {76, 69387, 3933}, {183, 69378, 32819}, {1975, 32828, 37688}, {3090, 32836, 32821}, {6390, 16239, 62362}, {7745, 17129, 50251}, {7810, 63922, 19695}, {7814, 69387, 5}, {7854, 18546, 33229}, {9466, 63924, 6656}, {11185, 69381, 7750}, {16924, 63933, 41624}, {32815, 32886, 69384}, {32824, 32885, 3525}, {32828, 52713, 1975}, {32834, 69378, 183}, {32868, 69382, 69377}, {32872, 69379, 34229}, {69377, 69382, 7773}

X(69423) = X(2)X(32)∩X(69)X(140)

Barycentrics   2*b^2*c^2 + 2*S^2 - SA^2 - 2*SB*SC : :

X(69423) lies on these lines: {2, 32}, {3, 32815}, {4, 32838}, {5, 32867}, {6, 32978}, {20, 7771}, {30, 69385}, {39, 37667}, {69, 140}, {76, 3523}, {99, 15717}, {115, 33023}, {141, 32970}, {183, 631}, {187, 32971}, {193, 7780}, {194, 33012}, {230, 16043}, {316, 5056}, {325, 3525}, {376, 32826}, {385, 31400}, {439, 3734}, {548, 67536}, {549, 6337}, {574, 6392}, {632, 7776}, {1007, 3526}, {1153, 7895}, {1235, 35486}, {1656, 32006}, {1975, 3524}, {1992, 31406}, {2996, 60220}, {3053, 32968}, {3054, 7784}, {3055, 63932}, {3090, 7750}, {3091, 14907}, {3146, 32870}, {3314, 33000}, {3522, 11185}, {3528, 32819}, {3530, 32868}, {3533, 32884}, {3619, 32954}, {3620, 3788}, {3767, 32990}, {3832, 7802}, {3933, 5054}, {3934, 21843}, {5013, 13468}, {5023, 14033}, {5067, 7773}, {5206, 32981}, {5207, 14693}, {5210, 33239}, {5254, 7610}, {5265, 64133}, {5286, 7824}, {5304, 7786}, {5319, 15482}, {5569, 7816}, {6179, 37665}, {6340, 7734}, {6390, 15720}, {6656, 62992}, {6683, 51171}, {6776, 10104}, {6921, 37670}, {7509, 68654}, {7526, 34883}, {7615, 65633}, {7735, 11285}, {7737, 32987}, {7745, 32975}, {7746, 32974}, {7747, 32991}, {7754, 63029}, {7759, 63077}, {7761, 32972}, {7762, 62993}, {7763, 10303}, {7764, 20080}, {7768, 61856}, {7769, 37668}, {7774, 33015}, {7777, 33003}, {7778, 32977}, {7782, 15692}, {7788, 15709}, {7789, 8556}, {7791, 17004}, {7792, 32960}, {7795, 32989}, {7796, 32835}, {7797, 33258}, {7798, 31450}, {7799, 15721}, {7803, 37689}, {7805, 55797}, {7823, 32999}, {7825, 52250}, {7828, 33202}, {7830, 32982}, {7831, 33180}, {7832, 33203}, {7836, 33206}, {7849, 55729}, {7860, 46935}, {7866, 63104}, {7868, 33189}, {7871, 61846}, {7879, 37690}, {7885, 32998}, {7893, 63083}, {7898, 32963}, {7904, 17006}, {7907, 16990}, {7917, 32898}, {7921, 31407}, {7928, 33248}, {7937, 33182}, {7945, 33262}, {8359, 23055}, {8860, 33190}, {9605, 63034}, {10299, 32886}, {10304, 69383}, {10513, 61848}, {11057, 61936}, {11168, 32985}, {11315, 32806}, {11316, 32805}, {11318, 23053}, {12108, 32875}, {13881, 32986}, {14001, 15271}, {14023, 31455}, {14037, 39266}, {14061, 33200}, {14064, 37637}, {14712, 32962}, {14853, 50685}, {14929, 16239}, {15031, 17578}, {15513, 35927}, {15597, 32984}, {15655, 68177}, {15701, 32896}, {15702, 32818}, {15705, 32893}, {15708, 32833}, {15719, 32892}, {16992, 17567}, {16999, 33055}, {17566, 45962}, {30435, 63041}, {31274, 54103}, {31276, 32964}, {31415, 63935}, {31489, 63928}, {32448, 40925}, {32817, 61814}, {32820, 32877}, {32821, 61836}, {32822, 61138}, {32823, 37647}, {32830, 61820}, {32831, 61834}, {32869, 61812}, {32872, 61791}, {32874, 61806}, {32876, 61832}, {32878, 61811}, {32888, 61807}, {32890, 61818}, {32891, 61835}, {32957, 46453}, {32965, 43448}, {33013, 66699}, {33185, 63121}, {33188, 63046}, {33226, 44518}, {33234, 63533}, {33238, 63534}, {33247, 53419}, {34505, 47061}, {37466, 40330}, {37512, 63955}, {37534, 55419}, {38440, 54111}, {38739, 46236}, {39590, 47102}, {42009, 55040}, {42060, 55041}, {42089, 69165}, {42092, 69157}, {42944, 69180}, {42945, 69186}, {44277, 68346}, {48913, 61906}, {53095, 63923}, {53857, 64982}, {55085, 63005}, {55104, 55448}, {55449, 63399}, {63025, 63950}, {68085, 68355}

X(69423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1078, 3785}, {2, 3785, 32816}, {2, 7793, 69208}, {2, 20065, 31404}, {3, 32828, 32815}, {3, 34229, 32828}, {4, 37688, 32838}, {20, 32832, 69382}, {69, 140, 32829}, {69, 32829, 32825}, {141, 44535, 32970}, {183, 631, 3926}, {325, 3525, 32839}, {376, 59635, 32826}, {385, 33001, 31400}, {549, 69381, 6337}, {632, 7776, 34803}, {1975, 69384, 46951}, {3053, 58446, 32968}, {3054, 7784, 32969}, {3090, 7750, 32827}, {3146, 32870, 69387}, {3524, 69384, 1975}, {3526, 7767, 1007}, {3528, 69386, 32819}, {3832, 32897, 53127}, {3934, 21843, 32973}, {6337, 69381, 32836}, {7749, 7800, 2}, {7771, 32832, 20}, {7780, 31401, 193}, {7802, 53127, 3832}, {7815, 34506, 69207}, {7815, 69207, 2}, {7824, 17008, 5286}, {7830, 43620, 32982}, {7904, 17006, 32961}, {7907, 16990, 53033}, {10303, 15589, 7763}, {11185, 43459, 3522}, {14023, 31455, 62988}, {15692, 69379, 7782}, {15717, 32834, 99}, {32823, 61867, 37647}, {32826, 32885, 59635}, {32827, 32883, 3090}, {32867, 64018, 5}, {32973, 55819, 21843}, {33189, 55732, 7868}, {37668, 55864, 7769}

X(69424) = X(2)X(99)∩X(3)X(69)

Barycentrics   2*S^2 - SA^2 - 2*SB*SC : :

X(69424) lies on these lines: {2, 99}, {3, 69}, {4, 32829}, {5, 32826}, {6, 32459}, {20, 316}, {23, 56435}, {30, 1007}, {32, 439}, {39, 32973}, {76, 3523}, {83, 33201}, {140, 32838}, {141, 33215}, {183, 3524}, {184, 65747}, {187, 193}, {194, 2021}, {230, 8716}, {232, 35940}, {302, 5334}, {303, 5335}, {311, 58846}, {315, 3522}, {317, 37460}, {325, 376}, {381, 34803}, {382, 32887}, {384, 31400}, {441, 30227}, {492, 9541}, {524, 5210}, {538, 21843}, {548, 7776}, {549, 34229}, {550, 32006}, {599, 47061}, {625, 34504}, {626, 33023}, {631, 1975}, {877, 40138}, {966, 21937}, {1003, 7736}, {1078, 15717}, {1285, 41624}, {1350, 59552}, {1352, 9734}, {1384, 1992}, {1506, 32979}, {1656, 32884}, {1657, 32889}, {2548, 32981}, {2996, 7746}, {3053, 3629}, {3054, 34505}, {3068, 35306}, {3069, 35305}, {3090, 32819}, {3091, 7769}, {3146, 7752}, {3314, 33008}, {3329, 33255}, {3525, 32822}, {3526, 32883}, {3528, 7750}, {3529, 7773}, {3530, 69381}, {3545, 37647}, {3552, 63018}, {3589, 5013}, {3618, 5024}, {3619, 8359}, {3620, 7801}, {3763, 7789}, {3767, 32989}, {3788, 32974}, {3793, 11008}, {3796, 4176}, {3815, 14033}, {3832, 62362}, {3972, 37665}, {3978, 54033}, {4232, 11059}, {4235, 41370}, {4255, 59538}, {4563, 64058}, {4846, 43705}, {5007, 51581}, {5008, 5032}, {5023, 6144}, {5054, 32885}, {5058, 6463}, {5062, 6462}, {5068, 32871}, {5085, 59548}, {5162, 20065}, {5167, 35687}, {5191, 10553}, {5206, 7758}, {5254, 32970}, {5281, 64133}, {5286, 7783}, {5304, 7757}, {5406, 8222}, {5407, 8223}, {5475, 63077}, {5485, 8860}, {5486, 63179}, {5585, 40341}, {5731, 69038}, {5739, 35276}, {5890, 51439}, {5921, 14981}, {5939, 64090}, {5971, 66368}, {5976, 7709}, {6389, 40349}, {6392, 7781}, {6394, 35912}, {6658, 45017}, {6671, 40922}, {6672, 40921}, {6676, 19583}, {7487, 32002}, {7694, 8781}, {7710, 54996}, {7735, 31859}, {7737, 32456}, {7738, 7807}, {7745, 33239}, {7748, 31275}, {7756, 32982}, {7762, 68516}, {7768, 62067}, {7771, 15589}, {7774, 13586}, {7775, 43618}, {7777, 33007}, {7778, 32986}, {7784, 33226}, {7785, 33244}, {7786, 33198}, {7788, 19708}, {7791, 7891}, {7792, 33191}, {7795, 32990}, {7796, 21734}, {7799, 7850}, {7800, 7863}, {7802, 50693}, {7803, 33181}, {7808, 31450}, {7809, 62120}, {7811, 10513}, {7813, 8588}, {7816, 31401}, {7819, 63120}, {7823, 33254}, {7828, 33203}, {7832, 33202}, {7836, 32965}, {7840, 66699}, {7841, 37690}, {7845, 47102}, {7847, 33180}, {7851, 33189}, {7860, 62110}, {7862, 32980}, {7871, 62083}, {7885, 33253}, {7890, 51587}, {7898, 33207}, {7899, 33200}, {7906, 33276}, {7912, 32997}, {7917, 62078}, {7918, 33182}, {7925, 33017}, {7934, 33210}, {7940, 33199}, {7941, 33268}, {7947, 33275}, {8182, 11160}, {8370, 62993}, {8598, 9770}, {8644, 58284}, {8722, 62174}, {8724, 11180}, {9145, 63646}, {9177, 61188}, {9605, 63011}, {9737, 31670}, {9741, 22329}, {9744, 21166}, {9752, 37459}, {9771, 18584}, {9855, 23334}, {10299, 32820}, {10303, 32832}, {10330, 54013}, {10565, 57518}, {10645, 69121}, {10646, 69120}, {11054, 11148}, {11057, 62094}, {11064, 37188}, {11159, 12040}, {11171, 18906}, {11174, 14039}, {11179, 50567}, {11184, 53418}, {11286, 63041}, {11288, 15048}, {11292, 32807}, {11361, 63083}, {11433, 35302}, {11480, 59540}, {11481, 59539}, {11676, 51580}, {12150, 14930}, {13665, 32806}, {13785, 32805}, {13881, 32977}, {14023, 15513}, {14035, 31404}, {14064, 63548}, {14075, 63000}, {14535, 66318}, {14555, 16436}, {14558, 35493}, {14614, 46453}, {14712, 33208}, {14826, 37457}, {14912, 51374}, {14929, 34200}, {14962, 35704}, {15022, 15031}, {15484, 66391}, {15631, 67630}, {15640, 48913}, {15693, 32892}, {15698, 32896}, {15702, 69386}, {15712, 32877}, {15720, 32886}, {16041, 44377}, {16431, 18141}, {16989, 33246}, {16990, 33273}, {17005, 33016}, {17008, 33274}, {17128, 33001}, {17321, 37599}, {17538, 32823}, {17549, 45962}, {17578, 32873}, {19661, 63022}, {20088, 68518}, {21309, 62995}, {21508, 37656}, {21511, 63003}, {21735, 32821}, {21736, 26468}, {22110, 66616}, {22236, 59542}, {22238, 59541}, {22253, 51123}, {22401, 28710}, {23053, 40727}, {23055, 52229}, {26288, 44365}, {26289, 44364}, {28406, 28436}, {28724, 56339}, {30435, 63073}, {30761, 48813}, {31276, 33012}, {31406, 68527}, {31455, 32987}, {31457, 55797}, {31489, 32983}, {32477, 55819}, {32834, 61820}, {32840, 43459}, {32868, 61814}, {32869, 61806}, {32870, 61842}, {32874, 61812}, {32875, 61138}, {32878, 61807}, {32881, 62102}, {32888, 61811}, {32890, 61795}, {32891, 62092}, {32893, 61825}, {32895, 62152}, {32897, 61848}, {32955, 39142}, {32969, 44518}, {32976, 63534}, {32978, 69139}, {32984, 53419}, {33004, 38907}, {33237, 63119}, {33238, 44519}, {33249, 63533}, {33258, 46226}, {33265, 63021}, {33266, 63017}, {33813, 46236}, {33929, 35298}, {35296, 37644}, {35303, 63105}, {35304, 63106}, {35486, 44146}, {35497, 51884}, {35902, 54075}, {36181, 47326}, {36748, 59555}, {37242, 61561}, {37340, 42129}, {37341, 42132}, {37458, 63155}, {37461, 54132}, {37534, 55448}, {38279, 47412}, {38747, 54103}, {38940, 54087}, {39099, 50967}, {40824, 54859}, {41622, 50370}, {42150, 69165}, {42151, 69157}, {42944, 69186}, {42945, 69180}, {44369, 50974}, {44401, 50571}, {44535, 63923}, {45375, 58804}, {45376, 58803}, {45420, 66443}, {46127, 51253}, {47077, 48945}, {47113, 63722}, {47286, 62992}, {47582, 52277}, {47596, 62299}, {51383, 61136}, {51386, 64100}, {51584, 63116}, {51589, 66763}, {52250, 69141}, {52275, 63084}, {52718, 61836}, {53096, 69209}, {54097, 65633}, {55104, 55419}, {55418, 63399}, {55812, 60200}, {55864, 69383}, {56429, 67717}, {57275, 67894}, {59373, 68718}, {61187, 62705}, {63440, 64711}, {63756, 69094}, {64982, 66767}

X(69424) = reflection of X(i) in X(j) for these {i,j}: {32827, 1007}, {37667, 21843}
X(69424) = isotomic conjugate of X(68564)
X(69424) = anticomplement of X(43620)
X(69424) = isotomic conjugate of the polar conjugate of X(1992)
X(69424) = isogonal conjugate of the polar conjugate of X(11059)
X(69424) = X(i)-Ceva conjugate of X(j) for these (i,j): {11059, 1992}, {64982, 69}
X(69424) = X(i)-isoconjugate of X(j) for these (i,j): {19, 21448}, {25, 55923}, {31, 68564}, {92, 39238}, {798, 65353}, {923, 52477}, {1096, 55977}, {1973, 5485}, {2489, 37216}, {14273, 36045}, {36128, 57467}
X(69424) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 68564}, {6, 21448}, {599, 5094}, {1992, 52290}, {2482, 52477}, {6337, 5485}, {6503, 55977}, {6505, 55923}, {11147, 4}, {22391, 39238}, {31654, 14273}, {31998, 65353}, {35133, 2501}, {53992, 58757}, {62568, 8754}
X(69424) = crossdifference of every pair of points on line {351, 2489}
X(69424) = barycentric product X(i)*X(j) for these {i,j}: {3, 11059}, {69, 1992}, {287, 51438}, {304, 36277}, {305, 1384}, {394, 58782}, {1444, 42724}, {1499, 4563}, {3926, 4232}, {4561, 4786}, {4592, 14207}, {6390, 52141}, {6791, 47389}, {8644, 52608}, {11165, 64982}, {13608, 66767}, {27088, 30786}
X(69424) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 68564}, {3, 21448}, {63, 55923}, {69, 5485}, {99, 65353}, {184, 39238}, {394, 55977}, {524, 52477}, {1384, 25}, {1499, 2501}, {1992, 4}, {2408, 68629}, {3292, 57467}, {4232, 393}, {4558, 1296}, {4563, 35179}, {4592, 37216}, {4786, 7649}, {6791, 8754}, {8644, 2489}, {9125, 14273}, {9126, 47230}, {11059, 264}, {11147, 52290}, {11165, 5094}, {14207, 24006}, {15471, 60428}, {27088, 468}, {30234, 6591}, {35266, 1990}, {36277, 19}, {41585, 27376}, {41614, 14262}, {42724, 41013}, {51438, 297}, {52141, 17983}, {53778, 53419}, {58782, 2052}, {61345, 68566}
X(69424) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 99, 32815}, {2, 8591, 7620}, {2, 32815, 69382}, {2, 53141, 671}, {3, 3926, 3785}, {3, 6337, 3926}, {3, 6390, 69}, {3, 9723, 68654}, {6, 32459, 32985}, {20, 7763, 32816}, {20, 63098, 316}, {69, 6337, 6390}, {69, 6390, 3926}, {140, 69378, 32838}, {141, 53095, 33215}, {183, 32817, 32836}, {183, 59634, 32817}, {187, 34511, 193}, {193, 35287, 187}, {315, 32831, 32825}, {316, 7763, 63098}, {316, 63098, 32816}, {325, 376, 64018}, {549, 69380, 34229}, {550, 69158, 32006}, {574, 2482, 69206}, {574, 69206, 2}, {620, 2549, 2}, {625, 34504, 43619}, {631, 1975, 32828}, {631, 52713, 37688}, {1975, 37688, 52713}, {1992, 11147, 27088}, {2482, 7618, 2}, {2548, 69171, 32981}, {3146, 32835, 7752}, {3522, 32831, 315}, {3524, 32817, 183}, {3524, 59634, 32836}, {3525, 32822, 59635}, {3525, 59635, 32867}, {3526, 69385, 32883}, {3528, 32818, 7750}, {5013, 59545, 14001}, {5024, 8369, 3618}, {5866, 6337, 40697}, {5866, 44180, 3}, {7618, 69206, 574}, {7737, 32456, 35927}, {7763, 7782, 20}, {7771, 32833, 15589}, {7781, 69207, 6392}, {7783, 16925, 5286}, {7789, 15815, 16043}, {7791, 7891, 53033}, {7795, 37512, 32990}, {7799, 14907, 37668}, {7816, 31401, 32971}, {7863, 15515, 7800}, {8182, 39785, 11160}, {9741, 22329, 66458}, {9885, 9886, 7615}, {10303, 69379, 32832}, {10304, 37668, 14907}, {11159, 12040, 63025}, {11165, 27088, 1992}, {15589, 15692, 7771}, {15717, 32830, 1078}, {31859, 35297, 7735}, {32826, 32839, 5}, {32837, 64018, 325}, {34229, 69380, 46951}, {34803, 67536, 381}, {35927, 62988, 7737}, {37688, 52713, 32828}, {40680, 68654, 3785}, {44377, 44526, 16041}, {52193, 52194, 11898}

X(69425) = X(2)X(1975)∩X(20)X(99)

Barycentrics   b^2*c^2 - 2*S^2 + SA^2 + 2*SB*SC : :

X(69425) lies on these lines: {2, 1975}, {3, 15589}, {4, 6390}, {5, 32822}, {6, 33201}, {20, 99}, {30, 32818}, {39, 33198}, {69, 3522}, {76, 3523}, {140, 52713}, {148, 32972}, {183, 15717}, {193, 3552}, {194, 5304}, {253, 34410}, {274, 17558}, {302, 22237}, {303, 22235}, {305, 10565}, {316, 32825}, {325, 3146}, {350, 5265}, {376, 3933}, {382, 32823}, {384, 37665}, {385, 439}, {549, 32874}, {626, 33210}, {631, 32834}, {1007, 3832}, {1078, 15692}, {1235, 58846}, {1285, 68513}, {1656, 32871}, {1909, 5281}, {2482, 69207}, {2549, 7863}, {3053, 63042}, {3091, 7763}, {3266, 4232}, {3314, 33023}, {3491, 35687}, {3516, 32000}, {3524, 32869}, {3525, 32870}, {3526, 32897}, {3528, 7767}, {3529, 7776}, {3543, 7799}, {3593, 43377}, {3595, 43376}, {3618, 59552}, {3620, 32965}, {3685, 17081}, {3734, 31400}, {3767, 33203}, {3785, 7782}, {3788, 15301}, {3793, 68528}, {3839, 7752}, {3945, 17103}, {3964, 6527}, {4576, 57008}, {5024, 16045}, {5054, 32893}, {5056, 11185}, {5059, 32006}, {5067, 32898}, {5286, 6680}, {5305, 33191}, {5334, 69165}, {5335, 69157}, {5395, 14031}, {5695, 59602}, {5921, 6393}, {5976, 20081}, {6392, 16925}, {7396, 34254}, {7398, 16276}, {7486, 7769}, {7618, 7815}, {7620, 39565}, {7664, 53857}, {7735, 33205}, {7736, 59546}, {7750, 10513}, {7751, 35022}, {7754, 32985}, {7758, 14148}, {7762, 33239}, {7768, 62110}, {7771, 61788}, {7773, 17578}, {7774, 32981}, {7777, 32979}, {7778, 33200}, {7779, 33244}, {7788, 62120}, {7790, 33182}, {7793, 9740}, {7795, 33202}, {7801, 53142}, {7809, 15640}, {7811, 32896}, {7816, 34511}, {7821, 43619}, {7835, 33183}, {7836, 32974}, {7839, 33255}, {7841, 53141}, {7871, 62160}, {7879, 33226}, {7881, 32986}, {7882, 47102}, {7887, 47287}, {7893, 33254}, {7895, 34504}, {7897, 32997}, {7900, 33193}, {7903, 43618}, {7906, 33007}, {7912, 8591}, {7925, 32980}, {7929, 33207}, {7939, 33253}, {7941, 33280}, {7947, 33017}, {8359, 18840}, {8369, 9741}, {8778, 56013}, {9605, 14039}, {9723, 14118}, {9770, 65630}, {10303, 32828}, {10323, 22241}, {10330, 64059}, {11057, 62122}, {11160, 33208}, {11165, 31406}, {11488, 59539}, {11489, 59540}, {12111, 51386}, {13331, 14037}, {14001, 31859}, {14023, 32456}, {14035, 43450}, {14063, 20094}, {14068, 63021}, {14069, 15048}, {14548, 52352}, {14615, 51348}, {14907, 62097}, {14929, 15696}, {14961, 28717}, {15022, 32873}, {15300, 65633}, {15574, 16661}, {15708, 46951}, {16044, 63077}, {17084, 24280}, {17576, 45962}, {20065, 35927}, {20080, 51374}, {20088, 33187}, {20105, 63048}, {20580, 43673}, {21166, 39647}, {27162, 56987}, {28425, 54075}, {30769, 37804}, {31304, 68354}, {32459, 63933}, {32827, 50688}, {32832, 55864}, {32838, 61856}, {32839, 46936}, {32872, 37688}, {32875, 62083}, {32876, 50691}, {32877, 43459}, {32880, 61791}, {32881, 50689}, {32882, 61804}, {32885, 61846}, {32892, 61805}, {32894, 61816}, {32895, 69402}, {32959, 43291}, {32970, 47286}, {32991, 63083}, {33014, 63046}, {33025, 63548}, {33242, 63633}, {33296, 37666}, {34229, 61820}, {34383, 63559}, {34403, 53050}, {37640, 59541}, {37641, 59542}, {37642, 59538}, {37647, 61914}, {37664, 56999}, {37671, 62063}, {37690, 44518}, {40123, 59343}, {40405, 45833}, {40925, 61132}, {40995, 44247}, {41136, 66398}, {42150, 69121}, {42151, 69120}, {43136, 66393}, {44377, 63533}, {48913, 62003}, {50248, 68518}, {51580, 68525}, {53127, 62362}, {54429, 62400}, {62992, 63923}, {63017, 68517}

X(69425) = isotomic conjugate of the polar conjugate of X(56013)
X(69425) = X(34233)-anticomplementary conjugate of X(5905)
X(69425) = X(40995)-Dao conjugate of X(26958)
X(69425) = barycentric product X(i)*X(j) for these {i,j}: {69, 56013}, {304, 8765}, {305, 8778}
X(69425) = barycentric quotient X(i)/X(j) for these {i,j}: {8765, 19}, {8778, 25}, {56013, 4}
X(69425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1975, 69379}, {2, 69379, 69383}, {3, 32817, 32830}, {3, 32830, 15589}, {4, 6390, 32831}, {4, 32831, 63098}, {20, 3926, 37668}, {69, 32820, 32840}, {99, 3926, 20}, {148, 32972, 63536}, {194, 32973, 5304}, {631, 69380, 32834}, {1007, 32819, 3832}, {1975, 6337, 2}, {1975, 59634, 6337}, {2549, 7863, 53033}, {2549, 53033, 33180}, {3146, 32841, 325}, {3522, 32840, 69}, {3525, 64093, 32870}, {3526, 69386, 32897}, {3785, 7782, 10304}, {3788, 43448, 33199}, {3926, 64018, 7796}, {5286, 69206, 33181}, {6392, 16925, 37689}, {7735, 59545, 33205}, {7738, 7789, 2}, {7752, 32826, 3839}, {7763, 32815, 3091}, {7769, 69382, 7486}, {7773, 67536, 17578}, {7781, 69206, 5286}, {7782, 32833, 3785}, {7789, 8716, 7738}, {7816, 34511, 69208}, {10513, 50693, 7750}, {11185, 32829, 5056}, {14031, 63018, 5395}, {14148, 69171, 7758}, {15022, 32873, 34803}, {20081, 32964, 37667}, {32826, 32837, 7752}, {32839, 69387, 46936}, {32872, 61834, 37688}, {32879, 50693, 10513}, {37667, 51579, 32964}

X(69426) = X(2)X(7765)∩X(3)X(76)

Barycentrics   2*b^2*c^2 - S^2 + SA^2 + 2*SB*SC : :
X(69426) = 4 X[3] - 3 X[12203], 3 X[384] - 2 X[5007], 4 X[384] - 3 X[12150], 5 X[384] - 3 X[63038], 4 X[5007] - 3 X[7760], 8 X[5007] - 9 X[12150], 10 X[5007] - 9 X[63038], 2 X[7760] - 3 X[12150], 5 X[7760] - 6 X[63038], 5 X[12150] - 4 X[63038], 2 X[6655] - 3 X[7883], 4 X[7794] - 3 X[7883], 4 X[7819] - 3 X[7827], 4 X[7829] - 5 X[19689], 4 X[7849] - 3 X[7924], 5 X[19693] - 3 X[34604], 2 X[39593] - 3 X[66322]

X(69426) lies on these lines: {2, 7765}, {3, 76}, {4, 7796}, {5, 7799}, {20, 7811}, {23, 8024}, {30, 7768}, {32, 20081}, {39, 17128}, {61, 42674}, {62, 42675}, {69, 3529}, {83, 194}, {95, 59287}, {115, 7836}, {140, 59634}, {141, 7847}, {148, 626}, {187, 17129}, {264, 35502}, {274, 5047}, {298, 16964}, {299, 16965}, {305, 1995}, {311, 35500}, {315, 3146}, {316, 3627}, {325, 546}, {340, 6240}, {350, 5563}, {381, 7814}, {382, 7788}, {384, 538}, {385, 7816}, {404, 18145}, {474, 18146}, {524, 19687}, {543, 6655}, {550, 37671}, {574, 31276}, {575, 12215}, {576, 18906}, {599, 7936}, {625, 7947}, {632, 64093}, {633, 11128}, {634, 11129}, {671, 5025}, {726, 12195}, {729, 63564}, {754, 6658}, {1003, 6179}, {1007, 3544}, {1232, 18354}, {1235, 14865}, {1236, 12086}, {1238, 32002}, {1494, 52130}, {1506, 14148}, {1657, 11057}, {1799, 7492}, {1909, 3746}, {2396, 46512}, {2482, 33259}, {2549, 3096}, {2896, 7756}, {2996, 53033}, {3090, 7763}, {3091, 3926}, {3266, 16042}, {3303, 64133}, {3314, 7748}, {3329, 32450}, {3518, 44146}, {3522, 32869}, {3523, 46951}, {3525, 6337}, {3552, 7751}, {3628, 6390}, {3767, 7835}, {3770, 54409}, {3785, 50693}, {3788, 14061}, {3832, 32825}, {3839, 32896}, {3843, 48913}, {3849, 19696}, {3934, 7783}, {3972, 7754}, {3978, 37338}, {4045, 46226}, {4195, 48838}, {4366, 69257}, {4479, 8666}, {5056, 32837}, {5072, 69158}, {5076, 7776}, {5088, 33939}, {5198, 58782}, {5201, 33769}, {5206, 68526}, {5254, 7832}, {5286, 7846}, {5309, 7892}, {5319, 14037}, {5355, 10583}, {5475, 7906}, {5485, 33189}, {6054, 40279}, {6645, 69255}, {7283, 20924}, {7496, 39998}, {7610, 51237}, {7615, 32963}, {7618, 33012}, {7620, 32980}, {7737, 7877}, {7738, 31268}, {7739, 16898}, {7745, 7905}, {7746, 7891}, {7747, 7779}, {7750, 15704}, {7753, 13571}, {7755, 19570}, {7757, 7770}, {7758, 7812}, {7759, 11361}, {7761, 32027}, {7764, 16044}, {7766, 69172}, {7767, 12103}, {7773, 7871}, {7775, 33018}, {7780, 13586}, {7785, 7813}, {7786, 22332}, {7787, 7798}, {7789, 7828}, {7790, 7795}, {7791, 31168}, {7793, 17131}, {7797, 7820}, {7804, 7839}, {7807, 14568}, {7810, 8591}, {7817, 14043}, {7818, 33019}, {7819, 7827}, {7821, 14041}, {7822, 7864}, {7823, 7855}, {7824, 9466}, {7825, 7897}, {7826, 14712}, {7829, 19689}, {7830, 63044}, {7831, 47287}, {7833, 7854}, {7837, 14034}, {7840, 7843}, {7841, 7922}, {7842, 7939}, {7844, 7945}, {7849, 7924}, {7850, 49136}, {7851, 7930}, {7856, 14001}, {7857, 69206}, {7858, 8370}, {7859, 15048}, {7861, 7931}, {7868, 7918}, {7869, 7933}, {7870, 7887}, {7872, 7938}, {7873, 33256}, {7874, 32457}, {7876, 10159}, {7878, 11055}, {7879, 7910}, {7880, 7901}, {7881, 7934}, {7884, 33217}, {7885, 7895}, {7888, 18546}, {7890, 20088}, {7894, 22253}, {7896, 7898}, {7900, 7916}, {7907, 41134}, {7908, 7912}, {7915, 7923}, {7925, 39565}, {7926, 65630}, {7940, 13881}, {7941, 39590}, {7946, 63931}, {8362, 59780}, {8369, 11054}, {8667, 33235}, {8716, 11285}, {9464, 14002}, {9605, 60855}, {9606, 66416}, {9607, 55767}, {9698, 33020}, {9723, 52712}, {9741, 32957}, {10302, 32480}, {10303, 32828}, {10304, 32892}, {10594, 54412}, {10723, 54393}, {10796, 32520}, {11164, 63950}, {11284, 57518}, {11319, 33955}, {11541, 67536}, {11591, 51383}, {12110, 32515}, {12156, 14033}, {13172, 32152}, {13355, 54223}, {13740, 16712}, {14023, 33007}, {14044, 31173}, {14045, 39563}, {14062, 63957}, {14066, 63956}, {14509, 67364}, {14614, 68527}, {14869, 37688}, {14907, 17538}, {14929, 62162}, {14994, 55606}, {15000, 22254}, {15019, 33798}, {15022, 32831}, {15301, 37512}, {15533, 66395}, {15589, 62097}, {15717, 32874}, {16043, 52691}, {16084, 52568}, {16711, 56983}, {16865, 34284}, {16924, 34511}, {16925, 63955}, {16926, 50179}, {16927, 50160}, {16928, 50174}, {16929, 50163}, {16930, 50184}, {16931, 50155}, {16950, 19568}, {16952, 42037}, {16992, 19526}, {17531, 18140}, {17572, 18135}, {17574, 37670}, {17688, 48840}, {17691, 48869}, {19693, 34604}, {20023, 37465}, {20398, 62348}, {20943, 25440}, {22165, 66424}, {22468, 40996}, {27109, 29479}, {28706, 44802}, {30092, 68700}, {32006, 62028}, {32151, 38730}, {32456, 68523}, {32816, 32840}, {32818, 61964}, {32826, 37668}, {32827, 32875}, {32829, 46936}, {32834, 61820}, {32838, 61863}, {32841, 69404}, {32868, 61798}, {32877, 49140}, {32878, 62083}, {32880, 62152}, {32882, 62078}, {32885, 55864}, {32893, 61834}, {32930, 41805}, {32992, 59546}, {33013, 39785}, {33233, 64019}, {33236, 63107}, {33250, 51224}, {33257, 63935}, {33268, 47101}, {33275, 34504}, {33827, 48864}, {34229, 61814}, {34283, 37503}, {34341, 37889}, {34386, 57008}, {34506, 52695}, {36165, 47288}, {37647, 61900}, {39099, 55718}, {39593, 66322}, {40022, 40916}, {40247, 51386}, {40706, 46854}, {40707, 46855}, {40908, 69212}, {45201, 46517}, {50009, 65711}, {51238, 53144}, {53348, 69178}, {55631, 60702}, {58212, 61727}, {60781, 69385}, {61807, 69384}, {61870, 69386}, {63926, 66391}, {63938, 66387}, {63939, 66328}, {63954, 68513}, {66421, 66455}

X(69426) = reflection of X(i) in X(j) for these {i,j}: {6655, 7794}, {7760, 384}, {33256, 7873}
X(69426) = anticomplement of X(7765)
X(69426) = barycentric product X(99)*X(31072)
X(69426) = barycentric quotient X(31072)/X(523)
X(69426) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 7796, 7809}, {4, 32833, 7796}, {5, 32820, 7799}, {76, 99, 1078}, {76, 1975, 99}, {76, 7771, 69381}, {76, 7782, 183}, {99, 43459, 7782}, {115, 7836, 7899}, {183, 7782, 43459}, {183, 43459, 1078}, {194, 3734, 83}, {194, 68525, 7772}, {316, 3933, 7917}, {381, 32821, 7814}, {382, 7788, 7860}, {384, 7760, 12150}, {599, 33234, 7936}, {671, 7909, 5025}, {1003, 63933, 6179}, {1232, 18354, 46724}, {1975, 69380, 76}, {2896, 20094, 7756}, {3314, 7748, 7911}, {3734, 7772, 68525}, {3926, 11185, 7752}, {3926, 69379, 11185}, {3933, 32819, 316}, {5025, 7801, 7909}, {5254, 7832, 7919}, {6337, 52713, 32832}, {6390, 59635, 7769}, {6655, 7794, 7883}, {7752, 11185, 15031}, {7757, 7770, 55085}, {7758, 14035, 7812}, {7763, 69378, 69387}, {7772, 68525, 83}, {7781, 17130, 2}, {7787, 20105, 7798}, {7789, 47286, 7828}, {7790, 7795, 7944}, {7807, 63923, 14568}, {7816, 63925, 35007}, {7821, 63922, 14041}, {7840, 14042, 7843}, {7863, 63924, 2}, {7869, 11648, 7933}, {7870, 34505, 9166}, {7879, 44526, 7910}, {7881, 44518, 7934}, {7888, 18546, 32966}, {7896, 65633, 7898}, {7908, 69141, 7912}, {7916, 62203, 7900}, {7946, 66419, 63931}, {8024, 16276, 33651}, {17131, 69171, 7793}, {19570, 33225, 7755}, {31859, 69139, 7786}, {32751, 32752, 14880}, {32815, 32830, 315}, {32817, 69378, 7763}, {32829, 69383, 53127}, {33250, 63928, 51224}, {35007, 63925, 385}

X(69427) = X(2)X(7765)∩X(20)X(76)

Barycentrics   2*b^2*c^2 + SA^2 + 2*SB*SC : :

X(69427) lies on these lines: {2, 7765}, {3, 46951}, {4, 7788}, {5, 1007}, {20, 76}, {30, 32892}, {69, 382}, {99, 15717}, {140, 32885}, {183, 3528}, {194, 33269}, {298, 5343}, {299, 5344}, {315, 17578}, {325, 3855}, {384, 63065}, {538, 32971}, {543, 33023}, {548, 32888}, {599, 33238}, {631, 1975}, {671, 33200}, {1078, 21734}, {2996, 7795}, {3090, 32820}, {3091, 7814}, {3146, 32869}, {3522, 32874}, {3525, 59634}, {3526, 6337}, {3529, 37671}, {3530, 32886}, {3543, 7768}, {3545, 32821}, {3620, 7748}, {3734, 5319}, {3760, 4317}, {3761, 4309}, {3832, 7796}, {3843, 3933}, {3853, 32006}, {3861, 7776}, {5056, 7799}, {5059, 7811}, {5067, 32817}, {5070, 6390}, {5254, 33221}, {5286, 7875}, {5485, 14069}, {6462, 31483}, {6776, 13108}, {6904, 18145}, {7486, 7763}, {7615, 7888}, {7620, 14063}, {7737, 63934}, {7747, 20080}, {7750, 49138}, {7751, 32981}, {7752, 32840}, {7755, 43681}, {7758, 32979}, {7764, 32991}, {7767, 17800}, {7771, 58188}, {7773, 32890}, {7780, 35927}, {7782, 61788}, {7789, 33222}, {7794, 32982}, {7801, 32972}, {7809, 50689}, {7816, 37667}, {7818, 54097}, {7836, 33277}, {7849, 32974}, {7854, 33272}, {7860, 50688}, {7866, 59780}, {7881, 63533}, {7933, 43448}, {7934, 63536}, {8357, 21356}, {8667, 33239}, {8716, 32978}, {9466, 32990}, {9606, 32968}, {9607, 69139}, {9656, 69093}, {9671, 69094}, {10691, 41927}, {11057, 49140}, {11159, 63926}, {13488, 52710}, {14001, 63923}, {14023, 14711}, {14033, 63933}, {14034, 63093}, {14037, 19570}, {14064, 34505}, {14066, 23334}, {14568, 33181}, {14929, 62038}, {16239, 32883}, {16924, 31407}, {17580, 18146}, {18546, 32980}, {20065, 37004}, {20081, 69208}, {22468, 52711}, {31276, 33258}, {31417, 62988}, {31470, 51122}, {32818, 32875}, {32819, 32878}, {32831, 61914}, {32832, 55864}, {32835, 53127}, {32867, 61867}, {32872, 61816}, {32882, 50692}, {32884, 48154}, {32887, 61894}, {32893, 61820}, {32894, 62102}, {32973, 63955}, {32975, 59546}, {32987, 34511}, {33004, 53142}, {33202, 55738}, {33217, 47286}, {33235, 63029}, {33248, 53033}, {33287, 41135}, {34803, 61905}, {37122, 44146}, {37668, 61982}, {37809, 60200}, {52854, 61044}, {61138, 69384}, {63034, 68527}, {63954, 68177}

X(69427) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 32815, 3785}, {76, 69379, 32815}, {1975, 52713, 32828}, {3090, 32820, 32837}, {3091, 32833, 32825}, {3832, 7796, 32816}, {3832, 32830, 7796}, {3926, 69378, 69382}, {6337, 64093, 32838}, {6390, 69385, 32839}, {7615, 7888, 52250}, {7796, 11185, 3832}, {11185, 32830, 32816}, {32817, 59635, 32829}, {32819, 69377, 64018}, {32827, 32877, 3933}, {32878, 64018, 69377}, {69378, 69380, 3926}

X(69428) = X(2)X(7765)∩X(4)X(69)

Barycentrics   2*b^2*c^2 + S^2 + SA^2 + 2*SB*SC : :

X(69428) lies on these lines: {2, 7765}, {4, 69}, {5, 32821}, {20, 46951}, {32, 14031}, {83, 6392}, {99, 3523}, {115, 7869}, {140, 1975}, {148, 7800}, {183, 550}, {194, 33020}, {298, 42159}, {299, 42162}, {325, 3851}, {377, 18145}, {382, 37671}, {384, 63955}, {385, 14034}, {443, 18146}, {538, 16924}, {543, 32965}, {546, 7788}, {597, 7770}, {599, 33229}, {626, 33290}, {671, 32974}, {754, 14068}, {1007, 61921}, {1078, 3522}, {1656, 7763}, {1657, 14907}, {2482, 33206}, {2548, 13571}, {2549, 31276}, {2996, 7790}, {3090, 7799}, {3091, 7796}, {3096, 43448}, {3146, 7811}, {3314, 14045}, {3357, 69178}, {3526, 59634}, {3533, 6337}, {3545, 7814}, {3619, 7918}, {3620, 7911}, {3734, 7755}, {3760, 5270}, {3761, 4857}, {3767, 7892}, {3785, 5059}, {3832, 7809}, {3839, 32892}, {3850, 3933}, {3854, 15031}, {3858, 7773}, {3926, 5056}, {4441, 56880}, {5032, 7760}, {5068, 7752}, {5073, 7750}, {5215, 16925}, {5254, 34573}, {5309, 16898}, {5319, 19570}, {5485, 7827}, {6179, 14033}, {6390, 55856}, {6656, 21358}, {7486, 32837}, {7615, 32966}, {7617, 33270}, {7620, 7883}, {7737, 17129}, {7739, 68522}, {7747, 63046}, {7748, 16990}, {7751, 14035}, {7754, 32455}, {7757, 32968}, {7758, 16044}, {7759, 14711}, {7764, 32962}, {7767, 62036}, {7769, 32817}, {7771, 21735}, {7772, 33269}, {7775, 32995}, {7776, 61970}, {7780, 33007}, {7782, 10299}, {7791, 9466}, {7794, 14063}, {7795, 7901}, {7801, 32961}, {7802, 15589}, {7803, 47286}, {7810, 32997}, {7812, 32979}, {7816, 17008}, {7818, 32996}, {7821, 33006}, {7822, 32457}, {7836, 43620}, {7841, 50991}, {7849, 33251}, {7854, 33017}, {7856, 33198}, {7858, 32983}, {7870, 32969}, {7873, 33279}, {7878, 11054}, {7879, 53419}, {7880, 33248}, {7881, 63534}, {7888, 32963}, {7895, 18424}, {7904, 43619}, {7906, 31415}, {7909, 32972}, {7916, 43457}, {7917, 32827}, {7922, 16041}, {7934, 63533}, {7936, 33238}, {7937, 18840}, {8024, 62937}, {8182, 33268}, {8352, 51189}, {8361, 59780}, {8370, 15534}, {8591, 33022}, {8667, 19687}, {9166, 33199}, {9698, 33261}, {9744, 13108}, {10303, 32885}, {11057, 33703}, {11128, 59396}, {11129, 59394}, {11303, 41119}, {11304, 41120}, {11361, 14023}, {12203, 33750}, {13468, 33235}, {14001, 14568}, {14061, 53033}, {14066, 44678}, {14537, 63934}, {15717, 32893}, {15720, 37688}, {15815, 47287}, {16063, 39998}, {16275, 41916}, {16921, 34511}, {17131, 20065}, {17143, 56879}, {18140, 37462}, {18362, 33277}, {19696, 47102}, {22329, 68527}, {22331, 66319}, {31168, 33025}, {32818, 69403}, {32829, 46935}, {32838, 61856}, {32872, 61791}, {32886, 43459}, {32888, 50691}, {32894, 50690}, {32896, 61924}, {32954, 40727}, {33221, 47005}, {33226, 42850}, {33239, 63029}, {33247, 55164}, {33280, 63935}, {33651, 52301}, {33940, 52709}, {34284, 37162}, {37809, 68524}, {38259, 60640}, {40022, 46336}, {42948, 59540}, {42949, 59539}, {45201, 62977}, {48913, 61964}, {53102, 60636}, {60145, 60216}, {60146, 60200}, {60219, 60642}, {60250, 60647}, {61919, 69158}, {62147, 67536}, {63048, 69172}, {63061, 69208}, {63093, 63925}, {63926, 66409}, {63938, 66408}

X(69428) = anticomplement of X(53096)
X(69428) = anticomplement of the isogonal conjugate of X(53102)
X(69428) = X(53102)-anticomplementary conjugate of X(8)
X(69428) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 69, 7860}, {76, 316, 69377}, {76, 11185, 315}, {76, 69378, 11185}, {115, 7869, 33283}, {637, 638, 43150}, {1656, 32820, 7763}, {1656, 69380, 32820}, {1975, 64093, 32832}, {3091, 32836, 7796}, {3734, 7755, 14037}, {3926, 69383, 69387}, {5068, 32825, 7752}, {5068, 32830, 32825}, {7763, 59635, 53127}, {7794, 18546, 14063}, {7854, 63922, 33017}, {7873, 63957, 33279}, {15589, 32826, 7802}, {17130, 63924, 2}, {19570, 68525, 5319}, {32815, 32834, 1078}, {32817, 69385, 7769}, {32819, 69381, 14907}, {32820, 59635, 1656}, {32822, 34229, 7782}, {32825, 69382, 5068}, {32826, 32868, 15589}, {32828, 69379, 99}, {32830, 69382, 7752}, {47286, 69139, 7803}, {52713, 69378, 76}, {59635, 69380, 7763}

X(69429) = X(2)X(7765)∩X(69)X(546)

Barycentrics   2*b^2*c^2 + 2*S^2 + SA^2 + 2*SB*SC : :

X(69429) lies on these lines: {2, 7765}, {3, 32815}, {4, 37671}, {5, 32825}, {69, 546}, {76, 3091}, {99, 61820}, {183, 3529}, {193, 63925}, {194, 33261}, {315, 50689}, {325, 3544}, {538, 32987}, {631, 32885}, {632, 6337}, {671, 33025}, {1007, 5079}, {1078, 32872}, {1656, 32837}, {1975, 3525}, {2996, 3934}, {3090, 3926}, {3146, 3785}, {3522, 32893}, {3533, 59634}, {3545, 32892}, {3627, 64018}, {3628, 32829}, {3832, 32874}, {3839, 7768}, {3854, 7809}, {3855, 7788}, {3857, 7776}, {3933, 5072}, {4208, 18146}, {5007, 32971}, {5055, 32896}, {5056, 32833}, {5067, 32820}, {5068, 7796}, {5071, 32821}, {5076, 7767}, {5177, 18145}, {5286, 68522}, {5395, 7805}, {5485, 32957}, {6390, 55857}, {6392, 7772}, {6655, 7620}, {7486, 7799}, {7615, 7794}, {7750, 62028}, {7751, 32979}, {7752, 69404}, {7758, 32991}, {7763, 46936}, {7771, 62083}, {7782, 61798}, {7801, 32988}, {7811, 17578}, {7814, 61936}, {7819, 40727}, {7843, 11160}, {7849, 60285}, {7860, 61982}, {8556, 33226}, {9466, 32974}, {10303, 32832}, {11057, 50691}, {11180, 40279}, {12103, 67536}, {12812, 32875}, {13468, 33239}, {14033, 22331}, {14568, 33198}, {14907, 49140}, {15022, 32830}, {15031, 32894}, {15589, 50688}, {16043, 34505}, {16924, 63028}, {17538, 32819}, {18140, 37436}, {18546, 32982}, {19570, 33269}, {19687, 63029}, {19696, 66699}, {20081, 31404}, {20943, 31418}, {31099, 39998}, {31276, 43448}, {32006, 61984}, {32456, 55819}, {32817, 32839}, {32818, 32877}, {32822, 37688}, {32827, 32888}, {32831, 53127}, {32870, 61848}, {32876, 61903}, {32883, 61870}, {32890, 61923}, {32968, 63923}, {32983, 63933}, {33012, 53142}, {33234, 42850}, {33242, 63107}, {33272, 63922}, {34609, 41927}, {34803, 61900}, {35007, 37667}, {40330, 61550}, {40344, 55729}, {63019, 68525}

X(69429) = anticomplement of X(31450)
X(69429) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 32836, 32825}, {76, 69382, 32816}, {76, 69383, 69382}, {1975, 69386, 32838}, {5068, 32869, 7796}, {7615, 7794, 32980}, {11185, 32834, 3785}, {32826, 32886, 183}, {32827, 32888, 69377}, {32828, 69378, 32815}, {52713, 59635, 3926}, {64093, 69378, 32828}, {69380, 69385, 32829}

X(69430) = X(2)X(32)∩X(4)X(7769)

Barycentrics   b^2*c^2 + 2*S^2 - 2*SA^2 + 2*SB*SC : :

X(69430) lies on these lines: {2, 32}, {3, 37647}, {4, 7769}, {5, 1975}, {6, 33249}, {20, 32839}, {39, 32961}, {69, 5067}, {76, 1007}, {99, 3091}, {114, 53765}, {115, 32963}, {140, 7773}, {148, 33011}, {183, 3628}, {187, 33000}, {194, 43620}, {264, 36612}, {274, 6933}, {302, 40693}, {303, 40694}, {316, 631}, {317, 7505}, {325, 1656}, {384, 31415}, {491, 10577}, {492, 10576}, {547, 3933}, {574, 14063}, {620, 14035}, {625, 7791}, {2549, 32966}, {3055, 11285}, {3146, 32871}, {3314, 16922}, {3522, 32898}, {3523, 7802}, {3524, 48913}, {3525, 7771}, {3526, 7750}, {3533, 7860}, {3545, 6337}, {3589, 33218}, {3618, 7942}, {3734, 32962}, {3767, 7777}, {3788, 7603}, {3815, 7803}, {3832, 62362}, {3851, 32819}, {3926, 5056}, {3934, 32999}, {3972, 32970}, {4045, 33283}, {5013, 33228}, {5025, 17005}, {5055, 32833}, {5068, 15031}, {5070, 7776}, {5071, 7799}, {5079, 69380}, {5206, 33206}, {5286, 14061}, {5305, 11163}, {5319, 63018}, {5475, 16925}, {6179, 62992}, {6622, 58782}, {6656, 31489}, {6721, 10358}, {6722, 7772}, {6931, 18140}, {7486, 7796}, {7569, 39113}, {7581, 32805}, {7582, 32806}, {7622, 33192}, {7735, 7858}, {7736, 7828}, {7737, 7907}, {7738, 32984}, {7745, 33233}, {7746, 7774}, {7747, 32964}, {7748, 33006}, {7755, 63017}, {7756, 32996}, {7758, 63021}, {7759, 17008}, {7760, 62988}, {7761, 33001}, {7762, 37637}, {7764, 33270}, {7767, 55856}, {7768, 32823}, {7770, 44377}, {7778, 32992}, {7786, 14064}, {7788, 15699}, {7789, 44543}, {7790, 31400}, {7795, 7925}, {7801, 66511}, {7816, 33016}, {7820, 33269}, {7821, 16990}, {7823, 16923}, {7825, 32965}, {7827, 63025}, {7830, 33012}, {7831, 32978}, {7832, 32968}, {7834, 31275}, {7835, 32971}, {7836, 33002}, {7838, 63048}, {7841, 9771}, {7842, 33008}, {7844, 9698}, {7847, 16041}, {7850, 61881}, {7851, 31406}, {7856, 37665}, {7859, 32951}, {7871, 61899}, {7872, 33290}, {7874, 16898}, {7877, 37667}, {7878, 32958}, {7879, 58446}, {7881, 22110}, {7885, 33015}, {7886, 16989}, {7888, 33009}, {7890, 12815}, {7891, 33013}, {7893, 17006}, {7903, 63046}, {7905, 9770}, {7906, 63955}, {7910, 33215}, {7911, 32990}, {7917, 15589}, {7918, 33285}, {7919, 33199}, {7930, 16045}, {7934, 16043}, {7935, 33258}, {7937, 32960}, {7940, 14001}, {7941, 14023}, {7945, 33020}, {8176, 33007}, {8227, 69038}, {8352, 44519}, {8361, 11174}, {8589, 33253}, {9744, 14880}, {9863, 10486}, {10303, 32884}, {10513, 32897}, {10588, 64133}, {10653, 62601}, {10654, 62600}, {10983, 39663}, {11057, 15702}, {11184, 13881}, {11318, 31467}, {11482, 44369}, {13880, 45421}, {13921, 45420}, {14062, 43619}, {14068, 43457}, {14069, 60855}, {14929, 55861}, {14994, 42786}, {15022, 32831}, {15513, 63956}, {15515, 32997}, {15703, 37671}, {15815, 33229}, {16044, 69206}, {16589, 33053}, {18022, 55079}, {18502, 37466}, {19695, 53095}, {19709, 59634}, {20105, 50570}, {31274, 69172}, {31407, 51171}, {31417, 33245}, {31859, 63534}, {32456, 33280}, {32807, 35813}, {32817, 61921}, {32820, 61919}, {32821, 35018}, {32822, 69403}, {32825, 32834}, {32830, 61914}, {32836, 61912}, {32837, 61924}, {32838, 37668}, {32873, 69402}, {32953, 63119}, {32974, 55797}, {32987, 53033}, {33014, 43618}, {33017, 37512}, {33204, 63931}, {33235, 53418}, {33244, 62203}, {33274, 66466}, {34254, 37990}, {34484, 44180}, {35297, 65630}, {37832, 69165}, {37835, 69157}, {42488, 69145}, {42489, 69137}, {43448, 52250}, {46951, 61906}, {52713, 69405}, {52718, 61889}, {61895, 69384}, {61964, 67536}, {64809, 69406}

X(69430) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7752, 315}, {2, 7785, 69207}, {2, 7912, 7800}, {2, 20065, 7749}, {2, 31404, 83}, {2, 32816, 1078}, {2, 69208, 7857}, {4, 34803, 7769}, {5, 7763, 11185}, {76, 3090, 53127}, {140, 7773, 14907}, {325, 1656, 32832}, {625, 31455, 7791}, {1007, 3090, 76}, {1007, 69385, 32818}, {1078, 7752, 32816}, {1078, 32816, 315}, {1506, 7862, 2}, {3090, 32818, 69385}, {3091, 32829, 99}, {3523, 32827, 7802}, {3525, 32006, 7771}, {3618, 32955, 7942}, {3785, 7809, 315}, {3788, 7603, 16924}, {3815, 7887, 7803}, {3926, 5056, 69387}, {5025, 17005, 31401}, {5055, 69158, 59635}, {5068, 32815, 15031}, {5068, 32835, 32815}, {5070, 7776, 37688}, {5286, 32988, 14061}, {7486, 63098, 32828}, {7736, 32969, 7828}, {7749, 7775, 20065}, {7774, 32998, 7746}, {7777, 32967, 3767}, {7823, 16923, 21843}, {7834, 31275, 33248}, {7925, 16921, 7795}, {7941, 17004, 14023}, {10303, 64018, 43459}, {14064, 62993, 7786}, {15022, 32831, 69382}, {15589, 46935, 32867}, {31400, 32972, 7790}, {32818, 69385, 76}, {32823, 34229, 7768}, {32823, 61886, 34229}, {32828, 63098, 7796}, {32884, 64018, 10303}, {32951, 63041, 7859}, {32961, 63083, 39}, {32968, 37690, 7832}, {32988, 63077, 5286}, {37668, 46936, 32838}, {43457, 69171, 14068}, {59635, 69158, 32833}

X(69431) = X(2)X(3108)∩X(3)X(315)

Barycentrics   b^2*c^2 - 2*S^2 + 2*SA^2 - 2*SB*SC : :

X(69431) lies on these lines: {2, 3108}, {3, 315}, {4, 7799}, {5, 32821}, {20, 7809}, {69, 575}, {76, 1007}, {83, 53033}, {99, 3146}, {140, 7788}, {183, 632}, {193, 7857}, {298, 42152}, {299, 42149}, {316, 3529}, {317, 3518}, {340, 3147}, {376, 7860}, {381, 32820}, {382, 59634}, {439, 41134}, {491, 6420}, {492, 6419}, {524, 33233}, {538, 32961}, {543, 32996}, {546, 1975}, {576, 51371}, {620, 7903}, {626, 53096}, {631, 7768}, {671, 32980}, {754, 32964}, {1078, 10303}, {1273, 37119}, {1506, 7908}, {1992, 33189}, {1995, 34254}, {2482, 33244}, {2548, 7836}, {2549, 7912}, {3091, 3926}, {3096, 31400}, {3303, 69254}, {3304, 69135}, {3314, 31401}, {3523, 7811}, {3526, 37671}, {3528, 11057}, {3544, 69378}, {3618, 7930}, {3619, 64713}, {3620, 5034}, {3627, 6390}, {3628, 3933}, {3767, 7906}, {3785, 7917}, {3788, 5007}, {3815, 7881}, {3832, 32824}, {3849, 33254}, {3934, 63083}, {5025, 34511}, {5056, 32836}, {5072, 69380}, {5079, 59635}, {5286, 7899}, {5309, 33248}, {6179, 32970}, {6243, 51383}, {6392, 14061}, {6393, 11477}, {6656, 22332}, {6680, 63017}, {6931, 18145}, {7486, 46951}, {7615, 33011}, {7618, 33260}, {7735, 7905}, {7736, 7832}, {7737, 7891}, {7738, 7934}, {7739, 7901}, {7746, 63925}, {7749, 7916}, {7753, 14037}, {7754, 44377}, {7756, 51587}, {7757, 14064}, {7759, 16925}, {7762, 22331}, {7765, 33283}, {7767, 14869}, {7771, 61814}, {7775, 7863}, {7777, 7795}, {7778, 7803}, {7779, 69207}, {7780, 33000}, {7781, 14063}, {7782, 17538}, {7785, 68517}, {7791, 7821}, {7800, 7897}, {7801, 16924}, {7802, 50693}, {7807, 9766}, {7810, 33012}, {7812, 32973}, {7813, 7862}, {7818, 32965}, {7819, 11163}, {7827, 32951}, {7828, 37690}, {7835, 69208}, {7837, 33245}, {7840, 7907}, {7841, 59546}, {7843, 33007}, {7846, 37665}, {7850, 61807}, {7854, 33001}, {7855, 17008}, {7858, 7870}, {7865, 33258}, {7868, 31406}, {7873, 33008}, {7874, 16989}, {7878, 14069}, {7880, 16898}, {7883, 32990}, {7887, 22110}, {7890, 63048}, {7893, 21843}, {7895, 16990}, {7900, 68518}, {7922, 16043}, {7936, 33215}, {7945, 63018}, {7946, 33259}, {7982, 69038}, {8716, 33229}, {8781, 38664}, {9300, 33217}, {9466, 32999}, {9607, 33219}, {10513, 61848}, {11184, 32992}, {11648, 33290}, {12150, 33181}, {12812, 64093}, {14042, 66466}, {14043, 63028}, {14148, 69141}, {14568, 32969}, {14712, 68526}, {14929, 61808}, {15022, 32830}, {15589, 32839}, {16051, 57518}, {16862, 37664}, {17128, 31415}, {17130, 32962}, {20080, 39764}, {20081, 43620}, {20399, 32458}, {30471, 42150}, {30472, 42151}, {31173, 33279}, {31417, 66413}, {31450, 33021}, {31492, 66417}, {32815, 32841}, {32817, 61964}, {32819, 61984}, {32826, 32876}, {32827, 50688}, {32828, 46936}, {32840, 69382}, {32869, 61914}, {32881, 62152}, {32885, 46935}, {32892, 61912}, {32896, 61936}, {32952, 59373}, {32954, 41624}, {32957, 63025}, {32959, 63034}, {32963, 63924}, {32967, 63955}, {33006, 39785}, {33025, 52691}, {33204, 34506}, {33222, 63006}, {33227, 63945}, {33249, 63933}, {33257, 44678}, {33271, 34504}, {33276, 47102}, {33277, 69162}, {33280, 63956}, {34229, 61870}, {34803, 60781}, {35297, 63932}, {36521, 66398}, {37647, 55857}, {37688, 55858}, {39113, 68660}, {43459, 61804}, {44175, 54046}, {45795, 57805}, {50567, 55721}, {51374, 55724}, {51396, 55583}, {51397, 55718}, {52713, 69406}, {62097, 64018}, {63941, 68516}

X(69431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 13571, 5319}, {2, 32825, 7796}, {5, 32821, 32833}, {325, 7763, 315}, {325, 69158, 7763}, {620, 7903, 20065}, {1007, 32818, 76}, {2482, 63931, 33244}, {3788, 69197, 7774}, {3926, 7752, 11185}, {3926, 63098, 7752}, {6337, 32823, 316}, {7749, 7916, 63046}, {7764, 7888, 2}, {7769, 7871, 69}, {7775, 7863, 14035}, {7776, 14907, 315}, {7777, 7947, 7795}, {7799, 7814, 4}, {7811, 62362, 3523}, {7836, 63021, 2548}, {7840, 7907, 14023}, {7858, 7870, 14001}, {7869, 9698, 2}, {7891, 7941, 7737}, {7895, 31455, 16990}, {7905, 7940, 7735}, {7906, 7925, 3767}, {7945, 63018, 69209}, {9770, 14001, 7858}, {15031, 69379, 11185}, {32816, 32831, 99}, {32829, 37668, 1078}, {33259, 41136, 7946}, {53033, 62988, 83}

X(69432) = X(2)X(3933)∩X(20)X(99)

Barycentrics   2*b^2*c^2 - S^2 + 2*SA^2 - 2*SB*SC : :

X(69432) lies on these lines: {2, 3933}, {3, 10513}, {4, 32840}, {20, 99}, {69, 3523}, {76, 5056}, {183, 32835}, {193, 7836}, {194, 33180}, {253, 52347}, {279, 69279}, {316, 32824}, {325, 3091}, {346, 17181}, {371, 1270}, {372, 1271}, {385, 33203}, {439, 7893}, {1007, 7486}, {1078, 15708}, {1235, 9464}, {1656, 32872}, {1975, 3543}, {2896, 66448}, {2996, 5503}, {3146, 7776}, {3314, 33202}, {3522, 6390}, {3525, 32873}, {3528, 14929}, {3593, 58866}, {3595, 8960}, {3620, 51371}, {3785, 7799}, {3788, 37689}, {3832, 32823}, {3839, 7871}, {3945, 25650}, {3964, 7488}, {4346, 17211}, {5032, 10583}, {5068, 32880}, {5182, 7793}, {5286, 7813}, {5304, 6680}, {6337, 7788}, {6392, 33199}, {7750, 62097}, {7752, 32836}, {7762, 33201}, {7763, 10303}, {7767, 15717}, {7768, 62067}, {7773, 61982}, {7774, 33198}, {7779, 32973}, {7782, 62094}, {7794, 31400}, {7795, 37665}, {7801, 69208}, {7807, 63042}, {7809, 32826}, {7811, 62059}, {7814, 69382}, {7819, 14930}, {7821, 43448}, {7841, 11148}, {7892, 51170}, {7895, 34511}, {7897, 32974}, {7916, 69206}, {7939, 33023}, {7941, 32979}, {8364, 14482}, {9167, 9740}, {9770, 69139}, {10565, 40123}, {13571, 51171}, {14001, 63091}, {14069, 63005}, {14118, 22241}, {14907, 62083}, {14986, 69093}, {15022, 32882}, {15697, 59634}, {15721, 37671}, {16925, 20080}, {17558, 45962}, {17578, 32822}, {20081, 32972}, {20105, 33283}, {22246, 66344}, {22253, 32951}, {25583, 39570}, {25592, 33091}, {31276, 63077}, {31404, 69197}, {31859, 33025}, {32006, 32820}, {32815, 50688}, {32827, 32875}, {32828, 46936}, {32829, 61856}, {32868, 53127}, {32869, 59635}, {32870, 34803}, {32871, 37688}, {32874, 61912}, {32876, 61783}, {32881, 61820}, {32893, 61897}, {32894, 61914}, {32895, 61842}, {32897, 37647}, {32954, 63097}, {32987, 63021}, {32989, 63046}, {33194, 63633}, {34229, 61863}, {37690, 63933}, {42998, 69121}, {42999, 69120}, {45795, 68354}, {46453, 63936}, {46951, 61906}, {63032, 69157}, {63033, 69165}

X(69432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 32821, 32831}, {69, 32831, 3523}, {76, 32825, 63098}, {76, 63098, 5056}, {183, 32835, 55864}, {193, 7836, 33181}, {325, 32830, 3091}, {1007, 32834, 7486}, {3146, 32879, 32817}, {3926, 7796, 37668}, {3926, 37668, 20}, {3933, 32818, 2}, {3933, 69158, 69377}, {5068, 32880, 52713}, {5304, 53033, 33183}, {7752, 32836, 69383}, {7752, 69383, 61936}, {7758, 7908, 53033}, {7758, 53033, 5304}, {7763, 15589, 10303}, {7776, 32817, 3146}, {7871, 32833, 32816}, {10513, 32841, 3}, {15022, 32882, 64093}, {18840, 31406, 2}, {32816, 32833, 69379}, {32816, 69379, 3839}, {32818, 69377, 69158}, {32823, 69380, 3832}, {32894, 61914, 69386}, {52718, 69377, 69381}, {69158, 69377, 2}

X(69433) = X(2)X(3933)∩X(20)X(64)

Barycentrics   2*b^2*c^2 + S^2 + 2*SA^2 - 2*SB*SC : :

X(69433) lies on these lines: {2, 3933}, {3, 32840}, {4, 10513}, {20, 64}, {76, 3091}, {99, 62097}, {183, 10303}, {193, 5039}, {194, 3620}, {279, 54433}, {315, 3543}, {325, 5056}, {346, 17170}, {384, 20080}, {385, 33181}, {394, 11348}, {599, 7738}, {631, 32841}, {1007, 46936}, {1078, 3523}, {1235, 7378}, {1270, 1588}, {1271, 1587}, {1285, 63936}, {2896, 11148}, {2996, 60180}, {3090, 32872}, {3146, 32880}, {3314, 6392}, {3424, 30270}, {3522, 7767}, {3526, 32873}, {3529, 14929}, {3665, 3974}, {3703, 7195}, {3760, 5274}, {3761, 5261}, {3785, 7782}, {3832, 7776}, {3839, 7788}, {3945, 33953}, {3964, 37126}, {4195, 62999}, {4208, 34284}, {4352, 5232}, {4454, 33865}, {5024, 55732}, {5059, 32822}, {5068, 32823}, {5129, 45962}, {5286, 7794}, {5304, 7795}, {6337, 15692}, {6381, 8165}, {6390, 15717}, {7396, 8024}, {7398, 40123}, {7486, 7796}, {7735, 33183}, {7751, 37689}, {7752, 46951}, {7758, 7808}, {7763, 55864}, {7768, 32815}, {7770, 63091}, {7779, 32971}, {7799, 15721}, {7801, 9740}, {7802, 32890}, {7809, 32892}, {7811, 15697}, {7813, 31400}, {7814, 67096}, {7819, 63005}, {7836, 33203}, {7855, 41750}, {7871, 61912}, {7879, 33025}, {7893, 32981}, {7895, 63955}, {7897, 32972}, {7917, 32827}, {7929, 33272}, {7939, 32982}, {7941, 32991}, {8367, 60143}, {9464, 41009}, {9466, 31404}, {10008, 32477}, {10481, 56264}, {10565, 26233}, {10996, 40995}, {11160, 20065}, {11185, 61982}, {11294, 32814}, {13571, 60285}, {14001, 63042}, {14037, 50248}, {14045, 43681}, {14069, 63097}, {14482, 55741}, {14907, 32824}, {14930, 16045}, {14986, 69094}, {15022, 69386}, {16895, 63123}, {16898, 51170}, {17233, 32034}, {17378, 51675}, {17691, 63001}, {18845, 66410}, {20081, 32974}, {22241, 22467}, {22253, 32956}, {25242, 29616}, {26166, 44149}, {31268, 66703}, {31276, 62988}, {32006, 50688}, {32092, 40333}, {32419, 43257}, {32421, 43256}, {32819, 50691}, {32820, 62067}, {32821, 32835}, {32825, 32832}, {32829, 61863}, {32837, 61844}, {32868, 69387}, {32874, 59635}, {32875, 58188}, {32877, 49140}, {32881, 61842}, {32886, 53127}, {32893, 61906}, {32896, 61781}, {32897, 34803}, {32973, 63046}, {32990, 63044}, {33182, 63933}, {33200, 47286}, {33247, 47287}, {34604, 63118}, {42998, 69106}, {42999, 69107}, {44519, 53141}, {46226, 51171}, {59634, 62059}

X(69433) = crossdifference of every pair of points on line {3804, 62176}
X(69433) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 32830, 20}, {76, 32816, 69383}, {76, 37668, 3091}, {183, 32831, 10303}, {194, 3620, 33202}, {315, 32836, 69379}, {315, 69379, 3543}, {325, 32834, 5056}, {3146, 32880, 69380}, {3314, 6392, 33180}, {3832, 32882, 52713}, {3926, 15589, 3523}, {3933, 69377, 2}, {3933, 69381, 32818}, {5068, 32894, 64093}, {7751, 53033, 37689}, {7767, 32817, 3522}, {7776, 52713, 3832}, {7788, 32869, 3839}, {7796, 32828, 63098}, {7836, 37667, 33203}, {9605, 18840, 2}, {15717, 32879, 6390}, {32816, 69383, 3091}, {32818, 69377, 69381}, {32818, 69381, 2}, {32821, 34229, 32835}, {32823, 64093, 5068}, {32828, 63098, 7486}, {32835, 34229, 61856}, {37668, 69383, 32816}, {69158, 69384, 2}

X(69434) = X(2)X(3933)∩X(4)X(99)

Barycentrics   2*S^2 - 2*SA^2 + SB*SC : :

X(69434) lies on these lines: {2, 3933}, {3, 32823}, {4, 99}, {5, 32817}, {6, 33189}, {32, 9167}, {39, 32951}, {69, 575}, {76, 5067}, {140, 37668}, {183, 3533}, {193, 33233}, {194, 32969}, {315, 3524}, {316, 17538}, {325, 631}, {376, 32816}, {385, 32977}, {632, 32898}, {1078, 15702}, {1235, 8893}, {1285, 16925}, {1656, 32830}, {1975, 3545}, {1992, 7857}, {2548, 14039}, {2549, 33292}, {3090, 3926}, {3091, 6390}, {3146, 32895}, {3147, 32001}, {3314, 32978}, {3518, 9723}, {3523, 7776}, {3526, 15589}, {3528, 7814}, {3529, 7773}, {3544, 11185}, {3567, 51386}, {3618, 32952}, {3619, 7909}, {3628, 32834}, {3767, 32958}, {3788, 7736}, {3815, 16045}, {3855, 32815}, {5013, 22110}, {5024, 33180}, {5055, 69383}, {5056, 32841}, {5068, 32881}, {5071, 7799}, {5286, 32955}, {5334, 59541}, {5335, 59542}, {5461, 7862}, {5866, 35475}, {6292, 7888}, {6392, 33249}, {6680, 63024}, {7486, 32840}, {7618, 7842}, {7735, 7764}, {7738, 7861}, {7750, 10299}, {7758, 62992}, {7760, 63104}, {7762, 32989}, {7767, 10303}, {7768, 61836}, {7770, 63077}, {7771, 61817}, {7774, 32970}, {7777, 14001}, {7778, 31400}, {7779, 33000}, {7781, 63533}, {7782, 11001}, {7783, 16041}, {7785, 32985}, {7787, 33224}, {7788, 15709}, {7789, 11184}, {7792, 33195}, {7795, 32957}, {7796, 34229}, {7802, 62092}, {7803, 32953}, {7807, 62988}, {7809, 19708}, {7811, 61822}, {7832, 63041}, {7836, 32968}, {7851, 41133}, {7853, 31450}, {7855, 63029}, {7858, 33236}, {7860, 62061}, {7863, 31415}, {7864, 7925}, {7870, 63025}, {7885, 33226}, {7891, 14033}, {7893, 33206}, {7897, 33001}, {7903, 21843}, {7905, 63034}, {7912, 32986}, {7930, 63119}, {7938, 16043}, {7939, 33012}, {7941, 32964}, {8596, 33006}, {8889, 62310}, {9741, 32984}, {10513, 55864}, {10595, 69038}, {11057, 62058}, {11163, 33197}, {14907, 61138}, {14929, 15720}, {14994, 63121}, {15048, 33199}, {15484, 33201}, {15699, 32869}, {15703, 32874}, {16923, 63046}, {16989, 33222}, {17005, 32975}, {18841, 33217}, {18907, 33205}, {20065, 33216}, {20081, 32998}, {21735, 64018}, {27269, 33048}, {30435, 33203}, {30769, 59766}, {31859, 32972}, {32000, 37119}, {32819, 61964}, {32820, 61921}, {32821, 32828}, {32824, 69403}, {32826, 41099}, {32827, 32889}, {32832, 60781}, {32833, 61899}, {32836, 61895}, {32838, 61881}, {32870, 55856}, {32885, 61884}, {32893, 61887}, {32896, 61902}, {32897, 55857}, {32954, 37665}, {32988, 47286}, {33003, 63044}, {33191, 69208}, {33245, 63017}, {33250, 51579}, {37640, 69165}, {37641, 69157}, {37671, 61859}, {37688, 61870}, {41136, 55823}, {42133, 59539}, {42134, 59540}, {43448, 59546}, {43532, 62932}, {44535, 50771}, {45017, 66389}, {46951, 61889}, {48913, 62029}, {54822, 60234}, {55797, 66417}, {59211, 62955}, {60143, 62881}, {69197, 69207}, {69387, 69405}

X(69434) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3933, 69384}, {2, 7881, 18840}, {2, 32818, 69377}, {2, 69158, 32818}, {2, 69381, 52718}, {3, 63098, 32823}, {5, 32831, 32817}, {39, 37690, 32951}, {69, 7769, 3525}, {76, 34803, 5067}, {183, 32839, 3533}, {325, 32829, 631}, {1007, 6337, 7752}, {1007, 7763, 4}, {1285, 39142, 16925}, {1656, 32830, 69386}, {3090, 3926, 52713}, {3091, 6390, 32822}, {3314, 32978, 55732}, {3788, 7736, 14069}, {3815, 53033, 16045}, {3933, 69384, 69377}, {5056, 32841, 69380}, {5286, 44377, 32955}, {6337, 7752, 4}, {7486, 32840, 64093}, {7752, 7763, 6337}, {7762, 32989, 46453}, {7778, 31400, 32956}, {7789, 11184, 31404}, {7795, 62993, 32957}, {7836, 63083, 32968}, {15589, 32871, 3526}, {32818, 69384, 3933}, {32821, 37647, 32828}, {32825, 32839, 183}, {32828, 37647, 61886}, {32835, 63098, 3}

X(69435) = X(2)X(9609)∩X(54)X(69)

Barycentrics   b^2*c^2 - 2*S^2 + 2*SA^2 - SB*SC : :

X(69435) lies on these lines: {2, 9606}, {3, 32825}, {4, 7799}, {5, 1007}, {20, 325}, {24, 52437}, {39, 33221}, {54, 69}, {76, 5067}, {99, 32823}, {141, 31492}, {183, 32835}, {194, 33248}, {315, 3528}, {316, 49138}, {381, 32824}, {382, 6390}, {384, 9770}, {385, 33262}, {439, 63932}, {524, 32989}, {538, 32969}, {548, 7776}, {858, 19583}, {1656, 32836}, {1975, 3832}, {1992, 7807}, {3090, 32833}, {3091, 32820}, {3146, 59634}, {3314, 33258}, {3523, 7788}, {3524, 7768}, {3526, 3933}, {3529, 7809}, {3530, 3785}, {3541, 52710}, {3618, 33217}, {3619, 7881}, {3628, 46951}, {3734, 31417}, {3788, 5319}, {3843, 32815}, {3853, 32827}, {3855, 7752}, {3861, 32826}, {5007, 33224}, {5055, 32896}, {5070, 32828}, {5286, 33218}, {6179, 63064}, {6340, 30769}, {6392, 44377}, {7486, 32830}, {7618, 7873}, {7735, 7906}, {7736, 7836}, {7738, 7933}, {7750, 21734}, {7751, 23055}, {7754, 63104}, {7757, 32951}, {7758, 32970}, {7759, 32985}, {7760, 33189}, {7764, 14001}, {7765, 7888}, {7767, 61811}, {7769, 61867}, {7770, 31407}, {7773, 17578}, {7774, 33225}, {7777, 33269}, {7778, 9607}, {7781, 16041}, {7782, 62113}, {7786, 41622}, {7789, 62988}, {7794, 32978}, {7795, 9698}, {7800, 31457}, {7801, 32968}, {7802, 62117}, {7811, 10299}, {7818, 33226}, {7821, 32986}, {7827, 32953}, {7832, 63119}, {7840, 32964}, {7849, 16043}, {7856, 33195}, {7858, 14039}, {7860, 17538}, {7863, 14033}, {7870, 14069}, {7871, 61138}, {7876, 7947}, {7877, 46453}, {7892, 63024}, {7905, 62996}, {7908, 31401}, {7909, 32956}, {7916, 21843}, {7925, 33277}, {7945, 51860}, {8357, 11165}, {8362, 31470}, {8366, 66763}, {8716, 32982}, {9715, 9723}, {9741, 33292}, {9766, 32973}, {10303, 37671}, {10513, 61816}, {11057, 62092}, {11147, 33014}, {11163, 33198}, {11184, 32987}, {11185, 61945}, {11285, 21356}, {11307, 42999}, {11308, 42998}, {13571, 63006}, {14023, 33216}, {14531, 51386}, {14568, 32958}, {14614, 33203}, {14907, 62066}, {14929, 61790}, {14981, 46236}, {15589, 61842}, {15696, 32891}, {15699, 32892}, {15717, 37668}, {16239, 32839}, {19695, 53142}, {22110, 32972}, {32810, 39387}, {32811, 39388}, {32819, 61982}, {32832, 61881}, {32834, 37647}, {32838, 48154}, {32840, 59635}, {32867, 61878}, {32869, 46936}, {32874, 46935}, {32875, 61911}, {32879, 69383}, {32884, 61876}, {32885, 55856}, {32887, 61853}, {32889, 61837}, {32960, 55738}, {32974, 59546}, {32976, 63955}, {32984, 39785}, {32988, 63923}, {33000, 63029}, {33015, 42850}, {33181, 41624}, {33197, 63022}, {34254, 40132}, {34505, 52250}, {35287, 63938}, {35486, 52149}, {37122, 63155}, {39143, 47286}, {43193, 59540}, {43194, 59539}, {52713, 69405}, {55085, 63109}, {55729, 55803}, {55732, 55799}, {55737, 55795}, {55739, 55792}, {55741, 55789}, {55747, 55787}, {55753, 55774}, {55757, 55771}, {55762, 55767}, {55812, 55825}, {61905, 64093}, {63077, 69139}, {63934, 69207}, {66607, 68660}, {69197, 69206}, {69379, 69402}

X(69435) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {325, 6337, 32006}, {325, 32831, 6337}, {627, 628, 14912}, {631, 7796, 69}, {631, 32818, 7796}, {1007, 3926, 69378}, {3788, 5319, 33222}, {3926, 69158, 1007}, {3933, 32829, 34229}, {7751, 32977, 23055}, {7758, 32970, 63034}, {7763, 7796, 631}, {7763, 32818, 69}, {7849, 31450, 16043}, {7881, 31400, 3619}, {7888, 34511, 14064}, {32825, 32837, 3}, {32841, 63098, 1975}

X(69436) = X(2)X(14482)∩X(4)X(325)

Barycentrics   2*b^2*c^2 - 2*S^2 + 2*SA^2 - SB*SC : :

X(69436) lies on these lines: {2, 14482}, {3, 10513}, {4, 325}, {5, 32840}, {69, 3431}, {76, 5067}, {99, 11001}, {183, 15702}, {186, 3964}, {193, 33191}, {194, 32951}, {253, 47090}, {305, 52299}, {315, 17538}, {316, 62042}, {376, 6390}, {524, 46453}, {538, 37690}, {574, 66448}, {631, 3933}, {632, 32873}, {1007, 5071}, {1078, 61817}, {1270, 6221}, {1271, 6398}, {1285, 69206}, {1992, 7835}, {2549, 9741}, {3090, 32830}, {3091, 32879}, {3525, 7763}, {3528, 6337}, {3529, 7776}, {3533, 32835}, {3544, 69378}, {3545, 63098}, {3734, 9770}, {3785, 32876}, {3855, 69379}, {4045, 7908}, {4121, 18950}, {4175, 11427}, {5070, 32872}, {5286, 32953}, {5304, 33197}, {5305, 33195}, {5485, 22110}, {6392, 32955}, {6704, 7795}, {7486, 32880}, {7618, 7848}, {7735, 7813}, {7736, 7801}, {7738, 7853}, {7750, 62092}, {7752, 61945}, {7754, 33189}, {7758, 33236}, {7766, 33224}, {7767, 10299}, {7768, 62061}, {7769, 61873}, {7774, 14039}, {7779, 32985}, {7782, 62096}, {7788, 19708}, {7794, 55732}, {7809, 62029}, {7811, 62058}, {7820, 63024}, {7822, 55774}, {7836, 14069}, {7850, 62090}, {7871, 32006}, {7881, 32956}, {7897, 32986}, {7906, 14001}, {7939, 33226}, {7947, 14064}, {8368, 63005}, {8369, 63091}, {10303, 32881}, {10304, 14929}, {10753, 50567}, {11185, 41106}, {11288, 63042}, {14501, 51898}, {14502, 51899}, {14930, 33237}, {14981, 15428}, {15048, 33196}, {16051, 62299}, {16925, 50248}, {16986, 32960}, {17129, 32977}, {18840, 31400}, {20080, 35297}, {20081, 32969}, {20105, 33248}, {22241, 35921}, {31859, 33190}, {32828, 60781}, {32829, 61867}, {32834, 61886}, {32836, 61899}, {32839, 52718}, {32869, 61895}, {32874, 61889}, {32875, 69406}, {32882, 46936}, {32893, 61884}, {32894, 46935}, {32895, 61856}, {32896, 61932}, {32952, 53033}, {32970, 63047}, {32983, 63021}, {33216, 63046}, {33220, 51170}, {34254, 38282}, {35022, 47102}, {37640, 69121}, {37641, 69120}, {37647, 46951}, {37671, 61822}, {37688, 61861}, {39142, 69207}, {41134, 50992}, {41135, 42010}, {42006, 60183}, {44381, 63933}, {45962, 50739}, {52079, 59539}, {52080, 59540}, {53127, 61904}, {59634, 62130}, {59635, 69405}, {60233, 60636}, {61921, 69383}

X(69436) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {325, 1975, 32827}, {325, 3926, 32817}, {325, 32817, 4}, {325, 32827, 32823}, {1007, 32833, 52713}, {1007, 52713, 5071}, {1975, 32823, 4}, {1975, 32825, 32823}, {3926, 32816, 32820}, {3926, 32818, 4}, {3926, 32821, 32818}, {3926, 32825, 1975}, {3933, 32831, 631}, {6390, 37668, 376}, {7763, 69377, 3525}, {32816, 32820, 32822}, {32816, 32822, 4}, {32817, 32818, 325}, {32818, 32823, 32825}, {32825, 32827, 325}, {32829, 69384, 61867}, {32830, 69158, 3090}, {32835, 69381, 3533}, {63098, 69380, 3545}

X(69437) = X(2)X(14482)∩X(3)X(69)

Barycentrics   2*b^2*c^2 - S^2 + 2*SA^2 - SB*SC : :

X(69437) = 5 X[1384] - 6 X[37809], 2 X[1384] - 3 X[68718], 4 X[37809] - 5 X[68718], 3 X[37809] - 5 X[69206], 3 X[68718] - 4 X[69206], 3 X[7778] - 2 X[7844], 4 X[7778] - 3 X[33240], X[7844] - 3 X[7908], 8 X[7844] - 9 X[33240], 8 X[7908] - 3 X[33240], 3 X[33191] - X[63042]

X(69437) lies on these lines: {2, 14482}, {3, 69}, {4, 32840}, {5, 32818}, {6, 7801}, {20, 32879}, {30, 32817}, {32, 6144}, {39, 3763}, {76, 1656}, {99, 3534}, {140, 32831}, {141, 5024}, {183, 5054}, {187, 40341}, {193, 8369}, {194, 7866}, {230, 63954}, {274, 50726}, {302, 42818}, {303, 42817}, {305, 339}, {315, 1657}, {316, 382}, {325, 381}, {345, 1565}, {348, 3695}, {376, 10513}, {385, 11288}, {394, 4175}, {491, 18512}, {492, 18510}, {524, 1384}, {538, 7778}, {546, 32823}, {547, 32869}, {549, 15589}, {574, 599}, {620, 8667}, {625, 34505}, {631, 32841}, {632, 32835}, {999, 69093}, {1003, 7779}, {1007, 5055}, {1078, 61811}, {1285, 66393}, {1350, 14981}, {2418, 57620}, {2482, 5210}, {2549, 51122}, {2895, 16436}, {3053, 7855}, {3148, 38909}, {3295, 69094}, {3314, 11287}, {3526, 7763}, {3589, 7795}, {3593, 13939}, {3595, 13886}, {3618, 22246}, {3620, 8359}, {3627, 32822}, {3628, 32834}, {3629, 7758}, {3630, 15655}, {3734, 9766}, {3761, 31479}, {3788, 63933}, {3793, 20080}, {3830, 32815}, {3843, 32816}, {3851, 32825}, {3934, 31467}, {5008, 15534}, {5013, 7794}, {5023, 7826}, {5056, 32880}, {5070, 32828}, {5072, 7752}, {5073, 32006}, {5076, 32819}, {5079, 59635}, {5094, 9464}, {5254, 33241}, {5304, 8368}, {5305, 53033}, {5485, 8355}, {5739, 21509}, {5858, 41621}, {5859, 41620}, {5861, 66443}, {5971, 47597}, {6292, 22332}, {6392, 8361}, {6661, 63017}, {7486, 32882}, {7618, 22165}, {7737, 50771}, {7750, 15696}, {7751, 58448}, {7754, 7806}, {7757, 7868}, {7761, 8716}, {7762, 68527}, {7764, 69139}, {7766, 33220}, {7768, 62100}, {7769, 55858}, {7770, 7906}, {7771, 15700}, {7773, 7871}, {7774, 11286}, {7781, 7784}, {7782, 62085}, {7783, 7879}, {7798, 7880}, {7800, 59546}, {7802, 62143}, {7807, 63048}, {7809, 38335}, {7810, 53095}, {7811, 14093}, {7814, 61946}, {7816, 7916}, {7818, 44526}, {7819, 51171}, {7821, 44518}, {7835, 14614}, {7839, 33217}, {7840, 11159}, {7841, 7897}, {7851, 7909}, {7854, 15815}, {7860, 49133}, {7869, 32450}, {7873, 44519}, {7882, 63938}, {7887, 7947}, {7888, 13881}, {7893, 33235}, {7901, 20105}, {7903, 65630}, {7917, 49137}, {7919, 11055}, {7939, 33234}, {8366, 63019}, {8721, 48881}, {8724, 50567}, {9300, 14535}, {9466, 31489}, {9737, 43150}, {9770, 59780}, {9909, 40123}, {11160, 27088}, {11171, 14994}, {11284, 56435}, {11317, 41136}, {11318, 47286}, {11343, 37656}, {12040, 42850}, {13188, 32458}, {14001, 43136}, {14023, 59545}, {14039, 63091}, {14269, 32827}, {14360, 31152}, {14501, 67676}, {14502, 67687}, {14615, 58846}, {14907, 15688}, {14928, 52090}, {14961, 20208}, {15069, 18860}, {15603, 35287}, {15681, 64018}, {15684, 67536}, {15693, 37671}, {15694, 32837}, {15699, 32874}, {15703, 34803}, {15905, 54075}, {15980, 40824}, {16418, 45962}, {16431, 32863}, {17129, 33233}, {17130, 69197}, {17131, 37637}, {17272, 37599}, {17343, 21937}, {17675, 27523}, {18440, 35002}, {18600, 56780}, {19661, 63064}, {19709, 69382}, {20065, 68513}, {20094, 66388}, {20806, 22121}, {20888, 31493}, {21514, 63003}, {22110, 40727}, {22236, 69107}, {22238, 69106}, {23039, 51386}, {23106, 46127}, {23115, 23133}, {30270, 48905}, {30761, 48838}, {31461, 69095}, {31477, 69261}, {32000, 64474}, {32001, 37458}, {32216, 62299}, {32515, 37071}, {32826, 62008}, {32829, 46219}, {32832, 55857}, {32838, 55860}, {32839, 55866}, {32868, 61892}, {32870, 48154}, {32871, 52718}, {32872, 61886}, {32873, 61867}, {32876, 61832}, {32877, 61919}, {32878, 61903}, {32881, 55864}, {32885, 61882}, {32890, 61923}, {32892, 61901}, {32893, 61885}, {32894, 46936}, {32895, 61863}, {32897, 55861}, {32898, 55862}, {33017, 47287}, {33191, 63042}, {33197, 63097}, {33246, 50248}, {34291, 52629}, {35022, 47101}, {35297, 63046}, {35302, 45794}, {36212, 37638}, {37344, 37644}, {37647, 61887}, {37690, 43291}, {37779, 52275}, {38744, 54103}, {38940, 45662}, {39099, 50962}, {40107, 52771}, {40825, 50253}, {41266, 47582}, {42988, 69157}, {42989, 69165}, {43448, 52229}, {44369, 51175}, {44377, 63955}, {44543, 63021}, {44715, 59257}, {48913, 61977}, {51439, 54048}, {53127, 61908}, {54714, 60202}, {55610, 59548}, {56967, 67400}, {59503, 69038}, {61905, 69385}, {61920, 64809}, {62988, 66415}, {63198, 69114}, {63199, 69115}, {66455, 66616}, {69149, 69201}

X(69437) = midpoint of X(32817) and X(37668)
X(69437) = reflection of X(i) in X(j) for these {i,j}: {1384, 69206}, {7778, 7908}
X(69437) = isotomic conjugate of X(68566)
X(69437) = isotomic conjugate of the polar conjugate of X(599)
X(69437) = isogonal conjugate of the polar conjugate of X(9464)
X(69437) = X(9464)-Ceva conjugate of X(599)
X(69437) = X(i)-isoconjugate of X(j) for these (i,j): {19, 1383}, {25, 55927}, {31, 68566}, {162, 46001}, {598, 1973}, {1096, 43697}, {8599, 32676}, {24019, 30491}
X(69437) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 68566}, {6, 1383}, {125, 46001}, {597, 10301}, {599, 4232}, {5181, 65007}, {6337, 598}, {6338, 64982}, {6503, 43697}, {6505, 55927}, {8542, 25}, {11165, 4}, {15526, 8599}, {17413, 2489}, {17416, 2501}, {17436, 8754}, {35071, 30491}, {52881, 51541}, {62590, 52692}, {62594, 23287}, {62604, 40826}, {62607, 18818}
X(69437) = crossdifference of every pair of points on line {2489, 46001}
X(69437) = barycentric product X(i)*X(j) for these {i,j}: {3, 9464}, {69, 599}, {287, 51397}, {304, 36263}, {305, 574}, {525, 9146}, {3267, 9145}, {3906, 4563}, {3908, 15413}, {3926, 5094}, {3933, 10130}, {6390, 42008}, {8288, 47389}, {17414, 52608}, {30786, 39785}, {32583, 45807}
X(69437) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 68566}, {3, 1383}, {63, 55927}, {69, 598}, {305, 40826}, {394, 43697}, {520, 30491}, {525, 8599}, {574, 25}, {599, 4}, {647, 46001}, {3906, 2501}, {3908, 1783}, {3917, 30489}, {3926, 64982}, {3933, 23297}, {4141, 8756}, {4558, 11636}, {4563, 35138}, {5094, 393}, {6390, 51541}, {8288, 8754}, {8541, 2207}, {9145, 112}, {9146, 648}, {9464, 264}, {10130, 32085}, {10510, 8744}, {11165, 4232}, {13857, 1990}, {14417, 23287}, {14961, 65007}, {15810, 10301}, {17414, 2489}, {19510, 5523}, {23288, 68629}, {30786, 18818}, {34897, 10511}, {36212, 52692}, {36263, 19}, {39785, 468}, {42007, 8753}, {42008, 17983}, {45807, 65008}, {51397, 297}, {62657, 44102}, {65747, 20380}
X(69437) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 3926, 6390}, {69, 6390, 3}, {76, 32821, 69158}, {76, 69158, 1656}, {141, 34511, 5024}, {187, 40341, 63950}, {193, 8369, 21309}, {194, 7881, 7866}, {325, 32833, 69380}, {325, 69380, 381}, {376, 10513, 14929}, {599, 39785, 11165}, {1007, 32836, 64093}, {1007, 64093, 5055}, {1238, 3926, 22241}, {1238, 45795, 3964}, {1384, 69206, 68718}, {1975, 7776, 382}, {1975, 7796, 7776}, {3053, 7855, 63936}, {3314, 31859, 11287}, {3734, 9766, 15484}, {3926, 3933, 3}, {3933, 6390, 69}, {3964, 22241, 3}, {6337, 7767, 3}, {7754, 7836, 32954}, {7758, 7789, 30435}, {7761, 14148, 8716}, {7763, 69381, 3526}, {7781, 7895, 7784}, {7789, 30435, 33242}, {7801, 7813, 6}, {7816, 7916, 63932}, {7855, 7863, 3053}, {7882, 69171, 63938}, {7947, 20081, 7887}, {20080, 32985, 3793}, {32818, 32830, 5}, {32818, 52713, 63098}, {32823, 69379, 546}, {32830, 63098, 52713}, {32831, 69377, 140}, {32835, 69384, 632}, {32871, 52718, 55859}, {52193, 52194, 6776}, {52713, 63098, 5}, {69120, 69121, 6}, {69157, 69180, 42988}, {69165, 69186, 42989}

X(69438) = X(2)X(14482)∩X(69)X(74)

Barycentrics   2*b^2*c^2 + 2*SA^2 - SB*SC : :

X(69438) lies on these lines: {2, 14482}, {3, 32840}, {4, 3933}, {20, 14929}, {30, 10513}, {39, 18840}, {69, 74}, {76, 1007}, {115, 5485}, {140, 32841}, {183, 631}, {186, 22241}, {193, 14039}, {194, 32956}, {253, 21312}, {279, 3695}, {315, 32822}, {316, 62017}, {325, 3545}, {339, 9464}, {346, 1565}, {385, 33191}, {524, 1285}, {538, 33196}, {599, 9741}, {620, 63029}, {1003, 20080}, {1056, 69093}, {1058, 69094}, {1078, 61807}, {1270, 26619}, {1271, 26620}, {1350, 15428}, {1975, 3529}, {2996, 33292}, {3091, 32880}, {3314, 33190}, {3329, 16045}, {3523, 32879}, {3524, 6390}, {3525, 32831}, {3528, 7767}, {3533, 7763}, {3537, 41005}, {3544, 69383}, {3619, 7757}, {3620, 31859}, {3628, 32872}, {3631, 8716}, {3760, 47743}, {3761, 8164}, {3785, 21735}, {3855, 7796}, {3964, 35921}, {3972, 11008}, {4234, 62999}, {5013, 55732}, {5056, 32882}, {5067, 32834}, {5071, 32869}, {5286, 33194}, {5305, 32952}, {5562, 34403}, {5860, 61389}, {5861, 61388}, {5971, 40132}, {6337, 7771}, {6392, 7881}, {6661, 51170}, {6722, 7908}, {7486, 32894}, {7709, 14994}, {7714, 40123}, {7735, 7801}, {7736, 7813}, {7738, 7794}, {7750, 32824}, {7752, 69403}, {7754, 14069}, {7758, 7804}, {7766, 14001}, {7768, 62147}, {7769, 52718}, {7772, 18841}, {7773, 32890}, {7779, 14033}, {7788, 15682}, {7795, 7798}, {7799, 15709}, {7809, 61983}, {7820, 63006}, {7821, 63533}, {7835, 63034}, {7836, 33189}, {7850, 62165}, {7870, 63104}, {7893, 33239}, {7897, 16041}, {7906, 32968}, {7929, 33247}, {7939, 33238}, {7947, 32969}, {8024, 8889}, {8182, 35022}, {8368, 63097}, {8369, 63042}, {9466, 62993}, {11147, 52886}, {11185, 41099}, {11286, 63091}, {12150, 62996}, {14064, 20081}, {14535, 37665}, {14711, 43620}, {15048, 33230}, {15482, 34511}, {15484, 59780}, {15698, 32896}, {15710, 59634}, {17129, 32970}, {17131, 31274}, {17205, 53665}, {20023, 52608}, {21356, 51397}, {22165, 53142}, {32006, 62028}, {32814, 35949}, {32816, 32877}, {32819, 62021}, {32821, 32828}, {32825, 59635}, {32827, 61980}, {32829, 61870}, {32832, 61881}, {32835, 61867}, {32837, 37688}, {32873, 46219}, {32874, 61899}, {32881, 61856}, {32892, 61915}, {32893, 61888}, {32985, 63046}, {33017, 35369}, {33195, 53033}, {33215, 63044}, {33224, 63048}, {33231, 37689}, {33237, 63005}, {33255, 50248}, {33272, 47287}, {33285, 47286}, {34803, 61889}, {36163, 47289}, {37640, 69120}, {37641, 69121}, {39785, 42850}, {40824, 43532}, {40995, 61113}, {41676, 52283}, {46144, 51429}, {46453, 69206}, {46951, 61895}, {47005, 63120}, {50994, 52691}, {55741, 60728}, {58090, 65652}, {61926, 69387}, {63109, 66703}

X(69438) = isotomic conjugate of X(39453)
X(69438) = X(31)-isoconjugate of X(39453)
X(69438) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 39453}, {11165, 43956}
X(69438) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 39453}, {599, 43956}
X(69438) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 32817, 376}, {69, 32833, 32817}, {76, 1007, 69386}, {76, 32818, 3090}, {315, 32822, 33703}, {325, 32836, 52713}, {325, 52713, 3545}, {1007, 69386, 3090}, {3785, 32875, 32820}, {3926, 69377, 631}, {3933, 32830, 4}, {3933, 69380, 37668}, {6390, 15589, 3524}, {6392, 7881, 32951}, {7763, 69384, 3533}, {7776, 69379, 4}, {7796, 69378, 32823}, {7908, 63955, 37690}, {32818, 69386, 1007}, {32823, 69378, 3855}, {32830, 37668, 69380}, {32831, 69381, 3525}, {32834, 69158, 5067}, {32869, 63098, 64093}, {37668, 69380, 4}, {63098, 64093, 5071}

X(69439) = X(2)X(14482)∩X(5)X(76)

Barycentrics   2*b^2*c^2 + S^2 + 2*SA^2 - SB*SC : :

X(69439) = 3 X[69] + X[32815], X[69] + 3 X[32836], 5 X[69] - X[64018], 7 X[69] + X[67536], 3 X[14929] + 2 X[32815], X[14929] + 6 X[32836], 5 X[14929] - 2 X[64018], 7 X[14929] + 2 X[67536], X[14929] + 2 X[69380], X[32815] - 9 X[32836], 5 X[32815] + 3 X[64018], 7 X[32815] - 3 X[67536], X[32815] - 3 X[69380], 15 X[32836] + X[64018], 21 X[32836] - X[67536], 3 X[32836] - X[69380], 7 X[64018] + 5 X[67536], X[64018] + 5 X[69380], X[67536] - 7 X[69380], 3 X[141] - 2 X[4045], 4 X[4045] - 3 X[15048], X[193] - 3 X[11286], 2 X[3734] - 3 X[59780], X[18907] - 3 X[59780], 2 X[7848] - 3 X[22165], 5 X[597] - 4 X[61046], 3 X[599] - X[2549], 5 X[3620] - 3 X[11287], 5 X[3763] - 3 X[7739], 3 X[10519] - 2 X[55167], 3 X[14033] + X[20080], X[14532] - 3 X[62174]

X(69439) lies on these lines: {2, 14482}, {3, 15589}, {4, 10513}, {5, 76}, {20, 32880}, {30, 69}, {32, 15480}, {85, 3695}, {99, 8703}, {115, 14711}, {140, 3926}, {141, 538}, {148, 66392}, {183, 549}, {193, 11286}, {194, 8362}, {230, 7801}, {253, 35513}, {305, 59766}, {312, 1565}, {315, 3627}, {316, 15687}, {339, 1368}, {343, 51389}, {381, 32869}, {382, 69379}, {384, 50248}, {385, 8369}, {395, 69121}, {396, 69120}, {397, 69106}, {398, 69107}, {495, 3761}, {496, 3760}, {511, 16983}, {523, 52629}, {524, 3734}, {543, 7848}, {546, 7776}, {547, 1007}, {548, 3785}, {550, 1975}, {574, 51123}, {597, 61046}, {599, 2549}, {620, 13468}, {626, 32457}, {631, 32840}, {632, 7763}, {754, 3630}, {1003, 3793}, {1078, 15712}, {1089, 3665}, {1111, 3703}, {1213, 48840}, {1232, 52347}, {1235, 1595}, {1270, 23273}, {1271, 23267}, {1316, 38940}, {1358, 6057}, {1384, 66393}, {1596, 44146}, {1597, 32000}, {1656, 32818}, {1657, 32822}, {1991, 66428}, {2407, 44649}, {2782, 14994}, {3091, 32882}, {3314, 33184}, {3363, 7840}, {3525, 32841}, {3526, 32831}, {3530, 6337}, {3564, 64653}, {3589, 7798}, {3620, 11287}, {3628, 32828}, {3629, 7804}, {3631, 7761}, {3763, 7739}, {3767, 33186}, {3788, 44381}, {3815, 7813}, {3820, 6381}, {3845, 7788}, {3850, 32816}, {3851, 32823}, {3853, 32006}, {3858, 7773}, {3886, 28915}, {3934, 15491}, {3945, 11354}, {3964, 7514}, {3972, 50251}, {4352, 56734}, {4357, 66675}, {4888, 48812}, {5024, 66418}, {5055, 32874}, {5056, 32894}, {5066, 32892}, {5067, 32872}, {5224, 48815}, {5232, 11359}, {5254, 7794}, {5286, 8364}, {5304, 33237}, {5305, 7795}, {5306, 7820}, {5468, 34094}, {5475, 50771}, {5860, 13763}, {5861, 13644}, {5874, 45406}, {5875, 45407}, {6031, 66373}, {6292, 9607}, {6392, 7866}, {6644, 22241}, {6656, 20081}, {6661, 7766}, {6680, 63925}, {6770, 52193}, {6773, 52194}, {7484, 41916}, {7735, 8368}, {7736, 8367}, {7737, 40341}, {7745, 7855}, {7750, 15704}, {7751, 7789}, {7754, 7819}, {7758, 69139}, {7762, 17128}, {7768, 32819}, {7769, 55859}, {7770, 63017}, {7771, 17504}, {7774, 66415}, {7778, 43291}, {7779, 8370}, {7780, 59545}, {7782, 62069}, {7799, 11539}, {7802, 62159}, {7807, 17129}, {7809, 23046}, {7811, 15686}, {7815, 59546}, {7816, 63928}, {7818, 53419}, {7821, 63534}, {7828, 33212}, {7832, 33211}, {7833, 47287}, {7835, 22329}, {7837, 53489}, {7845, 53418}, {7850, 33699}, {7854, 63548}, {7876, 20105}, {7879, 8357}, {7881, 8361}, {7893, 19687}, {7895, 63924}, {7897, 33228}, {7906, 32992}, {7908, 44377}, {7917, 61976}, {7919, 11054}, {7920, 66342}, {7929, 19695}, {7931, 19570}, {7939, 33229}, {7947, 33249}, {8353, 20094}, {8355, 40727}, {8356, 63044}, {8358, 51122}, {8359, 16990}, {8667, 69206}, {8728, 34284}, {9464, 30739}, {9606, 31239}, {9740, 46453}, {10124, 32837}, {10303, 32879}, {10519, 55167}, {10796, 64067}, {11057, 62154}, {11159, 11160}, {11165, 42850}, {11168, 12040}, {11288, 37667}, {11352, 31303}, {11812, 32896}, {12108, 32875}, {12812, 32888}, {14033, 20080}, {14039, 21309}, {14148, 15598}, {14501, 67677}, {14502, 67688}, {14532, 62174}, {14535, 63024}, {14552, 56963}, {15067, 51386}, {15271, 34511}, {15484, 66412}, {15533, 63945}, {15703, 32893}, {15810, 63654}, {16084, 33769}, {16239, 32829}, {16509, 22110}, {16987, 46226}, {17234, 48869}, {17245, 48864}, {17392, 24275}, {17527, 18135}, {17670, 40908}, {18139, 50154}, {18140, 51559}, {18494, 32001}, {19697, 30435}, {20065, 68177}, {20888, 31419}, {20911, 50042}, {21850, 51396}, {31995, 48806}, {32087, 48804}, {32458, 67268}, {32821, 32832}, {32824, 32890}, {32825, 32868}, {32826, 62026}, {32835, 46219}, {32838, 48154}, {32839, 55862}, {32867, 61877}, {32870, 55857}, {32871, 55866}, {32873, 61870}, {32876, 45760}, {32881, 61863}, {32885, 47599}, {32886, 61894}, {32897, 55860}, {32898, 61875}, {33198, 43136}, {33220, 63048}, {33458, 47857}, {33459, 47858}, {34380, 35930}, {35022, 46893}, {35369, 67270}, {35520, 44148}, {35954, 44367}, {37242, 61545}, {37647, 61885}, {37912, 46444}, {38112, 69038}, {40123, 66529}, {41005, 44149}, {41826, 50044}, {43448, 66394}, {43459, 61785}, {44340, 59657}, {48817, 62999}, {48913, 61963}, {49671, 68660}, {50168, 63056}, {52718, 55858}, {53127, 61910}, {53795, 67920}, {57619, 68455}, {63018, 66416}, {63093, 66318}, {63934, 69172}

X(69439) = midpoint of X(i) and X(j) for these {i,j}: {69, 69380}, {7737, 40341}, {11159, 11160}
X(69439) = reflection of X(i) in X(j) for these {i,j}: {3629, 7804}, {7761, 3631}, {7798, 3589}, {14929, 69}, {15048, 141}, {18907, 3734}, {22253, 63633}, {37242, 61545}, {64067, 10796}
X(69439) = complement of X(22253)
X(69439) = anticomplement of X(63633)
X(69439) = isotomic conjugate of the isogonal conjugate of X(5650)
X(69439) = barycentric product X(i)*X(j) for these {i,j}: {76, 5650}, {305, 33843}
X(69439) = barycentric quotient X(i)/X(j) for these {i,j}: {5650, 6}, {33843, 25}
X(69439) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 22253, 63633}, {69, 32836, 69380}, {76, 325, 64093}, {76, 3933, 5}, {76, 7796, 59635}, {183, 6390, 549}, {183, 32833, 6390}, {325, 64093, 5}, {1003, 63046, 3793}, {1975, 7767, 550}, {3314, 47286, 33184}, {3760, 69094, 496}, {3761, 69093, 495}, {3926, 69381, 140}, {3933, 64093, 325}, {5305, 7795, 33185}, {7776, 69378, 546}, {7778, 63955, 43291}, {7795, 63933, 5305}, {7801, 17131, 230}, {7813, 9466, 3815}, {7855, 17130, 7745}, {11168, 39785, 12040}, {14039, 63042, 21309}, {15589, 32817, 3}, {15589, 32830, 32817}, {16990, 31859, 8359}, {18907, 59780, 3734}, {32817, 69377, 15589}, {32818, 32834, 1656}, {32823, 69383, 3851}, {32825, 32868, 69385}, {32828, 69158, 3628}, {32830, 69377, 3}, {32831, 69384, 3526}, {32869, 37668, 52713}, {32874, 63098, 69386}, {37668, 52713, 381}, {63098, 69386, 5055}

X(69440) = X(2)X(14482)∩X(4)X(69)

Barycentrics   2*b^2*c^2 + 2*S^2 + 2*SA^2 - SB*SC : :

X(69440) lies on these lines: {2, 14482}, {4, 69}, {20, 32882}, {83, 63073}, {99, 19708}, {140, 32840}, {141, 33230}, {183, 3524}, {193, 53489}, {194, 32960}, {230, 33231}, {325, 5071}, {339, 7386}, {376, 15589}, {381, 10513}, {385, 14039}, {574, 9741}, {599, 5485}, {631, 6390}, {671, 50990}, {1007, 61899}, {1056, 3761}, {1058, 3760}, {1078, 61138}, {1089, 7195}, {1111, 3974}, {1270, 13785}, {1271, 13665}, {1285, 3734}, {1384, 9740}, {1656, 32872}, {1975, 3528}, {2482, 55823}, {2549, 14711}, {2996, 7879}, {3090, 3933}, {3091, 32894}, {3314, 33285}, {3523, 32880}, {3525, 3926}, {3526, 32841}, {3529, 7767}, {3533, 32831}, {3538, 41009}, {3543, 14929}, {3544, 32823}, {3545, 32874}, {3589, 63933}, {3593, 7376}, {3595, 7375}, {3620, 33190}, {3629, 69139}, {3631, 34505}, {3673, 32087}, {3763, 5286}, {3767, 32953}, {3785, 17538}, {3855, 7776}, {3964, 7550}, {4385, 11024}, {4648, 68938}, {5067, 32818}, {5254, 33232}, {5304, 63954}, {6144, 69208}, {6337, 61814}, {6392, 32956}, {7620, 22165}, {7735, 7820}, {7736, 9466}, {7738, 55732}, {7749, 39142}, {7750, 49138}, {7752, 69406}, {7754, 16045}, {7760, 63011}, {7763, 61867}, {7770, 51170}, {7771, 15715}, {7779, 32983}, {7788, 41106}, {7789, 33236}, {7790, 21356}, {7795, 32952}, {7796, 69385}, {7799, 61861}, {7801, 62992}, {7806, 14069}, {7809, 61959}, {7811, 62161}, {7813, 62993}, {7827, 63121}, {7836, 32959}, {7844, 63955}, {7881, 32955}, {7897, 32984}, {7906, 32975}, {7931, 32951}, {7947, 32976}, {8164, 69093}, {8370, 20080}, {8667, 46453}, {8716, 15598}, {10302, 60629}, {11001, 32815}, {11054, 51397}, {11286, 63042}, {11488, 69120}, {11489, 69121}, {11541, 32819}, {12243, 50567}, {14001, 17129}, {14033, 63046}, {14907, 62130}, {15533, 53418}, {15534, 18842}, {15702, 32833}, {16043, 20081}, {16239, 32873}, {17559, 18135}, {17582, 34284}, {18935, 42554}, {20023, 57817}, {27269, 33027}, {31276, 32957}, {32816, 32888}, {32820, 32877}, {32821, 32838}, {32824, 61787}, {32825, 32886}, {32827, 61967}, {32829, 52718}, {32832, 60781}, {32835, 61870}, {32837, 61865}, {32870, 61881}, {32879, 55864}, {32885, 37647}, {32893, 61895}, {32968, 63018}, {32986, 63044}, {33191, 37667}, {33197, 37689}, {33224, 63047}, {33237, 63097}, {34803, 61888}, {37670, 50739}, {41254, 65747}, {41927, 57518}, {46105, 60114}, {47743, 69094}, {51358, 52283}, {53127, 61913}, {54616, 60628}, {54637, 60200}, {60126, 60212}, {60183, 60285}, {60616, 60638}, {60641, 60643}, {60855, 62995}, {61886, 69158}, {61932, 69387}, {62042, 64018}, {63029, 69206}, {63091, 66415}, {63925, 69209}

X(69440) = anticomplement of the isogonal conjugate of X(54616)
X(69440) = X(54616)-anticomplementary conjugate of X(8)
X(69440) = crosspoint of X(5485) and X(18840)
X(69440) = crosssum of X(1384) and X(30435)
X(69440) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 22253, 14482}, {69, 76, 52713}, {69, 52713, 4}, {69, 69378, 316}, {76, 44149, 44146}, {76, 69377, 4}, {183, 32817, 3524}, {183, 32836, 32817}, {325, 46951, 69386}, {325, 69386, 5071}, {621, 622, 51537}, {3620, 47286, 33190}, {3785, 32822, 17538}, {3926, 69384, 3525}, {3933, 32834, 3090}, {7767, 69379, 3529}, {7776, 69383, 3855}, {15589, 32869, 69380}, {15589, 69380, 376}, {32000, 44146, 4}, {32818, 32828, 5067}, {32823, 59635, 3544}, {32829, 52718, 61873}, {32830, 69381, 631}, {32874, 37668, 64093}, {37668, 64093, 3545}, {52713, 69377, 69}

X(69441) = X(2)X(39)∩X(4)X(10513)

Barycentrics   2*b^2*c^2 + S^2 + 2*SA^2 : :

X(69441) lies on these lines: {2, 39}, {4, 10513}, {8, 64695}, {20, 7767}, {69, 3146}, {85, 346}, {99, 21734}, {183, 15717}, {193, 17128}, {253, 37201}, {279, 312}, {304, 43983}, {315, 17578}, {325, 5068}, {343, 34403}, {384, 63042}, {385, 33201}, {1007, 61914}, {1078, 32824}, {1219, 62697}, {1270, 3071}, {1271, 3070}, {1285, 63926}, {1975, 3522}, {2996, 3314}, {3053, 9740}, {3091, 3933}, {3523, 32817}, {3593, 43880}, {3595, 43879}, {3620, 7928}, {3760, 14986}, {3785, 50693}, {3832, 7773}, {3839, 7776}, {3854, 15031}, {4461, 20911}, {5056, 32818}, {5059, 7802}, {5261, 69093}, {5274, 69094}, {5304, 63933}, {5334, 69107}, {5335, 69106}, {5395, 7837}, {6337, 61820}, {6390, 10303}, {6527, 44149}, {7486, 69158}, {7620, 7825}, {7750, 15683}, {7752, 69404}, {7754, 33198}, {7768, 32826}, {7770, 14930}, {7779, 32979}, {7782, 62060}, {7788, 61985}, {7789, 37689}, {7794, 43448}, {7796, 69382}, {7809, 61972}, {7810, 53141}, {7811, 62148}, {7813, 31404}, {7823, 11160}, {7879, 33210}, {7881, 33199}, {7890, 17130}, {7897, 32980}, {7906, 32987}, {7917, 11185}, {7921, 32971}, {7947, 32988}, {8359, 11148}, {11173, 14035}, {14001, 63097}, {14031, 50248}, {14907, 62124}, {14929, 33703}, {14994, 62174}, {15022, 59635}, {15048, 18840}, {15815, 42850}, {16045, 22253}, {16986, 60259}, {17129, 32973}, {18134, 57826}, {21605, 42032}, {22241, 44802}, {26170, 53481}, {26218, 36413}, {30435, 59780}, {32006, 50687}, {32814, 43520}, {32820, 34229}, {32821, 69385}, {32825, 69387}, {32981, 63046}, {33023, 63044}, {33180, 47286}, {33205, 37667}, {33226, 47287}, {33283, 40824}, {34505, 63536}, {37665, 69139}, {37671, 62120}, {37688, 61848}, {39749, 51972}, {43459, 61778}, {44140, 54303}, {50692, 64018}, {51170, 68525}, {59545, 63029}, {59634, 61806}, {60200, 60262}, {60212, 60639}, {63079, 69180}, {63080, 69186}

X(69441) = barycentric product X(i)*X(j) for these {i,j}: {305, 11403}, {32793, 32794}
X(69441) = barycentric quotient X(i)/X(j) for these {i,j}: {11403, 25}, {32793, 63689}
X(69441) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 32830, 32840}, {2, 32831, 32873}, {2, 32840, 32841}, {2, 32879, 32831}, {2, 32880, 32830}, {2, 32881, 32829}, {2, 32882, 76}, {2, 32894, 32834}, {2, 32895, 7769}, {69, 69379, 3146}, {76, 3926, 32834}, {76, 7763, 46951}, {76, 32828, 32874}, {76, 32830, 2}, {76, 32832, 32868}, {76, 32833, 32828}, {76, 32834, 32894}, {76, 32836, 32830}, {76, 32840, 32872}, {76, 32869, 32882}, {76, 32875, 32893}, {76, 32877, 32831}, {76, 32880, 32840}, {76, 32890, 32835}, {325, 69383, 5068}, {1975, 15589, 3522}, {2996, 3314, 33200}, {3926, 7769, 32831}, {3926, 32828, 7769}, {3926, 32832, 32835}, {3926, 32834, 2}, {3926, 32839, 32891}, {3926, 32867, 7763}, {3926, 32868, 32832}, {3926, 32878, 76}, {3926, 32897, 32895}, {3926, 46951, 32867}, {3933, 52713, 3091}, {6390, 69384, 10303}, {7754, 33198, 63005}, {7763, 32867, 32898}, {7767, 32822, 20}, {7767, 69380, 32822}, {7769, 32828, 32897}, {7769, 32831, 32895}, {7769, 32833, 3926}, {7769, 32884, 32898}, {7769, 32895, 32873}, {7769, 32897, 2}, {7799, 32893, 2}, {32817, 69381, 3523}, {32818, 64093, 5056}, {32822, 69377, 7767}, {32828, 32830, 32879}, {32828, 32831, 2}, {32828, 32833, 32831}, {32828, 32836, 32877}, {32828, 32877, 32833}, {32828, 32879, 32873}, {32828, 32892, 76}, {32829, 32870, 2}, {32830, 32831, 32833}, {32830, 32834, 3926}, {32830, 32836, 32880}, {32830, 32869, 76}, {32830, 32874, 32831}, {32830, 32878, 32894}, {32830, 32888, 32881}, {32831, 32833, 32879}, {32831, 32834, 32897}, {32831, 32874, 32828}, {32831, 32897, 7769}, {32832, 32835, 2}, {32832, 32868, 32834}, {32833, 32834, 32895}, {32833, 32874, 2}, {32833, 32877, 32830}, {32833, 32879, 32840}, {32833, 32892, 32874}, {32834, 32835, 32832}, {32834, 32897, 32828}, {32836, 32868, 32890}, {32836, 32869, 2}, {32836, 32878, 3926}, {32836, 32882, 32840}, {32836, 32892, 32833}, {32838, 32875, 7799}, {32841, 32872, 2}, {32867, 32898, 2}, {32868, 32890, 3926}, {32869, 32874, 32892}, {32873, 32874, 32872}, {32873, 32879, 32841}, {32874, 32877, 32879}, {32874, 32897, 32834}, {32877, 32892, 32828}, {32878, 32890, 32868}, {32880, 32882, 2}, {32880, 32892, 32873}, {32884, 46951, 32828}, {32891, 32896, 3926}, {37668, 69378, 3832}, {59635, 63098, 15022}, {69158, 69386, 7486}, {69377, 69380, 20}

X(69442) = X(2)X(1975)∩X(4)X(7769)

Barycentrics   b^2*c^2 + 2*S^2 - 2*SA^2 - SB*SC : :

X(69442) lies on these lines: {2, 1975}, {3, 1007}, {4, 7769}, {5, 32826}, {6, 32989}, {20, 53017}, {24, 44180}, {32, 33216}, {39, 32970}, {54, 69}, {76, 3525}, {83, 33191}, {99, 3090}, {115, 32976}, {140, 3926}, {148, 32998}, {183, 10303}, {194, 33000}, {315, 3524}, {316, 3528}, {325, 3523}, {376, 7752}, {384, 62993}, {385, 33206}, {439, 7745}, {491, 8406}, {492, 8414}, {549, 3785}, {550, 32827}, {574, 14064}, {620, 7808}, {625, 33238}, {626, 7622}, {629, 60222}, {632, 32838}, {1003, 31404}, {1285, 7858}, {1506, 14033}, {1587, 15884}, {1588, 15883}, {1656, 32815}, {2548, 32985}, {2549, 32969}, {3053, 62988}, {3055, 32987}, {3091, 32871}, {3314, 33012}, {3522, 7773}, {3526, 6390}, {3530, 7776}, {3533, 32817}, {3552, 63083}, {3589, 31492}, {3618, 7807}, {3619, 11285}, {3628, 32884}, {3734, 32975}, {3763, 59552}, {3767, 32977}, {3788, 16043}, {3815, 32973}, {3933, 5054}, {5050, 10008}, {5056, 32819}, {5067, 11185}, {5286, 33233}, {5305, 12040}, {5319, 58448}, {5475, 33239}, {5569, 7882}, {5866, 7503}, {6179, 62996}, {6392, 37637}, {7618, 7748}, {7735, 7839}, {7736, 7787}, {7749, 23055}, {7750, 15717}, {7764, 21843}, {7767, 15720}, {7768, 61817}, {7771, 61814}, {7774, 33259}, {7777, 32964}, {7778, 32990}, {7786, 14069}, {7788, 15708}, {7790, 32955}, {7791, 37690}, {7792, 33203}, {7795, 32978}, {7799, 15702}, {7802, 21735}, {7803, 33189}, {7805, 63034}, {7806, 33262}, {7809, 15698}, {7811, 15719}, {7814, 61138}, {7816, 32983}, {7824, 51580}, {7825, 33247}, {7828, 32959}, {7832, 32960}, {7834, 31450}, {7835, 16045}, {7836, 33001}, {7846, 33197}, {7847, 33285}, {7856, 14482}, {7859, 32952}, {7860, 61787}, {7862, 16041}, {7871, 61822}, {7874, 31457}, {7881, 21356}, {7899, 33190}, {7905, 63064}, {7912, 33008}, {7925, 32965}, {7931, 33258}, {7940, 32951}, {8369, 31467}, {8781, 38748}, {9606, 51171}, {9608, 35296}, {9723, 19440}, {9737, 58883}, {9752, 10983}, {9770, 20065}, {10124, 32885}, {10299, 14907}, {10358, 14494}, {10513, 32895}, {11057, 15715}, {11147, 33007}, {11174, 33181}, {11184, 35287}, {11288, 31406}, {11539, 46951}, {11540, 32892}, {12108, 32889}, {14035, 17005}, {14929, 61810}, {14994, 50654}, {15031, 61921}, {15043, 51439}, {15515, 33226}, {15589, 32821}, {15694, 32836}, {15709, 32833}, {15721, 37671}, {15815, 32974}, {16196, 40680}, {16239, 32883}, {16589, 33049}, {16644, 59542}, {16645, 59541}, {17004, 33204}, {18840, 62892}, {20080, 55814}, {26877, 55469}, {26878, 55470}, {30761, 37339}, {31274, 53096}, {31276, 33003}, {31407, 53489}, {31415, 69171}, {31455, 32968}, {31489, 32971}, {32459, 32981}, {32820, 32834}, {32822, 61886}, {32824, 32867}, {32830, 37688}, {32840, 61848}, {32841, 61842}, {32868, 61855}, {32875, 61849}, {32876, 61840}, {32877, 61852}, {32886, 61858}, {32888, 45760}, {32896, 61847}, {32967, 63533}, {32972, 63548}, {32979, 51579}, {32980, 44526}, {32986, 37512}, {32988, 44518}, {33023, 53095}, {33217, 63120}, {33224, 69209}, {33249, 39143}, {35260, 65564}, {35297, 69208}, {35927, 65630}, {37118, 52710}, {37291, 45962}, {37612, 55418}, {37640, 62600}, {37641, 62601}, {37667, 44535}, {37668, 61820}, {38282, 54412}, {43028, 59539}, {43029, 59540}, {43459, 61807}, {44499, 59373}, {47355, 59548}, {48913, 62130}, {51373, 61132}, {52250, 53419}, {52713, 61867}, {52718, 61859}, {53127, 60781}, {55726, 55808}, {55732, 55804}, {55735, 55803}, {55741, 55801}, {55745, 55800}, {55754, 55795}, {55759, 55794}, {55762, 55793}, {55765, 55792}, {55767, 55790}, {55779, 55787}, {55780, 55785}, {55782, 55783}, {55811, 55823}, {61870, 69386}

X(69442) = X(60221)-Ceva conjugate of X(69)
X(69442) = crossdifference of every pair of points on line {8651, 55219}
X(69442) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1975, 69385}, {2, 6337, 69378}, {3, 1007, 32006}, {3, 32829, 1007}, {4, 7769, 34803}, {140, 3926, 34229}, {194, 33000, 62992}, {439, 63077, 7745}, {549, 69158, 3785}, {620, 31401, 14001}, {631, 7763, 69}, {631, 32818, 1078}, {632, 69380, 32838}, {1078, 7763, 32818}, {1078, 32818, 69}, {1975, 69385, 69378}, {3091, 32871, 37647}, {3523, 32835, 325}, {3526, 6390, 32828}, {3533, 32817, 32832}, {5286, 33233, 63104}, {6337, 69385, 1975}, {7786, 14069, 63119}, {7807, 31400, 3618}, {7832, 32960, 63121}, {10299, 32823, 14907}, {10303, 32831, 183}, {11285, 53033, 3619}, {14001, 31401, 63041}, {15717, 32873, 63098}, {15717, 63098, 7750}, {15815, 44377, 32974}, {31455, 69206, 32968}, {31489, 59545, 32971}, {32822, 61886, 69387}, {32824, 32867, 64093}, {32830, 55864, 37688}, {32884, 69382, 3628}, {33249, 43448, 39143}, {37637, 59546, 6392}, {45508, 45509, 14912}, {46219, 64093, 32867}

X(69443) = X(2)X(2418)∩X(4)X(99)

Barycentrics   2*S^2 - 2*SA^2 - SB*SC : :

X(69443) lies on these lines: {2, 2418}, {3, 14929}, {4, 99}, {5, 32822}, {6, 33191}, {20, 32823}, {30, 63098}, {39, 14069}, {69, 3431}, {76, 3525}, {140, 32830}, {148, 32984}, {183, 631}, {186, 9723}, {193, 35297}, {194, 32970}, {315, 3528}, {316, 11001}, {325, 376}, {385, 33216}, {393, 59555}, {439, 7762}, {538, 62992}, {549, 15589}, {620, 7735}, {1003, 62988}, {1078, 61814}, {1285, 7774}, {1384, 63091}, {1656, 69379}, {1975, 3090}, {1992, 41134}, {2482, 7737}, {2549, 33285}, {2996, 33249}, {3314, 33215}, {3329, 7891}, {3522, 7776}, {3523, 3933}, {3526, 32834}, {3529, 32816}, {3533, 32820}, {3545, 32815}, {3618, 7835}, {3628, 32871}, {3734, 62993}, {3767, 31274}, {3785, 10299}, {3788, 7738}, {3832, 32895}, {3855, 32819}, {5013, 32956}, {5056, 32873}, {5067, 7769}, {5070, 32898}, {5071, 11185}, {5210, 50771}, {5254, 32955}, {5286, 33189}, {5304, 11288}, {5305, 33203}, {5334, 59539}, {5335, 59540}, {5468, 64177}, {5866, 35473}, {5890, 51386}, {5971, 7493}, {5999, 15428}, {6392, 33233}, {6393, 14912}, {6527, 47090}, {6722, 7781}, {6776, 59552}, {7485, 22241}, {7618, 7761}, {7622, 42850}, {7709, 51373}, {7710, 18860}, {7736, 7804}, {7750, 21735}, {7754, 32989}, {7757, 33231}, {7766, 16925}, {7767, 15717}, {7768, 61787}, {7773, 33703}, {7775, 35022}, {7777, 14033}, {7778, 33190}, {7782, 17538}, {7783, 14064}, {7785, 33239}, {7788, 15698}, {7789, 16045}, {7792, 14482}, {7795, 15482}, {7796, 61138}, {7797, 33222}, {7802, 62113}, {7803, 32952}, {7809, 62130}, {7811, 15715}, {7813, 21843}, {7814, 49138}, {7822, 31450}, {7824, 55732}, {7836, 16043}, {7850, 62058}, {7862, 63533}, {7863, 31401}, {7870, 33230}, {7871, 62061}, {7881, 32990}, {7885, 33247}, {7897, 33008}, {7900, 33254}, {7906, 32964}, {7912, 33238}, {7925, 16041}, {7941, 33244}, {7945, 33221}, {7947, 32965}, {7967, 69038}, {8369, 37665}, {8370, 63077}, {8716, 43448}, {9146, 37645}, {9170, 36890}, {9605, 33181}, {9764, 62367}, {9766, 32459}, {10008, 21445}, {10124, 32893}, {10303, 32840}, {10513, 15692}, {11057, 62090}, {11147, 51224}, {11285, 18840}, {11539, 32874}, {14148, 63955}, {14535, 31406}, {14853, 59548}, {14907, 19708}, {14928, 51023}, {14981, 53015}, {15574, 44832}, {15655, 63940}, {15682, 32827}, {15694, 32869}, {15702, 32833}, {15709, 32836}, {15719, 37671}, {16239, 32897}, {16976, 40995}, {16989, 33224}, {17128, 32975}, {17129, 33206}, {18842, 35954}, {19583, 52299}, {20081, 33000}, {20094, 33006}, {22110, 44526}, {22253, 37689}, {25406, 51371}, {26255, 56435}, {26613, 63064}, {27269, 33049}, {30435, 33205}, {32000, 40697}, {32824, 32839}, {32826, 32889}, {32832, 61867}, {32838, 61870}, {32870, 46219}, {32872, 61863}, {32876, 61807}, {32879, 61834}, {32880, 61848}, {32885, 61866}, {32887, 61921}, {32891, 62066}, {32896, 61838}, {32966, 35369}, {32983, 63083}, {33007, 63021}, {33220, 51171}, {33235, 51579}, {33246, 63017}, {33255, 63018}, {33274, 63046}, {35925, 40824}, {36521, 66466}, {37350, 53141}, {38282, 44146}, {38748, 63722}, {39099, 63428}, {39785, 63029}, {40819, 47389}, {42998, 59542}, {42999, 59541}, {43453, 60657}, {46951, 61859}, {48913, 62019}, {51396, 54170}, {52094, 56603}, {52283, 59211}, {59545, 69208}, {60143, 60220}, {60176, 62932}, {60781, 69385}, {61128, 68654}, {61899, 69387}, {62310, 62960}, {63107, 64019}, {67531, 68152}, {68663, 69135}

X(69443) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6390, 32817}, {2, 32817, 52713}, {2, 69380, 69386}, {3, 32831, 32818}, {20, 69158, 32823}, {99, 1007, 4}, {99, 7763, 1007}, {140, 32830, 69384}, {193, 35297, 46453}, {620, 34511, 7735}, {631, 3926, 69377}, {1007, 6337, 99}, {1975, 32829, 3090}, {1975, 37647, 69382}, {2549, 37690, 33285}, {3523, 32841, 3933}, {3526, 32834, 52718}, {3788, 7738, 32951}, {5013, 53033, 32956}, {6337, 7763, 4}, {7736, 69206, 14039}, {7769, 69378, 5067}, {7774, 32985, 1285}, {7782, 32006, 17538}, {7789, 31400, 16045}, {8716, 44377, 43448}, {10303, 32840, 69381}, {11179, 51397, 69}, {11185, 34803, 5071}, {14482, 33197, 7792}, {32817, 69386, 69380}, {32824, 32839, 59635}, {32829, 69382, 37647}, {32839, 59635, 61886}, {32871, 69383, 3628}, {37647, 69382, 3090}, {69380, 69386, 52713}

X(69444) = X(2)X(9607)∩X(3)X(69)

Barycentrics   b^2*c^2 - 2*S^2 + 2*SA^2 + SB*SC : :

X(69444) lies on these lines: {2, 9607}, {3, 69}, {4, 7799}, {5, 32824}, {17, 37178}, {18, 37177}, {20, 32821}, {30, 32825}, {32, 62996}, {39, 63119}, {76, 3525}, {99, 3529}, {140, 32836}, {183, 32840}, {193, 22331}, {315, 17538}, {316, 11541}, {325, 3146}, {376, 7796}, {397, 63105}, {398, 63106}, {439, 524}, {538, 32970}, {546, 32815}, {549, 32896}, {574, 51587}, {631, 32833}, {632, 32828}, {1007, 1975}, {1078, 61807}, {1285, 7905}, {1992, 32973}, {2482, 14023}, {2996, 39143}, {3053, 11008}, {3090, 7763}, {3292, 4176}, {3470, 36890}, {3522, 7788}, {3526, 46951}, {3528, 7768}, {3533, 62362}, {3544, 11185}, {3618, 7789}, {3619, 5013}, {3620, 15815}, {3627, 32816}, {3628, 32829}, {3767, 14148}, {5007, 62995}, {5023, 20080}, {5076, 32827}, {5079, 69382}, {5159, 6340}, {5309, 33222}, {5319, 33224}, {6392, 63104}, {6461, 20806}, {7492, 40123}, {7618, 7854}, {7735, 7891}, {7736, 68525}, {7738, 7836}, {7750, 62097}, {7751, 33216}, {7752, 32822}, {7757, 14069}, {7758, 32985}, {7759, 33239}, {7760, 33191}, {7764, 14033}, {7769, 52713}, {7772, 7863}, {7773, 50688}, {7774, 68517}, {7776, 15704}, {7779, 68518}, {7781, 14064}, {7782, 62092}, {7794, 33215}, {7795, 53096}, {7801, 16043}, {7802, 62133}, {7809, 33703}, {7811, 21735}, {7818, 33247}, {7819, 63109}, {7821, 33238}, {7827, 32952}, {7840, 33244}, {7846, 14482}, {7856, 33197}, {7860, 11001}, {7866, 51123}, {7870, 9741}, {7871, 62146}, {7880, 33221}, {7888, 16041}, {7893, 68523}, {7907, 23055}, {7909, 33190}, {7916, 35022}, {7925, 63533}, {7946, 33254}, {8361, 51122}, {8362, 11165}, {8363, 31859}, {8369, 63022}, {8591, 33279}, {8716, 32974}, {8781, 38734}, {9766, 32981}, {9770, 14035}, {10303, 32830}, {10513, 62078}, {10541, 59552}, {11055, 33231}, {11057, 62113}, {11184, 32991}, {11291, 35822}, {11292, 35823}, {12103, 64018}, {12108, 32875}, {12322, 51401}, {12323, 51395}, {12811, 32891}, {13571, 33255}, {14037, 63024}, {14568, 32959}, {14614, 33205}, {14869, 69381}, {14907, 62084}, {14929, 62087}, {15022, 69379}, {15077, 43705}, {15589, 32879}, {15694, 32892}, {15717, 37671}, {16051, 19583}, {16924, 63025}, {16925, 63034}, {17128, 33261}, {17130, 32975}, {20081, 62992}, {21356, 32990}, {22110, 32980}, {23235, 46236}, {27088, 63936}, {28620, 37176}, {28628, 42697}, {28728, 65171}, {30552, 65711}, {32805, 43880}, {32806, 43879}, {32810, 41945}, {32811, 41946}, {32819, 50689}, {32823, 62028}, {32826, 61984}, {32832, 61870}, {32834, 61863}, {32835, 46936}, {32838, 55858}, {32839, 55857}, {32867, 55862}, {32868, 61858}, {32869, 55864}, {32874, 61856}, {32877, 61840}, {32878, 61852}, {32881, 69404}, {32885, 46219}, {32887, 61892}, {32889, 61903}, {32976, 63924}, {32977, 63955}, {32988, 34505}, {32989, 63933}, {33012, 42850}, {33201, 41624}, {33203, 63107}, {33225, 63006}, {33227, 63950}, {33234, 53142}, {33259, 63029}, {34828, 56339}, {35287, 50992}, {35927, 63932}, {36212, 46831}, {36836, 59539}, {36843, 59540}, {37172, 41101}, {37173, 41100}, {37460, 52149}, {37647, 69383}, {37668, 50693}, {37688, 61848}, {38282, 59559}, {38664, 62348}, {40824, 54873}, {45187, 51386}, {53097, 59548}, {55803, 60639}, {60219, 62427}, {61814, 69377}, {63041, 68522}, {63925, 69207}, {63940, 68528}, {64809, 69406}, {64950, 69094}

X(69444) = isotomic conjugate of X(36611)
X(69444) = isotomic conjugate of the polar conjugate of X(20080)
X(69444) = X(6340)-Ceva conjugate of X(69)
X(69444) = X(i)-isoconjugate of X(j) for these (i,j): {19, 36616}, {31, 36611}, {1096, 38263}, {1973, 38259}
X(69444) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 36611}, {6, 36616}, {193, 6353}, {6337, 38259}, {6503, 38263}, {59549, 63547}
X(69444) = barycentric product X(i)*X(j) for these {i,j}: {69, 20080}, {304, 16570}, {305, 5023}, {3926, 38282}, {4561, 59550}, {4563, 59549}, {6340, 51579}, {57799, 59559}
X(69444) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 36611}, {3, 36616}, {69, 38259}, {394, 38263}, {4558, 58097}, {5023, 25}, {16570, 19}, {20080, 4}, {38282, 393}, {51579, 6353}, {59549, 2501}, {59550, 7649}, {59559, 232}
X(69444) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {99, 32818, 32006}, {487, 488, 39899}, {1975, 32831, 1007}, {2996, 44377, 39143}, {3926, 6337, 69}, {3926, 6390, 6337}, {3964, 50572, 69}, {7763, 32817, 69378}, {7763, 69378, 34803}, {7863, 34511, 14001}, {7946, 52695, 33254}, {20080, 51579, 5023}, {32821, 59634, 20}, {32824, 32837, 5}, {32829, 69380, 69385}

X(69445) = X(2)X(55790)∩X(4)X(325)

Barycentrics   2*b^2*c^2 - 2*S^2 + 2*SA^2 + SB*SC : :

X(69445) lies on these lines: {2, 55790}, {3, 32840}, {4, 325}, {5, 32841}, {20, 32879}, {39, 55774}, {69, 3528}, {76, 3525}, {99, 17538}, {183, 61814}, {194, 14069}, {305, 38282}, {315, 11001}, {376, 3933}, {538, 33231}, {550, 10513}, {574, 55732}, {631, 6390}, {632, 32872}, {1007, 3544}, {1078, 3524}, {1092, 34403}, {1285, 7758}, {2548, 39785}, {3090, 32831}, {3520, 3964}, {3529, 37668}, {3533, 32834}, {3545, 69158}, {3785, 19708}, {3855, 63098}, {4175, 11206}, {5013, 18840}, {5067, 7763}, {5070, 32873}, {5071, 7799}, {5286, 32952}, {5305, 33197}, {5485, 13881}, {6392, 33189}, {7486, 32881}, {7512, 22241}, {7735, 7863}, {7738, 7801}, {7750, 62113}, {7752, 41106}, {7754, 33191}, {7767, 21735}, {7776, 33703}, {7779, 33239}, {7781, 33232}, {7787, 14039}, {7788, 62130}, {7795, 14148}, {7796, 49138}, {7805, 69206}, {7808, 34511}, {7809, 62019}, {7811, 62090}, {7819, 14482}, {7836, 32951}, {7839, 14001}, {7871, 62029}, {7881, 33190}, {7888, 63533}, {7891, 8859}, {7897, 33238}, {7906, 14033}, {7939, 33247}, {7947, 16041}, {8360, 11148}, {8362, 55741}, {10299, 15589}, {10303, 32880}, {11185, 61945}, {11541, 32006}, {14907, 62096}, {14929, 50693}, {14930, 19697}, {15048, 33194}, {15428, 30270}, {15702, 32836}, {15709, 32869}, {15715, 37671}, {17129, 33216}, {17130, 62993}, {20080, 33235}, {20081, 32970}, {22253, 33181}, {23040, 68654}, {30471, 43493}, {30472, 43494}, {31467, 59780}, {31859, 32956}, {32828, 61867}, {32829, 60781}, {32832, 61873}, {32835, 61886}, {32837, 59635}, {32874, 61859}, {32875, 61138}, {32876, 69382}, {32882, 55864}, {32893, 61866}, {32894, 61856}, {32895, 46935}, {32953, 53033}, {32955, 47286}, {32982, 47287}, {32985, 44367}, {33196, 51122}, {33242, 63005}, {34229, 61836}, {34254, 52299}, {39142, 62992}, {46453, 59545}, {46951, 52718}, {60128, 60143}, {62146, 64018}, {63091, 68527}, {68663, 69093}

X(69445) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {325, 1975, 32826}, {325, 32822, 4}, {325, 32824, 32822}, {1975, 3926, 32818}, {1975, 32818, 4}, {1975, 32826, 32822}, {3926, 32815, 32821}, {3926, 32817, 4}, {3926, 32820, 32817}, {3926, 32824, 325}, {6337, 32833, 69377}, {6337, 69377, 3524}, {6390, 32830, 631}, {7763, 52713, 5067}, {7801, 9741, 33230}, {32815, 32821, 32823}, {32815, 32823, 4}, {32817, 32818, 1975}, {32817, 32822, 32824}, {32824, 32826, 1975}, {32829, 69386, 60781}, {32831, 69380, 3090}, {32835, 64093, 61886}, {69158, 69379, 3545}

X(69446) = X(2)X(55790)∩X(76)X(140)

Barycentrics   2*b^2*c^2 - S^2 + 2*SA^2 + SB*SC : :

X(69446) = X[315] + 3 X[1975], X[315] - 3 X[3933], 5 X[315] - 9 X[7788], X[315] - 9 X[32833], 5 X[1975] + 3 X[7788], X[1975] + 3 X[32833], 5 X[3933] - 3 X[7788], X[3933] - 3 X[32833], X[7788] - 5 X[32833], 2 X[32] - 3 X[66393], 3 X[5305] - 4 X[6680], 2 X[5305] - 3 X[8368], 2 X[6680] - 3 X[7789], 8 X[6680] - 9 X[8368], 4 X[7789] - 3 X[8368], 4 X[626] - 3 X[66394], X[5254] - 3 X[7801], 3 X[5254] - 5 X[7867], 2 X[5254] - 3 X[8360], 9 X[7801] - 5 X[7867], 10 X[7867] - 9 X[8360], X[7754] - 3 X[8369], 5 X[7754] - 9 X[63065], 5 X[8369] - 3 X[63065], 5 X[7851] - 6 X[33213], 5 X[7881] - 3 X[33184], X[20065] - 3 X[66391]

X(69446) lies on these lines: {2, 55790}, {3, 15589}, {4, 32840}, {5, 1007}, {6, 19697}, {20, 14929}, {26, 22241}, {30, 315}, {32, 66393}, {69, 550}, {76, 140}, {99, 548}, {141, 7781}, {183, 3530}, {193, 68527}, {194, 7819}, {230, 7863}, {274, 50205}, {305, 6677}, {316, 62026}, {325, 546}, {339, 16196}, {381, 32818}, {382, 32822}, {397, 69120}, {398, 69121}, {468, 9464}, {524, 7816}, {538, 5305}, {543, 7895}, {547, 7799}, {549, 6337}, {626, 66394}, {632, 32828}, {1078, 12100}, {1235, 64474}, {1238, 11819}, {1506, 39785}, {1565, 7283}, {1656, 32831}, {2396, 57588}, {2548, 66412}, {2549, 66347}, {2896, 8354}, {2996, 11318}, {3053, 63926}, {3090, 32841}, {3091, 32879}, {3314, 8357}, {3523, 32880}, {3526, 32834}, {3529, 10513}, {3533, 32872}, {3552, 3793}, {3564, 35387}, {3589, 32450}, {3627, 7776}, {3628, 7763}, {3629, 69172}, {3630, 63935}, {3631, 7830}, {3695, 5088}, {3760, 15325}, {3785, 8703}, {3788, 43291}, {3815, 17130}, {3843, 32823}, {3845, 32816}, {3850, 11185}, {3851, 63098}, {3853, 7796}, {3858, 32825}, {3859, 7814}, {3861, 7773}, {3934, 14148}, {3964, 7526}, {4352, 17698}, {5013, 51123}, {5054, 32869}, {5055, 69383}, {5066, 7752}, {5070, 32835}, {5254, 7801}, {5286, 33185}, {5304, 33242}, {6248, 40927}, {6381, 47742}, {6392, 32954}, {6393, 61545}, {6655, 47287}, {6661, 7839}, {6675, 34284}, {6676, 8024}, {7738, 51122}, {7739, 66340}, {7745, 7813}, {7747, 50771}, {7749, 14711}, {7750, 12103}, {7751, 59545}, {7754, 8369}, {7758, 18907}, {7762, 68177}, {7768, 62144}, {7769, 48154}, {7771, 61790}, {7779, 19687}, {7780, 32459}, {7782, 34200}, {7783, 8359}, {7785, 66409}, {7787, 66318}, {7793, 27088}, {7794, 63548}, {7795, 8364}, {7800, 8358}, {7802, 62151}, {7803, 66344}, {7807, 20081}, {7809, 12101}, {7811, 15690}, {7821, 53419}, {7822, 9607}, {7836, 8361}, {7837, 66321}, {7846, 11055}, {7850, 62156}, {7851, 33213}, {7855, 63940}, {7864, 66326}, {7868, 66346}, {7871, 14893}, {7881, 33184}, {7882, 63941}, {7888, 63534}, {7892, 20105}, {7893, 33250}, {7897, 33229}, {7900, 66408}, {7903, 53418}, {7906, 8370}, {7912, 37350}, {7917, 62038}, {7929, 8353}, {7938, 66335}, {7939, 19695}, {7947, 33228}, {8355, 34505}, {8362, 31859}, {8367, 31406}, {8781, 38229}, {9741, 18840}, {10303, 32882}, {11057, 62139}, {11148, 33230}, {11257, 55167}, {11539, 46951}, {11542, 69157}, {11543, 69165}, {11591, 51386}, {12215, 44224}, {12812, 69387}, {13108, 56370}, {13571, 53489}, {14001, 22253}, {14039, 43136}, {14869, 34229}, {14907, 44245}, {14994, 59548}, {15171, 69094}, {15480, 35007}, {15687, 32826}, {15694, 32874}, {15699, 32837}, {15712, 32877}, {15713, 32892}, {16238, 28706}, {16239, 32832}, {16705, 50318}, {17128, 66415}, {17129, 35297}, {18135, 52264}, {18358, 51371}, {18990, 69093}, {19702, 63020}, {20023, 52261}, {20065, 66391}, {20080, 33239}, {20088, 66319}, {21258, 62426}, {21309, 33201}, {21358, 63654}, {21841, 44146}, {22110, 39565}, {22165, 34504}, {26233, 66370}, {31833, 52347}, {32000, 55571}, {32006, 62036}, {32134, 61624}, {32515, 35436}, {32827, 61988}, {32829, 55856}, {32838, 55859}, {32839, 55861}, {32867, 61876}, {32868, 61853}, {32870, 55858}, {32871, 55860}, {32873, 60781}, {32876, 61907}, {32878, 61837}, {32881, 46936}, {32883, 41992}, {32885, 61869}, {32888, 61852}, {32890, 44682}, {32893, 61864}, {32894, 55864}, {32897, 55866}, {32898, 61878}, {33186, 53033}, {33235, 63046}, {34254, 64852}, {34380, 50685}, {34803, 61900}, {35438, 49111}, {37647, 61894}, {37664, 50238}, {37911, 66767}, {41987, 48913}, {42147, 69107}, {42148, 69106}, {42788, 51373}, {42912, 59541}, {42913, 59542}, {43459, 61784}, {44377, 63924}, {45962, 50241}, {50567, 51524}, {50774, 63925}, {51884, 63441}, {54412, 64471}, {61510, 69038}, {61560, 62348}, {62041, 67536}, {62155, 64018}, {63928, 69171}, {63933, 69206}

X(69446) = midpoint of X(1975) and X(3933)
X(69446) = reflection of X(i) in X(j) for these {i,j}: {5305, 7789}, {8360, 7801}
X(69446) = barycentric product X(99)*X(55188)
X(69446) = barycentric quotient X(55188)/X(523)
X(69446) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 6390, 140}, {76, 32820, 6390}, {99, 7767, 548}, {194, 7819, 63633}, {1975, 32833, 3933}, {3788, 63923, 43291}, {3926, 69378, 69158}, {3926, 69380, 5}, {3934, 14148, 59546}, {5305, 7789, 8368}, {6337, 32836, 69381}, {6337, 69381, 549}, {7763, 64093, 3628}, {7776, 32815, 3627}, {7795, 15048, 8364}, {7836, 47286, 8361}, {7939, 20094, 19695}, {31406, 59780, 69139}, {31406, 69139, 8367}, {32817, 32830, 3}, {32818, 69379, 381}, {32822, 37668, 382}, {32831, 52713, 1656}, {32835, 69386, 5070}, {34511, 59780, 8367}, {34511, 69139, 31406}, {69158, 69378, 5}, {69158, 69380, 69378}

X(69447) = X(2)X(55790)∩X(76)X(631)

Barycentrics   2*b^2*c^2 + 2*SA^2 + SB*SC : :

X(69447) lies on these lines: {2, 55790}, {4, 3933}, {5, 32840}, {20, 32880}, {69, 3529}, {76, 631}, {99, 21735}, {183, 10299}, {194, 14482}, {315, 15682}, {325, 3855}, {376, 1975}, {382, 10513}, {538, 69209}, {1007, 61921}, {1078, 15698}, {1656, 32841}, {2996, 7881}, {3090, 3926}, {3518, 22241}, {3523, 32882}, {3524, 32869}, {3525, 6390}, {3526, 32872}, {3528, 15589}, {3533, 32820}, {3544, 63098}, {3545, 7752}, {3767, 33195}, {3964, 35500}, {5013, 9741}, {5056, 32879}, {5059, 14929}, {5067, 32831}, {5071, 69158}, {5254, 33196}, {5485, 7801}, {6292, 7738}, {6353, 8024}, {6392, 14069}, {7736, 17130}, {7750, 62147}, {7751, 46453}, {7754, 14039}, {7763, 61886}, {7767, 17538}, {7768, 67536}, {7782, 15710}, {7788, 32826}, {7789, 33197}, {7795, 7902}, {7799, 61895}, {7809, 61987}, {7811, 62135}, {7836, 32955}, {7847, 21356}, {7863, 62992}, {7864, 32956}, {7871, 11185}, {7906, 32983}, {7938, 33190}, {7947, 32984}, {8359, 60143}, {9464, 40132}, {9605, 59780}, {10302, 55737}, {10303, 32894}, {14001, 20081}, {14033, 20088}, {14711, 69207}, {14907, 62117}, {15702, 32874}, {15709, 46951}, {15719, 32892}, {16845, 34284}, {16898, 20105}, {17128, 63028}, {17129, 32985}, {19687, 20080}, {19697, 63005}, {20094, 33247}, {22253, 33198}, {31859, 32960}, {32006, 62021}, {32815, 32877}, {32816, 41099}, {32819, 32890}, {32821, 32875}, {32829, 61881}, {32832, 61870}, {32835, 60781}, {32837, 61889}, {32868, 37688}, {32870, 61873}, {32873, 55856}, {32878, 61807}, {32881, 46935}, {32893, 61861}, {32896, 61926}, {32951, 47286}, {33023, 47287}, {33226, 63044}, {33230, 52229}, {33236, 37689}, {33239, 63046}, {33242, 63097}, {37512, 42850}, {37637, 39142}, {37690, 63924}, {43542, 69157}, {43543, 69165}, {52718, 61859}, {53033, 63923}, {62171, 64018}, {63042, 68527}

X(69447) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 32822, 3529}, {76, 6337, 69384}, {76, 32817, 631}, {194, 16045, 14482}, {1975, 32836, 69377}, {1975, 69377, 376}, {2996, 7881, 33285}, {3926, 52713, 3090}, {3933, 69379, 4}, {3933, 69380, 69379}, {6337, 69384, 631}, {6390, 32834, 3525}, {7763, 69386, 61886}, {11185, 32823, 61964}, {32817, 69384, 6337}, {32818, 69378, 3545}, {32830, 69379, 3933}, {32830, 69380, 4}, {32831, 64093, 5067}, {32833, 69378, 32818}, {32875, 69382, 32821}, {69158, 69383, 5071}

X(69448) = X(2)X(55790)∩X(3)X(76)

Barycentrics   2*b^2*c^2 + S^2 + 2*SA^2 + SB*SC : :

X(69448) = 3 X[32869] - X[69377], 3 X[32869] + X[69379], 4 X[7808] - 3 X[9605], 2 X[7808] - 3 X[69139]

X(69448) lies on these lines: {2, 55790}, {3, 76}, {4, 10513}, {5, 32818}, {6, 17130}, {20, 32882}, {30, 32869}, {32, 14711}, {69, 382}, {140, 32817}, {148, 7879}, {274, 16853}, {315, 3830}, {316, 62008}, {325, 3851}, {350, 7373}, {381, 3933}, {384, 21309}, {385, 68527}, {538, 7808}, {546, 37668}, {549, 32874}, {550, 15589}, {599, 7748}, {626, 34505}, {999, 3760}, {1003, 17129}, {1007, 5079}, {1230, 37269}, {1235, 1597}, {1270, 23275}, {1271, 23269}, {1351, 18502}, {1384, 7751}, {1598, 44146}, {1656, 3926}, {1657, 7767}, {1909, 6767}, {2896, 5077}, {2996, 33184}, {3053, 17131}, {3090, 32840}, {3091, 32880}, {3146, 14929}, {3295, 3761}, {3523, 32894}, {3525, 32872}, {3526, 6390}, {3534, 3785}, {3620, 8357}, {3628, 32831}, {3734, 7805}, {3763, 7765}, {3793, 32981}, {3843, 7776}, {3850, 32823}, {3934, 5024}, {5013, 9466}, {5017, 37004}, {5020, 8024}, {5054, 6337}, {5055, 32833}, {5067, 32841}, {5070, 7763}, {5072, 32877}, {5073, 32819}, {5076, 32006}, {5304, 19697}, {5305, 33237}, {5339, 69107}, {5340, 69106}, {5485, 8360}, {5971, 30734}, {6381, 9709}, {6392, 7819}, {7486, 32879}, {7506, 22241}, {7667, 41916}, {7697, 10983}, {7735, 33242}, {7737, 63936}, {7738, 52229}, {7747, 40341}, {7750, 17800}, {7752, 19709}, {7754, 7787}, {7758, 15484}, {7768, 62023}, {7769, 55860}, {7770, 7839}, {7772, 14535}, {7773, 61970}, {7778, 63924}, {7781, 15271}, {7788, 14269}, {7789, 63955}, {7794, 44518}, {7795, 63923}, {7796, 61953}, {7799, 15703}, {7801, 13881}, {7802, 49139}, {7809, 61974}, {7810, 44519}, {7811, 15685}, {7815, 8716}, {7816, 8667}, {7846, 11054}, {7854, 44526}, {7855, 65630}, {7863, 37637}, {7866, 47286}, {7869, 32457}, {7871, 61948}, {7881, 11318}, {7895, 18546}, {7896, 63922}, {7900, 11317}, {7906, 44543}, {7929, 66388}, {8556, 37512}, {9464, 11284}, {9654, 69093}, {9669, 69094}, {9708, 20888}, {9909, 26233}, {11108, 34284}, {11159, 20065}, {11539, 32893}, {12215, 55705}, {13488, 32000}, {14034, 50248}, {14907, 62131}, {14994, 33878}, {15655, 69171}, {15681, 37671}, {15701, 59634}, {15720, 32824}, {15723, 32885}, {16045, 63633}, {16239, 32870}, {16408, 18135}, {16409, 18152}, {16412, 31060}, {16419, 39998}, {16863, 18140}, {16866, 16992}, {17571, 37670}, {18535, 54412}, {18906, 44456}, {19687, 63046}, {20105, 68522}, {21264, 31468}, {21358, 63651}, {22332, 31239}, {26166, 64585}, {26541, 37244}, {26563, 50044}, {31276, 31859}, {31467, 34511}, {32458, 38732}, {32820, 32832}, {32821, 61919}, {32825, 32890}, {32827, 61975}, {32829, 55857}, {32835, 55856}, {32837, 61887}, {32838, 55858}, {32839, 61878}, {32867, 61875}, {32871, 55861}, {32873, 61881}, {32875, 61911}, {32886, 61850}, {32888, 61811}, {32896, 61908}, {32897, 55859}, {32898, 61877}, {32965, 47287}, {33234, 63044}, {37491, 42554}, {37647, 61892}, {37688, 55863}, {42153, 69121}, {42156, 69120}, {43291, 53033}, {44149, 68346}, {49136, 64018}, {49137, 67536}, {52854, 53097}, {53127, 61903}, {55624, 60702}, {63925, 69172}, {69116, 69188}, {69117, 69182}

X(69448) = midpoint of X(69377) and X(69379)
X(69448) = reflection of X(9605) in X(69139)
X(69448) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 1975, 69381}, {76, 69380, 3}, {183, 1975, 7782}, {1975, 69381, 3}, {3734, 63933, 30435}, {3926, 64093, 1656}, {3933, 69378, 381}, {6390, 32828, 3526}, {7754, 11286, 43136}, {7754, 17128, 11286}, {7767, 32815, 1657}, {7770, 20081, 22253}, {7776, 11185, 3843}, {15589, 32822, 550}, {32817, 32834, 140}, {32818, 52713, 69383}, {32818, 69383, 5}, {32824, 32868, 34229}, {32830, 52713, 5}, {32830, 69383, 32818}, {32831, 69386, 3628}, {32833, 59635, 69158}, {32836, 69378, 3933}, {32869, 69379, 69377}, {59635, 69158, 5055}, {69380, 69381, 1975}

X(69449) = X(2)X(55790)∩X(4)X(69)

Barycentrics   2*b^2*c^2 + 2*S^2 + 2*SA^2 + SB*SC : :

X(69449) lies on these lines: {2, 55790}, {4, 69}, {20, 32894}, {83, 60627}, {99, 61138}, {140, 32872}, {141, 33232}, {183, 3528}, {194, 32957}, {230, 33236}, {325, 3544}, {339, 10996}, {376, 32874}, {546, 10513}, {597, 69139}, {631, 32834}, {1007, 69405}, {1056, 3760}, {1058, 3761}, {1078, 19708}, {1285, 7751}, {1656, 32840}, {1975, 3524}, {2548, 14711}, {2549, 55732}, {2996, 33190}, {3090, 32830}, {3091, 32882}, {3314, 33292}, {3525, 32817}, {3529, 15589}, {3533, 6390}, {3537, 41009}, {3545, 3933}, {3628, 32841}, {3673, 52709}, {3767, 32952}, {3785, 11001}, {3855, 37668}, {3926, 5067}, {5032, 7754}, {5056, 32880}, {5071, 32818}, {5254, 5485}, {5286, 47355}, {6337, 15702}, {6392, 16045}, {6680, 63955}, {7615, 7895}, {7735, 17130}, {7738, 9466}, {7739, 55774}, {7750, 11541}, {7752, 61932}, {7763, 60781}, {7767, 33703}, {7770, 63123}, {7776, 61964}, {7782, 15715}, {7788, 61967}, {7789, 33231}, {7794, 63533}, {7795, 32953}, {7799, 61888}, {7809, 61961}, {7811, 62049}, {7816, 63029}, {7836, 32958}, {7863, 39142}, {7909, 39143}, {7912, 51238}, {14033, 17129}, {14039, 17128}, {14069, 16984}, {14843, 54124}, {14907, 62146}, {14929, 17578}, {15534, 69208}, {15709, 32893}, {17538, 32815}, {17559, 34284}, {17582, 18135}, {19697, 63097}, {20081, 32968}, {31276, 32960}, {32455, 63933}, {32816, 32892}, {32819, 49138}, {32820, 32838}, {32821, 32877}, {32823, 32878}, {32824, 32886}, {32826, 37671}, {32831, 61886}, {32832, 61867}, {32833, 61899}, {32835, 61881}, {32870, 61870}, {32873, 55857}, {32879, 46936}, {32885, 61865}, {32956, 47286}, {32971, 63061}, {33197, 59780}, {33238, 63044}, {33750, 39646}, {34229, 61814}, {34505, 50991}, {34573, 63923}, {41916, 44442}, {43681, 60183}, {51581, 55816}, {54637, 60639}, {57811, 60114}, {59634, 61833}, {60250, 60629}, {61921, 63098}, {69387, 69406}

X(69449) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 52713, 4}, {76, 69378, 69377}, {183, 32822, 3528}, {1975, 46951, 69384}, {1975, 69384, 3524}, {3926, 69386, 5067}, {3933, 69383, 3545}, {5254, 18840, 33230}, {5485, 18840, 5254}, {32817, 32828, 3525}, {32818, 59635, 5071}, {32823, 69382, 61945}, {32824, 32886, 37688}, {32830, 64093, 3090}, {32834, 69380, 631}, {32836, 59635, 32818}, {32869, 69383, 3933}, {32874, 69379, 69381}, {52713, 69377, 69378}, {69377, 69378, 4}, {69379, 69381, 376}

X(69450) = X(2)X(99)∩X(3)X(315)

Barycentrics   b^2*c^2 + 2*S^2 - 2*SA^2 - 2*SB*SC : :

X(69450) lies on these lines: {2, 99}, {3, 315}, {4, 7769}, {6, 35297}, {20, 7694}, {32, 32964}, {39, 16925}, {69, 3431}, {76, 631}, {83, 31400}, {95, 40697}, {114, 9734}, {140, 1975}, {147, 8350}, {182, 38748}, {183, 549}, {186, 317}, {187, 7774}, {193, 5033}, {194, 33259}, {230, 31859}, {274, 6910}, {302, 10654}, {303, 10653}, {316, 376}, {381, 37647}, {384, 31401}, {385, 21843}, {439, 69208}, {491, 6396}, {492, 6200}, {538, 17008}, {550, 7773}, {568, 51440}, {598, 11147}, {621, 14144}, {622, 14145}, {625, 33017}, {626, 15515}, {637, 43144}, {638, 43141}, {754, 8588}, {877, 34349}, {1003, 3815}, {1078, 3523}, {1236, 62698}, {1352, 5152}, {1384, 41624}, {1506, 14035}, {1656, 32819}, {1992, 2030}, {2548, 3552}, {2896, 33022}, {3055, 44543}, {3091, 32839}, {3096, 32990}, {3098, 51396}, {3111, 15631}, {3147, 54412}, {3314, 33273}, {3329, 33246}, {3363, 11164}, {3448, 62336}, {3522, 7802}, {3525, 69378}, {3526, 59635}, {3528, 7814}, {3530, 3933}, {3533, 32822}, {3545, 67536}, {3576, 69038}, {3589, 33220}, {3618, 33191}, {3763, 66417}, {3767, 7783}, {3785, 7796}, {3788, 7791}, {3832, 32871}, {3934, 33001}, {3972, 7736}, {4590, 5486}, {5013, 7803}, {5023, 7762}, {5024, 7792}, {5050, 51374}, {5054, 37688}, {5056, 15031}, {5068, 32898}, {5077, 41133}, {5085, 6393}, {5204, 69135}, {5206, 7764}, {5210, 9766}, {5217, 69254}, {5218, 64133}, {5237, 69157}, {5238, 69165}, {5254, 33233}, {5286, 7857}, {5309, 58448}, {5351, 69137}, {5352, 69145}, {5395, 31407}, {5475, 32456}, {5485, 23053}, {5569, 39785}, {5866, 35921}, {5939, 8724}, {5976, 11171}, {5989, 61561}, {6292, 33258}, {6298, 43451}, {6299, 43452}, {6353, 58782}, {6656, 15815}, {6671, 43454}, {6672, 43455}, {6676, 59766}, {6680, 53096}, {6683, 16898}, {6770, 11132}, {6773, 11133}, {6776, 34473}, {6781, 7775}, {6795, 67603}, {6921, 18140}, {7485, 34254}, {7486, 32884}, {7493, 11059}, {7494, 57518}, {7495, 62299}, {7603, 33016}, {7735, 7757}, {7737, 7777}, {7738, 7828}, {7739, 7806}, {7745, 33235}, {7746, 32457}, {7747, 33244}, {7748, 32961}, {7749, 7781}, {7753, 33266}, {7754, 50774}, {7756, 7862}, {7758, 7793}, {7759, 15513}, {7761, 8589}, {7765, 33262}, {7767, 15712}, {7768, 10299}, {7770, 15491}, {7778, 8356}, {7785, 33014}, {7786, 14001}, {7788, 12100}, {7789, 11285}, {7795, 7824}, {7798, 63048}, {7800, 7836}, {7801, 16990}, {7804, 33255}, {7808, 14037}, {7809, 10304}, {7810, 7908}, {7811, 15692}, {7812, 35287}, {7813, 63046}, {7815, 7863}, {7816, 16924}, {7823, 33276}, {7825, 32997}, {7830, 7888}, {7831, 7870}, {7832, 16043}, {7833, 7925}, {7834, 31652}, {7840, 8182}, {7841, 44377}, {7842, 33253}, {7845, 47101}, {7846, 33181}, {7847, 7940}, {7850, 15698}, {7859, 14069}, {7860, 21735}, {7861, 33248}, {7864, 33245}, {7868, 8359}, {7871, 61138}, {7872, 33283}, {7885, 33275}, {7887, 63548}, {7892, 16987}, {7899, 32974}, {7900, 68514}, {7904, 7947}, {7906, 14023}, {7910, 33226}, {7911, 33023}, {7912, 33260}, {7917, 32825}, {7918, 32951}, {7926, 9770}, {7930, 32956}, {7931, 66414}, {7934, 32986}, {7942, 33189}, {7944, 33202}, {7945, 33021}, {8176, 52942}, {8352, 66616}, {8353, 44541}, {8369, 11174}, {8370, 31489}, {8592, 49788}, {8598, 11184}, {8716, 37637}, {8860, 52229}, {9675, 62987}, {9698, 69172}, {9730, 51439}, {9735, 51387}, {9736, 51388}, {9741, 11054}, {9753, 37459}, {9771, 11317}, {9855, 66466}, {9996, 35705}, {10303, 32828}, {10411, 11003}, {11001, 48913}, {11057, 19708}, {11163, 12040}, {11165, 22253}, {11178, 14928}, {11180, 45018}, {11361, 17005}, {12117, 57634}, {12150, 37665}, {12833, 67630}, {13219, 35493}, {14033, 62993}, {14039, 60855}, {14041, 43619}, {14148, 17131}, {14558, 61128}, {14568, 62992}, {14712, 63021}, {14929, 17504}, {15561, 37242}, {15693, 37671}, {15702, 52713}, {15708, 32836}, {15709, 69386}, {15720, 32820}, {15721, 46951}, {16371, 37664}, {16589, 33054}, {16772, 59542}, {16773, 59541}, {16836, 51386}, {16976, 41005}, {16992, 37298}, {17004, 63955}, {17128, 33015}, {17130, 33188}, {17508, 51371}, {18122, 51778}, {18800, 64942}, {18860, 37182}, {20477, 47090}, {21163, 51373}, {21166, 43461}, {21395, 44128}, {21445, 63722}, {22110, 35955}, {23267, 32806}, {23273, 32805}, {25303, 31452}, {26276, 56435}, {26686, 31448}, {27162, 56781}, {31404, 32981}, {31450, 33225}, {31467, 68527}, {31958, 50640}, {32824, 32834}, {32830, 61820}, {32838, 55864}, {32840, 61816}, {32841, 61804}, {32867, 61856}, {32869, 61825}, {32870, 61848}, {32873, 50693}, {32874, 61830}, {32883, 61863}, {32885, 61844}, {32887, 62097}, {32889, 58188}, {32895, 62060}, {32963, 69141}, {32971, 51579}, {32976, 63533}, {32996, 65633}, {32998, 39565}, {33006, 34504}, {33018, 45017}, {33193, 62203}, {33204, 63924}, {33205, 55085}, {33228, 44526}, {33229, 44519}, {33249, 44518}, {33250, 65630}, {33265, 43618}, {33280, 39590}, {33813, 37348}, {34284, 37291}, {34505, 47287}, {35520, 44134}, {35920, 59229}, {35922, 40877}, {35925, 46236}, {35927, 63077}, {35954, 42849}, {37450, 52771}, {37534, 55416}, {37804, 46336}, {37809, 63028}, {38437, 41008}, {39099, 54173}, {39113, 44837}, {39601, 63957}, {40280, 51383}, {40693, 62600}, {40694, 62601}, {40826, 44658}, {42490, 69180}, {42491, 69186}, {43120, 45508}, {43121, 45509}, {43527, 55774}, {45421, 66429}, {46127, 66074}, {47061, 55164}, {50659, 59695}, {51484, 63106}, {51485, 63105}, {52451, 54086}, {53418, 66387}, {54033, 63170}, {54568, 62932}, {54996, 64711}, {55104, 55470}, {55469, 63399}, {56370, 63424}, {57216, 64058}, {57275, 65151}, {61814, 69377}, {61932, 64809}, {63101, 68718}, {63935, 69197}, {66886, 68648}

X(69450) = barycentric quotient X(44202)/X(1637)
X(69450) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 99, 11185}, {2, 148, 43620}, {2, 8591, 7615}, {2, 11185, 53127}, {2, 32815, 69387}, {2, 43448, 14061}, {2, 53142, 671}, {3, 325, 14907}, {3, 7763, 315}, {3, 69158, 7750}, {20, 32829, 7752}, {69, 3524, 7771}, {99, 69387, 32815}, {140, 1975, 32832}, {183, 6390, 32833}, {194, 33259, 69207}, {325, 14907, 315}, {376, 1007, 316}, {385, 33274, 21843}, {549, 6390, 183}, {574, 620, 2}, {626, 15515, 32965}, {631, 6337, 76}, {631, 32817, 34229}, {1506, 69171, 14035}, {2482, 7622, 2}, {3522, 32816, 7802}, {3522, 32835, 32816}, {3523, 3926, 1078}, {3533, 32822, 69385}, {3785, 15717, 43459}, {3785, 32831, 7796}, {3788, 37512, 7791}, {3815, 32459, 1003}, {5013, 7807, 7803}, {5024, 11288, 7792}, {5054, 69380, 37688}, {5056, 32826, 15031}, {5206, 7764, 20065}, {5286, 32989, 7857}, {5475, 32456, 33007}, {6337, 34229, 32817}, {7736, 32985, 3972}, {7738, 32970, 7828}, {7752, 62362, 32829}, {7756, 7862, 14063}, {7761, 8589, 33008}, {7763, 14907, 325}, {7769, 7782, 4}, {7771, 7799, 69}, {7777, 13586, 7737}, {7778, 53095, 8356}, {7783, 7907, 3767}, {7796, 43459, 3785}, {7816, 31455, 16924}, {7820, 15482, 2}, {7824, 7891, 7795}, {7836, 33004, 7800}, {7844, 31274, 2}, {7847, 7940, 14064}, {8716, 37637, 47286}, {9741, 23055, 11054}, {10304, 63098, 64018}, {11361, 17005, 31415}, {12040, 27088, 11163}, {14039, 63041, 60855}, {14148, 34506, 17131}, {15692, 32837, 7811}, {15717, 32831, 3785}, {21843, 34511, 385}, {31400, 32973, 83}, {32815, 69387, 11185}, {32817, 34229, 76}, {32986, 37690, 7934}, {32990, 53033, 3096}, {33007, 63083, 5475}, {37688, 59634, 69380}, {52691, 64019, 2}, {55864, 69379, 32838}, {56435, 66368, 26276}, {61856, 69383, 32867}, {63098, 64018, 7809}

X(69451) = X(2)X(7765)∩X(20)X(99)

Barycentrics   b^2*c^2 - 2*S^2 + 2*SA^2 + 2*SB*SC : :

X(69451) lies on these lines: {2, 7765}, {3, 32820}, {4, 7799}, {5, 1975}, {20, 99}, {30, 32821}, {32, 14148}, {39, 16898}, {69, 3528}, {75, 31458}, {76, 631}, {115, 33277}, {183, 3530}, {187, 63934}, {194, 5319}, {298, 42150}, {299, 42151}, {305, 7493}, {316, 32818}, {317, 1273}, {325, 382}, {376, 7768}, {384, 34511}, {524, 33235}, {538, 16925}, {543, 7888}, {548, 3933}, {550, 7788}, {574, 33258}, {620, 33262}, {633, 14145}, {634, 14144}, {671, 32972}, {754, 33244}, {858, 34254}, {1007, 3855}, {1078, 15717}, {1569, 20081}, {1909, 31452}, {2482, 7751}, {2549, 7836}, {2996, 14061}, {3091, 32837}, {3146, 7809}, {3520, 44134}, {3522, 7811}, {3523, 32836}, {3526, 32832}, {3529, 7860}, {3552, 7758}, {3620, 65417}, {3734, 9698}, {3767, 7891}, {3785, 21734}, {3788, 33248}, {3832, 7752}, {3843, 32819}, {3853, 7773}, {3934, 31457}, {3964, 5879}, {5007, 33255}, {5067, 7769}, {5070, 59635}, {5206, 35022}, {5254, 33218}, {5286, 7835}, {5881, 69038}, {6179, 32985}, {6392, 7857}, {6393, 15069}, {6656, 8716}, {6781, 7916}, {6921, 18145}, {7486, 32829}, {7618, 33004}, {7620, 52250}, {7622, 33188}, {7737, 7906}, {7738, 7832}, {7739, 7892}, {7748, 15301}, {7750, 15696}, {7753, 14031}, {7754, 59545}, {7756, 7908}, {7757, 14001}, {7759, 33007}, {7760, 32973}, {7764, 14035}, {7767, 46853}, {7770, 9606}, {7771, 61138}, {7772, 14037}, {7774, 7816}, {7775, 14068}, {7776, 17800}, {7777, 31417}, {7783, 7795}, {7789, 7803}, {7790, 53033}, {7791, 7801}, {7794, 32965}, {7812, 32981}, {7813, 20065}, {7818, 32997}, {7819, 51123}, {7821, 33017}, {7827, 9741}, {7828, 33222}, {7840, 33257}, {7843, 33280}, {7850, 62117}, {7854, 33008}, {7855, 32456}, {7856, 33181}, {7858, 14033}, {7870, 14064}, {7871, 32006}, {7873, 33253}, {7878, 14039}, {7881, 63548}, {7883, 33023}, {7885, 43619}, {7899, 43448}, {7900, 43618}, {7907, 63955}, {7909, 32974}, {7912, 20094}, {7922, 32986}, {7936, 33226}, {7946, 33265}, {8591, 33019}, {8598, 63938}, {9466, 33001}, {9657, 69135}, {9670, 69254}, {9766, 19687}, {10303, 46951}, {10304, 32896}, {10411, 61755}, {11055, 33191}, {11057, 17538}, {11128, 22531}, {11129, 22532}, {11165, 31470}, {11307, 40922}, {11308, 40921}, {11648, 33283}, {12150, 33201}, {12215, 35424}, {13571, 68517}, {13586, 14023}, {14066, 66466}, {14568, 32970}, {14929, 62106}, {15031, 69402}, {15300, 33192}, {15589, 43459}, {15708, 32892}, {16043, 55738}, {16239, 64093}, {16712, 37176}, {16772, 59539}, {16773, 59540}, {16989, 32450}, {16990, 37512}, {17128, 31401}, {17129, 21843}, {17567, 18146}, {17578, 32816}, {18436, 51383}, {18546, 32963}, {18906, 44423}, {19690, 32480}, {19696, 44678}, {19702, 47352}, {20172, 31469}, {22468, 50572}, {27088, 63926}, {30471, 42152}, {30472, 42149}, {31407, 32971}, {31492, 69139}, {32823, 62021}, {32826, 61982}, {32827, 32876}, {32828, 55864}, {32834, 61842}, {32835, 61914}, {32839, 69383}, {32869, 61820}, {32874, 61834}, {32875, 58188}, {32879, 62102}, {32885, 61856}, {32893, 61848}, {32954, 51122}, {32989, 41134}, {33006, 63922}, {33014, 52695}, {33193, 63931}, {33198, 55085}, {33202, 52691}, {33208, 36521}, {33233, 63923}, {33249, 34505}, {33250, 63932}, {33254, 63935}, {33268, 47102}, {34604, 68520}, {34803, 69405}, {35007, 63093}, {35297, 63933}, {35497, 68654}, {37647, 61905}, {37664, 56997}, {37688, 55863}, {40132, 57518}, {41136, 68420}, {41624, 68527}, {42407, 45921}, {44518, 47287}, {51579, 52886}, {51860, 69209}, {52713, 61867}, {55744, 60278}, {61811, 69381}, {61881, 69385}, {63017, 69172}, {63928, 68516}, {63936, 68528}

X(69451) = reflection of X(14063) in X(7888)
X(69451) = anticomplement of X(69162)
X(69451) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 32820, 32833}, {20, 3926, 7796}, {20, 7796, 315}, {99, 3926, 315}, {99, 7796, 20}, {194, 33225, 5319}, {1975, 6390, 7763}, {1975, 7763, 11185}, {2482, 7751, 32964}, {3146, 32825, 7809}, {3734, 9698, 33269}, {5319, 69206, 33225}, {6337, 32817, 76}, {7769, 69378, 53127}, {7781, 7863, 2}, {7789, 9607, 33217}, {7789, 31859, 7803}, {7802, 37668, 315}, {7813, 69171, 20065}, {7873, 34504, 33253}, {7917, 64018, 315}, {9607, 33217, 7803}, {31859, 33217, 9607}, {32815, 32831, 7752}, {32820, 59634, 3}, {32829, 69379, 69387}

X(69452) = X(2)X(2418)∩X(20)X(64)

Barycentrics   2*b^2*c^2 - S^2 + 2*SA^2 + 2*SB*SC : :

X(69452) lies on these lines: {2, 2418}, {4, 32840}, {5, 32841}, {20, 64}, {23, 22241}, {30, 10513}, {76, 3523}, {86, 51675}, {99, 10304}, {140, 32872}, {147, 60140}, {183, 15692}, {187, 9740}, {193, 34604}, {194, 33198}, {274, 17554}, {279, 7283}, {315, 49135}, {316, 3543}, {325, 3839}, {346, 5088}, {384, 51170}, {439, 17129}, {485, 3595}, {486, 3593}, {538, 5304}, {599, 53141}, {625, 7620}, {1003, 63042}, {1007, 61936}, {1078, 61788}, {1270, 6561}, {1271, 6560}, {2996, 7836}, {3091, 3926}, {3146, 3933}, {3314, 33210}, {3522, 32880}, {3734, 37665}, {3760, 5265}, {3761, 5281}, {3763, 7738}, {3785, 62097}, {3832, 32818}, {3964, 7527}, {4232, 9464}, {4352, 17175}, {5056, 32820}, {5067, 32873}, {5068, 69158}, {5254, 33182}, {5286, 7820}, {5334, 34508}, {5335, 34509}, {5395, 13571}, {6337, 10303}, {6392, 7806}, {7486, 7763}, {7500, 41896}, {7754, 33201}, {7767, 50693}, {7771, 61781}, {7776, 17578}, {7782, 58188}, {7788, 15640}, {7789, 33183}, {7796, 32826}, {7799, 61924}, {7801, 43448}, {7809, 62003}, {7811, 62122}, {7844, 53033}, {7850, 49140}, {7881, 33200}, {7906, 32979}, {7931, 33180}, {7947, 32980}, {8024, 10565}, {8369, 63097}, {8591, 11161}, {10153, 60200}, {10519, 23235}, {11001, 14929}, {11064, 11348}, {11160, 14712}, {11164, 50992}, {11286, 14930}, {12215, 33748}, {14033, 63091}, {14037, 20105}, {14039, 22253}, {14045, 38259}, {14711, 21843}, {14907, 15697}, {14928, 66755}, {15301, 53142}, {15708, 32874}, {15717, 32882}, {15721, 46951}, {17130, 31400}, {17558, 34284}, {18600, 56987}, {20080, 33007}, {20081, 32973}, {20094, 33272}, {22235, 69157}, {22237, 69165}, {25242, 25243}, {25583, 66681}, {25650, 57826}, {26166, 64495}, {31415, 39785}, {32006, 50691}, {32459, 63029}, {32816, 61982}, {32819, 50688}, {32823, 50689}, {32827, 32896}, {32828, 55864}, {32829, 46935}, {32832, 61863}, {32835, 46936}, {32837, 61912}, {32877, 62110}, {32881, 61914}, {32883, 62362}, {32892, 61796}, {32893, 61846}, {32894, 61820}, {32971, 63018}, {32986, 47287}, {33187, 50248}, {33278, 35369}, {34505, 37690}, {34803, 61906}, {35927, 63046}, {37667, 50370}, {37671, 62094}, {37681, 62755}, {37689, 69206}, {42850, 53095}, {46453, 63954}, {51678, 62999}, {51898, 67687}, {51899, 67676}, {53127, 61897}, {58448, 63955}, {62160, 64018}, {62555, 63248}

X(69452) = crossdifference of every pair of points on line {8644, 62176}
X(69452) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {99, 15589, 10304}, {99, 32836, 15589}, {1975, 32830, 20}, {2996, 7836, 33199}, {3522, 32880, 69377}, {3832, 32879, 32818}, {3926, 11185, 63098}, {3926, 69379, 3091}, {3933, 32822, 3146}, {6337, 32834, 10303}, {6390, 52713, 2}, {6390, 69380, 52713}, {7763, 69383, 7486}, {9741, 59780, 2}, {11185, 63098, 3091}, {14039, 22253, 63005}, {15717, 32882, 69381}, {32815, 32833, 37668}, {32815, 37668, 3543}, {32817, 52713, 6390}, {32817, 69380, 2}, {32820, 69378, 32831}, {32826, 32875, 7796}, {32831, 69378, 5056}, {32835, 59635, 46936}, {32874, 59634, 15708}, {32894, 61820, 69384}, {63098, 69379, 11185}

X(69453) = X(2)X(2418)∩X(20)X(76)

Barycentrics   2*b^2*c^2 + S^2 + 2*SA^2 + 2*SB*SC : :

X(69453) lies on these lines: {2, 2418}, {4, 10513}, {5, 32840}, {20, 76}, {69, 3543}, {99, 15692}, {148, 3620}, {183, 10304}, {193, 22486}, {253, 44440}, {315, 50688}, {316, 62007}, {325, 3091}, {339, 61113}, {390, 3761}, {538, 37665}, {626, 63536}, {631, 32872}, {1007, 61924}, {1078, 32868}, {1270, 23259}, {1271, 23249}, {1285, 63954}, {1975, 3523}, {2996, 33180}, {3090, 32841}, {3146, 32882}, {3522, 32822}, {3600, 3760}, {3628, 32873}, {3734, 5304}, {3767, 33183}, {3832, 3933}, {3839, 7809}, {3854, 32823}, {3926, 5056}, {5059, 7767}, {5068, 32818}, {5129, 34284}, {5286, 7889}, {5343, 69107}, {5344, 69106}, {6337, 55864}, {6392, 16989}, {6995, 44146}, {7398, 8024}, {7486, 32831}, {7615, 7908}, {7737, 14711}, {7750, 49140}, {7763, 46936}, {7771, 62059}, {7776, 50689}, {7782, 61783}, {7788, 61989}, {7795, 32457}, {7796, 32877}, {7799, 61906}, {7811, 62166}, {7853, 43448}, {7892, 43681}, {7906, 32991}, {7939, 54097}, {7947, 52250}, {8370, 63091}, {9740, 17131}, {10303, 32828}, {11286, 63005}, {11361, 20080}, {13595, 22241}, {14033, 63042}, {14035, 50248}, {14039, 63097}, {14532, 46944}, {14929, 15682}, {14930, 22253}, {14994, 61044}, {15022, 32879}, {15031, 32890}, {15069, 53016}, {15640, 32892}, {15717, 69384}, {15721, 32893}, {16986, 33202}, {17129, 32981}, {17580, 18135}, {19768, 56986}, {20081, 32971}, {20105, 33269}, {22493, 36970}, {22494, 36969}, {32819, 49135}, {32820, 32835}, {32824, 32832}, {32826, 32878}, {32833, 61936}, {32837, 53127}, {32870, 61863}, {32888, 62083}, {32973, 63047}, {33215, 47287}, {33242, 60636}, {33263, 35369}, {33272, 63044}, {37647, 61897}, {37670, 50742}, {37671, 62160}, {37689, 63955}, {40330, 51397}, {41614, 60874}, {42850, 53141}, {43403, 69120}, {43404, 69121}, {43459, 58186}, {51898, 67688}, {51899, 67677}, {52718, 61848}, {53033, 63924}, {59634, 61844}, {60200, 62912}

X(69453) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 32815, 15589}, {76, 69379, 20}, {148, 3620, 33210}, {1975, 32834, 3523}, {3146, 32882, 69377}, {3522, 32894, 69381}, {3832, 32880, 3933}, {3926, 69383, 5056}, {5485, 59780, 2}, {6390, 69386, 2}, {6392, 17128, 33198}, {11185, 32836, 37668}, {11185, 37668, 3839}, {15022, 32879, 69158}, {15589, 32815, 20}, {15589, 69379, 32815}, {32817, 52713, 64093}, {32817, 64093, 2}, {32820, 69385, 32835}, {32822, 69381, 3522}, {32830, 69378, 3091}, {32831, 59635, 7486}, {32833, 69382, 63098}, {32835, 69385, 46935}, {52713, 69380, 2}, {63098, 69382, 61936}, {64093, 69380, 32817}




(Part 36 will be started in the future.)

This is the end of PART 35: Centers X(68001) - X(70000)

PART 1: Introduction and Centers X(1) - X(1000) PART 2: Centers X(1001) - X(3000) PART 3: Centers X(3001) - X(5000)
PART 4: Centers X(5001) - X(7000) PART 5: Centers X(7001) - X(10000) PART 6: Centers X(10001) - X(12000)
PART 7: Centers X(12001) - X(14000) PART 8: Centers X(14001) - X(16000) PART 9: Centers X(16001) - X(18000)
PART 10: Centers X(18001) - X(20000) PART 11: Centers X(20001) - X(22000) PART 12: Centers X(22001) - X(24000)
PART 13: Centers X(24001) - X(26000) PART 14: Centers X(26001) - X(28000) PART 15: Centers X(28001) - X(30000)
PART 16: Centers X(30001) - X(32000) PART 17: Centers X(32001) - X(34000) PART 18: Centers X(34001) - X(36000)
PART 19: Centers X(36001) - X(38000) PART 20: Centers X(38001) - X(40000) PART 21: Centers X(40001) - X(42000)
PART 22: Centers X(42001) - X(44000) PART 23: Centers X(44001) - X(46000) PART 24: Centers X(46001) - X(48000)
PART 25: Centers X(48001) - X(50000) PART 26: Centers X(50001) - X(52000) PART 27: Centers X(52001) - X(54000)
PART 28: Centers X(54001) - X(56000) PART 29: Centers X(56001) - X(58000) PART 30: Centers X(58001) - X(60000)
PART 31: Centers X(60001) - X(62000) PART 32: Centers X(62001) - X(64000) PART 33: Centers X(64001) - X(66000)
PART 34: Centers X(66001) - X(68000) PART 35: Centers X(68001) - X(70000) PART 36: Centers X(70001) - X(72000)