Clark Kimberling's

Encyclopedia of Triangle Centers - ETC

X(52001) = X(1)X(1362)∩X(33)X(42071)

X(52001) lies on the cubics K040 and K984 and these lines: {1, 1362}, {33, 42071}, {55, 20995}, {101, 10482}, {165, 36601}, {200, 14943}, {354, 658}, {518, 10025}, {1282, 7220}, {2389, 42064}, {11028, 30329}, {23612, 28071}

X(52001) = isogonal conjugate of X(14189)
X(52001) = X(i)-cross conjugate of X(j) for these (i,j): {294, 7077}, {672, 55}
X(52001) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14189}, {6, 40864}, {7, 9441}, {56, 33677}, {57, 10025}, {105, 36905}, {269, 28058}
X(52001) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 33677}, {3, 14189}, {9, 40864}, {5452, 10025}, {6600, 28058}, {36905, 39046}
X(52001) = cevapoint of X(i) and X(j) for these (i,j): {926, 3022}, {9440, 9441}
X(52001) = crosspoint of X(9442) and X(14943)
X(52001) = trilinear pole of line {657, 2293}
X(52001) = barycentric product X(i)*X(j) for these {i,j}: {1, 14943}, {9, 9442}
X(52001) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40864}, {6, 14189}, {9, 33677}, {41, 9441}, {55, 10025}, {220, 28058}, {672, 36905}, {9442, 85}, {14943, 75}

X(52002) = X(1)X(30)∩X(5)X(41496)

X(52002) lies on the cubic K040 and these lines: {1, 30}, {5, 41496}, {55, 43682}, {105, 476}, {354, 5196}, {405, 6757}, {518, 6742}, {523, 9404}, {1155, 38340}, {1621, 30690}, {1989, 3011}, {2688, 26700}, {3615, 51715}, {5205, 15455}, {7110, 47160}, {8818, 17718}, {17728, 44908}, {36026, 37080}

X(52002) = crossdifference of every pair of points on line {500, 9404}
X(52002) = X(i)-line conjugate of X(j) for these (i,j): {1, 500}, {523, 9404}

X(52003) = X(4)X(974)∩X(6)X(10574)

X(52003) lies on the cubic K934 and these lines: {4, 974}, {6, 10574}, {24, 154}, {25, 46372}, {30, 143}, {51, 40928}, {52, 44240}, {110, 2929}, {185, 235}, {427, 32184}, {511, 44247}, {578, 36153}, {1147, 1511}, {1154, 43615}, {1192, 5889}, {1204, 2781}, {1620, 2979}, {1992, 27082}, {2807, 51702}, {3357, 15151}, {3522, 44439}, {3567, 35490}, {5448, 9826}, {5622, 32345}, {5655, 25711}, {5663, 44235}, {5893, 44084}, {5907, 47296}, {6000, 44226}, {6293, 18913}, {6467, 8550}, {7687, 32137}, {9729, 16196}, {9730, 11585}, {10112, 22952}, {11412, 37487}, {11561, 16534}, {11591, 44673}, {11806, 45286}, {12006, 18388}, {12111, 26958}, {12300, 43608}, {13382, 14862}, {13491, 18390}, {13754, 16238}, {14831, 44268}, {14865, 43904}, {15045, 31282}, {15738, 26917}, {15739, 37118}, {16270, 20299}, {17818, 19149}, {17824, 43617}, {18404, 37481}, {19118, 34117}, {20771, 45735}, {21851, 32284}, {22467, 22966}, {22529, 43601}, {31725, 39571}, {31728, 51694}, {34146, 51734}, {35603, 44269}, {37490, 44259}, {50649, 51737}

X(52003) = midpoint of X(i) and X(j) for these {i,j}: {52, 44240}, {185, 235}, {6102, 37814}, {13491, 44271}, {14831, 44268}, {31728, 51694}
X(52003) = reflection of X(i) in X(j) for these {i,j}: {16196, 9729}, {49673, 12006}
X(52003) = X(250)-Ceva conjugate of X(1624)
X(52003) = X(775)-isoconjugate of X(22466)
X(52003) = X(2883)-Dao conjugate of X(22466)
X(52003) = barycentric product X(i)*X(j) for these {i,j}: {249, 46658}, {800, 22468}, {13567, 22467}
X(52003) = barycentric quotient X(i)/X(j) for these {i,j}: {800, 22466}, {22467, 801}, {22468, 40830}, {46658, 338}
X(52003) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {185, 44079, 36982}, {389, 40647, 12241}

X(52004) = X(1)X(926)∩X(101)X(7193)

X(52004) lies on the cubic K040 and these lines: {1, 926}, {101, 7193}, {518, 664}, {654, 9319}, {1026, 46793}, {1155, 37138}, {1362, 41353}, {1814, 7077}, {2283, 42079}, {2284, 20778}, {34253, 42071}

X(52004) = isogonal conjugate of X(14197)
X(52004) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14197}, {6, 46792}, {105, 28850}, {28143, 36146}
X(52004) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 14197}, {9, 46792}, {28143, 39014}, {28850, 39046}
X(52004) = barycentric product X(i)*X(j) for these {i,j}: {1, 46793}, {3912, 12032}
X(52004) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 46792}, {6, 14197}, {672, 28850}, {926, 28143}, {12032, 673}, {46793, 75}

X(52005) = X(1)X(5)∩X(3)X(513)

X(52005) lies on the cubic K258 and these lines: {1, 5}, {2, 14266}, {3, 513}, {39, 23980}, {40, 34464}, {517, 14260}, {631, 47045}, {978, 1052}, {1062, 10017}, {11499, 34040}, {16128, 33810}, {18340, 37821}, {22350, 35015}, {32612, 34467}, {37611, 47645}, {41343, 49997}

X(52005) = complement of X(14266)
X(52005) = complement of the isogonal conjugate of X(39173)
X(52005) = X(i)-complementary conjugate of X(j) for these (i,j): {913, 26011}, {1769, 15608}, {22350, 42423}, {32655, 3911}, {36052, 517}, {36106, 8677}, {39173, 10}
X(52005) = X(104)-Ceva conjugate of X(517)
X(52005) = X(3262)-Dao conjugate of X(26611)
X(52005) = crossdifference of every pair of points on line {654, 8609}
X(52005) = barycentric product X(908)*X(10090)
X(52005) = barycentric quotient X(10090)/X(34234)
X(52005) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 39173, 47081}, {6127, 6326, 34465}, {39173, 47051, 3}

X(52006) = X(1)X(7095)∩X(2)X(14265)

X(52006) lies on the cubic K258 and these lines: {1, 7095}, {2, 14265}, {3, 512}, {5, 39}, {49, 3203}, {182, 32540}, {446, 511}, {575, 34236}, {631, 47044}, {682, 3491}, {868, 36212}, {1352, 5661}, {5907, 40804}, {6132, 47082}, {7769, 37121}, {7821, 35088}, {15819, 46840}, {22401, 38974}, {34810, 37742}

X(52006) = midpoint of X(446) and X(23098)
X(52006) = complement of X(14265)
X(52006) = complement of the isogonal conjugate of X(34157)
X(52006) = X(i)-complementary conjugate of X(j) for these (i,j): {8773, 21531}, {32654, 16609}, {34157, 10}, {36051, 511}, {36105, 39469}
X(52006) = X(98)-Ceva conjugate of X(511)
X(52006) = X(325)-Dao conjugate of X(36790)
X(52006) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 34157, 47079}, {34157, 47049, 3}

X(52007) = X(1)X(84)∩X(8)X(44189)

X(52007) lies on the cubic K040 and these lines: {1, 84}, {8, 44189}, {40, 1256}, {55, 41081}, {65, 41084}, {105, 6081}, {189, 497}, {271, 960}, {280, 3057}, {518, 13138}, {521, 3239}, {1155, 37141}, {1210, 5908}, {2739, 8059}, {3586, 5923}, {5205, 44327}, {14986, 37744}, {18239, 18283}

X(52007) = X(9372)-Ceva conjugate of X(9376)
X(52007) = X(40)-isoconjugate of X(9372)
X(52007) = X(329)-Dao conjugate of X(39050)
X(52007) = crosspoint of X(9372) and X(9375)
X(52007) = crosssum of X(9370) and X(9371)
X(52007) = crossdifference of every pair of points on line {221, 14298}
X(52007) = X(i)-lineconjugate of X(j) for these (i,j): {1, 221}, {521, 14298}
X(52007) = barycentric product X(i)*X(j) for these {i,j}: {84, 40880}, {189, 9371}, {280, 34371}
X(52007) = barycentric quotient X(i)/X(j) for these {i,j}: {1436, 9372}, {9371, 329}, {34371, 347}, {40880, 322}
X(52007) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 51313, 84}, {40, 1256, 9376}

X(52008) = X(4)X(10117)∩X(6)X(32354)

X(52008) lies on the cubic K934 and these lines: {4, 10117}, {6, 32354}, {30, 6689}, {54, 7576}, {141, 7691}, {143, 10112}, {427, 20376}, {428, 32340}, {973, 11745}, {1147, 20424}, {1154, 31830}, {1503, 32332}, {1595, 32401}, {2917, 7487}, {3574, 3575}, {5447, 31833}, {6145, 13567}, {6288, 41587}, {6756, 10110}, {8254, 18475}, {9786, 32337}, {9825, 32348}, {10610, 11819}, {10619, 21637}, {11262, 47328}, {11576, 22483}, {12233, 32379}, {12362, 32396}, {13163, 20584}, {13490, 22804}, {15577, 32333}, {15800, 38321}, {23358, 37458}, {32260, 32352}, {32402, 39571}

X(52008) = midpoint of X(i) and X(j) for these {i,j}: {3574, 3575}, {10115, 45286}, {10610, 11819}
X(52008) = reflection of X(i) in X(j) for these {i,j}: {973, 11745}, {12362, 32396}, {20584, 13163}, {32348, 9825}
X(52008) = {X(3574),X(32391)}-harmonic conjugate of X(23292)

X(52009) = X(39)X(9469)∩X(98)X(9468)

X(52009) lies on the cubic K787 and these lines: {39, 9469}, {98, 9468}, {232, 17984}, {804, 2023}, {1691, 9418}, {3229, 41520}, {5976, 11672}, {9419, 36213}, {10352, 16069}, {43702, 47642}

X(52009) = isogonal conjugate of X(39941)
X(52009) = X(i)-cross conjugate of X(j) for these (i,j): {2, 232}, {14251, 511}
X(52009) = X(i)-isoconjugate of X(j) for these (i,j): {1, 39941}, {75, 51327}, {1821, 3511}, {1910, 25332}
X(52009) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 39941}, {206, 51327}, {3511, 40601}, {11672, 25332}
X(52009) = barycentric product X(511)*X(41520)
X(52009) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 39941}, {32, 51327}, {237, 3511}, {511, 25332}, {14251, 39092}, {41520, 290}

X(52010) = X(1)X(9405)∩X(5)X(113)

X(52010) lies on the cubic K258 and these lines: {1, 9405}, {2, 14264}, {3, 523}, {5, 113}, {30, 14254}, {39, 3163}, {74, 41512}, {140, 14670}, {395, 14817}, {396, 14816}, {402, 51425}, {541, 16535}, {549, 18285}, {550, 16168}, {631, 33927}, {1941, 3520}, {2088, 16310}, {3134, 13754}, {3184, 44240}, {5877, 18917}, {8901, 11806}, {11064, 39234}, {11585, 16177}, {12079, 39170}, {14385, 14611}, {14920, 37118}, {14933, 45147}, {16934, 51346}, {18030, 34153}, {18279, 36169}, {20771, 38605}, {32162, 51393}, {34351, 47222}, {38393, 39235}
X(52010) = midpoint of X(74) and X(41512)
X(52010) = reflection of X(14670) in X(140)
X(52010) = complement of X(14264)
X(52010) = complement of the isogonal conjugate of X(15454)
X(52010) = medial isogonal conjugate of X(39170)
X(52010) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 39170}, {1784, 46085}, {14910, 18593}, {15454, 10}, {36035, 16221}, {36053, 30}, {36114, 9033}
X(52010) = X(i)-Ceva conjugate of X(j) for these (i,j): {74, 30}, {41512, 523}
X(52010) = X(3260)-Dao conjugate of X(36789)
X(52010) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 15454, 47084}, {3, 34810, 15454}, {15454, 47050, 3}, {34810, 47050, 47084}

X(52011) = X(4)X(1562)∩X(112)X(376)

X(52011) lies on the cubic K245 and these lines: {4, 1562}, {74, 33630}, {112, 376}, {393, 3269}, {1075, 12251}, {2331, 18446}, {3462, 39575}, {5286, 51334}, {5523, 6761}, {6770, 36302}, {6773, 36303}, {7709, 41371}, {8743, 12252}, {9862, 41204}, {14361, 50188}, {16318, 40664}, {44216, 51358}, {44909, 51385}

X(52011) = polar conjugate of the isotomic conjugate of X(34186)
X(52011) = X(525)-Ceva conjugate of X(4)
X(52011) = X(63)-isoconjugate of X(34185)
X(52011) = X(i)-Dao conjugate of X(j) for these (i,j): {107, 648}, {3162, 34185}
X(52011) = barycentric product X(4)*X(34186)
X(52011) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 34185}, {34186, 69}
X(52011) = {X(1562),X(6529)}-harmonic conjugate of X(4)

X(52012) = X(2)X(3)∩X(46)X(2194)

See Ivan Pavlov, euclid 5507.

X(52012) lies on these lines: {2, 3}, {36, 40980}, {46, 2194}, {57, 1408}, {58, 3752}, {81, 5708}, {283, 51420}, {284, 501}, {517, 2360}, {1155, 1780}, {1437, 37532}, {1474, 41340}, {1790, 10202}, {1819, 37585}, {2193, 37697}, {2206, 24443}, {2287, 3927}, {2328, 3579}, {3285, 17054}, {3286, 5358}, {3913, 37547}, {5285, 33079}, {5709, 41608}, {5814, 24632}, {8666, 24446}, {8747, 14192}, {12699, 17188}, {15171, 51621}, {22136, 48917}, {22139, 48924}, {23171, 41227}, {24310, 42463}, {33097, 51624}, {37623, 51966}, {51687, 51699}

X(52012) = barycentric product X(57)*X(27412)
X(52012) = barycentric quotient X(27412)/X(312)
X(52012) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 28, 36011}, {3, 7535, 16290}, {3, 11347, 16415}, {3, 28446, 5428}, {21, 17581, 11108}, {28, 1817, 3}, {859, 16049, 15952}, {1325, 4225, 37227}, {7520, 37264, 3}, {11347, 37408, 3}, {11349, 37431, 16414}, {15771, 15772, 27}, {16368, 19285, 16848}, {17513, 37288, 21}, {33325, 37405, 21}, {37425, 46548, 20831}

X(52013) = ISOGONAL CONJUGATE OF X(390)

See Ivan Pavlov, euclid 5509.

X(52013) lies on these lines: {1, 738}, {6, 1362}, {33, 354}, {55, 1407}, {56, 220}, {57, 200}, {999, 4845}

X(52013) = isogonal conjugate of X(390)

X(52014) = X(3)X(51)∩X(4)X(1609)

See Angel Montesdeoca, euclid 5508.

X(52014) lies on these lines: {3,51}, {4,1609}, {24,6525}, {1181,8573}, {1859,11399}, {3517,13558}, {6146,8721}

X(52015) = X(1)X(41)∩X(2)X(40910)

X(52015) lies on the cubic K1291 and these lines: {1, 41}, {2, 40910}, {3, 142}, {8, 45765}, {21, 17175}, {32, 1279}, {55, 29571}, {56, 10481}, {58, 2191}, {85, 927}, {238, 4253}, {481, 30385}, {482, 30386}, {975, 41230}, {993, 24331}, {1002, 17745}, {1386, 5045}, {1621, 11349}, {1633, 6173}, {2182, 10177}, {2975, 30625}, {3008, 37580}, {3220, 38053}, {3361, 28017}, {3434, 24588}, {3616, 17170}, {3664, 7083}, {3685, 33942}, {3816, 19512}, {3822, 36526}, {3912, 26241}, {4224, 10582}, {4228, 4666}, {4262, 16484}, {4675, 16686}, {5020, 13405}, {5030, 15485}, {5228, 35273}, {5259, 17732}, {5263, 30110}, {5284, 29603}, {5542, 24320}, {6666, 12329}, {6745, 11284}, {7397, 26105}, {7677, 38859}, {8299, 25440}, {8301, 8715}, {9895, 15569}, {11019, 25514}, {14942, 17682}, {15507, 34928}, {16551, 21346}, {16845, 26939}, {19288, 25512}, {19309, 19868}, {20468, 38186}, {22654, 51723}, {24388, 40530}, {26261, 29830}, {26531, 50896}, {30130, 49470}, {38668, 41680}, {38902, 42884}

X(52015) = midpoint of X(i) and X(j) for these {i,j}: {1, 169}, {8225, 31546}
X(52015) = reflection of X(34847) in X(1125)
X(52015) = crossdifference of every pair of points on line {2254, 6586}
X(52015) = barycentric product X(1)*X(24596)
X(52015) = barycentric quotient X(24596)/X(75)
X(52015) = {X(142),X(1486)}-harmonic conjugate of X(24309)

X(52016) = X(3)X(34817)∩X(6)X(1196)

X(52016) lies on the cubic K1291 and these lines: {3, 34817}, {6, 1196}, {32, 15371}, {49, 11898}, {66, 542}, {68, 24206}, {69, 184}, {110, 193}, {114, 41770}, {125, 28408}, {140, 141}, {154, 16199}, {155, 159}, {156, 34380}, {157, 9737}, {160, 5171}, {206, 524}, {323, 12220}, {389, 12166}, {394, 11574}, {518, 14529}, {539, 11178}, {571, 1634}, {576, 9925}, {577, 20794}, {578, 1352}, {597, 13361}, {599, 5157}, {1092, 6776}, {1191, 42461}, {1350, 12164}, {1351, 7716}, {1353, 44489}, {1386, 5045}, {1503, 13346}, {1660, 34774}, {1843, 1993}, {1899, 28419}, {2393, 34966}, {2930, 9973}, {3044, 41274}, {3047, 32244}, {3098, 13754}, {3284, 23163}, {3292, 6467}, {3618, 5651}, {3619, 43650}, {3620, 5012}, {3629, 19136}, {3630, 19127}, {3818, 44665}, {3955, 5227}, {5039, 43977}, {5092, 47391}, {5093, 18350}, {5138, 41608}, {5654, 19130}, {5921, 19124}, {5965, 10274}, {5972, 32245}, {6053, 31670}, {6144, 18374}, {7193, 7289}, {7758, 15594}, {8541, 12272}, {8548, 39561}, {8909, 40067}, {9027, 41593}, {9028, 25353}, {9544, 19121}, {9676, 39893}, {9703, 19129}, {9820, 38317}, {9924, 37672}, {9936, 40107}, {10110, 12309}, {10282, 37488}, {10516, 12429}, {10519, 10984}, {10540, 44456}, {11411, 37515}, {11441, 12294}, {12038, 17508}, {12118, 29012}, {12163, 14810}, {12242, 14561}, {12293, 48889}, {13198, 32257}, {13352, 18440}, {14984, 19140}, {15083, 35707}, {15143, 51936}, {15316, 43130}, {15520, 16776}, {15577, 46730}, {16187, 47355}, {17702, 48884}, {18438, 50461}, {18445, 37511}, {18935, 37669}, {19122, 37784}, {19142, 22955}, {21637, 41614}, {21769, 23075}, {21850, 46261}, {22115, 39899}, {22152, 37893}, {22660, 48901}, {25406, 43652}, {26156, 45968}, {26206, 40673}, {26883, 51212}, {32217, 47446}, {32661, 41412}, {33872, 35222}, {34146, 46372}, {34801, 43725}, {35264, 44091}, {36212, 40947}, {37480, 46264}, {37495, 48662}, {37498, 39879}, {39874, 43574}, {40318, 44102}, {40341, 41615}, {41584, 44077}, {42022, 45726}, {43572, 50974}

X(52016) = midpoint of X(i) and X(j) for these {i,j}: {6, 19588}, {1350, 12164}, {1352, 6193}, {9925, 19139}, {37498, 39879}
X(52016) = reflection of X(i) in X(j) for these {i,j}: {68, 24206}, {182, 1147}, {576, 19139}, {12163, 14810}, {12293, 48889}, {19139, 41597}, {37488, 10282}, {46730, 15577}, {48901, 22660}
X(52016) = isogonal conjugate of X(47847)
X(52016) = isogonal conjugate of the polar conjugate of X(7754)
X(52016) = X(251)-Ceva conjugate of X(3)
X(52016) = X(1)-isoconjugate of X(47847)
X(52016) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 47847}, {3933, 8024}, {5139, 8651}
X(52016) = cevapoint of X(3167) and X(19597)
X(52016) = crosspoint of X(110) and X(47389)
X(52016) = crosssum of X(523) and X(2971)
X(52016) = barycentric product X(i)*X(j) for these {i,j}: {3, 7754}, {48, 18056}, {1176, 19568}
X(52016) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 47847}, {7754, 264}, {18056, 1969}, {19568, 1235}
X(52016) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 9306, 19137}, {6, 14913, 9813}, {49, 11898, 19131}, {69, 184, 19126}, {110, 193, 1974}, {394, 19459, 11574}, {3167, 19588, 6}, {3292, 6467, 20806}, {5921, 34148, 19124}, {6467, 20806, 11511}, {9544, 20080, 19121}, {9925, 41597, 576}, {14913, 34986, 6}

X(52017) = X(10)X(56)∩X(106)X(614)

X(52017) lies on the cubic K1291 and these lines: {10, 56}, {106, 614}, {517, 3098}, {529, 48826}, {3436, 19836}, {12609, 28037}, {24928, 30148}

X(52018) = X(10)X(55)∩X(58)X(614)

X(52018) lies on the cubic K1291 and these lines: {10, 55}, {58, 614}, {1054, 37231}, {1125, 36740}, {1386, 5045}, {7083, 12609}, {37582, 40656}

X(52019) = X(3)X(4549)∩X(22)X(74)

X(52019) lies on the cubic K1291 and these lines: {3, 4549}, {22, 74}, {24, 11439}, {26, 3357}, {30, 18382}, {186, 40914}, {378, 40909}, {525, 46608}, {541, 13289}, {548, 19908}, {550, 9938}, {2070, 3426}, {2071, 41465}, {2937, 11820}, {3098, 13754}, {3579, 39582}, {4550, 6644}, {5663, 15577}, {6800, 15136}, {7387, 44158}, {7514, 32600}, {7526, 7706}, {7530, 32223}, {7689, 8717}, {7691, 10323}, {10605, 32322}, {11411, 16661}, {11413, 16013}, {11414, 12359}, {11999, 13564}, {12082, 14852}, {12084, 32401}, {12293, 33524}, {12901, 37853}, {13861, 20191}, {15041, 19376}, {15080, 47391}, {16063, 32123}, {17702, 32305}, {17928, 32620}, {23306, 31152}, {35243, 37485}, {44259, 44754}, {44837, 45839}

X(52019) = midpoint of X(i) and X(j) for these {i,j}: {7689, 8717}, {34801, 35237}

X(52020) = X(1)X(3688)∩X(2)X(35892)

X(52020) lies on the cubic K089 and these lines: {1, 3688}, {2, 35892}, {6, 692}, {7, 1002}, {10, 4111}, {37, 4890}, {42, 181}, {43, 17065}, {51, 2187}, {55, 579}, {65, 1439}, {72, 50290}, {142, 354}, {209, 37593}, {210, 5257}, {213, 40934}, {238, 39543}, {239, 17049}, {256, 2663}, {284, 17798}, {291, 1045}, {511, 4649}, {513, 7277}, {518, 4026}, {583, 8053}, {584, 1631}, {594, 22279}, {672, 4343}, {674, 1100}, {757, 3110}, {869, 17053}, {872, 3122}, {894, 6007}, {942, 1738}, {970, 39551}, {1015, 1964}, {1086, 13476}, {1194, 1206}, {1213, 22271}, {1397, 44094}, {1401, 28017}, {1449, 3056}, {1475, 2293}, {1500, 2667}, {1682, 19767}, {1716, 3751}, {1740, 50584}, {1742, 14520}, {1843, 1973}, {1962, 3690}, {2171, 4516}, {2200, 20970}, {2209, 4274}, {2223, 2260}, {2274, 50596}, {2278, 4497}, {2294, 21804}, {2308, 23202}, {2309, 20456}, {2653, 23629}, {3022, 4336}, {3030, 3240}, {3041, 27547}, {3121, 21815}, {3247, 4517}, {3248, 46189}, {3475, 25521}, {3555, 49511}, {3662, 3873}, {3678, 25354}, {3681, 17248}, {3764, 4263}, {3819, 4038}, {3821, 3874}, {3879, 17792}, {3881, 49676}, {3888, 20090}, {3914, 39793}, {3943, 21865}, {3946, 20358}, {3957, 27678}, {3963, 25295}, {4014, 17365}, {4092, 21933}, {4093, 21827}, {4251, 23868}, {4253, 20992}, {4261, 4484}, {4270, 15494}, {4319, 39789}, {4335, 44421}, {4360, 14839}, {4430, 17236}, {4553, 17390}, {4667, 49537}, {4684, 34791}, {4849, 21892}, {5138, 37576}, {5650, 9345}, {5711, 50583}, {9038, 17344}, {9054, 17045}, {9441, 50658}, {9565, 20018}, {17018, 27624}, {17391, 25279}, {18164, 35338}, {19133, 40910}, {20229, 40983}, {20861, 20963}, {20959, 44085}, {21039, 21808}, {21299, 34283}, {21753, 21813}, {21803, 22167}, {21805, 22174}, {22343, 23634}, {23633, 23659}, {24473, 50091}, {25106, 46843}, {29633, 38485}, {33137, 35612}, {39773, 49495}, {41886, 49490}, {50591, 50626}

X(52020) = reflection of X(594) in X(22279)
X(52020) = isogonal conjugate of the isotomic conjugate of X(3925)
X(52020) = X(i)-Ceva conjugate of X(j) for these (i,j): {354, 21808}, {1020, 3709}, {21808, 21795}
X(52020) = X(i)-isoconjugate of X(j) for these (i,j): {21, 21453}, {29, 40443}, {81, 32008}, {86, 2346}, {274, 1174}, {284, 31618}, {286, 47487}, {333, 1170}, {1434, 6605}, {1803, 31623}, {2287, 10509}, {2328, 42311}, {3737, 6606}
X(52020) = X(i)-Dao conjugate of X(j) for these (i,j): {142, 314}, {274, 40606}, {310, 1212}, {354, 33297}, {2346, 40600}, {3925, 17143}, {21453, 40611}, {31618, 40590}, {32008, 40586}, {36908, 42311}
X(52020) = crosspoint of X(i) and X(j) for these (i,j): {6, 13476}, {37, 15320}, {42, 65}, {354, 1475}, {1827, 2293}
X(52020) = crosssum of X(i) and X(j) for these (i,j): {2, 1621}, {21, 86}, {81, 4184}, {2346, 32008}, {8299, 33295}, {21453, 40443}
X(52020) = crossdifference of every pair of points on line {918, 4560}
X(52020) = barycentric product X(i)*X(j) for these {i,j}: {1, 21808}, {6, 3925}, {7, 21795}, {10, 1475}, {37, 354}, {42, 142}, {57, 21039}, {65, 1212}, {73, 1855}, {181, 16713}, {210, 1418}, {213, 20880}, {226, 2293}, {306, 40983}, {513, 35310}, {523, 35326}, {661, 35338}, {756, 18164}, {872, 16708}, {1018, 48151}, {1020, 6608}, {1042, 51972}, {1214, 1827}, {1229, 1402}, {1233, 1918}, {1334, 10481}, {1400, 4847}, {1427, 3059}, {1441, 20229}, {1500, 17169}, {1826, 22053}, {2171, 17194}, {2488, 4552}, {3668, 8012}, {3709, 35312}, {4017, 35341}, {4551, 21127}, {4557, 21104}, {4559, 6362}, {4566, 10581}, {15320, 40606}, {18087, 21035}, {22079, 40149}
X(52020) = barycentric quotient X(i)/X(j) for these {i,j}: {42, 32008}, {65, 31618}, {142, 310}, {213, 2346}, {354, 274}, {1042, 10509}, {1212, 314}, {1229, 40072}, {1400, 21453}, {1402, 1170}, {1409, 40443}, {1427, 42311}, {1475, 86}, {1827, 31623}, {1855, 44130}, {1918, 1174}, {2200, 47487}, {2293, 333}, {2488, 4560}, {3925, 76}, {4559, 6606}, {4847, 28660}, {8012, 1043}, {10581, 7253}, {16713, 18021}, {18164, 873}, {20229, 21}, {20880, 6385}, {21039, 312}, {21127, 18155}, {21795, 8}, {21808, 75}, {22053, 17206}, {22079, 1812}, {35310, 668}, {35326, 99}, {35338, 799}, {35341, 7257}, {40606, 33297}, {40983, 27}, {48151, 7199}
X(52020) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3779, 3688}, {6, 21746, 3271}, {6, 37580, 2175}, {37, 20683, 7064}, {37, 22277, 20683}, {42, 3778, 2092}, {42, 40952, 181}, {209, 37593, 40966}, {872, 3122, 21796}, {2667, 21035, 1500}, {4336, 42447, 3022}, {4890, 20683, 37}, {4890, 22277, 7064}, {5257, 22312, 210}

X(52021) = X(2)X(13)∩X(3)X(51012)

X(52021) lies on the cubic K452 and these lines: {2, 13}, {3, 51012}, {6, 9885}, {30, 41044}, {99, 9113}, {187, 524}, {230, 22573}, {298, 51224}, {299, 41134}, {395, 543}, {511, 35304}, {531, 6782}, {532, 35297}, {533, 8594}, {542, 13349}, {1992, 11153}, {3363, 33474}, {5182, 36760}, {5460, 31710}, {5464, 8593}, {5472, 33475}, {5615, 14848}, {5617, 11296}, {6781, 44385}, {7685, 41047}, {8369, 51202}, {8724, 51015}, {9084, 9203}, {9114, 16530}, {9116, 16963}, {9123, 13305}, {9167, 44382}, {9761, 11159}, {9763, 41745}, {9855, 33377}, {9886, 51798}, {11145, 13859}, {11486, 22580}, {11543, 31696}, {14538, 35931}, {15534, 19781}, {16961, 22572}, {18581, 22576}, {21159, 25406}, {21402, 37341}, {22490, 23005}, {23303, 33477}, {25555, 37340}, {26613, 37786}, {30455, 51226}, {31173, 44383}, {32985, 51011}, {33007, 34508}, {35287, 51201}, {35690, 49861}, {35691, 49812}, {35696, 49948}, {35697, 49906}, {35932, 41022}, {41045, 44219}, {47863, 51484}, {49889, 49901}, {49933, 49953}

X(52021) = midpoint of X(i) and X(j) for these {i,j}: {2, 8595}, {16, 5463}, {99, 37785}, {298, 51224}, {2482, 9115}, {5464, 22998}, {5979, 51485}, {6779, 50858}, {9114, 22496}
X(52021) = reflection of X(i) in X(j) for these {i,j}: {5459, 6672}, {6108, 45880}, {22573, 230}, {22574, 395}, {31173, 44383}, {31696, 11543}, {31710, 5460}, {33517, 5459}, {41045, 44219}, {41047, 7685}
X(52021) = orthoptic-circle-of-Steiner-inellipse-inverse of X(9762)
X(52021) = circumcircle-of-outer-Napoleon-triangle-inverse of X(22492)
X(52021) = X(i)-isoconjugate of X(j) for these (i,j): {923, 42035}, {9202, 23894}
X(52021) = X(2482)-Dao conjugate of X(42035)
X(52021) = crossdifference of every pair of points on line {6137, 9178}
X(52021) = barycentric product X(i)*X(j) for these {i,j}: {524, 37786}, {3266, 41406}, {5468, 27551}
X(52021) = barycentric quotient X(i)/X(j) for these {i,j}: {524, 42035}, {5467, 9202}, {21466, 36307}, {27551, 5466}, {37786, 671}, {41406, 111}
X(52021) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 12155, 40671}, {9114, 16530, 22496}, {16961, 22578, 22572}, {47363, 47364, 22492}

X(52022) = X(2)X(14)∩X(3)X(51015)

X(52022) lies on the cubic K452 and these lines: {2, 14}, {3, 51015}, {6, 9886}, {30, 41045}, {99, 9112}, {187, 524}, {230, 22574}, {298, 41134}, {299, 51224}, {396, 543}, {511, 35303}, {530, 6783}, {532, 8595}, {533, 35297}, {542, 13350}, {1992, 11154}, {3363, 33475}, {3390, 35686}, {5182, 36759}, {5459, 31709}, {5463, 8593}, {5471, 33474}, {5611, 14848}, {5613, 11295}, {6781, 44384}, {7684, 41046}, {8369, 51205}, {8724, 51012}, {9084, 9202}, {9114, 16962}, {9116, 16529}, {9123, 13304}, {9167, 44383}, {9761, 41746}, {9763, 11159}, {9855, 33376}, {9885, 51798}, {11146, 13858}, {11485, 22579}, {11542, 31695}, {14539, 35932}, {15534, 19780}, {16960, 22571}, {18582, 22575}, {21158, 25406}, {21401, 37340}, {22489, 23004}, {23302, 33476}, {25555, 37341}, {26613, 37785}, {30454, 51226}, {31173, 44382}, {32985, 51014}, {33007, 34509}, {35287, 51204}, {35692, 49947}, {35693, 49905}, {35694, 49862}, {35695, 49813}, {35931, 41023}, {47864, 51485}, {49890, 49902}, {49934, 49952}

X(52022) = midpoint of X(i) and X(j) for these {i,j}: {2, 8594}, {15, 5464}, {99, 37786}, {299, 51224}, {2482, 9117}, {5463, 22997}, {5978, 51484}, {6780, 50855}, {9116, 22495}
X(52022) = reflection of X(i) in X(j) for these {i,j}: {5460, 6671}, {6109, 45879}, {22573, 396}, {22574, 230}, {31173, 44382}, {31695, 11542}, {31709, 5459}, {33518, 5460}, {41046, 7684}
X(52022) = orthoptic-circle-of-Steiner-inellipse-inverse of X(9760)
X(52022) = circumcircle-of-inner-Napoleon-triangle-inverse of X(22491)
X(52022) = X(i)-isoconjugate of X(j) for these (i,j): {923, 42036}, {9203, 23894}
X(52022) = X(2482)-Dao conjugate of X(42036)
X(52022) = crossdifference of every pair of points on line {6138, 9178}
X(52022) = barycentric product X(i)*X(j) for these {i,j}: {524, 37785}, {3266, 41407}, {5468, 27550}
X(52022) = barycentric quotient X(i)/X(j) for these {i,j}: {524, 42036}, {5467, 9203}, {21467, 36310}, {27550, 5466}, {37785, 671}, {41407, 111}
X(52022) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 12154, 40672}, {9116, 16529, 22495}, {16960, 22577, 22571}, {47361, 47362, 22491}

X(52023) = X(6)X(7)∩X(11)X(21346)

X(52023) lies on the cubic K1089 and these lines: {6, 7}, {11, 21346}, {34, 1892}, {37, 226}, {45, 8232}, {53, 342}, {56, 28081}, {57, 583}, {65, 21867}, {73, 3649}, {77, 17392}, {85, 141}, {142, 1212}, {212, 11246}, {218, 24779}, {223, 4328}, {241, 17245}, {269, 4675}, {279, 4648}, {307, 1213}, {347, 16777}, {348, 15668}, {354, 1827}, {388, 3242}, {481, 45704}, {580, 24470}, {581, 6147}, {594, 1441}, {664, 17390}, {908, 25067}, {946, 1547}, {990, 1062}, {1100, 43035}, {1104, 4298}, {1146, 16608}, {1229, 1233}, {1279, 12573}, {1362, 13476}, {1442, 43066}, {1445, 17337}, {1446, 18635}, {1453, 4355}, {1565, 24220}, {1699, 18216}, {1742, 8255}, {1836, 4319}, {1841, 5236}, {2257, 3772}, {2294, 6046}, {3120, 21955}, {3589, 41246}, {3664, 6610}, {3671, 3755}, {3672, 3782}, {3739, 9436}, {3756, 28087}, {3925, 21039}, {3936, 45744}, {4334, 25557}, {4353, 5717}, {4361, 6604}, {4364, 26125}, {4371, 32003}, {4395, 32007}, {4402, 32098}, {4554, 25660}, {4657, 40719}, {4851, 9312}, {4859, 16572}, {4864, 10106}, {4869, 43983}, {4916, 25718}, {5290, 7174}, {5434, 51422}, {6173, 7271}, {6707, 17095}, {7190, 17395}, {7228, 40862}, {7263, 39126}, {7365, 37674}, {8545, 17334}, {9440, 38454}, {10741, 45924}, {12848, 16885}, {15852, 21620}, {16732, 21933}, {17045, 17086}, {17078, 49738}, {17079, 17313}, {17246, 22464}, {17332, 17950}, {17369, 28739}, {17372, 25719}, {18634, 46835}, {20195, 51302}, {21239, 27471}, {21454, 40688}, {24471, 50011}, {24554, 31019}, {24796, 28017}, {25964, 30807}, {26669, 31053}, {35338, 41548}, {37633, 43036}, {38053, 42314}, {40892, 49733}

X(52023) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 43915}, {226, 21808}, {4551, 7178}
X(52023) = X(21808)-cross conjugate of X(3925)
X(52023) = X(i)-isoconjugate of X(j) for these (i,j): {21, 1174}, {58, 6605}, {81, 10482}, {284, 2346}, {1170, 2328}, {1172, 47487}, {1803, 4183}, {2194, 32008}, {2332, 40443}
X(52023) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 6605}, {21, 40606}, {142, 2287}, {333, 1212}, {1111, 18155}, {1170, 36908}, {1174, 40611}, {1214, 32008}, {2346, 40590}, {10482, 40586}
X(52023) = crosspoint of X(226) and X(1446)
X(52023) = crossdifference of every pair of points on line {926, 21789}
X(52023) = barycentric product X(i)*X(j) for these {i,j}: {7, 3925}, {10, 10481}, {12, 17169}, {65, 20880}, {85, 21808}, {142, 226}, {321, 1418}, {349, 1475}, {354, 1441}, {523, 35312}, {1018, 23599}, {1088, 21039}, {1212, 1446}, {1229, 1427}, {1233, 1400}, {2171, 16708}, {3668, 4847}, {4077, 35338}, {4552, 21104}, {4566, 6362}, {6354, 16713}, {6358, 18164}, {24002, 35310}, {40216, 43915}
X(52023) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 6605}, {42, 10482}, {65, 2346}, {73, 47487}, {142, 333}, {226, 32008}, {354, 21}, {1212, 2287}, {1233, 28660}, {1400, 1174}, {1418, 81}, {1427, 1170}, {1439, 40443}, {1446, 31618}, {1475, 284}, {1827, 4183}, {1855, 2322}, {2293, 2328}, {2488, 21789}, {3668, 21453}, {3925, 8}, {4566, 6606}, {4847, 1043}, {6362, 7253}, {10481, 86}, {16713, 7058}, {17169, 261}, {17194, 1098}, {18164, 2185}, {20880, 314}, {21039, 200}, {21104, 4560}, {21127, 1021}, {21795, 220}, {21808, 9}, {22053, 283}, {23599, 7199}, {35310, 644}, {35312, 99}, {35326, 5546}, {35338, 643}, {35341, 7259}, {40983, 2299}, {43915, 1621}, {48151, 3737}
X(52023) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 948, 6}, {7, 6180, 17365}, {7, 37800, 5228}, {142, 10481, 1418}, {226, 1427, 17056}, {226, 3668, 37}, {226, 6354, 4415}, {226, 16888, 41003}, {241, 21617, 17245}, {5228, 6180, 23144}, {5228, 37800, 17366}

X(52024) = X(1)X(9551)∩X(7)X(8049)

X(52024) lies on the Jerabek circumhyperbola of the intouch triangle, the, cubic K1089, and these lines: {1, 9551}, {7, 8049}, {42, 181}, {55, 34429}, {65, 1418}, {73, 3649}, {142, 39046}, {145, 37558}, {256, 13265}, {941, 43739}, {1064, 1537}, {1071, 3931}, {1100, 51659}, {1214, 15185}, {1284, 2594}, {1404, 2309}, {1409, 2293}, {1441, 39775}, {1442, 46153}, {1457, 39782}, {1818, 3696}, {1827, 1880}, {2171, 2667}, {3747, 21741}, {3870, 39773}, {3896, 39774}, {4068, 4559}, {4331, 45638}, {4334, 5586}, {4343, 14100}, {4424, 11570}, {4551, 29822}, {4552, 25295}, {10052, 24248}, {17077, 17135}, {21035, 21794}, {21859, 22279}, {39778, 46177}, {40591, 42443}

X(52024) = reflection of X(i) in X(j) for these {i,j}: {1, 50189}, {3588, 40600}
X(52024) = X(7)-Ceva conjugate of X(1400)
X(52024) = X(i)-isoconjugate of X(j) for these (i,j): {21, 8049}, {261, 40504}, {284, 39735}, {314, 34444}, {333, 39797}, {2185, 40515}, {2194, 40005}, {6577, 18155}
X(52024) = X(i)-Dao conjugate of X(j) for these (i,j): {8, 42}, {1214, 40005}, {8049, 40611}, {39735, 40590}
X(52024) = crosspoint of X(7) and X(17077)
X(52024) = crosssum of X(i) and X(j) for these (i,j): {1, 24220}, {663, 17197}
X(52024) = barycentric product X(i)*X(j) for these {i,j}: {7, 40586}, {42, 17077}, {56, 21070}, {57, 22271}, {65, 16552}, {73, 17911}, {181, 29767}, {225, 22126}, {226, 8053}, {1020, 50518}, {1400, 17135}, {1402, 18137}, {3294, 20614}, {4559, 8714}
X(52024) = barycentric quotient X(i)/X(j) for these {i,j}: {65, 39735}, {181, 40515}, {226, 40005}, {1400, 8049}, {1402, 39797}, {8053, 333}, {16552, 314}, {17077, 310}, {17135, 28660}, {17911, 44130}, {18137, 40072}, {20614, 40004}, {21070, 3596}, {22126, 332}, {22271, 312}, {29767, 18021}, {40586, 8}

X(52025) = X(1)X(224)∩X(2)X(272)

X(52025) lies on the Kiepert circumhyperbola of the anticomplementary triangle, the cubic K109, and these lines: {1, 224}, {2, 272}, {6, 50201}, {7, 17147}, {21, 32782}, {27, 18134}, {63, 69}, {78, 37179}, {141, 16368}, {194, 17778}, {326, 18651}, {329, 31015}, {343, 18642}, {379, 18139}, {579, 46885}, {908, 37185}, {1004, 37538}, {1211, 37323}, {1474, 28753}, {1726, 17776}, {1764, 37419}, {1848, 37181}, {2128, 40975}, {3305, 37169}, {3870, 5800}, {3882, 6515}, {3912, 18596}, {3936, 19645}, {3998, 41004}, {4197, 5333}, {4202, 19752}, {4259, 16465}, {4304, 48835}, {4313, 17676}, {4855, 37180}, {5271, 26130}, {5273, 30564}, {5279, 32858}, {5285, 35980}, {5739, 14021}, {5747, 27052}, {5905, 22021}, {6194, 37443}, {9965, 30579}, {10449, 26054}, {19716, 37266}, {20537, 36857}, {21376, 24683}, {30965, 37467}, {33171, 37175}

X(52025) = anticomplement of X(1751)
X(52025) = anticomplement of the isogonal conjugate of X(579)
X(52025) = anticomplement of the isotomic conjugate of X(18134)
X(52025) = isotomic conjugate of the polar conjugate of X(1714)
X(52025) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {28, 2997}, {31, 37652}, {48, 7538}, {108, 46107}, {209, 2895}, {579, 8}, {651, 8676}, {2149, 1331}, {2198, 1654}, {2352, 2}, {3190, 329}, {3868, 69}, {4306, 7}, {5125, 21270}, {8676, 37781}, {18134, 6327}, {22021, 1330}, {23800, 150}, {27396, 3436}, {40572, 72}, {41320, 5942}, {43060, 149}
X(52025) = X(i)-Ceva conjugate of X(j) for these (i,j): {27, 63}, {18134, 2}
X(52025) = X(19)-isoconjugate of X(39945)
X(52025) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 39945}, {306, 3998}
X(52025) = barycentric product X(i)*X(j) for these {i,j}: {69, 1714}, {86, 41507}
X(52025) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 39945}, {1714, 4}, {41507, 10}
X(52025) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 464, 63}, {306, 18650, 63}

X(52026) = X(1)X(227)∩X(2)X(515)

X(52026) lies on the cubic K758 and these lines: {1, 227}, {2, 515}, {3, 9}, {4, 3601}, {5, 5436}, {10, 6988}, {20, 908}, {30, 37713}, {33, 37380}, {35, 12705}, {40, 78}, {56, 33995}, {57, 6905}, {72, 9942}, {80, 20418}, {119, 6922}, {140, 5787}, {165, 5692}, {200, 3428}, {223, 46974}, {226, 50701}, {259, 8112}, {355, 5705}, {376, 5658}, {386, 22063}, {390, 946}, {404, 10884}, {474, 8726}, {517, 3158}, {550, 6259}, {581, 37554}, {631, 6245}, {912, 3928}, {938, 5882}, {944, 1210}, {950, 6848}, {960, 10268}, {990, 4256}, {1006, 7308}, {1012, 1750}, {1055, 46345}, {1064, 5269}, {1071, 15803}, {1125, 6864}, {1151, 19068}, {1152, 19067}, {1158, 35242}, {1376, 30503}, {1385, 6918}, {1394, 1745}, {1445, 18450}, {1449, 5396}, {1453, 37732}, {1519, 9580}, {1532, 3586}, {1538, 9668}, {1657, 22792}, {1697, 11491}, {1699, 5842}, {1706, 11499}, {1709, 5010}, {1870, 36636}, {2077, 10860}, {2096, 41561}, {2360, 13614}, {2829, 5660}, {2950, 33814}, {3207, 46830}, {3305, 37106}, {3306, 18444}, {3340, 21740}, {3341, 7114}, {3361, 12675}, {3452, 6987}, {3488, 7682}, {3516, 12136}, {3522, 6223}, {3523, 6705}, {3528, 12246}, {3553, 4255}, {3612, 5691}, {3651, 37551}, {3679, 5659}, {3868, 40249}, {3876, 9960}, {3895, 7982}, {3911, 5768}, {3929, 21165}, {4297, 6700}, {4652, 12528}, {4867, 7991}, {5126, 30283}, {5172, 30223}, {5204, 12680}, {5217, 12688}, {5251, 7987}, {5398, 16670}, {5437, 6911}, {5440, 6282}, {5450, 6986}, {5531, 22775}, {5534, 6762}, {5584, 18237}, {5603, 10389}, {5657, 46917}, {5709, 11523}, {5715, 11374}, {5719, 5805}, {5817, 50739}, {5881, 6734}, {5886, 38150}, {5923, 8808}, {5927, 16370}, {6253, 11375}, {6256, 6836}, {6409, 49234}, {6410, 49235}, {6684, 18231}, {6765, 22770}, {6826, 25525}, {6827, 30827}, {6828, 18492}, {6834, 9581}, {6835, 8227}, {6839, 31266}, {6840, 30852}, {6854, 41867}, {6855, 19925}, {6880, 31231}, {6883, 51780}, {6914, 18540}, {6915, 51683}, {6924, 37534}, {6934, 9579}, {6947, 20196}, {6954, 51755}, {6968, 51792}, {6970, 31190}, {6985, 37531}, {7070, 22350}, {7280, 10085}, {7415, 27398}, {7992, 16192}, {8583, 16293}, {9578, 10786}, {9612, 37468}, {9613, 15844}, {9614, 38038}, {9910, 37198}, {9943, 10270}, {10157, 16418}, {10164, 14647}, {10167, 16371}, {10310, 12565}, {10396, 12875}, {10531, 41864}, {10902, 11344}, {11012, 12687}, {11220, 13587}, {11227, 16417}, {11362, 20007}, {11496, 24644}, {12116, 50443}, {12196, 37479}, {12520, 25440}, {12608, 41869}, {12616, 31423}, {12666, 18254}, {12678, 15326}, {12679, 15338}, {12704, 41863}, {15696, 48664}, {15829, 45770}, {16132, 35979}, {16143, 35204}, {17057, 37714}, {17605, 36999}, {18283, 40836}, {18528, 22758}, {18634, 51775}, {19541, 24929}, {22341, 41403}, {24466, 46435}, {28381, 48897}, {31160, 34628}, {32141, 49163}, {37251, 37615}, {37282, 37561}, {37675, 46344}, {40945, 46021}, {41229, 49170}

X(52026) = midpoint of X(376) and X(5658)
X(52026) = reflection of X(14647) in X(10164)
X(52026) = Thomson-isogonal conjugate of X(57)
X(52026) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1490, 84}, {3, 5720, 9}, {3, 5777, 31424}, {3, 41854, 9841}, {40, 6261, 7971}, {78, 411, 40}, {404, 10884, 37526}, {944, 1512, 5727}, {944, 6927, 1210}, {1519, 37000, 9580}, {1750, 30282, 1012}, {2077, 50528, 10860}, {3149, 33597, 1}, {3523, 9799, 6705}, {4297, 6700, 6865}, {5440, 7580, 6282}, {5534, 11249, 6762}, {5703, 50700, 946}, {5709, 37700, 11523}, {5731, 35262, 3576}, {6261, 6796, 40}, {6905, 18446, 57}, {6911, 18443, 5437}, {10167, 16371, 21164}, {11374, 20420, 5715}, {11500, 37837, 1}, {12520, 25440, 37560}, {12671, 31424, 84}, {32555, 32556, 2182}

X(52027) = X(1)X(104)∩X(2)X(21164)

X(52027) lies on the cubic K758 and these lines: {1, 104}, {2, 21164}, {3, 9}, {4, 3911}, {5, 31190}, {10, 10270}, {20, 4652}, {21, 8726}, {35, 10085}, {36, 1709}, {40, 956}, {56, 12705}, {57, 1012}, {63, 6282}, {72, 18238}, {140, 6259}, {154, 392}, {165, 376}, {200, 2077}, {226, 2096}, {355, 38128}, {371, 19067}, {372, 19068}, {405, 37526}, {516, 45700}, {517, 3928}, {550, 5771}, {553, 5603}, {631, 5316}, {908, 6966}, {946, 3361}, {952, 34701}, {958, 37560}, {960, 17649}, {990, 4257}, {993, 30503}, {1000, 5882}, {1006, 10857}, {1071, 3601}, {1108, 4252}, {1145, 5881}, {1151, 49235}, {1152, 49234}, {1385, 7971}, {1394, 17102}, {1420, 12672}, {1466, 10396}, {1470, 30223}, {1512, 4316}, {1532, 31231}, {1587, 8987}, {1588, 13974}, {1656, 22792}, {1698, 6256}, {1699, 3582}, {1750, 6905}, {1753, 38870}, {2057, 38901}, {2256, 14597}, {2299, 4227}, {2801, 47375}, {2829, 5587}, {2932, 17857}, {2956, 10571}, {3306, 6912}, {3333, 11496}, {3359, 9623}, {3428, 10860}, {3515, 12136}, {3522, 9799}, {3523, 6223}, {3524, 5658}, {3526, 48664}, {3560, 37534}, {3577, 36279}, {3583, 31515}, {3586, 6938}, {3612, 15071}, {3624, 12608}, {3895, 38669}, {4189, 10884}, {4221, 10856}, {4292, 5715}, {4297, 10268}, {4304, 5768}, {4855, 12528}, {5122, 19541}, {5171, 12196}, {5204, 12688}, {5217, 12680}, {5234, 6684}, {5249, 6974}, {5267, 12520}, {5303, 9961}, {5432, 12678}, {5433, 12679}, {5435, 7682}, {5436, 9940}, {5437, 6913}, {5534, 26285}, {5693, 41389}, {5705, 6850}, {5722, 13226}, {5731, 35258}, {5745, 6916}, {5806, 37545}, {5811, 6700}, {5818, 44848}, {5886, 6173}, {5927, 16371}, {6257, 45553}, {6258, 45552}, {6261, 6875}, {6326, 51636}, {6692, 6939}, {6713, 46435}, {6762, 10306}, {6765, 11248}, {6796, 16192}, {6831, 9579}, {6833, 9612}, {6846, 12436}, {6901, 7989}, {6911, 18540}, {6914, 18443}, {6926, 12572}, {6945, 31224}, {6948, 51755}, {6950, 18446}, {7416, 23206}, {7701, 17653}, {7966, 30283}, {7991, 40256}, {8074, 18328}, {8166, 18483}, {8227, 41865}, {8583, 37561}, {8732, 37434}, {9613, 37002}, {9614, 10785}, {9948, 51576}, {9956, 40267}, {10156, 16857}, {10157, 16417}, {10165, 21151}, {10202, 28444}, {10267, 12687}, {10269, 12686}, {10470, 12547}, {10864, 11500}, {10902, 49170}, {11012, 12565}, {11220, 17549}, {11227, 16418}, {11372, 22753}, {11522, 11552}, {12115, 31434}, {12629, 49163}, {12651, 12704}, {12666, 20117}, {12676, 25522}, {13743, 37612}, {14872, 51380}, {16200, 24473}, {17606, 37001}, {18237, 31435}, {18242, 31423}, {18260, 18398}, {18481, 33899}, {18908, 46917}, {20418, 37704}, {21370, 36984}, {21669, 41547}, {22344, 37195}, {23708, 34789}, {24467, 37531}, {30389, 40257}, {30827, 37822}, {31730, 35514}, {33574, 38031}, {35262, 38693}, {38038, 50443}

X(52027) = midpoint of X(5603) and X(14646)
X(52027) = circumcircle-inverse of X(34143)
X(52027) = Thomson-isogonal conjugate of X(223)
X(52027) = crossdifference of every pair of points on line {6129, 46393}
X(52027) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 84, 1490}, {3, 5777, 5438}, {3, 7171, 5732}, {3, 7330, 936}, {3, 34862, 84}, {40, 12114, 12650}, {63, 6909, 6282}, {376, 21165, 165}, {631, 12246, 6260}, {956, 17613, 40}, {1158, 5450, 1}, {2096, 6935, 226}, {3358, 7171, 84}, {3359, 22758, 9623}, {3916, 37022, 40}, {4292, 6847, 5715}, {6950, 18446, 30282}, {7987, 7992, 6261}, {10167, 16370, 3576}, {10864, 35242, 11500}, {21151, 50739, 10165}, {30282, 30304, 18446}

X(52028) = X(2)X(154)∩X(6)X(64)

X(52028) lies on the cubic K907 and these lines: {2, 154}, {3, 9924}, {4, 32602}, {6, 64}, {20, 15583}, {66, 14528}, {69, 6696}, {182, 1498}, {206, 10541}, {221, 2330}, {235, 23300}, {264, 34808}, {511, 10606}, {895, 11598}, {1177, 14490}, {1192, 1843}, {1204, 12167}, {1350, 7691}, {1351, 3357}, {1352, 16196}, {1386, 7973}, {1428, 2192}, {1619, 17825}, {1974, 15811}, {2393, 31884}, {2781, 5102}, {2883, 3618}, {2935, 11579}, {3516, 6467}, {3523, 15585}, {3564, 37497}, {3589, 41735}, {3751, 12262}, {5050, 6000}, {5092, 17821}, {5093, 10250}, {5480, 5895}, {5878, 18583}, {5894, 51212}, {5921, 35602}, {5925, 31670}, {6225, 51171}, {6247, 6776}, {6759, 12017}, {6823, 46264}, {10117, 31860}, {10519, 23328}, {10605, 39588}, {10996, 17845}, {11381, 19118}, {11477, 12086}, {11744, 15118}, {12174, 21637}, {12283, 35477}, {12324, 34774}, {12791, 51741}, {13093, 34779}, {14216, 37476}, {14853, 15311}, {14927, 41362}, {15578, 34787}, {16010, 17847}, {17835, 32305}, {18405, 29012}, {18440, 20299}, {19357, 39874}, {20427, 21850}, {23042, 32063}, {23327, 51745}, {26937, 39871}, {34776, 34780}, {37201, 48905}

X(52028) = midpoint of X(5093) and X(35450)
X(52028) = reflection of X(i) in X(j) for these {i,j}: {154, 5085}, {5085, 10249}, {5093, 10250}, {10519, 23328}, {32063, 23042}
X(52028) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {182, 1498, 19132}, {1350, 44883, 8567}, {2935, 11579, 32276}, {5092, 39879, 17821}, {8549, 44883, 1350}, {36851, 44882, 17845}

X(52029) = X(1)X(41)∩X(8)X(76)

X(52029) lies on the cubic K1041 and these lines: {1, 41}, {8, 76}, {239, 335}, {257, 3057}, {292, 9472}, {517, 51929}, {666, 760}, {761, 919}, {869, 4475}, {960, 6559}, {984, 3802}, {1002, 5222}, {1027, 21385}, {1829, 36124}, {2113, 20670}, {2195, 3512}, {2223, 7291}, {2802, 36816}, {3010, 23694}, {3212, 34018}, {3497, 36057}, {3661, 3789}, {3783, 3864}, {4712, 24578}, {7179, 45974}, {16823, 31638}, {16830, 31637}, {20539, 20552}, {25050, 26582}, {25426, 49478}, {28043, 37555}, {28600, 29630}, {35628, 36796}

X(52029) = reflection of X(3799) in X(3789)
X(52029) = X(i)-cross conjugate of X(j) for these (i,j): {16514, 40773}, {30665, 3799}
X(52029) = X(i)-isoconjugate of X(j) for these (i,j): {241, 2344}, {518, 985}, {665, 4586}, {672, 14621}, {825, 918}, {870, 2223}, {871, 9455}, {1492, 2254}, {2284, 4817}, {3286, 40718}, {3509, 40764}, {3675, 5384}, {3912, 40746}, {4447, 40763}, {18206, 40747}
X(52029) = X(i)-Dao conjugate of X(j) for these (i,j): {518, 3789}, {2254, 38995}, {3263, 27481}, {3912, 19584}
X(52029) = cevapoint of X(984) and X(3783)
X(52029) = crosspoint of X(i) and X(j) for these (i,j): {291, 43751}, {1002, 2113}
X(52029) = crosssum of X(1001) and X(8301)
X(52029) = trilinear pole of line {1491, 2276}
X(52029) = barycentric product X(i)*X(j) for these {i,j}: {105, 3661}, {294, 7179}, {666, 1491}, {668, 29956}, {673, 984}, {788, 36803}, {824, 36086}, {869, 18031}, {1027, 3807}, {1438, 33931}, {1462, 3790}, {1469, 36796}, {2276, 2481}, {3250, 51560}, {3864, 6654}, {4505, 43929}, {4517, 34018}, {4522, 36146}, {6559, 7204}, {7146, 14942}, {7261, 40791}, {13576, 40773}, {18785, 30966}
X(52029) = barycentric quotient X(i)/X(j) for these {i,j}: {105, 14621}, {666, 789}, {673, 870}, {788, 665}, {869, 672}, {919, 1492}, {984, 3912}, {1027, 4817}, {1438, 985}, {1469, 241}, {1491, 918}, {2195, 2344}, {2276, 518}, {3250, 2254}, {3661, 3263}, {3736, 18206}, {3774, 20683}, {3781, 25083}, {3783, 17755}, {3799, 42720}, {3802, 27919}, {3862, 22116}, {3864, 40217}, {4481, 23829}, {4517, 3693}, {7146, 9436}, {7179, 40704}, {8852, 40764}, {16514, 8299}, {18031, 871}, {18785, 40718}, {18900, 9454}, {29956, 513}, {30966, 18157}, {32666, 825}, {36086, 4586}, {36803, 46132}, {40728, 2223}, {40773, 30941}, {40791, 4645}, {43921, 43266}, {51560, 37133}
X(52029) = {X(1),X(18785)}-harmonic conjugate of X(105)

X(52030) = X(1)X(3252)∩X(105)X(238)

X(52030) lies on the cubic K135 and these lines: {1, 3252}, {6, 51838}, {7, 14267}, {65, 30648}, {105, 238}, {239, 335}, {292, 1279}, {334, 32850}, {919, 41333}, {1001, 22116}, {1002, 40730}, {1015, 14665}, {1016, 14839}, {1024, 3572}, {1429, 1458}, {1509, 3110}, {1757, 18785}, {2113, 38989}, {2481, 41072}, {3286, 16876}, {4366, 40098}, {4514, 51859}, {4589, 4645}, {4649, 9505}, {5030, 16801}, {6654, 14621}, {7233, 14189}, {14942, 39914}, {16484, 40794}, {18792, 35333}, {20470, 34067}, {30669, 49704}, {35104, 36802}, {40724, 49537}

X(52030) = reflection of X(i) in X(j) for these {i,j}: {660, 9470}, {2113, 38989}
X(52030) = isogonal conjugate of X(8299)
X(52030) = isogonal conjugate of the complement of X(13576)
X(52030) = X(i)-cross conjugate of X(j) for these (i,j): {1, 105}, {6, 37128}, {512, 919}, {513, 660}, {2275, 34018}, {3271, 3572}, {4367, 927}, {21746, 694}, {40730, 292}, {44410, 35185}
X(52030) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8299}, {4, 20778}, {6, 17755}, {8, 51329}, {9, 34253}, {55, 39775}, {238, 518}, {239, 672}, {241, 3684}, {242, 1818}, {292, 27919}, {350, 2223}, {659, 1026}, {665, 3570}, {740, 3286}, {765, 38989}, {812, 2284}, {1025, 4435}, {1428, 3717}, {1429, 3693}, {1447, 2340}, {1458, 3685}, {1861, 7193}, {1914, 3912}, {1921, 9454}, {2110, 33701}, {2201, 25083}, {2210, 3263}, {2238, 18206}, {2254, 3573}, {2283, 3716}, {3252, 4366}, {3747, 30941}, {3932, 5009}, {4432, 34230}, {4447, 18786}, {5089, 20769}, {6184, 6654}, {8300, 22116}, {8632, 42720}, {9455, 18891}, {15507, 36819}, {18157, 41333}, {19557, 40781}, {20683, 33295}, {30940, 39258}, {39044, 40730}, {40217, 51328}
X(52030) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 8299}, {9, 17755}, {223, 39775}, {478, 34253}, {513, 38989}, {518, 9470}, {1921, 33675}, {3912, 36906}, {19557, 27919}, {20778, 36033}
X(52030) = cevapoint of X(i) and X(j) for these (i,j): {1, 291}, {292, 40730}, {513, 43921}, {875, 1015}, {1024, 3271}
X(52030) = trilinear pole of line {292, 659}
X(52030) = barycentric product X(i)*X(j) for these {i,j}: {57, 33676}, {75, 51866}, {105, 335}, {291, 673}, {292, 2481}, {294, 7233}, {334, 1438}, {337, 8751}, {666, 876}, {875, 36803}, {1027, 4562}, {1462, 4518}, {1911, 18031}, {3572, 51560}, {4444, 36086}, {4583, 43929}, {6185, 22116}, {6654, 30663}, {7077, 34018}, {13576, 37128}, {18785, 18827}, {24479, 40724}, {29956, 41072}, {40217, 51838}
X(52030) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17755}, {6, 8299}, {48, 20778}, {56, 34253}, {57, 39775}, {105, 239}, {238, 27919}, {291, 3912}, {292, 518}, {294, 3685}, {295, 25083}, {335, 3263}, {604, 51329}, {660, 42720}, {666, 874}, {673, 350}, {741, 18206}, {813, 1026}, {875, 665}, {876, 918}, {884, 4435}, {919, 3573}, {1015, 38989}, {1024, 3716}, {1027, 812}, {1416, 1429}, {1438, 238}, {1462, 1447}, {1911, 672}, {1922, 2223}, {2111, 33701}, {2195, 3684}, {2196, 1818}, {2481, 1921}, {3252, 4712}, {3572, 2254}, {4876, 3717}, {6654, 39044}, {7077, 3693}, {7233, 40704}, {8751, 242}, {10099, 24459}, {13576, 3948}, {14598, 9454}, {14942, 3975}, {18031, 18891}, {18268, 3286}, {18785, 740}, {18827, 18157}, {18897, 9455}, {22116, 4437}, {29956, 30665}, {30648, 40781}, {30663, 40217}, {32658, 7193}, {33676, 312}, {34018, 18033}, {34067, 2284}, {36057, 20769}, {36086, 3570}, {36796, 4087}, {37128, 30941}, {40724, 18037}, {40730, 6184}, {40754, 1281}, {43921, 27918}, {43929, 659}, {51560, 27853}, {51838, 6654}, {51856, 40730}, {51858, 2340}, {51866, 1}, {51987, 15507}

X(52031) = X(2)X(514)∩X(57)X(88)

X(52031) lies on the cubic K972 and these lines: {1, 1168}, {2, 514}, {9, 19618}, {43, 994}, {55, 14190}, {57, 88}, {63, 3257}, {92, 4997}, {106, 614}, {165, 901}, {354, 34230}, {517, 14260}, {903, 31164}, {908, 2397}, {1243, 34465}, {1320, 3577}, {1435, 7128}, {1465, 24029}, {1836, 19636}, {2051, 4080}, {3306, 4638}, {3576, 47645}, {4013, 29671}, {4219, 36125}, {4778, 47045}, {4792, 25415}, {4850, 30575}, {5119, 39148}, {5219, 14628}, {5256, 47056}, {5587, 36590}, {5919, 45247}, {17591, 36814}, {17595, 51908}, {17596, 17960}, {20568, 34020}, {23345, 34583}, {31224, 31227}, {36038, 46805}

X(52031) = X(4638)-Ceva conjugate of X(1022)
X(52031) = X(i)-cross conjugate of X(j) for these (i,j): {1465, 88}, {24028, 22464}, {42754, 1022}
X(52031) = X(i)-isoconjugate of X(j) for these (i,j): {6, 36944}, {44, 104}, {55, 40218}, {519, 909}, {900, 32641}, {902, 34234}, {1145, 41933}, {1309, 22086}, {1404, 51565}, {1635, 36037}, {1639, 2720}, {1795, 8756}, {1960, 13136}, {2251, 18816}, {2342, 3911}, {2401, 23344}, {2423, 17780}, {3285, 38955}, {3689, 34051}, {4358, 34858}, {4370, 10428}, {4768, 32669}, {4895, 37136}, {14418, 36110}, {14578, 38462}, {16082, 23202}, {17455, 40437}, {22356, 36123}
X(52031) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 36944}, {9, 45247}, {44, 40613}, {104, 40595}, {214, 2245}, {223, 40218}, {519, 23980}, {908, 51583}, {1145, 2325}, {1635, 3259}, {1639, 38981}, {3762, 46398}, {4358, 16586}, {8756, 25640}, {9460, 18816}, {14418, 39004}, {34234, 40594}
X(52031) = cevapoint of X(2183) and X(34586)
X(52031) = crosssum of X(44) and X(17455)
X(52031) = trilinear pole of line {517, 1769}
X(52031) = crossdifference of every pair of points on line {902, 4895}
X(52031) = barycentric product X(i)*X(j) for these {i,j}: {75, 14260}, {88, 908}, {106, 3262}, {269, 51984}, {517, 903}, {679, 1145}, {901, 36038}, {1022, 2397}, {1320, 22464}, {1465, 4997}, {1769, 4555}, {2183, 20568}, {3257, 10015}, {4618, 23757}, {4674, 17139}, {5376, 42754}
X(52031) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 36944}, {57, 40218}, {88, 34234}, {106, 104}, {517, 519}, {901, 36037}, {903, 18816}, {908, 4358}, {1022, 2401}, {1145, 4738}, {1168, 40437}, {1320, 51565}, {1457, 1319}, {1465, 3911}, {1769, 900}, {1785, 38462}, {1875, 1877}, {2183, 44}, {2397, 24004}, {2427, 1023}, {2804, 4768}, {3257, 13136}, {3262, 3264}, {3310, 1635}, {4246, 46541}, {4674, 38955}, {4792, 36921}, {4997, 36795}, {6336, 16082}, {6735, 4723}, {9456, 909}, {10015, 3762}, {14260, 1}, {14571, 8756}, {15507, 4432}, {16586, 51583}, {17139, 30939}, {17757, 3992}, {21801, 3943}, {22350, 5440}, {23838, 43728}, {23981, 23703}, {24028, 1145}, {32659, 14578}, {32665, 32641}, {34230, 36819}, {34586, 214}, {36058, 1795}, {36125, 36123}, {42753, 1647}, {42757, 23757}, {43922, 15635}, {46393, 1639}, {51377, 21805}, {51409, 4975}, {51423, 4742}, {51433, 4487}, {51984, 341}
X(52031) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {88, 40215, 57}, {88, 47058, 40215}

X(52032) = X(2)X(311)∩X(3)X(49)

X(52032) lies on the Kiepert circumhyperbola of the medial triangle, the cubic K612, and these lines: {2, 311}, {3, 49}, {5, 8800}, {6, 6503}, {39, 37649}, {51, 23181}, {52, 3133}, {97, 44180}, {99, 275}, {114, 136}, {216, 343}, {324, 14570}, {418, 44716}, {467, 14576}, {511, 3135}, {569, 16391}, {571, 1993}, {642, 8967}, {1370, 7710}, {1994, 4558}, {2482, 39013}, {3003, 6515}, {3051, 47406}, {3260, 43988}, {3313, 34452}, {4993, 18354}, {6337, 11427}, {6509, 11064}, {7499, 15819}, {7763, 11547}, {8961, 11091}, {8962, 13882}, {9967, 23195}, {13567, 34990}, {14806, 15066}, {15850, 37454}, {18315, 45832}, {19161, 36790}, {21243, 50648}, {26874, 41716}, {27365, 50947}, {28710, 37188}, {34986, 39805}

X(52032) = complement of X(5392)
X(52032) = complement of the isogonal conjugate of X(571)
X(52032) = complement of the isotomic conjugate of X(1993)
X(52032) = isotomic conjugate of the polar conjugate of X(52)
X(52032) = isogonal conjugate of the polar conjugate of X(39113)
X(52032) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 34825}, {19, 5449}, {24, 20305}, {31, 343}, {47, 141}, {48, 11585}, {163, 924}, {184, 18588}, {560, 7746}, {563, 3}, {571, 10}, {603, 18638}, {924, 21253}, {1147, 18589}, {1748, 21243}, {1973, 9722}, {1993, 2887}, {2148, 1216}, {2180, 1209}, {7763, 21235}, {9247, 577}, {18605, 3741}, {23995, 46184}, {30451, 34846}, {32739, 46389}, {34948, 116}, {34952, 8287}, {41679, 21259}, {44077, 226}, {44179, 626}
X(52032) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 343}, {99, 924}, {311, 5562}, {328, 45780}, {7763, 39113}, {39113, 52}, {40697, 8905}
X(52032) = X(216)-cross conjugate of X(40678)
X(52032) = X(i)-isoconjugate of X(j) for these (i,j): {4, 2168}, {19, 96}, {91, 8882}, {92, 41271}, {847, 2148}, {1820, 8884}, {1973, 34385}, {2165, 2190}, {2167, 14593}, {24006, 32692}
X(52032) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 343}, {5, 2165}, {6, 96}, {54, 577}, {134, 6753}, {135, 15422}, {216, 847}, {523, 47421}, {570, 1216}, {2168, 36033}, {5408, 16032}, {5409, 16037}, {6337, 34385}, {8882, 34116}, {14593, 40588}, {22391, 41271}
X(52032) = crosspoint of X(i) and X(j) for these (i,j): {2, 1993}, {467, 39114}, {7763, 9723}
X(52032) = crosssum of X(i) and X(j) for these (i,j): {6, 2165}, {3049, 41221}
X(52032) = crossdifference of every pair of points on line {2501, 34952}
X(52032) = barycentric product X(i)*X(j) for these {i,j}: {3, 39113}, {5, 9723}, {47, 18695}, {52, 69}, {216, 7763}, {304, 2180}, {311, 1147}, {317, 5562}, {343, 1993}, {394, 467}, {571, 28706}, {1273, 5961}, {3133, 20563}, {3926, 14576}, {6503, 39114}, {6563, 23181}, {16697, 42700}, {18605, 42698}, {31635, 44716}, {40678, 40697}, {44179, 44706}
X(52032) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 96}, {5, 847}, {24, 8884}, {47, 2190}, {48, 2168}, {51, 14593}, {52, 4}, {69, 34385}, {184, 41271}, {216, 2165}, {317, 8795}, {343, 5392}, {418, 2351}, {467, 2052}, {563, 2148}, {571, 8882}, {1147, 54}, {1154, 5962}, {1993, 275}, {2180, 19}, {3133, 24}, {5408, 16037}, {5409, 16032}, {5562, 68}, {5891, 51833}, {5961, 1141}, {6753, 15422}, {7763, 276}, {8905, 40698}, {9723, 95}, {11547, 8794}, {14570, 30450}, {14576, 393}, {18695, 20571}, {23181, 925}, {27374, 27367}, {30451, 2623}, {32661, 32692}, {39110, 14518}, {39113, 264}, {40678, 254}, {41679, 16813}, {44179, 40440}, {44706, 91}
X(52032) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1147, 15827, 3}, {5408, 5409, 1147}, {36212, 46832, 394}

X(52033) = X(1)X(1826)∩X(6)X(19)

X(52033) lies on the cubic K678 and these lines: {1, 1826}, {6, 19}, {33, 430}, {48, 21147}, {77, 24315}, {78, 21076}, {198, 1426}, {225, 3553}, {281, 1870}, {393, 8776}, {407, 4272}, {913, 32674}, {998, 1474}, {1068, 21077}, {1172, 17098}, {1411, 7129}, {1435, 1465}, {1718, 1723}, {1875, 3209}, {1953, 34036}, {2260, 2333}, {2594, 8736}, {3068, 8938}, {3069, 8942}, {3554, 8755}, {5115, 37238}, {5233, 17923}, {5236, 37800}, {5256, 5307}, {5317, 41505}, {5336, 7106}, {6505, 20930}, {6520, 36127}, {7079, 11108}, {11323, 37593}, {40149, 45126}

X(52033) = isogonal conjugate of X(6513)
X(52033) = polar conjugate of X(20570)
X(52033) = polar conjugate of the isotomic conjugate of X(46)
X(52033) = X(i)-Ceva conjugate of X(j) for these (i,j): {393, 19}, {1063, 33}
X(52033) = X(2178)-cross conjugate of X(19)
X(52033) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6513}, {2, 1069}, {3, 2994}, {4, 6512}, {48, 20570}, {63, 90}, {69, 2164}, {219, 7318}, {222, 36626}, {348, 7072}, {394, 7040}, {6332, 36082}, {6505, 7042}
X(52033) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 6513}, {63, 3926}, {90, 3162}, {1069, 32664}, {1249, 20570}, {2994, 36103}, {6512, 36033}, {23978, 24006}
X(52033) = cevapoint of X(8938) and X(8942)
X(52033) = crosspoint of X(7128) and X(36127)
X(52033) = crosssum of X(394) and X(6512)
X(52033) = barycentric product X(i)*X(j) for these {i,j}: {1, 1068}, {4, 46}, {19, 5905}, {25, 20930}, {27, 21853}, {28, 21077}, {34, 5552}, {65, 3559}, {92, 2178}, {158, 3157}, {225, 3193}, {318, 1406}, {393, 6505}, {653, 46389}, {1079, 7040}, {1783, 21188}, {1880, 31631}, {1897, 51650}, {6506, 7128}, {6511, 6520}
X(52033) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 20570}, {6, 6513}, {19, 2994}, {25, 90}, {31, 1069}, {33, 36626}, {34, 7318}, {46, 69}, {48, 6512}, {1068, 75}, {1096, 7040}, {1406, 77}, {1973, 2164}, {2178, 63}, {2212, 7072}, {3157, 326}, {3193, 332}, {3559, 314}, {5552, 3718}, {5905, 304}, {6505, 3926}, {6511, 1102}, {20930, 305}, {21077, 20336}, {21188, 15413}, {21853, 306}, {46389, 6332}, {51650, 4025}
X(52033) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 1880, 19}, {34, 2331, 19}, {607, 1841, 19}, {608, 14571, 19}

X(52034) = X(3)X(76)∩X(32)X(148)

X(52034) lies on the cubic K937 and these lines: {3, 76}, {32, 148}, {83, 115}, {147, 7752}, {182, 14651}, {194, 8178}, {315, 5984}, {542, 5207}, {543, 19570}, {620, 46226}, {671, 3407}, {804, 18105}, {1281, 33941}, {1691, 14568}, {1916, 7760}, {2854, 31998}, {3124, 13519}, {3818, 12177}, {3934, 8290}, {4577, 7668}, {5026, 24273}, {5103, 12216}, {5149, 8289}, {5171, 13172}, {5182, 7884}, {5368, 12206}, {5986, 33651}, {6055, 9302}, {6321, 12110}, {7061, 33940}, {7746, 10131}, {7751, 8782}, {7757, 42535}, {7769, 51872}, {7786, 44531}, {7790, 10347}, {7793, 20094}, {7796, 46236}, {7802, 9862}, {7808, 10353}, {7811, 11177}, {7822, 33015}, {7834, 10352}, {7844, 10334}, {7858, 12830}, {7859, 9478}, {7861, 10333}, {7902, 10345}, {7919, 11646}, {9878, 11648}, {10796, 38732}, {11599, 12194}, {12195, 13178}, {13193, 16278}, {13196, 33228}, {13885, 49266}, {13938, 49267}, {20081, 39603}, {22505, 48674}, {23514, 32135}, {23698, 43453}, {35705, 38224}, {39836, 41262}, {46294, 51441}

X(52034) = midpoint of X(32751) and X(32752)
X(52034) = reflection of X(i) in X(j) for these {i,j}: {99, 5152}, {5152, 5939}, {32528, 115}, {35464, 12042}
X(52034) = circumcircle-inverse of X(1078)
X(52034) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 5989, 99}, {98, 99, 1078}, {115, 4027, 83}, {5989, 12188, 76}, {7782, 13188, 99}, {7797, 11606, 115}, {12188, 14880, 98}

X(52035) = X(4)X(542)∩X(99)X(523)

X(52035) lies on the cubic K186 and these lines: {4, 542}, {98, 5968}, {99, 523}, {107, 35191}, {110, 5466}, {114, 36875}, {2407, 2793}, {2408, 48947}, {3564, 34174}, {4558, 48952}, {6033, 51258}, {6054, 16092}, {8593, 17948}, {9160, 39450}, {9166, 14995}, {9213, 14480}, {10555, 14683}, {10556, 24981}, {12117, 38679}, {14559, 34574}, {14977, 40866}, {35278, 50941}, {50641, 51405}

X(52035) = reflection of X(i) in X(j) for these {i,j}: {98, 34810}, {671, 9214}, {36875, 114}
X(52035) = antigonal image of X(36875)
X(52035) = symgonal image of X(34810)
X(52035) = X(i)-isoconjugate of X(j) for these (i,j): {351, 8773}, {690, 36051}, {896, 35364}, {2642, 2987}
X(52035) = X(i)-Dao conjugate of X(j) for these (i,j): {114, 690}, {351, 39072}, {868, 51429}, {2642, 39069}, {14417, 35067}, {15899, 35364}
X(52035) = trilinear pole of line {230, 4226}
X(52035) = barycentric product X(i)*X(j) for these {i,j}: {230, 892}, {671, 4226}, {691, 51481}, {1733, 36085}, {34174, 50941}
X(52035) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 35364}, {230, 690}, {460, 14273}, {691, 2987}, {892, 8781}, {1692, 351}, {3564, 14417}, {4226, 524}, {5477, 1649}, {6782, 9204}, {6783, 9205}, {8772, 2642}, {12829, 11183}, {32729, 32654}, {34174, 50942}, {36085, 8773}, {36142, 36051}, {42663, 21906}, {51481, 35522}
X(52035) = {X(895),X(48983)}-harmonic conjugate of X(671)

X(52036) = X(2)X(99)∩X(182)X(1511)

X(52036) lies on the cubic K304 and these lines: {2, 99}, {182, 1511}, {549, 44386}, {1296, 43656}, {1995, 9177}, {2088, 15066}, {2780, 4550}, {3849, 40115}, {5028, 36696}, {5118, 34417}, {5569, 46066}, {5651, 9129}, {7495, 47325}, {8588, 21395}, {8589, 35936}, {9734, 14662}, {10411, 11004}, {11171, 33900}, {11178, 36832}, {14654, 43461}, {15515, 38524}, {15921, 38698}, {18570, 38623}, {18580, 34840}, {20112, 46068}, {23699, 50008}, {26316, 34106}, {34013, 44420}, {35485, 38803}, {37470, 47049}, {44814, 46609}

X(52036) = reflection of X(i) in X(j) for these {i,j}: {15563, 15560}, {34010, 14650}
X(52036) = Brocard-circle-inverse of X(1511)
X(52036) = psi-transform of X(323)
X(52036) = {X(2),X(11185)}-harmonic conjugate of X(13162)

X(52037) = X(1)X(84)∩X(7)X(92)

X(52037) lies on the cubic K964 and these lines: {1, 84}, {4, 8810}, {7, 92}, {57, 282}, {63, 268}, {65, 7157}, {72, 856}, {226, 1439}, {280, 3868}, {285, 40431}, {304, 7055}, {306, 51368}, {942, 40836}, {971, 44695}, {1214, 41087}, {1412, 8059}, {1708, 7367}, {2188, 10167}, {2349, 37141}, {3341, 44547}, {4292, 51490}, {4641, 36049}, {5728, 7008}, {7129, 37543}, {8581, 39796}, {9118, 12664}, {13138, 16465}, {22124, 23122}

X(52037) = reflection of X(8807) in X(36908)
X(52037) = X(189)-Ceva conjugate of X(8808)
X(52037) = X(i)-cross conjugate of X(j) for these (i,j): {65, 1439}, {1901, 43724}
X(52037) = X(i)-isoconjugate of X(j) for these (i,j): {9, 3194}, {21, 2331}, {25, 27398}, {27, 7074}, {28, 2324}, {29, 198}, {33, 1817}, {40, 1172}, {55, 41083}, {81, 40971}, {107, 10397}, {112, 8058}, {162, 14298}, {196, 2328}, {208, 2287}, {221, 2322}, {223, 4183}, {227, 2326}, {270, 21871}, {281, 2360}, {284, 7952}, {322, 2204}, {329, 2299}, {333, 3195}, {347, 2332}, {393, 1819}, {607, 8822}, {1043, 3209}, {1474, 7080}, {2187, 31623}, {2189, 21075}, {2193, 47372}, {7078, 8748}, {7156, 41082}
X(52037) = X(i)-Dao conjugate of X(j) for these (i,j): {125, 14298}, {196, 36908}, {223, 41083}, {226, 329}, {478, 3194}, {2322, 3341}, {2324, 40591}, {2331, 40611}, {6505, 27398}, {7080, 51574}, {7952, 40590}, {8058, 34591}, {8808, 14361}, {10397, 38985}, {40586, 40971}, {47345, 47372}
X(52037) = cevapoint of X(65) and X(1903)
X(52037) = crosspoint of X(i) and X(j) for these (i,j): {63, 1032}, {189, 41081}
X(52037) = crosssum of X(i) and X(j) for these (i,j): {19, 1033}, {55, 7156}, {198, 2331}
X(52037) = barycentric product X(i)*X(j) for these {i,j}: {21, 6355}, {37, 34400}, {63, 8808}, {72, 1440}, {73, 309}, {77, 39130}, {84, 307}, {85, 41087}, {189, 1214}, {226, 41081}, {268, 1446}, {271, 3668}, {280, 1439}, {285, 6356}, {306, 1422}, {348, 1903}, {525, 37141}, {1231, 1436}, {1409, 44190}, {1413, 20336}, {1427, 44189}, {1433, 1441}, {1812, 13853}, {2357, 7182}, {2358, 3926}, {8059, 14208}, {13138, 17094}, {44327, 51640}
X(52037) = barycentric quotient X(i)/X(j) for these {i,j}: {42, 40971}, {56, 3194}, {57, 41083}, {63, 27398}, {65, 7952}, {71, 2324}, {72, 7080}, {73, 40}, {77, 8822}, {84, 29}, {189, 31623}, {201, 21075}, {222, 1817}, {225, 47372}, {228, 7074}, {255, 1819}, {268, 2287}, {271, 1043}, {282, 2322}, {307, 322}, {309, 44130}, {603, 2360}, {647, 14298}, {656, 8058}, {822, 10397}, {1042, 208}, {1214, 329}, {1400, 2331}, {1402, 3195}, {1409, 198}, {1410, 221}, {1413, 28}, {1422, 27}, {1425, 227}, {1427, 196}, {1433, 21}, {1436, 1172}, {1439, 347}, {1440, 286}, {1446, 40701}, {1903, 281}, {2188, 2328}, {2192, 4183}, {2197, 21871}, {2208, 2299}, {2357, 33}, {2358, 393}, {3668, 342}, {6355, 1441}, {6612, 1396}, {7118, 2332}, {7129, 8748}, {8059, 162}, {8808, 92}, {13138, 36797}, {13853, 40149}, {17094, 17896}, {18210, 38357}, {22341, 7078}, {34400, 274}, {37141, 648}, {39130, 318}, {40836, 1896}, {41081, 333}, {41086, 44695}, {41087, 9}, {46881, 13614}, {51640, 14837}
X(52037) = {X(63),X(41081)}-harmonic conjugate of X(268)

X(52038) = X(6)X(523)∩X(98)X(843)

X(52038) lies on the cubics K150, K624, K625, the curve Q101, and these lines: {6, 523}, {98, 843}, {99, 249}, {115, 512}, {187, 690}, {230, 3569}, {248, 51480}, {290, 35146}, {325, 24284}, {524, 35522}, {526, 15993}, {598, 30491}, {647, 5642}, {669, 51869}, {826, 33694}, {878, 42660}, {1569, 3906}, {1649, 51927}, {1976, 32741}, {2501, 6531}, {3566, 3568}, {3800, 42345}, {5027, 9418}, {9154, 10630}, {10097, 16092}, {11183, 18872}, {14273, 33919}, {14559, 32313}, {14824, 32540}, {18800, 23878}, {35912, 46981}

X(52038) = reflection of X(i) in X(j) for these {i,j}: {325, 24284}, {3569, 230}
X(52038) = X(9154)-Ceva conjugate of X(51441)
X(52038) = X(i)-isoconjugate of X(j) for these (i,j): {325, 36142}, {511, 36085}, {662, 5968}, {671, 23997}, {691, 1959}, {799, 51980}, {877, 36060}, {892, 1755}, {897, 2421}, {923, 2396}, {3405, 36827}, {5380, 17209}, {8430, 24041}, {14966, 46277}, {32729, 46238}, {36045, 51438}
X(52038) = X(i)-Dao conjugate of X(j) for these (i,j): {297, 48317}, {325, 23992}, {511, 38988}, {877, 1560}, {892, 36899}, {1084, 5968}, {1648, 50567}, {1649, 2799}, {2396, 2482}, {2421, 6593}, {3005, 8430}, {3569, 21905}, {31654, 51438}, {38996, 51980}
X(52038) = cevapoint of X(690) and X(11183)
X(52038) = crosspoint of X(2966) and X(9154)
X(52038) = crosssum of X(i) and X(j) for these (i,j): {3569, 9155}, {5968, 8430}
X(52038) = trilinear pole of line {351, 1648}
X(52038) = crossdifference of every pair of points on line {511, 2421}
X(52038) = barycentric product X(i)*X(j) for these {i,j}: {98, 690}, {187, 43665}, {287, 14273}, {290, 351}, {468, 879}, {523, 5967}, {524, 2395}, {878, 44146}, {1648, 2966}, {1649, 9154}, {1821, 2642}, {1976, 35522}, {2422, 3266}, {4235, 51404}, {5468, 51441}, {6531, 14417}, {11183, 36897}, {14355, 51479}, {20021, 22105}, {21906, 43187}, {34369, 50942}, {41173, 51429}
X(52038) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 892}, {187, 2421}, {351, 511}, {468, 877}, {512, 5968}, {524, 2396}, {669, 51980}, {690, 325}, {878, 895}, {879, 30786}, {922, 23997}, {1648, 2799}, {1649, 50567}, {1910, 36085}, {1976, 691}, {2395, 671}, {2422, 111}, {2642, 1959}, {3124, 8430}, {4750, 51370}, {5967, 99}, {9125, 51438}, {9155, 15631}, {11183, 5976}, {14273, 297}, {14417, 6393}, {14419, 51369}, {14424, 51371}, {14443, 51429}, {14567, 14966}, {14601, 32729}, {15630, 9178}, {21839, 42717}, {21906, 3569}, {22105, 20022}, {33919, 868}, {34369, 50941}, {43665, 18023}, {44102, 4230}, {44814, 51383}, {51404, 14977}, {51441, 5466}, {51869, 36827}
X(52038) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {879, 5652, 36822}, {879, 35906, 1640}, {35906, 48452, 48721}

X(52039) = X(2)X(13)∩X(14)X(476)

X(52039) lies on the cubic K625 and these lines: {2, 13}, {14, 476}, {15, 15360}, {18, 48353}, {30, 30465}, {110, 22998}, {187, 1648}, {299, 4590}, {300, 40826}, {395, 523}, {471, 36306}, {524, 30454}, {533, 35314}, {2770, 5995}, {3181, 35511}, {3628, 11555}, {5318, 46858}, {5334, 45778}, {5486, 36296}, {5640, 33480}, {5642, 9115}, {5648, 11081}, {6104, 13858}, {6780, 14185}, {6792, 41407}, {8836, 32906}, {8929, 42489}, {9117, 41586}, {9205, 50942}, {10217, 11486}, {11080, 16645}, {11489, 51270}, {11542, 46862}, {11581, 16967}, {14559, 30455}, {15441, 18581}, {15768, 42942}, {16268, 46078}, {16963, 36211}, {18823, 23895}, {19106, 46856}, {22826, 47352}, {22997, 44555}, {30460, 31945}, {34395, 38414}, {36186, 36968}, {36953, 44382}, {36970, 44466}, {37641, 40578}, {40158, 43543}, {42099, 46860}

X(52039) = X(i)-Ceva conjugate of X(j) for these (i,j): {13, 30454}, {36307, 13}
X(52039) = X(i)-cross conjugate of X(j) for these (i,j): {9115, 524}, {30454, 13}
X(52039) = X(i)-isoconjugate of X(j) for these (i,j): {15, 897}, {298, 923}, {470, 36060}, {671, 2151}, {1094, 36307}, {6137, 36085}, {6149, 36310}, {7316, 44688}, {9207, 32679}, {17402, 23894}, {23870, 36142}, {34394, 46277}, {36128, 44718}
X(52039) = X(i)-Dao conjugate of X(j) for these (i,j): {15, 6593}, {298, 2482}, {470, 1560}, {671, 40578}, {1648, 9204}, {1649, 30465}, {6137, 38988}, {9213, 38994}, {14993, 36310}, {23870, 23992}
X(52039) = crosspoint of X(13) and X(36307)
X(52039) = trilinear pole of line {690, 9117}
X(52039) = crossdifference of every pair of points on line {15, 6137}
X(52039) = X(15360)-line conjugate of X(15)
X(52039) = barycentric product X(i)*X(j) for these {i,j}: {13, 524}, {16, 43084}, {187, 300}, {468, 40709}, {476, 9205}, {671, 30454}, {690, 23895}, {2153, 14210}, {2482, 36307}, {3266, 3457}, {5468, 20578}, {5642, 36308}, {5995, 35522}, {6390, 8737}, {9115, 11119}, {9117, 11118}, {9204, 36839}, {14417, 36306}, {14559, 23871}, {17403, 51479}, {36296, 44146}, {36299, 36890}, {41586, 51275}, {44690, 51655}
X(52039) = barycentric quotient X(i)/X(j) for these {i,j}: {13, 671}, {187, 15}, {300, 18023}, {351, 6137}, {468, 470}, {524, 298}, {690, 23870}, {922, 2151}, {1648, 30465}, {1649, 9204}, {1989, 36310}, {2153, 897}, {3292, 44718}, {3457, 111}, {5467, 17402}, {5477, 6782}, {5642, 41887}, {5995, 691}, {6138, 9213}, {8737, 17983}, {9115, 618}, {9117, 533}, {9205, 3268}, {9206, 34574}, {11080, 36307}, {14559, 23896}, {14560, 9207}, {14567, 34394}, {20578, 5466}, {23200, 46112}, {23895, 892}, {30454, 524}, {36296, 895}, {36299, 9214}, {40709, 30786}, {41586, 33529}, {43084, 301}, {44102, 8739}
X(52039) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 21466, 13}, {13, 21466, 8014}, {395, 11537, 18777}, {11537, 18777, 36299}

X(52040) = X(2)X(14)∩X(13)X(476)

X(52040) lies on the cubic K625 and these lines: {2, 14}, {13, 476}, {16, 15360}, {17, 48355}, {30, 30468}, {110, 22997}, {187, 1648}, {298, 4590}, {301, 40826}, {396, 523}, {470, 36309}, {524, 30455}, {532, 35315}, {2770, 5994}, {3180, 35511}, {3628, 11556}, {5321, 46859}, {5335, 45779}, {5486, 36297}, {5640, 33481}, {5642, 9117}, {5648, 11086}, {6105, 13859}, {6779, 14187}, {6792, 41406}, {8838, 32908}, {8930, 42488}, {9115, 41586}, {9204, 50942}, {10218, 11485}, {11085, 16644}, {11488, 51277}, {11543, 46863}, {11582, 16966}, {14559, 30454}, {15442, 18582}, {15769, 42943}, {16267, 46074}, {16962, 36210}, {18823, 23896}, {19107, 46857}, {22827, 47352}, {22998, 44555}, {30463, 31945}, {34394, 38413}, {36185, 36967}, {36953, 44383}, {36969, 44462}, {37640, 40579}, {40159, 43542}, {42100, 46861}

X(52040) = X(i)-Ceva conjugate of X(j) for these (i,j): {14, 30455}, {36310, 14}
X(52040) = X(i)-cross conjugate of X(j) for these (i,j): {9117, 524}, {30455, 14}
X(52040) = X(i)-isoconjugate of X(j) for these (i,j): {16, 897}, {299, 923}, {471, 36060}, {671, 2152}, {1095, 36310}, {6138, 36085}, {6149, 36307}, {7316, 44689}, {9206, 32679}, {17403, 23894}, {23871, 36142}, {34395, 46277}, {36128, 44719}
X(52040) = X(i)-Dao conjugate of X(j) for these (i,j): {16, 6593}, {299, 2482}, {471, 1560}, {671, 40579}, {1648, 9205}, {1649, 30468}, {6138, 38988}, {9213, 38993}, {14993, 36307}, {23871, 23992}
X(52040) = crosspoint of X(14) and X(36310)
X(52040) = trilinear pole of line {690, 9115}
X(52040) = crossdifference of every pair of points on line {16, 6138}
X(52040) = X(15360)-line conjugate of X(16)
X(52040) = barycentric product X(i)*X(j) for these {i,j}: {14, 524}, {15, 43084}, {187, 301}, {468, 40710}, {476, 9204}, {671, 30455}, {690, 23896}, {2154, 14210}, {2482, 36310}, {3266, 3458}, {5468, 20579}, {5642, 36311}, {5994, 35522}, {6390, 8738}, {9115, 11117}, {9117, 11120}, {9205, 36840}, {14417, 36309}, {14559, 23870}, {17402, 51479}, {36297, 44146}, {36298, 36890}, {41586, 51268}, {44691, 51655}
X(52040) = barycentric quotient X(i)/X(j) for these {i,j}: {14, 671}, {187, 16}, {301, 18023}, {351, 6138}, {468, 471}, {524, 299}, {690, 23871}, {922, 2152}, {1648, 30468}, {1649, 9205}, {1989, 36307}, {2154, 897}, {3292, 44719}, {3458, 111}, {5467, 17403}, {5477, 6783}, {5642, 41888}, {5994, 691}, {6137, 9213}, {8738, 17983}, {9115, 532}, {9117, 619}, {9204, 3268}, {9207, 34574}, {11085, 36310}, {14559, 23895}, {14560, 9206}, {14567, 34395}, {20579, 5466}, {23200, 46113}, {23896, 892}, {30455, 524}, {36297, 895}, {36298, 9214}, {40710, 30786}, {41586, 33530}, {43084, 300}, {44102, 8740}
X(52040) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 21467, 14}, {14, 21467, 8015}, {396, 11549, 18776}, {11549, 18776, 36298}

X(52041) = X(2)X(2138)∩X(3)X(206)

X(52041) lies on the cubics K177 and K389 and these lines: {2, 2138}, {3, 206}, {6, 14376}, {22, 33652}, {24, 1297}, {25, 34427}, {32, 46767}, {127, 2207}, {155, 17974}, {394, 10316}, {1073, 30435}, {1214, 21774}, {1217, 7401}, {2353, 47413}, {2972, 44162}, {3053, 18876}, {3266, 28701}, {3346, 7487}, {3398, 14059}, {3926, 20806}, {3998, 22119}, {5481, 7509}, {6642, 40801}, {6760, 9821}, {22120, 34897}, {28406, 28708}, {28695, 31636}, {28704, 28732}, {37444, 51940}

X(52041) = isogonal conjugate of X(41361)
X(52041) = isogonal conjugate of the anticomplement of X(14376)
X(52041) = isotomic conjugate of the polar conjugate of X(34207)
X(52041) = isogonal conjugate of the polar conjugate of X(13575)
X(52041) = X(i)-Ceva conjugate of X(j) for these (i,j): {13575, 34207}, {40358, 6}
X(52041) = X(32)-cross conjugate of X(3)
X(52041) = X(i)-isoconjugate of X(j) for these (i,j): {1, 41361}, {4, 18596}, {19, 1370}, {25, 21582}, {33, 18629}, {63, 41766}, {75, 3162}, {92, 159}, {158, 23115}, {162, 47125}, {455, 39733}, {1096, 28419}, {1760, 17407}, {8793, 20883}, {18750, 33584}
X(52041) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 41361}, {6, 1370}, {125, 47125}, {159, 22391}, {206, 3162}, {1147, 23115}, {3162, 41766}, {6503, 28419}, {6505, 21582}, {18596, 36033}
X(52041) = cevapoint of X(i) and X(j) for these (i,j): {647, 47413}, {2972, 3049}
X(52041) = crosssum of X(i) and X(j) for these (i,j): {159, 3162}, {19595, 40938}
X(52041) = barycentric product X(i)*X(j) for these {i,j}: {3, 13575}, {48, 39733}, {69, 34207}, {184, 40009}, {1176, 39129}, {3265, 39417}, {3926, 40144}, {14376, 40358}, {18018, 39172}
X(52041) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 1370}, {6, 41361}, {25, 41766}, {32, 3162}, {48, 18596}, {63, 21582}, {184, 159}, {185, 41602}, {222, 18629}, {394, 28419}, {577, 23115}, {647, 47125}, {2353, 17407}, {10547, 8793}, {13575, 264}, {33579, 3091}, {33581, 33584}, {34207, 4}, {39129, 1235}, {39172, 22}, {39417, 107}, {39733, 1969}, {40009, 18022}, {40144, 393}, {40358, 17907}, {46765, 40357}, {46767, 8743}, {46769, 8879}
X(52041) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 35211, 3162}, {13575, 40358, 40144}, {20806, 28696, 23115}

X(52042) = X(2)X(27375)∩X(3)X(3202)

X(52042) lies on the cubic K1068 and these lines: {2, 27375}, {3, 3202}, {39, 3051}, {76, 4576}, {140, 143}, {211, 8362}, {512, 7830}, {596, 14839}, {732, 6665}, {1216, 13334}, {2782, 32142}, {2979, 7786}, {3111, 43459}, {3203, 41328}, {3819, 3934}, {4173, 7810}, {5562, 21163}, {5650, 31239}, {6101, 40108}, {7803, 34095}, {7824, 14962}, {7849, 51427}, {7852, 47638}, {7859, 11673}, {7999, 11257}, {11285, 41262}, {27370, 37125}, {32516, 44324}, {46283, 47648}

X(52042) = midpoint of X(i) and X(j) for these {i,j}: {1216, 13334}, {10627, 11272}
X(52042) = complement of X(27375)
X(52042) = complement of the isogonal conjugate of X(1078)
X(52042) = complement of the isotomic conjugate of X(33769)
X(52042) = medial-isogonal conjugate of X(1506)
X(52042) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 1506}, {82, 20965}, {255, 46394}, {662, 850}, {1078, 10}, {1629, 24005}, {3050, 16592}, {5012, 37}, {7668, 24040}, {10312, 16583}, {18042, 2}, {31296, 8287}, {33764, 141}, {33769, 2887}, {33778, 626}, {36794, 226}, {41296, 1215}, {41328, 16587}
X(52042) = X(27867)-Ceva conjugate of X(1634)
X(52042) = X(82)-isoconjugate of X(30505)
X(52042) = X(i)-Dao conjugate of X(j) for these (i,j): {141, 30505}, {18092, 20965}
X(52042) = crosspoint of X(i) and X(j) for these (i,j): {2, 33769}, {1634, 27867}
X(52042) = barycentric product X(i)*X(j) for these {i,j}: {141, 41328}, {1078, 8041}, {3203, 8024}, {3917, 37125}, {4175, 10312}, {5012, 7794}
X(52042) = barycentric quotient X(i)/X(j) for these {i,j}: {39, 30505}, {3203, 251}, {8041, 3613}, {37125, 46104}, {41328, 83}

X(52043) = X(2)X(330)∩X(10)X(38)

X(52043) lies on the cubics K994 and K1020 and these lines: {2, 330}, {6, 18044}, {9, 18065}, {10, 38}, {37, 18040}, {44, 18073}, {75, 4377}, {76, 321}, {81, 41240}, {141, 313}, {142, 30044}, {192, 17786}, {239, 668}, {244, 20340}, {274, 29610}, {312, 17230}, {314, 17287}, {320, 17790}, {334, 3263}, {335, 1921}, {344, 27267}, {350, 6542}, {514, 661}, {536, 4033}, {594, 1269}, {696, 21142}, {714, 2228}, {799, 1931}, {894, 4503}, {1018, 29716}, {1086, 3264}, {1100, 18046}, {1109, 20634}, {1111, 20432}, {1230, 21024}, {1278, 4110}, {1575, 27044}, {2210, 24294}, {2239, 4039}, {2345, 44147}, {3006, 20486}, {3009, 17793}, {3436, 36926}, {3501, 32933}, {3507, 32927}, {3596, 3662}, {3666, 18136}, {3701, 29674}, {3702, 49560}, {3739, 18143}, {3752, 18739}, {3760, 17294}, {3761, 17308}, {3770, 17289}, {3782, 30713}, {3821, 4710}, {3834, 18150}, {3963, 4357}, {3992, 49769}, {4022, 21238}, {4043, 17229}, {4044, 29594}, {4393, 24524}, {4440, 40875}, {4446, 31337}, {4469, 6385}, {4487, 49772}, {4494, 17274}, {4644, 41316}, {4715, 39996}, {4742, 49764}, {4851, 18147}, {4852, 29764}, {4975, 49767}, {4980, 20888}, {5016, 10449}, {5249, 30045}, {6646, 17787}, {11076, 15455}, {16610, 25125}, {16732, 21417}, {16814, 29396}, {16826, 18140}, {16997, 40745}, {17143, 29615}, {17144, 20055}, {17147, 20691}, {17148, 27095}, {17155, 46032}, {17184, 28654}, {17228, 30596}, {17231, 18137}, {17232, 20923}, {17233, 22016}, {17234, 29982}, {17244, 30830}, {17266, 30866}, {17279, 27252}, {17310, 18145}, {17316, 18135}, {17317, 25660}, {17336, 29542}, {17348, 29484}, {17351, 29423}, {17368, 34283}, {17374, 30939}, {17490, 40598}, {17789, 18159}, {17798, 26250}, {17863, 44153}, {18139, 29983}, {18152, 31027}, {18153, 31038}, {18743, 29572}, {19806, 37683}, {19810, 37653}, {19811, 26840}, {20173, 33780}, {20174, 32025}, {20332, 39914}, {20335, 30059}, {20352, 20358}, {20367, 29511}, {20440, 35544}, {20530, 27166}, {20532, 35538}, {20590, 43534}, {20648, 20939}, {21080, 25140}, {21257, 21330}, {21352, 30982}, {21868, 42051}, {22343, 36856}, {24215, 30054}, {24589, 29576}, {24598, 27091}, {25257, 42720}, {25303, 29586}, {26541, 26575}, {26594, 51481}, {27020, 40773}, {27255, 27285}, {27272, 27295}, {27321, 33129}, {27792, 44417}, {28660, 30965}, {28809, 29579}, {29561, 45751}, {29570, 30963}, {29571, 30850}, {29599, 30829}, {29967, 30034}, {29968, 30004}, {29988, 30026}, {30007, 30075}, {30989, 45782}, {34832, 42027}, {37096, 40071}, {37676, 41233}

X(52043) = midpoint of X(4033) and X(39995)
X(52043) = isogonal conjugate of X(34077)
X(52043) = isotomic conjugate of X(20332)
X(52043) = isotomic conjugate of the anticomplement of X(20343)
X(52043) = isotomic conjugate of the complement of X(20355)
X(52043) = isotomic conjugate of the isogonal conjugate of X(1575)
X(52043) = X(i)-Ceva conjugate of X(j) for these (i,j): {85, 20443}, {92, 20639}, {335, 321}, {1921, 3263}, {32020, 75}
X(52043) = X(i)-cross conjugate of X(j) for these (i,j): {20343, 2}, {20600, 1}
X(52043) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34077}, {6, 727}, {31, 20332}, {32, 3226}, {101, 23355}, {560, 32020}, {604, 8851}, {1333, 18793}, {1397, 36799}, {1919, 8709}, {1922, 3253}, {2206, 27809}
X(52043) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 20532}, {2, 20332}, {3, 34077}, {6, 17793}, {9, 727}, {37, 18793}, {238, 1575}, {239, 3948}, {292, 22116}, {726, 1575}, {1015, 23355}, {3161, 8851}, {3226, 6376}, {3253, 39028}, {6374, 32020}, {8632, 27846}, {8709, 9296}, {20340, 23566}, {20530, 20669}, {27809, 40603}
X(52043) = cevapoint of X(i) and X(j) for these (i,j): {2, 20355}, {21053, 21140}
X(52043) = X(52043) = crosspoint of X(i) and X(j) for these (i,j): {75, 32020}, {76, 334}
X(52043) = crosssum of X(i) and X(j) for these (i,j): {31, 21760}, {32, 2210}
X(52043) = trilinear pole of line {3837, 20908}
X(52043) = crossdifference of every pair of points on line {31, 1980}
X(52043) = barycentric product X(i)*X(j) for these {i,j}: {1, 35538}, {75, 726}, {76, 1575}, {190, 20908}, {312, 43040}, {313, 18792}, {334, 17793}, {561, 3009}, {668, 3837}, {693, 23354}, {799, 21053}, {1463, 3596}, {1502, 21760}, {1969, 20785}, {3264, 36814}, {6373, 6386}, {6382, 40881}, {6385, 21830}, {7035, 21140}, {17475, 18895}, {18022, 20777}, {18275, 40782}, {18891, 40155}, {20532, 32020}, {20663, 44172}, {27044, 40013}, {40367, 51864}
X(52043) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 727}, {2, 20332}, {6, 34077}, {8, 8851}, {10, 18793}, {75, 3226}, {76, 32020}, {312, 36799}, {321, 27809}, {350, 3253}, {513, 23355}, {668, 8709}, {726, 1}, {1463, 56}, {1575, 6}, {3009, 31}, {3837, 513}, {6373, 667}, {6382, 40844}, {8850, 1428}, {17475, 1914}, {17793, 238}, {18792, 58}, {19565, 40755}, {20532, 1575}, {20663, 2210}, {20671, 21760}, {20681, 3747}, {20690, 21830}, {20759, 20777}, {20777, 184}, {20785, 48}, {20908, 514}, {21053, 661}, {21140, 244}, {21760, 32}, {21830, 213}, {22092, 22383}, {23354, 100}, {24816, 1319}, {27044, 32911}, {35538, 75}, {36814, 106}, {40155, 1911}, {40782, 51919}, {40881, 2162}, {42766, 1769}, {43040, 57}, {48084, 35367}
X(52043) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 20913, 4359}, {76, 3661, 321}, {141, 313, 20891}, {239, 668, 25298}, {312, 20943, 31060}, {3596, 3662, 20892}, {3912, 3948, 4358}, {3912, 6381, 3948}, {4377, 17237, 75}, {6376, 20917, 2}, {17148, 27095, 27633}, {17230, 31060, 312}, {18040, 18133, 37}, {18144, 30473, 75}, {40013, 40603, 4359}

X(52044) = X(1)X(668)∩X(2)X(3121)

X(52044) lies on the cubic K323 and these lines: {1, 668}, {2, 3121}, {6, 19579}, {75, 291}, {239, 19567}, {308, 24437}, {350, 518}, {537, 4479}, {673, 2319}, {789, 8300}, {874, 2108}, {1015, 17030}, {1575, 19565}, {1909, 9263}, {1966, 24727}, {3759, 20457}, {9278, 18298}, {9466, 33908}, {14839, 17144}, {17027, 17149}, {17034, 18148}, {17731, 24731}, {17762, 20644}, {18891, 20456}, {20671, 32453}, {27076, 27255}, {29383, 29547}, {29742, 29757}, {30998, 31298}, {39360, 41144}

X(52044) = reflection of X(i) in X(j) for these {i,j}: {9263, 17448}, {24524, 668}
X(52044) = isotomic conjugate of X(18795)
X(52044) = isotomic conjugate of the isogonal conjugate of X(18794)
X(52044) = X(239)-Ceva conjugate of X(75)
X(52044) = X(i)-isoconjugate of X(j) for these (i,j): {31, 18795}, {1967, 16361}
X(52044) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 18795}, {334, 335}, {8290, 16361}
X(52044) = barycentric product X(i)*X(j) for these {i,j}: {75, 30667}, {76, 18794}, {350, 39918}, {1920, 16363}
X(52044) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 18795}, {385, 16361}, {16363, 893}, {18794, 6}, {30667, 1}, {39918, 291}
X(52044) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {668, 32020, 17793}, {9230, 24575, 75}, {17793, 32020, 30963}

X(52045) = X(2)X(489)∩X(3)X(9680)

X(52045) lies on the cubic K1193 and these lines: {2, 489}, {3, 9680}, {4, 43210}, {6, 3524}, {20, 43879}, {30, 590}, {140, 6453}, {371, 549}, {372, 12100}, {376, 3070}, {381, 5418}, {395, 42167}, {396, 42169}, {485, 3534}, {486, 6407}, {547, 6484}, {548, 8960}, {550, 35812}, {551, 49226}, {591, 8414}, {599, 49228}, {615, 5054}, {631, 6425}, {1152, 15692}, {1327, 8976}, {1328, 1656}, {1587, 19708}, {1588, 6429}, {1657, 41967}, {2043, 42165}, {2044, 42164}, {2066, 5298}, {2067, 4995}, {2482, 49212}, {3068, 6411}, {3069, 6437}, {3090, 10147}, {3091, 6488}, {3311, 15693}, {3312, 15700}, {3364, 15764}, {3522, 43411}, {3523, 3592}, {3525, 42573}, {3526, 6519}, {3528, 43256}, {3530, 6419}, {3533, 9693}, {3543, 42265}, {3544, 43522}, {3545, 6433}, {3584, 31499}, {3594, 15717}, {3653, 35775}, {3655, 49232}, {3679, 9615}, {3830, 42260}, {3839, 32785}, {3843, 10195}, {3845, 6486}, {5055, 6445}, {5066, 35821}, {5071, 23261}, {5073, 43503}, {5215, 6567}, {5304, 6444}, {5309, 9674}, {5414, 9663}, {5420, 15701}, {5642, 49216}, {6036, 49215}, {6055, 49266}, {6174, 48700}, {6199, 15707}, {6396, 17504}, {6398, 15706}, {6410, 15698}, {6412, 7585}, {6417, 15718}, {6420, 15712}, {6426, 10299}, {6431, 43888}, {6432, 42522}, {6436, 42643}, {6438, 43259}, {6439, 23273}, {6443, 26463}, {6447, 15720}, {6450, 15716}, {6451, 6560}, {6454, 44682}, {6468, 8252}, {6478, 13993}, {6480, 11539}, {6482, 16239}, {6496, 13903}, {6502, 9648}, {6565, 15699}, {7581, 15715}, {7583, 34200}, {7584, 11812}, {7739, 9600}, {7746, 49261}, {7810, 41491}, {7968, 50828}, {8703, 8981}, {8972, 42264}, {8980, 49214}, {8983, 50808}, {9300, 12963}, {9582, 31162}, {9585, 13947}, {9616, 25055}, {9683, 51519}, {9685, 49263}, {9690, 13785}, {9691, 13951}, {9692, 10303}, {10124, 10577}, {10819, 20126}, {11001, 23251}, {11241, 23328}, {11694, 12376}, {11737, 35787}, {12108, 35813}, {12306, 26516}, {13665, 15689}, {13786, 13850}, {13812, 37343}, {13882, 36733}, {13893, 34628}, {13902, 34632}, {13908, 38738}, {13910, 50965}, {13911, 50811}, {13912, 51705}, {13925, 15690}, {13935, 15719}, {13936, 50829}, {14241, 17538}, {14269, 42277}, {15513, 44647}, {15640, 42606}, {15683, 31412}, {15684, 42269}, {15685, 22644}, {15686, 35820}, {15687, 42266}, {15704, 42639}, {15711, 19117}, {15759, 35815}, {16963, 51728}, {17800, 43568}, {18586, 42920}, {18587, 42921}, {18762, 47598}, {19710, 51911}, {19711, 35771}, {22236, 36467}, {22238, 36449}, {22331, 31465}, {23249, 41954}, {23259, 42566}, {32786, 43258}, {32808, 32821}, {33699, 35786}, {34551, 36470}, {34552, 36453}, {35256, 41983}, {36445, 42942}, {36463, 42943}, {38071, 42225}, {38335, 42275}, {41099, 43408}, {41949, 41959}, {41957, 41985}, {42216, 45759}, {42276, 45384}, {42537, 50688}, {42642, 43517}, {43413, 43883}, {43566, 50692}, {43786, 49136}, {44635, 50810}, {49229, 51737}, {49233, 50821}, {49618, 50815}

X(52045) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1151, 41945}, {2, 41945, 3071}, {3, 9680, 41963}, {3, 32787, 41946}, {3, 41963, 31454}, {371, 549, 32788}, {376, 3070, 43209}, {376, 9540, 13846}, {376, 13846, 3070}, {1151, 42262, 43512}, {1327, 8976, 41952}, {1327, 15681, 42272}, {3069, 9542, 6437}, {3071, 41945, 42417}, {3530, 6419, 41964}, {3830, 42602, 42273}, {5055, 43254, 32789}, {5071, 43257, 23261}, {5418, 6449, 42258}, {5418, 42258, 42582}, {6200, 35255, 590}, {6409, 9540, 3070}, {6409, 13846, 376}, {6409, 42568, 9540}, {6433, 8253, 9541}, {6459, 42583, 3071}, {6496, 13903, 42261}, {6561, 43254, 5055}, {8253, 9541, 42283}, {8703, 8981, 35822}, {8703, 35822, 42259}, {8976, 15681, 1327}, {13846, 43209, 42572}, {15692, 19054, 1152}, {31454, 41946, 32787}, {41946, 43887, 31454}, {41952, 42272, 1327}, {42260, 42602, 3830}

X(52046) = X(2)X(490)∩X(3)X(9681)

X(52046) lies on the cubic K1193 and these lines: {2, 490}, {3, 9681}, {4, 43209}, {6, 3524}, {20, 43880}, {30, 615}, {140, 6454}, {371, 12100}, {372, 549}, {376, 3071}, {381, 5420}, {395, 42168}, {396, 42170}, {485, 6408}, {486, 3534}, {547, 6485}, {550, 35813}, {551, 49227}, {590, 5054}, {599, 49229}, {631, 6426}, {1151, 15692}, {1327, 1656}, {1328, 13951}, {1587, 6430}, {1588, 19708}, {1657, 41968}, {1991, 8406}, {2043, 42164}, {2044, 42165}, {2482, 49213}, {3068, 6438}, {3069, 6412}, {3090, 10148}, {3091, 6489}, {3311, 15700}, {3312, 15693}, {3390, 15764}, {3522, 43412}, {3523, 3594}, {3525, 17852}, {3526, 6522}, {3528, 43257}, {3530, 6420}, {3533, 10142}, {3543, 42262}, {3544, 43521}, {3545, 6434}, {3592, 15717}, {3653, 35774}, {3655, 49233}, {3830, 42261}, {3839, 32786}, {3843, 10194}, {3845, 6487}, {4995, 6502}, {5055, 6446}, {5066, 35820}, {5071, 23251}, {5073, 43504}, {5215, 6566}, {5298, 5414}, {5304, 6443}, {5418, 15701}, {5642, 49217}, {6036, 49214}, {6055, 49267}, {6174, 48701}, {6200, 17504}, {6221, 15706}, {6395, 15707}, {6409, 15698}, {6411, 7586}, {6418, 15718}, {6419, 15712}, {6425, 10299}, {6428, 9680}, {6431, 42523}, {6432, 43887}, {6435, 42644}, {6437, 43258}, {6440, 23267}, {6444, 26456}, {6448, 15720}, {6449, 15716}, {6452, 6561}, {6453, 44682}, {6469, 8253}, {6479, 13925}, {6481, 11539}, {6483, 16239}, {6497, 13961}, {6564, 15699}, {7582, 15715}, {7583, 11812}, {7584, 34200}, {7746, 49262}, {7810, 41490}, {7969, 50828}, {8703, 13966}, {8960, 14869}, {9300, 12968}, {9540, 15719}, {9541, 15710}, {10124, 10576}, {10820, 20126}, {11001, 23261}, {11242, 23328}, {11694, 12375}, {11737, 35786}, {12108, 35812}, {12305, 26521}, {13665, 43415}, {13666, 13932}, {13692, 37342}, {13785, 15689}, {13883, 50829}, {13934, 36719}, {13941, 42263}, {13947, 34628}, {13959, 34632}, {13967, 49215}, {13968, 38738}, {13971, 50808}, {13972, 50965}, {13973, 50811}, {13975, 51705}, {13993, 15690}, {14226, 17538}, {14269, 42274}, {15513, 44648}, {15640, 42607}, {15683, 42561}, {15684, 42268}, {15685, 22615}, {15686, 35821}, {15687, 42267}, {15704, 42640}, {15711, 19116}, {15759, 35814}, {17800, 43569}, {17851, 45384}, {18538, 47598}, {18586, 42921}, {18587, 42920}, {19710, 51910}, {19711, 35770}, {22236, 36450}, {22238, 36468}, {23249, 42567}, {23259, 41953}, {31414, 43409}, {31481, 49260}, {32785, 43259}, {32809, 32821}, {33699, 35787}, {34551, 36469}, {34552, 36452}, {35255, 41983}, {35739, 36439}, {36445, 42943}, {36463, 42942}, {38071, 42226}, {38335, 42276}, {41099, 43407}, {41950, 41960}, {41958, 41985}, {42215, 45759}, {42275, 45385}, {42538, 50688}, {42641, 43518}, {43414, 43884}, {43567, 50692}, {43785, 49136}, {44636, 50810}, {49228, 51737}, {49232, 50821}, {49619, 50815}

X(52046) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1152, 41946}, {2, 41946, 3070}, {3, 32788, 41945}, {372, 549, 32787}, {376, 3071, 43210}, {376, 13847, 3071}, {376, 13935, 13847}, {1152, 42265, 43511}, {1328, 13951, 41951}, {1328, 15681, 42271}, {3070, 41946, 42418}, {3523, 3594, 31454}, {3530, 6420, 41963}, {3830, 42603, 42270}, {5055, 43255, 32790}, {5071, 43256, 23251}, {5420, 6450, 42259}, {5420, 42259, 42583}, {6396, 35256, 615}, {6410, 13847, 376}, {6410, 13935, 3071}, {6410, 42569, 13935}, {6460, 42582, 3070}, {6497, 13961, 42260}, {6560, 43255, 5055}, {8703, 13966, 35823}, {8703, 35823, 42258}, {13847, 43210, 42573}, {13951, 15681, 1328}, {15692, 19053, 1151}, {41951, 42271, 1328}, {42261, 42603, 3830}

X(52047) = X(2)X(6221)∩X(30)X(371)

X(52047) lies on the cubic K1193 and these lines: {2, 6221}, {3, 19053}, {4, 6447}, {5, 1328}, {6, 8703}, {30, 371}, {140, 6453}, {372, 34200}, {376, 3311}, {381, 6459}, {485, 15687}, {486, 6429}, {519, 31439}, {546, 31454}, {547, 3071}, {548, 6419}, {549, 1151}, {550, 3592}, {590, 5066}, {615, 6480}, {631, 6519}, {632, 9680}, {1152, 45759}, {1327, 33699}, {1587, 15681}, {1588, 5054}, {1702, 3655}, {2067, 15170}, {3068, 3830}, {3069, 6445}, {3146, 31487}, {3299, 9649}, {3301, 9662}, {3312, 10304}, {3522, 6427}, {3523, 9693}, {3524, 6449}, {3525, 9692}, {3528, 6428}, {3534, 6199}, {3543, 14241}, {3594, 46853}, {3627, 43503}, {3628, 41963}, {3653, 9615}, {3654, 9616}, {3656, 9583}, {3839, 8976}, {3845, 6437}, {3850, 35812}, {3853, 8960}, {3860, 42283}, {3861, 43879}, {3933, 32808}, {5055, 9540}, {5306, 9675}, {5418, 15699}, {5874, 13692}, {6200, 12100}, {6396, 15759}, {6398, 19708}, {6409, 17504}, {6410, 15714}, {6411, 15711}, {6417, 15688}, {6418, 14093}, {6420, 33923}, {6433, 19711}, {6448, 21734}, {6451, 7586}, {6455, 7582}, {6456, 15710}, {6460, 15689}, {6468, 15713}, {6470, 42261}, {6474, 15703}, {6476, 11540}, {6478, 47599}, {6484, 41968}, {6496, 15705}, {6500, 42637}, {6560, 19710}, {6564, 12101}, {6565, 10109}, {7585, 11001}, {7880, 32419}, {8252, 42640}, {8375, 15048}, {8972, 41099}, {8997, 22566}, {9690, 15701}, {9691, 15694}, {10124, 42573}, {10576, 11737}, {10577, 47598}, {10817, 19051}, {13665, 15682}, {13712, 15534}, {13833, 49364}, {13835, 47352}, {13883, 28208}, {13886, 50687}, {13903, 14269}, {13908, 22515}, {13925, 14893}, {13935, 15700}, {13939, 15721}, {13951, 15702}, {13961, 15718}, {13968, 26614}, {14830, 19109}, {14869, 43255}, {14892, 42582}, {15683, 42522}, {15684, 42537}, {15685, 18512}, {15686, 19117}, {15691, 42259}, {15764, 36467}, {19058, 38730}, {19709, 23259}, {23046, 42265}, {23249, 43316}, {23251, 35404}, {23261, 38071}, {31412, 38335}, {31414, 49136}, {32790, 41949}, {34551, 42925}, {34552, 42924}, {35770, 41982}, {35775, 50824}, {35786, 41952}, {35815, 42271}, {36446, 36466}, {36448, 36464}, {42262, 43254}, {42270, 47478}, {42284, 42643}, {42575, 45384}, {43228, 51728}, {43291, 49261}, {43318, 43569}, {43343, 43439}, {43383, 43386}

X(52047) = midpoint of X(i) and X(j) for these {i,j}: {371, 41945}, {35822, 42258}
X(52047) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {549, 7584, 43212}, {1327, 42263, 33699}, {3068, 43257, 3830}, {3534, 6199, 19054}, {3534, 19054, 42216}, {3592, 9681, 550}, {3830, 43257, 42225}, {3845, 13846, 18538}, {6199, 9541, 42216}, {6200, 32788, 12100}, {6221, 13785, 43509}, {6221, 42215, 35255}, {6470, 43339, 42261}, {6561, 13846, 3845}, {7582, 9543, 6455}, {9541, 19054, 3534}, {12100, 32788, 35256}, {23261, 42602, 38071}, {35255, 42215, 18762}

X(52048) = X(2)X(6398)∩X(30)X(372)

X(52048) lies on the cubic K1193 and these lines: {2, 6398}, {3, 19054}, {4, 6448}, {5, 1327}, {6, 8703}, {30, 372}, {140, 6454}, {371, 34200}, {376, 3312}, {381, 6460}, {485, 6430}, {486, 15687}, {547, 3070}, {548, 6420}, {549, 1152}, {550, 3594}, {590, 6481}, {615, 5066}, {631, 6522}, {632, 42602}, {1151, 45759}, {1328, 33699}, {1587, 5054}, {1588, 15681}, {1703, 3655}, {3068, 6446}, {3069, 3830}, {3311, 9543}, {3522, 6428}, {3524, 6450}, {3528, 6427}, {3534, 6395}, {3543, 14226}, {3592, 46853}, {3627, 43504}, {3628, 41964}, {3839, 13951}, {3845, 6438}, {3850, 35813}, {3860, 42284}, {3861, 43880}, {3933, 32809}, {5055, 13935}, {5420, 15699}, {5875, 13812}, {6200, 15759}, {6221, 19708}, {6396, 12100}, {6409, 15714}, {6410, 17504}, {6412, 15711}, {6417, 14093}, {6418, 15688}, {6419, 33923}, {6434, 19711}, {6447, 21734}, {6452, 7585}, {6455, 15710}, {6456, 7581}, {6459, 15689}, {6469, 15713}, {6471, 42260}, {6475, 15703}, {6477, 11540}, {6479, 47599}, {6485, 41967}, {6497, 15705}, {6501, 42638}, {6502, 15170}, {6561, 19710}, {6564, 10109}, {6565, 12101}, {7586, 11001}, {7880, 32421}, {8253, 42639}, {8376, 15048}, {8960, 12108}, {8976, 15702}, {9540, 15700}, {9541, 15695}, {10124, 42572}, {10299, 31487}, {10576, 47598}, {10577, 11737}, {10818, 19052}, {13712, 47352}, {13769, 49361}, {13785, 15682}, {13835, 15534}, {13886, 15721}, {13903, 15718}, {13908, 26614}, {13936, 28208}, {13939, 50687}, {13941, 41099}, {13961, 14269}, {13968, 22515}, {13989, 22566}, {13993, 14893}, {14830, 19108}, {14869, 17852}, {14892, 42583}, {15683, 42523}, {15684, 42538}, {15685, 18510}, {15686, 19116}, {15691, 42258}, {15701, 18512}, {15709, 43536}, {15764, 36468}, {17851, 32785}, {19057, 38730}, {19709, 23249}, {23046, 42262}, {23251, 38071}, {23259, 43317}, {23261, 35404}, {31414, 46219}, {32789, 41950}, {34551, 42924}, {34552, 42925}, {35771, 41982}, {35774, 50824}, {35787, 41951}, {35814, 42272}, {36447, 36466}, {36448, 36465}, {38335, 42561}, {42265, 43255}, {42273, 47478}, {42283, 42644}, {42574, 45385}, {42791, 51728}, {43291, 49262}, {43319, 43568}, {43342, 43438}, {43382, 43387}

X(52048) = midpoint of X(i) and X(j) for these {i,j}: {372, 41946}, {35823, 42259}
X(52048) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {549, 7583, 43211}, {1328, 42264, 33699}, {3069, 43256, 3830}, {3534, 6395, 19053}, {3534, 19053, 42215}, {3830, 43256, 42226}, {3845, 13847, 18762}, {6396, 32787, 12100}, {6398, 13665, 43510}, {6398, 42216, 35256}, {6471, 43338, 42260}, {6560, 13847, 3845}, {12100, 32787, 35255}, {23251, 42603, 38071}, {35256, 42216, 18538}

X(52049) = X(2)X(21902)∩X(10)X(75)

X(52049) lies on the cubic K994 and these lines: {2, 21902}, {10, 75}, {37, 1655}, {38, 18152}, {190, 1966}, {274, 3842}, {310, 756}, {312, 17149}, {321, 27495}, {334, 3932}, {335, 3948}, {350, 518}, {514, 4079}, {536, 9295}, {537, 18145}, {538, 21830}, {561, 32925}, {668, 740}, {698, 3862}, {716, 4664}, {799, 17763}, {872, 33296}, {1215, 31008}, {1237, 6385}, {1757, 30940}, {1920, 3971}, {1961, 8033}, {1965, 32926}, {1975, 34247}, {1978, 3994}, {2664, 2669}, {3685, 35961}, {3696, 25280}, {3760, 49448}, {3774, 25264}, {3782, 30631}, {3896, 25286}, {3954, 4043}, {3967, 41318}, {3978, 41531}, {3995, 18059}, {4358, 18149}, {4415, 7018}, {4479, 50075}, {4531, 17144}, {4583, 18157}, {4687, 5283}, {6384, 18743}, {6542, 39719}, {8026, 20945}, {15569, 25303}, {16284, 39467}, {17143, 49457}, {17759, 21897}, {17787, 49514}, {18058, 46747}, {18064, 32930}, {18135, 24349}, {18137, 18148}, {18140, 24325}, {18146, 31178}, {18159, 20448}, {19804, 31322}, {20593, 49755}, {20913, 31323}, {20920, 20939}, {21788, 39916}, {24524, 49470}, {24732, 49509}, {30632, 33151}, {30964, 32931}, {32922, 39044}

X(52049) = reflection of X(i) in X(j) for these {i,j}: {75, 1921}, {1921, 6381}, {17759, 21897}, {19565, 37}
X(52049) = isotomic conjugate of X(2665)
X(52049) = isotomic conjugate of the isogonal conjugate of X(2664)
X(52049) = X(i)-Ceva conjugate of X(j) for these (i,j): {335, 75}, {3948, 20947}, {40874, 17759}
X(52049) = X(i)-isoconjugate of X(j) for these (i,j): {6, 51333}, {31, 2665}, {32, 39925}, {58, 2107}, {1911, 40769}, {8937, 18757}
X(52049) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 2665}, {6, 39056}, {9, 51333}, {10, 2107}, {81, 39057}, {239, 350}, {3248, 27854}, {6376, 39925}, {6651, 40769}
X(52049) = cevapoint of X(1655) and X(17794)
X(52049) = crosspoint of X(40874) and X(41535)
X(52049) = barycentric product X(i)*X(j) for these {i,j}: {10, 40874}, {37, 41535}, {75, 17759}, {76, 2664}, {310, 21897}, {313, 2106}, {321, 2669}, {334, 39916}, {335, 39028}, {561, 21788}, {1921, 40796}, {1969, 20796}, {4583, 27854}, {15148, 40071}, {44172, 51331}
X(52049) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 51333}, {2, 2665}, {37, 2107}, {75, 39925}, {239, 40769}, {313, 43685}, {1654, 8937}, {2106, 58}, {2664, 6}, {2669, 81}, {3797, 40798}, {15148, 1474}, {17759, 1}, {20533, 8934}, {20796, 48}, {21788, 31}, {21897, 42}, {27854, 659}, {39028, 239}, {39916, 238}, {40796, 292}, {40874, 86}, {41535, 274}, {51331, 2210}
X(52049) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {75, 20943, 21615}, {76, 984, 75}, {3596, 49518, 75}, {10009, 49493, 75}, {21615, 49447, 75}

X(52050) = X(1)X(908)∩X(9)X(36)

X(52050) lies on Jerabek circumhyperbola of the excentral triangle, the cubic K1293, and these lines: {1, 908}, {3, 45633}, {9, 36}, {40, 5692}, {63, 10090}, {78, 4302}, {80, 24392}, {165, 2950}, {191, 936}, {200, 3899}, {997, 21578}, {1420, 3646}, {1512, 3679}, {2057, 11010}, {2136, 5697}, {4677, 13144}, {4867, 7962}, {5119, 41389}, {5219, 26725}, {5506, 8583}, {5732, 15015}, {6326, 28459}, {7951, 31936}, {11249, 41229}, {16009, 16143}, {22083, 48099}, {25522, 37701}, {27131, 39778}, {31160, 38021}, {35238, 50528}

X(52050) = reflection of X(45633) in X(3)
X(52050) = excentral isogonal conjugate of X(3359)
X(52050) = X(i)-Ceva conjugate of X(j) for these (i,j): {997, 1}, {21578, 5541}, {31018, 9}

X(52051) = X(2)X(1340)∩X(6)X(30)

X(52051) lies on the minor axis of the Steiner ellipses, the cubic K1293, and these lines: {2, 1340}, {3, 39023}, {4, 14630}, {6, 30}, {20, 3558}, {115, 47366}, {376, 1380}, {1379, 7735}, {2029, 5309}, {3413, 6776}, {3557, 5286}, {6039, 9744}, {6189, 14907}, {7738, 14631}, {9862, 47368}, {9939, 39366}, {12243, 47367}, {12251, 13325}, {21036, 23195}, {22243, 51825}, {31863, 43448}

X(52051) = reflection of X(4) in X(46024)
X(52051) = crossdifference of every pair of points on line {5638, 8675}
X(52051) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 35914, 31862}, {1340, 31862, 2}, {1349, 47089, 2}, {2543, 51899, 2}

X(52052) = X(2)X(1341)∩X(6)X(30)

X(52052) lies on the major axis of the Steiner ellipses, cubic K1293, and these lines: {2, 1341}, {3, 39022}, {4, 14631}, {6, 30}, {20, 3557}, {115, 47365}, {376, 1379}, {1380, 7735}, {2028, 5309}, {3414, 6776}, {3558, 5286}, {6040, 9744}, {6190, 14907}, {7738, 14630}, {9862, 47367}, {9939, 39365}, {12243, 47368}, {12251, 13326}, {21032, 23195}, {22242, 51826}, {31862, 43448}

X(52052) = reflection of X(4) in X(46023)
X(52052) = crossdifference of every pair of points on line {5639, 8675}
X(52052) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 35913, 31863}, {1341, 31863, 2}, {1348, 47088, 2}, {2542, 51898, 2}

X(52053) = X(3)X(39162)∩X(20)X(39159)

X(52053) lies on the cubics K706, K758, K1293, and these lines: {3, 39162}, {20, 39159}, {30, 39163}, {376, 3413}, {550, 42411}, {1380, 32443}, {10304, 39158}

X(52053) = midpoint of X(i) and X(j) for these {i,j}: {20, 39159}, {39162, 42412}
X(52053) = reflection of X(i) in X(j) for these {i,j}: {39162, 3}, {40851, 39162}, {40852, 39159}
X(52053) = Thomson-isogonal conjugate of X(40989)
X(52053) = {X(3),X(42412)}-harmonic conjugate of X(40851)

X(52054) = X(3)X(39163)∩X(20)X(39158)

X(52054) lies on the cubics K706, K758, K1293, and these lines: {3, 39163}, {20, 39158}, {30, 39162}, {376, 3413}, {550, 42412}, {1380, 40989}, {10304, 39159}

X(52054) = midpoint of X(i) and X(j) for these {i,j}: {20, 39158}, {39163, 42411}
X(52054) = reflection of X(i) in X(j) for these {i,j}: {39163, 3}, {40851, 39158}, {40852, 39163}
X(52054) = Thomson-isogonal conjugate of X(40990)
X(52054) = {X(3),X(42411)}-harmonic conjugate of X(40852)

X(52055) = X(3)X(7712)∩X(74)X(381)

X(52055) lies on the cubic K727 and these lines: {3, 7712}, {4, 11999}, {6, 10620}, {30, 18387}, {64, 32349}, {74, 381}, {378, 2914}, {382, 3581}, {399, 12901}, {599, 3098}, {1511, 35495}, {1656, 3357}, {1657, 7691}, {1995, 18551}, {2070, 3426}, {3526, 15062}, {3579, 5692}, {3830, 48912}, {4550, 5054}, {5076, 32138}, {5092, 51797}, {5094, 19402}, {5663, 11935}, {5888, 15693}, {8705, 33878}, {9938, 12164}, {10546, 12041}, {10605, 19348}, {10606, 13289}, {10814, 26336}, {10815, 26346}, {11438, 15432}, {11440, 49136}, {11456, 16013}, {11472, 15041}, {12017, 19153}, {13093, 18364}, {13445, 15688}, {13665, 44592}, {13785, 44593}, {18439, 35496}, {18510, 19001}, {18512, 19002}, {18571, 20421}, {22585, 22978}, {31861, 41428}, {32124, 37958}, {37478, 49137}, {37517, 46202}, {40920, 44747}, {41617, 44456}, {45375, 45418}, {45376, 45419}

X(52055) = midpoint of X(i) and X(j) for these {i,j}: {3, 33887}, {7712, 11738}
X(52055) = reflection of X(i) in X(j) for these {i,j}: {10620, 43720}, {18550, 7703}, {51942, 12041}
X(52055) = {X(7703),X(18550)}-harmonic conjugate of X(381)

X(52056) = X(3)X(3447)∩X(30)X(74)

X(52056) lies on the cubic K811 and these lines: {3, 3447}, {20, 14254}, {30, 74}, {94, 38896}, {187, 1989}, {382, 39170}, {477, 38723}, {523, 23236}, {1511, 14731}, {1553, 7728}, {1657, 51254}, {2070, 5961}, {2777, 34192}, {3258, 38794}, {3534, 14583}, {6344, 13619}, {7471, 14643}, {7575, 18883}, {10295, 18384}, {10296, 18576}, {11001, 51835}, {12121, 16168}, {14560, 37477}, {14934, 41522}, {15061, 17511}, {16163, 38581}, {16340, 38700}, {17702, 38580}, {31676, 37949}, {31874, 32423}, {34310, 47327}

X(52056) = reflection of X(i) in X(j) for these {i,j}: {265, 476}, {7728, 36193}, {10733, 18319}, {14731, 1511}, {14993, 51345}, {17511, 38609}, {20957, 7471}, {38581, 16163}
X(52056) = barycentric product X(i)*X(j) for these {i,j}: {94, 32609}, {476, 45681}
X(52056) = barycentric quotient X(i)/X(j) for these {i,j}: {32609, 323}, {45681, 3268}
X(52056) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {265, 476, 14993}, {265, 51345, 476}, {7471, 20957, 14643}, {11586, 15743, 1989}, {16340, 38700, 38728}, {17511, 38609, 15061}

X(52057) = X(3)X(9530)∩X(4)X(74)

X(52057) lies on the cubic K811 and these lines: {3, 9530}, {4, 74}, {20, 38672}, {30, 34109}, {122, 140}, {187, 1990}, {376, 38686}, {389, 34980}, {550, 3184}, {631, 10714}, {1294, 3522}, {1656, 6716}, {1657, 23240}, {2501, 46997}, {2790, 10991}, {2797, 10992}, {2803, 10993}, {2811, 33520}, {2816, 11713}, {2822, 33521}, {2847, 34106}, {2848, 9409}, {3146, 23241}, {3154, 43911}, {3183, 9833}, {3515, 14703}, {3517, 14673}, {3523, 23239}, {3850, 49117}, {4240, 15063}, {5059, 34549}, {5073, 22337}, {9033, 12790}, {11430, 42873}, {14216, 34286}, {16111, 34334}, {16163, 35360}, {18400, 40664}, {21735, 38714}, {31510, 32417}, {32110, 39569}, {33923, 38621}, {44985, 49135}

X(52057) = midpoint of X(i) and X(j) for these {i,j}: {20, 38672}, {107, 5667}, {3146, 23241}, {23240, 38577}
X(52057) = reflection of X(i) in X(j) for these {i,j}: {122, 38605}, {133, 107}, {10745, 6716}, {34186, 34842}
X(52057) = polar-circle-inverse of X(14644)
X(52057) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6716, 10745, 36520}, {10990, 16240, 4}, {23239, 34186, 34842}

X(52058) = X(2)X(6)∩X(4)X(22120)

X(52058) lies on these lines: {2, 6}, {4, 22120}, {20, 8743}, {22, 15905}, {23, 232}, {25, 38292}, {30, 8744}, {32, 22467}, {39, 14118}, {50, 37978}, {83, 26166}, {110, 8779}, {111, 34570}, {112, 2071}, {147, 34137}, {186, 10317}, {187, 37941}, {216, 15246}, {237, 44090}, {251, 41890}, {315, 26170}, {393, 7391}, {577, 6636}, {648, 30737}, {800, 1627}, {858, 13573}, {1172, 37456}, {1180, 5065}, {1249, 1370}, {1369, 26159}, {1384, 15078}, {1554, 43460}, {1625, 13509}, {1968, 12086}, {1971, 35265}, {1990, 5189}, {2207, 3146}, {2211, 40236}, {3087, 7394}, {3108, 41891}, {3153, 5523}, {3162, 7396}, {3163, 10989}, {3172, 11413}, {3291, 34569}, {3424, 8549}, {4230, 51937}, {5158, 7496}, {5159, 39220}, {5702, 46336}, {5999, 14965}, {6103, 30745}, {6531, 46571}, {6748, 37349}, {6749, 7533}, {6997, 40065}, {7484, 15851}, {7488, 10316}, {7500, 8745}, {7503, 9605}, {7667, 8792}, {7710, 34117}, {7745, 34007}, {7754, 26226}, {7762, 26154}, {7772, 26216}, {8746, 20062}, {9209, 32320}, {9475, 37183}, {9909, 33636}, {10311, 13595}, {10312, 44802}, {10314, 16042}, {12384, 44704}, {14581, 37944}, {15013, 41676}, {16063, 40138}, {16308, 18365}, {17928, 30435}, {18860, 38699}, {18907, 38323}, {26218, 32830}, {28406, 51884}, {33630, 44442}, {36415, 39176}, {37254, 45786}, {37444, 41361}, {39643, 43605}, {41370, 44440}, {44102, 51412}

X(52058) = X(i)-isoconjugate of X(j) for these (i,j): {661, 2867}, {3708, 39297}
X(52058) = X(2867)-Dao conjugate of X(36830)
X(52058) = crossdifference of every pair of points on line {512, 5489}
X(52058) = barycentric product X(i)*X(j) for these {i,j}: {99, 2881}, {2407, 15292}, {3100, 4296}, {3220, 7270}, {4872, 5285}, {5002, 5003}, {5279, 7291}
X(52058) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 2867}, {250, 39297}, {1660, 10229}, {2881, 523}, {3220, 15314}, {15292, 2394}, {44757, 34136}, {44758, 34135}
X(52058) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {112, 14961, 2071}, {232, 3284, 10313}, {232, 10313, 23}, {577, 22240, 6636}, {8743, 23115, 20}, {10311, 15355, 13595}, {10316, 39575, 7488}, {15905, 45141, 22}

X(52059) = X(6)X(909)∩X(32)X(9456)

X(52059) lies on these lines: {6, 909}, {32, 9456}, {36, 40584}, {50, 7113}, {219, 8560}, {604, 2965}, {1015, 1333}, {1086, 4565}, {1407, 32655}, {1576, 22096}, {1983, 17455}, {2174, 21742}, {2197, 15109}, {4996, 35069}, {5063, 14827}, {5124, 13006}, {14936, 19302}, {18877, 39687}, {19297, 23980}, {21758, 22379}, {32659, 32739}

X(52059) = X(i)-isoconjugate of X(j) for these (i,j): {8, 34535}, {80, 18359}, {2161, 20566}, {2166, 41226}, {4858, 46649}, {14628, 36590}, {15065, 24624}, {18815, 36910}, {23592, 24026}
X(52059) = X(i)-Dao conjugate of X(j) for these (i,j): {312, 6149}, {758, 28654}, {3738, 23978}, {11597, 41226}, {20566, 40584}
X(52059) = crosssum of X(i) and X(j) for these (i,j): {6, 35221}, {594, 15065}
X(52059) = crossdifference of every pair of points on line {2804, 15065}
X(52059) = barycentric product X(i)*X(j) for these {i,j}: {7, 215}, {8, 41282}, {36, 36}, {56, 4996}, {57, 34544}, {59, 3025}, {60, 3028}, {202, 203}, {214, 16944}, {593, 35069}, {849, 4736}, {1262, 35128}, {1443, 2361}, {1983, 3960}, {3218, 7113}, {4242, 22379}, {4282, 18593}, {4585, 21758}, {17455, 40215}
X(52059) = barycentric quotient X(i)/X(j) for these {i,j}: {36, 20566}, {50, 41226}, {215, 8}, {604, 34535}, {1983, 36804}, {3025, 34387}, {3028, 34388}, {3724, 15065}, {4996, 3596}, {7113, 18359}, {23979, 23592}, {34544, 312}, {35069, 28654}, {35128, 23978}, {41282, 7}

X(52060) = X(145)X(4859)∩X(3161)X(24392)

See Ivan Pavlov, euclid 5514.

X(52060) lies on these lines: {145, 4859}, {3161, 24392}

X(52061) = X(100)X(4885)∩X(2254)X(4926)

See Ivan Pavlov, euclid 5514.

X(52061) lies on this line: X(52061) lies on these lines: {100, 4885}, {2254, 4926}, {11235, 45320}, {11934, 42552}

X(52062) = (name pending)

See Ivan Pavlov, euclid 5515.

X(52062) lies on this line: {222, 2911}

X(52063) = X(9)X(222)∩X(963)X(2910)

See Ivan Pavlov, euclid 5515.

X(52063) lies on these lines: {9, 222}, {963, 2910}

X(52064) = X(9)X(664)∩X(738)X(2125)

See Ivan Pavlov, euclid 5516.

X(52064) lies on the inconic with perspector X(9) and these lines: {9, 664}, {738, 2125}

X(52065) = X(37)X(668)∩X(213)X(904)

See Ivan Pavlov, euclid 5516.

X(52065) lies on the inconic with perspector X(37) and these lines: {37, 668}, {213, 904}

X(52066) = X(3)X(648)∩X(216)X(2972)

See Ivan Pavlov, euclid 5516.

X(52066) lies on the inconic with perspector X(3) and these lines: {3, 648}, {216, 2972}

X(52067) = X(6)X(99)∩X(39)X(373)

See Ivan Pavlov, euclid 5516.

X(52067) lies on the inconic with perspector X(6) and these lines: {6, 99}, {39, 373}, {194, 4576}

X(52068) = X(10)X(190)∩X(67)X(71)

See Ivan Pavlov, euclid 5516.

X(52068) lies on the inconic with perspector X(10) and these lines: {10, 190}, {67, 71}, {191, 3882}, {896, 3712}

X(52069) = EULER LINE INTERCEPT OF X(54)X(22660)

See Panchapakesan and César Lozada, euclid 5519.

X(52069) lies on these lines: {2, 3}, {54, 22660}, {113, 18475}, {141, 10733}, {146, 40640}, {311, 6148}, {343, 50435}, {524, 41716}, {541, 11562}, {542, 6467}, {1176, 11744}, {1503, 15305}, {1514, 15080}, {1531, 14389}, {1568, 11430}, {1989, 41890}, {2777, 12824}, {3058, 9627}, {3060, 16657}, {3098, 26156}, {3521, 37471}, {3580, 18390}, {4550, 18474}, {5092, 13202}, {5562, 13403}, {5655, 11597}, {5876, 44076}, {5889, 12241}, {5891, 17702}, {5907, 14516}, {6146, 12111}, {6247, 15062}, {7689, 26879}, {7745, 26216}, {9924, 34775}, {9971, 29181}, {10112, 45187}, {10263, 15807}, {10605, 18911}, {10625, 12897}, {10984, 22802}, {11232, 12022}, {11381, 44829}, {11439, 16655}, {11441, 19467}, {11442, 18396}, {11454, 26913}, {11459, 44665}, {11464, 51425}, {11750, 16659}, {12006, 34798}, {12134, 12289}, {12162, 34224}, {12163, 18912}, {12233, 13434}, {12278, 15056}, {12370, 18436}, {13219, 41005}, {13568, 15043}, {13851, 21243}, {15030, 18400}, {15048, 52058}, {15053, 37648}, {15060, 30522}, {15072, 15311}, {16194, 16658}, {18374, 44882}, {18451, 46818}, {19149, 43273}, {19153, 25406}, {21969, 32352}, {23324, 45303}, {26166, 32819}, {26206, 46264}, {26917, 44158}, {29012, 32062}, {31804, 43605}, {34796, 45298}, {37283, 51163}, {43595, 43818}

X(52069) = midpoint of X(i) and X(j) for these {i, j}: {381, 18564}, {1885, 7667}, {12225, 34603}, {18560, 44458}, {18563, 38321}
X(52069) = reflection of X(i) in X(j) for these (i, j): (2, 34664), (20, 7667), (3060, 16657), (6240, 38321), (7540, 3845), (7576, 381), (7667, 12362), (16658, 16194), (31829, 7734), (34603, 4), (34613, 3830), (38320, 3545), (38321, 5), (38322, 5066), (38323, 2), (44458, 3), (45968, 12022)
X(52069) = inverse of X(23) in 2nd Droz-Farny circle
X(52069) = intersection, other than A, B, C, of circumconics {{A, B, C, X(186), X(41890)}} and {{A, B, C, X(235), X(1989)}}
X(52069) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 16072, 2), (4, 35500, 5), (5, 18563, 6240), (5, 44249, 186), (378, 18531, 858), (381, 10201, 403), (381, 48411, 5), (550, 46030, 2070), (1596, 44239, 23), (1885, 12362, 20), (2043, 2044, 24), (3153, 7527, 427), (7526, 18404, 1594), (10018, 18560, 6240), (18403, 35500, 5133), (18403, 48411, 381), (18531, 49669, 378), (18533, 18537, 1995), (31180, 44441, 858), (37126, 50009, 6823)

X(52070) = EULER LINE INTERCEPT OF X(113)X(13367)

See Sriram Panchapakesan and César Lozada, euclid 5519.

X(52070) lies on these lines: {2, 3}, {113, 13367}, {143, 15807}, {511, 12897}, {541, 18128}, {1209, 12295}, {1503, 15074}, {1539, 5893}, {2777, 11557}, {3521, 13353}, {3564, 44439}, {3581, 18442}, {5448, 11430}, {5663, 6146}, {5876, 44665}, {5907, 17702}, {5944, 16252}, {5946, 13568}, {6102, 12241}, {6720, 7861}, {7354, 37729}, {7689, 18390}, {7728, 12228}, {9627, 15171}, {9729, 43577}, {9820, 43394}, {9826, 34584}, {10112, 12370}, {10113, 34826}, {10282, 46817}, {10483, 37696}, {10605, 18952}, {10619, 15063}, {11245, 43575}, {11381, 11750}, {11454, 26917}, {11468, 26913}, {11572, 18488}, {11744, 40441}, {11808, 13598}, {12022, 34783}, {12041, 49109}, {12111, 44076}, {12118, 15068}, {12134, 30522}, {12162, 21659}, {12278, 15058}, {12289, 15305}, {12825, 32423}, {13202, 37513}, {13382, 43573}, {13419, 46849}, {13474, 44407}, {13491, 15311}, {14516, 18435}, {14915, 44829}, {15062, 25739}, {16654, 32332}, {16655, 32137}, {18396, 32140}, {18439, 34224}, {18555, 41586}, {18914, 45957}, {19137, 48896}, {19467, 32139}, {21663, 43817}, {22466, 43689}, {22802, 34114}, {25738, 45788}, {26879, 43821}, {31815, 44413}, {32171, 51425}, {32271, 43585}, {34117, 48906}, {34514, 41362}, {34785, 46261}, {34796, 45967}, {36201, 44479}, {41008, 44138}, {41726, 50708}, {43865, 44158}, {44516, 46686}, {44870, 45286}

X(52070) = midpoint of X(i) and X(j) for these {i, j}: {3, 18560}, {4, 18563}, {382, 12225}, {1885, 12605}, {6240, 18562}, {7576, 18561}, {11381, 11750}, {12111, 44076}, {12162, 21659}, {18439, 34224}
X(52070) = reflection of X(i) in X(j) for these (i, j): (143, 15807), (550, 12362), (3575, 546), (3627, 13488), (6102, 12241), (6240, 31830), (7553, 3853), (11819, 4), (12134, 45959), (12370, 13403), (13419, 46849), (16655, 32137), (32358, 12370), (34783, 43588), (34798, 13568), (43577, 9729), (45286, 44870), (45957, 18914), (45971, 3850)
X(52070) = inverse of X(2070) in 2nd Droz-Farny circle
X(52070) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 16386, 548), (3, 34350, 550), (4, 5576, 546), (4, 7526, 5), (4, 7527, 5576), (4, 7564, 3845), (235, 44249, 1658), (381, 18562, 6240), (403, 34005, 3), (546, 10020, 403), (3628, 10212, 140), (5576, 18323, 4), (7527, 18323, 546), (9818, 44457, 6644), (10024, 14118, 140), (14130, 18403, 1594), (15331, 44235, 468), (16238, 44920, 5), (18364, 35487, 140), (44242, 46030, 24)

X(52071) = EULER LINE INTERCEPT OF X(52)X(43577)

See Sriram Panchapakesan and César Lozada, euclid 5519.

X(52071) lies on these lines: {2, 3}, {52, 43577}, {64, 11442}, {69, 31369}, {74, 12359}, {110, 2883}, {185, 10112}, {343, 5894}, {394, 5895}, {511, 40316}, {1092, 22802}, {1204, 3580}, {1498, 46818}, {1503, 12272}, {2777, 5562}, {2892, 16789}, {3060, 13568}, {3100, 7354}, {3357, 50434}, {3521, 37495}, {3933, 13219}, {4296, 6284}, {4846, 7592}, {5254, 41336}, {5448, 10564}, {5449, 43607}, {5480, 43815}, {5878, 11441}, {5893, 11064}, {6000, 14516}, {6146, 15072}, {6225, 40196}, {6241, 12282}, {6247, 13445}, {6509, 33553}, {6696, 23293}, {8549, 48905}, {8567, 37638}, {8743, 15075}, {9538, 18990}, {9730, 12897}, {10282, 16163}, {10539, 32111}, {10574, 12241}, {10575, 17702}, {10627, 34584}, {11381, 12058}, {11417, 42258}, {11418, 42259}, {11420, 42087}, {11421, 42088}, {11449, 16252}, {11454, 20725}, {11456, 12118}, {11457, 12293}, {11468, 44158}, {11750, 12235}, {12022, 40647}, {12111, 15311}, {12134, 12290}, {12220, 29181}, {12302, 20302}, {13391, 34798}, {13394, 51033}, {13491, 44076}, {13567, 22466}, {14915, 16659}, {15032, 43595}, {15043, 16657}, {15262, 42459}, {15466, 18848}, {16111, 46085}, {16226, 40240}, {16658, 45286}, {19121, 44882}, {20191, 43907}, {21659, 46850}, {22530, 52003}, {22660, 43574}, {27082, 35260}, {29323, 43130}, {30737, 32819}, {34146, 46442}, {41736, 46373}, {44829, 45237}

X(52071) = midpoint of X(i) and X(j) for these {i, j}: {1657, 18565}, {3529, 34797}, {12278, 12279}
X(52071) = reflection of X(i) in X(j) for these (i, j): (52, 43577), (1885, 31829), (3146, 3575), (5073, 11819), (11750, 14641), (12225, 20), (12290, 12134), (18560, 3), (18563, 550), (21659, 46850), (33703, 7553), (34224, 10575), (44076, 13491)
X(52071) = anticomplement of X(1885)
X(52071) = inverse of X(2071) in 2nd Droz-Farny circle
X(52071) = intersection, other than A, B, C, of circumconics {{A, B, C, X(69), X(30552)}} and {{A, B, C, X(74), X(45172)}}
X(52071) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 31726, 5), (3, 37197, 2), (4, 376, 3546), (4, 11413, 858), (20, 3146, 1370), (20, 7488, 550), (20, 37201, 22), (20, 44440, 3), (20, 50009, 2071), (376, 7505, 3), (382, 6642, 4), (548, 44452, 3), (550, 15761, 3), (1370, 1995, 858), (1657, 11414, 20), (1885, 31829, 2), (2071, 50009, 5), (3529, 35513, 20), (12086, 34007, 427), (12225, 38323, 858), (15704, 44239, 20), (18560, 44458, 3)

X(52072) = EULER LINE INTERCEPT OF X(79)X(1448)

Sriram Panchapakesan and César Lozada, euclid 5519.

X(52072) lies on these lines: {2, 3}, {79, 1448}, {758, 41733}, {1798, 11744}, {2771, 12825}, {2777, 12826}, {9627, 10543}, {15311, 41734}

X(52072) = inverse of X(1325) in 2nd Droz-Farny circle

X(52073) = EULER LINE INTERCEPT OF X(323)X(43818)

See Sriram Panchapakesan and César Lozada, euclid 5519.

X(52073) lies on these lines: {2, 3}, {323, 43818}, {1154, 12241}, {1209, 13851}, {1216, 13403}, {1503, 13470}, {2777, 44862}, {3521, 13339}, {3564, 31834}, {3580, 43821}, {4549, 39571}, {4550, 18381}, {5562, 12370}, {5876, 6146}, {5891, 21659}, {5944, 51425}, {7687, 32348}, {9627, 15172}, {10263, 16657}, {10982, 31815}, {11064, 43394}, {11459, 44076}, {11577, 15738}, {11591, 44665}, {11745, 13364}, {11750, 15030}, {11793, 17702}, {11801, 46085}, {12006, 13568}, {12022, 18436}, {12134, 15060}, {12163, 18952}, {12289, 15056}, {12897, 15644}, {13292, 43575}, {13391, 15807}, {13754, 43588}, {14128, 30522}, {15068, 19467}, {16836, 43577}, {18435, 34224}, {22261, 27353}, {22660, 32046}, {29012, 46849}, {32110, 35240}, {34380, 44439}, {37513, 43831}, {44407, 44870}

X(52073) = midpoint of X(i) and X(j) for these {i, j}: {5, 12605}, {550, 1885}, {1216, 13403}, {5562, 12370}, {5876, 6146}, {11819, 12225}, {12897, 15644}, {13470, 45959}, {18436, 32358}, {31834, 45970}
X(52073) = reflection of X(i) in X(j) for these (i, j): (6756, 3850), (13292, 43575), (13568, 12006), (31829, 3530), (31830, 5), (31833, 3628), (45971, 9825)
X(52073) = complement of the complement of X(18563)
X(52073) = inverse of X(5899) in 2nd Droz-Farny circle
X(52073) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 35481, 550), (5, 549, 6639), (5, 550, 24), (5, 18377, 546), (5, 23047, 3850), (5, 47339, 3861), (548, 44232, 1658), (1656, 18564, 6240), (3153, 35500, 5576), (7503, 18404, 5), (7505, 18560, 3575), (12605, 34664, 5), (16072, 34477, 547), (18377, 49671, 5)

X(52074) = X(1)X(24302)∩X(5)X(3244)

See Antreas Hatzipolakis and Peter Moses, euclid 5521.

X(52074) lies on these lines: {1, 24302}, {5, 3244}, {952, 7704}, {1482, 23958}, {1737, 17662}, {3623, 18549}, {10247, 13743}, {11545, 15079}, {33658, 33668}, {37735, 43731}

X(52075) = (name pending)

See Antreas Hatzipolakis and Peter Moses, euclid 5521.

X(52075) lies on the Mandart circle and these lines: { }

X(52076) = X(4)X(512)∩X(98)X(523)

X(52076) lies on the cubic K186 and these lines: {4, 512}, {23, 9979}, {98, 523}, {107, 685}, {316, 9517}, {525, 38664}, {850, 3448}, {1383, 2395}, {2799, 5999}, {2966, 47290}, {5466, 5967}, {6130, 38227}, {8029, 40820}, {9154, 10630}, {10555, 10561}, {11177, 23878}, {13574, 47254}, {14120, 51441}, {16230, 41204}, {31953, 43460}, {32120, 41254}, {38672, 38679}, {41377, 44427}, {46619, 47502}

X(52076) = reflection of X(43460) in X(31953)
X(52076) = X(i)-isoconjugate of X(j) for these (i,j): {67, 23997}, {1755, 17708}, {2157, 2421}
X(52076) = X(i)-Dao conjugate of X(j) for these (i,j): {511, 5099}, {2421, 40583}, {6333, 18311}, {17708, 36899}
X(52076) = crosspoint of X(685) and X(9154)
X(52076) = crosssum of X(684) and X(9155)
X(52076) = barycentric product X(i)*X(j) for these {i,j}: {23, 43665}, {98, 9979}, {290, 2492}, {316, 2395}, {879, 37765}, {2422, 40074}, {9154, 18311}, {9517, 16081}, {33752, 34536}
X(52076) = barycentric quotient X(i)/X(j) for these {i,j}: {23, 2421}, {98, 17708}, {316, 2396}, {879, 34897}, {2395, 67}, {2422, 3455}, {2492, 511}, {6531, 935}, {8744, 4230}, {9517, 36212}, {9979, 325}, {10561, 5968}, {18311, 50567}, {18374, 14966}, {21205, 51370}, {33752, 36790}, {37765, 877}, {42659, 3289}, {43665, 18019}

X(52077) = X(3)X(17818)∩X(4)X(155)

X(52077) lies on the cubic K163 and these lines: {3, 17818}, {4, 155}, {6, 1196}, {24, 12271}, {25, 34382}, {154, 41615}, {193, 11382}, {323, 7396}, {394, 1368}, {511, 1619}, {539, 16072}, {542, 17847}, {1147, 5892}, {1181, 31829}, {1498, 17837}, {1994, 7398}, {2063, 23291}, {2138, 15143}, {2979, 12163}, {5012, 47391}, {6000, 12164}, {6353, 40318}, {6503, 47195}, {6642, 21651}, {7387, 34750}, {8548, 10601}, {8780, 21313}, {9825, 12161}, {9908, 37486}, {9936, 44752}, {9937, 44077}, {10539, 12309}, {12058, 13754}, {12282, 17928}, {12429, 17814}, {14984, 33586}, {15068, 44920}, {15261, 40321}, {17810, 41714}, {17811, 34507}

X(52077) = reflection of X(i) in X(j) for these {i,j}: {25, 41619}, {394, 34966}, {12429, 18390}, {18451, 155}
X(52077) = X(393)-Ceva conjugate of X(3)
X(52077) = X(3926)-Dao conjugate of X(3964)
X(52077) = crosssum of X(647) and X(5139)
X(52077) = crossdifference of every pair of points on line {3566, 13400}
X(52077) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {155, 15316, 36747}, {1147, 19458, 36752}, {3167, 6391, 5020}, {6391, 19588, 40337}

X(52078) = X(1)X(3346)∩X(4)X(57)

X(52078) lies on the cubic K964 and these lines: {1, 3346}, {3, 9118}, {4, 57}, {7, 253}, {20, 7338}, {21, 37141}, {65, 7157}, {196, 42465}, {226, 8806}, {268, 9376}, {271, 377}, {281, 47851}, {282, 5746}, {515, 46881}, {1012, 8886}, {1249, 1394}, {1294, 4304}, {1422, 1448}, {1433, 5706}, {1935, 36049}, {4313, 51348}, {5930, 41086}, {14249, 44697}, {17102, 20264}, {28784, 37046}

X(52078) = X(37141)-Ceva conjugate of X(14331)
X(52078) = X(40933)-cross conjugate of X(5930)
X(52078) = X(i)-isoconjugate of X(j) for these (i,j): {21, 41088}, {55, 41082}, {1817, 30457}, {2155, 27398}, {2187, 5931}, {2360, 44692}, {14298, 46639}
X(52078) = X(i)-Dao conjugate of X(j) for these (i,j): {122, 8058}, {223, 41082}, {347, 1427}, {1819, 45248}, {27398, 45245}, {40611, 41088}
X(52078) = crosspoint of X(7) and X(44697)
X(52078) = barycentric product X(i)*X(j) for these {i,j}: {20, 8808}, {85, 41086}, {189, 5930}, {226, 41084}, {280, 36908}, {309, 30456}, {1440, 8804}, {1903, 33673}, {17898, 37141}, {18623, 39130}, {34404, 40933}
X(52078) = barycentric quotient X(i)/X(j) for these {i,j}: {20, 27398}, {57, 41082}, {189, 5931}, {1394, 1817}, {1400, 41088}, {1903, 44692}, {2357, 30457}, {3198, 2324}, {3213, 3194}, {5930, 329}, {6587, 8058}, {8059, 46639}, {8804, 7080}, {8808, 253}, {15905, 1819}, {18623, 8822}, {30456, 40}, {36908, 347}, {40933, 223}, {41084, 333}, {41086, 9}, {42658, 10397}, {44696, 41083}

X(52079) = X(2)X(42122)∩X(15)X(376)

X(52079) lies on the cubic K1233 and these lines: {2, 42122}, {3, 43870}, {4, 11480}, {5, 43869}, {6, 3528}, {13, 46333}, {14, 15719}, {15, 376}, {16, 21735}, {20, 11542}, {140, 43466}, {395, 15715}, {397, 43777}, {631, 10645}, {1656, 43630}, {3090, 5352}, {3091, 42130}, {3146, 42124}, {3522, 11485}, {3523, 42117}, {3524, 5334}, {3525, 5321}, {3529, 5238}, {3533, 42139}, {3534, 43465}, {3543, 42132}, {3544, 43029}, {3545, 36967}, {3832, 42144}, {3855, 19107}, {3861, 42950}, {5055, 43365}, {5056, 42136}, {5059, 42128}, {5067, 42133}, {5068, 43103}, {5071, 42093}, {5072, 42888}, {5073, 42627}, {5318, 11001}, {5335, 17538}, {5339, 43446}, {5343, 43028}, {5344, 43487}, {5365, 42773}, {10299, 11489}, {10303, 42125}, {10304, 11486}, {10654, 15698}, {11476, 35483}, {11481, 19708}, {11539, 43541}, {11541, 42094}, {11543, 15717}, {12100, 42818}, {12101, 43647}, {12103, 42815}, {12250, 17826}, {14891, 42917}, {15682, 42099}, {15683, 42137}, {15685, 42889}, {15689, 43631}, {15690, 42922}, {15692, 42121}, {15693, 42628}, {15695, 42633}, {15697, 42974}, {15702, 42154}, {15704, 42817}, {15705, 42975}, {15708, 43102}, {15709, 16967}, {15710, 37641}, {15712, 42816}, {15716, 42497}, {15721, 43417}, {16239, 42963}, {16241, 41099}, {16772, 42134}, {16809, 43770}, {16960, 42892}, {17578, 42146}, {18582, 33703}, {21734, 42115}, {22235, 42916}, {33604, 43332}, {33605, 42956}, {35409, 42430}, {36970, 42473}, {37835, 43772}, {41100, 42516}, {41108, 43002}, {41121, 42997}, {42086, 42980}, {42100, 42965}, {42101, 42490}, {42103, 42908}, {42109, 43403}, {42114, 43632}, {42118, 50693}, {42131, 42912}, {42138, 49135}, {42141, 42152}, {42156, 42683}, {42160, 42904}, {42501, 49873}, {42505, 43498}, {42509, 42778}, {42584, 42988}, {42589, 42910}, {42914, 42930}, {42932, 50687}, {42940, 43502}, {42989, 43496}, {43106, 49947}, {43228, 51945}, {43328, 43556}, {43555, 49824}

X(52079) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {15, 42091, 37640}, {15, 42529, 42091}, {3524, 5334, 43464}, {3524, 42942, 43482}, {3524, 43482, 43543}, {5238, 42090, 11488}, {5334, 43464, 43543}, {5334, 43482, 43778}, {10645, 42119, 631}, {11480, 42096, 42945}, {11480, 42098, 42687}, {11488, 42090, 3529}, {15717, 43243, 11543}, {36967, 42092, 42140}, {37640, 42529, 376}, {42087, 42687, 42098}, {42092, 42140, 3545}, {42099, 42142, 15682}, {42131, 42912, 42982}, {42132, 42585, 3543}, {42910, 43645, 42589}, {43028, 43105, 5343}, {43464, 43482, 5334}, {43543, 43778, 5334}

X(52080) = X(2)X(42123)∩X(16)X(376)

X(52080) lies on the cubic K1233 and these lines: {2, 42123}, {3, 43869}, {4, 11481}, {5, 43870}, {6, 3528}, {13, 15719}, {14, 46333}, {15, 21735}, {16, 376}, {20, 11543}, {140, 43465}, {396, 15715}, {398, 43778}, {631, 10646}, {1656, 43631}, {3090, 5351}, {3091, 42131}, {3146, 42121}, {3522, 11486}, {3523, 42118}, {3524, 5335}, {3525, 5318}, {3529, 5237}, {3533, 42142}, {3534, 43466}, {3543, 42129}, {3544, 43028}, {3545, 36968}, {3832, 42145}, {3855, 19106}, {3861, 42951}, {5055, 43364}, {5056, 42137}, {5059, 42125}, {5067, 42134}, {5068, 43102}, {5071, 42094}, {5072, 42889}, {5073, 42628}, {5321, 11001}, {5334, 17538}, {5340, 43447}, {5343, 43488}, {5344, 43029}, {5366, 42774}, {10299, 11488}, {10303, 42128}, {10304, 11485}, {10653, 15698}, {11475, 35483}, {11480, 19708}, {11539, 43540}, {11541, 42093}, {11542, 15717}, {12100, 42817}, {12101, 43648}, {12103, 42816}, {12250, 17827}, {14891, 42916}, {15682, 42100}, {15683, 42136}, {15685, 42888}, {15689, 43630}, {15690, 42923}, {15692, 42124}, {15693, 42627}, {15695, 42634}, {15697, 42975}, {15702, 42155}, {15704, 42818}, {15705, 42974}, {15708, 43103}, {15709, 16966}, {15710, 37640}, {15712, 42815}, {15716, 42496}, {15721, 43416}, {16239, 42962}, {16242, 41099}, {16773, 42133}, {16808, 43769}, {16961, 42893}, {17578, 42143}, {18581, 33703}, {21734, 42116}, {22237, 42917}, {33604, 42957}, {33605, 43333}, {35409, 42429}, {36969, 42472}, {37832, 43771}, {41101, 42517}, {41107, 43003}, {41122, 42996}, {42085, 42981}, {42099, 42964}, {42102, 42491}, {42106, 42909}, {42108, 43404}, {42111, 43633}, {42117, 50693}, {42130, 42913}, {42135, 49135}, {42140, 42149}, {42153, 42682}, {42161, 42905}, {42500, 49874}, {42504, 43497}, {42508, 42777}, {42585, 42989}, {42588, 42911}, {42915, 42931}, {42933, 50687}, {42941, 43501}, {42988, 43495}, {43105, 49948}, {43229, 51944}, {43329, 43557}, {43554, 49825}

X(52080) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {16, 42090, 37641}, {16, 42528, 42090}, {3524, 5335, 43463}, {3524, 42943, 43481}, {3524, 43481, 43542}, {5237, 42091, 11489}, {5335, 43463, 43542}, {5335, 43481, 43777}, {10646, 42120, 631}, {11481, 42095, 42686}, {11481, 42097, 42944}, {11489, 42091, 3529}, {15717, 43242, 11542}, {36968, 42089, 42141}, {37641, 42528, 376}, {42088, 42686, 42095}, {42089, 42141, 3545}, {42100, 42139, 15682}, {42129, 42584, 3543}, {42130, 42913, 42983}, {42911, 43646, 42588}, {43029, 43106, 5344}, {43463, 43481, 5335}, {43542, 43777, 5335}

X(52081) = X(2)X(9473)∩X(6)X(264)

X(52081) lies on the cubic K421 and these lines: {2, 9473}, {5, 47388}, {6, 264}, {69, 35906}, {83, 45910}, {98, 3407}, {141, 2966}, {384, 8861}, {1031, 20021}, {1352, 32545}, {3329, 9477}, {3589, 34369}, {5967, 51171}, {6655, 9474}, {7792, 41932}, {9230, 43187}, {14382, 42534}, {14561, 47741}, {15407, 43278}, {16989, 40820}, {39941, 44155}

X(52081) = X(i)-isoconjugate of X(j) for these (i,j): {1755, 43696}, {1959, 41533}
X(52081) = X(i)-Dao conjugate of X(j) for these (i,j): {1691, 36213}, {3569, 46669}, {36899, 43696}
X(52081) = cevapoint of X(5207) and X(19571)
X(52081) = trilinear pole of line {6660, 14316}
X(52081) = barycentric product X(i)*X(j) for these {i,j}: {98, 5207}, {290, 6660}, {1821, 19555}, {2966, 14316}, {3493, 14382}, {8783, 34238}, {18024, 19558}, {19559, 46273}, {19571, 36897}
X(52081) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 43696}, {1976, 41533}, {3493, 40810}, {5207, 325}, {6660, 511}, {14316, 2799}, {19555, 1959}, {19556, 9418}, {19558, 237}, {19559, 1755}, {19560, 9417}, {19571, 5976}, {19576, 36213}, {40820, 51244}, {51954, 14251}

X(52082) = X(4)X(8)∩X(19)X(208)

X(52082) lies on the cubic K696 and these lines: {4, 8}, {19, 208}, {25, 7952}, {28, 1068}, {33, 976}, {34, 49487}, {158, 6524}, {196, 1426}, {242, 4194}, {264, 1240}, {278, 961}, {281, 429}, {405, 51410}, {1598, 21664}, {1714, 1726}, {1785, 7713}, {1826, 20653}, {1838, 4196}, {1861, 36568}, {1882, 7235}, {1895, 37394}, {1897, 7718}, {2969, 4214}, {3100, 28104}, {3616, 16066}, {4186, 7071}, {4198, 7009}, {4206, 8747}, {4207, 39579}, {4222, 26378}, {5307, 27368}, {5521, 50932}, {5905, 41723}, {6197, 37414}, {7408, 20056}, {7412, 11383}, {7487, 45766}, {9780, 17927}, {10629, 44662}, {11396, 34231}, {11406, 37194}, {12679, 40953}, {14018, 24883}, {17677, 34730}, {21370, 23537}, {22479, 37305}, {23661, 26118}, {37245, 44696}, {37432, 40836}

X(52082) = barycentric product X(i)*X(j) for these {i,j}: {158, 10319}, {278, 2551}, {653, 47136}, {1118, 23600}
X(52082) = barycentric quotient X(i)/X(j) for these {i,j}: {2551, 345}, {10319, 326}, {23600, 1264}, {47136, 6332}
X(52082) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 7046, 5090}, {4, 41013, 7102}, {19, 225, 37384}, {33, 1842, 28076}, {278, 14257, 7103}, {1824, 1828, 1902}, {11396, 37226, 34231}, {39579, 39585, 4207}

X(52083) = X(2)X(32)∩X(6)X(19585)

X(52083) lies on the cubic K421 and these lines: {2, 32}, {6, 19585}, {20, 14133}, {31, 17743}, {194, 40382}, {384, 3051}, {1031, 20021}, {1316, 41520}, {3114, 7766}, {3117, 3972}, {3329, 14096}, {3407, 34396}, {3978, 5007}, {7754, 11335}, {7797, 14957}, {7806, 37988}, {7823, 33734}, {9229, 31390}, {11205, 32476}, {11338, 30435}, {12212, 14617}, {16989, 37190}

X(52083) = anticomplement of the isotomic conjugate of X(14617)
X(52083) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3407, 21278}, {14617, 6327}, {18898, 17165}
X(52083) = X(i)-Ceva conjugate of X(j) for these (i,j): {12212, 7766}, {14617, 2}
X(52083) = barycentric product X(183)*X(3498)
X(52083) = barycentric quotient X(3498)/X(262)
X(52083) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20088, 20022}, {32, 10337, 2896}, {83, 8623, 2}, {384, 3051, 40858}, {384, 10339, 10340}, {12206, 16985, 251}

X(52084) = X(1)X(41)∩X(9)X(1742)

X(52084) lies on the cubic K294 and these lines: {1, 41}, {9, 1742}, {44, 39344}, {57, 20662}, {218, 1046}, {220, 2943}, {241, 2348}, {269, 651}, {910, 9441}, {936, 24036}, {978, 16572}, {1447, 3008}, {1757, 9499}, {2310, 3731}, {5293, 16601}, {6559, 28850}, {7083, 34247}, {8074, 49772}, {8915, 12514}, {9259, 40133}, {16970, 38865}, {17966, 43065}, {20601, 37597}, {31638, 39775}

X(52084) = X(i)-Ceva conjugate of X(j) for these (i,j): {241, 1}, {2348, 1743}
X(52084) = X(14942)-Dao conjugate of X(36796)
X(52084) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {294, 9502, 1}, {2246, 9502, 294}

X(52085) = X(1)X(335)∩X(12)X(85)

X(52085) lies on the cubic K744 and these lines: {1, 335}, {8, 40217}, {10, 40098}, {12, 85}, {72, 660}, {76, 18034}, {291, 24443}, {876, 48396}, {986, 40155}, {1224, 40092}, {1281, 40724}, {1655, 6625}, {2292, 40796}, {3701, 4583}, {3868, 40730}, {4562, 35163}, {7187, 30663}, {16735, 37128}, {17789, 51859}, {18836, 44172}, {33299, 36906}, {33950, 51866}

X(52085) = X(3509)-cross conjugate of X(335)
X(52085) = X(i)-isoconjugate of X(j) for these (i,j): {238, 8852}, {1428, 7281}, {1914, 3512}, {2210, 7261}, {8300, 30648}, {14599, 40845}, {18036, 18892}, {24479, 51328}
X(52085) = X(i)-Dao conjugate of X(j) for these (i,j): {238, 8300}, {3512, 36906}, {8852, 9470}
X(52085) = trilinear pole of line {4071, 4458}
X(52085) = barycentric product X(i)*X(j) for these {i,j}: {1, 51859}, {291, 17789}, {334, 3509}, {335, 4645}, {1281, 40098}, {4071, 18827}, {4458, 4562}, {8868, 51868}, {17798, 18895}, {18037, 30663}, {18262, 44170}, {19554, 44172}, {20715, 40017}, {40217, 40724}
X(52085) = barycentric quotient X(i)/X(j) for these {i,j}: {291, 3512}, {292, 8852}, {334, 40845}, {335, 7261}, {1281, 4366}, {3509, 238}, {4071, 740}, {4458, 812}, {4562, 51614}, {4645, 239}, {4876, 7281}, {4987, 4974}, {5018, 1429}, {17789, 350}, {17798, 1914}, {18037, 39044}, {18262, 14599}, {18787, 41534}, {18895, 18036}, {19554, 2210}, {19557, 8300}, {19561, 51328}, {20715, 2238}, {20741, 7193}, {22116, 40781}, {27951, 27855}, {30663, 24479}, {30669, 7061}, {40724, 6654}, {40873, 18786}, {51859, 75}

X(52086) = X(1)X(33844)∩X(3)X(142)

X(52086) lies on the cubic K1122 and these lines: {1, 33844}, {3, 142}, {35, 33149}, {36, 50301}, {37, 750}, {55, 17301}, {65, 7225}, {75, 100}, {198, 4413}, {513, 2267}, {956, 3416}, {1086, 37586}, {1279, 37600}, {1308, 38884}, {1621, 17399}, {2214, 2260}, {2217, 47521}, {3008, 4471}, {3185, 37261}, {3220, 5251}, {3664, 4497}, {4026, 11112}, {4259, 4649}, {4268, 49537}, {4675, 17798}, {5135, 16468}, {8424, 17279}, {17278, 23868}, {17302, 29831}, {17400, 37449}, {17596, 30903}, {18042, 25279}, {25439, 32921}, {30810, 40560}, {30944, 44304}

X(52087) = X(2)X(573)∩X(37)X(65)

X(52087) lies on the cubic K253 and these lines: {2, 573}, {6, 34278}, {9, 27040}, {10, 3588}, {37, 65}, {190, 1240}, {429, 2354}, {672, 2670}, {992, 4271}, {1193, 1682}, {1211, 19608}, {1213, 2183}, {2238, 2347}, {2277, 16685}, {3161, 3730}, {4225, 37508}, {5816, 37191}, {13738, 37499}, {17751, 21061}, {17755, 20891}, {19734, 21483}, {21748, 22065}, {40590, 40611}

X(52087) = complement of X(20028)
X(52087) = complement of the isotomic conjugate of X(17751)
X(52087) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 1193}, {42, 12}, {213, 37662}, {572, 3739}, {1918, 21796}, {2975, 3741}, {14829, 21240}, {14973, 3454}, {17074, 17050}, {17751, 2887}, {20986, 1125}, {21061, 141}, {37558, 2886}, {51664, 17059}
X(52087) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 1193}, {17185, 2292}
X(52087) = X(i)-isoconjugate of X(j) for these (i,j): {2, 40453}, {961, 46880}, {2051, 2363}, {2298, 20028}, {14534, 34434}
X(52087) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 1193}, {960, 2051}, {32664, 40453}
X(52087) = crosspoint of X(i) and X(j) for these (i,j): {2, 17751}, {572, 21061}, {14829, 37558}
X(52087) = crossdifference of every pair of points on line {3737, 4581}
X(52087) = barycentric product X(i)*X(j) for these {i,j}: {226, 46879}, {572, 1211}, {960, 37558}, {1193, 17751}, {2092, 14829}, {2292, 2975}, {3666, 21061}, {11109, 22076}, {17074, 21033}, {18697, 20986}, {20617, 46877}
X(52087) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 40453}, {572, 14534}, {1193, 20028}, {2092, 2051}, {2269, 46880}, {3725, 34434}, {14829, 40827}, {17751, 1240}, {20986, 2363}, {21061, 30710}, {37558, 31643}, {46879, 333}

X(52088) = X(6)X(598)∩X(39)X(9878)

X(52088) lies on the cubic K914 and these lines: {6, 598}, {39, 9878}, {76, 542}, {99, 3534}, {147, 7799}, {148, 14537}, {316, 11645}, {512, 16175}, {543, 7837}, {2482, 3314}, {3849, 7779}, {4027, 7884}, {5026, 7937}, {5149, 47005}, {5152, 6054}, {5309, 32528}, {5461, 7875}, {5939, 10033}, {5989, 7809}, {6033, 14458}, {6034, 7878}, {7757, 33693}, {7771, 9774}, {7782, 8724}, {7840, 51932}, {7865, 8290}, {7926, 14931}, {7934, 8289}, {9143, 22254}, {9890, 51224}, {11152, 32452}, {19569, 20094}, {35356, 40877}

X(52088) = midpoint of X(19569) and X(20094)
X(52088) = reflection of X(i) in X(j) for these {i,j}: {148, 14537}, {9878, 39}, {11057, 99}, {14458, 6033}
X(52088) = isotomic conjugate of the isogonal conjugate of X(9999)
X(52088) = barycentric product X(76)*X(9999)
X(52088) = barycentric quotient X(9999)/X(6)

X(52089) = X(1)X(28391)∩X(6)X(57)

X(52089) lies on the cubic K135 and these lines: {1, 28391}, {6, 57}, {42, 41350}, {77, 17754}, {239, 9436}, {278, 36482}, {291, 5018}, {292, 24479}, {348, 16822}, {614, 1423}, {651, 3509}, {672, 2114}, {894, 40723}, {982, 6180}, {1214, 36483}, {4318, 6168}, {4853, 39959}, {7187, 9312}, {8270, 34492}, {18788, 51871}, {20980, 43051}, {21147, 34497}

X(52089) = X(i)-Ceva conjugate of X(j) for these (i,j): {291, 57}, {41352, 18788}
X(52089) = X(9)-isoconjugate of X(43747)
X(52089) = X(i)-Dao conjugate of X(j) for these (i,j): {350, 1447}, {478, 43747}
X(52089) = crosssum of X(3119) and X(4435)
X(52089) = barycentric product X(i)*X(j) for these {i,j}: {1, 41352}, {7, 18788}, {75, 51871}, {7233, 8932}
X(52089) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 43747}, {8932, 3685}, {18788, 8}, {41352, 75}, {51871, 1}
X(52089) = {X(6),X(7204)}-harmonic conjugate of X(57)

X(52090) = X(3)X(67)∩X(98)X(140)

X(52090) lies on the cubic K837 and these lines: {2, 51523}, {3, 67}, {4, 147}, {5, 6054}, {20, 51524}, {30, 23235}, {39, 6287}, {76, 35705}, {98, 140}, {99, 550}, {110, 15000}, {114, 1656}, {115, 3851}, {237, 41724}, {381, 36523}, {382, 543}, {546, 671}, {620, 15720}, {631, 11177}, {1352, 10007}, {1503, 35002}, {1657, 2794}, {2080, 3564}, {2784, 5882}, {3090, 38627}, {3091, 12243}, {3398, 8550}, {3448, 9155}, {3522, 9862}, {3523, 5984}, {3526, 6055}, {3529, 7946}, {3530, 41134}, {3628, 23234}, {3746, 12350}, {3818, 32447}, {3843, 38734}, {3855, 41135}, {3858, 14639}, {4027, 7892}, {4857, 12185}, {5026, 7795}, {5054, 38751}, {5055, 20398}, {5056, 14651}, {5059, 13172}, {5070, 38740}, {5073, 23698}, {5076, 12355}, {5079, 5461}, {5095, 22143}, {5182, 7819}, {5191, 14683}, {5270, 12184}, {5475, 48663}, {5477, 30435}, {5563, 12351}, {5613, 47519}, {5617, 47517}, {5655, 31854}, {5939, 7763}, {5965, 9301}, {5969, 7758}, {5986, 7495}, {5989, 32821}, {6034, 7772}, {6036, 46219}, {6390, 45018}, {6655, 11152}, {6756, 20774}, {6776, 26316}, {7697, 9744}, {7709, 9996}, {7755, 44534}, {7757, 40250}, {7813, 29012}, {7836, 8289}, {7838, 13111}, {7906, 14931}, {7910, 11257}, {8359, 11161}, {8369, 8593}, {8596, 50688}, {8721, 9821}, {8960, 35824}, {9772, 49111}, {9830, 34511}, {9890, 14023}, {10222, 50881}, {11006, 51522}, {11180, 40925}, {12117, 15704}, {12251, 40278}, {13464, 21636}, {14061, 35018}, {15300, 15681}, {15688, 36521}, {15712, 34473}, {16001, 41043}, {16002, 41042}, {16534, 18332}, {18800, 33237}, {20094, 49135}, {20975, 32255}, {21163, 43150}, {21166, 33923}, {23115, 39849}, {31274, 35021}, {32515, 43460}, {32825, 46236}, {37455, 45109}, {37914, 41586}, {38733, 39838}, {41672, 43136}, {46226, 49112}

X(52090) = midpoint of X(6033) and X(14692)
X(52090) = reflection of X(i) in X(j) for these {i,j}: {3, 14981}, {20, 51524}, {98, 51872}, {148, 22505}, {1657, 10992}, {5984, 12042}, {6033, 147}, {6321, 6033}, {8724, 48657}, {9862, 33813}, {11632, 6054}, {12188, 114}, {12243, 22566}, {14830, 8724}, {15681, 15300}, {38664, 5}, {38730, 13188}, {38733, 39838}, {38741, 99}, {47618, 7813}
X(52090) = anticomplement of X(51523)
X(52090) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 14981, 8724}, {3, 48657, 14981}, {5, 38664, 11632}, {98, 15561, 38739}, {98, 51872, 15561}, {99, 38741, 38731}, {114, 11623, 1656}, {114, 12188, 38224}, {147, 14692, 6321}, {1656, 11623, 38224}, {1656, 12188, 11623}, {1657, 10992, 38730}, {1657, 13188, 10992}, {6054, 38664, 5}, {6055, 20399, 3526}, {9862, 33813, 38742}

X(52091) = X(3)X(2065)∩X(69)X(523)

X(52091) lies on the cubic K257 and these lines: {3, 2065}, {6, 2987}, {69, 523}, {183, 36897}, {232, 2421}, {250, 19118}, {262, 5976}, {264, 670}, {511, 34157}, {842, 10425}, {877, 6530}, {1350, 34130}, {1351, 14253}, {3425, 42065}, {3563, 35575}, {5968, 51383}, {14356, 39374}, {18873, 36213}, {41517, 44453}, {44114, 51386}, {46888, 51543}, {51440, 51862}

X(52091) = isogonal conjugate of X(51820)
X(52091) = isotomic conjugate of X(14265)
X(52091) = isotomic conjugate of the anticomplement of X(52006)
X(52091) = isotomic conjugate of the isogonal conjugate of X(34157)
X(52091) = X(35142)-Ceva conjugate of X(325)
X(52091) = X(i)-cross conjugate of X(j) for these (i,j): {511, 2987}, {41167, 2421}, {47049, 35910}, {47079, 46787}, {52006, 2}
X(52091) = X(i)-isoconjugate of X(j) for these (i,j): {1, 51820}, {31, 14265}, {98, 8772}, {230, 1910}, {293, 460}, {336, 44099}, {1692, 1821}, {1733, 1976}, {17462, 41932}, {36036, 42663}
X(52091) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 14265}, {3, 51820}, {132, 460}, {230, 11672}, {511, 51335}, {1692, 40601}, {1733, 39040}, {2679, 42663}, {2974, 36212}, {5976, 51481}, {8623, 12829}, {47648, 47734}
X(52091) = cevapoint of X(511) and X(36790)
X(52091) = crosssum of X(230) and X(12829)
X(52091) = trilinear pole of line {3569, 36212}
X(52091) = barycentric product X(i)*X(j) for these {i,j}: {76, 34157}, {297, 43705}, {325, 2987}, {511, 8781}, {1959, 8773}, {2065, 32458}, {2396, 35364}, {2799, 10425}, {3563, 6393}, {6333, 32697}, {7799, 39374}, {35142, 36212}, {35910, 36891}, {36051, 46238}, {36790, 40428}, {42065, 44132}
X(52091) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14265}, {6, 51820}, {232, 460}, {237, 1692}, {297, 44145}, {325, 51481}, {511, 230}, {1755, 8772}, {1959, 1733}, {2065, 41932}, {2211, 44099}, {2421, 4226}, {2491, 42663}, {2987, 98}, {3563, 6531}, {8773, 1821}, {8781, 290}, {9155, 5477}, {10425, 2966}, {11672, 51335}, {23996, 17462}, {32654, 1976}, {32697, 685}, {34157, 6}, {35142, 16081}, {35364, 2395}, {35910, 36875}, {36051, 1910}, {36212, 3564}, {36213, 12829}, {36790, 114}, {39374, 1989}, {40428, 34536}, {40810, 47734}, {40812, 47741}, {42065, 248}, {43705, 287}, {46787, 34174}, {51229, 46039}, {51455, 48452}

X(52092) = X(6)X(41)∩X(40)X(78)

X(52092) lies on the cubic K1122 and these lines: {1, 33849}, {3, 47041}, {6, 41}, {40, 78}, {169, 8583}, {197, 1457}, {282, 18598}, {976, 3057}, {993, 21361}, {997, 1763}, {1042, 37257}, {1046, 15803}, {1106, 23154}, {1201, 8192}, {1610, 37694}, {1726, 10176}, {2187, 22350}, {3556, 22072}, {5440, 30269}, {17451, 46344}

X(52093) = X(2)X(13474)∩X(20)X(185)

X(52093) lies on the cubic K837 and these lines: {2, 13474}, {3, 6030}, {4, 5892}, {20, 185}, {24, 35237}, {30, 3567}, {51, 49135}, {52, 11001}, {64, 15577}, {140, 11455}, {143, 49137}, {154, 36983}, {376, 1216}, {381, 12046}, {382, 5640}, {389, 5059}, {548, 11459}, {550, 2979}, {578, 37944}, {631, 11439}, {1593, 12017}, {1598, 43584}, {1657, 5890}, {2930, 15054}, {3060, 3529}, {3090, 46852}, {3146, 10110}, {3357, 6636}, {3520, 15080}, {3522, 6000}, {3523, 11381}, {3525, 16194}, {3526, 11017}, {3528, 7998}, {3530, 33879}, {3534, 11412}, {3543, 9729}, {3627, 15045}, {3830, 15024}, {3832, 16836}, {3845, 11465}, {3853, 40280}, {5012, 12085}, {5054, 32137}, {5056, 17704}, {5067, 46849}, {5073, 9781}, {5446, 49138}, {5462, 15682}, {5562, 50693}, {5663, 15696}, {5878, 16063}, {5891, 21735}, {5907, 10304}, {5925, 41715}, {5943, 50688}, {5946, 49136}, {6102, 15681}, {6815, 51537}, {7387, 43601}, {7486, 46847}, {7502, 11468}, {7512, 8717}, {7516, 43613}, {7530, 43597}, {7556, 43604}, {7691, 35243}, {7999, 8703}, {9706, 37497}, {9730, 33703}, {10095, 15684}, {10299, 44299}, {10303, 44870}, {10323, 11440}, {10605, 33524}, {10627, 15689}, {10733, 16270}, {10984, 12086}, {11413, 19357}, {11438, 12087}, {11585, 18504}, {11591, 15688}, {11695, 50689}, {11793, 21734}, {12103, 34783}, {12161, 43576}, {12239, 42414}, {12240, 42413}, {12270, 16111}, {12273, 17854}, {12278, 44458}, {12281, 38788}, {12315, 15066}, {13336, 13596}, {13340, 45957}, {13434, 51739}, {13598, 49140}, {13630, 17800}, {13754, 17538}, {14449, 44903}, {14677, 15102}, {15030, 15717}, {15051, 17821}, {15055, 15738}, {15683, 45186}, {15686, 37484}, {15740, 41257}, {16618, 43607}, {18435, 33923}, {19347, 21312}, {21663, 38435}, {23039, 44245}, {32046, 35452}, {33534, 43603}, {33878, 35253}, {34005, 44882}, {34484, 37470}, {34513, 35498}, {35240, 48892}, {35268, 38448}, {37201, 50435}, {37480, 43605}, {38942, 44110}, {44866, 45014}

X(52093) = reflection of X(i) in X(j) for these {i,j}: {11439, 631}, {11444, 3522}, {15058, 3}
X(52093) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 8718, 26881}, {3, 12279, 15305}, {3, 12290, 15056}, {4, 20791, 15028}, {20, 15072, 5889}, {20, 46850, 15072}, {376, 10575, 12111}, {550, 6241, 2979}, {3528, 12162, 7998}, {3529, 40647, 3060}, {3534, 13491, 11412}, {8703, 18439, 7999}, {12279, 15056, 12290}, {12290, 15056, 15305}, {14641, 14855, 4}, {17704, 32062, 5056}

X(52094) = X(2)X(1637)∩X(30)X(99)

X(52094) lies on the cubic K953 and these lines: {2, 1637}, {23, 9141}, {30, 99}, {98, 1494}, {339, 49102}, {620, 48453}, {671, 34897}, {690, 36890}, {1272, 46236}, {2373, 36884}, {2482, 4235}, {5468, 5642}, {5649, 41134}, {6035, 9170}, {6148, 32458}, {7426, 51263}, {7763, 38939}, {23350, 45319}, {37668, 45772}, {40112, 50639}

X(52094) = reflection of X(4235) in X(2482)
X(52094) = isotomic conjugate of X(16092)
X(52094) = antitomic image of X(4235)
X(52094) = isotomic conjugate of the anticomplement of X(46986)
X(52094) = X(i)-cross conjugate of X(j) for these (i,j): {32313, 648}, {45662, 524}, {46986, 2}
X(52094) = X(i)-isoconjugate of X(j) for these (i,j): {31, 16092}, {111, 2247}, {542, 923}, {798, 50941}, {897, 5191}, {1640, 36142}, {1973, 51405}, {6041, 36085}, {6103, 36060}
X(52094) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 16092}, {524, 45662}, {542, 2482}, {1560, 6103}, {1640, 23992}, {1649, 51428}, {5191, 6593}, {6041, 38988}, {6337, 51405}, {31998, 50941}
X(52094) = cevapoint of X(524) and X(45662)
X(52094) = trilinear pole of line {524, 18311}
X(52094) = crossdifference of every pair of points on line {5191, 6041}
X(52094) = barycentric product X(i)*X(j) for these {i,j}: {99, 50942}, {524, 5641}, {690, 6035}, {842, 3266}, {5468, 14223}, {5649, 35522}, {36890, 51228}
X(52094) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 16092}, {69, 51405}, {99, 50941}, {187, 5191}, {351, 6041}, {468, 6103}, {524, 542}, {690, 1640}, {842, 111}, {896, 2247}, {1648, 51428}, {2482, 45662}, {4235, 7473}, {5468, 14999}, {5641, 671}, {5649, 691}, {5967, 34369}, {6035, 892}, {9717, 48451}, {14223, 5466}, {14559, 23968}, {14998, 9178}, {23350, 8430}, {35522, 18312}, {35909, 10097}, {36890, 51227}, {43084, 43087}, {45662, 23967}, {46157, 46154}, {46787, 5968}, {50942, 523}, {51228, 9214}
X(52094) = {X(2),X(46787)}-harmonic conjugate of X(51228)

X(52095) = X(2)X(3558)∩X(4)X(69)

X(52095) lies on the cubic K852 and these lines: {2, 3558}, {3, 6190}, {4, 69}, {20, 3413}, {193, 2543}, {194, 13325}, {1341, 3523}, {1380, 6178}, {2040, 7912}, {2542, 10519}, {3414, 20081}, {3926, 51898}, {6040, 7788}, {6189, 7776}, {6194, 13326}, {7767, 19659}, {7796, 14501}, {7917, 47365}, {7946, 39366}, {9821, 47367}, {20080, 35914}, {38596, 51876}

X(52095) = reflection of X(194) in X(13325)
X(52095) = anticomplement of X(3558)
X(52095) = anticomplement of the isogonal conjugate of X(6178)
X(52095) = isotomic conjugate of the isogonal conjugate of X(21032)
X(52095) = X(6178)-anticomplementary conjugate of X(8)
X(52095) = barycentric product X(76)*X(21032)
X(52095) = barycentric quotient X(21032)/X(6)

X(52096) = X(2)X(3557)∩X(4)X(69)

X(52096) lies on the cubic K852 and these lines: {2, 3557}, {3, 6189}, {4, 69}, {20, 3414}, {193, 2542}, {194, 13326}, {1340, 3523}, {1379, 6177}, {2039, 7912}, {2543, 10519}, {3413, 20081}, {3926, 51899}, {6039, 7788}, {6190, 7776}, {6194, 13325}, {7767, 19660}, {7796, 14502}, {7917, 47366}, {7946, 39365}, {9821, 47368}, {20080, 35913}, {38597, 51878}

X(52096) = reflection of X(194) in X(13326)
X(52096) = anticomplement of X(3557)
X(52096) = anticomplement of the isogonal conjugate of X(6177)
X(52096) = isotomic conjugate of the isogonal conjugate of X(21036)
X(52096) = X(6177)-anticomplementary conjugate of X(8)
X(52096) = barycentric product X(76)*X(21036)
X(52096) = barycentric quotient X(21036)/X(6)

X(52097) = X(3)X(77)∩X(4)X(8)

X(52097) lies on the cubic K401 and these lines: {1, 9786}, {2, 5908}, {3, 77}, {4, 8}, {20, 20211}, {40, 221}, {46, 46009}, {65, 40944}, {84, 34371}, {185, 1071}, {198, 15836}, {219, 2270}, {389, 5728}, {946, 2262}, {1427, 1715}, {1498, 1763}, {1753, 34048}, {2817, 5930}, {2823, 18239}, {3100, 19904}, {3182, 6282}, {5709, 11347}, {5889, 16465}, {6260, 40953}, {6848, 51413}, {7066, 7957}, {7358, 21075}, {10310, 34049}, {10360, 37613}, {10361, 41600}, {11249, 37310}, {12664, 44661}, {13737, 22770}, {17102, 40945}, {18641, 22076}, {18909, 41004}, {26884, 37623}, {30268, 41810}, {37046, 37531}

X(52097) = reflection of X(i) in X(j) for these {i,j}: {4, 5909}, {40953, 6260}, {51490, 3}
X(52097) = anticomplement of X(5908)
X(52097) = isotomic conjugate of the polar conjugate of X(40943)
X(52097) = X(i)-isoconjugate of X(j) for these (i,j): {84, 40396}, {947, 40836}, {7151, 40417}
X(52097) = X(i)-Dao conjugate of X(j) for these (i,j): {4, 946}, {189, 20262}, {7020, 17102}
X(52097) = crosspoint of X(329) and X(7013)
X(52097) = crosssum of X(i) and X(j) for these (i,j): {25, 7129}, {1436, 7008}
X(52097) = barycentric product X(i)*X(j) for these {i,j}: {69, 40943}, {322, 22063}, {329, 17102}, {7011, 23528}, {7013, 20262}, {40702, 40945}
X(52097) = barycentric quotient X(i)/X(j) for these {i,j}: {198, 40396}, {2262, 40836}, {17102, 189}, {20262, 7020}, {22063, 84}, {40943, 4}, {40945, 282}, {40957, 7008}
X(52097) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 223, 37413}, {185, 17441, 1071}, {10373, 14557, 4}

X(52098) = X(3)X(34437)∩X(4)X(542)

X(52098) lies on the cubic K914 and these lines: {3, 34437}, {4, 542}, {6, 12308}, {67, 5655}, {74, 17508}, {110, 3098}, {146, 29012}, {182, 4550}, {381, 25335}, {399, 511}, {575, 12162}, {1205, 11456}, {1974, 7722}, {2777, 36989}, {2781, 5609}, {2854, 37517}, {3448, 15019}, {5039, 14901}, {5092, 10620}, {5476, 25328}, {5621, 20190}, {5643, 25565}, {6000, 15141}, {6053, 32257}, {7723, 19126}, {7728, 48884}, {9140, 25566}, {9143, 19924}, {9976, 15520}, {10264, 38317}, {10510, 11645}, {10628, 19149}, {11422, 51882}, {11438, 38851}, {11477, 12316}, {11579, 25556}, {12121, 48879}, {12244, 48892}, {12317, 14561}, {12383, 29317}, {12902, 48895}, {13392, 21167}, {13858, 47068}, {13859, 47066}, {14683, 31670}, {14708, 19137}, {14787, 16003}, {14810, 32609}, {14982, 32272}, {14984, 15083}, {15054, 15462}, {15060, 32154}, {16534, 49116}, {17702, 48904}, {18440, 25336}, {18553, 32306}, {19131, 22584}, {19151, 19402}, {22234, 34155}, {22330, 39562}, {24206, 44834}, {25331, 39899}, {29323, 38790}, {32247, 40107}, {32423, 48901}, {34153, 48880}, {35422, 38653}, {38789, 48889}

X(52098) = midpoint of X(i) and X(j) for these {i,j}: {6, 12308}, {399, 51941}, {2930, 48679}, {9970, 14094}, {11477, 32254}, {14683, 31670}, {18440, 25336}
X(52098) = reflection of X(i) in X(j) for these {i,j}: {182, 19140}, {576, 9970}, {3098, 110}, {3448, 19130}, {9140, 25566}, {10620, 5092}, {11178, 5655}, {11579, 25556}, {12244, 48892}, {12584, 5609}, {12902, 48895}, {16010, 575}, {32247, 40107}, {32273, 32271}, {32305, 6593}, {32306, 18553}, {48879, 12121}, {48880, 34153}, {48884, 7728}, {49116, 16534}
X(52098) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {399, 48679, 2930}, {2930, 51941, 48679}, {6593, 32305, 182}, {11579, 25556, 39561}, {16010, 45016, 575}, {19140, 32305, 6593}

X(52099) = X(3)X(5888)∩X(6)X(3534)

X(52099) lies on the Feuerbach circumhyperbola of the tangential triangle and these lines: {3, 5888}, {6, 3534}, {20, 15047}, {30, 7693}, {74, 5898}, {155, 15696}, {159, 35450}, {195, 550}, {323, 15690}, {376, 399}, {548, 12112}, {1657, 15805}, {2929, 13564}, {2930, 3098}, {2935, 11202}, {3216, 16117}, {5054, 33534}, {5092, 35001}, {5663, 33544}, {5899, 41448}, {6451, 8939}, {6452, 8943}, {7492, 20421}, {7712, 15040}, {8547, 33878}, {8717, 32609}, {12103, 15037}, {12307, 43806}, {14093, 46945}, {15018, 19710}, {15038, 15686}, {15041, 41398}, {15066, 15695}, {15688, 35237}, {43829, 45045}

X(52099) = reflection of X(i) in X(j) for these {i,j}: {14926, 3}, {18551, 5888}
X(52099) = X(8703)-Ceva conjugate of X(3)
X(52099) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 18551, 5888}, {5888, 18551, 14926}

X(52100) = X(6)X(382)∩X(20)X(399)

X(52100) lies on on Feuerbach circumhyperbola of the tangential triangle and these lines: {1, 26200}, {3, 6030}, {4, 15047}, {6, 382}, {20, 399}, {23, 43807}, {24, 43846}, {25, 22948}, {30, 195}, {49, 35001}, {143, 43612}, {155, 1657}, {159, 12315}, {185, 37924}, {381, 15805}, {548, 12112}, {1181, 34563}, {1498, 3534}, {1533, 43821}, {1614, 35452}, {1656, 18488}, {1658, 11270}, {1993, 49139}, {2070, 2929}, {2777, 48669}, {2916, 12162}, {2917, 6000}, {2918, 18439}, {2930, 18436}, {2931, 2937}, {2935, 6759}, {3357, 44753}, {3529, 44755}, {3627, 15038}, {3853, 15037}, {5072, 15811}, {5073, 15087}, {5663, 5898}, {5878, 15141}, {5899, 13491}, {6101, 12308}, {6102, 37949}, {6241, 15086}, {6449, 8939}, {6450, 8943}, {7517, 45045}, {7592, 15684}, {9937, 12083}, {10540, 14641}, {11381, 34864}, {11455, 15084}, {11456, 17800}, {12038, 18859}, {12087, 45957}, {12161, 49134}, {12174, 44457}, {12291, 13391}, {13339, 46849}, {13474, 18374}, {13621, 43597}, {14130, 14915}, {14926, 45958}, {15052, 46853}, {15072, 18378}, {15681, 32139}, {15685, 16266}, {15696, 35237}, {15704, 50461}, {16936, 18451}, {17508, 33539}, {17814, 46945}, {18445, 49137}, {23039, 33542}, {32062, 37471}, {32063, 46373}, {36747, 49133}, {36752, 38335}, {37496, 43605}, {37958, 43604}, {40647, 44106}, {43689, 43719}, {43809, 46850}, {44082, 45735}

X(52100) = midpoint of X(6241) and X(15086)
X(52100) = reflection of X(i) in X(j) for these {i,j}: {3, 8718}, {382, 3521}, {3521, 44866}, {12307, 47748}, {33541, 3}
X(52100) = tangential-isogonal conjugate of X(45735)
X(52100) = X(550)-Ceva conjugate of X(3)
X(52100) = {X(8718),X(16835)}-harmonic conjugate of X(6030)

X(52101) = X(4)X(541)∩X(5)X(13474)

X(52101) lies on these lines: {4, 541}, {5, 13474}, {20, 4550}, {30, 18553}, {113, 31857}, {382, 6288}, {631, 8717}, {690, 18039}, {858, 16194}, {3426, 3521}, {3526, 35237}, {3830, 41586}, {3832, 4846}, {4549, 33703}, {5070, 11820}, {5169, 11455}, {5448, 32137}, {7579, 51403}, {7747, 45723}, {10193, 22249}, {11645, 16511}, {11799, 32062}, {13399, 14269}, {15696, 32620}, {18431, 23325}, {20299, 20396}, {20397, 44275}, {22802, 32393}, {40280, 44300}, {44407, 49669}

X(52101) = midpoint of X(3426) and X(7706)

X(52102) = X(2)X(14862)∩X(3)X(45185)

X(52102) lies on these lines: {2, 14862}, {3, 45185}, {4, 13399}, {5, 2883}, {20, 2888}, {24, 20417}, {30, 14864}, {64, 265}, {125, 12290}, {185, 15559}, {221, 31480}, {389, 1907}, {541, 18377}, {542, 9925}, {548, 1503}, {549, 44762}, {631, 5651}, {1204, 13419}, {1495, 43607}, {1498, 3526}, {1595, 13382}, {1597, 40240}, {1853, 3843}, {1906, 13474}, {3528, 9833}, {3530, 6696}, {3627, 15105}, {3832, 5878}, {3853, 15311}, {3855, 6225}, {3856, 5893}, {4550, 44862}, {5070, 12315}, {5656, 7486}, {5895, 18376}, {5925, 49134}, {6241, 18388}, {6285, 37720}, {7355, 37719}, {7464, 43895}, {7533, 43603}, {7689, 29012}, {9657, 10076}, {9670, 10060}, {9698, 32445}, {9934, 15057}, {9936, 13346}, {9968, 25555}, {10412, 20184}, {10575, 21243}, {10606, 15696}, {10619, 35475}, {10990, 34797}, {11202, 15717}, {11271, 12317}, {11438, 37122}, {11457, 13403}, {12112, 43608}, {12162, 37452}, {12250, 17578}, {12270, 15063}, {12964, 35812}, {12970, 35813}, {13202, 18394}, {13293, 23236}, {13630, 19130}, {14499, 32617}, {14500, 32616}, {15038, 17823}, {16239, 16252}, {16659, 21663}, {17800, 35450}, {18378, 20126}, {20427, 32064}, {23294, 51403}, {23328, 44682}, {26879, 32062}, {26883, 44673}, {31447, 40660}, {32138, 44407}, {32359, 32401}, {34514, 43577}, {34782, 46853}, {36253, 44271}

X(52102) = midpoint of X(i) and X(j) for these {i,j}: {64, 18381}, {3357, 14216}, {3627, 15105}, {6759, 12324}, {13093, 22802}, {20427, 34786}, {34780, 34785}
X(52102) = reflection of X(i) in X(j) for these {i,j}: {2883, 32767}, {6759, 25563}, {9968, 25555}, {10282, 6696}, {20299, 6247}, {44762, 50414}, {45185, 3}
X(52102) = anticomplement of X(14862)
X(52102) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {549, 44762, 50414}, {1853, 13093, 22802}, {6696, 10282, 10193}, {6759, 25563, 10182}, {10606, 34780, 34785}, {20427, 32064, 34786}

X(52103) = X(4)X(7883)∩X(5)X(1533)

X(52103) lies on these lines: {4, 7883}, {5, 1533}, {20, 5888}, {382, 14926}, {546, 16003}, {1512, 9856}, {1514, 3856}, {1531, 3861}, {3832, 5878}, {3843, 14852}, {3845, 41586}, {3855, 13603}, {13399, 23046}, {14915, 44300}, {15063, 46847}, {15559, 22970}, {18488, 20396}

X(52104) = X(3)X(539)∩X(20)X(68)

X(52104) lies on these lines: {2, 15083}, {3, 539}, {5, 389}, {20, 68}, {23, 43895}, {30, 14864}, {52, 15559}, {125, 18436}, {140, 41597}, {155, 3526}, {323, 43608}, {343, 40647}, {382, 9927}, {542, 1658}, {548, 44665}, {631, 1147}, {912, 4662}, {1092, 6699}, {1154, 20299}, {1209, 5890}, {1657, 13399}, {1906, 41587}, {1907, 5446}, {2072, 45187}, {3060, 18488}, {3520, 41724}, {3528, 12118}, {3530, 3564}, {3548, 20397}, {3580, 12162}, {3843, 14852}, {4301, 12259}, {4309, 10071}, {4317, 10055}, {4550, 39571}, {5067, 5654}, {5070, 12164}, {5181, 33556}, {5504, 15057}, {5562, 37452}, {5576, 14831}, {5889, 7703}, {5891, 26879}, {5965, 25563}, {6101, 10264}, {6193, 15717}, {6238, 37720}, {6241, 44866}, {6689, 7592}, {7352, 37719}, {7404, 11431}, {7503, 43573}, {7505, 16534}, {9588, 9928}, {9820, 16239}, {9925, 50977}, {9938, 22978}, {10112, 18570}, {10193, 50708}, {10564, 43607}, {10665, 35812}, {10666, 35813}, {11271, 34148}, {11430, 32358}, {11442, 45286}, {11459, 43817}, {12006, 24206}, {12107, 45185}, {12111, 44958}, {12293, 17800}, {12307, 15085}, {12429, 15696}, {12893, 23236}, {14516, 32110}, {14862, 44278}, {15063, 46085}, {15069, 19908}, {15873, 46852}, {16266, 23329}, {16511, 33749}, {16868, 45014}, {16881, 19130}, {17710, 34382}, {17712, 46728}, {17845, 44322}, {18356, 18400}, {18404, 36253}, {18445, 44516}, {18914, 44201}, {18917, 47525}, {20396, 33547}, {21844, 30714}, {22466, 34801}, {23294, 51392}, {31831, 43586}, {32140, 44407}, {33533, 44862}, {43599, 50009}, {43689, 45736}, {44213, 50414}

X(52104) = midpoint of X(i) and X(j) for these {i,j}: {68, 7689}, {1147, 11411}, {9927, 12163}, {32140, 46730}
X(52104) = reflection of X(i) in X(j) for these {i,j}: {155, 43839}, {1147, 20191}, {5448, 5449}, {5449, 12359}, {12038, 44158}, {41597, 140}, {45184, 1147}, {45185, 12107}
X(52104) = complement of X(15083)
X(52104) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {631, 9936, 1147}, {631, 11411, 9936}, {11411, 20191, 45184}

X(52105) = X(2)X(34213)∩X(3)X(11178)

X(52105) lies on the cubics K038 and K1295 and these lines: {2, 34213}, {3, 11178}, {30, 46657}, {187, 17416}, {352, 14901}, {3906, 4141}

X(52106) = X(39)X(48653)∩X(110)X(187)

X(52106) lies on the cubic K1295 and these lines: {39, 48653}, {110, 187}, {511, 8724}, {512, 1649}, {2854, 23992}, {10717, 10989}

X(52106) = crossdifference of every pair of points on line {8371, 11580}

X(52107) = X(2)X(34213)∩X(110)X(599)

X(521107) lies on the cubic K1295 and these lines: {2, 34213}, {110, 599}, {574, 6032}, {3849, 7492}, {6031, 7850}, {31762, 35499}

X(52107) = {X(6325),X(47596)}-harmonic conjugate of X(9829)

X(52108) = X(109)X(40558)∩X(124)X(522)

X(52108) lies on the curve Q000H3 (see Q000, Steiner deltoids)) and these lines: {109, 40558}, {124, 522}, {1309, 33650}, {2818, 39535}, {3326, 24026}, {4086, 7068}, {5057, 10538}, {10747, 51565}, {23104, 42455}

X(52108) = midpoint of X(1309) and X(33650)
X(52108) = reflection of X(i) in X(j) for these {i,j}: {109, 40558}, {2968, 15633}, {10017, 124}
X(52108) = X(2734)-isoconjugate of X(24027)
X(52108) = X(522)-Dao conjugate of X(2734)
X(52108) = crossdifference of every pair of points on line {2425, 23979}
X(52108) = barycentric product X(2818)*X(23978)
X(52108) = barycentric quotient X(i)/X(j) for these {i,j}: {1146, 2734}, {2818, 1262}

X(52109) = X(102)X(40558)∩X(117)X(515)

X(52109) lies on the curve Q000H3 ((see Q000, Steiner deltoids))) and these lines: {102, 40558}, {117, 515}, {151, 1309}, {2818, 39535}, {5176, 6073}

X(52109) = midpoint of X(151) and X(1309)
X(52109) = reflection of X(i) in X(j) for these {i,j}: {102, 40558}, {10017, 117}
X(52109) = X(515)-Dao conjugate of X(2734)
X(52109) = barycentric quotient X(23986)/X(2734)

X(52110) = X(137)X(1510)∩X(143)X(27357)

X(52110) lies on these lines: {137, 1510}, {143, 27357}, {14101, 44053}, {15907, 47065}, {25150, 33333}

X(52110) = reflection of X(35591) in X(137)
X(52110) = X(1510)-Dao conjugate of X(15907)
X(52110) = barycentric quotient X(39018)/X(15907)

X(52111) = X(10)X(11)∩X(1364)X(5687)

X(52111) lies on the Mandard circle and these lines: {10, 11}, {1364, 5687}, {2810, 3359}, {2841, 38501}, {3271, 25438}, {34139, 38695}

X(52112) = X(8)X(2829)∩X(11)X(123)

X(52112) lies on these lines: {2, 10271}, {8, 2829}, {11, 123}, {108, 1376}, {280, 1295}, {318, 25640}, {355, 50917}, {522, 7358}, {1359, 10944}, {2123, 40836}, {2817, 10914}, {2823, 17658}, {3434, 34188}, {5687, 49207}, {10525, 10746}, {10702, 10912}, {10715, 11235}, {10776, 13271}, {11719, 17614}, {11798, 34122}, {17649, 39130}, {18516, 33566}, {18519, 38592}

X(52112) = reflection of X(3318) in X(123)
X(52112) = anticomplement of X(10271)
X(52112) = X(189)-Ceva conjugate of X(3239)
X(52112) = X(i)-isoconjugate of X(j) for these (i,j): {109, 30239}, {1262, 8602}, {10309, 24027}
X(52112) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 30239}, {522, 10309}, {10310, 30201}
X(52112) = crosspoint of X(280) and X(4397)
X(52112) = barycentric product X(i)*X(j) for these {i,j}: {2057, 4858}, {4391, 30201}, {10310, 23978}
X(52112) = barycentric quotient X(i)/X(j) for these {i,j}: {650, 30239}, {1146, 10309}, {2057, 4564}, {2310, 8602}, {10310, 1262}, {30201, 651}

X(52113) = X(11)X(521)∩X(1259)X(6099)

X(52113) lies on the Mandart circle and these lines: {11, 521}, {1259, 6099}, {2720, 2750}, {2745, 6081}, {2968, 35604}, {6001, 24466}

X(52114) = X(8)X(1309)∩X(521)X(1364)

X(52114) lies on the Mandart circle and these lines: {8, 1309}, {521, 1364}, {522, 7358}, {2739, 2765}, {3239, 5514}, {4086, 7068}

X(52114) = X(8)-Ceva conjugate of X(14312)
X(52114) = X(i)-isoconjugate of X(j) for these (i,j): {2720, 36044}, {15405, 24033}, {32647, 37136}
X(52114) = X(i)-Dao conjugate of X(j) for these (i,j): {521, 15405}, {2720, 35580}, {14571, 23984}, {36044, 38981}
X(52114) = barycentric product X(23983)*X(25640)
X(52114) = barycentric quotient X(i)/X(j) for these {i,j}: {25640, 23984}, {35072, 15405}, {46393, 36044}

X(52115) = X(101)X(222)∩X(116)X(5514)

X(52115) lies on Privalov conic (see X(5452), the Mandart circle, and these lines: {101, 222}, {116, 5514}, {150, 26871}, {928, 1364}, {1071, 2809}, {1362, 30493}, {2810, 3359}, {3937, 24031}, {4091, 7117}, {5185, 26892}, {7215, 34591}

X(52115) = X(8)-Ceva conjugate of X(905)

X(52116) = X(8)X(2829)∩X(9)X(119)

X(52116) lies on the Mandart hyperbola, the Mandart circle, and these lines: {8, 2829}, {9, 119}, {11, 1158}, {104, 14646}, {210, 34293}, {1071, 1317}, {1387, 52027}, {1537, 44675}, {1768, 51785}, {4189, 22775}, {5289, 38759}, {6001, 24466}, {6700, 46684}, {6705, 38038}, {12608, 31235}, {12700, 24467}, {12705, 24465}, {14740, 18239}, {17661, 41559}, {21031, 40256}

X(52116) = reflection of X(i) in X(j) for these {i,j}: {11, 1158}, {18239, 14740}
X(52116) = X(8)-Ceva conjugate of X(44675)

X(52117) = X(109)X(6611)∩X(124)X(5514)

X(52117) lies on the Mandart circle and these lines: {109, 6611}, {124, 5514}, {208, 1361}, {1364, 3270}, {2800, 40953}, {2841, 38501}

X(52117) = X(8)-Ceva conjugate of X(6129)
X(52117) = barycentric product X(i)*X(j) for these {i,j}: {6129, 20296}, {16596, 34040}

X(52118) = COMPLEMENT OF X(43346)

See Antreas Hatzipolakis and César Lozada, euclid 5523.

X(52118) lies on the nine-point circle and these lines: {2, 43346}, {116, 26956}, {3270, 46660}, {10017, 26933}, {38357, 38966}

X(52118) = complement of X(43346)
X(52118) = complementary conjugate of the isogonal conjugate of X(43346)
X(52118) = center of the circumconic {{A, B, C, X(4), X(77)}}
X(52118) = Poncelet point of X(i) for these i: {77, 4295}

X(52119) = COMPLEMENT OF X(43354)

See Antreas Hatzipolakis and César Lozada, euclid 5523.

X(52119) lies on the nine-point circle and these lines: {2, 43354}, {4, 51760}, {11, 34973}, {3120, 35583}, {16188, 37959}, {34981, 38967}

X(52119) = midpoint of X(4) and X(51760)
X(52119) = complement of X(43354)
X(52119) = complementary conjugate of the isogonal conjugate of X(43354)
X(52119) = center of the circumconic {{A, B, C, X(4), X(12)}}
X(52119) = Poncelet point of X(i) for these i: {12, 442, 1234, 1865, 8818, 21675, 41493, 41501, 41508, 45926}

X(52120) = COMPLEMENT OF X(46963)

See Antreas Hatzipolakis and César Lozada, euclid 5523.

X(52120) lies on the nine-point circle and these lines: {2, 46963}, {3, 15240}, {4, 1288}, {22, 114}, {113, 5562}, {131, 18404}, {133, 35488}, {136, 16186}, {23323, 25641}

X(52120) = midpoint of X(4) and X(51761)
X(52120) = complement of X(46963)
X(52120) = complementary conjugate of the isogonal conjugate of X(46963)
X(52120) = X(i)-complementary conjugate of-X(j) for these (i, j): (656, 6640), (661, 1993)
X(52120) = center of the circumconic {{A, B, C, X(4), X(26)}}
X(52120) = inverse of X(1288) in polar circle
X(52120) = orthoassociate of X(1288)
X(52120) = Poncelet point of X(i) for these i: {26, 1485, 8746, 34225, 44128, 51761}

X(52121) = COMPLEMENT OF X(3465)

X(52121) lies on the cubic K1297 and these lines: {2, 3465}, {3, 10}, {58, 2907}, {113, 11813}, {519, 34050}, {522, 4823}, {860, 7004}, {1054, 1737}, {1076, 42456}, {1125, 45272}, {1210, 1785}, {1457, 31680}, {1789, 15777}, {1861, 45766}, {3075, 5174}, {3152, 6734}, {3218, 3464}, {9843, 16870}, {10916, 37591}, {11019, 51616}, {11709, 34589}, {16336, 51569}, {16388, 21616}, {21370, 29673}, {21621, 29655}

X(52121) = reflection of X(i) in X(j) for these {i,j}: {10, 50368}, {45272, 1125}
X(52121) = complement of X(3465)
X(52121) = circumcircle-inverse of X(23850)
X(52121) = complement of the isogonal conjugate of X(3466)
X(52121) = X(i)-complementary conjugate of X(j) for these (i,j): {3466, 10}, {34299, 3814}, {38934, 51569}
X(52121) = crossdifference of every pair of points on line {2174, 6589}
X(52121) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 6245, 14058}, {34822, 51755, 10}

X(52122) = COMPLEMENT OF X(3484)

X(52121) lies on the cubic K1297 and these lines: {2, 3484}, {3, 161}, {113, 16336}, {128, 51393}, {5667, 14918}, {6000, 20625}, {6368, 18314}, {10615, 46966}, {15780, 33664}

X(52122) = reflection of X(46966) in X(10615)
X(52122) = complement of X(3484)
X(52122) = complement of the isogonal conjugate of X(8439)
X(52122) = X(8439)-complementary conjugate of X(10)
X(52122) = crossdifference of every pair of points on line {14533, 16040}

X(52123) = X(1)X(2916)∩X(2)X(16555)

X(52123) lies on the conic {{A,B,C,X(1),X(2)}}, the cubic K655, and these lines: {1, 2916}, {2, 16555}, {46, 39959}, {141, 17744}, {291, 3336}, {484, 9941}, {985, 3337}, {1219, 4293}, {1224, 3826}, {1280, 3874}, {2896, 39722}, {3338, 39958}, {4089, 39724}, {5525, 30701}, {7313, 16502}

X(52123) = isogonal conjugate of X(17744)
X(52123) = isotomic conjugate of X(46747)
X(52123) = X(i)-cross conjugate of X(j) for these (i,j): {251, 7194}, {5299, 1}, {5322, 269}
X(52123) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17744}, {6, 33091}, {31, 46747}, {55, 28780}
X(52123) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 46747}, {3, 17744}, {9, 33091}, {223, 28780}
X(52123) = barycentric product X(1)*X(39728)
X(52123) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 33091}, {2, 46747}, {6, 17744}, {57, 28780}, {39728, 75}

X(52124) = X(4)X(195)∩X(6)X(110)

X(52324) lies on the cubic K441 and these lines: {2, 39562}, {4, 195}, {6, 110}, {22, 14984}, {74, 9938}, {125, 9976}, {155, 18394}, {265, 15068}, {323, 858}, {394, 9140}, {542, 1993}, {1511, 17928}, {1994, 9143}, {2931, 11464}, {2979, 5621}, {3047, 12310}, {3620, 32241}, {5422, 5642}, {5609, 36749}, {5622, 7485}, {5655, 39522}, {6800, 44493}, {7426, 37784}, {7464, 10620}, {7495, 32251}, {7592, 30714}, {8549, 17847}, {9544, 45082}, {10117, 15107}, {10706, 44413}, {10733, 17838}, {11438, 21649}, {11456, 17702}, {11800, 34417}, {11806, 37470}, {12161, 23236}, {12273, 19457}, {12375, 35832}, {12376, 35833}, {12383, 15032}, {12827, 31236}, {13171, 33878}, {13417, 37517}, {14094, 36747}, {14805, 15089}, {15020, 37514}, {15034, 36752}, {15037, 32609}, {15054, 37498}, {15122, 19403}, {15141, 32255}, {15329, 22143}, {16010, 23061}, {19140, 24981}, {19458, 38444}, {30744, 34966}, {41614, 47596}, {46818, 47571}

X(52124) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 895, 45237}, {110, 45237, 1995}, {323, 3448, 15106}, {399, 19504, 51882}, {1994, 9143, 45016}, {11004, 14683, 51882}, {11004, 51882, 19504}, {14683, 51882, 399}

X(52125) = X(4)X(110)∩X(111)X(925)

X(52125) lies on the Parry circle, the buics K441 and K884, and these lines: {4, 110}, {23, 13558}, {94, 34310}, {111, 925}, {184, 39120}, {352, 45938}, {353, 43448}, {393, 35360}, {5169, 7711}, {5191, 13556}, {5961, 7556}, {7519, 9999}, {7552, 47065}, {9138, 9979}, {41203, 46106}

X(52125) = psi-transform of X(9730)

X(52126) = X(8)X(191)∩X(21)X(60)

X(52126) lies oon the cubic K1240 and these lines: {2, 14526}, {8, 191}, {10, 1749}, {21, 60}, {30, 5086}, {35, 3219}, {46, 2475}, {63, 2894}, {72, 22936}, {79, 3218}, {100, 22937}, {404, 41542}, {517, 3652}, {518, 45065}, {758, 4861}, {956, 13465}, {1155, 35982}, {2771, 2975}, {3337, 11263}, {3681, 11849}, {3869, 13743}, {3878, 46816}, {4652, 16143}, {5057, 16160}, {5303, 33598}, {5730, 28453}, {6067, 17768}, {6175, 41697}, {6597, 11604}, {6690, 13995}, {6701, 27003}, {7098, 18977}, {9965, 10527}, {10266, 15910}, {11011, 16140}, {11680, 16159}, {12514, 15677}, {12769, 23015}, {12786, 23016}, {14552, 36626}, {16116, 24467}, {18253, 21031}, {21677, 33559}, {21842, 39778}, {26877, 49107}, {26878, 33862}, {31159, 37005}, {37286, 45392}

X(52126) = reflection of X(i) in X(j) for these {i,j}: {35, 3647}, {79, 25639}
X(52126) = anticomplement of X(14526)
X(52126) = X(i)-Dao conjugate of X(j) for these (i,j): {79, 13089}, {226, 5949}
X(52126) = barycentric product X(i)*X(j) for these {i,j}: {314, 50657}, {3337, 42033}, {4420, 26842}, {32851, 47054}
X(52126) = barycentric quotient X(i)/X(j) for these {i,j}: {11263, 43682}, {47054, 2006}, {50657, 65}
X(52126) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {63, 7701, 3648}, {72, 22936, 48698}

X(52127) = X(6)X(292)∩X(32)X(101)

X(52127) lies on the cubic K771 and these lines: {1, 1929}, {6, 292}, {31, 40733}, {32, 101}, {48, 34249}, {55, 16515}, {100, 17475}, {386, 1015}, {1107, 5293}, {1403, 21780}, {1438, 16524}, {1575, 3507}, {1914, 2109}, {2162, 20760}, {2223, 16514}, {3207, 16969}, {3230, 21781}, {3511, 21792}, {3570, 30667}, {4262, 8649}, {5168, 10987}, {6184, 41276}, {9454, 51921}, {10329, 36559}, {15624, 16525}, {16518, 31477}, {16520, 18611}, {16523, 41265}, {16685, 21790}, {16687, 21779}, {16826, 20179}, {24289, 35338}

X(52127) = isogonal conjugate of the isotomic conjugate of X(33888)
X(52127) = polar conjugate of the isotomic conjugate of X(20797)
X(52127) = X(i)-Ceva conjugate of X(j) for these (i,j): {1914, 6}, {2112, 20672}, {3009, 2176}, {18266, 18755}, {33888, 20797}
X(52127) = X(i)-isoconjugate of X(j) for these (i,j): {75, 2109}, {3766, 39420}
X(52127) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 2109}, {335, 18895}
X(52127) = crosssum of X(514) and X(35119)
X(52127) = crossdifference of every pair of points on line {812, 3837}
X(52127) = barycentric product X(i)*X(j) for these {i,j}: {1, 2108}, {4, 20797}, {6, 33888}, {101, 25381}, {292, 27920}, {21760, 33679}
X(52127) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 2109}, {2108, 75}, {20797, 69}, {25381, 3261}, {27920, 1921}, {33888, 76}
X(52127) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {101, 38865, 51328}, {2223, 16514, 17735}, {38865, 51328, 21793}

X(52128) = X(2)X(98)∩X(3)X(1625)

X(52128) lies on the cubic K783 and these lines: {2, 98}, {3, 1625}, {129, 1971}, {140, 34850}, {154, 1624}, {186, 11674}, {232, 511}, {315, 1092}, {389, 47421}, {401, 39682}, {418, 46093}, {578, 2548}, {682, 3491}, {684, 9420}, {852, 38999}, {1297, 3098}, {1511, 9517}, {1513, 9418}, {1634, 46185}, {1936, 39004}, {2081, 14696}, {2972, 3819}, {3425, 17970}, {5118, 38738}, {5907, 13334}, {6000, 47426}, {6747, 14129}, {6759, 8721}, {7785, 34148}, {8780, 38283}, {8841, 42671}, {9292, 34157}, {10282, 23208}, {11202, 14649}, {13188, 34360}, {13509, 49124}, {16089, 41204}, {16591, 39007}, {22103, 51455}, {22115, 35088}, {33983, 44508}, {39005, 47195}, {40817, 47642}, {47079, 47113}, {47239, 51733}

X(52128) = midpoint of X(3) and X(1625)
X(52128) = complement of the isogonal conjugate of X(19189)
X(52128) = psi-transform of X(37918)
X(52128) = X(i)-complementary conjugate of X(j) for these (i,j): {240, 1209}, {2148, 441}, {2190, 511}, {2616, 3150}, {8882, 16609}, {19189, 10}, {40440, 21531}, {41270, 1214}
X(52128) = X(i)-Ceva conjugate of X(j) for these (i,j): {3, 511}, {1303, 39469}, {11672, 36213}, {18831, 2799}, {23582, 14966}
X(52128) = X(i)-isoconjugate of X(j) for these (i,j): {98, 1956}, {1821, 1987}, {1910, 1972}, {14941, 36120}
X(52128) = X(i)-Dao conjugate of X(j) for these (i,j): {98, 39045}, {125, 6130}, {264, 297}, {290, 39081}, {511, 40804}, {1821, 39038}, {1972, 11672}, {1987, 40601}, {14941, 46094}, {38974, 43665}, {39073, 51960}
X(52128) = crosspoint of X(i) and X(j) for these (i,j): {2421, 18020}, {39287, 51862}
X(52128) = crosssum of X(2395) and X(20975)
X(52128) = crossdifference of every pair of points on line {879, 1987}
X(52128) = X(3)-lineconjugate of X(1987)
X(52128) = barycentric product X(i)*X(j) for these {i,j}: {237, 44137}, {325, 1971}, {401, 511}, {1955, 1959}, {2421, 6130}, {3289, 16089}, {18020, 38974}, {32545, 36790}, {36212, 41204}
X(52128) = barycentric quotient X(i)/X(j) for these {i,j}: {237, 1987}, {401, 290}, {511, 1972}, {1755, 1956}, {1955, 1821}, {1971, 98}, {3289, 14941}, {6130, 43665}, {9475, 51960}, {11672, 40804}, {32545, 34536}, {38974, 125}, {41204, 16081}, {41270, 1298}, {44137, 18024}

X(52129) = X(4)X(80)∩X(35)X(102)

X(52129) lies on the cubic K681 and these lines: {1, 1361}, {4, 80}, {35, 102}, {36, 109}, {46, 978}, {55, 38573}, {56, 38579}, {117, 7741}, {124, 7951}, {151, 1479}, {1478, 33650}, {1698, 3040}, {1780, 51966}, {1795, 5563}, {2093, 5400}, {2773, 19470}, {2779, 7727}, {2807, 15430}, {2817, 5697}, {3042, 3679}, {3583, 10740}, {3585, 10747}, {3738, 7972}, {3746, 38667}, {4316, 38777}, {5010, 38600}, {7280, 38607}, {10895, 38779}, {11700, 21842}, {11713, 37525}, {12016, 18398}, {12943, 38780}, {13532, 37710}, {14690, 37572}, {14795, 38565}

X(52129) = reflection of X(i) in X(j) for these {i,j}: {1, 1361}, {5903, 1845}, {38507, 11700}

X(52130) = X(3)X(74)∩X(4)X(5627)

X(521) lies on the cubic K846 and these lines: {3, 74}, {4, 5627}, {5, 40630}, {146, 39170}, {185, 34329}, {546, 12079}, {1304, 44879}, {1510, 14380}, {2132, 12290}, {3146, 14989}, {3529, 36875}, {3627, 34150}, {5890, 38933}, {6240, 17986}, {10125, 16243}, {10421, 34797}, {12317, 51254}, {13754, 15786}, {14865, 38937}, {16080, 16868}, {18439, 50467}, {34783, 50464}, {35007, 48451}, {35502, 35908}

X(52130) = barycentric product X(i)*X(j) for these {i,j}: {1539, 40384}, {2394, 30510}
X(52130) = barycentric quotient X(i)/X(j) for these {i,j}: {1539, 36789}, {30510, 2407}, {51544, 22451}
X(52130) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 3470, 14385}, {3, 14264, 3470}, {74, 3470, 3}, {74, 14264, 14385}, {14264, 14385, 39239}

X(52131) = X(6)X(1345)∩X(110)X(112)

X(52131) lies on the cubics K162, K190, and K1067, and these lines: {6, 1345}, {110, 112}, {458, 2593}, {512, 44123}, {525, 8115}, {1113, 3288}, {1312, 51404}, {2421, 39298}, {2433, 41941}, {2576, 7252}, {10097, 15461}

X(52131) = isogonal conjugate of X(50945)
X(52131) = isogonal conjugate of the isotomic conjugate of X(50944)
X(52131) = X(i)-Ceva conjugate of X(j) for these (i,j): {110, 44123}, {648, 1113}, {15461, 44125}, {39298, 3}, {41941, 15167}
X(52131) = X(i)-cross conjugate of X(j) for these (i,j): {512, 2575}, {647, 8106}, {3049, 42667}, {15167, 41941}, {44125, 15461}
X(52131) = X(i)-isoconjugate of X(j) for these (i,j): {1, 50945}, {662, 1313}, {799, 44126}, {811, 15166}, {1114, 2582}, {1577, 15460}, {1823, 2592}, {2574, 2581}, {2577, 22339}, {2578, 15165}, {2584, 46812}, {2587, 46814}, {2588, 8116}, {14208, 41942}
X(52131) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 50945}, {525, 2575}, {850, 8106}, {1084, 1313}, {1312, 2592}, {3267, 46811}, {15166, 17423}, {15167, 22339}, {38996, 44126}
X(52131) = cevapoint of X(3049) and X(42667)
X(52131) = crosspoint of X(i) and X(j) for these (i,j): {110, 8115}, {648, 1113}
X(52131) = crosssum of X(i) and X(j) for these (i,j): {523, 8105}, {525, 46814}, {647, 2574}
X(52131) = trilinear pole of line {15167, 20975}
X(52131) = crossdifference of every pair of points on line {125, 1313}
X(52131) = barycentric product X(i)*X(j) for these {i,j}: {6, 50944}, {99, 44125}, {110, 1312}, {523, 15461}, {525, 41941}, {648, 15167}, {1113, 2575}, {1304, 14500}, {1822, 2589}, {2576, 2583}, {2579, 2580}, {2585, 2586}, {8106, 8115}, {15164, 42667}, {22340, 44123}
X(52131) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 50945}, {512, 1313}, {669, 44126}, {1113, 15165}, {1312, 850}, {1576, 15460}, {2575, 22339}, {2576, 2581}, {2579, 2582}, {3049, 15166}, {8106, 2592}, {8115, 46810}, {9409, 14499}, {15167, 525}, {15461, 99}, {41941, 648}, {42667, 2574}, {44123, 1114}, {44125, 523}, {50944, 76}

X(52132) = X(6)X(1344)∩X(110)X(112)

X(52132) lies on the cubics K162, K190, and K1067, and these lines: {6, 1344}, {110, 112}, {458, 2592}, {512, 44124}, {525, 8116}, {1114, 3288}, {1313, 51404}, {2421, 39299}, {2433, 41942}, {2577, 7252}, {10097, 15460}

X(52132) = isogonal conjugate of X(50944)
X(52132) = isogonal conjugate of the isotomic conjugate of X(50945)
X(52132) = X(i)-Ceva conjugate of X(j) for these (i,j): {110, 44124}, {648, 1114}, {15460, 44126}, {39299, 3}, {41942, 15166}
X(52132) = X(i)-cross conjugate of X(j) for these (i,j): {512, 2574}, {647, 8105}, {3049, 42668}, {15166, 41942}, {44126, 15460}
X(52132) = X(i)-isoconjugate of X(j) for these (i,j): {1, 50944}, {662, 1312}, {799, 44125}, {811, 15167}, {1113, 2583}, {1577, 15461}, {1822, 2593}, {2575, 2580}, {2576, 22340}, {2579, 15164}, {2585, 46815}, {2586, 46811}, {2589, 8115}, {14208, 41941}
X(52132) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 50944}, {525, 2574}, {850, 8105}, {1084, 1312}, {1313, 2593}, {3267, 46814}, {15166, 22340}, {15167, 17423}, {38996, 44125}
X(52132) = cevapoint of X(3049) and X(42668)
X(52132) = crosspoint of X(i) and X(j) for these (i,j): {110, 8116}, {648, 1114}
X(52132) = crosssum of X(i) and X(j) for these (i,j): {523, 8106}, {525, 46811}, {647, 2575}
X(52132) = trilinear pole of line {15166, 20975}
X(52132) = crossdifference of every pair of points on line {125, 1312}
X(52132) = barycentric product X(i)*X(j) for these {i,j}: {6, 50945}, {99, 44126}, {110, 1313}, {523, 15460}, {525, 41942}, {648, 15166}, {1114, 2574}, {1304, 14499}, {1823, 2588}, {2577, 2582}, {2578, 2581}, {2584, 2587}, {8105, 8116}, {15165, 42668}, {22339, 44124}
X(52132) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 50944}, {512, 1312}, {669, 44125}, {1114, 15164}, {1313, 850}, {1576, 15461}, {2574, 22340}, {2577, 2580}, {2578, 2583}, {3049, 15167}, {8105, 2593}, {8116, 46813}, {9409, 14500}, {15166, 525}, {15460, 99}, {41942, 648}, {42668, 2575}, {44124, 1113}, {44126, 523}, {50945, 76}

X(52133) = X(1)X(257)∩X(2)X(31)

X(52133) lies on the cubic K1031 and these lines: {1, 257}, {2, 31}, {8, 41}, {9, 4518}, {10, 3407}, {21, 28660}, {25, 92}, {29, 2204}, {55, 312}, {56, 85}, {75, 1281}, {87, 17000}, {98, 30670}, {105, 789}, {183, 1001}, {189, 2208}, {333, 2194}, {825, 1311}, {958, 2053}, {1121, 4586}, {1220, 4426}, {1284, 3113}, {1492, 34234}, {1621, 26243}, {1952, 24806}, {2176, 5275}, {2218, 40011}, {2726, 30664}, {3056, 27958}, {3314, 33082}, {3329, 16468}, {3415, 26234}, {3570, 3789}, {3996, 4102}, {4026, 7792}, {4042, 42030}, {4384, 9746}, {4386, 5263}, {4514, 36488}, {4613, 6187}, {4649, 7766}, {4676, 17735}, {4679, 4997}, {4817, 28851}, {4966, 37671}, {5205, 34446}, {5276, 40728}, {5999, 6210}, {6186, 30690}, {7020, 7154}, {7875, 29633}, {8298, 32941}, {9282, 40766}, {10527, 41791}, {16372, 23407}, {16466, 37053}, {16681, 20875}, {16825, 33891}, {16986, 29637}, {16993, 39586}, {16994, 37608}, {22329, 49740}, {29641, 40435}, {29667, 40394}, {29840, 32853}, {30038, 48932}, {30179, 50308}, {31120, 32843}, {37658, 40739}, {39581, 40745}, {49516, 51435}, {49563, 50023}, {51314, 51443}

X(52133) = isogonal conjugate of X(1469)
X(52133) = isotomic conjugate of X(7179)
X(52133) = X(i)-Ceva conjugate of X(j) for these (i,j): {870, 14621}, {5384, 4613}
X(52133) = X(i)-cross conjugate of X(j) for these (i,j): {55, 40757}, {2344, 14621}, {3883, 8}
X(52133) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1469}, {6, 7146}, {7, 869}, {31, 7179}, {34, 3781}, {55, 7204}, {56, 984}, {57, 2276}, {59, 4475}, {65, 3736}, {85, 40728}, {109, 1491}, {269, 4517}, {307, 46503}, {604, 3661}, {651, 3250}, {664, 788}, {824, 1415}, {985, 12837}, {1025, 29956}, {1042, 3786}, {1106, 3790}, {1333, 16603}, {1397, 33931}, {1400, 40773}, {1402, 30966}, {1403, 45782}, {1408, 3773}, {1409, 31909}, {1411, 3792}, {1417, 4439}, {1428, 3864}, {1429, 3862}, {1431, 40790}, {1434, 3774}, {1458, 52029}, {2263, 45974}, {2279, 40784}, {3094, 7132}, {3799, 43924}, {3805, 29055}, {4481, 4559}, {4554, 46386}, {4572, 8630}, {6063, 18900}, {18784, 40797}, {30545, 40736}, {37137, 45882}, {41526, 51837}
X(52133) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 984}, {2, 7179}, {3, 1469}, {9, 7146}, {11, 1491}, {37, 16603}, {223, 7204}, {788, 39025}, {824, 1146}, {2276, 5452}, {2968, 4522}, {3161, 3661}, {3250, 38991}, {3736, 40602}, {3781, 11517}, {3789, 12837}, {3790, 6552}, {3792, 35204}, {4122, 6741}, {4475, 6615}, {4517, 6600}, {30966, 40605}, {40582, 40773}, {41886, 51836}
X(52133) = cevapoint of X(i) and X(j) for these (i,j): {1, 6210}, {11, 50347}, {55, 37658}
X(52133) = trilinear pole of line {522, 3063}
X(52133) = crossdifference of every pair of points on line {3250, 45882}
X(52133) = barycentric product X(i)*X(j) for these {i,j}: {8, 14621}, {9, 870}, {41, 871}, {75, 2344}, {312, 985}, {314, 40747}, {333, 40718}, {522, 4586}, {650, 789}, {663, 37133}, {825, 35519}, {1492, 4391}, {3056, 3114}, {3061, 3113}, {3063, 46132}, {3271, 5388}, {3407, 3705}, {3596, 40746}, {3699, 4817}, {3716, 37207}, {4076, 43266}, {4384, 40739}, {4435, 41072}, {4441, 40757}, {4451, 40745}, {4560, 4613}, {4858, 5384}, {7081, 40738}, {17787, 40763}, {20665, 46281}, {23597, 36801}, {30660, 40771}
X(52133) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 7146}, {2, 7179}, {6, 1469}, {8, 3661}, {9, 984}, {10, 16603}, {21, 40773}, {29, 31909}, {41, 869}, {55, 2276}, {57, 7204}, {219, 3781}, {220, 4517}, {284, 3736}, {294, 52029}, {312, 33931}, {333, 30966}, {346, 3790}, {522, 824}, {644, 3799}, {646, 4505}, {650, 1491}, {663, 3250}, {789, 4554}, {825, 109}, {870, 85}, {871, 20567}, {884, 29956}, {949, 45974}, {985, 57}, {1001, 40784}, {1492, 651}, {2170, 4475}, {2175, 40728}, {2204, 46503}, {2276, 12837}, {2287, 3786}, {2319, 45782}, {2321, 3773}, {2323, 3792}, {2325, 4439}, {2329, 40790}, {2344, 1}, {3056, 3094}, {3061, 51836}, {3063, 788}, {3239, 4522}, {3287, 3805}, {3684, 3783}, {3685, 3797}, {3686, 3775}, {3699, 3807}, {3700, 4122}, {3705, 3314}, {3707, 4407}, {3716, 4486}, {3737, 4481}, {3786, 4469}, {3886, 27474}, {4435, 30665}, {4586, 664}, {4613, 4552}, {4765, 4818}, {4817, 3676}, {4876, 3864}, {4944, 4951}, {5332, 18957}, {5384, 4564}, {7077, 3862}, {7155, 51837}, {8424, 40797}, {9447, 18900}, {14621, 7}, {20665, 3116}, {23597, 43041}, {30670, 37137}, {34069, 1415}, {37133, 4572}, {37658, 3789}, {40718, 226}, {40722, 17084}, {40738, 7249}, {40739, 27475}, {40745, 7176}, {40746, 56}, {40747, 65}, {40757, 1002}, {40763, 1432}, {43266, 1358}
X(52133) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {385, 40738, 40752}, {985, 40718, 14621}

X(52134) = X(1)X(21)∩X(48)X(75)

X(52134) lies on the cubic K1031 and these lines: {1, 21}, {2, 1429}, {8, 20769}, {19, 17868}, {41, 239}, {48, 75}, {69, 18162}, {86, 30035}, {92, 1973}, {101, 4384}, {144, 16503}, {190, 2267}, {192, 2268}, {219, 28287}, {284, 3875}, {304, 2167}, {329, 26626}, {536, 2278}, {560, 17445}, {572, 3729}, {584, 4852}, {604, 894}, {750, 40790}, {908, 16788}, {940, 51319}, {956, 23151}, {1055, 11329}, {1100, 43216}, {1334, 16367}, {1385, 25083}, {1423, 15988}, {1438, 36816}, {1748, 17442}, {1760, 1953}, {1930, 20879}, {2172, 14213}, {2174, 4361}, {2187, 3757}, {2223, 24264}, {2280, 2344}, {2304, 17143}, {2908, 4812}, {3056, 8424}, {3061, 3219}, {3204, 17348}, {3217, 17349}, {3218, 7146}, {3223, 38252}, {3262, 24315}, {3264, 30882}, {3501, 21495}, {3661, 4390}, {3662, 7225}, {3682, 19844}, {3905, 17147}, {3912, 33113}, {4112, 12263}, {4251, 16834}, {4268, 17351}, {4289, 50120}, {4359, 16822}, {4360, 11683}, {4363, 7113}, {4553, 52086}, {4564, 31225}, {4861, 7291}, {4876, 41423}, {5053, 50127}, {5176, 29659}, {5278, 30038}, {5279, 7269}, {5744, 17316}, {5773, 24591}, {6602, 25943}, {7081, 12197}, {8649, 31198}, {8931, 25306}, {9327, 29597}, {9454, 20172}, {11681, 29633}, {14621, 52083}, {14839, 37586}, {14953, 20244}, {16779, 20332}, {16781, 23538}, {16783, 17781}, {17379, 20348}, {17438, 44179}, {17760, 32933}, {17786, 18048}, {17859, 21406}, {18047, 20917}, {18671, 23491}, {18750, 34065}, {19554, 31317}, {19684, 30076}, {20352, 26232}, {20556, 48900}, {20557, 32772}, {21511, 37555}, {24203, 37086}, {24455, 30807}, {24460, 27907}, {26130, 28420}, {26222, 26971}, {26657, 28351}, {27944, 30854}, {29598, 30852}, {32774, 49612}, {37132, 37216}, {40214, 50106}, {44694, 46475}

X(52134) = isogonal conjugate of X(2186)
X(52134) = anticomplement of X(16603)
X(52134) = isogonal conjugate of the isotomic conjugate of X(3403)
X(52134) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {985, 2893}, {2344, 1330}, {40746, 2475}
X(52134) = X(3113)-Ceva conjugate of X(31)
X(52134) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2186}, {2, 263}, {4, 43718}, {6, 262}, {25, 42313}, {32, 327}, {39, 42299}, {51, 42300}, {53, 51444}, {75, 3402}, {76, 46319}, {98, 51543}, {141, 42288}, {523, 26714}, {882, 39681}, {1976, 46807}, {1987, 39682}, {2799, 32716}, {3569, 6037}, {7612, 51338}, {7735, 40803}, {14998, 36885}, {19222, 51997}
X(52134) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 2186}, {9, 262}, {75, 51580}, {182, 16567}, {206, 3402}, {263, 32664}, {327, 6376}, {661, 38997}, {3117, 51836}, {6505, 42313}, {17471, 24206}, {36033, 43718}, {39038, 39682}, {39040, 46807}
X(52134) = crosspoint of X(4564) and X(4586)
X(52134) = crosssum of X(2170) and X(3250)
X(52134) = barycentric product X(i)*X(j) for these {i,j}: {1, 183}, {6, 3403}, {31, 20023}, {48, 44144}, {58, 42711}, {63, 458}, {75, 182}, {82, 14994}, {304, 10311}, {326, 33971}, {394, 51315}, {561, 34396}, {662, 23878}, {799, 3288}, {1580, 8842}, {1910, 51373}, {1959, 46806}, {2349, 51372}, {3112, 14096}, {6784, 24037}, {46238, 51542}
X(52134) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 262}, {6, 2186}, {31, 263}, {32, 3402}, {48, 43718}, {63, 42313}, {75, 327}, {82, 42299}, {163, 26714}, {182, 1}, {183, 75}, {458, 92}, {560, 46319}, {1755, 51543}, {1955, 39682}, {1959, 46807}, {2167, 42300}, {2169, 51444}, {2715, 36132}, {3288, 661}, {3403, 76}, {6784, 2643}, {8842, 1934}, {9755, 4008}, {10311, 19}, {14096, 38}, {14994, 1930}, {20023, 561}, {23878, 1577}, {33971, 158}, {34396, 31}, {36084, 6037}, {39683, 1956}, {42711, 313}, {44144, 1969}, {46289, 42288}, {46806, 1821}, {51315, 2052}, {51372, 14206}, {51373, 46238}, {51542, 1910}
X(52134) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 63, 1959}, {1, 1580, 31}, {38, 8772, 1}, {48, 75, 1958}, {75, 18042, 48}, {1429, 2329, 2}, {3219, 26639, 3061}, {3661, 27950, 25940}, {4390, 25940, 3661}, {4393, 40744, 2280}, {5773, 28797, 24591}, {34055, 39731, 1973}, {45220, 45224, 1959}

X(52135) = X(1)X(257)∩X(38)X(256)

X(52135) lies on the cubics K766 and K863 and these lines: {1, 257}, {2, 40777}, {38, 256}, {75, 325}, {76, 18760}, {147, 29057}, {230, 29634}, {335, 40849}, {350, 1934}, {517, 3903}, {523, 25667}, {524, 4835}, {694, 3726}, {726, 1916}, {1178, 7191}, {1281, 41532}, {1581, 18208}, {1925, 18835}, {2481, 18829}, {3509, 41882}, {3865, 49613}, {4511, 29055}, {4518, 36897}, {7015, 35614}, {10026, 49509}, {17145, 50248}, {17731, 32010}, {26242, 40729}, {37592, 40432}

X(52135) = isotomic conjugate of X(7061)
X(52135) = isotomic conjugate of the isogonal conjugate of X(41532)
X(52135) = X(i)-Ceva conjugate of X(j) for these (i,j): {350, 40849}, {1934, 257}
X(52135) = X(18037)-cross conjugate of X(52085)
X(52135) = X(i)-isoconjugate of X(j) for these (i,j): {6, 41534}, {31, 7061}, {32, 40846}, {171, 8852}, {172, 3512}, {1580, 30648}, {1691, 24479}, {7122, 7261}
X(52135) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 7061}, {9, 41534}, {238, 1580}, {6376, 40846}, {30648, 39092}
X(52135) = barycentric product X(i)*X(j) for these {i,j}: {75, 40873}, {76, 41532}, {256, 17789}, {257, 4645}, {561, 41882}, {1281, 1916}, {1581, 18037}, {1934, 19557}, {3509, 7018}, {4071, 32010}, {4458, 27805}, {17493, 52085}, {17798, 44187}, {18786, 51859}, {18896, 19561}
X(52135) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 41534}, {2, 7061}, {75, 40846}, {256, 3512}, {257, 7261}, {694, 30648}, {893, 8852}, {1281, 385}, {1581, 24479}, {3509, 171}, {4071, 1215}, {4458, 4369}, {4645, 894}, {4987, 4697}, {5018, 7175}, {7018, 40845}, {8868, 51920}, {17789, 1909}, {17798, 172}, {18037, 1966}, {18038, 1933}, {19554, 7122}, {19557, 1580}, {19561, 1691}, {20715, 2295}, {20741, 3955}, {27805, 51614}, {27951, 14296}, {40873, 1}, {41532, 6}, {41882, 31}, {44187, 18036}, {52085, 30669}

X(52136) = X(1)X(257)∩X(6)X(75)

X(52136) lies on the cubic K1031 and these lines: {1, 257}, {6, 75}, {8, 40718}, {31, 19580}, {43, 40848}, {57, 7233}, {183, 16826}, {192, 2209}, {213, 3114}, {330, 985}, {1258, 17033}, {2176, 6376}, {2280, 2344}, {2998, 34248}, {3009, 16997}, {3212, 41526}, {3226, 4586}, {3329, 4384}, {3570, 40733}, {3661, 37678}, {3765, 24514}, {4279, 32453}, {8667, 29580}, {9361, 40742}, {11174, 16815}, {14614, 29584}, {15271, 29578}, {16998, 21352}, {17367, 37686}, {21385, 23597}, {33296, 33890}

X(52136) = isotomic conjugate of X(51837)
X(52136) = X(i)-Ceva conjugate of X(j) for these (i,j): {985, 14621}, {40745, 40722}
X(52136) = X(i)-isoconjugate of X(j) for these (i,j): {6, 45782}, {31, 51837}, {75, 40736}, {87, 2276}, {330, 869}, {788, 4598}, {932, 3250}, {984, 2162}, {1469, 2319}, {1491, 34071}, {2053, 7146}, {3661, 7121}, {3736, 16606}, {3862, 34252}, {3864, 51321}, {4517, 7153}, {6383, 18900}, {6384, 40728}, {18830, 46386}, {21759, 30966}, {22381, 31909}, {23493, 40773}, {40790, 51974}
X(52136) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 51837}, {9, 45782}, {75, 33931}, {206, 40736}, {824, 6377}, {1491, 40610}, {3061, 51836}, {3661, 40598}
X(52136) = crosspoint of X(4586) and X(5388)
X(52136) = crosssum of X(984) and X(45782)
X(52136) = trilinear pole of line {3835, 8640}
X(52136) = crossdifference of every pair of points on line {788, 45882}
X(52136) = barycentric product X(i)*X(j) for these {i,j}: {43, 870}, {192, 14621}, {789, 4083}, {871, 2209}, {985, 6376}, {1492, 20906}, {2344, 30545}, {3113, 41886}, {3114, 20284}, {3407, 33890}, {3835, 4586}, {4595, 4817}, {4613, 17217}, {5388, 6377}, {6382, 40746}, {8640, 46132}, {17752, 40738}, {20979, 37133}, {30963, 40756}, {31008, 40747}, {33296, 40718}, {40763, 41318}
X(52136) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 45782}, {2, 51837}, {32, 40736}, {43, 984}, {192, 3661}, {789, 18830}, {825, 34071}, {870, 6384}, {985, 87}, {1403, 1469}, {1423, 7146}, {1492, 932}, {2176, 2276}, {2209, 869}, {2344, 2319}, {3123, 4475}, {3212, 7179}, {3835, 824}, {3971, 3773}, {4083, 1491}, {4147, 4522}, {4586, 4598}, {4595, 3807}, {4970, 3775}, {6376, 33931}, {8640, 788}, {14621, 330}, {16468, 40783}, {18197, 4481}, {20284, 3094}, {20760, 3781}, {20979, 3250}, {21051, 4122}, {24533, 3805}, {27538, 3790}, {27644, 40773}, {33296, 30966}, {33890, 3314}, {36863, 4505}, {38832, 3736}, {40718, 42027}, {40738, 27447}, {40746, 2162}, {40747, 16606}, {41531, 3864}, {41886, 51836}, {51902, 40790}, {51973, 3862}
X(52136) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {385, 40763, 40722}, {870, 40747, 14621}

X(52137) = X(1)X(75)∩X(2)X(40776)

X(52137) lies on the cubic K766 and these lines: {1, 75}, {2, 40776}, {9, 34016}, {76, 6625}, {99, 3685}, {141, 18140}, {312, 8033}, {321, 873}, {335, 4639}, {350, 9505}, {668, 32846}, {670, 1921}, {726, 18827}, {799, 4358}, {894, 1509}, {1215, 2668}, {1577, 4960}, {1757, 17731}, {1920, 18021}, {1931, 6651}, {1999, 7304}, {2113, 17794}, {3751, 33932}, {3786, 33297}, {3797, 37128}, {3840, 32010}, {3912, 24479}, {3923, 17103}, {3948, 9510}, {4359, 33779}, {4567, 4590}, {4576, 29824}, {4589, 9470}, {4594, 41532}, {4625, 40704}, {4634, 20568}, {4654, 30545}, {6541, 40794}, {9279, 17166}, {17398, 25457}, {17778, 51857}, {18035, 20362}, {24241, 30966}, {27184, 30965}, {27705, 27954}, {32004, 33938}, {32020, 40849}, {33770, 33941}, {33931, 51356}, {36036, 36056}, {38485, 41851}, {39719, 41535}

X(52137) = reflection of X(9505) in X(49676)
X(52137) = isotomic conjugate of X(9278)
X(52137) = isogonal conjugate of the anticomplement of X(20548)
X(52137) = isogonal conjugate of the complement of X(20560)
X(52137) = isotomic conjugate of the anticomplement of X(20529)
X(52137) = isotomic conjugate of the complement of X(20538)
X(52137) = isotomic conjugate of the isogonal conjugate of X(1931)
X(52137) = X(46193)-anticomplementary conjugate of X(69)
X(52137) = X(i)-Ceva conjugate of X(j) for these (i,j): {350, 40874}, {40017, 274}
X(52137) = X(i)-cross conjugate of X(j) for these (i,j): {6542, 17731}, {20362, 1}, {20529, 2}, {20867, 6}
X(52137) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2054}, {31, 9278}, {32, 11599}, {42, 17962}, {101, 18001}, {213, 1929}, {512, 2702}, {669, 35148}, {740, 18263}, {798, 37135}, {1918, 6650}, {2200, 17982}, {2205, 18032}, {2206, 6543}, {2333, 17972}, {3747, 9506}, {4079, 17940}, {9505, 41333}, {18014, 32739}
X(52137) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 9278}, {6, 39042}, {9, 2054}, {37, 41841}, {42, 39041}, {239, 2238}, {661, 35080}, {1015, 18001}, {1929, 6626}, {1931, 20472}, {2702, 39054}, {3125, 27929}, {6376, 11599}, {6543, 40603}, {6650, 34021}, {17962, 40592}, {18014, 40619}, {20337, 20461}, {31998, 37135}
X(52137) = cevapoint of X(i) and X(j) for these (i,j): {1, 20371}, {2, 20538}, {6542, 20947}
X(52137) = crosspoint of X(4601) and X(4639)
X(52137) = barycentric product X(i)*X(j) for these {i,j}: {75, 17731}, {76, 1931}, {86, 20947}, {274, 6542}, {304, 423}, {310, 1757}, {561, 1326}, {670, 9508}, {693, 17934}, {799, 2786}, {873, 6541}, {4602, 5029}, {4623, 18004}, {4634, 28602}, {4639, 27929}, {6385, 17735}, {6651, 40017}, {17943, 40495}, {18035, 37128}, {20568, 31059}
X(52137) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2054}, {2, 9278}, {75, 11599}, {81, 17962}, {86, 1929}, {99, 37135}, {274, 6650}, {286, 17982}, {310, 18032}, {321, 6543}, {423, 19}, {513, 18001}, {662, 2702}, {693, 18014}, {799, 35148}, {1326, 31}, {1444, 17972}, {1757, 42}, {1931, 6}, {2786, 661}, {4623, 17930}, {5029, 798}, {6541, 756}, {6542, 37}, {6651, 2238}, {8298, 3747}, {9508, 512}, {17731, 1}, {17735, 213}, {17927, 1824}, {17934, 100}, {17943, 692}, {17976, 228}, {17990, 50487}, {18004, 4705}, {18035, 3948}, {18266, 1918}, {18268, 18263}, {18827, 9505}, {20693, 1500}, {20756, 20782}, {20947, 10}, {27929, 21832}, {28602, 4730}, {30940, 40725}, {31059, 44}, {33295, 40767}, {37128, 9506}, {38348, 4455}, {38814, 51332}, {39042, 20472}
X(52137) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {75, 51314, 274}, {4358, 16741, 799}, {18157, 30939, 30940}, {18157, 30940, 274}

X(52138) = X(1)X(75)∩X(6)X(17743)

X(52138) lies on the cubic K1031 and these lines: {1, 75}, {6, 17743}, {31, 799}, {76, 49482}, {82, 18832}, {171, 6384}, {183, 1001}, {238, 6376}, {334, 50289}, {561, 17469}, {668, 16468}, {673, 36630}, {902, 30964}, {1920, 17716}, {1965, 18056}, {2345, 20146}, {3508, 17336}, {3550, 34020}, {4479, 48805}, {4593, 51312}, {4676, 52049}, {5156, 18148}, {5989, 30545}, {7018, 29634}, {7290, 39044}, {8026, 32926}, {8616, 31008}, {10009, 50023}, {14621, 20917}, {15485, 18140}, {16681, 23375}, {16826, 24679}, {16974, 21608}, {17349, 25280}, {17370, 26959}, {17371, 27020}, {17379, 17787}, {17446, 23478}, {18058, 33788}, {20134, 24656}, {20158, 25298}, {20172, 21780}, {20179, 41240}, {20284, 30661}, {21760, 29423}, {23660, 30114}, {23857, 51621}, {25287, 32911}, {26230, 30632}, {29438, 29459}, {29838, 30660}, {30138, 40001}, {30141, 46747}

X(52138) = isotomic conjugate of X(51844)
X(52138) = isotomic conjugate of the isogonal conjugate of X(51291)
X(52138) = X(3113)-Ceva conjugate of X(75)
X(52138) = X(i)-isoconjugate of X(j) for these (i,j): {31, 51844}, {32, 43688}, {39, 51450}, {512, 25424}
X(52138) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 51844}, {3314, 51836}, {6376, 43688}, {25424, 39054}
X(52138) = crosssum of X(788) and X(38986)
X(52138) = barycentric product X(i)*X(j) for these {i,j}: {1, 41259}, {31, 10010}, {75, 7766}, {76, 51291}, {799, 25423}, {3112, 32449}, {3113, 10335}
X(52138) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 51844}, {75, 43688}, {82, 51450}, {662, 25424}, {7766, 1}, {10010, 561}, {10335, 51836}, {25423, 661}, {32449, 38}, {41259, 75}, {45680, 2642}, {51291, 6}
X(52138) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1740, 51907}, {1, 1966, 75}, {31, 18064, 17149}, {1965, 18056, 20945}, {17752, 39914, 6}, {30940, 32941, 17144}

X(52139) = X(3)X(10)∩X(6)X(31)

X(52139) lies on the cubics K430 and K1057 and these lines: {1, 16287}, {2, 16678}, {3, 10}, {6, 31}, {8, 16452}, {9, 3185}, {12, 37225}, {21, 1220}, {25, 1631}, {35, 3293}, {36, 16374}, {37, 1402}, {40, 22300}, {43, 5132}, {44, 20967}, {56, 37674}, {63, 22275}, {100, 333}, {165, 7416}, {171, 3286}, {181, 2245}, {183, 313}, {190, 11688}, {198, 45739}, {199, 20989}, {200, 15624}, {210, 228}, {220, 2200}, {375, 2183}, {404, 19874}, {405, 23383}, {474, 16828}, {498, 16455}, {572, 9562}, {612, 2352}, {669, 14321}, {692, 2328}, {756, 3724}, {846, 5143}, {851, 3925}, {859, 5251}, {960, 23846}, {1001, 16058}, {1005, 41260}, {1107, 2223}, {1125, 16286}, {1155, 22060}, {1284, 4415}, {1329, 13731}, {1397, 4268}, {1460, 36743}, {1486, 8804}, {1593, 1869}, {1621, 29822}, {1656, 39583}, {1698, 16453}, {1962, 21801}, {2333, 32561}, {2345, 23381}, {2533, 44408}, {2550, 37400}, {2594, 22076}, {2886, 4192}, {2975, 14829}, {3085, 13726}, {3158, 21384}, {3191, 42443}, {3198, 21867}, {3214, 5217}, {3624, 16296}, {3634, 16414}, {3741, 19341}, {3745, 16679}, {3752, 37575}, {3757, 16684}, {3826, 16056}, {3920, 16687}, {3941, 5269}, {4026, 4199}, {4036, 39199}, {4057, 4448}, {4068, 21871}, {4191, 4413}, {4203, 5263}, {4225, 5260}, {4271, 23638}, {4281, 33771}, {4383, 27631}, {4421, 4685}, {4423, 16373}, {4426, 23851}, {4428, 50300}, {4429, 37467}, {4436, 32932}, {4497, 37538}, {4640, 22325}, {4649, 18185}, {4682, 37609}, {4705, 11124}, {4999, 19513}, {5010, 31855}, {5220, 20760}, {5248, 48866}, {5268, 16778}, {5285, 8021}, {5314, 41243}, {5432, 30944}, {5584, 37195}, {5587, 7420}, {5687, 37057}, {5711, 19762}, {6187, 40519}, {6253, 37409}, {6600, 22312}, {6679, 19263}, {6690, 8731}, {7080, 37297}, {7119, 37908}, {7234, 23865}, {7354, 47521}, {7484, 37578}, {8013, 21012}, {8193, 37284}, {8679, 22097}, {8683, 22313}, {9591, 20840}, {9780, 16451}, {10198, 16290}, {10310, 15592}, {10319, 22282}, {10902, 15623}, {11491, 21672}, {11849, 35468}, {12514, 23844}, {13588, 27164}, {13738, 19744}, {14973, 21061}, {15246, 31073}, {15523, 20999}, {15626, 15931}, {15985, 44419}, {16064, 36475}, {16291, 19862}, {16294, 24987}, {16295, 24982}, {16370, 48832}, {16371, 19870}, {16372, 17798}, {16418, 48826}, {16606, 21856}, {16877, 40790}, {16878, 17022}, {17279, 23379}, {17303, 37079}, {17337, 28250}, {17747, 23398}, {18082, 51862}, {18235, 44416}, {19338, 26037}, {19339, 31330}, {19543, 26363}, {19648, 25639}, {19855, 37264}, {19857, 37034}, {19998, 37652}, {20617, 37558}, {20834, 20872}, {20835, 37577}, {20838, 21686}, {20988, 23854}, {21010, 23393}, {21321, 37662}, {21363, 38472}, {21674, 37247}, {21714, 39478}, {21860, 22281}, {22080, 51377}, {22297, 28043}, {23207, 51361}, {23339, 32777}, {23368, 33157}, {23369, 32779}, {23841, 35203}, {24703, 31394}, {24953, 27622}, {25992, 50717}, {26040, 37262}, {26244, 38871}, {26885, 35327}, {27714, 37259}, {28248, 37679}, {29444, 30109}, {31737, 48907}, {32942, 35992}, {33105, 40109}, {37246, 37579}, {40635, 40937}, {40910, 41239}

X(52139) = isogonal conjugate of X(20028)
X(52139) = isogonal conjugate of the anticomplement of X(52087)
X(52139) = isogonal conjugate of the isotomic conjugate of X(17751)
X(52139) = X(i)-Ceva conjugate of X(j) for these (i,j): {21, 37}, {1220, 6}, {2975, 21061}, {23617, 213}
X(52139) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20028}, {57, 46880}, {81, 2051}, {86, 34434}, {757, 51870}, {4357, 40453}
X(52139) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 20028}, {12, 1441}, {693, 34589}, {1193, 4357}, {2051, 40586}, {5452, 46880}, {21796, 26563}, {34434, 40600}, {40607, 51870}
X(52139) = crosspoint of X(i) and X(j) for these (i,j): {572, 2975}, {1252, 8707}, {8687, 15386}, {21061, 37558}
X(52139) = crosssum of X(i) and X(j) for these (i,j): {2, 20040}, {124, 3910}, {513, 17197}, {1086, 6371}, {2051, 34434}
X(52139) = crossdifference of every pair of points on line {514, 6589}
X(52139) = barycentric product X(i)*X(j) for these {i,j}: {1, 21061}, {6, 17751}, {9, 37558}, {10, 572}, {37, 2975}, {42, 14829}, {71, 11109}, {81, 14973}, {210, 17074}, {321, 20986}, {644, 51664}, {1018, 21173}, {1220, 52087}, {2287, 20617}, {4557, 17496}, {22118, 41013}, {26115, 34278}
X(52139) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 20028}, {42, 2051}, {55, 46880}, {213, 34434}, {572, 86}, {1500, 51870}, {2975, 274}, {11109, 44129}, {14829, 310}, {14973, 321}, {17751, 76}, {20617, 1446}, {20986, 81}, {21061, 75}, {21173, 7199}, {22118, 1444}, {23187, 15419}, {37558, 85}, {38344, 17219}, {51664, 24002}, {52087, 4357}
X(52139) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 16678, 20470}, {3, 958, 23361}, {9, 10434, 3185}, {35, 5247, 4267}, {42, 71, 22276}, {55, 1011, 8053}, {55, 20992, 3052}, {210, 228, 4557}, {612, 2352, 20990}, {958, 1376, 5737}, {1001, 23853, 18613}, {1698, 39578, 16453}, {3745, 40956, 16679}, {4640, 37619, 23845}, {8053, 15621, 55}, {16058, 23853, 1001}

X(52140) = X(2)X(514)∩X(63)X(88)

X(52140) lies on the cubic K973 and these lines: {2, 514}, {57, 3257}, {63, 88}, {106, 5272}, {165, 14193}, {200, 1320}, {312, 646}, {679, 39963}, {903, 31142}, {997, 1168}, {1318, 3872}, {1376, 14190}, {1699, 19634}, {3306, 40215}, {3742, 34230}, {3753, 14260}, {3880, 23705}, {3899, 4674}, {4080, 27131}, {4582, 30568}, {19636, 24703}, {23511, 47056}, {23838, 24498}, {31227, 31231}

X(52140) = X(679)-Ceva conjugate of X(1320)
X(52140) = X(i)-isoconjugate of X(j) for these (i,j): {44, 8686}, {1023, 37627}, {1120, 1404}, {1319, 40400}
X(52140) = X(i)-Dao conjugate of X(j) for these (i,j): {2087, 39771}, {2325, 4738}, {3911, 16594}, {8686, 40595}
X(52140) = barycentric product X(i)*X(j) for these {i,j}: {75, 45247}, {903, 3880}, {1266, 1320}, {1318, 20900}, {3596, 17109}, {4997, 16610}, {6548, 23705}
X(52140) = barycentric quotient X(i)/X(j) for these {i,j}: {106, 8686}, {1149, 1319}, {1320, 1120}, {1878, 1877}, {2316, 40400}, {3880, 519}, {4695, 40663}, {4997, 36805}, {6018, 17460}, {16610, 3911}, {17109, 56}, {17460, 1317}, {23345, 37627}, {23705, 17780}, {23832, 23703}, {23838, 23836}, {45247, 1}

X(52141) = X(2)X(99)∩X(23)X(11643)

X(52141) lies on the cubic K106 and these lines: {2, 99}, {23, 11643}, {98, 14694}, {125, 14833}, {230, 17948}, {468, 10416}, {524, 9225}, {597, 42007}, {598, 1995}, {691, 7426}, {892, 16317}, {895, 5642}, {1637, 14977}, {1641, 10754}, {1648, 11161}, {1992, 6791}, {2408, 9125}, {2502, 8593}, {3290, 46799}, {3291, 8859}, {5108, 50639}, {5466, 9123}, {5912, 18823}, {5913, 51224}, {5968, 47597}, {6055, 48983}, {6353, 8753}, {7883, 16055}, {8352, 24855}, {8430, 44560}, {8597, 39602}, {8598, 34169}, {8787, 46276}, {8860, 18818}, {9178, 11176}, {9214, 44556}, {9830, 20998}, {10097, 45336}, {11056, 16509}, {11059, 11165}, {11164, 11336}, {11580, 51541}, {14653, 38796}, {14908, 19661}, {15360, 32583}, {15466, 46111}, {15638, 37745}, {15810, 26257}, {15899, 37760}, {20063, 40343}, {23287, 23288}, {26255, 35278}, {32479, 40350}, {32525, 51798}, {37804, 41133}, {37907, 46783}, {43535, 46290}, {44569, 51405}, {46154, 47352}

X(52141) = X(18818)-Ceva conjugate of X(671)
X(52141) = X(27088)-cross conjugate of X(1992)
X(52141) = X(i)-isoconjugate of X(j) for these (i,j): {351, 37216}, {661, 2434}, {798, 2418}, {896, 21448}, {922, 5485}, {1296, 2642}, {1649, 36045}, {14210, 39238}, {17959, 51927}
X(52141) = X(i)-Dao conjugate of X(j) for these (i,j): {524, 11147}, {599, 39785}, {690, 35133}, {1649, 31654}, {2418, 31998}, {2434, 36830}, {5485, 39061}, {15477, 39238}, {15899, 21448}, {16051, 24855}
X(52141) = cevapoint of X(i) and X(j) for these (i,j): {1992, 27088}, {1995, 11580}, {6791, 9125}
X(52141) = trilinear pole of line {1499, 1992}
X(52141) = crossdifference of every pair of points on line {351, 21905}
X(52141) = barycentric product X(i)*X(j) for these {i,j}: {99, 2408}, {111, 11059}, {670, 2444}, {671, 1992}, {892, 1499}, {1384, 18023}, {4232, 30786}, {9154, 51438}, {11165, 18818}, {14207, 36085}, {36277, 46277}
X(52141) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 2418}, {110, 2434}, {111, 21448}, {671, 5485}, {691, 1296}, {892, 35179}, {1384, 187}, {1499, 690}, {1992, 524}, {2408, 523}, {2444, 512}, {4232, 468}, {4786, 4750}, {6791, 1648}, {8644, 351}, {9125, 1649}, {9126, 44814}, {11059, 3266}, {11165, 39785}, {11580, 10354}, {15471, 5095}, {15638, 6791}, {27088, 2482}, {30234, 14419}, {32740, 39238}, {35234, 31654}, {35266, 5642}, {36085, 37216}, {36277, 896}, {37745, 12036}, {42724, 42713}, {51438, 50567}
X(52141) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 111, 671}, {2, 671, 30786}, {2, 7665, 2482}, {2, 8591, 126}, {468, 15398, 10416}, {2482, 6719, 2}, {2502, 9169, 8593}, {9172, 10418, 2}

X(52142) = X(2)X(691)∩X(25)X(111)

X(52142) lies on the cubic K969 and these lines: {2, 691}, {4, 10415}, {6, 22259}, {22, 3447}, {23, 14246}, {25, 111}, {32, 41936}, {110, 10559}, {184, 10558}, {251, 9178}, {428, 51258}, {671, 5986}, {895, 3060}, {1383, 10630}, {1501, 32740}, {1629, 17983}, {1995, 15899}, {5012, 21460}, {5169, 10416}, {9306, 32583}, {9544, 10560}, {13567, 51938}, {13574, 14357}, {13595, 46783}, {14002, 15398}, {14163, 14583}, {15356, 51926}, {15477, 41404}, {19136, 51962}, {30786, 31074}, {31133, 42008}, {36417, 41937}

X(52142) = isogonal conjugate of the isotomic conjugate of X(14246)
X(52142) = X(i)-Ceva conjugate of X(j) for these (i,j): {691, 2492}, {10422, 111}, {10630, 32740}
X(52142) = X(42659)-cross conjugate of X(32729)
X(52142) = X(i)-isoconjugate of X(j) for these (i,j): {67, 14210}, {75, 14357}, {896, 18019}, {2157, 3266}, {7813, 37221}, {10415, 24038}
X(52142) = X(i)-Dao conjugate of X(j) for these (i,j): {67, 15477}, {69, 39169}, {187, 36792}, {206, 14357}, {3266, 40583}, {5099, 35522}, {15899, 18019}
X(52142) = cevapoint of X(32) and X(51962)
X(52142) = crossdifference of every pair of points on line {7813, 14417}
X(52142) = barycentric product X(i)*X(j) for these {i,j}: {6, 14246}, {23, 111}, {110, 10561}, {316, 32740}, {671, 18374}, {691, 2492}, {895, 8744}, {923, 16568}, {6593, 10630}, {7664, 41936}, {8753, 22151}, {9979, 32729}, {10317, 17983}, {10415, 36415}, {10555, 23357}, {14908, 37765}, {19626, 40074}, {20410, 41511}
X(52142) = barycentric quotient X(i)/X(j) for these {i,j}: {23, 3266}, {32, 14357}, {111, 18019}, {2492, 35522}, {6593, 36792}, {8744, 44146}, {8753, 46105}, {9517, 45807}, {10317, 6390}, {10555, 23962}, {10561, 850}, {14246, 76}, {14908, 34897}, {18374, 524}, {19626, 3455}, {32729, 17708}, {32740, 67}, {36415, 7664}, {41936, 10415}, {42659, 14417}
X(52142) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {184, 51980, 10558}, {10558, 32729, 184}

X(52143) = X(1)X(1437)∩X(6)X(25)

X(52143) lies on the cubic K430 and these lines: {1, 1437}, {6, 25}, {21, 1798}, {28, 44545}, {31, 18611}, {58, 3556}, {81, 3827}, {110, 1812}, {197, 22132}, {221, 1408}, {478, 17408}, {692, 5285}, {970, 10282}, {1333, 1402}, {1503, 6703}, {1801, 37577}, {2193, 3185}, {2328, 7169}, {3560, 6759}, {5743, 10192}, {5799, 7511}, {14555, 35260}, {16049, 41600}, {26884, 40959}, {32734, 34858}, {37547, 41608}, {40571, 41607}

X(52143) = circumcircle-of-inner-mixtilinear-triangle-inverse of X(45284)
X(52143) = isogonal conjugate of the isotomic conjugate of X(16049)
X(52143) = isogonal conjugate of the polar conjugate of X(41364)
X(52143) = X(i)-Ceva conjugate of X(j) for these (i,j): {21, 1333}, {1798, 6}
X(52143) = X(i)-isoconjugate of X(j) for these (i,j): {10, 8048}, {75, 43703}, {226, 34277}, {307, 43742}, {313, 3435}, {321, 42467}, {1577, 46640}, {14208, 40097}, {18697, 40454}
X(52143) = X(i)-Dao conjugate of X(j) for these (i,j): {56, 1441}, {123, 850}, {206, 43703}
X(52143) = crosspoint of X(16049) and X(41364)
X(52143) = crosssum of X(3910) and X(34588)
X(52143) = barycentric product X(i)*X(j) for these {i,j}: {3, 41364}, {6, 16049}, {21, 478}, {28, 22132}, {58, 1766}, {81, 197}, {86, 205}, {110, 6588}, {163, 21186}, {284, 21147}, {849, 21074}, {1169, 41600}, {1333, 3436}, {1812, 17408}, {2193, 14257}, {2206, 20928}, {41601, 41890}
X(52143) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 43703}, {197, 321}, {205, 10}, {478, 1441}, {1333, 8048}, {1576, 46640}, {1766, 313}, {2194, 34277}, {2204, 43742}, {2206, 42467}, {3436, 27801}, {6588, 850}, {16049, 76}, {17408, 40149}, {21147, 349}, {21186, 20948}, {22132, 20336}, {41364, 264}, {41600, 1228}

X(52144) = X(3)X(49)∩X(32)X(51)

X(52144) lies on the cubics K781 and K786 and these lines: {2, 13335}, {3, 49}, {6, 44200}, {22, 5171}, {23, 14669}, {25, 1611}, {32, 51}, {39, 13366}, {50, 2393}, {76, 35926}, {98, 401}, {110, 23700}, {112, 34854}, {125, 441}, {154, 5023}, {157, 571}, {186, 47213}, {187, 237}, {206, 8553}, {216, 14575}, {228, 20781}, {230, 460}, {248, 8779}, {263, 41413}, {297, 47200}, {418, 682}, {419, 21445}, {468, 14900}, {493, 39648}, {494, 39679}, {511, 1976}, {524, 51611}, {542, 22463}, {574, 44109}, {577, 6467}, {1194, 34870}, {1351, 10608}, {1384, 34417}, {1516, 1531}, {1576, 3003}, {1609, 1974}, {1624, 44896}, {1632, 44375}, {1692, 51335}, {1899, 37188}, {1993, 9737}, {2021, 14602}, {2080, 6660}, {2450, 2794}, {2965, 9969}, {3135, 15270}, {3155, 8854}, {3156, 8855}, {3284, 20975}, {3506, 35375}, {3785, 43653}, {4226, 14265}, {4558, 8681}, {4563, 19599}, {5007, 34565}, {5008, 44107}, {5012, 13334}, {5013, 11402}, {5041, 34566}, {5063, 40673}, {5188, 46546}, {5206, 41275}, {5210, 15504}, {5467, 32127}, {5651, 37344}, {5943, 37335}, {6457, 21655}, {6458, 21656}, {7473, 47207}, {7494, 37804}, {7499, 21248}, {7772, 44111}, {8754, 16310}, {9723, 52016}, {10282, 37114}, {10317, 34982}, {10547, 51477}, {10607, 19588}, {10991, 47526}, {11063, 18374}, {11257, 51350}, {12042, 21531}, {13851, 44231}, {14060, 22151}, {14826, 32973}, {14908, 32662}, {15004, 30435}, {15080, 34095}, {15391, 17970}, {15513, 44108}, {15815, 17809}, {17810, 22331}, {19125, 36751}, {19161, 37813}, {19358, 30398}, {19359, 30399}, {19459, 36748}, {19780, 44122}, {19781, 44083}, {20775, 22052}, {20777, 22055}, {22056, 22389}, {22401, 23606}, {26926, 34828}, {32152, 41237}, {35007, 44106}, {35282, 44887}, {35941, 39646}, {38608, 44334}, {40318, 45199}, {40320, 44079}, {41266, 44082}, {41336, 44084}, {44221, 51393}, {45921, 46184}

X(52144) = midpoint of X(50) and X(7669)
X(52144) = reflection of X(i) in X(j) for these {i,j}: {8754, 16310}, {45921, 46184}
X(52144) = isogonal conjugate of X(35142)
X(52144) = isogonal conjugate of the anticomplement of X(35067)
X(52144) = isogonal conjugate of the isotomic conjugate of X(3564)
X(52144) = isotomic conjugate of the polar conjugate of X(1692)
X(52144) = isogonal conjugate of the polar conjugate of X(230)
X(52144) = X(i)-Ceva conjugate of X(j) for these (i,j): {3, 47406}, {230, 1692}, {511, 8779}, {1976, 184}, {3563, 6}, {39644, 6467}, {43754, 647}, {51776, 22391}
X(52144) = X(i)-isoconjugate of X(j) for these (i,j): {1, 35142}, {4, 8773}, {19, 8781}, {75, 3563}, {92, 2987}, {158, 43705}, {240, 40428}, {264, 36051}, {523, 36105}, {811, 35364}, {1577, 32697}, {1581, 47736}, {1969, 32654}, {2065, 40703}, {10425, 24006}, {36119, 36891}, {36120, 52091}
X(52144) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 35142}, {4, 39072}, {6, 8781}, {76, 35067}, {92, 39069}, {114, 264}, {206, 3563}, {230, 44132}, {290, 34156}, {523, 39001}, {1147, 43705}, {1511, 36891}, {2987, 22391}, {3267, 41181}, {8773, 36033}, {17423, 35364}, {19576, 47736}, {39085, 40428}, {46094, 52091}
X(52144) = crosspoint of X(i) and X(j) for these (i,j): {3, 248}, {6, 3563}, {69, 51454}, {230, 3564}, {1976, 51820}, {2966, 47389}
X(52144) = crosssum of X(i) and X(j) for these (i,j): {2, 3564}, {4, 297}, {25, 41363}, {325, 52091}, {525, 868}, {2971, 3569}, {2987, 3563}
X(52144) = crossdifference of every pair of points on line {2, 2501}
X(52144) = X(13335)-lineconjugate of X(2)
X(52144) = barycentric product X(i)*X(j) for these {i,j}: {3, 230}, {6, 3564}, {48, 1733}, {63, 8772}, {69, 1692}, {98, 47406}, {114, 248}, {184, 51481}, {287, 51335}, {293, 17462}, {394, 460}, {577, 44145}, {647, 4226}, {895, 5477}, {2974, 32654}, {3284, 36875}, {3289, 14265}, {3563, 35067}, {3926, 44099}, {4563, 42663}, {6782, 36296}, {6783, 36297}, {12829, 36214}, {14919, 51431}, {36212, 51820}, {39072, 51454}
X(52144) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 8781}, {6, 35142}, {32, 3563}, {48, 8773}, {114, 44132}, {163, 36105}, {184, 2987}, {230, 264}, {248, 40428}, {460, 2052}, {577, 43705}, {1576, 32697}, {1691, 47736}, {1692, 4}, {1733, 1969}, {3049, 35364}, {3284, 36891}, {3289, 52091}, {3564, 76}, {4226, 6331}, {5477, 44146}, {8772, 92}, {9247, 36051}, {12829, 17984}, {14575, 32654}, {14585, 42065}, {14600, 2065}, {17462, 40703}, {32661, 10425}, {42663, 2501}, {44099, 393}, {44145, 18027}, {47406, 325}, {51335, 297}, {51431, 46106}, {51481, 18022}, {51820, 16081}
X(52144) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 3148, 51}, {39, 34396, 13366}, {157, 571, 1843}, {187, 5191, 1495}, {187, 42671, 237}, {216, 14575, 21637}, {237, 5191, 42671}, {237, 8623, 47638}, {237, 42671, 1495}, {577, 40947, 6467}, {1576, 3003, 44102}, {3053, 39653, 40319}, {3284, 20975, 21639}, {10132, 10133, 3167}, {20975, 23200, 3284}, {34396, 37457, 39}, {47195, 51458, 3292}

X(52145) = X(2)X(647)∩X(3)X(76)

X(52145) lies on the cubic K657 and these lines: {2, 647}, {3, 76}, {69, 15631}, {140, 51257}, {141, 51404}, {187, 4235}, {262, 46142}, {264, 842}, {287, 15066}, {305, 31614}, {385, 14966}, {879, 3111}, {1352, 51943}, {1821, 37210}, {2715, 2868}, {2966, 3972}, {3266, 3292}, {3734, 48452}, {5092, 35912}, {6785, 30226}, {7792, 44345}, {7804, 35906}, {8024, 51820}, {9154, 39296}, {9170, 11059}, {9717, 34336}, {11185, 34175}, {13137, 18906}, {14355, 17932}, {14608, 52038}, {16081, 21448}, {17984, 38227}, {26233, 47635}, {35522, 52094}

X(52145) = isogonal conjugate of X(51980)
X(52145) = isotomic conjugate of X(5968)
X(52145) = isotomic conjugate of the isogonal conjugate of X(5967)
X(52145) = X(9155)-cross conjugate of X(524)
X(52145) = X(i)-isoconjugate of X(j) for these (i,j): {1, 51980}, {31, 5968}, {111, 1755}, {163, 8430}, {232, 36060}, {237, 897}, {240, 14908}, {511, 923}, {671, 9417}, {1959, 32740}, {2491, 36085}, {3289, 36128}, {3405, 41272}, {3569, 36142}, {5547, 51653}, {9154, 42075}, {9178, 23997}, {9418, 46277}, {14966, 23894}, {19626, 46238}
X(52145) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 5968}, {3, 51980}, {111, 36899}, {115, 8430}, {232, 1560}, {237, 6593}, {511, 2482}, {524, 9155}, {671, 39058}, {1649, 44114}, {2491, 38988}, {3569, 23992}, {14908, 39085}, {17994, 48317}
X(52145) = cevapoint of X(524) and X(9155)
X(52145) = trilinear pole of line {524, 35522}
X(52145) = crossdifference of every pair of points on line {237, 2491}
X(52145) = barycentric product X(i)*X(j) for these {i,j}: {76, 5967}, {98, 3266}, {187, 18024}, {287, 44146}, {290, 524}, {670, 52038}, {685, 45807}, {690, 43187}, {896, 46273}, {1821, 14210}, {2966, 35522}, {5468, 43665}, {6390, 16081}, {6394, 37778}, {9154, 36792}, {14417, 22456}, {34536, 50567}, {46786, 52094}
X(52145) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 5968}, {6, 51980}, {98, 111}, {187, 237}, {248, 14908}, {287, 895}, {290, 671}, {293, 36060}, {351, 2491}, {468, 232}, {523, 8430}, {524, 511}, {690, 3569}, {879, 10097}, {896, 1755}, {922, 9417}, {1648, 44114}, {1821, 897}, {1910, 923}, {1976, 32740}, {2395, 9178}, {2482, 9155}, {2715, 32729}, {2966, 691}, {3266, 325}, {3292, 3289}, {4235, 4230}, {5026, 36213}, {5467, 14966}, {5468, 2421}, {5477, 51335}, {5967, 6}, {6390, 36212}, {6531, 8753}, {6629, 17209}, {7181, 43034}, {9154, 10630}, {9155, 11672}, {14210, 1959}, {14273, 17994}, {14417, 684}, {14567, 9418}, {14601, 19626}, {15628, 5547}, {16081, 17983}, {16741, 51369}, {18024, 18023}, {18311, 33752}, {18872, 14251}, {20021, 46154}, {21839, 5360}, {23889, 23997}, {34536, 9154}, {35282, 9475}, {35522, 2799}, {36036, 36085}, {36084, 36142}, {36120, 36128}, {36792, 50567}, {36822, 14609}, {36874, 14263}, {36890, 35910}, {37778, 6530}, {37858, 46783}, {42721, 42717}, {42760, 42751}, {43084, 14356}, {43187, 892}, {43665, 5466}, {44102, 2211}, {44146, 297}, {45672, 6786}, {45807, 6333}, {46273, 46277}, {46786, 16092}, {50567, 36790}, {50942, 23350}, {51655, 51653}, {51869, 41272}, {52038, 512}, {52076, 10561}, {52094, 46787}
X(52145) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 14382, 14265}, {98, 51259, 14265}

X(52146) = X(1)X(85)∩X(43)X(294)

X(52146) lies on the cubic K1026 and these lines: {1, 85}, {43, 294}, {87, 572}, {105, 21214}, {666, 3507}, {673, 1740}, {1416, 37608}, {1581, 2664}, {1716, 2195}, {1777, 37576}, {4335, 28071}, {6180, 37580}, {8926, 18793}, {8931, 16360}, {13576, 15971}, {16569, 31638}, {18758, 28391}, {18787, 43747}, {18792, 23605}, {19557, 35338}

X(52146) = X(39919)-cross conjugate of X(1)
X(52146) = X(i)-isoconjugate of X(j) for these (i,j): {1458, 39924}, {2223, 18299}
X(52146) = cevapoint of X(8844) and X(17792)
X(52146) = barycentric product X(i)*X(j) for these {i,j}: {105, 17760}, {673, 17792}, {1438, 51861}, {14942, 28391}, {18031, 18758}, {36796, 41350}, {45902, 51560}
X(52146) = barycentric quotient X(i)/X(j) for these {i,j}: {294, 39924}, {673, 18299}, {8844, 17755}, {17760, 3263}, {17792, 3912}, {18758, 672}, {28391, 9436}, {41350, 241}, {45902, 2254}

X(52147) = X(2)X(216)∩X(3)X(107)

X(52147) lies on the cubic K657 and these lines: {2, 216}, {3, 107}, {4, 373}, {76, 6331}, {92, 142}, {141, 51358}, {182, 450}, {275, 10601}, {277, 16082}, {297, 7790}, {340, 37644}, {376, 47392}, {436, 43650}, {443, 1896}, {468, 43976}, {631, 1093}, {648, 15066}, {852, 42329}, {1075, 11793}, {1304, 6795}, {1352, 51939}, {1629, 5020}, {1947, 3305}, {1948, 3306}, {3066, 37200}, {3168, 3917}, {3525, 13450}, {3537, 36876}, {3819, 51877}, {4240, 15080}, {4359, 7017}, {4993, 43752}, {5067, 44732}, {5085, 37070}, {5651, 41204}, {6335, 17740}, {6530, 30739}, {6820, 18928}, {7480, 9159}, {7998, 35360}, {11059, 44132}, {11064, 14615}, {11284, 33971}, {11433, 32001}, {14389, 34289}, {16081, 21448}, {17861, 24177}, {17924, 42764}, {18026, 31018}, {18027, 46328}, {20207, 26906}, {22112, 37124}, {22712, 47202}, {34417, 35474}, {37514, 51031}, {37643, 40814}, {40138, 44133}, {41760, 47296}, {42298, 46111}

X(52147) = isogonal conjugate of X(51990)
X(52147) = polar conjugate of X(3426)
X(52147) = isotomic conjugate of the isogonal conjugate of X(40138)
X(52147) = isotomic conjugate of the polar conjugate of X(47392)
X(52147) = polar conjugate of the isotomic conjugate of X(44133)
X(52147) = polar conjugate of the isogonal conjugate of X(376)
X(52147) = X(i)-cross conjugate of X(j) for these (i,j): {376, 44133}, {40138, 47392}
X(52147) = X(i)-isoconjugate of X(j) for these (i,j): {1, 51990}, {48, 3426}, {822, 9064}, {9247, 36889}
X(52147) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 51990}, {6, 16253}, {381, 5158}, {1249, 3426}, {10606, 40138}
X(52147) = cevapoint of X(376) and X(40138)
X(52147) = trilinear pole of line {1515, 9007}
X(52147) = barycentric product X(i)*X(j) for these {i,j}: {4, 44133}, {69, 47392}, {76, 40138}, {264, 376}, {6331, 9209}, {6528, 9007}, {18022, 26864}, {39263, 44134}
X(52147) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 3426}, {6, 51990}, {107, 9064}, {264, 36889}, {376, 3}, {1515, 6000}, {9007, 520}, {9209, 647}, {16253, 10606}, {26864, 184}, {39263, 4846}, {40138, 6}, {40348, 2351}, {44133, 69}, {47392, 4}
X(52147) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3164, 44436}, {2, 15466, 2052}, {2, 17907, 14165}, {2, 46106, 264}, {2, 46717, 46831}, {264, 15466, 46106}, {264, 46106, 2052}, {297, 37648, 43462}, {2052, 14165, 21447}

X(52148) = X(2)X(104)∩X(9)X(36)

X(52148) lies on the Feuerbach circumhyperbola of the medial triangle, the cubic K1265, and these lines: {1, 8668}, {2, 104}, {3, 392}, {9, 36}, {10, 56}, {35, 45036}, {55, 214}, {100, 1000}, {142, 20270}, {404, 5657}, {405, 5450}, {442, 499}, {443, 5253}, {574, 6184}, {856, 18642}, {958, 13747}, {960, 40293}, {993, 5316}, {997, 1470}, {999, 3306}, {1001, 10058}, {1125, 22768}, {1145, 1376}, {1466, 5730}, {1480, 35281}, {1512, 6911}, {2256, 51574}, {2975, 17567}, {3295, 51577}, {3304, 5883}, {3428, 16371}, {3647, 5204}, {3754, 18967}, {3877, 35238}, {4187, 12114}, {4188, 35239}, {4193, 18761}, {4423, 51569}, {4512, 46947}, {5193, 9623}, {5440, 6600}, {5563, 11530}, {5584, 19537}, {5687, 12640}, {7280, 51576}, {7987, 37284}, {8071, 11517}, {8583, 37561}, {10039, 10094}, {10200, 22760}, {10320, 25466}, {10914, 15347}, {10966, 25440}, {11012, 12120}, {11112, 22753}, {11344, 13624}, {11509, 30144}, {14793, 35204}, {16187, 38604}, {16203, 19860}, {16370, 43182}, {16408, 37535}, {16410, 51572}, {16417, 22765}, {17057, 31190}, {17502, 20835}, {21669, 26129}, {22759, 26364}, {26286, 37282}, {32612, 37248}, {36058, 47041}, {38902, 39048}

X(52148) = midpoint of X(7284) and X(52050)
X(52148) = complement of X(30513)
X(52148) = complement of the isogonal conjugate of X(1470)
X(52148) = X(i)-complementary conjugate of X(j) for these (i,j): {73, 30445}, {109, 9001}, {603, 1060}, {604, 17720}, {997, 1329}, {1470, 10}, {4227, 34831}, {9001, 124}, {11383, 20262}, {17740, 21244}, {26637, 21246}
X(52148) = X(100)-Ceva conjugate of X(9001)
X(52148) = X(998)-isoconjugate of X(1000)
X(52148) = crossdifference of every pair of points on line {26275, 40134}
X(52148) = barycentric product X(i)*X(j) for these {i,j}: {997, 3306}, {999, 17740}, {1470, 28808}, {3753, 26637}

X(52149) = X(2)X(6)∩X(30)X(1272)

X(52149) lies on the cubic K504 and these lines: {2, 6}, {30, 1272}, {114, 41721}, {186, 340}, {264, 48913}, {316, 35520}, {317, 18354}, {328, 1494}, {381, 44135}, {1989, 34827}, {2072, 40996}, {6390, 44265}, {7550, 7768}, {7577, 44134}, {7752, 49674}, {7811, 35921}, {9723, 18324}, {14570, 40885}, {18420, 32836}, {36212, 45312}, {40662, 40705}, {41008, 45799}, {47282, 51872}

X(52149) = reflection of X(i) in X(j) for these {i,j}: {1989, 34827}, {41626, 2}
X(52149) = isotomic conjugate of X(18316)
X(52149) = reflection of X(1272) in the De Longchamps axis
X(52149) = isotomic conjugate of the isogonal conjugate of X(3581)
X(52149) = X(31)-isoconjugate of X(18316)
X(52149) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 18316}, {3284, 51545}, {3431, 40604}
X(52149) = barycentric product X(i)*X(j) for these {i,j}: {76, 3581}, {323, 44135}, {340, 37638}, {381, 7799}, {1273, 4993}, {6148, 46808}, {6331, 14314}
X(52149) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 18316}, {323, 3431}, {340, 43530}, {381, 1989}, {1511, 51545}, {3581, 6}, {4993, 1141}, {6148, 46809}, {14165, 16263}, {14314, 647}, {18487, 14583}, {34417, 11060}, {37638, 265}, {44135, 94}, {46808, 5627}, {51544, 40355}
X(52149) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {298, 299, 323}, {340, 7799, 6148}, {1273, 6148, 7799}, {1494, 7809, 3260}

X(52150) = X(1)X(859)∩X(3)X(2051)

X(52150) lies on the conic {{A,B,C,X(1),X(6)}}, the cubics K430 and K1057, and these lines: {1, 859}, {3, 2051}, {6, 10457}, {21, 1220}, {28, 36123}, {36, 39949}, {58, 9563}, {86, 4225}, {87, 3286}, {333, 1222}, {958, 996}, {1027, 16695}, {1126, 4281}, {1468, 40496}, {2983, 4269}, {4216, 34262}, {4245, 50605}, {5737, 28383}, {7428, 39748}, {13738, 37674}, {16948, 37129}, {17588, 23375}, {41245, 46502}

X(52150) = isogonal conjugate of X(17751)
X(52150) = isogonal conjugate of the anticomplement of X(1193)
X(52150) = isogonal conjugate of the complement of X(20040)
X(52150) = isogonal conjugate of the isotomic conjugate of X(20028)
X(52150) = X(40453)-Ceva conjugate of X(34434)
X(52150) = X(i)-cross conjugate of X(j) for these (i,j): {1402, 1333}, {18191, 3733}, {20228, 81}
X(52150) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17751}, {2, 21061}, {8, 37558}, {10, 2975}, {37, 14829}, {72, 11109}, {86, 14973}, {313, 20986}, {321, 572}, {1018, 17496}, {1043, 20617}, {2321, 17074}, {3699, 51664}, {3952, 21173}, {21014, 31629}, {30710, 52087}
X(52150) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 17751}, {14829, 40589}, {14973, 40600}, {21061, 32664}
X(52150) = crosssum of X(10) and X(22299)
X(52150) = barycentric product X(i)*X(j) for these {i,j}: {6, 20028}, {56, 46880}, {58, 2051}, {81, 34434}, {593, 51870}, {3666, 40453}
X(52150) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 17751}, {31, 21061}, {58, 14829}, {213, 14973}, {604, 37558}, {1333, 2975}, {1408, 17074}, {1474, 11109}, {2051, 313}, {2206, 572}, {3733, 17496}, {16695, 27346}, {18191, 40624}, {20028, 76}, {34434, 321}, {40453, 30710}, {46880, 3596}, {51870, 28654}
X(52150) = {X(859),X(4267)}-harmonic conjugate of X(23361)

X(52151) = X(1)X(668)∩X(75)X(141)

X(52151) lies on the cubic K766 and these lines: {1, 668}, {2, 40789}, {69, 7155}, {75, 141}, {76, 4485}, {312, 20644}, {319, 25048}, {350, 6542}, {537, 4505}, {740, 24731}, {1111, 33935}, {2112, 3570}, {2481, 4479}, {3122, 5224}, {3912, 20947}, {3948, 10026}, {4010, 20538}, {4357, 21100}, {4986, 28611}, {6381, 49764}, {16284, 20943}, {17230, 33931}, {17250, 41683}, {17308, 30866}, {18052, 51863}, {18133, 24732}, {18149, 25748}, {18720, 18747}, {18795, 40874}, {29674, 33938}, {39997, 41314}

X(52151) = reflection of X(40848) in X(20532)
X(52151) = isotomic conjugate of the isogonal conjugate of X(2108)
X(52151) = X(i)-Ceva conjugate of X(j) for these (i,j): {350, 75}, {6542, 17762}, {52043, 6376}
X(52151) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2109}, {8632, 39420}
X(52151) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 2109}, {291, 335}
X(52151) = barycentric product X(i)*X(j) for these {i,j}: {75, 33888}, {76, 2108}, {334, 27920}, {668, 25381}, {726, 33679}, {1969, 20797}
X(52151) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2109}, {660, 39420}, {2108, 6}, {20797, 48}, {25381, 513}, {27920, 238}, {33679, 3226}, {33888, 1}
X(52151) = {X(76),X(49560)}-harmonic conjugate of X(17762)

X(52152) = X(2)X(523)∩X(3)X(111)

X(52152) lies on the cubic K657 and these lines: {2, 523}, {3, 111}, {6, 32583}, {76, 30786}, {373, 51980}, {381, 34320}, {691, 1995}, {895, 15066}, {1352, 51938}, {3291, 9177}, {5422, 10559}, {5467, 11580}, {5640, 36827}, {8585, 17964}, {8869, 26206}, {9176, 46589}, {9745, 32447}, {10558, 10601}, {10560, 15018}, {11284, 15899}, {15398, 40916}, {18911, 51405}, {30739, 51258}, {32216, 40727}, {32729, 35259}

X(52152) = reflection of X(46589) in X(9176)
X(52152) = X(896)-isoconjugate of X(9084)
X(52152) = X(9084)-Dao conjugate of X(15899)
X(52152) = crossdifference of every pair of points on line {187, 9125}
X(52152) = barycentric product X(671)*X(9027)
X(52152) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 9084}, {9027, 524}
X(52152) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 46783, 5968}, {21448, 45143, 111}

X(52153) = X(3)X(125)∩X(23)X(94)

X(52153) lies on the cubic K1170 and these lines: {3, 125}, {5, 47201}, {13, 1605}, {14, 1606}, {23, 94}, {24, 47204}, {25, 1989}, {32, 3124}, {51, 1576}, {74, 2132}, {107, 1141}, {110, 18114}, {111, 23969}, {184, 5158}, {186, 5627}, {228, 8606}, {237, 3455}, {328, 1799}, {418, 51477}, {468, 43090}, {686, 9409}, {759, 3437}, {878, 14582}, {1402, 6186}, {1495, 3003}, {1637, 14270}, {1995, 3425}, {2070, 14993}, {2353, 20897}, {3292, 42065}, {5201, 46155}, {5899, 51345}, {6130, 10412}, {6321, 50377}, {6660, 14830}, {7527, 18300}, {7575, 11657}, {7668, 36178}, {11063, 34448}, {11078, 34009}, {11092, 34008}, {11145, 41460}, {11146, 41459}, {12106, 14254}, {12192, 15080}, {14002, 30529}, {14592, 47205}, {14652, 34308}, {14687, 32235}, {14703, 14847}, {14865, 23956}, {14908, 32662}, {14919, 16186}, {15329, 34218}, {16230, 43088}, {18576, 31861}, {20999, 26700}, {21650, 50467}, {21663, 50464}, {23716, 38431}, {23717, 38432}, {32305, 46602}, {33581, 44082}, {34449, 37440}, {36829, 40604}, {37924, 52056}, {38413, 47482}, {38414, 47481}, {39380, 39381}, {39477, 42736}, {40102, 43084}, {41533, 44534}, {43082, 47203}, {43089, 47202}, {47004, 51479}

X(52153) = isogonal conjugate of X(340)
X(52153) = circumcircle-inverse of X(34310)
X(52153) = Dao-Moses-Telv circle-inverse of X(15550)
X(52153) = isogonal conjugate of the anticomplement of X(3284)
X(52153) = isogonal conjugate of the isotomic conjugate of X(265)
X(52153) = isotomic conjugate of the polar conjugate of X(11060)
X(52153) = isogonal conjugate of the polar conjugate of X(1989)
X(52153) = polar conjugate of the isotomic conjugate of X(50433)
X(52153) = X(i)-Ceva conjugate of X(j) for these (i,j): {265, 50433}, {476, 14582}, {1141, 1989}, {1989, 11060}, {5627, 6}
X(52153) = X(i)-isoconjugate of X(j) for these (i,j): {1, 340}, {19, 7799}, {27, 42701}, {50, 1969}, {63, 14165}, {75, 186}, {92, 323}, {95, 51801}, {162, 3268}, {264, 6149}, {276, 2290}, {319, 1870}, {320, 6198}, {526, 811}, {561, 34397}, {648, 32679}, {662, 44427}, {799, 47230}, {823, 8552}, {1154, 40440}, {1273, 2190}, {1442, 5081}, {1494, 35201}, {1577, 14590}, {1748, 37802}, {2088, 46254}, {2167, 14918}, {2349, 14920}, {2624, 6331}, {3219, 17923}, {4242, 4467}, {4511, 7282}, {5962, 44179}, {6148, 36119}, {10411, 24006}, {11107, 41804}, {14355, 40703}, {14591, 20948}, {14975, 40075}, {16186, 23999}, {17515, 40999}, {24019, 45792}, {24041, 35235}, {33805, 39176}, {36120, 51383}
X(52153) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 340}, {4, 15295}, {5, 1273}, {6, 7799}, {125, 3268}, {186, 206}, {264, 14993}, {323, 22391}, {526, 17423}, {562, 46604}, {1084, 44427}, {1511, 6148}, {3005, 35235}, {3162, 14165}, {3260, 39170}, {5962, 37864}, {14918, 40588}, {15450, 41078}, {34397, 40368}, {35071, 45792}, {38996, 47230}, {46094, 51383}
X(52153) = cevapoint of X(i) and X(j) for these (i,j): {51, 1495}, {3129, 3130}, {9409, 20975}
X(52153) = crosspoint of X(i) and X(j) for these (i,j): {6, 35372}, {74, 14910}, {265, 1989}, {1141, 11077}
X(52153) = crosssum of X(i) and X(j) for these (i,j): {2, 12383}, {30, 3580}, {186, 323}, {1154, 14918}, {35235, 44427}, {40631, 46064}
X(52153) = trilinear pole of line {217, 3049}
X(52153) = crossdifference of every pair of points on line {3268, 5664}
X(52153) = barycentric product X(i)*X(j) for these {i,j}: {3, 1989}, {4, 50433}, {5, 11077}, {6, 265}, {13, 36297}, {14, 36296}, {30, 11079}, {32, 328}, {48, 2166}, {53, 50463}, {69, 11060}, {94, 184}, {110, 14582}, {112, 43083}, {216, 1141}, {217, 46138}, {248, 14356}, {323, 14595}, {394, 18384}, {476, 647}, {523, 32662}, {525, 14560}, {577, 6344}, {656, 32678}, {661, 36061}, {810, 32680}, {822, 36129}, {906, 43082}, {1304, 18558}, {1576, 14592}, {1807, 2160}, {1990, 50464}, {2006, 8606}, {2161, 7100}, {2165, 5961}, {2351, 18883}, {2437, 14220}, {3003, 12028}, {3049, 35139}, {3284, 5627}, {3457, 40710}, {3458, 40709}, {4558, 15475}, {5158, 18316}, {8749, 51254}, {9409, 39290}, {10097, 14559}, {10217, 11086}, {10218, 11081}, {10412, 32661}, {11063, 15392}, {11064, 40355}, {11070, 50467}, {11071, 50461}, {11074, 20123}, {11075, 50462}, {11080, 50466}, {11082, 50469}, {11085, 50465}, {11087, 50468}, {14254, 18877}, {14380, 41392}, {14575, 20573}, {14583, 14919}, {14585, 18817}, {14908, 43084}, {14910, 39170}, {16186, 23588}, {18479, 43530}, {18557, 32715}, {20578, 38413}, {20579, 38414}, {20975, 39295}, {21354, 34304}, {23968, 35909}, {30529, 51477}, {32663, 34209}, {36298, 39377}, {36299, 39378}, {39201, 46456}
X(52153) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 7799}, {6, 340}, {25, 14165}, {32, 186}, {51, 14918}, {94, 18022}, {184, 323}, {216, 1273}, {217, 1154}, {228, 42701}, {265, 76}, {328, 1502}, {476, 6331}, {512, 44427}, {520, 45792}, {647, 3268}, {669, 47230}, {810, 32679}, {1141, 276}, {1495, 14920}, {1501, 34397}, {1576, 14590}, {1807, 33939}, {1989, 264}, {2166, 1969}, {2179, 51801}, {2351, 37802}, {3049, 526}, {3124, 35235}, {3284, 6148}, {3289, 51383}, {3457, 471}, {3458, 470}, {5961, 7763}, {6186, 17923}, {6344, 18027}, {7100, 20924}, {8606, 32851}, {9247, 6149}, {9406, 35201}, {9407, 39176}, {9409, 5664}, {11060, 4}, {11077, 95}, {11079, 1494}, {11084, 38428}, {11089, 38427}, {12028, 40832}, {14356, 44132}, {14560, 648}, {14574, 14591}, {14575, 50}, {14582, 850}, {14583, 46106}, {14585, 22115}, {14592, 44173}, {14595, 94}, {14600, 14355}, {15451, 41078}, {15475, 14618}, {16186, 23965}, {18384, 2052}, {18479, 37638}, {19627, 3043}, {20573, 44161}, {32661, 10411}, {32662, 99}, {32678, 811}, {36061, 799}, {36296, 299}, {36297, 298}, {39201, 8552}, {40355, 16080}, {40373, 19627}, {40981, 11062}, {43083, 3267}, {50433, 69}, {50463, 34386}, {50465, 11128}, {50466, 11129}, {50468, 11132}, {50469, 11133}
X(52153) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {265, 34310, 125}, {265, 39170, 18478}, {3457, 3458, 11060}, {11141, 11142, 1989}, {13558, 34310, 5961}, {36296, 36297, 50433}

X(52154) = X(3)X(14579)∩X(6)X(5055)

X(52154) lies on the conic {{A,B,C,X(1),X(6)}}, the cubic K936, and these lines: {3, 14579}, {6, 5055}, {25, 11063}, {50, 18362}, {111, 37637}, {230, 1383}, {251, 33886}, {393, 16328}, {588, 13846}, {589, 13847}, {599, 2987}, {1157, 8882}, {1989, 7746}, {2981, 16644}, {3054, 40103}, {3108, 7571}, {3470, 8749}, {5306, 39955}, {5309, 30537}, {6151, 16645}, {6749, 7577}, {7884, 39968}, {8553, 37956}, {13881, 14910}, {21358, 40802}, {30535, 47352}, {31489, 39389}, {37909, 47275}

X(52154) = isogonal conjugate of X(11004)
X(52154) = X(i)-cross conjugate of X(j) for these (i,j): {566, 6}, {44109, 4}
X(52154) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11004}, {63, 47485}, {92, 9703}
X(52154) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 11004}, {3162, 47485}, {9703, 22391}
X(52154) = cevapoint of X(3124) and X(18117)
X(52154) = barycentric product X(549)*X(19307)
X(52154) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 11004}, {25, 47485}, {184, 9703}

X(52155) = X(1)X(672)∩X(2)X(165)

X(52155) lies on on the Kiepert circumhyperbola of the excentral triangle, the cubic K078, and these lines: {1, 672}, {2, 165}, {3, 170}, {9, 1282}, {20, 34848}, {21, 8931}, {42, 16469}, {43, 55}, {57, 846}, {105, 2108}, {513, 24708}, {910, 15254}, {1001, 17754}, {1009, 2944}, {1011, 2947}, {1051, 9052}, {1054, 39954}, {1125, 17753}, {1738, 31200}, {2140, 3624}, {2238, 15601}, {2276, 7290}, {2280, 16468}, {2808, 3576}, {2954, 37319}, {3136, 41858}, {3523, 43158}, {3601, 39789}, {3685, 17026}, {3720, 10980}, {3741, 5273}, {3799, 23407}, {4209, 48925}, {4213, 5338}, {4312, 30949}, {4335, 27626}, {4375, 48547}, {4428, 38048}, {5537, 16373}, {5698, 20335}, {6822, 26040}, {7965, 47514}, {7987, 8835}, {9324, 35445}, {10268, 36670}, {16569, 31508}, {17794, 25728}, {21214, 32524}, {24047, 52015}, {29814, 30350}, {30946, 51090}

X(52155) = reflection of X(i) in X(j) for these {i,j}: {170, 47641}, {47641, 3}
X(52155) = X(i)-Ceva conjugate of X(j) for these (i,j): {1001, 1}, {17754, 43}, {42316, 165}
X(52155) = barycentric product X(1)*X(27484)
X(52155) = barycentric quotient X(27484)/X(75)
X(52155) = {X(2),X(35270)}-harmonic conjugate of X(165)

X(52156) = X(2)X(658)∩X(8)X(348)

X(52156) lies on the cubic K623 and these lines: {2, 658}, {7, 1360}, {8, 348}, {29, 1434}, {85, 23058}, {92, 1088}, {103, 516}, {189, 17093}, {279, 1146}, {312, 4554}, {333, 4573}, {1121, 17078}, {1311, 24016}, {1815, 18652}, {1861, 51364}, {1996, 50442}, {2338, 17095}, {2400, 4453}, {4626, 26932}, {7079, 7177}, {14004, 34398}, {14828, 36056}, {18359, 37780}, {18634, 23062}, {23618, 25019}, {34234, 37757}, {36605, 39351}

X(52156) = reflection of X(i) in X(j) for these {i,j}: {664, 3160}, {10405, 1146}
X(52156) = isotomic conjugate of X(40869)
X(52156) = antitomic image of X(10405)
X(52156) = isotomic conjugate of the complement of X(9436)
X(52156) = X(i)-cross conjugate of X(j) for these (i,j): {673, 35160}, {918, 664}, {26001, 75}, {36101, 18025}, {43035, 7}, {51364, 1088}
X(52156) = X(i)-isoconjugate of X(j) for these (i,j): {6, 41339}, {21, 51436}, {25, 51376}, {31, 40869}, {41, 516}, {55, 910}, {56, 51418}, {109, 46392}, {212, 1886}, {220, 1456}, {650, 2426}, {1253, 43035}, {1946, 41321}, {2175, 30807}, {2194, 17747}, {2195, 9502}, {2204, 51366}, {2212, 26006}, {2338, 42077}, {2398, 3063}, {9447, 35517}, {51435, 51858}
X(52156) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 51418}, {2, 40869}, {9, 41339}, {11, 46392}, {223, 910}, {516, 3160}, {676, 40615}, {1214, 17747}, {1886, 40837}, {2340, 45250}, {2398, 10001}, {6505, 51376}, {9502, 39063}, {17113, 43035}, {30807, 40593}, {36905, 50441}, {39053, 41321}, {40611, 51436}
X(52156) = cevapoint of X(i) and X(j) for these (i,j): {2, 9436}, {7, 43035}, {2254, 3119}, {36101, 43736}
X(52156) = trilinear pole of line {7, 522}
X(52156) = barycentric product X(i)*X(j) for these {i,j}: {7, 18025}, {75, 43736}, {85, 36101}, {103, 6063}, {331, 1815}, {664, 2400}, {911, 20567}, {2424, 4572}, {4998, 15634}, {7182, 36122}, {9503, 40704}, {24016, 35519}
X(52156) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 41339}, {2, 40869}, {7, 516}, {9, 51418}, {57, 910}, {63, 51376}, {85, 30807}, {103, 55}, {109, 2426}, {226, 17747}, {241, 9502}, {269, 1456}, {278, 1886}, {279, 43035}, {307, 51366}, {348, 26006}, {650, 46392}, {653, 41321}, {664, 2398}, {677, 3939}, {911, 41}, {1400, 51436}, {1434, 14953}, {1447, 51435}, {1456, 42077}, {1815, 219}, {2338, 220}, {2400, 522}, {2424, 663}, {3676, 676}, {3911, 51406}, {4554, 42719}, {4626, 23973}, {6063, 35517}, {9436, 50441}, {9503, 294}, {15634, 11}, {18025, 8}, {24016, 109}, {32668, 1415}, {36056, 212}, {36101, 9}, {36122, 33}, {36838, 24015}, {37787, 28345}, {43035, 23972}, {43736, 1}

X(52157) = X(2)X(38)∩X(7)X(190)

X(52157) lies on the cubic K623 and these lines: {2, 38}, {7, 190}, {8, 673}, {239, 49707}, {516, 3685}, {644, 1462}, {894, 4473}, {1086, 2345}, {3008, 4899}, {3662, 3729}, {3699, 26007}, {4000, 9055}, {4307, 4432}, {4318, 28982}, {4366, 17316}, {4370, 35578}, {4422, 5749}, {4648, 24358}, {4767, 31226}, {5222, 32029}, {5279, 16560}, {5308, 20131}, {5845, 21296}, {6542, 17765}, {6650, 29587}, {6651, 17244}, {6653, 17230}, {7155, 9442}, {10430, 19645}, {14621, 29569}, {17084, 18055}, {17089, 33946}, {17291, 24199}, {17302, 49528}, {17487, 41141}, {18037, 20917}, {19582, 28740}, {24231, 24821}, {26113, 26986}, {26582, 29611}, {28734, 28770}, {28739, 28778}, {28748, 28755}, {29573, 50289}, {29579, 41842}, {29600, 50307}, {29616, 41845}, {30566, 30857}, {30568, 30813}, {38186, 49499}, {38314, 39716}, {40538, 52030}, {40704, 40864}

X(52157) = reflection of X(i) in X(j) for these {i,j}: {4440, 4862}, {4488, 190}
X(52157) = X(9436)-Ceva conjugate of X(8)
X(52157) = X(6)-isoconjugate of X(9452)
X(52157) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 9452}, {6559, 14942}
X(52157) = barycentric product X(i)*X(j) for these {i,j}: {75, 9451}, {3263, 9453}
X(52157) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 9452}, {9451, 1}, {9453, 105}
X(52157) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {190, 36807, 16593}, {335, 17755, 24349}, {673, 4437, 8}, {3912, 17738, 20533}, {16593, 36807, 29627}

X(52158) = X(1)X(204)∩X(3)X(64)

X(52158) lies on the conic {{A,B,C,X(1),X(3)}}, the cubic K318, and these lines: {1, 204}, {3, 64}, {21, 77}, {28, 3345}, {29, 40960}, {40, 13618}, {58, 1433}, {78, 7070}, {102, 1301}, {219, 30457}, {253, 27402}, {282, 1172}, {283, 6061}, {332, 5931}, {405, 10380}, {459, 7498}, {1630, 2155}, {1780, 1795}, {2732, 46968}, {8747, 47441}, {26702, 36079}

X(52158) = isogonal conjugate of X(5930)
X(52158) = isogonal conjugate of the isotomic conjugate of X(5931)
X(52158) = X(i)-cross conjugate of X(j) for these (i,j): {56, 21}, {2192, 1172}, {2299, 284}, {6285, 1896}
X(52158) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5930}, {2, 30456}, {7, 3198}, {8, 40933}, {9, 36908}, {10, 1394}, {20, 65}, {37, 18623}, {40, 52078}, {42, 33673}, {57, 8804}, {71, 44697}, {72, 44696}, {73, 1895}, {108, 8057}, {109, 17898}, {154, 1441}, {201, 44698}, {204, 307}, {226, 610}, {227, 41084}, {306, 3213}, {347, 41086}, {608, 42699}, {651, 6587}, {934, 14308}, {1020, 14331}, {1214, 1249}, {1231, 3172}, {1400, 18750}, {1402, 14615}, {1409, 15466}, {1427, 27382}, {1439, 44695}, {1880, 37669}, {3344, 8807}, {3668, 7070}, {4551, 21172}, {6516, 44705}, {14249, 22341}, {15905, 40149}, {18026, 42658}, {18210, 44699}
X(52158) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 5930}, {11, 17898}, {20, 40602}, {226, 14092}, {307, 3343}, {478, 36908}, {5452, 8804}, {6587, 38991}, {8057, 38983}, {14308, 14714}, {14390, 40152}, {14615, 40605}, {18623, 40589}, {18750, 40582}, {30456, 32664}, {33673, 40592}
X(52158) = cevapoint of X(64) and X(19614)
X(52158) = crosssum of X(i) and X(j) for these (i,j): {73, 8803}, {3198, 30456}
X(52158) = barycentric product X(i)*X(j) for these {i,j}: {6, 5931}, {21, 2184}, {29, 1073}, {64, 333}, {81, 44692}, {86, 30457}, {253, 284}, {282, 41082}, {283, 459}, {314, 2155}, {332, 41489}, {522, 46639}, {663, 44326}, {1172, 19611}, {1301, 6332}, {2287, 8809}, {2299, 34403}, {6514, 6526}, {8748, 15394}, {13157, 35196}, {14642, 44130}, {19614, 31623}, {28660, 33581}, {47637, 47850}
X(52158) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 5930}, {21, 18750}, {28, 44697}, {29, 15466}, {31, 30456}, {41, 3198}, {55, 8804}, {56, 36908}, {58, 18623}, {64, 226}, {78, 42699}, {81, 33673}, {253, 349}, {283, 37669}, {284, 20}, {333, 14615}, {604, 40933}, {650, 17898}, {652, 8057}, {657, 14308}, {663, 6587}, {1073, 307}, {1172, 1895}, {1301, 653}, {1333, 1394}, {1436, 52078}, {1474, 44696}, {2155, 65}, {2184, 1441}, {2189, 44698}, {2194, 610}, {2203, 3213}, {2204, 204}, {2299, 1249}, {2328, 27382}, {2332, 44695}, {4636, 36841}, {5931, 76}, {7118, 41086}, {7252, 21172}, {8748, 14249}, {8809, 1446}, {14379, 40152}, {14642, 73}, {19611, 1231}, {19614, 1214}, {21789, 14331}, {30457, 10}, {33581, 1400}, {41082, 40702}, {41489, 225}, {44326, 4572}, {44692, 321}, {46639, 664}, {47437, 8803}

X(52159) = X(21)X(572)∩X(31)X(184)

X(52159) lies on the cubic K430 and these lines: {3, 52087}, {6, 10457}, {21, 572}, {31, 184}, {37, 48}, {101, 38869}, {154, 1011}, {284, 4224}, {478, 40613}, {573, 4216}, {662, 18137}, {692, 16872}, {992, 13724}, {1800, 37231}, {1958, 20891}, {2112, 25842}, {2179, 21743}, {2200, 21748}, {2328, 10438}, {2360, 37399}, {3588, 23361}, {4215, 44087}, {5110, 37259}, {7113, 16685}, {19757, 37052}, {20775, 44112}

X(52159) = isogonal conjugate of the isotomic conjugate of X(1764)
X(52159) = X(i)-Ceva conjugate of X(j) for these (i,j): {21, 31}, {572, 6}
X(52159) = X(75)-isoconjugate of X(43739)
X(52159) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 43739}, {1400, 1441}
X(52159) = crosspoint of X(662) and X(24027)
X(52159) = crosssum of X(i) and X(j) for these (i,j): {514, 40624}, {661, 24026}
X(52159) = crossdifference of every pair of points on line {4391, 21189}
X(52159) = barycentric product X(i)*X(j) for these {i,j}: {1, 23361}, {6, 1764}, {19, 23131}, {21, 40611}, {31, 20245}, {32, 21596}, {58, 22299}, {81, 3588}, {692, 23799}, {1193, 40455}, {1333, 22020}
X(52159) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 43739}, {1764, 76}, {3588, 321}, {20245, 561}, {21596, 1502}, {22020, 27801}, {22299, 313}, {23131, 304}, {23361, 75}, {23799, 40495}, {40455, 1240}, {40611, 1441}

X(52160) = X(1)X(147)∩X(6)X(7)

X(52160) lies on the cubics K323 and K623 and these lines: {1, 147}, {2, 9502}, {6, 7}, {8, 348}, {77, 28043}, {239, 9436}, {241, 1575}, {279, 291}, {518, 41352}, {1317, 31721}, {1323, 49772}, {1358, 5221}, {1362, 52029}, {1429, 2112}, {1447, 3008}, {2114, 20533}, {2246, 3598}, {2254, 43041}, {2272, 7291}, {2310, 3672}, {2550, 41354}, {2801, 45276}, {3485, 5543}, {3663, 9355}, {4872, 41339}, {9442, 41527}, {15251, 24203}, {15913, 36640}, {17014, 24712}, {17367, 40719}, {25281, 35312}, {28391, 40133}, {32040, 35160}

X(52160) = isogonal conjugate of X(2115)
X(52160) = X(i)-Ceva conjugate of X(j) for these (i,j): {239, 3212}, {9436, 7}, {43035, 3160}
X(52160) = X(1282)-cross conjugate of X(20533)
X(52160) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2115}, {9, 9500}, {55, 9499}
X(52160) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 2115}, {223, 9499}, {478, 9500}, {673, 14942}
X(52160) = cevapoint of X(1282) and X(2114)
X(52160) = barycentric product X(i)*X(j) for these {i,j}: {7, 20533}, {75, 2114}, {85, 1282}, {331, 20761}, {6063, 20672}, {7233, 27945}
X(52160) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 2115}, {56, 9500}, {57, 9499}, {1282, 9}, {2114, 1}, {20533, 8}, {20672, 55}, {20692, 210}, {20761, 219}, {27945, 3685}

X(52161) = X(1)X(85)∩X(43)X(57)

X(52161) lies on the cubic K1026 and these lines: {1, 85}, {7, 1045}, {43, 57}, {56, 2665}, {87, 1419}, {223, 5272}, {241, 2664}, {269, 1740}, {651, 9359}, {741, 4573}, {978, 16572}, {1025, 2108}, {1044, 34498}, {1416, 8300}, {1421, 29820}, {1447, 1458}, {1463, 18795}, {1758, 39344}, {2124, 21214}, {2663, 41246}, {3783, 9436}, {3979, 37736}, {4554, 17793}, {7168, 17082}, {9318, 39046}, {9499, 18786}, {16576, 23511}, {20762, 24578}, {33633, 36646}

X(52161) = X(i)-Ceva conjugate of X(j) for these (i,j): {1447, 57}, {1458, 1}
X(52161) = X(i)-cross conjugate of X(j) for these (i,j): {2110, 24578}, {33674, 1}
X(52161) = X(9)-isoconjugate of X(2111)
X(52161) = X(i)-Dao conjugate of X(j) for these (i,j): {291, 4518}, {478, 2111}
X(52161) = crossdifference of every pair of points on line {4435, 46388}
X(52161) = barycentric product X(i)*X(j) for these {i,j}: {7, 24578}, {57, 17794}, {85, 2110}, {226, 8849}, {241, 33674}, {273, 20762}, {1434, 20694}, {1447, 36906}
X(52161) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 2111}, {2110, 9}, {8849, 333}, {17794, 312}, {20694, 2321}, {20762, 78}, {24578, 8}, {33674, 36796}, {34253, 33701}, {36906, 4518}
X(52161) = {X(1362),X(43063)}-harmonic conjugate of X(291)

X(52162) = X(6)X(157)∩X(22)X(110)

X(52162) lies on the cubics K786 and K1001 and these lines: {2, 39096}, {3, 8925}, {6, 157}, {22, 110}, {25, 694}, {31, 1469}, {48, 3056}, {51, 12212}, {69, 19571}, {111, 31860}, {159, 41373}, {184, 3094}, {206, 2934}, {237, 2076}, {263, 46286}, {287, 35278}, {323, 9999}, {401, 1503}, {511, 3506}, {526, 2930}, {1184, 3124}, {1316, 20021}, {1495, 5104}, {1625, 11641}, {2420, 51240}, {2502, 41424}, {2794, 15595}, {2878, 20468}, {2881, 3569}, {2936, 5118}, {2987, 11477}, {3053, 38880}, {3098, 21512}, {3229, 35374}, {4048, 35926}, {4226, 25046}, {5027, 39469}, {5106, 41266}, {5116, 37457}, {5147, 20991}, {5621, 46130}, {5989, 25332}, {7665, 14467}, {9155, 31884}, {9208, 9411}, {9306, 35387}, {9479, 15588}, {10516, 41231}, {11328, 35424}, {11634, 38661}, {11646, 51431}, {16685, 16686}, {17809, 20976}, {17835, 38551}, {19459, 34980}, {21513, 34236}, {32661, 40121}, {34359, 39857}, {37183, 47619}, {40559, 47355}

X(52162) = reflection of X(6) in X(1576)
X(52162) = isogonal conjugate of X(9473)
X(52162) = isogonal conjugate of the anticomplement of X(36899)
X(52162) = isogonal conjugate of the isotomic conjugate of X(147)
X(52162) = X(i)-Ceva conjugate of X(j) for these (i,j): {511, 6}, {6660, 10329}, {42671, 154}
X(52162) = X(i)-isoconjugate of X(j) for these (i,j): {1, 9473}, {75, 34130}
X(52162) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 9473}, {98, 290}, {206, 34130}
X(52162) = crosssum of X(i) and X(j) for these (i,j): {2, 5984}, {523, 35088}
X(52162) = crossdifference of every pair of points on line {2799, 14316}
X(52162) = barycentric product X(i)*X(j) for these {i,j}: {1, 16559}, {6, 147}, {511, 36899}
X(52162) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 9473}, {32, 34130}, {147, 76}, {16559, 75}, {36899, 290}
X(52162) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 38873, 36790}, {1976, 51335, 6}, {5191, 51335, 1976}, {36790, 38873, 1350}

X(52163) = X(3)X(373)∩X(5)X(599)

X(52163) lies on the cubic K812 and these lines: {3, 373}, {5, 599}, {6, 5609}, {140, 14924}, {155, 11482}, {381, 15063}, {546, 14216}, {575, 6759}, {1995, 15033}, {3090, 44833}, {3091, 11411}, {3146, 45073}, {3574, 5072}, {3627, 9815}, {5079, 40912}, {5085, 37967}, {5446, 33540}, {5462, 13093}, {5640, 11472}, {5692, 7982}, {7530, 10541}, {9972, 17814}, {10606, 13364}, {11284, 14845}, {11935, 15039}, {12105, 31860}, {13352, 30734}, {14002, 37506}, {14848, 16534}, {15019, 18451}, {16619, 47352}, {17810, 49671}, {17825, 33532}, {27355, 44838}

X(52163) = midpoint of X(3531) and X(5544)

X(52164) = X(2)X(11)∩X(152)X(516)

X(52164) lies on the cubic K623 and these lines: {2, 11}, {7, 36905}, {8, 3177}, {145, 279}, {152, 516}, {190, 40609}, {200, 4388}, {244, 10580}, {518, 40868}, {519, 39353}, {1054, 11019}, {1146, 3617}, {1758, 26015}, {2082, 3501}, {2246, 30332}, {2263, 3870}, {2635, 3935}, {3210, 36845}, {3622, 17044}, {3693, 32850}, {3722, 10578}, {3732, 28915}, {4000, 46178}, {4440, 36221}, {4453, 45290}, {4847, 32932}, {5082, 36706}, {5175, 26036}, {5484, 20552}, {5687, 36652}, {5698, 51352}, {5853, 9436}, {6225, 20070}, {6646, 25722}, {7411, 20999}, {10431, 36855}, {10695, 38941}, {13405, 33109}, {15733, 17950}, {20015, 20211}, {23858, 36002}, {26246, 33131}, {31640, 46933}, {40483, 46932}

X(52164) = reflection of X(i) in X(j) for these {i,j}: {145, 664}, {14942, 50441}, {39351, 8}
X(52164) = anticomplement of X(14942)
X(52164) = anticomplement of the isogonal conjugate of X(1458)
X(52164) = anticomplement of the isotomic conjugate of X(9436)
X(52164) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 30807}, {7, 20556}, {31, 10025}, {56, 518}, {57, 20347}, {241, 69}, {518, 3436}, {603, 3100}, {604, 239}, {651, 3766}, {665, 37781}, {672, 329}, {883, 21301}, {911, 36101}, {934, 926}, {1025, 20295}, {1262, 883}, {1362, 20344}, {1408, 32922}, {1415, 918}, {1417, 24841}, {1458, 8}, {1462, 2481}, {1876, 4}, {2223, 144}, {2254, 33650}, {2283, 513}, {2284, 4462}, {2356, 5942}, {3286, 3869}, {3912, 21286}, {5236, 21270}, {9436, 6327}, {9454, 3177}, {9455, 21218}, {18206, 20245}, {24027, 2398}, {32668, 2400}, {32735, 885}, {34230, 5176}, {34253, 20345}, {34855, 3434}, {39775, 20554}, {40704, 315}, {41353, 21302}, {43042, 21293}, {51329, 17794}
X(52164) = X(i)-Ceva conjugate of X(j) for these (i,j): {5853, 145}, {9436, 2}, {14189, 3177}
X(52164) = crosssum of X(663) and X(35505)
X(52164) = barycentric product X(75)*X(52084)
X(52164) = barycentric quotient X(52084)/X(1)
X(52164) = {X(14942),X(50441)}-harmonic conjugate of X(2)

X(52165) = X(6)X(30)∩X(66)X(51964)

X(52165) lies on the cubics K429, K496, and K1169, and these lines: {6, 30}, {66, 51964}, {74, 40385}, {193, 43756}, {1974, 16240}, {3618, 34289}, {18533, 40387}

X(52165) = polar conjugate of the isotomic conjugate of X(51471)
X(52165) = barycentric product X(i)*X(j) for these {i,j}: {4, 51471}, {30, 40387}, {4846, 18533}, {34288, 37645}
X(52165) = barycentric quotient X(i)/X(j) for these {i,j}: {18533, 44134}, {37645, 32833}, {40387, 1494}, {51471, 69}

X(52166) = X(3)X(3163)∩X(6)X(74)

X(52166) lies on the cubic K1170 and these lines: {3, 3163}, {6, 74}, {19, 19302}, {25, 1989}, {50, 34178}, {115, 8573}, {186, 1138}, {230, 37962}, {393, 14579}, {457, 14993}, {566, 8792}, {1033, 8553}, {1249, 6188}, {1494, 48871}, {1560, 31489}, {1597, 6128}, {1609, 2079}, {1611, 2493}, {2070, 18487}, {2453, 47202}, {2930, 4230}, {3003, 14581}, {3053, 14910}, {3284, 18859}, {5523, 47275}, {6749, 13596}, {7545, 36430}, {8428, 16318}, {8743, 18573}, {8744, 16328}, {14130, 15860}, {15262, 18365}, {16310, 44272}, {21284, 41358}, {34288, 44274}, {35473, 40138}, {47189, 49123}

X(52166) = polar conjugate of X(40705)
X(52166) = polar conjugate of the isotomic conjugate of X(399)
X(52166) = X(i)-Ceva conjugate of X(j) for these (i,j): {186, 25}, {1990, 6}
X(52166) = X(i)-isoconjugate of X(j) for these (i,j): {48, 40705}, {63, 1138}, {2349, 20123}
X(52166) = X(i)-Dao conjugate of X(j) for these (i,j): {328, 1989}, {1138, 3162}, {1249, 40705}
X(52166) = crosssum of X(i) and X(j) for these (i,j): {6, 9919}, {520, 39008}
X(52166) = crossdifference of every pair of points on line {6699, 8552}
X(52166) = barycentric product X(i)*X(j) for these {i,j}: {4, 399}, {25, 1272}, {92, 19303}, {112, 14566}, {186, 14993}, {457, 1138}, {1117, 2914}, {3163, 40391}, {5667, 46036}, {11074, 14920}, {16077, 42656}
X(52166) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 40705}, {25, 1138}, {399, 69}, {457, 1272}, {1272, 305}, {1495, 20123}, {14566, 3267}, {14581, 11070}, {14993, 328}, {19303, 63}, {40391, 31621}, {42656, 9033}, {44084, 18781}
X(52166) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {112, 8749, 39176}, {112, 47228, 6}, {1609, 3018, 2079}, {8749, 39176, 6}, {9408, 51544, 6}, {18877, 47433, 6}, {39176, 47228, 8749}

X(52167) = X(1)X(2816)∩X(4)X(80)

X(52167) lies on the cubics K683 and K1051 and these lines: {1, 2816}, {4, 80}, {7, 1364}, {11, 653}, {19, 41499}, {65, 151}, {92, 1836}, {102, 3485}, {109, 278}, {117, 1788}, {124, 281}, {158, 12699}, {196, 497}, {243, 516}, {329, 3042}, {331, 33867}, {388, 2817}, {499, 8762}, {528, 1897}, {648, 19642}, {946, 1940}, {962, 1118}, {1148, 1479}, {1155, 17923}, {1785, 28194}, {1857, 9812}, {1895, 12701}, {2792, 5307}, {2818, 4295}, {3176, 5225}, {3476, 10696}, {3982, 51808}, {5081, 44663}, {5125, 37567}, {5174, 42379}, {5229, 13532}, {6331, 19643}, {6335, 17777}, {6336, 19636}, {6718, 17917}, {7541, 40663}, {7551, 18393}, {10740, 18391}, {11713, 37028}, {14690, 37417}, {35489, 50148}, {38573, 39542}

X(52168) = X(3)X(1495)∩X(24)X(9064)

X(52168) lies on the cubics K389, K855, and K1170, and these lines: {3, 1495}, {24, 9064}, {39, 51990}, {382, 50935}, {3003, 47649}, {3053, 14910}, {16334, 37934}, {40909, 51471}

X(52168) = isogonal conjugate of the anticomplement of X(51471)
X(52168) = barycentric product X(3426)*X(37645)
X(52168) = barycentric quotient X(i)/X(j) for these {i,j}: {37645, 44133}, {51990, 34801}

X(52169) = X(3)X(74)∩X(184)X(566)

X(52169) lies on the cubic K1170 and these lines: {3, 74}, {25, 1576}, {50, 1495}, {136, 12173}, {184, 566}, {186, 35372}, {237, 51240}, {381, 18576}, {3003, 34397}, {3018, 18384}, {3053, 35325}, {3185, 51235}, {3233, 51456}, {3830, 30685}, {5094, 9756}, {5877, 47201}, {8154, 37932}, {9409, 47230}, {9512, 36178}, {10295, 47148}, {11402, 20975}, {13558, 19504}, {15329, 45082}, {19165, 21284}, {32341, 43840}, {40114, 42671}, {41335, 44109}, {46602, 48679}

X(52169) = isogonal conjugate of the isotomic conjugate of X(12383)
X(52169) = X(186)-Ceva conjugate of X(6)
X(52169) = X(75)-isoconjugate of X(35372)
X(52169) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 35372}, {265, 328}
X(52169) = crosspoint of X(i) and X(j) for these (i,j): {112, 15395}, {10421, 40389}
X(52169) = crosssum of X(525) and X(3258)
X(52169) = crossdifference of every pair of points on line {1637, 6334}
X(52169) = barycentric product X(i)*X(j) for these {i,j}: {6, 12383}, {112, 38401}, {1511, 40389}, {3284, 10421}
X(52169) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 35372}, {12383, 76}, {34397, 40390}, {38401, 3267}

X(52170) = X(3)X(284)∩X(25)X(110)

X(52170) lies on the cubic K786 and these lines: {3, 248}, {25, 110}, {147, 385}, {155, 3511}, {184, 13558}, {394, 6638}, {399, 9517}, {511, 1971}, {525, 13188}, {1147, 3095}, {1297, 33878}, {2080, 13754}, {2871, 9861}, {2882, 19149}, {5989, 17932}, {6321, 16278}, {8909, 45489}, {8912, 12313}, {18440, 37074}, {22161, 23171}, {31850, 39839}, {34383, 39820}, {35389, 52016}

X(52170) = reflection of X(3) in X(32661)
X(52170) = isogonal conjugate of the polar conjugate of X(40867)
X(52170) = X(i)-Ceva conjugate of X(j) for these (i,j): {511, 3}, {1971, 6638}
X(52170) = X(287)-Dao conjugate of X(290)
X(52170) = barycentric product X(3)*X(40867)
X(52170) = barycentric quotient X(40867)/X(264)

X(52171) = X(2)X(19140)∩X(3)X(74)

X(52171) lies on the cubic K524 and these lines: {2, 19140}, {3, 74}, {6, 6032}, {23, 32235}, {125, 15018}, {323, 542}, {526, 3005}, {541, 12112}, {1176, 5888}, {1995, 51941}, {2781, 15107}, {2854, 23061}, {3060, 10752}, {3448, 9976}, {5505, 41743}, {5640, 9970}, {5651, 52098}, {6723, 46440}, {7565, 7687}, {7712, 45839}, {8549, 17847}, {9143, 46264}, {9463, 14901}, {10264, 14389}, {10545, 12824}, {11003, 32305}, {11061, 18911}, {11422, 11579}, {11673, 51234}, {12317, 37645}, {12584, 33884}, {15032, 16003}, {15037, 20379}, {15052, 15063}, {15305, 45019}, {15919, 36829}, {20404, 32730}, {25711, 43584}, {38397, 44493}, {41450, 52019}, {41603, 41617}

X(52171) = reflection of X(23) in X(32235)
X(52171) = circumcircle-inverse of X(15080)
X(52171) = crossdifference of every pair of points on line {1637, 7753}
X(52171) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {74, 110, 15080}, {110, 41462, 1511}, {125, 25556, 15018}, {399, 15066, 110}, {399, 15106, 15066}, {3448, 11004, 9976}, {15018, 51882, 25556}

X(52172) = X(4)X(51)∩X(30)X(133)

X(52172) lies on the cubic K427 and these lines: {4, 51}, {5, 12096}, {30, 133}, {53, 39563}, {136, 10151}, {186, 44992}, {381, 6760}, {382, 51342}, {403, 46424}, {546, 34147}, {1304, 20480}, {1539, 33641}, {1559, 18400}, {2777, 51385}, {3627, 34109}, {6526, 20427}, {10152, 40664}, {11251, 51393}, {14269, 42854}, {18554, 50687}, {22337, 33892}, {38625, 46031}

X(52172) = midpoint of X(i) and X(j) for these {i,j}: {4, 34170}, {186, 44992}, {6761, 51892}, {10152, 40664}
X(52172) = reflection of X(i) in X(j) for these {i,j}: {12096, 5}, {18809, 10151}, {38625, 46031}
X(52172) = polar-circle-inverse of X(6241)
X(52172) = barycentric product X(3357)*X(46106)
X(52172) = barycentric quotient X(3357)/X(14919)
X(52172) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 6761, 51892}, {34170, 51892, 6761}

X(52173) = X(4)X(5609)∩X(30)X(13530)

X(52173) lies on the cubics K025, K473, K811, and K952, and these lines: {4, 5609}, {30, 13530}, {187, 1989}, {549, 19307}, {8724, 18575}, {9158, 44266}

X(52173) = polar-circle-inverse of X(10294)
X(52173) = antigonal image of X(381)
X(52173) = symgonal image of X(549)
X(52173) = barycentric quotient X(i)/X(j) for these {i,j}: {381, 44555}, {36430, 15362}, {51544, 39239}

X(52174) = X(2)X(1296)∩X(25)X(187)

X(52174) lies on the cubics K969 and K1048 and these lines: {2, 1296}, {4, 32133}, {25, 187}, {184, 32740}, {574, 10355}, {1995, 14262}, {2434, 9306}, {5020, 33552}, {5943, 17979}, {6525, 51823}, {8542, 13493}, {8585, 10354}, {32648, 52142}

X(52174) = isogonal conjugate of the isotomic conjugate of X(14262)
X(52174) = X(75)-isoconjugate of X(13608)
X(52174) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 13608}, {3266, 10354}
X(52174) = barycentric product X(i)*X(j) for these {i,j}: {6, 14262}, {1995, 21448}, {5485, 19136}, {11185, 39238}, {13492, 13493}, {17968, 34241}
X(52174) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 13608}, {1995, 11059}, {14262, 76}, {19136, 1992}, {39238, 5486}
X(52174) = {X(25),X(21448)}-harmonic conjugate of X(38532)

X(52175) = X(1)X(7168)∩X(350)X(694)

X(52175) lies on the cubics K766 and K992 and these lines: {1, 7168}, {2, 40778}, {172, 19585}, {350, 694}, {726, 24576}, {1469, 17149}, {1909, 18273}, {8868, 17738}, {27890, 40790}, {39080, 39929}, {40731, 51920}

X(52175) = isotomic conjugate of the isogonal conjugate of X(51920)
X(52175) = X(i)-cross conjugate of X(j) for these (i,j): {335, 40846}, {1926, 1909}
X(52175) = X(i)-isoconjugate of X(j) for these (i,j): {6, 51979}, {32, 40849}, {694, 18274}, {893, 18278}, {904, 3510}, {1581, 30634}, {1927, 19581}, {1967, 19580}, {7104, 19565}, {8789, 18277}, {8875, 41882}, {9468, 19579}
X(52175) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 51979}, {6376, 40849}, {8290, 19580}, {18274, 39043}, {18277, 39030}, {18278, 40597}, {19576, 30634}, {19579, 39044}
X(52175) = barycentric product X(i)*X(j) for these {i,j}: {75, 39933}, {76, 51920}, {385, 30633}, {1920, 7168}, {3978, 24576}
X(52175) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 51979}, {75, 40849}, {171, 18278}, {385, 19580}, {894, 3510}, {1580, 18274}, {1691, 30634}, {1909, 19565}, {1920, 19567}, {1926, 18277}, {1966, 19579}, {3978, 19581}, {7061, 8875}, {7168, 893}, {8868, 41532}, {24576, 694}, {30633, 1916}, {39933, 1}, {51919, 904}, {51920, 6}

X(52176) = X(1)X(39917)∩X(10)X(27536)

X(52176) lies on the cubic K744 and K1026 and these lines: {1, 39917}, {10, 27436}, {43, 40731}, {172, 3510}, {256, 40778}, {1045, 17792}, {1655, 3971}, {1909, 18275}, {2664, 28391}, {6376, 34021}, {8868, 18795}, {8875, 18794}, {8931, 16360}, {17739, 18787}, {18784, 18786}

X(52176) = isogonal conjugate of X(18754)
X(52176) = X(i)-cross conjugate of X(j) for these (i,j): {257, 1}, {7168, 18795}
X(52176) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18754}, {6, 30661}, {43, 40768}, {172, 39917}, {238, 16362}, {2176, 40741}
X(52176) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 18754}, {9, 30661}, {9470, 16362}
X(52176) = trilinear pole of line {21834, 45882}
X(52176) = barycentric product X(i)*X(j) for these {i,j}: {330, 40795}, {16360, 40098}
X(52176) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 30661}, {6, 18754}, {87, 40741}, {256, 39917}, {292, 16362}, {2162, 40768}, {16360, 4366}, {40795, 192}

X(52177) = X(6)X(1987)∩X(51)X(107)

X(52177) lies on the cubic K223 and K786 and these lines: {6, 1987}, {25, 1988}, {51, 107}, {154, 36617}, {184, 9419}, {287, 401}, {394, 6638}, {1495, 26717}, {1576, 14533}, {17974, 23611}, {18877, 39231}, {32542, 52162}, {36213, 40804}

X(52177) = isogonal conjugate of X(16089)
X(52177) = isogonal conjugate of the anticomplement of X(46841)
X(52177) = isogonal conjugate of the isotomic conjugate of X(14941)
X(52177) = isogonal conjugate of the polar conjugate of X(1987)
X(52177) = X(1298)-Ceva conjugate of X(1987)
X(52177) = X(i)-cross conjugate of X(j) for these (i,j): {237, 184}, {248, 17970}
X(52177) = X(i)-isoconjugate of X(j) for these (i,j): {1, 16089}, {19, 44137}, {75, 41204}, {92, 401}, {264, 1955}, {276, 2313}, {811, 6130}, {1969, 1971}, {32428, 40440}, {32545, 40703}
X(52177) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 16089}, {6, 44137}, {206, 41204}, {401, 22391}, {6130, 17423}
X(52177) = cevapoint of X(i) and X(j) for these (i,j): {1970, 1971}, {34980, 39469}
X(52177) = crosspoint of X(1987) and X(14941)
X(52177) = crosssum of X(401) and X(41204)
X(52177) = trilinear pole of line {217, 39201}
X(52177) = barycentric product X(i)*X(j) for these {i,j}: {3, 1987}, {6, 14941}, {48, 1956}, {184, 1972}, {216, 1298}, {248, 40804}, {32542, 36214}, {39683, 43718}, {41208, 42293}
X(52177) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 44137}, {6, 16089}, {32, 41204}, {184, 401}, {217, 32428}, {1298, 276}, {1956, 1969}, {1972, 18022}, {1987, 264}, {3049, 6130}, {9247, 1955}, {14575, 1971}, {14600, 32545}, {14941, 76}, {32542, 17984}, {39683, 44144}, {40804, 44132}

X(52178) = X(1)X(40218)∩X(3)X(8)

X(52178) lies on the cubic K654 and K844 and these lines: {1, 40218}, {3, 8}, {4, 6075}, {84, 3667}, {221, 34051}, {936, 36819}, {2401, 21132}, {5265, 37136}, {12672, 15635}, {14986, 40450}

X(52178) = barycentric product X(34234)*X(44675)
X(52178) = barycentric quotient X(i)/X(j) for these {i,j}: {1537, 26611}, {44675, 908}
X(52178) = {X(104),X(14266)}-harmonic conjugate of X(36944)

X(52179) = X(23)X(110)∩X(67)X(98)

X(52179) lies on the cubics K433 and K503 and these lines: {23, 110}, {54, 33695}, {67, 98}, {76, 18332}, {526, 14355}, {1511, 10411}, {2986, 51228}, {3043, 14591}, {14270, 14385}, {22115, 51478}, {23236, 34174}, {40356, 48453}, {51457, 52094}

X(52179) = isogonal conjugate of X(43087)
X(52179) = X(i)-isoconjugate of X(j) for these (i,j): {1, 43087}, {94, 2247}, {542, 2166}, {1577, 23968}, {1640, 32680}, {18312, 32678}
X(52179) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 43087}, {542, 11597}, {18312, 18334}, {51472, 51847}
X(52179) = barycentric product X(i)*X(j) for these {i,j}: {50, 5641}, {323, 842}, {526, 5649}, {6035, 14270}, {10411, 14998}, {14355, 46787}, {14385, 51228}, {14590, 35909}, {50942, 51478}
X(52179) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 43087}, {50, 542}, {526, 18312}, {842, 94}, {1576, 23968}, {5641, 20573}, {5649, 35139}, {14270, 1640}, {14355, 46786}, {14385, 51227}, {14591, 7473}, {14998, 10412}, {19627, 5191}, {34397, 6103}, {35909, 14592}, {48453, 14254}, {51478, 50941}

X(52180) = X(1)X(474)∩X(518)X(36603)

See Ivan Pavlov and César Lozada, euclid 5531.

X(52180) lies on these lines: {1, 474}, {518, 36603}, {1054, 33795}, {4512, 17125}, {5853, 26718}, {16192, 47302}, {17127, 23511}, {17132, 25567}, {28530, 28655}

X(52180) = {X(7963), X(10563)}-harmonic conjugate of X(1)

X(52181) = X(1)X(6)∩X(519)X(6552)

See Ivan Pavlov and César Lozada, euclid 5531.

X(52181) lies on these lines: {1, 6}, {517, 45047}, {519, 6552}, {2098, 23511}, {5272, 16189}, {7963, 7991}, {7982, 46943}, {8056, 11531}, {11224, 21214}

X(52181) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 3973, 11260), (7991, 47623, 7963)

X(52182) = X(1)X(4004)∩X(3973)X(7991)

See Ivan Pavlov and César Lozada, euclid 5531.

X(52182) lies on these lines: {1, 4004}, {3680, 33795}, {3973, 7991}, {4902, 12640}

X(52182) = {X(7991), X(10563)}-harmonic conjugate of X(3973)

X(52183) = X(40)X(9519)∩X(986)X(1706)

See Ivan Pavlov and César Lozada, euclid 5531.

X(52183) lies on these lines: {40, 9519}, {986, 1706}, {1054, 3680}, {3913, 15600}, {4695, 5128}, {4859, 32157}, {7290, 24440}, {10563, 11260}, {11530, 17596}, {14923, 15839}

X(52184) = X(40)X(3973)∩X(101)X(604)

See Ivan Pavlov and César Lozada, euclid 5531.

X(52184) lies on these lines: {9, 10912}, {40, 3973}, {101, 604}, {7991, 48718}

X(52185) = ISOGONAL CONJUGATE OF X(12047)

See Ivan Pavlov and César Lozada, euclid 5530.

X(52185) lies on these lines: {1, 1825}, {3, 2594}, {10, 1800}, {29, 10572}, {35, 283}, {36, 40442}, {46, 77}, {55, 1069}, {65, 7100}, {78, 993}, {219, 2278}, {319, 332}, {581, 3422}, {1065, 45287}, {1433, 11507}, {1807, 2646}, {3295, 38248}

X(52185) = isogonal conjugate of X(12047)
X(52185) = crosspoint of X(i) and X(j) for these (i, j): {100, 48391}, {101, 4041}, {109, 23226}
X(52185) = X(10)-Dao conjugate of-X(23555)
X(52185) = X(i)-isoconjugate-of-X(j) for these {i, j}: {4, 37565}, {58, 23555}, {81, 50036}, {649, 18740}, {757, 21696}
X(52185) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (37, 23555), (42, 50036), (48, 37565), (100, 18740), (1500, 21696)
X(52185) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(3)}} and {{A, B, C, X(4), X(37741)}}
X(52185) = barycentric quotient X(i)/X(j) for these (i, j): (37, 23555), (42, 50036), (48, 37565), (100, 18740), (1500, 21696)
X(52185) = trilinear quotient X(i)/X(j) for these (i, j): (3, 37565), (10, 23555), (37, 50036), (190, 18740), (756, 21696)

X(52186) = ISOGONAL CONJUGATE OF X(499)

See Ivan Pavlov and César Lozada, euclid 5530.

X(52186) lies on these lines: {1, 5422}, {6, 11508}, {34, 5903}, {47, 11248}, {56, 7130}, {58, 16473}, {86, 498}, {255, 41442}, {269, 1079}, {1126, 16472}, {1220, 12647}, {1411, 1482}, {1772, 3157}, {3193, 11681}, {10039, 43531}, {10573, 36123}

X(52186) = isogonal conjugate of X(499)
X(52186) = crosspoint of X(100) and X(34948)
X(52186) = X(i)-isoconjugate-of-X(j) for these {i, j}: {4, 24467}, {7, 7082}, {90, 10052}
X(52186) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (41, 7082), (48, 24467)
X(52186) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(6)}} and {{A, B, C, X(3), X(80)}}
X(52186) = barycentric product X(8)*X(7130)
X(52186) = barycentric quotient X(i)/X(j) for these (i, j): (41, 7082), (48, 24467)
X(52186) = trilinear product X(9)*X(7130)
X(52186) = trilinear quotient X(i)/X(j) for these (i, j): (3, 24467), (46, 10052), (55, 7082)

X(52187) = ISOTOMIC CONJUGATE OF X(32936)

X(52187) lies on the conic {{A,B,C,X(2),X(6)}} and these lines: {2, 44133}, {6, 376}, {25, 40138}, {32, 36427}, {111, 3163}, {263, 40673}, {308, 46951}, {393, 5309}, {493, 19054}, {494, 19053}, {800, 46952}, {1249, 34818}, {1992, 40802}, {2987, 5032}, {3068, 41438}, {3069, 41437}, {3087, 15433}, {3108, 5158}, {5007, 37460}, {5286, 34288}, {5306, 8770}, {7735, 21448}, {8882, 13342}, {10304, 33871}, {12082, 14836}, {14853, 47433}, {14910, 33872}, {30435, 44273}, {30537, 31400}, {32837, 42407}, {37665, 39389}, {39662, 47228}, {39951, 43957}

X(52187) = isotomic conjugate of X(32836)
X(52187) = isotomic conjugate of the anticomplement of X(7739)
X(52187) = X(7739)-cross conjugate of X(2)
X(52187) = X(i)-isoconjugate of X(j) for these (i,j): {31, 32836}, {63, 1597}, {75, 33871}
X(52187) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 32836}, {206, 33871}, {1597, 3162}
X(52187) = cevapoint of X(37640) and X(37641)
X(52187) = crosssum of X(6) and X(46945)
X(52187) = trilinear pole of line {512, 9209}
X(52187) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 32836}, {25, 1597}, {32, 33871}

X(52188) = ISOTOMIC CONJUGATE OF X(46951)

X(52188) lies on the conic {{A,B,C,X(2),X(6)}} and these lines: {6, 3524}, {37, 36916}, {111, 37665}, {251, 33871}, {308, 32836}, {393, 7739}, {493, 19053}, {494, 19054}, {1383, 14930}, {1989, 5286}, {3068, 41437}, {3069, 41438}, {3087, 34818}, {3284, 39955}, {5032, 30535}, {5304, 39389}, {7736, 21448}, {7772, 8749}, {8770, 9300}

X(52188) = isotomic conjugate of X(46951)
X(52188) = X(i)-isoconjugate of X(j) for these (i,j): {31, 46951}, {63, 18535}, {999, 3305}, {3295, 3306}, {35281, 47965}
X(52188) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 46951}, {3162, 18535}
X(52188) = cevapoint of X(3087) and X(40138)
X(52188) = barycentric product X(1000)*X(3296)
X(52188) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 46951}, {25, 18535}, {1000, 42696}, {3296, 42697}, {34446, 3295}, {36916, 42032}

X(52189) = X(2)X(11643)∩X(4)X(542)

X(52189) lies on the cubic K481 and these lines: {2, 11643}, {4, 542}, {67, 316}, {111, 7533}, {598, 6593}, {691, 7574}, {2930, 11317}, {5189, 10416}, {7790, 15118}, {8352, 25328}, {9154, 45821}, {16063, 30786}, {16092, 18572}, {18818, 34319}, {43461, 50008}

X(52189) = barycentric product X(18472)*X(46111)
X(52189) = barycentric quotient X(18472)/X(3292)

X(52190) = X(2)X(98)∩X(67)X(51943)

X(52190) lies on the cubic K481 and these lines: {2, 98}, {67, 51943}, {248, 33565}, {265, 290}, {526, 43665}, {879, 11564}, {2387, 25739}, {2715, 43658}, {2854, 51259}, {10264, 44221}, {15081, 31415}

X(52190) = reflection of X(15920) in X(125)
X(52190) = psi-transform of X(37988)
X(52190) = X(1755)-isoconjugate of X(7578)
X(52190) = X(7578)-Dao conjugate of X(36899)
X(52190) = barycentric product X(i)*X(j) for these {i,j}: {287, 7577}, {290, 566}, {16081, 23039}, {18117, 43187}, {36829, 43665}
X(52190) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 7578}, {566, 511}, {7577, 297}, {18117, 3569}, {23039, 36212}, {36829, 2421}, {51391, 51389}
X(52190) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {98, 287, 14355}, {265, 36826, 34175}

X(52191) = X(2)X(67)∩X(4)X(94)

X(52191) lies on the cubic K481 and these lines: {2, 67}, {4, 94}, {23, 542}, {110, 34507}, {125, 15018}, {541, 10296}, {1209, 16534}, {1993, 16176}, {1994, 5095}, {1995, 32306}, {2781, 5189}, {2854, 37779}, {3060, 32273}, {3580, 27085}, {5169, 5476}, {5609, 7552}, {6515, 32255}, {7488, 30714}, {7492, 32233}, {7493, 9143}, {7496, 49116}, {7527, 16003}, {7533, 12824}, {7556, 23236}, {7565, 36253}, {7605, 20304}, {10990, 44829}, {11422, 41731}, {11564, 18474}, {14118, 20417}, {14389, 25329}, {14683, 15647}, {15118, 34545}, {16063, 32247}, {18125, 22336}, {25336, 37638}, {31133, 48679}, {34802, 37077}, {43578, 43651}

X(52191) = reflection of X(52171) in X(125)
X(52191) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3448, 11002, 265}, {9140, 9970, 5169}, {9140, 15019, 20301}, {11061, 43697, 34319}, {12824, 32274, 7533}

X(52192) = ISOGONAL CONJUGATE OF X(12584)

X(52192) lies on the cubics K481 and K930 and these lines: {5, 14246}, {23, 542}, {67, 842}, {316, 7574}, {6103, 8744}, {7495, 51456}, {7533, 43087}, {10301, 17986}, {10416, 23234}, {10561, 12077}

X(52192) = isogonal conjugate of X(12584)
X(52192) = isogonal conjugate of the anticomplement of X(20301)
X(52192) = X(32274)-cross conjugate of X(4)
X(52192) = X(1)-isoconjugate of X(12584)
X(52192) = X(3)-Dao conjugate of X(12584)
X(52192) = trilinear pole of line {1640, 2492}
X(52192) = barycentric quotient X(6)/X(12584)

X(52193) = ISOGONAL CONJUGATE OF X(51447)

X(52193) lies on the cubic K1278 and these lines: {2, 11485}, {3, 69}, {5, 303}, {6, 37341}, {13, 33458}, {14, 40335}, {15, 141}, {16, 524}, {30, 299}, {61, 3589}, {62, 3629}, {99, 44250}, {140, 302}, {193, 11486}, {298, 549}, {325, 5981}, {343, 465}, {394, 466}, {395, 533}, {396, 3642}, {398, 7867}, {462, 41001}, {491, 18585}, {492, 15765}, {511, 41034}, {525, 44712}, {530, 15300}, {531, 624}, {532, 42943}, {538, 6783}, {542, 36756}, {543, 33517}, {550, 634}, {590, 6304}, {599, 11480}, {615, 6300}, {616, 8703}, {620, 6782}, {623, 23302}, {625, 33518}, {627, 3530}, {630, 42599}, {635, 6671}, {636, 42147}, {637, 14813}, {638, 14814}, {698, 3105}, {1352, 41035}, {1503, 14539}, {1654, 21869}, {1975, 5873}, {3070, 22629}, {3071, 22600}, {3107, 23009}, {3130, 47582}, {3180, 11300}, {3181, 42913}, {3366, 23311}, {3367, 23312}, {3580, 11130}, {3620, 37172}, {3630, 10646}, {3631, 10645}, {3643, 42942}, {3763, 22236}, {3818, 47068}, {5318, 34509}, {5334, 11306}, {5335, 11296}, {5463, 22165}, {5613, 41017}, {5615, 34380}, {5700, 17768}, {5739, 21475}, {5859, 10653}, {5865, 31670}, {5872, 7763}, {5965, 13349}, {5978, 44219}, {5980, 11129}, {6144, 22238}, {6303, 32419}, {6307, 32421}, {6780, 22850}, {7615, 9763}, {7799, 47611}, {9735, 34507}, {9761, 42089}, {10109, 33612}, {10218, 11064}, {10654, 37351}, {11098, 49716}, {11127, 14389}, {11145, 37779}, {11290, 42925}, {11295, 42119}, {11298, 37640}, {11302, 37641}, {11303, 11542}, {11304, 34541}, {11305, 11488}, {11481, 40341}, {12322, 42279}, {12323, 42278}, {13350, 40107}, {14541, 48881}, {15764, 32808}, {15769, 35315}, {16241, 44383}, {16242, 33459}, {16961, 22496}, {16966, 33475}, {17300, 21898}, {18358, 37333}, {18583, 47519}, {19107, 36329}, {20080, 42115}, {21360, 36970}, {21850, 37332}, {22491, 42095}, {22492, 42094}, {22493, 33416}, {22494, 42100}, {22689, 41754}, {22856, 37835}, {22882, 33392}, {22883, 33394}, {23303, 34508}, {32461, 35314}, {33624, 42634}, {34009, 46818}, {34573, 34754}, {35932, 42123}, {36356, 49930}, {36357, 49928}, {36366, 43418}, {36368, 49952}, {36767, 42930}, {36929, 37728}, {37178, 51171}, {40668, 43086}, {40901, 42118}, {41020, 44882}, {41022, 51388}

X(52193) = midpoint of X(i) and X(j) for these {i,j}: {299, 617}, {36329, 50858}
X(52193) = reflection of X(i) in X(j) for these {i,j}: {14, 44382}, {395, 619}, {3181, 42913}, {5321, 624}, {6782, 620}, {33518, 625}, {35303, 5464}, {41017, 5613}, {47286, 11542}
X(52193) = isogonal conjugate of X(51447)
X(52193) = isotomic conjugate of X(38427)
X(52193) = anticomplement of X(11543)
X(52193) = isotomic conjugate of the polar conjugate of X(395)
X(52193) = isogonal conjugate of the polar conjugate of X(41001)
X(52193) = X(41001)-Ceva conjugate of X(395)
X(52193) = X(i)-isoconjugate of X(j) for these (i,j): {1, 51447}, {19, 6151}, {31, 38427}, {1973, 40706}
X(52193) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 38427}, {3, 51447}, {4, 619}, {6, 6151}, {186, 33527}, {340, 14922}, {395, 471}, {6337, 40706}
X(52193) = cevapoint of X(617) and X(628)
X(52193) = crosspoint of X(69) and X(40710)
X(52193) = crosssum of X(25) and X(8740)
X(52193) = crossdifference of every pair of points on line {2489, 8739}
X(52193) = barycentric product X(i)*X(j) for these {i,j}: {3, 41001}, {69, 395}, {265, 14921}, {287, 51387}, {328, 19295}, {462, 3926}, {525, 35315}, {533, 40709}, {619, 40710}, {3267, 35330}, {6672, 40711}, {9117, 30786}, {43086, 44719}
X(52193) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 38427}, {3, 6151}, {6, 51447}, {69, 40706}, {395, 4}, {462, 393}, {533, 470}, {619, 471}, {4558, 10410}, {6672, 472}, {8015, 8738}, {9117, 468}, {14921, 340}, {19295, 186}, {30459, 23715}, {32586, 34322}, {35315, 648}, {35330, 112}, {36296, 2381}, {36297, 16460}, {36305, 8742}, {40709, 11118}, {40710, 11120}, {41001, 264}, {44719, 38404}, {51387, 297}, {52153, 11089}
X(52193) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {15, 141, 37340}, {193, 37173, 11486}, {303, 621, 5}, {396, 3642, 37352}, {621, 628, 303}, {624, 5321, 31694}, {5464, 22997, 51160}, {5464, 51014, 9886}, {40710, 44719, 11064}

X(52194) = ISOGONAL CONJUGATE OF X(51446)

X(52194) lies on the cubic K1278 and these lines: {2, 11486}, {3, 69}, {5, 302}, {6, 37340}, {13, 40334}, {14, 33459}, {15, 524}, {16, 141}, {30, 298}, {61, 3629}, {62, 3589}, {140, 303}, {193, 11485}, {299, 549}, {325, 5980}, {343, 466}, {394, 465}, {395, 3643}, {396, 532}, {397, 7867}, {463, 41000}, {491, 15765}, {492, 18585}, {511, 41035}, {525, 44711}, {530, 623}, {531, 15300}, {533, 42942}, {538, 6782}, {542, 36755}, {543, 33518}, {550, 633}, {590, 6305}, {599, 11481}, {615, 6301}, {617, 8703}, {620, 6783}, {624, 23303}, {625, 33517}, {628, 3530}, {629, 42598}, {635, 42148}, {636, 6672}, {637, 14814}, {638, 14813}, {698, 3104}, {1352, 41034}, {1503, 14538}, {1654, 21898}, {1975, 5872}, {3070, 22627}, {3071, 22598}, {3106, 23000}, {3129, 47582}, {3180, 42912}, {3181, 11299}, {3391, 23311}, {3392, 23312}, {3580, 11131}, {3620, 37173}, {3630, 10645}, {3631, 10646}, {3642, 42943}, {3763, 22238}, {3818, 47066}, {5321, 34508}, {5334, 11295}, {5335, 11305}, {5464, 22165}, {5611, 34380}, {5617, 41016}, {5699, 17768}, {5739, 21476}, {5858, 10654}, {5864, 31670}, {5873, 7763}, {5965, 13350}, {5981, 11128}, {6144, 22236}, {6302, 32419}, {6306, 32421}, {6779, 22894}, {7615, 9761}, {7799, 47610}, {9736, 34507}, {9763, 42092}, {10109, 33613}, {10217, 11064}, {10653, 37352}, {11097, 49716}, {11126, 14389}, {11146, 37779}, {11289, 42924}, {11296, 42120}, {11297, 37641}, {11301, 37640}, {11303, 34540}, {11304, 11543}, {11306, 11489}, {11480, 40341}, {12322, 42278}, {12323, 42279}, {13349, 40107}, {14540, 48881}, {14907, 44250}, {15764, 32809}, {15768, 35314}, {16241, 33458}, {16242, 44382}, {16267, 36770}, {16960, 22495}, {16967, 33474}, {17300, 21869}, {18358, 37332}, {18583, 47517}, {19106, 35751}, {20080, 42116}, {21359, 36969}, {21850, 37333}, {22491, 42093}, {22492, 42098}, {22493, 42099}, {22494, 33417}, {22687, 41752}, {22900, 37832}, {22927, 33395}, {22928, 33393}, {23302, 34509}, {32460, 35315}, {33622, 42633}, {34008, 46818}, {34573, 34755}, {35741, 42170}, {35931, 42122}, {36348, 49929}, {36349, 49927}, {36366, 49953}, {36368, 43419}, {36768, 45879}, {36771, 38412}, {36928, 37728}, {37177, 51171}, {40667, 43085}, {40900, 42117}, {41021, 44882}, {41023, 51387}

X(52194) = midpoint of X(i) and X(j) for these {i,j}: {298, 616}, {35751, 50855}
X(52194) = reflection of X(i) in X(j) for these {i,j}: {13, 44383}, {396, 618}, {3180, 42912}, {5318, 623}, {6783, 620}, {33517, 625}, {35304, 5463}, {41016, 5617}, {45879, 36768}, {47286, 11543}
X(52194) = isogonal conjugate of X(51446)
X(52194) = isotomic conjugate of X(38428)
X(52194) = anticomplement of X(11542)
X(52194) = isotomic conjugate of the polar conjugate of X(396)
X(52194) = isogonal conjugate of the polar conjugate of X(41000)
X(52194) = X(41000)-Ceva conjugate of X(396)
X(52194) = X(i)-isoconjugate of X(j) for these (i,j): {1, 51446}, {19, 2981}, {31, 38428}, {1973, 40707}
X(52194) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 38428}, {3, 51446}, {4, 618}, {6, 2981}, {186, 33526}, {340, 14921}, {396, 470}, {6337, 40707}
X(52194) = cevapoint of X(616) and X(627)
X(52194) = crosspoint of X(69) and X(40709)
X(52194) = crosssum of X(25) and X(8739)
X(52194) = crossdifference of every pair of points on line {2489, 8740}
X(52194) = barycentric product X(i)*X(j) for these {i,j}: {3, 41000}, {69, 396}, {265, 14922}, {287, 51388}, {328, 19294}, {463, 3926}, {525, 35314}, {532, 40710}, {618, 40709}, {3267, 35329}, {6671, 40712}, {9115, 30786}, {43085, 44718}
X(52194) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 38428}, {3, 2981}, {6, 51446}, {69, 40707}, {396, 4}, {463, 393}, {532, 471}, {618, 470}, {4558, 10409}, {6671, 473}, {8014, 8737}, {9115, 468}, {14922, 340}, {19294, 186}, {30462, 23714}, {32585, 34321}, {35314, 648}, {35329, 112}, {36296, 16459}, {36297, 2380}, {36304, 8741}, {40709, 11119}, {40710, 11117}, {41000, 264}, {44718, 38403}, {51388, 297}, {52153, 11084}
X(52194) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {16, 141, 37341}, {193, 37172, 11485}, {302, 622, 5}, {395, 3643, 37351}, {622, 627, 302}, {623, 5318, 31693}, {5463, 22998, 51159}, {5463, 51011, 9885}, {40709, 44718, 11064}

X(52195) = X(2)X(9309)∩X(56)X(87)

X(52195) lies on the cubic K506 and these lines: {2, 9309}, {8, 3056}, {56, 87}, {144, 330}, {537, 31165}, {672, 16606}, {673, 27447}, {932, 38452}, {1201, 27499}, {2319, 37658}, {3271, 20258}, {4124, 20892}, {4384, 27497}, {6384, 30946}, {7209, 40864}, {16569, 20368}, {20348, 20358}, {20359, 20864}, {20978, 23493}, {23483, 30082}, {27627, 45197}

X(52195) = X(3452)-cross conjugate of X(3057)
X(52195) = X(i)-isoconjugate of X(j) for these (i,j): {43, 1476}, {192, 3451}, {1222, 1403}, {1423, 23617}, {2176, 40420}, {3212, 51476}, {20760, 40446}, {32017, 41526}
X(52195) = X(i)-Dao conjugate of X(j) for these (i,j): {2170, 3835}, {3212, 3452}, {3752, 6376}, {12640, 27538}
X(52195) = crosspoint of X(87) and X(7155)
X(52195) = crosssum of X(43) and X(1403)
X(52195) = barycentric product X(i)*X(j) for these {i,j}: {8, 27499}, {87, 3452}, {330, 3057}, {932, 21120}, {1201, 27424}, {2053, 26563}, {2162, 20895}, {2319, 3663}, {2347, 6384}, {3752, 7155}, {4598, 6615}, {6736, 7153}, {16606, 17183}, {18163, 42027}, {25268, 43931}
X(52195) = barycentric quotient X(i)/X(j) for these {i,j}: {87, 40420}, {1201, 1423}, {2053, 23617}, {2162, 1476}, {2319, 1222}, {2347, 43}, {3057, 192}, {3452, 6376}, {3663, 30545}, {3752, 3212}, {6363, 43051}, {6615, 3835}, {6736, 4110}, {7121, 3451}, {7155, 32017}, {17183, 31008}, {18163, 33296}, {20228, 1403}, {20895, 6382}, {21120, 20906}, {21809, 3971}, {22072, 22370}, {25268, 36863}, {27499, 7}
X(52195) = {X(2),X(27429)}-harmonic conjugate of X(27431)

X(52196) = X(2)X(6)∩X(55)X(332)

X(52196) lies on the cubic K506 and these lines: {2, 6}, {55, 332}, {56, 17206}, {65, 274}, {75, 35614}, {261, 2194}, {314, 3706}, {980, 45787}, {1444, 16678}, {4038, 10452}, {4872, 7018}, {8033, 10401}, {10371, 33297}, {10441, 10471}, {33296, 37614}

X(52196) = X(981)-isoconjugate of X(1400)
X(52196) = X(981)-Dao conjugate of X(40582)
X(52196) = barycentric product X(i)*X(j) for these {i,j}: {274, 35628}, {314, 980}, {2274, 28660}
X(52196) = barycentric quotient X(i)/X(j) for these {i,j}: {21, 981}, {980, 65}, {2274, 1400}, {35628, 37}

X(52197) = X(6)X(512)∩X(67)X(524)

X(52197) lies on the cubic K381 and these lines: {6, 512}, {50, 14908}, {67, 524}, {111, 12367}, {892, 3978}, {1990, 8753}, {2393, 3291}, {2854, 46783}, {3231, 42007}, {7827, 14246}, {14567, 18374}, {15477, 41336}, {18877, 32648}, {32741, 36696}, {33875, 41272}

X(52197) = midpoint of X(895) and X(36827)
X(52197) = isogonal conjugate of the isotomic conjugate of X(46783)
X(52197) = X(i)-isoconjugate of X(j) for these (i,j): {2770, 14210}, {3266, 36150}
X(52197) = X(2770)-Dao conjugate of X(15477)
X(52197) = crosssum of X(524) and X(9177)
X(52197) = X(67)-line conjugate of X(524)
X(52197) = barycentric product X(i)*X(j) for these {i,j}: {6, 46783}, {111, 2854}, {895, 44467}, {7482, 10097}, {9177, 10630}, {37858, 51980}
X(52197) = barycentric quotient X(i)/X(j) for these {i,j}: {2854, 3266}, {9177, 36792}, {19626, 32741}, {32740, 2770}, {41272, 36824}, {44467, 44146}, {46783, 76}, {51819, 34171}
X(52197) = {X(32729),X(32740)}-harmonic conjugate of X(18374)

X(52198) = X(6)X(691)∩X(316)X(524)

X(52198) lies on the cubic K305 and these lines: {6, 691}, {69, 44956}, {111, 352}, {187, 5166}, {316, 524}, {512, 10765}, {574, 12157}, {842, 35188}, {895, 5107}, {1351, 45143}, {1499, 10753}, {2549, 48945}, {5104, 11643}, {5913, 34806}, {5941, 11173}, {6792, 16188}, {8430, 10752}, {9127, 52141}, {10766, 21639}, {11477, 14263}, {14853, 46659}, {15398, 20977}, {23061, 41936}, {31670, 44946}, {34235, 51980}, {36827, 48653}

X(52198) = reflection of X(i) in X(j) for these {i,j}: {69, 44956}, {843, 6}, {44946, 31670}
X(52198) = reflection of X(10765) in the Brocard axis
X(52198) = Schoutte-circle-inverse of X(6091)
X(52198) = barycentric product X(111)*X(50639)
X(52198) = barycentric quotient X(50639)/X(3266)
X(52198) = {X(5107),X(17964)}-harmonic conjugate of X(895)

X(52199) = X(6)X(526)∩X(50)X(237)

X(52199) lies on the cubic K381 and these lines: {6, 526}, {50, 237}, {67, 868}, {297, 340}, {511, 34370}, {842, 5104}, {2421, 36790}, {2493, 2781}, {4230, 6593}, {5649, 51229}, {9161, 23969}, {9420, 43112}, {11646, 34174}, {11672, 14966}, {13330, 38939}, {18877, 32740}, {39469, 51980}

X(52199) = isogonal conjugate of X(46786)
X(52199) = isogonal conjugate of the isotomic conjugate of X(46787)
X(52199) = X(i)-isoconjugate of X(j) for these (i,j): {1, 46786}, {75, 34369}, {290, 2247}, {336, 6103}, {542, 1821}, {1577, 34761}, {1640, 36036}, {5191, 46273}, {18312, 36084}
X(52199) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 46786}, {206, 34369}, {542, 40601}, {1640, 2679}, {18312, 38987}
X(52199) = crossdifference of every pair of points on line {542, 18312}
X(52199) = barycentric product X(i)*X(j) for these {i,j}: {6, 46787}, {110, 23350}, {237, 5641}, {323, 34370}, {511, 842}, {1576, 34765}, {2421, 14998}, {2491, 6035}, {3569, 5649}, {4230, 35909}, {14223, 14966}, {14356, 52179}, {32112, 51263}, {34157, 34174}, {35910, 48453}, {38939, 40083}, {39265, 40080}, {46157, 51862}, {47110, 51472}, {51980, 52094}
X(52199) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 46786}, {32, 34369}, {237, 542}, {842, 290}, {1576, 34761}, {2211, 6103}, {2491, 1640}, {3569, 18312}, {5641, 18024}, {5649, 43187}, {9417, 2247}, {9418, 5191}, {9420, 45321}, {14966, 14999}, {14998, 43665}, {23350, 850}, {34370, 94}, {34765, 44173}, {34859, 35907}, {46787, 76}, {51822, 47105}, {51980, 16092}

X(52200) = X(1)X(30)∩X(5)X(523)

X(52200) lies on the cubic K165 and these lines: {1, 30}, {5, 523}, {11, 2166}, {12, 31522}, {265, 952}, {355, 47274}, {404, 46635}, {476, 953}, {1325, 45977}, {1482, 36154}, {2072, 50144}, {2687, 5253}, {5055, 50145}, {5603, 38514}, {5627, 18357}, {5690, 36155}, {5886, 47270}, {6741, 20304}, {6906, 46636}, {7575, 16332}, {8227, 47273}, {10272, 14985}, {10595, 36171}, {11849, 36167}, {14993, 47272}, {16305, 44282}, {26700, 38602}, {28174, 36158}, {37535, 46618}

X(52200) = midpoint of X(i) and X(j) for these {i,j}: {265, 6742}, {355, 47274}, {1482, 36154}, {31522, 42422}
X(52200) = reflection of X(i) in X(j) for these {i,j}: {3109, 5901}, {5690, 36155}, {6741, 20304}, {7575, 16332}, {14985, 10272}
X(52200) = X(6149)-isoconjugate of X(43655)
X(52200) = X(14993)-Dao conjugate of X(43655)
X(52200) = crossdifference of every pair of points on line {50, 9404}
X(52200) = barycentric quotient X(1989)/X(43655)
X(52200) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {476, 3615, 3109}, {18115, 45934, 5}, {18119, 45934, 18115}

X(52201) = X(13)X(2160)∩X(14)X(2161)

X(52201) lies on the cubic K1278 and these lines: {5, 6192}, {13, 2160}, {14, 2161}, {30, 1277}, {57, 554}, {58, 11073}, {71, 265}, {79, 3383}, {80, 3376}, {1807, 50469}, {7100, 50465}, {10217, 50462}, {19551, 33654}, {33653, 33655}, {39152, 46073}

X(52201) = isotomic conjugate of the polar conjugate of X(11073)
X(52201) = X(i)-isoconjugate of X(j) for these (i,j): {4, 5357}, {1870, 7150}, {3179, 6198}, {17923, 42624}
X(52201) = X(5357)-Dao conjugate of X(36033)
X(52201) = cevapoint of X(1277) and X(6192)
X(52201) = barycentric product X(i)*X(j) for these {i,j}: {69, 11073}, {39151, 40710}, {40709, 42680}
X(52201) = barycentric quotient X(i)/X(j) for these {i,j}: {48, 5357}, {11073, 4}, {36296, 42677}, {36297, 39150}, {39151, 471}, {41225, 17923}, {42680, 470}, {52153, 11072}
X(52201) = {X(39151),X(42680)}-harmonic conjugate of X(11073)

X(52202) = X(13)X(2161)∩X(14)X(2160)

X(52202) lies on the cubic K1278 and these lines: {5, 6191}, {13, 2161}, {14, 2160}, {30, 1276}, {57, 1081}, {58, 11072}, {71, 265}, {79, 3376}, {80, 3383}, {1251, 7052}, {1807, 50468}, {2306, 7126}, {7100, 50466}, {10218, 50462}, {39153, 46077}

X(52202) = isotomic conjugate of the polar conjugate of X(11072)
X(52202) = X(i)-isoconjugate of X(j) for these (i,j): {4, 5353}, {6198, 41225}
X(52202) = X(5353)-Dao conjugate of X(36033)
X(52202) = cevapoint of X(1276) and X(6191)
X(52202) = barycentric product X(i)*X(j) for these {i,j}: {69, 11072}, {39150, 40709}, {40710, 42677}
X(52202) = barycentric quotient X(i)/X(j) for these {i,j}: {48, 5353}, {3179, 17923}, {11072, 4}, {36296, 39151}, {36297, 42680}, {39150, 470}, {42623, 1870}, {42624, 6198}, {42677, 471}, {52153, 11073}
X(52202) = {X(39150),X(42677)}-harmonic conjugate of X(11072)

X(52203) = X(5)X(14)∩X(324)X(473)

X(52203) lies on the cubics K419b and K1278 and these lines: {2, 51268}, {5, 14}, {30, 8174}, {265, 10663}, {324, 473}, {343, 466}, {381, 36300}, {542, 16806}, {577, 11077}, {617, 32036}, {3166, 32423}, {3519, 36297}, {10218, 10662}, {11117, 19712}, {15444, 42148}, {16771, 31610}, {18282, 41089}, {19773, 19779}, {34389, 46754}, {36253, 40683}, {36304, 42974}, {40667, 42943}, {41998, 50465}

X(52203) = isogonal conjugate of X(10632)
X(52203) = isotomic conjugate of the polar conjugate of X(11087)
X(52203) = X(i)-cross conjugate of X(j) for these (i,j): {3, 10218}, {40682, 12028}, {44712, 3}
X(52203) = X(i)-isoconjugate of X(j) for these (i,j): {1, 10632}, {4, 35199}, {19, 11126}, {92, 11135}, {473, 2152}, {1095, 46926}, {1973, 11132}
X(52203) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 10632}, {6, 11126}, {473, 40579}, {6337, 11132}, {11135, 22391}, {35199, 36033}
X(52203) = cevapoint of X(17) and X(8174)
X(52203) = barycentric product X(i)*X(j) for these {i,j}: {14, 40712}, {17, 40710}, {69, 11087}, {301, 32585}, {328, 8603}, {3519, 8836}, {10218, 19779}, {11140, 50469}, {11600, 40709}, {34389, 36297}
X(52203) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 11126}, {6, 10632}, {14, 473}, {17, 471}, {48, 35199}, {69, 11132}, {184, 11135}, {265, 8838}, {3458, 10642}, {8603, 186}, {8836, 32002}, {10218, 16771}, {11085, 46926}, {11087, 4}, {11088, 3518}, {11600, 470}, {21461, 8740}, {32585, 16}, {32586, 10678}, {36296, 6104}, {36297, 61}, {36300, 6116}, {36304, 23714}, {40710, 302}, {40712, 299}, {46113, 3201}, {50433, 50468}, {50466, 11146}, {50469, 1994}, {51477, 8604}, {52153, 11083}
X(52203) = {X(17),X(11600)}-harmonic conjugate of X(11087)

X(52204) = X(5)X(13)∩X(324)X(472)

X(52204) lies on the cubics K419a and K1279 and these lines: {2, 51275}, {5, 13}, {30, 8175}, {265, 10664}, {324, 472}, {343, 465}, {381, 36301}, {542, 16807}, {577, 11077}, {616, 32037}, {3165, 32423}, {3519, 36296}, {10217, 10661}, {11118, 19713}, {15445, 42147}, {16770, 31610}, {18282, 41090}, {19772, 19778}, {34390, 46753}, {36253, 40682}, {36305, 42975}, {37850, 42619}, {40668, 42942}, {41997, 50466}

X(52204) = isogonal conjugate of X(10633)
X(52204) = isotomic conjugate of the polar conjugate of X(11082)
X(52204) = X(i)-cross conjugate of X(j) for these (i,j): {3, 10217}, {40683, 12028}, {44711, 3}
X(52204) = X(i)-isoconjugate of X(j) for these (i,j): {1, 10633}, {4, 35198}, {19, 11127}, {92, 11136}, {472, 2151}, {1094, 46925}, {1973, 11133}
X(52204) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 10633}, {6, 11127}, {472, 40578}, {6337, 11133}, {11136, 22391}, {35198, 36033}
X(52204) = cevapoint of X(18) and X(8175)
X(52204) = barycentric product X(i)*X(j) for these {i,j}: {13, 40711}, {18, 40709}, {69, 11082}, {300, 32586}, {328, 8604}, {3519, 8838}, {10217, 19778}, {11140, 50468}, {11601, 40710}, {34390, 36296}
X(52204) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 11127}, {6, 10633}, {13, 472}, {18, 470}, {48, 35198}, {69, 11133}, {184, 11136}, {265, 8836}, {3457, 10641}, {8604, 186}, {8838, 32002}, {10217, 16770}, {11080, 46925}, {11082, 4}, {11083, 3518}, {11601, 471}, {21462, 8739}, {32585, 10677}, {32586, 15}, {36296, 62}, {36297, 6105}, {36301, 6117}, {36305, 23715}, {40709, 303}, {40711, 298}, {46112, 3200}, {50433, 50469}, {50465, 11145}, {50468, 1994}, {51477, 8603}, {52153, 11088}
X(52204) = {X(18),X(11601)}-harmonic conjugate of X(11082)

X(52205) = ISOGONAL CONJUGATE OF X(4366)

X(52205) lies on the cubics K769 and K775 and these lines: {6, 40730}, {7, 24289}, {37, 40796}, {42, 9506}, {291, 518}, {292, 672}, {335, 350}, {385, 37207}, {660, 2238}, {694, 3509}, {741, 813}, {1458, 30657}, {1911, 1914}, {1977, 9111}, {2113, 35505}, {2276, 3252}, {3721, 52085}, {4562, 6542}, {4876, 40881}, {40217, 49509}, {40774, 40789}

X(52205) = isogonal conjugate of X(4366)
X(52205) = isogonal conjugate of the anticomplement of X(26582)
X(52205) = isogonal conjugate of the complement of X(6653)
X(52205) = isotomic conjugate of the isogonal conjugate of X(51856)
X(52205) = isogonal conjugate of the isotomic conjugate of X(40098)
X(52205) = X(i)-cross conjugate of X(j) for these (i,j): {1, 694}, {39, 37128}, {512, 660}, {1015, 3572}, {3252, 292}, {40155, 291}, {45882, 37207}
X(52205) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4366}, {2, 8300}, {6, 39044}, {75, 51328}, {81, 4368}, {100, 4375}, {101, 27855}, {105, 27919}, {238, 239}, {242, 20769}, {291, 6652}, {312, 12835}, {350, 1914}, {385, 18786}, {659, 3570}, {757, 35068}, {765, 35119}, {812, 3573}, {874, 8632}, {1428, 3975}, {1429, 3685}, {1447, 3684}, {1509, 4094}, {1580, 17493}, {1921, 2210}, {1929, 27926}, {2185, 3027}, {2238, 33295}, {3253, 17475}, {3747, 30940}, {3802, 14621}, {3948, 5009}, {4154, 40432}, {6651, 40767}, {6654, 8299}, {8298, 40725}, {9472, 27916}, {14599, 18891}, {16360, 30661}, {18892, 44169}, {18894, 44171}, {27950, 36815}, {39916, 40769}, {40099, 51903}
X(52205) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 4366}, {9, 39044}, {206, 51328}, {239, 9470}, {350, 36906}, {513, 35119}, {1015, 27855}, {4368, 40586}, {4375, 8054}, {6652, 39029}, {8300, 32664}, {17493, 39092}, {27919, 39046}, {35068, 40607}
X(52205) = cevapoint of X(i) and X(j) for these (i,j): {1, 18787}, {291, 40796}, {1015, 3572}
X(52205) = crosssum of X(i) and X(j) for these (i,j): {239, 27920}, {4375, 35119}, {8299, 17475}
X(52205) = trilinear pole of line {665, 3572}
X(52205) = barycentric product X(i)*X(j) for these {i,j}: {1, 30663}, {6, 40098}, {76, 51856}, {257, 30657}, {291, 291}, {292, 335}, {334, 1911}, {561, 18267}, {660, 876}, {694, 30669}, {741, 43534}, {813, 4444}, {875, 4583}, {904, 30642}, {1581, 18787}, {1922, 18895}, {3572, 4562}, {6645, 41517}, {7077, 7233}, {9505, 40794}, {14598, 44172}, {18897, 44170}, {22116, 52030}, {30648, 52085}, {30671, 37207}, {40217, 51866}
X(52205) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 39044}, {6, 4366}, {31, 8300}, {32, 51328}, {42, 4368}, {181, 3027}, {291, 350}, {292, 239}, {334, 18891}, {335, 1921}, {513, 27855}, {649, 4375}, {660, 874}, {672, 27919}, {694, 17493}, {741, 33295}, {813, 3570}, {869, 3802}, {872, 4094}, {875, 659}, {876, 3766}, {1015, 35119}, {1397, 12835}, {1500, 35068}, {1911, 238}, {1914, 6652}, {1922, 1914}, {1967, 18786}, {2196, 20769}, {3252, 17755}, {3572, 812}, {3862, 3797}, {4518, 4087}, {4562, 27853}, {4876, 3975}, {7077, 3685}, {7233, 18033}, {9506, 40725}, {14598, 2210}, {17735, 27926}, {18267, 31}, {18787, 1966}, {18893, 18892}, {18895, 44169}, {18897, 14599}, {20964, 4154}, {30657, 894}, {30663, 75}, {30669, 3978}, {30671, 4486}, {34067, 3573}, {37128, 30940}, {40098, 76}, {40155, 17793}, {40730, 8299}, {40796, 39028}, {41517, 40099}, {43534, 35544}, {44172, 44171}, {51856, 6}, {51858, 3684}, {51866, 6654}
X(52205) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {291, 36906, 1575}, {335, 39918, 350}, {813, 38874, 17735}, {3252, 40155, 2276}

X(52206) = X(2)X(45)∩X(55)X(106)

X(52206) lies on the cubic K577 and these lines: {2, 45}, {6, 2226}, {42, 34230}, {55, 106}, {57, 1022}, {222, 38828}, {614, 14190}, {679, 36275}, {901, 3052}, {982, 4674}, {995, 14260}, {1149, 17109}, {1320, 17597}, {3257, 4383}, {3752, 52031}, {8027, 23345}, {13744, 24046}, {16483, 39148}, {16569, 36814}, {16610, 52140}

X(52206) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 14260}, {88, 16610}, {4618, 23345}
X(52206) = X(20972)-cross conjugate of X(1149)
X(52206) = X(i)-isoconjugate of X(j) for these (i,j): {44, 1120}, {519, 40400}, {902, 36805}, {1023, 23836}, {1635, 6079}, {1811, 8756}, {2325, 8686}, {30731, 37627}
X(52206) = X(i)-Dao conjugate of X(j) for these (i,j): {1120, 40595}, {4358, 16594}, {16610, 36791}, {36805, 40594}
X(52206) = cevapoint of X(1149) and X(20972)
X(52206) = crosspoint of X(88) and X(2226)
X(52206) = crosssum of X(i) and X(j) for these (i,j): {44, 4370}, {1639, 35092}
X(52206) = trilinear pole of line {1149, 6085}
X(52206) = crossdifference of every pair of points on line {1960, 3689}
X(52206) = barycentric product X(i)*X(j) for these {i,j}: {7, 45247}, {57, 52140}, {75, 17109}, {88, 16610}, {106, 1266}, {679, 17460}, {901, 4927}, {903, 1149}, {2226, 16594}, {4555, 6085}, {4638, 21129}, {6548, 23832}
X(52206) = barycentric quotient X(i)/X(j) for these {i,j}: {88, 36805}, {106, 1120}, {901, 6079}, {1149, 519}, {1266, 3264}, {1417, 8686}, {1878, 38462}, {3880, 4723}, {4695, 3992}, {6085, 900}, {8660, 1960}, {9456, 40400}, {16594, 36791}, {16610, 4358}, {17109, 1}, {17460, 4738}, {20972, 4370}, {23205, 5440}, {23345, 23836}, {23832, 17780}, {36058, 1811}, {45247, 8}, {52140, 312}

X(52207) = X(1)X(99)∩X(2)X(694)

X(52207) lies on the cubic K769 and these lines: {1, 99}, {2, 694}, {7, 3027}, {37, 40786}, {86, 1281}, {291, 51578}, {334, 36934}, {335, 20362}, {350, 9505}, {518, 7061}, {846, 51867}, {3666, 9281}, {4589, 43534}, {5988, 30966}, {5992, 30941}, {6626, 21879}, {49544, 51370}

X(52207) = isotomic conjugate of the isogonal conjugate of X(51867)
X(52207) = X(335)-Ceva conjugate of X(18827)
X(52207) = X(i)-isoconjugate of X(j) for these (i,j): {740, 18757}, {2201, 15377}, {2238, 2248}, {3747, 13610}, {6625, 41333}
X(52207) = X(i)-Dao conjugate of X(j) for these (i,j): {86, 239}, {4010, 6627}
X(52207) = crossdifference of every pair of points on line {5027, 46390}
X(52207) = barycentric product X(i)*X(j) for these {i,j}: {75, 45783}, {76, 51867}, {334, 38814}, {335, 6626}, {337, 2905}, {741, 51857}, {846, 40017}, {1654, 18827}, {4584, 50451}, {4589, 21196}, {17084, 36800}, {17762, 37128}
X(52207) = barycentric quotient X(i)/X(j) for these {i,j}: {295, 15377}, {741, 2248}, {846, 2238}, {1654, 740}, {2905, 242}, {6626, 239}, {17084, 16609}, {17731, 39922}, {17762, 3948}, {18268, 18757}, {18755, 3747}, {18827, 6625}, {21085, 4037}, {21196, 4010}, {27691, 7235}, {27954, 4039}, {37128, 13610}, {38814, 238}, {40017, 51865}, {45783, 1}, {51857, 35544}, {51867, 6}

X(52208) = ISOGONAL CONJUGATE OF X(38814)

X(52208) lies on the cubic K769 and these lines: {37, 171}, {42, 21899}, {81, 21833}, {210, 20698}, {321, 1909}, {335, 20362}, {594, 1215}, {756, 2295}, {893, 21823}, {1100, 21353}, {1255, 40592}, {1824, 7119}, {2171, 15377}, {2653, 36934}, {4425, 23905}, {9281, 23928}, {21883, 35310}, {21887, 21902}, {27569, 40033}

X(52208) = isogonal conjugate of X(38814)
X(52208) = antitomic image of X(9278)
X(52208) = X(i)-cross conjugate of X(j) for these (i,j): {1, 37}, {2653, 6}, {23928, 10}
X(52208) = X(i)-isoconjugate of X(j) for these (i,j): {1, 38814}, {3, 2905}, {6, 6626}, {27, 22139}, {58, 1654}, {60, 27691}, {81, 846}, {86, 18755}, {110, 21196}, {163, 50451}, {238, 45783}, {239, 51867}, {284, 17084}, {593, 21085}, {757, 21879}, {849, 27569}, {1178, 27954}, {1326, 39921}, {1333, 17762}, {1790, 4213}, {2106, 8937}, {2206, 51857}, {3736, 40722}, {8025, 38836}, {14844, 40214}, {17731, 51332}, {40751, 40773}
X(52208) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 38814}, {9, 6626}, {10, 1654}, {37, 17762}, {115, 50451}, {244, 21196}, {846, 40586}, {2905, 36103}, {4075, 27569}, {9470, 45783}, {17084, 40590}, {18755, 40600}, {21879, 40607}, {40603, 51857}
X(52208) = cevapoint of X(i) and X(j) for these (i,j): {1, 13610}, {512, 21823}, {661, 21833}
X(52208) = crosspoint of X(6625) and X(13610)
X(52208) = crosssum of X(846) and X(18755)
X(52208) = trilinear pole of line {4705, 9279}
X(52208) = barycentric product X(i)*X(j) for these {i,j}: {10, 13610}, {37, 6625}, {42, 51865}, {92, 15377}, {313, 18757}, {321, 2248}, {756, 40164}, {40718, 40777}
X(52208) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 6626}, {6, 38814}, {10, 17762}, {19, 2905}, {37, 1654}, {42, 846}, {65, 17084}, {213, 18755}, {228, 22139}, {292, 45783}, {321, 51857}, {523, 50451}, {594, 27569}, {661, 21196}, {756, 21085}, {1500, 21879}, {1824, 4213}, {1911, 51867}, {2107, 8937}, {2171, 27691}, {2248, 81}, {2295, 27954}, {6625, 274}, {9278, 39921}, {13610, 86}, {15377, 63}, {18757, 58}, {21833, 6627}, {40164, 873}, {40747, 40722}, {40777, 30966}, {51865, 310}

X(52209) = ISOTOMIC CONJUGATE OF X(17755)

X(52209) lies on the cubic K769 and these lines: {1, 6185}, {2, 19897}, {37, 40788}, {105, 30664}, {226, 24479}, {239, 335}, {241, 292}, {291, 3008}, {334, 350}, {337, 6559}, {666, 2311}, {694, 41352}, {813, 20367}, {876, 885}, {2170, 35167}, {3252, 27475}, {3747, 36086}, {4384, 40217}, {4639, 17789}, {5098, 7192}, {6542, 13576}, {11349, 34067}, {14267, 17316}, {16823, 52085}, {16826, 40724}, {16831, 27922}, {17308, 40095}, {29571, 40794}, {34018, 43040}, {40747, 40754}

X(52209) = reflection of X(4562) in X(36906)
X(52209) = isotomic conjugate of X(17755)
X(52209) = isotomic conjugate of the complement of X(335)
X(52209) = isotomic conjugate of the isogonal conjugate of X(51866)
X(52209) = X(i)-cross conjugate of X(j) for these (i,j): {1, 18827}, {2, 673}, {514, 4562}, {661, 36086}, {2170, 876}, {3252, 291}, {4369, 34085}, {17451, 1581}, {17719, 39293}, {27942, 6650}, {47694, 41072}
X(52209) = X(i)-isoconjugate of X(j) for these (i,j): {6, 8299}, {9, 51329}, {19, 20778}, {31, 17755}, {41, 39775}, {55, 34253}, {238, 672}, {239, 2223}, {242, 20752}, {350, 9454}, {518, 1914}, {659, 2284}, {665, 3573}, {1026, 8632}, {1252, 38989}, {1428, 3693}, {1429, 2340}, {1458, 3684}, {1818, 2201}, {1911, 27919}, {1921, 9455}, {2210, 3912}, {2238, 3286}, {2283, 4435}, {2356, 20769}, {3252, 8300}, {3263, 14599}, {3747, 18206}, {3930, 5009}, {4366, 40730}, {5089, 7193}, {6654, 42079}, {19561, 40781}, {22116, 51328}, {30941, 41333}, {33295, 39258}
X(52209) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 17755}, {6, 20778}, {9, 8299}, {223, 34253}, {350, 33675}, {478, 51329}, {518, 36906}, {661, 38989}, {672, 9470}, {673, 27945}, {3160, 39775}, {6651, 27919}
X(52209) = cevapoint of X(i) and X(j) for these (i,j): {1, 18785}, {2, 335}, {105, 40754}, {244, 3572}, {291, 3252}, {885, 2170}
X(52209) = trilinear pole of line {291, 812}
X(52209) = barycentric product X(i)*X(j) for these {i,j}: {7, 33676}, {75, 52030}, {76, 51866}, {105, 334}, {291, 2481}, {292, 18031}, {335, 673}, {337, 36124}, {666, 4444}, {876, 51560}, {1027, 4583}, {1438, 18895}, {3572, 36803}, {4876, 34018}, {6185, 40217}, {6654, 40098}, {7233, 14942}, {13576, 18827}, {18785, 40017}, {36801, 43930}
X(52209) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 8299}, {2, 17755}, {3, 20778}, {7, 39775}, {56, 51329}, {57, 34253}, {105, 238}, {239, 27919}, {244, 38989}, {291, 518}, {292, 672}, {294, 3684}, {295, 1818}, {334, 3263}, {335, 3912}, {660, 1026}, {666, 3570}, {673, 239}, {741, 3286}, {813, 2284}, {876, 2254}, {885, 3716}, {1024, 4435}, {1027, 659}, {1416, 1428}, {1438, 1914}, {1462, 1429}, {1814, 20769}, {1911, 2223}, {1922, 9454}, {2196, 20752}, {2481, 350}, {3252, 6184}, {3572, 665}, {4444, 918}, {4518, 3717}, {4562, 42720}, {4876, 3693}, {6185, 6654}, {6654, 4366}, {7077, 2340}, {7233, 9436}, {8751, 2201}, {13576, 740}, {14598, 9455}, {14625, 4771}, {14942, 3685}, {18031, 1921}, {18785, 2238}, {18787, 4447}, {18827, 30941}, {22116, 4712}, {24479, 40781}, {28132, 4148}, {30663, 22116}, {33676, 8}, {34018, 10030}, {35352, 4088}, {36057, 7193}, {36086, 3573}, {36124, 242}, {36796, 3975}, {36803, 27853}, {36816, 4465}, {37128, 18206}, {40017, 18157}, {40098, 40217}, {40217, 4437}, {40724, 1281}, {40730, 42079}, {40754, 19557}, {43534, 3932}, {43921, 27846}, {43929, 8632}, {43930, 43041}, {51560, 874}, {51866, 6}, {52029, 3783}, {52030, 1}, {52146, 8844}
X(52209) = {X(43921),X(46798)}-harmonic conjugate of X(673)

X(52210) = X(2)X(11)∩X(57)X(649)

X(52210) lies on the cubic K577 and these lines: {2, 11}, {57, 649}, {222, 1462}, {354, 43921}, {885, 38371}, {910, 27918}, {927, 5435}, {1002, 40730}, {2246, 24407}, {3008, 8647}, {5091, 5222}, {5332, 51922}, {6557, 6634}, {9533, 40151}, {14197, 24600}, {16833, 36816}, {17127, 36086}, {24403, 40131}

X(52210) = X(673)-Ceva conjugate of X(3008)
X(52210) = X(i)-cross conjugate of X(j) for these (i,j): {2348, 105}, {2976, 927}, {20662, 1279}
X(52210) = X(i)-isoconjugate of X(j) for these (i,j): {672, 1280}, {1477, 3693}, {1810, 5089}, {2223, 36807}, {2254, 6078}, {2284, 35355}, {2340, 43760}
X(52210) = X(i)-Dao conjugate of X(j) for these (i,j): {518, 39048}, {3008, 4437}, {3717, 35111}, {3912, 16593}
X(52210) = cevapoint of X(i) and X(j) for these (i,j): {1279, 20662}, {2348, 3021}
X(52210) = crosspoint of X(673) and X(6185)
X(52210) = crosssum of X(672) and X(6184)
X(52210) = trilinear pole of line {1279, 6084}
X(52210) = crossdifference of every pair of points on line {665, 2340}
X(52210) = barycentric product X(i)*X(j) for these {i,j}: {666, 6084}, {673, 3008}, {1279, 2481}, {2348, 34018}, {6185, 16593}, {8659, 36803}, {48032, 51560}
X(52210) = barycentric quotient X(i)/X(j) for these {i,j}: {105, 1280}, {673, 36807}, {919, 6078}, {1027, 35355}, {1279, 518}, {1416, 1477}, {1462, 43760}, {2348, 3693}, {2976, 4925}, {3008, 3912}, {3021, 40609}, {5853, 3717}, {6084, 918}, {8647, 2340}, {8659, 665}, {16593, 4437}, {20662, 6184}, {20780, 1818}, {36057, 1810}, {48032, 2254}, {51419, 51390}
X(52210) = {X(673),X(6654)}-harmonic conjugate of X(2)

X(52211) = X(1)X(2162)∩X(7)X(350)

X(52211) lies on the cubic K769 and these lines: {1, 2162}, {7, 350}, {37, 40795}, {63, 2053}, {87, 982}, {291, 694}, {330, 20358}, {335, 20359}, {354, 39914}, {518, 2319}, {875, 43931}, {932, 11688}, {1920, 18830}, {3006, 27458}, {3494, 18208}, {3794, 18827}, {4598, 7081}, {8844, 41350}, {14199, 32939}, {17792, 27436}, {23493, 37596}, {32913, 34252}

X(52211) = X(335)-Ceva conjugate of X(40881)
X(52211) = X(17760)-cross conjugate of X(17792)
X(52211) = X(i)-isoconjugate of X(j) for these (i,j): {1403, 39924}, {2209, 18299}
X(52211) = cevapoint of X(17760) and X(27436)
X(52211) = barycentric product X(i)*X(j) for these {i,j}: {1, 27436}, {87, 17760}, {330, 17792}, {561, 18269}, {2162, 51861}, {6383, 18758}, {7155, 28391}, {18830, 45902}, {27424, 41350}
X(52211) = barycentric quotient X(i)/X(j) for these {i,j}: {330, 18299}, {2319, 39924}, {17760, 6376}, {17792, 192}, {18269, 31}, {18758, 2176}, {27436, 75}, {28391, 3212}, {41350, 1423}, {45902, 4083}, {51861, 6382}

X(52212) = X(1)X(47645)∩X(2)X(655)

X(52212) lies on the cubic K577 and these lines: {1, 47645}, {2, 655}, {7, 40215}, {33, 1807}, {55, 2222}, {57, 1020}, {80, 1537}, {92, 324}, {181, 994}, {226, 514}, {278, 7128}, {517, 35015}, {998, 1411}, {1168, 2099}, {1243, 45022}, {1465, 42754}, {1836, 14204}, {3434, 51562}, {5603, 40437}, {21452, 24029}, {23981, 42759}, {34232, 39756}

X(52212) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 14584}, {655, 10015}, {2006, 1465}
X(52212) = X(46393)-cross conjugate of X(2222)
X(52212) = X(i)-isoconjugate of X(j) for these (i,j): {104, 2323}, {654, 36037}, {909, 4511}, {1983, 43728}, {2342, 3218}, {2361, 34234}, {3738, 32641}, {4282, 38955}, {5081, 14578}, {7113, 51565}, {8648, 13136}, {32851, 34858}, {34544, 40437}
X(52212) = X(i)-Dao conjugate of X(j) for these (i,j): {654, 3259}, {2245, 4996}, {2323, 40613}, {3310, 51402}, {3904, 46398}, {3911, 51583}, {4511, 23980}, {16586, 32851}
X(52212) = cevapoint of X(3326) and X(46393)
X(52212) = crosspoint of X(i) and X(j) for these (i,j): {655, 23592}, {2006, 34535}
X(52212) = crosssum of X(i) and X(j) for these (i,j): {654, 35128}, {2323, 34544}
X(52212) = trilinear pole of line {1769, 35013}
X(52212) = barycentric product X(i)*X(j) for these {i,j}: {80, 22464}, {517, 18815}, {655, 10015}, {908, 2006}, {1411, 3262}, {1457, 20566}, {1465, 18359}, {1769, 35174}, {2222, 36038}, {3310, 46405}, {14628, 52031}, {16586, 34535}, {23592, 46398}
X(52212) = barycentric quotient X(i)/X(j) for these {i,j}: {80, 51565}, {517, 4511}, {655, 13136}, {908, 32851}, {1361, 34586}, {1411, 104}, {1457, 36}, {1465, 3218}, {1769, 3738}, {1785, 5081}, {1807, 1809}, {1875, 1870}, {2006, 34234}, {2183, 2323}, {2222, 36037}, {3259, 51402}, {3310, 654}, {6187, 2342}, {10015, 3904}, {14584, 36944}, {18359, 36795}, {18815, 18816}, {22464, 320}, {23706, 4242}, {24029, 4585}, {32675, 32641}, {34586, 4996}, {39534, 44428}

X(52213) = X(55)X(103)∩X(57)X(650)

X(52213) lies on the cubic K577 and these lines: {2, 658}, {55, 103}, {56, 2115}, {57, 650}, {196, 40154}, {226, 15634}, {394, 1252}, {911, 949}, {1025, 3693}, {1407, 24016}, {1815, 7123}, {2283, 2340}, {5089, 34855}, {5228, 9503}, {7177, 44357}, {40779, 43736}

X(52213) = X(36101)-Ceva conjugate of X(241)
X(52213) = X(672)-cross conjugate of X(103)
X(52213) = X(i)-isoconjugate of X(j) for these (i,j): {105, 40869}, {294, 516}, {673, 41339}, {884, 42719}, {910, 14942}, {927, 46392}, {1024, 2398}, {1456, 6559}, {2195, 30807}, {23696, 41321}, {28071, 43035}, {36124, 51376}
X(52213) = X(i)-Dao conjugate of X(j) for these (i,j): {8, 45250}, {30807, 39063}, {35517, 36905}, {39046, 40869}
X(52213) = cevapoint of X(672) and X(1362)
X(52213) = crosssum of X(910) and X(23972)
X(52213) = trilinear pole of line {926, 1458}
X(52213) = barycentric product X(i)*X(j) for these {i,j}: {103, 9436}, {241, 36101}, {518, 43736}, {672, 52156}, {677, 43042}, {883, 2424}, {911, 40704}, {1458, 18025}, {1815, 5236}, {2283, 2400}, {24016, 50333}
X(52213) = barycentric quotient X(i)/X(j) for these {i,j}: {103, 14942}, {241, 30807}, {672, 40869}, {677, 36802}, {911, 294}, {1025, 42719}, {1362, 50441}, {1458, 516}, {2223, 41339}, {2283, 2398}, {2338, 6559}, {2424, 885}, {9436, 35517}, {20752, 51376}, {24016, 927}, {32668, 36146}, {35505, 1566}, {36101, 36796}, {43736, 2481}, {46388, 46392}, {51329, 51435}, {52156, 18031}

X(52214) = X(2)X(371)∩X(16)X(396)

X(52214) lies on the cubic K1199 and these lines: {2, 371}, {13, 590}, {14, 42215}, {15, 36439}, {16, 396}, {17, 34552}, {30, 3389}, {61, 34551}, {62, 15765}, {372, 36468}, {381, 42204}, {397, 43211}, {485, 36463}, {1328, 42228}, {2042, 6419}, {2043, 42162}, {2044, 42150}, {2045, 35812}, {2046, 6453}, {2066, 36461}, {2067, 36460}, {2460, 45879}, {3311, 42565}, {3364, 16962}, {3365, 15764}, {3366, 12817}, {3390, 43228}, {3391, 41121}, {3392, 16268}, {5339, 18586}, {6200, 36455}, {6221, 36456}, {6449, 51924}, {6561, 36454}, {6564, 42134}, {6565, 42111}, {7584, 43107}, {8981, 16963}, {10653, 35822}, {11486, 36438}, {13846, 51854}, {13847, 43238}, {14813, 51727}, {16267, 42563}, {16772, 52047}, {16965, 50246}, {18585, 52045}, {19116, 42976}, {23302, 35255}, {32790, 42498}, {35775, 36462}, {36436, 42142}, {36447, 43255}, {36449, 42152}, {36466, 42265}, {37786, 41490}, {41108, 42235}, {41112, 42256}, {41963, 42592}, {42158, 42253}, {42222, 43544}, {42237, 42973}, {42251, 42488}, {42912, 51728}, {43479, 43526}, {43556, 43568}

X(52214) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {371, 51852, 51853}, {3389, 35731, 51855}, {6200, 36469, 36455}, {36463, 36464, 485}

X(52215) = X(2)X(372)∩X(16)X(396)

X(52215) lies on the cubic K1199 and these lines: {2, 372}, {13, 615}, {14, 42216}, {15, 36457}, {16, 396}, {17, 34551}, {30, 3390}, {61, 34552}, {62, 18585}, {371, 36449}, {381, 42203}, {397, 43212}, {486, 36445}, {1327, 42227}, {2041, 6420}, {2043, 42150}, {2044, 42162}, {2045, 6454}, {2046, 35813}, {2459, 45879}, {3312, 42564}, {3364, 41100}, {3365, 16962}, {3367, 12817}, {3389, 43228}, {3391, 16268}, {3392, 41121}, {5339, 18587}, {5414, 36443}, {6396, 36437}, {6398, 36438}, {6418, 51925}, {6502, 36442}, {6560, 36436}, {6564, 42111}, {6565, 42134}, {7583, 43107}, {10653, 35823}, {11486, 36456}, {13846, 43238}, {13847, 51852}, {13966, 16963}, {15764, 35739}, {15765, 52046}, {16267, 42562}, {16772, 52048}, {19117, 42976}, {23302, 35256}, {32789, 42498}, {35774, 36444}, {36448, 42262}, {36454, 42142}, {36464, 43254}, {36468, 42152}, {37786, 41491}, {41108, 42237}, {41112, 42254}, {41964, 42592}, {42158, 42251}, {42224, 43544}, {42235, 42973}, {42253, 42488}, {42943, 51728}, {43479, 43525}, {43556, 43569}

X(52215) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {372, 51854, 51855}, {6396, 36453, 36437}, {36445, 36447, 486}

X(52216) = X(2)X(372)∩X(14)X(615)

X(52216) lies on the cubic K1199 and these lines: {2, 372}, {13, 42216}, {14, 615}, {15, 395}, {16, 36439}, {18, 34552}, {30, 3365}, {61, 15765}, {62, 34551}, {371, 36467}, {381, 42205}, {398, 43212}, {486, 36463}, {1327, 42229}, {2042, 6420}, {2043, 42159}, {2044, 42151}, {2045, 35813}, {2046, 6454}, {2459, 45880}, {3312, 42562}, {3364, 43229}, {3366, 16267}, {3367, 41122}, {3389, 15764}, {3390, 16963}, {3392, 12816}, {5340, 18586}, {5414, 36461}, {6396, 36455}, {6398, 36456}, {6450, 51925}, {6502, 36460}, {6560, 36454}, {6564, 42114}, {6565, 42133}, {7583, 43100}, {10654, 35823}, {11485, 36438}, {13846, 43239}, {13847, 51853}, {13966, 16962}, {16268, 42564}, {16773, 52048}, {18585, 52046}, {19117, 42977}, {23303, 35256}, {32789, 42499}, {35774, 36462}, {36436, 42139}, {36446, 43254}, {36450, 42149}, {36466, 42262}, {37785, 41491}, {41107, 42238}, {41113, 42255}, {41964, 42593}, {42157, 42250}, {42223, 43545}, {42236, 42972}, {42252, 42489}, {43480, 43525}, {43557, 43569}

X(52216) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {372, 51855, 51854}, {6396, 36470, 36455}, {36463, 36465, 486}

X(52217) = X(2)X(371)∩X(14)X(590)

X(52217) lies on the cubic K1199 and these lines: {2, 371}, {5, 35730}, {13, 42215}, {14, 590}, {15, 395}, {16, 36457}, {18, 34551}, {30, 3364}, {61, 18585}, {62, 34552}, {140, 51727}, {372, 36450}, {381, 42206}, {398, 43211}, {485, 36445}, {1328, 42230}, {2041, 6419}, {2043, 42151}, {2044, 42159}, {2045, 6453}, {2046, 35812}, {2066, 36443}, {2067, 36442}, {2460, 45880}, {3311, 42563}, {3365, 43229}, {3366, 41122}, {3367, 16267}, {3389, 16963}, {3390, 41101}, {3391, 12816}, {5340, 18587}, {6200, 36437}, {6221, 36438}, {6417, 51924}, {6561, 36436}, {6564, 42133}, {6565, 42114}, {7584, 43100}, {8981, 16962}, {10654, 35822}, {11485, 36456}, {13846, 51855}, {13847, 43239}, {15765, 52045}, {16268, 42565}, {16773, 52047}, {19116, 42977}, {23303, 35255}, {32790, 42499}, {35775, 36444}, {36448, 42265}, {36454, 42139}, {36465, 43255}, {36467, 42149}, {37785, 41490}, {41107, 42236}, {41113, 42257}, {41963, 42593}, {42157, 42252}, {42221, 43545}, {42238, 42972}, {42250, 42489}, {42281, 50246}, {43480, 43526}, {43557, 43568}

X(52217) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {371, 51853, 51852}, {6200, 36452, 36437}, {36445, 36446, 485}

X(52218) = X(1)X(7125)∩X(6)X(41)

X(52218) lies on the cubic K180 and these lines: {1, 7125}, {6, 41}, {31, 7335}, {221, 7099}, {602, 36059}, {603, 2187}, {963, 2192}, {1208, 19354}, {1256, 8059}, {1406, 7366}, {1413, 20991}, {1420, 6261}, {1433, 1622}, {1496, 22341}, {2208, 22654}

X(52218) = X(84)-Ceva conjugate of X(603)
X(52218) = X(322)-Dao conjugate of X(7011)
X(52218) = crosspoint of X(8059) and X(24027)
X(52218) = crosssum of X(8058) and X(24026)
X(52218) = barycentric product X(222)*X(1753)
X(52218) = barycentric quotient X(1753)/X(7017)
X(52218) = {X(56),X(19365)}-harmonic conjugate of X(1471)

X(52219) = X(4)X(523)∩X(30)X(113)

X(52219) lies on the cubic K591 and these lines: {4, 523}, {5, 47852}, {30, 113}, {133, 10151}, {381, 46632}, {382, 14934}, {974, 12052}, {2777, 3154}, {3845, 14583}, {3858, 21315}, {3861, 21316}, {7471, 44967}, {7687, 12079}, {7728, 36184}, {10706, 14559}, {10721, 36164}, {10733, 14611}, {11657, 16240}, {12068, 36518}, {12295, 30221}, {12825, 16978}, {13473, 16933}, {15081, 40630}, {15687, 21269}, {16312, 47310}, {36169, 46686}

X(52219) = midpoint of X(i) and X(j) for these {i,j}: {4, 46045}, {382, 14934}, {3258, 13202}, {7471, 44967}, {7728, 36184}, {10721, 36164}, {10733, 14611}, {12825, 16978}
X(52219) = reflection of X(i) in X(j) for these {i,j}: {974, 12052}, {3233, 113}, {11657, 37984}, {12079, 7687}, {16163, 31945}, {21316, 3861}, {36169, 46686}
X(52219) = polar-circle-inverse of X(18808)
X(52219) = crosssum of X(74) and X(15035)
X(52219) = crossdifference of every pair of points on line {2433, 3284}

X(52220) = ISOTOMIC CONJUGATE OF X(11600)

X(52220) lies on the cubic K636 and these lines: {2, 19294}, {13, 298}, {69, 300}, {94, 11071}, {95, 303}, {265, 301}, {299, 6148}, {302, 8838}, {524, 21469}, {621, 1154}, {634, 15067}, {1989, 3181}, {6104, 11132}, {11555, 44776}, {16771, 36980}, {23895, 44361}

X(52220) = reflection of X(17403) in X(33530)
X(52220) = isotomic conjugate of X(11600)
X(52220) = anticomplement of X(19294)
X(52220) = isotomic conjugate of the anticomplement of X(33526)
X(52220) = isotomic conjugate of the isogonal conjugate of X(6104)
X(52220) = X(11084)-anticomplementary conjugate of X(192)
X(52220) = X(46138)-Ceva conjugate of X(300)
X(52220) = X(33526)-cross conjugate of X(2)
X(52220) = X(i)-isoconjugate of X(j) for these (i,j): {31, 11600}, {2151, 11087}, {2154, 8603}, {21461, 51806}
X(52220) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 11600}, {15, 11130}, {1154, 33530}, {8172, 11078}, {8603, 40581}, {10640, 11086}, {11087, 40578}, {37848, 40604}
X(52220) = cevapoint of X(i) and X(j) for these (i,j): {532, 33530}, {622, 37779}
X(52220) = barycentric product X(i)*X(j) for these {i,j}: {13, 11132}, {76, 6104}, {299, 8838}, {300, 11126}, {302, 11078}, {3201, 20573}, {7769, 11601}, {7799, 11581}
X(52220) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 11600}, {13, 11087}, {16, 8603}, {61, 11086}, {302, 11092}, {323, 37848}, {1994, 6105}, {3201, 50}, {6104, 6}, {8838, 14}, {10632, 8739}, {10678, 51891}, {11078, 17}, {11081, 21461}, {11083, 3458}, {11126, 15}, {11132, 298}, {11135, 34394}, {11145, 10677}, {11146, 36209}, {11581, 1989}, {11601, 2963}, {16463, 11060}, {16771, 36210}, {20411, 11062}, {23872, 23284}, {30529, 11582}, {33526, 19294}, {35199, 2151}, {36208, 51890}, {36211, 11139}, {41907, 39432}, {41999, 43968}, {50465, 32585}, {50468, 36297}
X(52220) = {X(69),X(16770)}-harmonic conjugate of X(300)

X(52221) = ISOTOMIC CONJUGATE OF X(11601)

X(52221) lies on the cubic K636 and these lines: {2, 19295}, {14, 299}, {69, 301}, {94, 11071}, {95, 302}, {265, 300}, {298, 6148}, {303, 8836}, {524, 21468}, {622, 1154}, {633, 15067}, {1989, 3180}, {6105, 11133}, {11556, 44777}, {16770, 36978}, {23896, 44362}

X(52221) = reflection of X(17402) in X(33529)
X(52221) = isotomic conjugate of X(11601)
X(52221) = anticomplement of X(19295)
X(52221) = isotomic conjugate of the anticomplement of X(33527)
X(52221) = isotomic conjugate of the isogonal conjugate of X(6105)
X(52221) = X(11089)-anticomplementary conjugate of X(192)
X(52221) = X(46138)-Ceva conjugate of X(301)
X(52221) = X(33527)-cross conjugate of X(2)
X(52221) = X(i)-isoconjugate of X(j) for these (i,j): {31, 11601}, {2152, 11082}, {2153, 8604}, {21462, 51805}
X(52221) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 11601}, {16, 11131}, {1154, 33529}, {8173, 11092}, {8604, 40580}, {10639, 11081}, {11082, 40579}, {37850, 40604}
X(52221) = cevapoint of X(i) and X(j) for these (i,j): {533, 33529}, {621, 37779}
X(52221) = barycentric product X(i)*X(j) for these {i,j}: {14, 11133}, {76, 6105}, {298, 8836}, {301, 11127}, {303, 11092}, {3200, 20573}, {7769, 11600}, {7799, 11582}
X(52221) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 11601}, {14, 11082}, {15, 8604}, {62, 11081}, {303, 11078}, {323, 37850}, {1994, 6104}, {3200, 50}, {6105, 6}, {8836, 13}, {10633, 8740}, {10677, 51890}, {11086, 21462}, {11088, 3457}, {11092, 18}, {11127, 16}, {11133, 299}, {11136, 34395}, {11145, 36208}, {11146, 10678}, {11582, 1989}, {11600, 2963}, {16464, 11060}, {16770, 36211}, {20412, 11062}, {23873, 23283}, {30529, 11581}, {33527, 19295}, {35198, 2152}, {36209, 51891}, {36210, 11138}, {41908, 39433}, {42000, 43967}, {50466, 32586}, {50469, 36296}
X(52221) = {X(69),X(16771)}-harmonic conjugate of X(301)

X(52222) = ISOGONAL CONJUGATE OF X(1981)

X(52222) lies on the Jerabek circumhyperbola circumhyperbola and these lines: {3, 822}, {4, 661}, {6, 810}, {65, 650}, {69, 15411}, {73, 652}, {74, 2249}, {290, 35145}, {296, 2637}, {905, 1439}, {1945, 2432}

X(52222) = isogonal conjugate of X(1981)
X(52222) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1981}, {2, 23353}, {25, 15418}, {108, 1944}, {109, 1948}, {112, 44150}, {162, 8680}, {190, 1430}, {243, 651}, {648, 851}, {653, 1936}, {664, 2202}, {811, 42669}, {1020, 15146}, {1783, 5088}, {1951, 18026}, {4554, 51726}, {6331, 44112}, {6335, 26884}, {6518, 36127}, {7360, 32714}, {9391, 23582}, {36797, 51647}
X(52222) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 1981}, {11, 1948}, {125, 8680}, {243, 38991}, {1944, 38983}, {2202, 39025}, {5088, 39006}, {6505, 15418}, {17423, 42669}, {23353, 32664}, {34591, 44150}
X(52222) = crosssum of X(2811) and X(5179)
X(52222) = trilinear pole of line {647, 3270}
X(52222) = crossdifference of every pair of points on line {243, 1430}
X(52222) = barycentric product X(i)*X(j) for these {i,j}: {296, 522}, {521, 1937}, {525, 2249}, {647, 35145}, {650, 40843}, {652, 1952}, {656, 37142}, {1942, 8062}, {1945, 6332}, {1949, 4391}
X(52222) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 1981}, {31, 23353}, {63, 15418}, {296, 664}, {647, 8680}, {650, 1948}, {652, 1944}, {656, 44150}, {663, 243}, {667, 1430}, {810, 851}, {1459, 5088}, {1937, 18026}, {1945, 653}, {1946, 1936}, {1949, 651}, {1952, 46404}, {2249, 648}, {3049, 42669}, {3063, 2202}, {5075, 41499}, {21761, 450}, {21789, 15146}, {22382, 40888}, {35145, 6331}, {36054, 6518}, {37142, 811}, {40843, 4554}

X(52223) = ISOGONAL CONJUGATE OF X(17811)

X(52223) lies on the conic {{A,B,C,X(2)X(6)}} and these lines: {2, 800}, {4, 41489}, {6, 20}, {25, 1249}, {32, 36413}, {37, 6554}, {42, 4319}, {111, 23591}, {193, 40802}, {194, 6339}, {216, 46952}, {251, 13342}, {253, 13567}, {263, 6467}, {264, 42373}, {308, 32834}, {393, 14249}, {493, 7585}, {494, 7586}, {694, 14772}, {967, 37666}, {1285, 38292}, {1400, 2082}, {1427, 4000}, {1885, 33893}, {1990, 34818}, {2433, 47122}, {2987, 51170}, {2998, 6392}, {3087, 8749}, {3108, 14930}, {3346, 9786}, {3522, 5065}, {3767, 51316}, {5305, 33630}, {5306, 36616}, {6908, 50653}, {7386, 37665}, {7735, 8770}, {8573, 17928}, {8882, 38808}, {8972, 41438}, {12362, 15851}, {13345, 14910}, {13941, 41437}, {16081, 43981}, {16303, 37945}, {17807, 40144}, {18907, 33702}, {21448, 37689}, {32831, 42407}, {40348, 41271}, {41891, 46432}

X(52223) = isogonal conjugate of X(17811)
X(52223) = isotomic conjugate of X(32830)
X(52223) = complement of X(30698)
X(52223) = polar conjugate of X(32000)
X(52223) = isogonal conjugate of the complement of X(11433)
X(52223) = isotomic conjugate of the anticomplement of X(5286)
X(52223) = isogonal conjugate of the isotomic conjugate of X(37874)
X(52223) = polar conjugate of the isotomic conjugate of X(15740)
X(52223) = X(i)-Ceva conjugate of X(j) for these (i,j): {15740, 40174}, {37874, 15740}, {40190, 40192}
X(52223) = X(i)-cross conjugate of X(j) for these (i,j): {5286, 2}, {8898, 19}, {17810, 4}
X(52223) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17811}, {2, 1496}, {31, 32830}, {38, 26224}, {48, 32000}, {63, 1593}, {75, 5065}, {92, 43652}
X(52223) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 32830}, {3, 17811}, {20, 40192}, {206, 5065}, {1249, 32000}, {1496, 32664}, {1593, 3162}, {22391, 43652}, {40182, 40187}
X(52223) = cevapoint of X(i) and X(j) for these (i,j): {6, 8573}, {8678, 14936}
X(52223) = crosssum of X(i) and X(j) for these (i,j): {6, 16936}, {5065, 43652}
X(52223) = trilinear pole of line {512, 6587}
X(52223) = barycentric product X(i)*X(j) for these {i,j}: {4, 15740}, {6, 37874}, {25, 40032}, {253, 40174}, {18840, 40190}
X(52223) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 32830}, {4, 32000}, {6, 17811}, {25, 1593}, {31, 1496}, {32, 5065}, {184, 43652}, {251, 26224}, {15740, 69}, {17810, 33537}, {37874, 76}, {39951, 40187}, {40032, 305}, {40174, 20}, {40190, 3618}

X(52224) = ISOGONAL CONJUGATE OF X(17825)

X(52224) lies on the conic {{A,B,C,X(2)X(6)}} and these lines: {2, 13341}, {6, 3523}, {25, 37665}, {37, 14986}, {216, 52187}, {251, 5065}, {308, 32830}, {493, 7586}, {494, 7585}, {3087, 33631}, {5286, 51316}, {5304, 39951}, {5421, 34288}, {6748, 11282}, {7736, 8770}, {7772, 36413}, {8882, 33871}, {8972, 41437}, {9300, 36616}, {9605, 41489}, {13342, 15717}, {13941, 41438}, {30535, 51170}, {32835, 42407}, {40802, 51171}

X(52224) = isogonal conjugate of X(17825)
X(52224) = isotomic conjugate of X(32834)
X(52224) = isotomic conjugate of the anticomplement of X(31400)
X(52224) = X(31400)-cross conjugate of X(2)
X(52224) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17825}, {31, 32834}, {63, 5198}, {75, 13342}, {2167, 27355}
X(52224) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 32834}, {3, 17825}, {206, 13342}, {3162, 5198}, {27355, 40588}
X(52224) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 32834}, {6, 17825}, {25, 5198}, {32, 13342}, {51, 27355}

X(52225) = X(1)X(1635)∩X(44)X(3251)

X(52225) lies on the cubic K228 and these lines: {1, 1635}, {44, 3251}, {513, 2087}, {519, 6544}, {679, 1022}, {765, 1023}, {812, 39704}, {2384, 2718}, {18822, 35168}

X(52225) = X(i)-isoconjugate of X(j) for these (i,j): {6, 34762}, {100, 51908}, {106, 6633}, {545, 901}, {1644, 4638}, {4555, 8649}, {5376, 14421}, {5548, 43038}, {9268, 14475}
X(52225) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 34762}, {214, 6633}, {545, 38979}, {8054, 51908}
X(52225) = trilinear pole of line {1635, 42084}
X(52225) = barycentric product X(i)*X(j) for these {i,j}: {1, 34764}, {1635, 35168}, {2384, 3762}
X(52225) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 34762}, {44, 6633}, {649, 51908}, {1635, 545}, {2087, 14475}, {2384, 3257}, {3251, 1644}, {34764, 75}, {42084, 33920}

X(52226) = X(1)X(659)∩X(239)X(4375)

X(52226) lies on the cubic K228 and these lines: {1, 659}, {239, 4375}, {1016, 1023}, {1022, 23892}, {1027, 52030}, {2382, 14665}, {14621, 23597}, {18822, 35172}

X(52226) = X(i)-isoconjugate of X(j) for these (i,j): {537, 813}, {660, 20331}
X(52226) = X(537)-Dao conjugate of X(40623)
X(52226) = barycentric product X(i)*X(j) for these {i,j}: {1, 47070}, {659, 18822}, {2382, 3766}
X(52226) = barycentric quotient X(i)/X(j) for these {i,j}: {659, 537}, {2382, 660}, {8632, 20331}, {18822, 4583}, {27846, 36848}, {47070, 75}

X(52227) = X(1)X(41)∩X(919)X(1027)

X(52227) lies on the cubic K228 and these lines: {1, 41}, {919, 1027}, {1022, 23890}, {1023, 5377}, {1024, 36086}, {14190, 51922}

X(52227) = X(i)-isoconjugate of X(j) for these (i,j): {665, 18821}, {840, 918}, {2254, 37131}
X(52227) = trilinear pole of line {2246, 51922}
X(52227) = barycentric product X(i)*X(j) for these {i,j}: {190, 51922}, {528, 36086}, {666, 2246}, {1438, 42722}
X(52227) = barycentric quotient X(i)/X(j) for these {i,j}: {919, 37131}, {2246, 918}, {32666, 840}, {36086, 18821}, {51922, 514}

X(52228) = X(1)X(2254)∩X(85)X(3762)

X(52228) lies on the cubic K228 and these lines: {1, 2254}, {85, 3762}, {514, 4089}, {518, 3126}, {840, 2725}, {1022, 23893}, {1023, 1025}, {4444, 52209}, {18821, 35167}

X(52228) = X(i)-isoconjugate of X(j) for these (i,j): {100, 51922}, {528, 919}, {1643, 5377}, {2246, 36086}
X(52228) = X(i)-Dao conjugate of X(j) for these (i,j): {528, 38980}, {2246, 38989}, {8054, 51922}, {17755, 42722}
X(52228) = crossdifference of every pair of points on line {2246, 51922}
X(52228) = barycentric product X(i)*X(j) for these {i,j}: {918, 37131}, {2254, 18821}
X(52228) = barycentric quotient X(i)/X(j) for these {i,j}: {649, 51922}, {665, 2246}, {840, 36086}, {2254, 528}, {3912, 42722}, {18821, 51560}, {37131, 666}

X(52229) = X(2)X(2418)∩X(30)X(511)

X(52229) lies on the cubic K1298 and these lines: {2, 2418}, {3, 33850}, {5, 7615}, {30, 511}, {39, 8367}, {69, 5077}, {76, 8359}, {99, 9136}, {115, 8355}, {140, 7622}, {148, 7840}, {187, 15300}, {194, 8370}, {230, 2482}, {325, 671}, {376, 9740}, {381, 7620}, {385, 8591}, {396, 36775}, {546, 7775}, {547, 7617}, {548, 7751}, {549, 7610}, {574, 11168}, {597, 3734}, {598, 11055}, {599, 2549}, {620, 44401}, {625, 36523}, {1003, 19661}, {1153, 11812}, {1641, 35606}, {1975, 5305}, {1992, 11159}, {2996, 32984}, {3363, 11163}, {3530, 34506}, {3627, 7758}, {3793, 51224}, {3830, 23334}, {3845, 9766}, {3850, 7764}, {3853, 7759}, {3926, 11318}, {3933, 7841}, {4045, 20582}, {4052, 50223}, {4754, 50235}, {4854, 34914}, {5032, 14033}, {5066, 8176}, {5254, 7801}, {5286, 33237}, {5309, 8368}, {5334, 11295}, {5335, 11296}, {5461, 10150}, {5569, 12100}, {5648, 16334}, {5913, 10717}, {6144, 43618}, {6392, 32985}, {6393, 11287}, {6772, 42035}, {6775, 42036}, {6781, 15480}, {6791, 37746}, {7426, 47293}, {7619, 10124}, {7737, 15534}, {7739, 47352}, {7754, 33007}, {7761, 22165}, {7762, 20105}, {7765, 8364}, {7767, 7833}, {7774, 11317}, {7779, 8596}, {7780, 33923}, {7789, 7817}, {7798, 8584}, {7799, 9166}, {7812, 32819}, {7813, 31173}, {7819, 7827}, {7843, 12102}, {7870, 8361}, {7880, 33213}, {7883, 8357}, {8182, 8667}, {8354, 37671}, {8356, 32474}, {8358, 14711}, {8592, 9887}, {8724, 9877}, {8859, 19570}, {9167, 41139}, {9607, 17130}, {9761, 11543}, {9763, 11542}, {9855, 20094}, {9872, 45012}, {9881, 50254}, {11164, 14614}, {11286, 12215}, {12036, 37745}, {13085, 32448}, {13108, 37345}, {13188, 37461}, {13639, 13644}, {13745, 39581}, {13759, 13763}, {14023, 15704}, {14041, 41136}, {14482, 14535}, {14568, 41134}, {14929, 15533}, {14930, 18842}, {15301, 32459}, {15690, 47101}, {15759, 46893}, {17708, 46338}, {19662, 51397}, {19687, 34604}, {19710, 47102}, {19911, 37459}, {22247, 44381}, {22566, 25486}, {23234, 39663}, {30435, 32822}, {32456, 50774}, {32824, 32954}, {32830, 33190}, {32833, 33184}, {33228, 41135}, {33474, 42628}, {33475, 42627}, {33699, 44678}, {35369, 40246}, {35923, 40996}, {36212, 44468}, {37631, 50218}, {37855, 41676}, {39580, 50163}, {39586, 50177}, {39587, 50171}, {39906, 43957}, {40341, 43619}, {42008, 47097}, {42023, 49261}, {42024, 49262}, {44392, 49215}, {44394, 49214}, {44569, 51389}, {49718, 50216}, {50154, 50167}, {50168, 50183}, {50169, 50184}

X(52229) = crossdifference of every pair of points on line {6, 8644}
X(52229) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5485, 40727}, {2, 9741, 11165}, {2, 11148, 9741}, {2, 11165, 12040}, {2, 40727, 16509}, {2, 51122, 51123}, {99, 11054, 22329}, {99, 22329, 27088}, {115, 22110, 8355}, {115, 39785, 22110}, {148, 7840, 8352}, {325, 671, 37350}, {385, 8591, 8598}, {1992, 11159, 18907}, {1992, 32815, 11159}, {5254, 7801, 8360}, {5485, 9741, 2}, {5485, 11148, 11165}, {5485, 11165, 16509}, {5485, 51122, 12040}, {6390, 47286, 43291}, {7610, 7618, 549}, {7610, 8716, 7618}, {7615, 11184, 5}, {7615, 34511, 11184}, {7617, 9771, 547}, {7620, 9770, 381}, {7622, 15597, 140}, {7779, 8596, 8597}, {7789, 7817, 8365}, {7799, 9166, 41133}, {8176, 18546, 20112}, {8176, 20112, 5066}, {8598, 47287, 8591}, {9741, 11148, 51122}, {9741, 11165, 51123}, {9741, 40727, 12040}, {11148, 40727, 51123}, {11159, 22253, 1992}, {11163, 11185, 3363}, {11165, 40727, 2}, {11165, 51122, 9741}, {11184, 34505, 7615}, {12040, 16509, 2}, {12040, 51123, 11165}, {14148, 32457, 44377}, {16509, 51123, 12040}, {20094, 44367, 9855}, {22253, 32815, 18907}, {33850, 33861, 3}, {34505, 34511, 5}, {40727, 51122, 11165}

X(52230) = X(6)X(1296)∩X(32)X(6233)

X(52230) lies on the circumcircle, the cubic K1299, and these lines: {6, 1296}, {32, 6233}, {98, 43674}, {99, 1285}, {110, 1384}, {111, 2444}, {187, 2709}, {512, 9136}, {574, 39639}, {691, 2030}, {1691, 13241}, {2770, 50566}, {6082, 34581}, {9202, 41406}, {9203, 41407}, {9871, 17968}, {11636, 41412}, {12074, 41413}, {18842, 34164}, {21448, 39236}, {28295, 37508}

X(52230) = reflection of X(9136) in the Brocard axis
X(52230) = Schoutte-circle-inverse of X(2709)
X(52230) = X(i)-isoconjugate of X(j) for these (i,j): {897, 12036}, {14210, 45143}
X(52230) = X(i)-Dao conjugate of X(j) for these (i,j): {6593, 12036}, {15477, 45143}
X(52230) = cevapoint of X(41406) and X(41407)
X(52230) = trilinear pole of line {6, 8644}
X(52230) = barycentric product X(110)*X(43674)
X(52230) = barycentric quotient X(i)/X(j) for these {i,j}: {187, 12036}, {1384, 37745}, {32740, 45143}, {43674, 850}

X(52231) = X(2)X(6)∩X(30)X(11162)

X(52231) lies on the cubic K1298 and these lines: {2, 6}, {30, 11162}, {110, 6093}, {1499, 8598}, {2709, 9084}, {4563, 11054}, {6082, 34581}, {8352, 40915}, {8593, 37860}, {11317, 38951}, {13857, 19662}, {15098, 33007}, {21766, 47061}, {34806, 52141}

X(52231) = midpoint of X(352) and X(14916)
X(52231) = reflection of X(i) in X(j) for these {i,j}: {2, 37745}, {5913, 9127}
X(52231) = anticomplement of X(37746)
X(52231) = orthoptic-circle-of-the-Steiner-inellipse-inverse of X(11184)
X(52231) = psi-transform of X(7618)
X(52231) = crossdifference of every pair of points on line {512, 22111}

X(52232) = X(2)X(523)∩X(30)X(111)

X(52232) lies on the cubic K1298 and these lines: {2, 523}, {30, 111}, {468, 8753}, {524, 32583}, {542, 51938}, {543, 31655}, {671, 858}, {691, 7426}, {895, 40112}, {5159, 10415}, {5648, 52197}, {6677, 8877}, {6791, 34806}, {7799, 30786}, {9140, 51405}, {9172, 36168}, {9759, 36170}, {9870, 31125}, {10418, 17964}, {11580, 46998}, {14694, 46633}, {15360, 36827}, {32729, 35266}, {40343, 46517}, {42008, 47097}, {44212, 52142}

X(52232) = midpoint of X(111) and X(34320)
X(52232) = reflection of X(36168) in X(9172)
X(52232) = X(i)-isoconjugate of X(j) for these (i,j): {896, 10102}, {10103, 23889}
X(52232) = X(10102)-Dao conjugate of X(15899)
X(52232) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 10102}, {9178, 10103}
X(52232) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 46783, 16092}, {691, 52141, 7426}, {47097, 51258, 42008}

X(52233) = X(6)X(512)∩X(111)X(352)

X(52233) lies on the cubic K1299 and these lines: {6, 512}, {111, 352}, {187, 13493}, {542, 34169}, {576, 14263}, {671, 39099}, {691, 2030}, {843, 36696}, {895, 10630}, {1992, 36877}, {3292, 41936}, {5107, 9027}, {6791, 34806}, {8586, 9872}, {11580, 36827}, {14246, 44499}, {15638, 16317}, {22111, 52152}, {47574, 51258}

X(52233) = midpoint of X(i) and X(j) for these {i,j}: {111, 52198}, {8586, 9872}
X(52233) = isogonal conjugate of X(37860)
X(52233) = X(i)-isoconjugate of X(j) for these (i,j): {1, 37860}, {896, 9487}, {9136, 14210}, {23889, 43667}
X(52233) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 37860}, {524, 39075}, {9136, 15477}, {9487, 15899}
X(52233) = barycentric product X(671)*X(9486)
X(52233) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 37860}, {111, 9487}, {9178, 43667}, {9486, 524}, {32740, 9136}

X(52234) = X(6)X(2780)∩X(110)X(9027)

X(52234) lies on the cubic K1299 and these lines: {6, 2780}, {110, 9027}, {895, 13493}, {2434, 6593}, {5468, 15303}, {8744, 13492}, {14559, 47545}

X(52234) = reflection of X(2434) in X(6593)
X(52234) = barycentric product X(i)*X(j) for these {i,j}: {524, 10102}, {5468, 10103}
X(52234) = barycentric quotient X(i)/X(j) for these {i,j}: {10102, 671}, {10103, 5466}

X(52235) = X(2)X(1499)∩X(524)X(46144)

X(52235) lies on the cubics K185 and K1298 and these lines: {2, 1499}, {524, 46144}, {671, 15638}, {6082, 34581}, {11162, 52229}, {26613, 37745}

X(52235) = reflection of X(17952) in X(37746)
X(52235) = barycentric quotient X(i)/X(j) for these {i,j}: {43674, 43667}, {52230, 9136}, {52233, 45143}

X(52236) = X(2)X(10748)∩X(67)X(39602)

X(52236) lies on the cubic K1298 and these lines: {2, 10748}, {67, 39602}, {115, 5094}, {125, 599}, {339, 9464}, {9140, 52231}, {10130, 30739}, {13377, 45163}, {42008, 47097}

X(52237) = X(2)X(23287)∩X(468)X(52229)

X(52237) lies on the cubic K1298 and these lines: {2, 23287}, {468, 52229}, {671, 9084}, {6236, 14653}, {26613, 39157}, {41720, 52231}

X(52238) = X(2)X(47455)∩X(6)X(23)

X(52238) lies on the cubic K1299 and these lines: {2, 47455}, {4, 47581}, {6, 23}, {30, 5050}, {69, 468}, {110, 9027}, {141, 47453}, {182, 7464}, {186, 249}, {187, 15596}, {193, 32113}, {206, 15531}, {323, 6593}, {385, 47561}, {524, 25321}, {575, 37946}, {576, 11464}, {597, 10989}, {691, 2030}, {858, 3618}, {895, 1495}, {1177, 41744}, {1350, 37952}, {1351, 7575}, {1353, 25338}, {1503, 25320}, {1691, 5166}, {1692, 36696}, {1974, 11188}, {1992, 7426}, {2070, 5093}, {2071, 5085}, {2452, 37906}, {2854, 18374}, {3153, 51742}, {3167, 47447}, {3580, 11061}, {3589, 30745}, {3619, 47454}, {3629, 32218}, {3751, 51693}, {5032, 37909}, {5095, 32223}, {5102, 23041}, {5159, 47456}, {5189, 47458}, {5467, 35298}, {5480, 10296}, {5622, 14915}, {5640, 19136}, {6144, 47448}, {6776, 11799}, {7468, 51735}, {7574, 18583}, {7693, 25488}, {8542, 10546}, {8675, 47263}, {9188, 9213}, {9605, 37905}, {10169, 34603}, {10295, 47571}, {10519, 44214}, {10545, 12039}, {10752, 32110}, {10754, 47326}, {11008, 47449}, {11160, 47556}, {11180, 47334}, {11402, 37904}, {11459, 44470}, {11477, 37957}, {11482, 12105}, {11579, 12112}, {11636, 32425}, {11649, 15520}, {12017, 37950}, {12251, 47583}, {15032, 44490}, {15303, 15360}, {15471, 32269}, {15534, 47445}, {16324, 36163}, {16836, 43815}, {17508, 37948}, {18325, 48906}, {18403, 38136}, {18440, 44961}, {18571, 33878}, {18579, 50967}, {18906, 47573}, {19121, 37929}, {19125, 37897}, {19140, 32599}, {19154, 34513}, {19459, 37973}, {20080, 47452}, {20806, 37977}, {27085, 41613}, {31884, 37941}, {32225, 41720}, {32255, 46818}, {32300, 51360}, {32740, 46783}, {34319, 44555}, {34397, 41616}, {35259, 37962}, {36181, 47322}, {37183, 46127}, {37491, 37920}, {37899, 47461}, {37900, 47460}, {37925, 39561}, {37927, 40825}, {37958, 44456}, {37984, 51537}, {39099, 47550}, {47171, 47560}, {47279, 47316}, {47280, 51170}, {47321, 51192}, {47332, 51023}, {47451, 47552}, {47462, 47630}, {47473, 50992}, {47495, 50999}

X(52238) = midpoint of X(i) and X(j) for these {i,j}: {2070, 5093}, {5032, 37909}, {15534, 47445}, {35265, 37784}, {37904, 47463}, {47447, 47541}
X(52238) = reflection of X(i) in X(j) for these {i,j}: {2, 47455}, {2071, 5085}, {5085, 51733}, {10519, 44214}, {18403, 38136}, {35265, 18374}, {47455, 47544}
X(52238) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 32217, 23}, {6, 51797, 8546}, {193, 37760, 32113}, {468, 32220, 69}, {858, 47457, 3618}, {3629, 32218, 47276}, {5095, 32223, 41721}, {7426, 47545, 1992}, {10295, 47571, 51212}, {32113, 47549, 193}

X(52239) = X(6)X(9966)∩X(115)X(599)

X(52239) lies on the cubic K1299 and these lines: {6, 9966}, {115, 599}, {574, 3124}, {2971, 8541}, {5107, 9027}, {6323, 13192}

X(52239) = isogonal conjugate of X(26613)
X(52239) = X(i)-isoconjugate of X(j) for these (i,j): {1, 26613}, {662, 9123}
X(52239) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 26613}, {1084, 9123}
X(52239) = trilinear pole of line {17414, 22260}
X(52239) = barycentric product X(523)*X(9124)
X(52239) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 26613}, {512, 9123}, {9124, 99}

X(52240) = X(1)X(647)∩X(2)X(3)

X(52240) lies on these lines: {1, 647}, {2, 3}, {1624, 25968}, {13869, 47248}, {47004, 47270}, {47175, 47272}, {47250, 47274}, {47254, 47273}

X(52240) = crossdifference of every pair of points on line {647, 851}
X(52240) = X(i)-lineconjugate of X(j) for these (i,j): {1, 647}, {2, 851}
X(52240) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21, 29, 1984}, {415, 416, 851}

X(52241) = X(1)X(672)∩X(2)X(3)

X(52241) lies on these lines: {1, 672}, {2, 3}, {8, 23407}, {40, 35270}, {42, 1453}, {72, 25082}, {238, 2268}, {958, 8299}, {1001, 41325}, {1104, 2276}, {1279, 5069}, {1621, 7085}, {1724, 4251}, {2182, 15254}, {2238, 4258}, {2550, 8053}, {3191, 40586}, {3286, 4648}, {3333, 3720}, {3361, 26102}, {3746, 50282}, {3789, 5302}, {3945, 37507}, {4292, 30949}, {4294, 13576}, {4307, 20992}, {4314, 30502}, {4441, 7283}, {4999, 30959}, {5022, 24512}, {5045, 29814}, {5105, 7290}, {5132, 37650}, {5134, 5144}, {5248, 40910}, {5258, 50316}, {5259, 17732}, {5276, 19761}, {5308, 37609}, {5436, 17754}, {9534, 27109}, {9776, 22060}, {12572, 30961}, {15624, 38057}, {16020, 37575}, {17759, 19851}, {22139, 37685}, {25001, 30273}, {37474, 37659}, {37502, 37681}, {37590, 39587}, {38052, 41430}

X(52241) = crossdifference of every pair of points on line {647, 4724}
X(52241) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1011, 37400}, {2, 4184, 37262}, {3, 405, 4223}, {3, 6913, 37412}, {3, 11108, 37272}, {405, 1009, 2}, {405, 19329, 51555}, {1010, 17687, 19288}, {1011, 37319, 2}, {4184, 4196, 37400}, {6986, 37048, 3}, {7523, 37284, 36510}, {8731, 16058, 2}, {13615, 21483, 4220}, {16373, 30944, 2}, {16439, 20835, 37261}, {24604, 37297, 3}

X(52242) = X(2)X(3)∩X(6)X(661)

X(52242) lies on these lines: {1, 42753}, {2, 3}, {6, 661}, {11, 20999}, {33, 35014}, {55, 35015}, {56, 1647}, {100, 15507}, {109, 38389}, {244, 28083}, {528, 23858}, {692, 45885}, {1001, 25378}, {1324, 3583}, {1376, 9458}, {1807, 15906}, {1862, 2968}, {1878, 46974}, {2098, 3938}, {2099, 34431}, {2222, 3259}, {2635, 26884}, {2933, 12953}, {2969, 15252}, {3835, 24279}, {3961, 5697}, {5091, 6161}, {5687, 17780}, {7741, 23850}, {9318, 24328}, {10896, 23843}, {12138, 38554}, {17798, 40109}, {19736, 19763}, {21382, 36197}, {21842, 29820}, {24329, 32931}, {26611, 29243}, {28602, 46611}, {29349, 35281}, {34361, 48269}, {38507, 42448}, {43728, 45145}

X(52242) = orthocentroidal-circle-inverse of X(867)
X(52242) = crossdifference of every pair of points on line {647, 758}
X(52242) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 867}, {2, 13589, 3}, {2, 16378, 4191}, {4, 28077, 37259}, {4, 46588, 25}, {2478, 19548, 37247}, {3141, 47403, 37041}, {3149, 4186, 28348}, {4192, 35996, 199}, {6915, 28029, 28349}, {7580, 37366, 4191}, {19515, 49127, 3}, {19546, 46549, 37449}, {33311, 45922, 867}, {33849, 36002, 851}

X(52243) = X(2)X(3)∩X(75)X(672)

X(52243) lies on these lines: {2, 3}, {75, 672}, {310, 19792}, {7085, 20172}, {17026, 19787}, {17027, 19791}, {19805, 24631}

X(52244) = X(2)X(3)∩X(10)X(501)

X(52244) lies on these lines: {2, 3}, {10, 501}, {86, 5883}, {392, 25533}, {1326, 37717}, {1330, 44396}, {1737, 2185}, {1834, 44378}, {5080, 40592}, {37715, 38814}

X(52244) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {404, 37152, 1010}, {409, 7483, 11110}, {3109, 37298, 21}

X(52245) = X(2)X(3)∩X(10)X(672)

X(52245) lies on these lines: {2, 3}, {6, 2550}, {7, 10477}, {10, 672}, {42, 5717}, {149, 19740}, {497, 19701}, {942, 20247}, {1438, 13576}, {1441, 1876}, {1742, 38052}, {1834, 24512}, {2268, 50302}, {2285, 50314}, {3419, 5808}, {3434, 19684}, {3555, 4968}, {3585, 19856}, {3720, 23536}, {3741, 4298}, {4314, 43223}, {4651, 5300}, {5283, 6184}, {8299, 25466}, {9612, 30961}, {10449, 10471}, {10453, 11037}, {10527, 19769}, {16713, 37507}, {17183, 48902}, {18698, 32118}, {19717, 33110}, {19732, 26040}, {19742, 20101}, {19755, 31418}, {19759, 30478}, {19762, 19843}, {25524, 30959}, {26770, 39570}, {28797, 37609}, {31995, 50632}, {37676, 49745}

X(52245) = anticomplement of X(16850)
X(52245) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 50694, 4199}, {377, 50408, 15971}, {379, 964, 405}, {442, 1009, 2}, {2049, 49130, 405}

X(52246) = X(2)X(3)∩X(10)X(4370)

X(52246) lies on these lines: {2, 3}, {10, 4370}, {495, 49746}, {519, 4415}, {752, 37715}, {3679, 4126}, {3695, 4908}, {4669, 5295}, {4677, 5814}, {5717, 51103}, {5722, 17274}, {9945, 30867}, {12019, 24723}, {12690, 26580}, {48836, 51415}

X(52246) = midpoint of X(i) and X(j) for these {i,j}: {3543, 23512}, {3679, 33095}
X(52246) = crosssum of X(4256) and X(4257)
X(52246) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5051, 17537, 51672}, {11113, 17532, 440}, {11114, 17577, 46487}, {36721, 36722, 1536}, {37144, 37145, 140}

X(52247) = ORTHOCENTROIDAL-CIRCLE-INVERSE OF X(401)

X(52247) lies on these lines: {2, 3}, {6, 17035}, {53, 3164}, {115, 40814}, {248, 7806}, {264, 1972}, {287, 3818}, {324, 18027}, {327, 3314}, {338, 9220}, {394, 7773}, {577, 32002}, {1879, 41760}, {1993, 7785}, {1994, 7921}, {2052, 9290}, {5158, 37765}, {5422, 7797}, {5480, 47740}, {6033, 40870}, {7752, 36212}, {7777, 11672}, {7851, 10601}, {7920, 34545}, {13509, 15018}, {14129, 46394}, {15595, 19130}, {17129, 45794}, {18424, 41254}, {24206, 42313}, {31276, 37636}, {31610, 36952}, {35142, 40812}, {40896, 41005}, {50188, 50718}

X(52247) = complement of X(51350)
X(52247) = orthocentroidal-circle-inverse of X(401)
X(52247) = polar conjugate of the isogonal conjugate of X(30258)
X(52247) = X(30258)-Dao conjugate of X(40805)
X(52247) = crosspoint of X(327) and X(18027)
X(52247) = crosssum of X(14585) and X(34396)
X(52247) = barycentric product X(i)*X(j) for these {i,j}: {264, 30258}, {311, 9792}
X(52247) = barycentric quotient X(i)/X(j) for these {i,j}: {9792, 54}, {30258, 3}, {43975, 19210}
X(52247) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 401}, {2, 16044, 41231}, {2, 35296, 7907}, {2, 40853, 3}, {5, 297, 2}, {53, 45198, 3164}, {868, 37988, 5117}, {1585, 1586, 436}, {1656, 37067, 2}, {7887, 37344, 2}, {32488, 32489, 6816}

X(52248) = ORTHOCENTROIDAL-CIRCLE-INVERSE OF X(412)

X(52248) lies on these lines: {1, 39531}, {2, 3}, {7, 6355}, {33, 37694}, {92, 946}, {158, 12047}, {162, 37530}, {226, 1895}, {243, 11375}, {281, 962}, {318, 908}, {653, 4295}, {1148, 39542}, {1699, 39585}, {1753, 3305}, {1836, 1940}, {1838, 3817}, {1844, 31803}, {1857, 3485}, {2322, 46019}, {2646, 42387}, {2907, 5788}, {5174, 5587}, {5226, 7952}, {5703, 44695}, {5747, 8748}, {5796, 51223}, {7013, 7282}, {8227, 17923}, {9955, 39529}, {16318, 45985}, {17605, 42385}, {37692, 51282}, {45141, 45991}

X(52248) = midpoint of X(4) and X(7531)
X(52248) = orthocentroidal-circle-inverse of X(412)
X(52248) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 412}, {4, 5, 5125}, {4, 406, 37420}, {4, 3089, 37390}, {4, 3091, 7541}, {4, 4213, 37194}, {4, 6622, 37384}, {4, 6824, 3559}, {4, 6846, 37279}, {4, 7498, 20}, {4, 7501, 46468}, {4, 7551, 3}, {4, 7554, 46467}, {4, 37028, 3146}, {4, 37055, 3575}, {5, 44225, 4}, {381, 7524, 4}, {469, 37448, 857}, {546, 7510, 4}, {946, 39574, 92}, {3832, 7518, 4}, {10151, 37239, 4}

X(52249) = ORTHOCENTROIDAL-CIRCLE-INVERSE OF X(436)

X(52249) lies on these lines: {2, 3}, {51, 43462}, {53, 23332}, {125, 2052}, {184, 14165}, {264, 21243}, {324, 23293}, {393, 23291}, {1075, 26879}, {1092, 43995}, {1249, 18950}, {1629, 11550}, {1853, 33971}, {1899, 11547}, {2979, 14918}, {3168, 6530}, {3462, 7592}, {6524, 37643}, {6750, 20299}, {8884, 18381}, {8887, 32767}, {11433, 41371}, {13450, 26917}, {14569, 51358}, {15466, 39569}, {23294, 44732}, {23295, 34845}, {26913, 46106}, {41588, 44704}, {44114, 44131}

X(52249) = orthocentroidal-circle-inverse of X(436)
X(52249) = barycentric product X(i)*X(j) for these {i,j}: {95, 27358}, {324, 19209}
X(52249) = barycentric quotient X(i)/X(j) for these {i,j}: {19209, 97}, {27358, 5}
X(52249) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 436}, {4, 38282, 6618}, {125, 6747, 2052}, {1585, 1586, 401}, {1899, 11547, 41204}, {6530, 13567, 3168}

X(52250) = ORTHOCENTROIDAL-CIRCLE-INVERSE OF X(439)

X(52250) lies on these lines: {2, 3}, {6, 39143}, {148, 32835}, {193, 13881}, {625, 32828}, {1007, 2996}, {1611, 8892}, {2896, 32870}, {3767, 51170}, {3926, 39565}, {5319, 5461}, {5921, 33684}, {6392, 7752}, {6722, 18845}, {7615, 7888}, {7746, 32827}, {7748, 32839}, {7751, 20080}, {7758, 18362}, {7761, 32867}, {7773, 37667}, {7795, 39601}, {7817, 31417}, {7821, 46951}, {7827, 31407}, {7854, 32885}, {7862, 32815}, {7895, 32868}, {7912, 32834}, {18424, 32826}, {18546, 32824}, {32825, 43681}, {32829, 38259}, {32884, 37512}, {34803, 44518}

X(52250) = orthocentroidal-circle-inverse of X(439)
X(52250) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 439}, {2, 3091, 32979}, {2, 3832, 32981}, {2, 3854, 14035}, {2, 5059, 33259}, {2, 5068, 32991}, {2, 14063, 33023}, {2, 14068, 33205}, {2, 17578, 32964}, {2, 32966, 32980}, {2, 32980, 32982}, {2, 32993, 3146}, {2, 32996, 3522}, {2, 33011, 5068}, {2, 33018, 33201}, {2, 33019, 15717}, {2, 33290, 33025}, {2, 50689, 3552}, {2, 50693, 33206}, {4, 3090, 10011}, {4, 32988, 2}, {4, 33233, 35927}, {4, 33249, 32989}, {5, 8355, 7866}, {5, 11318, 32968}, {5, 14064, 32987}, {5, 32972, 2}, {5, 32984, 32972}, {20, 32967, 2}, {381, 32969, 32973}, {382, 32977, 35287}, {1656, 16041, 32990}, {1656, 32990, 2}, {3090, 32974, 2}, {3090, 33228, 32974}, {3090, 33292, 11285}, {3091, 32961, 2}, {3091, 33181, 33016}, {3523, 32998, 2}, {3526, 37350, 33238}, {3545, 7887, 32971}, {3832, 33205, 14068}, {4208, 33053, 2}, {5025, 5056, 2}, {5025, 32999, 33202}, {5056, 33202, 32999}, {5072, 8361, 32983}, {5079, 33184, 32975}, {6655, 33270, 2}, {6856, 33038, 2}, {7486, 7791, 2}, {7824, 46936, 2}, {7887, 32971, 2}, {7933, 33009, 2}, {8364, 11318, 14064}, {11285, 33228, 33292}, {11285, 33292, 32974}, {11305, 11306, 11539}, {14041, 32998, 3523}, {14045, 33001, 33210}, {14062, 32967, 33000}, {14062, 33000, 20}, {14063, 33023, 32982}, {14063, 33188, 6655}, {14063, 33270, 33188}, {14064, 32968, 8364}, {14064, 32987, 2}, {14068, 33205, 32981}, {16044, 33277, 2}, {16921, 33180, 2}, {16923, 33279, 10304}, {16924, 33199, 2}, {32959, 41099, 19687}, {32963, 32966, 2}, {32967, 33006, 20}, {32969, 32973, 2}, {32972, 32987, 14064}, {32980, 33023, 14063}, {32988, 32989, 33249}, {32989, 33249, 2}, {32999, 33202, 2}, {33000, 33006, 14062}, {33001, 46935, 2}, {33002, 33283, 2}, {33013, 33248, 33198}, {33198, 33248, 2}, {33210, 46935, 33001}

X(52251) = ORTHOCENTROIDAL-CIRCLE-INVERSE OF X(441)

X(52253) lies on these lines: {2, 3}, {6, 15595}, {53, 6389}, {159, 23333}, {264, 6330}, {287, 18440}, {317, 15905}, {343, 44141}, {393, 41005}, {394, 7776}, {459, 47435}, {626, 17811}, {1853, 12202}, {2052, 41009}, {2548, 23292}, {3199, 6509}, {3618, 20204}, {3763, 14767}, {3767, 13567}, {5305, 11433}, {7759, 37672}, {7834, 17825}, {8573, 41770}, {9308, 39352}, {10601, 34836}, {13881, 26958}, {14341, 45327}, {15252, 17316}, {15341, 37648}, {17907, 45198}, {22135, 37649}, {25935, 45271}, {27377, 38292}, {32816, 37669}, {33582, 41761}, {37643, 43291}, {39899, 40867}, {40680, 42459}, {41145, 47353}, {42287, 51537}, {43620, 47296}

X(52251) = midpoint of X(37174) and X(37188)
X(52251) = complement of X(37188)
X(52251) = orthocentroidal-circle-inverse of X(441)
X(52251) = X(i)-complementary conjugate of X(j) for these (i,j): {40801, 18589}, {40823, 828}
X(52251) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 441}, {2, 297, 3}, {2, 3832, 11348}, {2, 5025, 41235}, {2, 37067, 3526}, {2, 37174, 37188}, {2, 37187, 6676}, {2, 37344, 32954}, {5, 44334, 2}, {470, 471, 37070}, {2454, 2455, 44228}, {5000, 5001, 37074}, {18420, 44340, 11286}, {35941, 40853, 1657}

X(52252) = ORTHOCENTROIDAL-CIRCLE-INVERSE OF X(451)

X(52252) lies on these lines: {1, 21911}, {2, 3}, {10, 1870}, {33, 3624}, {34, 1698}, {108, 5433}, {208, 31231}, {274, 1235}, {278, 1224}, {498, 34231}, {499, 7952}, {1068, 17917}, {1125, 1861}, {1172, 17398}, {1395, 33174}, {1398, 9708}, {1753, 8227}, {1845, 5445}, {1872, 11230}, {1876, 5044}, {1890, 38204}, {2356, 29637}, {3074, 21912}, {3192, 17749}, {3634, 46878}, {3753, 41722}, {3925, 41227}, {4413, 11398}, {5081, 27529}, {5089, 25068}, {5277, 10312}, {5330, 31948}, {7009, 19863}, {7046, 10527}, {7718, 19836}, {10157, 12136}, {10449, 28754}, {10785, 18283}, {11491, 25968}, {17904, 17920}, {17913, 17927}, {17923, 41013}, {18140, 44146}, {19881, 49542}, {20268, 20320}, {26540, 36742}, {33172, 44105}, {40971, 50443}, {44426, 48230}

X(52252) = orthocentroidal-circle-inverse of X(451)
X(52252) = polar conjugate of the isotomic conjugate of X(32863)
X(52252) = polar conjugate of the isogonal conjugate of X(5124)
X(52252) = X(5124)-cross conjugate of X(32863)
X(52252) = barycentric product X(i)*X(j) for these {i,j}: {4, 32863}, {92, 6763}, {264, 5124}
X(52252) = barycentric quotient X(i)/X(j) for these {i,j}: {5124, 3}, {6763, 63}, {32863, 69}
X(52252) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 451}, {2, 475, 4}, {2, 4200, 406}, {4, 631, 37441}, {4, 7521, 17562}, {5, 37305, 4}, {28, 427, 4}, {406, 475, 4200}, {406, 4200, 4}, {442, 26020, 1594}, {468, 1883, 4222}, {860, 11109, 4}, {1125, 1861, 6198}, {1593, 37372, 4}, {1594, 37117, 4}, {1883, 4222, 4}, {4185, 5094, 5142}, {4185, 5142, 4}, {4219, 37368, 4}, {5125, 5136, 4}, {6143, 11109, 451}, {14940, 17555, 451}, {15559, 36009, 4}, {24984, 37157, 6951}

X(52253) = ORTHOCENTROIDAL-CIRCLE-INVERSE OF X(467)

X(52253) lies on these lines: {2, 3}, {6, 324}, {53, 37649}, {182, 42400}, {184, 39530}, {264, 275}, {311, 394}, {317, 37636}, {343, 6748}, {1629, 6800}, {1994, 9308}, {2052, 5422}, {2979, 5647}, {3087, 6515}, {4993, 20477}, {4994, 11412}, {5012, 33971}, {5359, 6531}, {9777, 35360}, {10601, 46106}, {11064, 37873}, {11197, 23606}, {11547, 14389}, {12161, 14978}, {13434, 41365}, {14569, 18583}, {19188, 46724}, {27377, 45794}, {32046, 35719}, {34148, 45062}, {36752, 44732}

X(52253) = orthocentroidal-circle-inverse of X(467)
X(52253) = polar conjugate of the isogonal conjugate of X(569)
X(52253) = cevapoint of X(1993) and X(41931)
X(52253) = barycentric product X(264)*X(569)
X(52253) = barycentric quotient X(569)/X(3)
X(52253) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 467}, {2, 8613, 37068}, {4, 458, 41231}, {6, 41244, 324}, {264, 275, 1993}, {1585, 1586, 1594}, {2052, 36794, 5422}

X(52254) = ORTHOCENTROIDAL-CIRCLE-INVERSE OF X(1004)

X(52254) lies on these lines: {2, 3}, {11, 518}, {12, 51715}, {149, 51416}, {200, 10826}, {960, 25006}, {1836, 15297}, {1837, 3870}, {1998, 9581}, {2886, 3305}, {3678, 4847}, {3813, 3984}, {3814, 6745}, {3816, 31266}, {3825, 51706}, {3826, 35258}, {3847, 10404}, {3876, 24390}, {3925, 15254}, {3935, 12019}, {3957, 37730}, {4666, 11375}, {5057, 37787}, {5123, 51432}, {5231, 7741}, {5302, 7173}, {5729, 5905}, {6003, 26017}, {6734, 45120}, {7958, 24541}, {7965, 25973}, {8582, 12558}, {10395, 24391}, {10582, 37692}, {10707, 34894}, {11680, 18228}, {17188, 37649}, {18230, 33108}, {29817, 37737}, {30384, 51379}, {31146, 37720}, {31272, 37797}

X(52254) = complement of X(36003)
X(52254) = orthocentroidal-circle-inverse of X(1004)
X(52254) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 1004}, {2, 2475, 35985}, {2, 5046, 1005}, {2, 10431, 37309}, {5, 14022, 2}, {377, 4193, 50206}, {452, 6991, 47516}, {1532, 37374, 36002}, {2476, 31259, 8728}, {3091, 17554, 50736}, {4187, 8226, 37363}, {4187, 37363, 2}, {4193, 10883, 2}, {6831, 8226, 10883}, {8226, 37359, 14022}, {19512, 33302, 37449}

X(52255) = ORTHOCENTROIDAL-CIRCLE-INVERSE OF X(1005)

X(52255) lies on these lines: {2, 3}, {9, 5057}, {11, 5809}, {12, 3189}, {100, 7679}, {149, 954}, {218, 33139}, {226, 3873}, {329, 2886}, {1260, 33110}, {1708, 20292}, {1776, 5880}, {1864, 3838}, {3419, 3935}, {3434, 8543}, {3485, 3813}, {3486, 31936}, {3586, 3822}, {3869, 25006}, {3870, 5086}, {3925, 5698}, {4666, 10393}, {4847, 5904}, {5231, 9612}, {5249, 10394}, {5728, 31019}, {6601, 8232}, {6745, 7951}, {10382, 31266}, {10399, 11263}, {11373, 29817}, {17616, 27186}, {24387, 31146}, {40998, 41858}

X(52255) = orthocentroidal-circle-inverse of X(1005)
X(52255) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 1005}, {2, 377, 35985}, {2, 2475, 1004}, {4, 405, 11114}, {4, 6846, 6912}, {4, 6871, 46870}, {4, 8226, 10883}, {5, 37363, 2}, {442, 6990, 4193}, {442, 8226, 14022}, {442, 14022, 2}, {1004, 37301, 35977}, {2476, 10883, 2}, {2476, 11114, 4197}, {2476, 17577, 6932}

X(52256) = ORTHOCENTROIDAL-CIRCLE-INVERSE OF X(1008)

X(52256) lies on these lines: {1, 30953}, {2, 3}, {10, 3975}, {42, 5015}, {76, 10471}, {83, 16889}, {256, 3821}, {315, 37632}, {1330, 37676}, {1698, 2108}, {1714, 7803}, {2276, 16886}, {3670, 33940}, {3741, 13161}, {4279, 4660}, {4385, 31330}, {4388, 4972}, {4429, 17277}, {5254, 23897}, {7762, 40721}, {17304, 50614}, {20556, 26115}, {24259, 24851}, {26094, 30993}, {26098, 32773}, {28356, 32947}

X(52256) = orthocentroidal-circle-inverse of X(1008)
X(52256) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 1008}, {2, 26117, 16850}, {5, 37148, 2}, {13740, 37086, 37027}

X(52257) = ORTHOCENTROIDAL-CIRCLE-INVERSE OF X(1009)

X(52257) lies on these lines: {1, 20544}, {2, 3}, {10, 4517}, {11, 30811}, {43, 3191}, {72, 30961}, {220, 1329}, {625, 3110}, {626, 30945}, {942, 30985}, {1210, 20335}, {1479, 8299}, {1698, 3294}, {1714, 2238}, {2486, 5695}, {2548, 24512}, {3454, 19755}, {3813, 50316}, {3814, 5091}, {3821, 27691}, {3934, 10479}, {4876, 16886}, {5292, 37676}, {5305, 37657}, {5439, 30949}, {5687, 13576}, {7741, 29637}, {7776, 30941}, {7951, 16788}, {9612, 17754}, {12607, 50282}, {15888, 48830}, {19754, 20083}, {20256, 26132}, {20430, 26665}, {20556, 37590}, {24387, 50311}, {25448, 31187}, {30960, 33140}, {30962, 32816}

X(52257) = orthocentroidal-circle-inverse of X(1009)
X(52257) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 1009}, {2, 37193, 8731}, {4, 1009, 11355}, {6998, 37086, 13723}, {7741, 29637, 30959}, {36687, 37445, 37049}

X(52258) = ORTHOCENTROIDAL-CIRCLE-INVERSE OF X(1010)

X(52258) lies on these lines: {1, 3846}, {2, 3}, {8, 37715}, {10, 312}, {46, 24723}, {158, 264}, {314, 5224}, {333, 5292}, {386, 5233}, {387, 14555}, {581, 18465}, {612, 5015}, {756, 36568}, {940, 1330}, {942, 27184}, {975, 7270}, {986, 4425}, {999, 5484}, {1046, 4703}, {1098, 1724}, {1125, 17717}, {1150, 26064}, {1193, 25960}, {1210, 4357}, {1211, 10449}, {1329, 4026}, {1352, 25898}, {1468, 29845}, {1479, 5263}, {1698, 4429}, {1714, 17277}, {1834, 5743}, {1999, 5814}, {2322, 34266}, {2551, 27539}, {2886, 19853}, {3210, 50067}, {3454, 18134}, {3555, 29843}, {3623, 26758}, {3634, 17601}, {3662, 5439}, {3670, 4389}, {3695, 41839}, {3701, 29667}, {3746, 49746}, {3772, 16817}, {3814, 5143}, {3821, 24174}, {3831, 21257}, {3868, 26580}, {3944, 49598}, {3980, 24851}, {4085, 6048}, {4150, 4687}, {4388, 5711}, {4972, 9780}, {5247, 29635}, {5278, 24883}, {5297, 5300}, {5530, 50290}, {5587, 35635}, {5708, 26840}, {5741, 19767}, {5799, 9535}, {5955, 32932}, {7262, 8258}, {7741, 19863}, {9612, 10436}, {10591, 19866}, {11681, 26115}, {14534, 43531}, {16478, 29645}, {17046, 17306}, {17202, 18180}, {17592, 17748}, {17749, 48843}, {19701, 20337}, {19732, 25446}, {19804, 23537}, {19858, 25639}, {19874, 33108}, {20653, 32915}, {21935, 31339}, {24217, 50608}, {24443, 32776}, {24621, 50177}, {24931, 48863}, {24936, 30834}, {25645, 40430}, {26085, 37675}, {26625, 36742}, {27269, 31090}, {27410, 28836}, {28082, 32775}, {28612, 36250}, {33084, 35633}, {33121, 41229}, {33174, 46827}, {33932, 42714}, {37646, 49728}, {37683, 49716}

X(52258) = orthocentroidal-circle-inverse of X(1010)
X(52258) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 1010}, {2, 452, 37176}, {2, 2475, 16454}, {2, 2478, 13740}, {2, 3839, 50407}, {2, 4201, 474}, {2, 4205, 37039}, {2, 5046, 964}, {2, 5051, 16062}, {2, 5084, 13741}, {2, 5129, 13742}, {2, 5177, 37153}, {2, 13725, 19270}, {2, 13736, 6857}, {2, 13740, 37036}, {2, 16062, 33833}, {2, 17550, 33838}, {2, 17676, 404}, {2, 17680, 33035}, {2, 17685, 17688}, {2, 17689, 7907}, {2, 17697, 17698}, {2, 19278, 140}, {2, 22267, 17694}, {2, 26051, 16458}, {2, 26117, 3}, {2, 26601, 37445}, {2, 33029, 33821}, {2, 33736, 37097}, {2, 33832, 16917}, {2, 37150, 51604}, {2, 37162, 5192}, {2, 37164, 13728}, {2, 37314, 11110}, {2, 37339, 17567}, {2, 51681, 37298}, {3, 26117, 37038}, {5, 4205, 2}, {405, 11334, 21}, {442, 5084, 17671}, {452, 37176, 13735}, {474, 50056, 4201}, {1656, 19273, 2}, {1834, 5743, 9534}, {2475, 16454, 48816}, {2476, 2478, 469}, {4187, 13728, 2}, {5051, 16342, 37156}, {5051, 37314, 17555}, {5055, 50410, 2}, {16458, 17532, 26051}, {16904, 32961, 2}, {17567, 51665, 37339}, {19266, 19543, 404}

X(52259) = MIDPOINT OF X(3) AND X(15763)

X(52259) lies on these lines: {2, 3}, {10, 51698}, {58, 24884}, {975, 37729}, {1125, 9895}, {2828, 6713}, {2838, 6714}, {3002, 16604}, {3624, 9816}, {4999, 6708}, {5433, 37695}, {5810, 26958}, {6692, 6693}, {6707, 34847}, {11064, 18180}, {15172, 41230}, {15943, 34773}, {16100, 27195}, {16608, 42463}, {19372, 31231}, {23305, 49553}, {35466, 37697}, {36812, 51775}, {40533, 40561}

X(52259) = midpoint of X(i) and X(j) for these {i,j}: {3, 15763}, {5, 44220}, {10, 51698}, {28, 21530}, {381, 48370}, {442, 44253}, {18641, 44290}
X(52259) = complement of X(21530)
X(52259) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 28, 21530}, {2, 7521, 3}, {2, 7535, 5}, {2, 7561, 140}, {2, 19285, 8728}, {28, 31154, 31293}, {1125, 40530, 9895}, {2454, 2455, 44330}

X(52260) = MIDPOINT OF X(3) AND X(44225)

X(52260) lies on these lines: {2, 3}, {10, 51699}, {78, 37729}, {142, 40657}, {282, 7100}, {936, 37696}, {942, 9119}, {1125, 6708}, {1146, 8555}, {1698, 7070}, {2287, 22136}, {2816, 6711}, {3002, 46830}, {3624, 37695}, {3646, 9816}, {3812, 40535}, {3927, 27382}, {5713, 26958}, {6684, 44916}, {6693, 40532}, {6703, 9843}, {6706, 6707}, {8747, 18643}, {9895, 11221}, {13411, 15252}, {17073, 39585}, {18635, 49743}, {27398, 41014}, {44922, 51118}

X(52260) = midpoint of X(i) and X(j) for these {i,j}: {3, 44225}, {4, 44244}, {10, 51699}, {29, 18641}, {21530, 44290}
X(52260) = complement of X(18641)
X(52260) = X(40407)-complementary conjugate of X(18642)
X(52260) = X(14544)-Ceva conjugate of X(522)
X(52260) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 29, 18641}, {2, 3152, 31258}, {2, 7498, 3}, {2, 7515, 140}, {2, 7532, 5}, {2, 16416, 50409}, {29, 31155, 31294}, {29, 31258, 3152}, {2454, 2455, 44331}, {3152, 31258, 18641}, {14782, 14783, 37417}

X(52261) = MIDPOINT OF X(2) AND X(44215)

X(52261) lies on these lines: {2, 3}, {10, 51703}, {141, 51735}, {230, 3229}, {327, 37688}, {512, 31286}, {2387, 6680}, {3117, 5305}, {3564, 36213}, {3589, 34236}, {3978, 6390}, {5106, 43291}, {5943, 11272}, {5972, 14693}, {6683, 6688}, {7806, 19222}, {9155, 51481}, {9306, 10104}, {11064, 47638}, {12042, 42671}, {17414, 47173}, {20576, 23292}, {32448, 40814}, {32515, 36212}, {33873, 47582}, {51430, 52144}

X(52261) = midpoint of X(i) and X(j) for these {i,j}: {2, 44215}, {3, 44227}, {5, 44221}, {10, 51703}, {141, 51735}, {237, 21531}
X(52261) = complement of X(21531)
X(52261) = crossdifference of every pair of points on line {647, 21001}
X(52261) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 237, 21531}, {2, 11328, 5}, {2, 16925, 11333}, {2, 20885, 1368}, {2, 37465, 37988}, {5, 549, 35934}, {2454, 2455, 10684}, {21531, 44215, 237}

X(52262) = MIDPOINT OF X(2) AND X(44218)

X(52262) lies on these lines: {2, 3}, {10, 51707}, {66, 10249}, {68, 11425}, {141, 51739}, {182, 23329}, {343, 13352}, {389, 44158}, {511, 44201}, {517, 51718}, {567, 11245}, {570, 3018}, {578, 12359}, {620, 14767}, {1147, 31831}, {1154, 44683}, {1352, 47391}, {1503, 18475}, {1511, 18358}, {1899, 37506}, {2781, 3589}, {3580, 15033}, {3818, 11202}, {4846, 10606}, {5432, 37697}, {5433, 37696}, {5449, 12241}, {5462, 20191}, {5622, 10264}, {5890, 14389}, {5891, 11064}, {5907, 9820}, {5943, 44673}, {5946, 18583}, {6247, 44679}, {6689, 6696}, {7689, 12233}, {9729, 25563}, {9730, 37649}, {9813, 50977}, {9827, 11808}, {10165, 51692}, {10170, 14156}, {10192, 46261}, {10574, 43607}, {11402, 18917}, {11424, 41587}, {11426, 18951}, {11430, 21243}, {11605, 38699}, {12022, 23293}, {12042, 38616}, {12134, 13367}, {12370, 34826}, {12827, 32423}, {13434, 26879}, {13754, 23292}, {15030, 51425}, {15136, 22473}, {15325, 37729}, {16165, 35283}, {16235, 44817}, {16252, 44516}, {16655, 18488}, {16658, 26881}, {18474, 39242}, {18914, 32046}, {19220, 31455}, {23583, 44562}, {26937, 36752}, {31804, 32140}, {32300, 46267}, {34380, 41614}, {36989, 39884}, {37476, 40686}, {37483, 43653}, {37636, 43574}, {39522, 41588}, {41730, 50708}, {43608, 43651}

X(52262) = midpoint of X(i) and X(j) for these {i,j}: {2, 44218}, {3, 427}, {4, 44249}, {5, 18570}, {10, 51707}, {140, 44236}, {141, 51739}, {343, 13352}, {378, 15760}, {381, 44285}, {549, 44287}, {550, 44288}, {1368, 44274}, {2072, 44281}, {11430, 21243}, {11585, 44269}, {31133, 44261}, {31723, 44239}, {39242, 45303}
X(52262) = reflection of X(i) in X(j) for these {i,j}: {546, 13413}, {6676, 140}, {16618, 6676}, {46029, 3628}
X(52262) = complement of X(15760)
X(52262) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5, 44911}, {2, 378, 15760}, {2, 7527, 403}, {2, 9818, 5}, {2, 10257, 140}, {2, 37118, 10257}, {3, 5, 31833}, {3, 381, 18533}, {3, 1656, 6815}, {3, 3526, 7383}, {3, 3541, 23335}, {3, 5576, 3575}, {3, 14790, 550}, {3, 31723, 44239}, {3, 38321, 37931}, {3, 48411, 37347}, {4, 7542, 13383}, {5, 140, 16238}, {5, 549, 6644}, {5, 6644, 10127}, {5, 23323, 5066}, {5, 37814, 9825}, {5, 44452, 6677}, {5, 52070, 546}, {68, 11425, 43595}, {140, 546, 10020}, {140, 3853, 34577}, {140, 5066, 44234}, {140, 6677, 44452}, {140, 12103, 34004}, {140, 13383, 7542}, {140, 16197, 7568}, {140, 23336, 16196}, {140, 44232, 10125}, {381, 468, 44233}, {427, 44239, 31723}, {546, 10020, 21841}, {547, 44920, 5}, {550, 7568, 16197}, {578, 12359, 13292}, {631, 7404, 6642}, {1594, 14118, 12605}, {1596, 31861, 44804}, {3523, 7391, 44837}, {3524, 31133, 44261}, {3530, 9825, 37814}, {3575, 5576, 546}, {3628, 5498, 140}, {3850, 10125, 44232}, {5169, 10298, 7576}, {5576, 7499, 10096}, {6642, 7404, 5}, {6644, 18580, 549}, {6677, 44452, 16238}, {7488, 15559, 7553}, {7499, 31723, 25337}, {7499, 47090, 3}, {7503, 37119, 11585}, {7514, 18281, 1368}, {7552, 13596, 47096}, {7577, 52069, 10297}, {10024, 14130, 1885}, {10201, 31861, 1596}, {11585, 37119, 32144}, {11818, 18324, 37458}, {11819, 33332, 16198}, {12106, 34477, 37935}, {14782, 14783, 3147}, {15331, 50138, 31830}, {15760, 16387, 25337}, {15760, 44218, 378}, {15765, 18585, 34351}, {18323, 49669, 52070}, {18420, 31723, 44263}, {31664, 31665, 140}, {37077, 47332, 14893}, {37347, 37454, 547}, {37347, 48411, 37454}, {42807, 42808, 3515}

X(52263) = MIDPOINT OF X(2) AND X(44219)

X(52263) lies on these lines: {2, 3}, {114, 6108}, {230, 5471}, {299, 34380}, {395, 3564}, {396, 18583}, {511, 44382}, {576, 33458}, {619, 7685}, {624, 44377}, {630, 51754}, {1352, 16645}, {1353, 37641}, {1503, 6774}, {3054, 42143}, {3055, 6115}, {3589, 6771}, {3815, 11542}, {5460, 44401}, {5474, 41037}, {5476, 33475}, {5617, 18358}, {5978, 38227}, {6036, 6670}, {6109, 20253}, {6672, 41023}, {9749, 33378}, {11178, 33474}, {13349, 22797}, {13468, 34508}, {14561, 16644}, {15069, 49906}, {16627, 42148}, {16652, 33407}, {16773, 37825}, {18581, 37637}, {18582, 31489}, {20426, 52193}, {20429, 42943}, {21159, 36962}, {22511, 36776}, {23005, 42915}, {23514, 31710}, {24206, 44383}, {33459, 34507}, {33561, 44666}, {36959, 42164}, {37824, 42599}

X(52263) = midpoint of X(i) and X(j) for these {i,j}: {2, 44219}, {3, 41017}, {4, 44250}, {5, 44223}, {114, 6108}, {381, 35303}, {395, 5613}, {549, 44289}, {619, 7685}, {13349, 22797}, {20426, 52193}, {20429, 42943}, {31693, 37461}
X(52263) = reflection of X(43417) in X(20253)
X(52263) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(44462)
X(52263) = circumcircle-of-outer-Napoleon-triangle-inverse of X(36181)
X(52263) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 381, 44459}, {230, 6114, 11543}, {11302, 41041, 44461}

X(52264) = MIDPOINT OF X(10) AND X(51714)

X(52264) lies on these lines: {1, 47742}, {2, 3}, {8, 35272}, {10, 6691}, {46, 24954}, {56, 3820}, {58, 51415}, {72, 34753}, {100, 15172}, {329, 37545}, {495, 25524}, {496, 1376}, {499, 4413}, {551, 12640}, {908, 24470}, {936, 31190}, {942, 6692}, {950, 9945}, {952, 17614}, {1125, 1387}, {1213, 5053}, {1329, 18990}, {1385, 8582}, {1420, 1698}, {1467, 5791}, {1478, 31246}, {1483, 5554}, {1697, 3624}, {1706, 11373}, {2078, 7294}, {3086, 9709}, {3216, 37634}, {3306, 6147}, {3452, 37582}, {3614, 31263}, {3634, 4999}, {3670, 43055}, {3746, 6174}, {3753, 5901}, {3763, 5820}, {3812, 37737}, {3813, 10199}, {3816, 15171}, {3819, 34466}, {3833, 11281}, {3841, 20107}, {3898, 32157}, {3911, 5044}, {3927, 5435}, {4266, 17398}, {4317, 31141}, {4968, 37762}, {5030, 38930}, {5045, 6745}, {5121, 5266}, {5122, 12572}, {5253, 17757}, {5258, 5298}, {5275, 31406}, {5316, 31445}, {5437, 11374}, {5438, 5722}, {5439, 5719}, {5440, 12433}, {5482, 5943}, {5563, 21031}, {5660, 35010}, {5690, 19861}, {5770, 5780}, {5777, 13226}, {5832, 20195}, {5843, 26877}, {5883, 16137}, {5886, 49163}, {6259, 21164}, {6390, 18140}, {6667, 25639}, {6684, 31798}, {6690, 19862}, {6703, 20108}, {6705, 10157}, {6713, 9956}, {6736, 51788}, {7080, 7373}, {7288, 9708}, {7681, 31777}, {7956, 10310}, {8583, 26446}, {8666, 9711}, {8715, 15170}, {8981, 31473}, {9843, 24929}, {10247, 24558}, {10543, 15015}, {10827, 24953}, {11227, 18238}, {11248, 25893}, {11814, 24850}, {12019, 17647}, {12679, 16209}, {12699, 25522}, {12702, 26062}, {14829, 49718}, {15803, 20196}, {15934, 27383}, {15988, 51732}, {17612, 40263}, {17619, 18357}, {17749, 37646}, {18146, 32820}, {19860, 38028}, {20103, 34790}, {20789, 44675}, {23708, 34595}, {24046, 39544}, {24387, 49732}, {25681, 39542}, {25934, 36754}, {25973, 26470}, {26040, 31493}, {28174, 41012}, {31663, 40998}, {32153, 52148}, {34123, 51700}, {34612, 37720}, {34773, 35262}, {37522, 37663}, {37662, 49743}, {47038, 51126}

X(52264) = midpoint of X(i) and X(j) for these {i,j}: {10, 51714}, {404, 4187}, {5563, 21031}, {17614, 24982}
X(52264) = complement of X(4187)
X(52264) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 17527}, {2, 21, 17575}, {2, 140, 6675}, {2, 404, 4187}, {2, 405, 51559}, {2, 442, 3628}, {2, 443, 1656}, {2, 474, 5}, {2, 631, 11108}, {2, 3523, 17559}, {2, 4208, 5067}, {2, 5054, 50202}, {2, 6856, 5070}, {2, 6857, 16853}, {2, 6910, 16842}, {2, 6921, 405}, {2, 7483, 50205}, {2, 7824, 33034}, {2, 10303, 16845}, {2, 11319, 37051}, {2, 13747, 140}, {2, 15674, 17546}, {2, 16408, 8728}, {2, 16410, 6922}, {2, 16411, 8727}, {2, 16845, 16855}, {2, 16917, 32992}, {2, 17531, 442}, {2, 17535, 17529}, {2, 17566, 7483}, {2, 17567, 3}, {2, 17572, 4193}, {2, 17580, 3090}, {2, 17670, 8361}, {2, 17694, 7819}, {2, 31259, 16856}, {2, 32974, 33048}, {2, 32990, 33042}, {2, 33001, 33036}, {2, 33039, 32975}, {2, 33054, 7770}, {2, 36006, 17533}, {2, 37229, 50206}, {2, 37291, 17536}, {2, 44217, 15699}, {3, 1656, 6893}, {3, 5084, 50241}, {3, 17567, 17564}, {4, 16417, 17563}, {10, 6691, 15325}, {140, 11277, 11812}, {140, 50205, 7483}, {377, 17558, 1012}, {381, 6904, 50240}, {405, 6921, 549}, {474, 19525, 404}, {499, 4413, 31419}, {547, 50238, 2476}, {549, 51559, 405}, {631, 6848, 3}, {631, 6983, 1012}, {1012, 6983, 5}, {1125, 5836, 1387}, {1376, 10200, 496}, {1656, 6923, 5}, {2475, 17533, 3850}, {2478, 16371, 550}, {3090, 17580, 17528}, {3149, 6967, 37364}, {3523, 17559, 16418}, {3524, 5129, 17571}, {3526, 16863, 2}, {3634, 6681, 4999}, {3816, 25440, 15171}, {3843, 19706, 37435}, {4188, 11113, 548}, {4190, 17556, 3627}, {4193, 11112, 546}, {4193, 17572, 11112}, {5054, 16853, 6857}, {5054, 44284, 549}, {5187, 50239, 3845}, {5298, 50038, 5258}, {5439, 27385, 5719}, {6692, 6700, 942}, {6857, 16853, 50202}, {6872, 19537, 8703}, {6882, 45976, 37281}, {6891, 6918, 8727}, {6911, 6922, 20420}, {6931, 17532, 5}, {7483, 7504, 10021}, {7483, 13747, 17566}, {7483, 17566, 140}, {7483, 50205, 6675}, {12100, 50243, 4189}, {16374, 28238, 48930}, {16408, 50204, 474}, {16409, 19549, 16415}, {16862, 50203, 16408}, {17527, 17564, 3}, {17527, 50241, 5084}, {17536, 37291, 15670}, {25524, 26364, 495}, {46219, 50726, 2}, {47598, 50395, 2}

X(52265) = MIDPOINT OF X(3) AND X(6842)

X(52265) lies on these lines: {2, 3}, {10, 37837}, {11, 10902}, {12, 11012}, {35, 15908}, {40, 5432}, {72, 5771}, {78, 5690}, {165, 37692}, {226, 37623}, {355, 5705}, {389, 34466}, {495, 11249}, {496, 10267}, {498, 3428}, {515, 4999}, {517, 13411}, {580, 37662}, {581, 37646}, {936, 26446}, {938, 10246}, {946, 6690}, {952, 6734}, {956, 10786}, {960, 2800}, {993, 18242}, {1125, 7686}, {1210, 1385}, {1445, 31657}, {1482, 5703}, {1483, 12649}, {1728, 37526}, {1837, 3576}, {2095, 3487}, {2829, 5267}, {2886, 6796}, {3085, 22770}, {3585, 30264}, {3647, 21635}, {3649, 5535}, {3911, 9940}, {4303, 43043}, {4640, 12608}, {4679, 10270}, {5010, 11826}, {5218, 10306}, {5219, 5812}, {5248, 7681}, {5258, 37725}, {5326, 7688}, {5482, 15644}, {5536, 37731}, {5587, 24953}, {5657, 5730}, {5709, 11374}, {5718, 37530}, {5719, 24474}, {5720, 5791}, {5729, 21151}, {5745, 5777}, {5762, 21617}, {5763, 37584}, {5840, 33862}, {5842, 25639}, {5844, 34772}, {5901, 34353}, {6147, 37532}, {6261, 26066}, {6326, 21677}, {6666, 51489}, {6691, 9843}, {6713, 12019}, {7951, 11827}, {7982, 31436}, {7987, 10826}, {8068, 14794}, {8141, 24611}, {8227, 10268}, {8726, 31231}, {9955, 15911}, {10164, 21616}, {10192, 14925}, {10198, 22753}, {10202, 34753}, {10320, 40292}, {10526, 10592}, {10958, 37561}, {11281, 31870}, {11491, 24390}, {11499, 31419}, {11500, 26363}, {12433, 24299}, {12704, 17718}, {13226, 13369}, {13408, 45944}, {15171, 32613}, {15172, 37621}, {15252, 37565}, {15844, 18990}, {16208, 23708}, {18233, 18243}, {18253, 20117}, {20171, 51046}, {26285, 31777}, {26921, 37713}, {27385, 37562}, {28212, 38045}, {30389, 37721}, {30503, 31423}, {31424, 37822}, {34486, 37722}, {35466, 37732}, {41347, 49107}

X(52265) = midpoint of X(i) and X(j) for these {i,j}: {3, 6842}, {10, 51717}, {12, 11012}, {35, 15908}, {411, 6831}, {3585, 30264}, {6734, 33597}, {11491, 24390}
X(52265) = complement of X(6831)
X(52265) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 6922}, {2, 20, 6956}, {2, 411, 6831}, {2, 3149, 5}, {2, 6864, 1656}, {2, 6927, 6918}, {2, 6962, 3149}, {2, 6988, 3}, {2, 16293, 17527}, {2, 50695, 6860}, {2, 50700, 6855}, {3, 5, 31789}, {3, 381, 6868}, {3, 1656, 6827}, {3, 3526, 6891}, {3, 5054, 6961}, {3, 6825, 6907}, {3, 6863, 5}, {3, 6907, 31775}, {3, 6923, 550}, {3, 6958, 37364}, {3, 6971, 28459}, {3, 6980, 7491}, {5, 140, 6675}, {21, 6960, 1532}, {140, 11277, 3530}, {405, 6834, 5}, {442, 6905, 37281}, {549, 37424, 3}, {631, 6834, 405}, {631, 6906, 37298}, {631, 6908, 3}, {631, 13747, 140}, {632, 37364, 6958}, {1006, 6949, 4187}, {1656, 44229, 5}, {3523, 6916, 3}, {3523, 37437, 17549}, {3651, 6952, 37374}, {5705, 52026, 355}, {6825, 6954, 3}, {6835, 6865, 6928}, {6838, 6910, 1012}, {6848, 6857, 6913}, {6853, 6905, 442}, {6855, 50700, 381}, {6862, 6985, 8727}, {6875, 6941, 11113}, {6878, 6983, 16842}, {6880, 6889, 474}, {6883, 6959, 17527}, {6888, 36002, 37447}, {6937, 6942, 11112}, {6970, 6989, 16408}, {6980, 7491, 546}, {11249, 26487, 495}, {14782, 14783, 6857}, {14784, 14785, 6844}, {15908, 21155, 35}, {17566, 17576, 7483}

X(52266) = MIDPOINT OF X(3) AND X(41016)

X(52266) lies on these lines: {2, 3}, {15, 36765}, {114, 6109}, {230, 5472}, {298, 34380}, {395, 18583}, {396, 3564}, {511, 44383}, {576, 33459}, {618, 7684}, {623, 44377}, {629, 51753}, {1352, 16644}, {1353, 37640}, {1503, 6771}, {3054, 42146}, {3055, 6114}, {3589, 6774}, {3815, 11543}, {5459, 44401}, {5473, 41036}, {5476, 33474}, {5613, 18358}, {5979, 38227}, {6036, 6669}, {6108, 20252}, {6671, 41022}, {9750, 33379}, {11178, 33475}, {13350, 22796}, {13468, 34509}, {14538, 36770}, {14561, 16645}, {15069, 49905}, {16626, 42147}, {16653, 33406}, {16772, 37824}, {18581, 31489}, {18582, 37637}, {20425, 52194}, {20428, 42942}, {21158, 36961}, {23004, 42914}, {23514, 31709}, {24206, 44382}, {33458, 34507}, {33560, 44667}, {36771, 41406}, {36958, 42165}, {37825, 42598}

X(52266) = midpoint of X(i) and X(j) for these {i,j}: {3, 41016}, {114, 6109}, {381, 35304}, {396, 5617}, {618, 7684}, {13350, 22796}, {20425, 52194}, {20428, 42942}, {31694, 37461}
X(52266) = reflection of X(43416) in X(20252)
X(52266) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(44466)
X(52266) = circumcircle-of-inner-Napoleon-triangle-inverse of X(36181)
X(52266) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 381, 44463}, {230, 6115, 11542}, {11301, 41040, 44465}, {36757, 36763, 396}

X(52267) = MIDPOINT OF X(2) AND X(46856)

X(52268) lies on these lines: {2, 3}, {13, 5916}, {110, 41042}, {5318, 21466}, {5642, 22796}, {7703, 16808}, {7998, 33957}, {8014, 42973}, {9143, 48655}, {11078, 25154}, {11092, 14356}, {14995, 37786}, {20425, 44555}, {22493, 23061}, {22495, 41724}, {25164, 40854}, {36969, 52039}, {42735, 45147}

X(52267) = midpoint of X(2) and X(46856)
X(52267) = reflection of X(i) in X(j) for these {i,j}: {2, 46858}, {15768, 46862}, {46824, 2}, {46860, 15768}
X(52267) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 44462, 45662}, {46856, 46858, 46824}, {46860, 46862, 46824}

X(52268) = MIDPOINT OF X(2) AND X(46857)

X(52268) lies on these lines: {2, 3}, {14, 5917}, {110, 41043}, {5321, 21467}, {5642, 22797}, {7703, 16809}, {7998, 33958}, {8015, 42972}, {9143, 48656}, {11078, 14356}, {11092, 25164}, {14995, 37785}, {20426, 44555}, {22494, 23061}, {22496, 41724}, {25154, 40855}, {36970, 52040}, {42734, 45147}

X(52268) = midpoint of X(2) and X(46857)
X(52268) = reflection of X(i) in X(j) for these {i,j}: {2, 46859}, {15769, 46863}, {46825, 2}, {46861, 15769}
X(52268) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 44466, 45662}, {46857, 46859, 46825}, {46861, 46863, 46825}

X(52269) = MIDPOINT OF X(3830) AND X(28443)

X(52269) lies on these lines: {2, 3}, {78, 18492}, {79, 1156}, {81, 45924}, {100, 18406}, {115, 40129}, {758, 1699}, {938, 3649}, {946, 10707}, {962, 21677}, {1479, 8543}, {1776, 41697}, {2287, 32431}, {3485, 11238}, {3486, 11237}, {3647, 5705}, {3817, 26725}, {3829, 34695}, {3869, 31162}, {3871, 18517}, {3947, 10572}, {4870, 45230}, {5057, 6734}, {5086, 21075}, {5427, 12943}, {5441, 13411}, {5691, 35016}, {5698, 18253}, {5703, 10543}, {5715, 12528}, {6261, 38021}, {6701, 16143}, {7680, 34746}, {7682, 41557}, {7701, 16125}, {8261, 12688}, {9579, 41547}, {9612, 10122}, {9654, 15174}, {9669, 16137}, {9955, 33858}, {10129, 16132}, {10711, 12738}, {10724, 35204}, {10733, 16164}, {10893, 14450}, {11236, 25568}, {11263, 12571}, {11496, 31660}, {12047, 12563}, {12514, 50865}, {12617, 31159}, {12649, 45630}, {16133, 30311}, {16139, 22793}, {16159, 22798}, {17605, 33857}, {19925, 31160}, {21740, 51709}, {26332, 34605}, {31142, 40661}, {31871, 47319}, {34629, 37820}, {36990, 51729}, {40571, 50435}

X(52269) = midpoint of X(i) and X(j) for these {i,j}: {3830, 28443}, {15672, 50687}
X(52269) = reflection of X(i) in X(j) for these {i,j}: {376, 28465}, {15678, 28461}, {26725, 3817}, {28443, 44257}
X(52269) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6895, 37428}, {2, 37428, 6986}, {4, 5, 36002}, {4, 381, 17577}, {4, 3091, 6932}, {4, 3855, 6982}, {4, 6828, 411}, {4, 6841, 21}, {4, 6855, 50695}, {4, 6866, 2476}, {4, 6870, 6828}, {4, 6873, 6985}, {4, 10883, 6912}, {4, 11111, 3543}, {5, 3651, 31254}, {5, 6895, 6986}, {5, 6903, 17536}, {5, 37428, 2}, {21, 17577, 6175}, {21, 36002, 3651}, {381, 17556, 3091}, {381, 17577, 7548}, {442, 37433, 33557}, {546, 6831, 6894}, {546, 16160, 37230}, {2043, 2044, 6829}, {3091, 6836, 6991}, {3091, 37433, 442}, {3146, 15674, 44238}, {3543, 6837, 11111}, {6175, 17577, 46870}, {6828, 6932, 6991}, {6831, 6894, 6915}, {6831, 37230, 35979}, {6836, 6932, 411}, {6839, 8727, 6909}, {6845, 44229, 404}, {6900, 37356, 17531}, {6986, 36002, 411}, {11111, 15670, 21}, {16160, 37230, 21669}, {44258, 46028, 3}

X(52270) = X(2)X(3)∩X(36)X(920)

X(52270) lies on these lines: {2, 3}, {36, 920}, {56, 21740}, {104, 7742}, {515, 36152}, {944, 37579}, {1512, 6796}, {1699, 14794}, {1858, 37837}, {3485, 8071}, {3486, 8069}, {3869, 11249}, {3916, 5887}, {4996, 11415}, {5086, 11499}, {5172, 11500}, {5267, 12617}, {5398, 7592}, {5603, 26357}, {5884, 41547}, {6361, 12775}, {7280, 52027}, {8185, 37812}, {9723, 17139}, {10167, 32612}, {10698, 22770}, {11012, 12514}, {11248, 48363}, {12047, 14793}, {18446, 37583}, {22753, 37564}, {22766, 45230}, {40245, 45392}

X(52270) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 25, 37116}, {3, 859, 24}, {3, 3560, 20846}, {3, 6914, 37285}, {3, 6918, 19524}, {3, 7387, 37311}, {3, 37248, 1006}, {3, 37249, 631}, {3, 37282, 6940}, {3, 37302, 4}, {4, 6852, 6830}, {4, 6857, 6833}, {4, 6875, 6906}, {4, 6880, 6825}, {4, 6942, 411}, {4, 6949, 2476}, {20, 27086, 3}, {21, 411, 6868}, {21, 6828, 3560}, {21, 27653, 14016}, {1532, 7491, 4}, {3149, 3560, 4}, {6837, 50701, 4}, {6841, 37468, 4}, {6848, 6872, 4}, {6905, 6906, 6934}, {6905, 6941, 3149}, {6927, 6942, 6905}, {42789, 42790, 37414}

X(52271) = X(2)X(3)∩X(36)X(1473)

X(52271) lies on these lines: {2, 3}, {36, 1473}, {56, 3185}, {197, 5172}, {1470, 20470}, {2194, 4252}, {5204, 15494}, {5396, 9777}, {5398, 11402}, {7742, 15654}, {8192, 23361}, {9798, 36152}, {11396, 37565}, {18165, 19765}, {23206, 26866}, {23383, 26357}, {23846, 26437}

X(52271) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 859, 25}, {3, 3517, 37116}, {3, 9909, 37311}, {3, 13737, 37259}, {3, 13738, 37257}, {3, 28383, 11344}, {3, 37248, 37246}, {3, 37249, 7484}, {3, 37252, 37195}, {3, 37260, 3145}, {3, 37302, 1593}, {3, 50204, 16374}, {21, 404, 50408}, {21, 27621, 4185}, {22, 27086, 3}, {411, 28376, 4186}, {4216, 37300, 3}, {6905, 19256, 37366}, {11347, 16370, 37397}, {11349, 24608, 11347}, {19525, 37058, 3}, {23361, 37579, 8192}

X(52272) = X(2)X(3)∩X(36)X(5693)

X(52262) lies on these lines: {2, 3}, {36, 5693}, {56, 24475}, {355, 36152}, {952, 37579}, {960, 26286}, {1470, 24470}, {1724, 34465}, {5172, 11499}, {5289, 11249}, {5398, 12161}, {5730, 22765}, {5901, 26357}, {7686, 26285}, {7742, 32153}, {8069, 32141}, {8071, 37737}, {8227, 14794}, {9809, 18861}, {10395, 17010}, {10526, 51506}, {10680, 19907}, {11375, 14793}, {12019, 38722}, {13369, 32612}, {14792, 37692}, {18397, 37583}, {23961, 31828}, {35204, 37625}

X(52272) = midpoint of X(3) and X(37302)

X(52272) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 859, 26}, {3, 7489, 20846}, {3, 7506, 37116}, {3, 7517, 37311}, {3, 37249, 140}, {4, 27086, 3}, {5, 5428, 6914}, {405, 6911, 5}, {6861, 6917, 5}, {6928, 6959, 5}, {10883, 36003, 7580}, {19245, 37116, 7506}

X(52273) = X(2)X(3)∩X(36)X(846)

X(52273) lies on these lines: {2, 3}, {35, 3987}, {36, 846}, {55, 49487}, {56, 2292}, {58, 22076}, {993, 20999}, {999, 4392}, {1193, 22421}, {1283, 5010}, {1284, 1470}, {1324, 5251}, {2305, 52087}, {2932, 12746}, {3679, 23858}, {3917, 37469}, {5172, 52139}, {5260, 38903}, {5422, 9567}, {5563, 11533}, {7085, 11031}, {8235, 37561}, {9959, 32612}, {10269, 30285}, {11429, 22072}, {22350, 26890}

X(52273) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 21, 3145}, {3, 405, 37259}, {3, 859, 199}, {3, 11108, 20842}, {3, 16286, 19524}, {3, 16370, 16064}, {3, 16418, 11334}, {3, 16422, 4188}, {3, 20834, 37311}, {3, 20849, 21}, {3, 28383, 11337}, {3, 37244, 37257}, {3, 37248, 13738}, {3, 37249, 4191}, {21, 404, 26117}, {21, 4220, 855}, {21, 37311, 20834}, {4184, 27086, 3}, {4218, 17549, 3}, {6906, 7523, 47521}, {11334, 16418, 47523}, {20834, 37311, 3145}

X(52274) = X(2)X(3)∩X(32)X(32046)

X(52274) lies on these lines: {2, 3}, {32, 32046}, {39, 143}, {160, 3564}, {187, 11675}, {211, 13335}, {577, 19154}, {1147, 5171}, {1353, 20775}, {3003, 15074}, {3095, 14449}, {5158, 11255}, {5188, 10627}, {5210, 45938}, {5462, 13334}, {6101, 36212}, {8266, 48876}, {8721, 32140}, {9019, 50648}, {13363, 21163}, {23195, 41588}, {38110, 41328}, {41480, 47406}

X(52274) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 237, 5}, {3, 2937, 37183}, {3, 37114, 44221}, {3, 37344, 7516}, {7512, 35296, 3}, {37114, 37184, 3}

X(52275) = X(2)X(3)∩X(32)X(1993)

X(52275) lies on these lines: {2, 3}, {6, 2987}, {32, 1993}, {39, 5422}, {51, 9737}, {56, 26639}, {69, 1609}, {99, 40814}, {141, 8553}, {184, 13335}, {187, 15066}, {193, 3964}, {198, 26699}, {323, 1384}, {343, 7789}, {373, 9734}, {394, 1501}, {571, 20806}, {574, 21399}, {577, 26206}, {599, 11063}, {800, 40318}, {1975, 51481}, {1994, 30435}, {3003, 41614}, {3117, 23584}, {3618, 44180}, {3917, 5171}, {3926, 6515}, {3933, 45794}, {4996, 5222}, {5013, 10601}, {5023, 17811}, {5024, 15018}, {5210, 8627}, {5359, 13357}, {5651, 47113}, {5749, 7279}, {5866, 9721}, {6337, 6503}, {6389, 26156}, {6390, 37644}, {7758, 41628}, {7795, 37636}, {8069, 17316}, {8071, 26626}, {8263, 51611}, {8722, 21766}, {9155, 34396}, {9306, 52144}, {9308, 41678}, {9605, 34545}, {9777, 10983}, {11004, 21309}, {11162, 33900}, {11165, 40604}, {11188, 14060}, {13334, 43650}, {14792, 29598}, {14793, 17023}, {14809, 18310}, {14826, 39647}, {15109, 47355}, {15815, 17825}, {15817, 17257}, {15988, 36744}, {18311, 46616}, {22331, 37672}, {25007, 25440}, {26594, 38901}, {32459, 37648}, {32640, 35910}, {35264, 42671}, {35424, 36213}, {37809, 40112}, {39113, 42406}, {40802, 40825}, {40867, 40947}, {46319, 46807}, {47352, 50660}

X(52275) = X(5028)-Dao conjugate of X(7778)
X(52275) = crosssum of X(6) and X(44524)
X(52275) = crossdifference of every pair of points on line {647, 12075}
X(52275) = barycentric product X(99)*X(46953)
X(52275) = barycentric quotient X(46953)/X(523)
X(52275) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 384, 41231}, {2, 401, 41238}, {2, 3552, 401}, {2, 35296, 3}, {2, 40853, 5025}, {3, 25, 37183}, {3, 237, 22}, {3, 11328, 3148}, {3, 35302, 35296}, {3, 37344, 2}, {32, 36212, 1993}, {297, 7807, 2}, {1583, 1584, 5020}, {1599, 1600, 22}, {3148, 11328, 1995}, {3964, 8573, 193}, {7770, 37067, 2}, {11145, 11146, 7492}, {11291, 11292, 7404}, {21495, 21511, 36698}, {25883, 25936, 2}, {33235, 41235, 35941}, {35302, 37344, 3}, {37183, 37465, 25}, {37340, 37341, 14787}

X(52276) = X(2)X(3)∩X(32)X(7592)

X(52276) lies on these lines: {2, 3}, {32, 7592}, {50, 34117}, {187, 11456}, {216, 39588}, {511, 9723}, {1181, 3053}, {1199, 30435}, {1384, 15032}, {1498, 5023}, {1503, 8553}, {1609, 6776}, {4558, 44492}, {5013, 10982}, {5171, 5562}, {5206, 44437}, {6759, 52144}, {7669, 15581}, {8573, 14912}, {8721, 34224}, {9737, 45186}, {9924, 10608}, {10316, 35603}, {10984, 13335}, {12112, 15655}, {20477, 44145}, {20987, 37813}, {34781, 39647}, {39879, 44200}, {43976, 46724}, {44180, 51212}

X(52276) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 237, 24}, {3, 7387, 37183}, {3, 37344, 631}, {20, 35296, 3}

X(52277) = X(2)X(3)∩X(32)X(11402)

X(52277) lies on these lines: {2, 3}, {32, 11402}, {39, 9777}, {50, 19153}, {51, 5013}, {154, 5023}, {159, 7669}, {160, 1609}, {184, 3053}, {187, 26864}, {216, 12167}, {394, 5171}, {571, 19125}, {577, 19118}, {1384, 34396}, {1495, 5210}, {1843, 36751}, {1974, 36748}, {3003, 10602}, {3060, 10983}, {3796, 13335}, {5017, 46320}, {5158, 11405}, {5191, 15655}, {5206, 42671}, {5585, 41424}, {7789, 43653}, {8573, 20775}, {9723, 37491}, {9737, 33586}, {10601, 13334}, {11206, 39647}, {11480, 44122}, {11481, 44083}, {15270, 39653}, {15815, 17810}, {17809, 22331}, {26870, 39871}, {33878, 51335}, {40680, 41584}

X(52277) = X(7612)-Ceva conjugate of X(6)
X(52277) = X(1007)-Dao conjugate of X(1351)
X(52277) = crossdifference of every pair of points on line {647, 14341}
X(52277) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 237, 25}, {3, 9909, 37183}, {3, 37344, 7484}, {3, 41266, 3148}, {22, 35296, 3}, {154, 5023, 52144}, {159, 8553, 44200}, {160, 1609, 19459}, {237, 3148, 41266}, {237, 37457, 20897}, {237, 41275, 3}, {3148, 41266, 25}, {3155, 3156, 9909}

X(52278) = X(2)X(3)∩X(32)X(12161)

X(52278) lies on these lines: {2, 3}, {32, 12161}, {155, 3053}, {157, 32762}, {187, 15068}, {571, 19139}, {1216, 5171}, {1351, 9723}, {1352, 8553}, {1353, 8573}, {1609, 3564}, {1634, 9925}, {3003, 8548}, {3964, 34380}, {5023, 17814}, {5446, 9737}, {6503, 41588}, {10539, 52144}, {11063, 15069}, {13142, 52014}, {14853, 44180}, {16266, 36212}, {34990, 44469}, {46261, 47113}

X(52278) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 237, 26}, {3, 7517, 37183}, {3, 37344, 140}, {4, 35296, 3}, {37342, 37343, 50137}

X(52279) = X(2)X(3)∩X(32)X(5890)

X(52279) lies on these lines: {2, 3}, {32, 5890}, {39, 15033}, {50, 2781}, {74, 187}, {477, 47326}, {511, 4558}, {566, 51739}, {974, 41336}, {1503, 7669}, {1986, 10317}, {2351, 32064}, {3003, 5622}, {3053, 10605}, {3425, 7710}, {3566, 39201}, {5023, 10606}, {5191, 12112}, {5621, 11063}, {6000, 52144}, {6394, 44146}, {8553, 44883}, {8744, 9475}, {9210, 32231}, {9752, 45030}, {11645, 34218}, {14157, 42671}, {14961, 15472}, {14972, 35002}, {15032, 34396}, {18860, 43576}, {36212, 43574}, {36990, 37813}, {39874, 40947}, {41204, 41678}, {48881, 50669}

X(52279) = reflection of X(i) in X(j) for these {i,j}: {4, 2450}, {37183, 3}
X(52279) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 237, 186}, {3, 32444, 37457}, {186, 7464, 50401}, {2071, 35296, 3}, {35469, 35470, 7464}, {42671, 44437, 14157}, {42789, 42790, 297}

X(52280) = BARYCENTRIC PRODUCT X(264)*X(389)

X(52280) lies on these lines: {2, 3}, {6, 11547}, {51, 6530}, {53, 2052}, {64, 31366}, {92, 41883}, {125, 42400}, {129, 136}, {141, 37873}, {264, 343}, {275, 1971}, {317, 394}, {324, 3580}, {389, 6750}, {393, 11433}, {459, 8796}, {523, 50460}, {1503, 1629}, {1899, 33971}, {1941, 13142}, {1993, 27377}, {3060, 44704}, {3087, 11427}, {3168, 14569}, {5943, 39569}, {6146, 8884}, {6515, 9308}, {8612, 8795}, {8882, 45832}, {8966, 26868}, {9777, 41371}, {10601, 17907}, {11064, 32002}, {11245, 41204}, {11542, 51273}, {11543, 51266}, {11550, 16264}, {13367, 35717}, {14129, 46106}, {14249, 15873}, {14918, 37636}, {15004, 42873}, {15466, 37648}, {16080, 39284}, {16252, 51031}, {18464, 22115}, {19189, 23195}, {21243, 39530}, {26879, 44732}, {26905, 42329}, {27376, 40814}, {34545, 37766}, {34826, 35719}, {34836, 45198}, {36794, 37649}, {37638, 41244}, {39571, 41365}, {39871, 40867}, {41005, 46717}, {44138, 45793}

X(52280) = midpoint of X(472) and X(473)
X(52280) = complement of X(8613)
X(52280) = polar-circle-inverse of X(41202)
X(52280) = polar conjugate of X(40448)
X(52280) = complement of the isogonal conjugate of X(8612)
X(52280) = polar conjugate of the isotomic conjugate of X(45198)
X(52280) = polar conjugate of the isogonal conjugate of X(389)
X(52280) = X(45301)-anticomplementary conjugate of X(4329)
X(52280) = X(8612)-complementary conjugate of X(10)
X(52280) = X(i)-Ceva conjugate of X(j) for these (i,j): {264, 42441}, {275, 19170}
X(52280) = X(389)-cross conjugate of X(45198)
X(52280) = X(i)-isoconjugate of X(j) for these (i,j): {48, 40448}, {255, 40402}, {9247, 42333}
X(52280) = X(i)-Dao conjugate of X(j) for these (i,j): {343, 46832}, {394, 34836}, {1249, 40448}, {6523, 40402}
X(52280) = crosspoint of X(275) and X(2052)
X(52280) = crosssum of X(216) and X(577)
X(52280) = crossdifference of every pair of points on line {647, 34983}
X(52280) = barycentric product X(i)*X(j) for these {i,j}: {4, 45198}, {75, 45225}, {92, 45224}, {95, 6750}, {264, 389}, {275, 34836}, {311, 51887}, {324, 19170}, {2052, 46832}, {8795, 42441}
X(52280) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 40448}, {264, 42333}, {389, 3}, {393, 40402}, {6750, 5}, {19170, 97}, {34836, 343}, {42441, 5562}, {45198, 69}, {45224, 63}, {45225, 1}, {46832, 394}, {51887, 54}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 467, 297}, {2, 37174, 6820}, {4, 1596, 1559}, {4, 6619, 7378}, {4, 52249, 427}, {5, 26906, 2}, {25, 427, 1513}, {51, 6747, 6530}, {53, 13567, 2052}, {53, 43462, 51358}, {275, 14165, 23292}, {297, 458, 6656}, {427, 6755, 4}, {470, 471, 140}, {1585, 1586, 3}, {2052, 13567, 51358}, {2052, 43462, 13567}, {3535, 3536, 3525}, {6617, 52251, 2}, {6748, 23292, 275}, {11109, 17555, 4205}, {14918, 40684, 37636}, {34836, 46832, 45198}

X(52281) = BARYCENTRIC PRODUCT X(264)*X(575)

X(52281) lies on these lines: {2, 3}, {53, 597}, {83, 39284}, {107, 20192}, {141, 32002}, {264, 524}, {273, 50128}, {275, 671}, {287, 5480}, {316, 343}, {317, 599}, {318, 29617}, {324, 35318}, {340, 22165}, {394, 11185}, {542, 39530}, {543, 33843}, {598, 2052}, {648, 6749}, {1992, 3087}, {1993, 47286}, {1994, 22146}, {3564, 47740}, {3917, 35884}, {5032, 40065}, {5081, 29615}, {5254, 14153}, {5306, 6531}, {5459, 6116}, {5460, 6117}, {5476, 6530}, {7745, 40814}, {7762, 41628}, {7790, 37649}, {7817, 27371}, {7827, 27376}, {8744, 34545}, {8796, 18842}, {10311, 22329}, {10985, 37688}, {11160, 32000}, {11161, 20774}, {11179, 33971}, {11206, 46034}, {11427, 43448}, {11645, 16264}, {13449, 21243}, {13567, 41254}, {14848, 41371}, {15533, 44134}, {15595, 48889}, {16080, 45103}, {17035, 41005}, {17503, 43530}, {17907, 47352}, {20423, 44704}, {29181, 42313}, {32819, 36212}, {34505, 37672}, {35714, 40671}, {35715, 40672}, {36426, 41145}, {39884, 40867}, {39931, 41624}, {40684, 44146}, {41204, 50979}, {43462, 44569}

X(52281) = reflection of X(35937) in X(2)
X(52281) = polar conjugate of X(7608)
X(52281) = isotomic conjugate of the isogonal conjugate of X(10985)
X(52281) = polar conjugate of the isotomic conjugate of X(37688)
X(52281) = polar conjugate of the isogonal conjugate of X(575)
X(52281) = X(575)-cross conjugate of X(37688)
X(52281) = X(48)-isoconjugate of X(7608)
X(52281) = X(i)-Dao conjugate of X(j) for these (i,j): {1249, 7608}, {10985, 15582}
X(52281) = cevapoint of X(575) and X(10985)
X(52281) = barycentric product X(i)*X(j) for these {i,j}: {4, 37688}, {76, 10985}, {264, 575}
X(52281) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 7608}, {575, 3}, {10985, 6}, {37688, 69}
X(52281) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 33007, 35302}, {4, 458, 297}, {53, 597, 37765}, {264, 6748, 27377}, {441, 546, 52247}, {472, 473, 427}, {1585, 1586, 5094}, {11303, 11304, 7399}, {11331, 37174, 297}, {36794, 37765, 597}

X(52282) = BARYCENTRIC PRODUCT X(264)*X(576)

X(52282) lies on these lines: {2, 3}, {6, 32002}, {53, 317}, {76, 39284}, {232, 11163}, {264, 599}, {275, 598}, {287, 36990}, {316, 394}, {318, 29615}, {324, 44146}, {340, 15533}, {343, 11185}, {393, 1992}, {459, 41895}, {542, 33971}, {597, 6748}, {648, 15534}, {671, 2052}, {1249, 5032}, {1351, 40867}, {1625, 1993}, {1785, 29574}, {1990, 8584}, {1994, 8744}, {2207, 7812}, {3172, 34604}, {3199, 7775}, {5081, 29617}, {5207, 40802}, {5476, 39569}, {5485, 8796}, {5523, 14715}, {6515, 47286}, {6530, 20423}, {7282, 50128}, {7754, 41628}, {7773, 36212}, {7790, 10601}, {7879, 44142}, {8593, 20774}, {9306, 13449}, {9766, 35142}, {10516, 42313}, {11160, 32001}, {11178, 39530}, {11433, 43448}, {15595, 48901}, {16080, 17503}, {21849, 34854}, {22165, 44134}, {26958, 41254}, {31173, 33842}, {31687, 40671}, {31688, 40672}, {32064, 39908}, {33630, 51170}, {36794, 47352}, {38744, 40870}, {40814, 44518}, {40896, 40995}, {43530, 45103}

X(52282) = reflection of X(35941) in X(2)
X(52282) = polar conjugate of X(7607)
X(52282) = polar conjugate of the isogonal conjugate of X(576)
X(52282) = X(i)-isoconjugate of X(j) for these (i,j): {48, 7607}, {810, 35178}
X(52282) = X(i)-Dao conjugate of X(j) for these (i,j): {525, 35132}, {1249, 7607}, {35178, 39062}
X(52282) = barycentric product X(264)*X(576)
X(52282) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 7607}, {576, 3}, {648, 35178}
X(52282) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 21972, 44576}, {4, 297, 458}, {4, 37174, 297}, {4, 37855, 11317}, {53, 317, 9308}, {297, 458, 11331}, {382, 52251, 401}, {472, 473, 25}, {1585, 1586, 468}, {11303, 11304, 7395}, {40853, 52247, 3}

X(52283) = BARYCENTRIC PRODUCT X(264)*X(1350)

X(52283) lies on these lines: {2, 3}, {6, 32001}, {53, 3763}, {69, 648}, {76, 459}, {92, 26563}, {141, 393}, {264, 3619}, {275, 18841}, {278, 9312}, {281, 4357}, {287, 39874}, {315, 34412}, {317, 3618}, {318, 29611}, {340, 1992}, {343, 14361}, {394, 8743}, {524, 40138}, {599, 1990}, {1007, 47738}, {1119, 3662}, {1235, 2052}, {1350, 10002}, {1783, 23151}, {1785, 17284}, {1841, 25887}, {1897, 29616}, {2202, 25940}, {2207, 17811}, {2322, 5232}, {2996, 14952}, {3087, 3589}, {3168, 12251}, {3620, 9308}, {3661, 7046}, {3673, 17862}, {3912, 7952}, {3917, 34854}, {5081, 5222}, {5254, 26958}, {5286, 13567}, {5485, 16080}, {5523, 37638}, {5749, 7282}, {5921, 41374}, {6194, 45864}, {6331, 20023}, {6389, 40484}, {6524, 43653}, {6527, 20208}, {6530, 10519}, {6748, 47355}, {6749, 47352}, {7710, 45031}, {7803, 18928}, {8744, 15066}, {11064, 41370}, {11679, 18678}, {14927, 42287}, {15069, 15258}, {15262, 41614}, {15589, 16318}, {15905, 20204}, {16077, 30227}, {17023, 34231}, {17913, 25521}, {18842, 43530}, {19222, 43727}, {20200, 20477}, {20207, 37877}, {21356, 37765}, {23291, 39646}, {27377, 51171}, {28708, 44128}, {32817, 51389}, {32818, 36212}, {33971, 40330}, {37643, 40814}, {37668, 45141}, {40803, 42313}, {42854, 48881}, {43448, 47296}

X(52283) = isotomic conjugate of X(42287)
X(52283) = polar conjugate of X(3424)
X(52283) = complement of the isotomic conjugate of X(42352)
X(52283) = isotomic conjugate of the isogonal conjugate of X(45141)
X(52283) = isotomic conjugate of the polar conjugate of X(10002)
X(52283) = polar conjugate of the isotomic conjugate of X(37668)
X(52283) = polar conjugate of the isogonal conjugate of X(1350)
X(52283) = X(42352)-complementary conjugate of X(2887)
X(52283) = X(i)-cross conjugate of X(j) for these (i,j): {1350, 37668}, {45141, 10002}
X(52283) = X(i)-isoconjugate of X(j) for these (i,j): {31, 42287}, {48, 3424}
X(52283) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 42287}, {1249, 3424}, {6776, 7735}
X(52283) = cevapoint of X(1350) and X(45141)
X(52283) = crosspoint of X(2) and X(42352)
X(52283) = trilinear pole of line {1529, 14343}
X(52283) = barycentric product X(i)*X(j) for these {i,j}: {4, 37668}, {69, 10002}, {75, 23052}, {76, 45141}, {92, 51304}, {264, 1350}, {325, 45031}, {1529, 35140}, {12037, 23582}, {15466, 40813}
X(52283) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 42287}, {4, 3424}, {1350, 3}, {1529, 1503}, {7710, 6776}, {10002, 4}, {12037, 15526}, {14343, 8057}, {23052, 1}, {37668, 69}, {40813, 1073}, {45031, 98}, {45141, 6}, {47382, 47388}, {51304, 63}
X(52283) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20, 441}, {2, 297, 4}, {2, 467, 6819}, {2, 3146, 11348}, {2, 32974, 41235}, {2, 37067, 3525}, {2, 37174, 458}, {2, 37187, 6353}, {2, 37344, 14069}, {3, 44334, 2}, {4, 420, 6353}, {4, 3525, 37124}, {4, 38282, 419}, {69, 17907, 1249}, {141, 393, 32000}, {297, 458, 37174}, {297, 11331, 2}, {317, 3618, 40065}, {458, 37174, 4}, {470, 471, 4232}, {1585, 1586, 6995}, {3091, 37200, 4}, {3535, 3536, 6353}, {20208, 42459, 6527}

X(52284) = BARYCENTRIC PRODUCT X(264)*X(5024)

X(52284) lies on these lines: {2, 3}, {33, 7292}, {34, 5297}, {39, 15880}, {69, 45303}, {110, 15431}, {115, 39662}, {125, 10752}, {132, 14484}, {147, 14920}, {193, 7703}, {232, 15302}, {264, 3266}, {275, 43537}, {305, 32825}, {323, 39588}, {340, 15589}, {393, 39389}, {1131, 11474}, {1132, 11473}, {1235, 9464}, {1829, 46933}, {1843, 7998}, {1853, 8550}, {1990, 7736}, {2892, 15118}, {3087, 37689}, {3291, 33843}, {3292, 11180}, {3424, 43530}, {3448, 32234}, {3574, 18913}, {3619, 3867}, {3622, 5090}, {3767, 15820}, {3785, 11056}, {4678, 11396}, {5032, 5095}, {5265, 11392}, {5281, 11393}, {5304, 6103}, {5480, 37643}, {5486, 41603}, {5550, 49542}, {5640, 12294}, {5702, 14930}, {5921, 37645}, {6032, 37665}, {6336, 34337}, {6403, 33884}, {6747, 7608}, {6749, 7735}, {6776, 44109}, {7713, 19877}, {7718, 46934}, {7777, 43981}, {8541, 11160}, {8801, 17907}, {8878, 8892}, {9139, 10415}, {9191, 16230}, {9754, 35710}, {10511, 11605}, {10519, 51360}, {11002, 21852}, {11424, 43617}, {11433, 23332}, {11470, 15019}, {11475, 37775}, {11476, 37776}, {11550, 44108}, {12242, 32337}, {13394, 14927}, {14216, 43841}, {14826, 18553}, {15126, 23327}, {15131, 32274}, {15139, 51739}, {17983, 42008}, {18911, 51171}, {23291, 34565}, {23292, 32064}, {27371, 31400}, {32269, 51538}, {32815, 37804}, {32824, 34254}, {35259, 51537}, {35260, 36990}, {37638, 51212}, {37668, 44134}, {38294, 41136}, {39157, 40103}

X(52284) = midpoint of X(4) and X(35483)
X(52284) = orthocentroidal-circle-inverse of X(4232)
X(52284) = polar-circle-inverse of X(37904)
X(52284) = orthoptic-circle-of-Steiner-inellipse-inverse of X(37934)
X(52284) = polar conjugate of X(18842)
X(52284) = polar conjugate of the isotomic conjugate of X(21356)
X(52284) = polar conjugate of the isogonal conjugate of X(5024)
X(52284) = X(5024)-cross conjugate of X(21356)
X(52284) = X(48)-isoconjugate of X(18842)
X(52284) = X(1249)-Dao conjugate of X(18842)
X(52284) = barycentric product X(i)*X(j) for these {i,j}: {4, 21356}, {264, 5024}
X(52284) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 18842}, {5024, 3}, {21356, 69}
X(52284) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 4232}, {2, 427, 7378}, {2, 3146, 7493}, {2, 3522, 7495}, {2, 3832, 1995}, {2, 5169, 3091}, {2, 7378, 6995}, {2, 7391, 10565}, {2, 7408, 6353}, {2, 7409, 25}, {2, 7570, 46935}, {2, 10989, 10304}, {2, 15683, 47596}, {2, 16063, 3523}, {2, 31074, 7396}, {2, 31099, 20}, {2, 31100, 4208}, {2, 31105, 3543}, {2, 31106, 17558}, {2, 31107, 33202}, {2, 31857, 31099}, {2, 32982, 26257}, {2, 50687, 7426}, {4, 378, 49670}, {4, 631, 37458}, {4, 3516, 5059}, {4, 3533, 3517}, {4, 3545, 37984}, {4, 4232, 6995}, {4, 5094, 2}, {4, 6353, 10301}, {4, 8889, 5094}, {4, 10301, 7408}, {4, 35486, 7487}, {4, 37118, 37460}, {4, 37119, 35486}, {4, 49670, 3543}, {5, 16051, 2}, {25, 3520, 7492}, {25, 40916, 37977}, {378, 427, 31105}, {381, 5159, 40132}, {381, 30775, 2}, {427, 1594, 5169}, {427, 3541, 31099}, {427, 5094, 4}, {427, 8889, 2}, {427, 30739, 1595}, {427, 47097, 1597}, {631, 37454, 2}, {1370, 7495, 3522}, {1370, 31236, 2}, {1594, 3088, 3091}, {1597, 37984, 4}, {2476, 30776, 2}, {2478, 30770, 2}, {3088, 5169, 7378}, {3088, 8889, 30769}, {3090, 30739, 2}, {3091, 30769, 2}, {3523, 7396, 16063}, {3545, 47097, 2}, {4232, 7378, 4}, {5000, 5001, 3545}, {5064, 6353, 7408}, {5064, 10301, 4}, {5071, 32216, 2}, {5159, 40132, 2}, {6997, 30744, 2}, {7392, 30771, 2}, {7401, 30802, 2}, {7487, 37119, 10303}, {7493, 31133, 3146}, {16924, 30777, 2}, {30775, 40132, 5159}, {31152, 37454, 631}, {37118, 37460, 3523}

X(52285) = BARYCENTRIC PRODUCT X(264)*X(5041)

X(52285) lies on these lines: {2, 3}, {132, 11792}, {136, 45165}, {343, 48901}, {1503, 13366}, {1627, 43291}, {1829, 3626}, {1843, 3631}, {1848, 1862}, {1876, 3982}, {1993, 39884}, {2052, 14488}, {2979, 18358}, {3244, 12135}, {3410, 34380}, {3574, 16621}, {3629, 3867}, {3632, 5090}, {3636, 49542}, {5422, 38136}, {5480, 11245}, {6329, 44102}, {6748, 16318}, {8877, 51258}, {8878, 47286}, {9777, 32064}, {10169, 47461}, {10938, 11455}, {11008, 12167}, {11363, 15808}, {11396, 20050}, {11405, 46444}, {11442, 21850}, {11475, 43105}, {11476, 43106}, {12143, 32450}, {12834, 45298}, {15433, 15437}, {16654, 18388}, {16656, 43831}, {17809, 36990}, {18382, 41602}, {21243, 47582}, {23251, 34516}, {23261, 34515}, {23292, 44110}, {23332, 34417}, {29012, 37649}, {32239, 40342}, {33843, 40938}, {33880, 43457}, {43653, 48910}, {44078, 51739}, {44106, 47296}

X(52285) = reflection of X(i) in X(j) for these {i,j}: {6636, 11548}, {7499, 5133}
X(52285) = polar-circle-inverse of X(20063)
X(52285) = polar conjugate of the isotomic conjugate of X(34573)
X(52285) = polar conjugate of the isogonal conjugate of X(5041)
X(52285) = X(5041)-cross conjugate of X(34573)
X(52285) = X(63)-isoconjugate of X(34572)
X(52285) = X(i)-Dao conjugate of X(j) for these (i,j): {3162, 34572}, {7767, 34573}
X(52285) = crosssum of X(3) and X(22352)
X(52285) = barycentric product X(i)*X(j) for these {i,j}: {4, 34573}, {264, 5041}
X(52285) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 34572}, {5041, 3}, {34573, 69}
X(52285) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 427, 428}, {4, 1595, 3575}, {4, 1907, 1885}, {4, 5064, 427}, {4, 7378, 25}, {4, 7409, 5064}, {4, 7507, 1906}, {4, 8889, 7408}, {4, 15559, 6756}, {5, 7391, 7667}, {25, 5064, 7378}, {25, 5094, 38282}, {25, 7378, 427}, {381, 1370, 37439}, {427, 428, 468}, {427, 10301, 2}, {1368, 3845, 7394}, {1370, 37439, 43957}, {1906, 7507, 10019}, {3091, 44442, 7484}, {3843, 34609, 6997}, {3850, 10691, 37990}, {3853, 5169, 37899}, {3853, 6676, 34603}, {5000, 5001, 3628}, {5133, 6636, 11548}, {5169, 34603, 6676}, {5189, 37990, 10691}, {5480, 11550, 11245}, {6636, 11548, 7499}, {6676, 34603, 37899}, {6677, 31074, 47097}, {6995, 38282, 25}, {6997, 34609, 30739}, {7378, 7408, 8889}, {7391, 7667, 47095}, {7394, 31133, 1368}, {7408, 8889, 25}, {7519, 31236, 10154}

X(52286) = BARYCENTRIC PRODUCT X(264)*X(5058)

X(52286) lies on these lines: {2, 3}, {6, 6220}, {51, 3070}, {53, 5412}, {136, 8968}, {154, 23261}, {184, 3071}, {275, 14231}, {371, 8967}, {393, 5410}, {485, 8940}, {486, 10133}, {615, 44638}, {1163, 19042}, {1322, 42262}, {1495, 42283}, {1587, 9777}, {1588, 11402}, {1659, 2969}, {1824, 1850}, {1843, 44637}, {1899, 13749}, {1993, 49029}, {2052, 14238}, {3060, 6400}, {3068, 26376}, {3087, 5411}, {3167, 12601}, {3564, 13428}, {3580, 26438}, {3917, 12299}, {5413, 6748}, {6290, 11091}, {6460, 12172}, {6515, 49087}, {6561, 10132}, {6564, 45840}, {6751, 44633}, {7140, 14121}, {7583, 21464}, {7584, 13429}, {8036, 44648}, {8576, 14715}, {8754, 32787}, {8855, 35830}, {9722, 44193}, {11245, 45407}, {12322, 14826}, {13020, 23292}, {13439, 41588}, {13567, 14230}, {13748, 31383}, {17810, 23251}, {18130, 26919}, {23259, 26864}, {24243, 32806}, {34417, 42284}, {41762, 45401}

X(52286) = orthocentroidal-circle-inverse of X(32587)
X(52286) = polar conjugate of the isotomic conjugate of X(615)
X(52286) = polar conjugate of the isogonal conjugate of X(5058)
X(52286) = orthic isogonal conjugate of X(44638)
X(52286) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 44638}, {24243, 13933}, {44638, 44648}
X(52286) = X(5058)-cross conjugate of X(615)
X(52286) = X(63)-isoconjugate of X(589)
X(52286) = X(i)-Dao conjugate of X(j) for these (i,j): {69, 642}, {589, 3162}
X(52286) = crosspoint of X(4) and X(41516)
X(52286) = crosssum of X(3) and X(5409)
X(52286) = barycentric product X(i)*X(j) for these {i,j}: {4, 615}, {264, 5058}, {486, 44638}, {642, 41516}, {1586, 8036}, {6531, 51401}, {8741, 33395}, {8742, 33392}, {24243, 44648}
X(52286) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 589}, {615, 69}, {5058, 3}, {8036, 11091}, {44638, 491}, {44648, 487}, {51401, 6393}
X(52286) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 32587}, {4, 1585, 427}, {4, 3127, 5064}, {4, 3535, 3128}, {4, 19219, 5200}, {5, 8964, 2}, {3128, 3535, 5094}, {5200, 19219, 25}, {6806, 21737, 7484}

X(52287) = BARYCENTRIC PRODUCT X(264)*X(5062)

X(52287) lies on these lines: {2, 3}, {6, 6219}, {51, 3071}, {53, 5413}, {136, 45544}, {154, 23251}, {184, 3070}, {275, 14245}, {372, 12148}, {393, 5411}, {485, 10132}, {486, 8944}, {590, 44637}, {1162, 19041}, {1321, 42265}, {1495, 42284}, {1587, 11402}, {1588, 9777}, {1824, 1849}, {1843, 44638}, {1899, 13748}, {1993, 49028}, {2052, 14234}, {2969, 13390}, {3060, 6239}, {3069, 26375}, {3087, 5410}, {3167, 12602}, {3564, 13439}, {3580, 18539}, {3917, 12298}, {5412, 6748}, {6250, 8968}, {6289, 11090}, {6459, 12171}, {6515, 49086}, {6560, 10133}, {6565, 45841}, {6751, 44634}, {7090, 7140}, {7583, 13440}, {7584, 21463}, {8035, 44647}, {8577, 14715}, {8754, 32788}, {8854, 35831}, {9722, 44192}, {11245, 45406}, {12323, 14826}, {13019, 23292}, {13428, 41588}, {13567, 14233}, {13749, 31383}, {17810, 23261}, {18130, 26894}, {23249, 26864}, {24244, 32805}, {34417, 42283}, {41762, 45400}

X(52287) = orthocentroidal-circle-inverse of X(32588)
X(52287) = polar conjugate of the isotomic conjugate of X(590)
X(52287) = polar conjugate of the isogonal conjugate of X(5062)
X(52287) = orthic isogonal conjugate of X(44637)
X(52287) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 44637}, {24244, 13879}, {44637, 44647}
X(52287) = X(5062)-cross conjugate of X(590)
X(52287) = X(i)-isoconjugate of X(j) for these (i,j): {63, 588}, {75, 8825}
X(52287) = X(i)-Dao conjugate of X(j) for these (i,j): {69, 641}, {206, 8825}, {588, 3162}
X(52287) = crosspoint of X(4) and X(41515)
X(52287) = crosssum of X(i) and X(j) for these (i,j): {3, 5408}, {5406, 8910}
X(52287) = barycentric product X(i)*X(j) for these {i,j}: {4, 590}, {264, 5062}, {485, 44637}, {641, 41515}, {1585, 8035}, {6531, 51395}, {8741, 33393}, {8742, 33394}, {24244, 44647}
X(52287) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 588}, {32, 8825}, {590, 69}, {5062, 3}, {8035, 11090}, {44637, 492}, {44647, 488}, {51395, 6393}
X(52287) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 32588}, {4, 1586, 427}, {4, 3128, 5064}, {4, 3536, 3127}, {4, 5200, 25}, {3127, 3536, 5094}

X(52288) = BARYCENTRIC PRODUCT X(264)*X(5058)

X(52288) lies on these lines: {2, 3}, {6, 32000}, {53, 47355}, {69, 36794}, {76, 37669}, {83, 459}, {141, 3087}, {239, 7046}, {264, 1249}, {273, 5749}, {275, 18840}, {281, 17353}, {287, 14912}, {311, 28708}, {317, 3619}, {318, 5222}, {340, 21356}, {373, 34854}, {393, 3589}, {597, 40138}, {599, 6749}, {648, 5702}, {894, 1119}, {1073, 47435}, {1235, 11427}, {1785, 29598}, {1897, 17014}, {1990, 47352}, {1992, 44134}, {2052, 18841}, {2207, 17825}, {2322, 37681}, {3620, 27377}, {3763, 6748}, {3912, 34231}, {3978, 44144}, {5081, 29611}, {5286, 23292}, {5395, 38253}, {5485, 43530}, {6531, 7736}, {7745, 26958}, {7952, 17023}, {8743, 10601}, {9308, 51171}, {11175, 16081}, {14482, 41676}, {16080, 18842}, {16989, 39931}, {18311, 18808}, {26591, 46108}, {27376, 37873}, {32817, 36212}, {33630, 43981}, {35282, 45031}, {37645, 51481}, {37648, 41370}, {37649, 41361}, {41371, 43999}

X(52288) = polar conjugate of X(14484)
X(52288) = polar conjugate of the isotomic conjugate of X(15589)
X(52288) = polar conjugate of the isogonal conjugate of X(5085)
X(52288) = X(5085)-cross conjugate of X(15589)
X(52288) = X(48)-isoconjugate of X(14484)
X(52288) = X(i)-Dao conjugate of X(j) for these (i,j): {1249, 14484}, {7736, 10519}
X(52288) = barycentric product X(i)*X(j) for these {i,j}: {4, 15589}, {264, 5085}
X(52288) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 14484}, {5085, 3}, {15589, 69}
X(52288) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 458, 4}, {2, 3091, 52251}, {2, 3839, 44216}, {2, 11348, 441}, {2, 37174, 11331}, {2, 37188, 631}, {2, 40884, 3524}, {2, 52253, 6820}, {4, 3524, 35474}, {69, 36794, 40065}, {141, 3087, 32001}, {264, 3618, 1249}, {427, 6620, 4}, {1585, 1586, 7378}, {3535, 3536, 8889}, {5055, 44346, 2}

X(52289) = BARYCENTRIC PRODUCT X(264)*X(5092)

X(52289) lies on these lines: {2, 3}, {6, 44134}, {53, 51126}, {76, 11064}, {83, 16080}, {141, 340}, {264, 1990}, {273, 17368}, {275, 10159}, {287, 8550}, {315, 37638}, {317, 3763}, {318, 17367}, {343, 7768}, {373, 47204}, {575, 41145}, {597, 648}, {1235, 14389}, {2052, 43527}, {3087, 3619}, {3096, 43462}, {3199, 6704}, {3580, 7762}, {3618, 9308}, {3620, 40065}, {3815, 6531}, {5081, 17292}, {5092, 16264}, {5254, 41254}, {5702, 32000}, {5972, 6248}, {6103, 7792}, {6530, 38317}, {6688, 34854}, {6748, 34573}, {7282, 17291}, {7745, 47296}, {7750, 51372}, {7754, 37645}, {7812, 44569}, {7859, 14165}, {7875, 16318}, {8743, 51358}, {12203, 13394}, {14561, 44704}, {15258, 42287}, {15595, 18553}, {17907, 47355}, {18022, 41259}, {23292, 40814}, {25555, 42873}, {29579, 34231}, {32002, 51128}, {32819, 51389}, {32820, 36212}, {37765, 48310}, {38110, 41204}, {40684, 41366}, {44142, 46106}

X(52289) = orthocentroidal-circle-inverse of X(11331)
X(52289) = polar conjugate of X(14492)
X(52289) = isotomic conjugate of the polar conjugate of X(16264)
X(52289) = polar conjugate of the isotomic conjugate of X(37671)
X(52289) = polar conjugate of the isogonal conjugate of X(5092)
X(52289) = X(5092)-cross conjugate of X(37671)
X(52289) = X(48)-isoconjugate of X(14492)
X(52289) = X(1249)-Dao conjugate of X(14492)
X(52289) = barycentric product X(i)*X(j) for these {i,j}: {4, 37671}, {69, 16264}, {264, 5092}
X(52289) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 14492}, {5092, 3}, {16264, 4}, {37671, 69}
X(52289) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 11331}, {2, 381, 44576}, {2, 458, 297}, {2, 11348, 37188}, {2, 37188, 37067}, {2, 41238, 6656}, {2, 44575, 549}, {2, 46571, 41237}, {2, 52247, 44334}, {4, 11331, 297}, {141, 6749, 340}, {141, 36794, 27377}, {340, 6749, 27377}, {340, 36794, 6749}, {458, 11331, 4}, {470, 471, 427}, {470, 16250, 462}, {471, 16249, 463}, {1585, 1586, 5064}, {11289, 11290, 7399}

X(52290) = BARYCENTRIC PRODUCT X(264)*X(5210)

X(52290) lies on these lines: {2, 3}, {67, 41719}, {69, 5095}, {125, 35260}, {184, 43812}, {230, 40138}, {340, 1007}, {373, 6403}, {393, 3054}, {459, 7607}, {511, 21971}, {599, 15471}, {648, 23055}, {1112, 33884}, {1249, 6103}, {1649, 41357}, {1876, 31188}, {1899, 44108}, {1974, 16187}, {1990, 37637}, {2501, 9168}, {3055, 3087}, {3291, 39575}, {3316, 10881}, {3317, 10880}, {3634, 7718}, {4549, 12900}, {5466, 47627}, {5544, 12167}, {5642, 32234}, {5651, 19128}, {5702, 7735}, {6337, 37803}, {6723, 46264}, {6749, 31489}, {6776, 47296}, {7612, 16080}, {7713, 19878}, {7716, 51127}, {7998, 44084}, {8371, 47217}, {8550, 18950}, {8617, 35325}, {8744, 20481}, {8791, 40103}, {8972, 13937}, {9189, 44427}, {9464, 34336}, {9754, 45864}, {9927, 25712}, {10192, 23291}, {11059, 44146}, {11202, 18918}, {11216, 47446}, {11363, 19877}, {11427, 34565}, {12135, 46932}, {12294, 15082}, {13884, 13941}, {14494, 43530}, {14580, 39576}, {14912, 37643}, {15118, 18919}, {16774, 34774}, {17907, 17983}, {19119, 31267}, {19872, 49542}, {21356, 44102}, {22750, 37515}, {24855, 41370}, {25502, 40976}, {30247, 32133}, {32000, 37688}, {32223, 51212}, {32250, 45311}, {32818, 37804}, {34229, 44134}, {37687, 44105}, {38253, 43537}, {40112, 51179}, {44145, 52147}, {44569, 50974}

X(52290) = complement of X(30769)
X(52290) = polar conjugate of X(41895)
X(52290) = polar conjugate of the isotomic conjugate of X(11160)
X(52290) = polar conjugate of the isogonal conjugate of X(5210)
X(52290) = X(5210)-cross conjugate of X(11160)
X(52290) = X(48)-isoconjugate of X(41895)
X(52290) = X(1249)-Dao conjugate of X(41895)
X(52290) = crosssum of X(6) and X(17813)
X(52290) = barycentric product X(i)*X(j) for these {i,j}: {4, 11160}, {264, 5210}
X(52290) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 41895}, {5210, 3}, {11160, 69}
X(52290) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20, 5159}, {2, 468, 4}, {2, 4232, 5094}, {2, 6353, 8889}, {2, 7426, 30775}, {2, 7493, 16051}, {2, 10565, 30771}, {2, 11284, 5067}, {2, 37453, 38282}, {2, 38282, 6353}, {2, 40132, 3090}, {2, 47597, 5071}, {3, 37911, 2}, {3, 37984, 49670}, {4, 468, 6353}, {4, 3524, 35485}, {4, 3525, 37118}, {4, 38282, 468}, {5, 37460, 4}, {25, 37977, 44879}, {25, 40916, 3520}, {125, 35260, 39874}, {403, 35485, 4}, {468, 5094, 4232}, {470, 471, 37174}, {631, 7505, 6622}, {1657, 47629, 7396}, {3091, 37196, 4}, {3147, 14940, 3090}, {3147, 40132, 6353}, {3515, 5056, 4}, {3542, 37118, 4}, {4232, 5094, 4}, {5000, 5001, 3543}, {5059, 10019, 4}, {6353, 8889, 7714}, {6677, 7401, 40132}, {6804, 7542, 631}, {7426, 30775, 15682}, {7493, 16051, 376}, {7494, 46336, 10299}, {9909, 31856, 47315}, {9909, 47315, 5059}, {10154, 47629, 1657}, {14940, 38282, 40132}, {16042, 37977, 25}, {16419, 21313, 3516}, {34609, 37910, 49135}, {37984, 49670, 4}

X(52291) = BARYCENTRIC PRODUCT X(264)*X(6424)

X(52291) lies on these lines: {2, 3}, {6, 1163}, {19, 13456}, {51, 1587}, {154, 3071}, {184, 1588}, {242, 13387}, {275, 45102}, {371, 8970}, {393, 5412}, {590, 1164}, {637, 14826}, {1162, 26375}, {1165, 3087}, {1249, 5410}, {1495, 23259}, {1659, 1851}, {1843, 19042}, {1899, 5871}, {2052, 14244}, {3068, 45401}, {3069, 10133}, {3070, 17810}, {3167, 49029}, {5409, 49039}, {5411, 40065}, {5870, 31383}, {6406, 44079}, {6459, 10132}, {6515, 49322}, {6524, 8946}, {7009, 13386}, {7102, 14121}, {7581, 9777}, {7582, 11402}, {8780, 12601}, {9540, 45503}, {10192, 45863}, {10783, 11245}, {11206, 45406}, {11433, 45407}, {13020, 26331}, {13052, 33365}, {13429, 44077}, {13440, 14593}, {13567, 13749}, {21970, 26438}, {23249, 34417}, {23253, 44106}, {23263, 44082}, {23273, 26864}, {23292, 45441}, {24244, 32085}, {31860, 42284}, {41424, 42283}, {41588, 49087}, {43976, 45511}

X(52291) = orthocentroidal-circle-inverse of X(3128)
X(52291) = polar conjugate of X(5491)
X(52291) = polar conjugate of the isotomic conjugate of X(3069)
X(52291) = polar conjugate of the isogonal conjugate of X(6424)
X(52291) = orthic-isogonal conjugate of X(19041)
X(52291) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 19041}, {6524, 5200}
X(52291) = X(i)-cross conjugate of X(j) for these (i,j): {6424, 3069}, {44648, 41515}, {45479, 4}
X(52291) = X(i)-isoconjugate of X(j) for these (i,j): {48, 5491}, {63, 494}, {255, 24243}, {304, 26461}, {326, 8946}, {394, 19217}, {656, 1307}, {6464, 19215}, {34055, 45594}
X(52291) = X(i)-Dao conjugate of X(j) for these (i,j): {69, 33365}, {494, 3162}, {1249, 5491}, {1307, 40596}, {4176, 8222}, {6523, 24243}, {8946, 15259}
X(52291) = barycentric product X(i)*X(j) for these {i,j}: {4, 3069}, {107, 17432}, {158, 19216}, {264, 6424}, {393, 487}, {1093, 51905}, {1132, 19041}, {2052, 10133}, {2207, 46743}, {3536, 8038}, {5200, 26494}, {6524, 8223}, {39388, 41516}
X(52291) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 5491}, {25, 494}, {112, 1307}, {393, 24243}, {487, 3926}, {1096, 19217}, {1843, 45594}, {1974, 26461}, {2207, 8946}, {3069, 69}, {5200, 26503}, {6423, 42022}, {6424, 3}, {6562, 17431}, {8223, 4176}, {8948, 6464}, {10133, 394}, {17432, 3265}, {19041, 1271}, {19216, 326}, {45400, 45726}, {45401, 45414}, {45479, 45473}, {51905, 3964}
X(52291) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 3128}, {2, 6620, 5200}, {4, 25, 5200}, {4, 1585, 3127}, {4, 3535, 427}, {4, 3536, 32587}, {4, 6353, 1586}, {4, 19219, 25}, {235, 6618, 5200}, {428, 32588, 4}, {460, 6353, 5200}, {461, 37226, 5200}, {468, 32587, 3536}, {3079, 37197, 5200}, {3540, 36703, 7484}, {5412, 41516, 393}

X(52292) = BARYCENTRIC PRODUCT X(264)*X(8588)

X(52292) lies on these lines: {2, 3}, {67, 19153}, {76, 34336}, {125, 26864}, {141, 15471}, {232, 18573}, {599, 5095}, {648, 8860}, {1112, 7998}, {1560, 8778}, {1649, 2501}, {1829, 34595}, {1990, 3054}, {2052, 10185}, {2970, 52147}, {3055, 6749}, {3167, 41724}, {3172, 7749}, {3624, 11396}, {3763, 19118}, {5023, 39602}, {5090, 51073}, {5410, 8252}, {5411, 8253}, {5489, 9209}, {5642, 15069}, {5643, 8537}, {5650, 44084}, {5651, 34397}, {5702, 37689}, {5972, 6090}, {6103, 37637}, {6746, 11465}, {6800, 15059}, {7607, 9717}, {7608, 43530}, {7746, 8791}, {7755, 40126}, {8371, 47627}, {8550, 26869}, {8743, 20481}, {8780, 23293}, {9308, 17006}, {9745, 30435}, {10169, 47446}, {10182, 18396}, {10294, 44751}, {10418, 13881}, {10602, 15118}, {11216, 47276}, {11402, 26958}, {11405, 16511}, {11408, 43028}, {11409, 43029}, {11459, 13148}, {11477, 32225}, {12099, 15073}, {12135, 19877}, {12167, 47355}, {12315, 43608}, {12505, 30514}, {12828, 40107}, {13884, 32786}, {13937, 32785}, {14530, 23294}, {14852, 30714}, {14975, 17124}, {15066, 19504}, {15106, 34117}, {15271, 44090}, {15274, 47204}, {18553, 35259}, {21358, 44102}, {22112, 44080}, {23606, 37877}, {31201, 42856}, {32234, 38397}, {32250, 43273}, {32821, 37804}, {35265, 48662}, {37674, 44097}, {37682, 44086}, {37688, 44134}, {39575, 39576}

X(52292) = polar conjugate of X(17503)
X(52292) = polar conjugate of the isotomic conjugate of X(15533)
X(52292) = polar conjugate of the isogonal conjugate of X(8588)
X(52292) = X(8588)-cross conjugate of X(15533)
X(52292) = X(48)-isoconjugate of X(17503)
X(52292) = X(1249)-Dao conjugate of X(17503)
X(52292) = crosssum of X(6) and X(11216)
X(52292) = crossdifference of every pair of points on line {647, 44810}
X(52292) = barycentric product X(i)*X(j) for these {i,j}: {4, 15533}, {264, 8588}
X(52292) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 17503}, {8588, 3}, {15533, 69}
X(52292) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 468, 5094}, {2, 6676, 31255}, {2, 6677, 7539}, {2, 7493, 5159}, {2, 37453, 25}, {2, 38282, 427}, {2, 40132, 37454}, {4, 35486, 37934}, {4, 37934, 37196}, {5, 35486, 37196}, {5, 37934, 4}, {5, 47597, 30734}, {140, 1656, 7395}, {140, 6823, 3523}, {186, 5055, 18386}, {403, 5054, 11410}, {403, 35492, 4}, {427, 468, 4232}, {427, 7408, 5064}, {468, 5094, 25}, {468, 10301, 6353}, {549, 37984, 35485}, {1656, 1657, 10255}, {1656, 10018, 3515}, {1656, 45735, 3851}, {1657, 31856, 858}, {1995, 21284, 25}, {2045, 2046, 34664}, {3147, 3628, 7507}, {3515, 7395, 3516}, {3526, 7505, 1593}, {3533, 7383, 140}, {4232, 38282, 468}, {5000, 5001, 3830}, {5004, 5005, 35493}, {5064, 6353, 25}, {5094, 37453, 468}, {5159, 7493, 31152}, {5972, 37638, 6090}, {14002, 21844, 37969}, {15750, 30734, 25}, {32534, 45735, 3515}, {35485, 37984, 44438}, {35486, 37196, 15750}, {39575, 39576, 44467}

X(52293) = BARYCENTRIC PRODUCT X(264)*X(8589)

X(52293) lies on these lines: {2, 3}, {39, 44467}, {125, 8550}, {275, 10185}, {340, 37688}, {373, 1112}, {575, 45311}, {576, 44569}, {597, 5095}, {1235, 34336}, {1506, 1560}, {1829, 51073}, {1843, 51128}, {1974, 51127}, {1990, 3055}, {3054, 6749}, {3589, 15471}, {3624, 12135}, {3815, 6103}, {3917, 40929}, {5050, 40920}, {5090, 34595}, {5305, 9745}, {5643, 38079}, {5972, 18553}, {6403, 33879}, {6723, 25555}, {7607, 43530}, {7608, 16080}, {7703, 39884}, {7746, 24855}, {8541, 20582}, {8791, 31406}, {9730, 13148}, {10169, 47280}, {10990, 23328}, {11064, 34507}, {11363, 19878}, {11396, 19877}, {11405, 21356}, {12099, 50649}, {12506, 34113}, {13567, 34565}, {13881, 16317}, {13884, 32789}, {13937, 32790}, {14389, 15059}, {15018, 19504}, {16187, 44080}, {16318, 31489}, {17006, 27377}, {23332, 44108}, {32154, 44102}, {32234, 50979}, {32250, 47354}, {32820, 37804}, {34573, 41584}, {37687, 44097}, {38397, 40112}, {43831, 43903}

X(52293) = polar-circle-inverse of X(37907)
X(52293) = polar conjugate of X(45103)
X(52293) = polar conjugate of the isotomic conjugate of X(22165)
X(52293) = polar conjugate of the isogonal conjugate of X(8589)
X(52293) = X(8589)-cross conjugate of X(22165)
X(52293) = X(48)-isoconjugate of X(45103)
X(52293) = X(1249)-Dao conjugate of X(45103)
X(52293) = barycentric product X(i)*X(j) for these {i,j}: {4, 22165}, {264, 8589}
X(52293) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 45103}, {8589, 3}, {22165, 69}
X(52293) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1995, 37911}, {2, 5094, 468}, {2, 5159, 30739}, {2, 8889, 37453}, {2, 30744, 6676}, {2, 30745, 7495}, {2, 30771, 7499}, {2, 31236, 6677}, {5, 5498, 44240}, {25, 7409, 428}, {140, 1656, 7399}, {140, 3850, 37814}, {140, 10224, 550}, {140, 12362, 3523}, {427, 468, 10301}, {428, 468, 4232}, {428, 8889, 427}, {468, 5094, 427}, {1656, 6640, 140}, {1656, 11479, 5056}, {3516, 5056, 10019}, {3520, 16042, 25}, {3533, 7509, 140}, {3545, 11410, 13473}, {3628, 37119, 235}, {4232, 37453, 468}, {5000, 5001, 3845}, {6676, 47629, 16063}, {7426, 31857, 3627}, {7495, 30745, 1368}, {8889, 37453, 428}, {10297, 18580, 44285}, {14709, 14710, 15750}, {15765, 18585, 47333}, {16063, 30744, 47629}, {47612, 47613, 44249}

X(52294) = BARYCENTRIC PRODUCT X(264)*X(13338)

X(52294) lies on these lines: {2, 3}, {6, 14491}, {51, 14157}, {53, 52166}, {54, 44110}, {74, 13603}, {112, 10985}, {143, 43605}, {185, 38848}, {1154, 15052}, {1179, 15424}, {1199, 6759}, {1495, 15033}, {1614, 10110}, {1843, 2914}, {1870, 37602}, {1994, 10540}, {3060, 46261}, {3092, 6470}, {3093, 6471}, {3431, 41424}, {3455, 14639}, {3563, 12074}, {3567, 26883}, {5480, 19596}, {5621, 15081}, {5890, 12112}, {5891, 15107}, {5892, 10545}, {6000, 44106}, {6344, 32085}, {8164, 10833}, {8541, 15520}, {8744, 10311}, {9544, 39522}, {9626, 12571}, {9658, 10591}, {9673, 10590}, {10117, 23324}, {10312, 14581}, {10605, 31860}, {10625, 43614}, {10641, 16538}, {10642, 16539}, {11002, 18445}, {11224, 31948}, {11424, 26882}, {11438, 11455}, {11440, 46849}, {11456, 17810}, {11464, 44082}, {13364, 15018}, {13451, 15087}, {13567, 16658}, {13570, 32237}, {14094, 14831}, {14483, 44109}, {14855, 43584}, {14915, 15053}, {15516, 44102}, {15873, 34224}, {16621, 26879}, {16659, 43808}, {18532, 43699}, {18874, 37471}, {18954, 47743}, {20125, 21850}, {23267, 44599}, {23273, 44598}, {31412, 35777}, {34782, 43818}, {35264, 44413}, {35707, 38072}, {35776, 42561}, {37546, 38074}, {43598, 45186}

X(52294) = reflection of X(13595) in X(7545)
X(52294) = polar conjugate of the isotomic conjugate of X(15018)
X(52294) = polar conjugate of the isogonal conjugate of X(13338)
X(52294) = X(13338)-cross conjugate of X(15018)
X(52294) = X(63)-isoconjugate of X(30537)
X(52294) = X(3162)-Dao conjugate of X(30537)
X(52294) = barycentric product X(i)*X(j) for these {i,j}: {4, 15018}, {264, 13338}, {275, 13364}
X(52294) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 30537}, {13338, 3}, {13364, 343}, {15018, 69}
X(52294) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7530, 37925}, {2, 12083, 44832}, {3, 47486, 44880}, {4, 24, 14865}, {4, 25, 186}, {4, 186, 13596}, {4, 1598, 26863}, {4, 3089, 16868}, {4, 3517, 35478}, {4, 3518, 3520}, {4, 6143, 1595}, {4, 7487, 34797}, {4, 10594, 34484}, {4, 14940, 15559}, {4, 21844, 35502}, {4, 34484, 3518}, {4, 35473, 1597}, {4, 35482, 1907}, {4, 37943, 427}, {4, 44879, 1593}, {4, 47485, 378}, {5, 5899, 6636}, {5, 34603, 46450}, {22, 3545, 7550}, {23, 381, 35921}, {24, 1597, 35473}, {24, 5198, 4}, {24, 14865, 17506}, {24, 35473, 186}, {25, 186, 3518}, {25, 378, 47485}, {25, 1597, 24}, {25, 5198, 1597}, {25, 11410, 3517}, {25, 18535, 378}, {25, 44274, 37951}, {26, 3832, 35500}, {51, 14157, 15032}, {186, 13596, 3520}, {186, 14865, 35473}, {186, 34484, 25}, {186, 35473, 17506}, {376, 18534, 37946}, {378, 18535, 4}, {378, 47485, 186}, {403, 428, 4}, {546, 18378, 7488}, {549, 21308, 16042}, {1596, 7576, 4}, {1596, 10301, 7576}, {1597, 14865, 13596}, {1597, 35473, 14865}, {1598, 10594, 4}, {1906, 6240, 4}, {1906, 7715, 6240}, {1907, 10018, 35482}, {1995, 18534, 376}, {2070, 3845, 7527}, {3091, 7517, 7512}, {3091, 37913, 7514}, {3517, 11410, 44878}, {3517, 35502, 21844}, {3518, 13596, 186}, {3518, 17506, 24}, {3518, 44880, 47486}, {3543, 6644, 7464}, {3543, 14002, 6644}, {3575, 44803, 4}, {3627, 13621, 22467}, {3830, 12106, 2071}, {3843, 37440, 14118}, {3851, 17714, 37126}, {3853, 45735, 12086}, {5000, 5001, 7496}, {5020, 12082, 3524}, {5066, 37947, 3}, {5133, 37971, 7552}, {5198, 34484, 17506}, {5899, 6636, 12088}, {6759, 9781, 1199}, {7514, 7517, 37913}, {7514, 37913, 7512}, {7540, 46030, 3153}, {7556, 41099, 9818}, {10298, 51519, 37953}, {10311, 33885, 8744}, {10594, 26863, 3518}, {10985, 33842, 112}, {11410, 44878, 21844}, {14865, 17506, 3520}, {14865, 26863, 5198}, {15559, 21841, 14940}, {21308, 37924, 549}, {21844, 35502, 35478}, {21844, 44878, 186}, {25337, 50135, 2}, {26863, 34484, 4}, {31861, 51519, 10298}, {35472, 44879, 186}, {35502, 44878, 11410}, {37349, 46451, 5}, {37925, 44832, 12083}, {42789, 42790, 8703}, {51446, 51447, 1843}

X(52295) = BARYCENTRIC PRODUCT X(264)*X(13351)

X(52295) lies on these lines: {2, 3}, {6, 43808}, {49, 34514}, {51, 26917}, {52, 23293}, {54, 18381}, {93, 264}, {113, 11439}, {125, 3567}, {195, 18356}, {262, 14378}, {275, 25044}, {389, 23294}, {393, 31407}, {568, 13561}, {578, 25739}, {1092, 41171}, {1112, 20396}, {1157, 4994}, {1173, 17711}, {1199, 1899}, {1209, 2979}, {1235, 7796}, {1568, 15058}, {1614, 11550}, {1853, 7592}, {1870, 37719}, {1986, 15101}, {1993, 11271}, {1994, 25738}, {2548, 8744}, {2888, 16266}, {2914, 3448}, {3043, 23236}, {3060, 5449}, {3527, 5900}, {3574, 5890}, {3818, 43598}, {5448, 15305}, {5462, 26913}, {5476, 43836}, {5523, 9607}, {5640, 43817}, {5734, 31948}, {5889, 7703}, {6101, 6152}, {6102, 12300}, {6150, 35718}, {6198, 37720}, {6241, 18388}, {6242, 11412}, {6243, 34826}, {6403, 40107}, {6530, 45090}, {7699, 12290}, {7706, 43601}, {7722, 15100}, {7730, 14076}, {7749, 10986}, {7752, 44142}, {9300, 41366}, {9606, 27376}, {9698, 27371}, {9936, 11442}, {10116, 11422}, {10627, 11576}, {10641, 42488}, {10642, 42489}, {10880, 35812}, {10881, 35813}, {11004, 32358}, {11270, 23328}, {11393, 31452}, {11423, 12242}, {11424, 23325}, {11430, 11572}, {11431, 20303}, {11438, 43608}, {11444, 51392}, {11449, 45286}, {11457, 15032}, {11475, 42814}, {11476, 42813}, {12279, 44866}, {13403, 18394}, {13419, 26882}, {13470, 14805}, {13472, 38433}, {13568, 43607}, {14143, 35887}, {14644, 32743}, {14651, 39845}, {14853, 20300}, {14912, 23300}, {15063, 15102}, {15069, 39588}, {15099, 23301}, {15110, 38397}, {15114, 45237}, {15345, 35719}, {16252, 16658}, {17835, 23315}, {17847, 33565}, {18396, 43818}, {18474, 34148}, {18952, 34545}, {23292, 34224}, {23319, 47065}, {23332, 26879}, {24206, 46026}, {26881, 44516}, {32138, 34796}

X(52295) = midpoint of X(4) and X(35475)
X(52295) = complement of X(38435)
X(52295) = orthocentroidal-circle-inverse of X(3518)
X(52295) = polar-circle-inverse of X(37936)
X(52295) = complement of the isogonal conjugate of X(38433)
X(52295) = polar conjugate of the isogonal conjugate of X(13351)
X(52295) = X(38433)-complementary conjugate of X(10)
X(52295) = crosssum of X(i) and X(j) for these (i,j): {3, 9704}, {184, 10979}
X(52295) = barycentric product X(264)*X(13351)
X(52295) = barycentric quotient X(13351)/X(3)
X(52295) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 3518}, {2, 20, 47525}, {2, 14790, 7512}, {3, 4, 18559}, {4, 5, 44958}, {4, 1594, 7577}, {4, 3088, 13596}, {4, 3520, 34797}, {4, 3541, 3520}, {4, 6143, 24}, {4, 7505, 34484}, {4, 7577, 16868}, {4, 8889, 37119}, {4, 13619, 12173}, {4, 14940, 25}, {4, 21844, 3575}, {4, 35473, 6240}, {4, 35482, 378}, {4, 37119, 186}, {4, 37943, 10594}, {4, 44879, 7576}, {4, 47485, 6756}, {5, 427, 15559}, {5, 631, 14789}, {5, 858, 631}, {5, 1595, 1906}, {5, 1906, 403}, {5, 13371, 37452}, {5, 15559, 4}, {5, 37452, 2}, {5, 44958, 16868}, {20, 47525, 7512}, {24, 5094, 6143}, {51, 32767, 26917}, {140, 7576, 44879}, {378, 7507, 4}, {381, 12084, 34007}, {381, 35502, 4}, {382, 7579, 5}, {403, 1595, 4}, {427, 1594, 4}, {427, 10224, 35502}, {858, 45181, 186}, {1368, 14788, 3525}, {1593, 7547, 4}, {1594, 15559, 5}, {1597, 35488, 4}, {1656, 5064, 10594}, {1656, 10594, 37943}, {1656, 13154, 2}, {1656, 50138, 37353}, {2041, 2042, 7488}, {3518, 3520, 7512}, {3518, 14790, 34797}, {3528, 45179, 44958}, {3542, 7378, 4}, {3549, 7391, 12088}, {3574, 20299, 5890}, {3575, 37118, 21844}, {5000, 5001, 37353}, {5064, 5094, 37920}, {5064, 10594, 4}, {5133, 11585, 3090}, {5448, 18488, 15305}, {5576, 13154, 37353}, {5576, 13371, 2}, {5576, 37452, 5}, {6640, 11818, 44802}, {6756, 10018, 47485}, {7506, 31283, 2}, {7566, 30744, 6642}, {7577, 44958, 5}, {7699, 12290, 43831}, {10224, 33332, 381}, {11430, 11572, 12289}, {12173, 35477, 13619}, {13160, 23335, 376}, {13371, 50138, 13154}, {14789, 45181, 16868}, {14790, 47525, 20}, {18377, 44287, 14130}, {18386, 35490, 4}, {18560, 23047, 4}, {18586, 18587, 44278}, {23332, 45089, 26879}, {33703, 45177, 44958}, {37119, 45181, 14789}, {37938, 50138, 1656}, {44995, 45007, 44958}, {44996, 44998, 45001}

X(52296) = BARYCENTRIC PRODUCT X(264)*X(14806)

X(52296) lies on these lines: {2, 3}, {6, 26917}, {96, 43530}, {125, 7592}, {155, 23293}, {184, 32767}, {567, 45622}, {599, 8537}, {974, 12281}, {1181, 23294}, {1199, 26869}, {1209, 15066}, {1506, 8743}, {1614, 1853}, {1698, 41722}, {1968, 7603}, {1986, 15043}, {1993, 5449}, {2904, 5422}, {3043, 38724}, {3055, 27376}, {3567, 26958}, {3763, 6403}, {5523, 31401}, {5972, 44795}, {6241, 40686}, {6746, 15067}, {6800, 44516}, {7608, 43678}, {8252, 10881}, {8253, 10880}, {8739, 42489}, {8740, 42488}, {9707, 18381}, {9820, 11442}, {10312, 37637}, {10605, 43608}, {10632, 43029}, {10633, 43028}, {11202, 11572}, {11402, 43808}, {11412, 37638}, {11456, 20299}, {11457, 23332}, {11475, 42914}, {11476, 42915}, {11704, 15033}, {12242, 14076}, {12300, 15028}, {13367, 23325}, {13561, 18445}, {14528, 16000}, {14852, 34148}, {14864, 40276}, {15024, 43866}, {15032, 26944}, {15463, 20304}, {15739, 32339}, {18474, 43839}, {18912, 23292}, {18916, 43593}, {19128, 47355}, {19357, 25739}, {20806, 34507}, {23329, 43831}, {24206, 39588}, {26913, 36752}, {28706, 32821}, {31489, 39575}, {33565, 43908}, {34781, 41450}, {39113, 44134}, {44518, 50718}, {45089, 47296}

X(52296) = orthocentroidal-circle-inverse of X(10018)
X(52296) = polar conjugate of the isogonal conjugate of X(14806)
X(52296) = barycentric product X(264)*X(14806)
X(52296) = barycentric quotient X(14806)/X(3)
X(52296) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 10018}, {2, 1594, 24}, {2, 7544, 16238}, {2, 11585, 7509}, {2, 37444, 7542}, {3, 7547, 35480}, {3, 7577, 7547}, {3, 18386, 34797}, {3, 31283, 30744}, {4, 140, 32534}, {4, 3523, 10295}, {4, 3533, 35486}, {4, 5056, 35487}, {4, 7505, 21841}, {4, 10018, 24}, {4, 17506, 37196}, {4, 21841, 10594}, {4, 37118, 35477}, {5, 378, 35488}, {5, 6640, 17928}, {5, 37119, 378}, {25, 5070, 14940}, {381, 3520, 35490}, {403, 3541, 35502}, {427, 3628, 7505}, {427, 7505, 10594}, {427, 21841, 4}, {470, 471, 52253}, {549, 23047, 35471}, {631, 6240, 35472}, {858, 3549, 10323}, {1593, 5055, 16868}, {1594, 10018, 4}, {1656, 5094, 4}, {2045, 2046, 13160}, {3090, 3541, 403}, {3147, 7576, 24}, {3516, 3851, 4}, {3526, 7507, 186}, {3542, 8889, 15559}, {5054, 12173, 21844}, {5067, 8889, 3542}, {5498, 44263, 3}, {6143, 7577, 3}, {6639, 13371, 22}, {7506, 39504, 7566}, {7542, 37444, 44837}, {7569, 31282, 2}, {7570, 46935, 1656}, {7579, 45735, 7564}, {10024, 18281, 11413}, {14813, 14814, 6639}, {15720, 37196, 17506}, {35482, 44958, 1597}, {35484, 44959, 4}

X(52297) = BARYCENTRIC PRODUCT X(264)*X(15513)

X(52297) lies on these lines: {2, 3}, {33, 5326}, {34, 7294}, {125, 10192}, {141, 44102}, {184, 47296}, {232, 3054}, {343, 5972}, {590, 13937}, {615, 13884}, {1112, 3917}, {1495, 23332}, {1514, 11204}, {1698, 12135}, {1829, 19862}, {1843, 40670}, {1862, 31235}, {1974, 34573}, {2052, 11668}, {2501, 10190}, {2883, 43903}, {3055, 10311}, {3589, 8541}, {3618, 11405}, {3619, 19118}, {3634, 11363}, {3819, 44084}, {5095, 22165}, {5186, 31274}, {5410, 32786}, {5411, 32785}, {5412, 32790}, {5413, 32789}, {5550, 11396}, {6688, 47328}, {7749, 14581}, {8024, 34336}, {8167, 11383}, {8739, 23302}, {8740, 23303}, {10169, 32113}, {10189, 47217}, {10278, 47627}, {10625, 43823}, {11216, 47279}, {11245, 17809}, {11402, 37643}, {11550, 15448}, {11614, 33842}, {12133, 14855}, {12143, 31239}, {12834, 14389}, {13148, 38795}, {13366, 13567}, {14165, 14569}, {14975, 17122}, {15004, 23292}, {15067, 52000}, {15108, 18947}, {15151, 17853}, {15471, 50991}, {16318, 37637}, {17005, 27377}, {19596, 23300}, {23291, 26864}, {23328, 51403}, {26913, 48906}, {26917, 31804}, {26926, 31267}, {31253, 49542}, {34481, 47298}, {37633, 44097}, {37775, 43329}, {37776, 43328}, {39176, 40559}, {40938, 44467}, {41585, 48310}, {41587, 43839}

X(52297) = complement of X(30744)
X(52297) = polar conjugate of the isotomic conjugate of X(3630)
X(52297) = polar conjugate of the isogonal conjugate of X(15513)
X(52297) = X(15513)-cross conjugate of X(3630)
X(52297) = barycentric product X(i)*X(j) for these {i,j}: {4, 3630}, {264, 15513}
X(52297) = barycentric quotient X(i)/X(j) for these {i,j}: {3630, 69}, {15513, 3}
X(52297) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 22, 5159}, {2, 468, 427}, {2, 5020, 37454}, {2, 6353, 5094}, {2, 6676, 30739}, {2, 6677, 37439}, {2, 7493, 30771}, {2, 7494, 31255}, {2, 37453, 468}, {2, 38282, 25}, {2, 40132, 7539}, {5, 34477, 44249}, {5, 44234, 44211}, {25, 5094, 7378}, {25, 7378, 428}, {25, 37453, 38282}, {25, 38282, 468}, {140, 7505, 235}, {140, 37942, 378}, {378, 7505, 37942}, {378, 37942, 235}, {381, 35486, 37931}, {428, 468, 6353}, {428, 5094, 427}, {547, 37935, 4}, {631, 6623, 11410}, {1596, 11539, 37118}, {1656, 3147, 3575}, {2072, 34351, 44239}, {3090, 3515, 23047}, {5000, 5001, 3627}, {5004, 5005, 35497}, {5066, 37934, 35480}, {5071, 37460, 18386}, {5094, 6353, 428}, {6353, 7378, 25}, {6622, 10303, 3516}, {6623, 11410, 1885}, {6639, 16238, 7399}, {6676, 37911, 2}, {7493, 30771, 7667}, {7494, 31255, 43957}, {10018, 14940, 5}, {12100, 37984, 35481}, {12108, 44226, 35477}, {14869, 44960, 3520}, {15699, 37458, 7577}, {16239, 21841, 37119}, {21213, 37454, 427}, {21841, 37119, 1907}, {21844, 35487, 3627}, {34330, 44234, 5}, {37118, 37943, 1596}

X(52298) = BARYCENTRIC PRODUCT X(264)*X(15515)

X(52298) lies on these lines: {2, 3}, {125, 11402}, {275, 11668}, {599, 11405}, {1112, 11451}, {1180, 44467}, {1506, 3172}, {1698, 11396}, {1853, 26864}, {2904, 15047}, {3167, 23293}, {3763, 12167}, {5050, 26913}, {5090, 19862}, {5095, 51185}, {5326, 11393}, {5410, 8253}, {5411, 8252}, {5422, 15059}, {5550, 12135}, {5650, 47328}, {6090, 21243}, {6403, 44299}, {6697, 19125}, {6723, 12828}, {6746, 7999}, {7140, 17917}, {7294, 11392}, {7703, 35264}, {8541, 21358}, {8791, 39951}, {9308, 17005}, {9777, 26958}, {11363, 34595}, {11408, 43029}, {11409, 43028}, {11426, 26917}, {12174, 13399}, {14389, 40920}, {14975, 17125}, {15153, 19467}, {19118, 47355}, {19347, 23294}, {19357, 32767}, {22112, 44077}, {23292, 26869}, {31489, 45141}, {33565, 44731}, {34397, 43650}, {34469, 43831}, {34573, 41585}, {37679, 44097}, {37682, 44105}

X(52298) = polar conjugate of the isogonal conjugate of X(15515)
X(52298) = barycentric product X(264)*X(15515)
X(52298) = barycentric quotient X(15515)/X(3)
X(52298) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 427, 37453}, {2, 5094, 25}, {2, 5159, 31255}, {2, 8889, 468}, {2, 16051, 7499}, {2, 30744, 3}, {2, 30745, 7485}, {2, 30769, 7494}, {2, 30771, 7484}, {2, 30775, 44210}, {2, 31236, 5020}, {3, 7577, 18386}, {140, 7507, 15750}, {381, 37118, 11410}, {427, 468, 6995}, {427, 6995, 5064}, {427, 37453, 25}, {468, 5064, 25}, {468, 8889, 5064}, {1594, 3526, 3515}, {1656, 37119, 1593}, {5000, 5001, 3843}, {5064, 5094, 8889}, {5094, 37453, 427}, {6995, 8889, 427}, {7484, 30771, 32216}, {7577, 13619, 7547}, {7577, 31101, 427}

X(52299) = BARYCENTRIC PRODUCT X(264)*X(15815)

X(52299) lies on these lines: {2, 3}, {112, 15880}, {125, 11427}, {230, 40065}, {262, 38253}, {264, 34803}, {393, 31489}, {459, 14494}, {1007, 32000}, {1235, 8893}, {1249, 3815}, {1285, 15820}, {1829, 19877}, {1853, 39874}, {1870, 5268}, {1899, 44109}, {2052, 10155}, {2356, 25502}, {3087, 37637}, {3624, 7718}, {3763, 41585}, {3819, 6403}, {5090, 5550}, {5272, 6198}, {5622, 15113}, {5702, 9300}, {6036, 20774}, {6340, 7763}, {6688, 12294}, {6776, 23332}, {7713, 51073}, {7716, 51128}, {7998, 47328}, {8280, 13886}, {8281, 13939}, {8541, 21356}, {9342, 11383}, {9744, 41374}, {11160, 11405}, {11396, 46933}, {11430, 18918}, {11433, 34565}, {11550, 35260}, {12135, 46934}, {12828, 15059}, {13854, 39389}, {14826, 45303}, {14853, 26958}, {14912, 23291}, {15153, 18945}, {15302, 40938}, {17811, 39588}, {18388, 18931}, {21243, 37669}, {22110, 38294}, {23293, 37645}, {29639, 38295}, {30792, 44428}, {32001, 34229}, {32064, 44108}, {33522, 51360}, {34595, 49542}, {34986, 50974}, {37687, 44086}, {42391, 44541}

X(52299) = orthocentroidal-circle-inverse of X(38282)
X(52299) = polar-circle-inverse of X(47316)
X(52299) = polar conjugate of X(18845)
X(52299) = isotomic conjugate of the complement of X(41925)
X(52299) = polar conjugate of the isogonal conjugate of X(15815)
X(52299) = X(48)-isoconjugate of X(18845)
X(52299) = X(1249)-Dao conjugate of X(18845)
X(52299) = cevapoint of X(2) and X(41925)
X(52299) = barycentric product X(264)*X(15815)
X(52299) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 18845}, {15815, 3}
X(52299) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 38282}, {2, 427, 6353}, {2, 858, 7494}, {2, 1368, 631}, {2, 3091, 6677}, {2, 5094, 8889}, {2, 5133, 40132}, {2, 6995, 37453}, {2, 7378, 468}, {2, 7396, 6676}, {2, 8889, 4}, {2, 16419, 3533}, {2, 30744, 16051}, {2, 30769, 1368}, {2, 31074, 7493}, {4, 6143, 3525}, {4, 15702, 186}, {4, 35473, 11001}, {25, 427, 7409}, {125, 11427, 18950}, {378, 3545, 4}, {427, 6353, 4}, {427, 6676, 35480}, {427, 37453, 6995}, {436, 6619, 4}, {468, 7378, 7714}, {631, 1594, 4}, {631, 3090, 7399}, {1656, 3088, 6622}, {3088, 6622, 4}, {3090, 3541, 4}, {3529, 7547, 4}, {3535, 3536, 458}, {5000, 5001, 3832}, {5004, 5005, 38438}, {5094, 5159, 3541}, {6353, 8889, 427}, {6676, 7396, 376}, {6995, 37453, 6353}, {7378, 7714, 4}, {7493, 31074, 44442}, {10565, 34609, 3529}, {23291, 23292, 14912}, {31255, 37454, 2}

X(52300) = BARYCENTRIC PRODUCT X(264)*X(18472)

X(52300) lies on these lines: {2, 3}, {17, 37776}, {18, 37775}, {67, 19127}, {76, 7664}, {94, 7607}, {110, 34507}, {111, 7746}, {125, 15080}, {154, 3410}, {184, 41724}, {230, 18573}, {265, 34513}, {323, 44493}, {343, 9544}, {524, 9716}, {542, 38397}, {575, 32225}, {576, 15360}, {599, 11061}, {1209, 26882}, {1352, 35265}, {1383, 7745}, {1495, 18553}, {1503, 7712}, {1648, 39560}, {1649, 44451}, {2888, 9707}, {3053, 9745}, {3066, 7605}, {3448, 6800}, {3580, 8550}, {3620, 41719}, {3934, 10163}, {5012, 43810}, {5467, 7777}, {5480, 48912}, {5486, 37784}, {5640, 25555}, {5642, 40107}, {5971, 7763}, {5972, 7998}, {5987, 52090}, {6031, 7750}, {6032, 7747}, {6103, 22240}, {6689, 9781}, {7578, 7608}, {7703, 29012}, {7711, 7836}, {7749, 10418}, {7752, 26276}, {7755, 9465}, {7768, 26233}, {7806, 46127}, {7810, 9829}, {7812, 51541}, {8547, 25320}, {8743, 40583}, {9076, 14247}, {9143, 15069}, {9159, 22104}, {9464, 32820}, {9717, 16243}, {9970, 50977}, {9999, 32151}, {10192, 37636}, {10545, 38317}, {10546, 24206}, {10990, 11454}, {11002, 14389}, {11064, 33884}, {11412, 44516}, {11422, 41586}, {11459, 16534}, {11464, 30714}, {14169, 40709}, {14170, 40710}, {14643, 33533}, {14683, 26864}, {15056, 32348}, {15066, 17847}, {15072, 20417}, {15074, 45237}, {15513, 39602}, {16981, 47582}, {18312, 47263}, {20481, 44535}, {20791, 44673}, {21243, 26881}, {22352, 26913}, {25563, 52093}, {30516, 44500}, {34504, 42008}, {37688, 44135}, {41939, 44453}, {44010, 47173}, {44386, 47246}

X(52300) = complement of X(31857)
X(52300) = orthoptic-circle-of-Steiner-inellipse-inverse of X(18572)
X(52300) = polar conjugate of the isogonal conjugate of X(18472)
X(52300) = barycentric product X(i)*X(j) for these {i,j}: {264, 18472}, {524, 52189}
X(52300) = barycentric quotient X(i)/X(j) for these {i,j}: {18472, 3}, {52189, 671}
X(52300) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 22, 31074}, {2, 23, 5169}, {2, 25, 37353}, {2, 5189, 5094}, {2, 6636, 31101}, {2, 7492, 858}, {2, 7493, 23}, {2, 7494, 15246}, {2, 10565, 7391}, {2, 14002, 5}, {2, 16063, 30745}, {2, 20062, 8889}, {2, 37760, 1995}, {2, 37909, 381}, {2, 37913, 427}, {4, 3523, 14118}, {5, 7426, 14002}, {20, 50007, 3}, {22, 1995, 37972}, {22, 5094, 5189}, {26, 1656, 4}, {468, 6676, 7495}, {468, 7495, 2}, {550, 10024, 4}, {858, 44210, 7492}, {2045, 2046, 7550}, {3580, 13394, 11003}, {5000, 5001, 18559}, {5004, 5005, 18570}, {5068, 31304, 4}, {5094, 5189, 31074}, {6636, 30745, 16063}, {6800, 37638, 3448}, {7485, 37453, 2}, {7505, 47525, 37126}, {7542, 16618, 37118}, {7556, 14002, 23}, {7749, 10418, 39576}, {10018, 34002, 3523}, {14389, 32269, 11002}, {16063, 30745, 31101}, {16618, 37118, 20}, {37347, 44213, 47485}

X(52301) = BARYCENTRIC PRODUCT X(264)*X(21309)

X(52301) lies on these lines: {2, 3}, {53, 37689}, {98, 16240}, {145, 7713}, {193, 7716}, {232, 14930}, {340, 10513}, {393, 1383}, {1285, 40126}, {1395, 30652}, {1495, 14853}, {1503, 31860}, {1629, 6525}, {1829, 3623}, {1843, 11002}, {1974, 11003}, {1990, 5304}, {2052, 47586}, {2207, 5354}, {2212, 30653}, {2333, 19998}, {3066, 25406}, {3199, 9465}, {3291, 33842}, {3448, 32250}, {3617, 49542}, {3621, 7718}, {4661, 41611}, {5032, 15471}, {5095, 9143}, {5480, 35260}, {5921, 41724}, {6403, 16981}, {6553, 19993}, {6749, 37665}, {6776, 34417}, {6800, 51171}, {7712, 44091}, {7717, 20059}, {7737, 40350}, {8550, 11206}, {8796, 43537}, {9485, 14273}, {9543, 11473}, {10311, 40138}, {11064, 51538}, {11160, 41585}, {11180, 41586}, {11566, 20125}, {12135, 20052}, {12294, 33884}, {14561, 32237}, {14580, 33885}, {14683, 32234}, {14927, 37648}, {15302, 33843}, {16276, 32824}, {16654, 18931}, {18489, 32620}, {20080, 48912}, {21970, 39884}, {22112, 33750}, {23061, 51028}, {26233, 32834}, {31383, 44106}, {32085, 43981}, {32827, 37804}, {35259, 51212}, {35510, 41896}, {36990, 37643}, {37638, 51537}, {43530, 43951}

X(52301) = polar-circle-inverse of X(47311)
X(52301) = polar conjugate of the isogonal conjugate of X(21309)
X(52301) = barycentric product X(264)*X(21309)
X(52301) = barycentric quotient X(i)/X(j) for these {i,j}: {21309, 3}, {44839, 10519}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5059, 16063}, {2, 6995, 7408}, {2, 7408, 7409}, {2, 7492, 15717}, {2, 7519, 3146}, {2, 7533, 5068}, {2, 17578, 31099}, {2, 21734, 7496}, {2, 50689, 5169}, {4, 25, 4232}, {4, 3517, 3523}, {4, 3518, 35486}, {4, 4232, 2}, {4, 5094, 7378}, {4, 6353, 5094}, {4, 7714, 10301}, {4, 10301, 6995}, {4, 18533, 49670}, {4, 21841, 5056}, {4, 35483, 1597}, {4, 35486, 3088}, {4, 37458, 20}, {4, 37984, 3839}, {20, 1995, 2}, {20, 3091, 34664}, {22, 7398, 2}, {25, 428, 6353}, {25, 1596, 26255}, {25, 1598, 1995}, {25, 4186, 7438}, {25, 6756, 7493}, {25, 6995, 2}, {25, 7714, 6995}, {25, 10301, 4}, {25, 18494, 7426}, {382, 44212, 16051}, {428, 5094, 4}, {428, 6353, 7378}, {452, 4239, 2}, {1597, 37934, 35483}, {1598, 37458, 4}, {1995, 37900, 46336}, {3091, 7493, 2}, {3543, 26255, 2}, {3839, 7426, 2}, {4232, 6995, 4}, {5000, 5001, 3524}, {5056, 7495, 2}, {5056, 10565, 7495}, {5177, 26256, 2}, {5480, 41424, 35260}, {6353, 7378, 2}, {6995, 7378, 428}, {6995, 7487, 7519}, {6997, 7495, 5056}, {6997, 10565, 2}, {7500, 13595, 2}, {7500, 16063, 5059}, {7519, 14002, 2}, {11284, 37899, 376}, {18494, 37984, 4}, {26257, 32971, 2}, {30769, 50688, 31133}, {34484, 37122, 3089}, {35483, 37934, 10304}, {37900, 46336, 20}, {42789, 42790, 41463}

  4th tangent line ot two inconics: X(52302) - X(52343)  

Let ABC be a triangle, P₁ = x₁ : y₁ : z₁, P₂ = x₂ : y₂ : z₂ (barycentrics) two distinct points on the plane of ABC and not on the sidelines of ABC. Let q₁, q₂ be the inconics of ABC with perspectors P₁ and P₂, respectively.

There exists a 4th-common tangent line τ to q₁ and q₂, other than the sidelines of ABC. It can be deduced that:

• The trilinear pole of τ has barycentrics:
  
    Q(τ) = x₁*x₂*(y₁*z₂-y₂*z₁) : y₁*y₂*(x₂*z₁-x₁*z₂) : z₁*z₂*(x₁*y₂-x₂*y₁)
• The touchpoints T₁, T₂ of τ with q₁, q₂, respectively, are:
  
    T₁ = x₁*x₂^2*(y₁*z₂-y₂*z₁)^2 : y₁*y₂^2*(x₂*z₁-x₁*z₂)^2 : z₁*z₂^2*(x₁*y₂-x₂*y₁)^2
  
    T₂ = x₂*x₁^2*(y₂*z₁-y₁*z₂)^2 : y₂*y₁^2*(x₁*z₂-x₂*z₁)^2 : z₂*z₁^2*(x₂*y₁-x₁*y₂)^2

The preceding coordinates have the following equivalences leading to geometric constructions of τ, T₁ and T₂:

1. Q(τ) is the perspector of the circumconic of ABC passing through P₁ and P₂.

   Consequently, if U' and U" are two given points and q₀ is the circumconic {{A,B,C,U',U"}}, then, for any pair of points P', P" on q₀, the inconics with perspectors P', P" have the same 4th common tangent line and it is tripolar of the perspector of q₀.

2. T₁ is the perspector of the cevian-triangle-of-P₁ and the side-triangle-of-(cevian-triangle-of-P₁ and cevian-triangle-of-P₂), which are always`perspective. (Note: the side-triangle A_sB_sC_s of two triangles A'B'C' and A"B"C" has A-vertex A_s = B'C' ∩ B"C" and cyclically B_s and C_s).
   
   Symmetrically, T₂ is the perspector of the cevian-triangle-of-P₂ and the side-triangle-of-(cevian-triangle-of-P₁ and cevian-triangle-of-P₂).
   
   See centers X(41211) to X(41224), which are related to analogous perspectors.
3. T₁ is also the P₁-ceva conjugate of-Q(τ) and T₂ is also the P₂-ceva conjugate of-Q(τ), which means that T₁ is the perspector(cevian-of-P₁, anticevian-of-Q(τ)) and T₂ is the perspector(cevian-of-P₂, anticevian-of-Q(τ)).

For shortening, the inconic with perspector U will be denoted in this section as IwP(U).

(1, 2, 244, 1015, 513), (1, 3, 2638, 3270, 652), (1, 4, 2310, 3270, 650), (1, 5, 41211, 41218, 2600), (1, 6, 3248, 1015, 649), (1, 7, 2310, 3022, 650), (1, 8, 2310, 3271, 650), (1, 10, 2643, 3122, 661), (1, 11, 52302, 52303, 46384), (2, 3, 35071, 2972, 520), (2, 4, 115, 125, 523), (2, 5, 39019, 35442, 6368), (2, 6, 1084, 3124, 512), (2, 7, 1086, 11, 514), (2, 8, 1146, 11, 522), (2, 10, 115, 3120, 523), (2, 11, 35509, 52304, 52305), (3, 4, 20975, 34980, 647), (3, 5, 41212, 41219, 17434), (3, 6, 20975, 3269, 647), (3, 7, 41214, 15616, 52306), (3, 8, 41215, 41220, 52307), (3, 10, 52308, 52309, 52310), (3, 11, 52311, 52312, 52313), (4, 5, 24862, 41221, 12077), (4, 6, 34980, 3269, 647), (4, 7, 3270, 3022, 650), (4, 8, 3270, 3271, 650), (4, 10, 125, 3120, 523), (4, 11, 52314, 52315, 52316), (5, 6, 41213, 41222, 52317), (5, 7, 41216, 31889, 52318), (5, 8, 41217, 41223, 52319), (5, 10, 52320, 52321, 52322), (5, 11, 52323, 52324, 52325), (6, 7, 35505, 15615, 665), (6, 8, 35506, 41224, 52326), (6, 10, 52327, 52328, 42664), (6, 11, 52329, 52330, 52331), (7, 8, 3022, 3271, 650), (7, 10, 31890, 52332, 4841), (7, 11, 31891, 52333, 52334), (8, 10, 4092, 52335, 3700), (8, 11, 52336, 52337, 52338), (10, 11, 52339, 52340, 52341), (13, 14, 30465, 30468, 523), (15, 16, 52342, 52343, 526)

X(52302) = TOUCHPOINT OF IwP( X(1) ) AND THE 4th-COMMON-TANGENT OF IwP( X(1) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(11) ) at X(52303) and has trilinear pole X(46384).

X(52302) lies on the inconics with perspectors X(n) for n in {1, 1090} and these lines: {2310, 11193}

X(52302) = barycentric product X(1090)*X(35128)
X(52302) = trilinear product X(i)*X(j) for these {i, j}: {11, 52303}, {54, 52323}
X(52302) = trilinear square of X(46384)
X(52302) = crosssum of X(654) and X(2957)
X(52302) = X(1)-Ceva conjugate of-X(46384)

X(52303) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(1) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(1) ) at X(52302) and has trilinear pole X(46384).

X(52303) lies on the inconics with perspectors X(n) for n in {11, 59, 3025} and these lines: {1362, 34586}, {1397, 34179}, {3251, 42771}, {3328, 3675}

X(52303) = barycentric product X(i)*X(j) for these {i, j}: {11, 35128}, {1090, 34544}, {1146, 3025}
X(52303) = trilinear product X(i)*X(j) for these {i, j}: {59, 52302}, {215, 1090}, {654, 46384}
X(52303) = X(i)-Ceva conjugate of-X(j) for these (i, j): (11, 46384), (59, 654)

X(52304) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(2) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(2) ) at X(35509) and has trilinear pole X(52305).

X(52304) lies on the inconics with perspectors X(n) for n in {11, 3323, 4998} and these lines: {2, 35313}, {7, 10001}, {8, 6631}, {11, 885}, {56, 18343}, {1259, 47048}, {1317, 50441}, {1566, 3328}, {3126, 3675}, {4996, 8299}, {5532, 21132}, {6068, 16593}, {7336, 42462}, {14393, 35509}, {20504, 42770}, {35967, 52315}

X(52304) = complement of X(35313)
X(52304) = barycentric product X(i)*X(j) for these {i, j}: {11, 35094}, {918, 52305}, {1146, 3323}
X(52304) = trilinear product X(1090)*X(1362)
X(52304) = perspector of the circumconic {{A, B, C, X(918), X(52305)}}
X(52304) = center of the circumconic {{A, B, C, X(11), X(3035)}}
X(52304) = Cevapoint of X(918) and X(4998)
X(52304) = crosssum of X(918) and X(3271)
X(52304) = X(11)-Ceva conjugate of-X(52305)
X(52304) = X(i)-Dao conjugate of-X(j) for these (i, j): (918, 4998), (926, 6066)

X(52305) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(2) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(2) ) and IwP( X(11) ) at X(35509) and X(52304), respectively.

X(52305) lies on the inconics with perspectors X(n) for n in {885, 43042, 50333} and these lines: {30, 511}, {242, 44428}, {659, 3220}, {693, 40704}, {764, 23760}, {1090, 5532}, {1456, 30725}, {1738, 10015}, {2400, 43930}, {3328, 4542}, {3685, 3904}, {3717, 50333}, {3766, 7112}, {3776, 15280}, {4077, 23742}, {4105, 48117}, {4530, 14393}, {4867, 49276}, {6545, 23615}, {6546, 11124}, {6608, 47930}, {14475, 14476}, {14505, 23100}, {14887, 50351}, {23761, 40166}, {36236, 36802}, {44009, 45290}

X(52305) = complementary conjugate of X(35967)
X(52305) = barycentric product X(i)*X(j) for these {i, j}: {11, 918}, {241, 42455}, {518, 40166}, {665, 34387}, {666, 52304}, {693, 17435}
X(52305) = barycentric quotient X(i)/X(j) for these (i, j): (11, 666), (244, 36146), (514, 39293), (518, 31615), (650, 5377), (665, 59)
X(52305) = trilinear product X(i)*X(j) for these {i, j}: {11, 2254}, {241, 42462}, {244, 50333}, {514, 17435}, {518, 21132}, {522, 3675}
X(52305) = trilinear quotient X(i)/X(j) for these (i, j): (11, 36086), (241, 4619), (244, 32735), (522, 5377), (665, 2149), (693, 39293)
X(52305) = infinite point of tripolar of X(i) for these i: {11, 23989, 31611, 31619, 40166, 46395}
X(52305) = trilinear pole of the line {14393, 35509}
X(52305) = Cevapoint of X(i) and X(j) for these (i, j): {11, 885}, {1086, 2400}
X(52305) = crossdifference of every pair of points on line {X(6), X(59)}
X(52305) = crosspoint of X(i) and X(j) for these (i, j): {2, 35509}, {11, 52304}
X(52305) = crosssum of X(i) and X(j) for these (i, j): {11, 2283}, {125, 7437}, {1086, 2426}
X(52305) = X(1110)-anticomplementary conjugate of-X(14732)
X(52305) = X(i)-Ceva conjugate of-X(j) for these (i, j): (2, 35509), (4, 35967), (11, 52304), (514, 1566), (660, 46100)
X(52305) = X(i)-complementary conjugate of-X(j) for these (i, j): (1, 35967), (31, 35509), (1110, 1566)
X(52305) = X(i)-Dao conjugate of-X(j) for these (i, j): (11, 5377), (513, 32735), (514, 927), (522, 36802), (650, 666)
X(52305) = X(i)-isoconjugate-of-X(j) for these {i, j}: {59, 36086}, {109, 5377}, {294, 4619}, {666, 2149}, {692, 39293}
X(52305) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (11, 666), (244, 36146), (514, 39293), (518, 31615), (650, 5377)

X(52306) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(7) )

This 4th-common-tangent touches IwP( X(3) ) and IwP( X(7) ) at X(41214) and X(15616), respectively.

X(52306) lies on these lines: {241, 514}, {520, 647}, {649, 17412}, {654, 4282}, {656, 40628}, {822, 8677}, {3709, 46389}, {6586, 14298}, {33525, 50354}

X(52306) = barycentric product X(i)*X(j) for these {i, j}: {78, 50354}, {283, 23752}, {348, 33525}, {442, 23189}, {521, 942}, {525, 46882}
X(52306) = barycentric quotient X(i)/X(j) for these (i, j): (184, 15439), (521, 40422), (603, 36048), (649, 40573), (650, 40447), (652, 40435)
X(52306) = trilinear product X(i)*X(j) for these {i, j}: {77, 33525}, {219, 50354}, {514, 23207}, {520, 46884}, {521, 2260}, {522, 14597}
X(52306) = trilinear quotient X(i)/X(j) for these (i, j): (48, 15439), (216, 35320), (222, 36048), (513, 40573), (521, 40435), (522, 40447)
X(52306) = trilinear pole of the line {15616, 41214}
X(52306) = perspector of the circumconic {{A, B, C, X(3), X(7)}}
X(52306) = tripole of the tangent to incircle at X(15616)
X(52306) = crossdifference of every pair of points on line {X(4), X(12)}
X(52306) = crosspoint of X(i) and X(j) for these (i, j): {3, 41214}, {7, 15616}
X(52306) = X(i)-Ceva conjugate of-X(j) for these (i, j): (2, 39007), (3, 41214), (7, 15616), (81, 1364), (1214, 7004)
X(52306) = X(31)-complementary conjugate of-X(39007)
X(52306) = X(i)-Dao conjugate of-X(j) for these (i, j): (11, 40447), (442, 6335), (942, 4552)
X(52306) = X(i)-isoconjugate-of-X(j) for these {i, j}: {92, 15439}, {100, 40573}, {108, 40435}, {109, 40447}, {275, 35320}
X(52306) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (184, 15439), (521, 40422), (603, 36048), (649, 40573), (650, 40447)
X(52306) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (647, 652, 52307), (652, 1459, 36054)

X(52307) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(8) )

This 4th-common-tangent touches IwP( X(3) ) and IwP( X(8) ) at X(41215) and X(41220), respectively.

X(52307) lies on these lines: {101, 109}, {520, 647}, {522, 650}, {1214, 39470}, {1769, 3310}, {2431, 14578}, {2631, 18591}, {6589, 14298}, {7004, 7117}, {7069, 30692}, {7180, 46389}, {22055, 22346}, {43060, 51656}

X(52307) = isogonal conjugate of the polar conjugate of X(2804)
X(52307) = barycentric product X(i)*X(j) for these {i, j}: {3, 2804}, {8, 8677}, {63, 46393}, {78, 1769}, {100, 35014}, {212, 36038}
X(52307) = barycentric quotient X(i)/X(j) for these (i, j): (31, 36110), (32, 32702), (48, 37136), (55, 1309), (101, 39294), (184, 2720)
X(52307) = trilinear product X(i)*X(j) for these {i, j}: {3, 46393}, {9, 8677}, {48, 2804}, {78, 3310}, {101, 35014}, {212, 10015}
X(52307) = trilinear quotient X(i)/X(j) for these (i, j): (3, 37136), (6, 36110), (9, 1309), (31, 32702), (41, 14776), (48, 2720)
X(52307) = trilinear pole of the line {41215, 41220}
X(52307) = perspector of the circumconic {{A, B, C, X(3), X(8)}}
X(52307) = tripole of the tangent to Mandart inellipse at X(41220)
X(52307) = intersection, other than A, B, C, of circumconics {{A, B, C, X(100), X(1364)}} and {{A, B, C, X(101), X(3239)}}
X(52307) = Cevapoint of X(i) and X(j) for these (i, j): {72, 50039}, {219, 5548}
X(52307) = crossdifference of every pair of points on line {X(4), X(11)}
X(52307) = crosspoint of X(i) and X(j) for these (i, j): {3, 41215}, {8, 41220}
X(52307) = crosssum of X(i) and X(j) for these (i, j): {72, 21786}, {212, 44428}, {219, 30725}, {1364, 14571}
X(52307) = X(63)-beth conjugate of-X(39470)
X(52307) = X(i)-Ceva conjugate of-X(j) for these (i, j): (2, 39004), (3, 41215), (8, 41220), (219, 38353), (906, 47408)
X(52307) = X(31)-complementary conjugate of-X(39004)
X(52307) = X(i)-Dao conjugate of-X(j) for these (i, j): (11, 16082), (206, 32702), (1145, 6335)
X(52307) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2, 36110}, {4, 37136}, {34, 13136}, {57, 1309}, {75, 32702}
X(52307) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (31, 36110), (32, 32702), (48, 37136), (55, 1309), (101, 39294)
X(52307) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (647, 652, 52306), (652, 10397, 36054)

X(52308) = TOUCHPOINT OF IwP( X(3) ) AND THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(10) )

This 4th-common-tangent touches IwP( X(10) ) at X(52309) and has trilinear pole X(52310).

X(52308) lies on the inconics with perspectors X(n) for n in {3, 7141} and these lines: {}

X(52308) = X(3)-Ceva conjugate of-X(52310)

X(52309) = TOUCHPOINT OF IwP( X(10) ) AND THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(10) )

This 4th-common-tangent touches IwP( X(3) ) at X(52308) and has trilinear pole X(52310).

X(52309) lies on the inconic with perspector X(10) and these lines: {}

X(52309) = barycentric product X(1790)*X(52308)
X(52309) = trilinear product X(1437)*X(52308)
X(52309) = X(10)-Ceva conjugate of-X(52310)

X(52310) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(10) )

This 4th-common-tangent touches IwP( X(3) ) and IwP( X(10) ) at X(52308) and X(52309), respectively.

X(52310) lies on these lines: {520, 647}, {523, 661}, {649, 924}, {2294, 6587}, {2501, 46393}, {43708, 52222}

X(52310) = barycentric product X(i)*X(j) for these {i, j}: {72, 21189}, {124, 23067}, {306, 6589}, {512, 51612}, {520, 17555}, {525, 573}
X(52310) = barycentric quotient X(i)/X(j) for these (i, j): (42, 26704), (71, 44765), (228, 36050), (573, 648), (647, 13478), (652, 19607)
X(52310) = trilinear product X(i)*X(j) for these {i, j}: {71, 21189}, {72, 6589}, {521, 40590}, {523, 22134}, {525, 3185}, {573, 656}
X(52310) = trilinear quotient X(i)/X(j) for these (i, j): (37, 26704), (71, 36050), (72, 44765), (228, 32653), (521, 19607), (525, 2995)
X(52310) = trilinear pole of the line {52308, 52309}
X(52310) = perspector of the circumconic {{A, B, C, X(3), X(10)}}
X(52310) = Cevapoint of X(i) and X(j) for these (i, j): {71, 4559}, {656, 6332}
X(52310) = crossdifference of every pair of points on line {X(4), X(58)}
X(52310) = crosspoint of X(i) and X(j) for these (i, j): {3, 52308}, {10, 52309}
X(52310) = crosssum of X(i) and X(j) for these (i, j): {71, 4560}, {656, 32674}
X(52310) = X(i)-Ceva conjugate of-X(j) for these (i, j): (3, 52308), (10, 52309)
X(52310) = X(i)-Dao conjugate of-X(j) for these (i, j): (65, 653), (124, 27), (125, 13478)
X(52310) = X(i)-isoconjugate-of-X(j) for these {i, j}: {27, 36050}, {28, 44765}, {81, 26704}, {108, 19607}, {112, 2995}
X(52310) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (42, 26704), (71, 44765), (228, 36050), (573, 648), (647, 13478)

X(52311) = TOUCHPOINT OF IwP( X(3) ) AND THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(11) ) at X(52312) and has trilinear pole X(52313).

X(52311) lies on the inconic with perspector X(3) and these lines: {}

X(52311) = X(3)-Ceva conjugate of-X(52313)

X(52312) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(3) ) at X(52311) and has trilinear pole X(52313).

X(52312) lies on the inconic with perspector X(11) and these lines: {}

X(52312) = X(11)-Ceva conjugate of-X(52313)

X(52313) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(3) ) and IwP( X(11) ) at X(52311) and X(52312), respectively.

X(52313) lies on these lines: {520, 647}, {4530, 14393}

X(52313) = trilinear pole of the line {52311, 52312}
X(52313) = perspector of the circumconic {{A, B, C, X(3), X(11)}}
X(52313) = crossdifference of every pair of points on line {X(4), X(59)}
X(52313) = crosspoint of X(i) and X(j) for these (i, j): {3, 52311}, {11, 52312}
X(52313) = X(i)-Ceva conjugate of-X(j) for these (i, j): (3, 52311), (11, 52312)

X(52314) = TOUCHPOINT OF IwP( X(4) ) AND THE 4th-COMMON-TANGENT OF IwP( X(4) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(11) ) at X(52315) and has trilinear pole X(52316).

X(52314) lies on the orthic inconic and these lines: {3270, 11193}

X(52314) = trilinear product X(1090)*X(41215)
X(52314) = X(4)-Ceva conjugate of-X(52316)

X(52315) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(4) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(4) ) at X(52314) and has trilinear pole X(52316).

X(52315) lies on the inconics with perspectors X(n) for n in {11, 3326} and these lines: {11, 2401}, {1360, 1846}, {3259, 3328}, {5532, 23615}, {35967, 52304}

X(52315) = barycentric product X(1146)*X(3326)
X(52315) = X(11)-Ceva conjugate of-X(52316)

X(52316) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(4) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(4) ) and IwP( X(11) ) at X(52314) and X(52315), respectively.

X(52316) lies on the inconics with perspectors X(n) for n in {2401, 10015} and these lines: {230, 231}, {654, 7297}, {2183, 35013}, {4530, 14393}, {10015, 22464}, {21801, 42763}

X(52316) = complement of the isotomic conjugate of X(43353)
X(52316) = barycentric product X(i)*X(j) for these {i, j}: {11, 2804}, {517, 42455}, {522, 35015}, {523, 14010}, {908, 42462}, {1146, 10015}
X(52316) = barycentric quotient X(1146)/X(13136)
X(52316) = trilinear product X(i)*X(j) for these {i, j}: {11, 46393}, {517, 42462}, {650, 35015}, {661, 14010}, {1021, 42759}, {1090, 2427}
X(52316) = trilinear quotient X(i)/X(j) for these (i, j): (11, 37136), (517, 4619), (1090, 2401), (1146, 36037)
X(52316) = trilinear pole of the line {52314, 52315}
X(52316) = perspector of the circumconic {{A, B, C, X(4), X(11)}}
X(52316) = tripole of the tangent to orthic inconic at X(52314)
X(52316) = Cevapoint of X(11) and X(2401)
X(52316) = crossdifference of every pair of points on line {X(3), X(59)}
X(52316) = crosspoint of X(i) and X(j) for these (i, j): {4, 52314}, {11, 52315}
X(52316) = crosssum of X(11) and X(2427)
X(52316) = X(i)-Ceva conjugate of-X(j) for these (i, j): (4, 52314), (11, 52315), (514, 3259)
X(52316) = X(i)-Dao conjugate of-X(j) for these (i, j): (522, 13136), (1145, 31615)
X(52316) = X(i)-isoconjugate-of-X(j) for these {i, j}: {59, 37136}, {104, 4619}
X(52316) = X(1146)-reciprocal conjugate of-X(13136)
X(52316) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (42462, 46384, 52338), (52334, 52338, 46384)

X(52317) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(6) )

This 4th-common-tangent touches IwP( X(5) ) and IwP( X(6) ) at X(41213) and X(41222), respectively.

X(52317) lies on the inconic with perspector X(32692) and these lines: {4, 15422}, {6, 47331}, {53, 35361}, {137, 3124}, {187, 237}, {686, 2501}, {924, 6753}, {1510, 2623}, {2081, 2600}

X(52317) = reflection of X(i) in X(j) for these (i, j): (2623, 16040), (30451, 6753)
X(52317) = Gibert-circumtangential conjugate of X(32692)
X(52317) = barycentric product X(i)*X(j) for these {i, j}: {5, 924}, {24, 6368}, {47, 2618}, {51, 6563}, {52, 523}, {136, 23181}
X(52317) = barycentric quotient X(i)/X(j) for these (i, j): (5, 46134), (24, 18831), (32, 32692), (51, 925), (52, 99), (53, 30450)
X(52317) = trilinear product X(i)*X(j) for these {i, j}: {47, 12077}, {52, 661}, {467, 810}, {523, 2180}, {563, 23290}, {571, 2618}
X(52317) = trilinear quotient X(i)/X(j) for these (i, j): (31, 32692), (47, 18315), (51, 36145), (52, 662), (467, 811), (512, 2168)
X(52317) = trilinear pole of the line {41213, 41222}
X(52317) = perspector of the circumconic {{A, B, C, X(5), X(6)}}
X(52317) = tripole of the tangent to Brocard inellipse at X(41222)
X(52317) = intersection, other than A, B, C, of circumconics {{A, B, C, X(5), X(45938)}} and {{A, B, C, X(51), X(52032)}}
X(52317) = Cevapoint of X(i) and X(j) for these (i, j): {6, 32692}, {51, 23181}
X(52317) = crossdifference of every pair of points on line {X(2), X(54)}
X(52317) = crosspoint of X(i) and X(j) for these (i, j): {5, 41213}, {6, 41222}
X(52317) = X(i)-Ceva conjugate of-X(j) for these (i, j): (5, 41213), (6, 41222), (571, 47421)
X(52317) = X(i)-Dao conjugate of-X(j) for these (i, j): (115, 34385), (134, 1993), (135, 275), (137, 5392), (139, 264)
X(52317) = X(i)-isoconjugate-of-X(j) for these {i, j}: {75, 32692}, {91, 18315}, {95, 36145}, {96, 662}, {99, 2168}
X(52317) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (5, 46134), (24, 18831), (32, 32692), (51, 925), (52, 99)
X(52317) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2081, 12077, 17434), (6753, 30451, 14397)

X(52318) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(7) )

This 4th-common-tangent touches IwP( X(5) ) and IwP( X(7) ) at X(41216) and X(31889), respectively.

X(52318) lies on these lines: {241, 514}, {2081, 2600}

X(52318) = trilinear pole of the line {31889, 41216}
X(52318) = perspector of the circumconic {{A, B, C, X(5), X(7)}}
X(52318) = tripole of the tangent to incircle at X(31889)
X(52318) = crossdifference of every pair of points on line {X(54), X(55)}
X(52318) = crosspoint of X(i) and X(j) for these (i, j): {5, 41216}, {7, 31889}
X(52318) = X(i)-Ceva conjugate of-X(j) for these (i, j): (5, 41216), (7, 31889)
X(52318) = {X(2600), X(12077)}-harmonic conjugate of X(52319)

X(52319) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(8) )

This 4th-common-tangent touches IwP( X(5) ) and IwP( X(8) ) at X(41217) and X(41223), respectively.

X(52319) lies on these lines: {522, 650}, {2081, 2600}

X(52319) = trilinear pole of the line {41217, 41223}
X(52319) = perspector of the circumconic {{A, B, C, X(5), X(8)}}
X(52319) = tripole of the tangent to Mandart inellipse at X(41223)
X(52319) = crossdifference of every pair of points on line {X(54), X(56)}
X(52319) = crosspoint of X(i) and X(j) for these (i, j): {5, 41217}, {8, 41223}
X(52319) = X(i)-Ceva conjugate of-X(j) for these (i, j): (5, 41217), (8, 41223)
X(52319) = {X(2600), X(12077)}-harmonic conjugate of X(52318)

X(52320) = TOUCHPOINT OF IwP( X(5) ) AND THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(10) )

This 4th-common-tangent touches IwP( X(10) ) at X(52321) and has trilinear pole X(52322).

X(52320) lies on the inconic with perspector X(5) and these lines: {}

X(52320) = X(5)-Ceva conjugate of-X(52322)

X(52321) = TOUCHPOINT OF IwP( X(10) ) AND THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(10) )

This 4th-common-tangent touches IwP( X(5) ) at X(52320) and has trilinear pole X(52322).

X(52321) lies on the inconic with perspector X(10) and these lines: {3120, 8902}

X(52321) = X(10)-Ceva conjugate of-X(52322)

X(52322) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(10) )

This 4th-common-tangent touches IwP( X(5) ) and IwP( X(10) ) at X(52320) and X(52321), respectively.

X(52322) lies on these lines: {523, 661}, {2081, 2600}

X(52322) = barycentric product X(572)*X(18314)
X(52322) = barycentric quotient X(572)/X(18315)
X(52322) = trilinear product X(572)*X(2618)
X(52322) = trilinear quotient X(572)/X(36134)
X(52322) = trilinear pole of the line {52320, 52321}
X(52322) = perspector of the circumconic {{A, B, C, X(5), X(10)}}
X(52322) = crossdifference of every pair of points on line {X(54), X(58)}
X(52322) = crosspoint of X(i) and X(j) for these (i, j): {5, 52320}, {10, 52321}
X(52322) = X(i)-Ceva conjugate of-X(j) for these (i, j): (5, 52320), (10, 52321)
X(52322) = X(137)-Dao conjugate of-X(2051)
X(52322) = X(572)-reciprocal conjugate of-X(18315)

X(52323) = TOUCHPOINT OF IwP( X(5) ) AND THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(11) ) at X(52324) and has trilinear pole X(52325).

X(52323) lies on the inconic with perspector X(5) and these lines: {}

X(52323) = trilinear product X(i)*X(j) for these {i, j}: {5, 52302}, {1090, 41218}
X(52323) = X(5)-Ceva conjugate of-X(52325)

X(52324) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(5) ) at X(52323) and has trilinear pole X(52325).

X(52324) lies on the inconic with perspector X(11) and these lines: {}

X(52324) = trilinear product X(i)*X(j) for these {i, j}: {11, 41211}, {1087, 52303}
X(52324) = X(11)-Ceva conjugate of-X(52325)

X(52325) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(5) ) and IwP( X(11) ) at X(52323) and X(52324), respectively.

X(52325) lies on these lines: {2081, 2600}, {4530, 14393}

X(52325) = barycentric product X(11)*X(6369)
X(52325) = trilinear product X(i)*X(j) for these {i, j}: {5, 46384}, {11, 2600}
X(52325) = trilinear pole of the line {52323, 52324}
X(52325) = perspector of the circumconic {{A, B, C, X(5), X(11)}}
X(52325) = crossdifference of every pair of points on line {X(54), X(59)}
X(52325) = crosspoint of X(i) and X(j) for these (i, j): {5, 52323}, {11, 52324}
X(52325) = X(i)-Ceva conjugate of-X(j) for these (i, j): (5, 52323), (11, 52324)

X(52326) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(6) ) AND IwP( X(8) )

This 4th-common-tangent touches IwP( X(6) ) and IwP( X(8) ) at X(35506) and X(41224), respectively.

X(52326) lies on the inconic with perspector X(8687) and these lines: {187, 237}, {513, 6589}, {514, 25098}, {522, 650}, {652, 3063}, {654, 4282}, {661, 3310}, {784, 45745}, {905, 3798}, {1021, 4435}, {2804, 47135}, {3004, 3666}, {3835, 24782}, {4025, 28374}, {4077, 28116}, {4374, 26114}, {4394, 6586}, {4790, 43060}, {4979, 21828}, {4988, 21132}, {6085, 23751}, {7178, 23723}, {14739, 44416}, {17066, 25511}, {24002, 28025}, {24622, 27293}, {25084, 31286}, {27486, 27648}, {27674, 47785}, {28606, 47660}

X(52326) = reflection of X(7180) in X(6589)
X(52326) = isogonal conjugate of X(6648)
X(52326) = Gibert-circumtangential conjugate of X(8687)
X(52326) = barycentric product X(i)*X(j) for these {i, j}: {1, 17420}, {6, 3910}, {8, 6371}, {9, 48131}, {21, 50330}, {41, 4509}
X(52326) = barycentric quotient X(i)/X(j) for these (i, j): (31, 36098), (32, 8687), (41, 36147), (55, 8707), (513, 31643), (522, 1240)
X(52326) = trilinear product X(i)*X(j) for these {i, j}: {6, 17420}, {9, 6371}, {31, 3910}, {41, 3004}, {55, 48131}, {284, 50330}
X(52326) = trilinear quotient X(i)/X(j) for these (i, j): (6, 36098), (9, 8707), (31, 8687), (41, 32736), (55, 36147), (514, 31643)
X(52326) = trilinear pole of the line {35506, 41224}
X(52326) = perspector of the circumconic {{A, B, C, X(6), X(8)}}
X(52326) = center of the circumconic {{A, B, C, X(48131), X(50353)}}
X(52326) = tripole of the tangent to Brocard inellipse at X(35506)
X(52326) = tripole of the tangent to Mandart inellipse at X(41224)
X(52326) = intersection, other than A, B, C, of circumconics {{A, B, C, X(21), X(1682)}} and {{A, B, C, X(41), X(17108)}}
X(52326) = Cevapoint of X(i) and X(j) for these (i, j): {6, 8687}, {11, 46880}
X(52326) = crossdifference of every pair of points on line {X(2), X(12)}
X(52326) = crosspoint of X(i) and X(j) for these (i, j): {6, 35506}, {8, 41224}
X(52326) = crosssum of X(i) and X(j) for these (i, j): {6, 3910}, {1015, 1999}
X(52326) = X(1021)-beth conjugate of-X(4976)
X(52326) = X(i)-Ceva conjugate of-X(j) for these (i, j): (2, 38992), (6, 35506), (8, 41224), (21, 3271), (37, 2170)
X(52326) = X(31)-complementary conjugate of-X(38992)
X(52326) = X(i)-Dao conjugate of-X(j) for these (i, j): (11, 30710), (206, 8687), (960, 4552), (1015, 31643), (1146, 1240)
X(52326) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2, 36098}, {7, 36147}, {57, 8707}, {75, 8687}, {85, 32736}
X(52326) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (31, 36098), (32, 8687), (41, 36147), (55, 8707), (513, 31643)
X(52326) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (647, 649, 665), (649, 42664, 50511), (2488, 8653, 663)

X(52327) = TOUCHPOINT OF IwP( X(6) ) AND THE 4th-COMMON-TANGENT OF IwP( X(6) ) AND IwP( X(10) )

This 4th-common-tangent touches IwP( X(10) ) at X(52328) and has trilinear pole X(42664).

X(52327) lies on the Brocard inellipse, the inconic with perspector X(28654) and these lines: {1015, 20975}, {1977, 20982}

X(52327) = barycentric product X(i)*X(j) for these {i, j}: {313, 52328}, {834, 23282}
X(52327) = trilinear product X(i)*X(j) for these {i, j}: {321, 52328}, {1089, 39016}
X(52327) = barycentric square of X(47842)
X(52327) = perspector of the circumconic {{A, B, C, X(23282), X(42664)}}
X(52327) = touchpoint of the line {52327, 52328} and Brocard inellipse
X(52327) = X(834)-Dao conjugate of-X(593)

X(52328) = TOUCHPOINT OF IwP( X(10) ) AND THE 4th-COMMON-TANGENT OF IwP( X(6) ) AND IwP( X(10) )

This 4th-common-tangent touches IwP( X(6) ) at X(52327) and has trilinear pole X(42664).

X(52328) lies on the inconics with perspectors X(n) for n in {10, 2206} and these lines: {2170, 3122}, {3120, 7668}

X(52328) = barycentric product X(i)*X(j) for these {i, j}: {10, 39016}, {58, 52327}, {834, 42664}
X(52328) = trilinear product X(i)*X(j) for these {i, j}: {37, 39016}, {834, 50488}, {1333, 52327}
X(52328) = perspector of the circumconic {{A, B, C, X(8637), X(42664)}}
X(52328) = X(834)-Dao conjugate of-X(86)

X(52329) = TOUCHPOINT OF IwP( X(6) ) AND THE 4th-COMMON-TANGENT OF IwP( X(6) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(11) ) at X(52330) and has trilinear pole X(52331).

X(52329) lies on the Brocard inellipse and these lines: {}

X(52329) = trilinear product X(1090)*X(39017)
X(52329) = touchpoint of the line {52329, 52330} and Brocard inellipse
X(52329) = X(6)-Ceva conjugate of-X(52331)

X(52330) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(6) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(6) ) at X(52329) and has trilinear pole X(52331).

X(52330) lies on the inconic with perspector X(11) and these lines: {}

X(52330) = barycentric product X(i)*X(j) for these {i, j}: {11, 39017}, {59, 52329}, {928, 52331}
X(52330) = X(11)-Ceva conjugate of-X(52331)
X(52330) = X(928)-Dao conjugate of-X(4998)

X(52331) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(6) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(6) ) and IwP( X(11) ) at X(52329) and X(52330), respectively.

X(52331) lies on these lines: {187, 237}, {654, 1951}, {3310, 8776}, {4530, 14393}

X(52331) = barycentric product X(11)*X(928)
X(52331) = barycentric quotient X(928)/X(4998)
X(52331) = trilinear product X(928)*X(2170)
X(52331) = trilinear quotient X(928)/X(4564)
X(52331) = trilinear pole of the line {52329, 52330}
X(52331) = perspector of the circumconic {{A, B, C, X(6), X(11)}}
X(52331) = tripole of the tangent to Brocard inellipse at X(52329)
X(52331) = crossdifference of every pair of points on line {X(2), X(59)}
X(52331) = crosspoint of X(i) and X(j) for these (i, j): {6, 52329}, {11, 52330}
X(52331) = X(i)-Ceva conjugate of-X(j) for these (i, j): (6, 52329), (11, 52330)
X(52331) = X(929)-isoconjugate-of-X(4564)
X(52331) = X(928)-reciprocal conjugate of-X(4998)

X(52332) = TOUCHPOINT OF IwP( X(10) ) AND THE 4th-COMMON-TANGENT OF IwP( X(7) ) AND IwP( X(10) )

This 4th-common-tangent touches IwP( X(7) ) at X(31890) and has trilinear pole X(4841).

X(52332) lies on the inconic with perspector X(10) and these lines: {3120, 8287}, {3125, 52335}, {21043, 21950}

X(52332) = barycentric product X(1434)*X(31890)
X(52332) = trilinear product X(1014)*X(31890)
X(52332) = perspector of the circumconic {{A, B, C, X(4841), X(30723)}}

X(52333) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(7) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(7) ) at X(31891) and has trilinear pole X(52334).

X(52333) lies on the inconics with perspectors X(n) for n in {11, 3328, 5532} and these lines: {55, 40577}, {3328, 30573}, {5532, 42462}, {14393, 52337}

X(52333) = barycentric product X(i)*X(j) for these {i, j}: {11, 35091}, {1146, 3328}, {1275, 31891}
X(52333) = perspector of the circumconic {{A, B, C, X(1638), X(52334)}}
X(52333) = X(11)-Ceva conjugate of-X(52334)

X(52334) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(7) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(7) ) and IwP( X(11) ) at X(31891) and X(52333), respectively.

X(52334) lies on the inconic with perspector X(1638) and these lines: {241, 514}, {654, 5011}, {1639, 5199}, {4530, 14393}, {6366, 6603}

X(52334) = barycentric product X(i)*X(j) for these {i, j}: {11, 6366}, {514, 33573}, {527, 42462}, {1111, 14392}, {1146, 1638}, {1155, 42455}
X(52334) = barycentric quotient X(i)/X(j) for these (i, j): (11, 35157), (1055, 4619), (1638, 1275)
X(52334) = trilinear product X(i)*X(j) for these {i, j}: {513, 33573}, {1055, 42455}, {1086, 14392}, {1146, 14413}, {1155, 42462}
X(52334) = trilinear quotient X(i)/X(j) for these (i, j): (11, 37139), (1155, 4619), (1638, 7045)
X(52334) = trilinear pole of the line {31891, 52333}
X(52334) = perspector of the circumconic {{A, B, C, X(7), X(11)}}
X(52334) = tripole of the tangent to incircle at X(31891)
X(52334) = crossdifference of every pair of points on line {X(55), X(59)}
X(52334) = crosspoint of X(i) and X(j) for these (i, j): {7, 31891}, {11, 52333}
X(52334) = X(i)-Ceva conjugate of-X(j) for these (i, j): (7, 31891), (11, 52333), (514, 3328)
X(52334) = X(650)-Dao conjugate of-X(35157)
X(52334) = X(i)-isoconjugate-of-X(j) for these {i, j}: {59, 37139}, {1156, 4619}
X(52334) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (11, 35157), (1055, 4619)
X(52334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4530, 21132, 14393), (46384, 52316, 52338)

X(52335) = TOUCHPOINT OF IwP( X(10) ) AND THE 4th-COMMON-TANGENT OF IwP( X(8) ) AND IwP( X(10) )

This 4th-common-tangent touches IwP( X(8) ) at X(4092) and has trilinear pole X(3700).

X(52335) lies on the inconics with perspectors X(n) for n in {10, 1043, 4082} and these lines: {8, 2648}, {10, 4552}, {210, 6535}, {281, 4336}, {523, 4466}, {594, 21039}, {1042, 39130}, {1109, 2632}, {1146, 2310}, {1441, 21931}, {2254, 23771}, {2293, 23529}, {2321, 4069}, {2784, 14543}, {3119, 42069}, {3122, 21950}, {3125, 52332}, {3271, 4530}, {3939, 36910}, {4092, 4516}, {4858, 17059}, {8013, 40967}, {17216, 17886}, {17874, 21717}, {21014, 23902}, {21033, 21673}, {21043, 21961}, {21252, 42754}, {21677, 52068}, {21686, 42440}, {23978, 24026}, {24411, 37781}, {39350, 48628}

X(52335) = reflection of X(4466) in X(21045)
X(52335) = barycentric product X(i)*X(j) for these {i, j}: {8, 21044}, {10, 1146}, {11, 2321}, {37, 24026}, {42, 23978}, {71, 21666}
X(52335) = barycentric quotient X(i)/X(j) for these (i, j): (8, 4620), (10, 1275), (11, 1434), (37, 7045), (42, 1262), (115, 3668)
X(52335) = trilinear product X(i)*X(j) for these {i, j}: {2, 36197}, {8, 4516}, {9, 21044}, {10, 2310}, {11, 210}, {21, 4092}
X(52335) = trilinear quotient X(i)/X(j) for these (i, j): (10, 7045), (11, 1014), (37, 1262), (42, 24027), (65, 7339), (115, 1427)
X(52335) = perspector of the circumconic {{A, B, C, X(1577), X(3239)}}
X(52335) = Cevapoint of X(i) and X(j) for these (i, j): {657, 2322}, {661, 39130}
X(52335) = crossdifference of every pair of points on line {X(163), X(1461)}
X(52335) = crosssum of X(i) and X(j) for these (i, j): {650, 34043}, {661, 2360}
X(52335) = X(1146)-Ceva conjugate of-X(36197)
X(52335) = X(i)-Dao conjugate of-X(j) for these (i, j): (10, 7045), (11, 1414), (37, 1275), (115, 658), (136, 36118)
X(52335) = X(i)-isoconjugate-of-X(j) for these {i, j}: {21, 7339}, {58, 7045}, {59, 1014}, {81, 1262}, {86, 24027}
X(52335) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (8, 4620), (10, 1275), (11, 1434), (37, 7045), (42, 1262)
X(52335) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (10, 4552, 21914), (1146, 4081, 2310), (4092, 4516, 21044), (16732, 21945, 3120)

X(52336) = TOUCHPOINT OF IwP( X(8) ) AND THE 4th-COMMON-TANGENT OF IwP( X(8) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(11) ) at X(52337) and has trilinear pole X(52338).

X(52336) lies on the Mandart inellipse, the inconics with perspectors X(n) for n in {7336, 52337} and these lines: {3271, 11193}

X(52336) = barycentric product X(1086)*X(52337)
X(52336) = trilinear product X(244)*X(52337)
X(52336) = touchpoint of the line {52336, 52337} and Mandart inellipse
X(52336) = X(8)-Ceva conjugate of-X(52338)

X(52337) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(8) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(8) ) at X(52336) and has trilinear pole X(52338).

X(52337) lies on the inconics with perspectors X(n) for n in {11, 4076, 4542, 7336, 14027} and these lines: {3251, 35092}, {4124, 51402}, {4530, 4542}, {7336, 21132}, {14393, 52333}

X(52337) = barycentric product X(i)*X(j) for these {i, j}: {11, 35092}, {522, 14442}, {900, 52338}, {1016, 52336}, {1086, 4542}, {1146, 14027}
X(52337) = barycentric quotient X(1639)/X(6635)
X(52337) = trilinear product X(i)*X(j) for these {i, j}: {11, 42084}, {244, 4542}, {650, 14442}, {678, 7336}, {764, 4543}, {765, 52336}
X(52337) = perspector of the circumconic {{A, B, C, X(1639), X(14442)}}
X(52337) = X(900)-Dao conjugate of-X(4998)

X(52338) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(8) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(8) ) and IwP( X(11) ) at X(52336) and X(52337), respectively.

X(52338) lies on the inconics with perspectors X(n) for n in {1639, 30725} and these lines: {9, 24096}, {44, 900}, {142, 24098}, {226, 24115}, {514, 4887}, {522, 650}, {654, 1731}, {665, 8609}, {918, 1266}, {1638, 17067}, {2323, 4435}, {3452, 24113}, {4357, 24141}, {4530, 14393}, {6550, 14442}

X(52338) = barycentric product X(i)*X(j) for these {i, j}: {8, 6550}, {11, 900}, {44, 40166}, {244, 4768}, {514, 4530}, {519, 21132}
X(52338) = barycentric quotient X(i)/X(j) for these (i, j): (8, 6635), (11, 4555), (44, 31615), (55, 6551), (650, 5376), (663, 9268)
X(52338) = trilinear product X(i)*X(j) for these {i, j}: {9, 6550}, {11, 1635}, {44, 21132}, {244, 1639}, {312, 8661}, {513, 4530}
X(52338) = trilinear quotient X(i)/X(j) for these (i, j): (9, 6551), (11, 3257), (312, 6635), (519, 31615), (522, 5376), (646, 42372)
X(52338) = trilinear pole of the line {52336, 52337}
X(52338) = perspector of the circumconic {{A, B, C, X(8), X(11)}}
X(52338) = tripole of the tangent to Mandart inellipse at X(52336)
X(52338) = crossdifference of every pair of points on line {X(56), X(59)}
X(52338) = crosspoint of X(i) and X(j) for these (i, j): {8, 52336}, {11, 52337}
X(52338) = X(i)-Ceva conjugate of-X(j) for these (i, j): (8, 52336), (11, 52337), (522, 4542)
X(52338) = X(i)-Dao conjugate of-X(j) for these (i, j): (11, 5376), (214, 31615), (522, 4582), (650, 4555)
X(52338) = X(i)-isoconjugate-of-X(j) for these {i, j}: {57, 6551}, {59, 3257}, {106, 31615}, {109, 5376}, {604, 6635}
X(52338) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (8, 6635), (11, 4555), (44, 31615), (55, 6551), (650, 5376)
X(52338) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (42462, 46384, 52316), (46384, 52316, 52334)

X(52339) = TOUCHPOINT OF IwP( X(10) ) AND THE 4th-COMMON-TANGENT OF IwP( X(10) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(11) ) at X(52340) and has trilinear pole X(52341).

X(52339) lies on the inconic with perspector X(10) and these lines: {}

X(52340) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(10) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(10) ) at X(52339) and has trilinear pole X(52341).

X(52340) lies on the inconic with perspector X(11) and these lines: {}

X(52341) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(10) ) AND IwP( X(11) )

This 4th-common-tangent touches IwP( X(10) ) and IwP( X(11) ) at X(52339) and X(52340), respectively.

X(52341) lies on these lines: {523, 661}, {4530, 14393}

X(52341) = trilinear product X(11)*X(21894)
X(52341) = trilinear pole of the line {52339, 52340}
X(52341) = perspector of the circumconic {{A, B, C, X(10), X(11)}}
X(52341) = crossdifference of every pair of points on line {X(58), X(59)}
X(52341) = crosspoint of X(i) and X(j) for these (i, j): {10, 52339}, {11, 52340}
X(52341) = X(i)-Ceva conjugate of-X(j) for these (i, j): (10, 52339), (11, 52340)

X(52342) = TOUCHPOINT OF IwP( X(15) ) AND THE 4th-COMMON-TANGENT OF IwP( X(15) ) AND IwP( X(16) )

This 4th-common-tangent touches IwP( X(16) ) at X(52343) and has trilinear pole X(526).

X(52342) lies on the inconics with perspectors X(n) for n in {15, 300, 36208} and these lines: {2, 35316}, {16, 323}, {2088, 16186}, {3258, 15609}, {7998, 30261}, {9205, 30468}, {14816, 47055}, {41888, 52039}

X(52342) = complement of X(35316)
X(52342) = barycentric product X(i)*X(j) for these {i, j}: {15, 43962}, {299, 2088}, {300, 18334}, {323, 30468}, {471, 16186}, {526, 23871}
X(52342) = barycentric quotient X(i)/X(j) for these (i, j): (16, 39295), (526, 23896)
X(52342) = trilinear product X(i)*X(j) for these {i, j}: {1094, 30460}, {1095, 30465}
X(52342) = perspector of the circumconic {{A, B, C, X(526), X(17403)}}
X(52342) = center of the circumconic {{A, B, C, X(15), X(300)}}
X(52342) = crossdifference of every pair of points on line {X(476), X(5994)}
X(52342) = X(i)-Ceva conjugate of-X(j) for these (i, j): (15, 526), (300, 23871)
X(52342) = X(526)-Dao conjugate of-X(15)
X(52342) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (16, 39295), (526, 23896)
X(52342) = {X(2088), X(16186)}-harmonic conjugate of X(52343)

X(52343) = TOUCHPOINT OF IwP( X(16) ) AND THE 4th-COMMON-TANGENT OF IwP( X(15) ) AND IwP( X(16) )

This 4th-common-tangent touches IwP( X(15) ) at X(52342) and has trilinear pole X(526).

X(52343) lies on the inconics with perspectors X(n) for n in {16, 301, 36209} and these lines: {2, 35317}, {15, 323}, {2088, 16186}, {3258, 15610}, {7998, 30260}, {9204, 30465}, {14817, 47055}, {41887, 52040}

X(52343) = complement of X(35317)
X(52343) = barycentric product X(i)*X(j) for these {i, j}: {16, 43961}, {298, 2088}, {301, 18334}, {323, 30465}, {470, 16186}, {526, 23870}
X(52343) = barycentric quotient X(i)/X(j) for these (i, j): (15, 39295), (526, 23895)
X(52343) = trilinear product X(i)*X(j) for these {i, j}: {1094, 30468}, {1095, 30463}
X(52343) = perspector of the circumconic {{A, B, C, X(526), X(17402)}}
X(52343) = center of the circumconic {{A, B, C, X(16), X(301)}}
X(52343) = crossdifference of every pair of points on line {X(476), X(5995)}
X(52343) = X(i)-Ceva conjugate of-X(j) for these (i, j): (16, 526), (301, 23870)
X(52343) = X(526)-Dao conjugate of-X(16)
X(52343) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (15, 39295), (526, 23895)
X(52343) = {X(2088), X(16186)}-harmonic conjugate of X(52342)

X(52344) = X(1)-DAO CONJUGATE OF X(35)

X(52344) lies on these lines: {2, 36626}, {8, 79}, {10, 94}, {35, 18359}, {75, 3260}, {92, 46468}, {100, 1141}, {265, 5086}, {280, 10527}, {318, 860}, {321, 4420}, {328, 18815}, {333, 52126}, {346, 5552}, {1043, 3615}, {1109, 24851}, {1222, 4968}, {1733, 24883}, {1789, 10538}, {2322, 3559}, {2370, 26700}, {3153, 14213}, {3616, 20320}, {3648, 14206}, {4861, 6742}, {5015, 20887}, {6533, 24026}, {7081, 26266}, {9780, 36624}, {13746, 38336}, {17874, 26131}, {18025, 20880}, {19877, 36625}, {24390, 52200}, {33944, 34387}, {43533, 43682}

X(52344) = isogonal conjugate of X(1399)
X(52344) = isotomic conjugate of X(1442)
X(52344) = isotomic conjugate of the isogonal conjugate of X(7073)
X(52344) = X(52185)-anticomplementary conjugate of X(3648)
X(52344) = X(20565)-Ceva conjugate of X(30690)
X(52344) = X(i)-cross conjugate of X(j) for these (i,j): {2323, 18359}, {3686, 312}, {6734, 8}, {7110, 30690}, {40942, 92}
X(52344) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1399}, {6, 2003}, {28, 22342}, {31, 1442}, {32, 17095}, {35, 56}, {50, 2006}, {57, 2174}, {58, 2594}, {65, 17104}, {77, 14975}, {79, 2477}, {81, 21741}, {108, 23226}, {109, 2605}, {184, 7282}, {319, 1397}, {593, 21794}, {603, 6198}, {604, 3219}, {1042, 35193}, {1106, 4420}, {1333, 16577}, {1400, 40214}, {1408, 3678}, {1410, 11107}, {1411, 6149}, {1415, 14838}, {1427, 35192}, {1437, 1825}, {1461, 9404}, {2149, 7202}, {2206, 40999}, {3969, 16947}, {6186, 7279}, {7206, 7342}
X(52344) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 35}, {2, 1442}, {3, 1399}, {9, 2003}, {10, 2594}, {11, 2605}, {37, 16577}, {442, 500}, {650, 7202}, {1146, 14838}, {1411, 14993}, {2174, 5452}, {2968, 35057}, {3161, 3219}, {4420, 6552}, {4467, 40624}, {6149, 35204}, {6198, 7952}, {6376, 17095}, {7110, 47057}, {7186, 41886}, {9404, 35508}, {17104, 40602}, {21741, 40586}, {22342, 40591}, {23226, 38983}, {40214, 40582}, {40603, 40999}
X(52344) = cevapoint of X(i) and X(j) for these (i,j): {8, 27529}, {4985, 24026}
X(52344) = barycentric product X(i)*X(j) for these {i,j}: {8, 30690}, {9, 20565}, {75, 7110}, {76, 7073}, {79, 312}, {94, 4511}, {314, 8818}, {321, 3615}, {333, 6757}, {522, 15455}, {1043, 43682}, {1969, 8606}, {2160, 3596}, {2166, 32851}, {2361, 20573}, {4391, 6742}, {4397, 38340}, {6186, 28659}, {7017, 7100}
X(52344) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2003}, {2, 1442}, {6, 1399}, {8, 3219}, {9, 35}, {10, 16577}, {11, 7202}, {21, 40214}, {37, 2594}, {42, 21741}, {55, 2174}, {71, 22342}, {75, 17095}, {79, 57}, {92, 7282}, {94, 18815}, {281, 6198}, {284, 17104}, {312, 319}, {314, 34016}, {321, 40999}, {341, 42033}, {346, 4420}, {522, 14838}, {607, 14975}, {650, 2605}, {652, 23226}, {756, 21794}, {1789, 1790}, {1826, 1825}, {1859, 44095}, {1989, 1411}, {2160, 56}, {2166, 2006}, {2174, 2477}, {2287, 35193}, {2321, 3678}, {2322, 11107}, {2323, 6149}, {2328, 35192}, {2361, 50}, {3061, 7186}, {3219, 7279}, {3239, 35057}, {3596, 33939}, {3615, 81}, {3683, 17454}, {3686, 3647}, {3701, 3969}, {3702, 3578}, {3900, 9404}, {4086, 7265}, {4092, 21824}, {4183, 41502}, {4391, 4467}, {4511, 323}, {4516, 20982}, {6186, 604}, {6734, 16585}, {6742, 651}, {6757, 226}, {7073, 6}, {7100, 222}, {7110, 1}, {8606, 48}, {8818, 65}, {13486, 4565}, {15455, 664}, {18155, 16755}, {20565, 85}, {21011, 2599}, {21044, 2611}, {26700, 1461}, {30690, 7}, {33653, 2307}, {34922, 7128}, {35519, 18160}, {38340, 934}, {40937, 500}, {43682, 3668}
X(52344) = {X(79),X(6757)}-harmonic conjugate of X(30690)

X(52345) = X(4)-DAO CONJUGATE OF X(28)

X(52345) lies on these lines: {1, 27410}, {2, 51502}, {4, 8}, {10, 307}, {20, 3198}, {37, 6554}, {40, 45738}, {63, 1715}, {69, 21588}, {85, 4208}, {100, 1294}, {169, 21061}, {253, 322}, {306, 857}, {379, 4968}, {387, 17863}, {388, 43214}, {411, 7360}, {668, 35140}, {758, 17869}, {821, 3990}, {942, 20905}, {948, 31993}, {956, 7535}, {1089, 21060}, {1229, 10449}, {1249, 1895}, {1265, 20928}, {1944, 3562}, {2322, 5279}, {2901, 6765}, {2975, 37275}, {3091, 20921}, {3191, 3870}, {3346, 3998}, {3710, 8806}, {3757, 4223}, {3868, 17862}, {3876, 26591}, {3931, 25255}, {4858, 24391}, {5129, 30854}, {5179, 21070}, {5247, 26000}, {5552, 6350}, {5687, 15951}, {5730, 49687}, {7078, 28950}, {7291, 37088}, {7952, 27540}, {9307, 22028}, {12526, 17860}, {14543, 40660}, {15973, 30082}, {16062, 26563}, {17757, 21530}, {18663, 26050}, {20305, 21686}, {21620, 25935}, {22027, 31397}, {24987, 25017}, {26678, 41249}, {28787, 38955}, {28951, 41344}, {30031, 30059}, {30809, 46937}, {38298, 42714}

X(52345) = isotomic conjugate of the isogonal conjugate of X(3198)
X(52345) = polar conjugate of the isotomic conjugate of X(42699)
X(52345) = X(i)-Ceva conjugate of X(j) for these (i,j): {322, 306}, {341, 10}, {18750, 8804}, {20336, 321}
X(52345) = X(i)-isoconjugate of X(j) for these (i,j): {27, 14642}, {28, 19614}, {56, 52158}, {58, 64}, {81, 2155}, {86, 33581}, {253, 2206}, {649, 46639}, {1073, 1474}, {1301, 1459}, {1333, 2184}, {1397, 5931}, {1408, 44692}, {1412, 30457}, {1790, 41489}, {1919, 44326}, {2194, 8809}, {2203, 19611}, {2208, 41082}, {8747, 14379}, {21789, 36079}
X(52345) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 52158}, {4, 28}, {10, 64}, {37, 2184}, {81, 45245}, {122, 513}, {253, 40603}, {269, 1427}, {800, 18603}, {905, 39020}, {1019, 40616}, {1073, 51574}, {1214, 8809}, {1437, 45248}, {2155, 40586}, {3198, 18615}, {5375, 46639}, {5930, 41402}, {8804, 11347}, {9296, 44326}, {18180, 45249}, {19614, 40591}, {30457, 40599}, {33581, 40600}
X(52345) = crosspoint of X(i) and X(j) for these (i,j): {14615, 18750}, {20336, 42699}
X(52345) = crosssum of X(2155) and X(33581)
X(52345) = trilinear pole of line {6587, 17898}
X(52345) = barycentric product X(i)*X(j) for these {i,j}: {4, 42699}, {10, 18750}, {20, 321}, {37, 14615}, {72, 15466}, {75, 8804}, {76, 3198}, {154, 27801}, {190, 17898}, {204, 40071}, {306, 1895}, {312, 5930}, {313, 610}, {341, 36908}, {349, 7070}, {668, 6587}, {1231, 44695}, {1249, 20336}, {1394, 30713}, {1441, 27382}, {2321, 33673}, {3596, 30456}, {3701, 18623}, {3710, 44697}, {3998, 14249}, {4033, 21172}, {4036, 36841}, {4554, 14308}, {6335, 8057}, {37669, 41013}, {38808, 42698}
X(52345) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 52158}, {10, 2184}, {20, 81}, {37, 64}, {42, 2155}, {71, 19614}, {72, 1073}, {100, 46639}, {154, 1333}, {204, 1474}, {210, 30457}, {213, 33581}, {226, 8809}, {228, 14642}, {306, 19611}, {312, 5931}, {321, 253}, {329, 41082}, {610, 58}, {668, 44326}, {1020, 36079}, {1249, 28}, {1394, 1412}, {1562, 18210}, {1783, 1301}, {1824, 41489}, {1895, 27}, {2321, 44692}, {2883, 18603}, {3172, 2203}, {3198, 6}, {3990, 14379}, {3998, 15394}, {5930, 57}, {6525, 5317}, {6587, 513}, {7070, 284}, {7156, 2299}, {8057, 905}, {8804, 1}, {14308, 650}, {14331, 3737}, {14615, 274}, {15466, 286}, {15905, 1437}, {17898, 514}, {18623, 1014}, {18750, 86}, {20336, 34403}, {20580, 4131}, {21172, 1019}, {21871, 41088}, {27382, 21}, {27801, 41530}, {30456, 56}, {33673, 1434}, {35602, 18604}, {36908, 269}, {37669, 1444}, {40933, 1407}, {41013, 459}, {41086, 1436}, {42459, 18180}, {42658, 22383}, {42699, 69}, {44695, 1172}, {44696, 1396}, {44705, 6591}, {52078, 1422}
X(52345) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 5080, 5174}, {72, 41013, 321}, {322, 33672, 253}, {20235, 44150, 1446}, {21075, 39130, 306}

X(52346) = X(4)-DAO CONJUGATE OF X(34)

X(52346) lies on these lines: {3, 7360}, {4, 20921}, {7, 8}, {20, 3198}, {29, 33}, {100, 41904}, {145, 17862}, {200, 4385}, {280, 341}, {306, 37448}, {309, 326}, {321, 20007}, {331, 35517}, {452, 30854}, {519, 20320}, {653, 8899}, {1034, 1265}, {1097, 37669}, {1103, 24034}, {1259, 10538}, {1260, 7283}, {1427, 26050}, {1895, 15466}, {2057, 3699}, {3100, 27410}, {3146, 30807}, {3436, 37201}, {3661, 26550}, {3687, 37445}, {3718, 5931}, {4673, 23528}, {4692, 4882}, {4737, 6736}, {4858, 12625}, {5552, 33116}, {5930, 14615}, {6527, 33672}, {6734, 19804}, {6735, 44720}, {6737, 17860}, {6745, 46937}, {7081, 19310}, {10446, 43213}, {11679, 16054}, {17880, 21605}, {18743, 27383}, {20013, 48380}, {20928, 34414}, {24218, 24440}, {27385, 30829}

X(52346) = isotomic conjugate of X(8809)
X(52346) = isotomic conjugate of the isogonal conjugate of X(7070)
X(52346) = X(i)-Ceva conjugate of X(j) for these (i,j): {3718, 312}, {14615, 18750}
X(52346) = X(i)-cross conjugate of X(j) for these (i,j): {5930, 8}, {27382, 18750}
X(52346) = X(i)-isoconjugate of X(j) for these (i,j): {7, 33581}, {31, 8809}, {34, 19614}, {56, 64}, {57, 2155}, {222, 41489}, {253, 1397}, {278, 14642}, {604, 2184}, {608, 1073}, {663, 36079}, {1042, 52158}, {1106, 44692}, {1118, 14379}, {1395, 19611}, {1407, 30457}, {1413, 41088}, {1472, 10375}, {6526, 7335}, {7180, 46639}, {7337, 15394}, {8810, 47437}, {41280, 41530}
X(52346) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 64}, {2, 8809}, {4, 34}, {57, 45245}, {122, 4017}, {603, 45248}, {1393, 45249}, {2155, 5452}, {2184, 3161}, {3669, 40616}, {6552, 44692}, {7004, 21172}, {11517, 19614}, {17898, 38357}, {24771, 30457}, {39020, 51640}
X(52346) = barycentric product X(i)*X(j) for these {i,j}: {8, 18750}, {9, 14615}, {20, 312}, {29, 42699}, {75, 27382}, {76, 7070}, {78, 15466}, {154, 28659}, {304, 44695}, {305, 7156}, {314, 8804}, {318, 37669}, {341, 18623}, {345, 1895}, {346, 33673}, {610, 3596}, {645, 17898}, {646, 21172}, {668, 14331}, {799, 14308}, {1249, 3718}, {1265, 44697}, {3198, 28660}, {3719, 14249}, {4086, 36841}, {6587, 7257}
X(52346) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 8809}, {8, 2184}, {9, 64}, {20, 57}, {33, 41489}, {41, 33581}, {55, 2155}, {78, 1073}, {154, 604}, {200, 30457}, {204, 608}, {212, 14642}, {219, 19614}, {312, 253}, {318, 459}, {345, 19611}, {346, 44692}, {610, 56}, {643, 46639}, {651, 36079}, {1097, 18623}, {1249, 34}, {1394, 1407}, {1895, 278}, {2287, 52158}, {2289, 14379}, {2324, 41088}, {2345, 10375}, {3079, 3213}, {3172, 1395}, {3198, 1400}, {3213, 1398}, {3718, 34403}, {3719, 15394}, {5930, 1427}, {6060, 610}, {6587, 4017}, {7070, 6}, {7156, 25}, {7257, 44326}, {8057, 51640}, {8804, 65}, {14308, 661}, {14331, 513}, {14615, 85}, {15466, 273}, {15905, 603}, {17898, 7178}, {18623, 269}, {18750, 7}, {21172, 3669}, {27382, 1}, {27398, 41082}, {28659, 41530}, {30456, 1042}, {33673, 279}, {35602, 7125}, {36413, 1394}, {36841, 1414}, {37669, 77}, {40616, 7004}, {41084, 1422}, {42459, 1393}, {42699, 307}, {44695, 19}, {44696, 1435}, {44697, 1119}, {44698, 1396}, {44699, 7128}
X(52346) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 23661, 75}, {75, 16284, 1231}, {78, 318, 312}, {280, 7080, 345}, {388, 4012, 8}

X(52347) = X(5)-DAO CONJUGATE OF X(25)

X(52347) lies on these lines: {3, 69}, {5, 311}, {30, 44128}, {76, 7399}, {99, 18401}, {141, 570}, {160, 16789}, {183, 7499}, {216, 343}, {264, 305}, {274, 51495}, {315, 20477}, {317, 1975}, {340, 32820}, {394, 6389}, {441, 20806}, {524, 571}, {1368, 20819}, {1370, 6527}, {1591, 34391}, {1592, 34392}, {2871, 3313}, {3001, 23300}, {3148, 13562}, {3260, 6662}, {3265, 43083}, {3541, 32000}, {3631, 14806}, {5562, 6751}, {5576, 44135}, {6148, 32902}, {6515, 8573}, {6656, 28728}, {6815, 32830}, {7667, 7788}, {7750, 46724}, {7776, 14790}, {7796, 14615}, {7802, 38434}, {8362, 28710}, {8901, 34385}, {9722, 44388}, {11585, 20563}, {14570, 42459}, {18494, 32815}, {18533, 32001}, {19553, 45799}, {20208, 28419}, {26906, 40681}, {27364, 41221}, {27377, 44363}, {32002, 32819}, {32821, 44134}, {37183, 46442}, {37688, 37804}, {38321, 52149}, {40981, 41588}, {42487, 51386}

X(52347) = isotomic conjugate of X(8884)
X(52347) = isotomic conjugate of the anticomplement of X(10600)
X(52347) = isotomic conjugate of the isogonal conjugate of X(5562)
X(52347) = isotomic conjugate of the polar conjugate of X(343)
X(52347) = isogonal conjugate of the polar conjugate of X(28706)
X(52347) = X(i)-Ceva conjugate of X(j) for these (i,j): {305, 28706}, {670, 4143}, {28706, 343}
X(52347) = X(i)-cross conjugate of X(j) for these (i,j): {5562, 343}, {10600, 2}
X(52347) = X(i)-isoconjugate of X(j) for these (i,j): {19, 8882}, {25, 2190}, {31, 8884}, {54, 1096}, {163, 15422}, {275, 1973}, {393, 2148}, {560, 8795}, {798, 16813}, {1924, 42405}, {1974, 40440}, {2167, 2207}, {2168, 8745}, {2169, 6524}, {2616, 32713}, {2623, 24019}, {6520, 14533}, {7337, 44687}, {8794, 9247}, {19174, 46289}
X(52347) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 8884}, {4, 52032}, {5, 25}, {6, 8882}, {24, 343}, {39, 19174}, {54, 6503}, {95, 6338}, {115, 15422}, {115, 35441}, {130, 669}, {216, 393}, {275, 6337}, {394, 8883}, {512, 2972}, {525, 8901}, {2190, 6505}, {2207, 40588}, {2489, 15450}, {2501, 39019}, {2623, 35071}, {3575, 23292}, {6368, 41221}, {6374, 8795}, {6524, 14363}, {6525, 45249}, {6663, 14569}, {6753, 47421}, {9428, 42405}, {13567, 16035}, {14533, 37867}, {16813, 31998}, {34836, 51887}
X(52347) = crosspoint of X(i) and X(j) for these (i,j): {69, 20563}, {305, 3926}
X(52347) = crosssum of X(i) and X(j) for these (i,j): {25, 44077}, {1974, 2207}
X(52347) = barycentric product X(i)*X(j) for these {i,j}: {3, 28706}, {5, 3926}, {53, 4176}, {63, 18695}, {69, 343}, {76, 5562}, {216, 305}, {217, 40050}, {304, 44706}, {311, 394}, {324, 3964}, {326, 14213}, {418, 1502}, {670, 17434}, {1444, 42698}, {3265, 14570}, {3267, 23181}, {3718, 44708}, {4143, 35360}, {4563, 6368}, {4590, 35442}, {4609, 42293}, {16697, 20336}, {20563, 52032}, {34254, 41168}, {40071, 44709}, {40362, 44088}, {40824, 42353}
X(52347) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 8884}, {3, 8882}, {5, 393}, {51, 2207}, {52, 8745}, {53, 6524}, {63, 2190}, {69, 275}, {76, 8795}, {99, 16813}, {141, 19174}, {216, 25}, {217, 1974}, {255, 2148}, {264, 8794}, {304, 40440}, {305, 276}, {311, 2052}, {324, 1093}, {326, 2167}, {343, 4}, {394, 54}, {418, 32}, {520, 2623}, {523, 15422}, {670, 42405}, {1092, 14533}, {1273, 14165}, {1568, 1990}, {1625, 32713}, {1953, 1096}, {2617, 24019}, {3133, 36416}, {3265, 15412}, {3719, 44687}, {3926, 95}, {3964, 97}, {4176, 34386}, {4558, 933}, {4563, 18831}, {5562, 6}, {6368, 2501}, {6503, 8883}, {6507, 2169}, {6509, 16035}, {6514, 35196}, {6751, 42295}, {8798, 41489}, {13157, 6526}, {14213, 158}, {14569, 36434}, {14570, 107}, {14585, 14573}, {15451, 2489}, {15526, 8901}, {16697, 28}, {17167, 8747}, {17434, 512}, {18180, 5317}, {18695, 92}, {23181, 112}, {24018, 2616}, {28706, 264}, {30493, 608}, {35360, 6529}, {35442, 115}, {35602, 33629}, {36212, 19189}, {36412, 14569}, {36831, 32695}, {37669, 38808}, {39019, 41221}, {39113, 11547}, {40981, 36417}, {41168, 13854}, {42293, 669}, {42353, 7735}, {42445, 47328}, {42459, 6525}, {42698, 41013}, {44088, 1501}, {44706, 19}, {44707, 607}, {44708, 34}, {44709, 1474}, {44710, 7115}, {44711, 8739}, {44712, 8740}, {44713, 8737}, {44714, 8738}, {44715, 8749}, {44716, 232}, {45793, 13450}, {46394, 40981}, {46832, 51887}, {52032, 24}
X(52347) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 1238, 3933}, {69, 3926, 3964}, {69, 9723, 41008}, {69, 40697, 3}, {69, 50572, 40680}, {311, 1273, 39113}, {311, 39113, 5}, {487, 488, 18925}, {3933, 41005, 69}, {6390, 41008, 9723}

X(52348) = X(6)-DAO CONJUGATE OF X(17)

X(52348) lies on these lines: {2, 13}, {3, 49}, {15, 1993}, {18, 13579}, {22, 14538}, {51, 5615}, {61, 1994}, {62, 5422}, {69, 11516}, {97, 44711}, {249, 40157}, {301, 32037}, {302, 473}, {323, 10645}, {343, 466}, {465, 11064}, {472, 11476}, {511, 3131}, {627, 8839}, {629, 2912}, {1599, 3364}, {1600, 3365}, {2979, 14539}, {3060, 38431}, {3129, 47066}, {3132, 9306}, {3391, 15234}, {3392, 15233}, {3819, 13349}, {4558, 36297}, {5238, 11127}, {5351, 11145}, {6636, 14169}, {7998, 21159}, {9544, 14170}, {10601, 11486}, {10646, 15066}, {11004, 34754}, {11086, 33459}, {11143, 16771}, {11427, 37172}, {11480, 37672}, {11481, 17811}, {11515, 20806}, {13350, 34986}, {14540, 34009}, {17825, 22238}, {18470, 37638}, {22151, 46113}, {26881, 34008}, {34395, 35296}, {34545, 41477}, {37340, 37649}

X(52348) = isogonal conjugate of X(8741)
X(52348) = isotomic conjugate of the polar conjugate of X(61)
X(52348) = isogonal conjugate of the polar conjugate of X(302)
X(52348) = X(i)-Ceva conjugate of X(j) for these (i,j): {302, 61}, {40710, 44719}
X(52348) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8741}, {17, 19}, {92, 21461}, {158, 32585}, {1096, 40712}, {1973, 34389}, {2190, 36300}, {2962, 10641}, {3375, 8738}, {16806, 24006}
X(52348) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 8741}, {4, 10640}, {5, 36300}, {6, 17}, {471, 11130}, {1147, 32585}, {6337, 34389}, {6503, 40712}, {21461, 22391}
X(52348) = crossdifference of every pair of points on line {2501, 6137}
X(52348) = barycentric product X(i)*X(j) for these {i,j}: {3, 302}, {18, 44180}, {49, 34390}, {61, 69}, {298, 50468}, {394, 473}, {1994, 40711}, {3926, 10642}, {4558, 23872}, {7769, 32586}, {8838, 44718}, {11126, 40710}, {11132, 36297}, {11146, 40709}, {16771, 44719}, {50466, 52220}
X(52348) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 17}, {6, 8741}, {18, 93}, {49, 62}, {61, 4}, {69, 34389}, {184, 21461}, {216, 36300}, {302, 264}, {394, 40712}, {473, 2052}, {577, 32585}, {1994, 472}, {2965, 10641}, {4558, 32036}, {10642, 393}, {11083, 8737}, {11126, 471}, {11135, 8740}, {11137, 8739}, {11141, 8738}, {11146, 470}, {23872, 14618}, {32037, 38342}, {32586, 2963}, {32661, 16806}, {34390, 20572}, {36296, 11139}, {36297, 11087}, {40711, 11140}, {44180, 303}, {44719, 19779}, {46112, 8603}, {46113, 51890}, {50466, 11600}, {50468, 13}, {51546, 8742}
X(52348) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 44718, 44719}, {343, 466, 40710}, {343, 52194, 40712}, {466, 52194, 343}, {627, 19773, 33529}, {5408, 5409, 44718}, {9306, 9736, 3132}, {11126, 11146, 61}, {40710, 40712, 343}

X(52349) = X(6)-DAO CONJUGATE OF X(18)

X(52349) lies on these lines: {2, 14}, {3, 49}, {16, 1993}, {17, 13579}, {22, 14539}, {51, 5611}, {61, 5422}, {62, 1994}, {69, 11515}, {97, 44712}, {249, 40156}, {300, 32036}, {303, 472}, {323, 10646}, {343, 465}, {466, 11064}, {473, 11475}, {511, 3132}, {628, 8837}, {630, 2913}, {1599, 3389}, {1600, 3390}, {2979, 14538}, {3060, 38432}, {3130, 47068}, {3131, 9306}, {3366, 15234}, {3367, 15233}, {3819, 13350}, {4558, 36296}, {5237, 11126}, {5352, 11146}, {6636, 14170}, {7998, 21158}, {9544, 14169}, {10601, 11485}, {10645, 15066}, {11004, 34755}, {11081, 33458}, {11144, 16770}, {11427, 37173}, {11480, 17811}, {11481, 37672}, {11516, 20806}, {13349, 34986}, {14541, 34008}, {17825, 22236}, {18468, 37638}, {22151, 46112}, {26881, 34009}, {34394, 35296}, {34545, 41478}, {37341, 37649}

X(52349) = isogonal conjugate of X(8742)
X(52349) = isotomic conjugate of the polar conjugate of X(62)
X(52349) = isogonal conjugate of the polar conjugate of X(303)
X(52349) = X(i)-Ceva conjugate of X(j) for these (i,j): {303, 62}, {40709, 44718}
X(52349) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8742}, {18, 19}, {92, 21462}, {158, 32586}, {1096, 40711}, {1973, 34390}, {2190, 36301}, {2962, 10642}, {3384, 8737}, {16807, 24006}
X(52349) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 8742}, {4, 10639}, {5, 36301}, {6, 18}, {470, 11131}, {1147, 32586}, {6337, 34390}, {6503, 40711}, {21462, 22391}
X(52349) = crossdifference of every pair of points on line {2501, 6138}
X(52349) = barycentric product X(i)*X(j) for these {i,j}: {3, 303}, {17, 44180}, {49, 34389}, {62, 69}, {299, 50469}, {394, 472}, {1994, 40712}, {3926, 10641}, {4558, 23873}, {7769, 32585}, {8836, 44719}, {11127, 40709}, {11133, 36296}, {11145, 40710}, {16770, 44718}, {50465, 52221}
X(52349) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 18}, {6, 8742}, {17, 93}, {49, 61}, {62, 4}, {69, 34390}, {184, 21462}, {216, 36301}, {303, 264}, {394, 40711}, {472, 2052}, {577, 32586}, {1994, 473}, {2965, 10642}, {4558, 32037}, {10641, 393}, {11088, 8738}, {11127, 470}, {11134, 8740}, {11136, 8739}, {11142, 8737}, {11145, 471}, {23873, 14618}, {32036, 38342}, {32585, 2963}, {32661, 16807}, {34389, 20572}, {36296, 11082}, {36297, 11138}, {40712, 11140}, {44180, 302}, {44718, 19778}, {46112, 51891}, {46113, 8604}, {50465, 11601}, {50469, 14}, {51547, 8741}
X(52349) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 44719, 44718}, {343, 465, 40709}, {343, 52193, 40711}, {465, 52193, 343}, {628, 19772, 33530}, {5408, 5409, 44719}, {9306, 9735, 3131}, {11127, 11145, 62}, {40709, 40711, 343}

X(52350) = X(6)-DAO CONJUGATE OF X(24)

X(52350) lies on these lines: {2, 311}, {3, 68}, {69, 97}, {91, 19854}, {96, 631}, {182, 41271}, {276, 7763}, {394, 6389}, {427, 14593}, {441, 52041}, {571, 6515}, {847, 1217}, {925, 1297}, {1073, 11064}, {3135, 23181}, {3546, 40698}, {3548, 32132}, {4558, 45794}, {5449, 15827}, {5962, 18533}, {6388, 10318}, {6459, 13428}, {6460, 13439}, {6511, 26942}, {6512, 26932}, {6815, 45301}, {7667, 40348}, {7836, 46735}, {8906, 11585}, {9723, 37636}, {11427, 50572}, {13567, 47731}, {14376, 36212}, {14790, 34225}, {14919, 37669}, {16032, 45509}, {16037, 45508}, {18420, 34836}, {23335, 46200}, {28724, 37188}, {30450, 46927}, {34385, 37872}, {39113, 52253}

X(52350) = isogonal conjugate of X(8745)
X(52350) = isotomic conjugate of X(11547)
X(52350) = isotomic conjugate of the polar conjugate of X(68)
X(52350) = isogonal conjugate of the polar conjugate of X(20563)
X(52350) = polar conjugate of the isogonal conjugate of X(16391)
X(52350) = X(20563)-Ceva conjugate of X(68)
X(52350) = X(i)-cross conjugate of X(j) for these (i,j): {125, 3265}, {1092, 69}
X(52350) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8745}, {19, 24}, {25, 1748}, {31, 11547}, {47, 393}, {91, 36416}, {92, 44077}, {158, 571}, {162, 6753}, {317, 1973}, {563, 1093}, {823, 34952}, {924, 24019}, {1096, 1993}, {1147, 6520}, {2180, 8884}, {2190, 14576}, {2207, 44179}, {17881, 41937}, {24000, 47421}, {30451, 36126}
X(52350) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 11547}, {3, 8745}, {5, 14576}, {6, 24}, {125, 6753}, {136, 647}, {317, 6337}, {393, 34853}, {467, 52032}, {571, 1147}, {924, 35071}, {1147, 37867}, {1748, 6505}, {1993, 6503}, {2165, 3542}, {2207, 37864}, {3767, 41770}, {6338, 7763}, {14397, 38999}, {22391, 44077}, {30451, 46093}, {34116, 36416}
X(52350) = crosssum of X(6753) and X(47421)
X(52350) = crossdifference of every pair of points on line {6753, 34952}
X(52350) = barycentric product X(i)*X(j) for these {i,j}: {3, 20563}, {68, 69}, {91, 326}, {255, 20571}, {264, 16391}, {304, 1820}, {305, 2351}, {339, 44174}, {394, 5392}, {520, 46134}, {847, 3964}, {925, 3265}, {2165, 3926}, {4176, 14593}, {5562, 34385}, {11090, 11091}, {26922, 34391}, {32132, 40697}
X(52350) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 11547}, {3, 24}, {6, 8745}, {63, 1748}, {68, 4}, {69, 317}, {91, 158}, {96, 8884}, {125, 136}, {155, 35603}, {184, 44077}, {216, 14576}, {255, 47}, {326, 44179}, {343, 467}, {394, 1993}, {520, 924}, {571, 36416}, {577, 571}, {647, 6753}, {847, 1093}, {925, 107}, {1092, 1147}, {1636, 14397}, {1820, 19}, {2165, 393}, {2351, 25}, {3265, 6563}, {3269, 47421}, {3519, 14111}, {3926, 7763}, {3964, 9723}, {3998, 42700}, {4100, 563}, {4558, 41679}, {5392, 2052}, {5562, 52}, {6389, 41770}, {6394, 31635}, {6413, 5412}, {6414, 5413}, {11090, 1586}, {11091, 1585}, {13754, 52000}, {14593, 6524}, {15316, 34756}, {16391, 3}, {17879, 17881}, {18604, 18605}, {20563, 264}, {23224, 34948}, {26922, 371}, {30450, 15352}, {32132, 254}, {32320, 30451}, {32734, 32713}, {34385, 8795}, {34853, 3542}, {36145, 24019}, {37802, 14165}, {39201, 34952}, {43083, 43088}, {44174, 250}, {46134, 6528}, {51386, 51439}, {51394, 51393}
X(52350) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5392, 2165}, {2, 40697, 52032}, {5392, 37802, 2}, {11090, 11091, 68}, {13430, 13441, 20563}

X(52351) = X(6)-DAO CONJUGATE OF X(36)

X(52351) lies on these lines: {2, 2006}, {10, 21}, {63, 343}, {78, 1062}, {94, 7110}, {280, 5552}, {306, 1332}, {333, 39277}, {348, 6350}, {498, 15065}, {655, 908}, {914, 22128}, {1411, 19860}, {1791, 34851}, {2161, 2339}, {2222, 26703}, {3035, 14204}, {3264, 32851}, {3306, 7131}, {6335, 20566}, {6505, 41081}, {6703, 11069}, {6735, 40437}, {6745, 51562}, {6796, 7488}, {10573, 24597}, {14616, 19808}, {16067, 29857}, {16548, 46487}, {16594, 17279}, {17740, 33937}, {17776, 34277}, {24209, 33129}, {26231, 29828}, {27509, 30680}, {30787, 50752}, {30852, 52212}, {32849, 46785}, {35174, 37774}

X(52351) = isotomic conjugate of X(17923)
X(52351) = isotomic conjugate of the polar conjugate of X(80)
X(52351) = isogonal conjugate of the polar conjugate of X(20566)
X(52351) = X(20566)-Ceva conjugate of X(80)
X(52351) = X(i)-cross conjugate of X(j) for these (i,j): {5440, 69}, {14429, 4561}, {22059, 3}
X(52351) = X(i)-isoconjugate of X(j) for these (i,j): {4, 7113}, {6, 1870}, {19, 36}, {25, 3218}, {27, 3724}, {28, 2245}, {31, 17923}, {34, 2323}, {81, 44113}, {108, 654}, {162, 21828}, {186, 2160}, {214, 8752}, {225, 4282}, {278, 2361}, {284, 1835}, {320, 1973}, {604, 5081}, {607, 1443}, {608, 4511}, {649, 4242}, {653, 8648}, {758, 1474}, {860, 1333}, {909, 1845}, {913, 11570}, {1096, 22128}, {1172, 1464}, {1395, 32851}, {1400, 17515}, {1415, 44428}, {1897, 21758}, {1974, 20924}, {1983, 7649}, {2203, 3936}, {2204, 41804}, {2212, 17078}, {2299, 18593}, {3738, 32674}, {3960, 8750}, {4707, 32676}, {8739, 39153}, {8740, 39152}, {8756, 16944}, {13486, 47230}, {17455, 36125}, {30690, 34397}
X(52351) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 17923}, {6, 36}, {9, 1870}, {19, 15898}, {37, 860}, {125, 21828}, {226, 18593}, {281, 36909}, {320, 6337}, {654, 38983}, {758, 51574}, {1146, 44428}, {1835, 40590}, {1845, 23980}, {2245, 40591}, {2323, 11517}, {3161, 5081}, {3218, 6505}, {3738, 35072}, {3904, 40626}, {3960, 26932}, {4242, 5375}, {4453, 40618}, {4707, 15526}, {6503, 22128}, {7113, 36033}, {17515, 40582}, {21758, 34467}, {40586, 44113}
X(52351) = cevapoint of X(i) and X(j) for these (i,j): {3, 22123}, {71, 22350}, {306, 3977}
X(52351) = trilinear pole of line {72, 521}
X(52351) = crossdifference of every pair of points on line {8648, 21828}
X(52351) = barycentric product X(i)*X(j) for these {i,j}: {3, 20566}, {35, 328}, {63, 18359}, {69, 80}, {72, 14616}, {75, 1807}, {78, 18815}, {265, 319}, {304, 2161}, {305, 6187}, {306, 24624}, {307, 6740}, {337, 36815}, {345, 2006}, {348, 36910}, {521, 35174}, {525, 47318}, {652, 46405}, {655, 6332}, {759, 20336}, {905, 36804}, {1231, 2341}, {1411, 3718}, {1441, 1793}, {1444, 15065}, {2222, 35518}, {4025, 51562}, {34079, 40071}
X(52351) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1870}, {2, 17923}, {3, 36}, {8, 5081}, {10, 860}, {21, 17515}, {35, 186}, {42, 44113}, {48, 7113}, {63, 3218}, {65, 1835}, {69, 320}, {71, 2245}, {72, 758}, {73, 1464}, {77, 1443}, {78, 4511}, {80, 4}, {100, 4242}, {212, 2361}, {219, 2323}, {228, 3724}, {265, 79}, {304, 20924}, {305, 40075}, {306, 3936}, {307, 41804}, {319, 340}, {328, 20565}, {345, 32851}, {348, 17078}, {394, 22128}, {517, 1845}, {521, 3738}, {522, 44428}, {525, 4707}, {647, 21828}, {652, 654}, {655, 653}, {759, 28}, {905, 3960}, {906, 1983}, {912, 11570}, {1060, 4351}, {1128, 8119}, {1168, 36125}, {1214, 18593}, {1332, 4585}, {1411, 34}, {1565, 4089}, {1793, 21}, {1797, 40215}, {1807, 1}, {1946, 8648}, {2006, 278}, {2161, 19}, {2193, 4282}, {2222, 108}, {2341, 1172}, {3781, 3792}, {3916, 4973}, {3927, 4880}, {3940, 4867}, {3949, 4053}, {3977, 51583}, {4025, 4453}, {4064, 6370}, {4855, 4881}, {5440, 214}, {6187, 25}, {6332, 3904}, {6740, 29}, {7265, 44427}, {10215, 8120}, {14584, 1877}, {14616, 286}, {14628, 37790}, {15065, 41013}, {18359, 92}, {18815, 273}, {20336, 35550}, {20566, 264}, {20769, 27950}, {21054, 35235}, {21912, 51462}, {22123, 40584}, {22350, 34586}, {22356, 17455}, {22383, 21758}, {23071, 6126}, {23224, 22379}, {24624, 27}, {32675, 32674}, {34079, 1474}, {34857, 1824}, {35174, 18026}, {35194, 51801}, {36058, 16944}, {36061, 13486}, {36804, 6335}, {36815, 242}, {36910, 281}, {36927, 2907}, {38938, 5146}, {40437, 36123}, {45926, 1838}, {46405, 46404}, {46974, 11700}, {47318, 648}, {49280, 23884}, {51562, 1897}, {51975, 38462}, {52153, 6186}, {52201, 41225}, {52202, 3179}
X(52351) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 18359, 2006}, {2, 41226, 18359}, {2006, 36910, 18359}, {18359, 41226, 36910}

X(52352) = X(8)-DAO CONJUGATE OF X(10)

X(52352) lies on these lines: {1, 4234}, {2, 49734}, {8, 21}, {20, 18134}, {29, 4997}, {30, 25650}, {58, 643}, {69, 17576}, {75, 28627}, {81, 3623}, {86, 3445}, {99, 1434}, {100, 7419}, {145, 3052}, {190, 34772}, {284, 2325}, {312, 3601}, {320, 8822}, {386, 13735}, {452, 5233}, {903, 26729}, {950, 32851}, {979, 5331}, {993, 50625}, {1010, 1125}, {1098, 30606}, {1220, 37573}, {1420, 39126}, {1698, 11110}, {1834, 25459}, {1975, 14828}, {2475, 41878}, {2646, 3685}, {2975, 8053}, {3057, 3794}, {3146, 30828}, {3158, 44720}, {3161, 44722}, {3216, 33309}, {3522, 18141}, {3616, 4854}, {3621, 4921}, {3707, 4877}, {3847, 14011}, {3912, 35935}, {3936, 15680}, {3950, 33628}, {3984, 17336}, {4184, 5303}, {4189, 14829}, {4190, 17234}, {4195, 19765}, {4248, 4855}, {4255, 17697}, {4256, 13741}, {4258, 27523}, {4304, 7270}, {4314, 4514}, {4417, 6872}, {4427, 34195}, {4645, 15338}, {4647, 5426}, {4869, 50693}, {4975, 37616}, {5247, 50590}, {5263, 10448}, {5436, 19804}, {5438, 30829}, {7253, 28221}, {7283, 24929}, {7354, 29839}, {7415, 31730}, {7538, 20477}, {9534, 16418}, {9780, 17553}, {10107, 18165}, {10449, 16370}, {10609, 37227}, {11106, 14555}, {13588, 30947}, {14007, 34595}, {15670, 25446}, {15674, 31205}, {15952, 34773}, {16046, 17316}, {16050, 17367}, {16704, 20014}, {16865, 17277}, {17297, 37299}, {17524, 29766}, {17525, 49716}, {17588, 46933}, {17677, 25645}, {18139, 37256}, {19270, 48863}, {19532, 37502}, {19701, 51674}, {19768, 36489}, {24936, 50171}, {26117, 30832}, {27505, 28793}, {28461, 48877}, {34064, 37539}, {41816, 49728}, {49745, 51678}

X(52352) = X(86)-Ceva conjugate of X(333)
X(52352) = X(4162)-cross conjugate of X(30720)
X(52352) = X(i)-isoconjugate of X(j) for these (i,j): {10, 16945}, {37, 40151}, {42, 19604}, {65, 3445}, {213, 27818}, {226, 38266}, {604, 4052}, {661, 38828}, {1042, 3680}, {1293, 4017}, {1400, 8056}, {1402, 4373}, {2333, 27832}, {4849, 16079}, {7178, 34080}, {7180, 27834}, {7250, 31343}, {30572, 36042}
X(52352) = X(i)-Dao conjugate of X(j) for these (i,j): {8, 10}, {65, 45036}, {523, 3756}, {1293, 34961}, {3161, 4052}, {3445, 40602}, {4373, 40605}, {4859, 21949}, {5516, 30572}, {6626, 27818}, {7178, 40621}, {8056, 40582}, {19604, 40592}, {36830, 38828}, {40151, 40589}
X(52352) = cevapoint of X(i) and X(j) for these (i,j): {145, 4855}, {3158, 3161}
X(52352) = crosspoint of X(86) and X(41629)
X(52352) = barycentric product X(i)*X(j) for these {i,j}: {8, 41629}, {21, 18743}, {27, 44722}, {58, 44723}, {81, 44720}, {86, 3161}, {99, 4521}, {145, 333}, {261, 3950}, {274, 3158}, {312, 16948}, {314, 1743}, {345, 4248}, {643, 4462}, {645, 3667}, {799, 4162}, {1043, 5435}, {1434, 6555}, {1790, 44721}, {2287, 39126}, {3052, 28660}, {3596, 33628}, {4394, 7257}, {4404, 4612}, {4534, 4600}, {4546, 4573}, {4560, 43290}, {4567, 4939}, {4610, 44729}, {4620, 4953}, {4631, 4729}, {4848, 7058}, {4855, 31623}, {6064, 21950}, {7192, 30720}, {7256, 30719}, {7258, 51658}, {17197, 44724}, {18653, 44727}, {20818, 44130}
X(52352) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 4052}, {21, 8056}, {58, 40151}, {81, 19604}, {86, 27818}, {110, 38828}, {145, 226}, {284, 3445}, {314, 40014}, {333, 4373}, {643, 27834}, {1043, 6557}, {1333, 16945}, {1420, 1427}, {1444, 27832}, {1743, 65}, {2194, 38266}, {2287, 3680}, {3052, 1400}, {3158, 37}, {3161, 10}, {3667, 7178}, {3950, 12}, {4162, 661}, {4248, 278}, {4394, 4017}, {4462, 4077}, {4521, 523}, {4534, 3120}, {4546, 3700}, {4848, 6354}, {4849, 2171}, {4855, 1214}, {4856, 3649}, {4881, 18593}, {4936, 210}, {4939, 16732}, {4943, 14321}, {4953, 21044}, {5435, 3668}, {5546, 1293}, {6555, 2321}, {7259, 31343}, {8643, 7180}, {12640, 4415}, {14425, 30572}, {15519, 3950}, {16948, 57}, {18743, 1441}, {20818, 73}, {21950, 1365}, {30720, 3952}, {30941, 10029}, {33628, 56}, {39126, 1446}, {41629, 7}, {43290, 4552}, {44720, 321}, {44722, 306}, {44723, 313}, {44729, 4024}, {51658, 7216}
X(52352) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21, 1043, 333}, {145, 16948, 41629}, {3712, 10543, 8}

X(52353) = X(8)-DAO CONJUGATE OF X(21)

X(52353) lies on these lines: {1, 4723}, {2, 341}, {8, 392}, {10, 321}, {58, 16729}, {65, 3952}, {75, 46933}, {100, 28029}, {145, 4487}, {210, 17751}, {312, 3617}, {318, 36624}, {344, 10528}, {518, 29982}, {594, 42712}, {946, 30566}, {950, 49991}, {984, 20892}, {1191, 26688}, {1201, 24003}, {1220, 5297}, {1265, 5554}, {1329, 3006}, {1698, 4968}, {1722, 3891}, {1739, 24068}, {1997, 10529}, {2478, 5014}, {2551, 5016}, {2650, 4090}, {2885, 4884}, {2899, 3434}, {2901, 31855}, {2975, 5205}, {3175, 21896}, {3214, 3896}, {3244, 4738}, {3263, 6376}, {3436, 7386}, {3596, 24547}, {3616, 4737}, {3621, 46938}, {3622, 30829}, {3625, 4975}, {3626, 3902}, {3632, 4742}, {3634, 4692}, {3679, 3702}, {3696, 22016}, {3698, 3967}, {3699, 34772}, {3703, 9711}, {3714, 3983}, {3717, 20905}, {3718, 24993}, {3812, 17165}, {3823, 18133}, {3831, 46909}, {3869, 27538}, {3921, 5295}, {3932, 21031}, {3936, 21075}, {3953, 49993}, {3970, 4103}, {3991, 30730}, {3995, 4646}, {4009, 5836}, {4033, 42724}, {4037, 21868}, {4359, 4385}, {4397, 23757}, {4450, 12572}, {4462, 23764}, {4514, 37162}, {4518, 30082}, {4662, 17135}, {4673, 4678}, {4712, 30045}, {4767, 34195}, {4850, 26029}, {4939, 12640}, {5046, 32850}, {5086, 36926}, {5128, 25734}, {5250, 51284}, {5253, 9369}, {5260, 7081}, {5433, 37762}, {5552, 33113}, {6381, 20880}, {7080, 17776}, {8728, 40013}, {9330, 31359}, {9335, 34860}, {10005, 20946}, {11681, 29641}, {12648, 42020}, {14923, 19582}, {17490, 26046}, {17752, 26689}, {17757, 21530}, {17862, 25005}, {18156, 25278}, {19804, 46932}, {20653, 48648}, {20911, 33932}, {20947, 25280}, {21101, 21921}, {21674, 27705}, {23536, 24988}, {24171, 24183}, {24349, 30044}, {24440, 32925}, {25237, 40883}, {25610, 41269}, {25992, 26230}, {25994, 31087}, {26234, 33938}, {26770, 44798}, {28080, 30614}, {28996, 34040}, {30090, 31302}, {30615, 36500}, {30806, 33942}, {30807, 39570}, {31035, 37548}, {32911, 41261}

X(52353) = isotomic conjugate of the isogonal conjugate of X(4849)
X(52353) = X(i)-Ceva conjugate of X(j) for these (i,j): {1441, 321}, {18743, 3950}
X(52353) = X(i)-cross conjugate of X(j) for these (i,j): {4918, 145}, {21950, 4462}
X(52353) = X(i)-isoconjugate of X(j) for these (i,j): {21, 16945}, {58, 3445}, {81, 38266}, {284, 40151}, {1019, 34080}, {1293, 3733}, {1333, 8056}, {1408, 3680}, {2194, 19604}, {2204, 27832}, {2206, 4373}, {6557, 16947}, {7252, 38828}
X(52353) = X(i)-Dao conjugate of X(j) for these (i,j): {8, 21}, {10, 3445}, {37, 8056}, {58, 45036}, {1019, 40621}, {1214, 19604}, {3667, 18211}, {3737, 3756}, {3950, 37679}, {4373, 40603}, {4521, 16726}, {4849, 16688}, {16945, 40611}, {38266, 40586}, {40151, 40590}
X(52353) = crosspoint of X(44720) and X(44721)
X(52353) = trilinear pole of line {4404, 14321}
X(52353) = barycentric product X(i)*X(j) for these {i,j}: {10, 18743}, {65, 44723}, {75, 3950}, {76, 4849}, {145, 321}, {190, 4404}, {226, 44720}, {312, 4848}, {313, 1743}, {349, 3158}, {668, 14321}, {1089, 41629}, {1214, 44721}, {1420, 30713}, {1441, 3161}, {1446, 6555}, {1577, 43290}, {1978, 4729}, {2321, 39126}, {3052, 27801}, {3667, 4033}, {3701, 5435}, {3952, 4462}, {3971, 27496}, {3992, 31227}, {4077, 30720}, {4080, 4487}, {4394, 27808}, {4554, 44729}, {4918, 30710}, {7035, 21950}, {16732, 44724}, {16948, 28654}, {40149, 44722}
X(52353) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 8056}, {37, 3445}, {42, 38266}, {65, 40151}, {145, 81}, {226, 19604}, {307, 27832}, {313, 40014}, {321, 4373}, {1018, 1293}, {1089, 4052}, {1400, 16945}, {1420, 1412}, {1441, 27818}, {1743, 58}, {2321, 3680}, {2885, 16736}, {3052, 1333}, {3158, 284}, {3161, 21}, {3667, 1019}, {3701, 6557}, {3756, 16726}, {3950, 1}, {3952, 27834}, {4058, 10563}, {4162, 7252}, {4394, 3733}, {4404, 514}, {4462, 7192}, {4487, 16704}, {4504, 18200}, {4521, 3737}, {4534, 18191}, {4546, 1021}, {4551, 38828}, {4557, 34080}, {4729, 649}, {4848, 57}, {4849, 6}, {4855, 1790}, {4884, 16696}, {4891, 18166}, {4898, 4658}, {4899, 18206}, {4918, 3666}, {4936, 2328}, {4939, 17197}, {4949, 4840}, {5435, 1014}, {6555, 2287}, {12640, 18163}, {14321, 513}, {16948, 593}, {18743, 86}, {20818, 1437}, {21052, 27837}, {21950, 244}, {23764, 8042}, {30719, 7203}, {30720, 643}, {30730, 31343}, {33628, 849}, {39126, 1434}, {40621, 18211}, {41629, 757}, {43290, 662}, {44720, 333}, {44721, 31623}, {44722, 1812}, {44723, 314}, {44724, 4567}, {44729, 650}
X(52353) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 341, 4696}, {8, 46937, 4358}, {10, 3159, 3987}, {10, 3701, 321}, {10, 3971, 4642}, {10, 3992, 3701}, {10, 4066, 4714}, {10, 4075, 4424}, {10, 4125, 4647}, {145, 4487, 4935}, {145, 44720, 4487}, {1698, 4968, 24589}, {2551, 10327, 5016}, {3263, 6376, 26563}, {3263, 26563, 21432}, {3698, 3967, 17164}, {3714, 3983, 4651}, {4009, 5836, 25253}, {4385, 9780, 4359}, {18743, 44720, 145}

X(52354) = X(8)-DAO CONJUGATE OF X(29)

X(52354) lies on these lines: {2, 39589}, {8, 3586}, {10, 3120}, {20, 25734}, {40, 49991}, {63, 1265}, {72, 306}, {78, 3977}, {145, 1743}, {344, 11520}, {345, 3984}, {537, 23675}, {976, 35263}, {982, 25881}, {2650, 4078}, {3610, 3958}, {3704, 4005}, {3717, 3869}, {3932, 3962}, {3947, 27690}, {3951, 4001}, {3967, 21677}, {4069, 37558}, {4082, 17751}, {4115, 21073}, {4126, 5836}, {4358, 24391}, {4487, 12640}, {4488, 37435}, {4537, 21081}, {4696, 5837}, {4723, 11362}, {4756, 5086}, {4847, 25253}, {4849, 4918}, {4855, 44722}, {6743, 32929}, {6872, 25728}, {7270, 17781}, {10327, 12526}, {11523, 17776}, {12649, 30568}, {18249, 26227}, {19582, 26015}, {19860, 27549}, {24349, 24564}, {24982, 27538}, {24987, 32937}, {36500, 40998}, {43290, 44727}, {44720, 44721}

X(52354) = isotomic conjugate of the polar conjugate of X(3950)
X(52354) = X(307)-Ceva conjugate of X(306)
X(52354) = X(i)-isoconjugate of X(j) for these (i,j): {27, 38266}, {28, 3445}, {29, 16945}, {1172, 40151}, {1474, 8056}, {2203, 4373}, {2204, 27818}, {2299, 19604}, {17925, 34080}, {27834, 43925}
X(52354) = X(i)-Dao conjugate of X(j) for these (i,j): {8, 29}, {28, 45036}, {226, 19604}, {3445, 40591}, {8056, 51574}, {17925, 40621}
X(52354) = crosspoint of X(44720) and X(44722)
X(52354) = barycentric product X(i)*X(j) for these {i,j}: {69, 3950}, {72, 18743}, {73, 44723}, {145, 306}, {226, 44722}, {304, 4849}, {307, 3161}, {313, 20818}, {321, 4855}, {345, 4848}, {525, 43290}, {1214, 44720}, {1231, 3158}, {1332, 4404}, {1743, 20336}, {3052, 40071}, {3694, 39126}, {3695, 41629}, {3710, 5435}, {4466, 44724}, {4561, 14321}, {17094, 30720}, {40152, 44721}
X(52354) = barycentric quotient X(i)/X(j) for these {i,j}: {71, 3445}, {72, 8056}, {73, 40151}, {145, 27}, {228, 38266}, {306, 4373}, {307, 27818}, {1214, 19604}, {1409, 16945}, {1420, 1396}, {1743, 28}, {3052, 1474}, {3158, 1172}, {3161, 29}, {3667, 17925}, {3694, 3680}, {3695, 4052}, {3710, 6557}, {3950, 4}, {4248, 36419}, {4404, 17924}, {4546, 17926}, {4574, 1293}, {4729, 6591}, {4848, 278}, {4849, 19}, {4855, 81}, {4856, 31900}, {4884, 17171}, {4898, 31902}, {4899, 15149}, {4918, 1848}, {4924, 31922}, {4929, 31925}, {4936, 4183}, {6555, 2322}, {8643, 43925}, {14321, 7649}, {18743, 286}, {20336, 40014}, {20818, 58}, {21950, 2969}, {23067, 38828}, {30720, 36797}, {43290, 648}, {44720, 31623}, {44722, 333}, {44723, 44130}, {44729, 3064}
X(52354) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {72, 3695, 4101}, {72, 3710, 306}, {3695, 4101, 306}, {3710, 4101, 3695}

X(52355) = X(11)-DAO CONJUGATE OF X(28)

X(52355) lies on these lines: {37, 2485}, {72, 520}, {100, 2766}, {101, 2769}, {513, 28591}, {521, 6332}, {522, 650}, {523, 1577}, {525, 656}, {676, 48186}, {765, 39189}, {832, 48299}, {905, 20315}, {918, 23800}, {2483, 21390}, {2525, 21107}, {2804, 4397}, {3265, 14208}, {3267, 20336}, {3800, 50332}, {3904, 20293}, {3952, 23067}, {4017, 4088}, {4163, 8058}, {4391, 20294}, {4581, 47809}, {4768, 42337}, {4804, 27588}, {4985, 6362}, {7253, 15776}, {8057, 24018}, {8062, 44409}, {8672, 48047}, {10015, 20316}, {14430, 21119}, {15411, 23189}, {17420, 48278}, {21187, 41800}, {21189, 48272}, {34958, 48209}, {38469, 48290}, {47695, 48173}, {48069, 50495}

X(52355) = midpoint of X(i) and X(j) for these {i,j}: {656, 4064}, {3904, 20293}, {4017, 4088}, {4391, 20294}, {17420, 48278}, {21189, 48272}
X(52355) = reflection of X(i) in X(j) for these {i,j}: {905, 20315}, {10015, 20316}, {44409, 8062}, {48402, 47842}
X(52355) = isotomic conjugate of the polar conjugate of X(3700)
X(52355) = X(i)-complementary conjugate of X(j) for these (i,j): {8615, 11}, {15314, 21252}
X(52355) = X(i)-Ceva conjugate of X(j) for these (i,j): {8, 7068}, {78, 2968}, {100, 3704}, {318, 7358}, {645, 219}, {3718, 26932}, {3952, 72}, {4397, 4086}, {4552, 2321}, {5546, 3703}, {6332, 8611}, {14208, 525}, {36626, 6741}, {36797, 8}
X(52355) = X(i)-cross conjugate of X(j) for these (i,j): {4064, 4086}, {7068, 8}, {8611, 525}, {21044, 40161}
X(52355) = X(i)-isoconjugate of X(j) for these (i,j): {7, 32676}, {19, 4565}, {25, 1414}, {27, 1415}, {28, 109}, {34, 110}, {56, 162}, {57, 112}, {58, 108}, {77, 32713}, {81, 32674}, {99, 1395}, {101, 1396}, {107, 603}, {163, 278}, {222, 24019}, {250, 4017}, {273, 1576}, {284, 32714}, {604, 648}, {607, 4637}, {608, 662}, {643, 1398}, {651, 1474}, {653, 1333}, {658, 2204}, {664, 2203}, {685, 51653}, {811, 1397}, {859, 36110}, {933, 1393}, {934, 2299}, {1014, 8750}, {1019, 7115}, {1020, 2189}, {1106, 36797}, {1118, 4575}, {1172, 1461}, {1301, 1394}, {1304, 51656}, {1408, 1897}, {1412, 1783}, {1416, 4238}, {1417, 46541}, {1426, 4636}, {1435, 5546}, {1437, 36127}, {1451, 36077}, {1813, 5317}, {1835, 36069}, {1880, 4556}, {1973, 4573}, {1974, 4625}, {2149, 17925}, {2194, 36118}, {2206, 18026}, {2212, 4616}, {2332, 4617}, {3194, 8059}, {3213, 46639}, {3733, 7012}, {4183, 6614}, {4564, 43925}, {4570, 43923}, {4592, 7337}, {5323, 32691}, {5338, 5545}, {5379, 43924}, {6335, 16947}, {6357, 36131}, {6529, 7125}, {7128, 7252}, {7335, 36126}, {8747, 36059}, {15439, 46883}, {18020, 51642}, {23189, 24033}, {23964, 51640}, {32230, 51641}, {32651, 46884}, {36046, 43045}, {36104, 43034}, {44770, 51649}
X(52355) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 162}, {4, 6741}, {6, 4565}, {7, 15526}, {10, 108}, {11, 28}, {21, 7358}, {25, 40608}, {27, 1146}, {29, 2968}, {34, 244}, {37, 653}, {56, 125}, {57, 34591}, {58, 38983}, {81, 35072}, {86, 40626}, {107, 7952}, {109, 40591}, {110, 11517}, {112, 5452}, {115, 278}, {122, 44696}, {136, 1118}, {222, 35071}, {226, 934}, {250, 34961}, {273, 4858}, {286, 40624}, {331, 36901}, {521, 23189}, {525, 17094}, {603, 38985}, {608, 1084}, {647, 7178}, {648, 3161}, {650, 17925}, {651, 51574}, {656, 3737}, {859, 39004}, {905, 7192}, {1014, 26932}, {1015, 1396}, {1019, 40628}, {1119, 40622}, {1145, 4246}, {1172, 35508}, {1214, 36118}, {1354, 1650}, {1395, 38986}, {1397, 17423}, {1408, 34467}, {1412, 39006}, {1414, 6505}, {1434, 40618}, {1474, 38991}, {1783, 40599}, {1835, 38982}, {2203, 39025}, {2299, 14714}, {2972, 30493}, {3239, 4560}, {3756, 4248}, {4230, 50440}, {4233, 4904}, {4238, 40609}, {4240, 6739}, {4241, 50441}, {4573, 6337}, {5139, 7337}, {5323, 17421}, {6357, 39008}, {6388, 17081}, {6552, 36797}, {7335, 46093}, {8286, 13739}, {8747, 20620}, {8885, 13612}, {18026, 40603}, {18623, 39020}, {32674, 40586}, {32714, 40590}, {33504, 43045}, {37168, 51402}, {39000, 43034}, {43923, 50330}
X(52355) = cevapoint of X(3700) and X(14308)
X(52355) = crosspoint of X(i) and X(j) for these (i,j): {8, 36797}, {100, 1791}, {307, 4552}, {645, 3596}, {3701, 3952}, {6332, 35518}, {6335, 40422}
X(52355) = crosssum of X(i) and X(j) for these (i,j): {513, 1829}, {1397, 7180}, {1408, 3733}, {2203, 43925}, {2299, 7252}, {22383, 40956}
X(52355) = crossdifference of every pair of points on line {56, 608}
X(52355) = barycentric product X(i)*X(j) for these {i,j}: {8, 525}, {9, 14208}, {10, 6332}, {12, 15411}, {37, 35518}, {55, 3267}, {63, 4086}, {65, 15416}, {69, 3700}, {71, 35519}, {72, 4391}, {75, 8611}, {78, 1577}, {125, 645}, {210, 15413}, {212, 20948}, {219, 850}, {281, 3265}, {304, 4041}, {305, 3709}, {306, 522}, {307, 3239}, {312, 656}, {313, 652}, {318, 24018}, {321, 521}, {332, 4024}, {333, 4064}, {339, 5546}, {341, 51640}, {345, 523}, {346, 17094}, {514, 3710}, {520, 7017}, {643, 20902}, {646, 18210}, {647, 3596}, {648, 7068}, {650, 20336}, {661, 3718}, {663, 40071}, {693, 3694}, {810, 28659}, {905, 3701}, {1018, 17880}, {1214, 4397}, {1231, 3900}, {1259, 14618}, {1264, 2501}, {1265, 7178}, {1459, 30713}, {1565, 30730}, {1812, 4036}, {1857, 4143}, {1946, 27801}, {2318, 3261}, {2321, 4025}, {2968, 4552}, {3049, 40363}, {3682, 46110}, {3692, 4077}, {3695, 4560}, {3699, 4466}, {3703, 4580}, {3704, 15420}, {3708, 7257}, {3712, 14977}, {3719, 24006}, {3949, 18155}, {3952, 26932}, {3998, 44426}, {4033, 7004}, {4092, 4563}, {4103, 17219}, {4140, 7019}, {4171, 7182}, {4518, 24459}, {4561, 21044}, {4571, 16732}, {4574, 34387}, {4587, 21207}, {4997, 14429}, {5547, 45807}, {6057, 15419}, {6333, 15628}, {7117, 27808}, {7253, 26942}, {7359, 34767}, {14308, 34403}, {14638, 44695}, {15526, 36797}, {20294, 40161}, {23067, 23978}, {23090, 34388}, {23189, 28654}, {43728, 51367}
X(52355) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 4565}, {8, 648}, {9, 162}, {10, 653}, {11, 17925}, {33, 24019}, {37, 108}, {41, 32676}, {42, 32674}, {55, 112}, {63, 1414}, {65, 32714}, {69, 4573}, {71, 109}, {72, 651}, {73, 1461}, {77, 4637}, {78, 662}, {125, 7178}, {201, 1020}, {210, 1783}, {212, 163}, {219, 110}, {226, 36118}, {228, 1415}, {281, 107}, {283, 4556}, {304, 4625}, {306, 664}, {307, 658}, {312, 811}, {313, 46404}, {318, 823}, {321, 18026}, {332, 4610}, {345, 99}, {346, 36797}, {348, 4616}, {512, 608}, {513, 1396}, {520, 222}, {521, 81}, {522, 27}, {523, 278}, {525, 7}, {607, 32713}, {644, 5379}, {645, 18020}, {647, 56}, {650, 28}, {652, 58}, {656, 57}, {657, 2299}, {661, 34}, {663, 1474}, {684, 43034}, {798, 1395}, {810, 604}, {822, 603}, {850, 331}, {905, 1014}, {1018, 7012}, {1021, 270}, {1214, 934}, {1231, 4569}, {1259, 4558}, {1260, 5546}, {1264, 4563}, {1265, 645}, {1334, 8750}, {1364, 7254}, {1439, 4617}, {1441, 13149}, {1459, 1412}, {1565, 17096}, {1577, 273}, {1639, 37168}, {1792, 4612}, {1793, 37140}, {1826, 36127}, {1857, 6529}, {1946, 1333}, {2250, 36110}, {2289, 4575}, {2318, 101}, {2321, 1897}, {2325, 46541}, {2327, 4636}, {2335, 36077}, {2489, 7337}, {2501, 1118}, {2522, 5323}, {2525, 3665}, {2610, 1835}, {2631, 51656}, {2632, 51640}, {2968, 4560}, {3049, 1397}, {3063, 2203}, {3064, 8747}, {3125, 43923}, {3239, 29}, {3265, 348}, {3267, 6063}, {3270, 7252}, {3271, 43925}, {3596, 6331}, {3610, 14594}, {3682, 1813}, {3688, 35325}, {3690, 4559}, {3692, 643}, {3693, 4238}, {3694, 100}, {3695, 4552}, {3700, 4}, {3701, 6335}, {3703, 41676}, {3708, 4017}, {3709, 25}, {3710, 190}, {3712, 4235}, {3716, 31905}, {3718, 799}, {3719, 4592}, {3810, 31917}, {3900, 1172}, {3942, 7203}, {3949, 4551}, {3952, 46102}, {3954, 46152}, {3990, 36059}, {3998, 6516}, {4017, 1435}, {4019, 6649}, {4024, 225}, {4025, 1434}, {4036, 40149}, {4041, 19}, {4055, 32660}, {4064, 226}, {4077, 1847}, {4081, 17926}, {4086, 92}, {4088, 5236}, {4092, 2501}, {4105, 2332}, {4120, 1877}, {4130, 4183}, {4140, 7009}, {4143, 7055}, {4148, 14024}, {4157, 46543}, {4163, 2322}, {4171, 33}, {4391, 286}, {4397, 31623}, {4466, 3676}, {4516, 6591}, {4521, 4248}, {4522, 31909}, {4524, 607}, {4529, 14006}, {4551, 7128}, {4557, 7115}, {4561, 4620}, {4563, 7340}, {4571, 4567}, {4574, 59}, {4587, 4570}, {4705, 1880}, {4765, 31903}, {4820, 31902}, {4913, 31904}, {4976, 31900}, {5546, 250}, {6056, 32661}, {6332, 86}, {6587, 44696}, {7004, 1019}, {7017, 6528}, {7065, 32320}, {7068, 525}, {7117, 3733}, {7178, 1119}, {7180, 1398}, {7182, 4635}, {7253, 46103}, {7254, 7341}, {7257, 46254}, {7265, 7282}, {7359, 4240}, {8057, 18623}, {8058, 41083}, {8611, 1}, {8641, 2204}, {9033, 6357}, {9391, 51647}, {10097, 7316}, {10099, 1462}, {10397, 2360}, {14208, 85}, {14298, 3194}, {14308, 1249}, {14331, 44698}, {14395, 51420}, {14401, 1354}, {14417, 7181}, {14429, 3911}, {15411, 261}, {15416, 314}, {15419, 552}, {15526, 17094}, {15627, 1304}, {15628, 685}, {17094, 279}, {17434, 30493}, {17880, 7199}, {17898, 44697}, {18210, 3669}, {18344, 5317}, {20336, 4554}, {20902, 4077}, {20975, 7180}, {21031, 17906}, {21044, 7649}, {21107, 7195}, {21789, 2189}, {21801, 23706}, {21831, 7120}, {22080, 36075}, {22383, 1408}, {23067, 1262}, {23090, 60}, {23189, 593}, {23289, 40574}, {23616, 1367}, {24018, 77}, {24290, 1876}, {24459, 1447}, {26932, 7192}, {26942, 4566}, {30457, 1301}, {30681, 7256}, {30730, 15742}, {32320, 7335}, {34588, 16754}, {34591, 3737}, {35072, 23189}, {35518, 274}, {35519, 44129}, {36054, 1437}, {36197, 18344}, {36797, 23582}, {37754, 51641}, {40071, 4572}, {40161, 1305}, {40869, 4241}, {41087, 8059}, {44707, 1625}, {46382, 1451}, {48278, 17171}, {50333, 15149}, {50347, 31906}, {51640, 269}, {51641, 7099}
X(52355) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3265, 14208, 17094}, {4064, 14429, 656}

X(52356) = X(11)-DAO CONJUGATE OF X(36)

X(52356) lies on the cubic K230 and these lines: {5, 523}, {8, 4086}, {10, 522}, {80, 3738}, {265, 35053}, {281, 17926}, {476, 35056}, {513, 18480}, {656, 18395}, {693, 1441}, {996, 3907}, {1309, 2222}, {1577, 17057}, {2166, 36035}, {2321, 3239}, {2341, 15628}, {2400, 18815}, {2475, 39270}, {2804, 15065}, {3701, 4397}, {3737, 13746}, {4404, 45035}, {7951, 42768}, {8062, 30144}, {8836, 35052}, {8838, 35051}, {10272, 35054}, {14127, 43655}, {14616, 35141}, {15446, 23226}, {17931, 17934}, {23838, 36590}, {28132, 36910}, {35157, 35174}, {36802, 36804}, {37628, 40437}, {37630, 51562}, {41013, 44426}

X(52356) = reflection of X(35050) in X(5)
X(52356) = reflection of X(35050) in the Euler line
X(52356) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {11075, 37781}, {14147, 329}, {26743, 150}, {34921, 6224}
X(52356) = X(i)-Ceva conjugate of X(j) for these (i,j): {35174, 18359}, {36804, 36910}, {51562, 15065}
X(52356) = X(i)-cross conjugate of X(j) for these (i,j): {11, 36590}, {1639, 4391}, {2804, 522}
X(52356) = X(i)-isoconjugate of X(j) for these (i,j): {36, 109}, {50, 38340}, {57, 1983}, {110, 1464}, {163, 18593}, {603, 4242}, {604, 4585}, {651, 7113}, {654, 1262}, {655, 52059}, {692, 1443}, {934, 2361}, {1020, 4282}, {1101, 51645}, {1414, 3724}, {1415, 3218}, {1461, 2323}, {1511, 36064}, {1576, 41804}, {1835, 4575}, {1870, 36059}, {2149, 3960}, {2245, 4565}, {2624, 35049}, {2720, 34586}, {3028, 36069}, {3738, 24027}, {3904, 23979}, {4564, 21758}, {6126, 34921}, {6149, 26700}, {7012, 22379}, {7045, 8648}, {11700, 36040}, {16586, 32669}, {16944, 23703}, {17078, 32739}, {17923, 32660}, {22128, 32674}, {32719, 41801}, {41282, 51562}
X(52356) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 36}, {100, 36909}, {109, 15898}, {115, 18593}, {136, 1835}, {214, 51402}, {244, 1464}, {320, 40624}, {522, 3738}, {523, 51645}, {650, 3960}, {758, 6741}, {1086, 1443}, {1146, 3218}, {1577, 4453}, {1870, 20620}, {1983, 5452}, {2323, 35508}, {2361, 14714}, {2968, 4511}, {3028, 38982}, {3161, 4585}, {3700, 32679}, {3724, 40608}, {3756, 4881}, {4242, 7952}, {4351, 38964}, {4858, 41804}, {7113, 38991}, {8286, 27086}, {8648, 17115}, {10017, 11700}, {14993, 26700}, {17078, 40619}, {22128, 35072}, {34586, 38981}
X(52356) = cevapoint of X(i) and X(j) for these (i,j): {523, 36035}, {4041, 46393}, {4086, 4768}
X(52356) = crosspoint of X(i) and X(j) for these (i,j): {6740, 51562}, {14616, 32680}, {18359, 35174}, {20566, 36804}
X(52356) = crosssum of X(i) and X(j) for these (i,j): {2183, 42657}, {2624, 3724}, {7113, 8648}
X(52356) = trilinear pole of line {1146, 3700}
X(52356) = crossdifference of every pair of points on line {50, 7113}
X(52356) = barycentric product X(i)*X(j) for these {i,j}: {11, 36804}, {80, 4391}, {94, 35057}, {522, 18359}, {650, 20566}, {655, 24026}, {693, 36910}, {850, 2341}, {1146, 35174}, {1577, 6740}, {1793, 14618}, {1807, 46110}, {2006, 4397}, {2161, 35519}, {2222, 23978}, {2310, 46405}, {3239, 18815}, {3700, 14616}, {3762, 36590}, {4086, 24624}, {4560, 15065}, {4858, 51562}, {6741, 32680}, {11107, 14592}, {42033, 43082}
X(52356) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 4585}, {11, 3960}, {55, 1983}, {80, 651}, {115, 51645}, {281, 4242}, {476, 35049}, {514, 1443}, {521, 22128}, {522, 3218}, {523, 18593}, {650, 36}, {655, 7045}, {657, 2361}, {661, 1464}, {663, 7113}, {693, 17078}, {759, 4565}, {1146, 3738}, {1411, 1461}, {1577, 41804}, {1639, 214}, {1793, 4558}, {1807, 1813}, {1989, 26700}, {2006, 934}, {2161, 109}, {2166, 38340}, {2222, 1262}, {2310, 654}, {2341, 110}, {2501, 1835}, {2610, 3028}, {2804, 16586}, {3064, 1870}, {3239, 4511}, {3271, 21758}, {3700, 758}, {3709, 3724}, {3716, 27950}, {3762, 41801}, {3900, 2323}, {4041, 2245}, {4086, 3936}, {4092, 2610}, {4391, 320}, {4397, 32851}, {4516, 21828}, {4521, 4881}, {4768, 51583}, {4791, 36589}, {4820, 4880}, {4858, 4453}, {4895, 17455}, {4944, 4867}, {4976, 4973}, {5532, 46384}, {6187, 1415}, {6740, 662}, {6741, 32679}, {7117, 22379}, {8648, 52059}, {9404, 6149}, {10412, 43682}, {11075, 34921}, {11107, 14590}, {14616, 4573}, {14936, 8648}, {15065, 4552}, {17926, 17515}, {18359, 664}, {18815, 658}, {20566, 4554}, {21758, 41282}, {21789, 4282}, {23838, 40215}, {24026, 3904}, {24624, 1414}, {32675, 24027}, {34857, 4559}, {35057, 323}, {35174, 1275}, {35519, 20924}, {36590, 3257}, {36804, 4998}, {36910, 100}, {40166, 4089}, {40437, 37136}, {44426, 17923}, {46384, 3025}, {46393, 34586}, {51562, 4564}

X(52357) = X(12)-DAO CONJUGATE OF X(1)

X(52357) lies on these lines: {8, 4551}, {10, 12}, {56, 50605}, {76, 33298}, {201, 1089}, {221, 5774}, {225, 44143}, {227, 5295}, {307, 34388}, {388, 10479}, {519, 2594}, {596, 51975}, {1091, 1254}, {1210, 37592}, {1214, 3714}, {1420, 30942}, {1450, 19864}, {1737, 3670}, {2197, 2321}, {2975, 34589}, {3741, 10106}, {3778, 21935}, {3831, 3911}, {4357, 24982}, {4968, 6734}, {5219, 31339}, {5795, 26013}, {9578, 31330}, {10895, 48888}, {11679, 21147}, {11681, 41797}, {12527, 34831}, {14973, 20617}, {17751, 37558}, {18961, 48835}, {20718, 51870}, {21674, 21803}, {24310, 50037}, {25005, 27184}, {26125, 46933}, {37579, 48863}

X(52357) = X(75)-Ceva conjugate of X(6358)
X(52357) = X(i)-isoconjugate of X(j) for these (i,j): {21, 52150}, {60, 34434}, {1333, 46880}, {2051, 2150}, {2194, 20028}, {4267, 40453}
X(52357) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 12}, {37, 46880}, {1193, 4267}, {1214, 20028}, {3737, 34589}, {18163, 21796}, {40611, 52150}
X(52357) = crosspoint of X(75) and X(14829)
X(52357) = barycentric product X(i)*X(j) for these {i,j}: {12, 14829}, {85, 14973}, {226, 17751}, {312, 20617}, {321, 37558}, {349, 52139}, {572, 34388}, {1089, 17074}, {1441, 21061}, {2975, 6358}, {4033, 51664}, {11109, 26942}
X(52357) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 46880}, {12, 2051}, {226, 20028}, {572, 60}, {1400, 52150}, {2171, 34434}, {2975, 2185}, {10408, 34262}, {11109, 46103}, {14829, 261}, {14973, 9}, {17074, 757}, {17751, 333}, {20617, 57}, {20986, 2150}, {21061, 21}, {34589, 26856}, {37558, 81}, {51664, 1019}, {52087, 4267}, {52139, 284}
X(52357) = {X(12),X(65)}-harmonic conjugate of X(10408)

X(52358) = X(12)-DAO CONJUGATE OF X(37)

X(52358) lies on these lines: {2, 7}, {3, 27394}, {8, 73}, {10, 1042}, {12, 19874}, {56, 964}, {65, 22325}, {72, 37154}, {75, 17080}, {77, 11679}, {171, 5764}, {210, 26031}, {222, 1150}, {223, 5271}, {225, 4200}, {241, 16720}, {269, 18229}, {278, 17075}, {310, 4554}, {321, 1214}, {333, 651}, {348, 349}, {475, 7103}, {1211, 40999}, {1254, 49598}, {1407, 37660}, {1412, 27163}, {1427, 1441}, {1442, 1999}, {1458, 3741}, {1465, 4359}, {1471, 25496}, {1617, 24552}, {1758, 4418}, {1788, 26030}, {2003, 16704}, {2006, 39747}, {2051, 20367}, {2275, 3772}, {2295, 17056}, {2975, 11109}, {3187, 45126}, {3757, 4318}, {3868, 26028}, {3936, 26942}, {3947, 16828}, {3995, 16577}, {4077, 26983}, {4101, 7080}, {4192, 20256}, {4298, 19863}, {4306, 10479}, {4322, 50608}, {4417, 17137}, {4551, 4651}, {4640, 26095}, {4652, 27506}, {5261, 19853}, {5278, 34048}, {5290, 19858}, {5737, 6180}, {5830, 43056}, {6354, 41804}, {6358, 18593}, {6708, 30807}, {7175, 16738}, {7288, 17526}, {7677, 32942}, {8055, 28778}, {8270, 26227}, {9316, 32916}, {9364, 32918}, {10478, 17220}, {10888, 18655}, {11115, 37583}, {14829, 17074}, {16586, 20879}, {16698, 18601}, {16705, 17095}, {17084, 44350}, {17625, 46909}, {17751, 37558}, {18097, 27005}, {18134, 28803}, {18662, 48380}, {18743, 25082}, {19684, 37543}, {24778, 45204}, {25000, 41883}, {26126, 32636}, {28734, 28769}, {28755, 28765}, {28761, 28763}, {37759, 41808}, {41539, 46897}, {42289, 43223}, {45206, 48381}

X(52358) = isotomic conjugate of X(46880)
X(52358) = X(274)-Ceva conjugate of X(1441)
X(52358) = X(21061)-cross conjugate of X(17751)
X(52358) = X(i)-isoconjugate of X(j) for these (i,j): {9, 52150}, {31, 46880}, {41, 20028}, {284, 34434}, {2051, 2194}, {2150, 51870}, {2269, 40453}
X(52358) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 46880}, {12, 37}, {478, 52150}, {650, 34589}, {1193, 2269}, {1214, 2051}, {3057, 21796}, {3160, 20028}, {34434, 40590}
X(52358) = cevapoint of X(21061) and X(37558)
X(52358) = barycentric product X(i)*X(j) for these {i,j}: {7, 17751}, {75, 37558}, {85, 21061}, {226, 14829}, {307, 11109}, {314, 20617}, {321, 17074}, {349, 572}, {668, 51664}, {1441, 2975}, {4552, 17496}, {6063, 52139}
X(52358) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 46880}, {7, 20028}, {12, 51870}, {56, 52150}, {65, 34434}, {226, 2051}, {572, 284}, {961, 40453}, {2975, 21}, {11109, 29}, {14829, 333}, {14973, 210}, {17074, 81}, {17496, 4560}, {17751, 8}, {20617, 65}, {20986, 2194}, {21061, 9}, {21173, 3737}, {22118, 2193}, {23187, 23189}, {24237, 17197}, {27346, 27527}, {37558, 1}, {46879, 46889}, {51664, 513}, {52087, 2269}, {52139, 55}
X(52358) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 27339, 17077}, {2, 28739, 28776}, {321, 1214, 4552}, {1427, 31993, 1441}

X(52359) = X(19)-DAO CONJUGATE OF X(27)

X(52359) lies on these lines: {1, 197}, {3, 44662}, {4, 44670}, {10, 4523}, {12, 431}, {37, 2333}, {40, 3827}, {42, 65}, {46, 18732}, {55, 1829}, {72, 3704}, {78, 41600}, {210, 20653}, {517, 3811}, {692, 40660}, {774, 2183}, {910, 1973}, {976, 3057}, {986, 24476}, {1040, 22654}, {1062, 9798}, {1071, 8679}, {1376, 37613}, {1486, 7713}, {1763, 3556}, {1828, 6284}, {1834, 40965}, {1871, 7680}, {1872, 18242}, {1878, 12953}, {1900, 10895}, {1905, 3185}, {2809, 24391}, {2835, 5493}, {2933, 46974}, {3085, 40635}, {3436, 20243}, {3755, 35650}, {3812, 28600}, {3949, 4515}, {4329, 20914}, {5250, 41581}, {8758, 13738}, {9627, 11363}, {9645, 26309}, {9895, 10198}, {9943, 24683}, {10831, 11383}, {10833, 26378}, {11491, 41722}, {12711, 14557}, {15624, 37528}, {17102, 23361}, {20691, 21853}, {21148, 36103}, {30265, 45188}, {37566, 52218}, {37593, 42443}, {41538, 51377}

X(52359) = reflection of X(1872) in X(18242)
X(52359) = X(i)-Ceva conjugate of X(j) for these (i,j): {306, 37}, {4329, 21062}
X(52359) = X(i)-isoconjugate of X(j) for these (i,j): {58, 7219}, {81, 7097}, {86, 7169}, {270, 47344}, {1333, 40015}, {1444, 40169}
X(52359) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 7219}, {19, 27}, {37, 40015}, {7097, 40586}, {7169, 40600}
X(52359) = crosspoint of X(1763) and X(4329)
X(52359) = crosssum of X(i) and X(j) for these (i,j): {1, 37034}, {7097, 7169}
X(52359) = barycentric product X(i)*X(j) for these {i,j}: {1, 21062}, {10, 1763}, {37, 4329}, {42, 20914}, {65, 27540}, {72, 17903}, {306, 36103}, {321, 3556}, {1018, 21174}, {20336, 21148}, {22119, 41013}
X(52359) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 40015}, {37, 7219}, {42, 7097}, {213, 7169}, {1763, 86}, {2197, 47344}, {2333, 40169}, {3556, 81}, {4329, 274}, {17903, 286}, {20914, 310}, {21062, 75}, {21148, 28}, {21174, 7199}, {22119, 1444}, {27540, 314}, {36103, 27}
X(52359) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9627, 20989, 11363}, {12711, 14557, 42450}, {20691, 21861, 21853}, {20691, 21862, 21861}

X(52360) = X(21)-DAO CONJUGATE OF X(1)

X(52360) lies on these lines: {1, 42005}, {2, 13583}, {8, 13746}, {10, 643}, {11, 21}, {29, 270}, {35, 52244}, {86, 16749}, {99, 17095}, {100, 37158}, {110, 5086}, {229, 2475}, {349, 811}, {409, 2886}, {447, 32939}, {759, 24387}, {950, 2185}, {1010, 1125}, {1043, 3615}, {1212, 27415}, {1220, 36934}, {1479, 17512}, {1621, 37152}, {1631, 37405}, {2905, 11683}, {2975, 7424}, {3109, 24390}, {3152, 28755}, {3582, 4234}, {4861, 6740}, {5263, 17872}, {6739, 37157}, {7253, 17164}, {11101, 11680}, {13740, 25441}, {15680, 40592}, {23665, 32922}, {24541, 40430}, {25639, 37816}

X(52360) = X(75)-Ceva conjugate of X(333)
X(52360) = X(65)-isoconjugate of X(34435)
X(52360) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 21}, {34435, 40602}
X(52360) = cevapoint of X(1) and X(18865)
X(52360) = barycentric product X(i)*X(j) for these {i,j}: {29, 28754}, {75, 40582}, {229, 312}, {314, 1781}, {333, 2475}, {1043, 18625}
X(52360) = barycentric quotient X(i)/X(j) for these {i,j}: {229, 57}, {284, 34435}, {1781, 65}, {2475, 226}, {18625, 3668}, {28754, 307}, {40582, 1}
X(52360) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1098, 6734, 333}, {6734, 51382, 1098}

X(52361) = X(21)-DAO CONJUGATE OF X(9)

X(52361) lies on these lines: {2, 1029}, {12, 35991}, {21, 12615}, {27, 86}, {81, 1086}, {99, 33116}, {110, 20292}, {226, 662}, {229, 2475}, {320, 7058}, {333, 2160}, {354, 19642}, {379, 17103}, {409, 7354}, {501, 11263}, {553, 37756}, {593, 33129}, {673, 40164}, {757, 40940}, {1043, 17647}, {1098, 4292}, {1326, 33130}, {1478, 11116}, {1621, 5196}, {1626, 11101}, {1738, 6043}, {2363, 23536}, {2651, 11246}, {3752, 24617}, {3822, 52244}, {4297, 40430}, {5253, 37369}, {5267, 11110}, {5745, 6626}, {14534, 16706}, {14616, 17862}, {14829, 20913}, {17056, 38814}, {17190, 25507}, {17483, 37783}, {17605, 25533}, {18625, 40582}, {24378, 41245}, {24624, 27003}, {31019, 40214}, {37791, 40688}, {40605, 41878}

X(52361) = X(85)-Ceva conjugate of X(86)
X(52361) = X(1781)-cross conjugate of X(229)
X(52361) = X(37)-isoconjugate of X(34435)
X(52361) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 21}, {34435, 40589}
X(52361) = cevapoint of X(i) and X(j) for these (i,j): {229, 40582}, {1781, 2475}
X(52361) = barycentric product X(i)*X(j) for these {i,j}: {27, 28754}, {75, 229}, {85, 40582}, {86, 2475}, {274, 1781}, {333, 18625}
X(52361) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 34435}, {229, 1}, {1781, 37}, {2475, 10}, {18625, 226}, {28754, 306}, {40582, 9}, {41495, 41501}
X(52361) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2185, 5249, 86}, {5249, 18653, 2185}

X(52362) = X(21)-DAO CONJUGATE OF X(29)

X(52362) lies on these lines: {1, 21}, {3, 18210}, {8, 14544}, {29, 14206}, {34, 2000}, {40, 2071}, {72, 18447}, {73, 18123}, {77, 224}, {78, 1060}, {201, 1331}, {229, 1781}, {307, 1442}, {950, 37782}, {1038, 4855}, {1047, 2617}, {1062, 4652}, {1790, 18673}, {1870, 6734}, {2327, 18675}, {2475, 18625}, {2888, 6326}, {2893, 18631}, {3160, 17136}, {3218, 33178}, {3419, 32047}, {3434, 4347}, {3622, 27509}, {3916, 18455}, {3984, 25909}, {4228, 51698}, {4296, 5930}, {4351, 17647}, {4640, 9627}, {5175, 18624}, {5263, 21406}, {5314, 37613}, {5693, 11441}, {10884, 44243}, {11396, 37581}, {13739, 18721}, {14212, 39585}, {16049, 44661}, {17768, 38336}, {18446, 44665}, {18593, 35979}, {18719, 21376}, {19861, 26543}, {24883, 41557}, {26131, 41550}, {41571, 49743}

X(52362) = isotomic conjugate of the polar conjugate of X(1781)
X(52362) = X(i)-Ceva conjugate of X(j) for these (i,j): {307, 63}, {1442, 6505}, {3879, 8897}, {34772, 224}
X(52362) = X(4)-isoconjugate of X(34435)
X(52362) = X(i)-Dao conjugate of X(j) for these (i,j): {21, 29}, {34435, 36033}
X(52362) = crosssum of X(663) and X(3708)
X(52362) = crossdifference of every pair of points on line {661, 47235}
X(52362) = barycentric product X(i)*X(j) for these {i,j}: {1, 28754}, {63, 2475}, {69, 1781}, {78, 18625}, {229, 306}, {307, 40582}
X(52362) = barycentric quotient X(i)/X(j) for these {i,j}: {48, 34435}, {229, 27}, {1781, 4}, {2475, 92}, {18625, 273}, {28754, 75}, {40582, 29}

X(52363) = X(23)-DAO CONJUGATE OF X(4)

X(52363) lies on these lines: {3, 18125}, {20, 15582}, {22, 11061}, {23, 6593}, {67, 6636}, {69, 2916}, {110, 3313}, {159, 2892}, {511, 2914}, {542, 7512}, {895, 1176}, {2781, 41590}, {2931, 25406}, {3098, 12281}, {3518, 15462}, {3520, 18553}, {5181, 37929}, {5189, 19596}, {5562, 8718}, {5972, 46026}, {6698, 15246}, {7464, 16163}, {7488, 32233}, {7525, 32306}, {8907, 15581}, {9969, 27866}, {9970, 12088}, {12106, 45034}, {12310, 25320}, {12825, 33851}, {17714, 45016}, {22109, 32305}, {28408, 31099}, {32271, 37925}, {32274, 37126}, {41716, 52098}, {43697, 44260}

X(52363) = midpoint of X(i) and X(j) for these {i,j}: {110, 41464}, {2930, 34437}
X(52363) = reflection of X(46026) in X(5972)
X(52363) = isotomic conjugate of the polar conjugate of X(40583)
X(52363) = X(69)-Ceva conjugate of X(22151)
X(52363) = X(4)-Dao conjugate of X(23)
X(52363) = barycentric product X(i)*X(j) for these {i,j}: {69, 40583}, {316, 22121}, {5189, 22151}, {19596, 37804}
X(52363) = barycentric quotient X(i)/X(j) for these {i,j}: {5189, 46105}, {10317, 34437}, {19596, 8791}, {22121, 67}, {40583, 4}
X(52363) = {X(2916),X(2930)}-harmonic conjugate of X(34437)

X(52364) = X(28)-DAO CONJUGATE OF X(28)

X(52364) lies on these lines: {2, 3}, {8, 2893}, {9, 26064}, {10, 3101}, {72, 2895}, {153, 2828}, {226, 4296}, {329, 1330}, {347, 388}, {950, 3100}, {975, 9612}, {1228, 30737}, {1478, 25080}, {1654, 26939}, {1655, 18666}, {1762, 21671}, {1891, 18589}, {1901, 2303}, {2550, 41921}, {2838, 5300}, {2894, 20242}, {3419, 20243}, {3436, 32862}, {3487, 37635}, {3488, 9538}, {3616, 51698}, {3868, 41004}, {5175, 41915}, {5225, 17070}, {5279, 8804}, {5657, 9537}, {5691, 30265}, {6284, 41230}, {6360, 20060}, {7270, 20336}, {9263, 16100}, {9961, 12779}, {10477, 12220}, {13397, 43659}, {17778, 20009}, {17794, 18659}, {18397, 41329}, {18667, 20222}, {32849, 41507}, {39352, 39360}

X(52364) = reflection of X(i) in X(j) for these {i,j}: {2, 31154}, {20, 30267}, {28, 21530}, {9263, 16100}, {31293, 28}, {31294, 44292}, {44294, 18641}
X(52364) = complement of X(31293)
X(52364) = anticomplement of X(28)
X(52364) = orthocentroidal-circle-inverse of X(7557)
X(52364) = deLongchamps-circle-inverse of X(1325)
X(52364) = anticomplement of the isogonal conjugate of X(72)
X(52364) = anticomplement of the isotomic conjugate of X(20336)
X(52364) = isotomic conjugate of the polar conjugate of X(18685)
X(52364) = anticomplementary isogonal conjugate of X(3868)
X(52364) = X(28)-Dao conjugate of X(28)
X(52364) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 3868}, {2, 17220}, {3, 1}, {6, 3187}, {9, 92}, {10, 4}, {37, 5905}, {40, 1895}, {42, 193}, {48, 17147}, {59, 14544}, {63, 75}, {65, 12649}, {69, 17135}, {71, 2}, {72, 8}, {73, 145}, {75, 20242}, {77, 3873}, {78, 3869}, {97, 17221}, {100, 7253}, {101, 525}, {184, 17148}, {190, 850}, {201, 2475}, {210, 5942}, {213, 21216}, {219, 63}, {222, 3875}, {228, 192}, {255, 20222}, {268, 20223}, {271, 20220}, {283, 2975}, {295, 740}, {304, 17137}, {305, 17138}, {306, 69}, {307, 3434}, {313, 11442}, {321, 21270}, {326, 20243}, {332, 35614}, {345, 20245}, {348, 20244}, {394, 17134}, {525, 150}, {647, 4440}, {653, 23683}, {656, 149}, {668, 21300}, {692, 17498}, {810, 9263}, {895, 17162}, {906, 4560}, {947, 3562}, {1018, 4391}, {1020, 17896}, {1042, 11851}, {1073, 18655}, {1176, 17150}, {1214, 7}, {1220, 41723}, {1231, 21285}, {1252, 14543}, {1260, 45738}, {1293, 44409}, {1331, 523}, {1332, 7192}, {1334, 30694}, {1400, 30699}, {1409, 3210}, {1410, 17480}, {1439, 36845}, {1444, 17140}, {1459, 17154}, {1751, 2997}, {1790, 4360}, {1791, 17164}, {1794, 21}, {1796, 86}, {1797, 17160}, {1799, 17142}, {1807, 758}, {1812, 21273}, {1813, 4467}, {1826, 6515}, {1897, 520}, {2193, 18662}, {2197, 17778}, {2200, 194}, {2250, 48380}, {2259, 40571}, {2318, 144}, {2332, 46713}, {2333, 6392}, {2359, 81}, {2983, 27}, {3049, 21224}, {3504, 17157}, {3682, 20}, {3690, 1654}, {3692, 18750}, {3694, 329}, {3695, 1330}, {3710, 3436}, {3730, 17911}, {3926, 18659}, {3949, 2895}, {3952, 20293}, {3958, 41821}, {3990, 6360}, {3998, 4329}, {4019, 30660}, {4047, 41915}, {4055, 3164}, {4064, 3448}, {4456, 41361}, {4551, 521}, {4552, 46400}, {4557, 25259}, {4558, 17161}, {4561, 512}, {4563, 17159}, {4570, 110}, {4574, 514}, {4580, 25049}, {4592, 17166}, {6391, 17156}, {6539, 20289}, {7013, 20221}, {7015, 38}, {7019, 17153}, {7066, 3152}, {7100, 3874}, {7123, 19}, {7177, 17158}, {7591, 7057}, {8611, 37781}, {8750, 33294}, {8804, 14361}, {14208, 21293}, {14376, 18656}, {14919, 18661}, {15377, 20090}, {17206, 17143}, {17976, 13174}, {18082, 3060}, {20336, 6327}, {21017, 34163}, {21035, 8878}, {22076, 5484}, {22356, 30579}, {22370, 17149}, {22381, 330}, {23067, 522}, {23086, 17155}, {23604, 43740}, {26942, 2893}, {29014, 17925}, {32656, 31296}, {34055, 17141}, {34897, 18657}, {36057, 32922}, {40071, 315}, {40152, 347}, {40435, 286}, {40442, 34195}, {41013, 5906}, {41014, 2891}, {41087, 9965}, {41267, 10340}, {43072, 2292}, {44189, 20246}, {44717, 17136}, {47487, 1621}, {51366, 152}, {51574, 52025}, {52185, 3193}
X(52364) = X(i)-Ceva conjugate of X(j) for these (i,j): {7270, 8}, {20336, 2}, {21287, 2895}
X(52364) = X(18598)-cross conjugate of X(18632)
X(52364) = crosspoint of X(668) and X(23582)
X(52364) = crosssum of X(667) and X(3269)
X(52364) = barycentric product X(i)*X(j) for these {i,j}: {8, 18632}, {69, 18685}, {75, 18598}
X(52364) = barycentric quotient X(i)/X(j) for these {i,j}: {18598, 1}, {18632, 7}, {18685, 4}
X(52364) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 7557}, {2, 20, 7520}, {2, 3146, 4198}, {2, 15680, 17521}, {2, 31293, 28}, {3, 5142, 2}, {4, 13442, 37456}, {4, 26052, 5177}, {4, 37098, 2475}, {4, 37179, 2}, {5, 7523, 2}, {23, 31103, 2}, {28, 21530, 2}, {28, 31154, 21530}, {28, 31257, 52259}, {440, 13442, 21}, {442, 47512, 2}, {468, 30773, 2}, {851, 28380, 27652}, {851, 30974, 2}, {857, 24606, 2}, {857, 48890, 29}, {858, 26253, 2}, {1375, 30842, 2}, {2915, 30447, 451}, {3151, 3152, 20}, {5002, 5003, 37456}, {14953, 31046, 2}, {21530, 52259, 31257}, {24568, 25014, 2}, {25795, 25869, 2}, {25909, 25987, 2}, {25949, 26022, 2}, {26054, 26120, 2}, {26553, 26681, 2}, {26604, 26646, 2}, {26784, 26831, 2}, {26992, 27122, 2}, {27053, 27175, 2}, {29463, 46499, 29519}, {29725, 29778, 46497}, {30920, 33313, 2}, {31257, 52259, 2}

X(52365) = X(33)-DAO CONJUGATE OF X(33)

X(52365) lies on these lines: {1, 18652}, {2, 33}, {4, 23661}, {7, 2897}, {8, 20}, {10, 27505}, {22, 100}, {75, 1370}, {92, 10431}, {103, 41906}, {145, 4296}, {152, 329}, {318, 6836}, {347, 3875}, {377, 5174}, {475, 1062}, {497, 17862}, {516, 20223}, {858, 11680}, {944, 37404}, {956, 21312}, {1012, 11406}, {1043, 7169}, {1125, 9643}, {1295, 43347}, {1479, 20320}, {1610, 1619}, {1735, 18391}, {1824, 26118}, {1862, 37366}, {2968, 7580}, {2975, 11413}, {3151, 27484}, {3152, 3210}, {3164, 17759}, {3421, 35513}, {3436, 37201}, {3616, 9538}, {3869, 6225}, {3873, 6604}, {4441, 30737}, {4463, 42699}, {4847, 30265}, {5080, 44440}, {5081, 6925}, {5175, 41915}, {5256, 5807}, {5263, 24543}, {5687, 11414}, {6350, 7411}, {6360, 52164}, {6527, 17134}, {6827, 38462}, {6851, 41013}, {7071, 25907}, {7493, 33116}, {8144, 34120}, {9960, 12324}, {10394, 11433}, {11220, 26871}, {12220, 25304}, {14544, 18623}, {14956, 31623}, {17756, 22240}, {17776, 36018}, {20221, 33673}, {24611, 50698}, {25722, 33075}, {26872, 41228}, {34036, 45281}, {36496, 52082}, {36850, 44661}, {43363, 46964}

X(52365) = reflection of X(i) in X(j) for these {i,j}: {20, 36984}, {33, 34822}, {36985, 10}
X(52365) = anticomplement of X(33)
X(52365) = anticomplement of the isogonal conjugate of X(77)
X(52365) = anticomplement of the isotomic conjugate of X(7182)
X(52365) = isotomic conjugate of the isogonal conjugate of X(18621)
X(52365) = anticomplementary isogonal conjugate of X(5942)
X(52365) = X(33)-Dao conjugate of X(33)
X(52365) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 5942}, {3, 144}, {6, 30694}, {7, 4}, {48, 3177}, {56, 193}, {57, 5905}, {59, 3732}, {63, 329}, {69, 3436}, {73, 1654}, {77, 8}, {81, 92}, {85, 21270}, {109, 25259}, {184, 21218}, {219, 30695}, {222, 2}, {269, 12649}, {273, 5906}, {278, 6515}, {283, 45738}, {304, 21286}, {307, 1330}, {331, 317}, {348, 69}, {603, 192}, {604, 21216}, {608, 6392}, {651, 4391}, {658, 46400}, {664, 20293}, {905, 37781}, {934, 521}, {1014, 3868}, {1214, 2895}, {1231, 21287}, {1262, 651}, {1331, 4468}, {1332, 4462}, {1401, 8878}, {1407, 30699}, {1409, 1655}, {1412, 3187}, {1414, 7253}, {1434, 17220}, {1439, 2475}, {1444, 3869}, {1459, 39351}, {1790, 63}, {1791, 43216}, {1796, 17781}, {1797, 908}, {1798, 11683}, {1803, 9}, {1804, 20}, {1811, 41772}, {1812, 18750}, {1813, 514}, {1814, 30807}, {2189, 46713}, {2197, 46707}, {3937, 17036}, {4025, 33650}, {4131, 34188}, {4565, 525}, {4573, 850}, {4617, 17896}, {4625, 21300}, {6063, 11442}, {6516, 513}, {6517, 20294}, {7011, 20211}, {7013, 6223}, {7023, 11851}, {7053, 145}, {7055, 1370}, {7056, 3434}, {7099, 3210}, {7125, 6360}, {7177, 7}, {7182, 6327}, {7183, 4329}, {7316, 47286}, {7335, 3164}, {7339, 1897}, {8810, 6820}, {13388, 13387}, {13389, 13386}, {17094, 3448}, {17206, 20245}, {18623, 14361}, {22341, 18666}, {23067, 31290}, {23086, 20535}, {27832, 21296}, {30493, 17035}, {30682, 6604}, {32660, 21225}, {34051, 48380}, {34055, 20248}, {34400, 21279}, {36057, 10025}, {36059, 17494}, {40152, 3151}, {40442, 4416}, {40443, 3681}, {41081, 189}, {43034, 40867}, {44717, 190}, {47487, 30625}, {51640, 21221}
X(52365) = X(7182)-Ceva conjugate of X(2)
X(52365) = barycentric product X(i)*X(j) for these {i,j}: {76, 18621}, {305, 21058}
X(52365) = barycentric quotient X(i)/X(j) for these {i,j}: {18621, 6}, {21058, 25}
X(52365) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 10430, 189}, {33, 34822, 2}, {1040, 1861, 2}, {1370, 20243, 4329}

X(52366) = X(34)-DAO CONJUGATE OF X(34)

X(52366) lies on these lines: {1, 27505}, {2, 34}, {8, 20}, {21, 6350}, {22, 2975}, {58, 1771}, {69, 3827}, {92, 377}, {100, 11413}, {145, 3100}, {278, 24984}, {281, 24537}, {318, 6925}, {329, 1330}, {333, 16049}, {341, 1370}, {347, 4357}, {406, 1060}, {858, 11681}, {938, 27402}, {956, 11414}, {1220, 24543}, {1222, 35998}, {1305, 2365}, {1396, 24568}, {1398, 25947}, {1654, 3152}, {1783, 23115}, {1936, 7538}, {2840, 21290}, {2897, 32099}, {2968, 37022}, {3164, 21226}, {3194, 27405}, {3241, 9538}, {3244, 9643}, {3434, 20220}, {3486, 36740}, {3868, 26871}, {4292, 20223}, {5080, 37444}, {5081, 6836}, {5082, 35513}, {5130, 26118}, {5174, 10431}, {5176, 42020}, {5303, 38444}, {5342, 6835}, {5484, 6360}, {5657, 37404}, {5687, 21312}, {5703, 27535}, {5744, 7520}, {5758, 5906}, {6737, 30265}, {6850, 41013}, {6916, 23661}, {7102, 15971}, {7718, 28029}, {11415, 33650}, {17080, 25876}, {27378, 34231}, {27379, 46974}

X(52366) = reflection of X(i) in X(j) for these {i,j}: {20, 36986}, {34, 34823}
X(52366) = anticomplement of X(34)
X(52366) = anticomplement of the isogonal conjugate of X(78)
X(52366) = anticomplement of the isotomic conjugate of X(3718)
X(52366) = anticomplementary isogonal conjugate of X(12649)
X(52366) = X(34)-Dao conjugate of X(34)
X(52366) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 12649}, {3, 145}, {6, 30699}, {8, 4}, {9, 5905}, {21, 3868}, {41, 21216}, {48, 3210}, {55, 193}, {56, 11851}, {59, 1897}, {63, 7}, {69, 3434}, {71, 17778}, {72, 2475}, {77, 36845}, {78, 8}, {100, 521}, {190, 46400}, {200, 5942}, {212, 192}, {219, 2}, {220, 30694}, {222, 4452}, {268, 9965}, {271, 962}, {281, 6515}, {283, 1}, {284, 3187}, {304, 21285}, {305, 21280}, {306, 2893}, {312, 21270}, {314, 20242}, {318, 5906}, {332, 17135}, {333, 17220}, {345, 69}, {348, 6604}, {394, 347}, {521, 149}, {603, 17480}, {607, 6392}, {643, 7253}, {644, 4391}, {645, 850}, {648, 23683}, {651, 17896}, {652, 4440}, {895, 4442}, {906, 17496}, {1069, 10529}, {1176, 3891}, {1252, 651}, {1257, 5174}, {1259, 20}, {1260, 144}, {1264, 1370}, {1265, 3436}, {1331, 522}, {1332, 693}, {1444, 3873}, {1789, 3874}, {1790, 3875}, {1791, 65}, {1792, 3869}, {1793, 758}, {1794, 34772}, {1795, 38460}, {1796, 3879}, {1797, 1266}, {1802, 3177}, {1808, 740}, {1809, 517}, {1810, 5853}, {1811, 3880}, {1812, 75}, {1813, 4025}, {1815, 9436}, {1818, 52164}, {1946, 9263}, {2193, 17147}, {2287, 92}, {2289, 6360}, {2318, 1654}, {2327, 63}, {2338, 48381}, {2359, 1999}, {3270, 17036}, {3596, 11442}, {3682, 3152}, {3688, 8878}, {3692, 329}, {3694, 2895}, {3699, 20293}, {3710, 1330}, {3718, 6327}, {3719, 4329}, {3939, 25259}, {3990, 18667}, {3998, 2897}, {4102, 20289}, {4558, 4467}, {4561, 21302}, {4563, 4374}, {4564, 4566}, {4570, 14544}, {4571, 513}, {4587, 514}, {5546, 525}, {5547, 47286}, {5548, 10015}, {6056, 3164}, {6065, 3732}, {6099, 44428}, {6332, 150}, {6514, 17134}, {6516, 3900}, {7015, 29840}, {7017, 317}, {7257, 21300}, {8611, 21221}, {8805, 6820}, {14919, 41804}, {15627, 3580}, {15628, 51481}, {17206, 20244}, {22128, 41803}, {22356, 30577}, {22370, 20537}, {23189, 17154}, {27382, 14361}, {34055, 20247}, {34259, 388}, {35518, 21293}, {36797, 520}, {39167, 20076}, {39947, 39695}, {40399, 273}, {40442, 41575}, {40443, 30628}, {44189, 21279}, {44707, 17035}, {44717, 664}, {44722, 42020}, {45393, 912}, {46102, 18026}, {47487, 3870}, {51379, 153}
X(52366) = X(3718)-Ceva conjugate of X(2)
X(52366) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {34, 34823, 2}, {1038, 46878, 2}, {7270, 18750, 3436}, {46421, 46422, 3101}

X(52367) = X(3)-DAO CONJUGATE OF X(34441)

X(52367) lies on these lines: {1, 149}, {2, 35}, {3, 11680}, {4, 8}, {5, 100}, {7, 7702}, {10, 3583}, {11, 404}, {12, 528}, {20, 5450}, {21, 2886}, {30, 2975}, {31, 24883}, {32, 17737}, {36, 24387}, {38, 24851}, {40, 6840}, {55, 2476}, {56, 11235}, {58, 33142}, {60, 19642}, {63, 2894}, {69, 9047}, {76, 20553}, {78, 1699}, {79, 3874}, {83, 13576}, {104, 10943}, {145, 1478}, {146, 2779}, {148, 21226}, {150, 20244}, {153, 5881}, {214, 37735}, {225, 4318}, {312, 5300}, {315, 4441}, {316, 17143}, {377, 497}, {379, 28757}, {381, 5687}, {382, 956}, {386, 33107}, {388, 3241}, {390, 5177}, {405, 9668}, {411, 5842}, {442, 1621}, {443, 5550}, {474, 9669}, {496, 5253}, {498, 5141}, {499, 4188}, {515, 4861}, {516, 6734}, {519, 3585}, {535, 5288}, {546, 17757}, {550, 5303}, {631, 33862}, {662, 3615}, {673, 33839}, {908, 4420}, {938, 13750}, {942, 20292}, {944, 6923}, {946, 4511}, {958, 11114}, {964, 32773}, {976, 3944}, {986, 33094}, {993, 15680}, {999, 50239}, {1001, 4197}, {1054, 28096}, {1056, 20057}, {1058, 38314}, {1089, 33091}, {1125, 4857}, {1145, 18357}, {1154, 48877}, {1193, 33106}, {1259, 10883}, {1278, 7900}, {1320, 10944}, {1329, 34612}, {1330, 2891}, {1352, 25304}, {1376, 4193}, {1385, 6951}, {1468, 33141}, {1484, 37535}, {1698, 37162}, {1714, 17127}, {1724, 11330}, {1770, 3218}, {1836, 3868}, {1839, 5279}, {2077, 6972}, {2292, 33095}, {2320, 4305}, {2478, 2550}, {2548, 17756}, {2599, 4552}, {2802, 37710}, {2893, 17220}, {2901, 33093}, {2932, 45976}, {3006, 7283}, {3035, 7173}, {3058, 6175}, {3085, 6871}, {3086, 4190}, {3091, 5552}, {3153, 14213}, {3244, 5270}, {3295, 17532}, {3454, 33175}, {3476, 18961}, {3496, 21029}, {3501, 26074}, {3521, 38955}, {3576, 37163}, {3579, 6903}, {3586, 19860}, {3600, 11240}, {3614, 6154}, {3632, 18513}, {3670, 33102}, {3678, 26792}, {3679, 18514}, {3685, 7557}, {3701, 32850}, {3702, 7270}, {3705, 37456}, {3746, 3822}, {3754, 37702}, {3811, 31053}, {3813, 7354}, {3816, 17531}, {3817, 27385}, {3826, 17536}, {3829, 5433}, {3831, 32948}, {3832, 7080}, {3838, 37080}, {3854, 27525}, {3870, 9612}, {3872, 5691}, {3876, 24703}, {3877, 5794}, {3878, 47033}, {3885, 5252}, {3889, 10404}, {3895, 9578}, {3913, 10895}, {3914, 5262}, {3916, 28146}, {3923, 36568}, {3925, 5047}, {3935, 21077}, {3957, 13407}, {4000, 26099}, {4189, 4302}, {4200, 11393}, {4202, 32942}, {4209, 28734}, {4214, 12410}, {4292, 26015}, {4293, 10529}, {4295, 12649}, {4299, 45700}, {4304, 24541}, {4309, 10198}, {4324, 5267}, {4347, 37798}, {4355, 31146}, {4366, 33841}, {4413, 9671}, {4429, 5192}, {4514, 4968}, {4642, 37717}, {4645, 24523}, {4652, 5231}, {4847, 51118}, {4855, 8227}, {4872, 20880}, {4894, 33090}, {4972, 13740}, {4995, 6668}, {4996, 5840}, {4999, 15338}, {5010, 37291}, {5051, 5263}, {5084, 19877}, {5134, 16552}, {5154, 26364}, {5218, 6933}, {5223, 7997}, {5229, 20050}, {5247, 33136}, {5250, 9580}, {5255, 21935}, {5260, 11113}, {5261, 11239}, {5266, 33133}, {5274, 6904}, {5283, 9664}, {5284, 8728}, {5291, 7747}, {5292, 17126}, {5432, 7504}, {5434, 15679}, {5439, 18527}, {5440, 6900}, {5540, 26793}, {5556, 6601}, {5587, 13729}, {5603, 6917}, {5657, 6928}, {5705, 35258}, {5731, 6850}, {5734, 10532}, {5744, 6851}, {5748, 6849}, {5787, 9961}, {5805, 25722}, {5818, 6929}, {5880, 7671}, {5884, 49176}, {5886, 6901}, {5901, 10609}, {5904, 17484}, {5906, 48878}, {6246, 39776}, {6253, 36002}, {6327, 10449}, {6653, 16044}, {6655, 26801}, {6735, 19925}, {6737, 51423}, {6745, 12571}, {6796, 6960}, {6818, 26038}, {6821, 26103}, {6825, 37000}, {6826, 7704}, {6828, 11496}, {6830, 11248}, {6835, 9779}, {6836, 9778}, {6842, 11491}, {6845, 45630}, {6853, 32613}, {6872, 19843}, {6888, 45392}, {6902, 26446}, {6909, 11826}, {6911, 11928}, {6915, 7681}, {6921, 10589}, {6932, 11500}, {6937, 10267}, {6941, 11499}, {6943, 10310}, {6944, 10598}, {6945, 10893}, {6948, 10785}, {6952, 13199}, {6965, 9956}, {6980, 32141}, {6982, 10786}, {6985, 18499}, {6990, 11517}, {6999, 28797}, {7191, 23537}, {7280, 36004}, {7319, 30513}, {7377, 28813}, {7382, 34255}, {7394, 10327}, {7548, 7680}, {7613, 28080}, {7743, 17614}, {7745, 21956}, {7748, 16975}, {7785, 17759}, {7812, 37857}, {7951, 8715}, {7967, 33281}, {8256, 12764}, {8257, 9581}, {8666, 10483}, {9342, 17527}, {9397, 48304}, {9579, 24392}, {9613, 36846}, {9614, 19861}, {9655, 34605}, {9709, 17556}, {9809, 12528}, {9963, 15950}, {10129, 11374}, {10200, 17572}, {10266, 24298}, {10479, 33083}, {10528, 10590}, {10580, 16193}, {10584, 17567}, {10588, 34607}, {10593, 13747}, {10624, 24987}, {10742, 12531}, {10782, 18871}, {10826, 25005}, {10942, 38665}, {11109, 23541}, {11238, 25524}, {11322, 19755}, {11682, 31162}, {12047, 34772}, {12247, 25413}, {12513, 12943}, {12526, 50865}, {12532, 31803}, {12559, 41709}, {12572, 25006}, {12682, 12919}, {12690, 37730}, {13278, 26482}, {13589, 23850}, {13727, 23542}, {13743, 48680}, {14956, 51290}, {14986, 37435}, {16062, 24552}, {16086, 25253}, {16160, 48698}, {16370, 31493}, {16478, 33128}, {16547, 21065}, {16865, 19854}, {16910, 27248}, {17522, 31084}, {17533, 47742}, {17541, 26582}, {17550, 20172}, {17616, 50196}, {17647, 30384}, {17671, 24596}, {17680, 27097}, {17686, 26590}, {17751, 21282}, {17753, 21285}, {17889, 28082}, {18393, 22836}, {18398, 26842}, {18802, 38156}, {19767, 26098}, {19855, 31420}, {20076, 34625}, {20084, 37005}, {20085, 37706}, {20247, 33865}, {20293, 23104}, {20533, 28742}, {20919, 41591}, {21279, 31995}, {22791, 37230}, {24026, 34007}, {24042, 51433}, {24443, 24715}, {24850, 33119}, {24916, 25671}, {24936, 33111}, {25439, 37719}, {25525, 41864}, {26064, 31330}, {26105, 37462}, {29662, 37603}, {29817, 51706}, {29958, 38389}, {30171, 33168}, {30172, 32849}, {31164, 41863}, {31737, 38474}, {31777, 37374}, {32558, 35262}, {32949, 35633}, {33086, 50605}, {33089, 50044}, {33105, 37573}, {33771, 37693}, {34195, 39542}, {34773, 47032}, {35614, 48899}, {36175, 47270}, {36944, 38952}, {37371, 41227}, {37579, 37797}, {38460, 45287}, {38903, 52242}, {40273, 51409}, {51886, 51975}

X(52367) = midpoint of X(24468) and X(41869)
X(52367) = reflection of X(i) in X(j) for these {i,j}: {2, 31159}, {8, 5086}, {20, 11012}, {35, 25639}, {145, 11009}, {411, 15908}, {2975, 24390}, {3648, 52126}, {3871, 12}, {4324, 5267}, {6906, 26470}, {11010, 10}, {11015, 2646}, {11491, 6842}, {11849, 5}, {15338, 4999}, {20060, 3585}, {20066, 35}, {20084, 37005}, {34772, 12047}
X(52367) = isogonal conjugate of X(34441)
X(52367) = complement of X(20066)
X(52367) = anticomplement of X(35)
X(52367) = Fuhrmann-circle-inverse of X(5080)
X(52367) = circumcircle-of-anticomplementary-triangle-inverse of X(5180)
X(52367) = anticomplement of the isogonal conjugate of X(79)
X(52367) = anticomplement of the isotomic conjugate of X(20565)
X(52367) = isotomic conjugate of the isogonal conjugate of X(20988)
X(52367) = polar conjugate of the isogonal conjugate of X(22122)
X(52367) = anticomplementary isogonal conjugate of X(3648)
X(52367) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 3648}, {57, 41808}, {79, 8}, {94, 21277}, {1870, 12383}, {1989, 17484}, {2160, 2}, {2166, 5080}, {3218, 1272}, {3615, 3869}, {6186, 192}, {6742, 513}, {6757, 1330}, {7073, 144}, {7100, 20}, {7110, 329}, {7113, 18301}, {8818, 2895}, {13486, 523}, {15455, 20295}, {20565, 6327}, {26700, 522}, {30602, 14450}, {30690, 69}, {32678, 49274}, {35049, 17136}, {38340, 693}, {39152, 616}, {39153, 617}, {43682, 2893}
X(52367) = X(20565)-Ceva conjugate of X(2)
X(52367) = X(21065)-cross conjugate of X(20919)
X(52367) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34441}, {2160, 52062}
X(52367) = X(3)-Dao conjugate of X(34441)
X(52367) = cevapoint of X(20988) and X(22122)
X(52367) = barycentric product X(i)*X(j) for these {i,j}: {1, 20919}, {75, 16547}, {76, 20988}, {86, 21065}, {190, 21179}, {264, 22122}, {30710, 41591}
X(52367) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 34441}, {35, 52062}, {16547, 1}, {20919, 75}, {20988, 6}, {21065, 10}, {21179, 514}, {22122, 3}, {41591, 3666}, {41592, 16698}
X(52367) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 36250, 33155}, {2, 20066, 35}, {4, 8, 5080}, {4, 3434, 8}, {4, 5082, 3436}, {4, 12245, 10526}, {5, 100, 27529}, {8, 5180, 3869}, {8, 9812, 11415}, {10, 3583, 5046}, {35, 25639, 2}, {35, 31159, 25639}, {72, 5178, 8}, {72, 22793, 5057}, {79, 3874, 17483}, {149, 2475, 1}, {321, 5015, 8}, {355, 14923, 8}, {377, 497, 3616}, {381, 5687, 11681}, {442, 15171, 1621}, {496, 11112, 5253}, {958, 12953, 11114}, {958, 34706, 12953}, {962, 5175, 8}, {1376, 10896, 4193}, {1770, 10916, 3218}, {1836, 3868, 14450}, {2478, 2550, 9780}, {2550, 5225, 2478}, {2886, 6284, 21}, {2894, 37433, 63}, {3091, 17784, 5552}, {3419, 3869, 8}, {3419, 12699, 3869}, {3434, 3436, 5082}, {3436, 5082, 8}, {3841, 5259, 2}, {3869, 12699, 5180}, {3871, 17577, 12}, {4294, 31418, 2}, {4302, 26363, 4189}, {4324, 5267, 37299}, {4385, 5014, 8}, {4696, 5100, 8}, {4999, 15338, 17549}, {5046, 33110, 10}, {5057, 5178, 72}, {5081, 23528, 8}, {5176, 10914, 8}, {5253, 10707, 496}, {5295, 33075, 8}, {5794, 12701, 3877}, {6653, 16044, 26752}, {6850, 12116, 5731}, {6871, 20075, 3085}, {6952, 13199, 26285}, {7705, 37828, 9780}, {7741, 25440, 2}, {8666, 10483, 20067}, {10525, 37820, 4}, {10529, 31295, 4293}, {10593, 13747, 31272}, {10914, 18480, 5176}, {10944, 13463, 1320}, {11113, 31419, 5260}, {11681, 49719, 5687}, {12699, 18407, 4}, {12953, 31140, 958}, {13271, 13273, 1320}, {26060, 26127, 2}, {31140, 34706, 11114}, {45287, 49600, 38460}

X(52368) = X(36)-DAO CONJUGATE OF X(1)

X(52368) lies on these lines: {1, 149}, {2, 1718}, {8, 14544}, {21, 34977}, {86, 664}, {100, 33649}, {106, 7191}, {109, 283}, {244, 5262}, {323, 758}, {514, 24322}, {515, 3153}, {517, 43574}, {651, 12532}, {860, 1870}, {1125, 38458}, {1325, 12826}, {1421, 3616}, {3218, 4351}, {3869, 22136}, {3904, 6370}, {3920, 24222}, {4658, 17016}, {4861, 6742}, {4996, 11700}, {5080, 37798}, {5086, 18447}, {7286, 17768}, {9627, 11015}, {14988, 50461}, {17015, 23668}, {17080, 51236}, {17100, 24025}, {18689, 18697}, {20932, 49492}, {23669, 49487}, {33961, 39751}

X(52368) = X(75)-Ceva conjugate of X(3218)
X(52368) = X(i)-isoconjugate of X(j) for these (i,j): {80, 34442}, {759, 10693}
X(52368) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 36}, {10693, 34586}
X(52368) = crosspoint of X(75) and X(20920)
X(52368) = barycentric product X(i)*X(j) for these {i,j}: {36, 20920}, {75, 40584}, {320, 16548}, {1325, 3936}, {3218, 5080}, {4511, 37798}, {4585, 21180}, {20924, 20989}
X(52368) = barycentric quotient X(i)/X(j) for these {i,j}: {1325, 24624}, {2245, 10693}, {5080, 18359}, {7113, 34442}, {16548, 80}, {20920, 20566}, {20989, 2161}, {21066, 15065}, {22123, 1807}, {37798, 18815}, {40584, 1}
X(52368) = {X(11700),X(16586)}-harmonic conjugate of X(4996)

X(52369) = X(37)-DAO CONJUGATE OF X(28)

X(52369) lies on these lines: {10, 321}, {19, 46738}, {63, 3718}, {92, 341}, {190, 21376}, {226, 42715}, {306, 3610}, {312, 3305}, {329, 2835}, {612, 1215}, {668, 20929}, {1265, 26872}, {1930, 32782}, {2064, 14206}, {2321, 42707}, {2525, 14208}, {3263, 5249}, {3596, 14213}, {3678, 5739}, {3694, 40161}, {3695, 21015}, {3967, 43214}, {3969, 42710}, {4037, 21813}, {4357, 19835}, {4358, 40940}, {4362, 20663}, {4574, 23112}, {4710, 17874}, {4975, 49683}, {6358, 28654}, {7237, 21688}, {19684, 30142}, {19785, 33937}, {19811, 20879}, {20896, 40603}, {42698, 51367}

X(52369) = isotomic conjugate of the isogonal conjugate of X(3949)
X(52369) = isotomic conjugate of the polar conjugate of X(1089)
X(52369) = X(i)-Ceva conjugate of X(j) for these (i,j): {20336, 3695}, {30713, 28654}
X(52369) = X(i)-cross conjugate of X(j) for these (i,j): {125, 14208}, {3708, 4064}, {3949, 1089}, {21678, 10}
X(52369) = X(i)-isoconjugate of X(j) for these (i,j): {3, 36420}, {19, 849}, {25, 593}, {27, 2206}, {28, 1333}, {29, 16947}, {34, 2150}, {56, 2189}, {58, 1474}, {60, 608}, {81, 2203}, {110, 43925}, {112, 3733}, {184, 36419}, {249, 42067}, {250, 1015}, {270, 604}, {281, 7342}, {607, 7341}, {757, 1973}, {1014, 2204}, {1019, 32676}, {1106, 2326}, {1172, 1408}, {1175, 46890}, {1395, 2185}, {1396, 2194}, {1397, 46103}, {1398, 7054}, {1412, 2299}, {1437, 5317}, {1509, 1974}, {1565, 41937}, {1576, 17925}, {1977, 18020}, {2969, 23357}, {2973, 23963}, {3937, 23964}, {7215, 23975}, {7254, 32713}, {8034, 47443}, {22096, 23582}
X(52369) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 2189}, {6, 849}, {10, 1474}, {19, 4075}, {27, 40603}, {28, 37}, {58, 51574}, {226, 1412}, {244, 647}, {244, 43925}, {270, 3161}, {525, 3942}, {593, 6505}, {594, 4222}, {757, 6337}, {1019, 15526}, {1111, 23285}, {1214, 1396}, {1333, 40591}, {1973, 40607}, {2150, 11517}, {2203, 40586}, {2299, 40599}, {2326, 6552}, {3733, 34591}, {4858, 17925}, {36103, 36420}, {40937, 46883}
X(52369) = cevapoint of X(3708) and X(4064)
X(52369) = crosspoint of X(20336) and X(40071)
X(52369) = barycentric product X(i)*X(j) for these {i,j}: {10, 20336}, {12, 3718}, {37, 40071}, {63, 28654}, {69, 1089}, {71, 27801}, {72, 313}, {75, 3695}, {76, 3949}, {78, 34388}, {125, 7035}, {201, 3596}, {304, 594}, {305, 756}, {306, 321}, {307, 3701}, {312, 26942}, {326, 7141}, {339, 765}, {341, 6356}, {345, 6358}, {349, 3694}, {525, 4033}, {561, 3690}, {656, 27808}, {668, 4064}, {872, 40050}, {1016, 20902}, {1018, 3267}, {1214, 30713}, {1231, 2321}, {1441, 3710}, {1500, 40364}, {2197, 28659}, {3708, 31625}, {3952, 14208}, {4036, 4561}, {4103, 15413}, {4574, 20948}, {4601, 21046}, {4605, 15416}, {6057, 7182}, {15742, 17879}, {20618, 30693}
X(52369) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 849}, {8, 270}, {9, 2189}, {10, 28}, {12, 34}, {19, 36420}, {37, 1474}, {42, 2203}, {63, 593}, {69, 757}, {71, 1333}, {72, 58}, {73, 1408}, {77, 7341}, {78, 60}, {92, 36419}, {125, 244}, {181, 1395}, {201, 56}, {210, 2299}, {219, 2150}, {226, 1396}, {228, 2206}, {304, 1509}, {305, 873}, {306, 81}, {307, 1014}, {312, 46103}, {313, 286}, {321, 27}, {339, 1111}, {345, 2185}, {346, 2326}, {442, 46883}, {525, 1019}, {594, 19}, {603, 7342}, {656, 3733}, {661, 43925}, {756, 25}, {762, 2333}, {765, 250}, {872, 1974}, {1018, 112}, {1089, 4}, {1109, 2969}, {1214, 1412}, {1231, 1434}, {1254, 1398}, {1265, 1098}, {1332, 4556}, {1334, 2204}, {1409, 16947}, {1425, 1106}, {1500, 1973}, {1577, 17925}, {1826, 5317}, {2171, 608}, {2197, 604}, {2294, 46890}, {2318, 2194}, {2321, 1172}, {2632, 3937}, {2643, 42067}, {3267, 7199}, {3610, 2303}, {3682, 1437}, {3690, 31}, {3692, 7054}, {3694, 284}, {3695, 1}, {3701, 29}, {3708, 1015}, {3710, 21}, {3718, 261}, {3914, 4211}, {3948, 31905}, {3949, 6}, {3952, 162}, {3977, 30576}, {3992, 37168}, {3998, 1790}, {4001, 30581}, {4013, 36125}, {4024, 6591}, {4033, 648}, {4036, 7649}, {4037, 2201}, {4044, 31926}, {4064, 513}, {4066, 31902}, {4075, 4222}, {4082, 4183}, {4103, 1783}, {4158, 255}, {4424, 4247}, {4466, 16726}, {4515, 2332}, {4557, 32676}, {4571, 4636}, {4574, 163}, {4580, 39179}, {4605, 32714}, {4647, 31900}, {6057, 33}, {6354, 1435}, {6356, 269}, {6358, 278}, {6535, 1824}, {7019, 7303}, {7035, 18020}, {7064, 2212}, {7066, 603}, {7068, 7004}, {7101, 36421}, {7140, 1096}, {7141, 158}, {7182, 552}, {7206, 6198}, {8013, 2355}, {8611, 7252}, {14208, 7192}, {15526, 3942}, {15742, 24000}, {17094, 7203}, {17206, 763}, {17879, 1565}, {20234, 31917}, {20336, 86}, {20618, 738}, {20653, 1829}, {20902, 1086}, {20975, 3248}, {21015, 614}, {21021, 7119}, {21046, 3125}, {21074, 41364}, {21075, 3194}, {21081, 2906}, {21134, 764}, {21671, 1104}, {21674, 40985}, {21675, 1841}, {21678, 40941}, {21810, 2354}, {21859, 32674}, {23994, 2973}, {24018, 7254}, {24020, 7215}, {24459, 50456}, {26942, 57}, {27569, 2905}, {27801, 44129}, {27808, 811}, {28654, 92}, {30591, 46542}, {30713, 31623}, {31625, 46254}, {34388, 273}, {35309, 35325}, {37755, 1407}, {40071, 274}, {40521, 8750}, {41013, 8747}, {41508, 46886}, {42031, 31901}, {42698, 17167}
X(52369) = {X(3718),X(19799)}-harmonic conjugate of X(63)

X(52370) = X(7)-ISOCONJUGATE OF X(27)

X(52370) lies on these lines: {3, 20752}, {6, 37080}, {10, 21930}, {33, 210}, {37, 41538}, {41, 1253}, {42, 213}, {71, 73}, {72, 4574}, {78, 219}, {212, 1802}, {227, 4559}, {228, 2200}, {283, 1808}, {644, 3701}, {1402, 2198}, {1880, 21871}, {1946, 20781}, {2333, 51377}, {3230, 3924}, {3714, 4513}, {3916, 22164}, {4020, 22344}, {4652, 22163}, {4855, 22127}, {5440, 22126}, {7117, 22072}, {8012, 40957}, {9247, 23202}, {21809, 40977}, {22070, 22350}

X(52370) = isogonal conjugate of the isotomic conjugate of X(3694)
X(52370) = isogonal conjugate of the polar conjugate of X(210)
X(52370) = X(i)-Ceva conjugate of X(j) for these (i,j): {71, 228}, {219, 2318}, {220, 1334}, {4587, 1946}
X(52370) = X(i)-isoconjugate of X(j) for these (i,j): {4, 1434}, {7, 27}, {21, 1847}, {28, 85}, {29, 279}, {34, 274}, {56, 44129}, {57, 286}, {58, 331}, {75, 1396}, {81, 273}, {86, 278}, {92, 1014}, {108, 7199}, {162, 24002}, {225, 1509}, {264, 1412}, {269, 31623}, {270, 1446}, {307, 36419}, {310, 608}, {314, 1435}, {333, 1119}, {348, 8747}, {479, 2322}, {552, 1826}, {648, 3676}, {653, 7192}, {664, 17925}, {757, 40149}, {799, 43923}, {811, 3669}, {873, 1880}, {1019, 18026}, {1041, 16750}, {1088, 1172}, {1118, 17206}, {1395, 6385}, {1398, 28660}, {1407, 44130}, {1408, 1969}, {1414, 17924}, {1440, 41083}, {1474, 6063}, {1896, 7177}, {1897, 17096}, {2203, 20567}, {2969, 4620}, {3064, 4616}, {3668, 46103}, {3733, 46404}, {3737, 13149}, {4183, 23062}, {4248, 27818}, {4560, 36118}, {4565, 46107}, {4572, 43925}, {4573, 7649}, {4625, 6591}, {4626, 17926}, {4635, 18344}, {4637, 44426}, {5317, 7182}, {6331, 43924}, {6335, 7203}, {6628, 8736}, {7012, 16727}, {7056, 8748}, {7195, 40411}, {7233, 31905}, {15419, 36127}, {16947, 18022}, {17205, 46102}, {17219, 23984}, {18155, 32714}, {18600, 40446}
X(52370) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 44129}, {10, 331}, {85, 40591}, {125, 24002}, {206, 1396}, {264, 40599}, {273, 40586}, {274, 11517}, {278, 40600}, {286, 5452}, {1014, 22391}, {1434, 36033}, {1847, 40611}, {3669, 17423}, {6063, 51574}, {6600, 31623}, {7199, 38983}, {17096, 34467}, {17924, 40608}, {17925, 39025}, {24771, 44130}, {38996, 43923}, {40149, 40607}
X(52370) = crosspoint of X(i) and X(j) for these (i,j): {71, 2318}, {210, 3694}, {212, 219}, {220, 1802}
X(52370) = crosssum of X(i) and X(j) for these (i,j): {273, 278}, {279, 1847}, {1014, 1396}, {1172, 4233}, {3676, 17197}, {16727, 17096}
X(52370) = crossdifference of every pair of points on line {7192, 24002}
X(52370) = barycentric product X(i)*X(j) for these {i,j}: {1, 2318}, {3, 210}, {6, 3694}, {8, 228}, {9, 71}, {10, 212}, {21, 3690}, {31, 3710}, {33, 3682}, {37, 219}, {41, 306}, {42, 78}, {48, 2321}, {55, 72}, {63, 1334}, {65, 1260}, {73, 200}, {101, 8611}, {181, 1792}, {184, 3701}, {201, 2328}, {213, 345}, {220, 1214}, {222, 4515}, {226, 1802}, {268, 21871}, {281, 3990}, {283, 756}, {284, 3949}, {295, 4433}, {307, 1253}, {312, 2200}, {318, 4055}, {332, 872}, {346, 1409}, {480, 1439}, {512, 4571}, {521, 4557}, {594, 2193}, {603, 4082}, {607, 3998}, {644, 647}, {646, 3049}, {650, 4574}, {652, 1018}, {656, 3939}, {661, 4587}, {692, 52355}, {810, 3699}, {906, 3700}, {1231, 14827}, {1259, 1824}, {1265, 1402}, {1331, 4041}, {1332, 3709}, {1400, 3692}, {1410, 5423}, {1437, 6057}, {1444, 7064}, {1459, 4069}, {1500, 1812}, {1791, 40966}, {1794, 40967}, {1809, 51377}, {1813, 4171}, {1826, 2289}, {1918, 3718}, {1946, 3952}, {2171, 2327}, {2175, 20336}, {2188, 21075}, {2194, 3695}, {2196, 3985}, {2197, 2287}, {2324, 41087}, {2333, 3719}, {2359, 21033}, {3678, 8606}, {3900, 23067}, {3958, 33635}, {4047, 34820}, {4086, 32656}, {4095, 7116}, {4183, 7066}, {4524, 6516}, {6056, 41013}, {6065, 18210}, {6726, 7591}, {7046, 22341}, {7079, 40152}, {7368, 52037}, {9247, 30713}, {9447, 40071}, {14624, 22074}, {15628, 42702}, {21039, 47487}, {21859, 23090}, {22080, 32635}, {22381, 27538}, {22383, 30730}, {23189, 40521}, {36197, 44717}
X(52370) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 44129}, {32, 1396}, {37, 331}, {41, 27}, {42, 273}, {48, 1434}, {55, 286}, {71, 85}, {72, 6063}, {73, 1088}, {78, 310}, {184, 1014}, {200, 44130}, {210, 264}, {212, 86}, {213, 278}, {219, 274}, {220, 31623}, {228, 7}, {283, 873}, {306, 20567}, {345, 6385}, {644, 6331}, {647, 24002}, {652, 7199}, {669, 43923}, {810, 3676}, {872, 225}, {906, 4573}, {1018, 46404}, {1253, 29}, {1260, 314}, {1265, 40072}, {1331, 4625}, {1334, 92}, {1400, 1847}, {1402, 1119}, {1409, 279}, {1410, 479}, {1437, 552}, {1500, 40149}, {1792, 18021}, {1802, 333}, {1813, 4635}, {1918, 34}, {1946, 7192}, {2175, 28}, {2193, 1509}, {2197, 1446}, {2200, 57}, {2204, 36419}, {2205, 608}, {2212, 8747}, {2289, 17206}, {2318, 75}, {2321, 1969}, {2638, 17219}, {3049, 3669}, {3063, 17925}, {3682, 7182}, {3688, 16747}, {3690, 1441}, {3692, 28660}, {3694, 76}, {3701, 18022}, {3709, 17924}, {3710, 561}, {3939, 811}, {3949, 349}, {3990, 348}, {4041, 46107}, {4055, 77}, {4171, 46110}, {4433, 40717}, {4515, 7017}, {4516, 2973}, {4524, 44426}, {4557, 18026}, {4559, 13149}, {4571, 670}, {4574, 4554}, {4587, 799}, {6056, 1444}, {6066, 5379}, {6602, 2322}, {7064, 41013}, {7071, 1896}, {7109, 1880}, {7117, 16727}, {7124, 16750}, {8611, 3261}, {9247, 1412}, {9447, 1474}, {9448, 2203}, {14575, 1408}, {14827, 1172}, {20336, 41283}, {20665, 31917}, {20753, 33947}, {21871, 40701}, {22061, 7196}, {22074, 16705}, {22079, 17169}, {22341, 7056}, {22363, 7195}, {22364, 7248}, {22368, 18165}, {22369, 4059}, {22383, 17096}, {23067, 4569}, {32656, 1414}, {32660, 4637}, {36054, 15419}, {36059, 4616}, {39258, 5236}, {40972, 17171}, {52355, 40495}
X(52370) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {71, 3990, 1409}, {220, 7074, 607}

X(52371) = X(7)-ISOCONJUGATE OF X(36)

X(52371) lies on the cubics K497 and K498 and these lines: {1, 5}, {8, 28829}, {33, 7140}, {42, 1989}, {44, 2342}, {45, 55}, {64, 37567}, {88, 14191}, {89, 40269}, {100, 24433}, {103, 1155}, {200, 4069}, {210, 2328}, {220, 3119}, {528, 26611}, {759, 8694}, {963, 5204}, {997, 16594}, {1043, 3699}, {1168, 5048}, {1725, 18524}, {1793, 5302}, {1864, 40528}, {1936, 49712}, {2173, 20989}, {2265, 51235}, {2340, 42064}, {3021, 7218}, {3058, 3961}, {3214, 10118}, {3293, 38336}, {3465, 40663}, {3617, 27542}, {3716, 52356}, {3935, 14942}, {3938, 11238}, {4511, 4997}, {4792, 10703}, {4845, 41339}, {4860, 52013}, {4895, 46393}, {5160, 5524}, {5293, 10543}, {5297, 7474}, {7026, 36930}, {7043, 36931}, {7072, 7074}, {7220, 36815}, {9355, 33519}, {11075, 17796}, {11502, 17595}, {14629, 15898}, {17601, 24430}, {18360, 40263}, {18815, 21453}, {32286, 34893}, {35459, 38954}, {36737, 39150}, {36738, 39151}, {51805, 52202}, {51806, 52201}

X(52371) = isogonal conjugate of X(1443)
X(52371) = X(i)-Ceva conjugate of X(j) for these (i,j): {80, 2161}, {6740, 36910}, {36590, 9}
X(52371) = X(9629)-cross conjugate of X(7073)
X(52371) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1443}, {6, 17078}, {7, 36}, {56, 320}, {57, 3218}, {58, 41804}, {59, 4089}, {77, 1870}, {81, 18593}, {85, 7113}, {86, 1464}, {106, 41801}, {109, 4453}, {222, 17923}, {269, 4511}, {278, 22128}, {279, 2323}, {604, 20924}, {651, 3960}, {654, 658}, {758, 1014}, {934, 3738}, {1088, 2361}, {1227, 1417}, {1397, 40075}, {1407, 32851}, {1408, 35550}, {1412, 3936}, {1434, 2245}, {1439, 17515}, {1444, 1835}, {1446, 4282}, {1461, 3904}, {1983, 24002}, {2163, 36589}, {3669, 4585}, {3911, 40215}, {4554, 21758}, {4565, 4707}, {4569, 8648}, {4573, 21828}, {4881, 19604}, {5081, 7053}, {16586, 34051}, {18026, 22379}, {20566, 41282}, {37772, 37773}
X(52371) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 320}, {3, 1443}, {7, 15898}, {9, 17078}, {10, 41804}, {11, 4453}, {75, 36909}, {214, 41801}, {1464, 40600}, {3161, 20924}, {3218, 5452}, {3738, 14714}, {3904, 35508}, {3936, 40599}, {3960, 38991}, {4089, 6615}, {4511, 6600}, {5081, 23050}, {18593, 40586}, {24771, 32851}, {36589, 40587}, {38966, 44428}
X(52371) = cevapoint of X(i) and X(j) for these (i,j): {210, 3689}, {2310, 4895}
X(52371) = crosspoint of X(i) and X(j) for these (i,j): {80, 36910}, {2341, 6740}
X(52371) = crosssum of X(i) and X(j) for these (i,j): {1319, 1465}, {1464, 18593}, {3218, 4881}
X(52371) = trilinear pole of line {657, 1334}
X(52371) = crossdifference of every pair of points on line {654, 3960}
X(52371) = barycentric product X(i)*X(j) for these {i,j}: {1, 36910}, {8, 2161}, {9, 80}, {10, 2341}, {33, 52351}, {37, 6740}, {41, 20566}, {44, 36590}, {55, 18359}, {101, 52356}, {200, 2006}, {210, 24624}, {220, 18815}, {281, 1807}, {284, 15065}, {312, 6187}, {333, 34857}, {346, 1411}, {650, 51562}, {655, 3900}, {657, 35174}, {663, 36804}, {759, 2321}, {1168, 2325}, {1334, 14616}, {1793, 1826}, {1989, 4420}, {2222, 3239}, {2316, 51975}, {3701, 34079}, {4041, 47318}, {4397, 32675}, {4876, 36815}, {4997, 40172}, {7026, 7126}, {7043, 19551}, {7073, 41226}, {8641, 46405}
X(52371) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17078}, {6, 1443}, {8, 20924}, {9, 320}, {33, 17923}, {37, 41804}, {41, 36}, {42, 18593}, {44, 41801}, {45, 36589}, {55, 3218}, {80, 85}, {200, 32851}, {210, 3936}, {212, 22128}, {213, 1464}, {220, 4511}, {312, 40075}, {607, 1870}, {650, 4453}, {655, 4569}, {657, 3738}, {663, 3960}, {759, 1434}, {1253, 2323}, {1334, 758}, {1411, 279}, {1793, 17206}, {1807, 348}, {2006, 1088}, {2161, 7}, {2170, 4089}, {2175, 7113}, {2222, 658}, {2321, 35550}, {2325, 1227}, {2332, 17515}, {2333, 1835}, {2341, 86}, {3689, 51583}, {3711, 27757}, {3900, 3904}, {3939, 4585}, {4041, 4707}, {4079, 51645}, {4420, 7799}, {4814, 23884}, {6187, 57}, {6740, 274}, {7064, 4053}, {7079, 5081}, {8641, 654}, {14827, 2361}, {15065, 349}, {18359, 6063}, {20566, 20567}, {32675, 934}, {34079, 1014}, {34857, 226}, {35174, 46406}, {36590, 20568}, {36804, 4572}, {36815, 10030}, {36910, 75}, {40172, 3911}, {47318, 4625}, {51562, 4554}, {52351, 7182}, {52356, 3261}
X(52371) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {80, 1807, 1411}, {6187, 34857, 2161}, {7126, 19551, 2161}, {14629, 45272, 15898}, {34857, 40172, 6187}

X(52372) = X(8)-ISOCONJUGATE OF X(35)

X(52372) lies on the conic {{A,B,C,X(1),X(6)}}, the cubic K497, and these lines: {1, 30}, {6, 1406}, {12, 5018}, {56, 6186}, {58, 14158}, {65, 1126}, {86, 1443}, {106, 26700}, {582, 6149}, {939, 5217}, {996, 6757}, {1042, 1411}, {1120, 6742}, {1155, 1167}, {1220, 30690}, {1222, 4968}, {1399, 1427}, {1725, 48668}, {2279, 39690}, {2297, 7110}, {2334, 2650}, {2424, 48151}, {2687, 36064}, {3214, 32259}, {3468, 11246}, {3579, 18360}, {3647, 18593}, {4653, 51748}, {5378, 52085}, {6126, 12778}, {9277, 34502}, {13486, 16948}, {36052, 37582}, {36279, 52186}, {37129, 38340}, {37567, 42019}, {39152, 42677}, {39153, 42680}, {43531, 43682}, {51805, 52201}, {51806, 52202}

X(52372) = isogonal conjugate of X(4420)
X(52372) = X(i)-cross conjugate of X(j) for these (i,j): {3125, 3669}, {6186, 2160}, {42753, 43082}
X(52372) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4420}, {6, 42033}, {8, 35}, {9, 3219}, {10, 35193}, {21, 3678}, {41, 33939}, {55, 319}, {60, 7206}, {72, 11107}, {78, 6198}, {100, 35057}, {190, 9404}, {200, 1442}, {220, 17095}, {284, 3969}, {306, 41502}, {312, 2174}, {321, 35192}, {323, 36910}, {341, 1399}, {346, 2003}, {644, 14838}, {943, 31938}, {1043, 2594}, {1250, 40714}, {1260, 7282}, {1334, 34016}, {1792, 1825}, {2287, 16577}, {2321, 40214}, {2323, 41226}, {2328, 40999}, {2341, 42701}, {2605, 3699}, {3578, 33635}, {3647, 32635}, {3701, 17104}, {3718, 14975}, {3939, 4467}, {4102, 17454}, {4570, 6741}, {5546, 7265}, {7058, 21794}, {7161, 52126}, {10638, 40713}, {35194, 44687}, {44688, 46073}, {44689, 46077}
X(52372) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 4420}, {9, 42033}, {223, 319}, {478, 3219}, {1442, 6609}, {3160, 33939}, {3678, 40611}, {3969, 40590}, {4467, 40617}, {6741, 50330}, {8054, 35057}, {18160, 40615}, {36908, 40999}
X(52372) = trilinear pole of line {649, 14399}
X(52372) = crossdifference of every pair of points on line {9404, 35057}
X(52372) = barycentric product X(i)*X(j) for these {i,j}: {7, 2160}, {56, 30690}, {57, 79}, {58, 43682}, {85, 6186}, {269, 7110}, {278, 7100}, {279, 7073}, {513, 38340}, {514, 26700}, {554, 2306}, {604, 20565}, {1014, 8818}, {1081, 33654}, {1407, 52344}, {1412, 6757}, {1427, 3615}, {1443, 1989}, {1769, 47317}, {1847, 8606}, {3120, 35049}, {3669, 6742}, {3942, 34922}, {7178, 13486}, {15455, 43924}, {30602, 47057}
X(52372) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 42033}, {6, 4420}, {7, 33939}, {56, 3219}, {57, 319}, {65, 3969}, {79, 312}, {269, 17095}, {604, 35}, {608, 6198}, {649, 35057}, {667, 9404}, {1014, 34016}, {1042, 16577}, {1106, 2003}, {1333, 35193}, {1357, 7202}, {1397, 2174}, {1400, 3678}, {1407, 1442}, {1408, 40214}, {1411, 41226}, {1427, 40999}, {1435, 7282}, {1443, 7799}, {1464, 42701}, {1474, 11107}, {2160, 8}, {2171, 7206}, {2203, 41502}, {2206, 35192}, {2260, 31938}, {2306, 40714}, {3125, 6741}, {3669, 4467}, {3676, 18160}, {4017, 7265}, {6186, 9}, {6742, 646}, {6757, 30713}, {7073, 346}, {7100, 345}, {7110, 341}, {7203, 16755}, {8606, 3692}, {8818, 3701}, {13486, 645}, {16947, 17104}, {20565, 28659}, {26700, 190}, {30690, 3596}, {32636, 3578}, {33654, 40713}, {35049, 4600}, {38340, 668}, {43682, 313}, {43924, 14838}
X(52372) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {79, 7100, 7073}, {2306, 33654, 2160}

X(52373) = X(9)-ISOCONJUGATE OF X(29)

X(52373) lies on these lines: {6, 2155}, {19, 57}, {31, 56}, {48, 222}, {63, 348}, {65, 2357}, {71, 1214}, {73, 228}, {77, 22097}, {81, 934}, {223, 2183}, {226, 1020}, {241, 28274}, {269, 2215}, {279, 2282}, {347, 24310}, {354, 7004}, {658, 1821}, {1398, 1451}, {1400, 1427}, {1458, 2352}, {1461, 2003}, {1730, 43035}, {1826, 8808}, {1901, 13853}, {1943, 24435}, {2148, 33629}, {2360, 34043}, {3173, 22356}, {3182, 11471}, {3333, 3671}, {3937, 23204}, {5706, 37818}, {6354, 40160}, {7013, 10319}, {7114, 19349}, {9316, 37538}, {14547, 45963}, {20277, 40945}, {22053, 23207}, {22153, 23089}, {34032, 46009}, {43058, 46017}

X(52373) = isogonal conjugate of X(2322)
X(52373) = isogonal conjugate of the anticomplement of X(18643)
X(52373) = isotomic conjugate of the polar conjugate of X(1042)
X(52373) = isogonal conjugate of the polar conjugate of X(3668)
X(52373) = X(i)-Ceva conjugate of X(j) for these (i,j): {57, 1427}, {934, 1459}, {1020, 51640}, {1439, 73}, {3668, 1042}, {7053, 1410}, {7177, 1439}
X(52373) = X(i)-cross conjugate of X(j) for these (i,j): {1409, 73}, {1425, 1439}, {23620, 71}, {44093, 3}
X(52373) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2322}, {2, 4183}, {4, 2287}, {8, 1172}, {9, 29}, {10, 2326}, {19, 1043}, {21, 281}, {27, 200}, {28, 346}, {33, 333}, {41, 44130}, {55, 31623}, {58, 7101}, {72, 36421}, {75, 2332}, {78, 8748}, {81, 7046}, {86, 7079}, {92, 2328}, {100, 17926}, {112, 4397}, {158, 2327}, {162, 3239}, {210, 46103}, {219, 1896}, {220, 286}, {270, 2321}, {274, 7071}, {284, 318}, {312, 2299}, {314, 607}, {341, 1474}, {393, 1792}, {643, 3064}, {645, 18344}, {648, 3900}, {650, 36797}, {657, 811}, {1021, 1897}, {1098, 1826}, {1146, 5379}, {1253, 44129}, {1265, 5317}, {1396, 5423}, {1783, 7253}, {1812, 1857}, {1824, 7058}, {1863, 40403}, {2189, 3701}, {2194, 7017}, {2204, 3596}, {2212, 28660}, {2341, 5081}, {3692, 8747}, {4238, 28132}, {4319, 40411}, {4567, 42069}, {4578, 17925}, {4876, 14024}, {5546, 44426}, {5931, 7156}, {6061, 40149}, {6331, 8641}, {6335, 21789}, {6591, 7256}, {7008, 27398}, {7054, 41013}, {7110, 11107}, {7259, 7649}, {8021, 40447}, {13614, 40838}, {15149, 28071}, {15416, 32713}, {17515, 36910}, {17519, 36916}, {18020, 36197}, {36796, 37908}, {41502, 52344}
X(52373) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 2322}, {6, 1043}, {10, 7101}, {27, 6609}, {29, 478}, {92, 36908}, {125, 3239}, {206, 2332}, {223, 31623}, {226, 312}, {281, 40611}, {318, 40590}, {341, 51574}, {346, 40591}, {657, 17423}, {1021, 34467}, {1147, 2327}, {1214, 7017}, {1427, 15466}, {1826, 15267}, {2287, 36033}, {2328, 22391}, {3160, 44130}, {4183, 32664}, {4397, 34591}, {4988, 21666}, {7046, 40586}, {7079, 40600}, {7253, 39006}, {8054, 17926}, {17113, 44129}, {40622, 46110}, {40627, 42069}
X(52373) = cevapoint of X(1409) and X(1410)
X(52373) = crosspoint of X(i) and X(j) for these (i,j): {57, 222}, {63, 1073}, {1214, 52037}, {7053, 7177}
X(52373) = crosssum of X(i) and X(j) for these (i,j): {9, 281}, {19, 1249}, {657, 52335}, {2328, 2332}, {7046, 7079}
X(52373) = trilinear pole of line {810, 42658}
X(52373) = crossdifference of every pair of points on line {3239, 17926}
X(52373) = barycentric product X(i)*X(j) for these {i,j}: {1, 1439}, {3, 3668}, {7, 73}, {10, 7053}, {37, 7177}, {42, 7056}, {48, 1446}, {56, 307}, {57, 1214}, {58, 6356}, {63, 1427}, {65, 77}, {69, 1042}, {71, 279}, {72, 269}, {75, 1410}, {81, 37755}, {85, 1409}, {86, 1425}, {109, 17094}, {201, 1014}, {222, 226}, {223, 52037}, {225, 1804}, {228, 1088}, {273, 22341}, {278, 40152}, {283, 6046}, {284, 20618}, {286, 7138}, {306, 1407}, {321, 7099}, {326, 1426}, {332, 7143}, {348, 1400}, {479, 2318}, {520, 36118}, {525, 1461}, {603, 1441}, {604, 1231}, {647, 658}, {651, 51640}, {656, 934}, {738, 3694}, {810, 4569}, {822, 13149}, {905, 1020}, {1073, 36908}, {1106, 20336}, {1119, 3682}, {1254, 1444}, {1262, 4466}, {1332, 7216}, {1334, 30682}, {1402, 7182}, {1412, 26942}, {1434, 2197}, {1435, 3998}, {1459, 4566}, {1790, 6354}, {1803, 52023}, {1812, 7147}, {1813, 7178}, {1847, 3990}, {1880, 7183}, {2360, 6355}, {3049, 46406}, {3348, 8812}, {3676, 23067}, {3710, 7023}, {4017, 6516}, {4077, 36059}, {4306, 28786}, {4341, 28787}, {4561, 7250}, {4605, 7254}, {4617, 8611}, {6614, 52355}, {7011, 8808}, {7045, 18210}, {7125, 40149}, {7371, 7591}, {8057, 36079}, {10099, 41353}, {14256, 41087}, {18026, 51641}, {19611, 40933}, {23620, 30705}, {24018, 32714}, {40843, 51647}
X(52373) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 1043}, {6, 2322}, {7, 44130}, {31, 4183}, {32, 2332}, {34, 1896}, {37, 7101}, {42, 7046}, {48, 2287}, {56, 29}, {57, 31623}, {65, 318}, {71, 346}, {72, 341}, {73, 8}, {77, 314}, {109, 36797}, {184, 2328}, {201, 3701}, {213, 7079}, {222, 333}, {226, 7017}, {228, 200}, {255, 1792}, {269, 286}, {279, 44129}, {307, 3596}, {348, 28660}, {577, 2327}, {603, 21}, {604, 1172}, {608, 8748}, {647, 3239}, {649, 17926}, {656, 4397}, {658, 6331}, {810, 3900}, {906, 7259}, {934, 811}, {1020, 6335}, {1042, 4}, {1106, 28}, {1214, 312}, {1231, 28659}, {1254, 41013}, {1331, 7256}, {1332, 7258}, {1333, 2326}, {1397, 2299}, {1398, 8747}, {1399, 11107}, {1400, 281}, {1402, 33}, {1406, 3559}, {1407, 27}, {1408, 270}, {1409, 9}, {1410, 1}, {1412, 46103}, {1425, 10}, {1426, 158}, {1427, 92}, {1428, 14024}, {1437, 1098}, {1439, 75}, {1446, 1969}, {1459, 7253}, {1461, 648}, {1464, 5081}, {1474, 36421}, {1790, 7058}, {1804, 332}, {1813, 645}, {1918, 7071}, {2197, 2321}, {2200, 220}, {2318, 5423}, {3049, 657}, {3120, 21666}, {3122, 42069}, {3668, 264}, {3682, 1265}, {3690, 4082}, {3694, 30693}, {3990, 3692}, {4017, 44426}, {4055, 1260}, {4091, 15411}, {4303, 51978}, {4466, 23978}, {4574, 6558}, {4848, 44721}, {6356, 313}, {6516, 7257}, {6611, 41083}, {7011, 27398}, {7053, 86}, {7056, 310}, {7066, 3710}, {7099, 81}, {7125, 1812}, {7138, 72}, {7143, 225}, {7147, 40149}, {7177, 274}, {7178, 46110}, {7180, 3064}, {7182, 40072}, {7216, 17924}, {7250, 7649}, {7335, 283}, {7366, 1396}, {7591, 7027}, {16947, 2189}, {17094, 35519}, {18210, 24026}, {20618, 349}, {20975, 52335}, {22341, 78}, {22342, 4420}, {22345, 46877}, {22363, 4319}, {22383, 1021}, {23067, 3699}, {23620, 6554}, {24018, 15416}, {24027, 5379}, {26884, 15146}, {26942, 30713}, {32660, 5546}, {32714, 823}, {36059, 643}, {36118, 6528}, {36908, 15466}, {37755, 321}, {39791, 6734}, {40152, 345}, {40933, 1895}, {40934, 1863}, {51640, 4391}, {51641, 521}, {51642, 18344}, {51647, 1948}, {52037, 34404}
X(52373) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {56, 32065, 1457}, {57, 47848, 19}, {57, 51969, 7490}, {222, 7011, 48}, {222, 7053, 7099}, {1214, 1439, 37755}, {1214, 40152, 71}, {1410, 1425, 73}, {22341, 39791, 73}

X(52374) = X(9)-ISOCONJUGATE OF X(35)

X(52374) lies on the conic {{A,B,C,X(1),X(2)}} and these lines: {1, 30}, {2, 7110}, {7, 25417}, {57, 2160}, {81, 553}, {88, 26723}, {105, 5322}, {226, 1255}, {241, 42326}, {274, 17078}, {278, 18688}, {527, 40399}, {1002, 44661}, {1014, 40143}, {1022, 30726}, {1170, 43035}, {1219, 52344}, {1280, 6742}, {1358, 7313}, {1427, 1989}, {2003, 2982}, {2688, 36064}, {3175, 34892}, {3227, 19796}, {3578, 41804}, {3676, 21141}, {3911, 39962}, {6358, 39722}, {8056, 35466}, {8818, 17056}, {14844, 30571}, {15455, 36805}, {17301, 39948}, {18697, 50043}, {20565, 30710}, {26742, 44794}, {27789, 37635}, {30701, 42032}, {33633, 43261}, {48857, 51223}

X(52374) = isotomic conjugate of X(42033)
X(52374) = X(i)-cross conjugate of X(j) for these (i,j): {2160, 79}, {3120, 3676}, {21102, 36118}, {32636, 7}
X(52374) = X(i)-isoconjugate of X(j) for these (i,j): {6, 4420}, {8, 2174}, {9, 35}, {10, 35192}, {31, 42033}, {37, 35193}, {41, 319}, {55, 3219}, {71, 11107}, {72, 41502}, {100, 9404}, {101, 35057}, {200, 2003}, {210, 40214}, {219, 6198}, {220, 1442}, {284, 3678}, {345, 14975}, {346, 1399}, {644, 2605}, {1043, 21741}, {1098, 21794}, {1253, 17095}, {1802, 7282}, {1825, 2327}, {2150, 7206}, {2175, 33939}, {2194, 3969}, {2259, 31938}, {2287, 2594}, {2321, 17104}, {2322, 22342}, {2328, 16577}, {2361, 41226}, {3647, 33635}, {3939, 14838}, {6065, 7202}, {6149, 36910}, {17454, 32635}
X(52374) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 42033}, {9, 4420}, {35, 478}, {223, 3219}, {319, 3160}, {1015, 35057}, {1214, 3969}, {2003, 6609}, {3678, 40590}, {4467, 40615}, {4988, 6741}, {7265, 40622}, {8054, 9404}, {14838, 40617}, {14993, 36910}, {15267, 21794}, {16577, 36908}, {17095, 17113}, {18591, 31938}, {33939, 40593}, {35193, 40589}
X(52374) = cevapoint of X(i) and X(j) for these (i,j): {1086, 30724}, {2170, 50354}
X(52374) = trilinear pole of line {513, 11125}
X(52374) = barycentric product X(i)*X(j) for these {i,j}: {7, 79}, {56, 20565}, {57, 30690}, {81, 43682}, {85, 2160}, {269, 52344}, {273, 7100}, {279, 7110}, {514, 38340}, {554, 1081}, {693, 26700}, {1014, 6757}, {1088, 7073}, {1434, 8818}, {1443, 2166}, {1565, 34922}, {1989, 17078}, {3615, 3668}, {3669, 15455}, {3676, 6742}, {4077, 13486}, {6063, 6186}, {10015, 47317}, {16732, 35049}, {30602, 41808}
X(52374) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4420}, {2, 42033}, {7, 319}, {12, 7206}, {28, 11107}, {34, 6198}, {56, 35}, {57, 3219}, {58, 35193}, {65, 3678}, {79, 8}, {85, 33939}, {226, 3969}, {269, 1442}, {279, 17095}, {513, 35057}, {553, 3578}, {554, 40713}, {604, 2174}, {649, 9404}, {942, 31938}, {1042, 2594}, {1081, 40714}, {1106, 1399}, {1119, 7282}, {1333, 35192}, {1365, 21054}, {1393, 35194}, {1395, 14975}, {1407, 2003}, {1408, 17104}, {1410, 22342}, {1412, 40214}, {1426, 1825}, {1427, 16577}, {1434, 34016}, {1474, 41502}, {1789, 1792}, {1989, 36910}, {2006, 41226}, {2160, 9}, {3120, 6741}, {3337, 52126}, {3615, 1043}, {3668, 40999}, {3669, 14838}, {3676, 4467}, {6186, 55}, {6742, 3699}, {6757, 3701}, {7073, 200}, {7100, 78}, {7110, 346}, {7178, 7265}, {7248, 7186}, {8606, 1260}, {8818, 2321}, {13486, 643}, {15455, 646}, {17078, 7799}, {17096, 16755}, {18593, 42701}, {20565, 3596}, {24002, 18160}, {26700, 100}, {30690, 312}, {32636, 3647}, {34922, 15742}, {35049, 4567}, {37566, 41562}, {38340, 190}, {39152, 44688}, {39153, 44689}, {43082, 52356}, {43261, 4034}, {43682, 321}, {43924, 2605}, {47317, 13136}, {52344, 341}
X(52374) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {554, 1081, 79}, {4654, 47057, 37631}

X(52375) = X(10)-ISOCONJUGATE OF X(35)

X(52375) lies on these lines: {1, 2940}, {21, 36}, {56, 759}, {57, 1175}, {58, 14158}, {60, 3337}, {81, 24167}, {110, 3336}, {244, 849}, {261, 16709}, {265, 37251}, {267, 2611}, {284, 501}, {354, 15792}, {474, 24931}, {476, 2718}, {519, 35991}, {662, 3874}, {1014, 5358}, {1098, 4973}, {1210, 34301}, {1254, 24027}, {1325, 5563}, {1634, 16414}, {2189, 46883}, {2311, 2503}, {2646, 7073}, {3338, 9275}, {3453, 24046}, {3582, 37369}, {3746, 37294}, {4092, 13514}, {4188, 25650}, {4325, 7424}, {4653, 13624}, {4857, 5196}, {5127, 37582}, {5270, 37158}, {5322, 7469}, {6757, 8666}, {7110, 17581}, {8818, 11108}, {9563, 26884}, {13486, 16944}, {16817, 30690}, {17054, 28607}, {18398, 40214}, {21081, 33078}, {25055, 37032}, {25639, 52361}, {30598, 32014}, {30602, 40143}, {32783, 35983}, {35193, 37524}, {37259, 52153}, {37288, 40605}, {37960, 51883}

X(52375) = isogonal conjugate of X(3678)
X(52375) = isogonal conjugate of the complement of X(3874)
X(52375) = X(2170)-cross conjugate of X(1019)
X(52375) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3678}, {6, 3969}, {8, 2594}, {9, 16577}, {10, 35}, {12, 35193}, {37, 3219}, {42, 319}, {55, 40999}, {58, 7206}, {59, 6741}, {65, 4420}, {72, 6198}, {78, 1825}, {101, 7265}, {201, 11107}, {210, 1442}, {213, 33939}, {312, 21741}, {318, 22342}, {321, 2174}, {333, 21794}, {526, 51562}, {594, 40214}, {765, 2611}, {1016, 20982}, {1018, 14838}, {1089, 17104}, {1110, 17886}, {1252, 8287}, {1334, 17095}, {1399, 3701}, {1400, 42033}, {1500, 34016}, {2003, 2321}, {2161, 42701}, {2245, 41226}, {2318, 7282}, {2599, 44687}, {2605, 3952}, {2624, 36804}, {3615, 7144}, {4467, 4557}, {4551, 35057}, {4552, 9404}, {4567, 21824}, {4570, 21054}, {6149, 15065}, {6358, 35192}, {6539, 17454}, {14975, 20336}, {15742, 22094}, {21065, 52062}, {26942, 41502}
X(52375) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 3678}, {9, 3969}, {10, 7206}, {223, 40999}, {319, 40592}, {478, 16577}, {513, 2611}, {514, 17886}, {661, 8287}, {1015, 7265}, {3219, 40589}, {4420, 40602}, {6615, 6741}, {6626, 33939}, {7110, 42710}, {14993, 15065}, {15295, 34857}, {18160, 40620}, {21054, 50330}, {21824, 40627}, {40582, 42033}, {40584, 42701}
X(52375) = cevapoint of X(i) and X(j) for these (i,j): {56, 1393}, {244, 3733}, {2160, 6186}
X(52375) = crosspoint of X(i) and X(j) for these (i,j): {79, 30602}, {1014, 40143}
X(52375) = crosssum of X(i) and X(j) for these (i,j): {1, 2940}, {210, 21873}
X(52375) = trilinear pole of line {4840, 4979}
X(52375) = barycentric product X(i)*X(j) for these {i,j}: {11, 35049}, {27, 7100}, {57, 3615}, {58, 30690}, {60, 43682}, {79, 81}, {86, 2160}, {274, 6186}, {278, 1789}, {476, 3960}, {514, 13486}, {593, 6757}, {757, 8818}, {1014, 7110}, {1019, 6742}, {1333, 20565}, {1412, 52344}, {1434, 7073}, {3733, 15455}, {3737, 38340}, {4453, 32678}, {4560, 26700}, {14158, 21739}, {21758, 35139}, {22379, 46456}, {30602, 40592}
X(52375) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3969}, {6, 3678}, {21, 42033}, {36, 42701}, {37, 7206}, {56, 16577}, {57, 40999}, {58, 3219}, {79, 321}, {81, 319}, {86, 33939}, {244, 8287}, {284, 4420}, {476, 36804}, {513, 7265}, {604, 2594}, {608, 1825}, {757, 34016}, {759, 41226}, {764, 21141}, {849, 40214}, {1014, 17095}, {1015, 2611}, {1019, 4467}, {1086, 17886}, {1333, 35}, {1396, 7282}, {1397, 21741}, {1402, 21794}, {1408, 2003}, {1412, 1442}, {1474, 6198}, {1789, 345}, {1989, 15065}, {2150, 35193}, {2160, 10}, {2170, 6741}, {2189, 11107}, {2206, 2174}, {3122, 21824}, {3125, 21054}, {3248, 20982}, {3615, 312}, {3733, 14838}, {3960, 3268}, {4840, 23883}, {6186, 37}, {6742, 4033}, {6757, 28654}, {7073, 2321}, {7100, 306}, {7110, 3701}, {7192, 18160}, {7252, 35057}, {8606, 3694}, {8818, 1089}, {11060, 34857}, {13486, 190}, {14158, 17484}, {14844, 27569}, {15455, 27808}, {16947, 1399}, {20565, 27801}, {21741, 7144}, {21758, 526}, {22379, 8552}, {26700, 4552}, {30690, 313}, {32678, 51562}, {35049, 4998}, {43261, 42031}, {43682, 34388}, {46882, 31938}, {46883, 445}, {46890, 1844}, {52344, 30713}
X(52375) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21, 79, 51748}, {36, 229, 37816}, {32636, 51420, 58}

X(52376) = X(10)-ISOCONJUGATE OF X(39)

X(52376) lies on the conic {{A,B,C,X(1),X(2)}} these lines: {1, 82}, {2, 32}, {21, 1390}, {57, 7210}, {58, 291}, {81, 18167}, {86, 17192}, {88, 4599}, {105, 827}, {172, 30168}, {274, 1333}, {278, 31906}, {333, 39281}, {593, 39747}, {689, 713}, {757, 39950}, {1010, 1224}, {1022, 39179}, {1176, 5138}, {1255, 18098}, {2006, 18097}, {2224, 34072}, {2915, 51862}, {3227, 4577}, {4593, 32020}, {5291, 39722}, {16054, 18087}, {16082, 42396}, {17285, 33954}, {17500, 51500}, {18089, 26643}, {33955, 39724}

X(52376) = isogonal conjugate of X(3954)
X(52376) = isogonal conjugate of the complement of X(17141)
X(52376) = X(4599)-Ceva conjugate of X(39179)
X(52376) = X(i)-cross conjugate of X(j) for these (i,j): {172, 58}, {7191, 86}, {16735, 274}, {39179, 4599}
X(52376) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3954}, {2, 21035}, {3, 21016}, {6, 15523}, {10, 39}, {37, 38}, {42, 141}, {65, 33299}, {71, 427}, {72, 17442}, {75, 21814}, {76, 41267}, {100, 8061}, {101, 826}, {190, 3005}, {213, 1930}, {226, 3688}, {228, 20883}, {256, 16587}, {257, 40936}, {306, 1843}, {313, 3051}, {321, 1964}, {335, 4093}, {512, 4568}, {513, 35309}, {523, 46148}, {594, 17187}, {661, 4553}, {668, 2084}, {688, 1978}, {756, 16696}, {872, 16703}, {1018, 2530}, {1235, 2200}, {1334, 3665}, {1400, 3703}, {1401, 2321}, {1441, 40972}, {1500, 16887}, {1634, 4024}, {1826, 3917}, {1918, 8024}, {1923, 27801}, {2333, 3933}, {2525, 8750}, {2528, 4628}, {3108, 21038}, {3682, 27376}, {3690, 17171}, {3700, 46153}, {3930, 46149}, {3943, 46150}, {3952, 21123}, {4020, 41013}, {4033, 50521}, {4037, 46159}, {4053, 46160}, {4062, 46154}, {4064, 35325}, {4079, 4576}, {4088, 46163}, {4120, 46162}, {4557, 16892}, {4559, 48278}, {4570, 39691}, {4574, 21108}, {7018, 21752}, {8041, 18082}, {8611, 46152}, {16030, 21011}, {21022, 31613}, {21083, 21355}, {21818, 32010}, {23285, 32739}, {24290, 35333}, {27369, 40071}, {35334, 50330}
X(52376) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 3954}, {9, 15523}, {38, 40589}, {82, 21082}, {141, 40592}, {206, 21814}, {321, 41884}, {826, 1015}, {1930, 6626}, {2525, 26932}, {3703, 40582}, {4553, 36830}, {4568, 39054}, {8024, 34021}, {8054, 8061}, {21016, 36103}, {21035, 32664}, {21037, 21249}, {23285, 40619}, {33299, 40602}, {35309, 39026}, {39691, 50330}, {40620, 48084}
X(52376) = cevapoint of X(i) and X(j) for these (i,j): {6, 16687}, {81, 1333}, {82, 251}, {595, 4251}
X(52376) = crosssum of X(37) and X(21880)
X(52376) = trilinear pole of line {513, 1980}
X(52376) = crossdifference of every pair of points on line {3005, 8061}
X(52376) = barycentric product X(i)*X(j) for these {i,j}: {27, 34055}, {28, 1799}, {58, 3112}, {81, 83}, {82, 86}, {99, 18108}, {190, 39179}, {239, 39276}, {251, 274}, {286, 1176}, {308, 1333}, {310, 46289}, {513, 4577}, {514, 4599}, {649, 4593}, {662, 10566}, {667, 689}, {693, 827}, {757, 18082}, {905, 42396}, {1437, 46104}, {1444, 32085}, {1509, 18098}, {1919, 37204}, {1980, 42371}, {2185, 18097}, {2206, 18833}, {3115, 16717}, {3261, 34072}, {4164, 41209}, {4556, 18070}, {4603, 18111}, {4623, 18105}, {4628, 7199}, {4630, 40495}, {5262, 39281}, {6385, 46288}, {7303, 18099}, {7305, 16889}, {18089, 40408}, {18180, 39287}
X(52376) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 15523}, {6, 3954}, {19, 21016}, {21, 3703}, {27, 20883}, {28, 427}, {31, 21035}, {32, 21814}, {58, 38}, {81, 141}, {82, 10}, {83, 321}, {86, 1930}, {101, 35309}, {110, 4553}, {163, 46148}, {172, 16587}, {251, 37}, {274, 8024}, {284, 33299}, {286, 1235}, {308, 27801}, {513, 826}, {560, 41267}, {593, 16696}, {649, 8061}, {662, 4568}, {667, 3005}, {689, 6386}, {693, 23285}, {757, 16887}, {827, 100}, {849, 17187}, {905, 2525}, {1014, 3665}, {1019, 16892}, {1176, 72}, {1333, 39}, {1408, 1401}, {1437, 3917}, {1444, 3933}, {1474, 17442}, {1509, 16703}, {1799, 20336}, {1919, 2084}, {1980, 688}, {2194, 3688}, {2203, 1843}, {2206, 1964}, {2210, 4093}, {2530, 2528}, {3112, 313}, {3125, 39691}, {3733, 2530}, {3737, 48278}, {3803, 3806}, {4577, 668}, {4593, 1978}, {4599, 190}, {4628, 1018}, {4630, 692}, {4911, 17873}, {5007, 21817}, {5299, 17456}, {5317, 27376}, {7122, 40936}, {7191, 21249}, {7192, 48084}, {10547, 228}, {10566, 1577}, {14419, 14424}, {14885, 21880}, {16600, 21037}, {16696, 7794}, {16702, 7813}, {16706, 21425}, {16707, 42554}, {16735, 21248}, {16757, 23881}, {16948, 4884}, {17200, 20898}, {17469, 21038}, {18082, 1089}, {18097, 6358}, {18098, 594}, {18105, 4705}, {18108, 523}, {18167, 16893}, {20022, 42703}, {28724, 3998}, {32085, 41013}, {33632, 21874}, {34055, 306}, {34072, 101}, {38834, 21877}, {39179, 514}, {39276, 335}, {42396, 6335}, {46288, 213}, {46289, 42}, {51369, 51371}, {51420, 51360}, {51508, 3198}, {51906, 21833}

X(52377) = X(11)-ISOCONJUGATE OF X(36)

X(52377) lies on the Feuerbach circumhyperbola and these lines: {1, 59}, {4, 7012}, {7, 7045}, {8, 765}, {9, 1252}, {21, 4570}, {80, 23592}, {100, 46041}, {104, 1807}, {109, 15343}, {294, 2161}, {314, 4600}, {655, 885}, {901, 2222}, {1000, 34232}, {1110, 15175}, {1262, 7284}, {1320, 1411}, {2006, 3254}, {2320, 4564}, {2481, 18815}, {2648, 34857}, {6187, 9365}, {14888, 21132}, {24298, 45926}, {24457, 40577}, {28780, 43749}, {32675, 34075}, {36037, 43728}

X(52377) = X(i)-cross conjugate of X(j) for these (i,j): {44, 651}, {517, 100}, {1411, 2222}, {1731, 653}, {1807, 51562}, {2161, 655}, {2316, 37136}, {2342, 5548}, {2361, 101}, {12034, 3257}, {45272, 3699}, {51361, 644}
X(52377) = X(i)-isoconjugate of X(j) for these (i,j): {11, 36}, {55, 4089}, {80, 3025}, {106, 51402}, {244, 4511}, {320, 3271}, {513, 3738}, {514, 654}, {649, 3904}, {650, 3960}, {651, 46384}, {663, 4453}, {693, 8648}, {758, 18191}, {909, 46398}, {1015, 32851}, {1086, 2323}, {1111, 2361}, {1443, 2310}, {1459, 44428}, {1870, 7004}, {1983, 40166}, {2006, 35128}, {2170, 3218}, {2245, 17197}, {3937, 5081}, {4282, 16732}, {4391, 21758}, {4530, 40215}, {4560, 21828}, {4707, 7252}, {4858, 7113}, {7117, 17923}, {8735, 22128}, {14936, 17078}, {17219, 44113}, {17515, 18210}, {22379, 44426}
X(52377) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 15898}, {214, 51402}, {223, 4089}, {3738, 39026}, {3904, 5375}, {23980, 46398}, {24026, 36909}, {38991, 46384}
X(52377) = cevapoint of X(i) and X(j) for these (i,j): {1, 23703}, {55, 2427}, {101, 2361}, {109, 1319}, {1411, 2222}
X(52377) = trilinear pole of line {101, 650}
X(52377) = barycentric product X(i)*X(j) for these {i,j}: {59, 18359}, {80, 4564}, {100, 655}, {101, 35174}, {109, 36804}, {190, 2222}, {651, 51562}, {668, 32675}, {692, 46405}, {765, 2006}, {1016, 1411}, {1252, 18815}, {1807, 46102}, {2149, 20566}, {2161, 4998}, {3218, 46649}, {4511, 23592}, {4551, 47318}, {4619, 52356}, {4620, 34857}, {5376, 14584}, {7012, 52351}, {7045, 36910}, {9268, 14628}
X(52377) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 51402}, {57, 4089}, {59, 3218}, {80, 4858}, {100, 3904}, {101, 3738}, {109, 3960}, {517, 46398}, {651, 4453}, {655, 693}, {663, 46384}, {692, 654}, {759, 17197}, {765, 32851}, {1110, 2323}, {1252, 4511}, {1262, 1443}, {1411, 1086}, {1783, 44428}, {1807, 26932}, {2006, 1111}, {2149, 36}, {2161, 11}, {2222, 514}, {2361, 35128}, {4551, 4707}, {4564, 320}, {4998, 20924}, {6187, 2170}, {7012, 17923}, {7045, 17078}, {7113, 3025}, {7115, 1870}, {18359, 34387}, {18815, 23989}, {21859, 6370}, {23592, 18815}, {23990, 2361}, {32660, 22379}, {32675, 513}, {32739, 8648}, {34079, 18191}, {34857, 21044}, {35174, 3261}, {36804, 35519}, {36910, 24026}, {40172, 4530}, {46405, 40495}, {46649, 18359}, {47318, 18155}, {51562, 4391}, {52351, 17880}
X(52377) = {X(23703),X(37630)}-harmonic conjugate of X(2222)

X(52378) = X(11)-ISOCONJUGATE OF X(37)

X(52378) lies on these lines: {59, 3286}, {81, 4619}, {99, 32689}, {109, 691}, {110, 14733}, {163, 1019}, {226, 39295}, {249, 5060}, {250, 1101}, {333, 39294}, {349, 41174}, {648, 32707}, {651, 4556}, {653, 687}, {662, 32669}, {664, 2966}, {1025, 23889}, {1262, 17966}, {1813, 44769}, {2003, 9273}, {2149, 4564}, {2283, 17943}, {2406, 2407}, {2421, 2425}, {4558, 32643}, {4565, 4591}, {4590, 4620}, {4636, 36059}, {5467, 17942}, {7045, 7128}, {23582, 44331}, {32674, 32697}

X(52378) = isogonal conjugate of X(21044)
X(52378) = isogonal conjugate of the complement of X(17136)
X(52378) = isogonal conjugate of the isotomic conjugate of X(4620)
X(52378) = X(i)-cross conjugate of X(j) for these (i,j): {58, 1414}, {73, 664}, {81, 4556}, {284, 110}, {651, 4619}, {1400, 109}, {1412, 4565}, {1790, 662}, {2003, 651}, {2193, 4636}, {4225, 99}, {17209, 4584}, {37583, 934}, {40152, 1813}, {40214, 4629}
X(52378) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21044}, {2, 4516}, {7, 36197}, {8, 3125}, {9, 3120}, {10, 2170}, {11, 37}, {21, 115}, {29, 3708}, {33, 4466}, {41, 21207}, {42, 4858}, {55, 16732}, {57, 52335}, {65, 1146}, {72, 8735}, {81, 4092}, {125, 1172}, {210, 1086}, {213, 34387}, {225, 34591}, {226, 2310}, {244, 2321}, {257, 40608}, {261, 21833}, {270, 21046}, {281, 18210}, {284, 1109}, {312, 3122}, {314, 3124}, {321, 3271}, {333, 2643}, {338, 2194}, {339, 2204}, {512, 4391}, {513, 3700}, {514, 4041}, {521, 2501}, {522, 661}, {523, 650}, {525, 18344}, {594, 18191}, {643, 21131}, {646, 8034}, {647, 44426}, {649, 4086}, {652, 24006}, {656, 3064}, {657, 4077}, {663, 1577}, {693, 3709}, {756, 17197}, {764, 30730}, {798, 35519}, {810, 46110}, {850, 3063}, {885, 24290}, {1015, 3701}, {1018, 21132}, {1020, 23615}, {1024, 4088}, {1084, 40072}, {1111, 1334}, {1214, 42069}, {1358, 4515}, {1365, 2287}, {1400, 24026}, {1402, 23978}, {1409, 21666}, {1427, 4081}, {1441, 14936}, {1446, 3022}, {1812, 8754}, {1824, 26932}, {1826, 7004}, {1880, 2968}, {1896, 3269}, {1903, 38357}, {1946, 14618}, {2160, 6741}, {2185, 21043}, {2193, 2970}, {2250, 35015}, {2299, 20902}, {2333, 17880}, {2358, 7358}, {2489, 35518}, {2611, 7110}, {2632, 8748}, {2969, 3694}, {3119, 3668}, {3121, 3596}, {3239, 4017}, {3248, 30713}, {3270, 40149}, {3615, 21824}, {3676, 4171}, {3680, 21950}, {3737, 4024}, {3900, 7178}, {3954, 18101}, {4036, 7252}, {4049, 4895}, {4069, 6545}, {4079, 18155}, {4120, 23838}, {4163, 7216}, {4397, 7180}, {4415, 40528}, {4435, 35352}, {4451, 16592}, {4518, 39786}, {4524, 24002}, {4526, 35353}, {4530, 4674}, {4542, 30575}, {4551, 42462}, {4557, 40166}, {4559, 42455}, {4560, 4705}, {4612, 8029}, {4631, 22260}, {4843, 47915}, {5317, 7068}, {6057, 16726}, {6385, 7063}, {6591, 52355}, {7064, 16727}, {7073, 8287}, {7117, 41013}, {7649, 8611}, {11998, 51870}, {13576, 17435}, {14395, 18808}, {14432, 23894}, {15232, 38345}, {15320, 38358}, {20975, 31623}, {20982, 52344}, {21809, 40451}, {21828, 52356}, {23893, 30574}, {36795, 42752}
X(52378) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 21044}, {11, 40589}, {115, 40611}, {223, 16732}, {226, 20902}, {338, 1214}, {478, 3120}, {522, 36830}, {850, 10001}, {1109, 40590}, {1146, 40602}, {2970, 47345}, {3064, 40596}, {3160, 21207}, {3239, 34961}, {3700, 39026}, {4086, 5375}, {4092, 40586}, {4391, 39054}, {4516, 32664}, {4858, 40592}, {5452, 52335}, {6626, 34387}, {14618, 39053}, {23978, 40605}, {24026, 40582}, {31998, 35519}, {39052, 44426}, {39062, 46110}
X(52378) = cevapoint of X(i) and X(j) for these (i,j): {35, 101}, {56, 36075}, {58, 163}, {59, 2149}, {63, 3882}, {73, 32660}, {81, 651}, {109, 1400}, {110, 284}, {283, 5546}, {1408, 1415}, {1412, 4565}, {1813, 40152}, {2193, 36059}, {7122, 32739}
X(52378) = crosssum of X(i) and X(j) for these (i,j): {115, 3708}, {4516, 36197}
X(52378) = trilinear pole of line {109, 110}
X(52378) = barycentric product X(i)*X(j) for these {i,j}: {6, 4620}, {7, 4570}, {21, 7045}, {27, 44717}, {42, 7340}, {56, 4600}, {57, 4567}, {58, 4998}, {59, 86}, {65, 24041}, {73, 18020}, {77, 5379}, {81, 4564}, {99, 109}, {100, 1414}, {101, 4573}, {107, 6517}, {108, 4592}, {110, 664}, {162, 6516}, {163, 4554}, {190, 4565}, {226, 249}, {250, 307}, {274, 2149}, {284, 1275}, {314, 24027}, {333, 1262}, {349, 23357}, {604, 4601}, {643, 934}, {644, 4637}, {645, 1461}, {648, 1813}, {651, 662}, {653, 4558}, {658, 5546}, {692, 4625}, {765, 1014}, {799, 1415}, {811, 36059}, {1016, 1412}, {1019, 31615}, {1020, 4612}, {1042, 6064}, {1043, 7339}, {1101, 1441}, {1252, 1434}, {1400, 4590}, {1402, 24037}, {1408, 7035}, {1409, 46254}, {1444, 7012}, {1576, 4572}, {1790, 46102}, {1812, 7128}, {2701, 17933}, {3219, 35049}, {3286, 39293}, {3939, 4616}, {4552, 4556}, {4559, 4610}, {4560, 4619}, {4563, 32674}, {4566, 4636}, {4575, 18026}, {4577, 46153}, {4617, 7259}, {4622, 23703}, {4632, 36075}, {6331, 32660}, {6514, 23984}, {6614, 7256}, {7115, 17206}, {16947, 31625}, {17942, 35154}, {18645, 38809}, {22341, 23999}, {23582, 40152}, {23979, 28660}, {32661, 46404}
X(52378) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 21044}, {7, 21207}, {21, 24026}, {29, 21666}, {31, 4516}, {35, 6741}, {41, 36197}, {42, 4092}, {55, 52335}, {56, 3120}, {57, 16732}, {58, 11}, {59, 10}, {65, 1109}, {73, 125}, {81, 4858}, {86, 34387}, {99, 35519}, {100, 4086}, {101, 3700}, {108, 24006}, {109, 523}, {110, 522}, {112, 3064}, {162, 44426}, {163, 650}, {181, 21043}, {222, 4466}, {225, 2970}, {226, 338}, {249, 333}, {250, 29}, {283, 2968}, {284, 1146}, {307, 339}, {333, 23978}, {349, 23962}, {593, 17197}, {603, 18210}, {604, 3125}, {643, 4397}, {648, 46110}, {651, 1577}, {653, 14618}, {662, 4391}, {664, 850}, {692, 4041}, {765, 3701}, {849, 18191}, {859, 35015}, {906, 8611}, {934, 4077}, {1014, 1111}, {1016, 30713}, {1019, 40166}, {1042, 1365}, {1101, 21}, {1110, 210}, {1214, 20902}, {1252, 2321}, {1262, 226}, {1275, 349}, {1331, 52355}, {1333, 2170}, {1397, 3122}, {1399, 2611}, {1400, 115}, {1402, 2643}, {1408, 244}, {1409, 3708}, {1412, 1086}, {1414, 693}, {1415, 661}, {1434, 23989}, {1437, 7004}, {1441, 23994}, {1442, 17886}, {1444, 17880}, {1457, 42759}, {1461, 7178}, {1474, 8735}, {1576, 663}, {1617, 21945}, {1634, 48278}, {1790, 26932}, {1804, 17216}, {1813, 525}, {1819, 7358}, {2003, 8287}, {2149, 37}, {2193, 34591}, {2194, 2310}, {2197, 21046}, {2206, 3271}, {2283, 4088}, {2299, 42069}, {2328, 4081}, {2360, 38357}, {2420, 14400}, {2594, 21054}, {2701, 18013}, {3285, 4530}, {3682, 7068}, {3733, 21132}, {3737, 42455}, {4225, 124}, {4253, 21946}, {4551, 4036}, {4554, 20948}, {4556, 4560}, {4558, 6332}, {4559, 4024}, {4564, 321}, {4565, 514}, {4567, 312}, {4570, 8}, {4572, 44173}, {4573, 3261}, {4575, 521}, {4590, 28660}, {4592, 35518}, {4600, 3596}, {4601, 28659}, {4619, 4552}, {4620, 76}, {4625, 40495}, {4636, 7253}, {4637, 24002}, {4998, 313}, {5009, 4124}, {5379, 318}, {5467, 14432}, {5546, 3239}, {6065, 4082}, {6514, 23983}, {6516, 14208}, {6517, 3265}, {7012, 41013}, {7045, 1441}, {7098, 23673}, {7115, 1826}, {7122, 40608}, {7128, 40149}, {7180, 21131}, {7252, 42462}, {7253, 23104}, {7339, 3668}, {7340, 310}, {7341, 17205}, {14587, 35196}, {15386, 15232}, {16947, 1015}, {16948, 4939}, {17096, 23100}, {17942, 2785}, {18020, 44130}, {18191, 1090}, {21741, 21824}, {21786, 52341}, {21789, 23615}, {22341, 2632}, {23067, 4064}, {23346, 30574}, {23357, 284}, {23964, 8748}, {23979, 1400}, {23990, 1334}, {23995, 2194}, {24000, 1896}, {24027, 65}, {24037, 40072}, {24041, 314}, {31615, 4033}, {32660, 647}, {32661, 652}, {32674, 2501}, {32676, 18344}, {32739, 3709}, {33628, 4534}, {35049, 30690}, {36059, 656}, {36074, 4838}, {36075, 4988}, {37583, 8286}, {40152, 15526}, {44717, 306}, {46153, 826}, {47390, 283}
X(52378) = {X(5467),X(23346)}-harmonic conjugate of X(17942)

X(52379) = X(12)-ISOCONJUGATE OF X(32)

X(52379) lies on these lines: {8, 7257}, {10, 5209}, {21, 261}, {27, 310}, {41, 27958}, {58, 10471}, {75, 757}, {76, 799}, {81, 239}, {86, 1193}, {99, 2975}, {172, 19623}, {270, 14024}, {321, 1931}, {333, 3691}, {348, 4625}, {350, 6626}, {811, 17555}, {1434, 10030}, {1909, 17731}, {2106, 16827}, {3696, 14195}, {3878, 38477}, {4362, 40731}, {5278, 30022}, {6629, 20888}, {8033, 34284}, {11679, 28659}, {14829, 40827}, {16705, 16738}, {17379, 51314}, {17680, 40017}, {19684, 33779}, {24995, 30966}, {25303, 32004}, {27705, 27954}, {31997, 51356}, {33299, 36800}, {33764, 33935}, {33939, 42710}

X(52379) = isotomic conjugate of X(2171)
X(52379) = isotomic conjugate of the anticomplement of X(21233)
X(52379) = isotomic conjugate of the complement of X(21273)
X(52379) = isotomic conjugate of the isogonal conjugate of X(2185)
X(52379) = X(i)-Ceva conjugate of X(j) for these (i,j): {4590, 799}, {46254, 4610}
X(52379) = X(i)-cross conjugate of X(j) for these (i,j): {261, 873}, {314, 18021}, {333, 261}, {4560, 7257}, {4858, 18155}, {17185, 86}, {20882, 75}, {21233, 2}
X(52379) = X(i)-isoconjugate of X(j) for these (i,j): {6, 181}, {7, 7109}, {12, 32}, {25, 2197}, {31, 2171}, {37, 1402}, {41, 1254}, {42, 1400}, {56, 1500}, {57, 872}, {59, 3124}, {65, 213}, {73, 2333}, {109, 4079}, {184, 8736}, {201, 1973}, {220, 7143}, {225, 2200}, {226, 1918}, {228, 1880}, {512, 4559}, {560, 6358}, {594, 1397}, {604, 756}, {607, 1425}, {608, 3690}, {651, 50487}, {667, 21859}, {669, 4552}, {762, 1408}, {798, 4551}, {1016, 1356}, {1018, 51642}, {1037, 21813}, {1042, 1334}, {1084, 4998}, {1253, 7147}, {1275, 7063}, {1365, 23990}, {1395, 3949}, {1403, 6378}, {1407, 7064}, {1409, 1824}, {1415, 4705}, {1441, 2205}, {1501, 34388}, {1922, 7235}, {1974, 26942}, {2149, 2643}, {2175, 6354}, {2207, 7066}, {2212, 37755}, {2489, 23067}, {2971, 44717}, {3027, 51856}, {4092, 23979}, {4557, 7180}, {6046, 14827}, {6186, 21794}, {6535, 16947}, {7104, 7211}, {7115, 20975}, {7148, 41526}, {8687, 42661}, {18097, 41267}, {28654, 41280}, {32675, 42666}, {40147, 52024}
X(52379) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 1500}, {2, 2171}, {9, 181}, {11, 4079}, {12, 6376}, {37, 40605}, {42, 40582}, {65, 6626}, {115, 1577}, {201, 6337}, {213, 40602}, {226, 34021}, {650, 2643}, {661, 40625}, {756, 3161}, {872, 5452}, {905, 3708}, {1146, 4705}, {1254, 3160}, {1400, 40592}, {1402, 40589}, {1966, 3027}, {2197, 6505}, {3124, 6615}, {4017, 40620}, {4024, 40624}, {4551, 31998}, {4559, 39054}, {6354, 40593}, {6358, 6374}, {6631, 21859}, {7064, 24771}, {7147, 17113}, {7235, 39028}, {17419, 42661}, {20975, 40628}, {35128, 42666}, {38991, 50487}
X(52379) = cevapoint of X(i) and X(j) for these (i,j): {2, 21273}, {9, 3996}, {75, 14829}, {261, 7058}, {274, 17206}, {314, 333}, {4858, 18155}
X(52379) = trilinear pole of line {3716, 3737}
X(52379) = barycentric product X(i)*X(j) for these {i,j}: {1, 18021}, {8, 873}, {11, 24037}, {21, 310}, {58, 40072}, {60, 561}, {75, 261}, {76, 2185}, {81, 28660}, {85, 7058}, {86, 314}, {99, 18155}, {270, 305}, {274, 333}, {284, 6385}, {286, 332}, {304, 46103}, {312, 1509}, {341, 552}, {514, 4631}, {522, 4623}, {593, 28659}, {645, 7199}, {670, 3737}, {757, 3596}, {763, 30713}, {799, 4560}, {812, 36806}, {849, 40363}, {1098, 6063}, {1111, 6064}, {1444, 44130}, {1502, 2150}, {1812, 44129}, {2170, 34537}, {2189, 40364}, {3061, 7307}, {3261, 4612}, {3701, 6628}, {4391, 4610}, {4590, 4858}, {4601, 17197}, {4602, 7252}, {4625, 7253}, {4636, 40495}, {7054, 20567}, {7192, 7257}, {7258, 17096}, {7304, 27424}, {7340, 24026}, {12836, 14124}, {17185, 40827}, {17206, 31623}, {17880, 18020}, {20568, 30606}, {20882, 31620}, {24041, 34387}, {26932, 46254}, {30940, 36800}
X(52379) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 181}, {2, 2171}, {7, 1254}, {8, 756}, {9, 1500}, {11, 2643}, {21, 42}, {27, 1880}, {29, 1824}, {41, 7109}, {55, 872}, {58, 1402}, {60, 31}, {63, 2197}, {69, 201}, {75, 12}, {76, 6358}, {77, 1425}, {78, 3690}, {81, 1400}, {85, 6354}, {86, 65}, {92, 8736}, {99, 4551}, {190, 21859}, {200, 7064}, {249, 2149}, {261, 1}, {269, 7143}, {270, 25}, {274, 226}, {279, 7147}, {283, 228}, {284, 213}, {285, 2357}, {286, 225}, {304, 26942}, {310, 1441}, {312, 594}, {314, 10}, {318, 7140}, {326, 7066}, {332, 72}, {333, 37}, {341, 6057}, {345, 3949}, {348, 37755}, {350, 7235}, {522, 4705}, {552, 269}, {561, 34388}, {593, 604}, {643, 4557}, {645, 1018}, {646, 4103}, {650, 4079}, {662, 4559}, {663, 50487}, {757, 56}, {763, 1412}, {799, 4552}, {849, 1397}, {873, 7}, {1014, 1042}, {1019, 7180}, {1021, 3709}, {1043, 210}, {1088, 6046}, {1089, 6058}, {1098, 55}, {1111, 1365}, {1172, 2333}, {1434, 1427}, {1444, 73}, {1509, 57}, {1790, 1409}, {1792, 2318}, {1804, 7138}, {1812, 71}, {1909, 7211}, {2082, 21813}, {2150, 32}, {2170, 3124}, {2185, 6}, {2189, 1973}, {2193, 2200}, {2194, 1918}, {2287, 1334}, {2319, 6378}, {2321, 762}, {2326, 607}, {3219, 21794}, {3248, 1356}, {3596, 1089}, {3686, 21816}, {3687, 21810}, {3691, 21820}, {3699, 40521}, {3701, 6535}, {3702, 8013}, {3705, 7237}, {3706, 21699}, {3716, 4155}, {3718, 3695}, {3733, 51642}, {3737, 512}, {3738, 42666}, {3794, 3778}, {3904, 2610}, {3975, 4037}, {3996, 40607}, {4267, 3725}, {4391, 4024}, {4435, 46390}, {4453, 51645}, {4459, 21725}, {4554, 4605}, {4556, 1415}, {4560, 661}, {4573, 1020}, {4590, 4564}, {4592, 23067}, {4610, 651}, {4612, 101}, {4623, 664}, {4625, 4566}, {4631, 190}, {4636, 692}, {4858, 115}, {4985, 6367}, {5324, 40934}, {6061, 1253}, {6064, 765}, {6385, 349}, {6514, 3990}, {6628, 1014}, {6740, 34857}, {7004, 20975}, {7054, 41}, {7058, 9}, {7081, 21803}, {7155, 7148}, {7182, 6356}, {7192, 4017}, {7199, 7178}, {7203, 7250}, {7252, 798}, {7253, 4041}, {7256, 4069}, {7257, 3952}, {7258, 30730}, {7303, 1431}, {7304, 1423}, {7340, 7045}, {7341, 1106}, {8033, 4032}, {8822, 227}, {14024, 862}, {14570, 35307}, {15411, 8611}, {15419, 51640}, {16708, 52023}, {16709, 3649}, {16713, 21808}, {16738, 45208}, {16739, 41003}, {17096, 7216}, {17175, 39793}, {17183, 4642}, {17185, 2092}, {17194, 52020}, {17197, 3125}, {17206, 1214}, {17219, 18210}, {17277, 20616}, {17420, 42661}, {17515, 44113}, {17787, 21021}, {17880, 125}, {18020, 7012}, {18021, 75}, {18155, 523}, {18163, 21796}, {18169, 39780}, {18178, 21936}, {18191, 3122}, {20665, 21815}, {21044, 21833}, {21388, 22229}, {21610, 21958}, {23189, 810}, {24026, 4092}, {24037, 4998}, {24041, 59}, {26856, 2170}, {26932, 3708}, {27398, 21871}, {27527, 21834}, {27958, 2295}, {28659, 28654}, {28660, 321}, {30576, 1404}, {30593, 32636}, {30606, 44}, {30939, 40663}, {30940, 16609}, {31623, 1826}, {31631, 21853}, {32851, 4053}, {33295, 1284}, {34016, 16577}, {34387, 1109}, {34388, 1091}, {35518, 4064}, {35519, 4036}, {36806, 4562}, {37140, 32675}, {39044, 3027}, {39177, 2623}, {40072, 313}, {40166, 21131}, {40214, 21741}, {44129, 40149}, {44130, 41013}, {46103, 19}, {46254, 46102}, {46877, 40966}, {46882, 40978}, {48264, 50538}, {51978, 40967}, {52352, 4849}
X(52379) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 34016, 799}, {274, 1509, 873}

X(52380) = X(12)-ISOCONJUGATE OF X(36)

X(52380) lies on the Feuerbach circumhyperbola and these lines: {1, 60}, {4, 162}, {7, 757}, {8, 643}, {9, 1793}, {21, 4636}, {58, 79}, {80, 5127}, {104, 30576}, {256, 24436}, {283, 6598}, {584, 941}, {593, 7284}, {943, 1807}, {1156, 37140}, {1411, 17097}, {1937, 23692}, {2150, 40968}, {2161, 2298}, {2166, 24640}, {2185, 2320}, {2997, 14616}, {4653, 5424}, {6187, 43073}, {7235, 14194}, {7261, 21274}, {10308, 16948}, {18609, 51420}, {23838, 35055}, {39766, 39767}

X(52380) = X(i)-cross conjugate of X(j) for these (i,j): {1731, 333}, {2361, 284}, {45272, 29}
X(52380) = X(i)-isoconjugate of X(j) for these (i,j): {10, 1464}, {12, 36}, {37, 18593}, {42, 41804}, {57, 4053}, {65, 758}, {72, 1835}, {73, 860}, {80, 3028}, {100, 51645}, {109, 6370}, {181, 320}, {201, 1870}, {226, 2245}, {307, 44113}, {651, 2610}, {654, 4605}, {664, 42666}, {756, 1443}, {1254, 4511}, {1400, 3936}, {1402, 35550}, {1411, 4736}, {1425, 5081}, {1441, 3724}, {1500, 17078}, {2006, 35069}, {2171, 3218}, {2197, 17923}, {2323, 6354}, {3960, 21859}, {4552, 21828}, {4559, 4707}, {6358, 7113}, {8736, 22128}, {16186, 34922}
X(52380) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 6370}, {12, 15898}, {758, 40602}, {1089, 36909}, {2610, 38991}, {3936, 40582}, {4053, 5452}, {4736, 35204}, {8054, 51645}, {18593, 40589}, {35550, 40605}, {39025, 42666}, {40592, 41804}
X(52380) = cevapoint of X(i) and X(j) for these (i,j): {58, 51420}, {284, 2361}
X(52380) = trilinear pole of line {284, 650}
X(52380) = crossdifference of every pair of points on line {2610, 51645}
X(52380) = barycentric product X(i)*X(j) for these {i,j}: {21, 24624}, {27, 1793}, {60, 18359}, {80, 2185}, {81, 6740}, {86, 2341}, {261, 2161}, {270, 52351}, {284, 14616}, {314, 34079}, {333, 759}, {522, 37140}, {757, 36910}, {1098, 2006}, {1168, 30606}, {1411, 7058}, {1807, 46103}, {2150, 20566}, {3737, 47318}, {4391, 36069}, {4556, 52356}, {7054, 18815}, {30576, 36590}, {32671, 35519}
X(52380) = barycentric quotient X(i)/X(j) for these {i,j}: {21, 3936}, {55, 4053}, {58, 18593}, {60, 3218}, {80, 6358}, {81, 41804}, {261, 20924}, {270, 17923}, {284, 758}, {333, 35550}, {593, 1443}, {649, 51645}, {650, 6370}, {663, 2610}, {757, 17078}, {759, 226}, {1098, 32851}, {1172, 860}, {1333, 1464}, {1411, 6354}, {1474, 1835}, {1731, 31845}, {1793, 306}, {1807, 26942}, {2150, 36}, {2161, 12}, {2185, 320}, {2189, 1870}, {2194, 2245}, {2204, 44113}, {2222, 4605}, {2323, 4736}, {2326, 5081}, {2341, 10}, {2361, 35069}, {3063, 42666}, {3737, 4707}, {4636, 4585}, {6187, 2171}, {6740, 321}, {7054, 4511}, {7113, 3028}, {14616, 349}, {18178, 51465}, {18359, 34388}, {24624, 1441}, {30576, 41801}, {30606, 1227}, {32671, 109}, {34079, 65}, {35193, 42701}, {36069, 651}, {36910, 1089}, {37140, 664}
X(52380) = {X(6740),X(36927)}-harmonic conjugate of X(8)

X(52381) = X(19)-ISOCONJUGATE OF X(35)

X(52381) lies on these lines: {2, 7110}, {21, 36}, {63, 11064}, {77, 6513}, {78, 1060}, {81, 24780}, {94, 2006}, {280, 10527}, {345, 6349}, {499, 15474}, {905, 16697}, {1214, 52351}, {1790, 4466}, {1791, 25909}, {1812, 4001}, {2071, 51883}, {2160, 2339}, {2886, 52002}, {3305, 7131}, {3580, 47057}, {4666, 7073}, {4847, 6742}, {5743, 11076}, {6757, 19854}, {7361, 27339}, {8818, 31266}, {10072, 19785}, {15314, 38822}, {15455, 36795}, {17923, 20565}, {25639, 41496}, {26700, 26703}, {27509, 30679}, {36100, 38340}

X(52381) = isotomic conjugate of the polar conjugate of X(79)
X(52381) = isogonal conjugate of the polar conjugate of X(20565)
X(52381) = X(20565)-Ceva conjugate of X(79)
X(52381) = X(i)-cross conjugate of X(j) for these (i,j): {3916, 69}, {18607, 63}, {22058, 3}, {37565, 77}
X(52381) = X(i)-isoconjugate of X(j) for these (i,j): {2, 14975}, {4, 2174}, {6, 6198}, {19, 35}, {25, 3219}, {29, 21741}, {33, 2003}, {41, 7282}, {65, 41502}, {108, 9404}, {186, 2161}, {225, 35192}, {250, 21824}, {270, 21794}, {281, 1399}, {284, 1825}, {319, 1973}, {607, 1442}, {608, 4420}, {943, 44095}, {1172, 2594}, {1395, 42033}, {1400, 11107}, {1474, 3678}, {1783, 2605}, {1824, 40214}, {1826, 17104}, {1844, 2259}, {1880, 35193}, {1974, 33939}, {2203, 3969}, {2204, 40999}, {2212, 17095}, {2299, 16577}, {5379, 20982}, {7265, 32676}, {8739, 46073}, {8740, 46077}, {8748, 22342}, {8750, 14838}, {8882, 35194}, {18359, 34397}, {32674, 35057}
X(52381) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 35}, {9, 6198}, {186, 40584}, {226, 16577}, {319, 6337}, {445, 16585}, {451, 7110}, {647, 21054}, {1825, 40590}, {1844, 18591}, {2174, 36033}, {2605, 39006}, {3160, 7282}, {3219, 6505}, {3678, 51574}, {4467, 40618}, {7265, 15526}, {9404, 38983}, {11107, 40582}, {14838, 26932}, {14975, 32664}, {35057, 35072}, {40602, 41502}
X(52381) = cevapoint of X(i) and X(j) for these (i,j): {3, 22122}, {905, 4466}
X(52381) = crosssum of X(607) and X(44097)
X(52381) = barycentric product X(i)*X(j) for these {i,j}: {3, 20565}, {36, 328}, {63, 30690}, {69, 79}, {75, 7100}, {77, 52344}, {94, 22128}, {265, 320}, {304, 2160}, {305, 6186}, {307, 3615}, {348, 7110}, {905, 15455}, {1441, 1789}, {1444, 6757}, {1812, 43682}, {4025, 6742}, {6063, 8606}, {6332, 38340}, {7073, 7182}, {8818, 17206}, {13486, 14208}, {26700, 35518}, {40075, 52153}, {40716, 50462}
X(52381) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 6198}, {3, 35}, {7, 7282}, {21, 11107}, {31, 14975}, {36, 186}, {48, 2174}, {63, 3219}, {65, 1825}, {69, 319}, {72, 3678}, {73, 2594}, {77, 1442}, {78, 4420}, {79, 4}, {125, 21054}, {222, 2003}, {265, 80}, {283, 35193}, {284, 41502}, {304, 33939}, {306, 3969}, {307, 40999}, {320, 340}, {328, 20566}, {345, 42033}, {348, 17095}, {521, 35057}, {525, 7265}, {603, 1399}, {652, 9404}, {905, 14838}, {942, 1844}, {1062, 4354}, {1071, 41562}, {1214, 16577}, {1409, 21741}, {1437, 17104}, {1459, 2605}, {1789, 21}, {1790, 40214}, {2160, 19}, {2193, 35192}, {2197, 21794}, {2260, 44095}, {2523, 30600}, {3615, 29}, {3695, 7206}, {3708, 21824}, {3784, 7186}, {3916, 3647}, {3942, 7202}, {4001, 3578}, {4025, 4467}, {4292, 46468}, {4303, 500}, {4466, 8287}, {4707, 44427}, {5249, 445}, {6186, 25}, {6742, 1897}, {6757, 41013}, {7073, 33}, {7100, 1}, {7110, 281}, {8606, 55}, {8818, 1826}, {13486, 162}, {14844, 4213}, {15413, 18160}, {15419, 16755}, {15455, 6335}, {17206, 34016}, {17923, 14165}, {18210, 2611}, {18607, 16585}, {20565, 264}, {21828, 47230}, {22054, 17454}, {22128, 323}, {22341, 22342}, {23070, 35197}, {23224, 23226}, {26700, 108}, {30690, 92}, {38340, 653}, {43682, 40149}, {44706, 35194}, {50462, 484}, {52153, 6187}, {52202, 7150}, {52344, 318}, {52351, 41226}
X(52381) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 30690, 7110}, {2, 41808, 16585}

X(52382) = X(21)-ISOCONJUGATE OF X(35)

X(52382) lies on the cubic K360 and these lines: {1, 30}, {4, 36119}, {7, 40438}, {10, 6757}, {19, 1990}, {37, 8818}, {46, 50462}, {55, 37992}, {56, 759}, {57, 267}, {75, 3260}, {145, 6742}, {265, 18525}, {477, 36064}, {897, 9214}, {1247, 24789}, {1393, 4017}, {1406, 45923}, {1725, 22798}, {1837, 34301}, {1910, 35906}, {2153, 33654}, {2154, 2306}, {2166, 14254}, {2218, 6186}, {2363, 19785}, {2594, 6354}, {3017, 5221}, {3028, 5903}, {3336, 3471}, {3615, 3616}, {3648, 18625}, {4442, 34860}, {4934, 10896}, {6701, 18593}, {7178, 35347}, {8614, 14158}, {8773, 36891}, {11375, 14844}, {13610, 33149}, {15454, 36053}, {18398, 33642}, {18977, 51656}, {20840, 20988}, {23843, 52153}, {30690, 31359}

X(52382) = isogonal conjugate of X(35193)
X(52382) = X(43682)-Ceva conjugate of X(8818)
X(52382) = X(i)-cross conjugate of X(j) for these (i,j): {115, 7178}, {4017, 26700}, {6089, 476}, {42759, 10412}
X(52382) = X(i)-isoconjugate of X(j) for these (i,j): {1, 35193}, {2, 35192}, {3, 11107}, {8, 17104}, {9, 40214}, {21, 35}, {41, 34016}, {58, 4420}, {60, 3678}, {63, 41502}, {110, 35057}, {186, 1793}, {283, 6198}, {284, 3219}, {319, 2194}, {323, 2341}, {332, 14975}, {333, 2174}, {643, 2605}, {662, 9404}, {1043, 1399}, {1098, 2594}, {1101, 6741}, {1175, 31938}, {1333, 42033}, {1442, 2328}, {2003, 2287}, {2150, 3969}, {3467, 35195}, {4282, 41226}, {5546, 14838}, {6149, 6740}, {7054, 16577}, {7058, 21741}, {17190, 33635}, {23226, 36797}, {35194, 35196}
X(52382) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 35193}, {10, 4420}, {35, 40611}, {37, 42033}, {244, 35057}, {319, 1214}, {478, 40214}, {523, 6741}, {1084, 9404}, {1442, 36908}, {2594, 15267}, {3160, 34016}, {3162, 41502}, {3219, 40590}, {4467, 40622}, {6148, 18593}, {6740, 14993}, {11107, 36103}, {16755, 40615}, {32664, 35192}
X(52382) = cevapoint of X(1365) and X(4017)
X(52382) = crosspoint of X(2166) and X(5627)
X(52382) = crosssum of X(1511) and X(6149)
X(52382) = trilinear pole of line {661, 1637}
X(52382) = barycentric product X(i)*X(j) for these {i,j}: {1, 43682}, {7, 8818}, {57, 6757}, {65, 30690}, {79, 226}, {94, 1464}, {349, 6186}, {523, 38340}, {1109, 35049}, {1400, 20565}, {1427, 52344}, {1441, 2160}, {1446, 7073}, {1577, 26700}, {1989, 41804}, {2166, 18593}, {3615, 6354}, {3668, 7110}, {4017, 15455}, {4466, 34922}, {6742, 7178}, {7100, 40149}, {32680, 51645}, {36064, 41079}
X(52382) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 35193}, {7, 34016}, {10, 42033}, {12, 3969}, {19, 11107}, {25, 41502}, {31, 35192}, {37, 4420}, {56, 40214}, {65, 3219}, {79, 333}, {115, 6741}, {226, 319}, {512, 9404}, {604, 17104}, {661, 35057}, {1042, 2003}, {1254, 16577}, {1365, 8287}, {1400, 35}, {1402, 2174}, {1427, 1442}, {1441, 33939}, {1464, 323}, {1880, 6198}, {1989, 6740}, {2160, 21}, {2171, 3678}, {2294, 31938}, {3615, 7058}, {3649, 3578}, {3668, 17095}, {3676, 16755}, {4017, 14838}, {4077, 18160}, {6186, 284}, {6354, 40999}, {6742, 645}, {6757, 312}, {7073, 2287}, {7100, 1812}, {7110, 1043}, {7178, 4467}, {7180, 2605}, {8606, 2327}, {8818, 8}, {13486, 4612}, {15455, 7257}, {20565, 28660}, {20982, 3024}, {21773, 35195}, {26700, 662}, {30690, 314}, {32636, 17190}, {35049, 24041}, {36064, 44769}, {38340, 99}, {41804, 7799}, {43682, 75}, {51645, 32679}
X(52382) = {X(79),X(50148)}-harmonic conjugate of X(1)

X(52383) = X(21)-ISOCONJUGATE OF X(36)

X(52383) lies on the conic {{A,B,C,X(4),X(5)}} and these lines: {1, 5}, {4, 2190}, {10, 15065}, {19, 53}, {37, 21011}, {56, 39270}, {65, 3120}, {75, 311}, {82, 17500}, {109, 13273}, {115, 4559}, {117, 5348}, {158, 13450}, {181, 994}, {201, 41501}, {219, 31141}, {244, 20118}, {267, 1710}, {484, 1263}, {596, 51975}, {655, 897}, {759, 859}, {921, 8800}, {969, 10401}, {1054, 24914}, {1086, 12832}, {1141, 36078}, {1331, 13272}, {1399, 3585}, {1464, 51421}, {1772, 12619}, {1836, 34300}, {1879, 21768}, {1910, 32675}, {2165, 2168}, {2170, 35307}, {2216, 40449}, {2217, 27622}, {2218, 6187}, {2363, 24624}, {2962, 18395}, {4674, 40663}, {5292, 18962}, {6702, 43048}, {6740, 26095}, {7052, 36301}, {8769, 27364}, {8818, 21741}, {9654, 45931}, {14628, 42285}, {17638, 35015}, {18359, 23541}, {18827, 35174}, {18961, 34030}, {19628, 23592}, {20616, 23903}, {21770, 36412}, {22765, 38954}, {23404, 38938}, {25882, 52351}, {33136, 36920}, {33655, 36300}, {36927, 47318}, {37759, 51562}

X(52383) = X(i)-Ceva conjugate of X(j) for these (i,j): {23592, 4559}, {34535, 2171}
X(52383) = X(i)-cross conjugate of X(j) for these (i,j): {2171, 34535}, {3724, 8818}, {4642, 1168}, {6089, 3952}, {40663, 14584}, {42666, 4559}, {42759, 523}
X(52383) = X(i)-isoconjugate of X(j) for these (i,j): {2, 4282}, {3, 17515}, {21, 36}, {58, 4511}, {60, 758}, {81, 2323}, {86, 2361}, {99, 8648}, {110, 3738}, {163, 3904}, {186, 1789}, {215, 14616}, {261, 3724}, {283, 1870}, {284, 3218}, {320, 2194}, {333, 7113}, {645, 21758}, {654, 662}, {759, 4996}, {1098, 1464}, {1172, 22128}, {1333, 32851}, {1437, 5081}, {1443, 2328}, {1983, 4560}, {2150, 3936}, {2185, 2245}, {2193, 17923}, {2311, 27950}, {2316, 17191}, {2600, 18315}, {3615, 6149}, {3960, 5546}, {4242, 23189}, {4575, 44428}, {4585, 7252}, {4612, 21828}, {6369, 36134}, {7054, 18593}, {22379, 36797}, {24624, 34544}
X(52383) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 4511}, {21, 15898}, {36, 40611}, {37, 32851}, {115, 3904}, {136, 44428}, {137, 6369}, {244, 3738}, {320, 1214}, {654, 1084}, {1043, 36909}, {1273, 16577}, {1443, 36908}, {1464, 15267}, {2323, 40586}, {2361, 40600}, {3218, 40590}, {3615, 14993}, {4282, 32664}, {4453, 40622}, {4996, 34586}, {8648, 38986}, {17515, 36103}, {17923, 47345}
X(52383) = cevapoint of X(i) and X(j) for these (i,j): {12, 40663}, {115, 42666}, {523, 45260}, {3120, 30572}, {3724, 21741}
X(52383) = crosspoint of X(i) and X(j) for these (i,j): {1141, 2166}, {2006, 18815}
X(52383) = crosssum of X(i) and X(j) for these (i,j): {1, 34465}, {58, 51966}, {1154, 6149}, {2323, 2361}
X(52383) = trilinear pole of line {661, 2171}
X(52383) = crossdifference of every pair of points on line {654, 34544}
X(52383) = barycentric product X(i)*X(j) for these {i,j}: {10, 2006}, {12, 24624}, {37, 18815}, {57, 15065}, {65, 18359}, {80, 226}, {85, 34857}, {94, 2594}, {225, 52351}, {321, 1411}, {349, 6187}, {512, 46405}, {523, 655}, {661, 35174}, {758, 34535}, {759, 6358}, {850, 32675}, {1020, 52356}, {1400, 20566}, {1441, 2161}, {1577, 2222}, {1807, 40149}, {1989, 40999}, {2166, 16577}, {2171, 14616}, {3668, 36910}, {4017, 36804}, {4080, 14584}, {4674, 14628}, {6354, 6740}, {7178, 51562}, {18314, 36078}, {34079, 34388}, {38955, 52212}
X(52383) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 32851}, {12, 3936}, {19, 17515}, {31, 4282}, {37, 4511}, {42, 2323}, {65, 3218}, {73, 22128}, {80, 333}, {181, 2245}, {213, 2361}, {225, 17923}, {226, 320}, {349, 40075}, {512, 654}, {523, 3904}, {655, 99}, {661, 3738}, {759, 2185}, {798, 8648}, {1254, 18593}, {1284, 27950}, {1319, 17191}, {1400, 36}, {1402, 7113}, {1411, 81}, {1427, 1443}, {1441, 20924}, {1807, 1812}, {1826, 5081}, {1880, 1870}, {1989, 3615}, {2006, 86}, {2161, 21}, {2171, 758}, {2222, 662}, {2245, 4996}, {2341, 1098}, {2501, 44428}, {2594, 323}, {3668, 17078}, {3724, 34544}, {4017, 3960}, {4551, 4585}, {6187, 284}, {6354, 41804}, {6358, 35550}, {6740, 7058}, {7178, 4453}, {8736, 860}, {12077, 6369}, {14584, 16704}, {14628, 30939}, {15065, 312}, {18359, 314}, {18815, 274}, {20566, 28660}, {21741, 6149}, {24624, 261}, {32675, 110}, {34079, 60}, {34535, 14616}, {34857, 9}, {35174, 799}, {36078, 18315}, {36804, 7257}, {36910, 1043}, {40663, 51583}, {40999, 7799}, {42759, 46398}, {46405, 670}, {51562, 645}, {51642, 21758}, {52212, 17139}, {52351, 332}
X(52383) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 7951, 45944}, {80, 1411, 14584}, {80, 2006, 1411}, {17734, 38945, 5172}

X(52384) = X(21)-ISOCONJUGATE OF X(40)

X(52384) lies on these lines: {1, 84}, {4, 43058}, {10, 227}, {19, 56}, {33, 1035}, {34, 7008}, {37, 73}, {57, 37818}, {65, 2357}, {75, 280}, {158, 278}, {189, 31359}, {225, 1427}, {282, 8583}, {285, 40430}, {759, 8059}, {897, 37141}, {958, 30674}, {1104, 7151}, {1212, 7367}, {1319, 2208}, {1410, 1824}, {1426, 18210}, {1451, 7118}, {1468, 2219}, {1838, 7681}, {1880, 7157}, {2182, 7114}, {2188, 2646}, {3198, 22341}, {3668, 52078}, {6705, 17102}, {7335, 10537}, {8769, 9376}, {10200, 37695}, {12688, 51662}, {15836, 34052}, {20991, 22654}, {34434, 42549}

X(52384) = isotomic conjugate of the polar conjugate of X(2358)
X(52384) = polar conjugate of the isotomic conjugate of X(52037)
X(52384) = X(i)-Ceva conjugate of X(j) for these (i,j): {280, 52078}, {1440, 8808}, {8808, 1903}, {39130, 65}
X(52384) = X(i)-cross conjugate of X(j) for these (i,j): {1042, 65}, {1880, 1427}, {2357, 1903}, {23663, 16606}
X(52384) = X(i)-isoconjugate of X(j) for these (i,j): {4, 1819}, {6, 27398}, {8, 2360}, {9, 1817}, {21, 40}, {29, 7078}, {55, 8822}, {58, 7080}, {60, 21075}, {78, 3194}, {81, 2324}, {86, 7074}, {110, 8058}, {196, 2327}, {198, 333}, {208, 1792}, {219, 41083}, {221, 1043}, {223, 2287}, {227, 1098}, {283, 7952}, {284, 329}, {285, 1103}, {314, 2187}, {322, 2194}, {332, 3195}, {347, 2328}, {643, 6129}, {648, 10397}, {662, 14298}, {1434, 7368}, {1444, 40971}, {1812, 2331}, {2185, 21871}, {2322, 7011}, {3342, 13614}, {4183, 7013}, {4570, 38357}, {5546, 14837}, {7070, 41082}
X(52384) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 27398}, {10, 7080}, {40, 40611}, {223, 8822}, {227, 15267}, {244, 8058}, {322, 1214}, {329, 40590}, {347, 36908}, {478, 1817}, {1043, 3341}, {1084, 14298}, {1819, 36033}, {2324, 40586}, {6527, 8808}, {7074, 40600}, {17896, 40622}, {38357, 50330}
X(52384) = cevapoint of X(i) and X(j) for these (i,j): {4017, 18210}, {4516, 7180}
X(52384) = crosspoint of X(i) and X(j) for these (i,j): {1, 3346}, {84, 40836}, {278, 8809}, {1422, 1440}
X(52384) = crosssum of X(i) and X(j) for these (i,j): {1, 1498}, {40, 7078}, {219, 7070}, {2324, 7074}
X(52384) = barycentric product X(i)*X(j) for these {i,j}: {1, 8808}, {4, 52037}, {7, 1903}, {10, 1422}, {21, 13853}, {37, 1440}, {57, 39130}, {65, 189}, {69, 2358}, {84, 226}, {85, 2357}, {225, 41081}, {273, 41087}, {280, 1427}, {282, 3668}, {285, 6354}, {307, 7129}, {309, 1400}, {321, 1413}, {349, 2208}, {523, 37141}, {1042, 34404}, {1172, 6355}, {1214, 40836}, {1231, 7151}, {1402, 44190}, {1426, 44189}, {1433, 40149}, {1436, 1441}, {1439, 7003}, {1446, 2192}, {1577, 8059}, {1817, 7157}, {1824, 34400}, {2184, 52078}, {3701, 6612}, {4017, 44327}, {4077, 36049}, {7178, 13138}, {17094, 40117}
X(52384) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 27398}, {34, 41083}, {37, 7080}, {42, 2324}, {48, 1819}, {56, 1817}, {57, 8822}, {65, 329}, {84, 333}, {181, 21871}, {189, 314}, {213, 7074}, {226, 322}, {268, 1792}, {282, 1043}, {285, 7058}, {309, 28660}, {512, 14298}, {604, 2360}, {608, 3194}, {661, 8058}, {810, 10397}, {1042, 223}, {1400, 40}, {1402, 198}, {1409, 7078}, {1410, 7011}, {1413, 81}, {1422, 86}, {1426, 196}, {1427, 347}, {1433, 1812}, {1436, 21}, {1440, 274}, {1880, 7952}, {1903, 8}, {2171, 21075}, {2188, 2327}, {2192, 2287}, {2208, 284}, {2333, 40971}, {2357, 9}, {2358, 4}, {3125, 38357}, {3668, 40702}, {4017, 14837}, {4516, 5514}, {6355, 1231}, {6612, 1014}, {7008, 2322}, {7118, 2328}, {7129, 29}, {7151, 1172}, {7154, 4183}, {7178, 17896}, {7180, 6129}, {8059, 662}, {8808, 75}, {13138, 645}, {13853, 1441}, {18210, 16596}, {32652, 5546}, {36049, 643}, {37141, 99}, {39130, 312}, {40117, 36797}, {40836, 31623}, {41081, 332}, {41086, 27382}, {41087, 78}, {42549, 17183}, {44190, 40072}, {44327, 7257}, {52037, 69}, {52078, 18750}
X(52384) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 84, 2192}, {1, 41402, 1455}, {1, 47848, 221}, {84, 1422, 1413}, {1214, 5930, 227}

X(52385) = X(25)-ISOCONJUGATE OF X(29)

X(52385) lies on these lines: {2, 9119}, {7, 8}, {21, 1442}, {27, 1943}, {63, 77}, {72, 307}, {78, 7013}, {210, 40999}, {226, 17052}, {272, 18191}, {296, 332}, {306, 51368}, {326, 1259}, {329, 1032}, {343, 5249}, {347, 3869}, {379, 2262}, {517, 18655}, {651, 5279}, {664, 8822}, {758, 3668}, {857, 1903}, {912, 41004}, {916, 1071}, {927, 2749}, {1014, 1812}, {1020, 21078}, {1037, 1402}, {1264, 4176}, {1426, 10381}, {1813, 2327}, {1824, 5929}, {1876, 10477}, {1880, 40802}, {2003, 23130}, {2893, 7282}, {3419, 10400}, {3664, 18389}, {3682, 7138}, {3879, 16465}, {3896, 30619}, {3962, 41804}, {3998, 40152}, {4329, 6001}, {4552, 21871}, {5130, 10364}, {5273, 37669}, {5570, 24179}, {5693, 41010}, {5738, 44547}, {5739, 46017}, {5887, 41007}, {6180, 34377}, {7190, 11520}, {9965, 20110}, {14110, 17134}, {14872, 21270}, {15656, 18161}, {16368, 45126}, {28786, 28787}, {30621, 37593}, {34384, 44139}

X(52385) = isotomic conjugate of X(1896)
X(52385) = anticomplement of X(9119)
X(52385) = isotomic conjugate of the isogonal conjugate of X(22341)
X(52385) = isotomic conjugate of the polar conjugate of X(1214)
X(52385) = isogonal conjugate of the polar conjugate of X(1231)
X(52385) = X(i)-Ceva conjugate of X(j) for these (i,j): {69, 307}, {1231, 1214}, {6517, 4131}, {7183, 40152}, {34401, 226}
X(52385) = X(i)-cross conjugate of X(j) for these (i,j): {3682, 3998}, {7066, 40152}, {17216, 4131}, {22341, 1214}
X(52385) = X(i)-isoconjugate of X(j) for these (i,j): {4, 2299}, {6, 8748}, {9, 5317}, {19, 1172}, {21, 1096}, {25, 29}, {27, 607}, {28, 33}, {31, 1896}, {34, 4183}, {55, 8747}, {58, 1857}, {86, 6059}, {92, 2204}, {107, 663}, {112, 3064}, {158, 2194}, {162, 18344}, {270, 1824}, {278, 2332}, {281, 1474}, {283, 6524}, {284, 393}, {286, 2212}, {318, 2203}, {333, 2207}, {522, 32713}, {608, 2322}, {650, 24019}, {652, 6529}, {823, 3063}, {1039, 4206}, {1043, 7337}, {1118, 2328}, {1334, 36419}, {1396, 7079}, {1400, 36421}, {1826, 2189}, {1880, 2326}, {1946, 36126}, {1973, 31623}, {1974, 44130}, {2193, 6520}, {2321, 36420}, {2333, 46103}, {3194, 7008}, {4516, 24000}, {6514, 36434}, {6525, 52158}, {7007, 8885}, {7077, 34856}, {7154, 41083}, {14400, 32695}, {14569, 35196}, {17926, 32674}, {21044, 23964}, {21789, 36127}, {28660, 36417}, {32676, 44426}, {36124, 37908}, {40574, 41320}
X(52385) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 1896}, {4, 226}, {6, 1172}, {9, 8748}, {10, 1857}, {21, 6503}, {29, 6505}, {33, 40591}, {63, 3559}, {125, 18344}, {158, 1214}, {223, 8747}, {281, 51574}, {314, 6338}, {393, 40590}, {478, 5317}, {650, 35071}, {663, 38985}, {823, 10001}, {942, 1859}, {1096, 40611}, {1118, 36908}, {1147, 2194}, {1715, 40152}, {1946, 46093}, {2193, 37867}, {2204, 22391}, {2299, 36033}, {3064, 34591}, {4183, 11517}, {6059, 40600}, {6337, 31623}, {6520, 47345}, {6708, 42385}, {15352, 39060}, {15526, 44426}, {17926, 35072}, {36126, 39053}, {36421, 40582}
X(52385) = cevapoint of X(i) and X(j) for these (i,j): {65, 8807}, {3682, 40152}
X(52385) = crosspoint of X(i) and X(j) for these (i,j): {69, 326}, {7055, 7183}
X(52385) = crosssum of X(i) and X(j) for these (i,j): {25, 1096}, {4516, 18344}
X(52385) = crossdifference of every pair of points on line {3063, 18344}
X(52385) = barycentric product X(i)*X(j) for these {i,j}: {3, 1231}, {7, 3998}, {10, 7183}, {37, 7055}, {63, 307}, {65, 3926}, {69, 1214}, {71, 7182}, {72, 348}, {73, 304}, {75, 40152}, {76, 22341}, {77, 306}, {85, 3682}, {108, 4143}, {201, 17206}, {222, 20336}, {225, 1102}, {226, 326}, {255, 349}, {274, 7066}, {305, 1409}, {313, 7125}, {321, 1804}, {332, 37755}, {345, 1439}, {394, 1441}, {520, 4554}, {525, 6516}, {603, 40071}, {651, 3265}, {664, 24018}, {822, 4572}, {1259, 1446}, {1264, 1427}, {1332, 17094}, {1367, 4567}, {1444, 26942}, {1577, 6517}, {1792, 20618}, {1812, 6356}, {1813, 14208}, {1880, 4176}, {1978, 51641}, {2632, 4620}, {3267, 36059}, {3668, 3719}, {3694, 7056}, {3710, 7177}, {3964, 40149}, {3990, 6063}, {4055, 20567}, {4131, 4552}, {4551, 30805}, {4561, 51640}, {4564, 17216}, {7138, 28660}, {7335, 27801}, {15413, 23067}, {18604, 34388}, {27832, 52354}
X(52385) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 8748}, {2, 1896}, {3, 1172}, {21, 36421}, {37, 1857}, {48, 2299}, {56, 5317}, {57, 8747}, {63, 29}, {65, 393}, {69, 31623}, {71, 33}, {72, 281}, {73, 19}, {77, 27}, {78, 2322}, {108, 6529}, {109, 24019}, {184, 2204}, {201, 1826}, {212, 2332}, {213, 6059}, {219, 4183}, {222, 28}, {225, 6520}, {226, 158}, {228, 607}, {255, 284}, {283, 2326}, {304, 44130}, {306, 318}, {307, 92}, {326, 333}, {348, 286}, {394, 21}, {520, 650}, {521, 17926}, {525, 44426}, {577, 2194}, {603, 1474}, {647, 18344}, {651, 107}, {653, 36126}, {656, 3064}, {664, 823}, {822, 663}, {828, 11436}, {836, 30223}, {1014, 36419}, {1020, 36127}, {1092, 2193}, {1102, 332}, {1214, 4}, {1231, 264}, {1259, 2287}, {1332, 36797}, {1367, 16732}, {1400, 1096}, {1402, 2207}, {1408, 36420}, {1409, 25}, {1410, 608}, {1415, 32713}, {1425, 1880}, {1427, 1118}, {1429, 34856}, {1437, 2189}, {1439, 278}, {1441, 2052}, {1444, 46103}, {1790, 270}, {1804, 81}, {1813, 162}, {1880, 6524}, {2197, 1824}, {2200, 2212}, {2286, 4206}, {2289, 2328}, {2318, 7079}, {2632, 21044}, {3173, 30733}, {3265, 4391}, {3269, 4516}, {3682, 9}, {3694, 7046}, {3710, 7101}, {3719, 1043}, {3926, 314}, {3964, 1812}, {3990, 55}, {3998, 8}, {4047, 461}, {4055, 41}, {4091, 3737}, {4131, 4560}, {4143, 35518}, {4158, 3694}, {4303, 46884}, {4554, 6528}, {4620, 23999}, {6356, 40149}, {6505, 3559}, {6507, 283}, {6511, 3193}, {6514, 1098}, {6516, 648}, {6517, 662}, {6518, 15146}, {7011, 3194}, {7013, 41083}, {7053, 1396}, {7055, 274}, {7066, 37}, {7125, 58}, {7138, 1400}, {7182, 44129}, {7183, 86}, {7335, 1333}, {8807, 6523}, {14208, 46110}, {16730, 1364}, {17094, 17924}, {17216, 4858}, {18026, 15352}, {18210, 8735}, {18591, 1859}, {18592, 42385}, {18604, 60}, {20336, 7017}, {20752, 37908}, {20769, 14024}, {22057, 2082}, {22128, 17515}, {22129, 17519}, {22341, 6}, {23067, 1783}, {23144, 4233}, {23224, 7252}, {23620, 40987}, {24018, 522}, {26942, 41013}, {30456, 6525}, {30805, 18155}, {32320, 1946}, {32660, 32676}, {36054, 21789}, {36059, 112}, {37755, 225}, {39201, 3063}, {39791, 1841}, {40149, 1093}, {40152, 1}, {41087, 7008}, {41393, 1865}, {44717, 5379}, {51640, 7649}, {51641, 649}, {52037, 40836}
X(52385) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {63, 6505, 21482}, {72, 1439, 307}, {326, 7183, 1804}, {2893, 16091, 7282}

X(52386) = X(27)-ISOCONJUGATE OF X(28)

X(52386) lies on these lines: {2, 40422}, {6, 1260}, {10, 37}, {32, 2911}, {71, 228}, {72, 18591}, {78, 216}, {101, 38868}, {115, 41508}, {201, 2197}, {219, 1794}, {220, 2301}, {306, 18592}, {393, 7080}, {577, 1259}, {836, 3682}, {1018, 2324}, {1331, 22133}, {1818, 14597}, {1865, 17757}, {2281, 7109}, {2289, 35072}, {2295, 3553}, {3969, 25000}, {7368, 30457}, {17796, 41332}, {21031, 21860}, {21075, 21854}, {25091, 25964}

X(52386) = isogonal conjugate of X(36419)
X(52386) = isotomic conjugate of the polar conjugate of X(3690)
X(52386) = isogonal conjugate of the polar conjugate of X(3695)
X(52386) = polar conjugate of the isotomic conjugate of X(4158)
X(52386) = X(i)-Ceva conjugate of X(j) for these (i,j): {1252, 4574}, {3694, 3949}, {3695, 3690}
X(52386) = X(i)-isoconjugate of X(j) for these (i,j): {1, 36419}, {27, 28}, {29, 1396}, {34, 46103}, {75, 36420}, {81, 8747}, {86, 5317}, {107, 1019}, {158, 593}, {162, 17925}, {244, 23582}, {269, 36421}, {270, 278}, {273, 2189}, {286, 1474}, {393, 757}, {811, 43925}, {823, 3733}, {849, 2052}, {873, 2207}, {1014, 8748}, {1015, 23999}, {1086, 24000}, {1096, 1509}, {1111, 23964}, {1118, 2185}, {1119, 2326}, {1412, 1896}, {2203, 44129}, {3942, 32230}, {4211, 40411}, {7192, 24019}, {7199, 32713}, {7254, 36126}, {24033, 26856}, {34856, 37128}, {39179, 46151}, {40395, 46883}, {42067, 46254}
X(52386) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 36419}, {27, 40591}, {125, 17925}, {206, 36420}, {286, 51574}, {393, 40607}, {521, 26856}, {525, 23989}, {593, 1147}, {647, 2973}, {1019, 38985}, {1509, 6503}, {1565, 17434}, {1896, 40599}, {2052, 4075}, {5317, 40600}, {6600, 36421}, {7192, 35071}, {7254, 46093}, {8747, 40586}, {11517, 46103}, {17423, 43925}
X(52386) = crosspoint of X(i) and X(j) for these (i,j): {72, 40161}, {1252, 4574}, {3682, 3998}
X(52386) = crosssum of X(i) and X(j) for these (i,j): {6, 1612}, {1086, 17925}, {2969, 43925}, {5317, 8747}
X(52386) = crossdifference of every pair of points on line {3733, 17925}
X(52386) = barycentric product X(i)*X(j) for these {i,j}: {3, 3695}, {4, 4158}, {8, 7066}, {10, 3682}, {12, 1259}, {37, 3998}, {48, 52369}, {59, 7068}, {63, 3949}, {69, 3690}, {71, 306}, {72, 72}, {73, 3710}, {78, 201}, {181, 1264}, {219, 26942}, {228, 20336}, {255, 1089}, {307, 2318}, {313, 4055}, {321, 3990}, {326, 756}, {341, 7138}, {345, 2197}, {394, 594}, {520, 3952}, {525, 4574}, {577, 28654}, {765, 2632}, {822, 4033}, {1016, 3269}, {1018, 24018}, {1092, 7141}, {1110, 17879}, {1214, 3694}, {1252, 15526}, {1260, 6356}, {1265, 1425}, {1331, 4064}, {1367, 6065}, {1500, 3926}, {1804, 6057}, {2171, 3719}, {2200, 40071}, {2289, 6358}, {2321, 40152}, {2972, 15742}, {3265, 4557}, {3692, 37755}, {3701, 22341}, {3964, 7140}, {4091, 4103}, {4131, 40521}, {6056, 34388}, {7055, 7064}, {23067, 52355}, {23990, 36793}, {27808, 39201}, {40161, 51574}
X(52386) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 36419}, {32, 36420}, {42, 8747}, {71, 27}, {72, 286}, {125, 2973}, {181, 1118}, {201, 273}, {210, 1896}, {212, 270}, {213, 5317}, {219, 46103}, {220, 36421}, {228, 28}, {255, 757}, {306, 44129}, {326, 873}, {394, 1509}, {520, 7192}, {577, 593}, {594, 2052}, {647, 17925}, {756, 158}, {765, 23999}, {822, 1019}, {872, 1096}, {1018, 823}, {1110, 24000}, {1252, 23582}, {1259, 261}, {1264, 18021}, {1334, 8748}, {1409, 1396}, {1425, 1119}, {1500, 393}, {1802, 2326}, {1804, 552}, {2197, 278}, {2200, 1474}, {2289, 2185}, {2318, 29}, {2632, 1111}, {2972, 1565}, {3049, 43925}, {3269, 1086}, {3682, 86}, {3690, 4}, {3694, 31623}, {3695, 264}, {3710, 44130}, {3747, 34856}, {3949, 92}, {3952, 6528}, {3990, 81}, {3998, 274}, {4055, 58}, {4064, 46107}, {4158, 69}, {4557, 107}, {4574, 648}, {6056, 60}, {7064, 1857}, {7065, 1364}, {7066, 7}, {7068, 34387}, {7109, 2207}, {7138, 269}, {7140, 1093}, {7335, 7341}, {15526, 23989}, {18604, 763}, {20727, 31917}, {20975, 2969}, {22080, 31900}, {22341, 1014}, {22363, 4211}, {23990, 23964}, {24018, 7199}, {26942, 331}, {28654, 18027}, {32320, 7254}, {34980, 3937}, {35072, 26856}, {37754, 3942}, {37755, 1847}, {39201, 3733}, {40152, 1434}, {51641, 7203}, {52369, 1969}
X(52386) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 21858, 1834}, {72, 51574, 18591}

X(52387) = X(28)-ISOCONJUGATE OF X(28)

X(52387) lies on these lines: {1, 1257}, {10, 321}, {31, 3811}, {71, 72}, {78, 1794}, {100, 2939}, {200, 1005}, {201, 3695}, {255, 3719}, {345, 44706}, {872, 1245}, {1215, 3085}, {1259, 24031}, {2198, 3930}, {3556, 4557}, {3670, 20106}, {3682, 3998}, {3693, 44547}, {3842, 19855}, {3932, 21916}, {3952, 7080}, {3965, 45120}, {4073, 4294}, {4158, 7066}, {4303, 25083}, {4574, 7078}, {4712, 5904}, {5295, 40967}, {7283, 44694}

X(52387) = isotomic conjugate of the polar conjugate of X(3949)
X(52387) = isogonal conjugate of the polar conjugate of X(52369)
X(52387) = X(i)-Ceva conjugate of X(j) for these (i,j): {3710, 3695}, {52369, 3949}
X(52387) = X(3269)-cross conjugate of X(24018)
X(52387) = X(i)-isoconjugate of X(j) for these (i,j): {2, 36420}, {6, 36419}, {27, 1474}, {28, 28}, {34, 270}, {58, 8747}, {60, 1118}, {81, 5317}, {107, 3733}, {112, 17925}, {158, 849}, {244, 24000}, {250, 2969}, {261, 7337}, {278, 2189}, {286, 2203}, {393, 593}, {552, 6059}, {608, 46103}, {648, 43925}, {741, 34856}, {757, 1096}, {1015, 23582}, {1019, 24019}, {1086, 23964}, {1172, 1396}, {1407, 36421}, {1408, 1896}, {1412, 8748}, {1435, 2326}, {1509, 2207}, {1857, 7341}, {3248, 23999}, {3937, 32230}, {6529, 7254}, {7192, 32713}, {18020, 42067}, {23985, 26856}, {23989, 41937}, {40395, 46890}
X(52387) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 36419}, {10, 8747}, {27, 51574}, {28, 40591}, {158, 4075}, {270, 11517}, {525, 1111}, {757, 6503}, {849, 1147}, {873, 6338}, {942, 46883}, {1019, 35071}, {1096, 40607}, {3733, 38985}, {3942, 17434}, {5317, 40586}, {8299, 34856}, {8748, 40599}, {17925, 34591}, {24771, 36421}, {32664, 36420}
X(52387) = barycentric product X(i)*X(j) for these {i,j}: {3, 52369}, {10, 3998}, {12, 3719}, {63, 3695}, {69, 3949}, {71, 20336}, {72, 306}, {78, 26942}, {92, 4158}, {201, 345}, {228, 40071}, {255, 28654}, {304, 3690}, {307, 3694}, {312, 7066}, {313, 3990}, {321, 3682}, {326, 594}, {394, 1089}, {520, 4033}, {756, 3926}, {765, 15526}, {822, 27808}, {1016, 2632}, {1018, 3265}, {1102, 7140}, {1110, 36793}, {1214, 3710}, {1231, 2318}, {1252, 17879}, {1259, 6358}, {1264, 2171}, {1265, 37755}, {1332, 4064}, {2197, 3718}, {2289, 34388}, {3269, 7035}, {3692, 6356}, {3701, 40152}, {3952, 24018}, {4055, 27801}, {4103, 4131}, {4564, 7068}, {4574, 14208}, {6057, 7183}, {6507, 7141}, {22341, 30713}, {30805, 40521}
X(52387) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 36419}, {31, 36420}, {37, 8747}, {42, 5317}, {71, 28}, {72, 27}, {73, 1396}, {78, 46103}, {200, 36421}, {201, 278}, {210, 8748}, {212, 2189}, {219, 270}, {228, 1474}, {255, 593}, {306, 286}, {326, 1509}, {394, 757}, {520, 1019}, {577, 849}, {594, 158}, {656, 17925}, {756, 393}, {765, 23582}, {810, 43925}, {822, 3733}, {872, 2207}, {1016, 23999}, {1018, 107}, {1089, 2052}, {1110, 23964}, {1252, 24000}, {1259, 2185}, {1260, 2326}, {1425, 1435}, {1500, 1096}, {2171, 1118}, {2197, 34}, {2200, 2203}, {2238, 34856}, {2289, 60}, {2318, 1172}, {2321, 1896}, {2632, 1086}, {2972, 3942}, {3265, 7199}, {3269, 244}, {3682, 81}, {3690, 19}, {3694, 29}, {3695, 92}, {3708, 2969}, {3710, 31623}, {3719, 261}, {3926, 873}, {3949, 4}, {3952, 823}, {3958, 31900}, {3990, 58}, {3998, 86}, {4033, 6528}, {4047, 31903}, {4055, 1333}, {4064, 17924}, {4158, 63}, {4557, 24019}, {4574, 162}, {6056, 2150}, {6356, 1847}, {7066, 57}, {7068, 4858}, {7125, 7341}, {7138, 1407}, {7140, 6520}, {7141, 6521}, {7183, 552}, {15526, 1111}, {17216, 16727}, {17879, 23989}, {18591, 46883}, {20336, 44129}, {20902, 2973}, {21859, 36127}, {22341, 1412}, {23620, 4211}, {24018, 7192}, {24031, 26856}, {26942, 273}, {35309, 46151}, {37754, 3937}, {37755, 1119}, {40152, 1014}, {52369, 264}

X(52388) = X(28)-ISOCONJUGATE OF X(35)

X(52388) lies on these lines: {2, 582}, {3, 125}, {8, 6739}, {9, 46}, {10, 6757}, {40, 30447}, {63, 50462}, {72, 21912}, {78, 1060}, {318, 860}, {355, 36195}, {381, 30436}, {474, 24931}, {476, 36154}, {499, 8286}, {500, 3580}, {517, 27685}, {656, 44706}, {858, 48882}, {956, 26700}, {1063, 44113}, {1376, 34829}, {1479, 8287}, {1836, 14873}, {2166, 18395}, {3419, 34301}, {3579, 27553}, {3679, 50148}, {3708, 23899}, {3753, 27714}, {5657, 27686}, {5690, 36155}, {6361, 27554}, {7420, 20299}, {12699, 27555}, {13747, 25669}, {16297, 24206}, {16451, 23293}, {16452, 26913}, {16453, 21243}, {18749, 20565}, {20840, 32223}, {23555, 51462}, {23674, 33298}, {25015, 30690}, {25442, 37527}, {25646, 46623}, {26446, 27687}, {26958, 37284}, {27529, 37154}, {27721, 50810}, {37648, 50324}, {41586, 48907}, {48928, 51360}

X(52388) = isotomic conjugate of the polar conjugate of X(8818)
X(52388) = X(52344)-Ceva conjugate of X(6757)
X(52388) = X(i)-cross conjugate of X(j) for these (i,j): {3958, 306}, {18675, 226}
X(52388) = X(i)-isoconjugate of X(j) for these (i,j): {4, 17104}, {19, 40214}, {27, 2174}, {28, 35}, {29, 1399}, {34, 35193}, {56, 11107}, {57, 41502}, {58, 6198}, {60, 1825}, {86, 14975}, {107, 23226}, {112, 14838}, {162, 2605}, {186, 759}, {250, 2611}, {270, 2594}, {278, 35192}, {319, 2203}, {1172, 2003}, {1175, 1844}, {1442, 2299}, {1474, 3219}, {1973, 34016}, {2189, 16577}, {2194, 7282}, {2204, 17095}, {4467, 32676}, {9274, 35235}, {14616, 34397}, {16585, 40570}, {21741, 46103}, {22094, 24000}, {32671, 44427}, {37140, 47230}
X(52388) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 11107}, {6, 40214}, {10, 6198}, {35, 40591}, {125, 2605}, {186, 34586}, {226, 1442}, {340, 51583}, {445, 40937}, {500, 942}, {647, 8287}, {1214, 7282}, {3219, 51574}, {4467, 15526}, {5452, 41502}, {6337, 34016}, {7359, 14920}, {11517, 35193}, {14838, 34591}, {14975, 40600}, {16755, 40618}, {17104, 36033}, {23226, 38985}
X(52388) = cevapoint of X(125) and X(656)
X(52388) = crosssum of X(2605) and X(22094)
X(52388) = crossdifference of every pair of points on line {2605, 47230}
X(52388) = barycentric product X(i)*X(j) for these {i,j}: {63, 6757}, {69, 8818}, {71, 20565}, {72, 30690}, {78, 43682}, {79, 306}, {265, 3936}, {307, 7110}, {321, 7100}, {328, 2245}, {349, 8606}, {525, 6742}, {656, 15455}, {1214, 52344}, {1231, 7073}, {1789, 6358}, {2160, 20336}, {3615, 26942}, {6186, 40071}, {38340, 52355}
X(52388) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 40214}, {9, 11107}, {37, 6198}, {48, 17104}, {55, 41502}, {69, 34016}, {71, 35}, {72, 3219}, {73, 2003}, {79, 27}, {125, 8287}, {201, 16577}, {212, 35192}, {213, 14975}, {219, 35193}, {226, 7282}, {228, 2174}, {265, 24624}, {306, 319}, {307, 17095}, {442, 445}, {525, 4467}, {647, 2605}, {656, 14838}, {822, 23226}, {860, 14165}, {1214, 1442}, {1409, 1399}, {1789, 2185}, {1901, 46468}, {2160, 28}, {2171, 1825}, {2197, 2594}, {2245, 186}, {2294, 1844}, {3269, 22094}, {3615, 46103}, {3694, 4420}, {3695, 3969}, {3708, 2611}, {3710, 42033}, {3916, 17190}, {3936, 340}, {3949, 3678}, {3958, 3647}, {4025, 16755}, {4064, 7265}, {6186, 1474}, {6370, 44427}, {6739, 14920}, {6742, 648}, {6757, 92}, {7073, 1172}, {7100, 81}, {7110, 29}, {8606, 284}, {8611, 35057}, {8818, 4}, {14208, 18160}, {14844, 2905}, {15455, 811}, {18210, 7202}, {18591, 500}, {20336, 33939}, {20565, 44129}, {20727, 7186}, {20902, 17886}, {20975, 20982}, {21046, 21054}, {21134, 21141}, {22080, 17454}, {26942, 40999}, {30690, 286}, {32662, 36069}, {36061, 37140}, {40952, 44095}, {41014, 3578}, {42666, 47230}, {43261, 31901}, {43682, 273}, {52153, 34079}, {52344, 31623}

X(52389) = X(28)-ISOCONJUGATE OF X(40)

X(52389) lies on the cubic K345 and these lines: {1, 1073}, {2, 280}, {3, 9}, {10, 227}, {37, 46837}, {58, 36049}, {72, 856}, {78, 271}, {97, 44687}, {123, 6831}, {189, 24635}, {210, 22341}, {226, 8806}, {281, 3346}, {307, 6355}, {405, 7008}, {939, 1256}, {946, 16596}, {960, 46021}, {997, 2192}, {1038, 1422}, {1210, 2968}, {1297, 40117}, {1427, 20309}, {1440, 8813}, {1753, 3149}, {1793, 1800}, {2184, 3182}, {2188, 5440}, {3682, 3694}, {3710, 3998}, {3718, 3926}, {5266, 52041}, {5703, 24553}, {5705, 37565}, {6700, 7515}, {6762, 38290}, {6765, 38288}, {7003, 40937}, {7151, 37065}, {13138, 14919}, {14376, 37592}, {16573, 16583}, {17073, 19843}, {18446, 46881}, {18634, 20208}, {18876, 32652}, {20764, 34790}, {21075, 51368}, {25917, 40946}, {35072, 46830}, {36055, 40396}

X(52389) = isogonal conjugate of X(3194)
X(52389) = complement of X(1895)
X(52389) = complement of the isogonal conjugate of X(19614)
X(52389) = complement of the isotomic conjugate of X(19611)
X(52389) = isotomic conjugate of the anticomplement of X(46837)
X(52389) = isotomic conjugate of the polar conjugate of X(1903)
X(52389) = medial-isogonal conjugate of X(20308)
X(52389) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 20308}, {3, 2883}, {6, 20207}, {25, 20265}, {32, 46829}, {48, 36908}, {64, 5}, {184, 1249}, {253, 21243}, {520, 35968}, {1073, 141}, {1092, 31377}, {1301, 520}, {1974, 20313}, {2155, 226}, {2184, 20305}, {8798, 1209}, {11589, 113}, {14379, 3}, {14642, 2}, {15394, 1368}, {17510, 14390}, {19611, 2887}, {19614, 10}, {30457, 41883}, {33581, 6}, {34403, 626}, {39201, 39020}, {41489, 13567}, {42658, 13613}, {44181, 46185}, {46639, 30476}, {47437, 3343}, {52158, 34831}
X(52389) = X(i)-Ceva conjugate of X(j) for these (i,j): {280, 39130}, {41081, 41087}
X(52389) = X(i)-cross conjugate of X(j) for these (i,j): {37, 1214}, {73, 72}, {18675, 63}, {41087, 52037}, {46837, 2}
X(52389) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3194}, {4, 2360}, {6, 41083}, {19, 1817}, {21, 208}, {25, 8822}, {27, 198}, {28, 40}, {29, 221}, {58, 7952}, {81, 2331}, {86, 3195}, {112, 14837}, {162, 6129}, {196, 284}, {204, 41082}, {223, 1172}, {227, 270}, {286, 2187}, {322, 2203}, {329, 1474}, {333, 3209}, {342, 2194}, {347, 2299}, {608, 27398}, {1014, 40971}, {1118, 1819}, {1396, 2324}, {1437, 47372}, {1896, 7114}, {2199, 31623}, {2204, 40702}, {2322, 6611}, {2332, 14256}, {3342, 8885}, {4570, 38362}, {7011, 8748}, {7078, 8747}, {17896, 32676}, {41088, 44698}
X(52389) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 3194}, {6, 1817}, {9, 41083}, {10, 7952}, {29, 3341}, {40, 40591}, {125, 6129}, {196, 40590}, {208, 40611}, {226, 347}, {329, 51574}, {342, 1214}, {2331, 40586}, {2360, 36033}, {3195, 40600}, {3343, 41082}, {6505, 8822}, {14837, 34591}, {15526, 17896}, {38362, 50330}
X(52389) = cevapoint of X(1903) and X(41086)
X(52389) = crosspoint of X(i) and X(j) for these (i,j): {2, 19611}, {271, 280}
X(52389) = crosssum of X(i) and X(j) for these (i,j): {6, 204}, {208, 221}, {2331, 3195}
X(52389) = trilinear pole of line {520, 8611}
X(52389) = barycentric product X(i)*X(j) for these {i,j}: {8, 52037}, {10, 41081}, {63, 39130}, {65, 44189}, {69, 1903}, {71, 309}, {72, 189}, {73, 34404}, {75, 41087}, {78, 8808}, {84, 306}, {210, 34400}, {226, 271}, {228, 44190}, {268, 1441}, {280, 1214}, {282, 307}, {285, 26942}, {304, 2357}, {321, 1433}, {349, 2188}, {525, 13138}, {656, 44327}, {1231, 2192}, {1264, 2358}, {1422, 3710}, {1436, 20336}, {1440, 3694}, {1792, 13853}, {2208, 40071}, {2287, 6355}, {3265, 40117}, {3267, 32652}, {3998, 40836}, {7020, 40152}, {14208, 36049}, {34403, 41086}, {37141, 52355}
X(52389) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 41083}, {3, 1817}, {6, 3194}, {37, 7952}, {42, 2331}, {48, 2360}, {63, 8822}, {65, 196}, {71, 40}, {72, 329}, {73, 223}, {78, 27398}, {84, 27}, {189, 286}, {213, 3195}, {226, 342}, {228, 198}, {268, 21}, {271, 333}, {280, 31623}, {282, 29}, {285, 46103}, {306, 322}, {307, 40702}, {309, 44129}, {525, 17896}, {647, 6129}, {656, 14837}, {1073, 41082}, {1214, 347}, {1334, 40971}, {1400, 208}, {1402, 3209}, {1409, 221}, {1410, 6611}, {1413, 1396}, {1433, 81}, {1436, 28}, {1439, 14256}, {1441, 40701}, {1826, 47372}, {1903, 4}, {2188, 284}, {2192, 1172}, {2197, 227}, {2200, 2187}, {2208, 1474}, {2289, 1819}, {2318, 2324}, {2357, 19}, {2358, 1118}, {3125, 38362}, {3690, 21871}, {3694, 7080}, {3949, 21075}, {3990, 7078}, {6355, 1446}, {7003, 1896}, {7008, 8748}, {7118, 2299}, {7129, 8747}, {7151, 5317}, {7367, 4183}, {8611, 8058}, {8808, 273}, {13138, 648}, {22341, 7011}, {32652, 112}, {34404, 44130}, {36049, 162}, {39130, 92}, {40117, 107}, {40152, 7013}, {41081, 86}, {41086, 1249}, {41087, 1}, {44189, 314}, {44327, 811}, {52037, 7}, {52078, 44697}
X(52389) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 280, 40836}, {271, 41081, 1433}

X(52390) = X(29)-ISOCONJUGATE OF X(35)

X(52390) lies on the Jerabek circumhyperbola and these lines: {1, 74}, {3, 7100}, {4, 79}, {6, 1406}, {7, 8044}, {54, 3336}, {56, 34435}, {57, 1175}, {64, 7073}, {71, 41393}, {72, 21912}, {656, 14380}, {1020, 2599}, {1789, 4652}, {1903, 8818}, {2457, 3657}, {3426, 7986}, {3431, 18593}, {3485, 14844}, {3615, 37142}, {3649, 27555}, {3754, 6757}, {5903, 50148}, {6742, 14923}, {9275, 43700}, {9405, 13746}, {10693, 27685}, {11101, 13486}, {11270, 37571}, {13369, 34800}, {15453, 51645}, {22342, 37755}, {23070, 43704}, {31794, 44835}, {37732, 43855}, {48857, 51223}

X(52390) = isogonal conjugate of X(11107)
X(52390) = isogonal conjugate of the polar conjugate of X(43682)
X(52390) = X(i)-cross conjugate of X(j) for these (i,j): {3708, 51640}, {39791, 73}
X(52390) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11107}, {2, 41502}, {4, 35193}, {21, 6198}, {28, 4420}, {29, 35}, {92, 35192}, {162, 35057}, {186, 6740}, {250, 6741}, {270, 3678}, {281, 40214}, {314, 14975}, {318, 17104}, {319, 2299}, {607, 34016}, {648, 9404}, {1098, 1825}, {1172, 3219}, {1442, 4183}, {1474, 42033}, {2003, 2322}, {2174, 31623}, {2189, 3969}, {2204, 33939}, {2326, 16577}, {2328, 7282}, {2332, 17095}, {2605, 36797}
X(52390) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 11107}, {125, 35057}, {226, 319}, {942, 31938}, {1825, 15267}, {4420, 40591}, {6198, 40611}, {7282, 36908}, {22391, 35192}, {32664, 41502}, {35193, 36033}, {42033, 51574}
X(52390) = crosspoint of X(79) and X(7100)
X(52390) = crosssum of X(i) and X(j) for these (i,j): {35, 6198}, {186, 35201}
X(52390) = trilinear pole of line {647, 2631}
X(52390) = barycentric product X(i)*X(j) for these {i,j}: {3, 43682}, {73, 30690}, {77, 8818}, {79, 1214}, {125, 35049}, {222, 6757}, {226, 7100}, {265, 18593}, {307, 2160}, {525, 26700}, {656, 38340}, {1231, 6186}, {1409, 20565}, {1439, 7110}, {1446, 8606}, {1789, 6354}, {3615, 37755}, {6742, 51640}
X(52390) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 11107}, {31, 41502}, {48, 35193}, {71, 4420}, {72, 42033}, {73, 3219}, {77, 34016}, {79, 31623}, {184, 35192}, {201, 3969}, {307, 33939}, {603, 40214}, {647, 35057}, {810, 9404}, {1214, 319}, {1400, 6198}, {1409, 35}, {1410, 2003}, {1425, 16577}, {1427, 7282}, {1439, 17095}, {1789, 7058}, {1835, 14165}, {2160, 29}, {2197, 3678}, {3708, 6741}, {6186, 1172}, {6757, 7017}, {7073, 2322}, {7100, 333}, {8606, 2287}, {8818, 318}, {14582, 52356}, {17094, 18160}, {18591, 31938}, {18593, 340}, {26700, 648}, {30690, 44130}, {35049, 18020}, {37755, 40999}, {38340, 811}, {39791, 16585}, {43682, 264}, {50433, 1793}, {51640, 4467}, {51645, 44427}, {52153, 2341}
X(52390) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {79, 34301, 4}, {3649, 40669, 27555}, {7100, 50462, 3}

X(52391) = X(29)-ISOCONJUGATE OF X(36)

X(52391) lies on the Jerabek circumhyperbola and these lines: {1, 54}, {3, 201}, {4, 80}, {6, 1411}, {59, 7984}, {64, 37567}, {65, 3120}, {69, 17880}, {73, 18210}, {74, 484}, {290, 35174}, {655, 6740}, {758, 15065}, {759, 1175}, {1243, 45022}, {1246, 18815}, {1903, 21853}, {2006, 5292}, {2341, 46884}, {2595, 13746}, {2992, 36929}, {2993, 36928}, {3145, 14882}, {3431, 16577}, {4551, 43692}, {4552, 6224}, {7066, 43708}, {10260, 15530}, {10457, 46160}, {10950, 51879}, {11270, 37572}, {14528, 34471}, {23071, 43704}, {34259, 52351}, {40663, 50705}, {41329, 43712}, {41538, 43703}

X(52391) = isogonal conjugate of X(17515)
X(52391) = X(2631)-cross conjugate of X(1020)
X(52391) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17515}, {21, 1870}, {27, 2323}, {28, 4511}, {29, 36}, {58, 5081}, {60, 860}, {92, 4282}, {110, 44428}, {112, 3904}, {162, 3738}, {186, 3615}, {261, 44113}, {270, 758}, {284, 17923}, {286, 2361}, {320, 2299}, {648, 654}, {811, 8648}, {933, 6369}, {1098, 1835}, {1172, 3218}, {1443, 4183}, {1474, 32851}, {2189, 3936}, {2204, 20924}, {2245, 46103}, {2326, 18593}, {2332, 17078}, {3737, 4242}, {7113, 31623}, {8748, 22128}
X(52391) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 17515}, {10, 5081}, {29, 15898}, {125, 3738}, {226, 320}, {244, 44428}, {1835, 15267}, {1870, 40611}, {2600, 15450}, {3904, 34591}, {4282, 22391}, {4511, 40591}, {8648, 17423}, {17923, 40590}, {32851, 51574}
X(52391) = crosspoint of X(80) and X(1807)
X(52391) = crosssum of X(i) and X(j) for these (i,j): {36, 1870}, {186, 51801}
X(52391) = trilinear pole of line {647, 2197}
X(52391) = crossdifference of every pair of points on line {2600, 3738}
X(52391) = barycentric product X(i)*X(j) for these {i,j}: {65, 52351}, {71, 18815}, {72, 2006}, {73, 18359}, {80, 1214}, {94, 22342}, {201, 24624}, {222, 15065}, {226, 1807}, {265, 16577}, {306, 1411}, {307, 2161}, {328, 21741}, {348, 34857}, {525, 2222}, {647, 35174}, {655, 656}, {759, 26942}, {810, 46405}, {1231, 6187}, {1409, 20566}, {1439, 36910}, {1793, 6354}, {2197, 14616}, {2341, 6356}, {6740, 37755}, {14208, 32675}, {51562, 51640}
X(52391) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 17515}, {37, 5081}, {65, 17923}, {71, 4511}, {72, 32851}, {73, 3218}, {80, 31623}, {184, 4282}, {201, 3936}, {228, 2323}, {307, 20924}, {647, 3738}, {655, 811}, {656, 3904}, {661, 44428}, {759, 46103}, {810, 654}, {1214, 320}, {1231, 40075}, {1400, 1870}, {1409, 36}, {1411, 27}, {1425, 18593}, {1439, 17078}, {1793, 7058}, {1807, 333}, {1825, 14165}, {2006, 286}, {2161, 29}, {2171, 860}, {2197, 758}, {2200, 2361}, {2222, 648}, {2599, 14918}, {3049, 8648}, {4559, 4242}, {6187, 1172}, {15065, 7017}, {15451, 2600}, {16577, 340}, {18359, 44130}, {18815, 44129}, {21741, 186}, {22341, 22128}, {22342, 323}, {23067, 4585}, {26942, 35550}, {32675, 162}, {34079, 270}, {34857, 281}, {35174, 6331}, {37755, 41804}, {50433, 1789}, {51640, 4453}, {52351, 314}
X(52391) = {X(80),X(34300)}-harmonic conjugate of X(4)

X(52392) = X(33)-ISOCONJUGATE OF X(36)

X(52392) lies on cubic K660 and these lines: {1, 14516}, {7, 80}, {63, 343}, {69, 17880}, {77, 1060}, {81, 226}, {86, 39277}, {189, 5905}, {265, 1439}, {286, 7282}, {307, 1444}, {320, 18816}, {515, 22464}, {527, 655}, {535, 36589}, {759, 13395}, {969, 10401}, {1411, 7190}, {1813, 4466}, {1814, 9028}, {2161, 5845}, {2222, 9436}, {2250, 24029}, {2283, 24713}, {3738, 36038}, {4318, 29046}, {5219, 16554}, {8048, 34242}, {10446, 34300}, {12831, 14204}, {14584, 30617}, {20078, 41226}, {31164, 52212}, {31266, 36949}, {34393, 51565}

X(52392) = midpoint of X(5905) and X(37781)
X(52392) = reflection of X(i) in X(j) for these {i,j}: {63, 26932}, {651, 226}
X(52392) = isotomic conjugate of X(5081)
X(52392) = isotomic conjugate of the anticomplement of X(46974)
X(52392) = isotomic conjugate of the polar conjugate of X(2006)
X(52392) = X(2006)-anticomplementary conjugate of X(151)
X(52392) = X(14616)-Ceva conjugate of X(18815)
X(52392) = X(i)-cross conjugate of X(j) for these (i,j): {912, 63}, {1807, 52351}, {2850, 651}, {46974, 2}
X(52392) = X(i)-isoconjugate of X(j) for these (i,j): {4, 2361}, {19, 2323}, {21, 44113}, {25, 4511}, {29, 3724}, {31, 5081}, {33, 36}, {41, 17923}, {42, 17515}, {55, 1870}, {186, 7073}, {281, 7113}, {320, 2212}, {607, 3218}, {654, 1783}, {663, 4242}, {692, 44428}, {758, 2299}, {860, 2194}, {1172, 2245}, {1443, 7071}, {1464, 4183}, {1826, 4282}, {1835, 2328}, {1845, 2342}, {1897, 8648}, {1973, 32851}, {1983, 3064}, {2189, 4053}, {2204, 3936}, {2332, 18593}, {3738, 8750}, {34397, 52344}
X(52392) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 5081}, {6, 2323}, {33, 15898}, {223, 1870}, {226, 758}, {654, 39006}, {860, 1214}, {1086, 44428}, {1835, 36908}, {2361, 36033}, {3160, 17923}, {3738, 26932}, {3904, 40618}, {4511, 6505}, {6337, 32851}, {7046, 36909}, {8648, 34467}, {17515, 40592}, {40611, 44113}
X(52392) = cevapoint of X(i) and X(j) for these (i,j): {226, 515}, {26932, 39471}
X(52392) = crosssum of X(3724) and X(44113)
X(52392) = trilinear pole of line {905, 1214}
X(52392) = barycentric product X(i)*X(j) for these {i,j}: {7, 52351}, {63, 18815}, {69, 2006}, {77, 18359}, {80, 348}, {85, 1807}, {222, 20566}, {265, 17095}, {304, 1411}, {307, 24624}, {328, 2003}, {655, 4025}, {759, 1231}, {905, 35174}, {1214, 14616}, {1446, 1793}, {1459, 46405}, {2161, 7182}, {2222, 15413}, {7056, 36910}, {17094, 47318}
X(52392) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 5081}, {3, 2323}, {7, 17923}, {48, 2361}, {57, 1870}, {63, 4511}, {69, 32851}, {73, 2245}, {77, 3218}, {80, 281}, {81, 17515}, {201, 4053}, {222, 36}, {226, 860}, {265, 7110}, {307, 3936}, {348, 320}, {514, 44428}, {603, 7113}, {651, 4242}, {655, 1897}, {759, 1172}, {905, 3738}, {1214, 758}, {1231, 35550}, {1400, 44113}, {1409, 3724}, {1411, 19}, {1427, 1835}, {1437, 4282}, {1439, 18593}, {1459, 654}, {1465, 1845}, {1793, 2287}, {1804, 22128}, {1807, 9}, {2003, 186}, {2006, 4}, {2161, 33}, {2222, 1783}, {2341, 4183}, {4025, 3904}, {6187, 607}, {6516, 4585}, {6740, 2322}, {7056, 17078}, {7177, 1443}, {7182, 20924}, {7282, 14165}, {14584, 8756}, {14616, 31623}, {14628, 38462}, {17094, 4707}, {17095, 340}, {18359, 318}, {18815, 92}, {20566, 7017}, {22128, 4996}, {22383, 8648}, {23071, 26744}, {24624, 29}, {32675, 8750}, {34079, 2299}, {35174, 6335}, {36059, 1983}, {36910, 7046}, {47318, 36797}, {50433, 8606}, {52212, 1785}, {52351, 8}

X(52393) = X(35)-ISOCONJUGATE OF X(37)

X(52393) lies on these lines: {2, 8818}, {7, 40214}, {21, 36}, {57, 24624}, {60, 24470}, {81, 553}, {86, 17190}, {99, 18139}, {110, 11246}, {265, 28452}, {270, 31900}, {333, 2160}, {354, 5196}, {476, 840}, {497, 41504}, {554, 7051}, {593, 1086}, {662, 17483}, {757, 33150}, {1029, 8287}, {1081, 19373}, {1171, 17366}, {1262, 6354}, {1931, 26724}, {1989, 37646}, {2163, 23681}, {2185, 8025}, {2363, 19785}, {3100, 7073}, {3873, 6742}, {3972, 19684}, {4720, 17647}, {5745, 7110}, {10404, 37405}, {11076, 17012}, {13407, 37294}, {14377, 51311}, {17103, 32774}, {18601, 40432}, {19787, 20565}, {24617, 32911}, {32636, 37369}, {32680, 37222}

X(52393) = isotomic conjugate of X(3969)
X(52393) = X(35049)-Ceva conjugate of X(38340)
X(52393) = X(i)-cross conjugate of X(j) for these (i,j): {11, 7192}, {3337, 757}, {24470, 7}, {29821, 873}, {33135, 32010}
X(52393) = X(i)-isoconjugate of X(j) for these (i,j): {6, 3678}, {8, 21741}, {9, 2594}, {10, 2174}, {12, 35192}, {21, 21794}, {31, 3969}, {35, 37}, {41, 40999}, {42, 3219}, {50, 15065}, {55, 16577}, {71, 6198}, {201, 41502}, {210, 2003}, {213, 319}, {219, 1825}, {281, 22342}, {306, 14975}, {323, 34857}, {594, 17104}, {692, 7265}, {756, 40214}, {765, 20982}, {872, 34016}, {1018, 2605}, {1110, 8287}, {1252, 2611}, {1333, 7206}, {1334, 1442}, {1399, 2321}, {1400, 4420}, {1402, 42033}, {1918, 33939}, {2149, 6741}, {2171, 35193}, {2197, 11107}, {2624, 51562}, {3724, 41226}, {4551, 9404}, {4557, 14838}, {4559, 35057}, {4570, 21824}, {6187, 42701}, {7161, 50657}, {7343, 21864}, {14270, 36804}, {17886, 23990}
X(52393) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 3969}, {9, 3678}, {35, 40589}, {37, 7206}, {223, 16577}, {319, 6626}, {478, 2594}, {513, 20982}, {514, 8287}, {650, 6741}, {661, 2611}, {1086, 7265}, {3160, 40999}, {3219, 40592}, {4420, 40582}, {4467, 40620}, {4988, 21054}, {7110, 21081}, {21794, 40611}, {21824, 50330}, {33939, 34021}, {40605, 42033}, {40612, 42701}
X(52393) = cevapoint of X(i) and X(j) for these (i,j): {1, 583}, {79, 2160}, {1019, 1086}
X(52393) = trilinear pole of line {3737, 4960}
X(52393) = barycentric product X(i)*X(j) for these {i,j}: {7, 3615}, {58, 20565}, {79, 86}, {81, 30690}, {273, 1789}, {274, 2160}, {286, 7100}, {310, 6186}, {476, 4453}, {693, 13486}, {757, 6757}, {1014, 52344}, {1019, 15455}, {1434, 7110}, {1509, 8818}, {2185, 43682}, {3960, 32680}, {4560, 38340}, {4858, 35049}, {6742, 7192}, {14158, 40716}, {14844, 40164}, {17219, 34922}, {18155, 26700}
X(52393) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3678}, {2, 3969}, {7, 40999}, {10, 7206}, {11, 6741}, {21, 4420}, {28, 6198}, {34, 1825}, {56, 2594}, {57, 16577}, {58, 35}, {60, 35193}, {79, 10}, {81, 3219}, {86, 319}, {244, 2611}, {270, 11107}, {274, 33939}, {333, 42033}, {476, 51562}, {514, 7265}, {593, 40214}, {603, 22342}, {604, 21741}, {849, 17104}, {1014, 1442}, {1015, 20982}, {1019, 14838}, {1086, 8287}, {1111, 17886}, {1333, 2174}, {1393, 2599}, {1400, 21794}, {1408, 1399}, {1412, 2003}, {1434, 17095}, {1509, 34016}, {1789, 78}, {2150, 35192}, {2160, 37}, {2166, 15065}, {2189, 41502}, {2203, 14975}, {2594, 7144}, {3120, 21054}, {3125, 21824}, {3218, 42701}, {3615, 8}, {3733, 2605}, {3737, 35057}, {3937, 22094}, {3960, 32679}, {4453, 3268}, {4960, 23883}, {5358, 4354}, {6186, 42}, {6545, 21141}, {6742, 3952}, {6757, 1089}, {7073, 210}, {7100, 72}, {7110, 2321}, {7192, 4467}, {7199, 18160}, {7252, 9404}, {8025, 3578}, {8606, 2318}, {8818, 594}, {11076, 21864}, {13486, 100}, {14158, 484}, {14844, 21085}, {15455, 4033}, {16726, 7202}, {18180, 35194}, {20565, 313}, {21758, 2624}, {24624, 41226}, {26700, 4551}, {30581, 17190}, {30602, 502}, {30690, 321}, {32680, 36804}, {35049, 4564}, {38340, 4552}, {43682, 6358}, {46883, 1844}, {46890, 44095}, {52344, 3701}
X(52393) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {57, 43682, 38340}, {553, 18653, 81}

X(52394) = X(37)-ISOCONJUGATE OF X(39)

X(52394) lies on the conic {{A,B,C,X(2),X(7)}} and these lines: {2, 32}, {7, 1397}, {31, 75}, {58, 310}, {81, 335}, {86, 2206}, {199, 51862}, {273, 1395}, {314, 39281}, {673, 4599}, {675, 827}, {689, 727}, {871, 985}, {903, 4577}, {1088, 1106}, {1176, 1246}, {1268, 18082}, {1509, 39734}, {2296, 46289}, {3405, 21376}, {4593, 31002}, {4610, 7303}, {6384, 7121}, {6650, 33150}, {14621, 32774}, {15523, 33954}, {16889, 33730}, {17103, 40038}, {17141, 17150}, {17200, 29654}, {17763, 18099}, {18089, 27483}, {18824, 42371}, {33955, 39728}

X(52394) = isogonal conjugate of X(21035)
X(52394) = isotomic conjugate of X(15523)
X(52394) = polar conjugate of X(21016)
X(52394) = isogonal conjugate of the complement of X(17142)
X(52394) = isotomic conjugate of the anticomplement of X(29654)
X(52394) = isotomic conjugate of the complement of X(17150)
X(52394) = X(4593)-Ceva conjugate of X(39179)
X(52394) = X(i)-cross conjugate of X(j) for these (i,j): {171, 81}, {649, 4610}, {4911, 286}, {16706, 274}, {17200, 86}, {23790, 664}, {23791, 43190}, {24169, 39747}, {29654, 2}, {39179, 4593}
X(52394) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21035}, {2, 21814}, {6, 3954}, {10, 1964}, {31, 15523}, {37, 39}, {38, 42}, {48, 21016}, {65, 3688}, {71, 17442}, {72, 1843}, {75, 41267}, {100, 3005}, {101, 8061}, {141, 213}, {190, 2084}, {210, 1401}, {226, 40972}, {228, 427}, {256, 40936}, {257, 21752}, {291, 4093}, {313, 1923}, {321, 3051}, {512, 4553}, {649, 35309}, {661, 46148}, {668, 688}, {692, 826}, {756, 17187}, {798, 4568}, {872, 16887}, {893, 16587}, {1018, 21123}, {1400, 33299}, {1402, 3703}, {1500, 16696}, {1634, 4705}, {1824, 3917}, {1826, 4020}, {1918, 1930}, {2200, 20883}, {2205, 8024}, {2530, 4557}, {3108, 21817}, {3952, 50521}, {3990, 27376}, {4041, 46153}, {4576, 50487}, {4730, 46162}, {5360, 20021}, {6386, 9494}, {7109, 16703}, {8041, 18098}, {16030, 21807}, {16720, 40729}, {19606, 22028}, {20336, 27369}, {20683, 46149}, {20775, 41013}, {21355, 21880}, {21805, 46150}, {21818, 40432}, {21839, 46154}, {24290, 46163}, {27801, 41331}, {41272, 42713}
X(52394) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 15523}, {3, 21035}, {9, 3954}, {10, 41884}, {38, 40592}, {39, 40589}, {83, 21083}, {141, 6626}, {206, 41267}, {826, 1086}, {1015, 8061}, {1249, 21016}, {1930, 34021}, {2525, 40618}, {3005, 8054}, {3688, 40602}, {3703, 40605}, {4093, 39029}, {4553, 39054}, {4568, 31998}, {4988, 39691}, {5375, 35309}, {6292, 21038}, {16587, 40597}, {16892, 40620}, {21814, 32664}, {33299, 40582}, {36830, 46148}, {40625, 48278}
X(52394) = cevapoint of X(i) and X(j) for these (i,j): {2, 17150}, {6, 16683}, {58, 86}, {82, 83}, {1621, 32911}, {3120, 48101}
X(52394) = crosssum of X(37) and X(21881)
X(52394) = trilinear pole of line {514, 1919}
X(52394) = crossdifference of every pair of points on line {2084, 3005}
X(52394) = barycentric product X(i)*X(j) for these {i,j}: {27, 1799}, {58, 308}, {81, 3112}, {82, 274}, {83, 86}, {99, 10566}, {251, 310}, {261, 18097}, {286, 34055}, {350, 39276}, {513, 4593}, {514, 4577}, {649, 689}, {667, 37204}, {668, 39179}, {693, 4599}, {799, 18108}, {827, 3261}, {873, 18098}, {1176, 44129}, {1333, 18833}, {1509, 18082}, {1790, 46104}, {1919, 42371}, {2206, 40016}, {4025, 42396}, {4107, 41209}, {4594, 18111}, {4620, 18101}, {6385, 46289}, {17167, 39287}, {17168, 39289}, {17200, 40425}, {17206, 32085}, {18089, 40439}, {18091, 40409}, {19786, 39281}, {21110, 33515}, {34072, 40495}
X(52394) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3954}, {2, 15523}, {4, 21016}, {6, 21035}, {21, 33299}, {27, 427}, {28, 17442}, {31, 21814}, {32, 41267}, {58, 39}, {81, 38}, {82, 37}, {83, 10}, {86, 141}, {99, 4568}, {100, 35309}, {110, 46148}, {171, 16587}, {172, 40936}, {251, 42}, {274, 1930}, {284, 3688}, {286, 20883}, {308, 313}, {310, 8024}, {333, 3703}, {513, 8061}, {514, 826}, {593, 17187}, {649, 3005}, {662, 4553}, {667, 2084}, {689, 1978}, {757, 16696}, {827, 101}, {873, 16703}, {1019, 2530}, {1176, 71}, {1333, 1964}, {1412, 1401}, {1434, 3665}, {1437, 4020}, {1474, 1843}, {1509, 16887}, {1790, 3917}, {1799, 306}, {1914, 4093}, {1919, 688}, {2194, 40972}, {2206, 3051}, {3112, 321}, {3120, 39691}, {3261, 23285}, {3589, 21038}, {3733, 21123}, {4025, 2525}, {4556, 1634}, {4560, 48278}, {4565, 46153}, {4577, 190}, {4580, 4064}, {4591, 46162}, {4593, 668}, {4599, 100}, {4610, 4576}, {4628, 4557}, {4630, 32739}, {4750, 14424}, {4972, 21037}, {5299, 20969}, {6629, 7813}, {7122, 21752}, {7191, 17456}, {7192, 16892}, {7199, 48084}, {7293, 22077}, {8747, 27376}, {10547, 2200}, {10566, 523}, {16706, 21249}, {16707, 20898}, {16735, 21336}, {16887, 7794}, {16889, 16886}, {16890, 16894}, {16891, 16893}, {16892, 2528}, {17103, 16720}, {17186, 23208}, {17187, 8041}, {17200, 6292}, {17206, 3933}, {17469, 21817}, {17500, 21011}, {17925, 21108}, {18070, 4036}, {18082, 594}, {18083, 21028}, {18084, 21015}, {18086, 21031}, {18087, 3925}, {18088, 21029}, {18089, 21020}, {18091, 21024}, {18092, 21022}, {18095, 21023}, {18096, 20655}, {18097, 12}, {18098, 756}, {18099, 21021}, {18101, 21044}, {18102, 21025}, {18103, 20484}, {18105, 4079}, {18107, 21051}, {18108, 661}, {18109, 21026}, {18110, 21052}, {18111, 2533}, {18112, 21057}, {18113, 21950}, {18653, 51360}, {18703, 21692}, {18709, 21678}, {18833, 27801}, {20964, 21818}, {21178, 23881}, {27067, 20653}, {28724, 3682}, {32085, 1826}, {33940, 21425}, {33955, 17192}, {34055, 72}, {34072, 692}, {34294, 21043}, {37204, 6386}, {38946, 21064}, {39179, 513}, {39276, 291}, {41284, 18097}, {41629, 4884}, {41884, 21083}, {42396, 1897}, {44129, 1235}, {46288, 1918}, {46289, 213}, {47652, 21125}, {48060, 3806}, {51370, 51371}

X(52395) = X(38)-ISOCONJUGATE OF X(39)

X(52395) lies on these lines: {2, 40425}, {6, 4577}, {32, 26192}, {82, 20964}, {83, 316}, {141, 40000}, {251, 308}, {264, 42396}, {384, 28677}, {428, 32085}, {689, 39449}, {733, 8265}, {827, 12150}, {1031, 9483}, {1176, 42299}, {2980, 7856}, {3329, 46228}, {5480, 8928}, {6664, 7760}, {6665, 35137}, {7787, 9233}, {7827, 38946}, {7878, 14247}, {16095, 18907}, {18082, 21094}, {18097, 41284}, {20981, 39179}, {20990, 36081}, {26195, 28672}

X(52395) = isogonal conjugate of X(8041)
X(52395) = isotomic conjugate of X(7794)
X(52395) = isogonal conjugate of the complement of X(33798)
X(52395) = isotomic conjugate of the anticomplement of X(7829)
X(52395) = isotomic conjugate of the complement of X(7760)
X(52395) = X(i)-cross conjugate of X(j) for these (i,j): {7829, 2}, {18105, 4577}, {31299, 99}
X(52395) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8041}, {31, 7794}, {38, 39}, {41, 41285}, {141, 1964}, {163, 2528}, {427, 4020}, {799, 2531}, {1101, 15449}, {1401, 33299}, {1634, 8061}, {1923, 8024}, {1930, 3051}, {1973, 4175}, {2084, 4576}, {2530, 46148}, {3665, 40972}, {3917, 17442}, {3954, 17187}, {4553, 21123}, {4568, 50521}, {16696, 21035}, {16703, 41267}, {16887, 21814}, {20775, 20883}
X(52395) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 7794}, {3, 8041}, {115, 2528}, {141, 41884}, {523, 15449}, {2531, 38996}, {3160, 41285}, {4175, 6337}
X(52395) = cevapoint of X(i) and X(j) for these (i,j): {2, 7760}, {6, 35222}, {32, 5012}, {83, 251}
X(52395) = crosspoint of X(83) and X(40163)
X(52395) = crosssum of X(i) and X(j) for these (i,j): {2, 46715}, {6, 35214}
X(52395) = trilinear pole of line {5027, 7927}
X(52395) = barycentric product X(i)*X(j) for these {i,j}: {8, 41284}, {82, 3112}, {83, 83}, {251, 308}, {689, 18105}, {1176, 46104}, {1799, 32085}, {4580, 42396}, {4599, 18070}, {17500, 39287}, {18833, 46289}, {30505, 41296}, {40016, 46288}, {40163, 41884}
X(52395) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 7794}, {6, 8041}, {7, 41285}, {69, 4175}, {82, 38}, {83, 141}, {115, 15449}, {251, 39}, {308, 8024}, {428, 28666}, {523, 2528}, {669, 2531}, {827, 1634}, {1176, 3917}, {1799, 3933}, {3112, 1930}, {4577, 4576}, {4580, 2525}, {4628, 46148}, {5012, 52042}, {7760, 6665}, {8024, 14125}, {10547, 20775}, {10566, 16892}, {16890, 16893}, {18082, 15523}, {18098, 3954}, {18105, 3005}, {18108, 2530}, {20022, 51371}, {22105, 14424}, {32085, 427}, {33632, 3787}, {34294, 39691}, {40163, 33665}, {41284, 7}, {42037, 8362}, {42396, 41676}, {46104, 1235}, {46288, 3051}, {46289, 1964}
X(52395) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {83, 41884, 3589}, {83, 46288, 41296}, {40000, 40850, 141}

X(52396) = X(25)-ISOCONJUGATE OF X(28)

X(52396) lies on these lines: {10, 75}, {58, 3879}, {63, 69}, {86, 2983}, {101, 2366}, {190, 4150}, {201, 307}, {319, 1043}, {326, 1264}, {332, 1794}, {579, 3912}, {714, 23663}, {1210, 18147}, {1265, 19611}, {1918, 4028}, {2893, 7283}, {3006, 17220}, {3610, 4019}, {3668, 35550}, {3701, 40999}, {3948, 24005}, {4358, 5740}, {4427, 20291}, {5271, 19793}, {5736, 33113}, {5738, 17776}, {6527, 52366}, {6734, 44140}, {7270, 8822}, {17862, 24986}, {18734, 25083}, {41004, 41507}

X(52396) = isotomic conjugate of X(8747)
X(52396) = isotomic conjugate of the isogonal conjugate of X(3682)
X(52396) = isotomic conjugate of the polar conjugate of X(306)
X(52396) = isogonal conjugate of the polar conjugate of X(40071)
X(52396) = X(i)-Ceva conjugate of X(j) for these (i,j): {3718, 20336}, {3926, 3998}, {4600, 4561}, {40071, 306}
X(52396) = X(i)-cross conjugate of X(j) for these (i,j): {3682, 306}, {17216, 3265}
X(52396) = X(i)-isoconjugate of X(j) for these (i,j): {4, 2203}, {6, 5317}, {19, 1474}, {21, 7337}, {25, 28}, {27, 1973}, {29, 1395}, {31, 8747}, {34, 2299}, {37, 36420}, {58, 1096}, {81, 2207}, {107, 667}, {112, 6591}, {158, 2206}, {213, 36419}, {274, 36417}, {278, 2204}, {286, 1974}, {393, 1333}, {513, 32713}, {604, 8748}, {607, 1396}, {608, 1172}, {649, 24019}, {823, 1919}, {1014, 6059}, {1118, 2194}, {1397, 1896}, {1398, 4183}, {1408, 1857}, {1435, 2332}, {1437, 6524}, {1783, 43925}, {1841, 40570}, {1880, 2189}, {1911, 34856}, {1980, 6528}, {3121, 23582}, {3122, 24000}, {3125, 23964}, {3194, 7151}, {4206, 51686}, {5379, 42067}, {6529, 22383}, {7649, 32676}, {14399, 32695}, {16732, 41937}, {18604, 36434}
X(52396) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 8747}, {6, 1474}, {9, 5317}, {10, 1096}, {19, 51574}, {25, 40591}, {27, 6337}, {28, 6505}, {34, 226}, {37, 393}, {58, 6503}, {86, 6338}, {107, 6631}, {158, 40603}, {525, 3120}, {649, 35071}, {667, 38985}, {823, 9296}, {1118, 1214}, {1147, 2206}, {1714, 3998}, {1842, 40940}, {1851, 16583}, {2203, 36033}, {2207, 40586}, {2299, 11517}, {3161, 8748}, {3554, 24005}, {5375, 24019}, {6591, 34591}, {6626, 36419}, {6651, 34856}, {7337, 40611}, {7649, 15526}, {17925, 40618}, {32713, 39026}, {36420, 40589}, {39006, 43925}
X(52396) = cevapoint of X(3265) and X(17216)
X(52396) = crosspoint of X(i) and X(j) for these (i,j): {1264, 3718}, {4561, 4600}
X(52396) = crosssum of X(1395) and X(7337)
X(52396) = trilinear pole of line {3265, 24018}
X(52396) = barycentric product X(i)*X(j) for these {i,j}: {3, 40071}, {10, 3926}, {63, 20336}, {69, 306}, {71, 305}, {72, 304}, {75, 3998}, {76, 3682}, {78, 1231}, {190, 3265}, {226, 1264}, {228, 40364}, {255, 27801}, {307, 345}, {313, 394}, {321, 326}, {332, 26942}, {348, 3710}, {349, 1259}, {520, 1978}, {525, 4561}, {561, 3990}, {668, 24018}, {822, 6386}, {1016, 17216}, {1102, 41013}, {1214, 3718}, {1331, 3267}, {1332, 14208}, {1441, 3719}, {1444, 52369}, {1502, 4055}, {1804, 30713}, {1826, 4176}, {1897, 4143}, {2200, 40050}, {2321, 7055}, {2632, 4601}, {3596, 40152}, {3694, 7182}, {3695, 17206}, {3701, 7183}, {3952, 30805}, {4019, 7019}, {4033, 4131}, {4064, 4563}, {4091, 27808}, {4158, 44129}, {4567, 17879}, {4570, 36793}, {4600, 15526}, {4620, 7068}, {6514, 34388}, {7066, 28660}, {19611, 42699}, {21046, 47389}, {22341, 28659}
X(52396) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 5317}, {2, 8747}, {3, 1474}, {8, 8748}, {10, 393}, {37, 1096}, {42, 2207}, {48, 2203}, {58, 36420}, {63, 28}, {69, 27}, {71, 25}, {72, 19}, {73, 608}, {77, 1396}, {78, 1172}, {86, 36419}, {100, 24019}, {101, 32713}, {190, 107}, {201, 1880}, {212, 2204}, {219, 2299}, {226, 1118}, {228, 1973}, {239, 34856}, {255, 1333}, {283, 2189}, {304, 286}, {305, 44129}, {306, 4}, {307, 278}, {312, 1896}, {313, 2052}, {321, 158}, {326, 81}, {332, 46103}, {345, 29}, {394, 58}, {440, 1842}, {520, 649}, {525, 7649}, {577, 2206}, {656, 6591}, {668, 823}, {822, 667}, {906, 32676}, {1043, 36421}, {1102, 1444}, {1214, 34}, {1231, 273}, {1259, 284}, {1260, 2332}, {1264, 333}, {1265, 2322}, {1331, 112}, {1332, 162}, {1334, 6059}, {1400, 7337}, {1409, 1395}, {1439, 1435}, {1459, 43925}, {1792, 2326}, {1794, 40570}, {1804, 1412}, {1812, 270}, {1826, 6524}, {1897, 6529}, {1918, 36417}, {1978, 6528}, {2200, 1974}, {2289, 2194}, {2318, 607}, {2321, 1857}, {2525, 21108}, {2632, 3125}, {3265, 514}, {3267, 46107}, {3269, 3122}, {3610, 7102}, {3682, 6}, {3690, 2333}, {3692, 4183}, {3694, 33}, {3695, 1826}, {3710, 281}, {3718, 31623}, {3719, 21}, {3926, 86}, {3933, 17171}, {3949, 1824}, {3958, 2355}, {3964, 1790}, {3977, 37168}, {3990, 31}, {3998, 1}, {4001, 31900}, {4019, 7009}, {4025, 17925}, {4047, 5338}, {4055, 32}, {4064, 2501}, {4091, 3733}, {4131, 1019}, {4143, 4025}, {4158, 71}, {4176, 17206}, {4303, 46890}, {4466, 2969}, {4552, 36127}, {4561, 648}, {4567, 24000}, {4568, 46151}, {4570, 23964}, {4574, 8750}, {4600, 23582}, {4601, 23999}, {5227, 4206}, {5489, 21131}, {6335, 36126}, {6507, 1437}, {6514, 60}, {6517, 4565}, {7055, 1434}, {7066, 1400}, {7068, 21044}, {7125, 1408}, {7183, 1014}, {7289, 4211}, {7335, 16947}, {8611, 18344}, {8804, 6525}, {8896, 37384}, {14208, 17924}, {15523, 27376}, {15526, 3120}, {17216, 1086}, {17879, 16732}, {18589, 1851}, {18604, 849}, {18607, 46883}, {20336, 92}, {20580, 21172}, {21011, 14569}, {21046, 8754}, {22057, 16502}, {22076, 2354}, {22341, 604}, {23067, 32674}, {23616, 21134}, {23974, 17216}, {24018, 513}, {26872, 37383}, {26942, 225}, {28787, 46886}, {30805, 7192}, {36793, 21207}, {37669, 44698}, {37755, 1426}, {39201, 1919}, {40071, 264}, {40152, 56}, {41013, 6520}, {41014, 1839}, {41077, 11125}, {41087, 7151}, {42699, 1895}, {42706, 39585}, {42711, 51315}, {51366, 1886}, {51367, 1785}, {51386, 17209}, {51640, 43923}, {52347, 17167}, {52355, 3064}, {52369, 41013}
X(52396) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {190, 4150, 8804}, {307, 3710, 20336}, {1264, 3926, 326}

X(52397) = ANTICOMPLEMENT OF X(428)

X(52397) lies on these lines: {2, 3}, {51, 29317}, {52, 17712}, {98, 20185}, {99, 16275}, {125, 48920}, {141, 15321}, {159, 41736}, {184, 48898}, {305, 7802}, {343, 48881}, {394, 46818}, {395, 11421}, {396, 11420}, {511, 41628}, {524, 12220}, {528, 20243}, {539, 10625}, {542, 12058}, {551, 34657}, {597, 19121}, {612, 10483}, {754, 19568}, {925, 29011}, {930, 1297}, {933, 34168}, {1184, 44526}, {1194, 7756}, {1216, 16659}, {1232, 30737}, {1238, 7788}, {1291, 2697}, {1350, 11442}, {1369, 3933}, {1503, 2979}, {1627, 5254}, {1899, 48873}, {1993, 46264}, {1994, 48906}, {2549, 5359}, {2965, 5306}, {3058, 3100}, {3060, 29181}, {3098, 11550}, {3101, 34612}, {3164, 7837}, {3241, 34634}, {3313, 46442}, {3410, 48876}, {3448, 48874}, {3565, 9076}, {3580, 48880}, {3616, 51719}, {3618, 51745}, {3679, 34668}, {3819, 29323}, {3828, 34633}, {3829, 34655}, {3917, 29012}, {3920, 7354}, {4296, 5434}, {4316, 5322}, {4324, 5310}, {4329, 34611}, {5012, 44882}, {5422, 31670}, {5891, 16658}, {5913, 34481}, {5986, 38741}, {6030, 13394}, {6247, 7691}, {6284, 7191}, {6393, 33796}, {6563, 25423}, {7750, 8024}, {7753, 34661}, {7783, 8878}, {7811, 18018}, {7816, 21248}, {7830, 8891}, {7893, 40904}, {8280, 51911}, {8281, 51910}, {8584, 11416}, {8718, 22660}, {8792, 22121}, {9300, 13351}, {9306, 48896}, {9538, 15170}, {10601, 48910}, {11002, 45298}, {11064, 26881}, {11417, 32787}, {11418, 32788}, {11444, 16655}, {11457, 37486}, {13203, 41602}, {13567, 15107}, {14389, 22352}, {14516, 15644}, {14540, 40712}, {14541, 40711}, {15048, 34482}, {15056, 16621}, {15066, 31383}, {15080, 23292}, {15171, 17024}, {16264, 40684}, {18400, 36987}, {18911, 33586}, {18990, 29815}, {19924, 21969}, {20049, 34667}, {20806, 31166}, {21243, 48885}, {21850, 34545}, {26913, 32269}, {29263, 40998}, {31145, 34656}, {32819, 39998}, {33878, 45794}, {34666, 49732}, {35283, 44299}, {40634, 43768}, {42459, 52058}, {43650, 48901}, {48891, 51360}, {49719, 52365}

X(52397) = midpoint of X(1657) and X(18564)
X(52397) = reflection of X(i) in X(j) for these {i,j}: {2, 7667}, {428, 10691}, {3241, 34634}, {3543, 34664}, {7540, 549}, {7553, 10127}, {7576, 3}, {15683, 34614}, {16658, 5891}, {18560, 18564}, {20049, 34667}, {31145, 34656}, {34603, 2}, {34613, 381}, {34633, 3828}, {34655, 3829}, {34657, 551}, {34661, 7753}, {34666, 49732}, {34668, 3679}, {38320, 10304}, {38321, 8703}, {38322, 34200}, {38323, 376}, {44458, 3534}
X(52397) = anticomplement of X(428)
X(52397) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(44450)
X(52397) = de Longchamps circle inverse of X(20063)
X(52397) = anticomplement of the isogonal conjugate of X(41435)
X(52397) = isotomic conjugate of the polar conjugate of X(41366)
X(52397) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {48, 51860}, {63, 40002}, {3108, 5905}, {7953, 7253}, {10159, 21270}, {35137, 21300}, {41435, 8}
X(52397) = X(7768)-Ceva conjugate of X(141)
X(52397) = X(14378)-Dao conjugate of X(15321)
X(52397) = crosspoint of X(23582) and X(35137)
X(52397) = crosssum of X(3269) and X(8664)
X(52397) = barycentric product X(69)*X(41366)
X(52397) = barycentric quotient X(41366)/X(4)
X(52397) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5064, 5133}, {2, 7391, 5064}, {2, 7500, 7714}, {2, 7714, 1995}, {2, 9909, 7426}, {2, 44442, 31133}, {3, 5064, 2}, {3, 7391, 5133}, {4, 20, 33524}, {4, 7485, 37990}, {20, 1370, 22}, {20, 2071, 44239}, {20, 12225, 52071}, {20, 21312, 16386}, {20, 37444, 11414}, {20, 52364, 35998}, {22, 1370, 858}, {22, 30744, 7493}, {25, 1657, 20062}, {25, 20062, 37900}, {140, 52285, 37353}, {376, 44442, 2}, {382, 7484, 7394}, {427, 550, 6636}, {427, 6636, 7495}, {428, 7667, 10691}, {428, 10691, 2}, {548, 47095, 5169}, {550, 5189, 7495}, {1370, 7493, 7396}, {1370, 33524, 37990}, {1657, 16063, 37900}, {2043, 2044, 10323}, {3098, 11550, 37636}, {3529, 7386, 7500}, {3627, 37439, 37349}, {5002, 5003, 5}, {5004, 5005, 3518}, {5189, 6636, 427}, {6676, 46517, 31074}, {7386, 7500, 1995}, {7386, 7714, 2}, {7396, 7493, 30744}, {7396, 30744, 858}, {7485, 33524, 22}, {7488, 44450, 549}, {7492, 31074, 6676}, {7494, 31099, 31236}, {7496, 37349, 37439}, {7553, 43934, 10127}, {7667, 34614, 34658}, {9909, 31152, 2}, {10127, 43934, 631}, {10323, 14790, 13160}, {11112, 11113, 48815}, {11114, 17579, 48813}, {12082, 18531, 47096}, {12083, 14791, 403}, {12103, 46517, 7492}, {16063, 20062, 25}, {17579, 51669, 11112}, {20405, 20406, 5159}, {31101, 37913, 468}

X(52398) = ANTICOMPLEMENT OF X(1598)

X(53398) lies on these lines: {2, 3}, {69, 14216}, {84, 50861}, {193, 18914}, {389, 41256}, {394, 34781}, {511, 18909}, {578, 25406}, {1056, 4296}, {1058, 3100}, {1092, 11206}, {1192, 48872}, {1217, 33971}, {1249, 23115}, {1350, 6247}, {1352, 13348}, {3421, 52366}, {3426, 11469}, {3618, 37515}, {3926, 20477}, {3933, 6527}, {5082, 52365}, {5447, 11487}, {5562, 12324}, {5709, 26929}, {5800, 36746}, {5921, 34780}, {6193, 37483}, {6696, 48881}, {6759, 37669}, {6776, 37498}, {7330, 26939}, {9729, 31670}, {9786, 29181}, {9815, 17704}, {9833, 14927}, {10110, 18928}, {10625, 11411}, {10629, 37552}, {10984, 11427}, {11412, 12058}, {11425, 44882}, {11433, 45186}, {11438, 41257}, {11677, 30265}, {11745, 48910}, {12163, 12318}, {12250, 15438}, {12256, 12320}, {12257, 12321}, {12319, 16111}, {13203, 16163}, {13336, 19121}, {13346, 18925}, {13347, 14561}, {13391, 18951}, {14826, 16655}, {14853, 37514}, {14912, 36747}, {15583, 17822}, {16621, 17811}, {16836, 51538}, {17834, 18913}, {17845, 34944}, {18488, 41464}, {18917, 37484}, {18919, 44492}, {18931, 46730}, {22528, 41465}, {22660, 35237}, {25712, 34782}, {31383, 43652}, {32064, 36987}, {36844, 36984}, {36988, 41761}, {38738, 39842}, {38749, 39813}, {42329, 44443}

X(52398) = reflection of X(i) in X(j) for these {i,j}: {4, 6643}, {7487, 3}
X(52398) = anticomplement of X(1598)
X(52398) = deLongchamps-circle-inverse of X(37945)
X(52398) = X(46952)-anticomplementary conjugate of X(5905)
X(52398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20, 11414}, {3, 4, 6803}, {3, 382, 9825}, {3, 1595, 2}, {3, 7404, 631}, {3, 34938, 4}, {4, 20, 35513}, {4, 376, 10996}, {4, 631, 7392}, {4, 3537, 6815}, {4, 3538, 2}, {4, 7386, 6804}, {20, 1370, 4}, {20, 3522, 44239}, {20, 11413, 376}, {20, 12225, 3529}, {20, 30552, 17538}, {20, 37444, 37201}, {376, 3529, 35471}, {382, 18537, 4}, {427, 37198, 7400}, {631, 7714, 6642}, {1368, 39568, 3089}, {1370, 37201, 37444}, {3090, 3529, 37946}, {3146, 6816, 4}, {3146, 16063, 6816}, {3522, 6815, 3537}, {3522, 7391, 6815}, {3529, 47528, 4}, {3541, 10323, 7494}, {3546, 7387, 6353}, {3547, 23335, 8889}, {5002, 5003, 6997}, {6815, 7391, 4}, {9715, 47090, 3523}, {10996, 44442, 4}, {13346, 46264, 18925}, {14216, 15644, 69}, {14784, 14785, 39568}, {17704, 48901, 9815}, {23335, 35243, 3547}, {31152, 34614, 34621}, {37201, 37444, 4}

X(52399) = ANTICOMPLEMENT OF X(2043)

X(52399) lies on these lines: {2, 3}, {13, 42255}, {14, 42256}, {15, 9541}, {298, 490}, {299, 489}, {395, 42259}, {396, 42258}, {397, 36449}, {398, 36450}, {487, 617}, {488, 616}, {590, 42087}, {615, 42088}, {627, 13678}, {628, 13798}, {1151, 42942}, {1152, 42943}, {1587, 3364}, {1588, 3390}, {3068, 42119}, {3069, 42120}, {3070, 42154}, {3071, 42155}, {3180, 43134}, {3181, 43133}, {3367, 23263}, {3391, 23253}, {5318, 42218}, {5321, 42219}, {5334, 6560}, {5335, 6561}, {5339, 42252}, {5340, 42251}, {5464, 35742}, {5473, 33440}, {5474, 33443}, {5870, 6269}, {5871, 6270}, {6200, 51855}, {6396, 51852}, {6409, 42626}, {6410, 42625}, {6459, 37640}, {6460, 37641}, {6564, 42172}, {6565, 42173}, {6770, 26441}, {6773, 8982}, {7583, 43482}, {7584, 43481}, {7690, 48722}, {7692, 48725}, {8252, 42169}, {8253, 42168}, {9540, 36967}, {13340, 34553}, {13665, 42200}, {13785, 42201}, {13846, 42253}, {13847, 42250}, {13935, 35739}, {16644, 35740}, {16645, 42239}, {16808, 42196}, {16809, 42197}, {16964, 42248}, {16965, 42247}, {19106, 42188}, {19107, 42189}, {22236, 36468}, {22238, 36467}, {23249, 42085}, {23251, 42240}, {23259, 42086}, {23261, 42241}, {23267, 42117}, {23273, 42118}, {32787, 42147}, {32788, 42148}, {35730, 41943}, {35820, 42243}, {35821, 42244}, {35822, 42228}, {35823, 42229}, {36440, 50808}, {36444, 50865}, {36446, 43770}, {36447, 43769}, {36969, 42236}, {36970, 42237}, {36993, 45511}, {36995, 45510}, {40693, 43257}, {40694, 43256}, {42093, 42193}, {42094, 42192}, {42096, 42284}, {42097, 42283}, {42099, 42190}, {42100, 42187}, {42122, 43509}, {42123, 43510}, {42125, 42223}, {42126, 42213}, {42127, 42212}, {42128, 42222}, {42130, 42214}, {42131, 42211}, {42133, 42276}, {42134, 42275}, {42153, 43209}, {42156, 43210}, {42157, 42249}, {42158, 42246}, {42232, 52217}, {42233, 52215}, {42242, 42267}, {42245, 42266}, {42271, 42941}, {42272, 42940}, {42511, 51727}

X(52399) = midpoint of X(2041) and X(2044)
X(52399) = reflection of X(i) in X(j) for these {i,j}: {2, 36455}, {4, 18587}, {20, 2044}, {2043, 18585}, {36454, 3830}
X(52399) = anticomplement of X(2043)
X(52399) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2044, 35732}, {3, 2042, 3523}, {3, 2044, 2}, {3, 2045, 15717}, {3, 3534, 34552}, {3, 14813, 631}, {3, 15765, 3524}, {3, 18587, 36455}, {4, 20, 35732}, {4, 376, 2044}, {4, 2041, 42282}, {4, 3529, 2042}, {376, 36445, 2}, {382, 42280, 4}, {2041, 18587, 2}, {2043, 18585, 2}, {2043, 36455, 18585}, {2044, 2046, 36454}, {2044, 36437, 14813}, {2044, 36455, 4}, {2044, 42278, 15683}, {2045, 3861, 3091}, {2046, 15765, 2}, {2046, 33703, 42282}, {3524, 36454, 2}, {3534, 36455, 3839}, {3830, 15765, 4}, {3830, 42278, 15765}, {11001, 36445, 15692}, {14813, 34552, 36437}, {14813, 49134, 3529}, {15764, 17504, 36437}, {18585, 34551, 2044}, {18587, 36436, 42282}, {18587, 36466, 5}, {18587, 42280, 36436}, {19708, 35737, 15764}, {35739, 42254, 13935}, {36436, 36463, 36466}

X(52400) = ANTICOMPLEMENT OF X(2044)

X(52400) lies on these lines: {2, 3}, {13, 42257}, {14, 42254}, {16, 9541}, {298, 489}, {299, 490}, {395, 42258}, {396, 42259}, {397, 36468}, {398, 36467}, {487, 616}, {488, 617}, {590, 42088}, {615, 42087}, {627, 13798}, {628, 13678}, {1131, 51925}, {1132, 51924}, {1151, 42943}, {1152, 42942}, {1587, 3389}, {1588, 3365}, {3068, 42120}, {3069, 42119}, {3070, 42155}, {3071, 42154}, {3180, 43133}, {3181, 43134}, {3366, 23253}, {3392, 23263}, {5318, 42220}, {5321, 42217}, {5334, 6561}, {5335, 6560}, {5339, 42250}, {5340, 42253}, {5473, 33441}, {5474, 33442}, {5870, 6268}, {5871, 6271}, {6200, 51854}, {6396, 51853}, {6409, 42625}, {6410, 42626}, {6459, 37641}, {6460, 37640}, {6564, 42174}, {6565, 42171}, {6770, 8982}, {6773, 26441}, {7583, 43481}, {7584, 43482}, {7690, 48724}, {7692, 48723}, {8252, 42167}, {8253, 42170}, {9540, 36968}, {13340, 34555}, {13665, 42202}, {13785, 42199}, {13846, 42252}, {13847, 42251}, {13935, 36967}, {16644, 42241}, {16645, 42240}, {16808, 42198}, {16809, 42195}, {16964, 42246}, {16965, 42249}, {19106, 42190}, {19107, 42187}, {22236, 36449}, {22238, 36450}, {23249, 35731}, {23251, 35740}, {23259, 42085}, {23261, 42239}, {23267, 42118}, {23273, 42117}, {32787, 42148}, {32788, 42147}, {35739, 42529}, {35820, 42245}, {35821, 42242}, {35822, 42230}, {35823, 42227}, {36458, 50808}, {36462, 50865}, {36464, 43769}, {36465, 43770}, {36969, 42238}, {36970, 42235}, {36993, 45510}, {36995, 45511}, {40693, 43256}, {40694, 43257}, {42093, 42191}, {42094, 42194}, {42096, 42283}, {42097, 42284}, {42099, 42188}, {42100, 42189}, {42122, 43510}, {42123, 43509}, {42125, 42221}, {42126, 42211}, {42127, 42214}, {42128, 42224}, {42130, 42212}, {42131, 42213}, {42133, 42275}, {42134, 42276}, {42153, 43210}, {42156, 43209}, {42157, 42247}, {42158, 42248}, {42231, 52216}, {42234, 52214}, {42243, 42266}, {42244, 42267}, {42271, 42940}, {42272, 42941}

X(52400) = midpoint of X(2042) and X(2043)
X(52400) = reflection of X(i) in X(j) for these {i,j}: {2, 36437}, {4, 18586}, {20, 2043}, {2044, 15765}, {18585, 35738}, {36436, 3830}
X(52400) = anticomplement of X(2044)
X(52400) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2043, 42282}, {3, 2041, 3523}, {3, 2043, 2}, {3, 2046, 15717}, {3, 3534, 34551}, {3, 14814, 631}, {3, 18585, 3524}, {3, 18586, 36437}, {4, 20, 42282}, {4, 376, 2043}, {4, 2042, 35732}, {4, 3529, 2041}, {376, 36463, 2}, {381, 15764, 15709}, {382, 42281, 4}, {2042, 18586, 2}, {2043, 2045, 36436}, {2043, 36437, 4}, {2043, 36455, 14814}, {2043, 42279, 15683}, {2044, 15765, 2}, {2044, 36437, 15765}, {2045, 18585, 2}, {2045, 33703, 35732}, {2046, 3861, 3091}, {3524, 36436, 2}, {3534, 36437, 3839}, {3830, 18585, 4}, {3830, 42279, 18585}, {11001, 36463, 15692}, {14814, 34551, 36455}, {14814, 49134, 3529}, {15688, 15764, 376}, {15765, 34552, 2043}, {18586, 36448, 5}, {18586, 36454, 35732}, {18586, 42281, 36454}, {36445, 36454, 36448}

X(52401) = ANTICOMPLEMENT OF X(2045)

X(52401) lies on these lines: {2, 3}, {13, 42246}, {14, 42249}, {15, 9540}, {16, 13935}, {17, 42247}, {18, 42248}, {302, 490}, {303, 489}, {485, 5334}, {486, 5335}, {487, 628}, {488, 627}, {590, 42147}, {615, 42148}, {1151, 16772}, {1152, 16773}, {1587, 3390}, {1588, 3364}, {3070, 42153}, {3071, 42156}, {3304, 36459}, {3316, 42117}, {3317, 42118}, {3366, 42085}, {3367, 18582}, {3391, 18581}, {3392, 42086}, {3592, 36468}, {3594, 36467}, {4857, 36443}, {5270, 36442}, {5318, 42217}, {5321, 42220}, {5339, 42253}, {5340, 42250}, {5870, 33358}, {5871, 33359}, {6201, 51754}, {6202, 51753}, {6409, 42490}, {6410, 42491}, {6453, 36469}, {6454, 36470}, {6459, 11488}, {6460, 11489}, {6564, 42242}, {6565, 42245}, {8252, 42241}, {8253, 42240}, {8960, 52217}, {9541, 42255}, {10576, 42243}, {10577, 42244}, {11522, 36444}, {11542, 23273}, {11543, 23267}, {12974, 49106}, {12975, 49105}, {13678, 33404}, {13798, 33405}, {16644, 42251}, {16645, 42252}, {16808, 42187}, {16809, 42190}, {16964, 42257}, {16965, 42254}, {16966, 42188}, {16967, 42189}, {19106, 42195}, {19107, 42198}, {22236, 31454}, {23251, 42239}, {23261, 35740}, {23302, 42218}, {23303, 42219}, {30389, 36462}, {32785, 42119}, {32786, 42120}, {32789, 42087}, {32790, 42088}, {33416, 42197}, {33417, 42196}, {34089, 43630}, {34091, 43631}, {34555, 37481}, {35739, 42256}, {35786, 42175}, {35787, 42178}, {35812, 42228}, {35813, 42229}, {35820, 42171}, {35821, 42174}, {36440, 43174}, {42093, 42194}, {42094, 42191}, {42095, 42284}, {42098, 42283}, {42125, 42214}, {42126, 42224}, {42127, 42221}, {42128, 42211}, {42129, 42213}, {42132, 42212}, {42133, 42277}, {42134, 42274}, {42167, 42193}, {42170, 42192}, {42179, 42919}, {42180, 42915}, {42181, 42914}, {42182, 42918}, {42231, 43404}, {42234, 43403}, {42235, 42813}, {42236, 42488}, {42237, 42489}, {42238, 42814}

X(52401) = reflection of X(10303) in X(2046)
X(52401) = anticomplement of X(2045)
X(52401) = orthocentroidal-circle-inverse of X(42282)
X(52401) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 42282}, {3, 1656, 34551}, {3, 2041, 20}, {3, 2043, 3522}, {3, 2046, 2}, {3, 14814, 376}, {3, 18585, 4}, {3, 42279, 36439}, {4, 631, 2041}, {4, 2044, 35732}, {4, 3090, 2043}, {5, 2041, 2}, {5, 18585, 2041}, {20, 2041, 42282}, {381, 42281, 4}, {546, 18586, 4}, {1656, 42281, 41106}, {2041, 2044, 5}, {2041, 2046, 631}, {2041, 42279, 7486}, {2043, 2044, 36454}, {2043, 3850, 3091}, {2044, 18585, 2}, {2046, 36455, 3}, {3523, 36455, 42282}, {3851, 18585, 3525}, {5079, 18587, 2045}, {14784, 14785, 15765}, {14813, 36454, 35732}, {14814, 19709, 3090}, {18585, 36439, 18587}, {35738, 42278, 36463}, {36439, 36455, 15692}

X(52402) = ANTICOMPLEMENT OF X(2046)

X(52402) lies on these lines: {2, 3}, {13, 42248}, {14, 42247}, {15, 13935}, {16, 9540}, {17, 42249}, {18, 42246}, {302, 489}, {303, 490}, {485, 5335}, {486, 5334}, {487, 627}, {488, 628}, {590, 42148}, {615, 42147}, {1151, 16773}, {1152, 16772}, {1587, 3365}, {1588, 3389}, {3070, 42156}, {3071, 42153}, {3303, 36451}, {3304, 36441}, {3316, 42118}, {3317, 42117}, {3366, 18582}, {3367, 42085}, {3391, 42086}, {3392, 18581}, {3592, 36450}, {3594, 36449}, {4857, 36461}, {5270, 36460}, {5318, 42219}, {5321, 42218}, {5339, 42251}, {5340, 42252}, {5870, 33360}, {5871, 33361}, {6201, 51753}, {6202, 51754}, {6409, 42491}, {6410, 42490}, {6453, 36452}, {6454, 36453}, {6459, 11489}, {6460, 11488}, {6564, 42244}, {6565, 42243}, {8252, 42239}, {8253, 35740}, {8960, 52214}, {9541, 42254}, {10576, 42245}, {10577, 42242}, {11522, 36462}, {11542, 23267}, {11543, 23273}, {12974, 49105}, {12975, 49106}, {13678, 33405}, {13798, 33404}, {16644, 42253}, {16645, 42250}, {16808, 42189}, {16809, 42188}, {16964, 42255}, {16965, 42256}, {16966, 42190}, {16967, 42187}, {19106, 42197}, {19107, 42196}, {20070, 34560}, {22238, 31454}, {23251, 42241}, {23261, 42240}, {23302, 42220}, {23303, 42217}, {30389, 36444}, {32785, 42120}, {32786, 42119}, {32789, 42088}, {32790, 42087}, {33416, 42195}, {33417, 42198}, {34089, 43631}, {34091, 43630}, {34553, 37481}, {35731, 42151}, {35786, 42177}, {35787, 42176}, {35812, 42230}, {35813, 42227}, {35820, 42173}, {35821, 42172}, {36458, 43174}, {42093, 42192}, {42094, 42193}, {42095, 42283}, {42098, 42284}, {42125, 42212}, {42126, 42222}, {42127, 42223}, {42128, 42213}, {42129, 42211}, {42132, 42214}, {42133, 42274}, {42134, 42277}, {42168, 42191}, {42169, 42194}, {42179, 42914}, {42180, 42918}, {42181, 42919}, {42182, 42915}, {42232, 43404}, {42233, 43403}, {42235, 42489}, {42236, 42814}, {42237, 42813}, {42238, 42488}, {43540, 51925}, {43541, 51924}

X(52402) = reflection of X(10303) in X(2045)
X(52402) = anticomplement of X(2046)
X(52402) = orthocentroidal-circle-inverse of X(35732)
X(52402) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 35732}, {3, 1656, 34552}, {3, 2042, 20}, {3, 2044, 3522}, {3, 2045, 2}, {3, 14813, 376}, {3, 15765, 4}, {3, 42278, 36457}, {4, 631, 2042}, {4, 2043, 42282}, {4, 3090, 2044}, {5, 2042, 2}, {5, 15765, 2042}, {20, 2042, 35732}, {381, 42280, 4}, {546, 18587, 4}, {1656, 42280, 41106}, {2042, 2043, 5}, {2042, 2045, 631}, {2042, 42278, 7486}, {2043, 2044, 36436}, {2043, 15765, 2}, {2044, 3850, 3091}, {2045, 36437, 3}, {3523, 36437, 35732}, {3851, 15765, 3525}, {5079, 18586, 2046}, {12103, 35738, 42281}, {14784, 14785, 18585}, {14813, 19709, 3090}, {14814, 15765, 35738}, {14814, 35738, 2044}, {14814, 36436, 42282}, {15765, 36457, 18586}, {36437, 36457, 15692}

X(52403) = ANTICOMPLEMENT OF X(2071)

X(52403) lies on these lines: {2, 3}, {51, 13446}, {54, 12897}, {110, 51403}, {113, 43574}, {125, 13445}, {143, 3521}, {146, 7731}, {185, 22533}, {323, 1514}, {388, 10149}, {539, 14094}, {1154, 7728}, {1209, 43613}, {1291, 16337}, {1478, 9539}, {1498, 34799}, {1503, 37784}, {1533, 9934}, {1539, 13391}, {2883, 17824}, {2888, 12162}, {3410, 15305}, {3448, 6000}, {3580, 15311}, {3583, 4351}, {3585, 4354}, {3622, 51713}, {3623, 47471}, {5160, 5229}, {5225, 7286}, {5622, 29012}, {5731, 51701}, {5889, 22802}, {6288, 32137}, {8705, 51538}, {9140, 13399}, {9927, 12290}, {10152, 14918}, {10419, 14989}, {10540, 12383}, {11002, 32411}, {11455, 18474}, {11459, 15108}, {11550, 18392}, {11649, 31670}, {12028, 30529}, {12241, 43838}, {12278, 26883}, {12295, 44407}, {12827, 13202}, {13203, 32269}, {13219, 44146}, {13353, 15807}, {13491, 43808}, {13598, 48914}, {14157, 17702}, {14683, 32111}, {14712, 44969}, {14731, 34549}, {14915, 25739}, {14927, 32217}, {15055, 44673}, {15072, 18390}, {16194, 41171}, {16534, 43572}, {16657, 34545}, {18128, 44866}, {18550, 45972}, {18933, 50434}, {20070, 47321}, {20125, 40111}, {22466, 52003}, {23061, 38791}, {25406, 51733}, {29181, 51998}, {30690, 51883}, {32218, 48872}, {32220, 41735}, {32608, 38790}, {34170, 46106}, {34796, 37489}, {36990, 41614}, {37643, 40196}, {40647, 43816}, {43576, 46686}, {44443, 47282}, {44519, 47186}, {47571, 51170}, {51171, 51742}

X(52403) = midpoint of X(i) and X(j) for these {i,j}: {382, 5899}, {10296, 37945}, {18325, 31726}, {32608, 38790}
X(52403) = reflection of X(i) in X(j) for these {i,j}: {3, 11563}, {4, 31726}, {5, 11558}, {20, 186}, {23, 47096}, {110, 51403}, {186, 11799}, {550, 10096}, {858, 10151}, {1291, 16337}, {2070, 43893}, {2071, 403}, {2072, 47336}, {3153, 4}, {3448, 50435}, {5189, 3153}, {7464, 2072}, {10295, 37971}, {10540, 51548}, {12383, 10540}, {13445, 125}, {13619, 2070}, {16386, 468}, {18403, 44283}, {18859, 5}, {20063, 37945}, {31726, 44267}, {34152, 44961}, {35452, 37938}, {37900, 47094}, {37944, 858}, {37950, 46031}, {43574, 113}, {43576, 51392}, {46450, 18403}, {47090, 37984}, {47094, 47338}, {47337, 44912}, {48914, 13598}, {51392, 46686}
X(52403) = anticomplement of X(2071)
X(52403) = circumcircle-inverse of X(22467)
X(52403) = polar-circle-inverse of X(1885)
X(52403) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(6677)
X(52403) = orthoptic-circle-of-the-Steiner-circumellipse inverse of X(25)
X(52403) = circumcircle-of-anticomplementary-triangle-inverse of X(3146)
X(52403) = deLongchamps-circle-inverse of X(11413)
X(52403) = Stammler-circle-inverse of X(18378)
X(52403) = Stammler-circles-radical-circle-inverse of X(4)
X(52403) = anticomplement of the isogonal conjugate of X(11744)
X(52403) = anticomplement of the isotomic conjugate of X(51967)
X(52403) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {19, 51968}, {11744, 8}, {22239, 7253}, {48373, 7192}, {51967, 6327}
X(52403) = X(51967)-Ceva conjugate of X(2)
X(52403) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 381, 50140}, {3, 11563, 37943}, {3, 44271, 4}, {4, 3529, 18404}, {4, 7544, 50689}, {4, 15682, 31723}, {4, 18420, 3839}, {4, 33703, 18569}, {4, 44440, 2}, {4, 46450, 18403}, {4, 50009, 34007}, {20, 3091, 3546}, {20, 6623, 2}, {20, 11799, 37760}, {20, 46451, 186}, {23, 7500, 37901}, {26, 44288, 18531}, {186, 11799, 46451}, {186, 44958, 37943}, {186, 46451, 37760}, {235, 22467, 21451}, {235, 52071, 22467}, {381, 35452, 37938}, {382, 44279, 4}, {403, 2071, 2}, {468, 16386, 37941}, {550, 10096, 37955}, {858, 37962, 2}, {1113, 1114, 22467}, {1596, 38323, 13595}, {2072, 7464, 44450}, {3091, 44450, 2072}, {3543, 7500, 3146}, {3830, 44263, 4}, {3839, 18420, 7533}, {5004, 5005, 37928}, {7464, 47336, 3091}, {7488, 45179, 2}, {7527, 15760, 2}, {7574, 47332, 18537}, {7574, 47335, 47528}, {10295, 37971, 37940}, {10296, 47309, 17578}, {10750, 10751, 3627}, {10750, 15156, 14807}, {10751, 15157, 14808}, {12088, 18563, 20}, {13383, 35491, 38448}, {14807, 14808, 3146}, {15154, 15155, 18378}, {16386, 37941, 3522}, {17578, 20063, 10296}, {18325, 44267, 4}, {18403, 31726, 44283}, {18403, 44246, 18531}, {18403, 44283, 4}, {18403, 46450, 3153}, {30745, 37984, 5068}, {37760, 37901, 37913}, {37948, 44452, 3523}, {44276, 50009, 37349}, {44911, 47332, 403}, {47752, 47753, 3518}

X(52404) = ANTICOMPLEMENT OF X(3088)

X(52404) lies on these lines: {2, 3}, {40, 27509}, {69, 1498}, {76, 6527}, {141, 15811}, {193, 1181}, {280, 4385}, {343, 12324}, {347, 3673}, {800, 5286}, {1350, 2883}, {1351, 17040}, {1614, 41619}, {3060, 46363}, {3098, 11821}, {3164, 6392}, {3346, 47633}, {4296, 14986}, {4357, 12705}, {5562, 5656}, {5878, 46728}, {5893, 48881}, {5907, 10519}, {5921, 34781}, {5925, 40196}, {6223, 28739}, {6225, 33522}, {6523, 15466}, {6776, 10112}, {7080, 52365}, {7592, 51170}, {8743, 36413}, {9786, 15740}, {9841, 20266}, {9914, 37485}, {10516, 16656}, {10982, 51171}, {10984, 19121}, {11381, 43653}, {11444, 12058}, {11456, 20080}, {12220, 45186}, {12233, 51212}, {12241, 25406}, {12359, 35237}, {12429, 39874}, {13093, 44683}, {13142, 14912}, {13598, 14853}, {14826, 26883}, {15069, 44762}, {15644, 32605}, {15873, 18928}, {16252, 37669}, {16936, 26958}, {18913, 46850}, {18945, 46264}, {30737, 32834}, {35260, 35602}, {41361, 42458}, {41362, 48905}, {41602, 45813}

X(52404) = reflection of X(3088) in X(3547)
X(52404) = anticomplement of X(3088)
X(52404) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1596, 6804}, {3, 3089, 2}, {3, 6677, 631}, {3, 35513, 20}, {4, 376, 12362}, {4, 631, 11479}, {4, 7399, 3091}, {4, 7400, 2}, {4, 11414, 20}, {20, 3091, 1370}, {20, 3523, 11413}, {20, 3543, 12225}, {20, 7488, 3522}, {20, 10304, 30552}, {20, 10565, 3}, {20, 44440, 3146}, {22, 37201, 20}, {235, 37198, 7386}, {550, 44277, 3}, {1370, 7493, 40916}, {1370, 33524, 20}, {1596, 6804, 3091}, {1598, 6803, 7398}, {3088, 3547, 2}, {3529, 44239, 20}, {6823, 39568, 4}, {7400, 34621, 4}, {7493, 11413, 3523}, {9715, 32534, 7488}, {20062, 34007, 3146}, {30552, 38444, 10304}

X(52405) = X(1)X(6)∩X(71)X(74)

X(52405) lies on these lines: {1, 6}, {10, 7110}, {35, 2174}, {48, 3730}, {63, 15066}, {71, 74}, {81, 5325}, {86, 25589}, {198, 35239}, {200, 7072}, {284, 1334}, {323, 1442}, {394, 3929}, {517, 16547}, {527, 37659}, {572, 22356}, {579, 9310}, {583, 5563}, {584, 3746}, {594, 7359}, {610, 3587}, {644, 2287}, {645, 17787}, {672, 41431}, {1201, 4284}, {1332, 4416}, {1495, 3690}, {1731, 17452}, {1781, 21853}, {1802, 4262}, {1944, 17116}, {2175, 4517}, {2269, 41432}, {2289, 6602}, {2303, 3997}, {2318, 2328}, {2330, 7064}, {2338, 33634}, {2340, 41457}, {3098, 3220}, {3197, 41456}, {3217, 4266}, {3452, 26723}, {3579, 16553}, {3620, 27509}, {3631, 26932}, {3678, 6198}, {3713, 4873}, {3915, 5037}, {3928, 17811}, {4007, 4513}, {4034, 4914}, {4858, 17117}, {4890, 8539}, {5086, 21065}, {5092, 7193}, {5228, 20195}, {5273, 14996}, {5314, 15080}, {5341, 21864}, {5707, 31446}, {5745, 37633}, {5750, 24564}, {6173, 25878}, {7085, 26864}, {7293, 41462}, {7330, 37483}, {8693, 26702}, {11009, 17443}, {11464, 26915}, {14997, 18228}, {15018, 27065}, {15032, 26878}, {15068, 26921}, {15817, 37601}, {15988, 50093}, {16548, 21871}, {16603, 25651}, {17274, 26657}, {17296, 23151}, {17781, 52374}, {18253, 37559}, {21059, 41276}, {21236, 30848}, {22052, 35072}, {22054, 24047}, {24320, 33878}, {25006, 40869}, {26872, 37643}, {26890, 44109}, {26893, 34417}, {26939, 39874}, {26942, 47296}, {29574, 41610}, {31445, 51340}, {33774, 38825}, {35192, 35193}, {40659, 41339}, {40910, 41454}

X(52405) = isogonal conjugate of X(52374)
X(52405) = complement of the isotomic conjugate of X(43741)
X(52405) = isogonal conjugate of the isotomic conjugate of X(42033)
X(52405) = X(43741)-complementary conjugate of X(2887)
X(52405) = X(i)-Ceva conjugate of X(j) for these (i,j): {3219, 35}, {4570, 3939}, {32635, 55}
X(52405) = X(i)-isoconjugate of X(j) for these (i,j): {1, 52374}, {2, 52372}, {7, 2160}, {27, 52390}, {34, 52381}, {56, 30690}, {57, 79}, {58, 43682}, {65, 52393}, {81, 52382}, {85, 6186}, {226, 52375}, {269, 7110}, {278, 7100}, {279, 7073}, {513, 38340}, {514, 26700}, {554, 2306}, {604, 20565}, {1014, 8818}, {1081, 33654}, {1396, 52388}, {1407, 52344}, {1412, 6757}, {1427, 3615}, {1443, 1989}, {1769, 47317}, {1847, 8606}, {3120, 35049}, {3669, 6742}, {3942, 34922}, {7178, 13486}, {15455, 43924}, {30602, 47057}
X(52405) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 30690}, {3, 52374}, {10, 43682}, {79, 5452}, {94, 36909}, {1443, 34544}, {3161, 20565}, {3700, 21207}, {6600, 7110}, {6757, 40599}, {8287, 24002}, {11517, 52381}, {17078, 40604}, {24771, 52344}, {32664, 52372}, {38340, 39026}, {40586, 52382}, {40602, 52393}
X(52405) = crosspoint of X(i) and X(j) for these (i,j): {2, 43741}, {3219, 4420}
X(52405) = crosssum of X(i) and X(j) for these (i,j): {1086, 30724}, {2160, 52372}, {2170, 50354}
X(52405) = crossdifference of every pair of points on line {513, 11125}
X(52405) = barycentric product X(i)*X(j) for these {i,j}: {1, 4420}, {6, 42033}, {8, 35}, {9, 3219}, {10, 35193}, {21, 3678}, {41, 33939}, {55, 319}, {60, 7206}, {72, 11107}, {78, 6198}, {100, 35057}, {190, 9404}, {200, 1442}, {220, 17095}, {284, 3969}, {306, 41502}, {312, 2174}, {321, 35192}, {323, 36910}, {341, 1399}, {346, 2003}, {644, 14838}, {943, 31938}, {1043, 2594}, {1250, 40714}, {1260, 7282}, {1334, 34016}, {1792, 1825}, {2287, 16577}, {2321, 40214}, {2323, 41226}, {2328, 40999}, {2341, 42701}, {2605, 3699}, {3578, 33635}, {3647, 32635}, {3701, 17104}, {3718, 14975}, {3939, 4467}, {4102, 17454}, {4570, 6741}, {5546, 7265}, {7058, 21794}, {7161, 52126}, {10638, 40713}, {35194, 44687}, {44688, 46073}, {44689, 46077}
X(52405) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 52374}, {8, 20565}, {9, 30690}, {31, 52372}, {35, 7}, {37, 43682}, {41, 2160}, {42, 52382}, {55, 79}, {101, 38340}, {200, 52344}, {210, 6757}, {212, 7100}, {219, 52381}, {220, 7110}, {228, 52390}, {284, 52393}, {319, 6063}, {323, 17078}, {644, 15455}, {692, 26700}, {1250, 554}, {1253, 7073}, {1334, 8818}, {1399, 269}, {1442, 1088}, {2003, 279}, {2174, 57}, {2175, 6186}, {2194, 52375}, {2318, 52388}, {2328, 3615}, {2594, 3668}, {2605, 3676}, {3219, 85}, {3678, 1441}, {3939, 6742}, {3969, 349}, {4420, 75}, {6149, 1443}, {6198, 273}, {6741, 21207}, {7186, 7185}, {7206, 34388}, {9404, 514}, {10638, 1081}, {11107, 286}, {14838, 24002}, {14975, 34}, {16577, 1446}, {17104, 1014}, {17454, 553}, {21741, 1427}, {21794, 6354}, {22342, 1439}, {32641, 47317}, {33939, 20567}, {35057, 693}, {35192, 81}, {35193, 86}, {36910, 94}, {40214, 1434}, {41502, 27}, {42033, 76}, {52371, 2166}
X(52405) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 3230, 16488}, {9, 219, 2323}, {218, 2256, 1449}, {219, 220, 9}, {644, 2287, 2321}, {2176, 2273, 16470}, {3690, 26885, 5285}, {30556, 30557, 5904}

X(52406) = X(2)X(30701)∩X(8)X(40962)

X(52406) lies on these lines: {2, 30701}, {8, 40962}, {43, 37042}, {75, 23681}, {92, 42709}, {190, 1763}, {200, 341}, {304, 305}, {312, 2321}, {322, 2064}, {345, 3694}, {668, 18750}, {1264, 44189}, {1427, 40883}, {2999, 4360}, {3699, 4123}, {3952, 20243}, {3961, 4737}, {4033, 20928}, {4095, 11679}, {4417, 46738}, {5795, 44720}, {17143, 19814}, {18152, 28659}, {20106, 33937}, {21621, 49773}, {34403, 42699}, {37266, 44153}

X(52406) = isotomic conjugate of X(1435)
X(52406) = isotomic conjugate of the isogonal conjugate of X(3692)
X(52406) = isotomic conjugate of the polar conjugate of X(341)
X(52406) = X(i)-cross conjugate of X(j) for these (i,j): {1265, 3718}, {3692, 341}
X(52406) = X(i)-isoconjugate of X(j) for these (i,j): {6, 1398}, {19, 1106}, {25, 1407}, {31, 1435}, {32, 1119}, {33, 7366}, {34, 604}, {56, 608}, {57, 1395}, {112, 7250}, {222, 7337}, {225, 16947}, {269, 1973}, {278, 1397}, {279, 1974}, {331, 41280}, {560, 1847}, {607, 7023}, {667, 32714}, {738, 2212}, {1042, 1474}, {1096, 7099}, {1262, 42067}, {1333, 1426}, {1357, 7115}, {1396, 1402}, {1408, 1880}, {1410, 5317}, {1413, 3209}, {1415, 43923}, {1425, 36420}, {1427, 2203}, {1919, 36118}, {1980, 13149}, {2189, 7143}, {2207, 7053}, {2969, 23979}, {3195, 6612}, {3248, 7128}, {3937, 23985}, {6611, 7151}, {7056, 36417}, {7216, 32676}, {7342, 8736}, {22096, 23984}, {32674, 43924}
X(52406) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 608}, {2, 1435}, {6, 1106}, {9, 1398}, {19, 6552}, {25, 24771}, {34, 3161}, {37, 1426}, {244, 3239}, {269, 6337}, {345, 28039}, {604, 11517}, {649, 7358}, {656, 1015}, {1042, 51574}, {1119, 6376}, {1146, 43923}, {1357, 40628}, {1395, 5452}, {1396, 40605}, {1407, 6505}, {1847, 6374}, {1973, 6600}, {2207, 23050}, {2968, 6591}, {3669, 40626}, {3772, 36570}, {6338, 7177}, {6503, 7099}, {6631, 32714}, {7216, 15526}, {7250, 34591}, {9296, 36118}, {35072, 43924}, {40618, 43932}
X(52406) = cevapoint of X(i) and X(j) for these (i,j): {306, 42699}, {1265, 30681}
X(52406) = barycentric product X(i)*X(j) for these {i,j}: {8, 3718}, {69, 341}, {75, 1265}, {76, 3692}, {78, 3596}, {85, 30681}, {190, 15416}, {200, 305}, {212, 40363}, {219, 28659}, {220, 40364}, {304, 346}, {312, 345}, {313, 1792}, {314, 3710}, {318, 1264}, {332, 3701}, {348, 30693}, {525, 7258}, {561, 1260}, {646, 6332}, {1043, 20336}, {1253, 40050}, {1502, 1802}, {1812, 30713}, {2287, 40071}, {2318, 40072}, {2327, 27801}, {2968, 7035}, {3267, 7259}, {3694, 28660}, {3699, 35518}, {3719, 7017}, {3926, 7101}, {4033, 15411}, {4076, 17880}, {4397, 4561}, {4571, 35519}, {5423, 7182}, {6558, 15413}, {7058, 52369}, {7256, 14208}, {7257, 52355}, {31625, 34591}
X(52406) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1398}, {2, 1435}, {3, 1106}, {8, 34}, {9, 608}, {10, 1426}, {33, 7337}, {55, 1395}, {63, 1407}, {69, 269}, {72, 1042}, {75, 1119}, {76, 1847}, {77, 7023}, {78, 56}, {190, 32714}, {200, 25}, {201, 7143}, {212, 1397}, {219, 604}, {220, 1973}, {222, 7366}, {271, 1413}, {283, 1408}, {304, 279}, {305, 1088}, {306, 1427}, {312, 278}, {318, 1118}, {326, 7053}, {332, 1014}, {333, 1396}, {341, 4}, {345, 57}, {346, 19}, {348, 738}, {394, 7099}, {480, 2212}, {521, 43924}, {522, 43923}, {525, 7216}, {644, 32674}, {646, 653}, {656, 7250}, {668, 36118}, {728, 607}, {1016, 7128}, {1021, 43925}, {1043, 28}, {1253, 1974}, {1259, 603}, {1260, 31}, {1264, 77}, {1265, 1}, {1332, 1461}, {1792, 58}, {1802, 32}, {1812, 1412}, {1978, 13149}, {2193, 16947}, {2287, 1474}, {2310, 42067}, {2318, 1402}, {2321, 1880}, {2322, 5317}, {2324, 3209}, {2326, 36420}, {2327, 1333}, {2328, 2203}, {2638, 22096}, {2968, 244}, {3059, 40983}, {3239, 6591}, {3270, 3248}, {3596, 273}, {3610, 8898}, {3682, 1410}, {3692, 6}, {3694, 1400}, {3695, 1254}, {3699, 108}, {3701, 225}, {3710, 65}, {3717, 1876}, {3718, 7}, {3719, 222}, {3926, 7177}, {3965, 2354}, {3998, 52373}, {4012, 40987}, {4025, 43932}, {4076, 7012}, {4082, 1824}, {4158, 7138}, {4163, 18344}, {4171, 2489}, {4385, 7103}, {4397, 7649}, {4515, 2333}, {4561, 934}, {4563, 4637}, {4571, 109}, {4578, 8750}, {4587, 1415}, {4723, 1877}, {5423, 33}, {6332, 3669}, {6516, 6614}, {6558, 1783}, {6559, 8751}, {6735, 1875}, {6736, 1828}, {6737, 40985}, {7004, 1357}, {7046, 1096}, {7079, 2207}, {7080, 208}, {7101, 393}, {7182, 479}, {7256, 162}, {7258, 648}, {7259, 112}, {7360, 1430}, {8611, 7180}, {15411, 1019}, {15416, 514}, {15742, 24033}, {17880, 1358}, {19799, 7365}, {20336, 3668}, {23983, 3942}, {24026, 2969}, {24031, 3937}, {26942, 7147}, {27382, 3213}, {27509, 28017}, {28659, 331}, {30681, 9}, {30693, 281}, {30713, 40149}, {34591, 1015}, {35518, 3676}, {40071, 1446}, {41081, 6612}, {42699, 36908}, {44189, 1422}, {44722, 1420}, {51379, 1457}, {51978, 46883}, {52346, 44696}, {52355, 4017}, {52369, 6354}, {52387, 1425}, {52396, 1439}
X(52406) = {X(306),X(19799)}-harmonic conjugate of X(304)

X(52407) = X(1)X(1399)∩X(3)X(73)

X(52407) lies on these lines: {1, 1399}, {3, 73}, {5, 1935}, {6, 41442}, {30, 1936}, {31, 999}, {34, 37532}, {35, 5399}, {36, 1464}, {47, 56}, {57, 5398}, {58, 942}, {63, 1060}, {77, 21165}, {109, 517}, {140, 3074}, {171, 495}, {184, 23206}, {221, 11249}, {238, 15325}, {283, 1789}, {355, 1771}, {496, 3073}, {499, 7299}, {580, 1427}, {582, 15803}, {595, 24928}, {601, 1496}, {602, 1106}, {651, 6905}, {750, 31479}, {758, 11700}, {859, 26884}, {906, 20752}, {912, 1795}, {1006, 17074}, {1038, 26921}, {1056, 17126}, {1069, 1433}, {1071, 3561}, {1147, 7335}, {1216, 7066}, {1324, 8679}, {1331, 5440}, {1364, 13754}, {1394, 5709}, {1409, 22118}, {1413, 37498}, {1414, 5088}, {1437, 18604}, {1451, 5708}, {1457, 22765}, {1458, 41345}, {1459, 4091}, {1466, 36754}, {1478, 5348}, {1497, 7373}, {1777, 12699}, {1822, 34593}, {1823, 34592}, {1870, 3218}, {2003, 5396}, {2169, 19210}, {2193, 14597}, {2252, 22123}, {2654, 13743}, {2771, 45272}, {2964, 5563}, {3072, 18990}, {3076, 3311}, {3077, 3312}, {3149, 8757}, {3211, 15905}, {3560, 41344}, {3562, 6906}, {4282, 7113}, {5122, 6610}, {5770, 34231}, {5841, 51421}, {5903, 18360}, {6638, 20764}, {6882, 43043}, {6911, 34048}, {6924, 37694}, {7004, 18455}, {7070, 7171}, {7074, 35238}, {7330, 37696}, {8614, 37564}, {9316, 36279}, {9370, 11499}, {10267, 34046}, {10526, 34030}, {10571, 26286}, {10680, 34040}, {11012, 34043}, {11334, 26892}, {11374, 37522}, {11507, 37698}, {12515, 45269}, {14529, 15654}, {14547, 51340}, {16374, 26890}, {18447, 44706}, {19624, 44858}, {21620, 50749}, {22457, 22458}, {23202, 23205}, {24430, 37729}, {24929, 37469}, {25405, 40091}, {32047, 37591}, {37541, 44414}, {37607, 37737}

X(52407) = reflection of X(1807) in X(46974)
X(52407) = isotomic conjugate of the polar conjugate of X(7113)
X(52407) = isogonal conjugate of the polar conjugate of X(3218)
X(52407) = X(i)-Ceva conjugate of X(j) for these (i,j): {1795, 3}, {1797, 48}, {3218, 7113}
X(52407) = X(i)-isoconjugate of X(j) for these (i,j): {4, 80}, {19, 18359}, {25, 20566}, {28, 15065}, {29, 52383}, {33, 18815}, {35, 6344}, {92, 2161}, {108, 52356}, {158, 1807}, {225, 6740}, {264, 6187}, {273, 52371}, {278, 36910}, {281, 2006}, {286, 34857}, {318, 1411}, {319, 18384}, {393, 52351}, {655, 3064}, {759, 41013}, {1168, 38462}, {1785, 40437}, {1824, 14616}, {1826, 24624}, {1857, 52392}, {1877, 36590}, {1896, 52391}, {2166, 6198}, {2222, 44426}, {2341, 40149}, {2501, 47318}, {6591, 36804}, {7649, 51562}, {18344, 35174}, {32675, 46110}, {36125, 51975}
X(52407) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 18359}, {44, 46109}, {80, 36033}, {92, 40584}, {264, 40612}, {318, 35204}, {1147, 1807}, {2161, 22391}, {5081, 6149}, {6198, 11597}, {6505, 20566}, {7113, 37770}, {15065, 40591}, {34586, 41013}, {35128, 46110}, {38983, 52356}, {38984, 44426}
X(52407) = crosspoint of X(i) and X(j) for these (i,j): {1437, 36058}, {52381, 52392}
X(52407) = crosssum of X(i) and X(j) for these (i,j): {1, 1727}, {1824, 14571}, {38462, 41013}
X(52407) = crossdifference of every pair of points on line {1826, 3064}
X(52407) = barycentric product X(i)*X(j) for these {i,j}: {1, 22128}, {3, 3218}, {36, 63}, {48, 320}, {69, 7113}, {77, 2323}, {184, 20924}, {190, 22379}, {212, 17078}, {214, 1797}, {219, 1443}, {222, 4511}, {255, 17923}, {283, 18593}, {295, 27950}, {307, 4282}, {323, 7100}, {348, 2361}, {394, 1870}, {603, 32851}, {654, 6516}, {758, 1790}, {860, 18604}, {906, 4453}, {1227, 32659}, {1331, 3960}, {1437, 3936}, {1444, 2245}, {1459, 4585}, {1464, 1812}, {1795, 16586}, {1796, 4973}, {1813, 3738}, {1835, 6514}, {1983, 4025}, {2193, 41804}, {3724, 17206}, {3904, 36059}, {3977, 16944}, {4091, 4242}, {4561, 21758}, {4575, 4707}, {4592, 21828}, {5081, 7125}, {5440, 40215}, {6149, 52381}, {8552, 13486}, {9247, 40075}, {17515, 40152}, {22115, 30690}, {34544, 52392}, {36058, 51583}, {39152, 44719}, {39153, 44718}
X(52407) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 18359}, {36, 92}, {48, 80}, {50, 6198}, {63, 20566}, {71, 15065}, {184, 2161}, {212, 36910}, {214, 46109}, {222, 18815}, {255, 52351}, {320, 1969}, {577, 1807}, {603, 2006}, {652, 52356}, {654, 44426}, {906, 51562}, {1331, 36804}, {1409, 52383}, {1437, 24624}, {1443, 331}, {1464, 40149}, {1790, 14616}, {1813, 35174}, {1870, 2052}, {1983, 1897}, {2160, 6344}, {2193, 6740}, {2200, 34857}, {2245, 41013}, {2323, 318}, {2361, 281}, {3218, 264}, {3724, 1826}, {3738, 46110}, {3960, 46107}, {4053, 7141}, {4282, 29}, {4511, 7017}, {4575, 47318}, {6516, 46405}, {7100, 94}, {7113, 4}, {7125, 52392}, {8648, 3064}, {9247, 6187}, {13486, 46456}, {14578, 40437}, {14597, 45926}, {16944, 6336}, {17455, 38462}, {19627, 14975}, {20924, 18022}, {21758, 7649}, {21828, 24006}, {22115, 3219}, {22128, 75}, {22356, 51975}, {22379, 514}, {27950, 40717}, {30690, 18817}, {32659, 1168}, {32660, 2222}, {34544, 5081}, {36059, 655}, {46112, 46077}, {46113, 46073}, {52059, 1870}
X(52407) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 23070, 73}, {3, 23071, 22350}, {3, 23072, 3157}, {36, 6149, 2361}, {255, 603, 3}, {601, 1496, 3295}, {1935, 3075, 5}, {4303, 22361, 3}

X(52408) = X(1)X(2361)∩X(3)X(73)

X(52408) = isotomic conjugate of the polar conjugate of X(2174)
X(52408) = isogonal conjugate of the polar conjugate of X(3219)
X(52408) = X(i)-Ceva conjugate of X(j) for these (i,j): {1794, 3}, {1796, 48}, {3219, 2174}, {35193, 35}
X(52408) = X(22342)-cross conjugate of X(35)
X(52408) = X(i)-isoconjugate of X(j) for these (i,j): {4, 79}, {11, 34922}, {19, 30690}, {25, 20565}, {27, 8818}, {28, 6757}, {29, 52382}, {34, 52344}, {36, 6344}, {92, 2160}, {158, 7100}, {225, 3615}, {264, 6186}, {273, 7073}, {278, 7110}, {281, 52374}, {318, 52372}, {320, 18384}, {393, 52381}, {451, 30602}, {1172, 43682}, {1826, 52393}, {1870, 2166}, {1896, 52390}, {1989, 17923}, {3064, 38340}, {4242, 43082}, {6591, 15455}, {6742, 7649}, {8747, 52388}, {13486, 24006}, {21828, 46456}, {26700, 44426}, {41013, 52375}
X(52408) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 30690}, {79, 36033}, {1147, 7100}, {1870, 11597}, {2160, 22391}, {6344, 15898}, {6505, 20565}, {6757, 40591}, {8287, 46107}, {11517, 52344}, {17923, 34544}
X(52408) = barycentric product X(i)*X(j) for these {i,j}: {3, 3219}, {35, 63}, {48, 319}, {69, 2174}, {72, 40214}, {78, 2003}, {97, 35194}, {184, 33939}, {190, 23226}, {212, 17095}, {219, 1442}, {222, 4420}, {228, 34016}, {283, 16577}, {306, 17104}, {307, 35192}, {323, 1807}, {332, 21741}, {333, 22342}, {345, 1399}, {394, 6198}, {603, 42033}, {906, 4467}, {1214, 35193}, {1331, 14838}, {1332, 2605}, {1437, 3969}, {1790, 3678}, {1794, 16585}, {1796, 3647}, {1812, 2594}, {1813, 35057}, {1825, 6514}, {2193, 40999}, {2289, 7282}, {3926, 14975}, {4567, 22094}, {4575, 7265}, {6149, 52351}, {6516, 9404}, {11107, 40152}, {18160, 32656}, {18359, 22115}, {41502, 52385}, {44718, 46073}, {44719, 46077}
X(52408) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 30690}, {35, 92}, {48, 79}, {50, 1870}, {63, 20565}, {71, 6757}, {73, 43682}, {184, 2160}, {212, 7110}, {219, 52344}, {228, 8818}, {255, 52381}, {319, 1969}, {577, 7100}, {603, 52374}, {906, 6742}, {1331, 15455}, {1399, 278}, {1409, 52382}, {1437, 52393}, {1442, 331}, {1807, 94}, {2003, 273}, {2149, 34922}, {2161, 6344}, {2174, 4}, {2193, 3615}, {2594, 40149}, {2605, 17924}, {3219, 264}, {3990, 52388}, {4420, 7017}, {6149, 17923}, {6198, 2052}, {7202, 2973}, {9247, 6186}, {9404, 44426}, {14838, 46107}, {14975, 393}, {17104, 27}, {18359, 18817}, {21741, 225}, {21824, 2970}, {22094, 16732}, {22115, 3218}, {22342, 226}, {23226, 514}, {32660, 26700}, {32661, 13486}, {33939, 18022}, {35057, 46110}, {35192, 29}, {35193, 31623}, {35194, 324}, {36059, 38340}, {40214, 286}, {41502, 1896}, {46112, 39152}, {46113, 39153}
X(52408) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 22117, 3157}, {3, 22161, 11573}, {3, 23070, 4303}, {3, 23071, 73}, {35, 2003, 500}, {35, 6149, 1399}, {212, 255, 3}, {283, 1331, 72}, {602, 1496, 999}, {1936, 3074, 5}, {3219, 6198, 35194}, {22350, 22361, 3}

X(52409) = X(8)X(80)∩X(10)X(94)

X(52409) lies on these lines: {8, 80}, {10, 94}, {75, 311}, {100, 43655}, {280, 5552}, {318, 7141}, {346, 27546}, {759, 7081}, {1043, 3699}, {1219, 2006}, {1222, 1411}, {1807, 4358}, {2222, 2370}, {3153, 5080}, {3992, 32849}, {4385, 45926}, {4723, 36590}, {4768, 52356}, {5176, 38954}, {7026, 44690}, {7043, 44691}, {7080, 36626}, {12532, 38955}, {18025, 30806}, {21296, 52392}

X(52409) = isotomic conjugate of X(1443)
X(52409) = isotomic conjugate of the isogonal conjugate of X(52371)
X(52409) = X(20566)-Ceva conjugate of X(18359)
X(52409) = X(i)-cross conjugate of X(j) for these (i,j): {2325, 312}, {6735, 8}, {36910, 18359}
X(52409) = X(i)-isoconjugate of X(j) for these (i,j): {31, 1443}, {32, 17078}, {36, 56}, {50, 52374}, {57, 7113}, {58, 1464}, {80, 41282}, {108, 22379}, {214, 1417}, {269, 2361}, {320, 1397}, {603, 1870}, {604, 3218}, {608, 22128}, {651, 21758}, {654, 1461}, {758, 1408}, {934, 8648}, {1014, 3724}, {1106, 4511}, {1319, 16944}, {1333, 18593}, {1404, 40215}, {1407, 2323}, {1410, 17515}, {1412, 2245}, {1415, 3960}, {1427, 4282}, {1437, 1835}, {1983, 3669}, {2006, 52059}, {2206, 41804}, {3936, 16947}, {4565, 21828}, {4881, 16945}, {6149, 52372}, {7051, 19373}, {40075, 41280}
X(52409) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 36}, {1, 36909}, {2, 1443}, {8, 4881}, {10, 1464}, {37, 18593}, {56, 15898}, {654, 35508}, {1145, 34586}, {1146, 3960}, {1577, 4089}, {1870, 7952}, {2245, 40599}, {2323, 24771}, {2361, 6600}, {2968, 3738}, {3161, 3218}, {4453, 40624}, {4511, 6552}, {5452, 7113}, {6376, 17078}, {8648, 14714}, {14993, 52372}, {21758, 38991}, {22379, 38983}, {40603, 41804}
X(52409) = cevapoint of X(i) and X(j) for these (i,j): {3701, 4723}, {4768, 24026}
X(52409) = trilinear pole of line {2321, 3239}
X(52409) = barycentric product X(i)*X(j) for these {i,j}: {8, 18359}, {9, 20566}, {75, 36910}, {76, 52371}, {80, 312}, {94, 4420}, {190, 52356}, {313, 2341}, {318, 52351}, {321, 6740}, {333, 15065}, {341, 2006}, {346, 18815}, {522, 36804}, {655, 4397}, {759, 30713}, {1807, 7017}, {2161, 3596}, {2166, 42033}, {2321, 14616}, {3239, 35174}, {3701, 24624}, {3900, 46405}, {4086, 47318}, {4358, 36590}, {4391, 51562}, {4997, 51975}, {6187, 28659}, {7101, 52392}, {23978, 52377}, {28654, 52380}, {28660, 34857}, {41226, 52344}
X(52409) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 1443}, {8, 3218}, {9, 36}, {10, 18593}, {37, 1464}, {55, 7113}, {75, 17078}, {78, 22128}, {80, 57}, {200, 2323}, {210, 2245}, {220, 2361}, {281, 1870}, {312, 320}, {318, 17923}, {321, 41804}, {341, 32851}, {346, 4511}, {522, 3960}, {652, 22379}, {655, 934}, {657, 8648}, {663, 21758}, {759, 1412}, {1320, 40215}, {1334, 3724}, {1411, 1407}, {1793, 1790}, {1807, 222}, {1826, 1835}, {1989, 52372}, {2006, 269}, {2161, 56}, {2166, 52374}, {2222, 1461}, {2316, 16944}, {2321, 758}, {2322, 17515}, {2325, 214}, {2328, 4282}, {2341, 58}, {2361, 52059}, {3161, 4881}, {3239, 3738}, {3596, 20924}, {3685, 27950}, {3686, 4973}, {3689, 17455}, {3699, 4585}, {3701, 3936}, {3900, 654}, {3939, 1983}, {4007, 4880}, {4024, 51645}, {4041, 21828}, {4053, 3028}, {4086, 4707}, {4358, 41801}, {4391, 4453}, {4397, 3904}, {4420, 323}, {4671, 36589}, {4723, 51583}, {4858, 4089}, {4873, 4867}, {6057, 4053}, {6187, 604}, {6735, 16586}, {6740, 81}, {7026, 37772}, {7043, 37773}, {7101, 5081}, {7113, 41282}, {7126, 19373}, {14147, 34921}, {14616, 1434}, {15065, 226}, {18359, 7}, {18815, 279}, {19551, 7051}, {20566, 85}, {23615, 46384}, {24624, 1014}, {28659, 40075}, {30713, 35550}, {34079, 1408}, {34857, 1400}, {35174, 658}, {36590, 88}, {36804, 664}, {36815, 1429}, {36910, 1}, {40172, 1404}, {40437, 34051}, {41226, 1442}, {46405, 4569}, {47318, 1414}, {51562, 651}, {51975, 3911}, {52351, 77}, {52356, 514}, {52371, 6}, {52377, 1262}, {52380, 593}, {52383, 1427}, {52391, 52373}, {52392, 7177}
X(52409) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {80, 15065, 18359}, {15065, 51975, 80}

X(52410) = X(6)X(1604)∩X(32)X(604)

X(52410) lies on these lines: {6, 1604}, {32, 604}, {39, 2286}, {48, 5065}, {56, 478}, {101, 23185}, {109, 21769}, {269, 40746}, {279, 52376}, {560, 1397}, {571, 52059}, {574, 2197}, {577, 7113}, {603, 2300}, {608, 1015}, {713, 934}, {1333, 1407}, {1400, 5042}, {1415, 16946}, {1436, 14936}, {1455, 20227}, {1461, 34077}, {1470, 2092}, {1950, 2241}, {1974, 22096}, {2174, 33871}, {2178, 23980}, {2242, 2285}, {7117, 46432}, {8736, 9651}, {9306, 23086}, {14642, 39687}, {34880, 40590}

X(52410) = isogonal conjugate of the complement of X(46716)
X(52410) = isogonal conjugate of the isotomic conjugate of X(1407)
X(52410) = isogonal conjugate of the polar conjugate of X(1398)
X(52410) = X(1106)-Ceva conjugate of X(1397)
X(52410) = X(i)-isoconjugate of X(j) for these (i,j): {2, 341}, {7, 30693}, {8, 312}, {9, 3596}, {21, 30713}, {41, 40363}, {55, 28659}, {69, 7101}, {75, 346}, {76, 200}, {78, 7017}, {85, 5423}, {92, 1265}, {190, 4397}, {210, 28660}, {220, 561}, {264, 3692}, {273, 30681}, {274, 4082}, {281, 3718}, {304, 7046}, {305, 7079}, {310, 4515}, {313, 2287}, {314, 2321}, {318, 345}, {321, 1043}, {333, 3701}, {480, 20567}, {522, 646}, {523, 7258}, {556, 7027}, {644, 35519}, {645, 4086}, {657, 6386}, {668, 3239}, {670, 4171}, {693, 6558}, {728, 6063}, {765, 23978}, {850, 7259}, {1016, 24026}, {1021, 27808}, {1089, 7058}, {1098, 28654}, {1146, 7035}, {1240, 3965}, {1253, 1502}, {1260, 1969}, {1334, 40072}, {1577, 7256}, {1802, 18022}, {1897, 15416}, {1928, 14827}, {1978, 3900}, {2310, 31625}, {2322, 20336}, {2328, 27801}, {3261, 4578}, {3263, 6559}, {3680, 44723}, {3694, 44130}, {3699, 4391}, {3700, 7257}, {3702, 4102}, {3710, 31623}, {3717, 36796}, {3975, 4518}, {4033, 7253}, {4076, 4858}, {4087, 4876}, {4110, 7155}, {4130, 4572}, {4148, 4583}, {4163, 4554}, {4183, 40071}, {4451, 17787}, {4524, 4602}, {4571, 46110}, {4582, 4768}, {4601, 52335}, {4723, 4997}, {6057, 52379}, {6555, 40014}, {6556, 18743}, {6557, 44720}, {6602, 41283}, {6632, 42455}, {6735, 36795}, {6736, 32017}, {7071, 40364}, {7080, 34404}, {9447, 44159}, {18155, 30730}, {23104, 31615}, {27424, 27538}, {42033, 52344}
X(52410) = X(i)-Dao conjugate of X(j) for these (i,j): {76, 6609}, {206, 346}, {220, 40368}, {223, 28659}, {341, 32664}, {478, 3596}, {513, 23978}, {1265, 22391}, {1502, 17113}, {3160, 40363}, {14827, 40369}, {15267, 28654}, {15416, 34467}, {27801, 36908}, {30713, 40611}
X(52410) = cevapoint of X(1919) and X(22096)
X(52410) = crosspoint of X(i) and X(j) for these (i,j): {1106, 7366}, {1398, 1407}
X(52410) = crosssum of X(i) and X(j) for these (i,j): {6, 1603}, {341, 30693}, {346, 1265}, {1146, 4397}
X(52410) = crossdifference of every pair of points on line {4397, 42337}
X(52410) = barycentric product X(i)*X(j) for these {i,j}: {1, 1106}, {3, 1398}, {6, 1407}, {7, 1397}, {9, 7366}, {12, 7342}, {19, 7099}, {25, 7053}, {28, 1410}, {31, 269}, {32, 279}, {34, 603}, {41, 738}, {48, 1435}, {55, 7023}, {56, 56}, {57, 604}, {58, 1042}, {59, 1357}, {60, 7143}, {65, 1408}, {77, 1395}, {109, 43924}, {110, 7250}, {163, 7216}, {181, 7341}, {184, 1119}, {198, 6612}, {221, 1413}, {222, 608}, {226, 16947}, {244, 24027}, {479, 2175}, {560, 1088}, {649, 1461}, {658, 1919}, {663, 6614}, {667, 934}, {669, 4616}, {692, 43932}, {798, 4637}, {849, 1254}, {1014, 1402}, {1015, 1262}, {1086, 23979}, {1118, 7335}, {1275, 1977}, {1319, 1417}, {1333, 1427}, {1356, 7340}, {1396, 1409}, {1399, 52372}, {1400, 1412}, {1414, 51642}, {1415, 3669}, {1416, 1458}, {1420, 16945}, {1422, 2199}, {1426, 1437}, {1436, 6611}, {1439, 2203}, {1472, 4320}, {1474, 52373}, {1804, 7337}, {1847, 9247}, {1924, 4635}, {1973, 7177}, {1974, 7056}, {1980, 4569}, {2150, 7147}, {2206, 3668}, {3063, 4617}, {3248, 7045}, {3271, 7339}, {3450, 17114}, {4565, 7180}, {4619, 21143}, {6063, 41280}, {6066, 41292}, {7153, 41526}, {9447, 23062}, {14936, 23971}, {22383, 32714}, {36059, 43923}, {41281, 41283}, {41286, 41287}, {41288, 41289}
X(52410) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 40363}, {31, 341}, {32, 346}, {41, 30693}, {56, 3596}, {57, 28659}, {163, 7258}, {184, 1265}, {269, 561}, {279, 1502}, {479, 41283}, {560, 200}, {603, 3718}, {604, 312}, {608, 7017}, {667, 4397}, {738, 20567}, {934, 6386}, {1014, 40072}, {1015, 23978}, {1042, 313}, {1088, 1928}, {1106, 75}, {1119, 18022}, {1262, 31625}, {1356, 4092}, {1357, 34387}, {1395, 318}, {1397, 8}, {1398, 264}, {1400, 30713}, {1402, 3701}, {1407, 76}, {1408, 314}, {1410, 20336}, {1412, 28660}, {1415, 646}, {1427, 27801}, {1428, 4087}, {1435, 1969}, {1461, 1978}, {1501, 220}, {1576, 7256}, {1917, 1253}, {1918, 4082}, {1919, 3239}, {1924, 4171}, {1973, 7101}, {1974, 7046}, {1977, 1146}, {1980, 3900}, {2175, 5423}, {2205, 4515}, {2206, 1043}, {3248, 24026}, {3249, 42462}, {4616, 4609}, {4637, 4602}, {6063, 44159}, {6612, 44190}, {6614, 4572}, {7023, 6063}, {7053, 305}, {7056, 40050}, {7099, 304}, {7143, 34388}, {7177, 40364}, {7216, 20948}, {7250, 850}, {7335, 1264}, {7341, 18021}, {7342, 261}, {7366, 85}, {8027, 42455}, {9233, 14827}, {9247, 3692}, {9426, 4524}, {9447, 728}, {9448, 480}, {14575, 1260}, {16947, 333}, {22096, 2968}, {22383, 15416}, {23979, 1016}, {24027, 7035}, {32739, 6558}, {38859, 40088}, {41280, 55}, {41281, 2175}, {41286, 9448}, {41526, 4110}, {42067, 21666}, {43924, 35519}, {43932, 40495}, {44162, 7071}, {51642, 4086}, {52373, 40071}
X(52410) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {48, 5065, 14827}, {56, 478, 17053}, {109, 38855, 21769}, {604, 2199, 20228}

X(52411) = X(1)X(1950)∩X(3)X(2197)

X(52411) lies on these lines: {1, 1950}, {3, 2197}, {6, 909}, {7, 4565}, {31, 2208}, {32, 604}, {41, 5065}, {48, 577}, {56, 608}, {73, 22054}, {172, 2285}, {184, 23197}, {221, 37519}, {222, 1790}, {255, 22074}, {478, 2178}, {571, 7113}, {607, 1436}, {610, 1951}, {906, 20818}, {1319, 5301}, {1395, 40956}, {1397, 2206}, {1400, 5019}, {1404, 16946}, {1405, 5042}, {1455, 1880}, {1968, 7120}, {2171, 2242}, {2174, 5063}, {2289, 20752}, {2327, 22127}, {3215, 3990}, {4282, 10571}, {5193, 16488}, {7053, 32658}, {7124, 15905}, {7354, 8736}, {10607, 23144}, {14575, 22096}, {18954, 46010}, {21794, 31451}, {21933, 51422}, {32659, 32660}, {33882, 38296}, {34046, 37504}

X(52411) = isogonal conjugate of X(7017)
X(52411) = isogonal conjugate of the isotomic conjugate of X(222)
X(52411) = isotomic conjugate of the polar conjugate of X(1397)
X(52411) = isogonal conjugate of the polar conjugate of X(56)
X(52411) = polar conjugate of the isotomic conjugate of X(7335)
X(52411) = X(i)-Ceva conjugate of X(j) for these (i,j): {56, 1397}, {222, 7335}, {603, 184}, {1333, 604}, {1415, 22383}, {44717, 36059}
X(52411) = X(i)-cross conjugate of X(j) for these (i,j): {9247, 184}, {23196, 3}
X(52411) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7017}, {2, 318}, {4, 312}, {7, 7101}, {8, 92}, {9, 264}, {10, 31623}, {19, 3596}, {25, 28659}, {27, 3701}, {28, 30713}, {29, 321}, {33, 76}, {37, 44130}, {41, 18022}, {55, 1969}, {75, 281}, {78, 2052}, {85, 7046}, {100, 46110}, {158, 345}, {190, 44426}, {200, 331}, {210, 44129}, {212, 18027}, {270, 28654}, {273, 346}, {276, 7069}, {278, 341}, {286, 2321}, {304, 1857}, {306, 1896}, {313, 1172}, {314, 1826}, {322, 7003}, {324, 44687}, {329, 7020}, {333, 41013}, {349, 4183}, {393, 3718}, {459, 52346}, {461, 40023}, {470, 44690}, {471, 44691}, {522, 6335}, {561, 607}, {643, 14618}, {644, 46107}, {645, 24006}, {646, 7649}, {648, 4086}, {653, 4397}, {668, 3064}, {811, 3700}, {823, 52355}, {1043, 40149}, {1089, 46103}, {1093, 3719}, {1119, 30693}, {1259, 6521}, {1264, 6520}, {1320, 46109}, {1441, 2322}, {1502, 2212}, {1577, 36797}, {1783, 35519}, {1785, 36795}, {1824, 28660}, {1847, 5423}, {1861, 36796}, {1897, 4391}, {1973, 40363}, {1978, 18344}, {2185, 7141}, {2299, 27801}, {2326, 34388}, {2333, 40072}, {2501, 7257}, {3239, 18026}, {3699, 17924}, {3900, 46404}, {4041, 6331}, {4092, 46254}, {4123, 43678}, {4163, 13149}, {4564, 21666}, {4723, 6336}, {4858, 15742}, {4876, 40717}, {4997, 38462}, {5081, 18359}, {6059, 40364}, {6063, 7079}, {6528, 8611}, {6734, 40447}, {6735, 16082}, {7012, 23978}, {7035, 8735}, {7071, 20567}, {7140, 52379}, {7156, 41530}, {7952, 34404}, {8056, 44721}, {8748, 20336}, {9447, 44161}, {13426, 46745}, {13454, 46744}, {14942, 46108}, {15416, 36127}, {15466, 44692}, {15628, 40703}, {16081, 44694}, {20928, 43742}, {24026, 46102}, {30710, 46878}, {33299, 46104}, {33672, 40838}, {36804, 44428}, {40971, 44190}, {44189, 47372}, {44693, 46106}
X(52411) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 7017}, {6, 3596}, {8, 22391}, {206, 281}, {223, 1969}, {226, 27801}, {264, 478}, {312, 36033}, {318, 32664}, {331, 6609}, {345, 1147}, {607, 40368}, {1264, 37867}, {3160, 18022}, {3700, 17423}, {4391, 34467}, {6337, 40363}, {6505, 28659}, {8054, 46110}, {18027, 40837}, {30713, 40591}, {35519, 39006}, {40589, 44130}
X(52411) = cevapoint of X(3) and X(23159)
X(52411) = crosspoint of X(i) and X(j) for these (i,j): {56, 222}, {603, 7099}, {1415, 23979}, {36059, 44717}
X(52411) = crosssum of X(i) and X(j) for these (i,j): {8, 281}, {318, 7101}, {4391, 23978}, {8735, 44426}
X(52411) = crossdifference of every pair of points on line {2804, 4397}
X(52411) = barycentric product X(i)*X(j) for these {i,j}: {1, 603}, {3, 56}, {4, 7335}, {6, 222}, {7, 184}, {9, 7099}, {19, 7125}, {21, 1410}, {25, 1804}, {28, 22341}, {31, 77}, {32, 348}, {34, 255}, {41, 7177}, {48, 57}, {54, 30493}, {55, 7053}, {58, 73}, {59, 3937}, {60, 1425}, {63, 604}, {65, 1437}, {69, 1397}, {71, 1412}, {72, 1408}, {78, 1106}, {81, 1409}, {84, 7114}, {85, 9247}, {108, 23224}, {109, 1459}, {162, 51641}, {163, 51640}, {201, 849}, {212, 269}, {219, 1407}, {221, 1433}, {228, 1014}, {241, 32658}, {248, 43034}, {268, 6611}, {270, 7138}, {278, 577}, {283, 1042}, {284, 52373}, {293, 51653}, {295, 1428}, {296, 26884}, {305, 41280}, {306, 16947}, {307, 2206}, {326, 1395}, {331, 14585}, {394, 608}, {513, 36059}, {514, 32660}, {560, 7182}, {593, 2197}, {647, 4565}, {649, 1813}, {651, 22383}, {652, 1461}, {667, 6516}, {738, 1802}, {810, 1414}, {905, 1415}, {906, 3669}, {927, 23225}, {934, 1946}, {961, 22345}, {1015, 44717}, {1037, 1473}, {1038, 1472}, {1069, 1406}, {1092, 1118}, {1119, 6056}, {1175, 39791}, {1176, 1401}, {1214, 1333}, {1259, 1398}, {1260, 7023}, {1262, 7117}, {1319, 36058}, {1331, 43924}, {1355, 47388}, {1356, 47389}, {1363, 32230}, {1365, 47390}, {1393, 2169}, {1394, 19614}, {1396, 3990}, {1399, 7100}, {1400, 1790}, {1402, 1444}, {1403, 23086}, {1404, 1797}, {1413, 7078}, {1416, 1818}, {1417, 5440}, {1423, 15373}, {1427, 2193}, {1429, 2196}, {1431, 3955}, {1434, 2200}, {1435, 2289}, {1436, 7011}, {1439, 2194}, {1455, 36055}, {1456, 36056}, {1457, 1795}, {1458, 36057}, {1462, 20752}, {1465, 14578}, {1474, 40152}, {1475, 1803}, {1476, 22344}, {1477, 20780}, {1576, 17094}, {1880, 18604}, {1973, 7183}, {1974, 7055}, {2067, 6502}, {2148, 44708}, {2149, 3942}, {2150, 37755}, {2175, 7056}, {2199, 41081}, {2203, 52385}, {2208, 7013}, {2221, 2286}, {2222, 22379}, {2720, 8677}, {2982, 14597}, {3049, 4573}, {3270, 7339}, {3292, 7316}, {3450, 23154}, {3665, 10547}, {3676, 32656}, {3690, 7341}, {3692, 7366}, {3695, 7342}, {3733, 23067}, {3911, 32659}, {3964, 7337}, {4017, 4575}, {4091, 32674}, {4558, 7180}, {4559, 7254}, {4592, 51642}, {4855, 16945}, {4998, 22096}, {5061, 17971}, {6063, 14575}, {6357, 18877}, {7004, 24027}, {7116, 7175}, {7178, 32661}, {7181, 14908}, {7251, 14376}, {8686, 23205}, {14379, 44696}, {14642, 18623}, {14827, 30682}, {15375, 42461}, {15389, 17082}, {17081, 40319}, {17104, 52390}, {20818, 40151}, {22093, 29055}, {22342, 52375}, {23226, 26700}, {23979, 26932}, {32651, 52306}, {32657, 43035}, {32714, 36054}, {35200, 51656}, {36052, 51651}, {36060, 51655}, {40050, 41281}, {40360, 41286}, {40373, 41283}, {41279, 46089}
X(52411) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 3596}, {6, 7017}, {7, 18022}, {31, 318}, {32, 281}, {41, 7101}, {48, 312}, {56, 264}, {57, 1969}, {58, 44130}, {63, 28659}, {69, 40363}, {71, 30713}, {73, 313}, {77, 561}, {181, 7141}, {184, 8}, {212, 341}, {222, 76}, {228, 3701}, {255, 3718}, {278, 18027}, {305, 44159}, {348, 1502}, {560, 33}, {577, 345}, {603, 75}, {604, 92}, {608, 2052}, {649, 46110}, {667, 44426}, {810, 4086}, {906, 646}, {1092, 1264}, {1106, 273}, {1214, 27801}, {1333, 31623}, {1356, 8754}, {1357, 2973}, {1395, 158}, {1397, 4}, {1401, 1235}, {1402, 41013}, {1404, 46109}, {1407, 331}, {1408, 286}, {1409, 321}, {1410, 1441}, {1412, 44129}, {1415, 6335}, {1425, 34388}, {1428, 40717}, {1437, 314}, {1444, 40072}, {1459, 35519}, {1461, 46404}, {1501, 607}, {1576, 36797}, {1790, 28660}, {1802, 30693}, {1804, 305}, {1813, 1978}, {1917, 2212}, {1919, 3064}, {1946, 4397}, {1974, 1857}, {1977, 8735}, {1980, 18344}, {2175, 7046}, {2197, 28654}, {2200, 2321}, {2203, 1896}, {2206, 29}, {2208, 7020}, {3049, 3700}, {3052, 44721}, {3271, 21666}, {3937, 34387}, {4055, 3710}, {4100, 3719}, {4173, 4178}, {4565, 6331}, {4575, 7257}, {6056, 1265}, {6063, 44161}, {6516, 6386}, {6611, 40701}, {7053, 6063}, {7055, 40050}, {7056, 41283}, {7099, 85}, {7114, 322}, {7117, 23978}, {7125, 304}, {7177, 20567}, {7180, 14618}, {7182, 1928}, {7183, 40364}, {7193, 4087}, {7251, 17907}, {7316, 46111}, {7335, 69}, {7337, 1093}, {7366, 1847}, {9247, 9}, {9447, 7079}, {9448, 7071}, {14575, 55}, {14578, 36795}, {14585, 219}, {14600, 15628}, {15373, 27424}, {16947, 27}, {17094, 44173}, {20775, 3703}, {20818, 44723}, {22096, 11}, {22341, 20336}, {22344, 20895}, {22383, 4391}, {23067, 27808}, {23196, 1329}, {23200, 3712}, {23201, 3702}, {23202, 4723}, {23220, 2804}, {23224, 35518}, {23225, 50333}, {23606, 1259}, {23979, 46102}, {26920, 13461}, {30493, 311}, {32656, 3699}, {32658, 36796}, {32659, 4997}, {32660, 190}, {32661, 645}, {34980, 7068}, {36054, 15416}, {36059, 668}, {39201, 52355}, {39791, 1234}, {40152, 40071}, {40373, 2175}, {41280, 25}, {41281, 1974}, {41286, 44162}, {43034, 44132}, {43924, 46107}, {44088, 44707}, {44162, 6059}, {44717, 31625}, {47390, 6064}, {51640, 20948}, {51641, 14208}, {51642, 24006}, {51653, 40703}, {52373, 349}
X(52411) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 2286, 2197}, {3, 22132, 22071}, {48, 603, 1409}

X(52412) = X(2)X(92)∩X(10)X(29)

X(52412) lies on these lines: {2, 92}, {4, 2355}, {8, 7498}, {9, 1748}, {10, 29}, {12, 1940}, {19, 469}, {27, 1268}, {28, 5260}, {35, 11107}, {37, 18679}, {40, 52248}, {63, 7079}, {81, 1783}, {100, 4183}, {101, 21072}, {158, 498}, {162, 171}, {165, 39531}, {196, 5226}, {226, 653}, {240, 756}, {242, 427}, {243, 5432}, {286, 28653}, {306, 2322}, {318, 406}, {343, 1944}, {345, 7101}, {355, 7531}, {412, 6684}, {423, 7119}, {445, 3219}, {451, 41013}, {468, 7009}, {475, 5342}, {517, 7551}, {607, 3661}, {608, 17368}, {750, 1957}, {857, 3101}, {894, 25977}, {908, 30686}, {958, 37253}, {993, 37304}, {1013, 1376}, {1096, 5268}, {1118, 10588}, {1146, 23292}, {1148, 11374}, {1430, 17122}, {1585, 7090}, {1586, 14121}, {1698, 5125}, {1762, 20305}, {1767, 8545}, {1784, 3584}, {1824, 4213}, {1838, 3634}, {1844, 3678}, {1848, 8756}, {1851, 8889}, {1855, 26003}, {1857, 5218}, {1859, 3740}, {1861, 14004}, {1867, 3144}, {1877, 5294}, {1891, 4248}, {1895, 3085}, {1943, 11064}, {2331, 5287}, {2333, 31909}, {2994, 37645}, {3176, 5703}, {3452, 30687}, {3535, 13387}, {3536, 13386}, {3820, 37321}, {3925, 37371}, {3969, 4420}, {4359, 46108}, {4413, 35994}, {5047, 41227}, {5081, 5136}, {5090, 37055}, {5130, 16066}, {5235, 44734}, {5249, 25993}, {5251, 17515}, {5261, 44696}, {5281, 44695}, {5743, 41234}, {5744, 37276}, {6335, 20566}, {6353, 7102}, {6354, 47296}, {6513, 44179}, {7017, 32851}, {7110, 40447}, {7359, 26942}, {7466, 26251}, {7510, 38042}, {7518, 46933}, {7541, 10175}, {7719, 17308}, {8232, 40837}, {9708, 37393}, {11231, 39529}, {11341, 29576}, {14006, 32917}, {14024, 15523}, {14165, 16577}, {14571, 44307}, {16585, 17095}, {16706, 20883}, {17605, 52167}, {17902, 17916}, {17906, 33133}, {17911, 17912}, {18589, 30841}, {20205, 34234}, {21016, 31905}, {21028, 26885}, {23521, 33150}, {24603, 37389}, {25647, 41340}, {26066, 37235}, {26872, 27382}, {27132, 52288}, {27287, 46835}, {33108, 37372}, {34591, 45224}, {46110, 47794}

X(52412) = isotomic conjugate of X(52381)
X(52412) = polar conjugate of X(79)
X(52412) = polar conjugate of the isotomic conjugate of X(319)
X(52412) = polar conjugate of the isogonal conjugate of X(35)
X(52412) = X(i)-complementary conjugate of X(j) for these (i,j): {3469, 141}, {34303, 21236}
X(52412) = X(40447)-Ceva conjugate of X(92)
X(52412) = X(i)-cross conjugate of X(j) for these (i,j): {35, 319}, {445, 92}, {6198, 7282}, {16577, 3219}
X(52412) = X(i)-isoconjugate of X(j) for these (i,j): {3, 2160}, {6, 7100}, {31, 52381}, {48, 79}, {57, 8606}, {63, 6186}, {71, 52375}, {184, 30690}, {212, 52374}, {219, 52372}, {222, 7073}, {228, 52393}, {265, 7113}, {284, 52390}, {603, 7110}, {647, 13486}, {652, 26700}, {1333, 52388}, {1400, 1789}, {1409, 3615}, {1437, 8818}, {1870, 50433}, {1946, 38340}, {2193, 52382}, {3218, 52153}, {6742, 22383}, {9247, 20565}, {14395, 36064}, {19302, 50462}, {21828, 36061}, {36296, 39152}, {36297, 39153}, {42623, 52201}
X(52412) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 52381}, {9, 7100}, {35, 22122}, {37, 52388}, {79, 1249}, {905, 8287}, {1100, 3916}, {1789, 40582}, {2160, 36103}, {3162, 6186}, {4466, 14838}, {5249, 18607}, {5452, 8606}, {7110, 7952}, {13486, 39052}, {16221, 21828}, {22058, 25639}, {22128, 40604}, {38340, 39053}, {40590, 52390}, {40837, 52374}, {47345, 52382}
X(52412) = cevapoint of X(281) and X(451)
X(52412) = crosssum of X(3) and X(23165)
X(52412) = trilinear pole of line {7265, 35057}
X(52412) = barycentric product X(i)*X(j) for these {i,j}: {4, 319}, {8, 7282}, {19, 33939}, {27, 3969}, {29, 40999}, {35, 264}, {75, 6198}, {80, 340}, {92, 3219}, {186, 20566}, {273, 4420}, {278, 42033}, {281, 17095}, {286, 3678}, {314, 1825}, {318, 1442}, {349, 41502}, {445, 40435}, {561, 14975}, {648, 7265}, {1441, 11107}, {1783, 18160}, {1826, 34016}, {1844, 40422}, {1897, 4467}, {1969, 2174}, {2003, 7017}, {2594, 44130}, {5379, 17886}, {6335, 14838}, {9404, 46404}, {14165, 52351}, {16577, 31623}, {16585, 40447}, {17923, 41226}, {18020, 21054}, {18026, 35057}, {21824, 46254}, {35194, 40440}, {44427, 47318}
X(52412) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 7100}, {2, 52381}, {4, 79}, {10, 52388}, {19, 2160}, {21, 1789}, {25, 6186}, {27, 52393}, {28, 52375}, {29, 3615}, {33, 7073}, {34, 52372}, {35, 3}, {55, 8606}, {65, 52390}, {80, 265}, {92, 30690}, {108, 26700}, {162, 13486}, {186, 36}, {225, 52382}, {264, 20565}, {278, 52374}, {281, 7110}, {318, 52344}, {319, 69}, {323, 22128}, {340, 320}, {445, 5249}, {484, 50462}, {500, 4303}, {653, 38340}, {1399, 603}, {1442, 77}, {1825, 65}, {1826, 8818}, {1844, 942}, {1897, 6742}, {2003, 222}, {2174, 48}, {2594, 73}, {2605, 1459}, {2611, 18210}, {3219, 63}, {3578, 4001}, {3647, 3916}, {3678, 72}, {3969, 306}, {4213, 14844}, {4354, 1062}, {4420, 78}, {4467, 4025}, {6187, 52153}, {6198, 1}, {6335, 15455}, {7150, 52202}, {7186, 3784}, {7202, 3942}, {7206, 3695}, {7265, 525}, {7282, 7}, {8287, 4466}, {9404, 652}, {11107, 21}, {14165, 17923}, {14838, 905}, {14975, 31}, {16577, 1214}, {16585, 18607}, {16755, 15419}, {17095, 348}, {17104, 1437}, {17454, 22054}, {18160, 15413}, {20566, 328}, {21054, 125}, {21741, 1409}, {21794, 2197}, {21824, 3708}, {22342, 22341}, {23226, 23224}, {30600, 2523}, {33939, 304}, {34016, 17206}, {35057, 521}, {35192, 2193}, {35193, 283}, {35194, 44706}, {35197, 23070}, {40149, 43682}, {40214, 1790}, {40999, 307}, {41013, 6757}, {41226, 52351}, {41502, 284}, {41562, 1071}, {42033, 345}, {44095, 2260}, {44427, 4707}, {46468, 4292}, {47230, 21828}
X(52412) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 92, 17923}, {2, 281, 92}, {2, 6360, 17073}, {10, 29, 5174}, {171, 7076, 162}, {445, 3219, 7282}, {468, 7140, 7009}, {1698, 39585, 5125}, {4213, 17927, 1824}, {6684, 39574, 412}

X(52413) = X(4)X(29046)∩X(6)X(19)

X(52413) lies on these lines: {4, 29046}, {6, 19}, {9, 1063}, {25, 28658}, {28, 1203}, {33, 44105}, {36, 4282}, {44, 5089}, {47, 579}, {48, 10571}, {58, 1474}, {73, 2302}, {79, 1172}, {81, 1848}, {92, 3758}, {109, 2252}, {216, 1950}, {222, 1435}, {278, 2003}, {320, 17923}, {513, 1430}, {517, 22123}, {613, 23052}, {651, 5236}, {758, 1870}, {894, 20883}, {909, 32667}, {913, 32669}, {971, 36742}, {1395, 3192}, {1400, 2148}, {1404, 32674}, {1415, 8607}, {1464, 7113}, {1783, 8756}, {1842, 3194}, {1861, 5847}, {1871, 36750}, {1877, 8755}, {1951, 3284}, {1973, 5299}, {1990, 8735}, {2077, 22059}, {2149, 2183}, {2210, 32676}, {2245, 2361}, {2299, 2308}, {2355, 44097}, {2356, 8750}, {3002, 32756}, {3271, 51726}, {5142, 37559}, {5280, 17442}, {5338, 44086}, {5410, 34121}, {5411, 34125}, {6748, 8736}, {7583, 16027}, {7584, 16033}, {7719, 16670}, {14936, 40135}, {21758, 46384}, {42067, 44102}

X(52413) = isogonal conjugate of X(52351)
X(52413) = polar conjugate of X(20566)
X(52413) = isogonal conjugate of the isotomic conjugate of X(17923)
X(52413) = polar conjugate of the isotomic conjugate of X(36)
X(52413) = X(i)-Ceva conjugate of X(j) for these (i,j): {17923, 36}, {36125, 25}
X(52413) = X(44113)-cross conjugate of X(1870)
X(52413) = X(i)-isoconjugate of X(j) for these (i,j): {1, 52351}, {2, 1807}, {3, 18359}, {9, 52392}, {48, 20566}, {63, 80}, {69, 2161}, {71, 14616}, {72, 24624}, {77, 36910}, {78, 2006}, {219, 18815}, {226, 1793}, {265, 3219}, {304, 6187}, {306, 759}, {307, 2341}, {328, 2174}, {333, 52391}, {345, 1411}, {348, 52371}, {521, 655}, {652, 35174}, {656, 47318}, {905, 51562}, {1168, 3977}, {1214, 6740}, {1459, 36804}, {1790, 15065}, {1797, 51975}, {1809, 52212}, {1812, 52383}, {1813, 52356}, {1946, 46405}, {2222, 6332}, {4064, 37140}, {7100, 41226}, {7265, 36061}, {17206, 34857}, {20336, 34079}, {26932, 52377}, {26942, 52380}, {32675, 35518}, {33939, 52153}, {40709, 46077}, {40710, 46073}
X(52413) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 52351}, {69, 40584}, {80, 3162}, {304, 40612}, {306, 34586}, {345, 35204}, {478, 52392}, {1249, 20566}, {1807, 32664}, {4391, 13999}, {6332, 38984}, {7265, 16221}, {18359, 36103}, {20336, 35069}, {35128, 35518}, {39053, 46405}, {40071, 51583}, {40596, 47318}
X(52413) = crosspoint of X(i) and X(j) for these (i,j): {27, 36123}, {1474, 8752}, {7128, 36110}
X(52413) = crosssum of X(i) and X(j) for these (i,j): {3, 22123}, {71, 22350}, {306, 3977}
X(52413) = trilinear pole of line {8648, 21828}
X(52413) = crossdifference of every pair of points on line {72, 521}
X(52413) = barycentric product X(i)*X(j) for these {i,j}: {1, 1870}, {4, 36}, {6, 17923}, {19, 3218}, {21, 1835}, {25, 320}, {27, 2245}, {28, 758}, {29, 1464}, {33, 1443}, {34, 4511}, {56, 5081}, {58, 860}, {65, 17515}, {79, 186}, {86, 44113}, {92, 7113}, {104, 1845}, {108, 3738}, {109, 44428}, {112, 4707}, {214, 36125}, {273, 2361}, {278, 2323}, {286, 3724}, {340, 6186}, {393, 22128}, {513, 4242}, {607, 17078}, {608, 32851}, {648, 21828}, {653, 654}, {915, 11570}, {1061, 4351}, {1130, 8119}, {1172, 18593}, {1474, 3936}, {1783, 3960}, {1973, 20924}, {1974, 40075}, {1983, 17924}, {2203, 35550}, {2299, 41804}, {2906, 39149}, {3904, 32674}, {4282, 40149}, {4453, 8750}, {4585, 6591}, {5146, 39166}, {6335, 21758}, {6336, 17455}, {8120, 10231}, {8648, 18026}, {8752, 51583}, {8756, 40215}, {11700, 36121}, {16944, 38462}, {20565, 34397}, {34586, 36123}
X(52413) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 20566}, {6, 52351}, {19, 18359}, {25, 80}, {28, 14616}, {31, 1807}, {34, 18815}, {36, 69}, {56, 52392}, {79, 328}, {108, 35174}, {112, 47318}, {186, 319}, {320, 305}, {607, 36910}, {608, 2006}, {653, 46405}, {654, 6332}, {758, 20336}, {860, 313}, {1395, 1411}, {1402, 52391}, {1443, 7182}, {1464, 307}, {1474, 24624}, {1783, 36804}, {1824, 15065}, {1835, 1441}, {1845, 3262}, {1870, 75}, {1973, 2161}, {1974, 6187}, {1983, 1332}, {2194, 1793}, {2203, 759}, {2204, 2341}, {2212, 52371}, {2245, 306}, {2299, 6740}, {2323, 345}, {2361, 78}, {3218, 304}, {3724, 72}, {3738, 35518}, {3936, 40071}, {3960, 15413}, {4053, 52369}, {4242, 668}, {4282, 1812}, {4511, 3718}, {4707, 3267}, {5081, 3596}, {6186, 265}, {7113, 63}, {8648, 521}, {8750, 51562}, {17455, 3977}, {17515, 314}, {17923, 76}, {18344, 52356}, {18593, 1231}, {20924, 40364}, {21758, 905}, {21828, 525}, {22128, 3926}, {22379, 4131}, {32674, 655}, {34397, 35}, {40075, 40050}, {42666, 4064}, {44113, 10}, {44428, 35519}, {47230, 7265}, {52059, 22128}
X(52413) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 221, 19350}, {6, 478, 2261}, {6, 608, 19}

X(52414) = X(4)X(3648)∩X(19)X(27)

X(52414) lies on these lines: {4, 3648}, {19, 27}, {28, 11684}, {29, 191}, {158, 46263}, {162, 240}, {186, 42701}, {342, 1708}, {445, 3219}, {758, 17515}, {822, 1577}, {823, 36102}, {920, 1895}, {1096, 16570}, {1725, 36034}, {1749, 1784}, {1821, 2157}, {1844, 3647}, {1955, 2632}, {1959, 24041}, {2074, 48698}, {2167, 17438}, {2173, 2349}, {2897, 6350}, {3218, 14920}, {3650, 15763}, {4467, 44427}, {5535, 7541}, {5692, 37304}, {6149, 35201}, {14590, 16585}, {15149, 39767}, {16546, 18694}, {17484, 37799}, {17768, 37371}, {17917, 23958}, {18253, 25987}, {19919, 44225}, {23052, 36277}, {36046, 36095}, {36085, 36105}, {46468, 52126}

X(52414) = polar conjugate of X(2166)
X(52414) = polar conjugate of the isogonal conjugate of X(6149)
X(52414) = X(2290)-cross conjugate of X(6149)
X(52414) = X(i)-isoconjugate of X(j) for these (i,j): {2, 52153}, {3, 1989}, {4, 50433}, {5, 11077}, {6, 265}, {13, 36297}, {14, 36296}, {30, 11079}, {32, 328}, {48, 2166}, {53, 50463}, {69, 11060}, {94, 184}, {110, 14582}, {112, 43083}, {216, 1141}, {217, 46138}, {248, 14356}, {323, 14595}, {394, 18384}, {476, 647}, {523, 32662}, {525, 14560}, {577, 6344}, {656, 32678}, {661, 36061}, {810, 32680}, {822, 36129}, {906, 43082}, {1304, 18558}, {1576, 14592}, {1807, 2160}, {1990, 50464}, {2006, 8606}, {2161, 7100}, {2165, 5961}, {2341, 52390}, {2351, 18883}, {2437, 14220}, {3003, 12028}, {3049, 35139}, {3284, 5627}, {3457, 40710}, {3458, 40709}, {4558, 15475}, {5158, 18316}, {6186, 52351}, {6187, 52381}, {8749, 51254}, {9409, 39290}, {10097, 14559}, {10217, 11086}, {10218, 11081}, {10412, 32661}, {11063, 15392}, {11064, 40355}, {11070, 50467}, {11071, 50461}, {11072, 52201}, {11073, 52202}, {11074, 20123}, {11075, 50462}, {11080, 50466}, {11082, 50469}, {11083, 52203}, {11084, 52194}, {11085, 50465}, {11087, 50468}, {11088, 52204}, {11089, 52193}, {14254, 18877}, {14380, 41392}, {14575, 20573}, {14583, 14919}, {14585, 18817}, {14908, 43084}, {14910, 39170}, {16186, 23588}, {18479, 43530}, {18557, 32715}, {20578, 38413}, {20579, 38414}, {20975, 39295}, {21354, 34304}, {23968, 35909}, {30529, 51477}, {32663, 34209}, {34079, 52388}, {36298, 39377}, {36299, 39378}, {39201, 46456}
X(52414) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 34544}, {9, 265}, {48, 11597}, {63, 40604}, {244, 14582}, {328, 6376}, {476, 39052}, {656, 18334}, {661, 16221}, {1249, 2166}, {1953, 18402}, {1989, 36103}, {2314, 12095}, {4858, 14592}, {5190, 43082}, {5664, 20902}, {7100, 40584}, {14206, 14918}, {14213, 14920}, {14356, 39039}, {32664, 52153}, {32678, 40596}, {32680, 39062}, {34591, 43083}, {35069, 52388}, {36033, 50433}, {36061, 36830}, {40612, 52381}
X(52414) = cevapoint of X(i) and X(j) for these (i,j): {1725, 2173}, {2290, 51801}
X(52414) = crosspoint of X(2167) and X(2349)
X(52414) = crosssum of X(i) and X(j) for these (i,j): {1953, 2173}, {2631, 3708}
X(52414) = trilinear pole of line {32679, 35201}
X(52414) = barycentric product X(i)*X(j) for these {i,j}: {1, 340}, {19, 7799}, {27, 42701}, {50, 1969}, {63, 14165}, {75, 186}, {92, 323}, {95, 51801}, {162, 3268}, {264, 6149}, {276, 2290}, {319, 1870}, {320, 6198}, {526, 811}, {561, 34397}, {648, 32679}, {662, 44427}, {799, 47230}, {823, 8552}, {1154, 40440}, {1273, 2190}, {1442, 5081}, {1494, 35201}, {1577, 14590}, {1748, 37802}, {2088, 46254}, {2167, 14918}, {2349, 14920}, {2624, 6331}, {3219, 17923}, {4242, 4467}, {4511, 7282}, {5962, 44179}, {6148, 36119}, {10411, 24006}, {11107, 41804}, {14355, 40703}, {14591, 20948}, {14975, 40075}, {16186, 23999}, {17515, 40999}, {24019, 45792}, {24041, 35235}, {33805, 39176}, {36120, 51383}
X(52414) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 265}, {4, 2166}, {19, 1989}, {31, 52153}, {35, 1807}, {36, 7100}, {47, 5961}, {48, 50433}, {50, 48}, {75, 328}, {92, 94}, {107, 36129}, {110, 36061}, {112, 32678}, {158, 6344}, {162, 476}, {163, 32662}, {186, 1}, {240, 14356}, {323, 63}, {340, 75}, {526, 656}, {562, 2962}, {648, 32680}, {656, 43083}, {661, 14582}, {758, 52388}, {811, 35139}, {823, 46456}, {860, 6757}, {1094, 50466}, {1095, 50465}, {1096, 18384}, {1154, 44706}, {1273, 18695}, {1442, 52392}, {1464, 52390}, {1577, 14592}, {1725, 39170}, {1748, 18883}, {1784, 14254}, {1825, 52383}, {1835, 52382}, {1844, 45926}, {1870, 79}, {1969, 20573}, {1973, 11060}, {1986, 1725}, {2088, 3708}, {2148, 11077}, {2151, 36297}, {2152, 36296}, {2159, 11079}, {2169, 50463}, {2190, 1141}, {2290, 216}, {2361, 8606}, {2594, 52391}, {2624, 647}, {2631, 18558}, {2914, 1749}, {3043, 6149}, {3218, 52381}, {3219, 52351}, {3268, 14208}, {3581, 18477}, {4242, 6742}, {5081, 52344}, {5353, 52201}, {5357, 52202}, {5962, 91}, {6126, 50462}, {6149, 3}, {6198, 80}, {7282, 18815}, {7649, 43082}, {7799, 304}, {8552, 24018}, {8739, 2154}, {8740, 2153}, {10411, 4592}, {11062, 1953}, {11107, 6740}, {14165, 92}, {14270, 810}, {14355, 293}, {14385, 35200}, {14590, 662}, {14591, 163}, {14918, 14213}, {14920, 14206}, {14975, 6187}, {16186, 2632}, {17515, 3615}, {17923, 30690}, {18477, 18478}, {19627, 9247}, {22115, 255}, {24006, 10412}, {32676, 14560}, {32679, 525}, {34210, 36062}, {34397, 31}, {35193, 1793}, {35198, 50469}, {35199, 50468}, {35200, 50464}, {35201, 30}, {35235, 1109}, {36053, 12028}, {36063, 34209}, {36119, 5627}, {36130, 43707}, {38936, 36053}, {39176, 2173}, {40440, 46138}, {41502, 2341}, {42701, 306}, {44067, 1822}, {44068, 1823}, {44427, 1577}, {47230, 661}, {51801, 5}, {51802, 50461}, {51804, 15392}, {51805, 10217}, {51806, 10218}
X(52414) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {63, 1748, 92}, {240, 896, 162}, {1844, 3647, 11107}, {2580, 2581, 92}

X(52415) = X(4)X(94)∩X(24)X(136)

X(52415) lies on these lines: {4, 94}, {24, 136}, {52, 14111}, {53, 112}, {107, 34310}, {133, 14583}, {186, 52056}, {378, 14356}, {403, 476}, {1552, 5627}, {1598, 31676}, {2904, 34756}, {5890, 18380}, {6530, 14560}, {7487, 30529}, {8754, 12140}, {8796, 18316}, {10151, 34209}, {10594, 52153}, {10688, 18403}, {11744, 43707}, {12918, 47076}, {13556, 15329}, {14980, 37943}, {16229, 51479}, {35488, 39170}, {35490, 51254}, {44145, 46155}

X(52415) = polar conjugate of X(37802)
X(52415) = polar conjugate of the isotomic conjugate of X(18883)
X(52415) = X(52000)-cross conjugate of X(24)
X(52415) = X(i)-isoconjugate of X(j) for these (i,j): {48, 37802}, {68, 6149}, {91, 22115}, {255, 5962}, {323, 1820}, {8552, 36145}
X(52415) = X(i)-Dao conjugate of X(j) for these (i,j): {68, 14993}, {135, 526}, {139, 41078}, {1249, 37802}, {2351, 15295}, {5962, 6523}, {8552, 39013}, {22115, 34116}
X(52415) = trilinear pole of line {6753, 14576}
X(52415) = barycentric product X(i)*X(j) for these {i,j}: {4, 18883}, {24, 94}, {136, 39295}, {265, 11547}, {317, 1989}, {328, 8745}, {467, 1141}, {571, 18817}, {648, 43088}, {924, 46456}, {1748, 2166}, {1993, 6344}, {2052, 5961}, {6753, 35139}, {7763, 18384}, {10412, 41679}, {14111, 30529}, {14576, 46138}, {20573, 44077}, {40427, 52000}
X(52415) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 37802}, {24, 323}, {94, 20563}, {265, 52350}, {317, 7799}, {393, 5962}, {467, 1273}, {571, 22115}, {924, 8552}, {1989, 68}, {5961, 394}, {6344, 5392}, {6563, 45792}, {6753, 526}, {8745, 186}, {11060, 2351}, {11547, 340}, {14576, 1154}, {18384, 2165}, {18883, 69}, {41679, 10411}, {43088, 525}, {44077, 50}, {46456, 46134}, {47421, 16186}, {50433, 16391}, {52000, 34834}
X(52415) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 2970, 44795}, {4, 6344, 265}, {265, 18384, 6344}

X(52416) = X(4)X(49)∩X(24)X(52)

X(52416) lies on these lines: {4, 49}, {24, 52}, {25, 9703}, {30, 15463}, {54, 1594}, {110, 403}, {184, 378}, {186, 323}, {215, 1870}, {232, 9696}, {343, 10018}, {468, 40111}, {511, 37932}, {524, 19128}, {567, 7577}, {569, 52296}, {578, 7547}, {924, 15423}, {1092, 32534}, {1493, 6746}, {1495, 48914}, {1514, 14157}, {1593, 9704}, {1614, 2883}, {2070, 19504}, {2071, 44573}, {2072, 12228}, {2477, 6198}, {3047, 12292}, {3147, 45794}, {3200, 8739}, {3201, 8740}, {3520, 13491}, {3575, 20424}, {5012, 37118}, {5962, 14165}, {6143, 13353}, {6240, 34148}, {6640, 34114}, {6759, 35490}, {9603, 39575}, {9706, 15559}, {10274, 13367}, {10295, 43574}, {10539, 35488}, {11935, 18494}, {13352, 35480}, {13619, 37477}, {13754, 37970}, {14984, 37951}, {15362, 37943}, {16868, 18350}, {18445, 44269}, {22750, 44959}, {23236, 45181}, {32046, 37119}, {32609, 37917}, {34797, 37495}, {35487, 43598}, {37954, 50461}, {38534, 44452}, {38851, 51742}, {44076, 45177}

X(52416) = X(14165)-Ceva conjugate of X(50)
X(52416) = X(i)-isoconjugate of X(j) for these (i,j): {68, 2166}, {91, 265}, {94, 1820}, {14592, 36145}, {20571, 52153}
X(52416) = X(i)-Dao conjugate of X(j) for these (i,j): {68, 11597}, {135, 10412}, {265, 34116}, {14592, 39013}, {20563, 40604}
X(52416) = barycentric product X(i)*X(j) for these {i,j}: {24, 323}, {50, 317}, {186, 1993}, {340, 571}, {526, 41679}, {648, 44808}, {924, 14590}, {1147, 14165}, {1748, 6149}, {3043, 18883}, {6563, 14591}, {6753, 10411}, {7763, 34397}, {7799, 44077}, {11547, 22115}
X(52416) = barycentric quotient X(i)/X(j) for these {i,j}: {24, 94}, {50, 68}, {186, 5392}, {317, 20573}, {323, 20563}, {571, 265}, {924, 14592}, {1993, 328}, {3043, 37802}, {6753, 10412}, {8745, 6344}, {11547, 18817}, {14590, 46134}, {14591, 925}, {19627, 2351}, {22115, 52350}, {30451, 43083}, {34397, 2165}, {34952, 14582}, {36423, 5962}, {41679, 35139}, {44077, 1989}, {44808, 525}
X(52416) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {186, 3043, 22115}, {22115, 34397, 186}, {44067, 44068, 1986}, {51393, 52000, 24}

X(52417) = X(4)X(54)∩X(24)X(195)

X(52417) lies on the cubic K934 and these lines: {4, 54}, {24, 195}, {26, 12226}, {49, 143}, {110, 539}, {156, 44958}, {186, 323}, {235, 36966}, {403, 32423}, {468, 50708}, {562, 14918}, {933, 15907}, {1112, 47117}, {1147, 15801}, {1209, 14940}, {1594, 8254}, {1658, 22815}, {2888, 7505}, {2917, 35603}, {3047, 10113}, {3147, 12325}, {3153, 12228}, {3200, 10633}, {3201, 10632}, {3515, 12316}, {3517, 12175}, {3520, 10610}, {3567, 12234}, {3575, 22051}, {5965, 19128}, {6143, 6689}, {6150, 24772}, {6198, 47378}, {6240, 20424}, {6288, 16868}, {6403, 19150}, {6515, 11271}, {7488, 12606}, {7512, 12363}, {7691, 21844}, {7722, 37970}, {9704, 10594}, {9707, 19468}, {10018, 21230}, {10203, 32348}, {10282, 21660}, {10605, 17824}, {10628, 21663}, {10880, 12971}, {10881, 12965}, {11576, 34484}, {11649, 27085}, {11808, 44106}, {12291, 26882}, {12307, 32534}, {13445, 35475}, {13619, 15463}, {14156, 27866}, {15073, 32367}, {15137, 21284}, {15800, 34797}, {18374, 44668}, {19457, 32349}, {19504, 37932}, {20585, 44803}, {21357, 52297}, {32046, 52295}, {32226, 52000}, {32377, 43808}, {33565, 38534}, {35488, 48675}, {35489, 43574}, {44077, 47485}

X(52417) = midpoint of X(i) and X(j) for these {i,j}: {186, 2914}, {10540, 15089}, {10619, 51403}
X(52417) = X(2052)-Ceva conjugate of X(36423)
X(52417) = X(i)-isoconjugate of X(j) for these (i,j): {265, 2962}, {2166, 3519}, {14592, 36148}
X(52417) = X(i)-Dao conjugate of X(j) for these (i,j): {3519, 11597}, {6368, 35591}, {14592, 39018}, {18402, 25043}
X(52417) = crossdifference of every pair of points on line {14582, 17434}
X(52417) = barycentric product X(i)*X(j) for these {i,j}: {49, 14165}, {50, 32002}, {186, 1994}, {323, 3518}, {340, 2965}, {648, 44809}, {1510, 14590}, {3043, 30529}, {7769, 34397}, {10632, 11127}, {10633, 11126}, {14591, 41298}, {14918, 25044}
X(52417) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 3519}, {186, 11140}, {1510, 14592}, {1994, 328}, {2965, 265}, {3518, 94}, {11062, 25043}, {11135, 52204}, {11136, 52203}, {14165, 20572}, {14590, 46139}, {14591, 930}, {19627, 51477}, {32002, 20573}, {34397, 2963}, {36423, 562}, {44809, 525}
X(52417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {24, 195, 6242}, {54, 1614, 10619}, {54, 32379, 12254}, {1986, 11702, 2914}, {2914, 11597, 3043}, {3574, 10619, 13403}, {10610, 12300, 3520}, {44067, 44068, 2914}

X(52418) = X(4)X(6)∩X(23)X(232)

X(52418) lies on these lines: {4, 6}, {23, 232}, {25, 9407}, {26, 15905}, {50, 186}, {107, 32730}, {112, 3003}, {216, 14118}, {297, 22151}, {323, 340}, {403, 16310}, {419, 46151}, {420, 18371}, {421, 2501}, {566, 3520}, {571, 41758}, {577, 7488}, {648, 37778}, {1177, 43717}, {1180, 33871}, {1825, 21741}, {1968, 5158}, {1994, 11547}, {2052, 7578}, {2493, 37777}, {3260, 41253}, {3289, 41203}, {5063, 39575}, {7505, 46262}, {7517, 38292}, {8739, 34394}, {8740, 34395}, {8749, 14581}, {9308, 44135}, {9465, 10311}, {11063, 16328}, {14570, 44328}, {14591, 38936}, {14865, 41335}, {14910, 37951}, {16318, 34978}, {18300, 50433}, {18563, 42459}, {32713, 34854}, {32715, 40388}, {35296, 41679}, {44228, 45016}, {44375, 44893}, {46432, 51936}, {47162, 47336}

X(52418) = polar conjugate of X(328)
X(52418) = isogonal conjugate of the isotomic conjugate of X(14165)
X(52418) = polar conjugate of the isotomic conjugate of X(186)
X(52418) = polar conjugate of the isogonal conjugate of X(34397)
X(52418) = X(14165)-Ceva conjugate of X(186)
X(52418) = X(i)-cross conjugate of X(j) for these (i,j): {2088, 47230}, {34397, 186}
X(52418) = X(i)-isoconjugate of X(j) for these (i,j): {48, 328}, {63, 265}, {75, 50433}, {94, 255}, {304, 52153}, {326, 1989}, {394, 2166}, {476, 24018}, {520, 32680}, {525, 36061}, {662, 43083}, {822, 35139}, {1102, 18384}, {1807, 52381}, {2349, 51254}, {2632, 39295}, {3265, 32678}, {4100, 18817}, {4575, 14592}, {4592, 14582}, {6344, 6507}, {7100, 52351}, {11077, 18695}, {14206, 50464}, {14208, 32662}, {14213, 50463}, {18557, 36034}
X(52418) = X(i)-Dao conjugate of X(j) for these (i,j): {94, 6523}, {136, 14592}, {206, 50433}, {265, 3162}, {326, 34544}, {328, 1249}, {343, 18402}, {394, 11597}, {525, 16221}, {1084, 43083}, {1989, 15259}, {3258, 18557}, {3265, 18334}, {3926, 40604}, {5139, 14582}, {5664, 36793}, {14920, 28706}
X(52418) = cevapoint of X(i) and X(j) for these (i,j): {2088, 47230}, {14581, 44084}
X(52418) = crosspoint of X(i) and X(j) for these (i,j): {8749, 8882}, {23964, 32695}
X(52418) = crosssum of X(i) and X(j) for these (i,j): {343, 11064}, {1650, 18558}, {15526, 41077}
X(52418) = trilinear pole of line {14270, 47230}
X(52418) = crossdifference of every pair of points on line {520, 5489}
X(52418) = barycentric product X(i)*X(j) for these {i,j}: {4, 186}, {6, 14165}, {24, 5962}, {25, 340}, {50, 2052}, {94, 36423}, {107, 526}, {112, 44427}, {158, 6149}, {250, 35235}, {264, 34397}, {275, 11062}, {323, 393}, {403, 38936}, {470, 8740}, {471, 8739}, {562, 3518}, {648, 47230}, {823, 2624}, {1093, 22115}, {1154, 8884}, {1300, 1986}, {1825, 17515}, {1835, 11107}, {1870, 6198}, {2081, 16813}, {2088, 23582}, {2190, 51801}, {2207, 7799}, {2501, 14590}, {3043, 6344}, {3268, 32713}, {3581, 16263}, {5317, 42701}, {5664, 32695}, {6528, 14270}, {6529, 8552}, {6530, 14355}, {8745, 37802}, {8749, 14920}, {8882, 14918}, {14222, 15329}, {14591, 14618}, {16080, 39176}, {16186, 32230}, {18027, 19627}, {24019, 32679}, {35201, 36119}
X(52418) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 328}, {25, 265}, {32, 50433}, {50, 394}, {107, 35139}, {186, 69}, {323, 3926}, {340, 305}, {393, 94}, {512, 43083}, {526, 3265}, {1093, 18817}, {1096, 2166}, {1154, 52347}, {1495, 51254}, {1637, 18557}, {1974, 52153}, {2052, 20573}, {2088, 15526}, {2207, 1989}, {2489, 14582}, {2501, 14592}, {2624, 24018}, {5962, 20563}, {6149, 326}, {6524, 6344}, {6529, 46456}, {8552, 4143}, {8739, 40710}, {8740, 40709}, {8745, 18883}, {8884, 46138}, {11062, 343}, {14165, 76}, {14270, 520}, {14355, 6394}, {14398, 18558}, {14590, 4563}, {14591, 4558}, {14918, 28706}, {14975, 1807}, {19627, 577}, {22115, 3964}, {23964, 39295}, {24019, 32680}, {32676, 36061}, {32695, 39290}, {32713, 476}, {34397, 3}, {34416, 18479}, {34417, 18478}, {34854, 14356}, {35235, 339}, {36417, 11060}, {36423, 323}, {39176, 11064}, {40352, 50464}, {40354, 11079}, {44077, 5961}, {44084, 39170}, {44113, 52388}, {44427, 3267}, {47230, 525}, {51801, 18695}
X(52418) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 1990, 15262}, {6, 8743, 40138}, {6, 8745, 393}, {6, 8746, 1249}, {50, 11062, 186}, {3087, 40138, 5286}, {8744, 15262, 1990}, {11062, 39176, 50}, {34854, 44102, 32713}

X(52419) = X(2)X(7)∩X(40)X(176)

X(52419) lies on these lines: {1, 34216}, {2, 7}, {3, 39794}, {40, 176}, {46, 482}, {65, 26504}, {77, 2066}, {109, 9098}, {175, 3333}, {347, 16663}, {481, 3338}, {484, 1371}, {651, 51842}, {1124, 10252}, {1267, 7183}, {1373, 3336}, {1374, 3337}, {1434, 13460}, {1697, 17805}, {1804, 3083}, {3160, 51955}, {5119, 31538}, {5128, 31601}, {5708, 39795}, {6212, 7177}, {7029, 7176}, {7190, 13388}, {11010, 17806}, {31539, 51816}

X(52419) = isogonal conjugate of X(13456)
X(52419) = X(13453)-Ceva conjugate of X(3083)
X(52419) = X(i)-cross conjugate of X(j) for these (i,j): {1124, 3083}, {10252, 13389}
X(52419) = X(i)-isoconjugate of X(j) for these (i,j): {1, 13456}, {6, 13454}, {33, 6213}, {55, 1123}, {200, 13438}, {220, 13437}, {281, 34121}, {607, 13387}, {650, 6135}, {1335, 1857}, {2207, 13458}, {2212, 46745}, {5391, 6059}
X(52419) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 13456}, {9, 13454}, {223, 1123}, {6609, 13438}, {7090, 13388}
X(52419) = cevapoint of X(57) and X(34494)
X(52419) = barycentric product X(i)*X(j) for these {i,j}: {1, 13453}, {7, 3083}, {57, 1267}, {77, 13386}, {85, 1124}, {222, 46744}, {269, 13425}, {326, 13459}, {348, 6212}, {605, 6063}, {664, 6364}, {1336, 7183}, {3926, 13460}, {4564, 22107}, {7182, 34125}, {34401, 38003}
X(52419) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 13454}, {6, 13456}, {57, 1123}, {77, 13387}, {109, 6135}, {222, 6213}, {269, 13437}, {326, 13458}, {348, 46745}, {603, 34121}, {605, 55}, {1124, 9}, {1267, 312}, {1407, 13438}, {1804, 3084}, {3083, 8}, {6212, 281}, {6364, 522}, {6502, 7133}, {7125, 1335}, {7183, 5391}, {7335, 606}, {13386, 318}, {13389, 7090}, {13425, 341}, {13453, 75}, {13459, 158}, {13460, 393}, {14440, 14430}, {22107, 4858}, {34125, 33}, {46744, 7017}
X(52419) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 34494, 176}, {57, 6204, 1445}

X(52420) = X(2)X(7)∩X(40)X(175)

X(52420) lies on these lines: {1, 34215}, {2, 7}, {3, 39795}, {40, 175}, {46, 481}, {65, 26495}, {77, 5414}, {109, 9099}, {176, 3333}, {347, 16662}, {482, 3338}, {484, 1372}, {651, 51841}, {1335, 10253}, {1373, 3337}, {1374, 3336}, {1434, 13438}, {1697, 17802}, {1804, 3084}, {3160, 51957}, {5119, 31539}, {5128, 31602}, {5391, 7183}, {5708, 39794}, {6213, 7177}, {7030, 7176}, {7190, 13389}, {11010, 17803}, {21169, 34494}, {31538, 51816}

X(52420) = isogonal conjugate of X(13427)
X(52420) = X(13436)-Ceva conjugate of X(3084)
X(52420) = X(i)-cross conjugate of X(j) for these (i,j): {1335, 3084}, {10253, 13388}
X(52420) = X(i)-isoconjugate of X(j) for these (i,j): {1, 13427}, {6, 13426}, {33, 6212}, {55, 1336}, {200, 13460}, {220, 13459}, {281, 34125}, {607, 13386}, {650, 6136}, {1124, 1857}, {1267, 6059}, {2207, 13425}, {2212, 46744}
X(52420) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 13427}, {9, 13426}, {223, 1336}, {6609, 13460}, {13389, 14121}
X(52420) = cevapoint of X(57) and X(34495)
X(52420) = barycentric product X(i)*X(j) for these {i,j}: {1, 13436}, {7, 3084}, {57, 5391}, {77, 13387}, {85, 1335}, {222, 46745}, {269, 13458}, {326, 13437}, {348, 6213}, {606, 6063}, {664, 6365}, {1123, 7183}, {3926, 13438}, {4564, 22106}, {7182, 34121}
X(52420) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 13426}, {6, 13427}, {57, 1336}, {77, 13386}, {109, 6136}, {222, 6212}, {269, 13459}, {326, 13425}, {348, 46744}, {603, 34125}, {606, 55}, {1335, 9}, {1407, 13460}, {1804, 3083}, {2067, 42013}, {3084, 8}, {5391, 312}, {6213, 281}, {6365, 522}, {7125, 1124}, {7183, 1267}, {7335, 605}, {13387, 318}, {13388, 14121}, {13436, 75}, {13437, 158}, {13438, 393}, {13458, 341}, {14445, 14430}, {22106, 4858}, {34121, 33}, {46745, 7017}
X(52420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 34495, 175}, {57, 6203, 1445}

X(52421) = X(8)X(7055)∩X(63)X(7112)

X(52421) lies on these lines: {8, 7055}, {63, 7112}, {75, 1088}, {76, 18737}, {85, 553}, {226, 27478}, {274, 1231}, {310, 4572}, {321, 4554}, {348, 33935}, {552, 7176}, {883, 4651}, {1494, 4569}, {1943, 4573}, {2064, 46238}, {2985, 20911}, {4359, 40704}, {4980, 37780}, {4998, 7081}, {6358, 7196}, {7056, 42696}, {7799, 16577}, {17163, 35312}, {18033, 33940}, {19804, 21609}, {31627, 42029}, {33298, 33941}

X(52421) = isotomic conjugate of X(7073)
X(52421) = isotomic conjugate of the isogonal conjugate of X(1442)
X(52421) = X(i)-cross conjugate of X(j) for these (i,j): {319, 33939}, {40999, 17095}
X(52421) = X(i)-isoconjugate of X(j) for these (i,j): {25, 8606}, {31, 7073}, {32, 7110}, {41, 2160}, {55, 6186}, {79, 2175}, {560, 52344}, {1253, 52372}, {1918, 3615}, {2212, 7100}, {2323, 11060}, {8641, 26700}, {9447, 30690}, {9448, 20565}, {14827, 52374}
X(52421) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 7073}, {79, 40593}, {223, 6186}, {663, 8287}, {2160, 3160}, {2361, 40604}, {3615, 34021}, {3700, 36197}, {4516, 14838}, {5249, 14547}, {6374, 52344}, {6376, 7110}, {6505, 8606}, {16577, 21807}, {17113, 52372}
X(52421) = cevapoint of X(i) and X(j) for these (i,j): {8, 28755}, {319, 17095}
X(52421) = barycentric product X(i)*X(j) for these {i,j}: {7, 33939}, {35, 20567}, {75, 17095}, {76, 1442}, {85, 319}, {274, 40999}, {304, 7282}, {310, 16577}, {561, 2003}, {664, 18160}, {1088, 42033}, {1399, 1502}, {1441, 34016}, {2174, 41283}, {2594, 6385}, {3219, 6063}, {4467, 4554}, {4572, 14838}, {4620, 17886}, {4625, 7265}, {7799, 18815}, {35057, 46406}
X(52421) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 7073}, {7, 2160}, {35, 41}, {57, 6186}, {63, 8606}, {75, 7110}, {76, 52344}, {85, 79}, {274, 3615}, {279, 52372}, {319, 9}, {323, 2361}, {348, 7100}, {349, 6757}, {445, 1859}, {658, 26700}, {1088, 52374}, {1231, 52388}, {1399, 32}, {1411, 11060}, {1434, 52375}, {1441, 8818}, {1442, 6}, {1446, 52382}, {1825, 2333}, {2003, 31}, {2174, 2175}, {2594, 213}, {2605, 3063}, {3219, 55}, {3578, 3683}, {3678, 1334}, {3969, 210}, {4420, 220}, {4467, 650}, {4554, 6742}, {4569, 38340}, {4572, 15455}, {4573, 13486}, {6063, 30690}, {6198, 607}, {6741, 36197}, {7182, 52381}, {7186, 20665}, {7202, 3271}, {7265, 4041}, {7279, 2174}, {7282, 19}, {7799, 4511}, {8287, 4516}, {9404, 8641}, {11107, 2332}, {14838, 663}, {16577, 42}, {16585, 14547}, {16755, 3737}, {17095, 1}, {17206, 1789}, {17886, 21044}, {18160, 522}, {18815, 1989}, {20567, 20565}, {21741, 1918}, {21794, 872}, {22342, 2200}, {33939, 8}, {34016, 21}, {35057, 657}, {40214, 2194}, {40999, 37}, {41226, 52371}, {42033, 200}
X(52421) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {75, 7182, 6063}, {75, 20935, 17860}

X(52422) = X(2)X(85)∩X(7)X(12)

X(52422) lies on these lines: {1, 25719}, {2, 85}, {5, 17170}, {7, 12}, {8, 5543}, {10, 6604}, {57, 24603}, {65, 3789}, {69, 6734}, {75, 7080}, {76, 345}, {77, 19372}, {150, 5818}, {226, 17308}, {281, 331}, {304, 28808}, {312, 32834}, {344, 349}, {347, 17322}, {388, 1447}, {474, 6516}, {498, 1111}, {551, 25716}, {631, 5088}, {664, 3616}, {938, 14828}, {966, 41246}, {1125, 9312}, {1210, 14548}, {1231, 21609}, {1323, 19862}, {1434, 5435}, {1441, 17321}, {1565, 1656}, {1698, 9436}, {2345, 10030}, {2476, 41826}, {2898, 8167}, {3085, 3673}, {3090, 17181}, {3091, 4872}, {3160, 5550}, {3212, 3485}, {3306, 7183}, {3476, 36487}, {3598, 5261}, {3619, 21617}, {3634, 10481}, {3672, 17895}, {3674, 5219}, {3926, 20925}, {3947, 10521}, {4059, 24914}, {4328, 4967}, {4400, 5839}, {4403, 31455}, {4461, 40023}, {4470, 40862}, {4643, 14564}, {4644, 9596}, {4748, 17950}, {4751, 8732}, {4798, 6610}, {4911, 10590}, {5226, 33949}, {5260, 38859}, {5283, 43063}, {5433, 7223}, {5552, 20880}, {5657, 17753}, {5714, 33865}, {6063, 18140}, {6666, 42309}, {7110, 30705}, {7176, 7288}, {7179, 7195}, {7190, 42696}, {7198, 11237}, {7209, 27432}, {7264, 10056}, {7270, 15589}, {7278, 10072}, {8232, 17289}, {8582, 10436}, {9776, 50559}, {10527, 30806}, {11375, 43037}, {12848, 17256}, {17084, 17090}, {17227, 30275}, {17327, 52023}, {17675, 20328}, {19872, 21314}, {19877, 32086}, {19878, 43186}, {20911, 28795}, {20924, 32832}, {21044, 26101}, {24477, 36854}, {24701, 36673}, {24774, 41785}, {24805, 30617}, {25055, 25723}, {25583, 26364}, {25718, 38314}, {26150, 41352}, {27816, 27818}, {29603, 43035}, {31598, 41003}, {32007, 46933}, {32830, 32851}, {32874, 42033}, {33116, 33780}, {37990, 39732}, {42032, 46951}, {46932, 51351}

X(52422) = X(3305)-cross conjugate of X(42696)
X(52422) = X(i)-isoconjugate of X(j) for these (i,j): {41, 3296}, {2212, 30679}
X(52422) = X(i)-Dao conjugate of X(j) for these (i,j): {3160, 3296}, {3715, 16777}
X(52422) = cevapoint of X(3305) and X(7190)
X(52422) = barycentric product X(i)*X(j) for these {i,j}: {7, 42696}, {75, 7190}, {85, 3305}, {279, 42032}, {664, 48268}, {3295, 6063}, {4554, 47965}, {4572, 48340}
X(52422) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 3296}, {348, 30679}, {3295, 55}, {3305, 9}, {3697, 210}, {4917, 3158}, {7190, 1}, {42032, 346}, {42696, 8}, {46951, 28808}, {47965, 650}, {48268, 522}, {48340, 663}, {51572, 3715}
X(52422) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 85, 348}, {2, 279, 17095}, {2, 30694, 1212}, {7, 9780, 33298}, {10, 40719, 6604}, {85, 348, 17079}, {85, 17078, 43983}, {85, 17095, 279}, {279, 17095, 348}, {304, 32828, 28808}, {3598, 5261, 7247}, {3616, 31994, 664}, {5433, 7223, 17081}, {6706, 46835, 2}, {7195, 10588, 7179}

X(52423) = X(1)X(6883)∩X(6)X(57)

X(52423) lies on these lines: {1, 6883}, {2, 2323}, {6, 57}, {9, 10601}, {31, 3256}, {35, 582}, {36, 5396}, {40, 37514}, {42, 2078}, {46, 16472}, {51, 3220}, {56, 5313}, {63, 5422}, {65, 1203}, {81, 3911}, {84, 10982}, {109, 2308}, {181, 1428}, {182, 5285}, {200, 45728}, {219, 7308}, {226, 26723}, {239, 6358}, {241, 47057}, {373, 26885}, {386, 1451}, {394, 5437}, {553, 651}, {559, 5353}, {575, 3955}, {576, 3784}, {611, 3677}, {613, 5269}, {940, 31231}, {942, 23071}, {999, 39523}, {1062, 10399}, {1082, 5357}, {1170, 43035}, {1171, 4565}, {1193, 18772}, {1319, 16474}, {1386, 41539}, {1401, 19369}, {1404, 1412}, {1421, 5173}, {1429, 5280}, {1445, 45126}, {1450, 5193}, {1473, 9777}, {1708, 5256}, {1876, 44097}, {1943, 17121}, {1947, 36794}, {1993, 3306}, {1994, 22128}, {2006, 2982}, {2099, 5315}, {2256, 25430}, {3060, 7293}, {3218, 26740}, {3219, 15018}, {3299, 13388}, {3301, 13389}, {3338, 16473}, {3340, 16466}, {3589, 26942}, {3601, 36745}, {3937, 34565}, {4383, 5219}, {4552, 45222}, {4654, 5228}, {4663, 17625}, {5050, 37581}, {5122, 51340}, {5221, 34043}, {5226, 14997}, {5262, 15556}, {5299, 7146}, {5312, 37579}, {5435, 37685}, {5706, 9581}, {5709, 36752}, {5902, 37697}, {5943, 7193}, {7011, 15851}, {7074, 10389}, {7078, 11518}, {7171, 44413}, {7269, 27065}, {8270, 16475}, {11427, 20266}, {13329, 14547}, {13366, 26884}, {15004, 26892}, {15803, 36742}, {16577, 17011}, {17020, 43048}, {17077, 19717}, {18162, 21361}, {18206, 18602}, {24431, 41700}, {24914, 37559}, {26893, 43650}, {33178, 44547}, {36747, 37534}, {36749, 37612}, {36750, 37582}, {36753, 37532}, {37498, 37526}

X(52423) = X(7269)-Ceva conjugate of X(3746)
X(52423) = X(9)-isoconjugate of X(5557)
X(52423) = X(478)-Dao conjugate of X(5557)
X(52423) = crosssum of X(37) and X(21012)
X(52423) = crossdifference of every pair of points on line {3900, 13401}
X(52423) = barycentric product X(i)*X(j) for these {i,j}: {1, 7269}, {7, 3746}, {56, 5564}, {57, 27065}, {651, 48003}, {664, 48306}, {1014, 4015}
X(52423) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 5557}, {3746, 8}, {4015, 3701}, {5564, 3596}, {7269, 75}, {27065, 312}, {48003, 4391}, {48306, 522}
X(52423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 57, 2003}, {51, 26889, 3220}, {219, 17825, 7308}, {386, 1451, 37583}, {1994, 27003, 22128}, {4383, 37543, 5219}, {5228, 34048, 4654}, {17011, 37787, 16577}

X(52424) = X(2)X(219)∩X(6)X(57)

X(52424) lies on these lines: {1, 5920}, {2, 219}, {3, 1451}, {6, 57}, {7, 32911}, {9, 17825}, {25, 26889}, {31, 37541}, {34, 37544}, {42, 1471}, {48, 37269}, {51, 1473}, {55, 13329}, {56, 181}, {58, 1466}, {63, 10601}, {65, 16466}, {81, 5435}, {109, 41422}, {171, 613}, {182, 37581}, {218, 226}, {220, 7308}, {221, 1203}, {241, 45126}, {278, 607}, {394, 3306}, {458, 1947}, {553, 6180}, {611, 982}, {614, 5173}, {651, 21454}, {748, 42289}, {940, 3911}, {942, 7078}, {950, 37537}, {990, 1864}, {999, 1450}, {1040, 5728}, {1124, 13388}, {1191, 3340}, {1210, 5706}, {1214, 1445}, {1335, 13389}, {1351, 3784}, {1376, 45728}, {1386, 8270}, {1393, 19349}, {1402, 40732}, {1428, 1460}, {1433, 3075}, {1497, 10306}, {1708, 3666}, {1714, 15844}, {1788, 5711}, {1817, 46882}, {1876, 44086}, {1943, 3759}, {1993, 23140}, {2099, 16483}, {2256, 17022}, {2260, 7011}, {2262, 21370}, {2308, 9316}, {2323, 5437}, {2334, 51773}, {2911, 5219}, {2947, 3149}, {3052, 3256}, {3157, 5708}, {3218, 5422}, {3220, 17810}, {3295, 7086}, {3305, 7190}, {3336, 16472}, {3337, 16473}, {3361, 34046}, {3527, 26928}, {3556, 45979}, {3682, 16410}, {3690, 22112}, {3751, 17625}, {3781, 16419}, {3937, 15004}, {3955, 5050}, {4255, 37583}, {4298, 9370}, {4361, 6358}, {4848, 5710}, {5020, 7193}, {5085, 5285}, {5226, 37680}, {5315, 18421}, {5709, 37514}, {5729, 24430}, {5928, 12610}, {5943, 24320}, {6354, 17366}, {6357, 7365}, {6692, 25934}, {7085, 43650}, {7132, 43071}, {7146, 16502}, {7191, 7672}, {7225, 28387}, {7248, 19369}, {7293, 33586}, {7484, 26893}, {7677, 17018}, {8257, 25091}, {8757, 24470}, {9777, 26866}, {10478, 24618}, {11284, 26885}, {11343, 41243}, {11402, 26884}, {11424, 26927}, {11433, 26932}, {12594, 21342}, {13462, 16474}, {14555, 23151}, {14853, 26929}, {15556, 37549}, {15803, 36746}, {15804, 40958}, {15805, 26921}, {16409, 17976}, {16412, 22127}, {16432, 32590}, {16433, 32592}, {16435, 24310}, {16577, 20182}, {17012, 17080}, {17074, 37685}, {17077, 19684}, {17595, 26740}, {17599, 41712}, {18928, 27509}, {20266, 23292}, {20905, 28950}, {21361, 24328}, {22153, 37272}, {23958, 34545}, {26723, 37695}, {28606, 37787}, {28997, 41241}, {31231, 37674}, {34051, 44794}, {36059, 44104}, {36742, 37582}, {36747, 37612}, {36750, 37545}, {36752, 37532}, {37498, 37534}, {37642, 43043}

X(52424) = X(7190)-Ceva conjugate of X(3295)
X(52424) = X(i)-isoconjugate of X(j) for these (i,j): {9, 3296}, {33, 30679}, {3872, 52188}
X(52424) = X(478)-Dao conjugate of X(3296)
X(52424) = crosssum of X(1146) and X(4820)
X(52424) = crossdifference of every pair of points on line {3900, 4976}
X(52424) = barycentric product X(i)*X(j) for these {i,j}: {1, 7190}, {7, 3295}, {56, 42696}, {57, 3305}, {109, 48268}, {651, 47965}, {664, 48340}, {1014, 3697}, {1407, 42032}, {4917, 19604}
X(52424) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 3296}, {222, 30679}, {3295, 8}, {3305, 312}, {3697, 3701}, {4917, 44720}, {7190, 75}, {42696, 3596}, {47965, 4391}, {48268, 35519}, {48340, 522}
X(52424) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 57, 222}, {6, 1407, 2003}, {7, 32911, 34048}, {42, 1471, 1617}, {57, 2003, 1407}, {57, 2999, 1465}, {65, 16466, 34040}, {942, 36754, 7078}, {1203, 3339, 221}, {1407, 2003, 222}, {1445, 5256, 1214}, {2323, 5437, 17811}, {4383, 5228, 226}, {5708, 37509, 3157}, {9777, 26866, 26892}

X(52425) = X(1)X(1729)∩X(31)X(32)

X(52425) lies on these lines: {1, 1729}, {3, 906}, {6, 2197}, {8, 5546}, {31, 32}, {48, 577}, {55, 607}, {101, 4291}, {184, 2200}, {198, 608}, {212, 1802}, {219, 283}, {220, 2342}, {221, 1415}, {222, 1803}, {255, 20752}, {315, 27516}, {571, 2174}, {604, 5065}, {610, 1950}, {692, 34078}, {919, 2724}, {1042, 1055}, {1253, 7118}, {1334, 16283}, {1451, 2266}, {1783, 11491}, {1914, 2082}, {1968, 2202}, {2170, 2241}, {2175, 9448}, {2189, 2335}, {2242, 2650}, {2264, 5301}, {2286, 15905}, {2289, 22074}, {2293, 40957}, {2347, 16946}, {2876, 23852}, {3002, 36152}, {5019, 21748}, {5063, 7113}, {5452, 32561}, {6284, 8735}, {7335, 32657}, {8300, 10801}, {14585, 39687}, {17046, 25886}, {17905, 37000}, {20760, 20799}, {20818, 22118}, {23383, 44081}, {23843, 39690}

X(52425) = isogonal conjugate of X(331)
X(52425) = isogonal conjugate of the isotomic conjugate of X(219)
X(52425) = isotomic conjugate of the polar conjugate of X(2175)
X(52425) = isogonal conjugate of the polar conjugate of X(55)
X(52425) = polar conjugate of the isotomic conjugate of X(6056)
X(52425) = X(i)-Ceva conjugate of X(j) for these (i,j): {48, 184}, {55, 2175}, {219, 6056}, {906, 1946}, {2193, 212}, {7115, 692}, {14578, 23202}
X(52425) = X(i)-cross conjugate of X(j) for these (i,j): {2200, 41}, {22368, 3}
X(52425) = X(i)-isoconjugate of X(j) for these (i,j): {1, 331}, {2, 273}, {4, 85}, {7, 92}, {8, 1847}, {19, 6063}, {25, 20567}, {27, 1441}, {28, 349}, {29, 1446}, {34, 76}, {56, 1969}, {57, 264}, {65, 44129}, {75, 278}, {77, 2052}, {84, 40701}, {86, 40149}, {108, 3261}, {158, 348}, {189, 342}, {196, 309}, {208, 44190}, {225, 274}, {226, 286}, {253, 44697}, {269, 7017}, {276, 1393}, {279, 318}, {281, 1088}, {304, 1118}, {310, 1880}, {312, 1119}, {313, 1396}, {393, 7182}, {459, 33673}, {479, 7101}, {513, 46404}, {514, 18026}, {522, 13149}, {561, 608}, {603, 18027}, {604, 18022}, {648, 4077}, {651, 46107}, {653, 693}, {658, 44426}, {664, 17924}, {811, 7178}, {823, 17094}, {873, 8736}, {903, 37790}, {934, 46110}, {1093, 7183}, {1111, 46102}, {1121, 38461}, {1226, 40397}, {1231, 8747}, {1365, 46254}, {1395, 1502}, {1398, 28659}, {1414, 14618}, {1426, 28660}, {1427, 44130}, {1434, 41013}, {1435, 3596}, {1804, 6521}, {1848, 31643}, {1855, 42311}, {1861, 34018}, {1874, 40017}, {1876, 18031}, {1877, 20568}, {1897, 24002}, {1973, 41283}, {1978, 43923}, {2481, 5236}, {2501, 4625}, {2973, 4564}, {3064, 4569}, {3213, 41530}, {3668, 31623}, {3676, 6335}, {4017, 6331}, {4391, 36118}, {4554, 7649}, {4572, 6591}, {4573, 24006}, {6520, 7055}, {6528, 51640}, {7012, 23989}, {7020, 14256}, {7046, 23062}, {7128, 34387}, {7210, 43678}, {7282, 30690}, {7337, 40364}, {8795, 44708}, {8809, 15466}, {13437, 46744}, {13459, 46745}, {15413, 36127}, {16082, 22464}, {16747, 18097}, {17880, 23984}, {17923, 18815}, {18344, 46406}, {24032, 26932}, {26563, 40446}, {32674, 40495}, {32714, 35519}, {34400, 47372}, {36124, 40704}, {40702, 40836}, {46111, 51655}
X(52425) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 1969}, {3, 331}, {6, 6063}, {7, 22391}, {76, 11517}, {85, 36033}, {206, 278}, {264, 5452}, {273, 32664}, {348, 1147}, {349, 40591}, {608, 40368}, {3161, 18022}, {3261, 38983}, {6331, 34961}, {6337, 41283}, {6505, 20567}, {6600, 7017}, {7055, 37867}, {7178, 17423}, {7952, 18027}, {14618, 40608}, {14714, 46110}, {17924, 39025}, {24002, 34467}, {35072, 40495}, {38991, 46107}, {39026, 46404}, {40149, 40600}, {40602, 44129}, {44132, 50440}
X(52425) = cevapoint of X(3) and X(23078)
X(52425) = crosspoint of X(i) and X(j) for these (i,j): {48, 212}, {55, 219}, {692, 7115}
X(52425) = crosssum of X(i) and X(j) for these (i,j): {1, 1729}, {4, 17905}, {7, 278}, {92, 273}, {693, 26932}, {2973, 17924}, {4077, 4858}, {10015, 38372}, {23989, 24002}, {34387, 46110}
X(52425) = crossdifference of every pair of points on line {693, 17094}
X(52425) = barycentric product X(i)*X(j) for these {i,j}: {1, 212}, {3, 55}, {4, 6056}, {6, 219}, {8, 184}, {9, 48}, {19, 2289}, {21, 228}, {25, 1259}, {29, 4055}, {31, 78}, {32, 345}, {33, 255}, {35, 8606}, {37, 2193}, {40, 2188}, {41, 63}, {42, 283}, {54, 44707}, {56, 1260}, {57, 1802}, {58, 2318}, {59, 3270}, {60, 3690}, {69, 2175}, {71, 284}, {72, 2194}, {73, 2328}, {77, 1253}, {81, 52370}, {100, 1946}, {101, 652}, {163, 8611}, {197, 39167}, {198, 268}, {200, 603}, {210, 1437}, {213, 1812}, {220, 222}, {271, 2187}, {281, 577}, {294, 20752}, {304, 9447}, {305, 9448}, {312, 9247}, {326, 2212}, {332, 1918}, {333, 2200}, {348, 14827}, {394, 607}, {480, 7053}, {521, 692}, {522, 32656}, {560, 3718}, {604, 3692}, {643, 810}, {644, 22383}, {645, 3049}, {647, 5546}, {649, 4587}, {650, 906}, {657, 1813}, {663, 1331}, {667, 4571}, {728, 7099}, {911, 51376}, {943, 23207}, {947, 40945}, {983, 20753}, {1036, 7085}, {1040, 7084}, {1092, 1857}, {1110, 7004}, {1172, 3990}, {1176, 3688}, {1252, 7117}, {1261, 22344}, {1264, 1974}, {1265, 1397}, {1318, 22371}, {1320, 23202}, {1332, 3063}, {1333, 3694}, {1334, 1790}, {1400, 2327}, {1402, 1792}, {1409, 2287}, {1425, 6061}, {1433, 7074}, {1459, 3939}, {1565, 6066}, {1576, 52355}, {1783, 36054}, {1791, 20967}, {1793, 3724}, {1794, 14547}, {1798, 40966}, {1803, 8012}, {1804, 7071}, {1807, 2361}, {1808, 3747}, {1810, 8647}, {1818, 2195}, {1819, 2357}, {1973, 3719}, {2053, 20760}, {2066, 5414}, {2115, 20761}, {2149, 34591}, {2150, 3949}, {2169, 7069}, {2189, 52386}, {2192, 7078}, {2196, 3684}, {2197, 7054}, {2204, 3998}, {2206, 3710}, {2269, 2359}, {2293, 47487}, {2298, 22074}, {2299, 3682}, {2316, 22356}, {2325, 32659}, {2329, 7116}, {2330, 7015}, {2332, 40152}, {2333, 6514}, {2340, 36057}, {2342, 22350}, {2346, 22079}, {2638, 7012}, {3157, 7072}, {3208, 15373}, {3239, 32660}, {3284, 15627}, {3289, 15628}, {3292, 5547}, {3596, 14575}, {3689, 36058}, {3693, 32658}, {3700, 32661}, {3703, 10547}, {3709, 4558}, {3712, 14908}, {3900, 36059}, {3937, 6065}, {3964, 6059}, {4041, 4575}, {4076, 22096}, {4092, 47390}, {4183, 22341}, {4548, 14376}, {4557, 23189}, {4559, 23090}, {4574, 7252}, {5548, 22086}, {6332, 32739}, {6510, 18889}, {6516, 8641}, {6602, 7177}, {7011, 7367}, {7016, 26885}, {7017, 14585}, {7046, 7335}, {7062, 47388}, {7063, 47389}, {7065, 32230}, {7070, 19614}, {7077, 7193}, {7079, 7125}, {7115, 35072}, {7123, 7124}, {7359, 18877}, {8851, 20777}, {10397, 36049}, {10482, 22053}, {14379, 44695}, {14418, 32665}, {14642, 27382}, {14936, 44717}, {15374, 42460}, {15905, 30457}, {19354, 42019}, {20769, 51858}, {21789, 23067}, {22054, 33635}, {22072, 51476}, {22131, 40141}, {22368, 40419}, {23146, 40523}, {23201, 32635}, {23225, 36802}, {23990, 26932}, {32641, 52307}, {32657, 40869}, {34055, 40972}, {34858, 51379}, {36055, 51361}, {36056, 41339}, {36797, 39201}, {39687, 46102}, {40363, 40373}
X(52425) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 6063}, {6, 331}, {8, 18022}, {9, 1969}, {31, 273}, {32, 278}, {41, 92}, {48, 85}, {55, 264}, {63, 20567}, {69, 41283}, {71, 349}, {78, 561}, {101, 46404}, {184, 7}, {198, 40701}, {212, 75}, {213, 40149}, {219, 76}, {220, 7017}, {228, 1441}, {255, 7182}, {268, 44190}, {281, 18027}, {283, 310}, {284, 44129}, {305, 41287}, {345, 1502}, {521, 40495}, {560, 34}, {577, 348}, {603, 1088}, {604, 1847}, {607, 2052}, {652, 3261}, {657, 46110}, {663, 46107}, {692, 18026}, {810, 4077}, {906, 4554}, {1092, 7055}, {1253, 318}, {1259, 305}, {1260, 3596}, {1264, 40050}, {1265, 40363}, {1331, 4572}, {1397, 1119}, {1409, 1446}, {1415, 13149}, {1501, 608}, {1792, 40072}, {1802, 312}, {1812, 6385}, {1813, 46406}, {1917, 1395}, {1918, 225}, {1946, 693}, {1974, 1118}, {1980, 43923}, {2175, 4}, {2187, 342}, {2188, 309}, {2193, 274}, {2194, 286}, {2200, 226}, {2205, 1880}, {2212, 158}, {2251, 37790}, {2289, 304}, {2318, 313}, {2327, 28660}, {2328, 44130}, {2638, 17880}, {3022, 21666}, {3049, 7178}, {3063, 17924}, {3270, 34387}, {3271, 2973}, {3596, 44161}, {3688, 1235}, {3690, 34388}, {3692, 28659}, {3694, 27801}, {3709, 14618}, {3718, 1928}, {3719, 40364}, {3955, 7205}, {3990, 1231}, {4055, 307}, {4100, 7183}, {4173, 7217}, {4548, 17907}, {4571, 6386}, {4575, 4625}, {4587, 1978}, {5546, 6331}, {5547, 46111}, {6056, 69}, {6059, 1093}, {6066, 15742}, {6602, 7101}, {7063, 8754}, {7064, 7141}, {7099, 23062}, {7109, 8736}, {7117, 23989}, {7193, 18033}, {7335, 7056}, {8606, 20565}, {8611, 20948}, {8641, 44426}, {9247, 57}, {9447, 19}, {9448, 25}, {9454, 5236}, {9455, 1876}, {9459, 1877}, {10316, 17076}, {10317, 17088}, {14575, 56}, {14585, 222}, {14827, 281}, {15373, 7209}, {20752, 40704}, {20753, 33930}, {20775, 3665}, {22074, 20911}, {22079, 20880}, {22096, 1358}, {22117, 50560}, {22368, 2886}, {22383, 24002}, {23200, 7181}, {23211, 17447}, {23225, 43042}, {23606, 1804}, {23990, 46102}, {32656, 664}, {32657, 52156}, {32658, 34018}, {32660, 658}, {32661, 4573}, {32739, 653}, {34980, 1367}, {36054, 15413}, {36059, 4569}, {39201, 17094}, {39687, 26932}, {40050, 41289}, {40360, 41290}, {40373, 1397}, {40972, 20883}, {41280, 1398}, {44088, 30493}, {44162, 7337}, {44707, 311}, {47390, 7340}, {52355, 44173}, {52370, 321}
X(52425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 7124, 7117}, {3, 22131, 22070}, {32, 14827, 41}, {212, 1802, 52370}, {2200, 9247, 184}

X(52426) = X(1)X(4291)∩X(31)X(32)

X(52426) lies on these lines: {1, 4291}, {6, 5172}, {31, 32}, {36, 1983}, {48, 571}, {50, 7113}, {112, 2202}, {187, 7117}, {577, 604}, {672, 906}, {1055, 1415}, {1195, 41332}, {1333, 21748}, {1914, 1951}, {1946, 3063}, {2082, 7031}, {2174, 2965}, {2183, 32655}, {2193, 2269}, {2220, 2347}, {2245, 47417}, {2323, 4282}, {3053, 7124}, {3684, 5546}, {5277, 33105}, {5301, 40968}, {8606, 14547}, {9454, 23990}, {19554, 51659}, {20065, 27516}, {25886, 31240}

X(52426) = isogonal conjugate of the isotomic conjugate of X(2323)
X(52426) = X(i)-Ceva conjugate of X(j) for these (i,j): {1983, 8648}, {4282, 2361}, {32655, 31}, {32677, 184}
X(52426) = X(i)-isoconjugate of X(j) for these (i,j): {2, 18815}, {7, 18359}, {57, 20566}, {75, 2006}, {76, 1411}, {80, 85}, {92, 52392}, {94, 1442}, {226, 14616}, {273, 52351}, {274, 52383}, {320, 34535}, {331, 1807}, {349, 759}, {513, 46405}, {514, 35174}, {655, 693}, {658, 52356}, {903, 14628}, {1088, 36910}, {1399, 20573}, {1434, 15065}, {1441, 24624}, {1446, 6740}, {2161, 6063}, {2166, 17095}, {2222, 3261}, {3676, 36804}, {4077, 47318}, {6187, 20567}, {14584, 20568}, {17791, 26743}, {18816, 52212}, {23989, 52377}, {24002, 51562}, {32675, 40495}, {44129, 52391}
X(52426) = X(i)-Dao conjugate of X(j) for these (i,j): {76, 35204}, {206, 2006}, {349, 34586}, {3261, 38984}, {5452, 20566}, {6063, 40584}, {6149, 20924}, {11597, 17095}, {18815, 32664}, {20567, 40612}, {22391, 52392}, {35128, 40495}, {39026, 46405}
X(52426) = crosspoint of X(i) and X(j) for these (i,j): {284, 2342}, {913, 2299}, {2161, 7073}, {2361, 7113}
X(52426) = crosssum of X(i) and X(j) for these (i,j): {7, 37798}, {226, 22464}, {307, 914}, {1442, 3218}, {2006, 52392}, {4707, 21207}, {4858, 36038}, {18359, 18815}
X(52426) = crossdifference of every pair of points on line {693, 1441}
X(52426) = barycentric product X(i)*X(j) for these {i,j}: {1, 2361}, {6, 2323}, {9, 7113}, {21, 3724}, {31, 4511}, {32, 32851}, {36, 55}, {37, 4282}, {41, 3218}, {50, 7110}, {80, 215}, {100, 8648}, {101, 654}, {184, 5081}, {186, 8606}, {212, 1870}, {228, 17515}, {283, 44113}, {284, 2245}, {320, 2175}, {607, 22128}, {644, 21758}, {650, 1983}, {692, 3738}, {758, 2194}, {1253, 1443}, {1464, 2328}, {1946, 4242}, {2150, 4053}, {2161, 34544}, {2316, 17455}, {2342, 34586}, {3063, 4585}, {3689, 16944}, {3904, 32739}, {4089, 6066}, {4636, 42666}, {4996, 6187}, {5546, 21828}, {6149, 7073}, {9447, 20924}, {9448, 40075}, {14827, 17078}, {19302, 26744}, {27950, 51858}, {32656, 44428}, {36910, 52059}
X(52426) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 18815}, {32, 2006}, {36, 6063}, {41, 18359}, {50, 17095}, {55, 20566}, {101, 46405}, {184, 52392}, {215, 320}, {320, 41283}, {560, 1411}, {654, 3261}, {692, 35174}, {1918, 52383}, {1983, 4554}, {2175, 80}, {2194, 14616}, {2245, 349}, {2251, 14628}, {2323, 76}, {2361, 75}, {3218, 20567}, {3724, 1441}, {3738, 40495}, {4282, 274}, {4511, 561}, {4996, 40075}, {5081, 18022}, {7110, 20573}, {7113, 85}, {8606, 328}, {8641, 52356}, {8648, 693}, {9447, 2161}, {9448, 6187}, {9459, 14584}, {14827, 36910}, {19627, 2003}, {21758, 24002}, {32739, 655}, {32851, 1502}, {34397, 7282}, {34544, 20924}, {40075, 41287}, {52059, 17078}
X(52426) = {X(1914),X(1951)}-harmonic conjugate of X(2170)

X(52427) = X(1)X(24)∩X(3)X(34)

X(52427) lies on these lines: {1, 24}, {2, 11393}, {3, 34}, {4, 35}, {11, 468}, {12, 3575}, {19, 25}, {21, 34851}, {22, 1040}, {23, 3100}, {26, 1062}, {28, 4276}, {31, 1182}, {36, 186}, {41, 41320}, {42, 8882}, {51, 11429}, {56, 3515}, {65, 20837}, {92, 37295}, {100, 1861}, {102, 1457}, {108, 2078}, {154, 19354}, {162, 2073}, {184, 11436}, {185, 26888}, {199, 23207}, {204, 1609}, {208, 37579}, {212, 573}, {215, 34397}, {221, 1192}, {225, 7412}, {232, 1914}, {235, 6284}, {242, 243}, {278, 15931}, {284, 2189}, {297, 26629}, {378, 5010}, {390, 4232}, {403, 3583}, {406, 5248}, {427, 5432}, {428, 4995}, {429, 1852}, {444, 21321}, {451, 5259}, {475, 25440}, {495, 37458}, {497, 6353}, {499, 3147}, {607, 4258}, {608, 37499}, {614, 10832}, {650, 1946}, {759, 2074}, {855, 10017}, {860, 13999}, {902, 8750}, {915, 1785}, {919, 9085}, {950, 30733}, {1030, 1841}, {1038, 17928}, {1060, 6644}, {1068, 14798}, {1079, 7163}, {1110, 2356}, {1155, 1876}, {1204, 7355}, {1250, 10641}, {1319, 10016}, {1364, 26884}, {1398, 5204}, {1435, 37578}, {1474, 2269}, {1478, 18533}, {1479, 3542}, {1495, 3270}, {1503, 26956}, {1593, 5217}, {1621, 30687}, {1753, 11248}, {1825, 6197}, {1829, 2646}, {1830, 5537}, {1838, 7414}, {1843, 2330}, {1848, 4231}, {1858, 32126}, {1872, 11849}, {1875, 5172}, {1877, 2077}, {1885, 15338}, {1887, 14882}, {1890, 7466}, {1892, 17718}, {1902, 37568}, {1905, 24929}, {1974, 3056}, {1995, 9817}, {2066, 5413}, {2070, 18455}, {2164, 7008}, {2223, 34179}, {2245, 2361}, {2268, 44103}, {2276, 10311}, {2291, 40116}, {2331, 8573}, {2342, 14776}, {3035, 26020}, {3052, 3195}, {3057, 11363}, {3060, 9637}, {3083, 15204}, {3084, 15203}, {3085, 7487}, {3089, 4294}, {3144, 35206}, {3145, 40946}, {3157, 37489}, {3220, 7004}, {3295, 3517}, {3299, 10881}, {3301, 10880}, {3518, 3746}, {3584, 7576}, {3585, 6240}, {3601, 7713}, {3614, 23047}, {4183, 7110}, {4185, 19760}, {4225, 45231}, {4242, 17923}, {4265, 17603}, {4293, 37460}, {4296, 22467}, {4316, 10295}, {4324, 18560}, {5081, 17515}, {5186, 15452}, {5225, 6622}, {5280, 10312}, {5281, 6995}, {5299, 39575}, {5320, 11435}, {5322, 21213}, {5353, 10632}, {5357, 10633}, {5370, 21284}, {5410, 19037}, {5411, 19038}, {5412, 5414}, {5563, 44879}, {6056, 26893}, {6152, 47378}, {6187, 8756}, {6238, 10539}, {6242, 35197}, {6285, 26883}, {6641, 26904}, {6690, 25985}, {7031, 8743}, {7079, 32561}, {7127, 8740}, {7280, 32534}, {7292, 37977}, {7501, 34231}, {7503, 19372}, {7505, 7741}, {7506, 37696}, {8144, 37440}, {8540, 44102}, {8554, 14936}, {8758, 14667}, {9371, 20872}, {9638, 26882}, {9643, 9645}, {9786, 19349}, {10391, 41611}, {10483, 35471}, {10589, 38282}, {10638, 10642}, {10895, 12173}, {10986, 16785}, {11189, 44082}, {11206, 18922}, {11396, 34471}, {11400, 22479}, {11428, 40952}, {11471, 37601}, {12047, 31384}, {12106, 37729}, {12428, 41587}, {12943, 37196}, {12953, 37197}, {13367, 19365}, {13884, 19030}, {13937, 19029}, {15171, 21841}, {15325, 37935}, {15326, 37931}, {17985, 52167}, {18447, 45735}, {18494, 31479}, {18513, 35480}, {18514, 35488}, {19842, 37087}, {20838, 22341}, {26885, 44707}, {31452, 37122}, {32047, 37814}, {35015, 46588}, {37564, 40985}, {37587, 44878}, {39873, 41584}

X(52427) = isogonal conjugate of X(52392)
X(52427) = Stevanovic-circle-inverse of X(32756)
X(52427) = isogonal conjugate of the isotomic conjugate of X(5081)
X(52427) = polar conjugate of the isotomic conjugate of X(2323)
X(52427) = X(i)-Ceva conjugate of X(j) for these (i,j): {915, 19}, {2766, 650}, {5081, 2323}, {36121, 6}
X(52427) = X(3724)-cross conjugate of X(2361)
X(52427) = X(i)-isoconjugate of X(j) for these (i,j): {1, 52392}, {3, 18815}, {7, 1807}, {57, 52351}, {63, 2006}, {69, 1411}, {73, 14616}, {77, 80}, {86, 52391}, {222, 18359}, {265, 1442}, {307, 759}, {328, 1399}, {348, 2161}, {603, 20566}, {655, 905}, {1214, 24624}, {1231, 34079}, {1439, 6740}, {1444, 52383}, {1459, 35174}, {1565, 52377}, {1793, 3668}, {1797, 14628}, {2222, 4025}, {6187, 7182}, {6356, 52380}, {7056, 52371}, {7177, 36910}, {15413, 32675}, {22128, 34535}, {22383, 46405}, {47318, 51640}
X(52427) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 52392}, {69, 35204}, {307, 34586}, {348, 40584}, {693, 13999}, {860, 35516}, {1231, 35069}, {2006, 3162}, {4025, 38984}, {5452, 52351}, {7182, 40612}, {7952, 20566}, {15413, 35128}, {18815, 36103}, {38966, 52356}, {40600, 52391}
X(52427) = cevapoint of X(3724) and X(44113)
X(52427) = crosspoint of X(i) and X(j) for these (i,j): {102, 284}, {7115, 36067}
X(52427) = crosssum of X(i) and X(j) for these (i,j): {226, 515}, {26932, 39471}
X(52427) = crossdifference of every pair of points on line {905, 1214}
X(52427) = barycentric product X(i)*X(j) for these {i,j}: {4, 2323}, {6, 5081}, {9, 1870}, {19, 4511}, {25, 32851}, {29, 2245}, {33, 3218}, {36, 281}, {37, 17515}, {55, 17923}, {92, 2361}, {101, 44428}, {186, 7110}, {270, 4053}, {284, 860}, {318, 7113}, {320, 607}, {333, 44113}, {650, 4242}, {654, 1897}, {758, 1172}, {1443, 7079}, {1464, 2322}, {1783, 3738}, {1835, 2287}, {1857, 22128}, {1983, 44426}, {2204, 35550}, {2212, 20924}, {2299, 3936}, {2332, 41804}, {3724, 31623}, {3904, 8750}, {4183, 18593}, {4282, 41013}, {4585, 18344}, {6335, 8648}, {7071, 17078}, {8606, 14165}, {21828, 36797}
X(52427) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 52392}, {19, 18815}, {25, 2006}, {33, 18359}, {36, 348}, {41, 1807}, {55, 52351}, {186, 17095}, {213, 52391}, {215, 22128}, {281, 20566}, {607, 80}, {654, 4025}, {758, 1231}, {860, 349}, {1172, 14616}, {1783, 35174}, {1835, 1446}, {1870, 85}, {1897, 46405}, {1973, 1411}, {1983, 6516}, {2204, 759}, {2212, 2161}, {2245, 307}, {2299, 24624}, {2323, 69}, {2332, 6740}, {2333, 52383}, {2361, 63}, {3218, 7182}, {3724, 1214}, {3738, 15413}, {4242, 4554}, {4282, 1444}, {4511, 304}, {5081, 76}, {7071, 36910}, {7110, 328}, {7113, 77}, {8648, 905}, {8750, 655}, {17515, 274}, {17923, 6063}, {21828, 17094}, {22128, 7055}, {32851, 305}, {34397, 2003}, {44113, 226}, {44428, 3261}
X(52427) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 11398, 34}, {25, 55, 33}, {25, 11383, 19}, {55, 10833, 4319}, {55, 20989, 51361}, {186, 1870, 36}, {1398, 15750, 5204}, {1495, 3270, 10535}, {3085, 7487, 11392}, {3295, 3517, 11399}, {7412, 41227, 225}

X(52428) = X(1)X(104)∩X(6)X(31)

X(52428) lies on these lines: {1, 104}, {3, 1057}, {6, 31}, {11, 750}, {33, 1395}, {35, 602}, {40, 1451}, {47, 3746}, {48, 44121}, {58, 1697}, {171, 497}, {238, 5218}, {255, 3295}, {354, 9316}, {390, 1936}, {595, 3601}, {612, 7069}, {748, 5432}, {756, 7082}, {950, 5264}, {976, 1858}, {984, 1776}, {991, 2078}, {999, 1480}, {1001, 25934}, {1058, 3075}, {1064, 8069}, {1106, 3304}, {1155, 1471}, {1200, 5452}, {1201, 22768}, {1279, 17603}, {1386, 9371}, {1399, 1496}, {1457, 34042}, {1458, 33925}, {1468, 3057}, {1617, 22053}, {1777, 21620}, {2187, 20986}, {2192, 7050}, {2324, 4512}, {2646, 3915}, {2654, 5711}, {3010, 21010}, {3058, 5348}, {3072, 4294}, {3073, 3085}, {3076, 35808}, {3077, 35809}, {3306, 35281}, {3486, 5255}, {3744, 10391}, {3816, 25938}, {3920, 24430}, {4300, 37579}, {5266, 12711}, {5281, 17127}, {6690, 25885}, {7084, 14935}, {9310, 35326}, {10459, 22760}, {10535, 21767}, {10589, 17122}, {10624, 37530}, {12053, 37522}, {13329, 35445}, {13384, 40091}, {14100, 51361}, {15845, 37634}, {16283, 20229}, {16466, 22072}, {17602, 38357}, {17720, 35015}, {33587, 42448}, {40967, 42012}

X(52428) = isogonal conjugate of the isotomic conjugate of X(3872)
X(52428) = X(i)-Ceva conjugate of X(j) for these (i,j): {2364, 41}, {36090, 46393}
X(52428) = X(i)-isoconjugate of X(j) for these (i,j): {7, 1000}, {278, 30680}, {279, 36916}, {3676, 51564}, {6063, 34446}
X(52428) = X(75)-Dao conjugate of X(52148)
X(52428) = crosssum of X(i) and X(j) for these (i,j): {2, 12648}, {226, 4424}
X(52428) = crossdifference of every pair of points on line {514, 46393}
X(52428) = barycentric product X(i)*X(j) for these {i,j}: {6, 3872}, {9, 999}, {31, 28808}, {33, 22129}, {41, 42697}, {55, 3306}, {71, 17519}, {284, 3753}, {650, 35281}, {1253, 17079}, {2175, 20925}, {2194, 4054}, {2364, 40587}
X(52428) = barycentric quotient X(i)/X(j) for these {i,j}: {41, 1000}, {212, 30680}, {999, 85}, {1253, 36916}, {3306, 6063}, {3753, 349}, {3872, 76}, {9447, 34446}, {17519, 44129}, {20925, 41283}, {22129, 7182}, {28808, 561}, {35281, 4554}, {42697, 20567}
X(52428) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 601, 603}, {31, 55, 212}, {31, 902, 21059}, {31, 1253, 2361}, {55, 2361, 1253}, {390, 17126, 1936}, {612, 30223, 7069}, {902, 2293, 55}, {1253, 2361, 212}, {1399, 3303, 1496}

X(52429) = X(1)X(631)∩X(44)X(55)

X(52429) lies on these lines: {1, 631}, {33, 8756}, {42, 52188}, {44, 55}, {64, 37568}, {103, 35445}, {200, 2325}, {212, 51476}, {220, 3689}, {612, 7073}, {678, 1253}, {963, 5217}, {1043, 44720}, {1155, 52013}, {2177, 8557}, {2328, 3158}, {3872, 30608}, {3935, 30680}, {4105, 4543}, {4319, 52371}, {4792, 7962}, {5524, 7220}, {7050, 7074}, {7218, 14556}, {11752, 36738}, {11789, 36737}, {14942, 51564}, {28043, 42064}

X(52429) = X(i)-isoconjugate of X(j) for these (i,j): {6, 17079}, {7, 999}, {56, 42697}, {57, 3306}, {109, 21183}, {269, 3872}, {278, 22129}, {604, 20925}, {1014, 3753}, {1407, 28808}, {1412, 4054}, {1439, 17519}, {2163, 36595}, {3676, 35281}
X(52429) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 42697}, {9, 17079}, {11, 21183}, {3161, 20925}, {3306, 5452}, {3872, 6600}, {4054, 40599}, {24771, 28808}, {36595, 40587}
X(52429) = cevapoint of X(2310) and X(4814)
X(52429) = crosspoint of X(1000) and X(36916)
X(52429) = trilinear pole of line {657, 4895}
X(52429) = barycentric product X(i)*X(j) for these {i,j}: {1, 36916}, {9, 1000}, {33, 30680}, {44, 36596}, {312, 34446}, {650, 51564}
X(52429) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17079}, {8, 20925}, {9, 42697}, {41, 999}, {45, 36595}, {55, 3306}, {200, 28808}, {210, 4054}, {212, 22129}, {220, 3872}, {650, 21183}, {1000, 85}, {1334, 3753}, {2332, 17519}, {30680, 7182}, {34446, 57}, {36596, 20568}, {36916, 75}, {51564, 4554}

X(52430) = X(1)X(1748)∩X(31)X(48)

X(52430) lies on these lines: {1, 1748}, {3, 22069}, {31, 48}, {38, 8766}, {63, 293}, {92, 1954}, {158, 829}, {162, 51299}, {184, 23197}, {204, 32676}, {212, 2638}, {255, 820}, {336, 33764}, {577, 4055}, {603, 7138}, {610, 1096}, {896, 6508}, {1106, 7114}, {1253, 2188}, {1415, 44087}, {1468, 2658}, {2173, 2181}, {2187, 42078}, {2210, 7124}, {2286, 7122}, {2308, 22063}, {4020, 26946}, {8772, 18671}, {15440, 30273}, {17453, 42075}, {22342, 36033}, {23201, 23231}, {36059, 38985}

X(52430) = isogonal conjugate of the anticomplement of X(828)
X(52430) = isogonal conjugate of the isotomic conjugate of X(255)
X(52430) = isotomic conjugate of the polar conjugate of X(9247)
X(52430) = isogonal conjugate of the polar conjugate of X(48)
X(52430) = polar conjugate of the isotomic conjugate of X(4100)
X(52430) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 563}, {48, 9247}, {255, 4100}, {2169, 48}, {4575, 822}, {24000, 163}, {24027, 32660}, {36114, 2631}
X(52430) = X(i)-isoconjugate of X(j) for these (i,j): {2, 2052}, {4, 264}, {5, 8795}, {6, 18027}, {19, 1969}, {25, 18022}, {53, 276}, {63, 6521}, {69, 1093}, {75, 158}, {76, 393}, {93, 32002}, {94, 14165}, {95, 13450}, {107, 850}, {186, 18817}, {225, 44130}, {253, 14249}, {273, 318}, {275, 324}, {278, 7017}, {281, 331}, {286, 41013}, {290, 6530}, {297, 16081}, {304, 6520}, {305, 6524}, {308, 27376}, {309, 47372}, {311, 8884}, {313, 8747}, {317, 847}, {327, 33971}, {338, 23582}, {339, 32230}, {340, 6344}, {342, 7020}, {343, 8794}, {349, 8748}, {427, 46104}, {459, 15466}, {468, 46111}, {523, 6528}, {525, 15352}, {561, 1096}, {648, 14618}, {653, 46110}, {671, 37778}, {811, 24006}, {821, 17858}, {823, 1577}, {1105, 44131}, {1109, 23999}, {1118, 3596}, {1235, 32085}, {1300, 44138}, {1441, 1896}, {1502, 2207}, {1826, 44129}, {1847, 7101}, {1857, 6063}, {1897, 46107}, {1974, 44161}, {2501, 6331}, {2592, 46815}, {2593, 46812}, {2970, 18020}, {2973, 15742}, {2996, 21447}, {3064, 46404}, {3267, 6529}, {3518, 20572}, {3542, 46746}, {5317, 27801}, {5392, 11547}, {6059, 41283}, {6335, 17924}, {6336, 46109}, {6525, 41530}, {6526, 14615}, {6531, 44132}, {7003, 40701}, {7337, 40363}, {9381, 37766}, {12077, 42405}, {14208, 36126}, {14569, 34384}, {14978, 39286}, {15459, 41079}, {15526, 34538}, {16080, 46106}, {16230, 22456}, {16263, 44135}, {16813, 18314}, {17434, 42401}, {17879, 24021}, {17907, 43678}, {17983, 44146}, {18024, 34854}, {18026, 44426}, {18831, 23290}, {20948, 24019}, {23590, 36793}, {23962, 23964}, {23978, 23984}, {23994, 24000}, {24026, 24032}, {28654, 36419}, {31623, 40149}, {32713, 44173}, {34170, 51967}, {34536, 36426}, {35142, 44145}, {35519, 36127}, {36120, 40703}, {36417, 40362}, {36889, 47392}, {37174, 42298}, {37765, 46105}, {39284, 40684}, {40009, 41766}, {40410, 44732}, {42293, 42369}, {44427, 46456}
X(52430) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 1969}, {9, 18027}, {75, 1147}, {92, 22391}, {130, 2618}, {158, 206}, {264, 36033}, {304, 37867}, {520, 17879}, {561, 6503}, {850, 38985}, {1096, 40368}, {1928, 6338}, {2052, 32664}, {3162, 6521}, {6505, 18022}, {14208, 46093}, {17423, 24006}, {20948, 35071}, {34467, 46107}, {40703, 46094}
X(52430) = crosspoint of X(i) and X(j) for these (i,j): {1, 1820}, {48, 255}, {163, 24000}, {577, 7335}, {24027, 32660}
X(52430) = crosssum of X(i) and X(j) for these (i,j): {1, 1748}, {2, 5906}, {4, 17902}, {75, 33808}, {92, 158}, {1577, 2632}, {2618, 20902}, {14618, 21666}, {24026, 46110}
X(52430) = crossdifference of every pair of points on line {1577, 46110}
X(52430) = barycentric product X(i)*X(j) for these {i,j}: {1, 577}, {3, 48}, {4, 4100}, {6, 255}, {9, 7335}, {19, 1092}, {25, 6507}, {31, 394}, {32, 326}, {41, 1804}, {42, 18604}, {55, 7125}, {56, 2289}, {57, 6056}, {58, 3990}, {63, 184}, {68, 563}, {69, 9247}, {71, 1437}, {73, 2193}, {75, 14585}, {81, 4055}, {92, 23606}, {101, 23224}, {109, 36054}, {110, 822}, {162, 32320}, {163, 520}, {212, 222}, {216, 2169}, {219, 603}, {228, 1790}, {250, 37754}, {268, 7114}, {283, 1409}, {284, 22341}, {293, 3289}, {304, 14575}, {418, 2167}, {521, 32660}, {560, 3926}, {604, 1259}, {605, 1335}, {606, 1124}, {610, 14379}, {647, 4575}, {652, 36059}, {656, 32661}, {662, 39201}, {692, 4091}, {810, 4558}, {820, 41890}, {849, 52386}, {905, 32656}, {906, 1459}, {1101, 3269}, {1102, 1974}, {1147, 1820}, {1176, 4020}, {1260, 7099}, {1262, 2638}, {1331, 22383}, {1333, 3682}, {1364, 2149}, {1397, 3719}, {1402, 6514}, {1410, 2327}, {1444, 2200}, {1576, 24018}, {1636, 36034}, {1755, 17974}, {1794, 14597}, {1796, 23201}, {1797, 23202}, {1802, 7053}, {1803, 22079}, {1813, 1946}, {1818, 32658}, {1953, 19210}, {1964, 28724}, {1973, 3964}, {2148, 5562}, {2150, 7066}, {2155, 35602}, {2159, 51394}, {2175, 7183}, {2188, 7011}, {2194, 40152}, {2196, 7193}, {2206, 3998}, {2290, 50463}, {2315, 5504}, {2359, 22345}, {2617, 46088}, {2632, 23357}, {3049, 4592}, {3063, 6517}, {3284, 35200}, {3292, 36060}, {3708, 47390}, {3955, 7116}, {4131, 32739}, {5440, 32659}, {5546, 51641}, {6149, 50433}, {6394, 9417}, {7045, 39687}, {7054, 7138}, {7055, 9447}, {10607, 38252}, {14533, 44706}, {14578, 22350}, {15373, 20760}, {15526, 23995}, {15905, 19614}, {17434, 36134}, {17879, 23963}, {20752, 36057}, {20775, 34055}, {22356, 36058}, {23582, 42080}, {23979, 24031}, {24000, 35071}, {24020, 41937}, {24027, 35072}, {36148, 37084}, {40364, 40373}
X(52430) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 18027}, {3, 1969}, {25, 6521}, {31, 2052}, {32, 158}, {48, 264}, {63, 18022}, {163, 6528}, {184, 92}, {212, 7017}, {255, 76}, {304, 44161}, {326, 1502}, {394, 561}, {418, 14213}, {520, 20948}, {560, 393}, {563, 317}, {577, 75}, {603, 331}, {810, 14618}, {822, 850}, {922, 37778}, {1092, 304}, {1102, 40050}, {1259, 28659}, {1437, 44129}, {1501, 1096}, {1576, 823}, {1804, 20567}, {1917, 2207}, {1923, 27376}, {1946, 46110}, {1973, 1093}, {1974, 6520}, {2148, 8795}, {2169, 276}, {2179, 13450}, {2193, 44130}, {2200, 41013}, {2289, 3596}, {2315, 44138}, {2632, 23962}, {2638, 23978}, {3049, 24006}, {3269, 23994}, {3289, 40703}, {3682, 27801}, {3719, 40363}, {3926, 1928}, {3964, 40364}, {3990, 313}, {4020, 1235}, {4055, 321}, {4091, 40495}, {4100, 69}, {4575, 6331}, {6056, 312}, {6507, 305}, {6514, 40072}, {7114, 40701}, {7125, 6063}, {7183, 41283}, {7335, 85}, {9247, 4}, {9417, 6530}, {9447, 1857}, {14533, 40440}, {14574, 24019}, {14575, 19}, {14585, 1}, {14600, 36120}, {17974, 46273}, {18604, 310}, {20775, 20883}, {22341, 349}, {22383, 46107}, {23202, 46109}, {23224, 3261}, {23357, 23999}, {23606, 63}, {23963, 24000}, {23979, 24032}, {23995, 23582}, {24018, 44173}, {28724, 18833}, {32320, 14208}, {32656, 6335}, {32660, 18026}, {32661, 811}, {32676, 15352}, {34396, 51315}, {34980, 20902}, {35071, 17879}, {36054, 35519}, {36059, 46404}, {36060, 46111}, {36134, 42405}, {36433, 255}, {37754, 339}, {39201, 1577}, {39687, 24026}, {40373, 1973}, {41937, 24021}, {42075, 36426}, {42080, 15526}, {42293, 2618}, {44088, 1953}, {46394, 1087}, {47390, 46254}, {51394, 46234}
X(52430) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 22130, 22069}, {577, 6056, 4055}, {1954, 1955, 92}

X(52431) = X(6)X(1411)∩X(19)X(34242)

X(524) lies on the cubic K498 and these lines: {6, 1411}, {19, 34242}, {31, 2875}, {37, 14737}, {42, 692}, {48, 216}, {71, 906}, {80, 1172}, {212, 3270}, {219, 1807}, {222, 3942}, {306, 1332}, {519, 2323}, {759, 36080}, {1409, 50433}, {1814, 9028}, {1949, 2253}, {2006, 2982}, {2183, 32675}, {2192, 52371}, {2219, 52383}, {2222, 32726}, {2252, 14578}, {2273, 4263}, {3187, 18359}, {3938, 40141}, {8677, 22086}, {20017, 41226}, {20566, 47318}, {21252, 24892}, {21293, 33137}, {22123, 23071}, {22350, 22356}, {36927, 40582}

X(52431) = isogonal conjugate of X(17923)
X(52431) = isogonal conjugate of the isotomic conjugate of X(52351)
X(52431) = isotomic conjugate of the polar conjugate of X(6187)
X(52431) = isogonal conjugate of the polar conjugate of X(80)
X(52431) = X(i)-Ceva conjugate of X(j) for these (i,j): {80, 6187}, {2341, 2161}
X(52431) = X(23202)-cross conjugate of X(3)
X(52431) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17923}, {2, 1870}, {4, 3218}, {19, 320}, {25, 20924}, {27, 758}, {28, 3936}, {29, 18593}, {33, 17078}, {34, 32851}, {36, 92}, {57, 5081}, {81, 860}, {108, 3904}, {158, 22128}, {162, 4707}, {186, 30690}, {214, 6336}, {226, 17515}, {264, 7113}, {273, 2323}, {274, 44113}, {278, 4511}, {281, 1443}, {286, 2245}, {331, 2361}, {333, 1835}, {340, 2160}, {470, 39153}, {471, 39152}, {514, 4242}, {651, 44428}, {653, 3738}, {654, 18026}, {811, 21828}, {1172, 41804}, {1227, 8752}, {1464, 31623}, {1474, 35550}, {1783, 4453}, {1845, 34234}, {1897, 3960}, {1973, 40075}, {1983, 46107}, {3724, 44129}, {4585, 7649}, {7100, 14165}, {8119, 8126}, {8120, 8125}, {8648, 46404}, {11570, 37203}, {13486, 44427}, {16082, 34586}, {16586, 36123}, {16944, 46109}, {36125, 51583}, {38462, 40215}
X(52431) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 17923}, {6, 320}, {36, 22391}, {92, 15898}, {125, 4707}, {860, 40586}, {1147, 22128}, {1870, 32664}, {3218, 36033}, {3904, 38983}, {3936, 40591}, {3960, 34467}, {4453, 39006}, {5081, 5452}, {6337, 40075}, {6505, 20924}, {7017, 36909}, {11517, 32851}, {17423, 21828}, {35550, 51574}, {38991, 44428}
X(52431) = cevapoint of X(i) and X(j) for these (i,j): {3, 23166}, {71, 22356}, {7117, 22086}
X(52431) = crosspoint of X(i) and X(j) for these (i,j): {80, 52351}, {1795, 1797}
X(52431) = crosssum of X(i) and X(j) for these (i,j): {1785, 8756}, {14400, 35015}
X(52431) = trilinear pole of line {228, 1946}
X(52431) = crossdifference of every pair of points on line {1845, 3738}
X(52431) = barycentric product X(i)*X(j) for these {i,j}: {1, 1807}, {3, 80}, {6, 52351}, {21, 52391}, {35, 265}, {48, 18359}, {55, 52392}, {63, 2161}, {65, 1793}, {69, 6187}, {71, 24624}, {72, 759}, {73, 6740}, {77, 52371}, {78, 1411}, {184, 20566}, {201, 52380}, {212, 18815}, {219, 2006}, {222, 36910}, {228, 14616}, {283, 52383}, {295, 36815}, {306, 34079}, {319, 52153}, {521, 2222}, {647, 47318}, {652, 655}, {1168, 5440}, {1214, 2341}, {1437, 15065}, {1444, 34857}, {1459, 51562}, {1794, 45926}, {1946, 35174}, {4064, 36069}, {6332, 32675}, {7004, 52377}, {7150, 52201}, {7265, 32662}, {9273, 21046}, {22350, 40437}, {22383, 36804}, {36058, 51975}, {36059, 52356}
X(52431) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 320}, {6, 17923}, {31, 1870}, {35, 340}, {42, 860}, {48, 3218}, {55, 5081}, {63, 20924}, {69, 40075}, {71, 3936}, {72, 35550}, {73, 41804}, {80, 264}, {184, 36}, {212, 4511}, {219, 32851}, {222, 17078}, {228, 758}, {265, 20565}, {577, 22128}, {603, 1443}, {647, 4707}, {652, 3904}, {655, 46404}, {663, 44428}, {692, 4242}, {759, 286}, {906, 4585}, {1402, 1835}, {1409, 18593}, {1411, 273}, {1459, 4453}, {1793, 314}, {1807, 75}, {1918, 44113}, {1946, 3738}, {2006, 331}, {2161, 92}, {2194, 17515}, {2200, 2245}, {2222, 18026}, {2341, 31623}, {3049, 21828}, {3937, 4089}, {5440, 1227}, {6187, 4}, {6740, 44130}, {9247, 7113}, {18359, 1969}, {20566, 18022}, {22356, 51583}, {22383, 3960}, {23201, 4973}, {23202, 214}, {24624, 44129}, {32659, 40215}, {32675, 653}, {34079, 27}, {34857, 41013}, {36815, 40717}, {36910, 7017}, {40172, 38462}, {46160, 16747}, {47318, 6331}, {50433, 52381}, {52153, 79}, {52351, 76}, {52371, 318}, {52391, 1441}, {52392, 6063}

X(52432) = X(4)X(54)∩X(24)X(52)

X(52432) lies on these lines: {3, 18532}, {4, 54}, {24, 52}, {25, 49}, {69, 3147}, {70, 125}, {110, 3542}, {156, 235}, {182, 37119}, {185, 44269}, {186, 1092}, {206, 9707}, {215, 11398}, {378, 10575}, {403, 9927}, {421, 847}, {427, 32046}, {567, 7507}, {569, 1594}, {576, 1974}, {933, 45135}, {1112, 20773}, {1598, 9704}, {1986, 12893}, {2477, 11399}, {3043, 15034}, {3053, 41759}, {3088, 11003}, {3089, 9544}, {3135, 19210}, {3146, 15472}, {3515, 22115}, {3517, 9703}, {3520, 10984}, {3541, 5012}, {3580, 46443}, {5094, 13353}, {5651, 14940}, {6143, 43650}, {6146, 45179}, {6240, 13352}, {6243, 11597}, {7487, 9545}, {7502, 12363}, {7505, 9306}, {7592, 45110}, {7689, 37970}, {7713, 9622}, {8745, 34756}, {8901, 34449}, {9677, 11473}, {9706, 37122}, {10018, 37636}, {10282, 44078}, {10540, 37197}, {10594, 44080}, {11449, 45172}, {11456, 36982}, {11457, 44679}, {12173, 37472}, {12228, 18569}, {13198, 14216}, {13336, 37118}, {13346, 15463}, {13371, 34114}, {15575, 21397}, {16266, 21213}, {16868, 41171}, {18374, 34787}, {18436, 37954}, {18533, 34148}, {19141, 40318}, {19154, 46444}, {21284, 37484}, {21844, 43652}, {32138, 44281}, {32534, 43898}, {35488, 46261}, {35503, 37480}, {37196, 37495}, {37478, 41590}

X(52432) = isotomic conjugate of the polar conjugate of X(36416)
X(52432) = X(i)-Ceva conjugate of X(j) for these (i,j): {11547, 571}, {18020, 41679}
X(52432) = X(i)-cross conjugate of X(j) for these (i,j): {3133, 24}, {39013, 15423}
X(52432) = X(i)-isoconjugate of X(j) for these (i,j): {68, 91}, {1820, 5392}, {2351, 20571}
X(52432) = X(i)-Dao conjugate of X(j) for these (i,j): {68, 34116}, {125, 924}, {134, 6368}, {577, 52350}
X(52432) = crosspoint of X(18020) and X(41679)
X(52432) = barycentric product X(i)*X(j) for these {i,j}: {24, 1993}, {47, 1748}, {69, 36416}, {110, 15423}, {249, 34338}, {275, 3133}, {317, 571}, {924, 41679}, {1147, 11547}, {4590, 6754}, {7763, 44077}, {8745, 9723}, {18020, 39013}
X(52432) = barycentric quotient X(i)/X(j) for these {i,j}: {24, 5392}, {571, 68}, {1147, 52350}, {1748, 20571}, {1993, 20563}, {3133, 343}, {6754, 115}, {8745, 847}, {15423, 850}, {34338, 338}, {35603, 39116}, {36416, 4}, {39013, 125}, {41679, 46134}, {44077, 2165}
X(52432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {24, 2904, 52}, {24, 35603, 52000}, {54, 1614, 18925}, {578, 6759, 21659}, {578, 10274, 184}, {1147, 44077, 24}, {15463, 35471, 13346}

X(52433) = X(3)X(6)∩X(53)X(10312)

X(52433) lies on these lines: {3, 6}, {53, 10312}, {112, 6748}, {193, 45795}, {230, 251}, {232, 8792}, {1501, 9604}, {1627, 3815}, {1879, 7755}, {1989, 13490}, {1990, 8882}, {2174, 21743}, {2493, 40583}, {3018, 40136}, {3567, 37813}, {3767, 18353}, {4558, 32455}, {5304, 20062}, {5306, 34603}, {7394, 7735}, {7669, 9969}, {8744, 14577}, {9722, 18907}, {9971, 40947}, {18374, 40981}

X(52433) = isogonal conjugate of the isotomic conjugate of X(34545)
X(52433) = isogonal conjugate of the polar conjugate of X(34484)
X(52433) = X(34567)-Ceva conjugate of X(184)
X(52433) = X(i)-isoconjugate of X(j) for these (i,j): {92, 34483}, {1577, 20189}, {20879, 34110}
X(52433) = X(22391)-Dao conjugate of X(34483)
X(52433) = crosspoint of X(34484) and X(34545)
X(52433) = crosssum of X(i) and X(j) for these (i,j): {2, 15108}, {6, 13564}
X(52433) = barycentric product X(i)*X(j) for these {i,j}: {3, 34484}, {6, 34545}, {54, 10095}, {110, 20188}, {140, 39168}, {186, 31676}, {1173, 36153}, {34567, 46084}
X(52433) = barycentric quotient X(i)/X(j) for these {i,j}: {184, 34483}, {1576, 20189}, {10095, 311}, {20188, 850}, {31676, 328}, {34484, 264}, {34545, 76}, {36153, 1232}, {39168, 40410}
X(52433) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 32, 2965}, {6, 1609, 566}, {6, 2965, 50}, {6, 8553, 13351}, {6, 11063, 570}, {6, 13345, 13338}, {6, 15109, 5421}, {6, 33886, 13345}, {32, 5008, 10317}, {32, 13338, 50}, {32, 13345, 6}, {32, 33886, 13338}, {187, 5421, 15109}, {371, 372, 37484}, {570, 5007, 6}, {577, 33872, 6}, {1609, 30435, 6}, {2965, 13338, 6}, {5063, 13342, 6}

X(52434) = X(31)X(184)∩X(110)X(238)

X(52434) lies on these lines: {1, 21326}, {6, 20962}, {31, 184}, {36, 22128}, {38, 3955}, {42, 2317}, {54, 3072}, {55, 4287}, {110, 238}, {171, 5012}, {181, 13366}, {182, 750}, {215, 2361}, {244, 1428}, {323, 3792}, {602, 1147}, {667, 838}, {692, 902}, {748, 9306}, {756, 26890}, {849, 9563}, {896, 7193}, {1106, 7335}, {1193, 1437}, {1203, 17104}, {1254, 19365}, {1395, 44077}, {1400, 44117}, {1404, 6187}, {1458, 36059}, {1460, 11402}, {1495, 3271}, {1614, 3073}, {1962, 17440}, {1977, 14567}, {1993, 5329}, {2003, 5322}, {2194, 2260}, {2206, 18892}, {2212, 44080}, {2223, 23202}, {2225, 19554}, {2310, 10535}, {3120, 5137}, {3924, 14529}, {4358, 5150}, {4579, 32927}, {4850, 5197}, {5363, 11422}, {5371, 21757}, {5651, 17125}, {6636, 7186}, {6800, 7295}, {7083, 26864}, {7299, 9652}, {9454, 32739}, {9459, 32719}, {9544, 17127}, {10448, 13323}, {11003, 17126}, {17124, 43650}, {17977, 32848}, {20665, 32664}, {20958, 51377}, {20959, 40952}, {20967, 23201}, {33105, 37527}, {34148, 37570}, {35505, 39689}, {40958, 44087}

X(52434) = isogonal conjugate of X(20566)
X(52434) = isogonal conjugate of the isotomic conjugate of X(36)
X(52434) = X(i)-Ceva conjugate of X(j) for these (i,j): {9456, 32}, {16944, 52059}, {34858, 31}
X(52434) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20566}, {2, 18359}, {8, 18815}, {10, 14616}, {75, 80}, {76, 2161}, {85, 36910}, {86, 15065}, {92, 52351}, {94, 3219}, {264, 1807}, {300, 46077}, {301, 46073}, {310, 34857}, {312, 2006}, {313, 759}, {314, 52383}, {318, 52392}, {319, 2166}, {321, 24624}, {328, 6198}, {334, 36815}, {349, 2341}, {514, 36804}, {522, 35174}, {561, 6187}, {650, 46405}, {655, 4391}, {664, 52356}, {693, 51562}, {903, 51975}, {1168, 3264}, {1411, 3596}, {1441, 6740}, {1577, 47318}, {1989, 33939}, {2174, 20573}, {2222, 35519}, {3262, 40437}, {4997, 14628}, {6063, 52371}, {7265, 32680}, {27801, 34079}, {30690, 41226}, {32851, 34535}, {34387, 52377}, {34388, 52380}, {35194, 46138}, {36795, 52212}, {40422, 45926}, {44130, 52391}
X(52434) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 20566}, {76, 40584}, {80, 206}, {313, 34586}, {319, 11597}, {561, 40612}, {3596, 35204}, {6187, 40368}, {7113, 21587}, {15065, 40600}, {18359, 32664}, {21237, 21411}, {22391, 52351}, {27801, 35069}, {33939, 34544}, {35519, 38984}, {39025, 52356}
X(52434) = crosspoint of X(i) and X(j) for these (i,j): {6, 34442}, {58, 909}, {1411, 2160}, {9456, 16944}, {24027, 32669}
X(52434) = crosssum of X(i) and X(j) for these (i,j): {1, 21368}, {2, 5080}, {8, 32849}, {10, 908}, {37, 22288}, {75, 17791}, {80, 52351}, {313, 3264}, {3219, 4511}, {4358, 51975}
X(52434) = crossdifference of every pair of points on line {321, 4391}
X(52434) = barycentric product X(i)*X(j) for these {i,j}: {1, 7113}, {6, 36}, {25, 22128}, {31, 3218}, {32, 320}, {41, 1443}, {44, 16944}, {48, 1870}, {50, 79}, {56, 2323}, {57, 2361}, {58, 2245}, {65, 4282}, {80, 52059}, {81, 3724}, {100, 21758}, {106, 17455}, {109, 654}, {110, 21828}, {184, 17923}, {214, 9456}, {215, 2006}, {284, 1464}, {323, 6186}, {513, 1983}, {560, 20924}, {604, 4511}, {651, 8648}, {667, 4585}, {692, 3960}, {758, 1333}, {849, 4053}, {902, 40215}, {909, 34586}, {1397, 32851}, {1409, 17515}, {1411, 34544}, {1415, 3738}, {1501, 40075}, {1576, 4707}, {1783, 22379}, {1790, 44113}, {1835, 2193}, {1845, 14578}, {1911, 27950}, {2151, 39153}, {2152, 39152}, {2160, 6149}, {2175, 17078}, {2194, 18593}, {2206, 3936}, {2624, 13486}, {3792, 40746}, {4089, 23990}, {4242, 22383}, {4453, 32739}, {4556, 42666}, {4867, 28607}, {4880, 34819}, {4881, 38266}, {4973, 28615}, {5353, 42623}, {6126, 19302}, {7127, 19373}, {11570, 32655}, {11700, 32677}, {16586, 34858}, {19619, 51235}, {19627, 20565}, {21773, 51803}, {32660, 44428}, {34397, 52381}, {34442, 40584}, {36910, 41282}
X(52434) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 20566}, {31, 18359}, {32, 80}, {36, 76}, {50, 319}, {79, 20573}, {109, 46405}, {184, 52351}, {213, 15065}, {215, 32851}, {320, 1502}, {560, 2161}, {604, 18815}, {654, 35519}, {692, 36804}, {758, 27801}, {1333, 14616}, {1397, 2006}, {1415, 35174}, {1443, 20567}, {1464, 349}, {1501, 6187}, {1576, 47318}, {1870, 1969}, {1983, 668}, {2175, 36910}, {2205, 34857}, {2206, 24624}, {2245, 313}, {2251, 51975}, {2323, 3596}, {2361, 312}, {3063, 52356}, {3218, 561}, {3724, 321}, {3960, 40495}, {4282, 314}, {4511, 28659}, {4585, 6386}, {4707, 44173}, {6149, 33939}, {6186, 94}, {7113, 75}, {8648, 4391}, {9247, 1807}, {9447, 52371}, {14270, 7265}, {14599, 36815}, {16944, 20568}, {17078, 41283}, {17455, 3264}, {17923, 18022}, {19627, 35}, {20924, 1928}, {21758, 693}, {21828, 850}, {22128, 305}, {22379, 15413}, {27950, 18891}, {32739, 51562}, {32851, 40363}, {40075, 40362}, {41282, 17078}, {52059, 320}
X(52434) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 21368, 21326}, {6, 20989, 20962}, {184, 1397, 31}, {1428, 26884, 244}, {1977, 14567, 14599}, {2361, 7113, 3724}, {20986, 44085, 42}

X(52435) = X(2)X(51458)∩X(3)X(19362)

X(52435) lies on these lines: {2, 51458}, {3, 19362}, {6, 2351}, {25, 1576}, {39, 13366}, {50, 3135}, {54, 15653}, {155, 16391}, {184, 418}, {237, 20968}, {571, 44077}, {933, 2052}, {1501, 9419}, {1993, 51776}, {2055, 19467}, {2120, 3567}, {3051, 22391}, {3060, 13558}, {3133, 34116}, {3148, 42442}, {3167, 6503}, {5392, 41205}, {5654, 13557}, {7577, 8154}, {10132, 10665}, {10133, 10666}, {15257, 52277}, {15512, 37493}, {19210, 19357}, {31381, 32046}, {34986, 47195}, {37893, 43653}, {40820, 43754}

X(52435) = isogonal conjugate of the isotomic conjugate of X(1147)
X(52435) = isogonal conjugate of the polar conjugate of X(571)
X(52435) = X(i)-Ceva conjugate of X(j) for these (i,j): {6, 14585}, {933, 52317}, {1166, 14533}, {1576, 34952}, {44174, 32661}
X(52435) = X(i)-isoconjugate of X(j) for these (i,j): {4, 20571}, {75, 847}, {91, 264}, {92, 5392}, {158, 20563}, {561, 14593}, {1577, 30450}, {1820, 18027}, {1969, 2165}, {6521, 52350}, {24006, 46134}
X(52435) = X(i)-Dao conjugate of X(j) for these (i,j): {76, 577}, {206, 847}, {264, 34116}, {1147, 20563}, {1216, 1225}, {5392, 22391}, {14593, 40368}, {15415, 47421}, {20571, 36033}
X(52435) = crosspoint of X(i) and X(j) for these (i,j): {6, 24}, {571, 1147}, {14586, 47390}, {32661, 44174}
X(52435) = crosssum of X(i) and X(j) for these (i,j): {2, 68}, {136, 14618}, {847, 5392}, {2970, 18314}, {34391, 34392}
X(52435) = crossdifference of every pair of points on line {6334, 14618}
X(52435) = barycentric product X(i)*X(j) for these {i,j}: {1, 563}, {3, 571}, {6, 1147}, {24, 577}, {32, 9723}, {47, 48}, {50, 5961}, {52, 14533}, {110, 30451}, {184, 1993}, {228, 18605}, {237, 51776}, {317, 14585}, {371, 26920}, {372, 8911}, {394, 44077}, {906, 34948}, {924, 32661}, {1092, 8745}, {2169, 2180}, {4558, 34952}, {7763, 14575}, {9247, 44179}, {11547, 23606}, {14576, 19210}, {14600, 51439}, {15958, 52317}, {16391, 36416}, {18877, 51393}, {32662, 44808}, {39013, 44174}, {39201, 41679}, {47390, 47421}
X(52435) = barycentric quotient X(i)/X(j) for these {i,j}: {24, 18027}, {32, 847}, {47, 1969}, {48, 20571}, {184, 5392}, {563, 75}, {571, 264}, {577, 20563}, {1147, 76}, {1501, 14593}, {1576, 30450}, {1993, 18022}, {5961, 20573}, {7763, 44161}, {8911, 34392}, {9247, 91}, {9723, 1502}, {14533, 34385}, {14575, 2165}, {14585, 68}, {19627, 5962}, {23606, 52350}, {26920, 34391}, {30451, 850}, {32661, 46134}, {34952, 14618}, {36433, 16391}, {44077, 2052}, {51776, 18024}
X(52435) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {184, 577, 23195}, {14575, 23606, 184}

X(52436) = X(6)X(26)∩X(24)X(47421)

X(52436) lies on these lines: {6, 26}, {24, 47421}, {25, 41271}, {32, 184}, {50, 9603}, {155, 3053}, {159, 40825}, {187, 5562}, {230, 12134}, {571, 1147}, {1352, 1691}, {1609, 9908}, {1692, 1843}, {1915, 2548}, {1970, 7737}, {1971, 3767}, {2030, 43130}, {2353, 14600}, {2715, 14265}, {2909, 27369}, {3202, 14573}, {3574, 7747}, {5889, 39839}, {6423, 44192}, {6424, 44193}, {8553, 42445}, {9233, 36433}, {9418, 20968}, {9700, 20859}, {9781, 40633}, {10539, 45938}, {14574, 44162}, {32379, 32445}, {35007, 43844}, {42295, 44527}

X(52436) = isogonal conjugate of the isotomic conjugate of X(571)
X(52436) = isogonal conjugate of the polar conjugate of X(44077)
X(52436) = X(25)-Ceva conjugate of X(14575)
X(52436) = X(i)-isoconjugate of X(j) for these (i,j): {2, 20571}, {68, 1969}, {75, 5392}, {76, 91}, {92, 20563}, {304, 847}, {561, 2165}, {925, 20948}, {1577, 46134}, {1820, 18022}, {14208, 30450}, {14213, 34385}, {14593, 40364}, {36145, 44173}
X(52436) = X(i)-Dao conjugate of X(j) for these (i,j): {76, 34116}, {206, 5392}, {305, 577}, {2165, 40368}, {20563, 22391}, {20571, 32664}, {39013, 44173}
X(52436) = crosspoint of X(i) and X(j) for these (i,j): {25, 8745}, {571, 44077}
X(52436) = crosssum of X(i) and X(j) for these (i,j): {2, 44128}, {69, 52350}, {3267, 23962}, {5392, 20563}
X(52436) = crossdifference of every pair of points on line {850, 15415}
X(52436) = barycentric product X(i)*X(j) for these {i,j}: {3, 44077}, {6, 571}, {19, 563}, {24, 184}, {25, 1147}, {31, 47}, {32, 1993}, {110, 34952}, {112, 30451}, {213, 18605}, {317, 14575}, {560, 44179}, {577, 8745}, {692, 34948}, {924, 1576}, {1501, 7763}, {1748, 9247}, {1974, 9723}, {2148, 2180}, {2211, 51776}, {3049, 41679}, {3133, 41271}, {5412, 8911}, {5413, 26920}, {5961, 34397}, {6563, 14574}, {6753, 32661}, {6754, 44174}, {9418, 31635}, {11547, 14585}, {14397, 32640}, {14533, 14576}, {14560, 44808}, {14573, 39113}, {14586, 52317}, {14601, 51439}, {18883, 19627}, {23357, 47421}, {40352, 51393}
X(52436) = barycentric quotient X(i)/X(j) for these {i,j}: {24, 18022}, {31, 20571}, {32, 5392}, {47, 561}, {184, 20563}, {317, 44161}, {560, 91}, {563, 304}, {571, 76}, {924, 44173}, {1147, 305}, {1501, 2165}, {1576, 46134}, {1974, 847}, {1993, 1502}, {7763, 40362}, {8745, 18027}, {9723, 40050}, {14573, 96}, {14574, 925}, {14575, 68}, {14585, 52350}, {18605, 6385}, {19627, 37802}, {30451, 3267}, {34948, 40495}, {34952, 850}, {40373, 2351}, {44077, 264}, {44162, 14593}, {44179, 1928}, {47421, 23962}, {52317, 15415}
X(52436) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 19627, 14585}, {1501, 14585, 32}, {9418, 40373, 20968}

X(52437) = X(3)X(69)∩X(23)X(325)

X(52437) lies on these lines: {3, 69}, {23, 325}, {26, 32825}, {50, 323}, {76, 7550}, {95, 252}, {99, 477}, {141, 50660}, {183, 7496}, {186, 340}, {264, 14865}, {298, 11146}, {299, 11145}, {311, 35500}, {316, 37946}, {317, 3518}, {319, 7279}, {320, 4996}, {328, 12028}, {524, 11063}, {895, 43705}, {1007, 1995}, {1078, 44148}, {1272, 2071}, {1494, 37948}, {1599, 32810}, {1600, 32811}, {1609, 11008}, {1634, 37183}, {1975, 7527}, {1992, 52275}, {3001, 9145}, {3003, 41617}, {3265, 15414}, {3268, 15470}, {3284, 4558}, {3292, 47390}, {3520, 44134}, {3631, 15109}, {5201, 35298}, {6503, 37669}, {6636, 7788}, {6644, 32837}, {7488, 32821}, {7492, 37668}, {7512, 7796}, {7514, 32836}, {7526, 32824}, {7530, 32816}, {7556, 32818}, {7782, 44133}, {7809, 37925}, {7811, 44832}, {7814, 34484}, {8553, 20080}, {8681, 14060}, {9146, 15365}, {9155, 52238}, {9737, 11188}, {10411, 14385}, {10607, 20806}, {11284, 34803}, {12082, 32006}, {12086, 44136}, {12088, 44128}, {12584, 51397}, {13337, 15018}, {14118, 32820}, {14570, 40888}, {14919, 43755}, {15246, 37671}, {17271, 37293}, {22087, 22143}, {23061, 51439}, {26863, 32002}, {31861, 32815}, {32000, 35475}, {32001, 44879}, {32833, 35921}, {34229, 40916}, {34990, 37784}, {35520, 46423}, {36163, 45921}, {36829, 37980}, {36896, 44769}, {39113, 44802}, {44377, 44533}, {44718, 46112}, {44719, 46113}, {45198, 45799}

X(52437) = isogonal conjugate of X(18384)
X(52437) = isotomic conjugate of X(6344)
X(52437) = isotomic conjugate of the isogonal conjugate of X(22115)
X(52437) = isotomic conjugate of the polar conjugate of X(323)
X(52437) = isogonal conjugate of the polar conjugate of X(7799)
X(52437) = X(7799)-Ceva conjugate of X(323)
X(52437) = X(i)-cross conjugate of X(j) for these (i,j): {16186, 8552}, {22115, 323}
X(52437) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18384}, {19, 1989}, {25, 2166}, {31, 6344}, {92, 11060}, {94, 1973}, {158, 52153}, {162, 15475}, {265, 1096}, {512, 36129}, {560, 18817}, {798, 46456}, {1141, 2181}, {1784, 40355}, {2153, 8738}, {2154, 8737}, {2489, 32680}, {2501, 32678}, {6520, 50433}, {8750, 43082}, {10412, 32676}, {14560, 24006}, {14582, 24019}, {14583, 36119}
X(52437) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 6344}, {3, 18384}, {4, 40604}, {6, 1989}, {19, 34544}, {25, 11597}, {94, 6337}, {125, 15475}, {265, 6503}, {323, 37943}, {328, 6338}, {403, 3580}, {462, 33527}, {463, 33526}, {1147, 52153}, {1511, 14583}, {1990, 3284}, {2088, 47236}, {2166, 6505}, {2501, 18334}, {2970, 5664}, {6374, 18817}, {8737, 40581}, {8738, 40580}, {10412, 15526}, {11060, 22391}, {13450, 14920}, {14569, 18402}, {14582, 35071}, {15595, 43089}, {17433, 51513}, {26932, 43082}, {31998, 46456}, {36129, 39054}, {37867, 50433}
X(52437) = cevapoint of X(8552) and X(16186)
X(52437) = crosspoint of X(34384) and X(40832)
X(52437) = crosssum of X(462) and X(463)
X(52437) = crossdifference of every pair of points on line {2489, 3199}
X(52437) = barycentric product X(i)*X(j) for these {i,j}: {3, 7799}, {50, 305}, {69, 323}, {76, 22115}, {97, 1273}, {99, 8552}, {110, 45792}, {186, 3926}, {287, 51383}, {298, 44719}, {299, 44718}, {304, 6149}, {319, 22128}, {340, 394}, {525, 10411}, {526, 4563}, {1154, 34386}, {1444, 42701}, {2088, 47389}, {3265, 14590}, {3268, 4558}, {3964, 14165}, {4590, 16186}, {4592, 32679}, {6148, 14919}, {6393, 14355}, {9723, 37802}, {11128, 50466}, {11129, 50465}, {19627, 40050}, {45807, 51478}
X(52437) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 6344}, {3, 1989}, {6, 18384}, {15, 8738}, {16, 8737}, {50, 25}, {63, 2166}, {69, 94}, {76, 18817}, {97, 1141}, {99, 46456}, {184, 11060}, {186, 393}, {305, 20573}, {323, 4}, {340, 2052}, {394, 265}, {441, 43089}, {520, 14582}, {525, 10412}, {526, 2501}, {577, 52153}, {647, 15475}, {662, 36129}, {905, 43082}, {1092, 50433}, {1154, 53}, {1273, 324}, {1511, 1990}, {2081, 51513}, {2088, 8754}, {2290, 2181}, {3265, 14592}, {3268, 14618}, {3284, 14583}, {3926, 328}, {4558, 476}, {4563, 35139}, {4575, 32678}, {4592, 32680}, {6148, 46106}, {6149, 19}, {6390, 43084}, {7799, 264}, {8552, 523}, {9723, 18883}, {10411, 648}, {11062, 14569}, {11064, 14254}, {11145, 46925}, {11146, 46926}, {14165, 1093}, {14270, 2489}, {14355, 6531}, {14385, 8749}, {14417, 51479}, {14590, 107}, {14591, 32713}, {14918, 13450}, {14919, 5627}, {16186, 115}, {17402, 36309}, {17403, 36306}, {18877, 40355}, {19210, 11077}, {19294, 463}, {19295, 462}, {19627, 1974}, {22115, 6}, {22128, 79}, {32661, 14560}, {32679, 24006}, {34386, 46138}, {34397, 2207}, {34834, 403}, {36212, 14356}, {37802, 847}, {39371, 51965}, {40604, 37943}, {41078, 23290}, {42701, 41013}, {43704, 11071}, {44180, 30529}, {44718, 14}, {44719, 13}, {44808, 6753}, {44814, 14273}, {45792, 850}, {46112, 3458}, {46113, 3457}, {50433, 14595}, {50465, 11080}, {50466, 11085}, {51383, 297}
X(52437) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 9723, 44180}, {340, 1273, 52149}, {340, 7799, 1273}, {487, 488, 18917}, {1238, 41008, 69}, {1273, 6148, 340}, {3964, 9723, 69}, {4558, 36212, 22151}, {6148, 7799, 52149}, {14368, 14369, 186}

X(52438) = X(6)X(1511)∩X(32)X(184)

X(52438) lies on these lines: {6, 1511}, {32, 184}, {39, 51394}, {249, 7757}, {378, 11653}, {682, 2909}, {1691, 11179}, {1692, 40673}, {1970, 3767}, {1971, 7737}, {3003, 18475}, {3202, 44162}, {5052, 44102}, {5309, 32761}, {5642, 7753}, {7745, 51425}, {7747, 51403}, {8573, 14533}, {9604, 41336}, {10540, 18373}, {14574, 14575}, {19626, 40823}, {32621, 40825}

X(52438) = isogonal conjugate of the isotomic conjugate of X(5063)
X(52438) = isogonal conjugate of the polar conjugate of X(44080)
X(52438) = X(i)-isoconjugate of X(j) for these (i,j): {75, 34289}, {561, 34288}, {1302, 20948}, {1969, 4846}, {36149, 44173}
X(52438) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 34289}, {34288, 40368}
X(52438) = crosspoint of X(5063) and X(44080)
X(52438) = barycentric product X(i)*X(j) for these {i,j}: {3, 44080}, {6, 5063}, {32, 15066}, {110, 42660}, {184, 378}, {237, 11653}, {1501, 32833}, {1576, 8675}, {10564, 40352}, {14574, 30474}, {14575, 44134}
X(52438) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 34289}, {378, 18022}, {1501, 34288}, {5063, 76}, {8675, 44173}, {11653, 18024}, {14574, 1302}, {14575, 4846}, {15066, 1502}, {32833, 40362}, {42660, 850}, {44080, 264}, {44134, 44161}
X(52438) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 19627, 1501}, {1501, 14585, 19627}

X(52439) = X(3)X(6530)∩X(25)X(393)

X(52439) lies on these lines: {3, 6530}, {4, 46741}, {6, 34854}, {25, 393}, {107, 2374}, {159, 1990}, {264, 5020}, {297, 37491}, {648, 19588}, {1093, 1598}, {1249, 19459}, {1593, 10002}, {1974, 2207}, {1995, 43981}, {3172, 14575}, {3199, 33578}, {3964, 15143}, {8743, 19125}, {8745, 19118}, {8746, 19153}, {9909, 37765}, {10790, 41365}, {15259, 41762}, {15905, 44096}, {16264, 18535}, {18534, 34334}, {20975, 41489}, {34208, 37777}, {37488, 39569}, {37921, 47158}, {39879, 41204}

X(52439) = isogonal conjugate of X(4176)
X(52439) = isogonal conjugate of the isotomic conjugate of X(6524)
X(52439) = isogonal conjugate of the polar conjugate of X(36434)
X(52439) = polar conjugate of the isotomic conjugate of X(2207)
X(52439) = polar conjugate of the isogonal conjugate of X(36417)
X(52439) = X(6524)-Ceva conjugate of X(2207)
X(52439) = X(36417)-cross conjugate of X(2207)
X(52439) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4176}, {2, 1102}, {63, 3926}, {69, 326}, {75, 3964}, {76, 6507}, {77, 1264}, {78, 7055}, {255, 305}, {304, 394}, {332, 52385}, {336, 51386}, {345, 7183}, {348, 3719}, {561, 1092}, {577, 40364}, {662, 4143}, {873, 4158}, {1231, 6514}, {1259, 7182}, {1332, 30805}, {1444, 52396}, {1502, 4100}, {1804, 3718}, {1928, 23606}, {2632, 47389}, {2972, 24037}, {3265, 4592}, {3998, 17206}, {4131, 4561}, {4563, 24018}, {4602, 32320}, {6517, 35518}, {7035, 7215}, {18020, 24020}, {18604, 40071}, {34537, 37754}
X(52439) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 4176}, {69, 15259}, {206, 3964}, {305, 6523}, {512, 2972}, {1084, 4143}, {1092, 40368}, {1102, 32664}, {3162, 3926}, {3265, 5139}, {14713, 44141}, {23606, 40369}
X(52439) = crosspoint of X(6524) and X(36434)
X(52439) = barycentric product X(i)*X(j) for these {i,j}: {3, 36434}, {4, 2207}, {6, 6524}, {19, 1096}, {25, 393}, {31, 6520}, {32, 1093}, {107, 2489}, {125, 23975}, {158, 1973}, {264, 36417}, {278, 6059}, {281, 7337}, {512, 6529}, {560, 6521}, {607, 1118}, {608, 1857}, {669, 15352}, {798, 36126}, {1824, 5317}, {1974, 2052}, {2333, 8747}, {2501, 32713}, {2970, 41937}, {2971, 23582}, {3124, 32230}, {3172, 6526}, {3199, 8884}, {3708, 24022}, {6525, 41489}, {6531, 34854}, {7140, 36420}, {8745, 14593}, {8754, 23964}, {8794, 40981}, {8882, 14569}, {17994, 20031}, {18027, 44162}, {20975, 23590}, {23985, 42069}, {40144, 41766}
X(52439) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 4176}, {25, 3926}, {31, 1102}, {32, 3964}, {158, 40364}, {393, 305}, {512, 4143}, {560, 6507}, {607, 1264}, {608, 7055}, {1084, 2972}, {1093, 1502}, {1096, 304}, {1395, 7183}, {1501, 1092}, {1917, 4100}, {1973, 326}, {1974, 394}, {1977, 7215}, {2052, 40050}, {2207, 69}, {2211, 51386}, {2212, 3719}, {2333, 52396}, {2489, 3265}, {2971, 15526}, {3080, 22401}, {3199, 52347}, {4117, 37754}, {6059, 345}, {6520, 561}, {6521, 1928}, {6524, 76}, {6529, 670}, {7109, 4158}, {7337, 348}, {8029, 23107}, {8754, 36793}, {9233, 23606}, {9426, 32320}, {9427, 34980}, {14569, 28706}, {15352, 4609}, {18027, 40360}, {20975, 23974}, {22260, 23616}, {23216, 35071}, {23964, 47389}, {23975, 18020}, {24022, 46254}, {32230, 34537}, {32713, 4563}, {34854, 6393}, {36126, 4602}, {36417, 3}, {36434, 264}, {42068, 3269}, {42295, 44141}, {44162, 577}
X(52439) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 1033, 40947}, {8745, 32713, 19118}

X(52440) = X(3)X(7163)∩X(31)X(56)

X(52440) lies on these lines: {1, 18360}, {3, 7163}, {12, 17122}, {31, 56}, {36, 1464}, {50, 7113}, {58, 14158}, {65, 15955}, {109, 1319}, {171, 5434}, {222, 1470}, {238, 5298}, {244, 51656}, {255, 5204}, {354, 37469}, {601, 3304}, {604, 28607}, {692, 23205}, {750, 11237}, {849, 1408}, {1193, 8614}, {1394, 5573}, {1411, 1455}, {1414, 1447}, {1415, 2251}, {1428, 3733}, {1454, 4320}, {1458, 3446}, {1468, 5221}, {1469, 7236}, {1496, 5217}, {1777, 11376}, {1914, 36075}, {1935, 5433}, {1936, 15326}, {2099, 9316}, {3075, 7354}, {3157, 40293}, {3230, 36074}, {3417, 26914}, {3649, 37607}, {3880, 35281}, {4292, 51751}, {4293, 5348}, {4303, 37564}, {4973, 18593}, {7286, 18201}, {7288, 7299}, {9363, 10944}, {9364, 40663}, {9456, 32669}, {10404, 37522}, {10571, 34880}, {11509, 34046}, {11510, 21000}, {19373, 42623}, {25938, 31141}, {32668, 34068}, {51236, 51651}

X(52440) = isogonal conjugate of the isotomic conjugate of X(1443)
X(52440) = X(i)-isoconjugate of X(j) for these (i,j): {2, 36910}, {8, 80}, {9, 18359}, {10, 6740}, {21, 15065}, {55, 20566}, {75, 52371}, {100, 52356}, {200, 18815}, {210, 14616}, {281, 52351}, {312, 2161}, {314, 34857}, {318, 1807}, {321, 2341}, {341, 1411}, {346, 2006}, {519, 36590}, {522, 51562}, {650, 36804}, {655, 3239}, {657, 46405}, {759, 3701}, {1043, 52383}, {1089, 52380}, {1168, 4723}, {1320, 51975}, {1793, 41013}, {1989, 42033}, {2166, 4420}, {2222, 4397}, {2321, 24624}, {3596, 6187}, {3700, 47318}, {3900, 35174}, {4518, 36815}, {6735, 40437}, {7026, 7043}, {7046, 52392}, {7110, 41226}, {24026, 52377}, {30713, 34079}, {36926, 36935}, {36927, 36934}, {36930, 36937}, {36931, 36938}, {44690, 46077}, {44691, 46073}
X(52440) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 52371}, {223, 20566}, {312, 40584}, {341, 35204}, {478, 18359}, {3596, 40612}, {3701, 34586}, {4397, 38984}, {4420, 11597}, {6609, 18815}, {8054, 52356}, {15065, 40611}, {30713, 35069}, {32664, 36910}, {34544, 42033}
X(52440) = crosspoint of X(i) and X(j) for these (i,j): {1014, 34051}, {1408, 1417}
X(52440) = crosssum of X(i) and X(j) for these (i,j): {3701, 4723}, {4768, 24026}
X(52440) = crossdifference of every pair of points on line {2321, 3239}
X(52440) = barycentric product X(i)*X(j) for these {i,j}: {6, 1443}, {7, 7113}, {31, 17078}, {34, 22128}, {36, 57}, {56, 3218}, {58, 18593}, {81, 1464}, {109, 3960}, {222, 1870}, {269, 2323}, {279, 2361}, {320, 604}, {323, 52372}, {603, 17923}, {653, 22379}, {654, 934}, {658, 8648}, {664, 21758}, {758, 1412}, {1014, 2245}, {1106, 32851}, {1319, 40215}, {1333, 41804}, {1397, 20924}, {1407, 4511}, {1408, 3936}, {1414, 21828}, {1415, 4453}, {1417, 51583}, {1434, 3724}, {1461, 3738}, {1790, 1835}, {1983, 3676}, {2149, 4089}, {3668, 4282}, {3911, 16944}, {4053, 7341}, {4556, 51645}, {4585, 43924}, {4881, 40151}, {5081, 7099}, {6149, 52374}, {7051, 37773}, {9456, 41801}, {16947, 35550}, {17515, 52373}, {18359, 41282}, {18815, 52059}, {19373, 37772}, {28607, 36589}, {34051, 34586}
X(52440) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 36910}, {32, 52371}, {36, 312}, {50, 4420}, {56, 18359}, {57, 20566}, {109, 36804}, {320, 28659}, {603, 52351}, {604, 80}, {649, 52356}, {654, 4397}, {758, 30713}, {934, 46405}, {1106, 2006}, {1333, 6740}, {1397, 2161}, {1399, 41226}, {1400, 15065}, {1404, 51975}, {1407, 18815}, {1408, 24624}, {1412, 14616}, {1415, 51562}, {1443, 76}, {1461, 35174}, {1464, 321}, {1870, 7017}, {1983, 3699}, {2206, 2341}, {2245, 3701}, {2323, 341}, {2361, 346}, {3218, 3596}, {3724, 2321}, {3960, 35519}, {4282, 1043}, {4881, 44723}, {6149, 42033}, {7099, 52392}, {7113, 8}, {8648, 3239}, {9456, 36590}, {16944, 4997}, {16947, 759}, {17078, 561}, {17455, 4723}, {18593, 313}, {20924, 40363}, {21758, 522}, {21828, 4086}, {22128, 3718}, {22379, 6332}, {23979, 52377}, {27950, 4087}, {41282, 3218}, {41804, 27801}, {46384, 23104}, {52059, 4511}, {52372, 94}
X(52440) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {56, 603, 1399}, {603, 1106, 56}, {1455, 18838, 1411}

X(52441) = ISOGONAL CONJUGATE OF X(26883)

See Ivan Pavlov, euclid 5545.

X(52441) lies on these lines: {216, 3523}, {376, 5562}, {3146, 8798}

X(52441) = isogonal conjugate of X(26883)

X(52442) = X(1442)X(4357)∩X(3219)X(3687)

See Ivan Pavlov, euclid 5545.

X(52442) lies on these lines: {1442, 4357}, {3219, 3687}

X(52442) = cross conjugate of X(2975) and X(2)

X(52443) = X(264)X(3854)∩X(307)X(25718)

See Ivan Pavlov, euclid 5545.

X(52443) lies on these lines: {264, 3854}, {307, 25718}

X(52443) = cross conjugate of X(3146) and X(2)

  A locus related to K025: X(52444) - X(52447)  

Let ABC be a triangle and P a point on its plane. Let A'B'C' be the cevian triangle of P, A" the reflection of A' in A. Let A* be the orthogonal projection of A" in BC and build B*, C* cyclically.

Then the locus of P such that ABC, A*B*C are perspective is the cubic K025 and the locus of the perspectors Q(P) is also K025.

The appearance of (i, j) in the following list means that Q( X(i) ) = X(j):

(4, 4), (30, 265), (265, 30), (316, 671), (671, 316), (1263, 19552), (1300, 5962), (5080, 11604), (5134, 39993), (5203, 41521), (5523, 11605), (5962, 1300), (10152, 34170), (11604, 5080), (11605, 5523), (11703, 52444), (19552, 1263), (34150, 39985), (34169, 34171), (34170, 10152), (34171, 34169), (34173, 47104), (34174, 34175), (34175, 34174), (34239, 34240), (34240, 34239), (38945, 39992), (38946, 52445), (38947, 52446), (38949, 39990), (38952, 39991), (39158, 39159), (39159, 39158), (39985, 34150), (39990, 38949), (39991, 38952), (39992, 38945), (39993, 5134), (41521, 5203), (42809, 42810), (42810, 42809), (47103, 52447), (47104, 34173), (47105, 47110), (47110, 47105), (52444, 11703), (52445, 38946), (52446, 38947), (52447, 47103)

Some additional properties:

• Transformation P → Q(P) is an isoconjugation.
• Q(P) = the antigonal conjugate of - the circumcircle inverse of - the antigonal conjugate of - P.

X(52444) = X(4)X(2889)∩X(30)X(20189)

See César Lozada, euclid 5551.

X(52444) lies on the cubic K025 and these lines: {4, 2889}, {30, 20189}, {140, 34110}, {10627, 11538}, {34174, 39989}

X(52444) = anticomplement of the circumperp conjugate of X(33542)
X(52444) = antigonal conjugate of X(5899)
X(52444) = inverse of X(11817) in polar circle
X(52444) = orthoassociate of X(11817)

X(52445) = X(4)X(18125)∩X(30)X(1287)

See César Lozada, euclid 5551.

X(52445) lies on the cubic K025 and these lines: {4, 18125}, {30, 1287}, {827, 5133}, {1263, 34174}, {6636, 46654}, {19552, 34175}

X(52445) = reflection of X(i) in X(j) for these (i, j): (827, 5133), (6636, 46654)
X(52445) = antigonal conjugate of X(6636)
X(52445) = syngonal image of X(5133)

X(52446) = X(4)X(18321)∩X(30)X(2698)

See César Lozada, euclid 5551.

X(52446) lies on the cubic K025 and these lines: {4, 18321}, {30, 2698}, {32, 2086}, {237, 2679}, {446, 34157}, {805, 21531}, {1316, 40077}, {2387, 37841}, {14251, 44114}, {14957, 41520}

X(52446) = reflection of X(i) in X(j) for these (i, j): (237, 2679), (805, 21531)
X(52446) = antigonal conjugate of X(237)
X(52446) = syngonal image of X(21531)

X(52447) = X(4)X(541)∩X(30)X(9060)

See César Lozada, euclid 5551.

X(52447) lies on the cubic K025 and these lines: {4, 541}, {30, 9060}, {316, 1494}, {381, 1302}, {671, 39985}, {3839, 50935}, {6128, 52187}, {11605, 47111}, {34175, 38951}, {39263, 44750}

X(52447) = reflection of X(1302) in X(381)
X(52447) = isogonal conjugate of the circumperp conjugate of X(11820)
X(52447) = antigonal conjugate of X(376)
X(52447) = X(376)-reciprocal conjugate of-X(40112)
X(52447) = syngonal image of X(381)
X(52447) = barycentric quotient X(376)/X(40112)

  A locus related to K616: X(52448) - X(52456)  

Let ABC be a triangle and P a point on its plane. Let A'B'C' be the cevian triangle of P, A" the midpoint of AA'. Let A* be the orthogonal projection of A" in BC and build B*, C* cyclically.

Then the locus of P such that ABC, A*B*C are perspective is the cubic K616 and the locus of the perspectors Q(P) is the cubic q' = anticomplement-of-K1265, given by:

  q': ∑[(SB*(S^2+(2*SB+SC)*SC)*y - SC*(S^2+(SB+2*SC)*SB)*z)*y*z] + (SB-SC)*(SC-SA)*(SA-SB) *x*y*z = 0

q' circumscribes ABC and passes through X(n) for these n: {2, 4, 3434, 4846, 11185, 14593, 30513, 34029, 41370, 52448, 52449, 52450, 52451, 52452, 52453, 52454, 52455, 52456}.

The appearance of (i, j) in the following list means that Q( X(i) ) = X(j):

(4, 4), (69, 2), (376, 4846), (1249, 52448), (3421, 30513), (5485, 11185), (6601, 3434), (9214, 52449), (34208, 14593), (36874, 52450), (36875, 52451), (36876, 52452), (36877, 52453), (36878, 52454), (51830, 52455), (51831, 41370), (51832, 52456)

X(52448) = X(2)X(107)∩X(4)X(51)

See César Lozada, euclid 5552.

X(52448) lies on the cubic K1301 and these lines: {2, 107}, {4, 51}, {22, 17907}, {25, 3425}, {30, 41372}, {112, 22391}, {154, 15274}, {161, 41170}, {184, 41371}, {251, 393}, {264, 6997}, {275, 14853}, {297, 33586}, {315, 27373}, {317, 3060}, {324, 7394}, {381, 51385}, {394, 44704}, {403, 27362}, {428, 14495}, {858, 46927}, {1249, 8779}, {1370, 15466}, {1619, 45279}, {1853, 42854}, {3079, 35260}, {3091, 6523}, {3434, 7017}, {3522, 34286}, {3832, 6526}, {5012, 32713}, {6353, 14165}, {6528, 11185}, {6529, 41370}, {6618, 11427}, {6619, 37643}, {6747, 34417}, {6756, 41365}, {6820, 51212}, {7378, 45063}, {7386, 52147}, {7391, 46106}, {7737, 41368}, {7774, 40887}, {8795, 43726}, {8884, 37122}, {9308, 46700}, {11204, 52057}, {11402, 42873}, {11442, 35360}, {11610, 17409}, {17810, 52280}, {18027, 46104}, {18437, 42453}, {20232, 23591}, {20859, 51334}, {23292, 37070}, {23590, 45280}, {26333, 47372}, {28144, 47147}, {30506, 44176}, {31383, 41204}, {33522, 52283}, {34029, 36127}, {40686, 42457}, {41580, 41761}, {41588, 44228}, {44134, 46151}

X(52448) = polar conjugate of X(14376)
X(52448) = Cevapoint of X(i) and X(j) for these (i, j): {107, 23590}, {393, 18027}
X(52448) = crosspoint of X(i) and X(j) for these (i, j): {22, 41580}, {107, 33294}, {315, 41761}
X(52448) = crosssum of X(393) and X(14585)
X(52448) = X(i)-Dao conjugate of-X(j) for these (i, j): (32, 577), (127, 520), (427, 3917)
X(52448) = X(i)-isoconjugate-of-X(j) for these {i, j}: {48, 14376}, {66, 255}, {326, 2353}, {394, 2156}, {822, 44766}
X(52448) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (4, 14376), (22, 394), (53, 41168), (107, 44766), (206, 577)
X(52448) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(21458)}} and {{A, B, C, X(3), X(14216)}}
X(52448) = barycentric product X(i)*X(j) for these {i, j}: {4, 17907}, {22, 2052}, {83, 41375}, {107, 33294}, {127, 32230}, {158, 1760}
X(52448) = barycentric quotient X(i)/X(j) for these (i, j): (4, 14376), (22, 394), (53, 41168), (107, 44766), (206, 577), (315, 3926)
X(52448) = trilinear product X(i)*X(j) for these {i, j}: {19, 17907}, {22, 158}, {82, 41375}, {92, 8743}, {315, 1096}, {393, 1760}
X(52448) = trilinear quotient X(i)/X(j) for these (i, j): (22, 255), (92, 14376), (158, 66), (206, 52430), (315, 326), (393, 2156)
X(52448) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 6525, 107), (4, 1075, 14216), (4, 3168, 1899), (4, 6524, 2052), (4, 14361, 32064), (4, 36876, 34170), (25, 6530, 11547), (393, 6995, 1629), (428, 14569, 33971), (6525, 10002, 2), (14249, 34170, 36876), (14361, 32064, 51939), (51877, 51939, 14361)

X(52449) = X(2)X(476)∩X(4)X(94)

See César Lozada, euclid 5552.

X(52449) lies on the cubic K1301 and these lines: {2, 476}, {4, 94}, {13, 21467}, {14, 21466}, {23, 17709}, {51, 11751}, {69, 13485}, {79, 38938}, {328, 41896}, {376, 52056}, {381, 34209}, {393, 23964}, {621, 35316}, {622, 35317}, {1383, 1989}, {1995, 51847}, {3091, 39170}, {3146, 51254}, {3524, 51345}, {3545, 14993}, {3580, 18867}, {3839, 5627}, {4240, 37766}, {4846, 43707}, {5169, 43090}, {7394, 43089}, {7533, 43087}, {7545, 31676}, {7605, 46512}, {7693, 10412}, {9143, 9214}, {10555, 14246}, {10688, 31723}, {10706, 41390}, {11003, 14560}, {11078, 46857}, {11092, 46856}, {11185, 35139}, {13448, 47146}, {13574, 43084}, {14002, 30529}, {14389, 18121}, {23588, 35906}, {36181, 39295}

X(52449) = crosspoint of X(23) and X(12824)
X(52449) = X(i)-isoconjugate-of-X(j) for these {i, j}: {67, 6149}, {323, 2157}
X(52449) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (23, 323), (94, 18019), (265, 34897), (316, 7799), (476, 17708)
X(52449) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(9979)}} and {{A, B, C, X(4), X(23)}}
X(52449) = barycentric product X(i)*X(j) for these {i, j}: {23, 94}, {265, 37765}, {316, 1989}, {328, 8744}
X(52449) = barycentric quotient X(i)/X(j) for these (i, j): (23, 323), (94, 18019), (265, 34897), (316, 7799), (476, 17708)
X(52449) = trilinear product X(i)*X(j) for these {i, j}: {23, 2166}, {1989, 16568}
X(52449) = trilinear quotient X(i)/X(j) for these (i, j): (23, 6149), (1989, 2157)
X(52449) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 51835, 265), (265, 14254, 51835), (476, 14356, 2)

X(52450) = X(2)X(99)∩X(4)X(1499)

See César Lozada, euclid 5552.

X(52450) lies on the cubic K1301 and these lines: {2, 99}, {4, 1499}, {6, 9214}, {30, 45143}, {193, 892}, {230, 4226}, {316, 45291}, {376, 5915}, {691, 36181}, {1316, 51258}, {1992, 17948}, {2395, 8430}, {3124, 10556}, {5108, 34505}, {5254, 41939}, {5304, 9154}, {5468, 47286}, {5477, 52035}, {5485, 14916}, {5968, 7736}, {6531, 34761}, {6620, 32729}, {6776, 48983}, {6791, 9880}, {7735, 16092}, {7737, 17964}, {8753, 14593}, {9169, 15638}, {10555, 39024}, {14568, 34245}, {14957, 36821}, {17983, 40138}, {18023, 44152}, {20423, 52233}, {23967, 37689}, {36874, 52038}, {43291, 45662}, {51431, 51820}

X(52450) = anticomplement of X(47047)
X(52450) = crossdifference of every pair of points on line {X(351), X(3292)}
X(52450) = crosspoint of X(230) and X(5477)
X(52450) = X(671)-daleth conjugate of-X(111)
X(52450) = X(i)-Dao conjugate of-X(j) for these (i, j): (114, 524), (230, 50567)
X(52450) = X(i)-isoconjugate-of-X(j) for these {i, j}: {187, 8773}, {524, 36051}, {896, 2987}, {922, 8781}
X(52450) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (111, 2987), (114, 50567), (230, 524), (460, 468), (671, 8781)
X(52450) = perspector of the circumconic {{A, B, C, X(892), X(17983)}}
X(52450) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(230)}} and {{A, B, C, X(4), X(99)}}
X(52450) = barycentric product X(i)*X(j) for these {i, j}: {111, 51481}, {114, 9154}, {230, 671}, {460, 30786}, {523, 52035}, {895, 44145}
X(52450) = barycentric quotient X(i)/X(j) for these (i, j): (111, 2987), (114, 50567), (230, 524), (460, 468), (671, 8781), (691, 10425)
X(52450) = trilinear product X(i)*X(j) for these {i, j}: {111, 1733}, {230, 897}, {661, 52035}, {671, 8772}, {923, 51481}, {1692, 46277}
X(52450) = trilinear quotient X(i)/X(j) for these (i, j): (111, 36051), (230, 896), (671, 8773), (897, 2987), (923, 32654), (1692, 922)
X(52450) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 36877, 34169), (115, 35606, 2), (14263, 34169, 36877)

X(52451) = X(2)X(98)∩X(4)X(512)

See César Lozada, euclid 5552.

X(52451) lies on the cubic K1301 and these lines: {2, 98}, {4, 512}, {5, 34953}, {6, 36183}, {30, 36822}, {69, 15631}, {290, 4846}, {315, 14382}, {317, 22456}, {511, 51259}, {1503, 7418}, {1513, 46777}, {1992, 37858}, {2395, 6792}, {2421, 3564}, {2433, 17986}, {2548, 45901}, {2715, 36472}, {3016, 43620}, {3580, 15329}, {3818, 15630}, {5627, 35906}, {5663, 36826}, {5890, 18304}, {6531, 41370}, {7737, 48452}, {10605, 34360}, {10684, 36998}, {11457, 33967}, {12824, 41512}, {12828, 51821}, {14853, 51980}, {15915, 46264}, {16083, 30226}, {18337, 18347}, {18912, 34156}, {26869, 44889}, {32112, 36875}, {32545, 39571}, {35934, 51869}, {37644, 46786}

X(52451) = anticomplement of X(47049)
X(52451) = crossdifference of every pair of points on line {X(3289), X(3569)}
X(52451) = X(113)-Dao conjugate of-X(511)
X(52451) = X(i)-isoconjugate-of-X(j) for these {i, j}: {240, 5504}, {511, 36053}, {684, 36114}
X(52451) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (98, 2986), (113, 51389), (248, 5504), (290, 40832), (403, 297)
X(52451) = perspector of the circumconic {{A, B, C, X(2966), X(16081)}}
X(52451) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(403)}} and {{A, B, C, X(3), X(13198)}}
X(52451) = barycentric product X(i)*X(j) for these {i, j}: {98, 3580}, {248, 44138}, {287, 403}, {290, 3003}, {685, 6334}, {686, 22456}
X(52451) = barycentric quotient X(i)/X(j) for these (i, j): (98, 2986), (113, 51389), (248, 5504), (290, 40832), (403, 297), (685, 687)
X(52451) = trilinear product X(i)*X(j) for these {i, j}: {98, 1725}, {293, 403}, {336, 44084}, {1821, 3003}
X(52451) = trilinear quotient X(i)/X(j) for these (i, j): (98, 36053), (293, 5504), (403, 240), (685, 36114), (1725, 511), (1821, 2986)
X(52451) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 6776, 15920), (4, 36874, 34175), (98, 287, 11653), (98, 14355, 35912), (98, 52190, 20021), (5967, 35912, 14355), (5967, 51820, 34761), (14265, 34175, 36874)

X(52452) = X(2)X(133)∩X(4)X(3426)

See César Lozada, euclid 5552.

X(52452) lies on the cubic K1301 and these lines: {2, 133}, {4, 3426}, {53, 52223}, {107, 16253}, {253, 3839}, {1249, 1559}, {1515, 36876}, {3087, 51990}, {3091, 3346}, {3543, 16251}, {3832, 15318}, {5056, 51348}, {6525, 10152}, {7487, 52168}, {11185, 35140}, {15319, 50689}, {18483, 39130}, {18554, 50687}, {31361, 50688}

X(52452) = X(i)-Dao conjugate of-X(j) for these (i, j): (4, 376), (122, 9007)
X(52452) = X(376)-isoconjugate-of-X(19614)
X(52452) = X(1249)-reciprocal conjugate of-X(376)
X(52452) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(1559)}} and {{A, B, C, X(3), X(13093)}}
X(52452) = barycentric product X(1249)*X(36889)
X(52452) = barycentric quotient X(1249)/X(376)
X(52452) = trilinear product X(i)*X(j) for these {i, j}: {204, 36889}, {1895, 3426}
X(52452) = trilinear quotient X(i)/X(j) for these (i, j): (204, 26864), (1895, 376)

X(52453) = X(2)X(1296)∩X(4)X(30230)

See César Lozada, euclid 5552.

X(52453) lies on the cubic K1301 and these lines: {2, 1296}, {4, 30230}, {670, 11185}

X(52453) = X(126)-Dao conjugate of-X(9027)
X(52453) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(47286)}} and {{A, B, C, X(4), X(670)}}

X(52454) = X(2)X(2374)∩X(4)X(193)

See César Lozada, euclid 5552.

X(52454) lies on the cubic K1301 and these lines: {2, 2374}, {4, 193}, {24, 5940}, {30, 40809}, {5395, 15369}, {6340, 7378}, {6995, 8770}, {9822, 32979}, {11185, 35136}

X(52454) = X(1384)-reciprocal conjugate of-X(3167)
X(52454) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(47286)}} and {{A, B, C, X(4), X(2374)}}
X(52454) = barycentric quotient X(1384)/X(3167)
X(52454) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 14248, 2996), (4, 34208, 5203), (5203, 14248, 34208), (5203, 34208, 2996)

X(52455) = X(2)X(15613)∩X(4)X(59433)

See César Lozada, euclid 5552.

X(52455) lies on the cubic K1301 and these lines: {2, 15613}, {4, 5943}, {1370, 39978}, {3108, 14930}

X(52455) = intersection, other than A, B, C, of circumconics {{A, B, C, X(3), X(9815)}} and {{A, B, C, X(4), X(3108)}}

X(52456) = X(2)X(11)∩X(4)X(885)

See César Lozada, euclid 5552.

X(52456) lies on the cubic K1301 and these lines: {2, 11}, {4, 885}, {1992, 38895}, {2481, 11185}, {5377, 43740}, {8751, 41370}, {10527, 16448}, {16450, 24390}

X(52456) = anticomplement of X(47048)
X(52456) = crossdifference of every pair of points on line {X(665), X(20752)}
X(52456) = crosssum of X(885) and X(42758)
X(52456) = X(119)-Dao conjugate of-X(518)
X(52456) = X(i)-isoconjugate-of-X(j) for these {i, j}: {518, 36052}, {672, 2990}, {913, 25083}, {915, 1818}
X(52456) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (105, 2990), (119, 51390), (294, 45393), (912, 25083), (919, 6099)
X(52456) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(1737)}} and {{A, B, C, X(4), X(100)}}
X(52456) = barycentric product X(i)*X(j) for these {i, j}: {105, 48380}, {673, 1737}, {914, 36124}
X(52456) = barycentric quotient X(i)/X(j) for these (i, j): (105, 2990), (119, 51390), (294, 45393), (912, 25083), (919, 6099), (1438, 36052)
X(52456) = trilinear product X(i)*X(j) for these {i, j}: {105, 1737}, {673, 8609}, {912, 36124}, {914, 8751}, {1438, 48380}
X(52456) = trilinear quotient X(i)/X(j) for these (i, j): (105, 36052), (673, 2990), (912, 1818), (914, 25083), (1438, 32655)

X(52457) = X(2)X(7)∩X(3)X(1633)

See Angel Montesdeoca, euclid 5554.

X(52457) lies on these lines: {1,41570}, {2,7}, {3,1633}, {4,5784}, {8,3254}, {11,42014}, {37,7961}, {69,37788}, {100,36976}, {141,281}, {219,4000}, {220,277}, {242,10519}, {278,17811}, {284,17183}, {320,30854}, {390,4511}, {443,960}, {480,8730}, {497,15733}, {516,997}, {517,2550}, {518,3421}, {528,5289}, {545,36916}, {599,1146}, {936,5735}, {942,2551}, {956,38055}, {958,25557}, {962,15829}, {965,5829}, {971,6827}, {999,38053}, {1001,8255}, {1006,2096}, {1119,37805}, {1212,4675}, {1329,5220}, {1376,38454}, {1467,12527}, {1479,5696}, {1737,5223}, {1851,3917}, {2093,38052}, {2095,3820}, {2182,7397}, {2323,5222}, {2324,3663}, {2478,10394}, {2801,5768}, {3035,25606}, {3059,4863}, {3061,17170}, {3174,10388}, {3474,37270}, {3485,19520}, {3826,33558}, {4187,5729}, {4231,7717}, {4413,36971}, {4419,37597}, {4643,34852}, {4648,40937}, {4679,17603}, {5233,28956}, {5528,20066}, {5732,6987}, {5759,6905}, {5762,6911}, {5766,27383}, {5779,6882}, {5785,6843}, {5795,11518}, {5811,6245}, {5815,24391}, {5817,6830}, {5832,6854}, {5853,7962}, {6554,26932}, {6603,17301}, {6745,47375}, {6835,45039}, {6840,36991}, {6847,21616}, {6857,15254}, {6859,38108}, {6865,9942}, {6880,21168}, {6883,31657}, {6904,11415}, {6947,36996}, {6954,31658}, {6970,37623}, {7999,31387}, {8726,12572}, {8728,36279}, {9785,12437}, {10177,11018}, {10383,40998}, {10941,46934}, {12436,30424}, {12514,37407}, {12915,15185}, {14256,36888}, {14555,28930}, {15346,37363}, {15726,24703}, {15823,16845}, {15842,16112}, {16054,17139}, {17227,37774}, {17237,46835}, {17272,20262}, {17334,34524}, {17392,34522}, {17658,40659}, {18250,43180}, {22464,25930}, {24477,41555}, {24929,47357}, {25934,34042}, {28452,31671}, {30295,35977}, {34371,47595}, {37611,43161}, {37659,37800}

X(52457) = midpoint of X(i) and X(j) for these {i,j}: {7,329}, {3059,17642}
X(52457) = reflection of X(i) in X(j) for these {i,j}: {9,3452}, {57,142}, {12848,8257}, {15185,12915}, {17658,40659}, {25606,3035}, {43161,37611}

X(52458) = POLAR-CIRCLE-INVERSE OF X(52459)

X(52458) lies on the cubic K1300 and these lines: {4, 2799}, {132, 46413}, {1503, 1562}, {2409, 6330}

X(52458) = polar-circle-inverse of X(52459)

X(52459) = POLAR-CIRCLE-INVERSE OF X(52458)

X(52459) lies on the Kiepert circumhyperbola and these lines: {2, 6333}, {4, 2799}, {98, 525}, {115, 43673}, {339, 43665}, {671, 46145}, {690, 3424}, {2786, 3429}

X(52459) = reflection of X(43673) in X(115)
X(52459) = polar-circle-inverse of X(52458)
X(52459) = antigonal image of X(43673)
X(52459) = antitomic image of X(43673)
X(52459) = X(163)-isoconjugate of X(2794)
X(52459) = X(115)-Dao conjugate of X(2794)
X(52459) = barycentric product X(i)*X(j) for these {i,j}: {523, 46145}, {850, 2710}
X(52459) = barycentric quotient X(i)/X(j) for these {i,j}: {523, 2794}, {879, 41175}, {2710, 110}, {46145, 99}

X(52460) = POLAR-CIRCLE-INVERSE OF X(76)

X(52460) lies on the cubic K1300 and these lines: {2, 32529}, {4, 69}, {6, 30496}, {19, 41532}, {24, 35375}, {25, 2076}, {112, 16385}, {132, 2679}, {187, 27369}, {232, 16068}, {419, 9467}, {427, 5103}, {468, 42068}, {512, 1692}, {732, 5186}, {1593, 47619}, {1597, 35458}, {1691, 1968}, {2386, 43449}, {2456, 19124}, {5111, 8541}, {5139, 33330}, {5480, 40951}, {5969, 32527}, {6071, 51434}, {6531, 34238}, {6655, 11574}, {6995, 11673}, {7378, 33873}, {9822, 16044}, {11470, 18321}, {12167, 15514}, {12220, 33019}, {19118, 35006}, {34214, 34854}, {35060, 37337}, {37511, 40279}, {40236, 51412}, {44099, 46627}

X(52460) = midpoint of X(5207) and X(49122)
X(52460) = reflection of X(48445) in X(6)
X(52460) = isogonal conjugate of X(8858)
X(52460) = complement of X(32529)
X(52460) = polar-circle-inverse of X(76)
X(52460) = polar conjugate of the isotomic conjugate of X(3229)
X(52460) = polar conjugate of the isogonal conjugate of X(32748)
X(52460) = X(419)-Ceva conjugate of X(232)
X(52460) = X(32748)-cross conjugate of X(3229)
X(52460) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8858}, {63, 3225}, {69, 43761}, {304, 699}
X(52460) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 8858}, {69, 39080}, {2086, 24284}, {3162, 3225}, {40708, 40810}
X(52460) = crosspoint of X(4) and X(17980)
X(52460) = crosssum of X(3) and X(12215)
X(52460) = crossdifference of every pair of points on line {69, 3049}
X(52460) = X(i)-line conjugate of X(j) for these (i,j): {4, 69}, {512, 3049}
X(52460) = barycentric product X(i)*X(j) for these {i,j}: {4, 3229}, {19, 2227}, {25, 698}, {92, 51907}, {264, 32748}, {297, 32540}, {419, 47648}, {420, 51248}, {468, 36821}, {1974, 35524}, {2501, 41337}, {6331, 9429}, {17980, 39080}
X(52460) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8858}, {25, 3225}, {698, 305}, {1973, 43761}, {1974, 699}, {2227, 304}, {3229, 69}, {9429, 647}, {32540, 287}, {32748, 3}, {35524, 40050}, {36821, 30786}, {41337, 4563}, {44089, 32544}, {44090, 8864}, {47648, 40708}, {51322, 12215}, {51907, 63}
X(52460) = {X(12294),X(40325)}-harmonic conjugate of X(1843)

X(52461) = POLAR-CIRCLE-INVERSE OF X(321)

X(52461) lies on the cubic K1300 and these lines: {4, 8}, {19, 38472}, {25, 5078}, {132, 3259}, {242, 46510}, {468, 42067}, {513, 1430}, {608, 5061}, {1503, 38389}, {1548, 2818}, {1848, 50362}, {1862, 5846}, {5710, 17516}, {5799, 42448}

X(52461) = polar-circle-inverse of X(321)
X(52461) = crosspoint of X(4) and X(17981)
X(52461) = crosssum of X(3) and X(17977)
X(52461) = crossdifference of every pair of points on line {72, 22383}
X(52461) = X(i)-lineconjugate of X(j) for these (i,j): {4, 72}, {513, 22383}

X(52462) = POLAR-CIRCLE-INVERSE OF X(1916)

X(52462) lies on the cubics K699 and K1300 these lines: {4, 147}, {25, 5989}, {98, 47643}, {99, 11325}, {242, 1281}, {419, 8290}, {428, 42068}, {468, 48439}, {804, 12829}, {1503, 48445}, {1569, 33874}, {1593, 38654}, {1899, 3981}, {1974, 19120}, {3186, 8782}, {4027, 39927}, {5976, 8783}, {7009, 8857}, {8856, 32085}, {11606, 37892}, {12042, 35476}, {41533, 51246}

X(52462) = polar-circle-inverse of X(1916)
X(52462) = Neuberg-circles-radical-circle-inverse of X(147)
X(52462) = polar conjugate of the isotomic conjugate of X(39080)
X(52462) = polar conjugate of the isogonal conjugate of X(51322)
X(52462) = X(51322)-cross conjugate of X(39080)
X(52462) = X(i)-isoconjugate of X(j) for these (i,j): {63, 51992}, {1967, 8858}, {36214, 43761}
X(52462) = X(i)-Dao conjugate of X(j) for these (i,j): {69, 3229}, {305, 35540}, {647, 2086}, {3162, 51992}, {8290, 8858}, {36214, 39080}
X(52462) = crosspoint of X(4) and X(17984)
X(52462) = crosssum of X(3) and X(17970)
X(52462) = X(4)-line conjugate of X(36214)
X(52462) = barycentric product X(i)*X(j) for these {i,j}: {4, 39080}, {92, 51912}, {264, 51322}, {419, 698}, {3229, 17984}, {35524, 44089}
X(52462) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 51992}, {385, 8858}, {419, 3225}, {698, 40708}, {3229, 36214}, {32540, 15391}, {32748, 17970}, {39080, 69}, {44089, 699}, {51322, 3}, {51912, 63}
X(52462) = {X(5186),X(32527)}-harmonic conjugate of X(4)

X(52463) = POLAR-CIRCLE-INVERSE OF X(2052)

X(52463) lies on the bucis K417 and K1300 and these lines: {1, 7016}, {4, 51}, {132, 35579}, {155, 38281}, {417, 12096}, {468, 34980}, {511, 34186}, {520, 647}, {1092, 2055}, {1942, 47195}, {5651, 14642}, {47638, 52177}

X(52463) = polar-circle-inverse of X(2052)
X(52463) = Moses-radical-circle-inverse of X(46841)
X(52463) = X(450)-Ceva conjugate of X(47195)
X(52463) = crosspoint of X(4) and X(1942)
X(52463) = crosssum of X(i) and X(j) for these (i,j): {3, 450}, {851, 1148}, {2501, 35236}
X(52463) = crossdifference of every pair of points on line {4, 32320}
X(52463) = X(i)-line conjugate of X(j) for these (i,j): {51, 4}, {520, 32320}

X(52464) = POLAR-CIRCLE-INVERSE OF X(2394)

X(52464) lies on the cubic K1300 and these lines: {4, 523}, {30, 1990}, {132, 468}, {133, 5099}, {427, 14583}, {1503, 5095}, {3154, 47204}, {3233, 14920}, {6530, 10295}, {9530, 37987}, {14560, 15472}, {14847, 47296}, {42854, 47285}, {51456, 51965}

X(52464) = polar-circle-inverse of X(2394)
X(52464) = crossdifference of every pair of points on line {3284, 14380}
X(52464) = X(i)-line conjugate of X(j) for these (i,j): {4, 14380}, {30, 3284}
X(52464) = {X(3258),X(16240)}-harmonic conjugate of X(468)

X(52465) = POLAR-CIRCLE-INVERSE OF X(2592)

X(52465) lies on the orthosymmedial circle, the cubic K1300, and these lines: {2, 24650}, {4, 2574}, {6, 1345}, {25, 13414}, {51, 125}, {389, 43395}, {468, 44125}, {511, 1312}, {1113, 15107}, {1114, 19128}, {1313, 44084}, {1347, 5640}, {1503, 44126}, {3580, 46698}, {13434, 14375}, {14500, 50649}, {14853, 31955}, {25407, 51212}

X(52465) = midpoint of X(i) and X(j) for these {i,j}: {4, 31954}, {25407, 51212}
X(52465) = reflection of X(46166) in X(1312)
X(52465) = complement of X(25408)
X(52465) = polar-circle-inverse of X(2592)
X(52465) = crosspoint of X(4) and X(16071)
X(52465) = crosssum of X(3) and X(13415)
X(52465) = X(4)-line conjugate of X(2574)

X(52466) = POLAR-CIRCLE-INVERSE OF X(2593)

X(52466) lies on the orthosymmedial circle, the cubic K1300, and these lines: {2, 24651}, {4, 2575}, {6, 1344}, {25, 13415}, {51, 125}, {389, 43396}, {468, 44126}, {511, 1313}, {1113, 19128}, {1114, 15107}, {1312, 44084}, {1346, 5640}, {1503, 44125}, {3580, 46699}, {13434, 14374}, {14499, 50649}, {14853, 31954}, {25408, 51212}

X(52466) = midpoint of X(i) and X(j) for these {i,j}: {4, 31955}, {25408, 51212}
X(52466) = reflection of X(46167) in X(1313)
X(52466) = complement of X(25407)
X(52466) = polar-circle-inverse of X(2593)
X(52466) = crosspoint of X(4) and X(16070)
X(52466) = crosssum of X(3) and X(13414)
X(52466) = X(4)-lineconjugate of X(2575)

X(52467) = POLAR-CIRCLE-INVERSE OF X(5466)

X(52467) lies on the cubic K1300 and these lines: {4, 1499}, {132, 31654}, {468, 524}, {1560, 46659}, {5094, 9169}, {5108, 52292}, {14916, 52290}, {32525, 52293}

X(52467) = polar-circle-inverse of X(5466)
X(52467) = crossdifference of every pair of points on line {3292, 10097}
X(52467) = X(i)-line conjugate of X(j) for these (i,j): {4, 10097}, {468, 3292}
X(52467) = barycentric product X(5095)*X(9166)
X(52467) = barycentric quotient X(5095)/X(9164)

X(52468) = POLAR-CIRCLE-INVERSE OF X(11599)

X(52468) lies on the cubic K1300 and these lines: {4, 2784}, {27, 295}, {33, 43}, {101, 2901}, {242, 740}, {430, 1862}, {468, 5205}, {6541, 17927}, {20096, 52082}

X(52468) = polar-circle-inverse of X(11599)
X(52468) = polar conjugate of the isotomic conjugate of X(6651)
X(52468) = orthic-isogonal conjugate of X(242)
X(52468) = X(4)-Ceva conjugate of X(242)
X(52468) = X(i)-isoconjugate of X(j) for these (i,j): {3, 9505}, {63, 9506}, {291, 17972}, {295, 1929}, {304, 18263}, {2196, 6650}
X(52468) = X(i)-Dao conjugate of X(j) for these (i,j): {69, 239}, {337, 41841}, {3162, 9506}, {9505, 36103}, {9508, 18210}, {17972, 39029}
X(52468) = crosspoint of X(4) and X(17927)
X(52468) = crosssum of X(3) and X(17972)
X(52468) = barycentric product X(i)*X(j) for these {i,j}: {4, 6651}, {25, 18035}, {92, 8298}, {239, 17927}, {242, 6542}, {423, 740}, {862, 52137}, {1897, 27929}, {2201, 20947}, {4213, 39922}, {6335, 38348}, {6541, 31905}, {17735, 40717}
X(52468) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 9505}, {25, 9506}, {242, 6650}, {423, 18827}, {862, 9278}, {1914, 17972}, {1974, 18263}, {2201, 1929}, {6542, 337}, {6651, 69}, {8298, 63}, {17735, 295}, {17927, 335}, {18035, 305}, {18266, 2196}, {27929, 4025}, {38348, 905}

X(52469) = POLAR-CIRCLE-INVERSE OF X(14223)

X(52469) lies on the cubic K1300 and these lines: {4, 690}, {125, 468}, {132, 5095}, {542, 6103}, {1550, 17986}, {4240, 9140}, {12828, 16240}, {18440, 51457}

X(52469) = polar-circle-inverse of X(14223)
X(52469) = X(16080)-Ceva conjugate of X(6103)
X(52469) = crosspoint of X(4) and X(17986)
X(52469) = X(4)-line conjugate of X(35909)

X(52470) = X(3)X(6524)∩X(25)X(6526)

See Angel Montesdeoca, euclid 5559.

X(52470) lies on these lines: {3,6524}, {4,18532}, {25,6526}, {186,1093}, {6530,14118}, {11414,18850}

X(52470) = {X(3),X(6524)}-harmonic conjugate of X(52634)

X(52471) = X(4)X(69)∩X(6)X(512)

X(52471) lies on the orthosymmedial circle, the cubic K955, and these lines: {4, 69}, {6, 512}, {51, 6792}, {115, 2882}, {141, 3143}, {187, 6375}, {206, 32761}, {647, 11332}, {1351, 18321}, {1570, 39125}, {1974, 46592}, {2679, 16188}, {3060, 45291}, {3098, 7418}, {3111, 3589}, {3849, 13239}, {5092, 37991}, {5480, 31850}, {5661, 45900}, {5972, 42068}, {6467, 35902}, {8617, 34098}, {8704, 12508}, {11574, 35923}, {14984, 22515}, {32442, 38010}, {32484, 44423}, {35424, 46627}

X(52471) = midpoint of X(i) and X(j) for these {i,j}: {316, 49122}, {1351, 18321}
X(52471) = reflection of X(31850) in X(5480)
X(52471) = polar-circle-inverse of X(44146)
X(52471) = crossdifference of every pair of points on line {524, 3049}

X(52472) = X(2)X(476)∩X(4)X(523)

X(524) lies on the cubic K955 and these lines: {2, 476}, {4, 523}, {20, 31990}, {30, 2407}, {112, 393}, {133, 5139}, {146, 148}, {338, 43089}, {376, 14995}, {381, 47285}, {2395, 8430}, {2409, 6530}, {2799, 34174}, {3060, 11751}, {4232, 9752}, {4240, 16319}, {5891, 16262}, {5967, 14853}, {10723, 52035}, {12383, 14559}, {16312, 44228}, {28438, 44441}, {32827, 35520}, {35235, 47146}

X(52472) = midpoint of X(10723) and X(52035)
X(52472) = reflection of X(i) in X(j) for these {i,j}: {376, 14995}, {12383, 14559}
X(52472) = X(2966)-Ceva conjugate of X(1637)
X(52472) = barycentric product X(1550)*X(51228)
X(52472) = barycentric quotient X(1550)/X(51227)
X(52472) = {X(14731),X(52449)}-harmonic conjugate of X(2)

---

X(52473) = X(2)X(98)∩X(4)X(3566)

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

X(52734) lies on the cubic K955 and these lines: {2, 98}, {4, 3566}, {193, 16188}, {3580, 30512}, {18918, 30227}, {34174, 52450}, {46980, 50974}

X(52473) = X(16080)-Ceva conjugate of X(230)
X(52473) = crosspoint of X(4) and X(36875)

---

X(52474) = X(4)X(524)∩X(30)X(48945)

Barycentrics    (5*a^2 - b^2 - c^2)*(a^10 - 5*a^8*b^2 + 10*a^6*b^4 + 4*a^4*b^6 - 11*a^2*b^8 + b^10 - 5*a^8*c^2 + 13*a^6*b^2*c^2 - 24*a^4*b^4*c^2 + 31*a^2*b^6*c^2 + b^8*c^2 + 10*a^6*c^4 - 24*a^4*b^2*c^4 - 24*a^2*b^4*c^4 - 2*b^6*c^4 + 4*a^4*c^6 + 31*a^2*b^2*c^6 - 2*b^4*c^6 - 11*a^2*c^8 + b^2*c^8 + c^10) : :
X(52474) = 2 X[9169] - 3 X[14853]

X(52474) lies on the cubic K955 and these lines: {4, 524}, {30, 48945}, {511, 14916}, {542, 10734}, {1499, 1992}, {5108, 50967}, {6792, 16188}, {9169, 14853}, {18346, 50962}, {38940, 51028}, {43964, 50955}

X(52474) = midpoint of X(i) and X(j) for these {i,j}: {18346, 50962}, {38940, 51028}
X(52474) = reflection of X(i) in X(j) for these {i,j}: {6792, 20423}, {38951, 46959}, {50955, 43964}, {50967, 5108}

---

X(52475) = X(4)X(523)∩X(30)X(6334)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-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)*(-a^4 - a^2*b^2 + 2*b^4 + 2*a^2*c^2 - b^2*c^2 - c^4) : :
X(52475) = 3 X[403] - X[44427]

X(52475) lies on these lines: {4, 523}, {30, 6334}, {74, 3566}, {99, 1304}, {113, 525}, {115, 2501}, {403, 44427}, {468, 690}, {512, 12133}, {924, 15738}, {1494, 36196}, {5186, 14052}, {9144, 51227}, {9180, 16080}, {10152, 44969}, {10297, 44921}, {10603, 34767}, {14270, 44281}, {16230, 37984}, {35522, 44146}, {36890, 46986}

X(52475) = reflection of X(i) in X(j) for these {i,j}: {10297, 44921}, {16230, 37984}
X(52475) = X(i)-isoconjugate of X(j) for these (i,j): {2407, 36060}, {3284, 36085}, {4575, 9214}, {11064, 36142}
X(52475) = X(i)-Dao conjugate of X(j) for these (i,j): {30, 48317}, {136, 9214}, {1560, 2407}, {1649, 9033}, {3284, 38988}, {9409, 21905}, {11064, 23992}
X(52475) = trilinear pole of line {1648, 14273}
X(52475) = barycentric product X(i)*X(j) for these {i,j}: {468, 2394}, {524, 18808}, {690, 16080}, {1494, 14273}, {1648, 16077}, {2433, 44146}, {2501, 36890}, {4235, 12079}, {8749, 35522}, {9717, 14618}, {14380, 37778}, {17986, 50942}
X(52475) = barycentric quotient X(i)/X(j) for these {i,j}: {351, 3284}, {468, 2407}, {690, 11064}, {1648, 9033}, {2394, 30786}, {2433, 895}, {2501, 9214}, {2682, 14401}, {8749, 691}, {9717, 4558}, {12079, 14977}, {14273, 30}, {16080, 892}, {17986, 50941}, {18808, 671}, {21906, 9409}, {36119, 36085}, {36890, 4563}, {40354, 32729}, {44102, 2420}, {52038, 35912}

---

X(52476) = X(4)X(3566)∩X(107)X(10425)

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

X(52476) lies on these lines: {4, 3566}, {107, 10425}, {132, 36170}, {136, 2501}, {523, 3563}, {690, 5203}, {1112, 14052}, {3564, 16230}, {7472, 32697}

X(52476) = X(255)-isoconjugate of X(52035)
X(52476) = X(i)-Dao conjugate of X(j) for these (i,j): {3564, 48317}, {6523, 52035}
X(52476) = barycentric product X(i)*X(j) for these {i,j}: {14273, 35142}, {35364, 37778}
X(52476) = barycentric quotient X(i)/X(j) for these {i,j}: {393, 52035}, {14273, 3564}

---

X(52477) = X(4)X(524)∩X(297)X(17952)

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

X(52477) lies on these lines: {4, 524}, {297, 17952}, {340, 35179}, {468, 2482}, {690, 2501}, {1296, 10295}, {4232, 10603}, {5094, 21448}, {5523, 13492}, {6103, 17968}, {15471, 20380}, {16183, 52229}, {17983, 41909}, {32648, 47242}, {36792, 44146}

X(52477) = polar conjugate of X(52141)
X(52477) = X(i)-isoconjugate of X(j) for these (i,j): {48, 52141}, {895, 36277}, {1992, 36060}, {2408, 4575}, {2444, 4592}
X(52477) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 2408}, {1249, 52141}, {1499, 48317}, {1560, 1992}, {2444, 5139}, {10354, 41614}
X(52477) = cevapoint of X(8352) and X(47286)
X(52477) = trilinear pole of line {1649, 14273}
X(52477) = barycentric product X(i)*X(j) for these {i,j}: {468, 5485}, {2418, 2501}, {2434, 14618}, {14273, 35179}, {21448, 44146}, {32133, 37855}
X(52477) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 52141}, {468, 1992}, {2418, 4563}, {2434, 4558}, {2489, 2444}, {2501, 2408}, {5095, 27088}, {5485, 30786}, {14273, 1499}, {21448, 895}, {39238, 14908}, {44102, 1384}, {44146, 11059}

---

X(52478) = X(1)X(513)∩X(36)X(4674)

Barycentrics    a*(a + b - 2*c)*(a - 2*b + c)*(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(52478) lies on these lines: {1, 513}, {36, 4674}, {65, 16944}, {88, 5126}, {104, 517}, {106, 1168}, {515, 38950}, {518, 39154}, {903, 5088}, {910, 32665}, {952, 6073}, {1145, 22102}, {1155, 4792}, {1318, 1870}, {1387, 3259}, {1417, 18838}, {2316, 43065}, {2841, 13756}, {3924, 17109}, {4622, 51369}, {5048, 10703}, {10246, 52031}, {14193, 35271}, {15906, 24201}, {19636, 21578}, {28204, 36590}, {35013, 39758}, {40215, 50194}, {46790, 50843}

X(52478) = midpoint of X(901) and X(1320)
X(52478) = reflection of X(i) in X(j) for these {i,j}: {1145, 22102}, {3259, 1387}, {15906, 24201}
X(52478) = X(i)-isoconjugate of X(j) for these (i,j): {101, 50943}, {519, 953}, {902, 46136}, {23703, 46041}, {23757, 35011}
X(52478) = X(i)-Dao conjugate of X(j) for these (i,j): {1015, 50943}, {38462, 39535}, {40594, 46136}
X(52478) = cevapoint of X(6075) and X(35013)
X(52478) = barycentric product X(i)*X(j) for these {i,j}: {88, 952}, {903, 2265}, {1320, 43043}, {5376, 6075}
X(52478) = barycentric quotient X(i)/X(j) for these {i,j}: {88, 46136}, {513, 50943}, {952, 4358}, {2265, 519}, {9456, 953}
X(52478) = {X(901),X(10428)}-harmonic conjugate of X(36058)

---

X(52479) = X(1)X(1769)∩X(100)X(517)

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

X(52479) lies on these lines: {1, 1769}, {100, 517}, {104, 35011}, {214, 23703}, {900, 36944}, {1145, 17780}, {1317, 1846}, {1320, 1807}, {4597, 5088}, {41801, 50843}

X(52479) = reflection of X(i) in X(j) for these {i,j}: {23703, 214}, {46041, 37629}
X(52479) = X(i)-isoconjugate of X(j) for these (i,j): {88, 2265}, {106, 952}, {2316, 43043}, {6075, 9268}
X(52479) = X(214)-Dao conjugate of X(952)
X(52479) = crosssum of X(6075) and X(35013)
X(52479) = barycentric product X(i)*X(j) for these {i,j}: {44, 46136}, {100, 50943}, {953, 4358}
X(52479) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 952}, {902, 2265}, {953, 88}, {1319, 43043}, {2087, 6075}, {46136, 20568}, {50943, 693}

---

X(52480) = X(1)X(514)∩X(104)X(927)

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

X(52480) lies on these lines: {1, 514}, {104, 927}, {105, 929}, {239, 34906}, {516, 20367}, {517, 1952}, {1737, 13576}, {3086, 14267}, {6559, 41391}, {8074, 18785}, {15507, 46792}, {33845, 36796}

X(52480) = {X(927),X(2481)}-harmonic conjugate of X(5088)

---

X(52481) = X(1)X(2575)∩X(3)X(2574)

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

X(52481) lies on these lines: {1, 2575}, {3, 2574}, {21, 104}, {72, 15460}, {405, 13414}, {517, 1114}, {1113, 51420}, {1823, 34592}, {5694, 44068}, {5752, 24651}, {7580, 32614}, {25408, 37482}, {34593, 45272}

---

X(52482) = X(1)X(2574)∩X(3)X(2575)

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

X(52482) lies on these lines: {1, 2574}, {3, 2575}, {21, 104}, {72, 15461}, {405, 13415}, {517, 1113}, {1114, 51420}, {1822, 34593}, {5694, 44067}, {5752, 24650}, {7580, 32615}, {25407, 37482}, {34592, 45272}

---

X(52483) = X(2)X(691)∩X(4)X(542)

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

X(52483) lies on these lines: {2, 691}, {4, 542}, {6, 34169}, {30, 5968}, {111, 7737}, {316, 5468}, {381, 16092}, {512, 598}, {892, 11185}, {1995, 52232}, {3434, 5380}, {3845, 51258}, {5475, 17964}, {7394, 8877}, {7417, 51224}, {7745, 14263}, {7753, 14609}, {9143, 10560}, {9154, 52451}, {10630, 52449}, {11179, 21460}, {11317, 17948}, {14908, 44275}, {14995, 36196}, {15484, 45143}, {16279, 46045}, {26255, 35278}, {31105, 42008}, {31415, 41404}, {34574, 52453}, {43576, 47049}, {47353, 51405}

X(52483) = X(15303)-cross conjugate of X(7426)
X(52483) = X(896)-isoconjugate of X(5505)
X(52483) = X(5505)-Dao conjugate of X(15899)
X(52483) = cevapoint of X(7426) and X(15303)
X(52483) = barycentric product X(i)*X(j) for these {i,j}: {671, 7426}, {38951, 52141}
X(52483) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 5505}, {7426, 524}, {15303, 2482}, {41583, 7813}
X(52483) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 9214, 671}, {671, 14246, 9214}

---

X(52484) = X(2)X(1296)∩X(4)X(524)

Barycentrics    (a^2 + b^2 - 5*c^2)*(a^2 - 5*b^2 + c^2)*(7*a^6 - a^4*b^2 - 7*a^2*b^4 + b^6 - a^4*c^2 + 18*a^2*b^2*c^2 - b^4*c^2 - 7*a^2*c^4 - b^2*c^4 + c^6) : :

X(52484) lies on the cubic K1301 and these lines: {2, 1296}, {4, 524}, {30, 21448}, {2418, 32815}, {4846, 52450}, {5476, 17979}, {7617, 38801}, {7737, 17968}, {10354, 22338}, {11179, 39238}, {11185, 35179}, {11317, 17952}, {15484, 39237}, {15638, 36877}, {31099, 32133}, {37863, 46950}

X(52484) = reflection of X(1296) in X(15304)
X(52484) = barycentric product X(5485)*X(26255)
X(52484) = barycentric quotient X(i)/X(j) for these {i,j}: {26255, 1992}, {47545, 27088}
X(52484) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 5485, 38951}, {4, 14262, 34165}, {5485, 38951, 34165}, {14262, 38951, 5485}

---

X(52485) = X(2)X(107)∩X(3)X(46115)

Barycentrics    (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^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(52495) lies on the cubic K1301 and these lines: {2, 107}, {3, 46115}, {4, 525}, {5, 40512}, {20, 23582}, {30, 51937}, {393, 2697}, {1249, 15639}, {3146, 15351}, {3524, 40506}, {3839, 31621}, {4240, 11064}, {4846, 41370}, {6529, 44988}, {6793, 35912}, {6995, 51343}, {11185, 35140}, {14853, 15407}, {18533, 32649}, {34211, 44704}, {45289, 46106}, {46270, 50687}

X(52485) = polar conjugate of the isogonal conjugate of X(51937)
X(52485) = X(i)-isoconjugate of X(j) for these (i,j): {74, 8766}, {441, 2159}, {1503, 35200}, {2312, 14919}, {2349, 8779}, {36131, 39473}
X(52485) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 1503}, {441, 3163}, {39008, 39473}
X(52485) = trilinear pole of line {1990, 9033}
X(52485) = barycentric product X(i)*X(j) for these {i,j}: {30, 6330}, {264, 51937}, {1297, 46106}, {1990, 35140}, {3260, 43717}, {4240, 43673}, {8767, 14206}, {41079, 44770}, {47105, 51228}
X(52485) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 441}, {1297, 14919}, {1495, 8779}, {1990, 1503}, {2173, 8766}, {4240, 34211}, {6330, 1494}, {8767, 2349}, {9033, 39473}, {9214, 36894}, {14581, 42671}, {16240, 6793}, {32649, 32640}, {32687, 1304}, {34212, 14380}, {35906, 34156}, {36046, 36034}, {39265, 35910}, {43673, 34767}, {43717, 74}, {44770, 44769}, {46106, 30737}, {47105, 51227}, {48453, 40080}, {51937, 3}

---

X(52486) = X(2)X(112)∩X(4)X(850)

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

X(52486) lies on the cubic K1301 and these lines: {2, 112}, {4, 850}, {297, 36823}, {315, 18020}, {317, 5641}, {325, 4230}, {327, 44134}, {2857, 10423}, {6528, 11185}

X(52486) = polar conjugate of the isogonal conjugate of X(36823)
X(52486) = X(i)-isoconjugate of X(j) for these (i,j): {248, 18669}, {293, 2393}, {1910, 14961}, {14600, 20884}, {36084, 42665}
X(52486) = X(i)-Dao conjugate of X(j) for these (i,j): {132, 2393}, {11672, 14961}, {18669, 39039}, {38970, 47138}, {38987, 42665}
X(52486) = trilinear pole of line {232, 2799}
X(52486) = barycentric product X(i)*X(j) for these {i,j}: {232, 46140}, {240, 37220}, {264, 36823}, {297, 2373}, {1177, 44132}, {36884, 37765}
X(52486) = barycentric quotient X(i)/X(j) for these {i,j}: {232, 2393}, {240, 18669}, {297, 858}, {511, 14961}, {1177, 248}, {2373, 287}, {3569, 42665}, {6530, 5523}, {10423, 2715}, {16230, 47138}, {18876, 17974}, {34854, 14580}, {36095, 36084}, {36823, 3}, {36884, 34897}, {37220, 336}, {40703, 20884}, {44132, 1236}, {51823, 5967}, {51980, 34158}

---

X(52487) = X(2)X(131)∩X(4)X(3580)

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

X(52487) lies on the cubic K1301 and these lines: {2, 131}, {4, 3580}, {5, 254}, {225, 10826}, {378, 18850}, {381, 33494}, {393, 403}, {847, 3091}, {1105, 3541}, {1179, 3089}, {1217, 1594}, {3542, 8884}, {3545, 34208}, {6526, 35488}, {6531, 41370}, {6564, 41516}, {6565, 41515}, {7547, 18855}, {8889, 18852}, {9721, 47735}, {11185, 35142}, {16263, 18533}, {18560, 18846}

X(52487) = isogonal conjugate of X(47391)
X(52487) = polar conjugate of X(37645)
X(52487) = X(381)-cross conjugate of X(4)
X(52487) = X(i)-isoconjugate of X(j) for these (i,j): {1, 47391}, {48, 37645}, {255, 18533}
X(52487) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 47391}, {1249, 37645}, {6523, 18533}
X(52487) = barycentric product X(2052)*X(34801)
X(52487) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 37645}, {6, 47391}, {393, 18533}, {34288, 51471}, {34801, 394}

---

X(52488) = X(2)X(74)∩X(4)X(523)

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

X(52488) lies on the cubic K1301 and these lines: {2, 74}, {4, 523}, {20, 14385}, {30, 9717}, {325, 36890}, {376, 3233}, {381, 12079}, {1304, 18533}, {1494, 11185}, {1992, 38894}, {2411, 39985}, {3091, 52130}, {3146, 3470}, {3541, 38937}, {3543, 14731}, {3545, 40630}, {3839, 5627}, {6623, 16221}, {7422, 32111}, {7737, 48451}, {8749, 41370}, {11251, 47146}, {11751, 15305}, {14593, 52452}, {16933, 44438}, {35278, 35485}, {40352, 49669}, {46147, 50008}

X(52488) = anticomplement of X(47050)
X(52488) = X(30)-Dao conjugate of X(50935)
X(52488) = barycentric product X(1494)*X(16303)
X(52488) = barycentric quotient X(16303)/X(30)
X(52488) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 36875, 34150}, {1552, 35908, 4}, {14264, 34150, 36875}

---

X(52489) = X(2)X(108)∩X(4)X(65)

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

X(52489) lies on the cubic K1301 and these lines: {2, 108}, {4, 65}, {208, 1737}, {388, 5136}, {1062, 6827}, {1708, 1861}, {3436, 28950}, {6826, 44696}, {11185, 18026}, {14593, 52456}

X(52489) = barycentric quotient X(41604)/X(2193)

---

X(52490) = X(4)X(1499)∩X(111)X(186)

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

X(52490) lies on the cubic K1302 and these lines: {4, 1499}, {111, 186}, {112, 17964}, {378, 45143}, {671, 44146}, {5140, 8744}, {7482, 44467}, {37777, 41936}, {41377, 48983}

X(52490) = polar conjugate of the isotomic conjugate of X(46783)
X(52490) = polar conjugate of the isogonal conjugate of X(52197)
X(52490) = X(52197)-cross conjugate of X(46783)
X(52490) = X(6390)-isoconjugate of X(36150)
X(52490) = barycentric product X(i)*X(j) for these {i,j}: {4, 46783}, {264, 52197}, {671, 44467}, {2854, 17983}, {5466, 7482}
X(52490) = barycentric quotient X(i)/X(j) for these {i,j}: {2854, 6390}, {7482, 5468}, {8753, 2770}, {44467, 524}, {46783, 69}, {52197, 3}

---

X(52491) = X(4)X(512)∩X(74)X(290)

Barycentrics    b^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)*(-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(52491) lies on the cubic K1301 and these lines: {4, 512}, {74, 290}, {98, 186}, {112, 48452}, {378, 36822}, {403, 51441}, {419, 685}, {1235, 14382}, {1550, 47105}, {5191, 7473}, {5523, 51404}, {6344, 6531}, {8753, 9154}, {9862, 16083}, {11464, 35912}, {34369, 35907}

X(52491) = polar conjugate of X(46787)
X(52491) = polar conjugate of the isotomic conjugate of X(46786)
X(52491) = polar conjugate of the isogonal conjugate of X(34369)
X(52491) = X(34369)-cross conjugate of X(46786)
X(52491) = X(i)-isoconjugate of X(j) for these (i,j): {48, 46787}, {63, 52199}, {4575, 23350}, {23997, 35909}
X(52491) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 23350}, {511, 42426}, {1249, 46787}, {3162, 52199}, {23967, 36212}
X(52491) = trilinear pole of line {6041, 6103}
X(52491) = barycentric product X(i)*X(j) for these {i,j}: {4, 46786}, {264, 34369}, {290, 6103}, {542, 16081}, {685, 18312}, {1640, 22456}, {7473, 43665}, {14618, 34761}, {41174, 51428}
X(52491) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 46787}, {25, 52199}, {542, 36212}, {685, 5649}, {879, 35911}, {1640, 684}, {2395, 35909}, {2501, 23350}, {5191, 3289}, {6041, 39469}, {6103, 511}, {6531, 842}, {7473, 2421}, {14618, 34765}, {16081, 5641}, {17986, 35910}, {18312, 6333}, {18384, 34370}, {22456, 6035}, {34369, 3}, {34761, 4558}, {35907, 4230}, {46786, 69}, {51428, 41172}, {51963, 40080}
X(52491) = {X(290),X(22456)}-harmonic conjugate of X(44146)

---

X(52492) = X(4)X(690)∩X(74)X(8753)

Barycentrics    a^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^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(52492) lies on the cubic K1302 and these lines: {4, 690}, {74, 8753}, {112, 186}, {877, 50567}, {2967, 4230}, {5641, 6528}, {6344, 46105}, {17994, 23350}, {51334, 52199}

X(52492) = polar conjugate of X(46786)
X(52492) = polar conjugate of the isotomic conjugate of X(46787)
X(52492) = polar conjugate of the isogonal conjugate of X(52199)
X(52492) = X(52199)-cross conjugate of X(46787)
X(52492) = X(i)-isoconjugate of X(j) for these (i,j): {48, 46786}, {63, 34369}, {287, 2247}, {293, 542}, {336, 5191}, {656, 34761}
X(52492) = X(i)-Dao conjugate of X(j) for these (i,j): {132, 542}, {1249, 46786}, {3162, 34369}, {18312, 38970}, {34761, 40596}
X(52492) = trilinear pole of line {232, 23350}
X(52492) = barycentric product X(i)*X(j) for these {i,j}: {4, 46787}, {112, 34765}, {232, 5641}, {264, 52199}, {297, 842}, {340, 34370}, {648, 23350}, {877, 14998}, {4230, 14223}, {5649, 16230}, {6035, 17994}, {35908, 51228}
X(52492) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 46786}, {25, 34369}, {112, 34761}, {232, 542}, {842, 287}, {2211, 5191}, {4230, 14999}, {5649, 17932}, {5968, 51405}, {14998, 879}, {16230, 18312}, {17994, 1640}, {23350, 525}, {34370, 265}, {34765, 3267}, {34854, 6103}, {35908, 51227}, {46787, 69}, {48453, 35912}, {52199, 3}

---

X(52493) = X(4)X(523)∩X(30)X(44715)

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)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4)*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 + 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - b^6*c^2 - 3*a^4*c^4 - 2*a^2*b^2*c^4 + 4*b^4*c^4 + 3*a^2*c^6 - b^2*c^6 - c^8) : :
X(52493) = 3 X[186] - 4 X[47215]

X(52493) lies on the cubic K1302 and these lines: {4, 523}, {30, 44715}, {74, 186}, {112, 48451}, {250, 14094}, {378, 9717}, {403, 12079}, {1494, 44146}, {1515, 47323}, {1553, 35520}, {3331, 16328}, {3470, 14865}, {3484, 46090}, {3518, 52130}, {3520, 14385}, {4240, 46632}, {5627, 6344}, {5663, 7480}, {6070, 18809}, {6241, 39174}, {7464, 14919}, {8744, 8749}, {8753, 9139}, {10152, 10421}, {10295, 46147}, {10419, 39465}, {11079, 52418}, {11251, 34209}, {13509, 18877}, {13596, 39239}, {13754, 36831}, {15262, 40353}, {18507, 21269}, {37943, 40630}, {47223, 50401}

X(52493) = polar conjugate of X(46789)
X(52493) = polar conjugate of the isotomic conjugate of X(46788)
X(52493) = X(i)-isoconjugate of X(j) for these (i,j): {30, 36062}, {48, 46789}, {2631, 30528}, {3284, 36102}, {11064, 36151}, {14206, 32663}, {36130, 51394}
X(52493) = X(i)-Dao conjugate of X(j) for these (i,j): {30, 18809}, {1249, 46789}
X(52493) = crosssum of X(30) and X(32162)
X(52493) = crossdifference of every pair of points on line {3284, 14401}
X(52493) = barycentric product X(i)*X(j) for these {i,j}: {4, 46788}, {1494, 47228}, {2349, 36063}, {2394, 7480}, {5663, 16080}, {8749, 35520}, {11251, 40384}
X(52493) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 46789}, {1304, 30528}, {2159, 36062}, {2433, 14220}, {5663, 11064}, {7480, 2407}, {8749, 477}, {11251, 36789}, {36063, 14206}, {36119, 36102}, {40352, 32663}, {46788, 69}, {47228, 30}
X(52493) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {74, 1304, 186}, {1552, 17986, 34150}, {1552, 34150, 4}, {10152, 10421, 14989}, {14385, 38937, 3520}

---

X(52494) = X(4)X(526)∩X(107)X(186)

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

X(52494) lies on the cubic K1302 and these lines: {4, 526}, {74, 6344}, {107, 186}, {1464, 36130}, {1511, 4240}, {14249, 51475}, {14910, 32650}, {15463, 34210}, {32663, 41368}

X(52494) = polar conjugate of X(46788)
X(52494) = polar conjugate of the isotomic conjugate of X(46789)
X(52494) = X(i)-isoconjugate of X(j) for these (i,j): {48, 46788}, {5663, 35200}
X(52494) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 5663}, {1249, 46788}
X(52494) = barycentric product X(i)*X(j) for these {i,j}: {4, 46789}, {477, 46106}, {1784, 36102}, {14206, 36130}, {14920, 43707}
X(52494) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 46788}, {477, 14919}, {1990, 5663}, {36130, 2349}, {36151, 35200}, {46106, 35520}, {46789, 69}, {51965, 39986}

---

X(52495) = X(1)X(210)∩X(58)X(1100)

Barycentrics    a (4 a^3+13 a^2 (b+c)+2 a (6 b^2+11 b c+6 c^2)+3 (b+c)^3) : :
X(52495) = (rR-4s^2) X[1] + 3rR X[210]

See Angel Montesdeoca, euclid 5560.

X(52495) lies on these lines: {1,210}, {3,41456}, {58,1100}, {942,1817}, {970,15178}, {1449,31445}, {4255,16884}, {5122,25417}, {5439,17013}, {5806,15836}, {8678,32195}

---

X(52496) = X(2)X(2418)∩X(23)X(1296)

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

X(52496) lies on the cubic K1303 and these lines: {2, 2418}, {23, 1296}, {110, 17979}, {5971, 35179}, {6636, 38532}, {9139, 14919}, {10989, 38951}, {11580, 17968}, {14262, 16042}, {15018, 39238}

X(52496) = barycentric product X(i)*X(j) for these {i,j}: {2780, 35179}, {5485, 41617}
X(52496) = barycentric quotient X(i)/X(j) for these {i,j}: {1296, 2696}, {2780, 1499}, {37962, 4232}, {41617, 1992}, {41618, 15471}

---

X(52497) = X(2)X(1990)∩X(23)X(2693)

Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 + 4*a^2*c^2 + 4*b^2*c^2 - 5*c^4)*(a^4 + 4*a^2*b^2 - 5*b^4 - 2*a^2*c^2 + 4*b^2*c^2 + c^4)*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + 11*a^4*b^2*c^2 - 6*a^2*b^4*c^2 - 3*b^6*c^2 - 6*a^2*b^2*c^4 + 8*b^4*c^4 + 2*a^2*c^6 - 3*b^2*c^6 - c^8) : :

X(52497) lies on the cubic K1303 and these lines: {2, 1990}, {23, 2693}, {111, 46788}, {323, 40386}, {1995, 3426}, {3266, 46789}, {9060, 37952}, {31099, 52452}

X(52497) = X(9209)-Dao conjugate of X(46436)
X(52497) = barycentric product X(7464)*X(36889)
X(52497) = barycentric quotient X(i)/X(j) for these {i,j}: {3426, 10293}, {7464, 376}, {40114, 26864}, {47103, 39263}

---

X(52498) = X(2)X(14618)∩X(23)X(1300)

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

X(52498) lies on the cubic K1303 and these lines: {2, 14618}, {23, 1300}, {94, 323}, {477, 39986}, {687, 46106}, {3266, 40832}, {10420, 36188}, {11464, 15454}

X(52498) = X(2315)-isoconjugate of X(32710)
X(52498) = X(3003)-Dao conjugate of X(25641)
X(52498) = barycentric product X(3018)*X(40832)
X(52498) = barycentric quotient X(i)/X(j) for these {i,j}: {1300, 32710}, {3018, 3003}, {7471, 15329}, {15328, 15453}, {15454, 15469}, {17702, 13754}, {34150, 14264}, {39375, 51349}

---

X(52499) = X(2)X(905)∩X(23)X(104)

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

X(52499) lies on the cubic K1303 and these lines: {2, 905}, {23, 104}, {94, 37798}, {149, 14198}, {13136, 14919}, {16082, 21907}, {16704, 41933}, {21739, 34234}

X(52499) = X(2183)-isoconjugate of X(2687)
X(52499) = barycentric product X(2771)*X(18816)
X(52499) = barycentric quotient X(i)/X(j) for these {i,j}: {104, 2687}, {2771, 517}, {14266, 39991}, {18816, 46141}, {37966, 4246}, {43728, 14224}, {47086, 47081}

---

X(52500) = X(2)X(47232)∩X(23)X(2766)

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

X(52500) lies on the cubic K1303 and these lines: {2, 47232}, {21, 2968}, {23, 2766}, {63, 2895}, {100, 1791}, {345, 27543}, {348, 17740}, {1812, 23983}, {3266, 46784}, {10693, 34259}, {16586, 52381}, {23978, 31623}, {32849, 46785}, {34188, 39990}, {36100, 46487}

X(52500) = isotomic conjugate of X(37798)
X(52500) = X(2323)-cross conjugate of X(8)
X(52500) = X(i)-isoconjugate of X(j) for these (i,j): {31, 37798}, {34, 22123}, {56, 16548}, {57, 20989}, {109, 47227}, {604, 5080}, {1325, 1400}, {1397, 20920}, {1408, 21066}, {1411, 40584}, {1415, 21180}, {2850, 32674}
X(52500) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 16548}, {2, 37798}, {11, 47227}, {1146, 21180}, {1325, 40582}, {2850, 35072}, {3161, 5080}, {5452, 20989}, {11517, 22123}, {35204, 40584}
X(52500) = cevapoint of X(6) and X(51638)
X(52500) = trilinear pole of line {521, 960}
X(52500) = barycentric product X(i)*X(j) for these {i,j}: {314, 10693}, {2766, 35518}, {3596, 34442}
X(52500) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 37798}, {8, 5080}, {9, 16548}, {21, 1325}, {55, 20989}, {219, 22123}, {312, 20920}, {521, 2850}, {522, 21180}, {650, 47227}, {2321, 21066}, {2323, 40584}, {2766, 108}, {4511, 52368}, {10693, 65}, {34442, 56}, {39990, 14257}, {51470, 5172}

---

X(52501) = X(2)X(2492)∩X(23)X(99)

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

X(52501) lies on the cubic K1303 and these lines: {2, 2492}, {23, 99}, {111, 18019}, {4235, 31128}, {5468, 6593}, {14360, 34171}

X(52501) = isogonal conjugate of X(52197)
X(52501) = isotomic conjugate of X(46783)
X(52501) = polar conjugate of X(52490)
X(52501) = X(9177)-cross conjugate of X(524)
X(52501) = X(i)-isoconjugate of X(j) for these (i,j): {1, 52197}, {31, 46783}, {48, 52490}, {923, 2854}, {36060, 44467}
X(52501) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 46783}, {3, 52197}, {524, 9177}, {1249, 52490}, {1560, 44467}, {2482, 2854}
X(52501) = cevapoint of X(524) and X(9177)
X(52501) = barycentric product X(2770)*X(3266)
X(52501) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 46783}, {4, 52490}, {6, 52197}, {468, 44467}, {524, 2854}, {2482, 9177}, {2770, 111}, {4235, 7482}, {32741, 32740}, {34171, 14263}, {36150, 923}, {36824, 46154}, {47077, 47078}, {52145, 37858}

---

X(52502) = X(2)X(47227)∩X(23)X(100)

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

X(52502) lies on the cubic K1303 and these lines: {2, 47227}, {23, 100}, {3266, 4554}, {4238, 34337}, {18019, 21907}, {20344, 47104}

X(52502) = isotomic conjugate of X(46784)
X(52502) = X(i)-isoconjugate of X(j) for these (i,j): {31, 46784}, {1438, 2836}, {36057, 47232}
X(52502) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 46784}, {2836, 6184}, {20621, 47232}
X(52502) = barycentric product X(2752)*X(3263)
X(52502) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 46784}, {518, 2836}, {2752, 105}, {4238, 7476}, {5089, 47232}, {47080, 47083}, {47104, 14267}

---

X(52503) = X(23)X(9070)∩X(662)X(3936)

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

X(52503) lies on the cubic K1303 and these lines: {23, 9070}, {662, 3936}, {4552, 32849}

X(52503) = isotomic conjugate of X(46785)
X(52503) = X(31)-isoconjugate of X(46785)
X(52503) = X(2)-Dao conjugate of X(46785)
X(52503) = barycentric product X(12030)*X(35550)
X(52503) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 46785}, {4242, 37964}, {12030, 759}

---

X(52504) = X(2)X(311)∩X(68)X(11459)

Barycentrics    b^2*c^2*(a^4 - 2*a^2*b^2 + b^4 - 2*b^2*c^2 + c^4)*(a^4 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(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(52504) lies on the cubic K1303 and these lines: {2, 311}, {20, 40698}, {23, 925}, {68, 11459}, {91, 52344}, {96, 14118}, {403, 34333}, {847, 3091}, {3153, 5962}, {6334, 14618}, {7394, 14593}, {8906, 37444}, {10594, 46200}, {14919, 52498}, {18537, 51833}, {30450, 46106}, {37644, 47731}

X(52504) = isotomic conjugate of X(52505)
X(52504) = X(13754)-cross conjugate of X(44138)
X(52504) = X(i)-isoconjugate of X(j) for these (i,j): {47, 14910}, {563, 1300}, {571, 36053}, {30451, 36114}
X(52504) = X(i)-Dao conjugate of X(j) for these (i,j): {113, 571}, {924, 39021}, {1993, 34834}, {2088, 44808}, {2165, 15478}, {3003, 51393}, {6753, 16178}, {12095, 16310}, {14910, 34853}, {30451, 39005}
X(52504) = crossdifference of every pair of points on line {34952, 52435}
X(52504) = barycentric product X(i)*X(j) for these {i,j}: {68, 44138}, {403, 20563}, {1725, 20571}, {3580, 5392}, {6334, 30450}
X(52504) = barycentric quotient X(i)/X(j) for these {i,j}: {68, 5504}, {91, 36053}, {113, 51393}, {131, 12095}, {403, 24}, {686, 30451}, {847, 1300}, {925, 10420}, {1725, 47}, {1986, 52416}, {2165, 14910}, {2315, 563}, {3003, 571}, {3580, 1993}, {5392, 2986}, {5962, 38936}, {13754, 1147}, {16172, 34756}, {16237, 41679}, {18609, 18605}, {21731, 34952}, {30450, 687}, {34853, 15478}, {39170, 5961}, {44084, 44077}, {44138, 317}, {46085, 45780}, {46134, 18878}, {47236, 6753}, {52000, 52432}
X(52504) = {X(5392),X(39116)}-harmonic conjugate of X(2)

---

X(52505) = X(2)X(2986)∩X(20)X(254)

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

X(52505) lies on the cubics K499 and K1303 and these lines: on lines {2, 2986}, {20, 254}, {23, 3563}, {24, 34338}, {54, 5504}, {323, 43755}, {687, 46106}, {10419, 37941}, {11547, 41679}, {15421, 44427}, {26166, 40832}, {32708, 35296}

X(52505) = isotomic conjugate of X(52504)
X(52505) = X(i)-cross conjugate of X(j) for these (i,j): {12095, 9723}, {45780, 317}
X(52505) = X(i)-isoconjugate of X(j) for these (i,j): {91, 3003}, {403, 1820}, {847, 2315}, {1725, 2165}
X(52505) = X(i)-Dao conjugate of X(j) for these (i,j): {135, 47236}, {577, 13754}, {3003, 34116}
X(52505) = cevapoint of X(i) and X(j) for these (i,j): {6, 13557}, {571, 51393}, {3003, 10539}, {14910, 15478}
X(52505) = crosspoint of X(687) and X(18879)
X(52505) = trilinear pole of line {924, 1147}
X(52505) = barycentric product X(i)*X(j) for these {i,j}: {317, 5504}, {571, 40832}, {924, 18878}, {1300, 9723}, {1993, 2986}, {6563, 10420}, {7763, 14910}, {15421, 41679}, {36053, 44179}, {40423, 51393}
X(52505) = barycentric quotient X(i)/X(j) for these {i,j}: {24, 403}, {47, 1725}, {317, 44138}, {563, 2315}, {571, 3003}, {687, 30450}, {1147, 13754}, {1300, 847}, {1993, 3580}, {2986, 5392}, {5504, 68}, {5961, 39170}, {6753, 47236}, {10420, 925}, {12095, 131}, {14910, 2165}, {15478, 34853}, {18605, 18609}, {18878, 46134}, {30451, 686}, {34756, 16172}, {34952, 21731}, {36053, 91}, {38936, 5962}, {41679, 16237}, {44077, 44084}, {45780, 46085}, {51393, 113}, {52416, 1986}, {52432, 52000}

---

X(52506) = ADAMS-CIRCLE-INVERSE OF X(36)

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

X(52506) lies on these lines: {1, 3}, {4, 42311}, {7, 2808}, {279, 14520}, {938, 2898}, {2389, 5880}, {2809, 30329}, {3022, 39792}, {3781, 5542}, {5728, 42309}, {11036, 27253}, {14189, 23839}, {23840, 34381}, {33765, 36028}

X(52506) = Adams-circle-inverse of X(36)

---

X(52507) = ADAMS-CIRCLE-INVERSE OF X(390)

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

X(52507) lies on these lines: {1, 7}, {55, 27475}, {242, 1827}, {518, 10025}, {673, 41339}, {4919, 5853}, {5144, 30502}, {5274, 24600}, {20533, 25943}

X(52507) = reflection of X(14189) in X(1)
X(52507) = Adams-circle-inverse of X(390)

---

X(52508) = ADAMS-CIRCLE-INVERSE OF X(934)

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

X(52508) lies on these lines: {1, 103}, {7, 2808}, {101, 38859}, {279, 1362}, {295, 7233}, {517, 14189}, {2809, 7672}, {4566, 34018}, {6604, 41789}

X(52508) = Adams-circle-inverse of X(934)

---

X(52509) = ADAMS-CIRCLE-INVERSE OF X(1156)

Barycentrics    a*(a^6*b^2 - 5*a^5*b^3 + 10*a^4*b^4 - 10*a^3*b^5 + 5*a^2*b^6 - a*b^7 - a^6*b*c + 4*a^5*b^2*c - 5*a^4*b^3*c + 2*a^3*b^4*c - a^2*b^5*c + 2*a*b^6*c - b^7*c + a^6*c^2 + 4*a^5*b*c^2 - 9*a^4*b^2*c^2 + 8*a^3*b^3*c^2 - 7*a^2*b^4*c^2 + 3*b^6*c^2 - 5*a^5*c^3 - 5*a^4*b*c^3 + 8*a^3*b^2*c^3 + 6*a^2*b^3*c^3 - a*b^4*c^3 - 3*b^5*c^3 + 10*a^4*c^4 + 2*a^3*b*c^4 - 7*a^2*b^2*c^4 - a*b^3*c^4 + 2*b^4*c^4 - 10*a^3*c^5 - a^2*b*c^5 - 3*b^3*c^5 + 5*a^2*c^6 + 2*a*b*c^6 + 3*b^2*c^6 - a*c^7 - b*c^7) : :
X(52509) = X[7] + 2 X[3022], X[7672] - 4 X[11028], 5 X[11025] - 8 X[14760]

X(52509) lies on these lines: {1, 651}, {7, 3022}, {101, 2346}, {354, 658}, {2808, 11038}, {3681, 28070}, {7672, 11028}, {11025, 14760}, {14189, 15726}

X(52509) = Adams-circle-inverse of X(1156)

---

X(52510) = ADAMS-CIRCLE-INVERSE OF X(1319)

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

X(52510) lies on these lines: {1, 3}, {8, 30494}, {1002, 1323}, {1362, 21314}, {2808, 4312}, {2809, 7672}, {3247, 43915}, {29350, 43930}

X(52510) = midpoint of X(7672) and X(23839)
X(52510) = Adams-circle-inverse of X(1319)
X(52510) = {X(1),X(40)}-harmonic conjugate of X(30502)

---

X(52511) = ADAMS-CIRCLE-INVERSE OF X(1323)

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

X(52511) lies on these lines: {1, 7}, {9, 6706}, {85, 5223}, {144, 32086}, {165, 33765}, {1088, 10980}, {1111, 10398}, {1565, 38036}, {3062, 42311}, {3673, 30330}, {6063, 35613}, {9436, 38052}, {11495, 30502}, {17078, 38024}, {21454, 24600}, {31507, 43750}

X(52511) = Adams-circle-inverse of X(1323)
X(52511) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 279, 5542}, {7, 10004, 43180}, {7, 42309, 1}

---

X(52512) = X(2)X(1235)∩X(23)X(1289)

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

X(52512) lies on the cubic K1303 and these lines: {2, 1235}, {23, 1289}, {66, 11188}, {323, 44766}, {935, 37978}, {2353, 26179}, {3267, 23881}, {3448, 34237}, {5189, 13573}, {7391, 17407}, {30737, 44183}, {46786, 52505}, {46787, 52504}

X(52512) = X(39436)-anticomplementary conjugate of X(21288)
X(52512) = X(2393)-cross conjugate of X(1236)
X(52512) = X(i)-isoconjugate of X(j) for these (i,j): {1177, 2172}, {2373, 17453}, {20968, 37220}
X(52512) = X(i)-Dao conjugate of X(j) for these (i,j): {2485, 38971}, {5181, 10316}
X(52512) = barycentric product X(i)*X(j) for these {i,j}: {66, 1236}, {858, 18018}, {2393, 40421}, {18669, 46244}
X(52512) = barycentric quotient X(i)/X(j) for these {i,j}: {66, 1177}, {858, 22}, {1236, 315}, {1289, 10423}, {2393, 206}, {5523, 8743}, {14376, 18876}, {14580, 17409}, {14961, 10316}, {18018, 2373}, {18669, 2172}, {20884, 1760}, {21017, 4456}, {34138, 36823}, {39269, 11605}, {40421, 46140}, {41603, 41580}, {46244, 37220}, {47138, 2485}

---

X(52513) = X(2)X(112)∩X(23)X(10423)

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

X(52513) lies on the cubic K1303 and these lines: {2, 112}, {23, 10423}, {193, 46767}, {251, 46140}, {1176, 1177}, {4611, 34254}, {10422, 46783}, {46786, 52504}, {46787, 52505}

X(52513) = X(46140)-Ceva conjugate of X(1177)
X(52513) = X(i)-isoconjugate of X(j) for these (i,j): {66, 18669}, {858, 2156}, {2353, 20884}
X(52513) = X(i)-Dao conjugate of X(j) for these (i,j): {32, 2393}, {127, 47138}
X(52513) = trilinear pole of line {206, 8673}
X(52513) = barycentric product X(i)*X(j) for these {i,j}: {22, 2373}, {206, 46140}, {315, 1177}, {2172, 37220}, {17907, 18876}, {31636, 36823}
X(52513) = barycentric quotient X(i)/X(j) for these {i,j}: {22, 858}, {206, 2393}, {315, 1236}, {1177, 66}, {1760, 20884}, {2172, 18669}, {2373, 18018}, {2485, 47138}, {4456, 21017}, {8743, 5523}, {10316, 14961}, {10423, 1289}, {11605, 39269}, {17409, 14580}, {18876, 14376}, {36823, 34138}, {37220, 46244}, {41580, 41603}, {46140, 40421}

---

X(52514) = X(2)X(253)∩X(23)X(1301)

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

X(52514) lies on the cubic K1303 and these lines: {2, 253}, {23, 1301}, {64, 22467}, {323, 46639}, {394, 41894}, {1304, 5896}, {1993, 3343}, {3146, 41085}, {3266, 44326}, {3580, 46065}, {6820, 40839}, {11413, 31942}, {12086, 39268}, {14118, 14379}, {15066, 15394}, {23964, 44436}, {46788, 52505}, {46789, 52504}
on K1303

X(52514) = X(15262)-cross conjugate of X(2071)
X(52514) = X(610)-isoconjugate of X(11744)
X(52514) = X(i)-Dao conjugate of X(j) for these (i,j): {6587, 16177}, {11744, 14092}, {14385, 15291}, {14390, 40082}
X(52514) = barycentric product X(i)*X(j) for these {i,j}: {253, 2071}, {15262, 34403}, {15394, 34170}, {44326, 46425}
X(52514) = barycentric quotient X(i)/X(j) for these {i,j}: {64, 11744}, {253, 51967}, {1301, 22239}, {2071, 20}, {11589, 51346}, {14379, 40082}, {15262, 1249}, {34170, 14249}, {38937, 10152}, {46425, 6587}, {46639, 48373}

---

X(52515) = X(2)X(2501)∩X(23)X(3563)

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

X(52515) lies on the cubic K1303 and these lines: {2, 2501}, {23, 3563}, {94, 3266}, {98, 44768}, {111, 323}, {232, 32697}, {5968, 35364}, {18019, 52504}

X(52515) = X(230)-Dao conjugate of X(16188)
X(52515) = barycentric product X(i)*X(j) for these {i,j}: {2493, 8781}, {14221, 35364}, {14984, 35142}, {34175, 52091}
X(52515) = barycentric quotient X(i)/X(j) for these {i,j}: {2493, 230}, {3563, 40118}, {7468, 4226}, {14984, 3564}, {34157, 40083}, {34175, 14265}, {35364, 51480}, {38939, 34174}
X(52515) = {X(2987),X(10425)}-harmonic conjugate of X(323)

---

X(52516) = X(2)X(41678)∩X(23)X(22239)

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

X(52516) lies on the cubic K1303 and these lines: {2, 41678}, {23, 22239}, {107, 1105}, {323, 48373}, {1032, 6515}, {11744, 13203}, {34403, 36793}, {46788, 52504}, {46789, 52505}
on K1303

X(52516) = X(i)-isoconjugate of X(j) for these (i,j): {2071, 2155}, {15262, 19614}
X(52516) = X(i)-Dao conjugate of X(j) for these (i,j): {4, 15262}, {122, 46425}, {2071, 45245}
X(52516) = trilinear pole of line {2883, 8057}
X(52516) = barycentric product X(i)*X(j) for these {i,j}: {20, 51967}, {11744, 14615}
X(52516) = barycentric quotient X(i)/X(j) for these {i,j}: {20, 2071}, {1249, 15262}, {6587, 46425}, {10152, 38937}, {11744, 64}, {14249, 34170}, {22239, 1301}, {40082, 14379}, {48373, 46639}, {51346, 11589}, {51967, 253}

---

X(52517) = X(2)X(2170)∩X(8)X(2310)

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

X(52517) lies on the cubic K1303 and these lines: {2, 2170}, {8, 2310}, {85, 1086}, {92, 8735}, {189, 3210}, {239, 34234}, {312, 1146}, {335, 46805}, {536, 1121}, {664, 3752}, {1220, 5836}, {1311, 38460}, {2082, 17743}, {3872, 52133}, {3880, 14942}, {4518, 6735}, {4997, 34852}, {14936, 35957}, {31640, 42339}, {34371, 40862}, {35101, 40845}

X(52517) = midpoint of X(3210) and X(39351)
X(52517) = reflection of X(i) in X(j) for these {i,j}: {312, 1146}, {664, 3752}
X(52517) = isotomic conjugate of X(40862)
X(52517) = antitomic image of X(312)
X(52517) = isotomic conjugate of the anticomplement of X(40880)
X(52517) = X(40880)-cross conjugate of X(2)
X(52517) = X(i)-isoconjugate of X(j) for these (i,j): {6, 9364}, {31, 40862}, {604, 5205}, {1397, 40875}, {1409, 15150}
X(52517) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 40862}, {9, 9364}, {3161, 5205}
X(52517) = cevapoint of X(3752) and X(34371)
X(52517) = trilinear pole of line {522, 3452}
X(52517) = barycentric product X(i)*X(j) for these {i,j}: {75, 9365}, {3596, 9432}
X(52517) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 9364}, {2, 40862}, {8, 5205}, {29, 15150}, {312, 40875}, {9365, 1}, {9432, 56}

---

X(52518) = X(51)X(64)∩X(54)X(154)

Barycentrics    a^2*(a^4 - 6*a^2*b^2 + 5*b^4 - 2*a^2*c^2 - 6*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - 6*a^2*c^2 - 6*b^2*c^2 + 5*c^4) : :
X(52518) = 11 X[5072] - 3 X[11850]

See Antreas Hatzipolakis and Peter Moses, euclid 5564.

X(52518) lies on the Jerabek circumhyperbola.and these lines: {3, 5943}, {4, 11431}, {5, 42021}, {6, 5198}, {25, 14528}, {51, 64}, {54, 154}, {67, 7507}, {68, 546}, {69, 3091}, {72, 4853}, {73, 3304}, {74, 9781}, {185, 22334}, {235, 5486}, {265, 38791}, {381, 3519}, {382, 14861}, {389, 3426}, {576, 6391}, {578, 41424}, {1173, 1181}, {1176, 51730}, {1192, 43713}, {1350, 41435}, {1439, 7271}, {1498, 3527}, {1593, 3532}, {1597, 43719}, {1598, 17809}, {1656, 26861}, {3060, 33537}, {3090, 15873}, {3146, 15740}, {3431, 3518}, {3521, 5076}, {3523, 16935}, {3525, 14924}, {3529, 11745}, {3531, 11432}, {3567, 16835}, {3574, 13622}, {3592, 6413}, {3594, 6414}, {3627, 4846}, {3628, 52163}, {5072, 11850}, {5102, 15801}, {5504, 13861}, {5646, 15644}, {5895, 35512}, {6241, 13603}, {6415, 6419}, {6416, 6420}, {6912, 34259}, {7393, 12002}, {7530, 40441}, {7545, 16867}, {7687, 45733}, {7716, 43725}, {8567, 11270}, {8717, 37514}, {8795, 14249}, {9777, 15811}, {9827, 17834}, {9969, 34817}, {10095, 37475}, {10601, 33524}, {10605, 13452}, {10606, 44763}, {11381, 14490}, {11423, 34567}, {11426, 44731}, {11455, 46851}, {11456, 14491}, {11482, 38263}, {12163, 13451}, {12164, 13570}, {12233, 45011}, {12290, 46848}, {12679, 15320}, {13160, 38072}, {13472, 26863}, {13595, 45248}, {13623, 49136}, {14853, 17040}, {15077, 50689}, {15083, 44863}, {15316, 41597}, {17714, 37476}, {18368, 44958}, {20421, 35475}, {22236, 32585}, {22238, 32586}, {31366, 52448}, {31371, 50688}, {31861, 43689}, {34207, 52028}, {36749, 43704}, {36990, 43726}, {38005, 51745}, {43831, 43834}, {46865, 51739}

X(52518) = midpoint of X(4) and X(11431)
X(52518) = isogonal conjugate of X(3523)
X(52518) = isogonal conjugate of the anticomplement of X(3090)
X(52518) = isogonal conjugate of the complement of X(3832)
X(52518) = isotomic conjugate of the anticomplement of X(13341)
X(52518) = X(i)-cross conjugate of X(j) for these (i,j): {9777, 6}, {13341, 2}, {15811, 64}
X(52518) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3523}, {63, 40065}, {75, 17809}
X(52518) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 3523}, {206, 17809}, {3162, 40065}
X(52518) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 3523}, {25, 40065}, {32, 17809}, {34818, 11282}

---

X(52519) = X(76)X(3855)∩X(83)X(3529)

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

See Antreas Hatzipolakis and Peter Moses, euclid 5564.

X(52519) lies on the Kiepert circumhyperbola.and these lines: {2, 38136}, {76, 3855}, {83, 3529}, {382, 5395}, {546, 2996}, {3406, 41400}, {3528, 18841}, {3544, 18840}, {3545, 10302}, {5480, 7612}, {7710, 14492}, {8796, 52285}, {9748, 43537}, {9754, 10185}, {9755, 47586}, {10299, 43527}, {14269, 41895}, {14458, 14912}, {18845, 50688}

---

X(52520) = X(3)X(6)∩X(69)X(185)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^6*b^2 - a^4*b^4 - a^2*b^6 + b^8 + a^6*c^2 + 4*a^4*b^2*c^2 + 5*a^2*b^4*c^2 - 2*b^6*c^2 - a^4*c^4 + 5*a^2*b^2*c^4 + 2*b^4*c^4 - a^2*c^6 - 2*b^2*c^6 + c^8) : :
Barycentrics    Cos[A]*Sin[A]*((Cos[A] + Cos[B - C]*Cos[2*w])*Sin[A] - 2*Sin[2*w]) : :
X(52520) = 3 X[3] - X[9967], 5 X[3] - X[18438], 2 X[182] - 3 X[16836], X[1351] - 3 X[9730], 2 X[3098] + X[21851], X[3313] - 3 X[31884], 3 X[5050] - 2 X[44495], 3 X[5085] - 4 X[17704], 3 X[5085] - X[50649], 2 X[9967] - 3 X[11574], 5 X[9967] - 3 X[18438], X[9967] + 3 X[37511], X[11477] - 4 X[15012], 5 X[11574] - 2 X[18438], X[11574] + 2 X[37511], 2 X[13348] - 3 X[31884], 2 X[13348] + X[37473], 4 X[17704] - X[50649], X[18438] + 5 X[37511], 3 X[31884] + X[37473], 5 X[37481] - X[44456], 3 X[51] - X[51212], X[193] - 5 X[10574], 3 X[376] + X[6403], X[1205] - 3 X[15055], 2 X[5480] - 3 X[5943], 5 X[3522] - X[12220], 4 X[11695] - 3 X[14561], 5 X[3620] - X[12111], 3 X[3917] - X[41716], X[5562] - 3 X[10519], 3 X[5892] - 2 X[18583], X[5921] + 3 X[15072], X[6467] - 3 X[25406], 3 X[9971] + X[48872], 3 X[10516] - 2 X[44870], 3 X[21849] - 4 X[32191], X[10752] - 3 X[16223], 3 X[11188] + X[14927], 3 X[15030] - 5 X[40330], 3 X[16776] - X[51163], 3 X[20791] - X[40673], 3 X[29959] - X[36990], 3 X[32062] - 5 X[51537]

See Antreas Hatzipolakis and Peter Moses, euclid 5564.

X(52520) lies on these lines: {2, 12058}, {3, 6}, {4, 9822}, {20, 1843}, {30, 51994}, {51, 7386}, {69, 185}, {141, 2883}, {193, 10574}, {373, 16051}, {376, 6403}, {542, 17855}, {1038, 3056}, {1040, 1469}, {1112, 43957}, {1176, 13367}, {1181, 52016}, {1205, 15055}, {1216, 16197}, {1352, 6000}, {1368, 5480}, {1503, 14913}, {1660, 15577}, {1974, 17928}, {2393, 44241}, {2781, 3819}, {2807, 49511}, {3522, 12220}, {3524, 52000}, {3537, 5890}, {3538, 3567}, {3546, 11695}, {3547, 11793}, {3548, 38317}, {3564, 40647}, {3589, 16196}, {3620, 12111}, {3818, 13474}, {3917, 7494}, {5462, 21850}, {5562, 7400}, {5663, 32257}, {5892, 18583}, {5921, 15072}, {6467, 25406}, {6643, 10110}, {6688, 30771}, {6776, 8681}, {7667, 47328}, {7716, 39568}, {9306, 19149}, {9813, 21312}, {9969, 12362}, {9971, 48872}, {10257, 47581}, {10263, 21852}, {10516, 44870}, {10575, 18440}, {10691, 21849}, {10752, 16223}, {11188, 14927}, {11413, 19124}, {11444, 32605}, {11470, 26206}, {11585, 19130}, {11649, 33751}, {11806, 14984}, {12605, 29317}, {13198, 22352}, {13630, 34380}, {13754, 48876}, {14831, 50967}, {14915, 39884}, {15030, 40330}, {15760, 24206}, {16661, 41464}, {16776, 51163}, {16976, 47457}, {17710, 40929}, {17792, 34823}, {17811, 41580}, {18404, 48895}, {18531, 48901}, {19121, 22467}, {19125, 35602}, {20791, 40673}, {20806, 43652}, {21531, 44924}, {22240, 44073}, {22970, 26156}, {29012, 43130}, {29323, 43129}, {29957, 37613}, {29959, 36990}, {30739, 44084}, {32062, 51537}, {32237, 44260}, {34382, 48906}, {34778, 46373}, {34785, 46264}, {36790, 46832}, {37182, 51412}, {41714, 48898}, {44091, 44802}, {44240, 48892}, {44247, 44668}, {44249, 48885}

X(52520) = midpoint of X(i) and X(j) for these {i,j}: {3, 37511}, {20, 1843}, {52, 33878}, {69, 185}, {1350, 19161}, {3313, 37473}, {10575, 18440}, {14831, 50967}, {14913, 46850}, {15644, 21851}, {17710, 40929}, {41714, 48898}
X(52520) = reflection of X(i) in X(j) for these {i,j}: {4, 9822}, {6, 9729}, {3313, 13348}, {5907, 141}, {10263, 21852}, {11574, 3}, {13474, 3818}, {13598, 9969}, {15644, 3098}, {21850, 5462}, {31670, 10110}, {44479, 5092}
X(52520) = complement of X(12294)
X(52520) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 19131, 5092}, {3313, 31884, 13348}, {15644, 16836, 11430}, {31884, 37473, 3313}

---

X(52521) = X(191)X(1050)∩X(21061)X(21936)

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

See Angel Montesdeoca, euclid 5567.

X(52521) lies on these lines: {191,1050}, {21061,21936}

   LACHANCE-MOSES-BRIANCHON POINTS: X(52522) - X(52523)  

This preamble is based on notes from Michael Lachance and Peter Moses, November, 2022.

In the plane of a triangle ABC, suppose that

A' = a₁ : a₂ : a₃
B' = b₁ : b₂ : b₃
C' = c₁ : c₂ : c₃

are points on the circumcircle.
Let A_b = AB∩B'C' = (c²b₁ + a²b₃)c₁ : a²b₃c₂ : 0, and define B_c and C_a cyclically.

Let A_c = AC∩B'C' = (b²c₁ + a²c₂)b₁ : 0 : a²b₃c₂, and define B_a and C_b cyclically.

Then the lines B_aC_a, C_bA_b, A_cB_c concur in the Brianchon point of the hexagon {{H(A_b, B_a, B_c, C_b, C_b, A_c, A_b)}}. This point, denoted by LMB(A,B,C,A',B',C'), is given by the following barycentrics:

a₂a₃b₁c₁(b²b₃c₁ + c²b₁c₂ + a²b₃c₂) : :

The following conic is inscribed in ABC:

a⁴ b₃^2 c₂^2 (c² a₂ + b² a₃)² x² + (cyclic) - 2 b² c² a₂ a₃ b₁ c₁ (a² b₃ + c² b₁)(a² c₂ + b² c₁) y z - (cyclic) = 0

In particular, if A'B'C' is the circumtangential triangle, then LMB(A,B,C,A',B',C') = X(52522), and the inscribed conic is the Kiepert parabola. If A'B'C' is the circumnormal triangle, then LMB(A,B,C,A',B',C') = X(52523).

X(52522) = 1ST LACHANCE-MOSES-BRIANCHON POINT

Barycentrics    Csc[B - C]*Sin[(B - C)/3] : :

X(52522) lies on these lines: {2, 14}, {110, 13593}

X(52522) = X(661)-isoconjugate of X(14146)
X(52522) = X(i)-Dao conjugate of X(j) for these (i,j): {523, 16271}, {14146, X(52522) = 36830}
X(52522) = barycentric product X(99)*X(5637)
X(52522) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 14146}, {5637, 523}

---

X(52523) = 2ND LACHANCE-MOSES-BRIANCHON POINT

Barycentrics    Cos[(B - C)/3]*Sec[B - C] : :

X(52523) lies on this line: {15, 15412}

X(52523) = barycentric product X(95)*X(3272)
X(52523) = barycentric quotient X(3272)/X(5)

---

X(52524) = X(1)X(79)∩X(4)X(386)

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

See Ivan Pavlov, euclid 5571.

X(52524) lies on these lines: {1,79}, {4,386}, {34,46468}, {40,846}, {42,31673}, {43,18492}, {58,21669}, {165,48915}, {381,3216}, {546,5400}, {580,6912}, {581,3146}, {936,49734}, {946,1201}, {962,33100}, {990,15938}, {991,3529}, {1012,4252}, {1043,27792}, {1046,7701}, {1064,51118}, {1193,18483}, {1389,23838}, {1657,50317}, {1699,46704}, {1724,37234}, {1754,3560}, {1765,5165}, {1770,37558}, {1834,37447}, {2475,45924}, {2654,4292}, {3191,22010}, {3214,50796}, {3293,18480}, {3543,19767}, {3545,17749}, {3576,37425}, {3585,4551}, {3627,5396}, {3651,4653}, {3679,48887}, {3853,22392}, {4080,34772}, {4301,48941}, {4853,49716}, {4915,49718}, {5492,48883}, {5587,6048}, {5691,37529}, {5713,6925}, {5881,48937}, {5882,48923}, {5903,24430}, {6097,7280}, {6361,30116}, {6684,50420}, {6831,33810}, {6913,37537}, {6920,13329}, {7982,16496}, {7991,48882}, {8227,15973}, {8583,50169}, {9623,49728}, {9955,49997}, {10222,48907}, {10246,48926}, {10459,50422}, {11001,48855}, {11459,50599}, {11522,48931}, {13744,18180}, {14157,17104}, {14636,35242}, {15030,50597}, {15488,37331}, {15979,37554}, {16160,45926}, {16200,48909}, {16475,48922}, {17188,37157}, {18357,31855}, {19765,37411}, {19860,49735}, {19861,50171}, {21214,38021}, {29821,46975}, {31423,50418}, {34627,50575}, {34648,50587}, {37406,37693}, {41343,45977}, {48917,48939}

X(52524) = midpoint of X(962) and X(50419)
X(52524) = reflection of X(i) in X(j) for these {i,j}: {1, 48903}, {40, 9840}, {5881, 48937}, {7991, 48882}, {15971, 946}, {37425, 48894}, {48897, 1}, {48907, 10222}, {48915, 48930}, {48916, 5453}, {48917, 48939}, {48923, 5882}, {48941, 4301}
X(52524) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10543, 11553, 1}, {14636, 48919, 35242}, {37425, 48894, 3576}, {48915, 48930, 165}

---

X(52525) = X(3)X(74)∩X(4)X(569)

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

See Ivan Pavlov, euclid 5571.

X(52525) lies on the circumconic {{A,B,C,X(74),X(1179)}} and these lines: {2,6759}, {3,74}, {4,569}, {5,14157}, {20,184}, {22,1181}, {23,389}, {24,10574}, {25,15043}, {26,5890}, {49,550}, {52,12088}, {54,3521}, {97,14152}, {113,27866}, {140,10540}, {143,5899}, {154,17928}, {155,2979}, {182,3091}, {185,7488}, {186,40647}, {195,47748}, {206,6816}, {215,15338}, {217,10313}, {235,19128}, {323,15644}, {376,1147}, {378,12279}, {381,43651}, {382,15033}, {468,22750}, {546,13353}, {548,9705}, {549,18350}, {567,3627}, {568,17714}, {578,3146}, {631,10539}, {1060,9638}, {1092,3522}, {1105,38808}, {1176,6145}, {1199,5446}, {1204,10298}, {1437,6909}, {1495,9729}, {1498,3796}, {1595,14389}, {1598,5422}, {1618,23844}, {1660,15740}, {1754,38850}, {1885,34397}, {1993,11414}, {1994,12087}, {1995,15028}, {2070,13630}, {2071,13367}, {2477,15326}, {2883,52069}, {2937,6102}, {3043,16111}, {3044,38738}, {3045,24466}, {3047,16163}, {3060,7387}, {3090,13336}, {3153,43831}, {3167,37198}, {3200,42433}, {3201,42434}, {3205,36968}, {3206,36967}, {3292,13348}, {3518,9730}, {3520,10575}, {3523,9306}, {3529,13352}, {3534,9704}, {3542,18911}, {3543,11424}, {3547,11442}, {3549,11457}, {3567,7517}, {3580,18914}, {3581,43596}, {3628,13339}, {3819,45308}, {4846,35471}, {5050,5198}, {5056,43650}, {5092,15052}, {5133,16655}, {5204,9667}, {5217,9652}, {5447,44832}, {5449,7552}, {5462,34484}, {5562,6636}, {5622,20301}, {5640,10594}, {5651,10303}, {5654,47528}, {5907,22352}, {5921,19126}, {5946,18378}, {6030,7512}, {6143,44516}, {6146,50435}, {6240,41482}, {6247,13394}, {6560,9677}, {6639,23294}, {6644,26882}, {6689,18488}, {6696,15139}, {6776,19121}, {6781,9697}, {6815,11206}, {6823,14516}, {6888,37527}, {7395,32063}, {7403,16658}, {7464,14641}, {7485,17814}, {7492,46728}, {7493,18909}, {7502,34783}, {7505,26913}, {7506,15045}, {7509,15056}, {7514,15058}, {7525,18436}, {7526,12290}, {7527,11381}, {7529,11451}, {7530,9781}, {7544,31383}, {7545,15026}, {7566,36990}, {7712,11438}, {8703,43572}, {8717,17538}, {9591,31732}, {9626,31728}, {9703,15696}, {9818,11439}, {9934,19506}, {10024,25739}, {10095,15037}, {10110,34545}, {10113,11565}, {10117,41589}, {10125,15061}, {10176,43609}, {10201,26917}, {10263,15087}, {10264,34577}, {10282,22467}, {10304,43652}, {10545,13861}, {10605,38444}, {10610,14130}, {10625,23061}, {10627,50461}, {10721,12228}, {10722,39805}, {10723,39834}, {10733,13198}, {11134,42147}, {11137,42148}, {11188,15581}, {11220,47371}, {11402,39568}, {11412,18445}, {11413,19357}, {11422,12082}, {11423,36749}, {11430,12086}, {11563,43821}, {11793,15246}, {12006,13621}, {12038,14855}, {12083,12161}, {12103,37477}, {12107,45956}, {12112,35500}, {12162,35921}, {12163,44837}, {12203,40643}, {12226,15801}, {12241,47096}, {12307,34006}, {12834,52294}, {12897,43818}, {13363,18369}, {13364,15047}, {13366,13598}, {13383,26879}, {13403,52403}, {13406,14644}, {13470,18403}, {13619,43577}, {14118,15062}, {14133,37336}, {14249,32713}, {14530,35264}, {14627,37924}, {14677,43391}, {14940,15059}, {15012,32237}, {15038,36153}, {15078,17821}, {15531,44492}, {15704,37495}, {15760,34224}, {16197,37636}, {16252,41738}, {16618,41724}, {16621,37649}, {16836,35265}, {16881,37947}, {17506,43604}, {17712,51392}, {18128,43808}, {18356,44262}, {18374,43815}, {18442,47360}, {18504,43273}, {18859,43394}, {18925,37201}, {19131,39874}, {19467,44440}, {20190,43811}, {21308,32205}, {21663,38448}, {22112,46936}, {22240,39643}, {23329,40276}, {26887,26897}, {31099,43841}, {33524,37498}, {33923,40111}, {34354,46089}, {34603,45089}, {34782,38323}, {34864,45959}, {35268,38435}, {35475,39242}, {35491,52416}, {35502,37506}, {35707,37473}, {37440,37481}, {37444,46264}, {37480,50693}, {37645,52398}, {37943,43817}, {43575,43893}, {43597,45735}, {43816,46451}, {44866,52417}, {46817,50143}

X(52525) = midpoint of X(i) and X(j) for these {i,j}: {54, 8718}, {195, 47748}
X(52525) = reflection of X(i) in X(j) for these {i,j}: {7691, 7512}, {14130, 10610}, {15062, 14118}, {18488, 6689}
X(52525) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 11441, 11444}, {3, 11456, 12111}, {3, 13491, 74}, {3, 15068, 7999}, {3, 1614, 110}, {3, 32139, 11459}, {3, 32171, 15035}, {3, 399, 11591}, {3, 6241, 11440}, {3, 7999, 41462}, {3, 9707, 11449}, {4, 5012, 13434}, {20, 184, 34148}, {20, 9545, 13346}, {24, 10574, 15053}, {49, 550, 43574}, {52, 12088, 15107}, {140, 10540, 43598}, {155, 10323, 2979}, {156, 32210, 15132}, {182, 26883, 3091}, {184, 13346, 9545}, {184, 34148, 9706}, {186, 40647, 43601}, {323, 16661, 15644}, {382, 32046, 15033}, {1199, 37925, 5446}, {1495, 9729, 44802}, {1498, 3796, 7503}, {1498, 7503, 15305}, {1994, 12087, 45186}, {1995, 37514, 15028}, {3146, 11003, 578}, {3520, 10575, 13445}, {3520, 18475, 51033}, {5899, 43845, 143}, {5946, 18378, 38848}, {6030, 7691, 7512}, {6241, 11440, 15054}, {6636, 43605, 5562}, {6759, 10984, 2}, {7387, 7592, 3060}, {7509, 18451, 15056}, {7530, 36753, 9781}, {9545, 13346, 34148}, {9781, 36753, 15019}, {10574, 15053, 43603}, {10574, 26881, 24}, {10575, 18475, 3520}, {10594, 36752, 5640}, {12088, 15032, 52}, {12111, 15080, 3}, {13336, 46261, 3090}, {13367, 46850, 2071}, {13445, 51033, 3520}, {13861, 15024, 10545}, {15704, 37495, 43576}, {21659, 50009, 10733}, {32138, 34513, 3}, {32616, 32617, 15100}, {35268, 46730, 38435}, {38848, 43600, 5946}, {43831, 44829, 3153}

   Centers of inscribed conics: X(52526) - X(52548)  

This preamble is contributed by Peter Moses, December 9, 2022.

In the preamble just before X(52522), if A'B'C' is the cevian triangle of a point P = p : q : r, then the center of the inconic is given by

p (b^2c^2 p (q+r) + a (c^2q^2 + b^2r^2) : :

The inconic is here given the name LMB-cevian-inconic of P. See also the preamble just before X(52549).

X(52526) = X(5)X(51)∩X(39)X(233)

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

X(52526) lies on these lines: {2, 51255}, {3, 31867}, {5, 51}, {39, 233}, {128, 10277}, {549, 15848}, {1594, 14918}, {3613, 6243}, {3670, 8068}, {3767, 43843}, {6689, 8901}, {7668, 13353}, {10600, 15760}, {14788, 51481}, {25150, 30533}, {37486, 37988}

X(52526) = complement of X(51255)
X(52526) = nine-point-circle-inverse of X(11583)
X(52526) = complement of the isogonal conjugate of X(40449)
X(52526) = X(i)-complementary conjugate of X(j) for these (i,j): {2216, 140}, {40393, 21231}, {40449, 10}
X(52526) = X(50947)-Ceva conjugate of X(523)
X(52526) = center of LMB-cevian-inconic of X(5)

---

X(52527) = X(2)X(17114)∩X(9)X(39)

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

X(52527) lies on these lines: {2, 17114}, {3, 34807}, {9, 39}, {121, 124}, {519, 960}, {3452, 46827}, {3740, 35682}, {5044, 15310}, {18228, 30830}, {19582, 23638}, {25079, 29958}, {25135, 39589}, {27538, 50621}, {28265, 29418}, {28271, 29529}

X(52527) = complement of X(17114)
X(52527) = X(2985)-complementary conjugate of X(142)
X(52527) = center of LMB-cevian-inconic of X(8)

---

X(52528) = X(8)X(9)∩X(39)X(1212)

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

X(52528) lies on these lines: {8, 9}, {39, 1212}, {142, 7195}, {519, 30618}, {527, 30617}, {908, 26793}, {960, 2809}, {1329, 5199}, {2329, 49771}, {2348, 6737}, {2391, 3452}, {3061, 40869}, {3212, 17353}, {3911, 26690}, {5273, 24599}, {5325, 16833}, {5836, 17355}, {6666, 43037}, {6684, 24036}, {8074, 25066}, {16572, 24391}, {17338, 41781}, {27541, 30827}

X(52528) = midpoint of X(2082) and X(51972)
X(52528) = X(i)-complementary conjugate of X(j) for these (i,j): {1222, 17046}, {1261, 141}, {1476, 21258}, {3451, 11019}, {23617, 2886}, {32017, 17047}, {51476, 142}
X(52528) = X(21272)-Ceva conjugate of X(3900)
X(52528) = crossdifference of every pair of points on line {23865, 43924}
X(52528) = {X(9),X(41006)}-harmonic conjugate of X(5795)
X(52528) = center of LMB-cevian-inconic of X(9)

---

X(52529) = X(1)X(27041)∩X(10)X(321)

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

X(52529) lies on these lines: {1, 27041}, {10, 321}, {39, 1213}, {386, 26772}, {1125, 27042}, {1211, 4187}, {1329, 3454}, {3678, 21238}, {3813, 35104}, {10479, 27081}, {13740, 29398}, {14963, 20262}, {20966, 24068}, {24046, 27792}, {25688, 41329}

X(52529) = X(40394)-complementary conjugate of X(3739)
X(52529) = X(3909)-Ceva conjugate of X(522)
X(52529) = center of LMB-cevian-inconic of X(10)

---

X(52530) = X(6)X(1210)∩X(9)X(1714)

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

X(52530) lies on these lines: {6, 1210}, {9, 1714}, {31, 1855}, {39, 1212}, {393, 1712}, {580, 5179}, {774, 1886}, {1104, 1146}, {1453, 23058}, {1842, 2312}, {2083, 2385}, {3772, 46829}, {5230, 7079}, {8558, 23537}, {16968, 34851}, {21044, 40950}, {25090, 40937}, {37646, 46830}

X(52530) = complement of the isotomic conjugate of X(8748)
X(52530) = X(i)-complementary conjugate of X(j) for these (i,j): {25, 18642}, {28, 18639}, {607, 21530}, {1096, 17052}, {1172, 1368}, {1395, 18643}, {1474, 34822}, {1857, 21245}, {1896, 626}, {1973, 18641}, {1974, 18592}, {2194, 6389}, {2203, 17073}, {2204, 3}, {2207, 442}, {2212, 440}, {2299, 18589}, {2332, 34823}, {3063, 122}, {5317, 2886}, {6059, 1211}, {7337, 18635}, {8747, 17046}, {8748, 2887}, {18344, 127}, {24019, 17072}, {32713, 4885}, {36417, 2092}, {36420, 3742}, {41937, 34977}
X(52530) = X(44766)-Ceva conjugate of X(522)
X(52530) = crosspoint of X(2) and X(8748)
X(52530) = crosssum of X(6) and X(40152)
X(52530) = {X(16583),X(20311)}-harmonic conjugate of X(1212)
X(52530) = center of LMB-cevian-inconic of X(19)

---

X(52531) = X(3)X(24269)∩X(5)X(10)

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

X(52531) lies on these lines: {3, 24269}, {5, 10}, {39, 40937}, {3702, 3871}, {3735, 3767}, {4647, 18235}, {4999, 16579}, {5745, 15349}, {24880, 26066}, {24931, 25681}, {24937, 28628}

X(52531) = X(i)-complementary conjugate of X(j) for these (i,j): {1169, 17056}, {1220, 34829}, {1798, 18641}, {2185, 51571}, {2363, 442}, {4636, 50330}, {14534, 17052}
X(52531) = center of LMB-cevian-inconic of X(21)

---

X(52532) = X(2)X(2353)∩X(5)X(182)

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

X(52532) lies on these lines: {2, 2353}, {3, 24270}, {5, 182}, {39, 427}, {114, 11585}, {626, 11574}, {858, 7763}, {1368, 3788}, {1594, 9744}, {1899, 2450}, {3934, 14917}, {5133, 7803}, {6720, 44884}, {7776, 45921}, {7852, 37439}, {7874, 30739}, {8149, 21536}, {9737, 23335}, {14064, 41256}, {14376, 30794}, {20775, 46508}, {28405, 41761}, {28417, 33314}, {28674, 28684}, {28697, 39857}, {28723, 39842}

X(52532) = complement of X(2353)
X(52532) = complement of the isogonal conjugate of X(315)
X(52532) = complement of the isotomic conjugate of X(40073)
X(52532) = medial-isogonal conjugate of X(32)
X(52532) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 32}, {2, 16580}, {22, 37}, {75, 427}, {76, 16607}, {85, 18636}, {86, 40959}, {162, 47125}, {206, 16584}, {315, 10}, {561, 6697}, {662, 3265}, {811, 8673}, {897, 47298}, {1760, 2}, {2172, 39}, {2485, 16592}, {3112, 9969}, {3313, 16587}, {4123, 9}, {4150, 1211}, {4456, 16589}, {4463, 1213}, {4599, 23881}, {4611, 14838}, {7210, 1}, {8673, 16573}, {8743, 16583}, {16757, 1086}, {17076, 142}, {17453, 8265}, {17907, 226}, {20641, 141}, {20806, 1214}, {21178, 11}, {31636, 16609}, {33294, 8287}, {34254, 18589}, {40073, 2887}, {52448, 24005}
X(52532) = X(4576)-Ceva conjugate of X(23881)
X(52532) = X(6389)-Dao conjugate of X(16277)
X(52532) = crosspoint of X(i) and X(j) for these (i,j): {2, 40073}, {28405, 41009}
X(52532) = crosssum of X(6) and X(40146)
X(52532) = center of LMB-cevian-inconic of X(22)
X(52532) = barycentric product X(i)*X(j) for these {i,j}: {141, 41761}, {427, 28405}, {1632, 23881}, {3313, 41760}, {6389, 41375}, {40938, 41009}
X(52532) = barycentric quotient X(i)/X(j) for these {i,j}: {1899, 40404}, {3767, 16277}, {28405, 1799}, {40947, 46765}, {41761, 83}

---

X(52533) = X(2)X(10415)∩X(5)X(542)

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

X(52533) lies on these lines: {2, 10415}, {3, 16188}, {5, 542}, {626, 40553}, {3767, 6792}, {3906, 3934}, {5099, 14246}, {6719, 7886}, {7797, 11638}, {7828, 16042}, {11628, 52300}

X(52533) = complement of X(14357)
X(52533) = complement of the isogonal conjugate of X(14246)
X(52533) = center of LMB-cevian-inconic of X(23)
X(52533) = X(i)-complementary conjugate of X(j) for these (i,j): {23, 16597}, {111, 16581}, {671, 21234}, {897, 858}, {923, 187}, {10561, 8287}, {14246, 10}, {16568, 126}, {36142, 14417}, {52142, 37}

---

X(52534) = X(2)X(16391)∩X(5)X(578)

Barycentrics    (a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4)*(a^10*b^2 - 3*a^8*b^4 + 4*a^6*b^6 - 4*a^4*b^8 + 3*a^2*b^10 - b^12 + a^10*c^2 + 2*a^8*b^2*c^2 - 4*a^6*b^4*c^2 + 4*a^4*b^6*c^2 - 9*a^2*b^8*c^2 + 6*b^10*c^2 - 3*a^8*c^4 - 4*a^6*b^2*c^4 + 6*a^2*b^6*c^4 - 15*b^8*c^4 + 4*a^6*c^6 + 4*a^4*b^2*c^6 + 6*a^2*b^4*c^6 + 20*b^6*c^6 - 4*a^4*c^8 - 9*a^2*b^2*c^8 - 15*b^4*c^8 + 3*a^2*c^10 + 6*b^2*c^10 - c^12) : :

X(52534) lies on these lines: {2, 16391}, {3, 44207}, {5, 578}, {24, 27362}, {39, 233}, {133, 136}, {511, 16390}, {2450, 9833}, {7399, 52032}, {8901, 20303}, {11585, 12095}, {23333, 31804}, {27352, 37813}, {33967, 52010}

X(52534) = complement of X(16391)
X(52534) = center of LMB-cevian-inconic of X(24)
X(52534) = X(i)-complementary conjugate of X(j) for these (i,j): {158, 11585}, {393, 18588}, {1093, 34825}, {1096, 577}, {1748, 6389}, {6520, 343}, {6753, 16595}, {8745, 1214}, {11547, 18589}, {24000, 34844}, {36126, 924}, {44077, 828}

---

X(52535) = X(1)X(7242)∩X(2)X(27801)

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

X(52535) lies on these lines: {1, 7242}, {2, 27801}, {6, 22458}, {31, 40370}, {32, 2172}, {37, 10469}, {39, 712}, {216, 7561}, {561, 25506}, {766, 2085}, {942, 1015}, {1193, 40986}, {2275, 3670}, {3121, 20963}, {3954, 8620}, {6377, 16586}, {14620, 41268}, {17486, 26108}, {20861, 21815}, {21327, 28245}, {22420, 23639}, {27642, 33935}, {28397, 33934}

X(52535) = midpoint of X(2085) and X(40935)
X(52535) = complement of X(27801)
X(52535) = complement of the isotomic conjugate of X(1333)
X(52535) = isogonal conjugate of the isotomic conjugate of X(18179)
X(52535) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 21245}, {32, 3454}, {58, 626}, {81, 21235}, {86, 40379}, {110, 21262}, {163, 21260}, {310, 40380}, {560, 1211}, {667, 21253}, {849, 21240}, {1333, 2887}, {1397, 17052}, {1408, 17046}, {1412, 17047}, {1474, 21243}, {1501, 1213}, {1576, 3835}, {1917, 16589}, {1919, 125}, {1980, 8287}, {2194, 21244}, {2203, 20305}, {2206, 141}, {4556, 23301}, {7342, 17050}, {9233, 21838}, {9247, 21530}, {9426, 6627}, {9448, 38930}, {14574, 514}, {14575, 440}, {14599, 45162}, {16947, 2886}, {18892, 46842}, {18894, 35068}, {32739, 31946}, {41280, 17056}
X(52535) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 21245}, {11794, 513}
X(52535) = X(2)-Dao conjugate of X(21245)
X(52535) = crosspoint of X(2) and X(1333)
X(52535) = crosssum of X(6) and X(321)
X(52535) = crossdifference of every pair of points on line {21005, 48383}
X(52535) = center of LMB-cevian-inconic of X(31)
X(52535) = barycentric product X(i)*X(j) for these {i,j}: {6, 18179}, {28, 22420}, {31, 17211}, {58, 4137}, {81, 23639}, {849, 20655}, {1333, 21245}, {2206, 21421}
X(52535) = barycentric quotient X(i)/X(j) for these {i,j}: {4137, 313}, {17211, 561}, {18179, 76}, {21245, 27801}, {22420, 20336}, {23639, 321}

---

X(52536) = X(2)X(1239)∩X(3)X(34811)

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

X(52536) lies on these lines: {2, 1239}, {3, 34811}, {32, 682}, {39, 698}, {115, 52462}, {211, 18899}, {468, 1196}, {1084, 5007}, {1194, 7829}, {1501, 18796}, {2387, 14820}, {3117, 7772}, {3118, 32748}, {3124, 42442}, {3229, 7794}, {3456, 51948}, {6292, 45210}, {7745, 51906}, {7856, 9465}, {7909, 32526}, {11326, 41272}, {14990, 46305}, {40376, 40377}

X(52536) = midpoint of X(14820) and X(44164)
X(52536) = complement of the isotomic conjugate of X(46288)
X(52536) = X(i)-complementary conjugate of X(j) for these (i,j): {82, 40379}, {251, 21235}, {560, 21248}, {827, 21263}, {1501, 21249}, {1917, 6292}, {1924, 46654}, {3112, 40380}, {4630, 42327}, {9233, 16587}, {34072, 23301}, {46288, 2887}, {46289, 626}
X(52536) = X(41676)-Ceva conjugate of X(512)
X(52536) = crosspoint of X(2) and X(46288)
X(52536) = crosssum of X(6) and X(8024)
X(52536) = crossdifference of every pair of points on line {3267, 21006}
X(52536) = center of LMB-cevian-inconic of X(32)
X(52536) = {X(39),X(6375)}-harmonic conjugate of X(7889)

---

X(52537) = X(1)X(88)∩X(2)X(36909)

Barycentrics    a*(a^2 - b^2 + b*c - c^2)*(a^3*b + a^2*b^2 - a*b^3 - b^4 + a^3*c - 6*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(52537) lies on these lines: {1, 88}, {2, 36909}, {10, 51402}, {522, 1125}, {1212, 35113}, {1385, 38617}, {2801, 52005}, {3825, 51889}, {4251, 34544}, {4973, 46820}, {14584, 43048}, {25436, 35587}, {37535, 43809}

X(52537) = complement of X(51975)
X(52537) = complement of the isogonal conjugate of X(16944)
X(52537) = X(i)-complementary conjugate of X(j) for these (i,j): {36, 121}, {88, 21237}, {106, 3814}, {1417, 1737}, {7113, 16594}, {9456, 908}, {16944, 10}, {32719, 1639}, {40215, 141}, {41935, 37691}, {52434, 4370}, {52440, 1145}
X(52537) = X(i)-Ceva conjugate of X(j) for these (i,j): {39697, 758}, {51562, 3738}
X(52537) = X(21630)-Dao conjugate of X(36909)
X(52537) = crosspoint of X(679) and X(21907)
X(52537) = crosssum of X(i) and X(j) for these (i,j): {6, 40172}, {678, 17796}
X(52537) = crossdifference of every pair of points on line {1635, 19297}
X(52537) = center of LMB-cevian-inconic of X(36)
X(52537) = barycentric product X(3218)*X(21630)
X(52537) = barycentric quotient X(21630)/X(18359)

---

X(52538) = X(1)X(21024)∩X(2)X(39)

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(52538) lies on these lines: {1, 21024}, {2, 39}, {3, 34812}, {6, 10479}, {8, 20970}, {9, 46}, {10, 213}, {21, 24275}, {32, 964}, {37, 1089}, {115, 1281}, {187, 11115}, {594, 3293}, {740, 6155}, {966, 4274}, {1010, 5277}, {1107, 19863}, {1211, 17308}, {1215, 3954}, {1220, 5291}, {1224, 1929}, {1475, 31241}, {1500, 26115}, {1574, 26030}, {1909, 27274}, {1966, 17289}, {2049, 5275}, {2092, 2345}, {2240, 26251}, {2241, 24552}, {3053, 16394}, {3125, 49598}, {3589, 29433}, {3634, 20331}, {3691, 30970}, {3741, 20963}, {3743, 4037}, {3760, 4657}, {3831, 5750}, {3936, 17292}, {3985, 21816}, {3995, 7230}, {4065, 24044}, {4357, 4721}, {4754, 16887}, {5021, 37660}, {5109, 17398}, {5206, 16393}, {5254, 13728}, {5263, 41333}, {5305, 50318}, {5306, 50323}, {5364, 26061}, {6292, 17672}, {6537, 27064}, {6675, 24956}, {6703, 29456}, {7735, 37037}, {7748, 17676}, {10455, 15985}, {11648, 50321}, {14005, 37675}, {16583, 31993}, {16604, 19864}, {16818, 21264}, {16819, 39044}, {16928, 16993}, {16971, 50608}, {16975, 23447}, {16992, 25497}, {17056, 17284}, {17499, 30966}, {17904, 44143}, {19856, 35068}, {19870, 25614}, {19875, 25610}, {21021, 28594}, {21044, 27714}, {21071, 43223}, {21753, 31330}, {23897, 36478}, {24512, 50605}, {24774, 24778}, {24960, 46381}, {24989, 27371}, {25526, 40750}, {25683, 33034}, {26987, 40479}, {27026, 27076}, {27041, 27044}, {27050, 27071}, {31264, 33299}, {33297, 40721}, {44518, 50056}, {45882, 47837}, {51583, 51586}

X(52538) = complement of X(16705)
X(52538) = complement of the isotomic conjugate of X(14624)
X(52538) = X(i)-complementary conjugate of X(j) for these (i,j): {42, 51571}, {798, 15611}, {961, 17050}, {1220, 21240}, {1924, 39015}, {2298, 3741}, {8707, 42327}, {14624, 2887}, {32736, 4369}, {36098, 17066}, {36147, 512}
X(52538) = crosspoint of X(2) and X(14624)
X(52538) = crosssum of X(6) and X(40153)
X(52538) = crossdifference of every pair of points on line {669, 2605}
X(52538) = center of LMB-cevian-inconic of X(37)
X(52538) = barycentric product X(10)*X(32772)
X(52538) = barycentric quotient X(32772)/X(86)
X(52538) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 76, 25499}, {2, 26035, 39}, {2, 27040, 16589}, {1010, 26244, 5277}, {1213, 17369, 2245}, {1698, 46196, 1213}, {16589, 21838, 5283}

---

X(52539) = X(2)X(39)∩X(37)X(4075)

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

X(52539) lies on these lines: {2, 39}, {37, 4075}, {45, 2092}, {187, 17524}, {1213, 1574}, {1500, 2238}, {1573, 21024}, {2176, 20970}, {2276, 46196}, {2388, 21700}, {4043, 7230}, {5277, 35978}, {6155, 21820}, {16588, 38930}

X(52539) = X(i)-complementary conjugate of X(j) for these (i,j): {560, 41820}, {798, 46660}, {1126, 21240}, {1924, 35076}, {6539, 21235}, {6540, 21263}, {8701, 42327}, {28615, 3741}, {37212, 23301}
X(52539) = X(6540)-Ceva conjugate of X(512)
X(52539) = crosspoint of X(1171) and X(39964)
X(52539) = crosssum of X(i) and X(j) for these (i,j): {6, 8025}, {1213, 17147}
X(52539) = center of LMB-cevian-inconic of X(42)
X(52539) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5283, 27040, 16589}, {16589, 21838, 39}

---

X(52540) = X(3)X(6)∩X(97)X(12234)

Barycentrics    a^2*(a^12*b^2 - 6*a^10*b^4 + 15*a^8*b^6 - 20*a^6*b^8 + 15*a^4*b^10 - 6*a^2*b^12 + b^14 + a^12*c^2 - 10*a^10*b^2*c^2 + 24*a^8*b^4*c^2 - 17*a^6*b^6*c^2 - 8*a^4*b^8*c^2 + 15*a^2*b^10*c^2 - 5*b^12*c^2 - 6*a^10*c^4 + 24*a^8*b^2*c^4 - 16*a^6*b^4*c^4 - 7*a^4*b^6*c^4 - 4*a^2*b^8*c^4 + 9*b^10*c^4 + 15*a^8*c^6 - 17*a^6*b^2*c^6 - 7*a^4*b^4*c^6 - 10*a^2*b^6*c^6 - 5*b^8*c^6 - 20*a^6*c^8 - 8*a^4*b^2*c^8 - 4*a^2*b^4*c^8 - 5*b^6*c^8 + 15*a^4*c^10 + 15*a^2*b^2*c^10 + 9*b^4*c^10 - 6*a^2*c^12 - 5*b^2*c^12 + c^14) : :

X(52540) lies on these lines: {3, 6}, {97, 12234}, {140, 15345}, {186, 25042}, {288, 1157}, {1989, 14938}, {2963, 22268}, {3628, 8902}, {6150, 36153}, {10115, 34292}, {10126, 50476}, {12006, 18016}, {12242, 46025}, {13366, 25044}, {14143, 32165}, {15047, 15770}, {15325, 51879}, {16030, 40632}, {42400, 52295}

X(52540) = crosspoint of X(i) and X(j) for these (i,j): {54, 43666}, {1173, 3459}
X(52540) = crosssum of X(i) and X(j) for these (i,j): {5, 14627}, {6, 3078}, {140, 195}
X(52540) = center of LMB-cevian-inconic of X(54)

---

X(52541) = X(1)X(474)∩X(2)X(341)

Barycentrics    a*(a^2*b + 2*a*b^2 + b^3 + a^2*c - 4*a*b*c - b^2*c + 2*a*c^2 - b*c^2 + c^3) : :
X(52541) = 3 X[2] + X[17480], 2 X[24046] + X[45219], 3 X[4903] - 11 X[5550]

X(52541) lies on these lines: {1, 474}, {2, 341}, {3, 1279}, {6, 3333}, {8, 16610}, {10, 16602}, {11, 1883}, {12, 23675}, {21, 16700}, {25, 34}, {31, 32636}, {36, 2920}, {37, 39}, {38, 25917}, {40, 1616}, {42, 17609}, {43, 34791}, {45, 3646}, {46, 16483}, {55, 28011}, {57, 1191}, {58, 16726}, {65, 244}, {72, 3953}, {78, 17597}, {106, 24928}, {192, 26111}, {210, 27627}, {227, 1319}, {241, 5265}, {312, 26093}, {330, 25994}, {354, 1193}, {377, 17721}, {386, 5045}, {392, 3670}, {404, 3744}, {475, 3086}, {496, 23537}, {497, 28016}, {517, 24046}, {518, 978}, {519, 21896}, {551, 3931}, {595, 37582}, {910, 16502}, {936, 3242}, {939, 2191}, {942, 995}, {946, 1086}, {958, 5272}, {960, 982}, {988, 1001}, {1010, 16736}, {1015, 16583}, {1054, 37588}, {1100, 2271}, {1106, 1456}, {1108, 1148}, {1149, 3057}, {1155, 3915}, {1196, 9367}, {1210, 3756}, {1212, 2275}, {1265, 25879}, {1329, 5121}, {1385, 30117}, {1386, 37607}, {1401, 42450}, {1407, 2122}, {1418, 3361}, {1420, 1465}, {1457, 37566}, {1575, 4515}, {1647, 21935}, {1697, 16486}, {1698, 31197}, {1718, 5563}, {1722, 12513}, {1738, 3813}, {1834, 11019}, {1837, 28074}, {2277, 38053}, {2292, 4003}, {2390, 17114}, {2646, 28082}, {2886, 24178}, {2975, 7292}, {3011, 5433}, {3052, 15803}, {3207, 16780}, {3216, 3555}, {3230, 21872}, {3306, 5710}, {3315, 34772}, {3337, 5315}, {3338, 16466}, {3486, 28080}, {3554, 40943}, {3576, 8572}, {3579, 40091}, {3616, 3666}, {3622, 4850}, {3660, 10571}, {3669, 23764}, {3677, 8583}, {3681, 27625}, {3696, 50608}, {3702, 42051}, {3714, 3840}, {3743, 15808}, {3755, 21625}, {3782, 41012}, {3811, 4864}, {3816, 13161}, {3836, 49613}, {3868, 3999}, {3869, 28370}, {3878, 24167}, {3893, 4695}, {3914, 37722}, {3967, 25079}, {3998, 26747}, {4000, 14986}, {4385, 25492}, {4432, 8720}, {4642, 5919}, {4661, 27645}, {4662, 16569}, {4666, 19765}, {4673, 17490}, {4692, 19847}, {4883, 19767}, {4903, 5550}, {4966, 21857}, {4968, 26094}, {5022, 16970}, {5044, 49515}, {5069, 16845}, {5211, 7270}, {5230, 17728}, {5248, 37599}, {5250, 17595}, {5253, 7191}, {5266, 30148}, {5293, 49465}, {5302, 17123}, {5313, 50190}, {5432, 28027}, {5542, 21796}, {5794, 36574}, {5886, 24159}, {5955, 48803}, {6051, 25055}, {6129, 23757}, {6762, 23511}, {10179, 37598}, {10476, 21769}, {10527, 24789}, {10582, 37059}, {10586, 19785}, {11010, 16489}, {11110, 16696}, {11115, 16753}, {11230, 24160}, {12053, 24177}, {12672, 32486}, {15349, 21254}, {15829, 46943}, {15955, 51788}, {16605, 16825}, {16685, 21866}, {16823, 37596}, {17071, 24025}, {17102, 44675}, {17278, 19843}, {17588, 18601}, {17606, 28096}, {17622, 45269}, {17642, 22072}, {17724, 27385}, {17749, 34790}, {19791, 27166}, {19858, 31238}, {19861, 25939}, {19868, 31198}, {20312, 24005}, {20323, 49487}, {21000, 35242}, {21075, 51415}, {21147, 41426}, {21620, 37662}, {21868, 49560}, {21892, 49511}, {21949, 24390}, {23681, 50443}, {24162, 41003}, {24239, 25466}, {25681, 33144}, {25918, 26274}, {26029, 31233}, {26728, 37737}, {28358, 28628}, {28606, 46934}, {29820, 51715}, {30271, 37819}, {31165, 42040}, {31884, 35657}, {32937, 34860}, {32942, 50054}, {37554, 38315}, {37573, 42819}, {39584, 49486}, {49682, 51714}

X(52541) = midpoint of X(i) and X(j) for these {i,j}: {1, 24440}, {341, 17480}, {978, 3976}
X(52541) = complement of X(341)
X(52541) = complement of the isogonal conjugate of X(1106)
X(52541) = complement of the isotomic conjugate of X(269)
X(52541) = polar conjugate of the isogonal conjugate of X(23222)
X(52541) = X(i)-complementary conjugate of X(j) for these (i,j): {32, 6554}, {56, 1329}, {57, 21244}, {59, 3038}, {184, 42018}, {269, 2887}, {279, 626}, {479, 17047}, {603, 34823}, {604, 3452}, {608, 41883}, {658, 21262}, {667, 5514}, {738, 17046}, {934, 21260}, {1042, 3454}, {1088, 21235}, {1106, 10}, {1119, 21243}, {1262, 27076}, {1357, 46100}, {1395, 20262}, {1397, 9}, {1398, 5}, {1407, 141}, {1408, 960}, {1410, 21530}, {1412, 21246}, {1413, 20306}, {1415, 20317}, {1417, 5123}, {1427, 21245}, {1435, 20305}, {1461, 3835}, {1919, 13609}, {1980, 35508}, {2175, 5574}, {4616, 23301}, {4635, 21263}, {4637, 42327}, {6612, 21239}, {6614, 17072}, {7023, 2886}, {7053, 1368}, {7099, 18589}, {7143, 34829}, {7216, 21253}, {7250, 125}, {7337, 15849}, {7340, 3037}, {7342, 4999}, {7366, 142}, {8027, 34530}, {16947, 5745}, {23979, 4422}, {24027, 24003}, {41280, 16588}, {43924, 124}, {43932, 21252}, {52410, 2}
X(52541) = X(1897)-Ceva conjugate of X(513)
X(52541) = X(3942)-Dao conjugate of X(4025)
X(52541) = crosspoint of X(i) and X(j) for these (i,j): {2, 269}, {28, 39949}, {278, 19604}
X(52541) = crosssum of X(i) and X(j) for these (i,j): {1, 5687}, {6, 200}, {72, 3293}, {219, 3158}
X(52541) = crossdifference of every pair of points on line {4057, 4394}
X(52541) = center of LMB-cevian-inconic of X(56)
X(52541) = barycentric product X(i)*X(j) for these {i,j}: {1, 24177}, {57, 12053}, {264, 23222}
X(52541) = barycentric quotient X(i)/X(j) for these {i,j}: {12053, 312}, {23222, 3}, {24177, 75}
X(52541) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1739, 10914}, {1, 3752, 4646}, {1, 5573, 17054}, {1, 8056, 1706}, {1, 11512, 1376}, {1, 17063, 3812}, {1, 24174, 5836}, {1, 45047, 5438}, {2, 17480, 341}, {38, 28352, 25917}, {42, 46190, 17609}, {56, 614, 1104}, {72, 3953, 21342}, {244, 1201, 65}, {386, 5045, 49478}, {946, 24171, 1086}, {982, 21214, 960}, {1015, 16583, 40133}, {1125, 37592, 37}, {1149, 24443, 3057}, {1319, 32577, 15854}, {2275, 3290, 1212}, {3216, 3555, 4849}, {3216, 4694, 3555}, {3361, 7290, 4252}, {3622, 4850, 37548}, {3924, 32577, 1319}, {3953, 49997, 72}, {4968, 26094, 30818}, {5253, 7191, 37539}, {17053, 20227, 37}, {23536, 28018, 11}

---

X(52542) = X(1)X(142)∩X(2)X(51972)

Barycentrics    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(52542) lies on these lines: {1, 142}, {2, 51972}, {6, 10481}, {39, 1212}, {57, 279}, {141, 6743}, {218, 527}, {220, 3663}, {241, 40940}, {348, 24600}, {673, 3674}, {910, 10521}, {942, 2809}, {948, 2999}, {950, 51400}, {1086, 15730}, {1210, 20269}, {1323, 17366}, {2340, 26978}, {3160, 8732}, {3673, 40869}, {3772, 29571}, {3827, 37544}, {4298, 51150}, {4850, 24555}, {5228, 24177}, {5308, 41867}, {5543, 30275}, {5795, 24249}, {6666, 16601}, {6692, 17048}, {6706, 13405}, {6738, 21258}, {9317, 10106}, {9776, 17014}, {13411, 24774}, {13464, 17761}, {17353, 25242}, {17367, 27340}, {18483, 40690}, {19785, 25930}, {24635, 26723}, {26015, 27006}, {31540, 45500}, {31541, 45501}, {37642, 51302}, {41006, 41785}

X(52542) = complement of X(51972)
X(52542) = complement of the isotomic conjugate of X(10509)
X(52542) = X(i)-complementary conjugate of X(j) for these (i,j): {1170, 1329}, {1803, 34823}, {3063, 38973}, {7023, 45226}, {10509, 2887}, {21453, 21244}, {42311, 626}
X(52542) = X(35312)-Ceva conjugate of X(513)
X(52542) = crosspoint of X(2) and X(10509)
X(52542) = crosssum of X(6) and X(8012)
X(52542) = crossdifference of every pair of points on line {4105, 23865}
X(52542) = center of LMB-cevian-inconic of X(57)
X(52542) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 277, 142}, {3008, 37597, 5745}

---

X(52543) = X(3)X(6)∩X(4)X(46831)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^10*b^2 - 5*a^8*b^4 + 10*a^6*b^6 - 10*a^4*b^8 + 5*a^2*b^10 - b^12 + a^10*c^2 + 10*a^8*b^2*c^2 - 10*a^6*b^4*c^2 - 8*a^4*b^6*c^2 + a^2*b^8*c^2 + 6*b^10*c^2 - 5*a^8*c^4 - 10*a^6*b^2*c^4 + 36*a^4*b^4*c^4 - 6*a^2*b^6*c^4 - 15*b^8*c^4 + 10*a^6*c^6 - 8*a^4*b^2*c^6 - 6*a^2*b^4*c^6 + 20*b^6*c^6 - 10*a^4*c^8 + a^2*b^2*c^8 - 15*b^4*c^8 + 5*a^2*c^10 + 6*b^2*c^10 - c^12) : :

X(52543) lies on these lines: {3, 6}, {4, 46831}, {20, 6509}, {24, 12096}, {25, 14379}, {30, 33553}, {64, 28782}, {84, 35072}, {122, 235}, {393, 3346}, {1073, 15811}, {1204, 11589}, {2972, 8798}, {3146, 44436}, {3199, 20329}, {3522, 46832}, {6247, 15526}, {6353, 31377}, {6908, 18592}, {9914, 33581}, {13474, 14059}, {19614, 47432}, {26880, 37198}, {26883, 34147}, {31504, 51030}

X(52543) = X(i)-complementary conjugate of X(j) for these (i,j): {64, 20308}, {810, 13613}, {2155, 20207}, {14642, 36908}, {19614, 2883}
X(52543) = crosspoint of X(3) and X(3346)
X(52543) = crosssum of X(i) and X(j) for these (i,j): {4, 1498}, {6, 3079}, {20, 11441}
X(52543) = center of LMB-cevian-inconic of X(64)
X(52543) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 30258, 17704}, {3, 34815, 1192}, {2972, 11381, 8798}

---

X(52544) = X(1)X(442)∩X(3)X(6)

Barycentrics    a^2*(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(52544) lies on these lines: {1, 442}, {2, 51978}, {3, 6}, {34, 407}, {37, 3682}, {41, 39690}, {42, 65}, {43, 26066}, {46, 5312}, {55, 22076}, {56, 40952}, {60, 37311}, {78, 1211}, {81, 35979}, {224, 5256}, {377, 5712}, {387, 6889}, {408, 19366}, {440, 10393}, {810, 8578}, {936, 1213}, {940, 37229}, {960, 4199}, {997, 4205}, {1035, 8614}, {1064, 14110}, {1066, 49478}, {1104, 1193}, {1191, 40984}, {1212, 2238}, {1284, 42443}, {1451, 20966}, {1453, 3612}, {1490, 1901}, {1772, 13750}, {1780, 37286}, {1792, 15988}, {1837, 3142}, {1864, 37324}, {2194, 3145}, {2292, 44782}, {2299, 20832}, {3120, 11553}, {3136, 11375}, {3191, 4415}, {3192, 37194}, {3194, 7414}, {3216, 7483}, {3330, 12664}, {3416, 3811}, {3454, 22836}, {3755, 12609}, {3936, 7270}, {4204, 25917}, {4511, 5051}, {5217, 22080}, {5292, 50317}, {5496, 36250}, {5713, 6917}, {5717, 17647}, {5720, 50036}, {6831, 37662}, {7078, 11507}, {9895, 17443}, {10381, 12635}, {10822, 52139}, {11112, 49744}, {15621, 50583}, {16471, 37284}, {17528, 48842}, {18165, 28258}, {18178, 48909}, {18391, 37154}, {19260, 50619}, {19765, 37228}, {20018, 26057}, {20967, 42450}, {21746, 23383}, {22392, 37374}, {25080, 40661}, {30444, 45770}, {34259, 37175}, {34526, 38930}, {37230, 45924}, {37259, 44087}, {37438, 48847}, {37600, 40958}, {40602, 45236}, {44238, 48897}, {46877, 49728}, {48855, 50740}

X(52544) = complement of X(51978)
X(52544) = X(i)-complementary conjugate of X(j) for these (i,j): {2982, 21246}, {32651, 4369}, {36048, 512}
X(52544) = X(i)-Ceva conjugate of X(j) for these (i,j): {32651, 647}, {43344, 512}
X(52544) = X(1234)-Dao conjugate of X(25080)
X(52544) = crosspoint of X(1) and X(1175)
X(52544) = crosssum of X(i) and X(j) for these (i,j): {1, 442}, {6, 8021}, {21, 46441}
X(52544) = crossdifference of every pair of points on line {523, 1021}
X(52544) = center of LMB-cevian-inconic of X(65)
X(52544) = barycentric product X(i)*X(j) for these {i,j}: {1, 25080}, {57, 40661}
X(52544) = barycentric quotient X(i)/X(j) for these {i,j}: {25080, 75}, {40661, 312}
X(52544) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 10974, 2245}, {386, 581, 6}, {1193, 14547, 1104}, {3120, 31880, 11553}, {4276, 41329, 48882}, {5752, 19763, 4271}

---

X(52545) = X(2)X(40325)∩X(3)X(6)

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

X(52545) lies on these lines: {2, 40325}, {3, 6}, {394, 39653}, {439, 12220}, {682, 36212}, {1843, 32973}, {2386, 3788}, {3926, 8681}, {5140, 7887}, {5562, 39647}, {6337, 6467}, {6461, 10602}, {7386, 11059}, {7400, 9752}, {7789, 14913}, {9822, 14001}, {16196, 34841}, {16925, 51412}

X(52545) = complement of X(40325)
X(52545) = X(i)-complementary conjugate of X(j) for these (i,j): {40405, 226}, {40413, 24005}
X(52545) = crosssum of X(i) and X(j) for these (i,j): {6, 3080}, {25, 46444}, {1368, 19588}
X(52545) = center of LMB-cevian-inconic of X(69)
X(52545) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 50666, 22401}, {12360, 12361, 52520}

---

X(52546) = X(2)X(23097)∩X(3)X(6)

Barycentrics    a^2*(a^12*b^2 - 6*a^10*b^4 + 15*a^8*b^6 - 20*a^6*b^8 + 15*a^4*b^10 - 6*a^2*b^12 + b^14 + a^12*c^2 + 6*a^10*b^2*c^2 - 12*a^8*b^4*c^2 - 5*a^6*b^6*c^2 + 12*a^4*b^8*c^2 + 3*a^2*b^10*c^2 - 5*b^12*c^2 - 6*a^10*c^4 - 12*a^8*b^2*c^4 + 48*a^6*b^4*c^4 - 27*a^4*b^6*c^4 - 12*a^2*b^8*c^4 + 9*b^10*c^4 + 15*a^8*c^6 - 5*a^6*b^2*c^6 - 27*a^4*b^4*c^6 + 30*a^2*b^6*c^6 - 5*b^8*c^6 - 20*a^6*c^8 + 12*a^4*b^2*c^8 - 12*a^2*b^4*c^8 - 5*b^6*c^8 + 15*a^4*c^10 + 3*a^2*b^2*c^10 + 9*b^4*c^10 - 6*a^2*c^12 - 5*b^2*c^12 + c^14) : :
X(52546) = X[14264] - 3 X[16186], X[14264] + 3 X[46585]

X(52546) lies on these lines: {2, 23097}, {3, 6}, {4, 47215}, {30, 47055}, {512, 23108}, {1495, 14385}, {1989, 18317}, {2132, 40384}, {3470, 12112}, {6000, 14264}, {7422, 47213}, {9003, 20417}, {14670, 14915}, {16163, 34834}, {22104, 39234}, {34178, 35372}

X(52546) = midpoint of X(16186) and X(46585)
X(52546) = complement of X(23097)
X(52546) = crosspoint of X(74) and X(1138)
X(52546) = crosssum of X(i) and X(j) for these (i,j): {6, 3081}, {30, 399}
X(52546) = center of LMB-cevian-inconic of X(74)
X(52546) = {X(3),X(18114)}-harmonic conjugate of X(16836)

---

X(52547) = X(1)X(25506)∩X(10)X(39)

Barycentrics    a^3*b^4 + b^4*c^3 + a^3*c^4 + b^3*c^4 : :

X(52547) lies on these lines: {1, 25506}, {2, 40935}, {8, 26108}, {10, 39}, {75, 40367}, {561, 7242}, {700, 2085}, {1502, 33788}, {1917, 30891}, {1964, 25619}, {2162, 27803}, {3778, 18148}, {3831, 3844}, {4493, 28659}, {8619, 21412}, {10479, 32784}, {17792, 24256}, {20541, 37890}, {20542, 39080}, {25141, 29960}, {30964, 31330}

X(52547) = midpoint of X(2085) and X(27801)
X(52547) = complement of X(40935)
X(52547) = X(i)-complementary conjugate of X(j) for these (i,j): {983, 21838}, {7033, 1213}, {7034, 3454}, {7255, 1015}, {7307, 3739}, {9063, 788}, {17743, 16589}, {38810, 2}, {38813, 16584}, {38840, 32664}, {40415, 37}, {40834, 3912}, {40835, 4357}
X(52547) = crosssum of X(6) and X(8022)
X(52547) = center of LMB-cevian-inconic of X(75)
X(52547) = {X(21238),X(25122)}-harmonic conjugate of X(3831)

---

X(52548) = X(1)X(7267)∩X(39)X(712)

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

X(52548) lies on these lines: {1, 7267}, {39, 712}, {1125, 24384}, {1509, 2134}, {3636, 24185}, {3739, 3993}, {3743, 25136}, {4359, 29609}, {4657, 24931}, {4850, 24897}, {6155, 8682}, {10180, 36812}, {21879, 46913}, {24945, 28606}, {25468, 27577}, {25499, 46904}, {29590, 49753}

X(52548) = midpoint of X(6155) and X(16705)
X(52548) = X(i)-complementary conjugate of X(j) for these (i,j): {4556, 50497}, {40408, 3454}, {40439, 21245}
X(52548) = crosssum of X(6) and X(21820)
X(52548) = center of LMB-cevian-inconic of X(81)

   Perspectors of inscribed conics: X(52549) - X(52561)  

This preamble is contributed by Peter Moses, December 9, 2022.

Referring to the preamble just before X(52526), the perspector of the LMB-cevian-inconic is given by

b^2*c^2*p*(b^2*p + a^2*q)*(c^2*p + a^2*r) : :

See also the preamble just before X(52549).

X(52549) = X(6)X(145)∩X(57)X(312)

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

X(52549) lies on these lines: {6, 145}, {9, 30693}, {19, 7101}, {55, 1261}, {57, 312}, {284, 2325}, {673, 17787}, {893, 3693}, {1024, 4529}, {2291, 8706}, {2316, 2321}, {6169, 6559}, {7077, 40528}, {17355, 32942}

X(52549) = isotomic conjugate of the anticomplement of X(52528)
X(52549) = X(32017)-Ceva conjugate of X(1222)
X(52549) = X(i)-cross conjugate of X(j) for these (i,j): {9, 23617}, {1261, 1222}, {3900, 3699}, {19589, 14942}, {52528, 2}
X(52549) = X(i)-isoconjugate of X(j) for these (i,j): {6, 1122}, {7, 20228}, {56, 3752}, {57, 1201}, {109, 48334}, {190, 42336}, {221, 42549}, {222, 1828}, {269, 2347}, {278, 22344}, {604, 3663}, {651, 6363}, {1014, 21796}, {1042, 18163}, {1106, 3452}, {1397, 26563}, {1402, 18600}, {1403, 27499}, {1407, 3057}, {1408, 4415}, {1412, 4642}, {1417, 51415}, {1435, 22072}, {1461, 6615}, {1743, 46367}, {3052, 45205}, {3669, 23845}, {6736, 7366}, {7177, 40982}, {16945, 45204}, {20895, 52410}, {21362, 43924}, {23113, 43923}, {40151, 45219}
X(52549) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 3752}, {8, 45204}, {9, 1122}, {11, 48334}, {1201, 5452}, {2347, 6600}, {2968, 21120}, {3057, 24771}, {3161, 3663}, {3341, 42549}, {3452, 6552}, {4642, 40599}, {6363, 38991}, {6615, 35508}, {18600, 40605}, {24151, 45205}
X(52549) = cevapoint of X(i) and X(j) for these (i,j): {9, 346}, {728, 6555}
X(52549) = trilinear pole of line {663, 4163}
X(52549) = perspector of LMB-cevian-inconic of X(9)
X(52549) = crossdifference of every pair of points on line {6363, 42336}
X(52549) = barycentric product X(i)*X(j) for these {i,j}: {8, 1222}, {9, 32017}, {75, 1261}, {312, 23617}, {341, 1476}, {346, 40420}, {522, 8706}, {1265, 40446}, {3596, 51476}, {4076, 40451}, {4163, 6613}, {7035, 40528}
X(52549) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1122}, {8, 3663}, {9, 3752}, {33, 1828}, {41, 20228}, {55, 1201}, {200, 3057}, {210, 4642}, {212, 22344}, {220, 2347}, {282, 42549}, {312, 26563}, {333, 18600}, {341, 20895}, {346, 3452}, {644, 21362}, {646, 21580}, {650, 48334}, {663, 6363}, {667, 42336}, {1043, 17183}, {1222, 7}, {1260, 22072}, {1261, 1}, {1334, 21796}, {1476, 269}, {2287, 18163}, {2319, 27499}, {2321, 4415}, {2325, 51415}, {3158, 45219}, {3161, 45204}, {3239, 21120}, {3445, 46367}, {3451, 1407}, {3699, 21272}, {3900, 6615}, {3939, 23845}, {4082, 21031}, {4163, 42337}, {4515, 21809}, {4529, 28006}, {4546, 14284}, {4587, 23113}, {5423, 6736}, {6555, 12640}, {6558, 25268}, {6613, 4626}, {7071, 40982}, {8056, 45205}, {8706, 664}, {23617, 57}, {32017, 85}, {40420, 279}, {40446, 1119}, {40451, 1358}, {40528, 244}, {51476, 56}

---

X(52550) = X(6)X(7058)∩X(75)X(757)

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

X(52550) lies on these lines: {6, 7058}, {75, 757}, {86, 7018}, {261, 958}, {264, 4185}, {312, 2185}, {552, 6063}, {1240, 20566}, {2217, 40452}, {4623, 18036}, {7033, 7305}, {19623, 30710}

X(52550) = isotomic conjugate of the anticomplement of X(52531)
X(52550) = X(i)-cross conjugate of X(j) for these (i,j): {21, 14534}, {4560, 4631}, {52531, 2}
X(52550) = X(i)-isoconjugate of X(j) for these (i,j): {65, 3725}, {73, 44092}, {109, 42661}, {181, 1193}, {604, 21810}, {872, 24471}, {1042, 40966}, {1254, 20967}, {1397, 20653}, {1400, 2092}, {1402, 2292}, {1425, 40976}, {1918, 41003}, {2171, 2300}, {2197, 2354}, {2205, 45196}, {3674, 7109}
X(52550) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 42661}, {2092, 40582}, {2292, 40605}, {3161, 21810}, {3725, 40602}, {34021, 41003}, {40625, 50330}
X(52550) = cevapoint of X(i) and X(j) for these (i,j): {21, 7058}, {261, 314}, {14534, 40452}
X(52550) = perspector of LMB-cevian-inconic of X(21)
X(52550) = barycentric product X(i)*X(j) for these {i,j}: {21, 40827}, {261, 30710}, {314, 14534}, {1169, 40072}, {1220, 52379}, {1240, 2185}, {2298, 18021}, {2363, 28660}, {4581, 4631}, {7058, 31643}
X(52550) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 21810}, {21, 2092}, {60, 2300}, {261, 3666}, {270, 2354}, {274, 41003}, {284, 3725}, {310, 45196}, {312, 20653}, {314, 1211}, {333, 2292}, {650, 42661}, {873, 3674}, {1043, 21033}, {1098, 2269}, {1169, 1402}, {1172, 44092}, {1220, 2171}, {1240, 6358}, {1509, 24471}, {1791, 2197}, {1798, 1409}, {1812, 22076}, {2185, 1193}, {2287, 40966}, {2298, 181}, {2326, 40976}, {2363, 1400}, {3687, 6042}, {4560, 50330}, {7054, 20967}, {7058, 960}, {8707, 21859}, {14534, 65}, {18021, 20911}, {18155, 21124}, {19607, 42550}, {28660, 18697}, {30710, 12}, {31623, 429}, {31643, 6354}, {40072, 1228}, {40452, 40590}, {40827, 1441}, {46103, 1829}, {52379, 4357}

---

X(52551) = X(6)X(598)∩X(69)X(892)

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

X(52551) lies on these lines: {6, 598}, {69, 892}, {99, 14995}, {111, 7806}, {141, 17948}, {183, 51926}, {264, 2970}, {308, 9178}, {316, 10510}, {393, 23582}, {599, 39061}, {3818, 48983}, {7664, 10555}, {7777, 42008}, {7790, 10630}, {7833, 41404}, {8914, 15069}, {10415, 40826}, {10416, 22258}, {11185, 34574}, {16092, 37688}, {34810, 45774}, {51171, 52450}

X(52551) = isotomic conjugate of X(14357)
X(52551) = isotomic conjugate of the anticomplement of X(52533)
X(52551) = isotomic conjugate of the isogonal conjugate of X(14246)
X(52551) = X(i)-cross conjugate of X(j) for these (i,j): {23, 671}, {5099, 9979}, {7664, 316}, {52533, 2}
X(52551) = X(i)-isoconjugate of X(j) for these (i,j): {31, 14357}, {67, 922}, {187, 2157}, {896, 3455}
X(52551) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 14357}, {67, 39061}, {184, 39169}, {187, 39689}, {187, 40583}, {351, 5099}, {858, 47426}, {2492, 23992}, {3455, 15899}, {7664, 7813}
X(52551) = cevapoint of X(i) and X(j) for these (i,j): {316, 7664}, {671, 10416}, {5099, 9979}
X(52551) = trilinear pole of line {316, 9979}
X(52551) = perspector of LMB-cevian-inconic of X(23)
X(52551) = barycentric product X(i)*X(j) for these {i,j}: {23, 18023}, {76, 14246}, {111, 40074}, {316, 671}, {670, 10561}, {892, 9979}, {897, 20944}, {1502, 52142}, {4590, 10555}, {16568, 46277}, {17983, 37804}, {22151, 46111}, {30786, 37765}
X(52551) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14357}, {23, 187}, {111, 3455}, {316, 524}, {671, 67}, {892, 17708}, {897, 2157}, {2492, 351}, {5099, 23992}, {6593, 39689}, {7664, 2482}, {8744, 44102}, {9979, 690}, {10317, 23200}, {10416, 15900}, {10555, 115}, {10561, 512}, {14246, 6}, {16568, 896}, {17088, 7181}, {17983, 8791}, {18023, 18019}, {18311, 1649}, {18374, 14567}, {18818, 10511}, {20944, 14210}, {21094, 4062}, {21205, 4750}, {22151, 3292}, {30786, 34897}, {37765, 468}, {37804, 6390}, {40074, 3266}, {46111, 46105}, {52076, 52038}, {52142, 32}

---

X(52552) = X(6)X(2986)∩X(69)X(850)

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

X(52552) lies on the cubic K257 and these lines: {6, 2986}, {69, 850}, {76, 40423}, {99, 264}, {2407, 46106}, {3260, 10564}, {6148, 39375}, {6795, 10419}, {14615, 18878}, {14911, 34178}, {40427, 40879}, {40705, 44133}

X(52552) = isogonal conjugate of X(51821)
X(52552) = isotomic conjugate of X(14264)
X(52552) = isotomic conjugate of the anticomplement of X(52010)
X(52552) = isotomic conjugate of the isogonal conjugate of X(15454)
X(52552) = X(i)-cross conjugate of X(j) for these (i,j): {30, 2986}, {47084, 46789}, {52010, 2}
X(52552) = X(i)-isoconjugate of X(j) for these (i,j): {1, 51821}, {31, 14264}, {686, 36131}, {1725, 40352}, {2159, 3003}, {2315, 8749}, {21731, 36034}, {35200, 44084}
X(52552) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 14264}, {3, 51821}, {133, 44084}, {686, 39008}, {1986, 14918}, {3003, 3163}, {3258, 21731}, {11064, 34333}
X(52552) = cevapoint of X(i) and X(j) for these (i,j): {30, 36789}, {2986, 14911}, {3260, 6148}
X(52552) = trilinear pole of line {11064, 41079}
X(52552) = perspector of LMB-cevian-inconic of X(30)
X(52552) = barycentric product X(i)*X(j) for these {i,j}: {30, 40832}, {76, 15454}, {305, 51965}, {2986, 3260}, {6148, 40427}, {7799, 39375}, {18878, 41079}, {20573, 39371}, {36053, 46234}, {36789, 40423}, {39988, 46789}
X(52552) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14264}, {6, 51821}, {30, 3003}, {687, 1304}, {1300, 8749}, {1637, 21731}, {1990, 44084}, {2407, 15329}, {2986, 74}, {3260, 3580}, {5504, 18877}, {6148, 34834}, {9033, 686}, {10419, 40353}, {10420, 32640}, {11064, 13754}, {12028, 11079}, {14206, 1725}, {14910, 40352}, {14911, 36896}, {14920, 1986}, {15328, 2433}, {15421, 14380}, {15454, 6}, {16163, 47405}, {18878, 44769}, {32708, 32715}, {36053, 2159}, {36114, 36131}, {36789, 113}, {39371, 50}, {39375, 1989}, {39988, 46788}, {40423, 40384}, {40427, 5627}, {40832, 1494}, {46106, 403}, {46789, 39985}, {51456, 48451}, {51895, 51964}, {51965, 25}, {52498, 34150}

---

X(52553) = X(1)X(679)∩X(6)X(88)

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

X(52553) lies on the cubic K970 and these lines: {1, 679}, {2, 8046}, {6, 88}, {7, 528}, {9, 3257}, {37, 51908}, {45, 9326}, {69, 8047}, {75, 4555}, {86, 4833}, {106, 13396}, {142, 46790}, {269, 7045}, {320, 4089}, {894, 46795}, {940, 47056}, {1168, 39704}, {2226, 4850}, {2320, 34431}, {3218, 4585}, {3306, 4638}, {3664, 6549}, {4080, 17300}, {4389, 14260}, {4792, 49490}, {4945, 17313}, {4997, 17234}, {9460, 40587}, {14190, 42819}, {17352, 31227}, {17601, 41461}, {17953, 26806}, {24868, 33151}, {26749, 50116}, {30575, 37633}

X(52553) = isogonal conjugate of X(40172)
X(52553) = isotomic conjugate of X(51975)
X(52553) = isotomic conjugate of the anticomplement of X(52537)
X(52553) = isotomic conjugate of the isogonal conjugate of X(16944)
X(52553) = X(4555)-Ceva conjugate of X(4453)
X(52553) = X(i)-cross conjugate of X(j) for these (i,j): {36, 88}, {214, 3218}, {3738, 3257}, {16586, 17078}, {52537, 2}
X(52553) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40172}, {31, 51975}, {41, 14628}, {44, 2161}, {55, 14584}, {80, 902}, {519, 6187}, {678, 1168}, {759, 21805}, {1319, 52371}, {1404, 36910}, {1411, 3689}, {1639, 32675}, {1960, 51562}, {2222, 4895}, {2251, 18359}, {3943, 34079}, {8756, 52431}, {9459, 20566}, {14407, 47318}
X(52553) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 51975}, {3, 40172}, {44, 4370}, {44, 40584}, {80, 40594}, {88, 36909}, {223, 14584}, {519, 40612}, {908, 1145}, {1639, 35128}, {2161, 40595}, {3160, 14628}, {3689, 35204}, {3936, 4738}, {3943, 35069}, {3992, 51583}, {4895, 38984}, {9460, 18359}, {21805, 34586}
X(52553) = cevapoint of X(i) and X(j) for these (i,j): {1, 52031}, {214, 3218}, {1320, 40594}
X(52553) = crosssum of X(i) and X(j) for these (i,j): {678, 21805}, {1960, 42084}
X(52553) = trilinear pole of line {3218, 3960}
X(52553) = perspector of LMB-cevian-inconic of X(36)
X(52553) = barycentric product X(i)*X(j) for these {i,j}: {36, 20568}, {75, 40215}, {76, 16944}, {88, 320}, {106, 20924}, {679, 51583}, {903, 3218}, {1227, 2226}, {1320, 17078}, {1443, 4997}, {3257, 4453}, {3960, 4555}, {4089, 5376}, {4585, 6548}, {4622, 4707}, {4634, 21828}, {4867, 40833}, {9456, 40075}
X(52553) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 51975}, {6, 40172}, {7, 14628}, {36, 44}, {57, 14584}, {88, 80}, {106, 2161}, {214, 4370}, {320, 4358}, {654, 4895}, {758, 3943}, {903, 18359}, {1227, 36791}, {1320, 36910}, {1443, 3911}, {1797, 1807}, {1870, 8756}, {1983, 23344}, {2226, 1168}, {2245, 21805}, {2316, 52371}, {2323, 3689}, {3218, 519}, {3257, 51562}, {3738, 1639}, {3904, 4768}, {3936, 3992}, {3960, 900}, {4080, 15065}, {4453, 3762}, {4511, 2325}, {4555, 36804}, {4585, 17780}, {4622, 47318}, {4867, 4908}, {4880, 4727}, {4973, 4969}, {4997, 52409}, {7113, 902}, {9456, 6187}, {16586, 1145}, {16944, 6}, {17455, 678}, {17923, 38462}, {18593, 40663}, {20568, 20566}, {20924, 3264}, {21758, 1960}, {21828, 4730}, {22128, 5440}, {22379, 22086}, {27950, 4432}, {32851, 4723}, {36058, 52431}, {40215, 1}, {40594, 36909}, {51583, 4738}, {52407, 22356}, {52434, 2251}, {52440, 1404}

---

X(52554) = X(2)X(6664)∩X(6)X(1627)

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

X(52554) lies on these lines: {2, 6664}, {6, 1627}, {69, 31068}, {76, 3763}, {599, 31360}, {755, 7953}, {2353, 5013}, {2916, 14370}, {3094, 11360}, {3313, 31506}, {3589, 4576}, {3954, 18183}, {4074, 47355}, {5116, 38826}, {9484, 28667}, {13331, 52042}, {14970, 18092}, {20987, 41443}, {23642, 46154}, {46906, 51128}

X(52554) = X(i)-cross conjugate of X(j) for these (i,j): {39, 3108}, {512, 4576}, {38303, 46154}
X(52554) = X(i)-isoconjugate of X(j) for these (i,j): {82, 3589}, {83, 17469}, {428, 34055}, {3112, 5007}, {4593, 8664}, {4599, 7927}, {4628, 48152}, {17200, 18098}, {17457, 52395}, {18062, 18105}, {21802, 52394}, {39998, 46289}
X(52554) = X(i)-Dao conjugate of X(j) for these (i,j): {39, 39998}, {141, 3589}, {3124, 7927}, {5007, 34452}, {6665, 42554}, {11205, 52042}, {40938, 44142}
X(52554) = cevapoint of X(i) and X(j) for these (i,j): {39, 8041}, {141, 6665}
X(52554) = crosspoint of X(3108) and X(10159)
X(52554) = crosssum of X(3589) and X(5007)
X(52554) = trilinear pole of line {3005, 8711}
X(52554) = perspector of LMB-cevian-inconic of X(39)
X(52554) = barycentric product X(i)*X(j) for these {i,j}: {39, 10159}, {110, 31067}, {141, 3108}, {427, 41435}, {826, 7953}, {1634, 31065}, {3005, 35137}, {8041, 40425}, {31068, 46154}
X(52554) = barycentric quotient X(i)/X(j) for these {i,j}: {39, 3589}, {141, 39998}, {427, 44142}, {688, 8664}, {1401, 7198}, {1634, 10330}, {1843, 428}, {1964, 17469}, {2530, 48152}, {3005, 7927}, {3051, 5007}, {3108, 83}, {3688, 4030}, {3917, 7767}, {7794, 42554}, {7953, 4577}, {8041, 6292}, {10159, 308}, {16696, 16707}, {17187, 17200}, {20775, 22352}, {21123, 48101}, {21814, 21802}, {27369, 44091}, {31067, 850}, {35137, 689}, {41435, 1799}

---

X(52555) = X(2)X(594)∩X(6)X(595)

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

X(52555) lies on the conic {{A,B,C,X(2),X(6)}} and these lines: {1, 39798}, {2, 594}, {6, 595}, {9, 39974}, {37, 762}, {42, 4068}, {45, 941}, {111, 8701}, {213, 28625}, {220, 45129}, {967, 1796}, {1000, 52188}, {1018, 1100}, {1030, 2248}, {1169, 3285}, {1171, 17735}, {1218, 32018}, {1400, 21794}, {1449, 39984}, {2054, 4557}, {2276, 2350}, {2295, 16884}, {2321, 6538}, {2891, 24967}, {3228, 6540}, {3444, 18755}, {3458, 42624}, {3572, 50344}, {3943, 14624}, {3963, 17318}, {4033, 17319}, {4272, 52539}, {7109, 39961}, {9281, 20693}, {16672, 39983}, {17362, 20051}, {18082, 27804}, {18166, 37128}, {30940, 32014}, {39957, 49509}

X(52555) = isogonal conjugate of X(8025)
X(52555) = isogonal conjugate of the anticomplement of X(41809)
X(52555) = isogonal conjugate of the complement of X(43990)
X(52555) = isotomic conjugate of the anticomplement of X(52539)
X(52555) = isogonal conjugate of the isotomic conjugate of X(6539)
X(52555) = X(37212)-Ceva conjugate of X(50344)
X(52555) = X(i)-cross conjugate of X(j) for these (i,j): {42, 1126}, {649, 1018}, {4079, 4557}, {4272, 6}, {52539, 2}
X(52555) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8025}, {6, 16709}, {10, 30581}, {21, 553}, {27, 3916}, {28, 4001}, {37, 30593}, {58, 4359}, {63, 31900}, {79, 17190}, {81, 1125}, {86, 1100}, {99, 4979}, {110, 4978}, {274, 2308}, {286, 22054}, {333, 32636}, {593, 4647}, {643, 30724}, {662, 4977}, {757, 1213}, {763, 8013}, {799, 50512}, {849, 1230}, {873, 20970}, {1014, 3686}, {1019, 4427}, {1171, 6533}, {1269, 1333}, {1332, 46542}, {1412, 3702}, {1414, 4976}, {1434, 3683}, {1444, 1839}, {1509, 1962}, {2185, 3649}, {2355, 17206}, {3578, 52375}, {3647, 52393}, {4556, 30591}, {4565, 4985}, {4610, 4983}, {4622, 4984}, {4637, 4990}, {4697, 40432}, {4973, 24624}, {4974, 37128}, {6628, 21816}, {7192, 35342}, {7199, 35327}, {7203, 30729}, {18155, 36075}, {23201, 44129}, {35339, 48580}, {39949, 45222}
X(52555) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 8025}, {9, 16709}, {10, 4359}, {37, 1269}, {244, 4978}, {553, 40611}, {1084, 4977}, {1100, 40600}, {1125, 40586}, {1213, 40607}, {1230, 4075}, {3162, 31900}, {3702, 40599}, {4001, 40591}, {4976, 40608}, {4979, 38986}, {30593, 40589}, {38996, 50512}
X(52555) = cevapoint of X(i) and X(j) for these (i,j): {42, 1500}, {1126, 38836}
X(52555) = crosspoint of X(1126) and X(1255)
X(52555) = crosssum of X(i) and X(j) for these (i,j): {1100, 1125}, {1213, 4065}
X(52555) = crossdifference of every pair of points on line {4973, 4974}
X(52555) = perspector of LMB-cevian-inconic of X(42)
X(52555) = barycentric product X(i)*X(j) for these {i,j}: {6, 6539}, {10, 1126}, {37, 1255}, {42, 1268}, {58, 6538}, {65, 32635}, {101, 31010}, {213, 32018}, {226, 33635}, {321, 28615}, {512, 6540}, {523, 8701}, {594, 1171}, {661, 37212}, {756, 40438}, {1018, 47947}, {1100, 30582}, {1400, 4102}, {1500, 32014}, {1796, 1826}, {2308, 30594}, {3952, 50344}, {4024, 4629}, {4079, 4632}, {4557, 4608}, {4596, 4705}, {28625, 43260}
X(52555) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 16709}, {6, 8025}, {10, 1269}, {25, 31900}, {37, 4359}, {42, 1125}, {58, 30593}, {71, 4001}, {181, 3649}, {210, 3702}, {213, 1100}, {228, 3916}, {512, 4977}, {594, 1230}, {661, 4978}, {669, 50512}, {756, 4647}, {798, 4979}, {872, 1962}, {1126, 86}, {1171, 1509}, {1203, 45221}, {1255, 274}, {1268, 310}, {1333, 30581}, {1334, 3686}, {1400, 553}, {1402, 32636}, {1500, 1213}, {1796, 17206}, {1918, 2308}, {1962, 6533}, {2174, 17190}, {2200, 22054}, {2333, 1839}, {3690, 41014}, {3709, 4976}, {3724, 4973}, {3747, 4974}, {4041, 4985}, {4079, 4988}, {4102, 28660}, {4272, 41820}, {4524, 4990}, {4557, 4427}, {4596, 4623}, {4629, 4610}, {4705, 30591}, {6538, 313}, {6539, 76}, {6540, 670}, {7064, 4046}, {7109, 20970}, {7140, 44143}, {7180, 30724}, {8701, 99}, {14404, 30592}, {14407, 4984}, {20683, 4966}, {20964, 4697}, {21805, 4975}, {28615, 81}, {30582, 32018}, {31010, 3261}, {32018, 6385}, {32635, 314}, {33635, 333}, {37212, 799}, {38836, 6626}, {40438, 873}, {47947, 7199}, {50344, 7192}, {50487, 4983}, {50491, 4992}, {51377, 51409}
X(52555) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1126, 28615, 6}, {1126, 33635, 28615}

---

X(52556) = X(2)X(2226)∩X(6)X(644)

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

X(52556) lies on the cubic K220 and these lines: {2, 2226}, {6, 644}, {7, 16594}, {9, 649}, {44, 30731}, {89, 17350}, {346, 35092}, {902, 8028}, {1023, 1404}, {1252, 16946}, {2384, 6079}, {6600, 34446}, {26685, 51583}

X(52556) = isogonal conjugate of X(52206)
X(52556) = X(i)-cross conjugate of X(j) for these (i,j): {44, 40400}, {55, 36944}, {3251, 17780}, {4152, 519}
X(52556) = X(i)-isoconjugate of X(j) for these (i,j): {1, 52206}, {2, 17109}, {56, 52140}, {57, 45247}, {88, 1149}, {106, 16610}, {679, 20972}, {1022, 23832}, {1266, 9456}, {1797, 1878}, {2226, 17460}, {3257, 6085}, {4927, 32665}, {6336, 23205}, {20900, 41935}
X(52556) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 52140}, {3, 52206}, {214, 16610}, {519, 16594}, {1266, 4370}, {1647, 21129}, {4927, 35092}, {5452, 45247}, {17109, 32664}
X(52556) = cevapoint of X(i) and X(j) for these (i,j): {44, 4370}, {1639, 35092}
X(52556) = crosssum of X(1149) and X(20972)
X(52556) = trilinear pole of line {1960, 3689}
X(52556) = crossdifference of every pair of points on line {1149, 6085}
X(52556) = perspector of LMB-cevian-inconic of X(44)
X(52556) = barycentric product X(i)*X(j) for these {i,j}: {44, 36805}, {519, 1120}, {900, 6079}, {1811, 38462}, {4358, 40400}, {4723, 8686}, {17780, 23836}
X(52556) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 52206}, {9, 52140}, {31, 17109}, {44, 16610}, {55, 45247}, {519, 1266}, {678, 17460}, {900, 4927}, {902, 1149}, {1017, 20972}, {1120, 903}, {1960, 6085}, {3689, 3880}, {4370, 16594}, {4738, 20900}, {6079, 4555}, {6544, 21129}, {16704, 16711}, {21805, 4695}, {22371, 22082}, {23202, 23205}, {23344, 23832}, {23836, 6548}, {36805, 20568}, {40400, 88}, {42070, 5151}

---

X(52557) = X(6)X(1511)∩X(69)X(43755)

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

X(52557) lies on the cubic K260 and these lines: {6, 1511}, {69, 43755}, {571, 23357}, {577, 18334}, {1249, 32708}, {2623, 14533}, {2986, 7578}, {3431, 10419}, {5063, 14385}, {9723, 34834}, {10420, 32730}, {14591, 38936}, {15454, 32761}, {19627, 39371}, {48906, 51456}

X(52557) = isogonal conjugate of the polar conjugate of X(38936)
X(52557) = X(43755)-Ceva conjugate of X(526)
X(52557) = X(i)-cross conjugate of X(j) for these (i,j): {50, 14910}, {184, 14385}
X(52557) = X(i)-isoconjugate of X(j) for these (i,j): {92, 39170}, {94, 1725}, {1577, 41512}, {2166, 3580}, {2315, 18817}, {6334, 36129}
X(52557) = X(i)-Dao conjugate of X(j) for these (i,j): {3580, 11597}, {22391, 39170}
X(52557) = crosssum of X(3003) and X(11557)
X(52557) = perspector of LMB-cevian-inconic of X(50)
X(52557) = barycentric product X(i)*X(j) for these {i,j}: {3, 38936}, {50, 2986}, {74, 39371}, {110, 15470}, {186, 5504}, {323, 14910}, {526, 10420}, {1300, 22115}, {1511, 10419}, {2088, 18879}, {3043, 12028}, {6149, 36053}, {8552, 32708}, {14270, 18878}, {14385, 15454}, {14591, 15421}, {19627, 40832}, {34210, 39986}, {43755, 47230}, {51456, 52179}
X(52557) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 3580}, {184, 39170}, {186, 44138}, {1300, 18817}, {1576, 41512}, {2986, 20573}, {5504, 328}, {10420, 35139}, {14591, 16237}, {14910, 94}, {15470, 850}, {19627, 3003}, {32708, 46456}, {34397, 403}, {38936, 264}, {39371, 3260}

---

X(52558) = X(1)X(757)∩X(6)X(593)

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

X(52558) lies on the conic {{A,B,C,X(1),X(6)}} and these lines: {1, 757}, {6, 593}, {56, 7341}, {86, 4683}, {106, 6578}, {261, 32014}, {292, 4629}, {763, 39949}, {1014, 52372}, {1126, 1326}, {1220, 1268}, {1255, 1963}, {1333, 25426}, {4556, 17962}

X(52558) = isogonal conjugate of X(8013)
X(52558) = X(i)-cross conjugate of X(j) for these (i,j): {58, 1171}, {1203, 81}, {3733, 4556}
X(52558) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8013}, {2, 21816}, {10, 1962}, {12, 3683}, {37, 1213}, {42, 4647}, {65, 4046}, {72, 430}, {100, 6367}, {181, 3702}, {210, 3649}, {213, 1230}, {228, 44143}, {321, 20970}, {594, 1100}, {661, 4115}, {668, 8663}, {756, 1125}, {762, 8025}, {872, 1269}, {1018, 4988}, {1089, 2308}, {1500, 4359}, {1824, 41014}, {1826, 3958}, {1839, 3949}, {2171, 3686}, {2355, 3695}, {3916, 7140}, {3952, 4983}, {4024, 35342}, {4036, 35327}, {4103, 4979}, {4427, 4705}, {4557, 30591}, {4970, 7148}, {4976, 21859}, {4977, 40521}, {6057, 32636}, {7141, 23201}, {22080, 41013}
X(52558) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 8013}, {1213, 40589}, {1230, 6626}, {4046, 40602}, {4115, 36830}, {4647, 40592}, {6367, 8054}, {21816, 32664}
X(52558) = cevapoint of X(58) and X(593)
X(52558) = perspector of LMB-cevian-inconic of X(58)
X(52558) = barycentric product X(i)*X(j) for these {i,j}: {58, 32014}, {81, 40438}, {86, 1171}, {514, 6578}, {552, 33635}, {593, 1268}, {757, 1255}, {849, 32018}, {873, 28615}, {1019, 4596}, {1126, 1509}, {3733, 4632}, {4102, 7341}, {4556, 4608}, {4610, 50344}, {4629, 7192}
X(52558) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8013}, {27, 44143}, {31, 21816}, {58, 1213}, {60, 3686}, {81, 4647}, {86, 1230}, {110, 4115}, {284, 4046}, {593, 1125}, {649, 6367}, {757, 4359}, {763, 16709}, {849, 1100}, {1019, 30591}, {1126, 594}, {1171, 10}, {1255, 1089}, {1268, 28654}, {1333, 1962}, {1412, 3649}, {1437, 3958}, {1474, 430}, {1509, 1269}, {1790, 41014}, {1796, 3695}, {1919, 8663}, {2150, 3683}, {2185, 3702}, {2206, 20970}, {3733, 4988}, {4556, 4427}, {4596, 4033}, {4629, 3952}, {4632, 27808}, {4636, 30729}, {6578, 190}, {7341, 553}, {8701, 4103}, {17209, 51417}, {28615, 756}, {30576, 4975}, {30581, 6533}, {32014, 313}, {33635, 6057}, {40438, 321}, {47947, 4036}, {50344, 4024}

---

X(52559) = X(1)X(15394)∩X(4)X(253)

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

X(52559) lies on the Jerabek circumhyperbola and these lines: {3, 15394}, {4, 253}, {6, 1073}, {64, 2130}, {65, 44692}, {69, 16096}, {1439, 19611}, {1903, 2184}, {3532, 15400}, {6527, 13155}, {8798, 52518}, {8809, 8811}, {11589, 43713}, {14379, 14528}, {15077, 40996}, {15740, 34403}, {20208, 20265}, {32000, 47435}, {38292, 46639}, {41489, 43717}

X(52559) = isogonal conjugate of X(3079)
X(52559) = isotomic conjugate of the anticomplement of X(52543)
X(52559) = X(i)-cross conjugate of X(j) for these (i,j): {64, 1073}, {52543, 2}
X(52559) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3079}, {19, 36413}, {20, 204}, {25, 1097}, {33, 7338}, {34, 6060}, {154, 1895}, {610, 1249}, {1394, 44695}, {3172, 18750}, {3198, 44698}, {3213, 27382}, {7070, 44696}, {7156, 18623}
X(52559) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 3079}, {6, 36413}, {20, 3343}, {1073, 6616}, {1097, 6505}, {1249, 14092}, {6060, 11517}, {13155, 20207}, {14249, 40839}, {14390, 15905}, {17434, 39020}, {23608, 45248}
X(52559) = crosssum of X(25) and X(17833)
X(52559) = perspector of LMB-cevian-inconic of X(64)
X(52559) = barycentric product X(i)*X(j) for these {i,j}: {64, 34403}, {253, 1073}, {459, 15394}, {1301, 14638}, {2184, 19611}, {3926, 31942}, {14572, 15400}, {14642, 41530}
X(52559) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 36413}, {6, 3079}, {63, 1097}, {64, 1249}, {219, 6060}, {222, 7338}, {253, 15466}, {459, 14249}, {1073, 20}, {2155, 204}, {2184, 1895}, {2972, 39020}, {3343, 6616}, {8798, 42459}, {8809, 44697}, {14379, 15905}, {14642, 154}, {15394, 37669}, {15400, 40170}, {15905, 23608}, {19611, 18750}, {19614, 610}, {30457, 44695}, {31942, 393}, {33581, 3172}, {34403, 14615}, {41489, 6525}, {46831, 13155}
X(52559) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {253, 14362, 4}, {15394, 46351, 3}

---

X(52560) = X(3)X(7)∩X(6)X(278)

Barycentrics    (a + b - c)^2*(a - b + c)^2*(b + c)*(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(52560) lies on the Jerabek circumhyperbola and these lines: {3, 7}, {4, 42447}, {6, 278}, {64, 4295}, {69, 6063}, {71, 226}, {72, 1441}, {73, 3668}, {248, 32651}, {273, 1243}, {329, 40435}, {948, 40572}, {1020, 2260}, {1119, 1425}, {1446, 28787}, {2893, 18123}, {15556, 43708}, {15740, 17753}, {17139, 40412}, {41004, 43724}

X(52560) = isogonal conjugate of X(8021)
X(52560) = isotomic conjugate of X(51978)
X(52560) = isotomic conjugate of the anticomplement of X(52544)
X(52560) = X(i)-cross conjugate of X(j) for these (i,j): {65, 2982}, {513, 1020}, {52544, 2}
X(52560) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8021}, {9, 46882}, {21, 14547}, {29, 23207}, {31, 51978}, {60, 40967}, {219, 46884}, {283, 1859}, {284, 40937}, {662, 33525}, {942, 2328}, {1043, 40956}, {1098, 40952}, {1260, 46883}, {1841, 2327}, {2194, 6734}, {2260, 2287}, {2294, 7054}, {2322, 14597}, {2326, 18591}, {2332, 18607}, {3692, 46890}, {4183, 4303}, {7058, 40978}
X(52560) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 51978}, {3, 8021}, {478, 46882}, {942, 36908}, {1084, 33525}, {1214, 6734}, {14547, 40611}, {15267, 40952}, {40590, 40937}
X(52560) = cevapoint of X(i) and X(j) for these (i,j): {65, 6354}, {226, 15556}, {1425, 1427}
X(52560) = crosssum of X(14547) and X(23207)
X(52560) = trilinear pole of line {647, 7178}
X(52560) = perspector of LMB-cevian-inconic of X(65)
X(52560) = barycentric product X(i)*X(j) for these {i,j}: {307, 40573}, {850, 32651}, {943, 1446}, {1427, 40422}, {1439, 40447}, {1441, 2982}, {1577, 36048}, {3668, 40435}, {6354, 40412}, {6356, 40395}
X(52560) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 51978}, {6, 8021}, {34, 46884}, {56, 46882}, {65, 40937}, {226, 6734}, {512, 33525}, {943, 2287}, {1042, 2260}, {1175, 7054}, {1254, 2294}, {1398, 46890}, {1400, 14547}, {1409, 23207}, {1410, 14597}, {1425, 18591}, {1426, 1841}, {1427, 942}, {1435, 46883}, {1439, 18607}, {1794, 2327}, {1880, 1859}, {2171, 40967}, {2259, 2328}, {2982, 21}, {3668, 5249}, {4341, 46885}, {6354, 442}, {7216, 50354}, {14775, 17926}, {15439, 5546}, {16577, 31938}, {32651, 110}, {36048, 662}, {40412, 7058}, {40435, 1043}, {40573, 29}, {52373, 4303}

---

X(52561) = X(3)X(4158)∩X(4)X(346)

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

X(52561) lies on the Jerabek circumhyperbola and these lines: {3, 4158}, {4, 346}, {6, 1260}, {65, 3694}, {73, 51574}, {74, 29163}, {306, 28786}, {951, 2213}, {1257, 51223}, {1439, 3998}, {3610, 15232}, {3949, 43708}, {4574, 43693}, {8814, 21454}

X(52561) = X(71)-cross conjugate of X(2983)
X(52561) = X(i)-isoconjugate of X(j) for these (i,j): {27, 1104}, {28, 40940}, {81, 1842}, {162, 29162}, {950, 1396}, {1474, 17863}, {5317, 18650}, {18673, 36419}
X(52561) = X(i)-Dao conjugate of X(j) for these (i,j): {125, 29162}, {1842, 40586}, {17863, 51574}, {40591, 40940}
X(52561) = cevapoint of X(71) and X(52386)
X(52561) = perspector of LMB-cevian-inconic of X(71)
X(52561) = barycentric product X(i)*X(j) for these {i,j}: {72, 1257}, {306, 2983}, {525, 29163}, {951, 3710}, {3682, 40445}, {40414, 52386}, {40431, 52387}
X(52561) = barycentric quotient X(i)/X(j) for these {i,j}: {42, 1842}, {71, 40940}, {72, 17863}, {228, 1104}, {647, 29162}, {1257, 286}, {2318, 950}, {2983, 27}, {3682, 18650}, {3690, 1834}, {4574, 14543}, {29163, 648}, {52370, 2264}, {52386, 440}

---

X(52562) = X(8)X(76)∩X(72)X(28849)

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

X(52562) lies on these lines: {8, 76}, {72, 28849}, {200, 3208}, {220, 3022}, {480, 7064}, {766, 23630}, {1212, 2389}, {3057, 3059}, {3271, 42014}, {3340, 3779}, {3954, 23667}, {4531, 23638}, {4847, 20257}, {5369, 50626}, {17451, 21746}, {20683, 28043}, {21039, 42447}, {22070, 46177}

X(52562) = reflection of X(39789) in X(1212)
X(52562) = X(8)-Ceva conjugate of X(40997)
X(52562) = X(i)-isoconjugate of X(j) for these (i,j): {269, 40419}, {1088, 3449}
X(52562) = X(i)-Dao conjugate of X(j) for these (i,j): {7, 2886}, {6600, 40419}
X(52562) = crosspoint of X(8) and X(220)
X(52562) = crosssum of X(56) and X(279)
X(52562) = barycentric product X(i)*X(j) for these {i,j}: {8, 16588}, {55, 40997}, {200, 17451}, {210, 16699}, {220, 2886}, {314, 21819}, {346, 21746}, {1253, 20236}, {2287, 21804}, {2328, 21029}, {3239, 46177}, {3596, 9449}, {4515, 18165}, {7017, 22368}, {7046, 22070}
X(52562) = barycentric quotient X(i)/X(j) for these {i,j}: {220, 40419}, {9449, 56}, {14827, 3449}, {16588, 7}, {17451, 1088}, {21746, 279}, {21804, 1446}, {21819, 65}, {22070, 7056}, {22368, 222}, {40997, 6063}, {46177, 658}

---

X(52563) = X(7)X(145)∩X(76)X(85)

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

X(52563) lies on these lines: {1, 7195}, {2, 52528}, {7, 145}, {10, 3665}, {56, 1323}, {57, 279}, {65, 1358}, {76, 85}, {77, 34489}, {142, 17451}, {269, 21147}, {348, 3911}, {519, 30617}, {527, 30616}, {553, 16834}, {764, 3676}, {946, 1111}, {950, 17170}, {1122, 3057}, {1210, 1565}, {1401, 20358}, {1420, 3160}, {1432, 34018}, {1434, 4031}, {1441, 21432}, {1446, 14554}, {1449, 28079}, {2099, 24796}, {3119, 32446}, {3212, 4848}, {3247, 28015}, {3304, 10520}, {3339, 21314}, {3452, 24994}, {3662, 20535}, {3668, 3827}, {3671, 4059}, {3672, 37556}, {3673, 12053}, {3946, 28017}, {3982, 29605}, {4000, 18725}, {4035, 36503}, {4056, 31673}, {4298, 7223}, {4315, 7198}, {4862, 11531}, {5226, 30833}, {6549, 14260}, {6737, 47595}, {7176, 17089}, {7179, 17090}, {7243, 34284}, {8074, 20269}, {9311, 41777}, {9578, 31994}, {10436, 31598}, {12640, 21272}, {15950, 24805}, {16749, 17197}, {17205, 18180}, {18163, 18600}, {24213, 41007}, {24798, 40663}, {26125, 27271}, {28391, 43063}, {30036, 30097}, {41006, 51400}, {42309, 51190}, {45204, 45205}

X(52563) = reflection of X(2082) in X(52542)
X(52563) = isotomic conjugate of X(52549)
X(52563) = anticomplement of X(52528)
X(52563) = X(i)-Ceva conjugate of X(j) for these (i,j): {85, 26563}, {4569, 3676}
X(52563) = X(3752)-cross conjugate of X(3663)
X(52563) = X(i)-isoconjugate of X(j) for these (i,j): {6, 1261}, {9, 51476}, {31, 52549}, {41, 1222}, {55, 23617}, {200, 3451}, {220, 1476}, {1252, 40528}, {1253, 40420}, {1802, 40446}, {2175, 32017}, {3063, 8706}
X(52563) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 52549}, {9, 1261}, {9, 3452}, {223, 23617}, {346, 3752}, {478, 51476}, {661, 40528}, {728, 12640}, {1222, 3160}, {2170, 3900}, {3451, 6609}, {8706, 10001}, {17113, 40420}, {19589, 19593}, {32017, 40593}
X(52563) = cevapoint of X(i) and X(j) for these (i,j): {1122, 3752}, {3663, 45204}
X(52563) = crosspoint of X(85) and X(279)
X(52563) = crosssum of X(41) and X(220)
X(52563) = barycentric product X(i)*X(j) for these {i,j}: {7, 3663}, {57, 26563}, {75, 1122}, {85, 3752}, {226, 18600}, {269, 20895}, {279, 3452}, {479, 6736}, {658, 21120}, {1088, 3057}, {1201, 6063}, {1434, 4415}, {1446, 18163}, {1828, 7182}, {3668, 17183}, {3669, 21580}, {3676, 21272}, {4554, 48334}, {4569, 6615}, {4572, 6363}, {4626, 42337}, {6386, 42336}, {18743, 45205}, {20228, 20567}, {21362, 24002}, {27499, 30545}, {27818, 45204}, {40702, 42549}
X(52563) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1261}, {2, 52549}, {7, 1222}, {56, 51476}, {57, 23617}, {85, 32017}, {244, 40528}, {269, 1476}, {279, 40420}, {664, 8706}, {1119, 40446}, {1122, 1}, {1201, 55}, {1358, 40451}, {1407, 3451}, {1828, 33}, {2347, 220}, {3057, 200}, {3452, 346}, {3663, 8}, {3752, 9}, {4415, 2321}, {4626, 6613}, {4642, 210}, {6363, 663}, {6615, 3900}, {6736, 5423}, {12640, 6555}, {14284, 4546}, {17183, 1043}, {18163, 2287}, {18600, 333}, {20228, 41}, {20895, 341}, {21031, 4082}, {21120, 3239}, {21272, 3699}, {21362, 644}, {21580, 646}, {21796, 1334}, {21809, 4515}, {22072, 1260}, {22344, 212}, {23113, 4587}, {23845, 3939}, {25268, 6558}, {26563, 312}, {27499, 2319}, {28006, 4529}, {40982, 7071}, {42336, 667}, {42337, 4163}, {42549, 282}, {45204, 3161}, {45205, 8056}, {45219, 3158}, {46367, 3445}, {48334, 650}, {51415, 2325}
X(52563) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 9312, 10106}, {7, 27818, 43983}, {57, 47444, 279}, {65, 1358, 10481}, {85, 3674, 226}, {279, 14256, 738}, {1323, 10521, 56}, {3160, 3598, 1420}, {3212, 7185, 9436}, {3212, 9436, 4848}, {3665, 43037, 10}

---

X(52564) = X(1)X(596)∩X(56)X(58)

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

X(52564) lies on these lines: {1, 596}, {2, 52529}, {6, 16414}, {8, 26819}, {10, 16738}, {56, 58}, {76, 86}, {77, 44709}, {81, 386}, {333, 17749}, {387, 26818}, {501, 5009}, {595, 3286}, {741, 831}, {942, 16726}, {978, 18192}, {1125, 18169}, {1178, 52375}, {1193, 17191}, {1964, 3874}, {2275, 14964}, {2392, 14815}, {2901, 30939}, {3216, 16704}, {3670, 18601}, {3674, 17205}, {3736, 3913}, {3778, 31737}, {4257, 15080}, {4278, 38832}, {5563, 10457}, {10449, 17178}, {10458, 28619}, {10479, 27163}, {16700, 18180}, {17197, 23537}, {17524, 40091}, {18185, 33771}, {18646, 30117}, {19767, 26860}, {25526, 30116}, {30115, 42705}, {35650, 51651}, {35978, 37610}, {42025, 48855}, {45763, 46877}

X(52564) = anticomplement of X(52529)
X(52564) = X(i)-Ceva conjugate of X(j) for these (i,j): {86, 17184}, {664, 1019}
X(52564) = X(i)-isoconjugate of X(j) for these (i,j): {37, 40394}, {1089, 3453}
X(52564) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 3454}, {522, 18191}, {40394, 40589}
X(52564) = crosspoint of X(86) and X(593)
X(52564) = crosssum of X(42) and X(594)
X(52564) = barycentric product X(i)*X(j) for these {i,j}: {1, 18601}, {27, 11573}, {58, 17184}, {81, 3670}, {593, 3454}, {757, 4016}, {849, 20896}, {873, 40986}, {1019, 3909}, {1509, 20966}, {4556, 21121}, {23197, 44129}
X(52564) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 40394}, {3454, 28654}, {3670, 321}, {3909, 4033}, {4016, 1089}, {11573, 306}, {17184, 313}, {18601, 75}, {20966, 594}, {22073, 3695}, {23197, 71}, {40986, 756}
X(52564) = {X(16700),X(18180)}-harmonic conjugate of X(24046)

---

X(52565) = X(69)X(73)∩X(76)X(85)

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

X(52565) lies on these lines: {2, 52530}, {69, 73}, {76, 85}, {109, 2366}, {306, 1231}, {332, 951}, {333, 17095}, {345, 34403}, {664, 5930}, {1102, 3719}, {1400, 7131}, {1412, 7364}, {1446, 3936}, {1792, 6516}, {3682, 17216}, {3710, 50563}, {4306, 9436}, {4417, 40702}, {6394, 6517}, {6514, 17206}, {7066, 52385}, {7283, 16091}, {14963, 36907}, {18589, 20727}, {18695, 23581}, {20235, 51608}, {28795, 52358}

X(52565) = isotomic conjugate of X(8748)
X(52565) = anticomplement of X(52530)
X(52565) = isotomic conjugate of the isogonal conjugate of X(40152)
X(52565) = isotomic conjugate of the polar conjugate of X(307)
X(52565) = X(i)-Ceva conjugate of X(j) for these (i,j): {304, 1231}, {7055, 52385}
X(52565) = X(i)-cross conjugate of X(j) for these (i,j): {3998, 52396}, {40152, 307}
X(52565) = X(i)-isoconjugate of X(j) for these (i,j): {4, 2204}, {19, 2299}, {21, 2207}, {25, 1172}, {27, 2212}, {28, 607}, {29, 1973}, {31, 8748}, {32, 1896}, {33, 1474}, {34, 2332}, {41, 8747}, {55, 5317}, {81, 6059}, {107, 3063}, {112, 18344}, {210, 36420}, {270, 2333}, {281, 2203}, {284, 1096}, {314, 36417}, {393, 2194}, {608, 4183}, {650, 32713}, {663, 24019}, {1333, 1857}, {1395, 2322}, {1396, 7071}, {1402, 36421}, {1812, 52439}, {1824, 2189}, {1859, 40570}, {1946, 6529}, {1974, 31623}, {2193, 6524}, {2287, 7337}, {3064, 32676}, {3194, 7154}, {4516, 23964}, {8751, 37908}, {34856, 51858}
X(52565) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 8748}, {6, 2299}, {19, 226}, {29, 6337}, {33, 51574}, {37, 1857}, {72, 41320}, {107, 10001}, {223, 5317}, {284, 6503}, {333, 6338}, {393, 1214}, {525, 21044}, {607, 40591}, {663, 35071}, {1096, 40590}, {1172, 6505}, {1779, 52385}, {1896, 6376}, {2204, 36033}, {2207, 40611}, {2332, 11517}, {3063, 38985}, {3064, 15526}, {3160, 8747}, {6059, 40586}, {6524, 47345}, {6529, 39053}, {17926, 40626}, {18344, 34591}, {24005, 30223}, {36126, 39060}, {36421, 40605}
X(52565) = cevapoint of X(i) and X(j) for these (i,j): {3998, 52385}, {17216, 24018}
X(52565) = crosspoint of X(304) and X(3926)
X(52565) = crosssum of X(1973) and X(2207)
X(52565) = barycentric product X(i)*X(j) for these {i,j}: {7, 52396}, {10, 7055}, {63, 1231}, {69, 307}, {72, 7182}, {73, 305}, {75, 52385}, {76, 40152}, {77, 20336}, {85, 3998}, {222, 40071}, {225, 4176}, {226, 3926}, {304, 1214}, {306, 348}, {310, 7066}, {313, 1804}, {321, 7183}, {326, 1441}, {332, 6356}, {349, 394}, {520, 4572}, {561, 22341}, {653, 4143}, {664, 3265}, {850, 6517}, {1102, 40149}, {1264, 3668}, {1367, 4600}, {1409, 40364}, {1439, 3718}, {1446, 3719}, {1813, 3267}, {3682, 6063}, {3710, 7056}, {3990, 20567}, {4055, 41283}, {4552, 30805}, {4554, 24018}, {4561, 17094}, {4620, 15526}, {4998, 17216}, {6386, 51640}, {6516, 14208}, {7125, 27801}, {7138, 40072}, {17206, 26942}, {36793, 52378}
X(52565) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 8748}, {3, 2299}, {7, 8747}, {10, 1857}, {42, 6059}, {48, 2204}, {57, 5317}, {63, 1172}, {65, 1096}, {69, 29}, {71, 607}, {72, 33}, {73, 25}, {75, 1896}, {77, 28}, {78, 4183}, {109, 32713}, {201, 1824}, {219, 2332}, {222, 1474}, {225, 6524}, {226, 393}, {228, 2212}, {255, 2194}, {304, 31623}, {305, 44130}, {306, 281}, {307, 4}, {326, 21}, {333, 36421}, {345, 2322}, {348, 27}, {349, 2052}, {394, 284}, {520, 663}, {525, 3064}, {603, 2203}, {651, 24019}, {653, 6529}, {656, 18344}, {664, 107}, {822, 3063}, {1038, 4206}, {1042, 7337}, {1102, 1812}, {1214, 19}, {1231, 92}, {1259, 2328}, {1264, 1043}, {1367, 3120}, {1400, 2207}, {1409, 1973}, {1410, 1395}, {1412, 36420}, {1434, 36419}, {1439, 34}, {1441, 158}, {1444, 270}, {1447, 34856}, {1790, 2189}, {1804, 58}, {1812, 2326}, {1813, 112}, {1818, 37908}, {2197, 2333}, {2318, 7071}, {2632, 4516}, {3265, 522}, {3267, 46110}, {3668, 1118}, {3682, 55}, {3694, 7079}, {3710, 7046}, {3719, 2287}, {3926, 333}, {3964, 283}, {3990, 41}, {3998, 9}, {4055, 2175}, {4091, 7252}, {4101, 461}, {4131, 3737}, {4143, 6332}, {4158, 2318}, {4176, 332}, {4466, 8735}, {4554, 823}, {4561, 36797}, {4566, 36127}, {4572, 6528}, {4620, 23582}, {5930, 6525}, {6332, 17926}, {6356, 225}, {6507, 2193}, {6514, 7054}, {6516, 162}, {6517, 110}, {7013, 3194}, {7055, 86}, {7066, 42}, {7068, 52335}, {7125, 1333}, {7138, 1402}, {7177, 1396}, {7182, 286}, {7183, 81}, {7335, 2206}, {14208, 44426}, {15394, 52158}, {15526, 21044}, {17094, 7649}, {17206, 46103}, {17216, 11}, {17441, 40987}, {18026, 36126}, {18604, 2150}, {18607, 46884}, {20336, 318}, {20580, 14331}, {22057, 7083}, {22076, 40976}, {22341, 31}, {22342, 14975}, {23067, 8750}, {24018, 650}, {26942, 1826}, {30805, 4560}, {34401, 837}, {36059, 32676}, {37755, 1880}, {40071, 7017}, {40149, 6520}, {40152, 6}, {41077, 14400}, {41087, 7154}, {46404, 15352}, {47344, 40169}, {51368, 8755}, {51574, 41320}, {51640, 667}, {51664, 6591}, {52037, 7129}, {52373, 608}, {52378, 23964}, {52385, 1}, {52386, 1334}, {52387, 210}, {52389, 7008}, {52396, 8}
X(52565) = {X(3926),X(7055)}-harmonic conjugate of X(7183)

---

X(52566) = X(3)X(64)∩X(76)X(253)

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

X(52566) lies on the cubic K1065 and these lines: {3, 64}, {4, 9307}, {39, 45207}, {76, 253}, {459, 6622}, {1093, 6526}, {2207, 31942}, {2883, 41005}, {2972, 30443}, {13381, 37942}, {14642, 39643}, {31978, 46831}, {32602, 34815}, {43917, 44960}, {44226, 45195}

X(52566) = X(253)-Ceva conjugate of X(13567)
X(52566) = X(44079)-cross conjugate of X(800)
X(52566) = X(i)-isoconjugate of X(j) for these (i,j): {20, 775}, {610, 801}, {821, 35602}, {18750, 41890}
X(52566) = X(i)-Dao conjugate of X(j) for these (i,j): {20, 2883}, {801, 14092}, {3269, 20580}, {6509, 14615}, {14091, 15466}
X(52566) = crosspoint of X(64) and X(6526)
X(52566) = crosssum of X(i) and X(j) for these (i,j): {20, 35602}, {154, 36413}
X(52566) = barycentric product X(i)*X(j) for these {i,j}: {64, 13567}, {185, 459}, {235, 1073}, {253, 800}, {774, 2184}, {2155, 17858}, {6509, 6526}, {13157, 16035}, {14642, 44131}, {31942, 45200}, {34403, 44079}, {41005, 41489}
X(52566) = barycentric quotient X(i)/X(j) for these {i,j}: {64, 801}, {185, 37669}, {235, 15466}, {253, 40830}, {774, 18750}, {800, 20}, {1624, 36841}, {2155, 775}, {13567, 14615}, {33581, 41890}, {41489, 1105}, {44079, 1249}

---

X(52567) = X(1)X(3)∩X(181)X(201)

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

X(52567) lies on the cubic K1065 and these lines: {1, 3}, {2, 52531}, {10, 7235}, {12, 1089}, {38, 50626}, {76, 1441}, {181, 201}, {226, 3178}, {774, 21746}, {1254, 1500}, {1284, 3743}, {1409, 12089}, {1733, 15973}, {1756, 9959}, {1829, 20967}, {1834, 4516}, {2292, 22076}, {2650, 18210}, {3125, 40986}, {3485, 25650}, {3597, 40149}, {3704, 18697}, {3896, 50582}, {3926, 24282}, {3947, 4135}, {4092, 21686}, {4552, 11611}, {4646, 40965}, {5290, 49445}, {6042, 21810}, {7143, 37755}, {7179, 33935}, {7211, 10408}, {10544, 11031}, {11375, 25645}, {12723, 15852}, {16579, 28265}, {17447, 24391}, {17611, 48883}, {21807, 21935}, {24914, 25441}, {34920, 43074}

X(52567) = isotomic conjugate of X(52550)
X(52567) = anticomplement of X(52531)
X(52567) = X(1441)-Ceva conjugate of X(1211)
X(52567) = X(i)-isoconjugate of X(j) for these (i,j): {21, 2363}, {29, 1798}, {31, 52550}, {60, 1220}, {270, 1791}, {284, 14534}, {333, 1169}, {961, 1098}, {2150, 30710}, {2185, 2298}, {2359, 46103}, {4581, 4636}
X(52567) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 52550}, {21, 960}, {65, 40452}, {261, 1211}, {314, 3666}, {961, 15267}, {2092, 7058}, {2185, 52087}, {2363, 40611}, {3125, 4560}, {14534, 40590}
X(52567) = crosspoint of X(i) and X(j) for these (i,j): {12, 65}, {429, 2292}, {1441, 6354}
X(52567) = crosssum of X(i) and X(j) for these (i,j): {21, 60}, {1798, 2363}, {2194, 7054}
X(52567) = barycentric product X(i)*X(j) for these {i,j}: {7, 21810}, {12, 3666}, {37, 41003}, {42, 45196}, {57, 20653}, {65, 1211}, {181, 20911}, {201, 1848}, {226, 2292}, {349, 3725}, {429, 1214}, {594, 24471}, {756, 3674}, {960, 6354}, {1193, 6358}, {1228, 1402}, {1231, 44092}, {1254, 3687}, {1400, 18697}, {1427, 3704}, {1441, 2092}, {1446, 40966}, {1829, 26942}, {2171, 4357}, {2300, 34388}, {3004, 21859}, {3668, 21033}, {3965, 6046}, {4551, 21124}, {4552, 50330}, {4554, 42661}, {4605, 17420}, {22076, 40149}, {37755, 46878}
X(52567) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 52550}, {12, 30710}, {65, 14534}, {181, 2298}, {429, 31623}, {960, 7058}, {1193, 2185}, {1211, 314}, {1228, 40072}, {1400, 2363}, {1402, 1169}, {1409, 1798}, {1441, 40827}, {1829, 46103}, {2092, 21}, {2171, 1220}, {2197, 1791}, {2269, 1098}, {2292, 333}, {2300, 60}, {2354, 270}, {3666, 261}, {3674, 873}, {3725, 284}, {4357, 52379}, {6042, 3687}, {6354, 31643}, {6358, 1240}, {18697, 28660}, {20653, 312}, {20911, 18021}, {20967, 7054}, {21033, 1043}, {21124, 18155}, {21810, 8}, {21859, 8707}, {22076, 1812}, {24471, 1509}, {40590, 40452}, {40966, 2287}, {40976, 2326}, {41003, 274}, {42550, 19607}, {42661, 650}, {44092, 1172}, {45196, 310}, {50330, 4560}
X(52567) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 46, 37527}, {1834, 42440, 4516}, {3743, 35650, 1284}

---

X(52568) = X(2)X(1239)∩X(76)X(141)

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

X(52568) lies on these lines: {2, 1239}, {39, 35540}, {67, 670}, {76, 141}, {83, 9230}, {99, 8783}, {305, 858}, {626, 35524}, {706, 44164}, {1235, 51371}, {1928, 44170}, {3051, 3978}, {3266, 7764}, {7794, 8024}, {10159, 40016}, {16893, 40359}, {16988, 26192}, {18891, 33940}, {23105, 44173}, {28706, 32458}, {32820, 36157}

X(52568) = isotomic conjugate of X(46288)
X(52568) = anticomplement of X(52536)
X(52568) = isotomic conjugate of the isogonal conjugate of X(8024)
X(52568) = X(1502)-Ceva conjugate of X(8024)
X(52568) = X(i)-cross conjugate of X(j) for these (i,j): {16893, 141}, {39691, 23285}
X(52568) = X(i)-isoconjugate of X(j) for these (i,j): {31, 46288}, {32, 46289}, {82, 1501}, {83, 1917}, {251, 560}, {669, 34072}, {798, 4630}, {827, 1924}, {1973, 10547}, {1980, 4628}, {3112, 9233}, {4599, 9426}, {18902, 43763}, {23995, 51906}, {34055, 44162}
X(52568) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 46288}, {32, 39}, {76, 38834}, {141, 1501}, {251, 6374}, {339, 512}, {560, 40585}, {669, 15449}, {827, 9428}, {1974, 40938}, {3051, 6665}, {3124, 9426}, {4630, 31998}, {6337, 10547}, {6376, 46289}, {7664, 18374}, {9233, 34452}, {10335, 43977}, {18105, 36901}, {18314, 51906}, {18902, 36213}
X(52568) = cevapoint of X(23285) and X(39691)
X(52568) = crosspoint of X(1502) and X(40362)
X(52568) = crosssum of X(1501) and X(9233)
X(52568) = trilinear pole of line {2528, 23285}
X(52568) = crossdifference of every pair of points on line {9426, 18902}
X(52568) = barycentric product X(i)*X(j) for these {i,j}: {38, 1928}, {39, 40362}, {76, 8024}, {141, 1502}, {305, 1235}, {427, 40050}, {561, 1930}, {670, 23285}, {732, 44160}, {826, 4609}, {1401, 44159}, {1843, 40360}, {2528, 42371}, {3051, 40359}, {3665, 40363}, {3688, 41287}, {3703, 41283}, {3917, 44161}, {3933, 18022}, {4576, 44173}, {6386, 48084}, {7794, 40016}, {16703, 27801}, {16893, 44165}, {18024, 51371}, {18896, 35540}, {20883, 40364}, {39691, 44168}
X(52568) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 46288}, {38, 560}, {39, 1501}, {69, 10547}, {75, 46289}, {76, 251}, {99, 4630}, {141, 32}, {305, 1176}, {327, 42288}, {338, 51906}, {427, 1974}, {561, 82}, {670, 827}, {732, 14602}, {799, 34072}, {826, 669}, {850, 18105}, {1235, 25}, {1401, 41280}, {1502, 83}, {1634, 14574}, {1843, 44162}, {1928, 3112}, {1930, 31}, {1964, 1917}, {1978, 4628}, {2525, 3049}, {2528, 688}, {2530, 1980}, {3005, 9426}, {3051, 9233}, {3118, 8023}, {3313, 20968}, {3314, 43977}, {3665, 1397}, {3688, 9448}, {3703, 2175}, {3917, 14575}, {3933, 184}, {3954, 2205}, {4175, 20775}, {4568, 32739}, {4576, 1576}, {4602, 4599}, {4609, 4577}, {6374, 38834}, {6385, 52376}, {7794, 3051}, {7813, 14567}, {8024, 6}, {8039, 16890}, {8041, 41331}, {8061, 1924}, {8623, 18902}, {14125, 8041}, {14615, 51508}, {14994, 34396}, {15523, 1918}, {16030, 14573}, {16703, 1333}, {16747, 2203}, {16887, 2206}, {16892, 1919}, {16893, 8265}, {18022, 32085}, {18896, 733}, {20021, 14601}, {20775, 40373}, {20883, 1973}, {23208, 40372}, {23285, 512}, {23962, 34294}, {27376, 36417}, {27801, 18098}, {28677, 10329}, {31125, 32740}, {33299, 9447}, {35540, 1691}, {39691, 1084}, {40016, 52395}, {40035, 14885}, {40043, 46287}, {40050, 1799}, {40359, 40016}, {40362, 308}, {40364, 34055}, {40421, 16277}, {40495, 18108}, {42551, 51951}, {42554, 5007}, {44160, 14970}, {44161, 46104}, {44163, 3115}, {46154, 19626}, {48084, 667}, {51360, 9407}, {51371, 237}
X(52568) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 6374, 3096}, {76, 47846, 7790}, {1502, 40050, 76}

---

X(52569) = X(8)X(79)∩X(76)X(20565)

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

X(52569) lies on the cubic K1066 and these lines: {8, 79}, {76, 20565}, {261, 16709}, {3615, 17586}, {3874, 48877}, {4973, 6533}, {5625, 7100}, {6554, 7110}, {6701, 14206}, {42285, 52382}, {43531, 43682}

X(52569) = X(20565)-Ceva conjugate of X(4359)
X(52569) = X(i)-isoconjugate of X(j) for these (i,j): {35, 1126}, {1255, 2174}, {1399, 32635}, {2003, 33635}, {2605, 8701}, {3219, 28615}, {40214, 52555}
X(52569) = X(i)-Dao conjugate of X(j) for these (i,j): {35, 3647}, {79, 33670}, {1125, 3678}, {1213, 3219}, {14838, 35076}
X(52569) = barycentric product X(i)*X(j) for these {i,j}: {79, 4359}, {94, 4973}, {553, 52344}, {1100, 20565}, {1125, 30690}, {1230, 52375}, {1269, 2160}, {3702, 52374}, {4647, 52393}, {4977, 15455}, {4978, 6742}, {4985, 38340}, {6757, 8025}, {8818, 16709}
X(52569) = barycentric quotient X(i)/X(j) for these {i,j}: {79, 1255}, {553, 1442}, {1100, 35}, {1125, 3219}, {1213, 3678}, {1269, 33939}, {1839, 6198}, {2160, 1126}, {2308, 2174}, {3649, 16577}, {3683, 52405}, {3686, 4420}, {3702, 42033}, {4359, 319}, {4647, 3969}, {4973, 323}, {4976, 35057}, {4977, 14838}, {4978, 4467}, {4979, 2605}, {6186, 28615}, {6533, 3578}, {6742, 37212}, {6757, 6539}, {7073, 33635}, {7100, 1796}, {7110, 32635}, {13486, 4629}, {15455, 6540}, {16709, 34016}, {20565, 32018}, {22054, 52408}, {30591, 7265}, {30690, 1268}, {32636, 2003}, {52344, 4102}, {52375, 1171}, {52393, 40438}
X(52569) = {X(79),X(30690)}-harmonic conjugate of X(6757)

---

X(52570) = X(6)X(76)∩X(251)X(6179)

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

X(52570) lies on these lines: {6, 76}, {251, 6179}, {274, 27005}, {305, 39668}, {315, 17500}, {670, 10159}, {1078, 51862}, {1799, 1995}, {2896, 26192}, {3096, 16890}, {3934, 51906}, {3972, 38834}, {5007, 39998}, {6292, 26198}, {6656, 34294}, {6704, 35540}, {7768, 20022}, {7780, 26235}, {7814, 45093}, {10130, 16055}, {14247, 41296}, {14970, 31622}, {18140, 27067}, {39999, 40003}, {40016, 41259}, {44091, 44142}

X(52570) = isotomic conjugate of X(52554)
X(52570) = X(308)-Ceva conjugate of X(39998)
X(52570) = X(i)-isoconjugate of X(j) for these (i,j): {31, 52554}, {1923, 10159}, {1964, 3108}, {2084, 7953}
X(52570) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 52554}, {39, 6292}, {512, 51906}, {3005, 15527}, {3108, 41884}, {3589, 8041}, {6665, 39998}, {31067, 36901}, {31128, 38303}
X(52570) = cevapoint of X(3589) and X(39998)
X(52570) = barycentric product X(i)*X(j) for these {i,j}: {83, 39998}, {308, 3589}, {689, 7927}, {1799, 44142}, {5007, 40016}, {7767, 46104}, {8664, 42371}, {17469, 18833}, {18062, 18070}, {42554, 52395}
X(52570) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 52554}, {83, 3108}, {308, 10159}, {428, 1843}, {689, 35137}, {850, 31067}, {1799, 41435}, {3589, 39}, {4030, 3688}, {4577, 7953}, {5007, 3051}, {6292, 8041}, {7198, 1401}, {7767, 3917}, {7927, 3005}, {8664, 688}, {10330, 1634}, {16707, 16696}, {17200, 17187}, {17469, 1964}, {21802, 21814}, {22352, 20775}, {39998, 141}, {42554, 7794}, {44091, 27369}, {44142, 427}, {48101, 21123}, {48152, 2530}
X(52570) = {X(83),X(308)}-harmonic conjugate of X(76)

---

X(52571) = X(8)X(20)∩X(76)X(309)

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^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(52571) lies on the cubic K1066 and these lines: {4, 18725}, {8, 20}, {34, 1256}, {76, 309}, {241, 8808}, {282, 6554}, {514, 16388}, {946, 21328}, {1436, 8074}, {1622, 22654}, {8726, 41006}, {34823, 52389}

X(52571) = X(309)-Ceva conjugate of X(17862)
X(52571) = X(37566)-cross conjugate of X(1210)
X(52571) = X(i)-isoconjugate of X(j) for these (i,j): {40, 1167}, {198, 40399}, {2187, 40424}
X(52571) = X(40)-Dao conjugate of X(6260)
X(52571) = barycentric product X(i)*X(j) for these {i,j}: {84, 17862}, {189, 1210}, {309, 1108}, {1226, 1436}, {34404, 37566}, {40958, 44190}
X(52571) = barycentric quotient X(i)/X(j) for these {i,j}: {84, 40399}, {189, 40424}, {1108, 40}, {1210, 329}, {1436, 1167}, {1864, 2324}, {17862, 322}, {21933, 21075}, {37566, 223}, {40836, 40444}, {40958, 198}

---

X(52572) = X(2)X(39)∩X(75)X(596)

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

X(52572) lies on these lines: {2, 39}, {75, 596}, {99, 9108}, {304, 16708}, {314, 3296}, {350, 17175}, {1125, 1269}, {1434, 6063}, {1509, 18166}, {1909, 3293}, {1978, 32018}, {4043, 18157}, {4441, 17169}, {4479, 17180}, {6385, 10009}, {10471, 33947}, {12263, 18792}, {14377, 14964}, {16696, 17030}, {16703, 33935}, {16727, 20911}, {17143, 30941}, {17205, 20888}, {18171, 26801}, {21281, 33297}, {29742, 39950}, {30939, 50116}, {31922, 44129}, {34282, 50597}, {52379, 52393}

X(52572) = isotomic conjugate of X(52555)
X(52572) = anticomplement of X(52539)
X(52572) = isotomic conjugate of the isogonal conjugate of X(8025)
X(52572) = X(i)-Ceva conjugate of X(j) for these (i,j): {310, 1269}, {1978, 7199}
X(52572) = X(4359)-cross conjugate of X(16709)
X(52572) = X(i)-isoconjugate of X(j) for these (i,j): {31, 52555}, {42, 28615}, {213, 1126}, {560, 6539}, {669, 37212}, {798, 8701}, {872, 1171}, {1255, 1918}, {1268, 2205}, {1402, 33635}, {1924, 6540}, {4629, 50487}, {7109, 40438}
X(52572) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 52555}, {42, 1213}, {86, 38836}, {213, 3647}, {512, 35076}, {649, 16726}, {1125, 1500}, {1126, 6626}, {1255, 34021}, {3120, 4079}, {4272, 41809}, {6374, 6539}, {6540, 9428}, {8701, 31998}, {28615, 40592}, {33635, 40605}, {40620, 50344}
X(52572) = cevapoint of X(1269) and X(4359)
X(52572) = crosssum of X(1918) and X(7109)
X(52572) = barycentric product X(i)*X(j) for these {i,j}: {75, 16709}, {76, 8025}, {86, 1269}, {274, 4359}, {305, 31900}, {310, 1125}, {313, 30593}, {553, 28660}, {670, 4977}, {799, 4978}, {873, 4647}, {1100, 6385}, {1230, 1509}, {3649, 18021}, {4001, 44129}, {4602, 4979}, {4609, 50512}, {4623, 30591}, {4625, 4985}, {27801, 30581}, {32636, 40072}
X(52572) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 52555}, {76, 6539}, {81, 28615}, {86, 1126}, {99, 8701}, {274, 1255}, {310, 1268}, {313, 6538}, {314, 32635}, {333, 33635}, {553, 1400}, {670, 6540}, {799, 37212}, {873, 40438}, {1100, 213}, {1125, 42}, {1213, 1500}, {1230, 594}, {1269, 10}, {1509, 1171}, {1839, 2333}, {1962, 872}, {2308, 1918}, {3261, 31010}, {3649, 181}, {3686, 1334}, {3702, 210}, {3916, 228}, {4001, 71}, {4046, 7064}, {4359, 37}, {4427, 4557}, {4610, 4629}, {4623, 4596}, {4647, 756}, {4697, 20964}, {4966, 20683}, {4973, 3724}, {4974, 3747}, {4975, 21805}, {4976, 3709}, {4977, 512}, {4978, 661}, {4979, 798}, {4983, 50487}, {4984, 14407}, {4985, 4041}, {4988, 4079}, {4990, 4524}, {4992, 50491}, {6385, 32018}, {6533, 1962}, {6626, 38836}, {6628, 52558}, {7192, 50344}, {7199, 47947}, {8025, 6}, {16709, 1}, {17190, 2174}, {17206, 1796}, {20970, 7109}, {22054, 2200}, {28660, 4102}, {30581, 1333}, {30591, 4705}, {30592, 14404}, {30593, 58}, {30724, 7180}, {31900, 25}, {32018, 30582}, {32636, 1402}, {41014, 3690}, {41820, 4272}, {44143, 7140}, {45221, 1203}, {50512, 669}, {51409, 51377}
X(52572) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {274, 310, 76}, {16705, 16748, 274}, {18600, 34284, 274}

---

X(52573) = X(10)X(330)∩X(87)X(1125)

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

X(52573) lies on these lines: {10, 330}, {39, 40881}, {76, 3840}, {87, 1125}, {551, 23493}, {596, 42027}, {4298, 7153}, {6381, 33789}, {6383, 24182}, {7155, 24231}, {16604, 34832}, {20899, 24165}, {29974, 46827}, {39914, 50307}

X(52573) = X(i)-Dao conjugate of X(j) for these (i,j): {43, 34832}, {3248, 8640}
X(52573) = barycentric product X(i)*X(j) for these {i,j}: {330, 24165}, {4598, 48406}, {6384, 16604}, {16710, 42027}, {21128, 32039}
X(52573) = barycentric quotient X(i)/X(j) for these {i,j}: {16604, 43}, {16710, 33296}, {20899, 8026}, {21128, 23886}, {21757, 2209}, {24165, 192}, {32039, 35572}, {48406, 3835}

---

X(52574) = X(8)X(903)∩X(10)X(6549)

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

X(52574) lies on the cubic K1069 and these lines: {8, 903}, {10, 6549}, {76, 1978}, {85, 20949}, {145, 4555}, {348, 27814}, {679, 42697}, {764, 6548}, {3241, 9460}, {3616, 27922}, {4945, 29579}, {16711, 52206}, {16816, 42026}, {17244, 46795}

X(52574) = isotomic conjugate of X(52556)
X(52574) = isotomic conjugate of the isogonal conjugate of X(52206)
X(52574) = X(16594)-cross conjugate of X(1266)
X(52574) = X(i)-isoconjugate of X(j) for these (i,j): {31, 52556}, {902, 40400}, {1120, 2251}, {9459, 36805}
X(52574) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 52556}, {44, 16594}, {1120, 9460}, {2087, 3251}, {2325, 4152}, {4370, 16610}, {40400, 40594}
X(52574) = cevapoint of X(1266) and X(16594)
X(52574) = crosssum of X(1017) and X(2251)
X(52574) = trilinear pole of line {1266, 4927}
X(52574) = barycentric product X(i)*X(j) for these {i,j}: {76, 52206}, {85, 52140}, {561, 17109}, {679, 20900}, {903, 1266}, {4080, 16711}, {4555, 4927}, {6063, 45247}, {16610, 20568}
X(52574) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 52556}, {88, 40400}, {903, 1120}, {1149, 902}, {1266, 519}, {3880, 3689}, {4555, 6079}, {4695, 21805}, {4927, 900}, {5151, 42070}, {6085, 1960}, {6548, 23836}, {16594, 4370}, {16610, 44}, {16711, 16704}, {17109, 31}, {17460, 678}, {20568, 36805}, {20900, 4738}, {20972, 1017}, {21129, 6544}, {22082, 22371}, {23205, 23202}, {23832, 23344}, {45247, 55}, {52140, 9}, {52206, 6}

---

X(52575) = X(65)X(290)∩X(76)X(331)

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

X(52575) lies on these lines: {65, 290}, {76, 331}, {92, 20926}, {108, 2367}, {264, 1441}, {276, 1214}, {308, 1880}, {653, 37219}, {1882, 40717}, {4554, 40832}, {18022, 40828}, {42333, 52385}, {44129, 46404}

X(52575) = isotomic conjugate of X(2193)
X(52575) = polar conjugate of X(2194)
X(52575) = isotomic conjugate of the isogonal conjugate of X(40149)
X(52575) = polar conjugate of the isogonal conjugate of X(1441)
X(52575) = X(1969)-Ceva conjugate of X(349)
X(52575) = X(21930)-cross conjugate of X(10)
X(52575) = X(i)-isoconjugate of X(j) for these (i,j): {21, 9247}, {29, 14585}, {31, 2193}, {32, 283}, {41, 1437}, {48, 2194}, {58, 52425}, {60, 2200}, {163, 1946}, {184, 284}, {212, 1333}, {217, 35196}, {219, 2206}, {228, 2150}, {255, 2204}, {332, 1501}, {333, 14575}, {560, 1812}, {577, 2299}, {652, 1576}, {663, 32661}, {849, 52370}, {1172, 52430}, {1260, 16947}, {1397, 2327}, {1408, 1802}, {1444, 9447}, {1474, 6056}, {1790, 2175}, {1808, 14599}, {1974, 6514}, {2189, 4055}, {2203, 2289}, {2212, 18604}, {2328, 52411}, {2332, 7335}, {3049, 4636}, {3063, 4575}, {6332, 14574}, {7252, 32656}, {8748, 23606}, {9448, 17206}, {21789, 32660}, {23189, 32739}, {28660, 40373}, {32676, 36054}
X(52575) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 2193}, {10, 52425}, {31, 47345}, {37, 212}, {48, 1214}, {110, 39060}, {115, 1946}, {136, 3063}, {163, 39053}, {184, 40590}, {219, 40603}, {226, 577}, {283, 6376}, {521, 36901}, {652, 4858}, {1249, 2194}, {1333, 40837}, {1437, 3160}, {1441, 23171}, {1790, 40593}, {1812, 6374}, {2204, 6523}, {3666, 22074}, {4075, 52370}, {4575, 10001}, {6056, 51574}, {9247, 40611}, {15526, 36054}, {16584, 20753}, {16732, 52306}, {22383, 40622}, {23090, 40624}, {23189, 40619}, {23207, 40937}, {36908, 52411}
X(52575) = cevapoint of X(1441) and X(40149)
X(52575) = crosspoint of X(1969) and X(18027)
X(52575) = crosssum of X(9247) and X(14585)
X(52575) = barycentric product X(i)*X(j) for these {i,j}: {65, 18022}, {76, 40149}, {92, 349}, {108, 44173}, {225, 561}, {226, 1969}, {264, 1441}, {273, 313}, {278, 27801}, {286, 34388}, {321, 331}, {653, 20948}, {850, 18026}, {1214, 18027}, {1231, 2052}, {1402, 44161}, {1426, 40363}, {1446, 7017}, {1502, 1880}, {1577, 46404}, {1824, 41283}, {1826, 20567}, {1847, 30713}, {1874, 44172}, {4554, 14618}, {4572, 24006}, {6063, 41013}, {6358, 44129}, {6385, 8736}
X(52575) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2193}, {4, 2194}, {7, 1437}, {10, 212}, {12, 228}, {27, 2150}, {34, 2206}, {37, 52425}, {65, 184}, {72, 6056}, {73, 52430}, {75, 283}, {76, 1812}, {85, 1790}, {92, 284}, {108, 1576}, {158, 2299}, {201, 4055}, {225, 31}, {226, 48}, {264, 21}, {273, 58}, {278, 1333}, {286, 60}, {304, 6514}, {306, 2289}, {307, 255}, {312, 2327}, {313, 78}, {318, 2328}, {321, 219}, {322, 1819}, {331, 81}, {334, 1808}, {342, 2360}, {348, 18604}, {349, 63}, {393, 2204}, {429, 20967}, {442, 23207}, {523, 1946}, {525, 36054}, {561, 332}, {594, 52370}, {651, 32661}, {653, 163}, {664, 4575}, {693, 23189}, {811, 4636}, {850, 521}, {860, 2361}, {1020, 32660}, {1089, 2318}, {1118, 2203}, {1119, 1408}, {1211, 22074}, {1214, 577}, {1231, 394}, {1400, 9247}, {1402, 14575}, {1409, 14585}, {1426, 1397}, {1427, 52411}, {1435, 16947}, {1439, 7335}, {1441, 3}, {1446, 222}, {1577, 652}, {1824, 2175}, {1826, 41}, {1835, 52434}, {1847, 1412}, {1874, 2210}, {1880, 32}, {1882, 5320}, {1969, 333}, {2052, 1172}, {2171, 2200}, {2321, 1802}, {2333, 9447}, {2501, 3063}, {2887, 20753}, {2970, 4516}, {2973, 18191}, {3596, 1792}, {3649, 23201}, {3668, 603}, {3701, 1260}, {3925, 22079}, {4033, 4587}, {4077, 1459}, {4391, 23090}, {4551, 32656}, {4552, 906}, {4554, 4558}, {4566, 36059}, {4572, 4592}, {6046, 1410}, {6063, 1444}, {6331, 4612}, {6335, 5546}, {6354, 1409}, {6356, 22341}, {6358, 71}, {6521, 8748}, {6757, 8606}, {7017, 2287}, {7141, 210}, {7178, 22383}, {7282, 17104}, {8736, 213}, {13149, 4565}, {14257, 52143}, {14618, 650}, {16732, 7117}, {17094, 23224}, {17869, 19354}, {17923, 4282}, {17924, 7252}, {18022, 314}, {18026, 110}, {18027, 31623}, {18698, 22361}, {20336, 1259}, {20565, 1789}, {20566, 1793}, {20567, 17206}, {20930, 1800}, {20948, 6332}, {21016, 40972}, {21207, 7004}, {21438, 23145}, {21804, 22368}, {22341, 23606}, {24002, 7254}, {24006, 663}, {26942, 3990}, {27801, 345}, {27808, 4571}, {28654, 3694}, {30713, 3692}, {31623, 7054}, {31643, 1798}, {34388, 72}, {36127, 32676}, {37790, 3285}, {37799, 19622}, {39130, 2188}, {40071, 3719}, {40149, 6}, {40152, 4100}, {40440, 35196}, {40663, 23202}, {40701, 1817}, {40999, 52408}, {41003, 22345}, {41013, 55}, {41079, 14395}, {41804, 52407}, {44129, 2185}, {44130, 1098}, {44143, 3683}, {44161, 40072}, {44173, 35518}, {44426, 21789}, {45196, 22097}, {46107, 3737}, {46110, 1021}, {46404, 662}, {52358, 22118}, {52385, 1092}, {52412, 35192}, {52489, 41604}

---

X(52576) = X(10)X(75)∩X(190)X(35468)

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

X(52576) lies on these lines: {10, 75}, {190, 35468}, {312, 21081}, {314, 41814}, {595, 4039}, {668, 41822}, {1089, 6535}, {1228, 21207}, {1230, 4647}, {2292, 21684}, {2901, 4710}, {3701, 6757}, {3743, 3948}, {3992, 21674}, {4066, 30713}, {4658, 28660}, {4692, 27714}, {11263, 27792}, {14815, 52529}, {20634, 20661}

X(52576) = reflection of X(14815) in X(52529)
X(52576) = isotomic conjugate of X(52558)
X(52576) = isotomic conjugate of the isogonal conjugate of X(8013)
X(52576) = X(313)-Ceva conjugate of X(1230)
X(52576) = X(i)-isoconjugate of X(j) for these (i,j): {31, 52558}, {593, 28615}, {667, 6578}, {849, 1126}, {1171, 1333}, {2206, 40438}, {7342, 32635}
X(52576) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 52558}, {37, 1171}, {58, 1125}, {512, 21709}, {593, 1213}, {849, 3647}, {1126, 4075}, {1203, 3743}, {3120, 3733}, {6578, 6631}, {40438, 40603}
X(52576) = crosspoint of X(313) and X(28654)
X(52576) = barycentric product X(i)*X(j) for these {i,j}: {10, 1230}, {76, 8013}, {306, 44143}, {313, 1213}, {321, 4647}, {349, 4046}, {430, 40071}, {561, 21816}, {594, 1269}, {850, 4115}, {1089, 4359}, {1125, 28654}, {1962, 27801}, {1978, 6367}, {3649, 30713}, {3686, 34388}, {3702, 6358}, {4001, 7141}, {4033, 30591}, {4988, 27808}
X(52576) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 52558}, {10, 1171}, {190, 6578}, {313, 32014}, {321, 40438}, {430, 1474}, {553, 7341}, {594, 1126}, {756, 28615}, {1089, 1255}, {1100, 849}, {1125, 593}, {1213, 58}, {1230, 86}, {1269, 1509}, {1962, 1333}, {3649, 1412}, {3683, 2150}, {3686, 60}, {3695, 1796}, {3702, 2185}, {3952, 4629}, {3958, 1437}, {4024, 50344}, {4033, 4596}, {4036, 47947}, {4046, 284}, {4103, 8701}, {4115, 110}, {4359, 757}, {4427, 4556}, {4647, 81}, {4975, 30576}, {4988, 3733}, {6057, 33635}, {6367, 649}, {6533, 30581}, {6535, 52555}, {8013, 6}, {8663, 1919}, {16709, 763}, {20970, 2206}, {21816, 31}, {27808, 4632}, {28654, 1268}, {30591, 1019}, {30729, 4636}, {41014, 1790}, {44143, 27}, {51417, 17209}

---

X(52577) = X(19)X(46)∩X(76)X(92)

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

X(52577) lies on these lines: {4, 28849}, {19, 46}, {76, 92}, {225, 2333}, {226, 23619}, {242, 50290}, {1089, 1826}, {1824, 20683}, {1828, 8735}, {1842, 1886}, {1851, 2082}, {1855, 17442}, {1865, 50033}, {2083, 2385}, {2201, 4251}, {3755, 23621}, {5930, 42669}, {17671, 18589}, {21207, 36907}, {40934, 40987}

X(52577) = reflection of X(2083) in X(52530)
X(52577) = isotomic conjugate of the isogonal conjugate of X(8020)
X(52577) = polar conjugate of the isotomic conjugate of X(3914)
X(52577) = polar conjugate of the isogonal conjugate of X(40934)
X(52577) = X(40934)-cross conjugate of X(3914)
X(52577) = X(i)-isoconjugate of X(j) for these (i,j): {3, 40403}, {255, 40411}, {283, 7131}, {1037, 1812}, {1041, 6514}, {1437, 30701}, {1444, 7123}, {2193, 8817}, {4575, 48070}, {7084, 17206}, {8269, 23090}
X(52577) = X(i)-Dao conjugate of X(j) for these (i,j): {63, 18589}, {136, 48070}, {1444, 15487}, {3926, 16583}, {6523, 40411}, {6554, 17206}, {8817, 47345}, {36103, 40403}
X(52577) = cevapoint of X(8020) and X(40934)
X(52577) = crosspoint of X(92) and X(393)
X(52577) = crosssum of X(i) and X(j) for these (i,j): {48, 394}, {1790, 2327}
X(52577) = barycentric product X(i)*X(j) for these {i,j}: {4, 3914}, {10, 1851}, {76, 8020}, {92, 16583}, {158, 17441}, {225, 497}, {264, 40934}, {273, 40965}, {318, 40961}, {393, 18589}, {614, 41013}, {1089, 4211}, {1093, 22057}, {1096, 20235}, {1441, 40987}, {1633, 24006}, {1824, 3673}, {1826, 4000}, {1863, 3668}, {1897, 48403}, {1969, 21750}, {2052, 23620}, {2082, 40149}, {2501, 3732}, {8747, 21015}, {18084, 27376}, {21813, 44129}
X(52577) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 40403}, {225, 8817}, {393, 40411}, {497, 332}, {614, 1444}, {1633, 4592}, {1826, 30701}, {1851, 86}, {1863, 1043}, {1880, 7131}, {2082, 1812}, {2333, 7123}, {2501, 48070}, {3732, 4563}, {3914, 69}, {4000, 17206}, {4211, 757}, {4319, 1792}, {7083, 283}, {7124, 6514}, {8020, 6}, {16502, 1790}, {16583, 63}, {17441, 326}, {18589, 3926}, {21015, 52396}, {21107, 30805}, {21750, 48}, {21813, 71}, {22057, 3964}, {22363, 255}, {23620, 394}, {30706, 2327}, {40934, 3}, {40961, 77}, {40965, 78}, {40987, 21}, {48398, 15419}, {48403, 4025}, {50490, 1459}
X(52577) = {X(1842),X(1886)}-harmonic conjugate of X(1973)

---

X(52578) = X(4)X(69)∩X(25)X(1105)

Barycentrics    b^2*c^2*(-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 : :

X(52578) lies on these lines: {2, 52543}, {3, 47392}, {4, 69}, {20, 6525}, {25, 1105}, {30, 1093}, {107, 11413}, {305, 1529}, {324, 17578}, {648, 1498}, {1657, 34334}, {1895, 33673}, {1896, 10431}, {1941, 26883}, {2052, 3146}, {2883, 44704}, {2972, 52441}, {3168, 46850}, {3522, 52147}, {5059, 46106}, {5064, 14860}, {6223, 18026}, {6523, 46927}, {6526, 7396}, {6529, 23115}, {6616, 37669}, {8884, 18534}, {9308, 15811}, {11414, 41372}, {11819, 16263}, {12279, 35360}, {13149, 42451}, {13380, 20975}, {13450, 33703}, {15005, 40196}, {15682, 44732}, {17907, 52404}, {31623, 37434}, {32602, 34808}, {34170, 37444}, {34621, 37765}, {36876, 52398}, {37201, 52448}, {39568, 52439}, {40684, 50689}

X(52578) = isotomic conjugate of X(52559)
X(52578) = anticomplement of X(52543)
X(52578) = isotomic conjugate of the isogonal conjugate of X(3079)
X(52578) = polar conjugate of the isogonal conjugate of X(36413)
X(52578) = X(14615)-Ceva conjugate of X(15466)
X(52578) = X(13155)-cross conjugate of X(20)
X(52578) = X(i)-isoconjugate of X(j) for these (i,j): {31, 52559}, {64, 19614}, {255, 31942}, {1073, 2155}, {2184, 14642}, {19611, 33581}
X(52578) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 52559}, {4, 64}, {1073, 45245}, {2972, 8057}, {6523, 31942}, {8798, 45249}, {14379, 45248}
X(52578) = cevapoint of X(i) and X(j) for these (i,j): {20, 6616}, {3079, 36413}
X(52578) = barycentric product X(i)*X(j) for these {i,j}: {20, 15466}, {76, 3079}, {92, 1097}, {264, 36413}, {331, 6060}, {1249, 14615}, {1895, 18750}, {6616, 47633}, {7017, 7338}, {14249, 37669}, {44697, 52346}
X(52578) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 52559}, {20, 1073}, {154, 14642}, {204, 2155}, {393, 31942}, {610, 19614}, {1097, 63}, {1249, 64}, {1895, 2184}, {3079, 6}, {3172, 33581}, {6060, 219}, {6525, 41489}, {6616, 3343}, {7338, 222}, {13155, 46831}, {14249, 459}, {14615, 34403}, {15466, 253}, {15905, 14379}, {18750, 19611}, {23608, 15905}, {36413, 3}, {37669, 15394}, {39020, 2972}, {40170, 15400}, {42459, 8798}, {44695, 30457}, {44697, 8809}
X(52578) = {X(20),X(14249)}-harmonic conjugate of X(15466)

---

X(52579) = X(2)X(52548)∩X(76)X(321)

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

X(52579) lies on these lines: {2, 52548}, {8, 21839}, {10, 4037}, {37, 1698}, {75, 25458}, {76, 321}, {115, 20653}, {213, 5295}, {312, 25661}, {335, 32018}, {594, 762}, {740, 6155}, {756, 7230}, {2321, 3178}, {3125, 4647}, {3626, 4115}, {3634, 24051}, {3679, 21879}, {3706, 20963}, {3721, 42031}, {3727, 4717}, {3970, 19584}, {4006, 4053}, {4109, 20685}, {4431, 27697}, {4665, 42713}, {4705, 23099}, {4714, 21025}, {4967, 27565}, {5283, 21883}, {6535, 21718}, {6537, 8013}, {9780, 24075}, {16589, 21020}, {16831, 31993}, {16886, 20708}, {17163, 27040}, {24081, 24603}, {24275, 27368}, {27569, 33932}, {27808, 43685}, {29569, 31025}

X(52579) = reflection of X(6155) in X(52538)
X(52579) = anticomplement of X(52548)
X(52579) = isotomic conjugate of the isogonal conjugate of X(21820)
X(52579) = X(i)-Ceva conjugate of X(j) for these (i,j): {21020, 21699}, {27808, 4705}
X(52579) = X(i)-isoconjugate of X(j) for these (i,j): {58, 40408}, {593, 40433}, {849, 32009}, {1333, 40439}, {4556, 50520}
X(52579) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 40408}, {37, 40439}, {81, 3739}, {1509, 16589}, {3121, 3733}, {4075, 32009}
X(52579) = crosspoint of X(321) and X(594)
X(52579) = crosssum of X(593) and X(1333)
X(52579) = barycentric product X(i)*X(j) for these {i,j}: {10, 21020}, {12, 3706}, {75, 21699}, {76, 21820}, {313, 2667}, {321, 16589}, {594, 3739}, {668, 50538}, {756, 20888}, {762, 16748}, {1089, 3720}, {1441, 4111}, {3691, 6358}, {3701, 39793}, {3952, 48393}, {4036, 4436}, {4059, 6057}, {4103, 47672}, {6535, 17175}, {7141, 22060}, {20963, 28654}, {21753, 27801}, {27808, 50497}, {40975, 52369}
X(52579) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 40439}, {37, 40408}, {594, 32009}, {756, 40433}, {2667, 58}, {3691, 2185}, {3706, 261}, {3720, 757}, {3739, 1509}, {4059, 552}, {4111, 21}, {4705, 50520}, {16589, 81}, {17175, 6628}, {18166, 763}, {20888, 873}, {20963, 593}, {21020, 86}, {21699, 1}, {21753, 1333}, {21820, 6}, {22369, 1437}, {39793, 1014}, {40521, 8708}, {48393, 7192}, {50497, 3733}, {50538, 513}
X(52579) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 4037, 21816}, {10, 24044, 4037}, {594, 1089, 762}, {1698, 24049, 37}, {4647, 21024, 3125}

---

X(52580) = X(6)X(76)∩X(25)X(251)

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

X(52580) lies on these lines: {3, 38834}, {4, 10549}, {6, 76}, {25, 251}, {32, 51862}, {315, 16285}, {1176, 40825}, {1184, 1799}, {1613, 7879}, {2207, 32085}, {3051, 7762}, {5007, 9149}, {5276, 27067}, {5354, 26257}, {7745, 34294}, {10339, 20088}, {10359, 43843}, {11338, 40016}, {12252, 48262}, {16950, 34482}, {18105, 23099}, {21459, 27376}, {27005, 33854}, {32027, 38303}

X(52580) = X(83)-Ceva conjugate of X(6656)
X(52580) = X(i)-isoconjugate of X(j) for these (i,j): {1241, 1964}, {2084, 35567}
X(52580) = X(i)-Dao conjugate of X(j) for these (i,j): {141, 21248}, {1241, 41884}, {8024, 37891}, {8041, 11574}
X(52580) = crosssum of X(i) and X(j) for these (i,j): {39, 7794}, {141, 4074}
X(52580) = crossdifference of every pair of points on line {688, 2525}
X(52580) = barycentric product X(i)*X(j) for these {i,j}: {82, 17446}, {83, 1194}, {251, 6656}, {827, 47126}, {2514, 4577}, {11574, 32085}, {16735, 18098}, {23642, 52395}
X(52580) = barycentric quotient X(i)/X(j) for these {i,j}: {83, 1241}, {1194, 141}, {2514, 826}, {4577, 35567}, {6656, 8024}, {11574, 3933}, {16735, 16703}, {17446, 1930}, {22424, 4175}, {23642, 7794}, {47126, 23285}
X(52580) = {X(83),X(308)}-harmonic conjugate of X(7770)

---

X(52581) = X(64)X(290)∩X(76)X(459)

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 + 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) : :

X(52581) lies on these lines: {64, 290}, {76, 459}, {253, 264}, {276, 1073}, {300, 44702}, {301, 44703}, {305, 40830}, {308, 41489}, {339, 1093}, {349, 7017}, {1301, 2367}, {13157, 18027}, {14572, 26166}, {14618, 23107}, {15077, 32001}, {15394, 42333}, {16096, 41009}, {17907, 32834}, {18018, 34407}, {32830, 46927}, {34170, 51884}, {34385, 43752}, {34861, 47409}, {40832, 44326}

X(52581) = isotomic conjugate of X(15905)
X(52581) = polar conjugate of X(154)
X(52581) = isotomic conjugate of the complement of X(32001)
X(52581) = isotomic conjugate of the isogonal conjugate of X(459)
X(52581) = polar conjugate of the isotomic conjugate of X(41530)
X(52581) = polar conjugate of the isogonal conjugate of X(253)
X(52581) = X(41530)-Ceva conjugate of X(264)
X(52581) = X(i)-cross conjugate of X(j) for these (i,j): {253, 41530}, {2052, 264}, {13157, 253}, {44131, 18022}
X(52581) = X(i)-isoconjugate of X(j) for these (i,j): {20, 9247}, {31, 15905}, {48, 154}, {163, 42658}, {184, 610}, {204, 577}, {255, 3172}, {560, 37669}, {1249, 52430}, {1394, 52425}, {1562, 23995}, {1895, 14585}, {1973, 35602}, {3213, 6056}, {4020, 51508}, {4100, 6525}, {7070, 52411}, {7156, 7335}, {14575, 18750}
X(52581) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 15905}, {6, 40839}, {115, 42658}, {122, 23285}, {154, 1249}, {184, 14092}, {525, 47409}, {577, 3343}, {1562, 18314}, {3172, 6523}, {6337, 35602}, {6374, 37669}, {8057, 36901}, {14390, 23606}
X(52581) = cevapoint of X(i) and X(j) for these (i,j): {2, 32001}, {253, 459}, {339, 14618}, {14638, 36793}, {23978, 46107}
X(52581) = barycentric product X(i)*X(j) for these {i,j}: {4, 41530}, {64, 18022}, {76, 459}, {253, 264}, {276, 13157}, {305, 6526}, {339, 44181}, {1073, 18027}, {1301, 44173}, {1502, 41489}, {1969, 2184}, {2052, 34403}, {14618, 44326}, {14638, 15352}, {33581, 44161}
X(52581) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 15905}, {4, 154}, {64, 184}, {69, 35602}, {76, 37669}, {92, 610}, {158, 204}, {253, 3}, {264, 20}, {273, 1394}, {275, 33629}, {318, 7070}, {324, 42459}, {331, 18623}, {338, 1562}, {339, 122}, {393, 3172}, {459, 6}, {523, 42658}, {850, 8057}, {1073, 577}, {1093, 6525}, {1301, 1576}, {1969, 18750}, {2052, 1249}, {2155, 9247}, {2184, 48}, {3267, 20580}, {5931, 283}, {6331, 36841}, {6526, 25}, {7017, 27382}, {8795, 38808}, {8798, 418}, {8809, 603}, {13157, 216}, {14249, 3079}, {14379, 23606}, {14572, 38292}, {14618, 6587}, {14642, 14585}, {15394, 1092}, {15466, 36413}, {15526, 47409}, {16080, 15291}, {18022, 14615}, {18027, 15466}, {19611, 255}, {19614, 52430}, {23978, 40616}, {27801, 42699}, {30457, 52425}, {31942, 33581}, {32001, 45248}, {32085, 51508}, {33581, 14575}, {34403, 394}, {39268, 1660}, {40149, 30456}, {41013, 3198}, {41079, 14345}, {41489, 32}, {41530, 69}, {42465, 28781}, {44131, 2883}, {44181, 250}, {44326, 4558}, {44692, 212}, {46107, 21172}, {46110, 14331}, {46639, 32661}, {47435, 6617}, {52559, 14379}
X(52581) = {X(76),X(41530)}-harmonic conjugate of X(34403)

---

X(52582) = X(2)X(254)∩X(4)X(47731)

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

X(52582) lies on the Kiepert circumhyperbola and these lines: {2, 254}, {4, 47731}, {68, 6504}, {76, 46746}, {98, 39416}, {275, 34756}, {1594, 39114}, {2052, 16172}, {2986, 15316}, {5392, 8800}, {5462, 14593}, {5962, 13579}, {39109, 40393}

X(52582) = X(46746)-Ceva conjugate of X(5392)
X(52582) = X(i)-cross conjugate of X(j) for these (i,j): {68, 847}, {8800, 254}
X(52582) = X(i)-isoconjugate of X(j) for these (i,j): {47, 155}, {255, 35603}, {563, 6515}, {920, 1147}, {33808, 52435}
X(52582) = X(i)-Dao conjugate of X(j) for these (i,j): {155, 34853}, {6523, 35603}
X(52582) = cevapoint of X(68) and X(32132)
X(52582) = barycentric product X(i)*X(j) for these {i,j}: {254, 5392}, {847, 6504}, {850, 39416}, {2052, 32132}, {2165, 46746}, {34385, 41536}
X(52582) = barycentric quotient X(i)/X(j) for these {i,j}: {68, 6503}, {254, 1993}, {393, 35603}, {847, 6515}, {2165, 155}, {5392, 40697}, {6504, 9723}, {8800, 52032}, {14593, 1609}, {32132, 394}, {39109, 571}, {39416, 110}, {41536, 52}, {46746, 7763}, {47731, 454}, {47732, 3133}

---

X(52583) = X(2)X(2138)∩X(4)X(9914)

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

X(52583) lies on the Kiepert circumhyperbola and these lines: {2, 2138}, {4, 9914}, {10, 23050}, {19, 36907}, {25, 40178}, {76, 17907}, {83, 41370}, {98, 3542}, {112, 28406}, {226, 20613}, {262, 3541}, {297, 6504}, {321, 17903}, {393, 43678}, {1249, 18840}, {2489, 23107}, {2986, 50188}, {2996, 5523}, {3088, 14484}, {3089, 3424}, {3399, 41371}, {3547, 39172}, {6143, 10155}, {7383, 40448}, {7505, 7612}, {10159, 44134}, {13579, 37174}, {14494, 37119}, {31363, 33579}, {34289, 44142}

X(52583) = isogonal conjugate of X(23115)
X(52583) = isotomic conjugate of X(28419)
X(52583) = polar conjugate of X(1370)
X(52583) = isotomic conjugate of the isogonal conjugate of X(40144)
X(52583) = polar conjugate of the isotomic conjugate of X(13575)
X(52583) = polar conjugate of the isogonal conjugate of X(34207)
X(52583) = X(40009)-Ceva conjugate of X(4)
X(52583) = X(i)-cross conjugate of X(j) for these (i,j): {127, 14618}, {2207, 4}, {13854, 393}, {34207, 13575}, {34944, 253}
X(52583) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23115}, {3, 18596}, {31, 28419}, {48, 1370}, {63, 159}, {184, 21582}, {212, 18629}, {255, 41361}, {326, 3162}, {4575, 47125}, {6507, 41766}
X(52583) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 28419}, {3, 23115}, {136, 47125}, {159, 3162}, {1249, 1370}, {3162, 15259}, {6523, 41361}, {14091, 41602}, {18596, 36103}, {18629, 40837}
X(52583) = cevapoint of X(i) and X(j) for these (i,j): {125, 2489}, {34207, 40144}
X(52583) = barycentric product X(i)*X(j) for these {i,j}: {4, 13575}, {19, 39733}, {25, 40009}, {76, 40144}, {264, 34207}, {850, 39417}, {2052, 52041}, {32085, 39129}, {40358, 43678}
X(52583) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 28419}, {4, 1370}, {6, 23115}, {19, 18596}, {25, 159}, {92, 21582}, {235, 41602}, {278, 18629}, {393, 41361}, {2207, 3162}, {2501, 47125}, {6524, 41766}, {13575, 69}, {34207, 3}, {39129, 3933}, {39417, 110}, {39733, 304}, {40009, 305}, {40144, 6}, {40358, 20806}, {46767, 10316}, {46769, 22135}, {52041, 394}
X(52583) = {X(2),X(13575)}-harmonic conjugate of X(40185)

---

X(52584) = X(2)X(14618)∩X(3)X(512)

Barycentrics    a^2*(b - c)*(b + c)*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4) : :
X(52584) = X[647] + 2 X[8552]

X(52584) lies on these lines: {2, 14618}, {3, 512}, {4, 44918}, {5, 16229}, {30, 47221}, {39, 2506}, {122, 38971}, {127, 35071}, {216, 18311}, {441, 525}, {520, 6760}, {523, 7663}, {684, 8673}, {850, 15412}, {924, 12095}, {1214, 7178}, {2072, 39503}, {2485, 2799}, {2489, 44817}, {3546, 30735}, {3566, 6132}, {4091, 52310}, {4108, 7494}, {4558, 43755}, {5664, 6509}, {5996, 7386}, {6563, 6753}, {6640, 23105}, {7630, 23285}, {8562, 38401}, {9517, 39228}, {11585, 34964}, {15451, 22089}, {15526, 18334}, {18314, 30476}, {18876, 35911}, {22159, 23115}, {23171, 39541}, {28407, 43665}, {34958, 37565}, {45681, 46832}

X(52584) = midpoint of X(i) and X(j) for these {i,j}: {684, 39201}, {850, 15412}, {4091, 52310}, {15451, 22089}
X(52584) = reflection of X(i) in X(j) for these {i,j}: {4, 44918}, {2489, 44817}, {2506, 7648}, {16229, 5}, {18314, 30476}, {42658, 39228}
X(52584) = isotomic conjugate of X(30450)
X(52584) = complement of X(14618)
X(52584) = complement of the isogonal conjugate of X(32661)
X(52584) = complement of the isotomic conjugate of X(4558)
X(52584) = isotomic conjugate of the isogonal conjugate of X(30451)
X(52584) = isotomic conjugate of the polar conjugate of X(924)
X(52584) = isogonal conjugate of the polar conjugate of X(6563)
X(52584) = X(46199)-anticomplementary conjugate of X(21294)
X(52584) = X(i)-complementary conjugate of X(j) for these (i,j): {3, 21253}, {48, 125}, {110, 20305}, {163, 5}, {184, 8287}, {249, 21259}, {255, 127}, {560, 6388}, {563, 136}, {577, 34846}, {662, 21243}, {906, 3454}, {1101, 30476}, {1331, 21245}, {1437, 116}, {1576, 226}, {1790, 21252}, {2193, 124}, {2315, 16221}, {3049, 24040}, {4020, 46654}, {4100, 122}, {4558, 2887}, {4563, 21235}, {4575, 141}, {4592, 626}, {9247, 115}, {9417, 41181}, {14574, 16583}, {14575, 16592}, {14585, 16573}, {15958, 21231}, {23357, 8062}, {23606, 16595}, {23889, 34517}, {23963, 16612}, {23995, 525}, {32656, 1211}, {32660, 442}, {32661, 10}, {32676, 13567}, {32739, 50036}, {34072, 5943}, {36059, 17052}, {36061, 34827}, {36134, 14767}, {36145, 5449}, {36148, 34826}, {47389, 21263}, {47390, 4369}, {52411, 8286}, {52430, 15526}
X(52584) = X(i)-Ceva conjugate of X(j) for these (i,j): {850, 520}, {6563, 924}, {9133, 8681}, {15412, 525}, {18878, 13754}, {41679, 1993}
X(52584) = X(i)-cross conjugate of X(j) for these (i,j): {30451, 924}, {47421, 3}
X(52584) = X(i)-isoconjugate of X(j) for these (i,j): {4, 36145}, {19, 925}, {31, 30450}, {68, 24019}, {91, 112}, {92, 32734}, {107, 1820}, {162, 2165}, {163, 847}, {662, 14593}, {823, 2351}, {920, 39416}, {1973, 46134}, {2168, 35360}, {4593, 27367}, {5392, 32676}, {5962, 32678}, {36149, 51833}
X(52584) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 30450}, {4, 39013}, {5, 47421}, {6, 925}, {68, 35071}, {91, 34591}, {110, 577}, {112, 34116}, {115, 847}, {125, 2165}, {134, 14576}, {135, 393}, {139, 13450}, {343, 14570}, {924, 6753}, {1084, 14593}, {1820, 38985}, {3003, 35588}, {5392, 15526}, {5962, 18334}, {6337, 46134}, {22391, 32734}, {36033, 36145}
X(52584) = crosspoint of X(i) and X(j) for these (i,j): {2, 4558}, {95, 4563}, {1993, 41679}
X(52584) = crosssum of X(i) and X(j) for these (i,j): {6, 2501}, {51, 2489}, {523, 1899}, {6753, 50647}, {7649, 41011}, {39383, 39384}
X(52584) = crossdifference of every pair of points on line {25, 53}
X(52584) = barycentric product X(i)*X(j) for these {i,j}: {3, 6563}, {24, 3265}, {47, 14208}, {69, 924}, {76, 30451}, {305, 34952}, {317, 520}, {328, 44808}, {523, 9723}, {525, 1993}, {563, 20948}, {571, 3267}, {647, 7763}, {656, 44179}, {684, 31635}, {850, 1147}, {879, 51439}, {905, 42700}, {1748, 24018}, {2799, 51776}, {3268, 5961}, {3926, 6753}, {4143, 8745}, {4563, 47421}, {4575, 17881}, {6334, 52505}, {8552, 18883}, {14576, 15414}, {15412, 52032}, {15423, 52350}, {15526, 41679}, {20336, 34948}, {23286, 39113}, {34386, 52317}, {34767, 51393}, {43088, 52437}, {44173, 52435}
X(52584) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 30450}, {3, 925}, {24, 107}, {47, 162}, {48, 36145}, {52, 35360}, {69, 46134}, {184, 32734}, {317, 6528}, {512, 14593}, {520, 68}, {523, 847}, {525, 5392}, {526, 5962}, {563, 163}, {571, 112}, {647, 2165}, {656, 91}, {688, 27367}, {822, 1820}, {924, 4}, {1147, 110}, {1748, 823}, {1993, 648}, {3265, 20563}, {5961, 476}, {6334, 52504}, {6563, 264}, {6753, 393}, {7763, 6331}, {8552, 37802}, {8675, 51833}, {8745, 6529}, {8911, 39384}, {9723, 99}, {11547, 15352}, {12095, 30512}, {14208, 20571}, {14397, 1990}, {14533, 32692}, {15423, 11547}, {18883, 46456}, {23286, 96}, {26920, 39383}, {30451, 6}, {31635, 22456}, {34948, 28}, {34952, 25}, {39013, 6753}, {39201, 2351}, {41679, 23582}, {42700, 6335}, {43088, 6344}, {44077, 32713}, {44179, 811}, {44808, 186}, {47421, 2501}, {51393, 4240}, {51439, 877}, {51776, 2966}, {52032, 14570}, {52317, 53}, {52435, 1576}, {52505, 687}
X(52584) = {X(46811),X(46814)}-harmonic conjugate of X(6334)

---

X(52585) = X(2)X(2416)∩X(4)X(42658)

Barycentrics    (b - c)*(b + c)*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 - 3*a^6*c^2 + a^2*b^4*c^2 + 2*b^6*c^2 + 3*a^4*c^4 + a^2*b^2*c^4 - 4*b^4*c^4 - a^2*c^6 + 2*b^2*c^6) : :
X(52585) = 3 X[647] - X[15412], 3 X[14618] + X[15412], 3 X[37943] - X[47225]

X(52585) lies on these lines: {2, 2416}, {4, 42658}, {5, 8673}, {512, 6130}, {523, 37942}, {525, 3239}, {647, 14165}, {924, 34964}, {2165, 18312}, {2485, 18314}, {2799, 47125}, {2848, 44705}, {9517, 39510}, {16040, 41300}, {16229, 39201}, {18557, 18883}, {37943, 47225}, {44560, 46425}

X(52585) = midpoint of X(i) and X(j) for these {i,j}: {4, 42658}, {647, 14618}, {16229, 39201}
X(52585) = reflection of X(44918) in X(39510)
X(52585) = isotomic conjugate of X(30441)
X(52585) = complement of the isogonal conjugate of X(6529)
X(52585) = complement of the isotomic conjugate of X(15352)
X(52585) = isotomic conjugate of the isogonal conjugate of X(30442)
X(52585) = X(i)-complementary conjugate of X(j) for these (i,j): {19, 122}, {25, 16595}, {107, 18589}, {158, 127}, {162, 6389}, {393, 34846}, {823, 1368}, {1093, 21253}, {1096, 15526}, {1973, 35071}, {2179, 38976}, {2207, 16573}, {5317, 2968}, {6520, 125}, {6524, 8287}, {6529, 10}, {8748, 123}, {15352, 2887}, {23590, 8062}, {23975, 16612}, {24019, 3}, {24021, 30476}, {24022, 525}, {32230, 4369}, {32676, 6509}, {32713, 1214}, {34538, 21259}, {36126, 141}, {36127, 18642}, {52439, 16592}
X(52585) = X(i)-isoconjugate of X(j) for these (i,j): {31, 30441}, {162, 18890}, {163, 15318}, {662, 32319}
X(52585) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 30441}, {53, 35360}, {115, 15318}, {125, 18890}, {1084, 32319}
X(52585) = crosspoint of X(2) and X(15352)
X(52585) = crosssum of X(6) and X(32320)
X(52585) = crossdifference of every pair of points on line {154, 160}
X(52585) = barycentric product X(i)*X(j) for these {i,j}: {76, 30442}, {523, 20477}, {850, 6759}, {3267, 51936}
X(52585) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 30441}, {512, 32319}, {523, 15318}, {647, 18890}, {6759, 110}, {14363, 35360}, {20477, 99}, {23286, 14371}, {30442, 6}, {51936, 112}

---

X(52586) = X(10)X(50488)∩X(514)X(661)

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

X(52586) lies on these lines: {10, 50488}, {514, 661}, {1019, 26983}, {6371, 31946}, {14288, 47842}, {18197, 27045}, {21051, 50493}, {21099, 24087}, {30023, 48196}, {30094, 48011}, {31003, 39548}, {48350, 50329}

X(52586) = X(i)-complementary conjugate of X(j) for these (i,j): {37, 5515}, {213, 39016}, {835, 3739}, {2214, 244}, {37218, 3741}, {43531, 17761}

---

X(52587) = X(6)X(14331)∩X(226)X(21174)

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

X(52587) lies on these lines: {6, 14331}, {226, 21174}, {520, 6587}, {522, 6589}, {661, 7649}, {676, 14321}, {1459, 6590}, {1769, 48269}, {2509, 3239}, {3553, 10397}, {3700, 6129}, {3756, 46101}, {11125, 47765}, {14837, 46396}

X(52587) = complement of the isotomic conjugate of X(36127)
X(52587) = X(i)-complementary conjugate of X(j) for these (i,j): {25, 123}, {108, 1368}, {1096, 124}, {1118, 21252}, {1395, 2968}, {1402, 122}, {1415, 6389}, {1880, 127}, {1973, 16596}, {1974, 35072}, {2203, 34588}, {2207, 26932}, {2212, 40616}, {6059, 5514}, {7337, 11}, {8750, 34823}, {23985, 4885}, {24019, 21246}, {24033, 17072}, {32674, 18589}, {32676, 34851}, {32713, 960}, {32714, 18639}, {36127, 2887}, {52439, 6506}
X(52587) = crosspoint of X(2) and X(36127)
X(52587) = crossdifference of every pair of points on line {1498, 3428}
X(52587) = {X(3239),X(21172)}-harmonic conjugate of X(6588)

---

X(52588) = X(6)X(8673)∩X(32)X(42658)

Barycentrics    a^2*(b - c)*(b + c)*(a^6 + a^4*b^2 - a^2*b^4 - b^6 + a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 + b^2*c^4 - c^6) : :
X(52588) = 2 X[2485] + X[7652], X[647] + 2 X[7651]

X(52588) lies on these lines: {6, 8673}, {32, 42658}, {115, 33504}, {512, 1692}, {525, 2485}, {647, 826}, {669, 2474}, {690, 2519}, {924, 22159}, {1084, 39008}, {1196, 14401}, {2492, 3566}, {2799, 7648}, {3569, 30442}, {4171, 21831}, {7950, 47133}, {39000, 47421}

X(52588) = midpoint of X(i) and X(j) for these {i,j}: {669, 2474}, {2485, 2506}
X(52588) = reflection of X(7652) in X(2506)
X(52588) = complement of the isotomic conjugate of X(32713)
X(52588) = isogonal conjugate of the isotomic conjugate of X(47125)
X(52588) = X(i)-complementary conjugate of X(j) for these (i,j): {107, 21235}, {560, 122}, {823, 40379}, {1501, 16595}, {1917, 35071}, {1973, 127}, {1974, 34846}, {2207, 21253}, {23582, 21263}, {23964, 42327}, {24000, 23301}, {24019, 626}, {32676, 1368}, {32713, 2887}, {36417, 8287}, {41937, 4369}, {44162, 16573}
X(52588) = X(i)-Ceva conjugate of X(j) for these (i,j): {525, 512}, {2485, 647}, {2506, 2519}, {2509, 661}
X(52588) = X(i)-isoconjugate of X(j) for these (i,j): {110, 39733}, {163, 40009}, {304, 39417}, {662, 13575}, {799, 34207}, {811, 52041}, {4599, 39129}
X(52588) = X(i)-Dao conjugate of X(j) for these (i,j): {25, 648}, {115, 40009}, {244, 39733}, {1084, 13575}, {3124, 39129}, {17423, 52041}, {34207, 38996}
X(52588) = crosspoint of X(2) and X(32713)
X(52588) = crosssum of X(i) and X(j) for these (i,j): {6, 3265}, {112, 39417}
X(52588) = crossdifference of every pair of points on line {22, 69}
X(52588) = barycentric product X(i)*X(j) for these {i,j}: {6, 47125}, {159, 523}, {512, 1370}, {520, 41766}, {525, 3162}, {647, 41361}, {661, 18596}, {798, 21582}, {826, 8793}, {2489, 28419}, {2501, 23115}, {3709, 18629}, {8057, 33584}, {8673, 17407}, {23881, 46766}
X(52588) = barycentric quotient X(i)/X(j) for these {i,j}: {159, 99}, {512, 13575}, {523, 40009}, {661, 39733}, {669, 34207}, {1370, 670}, {1974, 39417}, {3005, 39129}, {3049, 52041}, {3162, 648}, {8793, 4577}, {18596, 799}, {21582, 4602}, {23115, 4563}, {41361, 6331}, {41766, 6528}, {47125, 76}

---

X(52589) = X(2)X(40495)∩X(39)X(905)

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

X(52589) lies on these lines: {2, 40495}, {6, 22160}, {32, 1946}, {39, 905}, {513, 14991}, {574, 22091}, {647, 826}, {650, 784}, {663, 21837}, {798, 8637}, {810, 46388}, {824, 6586}, {838, 2084}, {1015, 39014}, {1084, 35090}, {1196, 40134}, {1500, 3900}, {1734, 2276}, {2491, 42653}, {3709, 50507}, {4391, 5283}, {6589, 23877}, {7255, 30912}, {8714, 25092}, {21901, 29066}, {24285, 25098}

X(52589) = midpoint of X(2084) and X(46386)
X(52589) = complement of X(40495)
X(52589) = complement of the isotomic conjugate of X(692)
X(52589) = polar conjugate of the isogonal conjugate of X(23228)
X(52589) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 21252}, {32, 116}, {100, 21235}, {101, 626}, {109, 17047}, {163, 21240}, {190, 40379}, {213, 21253}, {560, 11}, {692, 2887}, {1110, 21260}, {1252, 21262}, {1397, 17059}, {1415, 17046}, {1501, 1086}, {1576, 3741}, {1917, 1015}, {1918, 125}, {1978, 40380}, {2175, 124}, {2200, 127}, {2205, 8287}, {4567, 21263}, {4570, 23301}, {8750, 21243}, {9233, 6377}, {9447, 26932}, {9448, 1146}, {9459, 3259}, {14573, 44311}, {14574, 1125}, {14575, 2968}, {18892, 38989}, {18893, 39786}, {18894, 35119}, {23963, 21196}, {23979, 46399}, {23990, 3835}, {32656, 1368}, {32660, 18639}, {32666, 20544}, {32719, 21241}, {32739, 141}, {41267, 46654}, {41280, 3756}, {41281, 16614}, {46288, 44312}
X(52589) = X(2)-Ceva conjugate of X(21252)
X(52589) = X(2)-Dao conjugate of X(21252)
X(52589) = crosspoint of X(2) and X(692)
X(52589) = crosssum of X(6) and X(693)
X(52589) = crossdifference of every pair of points on line {22, 1602}
X(52589) = barycentric product X(i)*X(j) for these {i,j}: {31, 48272}, {100, 23646}, {101, 21339}, {264, 23228}, {667, 32862}, {692, 21252}, {1783, 22432}, {4557, 18181}, {21429, 32739}
X(52589) = barycentric quotient X(i)/X(j) for these {i,j}: {21252, 40495}, {21339, 3261}, {22432, 15413}, {23228, 3}, {23646, 693}, {32862, 6386}, {48272, 561}

---

X(52590) = X(2)X(44173)∩X(39)X(525)

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

X(52590) lies on these lines: {2, 44173}, {32, 39201}, {39, 525}, {115, 34984}, {187, 39228}, {512, 2021}, {574, 22089}, {647, 826}, {669, 2531}, {690, 2524}, {778, 42291}, {1084, 18334}, {1196, 9209}, {2485, 2799}, {2507, 23878}, {3049, 19627}, {3117, 9210}, {3199, 44705}, {7927, 45907}, {9494, 17415}, {10190, 40377}, {14270, 37085}, {39510, 39565}

X(52590) = midpoint of X(i) and X(j) for these {i,j}: {669, 2531}, {9494, 17415}
X(52590) = complement of X(44173)
X(52590) = complement of the isogonal conjugate of X(14574)
X(52590) = complement of the isotomic conjugate of X(1576)
X(52590) = X(i)-complementary conjugate of X(j) for these (i,j): {32, 21253}, {110, 21235}, {163, 626}, {249, 21263}, {560, 125}, {662, 40379}, {799, 40380}, {1101, 23301}, {1501, 8287}, {1576, 2887}, {1917, 115}, {1923, 46654}, {2206, 21252}, {4630, 21238}, {9233, 16592}, {9247, 127}, {9417, 36471}, {9426, 24040}, {14574, 10}, {14575, 34846}, {23357, 42327}, {23963, 4369}, {23995, 512}, {32676, 21243}, {32729, 21256}, {32739, 21245}, {40373, 16573}, {41280, 8286}, {41281, 16613}
X(52590) = X(i)-Ceva conjugate of X(j) for these (i,j): {15412, 512}, {31065, 688}, {43673, 39469}
X(52590) = X(i)-isoconjugate of X(j) for these (i,j): {163, 44185}, {662, 44176}, {799, 2980}, {4593, 27366}
X(52590) = X(i)-Dao conjugate of X(j) for these (i,j): {51, 14570}, {115, 44185}, {1084, 44176}, {2980, 38996}, {34845, 46726}
X(52590) = crosspoint of X(2) and X(1576)
X(52590) = crosssum of X(6) and X(850)
X(52590) = crossdifference of every pair of points on line {22, 157}
X(52590) = barycentric product X(i)*X(j) for these {i,j}: {160, 523}, {512, 2979}, {647, 39575}, {669, 7796}, {692, 18188}, {850, 3202}, {2623, 41480}, {15412, 40588}
X(52590) = barycentric quotient X(i)/X(j) for these {i,j}: {160, 99}, {512, 44176}, {523, 44185}, {669, 2980}, {688, 27366}, {2979, 670}, {3202, 110}, {7796, 4609}, {15897, 35360}, {18188, 40495}, {39575, 6331}, {40588, 14570}

---

X(52591) = X(39)X(826)∩X(187)X(237)

Barycentrics    a^2*(b - c)*(b + c)*(b^2 + c^2)*(a^4 - a^2*b^2 - a^2*c^2 - b^2*c^2) : :
X(52591) = X[9491] + 3 X[17414]

X(52591) lies on these lines: {5, 39799}, {39, 826}, {187, 237}, {2491, 7927}, {2524, 3906}, {2531, 8711}, {5007, 37085}, {5099, 35971}, {7630, 23285}, {15412, 43665}, {15449, 18334}, {38974, 46654}

X(52591) = midpoint of X(i) and X(j) for these {i,j}: {3005, 9494}, {15412, 43665}
X(52591) = complement of the isotomic conjugate of X(1634)
X(52591) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 7668}, {39, 21253}, {110, 21238}, {163, 3934}, {560, 3124}, {688, 24040}, {1333, 44312}, {1576, 1215}, {1634, 2887}, {1923, 115}, {1964, 125}, {3051, 8287}, {4020, 127}, {4575, 11574}, {4576, 21235}, {9247, 339}, {14574, 16600}, {17187, 21252}, {20775, 34846}, {23357, 8060}, {23990, 29512}, {23995, 826}, {24037, 42291}, {24041, 688}, {32676, 5943}, {35325, 20305}, {36827, 21256}, {41331, 16592}, {46148, 21245}, {52430, 47413}
X(52591) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 7668}, {15412, 826}
X(52591) = X(i)-isoconjugate of X(j) for these (i,j): {82, 11794}, {662, 30505}, {3613, 4599}, {4593, 27375}, {18070, 27867}
X(52591) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 7668}, {141, 11794}, {1084, 30505}, {3124, 3613}
X(52591) = crosspoint of X(i) and X(j) for these (i,j): {2, 1634}, {110, 39968}, {3050, 31296}
X(52591) = crosssum of X(i) and X(j) for these (i,j): {523, 20965}, {826, 1506}
X(52591) = crossdifference of every pair of points on line {2, 3613}
X(52591) = barycentric product X(i)*X(j) for these {i,j}: {39, 31296}, {141, 3050}, {523, 41328}, {647, 37125}, {688, 33769}, {826, 5012}, {850, 3203}, {1078, 3005}, {1634, 7668}, {2084, 33764}, {2525, 10312}, {8061, 18042}, {38352, 41676}
X(52591) = barycentric quotient X(i)/X(j) for these {i,j}: {39, 11794}, {512, 30505}, {688, 27375}, {1078, 689}, {3005, 3613}, {3050, 83}, {3203, 110}, {5012, 4577}, {10312, 42396}, {18042, 4593}, {27370, 35360}, {31296, 308}, {33764, 37204}, {33769, 42371}, {37125, 6331}, {38352, 4580}, {41328, 99}, {52042, 4576}

---

X(52592) = X(10)X(21901)∩X(37)X(4151)

Barycentrics    a^2*(b - c)*(b + c)*(a^2*b - a*b^2 + a^2*c - b^2*c - a*c^2 - b*c^2) : :
X(52592) = 3 X[14407] + X[40627], X[23807] - 5 X[31209]

X(52592) lies on these lines: {10, 21901}, {32, 21789}, {37, 4151}, {39, 14838}, {115, 1566}, {512, 798}, {514, 6586}, {647, 4988}, {650, 784}, {661, 665}, {810, 10581}, {1015, 38980}, {1084, 35092}, {1500, 4041}, {1573, 3907}, {1577, 16589}, {1924, 1960}, {2084, 46390}, {3124, 38990}, {3766, 27045}, {4560, 5283}, {4770, 22229}, {6544, 21838}, {16592, 40629}, {21800, 21832}, {23807, 31209}, {39003, 47421}

X(52592) = midpoint of X(21832) and X(21836)
X(52592) = complement of the isotomic conjugate of X(4557)
X(52592) = X(i)-complementary conjugate of X(j) for these (i,j): {32, 17761}, {42, 21252}, {101, 21240}, {213, 116}, {560, 244}, {692, 3741}, {765, 23301}, {872, 125}, {1016, 21263}, {1018, 626}, {1110, 512}, {1252, 42327}, {1402, 17059}, {1415, 17050}, {1500, 21253}, {1918, 11}, {1924, 6547}, {2149, 17066}, {2175, 34589}, {2205, 1086}, {3952, 21235}, {4033, 40379}, {4551, 17047}, {4557, 2887}, {4559, 17046}, {7109, 8287}, {9447, 4858}, {9459, 34590}, {23990, 4369}, {32739, 3739}
X(52592) = X(i)-Ceva conjugate of X(j) for these (i,j): {514, 512}, {6586, 647}
X(52592) = X(i)-isoconjugate of X(j) for these (i,j): {99, 39797}, {110, 39735}, {163, 40005}, {274, 6577}, {662, 8049}, {799, 34444}, {4610, 40504}, {4623, 40147}
X(52592) = X(i)-Dao conjugate of X(j) for these (i,j): {42, 190}, {115, 40005}, {244, 39735}, {1084, 8049}, {34444, 38996}, {38986, 39797}
X(52592) = crosspoint of X(i) and X(j) for these (i,j): {2, 4557}, {514, 8714}
X(52592) = crosssum of X(i) and X(j) for these (i,j): {6, 7192}, {101, 6577}
X(52592) = crossdifference of every pair of points on line {86, 1621}
X(52592) = barycentric product X(i)*X(j) for these {i,j}: {42, 8714}, {65, 50518}, {512, 17135}, {513, 22271}, {514, 40586}, {522, 52024}, {523, 8053}, {647, 17911}, {649, 21070}, {661, 16552}, {798, 18137}, {2501, 22126}, {3709, 17077}, {4079, 29767}
X(52592) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 8049}, {523, 40005}, {661, 39735}, {669, 34444}, {798, 39797}, {1918, 6577}, {4079, 40515}, {8053, 99}, {8714, 310}, {16552, 799}, {17135, 670}, {17911, 6331}, {18137, 4602}, {21070, 1978}, {22126, 4563}, {22271, 668}, {40586, 190}, {50487, 40504}, {50518, 314}, {52024, 664}
X(52592) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2084, 46390, 50493}, {4041, 21837, 1500}

---

X(52593) = X(514)X(1639)∩X(650)X(900)

Barycentrics    (a - 2*b - 2*c)*(5*a - b - c)*(b - c) : :
X(52593) = 2 X[1639] + X[47777], 4 X[1639] - X[47881], 2 X[45315] + X[47770], 2 X[45670] + X[47756], 2 X[47777] + X[47881], 5 X[650] + 4 X[14321], X[650] + 2 X[47765], 7 X[650] + 2 X[48269], 2 X[14321] - 5 X[47765], 14 X[14321] - 5 X[48269], 7 X[47765] - X[48269], 4 X[2490] - X[48576], 2 X[3239] + X[47876], 4 X[4521] - X[47767], 8 X[4521] + X[48026], 2 X[47767] + X[48026], 5 X[4893] + X[4931], 2 X[4893] + X[4944], and many others

X(52593) lies on these lines: {2, 28910}, {11, 35092}, {513, 6544}, {514, 1639}, {650, 900}, {661, 28220}, {1566, 15614}, {2490, 48576}, {3239, 28169}, {3711, 4775}, {4521, 47767}, {4776, 31992}, {4777, 4800}, {4976, 14350}, {4988, 28151}, {5219, 23598}, {7988, 28537}, {14425, 47764}, {17718, 47822}, {25666, 47754}, {27929, 28886}, {28209, 30792}, {29370, 48180}, {30563, 47763}, {30565, 47880}, {31250, 47891}, {31287, 47755}, {44567, 47769}, {45326, 48563}, {45661, 48560}, {47960, 48557}, {48095, 48550}, {48166, 48193}

X(52593) = midpoint of X(4776) and X(31992)
X(52593) = reflection of X(i) in X(j) for these {i,j}: {43052, 23598}, {47761, 45684}
X(52593) = complement of the isotomic conjugate of X(4767)
X(52593) = X(i)-complementary conjugate of X(j) for these (i,j): {45, 116}, {100, 21242}, {101, 34824}, {109, 17051}, {692, 551}, {1110, 4777}, {1252, 47779}, {1405, 4904}, {2099, 17059}, {2177, 11}, {3679, 21252}, {3711, 124}, {4273, 17761}, {4752, 141}, {4767, 2887}, {4814, 46100}, {32739, 4850}
X(52593) = X(514)-Ceva conjugate of X(4777)
X(52593) = X(i)-isoconjugate of X(j) for these (i,j): {89, 6014}, {4588, 39963}, {4604, 41436}, {34073, 36588}
X(52593) = X(190)-Dao conjugate of X(3679)
X(52593) = crosspoint of X(i) and X(j) for these (i,j): {2, 4767}, {514, 6006}
X(52593) = crosssum of X(101) and X(6014)
X(52593) = crossdifference of every pair of points on line {999, 2163}
X(52593) = barycentric product X(i)*X(j) for these {i,j}: {514, 36911}, {522, 16236}, {900, 36593}, {3241, 4777}, {3679, 6006}, {4029, 47683}, {4791, 16670}, {4893, 30829}
X(52593) = barycentric quotient X(i)/X(j) for these {i,j}: {2177, 6014}, {3241, 4597}, {4775, 41436}, {4777, 36588}, {4791, 40029}, {4814, 4900}, {4893, 39963}, {6006, 39704}, {8656, 2163}, {16236, 664}, {16670, 4604}, {36593, 4555}, {36911, 190}
X(52593) = {X(1639),X(47777)}-harmonic conjugate of X(47881)

---

X(52594) = X(9)X(4040)∩X(220)X(663)

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

X(52594) lies on these lines: {9, 4040}, {220, 663}, {514, 1212}, {650, 6362}, {657, 21789}, {905, 918}, {1146, 4147}, {3693, 44448}, {3900, 10581}, {4130, 4990}, {4406, 26059}, {4449, 34522}, {4515, 4546}, {6554, 47793}, {6589, 23877}, {6603, 48294}, {8676, 46388}, {9367, 40464}, {16601, 21185}, {32561, 48387}, {35128, 35508}, {46835, 47794}

X(52594) = complement of the isotomic conjugate of X(3939)
X(52594) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 17059}, {32, 4904}, {41, 116}, {55, 21252}, {100, 17047}, {101, 17046}, {163, 17050}, {560, 3756}, {644, 626}, {646, 40379}, {692, 2886}, {906, 18639}, {1110, 17072}, {1253, 124}, {1334, 21253}, {1415, 21258}, {1576, 3742}, {1917, 16614}, {1918, 8286}, {2149, 46399}, {2175, 11}, {2205, 17058}, {3699, 21235}, {3939, 2887}, {5546, 21240}, {6065, 21260}, {6066, 513}, {8641, 46100}, {9447, 1086}, {9448, 1015}, {14827, 26932}, {23990, 4885}, {32656, 34822}, {32739, 142}, {41280, 17071}, {52370, 127}
X(52594) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 17059}, {4560, 3900}
X(52594) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 17059}, {210, 4552}
X(52594) = crosspoint of X(2) and X(3939)
X(52594) = crosssum of X(6) and X(3676)
X(52594) = crossdifference of every pair of points on line {1486, 1617}
X(52594) = barycentric product X(i)*X(j) for these {i,j}: {200, 4905}, {220, 47676}, {650, 25082}, {657, 17234}, {1021, 3970}, {3239, 4253}, {3873, 3900}, {3939, 17059}, {3941, 4397}, {4130, 17092}, {4560, 40599}, {5546, 21946}, {6065, 23761}, {7253, 22277}, {8641, 33933}
X(52594) = barycentric quotient X(i)/X(j) for these {i,j}: {3873, 4569}, {3941, 934}, {4253, 658}, {4905, 1088}, {17092, 36838}, {17234, 46406}, {22277, 4566}, {25082, 4554}, {40599, 4552}

---

X(52595) = X(6)X(4091)∩X(905)X(3910)

Barycentrics    a^2*(b - c)*(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(52595) lies on these lines: {6, 4091}, {824, 6586}, {905, 3910}, {1459, 48387}, {3310, 3669}, {3666, 6332}, {3752, 14837}, {6129, 42337}, {8676, 22090}, {23880, 24782}, {29162, 43051}

X(52595) = complement of the isotomic conjugate of X(1461)
X(52595) = X(i)-complementary conjugate of X(j) for these (i,j): {32, 5514}, {109, 21244}, {560, 13609}, {604, 124}, {658, 21235}, {934, 626}, {1042, 21253}, {1106, 116}, {1262, 21260}, {1397, 26932}, {1407, 21252}, {1410, 127}, {1415, 1329}, {1461, 2887}, {1501, 35508}, {1977, 34530}, {2199, 46663}, {4569, 40379}, {4617, 17047}, {6614, 17046}, {7045, 21262}, {7366, 17059}, {9247, 40616}, {16947, 34589}, {23979, 513}, {24027, 3835}, {32660, 34823}, {32714, 21243}, {52410, 11}, {52411, 123}
X(52595) = crosspoint of X(2) and X(1461)
X(52595) = crosssum of X(6) and X(3239)
X(52595) = crossdifference of every pair of points on line {197, 1604}

---

X(52596) = X(106)X(2723)∩X(514)X(676)

Barycentrics    -((b - c)*(5*a^3 - 3*a^2*b - a*b^2 - b^3 - 3*a^2*c + 2*a*b*c + b^2*c - a*c^2 + b*c^2 - c^3)) : :
X(52596) = 3 X[551] + X[20517], 3 X[1638] + X[4162], 5 X[3616] - X[6332], X[4041] - 3 X[46919], X[4449] + 3 X[47800], 3 X[8643] + X[48398], 3 X[14413] - X[30719], 3 X[25055] - X[45683], 9 X[25055] - X[48272], 3 X[45683] - X[48272], 3 X[26275] + X[48346], 3 X[47757] + X[48322], 3 X[47758] + X[48338], 3 X[47801] + X[48334]

X(52596) lies on these lines: {1, 14837}, {2, 4163}, {106, 2723}, {513, 30723}, {514, 676}, {522, 905}, {551, 20517}, {663, 3676}, {1459, 4040}, {1638, 4162}, {1769, 4905}, {3333, 4091}, {3616, 6332}, {3667, 14353}, {3756, 6547}, {3900, 7658}, {4041, 46919}, {4449, 47800}, {4521, 6588}, {7649, 28147}, {8643, 48398}, {8710, 17072}, {11125, 28169}, {13462, 42756}, {14413, 30719}, {14414, 21186}, {21188, 28292}, {23696, 28011}, {25055, 45683}, {26275, 48346}, {28878, 48099}, {47757, 48322}, {47758, 48338}, {47801, 48334}

X(52596) = midpoint of X(i) and X(j) for these {i,j}: {1, 14837}, {663, 3676}, {1459, 7661}, {6129, 21172}, {21188, 48294}
X(52596) = complement of X(4163)
X(52596) = complement of the isogonal conjugate of X(6614)
X(52596) = complement of the isotomic conjugate of X(4626)
X(52596) = X(i)-complementary conjugate of X(j) for these (i,j): {56, 5514}, {269, 124}, {479, 21252}, {603, 40616}, {604, 13609}, {658, 21244}, {692, 5574}, {738, 116}, {934, 1329}, {1106, 1146}, {1262, 20317}, {1357, 34530}, {1397, 35508}, {1398, 6506}, {1407, 26932}, {1408, 34591}, {1415, 6554}, {1461, 3452}, {2175, 17426}, {4617, 141}, {4626, 2887}, {4637, 21246}, {6614, 10}, {7023, 11}, {7053, 123}, {7099, 16596}, {7339, 513}, {7366, 1086}, {23971, 4885}, {24013, 17072}, {24027, 4521}, {32714, 41883}, {36059, 42018}, {36838, 626}, {43932, 46100}
X(52596) = X(1418)-Dao conjugate of X(35312)
X(52596) = cevapoint of X(6129) and X(17427)
X(52596) = crosspoint of X(2) and X(4626)
X(52596) = crosssum of X(6) and X(4105)
X(52596) = crossdifference of every pair of points on line {198, 1615}
X(52596) = barycentric product X(i)*X(j) for these {i,j}: {514, 9778}, {4391, 34033}
X(52596) = barycentric quotient X(i)/X(j) for these {i,j}: {9778, 190}, {34033, 651}

---

X(52597) = X(116)X(38974)∩X(514)X(6589)

Barycentrics    a^2*(b - c)*(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(52597) lies on these lines: {116, 38974}, {514, 6589}, {647, 8045}, {810, 2774}, {905, 3666}, {1019, 21828}, {1086, 18334}, {3310, 48003}, {4359, 23685}, {4707, 16754}, {7180, 15309}, {7265, 28606}, {23875, 25098}, {24900, 47678}, {43060, 48064}

X(52597) = complement of the isotomic conjugate of X(4556)
X(52597) = X(i)-complementary conjugate of X(j) for these (i,j): {58, 21253}, {110, 21245}, {163, 3454}, {249, 21260}, {560, 6627}, {593, 21252}, {849, 116}, {1101, 3835}, {1333, 125}, {1415, 34829}, {1437, 127}, {1576, 1211}, {1919, 24040}, {1980, 23991}, {2150, 124}, {2206, 8287}, {4556, 2887}, {4610, 21235}, {4623, 40379}, {4636, 21244}, {7342, 4904}, {14574, 16589}, {16947, 8286}, {23357, 513}, {23963, 650}, {23995, 514}, {24041, 21262}, {32661, 21530}, {32671, 3814}, {36069, 21237}, {41937, 14298}
X(52597) = X(i)-Ceva conjugate of X(j) for these (i,j): {7372, 513}, {15412, 514}
X(52597) = X(14570)-Dao conjugate of X(17167)
X(52597) = crosspoint of X(2) and X(4556)
X(52597) = crosssum of X(6) and X(4024)
X(52597) = crossdifference of every pair of points on line {199, 20989}
X(52597) = barycentric product X(850)*X(9563)
X(52597) = barycentric quotient X(9563)/X(110)
X(52597) = {X(905),X(3666)}-harmonic conjugate of X(21192)

---

X(52598) = X(2)X(2489)∩X(141)X(924)

Barycentrics    (b - c)*(b + c)*(-a^2 + b^2 + c^2)*(-a^4 - a^2*b^2 - a^2*c^2 + 2*b^2*c^2) : :
X(52598) = 3 X[647] - X[4580], 3 X[3267] + X[4580]

X(52598) lies on these lines: {2, 2489}, {3, 9489}, {127, 5099}, {141, 924}, {512, 48044}, {520, 24284}, {523, 4885}, {647, 3267}, {850, 47133}, {2485, 35522}, {2799, 47125}, {3221, 11574}, {3788, 7631}, {7494, 8644}, {7630, 23285}, {7816, 18313}, {9426, 19126}, {14376, 15421}, {15526, 35078}

X(52598) = midpoint of X(i) and X(j) for these {i,j}: {647, 3267}, {850, 47133}, {2485, 35522}
X(52598) = complement of X(2489)
X(52598) = complement of the isogonal conjugate of X(4563)
X(52598) = medial-isogonal conjugate of X(6388)
X(52598) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 6388}, {3, 16592}, {48, 1084}, {63, 115}, {69, 8287}, {72, 6627}, {77, 17058}, {99, 226}, {110, 16583}, {162, 3767}, {163, 1196}, {190, 50036}, {222, 16613}, {249, 16612}, {304, 125}, {305, 21253}, {326, 15526}, {332, 26932}, {333, 6506}, {348, 8286}, {394, 16573}, {525, 24040}, {643, 46835}, {645, 20262}, {648, 24005}, {656, 23991}, {662, 6}, {670, 20305}, {799, 5}, {811, 13567}, {906, 21838}, {1101, 2485}, {1102, 122}, {1331, 16589}, {1332, 1213}, {1414, 3772}, {1437, 6377}, {1444, 1086}, {1790, 1015}, {1792, 13609}, {1812, 1146}, {1813, 2092}, {1959, 41181}, {2167, 47421}, {2327, 35508}, {3926, 34846}, {3964, 16595}, {4020, 35971}, {4556, 40941}, {4558, 37}, {4561, 1211}, {4563, 10}, {4565, 20227}, {4567, 3239}, {4570, 2509}, {4571, 38930}, {4573, 1210}, {4575, 39}, {4589, 26012}, {4590, 8062}, {4592, 2}, {4593, 5943}, {4599, 5305}, {4601, 20316}, {4602, 21243}, {4610, 942}, {4612, 40942}, {4620, 521}, {4623, 34830}, {4625, 16608}, {4631, 34831}, {4637, 17054}, {5546, 20310}, {6507, 35071}, {6514, 35072}, {6516, 17056}, {6517, 18592}, {7257, 41883}, {17206, 11}, {17932, 16609}, {24037, 30476}, {24039, 5181}, {24041, 525}, {32661, 16584}, {34055, 3124}, {34537, 21259}, {36085, 43291}, {37216, 43620}, {46254, 520}, {47389, 4369}, {52378, 6588}
X(52598) = X(163)-isoconjugate of X(47847)
X(52598) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 47847}, {3933, 4576}
X(52598) = crossdifference of every pair of points on line {3053, 27369}
X(52598) = barycentric product X(i)*X(j) for these {i,j}: {525, 7754}, {656, 18056}, {850, 52016}, {4580, 19568}
X(52598) = barycentric quotient X(i)/X(j) for these {i,j}: {523, 47847}, {7754, 648}, {18056, 811}, {19568, 41676}, {52016, 110}

---

X(52599) = X(2)X(17925)∩X(3)X(6002)

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

X(52599) lies on these lines: {2, 17925}, {3, 6002}, {127, 1566}, {339, 16573}, {440, 6544}, {514, 20315}, {525, 656}, {650, 1577}, {905, 14208}, {4129, 4521}, {4988, 27731}, {6356, 7216}, {7254, 24560}, {15526, 35092}, {17899, 20948}, {24018, 40628}, {29013, 50329}, {29162, 31946}, {34591, 39003}, {34846, 40629}

X(52599) = complement of X(17925)
X(52599) = complement of the isogonal conjugate of X(4574)
X(52599) = isotomic conjugate of the polar conjugate of X(50329)
X(52599) = X(i)-complementary conjugate of X(j) for these (i,j): {3, 17761}, {48, 244}, {71, 11}, {72, 116}, {73, 4904}, {100, 34830}, {101, 942}, {212, 4858}, {219, 34589}, {228, 1086}, {306, 21252}, {644, 34831}, {692, 40940}, {765, 30476}, {810, 6547}, {872, 6388}, {906, 1125}, {1016, 21259}, {1018, 5}, {1110, 525}, {1214, 17059}, {1252, 8062}, {1331, 3739}, {1332, 3741}, {1334, 6506}, {1409, 3756}, {1802, 34591}, {1813, 3742}, {2197, 8286}, {2200, 1015}, {2289, 34588}, {2318, 26932}, {3690, 8287}, {3694, 124}, {3695, 21253}, {3939, 6708}, {3949, 125}, {3952, 20305}, {3958, 46660}, {3990, 2968}, {4033, 21243}, {4069, 41883}, {4551, 16608}, {4557, 226}, {4559, 1210}, {4561, 21240}, {4571, 21246}, {4574, 10}, {4575, 17045}, {4587, 960}, {6516, 17050}, {7084, 17877}, {8611, 46100}, {22061, 40608}, {22356, 34590}, {23067, 142}, {23990, 16612}, {32656, 3666}, {32739, 40941}, {36059, 3946}, {52370, 1146}, {52386, 34846}, {52387, 127}
X(52599) = X(514)-Ceva conjugate of X(525)
X(52599) = X(i)-isoconjugate of X(j) for these (i,j): {28, 29014}, {1783, 15376}, {32676, 39700}
X(52599) = X(i)-Dao conjugate of X(j) for these (i,j): {190, 306}, {15376, 39006}, {15526, 39700}, {29014, 40591}
X(52599) = crosspoint of X(514) and X(29013)
X(52599) = crosssum of X(i) and X(j) for these (i,j): {6, 43925}, {101, 29014}
X(52599) = crossdifference of every pair of points on line {1474, 2352}
X(52599) = barycentric product X(i)*X(j) for these {i,j}: {69, 50329}, {306, 29013}, {525, 3187}, {656, 18147}, {850, 42463}, {1724, 14208}, {2901, 4025}, {3267, 5301}
X(52599) = barycentric quotient X(i)/X(j) for these {i,j}: {71, 29014}, {525, 39700}, {1459, 15376}, {1724, 162}, {2901, 1897}, {3187, 648}, {5301, 112}, {18147, 811}, {29013, 27}, {42463, 110}, {50329, 4}

---

X(52600) = X(2)X(2416)∩X(4)X(47225)

Barycentrics    a^2*(b - c)*(b + c)*(a^12 - 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 + 4*a^2*b^10 - b^12 - 4*a^10*c^2 + 13*a^8*b^2*c^2 - 13*a^6*b^4*c^2 + a^4*b^6*c^2 + 5*a^2*b^8*c^2 - 2*b^10*c^2 + 5*a^8*c^4 - 13*a^6*b^2*c^4 + 18*a^4*b^4*c^4 - 9*a^2*b^6*c^4 - b^8*c^4 + a^4*b^2*c^6 - 9*a^2*b^4*c^6 + 8*b^6*c^6 - 5*a^4*c^8 + 5*a^2*b^2*c^8 - b^4*c^8 + 4*a^2*c^10 - 2*b^2*c^10 - c^12) : :

X(52600) lies on these lines: {2, 2416}, {4, 47225}, {39, 2485}, {520, 11430}, {525, 23292}, {647, 2394}, {2133, 52546}, {5024, 34212}, {8552, 14566}, {8673, 18570}, {8675, 51739}, {14376, 15421}, {35473, 42658}, {42733, 47221}

X(52600) = complement of the isotomic conjugate of X(34568)
X(52600) = X(i)-complementary conjugate of X(j) for these (i,j): {2159, 16177}, {34568, 2887}, {36131, 113}, {40353, 34846}
X(52600) = crosspoint of X(2) and X(34568)
X(52600) = crosssum of X(6) and X(14401)

---

X(52601) = X(1)X(2533)∩X(2)X(4705)

Barycentrics    (b - c)*(a^3 + a*b*c + b^2*c + b*c^2) : :
X(52601) = 3 X[2] + X[17166], X[650] - 3 X[48564], 2 X[31288] - 3 X[48564], X[659] - 3 X[47818], X[659] + 3 X[47889], X[4978] + 3 X[47818], X[4978] - 3 X[47889], X[661] - 3 X[47839], X[663] + 3 X[4379], 3 X[4379] - X[50352], X[667] - 3 X[47820], X[693] + 3 X[47820], X[1491] - 3 X[47795], X[1577] - 3 X[47833], X[4367] + 3 X[47833], X[1734] - 3 X[47823], 3 X[47823] + X[48301], X[2254] - 3 X[48569], X[48305] + 3 X[48569], and many others

X(52601) lies on these lines: {1, 2533}, {2, 4705}, {11, 5099}, {140, 44824}, {350, 16737}, {512, 4369}, {513, 11813}, {514, 1125}, {519, 45332}, {523, 8043}, {525, 49290}, {649, 48273}, {650, 31288}, {656, 39547}, {659, 4978}, {661, 47839}, {663, 3720}, {667, 693}, {676, 29142}, {784, 905}, {812, 50512}, {814, 4823}, {826, 4458}, {830, 3837}, {832, 47843}, {838, 50516}, {1015, 35078}, {1019, 4010}, {1193, 4449}, {1491, 19863}, {1577, 2787}, {1734, 47823}, {1960, 29051}, {2254, 48305}, {2487, 4843}, {2530, 19894}, {3624, 4824}, {3669, 48220}, {3700, 29090}, {3716, 6372}, {3733, 30591}, {3741, 17072}, {3762, 47872}, {3777, 48234}, {3801, 29154}, {3803, 48089}, {3887, 48233}, {3900, 15584}, {3907, 48328}, {4040, 21146}, {4041, 24924}, {4063, 48279}, {4083, 48295}, {4106, 50515}, {4122, 29292}, {4142, 29312}, {4151, 9508}, {4160, 21051}, {4170, 4784}, {4378, 4391}, {4401, 29362}, {4448, 47970}, {4455, 7199}, {4490, 25512}, {4504, 29268}, {4560, 14419}, {4728, 50523}, {4730, 47836}, {4761, 4879}, {4762, 6050}, {4782, 29302}, {4791, 29324}, {4794, 29246}, {4801, 47804}, {4806, 15309}, {4808, 47809}, {4822, 31148}, {4834, 47762}, {4885, 8678}, {4977, 48058}, {4983, 7192}, {6004, 24720}, {6133, 28147}, {6367, 21196}, {7178, 29094}, {7234, 18154}, {7265, 50342}, {8062, 8672}, {9422, 17793}, {9505, 38220}, {14349, 47841}, {14432, 23755}, {15283, 18344}, {17212, 50451}, {20470, 48387}, {20517, 29017}, {21003, 48152}, {21301, 26985}, {21302, 29824}, {23770, 29098}, {25666, 48005}, {28840, 48053}, {29013, 48090}, {29029, 48403}, {29047, 48405}, {29066, 48221}, {29086, 48396}, {29102, 48299}, {29110, 47788}, {29128, 47712}, {29148, 48202}, {29186, 48098}, {29200, 49288}, {29224, 47887}, {29328, 48064}, {29366, 48294}, {29637, 35352}, {30795, 47816}, {30835, 47912}, {31147, 50526}, {31251, 47814}, {31286, 50504}, {43067, 48099}, {45320, 50517}, {47203, 48288}, {47697, 47819}, {47715, 50340}, {47716, 48103}, {47718, 48223}, {47719, 47798}, {47720, 47771}, {47760, 47956}, {47761, 50501}, {47799, 48402}, {47803, 47965}, {47812, 48150}, {47813, 48131}, {47821, 47949}, {47822, 47959}, {47827, 48407}, {47832, 48144}, {47838, 48024}, {47842, 48207}, {47844, 48209}, {47888, 47975}, {47893, 48409}, {47918, 48553}, {47921, 48561}, {47967, 48197}, {48012, 48218}, {48080, 48570}, {48108, 48351}, {48122, 48578}, {48237, 48410}, {48246, 50345}, {48248, 48406}, {48253, 48336}, {48265, 48320}, {48282, 49997}, {48321, 48392}, {48339, 48573}, {48367, 48579}, {48556, 50328}, {48563, 50508}, {50488, 50524}

X(52601) = midpoint of X(i) and X(j) for these {i,j}: {1, 2533}, {649, 48273}, {656, 39547}, {659, 4978}, {663, 50352}, {667, 693}, {905, 7662}, {1019, 4010}, {1577, 4367}, {1734, 48301}, {2254, 48305}, {2530, 47694}, {3733, 30591}, {3762, 48323}, {3801, 47682}, {3803, 48089}, {4040, 21146}, {4041, 48291}, {4063, 48279}, {4106, 50515}, {4170, 4784}, {4378, 4391}, {4455, 7199}, {4458, 8045}, {4560, 48393}, {4705, 17166}, {4761, 4879}, {4791, 48343}, {4983, 7192}, {7178, 48290}, {7265, 50342}, {14419, 47834}, {21003, 48152}, {43067, 48099}, {47715, 50340}, {47716, 48103}, {47818, 47889}, {47844, 50330}, {48098, 48331}, {48108, 48351}, {48144, 48267}, {48248, 48406}, {48265, 48320}, {48288, 50457}, {48321, 48392}, {48339, 50355}, {50488, 50524}
X(52601) = reflection of X(i) in X(j) for these {i,j}: {650, 31288}, {2530, 19947}, {21260, 4885}, {44824, 140}, {48005, 25666}, {50504, 31286}
X(52601) = complement of X(4705)
X(52601) = complement of the isotomic conjugate of X(4623)
X(52601) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 6627}, {58, 115}, {60, 1146}, {81, 8287}, {86, 125}, {99, 3454}, {101, 6537}, {110, 1213}, {162, 50036}, {163, 16589}, {249, 514}, {250, 3239}, {261, 124}, {270, 6506}, {274, 21253}, {513, 24040}, {552, 17059}, {593, 1086}, {649, 23991}, {662, 1211}, {664, 34829}, {757, 11}, {763, 17761}, {799, 21245}, {849, 1015}, {873, 21252}, {1014, 8286}, {1098, 5514}, {1101, 650}, {1326, 41180}, {1333, 16592}, {1408, 16613}, {1412, 17058}, {1414, 442}, {1437, 16573}, {1444, 34846}, {1474, 6388}, {1509, 116}, {1576, 21838}, {1790, 15526}, {2185, 26932}, {2206, 1084}, {4556, 2}, {4558, 440}, {4565, 17056}, {4567, 4129}, {4570, 661}, {4573, 17052}, {4575, 18591}, {4590, 3835}, {4591, 3936}, {4592, 21530}, {4600, 31946}, {4603, 46826}, {4610, 141}, {4612, 3452}, {4623, 2887}, {4629, 41809}, {4631, 21244}, {4636, 9}, {4637, 18635}, {5546, 38930}, {6578, 3634}, {6629, 5099}, {7054, 13609}, {7305, 3271}, {7340, 17072}, {7341, 3756}, {7342, 16614}, {13486, 5949}, {16887, 46654}, {17187, 15449}, {17200, 46665}, {17206, 127}, {17209, 35088}, {17930, 20546}, {17940, 10026}, {18020, 20316}, {18604, 16595}, {23357, 6586}, {24000, 14298}, {24037, 21260}, {24041, 513}, {30593, 46660}, {32671, 49758}, {34537, 21262}, {36066, 3836}, {36069, 44}, {37140, 908}, {44709, 39019}, {51370, 36471}, {52378, 1577}, {52394, 7668}, {52558, 3120}
X(52601) = X(i)-Dao conjugate of X(j) for these (i,j): {4425, 21295}, {4576, 16696}
X(52601) = crosspoint of X(2) and X(4623)
X(52601) = crosssum of X(6) and X(50487)
X(52601) = crossdifference of every pair of points on line {1030, 17735}
X(52601) = barycentric product X(i)*X(j) for these {i,j}: {513, 3770}, {514, 4418}, {693, 5277}
X(52601) = barycentric quotient X(i)/X(j) for these {i,j}: {3770, 668}, {4418, 190}, {5277, 100}
X(52601) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 17166, 4705}, {650, 48564, 31288}, {659, 47889, 4978}, {663, 4379, 50352}, {693, 47820, 667}, {2530, 47796, 19947}, {4041, 24924, 47837}, {4170, 48568, 4784}, {4367, 47833, 1577}, {4378, 47875, 4391}, {4560, 47834, 48393}, {4978, 47818, 659}, {7192, 47840, 4983}, {14419, 48393, 4560}, {47694, 47796, 2530}, {47823, 48301, 1734}, {47832, 48144, 48267}, {47837, 48291, 4041}, {47844, 48209, 50330}, {47872, 48323, 3762}, {48305, 48569, 2254}, {48339, 48573, 50355}

---

X(52602) = X(2)X(4079)∩X(3)X(23399)

Barycentrics    (b - c)*(a^3*b + a^3*c + a^2*b*c + b^2*c^2) : :
X(52602) = 3 X[2] + X[17159], X[21348] - 3 X[47761], X[20906] + 3 X[47762], X[20908] + 3 X[48566], X[21225] - 5 X[27013], X[21834] - 5 X[24924], X[47130] + 3 X[47758]

X(52602) lies on these lines: {2, 4079}, {3, 23399}, {116, 5099}, {239, 17212}, {512, 17066}, {513, 3739}, {514, 50014}, {523, 2487}, {649, 3261}, {665, 786}, {768, 4025}, {798, 4374}, {802, 4411}, {824, 21192}, {834, 21191}, {1086, 35078}, {3666, 21348}, {3733, 4107}, {3798, 28623}, {3835, 44316}, {4036, 24287}, {4359, 20906}, {4406, 20979}, {4785, 45659}, {6586, 31286}, {7199, 21832}, {8060, 24285}, {16737, 18196}, {17072, 21262}, {20367, 21390}, {20908, 48566}, {21225, 27013}, {21834, 24924}, {30765, 50332}, {47130, 47758}

X(52602) = midpoint of X(i) and X(j) for these {i,j}: {649, 3261}, {798, 4374}, {4025, 47129}, {4079, 17159}, {4406, 20979}, {7199, 21832}
X(52602) = reflection of X(i) in X(j) for these {i,j}: {6586, 31286}, {42327, 17066}
X(52602) = complement of X(4079)
X(52602) = complement of the isogonal conjugate of X(4610)
X(52602) = medial-isogonal conjugate of X(6627)
X(52602) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 6627}, {28, 6388}, {58, 16592}, {81, 115}, {86, 8287}, {99, 1211}, {100, 6537}, {110, 16589}, {163, 21838}, {249, 650}, {250, 2509}, {261, 26932}, {274, 125}, {310, 21253}, {513, 23991}, {514, 24040}, {552, 4904}, {593, 1015}, {643, 38930}, {648, 50036}, {662, 1213}, {670, 21245}, {757, 1086}, {763, 244}, {799, 3454}, {849, 6377}, {873, 116}, {1014, 17058}, {1098, 13609}, {1101, 6586}, {1333, 1084}, {1412, 16613}, {1414, 17056}, {1434, 8286}, {1444, 15526}, {1509, 11}, {1790, 16573}, {1931, 41180}, {2185, 1146}, {4554, 34829}, {4556, 37}, {4558, 18591}, {4563, 21530}, {4565, 2092}, {4567, 661}, {4573, 442}, {4590, 513}, {4592, 440}, {4594, 46826}, {4596, 41809}, {4600, 4129}, {4601, 31946}, {4610, 10}, {4612, 9}, {4616, 18635}, {4622, 3936}, {4623, 141}, {4625, 17052}, {4631, 1329}, {4636, 1212}, {4637, 1834}, {6064, 20317}, {6578, 44307}, {6628, 17761}, {7054, 35508}, {7058, 5514}, {7304, 5518}, {7305, 21138}, {7340, 4885}, {16696, 15449}, {16697, 39019}, {16702, 23992}, {16703, 46654}, {16707, 46665}, {16741, 5099}, {17206, 34846}, {17929, 44396}, {17930, 20337}, {18604, 35071}, {18605, 39013}, {18609, 39021}, {19623, 41179}, {23582, 14298}, {24037, 3835}, {24041, 514}, {26856, 34530}, {30576, 35092}, {30581, 35076}, {34537, 21260}, {36066, 3912}, {36069, 49758}, {37140, 44}, {46103, 6506}, {46254, 20316}, {51369, 35088}, {52376, 3124}, {52379, 124}, {52558, 16726}
X(52602) = X(4576)-Dao conjugate of X(16887)
X(52602) = crossdifference of every pair of points on line {18755, 21788}
X(52602) = barycentric product X(i)*X(j) for these {i,j}: {513, 18059}, {514, 17499}
X(52602) = barycentric quotient X(i)/X(j) for these {i,j}: {17499, 190}, {18059, 668}
X(52602) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 17159, 4079}, {17072, 48044, 21262}

---

X(52603) = X(3)X(54)∩X(110)X(351)

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

X(52603) lies on these lines: {2, 45943}, {3, 54}, {110, 351}, {186, 38936}, {250, 4240}, {323, 14355}, {476, 2407}, {523, 31941}, {532, 46824}, {533, 46825}, {539, 8154}, {562, 12044}, {925, 12092}, {930, 933}, {1291, 46590}, {1511, 34210}, {1994, 18114}, {3153, 51254}, {4226, 14366}, {4230, 40173}, {5899, 38539}, {6149, 17104}, {7669, 52124}, {7722, 13496}, {8016, 34394}, {8017, 34395}, {8562, 43969}, {9766, 47596}, {10411, 44814}, {11597, 38897}, {13398, 46963}, {13557, 50435}, {14185, 23871}, {14187, 23870}, {14270, 51478}, {14385, 22115}, {14611, 35345}, {14670, 50461}, {14889, 39170}, {14966, 23357}, {15080, 15919}, {17708, 34761}, {19294, 48354}, {19295, 48356}, {32423, 50474}, {41724, 51458}, {43083, 43965}

X(52603) = reflection of X(38897) in X(11597)
X(52603) = isogonal conjugate of X(10412)
X(52603) = isogonal conjugate of the anticomplement of X(8562)
X(52603) = isogonal conjugate of the isotomic conjugate of X(10411)
X(52603) = isotomic conjugate of the polar conjugate of X(14591)
X(52603) = isogonal conjugate of the polar conjugate of X(14590)
X(52603) = X(36134)-anticomplementary conjugate of X(14731)
X(52603) = X(i)-Ceva conjugate of X(j) for these (i,j): {250, 1511}, {933, 43969}, {10420, 110}, {14590, 14591}, {18879, 52557}, {44769, 32661}
X(52603) = X(i)-cross conjugate of X(j) for these (i,j): {11597, 14587}, {14270, 50}, {44808, 186}
X(52603) = X(i)-isoconjugate of X(j) for these (i,j): {1, 10412}, {10, 43082}, {19, 14592}, {75, 15475}, {91, 43088}, {92, 14582}, {94, 661}, {115, 32680}, {125, 36129}, {158, 43083}, {265, 24006}, {338, 32678}, {476, 1109}, {523, 2166}, {656, 6344}, {798, 20573}, {810, 18817}, {897, 51479}, {1141, 2618}, {1577, 1989}, {2643, 35139}, {2970, 36061}, {3708, 46456}, {5627, 36035}, {6070, 36047}, {11060, 20948}, {14208, 18384}, {14560, 23994}, {23894, 43084}, {52356, 52382}
X(52603) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 10412}, {6, 14592}, {50, 24978}, {94, 36830}, {137, 35591}, {206, 15475}, {338, 18334}, {523, 11597}, {850, 40604}, {1147, 43083}, {1577, 34544}, {2970, 16221}, {3284, 41079}, {6070, 35581}, {6344, 40596}, {6593, 51479}, {14582, 22391}, {18402, 23290}, {18817, 39062}, {20573, 31998}, {34116, 43088}
X(52603) = cevapoint of X(i) and X(j) for these (i,j): {50, 14270}, {526, 1154}, {3003, 6140}
X(52603) = crosspoint of X(i) and X(j) for these (i,j): {110, 1291}, {687, 18831}, {10411, 14590}, {35139, 38342}
X(52603) = crosssum of X(i) and X(j) for these (i,j): {523, 45147}, {686, 15451}, {14582, 15475}
X(52603) = trilinear pole of line {50, 18334}
X(52603) = crossdifference of every pair of points on line {115, 12077}
X(52603) = barycentric product X(i)*X(j) for these {i,j}: {3, 14590}, {6, 10411}, {15, 17403}, {16, 17402}, {50, 99}, {69, 14591}, {110, 323}, {112, 52437}, {186, 4558}, {249, 526}, {250, 8552}, {340, 32661}, {524, 51478}, {648, 22115}, {662, 6149}, {670, 19627}, {1101, 32679}, {1154, 18315}, {1273, 14586}, {1291, 40604}, {1511, 44769}, {1576, 7799}, {1986, 43755}, {2407, 14385}, {2421, 14355}, {2624, 24041}, {2715, 51383}, {3268, 23357}, {4563, 34397}, {4575, 52414}, {4585, 17104}, {4590, 14270}, {6148, 32640}, {8115, 44068}, {8116, 44067}, {10409, 19294}, {10410, 19295}, {10420, 34834}, {14587, 41078}, {14918, 15958}, {14999, 52179}, {16186, 47443}, {44427, 47390}
X(52603) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 14592}, {6, 10412}, {32, 15475}, {50, 523}, {99, 20573}, {110, 94}, {112, 6344}, {163, 2166}, {184, 14582}, {186, 14618}, {187, 51479}, {249, 35139}, {250, 46456}, {323, 850}, {526, 338}, {571, 43088}, {577, 43083}, {648, 18817}, {1101, 32680}, {1154, 18314}, {1273, 15415}, {1333, 43082}, {1511, 41079}, {1576, 1989}, {1983, 6757}, {2088, 23105}, {2290, 2618}, {2420, 14254}, {2439, 25043}, {2624, 1109}, {3043, 44427}, {3268, 23962}, {4558, 328}, {5467, 43084}, {6149, 1577}, {7799, 44173}, {8552, 339}, {10411, 76}, {10420, 40427}, {11062, 23290}, {11597, 24978}, {14270, 115}, {14355, 43665}, {14385, 2394}, {14574, 11060}, {14586, 1141}, {14590, 264}, {14591, 4}, {14966, 14356}, {17402, 301}, {17403, 300}, {18315, 46138}, {19627, 512}, {22115, 525}, {23357, 476}, {23963, 14560}, {23995, 32678}, {25044, 2413}, {32640, 5627}, {32661, 265}, {32679, 23994}, {34394, 20579}, {34395, 20578}, {34397, 2501}, {35192, 52356}, {44067, 2593}, {44068, 2592}, {47230, 2970}, {51394, 18557}, {51478, 671}, {52179, 14223}, {52437, 3267}, {52557, 15328}
X(52603) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 4558, 36829}, {110, 15329, 30510}, {110, 47053, 15329}, {5467, 15329, 47053}

---

X(52604) = X(3)X(9530)∩X(4)X(7668)

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

X(52604) lies on these lines: {3, 9530}, {4, 7668}, {6, 1987}, {25, 1989}, {50, 44096}, {53, 40981}, {107, 1624}, {112, 1576}, {133, 43919}, {159, 1033}, {160, 1249}, {162, 23845}, {206, 51936}, {237, 1990}, {264, 35222}, {297, 5201}, {338, 47202}, {523, 37937}, {524, 15143}, {648, 1634}, {1593, 5621}, {1598, 11623}, {1609, 52439}, {1968, 19136}, {2211, 51735}, {2967, 34990}, {3003, 34854}, {3172, 19153}, {5467, 41679}, {7480, 46995}, {8266, 17907}, {14192, 20470}, {14570, 23181}, {14581, 21177}, {15897, 41334}, {16813, 23286}, {18105, 32696}, {20975, 47228}, {32675, 32676}, {34777, 41489}, {37777, 47238}, {44889, 47204}, {50401, 52464}

X(52604) = midpoint of X(41678) and X(46151)
X(52604) = isogonal conjugate of the anticomplement of X(17434)
X(52604) = isogonal conjugate of the isotomic conjugate of X(35360)
X(52604) = polar conjugate of the isotomic conjugate of X(1625)
X(52604) = X(i)-Ceva conjugate of X(j) for these (i,j): {250, 14577}, {14560, 23347}, {32230, 6}, {35360, 1625}
X(52604) = X(i)-cross conjugate of X(j) for these (i,j): {15451, 51}, {42293, 6}, {51513, 25}
X(52604) = X(i)-isoconjugate of X(j) for these (i,j): {19, 15414}, {54, 14208}, {63, 15412}, {69, 2616}, {75, 23286}, {95, 656}, {97, 1577}, {275, 24018}, {276, 822}, {304, 2623}, {339, 36134}, {520, 40440}, {525, 2167}, {661, 34386}, {810, 34384}, {850, 2169}, {933, 17879}, {1102, 15422}, {1969, 46088}, {2148, 3267}, {2190, 3265}, {2632, 18831}, {4592, 8901}, {14533, 20948}, {15958, 23994}, {17094, 44687}, {18315, 20902}, {20879, 39181}, {26942, 39177}, {37754, 42405}
X(52604) = X(i)-Dao conjugate of X(j) for these (i,j): {5, 3265}, {6, 15414}, {95, 40596}, {130, 2972}, {137, 339}, {206, 23286}, {216, 3267}, {525, 40588}, {850, 14363}, {3162, 15412}, {3268, 18402}, {5139, 8901}, {15450, 15526}, {23107, 35441}, {34384, 39062}, {34386, 36830}, {36793, 39019}
X(52604) = cevapoint of X(i) and X(j) for these (i,j): {51, 15451}, {12077, 27371}, {13366, 39201}
X(52604) = crosspoint of X(107) and X(112)
X(52604) = crosssum of X(i) and X(j) for these (i,j): {2, 44003}, {520, 525}, {523, 52585}
X(52604) = trilinear pole of line {51, 217}
X(52604) = crossdifference of every pair of points on line {8552, 15526}
X(52604) = barycentric product X(i)*X(j) for these {i,j}: {4, 1625}, {5, 112}, {6, 35360}, {19, 2617}, {25, 14570}, {51, 648}, {53, 110}, {99, 3199}, {107, 216}, {162, 1953}, {217, 6528}, {249, 51513}, {250, 12077}, {270, 35307}, {324, 1576}, {343, 32713}, {393, 23181}, {418, 15352}, {467, 32734}, {476, 11062}, {662, 2181}, {811, 2179}, {827, 27371}, {925, 14576}, {930, 14577}, {933, 36412}, {1173, 35318}, {1301, 42459}, {1303, 27359}, {1568, 32695}, {1783, 18180}, {1990, 36831}, {2290, 36129}, {2715, 39569}, {4558, 14569}, {5562, 6529}, {5994, 6116}, {5995, 6117}, {6331, 40981}, {6368, 23964}, {6570, 8887}, {8750, 17167}, {13450, 32661}, {14129, 32737}, {14213, 32676}, {14560, 14918}, {15451, 23582}, {17434, 32230}, {17500, 35325}, {20031, 44716}, {23290, 23357}, {24019, 44706}, {26714, 39530}, {32085, 35319}, {32678, 51801}, {35320, 46884}, {41221, 47443}, {44770, 51363}
X(52604) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 15414}, {5, 3267}, {25, 15412}, {32, 23286}, {51, 525}, {53, 850}, {107, 276}, {110, 34386}, {112, 95}, {216, 3265}, {217, 520}, {324, 44173}, {648, 34384}, {1154, 45792}, {1576, 97}, {1625, 69}, {1953, 14208}, {1973, 2616}, {1974, 2623}, {2179, 656}, {2181, 1577}, {2489, 8901}, {2617, 304}, {3199, 523}, {5562, 4143}, {6368, 36793}, {6529, 8795}, {8798, 14638}, {11062, 3268}, {12077, 339}, {14569, 14618}, {14570, 305}, {14574, 14533}, {14575, 46088}, {14576, 6563}, {14577, 41298}, {15451, 15526}, {18180, 15413}, {23181, 3926}, {23290, 23962}, {23347, 43768}, {23963, 15958}, {23964, 18831}, {24019, 40440}, {27359, 42331}, {27371, 23285}, {32230, 42405}, {32676, 2167}, {32713, 275}, {34859, 19189}, {35318, 1232}, {35319, 3933}, {35360, 76}, {35442, 23107}, {40981, 647}, {41586, 45807}, {41937, 933}, {42293, 2972}, {42396, 41488}, {44088, 32320}, {44709, 30805}, {51513, 338}, {52439, 15422}
X(52604) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {112, 32713, 1576}, {648, 4230, 1634}, {1576, 32713, 23347}

---

X(52605) = X(99)X(16807)∩X(110)X(351)

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^2 - b^2 - c^2 - 2*Sqrt[3]*S) : :

X(52605) lies on these lines: {99, 16807}, {110, 351}, {249, 5994}, {338, 40854}, {648, 23895}, {1994, 11083}, {7669, 14170}, {11130, 34990}, {14570, 23896}, {18315, 38413}, {34395, 37785}, {40580, 52349}

X(52605) = X(23896)-Ceva conjugate of X(17403)
X(52605) = X(i)-isoconjugate of X(j) for these (i,j): {17, 661}, {656, 8741}, {798, 34389}, {1109, 16806}, {1577, 21461}, {2616, 36300}, {2643, 32036}, {3375, 20579}, {24006, 32585}
X(52605) = X(i)-Dao conjugate of X(j) for these (i,j): {17, 36830}, {523, 10640}, {8741, 40596}, {11130, 23871}, {15609, 30465}, {31998, 34389}
X(52605) = trilinear pole of line {61, 1994}
X(52605) = barycentric product X(i)*X(j) for these {i,j}: {61, 99}, {110, 302}, {249, 23872}, {473, 4558}, {648, 52348}, {1994, 32037}, {4563, 10642}, {5994, 11132}, {6671, 10409}, {7769, 16807}, {8838, 17402}, {11126, 23896}, {11146, 23895}, {16771, 17403}
X(52605) = barycentric quotient X(i)/X(j) for these {i,j}: {61, 523}, {99, 34389}, {110, 17}, {112, 8741}, {249, 32036}, {302, 850}, {473, 14618}, {1576, 21461}, {1625, 36300}, {1994, 23873}, {4558, 40712}, {5994, 11087}, {5995, 11139}, {6104, 23283}, {10642, 2501}, {11083, 20578}, {11126, 23871}, {11135, 6138}, {11137, 6137}, {11141, 20579}, {11146, 23870}, {16807, 2963}, {17403, 19779}, {23357, 16806}, {23872, 338}, {32037, 11140}, {32661, 32585}, {35329, 36304}, {38413, 52203}, {52348, 525}
X(52605) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 17402, 17403}, {14570, 35314, 32036}, {23896, 32036, 14570}

---

X(52606) = X(99)X(16806)∩X(110)X(351)

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^2 - b^2 - c^2 + 2*Sqrt[3]*S) : :

X(52606) lies on these lines: {99, 16806}, {110, 351}, {249, 5995}, {338, 40855}, {648, 23896}, {1994, 11088}, {7669, 14169}, {11131, 34990}, {14570, 23895}, {18315, 38414}, {34394, 37786}, {40581, 52348}

X(52606) = X(23895)-Ceva conjugate of X(17402)
X(52606) = X(i)-isoconjugate of X(j) for these (i,j): {18, 661}, {656, 8742}, {798, 34390}, {1109, 16807}, {1577, 21462}, {2616, 36301}, {2643, 32037}, {3384, 20578}, {24006, 32586}
X(52606) = X(i)-Dao conjugate of X(j) for these (i,j): {18, 36830}, {523, 10639}, {8742, 40596}, {11131, 23870}, {15610, 30468}, {31998, 34390}
X(52606) = trilinear pole of line {62, 1994}
X(52606) = barycentric product X(i)*X(j) for these {i,j}: {62, 99}, {110, 303}, {249, 23873}, {472, 4558}, {648, 52349}, {1994, 32036}, {4563, 10641}, {5995, 11133}, {6672, 10410}, {7769, 16806}, {8836, 17403}, {11127, 23895}, {11145, 23896}, {16770, 17402}
X(52606) = barycentric quotient X(i)/X(j) for these {i,j}: {62, 523}, {99, 34390}, {110, 18}, {112, 8742}, {249, 32037}, {303, 850}, {472, 14618}, {1576, 21462}, {1625, 36301}, {1994, 23872}, {4558, 40711}, {5994, 11138}, {5995, 11082}, {6105, 23284}, {10641, 2501}, {11088, 20579}, {11127, 23870}, {11134, 6138}, {11136, 6137}, {11142, 20578}, {11145, 23871}, {16806, 2963}, {17402, 19778}, {23357, 16807}, {23873, 338}, {32036, 11140}, {32661, 32586}, {35330, 36305}, {38414, 52204}, {52349, 525}
X(52606) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 17403, 17402}, {14570, 35315, 32037}, {23895, 32037, 14570}

---

X(52607) = X(4)X(774)∩X(108)X(676)

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

X(52607) lies on these lines: {4, 774}, {108, 676}, {112, 32651}, {196, 18676}, {241, 37805}, {278, 2006}, {281, 948}, {651, 653}, {823, 17926}, {1148, 7138}, {1427, 16732}, {1826, 3668}, {4341, 52033}, {4551, 4605}, {4554, 6335}, {7128, 17925}, {8755, 43035}, {17086, 17918}, {17903, 40837}, {18026, 32038}, {37770, 37798}

X(52607) = isogonal conjugate of X(23090)
X(52607) = isotomic conjugate of X(15411)
X(52607) = polar conjugate of X(7253)
X(52607) = polar conjugate of the isotomic conjugate of X(4566)
X(52607) = X(i)-Ceva conjugate of X(j) for these (i,j): {653, 1020}, {7045, 1068}, {7128, 278}, {23984, 6354}, {24032, 4}
X(52607) = X(i)-cross conjugate of X(j) for these (i,j): {523, 3668}, {647, 65}, {661, 4}, {1637, 52383}, {2501, 225}, {6354, 23984}, {6587, 10}, {7178, 40149}, {12077, 52382}
X(52607) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23090}, {3, 1021}, {9, 23189}, {21, 652}, {29, 36054}, {31, 15411}, {48, 7253}, {60, 8611}, {63, 21789}, {78, 7252}, {110, 34591}, {112, 24031}, {162, 35072}, {163, 2968}, {200, 7254}, {212, 4560}, {219, 3737}, {255, 17926}, {283, 650}, {284, 521}, {285, 10397}, {332, 3063}, {333, 1946}, {513, 2327}, {520, 2326}, {522, 2193}, {643, 7117}, {647, 1098}, {648, 2638}, {649, 1792}, {654, 1793}, {656, 7054}, {657, 1444}, {662, 3270}, {663, 1812}, {810, 7058}, {811, 39687}, {905, 2328}, {1019, 1260}, {1043, 22383}, {1146, 4575}, {1253, 15419}, {1437, 3239}, {1459, 2287}, {1789, 9404}, {1790, 3900}, {1802, 7192}, {1808, 4435}, {2150, 52355}, {2194, 6332}, {2206, 15416}, {2310, 4558}, {2322, 23224}, {2332, 4131}, {3692, 3733}, {3937, 7259}, {4091, 4183}, {4587, 18191}, {4592, 14936}, {5546, 7004}, {6061, 51664}, {6514, 18344}, {7258, 22096}, {8641, 17206}, {8750, 16731}, {18155, 52425}, {23983, 32676}, {24026, 32661}, {39177, 44707}
X(52607) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 15411}, {3, 23090}, {115, 2968}, {122, 40616}, {125, 35072}, {136, 1146}, {244, 34591}, {314, 39060}, {332, 10001}, {333, 39053}, {478, 23189}, {521, 40590}, {522, 47345}, {647, 15267}, {652, 40611}, {905, 36908}, {1021, 36103}, {1084, 3270}, {1098, 39052}, {1214, 6332}, {1249, 7253}, {1792, 5375}, {2327, 39026}, {3162, 21789}, {4560, 40837}, {5139, 14936}, {6523, 17926}, {6609, 7254}, {7054, 40596}, {7058, 39062}, {15416, 40603}, {15419, 17113}, {15526, 23983}, {16731, 26932}, {17219, 40615}, {17423, 39687}, {24031, 34591}, {26932, 40622}
X(52607) = cevapoint of X(i) and X(j) for these (i,j): {65, 647}, {225, 2501}, {523, 1826}, {650, 40950}, {661, 1254}, {1427, 7178}, {1841, 7649}, {1842, 6591}, {3064, 42385}
X(52607) = crosspoint of X(13149) and X(36118)
X(52607) = crosssum of X(652) and X(36054)
X(52607) = trilinear pole of line {65, 225}
X(52607) = crossdifference of every pair of points on line {3270, 35072}
X(52607) = barycentric product X(i)*X(j) for these {i,j}: {4, 4566}, {10, 36118}, {27, 4605}, {37, 13149}, {65, 18026}, {92, 1020}, {107, 6356}, {108, 1441}, {225, 664}, {226, 653}, {273, 4551}, {278, 4552}, {307, 36127}, {321, 32714}, {331, 4559}, {349, 32674}, {525, 23984}, {648, 6354}, {651, 40149}, {656, 24032}, {658, 1826}, {668, 1426}, {811, 1254}, {823, 37755}, {934, 41013}, {1018, 1847}, {1119, 3952}, {1262, 14618}, {1275, 2501}, {1398, 27808}, {1400, 46404}, {1415, 52575}, {1425, 6528}, {1427, 6335}, {1435, 4033}, {1446, 1783}, {1577, 7128}, {1824, 4569}, {1835, 35174}, {1880, 4554}, {1897, 3668}, {2333, 46406}, {3267, 23985}, {4077, 7012}, {4573, 8736}, {4616, 7140}, {6046, 36797}, {7045, 24006}, {7178, 46102}, {14208, 24033}
X(52607) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 15411}, {4, 7253}, {6, 23090}, {12, 52355}, {19, 1021}, {25, 21789}, {34, 3737}, {56, 23189}, {65, 521}, {100, 1792}, {101, 2327}, {108, 21}, {109, 283}, {112, 7054}, {162, 1098}, {225, 522}, {226, 6332}, {273, 18155}, {278, 4560}, {279, 15419}, {321, 15416}, {393, 17926}, {430, 4990}, {512, 3270}, {523, 2968}, {525, 23983}, {608, 7252}, {647, 35072}, {648, 7058}, {651, 1812}, {653, 333}, {656, 24031}, {658, 17206}, {661, 34591}, {664, 332}, {810, 2638}, {905, 16731}, {934, 1444}, {1018, 3692}, {1020, 63}, {1042, 1459}, {1119, 7192}, {1254, 656}, {1262, 4558}, {1275, 4563}, {1398, 3733}, {1400, 652}, {1402, 1946}, {1407, 7254}, {1409, 36054}, {1410, 23224}, {1415, 2193}, {1425, 520}, {1426, 513}, {1427, 905}, {1435, 1019}, {1439, 4131}, {1441, 35518}, {1446, 15413}, {1461, 1790}, {1783, 2287}, {1813, 6514}, {1824, 3900}, {1825, 35057}, {1826, 3239}, {1835, 3738}, {1840, 4529}, {1847, 7199}, {1874, 3716}, {1880, 650}, {1897, 1043}, {2171, 8611}, {2222, 1793}, {2333, 657}, {2489, 14936}, {2501, 1146}, {3049, 39687}, {3668, 4025}, {3676, 17219}, {3952, 1265}, {4017, 7004}, {4033, 52406}, {4077, 17880}, {4551, 78}, {4552, 345}, {4557, 1260}, {4559, 219}, {4566, 69}, {4605, 306}, {6046, 17094}, {6354, 525}, {6356, 3265}, {6529, 36421}, {6587, 40616}, {7012, 643}, {7045, 4592}, {7103, 47844}, {7115, 5546}, {7128, 662}, {7147, 51664}, {7178, 26932}, {7180, 7117}, {7216, 3942}, {7250, 3937}, {8736, 3700}, {8750, 2328}, {8898, 2522}, {10376, 51644}, {13149, 274}, {14618, 23978}, {15742, 7256}, {17925, 26856}, {18026, 314}, {21859, 3694}, {23067, 1259}, {23979, 32661}, {23984, 648}, {23985, 112}, {24006, 24026}, {24019, 2326}, {24027, 4575}, {24032, 811}, {24033, 162}, {26700, 1789}, {30730, 30681}, {32674, 284}, {32714, 81}, {36118, 86}, {36127, 29}, {37755, 24018}, {39534, 14010}, {40149, 4391}, {41013, 4397}, {43923, 18191}, {46102, 645}, {46404, 28660}, {51421, 39471}, {52373, 4091}
X(52607) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {653, 36118, 32714}, {823, 41678, 17926}

---

X(52608) = X(3)X(8858)∩X(75)X(23672)

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

X(52608) lies on these lines: {3, 8858}, {75, 23672}, {76, 2023}, {99, 670}, {110, 35567}, {274, 30748}, {305, 339}, {310, 30751}, {325, 16084}, {668, 22280}, {689, 907}, {799, 3732}, {1289, 42297}, {1502, 32833}, {2396, 4609}, {3926, 28438}, {3933, 36214}, {4554, 4602}, {4563, 24284}, {4590, 4611}, {4635, 7258}, {6374, 31859}, {6572, 7953}, {9428, 18829}, {17932, 47389}

X(52608) = isotomic conjugate of X(2489)
X(52608) = isotomic conjugate of the anticomplement of X(52598)
X(52608) = isotomic conjugate of the isogonal conjugate of X(4563)
X(52608) = isotomic conjugate of the polar conjugate of X(670)
X(52608) = isogonal conjugate of the polar conjugate of X(4609)
X(52608) = X(i)-Ceva conjugate of X(j) for these (i,j): {4609, 670}, {44168, 40050}
X(52608) = X(i)-cross conjugate of X(j) for these (i,j): {647, 69}, {2519, 4}, {2524, 3}, {3267, 305}, {3926, 47389}, {4563, 670}, {28412, 44183}, {28419, 18020}, {34254, 4590}, {52598, 2}
X(52608) = X(i)-isoconjugate of X(j) for these (i,j): {4, 1924}, {19, 669}, {25, 798}, {31, 2489}, {92, 9426}, {162, 1084}, {163, 2971}, {512, 1973}, {560, 2501}, {607, 51641}, {648, 4117}, {656, 36417}, {661, 1974}, {662, 42068}, {667, 2333}, {810, 2207}, {811, 9427}, {822, 52439}, {823, 23216}, {872, 43925}, {1096, 3049}, {1395, 3709}, {1474, 50487}, {1501, 24006}, {1577, 44162}, {1824, 1919}, {1826, 1980}, {1917, 14618}, {1918, 6591}, {2203, 4079}, {2205, 7649}, {2212, 7180}, {3121, 8750}, {3124, 32676}, {23610, 46254}, {36035, 40351}
X(52608) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 2489}, {4, 9428}, {6, 669}, {25, 31998}, {69, 8651}, {115, 2971}, {125, 1084}, {512, 6337}, {647, 6338}, {647, 22260}, {798, 6505}, {1084, 42068}, {1824, 9296}, {1924, 36033}, {1973, 39054}, {1974, 36830}, {2207, 39062}, {2333, 6631}, {2501, 6374}, {2519, 3926}, {3049, 6503}, {3121, 26932}, {3122, 40618}, {3124, 15526}, {5976, 17994}, {6388, 47430}, {6390, 21905}, {6591, 34021}, {8029, 23285}, {8754, 36901}, {9426, 22391}, {9427, 17423}, {35067, 42663}, {36417, 40596}, {40620, 42067}, {50487, 51574}
X(52608) = cevapoint of X(i) and X(j) for these (i,j): {3, 22159}, {69, 647}, {305, 3267}, {525, 3933}, {3049, 23221}
X(52608) = crosssum of X(9427) and X(23099)
X(52608) = trilinear pole of line {69, 305}
X(52608) = crossdifference of every pair of points on line {1084, 23216}
X(52608) = barycentric product X(i)*X(j) for these {i,j}: {3, 4609}, {63, 4602}, {69, 670}, {76, 4563}, {99, 305}, {110, 40050}, {304, 799}, {310, 4561}, {332, 4572}, {339, 31614}, {525, 34537}, {561, 4592}, {647, 44168}, {662, 40364}, {689, 3933}, {850, 47389}, {880, 40708}, {1231, 4631}, {1332, 6385}, {1444, 6386}, {1502, 4558}, {1576, 40360}, {1928, 4575}, {1978, 17206}, {3267, 4590}, {3718, 4625}, {3917, 42371}, {3926, 6331}, {4176, 6528}, {4601, 15413}, {4610, 40071}, {4623, 20336}, {4635, 52406}, {6393, 43187}, {6516, 40072}, {7182, 7257}, {14208, 24037}, {15419, 31625}, {32661, 40362}, {35567, 45201}, {41009, 42297}, {46810, 46813}
X(52608) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2489}, {3, 669}, {48, 1924}, {63, 798}, {69, 512}, {72, 50487}, {76, 2501}, {77, 51641}, {99, 25}, {107, 52439}, {110, 1974}, {112, 36417}, {125, 22260}, {184, 9426}, {190, 2333}, {274, 6591}, {276, 15422}, {287, 2422}, {304, 661}, {305, 523}, {306, 4079}, {310, 7649}, {311, 51513}, {314, 18344}, {325, 17994}, {326, 810}, {328, 15475}, {332, 663}, {339, 8029}, {345, 3709}, {348, 7180}, {394, 3049}, {512, 42068}, {523, 2971}, {525, 3124}, {561, 24006}, {643, 2212}, {645, 607}, {647, 1084}, {648, 2207}, {662, 1973}, {668, 1824}, {670, 4}, {689, 32085}, {799, 19}, {810, 4117}, {811, 1096}, {850, 8754}, {874, 862}, {877, 34854}, {879, 15630}, {880, 419}, {892, 8753}, {905, 3121}, {906, 2205}, {1102, 822}, {1265, 4524}, {1331, 1918}, {1332, 213}, {1414, 1395}, {1437, 1980}, {1444, 667}, {1502, 14618}, {1509, 43925}, {1565, 8034}, {1576, 44162}, {1634, 27369}, {1790, 1919}, {1792, 8641}, {1799, 18105}, {1812, 3063}, {1978, 1826}, {2396, 232}, {2407, 14581}, {2421, 2211}, {3049, 9427}, {3265, 20975}, {3266, 14273}, {3267, 115}, {3564, 42663}, {3718, 4041}, {3732, 8020}, {3785, 3804}, {3917, 688}, {3926, 647}, {3933, 3005}, {3964, 39201}, {3977, 14407}, {4025, 3122}, {4143, 3269}, {4176, 520}, {4226, 44099}, {4554, 1880}, {4558, 32}, {4561, 42}, {4563, 6}, {4569, 1426}, {4572, 225}, {4573, 608}, {4574, 7109}, {4575, 560}, {4576, 1843}, {4580, 51906}, {4590, 112}, {4592, 31}, {4600, 8750}, {4601, 1783}, {4602, 92}, {4609, 264}, {4610, 1474}, {4611, 17409}, {4612, 2204}, {4615, 8752}, {4616, 1398}, {4620, 32674}, {4623, 28}, {4625, 34}, {4631, 1172}, {4634, 36125}, {4635, 1435}, {5468, 44102}, {6331, 393}, {6333, 44114}, {6337, 8651}, {6338, 2519}, {6385, 17924}, {6386, 41013}, {6390, 351}, {6393, 3569}, {6394, 878}, {6516, 1402}, {6528, 6524}, {7056, 7250}, {7182, 4017}, {7192, 42067}, {7254, 1977}, {7256, 7071}, {7257, 33}, {7258, 7079}, {7763, 6753}, {7767, 8664}, {7799, 47230}, {9146, 8541}, {9723, 34952}, {10330, 44091}, {10411, 34397}, {11064, 14398}, {12215, 5027}, {14208, 2643}, {14417, 21906}, {14570, 3199}, {14615, 44705}, {15352, 36434}, {15411, 14936}, {15413, 3125}, {15416, 36197}, {15419, 1015}, {15958, 14573}, {16084, 47206}, {17170, 50490}, {17206, 649}, {17932, 1976}, {17941, 44089}, {18020, 32713}, {18829, 17980}, {20336, 4705}, {20775, 9494}, {20794, 9491}, {20975, 23099}, {22159, 38996}, {23092, 21762}, {23181, 40981}, {23342, 46522}, {24037, 162}, {24041, 32676}, {24284, 2086}, {25098, 21835}, {27808, 7140}, {28419, 52588}, {28660, 3064}, {28706, 12077}, {30786, 9178}, {31614, 250}, {32640, 40351}, {32661, 1501}, {34211, 51437}, {34254, 2485}, {34386, 2623}, {34537, 648}, {35136, 14248}, {35139, 18384}, {35518, 4516}, {36212, 2491}, {36214, 881}, {36797, 6059}, {36841, 3172}, {36895, 20186}, {37804, 2492}, {39201, 23216}, {40050, 850}, {40071, 4024}, {40072, 44426}, {40360, 44173}, {40364, 1577}, {40708, 882}, {41014, 8663}, {41174, 20031}, {42371, 46104}, {43187, 6531}, {43754, 14601}, {44168, 6331}, {44173, 2970}, {44326, 41489}, {44769, 40354}, {45201, 2514}, {45792, 2088}, {45807, 1648}, {46134, 14593}, {46254, 24019}, {46810, 8105}, {46813, 8106}, {47389, 110}, {47390, 14574}, {47443, 41937}, {51386, 39469}, {52347, 15451}, {52406, 4171}, {52437, 14270}

---

X(52609) = X(69)X(337)∩X(100)X(190)

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

X(52609) lies on these lines: {69, 337}, {100, 190}, {306, 4466}, {313, 45744}, {344, 25097}, {345, 52351}, {594, 27688}, {646, 2397}, {648, 7256}, {1089, 50107}, {1265, 52387}, {1332, 4561}, {3210, 26142}, {3694, 20336}, {3882, 4568}, {3969, 27726}, {3991, 42724}, {4033, 4552}, {4103, 22003}, {4671, 27290}, {7035, 7258}, {7257, 14570}, {9055, 28283}, {20248, 29711}, {24004, 25268}, {25241, 30473}, {25254, 40603}

X(52609) = isogonal conjugate of X(43925)
X(52609) = isotomic conjugate of X(17925)
X(52609) = isotomic conjugate of the anticomplement of X(52599)
X(52609) = isotomic conjugate of the isogonal conjugate of X(4574)
X(52609) = isotomic conjugate of the polar conjugate of X(3952)
X(52609) = isogonal conjugate of the polar conjugate of X(27808)
X(52609) = X(29014)-anticomplementary conjugate of X(149)
X(52609) = X(i)-Ceva conjugate of X(j) for these (i,j): {646, 4033}, {7035, 1265}, {27808, 3952}
X(52609) = X(i)-cross conjugate of X(j) for these (i,j): {525, 306}, {647, 72}, {656, 69}, {4574, 3952}, {52355, 20336}, {52599, 2}
X(52609) = X(i)-isoconjugate of X(j) for these (i,j): {1, 43925}, {19, 3733}, {25, 1019}, {27, 667}, {28, 649}, {31, 17925}, {34, 7252}, {58, 6591}, {112, 244}, {162, 1015}, {163, 2969}, {270, 7180}, {284, 43923}, {286, 1919}, {513, 1474}, {514, 2203}, {607, 7203}, {608, 3737}, {648, 3248}, {656, 36420}, {662, 42067}, {663, 1396}, {757, 2489}, {810, 36419}, {811, 1977}, {823, 22096}, {849, 2501}, {875, 31905}, {1021, 1398}, {1086, 32676}, {1096, 7254}, {1106, 17926}, {1172, 43924}, {1333, 7649}, {1395, 4560}, {1408, 3064}, {1412, 18344}, {1435, 21789}, {1459, 5317}, {1843, 39179}, {1973, 7192}, {1974, 7199}, {1980, 44129}, {2189, 4017}, {2204, 3676}, {2206, 17924}, {2212, 17096}, {2299, 3669}, {2326, 7250}, {2332, 43932}, {3937, 24019}, {3942, 32713}, {4817, 46503}, {5379, 21143}, {7121, 17921}, {8747, 22383}, {8750, 16726}, {16947, 44426}, {18191, 32674}, {28615, 46542}, {40570, 50354}, {46103, 51641}
X(52609) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 17925}, {3, 43925}, {6, 3733}, {10, 6591}, {27, 6631}, {28, 5375}, {37, 7649}, {72, 43060}, {115, 2969}, {125, 1015}, {226, 3669}, {244, 34591}, {286, 9296}, {306, 29013}, {513, 51574}, {649, 40591}, {1019, 6505}, {1084, 42067}, {1086, 15526}, {1213, 46542}, {1474, 39026}, {1977, 17423}, {2189, 34961}, {2489, 40607}, {2501, 4075}, {2973, 36901}, {3937, 35071}, {6337, 7192}, {6338, 15419}, {6503, 7254}, {6552, 17926}, {6741, 8735}, {7252, 11517}, {16583, 48398}, {16726, 26932}, {17197, 40626}, {17205, 40618}, {17921, 40598}, {17924, 40603}, {18191, 35072}, {18344, 40599}, {29162, 40940}, {36419, 39062}, {36420, 40596}, {40590, 43923}
X(52609) = cevapoint of X(i) and X(j) for these (i,j): {72, 647}, {306, 525}, {656, 3949}, {905, 4001}, {3694, 52355}, {29013, 40940}
X(52609) = crosssum of X(1977) and X(8034)
X(52609) = trilinear pole of line {72, 306}
X(52609) = crossdifference of every pair of points on line {1015, 22096}
X(52609) = barycentric product X(i)*X(j) for these {i,j}: {3, 27808}, {10, 4561}, {63, 4033}, {69, 3952}, {71, 1978}, {72, 668}, {76, 4574}, {99, 3695}, {100, 20336}, {101, 40071}, {190, 306}, {201, 7257}, {228, 6386}, {304, 1018}, {305, 4557}, {307, 3699}, {313, 1331}, {321, 1332}, {345, 4552}, {348, 30730}, {349, 4587}, {525, 1016}, {594, 4563}, {644, 1231}, {645, 26942}, {646, 1214}, {647, 31625}, {656, 7035}, {662, 52369}, {664, 3710}, {670, 3690}, {765, 14208}, {799, 3949}, {811, 52387}, {906, 27801}, {1020, 52406}, {1089, 4592}, {1252, 3267}, {1265, 4566}, {1441, 4571}, {1813, 30713}, {1897, 52396}, {2318, 4572}, {3265, 15742}, {3596, 23067}, {3610, 37215}, {3694, 4554}, {3701, 6516}, {3718, 4551}, {3998, 6335}, {4019, 27805}, {4064, 4600}, {4069, 7182}, {4076, 17094}, {4103, 17206}, {4158, 6528}, {4466, 6632}, {4558, 28654}, {4998, 52355}, {6331, 52386}, {6356, 7256}, {6540, 41014}, {7258, 37755}, {13136, 51367}
X(52609) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 17925}, {3, 3733}, {6, 43925}, {10, 7649}, {37, 6591}, {63, 1019}, {65, 43923}, {69, 7192}, {71, 649}, {72, 513}, {73, 43924}, {77, 7203}, {78, 3737}, {100, 28}, {101, 1474}, {112, 36420}, {190, 27}, {192, 17921}, {201, 4017}, {210, 18344}, {219, 7252}, {228, 667}, {304, 7199}, {306, 514}, {307, 3676}, {313, 46107}, {321, 17924}, {345, 4560}, {346, 17926}, {348, 17096}, {394, 7254}, {440, 29162}, {512, 42067}, {520, 3937}, {521, 18191}, {523, 2969}, {525, 1086}, {594, 2501}, {643, 270}, {644, 1172}, {645, 46103}, {646, 31623}, {647, 1015}, {648, 36419}, {651, 1396}, {656, 244}, {668, 286}, {692, 2203}, {765, 162}, {810, 3248}, {850, 2973}, {879, 43920}, {895, 43926}, {905, 16726}, {906, 1333}, {1016, 648}, {1018, 19}, {1020, 1435}, {1089, 24006}, {1110, 32676}, {1125, 46542}, {1214, 3669}, {1231, 24002}, {1252, 112}, {1259, 23189}, {1260, 21789}, {1265, 7253}, {1331, 58}, {1332, 81}, {1425, 7250}, {1439, 43932}, {1500, 2489}, {1633, 4211}, {1783, 5317}, {1813, 1412}, {1897, 8747}, {1978, 44129}, {2197, 7180}, {2200, 1919}, {2318, 663}, {2321, 3064}, {3049, 1977}, {3265, 1565}, {3267, 23989}, {3570, 31905}, {3610, 6590}, {3682, 1459}, {3690, 512}, {3692, 1021}, {3694, 650}, {3695, 523}, {3699, 29}, {3700, 8735}, {3701, 44426}, {3710, 522}, {3718, 18155}, {3807, 31909}, {3926, 15419}, {3927, 4840}, {3939, 2299}, {3940, 4833}, {3942, 8042}, {3949, 661}, {3952, 4}, {3958, 4979}, {3990, 22383}, {3995, 17922}, {3998, 905}, {4019, 4369}, {4025, 17205}, {4033, 92}, {4047, 4790}, {4064, 3120}, {4069, 33}, {4076, 36797}, {4101, 4778}, {4103, 1826}, {4115, 1839}, {4158, 520}, {4169, 8756}, {4427, 31900}, {4466, 6545}, {4551, 34}, {4552, 278}, {4557, 25}, {4558, 593}, {4559, 608}, {4561, 86}, {4563, 1509}, {4566, 1119}, {4568, 17171}, {4571, 21}, {4574, 6}, {4575, 849}, {4578, 4183}, {4587, 284}, {4592, 757}, {4756, 31902}, {5546, 2189}, {6332, 17197}, {6516, 1014}, {6558, 2322}, {7035, 811}, {7259, 2326}, {8611, 2170}, {10099, 43921}, {14208, 1111}, {14429, 1647}, {15411, 26856}, {15413, 16727}, {15523, 21108}, {15742, 107}, {17094, 1358}, {17757, 39534}, {17780, 37168}, {18210, 764}, {18589, 48398}, {20336, 693}, {20760, 16695}, {20769, 50456}, {20975, 8034}, {21015, 48403}, {21046, 21131}, {21859, 1880}, {22061, 20981}, {22076, 6371}, {22080, 50512}, {22370, 18197}, {23067, 56}, {23230, 8650}, {24018, 3942}, {24459, 27918}, {25098, 16742}, {26942, 7178}, {27808, 264}, {28654, 14618}, {30713, 46110}, {30730, 281}, {31625, 6331}, {32656, 2206}, {32660, 16947}, {33946, 31917}, {34055, 39179}, {35309, 17442}, {36059, 1408}, {37755, 7216}, {38955, 43933}, {39201, 22096}, {40071, 3261}, {40521, 1824}, {41014, 4977}, {42705, 47796}, {42706, 23882}, {42720, 15149}, {43290, 4248}, {44717, 4565}, {51366, 676}, {51367, 10015}, {51574, 43060}, {52354, 3667}, {52355, 11}, {52369, 1577}, {52370, 3063}, {52386, 647}, {52387, 656}, {52396, 4025}
X(52609) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3949, 4019, 69}, {4552, 30730, 4033}

---

X(52610) = X(3)X(296)∩X(101)X(36079)

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

X(52610) lies on these lines: {3, 296}, {101, 36079}, {109, 692}, {221, 2835}, {222, 3942}, {520, 4557}, {521, 35338}, {525, 4552}, {651, 653}, {662, 7045}, {906, 1813}, {916, 1458}, {934, 36080}, {1020, 4559}, {1106, 42463}, {1254, 3157}, {1407, 3173}, {1409, 1439}, {2196, 7053}, {4565, 32651}, {7011, 23112}, {7013, 22134}, {23128, 38877}

X(52610) = isogonal conjugate of X(17926)
X(52610) = isogonal conjugate of the polar conjugate of X(4566)
X(52610) = X(i)-Ceva conjugate of X(j) for these (i,j): {651, 1020}, {1275, 6356}, {7045, 3}, {7128, 7011}, {24032, 20764}
X(52610) = X(i)-cross conjugate of X(j) for these (i,j): {647, 73}, {822, 3}, {32320, 22341}, {51664, 222}
X(52610) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17926}, {4, 1021}, {19, 7253}, {21, 3064}, {27, 3900}, {28, 3239}, {29, 650}, {33, 4560}, {92, 21789}, {107, 34591}, {112, 24026}, {158, 23090}, {162, 1146}, {163, 21666}, {200, 17925}, {270, 3700}, {281, 3737}, {284, 44426}, {286, 657}, {318, 7252}, {333, 18344}, {341, 43925}, {513, 2322}, {514, 4183}, {521, 8748}, {522, 1172}, {523, 2326}, {607, 18155}, {643, 8735}, {648, 2310}, {652, 1896}, {656, 36421}, {662, 42069}, {663, 31623}, {693, 2332}, {811, 14936}, {823, 3270}, {1019, 7046}, {1043, 6591}, {1096, 15411}, {1098, 2501}, {1396, 4163}, {1474, 4397}, {2170, 36797}, {2189, 4086}, {2194, 46110}, {2204, 35519}, {2287, 7649}, {2299, 4391}, {2328, 17924}, {2341, 44428}, {2638, 15352}, {2968, 24019}, {2969, 7259}, {3063, 44130}, {3733, 7101}, {4041, 46103}, {5379, 42462}, {6529, 24031}, {7054, 24006}, {7071, 7199}, {7079, 7192}, {7258, 42067}, {8641, 44129}, {17115, 40411}, {23978, 32676}, {35072, 36126}
X(52610) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 17926}, {6, 7253}, {115, 21666}, {125, 1146}, {226, 4391}, {1021, 36033}, {1084, 42069}, {1147, 23090}, {1214, 46110}, {2322, 39026}, {2501, 15267}, {2968, 35071}, {3064, 40611}, {3239, 40591}, {4397, 51574}, {6503, 15411}, {6609, 17925}, {10001, 44130}, {14936, 17423}, {15526, 23978}, {17924, 36908}, {21789, 22391}, {24026, 34591}, {34591, 38985}, {35072, 46093}, {36421, 40596}, {40590, 44426}
X(52610) = cevapoint of X(i) and X(j) for these (i,j): {71, 520}, {73, 647}, {652, 40946}, {822, 7138}, {1459, 14597}, {3049, 22363}, {22341, 32320}, {22361, 36054}, {37755, 51664}
X(52610) = crosspoint of X(651) and X(1813)
X(52610) = crosssum of X(650) and X(3064)
X(52610) = trilinear pole of line {73, 228}
X(52610) = crossdifference of every pair of points on line {1146, 3270}
X(52610) = barycentric product X(i)*X(j) for these {i,j}: {3, 4566}, {7, 23067}, {59, 17094}, {63, 1020}, {65, 6516}, {71, 658}, {72, 934}, {73, 664}, {77, 4551}, {99, 1425}, {100, 1439}, {108, 52385}, {109, 307}, {110, 6356}, {190, 52373}, {201, 1414}, {222, 4552}, {225, 6517}, {226, 1813}, {228, 4569}, {279, 4574}, {306, 1461}, {348, 4559}, {349, 32660}, {525, 1262}, {647, 1275}, {651, 1214}, {653, 40152}, {656, 7045}, {662, 37755}, {668, 1410}, {811, 7138}, {906, 1446}, {1018, 7177}, {1042, 4561}, {1231, 1415}, {1254, 4592}, {1331, 3668}, {1332, 1427}, {1409, 4554}, {1441, 36059}, {1790, 4605}, {2197, 4573}, {2200, 46406}, {2318, 4626}, {3267, 23979}, {3682, 36118}, {3690, 4616}, {3694, 4617}, {3710, 6614}, {3949, 4637}, {3952, 7053}, {3990, 13149}, {3998, 32714}, {4033, 7099}, {4143, 23985}, {4466, 4619}, {4557, 7056}, {4558, 6354}, {4564, 51664}, {4565, 26942}, {5546, 20618}, {7128, 24018}, {7178, 44717}, {7339, 52355}, {8269, 17441}, {14208, 24027}, {18026, 22341}, {24016, 51366}, {32674, 52565}, {36838, 52370}
X(52610) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 7253}, {6, 17926}, {48, 1021}, {59, 36797}, {65, 44426}, {71, 3239}, {72, 4397}, {73, 522}, {77, 18155}, {101, 2322}, {108, 1896}, {109, 29}, {112, 36421}, {163, 2326}, {184, 21789}, {201, 4086}, {222, 4560}, {226, 46110}, {228, 3900}, {307, 35519}, {394, 15411}, {512, 42069}, {520, 2968}, {523, 21666}, {525, 23978}, {577, 23090}, {603, 3737}, {647, 1146}, {651, 31623}, {656, 24026}, {658, 44129}, {664, 44130}, {692, 4183}, {810, 2310}, {822, 34591}, {906, 2287}, {934, 286}, {1018, 7101}, {1020, 92}, {1042, 7649}, {1214, 4391}, {1254, 24006}, {1262, 648}, {1275, 6331}, {1331, 1043}, {1400, 3064}, {1402, 18344}, {1407, 17925}, {1409, 650}, {1410, 513}, {1415, 1172}, {1425, 523}, {1427, 17924}, {1439, 693}, {1461, 27}, {1464, 44428}, {1813, 333}, {2197, 3700}, {2200, 657}, {2318, 4163}, {3049, 14936}, {3668, 46107}, {3942, 40213}, {3998, 15416}, {4551, 318}, {4552, 7017}, {4557, 7046}, {4558, 7058}, {4559, 281}, {4565, 46103}, {4566, 264}, {4574, 346}, {4575, 1098}, {6354, 14618}, {6356, 850}, {6516, 314}, {6517, 332}, {7045, 811}, {7053, 7192}, {7066, 52355}, {7099, 1019}, {7128, 823}, {7138, 656}, {7177, 7199}, {7180, 8735}, {7250, 2969}, {7254, 26856}, {7335, 23189}, {8677, 14010}, {17094, 34387}, {18210, 42455}, {22061, 4529}, {22080, 4990}, {22341, 521}, {22342, 35057}, {22363, 17115}, {23067, 8}, {23979, 112}, {23984, 15352}, {23985, 6529}, {24027, 162}, {24033, 36126}, {32320, 35072}, {32651, 40395}, {32656, 2328}, {32660, 284}, {32661, 7054}, {32674, 8748}, {32739, 2332}, {36059, 21}, {37755, 1577}, {39201, 3270}, {40152, 6332}, {44717, 645}, {51640, 7004}, {51664, 4858}, {52370, 4130}, {52373, 514}, {52385, 35518}, {52391, 52356}, {52410, 43925}, {52411, 7252}

---

X(52611) = X(561)X(30953)∩X(668)X(46132)

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

X(52611) lies on these lines: {561, 30953}, {668, 46132}, {789, 17996}, {799, 4609}, {825, 9063}, {871, 31002}, {1921, 27238}, {1978, 7239}, {4583, 6386}, {4586, 4593}, {4602, 33946}, {6385, 40017}

X(52611) = isotomic conjugate of X(46386)
X(52611) = isotomic conjugate of the isogonal conjugate of X(37133)
X(52611) = X(9063)-Ceva conjugate of X(789)
X(52611) = X(3250)-cross conjugate of X(75)
X(52611) = X(i)-isoconjugate of X(j) for these (i,j): {6, 8630}, {31, 46386}, {32, 788}, {560, 3250}, {649, 18900}, {667, 40728}, {824, 1917}, {869, 1919}, {1491, 1501}, {1924, 3736}, {1927, 30654}, {1980, 2276}, {3049, 46503}, {4486, 18893}, {8640, 40736}, {9426, 40773}, {9455, 29956}, {14604, 30639}, {18892, 30671}, {18897, 30665}
X(52611) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 46386}, {9, 8630}, {788, 6376}, {869, 9296}, {3250, 6374}, {3736, 9428}, {5375, 18900}, {6631, 40728}, {16584, 17415}, {39338, 46132}
X(52611) = cevapoint of X(i) and X(j) for these (i,j): {75, 3250}, {514, 21443}, {4486, 20433}
X(52611) = trilinear pole of line {75, 700}
X(52611) = barycentric product X(i)*X(j) for these {i,j}: {75, 46132}, {76, 37133}, {561, 789}, {668, 871}, {825, 40362}, {870, 6386}, {1492, 1928}, {1502, 4586}, {2887, 9063}, {3261, 5388}, {4609, 40718}, {18891, 41072}, {30664, 44171}, {37207, 44169}
X(52611) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 8630}, {2, 46386}, {75, 788}, {76, 3250}, {100, 18900}, {190, 40728}, {561, 1491}, {668, 869}, {670, 3736}, {789, 31}, {811, 46503}, {825, 1501}, {870, 667}, {871, 513}, {985, 1980}, {1492, 560}, {1502, 824}, {1920, 45882}, {1978, 2276}, {2887, 17415}, {3778, 9006}, {3978, 30654}, {4033, 3774}, {4358, 14436}, {4572, 1469}, {4586, 32}, {4598, 40736}, {4602, 40773}, {4609, 30966}, {4613, 1918}, {4817, 1977}, {5388, 101}, {6385, 4481}, {6386, 984}, {9063, 40415}, {14621, 1919}, {18031, 29956}, {18891, 30665}, {18895, 30671}, {27853, 16514}, {30664, 14598}, {33946, 3117}, {34069, 1917}, {37133, 6}, {37207, 1922}, {40363, 4522}, {40495, 4475}, {40718, 669}, {40747, 1924}, {41072, 1911}, {43266, 3249}, {43289, 40370}, {44169, 4486}, {44170, 23596}, {46132, 1}

---

X(52612) = X(99)X(670)∩X(274)X(7200)

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

X(52612) lies on these lines: {99, 670}, {274, 7200}, {310, 21207}, {514, 7260}, {668, 2533}, {689, 43076}, {799, 4602}, {873, 18192}, {1978, 4632}, {2665, 17175}, {4107, 4610}, {4568, 4639}, {4573, 4631}, {4615, 34537}, {5209, 41535}, {7257, 51563}, {37204, 37205}

X(52612) = isotomic conjugate of X(4079)
X(52612) = isotomic conjugate of the anticomplement of X(52602)
X(52612) = isotomic conjugate of the complement of X(17159)
X(52612) = isotomic conjugate of the isogonal conjugate of X(4610)
X(52612) = X(i)-cross conjugate of X(j) for these (i,j): {649, 86}, {799, 4623}, {3261, 310}, {16737, 274}, {18196, 81}, {23572, 58}, {29767, 4600}, {52379, 24037}, {52602, 2}
X(52612) = X(i)-isoconjugate of X(j) for these (i,j): {6, 50487}, {10, 1924}, {31, 4079}, {32, 4705}, {37, 669}, {42, 798}, {99, 52065}, {100, 1084}, {181, 3063}, {190, 4117}, {213, 512}, {228, 2489}, {321, 9426}, {513, 7109}, {523, 2205}, {560, 4024}, {594, 1980}, {644, 1356}, {649, 872}, {651, 7063}, {661, 1918}, {667, 1500}, {668, 9427}, {688, 18098}, {692, 3124}, {756, 1919}, {810, 2333}, {906, 2971}, {1332, 42068}, {1334, 51641}, {1402, 3709}, {1501, 4036}, {1576, 21833}, {1824, 3049}, {1911, 46390}, {1922, 4155}, {1977, 40521}, {2281, 50494}, {2422, 5360}, {2643, 32739}, {3121, 4557}, {4567, 23099}, {4601, 23610}, {6335, 23216}, {6378, 8640}, {7234, 40729}, {8663, 28615}, {18105, 21814}, {21759, 50491}
X(52612) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 4079}, {9, 50487}, {10, 9428}, {42, 31998}, {76, 21056}, {181, 10001}, {213, 39054}, {512, 6626}, {661, 34021}, {669, 40589}, {756, 9296}, {798, 40592}, {799, 21887}, {872, 5375}, {1084, 8054}, {1086, 3124}, {1213, 8663}, {1500, 6631}, {1918, 36830}, {2333, 39062}, {2643, 40619}, {2971, 5190}, {3122, 40620}, {3709, 40605}, {4024, 6374}, {4155, 39028}, {4705, 6376}, {4858, 21833}, {4988, 22260}, {6651, 46390}, {7063, 38991}, {7109, 39026}, {16592, 21823}, {20975, 40618}, {21043, 36901}, {23099, 40627}, {38986, 52065}
X(52612) = cevapoint of X(i) and X(j) for these (i,j): {2, 17159}, {75, 18154}, {86, 649}, {310, 3261}, {514, 16887}, {670, 799}, {1019, 17175}, {4025, 18648}, {4623, 4631}, {7192, 16738}, {7199, 16739}
X(52612) = trilinear pole of line {86, 310}
X(52612) = crossdifference of every pair of points on line {1084, 4117}
X(52612) = barycentric product X(i)*X(j) for these {i,j}: {58, 4609}, {75, 4623}, {76, 4610}, {81, 4602}, {85, 4631}, {86, 670}, {99, 310}, {261, 4572}, {274, 799}, {314, 4625}, {514, 34537}, {649, 44168}, {662, 6385}, {664, 18021}, {668, 873}, {689, 16887}, {693, 24037}, {757, 6386}, {1414, 40072}, {1502, 4556}, {1509, 1978}, {3261, 4590}, {4554, 52379}, {4563, 44129}, {4573, 28660}, {4593, 16703}, {4601, 7199}, {4612, 20567}, {4632, 52572}, {4634, 30939}, {4636, 41283}, {4639, 30940}, {6331, 17206}, {6628, 27808}, {7058, 46406}, {7260, 8033}, {7307, 33946}, {7340, 35519}, {10030, 36806}, {15413, 46254}, {16696, 37204}, {17187, 42371}, {18891, 36066}, {21207, 31614}, {24041, 40495}, {43187, 51370}, {46107, 47389}
X(52612) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 50487}, {2, 4079}, {27, 2489}, {58, 669}, {75, 4705}, {76, 4024}, {81, 798}, {86, 512}, {99, 42}, {100, 872}, {101, 7109}, {110, 1918}, {163, 2205}, {190, 1500}, {239, 46390}, {249, 32739}, {261, 663}, {274, 661}, {305, 4064}, {310, 523}, {314, 4041}, {320, 42666}, {333, 3709}, {350, 4155}, {514, 3124}, {552, 43924}, {561, 4036}, {593, 1919}, {645, 1334}, {648, 2333}, {649, 1084}, {662, 213}, {663, 7063}, {664, 181}, {667, 4117}, {668, 756}, {670, 10}, {689, 18082}, {693, 2643}, {757, 667}, {798, 52065}, {799, 37}, {811, 1824}, {849, 1980}, {850, 21043}, {873, 513}, {880, 4039}, {1010, 50494}, {1014, 51641}, {1019, 3121}, {1043, 4524}, {1098, 8641}, {1125, 8663}, {1269, 6367}, {1333, 1924}, {1414, 1402}, {1434, 7180}, {1444, 810}, {1509, 649}, {1577, 21833}, {1634, 41267}, {1790, 3049}, {1919, 9427}, {1978, 594}, {2185, 3063}, {2206, 9426}, {3120, 22260}, {3122, 23099}, {3261, 115}, {3267, 21046}, {3699, 7064}, {3732, 21813}, {3798, 47430}, {4025, 20975}, {4033, 762}, {4107, 2086}, {4357, 42661}, {4369, 21823}, {4374, 21725}, {4554, 2171}, {4556, 32}, {4558, 2200}, {4561, 3690}, {4563, 71}, {4569, 1254}, {4572, 12}, {4573, 1400}, {4576, 21035}, {4590, 101}, {4592, 228}, {4593, 18098}, {4598, 6378}, {4600, 4557}, {4601, 1018}, {4602, 321}, {4603, 40729}, {4609, 313}, {4610, 6}, {4611, 21034}, {4612, 41}, {4616, 1042}, {4620, 4559}, {4623, 1}, {4625, 65}, {4626, 7143}, {4631, 9}, {4632, 52555}, {4634, 4674}, {4635, 1427}, {4636, 2175}, {4750, 21906}, {5209, 17989}, {5333, 4826}, {6064, 3939}, {6331, 1826}, {6374, 21056}, {6385, 1577}, {6386, 1089}, {6628, 3733}, {6629, 351}, {7035, 40521}, {7058, 657}, {7192, 3122}, {7199, 3125}, {7257, 210}, {7258, 4515}, {7304, 20979}, {7340, 109}, {7649, 2971}, {10566, 51906}, {15413, 3708}, {16696, 2084}, {16703, 8061}, {16704, 14407}, {16709, 4983}, {16737, 16592}, {16738, 40627}, {16739, 50330}, {16741, 2642}, {16755, 20982}, {16887, 3005}, {17103, 7234}, {17143, 21727}, {17175, 50497}, {17187, 688}, {17200, 8664}, {17205, 8034}, {17206, 647}, {17209, 2491}, {17212, 4128}, {17731, 17990}, {17930, 2054}, {18020, 8750}, {18021, 522}, {18062, 21802}, {18064, 22322}, {18155, 4516}, {18157, 24290}, {18160, 21824}, {18197, 21835}, {18200, 21755}, {18653, 14398}, {18830, 7148}, {20888, 50538}, {20924, 2610}, {21207, 8029}, {23788, 42752}, {23989, 21131}, {24037, 100}, {24039, 21839}, {24041, 692}, {26840, 17411}, {27808, 6535}, {27853, 4037}, {28660, 3700}, {29767, 52592}, {30593, 50512}, {30805, 3269}, {30939, 4730}, {30940, 21832}, {31008, 21834}, {31614, 4570}, {31625, 4103}, {33295, 4455}, {33296, 50491}, {33955, 50486}, {34022, 21836}, {34537, 190}, {35519, 4092}, {36066, 1911}, {36806, 4876}, {36838, 7147}, {36860, 20691}, {39915, 9402}, {40034, 21720}, {40072, 4086}, {40075, 6370}, {40089, 21714}, {40495, 1109}, {42028, 4832}, {43924, 1356}, {44129, 2501}, {44154, 48395}, {44168, 1978}, {46107, 8754}, {46254, 1783}, {46404, 8736}, {46406, 6354}, {47389, 1331}, {51370, 3569}, {52379, 650}, {52394, 18105}, {52572, 4988}

---

X(52613) = X(2)X(2416)∩X(3)X(2435)

Barycentrics    a^2*(b - c)*(b + c)*(a^2 - b^2 - c^2)^3 : :
X(52613) = X[647] - 4 X[8552], 2 X[3265] + X[41077], X[2489] - 4 X[7630], 2 X[18314] - 3 X[31174], 5 X[31277] - 2 X[41079]

X(52613) lies on these lines: {2, 2416}, {3, 2435}, {30, 47225}, {39, 52588}, {99, 39062}, {110, 2764}, {249, 4558}, {441, 525}, {512, 684}, {520, 4091}, {523, 21668}, {526, 44680}, {2485, 52595}, {2489, 2799}, {2797, 16229}, {3267, 7799}, {3566, 46953}, {3926, 23107}, {5562, 23103}, {5664, 52032}, {7215, 37754}, {14618, 30476}, {15352, 30441}, {15421, 52350}, {15423, 44427}, {18314, 31174}, {30211, 40494}, {31277, 41079}

X(52613) = midpoint of X(i) and X(j) for these {i,j}: {684, 22089}, {3265, 20580}
X(52613) = reflection of X(i) in X(j) for these {i,j}: {647, 52584}, {14618, 30476}, {41077, 20580}, {42658, 3}, {52584, 8552}
X(52613) = isogonal conjugate of X(6529)
X(52613) = isotomic conjugate of X(15352)
X(52613) = anticomplement of X(52585)
X(52613) = anticomplement of the isotomic conjugate of X(30441)
X(52613) = complement of the isotomic conjugate of X(46639)
X(52613) = isotomic conjugate of the isogonal conjugate of X(32320)
X(52613) = isogonal conjugate of the isotomic conjugate of X(4143)
X(52613) = isotomic conjugate of the polar conjugate of X(520)
X(52613) = isogonal conjugate of the polar conjugate of X(3265)
X(52613) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {15318, 21294}, {30441, 6327}, {32319, 21221}
X(52613) = X(i)-complementary conjugate of X(j) for these (i,j): {48, 35968}, {64, 21253}, {112, 20308}, {163, 2883}, {1301, 20305}, {1576, 36908}, {2155, 125}, {9247, 39020}, {14642, 34846}, {19614, 127}, {32676, 20207}, {33581, 8287}, {44326, 21235}, {46639, 2887}
X(52613) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 5562}, {1102, 7215}, {2416, 41077}, {3265, 520}, {3964, 2972}, {4176, 34980}, {4558, 394}, {6331, 69}, {15414, 3265}, {30441, 2}, {52350, 15526}, {52608, 44141}
X(52613) = X(i)-cross conjugate of X(j) for these (i,j): {2972, 3964}, {3269, 16391}, {32320, 520}, {34980, 4176}, {35071, 1092}, {37754, 4158}, {47409, 3}
X(52613) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6529}, {4, 24019}, {6, 36126}, {19, 107}, {25, 823}, {31, 15352}, {92, 32713}, {108, 8748}, {110, 6520}, {112, 158}, {162, 393}, {163, 1093}, {240, 20031}, {525, 24022}, {647, 24021}, {648, 1096}, {656, 23590}, {661, 32230}, {662, 6524}, {799, 52439}, {810, 34538}, {811, 2207}, {1172, 36127}, {1576, 6521}, {1783, 8747}, {1784, 32695}, {1896, 32674}, {1897, 5317}, {1973, 6528}, {2052, 32676}, {2181, 16813}, {2489, 23999}, {2501, 24000}, {4592, 36434}, {6530, 36104}, {8767, 23977}, {14208, 23975}, {16318, 36092}, {17926, 24033}, {23964, 24006}, {24024, 43717}, {36129, 52418}
X(52613) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 15352}, {3, 6529}, {4, 35071}, {6, 107}, {6, 46093}, {9, 36126}, {19, 38985}, {53, 2972}, {110, 37867}, {112, 1147}, {115, 1093}, {125, 393}, {130, 3199}, {158, 34591}, {235, 3269}, {244, 6520}, {520, 647}, {521, 17926}, {523, 17434}, {525, 14618}, {648, 6503}, {823, 6505}, {1084, 6524}, {1301, 14390}, {1896, 35072}, {1990, 38999}, {2052, 15526}, {2207, 17423}, {4858, 6521}, {5139, 36434}, {5317, 34467}, {6331, 6338}, {6337, 6528}, {6388, 21447}, {6523, 13613}, {6530, 39000}, {8747, 39006}, {8748, 38983}, {13450, 39019}, {13567, 41678}, {14249, 39020}, {14569, 15450}, {20031, 39085}, {22391, 32713}, {23290, 35441}, {23590, 40596}, {23977, 39071}, {24019, 36033}, {24021, 39052}, {32230, 36830}, {34538, 39062}, {38996, 52439}, {41181, 44145}
X(52613) = crosspoint of X(i) and X(j) for these (i,j): {2, 46639}, {69, 6331}, {99, 34386}, {394, 4558}, {3265, 4143}
X(52613) = crosssum of X(i) and X(j) for these (i,j): {6, 6587}, {25, 3049}, {393, 2501}, {512, 3199}, {523, 1853}, {2489, 44079}
X(52613) = trilinear pole of line {2972, 34980}
X(52613) = crossdifference of every pair of points on line {25, 393}
X(52613) = barycentric product X(i)*X(j) for these {i,j}: {3, 3265}, {6, 4143}, {63, 24018}, {69, 520}, {71, 30805}, {72, 4131}, {76, 32320}, {99, 2972}, {112, 23974}, {162, 24020}, {216, 15414}, {249, 23616}, {255, 14208}, {304, 822}, {305, 39201}, {306, 4091}, {326, 656}, {394, 525}, {512, 4176}, {521, 52385}, {523, 3964}, {577, 3267}, {647, 3926}, {652, 52565}, {661, 1102}, {670, 34980}, {684, 6394}, {799, 37754}, {850, 1092}, {879, 51386}, {905, 3998}, {1073, 20580}, {1231, 36054}, {1259, 17094}, {1331, 17216}, {1459, 52396}, {1577, 6507}, {1804, 52355}, {2416, 44436}, {2525, 28724}, {2632, 4592}, {3269, 4563}, {3682, 4025}, {3718, 51640}, {3719, 51664}, {3952, 7215}, {3990, 15413}, {4100, 20948}, {4158, 7192}, {4558, 15526}, {4575, 17879}, {6331, 35071}, {6332, 40152}, {6333, 17974}, {6335, 16730}, {6563, 16391}, {7183, 8611}, {8057, 15394}, {14638, 15905}, {14919, 41077}, {15419, 52386}, {17434, 34386}, {20336, 23224}, {22341, 35518}, {23107, 23357}, {23286, 52347}, {23606, 44173}, {23983, 52610}, {28706, 46088}, {32661, 36793}, {34767, 51394}, {43083, 52437}, {44326, 47409}, {45792, 50433}, {46811, 46814}, {52350, 52584}
X(52613) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 36126}, {2, 15352}, {3, 107}, {6, 6529}, {48, 24019}, {63, 823}, {69, 6528}, {73, 36127}, {97, 16813}, {110, 32230}, {112, 23590}, {162, 24021}, {184, 32713}, {248, 20031}, {255, 162}, {276, 42401}, {326, 811}, {394, 648}, {417, 1624}, {418, 52604}, {426, 1632}, {512, 6524}, {520, 4}, {521, 1896}, {523, 1093}, {525, 2052}, {577, 112}, {647, 393}, {648, 34538}, {652, 8748}, {656, 158}, {661, 6520}, {669, 52439}, {684, 6530}, {810, 1096}, {822, 19}, {1092, 110}, {1102, 799}, {1259, 36797}, {1459, 8747}, {1577, 6521}, {1636, 1990}, {2489, 36434}, {2584, 2587}, {2585, 2586}, {2632, 24006}, {2972, 523}, {3049, 2207}, {3265, 264}, {3267, 18027}, {3269, 2501}, {3682, 1897}, {3917, 46151}, {3926, 6331}, {3964, 99}, {3990, 1783}, {3998, 6335}, {4055, 8750}, {4091, 27}, {4100, 163}, {4131, 286}, {4143, 76}, {4158, 3952}, {4176, 670}, {4558, 23582}, {4575, 24000}, {4592, 23999}, {5489, 2970}, {5562, 35360}, {6368, 13450}, {6394, 22456}, {6507, 662}, {6509, 41678}, {7215, 7192}, {7254, 36419}, {8057, 14249}, {8552, 14165}, {8673, 52448}, {8766, 24024}, {8779, 23977}, {9007, 47392}, {14379, 1301}, {14417, 37778}, {14638, 52581}, {14919, 15459}, {15412, 8794}, {15414, 276}, {15451, 14569}, {15526, 14618}, {16391, 925}, {16730, 905}, {17216, 46107}, {17434, 53}, {17974, 685}, {18877, 32695}, {19210, 933}, {20580, 15466}, {22341, 108}, {22383, 5317}, {22384, 34856}, {23090, 36421}, {23103, 2972}, {23107, 23962}, {23224, 28}, {23286, 8884}, {23606, 1576}, {23613, 34980}, {23616, 338}, {23974, 3267}, {24018, 92}, {24020, 14208}, {28724, 42396}, {30451, 8745}, {30805, 44129}, {32320, 6}, {32661, 23964}, {32676, 24022}, {34386, 42405}, {34980, 512}, {35071, 647}, {35072, 17926}, {35442, 23290}, {36054, 1172}, {37084, 3518}, {37754, 661}, {39201, 25}, {39469, 34854}, {40082, 22239}, {40152, 653}, {41077, 46106}, {41219, 15451}, {42080, 810}, {42293, 3199}, {42658, 6525}, {43083, 6344}, {44436, 2404}, {46088, 8882}, {46811, 46815}, {46814, 46812}, {47194, 43976}, {47409, 6587}, {51386, 877}, {51394, 4240}, {51640, 34}, {52350, 30450}, {52385, 18026}, {52430, 32676}, {52565, 46404}, {52584, 11547}, {52610, 23984}
X(52613) = {X(46811),X(46814)}-harmonic conjugate of X(41077)

---

X(52614) = X(37)X(676)∩X(522)X(650)

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

X(52614) lies on these lines: {37, 676}, {522, 650}, {657, 4105}, {665, 1642}, {918, 36905}, {926, 46388}, {928, 3730}, {1212, 6366}, {1638, 16578}, {2310, 3119}, {2340, 14411}, {2499, 4079}, {2821, 14825}, {3676, 21348}, {3900, 10581}, {3904, 25082}, {4171, 6608}, {4515, 4528}, {7658, 25076}, {10015, 16601}, {21808, 30691}, {25268, 30610}

X(52614) = reflection of X(10581) in X(52594)
X(52614) = X(i)-complementary conjugate of X(j) for these (i,j): {9442, 21252}, {32739, 36905}, {52001, 124}
X(52614) = X(i)-Ceva conjugate of X(j) for these (i,j): {2284, 2340}, {28071, 3022}, {28132, 3900}, {36802, 52562}, {36803, 8}
X(52614) = X(i)-isoconjugate of X(j) for these (i,j): {7, 36146}, {56, 34085}, {57, 927}, {85, 32735}, {105, 658}, {109, 34018}, {269, 666}, {279, 36086}, {294, 4626}, {604, 46135}, {664, 1462}, {673, 934}, {738, 36802}, {919, 1088}, {1027, 1275}, {1106, 36803}, {1407, 51560}, {1416, 4554}, {1438, 4569}, {1461, 2481}, {1814, 36118}, {2195, 36838}, {3669, 39293}, {4564, 43930}, {4616, 18785}, {4617, 14942}, {4637, 13576}, {6185, 41353}, {6614, 36796}, {9503, 23973}, {13149, 36057}, {17093, 36041}, {24013, 28132}, {31637, 32714}
X(52614) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 34085}, {7, 39014}, {11, 34018}, {279, 38989}, {658, 39046}, {665, 926}, {666, 6600}, {673, 14714}, {927, 5452}, {1088, 38980}, {1462, 39025}, {2481, 35508}, {2968, 18031}, {3126, 24002}, {3161, 46135}, {3900, 28132}, {4554, 40609}, {4569, 6184}, {5519, 17093}, {6552, 36803}, {13149, 20621}, {17755, 46406}, {24015, 39077}, {24771, 51560}, {36838, 39063}
X(52614) = crosspoint of X(i) and X(j) for these (i,j): {8, 36803}, {644, 28071}, {2284, 2340}, {2338, 3939}, {3900, 28132}
X(52614) = crosssum of X(i) and X(j) for these (i,j): {1462, 43930}, {3669, 34855}, {3676, 43035}
X(52614) = crossdifference of every pair of points on line {56, 105}
X(52614) = barycentric product X(i)*X(j) for these {i,j}: {8, 926}, {55, 50333}, {200, 2254}, {220, 918}, {241, 4130}, {312, 46388}, {346, 665}, {480, 43042}, {518, 3900}, {522, 2340}, {644, 17435}, {650, 3693}, {657, 3912}, {663, 3717}, {672, 3239}, {883, 3022}, {1021, 3930}, {1025, 3119}, {1026, 2310}, {1146, 2284}, {1458, 4163}, {2223, 4397}, {2283, 4081}, {2287, 24290}, {2328, 4088}, {3126, 28071}, {3252, 4148}, {3263, 8641}, {3596, 8638}, {3675, 4578}, {3932, 21789}, {4105, 9436}, {4171, 18206}, {4524, 30941}, {4528, 34230}, {6065, 52305}, {6184, 28132}, {7253, 20683}, {14936, 42720}, {24010, 41353}, {36803, 39014}, {37908, 52355}
X(52614) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 46135}, {9, 34085}, {41, 36146}, {55, 927}, {200, 51560}, {220, 666}, {241, 36838}, {346, 36803}, {480, 36802}, {518, 4569}, {650, 34018}, {657, 673}, {665, 279}, {672, 658}, {926, 7}, {1253, 36086}, {1458, 4626}, {2175, 32735}, {2223, 934}, {2254, 1088}, {2284, 1275}, {2340, 664}, {2356, 36118}, {3022, 885}, {3063, 1462}, {3239, 18031}, {3271, 43930}, {3286, 4616}, {3693, 4554}, {3717, 4572}, {3900, 2481}, {3912, 46406}, {3939, 39293}, {4105, 14942}, {4130, 36796}, {4524, 13576}, {5089, 13149}, {8638, 56}, {8641, 105}, {9454, 1461}, {9502, 24015}, {14411, 5723}, {14827, 919}, {17435, 24002}, {18206, 4635}, {20683, 4566}, {23225, 7053}, {24290, 1446}, {35508, 28132}, {39014, 665}, {39258, 1020}, {41353, 24011}, {42079, 41353}, {46388, 57}, {50333, 6063}

---

X(52615) = X(2)X(52586)∩X(239)X(514)

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

X(52615) lies on these lines: {2, 52586}, {239, 514}, {386, 50488}, {512, 16695}, {663, 5216}, {669, 39548}, {798, 4481}, {834, 8637}, {3733, 5009}, {3737, 4401}, {3768, 48051}, {4129, 27527}, {4840, 8672}, {6002, 29402}, {16754, 48335}, {18155, 29013}, {20979, 48054}, {27673, 47795}, {50455, 50456}

X(52615) = midpoint of X(1019) and X(18197)
X(52615) = anticomplement of X(52586)
X(52615) = X(i)-isoconjugate of X(j) for these (i,j): {37, 835}, {42, 37218}, {1018, 43531}, {2214, 3952}
X(52615) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 39016}, {834, 42664}, {835, 40589}, {27808, 41849}, {37218, 40592}
X(52615) = crossdifference of every pair of points on line {42, 594}
X(52615) = barycentric product X(i)*X(j) for these {i,j}: {58, 45746}, {81, 14349}, {86, 834}, {310, 8637}, {386, 7192}, {469, 7254}, {593, 23879}, {757, 47842}, {873, 50488}, {1019, 28606}, {1509, 42664}, {3733, 5224}, {3876, 7203}, {7252, 33949}, {15419, 44103}
X(52615) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 835}, {81, 37218}, {386, 3952}, {834, 10}, {3733, 43531}, {5224, 27808}, {8637, 42}, {14349, 321}, {23879, 28654}, {28606, 4033}, {39016, 42664}, {42664, 594}, {45746, 313}, {47842, 1089}, {50488, 756}
X(52615) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1019, 4063, 4560}, {1019, 4960, 48144}

---

X(52616) = X(2)X(52587)∩X(99)X(2762)

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

X(52616) lies on these lines: {2, 52587}, {99, 2762}, {333, 14331}, {520, 3265}, {522, 4087}, {652, 6332}, {656, 4025}, {3699, 4998}, {4163, 29037}, {4357, 21174}, {4397, 4467}, {4561, 6517}, {4897, 50333}, {7192, 20294}, {7649, 25008}, {20293, 45746}, {20315, 24560}, {22441, 46386}

X(52616) = isotomic conjugate of X(36127)
X(52616) = anticomplement of X(52587)
X(52616) = isotomic conjugate of the polar conjugate of X(6332)
X(52616) = X(i)-Ceva conjugate of X(j) for these (i,j): {3926, 23983}, {4561, 326}, {4572, 304}
X(52616) = X(i)-cross conjugate of X(j) for these (i,j): {3265, 35518}, {23983, 3926}, {24031, 3719}
X(52616) = X(i)-isoconjugate of X(j) for these (i,j): {19, 32674}, {25, 108}, {31, 36127}, {34, 8750}, {65, 32713}, {100, 7337}, {107, 1402}, {109, 1096}, {112, 1880}, {225, 32676}, {393, 1415}, {607, 32714}, {608, 1783}, {650, 23985}, {651, 2207}, {653, 1973}, {663, 24033}, {692, 1118}, {934, 6059}, {1395, 1897}, {1400, 24019}, {1409, 6529}, {1875, 14776}, {1974, 18026}, {2204, 52607}, {2212, 36118}, {3063, 23984}, {3209, 40117}, {4554, 36417}, {4559, 5317}, {6516, 52439}, {6520, 32660}, {6524, 36059}, {6591, 7115}, {8755, 32667}, {14571, 32702}, {17408, 40097}, {21859, 36420}
X(52616) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 36127}, {4, 40626}, {6, 32674}, {11, 1096}, {19, 35072}, {25, 38983}, {33, 7358}, {34, 26932}, {107, 40605}, {108, 6505}, {109, 6503}, {158, 40624}, {225, 15526}, {278, 40618}, {393, 1146}, {521, 663}, {608, 39006}, {653, 6337}, {656, 18344}, {664, 6338}, {905, 7649}, {1086, 1118}, {1395, 34467}, {1400, 35071}, {1402, 38985}, {1857, 2968}, {1880, 34591}, {2207, 38991}, {3064, 3239}, {6059, 14714}, {6129, 24018}, {6506, 52033}, {6524, 20620}, {6591, 40628}, {7337, 8054}, {8747, 40625}, {8750, 11517}, {10001, 23984}, {24019, 40582}, {32660, 37867}, {32713, 40602}
X(52616) = crosspoint of X(i) and X(j) for these (i,j): {304, 4572}, {3718, 4561}
X(52616) = crossdifference of every pair of points on line {1973, 2207}
X(52616) = barycentric product X(i)*X(j) for these {i,j}: {8, 30805}, {29, 4143}, {63, 35518}, {69, 6332}, {77, 15416}, {78, 15413}, {283, 3267}, {304, 521}, {305, 652}, {307, 15411}, {312, 4131}, {314, 24018}, {326, 4391}, {332, 525}, {333, 3265}, {345, 4025}, {394, 35519}, {514, 1264}, {520, 28660}, {522, 3926}, {561, 36054}, {645, 17216}, {664, 23983}, {693, 3719}, {822, 40072}, {850, 6514}, {905, 3718}, {1102, 44426}, {1259, 3261}, {1332, 17880}, {1364, 1978}, {1812, 14208}, {1946, 40364}, {2289, 40495}, {2632, 4631}, {3064, 4176}, {3239, 7055}, {3596, 4091}, {3710, 15419}, {3964, 46110}, {3998, 18155}, {4397, 7183}, {4554, 24031}, {4560, 52396}, {4561, 26932}, {4572, 35072}, {4610, 7068}, {4612, 17879}, {4636, 36793}, {5931, 20580}, {6517, 23978}, {7253, 52565}, {17206, 52355}, {17219, 52609}, {23189, 40071}, {23224, 28659}
X(52616) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 36127}, {3, 32674}, {21, 24019}, {29, 6529}, {63, 108}, {69, 653}, {77, 32714}, {78, 1783}, {109, 23985}, {219, 8750}, {255, 1415}, {271, 40117}, {283, 112}, {284, 32713}, {304, 18026}, {305, 46404}, {307, 52607}, {314, 823}, {326, 651}, {332, 648}, {333, 107}, {345, 1897}, {348, 36118}, {394, 109}, {514, 1118}, {520, 1400}, {521, 19}, {522, 393}, {525, 225}, {649, 7337}, {650, 1096}, {651, 24033}, {652, 25}, {656, 1880}, {657, 6059}, {663, 2207}, {664, 23984}, {822, 1402}, {905, 34}, {1092, 32660}, {1102, 6516}, {1259, 101}, {1264, 190}, {1331, 7115}, {1332, 7012}, {1364, 649}, {1459, 608}, {1795, 32702}, {1804, 1461}, {1812, 162}, {1946, 1973}, {2193, 32676}, {2289, 692}, {2638, 3063}, {2968, 3064}, {3064, 6524}, {3239, 1857}, {3265, 226}, {3682, 4559}, {3718, 6335}, {3719, 100}, {3737, 5317}, {3926, 664}, {3942, 43923}, {3964, 1813}, {3998, 4551}, {4025, 278}, {4064, 8736}, {4091, 56}, {4131, 57}, {4143, 307}, {4391, 158}, {4554, 24032}, {4560, 8747}, {4561, 46102}, {4612, 24000}, {4631, 23999}, {4636, 23964}, {6056, 32739}, {6332, 4}, {6364, 13460}, {6365, 13438}, {6507, 36059}, {6512, 36082}, {6514, 110}, {6516, 7128}, {6517, 1262}, {6518, 23353}, {7004, 6591}, {7055, 658}, {7068, 4024}, {7182, 13149}, {7183, 934}, {7253, 8748}, {8611, 1824}, {10397, 3195}, {14208, 40149}, {14331, 6525}, {15411, 29}, {15413, 273}, {15416, 318}, {16730, 51640}, {16731, 3737}, {17216, 7178}, {17219, 17925}, {17880, 17924}, {20580, 5930}, {22383, 1395}, {23090, 2299}, {23189, 1474}, {23224, 604}, {23696, 8751}, {23983, 522}, {24018, 65}, {24031, 650}, {24459, 1874}, {26932, 7649}, {28660, 6528}, {30805, 7}, {31623, 36126}, {34591, 18344}, {35072, 663}, {35518, 92}, {35519, 2052}, {36054, 31}, {36055, 32667}, {39471, 8755}, {44130, 15352}, {44426, 6520}, {46110, 1093}, {48278, 27376}, {51664, 1426}, {52355, 1826}, {52385, 1020}, {52387, 21859}, {52396, 4552}, {52565, 4566}
X(52616) = {X(3265),X(4131)}-harmonic conjugate of X(30805)

---

X(52617) = X(2)X(52588)∩X(99)X(2867)

Barycentrics    b^2*(b - c)*c^2*(b + c)*(-a^2 + b^2 + c^2)^2 : :
X(52617) = X[3267] + 2 X[45807]

X(52617) lies on these lines: {2, 52588}, {69, 8673}, {76, 43673}, {99, 2867}, {305, 34767}, {339, 34953}, {512, 16084}, {525, 3267}, {670, 16077}, {826, 850}, {2474, 23301}, {3566, 35522}, {3785, 42658}, {3926, 23107}, {4563, 17708}, {17932, 47389}, {32828, 52585}

X(52617) = reflection of X(2474) in X(23301)
X(52617) = isotomic conjugate of X(32713)
X(52617) = anticomplement of X(52588)
X(52617) = isotomic conjugate of the isogonal conjugate of X(3265)
X(52617) = isotomic conjugate of the polar conjugate of X(3267)
X(52617) = polar conjugate of the isogonal conjugate of X(4143)
X(52617) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {13575, 21221}, {34207, 21220}, {39417, 21216}, {39733, 3448}, {40009, 21294}
X(52617) = X(i)-Ceva conjugate of X(j) for these (i,j): {670, 305}, {1502, 23974}, {52608, 3926}
X(52617) = X(i)-cross conjugate of X(j) for these (i,j): {3265, 3267}, {23974, 1502}
X(52617) = X(i)-isoconjugate of X(j) for these (i,j): {25, 32676}, {31, 32713}, {32, 24019}, {107, 560}, {112, 1973}, {158, 14574}, {162, 1974}, {163, 2207}, {661, 41937}, {662, 36417}, {669, 24000}, {798, 23964}, {811, 44162}, {822, 23975}, {823, 1501}, {1096, 1576}, {1910, 34859}, {1917, 6528}, {1924, 23582}, {2203, 8750}, {2204, 32674}, {2211, 36104}, {4575, 52439}, {5317, 32739}, {6529, 9247}, {9406, 32695}, {9417, 20031}, {9426, 23999}, {14575, 36126}, {14581, 36131}, {24001, 40351}, {24022, 39201}, {36046, 51437}
X(52617) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 32713}, {25, 15526}, {32, 35071}, {107, 6374}, {110, 6338}, {112, 6337}, {115, 2207}, {125, 1974}, {136, 52439}, {338, 14569}, {339, 27376}, {393, 36901}, {512, 525}, {560, 38985}, {647, 2489}, {1084, 36417}, {1096, 4858}, {1147, 14574}, {1474, 40618}, {1576, 6503}, {1692, 41181}, {1973, 34591}, {2203, 26932}, {2204, 35072}, {2211, 39000}, {2299, 40626}, {2445, 15595}, {2485, 3265}, {2501, 23285}, {2972, 40981}, {3049, 17434}, {3080, 20975}, {3172, 39020}, {3199, 39019}, {5317, 40619}, {6059, 6741}, {6376, 24019}, {6388, 19118}, {6505, 32676}, {7337, 40622}, {9407, 38999}, {9410, 32695}, {9428, 23582}, {11672, 34859}, {14398, 14401}, {14575, 46093}, {14581, 39008}, {17423, 44162}, {20031, 39058}, {23300, 36793}, {23964, 31998}, {33504, 51437}, {34854, 35088}, {36420, 40620}, {36830, 41937}, {52032, 52604}
X(52617) = cevapoint of X(3265) and X(4143)
X(52617) = crosspoint of X(i) and X(j) for these (i,j): {305, 670}, {40050, 52608}
X(52617) = crosssum of X(669) and X(1974)
X(52617) = trilinear pole of line {15526, 23974}
X(52617) = crossdifference of every pair of points on line {1501, 1974}
X(52617) = barycentric product X(i)*X(j) for these {i,j}: {69, 3267}, {76, 3265}, {99, 36793}, {125, 52608}, {264, 4143}, {304, 14208}, {305, 525}, {311, 15414}, {313, 30805}, {326, 20948}, {328, 45792}, {339, 4563}, {394, 44173}, {520, 1502}, {561, 24018}, {647, 40050}, {656, 40364}, {670, 15526}, {799, 17879}, {822, 1928}, {850, 3926}, {1231, 35518}, {1978, 17216}, {2632, 4602}, {3049, 40360}, {3261, 52396}, {3269, 4609}, {3998, 40495}, {4025, 40071}, {4131, 27801}, {4176, 14618}, {5489, 34537}, {6528, 23974}, {14615, 14638}, {15413, 20336}, {18020, 23107}, {20580, 41530}, {30786, 45807}, {32320, 44161}, {35519, 52565}, {39201, 40362}
X(52617) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 32713}, {63, 32676}, {69, 112}, {75, 24019}, {76, 107}, {99, 23964}, {107, 23975}, {110, 41937}, {125, 2489}, {264, 6529}, {287, 32696}, {290, 20031}, {304, 162}, {305, 648}, {306, 8750}, {307, 32674}, {326, 163}, {336, 36104}, {339, 2501}, {343, 52604}, {349, 36127}, {394, 1576}, {441, 2445}, {511, 34859}, {512, 36417}, {520, 32}, {521, 2204}, {523, 2207}, {525, 25}, {561, 823}, {577, 14574}, {647, 1974}, {656, 1973}, {670, 23582}, {684, 2211}, {693, 5317}, {799, 24000}, {822, 560}, {823, 24022}, {850, 393}, {905, 2203}, {1102, 4575}, {1231, 108}, {1264, 5546}, {1367, 7180}, {1494, 32695}, {1502, 6528}, {1565, 43925}, {1577, 1096}, {1636, 9407}, {1650, 14398}, {1969, 36126}, {2419, 43717}, {2501, 52439}, {2525, 1843}, {2632, 798}, {2799, 34854}, {2972, 3049}, {3049, 44162}, {3261, 8747}, {3265, 6}, {3267, 4}, {3268, 52418}, {3269, 669}, {3682, 32739}, {3700, 6059}, {3926, 110}, {3933, 35325}, {3964, 32661}, {3998, 692}, {4025, 1474}, {4064, 2333}, {4091, 2206}, {4131, 1333}, {4143, 3}, {4176, 4558}, {4563, 250}, {4566, 23985}, {4602, 23999}, {5489, 3124}, {6331, 32230}, {6332, 2299}, {6333, 232}, {6334, 44084}, {6368, 3199}, {6393, 4230}, {6394, 2715}, {6528, 23590}, {6563, 8745}, {7055, 4565}, {7068, 3709}, {7178, 7337}, {7192, 36420}, {8024, 46151}, {8057, 3172}, {8552, 34397}, {8611, 2212}, {8673, 17409}, {9033, 14581}, {11064, 23347}, {14208, 19}, {14380, 40354}, {14417, 44102}, {14592, 18384}, {14618, 6524}, {14638, 64}, {14919, 32715}, {14977, 8753}, {15413, 28}, {15414, 54}, {15415, 13450}, {15416, 4183}, {15526, 512}, {17094, 608}, {17216, 649}, {17434, 40981}, {17879, 661}, {18022, 15352}, {18314, 14569}, {18557, 14583}, {20336, 1783}, {20580, 154}, {20948, 158}, {23107, 125}, {23285, 27376}, {23616, 20975}, {23881, 27373}, {23974, 520}, {23983, 21789}, {24018, 31}, {24020, 822}, {24284, 44089}, {28706, 35360}, {28724, 4630}, {30737, 23977}, {30805, 58}, {32320, 14575}, {34384, 16813}, {34386, 933}, {34403, 1301}, {34767, 8749}, {35140, 32687}, {35518, 1172}, {35519, 8748}, {36793, 523}, {39201, 1501}, {39473, 42671}, {40050, 6331}, {40071, 1897}, {40364, 811}, {41077, 1495}, {43083, 11060}, {44173, 2052}, {44326, 15384}, {45792, 186}, {45807, 468}, {46088, 14573}, {46811, 44123}, {46814, 44124}, {47194, 40825}, {47389, 47443}, {51386, 14966}, {51664, 1395}, {52347, 1625}, {52350, 32734}, {52355, 607}, {52385, 1415}, {52396, 101}, {52437, 14591}, {52565, 109}, {52584, 44077}, {52608, 18020}

---

X(52618) = X(2)X(52591)∩X(76)X(826)

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

X(52618) lies on the Lemoine asymptotic hyperbola (see below) and these lines: {2, 52591}, {76, 826}, {83, 2422}, {308, 9178}, {316, 512}, {523, 14603}, {689, 691}, {804, 18105}, {827, 22456}, {876, 18833}, {881, 44165}, {1237, 1577}, {2489, 4580}, {3005, 42291}, {3261, 23790}, {4079, 4129}, {4577, 14560}, {7809, 23285}, {7878, 37085}, {7927, 14295}, {16921, 39799}, {18829, 42371}, {22105, 46001}, {23105, 52570}, {23582, 42396}, {31646, 51906}, {33803, 43187}

The Lemoine asymptotic hyperbola is described by Paul Yiu: Geometry of the Triangle, 2016, page 530.

X(52618) = reflection of X(i) in X(j) for these {i,j}: {3005, 42291}, {43665, 18314}
X(52618) = isotomic conjugate of X(1634)
X(52618) = anticomplement of X(52591)
X(52618) = polar conjugate of X(35325)
X(52618) = isotomic conjugate of the anticomplement of X(7668)
X(52618) = isotomic conjugate of the complement of X(25051)
X(52618) = polar conjugate of the isogonal conjugate of X(4580)
X(52618) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {4599, 8266}, {11794, 21289}, {30505, 21221}, {34072, 40642}
X(52618) = X(i)-Ceva conjugate of X(j) for these (i,j): {689, 308}, {40016, 34294}, {42371, 76}, {42396, 264}
X(52618) = X(i)-cross conjugate of X(j) for these (i,j): {115, 76}, {804, 43665}, {1084, 40162}, {2486, 321}, {2514, 2501}, {3143, 671}, {7668, 2}, {23962, 264}, {34294, 40016}, {34981, 5392}, {38393, 94}, {38394, 11140}
X(52618) = X(i)-isoconjugate of X(j) for these (i,j): {31, 1634}, {38, 1576}, {39, 163}, {48, 35325}, {99, 1923}, {110, 1964}, {112, 4020}, {162, 20775}, {249, 2084}, {560, 4576}, {662, 3051}, {688, 24041}, {692, 17187}, {799, 41331}, {826, 23995}, {922, 36827}, {1101, 3005}, {1333, 46148}, {1843, 4575}, {1930, 14574}, {1933, 46161}, {2148, 35319}, {2194, 46153}, {2206, 4553}, {2236, 17938}, {3404, 14966}, {3917, 32676}, {4556, 21814}, {4565, 40972}, {4570, 50521}, {4592, 27369}, {8041, 34072}, {8061, 23357}, {9247, 41676}, {9494, 24037}, {16696, 32739}, {17442, 32661}, {23889, 41272}, {23997, 51869}, {46151, 52430}
X(52618) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 1634}, {37, 46148}, {38, 4858}, {39, 115}, {110, 41884}, {125, 20775}, {127, 23208}, {136, 1843}, {141, 36901}, {216, 35319}, {244, 1964}, {339, 7794}, {512, 9494}, {523, 3005}, {688, 3005}, {826, 18314}, {1084, 3051}, {1086, 17187}, {1214, 46153}, {1249, 35325}, {1401, 40622}, {1923, 38986}, {2525, 23285}, {3688, 6741}, {3787, 15525}, {3917, 15526}, {4020, 34591}, {4553, 40603}, {4576, 6374}, {4988, 21123}, {5007, 51906}, {5139, 27369}, {8041, 15449}, {8623, 35078}, {11205, 15527}, {16696, 40619}, {20965, 34294}, {35336, 40695}, {35337, 40696}, {36827, 39061}, {38996, 41331}, {50330, 50521}
X(52618) = cevapoint of X(i) and X(j) for these (i,j): {2, 25051}, {523, 850}
X(52618) = crosspoint of X(i) and X(j) for these (i,j): {308, 689}, {670, 31630}, {4577, 39287}, {42396, 52395}
X(52618) = crosssum of X(i) and X(j) for these (i,j): {39, 52591}, {688, 3051}
X(52618) = trilinear pole of line {338, 3124}
X(52618) = crossdifference of every pair of points on line {3051, 20775}
X(52618) = barycentric product X(i)*X(j) for these {i,j}: {75, 18070}, {82, 20948}, {83, 850}, {115, 689}, {251, 44173}, {264, 4580}, {308, 523}, {311, 39182}, {313, 10566}, {338, 4577}, {339, 42396}, {512, 40016}, {525, 46104}, {661, 18833}, {670, 34294}, {827, 23962}, {1109, 4593}, {1502, 18105}, {1577, 3112}, {1799, 14618}, {2643, 37204}, {3124, 42371}, {3261, 18082}, {3267, 32085}, {4599, 23994}, {4609, 51906}, {12077, 41488}, {14295, 14970}, {18023, 22105}, {18097, 35519}, {18098, 40495}, {18108, 27801}, {18314, 39287}, {20022, 43665}, {23285, 52395}, {31065, 52570}
X(52618) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 1634}, {4, 35325}, {5, 35319}, {10, 46148}, {76, 4576}, {82, 163}, {83, 110}, {94, 46155}, {115, 3005}, {226, 46153}, {251, 1576}, {264, 41676}, {308, 99}, {313, 4568}, {321, 4553}, {338, 826}, {339, 2525}, {512, 3051}, {514, 17187}, {523, 39}, {525, 3917}, {623, 35336}, {624, 35337}, {647, 20775}, {656, 4020}, {661, 1964}, {669, 41331}, {671, 36827}, {689, 4590}, {693, 16696}, {733, 17938}, {798, 1923}, {804, 8623}, {826, 8041}, {827, 23357}, {850, 141}, {1084, 9494}, {1089, 35309}, {1109, 8061}, {1176, 32661}, {1577, 38}, {1799, 4558}, {1916, 46161}, {2052, 46151}, {2394, 46147}, {2395, 51869}, {2485, 23208}, {2489, 27369}, {2501, 1843}, {2592, 46167}, {2593, 46166}, {2643, 2084}, {3050, 3203}, {3112, 662}, {3120, 21123}, {3124, 688}, {3125, 50521}, {3261, 16887}, {3267, 3933}, {3405, 23997}, {3566, 3787}, {3700, 3688}, {4024, 21035}, {4036, 3954}, {4041, 40972}, {4049, 46150}, {4079, 41267}, {4080, 46162}, {4086, 33299}, {4444, 46159}, {4577, 249}, {4580, 3}, {4593, 24041}, {4599, 1101}, {4630, 23963}, {4705, 21814}, {5466, 46154}, {7178, 1401}, {7668, 52591}, {7927, 11205}, {8599, 30489}, {9178, 41272}, {9979, 9019}, {10130, 9145}, {10566, 58}, {13576, 46163}, {14223, 46157}, {14295, 732}, {14618, 427}, {14970, 805}, {15412, 16030}, {16732, 2530}, {17500, 1625}, {18010, 2076}, {18070, 1}, {18082, 101}, {18097, 109}, {18098, 692}, {18101, 7252}, {18105, 32}, {18107, 38832}, {18108, 1333}, {18833, 799}, {20022, 2421}, {20948, 1930}, {21207, 16892}, {21459, 46592}, {22105, 187}, {23105, 39691}, {23285, 7794}, {23290, 27371}, {23878, 14096}, {23962, 23285}, {24006, 17442}, {26546, 41582}, {27712, 18183}, {31065, 52554}, {31296, 41328}, {32085, 112}, {33294, 3313}, {34055, 4575}, {34072, 23995}, {34294, 512}, {35522, 7813}, {37204, 24037}, {39179, 849}, {39182, 54}, {39287, 18315}, {40016, 670}, {40149, 46152}, {40425, 7953}, {40495, 16703}, {41079, 51360}, {42299, 26714}, {42371, 34537}, {42396, 250}, {43665, 20021}, {43673, 46164}, {44173, 8024}, {46104, 648}, {46107, 17171}, {46288, 14574}, {47126, 23642}, {51862, 14966}, {51906, 669}, {52394, 4556}, {52395, 827}, {52570, 10330}

---

X(52619) = X(2)X(52592)∩X(75)X(4151)

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

X(52619) lies on these lines: {2, 52592}, {75, 4151}, {76, 1577}, {99, 927}, {274, 4560}, {310, 6548}, {512, 4374}, {514, 1921}, {670, 4555}, {693, 784}, {799, 37143}, {850, 4608}, {876, 18833}, {1019, 4817}, {1509, 7254}, {3676, 18033}, {3732, 31624}, {3766, 6372}, {4083, 4411}, {4406, 6005}, {4408, 29198}, {4583, 4602}, {8714, 20954}, {14296, 52601}, {15417, 15419}, {17930, 52612}, {18146, 45324}, {21300, 33297}, {21901, 27255}, {23100, 52572}, {30870, 50334}, {32028, 36803}, {35367, 48406}

X(52619) = midpoint of X(693) and X(23807)
X(52619) = isotomic conjugate of X(4557)
X(52619) = anticomplement of X(52592)
X(52619) = isotomic conjugate of the isogonal conjugate of X(7192)
X(52619) = polar conjugate of the isogonal conjugate of X(15419)
X(52619) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6577, 1655}, {8049, 21221}, {34444, 21220}, {39735, 3448}, {39797, 148}, {40005, 21294}
X(52619) = X(i)-Ceva conjugate of X(j) for these (i,j): {670, 310}, {4572, 16739}, {4602, 76}, {4609, 6385}, {4625, 6063}, {6385, 16727}, {52612, 52572}
X(52619) = X(i)-cross conjugate of X(j) for these (i,j): {244, 6383}, {693, 7199}, {1111, 76}, {1565, 1509}, {3004, 24002}, {3777, 1019}, {8034, 1086}, {16727, 6385}, {17198, 86}, {18181, 81}, {40619, 75}, {48400, 17924}
X(52619) = X(i)-isoconjugate of X(j) for these (i,j): {31, 4557}, {32, 1018}, {37, 32739}, {41, 4559}, {42, 692}, {100, 1918}, {101, 213}, {110, 872}, {163, 1500}, {190, 2205}, {228, 8750}, {512, 1110}, {560, 3952}, {661, 23990}, {662, 7109}, {669, 765}, {756, 1576}, {798, 1252}, {813, 41333}, {825, 3774}, {906, 2333}, {919, 39258}, {1016, 1924}, {1020, 14827}, {1089, 14574}, {1334, 1415}, {1397, 4069}, {1402, 3939}, {1501, 4033}, {1783, 2200}, {1824, 32656}, {1917, 27808}, {1973, 4574}, {2149, 3709}, {2175, 4551}, {2206, 40521}, {2212, 23067}, {3690, 32676}, {3725, 32736}, {3747, 34067}, {4017, 6066}, {4171, 23979}, {4524, 24027}, {4531, 8685}, {4552, 9447}, {4570, 50487}, {4613, 18900}, {4621, 21751}, {4628, 21814}, {6065, 51641}, {7035, 9426}, {15378, 21837}, {20683, 32666}, {21805, 32719}, {32674, 52370}, {35309, 46288}, {36039, 51436}
X(52619) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 4557}, {6, 40620}, {37, 40619}, {42, 1086}, {55, 40625}, {71, 40618}, {100, 34021}, {101, 6626}, {115, 1500}, {181, 40622}, {210, 40624}, {213, 1015}, {228, 26932}, {244, 872}, {512, 514}, {513, 669}, {522, 4524}, {594, 36901}, {650, 3709}, {661, 798}, {692, 40592}, {756, 4858}, {1016, 9428}, {1018, 6376}, {1084, 7109}, {1110, 39054}, {1111, 21808}, {1146, 1334}, {1252, 31998}, {1400, 40615}, {1402, 40617}, {1565, 23620}, {1566, 51436}, {1577, 4041}, {1918, 8054}, {2200, 39006}, {2308, 16726}, {2318, 40626}, {2333, 5190}, {3160, 4559}, {3690, 15526}, {3747, 35119}, {3939, 40605}, {3952, 6374}, {4079, 4988}, {4551, 40593}, {4574, 6337}, {6066, 34961}, {6544, 14407}, {6741, 7064}, {16592, 20964}, {16742, 45216}, {17197, 20967}, {17205, 20963}, {17990, 27929}, {20666, 41180}, {20683, 35094}, {20970, 35076}, {23990, 36830}, {32739, 40589}, {35072, 52370}, {38980, 39258}, {40521, 40603}, {40623, 41333}, {46398, 51377}, {50330, 50487}
X(52619) = cevapoint of X(i) and X(j) for these (i,j): {6, 23389}, {514, 8714}, {693, 3261}, {1086, 8034}, {7192, 15419}, {7199, 18155}
X(52619) = crosspoint of X(i) and X(j) for these (i,j): {310, 670}, {873, 4625}, {4609, 6385}
X(52619) = crosssum of X(i) and X(j) for these (i,j): {669, 1918}, {2205, 9426}
X(52619) = trilinear pole of line {1086, 16727}
X(52619) = crossdifference of every pair of points on line {1918, 2205}
X(52619) = barycentric product X(i)*X(j) for these {i,j}: {75, 7199}, {76, 7192}, {81, 40495}, {85, 18155}, {86, 3261}, {99, 23989}, {244, 4602}, {264, 15419}, {274, 693}, {286, 15413}, {305, 17925}, {310, 514}, {314, 24002}, {513, 6385}, {561, 1019}, {593, 44173}, {668, 16727}, {670, 1086}, {757, 20948}, {799, 1111}, {850, 1509}, {871, 4481}, {873, 1577}, {1015, 4609}, {1434, 35519}, {1502, 3733}, {1565, 6331}, {1978, 17205}, {2296, 23594}, {2969, 52608}, {2973, 4563}, {3004, 40827}, {3120, 52612}, {3596, 17096}, {3669, 40072}, {3676, 28660}, {3737, 20567}, {3766, 40017}, {3801, 7307}, {4025, 44129}, {4077, 52379}, {4560, 6063}, {4572, 17197}, {4573, 34387}, {4600, 23100}, {4608, 52572}, {4610, 21207}, {4616, 23978}, {4623, 16732}, {4625, 4858}, {4635, 24026}, {6383, 17217}, {6386, 16726}, {7018, 16737}, {7178, 18021}, {7203, 28659}, {7252, 41283}, {7254, 18022}, {8034, 44168}, {16755, 20565}, {17198, 31624}, {17206, 46107}, {17212, 44187}, {17219, 46404}, {18031, 23829}, {18077, 40038}, {20954, 40004}, {40050, 43925}, {44172, 50456}
X(52619) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 4557}, {7, 4559}, {11, 3709}, {27, 8750}, {58, 32739}, {69, 4574}, {75, 1018}, {76, 3952}, {81, 692}, {85, 4551}, {86, 101}, {99, 1252}, {110, 23990}, {244, 798}, {261, 5546}, {274, 100}, {286, 1783}, {305, 52609}, {310, 190}, {312, 4069}, {313, 4103}, {314, 644}, {321, 40521}, {332, 4587}, {333, 3939}, {348, 23067}, {512, 7109}, {513, 213}, {514, 42}, {521, 52370}, {522, 1334}, {523, 1500}, {525, 3690}, {552, 4565}, {561, 4033}, {593, 1576}, {645, 6065}, {649, 1918}, {659, 41333}, {661, 872}, {662, 1110}, {667, 2205}, {670, 1016}, {676, 51436}, {693, 37}, {757, 163}, {764, 3121}, {799, 765}, {812, 3747}, {850, 594}, {873, 662}, {905, 228}, {918, 20683}, {1014, 1415}, {1015, 669}, {1019, 31}, {1021, 1253}, {1086, 512}, {1088, 1020}, {1111, 661}, {1146, 4524}, {1269, 4115}, {1358, 7180}, {1414, 2149}, {1434, 109}, {1441, 21859}, {1444, 906}, {1459, 2200}, {1491, 3774}, {1502, 27808}, {1509, 110}, {1565, 647}, {1577, 756}, {1647, 14407}, {1790, 32656}, {1930, 35309}, {1977, 9426}, {2254, 39258}, {2530, 21814}, {2969, 2489}, {2973, 2501}, {3004, 2092}, {3120, 4079}, {3125, 50487}, {3248, 1924}, {3261, 10}, {3264, 4169}, {3265, 52386}, {3267, 3695}, {3596, 30730}, {3669, 1402}, {3676, 1400}, {3700, 7064}, {3733, 32}, {3737, 41}, {3762, 21805}, {3766, 2238}, {3776, 3778}, {3777, 16584}, {3810, 20684}, {3837, 21830}, {3910, 40966}, {3937, 3049}, {3942, 810}, {3960, 3724}, {4025, 71}, {4036, 762}, {4077, 2171}, {4089, 21828}, {4091, 4055}, {4131, 3990}, {4143, 4158}, {4369, 20964}, {4374, 2295}, {4391, 210}, {4397, 4515}, {4406, 3997}, {4453, 2245}, {4462, 4849}, {4468, 4878}, {4481, 869}, {4509, 2292}, {4560, 55}, {4573, 59}, {4602, 7035}, {4608, 52555}, {4609, 31625}, {4610, 4570}, {4615, 9268}, {4616, 1262}, {4623, 4567}, {4625, 4564}, {4634, 5376}, {4635, 7045}, {4637, 24027}, {4639, 5378}, {4801, 37593}, {4858, 4041}, {4957, 4770}, {4977, 20970}, {4978, 1962}, {5546, 6066}, {6063, 4552}, {6331, 15742}, {6332, 2318}, {6362, 21795}, {6372, 21753}, {6385, 668}, {6545, 3122}, {6628, 4556}, {7056, 52610}, {7178, 181}, {7192, 6}, {7199, 1}, {7200, 7234}, {7203, 604}, {7215, 32320}, {7252, 2175}, {7253, 220}, {7254, 184}, {7649, 2333}, {8025, 35327}, {8033, 4579}, {8034, 1084}, {8042, 3248}, {8714, 40586}, {10015, 51377}, {14208, 3949}, {14534, 32736}, {14618, 7140}, {14953, 2426}, {15411, 1260}, {15413, 72}, {15417, 405}, {15419, 3}, {16703, 4553}, {16704, 23344}, {16708, 35338}, {16709, 35342}, {16711, 23832}, {16726, 667}, {16727, 513}, {16732, 4705}, {16737, 171}, {16739, 3882}, {16742, 8640}, {16748, 4436}, {16750, 1633}, {16751, 15624}, {16754, 3185}, {16755, 35}, {16887, 46148}, {16892, 21035}, {17094, 2197}, {17096, 56}, {17139, 2427}, {17169, 35326}, {17197, 663}, {17198, 6586}, {17205, 649}, {17206, 1331}, {17212, 172}, {17217, 2176}, {17218, 9310}, {17219, 652}, {17302, 23861}, {17463, 21837}, {17496, 52139}, {17498, 12329}, {17880, 8611}, {17896, 21871}, {17924, 1824}, {17925, 25}, {17926, 7071}, {18021, 645}, {18077, 3961}, {18155, 9}, {18157, 1026}, {18160, 3678}, {18181, 52589}, {18191, 3063}, {18197, 2209}, {18200, 7122}, {18600, 23845}, {18827, 813}, {20880, 35310}, {20906, 20691}, {20948, 1089}, {20949, 3293}, {20954, 3294}, {21104, 52020}, {21123, 41267}, {21138, 50491}, {21178, 4456}, {21207, 4024}, {21606, 31855}, {21789, 14827}, {23100, 3120}, {23189, 52425}, {23594, 31330}, {23772, 22229}, {23788, 2183}, {23799, 3588}, {23800, 2198}, {23807, 21877}, {23824, 20979}, {23829, 672}, {23989, 523}, {24002, 65}, {24026, 4171}, {26546, 21867}, {26822, 20990}, {26824, 4068}, {26842, 21784}, {26856, 21789}, {27918, 4455}, {28660, 3699}, {29162, 40984}, {30591, 21816}, {30805, 3682}, {30939, 1023}, {30940, 3573}, {30941, 2284}, {32014, 8701}, {33930, 7239}, {33944, 21383}, {34387, 3700}, {35518, 3694}, {35519, 2321}, {35560, 28593}, {36038, 21801}, {36419, 32713}, {37128, 34067}, {39179, 46289}, {39747, 40519}, {40017, 660}, {40072, 646}, {40166, 4516}, {40213, 2310}, {40495, 321}, {40827, 8707}, {40834, 8684}, {42455, 36197}, {43920, 2422}, {43925, 1974}, {43926, 32740}, {43931, 21759}, {44129, 1897}, {44173, 28654}, {46107, 1826}, {47672, 2667}, {47676, 22277}, {47683, 2177}, {47871, 39688}, {48084, 3954}, {48131, 3725}, {48152, 21802}, {48393, 21820}, {48398, 40934}, {48403, 21813}, {48406, 21827}, {48415, 22172}, {48422, 4735}, {48629, 7287}, {50354, 40978}, {50451, 21879}, {50456, 2210}, {50514, 21751}, {52379, 643}, {52394, 4628}, {52572, 4427}, {52612, 4600}

---

X(52620) = X(2)X(28910)∩X(89)X(47763)

Barycentrics    (-2*a + b - 2*c)*(b - c)*(2*a + 2*b - c) : :
X(52620) = 2 X[31148] + X[48422], X[31150] - 4 X[47758], 5 X[693] + 4 X[4897], X[693] + 2 X[47755], X[693] - 4 X[47891], 2 X[4897] - 5 X[47755], X[4897] + 5 X[47891], X[47755] + 2 X[47891], 4 X[1638] - X[48548], 10 X[3676] - X[47981], 8 X[3676] + X[48107], 4 X[3676] - X[48550], 4 X[47981] + 5 X[48107], 2 X[47981] - 5 X[48550], X[48107] + 2 X[48550], 8 X[4025] + X[47655], 10 X[4369] - X[48117], 4 X[4369] - X[48557], 2 X[48117] - 5 X[48557], X[4608] - 10 X[43067], 4 X[4608] + 5 X[47657], X[4608] + 5 X[47894], 8 X[43067] + X[47657], 2 X[43067] + X[47894], X[47657] - 4 X[47894], 2 X[4809] + X[48108], 4 X[4932] + 5 X[48421], 2 X[4979] + 7 X[48420], X[7192] + 2 X[47754], 8 X[7653] + X[49302], 5 X[31209] + 4 X[49296], X[47652] + 2 X[48576], X[47666] - 4 X[47882], X[47676] + 2 X[47767], 2 X[47791] + X[48434], X[47871] + 2 X[48574], 4 X[47874] - X[49272], X[48175] - 4 X[48245], 4 X[48276] + 5 X[48433], 7 X[48432] + 2 X[49282], 2 X[48563] + X[48571]

X(52620) lies on these lines: {2, 28910}, {89, 47763}, {100, 4378}, {244, 6549}, {513, 6548}, {514, 1635}, {693, 900}, {927, 4588}, {1638, 48548}, {2320, 14154}, {3306, 23598}, {3676, 47981}, {4025, 28169}, {4369, 48117}, {4444, 4776}, {4604, 37143}, {4608, 28151}, {4777, 47780}, {4809, 48108}, {4817, 48577}, {4932, 48421}, {4979, 48420}, {6006, 21183}, {6544, 28851}, {7192, 28220}, {7653, 49302}, {14435, 21116}, {17925, 48580}, {26275, 28209}, {27836, 48079}, {29370, 48253}, {30608, 31992}, {31209, 49296}, {47652, 48576}, {47666, 47882}, {47676, 47767}, {47791, 48434}, {47871, 48574}, {47874, 49272}, {48175, 48245}, {48276, 48433}, {48432, 49282}, {48563, 48571}

X(52620) = midpoint of X(i) and X(j) for these {i,j}: {14435, 21116}, {23598, 48320}
X(52620) = reflection of X(i) in X(j) for these {i,j}: {4776, 14475}, {31992, 47761}
X(52620) = isotomic conjugate of X(4767)
X(52620) = anticomplement of X(52593)
X(52620) = X(4597)-Ceva conjugate of X(39704)
X(52620) = X(i)-cross conjugate of X(j) for these (i,j): {28209, 514}, {44435, 693}
X(52620) = X(i)-isoconjugate of X(j) for these (i,j): {6, 4752}, {31, 4767}, {45, 101}, {59, 4814}, {100, 2177}, {109, 3711}, {644, 1405}, {692, 3679}, {765, 4775}, {1018, 4273}, {1110, 4777}, {1252, 4893}, {1415, 4873}, {1576, 4125}, {2099, 3939}, {2149, 4944}, {3940, 8750}, {4557, 4653}, {4570, 4770}, {4671, 32739}, {4693, 34067}, {4791, 23990}, {4792, 23344}, {4908, 32665}, {4937, 32718}
X(52620) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 4767}, {9, 4752}, {11, 3711}, {45, 1015}, {513, 4775}, {514, 4777}, {650, 4944}, {661, 4893}, {1086, 3679}, {1146, 4873}, {2099, 40617}, {2177, 8054}, {3940, 26932}, {4125, 4858}, {4671, 40619}, {4693, 35119}, {4720, 40625}, {4770, 50330}, {4814, 6615}, {4908, 35092}, {4931, 4988}, {5219, 40615}, {5235, 40620}, {46398, 51362}
X(52620) = cevapoint of X(514) and X(6006)
X(52620) = crosspoint of X(4597) and X(39704)
X(52620) = crosssum of X(i) and X(j) for these (i,j): {2177, 4775}, {8656, 21747}
X(52620) = trilinear pole of line {1086, 2087}
X(52620) = crossdifference of every pair of points on line {2177, 21754}
X(52620) = barycentric product X(i)*X(j) for these {i,j}: {89, 693}, {513, 20569}, {514, 39704}, {900, 40833}, {1086, 4597}, {1111, 4604}, {2163, 3261}, {2320, 24002}, {3676, 30608}, {4588, 23989}, {7192, 30588}, {28607, 40495}, {40426, 44435}
X(52620) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4752}, {2, 4767}, {11, 4944}, {89, 100}, {244, 4893}, {513, 45}, {514, 3679}, {522, 4873}, {649, 2177}, {650, 3711}, {693, 4671}, {812, 4693}, {900, 4908}, {905, 3940}, {1015, 4775}, {1019, 4653}, {1022, 4792}, {1086, 4777}, {1111, 4791}, {1358, 43052}, {1565, 49280}, {1577, 4125}, {2163, 101}, {2170, 4814}, {2320, 644}, {2364, 3939}, {2401, 36921}, {3120, 4931}, {3125, 4770}, {3669, 2099}, {3676, 5219}, {3733, 4273}, {3960, 4867}, {4089, 23884}, {4453, 27757}, {4560, 4720}, {4588, 1252}, {4597, 1016}, {4604, 765}, {4728, 4937}, {4750, 4933}, {4978, 4717}, {5549, 6065}, {6006, 36911}, {6548, 4945}, {6549, 23598}, {7192, 5235}, {7200, 4774}, {7208, 4844}, {9002, 20973}, {10015, 51362}, {16726, 4833}, {17205, 47683}, {20569, 668}, {27918, 4800}, {28209, 16590}, {28607, 692}, {28658, 4557}, {30588, 3952}, {30589, 4756}, {30608, 3699}, {30722, 39782}, {30724, 4870}, {30725, 36920}, {34073, 1110}, {39428, 6017}, {39704, 190}, {40426, 9059}, {40833, 4555}, {43924, 1405}, {47683, 4803}, {47754, 27739}, {47755, 27754}, {47776, 4954}, {47871, 4956}, {47891, 27747}, {48335, 17461}, {48550, 27776}, {48580, 17553}, {50453, 21042}
X(52620) = {X(47755),X(47891)}-harmonic conjugate of X(693)

---

X(52621) = X(2)X(52594)∩X(85)X(514)

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

X(52621) lies on these lines: {2, 52594}, {7, 4406}, {75, 30181}, {85, 514}, {331, 46110}, {348, 47796}, {664, 4449}, {693, 6362}, {918, 3261}, {1088, 21183}, {1111, 19594}, {1434, 16737}, {1577, 23599}, {3673, 21185}, {3676, 18033}, {4040, 40719}, {4077, 23877}, {4163, 20907}, {4379, 7196}, {4569, 35174}, {6604, 21302}, {9312, 48282}, {9436, 50337}, {10481, 23789}, {15417, 17096}, {17072, 33298}, {17095, 47795}, {20954, 31605}, {47793, 52422}

X(52621) = isotomic conjugate of X(3939)
X(52621) = anticomplement of X(52594)
X(52621) = isotomic conjugate of the anticomplement of X(17059)
X(52621) = isotomic conjugate of the isogonal conjugate of X(3676)
X(52621) = X(i)-Ceva conjugate of X(j) for these (i,j): {4572, 6063}, {46406, 85}
X(52621) = X(i)-cross conjugate of X(j) for these (i,j): {693, 3261}, {1111, 85}, {4077, 24002}, {17059, 2}, {21118, 514}, {23615, 4858}, {34387, 331}
X(52621) = X(i)-isoconjugate of X(j) for these (i,j): {9, 32739}, {31, 3939}, {32, 644}, {33, 32656}, {41, 101}, {55, 692}, {59, 8641}, {100, 2175}, {109, 1253}, {112, 52370}, {163, 1334}, {190, 9447}, {210, 1576}, {212, 8750}, {213, 5546}, {220, 1415}, {513, 6066}, {560, 3699}, {607, 906}, {643, 1918}, {645, 2205}, {646, 1501}, {650, 23990}, {651, 14827}, {657, 2149}, {663, 1110}, {667, 6065}, {668, 9448}, {872, 4636}, {1252, 3063}, {1331, 2212}, {1397, 4578}, {1461, 6602}, {1783, 52425}, {1802, 32674}, {1973, 4587}, {1974, 4571}, {1980, 4076}, {2194, 4557}, {2204, 4574}, {2206, 4069}, {2251, 5548}, {2318, 32676}, {2340, 32666}, {3573, 18265}, {3689, 32719}, {3701, 14574}, {4105, 24027}, {4130, 23979}, {4258, 34074}, {4517, 34069}, {4612, 7109}, {4628, 40972}, {5377, 8638}, {7071, 36059}, {7074, 32652}, {7079, 32660}, {9455, 36802}, {18892, 36801}, {20967, 32736}, {32642, 41339}
X(52621) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 3939}, {6, 40615}, {9, 40619}, {11, 1253}, {31, 40617}, {41, 1015}, {42, 40622}, {55, 1086}, {100, 40593}, {101, 3160}, {109, 17113}, {115, 1334}, {200, 40624}, {210, 4858}, {212, 26932}, {219, 40618}, {220, 1146}, {223, 692}, {284, 40620}, {478, 32739}, {480, 2968}, {514, 663}, {522, 4105}, {607, 5190}, {643, 34021}, {644, 6376}, {650, 657}, {661, 3063}, {1111, 1212}, {1214, 4557}, {1252, 10001}, {1260, 40626}, {1565, 7124}, {1577, 3900}, {1802, 35072}, {2175, 8054}, {2212, 5521}, {2284, 36905}, {2318, 15526}, {2321, 36901}, {2328, 40625}, {2330, 16592}, {2340, 35094}, {3119, 8551}, {3669, 8643}, {3699, 6374}, {3709, 4988}, {4069, 40603}, {4587, 6337}, {5546, 6626}, {5548, 9460}, {6065, 6631}, {6066, 39026}, {6602, 35508}, {6615, 8641}, {7071, 20620}, {7074, 16596}, {8750, 40837}, {14827, 38991}, {16732, 40967}, {34591, 52370}, {39006, 52425}, {40499, 41771}
X(52621) = cevapoint of X(i) and X(j) for these (i,j): {514, 21185}, {522, 4468}, {693, 24002}, {1111, 23100}, {4858, 23615}
X(52621) = crosspoint of X(i) and X(j) for these (i,j): {4554, 31618}, {4572, 6063}
X(52621) = crosssum of X(3063) and X(20229)
X(52621) = trilinear pole of line {4858, 23989}
X(52621) = crossdifference of every pair of points on line {2175, 9447}
X(52621) = barycentric product X(i)*X(j) for these {i,j}: {7, 3261}, {11, 46406}, {57, 40495}, {75, 24002}, {76, 3676}, {85, 693}, {273, 15413}, {274, 4077}, {279, 35519}, {310, 7178}, {313, 17096}, {331, 4025}, {348, 46107}, {349, 7192}, {513, 20567}, {514, 6063}, {561, 3669}, {649, 41283}, {658, 34387}, {664, 23989}, {850, 1434}, {1014, 20948}, {1086, 4572}, {1088, 4391}, {1111, 4554}, {1358, 1978}, {1365, 52612}, {1412, 44173}, {1441, 7199}, {1446, 18155}, {1502, 43924}, {1565, 46404}, {1847, 35518}, {1919, 41287}, {4017, 6385}, {4089, 46405}, {4397, 23062}, {4444, 18033}, {4509, 31643}, {4569, 4858}, {4573, 21207}, {4625, 16732}, {4626, 23978}, {4998, 23100}, {7056, 46110}, {7182, 17924}, {7203, 27801}, {7209, 20906}, {7216, 40072}, {13149, 17880}, {17094, 44129}, {18031, 43042}, {18895, 43041}, {23104, 23586}, {24026, 36838}, {28659, 43932}, {40364, 43923}
X(52621) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3939}, {7, 101}, {11, 657}, {56, 32739}, {57, 692}, {69, 4587}, {75, 644}, {76, 3699}, {77, 906}, {85, 100}, {86, 5546}, {101, 6066}, {109, 23990}, {190, 6065}, {222, 32656}, {226, 4557}, {244, 3063}, {269, 1415}, {273, 1783}, {274, 643}, {278, 8750}, {279, 109}, {304, 4571}, {307, 4574}, {310, 645}, {312, 4578}, {313, 30730}, {314, 7259}, {321, 4069}, {331, 1897}, {348, 1331}, {349, 3952}, {479, 1461}, {513, 41}, {514, 55}, {521, 1802}, {522, 220}, {523, 1334}, {525, 2318}, {552, 4556}, {553, 35327}, {555, 6733}, {561, 646}, {649, 2175}, {650, 1253}, {651, 1110}, {656, 52370}, {658, 59}, {663, 14827}, {664, 1252}, {667, 9447}, {693, 9}, {824, 4517}, {850, 2321}, {873, 4612}, {876, 51858}, {903, 5548}, {905, 212}, {918, 2340}, {934, 2149}, {1014, 163}, {1019, 2194}, {1086, 663}, {1088, 651}, {1111, 650}, {1119, 32674}, {1146, 4105}, {1269, 30729}, {1357, 1919}, {1358, 649}, {1365, 4079}, {1396, 32676}, {1412, 1576}, {1422, 32652}, {1434, 110}, {1440, 36049}, {1441, 1018}, {1443, 1983}, {1446, 4551}, {1459, 52425}, {1462, 32666}, {1509, 4636}, {1565, 652}, {1577, 210}, {1847, 108}, {1919, 9448}, {1978, 4076}, {2170, 8641}, {2400, 2338}, {2401, 2342}, {2530, 40972}, {2973, 3064}, {3004, 2269}, {3064, 7071}, {3120, 3709}, {3239, 480}, {3261, 8}, {3264, 30731}, {3267, 3710}, {3572, 18265}, {3596, 6558}, {3662, 40499}, {3665, 46148}, {3668, 4559}, {3669, 31}, {3675, 46388}, {3676, 6}, {3762, 3689}, {3766, 3684}, {3776, 3056}, {3777, 20665}, {3801, 20684}, {3900, 6602}, {3911, 23344}, {3942, 1946}, {3960, 2361}, {4017, 213}, {4024, 7064}, {4025, 219}, {4077, 37}, {4086, 4515}, {4089, 654}, {4091, 6056}, {4106, 3217}, {4131, 2289}, {4369, 2330}, {4374, 2329}, {4391, 200}, {4397, 728}, {4411, 4390}, {4444, 7077}, {4449, 16283}, {4453, 2323}, {4462, 3158}, {4467, 52405}, {4468, 6600}, {4509, 960}, {4554, 765}, {4560, 2328}, {4569, 4564}, {4572, 1016}, {4573, 4570}, {4608, 33635}, {4616, 52378}, {4617, 24027}, {4625, 4567}, {4626, 1262}, {4778, 4258}, {4791, 3711}, {4801, 4512}, {4823, 3715}, {4858, 3900}, {4957, 4814}, {4978, 3683}, {6063, 190}, {6084, 8647}, {6332, 1260}, {6358, 40521}, {6362, 8012}, {6385, 7257}, {6545, 3271}, {6548, 2316}, {6591, 2212}, {6608, 8551}, {6614, 23979}, {7053, 32660}, {7056, 1813}, {7177, 36059}, {7178, 42}, {7180, 1918}, {7182, 1332}, {7192, 284}, {7196, 4579}, {7199, 21}, {7203, 1333}, {7205, 18047}, {7209, 932}, {7212, 3747}, {7216, 1402}, {7233, 813}, {7649, 607}, {8058, 7368}, {9436, 2284}, {10030, 3573}, {10481, 35326}, {13149, 7012}, {14208, 3694}, {14837, 7074}, {15413, 78}, {15419, 283}, {16727, 3737}, {16732, 4041}, {16755, 35193}, {16892, 3688}, {16947, 14574}, {17059, 52594}, {17079, 35281}, {17089, 41405}, {17094, 71}, {17096, 58}, {17197, 21789}, {17205, 7252}, {17219, 23090}, {17896, 2324}, {17924, 33}, {17925, 2299}, {18031, 36802}, {18033, 3570}, {18155, 2287}, {18160, 4420}, {18895, 36801}, {19604, 34080}, {20567, 668}, {20880, 35341}, {20906, 3208}, {20907, 4513}, {20948, 3701}, {20949, 3871}, {21044, 4524}, {21104, 2293}, {21118, 16588}, {21124, 40966}, {21132, 14936}, {21185, 5452}, {21207, 3700}, {22464, 2427}, {23062, 934}, {23100, 11}, {23104, 23970}, {23595, 1859}, {23599, 354}, {23615, 35508}, {23726, 42447}, {23748, 39789}, {23807, 7075}, {23978, 4163}, {23989, 522}, {24002, 1}, {24026, 4130}, {26721, 40141}, {27818, 1293}, {28660, 7256}, {30719, 3052}, {30722, 21747}, {30724, 2308}, {30725, 902}, {30805, 1259}, {31605, 218}, {31643, 36147}, {34018, 36086}, {34085, 5377}, {34387, 3239}, {34388, 4103}, {35160, 6078}, {35348, 18889}, {35518, 3692}, {35519, 346}, {36118, 7115}, {36621, 8699}, {36838, 7045}, {39704, 5549}, {39771, 1017}, {40014, 31343}, {40072, 7258}, {40166, 2310}, {40495, 312}, {40617, 8643}, {40704, 1026}, {41283, 1978}, {42455, 3119}, {42462, 3022}, {43035, 2426}, {43041, 1914}, {43042, 672}, {43049, 21059}, {43050, 19624}, {43051, 2209}, {43052, 2177}, {43067, 2268}, {43736, 36039}, {43762, 2742}, {43923, 1973}, {43924, 32}, {43930, 1438}, {43932, 604}, {44129, 36797}, {44173, 30713}, {44426, 7079}, {44435, 4266}, {46107, 281}, {46110, 7046}, {46404, 15742}, {46406, 4998}, {47995, 4254}, {48084, 33299}, {48131, 20967}, {48151, 20229}, {48398, 7083}, {51641, 2205}, {51658, 2198}, {51664, 228}, {52156, 677}, {52563, 23845}, {52612, 6064}

---

X(52622) = X(2)X(52595)∩X(75)X(14837)

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

X(52622) lies on these lines: {2, 52595}, {75, 14837}, {312, 6332}, {341, 4163}, {514, 21611}, {646, 36804}, {824, 1577}, {1089, 48272}, {3267, 14638}, {3700, 3910}, {4091, 14829}, {4397, 42337}, {4462, 47769}, {7178, 21438}, {10015, 23685}, {18003, 48267}, {20316, 21050}, {20942, 45683}

X(52622) = isotomic conjugate of X(1461)
X(52622) = anticomplement of X(52595)
X(52622) = isotomic conjugate of the isogonal conjugate of X(3239)
X(52622) = X(i)-Ceva conjugate of X(j) for these (i,j): {646, 30713}, {1978, 3596}
X(52622) = X(i)-cross conjugate of X(j) for these (i,j): {4397, 35519}, {42462, 24026}
X(52622) = X(i)-isoconjugate of X(j) for these (i,j): {31, 1461}, {32, 934}, {34, 32660}, {41, 6614}, {56, 1415}, {100, 52410}, {101, 1106}, {108, 52411}, {109, 604}, {112, 1410}, {163, 1042}, {184, 32714}, {269, 32739}, {513, 23979}, {560, 658}, {603, 32674}, {608, 36059}, {649, 24027}, {651, 1397}, {667, 1262}, {692, 1407}, {906, 1398}, {1020, 2206}, {1275, 1980}, {1395, 1813}, {1402, 4565}, {1408, 4559}, {1426, 32661}, {1427, 1576}, {1435, 32656}, {1446, 14574}, {1455, 32643}, {1457, 32669}, {1501, 4569}, {1917, 46406}, {1918, 4637}, {1919, 7045}, {2149, 43924}, {2175, 4617}, {2199, 8059}, {2203, 52610}, {2205, 4616}, {3063, 7339}, {3248, 4619}, {3939, 7366}, {4551, 16947}, {4554, 41280}, {4626, 9447}, {6610, 32728}, {6611, 32652}, {7099, 8750}, {7342, 21859}, {8641, 23971}, {9247, 36118}, {9448, 36838}, {13149, 14575}, {23224, 23985}, {23990, 43932}, {32651, 40956}, {32675, 52440}, {32676, 52373}, {51641, 52378}
X(52622) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 1415}, {2, 1461}, {6, 2968}, {11, 604}, {31, 35508}, {32, 14714}, {48, 7358}, {56, 1146}, {57, 40624}, {101, 6552}, {109, 3161}, {115, 1042}, {222, 40626}, {269, 40619}, {522, 649}, {603, 35072}, {608, 20620}, {650, 43924}, {656, 22383}, {658, 6374}, {692, 24771}, {934, 6376}, {1015, 1106}, {1020, 40603}, {1086, 1407}, {1262, 6631}, {1397, 38991}, {1398, 5190}, {1400, 6741}, {1404, 51402}, {1410, 34591}, {1412, 40625}, {1427, 4858}, {1459, 3239}, {1577, 3669}, {1919, 17115}, {1973, 38966}, {2262, 24026}, {3063, 6608}, {3160, 6614}, {3310, 23757}, {3668, 36901}, {4565, 40605}, {4617, 40593}, {4637, 34021}, {4988, 7250}, {5375, 24027}, {6600, 32739}, {6611, 16596}, {7023, 40615}, {7045, 9296}, {7053, 40618}, {7099, 26932}, {7339, 10001}, {7366, 40617}, {7952, 32674}, {8054, 52410}, {11517, 32660}, {15526, 52373}, {23979, 39026}, {32675, 36909}, {35128, 52440}, {38972, 51655}, {38983, 52411}
X(52622) = cevapoint of X(24026) and X(42462)
X(52622) = crosspoint of X(1978) and X(3596)
X(52622) = crosssum of X(1397) and X(1919)
X(52622) = trilinear pole of line {23978, 24026}
X(52622) = crossdifference of every pair of points on line {560, 1397}
X(52622) = barycentric product X(i)*X(j) for these {i,j}: {8, 35519}, {75, 4397}, {76, 3239}, {92, 15416}, {190, 23978}, {200, 40495}, {312, 4391}, {313, 7253}, {314, 4086}, {318, 35518}, {341, 693}, {345, 46110}, {346, 3261}, {522, 3596}, {561, 3900}, {646, 4858}, {650, 28659}, {657, 1502}, {663, 40363}, {668, 24026}, {670, 52335}, {850, 1043}, {1021, 27801}, {1146, 1978}, {1265, 46107}, {1928, 8641}, {2287, 20948}, {2310, 6386}, {2322, 3267}, {2328, 44173}, {3699, 34387}, {3700, 28660}, {3701, 18155}, {3718, 44426}, {4041, 40072}, {4081, 4572}, {4105, 41283}, {4130, 20567}, {4148, 18895}, {4163, 6063}, {4171, 6385}, {4529, 44187}, {4560, 30713}, {4561, 21666}, {4602, 36197}, {4998, 23104}, {6332, 7017}, {6558, 23989}, {7035, 42455}, {7101, 15413}, {7256, 21207}, {7258, 16732}, {17924, 52406}, {17926, 40071}, {23970, 46406}, {24002, 30693}, {31625, 42462}, {44130, 52355}
X(52622) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 1461}, {7, 6614}, {8, 109}, {9, 1415}, {11, 43924}, {75, 934}, {76, 658}, {78, 36059}, {85, 4617}, {92, 32714}, {100, 24027}, {101, 23979}, {190, 1262}, {200, 692}, {219, 32660}, {220, 32739}, {253, 36079}, {264, 36118}, {274, 4637}, {280, 8059}, {281, 32674}, {306, 52610}, {310, 4616}, {312, 651}, {313, 4566}, {314, 1414}, {318, 108}, {321, 1020}, {333, 4565}, {341, 100}, {345, 1813}, {346, 101}, {513, 1106}, {514, 1407}, {521, 603}, {522, 56}, {523, 1042}, {525, 52373}, {561, 4569}, {644, 2149}, {645, 52378}, {646, 4564}, {649, 52410}, {650, 604}, {652, 52411}, {656, 1410}, {657, 32}, {658, 23971}, {663, 1397}, {664, 7339}, {668, 7045}, {693, 269}, {850, 3668}, {885, 1416}, {905, 7099}, {1016, 4619}, {1021, 1333}, {1043, 110}, {1090, 764}, {1111, 43932}, {1146, 649}, {1260, 32656}, {1264, 6517}, {1265, 1331}, {1502, 46406}, {1577, 1427}, {1639, 1404}, {1792, 4575}, {1969, 13149}, {1978, 1275}, {2287, 163}, {2310, 667}, {2321, 4559}, {2322, 112}, {2327, 32661}, {2328, 1576}, {2517, 4320}, {2804, 1457}, {2968, 1459}, {3064, 608}, {3119, 3063}, {3120, 7250}, {3239, 6}, {3261, 279}, {3263, 41353}, {3596, 664}, {3669, 7366}, {3676, 7023}, {3686, 36075}, {3692, 906}, {3699, 59}, {3700, 1400}, {3701, 4551}, {3703, 46153}, {3710, 23067}, {3716, 1428}, {3717, 2283}, {3718, 6516}, {3737, 1408}, {3738, 52440}, {3810, 7248}, {3900, 31}, {4007, 36074}, {4025, 7053}, {4036, 1254}, {4041, 1402}, {4064, 1425}, {4081, 663}, {4082, 4557}, {4086, 65}, {4105, 2175}, {4130, 41}, {4147, 1403}, {4148, 1914}, {4163, 55}, {4171, 213}, {4183, 32676}, {4391, 57}, {4397, 1}, {4451, 29055}, {4477, 7122}, {4516, 51641}, {4522, 1469}, {4524, 1918}, {4528, 902}, {4529, 172}, {4546, 3052}, {4560, 1412}, {4569, 24013}, {4578, 1110}, {4723, 23703}, {4768, 1319}, {4811, 3361}, {4845, 32728}, {4858, 3669}, {4939, 51656}, {4944, 1405}, {4953, 8643}, {4985, 32636}, {4990, 2308}, {5423, 3939}, {6063, 4626}, {6332, 222}, {6335, 7128}, {6385, 4635}, {6556, 1293}, {6557, 38828}, {6558, 1252}, {6559, 919}, {6735, 23981}, {6736, 23845}, {6745, 23346}, {7017, 653}, {7027, 6733}, {7046, 8750}, {7058, 4556}, {7101, 1783}, {7252, 16947}, {7253, 58}, {7256, 4570}, {7258, 4567}, {7649, 1398}, {8058, 221}, {8611, 1409}, {8641, 560}, {14208, 1439}, {14298, 2199}, {14302, 1035}, {14304, 1455}, {14312, 51660}, {14427, 2251}, {14837, 6611}, {14936, 1919}, {14942, 32735}, {15411, 1790}, {15413, 7177}, {15416, 63}, {15629, 32643}, {16732, 7216}, {17898, 40933}, {17924, 1435}, {17926, 1474}, {18025, 24016}, {18155, 1014}, {18344, 1395}, {20294, 4306}, {20567, 36838}, {20948, 1446}, {20954, 38859}, {21044, 7180}, {21119, 17114}, {21132, 1357}, {21666, 7649}, {21789, 2206}, {23104, 11}, {23615, 3271}, {23838, 1417}, {23970, 657}, {23978, 514}, {23983, 4091}, {24002, 738}, {24006, 1426}, {24010, 8641}, {24026, 513}, {24031, 23224}, {28071, 32666}, {28132, 1438}, {28654, 4605}, {28659, 4554}, {28660, 4573}, {30681, 4587}, {30693, 644}, {30713, 4552}, {34387, 3676}, {34404, 37141}, {34591, 22383}, {35057, 1399}, {35517, 23973}, {35518, 77}, {35519, 7}, {36101, 32668}, {36197, 798}, {36626, 36082}, {36795, 37136}, {36796, 36146}, {36910, 32675}, {40072, 4625}, {40213, 16726}, {40363, 4572}, {40422, 36048}, {40435, 32651}, {40495, 1088}, {41798, 36141}, {42337, 1201}, {42455, 244}, {42462, 1015}, {44426, 34}, {44448, 1617}, {46107, 1119}, {46110, 278}, {46406, 23586}, {48278, 1401}, {50333, 1458}, {51565, 2720}, {51972, 35326}, {52335, 512}, {52344, 26700}, {52355, 73}, {52356, 1411}, {52406, 1332}, {52409, 2222}

---

X(52623) = X(2)X(52597)∩X(75)X(21192)

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

X(52623) lies on these lines: {2, 52597}, {10, 21050}, {75, 21192}, {101, 22456}, {190, 32680}, {321, 4391}, {514, 17789}, {693, 47678}, {824, 35559}, {850, 1577}, {905, 31993}, {2533, 18003}, {2774, 21300}, {2901, 21831}, {3175, 45664}, {3239, 46110}, {3261, 4823}, {3900, 5295}, {4036, 23282}, {4064, 4086}, {4079, 4129}, {14838, 17899}, {17496, 31025}, {20907, 21188}, {20948, 44173}, {21438, 23875}, {22043, 23886}, {23100, 23596}, {23105, 52576}

X(52623) = midpoint of X(4391) and X(23685)
X(52623) = isotomic conjugate of X(4556)
X(52623) = anticomplement of X(52597)
X(52623) = isotomic conjugate of the isogonal conjugate of X(4024)
X(52623) = polar conjugate of the isogonal conjugate of X(4064)
X(52623) = X(i)-Ceva conjugate of X(j) for these (i,j): {850, 4036}, {1978, 313}, {27801, 21207}, {27808, 52576}, {28654, 338}
X(52623) = X(i)-cross conjugate of X(j) for these (i,j): {338, 28654}, {20902, 6358}, {21046, 10}, {21131, 1109}
X(52623) = X(i)-isoconjugate of X(j) for these (i,j): {28, 32661}, {31, 4556}, {36, 32671}, {58, 163}, {60, 1415}, {81, 1576}, {101, 849}, {109, 2150}, {110, 1333}, {112, 1437}, {249, 667}, {250, 22383}, {270, 32660}, {274, 14574}, {513, 23357}, {514, 23995}, {560, 4610}, {593, 692}, {604, 4636}, {643, 16947}, {644, 7342}, {649, 1101}, {662, 2206}, {693, 23963}, {757, 32739}, {1397, 4612}, {1408, 5546}, {1474, 4575}, {1501, 4623}, {1790, 32676}, {1917, 52612}, {1919, 24041}, {1980, 4590}, {2189, 36059}, {2194, 4565}, {2203, 4558}, {4131, 41937}, {4630, 16696}, {4631, 41280}, {5006, 17939}, {6591, 47390}, {7113, 36069}, {9274, 21828}, {14586, 18180}, {14599, 36066}, {16702, 32729}, {17187, 34072}, {18604, 32713}, {18605, 32734}, {23224, 23964}, {30576, 32719}, {32640, 51420}, {37140, 52434}
X(52623) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 4556}, {10, 163}, {11, 2150}, {37, 110}, {58, 115}, {60, 1146}, {81, 4858}, {86, 36901}, {101, 4075}, {127, 17186}, {136, 1474}, {244, 1333}, {249, 6631}, {284, 6741}, {338, 17167}, {339, 16887}, {514, 18314}, {523, 649}, {525, 4091}, {593, 1086}, {647, 1459}, {662, 40603}, {757, 40619}, {849, 1015}, {1084, 2206}, {1101, 5375}, {1214, 4565}, {1412, 40622}, {1437, 34591}, {1576, 40586}, {1790, 15526}, {1919, 3005}, {2185, 40624}, {2189, 20620}, {2260, 52119}, {2968, 7054}, {3161, 4636}, {3733, 4988}, {4025, 23285}, {4575, 51574}, {4610, 6374}, {7113, 38982}, {7341, 40615}, {9296, 24041}, {14599, 38978}, {15449, 17187}, {15898, 32671}, {17209, 35088}, {20970, 21709}, {23357, 39026}, {32661, 40591}, {32739, 40607}, {39019, 44709}
X(52623) = cevapoint of X(i) and X(j) for these (i,j): {10, 21092}, {1109, 21131}, {4024, 4064}
X(52623) = crosspoint of X(i) and X(j) for these (i,j): {313, 1978}, {850, 20948}, {1897, 18082}
X(52623) = crosssum of X(i) and X(j) for these (i,j): {1459, 17187}, {1919, 2206}
X(52623) = trilinear pole of line {338, 1109}
X(52623) = crossdifference of every pair of points on line {2206, 18892}
X(52623) = barycentric product X(i)*X(j) for these {i,j}: {10, 850}, {12, 35519}, {37, 20948}, {42, 44173}, {75, 4036}, {76, 4024}, {100, 23994}, {101, 23962}, {115, 1978}, {190, 338}, {264, 4064}, {306, 14618}, {313, 523}, {321, 1577}, {339, 1897}, {349, 3700}, {514, 28654}, {522, 34388}, {561, 4705}, {594, 3261}, {661, 27801}, {668, 1109}, {670, 21043}, {693, 1089}, {756, 40495}, {1230, 31010}, {1441, 4086}, {1502, 4079}, {1826, 3267}, {1928, 50487}, {2501, 40071}, {2643, 6386}, {2970, 4561}, {3120, 27808}, {3695, 46107}, {3701, 4077}, {3952, 21207}, {4025, 7141}, {4033, 16732}, {4092, 4572}, {4103, 23989}, {4155, 44172}, {4391, 6358}, {4600, 23105}, {4602, 21833}, {4605, 23978}, {4608, 52576}, {6331, 21046}, {6335, 20902}, {6370, 20566}, {7178, 30713}, {8611, 52575}, {14208, 41013}, {17924, 52369}, {18082, 23285}, {20336, 24006}, {21056, 40162}, {21131, 31625}, {26942, 46110}, {35352, 35544}, {44170, 46390}
X(52623) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 4556}, {8, 4636}, {10, 110}, {12, 109}, {37, 163}, {42, 1576}, {71, 32661}, {72, 4575}, {76, 4610}, {80, 36069}, {100, 1101}, {101, 23357}, {115, 649}, {125, 1459}, {190, 249}, {201, 36059}, {226, 4565}, {306, 4558}, {312, 4612}, {313, 99}, {321, 662}, {334, 36066}, {338, 514}, {339, 4025}, {349, 4573}, {512, 2206}, {513, 849}, {514, 593}, {522, 60}, {523, 58}, {525, 1790}, {561, 4623}, {594, 101}, {650, 2150}, {656, 1437}, {661, 1333}, {668, 24041}, {692, 23995}, {693, 757}, {756, 692}, {826, 17187}, {850, 86}, {1089, 100}, {1109, 513}, {1268, 6578}, {1331, 47390}, {1365, 43924}, {1441, 1414}, {1446, 4637}, {1500, 32739}, {1502, 52612}, {1577, 81}, {1824, 32676}, {1826, 112}, {1897, 250}, {1918, 14574}, {1978, 4590}, {2161, 32671}, {2171, 1415}, {2197, 32660}, {2321, 5546}, {2485, 17186}, {2501, 1474}, {2610, 7113}, {2618, 18180}, {2632, 23224}, {2643, 667}, {2799, 17209}, {2970, 7649}, {3064, 2189}, {3120, 3733}, {3124, 1919}, {3239, 7054}, {3261, 1509}, {3267, 17206}, {3676, 7341}, {3690, 32656}, {3695, 1331}, {3700, 284}, {3701, 643}, {3708, 22383}, {3762, 30576}, {3949, 906}, {3952, 4570}, {4010, 5009}, {4013, 901}, {4017, 1408}, {4024, 6}, {4033, 4567}, {4036, 1}, {4041, 2194}, {4053, 1983}, {4062, 5467}, {4064, 3}, {4077, 1014}, {4079, 32}, {4080, 4591}, {4086, 21}, {4088, 3286}, {4092, 663}, {4103, 1252}, {4120, 3285}, {4122, 3736}, {4150, 4611}, {4155, 2210}, {4163, 6061}, {4391, 2185}, {4397, 1098}, {4404, 16948}, {4466, 7254}, {4552, 52378}, {4572, 7340}, {4605, 1262}, {4608, 52558}, {4705, 31}, {4806, 34476}, {4931, 4273}, {4978, 30581}, {6046, 6614}, {6057, 3939}, {6354, 1461}, {6358, 651}, {6367, 2308}, {6368, 44709}, {6370, 36}, {6386, 24037}, {6535, 4557}, {6538, 8701}, {6539, 4629}, {6541, 17943}, {6543, 2702}, {6757, 13486}, {7140, 8750}, {7141, 1897}, {7178, 1412}, {7180, 16947}, {7199, 763}, {7265, 40214}, {8013, 35327}, {8029, 3122}, {8611, 2193}, {8736, 32674}, {11599, 17940}, {14208, 1444}, {14321, 33628}, {14618, 27}, {15523, 1634}, {15526, 4091}, {16732, 1019}, {17879, 4131}, {18004, 1326}, {18070, 52376}, {18082, 827}, {18098, 34072}, {18314, 17167}, {18359, 37140}, {20336, 4592}, {20902, 905}, {20948, 274}, {21011, 1625}, {21012, 35324}, {21016, 35325}, {21043, 512}, {21044, 7252}, {21046, 647}, {21050, 9306}, {21051, 38832}, {21054, 2605}, {21056, 1613}, {21089, 9218}, {21092, 36830}, {21121, 52564}, {21124, 40153}, {21131, 1015}, {21134, 3937}, {21207, 7192}, {21682, 45235}, {21714, 40091}, {21720, 16685}, {21723, 42653}, {21728, 23861}, {21833, 798}, {21859, 2149}, {21947, 44410}, {23105, 3120}, {23282, 386}, {23285, 16887}, {23962, 3261}, {23994, 693}, {24006, 28}, {24018, 18604}, {26942, 1813}, {27801, 799}, {27808, 4600}, {28654, 190}, {28659, 4631}, {29037, 19655}, {30713, 645}, {31010, 1171}, {32739, 23963}, {34388, 664}, {35352, 741}, {35519, 261}, {35522, 6629}, {36035, 51420}, {36793, 30805}, {39691, 21123}, {40071, 4563}, {40495, 873}, {40521, 1110}, {41013, 162}, {41079, 18653}, {42666, 52434}, {42713, 23889}, {43924, 7342}, {44173, 310}, {44426, 270}, {46110, 46103}, {46390, 14599}, {47318, 9273}, {48395, 44119}, {48407, 39673}, {50487, 560}, {51663, 52440}, {52335, 21789}, {52355, 283}, {52356, 52380}, {52369, 1332}, {52576, 4427}
X(52623) = {X(321),X(4391)}-harmonic conjugate of X(7265)

---

X(52624) = X(2)X(2416)∩X(76)X(2394)

Barycentrics    b^2*(b - c)*c^2*(b + c)*(-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(52624) lies on these lines: {2, 2416}, {76, 2394}, {343, 525}, {647, 15421}, {684, 39509}, {1553, 23097}, {1637, 5664}, {2489, 41361}, {3265, 14566}, {4558, 30528}, {5254, 47125}, {14618, 15466}, {20580, 45681}

X(52624) = isotomic conjugate of X(34568)
X(52624) = anticomplement of X(52600)
X(52624) = isotomic conjugate of the isogonal conjugate of X(14401)
X(52624) = X(6331)-Ceva conjugate of X(3260)
X(52624) = X(39008)-cross conjugate of X(16163)
X(52624) = X(i)-isoconjugate of X(j) for these (i,j): {31, 34568}, {74, 36131}, {162, 40353}, {1304, 2159}, {2349, 32715}, {8749, 36034}, {32640, 36119}, {32676, 40384}, {32695, 35200}
X(52624) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 34568}, {6, 1650}, {30, 112}, {74, 39008}, {125, 40353}, {133, 32695}, {647, 9033}, {1304, 3163}, {1511, 32640}, {3258, 8749}, {14380, 14401}, {15526, 40384}, {18877, 38999}
X(52624) = crosspoint of X(3260) and X(6331)
X(52624) = crosssum of X(3049) and X(40352)
X(52624) = crossdifference of every pair of points on line {9407, 32715}
X(52624) = barycentric product X(i)*X(j) for these {i,j}: {76, 14401}, {339, 3233}, {525, 36789}, {850, 16163}, {1099, 14208}, {2631, 46234}, {3163, 3267}, {3260, 9033}, {3265, 34334}, {6331, 39008}, {11064, 41079}, {14920, 18557}, {23097, 34767}, {41077, 46106}
X(52624) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 34568}, {30, 1304}, {525, 40384}, {647, 40353}, {1099, 162}, {1495, 32715}, {1553, 7480}, {1568, 36831}, {1636, 18877}, {1637, 8749}, {1650, 14380}, {1990, 32695}, {2173, 36131}, {2631, 2159}, {3081, 23347}, {3163, 112}, {3233, 250}, {3260, 16077}, {3267, 31621}, {3284, 32640}, {9033, 74}, {9409, 40352}, {11064, 44769}, {14345, 15291}, {14398, 40354}, {14401, 6}, {14566, 40391}, {16163, 110}, {16240, 32713}, {18558, 11079}, {23097, 4240}, {34334, 107}, {36035, 36119}, {36789, 648}, {38956, 1301}, {39008, 647}, {41077, 14919}, {41079, 16080}, {42074, 32676}, {46106, 15459}, {47071, 41433}

---

X(52625) = X(76)X(141)∩X(115)X(125)

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

X(52625) lies on these lines: {2, 30229}, {6, 6787}, {76, 141}, {115, 125}, {148, 9998}, {217, 39764}, {230, 47044}, {512, 2086}, {574, 33705}, {671, 694}, {1645, 39010}, {2679, 21905}, {3231, 5118}, {6071, 21906}, {6786, 52067}, {9225, 10411}, {14041, 36881}, {34359, 43448}

X(52625) = isotomic conjugate of the isogonal conjugate of X(1645)
X(52625) = X(i)-Ceva conjugate of X(j) for these (i,j): {3231, 888}, {30736, 9148}, {34087, 523}, {46522, 887}
X(52625) = X(i)-isoconjugate of X(j) for these (i,j): {99, 36133}, {163, 886}, {662, 9150}, {729, 24037}, {799, 32717}, {1101, 34087}, {3228, 24041}, {4590, 37132}
X(52625) = X(i)-Dao conjugate of X(j) for these (i,j): {99, 39010}, {115, 886}, {512, 729}, {523, 34087}, {888, 3231}, {1084, 9150}, {1645, 23342}, {3005, 3228}, {4590, 38998}, {14608, 21905}, {32717, 38996}, {34537, 35073}, {36133, 38986}
X(52625) = crosspoint of X(i) and X(j) for these (i,j): {523, 34087}, {888, 3231}, {9148, 30736}
X(52625) = crosssum of X(i) and X(j) for these (i,j): {32, 5118}, {110, 33875}, {3228, 9150}
X(52625) = crossdifference of every pair of points on line {110, 4590}
X(52625) = barycentric product X(i)*X(j) for these {i,j}: {76, 1645}, {115, 3231}, {125, 46522}, {338, 33875}, {512, 9148}, {523, 888}, {538, 3124}, {850, 887}, {1084, 30736}, {1648, 14609}, {2234, 2643}, {4609, 33918}, {5118, 8029}, {6786, 51441}, {22260, 23342}, {34087, 39010}, {36822, 44114}
X(52625) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 34087}, {512, 9150}, {523, 886}, {538, 34537}, {669, 32717}, {798, 36133}, {887, 110}, {888, 99}, {1084, 729}, {1645, 6}, {2086, 51510}, {2234, 24037}, {3124, 3228}, {3231, 4590}, {5118, 31614}, {9148, 670}, {14406, 1634}, {21906, 14608}, {30736, 44168}, {33875, 249}, {33918, 669}, {39010, 3231}, {46522, 18020}
X(52625) = {X(6787),X(14700)}-harmonic conjugate of X(6)

---

X(52626) = X(76)X(321)∩X(514)X(2087)

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

X(52626) lies on these lines: {37, 46894}, {76, 321}, {244, 4403}, {335, 20568}, {514, 2087}, {536, 23891}, {764, 1647}, {918, 1086}, {1015, 21139}, {1212, 16602}, {1646, 14433}, {3230, 36816}, {3679, 49493}, {3994, 8031}, {4014, 23676}, {4049, 4444}, {4384, 17595}, {4424, 24399}, {4516, 21142}, {4530, 6547}, {7200, 21208}, {9318, 24281}, {19945, 30583}, {19950, 19976}, {20924, 30967}, {21129, 35092}, {21210, 23774}, {24403, 36226}

X(52626) = reflection of X(2087) in X(27918)
X(52626) = isotomic conjugate of X(5381)
X(52626) = isotomic conjugate of the isogonal conjugate of X(1646)
X(52626) = X(i)-Ceva conjugate of X(j) for these (i,j): {536, 4728}, {6381, 14431}, {43037, 891}
X(52626) = X(i)-isoconjugate of X(j) for these (i,j): {31, 5381}, {100, 34075}, {101, 898}, {190, 32718}, {692, 4607}, {739, 765}, {889, 32739}, {1110, 3227}, {1252, 37129}, {2149, 36798}, {6632, 23349}, {23990, 31002}
X(52626) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 5381}, {6, 14434}, {100, 39011}, {513, 739}, {514, 3227}, {650, 36798}, {661, 37129}, {765, 40614}, {889, 40619}, {891, 3230}, {898, 1015}, {1016, 13466}, {1086, 4607}, {1646, 23343}, {4988, 41683}, {6544, 36872}, {8054, 34075}
X(52626) = crosspoint of X(536) and X(4728)
X(52626) = crosssum of X(739) and X(34075)
X(52626) = crossdifference of every pair of points on line {692, 1252}
X(52626) = barycentric product X(i)*X(j) for these {i,j}: {11, 43037}, {75, 19945}, {76, 1646}, {244, 6381}, {514, 4728}, {536, 1086}, {693, 891}, {764, 41314}, {890, 40495}, {899, 1111}, {1015, 35543}, {1358, 4009}, {2401, 42764}, {3230, 23989}, {3261, 3768}, {3676, 14430}, {3994, 17205}, {4444, 14433}, {4526, 24002}, {4608, 30592}, {6386, 33917}, {6545, 23891}, {6548, 30583}, {7192, 14431}
X(52626) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 5381}, {11, 36798}, {244, 37129}, {513, 898}, {514, 4607}, {536, 1016}, {649, 34075}, {667, 32718}, {693, 889}, {764, 43928}, {890, 692}, {891, 100}, {899, 765}, {1015, 739}, {1086, 3227}, {1111, 31002}, {1646, 6}, {1647, 36872}, {3120, 41683}, {3230, 1252}, {3768, 101}, {4009, 4076}, {4526, 644}, {4728, 190}, {6381, 7035}, {8027, 23349}, {14404, 4557}, {14430, 3699}, {14431, 3952}, {14433, 3570}, {14434, 23343}, {14437, 1023}, {14441, 890}, {19945, 1}, {21143, 23892}, {23891, 6632}, {28603, 4767}, {30583, 17780}, {30592, 4427}, {33917, 667}, {35543, 31625}, {39011, 3230}, {42764, 2397}, {43037, 4998}
X(52626) = {X(1111),X(21138)}-harmonic conjugate of X(3125)

---

X(52627) = X(75)X(21198)∩X(312)X(514)

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

X(52627) lies on these lines: {75, 21198}, {76, 3261}, {312, 514}, {321, 4391}, {1089, 21132}, {1230, 1577}, {3762, 4120}, {4079, 21070}, {4169, 14437}, {4358, 46781}, {4385, 21201}, {4543, 4738}, {4975, 30573}, {33168, 47793}

X(52627) = isotomic conjugate of X(4638)
X(52627) = isotomic conjugate of the isogonal conjugate of X(6544)
X(52627) = X(1978)-Ceva conjugate of X(3264)
X(52627) = X(i)-isoconjugate of X(j) for these (i,j): {31, 4638}, {32, 4618}, {88, 32719}, {100, 41935}, {106, 32665}, {679, 32739}, {692, 2226}, {901, 9456}, {1318, 1415}, {1417, 5548}, {1576, 30575}, {2251, 39414}
X(52627) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 4638}, {6, 1647}, {101, 519}, {106, 35092}, {214, 32665}, {649, 900}, {679, 40619}, {901, 4370}, {1086, 2226}, {1146, 1318}, {2087, 17109}, {2316, 51402}, {4618, 6376}, {4858, 30575}, {6544, 23345}, {8054, 41935}, {9456, 38979}, {9460, 39414}
X(52627) = crosspoint of X(1978) and X(3264)
X(52627) = crosssum of X(32719) and X(32739)
X(52627) = crossdifference of every pair of points on line {9459, 32719}
X(52627) = barycentric product X(i)*X(j) for these {i,j}: {76, 6544}, {514, 36791}, {561, 3251}, {678, 40495}, {693, 4738}, {900, 3264}, {1317, 35519}, {1577, 16729}, {1978, 35092}, {3261, 4370}, {3596, 39771}, {3762, 4358}, {4542, 4572}, {4543, 6063}, {6386, 42084}, {14442, 31625}
X(52627) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 4638}, {44, 32665}, {75, 4618}, {514, 2226}, {519, 901}, {522, 1318}, {649, 41935}, {678, 692}, {693, 679}, {900, 106}, {902, 32719}, {903, 39414}, {1017, 32739}, {1317, 109}, {1577, 30575}, {1635, 9456}, {1639, 2316}, {1647, 23345}, {2325, 5548}, {3251, 31}, {3264, 4555}, {3762, 88}, {4152, 3939}, {4358, 3257}, {4370, 101}, {4542, 663}, {4543, 55}, {4738, 100}, {4768, 1320}, {6544, 6}, {6550, 43922}, {8028, 23344}, {14027, 43924}, {14442, 1015}, {16704, 4591}, {16729, 662}, {17780, 9268}, {21129, 52206}, {22086, 32659}, {22371, 32656}, {23757, 14260}, {24004, 5376}, {30939, 4622}, {33922, 902}, {35092, 649}, {36791, 190}, {36924, 6014}, {39771, 56}, {42070, 8750}, {42084, 667}

---

X(52628) = X(2)X(39)∩X(115)X(127)

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

X(52628) lies on these lines: {2, 39}, {32, 40856}, {111, 18019}, {115, 127}, {183, 35936}, {187, 4235}, {264, 18424}, {287, 14901}, {290, 48982}, {468, 14357}, {523, 3143}, {524, 43084}, {525, 2088}, {625, 1236}, {826, 44114}, {868, 5489}, {1235, 39565}, {2394, 43665}, {2974, 38738}, {3260, 31173}, {5099, 48317}, {5305, 44338}, {5306, 44649}, {5475, 44155}, {5891, 18390}, {6388, 36793}, {6785, 18304}, {7422, 14265}, {7615, 44135}, {7748, 35923}, {7813, 13162}, {10104, 18570}, {16186, 36189}, {17131, 40879}, {18311, 21906}, {18374, 36820}, {19626, 46140}, {31693, 43085}, {31694, 43086}, {35088, 39021}, {37347, 45943}

X(52628) = isotomic conjugate of the isogonal conjugate of X(1648)
X(52628) = X(i)-complementary conjugate of X(j) for these (i,j): {661, 15116}, {798, 1560}, {1177, 4369}, {2373, 42327}, {37220, 23301}, {46140, 21263}
X(52628) = X(i)-Ceva conjugate of X(j) for these (i,j): {3266, 35522}, {18019, 523}, {18023, 850}, {44146, 690}, {46140, 512}
X(52628) = X(i)-isoconjugate of X(j) for these (i,j): {110, 36142}, {111, 1101}, {163, 691}, {249, 923}, {250, 36060}, {662, 32729}, {671, 23995}, {798, 45773}, {897, 23357}, {922, 34539}, {1576, 36085}, {4117, 42370}, {19626, 24037}, {23963, 46277}, {24041, 32740}, {32678, 51478}, {34072, 36827}, {36128, 47390}
X(52628) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 1649}, {23, 2492}, {32, 21905}, {110, 23992}, {111, 523}, {112, 48317}, {115, 691}, {187, 690}, {244, 36142}, {249, 2482}, {250, 1560}, {512, 19626}, {647, 895}, {671, 18314}, {892, 36901}, {1084, 32729}, {1576, 38988}, {1648, 5467}, {3005, 32740}, {4858, 36085}, {6593, 23357}, {14273, 37777}, {14417, 22151}, {15449, 36827}, {17416, 32583}, {17436, 42007}, {18334, 51478}, {23285, 30786}, {31998, 45773}, {34539, 39061}
X(52628) = crosspoint of X(i) and X(j) for these (i,j): {468, 22105}, {850, 18023}, {3266, 35522}, {14618, 46105}
X(52628) = crosssum of X(i) and X(j) for these (i,j): {895, 36827}, {1576, 14567}, {10317, 32661}, {32729, 32740}
X(52628) = trilinear pole of line {33919, 51429}
X(52628) = crossdifference of every pair of points on line {669, 1576}
X(52628) = barycentric product X(i)*X(j) for these {i,j}: {76, 1648}, {115, 3266}, {125, 44146}, {187, 23962}, {290, 51429}, {338, 524}, {339, 468}, {351, 44173}, {523, 35522}, {670, 33919}, {690, 850}, {868, 52145}, {896, 23994}, {1109, 14210}, {1502, 21906}, {2501, 45807}, {2642, 20948}, {2970, 6390}, {3267, 14273}, {3268, 51479}, {4062, 21207}, {5099, 18019}, {5468, 23105}, {10412, 45808}, {14417, 14618}, {15526, 37778}, {16732, 42713}, {18023, 23992}, {18312, 50942}, {22105, 23285}, {34336, 51258}, {34537, 42344}
X(52628) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 45773}, {115, 111}, {125, 895}, {187, 23357}, {338, 671}, {339, 30786}, {351, 1576}, {468, 250}, {512, 32729}, {523, 691}, {524, 249}, {526, 51478}, {661, 36142}, {671, 34539}, {690, 110}, {826, 36827}, {850, 892}, {868, 5968}, {896, 1101}, {922, 23995}, {1084, 19626}, {1109, 897}, {1365, 7316}, {1577, 36085}, {1645, 41294}, {1648, 6}, {1649, 5467}, {2642, 163}, {2643, 923}, {2682, 1495}, {2970, 17983}, {3124, 32740}, {3266, 4590}, {3292, 47390}, {3708, 36060}, {3906, 32583}, {4036, 5380}, {4062, 4570}, {4092, 5547}, {4235, 47443}, {4750, 4556}, {5099, 23}, {5466, 34574}, {8029, 9178}, {8288, 42007}, {8371, 23348}, {8754, 8753}, {9204, 17402}, {9205, 17403}, {12079, 9139}, {14210, 24041}, {14273, 112}, {14417, 4558}, {14423, 9171}, {14424, 1634}, {14432, 4636}, {14443, 351}, {14444, 39689}, {14567, 23963}, {18312, 50941}, {20578, 9206}, {20579, 9207}, {20975, 14908}, {21906, 32}, {22105, 827}, {23105, 5466}, {23287, 11636}, {23962, 18023}, {23992, 187}, {23994, 46277}, {33919, 512}, {33921, 9181}, {34537, 42370}, {35522, 99}, {37778, 23582}, {38395, 35265}, {39691, 46154}, {42344, 3124}, {42713, 4567}, {43084, 39295}, {44114, 51980}, {44146, 18020}, {44814, 52603}, {45807, 4563}, {45808, 10411}, {46462, 1380}, {46463, 1379}, {47415, 10317}, {48317, 37777}, {50942, 5649}, {51258, 15398}, {51429, 511}, {51479, 476}, {51480, 35191}, {52038, 2715}, {52475, 1304}

---

X(52629) = X(69)X(512)∩X(141)X(525)

Barycentrics    b^2*(b - c)*c^2*(b + c)*(-2*a^2 + b^2 + c^2)^2 : :
X(52629) = 3 X[599] - X[10097], X[35522] - 3 X[45807], 3 X[7652] - 5 X[31277]

X(52629) lies on these lines: {69, 512}, {76, 850}, {99, 20404}, {141, 525}, {520, 5486}, {599, 10097}, {647, 7801}, {690, 5181}, {826, 22260}, {1649, 6077}, {3266, 34763}, {5468, 11183}, {5652, 38940}, {6041, 7820}, {6390, 44814}, {7652, 7746}, {7795, 8574}, {17131, 47229}, {30209, 34507}, {36890, 52145}

X(52629) = isotomic conjugate of X(34574)
X(52629) = isotomic conjugate of the isogonal conjugate of X(1649)
X(52629) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {13574, 21221}, {22259, 21220}
X(52629) = X(i)-Ceva conjugate of X(j) for these (i,j): {670, 3266}, {850, 35522}
X(52629) = X(i)-isoconjugate of X(j) for these (i,j): {31, 34574}, {111, 36142}, {163, 10630}, {662, 41936}, {691, 923}, {798, 34539}, {897, 32729}, {15398, 32676}, {32740, 36085}
X(52629) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 34574}, {6, 1648}, {110, 524}, {111, 23992}, {115, 10630}, {512, 690}, {691, 2482}, {1084, 41936}, {1649, 9178}, {2492, 10561}, {6593, 32729}, {8753, 48317}, {15398, 15526}, {15900, 39413}, {21906, 51819}, {31998, 34539}, {32740, 38988}
X(52629) = crosspoint of X(i) and X(j) for these (i,j): {670, 3266}, {850, 35522}
X(52629) = crosssum of X(i) and X(j) for these (i,j): {669, 32740}, {1576, 32729}
X(52629) = crossdifference of every pair of points on line {14567, 18374}
X(52629) = barycentric product X(i)*X(j) for these {i,j}: {76, 1649}, {468, 45807}, {523, 36792}, {524, 35522}, {525, 34336}, {670, 23992}, {690, 3266}, {850, 2482}, {1577, 24038}, {3261, 52068}, {3267, 5095}, {4036, 16733}, {5466, 23106}, {14417, 44146}, {14443, 34537}, {18023, 33915}, {20948, 42081}, {39689, 44173}, {43084, 45808}
X(52629) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 34574}, {67, 39413}, {99, 34539}, {187, 32729}, {351, 32740}, {512, 41936}, {523, 10630}, {524, 691}, {525, 15398}, {690, 111}, {896, 36142}, {1366, 4565}, {1641, 23348}, {1648, 9178}, {1649, 6}, {2482, 110}, {2642, 923}, {3266, 892}, {5095, 112}, {5099, 10561}, {7067, 5546}, {7813, 36827}, {8030, 5467}, {14210, 36085}, {14273, 8753}, {14417, 895}, {14424, 46154}, {14443, 3124}, {14444, 351}, {18311, 14246}, {20380, 11636}, {20382, 46001}, {21905, 51819}, {23106, 5468}, {23992, 512}, {24038, 662}, {30454, 5995}, {30455, 5994}, {33915, 187}, {33921, 17964}, {34336, 648}, {35522, 671}, {36792, 99}, {39689, 1576}, {39785, 32583}, {41176, 9171}, {42081, 163}, {42713, 5380}, {45807, 30786}, {46049, 21906}, {51429, 8430}, {52039, 9206}, {52040, 9207}, {52068, 101}

---

X(52630) = X(2)X(32)∩X(3)X(33695)

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

X(52630) lies on these lines: {2, 32}, {3, 33695}, {6, 35936}, {22, 38526}, {23, 14246}, {50, 45331}, {76, 40856}, {99, 112}, {110, 9208}, {147, 14676}, {249, 2420}, {250, 46592}, {316, 10317}, {394, 48871}, {576, 15033}, {691, 5467}, {729, 38880}, {827, 17997}, {1287, 11635}, {1968, 40890}, {2396, 40173}, {2966, 14592}, {3095, 18570}, {3506, 13210}, {3972, 40879}, {4226, 14884}, {4230, 47443}, {4567, 42720}, {5118, 9218}, {6787, 44127}, {7468, 52603}, {7664, 36415}, {7767, 44338}, {7802, 10316}, {9408, 36790}, {9737, 35473}, {13193, 36213}, {16175, 51240}, {18114, 35921}, {18472, 21395}, {18879, 32697}, {34245, 36830}, {35345, 47291}, {37671, 44649}, {43084, 44376}

X(52630) = X(1287)-anticomplementary conjugate of X(21294)
X(52630) = X(i)-Ceva conjugate of X(j) for these (i,j): {892, 110}, {4590, 7664}
X(52630) = X(i)-cross conjugate of X(j) for these (i,j): {2492, 23}, {9517, 316}
X(52630) = X(i)-isoconjugate of X(j) for these (i,j): {67, 661}, {523, 2157}, {656, 8791}, {798, 18019}, {810, 46105}, {935, 3708}, {1577, 3455}, {2642, 10415}, {2643, 17708}, {3005, 37221}, {8061, 9076}, {14357, 23894}
X(52630) = X(i)-Dao conjugate of X(j) for these (i,j): {67, 36830}, {115, 5099}, {187, 690}, {523, 40583}, {858, 47138}, {7664, 23285}, {8791, 40596}, {10097, 39169}, {18019, 31998}, {39062, 46105}
X(52630) = cevapoint of X(i) and X(j) for these (i,j): {23, 2492}, {9517, 10317}
X(52630) = crosspoint of X(i) and X(j) for these (i,j): {691, 827}, {4590, 45773}
X(52630) = crosssum of X(i) and X(j) for these (i,j): {523, 32193}, {690, 826}, {3124, 33919}
X(52630) = trilinear pole of line {23, 6593}
X(52630) = crossdifference of every pair of points on line {3005, 8288}
X(52630) = barycentric product X(i)*X(j) for these {i,j}: {23, 99}, {110, 316}, {112, 37804}, {163, 20944}, {249, 9979}, {648, 22151}, {662, 16568}, {670, 18374}, {691, 7664}, {892, 6593}, {1576, 40074}, {2492, 4590}, {4556, 21094}, {4558, 37765}, {4563, 8744}, {4570, 21205}, {4577, 9019}, {4599, 18715}, {4611, 37801}, {5099, 45773}, {5467, 52551}, {5468, 14246}, {5546, 17088}, {6331, 10317}, {9517, 18020}, {10411, 52449}, {10510, 35138}, {12824, 18878}, {16077, 16165}
X(52630) = barycentric quotient X(i)/X(j) for these {i,j}: {23, 523}, {99, 18019}, {110, 67}, {112, 8791}, {163, 2157}, {249, 17708}, {250, 935}, {316, 850}, {648, 46105}, {691, 10415}, {827, 9076}, {1576, 3455}, {2492, 115}, {4558, 34897}, {4599, 37221}, {5467, 14357}, {6593, 690}, {7664, 35522}, {8744, 2501}, {9019, 826}, {9517, 125}, {9979, 338}, {10317, 647}, {10510, 3906}, {11636, 10511}, {14246, 5466}, {16165, 9033}, {16568, 1577}, {18374, 512}, {20944, 20948}, {21205, 21207}, {22151, 525}, {33752, 868}, {35138, 10512}, {36415, 2492}, {37765, 14618}, {37804, 3267}, {40074, 44173}, {42659, 20975}, {52142, 9178}, {52449, 10412}
X(52630) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {112, 4611, 99}, {249, 2421, 10411}, {2420, 2421, 249}, {39298, 39299, 99}

---

X(52631) = X(262)X(1499)∩X(263)X(9009)

Barycentrics    a^2*(b - c)*(b + c)*(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(52631) lies on the Lemoine asymptotic hyperbola and these lines: {262, 1499}, {263, 9009}, {512, 2021}, {523, 3569}, {669, 2422}, {688, 2489}, {691, 26714}, {876, 2186}, {3049, 18105}, {3124, 51441}, {14398, 46001}, {25423, 39680}, {32696, 32716}, {35364, 43718}

X(52631) = X(26714)-Ceva conjugate of X(263)
X(52631) = X(17415)-cross conjugate of X(523)
X(52631) = X(i)-isoconjugate of X(j) for these (i,j): {99, 52134}, {110, 3403}, {163, 20023}, {182, 799}, {183, 662}, {458, 4592}, {3288, 24037}, {4556, 42711}, {4575, 44144}, {4593, 14096}, {4599, 14994}, {4602, 34396}, {23878, 24041}, {36084, 51373}
X(52631) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 20023}, {136, 44144}, {182, 38996}, {183, 1084}, {244, 3403}, {458, 5139}, {512, 3288}, {3005, 23878}, {3124, 14994}, {38986, 52134}, {38987, 51373}
X(52631) = cevapoint of X(512) and X(50550)
X(52631) = crosspoint of X(263) and X(26714)
X(52631) = crosssum of X(i) and X(j) for these (i,j): {183, 23878}, {3288, 14096}
X(52631) = crossdifference of every pair of points on line {182, 183}
X(52631) = barycentric product X(i)*X(j) for these {i,j}: {115, 26714}, {262, 512}, {263, 523}, {327, 669}, {661, 2186}, {826, 42288}, {850, 46319}, {868, 32716}, {1577, 3402}, {2395, 51543}, {2422, 46807}, {2489, 42313}, {2501, 43718}, {3005, 42299}, {6037, 44114}, {51444, 51513}
X(52631) = barycentric quotient X(i)/X(j) for these {i,j}: {262, 670}, {263, 99}, {327, 4609}, {512, 183}, {523, 20023}, {661, 3403}, {669, 182}, {688, 14096}, {798, 52134}, {882, 8842}, {1084, 3288}, {2186, 799}, {2422, 46806}, {2489, 458}, {2501, 44144}, {3005, 14994}, {3124, 23878}, {3402, 662}, {3569, 51373}, {4705, 42711}, {9426, 34396}, {14398, 51372}, {23099, 6784}, {26714, 4590}, {42288, 4577}, {42299, 689}, {42313, 52608}, {43718, 4563}, {46319, 110}, {50550, 51580}, {51543, 2396}

---

X(52632) = X(76)X(850)∩X(111)X(2367)

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

X(52632) lies on these lines: {76, 850}, {111, 2367}, {264, 8430}, {290, 671}, {308, 9178}, {523, 40826}, {691, 22456}, {892, 10412}, {897, 37219}, {1502, 44173}, {2395, 14568}, {3972, 47229}, {7801, 31072}, {7870, 30476}, {7940, 31277}, {9214, 44155}, {9517, 52189}, {9979, 10512}, {10555, 23962}, {10556, 46247}, {14295, 18007}, {16092, 52145}, {18023, 23288}, {30786, 40832}

X(52632) = isotomic conjugate of X(5467)
X(52632) = isotomic conjugate of the isogonal conjugate of X(5466)
X(52632) = polar conjugate of the isogonal conjugate of X(14977)
X(52632) = X(i)-cross conjugate of X(j) for these (i,j): {35522, 850}, {51258, 671}
X(52632) = X(i)-isoconjugate of X(j) for these (i,j): {31, 5467}, {32, 23889}, {110, 922}, {162, 23200}, {163, 187}, {351, 1101}, {560, 5468}, {662, 14567}, {690, 23995}, {896, 1576}, {1501, 24039}, {2642, 23357}, {3292, 32676}, {4235, 9247}, {4575, 44102}, {14210, 14574}, {16702, 32739}, {32729, 42081}, {36142, 39689}
X(52632) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 5467}, {110, 39061}, {115, 187}, {125, 23200}, {136, 44102}, {244, 922}, {338, 41586}, {339, 7813}, {351, 523}, {524, 36901}, {690, 18314}, {896, 4858}, {1084, 14567}, {1576, 15899}, {3292, 15526}, {5468, 6374}, {5664, 44814}, {6376, 23889}, {9155, 35088}, {14417, 23285}, {14574, 15477}, {16702, 40619}, {23992, 39689}, {38971, 47426}
X(52632) = cevapoint of X(i) and X(j) for these (i,j): {523, 9979}, {850, 35522}, {5466, 14977}
X(52632) = trilinear pole of line {338, 850}
X(52632) = crossdifference of every pair of points on line {14567, 23200}
X(52632) = barycentric product X(i)*X(j) for these {i,j}: {76, 5466}, {111, 44173}, {264, 14977}, {338, 892}, {523, 18023}, {525, 46111}, {561, 23894}, {671, 850}, {691, 23962}, {897, 20948}, {1502, 9178}, {1577, 46277}, {3267, 17983}, {6331, 51258}, {8430, 18024}, {9213, 20573}, {10097, 18022}, {14618, 30786}, {23288, 40826}, {23994, 36085}
X(52632) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 5467}, {75, 23889}, {76, 5468}, {94, 14559}, {111, 1576}, {115, 351}, {264, 4235}, {338, 690}, {339, 14417}, {512, 14567}, {523, 187}, {525, 3292}, {561, 24039}, {647, 23200}, {661, 922}, {671, 110}, {690, 39689}, {691, 23357}, {693, 16702}, {850, 524}, {892, 249}, {895, 32661}, {897, 163}, {1109, 2642}, {1577, 896}, {2394, 9717}, {2408, 1384}, {2501, 44102}, {2799, 9155}, {2970, 14273}, {3261, 6629}, {3267, 6390}, {4036, 21839}, {4077, 51653}, {5466, 6}, {5485, 2434}, {5968, 14966}, {8029, 21906}, {8430, 237}, {9139, 32640}, {9154, 2715}, {9178, 32}, {9180, 48450}, {9213, 50}, {9214, 2420}, {9979, 6593}, {10097, 184}, {10555, 2492}, {10561, 18374}, {10562, 39231}, {10630, 32729}, {14295, 5026}, {14618, 468}, {14977, 3}, {16732, 14419}, {17948, 9181}, {17983, 112}, {18007, 2502}, {18023, 99}, {18312, 45662}, {18314, 41586}, {18818, 11636}, {20948, 14210}, {21207, 4750}, {23105, 1648}, {23285, 7813}, {23288, 574}, {23894, 31}, {23962, 35522}, {27801, 42721}, {30786, 4558}, {31125, 1634}, {32729, 23963}, {32740, 14574}, {34246, 51927}, {35522, 2482}, {36085, 1101}, {36128, 32676}, {36142, 23995}, {36307, 5995}, {36310, 5994}, {40495, 16741}, {41079, 5642}, {42008, 9145}, {43665, 5967}, {44173, 3266}, {46111, 648}, {46277, 662}, {47138, 47426}, {48983, 46249}, {51258, 647}

---

X(52633) = X(11)X(244)∩X(75)X(982)

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

X(52633) lies on these lines: {2, 24413}, {11, 244}, {38, 25350}, {75, 982}, {192, 24420}, {194, 24421}, {239, 36817}, {876, 27846}, {894, 24426}, {1575, 40155}, {1646, 8027}, {3248, 6377}, {16614, 38986}, {19950, 24046}, {19951, 24174}, {19957, 24168}, {19962, 24167}, {21035, 25811}, {24423, 27912}

X(52633) = isogonal conjugate of the isotomic conjugate of X(21140)
X(52633) = X(20640)-complementary conjugate of X(24003)
X(52633) = X(i)-Ceva conjugate of X(j) for these (i,j): {334, 21123}, {876, 21143}, {3009, 6373}
X(52633) = X(i)-isoconjugate of X(j) for these (i,j): {100, 8709}, {727, 7035}, {765, 3226}, {1016, 20332}, {1252, 32020}, {3253, 5378}, {4564, 36799}, {4567, 27809}, {4600, 18793}, {4998, 8851}, {31625, 34077}
X(52633) = X(i)-Dao conjugate of X(j) for these (i,j): {350, 3837}, {513, 3226}, {661, 32020}, {874, 27846}, {3009, 6373}, {7035, 17793}, {8054, 8709}, {18793, 50497}, {20532, 31625}, {27809, 40627}
X(52633) = crosspoint of X(i) and X(j) for these (i,j): {291, 649}, {3009, 6373}
X(52633) = crosssum of X(i) and X(j) for these (i,j): {190, 238}, {3226, 8709}
X(52633) = crossdifference of every pair of points on line {101, 1016}
X(52633) = barycentric product X(i)*X(j) for these {i,j}: {6, 21140}, {244, 1575}, {514, 6373}, {649, 3837}, {667, 20908}, {726, 1015}, {1086, 3009}, {1111, 21760}, {1463, 2170}, {1977, 35538}, {2087, 36814}, {2423, 42766}, {2969, 20785}, {3123, 40881}, {3125, 18792}, {3248, 52043}, {3271, 43040}, {3733, 21053}, {7649, 22092}, {17205, 21830}, {21143, 23354}, {27918, 40155}
X(52633) = barycentric quotient X(i)/X(j) for these {i,j}: {244, 32020}, {649, 8709}, {726, 31625}, {1015, 3226}, {1575, 7035}, {1977, 727}, {3009, 1016}, {3121, 18793}, {3122, 27809}, {3123, 40844}, {3248, 20332}, {3249, 23355}, {3271, 36799}, {3837, 1978}, {6373, 190}, {18792, 4601}, {20908, 6386}, {21053, 27808}, {21140, 76}, {21760, 765}, {22092, 4561}, {38367, 3573}

---

X(52634) = {X(3), X(6524)}-HARMONIC CONJUGATE OF X(52470)

Barycentrics    2*Sin[2*A] - Tan[A]^3 : :
Barycentrics    a^2*(a^2 + b^2 - c^2)^3*(a^2 - b^2 + c^2)^3*((-a^2 + b^2 + c^2)^4 - 4*b^2*c^2*S^2) : :

X(52634) lies on these lines: {3, 6524}, {1093, 3520}, {6530, 22467}, {15750, 52439}, {21213, 34286}

X(52634) = {X(3),X(6524)}-harmonic conjugate of X(52470)

---

X(52635) = X(6)X(57)∩X(31)X(184)

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

X(52635) lies on the cubic K997 and these lines: {6, 57}, {9, 22163}, {31, 184}, {32, 603}, {36, 17966}, {37, 17625}, {39, 73}, {41, 1106}, {48, 5065}, {56, 213}, {65, 20963}, {109, 1914}, {221, 16502}, {226, 24512}, {230, 43043}, {241, 18206}, {295, 51956}, {388, 17750}, {572, 3955}, {573, 3784}, {649, 854}, {651, 1447}, {672, 1362}, {893, 3451}, {909, 1945}, {911, 32668}, {1015, 1457}, {1042, 1475}, {1055, 21742}, {1107, 37558}, {1319, 3230}, {1334, 4322}, {1394, 16780}, {1400, 1401}, {1403, 9315}, {1415, 2251}, {1416, 1428}, {1420, 2176}, {1429, 2111}, {1464, 43039}, {1466, 2271}, {1575, 4551}, {1758, 2323}, {1766, 35645}, {2078, 17735}, {2099, 16971}, {2149, 7113}, {2199, 16946}, {2238, 3911}, {2272, 14936}, {2273, 2286}, {2275, 10571}, {2280, 9316}, {2285, 10473}, {2295, 10106}, {2329, 9363}, {2333, 23630}, {2348, 35326}, {3125, 18838}, {3290, 3660}, {3554, 21767}, {3684, 9364}, {3726, 5083}, {3780, 4848}, {3990, 36743}, {3997, 4315}, {4253, 4306}, {4554, 39930}, {5089, 17435}, {5193, 9259}, {5276, 17074}, {5283, 37523}, {5299, 34043}, {5435, 37657}, {6180, 40719}, {7117, 8776}, {7248, 41264}, {8270, 16973}, {9436, 34253}, {9456, 36141}, {9599, 34029}, {16583, 37566}, {16778, 22134}, {16781, 34040}, {16968, 34489}, {16975, 24806}, {17594, 17613}, {18783, 21760}, {20227, 30456}, {21744, 30493}, {22164, 25066}, {31231, 37673}, {32735, 41934}, {33863, 37583}

X(52635) = isogonal conjugate of X(36796)
X(52635) = isogonal conjugate of the isotomic conjugate of X(241)
X(52635) = isogonal conjugate of the polar conjugate of X(1876)
X(52635) = X(i)-Ceva conjugate of X(j) for these (i,j): {1458, 2223}, {1462, 56}, {24016, 8641}, {32735, 667}
X(52635) = X(9454)-cross conjugate of X(2223)
X(52635) = X(i)-isoconjugate of X(j) for these (i,j): {1, 36796}, {2, 14942}, {7, 6559}, {8, 673}, {9, 2481}, {55, 18031}, {75, 294}, {76, 2195}, {85, 28071}, {105, 312}, {190, 885}, {200, 34018}, {239, 33676}, {281, 31637}, {314, 18785}, {318, 1814}, {333, 13576}, {341, 1462}, {345, 36124}, {514, 36802}, {522, 666}, {646, 1027}, {650, 51560}, {657, 46135}, {663, 36803}, {664, 28132}, {668, 1024}, {884, 1978}, {919, 35519}, {927, 3239}, {1146, 39293}, {1438, 3596}, {3685, 52209}, {3717, 6185}, {3718, 8751}, {3900, 34085}, {3975, 52030}, {4087, 51866}, {4391, 36086}, {4397, 36146}, {4518, 6654}, {4858, 5377}, {6335, 23696}, {6558, 43930}, {6601, 31638}, {7017, 36057}, {14330, 41075}, {32735, 52622}, {36798, 36816}
X(52635) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 36796}, {76, 39063}, {206, 294}, {223, 18031}, {312, 39046}, {478, 2481}, {561, 36905}, {2238, 4087}, {3126, 23978}, {3596, 6184}, {4391, 38989}, {4397, 39014}, {6609, 34018}, {7017, 20621}, {14942, 32664}, {17755, 28659}, {28132, 39025}, {35519, 38980}
X(52635) = crosspoint of X(i) and X(j) for these (i,j): {6, 911}, {56, 1462}, {241, 1876}, {1262, 32735}, {24027, 32668}
X(52635) = crosssum of X(i) and X(j) for these (i,j): {2, 30807}, {8, 3693}, {75, 33677}, {312, 3975}, {650, 4124}, {1146, 50333}, {6559, 14942}
X(52635) = trilinear pole of line {8638, 23225}
X(52635) = crossdifference of every pair of points on line {8, 885}
X(52635) = barycentric product X(i)*X(j) for these {i,j}: {1, 1458}, {3, 1876}, {6, 241}, {7, 2223}, {31, 9436}, {32, 40704}, {34, 1818}, {48, 5236}, {55, 34855}, {56, 518}, {57, 672}, {59, 3675}, {65, 3286}, {77, 2356}, {85, 9454}, {105, 1362}, {109, 2254}, {222, 5089}, {269, 2340}, {278, 20752}, {291, 51329}, {292, 34253}, {513, 2283}, {603, 1861}, {604, 3912}, {608, 25083}, {649, 1025}, {651, 665}, {658, 46388}, {663, 41353}, {667, 883}, {692, 43042}, {910, 52213}, {911, 39063}, {918, 1415}, {926, 934}, {1014, 20683}, {1026, 43924}, {1106, 3717}, {1262, 17435}, {1319, 34230}, {1397, 3263}, {1400, 18206}, {1402, 30941}, {1407, 3693}, {1408, 3932}, {1409, 15149}, {1412, 3930}, {1416, 4712}, {1428, 22116}, {1429, 3252}, {1431, 4447}, {1434, 39258}, {1439, 37908}, {1447, 40730}, {1457, 36819}, {1462, 6184}, {1911, 39775}, {2284, 3669}, {2428, 43049}, {2720, 42758}, {3126, 32735}, {3361, 14626}, {3513, 3514}, {4565, 24290}, {4569, 8638}, {4617, 52614}, {4899, 16945}, {6063, 9455}, {6168, 9315}, {9439, 41355}, {18026, 23225}, {20662, 43760}, {34018, 39686}, {46108, 52411}
X(52635) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 36796}, {31, 14942}, {32, 294}, {41, 6559}, {56, 2481}, {57, 18031}, {109, 51560}, {241, 76}, {518, 3596}, {560, 2195}, {603, 31637}, {604, 673}, {651, 36803}, {665, 4391}, {667, 885}, {672, 312}, {692, 36802}, {883, 6386}, {926, 4397}, {934, 46135}, {1025, 1978}, {1362, 3263}, {1395, 36124}, {1397, 105}, {1402, 13576}, {1407, 34018}, {1415, 666}, {1458, 75}, {1461, 34085}, {1818, 3718}, {1876, 264}, {1911, 33676}, {1919, 1024}, {1980, 884}, {2175, 28071}, {2223, 8}, {2254, 35519}, {2283, 668}, {2284, 646}, {2340, 341}, {2356, 318}, {3063, 28132}, {3263, 40363}, {3286, 314}, {3675, 34387}, {3912, 28659}, {3930, 30713}, {5089, 7017}, {5236, 1969}, {8299, 4087}, {8638, 3900}, {9436, 561}, {9454, 9}, {9455, 55}, {17435, 23978}, {18206, 28660}, {20683, 3701}, {20752, 345}, {23225, 521}, {24027, 39293}, {30941, 40072}, {34253, 1921}, {34855, 6063}, {39258, 2321}, {39686, 3693}, {39775, 18891}, {40704, 1502}, {40730, 4518}, {41353, 4572}, {42079, 3717}, {43042, 40495}, {46388, 3239}, {51329, 350}, {52410, 1462}, {52411, 1814}
X(52635) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {604, 1409, 2300}, {1319, 4559, 3230}

---

X(52636) = X(6)X(194)∩X(75)X(337)

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

X(52636) lies on the cubic K7643 and these lines: {6, 194}, {39, 37891}, {75, 337}, {76, 50666}, {99, 37893}, {141, 9229}, {216, 5976}, {264, 305}, {702, 19568}, {710, 7762}, {1194, 7792}, {3001, 44166}, {3313, 3852}, {3933, 40035}, {6374, 19602}, {6467, 34383}, {7019, 17788}, {7774, 15437}, {7783, 14950}, {9230, 36214}, {11596, 40405}, {16276, 44089}

X(52636) = isotomic conjugate of X(51246)
X(52636) = isotomic conjugate of the isogonal conjugate of X(3491)
X(52636) = X(i)-Ceva conjugate of X(j) for these (i,j): {9230, 141}, {36214, 325}
X(52636) = X(i)-isoconjugate of X(j) for these (i,j): {31, 51246}, {560, 43715}
X(52636) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 51246}, {6374, 43715}
X(52636) = barycentric product X(i)*X(j) for these {i,j}: {76, 3491}, {305, 51988}, {1502, 51950}
X(52636) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 51246}, {76, 43715}, {3491, 6}, {51950, 32}, {51988, 25}
X(52636) = {X(40073),X(50645)}-harmonic conjugate of X(325)

---

X(52637) = X(2)X(308)∩X(69)X(194)

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

X(52637) lies on the cubic K743 and these lines: {2, 308}, {4, 45211}, {6, 19585}, {39, 8790}, {69, 194}, {75, 2275}, {141, 40858}, {385, 22062}, {1031, 14570}, {1176, 16985}, {1369, 7779}, {1370, 3164}, {1655, 18133}, {2276, 17149}, {2998, 3618}, {3117, 6374}, {3978, 45210}, {7736, 19583}, {7757, 39361}, {7783, 20775}, {7839, 10339}, {8267, 40002}, {14360, 31088}, {16990, 41916}, {17135, 17148}, {17788, 20934}, {20794, 31859}, {21035, 21226}, {30229, 46274}, {31036, 46714}, {32747, 44152}, {38817, 51958}, {39346, 39355}, {40697, 40807}, {43715, 43718}, {46718, 46726}

X(52637) = reflection of X(25054) in X(39939)
X(52637) = isotomic conjugate of X(39953)
X(52637) = anticomplement of X(308)
X(52637) = anticomplement of the isogonal conjugate of X(3051)
X(52637) = anticomplement of the isotomic conjugate of X(39)
X(52637) = isotomic conjugate of the isogonal conjugate of X(3499)
X(52637) = polar conjugate of the isogonal conjugate of X(23174)
X(52637) = X(i)-Ceva conjugate of X(j) for these (i,j): {39, 2}, {9230, 69}, {38817, 3499}
X(52637) = X(31)-isoconjugate of X(39953)
X(52637) = X(2)-Dao conjugate of X(39953)
X(52637) = cevapoint of X(3499) and X(23174)
X(52637) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 21278}, {31, 76}, {32, 17165}, {38, 315}, {39, 6327}, {48, 12220}, {82, 33769}, {141, 21275}, {560, 6}, {662, 688}, {667, 25049}, {688, 21221}, {798, 25051}, {810, 25053}, {922, 25052}, {1333, 17142}, {1397, 20247}, {1401, 21285}, {1501, 17489}, {1634, 17217}, {1843, 21270}, {1917, 8267}, {1918, 3770}, {1919, 25048}, {1923, 2}, {1924, 25047}, {1927, 732}, {1930, 33796}, {1964, 69}, {1967, 20022}, {1973, 3060}, {2084, 3448}, {2175, 20248}, {2205, 17499}, {2206, 17141}, {3005, 21294}, {3051, 8}, {3404, 14957}, {3688, 21286}, {4020, 1370}, {4553, 21304}, {4576, 21305}, {9236, 20859}, {9247, 22}, {9406, 25045}, {9417, 25046}, {9454, 25050}, {9494, 21220}, {16696, 17138}, {17187, 17137}, {17442, 11442}, {19606, 21299}, {20775, 4329}, {21035, 21287}, {21123, 21293}, {21814, 1330}, {23208, 21288}, {23995, 10330}, {27369, 5905}, {35325, 21300}, {40972, 3436}, {41267, 2895}, {41272, 17491}, {41331, 192}, {46148, 21301}, {46154, 21298}, {46289, 33798}, {50521, 150}
X(52637) = barycentric product X(i)*X(j) for these {i,j}: {76, 3499}, {141, 38817}, {264, 23174}, {8024, 51958}
X(52637) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 39953}, {3499, 6}, {23174, 3}, {38817, 83}, {51958, 251}
X(52637) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {194, 2896, 10340}, {9229, 32476, 23642}

---

X(52638) = X(1)X(3526)∩X(2)X(3689)

Barycentrics    4*a^3 - 5*a^2*b - a*b^2 + 2*b^3 - 5*a^2*c + 2*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2 + 2*c^3 : :
X(52638) = X[5561] + 3 X[51817]

X(52638) lies on the cubic K574 and these lines: {1, 3526}, {2, 3689}, {7, 1155}, {11, 30331}, {12, 4304}, {35, 16117}, {37, 650}, {55, 1538}, {65, 5719}, {80, 3584}, {140, 17609}, {142, 6174}, {210, 6690}, {226, 4995}, {354, 3911}, {495, 21578}, {498, 5722}, {517, 39782}, {551, 1145}, {944, 2646}, {1000, 5048}, {1125, 4002}, {1317, 10265}, {1319, 3653}, {1387, 5919}, {1737, 15935}, {1836, 5281}, {2293, 45885}, {3011, 17366}, {3057, 13411}, {3059, 6666}, {3158, 31245}, {3246, 37651}, {3295, 23708}, {3303, 37704}, {3579, 11552}, {3601, 5726}, {3614, 4314}, {3616, 33895}, {3666, 17725}, {3683, 31018}, {3742, 51378}, {3744, 17722}, {3893, 24541}, {3922, 11281}, {3947, 15338}, {3983, 6675}, {4003, 17724}, {4021, 17602}, {4031, 10164}, {4315, 15888}, {4413, 6600}, {4421, 31266}, {4428, 30852}, {4640, 17484}, {4660, 30823}, {4689, 17719}, {4860, 11034}, {4870, 5119}, {5054, 51816}, {5326, 11019}, {5425, 50821}, {5440, 10197}, {5444, 51788}, {5561, 28146}, {8255, 50573}, {10039, 37728}, {10578, 17728}, {11230, 12331}, {11237, 30282}, {11374, 31452}, {11375, 30305}, {13226, 17603}, {14563, 40663}, {15702, 18490}, {16602, 29689}, {16610, 29675}, {17293, 29828}, {17594, 17783}, {18393, 51787}, {19862, 34501}, {21870, 35466}, {25055, 40587}, {26446, 44840}, {27529, 51715}, {27741, 28538}, {29571, 31192}, {30147, 37829}, {30284, 41701}, {32537, 51683}, {33176, 45081}, {34471, 51784}, {44319, 49537}

X(52638) = crossdifference of every pair of points on line {36, 17425}
X(52638) = barycentric product X(8)*X(14564)
X(52638) = barycentric quotient X(14564)/X(7)
X(52638) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {498, 37080, 17606}, {3911, 13405, 37703}, {3911, 37703, 354}, {5218, 17718, 1155}, {5432, 13405, 354}, {5432, 37703, 3911}, {11374, 31452, 37568}

---

X(52639) = X(1)X(523)∩X(58)X(79)

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

X(52639) lies on the cubic K915 and these lines: {1, 523}, {58, 79}, {80, 502}, {229, 759}, {546, 45926}, {2341, 17745}, {3244, 6740}

X(52639) = barycentric product X(14616)*X(15586)
X(52639) = barycentric quotient X(15586)/X(758)

---

X(52640) = X(1)X(522)∩X(29)X(58)

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

X(52640) lies on the cubic K915 and these lines: {1, 522}, {10, 36037}, {29, 58}, {36, 7456}, {79, 104}, {226, 34299}, {909, 36149}, {1012, 39175}, {1784, 51654}, {1809, 27385}, {2695, 2720}, {3244, 51565}, {3284, 7359}, {4304, 37136}, {15501, 36921}

X(52640) = X(i)-isoconjugate of X(j) for these (i,j): {74, 517}, {908, 2159}, {1457, 44693}, {1465, 15627}, {1785, 35200}, {2183, 2349}, {3262, 40352}, {4246, 14380}, {14571, 14919}, {22350, 36119}
X(52640) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 1785}, {908, 3163}, {1511, 22350}, {6735, 6739}
X(52640) = trilinear pole of line {2173, 14400}
X(52640) = barycentric product X(i)*X(j) for these {i,j}: {30, 34234}, {104, 14206}, {909, 3260}, {1795, 46106}, {2173, 18816}, {6357, 51565}, {11064, 36123}, {11125, 13136}, {18653, 38955}, {34858, 46234}, {36795, 51654}
X(52640) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 908}, {104, 2349}, {909, 74}, {1495, 2183}, {1795, 14919}, {1990, 1785}, {2173, 517}, {2342, 15627}, {3284, 22350}, {6357, 22464}, {7359, 6735}, {11125, 10015}, {14206, 3262}, {14399, 1769}, {14400, 2804}, {14578, 35200}, {18653, 17139}, {18816, 33805}, {34234, 1494}, {34858, 2159}, {36123, 16080}, {51654, 1465}

---

X(52641) = X(4)X(512)∩X(24)X(98)

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

X(52641) lies on the cubic K620 and these lines: {4, 512}, {24, 98}, {64, 290}, {76, 2710}, {235, 51441}, {264, 14382}, {287, 8549}, {1093, 2207}, {1235, 52145}, {1503, 51257}, {1593, 36822}, {1968, 48452}, {2052, 34536}, {2409, 34156}, {10282, 35912}, {12324, 36893}, {20968, 32545}, {23977, 51437}, {27376, 51404}

X(52641) = polar conjugate of the isogonal conjugate of X(51963)
X(52641) = X(34536)-Ceva conjugate of X(6531)
X(52641) = X(i)-isoconjugate of X(j) for these (i,j): {255, 39265}, {326, 51822}, {2435, 23997}, {15407, 23996}
X(52641) = X(i)-Dao conjugate of X(j) for these (i,j): {232, 36790}, {511, 50938}, {6523, 39265}, {15259, 51822}, {15595, 51386}, {23976, 36212}
X(52641) = barycentric product X(i)*X(j) for these {i,j}: {25, 51257}, {132, 34536}, {264, 51963}, {290, 16318}, {1503, 16081}, {2052, 34156}, {2409, 43665}, {6531, 30737}, {18024, 51437}
X(52641) = barycentric quotient X(i)/X(j) for these {i,j}: {132, 36790}, {393, 39265}, {441, 51386}, {1503, 36212}, {2207, 51822}, {2395, 2435}, {2409, 2421}, {2445, 14966}, {6531, 1297}, {16081, 35140}, {16318, 511}, {20031, 44770}, {23977, 4230}, {30737, 6393}, {34156, 394}, {41932, 15407}, {42671, 3289}, {43665, 2419}, {51257, 305}, {51363, 44716}, {51437, 237}, {51963, 3}
X(52641) = {X(4),X(52491)}-harmonic conjugate of X(14265)

---

X(52642) = X(2)X(34394)∩X(5)X(15)

Barycentrics    Sqrt[3]*a^2*(a^4 - a^2*b^2 - a^2*c^2 - 2*b^2*c^2) + 2*(a^4 - 4*a^2*b^2 + 2*b^4 - 4*a^2*c^2 - 4*b^2*c^2 + 2*c^4)*S : :

X(52642) lies on the cubic K1209 and these lines: {2, 34394}, {5, 15}, {13, 42788}, {16, 31455}, {17, 262}, {39, 46054}, {61, 7746}, {62, 3815}, {1506, 6771}, {1656, 46053}, {2548, 36760}, {5013, 46854}, {5254, 22846}, {7747, 49106}, {7749, 36759}, {7769, 22687}, {7891, 42674}, {7942, 16967}, {11304, 16241}, {11307, 33417}, {13083, 33013}, {13350, 39590}, {16921, 22689}, {22236, 47860}, {23005, 36771}, {31704, 42098}, {36766, 37825}, {37332, 37832}, {39565, 46855}, {41094, 42581}, {42936, 47517}

---

X(52643) = X(2)X(34395)∩X(5)X(16)

Barycentrics    Sqrt[3]*a^2*(a^4 - a^2*b^2 - a^2*c^2 - 2*b^2*c^2) - 2*(a^4 - 4*a^2*b^2 + 2*b^4 - 4*a^2*c^2 - 4*b^2*c^2 + 2*c^4)*S : :

X(52643) lies on the cubic K1209 and these lines: {2, 34395}, {5, 16}, {14, 42788}, {15, 31455}, {18, 262}, {39, 46053}, {61, 3815}, {62, 7746}, {1506, 6774}, {1656, 46054}, {2548, 36759}, {5013, 46855}, {5254, 22891}, {7747, 49105}, {7749, 36760}, {7769, 22689}, {7891, 42675}, {7942, 16966}, {11290, 36766}, {11303, 16242}, {11308, 33416}, {13084, 33013}, {13349, 39590}, {16921, 22687}, {22238, 47859}, {23004, 41035}, {31703, 42095}, {37333, 37835}, {39565, 46854}, {41098, 42580}, {42937, 47519}

---

X(52644) = X(3)X(13)∩X(5)X(9735)

Barycentrics    Sqrt[3]*a^2*(a^2*b^2 - b^4 + a^2*c^2 - c^4) - 2*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4)*S : :
X(52644) = 2 X[5] + 3 X[9735], 4 X[140] + X[47068], 6 X[549] - X[47066], 8 X[3530] - 3 X[9736], 9 X[5054] + X[5865], X[5864] - 11 X[15720]

X(52644) lies on the cubic K1206 and these lines: {3, 13}, {5, 9735}, {6, 49105}, {15, 31455}, {140, 47068}, {182, 16772}, {381, 49902}, {511, 631}, {549, 47066}, {576, 16773}, {628, 34507}, {635, 3788}, {1351, 42491}, {3106, 31457}, {3530, 9736}, {5054, 5865}, {5351, 20425}, {5864, 15720}, {6671, 7834}, {6774, 22236}, {7748, 16631}, {11308, 38317}, {13102, 42580}, {13349, 40693}, {16628, 16967}, {16966, 22843}, {20426, 42937}, {31652, 43454}, {36756, 42092}, {42598, 44250}, {43141, 52402}, {43144, 52401}

X(52644) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 16629, 42433}, {3, 42490, 49106}, {3, 43238, 6771}

---

X(52645) = X(3)X(14)∩X(5)X(9736)

Barycentrics    Sqrt[3]*a^2*(a^2*b^2 - b^4 + a^2*c^2 - c^4) + 2*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4)*S : :
X(52645) = 2 X[5] + 3 X[9736], 4 X[140] + X[47066], 6 X[549] - X[47068], 8 X[3530] - 3 X[9735], 9 X[5054] + X[5864], X[5865] - 11 X[15720]

X(52645) lies on the cubic K1209 and these lines: {3, 14}, {5, 9736}, {6, 49106}, {16, 31455}, {140, 47066}, {182, 16773}, {381, 49901}, {511, 631}, {549, 47068}, {576, 16772}, {627, 34507}, {636, 3788}, {1351, 42490}, {1656, 36770}, {3107, 31457}, {3530, 9735}, {5054, 5864}, {5352, 20426}, {5865, 15720}, {6672, 7834}, {6771, 22238}, {7748, 16630}, {11307, 38317}, {13103, 42581}, {13350, 40694}, {16629, 16966}, {16967, 22890}, {20425, 42936}, {31652, 43455}, {36755, 42089}, {43141, 52401}, {43144, 52402}

X(52645) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 16628, 42434}, {3, 42491, 49105}, {3, 43239, 6774}

---

X(52646) = X(4)X(523)∩X(24)X(64)

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)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4)*(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(52646) lies on the cubic K620 and these lines: {3, 1304}, {4, 523}, {24, 64}, {133, 39376}, {235, 12079}, {378, 14385}, {382, 10152}, {1073, 15404}, {1093, 2970}, {1498, 34185}, {1593, 9717}, {1968, 48451}, {1975, 16077}, {2207, 8749}, {2442, 51964}, {3172, 32695}, {3426, 40384}, {3470, 35502}, {5627, 48374}, {5879, 12084}, {6000, 39174}, {10421, 44438}, {10594, 52130}, {11456, 32715}, {12111, 36831}, {14989, 35490}, {15030, 35910}, {15459, 41372}, {17703, 44958}, {18279, 18809}, {37196, 46147}

X(52646) = polar conjugate of the isogonal conjugate of X(51964)
X(52646) = X(40384)-Ceva conjugate of X(8749)
X(52646) = X(i)-isoconjugate of X(j) for these (i,j): {1099, 15404}, {2430, 24001}
X(52646) = X(i)-Dao conjugate of X(j) for these (i,j): {30, 50937}, {1990, 36789}, {6000, 40948}
X(52646) = crosssum of X(i) and X(j) for these (i,j): {30, 38605}, {16163, 51394}
X(52646) = crossdifference of every pair of points on line {3284, 14345}
X(52646) = barycentric product X(i)*X(j) for these {i,j}: {74, 51358}, {133, 40384}, {264, 51964}, {2052, 39174}, {2394, 46587}, {2404, 14380}, {2442, 34767}, {6000, 16080}, {14165, 39376}, {14919, 51385}
X(52646) = barycentric quotient X(i)/X(j) for these {i,j}: {133, 36789}, {2433, 43701}, {2442, 4240}, {6000, 11064}, {8749, 1294}, {14380, 2416}, {39174, 394}, {40353, 15404}, {46587, 2407}, {47433, 16163}, {51358, 3260}, {51385, 46106}, {51964, 3}
X(52646) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 14264, 35908}, {4, 52493, 14264}, {1304, 38937, 3}

---

X(52647) = X(5)X(298)∩X(14)X(16)

Barycentrics    Sqrt[3]*(2*a^8 - 9*a^6*b^2 + 9*a^4*b^4 - 5*a^2*b^6 + 3*b^8 - 9*a^6*c^2 - 6*a^4*b^2*c^2 + 7*a^2*b^4*c^2 - 10*b^6*c^2 + 9*a^4*c^4 + 7*a^2*b^2*c^4 + 14*b^4*c^4 - 5*a^2*c^6 - 10*b^2*c^6 + 3*c^8) + 2*(2*a^6 - 11*a^4*b^2 + 10*a^2*b^4 - b^6 - 11*a^4*c^2 + b^4*c^2 + 10*a^2*c^4 + b^2*c^4 - c^6)*S : :
X(52647) = 3 X[395] - X[47611], X[617] + 3 X[20426], X[617] - 3 X[44223], 3 X[7685] - X[22797], 3 X[37785] + X[48656], 3 X[44289] - X[48656]

X(52647) lies on the cubic K1209 and these lines: {5, 298}, {14, 16}, {140, 47068}, {141, 547}, {262, 52266}, {383, 5984}, {385, 44219}, {511, 6670}, {617, 20426}, {5066, 33459}, {6108, 12829}, {7685, 22797}, {20423, 42910}, {25187, 32515}, {37785, 44289}, {43333, 48910}, {48655, 51487}

X(52647) = midpoint of X(i) and X(j) for these {i,j}: {20426, 44223}, {37785, 44289}

---

X(52648) = X(5)X(299)∩X(13)X(15)

Barycentrics    Sqrt[3]*(2*a^8 - 9*a^6*b^2 + 9*a^4*b^4 - 5*a^2*b^6 + 3*b^8 - 9*a^6*c^2 - 6*a^4*b^2*c^2 + 7*a^2*b^4*c^2 - 10*b^6*c^2 + 9*a^4*c^4 + 7*a^2*b^2*c^4 + 14*b^4*c^4 - 5*a^2*c^6 - 10*b^2*c^6 + 3*c^8) - 2*(2*a^6 - 11*a^4*b^2 + 10*a^2*b^4 - b^6 - 11*a^4*c^2 + b^4*c^2 + 10*a^2*c^4 + b^2*c^4 - c^6)*S : :
X(52648) = 3 X[396] - X[47610], X[616] + 3 X[20425], 3 X[7684] - X[22796], 3 X[37786] + X[48655]

X(52648) lies on the cubic K1209 and these lines: {5, 299}, {13, 15}, {140, 47066}, {141, 547}, {262, 52263}, {511, 6669}, {616, 20425}, {1080, 5984}, {5066, 33458}, {6109, 12829}, {7684, 22796}, {20423, 42911}, {25183, 32515}, {37786, 48655}, {43332, 48910}, {48656, 51486}

---

X(52649) = X(2)X(3)∩X(531)X(22566)

Barycentrics    2*a^8 + 9*a^6*b^2 - 13*a^4*b^4 + 9*a^2*b^6 - 7*b^8 + 9*a^6*c^2 + 2*a^4*b^2*c^2 - 7*a^2*b^4*c^2 + 18*b^6*c^2 - 13*a^4*c^4 - 7*a^2*b^2*c^4 - 22*b^4*c^4 + 9*a^2*c^6 + 18*b^2*c^6 - 7*c^8 - 2*Sqrt[3]*(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)*S : :
X(52649) = 2 X[546] + X[41035], X[35304] + 3 X[41016], 3 X[16267] - X[36383], X[25154] - 3 X[41036]

X(52649) lies on these lines: {2, 3}, {531, 22566}, {542, 7684}, {623, 19924}, {624, 25561}, {2548, 41113}, {3767, 41112}, {5459, 11645}, {5460, 19130}, {5872, 15534}, {6299, 8176}, {9762, 51872}, {14881, 33482}, {16267, 36383}, {22495, 36363}, {22570, 22576}, {22797, 33476}, {22803, 33485}, {25154, 41036}, {32907, 51523}, {36319, 51486}, {37786, 48655}, {41023, 49102}, {43291, 43416}

X(52649) = midpoint of X(i) and X(j) for these {i,j}: {381, 1080}, {3830, 35931}, {22495, 36363}, {37786, 48655}
X(52649) = reflection of X(i) in X(j) for these {i,j}: {549, 52266}, {31693, 5066}
X(52649) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 3845, 44289}

---

X(52650) = X(2)X(3)∩X(13)X(38230)

Barycentrics    Sqrt[3]*(a^2 + b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) - 2*(4*a^4 - 5*a^2*b^2 + b^4 - 5*a^2*c^2 - 2*b^2*c^2 + c^4)*S : :
X(52650) = 2 X[140] + X[41035], 3 X[35304] + X[41016], X[5978] - 3 X[15561], X[20428] - 3 X[36765], 3 X[36765] + X[36967]

X(52650) lies on these lines: {2, 3}, {13, 38230}, {15, 5617}, {182, 33391}, {187, 6115}, {298, 5611}, {302, 20426}, {395, 3107}, {396, 11136}, {511, 618}, {542, 45879}, {616, 20425}, {619, 24206}, {620, 623}, {635, 7880}, {1353, 42633}, {1503, 33388}, {2080, 5979}, {2782, 6109}, {3564, 42912}, {3642, 9735}, {5238, 37824}, {5318, 20252}, {5476, 13084}, {5613, 16241}, {5872, 22236}, {5873, 42152}, {5978, 15561}, {6108, 14693}, {6669, 44667}, {6671, 6771}, {6695, 49105}, {6774, 33478}, {11178, 13083}, {11542, 19780}, {13102, 14144}, {13350, 41022}, {16626, 42157}, {16627, 42488}, {16965, 36958}, {16967, 23004}, {18582, 21843}, {18583, 42913}, {20253, 37835}, {20428, 36765}, {20429, 30559}, {22507, 22997}, {22796, 44666}, {32428, 51219}

X(52650) = midpoint of X(i) and X(j) for these {i,j}: {3, 1080}, {15, 5617}, {298, 5611}, {381, 35931}, {616, 20425}, {2080, 5979}, {20428, 36967}, {22507, 22997}
X(52650) = reflection of X(i) in X(j) for these {i,j}: {5, 52266}, {5318, 20252}, {6108, 14693}, {6771, 6671}, {31693, 547}, {44223, 37459}
X(52650) = circumcircle-of-inner-Napoleon-triangle-inverse of X(44466)
X(52650) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 38431, 32460}, {3, 11307, 140}, {5, 549, 44223}, {31693, 35304, 8598}, {36765, 36967, 20428}

---

X(52651) = X(1)X(2653)∩X(9)X(43)

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

X(52651) lies on the cubic K1038 and these lines: {1, 2653}, {2, 694}, {6, 6043}, {9, 43}, {33, 862}, {37, 20684}, {38, 661}, {75, 40849}, {115, 3735}, {181, 2295}, {192, 40099}, {210, 20691}, {226, 3721}, {257, 312}, {612, 41532}, {756, 35309}, {805, 25819}, {904, 4426}, {979, 23447}, {982, 16592}, {984, 50440}, {1084, 4443}, {1107, 45240}, {1178, 5105}, {1213, 3863}, {1432, 25430}, {1581, 19584}, {2162, 21755}, {2229, 20966}, {2245, 40401}, {2273, 7104}, {2319, 50613}, {2321, 3971}, {2341, 2503}, {3029, 3923}, {3178, 22202}, {3501, 9560}, {3550, 20666}, {3727, 23903}, {3778, 16606}, {3865, 16589}, {3938, 21341}, {3954, 4109}, {3959, 23897}, {3961, 21383}, {4016, 8818}, {4096, 43265}, {4383, 40432}, {4386, 41882}, {4419, 35144}, {6685, 20861}, {7116, 39690}, {7239, 16587}, {7249, 17056}, {10026, 49509}, {16975, 20982}, {17144, 39917}, {17754, 20461}, {18591, 51986}, {18904, 22230}, {21035, 21902}, {21954, 31993}, {22171, 29653}, {22200, 43223}, {23639, 52538}, {24504, 37134}, {27439, 27447}, {27538, 35068}, {27966, 33890}, {36571, 44103}

X(52651) = isotomic conjugate of X(8033)
X(52651) = isogonal conjugate of the anticomplement of X(46826)
X(52651) = isotomic conjugate of the isogonal conjugate of X(40729)
X(52651) = X(14970)-Ceva conjugate of X(17493)
X(52651) = X(i)-cross conjugate of X(j) for these (i,j): {661, 27805}, {2292, 37}, {3728, 10}, {23668, 65}, {23928, 1}, {50440, 2238}, {51464, 13576}
X(52651) = X(i)-isoconjugate of X(j) for these (i,j): {6, 17103}, {21, 7175}, {27, 3955}, {31, 8033}, {56, 27958}, {58, 894}, {60, 4032}, {81, 171}, {86, 172}, {99, 20981}, {100, 18200}, {101, 17212}, {110, 4369}, {163, 4374}, {222, 14006}, {274, 7122}, {284, 7176}, {385, 741}, {593, 1215}, {648, 22093}, {662, 4367}, {692, 16737}, {757, 2295}, {763, 21803}, {849, 3963}, {1014, 2329}, {1019, 4579}, {1171, 4697}, {1178, 6645}, {1333, 1909}, {1408, 17787}, {1412, 7081}, {1414, 3287}, {1434, 2330}, {1444, 7119}, {1509, 20964}, {1580, 37128}, {1634, 18111}, {1691, 18827}, {1790, 7009}, {1920, 2206}, {1933, 40017}, {1966, 18268}, {2194, 7196}, {2236, 39276}, {2363, 28369}, {2533, 4556}, {3572, 17941}, {3733, 18047}, {3736, 40745}, {3907, 4565}, {4095, 7341}, {4128, 4590}, {4164, 4584}, {4459, 52378}, {4477, 4637}, {4570, 7200}, {4591, 4922}, {4610, 7234}, {5009, 30669}, {6649, 7252}, {7187, 38813}, {7305, 18905}, {14621, 40731}, {16592, 24041}, {21755, 24037}, {22373, 46254}
X(52651) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 27958}, {2, 8033}, {9, 17103}, {10, 894}, {37, 1909}, {75, 27891}, {115, 4374}, {171, 40586}, {172, 40600}, {244, 4369}, {385, 8299}, {512, 21755}, {960, 28369}, {1015, 17212}, {1084, 4367}, {1086, 16737}, {1214, 7196}, {1920, 40603}, {1966, 35068}, {2295, 40607}, {3005, 16592}, {3023, 3709}, {3287, 40608}, {3739, 4754}, {3963, 4075}, {7081, 40599}, {7175, 40611}, {7176, 40590}, {7200, 50330}, {8054, 18200}, {9467, 18268}, {20981, 38986}, {37128, 39092}
X(52651) = cevapoint of X(661) and X(3124)
X(52651) = crosspoint of X(i) and X(j) for these (i,j): {256, 257}, {16606, 52208}
X(52651) = crosssum of X(i) and X(j) for these (i,j): {171, 172}, {284, 51330}, {4367, 16592}, {27644, 38814}
X(52651) = trilinear pole of line {4041, 4155}
X(52651) = crossdifference of every pair of points on line {1580, 4367}
X(52651) = barycentric product X(i)*X(j) for these {i,j}: {10, 256}, {37, 257}, {42, 7018}, {65, 4451}, {76, 40729}, {210, 7249}, {213, 44187}, {313, 904}, {321, 893}, {523, 3903}, {594, 40432}, {661, 27805}, {694, 3948}, {740, 1581}, {756, 32010}, {862, 40708}, {874, 882}, {1089, 1178}, {1431, 3701}, {1432, 2321}, {1824, 7019}, {1916, 2238}, {1934, 3747}, {1967, 35544}, {2295, 40099}, {3700, 37137}, {3773, 40763}, {4024, 4603}, {4079, 7260}, {4086, 29055}, {4122, 30670}, {4155, 18829}, {4594, 4705}, {6535, 7303}, {7015, 41013}, {7104, 27801}, {18786, 43534}, {18896, 41333}, {20691, 27447}, {36897, 50440}
X(52651) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17103}, {2, 8033}, {9, 27958}, {10, 1909}, {33, 14006}, {37, 894}, {42, 171}, {65, 7176}, {210, 7081}, {213, 172}, {226, 7196}, {228, 3955}, {256, 86}, {257, 274}, {321, 1920}, {512, 4367}, {513, 17212}, {514, 16737}, {523, 4374}, {594, 3963}, {649, 18200}, {661, 4369}, {694, 37128}, {733, 39276}, {740, 1966}, {756, 1215}, {762, 21021}, {798, 20981}, {805, 36066}, {810, 22093}, {862, 419}, {869, 40731}, {872, 20964}, {874, 880}, {881, 875}, {882, 876}, {893, 81}, {904, 58}, {1018, 18047}, {1084, 21755}, {1089, 1237}, {1178, 757}, {1334, 2329}, {1400, 7175}, {1431, 1014}, {1432, 1434}, {1441, 7205}, {1500, 2295}, {1581, 18827}, {1824, 7009}, {1916, 40017}, {1918, 7122}, {1962, 4697}, {1967, 741}, {2092, 28369}, {2171, 4032}, {2238, 385}, {2295, 6645}, {2321, 17787}, {2333, 7119}, {3124, 16592}, {3125, 7200}, {3573, 17941}, {3709, 3287}, {3721, 7187}, {3728, 51575}, {3747, 1580}, {3778, 7184}, {3865, 33947}, {3903, 99}, {3948, 3978}, {3949, 4019}, {3954, 16720}, {3971, 41318}, {4010, 14296}, {4041, 3907}, {4093, 2236}, {4094, 4154}, {4128, 7207}, {4155, 804}, {4171, 4529}, {4365, 7244}, {4451, 314}, {4455, 4164}, {4516, 4459}, {4524, 4477}, {4551, 6649}, {4557, 4579}, {4594, 4623}, {4603, 4610}, {4705, 2533}, {4729, 4504}, {4730, 4922}, {4770, 4774}, {4838, 4842}, {6376, 27891}, {7015, 1444}, {7018, 310}, {7104, 1333}, {7116, 1790}, {7260, 52612}, {7303, 6628}, {9468, 18268}, {16589, 4754}, {17493, 30940}, {18786, 33295}, {20683, 4447}, {20691, 17752}, {21700, 27880}, {21805, 4434}, {21810, 27697}, {21832, 4107}, {21839, 7267}, {21879, 27954}, {22172, 7240}, {27805, 799}, {29055, 1414}, {32010, 873}, {35544, 1926}, {37137, 4573}, {40432, 1509}, {40608, 3023}, {40729, 6}, {40747, 40745}, {40835, 7307}, {40966, 18235}, {41333, 1691}, {44092, 444}, {44187, 6385}, {50440, 5976}, {50487, 7234}, {50491, 24533}
X(52651) = {X(27805),X(32010)}-harmonic conjugate of X(2)

---

X(52652) = X(2)X(893)∩X(6)X(75)

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

X(52652) lies on the cubic K1014 and these lines: {2, 893}, {6, 75}, {8, 7077}, {9, 3596}, {19, 264}, {43, 2258}, {55, 312}, {57, 6063}, {76, 3496}, {284, 314}, {309, 1436}, {321, 2205}, {333, 20665}, {673, 871}, {789, 2291}, {869, 5263}, {874, 27474}, {909, 4586}, {940, 40752}, {985, 4362}, {1821, 14382}, {1920, 24586}, {2160, 20565}, {2161, 20566}, {2164, 20570}, {2195, 36796}, {2259, 40422}, {2299, 14006}, {2319, 11679}, {3061, 36800}, {3114, 16609}, {3403, 4384}, {3886, 40739}, {3923, 4279}, {6630, 40740}, {11051, 44186}, {16826, 24679}, {17795, 24342}, {19302, 40716}, {20605, 21369}, {28926, 28934}, {35167, 41072}, {40001, 46750}, {40763, 51575}

X(52652) = isotomic conjugate of X(7146)
X(52652) = isotomic conjugate of the isogonal conjugate of X(2344)
X(52652) = X(i)-cross conjugate of X(j) for these (i,j): {9, 40739}, {4913, 7257}, {45755, 3699}, {52133, 870}
X(52652) = X(i)-isoconjugate of X(j) for these (i,j): {6, 1469}, {7, 40728}, {31, 7146}, {32, 7179}, {41, 7204}, {56, 2276}, {57, 869}, {85, 18900}, {109, 3250}, {604, 984}, {608, 3781}, {651, 788}, {664, 46386}, {1014, 3774}, {1214, 46503}, {1397, 3661}, {1400, 3736}, {1402, 40773}, {1407, 4517}, {1415, 1491}, {1428, 3862}, {2149, 4475}, {2206, 16603}, {2283, 29956}, {3116, 7132}, {3212, 40736}, {3773, 16947}, {3790, 52410}, {4554, 8630}, {12837, 40746}, {29055, 45882}, {40732, 42290}, {41526, 45782}, {52029, 52635}
X(52652) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 2276}, {2, 7146}, {9, 1469}, {11, 3250}, {650, 4475}, {788, 38991}, {824, 40624}, {869, 5452}, {984, 3161}, {1146, 1491}, {3094, 41886}, {3160, 7204}, {3736, 40582}, {4481, 40625}, {4517, 24771}, {6376, 7179}, {12837, 19584}, {16603, 40603}, {39025, 46386}, {40605, 40773}
X(52652) = cevapoint of X(9) and X(3886)
X(52652) = trilinear pole of line {663, 3716}
X(52652) = barycentric product X(i)*X(j) for these {i,j}: {8, 870}, {55, 871}, {75, 52133}, {76, 2344}, {312, 14621}, {314, 40718}, {522, 789}, {646, 4817}, {650, 37133}, {663, 46132}, {985, 3596}, {1492, 35519}, {2170, 5388}, {3056, 46281}, {3061, 3114}, {3063, 52611}, {3113, 3705}, {3716, 41072}, {4391, 4586}, {4441, 40739}, {4613, 18155}, {5384, 34387}, {17787, 40738}, {21615, 40757}, {27424, 52136}, {28659, 40746}, {28660, 40747}
X(52652) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1469}, {2, 7146}, {7, 7204}, {8, 984}, {9, 2276}, {11, 4475}, {21, 3736}, {41, 40728}, {55, 869}, {75, 7179}, {78, 3781}, {200, 4517}, {312, 3661}, {314, 30966}, {321, 16603}, {333, 40773}, {341, 3790}, {522, 1491}, {646, 3807}, {650, 3250}, {663, 788}, {789, 664}, {825, 1415}, {870, 7}, {871, 6063}, {984, 12837}, {985, 56}, {1024, 29956}, {1043, 3786}, {1334, 3774}, {1492, 109}, {2175, 18900}, {2299, 46503}, {2344, 6}, {3056, 3116}, {3061, 3094}, {3063, 46386}, {3287, 45882}, {3407, 7132}, {3596, 33931}, {3684, 16514}, {3685, 3783}, {3699, 3799}, {3701, 3773}, {3702, 3775}, {3705, 51836}, {3716, 30665}, {3786, 4476}, {3886, 3789}, {3902, 4407}, {3907, 3805}, {3975, 3797}, {4086, 4122}, {4384, 40784}, {4391, 824}, {4397, 4522}, {4511, 3792}, {4518, 3864}, {4560, 4481}, {4586, 651}, {4613, 4551}, {4723, 4439}, {4811, 4818}, {4817, 3669}, {4876, 3862}, {5384, 59}, {7081, 40790}, {7155, 45782}, {14024, 17569}, {14621, 57}, {14942, 52029}, {17739, 40797}, {20665, 3117}, {27424, 51837}, {28809, 27474}, {30670, 29055}, {31623, 31909}, {37133, 4554}, {40718, 65}, {40738, 1432}, {40739, 1002}, {40745, 7175}, {40746, 604}, {40747, 1400}, {40757, 2279}, {40763, 1431}, {40771, 18784}, {46132, 4572}, {52133, 1}, {52136, 1423}

---

X(52653) = X(7)X(21)∩X(8)X(9)

Barycentrics    (a - b - c)*(5*a^2 + 2*a*b + b^2 + 2*a*c - 2*b*c + c^2) : :
X(52653) = 2 X[1] + X[144], X[1] + 2 X[51090], X[144] - 4 X[51090], 3 X[2] - 4 X[38059], 5 X[2] - 4 X[38204], X[2] + 2 X[50836], 7 X[2] - 4 X[51100], 3 X[7988] - 2 X[38151], 3 X[9779] - 4 X[38037], 5 X[38052] - 6 X[38204], X[38052] + 3 X[50836], 7 X[38052] - 6 X[51100], 5 X[38059] - 3 X[38204], 2 X[38059] + 3 X[50836], 7 X[38059] - 3 X[51100], 2 X[38204] + 5 X[50836], 7 X[38204] - 5 X[51100], 7 X[50836] + 2 X[51100], X[7] - 4 X[1001], 2 X[7] - 5 X[3616], X[7] + 2 X[5698], 5 X[7] - 8 X[25557], 8 X[1001] - 5 X[3616], 2 X[1001] + X[5698], 5 X[1001] - 2 X[25557], 5 X[3616] + 4 X[5698], 25 X[3616] - 16 X[25557], 5 X[3616] - 4 X[38053], X[3648] + 2 X[16133], 5 X[5698] + 4 X[25557], 4 X[25557] - 5 X[38053], X[8] - 4 X[9], X[8] + 2 X[390], 5 X[8] - 8 X[24393], 2 X[9] + X[390], 5 X[9] - 2 X[24393], 5 X[390] + 4 X[24393], 5 X[5686] - 4 X[24393], X[20] + 2 X[11372], 2 X[72] + X[30628], 8 X[142] - 11 X[5550], X[145] + 2 X[5223], X[145] - 4 X[30331], X[5223] + 2 X[30331], X[962] + 2 X[5759], X[3241] + 2 X[6172], X[3241] - 4 X[47357], X[6172] + 2 X[47357], and many others

X(52653) lies on the cubic K1084 and these lines: {1, 144}, {2, 165}, {7, 21}, {8, 9}, {20, 11372}, {37, 4344}, {40, 5129}, {55, 18228}, {63, 10580}, {72, 30628}, {78, 4326}, {142, 5550}, {145, 5223}, {210, 10385}, {238, 5222}, {242, 28120}, {329, 954}, {354, 28610}, {392, 971}, {404, 11495}, {405, 962}, {480, 3871}, {497, 3683}, {517, 21168}, {518, 1992}, {527, 11038}, {528, 38057}, {673, 20156}, {740, 27484}, {758, 15933}, {858, 47470}, {938, 12514}, {944, 5779}, {946, 17558}, {952, 51516}, {958, 9785}, {960, 4313}, {966, 49484}, {1058, 31445}, {1125, 4312}, {1156, 6224}, {1193, 4335}, {1279, 4419}, {1320, 6068}, {1385, 36996}, {1456, 3160}, {1458, 24708}, {1482, 50243}, {1890, 4194}, {2320, 34919}, {2328, 17127}, {2476, 42356}, {2478, 2550}, {2551, 15837}, {2951, 3522}, {2975, 42884}, {3059, 3876}, {3062, 4297}, {3174, 4420}, {3219, 36845}, {3242, 51144}, {3243, 20057}, {3295, 5815}, {3305, 17784}, {3452, 5281}, {3474, 4423}, {3576, 50742}, {3579, 17559}, {3599, 31627}, {3617, 17339}, {3622, 5542}, {3623, 43179}, {3646, 17580}, {3672, 7290}, {3731, 39587}, {3740, 34607}, {3753, 34632}, {3755, 15601}, {3826, 4193}, {3868, 5572}, {3869, 5728}, {3878, 18412}, {3890, 20789}, {3897, 42819}, {3923, 7229}, {4208, 41869}, {4229, 24557}, {4295, 5259}, {4304, 5785}, {4307, 5308}, {4308, 8581}, {4314, 20007}, {4343, 19767}, {4349, 29624}, {4353, 16487}, {4356, 16469}, {4428, 25568}, {4432, 20533}, {4460, 49462}, {4488, 24349}, {4511, 7675}, {4640, 5435}, {4645, 29627}, {4666, 9965}, {4676, 5749}, {4679, 5218}, {5084, 31658}, {5085, 35280}, {5177, 10248}, {5220, 20050}, {5226, 24703}, {5234, 12575}, {5239, 30338}, {5240, 30339}, {5248, 5703}, {5251, 30305}, {5263, 5296}, {5274, 5745}, {5284, 9776}, {5316, 35445}, {5325, 24392}, {5330, 42871}, {5603, 5762}, {5657, 34629}, {5695, 32087}, {5722, 5775}, {5732, 17576}, {5735, 24541}, {5805, 6857}, {5811, 10267}, {5817, 11113}, {5833, 9614}, {5843, 10246}, {5852, 51099}, {5880, 6910}, {5886, 50739}, {6361, 11024}, {6557, 52133}, {6651, 41845}, {6666, 6919}, {6675, 31671}, {6764, 41229}, {6856, 18482}, {6872, 36991}, {7081, 8055}, {7172, 30568}, {7308, 46916}, {7987, 43182}, {9581, 18231}, {10179, 34610}, {10303, 25522}, {10453, 17185}, {10582, 21454}, {10861, 15726}, {11227, 14646}, {11344, 27381}, {11526, 50573}, {12526, 30330}, {12630, 15481}, {12672, 51489}, {12699, 16845}, {12705, 37423}, {12732, 45116}, {14022, 33108}, {14986, 31424}, {15485, 16020}, {15587, 25917}, {15717, 43151}, {16370, 21151}, {16484, 24695}, {16823, 24280}, {16857, 28174}, {16858, 38454}, {16866, 22791}, {17194, 29814}, {17320, 38046}, {17324, 30424}, {17571, 31657}, {17687, 48944}, {19526, 20330}, {19860, 43166}, {19875, 38201}, {20075, 27065}, {20078, 29817}, {20103, 31508}, {21214, 25570}, {24477, 49736}, {24558, 43177}, {24564, 37435}, {25055, 38054}, {25101, 39570}, {25728, 49466}, {27481, 50129}, {28534, 38025}, {28566, 41313}, {28581, 37654}, {29181, 49735}, {30333, 30557}, {30334, 30556}, {30389, 43176}, {31394, 52241}, {32086, 42309}, {32558, 38060}, {37161, 51118}, {38041, 51514}, {38043, 38107}, {38108, 38149}, {38113, 38121}, {39567, 49446}, {41011, 41825}, {48661, 50205}, {50834, 51093}, {50837, 51071}, {50995, 51192}, {50996, 51001}, {50997, 50999}, {51000, 51191}, {51053, 51054}, {51068, 51102}, {51098, 51110}

X(52653) = midpoint of X(i) and X(j) for these {i,j}: {390, 5686}, {5698, 38053}, {6172, 8236}
X(52653) = reflection of X(i) in X(j) for these {i,j}: {7, 38053}, {8, 5686}, {3241, 8236}, {5686, 9}, {8236, 47357}, {11038, 38316}, {21151, 38031}, {38052, 38059}, {38053, 1001}, {38107, 38043}, {38121, 38113}, {38149, 38108}, {51514, 38041}
X(52653) = anticomplement of X(38052)
X(52653) = crossdifference of every pair of points on line {3709, 43924}
X(52653) = barycentric product X(i)*X(j) for these {i,j}: {8, 17014}, {312, 16469}, {333, 4356}
X(52653) = barycentric quotient X(i)/X(j) for these {i,j}: {4356, 226}, {16469, 57}, {17014, 7}
X(52653) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 51090, 144}, {7, 1001, 3616}, {9, 390, 8}, {329, 1621, 10578}, {346, 3883, 8}, {391, 3886, 8}, {391, 4779, 3886}, {392, 11111, 5731}, {452, 5250, 8}, {497, 3683, 5273}, {960, 14100, 41228}, {1001, 5698, 7}, {2550, 15254, 18230}, {2550, 18230, 9780}, {3622, 20059, 5542}, {3646, 31730, 17580}, {3755, 15601, 37681}, {3923, 39581, 7229}, {4356, 16469, 17014}, {4512, 40998, 2}, {4640, 26105, 5435}, {4679, 5218, 5328}, {5223, 30331, 145}, {5284, 44447, 9776}, {5698, 8543, 11415}, {6172, 47357, 3241}, {6361, 11108, 11024}, {6666, 40333, 19877}, {11038, 38316, 38314}, {15254, 30332, 9780}, {15485, 24248, 16020}, {16823, 24280, 31995}, {18230, 30332, 2550}, {24313, 24314, 2321}, {38052, 38059, 2}

---

X(52654) = X(1)X(1575)∩X(43)X(81)

Barycentrics    a*(a*b + 2*b^2 - a*c + b*c)*(a*b - a*c - b*c - 2*c^2) : :
X(52654) = X[1] - 4 X[16604], X[1] + 2 X[46032], 2 X[16604] + X[46032], 2 X[10] + X[330], X[32020] - 4 X[40533], 5 X[1698] - 2 X[6376], 7 X[9780] - X[21219]

X(52654) lies on the conic {{A,B,C,X(1),X(2)}}, the cubic K1038, and these lines: {1, 1575}, {2, 726}, {8, 38247}, {9, 1929}, {10, 330}, {42, 25417}, {43, 81}, {57, 1463}, {75, 32020}, {87, 18793}, {88, 36263}, {89, 899}, {105, 2108}, {274, 1698}, {278, 17927}, {279, 17090}, {291, 3789}, {978, 1258}, {985, 16468}, {1002, 3783}, {1022, 48213}, {1125, 39738}, {1255, 26102}, {2276, 30571}, {3227, 3679}, {3624, 32009}, {3634, 39736}, {3720, 27789}, {3741, 39694}, {3840, 39703}, {4893, 43928}, {9780, 21219}, {10789, 33854}, {16602, 49692}, {17793, 49532}, {17795, 40756}, {18194, 39798}, {19875, 36871}, {20530, 49493}, {24342, 40738}, {25350, 32784}, {25381, 47828}, {25430, 25502}, {25583, 29633}, {26037, 39747}, {26077, 42027}, {29637, 30701}, {30942, 39698}, {30963, 49445}, {30998, 50117}, {31330, 35058}, {33062, 49545}, {34914, 35101}, {36634, 39980}, {37603, 52376}, {39948, 42043}

X(52654) = isogonal conjugate of X(16468)
X(52654) = isotomic conjugate of X(30963)
X(52654) = isotomic conjugate of the isogonal conjugate of X(40735)
X(52654) = X(i)-cross conjugate of X(j) for these (i,j): {984, 1}, {27483, 40775}
X(52654) = X(i)-isoconjugate of X(j) for these (i,j): {1, 16468}, {2, 21793}, {4, 23095}, {6, 4393}, {10, 34476}, {31, 30963}, {32, 10009}, {43, 40753}, {58, 3993}, {71, 31912}, {81, 21904}, {100, 4782}, {101, 4785}, {106, 4759}, {110, 4806}, {901, 45314}, {985, 3795}, {1126, 4991}, {2176, 40720}, {14621, 40733}, {27481, 40746}
X(52654) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 30963}, {3, 16468}, {9, 4393}, {10, 3993}, {214, 4759}, {244, 4806}, {1015, 4785}, {3647, 4991}, {3789, 3795}, {4782, 8054}, {6376, 10009}, {19584, 27481}, {21793, 32664}, {21904, 40586}, {23095, 36033}, {38979, 45314}
X(52654) = cevapoint of X(244) and X(1491)
X(52654) = trilinear pole of line {513, 4826}
X(52654) = barycentric product X(i)*X(j) for these {i,j}: {1, 27494}, {76, 40735}, {81, 34475}, {321, 51449}, {330, 40780}, {693, 43077}, {40756, 51837}
X(52654) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4393}, {2, 30963}, {6, 16468}, {28, 31912}, {31, 21793}, {37, 3993}, {42, 21904}, {44, 4759}, {48, 23095}, {75, 10009}, {87, 40720}, {513, 4785}, {649, 4782}, {661, 4806}, {869, 40733}, {984, 27481}, {1100, 4991}, {1333, 34476}, {1635, 45314}, {2162, 40753}, {2276, 3795}, {27494, 75}, {34475, 321}, {40735, 6}, {40756, 52136}, {40780, 192}, {43077, 100}, {51449, 81}
X(52654) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1575, 28600, 3795}, {3795, 28600, 1}, {16604, 46032, 1}

---

X(52655) = X(1)X(893)∩X(9)X(87)

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

X(52655) lies on the cubics K1015 and K1038 and these lines: {1, 893}, {2, 330}, {9, 87}, {31, 172}, {37, 7155}, {48, 34249}, {56, 21001}, {75, 52175}, {194, 41318}, {694, 3056}, {743, 932}, {869, 19586}, {1469, 19603}, {1914, 11328}, {1964, 45209}, {2056, 10799}, {2276, 3117}, {2277, 45197}, {2300, 21759}, {3051, 7296}, {4357, 27436}, {5106, 10987}, {6375, 18194}, {8620, 30650}, {16826, 39914}, {16969, 20471}, {17230, 20868}, {17248, 27444}, {18830, 43095}, {19522, 20606}, {25429, 40731}, {27188, 34086}, {27481, 40773}

X(52655) = isogonal conjugate of X(52136)
X(52655) = isotomic conjugate of the isogonal conjugate of X(40736)
X(52655) = isogonal conjugate of the isotomic conjugate of X(51837)
X(52655) = X(i)-cross conjugate of X(j) for these (i,j): {984, 2276}, {3116, 1469}
X(52655) = X(i)-isoconjugate of X(j) for these (i,j): {1, 52136}, {43, 14621}, {192, 985}, {789, 20979}, {825, 20906}, {870, 2176}, {1423, 52133}, {1492, 3835}, {2344, 3212}, {3113, 20284}, {3407, 41886}, {4083, 4586}, {4393, 40756}, {4613, 18197}, {5384, 21138}, {5388, 38986}, {6376, 40746}, {8640, 37133}, {17752, 40763}, {27644, 40718}, {33296, 40747}, {40738, 51902}
X(52655) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 52136}, {192, 3789}, {3835, 38995}, {6376, 19584}, {6382, 27481}, {19602, 33890}
X(52655) = cevapoint of X(984) and X(45782)
X(52655) = trilinear pole of line {788, 45882}
X(52655) = crossdifference of every pair of points on line {3835, 8640}
X(52655) = barycentric product X(i)*X(j) for these {i,j}: {1, 45782}, {6, 51837}, {76, 40736}, {87, 984}, {330, 2276}, {788, 18830}, {824, 34071}, {869, 6384}, {932, 1491}, {1469, 7155}, {2053, 7179}, {2162, 3661}, {2319, 7146}, {3250, 4598}, {3736, 42027}, {3799, 43931}, {3862, 39914}, {3864, 34252}, {6383, 40728}, {7121, 33931}, {16606, 40773}, {23493, 30966}
X(52655) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 52136}, {87, 870}, {788, 4083}, {869, 43}, {932, 789}, {984, 6376}, {1469, 3212}, {1491, 20906}, {2053, 52133}, {2162, 14621}, {2276, 192}, {3094, 33890}, {3116, 41886}, {3117, 20284}, {3250, 3835}, {3661, 6382}, {3736, 33296}, {3774, 20691}, {3799, 36863}, {3862, 40848}, {4517, 27538}, {4598, 37133}, {5383, 5388}, {6384, 871}, {7121, 985}, {7146, 30545}, {8630, 8640}, {14436, 14408}, {18830, 46132}, {18900, 2209}, {21759, 40747}, {23493, 40718}, {34071, 4586}, {40728, 2176}, {40735, 40756}, {40736, 6}, {40773, 31008}, {40783, 30963}, {40790, 41318}, {45782, 75}, {46386, 20979}, {51837, 76}, {51974, 40738}
X(52655) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3229, 20284}, {2053, 2162, 51321}, {3117, 40790, 2276}, {16606, 40881, 330}, {31641, 31642, 1909}

---

X(52656) = X(9)X(87)∩X(37)X(513)

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

X(52656) lies on the cubic K1038 and these lines: {1, 9470}, {2, 52205}, {6, 660}, {9, 87}, {37, 513}, {38, 25811}, {75, 141}, {192, 4562}, {291, 3789}, {984, 22116}, {1575, 40155}, {2276, 18795}, {3242, 52030}, {3551, 4876}, {3726, 20335}, {4639, 34022}, {4699, 40095}, {4941, 51058}, {6646, 30669}, {7077, 16973}, {7245, 17274}, {16710, 18827}, {16997, 37207}, {17149, 51868}, {17333, 43262}, {18792, 21830}, {19951, 20271}, {19953, 38406}, {24512, 40730}, {27184, 51859}, {30963, 39918}, {34067, 34247}, {39798, 40607}, {41527, 52209}

X(52656) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 22116}, {105, 20340}, {673, 20549}, {1438, 20530}, {1575, 20540}, {3009, 120}, {21760, 16593}, {36086, 6373}
X(52656) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 22116}, {335, 726}, {660, 6373}
X(52656) = X(i)-cross conjugate of X(j) for these (i,j): {20366, 40881}, {52633, 876}
X(52656) = X(i)-isoconjugate of X(j) for these (i,j): {6, 3253}, {238, 20332}, {239, 727}, {350, 34077}, {1428, 36799}, {1429, 8851}, {1914, 3226}, {2210, 32020}, {3570, 23355}, {5009, 27809}, {8632, 8709}
X(52656) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 22116}, {9, 3253}, {239, 17793}, {335, 33679}, {350, 20532}, {1575, 39044}, {3226, 36906}, {3837, 35119}, {4375, 27846}, {9470, 20332}
X(52656) = crosspoint of X(335) and X(30663)
X(52656) = crosssum of X(i) and X(j) for these (i,j): {238, 17475}, {1914, 8300}
X(52656) = barycentric product X(i)*X(j) for these {i,j}: {75, 40155}, {291, 726}, {292, 52043}, {334, 3009}, {335, 1575}, {660, 3837}, {813, 20908}, {876, 23354}, {1463, 4518}, {1911, 35538}, {4583, 6373}, {4584, 21053}, {4876, 43040}, {5378, 21140}, {17475, 40098}, {17793, 30663}, {18792, 43534}, {18895, 21760}, {21830, 40017}, {40848, 40881}
X(52656) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3253}, {291, 3226}, {292, 20332}, {335, 32020}, {660, 8709}, {726, 350}, {875, 23355}, {1463, 1447}, {1575, 239}, {1911, 727}, {1922, 34077}, {3009, 238}, {3837, 3766}, {4876, 36799}, {6373, 659}, {7077, 8851}, {17475, 4366}, {17793, 39044}, {18792, 33295}, {20663, 8300}, {20671, 17475}, {20681, 4368}, {20777, 7193}, {20785, 20769}, {21760, 1914}, {21830, 2238}, {23354, 874}, {35538, 18891}, {36814, 27922}, {40155, 1}, {40848, 40844}, {40881, 39914}, {43040, 10030}, {51864, 51321}, {52043, 1921}, {52633, 27846}
X(52656) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {984, 30663, 22116}, {36906, 40796, 2276}

---

X(52657) = X(6)X(43)∩X(76)X(85)

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

X(52657) lies on the cubics K1012 and K1036 and these lines: {3, 8866}, {6, 43}, {76, 85}, {262, 43687}, {978, 21876}, {982, 3094}, {984, 9285}, {1212, 25137}, {1215, 19222}, {2887, 3061}, {3662, 20684}, {3888, 20665}, {3925, 4534}, {4028, 41570}, {4051, 32865}, {4438, 24484}, {4562, 7033}, {4876, 33144}, {15310, 39250}, {16584, 41886}, {18161, 20541}, {20262, 41796}, {20459, 25306}, {20974, 33117}, {21827, 24456}, {21954, 24174}, {23636, 29641}, {27264, 41876}, {30822, 30825}, {32931, 40585}

X(52657) = complement of the isogonal conjugate of X(34247)
X(52657) = complement of the isotomic conjugate of X(32937)
X(52657) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 982}, {100, 25128}, {1918, 23447}, {1973, 28087}, {2175, 9367}, {2209, 14823}, {3501, 141}, {13588, 3741}, {17072, 21252}, {17786, 626}, {21348, 116}, {21958, 21253}, {22229, 8287}, {23655, 11}, {23864, 34589}, {32739, 25098}, {32937, 2887}, {34247, 10}, {51949, 2}, {51986, 17062}
X(52657) = X(2)-Ceva conjugate of X(982)
X(52657) = X(983)-isoconjugate of X(3500)
X(52657) = X(2)-Dao conjugate of X(982)
X(52657) = crosspoint of X(2) and X(32937)
X(52657) = barycentric product X(i)*X(j) for these {i,j}: {1, 51840}, {982, 32937}, {2275, 17786}, {2887, 13588}, {3501, 3662}, {3888, 17072}, {21348, 33946}, {33930, 34247}
X(52657) = barycentric quotient X(i)/X(j) for these {i,j}: {2275, 3500}, {3501, 17743}, {13588, 40415}, {32937, 7033}, {34247, 983}, {51840, 75}
X(52657) = {X(3094),X(18905)}-harmonic conjugate of X(982)

---

X(52658) = X(2)X(51)∩X(3)X(8925)

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

X(52658) lies on the cubic K1012 and these lines: {2, 51}, {3, 8925}, {6, 8623}, {69, 3978}, {76, 25332}, {126, 46656}, {141, 21531}, {182, 14096}, {183, 6784}, {211, 7914}, {237, 3098}, {512, 7761}, {599, 34383}, {694, 44453}, {984, 1469}, {1350, 11328}, {1352, 37190}, {2076, 41278}, {2387, 7865}, {3056, 41886}, {3094, 3117}, {3620, 32746}, {3739, 17792}, {3788, 52042}, {3818, 14957}, {4173, 7879}, {5039, 20965}, {5092, 52144}, {5447, 37466}, {6292, 41262}, {6310, 32974}, {6389, 11574}, {7467, 9306}, {7795, 35704}, {7924, 9879}, {9999, 15080}, {11325, 14134}, {13240, 15271}, {14135, 33023}, {14265, 20021}, {14810, 37184}, {14907, 32442}, {14994, 20023}, {21243, 21248}, {24206, 37988}, {34359, 39009}, {35296, 43152}, {39092, 40810}, {40802, 46316}, {44507, 46310}, {46518, 48880}

X(52658) = reflection of X(263) in X(34236)
X(52658) = complement of X(263)
X(52658) = anticomplement of X(34236)
X(52658) = complement of the isogonal conjugate of X(183)
X(52658) = complement of the isotomic conjugate of X(20023)
X(52658) = medial-isogonal conjugate of X(3815)
X(52658) = psi-transform of X(9772)
X(52658) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 3815}, {2, 16603}, {31, 3117}, {75, 24206}, {182, 37}, {183, 10}, {326, 42353}, {458, 226}, {662, 23878}, {3113, 34236}, {3288, 16592}, {3403, 141}, {10311, 16583}, {14096, 16587}, {14994, 21249}, {20023, 2887}, {23878, 8287}, {23997, 33569}, {33971, 24005}, {34396, 16584}, {36085, 45336}, {42711, 3454}, {44144, 20305}, {46806, 16609}, {51315, 13567}, {52134, 2}
X(52658) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 3117}, {290, 9865}, {670, 23878}, {3794, 51836}
X(52658) = X(i)-isoconjugate of X(j) for these (i,j): {263, 3113}, {2186, 3407}, {3114, 3402}, {46281, 46319}
X(52658) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 3117}, {262, 19602}, {327, 10335}, {512, 6784}, {3114, 51580}
X(52658) = crosspoint of X(2) and X(20023)
X(52658) = crosssum of X(i) and X(j) for these (i,j): {6, 46319}, {263, 51997}
X(52658) = barycentric product X(i)*X(j) for these {i,j}: {182, 3314}, {183, 3094}, {3116, 3403}, {3117, 20023}, {51836, 52134}
X(52658) = barycentric quotient X(i)/X(j) for these {i,j}: {182, 3407}, {183, 3114}, {3094, 262}, {3116, 2186}, {3117, 263}, {3314, 327}, {3403, 46281}, {14096, 14617}, {17415, 52631}, {18899, 46319}, {34396, 18898}, {43977, 42288}, {52134, 3113}
X(52658) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 263, 34236}, {2, 33884, 11673}, {2, 34095, 47638}, {3917, 47638, 34095}

---

X(52659) = X(1)X(5)∩X(7)X(88)

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

X(52659) lies on the cubic K453 and these lines: {1, 5}, {2, 222}, {7, 88}, {44, 3911}, {57, 21362}, {73, 4187}, {109, 3035}, {118, 10017}, {124, 25882}, {140, 1935}, {214, 51422}, {221, 26364}, {223, 30827}, {226, 1086}, {227, 21616}, {241, 40629}, {278, 5748}, {440, 16595}, {499, 9370}, {519, 36914}, {527, 43068}, {603, 13747}, {664, 4997}, {908, 1465}, {948, 43057}, {1054, 24465}, {1212, 5316}, {1214, 3452}, {1329, 10571}, {1361, 45122}, {1362, 3660}, {1376, 34029}, {1457, 17757}, {1466, 16415}, {1532, 22350}, {1537, 24028}, {1638, 39063}, {1639, 3762}, {1647, 41556}, {1745, 6922}, {1877, 5440}, {2003, 37634}, {2254, 10427}, {2635, 37374}, {3008, 15730}, {3030, 13601}, {3074, 52265}, {3157, 6959}, {3160, 5328}, {3173, 4383}, {3454, 6700}, {3562, 6979}, {3756, 5083}, {3814, 51421}, {3820, 24806}, {4031, 14564}, {4358, 37790}, {4370, 19618}, {4552, 30566}, {4945, 41803}, {5125, 27383}, {5226, 19785}, {5233, 33298}, {5308, 34056}, {5552, 34040}, {5741, 26942}, {6073, 39756}, {6174, 23703}, {6357, 40612}, {6505, 30852}, {6745, 50441}, {6834, 7078}, {6944, 41344}, {6958, 8757}, {7004, 13257}, {7066, 34466}, {8287, 17056}, {9371, 38385}, {9502, 33573}, {9940, 39007}, {10015, 42762}, {10200, 34046}, {10481, 40615}, {15507, 23981}, {16593, 43050}, {17080, 27131}, {17527, 37523}, {18340, 38757}, {21147, 25681}, {21635, 24025}, {28386, 41426}, {28741, 30834}, {28996, 33113}, {35110, 43035}, {37541, 47522}, {37680, 37797}, {51236, 51506}, {52031, 52212}

X(52659) = complement of X(34234)
X(52659) = complement of the isogonal conjugate of X(2183)
X(52659) = complement of the isotomic conjugate of X(908)
X(52659) = isotomic conjugate of the polar conjugate of X(1846)
X(52659) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 517}, {25, 26011}, {31, 3911}, {32, 8609}, {213, 2245}, {517, 141}, {604, 44675}, {859, 3739}, {908, 2887}, {1333, 15325}, {1409, 856}, {1415, 2804}, {1457, 142}, {1465, 2886}, {1769, 116}, {1783, 8677}, {1785, 20305}, {1875, 16608}, {2183, 10}, {2397, 21260}, {2423, 33646}, {2427, 513}, {3262, 626}, {3310, 11}, {4246, 30476}, {6735, 21244}, {9456, 1387}, {10015, 21252}, {14260, 3834}, {14571, 5}, {15507, 20333}, {17139, 21240}, {17757, 21245}, {21801, 3454}, {22350, 18589}, {22383, 35014}, {22464, 17046}, {23706, 46396}, {23979, 34345}, {23980, 119}, {23981, 4885}, {24029, 17072}, {32655, 6713}, {46393, 124}, {47408, 42423}, {47434, 25640}, {51377, 1211}, {51381, 20542}, {51987, 518}, {52031, 21241}, {52307, 123}
X(52659) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 3911}, {7, 517}, {651, 30725}, {664, 2804}, {4998, 23981}, {41801, 1317}
X(52659) = X(i)-isoconjugate of X(j) for these (i,j): {9, 10428}, {88, 2342}, {104, 2316}, {909, 1320}, {1809, 8752}, {4997, 34858}, {9456, 51565}, {23838, 32641}, {32665, 43728}
X(52659) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 3911}, {11, 3310}, {478, 10428}, {1320, 23980}, {2316, 40613}, {4370, 51565}, {4997, 16586}, {35092, 43728}
X(52659) = crosspoint of X(i) and X(j) for these (i,j): {2, 908}, {3911, 14628}
X(52659) = crosssum of X(6) and X(909)
X(52659) = crossdifference of every pair of points on line {654, 2342}
X(52659) = barycentric product X(i)*X(j) for these {i,j}: {7, 1145}, {69, 1846}, {519, 22464}, {664, 23757}, {908, 3911}, {1319, 3262}, {1434, 21942}, {1457, 3264}, {1465, 4358}, {2397, 30725}, {3259, 4998}, {3762, 24029}, {14628, 16586}, {17139, 40663}, {23703, 36038}, {26611, 40218}, {51583, 52212}
X(52659) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 10428}, {517, 1320}, {519, 51565}, {900, 43728}, {902, 2342}, {908, 4997}, {1145, 8}, {1317, 36944}, {1319, 104}, {1361, 14260}, {1404, 909}, {1457, 106}, {1465, 88}, {1769, 23838}, {1846, 4}, {1875, 36125}, {1877, 36123}, {2183, 2316}, {2397, 4582}, {2427, 5548}, {3259, 11}, {3911, 34234}, {4358, 36795}, {5440, 1809}, {12832, 14266}, {14260, 1318}, {14584, 40437}, {21942, 2321}, {22464, 903}, {23703, 36037}, {23757, 522}, {23981, 901}, {24029, 3257}, {30725, 2401}, {34346, 38541}, {36920, 36921}, {37790, 16082}, {40663, 38955}, {47420, 7117}
X(52659) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 651, 43043}, {226, 43048, 1086}, {908, 16586, 26611}, {2006, 5219, 37691}, {5723, 37691, 2006}, {21635, 24025, 38357}

---

X(52660) = X(2)X(698)∩X(3)X(46286)

Barycentrics    a^2*(a^2*b^2 + 2*b^4 - a^2*c^2 + b^2*c^2)*(a^2*b^2 - a^2*c^2 - b^2*c^2 - 2*c^4) : :
X(52660) = X[6] - 4 X[6375], 2 X[141] + X[2998], 5 X[3763] - 2 X[6374], 7 X[3619] - X[32747]

X(52660) lies on the conic {{A,B,C,X(2),X(6)}}, the cubic K1012, and these lines: {2, 698}, {3, 46286}, {6, 3229}, {25, 2076}, {37, 19584}, {42, 19586}, {69, 38262}, {111, 12149}, {141, 2998}, {251, 1613}, {308, 3763}, {393, 420}, {599, 3228}, {694, 44453}, {982, 16606}, {1383, 3231}, {1403, 51921}, {1976, 32540}, {2395, 9210}, {3051, 39955}, {3619, 32747}, {6379, 9462}, {9208, 14606}, {10161, 33237}, {11186, 34204}, {11328, 18898}, {11331, 16081}, {17230, 27809}, {18372, 37638}, {31884, 36616}, {39968, 47355}

X(52660) = isogonal conjugate of X(7766)
X(52660) = isotomic conjugate of X(41259)
X(52660) = isogonal conjugate of the anticomplement of X(3314)
X(52660) = isogonal conjugate of the isotomic conjugate of X(43688)
X(52660) = X(3094)-cross conjugate of X(6)
X(52660) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7766}, {2, 51291}, {6, 52138}, {31, 41259}, {82, 32449}, {560, 10010}, {662, 25423}, {36085, 45680}
X(52660) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 41259}, {3, 7766}, {9, 52138}, {141, 32449}, {1084, 25423}, {6374, 10010}, {10335, 19602}, {32664, 51291}, {38988, 45680}
X(52660) = cevapoint of X(i) and X(j) for these (i,j): {3124, 50549}, {3250, 6377}
X(52660) = crossdifference of every pair of points on line {25423, 45680}
X(52660) = barycentric product X(i)*X(j) for these {i,j}: {1, 51844}, {6, 43688}, {141, 51450}, {523, 25424}
X(52660) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 52138}, {2, 41259}, {6, 7766}, {31, 51291}, {39, 32449}, {76, 10010}, {351, 45680}, {512, 25423}, {3094, 10335}, {25424, 99}, {43688, 76}, {51450, 83}, {51844, 75}

---

X(52661) = X(4)X(51)∩X(107)X(186)

Barycentrics    b^2*c^2*(-a^2 + b^2 - c^2)^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) : :
X(52661) = 2 X[34334] + X[46106], 3 X[37943] - 2 X[47215]

X(52661) lies on the cubic K1302 and these lines: {4, 51}, {5, 6662}, {30, 34334}, {74, 40664}, {107, 186}, {112, 41368}, {133, 51403}, {158, 52382}, {235, 45195}, {264, 3545}, {324, 381}, {376, 15466}, {378, 41372}, {393, 33885}, {403, 523}, {436, 15033}, {450, 43574}, {468, 41522}, {847, 6526}, {1138, 14165}, {1148, 19368}, {1157, 16813}, {1199, 51031}, {1300, 22239}, {1559, 32111}, {1568, 3260}, {1596, 14569}, {1629, 52294}, {1990, 47433}, {2970, 10151}, {3520, 13489}, {3524, 52147}, {3529, 52578}, {3542, 6523}, {3850, 14978}, {4240, 15454}, {5627, 6344}, {5641, 6528}, {6525, 18533}, {6529, 8744}, {7049, 11461}, {8753, 9154}, {8795, 14483}, {8884, 11815}, {10594, 41365}, {11251, 14254}, {11799, 16104}, {13754, 35360}, {14157, 32230}, {14387, 18027}, {14581, 35906}, {33971, 45819}, {35480, 46255}, {35717, 40240}, {36434, 41361}, {37766, 46451}, {37984, 46081}, {40887, 41676}, {43976, 44556}, {44228, 51481}, {44673, 47204}

X(52661) = polar conjugate of X(14919)
X(52661) = polar conjugate of the isotomic conjugate of X(46106)
X(52661) = polar conjugate of the isogonal conjugate of X(1990)
X(52661) = X(6344)-Ceva conjugate of X(13450)
X(52661) = X(i)-cross conjugate of X(j) for these (i,j): {133, 4}, {1990, 46106}, {51403, 30}
X(52661) = X(i)-isoconjugate of X(j) for these (i,j): {3, 35200}, {48, 14919}, {63, 18877}, {74, 255}, {326, 40352}, {394, 2159}, {520, 36034}, {577, 2349}, {822, 44769}, {1092, 36119}, {1102, 40354}, {1494, 52430}, {2169, 44715}, {4100, 16080}, {4575, 14380}, {6149, 50464}, {6507, 8749}, {7125, 15627}, {7335, 44693}, {14585, 33805}, {24018, 32640}, {36131, 52613}, {42080, 42308}, {44706, 46090}
X(52661) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 133}, {30, 51394}, {74, 6523}, {136, 14380}, {394, 3163}, {520, 3258}, {1092, 1511}, {1249, 14919}, {1259, 6739}, {1990, 44436}, {3162, 18877}, {14363, 44715}, {14918, 52437}, {14993, 50464}, {15259, 40352}, {35200, 36103}, {39008, 52613}, {39174, 50937}
X(52661) = cevapoint of X(30) and X(51425)
X(52661) = crossdifference of every pair of points on line {577, 32320}
X(52661) = barycentric product X(i)*X(j) for these {i,j}: {4, 46106}, {30, 2052}, {53, 43752}, {92, 1784}, {107, 41079}, {158, 14206}, {264, 1990}, {393, 3260}, {823, 36035}, {1093, 11064}, {1096, 46234}, {1495, 18027}, {1568, 8794}, {1637, 6528}, {1650, 34538}, {4240, 14618}, {6344, 14920}, {9033, 15352}, {9214, 37778}, {13450, 43768}, {14165, 14254}, {14581, 18022}, {16080, 34334}, {18817, 39176}, {24001, 24006}, {44138, 51965}
X(52661) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 14919}, {19, 35200}, {25, 18877}, {30, 394}, {53, 44715}, {107, 44769}, {133, 44436}, {158, 2349}, {393, 74}, {1093, 16080}, {1096, 2159}, {1495, 577}, {1637, 520}, {1784, 63}, {1857, 15627}, {1989, 50464}, {1990, 3}, {2052, 1494}, {2173, 255}, {2207, 40352}, {2501, 14380}, {3163, 51394}, {3260, 3926}, {3284, 1092}, {4240, 4558}, {6110, 44718}, {6111, 44719}, {6357, 1804}, {6520, 36119}, {6524, 8749}, {6525, 15291}, {6529, 1304}, {6530, 35910}, {7359, 1259}, {8737, 39377}, {8738, 39378}, {8882, 46090}, {9033, 52613}, {9406, 52430}, {9407, 14585}, {9409, 32320}, {11064, 3964}, {11125, 4091}, {14206, 326}, {14398, 39201}, {14399, 23224}, {14581, 184}, {14583, 50433}, {14618, 34767}, {14920, 52437}, {15352, 16077}, {16240, 3284}, {18384, 11079}, {23347, 32661}, {24001, 4592}, {24019, 36034}, {27376, 46147}, {32713, 32640}, {34334, 11064}, {34538, 42308}, {35906, 17974}, {35907, 51262}, {36035, 24018}, {37778, 36890}, {39176, 22115}, {41079, 3265}, {43752, 34386}, {46106, 69}, {51382, 6514}, {51389, 51386}, {51403, 6509}, {51420, 18604}, {51425, 6503}, {51654, 7125}, {51965, 5504}, {52418, 14385}, {52439, 40354}
X(52661) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1075, 6241}, {4, 1093, 13450}, {4, 3168, 5890}, {4, 6761, 25739}, {4, 13450, 44732}, {1093, 14249, 4}, {1093, 47392, 2052}, {2052, 14249, 47392}, {2052, 47392, 4}, {6524, 36876, 4}, {6530, 51385, 403}

---

X(52662) = X(10)X(75)∩X(257)X(594)

Barycentrics    b*c*(-(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(52662) lies on the cubic K738 and these lines: {6, 17743}, {10, 75}, {257, 594}, {312, 30179}, {325, 3932}, {518, 52151}, {668, 1757}, {698, 40848}, {740, 4505}, {1107, 17289}, {1502, 7034}, {1654, 17787}, {2345, 41838}, {3061, 17233}, {3773, 17762}, {3992, 46238}, {4283, 18148}, {17370, 27324}, {17371, 27274}, {17789, 17949}, {17793, 32922}, {18047, 19554}, {18050, 46747}, {20917, 31317}, {21608, 46738}, {24502, 29423}, {27733, 35117}, {33888, 52043}, {41318, 51861}, {49755, 49781}

X(52662) = isotomic conjugate of X(7166)
X(52662) = isotomic conjugate of the isogonal conjugate of X(3507)
X(52662) = X(31)-isoconjugate of X(7166)
X(52662) = X(2)-Dao conjugate of X(7166)
X(52662) = barycentric product X(i)*X(j) for these {i,j}: {75, 33889}, {76, 3507}, {561, 51921}
X(52662) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 7166}, {3507, 6}, {33889, 1}, {51921, 31}

---

X(52663) = X(6)X(281)∩X(9)X(48)

Barycentrics    a*(a - b - c)*(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(52663) lies on the cubic K453 and these lines: {2, 222}, {3, 46015}, {6, 281}, {9, 48}, {44, 14578}, {200, 212}, {219, 346}, {220, 4370}, {282, 1743}, {294, 28132}, {519, 2323}, {645, 1812}, {666, 1814}, {672, 1949}, {860, 3330}, {908, 52392}, {918, 2401}, {1309, 32726}, {1708, 2184}, {1809, 2193}, {1864, 2194}, {1952, 39294}, {2003, 20205}, {2161, 52479}, {2182, 6001}, {2183, 6905}, {2316, 3738}, {2338, 37628}, {2423, 40400}, {2431, 39471}, {2982, 16082}, {2990, 18359}, {4238, 37142}, {4511, 5548}, {5776, 7152}, {5845, 52457}, {6554, 14266}, {7046, 19354}, {7110, 20262}, {7367, 39175}, {8257, 21446}, {15629, 46391}, {15733, 28071}, {16670, 19605}, {17745, 52640}, {17917, 34032}, {26890, 32736}, {27539, 28935}, {36037, 41798}, {36101, 37136}, {40149, 40397}

X(52663) = isogonal conjugate of X(1465)
X(52663) = complement of X(36918)
X(52663) = isogonal conjugate of the isotomic conjugate of X(36795)
X(52663) = polar conjugate of the isotomic conjugate of X(1809)
X(52663) = X(47645)-complementary conjugate of X(2886)
X(52663) = X(i)-Ceva conjugate of X(j) for these (i,j): {13136, 43728}, {34234, 104}, {36795, 1809}, {37136, 37628}
X(52663) = X(i)-cross conjugate of X(j) for these (i,j): {44, 9}, {1639, 644}, {2342, 104}, {2361, 21}, {3689, 36944}, {51361, 1}, {52371, 1320}
X(52663) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1465}, {2, 1457}, {6, 22464}, {7, 2183}, {36, 52212}, {56, 908}, {57, 517}, {59, 42754}, {63, 1875}, {77, 14571}, {109, 10015}, {222, 1785}, {226, 859}, {278, 22350}, {513, 24029}, {514, 23981}, {604, 3262}, {651, 1769}, {653, 8677}, {664, 3310}, {738, 51380}, {905, 23706}, {934, 46393}, {951, 51410}, {1014, 21801}, {1262, 35015}, {1319, 52031}, {1361, 34234}, {1400, 17139}, {1407, 6735}, {1411, 16586}, {1412, 17757}, {1415, 36038}, {1416, 51390}, {1434, 51377}, {1435, 51379}, {1461, 2804}, {1477, 51419}, {1797, 1846}, {1813, 39534}, {2006, 34586}, {2397, 43924}, {2427, 3676}, {3911, 14260}, {4246, 51664}, {4559, 23788}, {4564, 42753}, {4620, 42752}, {7091, 51413}, {7128, 35014}, {9436, 51987}, {14733, 42762}, {17107, 51378}, {23220, 46404}, {24028, 34051}, {35012, 39294}, {36118, 52307}, {36146, 42758}, {37136, 42757}, {40151, 51433}, {42759, 52378}
X(52663) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 908}, {3, 1465}, {9, 22464}, {11, 10015}, {220, 51378}, {517, 5452}, {1145, 26611}, {1146, 36038}, {1457, 32664}, {1769, 38991}, {1875, 3162}, {2804, 35508}, {3126, 42770}, {3161, 3262}, {3310, 39025}, {6615, 42754}, {6735, 24771}, {14714, 46393}, {15898, 52212}, {16586, 35204}, {17139, 40582}, {17757, 40599}, {24029, 39026}, {39014, 42758}, {40609, 51390}
X(52663) = cevapoint of X(i) and X(j) for these (i,j): {1, 44425}, {6, 2182}, {9, 2323}, {220, 3689}, {3119, 4895}, {34591, 46391}
X(52663) = crosspoint of X(i) and X(j) for these (i,j): {1320, 36100}, {34234, 51565}
X(52663) = crosssum of X(i) and X(j) for these (i,j): {6, 51236}, {1319, 2182}, {1457, 2183}, {3310, 42753}, {35014, 46393}
X(52663) = trilinear pole of line {55, 1946}
X(52663) = crossdifference of every pair of points on line {1361, 1769}
X(52663) = barycentric product X(i)*X(j) for these {i,j}: {1, 51565}, {4, 1809}, {6, 36795}, {8, 104}, {9, 34234}, {21, 38955}, {55, 18816}, {75, 2342}, {78, 36123}, {100, 43728}, {219, 16082}, {280, 15501}, {312, 909}, {318, 1795}, {333, 2250}, {346, 34051}, {521, 1309}, {522, 36037}, {644, 2401}, {646, 2423}, {650, 13136}, {1320, 36944}, {1897, 37628}, {2320, 36921}, {2720, 4397}, {3239, 37136}, {3596, 34858}, {4076, 15635}, {4391, 32641}, {4511, 40437}, {4571, 43933}, {4723, 10428}, {7017, 14578}, {14266, 45393}, {14776, 35518}, {14942, 36819}, {15416, 32702}, {32669, 52622}, {34591, 39294}, {36798, 45145}, {44693, 52640}
X(52663) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 22464}, {6, 1465}, {8, 3262}, {9, 908}, {21, 17139}, {25, 1875}, {31, 1457}, {33, 1785}, {41, 2183}, {55, 517}, {101, 24029}, {104, 7}, {200, 6735}, {210, 17757}, {212, 22350}, {480, 51380}, {522, 36038}, {607, 14571}, {644, 2397}, {650, 10015}, {657, 46393}, {663, 1769}, {692, 23981}, {909, 57}, {926, 42758}, {1260, 51379}, {1309, 18026}, {1334, 21801}, {1795, 77}, {1809, 69}, {1864, 1532}, {1946, 8677}, {2161, 52212}, {2170, 42754}, {2194, 859}, {2250, 226}, {2264, 51410}, {2310, 35015}, {2316, 52031}, {2323, 16586}, {2342, 1}, {2348, 51419}, {2361, 34586}, {2401, 24002}, {2423, 3669}, {2720, 934}, {2900, 51432}, {3059, 51416}, {3063, 3310}, {3158, 51433}, {3270, 35014}, {3271, 42753}, {3683, 51409}, {3684, 51381}, {3689, 1145}, {3693, 51390}, {3694, 51367}, {3711, 51362}, {3737, 23788}, {3900, 2804}, {3965, 51407}, {4512, 51423}, {4516, 42759}, {4526, 42764}, {4895, 23757}, {6600, 51378}, {8750, 23706}, {13136, 4554}, {14578, 222}, {14776, 108}, {15501, 347}, {15635, 1358}, {16082, 331}, {17435, 42770}, {18344, 39534}, {18816, 6063}, {30223, 1519}, {32641, 651}, {32669, 1461}, {32702, 32714}, {34051, 279}, {34234, 85}, {34858, 56}, {36037, 664}, {36110, 36118}, {36123, 273}, {36795, 76}, {36819, 9436}, {37136, 658}, {37628, 4025}, {38955, 1441}, {40437, 18815}, {41933, 34051}, {43728, 693}, {45145, 43037}, {51565, 75}, {51824, 18838}, {52427, 1845}
X(52663) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {909, 2250, 104}, {2265, 34591, 9}, {39146, 39147, 104}

---

X(52664) = X(2)X(37)∩X(6)X(7033)

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

X(52664) lies on the cubic K738 and these lines: {2, 37}, {6, 7033}, {76, 19584}, {190, 1755}, {290, 4562}, {341, 21608}, {513, 21611}, {594, 7018}, {698, 3862}, {730, 4385}, {1215, 3510}, {1502, 17786}, {1920, 21101}, {1926, 1978}, {1964, 32926}, {2186, 27424}, {2227, 3994}, {2236, 17763}, {2321, 20528}, {3116, 32925}, {3726, 18149}, {3807, 35544}, {3948, 40848}, {3967, 17792}, {3978, 4876}, {3985, 36863}, {16284, 20936}, {17445, 32942}, {18050, 33938}, {18895, 36906}, {20643, 40845}, {27808, 49753}

X(52664) = isotomic conjugate of X(7167)
X(52664) = isotomic conjugate of the isogonal conjugate of X(3508)
X(52664) = X(4876)-Ceva conjugate of X(17786)
X(52664) = X(i)-isoconjugate of X(j) for these (i,j): {31, 7167}, {2206, 43686}
X(52664) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 7167}, {40603, 43686}
X(52664) = barycentric product X(i)*X(j) for these {i,j}: {76, 3508}, {561, 51928}
X(52664) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 7167}, {321, 43686}, {3508, 6}, {32937, 8927}, {39940, 1429}, {51928, 31}, {51935, 1428}

---

X(52665) = X(8)X(144)∩X(165)X(210)

Barycentrics    a*(3*a^5 + 3*a^4*b - 18*a^3*b^2 + 6*a^2*b^3 + 15*a*b^4 - 9*b^5 + 3*a^4*c + 4*a^3*b*c + 10*a^2*b^2*c - 12*a*b^3*c - 5*b^4*c - 18*a^3*c^2 + 10*a^2*b*c^2 - 6*a*b^2*c^2 + 14*b^3*c^2 + 6*a^2*c^3 - 12*a*b*c^3 + 14*b^2*c^3 + 15*a*c^4 - 5*b*c^4 - 9*c^5) : :
X(52665) = X[1] - 4 X[5779], 4 X[7] - 7 X[7989], 7 X[7989] - 8 X[38158], 2 X[144] + X[5691], X[3062] + 2 X[5223], 2 X[3062] + X[7991], 4 X[5223] - X[7991], 8 X[9] - 5 X[7987], 4 X[5817] - 3 X[7988], 2 X[355] + X[41705], 4 X[1156] - X[7993], 5 X[1698] - 2 X[36996], 5 X[1698] - 4 X[38123], X[2951] - 4 X[5220], 4 X[11372] - X[11531], 2 X[4312] - 5 X[37714], 5 X[37714] - 4 X[38149], 4 X[6172] - X[34628], 5 X[8227] - 4 X[38041], 7 X[9588] - 4 X[43182], 4 X[9947] - X[31391], 7 X[19876] - 8 X[38179], 4 X[19925] - X[20059], 5 X[30308] - 4 X[38036], 7 X[30389] - 8 X[38031], 8 X[38043] - 7 X[51110], 4 X[38121] - 5 X[51066]

X(52665) lies on the cubic K1084 and these lines: {1, 5779}, {2, 24645}, {7, 7989}, {8, 144}, {9, 3207}, {57, 30291}, {165, 210}, {226, 5817}, {355, 41705}, {518, 11224}, {952, 9819}, {1156, 7993}, {1698, 36996}, {1699, 5850}, {2801, 16858}, {2951, 5220}, {3339, 5587}, {3361, 5777}, {3486, 30337}, {3576, 51516}, {3962, 11372}, {4312, 4848}, {4866, 5657}, {5234, 12528}, {5273, 5785}, {5531, 41166}, {5660, 11407}, {5731, 9851}, {5784, 16209}, {5851, 38200}, {5852, 7997}, {5886, 41870}, {6172, 34628}, {8227, 38041}, {8236, 18452}, {9588, 43182}, {9947, 31391}, {10861, 15064}, {11038, 30330}, {12709, 18412}, {15079, 38107}, {19876, 38179}, {19925, 20059}, {30308, 38036}, {30389, 38031}, {38043, 51110}, {38121, 51066}, {38140, 51514}

X(52665) = reflection of X(i) in X(j) for these {i,j}: {7, 38158}, {3576, 51516}, {4312, 38149}, {10861, 15064}, {11224, 24644}, {36996, 38123}, {51514, 38140}
X(52665) = {X(3062),X(5223)}-harmonic conjugate of X(7991)

---

X(52666) = X(2)X(6411)∩X(4)X(371)

Barycentrics    7*a^4 - 2*a^2*b^2 - 5*b^4 - 2*a^2*c^2 + 10*b^2*c^2 - 5*c^4 - 4*a^2*S : :
X(52666) = 5 X[3069] - 4 X[6398], 3 X[3069] - 4 X[13785], 3 X[6398] - 5 X[13785], 2 X[6398] - 5 X[23259], 2 X[13785] - 3 X[23259]

X(52666) lies on the cubic K1197 and these lines: {2, 6411}, {3, 23263}, {4, 371}, {5, 6455}, {6, 3543}, {20, 615}, {30, 3069}, {372, 23275}, {376, 6565}, {381, 6445}, {382, 1588}, {486, 3529}, {546, 6519}, {547, 6451}, {590, 3839}, {631, 42266}, {1131, 3592}, {1132, 1152}, {1151, 3832}, {1327, 43795}, {1328, 6396}, {1587, 3627}, {1657, 13935}, {2043, 42217}, {2044, 42218}, {3070, 17578}, {3071, 3146}, {3090, 35787}, {3091, 8253}, {3311, 3853}, {3316, 6453}, {3520, 35777}, {3522, 42262}, {3523, 42270}, {3524, 42274}, {3528, 10577}, {3534, 18762}, {3545, 6200}, {3734, 5590}, {3830, 18512}, {3843, 9691}, {3845, 6221}, {3850, 6449}, {3854, 42582}, {3855, 5418}, {3861, 8976}, {5056, 6409}, {5073, 7584}, {5076, 7583}, {5410, 13473}, {5420, 17538}, {5860, 33457}, {5861, 44678}, {6199, 38335}, {6395, 42538}, {6412, 42605}, {6419, 23269}, {6426, 42541}, {6429, 43413}, {6438, 43209}, {6441, 42540}, {6452, 15686}, {6480, 42602}, {6484, 10195}, {6493, 43377}, {6560, 15682}, {7581, 22644}, {7582, 35820}, {7585, 42284}, {8252, 10304}, {8972, 41945}, {9543, 41961}, {9615, 12571}, {9647, 10591}, {9660, 10590}, {9682, 52294}, {9683, 35500}, {9690, 43211}, {10146, 13961}, {10299, 12819}, {11541, 42267}, {12102, 42570}, {12240, 12279}, {12322, 17128}, {12510, 45024}, {12601, 49038}, {13665, 15687}, {13834, 46453}, {13836, 33878}, {13902, 18483}, {13939, 42261}, {13941, 15683}, {13951, 15704}, {13966, 17800}, {14226, 43792}, {14269, 18538}, {15640, 32788}, {15681, 35256}, {15684, 18510}, {15692, 32790}, {15697, 42577}, {15698, 43788}, {15709, 43337}, {15717, 42583}, {17852, 49140}, {18586, 42136}, {18587, 42137}, {19065, 41869}, {19066, 31673}, {23251, 50688}, {26361, 35949}, {26617, 32807}, {32488, 33365}, {35409, 42574}, {35823, 42276}, {36436, 42086}, {36437, 42133}, {36445, 37832}, {36450, 42588}, {36454, 42085}, {36455, 42134}, {36463, 37835}, {36468, 42589}, {39876, 48901}, {41099, 42277}, {41957, 42522}, {41987, 43313}, {42090, 42187}, {42091, 42188}, {42119, 42194}, {42120, 42193}, {42140, 52400}, {42141, 52399}, {42259, 49135}, {42265, 43512}, {42272, 50691}, {42525, 43314}, {42575, 45384}, {42606, 43383}, {43316, 43799}, {43317, 49134}, {43511, 50692}, {43561, 50693}, {45406, 48477}, {49820, 49821}

X(52666) = reflection of X(3069) in X(23259)
X(52666) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 43508, 42283}, {4, 6459, 31412}, {4, 6561, 3068}, {4, 13886, 35786}, {5, 43408, 42638}, {20, 23261, 42561}, {372, 33703, 42414}, {376, 6565, 32786}, {381, 9541, 32785}, {381, 42225, 9541}, {486, 3529, 42637}, {1132, 5059, 1152}, {3068, 6561, 6459}, {3071, 3146, 6460}, {3071, 42264, 7586}, {3146, 7586, 42264}, {3311, 3853, 23253}, {3311, 23253, 31414}, {3830, 42215, 23249}, {5073, 7584, 43407}, {6411, 42263, 43210}, {6560, 23273, 19053}, {6565, 42275, 376}, {7585, 50687, 42284}, {7586, 42264, 6460}, {13939, 49138, 42261}, {15682, 23273, 6560}, {15684, 18510, 42226}, {22615, 35821, 4}, {23249, 42215, 19054}, {23261, 42271, 20}, {23275, 33703, 372}, {35787, 42260, 3090}, {41099, 43509, 42277}, {42263, 42283, 2}, {42266, 42268, 631}, {43512, 50689, 42265}

---

X(52667) = X(2)X(6412)∩X(4)X(372)

Barycentrics    7*a^4 - 2*a^2*b^2 - 5*b^4 - 2*a^2*c^2 + 10*b^2*c^2 - 5*c^4 + 4*a^2*S : :
X(52667) = 5 X[3068] - 4 X[6221], 3 X[3068] - 2 X[9541], 3 X[3068] - 4 X[13665], 6 X[6221] - 5 X[9541], 3 X[6221] - 5 X[13665], 2 X[6221] - 5 X[23249], X[9541] - 3 X[23249], 2 X[13665] - 3 X[23249]

X(52667) lies on the cubic K1197 and these lines: {2, 6412}, {3, 23253}, {4, 372}, {5, 6456}, {6, 3543}, {20, 590}, {30, 3068}, {371, 23269}, {376, 6564}, {381, 6446}, {382, 1587}, {485, 3529}, {546, 6522}, {547, 6452}, {615, 3839}, {631, 42267}, {1131, 1151}, {1132, 3594}, {1152, 3832}, {1327, 6200}, {1328, 43796}, {1588, 3627}, {1657, 9540}, {2043, 42220}, {2044, 42219}, {3070, 3146}, {3071, 17578}, {3090, 35786}, {3091, 8252}, {3312, 3853}, {3317, 6454}, {3520, 35776}, {3522, 42265}, {3523, 42273}, {3524, 42277}, {3528, 10576}, {3534, 18538}, {3545, 6396}, {3734, 5591}, {3830, 18510}, {3845, 6398}, {3850, 6450}, {3854, 42583}, {3855, 5420}, {3861, 13951}, {5056, 6410}, {5073, 7583}, {5076, 7584}, {5411, 13473}, {5418, 17538}, {5860, 44678}, {5861, 33456}, {6199, 42537}, {6395, 38335}, {6411, 42604}, {6420, 23275}, {6425, 42542}, {6430, 43414}, {6437, 43210}, {6442, 42539}, {6451, 15686}, {6481, 42603}, {6485, 10194}, {6492, 43376}, {6561, 15682}, {7464, 9682}, {7581, 35821}, {7582, 22615}, {7586, 42283}, {8253, 10304}, {8972, 15683}, {8976, 15704}, {8981, 17800}, {9583, 28172}, {9616, 28158}, {10145, 13903}, {10299, 12818}, {11541, 42266}, {12102, 42571}, {12239, 12279}, {12323, 17128}, {12509, 45023}, {12602, 49039}, {12953, 31408}, {13711, 46453}, {13713, 33878}, {13785, 15687}, {13886, 42260}, {13941, 41946}, {13959, 18483}, {14241, 43791}, {14269, 18762}, {15640, 32787}, {15681, 35255}, {15684, 18512}, {15692, 32789}, {15697, 42576}, {15698, 43787}, {15709, 43336}, {15717, 42582}, {18586, 42137}, {18587, 42136}, {19065, 31673}, {19066, 41869}, {21736, 26330}, {23261, 50688}, {26362, 35948}, {31403, 44526}, {32489, 33364}, {35409, 42575}, {35822, 42275}, {36436, 42085}, {36437, 42134}, {36445, 37835}, {36449, 42589}, {36454, 42086}, {36455, 42133}, {36463, 37832}, {36467, 42588}, {39875, 48901}, {41099, 42274}, {41958, 42523}, {41987, 43312}, {42090, 42189}, {42091, 42190}, {42119, 42192}, {42120, 42191}, {42140, 52399}, {42141, 52400}, {42258, 49135}, {42262, 43511}, {42271, 50691}, {42524, 43315}, {42574, 45385}, {42607, 43382}, {43212, 43415}, {43316, 49134}, {43317, 43800}, {43512, 50692}, {43560, 50693}, {45407, 48476}, {49822, 49823}

X(52667) = reflection of X(i) in X(j) for these {i,j}: {3068, 23249}, {9541, 13665}
X(52667) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 43507, 42284}, {4, 6460, 42561}, {4, 6560, 3069}, {4, 13939, 35787}, {5, 43407, 42637}, {20, 23251, 31412}, {371, 23269, 31414}, {371, 33703, 42413}, {376, 6564, 32785}, {485, 3529, 42638}, {1131, 5059, 1151}, {3069, 6560, 6460}, {3070, 3146, 6459}, {3070, 42263, 7585}, {3146, 7585, 42263}, {3312, 3853, 23263}, {3830, 42216, 23259}, {5073, 7583, 43408}, {6412, 42264, 43209}, {6561, 23267, 19054}, {6564, 42276, 376}, {7585, 42263, 6459}, {7586, 50687, 42283}, {9541, 13665, 3068}, {9541, 23249, 13665}, {13886, 49138, 42260}, {15682, 23267, 6561}, {15684, 18512, 42225}, {22644, 35820, 4}, {23251, 42272, 20}, {23259, 42216, 19053}, {23269, 33703, 371}, {31414, 42413, 371}, {35786, 42261, 3090}, {41099, 43510, 42274}, {42264, 42284, 2}, {42267, 42269, 631}, {43511, 50689, 42262}

---

X(52668) = X(6)X(110)∩X(32)X(23357)

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

X(52668) lies on the cubic K835 and these lines: {6, 110}, {32, 23357}, {50, 52603}, {54, 2623}, {182, 32583}, {184, 10558}, {323, 2088}, {352, 32599}, {353, 52198}, {671, 7578}, {691, 10560}, {1648, 9225}, {2436, 34210}, {2770, 5486}, {3117, 19626}, {5012, 10559}, {5038, 41939}, {5050, 52152}, {5286, 52450}, {5967, 10415}, {8550, 51938}, {8877, 17809}, {9213, 14355}, {9745, 9775}, {11179, 34320}, {11427, 36894}, {14389, 51405}, {14591, 34397}, {14908, 34396}, {15073, 35901}, {33871, 40353}, {50979, 52232}

X(52668) = isogonal conjugate of X(43084)
X(52668) = X(9139)-Ceva conjugate of X(14908)
X(52668) = X(i)-isoconjugate of X(j) for these (i,j): {1, 43084}, {94, 896}, {524, 2166}, {662, 51479}, {690, 32680}, {922, 20573}, {1577, 14559}, {1989, 14210}, {2642, 35139}, {10412, 23889}, {14417, 36129}, {15475, 24039}, {32678, 35522}
X(52668) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 43084}, {94, 15899}, {524, 11597}, {1084, 51479}, {1989, 15477}, {3266, 40604}, {14210, 34544}, {18334, 35522}, {20573, 39061}
X(52668) = trilinear pole of line {50, 14270}
X(52668) = crossdifference of every pair of points on line {690, 41586}
X(52668) = barycentric product X(i)*X(j) for these {i,j}: {50, 671}, {110, 9213}, {111, 323}, {186, 895}, {340, 14908}, {523, 51478}, {526, 691}, {892, 14270}, {897, 6149}, {1511, 9139}, {2624, 36085}, {3268, 32729}, {5466, 52603}, {5968, 14355}, {7799, 32740}, {8753, 52437}, {9178, 10411}, {9214, 14385}, {10097, 14590}, {14591, 14977}, {16092, 52179}, {17983, 22115}, {18023, 19627}, {30786, 34397}, {32679, 36142}, {34574, 44814}, {36060, 52414}
X(52668) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 43084}, {50, 524}, {111, 94}, {186, 44146}, {323, 3266}, {512, 51479}, {526, 35522}, {671, 20573}, {691, 35139}, {895, 328}, {923, 2166}, {1576, 14559}, {2088, 52628}, {6149, 14210}, {8552, 45807}, {8753, 6344}, {9178, 10412}, {9213, 850}, {10097, 14592}, {14270, 690}, {14355, 52145}, {14385, 36890}, {14591, 4235}, {14908, 265}, {17983, 18817}, {19626, 11060}, {19627, 187}, {22115, 6390}, {32729, 476}, {32740, 1989}, {34394, 52040}, {34395, 52039}, {34397, 468}, {36142, 32680}, {44814, 52629}, {51478, 99}, {51980, 14356}, {52142, 52449}, {52179, 52094}, {52418, 37778}, {52603, 5468}
X(52668) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {184, 10558, 52142}, {184, 51980, 32729}, {10558, 32729, 51980}, {10560, 11003, 691}, {32729, 51980, 52142}

---

X(52669) = X(2)X(1692)∩X(30)X(182)

Barycentrics    7*a^6 - 2*a^4*b^2 + 4*a^2*b^4 + b^6 - 2*a^4*c^2 - 6*a^2*b^2*c^2 + b^4*c^2 + 4*a^2*c^4 + b^2*c^4 + c^6 : :
X(52669) = 2 X[575] + X[37466], 5 X[3618] - X[16041], X[35927] + 7 X[51171], X[11318] - 3 X[47352]

X(52669) lies on the cubic K1197 and these lines: {2, 1692}, {6, 620}, {30, 182}, {39, 7618}, {524, 3788}, {575, 37466}, {598, 5395}, {754, 40825}, {1691, 47101}, {2030, 7761}, {3618, 5475}, {3818, 15092}, {3972, 35927}, {5028, 35297}, {5032, 7763}, {5033, 8356}, {5182, 5309}, {5461, 47353}, {6337, 11156}, {7817, 11179}, {7834, 8176}, {7874, 21356}, {13354, 38064}, {14568, 39141}, {14848, 50684}, {15534, 51371}, {18860, 37809}, {35377, 50977}, {42510, 51019}, {42511, 51017}, {50659, 51185}

X(52669) = midpoint of X(6) and X(11288)

---

X(52670) = X(4)X(13)∩X(5)X(53)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^2 - b^2 - c^2 + 2*Sqrt[3]*S) : :

X(52670) lies on the cubic K416 and these lines: {4, 13}, {5, 53}, {17, 473}, {52, 20411}, {62, 472}, {143, 20412}, {235, 35715}, {264, 635}, {297, 636}, {324, 33529}, {393, 40694}, {397, 37505}, {463, 22482}, {467, 33530}, {470, 42488}, {1179, 8741}, {2913, 10641}, {3518, 6105}, {5318, 13403}, {6111, 42148}, {6530, 51754}, {6672, 51220}, {8884, 38944}, {8930, 36309}, {13350, 51219}, {27376, 36252}, {34509, 52282}, {36302, 42990}, {46924, 51264}
on K416

X(52670) = X(20412)-cross conjugate of X(6117)
X(52670) = X(i)-isoconjugate of X(j) for these (i,j): {18, 2169}, {2148, 40711}, {2167, 32586}
X(52670) = X(i)-Dao conjugate of X(j) for these (i,j): {18, 14363}, {97, 10639}, {216, 40711}, {32586, 40588}
X(52670) = barycentric product X(i)*X(j) for these {i,j}: {5, 472}, {17, 14129}, {53, 303}, {62, 324}, {311, 10641}, {6116, 8836}, {6117, 16770}, {13450, 52349}, {14577, 34389}, {23290, 52606}, {23873, 35360}, {32002, 36300}, {33529, 46925}
X(52670) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 40711}, {51, 32586}, {53, 18}, {62, 97}, {143, 52348}, {303, 34386}, {324, 34390}, {472, 95}, {3199, 21462}, {6117, 19778}, {8741, 252}, {10641, 54}, {14129, 302}, {14569, 8742}, {14577, 61}, {20412, 11131}, {35360, 32037}, {36300, 3519}, {46925, 51275}, {52604, 16807}
X(52670) = {X(53),X(6116)}-harmonic conjugate of X(6117)

---

X(52671) = X(4)X(14)∩X(5)X(53)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^2 - b^2 - c^2 - 2*Sqrt[3]*S) : :

X(52671) lies on the cubic K416 and these lines: {4, 14}, {5, 53}, {18, 472}, {52, 20412}, {61, 473}, {143, 20411}, {235, 35714}, {264, 636}, {297, 635}, {324, 33530}, {393, 40693}, {398, 37505}, {462, 22481}, {467, 33529}, {471, 42489}, {1179, 8742}, {2912, 10642}, {3518, 6104}, {5321, 13403}, {6110, 42147}, {6530, 51753}, {6671, 51219}, {8884, 38943}, {8929, 36306}, {13349, 51220}, {27376, 36251}, {34508, 52282}, {36303, 42991}, {46924, 51271}

X(52671) = X(20411)-cross conjugate of X(6116)
X(52671) = X(i)-isoconjugate of X(j) for these (i,j): {17, 2169}, {2148, 40712}, {2167, 32585}
X(52671) = X(i)-Dao conjugate of X(j) for these (i,j): {17, 14363}, {97, 10640}, {216, 40712}, {32585, 40588}
X(52671) = barycentric product X(i)*X(j) for these {i,j}: {5, 473}, {18, 14129}, {53, 302}, {61, 324}, {311, 10642}, {6116, 16771}, {6117, 8838}, {13450, 52348}, {14577, 34390}, {23290, 52605}, {23872, 35360}, {32002, 36301}, {33530, 46926}
X(52671) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 40712}, {51, 32585}, {53, 17}, {61, 97}, {143, 52349}, {302, 34386}, {324, 34389}, {473, 95}, {3199, 21461}, {6116, 19779}, {8742, 252}, {10642, 54}, {14129, 303}, {14569, 8741}, {14577, 62}, {20411, 11130}, {35360, 32036}, {36301, 3519}, {46926, 51268}, {52604, 16806}
X(52671) = {X(53),X(6117)}-harmonic conjugate of X(6116)

---

X(52672) = X(2)X(647)∩X(4)X(32)

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

X(52672) lies on the cubic K835 and these lines: {2, 647}, {4, 32}, {6, 36183}, {39, 51259}, {230, 7418}, {287, 37645}, {878, 46615}, {879, 6794}, {1236, 14961}, {1632, 51818}, {2549, 37991}, {2697, 2715}, {2966, 14907}, {3143, 14898}, {5286, 14265}, {5523, 46592}, {5967, 10415}, {6776, 51943}, {7736, 36875}, {7763, 14376}, {7803, 14382}, {8430, 52076}, {9463, 20021}, {10097, 16092}, {14609, 36874}, {15048, 36157}, {15630, 52471}, {17907, 22456}, {34175, 43448}, {41361, 52641}

X(52672) = isogonal conjugate of X(36823)
X(52672) = polar conjugate of X(52486)
X(52672) = X(i)-isoconjugate of X(j) for these (i,j): {1, 36823}, {48, 52486}, {237, 37220}, {240, 18876}, {684, 36095}, {1177, 1959}, {1755, 2373}, {9417, 46140}
X(52672) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 36823}, {1249, 52486}, {2373, 36899}, {2799, 38971}, {5181, 36212}, {14961, 50567}, {15900, 36884}, {18876, 39085}, {39058, 46140}
X(52672) = crosspoint of X(9154) and X(16081)
X(52672) = crosssum of X(3289) and X(9155)
X(52672) = trilinear pole of line {2393, 47138}
X(52672) = crossdifference of every pair of points on line {237, 684}
X(52672) = barycentric product X(i)*X(j) for these {i,j}: {98, 858}, {287, 5523}, {290, 2393}, {1236, 1976}, {1821, 18669}, {1910, 20884}, {2966, 47138}, {5181, 9154}, {11610, 52512}, {14961, 16081}, {22456, 42665}
X(52672) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 52486}, {6, 36823}, {67, 36884}, {98, 2373}, {248, 18876}, {290, 46140}, {858, 325}, {1821, 37220}, {1976, 1177}, {2393, 511}, {5181, 50567}, {5523, 297}, {11610, 52513}, {14580, 232}, {14961, 36212}, {17172, 51370}, {18669, 1959}, {19510, 51397}, {20021, 46165}, {20884, 46238}, {32696, 10423}, {36104, 36095}, {42665, 684}, {46592, 4230}, {47138, 2799}, {47426, 9155}, {51962, 51980}
X(52672) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 51404, 52451}, {35912, 51963, 2715}

---

X(52673) = X(2)X(7)∩X(4)X(916)

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

X(52673) lies on the cubics K267 and K610 and these lines: {2, 7}, {4, 916}, {48, 36023}, {65, 21922}, {69, 349}, {71, 15669}, {72, 1441}, {73, 347}, {273, 3868}, {442, 40999}, {1446, 52385}, {1490, 18655}, {1728, 24179}, {3668, 3682}, {4329, 5758}, {4552, 22021}, {5812, 37536}, {6356, 41804}, {9119, 30807}, {9612, 48888}, {17134, 18446}, {17751, 34388}, {17861, 18397}, {17863, 44547}, {18631, 37798}, {40571, 40573}

X(52673) = anticomplement of the isogonal conjugate of X(40573)
X(52673) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {278, 2894}, {2982, 20}, {14775, 37781}, {40395, 3869}, {40435, 52366}, {40447, 3436}, {40573, 8}, {52560, 2897}
X(52673) = X(286)-Ceva conjugate of X(1441)
X(52673) = X(15443)-cross conjugate of X(41342)
X(52673) = X(i)-isoconjugate of X(j) for these (i,j): {6, 43729}, {58, 41509}
X(52673) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 43729}, {10, 41509}, {72, 26942}
X(52673) = cevapoint of X(3191) and X(41342)
X(52673) = barycentric product X(i)*X(j) for these {i,j}: {75, 41342}, {85, 3191}, {274, 15443}, {307, 37279}, {349, 580}, {1231, 41227}
X(52673) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 43729}, {37, 41509}, {580, 284}, {3191, 9}, {15443, 37}, {37279, 29}, {41227, 1172}, {41342, 1}, {45038, 40937}, {46887, 46882}

---

X(52674) = X(3)X(315)∩X(4)X(12177)

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

X(52674) lies on the cubic K835 and these lines: {3, 315}, {4, 12177}, {6, 8370}, {20, 22503}, {76, 31958}, {83, 3767}, {99, 7737}, {194, 14035}, {316, 46264}, {491, 22718}, {492, 22716}, {850, 5652}, {1007, 9862}, {2548, 38907}, {2549, 39101}, {3096, 33001}, {3398, 32832}, {5038, 7803}, {5654, 17932}, {5939, 6033}, {6776, 39266}, {6782, 22687}, {6783, 22689}, {7752, 22664}, {7762, 51438}, {7785, 22498}, {7792, 14535}, {7841, 44380}, {8149, 46321}, {8598, 9766}, {11159, 41624}, {12252, 32006}, {13330, 35700}, {14712, 33208}, {33244, 45017}, {35702, 50720}, {35703, 50719}

---

X(52675) = X(49)X(143)∩X(52)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 - 2*a^2*c^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(52675) lies on the Feuerbach circumhyperbola of the orthic triangle, the cubic K416, and these lines: {3, 16880}, {4, 13585}, {5, 2914}, {6, 43808}, {24, 11935}, {49, 143}, {52, 186}, {54, 13289}, {93, 648}, {155, 16868}, {185, 1199}, {193, 7505}, {195, 13420}, {567, 1986}, {578, 16013}, {1173, 12234}, {1593, 33887}, {1843, 5097}, {2904, 7577}, {2965, 18212}, {5095, 25555}, {5895, 7592}, {6143, 15047}, {7576, 36966}, {7722, 11560}, {9972, 34155}, {10113, 43605}, {11702, 18350}, {11817, 52294}, {12161, 44958}, {12897, 46027}, {13202, 13403}, {13292, 45181}, {13596, 22948}, {15463, 22962}, {15472, 43612}, {15741, 35471}, {18559, 36749}, {34797, 40909}, {35475, 43807}, {43602, 43846}

X(52675) = orthic-isogonal conjugate of X(3518)
X(52675) = X(4)-Ceva conjugate of X(3518)
X(52675) = X(2962)-isoconjugate of X(15002)
X(52675) = X(69)-Dao conjugate of X(1994)
X(52675) = crosspoint of X(4) and X(14940)
X(52675) = crosssum of X(3) and X(15002)
X(52675) = barycentric product X(1994)*X(14940)
X(52675) = barycentric quotient X(i)/X(j) for these {i,j}: {2965, 15002}, {3518, 13585}, {14940, 11140}
X(52675) = {X(143),X(52417)}-harmonic conjugate of X(3518)

---

X(52676) = X(2)X(581)∩X(3)X(63)

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

X(52676) lies on the Kiepert circumhyperbola of the anticomplementary triangle, the cubic K610, and these lines: {1, 1441}, {2, 581}, {3, 63}, {4, 52025}, {100, 1715}, {286, 7513}, {307, 4303}, {411, 1764}, {572, 2287}, {580, 40571}, {914, 6734}, {965, 1212}, {1490, 19645}, {1754, 3811}, {1765, 12528}, {1818, 34823}, {2128, 19782}, {2360, 36018}, {3149, 34461}, {3870, 5706}, {3882, 5889}, {5534, 5767}, {6505, 41930}, {6835, 24220}, {7538, 46717}, {8897, 10441}, {10381, 37409}, {16346, 19861}, {17147, 34772}, {18443, 37151}, {36746, 37065}, {37106, 46877}

X(52676) = anticomplement of the isogonal conjugate of X(580)
X(52676) = isotomic conjugate of the polar conjugate of X(1713)
X(52676) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {580, 8}, {3191, 1330}, {37279, 21270}, {41227, 4}, {41342, 2893}, {46887, 2894}
X(52676) = X(286)-Ceva conjugate of X(63)
X(52676) = X(19)-isoconjugate of X(39944)
X(52676) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 39944}, {72, 3682}
X(52676) = barycentric product X(i)*X(j) for these {i,j}: {69, 1713}, {274, 41510}
X(52676) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 39944}, {1713, 4}, {41510, 37}

---

X(52677) = X(54)X(125)∩X(93)X(186)

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

X(52677) lies on the cubics K467 and K563 and these lines: {4, 1157}, {5, 933}, {54, 125}, {93, 186}, {95, 26166}, {96, 275}, {140, 6760}, {1141, 3520}, {3462, 32904}, {3484, 20299}, {3575, 19169}, {7577, 25044}, {7769, 18831}, {13561, 50463}, {14516, 18315}, {18284, 22467}, {19176, 38605}, {19210, 37938}, {25739, 46089}, {33664, 40448}, {35728, 44977}

X(52677) = isogonal conjugate of X(6798)
X(52677) = polar-circle-inverse of X(16337)
X(52677) = antigonal image of X(6801)
X(52677) = X(264)-Ceva conjugate of X(275)
X(52677) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6798}, {3432, 44706}
X(52677) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 54}, {3, 6798}
X(52677) = barycentric product X(i)*X(j) for these {i,j}: {275, 2888}, {8795, 45800}
X(52677) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6798}, {2888, 343}, {8565, 418}, {8882, 3432}, {45800, 5562}

---

X(52678) = X(6)X(41404)∩X(187)X(9225)

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

X(52678) lies on the cubics K751 and K1156 and these lines: {6, 41404}, {187, 9225}, {524, 8591}, {843, 20998}, {2502, 9217}, {5024, 18872}, {5524, 21839}, {33999, 51927}

X(52678) = isogonal conjugate of X(8591)
X(52678) = isogonal conjugate of the anticomplement of X(671)
X(52678) = isogonal conjugate of the complement of X(8596)
X(52678) = isogonal conjugate of the isotomic conjugate of X(46275)
X(52678) = X(111)-cross conjugate of X(6)
X(52678) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8591}, {2, 39339}, {75, 46276}, {896, 39061}, {897, 38239}, {14210, 41404}
X(52678) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 8591}, {206, 46276}, {6593, 38239}, {15477, 41404}, {15899, 39061}, {32664, 39339}
X(52678) = barycentric product X(6)*X(46275)
X(52678) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8591}, {31, 39339}, {32, 46276}, {111, 39061}, {187, 38239}, {32740, 41404}, {46275, 76}
X(52678) = {X(187),X(41449)}-harmonic conjugate of X(46276)

---

X(52679) = X(1)X(111)∩X(2)X(5540)

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

X(52679) lies on the Bevan circle, the cubic K1156, and these lines: {1, 111}, {2, 5540}, {6, 1054}, {9, 2503}, {43, 5213}, {57, 1366}, {573, 1768}, {1757, 41317}, {4062, 5525}, {5524, 21839}, {5541, 21832}, {7580, 39156}, {21873, 24052}, {24038, 37210}

X(52679) = reflection of X(1) in X(8691)
X(52679) = X(524)-Ceva conjugate of X(1)
X(52679) = X(671)-Dao conjugate of X(897)

---

X(52680) = ISOGONAL CONJUGATE OF X(4674)

Barycentrics    a*(a + b)*(2*a - b - c)*(a + c) : :
X(52680) = X[1] + 2 X[896], 4 X[1125] - X[17491], 5 X[3616] + X[31301], 7 X[3624] - 4 X[4892], 10 X[31280] - 13 X[34595]

X(52680) lies on these lines: {1, 21}, {2, 4257}, {3, 1724}, {6, 16370}, {8, 17539}, {9, 609}, {10, 11115}, {20, 1714}, {28, 8056}, {30, 35466}, {32, 16552}, {35, 3293}, {36, 238}, {40, 15952}, {43, 4184}, {44, 2251}, {46, 37227}, {56, 20615}, {57, 36011}, {60, 37616}, {79, 24161}, {86, 2163}, {99, 2382}, {100, 31855}, {110, 2718}, {112, 2751}, {162, 17515}, {165, 4221}, {171, 5251}, {172, 3294}, {187, 2238}, {210, 37589}, {214, 30576}, {244, 4973}, {261, 10455}, {267, 1247}, {272, 18655}, {284, 1743}, {286, 17885}, {333, 3550}, {384, 29433}, {386, 4189}, {387, 17576}, {404, 37687}, {405, 4252}, {442, 24902}, {484, 759}, {500, 5428}, {516, 50759}, {517, 18191}, {519, 902}, {529, 24222}, {540, 3936}, {549, 37663}, {551, 8025}, {577, 1713}, {580, 6906}, {581, 6875}, {602, 5450}, {672, 14964}, {673, 46498}, {679, 4622}, {741, 898}, {751, 3736}, {765, 1110}, {849, 1098}, {940, 16418}, {950, 22361}, {956, 3052}, {958, 5264}, {978, 4225}, {991, 37106}, {995, 17127}, {997, 2206}, {1006, 37469}, {1010, 1698}, {1012, 1754}, {1014, 7271}, {1018, 5291}, {1043, 3632}, {1054, 1325}, {1104, 3670}, {1125, 6536}, {1150, 48863}, {1155, 1739}, {1193, 5267}, {1201, 52564}, {1211, 49723}, {1227, 4432}, {1279, 4694}, {1326, 5150}, {1330, 25645}, {1384, 37658}, {1399, 37558}, {1408, 1420}, {1412, 13462}, {1437, 37618}, {1464, 5427}, {1722, 5358}, {1723, 2193}, {1730, 37397}, {1736, 46974}, {1759, 16968}, {1764, 49128}, {1776, 45272}, {1817, 23511}, {1877, 3911}, {1914, 45751}, {1935, 37583}, {1961, 33774}, {2183, 51637}, {2194, 3576}, {2243, 21372}, {2250, 2341}, {2287, 3973}, {2299, 4227}, {2303, 3731}, {2475, 24880}, {2647, 15932}, {2999, 27174}, {3008, 14953}, {3017, 15677}, {3065, 52380}, {3073, 11012}, {3110, 24494}, {3120, 50757}, {3218, 18173}, {3219, 30115}, {3246, 16726}, {3336, 11101}, {3337, 37816}, {3361, 5323}, {3454, 25669}, {3560, 37530}, {3579, 3987}, {3582, 51382}, {3583, 14956}, {3586, 51281}, {3616, 31301}, {3624, 4892}, {3633, 52352}, {3634, 17589}, {3750, 16474}, {3751, 33538}, {3786, 34476}, {3928, 16485}, {3953, 4906}, {3992, 4434}, {4188, 17749}, {4195, 10479}, {4228, 5272}, {4255, 19535}, {4256, 17549}, {4262, 37657}, {4273, 16670}, {4281, 5312}, {4292, 37113}, {4302, 33137}, {4340, 17558}, {4418, 46895}, {4424, 4640}, {4426, 16549}, {4427, 39766}, {4438, 4680}, {4551, 5172}, {4636, 34600}, {4641, 24929}, {4647, 24850}, {4648, 17561}, {4650, 5902}, {4652, 17521}, {4677, 4720}, {4723, 16729}, {4760, 8682}, {4862, 8822}, {4881, 45763}, {5021, 16783}, {5030, 33854}, {5046, 45939}, {5051, 6693}, {5060, 5540}, {5080, 17734}, {5119, 18163}, {5122, 16610}, {5134, 17737}, {5235, 19875}, {5255, 5258}, {5259, 37607}, {5271, 17587}, {5277, 46196}, {5278, 16393}, {5288, 37588}, {5292, 6872}, {5294, 19867}, {5315, 19247}, {5333, 17553}, {5396, 7508}, {5398, 6914}, {5400, 6905}, {5453, 12104}, {5492, 22936}, {5692, 7262}, {5697, 18178}, {5712, 50739}, {5737, 16394}, {5903, 18180}, {6126, 16164}, {6175, 31204}, {6626, 17210}, {6629, 30941}, {6645, 29383}, {6675, 49745}, {6703, 13745}, {6909, 13329}, {7031, 21384}, {7288, 34029}, {7419, 21214}, {7424, 37718}, {7428, 39748}, {7483, 37693}, {7741, 37357}, {7951, 47515}, {8690, 12029}, {8692, 40726}, {9348, 16475}, {9350, 25440}, {9612, 25516}, {9708, 37540}, {10304, 37681}, {10470, 13323}, {11111, 37642}, {11113, 37646}, {11194, 16483}, {11319, 50605}, {11354, 37660}, {11813, 17174}, {13408, 16617}, {13486, 14158}, {13586, 20142}, {13587, 37680}, {13588, 16569}, {13735, 14829}, {13739, 46883}, {13743, 52524}, {14007, 19872}, {14584, 23703}, {14996, 48855}, {15228, 24715}, {15670, 17056}, {15672, 37635}, {15673, 37631}, {15674, 26131}, {15676, 24936}, {15680, 24883}, {16046, 16833}, {16050, 17284}, {16054, 31183}, {16118, 37369}, {16192, 37402}, {16342, 43531}, {16371, 37679}, {16471, 36746}, {16484, 18166}, {16487, 18186}, {16705, 17200}, {16724, 24452}, {16738, 49482}, {16832, 26643}, {16858, 37633}, {16887, 29660}, {16916, 29438}, {16917, 29460}, {16918, 29440}, {16920, 29455}, {16957, 29448}, {16975, 21793}, {17010, 22350}, {17034, 17692}, {17103, 17175}, {17126, 30116}, {17147, 49683}, {17167, 18393}, {17197, 30384}, {17206, 33953}, {17259, 19290}, {17327, 51590}, {17512, 17596}, {17532, 31187}, {17542, 37682}, {17557, 31280}, {17571, 19765}, {17676, 20083}, {17677, 41806}, {18164, 51816}, {18593, 51654}, {19276, 19732}, {19744, 51602}, {19762, 28348}, {20077, 25650}, {22765, 32486}, {23205, 46513}, {24174, 37524}, {24440, 37572}, {24597, 48837}, {24931, 26064}, {25430, 37322}, {25441, 26117}, {26723, 26830}, {26725, 33097}, {26803, 41785}, {26841, 27097}, {26860, 38314}, {27185, 30110}, {28443, 51340}, {28453, 45923}, {29473, 33821}, {29767, 32941}, {29858, 30984}, {31037, 50215}, {31445, 37539}, {31649, 48903}, {33096, 37701}, {33954, 34016}, {37298, 37662}, {37552, 41229}, {37666, 50742}, {40096, 44671}, {41629, 51093}, {41809, 49729}, {42025, 51110}, {42028, 51105}

X(52680) = midpoint of X(4427) and X(39766)
X(52680) = reflection of X(3120) in X(50757)
X(52680) = isogonal conjugate of X(4674)
X(52680) = isogonal conjugate of the anticomplement of X(34587)
X(52680) = isogonal conjugate of the isotomic conjugate of X(30939)
X(52680) = X(i)-Ceva conjugate of X(j) for these (i,j): {86, 17191}, {759, 1}, {4622, 1019}, {30576, 3285}
X(52680) = X(i)-cross conjugate of X(j) for these (i,j): {44, 16704}, {214, 1}, {900, 23703}, {902, 3285}, {1319, 37168}, {1960, 1023}
X(52680) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4674}, {6, 4080}, {10, 106}, {37, 88}, {42, 903}, {44, 30575}, {58, 4013}, {65, 1320}, {71, 6336}, {72, 36125}, {101, 4049}, {213, 20568}, {226, 2316}, {306, 8752}, {321, 9456}, {512, 4555}, {523, 901}, {661, 3257}, {679, 21805}, {758, 1168}, {850, 32719}, {1018, 1022}, {1318, 40663}, {1400, 4997}, {1417, 3701}, {1464, 36590}, {1577, 32665}, {1797, 1826}, {2226, 3943}, {2250, 52031}, {3120, 9268}, {3125, 5376}, {3952, 23345}, {4024, 4591}, {4079, 4615}, {4120, 4638}, {4404, 36042}, {4551, 23838}, {4557, 6548}, {4582, 7180}, {4618, 4730}, {4622, 4705}, {4634, 50487}, {4945, 28658}, {5548, 7178}, {6635, 8034}, {10428, 17757}, {13576, 34230}, {14260, 38955}, {15065, 16944}, {18082, 46150}, {18793, 36814}, {32686, 50453}, {34857, 52553}, {36058, 41013}, {36091, 48350}, {38950, 43692}
X(52680) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 4674}, {9, 4080}, {10, 214}, {10, 4013}, {44, 3936}, {88, 40589}, {321, 4370}, {519, 3992}, {523, 38979}, {903, 40592}, {1015, 4049}, {1320, 40602}, {1441, 52659}, {1577, 35092}, {3257, 36830}, {3834, 21026}, {3936, 35550}, {4086, 51402}, {4125, 36912}, {4404, 5516}, {4555, 39054}, {4997, 40582}, {6544, 16732}, {6626, 20568}, {20619, 41013}, {30575, 40595}
X(52680) = cevapoint of X(44) and X(902)
X(52680) = crosspoint of X(i) and X(j) for these (i,j): {86, 24624}, {4567, 4622}, {24041, 37140}
X(52680) = crosssum of X(i) and X(j) for these (i,j): {37, 21805}, {42, 2245}, {2610, 2643}, {3125, 4730}
X(52680) = trilinear pole of line {1635, 17455}
X(52680) = crossdifference of every pair of points on line {37, 661}
X(52680) = barycentric product X(i)*X(j) for these {i,j}: {1, 16704}, {6, 30939}, {10, 30576}, {21, 3911}, {27, 5440}, {28, 3977}, {44, 86}, {58, 4358}, {63, 37168}, {65, 30606}, {75, 3285}, {80, 17191}, {81, 519}, {99, 1635}, {106, 16729}, {110, 3762}, {214, 24624}, {274, 902}, {283, 37790}, {286, 22356}, {310, 2251}, {314, 1404}, {333, 1319}, {593, 3992}, {643, 30725}, {662, 900}, {757, 3943}, {759, 51583}, {799, 1960}, {811, 22086}, {905, 46541}, {1014, 2325}, {1019, 17780}, {1023, 7192}, {1171, 4975}, {1227, 34079}, {1333, 3264}, {1412, 4723}, {1414, 1639}, {1434, 3689}, {1437, 46109}, {1444, 8756}, {1509, 21805}, {1647, 4567}, {1790, 38462}, {1812, 1877}, {1929, 31059}, {2087, 4600}, {2185, 40663}, {2341, 41801}, {3251, 4615}, {3733, 24004}, {4432, 37128}, {4434, 40432}, {4448, 4584}, {4528, 4637}, {4560, 23703}, {4565, 4768}, {4573, 4895}, {4596, 4984}, {4603, 4922}, {4610, 4730}, {4612, 30572}, {4614, 4773}, {4616, 14427}, {4622, 6544}, {4623, 14407}, {4702, 42302}, {4969, 40438}, {6385, 9459}, {7199, 23344}, {7203, 30731}, {14616, 17455}, {17195, 40400}, {23202, 44129}
X(52680) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4080}, {6, 4674}, {21, 4997}, {28, 6336}, {37, 4013}, {44, 10}, {58, 88}, {81, 903}, {86, 20568}, {106, 30575}, {110, 3257}, {163, 901}, {214, 3936}, {284, 1320}, {513, 4049}, {519, 321}, {643, 4582}, {662, 4555}, {678, 3943}, {859, 52031}, {900, 1577}, {902, 37}, {1017, 21805}, {1019, 6548}, {1023, 3952}, {1319, 226}, {1333, 106}, {1404, 65}, {1437, 1797}, {1474, 36125}, {1576, 32665}, {1635, 523}, {1639, 4086}, {1647, 16732}, {1877, 40149}, {1960, 661}, {2087, 3120}, {2194, 2316}, {2203, 8752}, {2206, 9456}, {2251, 42}, {2325, 3701}, {2341, 36590}, {3251, 4120}, {3264, 27801}, {3285, 1}, {3689, 2321}, {3733, 1022}, {3762, 850}, {3911, 1441}, {3943, 1089}, {3977, 20336}, {3992, 28654}, {4120, 4036}, {4273, 4792}, {4358, 313}, {4370, 3992}, {4432, 3948}, {4434, 3963}, {4556, 4622}, {4570, 5376}, {4591, 4618}, {4610, 4634}, {4653, 4945}, {4702, 4044}, {4723, 30713}, {4727, 4066}, {4730, 4024}, {4773, 4815}, {4833, 23598}, {4895, 3700}, {4908, 4125}, {4969, 4647}, {4975, 1230}, {4984, 30591}, {5440, 306}, {7252, 23838}, {8756, 41013}, {9459, 213}, {14407, 4705}, {14408, 21051}, {14418, 52355}, {14425, 4404}, {14437, 14431}, {14439, 3932}, {16704, 75}, {16723, 20893}, {16726, 6549}, {16729, 3264}, {16947, 1417}, {16948, 31227}, {17191, 320}, {17455, 758}, {17780, 4033}, {20972, 4695}, {21805, 594}, {22086, 656}, {22356, 72}, {23202, 71}, {23344, 1018}, {23703, 4552}, {24004, 27808}, {30576, 86}, {30606, 314}, {30725, 4077}, {30939, 76}, {31059, 20947}, {34079, 1168}, {37168, 92}, {39251, 4026}, {40663, 6358}, {46541, 6335}, {51583, 35550}
X(52680) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1707, 49500}, {3, 1724, 3216}, {21, 58, 1}, {21, 81, 4653}, {21, 16948, 58}, {31, 993, 1}, {35, 5247, 3293}, {36, 238, 49997}, {38, 49480, 1}, {58, 993, 18169}, {58, 4653, 81}, {63, 37817, 1}, {81, 4653, 1}, {187, 2238, 35342}, {238, 3286, 18792}, {284, 1778, 1743}, {405, 4252, 37522}, {595, 2975, 1}, {846, 5429, 1}, {859, 3286, 36}, {956, 3052, 37610}, {1104, 3916, 3670}, {1468, 5248, 1}, {2243, 49758, 21372}, {2303, 4877, 3731}, {2650, 35016, 1}, {3915, 8666, 1}, {4184, 4276, 5010}, {4225, 4278, 7280}, {4225, 27660, 978}, {4267, 17524, 35}, {4281, 37296, 37574}, {5291, 17735, 1018}, {8822, 17189, 4862}, {11110, 25526, 3624}, {16050, 24632, 17284}, {17549, 32911, 4256}

---

X(52681) = X(3)X(161)∩X(5)X(11701)

Barycentrics    -2*a^16+8*a^14*(b^2+c^2)+(b^2-c^2)^6*(b^2+c^2)^2-a^2*(b-c)^4*(b+c)^4*(b^2+c^2)*(2*b^4+3*b^2*c^2+2*c^4)+2*a^10*(b^2+c^2)*(7*b^4+4*b^2*c^2+7*c^4)+a^6*(b-c)^2*(b+c)^2*(b^2+c^2)*(12*b^4-b^2*c^2+12*c^4)-a^12*(13*b^4+22*b^2*c^2+13*c^4)-a^4*(b^2-c^2)^2*(3*b^8-5*b^6*c^2-10*b^4*c^4-5*b^2*c^6+3*c^8)-a^8*(15*b^8+5*b^6*c^2+14*b^4*c^4+5*b^2*c^6+15*c^8) : :

See Ivan Pavlov, euclid 5580.

X(52681) lies on these lines: {2,32904}, {3,161}, {5,11701}, {30,14143}, {381,20414}, {382,31392}, {550,15619}, {1510,45959}, {3519,35449}, {3574,8154}, {3845,32536}, {6146,52540}, {6150,22804}, {8703,35721}, {12254,25042}, {15345,27868}, {15704,35885}, {35717,52295}

X(52681) = midpoint of X(i) in X(j) for these {i,j}: {5, 35728}, {550, 15619}
X(52681) = reflection of X(15345) in X(27868)
X(52681) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {27868, 32423, 15345}

---

X(52682) = X(3)X(142)∩X(4)X(653)

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

See Ivan Pavlov, euclid 5580.

X(52682) lies on these lines: {1,11661}, {3,142}, {4,653}, {5,5698}, {7,944}, {9,9956}, {40,17528}, {46,11372}, {55,11218}, {65,971}, {144,38149}, {355,527}, {377,20070}, {381,28534}, {382,7686}, {390,6934}, {442,5759}, {515,1159}, {517,5735}, {518,11898}, {528,1482}, {962,11112}, {1155,1699}, {1385,6173}, {1454,15299}, {1483,51099}, {1656,15254}, {1657,43178}, {1836,11502}, {2095,37820}, {2550,5690}, {2646,38036}, {2801,4757}, {3062,5560}, {3474,8727}, {3577,28160}, {3579,5715}, {3824,10268}, {4295,20420}, {4338,12688}, {5135,38143}, {5220,5790}, {5536,31140}, {5542,13607}, {5603,30332}, {5696,37625}, {5708,48482}, {5779,13465}, {5806,41869}, {5812,9709}, {5818,6172}, {5833,15823}, {5842,15934}, {5882,43180}, {5901,47357}, {5902,36999}, {5903,36971}, {6244,37240}, {6684,50740}, {6830,30312}, {6847,15911}, {6862,38037}, {6880,38039}, {6905,8543}, {6906,30295}, {6941,30311}, {6978,38131}, {6984,21168}, {7580,20292}, {7956,9812}, {7967,30340}, {7982,44785}, {8226,44447}, {8227,50836}, {8545,9654}, {9580,17603}, {10246,25557}, {10310,49177}, {10427,22791}, {10894,16125}, {10896,51768}, {10943,41555}, {11499,16159}, {11522,37600}, {12650,31776}, {12702,15346}, {13750,14100}, {15446,15909}, {15733,24474}, {16118,37001}, {18481,43177}, {20195,38172}, {22765,42842}, {26066,38108}, {30275,37737}, {31162,50371}, {31423,31658}, {31493,37623}, {31657,43161}, {31673,48664}, {31822,34339}, {37281,52457}, {37298,38073}, {38030,43175}, {38053,51700}, {39542,50701}

X(52682) = midpoint of X(5696) and X(37625)
X(52682) = reflection of X(i) in X(j) for these {i,j}: {3, 5880}, {390, 20330}, {1657, 43178}, {5698, 5}, {5882, 43180}, {11372, 18482}, {18481, 43177}, {43161, 31657}
X(52682) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): {516, 5880, 3}

---

X(52683) = X(3)X(10)∩X(20)X(1145)

Barycentrics    -3*a^7+5*a^6*(b+c)+2*(b-c)^4*(b+c)^3+a^2*(b-c)^2*(b+c)*(b^2-16*b*c+c^2)+2*a^5*(b^2-11*b*c+c^2)-4*a*(b^2-c^2)^2*(b^2-3*b*c+c^2)+5*a^3*(b-c)^2*(b^2+4*b*c+c^2)-2*a^4*(b+c)*(4*b^2-11*b*c+4*c^2) : :

See Ivan Pavlov, euclid 5580.

X(52683) lies on these lines: {1,1538}, {3,10}, {8,12246}, {20,1145}, {30,34711}, {84,51781}, {153,5730}, {165,37829}, {496,944}, {517,18239}, {519,6259}, {952,12667}, {971,5881}, {1319,9581}, {1482,6256}, {1490,28204}, {1657,35460}, {1837,30283}, {2098,41698}, {2800,48664}, {2829,12702}, {3057,5691}, {3679,34862}, {5082,6925}, {5176,37022}, {5697,37001}, {5727,37566}, {5731,13747}, {5806,12128}, {6001,12645}, {6068,11827}, {6260,37727}, {6261,18526}, {6834,12019}, {6850,40587}, {6959,34773}, {7966,31795}, {7982,22792}, {7991,44784}, {9799,34627}, {10246,18242}, {10247,12608}, {10310,12751}, {10786,37606}, {10864,37712}, {11113,50864}, {11522,51789}, {12650,18480}, {12680,37711}, {12684,50798}, {12688,37708}, {12747,37726}, {13607,18530}, {15079,37006}, {15347,38455}, {19541,45287}, {21627,28236}, {28224,37406}, {30332,36991}, {31793,46677}, {34740,37623}, {34880,44425}, {40262,50811}

X(52683) = reflection of X(i) in X(j) for these {i,j}: {1482, 6256}, {7982, 22792}, {12650, 18480}, {18526, 6261}, {37727, 6260}
X(52683) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4297, 37828, 3}, {5691, 37709, 9856}

---

X(52684) = X(3)X(9)∩X(4)X(8545)

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

See Ivan Pavlov, euclid 5580.

X(52684) lies on these lines: {3,9}, {4,8545}, {7,6848}, {40,6068}, {63,5658}, {142,6944}, {144,37421}, {515,6930}, {516,6256}, {527,5709}, {1071,5729}, {1420,15299}, {1445,6927}, {1467,10398}, {1697,5252}, {1708,41561}, {1709,35445}, {2078,30223}, {2801,6261}, {3062,7162}, {5084,5817}, {5220,6001}, {5223,5693}, {5534,15733}, {5696,17857}, {5698,12667}, {5735,37826}, {5766,29007}, {5805,9612}, {5880,18242}, {6172,6223}, {6594,6796}, {6834,30379}, {6905,8544}, {6970,8257}, {8757,9121}, {10394,18446}, {11248,47375}, {11500,15726}, {12114,15254}, {12526,31798}, {12650,37234}, {14872,42014}, {15071,41700}, {15239,37584}, {15296,16112}, {15297,37837}, {17527,38108}, {17564,37526}, {17567,21151}, {23340,43166}, {38122,52264}

X(52684) = midpoint of X(5698) and X(12667)
X(52684) = reflection of X(i) in X(j) for these {i,j}: {3358, 9}, {5880, 18242}, {12114, 15254}, {43178, 6796}
X(52684) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9, 971, 3358}, {8257, 43177, 37534}, {32555, 32556, 32625}

---

X(52685) = X(2)X(3)∩X(511)X(5884)

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

See Ivan Pavlov, euclid 5580.

X(52685) lies on these lines: {2,3}, {511,5884}, {540,49609}, {542,11362}, {986,11246}, {1768,48883}, {3817,46975}, {15349,38456}, {19925,46976}, {46483,48917}

X(52685) = midpoint of X(46483) and X(48917)
X(52685) = reflection of X(46976) in X(19925)

---

X(52686) = X(3)X(74)∩X(4)X(13)

Barycentrics    a^2*(-(b^2*c^2*(S+sqrt[3]*SB)^2*(S+sqrt[3]*SC)^2)+(S+sqrt[3]*SA)^2*(b^2*SB*(S+sqrt[3]*SB)^2+2*S^2*(S+sqrt[3]*SB)*(S+sqrt[3]*SC)+c^2*SC*(S+sqrt[3]*SC)^2)) : :

See Ivan Pavlov, euclid 5580.

X(52686) lies on these lines: {3,74}, {4,13}, {15,12112}, {30,11127}, {62,15032}, {184,35470}, {376,44719}, {511,30485}, {1181,22238}, {1498,36836}, {2781,14173}, {3129,14157}, {3130,5890}, {3165,5237}, {5352,41090}, {5612,36967}, {5668,46112}, {5865,12082}, {6000,35469}, {10540,48365}, {11126,18445}, {12317,40709}, {13754,34008}, {14816,37776}, {22236,41476}

X(52686) = midpoint of X(15) and X(36963)

---

X(52687) = X(3)X(74)∩X(4)X(14)

Barycentrics    a^2*(-(b^2*c^2*(S-sqrt[3]*SB)^2*(S-sqrt[3]*SC)^2)+(S-sqrt[3]*SA)^2*(b^2*SB*(S-sqrt[3]*SB)^2+2*S^2*(S-sqrt[3]*SB)*(S-sqrt[3]*SC)+c^2*SC*(S-sqrt[3]*SC)^2)) : :

See Ivan Pavlov, euclid 5580.

X(52687) lies on these lines: {3,74}, {4,14}, {16,12112}, {30,11126}, {61,15032}, {184,35469}, {376,44718}, {511,30486}, {1181,22236}, {1498,36843}, {2781,14179}, {3129,5890}, {3130,14157}, {3166,5238}, {5351,41089}, {5616,36968}, {5669,46113}, {5864,12082}, {6000,35470}, {10540,48366}, {11127,18445}, {12317,40710}, {13754,34009}, {14817,37775}

X(52687) = midpoint of X(16) and X(36964)

---

X(52688) = X(2)X(3)∩X(61)X(6773)

Barycentrics    -(a^4*(sqrt[3]*S+SA)^2)+b^2*(sqrt[3]*S+SB)^2*SC+2*S^2*(sqrt[3]*S+SB)*(sqrt[3]*S+SC)+c^2*SB*(sqrt[3]*S+SC)^2 : :

See Ivan Pavlov, euclid 5580.

X(52688) lies on these lines: {2,3}, {16,51753}, {17,33421}, {61,6773}, {298,5865}, {398,9606}, {511,627}, {533,35689}, {617,37824}, {618,14540}, {621,7763}, {622,1078}, {628,1352}, {633,5617}, {635,14539}, {1503,16772}, {1506,22512}, {3107,40694}, {3412,41021}, {5206,22513}, {5238,22532}, {5334,31400}, {5335,19780}, {5339,31492}, {5478,42431}, {5480,16773}, {5611,5872}, {5868,36836}, {5869,16644}, {6115,9862}, {6695,21157}, {6770,42152}, {7684,16965}, {7685,42489}, {7832,36756}, {8260,14853}, {10653,22531}, {14144,16626}, {14651,36251}, {19130,49105}, {22568,23235}, {29012,49106}, {30559,42813}, {36303,39575}, {36759,40693}, {36962,42581}, {36990,42490}, {38751,41060}, {41036,43633}, {41038,43194}, {42434,44666}

X(52688) = reflection of X(22532) in X(5238)
X(52688) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1080, 4}, {3, 11307, 631}, {5238, 41022, 22532}, {5617, 47068, 633}

---

X(52689) = X(2)X(3)∩X(62)X(6770)

Barycentrics    -(a^4*(-sqrt[3]*S+SA)^2)+b^2*(-sqrt[3]*S+SB)^2*SC+2*S^2*(-sqrt[3]*S+SB)*(-sqrt[3]*S+SC)+c^2*SB*(-sqrt[3]*S+SC)^2 : :

See Ivan Pavlov, euclid 5580.

X(52689) lies on these lines: {2,3}, {15,51754}, {18,33420}, {62,6770}, {299,5864}, {397,9606}, {511,628}, {532,35688}, {616,37825}, {619,14541}, {621,1078}, {622,7763}, {627,1352}, {634,5613}, {636,14538}, {1503,16773}, {1506,22513}, {3106,40693}, {3411,41020}, {5206,22512}, {5237,22531}, {5334,19781}, {5335,31400}, {5340,31492}, {5479,42432}, {5480,16772}, {5615,5873}, {5868,16645}, {5869,36843}, {6114,9862}, {6694,21156}, {6773,42149}, {7684,42488}, {7685,16964}, {7832,36755}, {8259,14853}, {10654,22532}, {14145,16627}, {14651,36252}, {19130,49106}, {22570,23235}, {29012,49105}, {30560,42814}, {36302,39575}, {36760,40694}, {36961,42580}, {36990,42491}, {38751,41061}, {41037,43632}, {41039,43193}, {42433,44667}

X(52689) = reflection of X(22531) in X(5237)
X(52689) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 11308, 631}, {3, 383, 4}, {5613, 47066, 634}

---

X(52690) = X(3)X(37)∩X(19)X(511)

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

See Ivan Pavlov, euclid 5580.

X(52690) lies on these lines: {3,37}, {19,511}, {46,1423}, {65,611}, {182,2285}, {576,2082}, {607,44492}, {920,4459}, {1086,37532}, {1350,5341}, {1753,15488}, {3072,24445}, {3509,7350}, {4363,26921}, {5085,5356}, {5102,7300}, {5762,5880}, {7297,11477}, {10519,27059}, {12514,15973}, {14853,26998}, {42067,45186}, {44469,52413}

---

X(52691) = X(2)X(99)∩X(3)X(7827)

Barycentrics    2*a^4 - 5*a^2*b^2 - b^4 - 5*a^2*c^2 + b^2*c^2 - c^4 : :
X(52691) = 8 X[39] + X[7802], 4 X[39] - X[7812], 10 X[39] - X[7823], 2 X[39] + X[7833], 14 X[39] - 5 X[7921], X[7802] + 2 X[7812], 5 X[7802] + 4 X[7823], X[7802] - 4 X[7833], 7 X[7802] + 20 X[7921], 5 X[7812] - 2 X[7823], X[7812] + 2 X[7833], 7 X[7812] - 10 X[7921], X[7823] + 5 X[7833], 7 X[7823] - 25 X[7921], 7 X[7833] + 5 X[7921], X[76] - 4 X[8359], X[194] + 2 X[7810], 2 X[7757] + X[7811], X[7757] + 2 X[8356], X[7811] - 4 X[8356], 2 X[7753] + X[33264], 5 X[7786] - 2 X[8370], 8 X[7830] + X[7877], 4 X[7830] - X[9939], X[7877] + 2 X[9939], 5 X[7904] + 4 X[32450], X[8353] + 2 X[9300], 4 X[8354] - X[11057], 2 X[8354] + X[41624], X[11057] + 2 X[41624], 8 X[8358] + X[11055], 4 X[8358] - X[37671], X[11055] + 2 X[37671], 4 X[8367] - X[32819], X[9873] + 8 X[32516], X[11257] + 2 X[37345], X[11361] - 4 X[44562]

X(52691) lies on these lines: {2, 99}, {3, 7827}, {6, 35955}, {15, 35932}, {16, 35931}, {30, 262}, {39, 3849}, {76, 8359}, {83, 33007}, {182, 376}, {183, 11054}, {194, 7810}, {316, 5024}, {378, 37765}, {381, 43461}, {384, 34504}, {524, 3094}, {530, 3107}, {531, 3106}, {538, 10335}, {541, 15920}, {542, 7709}, {550, 7878}, {597, 3972}, {599, 7831}, {691, 50149}, {1003, 5116}, {1078, 7738}, {1992, 14907}, {2548, 33192}, {3096, 7783}, {3314, 39785}, {3329, 9855}, {3523, 51237}, {3524, 9734}, {3589, 35954}, {3815, 8352}, {5007, 33275}, {5013, 7752}, {5025, 31652}, {5028, 5032}, {5063, 52630}, {5215, 7817}, {5254, 15597}, {5286, 43459}, {5309, 5569}, {5475, 8597}, {5476, 11676}, {6034, 7606}, {6054, 37242}, {6179, 9607}, {6337, 7944}, {6390, 7937}, {6655, 7775}, {6656, 7870}, {6661, 48310}, {7472, 50147}, {7610, 14568}, {7735, 47061}, {7739, 8182}, {7748, 33013}, {7753, 33264}, {7755, 33022}, {7760, 32965}, {7761, 7840}, {7763, 33190}, {7765, 33004}, {7769, 11318}, {7771, 15048}, {7772, 33260}, {7777, 31173}, {7781, 33021}, {7782, 7846}, {7786, 8370}, {7791, 7796}, {7792, 27088}, {7797, 15515}, {7798, 44367}, {7799, 11165}, {7803, 32985}, {7806, 8589}, {7809, 9770}, {7814, 8357}, {7818, 41136}, {7828, 15815}, {7829, 33014}, {7830, 7877}, {7858, 22332}, {7859, 33237}, {7861, 10150}, {7875, 32456}, {7884, 35297}, {7888, 19690}, {7902, 33259}, {7904, 32450}, {7918, 8360}, {7920, 15513}, {7934, 22110}, {8176, 14041}, {8353, 9300}, {8354, 11057}, {8355, 37647}, {8358, 11055}, {8367, 32819}, {8703, 19661}, {8716, 21358}, {8719, 38072}, {9606, 19695}, {9698, 33019}, {9741, 21356}, {9771, 33228}, {9873, 32516}, {9996, 48657}, {10033, 11257}, {10166, 41939}, {10168, 35925}, {10753, 11179}, {11147, 33191}, {11148, 32836}, {11152, 43535}, {11159, 11174}, {11164, 11286}, {11168, 47286}, {11285, 34505}, {11299, 13084}, {11300, 13083}, {11317, 42849}, {11361, 32479}, {11645, 34624}, {12040, 33184}, {14063, 31450}, {14355, 39497}, {14848, 39656}, {16921, 47617}, {16989, 37809}, {19697, 43527}, {23334, 33272}, {25555, 35951}, {28562, 34645}, {30229, 41143}, {31401, 33006}, {31457, 32967}, {35936, 50660}, {39593, 46893}, {44541, 51185}, {47005, 51848}, {47288, 50146}, {51012, 51485}, {51015, 51484}

X(52691) = midpoint of X(i) and X(j) for these {i,j}: {2, 32480}, {10033, 11257}, {11152, 43535}
X(52691) = reflection of X(10033) in X(37345)
X(52691) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2482, 7835}, {2, 2549, 671}, {2, 7618, 41134}, {2, 8591, 3734}, {2, 41135, 7617}, {6, 35955, 51224}, {39, 7833, 7812}, {597, 8598, 3972}, {671, 41134, 9877}, {2482, 4045, 2}, {5013, 7847, 7752}, {5024, 5077, 11163}, {5077, 11163, 316}, {5309, 5569, 8859}, {7617, 11648, 41135}, {7619, 14971, 2}, {7757, 8356, 7811}, {7791, 34511, 7883}, {7812, 7833, 7802}, {7817, 33274, 7857}, {7817, 37512, 33274}, {7864, 33274, 7817}, {7864, 37512, 7857}, {7883, 34511, 7796}, {8354, 41624, 11057}, {8859, 33273, 5569}, {9741, 21356, 32833}, {12040, 33184, 41133}, {22332, 33234, 7858}, {42849, 44526, 11317}

---

X(52692) = X(2)X(11594)∩X(6)X(23)

Barycentrics    a^2*(2*a^2 + 2*b^2 - c^2)*(2*a^2 - b^2 + 2*c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4) : :
X(52692) = X[23] + 2 X[16308], X[16313] - 4 X[47316]

X(52692) lies on these lines: {2, 11594}, {6, 23}, {30, 262}, {94, 10511}, {186, 40801}, {237, 5968}, {250, 37969}, {264, 468}, {325, 5112}, {351, 523}, {842, 2080}, {858, 3613}, {1513, 14356}, {2021, 37927}, {2070, 3425}, {2491, 8430}, {5189, 45090}, {5640, 51735}, {7610, 37907}, {7792, 16324}, {8598, 9177}, {9139, 40352}, {9307, 37897}, {9832, 32224}, {10989, 42849}, {11181, 37953}, {11632, 44266}, {11654, 41517}, {13481, 17008}, {14084, 26613}, {14765, 37900}, {16092, 18818}, {16313, 47316}, {32218, 47245}, {35138, 46142}, {36897, 37906}

X(52692) = X(i)-isoconjugate of X(j) for these (i,j): {98, 36263}, {293, 5094}, {336, 8541}, {574, 1821}, {599, 1910}, {3404, 10130}, {3906, 36084}, {17414, 36036}
X(52692) = X(i)-Dao conjugate of X(j) for these (i,j): {132, 5094}, {574, 40601}, {599, 11672}, {2679, 17414}, {3906, 38987}, {5976, 9464}
X(52692) = trilinear pole of line {3569, 33752}
X(52692) = crossdifference of every pair of points on line {574, 3906}
X(52692) = barycentric product X(i)*X(j) for these {i,j}: {237, 40826}, {297, 43697}, {325, 1383}, {511, 598}, {877, 30491}, {2396, 46001}, {2421, 8599}, {2799, 11636}, {3569, 35138}, {5968, 51541}, {9155, 18818}, {20022, 30489}, {23297, 51862}
X(52692) = barycentric quotient X(i)/X(j) for these {i,j}: {232, 5094}, {237, 574}, {325, 9464}, {511, 599}, {598, 290}, {1383, 98}, {1755, 36263}, {2211, 8541}, {2421, 9146}, {2491, 17414}, {3569, 3906}, {5968, 42008}, {8430, 23288}, {8599, 43665}, {9155, 39785}, {11636, 2966}, {14966, 9145}, {30489, 20021}, {30491, 879}, {35138, 43187}, {36790, 51397}, {40826, 18024}, {43697, 287}, {44114, 8288}, {46001, 2395}, {51541, 52145}, {51862, 10130}, {51980, 42007}
X(52692) = {X(7426),X(50150)}-harmonic conjugate of X(22329)

---

X(52693) = X(6)X(110)∩X(39)X(38523)

Barycentrics    a^2*(2*a^6*b^2 - 4*a^4*b^4 + a^2*b^6 + b^8 + 2*a^6*c^2 - a^4*b^2*c^2 + 2*a^2*b^4*c^2 - b^6*c^2 - 4*a^4*c^4 + 2*a^2*b^2*c^4 - b^4*c^4 + a^2*c^6 - b^2*c^6 + c^8) : :
X(52693) = 4 X[39] - X[38523], 2 X[3095] + X[38520]

X(52693) lies on these lines: {6, 110}, {39, 38523}, {125, 7777}, {262, 542}, {511, 15915}, {690, 7757}, {1511, 11842}, {2421, 7998}, {3095, 38520}, {5466, 9171}, {5467, 15080}, {5663, 32444}, {5972, 7806}, {7753, 50711}, {9140, 11163}, {9159, 50149}, {9463, 21906}, {9513, 43718}, {11002, 44114}, {11654, 18872}, {13331, 45914}, {35265, 44127}

X(52693) = X(7757)-line conjugate of X(690)
X(52693) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 5968, 5640}, {110, 895, 46298}, {2421, 46127, 7998}

---

X(52694) = X(6)X(98)∩X(115)X(6785)

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

X(52694) lies on these lines: {6, 98}, {115, 6785}, {287, 35901}, {290, 671}, {353, 5967}, {542, 34235}, {2549, 52451}, {6792, 20021}, {11646, 52190}, {20423, 52672}

---

X(52695) = X(2)X(99)∩X(3)X(11177)

Barycentrics    7*a^4 - 7*a^2*b^2 + b^4 - 7*a^2*c^2 + 5*b^2*c^2 + c^4 : :
X(52695) = X(52694) = X[2] + 2 X[99], 7 X[2] - 4 X[115], 4 X[2] - X[148], 5 X[2] - 8 X[620], 5 X[2] - 2 X[671], X[2] - 4 X[2482], 11 X[2] - 8 X[5461], 19 X[2] - 16 X[6722], 2 X[2] + X[8591], 7 X[2] - X[8596], 3 X[2] - 4 X[9167], 13 X[2] - 10 X[14061], 5 X[2] - 4 X[14971], 5 X[2] + 4 X[15300], 5 X[2] + X[20094], 13 X[2] - 16 X[22247], 17 X[2] - 20 X[31274], X[2] - 16 X[35022], 13 X[2] - X[35369], X[2] + 8 X[36521], 17 X[2] - 8 X[36523], 43 X[2] - 16 X[41147], 91 X[2] - 64 X[41148], 59 X[2] - 32 X[41154], 7 X[99] + 2 X[115], 8 X[99] + X[148], 5 X[99] + 4 X[620], 5 X[99] + X[671], X[99] + 2 X[2482], 11 X[99] + 4 X[5461], 19 X[99] + 8 X[6722], 4 X[99] - X[8591], 14 X[99] + X[8596], 3 X[99] + X[9166], 3 X[99] + 2 X[9167], 13 X[99] + 5 X[14061], 5 X[99] + 2 X[14971], 5 X[99] - 2 X[15300], 10 X[99] - X[20094], 13 X[99] + 8 X[22247], 17 X[99] + 10 X[31274], X[99] + 8 X[35022], 26 X[99] + X[35369], X[99] - 4 X[36521], 17 X[99] + 4 X[36523], 4 X[99] + X[41135], 43 X[99] + 8 X[41147], 91 X[99] + 32 X[41148], 59 X[99] + 16 X[41154], 16 X[115] - 7 X[148], and many others

X(52695) lies on the cubic K1306 and these lines: {2, 99}, {3, 11177}, {20, 6054}, {30, 38743}, {69, 51798}, {76, 5569}, {98, 15692}, {114, 3543}, {147, 376}, {187, 44367}, {193, 18800}, {194, 9741}, {315, 45017}, {325, 9855}, {381, 13172}, {384, 9606}, {385, 27088}, {524, 2076}, {538, 26613}, {542, 10304}, {547, 12355}, {549, 12243}, {551, 13174}, {598, 19686}, {616, 5464}, {617, 5463}, {618, 9114}, {619, 9116}, {631, 11632}, {1003, 8290}, {1698, 50884}, {1916, 14039}, {1975, 7610}, {1992, 5026}, {2782, 3524}, {2795, 15672}, {2796, 25055}, {2896, 7782}, {2936, 6636}, {3091, 10992}, {3180, 52022}, {3181, 52021}, {3241, 9881}, {3314, 35955}, {3329, 35954}, {3363, 17005}, {3522, 14981}, {3523, 6055}, {3528, 52090}, {3534, 51872}, {3545, 15561}, {3552, 13571}, {3616, 50886}, {3617, 50885}, {3620, 11161}, {3623, 50888}, {3679, 51578}, {3828, 9875}, {3832, 20399}, {3839, 23234}, {3849, 7799}, {3926, 9939}, {4027, 33266}, {4590, 46275}, {5032, 5039}, {5054, 14651}, {5056, 38751}, {5059, 38745}, {5066, 38733}, {5071, 6321}, {5215, 14568}, {5395, 5503}, {5485, 33216}, {5939, 42850}, {5969, 13331}, {5976, 11147}, {5982, 22576}, {5983, 22575}, {5989, 33273}, {6033, 11001}, {6036, 15721}, {6337, 7785}, {6390, 7840}, {6655, 7870}, {6658, 7775}, {6669, 22577}, {6670, 22578}, {6777, 36331}, {6778, 35750}, {7486, 38734}, {7492, 34013}, {7757, 8782}, {7766, 37809}, {7777, 11159}, {7779, 32456}, {7783, 8369}, {7784, 7833}, {7793, 9740}, {7827, 33225}, {7841, 7891}, {7863, 7883}, {7864, 8366}, {7888, 19691}, {7898, 35705}, {7907, 34505}, {7912, 33192}, {7923, 8365}, {7925, 8352}, {7932, 33197}, {7945, 33190}, {7946, 33254}, {8182, 32833}, {8289, 32817}, {8356, 9878}, {8359, 46226}, {8370, 12040}, {8593, 11160}, {8597, 22110}, {8703, 9862}, {8716, 33246}, {8859, 19570}, {8997, 19058}, {9143, 11006}, {9164, 17948}, {9300, 44532}, {9761, 30472}, {9763, 30471}, {9771, 33013}, {9830, 21356}, {9884, 31145}, {10150, 39563}, {10299, 51523}, {10352, 12191}, {10353, 12150}, {10385, 12351}, {10488, 22165}, {10706, 33512}, {10991, 21734}, {11050, 12347}, {11164, 11184}, {11606, 47005}, {12042, 15698}, {12100, 12188}, {12177, 50967}, {13989, 19057}, {14036, 47352}, {14038, 22332}, {14041, 41133}, {14148, 50248}, {14645, 33684}, {14830, 19708}, {14831, 39807}, {14916, 35356}, {15078, 39803}, {15682, 22566}, {15683, 38738}, {15686, 38744}, {15688, 38635}, {15697, 38736}, {15699, 38732}, {15702, 38750}, {15705, 34473}, {15708, 38748}, {15709, 38224}, {15717, 38664}, {15810, 39091}, {16990, 47061}, {18840, 33215}, {19053, 49266}, {19054, 49267}, {19661, 51123}, {19689, 43527}, {19862, 50887}, {20112, 32819}, {22329, 32459}, {22515, 41106}, {23334, 32837}, {24714, 36224}, {25486, 32831}, {31173, 40246}, {32528, 32986}, {32552, 35751}, {32553, 36329}, {32820, 33276}, {32821, 33268}, {32825, 33214}, {32967, 51238}, {34628, 50879}, {34632, 50881}, {34638, 50882}, {35698, 42602}, {35699, 42603}, {35749, 49816}, {36174, 46986}, {36327, 36767}, {37901, 47326}, {39356, 44372}, {39358, 40866}, {43256, 50720}, {43257, 50719}

X(52695) = midpoint of X(i) and X(j) for these {i,j}: {99, 41134}, {8591, 41135}, {14971, 15300}, {26614, 51524}, {33265, 41136}
X(52695) = reflection of X(i) in X(j) for these {i,j}: {2, 41134}, {148, 41135}, {671, 14971}, {3545, 15561}, {3839, 23234}, {5032, 5182}, {8859, 35297}, {9166, 9167}, {10304, 21166}, {11632, 26614}, {14041, 41133}, {14568, 5215}, {14651, 5054}, {14971, 620}, {19570, 8859}, {38732, 15699}, {39563, 10150}, {41134, 2482}, {41135, 2}, {41136, 7799}
X(52695) = anticomplement of X(9166)
X(52695) = anticomplement of the isotomic conjugate of X(9164)
X(52695) = X(9164)-anticomplementary conjugate of X(6327)
X(52695) = X(9164)-Ceva conjugate of X(2)
X(52695) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 99, 8591}, {2, 8591, 148}, {2, 8596, 115}, {2, 20094, 671}, {99, 620, 20094}, {99, 671, 15300}, {99, 2482, 2}, {99, 15483, 14931}, {114, 12117, 3543}, {376, 8724, 147}, {549, 13188, 12243}, {618, 9114, 51483}, {619, 9116, 51482}, {620, 671, 2}, {620, 15300, 671}, {671, 15300, 20094}, {2482, 15300, 620}, {2482, 36521, 99}, {3552, 34511, 34604}, {3926, 33208, 9939}, {6390, 8598, 7840}, {7664, 10717, 2}, {7840, 8598, 14712}, {7870, 34504, 6655}, {8593, 50567, 11160}, {8703, 48657, 9862}, {8724, 33813, 376}, {9166, 9167, 2}, {9166, 41134, 9167}, {9881, 11711, 3241}, {9885, 9886, 9890}, {11164, 11184, 11361}, {12351, 15452, 10385}, {14061, 22247, 2}, {15300, 20094, 8591}, {18800, 50639, 193}, {22566, 38730, 15682}, {32456, 39785, 51224}, {34511, 34604, 13571}, {35022, 36521, 2482}, {38750, 49102, 15702}, {39103, 39105, 36521}, {39785, 51224, 7779}

---

X(52696) = X(2)X(32)∩X(3)X(14885)

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

X(52696) lies on the cubics K792 and K1306 and these lines: {2, 32}, {3, 14885}, {187, 827}, {733, 2080}, {1384, 41295}, {2079, 51862}, {4577, 13586}, {5206, 14247}, {6781, 38946}, {38663, 51240}

X(52696) = isogonal conjugate of X(33666)
X(52696) = circumcircle-inverse of X(14885)
X(52696) = orthoptic-circle-of-the-Steiner-circumellipse-inverse of X(8878)
X(52696) = psi-transform of X(10329)
X(52696) = X(1)-isoconjugate of X(33666)
X(52696) = X(3)-Dao conjugate of X(33666)
X(52696) = barycentric quotient X(6)/X(33666)
X(52696) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 9481, 46227}, {187, 827, 46228}

---

X(52697) = X(3)X(19140)∩X(6)X(110)

Barycentrics    a^2*(3*a^6 - a^4*b^2 - 3*a^2*b^4 + b^6 - 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(52697) = X[3] + 2 X[19140], 2 X[3] + X[51941], 4 X[19140] - X[51941], X[6] + 2 X[110], 5 X[6] - 2 X[895], 2 X[6] + X[2930], X[6] - 4 X[6593], 5 X[110] + X[895], 4 X[110] - X[2930], X[110] + 2 X[6593], 4 X[895] + 5 X[2930], X[895] - 10 X[6593], X[2930] + 8 X[6593], 2 X[67] - 5 X[3763], X[67] - 4 X[5972], 2 X[67] + X[25336], 5 X[3763] - 8 X[5972], 5 X[3763] + X[25336], 8 X[5972] + X[25336], 2 X[69] + X[16176], X[69] + 2 X[25329], X[16176] - 4 X[25329], 2 X[113] + X[32233], 4 X[113] - X[36990], 2 X[32233] + X[36990], 4 X[125] - X[25335], 4 X[125] - 7 X[47355], X[25335] - 7 X[47355], 2 X[141] + X[11061], X[146] + 2 X[44882], X[155] + 2 X[19138], 2 X[182] + X[399], 4 X[182] - X[16010], 2 X[399] + X[16010], X[193] - 4 X[41595], 4 X[206] - X[10117], 2 X[206] + X[15141], X[10117] + 2 X[15141], X[323] + 2 X[32217], X[25330] - 3 X[47352], 2 X[576] + 7 X[15039], 2 X[597] + X[9143], X[599] - 4 X[5642], X[599] + 2 X[34319], 4 X[5642] + X[25331], 2 X[5642] + X[34319], 2 X[1177] + X[17847], 2 X[1177] - 5 X[19132], X[17847] + 5 X[19132], and many others

X(52697) = X(52697) lies on the cubic K1306 and these lines: {3, 19140}, {6, 110}, {67, 3763}, {69, 16176}, {113, 32233}, {125, 19125}, {141, 11061}, {146, 44882}, {155, 19138}, {182, 399}, {193, 41595}, {206, 2916}, {249, 22259}, {323, 32217}, {511, 2070}, {524, 25321}, {542, 5050}, {576, 15039}, {597, 7605}, {599, 5642}, {1177, 17847}, {1350, 1511}, {1351, 12584}, {1352, 10272}, {1386, 2948}, {1495, 10510}, {1576, 9155}, {1974, 19504}, {2453, 51430}, {2771, 38029}, {2781, 15035}, {2836, 16475}, {2892, 34774}, {2931, 19139}, {2935, 19149}, {3098, 15040}, {3167, 5648}, {3242, 11720}, {3448, 3589}, {3564, 44282}, {3580, 47453}, {3618, 14683}, {3830, 25566}, {3844, 32261}, {5026, 15342}, {5085, 5621}, {5092, 10620}, {5093, 34155}, {5095, 6144}, {5102, 14984}, {5116, 38661}, {5181, 40341}, {5480, 12383}, {5596, 23315}, {5609, 11579}, {5650, 15106}, {5655, 43273}, {5898, 19150}, {5987, 11174}, {6034, 50711}, {6776, 20125}, {7669, 46127}, {7728, 48905}, {7998, 19127}, {8556, 9769}, {8705, 19596}, {8998, 32253}, {9027, 44102}, {9517, 14396}, {9924, 13248}, {9971, 45082}, {9976, 32254}, {10387, 32290}, {10541, 14094}, {10628, 23042}, {10752, 15034}, {11004, 37827}, {11284, 32235}, {11402, 51185}, {11485, 13858}, {11486, 13859}, {11559, 19151}, {11898, 41731}, {12017, 12308}, {12121, 32271}, {12375, 19146}, {12376, 19145}, {12902, 19130}, {13169, 50993}, {13363, 43811}, {13392, 48876}, {13990, 32252}, {14561, 32423}, {14982, 16534}, {15041, 17508}, {15118, 24981}, {15140, 20987}, {15533, 41720}, {16163, 48872}, {16261, 51739}, {19459, 34470}, {21637, 32251}, {24206, 32306}, {29012, 38789}, {31670, 34153}, {32242, 44000}, {32255, 51171}, {33884, 51882}, {38702, 38861}, {38790, 48898}, {38794, 49116}, {39484, 47353}, {39561, 39562}, {40825, 46301}, {40916, 52171}, {41721, 47448}, {47452, 47558}

X(52697) = midpoint of X(i) and X(j) for these {i,j}: {599, 25331}, {9143, 25320}, {22151, 35265}, {32609, 45016}
X(52697) = reflection of X(i) in X(j) for these {i,j}: {5085, 15462}, {5093, 34155}, {5621, 5085}, {10516, 14643}, {15041, 17508}, {19596, 35265}, {25320, 597}, {25331, 34319}, {31884, 15035}, {38724, 38317}, {39562, 39561}
X(52697) = crossdifference of every pair of points on line {690, 45801}
X(52697) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 19140, 51941}, {6, 110, 2930}, {67, 5972, 3763}, {69, 25329, 16176}, {110, 6593, 6}, {110, 9129, 46276}, {113, 32233, 36990}, {182, 399, 16010}, {206, 15141, 10117}, {1511, 9970, 1350}, {3618, 14683, 25328}, {3763, 25336, 67}, {5092, 52098, 10620}, {5642, 34319, 599}, {5648, 15303, 15534}, {10752, 15034, 33851}, {11720, 32278, 3242}, {12017, 12308, 32305}, {12121, 32271, 48910}, {12584, 25556, 1351}, {15040, 48679, 3098}, {17847, 19132, 1177}, {20976, 32740, 6}, {25335, 47355, 125}

---

X(52698) = X(3)X(111)∩X(5)X(14654)

Barycentrics    a^2*(3*a^8 - 10*a^6*b^2 + 2*a^4*b^4 + 10*a^2*b^6 - 5*b^8 - 10*a^6*c^2 + 35*a^4*b^2*c^2 - 25*a^2*b^4*c^2 + 20*b^6*c^2 + 2*a^4*c^4 - 25*a^2*b^2*c^4 - 22*b^4*c^4 + 10*a^2*c^6 + 20*b^2*c^6 - 5*c^8) : :
X(52698) = X[3] + 2 X[111], 5 X[3] - 2 X[1296], 2 X[3] + X[11258], X[3] - 4 X[14650], 4 X[3] - X[38593], 7 X[3] - 4 X[38623], 7 X[3] + 2 X[38675], 11 X[3] - 2 X[38688], 3 X[3] - 2 X[38716], 5 X[3] + 4 X[51535], 5 X[111] + X[1296], 4 X[111] - X[11258], X[111] + 2 X[14650], 8 X[111] + X[38593], 7 X[111] + 2 X[38623], 7 X[111] - X[38675], 11 X[111] + X[38688], 3 X[111] + X[38716], 5 X[111] - 2 X[51535], 4 X[1296] + 5 X[11258], X[1296] - 10 X[14650], 8 X[1296] - 5 X[38593], 7 X[1296] - 10 X[38623], 7 X[1296] + 5 X[38675], 11 X[1296] - 5 X[38688], X[1296] - 5 X[38698], 3 X[1296] - 5 X[38716], X[1296] + 2 X[51535], X[11258] + 8 X[14650], 2 X[11258] + X[38593], 7 X[11258] + 8 X[38623], 7 X[11258] - 4 X[38675], 11 X[11258] + 4 X[38688], X[11258] + 4 X[38698], 3 X[11258] + 4 X[38716], 5 X[11258] - 8 X[51535], 16 X[14650] - X[38593], 7 X[14650] - X[38623], 14 X[14650] + X[38675], 22 X[14650] - X[38688], 6 X[14650] - X[38716], 5 X[14650] + X[51535], 7 X[38593] - 16 X[38623], 7 X[38593] + 8 X[38675], 11 X[38593] - 8 X[38688], X[38593] - 8 X[38698], and many others

X(52698) lies on these lines: {3, 111}, {5, 14654}, {20, 38800}, {30, 38799}, {126, 3526}, {140, 14360}, {381, 9172}, {382, 5512}, {399, 9129}, {543, 5054}, {631, 20099}, {1351, 28662}, {1482, 11721}, {1656, 6719}, {1657, 22338}, {1995, 37811}, {2070, 15563}, {2780, 15041}, {2854, 5050}, {3843, 10734}, {5055, 32424}, {5070, 40340}, {5072, 38807}, {5093, 36696}, {6455, 11835}, {6456, 11836}, {9177, 44533}, {9179, 38580}, {10704, 37624}, {10717, 15694}, {11171, 33900}, {13310, 50381}, {14657, 45735}, {14684, 15916}, {14810, 37751}, {15688, 38798}, {15696, 38797}, {15720, 40556}, {15922, 33980}, {17800, 44987}, {34200, 37749}, {38335, 38802}, {38801, 49137}

X(52698) = midpoint of X(i) and X(j) for these {i,j}: {111, 38698}, {14666, 38796}
X(52698) = reflection of X(i) in X(j) for these {i,j}: {3, 38698}, {381, 38796}, {5093, 36696}, {38698, 14650}, {38796, 9172}
X(52698) = circumcircle-inverse of X(51535)
X(52698) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 111, 11258}, {3, 11258, 38593}, {111, 1296, 51535}, {111, 14650, 3}, {126, 38806, 3526}, {6719, 10748, 1656}, {9172, 14666, 381}

---

X(52699) = X(6)X(110)∩X(74)X(182)

Barycentrics    a^2*(3*a^6 - 2*a^4*b^2 - 3*a^2*b^4 + 2*b^6 - 2*a^4*c^2 + 5*a^2*b^2*c^2 - b^4*c^2 - 3*a^2*c^4 - b^2*c^4 + 2*c^6) : :
X(52699) = 4 X[2] - X[13169], X[2] + 2 X[15303], 2 X[2] + X[41720], X[13169] + 8 X[15303], X[13169] + 4 X[25321], X[13169] + 2 X[41720], 4 X[15303] - X[41720], 2 X[3] + X[10752], 2 X[6] + X[110], 4 X[6] - X[895], 5 X[6] + X[2930], X[6] + 2 X[6593], 2 X[110] + X[895], 5 X[110] - 2 X[2930], X[110] - 4 X[6593], 5 X[895] + 4 X[2930], X[895] + 8 X[6593], X[895] + 4 X[52697], X[2930] - 10 X[6593], X[2930] - 5 X[52697], 2 X[9129] + X[10765], X[11188] - 4 X[41670], X[67] - 4 X[3589], 2 X[67] - 5 X[15059], X[67] + 2 X[25329], 8 X[3589] - 5 X[15059], 2 X[3589] + X[25329], 5 X[15059] + 4 X[25329], X[69] + 2 X[5095], X[69] - 4 X[5972], X[5095] + 2 X[5972], X[74] - 4 X[182], X[74] + 2 X[9970], X[74] + 8 X[25556], 2 X[182] + X[9970], X[182] + 2 X[25556], X[9970] - 4 X[25556], 2 X[113] + X[6776], 4 X[113] - X[41737], 2 X[6776] + X[41737], 2 X[125] - 5 X[3618], 2 X[125] + X[11061], X[125] - 4 X[32300], 5 X[3618] + X[11061], 5 X[3618] - 8 X[32300], X[11061] + 8 X[32300], 4 X[141] - X[32244], X[141] + 2 X[41595], X[32244] + 8 X[41595], 2 X[184] + X[41743], and many others

X(52699) lies on the cubic K1307 and these lines: {2, 9769}, {3, 10752}, {6, 110}, {67, 3589}, {69, 5095}, {74, 182}, {113, 6776}, {125, 3618}, {141, 32244}, {184, 41743}, {186, 249}, {193, 5181}, {206, 13248}, {265, 18583}, {394, 41618}, {468, 41721}, {524, 47243}, {542, 3545}, {568, 19138}, {575, 11579}, {576, 15034}, {597, 9140}, {690, 5182}, {1112, 19118}, {1176, 1177}, {1350, 15051}, {1351, 1511}, {1352, 32234}, {1353, 10272}, {1386, 7984}, {1428, 32289}, {1576, 14060}, {1692, 5166}, {1843, 41671}, {1986, 41716}, {1992, 5642}, {1993, 41612}, {2330, 32290}, {2393, 35265}, {2452, 51430}, {2777, 25406}, {2781, 5085}, {3098, 15036}, {3292, 41617}, {3448, 15118}, {3564, 14643}, {3580, 47457}, {3619, 32257}, {3751, 11720}, {3763, 16176}, {4558, 9155}, {5050, 5622}, {5093, 13321}, {5097, 12584}, {5465, 8593}, {5480, 10733}, {5505, 12039}, {5648, 8584}, {5655, 50979}, {5888, 37283}, {6090, 41614}, {6091, 47426}, {6329, 25328}, {6698, 47355}, {6699, 32247}, {6800, 12824}, {7728, 48906}, {8262, 47453}, {8547, 25489}, {8550, 14982}, {8705, 11416}, {9027, 37784}, {9138, 9188}, {9142, 35357}, {9144, 18800}, {9177, 40078}, {9517, 14398}, {9730, 43815}, {9822, 32260}, {9967, 11557}, {9976, 15516}, {10250, 51023}, {10264, 51732}, {10510, 15107}, {10519, 38793}, {10541, 15021}, {10602, 20772}, {10706, 11179}, {10721, 32271}, {10819, 35841}, {10820, 35840}, {11002, 19136}, {11064, 15471}, {11216, 35264}, {11477, 15020}, {11511, 35268}, {11574, 13417}, {11597, 12596}, {11723, 39898}, {12017, 12041}, {12121, 21850}, {12177, 22265}, {12220, 40949}, {12367, 15826}, {12368, 39870}, {12900, 32275}, {13198, 21637}, {13202, 14927}, {13203, 41256}, {13212, 51741}, {13363, 15132}, {13910, 32253}, {13972, 32252}, {14653, 19661}, {14853, 17702}, {14999, 47550}, {15040, 44456}, {15054, 51941}, {15061, 38110}, {15072, 34117}, {15128, 26926}, {15140, 19121}, {15141, 26206}, {15360, 47544}, {15647, 19132}, {15988, 41606}, {16163, 51212}, {18358, 32272}, {19059, 19146}, {19060, 19145}, {19504, 20806}, {20304, 32306}, {23200, 38225}, {23315, 32264}, {24206, 41731}, {24981, 32255}, {25330, 51185}, {25331, 47352}, {25336, 40342}, {25711, 37506}, {31884, 38446}, {32250, 51537}, {32298, 49524}, {32305, 50664}, {37760, 41583}, {38520, 44507}, {38650, 40825}, {38794, 48876}, {39764, 46301}, {40112, 47545}, {41724, 47458}, {43602, 43810}, {46512, 48540}, {47454, 47558}, {50435, 51742}

X(52699) = midpoint of X(i) and X(j) for these {i,j}: {2, 25321}, {6, 52697}, {5050, 45016}, {5093, 32609}, {22151, 52238}
X(52699) = reflection of X(i) in X(j) for these {i,j}: {110, 52697}, {5622, 5050}, {10519, 38793}, {14644, 14561}, {15035, 15462}, {15055, 5085}, {15061, 38110}, {25321, 15303}, {41720, 25321}, {52238, 44102}, {52697, 6593}
X(52699) = 1st-Lemoine-circle-inverse of X(11636)
X(52699) = 2nd-Lemoine-circle-inverse of X(10765)
X(52699) = crossdifference of every pair of points on line {690, 44915}
X(52699) = X(5182)-lineconjugate of X(690)
X(52699) = barycentric product X(249)*X(14120)
X(52699) = barycentric quotient X(14120)/X(338)
X(52699) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 15303, 41720}, {2, 41720, 13169}, {6, 110, 895}, {6, 6593, 110}, {6, 32740, 39024}, {6, 39689, 10765}, {67, 3589, 15059}, {110, 41670, 35904}, {113, 6776, 41737}, {125, 32300, 3618}, {182, 9970, 74}, {182, 25556, 9970}, {206, 13248, 38885}, {575, 19140, 11579}, {597, 34319, 9140}, {895, 35904, 11188}, {1386, 32278, 7984}, {3448, 51171, 15118}, {3589, 25329, 67}, {3618, 11061, 125}, {5095, 5972, 69}, {5465, 8593, 14833}, {5480, 32233, 10733}, {9129, 39689, 110}, {10510, 32217, 15107}, {11064, 15471, 32220}, {11579, 19140, 14094}, {12017, 48679, 12041}, {12039, 16510, 5505}, {12900, 32275, 40330}, {21637, 34470, 13198}, {32271, 46264, 10721}

---

X(52700) = X(39)X(512)∩X(111)X(694)

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

X(52700) lies on the cubic K1308 and these lines: {39, 512}, {111, 694}, {524, 1916}, {4108, 41939}, {11654, 41517}, {36897, 52694}

X(52700) = X(i)-isoconjugate of X(j) for these (i,j): {843, 1966}, {1580, 18823}
X(52700) = X(i)-Dao conjugate of X(j) for these (i,j): {843, 9467}, {3978, 35087}, {18823, 39092}
X(52700) = crossdifference of every pair of points on line {385, 11183}
X(52700) = barycentric product X(i)*X(j) for these {i,j}: {543, 694}, {805, 8371}, {882, 9182}, {1916, 2502}, {9171, 18829}, {9468, 45809}, {17948, 18872}
X(52700) = barycentric quotient X(i)/X(j) for these {i,j}: {543, 3978}, {694, 18823}, {805, 9170}, {882, 9180}, {2502, 385}, {8371, 14295}, {9171, 804}, {9181, 17941}, {9182, 880}, {9468, 843}, {18872, 51226}, {45809, 14603}

---

X(52701) = X(2)X(52693)∩X(35146)X(51224)

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

X(52701) lies on the cubics K394 and K1308 and these lines: {2, 52693}, {35146, 51224}, {36897, 52694}

---

X(52702) = X(3)X(111)∩X(543)X(7697)

Barycentrics    a^2*(4*a^10*b^2 - 9*a^8*b^4 + 2*a^6*b^6 + 10*a^4*b^8 - 6*a^2*b^10 - b^12 + 4*a^10*c^2 - 31*a^8*b^2*c^2 + 38*a^6*b^4*c^2 - 20*a^4*b^6*c^2 + 35*a^2*b^8*c^2 + 2*b^10*c^2 - 9*a^8*c^4 + 38*a^6*b^2*c^4 - 33*a^4*b^4*c^4 - 18*a^2*b^6*c^4 - b^8*c^4 + 2*a^6*c^6 - 20*a^4*b^2*c^6 - 18*a^2*b^4*c^6 - 8*b^6*c^6 + 10*a^4*c^8 + 35*a^2*b^2*c^8 - b^4*c^8 - 6*a^2*c^10 + 2*b^2*c^10 - c^12) : :
X(52702) = X[11258] + 2 X[38651], X[38524] + 2 X[51535]

X(52702) lies on these lines: {3, 111}, {543, 7697}, {2854, 32447}, {3124, 37811}, {11842, 28662}

---

X(52703) = X(2)X(1990)∩X(3)X(6)

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

See Ivan Pavlov, euclid 5588.

X(52703) lies on the circumconic {{A,B,C,X(3),X(3545)}} and these lines: {2,1990}, {3,6}, {20,6749}, {23,16328}, {45,17102}, {51,26909}, {53,3091}, {157,19132}, {232,11284}, {233,5079}, {393,3090}, {441,47352}, {458,47383}, {465,49948}, {466,49947}, {546,42459}, {597,37188}, {599,40996}, {631,40138}, {648,37067}, {895,47406}, {1249,3525}, {1368,34288}, {1589,32788}, {1590,32787}, {1995,11062}, {2972,5646}, {3087,3529}, {3146,6748}, {3163,5054}, {3524,5702}, {3589,40680}, {3618,34828}, {3763,41005}, {3815,16051}, {5055,18487}, {5072,36412}, {5159,16303}, {5306,7494}, {5319,16197}, {5480,26870}, {6144,41008}, {6389,47355}, {6416,8908}, {6641,17809}, {6676,14836}, {7386,9300}, {7514,52166}, {7550,8743}, {7716,20897}, {7753,18536}, {8541,52277}, {8716,35928}, {8745,35500}, {9607,10996}, {9777,26907}, {10249,14634}, {10297,47322}, {14576,34818}, {14919,51544}, {15004,26865}, {15526,21358}, {15698,36427}, {16196,31492}, {16310,37637}, {16675,42018}, {17330,25876}, {17392,25932}, {17538,40065}, {17810,26898}, {17825,46832}, {18323,47275}, {19709,36430}, {20819,34817}, {21637,44200}, {33924,45141}, {38072,44231}, {40916,47228}, {41614,52437}, {44260,44522}, {44535,46262}

X(52703) = barycentric product X(3)*X(3545)
X(52703) = barycentric quotient X(3545)/X(264)
X(52703) = center of anticevian intersection conic of X(3)
X(52703) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 5158, 6}, {6, 15815, 5063}, {6, 216, 36751}, {6, 36751, 36748}, {6, 566, 5013}, {61, 62, 11426}, {216, 5158, 3}, {577, 15851, 6}, {577, 5158, 15860}, {5158, 15860, 15851}, {22236, 22238, 11425}

---

X(52704) = X(2)X(340)∩X(5)X(53)

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

See Ivan Pavlov, euclid 5588.

X(52704) lies on the circumconics {{A,B,C,X(5),X(5054)}}, {{A,B,C,X(53),X(43530)}} and these lines: {2,340}, {5,53}, {6,5070}, {382,10979}, {547,1990}, {566,39565}, {577,3526}, {631,10986}, {632,6749}, {1656,5158}, {3003,7603}, {3163,15699}, {3530,6748}, {3628,15860}, {3843,36751}, {5055,18487}, {6709,27377}, {7737,46216}, {7765,9722}, {11063,14537}, {31489,33871}, {40138,46936}

X(52704) = X(2167)-isoconjugate-of-X(14491)
X(52704) = barycentric product X(5)*X(5054)
X(52704) = barycentric quotient X(i)/X(j) for these (i, j): {51, 14491}, {5054, 95}
X(52704) = center of anticevian intersection conic of X(5)

---

X(52705) = X(1)X(6)∩X(2)X(1323)

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

See Ivan Pavlov, euclid 5588.

X(52705) lies on the circumconic {{A,B,C,X(44),X(15734)}} and these lines: {1,6}, {2,1323}, {10,11200}, {36,15288}, {101,30392}, {142,21314}, {165,5011}, {169,7987}, {214,52084}, {672,18421}, {1146,19875}, {1698,41006}, {2170,9819}, {2348,13384}, {3119,36835}, {3306,24635}, {3339,17451}, {3496,51576}, {3624,6554}, {3693,4915}, {3730,11531}, {3991,11519}, {4512,28125}, {4853,25082}, {4861,4936}, {4866,33299}, {4875,4882}, {4888,7960}, {4900,40779}, {5030,32578}, {5179,7988}, {5273,50114}, {5328,29571}, {5540,30282}, {5745,31326}, {6173,20121}, {7280,32561}, {8012,30350}, {9312,31269}, {9592,49758}, {9623,24036}, {13462,40131}, {17151,26059}, {19862,27541}, {21153,47621}, {25055,40869}, {25716,32008}, {25930,35595}, {30478,52528}, {31169,40719}, {34595,46835}, {35293,41798}

X(52705) = center of anticevian intersection conic of X(9)
X(52705) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {{9, 34522, 1}, {142, 42050, 21314}, {1212, 34522, 9}, {5239, 5240, 38316}}

---

X(52706) = X(2)X(44)∩X(6)X(3624)

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

See Ivan Pavlov, euclid 5588.

X(52706) lies on the circumconics {{A,B,C,X(10),X(25055)}}, [{A,B,C,X(37),X(89)}} and these lines: {1,50082}, {2,44}, {6,3624}, {8,50123}, {9,3336}, {10,37}, {45,1698}, {145,3723}, {192,28633}, {193,28640}, {391,16668}, {519,39260}, {536,29576}, {551,4969}, {573,9955}, {748,4290}, {756,4735}, {894,31248}, {966,1100}, {1125,4700}, {1266,4364}, {1268,17261}, {1573,8610}, {1654,28639}, {2325,3828}, {2345,46932}, {3247,4668}, {3622,50131}, {3626,50113}, {3632,16777}, {3634,17369}, {3636,3686}, {3661,4755}, {3679,4727}, {3707,19862}, {3720,4285}, {3739,4389}, {3741,39974}, {3753,21864}, {3763,36404}, {3922,21853}, {3962,21873}, {3995,27797}, {4004,21863}, {4007,16674}, {4273,17557}, {4357,31238}, {4370,25351}, {4384,41311}, {4416,6707}, {4423,37503}, {4472,50093}, {4480,10022}, {4678,17299}, {4687,17230}, {4690,16826}, {4698,5224}, {4701,17388}, {4702,48809}, {4718,4967}, {4725,29570}, {4739,17247}, {4751,17235}, {4908,16676}, {5109,27627}, {5251,19297}, {5296,16814}, {5550,37654}, {5750,15492}, {5816,18481}, {8040,37593}, {9780,17281}, {15254,19856}, {15668,17344}, {16669,17398}, {16670,34595}, {16815,17382}, {16831,17251}, {16832,17325}, {17023,25358}, {17229,27268}, {17252,17376}, {17259,17384}, {17260,17385}, {17271,29578}, {17277,25498}, {17278,26104}, {17308,41310}, {17322,17348}, {17326,17356}, {17327,17357}, {17346,29612}, {17351,28653}, {17359,29610}, {17360,29595}, {17372,29619}, {19742,41850}, {21699,22172}, {21904,29822}, {24357,31322}, {24589,39995}, {29569,50081}, {29586,50124}, {29593,51488}, {29596,31285}, {29599,48639}, {29624,50076}, {29825,37673}, {30598,37677}, {32431,33697}, {36479,49703}, {36480,49699}, {37499,41869}, {49701,51034}, {50084,51353}, {50115,51073}

X(52706) = complement of X(41847)
X(52706) = barycentric product X(i)*X(j) for these (i, j): {10, 25055}, {3952, 28220}
X(52706) = barycentric quotient X(i)/X(j) for these (i, j): {25055, 86}, {28220, 7192}
X(52706) = center of anticevian intersection conic of X(10)
X(52706) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 17250, 3834}, {2, 17256, 4670}, {2, 4708, 17237}, {2, 4748, 4675}, {10, 3986, 4029}, {10, 4029, 594}, {594, 3986, 37}, {1125, 17330, 16666}, {1213, 5257, 37}, {3679, 16672, 4727}, {3834, 17250, 17237}, {3834, 4708, 17250}, {4364, 24603, 4688}, {4690, 16826, 50125}, {4698, 5224, 17231}, {5296, 17303, 16814}, {16826, 31144, 4690}, {16831, 17251, 17374}, {25358, 49731, 17023}

---

X(52707) = X(20)X(1249)∩X(1990)X(3543)

Barycentrics    (13*a^4-11*(b^2-c^2)^2-2*a^2*(b^2+c^2))*(3*a^4-(b^2-c^2)^2-2*a^2*(b^2+c^2)) : :

See Ivan Pavlov, euclid 5588.

X(52707) lies on the circumconic {{A,B,C,X(20),X(50687)}} and these lines: {20,1249}, {1033,35497}, {1990,3543}, {3091,15860}, {3163,10304}, {3839,5702}, {5158,46935}, {5304,40350}, {17578,33630}, {38292,50692}

X(52707) = barycentric product X(20)*X(50687)
X(52707) = barycentric quotient X(50687)/X(253)
X(52707) = center of anticevian intersection conic of X(20)

---

X(52708) = X(2)X(3761)∩X(10)X(37)

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

See Ivan Pavlov, euclid 5588.

X(52708) lies on the circumconic {{A,B,C,X(37),X(36871)}} and these lines: {2,3761}, {10,37}, {39,51073}, {76,4751}, {551,1573}, {1015,19883}, {1107,19862}, {1125,16975}, {1212,25089}, {1698,17756}, {2275,19878}, {2276,3828}, {3290,48853}, {3634,5283}, {3879,24944}, {4035,29600}, {4044,31025}, {4084,21879}, {4129,22222}, {4699,20888}, {4850,24603}, {5179,7951}, {5251,37675}, {5299,17536}, {5316,5718}, {7031,16859}, {8074,25072}, {15808,17448}, {16828,27040}, {16885,17750}, {17275,39983}, {21816,21951}, {25280,32009}, {27784,41015}, {29594,44307}, {30950,45751}

X(52708) = center of anticevian intersection conic of X(37)
X(52708) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1500, 25614, 10}

---

X(52709) = X(2)X(1266)∩X(7)X(8)

Barycentrics    -a^2+b^2-10*b*c+c^2 : :

See Ivan Pavlov, euclid 5589.

X(52709) lies on the circumconics {{A,B,C,X(7),X(39963)}}, {{A,B,C,X(8),X(4915)}}, {{A,B,C,X(65),X(4731)}}, {{A,B,C,X(85),X(36588)}} and these lines: {1,50108}, {2,1266}, {7,8}, {10,4346}, {142,4461}, {144,3707}, {190,31722}, {200,41929}, {239,5032}, {329,20879}, {346,24199}, {536,5308}, {597,4363}, {894,4402}, {966,4739}, {1086,21358}, {1125,3672}, {1219,7185}, {1278,29569}, {1443,3872}, {1698,3663}, {1738,5772}, {2345,7263}, {3241,17160}, {3244,3945}, {3616,5695}, {3617,17274}, {3621,50099}, {3622,4452}, {3623,3875}, {3632,4896}, {3633,3664}, {3679,4887}, {3686,20059}, {3729,18230}, {3879,20014}, {3943,31139}, {4000,7229}, {4054,5328}, {4307,50017}, {4357,4373}, {4359,18228}, {4361,7222}, {4371,17365}, {4384,4454}, {4389,9780}, {4405,5839}, {4419,4688}, {4431,4869}, {4440,31349}, {4441,30947}, {4470,17301}, {4479,26103}, {4488,17277}, {4644,4969}, {4647,11037}, {4648,4686}, {4665,50991}, {4675,4727}, {4691,17272}, {4699,5296}, {4702,38314}, {4726,17314}, {4738,45116}, {4740,17316}, {4747,16834}, {4748,49747}, {4772,17257}, {4821,26806}, {4847,36595}, {4862,5232}, {4980,34255}, {5231,22464}, {5271,28610}, {5550,17320}, {5749,17116}, {5815,28612}, {5905,41915}, {6173,29616}, {9776,28605}, {15956,50044}, {16815,20073}, {16832,17132}, {17253,28635}, {17262,31285}, {17270,45789}, {17281,31243}, {17292,48627}, {17318,49733}, {17354,31189}, {17378,20050}, {17382,26039}, {19789,29833}, {20053,50088}, {24599,50127}, {24715,48849}, {28313,29602}, {29627,50107}, {31183,50118}, {36479,51060}

X(52709) = anticomplement of X(16676)
X(52709) = barycentric product X(i)*X(j) for these (i, j): {85, 4915}, {274, 4731}
X(52709) = barycentric quotient X(i)/X(j) for these (i, j): {4731, 37}, {4915, 9}
X(52709) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 32087, 32099}, {7, 75, 32087}, {8, 21296, 17360}, {8, 31995, 42697}, {8, 42697, 7}, {75, 31995, 7}, {75, 42697, 8}, {75, 7321, 42696}, {3945, 17151, 4460}, {4000, 17118, 7229}, {4384, 4454, 6172}, {4384, 50119, 4454}, {7321, 21296, 7}, {7321, 42696, 21296}, {17119, 49727, 4644}, {17360, 42696, 8}

---

X(52710) = X(4)X(69)∩X(53)X(3620)

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

See Ivan Pavlov, euclid 5589.

X(52710) lies on the circumconics {{A,B,C,X(2),X(44133)}}, {{A,B,C,X(4),X(1597)}}, {{A,B,C,X(76),X(32836)}} and these lines: {2,1990}, {4,69}, {25,41927}, {53,3620}, {95,10299}, {99,35483}, {140,40680}, {141,43981}, {183,4232}, {193,6749}, {253,5056}, {273,42696}, {297,21356}, {318,42697}, {325,52284}, {378,52437}, {381,40996}, {393,3619}, {458,1992}, {468,34229}, {599,37174}, {648,5702}, {1007,5094}, {1249,31886}, {1273,8889}, {1494,3545}, {1585,32811}, {1586,32810}, {1596,46951}, {1597,32836}, {1656,41005}, {1657,41008}, {3087,11008}, {3516,9723}, {3522,20477}, {3523,6527}, {3618,9308}, {3785,37458}, {3851,40995}, {3964,32824}, {6748,20080}, {6995,37671}, {7378,7788}, {11433,40684}, {14920,48540}, {15589,52301}, {18489,34334}, {21735,46724}, {30737,46336}, {32825,52347}, {32885,37942}, {34803,35520}, {35477,44180}, {40132,47204}, {50990,52282}, {50992,52281}

X(52710) = X(48)-isoconjugate-of-X(52187)
X(52710) = barycentric product X(i)*X(j) for these (i, j): {4, 32836}, {76, 1597}, {18022, 33871}
X(52710) = barycentric quotient X(i)/X(j) for these (i, j): {4, 52187}, {1597, 6}, {32836, 69}, {33871, 184}
X(52710) = anticomplement of center of anticevian intersection conic of X(3)
X(52710) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 32000, 44134}, {4, 44134, 69}, {264, 1235, 44135}, {264, 32000, 69}, {264, 44134, 4}, {9308, 52289, 40138}, {40138, 52289, 3618}

---

X(52711) = X(2)X(5702)∩X(20)X(64)

Barycentrics    -9*a^8+16*a^6*(b^2+c^2)-24*a^2*(b^2-c^2)^2*(b^2+c^2)+a^4*(6*b^4-44*b^2*c^2+6*c^4)+(b^2-c^2)^2*(11*b^4+26*b^2*c^2+11*c^4) : :

See Ivan Pavlov, euclid 5589.

X(52711) lies on these lines: {2,5702}, {20,64}, {340,3543}, {524,11348}, {631,40995}, {858,10513}, {1494,10304}, {3091,44134}, {3832,32000}, {3964,35497}, {7802,38440}, {15640,36889}, {15717,35510}, {17578,32001}, {20213,41914}, {20218,42459}, {21734,41008}, {30769,37668}, {35493,52437}

X(52711) = anticomplement of X(5702)
X(52711) = X(14490)-anticomplementary conjugate of X(5905)

---

X(52712) = X(2)X(340)∩X(3)X(95)

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

See Ivan Pavlov, euclid 5589.

X(52712) lies on these lines: {2,340}, {3,95}, {69,575}, {76,7550}, {140,40996}, {183,11059}, {317,3090}, {631,44134}, {632,45198}, {648,37067}, {1078,44133}, {1494,5054}, {3091,32002}, {3260,7771}, {3628,41008}, {5079,40410}, {7782,44135}, {11423,19166}, {14767,43980}, {14869,41005}, {15719,36889}, {34229,37803}, {35475,44131}, {37871,46717}, {41760,50660}

X(52712) = anticomplement of center of anticevian intersection conic of X(5)

---

X(52713) = X(2)X(2418)∩X(4)X(69)

Barycentrics    -a^4+b^4-10*b^2*c^2+c^4 : :

See Ivan Pavlov, euclid 5589.

X(52713) lies on the circumconic {{A,B,C,X(4),X(11284)}} and these lines: {2,2418}, {3,32822}, {4,69}, {5,32818}, {30,15589}, {39,32957}, {99,3524}, {115,33285}, {141,33190}, {148,16990}, {183,376}, {193,8370}, {194,32968}, {230,33191}, {274,17559}, {310,6822}, {325,3545}, {338,36883}, {350,1056}, {381,32869}, {385,1285}, {388,3760}, {443,18135}, {497,3761}, {538,7736}, {543,42850}, {549,32893}, {599,7620}, {625,7615}, {626,33292}, {631,1975}, {632,32897}, {671,19662}, {1003,37667}, {1007,5071}, {1058,1909}, {1078,3528}, {1350,46034}, {1656,32831}, {1995,22241}, {2481,21290}, {2549,9466}, {2550,6381}, {2551,20888}, {2896,33238}, {2996,6656}, {3090,3926}, {3091,3933}, {3096,33232}, {3146,7767}, {3314,16041}, {3421,4441}, {3523,32872}, {3525,6337}, {3526,32870}, {3529,3785}, {3533,32824}, {3544,7752}, {3589,5286}, {3593,7388}, {3595,7389}, {3618,42852}, {3619,7790}, {3620,7841}, {3628,32835}, {3673,24993}, {3734,7735}, {3763,5254}, {3767,7820}, {3788,32958}, {3793,9740}, {3830,14929}, {3832,7776}, {3839,10513}, {3855,32816}, {3934,7738}, {4385,32087}, {4576,44833}, {5032,18842}, {5056,32840}, {5067,7763}, {5068,32880}, {5084,34284}, {5088,28808}, {5232,17677}, {5304,11286}, {5305,33198}, {5475,14711}, {5523,37638}, {5976,43532}, {6144,7745}, {6392,7770}, {6804,26166}, {6821,18152}, {7283,52422}, {7375,32807}, {7386,39998}, {7391,41916}, {7392,8024}, {7486,32841}, {7550,9723}, {7610,32459}, {7618,15301}, {7737,17131}, {7746,32959}, {7750,32826}, {7754,32971}, {7762,32979}, {7771,19708}, {7773,32878}, {7774,32983}, {7779,33016}, {7783,32978}, {7788,32827}, {7789,33189}, {7793,33239}, {7795,7844}, {7799,34803}, {7801,31275}, {7802,11541}, {7806,14001}, {7810,43619}, {7812,11008}, {7813,9770}, {7828,32952}, {7832,32953}, {7835,33231}, {7836,32969}, {7839,33269}, {7851,33194}, {7857,33236}, {7868,33196}, {7879,32982}, {7881,32972}, {7891,32977}, {7893,14068}, {7897,33006}, {7899,39143}, {7904,33247}, {7906,32962}, {7929,33279}, {7931,14064}, {7939,32996}, {7941,32995}, {7945,50570}, {7947,32963}, {8369,37689}, {8591,47061}, {8744,41614}, {10299,32886}, {10996,41009}, {11001,14907}, {11064,52288}, {11111,37670}, {11160,11317}, {11174,14482}, {13881,32955}, {14035,17129}, {14360,46336}, {14928,25406}, {15428,37182}, {15533,23334}, {15574,37925}, {15682,37671}, {15709,32885}, {16043,31276}, {16924,20081}, {16989,19570}, {17004,33216}, {17008,32985}, {17582,18140}, {18841,43681}, {20094,33008}, {20105,33020}, {22253,37665}, {23053,41134}, {27269,33026}, {30737,35513}, {32820,32829}, {32825,32877}, {32837,37647}, {33221,46226}, {36874,52145}, {37664,50741}

X(52713) = anticomplement of X(5024)
X(52713) = X(18842)-anticomplementary conjugate of X(8)
X(52713) = barycentric product X(76)*X(11284)
X(52713) = barycentric quotient X(11284)/X(6)
X(52713) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 32830, 32818}, {69, 11185, 4}, {69, 32006, 7850}, {76, 11185, 69}, {99, 34229, 3524}, {141, 34505, 43448}, {141, 43448, 33190}, {148, 16990, 32986}, {183, 32815, 376}, {385, 14033, 1285}, {1003, 37667, 46453}, {1975, 32828, 631}, {3091, 3933, 32823}, {3619, 7790, 33230}, {3734, 7735, 14039}, {3785, 32819, 3529}, {3934, 7738, 32960}, {6337, 32832, 3525}, {7750, 32826, 33703}, {7801, 43620, 37690}, {7813, 31415, 9770}, {7891, 32977, 39142}, {32815, 46951, 183}

---

X(52714) = X(2)X(4480)∩X(7)X(145)

Barycentrics    -9*a^2+11*b^2-26*b*c+11*c^2+2*a*(b+c) : :

See Ivan Pavlov, euclid 5589.

X(52714) lies on these lines: {2,4480}, {7,145}, {10,4902}, {320,31145}, {1996,51351}, {3616,4862}, {3617,17271}, {3622,4346}, {3623,4896}, {3834,4454}, {4440,29621}, {4678,31995}, {4888,20057}, {5905,24184}, {7228,26104}, {7321,17250}, {17274,46933}, {17321,39707}, {20052,21296}, {20072,24599}, {30424,39567}, {46934,50116}

X(52714) = anticomplement of X(31722)
X(52714) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 4373, 32093}, {4373, 32093, 32105}

---

X(52715) = X(2)X(1323)∩X(7)X(8)

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

See Ivan Pavlov, euclid 5589.

X(52715) lies on these lines: {2,1323}, {7,8}, {10,21314}, {57,41926}, {279,9780}, {348,19877}, {664,31721}, {3160,5550}, {3241,40719}, {3598,50310}, {3616,9312}, {3617,10481}, {3665,27818}, {3679,20121}, {4054,29616}, {5195,9812}, {5219,29627}, {5261,52563}, {5435,7223}, {5556,33867}, {5726,39570}, {7181,31188}, {20050,25719}, {20057,25718}, {20195,41006}, {21454,50095}

X(52715) = anticomplement of center of anticevian intersection conic of X(9)
X(52715) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 32086, 32098}, {8, 85, 32086}, {85, 31994, 8}, {3160, 52422, 5550}

---

X(52716) = X(1)X(75)∩X(2)X(3761)

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

See Ivan Pavlov, euclid 5589.

X(52716) lies on the circumconic {{A,B,C,X(86),X(36871)}} and these lines: {1,75}, {2,3761}, {10,25278}, {36,16992}, {76,3624}, {85,3361}, {194,27268}, {239,14996}, {321,29597}, {330,16819}, {350,25055}, {551,4441}, {609,16998}, {668,19875}, {870,2163}, {1107,31238}, {1125,3760}, {1150,3306}, {1698,1909}, {1975,5259}, {3230,4363}, {3263,48854}, {3294,25728}, {3501,29383}, {3616,20888}, {3632,25303}, {3739,16975}, {4358,16831}, {4359,16834}, {4361,16971}, {4472,4494}, {4479,51110}, {4671,16826}, {4681,25130}, {4692,30758}, {4696,25585}, {4698,5283}, {4699,16829}, {4704,25264}, {4968,33942}, {5088,5144}, {5299,11321}, {5313,37632}, {7031,16915}, {7176,16817}, {7278,28611}, {9331,17759}, {9336,26801}, {9534,32099}, {16552,21371}, {16784,20172}, {16827,17120}, {16833,19804}, {16887,31339}, {18135,19862}, {18140,34595}, {19871,28653}, {20036,30712}, {20568,39044}, {20913,29603}, {20923,31312}, {25392,25694}, {26035,30110}, {26234,37817}, {27020,27318}, {28605,29580}, {29827,31002}

X(52716) = anticomplement of center of anticevian intersection conic of X(37)
X(52716) = barycentric product X(668)*X(48577)
X(52716) = barycentric quotient X(48577)/X(513)
X(52716) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 274, 32092}, {1, 32092, 32104}, {274, 31997, 1}

---

X(52717) = (name pending)

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

See Ivan Pavlov, euclid 5594.

X(52717) lies on the circumconics {{A,B,C,X(2),X(3618)}}, {{A,B,C,X(4),X(632)}}, {{A,B,C,X(6),X(11482)}}, {{A,B,C,X(17),X(11489)}}, {{A,B,C,X(18),X(11488)}}, {{A,B,C,X(66),X(14494)}}, {{A,B,C,X(69),X(7608)} and this line: {632,3618}

---

X(52718) = X(2)X(3933)∩X(99)X(631)

Barycentrics    7*a^4+5*b^4-18*b^2*c^2+5*c^4-12*a^2*(b^2+c^2) : :

See Ivan Pavlov, euclid 5594.

X(52718) lies on these lines: {2,3933}, {3,32870}, {4,32838}, {5,32897}, {69,15520}, {76,3533}, {83,23055}, {99,631}, {141,32958}, {183,5067}, {230,32957}, {302,43446}, {303,43447}, {315,3090}, {325,32883}, {632,32830}, {1078,3545}, {1235,52290}, {1285,32987}, {1975,15702}, {3054,32959}, {3525,32817}, {3526,32834}, {3544,7750}, {3628,15589}, {3785,5071}, {3934,33189}, {5070,37668}, {5461,7815}, {5485,7783}, {6337,15709}, {6390,32872}, {6680,16045}, {7486,7767}, {7746,32956}, {7749,33191}, {7771,33703}, {7776,46936}, {7782,15719}, {7806,18841}, {7831,39143}, {7867,32955}, {10124,32874}, {10299,11185}, {11539,32893}, {13468,31404}, {14069,37637}, {15271,32951}, {16239,32835}, {16508,33215}, {16924,46453}, {16990,32976}, {17004,32968}, {17006,32970}, {17008,32975}, {19708,32826}, {23053,33197}, {31276,32977}, {32001,52296}, {32820,32886}, {32821,32884}, {32831,46219}

X(52718) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {183, 32867, 5067}, {183, 5067, 32823}, {32838, 37688, 4}

---

X(52719) = X(6)X(3917)∩X(25)X(575)

Barycentrics    a^2*(7*a^4+5*b^4-18*b^2*c^2+5*c^4-12*a^2*(b^2+c^2)) : :

See Ivan Pavlov, euclid 5594.

X(52719) lies on these lines: {3,13421}, {6,3917}, {22,5050}, {25,575}, {110,5020}, {394,15516}, {597,7539}, {1351,15246}, {1899,6329}, {1993,33879}, {2056,21448}, {3167,15018}, {3292,14924}, {3526,41628}, {3527,37925}, {5012,20850}, {5085,34565}, {6090,10601}, {6515,51732}, {6997,50979}, {7391,14848}, {7485,11482}, {7529,36153}, {9818,34783}, {9909,15019}, {10541,21969}, {11245,46442}, {11284,13366}, {11423,11484}, {11432,35921}, {13595,26864}, {15029,19709}, {15038,44457}, {15118,25335}, {17811,44111}, {21312,36752}, {22112,34566}, {33586,50664}, {45311,52298}

X(52719) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 5422, 5644}, {5050, 34545, 9777}

---

X(52720) = X(2)X(525)∩X(122)X(125)

Barycentrics    (2*a^8-2*(b^2+c^2)*a^6-(3*b^4-8*b^2*c^2+3*c^4)*a^4+4*(b^4-c^4)*(b^2-c^2)*a^2-(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^2)*(-a^2+b^2+c^2)*(b^2-c^2) : :
X(52720) = 2 X[2] + X[23616], X[2] + 2 X[38240], 5 X[2] - X[45292], X[14401] + 2 X[34767], X[14401] + 4 X[38240], 5 X[14401] - 2 X[45292], X[23616] - 4 X[38240], 5 X[23616] + 2 X[45292], 5 X[34767] + X[45292], 10 X[38240] + X[45292], 2 X[3268] + X[14391]

X(52720) lies on the cubic K1310 and these lines: {2, 525}, {122, 125}, {520, 5650}, {599, 9007}, {1494, 42307}, {1636, 41145}, {1641, 39474}, {1651, 42733}, {3265, 37638}, {3268, 14391}, {5489, 22264}, {11049, 47071}, {11050, 18556}, {11472, 32120}, {15066, 32320}, {39062, 42308}

X(52720) = midpoint of X(i) and X(j) for these {i,j}: {2, 34767}, {14401, 23616}
X(52720) = reflection of X(i) in X(j) for these {i,j}: {14401, 2}, {23616, 34767}, {34767, 38240}, {47071, 11049}
X(52720) = isotomic conjugate of the polar conjugate of X(42733)
X(52720) = tripolar centroid for these (i,j): {1494, 16076}
X(52720) = X(16075)-isoconjugate of X(36131)
X(52720) = X(i)-Dao conjugate of X(j) for these (i,j): {14401, 47071}, {16075, 39008}
X(52720) = crossdifference of every pair of points on line {112, 1495}
X(52720) = barycentric product X(i)*X(j) for these {i,j}: {69, 42733}, {1651, 34767}, {3265, 47204}, {9033, 16076}
X(52720) = barycentric quotient X(i)/X(j) for these {i,j}: {1650, 47071}, {1651, 4240}, {9033, 16075}, {14380, 41433}, {16076, 16077}, {42733, 4}, {47204, 107}
X(52720) = {X(2),X(38240)}-harmonic conjugate of X(23616)

---

X(52721) = X(2)X(512)∩X(351)X(865)

Barycentrics    a^2*(b^2 - c^2)*(a^4*b^4 - 4*a^4*b^2*c^2 + 2*a^2*b^4*c^2 + a^4*c^4 + 2*a^2*b^2*c^4 - 2*b^4*c^4)::
X(52721) = 2 X[2] + X[23610], X[2] + 2 X[38237], 5 X[2] - X[44007], X[23610] - 4 X[38237], 5 X[23610] + 2 X[44007], 10 X[38237] + X[44007], 4 X[11176] - X[14406], 5 X[14061] + 4 X[38017]

X(52721) lies on the cubic K1310 and these lines: {2, 512}, {351, 865}, {511, 11633}, {597, 9009}, {881, 17414}, {3221, 5640}, {9172, 45680}, {9402, 27811}, {11176, 14406}, {14061, 38017}, {20965, 38366}

X(52721) = tripolar centroid of X(3228)
X(52721) = crossdifference of every pair of points on line {99, 3231}
X(52721) = {X(2),X(38237)}-harmonic conjugate of X(23610)

---

X(52722) = X(2)X(1341)∩X(351)X(690)

Barycentrics    (b^2 - c^2)*(-2*a^2 + b^2 + c^2)*(-2*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(52722) = 5 X[2] - X[45297], X[13722] + 2 X[30509], 5 X[13722] - 2 X[45297], 5 X[30509] + X[45297]

X(52722) lies on the Steiner major axis, the cubic K1310, and these lines: {2, 1341}, {351, 690}, {523, 13636}, {524, 46463}, {599, 41880}, {1316, 6142}, {1380, 2770}, {3414, 9168}, {4590, 6190}, {5108, 6141}, {5466, 22245}, {5639, 5652}, {6189, 18823}, {9164, 39022}, {34312, 51826}, {35511, 39366}

X(52722) = midpoint of X(2) and X(30509)
X(52722) = reflection of X(13722) in X(2)
X(52722) = tripolar centroid of X(6189)
X(52722) = X(22245)-Ceva conjugate of X(39023)
X(52722) = X(46463)-cross conjugate of X(690)
X(52722) = X(i)-isoconjugate of X(j) for these (i,j): {897, 1379}, {923, 6190}, {3414, 36142}, {5639, 36085}
X(52722) = X(i)-Dao conjugate of X(j) for these (i,j): {111, 39068}, {671, 39023}, {690, 46463}, {1379, 6593}, {1649, 13722}, {2482, 6190}, {3414, 23992}, {5466, 13722}, {5639, 38988}
X(52722) = cevapoint of X(690) and X(46463)
X(52722) = crosspoint of X(6190) and X(22244)
X(52722) = crosssum of X(i) and X(j) for these (i,j): {5638, 22242}, {14899, 35607}
X(52722) = trilinear pole of line {690, 46462}
X(52722) = crossdifference of every pair of points on line {111, 1379}
X(52722) = X(5652)-line conjugate of X(5639)
X(52722) = barycentric product X(i)*X(j) for these {i,j}: {99, 46462}, {524, 3413}, {690, 6189}, {1380, 35522}, {3266, 5638}, {5468, 13636}
X(52722) = barycentric quotient X(i)/X(j) for these {i,j}: {187, 1379}, {351, 5639}, {524, 6190}, {690, 3414}, {1380, 691}, {1648, 13722}, {3413, 671}, {5638, 111}, {6189, 892}, {13636, 5466}, {23992, 46463}, {46462, 523}, {46463, 39022}

---

X(52723) = X(2)X(1340)∩X(351)X(690)

Barycentrics    (b^2 - c^2)*(-2*a^2 + b^2 + c^2)*(-2*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(52723) = 5 X[2] - X[45296], X[13636] + 2 X[30508], 5 X[13636] - 2 X[45296], 5 X[30508] + X[45296]

X(52723) lies on the Steiner minor axis, the cubic K1310, and these lines: {2, 1340}, {351, 690}, {523, 13722}, {524, 46462}, {599, 41881}, {1316, 6141}, {1379, 2770}, {3413, 9168}, {4590, 6189}, {5108, 6142}, {5466, 22244}, {5638, 5652}, {6190, 18823}, {9164, 39023}, {34312, 51825}, {35511, 39365}

X(52723) = midpoint of X(2) and X(30508)
X(52723) = reflection of X(13636) in X(2)
X(52723) = tripolar centroid of X(6190)
X(52723) = X(22244)-Ceva conjugate of X(39022)
X(52723) = X(46462)-cross conjugate of X(690)
X(52723) = X(i)-isoconjugate of X(j) for these (i,j): {897, 1380}, {923, 6189}, {3413, 36142}, {5638, 36085}
X(52723) = X(i)-Dao conjugate of X(j) for these (i,j): {111, 39067}, {671, 39022}, {690, 46462}, {1380, 6593}, {1649, 13636}, {2482, 6189}, {3413, 23992}, {5466, 13636}, {5638, 38988}
X(52723) = cevapoint of X(690) and X(46462)
X(52723) = crosspoint of X(6189) and X(22245)
X(52723) = crosssum of X(i) and X(j) for these (i,j): {5639, 22243}, {35608, 35609}
X(52723) = trilinear pole of line {690, 46463}
X(52723) = crossdifference of every pair of points on line {111, 1380}
X(52723) = X(5652)-line conjugate of X(5638)
X(52723) = barycentric product X(i)*X(j) for these {i,j}: {99, 46463}, {524, 3414}, {690, 6190}, {1379, 35522}, {3266, 5639}, {5468, 13722}
X(52723) = barycentric quotient X(i)/X(j) for these {i,j}: {187, 1380}, {351, 5638}, {524, 6189}, {690, 3413}, {1379, 691}, {1648, 13636}, {3414, 671}, {5639, 111}, {6190, 892}, {13722, 5466}, {23992, 46462}, {46462, 39023}, {46463, 523}

---

X(52724) = X(2)X(900)∩X(519)X(45684)

Barycentrics    (2*a - b - c)*(b - c)*(2*a^4 - 4*a^3*b - 6*a^2*b^2 + 8*a*b^3 - b^4 - 4*a^3*c + 24*a^2*b*c - 12*a*b^2*c - 4*b^3*c - 6*a^2*c^2 - 12*a*b*c^2 + 12*b^2*c^2 + 8*a*c^3 - 4*b*c^3 - c^4)::
X(52724) = 5 X[2] - X[45295], 2 X[2] + X[46050], 5 X[34764] + X[45295], 2 X[45295] + 5 X[46050]

X(52724) lies on the cubic K1310 and these lines: {2, 900}, {519, 45684}, {1644, 33922}, {1647, 2087}, {6550, 14475}, {9271, 17780}

X(52724) = midpoint of X(i) and X(j) for these {i,j}: {2, 34764}, {1647, 6544}
X(52724) = eflection of X(46050) in X(34764)
X(52724) = tripolar centroid of X(35168)
X(52724) = crossdifference of every pair of points on line {901, 8649}

---

X(52725) = X(2)X(3)∩X(8182)X(33900)

Barycentrics    a^2 (a^8-24 a^4 b^2 c^2-4 a^6 (b^2+c^2)+a^2 (4 b^6+21 b^4 c^2+21 b^2 c^4+4 c^6)-b^8+11 b^6 c^2-12 b^4 c^4+11 b^2 c^6-c^8) : :

See Kadir Altintas and Angel Montesdeoca, euclid 5598.

X(52725) lies on these lines: {2,3}, {8182,33900}, {8722,33962}

X(52725) = reflection of X(10204) in X(3)

  Clawson circles: X(52726) - X(52740)  

This preamble and centers X(52726)-X(52740) were contributed by César Eliud Lozada, January 4, 2023.

Let 𝓁₁, 𝓁₂, 𝓁₃, 𝓁₄ be four lines, no three of them being concurrent. Let T₁ be the triangle bounded by lines 𝓁₂, 𝓁₃, 𝓁₄, and denote T₂, T₃, T₄ similarly. Then the circumcenters of these four triangles lie on a circle Ω, here named the Clawson circle of the given lines. Moreover, the circumcircles of the four triangles have a common point 𝒬. (J. W. Clawson, #2898, The American Mathematical Monthly, Vol. 29, No. 5 (May, 1922), pp. 230-231.)

The center of the circle Ω is the point QL-P5 in the Encyclopedia of Quadri-Figures.

In this section, three of the lines are the sidelines of a reference triangle ABC and the fourth line is given. Assume this last line is the polar trilinear of P = x : y : z, then the center of Ω is given by:

  Ω_o(P) = a^2*(x*(y^2-z^2)*(-a^2+b^2+c^2)+y*z*(-2*(b^2-c^2)*x+(a^2-c^2)*y-(a^2-b^2)*z)-(-a^2+b^2+2*c^2)*x^2*y+(-a^2+2*b^2+c^2)*z*x^2) : :

and the point 𝒬, obviously on the circumcircle of ABC, is:

  𝒬(P) = a^2*(x-y)*(x-z) : :, which corresponds to the isogonal conjugate of - the infinite point of - the tripolar of - the isotomic conjugate of P.

The appearance of (i, j) in the following list means that Ω_o(X(i)) = X(j):

(1, 52726), (3, 52737), (4, 44810), (5, 52738), (6, 14270), (7, 52730), (8, 52739), (9, 52740), (10, 44827), (25, 25644), (69, 8552), (76, 44826), (94, 14809), (107, 52734), (110, 52727), (111, 44821), (190, 52732), (278, 44807), (344, 44805), (385, 39495), (393, 44818), (468, 33752), (476, 52729), (524, 44814), (648, 7740), (651, 52731), (658, 52733), (694, 52728), (930, 52736), (1992, 9126), (1993, 44808), (1994, 44809), (1995, 44806), (2394, 12041), (2395, 12042), (2396, 33813), (2397, 33814), (2398, 38599), (2399, 38600), (2400, 38601), (2401, 38602), (2402, 38603), (2403, 38604), (2404, 38605), (2405, 38606), (2406, 38607), (2407, 1511), (2408, 14650), (2409, 38608), (2410, 38609), (2411, 38610), (2413, 38618), (2414, 38619), (2415, 38620), (2416, 38621), (2417, 38622), (2418, 38623), (2419, 38624), (3266, 44822), (4232, 44820), (5968, 11616), (6353, 44817), (6515, 44816), (8791, 18312), (9214, 9126), (10015, 52735), (11059, 11616), (16092, 9175), (33116, 44812), (35910, 44826), (37638, 14314), (37779, 8562), (46786, 44821), (46787, 44822), (46788, 14809), (50941, 38611), (50942, 38613), (50943, 38617), (50944, 35231), (50945, 35232), (51228, 44814), (52141, 9175)

X(52726) = CENTER OF THE CLAWSON CIRCLE OF ABC + ANTIORTHIC AXIS {X(44), X(513)}

Barycentrics    a^2*(b-c)*(a^4-(b+c)*a^3-(b-c)^2*a^2+(b^2-c^2)*(b-c)*a+b^2*c^2) : :
X(52726) = X(4091)+3*X(8643)

X(52726) lies on these lines: {1, 8648}, {3, 3887}, {35, 4895}, {36, 2254}, {56, 3960}, {101, 34905}, {514, 659}, {521, 39225}, {522, 34948}, {900, 38602}, {926, 38599}, {928, 1960}, {993, 3716}, {995, 22384}, {1946, 48294}, {2804, 44812}, {2814, 11249}, {2821, 52730}, {2827, 22775}, {2975, 3762}, {3309, 39476}, {3667, 23224}, {3737, 14299}, {3738, 39200}, {3746, 23057}, {3900, 39227}, {4057, 23187}, {4091, 8643}, {5258, 14430}, {5563, 14413}, {6003, 39199}, {6362, 44811}, {6366, 44805}, {8674, 39478}, {15313, 39226}, {22091, 48018}, {23226, 48307}, {25569, 42670}, {25901, 48218}, {35057, 48383}, {38469, 39480}, {39577, 48322}, {42325, 44408}

X(52726) = midpoint of X(4057) and X(23187)
X(52726) = reflection of X(i) in X(j) for these (i, j): (39210, 34948), (44827, 52739), (48386, 39227)
X(52726) = X(21)-beth conjugate of-X(8648)

---

X(52727) = CENTER OF THE CLAWSON CIRCLE OF ABC + BROCARD AXIS {X(3), X(6)}

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

X(52727) lies on these lines: {3, 66}, {50, 11674}, {147, 19165}, {237, 5967}, {804, 11616}, {2453, 37991}, {6132, 52128}, {9744, 16320}, {41172, 44668}

---

X(52728) = CENTER OF THE CLAWSON CIRCLE OF ABC + BROCARD LINE {X(39), X(512)}

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

X(52728) lies on these lines: {3, 512}, {237, 11182}, {549, 5926}, {669, 11183}, {690, 9420}, {804, 12042}, {5201, 9171}, {6292, 9494}, {7638, 39469}, {8266, 22260}, {11328, 45692}, {23878, 44823}, {30217, 44680}, {34290, 37184}, {37446, 39503}

X(52728) = reflection of X(9494) in X(39481)
X(52728) = {X(669), X(14096)}-harmonic conjugate of X(11183)

---

X(52729) = CENTER OF THE CLAWSON CIRCLE OF ABC + FERMAT AXIS {X(6), X(13)}

Barycentrics    a^2*(2*a^20-9*(b^2+c^2)*a^18+(17*b^4+32*b^2*c^2+17*c^4)*a^16-(b^2+c^2)*(19*b^4+27*b^2*c^2+19*c^4)*a^14+2*(7*b^8+7*c^8+(21*b^4+19*b^2*c^2+21*c^4)*b^2*c^2)*a^12-(b^2+c^2)*(b^8+c^8+(36*b^4-23*b^2*c^2+36*c^4)*b^2*c^2)*a^10-(16*b^12+16*c^12-(38*b^8+38*c^8-(b^4+4*b^2*c^2+c^4)*b^2*c^2)*b^2*c^2)*a^8+(b^2+c^2)*(23*b^12+23*c^12-(65*b^8+65*c^8-(91*b^4-101*b^2*c^2+91*c^4)*b^2*c^2)*b^2*c^2)*a^6-(b^2-c^2)^2*(16*b^12+16*c^12+(15*b^4+4*b^2*c^2+15*c^4)*b^4*c^4)*a^4+(b^4-c^4)*(b^2-c^2)*(6*b^12+6*c^12-(4*b^8+4*c^8-(b^4+3*b^2*c^2+c^4)*b^2*c^2)*b^2*c^2)*a^2-(b^2-c^2)^4*(b^12+c^12+(2*b^4+2*c^4-(b^2-c^2)*b*c)*(2*b^4+2*c^4+(b^2-c^2)*b*c)*b^2*c^2)) : :

X(52729) lies on these lines: {6, 74}, {9175, 20403}, {12042, 14356}, {24975, 46981}

---

X(52730) = CENTER OF THE CLAWSON CIRCLE OF ABC + GERGONNE LINE {X(241), X(514)}

Barycentrics    a^2*(b-c)*(2*a^4-(b+c)*a^3-3*(b^2+c^2)*a^2+(b+c)^3*a+b^4-2*(b^2+c^2)*b*c+c^4) : :
X(52730) = 3*X(4091)+X(4105)

X(52730) lies on these lines: {3, 928}, {512, 39476}, {513, 44805}, {676, 37582}, {926, 38601}, {1155, 10015}, {2821, 52726}, {3336, 30691}, {3579, 6366}, {3916, 50333}, {3960, 6139}, {4091, 4105}, {4973, 48286}, {8677, 38607}, {9521, 37623}, {22080, 32679}, {24047, 52614}, {42325, 50512}

X(52730) = reflection of X(52739) in X(3)

---

X(52731) = CENTER OF THE CLAWSON CIRCLE OF ABC + IO LINE {X(1), X(3)}

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

X(52731) lies on these lines: {3, 960}, {900, 33814}, {34346, 34913}

---

X(52732) = CENTER OF THE CLAWSON CIRCLE OF ABC + NAGEL LINE {X(1), X(2)}

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

X(52732) lies on these lines: {40, 22837}, {2390, 11204}, {6085, 38620}

---

X(52733) = CENTER OF THE CLAWSON CIRCLE OF ABC + SODDY LINE {X(1), X(7)}

Barycentrics    a^2*(2*a^11-3*(b+c)*a^10-2*(b-c)^2*a^9-(b+c)*(b^2-10*b*c+c^2)*a^8+2*(4*b^4+4*c^4-(6*b^2+5*b*c+6*c^2)*b*c)*a^7+(b+c)*(6*b^4+6*c^4-(16*b^2-23*b*c+16*c^2)*b*c)*a^6-4*(b^3-c^3)*(b-c)*(4*b^2+b*c+4*c^2)*a^5+(b^2-c^2)*(b-c)*(2*b^4+2*c^4+(8*b^2+17*b*c+8*c^2)*b*c)*a^4+2*(3*b^6+3*c^6+2*(2*b^4+2*c^4+(2*b^2-3*b*c+2*c^2)*b*c)*b*c)*(b-c)^2*a^3-(b^2-c^2)*(b-c)*(3*b^6+3*c^6+2*(3*b^4-b^2*c^2+3*c^4)*b*c)*a^2+2*(b^4-c^4)*(b^2-c^2)^3*a+(b^2-c^2)*(b-c)^3*(-b^6-c^6-2*(b^4+c^4+(2*b^2+b*c+2*c^2)*b*c)*b*c)) : :

X(52733) lies on these lines: {926, 38599}

---

X(52734) = CENTER OF THE CLAWSON CIRCLE OF ABC + VAN AUBEL LINE {X(4), X(6)}

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

X(52734) lies on these lines: {182, 3357}, {2881, 25644}

---

X(52735) = CENTER OF THE CLAWSON CIRCLE OF ABC + SHERMAN LINE {X(3259), X(3326)}

Barycentrics    a^2*(a^2-b^2+b*c-c^2)*(2*a^12-6*(b+c)*a^11+2*(b^2+13*b*c+c^2)*a^10+4*(b+c)*(4*b^2-13*b*c+4*c^2)*a^9-(25*b^4+25*c^4+2*(9*b^2-58*b*c+9*c^2)*b*c)*a^8-2*(b+c)*(2*b^4+2*c^4-(57*b^2-113*b*c+57*c^2)*b*c)*a^7+(40*b^6+40*c^6-(95*b^4+95*c^4+4*(25*b^2-78*b*c+25*c^2)*b*c)*b*c)*a^6-6*(b^2-c^2)*(b-c)*(4*b^4+4*c^4+(13*b^2-32*b*c+13*c^2)*b*c)*a^5-(20*b^6+20*c^6-(93*b^4+93*c^4+2*(24*b^2-97*b*c+24*c^2)*b*c)*b*c)*(b-c)^2*a^4+2*(b^2-c^2)*(b-c)^3*(13*b^4+13*c^4+5*(3*b^2-7*b*c+3*c^2)*b*c)*a^3-(b^2-c^2)^2*(2*b^6+2*c^6+(39*b^4+39*c^4-2*(69*b^2-98*b*c+69*c^2)*b*c)*b*c)*a^2-2*(b^2-c^2)^3*(b-c)^3*(4*b^2-5*b*c+4*c^2)*a+(b^2-c^2)^3*(b-c)*(b^3+c^3)*(3*b^2-4*b*c+3*c^2)) : :

X(52735) lies on these lines: {3, 102}, {38617, 45949}

---

X(52736) = CENTER OF THE CLAWSON CIRCLE OF ABC + NAPOLEON AXIS {X(6), X(17)}

Barycentrics    a^2*(2*a^20-15*(b^2+c^2)*a^18+(47*b^4+92*b^2*c^2+47*c^4)*a^16-(b^2+c^2)*(77*b^4+153*b^2*c^2+77*c^4)*a^14+2*(30*b^8+30*c^8+(161*b^4+197*b^2*c^2+161*c^4)*b^2*c^2)*a^12+(b^2+c^2)*(9*b^8+9*c^8-(320*b^4-b^2*c^2+320*c^4)*b^2*c^2)*a^10-(70*b^12+70*c^12-(264*b^8+264*c^8+(91*b^4+132*b^2*c^2+91*c^4)*b^2*c^2)*b^2*c^2)*a^8+(b^2+c^2)*(73*b^12+73*c^12-(283*b^8+283*c^8-3*(123*b^4-133*b^2*c^2+123*c^4)*b^2*c^2)*b^2*c^2)*a^6-(b^2-c^2)^2*(38*b^12+38*c^12-(44*b^8+44*c^8+(17*b^4+12*b^2*c^2+17*c^4)*b^2*c^2)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)*(10*b^12+10*c^12-(24*b^8+24*c^8-(11*b^4+9*b^2*c^2+11*c^4)*b^2*c^2)*b^2*c^2)*a^2-(b^2-c^2)^4*(b^12+c^12+(b^2-c^2)^2*(2*b^4-9*b^2*c^2+2*c^4)*b^2*c^2)) : :

X(52736) lies on these lines: {1350, 7691}

---

X(52737) = CENTER OF THE CLAWSON CIRCLE OF ABC + TRIPOLAR OF X(3) {X(520), X(647)}

Barycentrics    a^2*(a^12-3*(b^2+c^2)*a^10+(2*b^4+13*b^2*c^2+2*c^4)*a^8+(b^2+c^2)*(2*b^4-15*b^2*c^2+2*c^4)*a^6-(b^4+3*b^2*c^2+c^4)*(3*b^4-8*b^2*c^2+3*c^4)*a^4+(b^4-c^4)*(b^2-c^2)*(b^4+5*b^2*c^2+c^4)*a^2-2*(b^2-c^2)^2*b^4*c^4)*(b^2-c^2) : :
X(52737) = 3*X(25644)-2*X(44806)

X(52737) lies on these lines: {3, 2848}, {523, 14634}, {2799, 39841}, {2881, 25644}, {6086, 38621}, {9033, 13293}, {11413, 41077}

X(52737) = reflection of X(52738) in X(44810)

---

X(52738) = CENTER OF THE CLAWSON CIRCLE OF ABC + TRIPOLAR OF X(5) {X(2081), X(2600)}

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

X(52738) lies on these lines: {2848, 9409}, {2881, 25644}

X(52738) = reflection of X(i) in X(j) for these (i, j): (33752, 44806), (52737, 44810)

---

X(52739) = CENTER OF THE CLAWSON CIRCLE OF ABC + TRIPOLAR OF X(8) {X(522), X(650)}

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

X(52739) lies on these lines: {1, 30691}, {3, 928}, {101, 52614}, {512, 48386}, {520, 39226}, {676, 24929}, {900, 22935}, {926, 38599}, {1385, 6366}, {1960, 3887}, {2646, 10015}, {5440, 50333}, {8578, 22437}, {8676, 39227}, {8677, 38600}

X(52739) = midpoint of X(44827) and X(52726)
X(52739) = reflection of X(52730) in X(3)

---

X(52740) = CENTER OF THE CLAWSON CIRCLE OF ABC + TRIPOLAR OF X(9) {X(650), X(663)}

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

X(52740) lies on these lines: {2812, 11248}, {3738, 12332}, {8677, 38607}

---

X(52741) = X(3)X(46604)∩X(20)X(50210)

Barycentrics    a^2*(2*a^22-18*(b^2+c^2)*a^20+2*(38*b^4+63*b^2*c^2+38*c^4)*a^18-2*(b^2+c^2)*(100*b^4+93*b^2*c^2+100*c^4)*a^16+(366*b^8+366*c^8+5*(133*b^4+156*b^2*c^2+133*c^4)*b^2*c^2)*a^14-2*(b^2+c^2)*(245*b^8+245*c^8+2*(43*b^4+146*b^2*c^2+43*c^4)*b^2*c^2)*a^12+5*(98*b^12+98*c^12+(8*b^4+13*b^2*c^2+8*c^4)*(7*b^4-4*b^2*c^2+7*c^4)*b^2*c^2)*a^10-(b^2+c^2)*(366*b^12+366*c^12-(545*b^8+545*c^8-(563*b^4-552*b^2*c^2+563*c^4)*b^2*c^2)*b^2*c^2)*a^8+(200*b^12+200*c^12+(41*b^8+41*c^8+2*(21*b^4+20*b^2*c^2+21*c^4)*b^2*c^2)*b^2*c^2)*(b^2-c^2)^2*a^6-2*(b^4-c^4)*(b^2-c^2)*(38*b^12+38*c^12-(87*b^8+87*c^8-8*(12*b^4-11*b^2*c^2+12*c^4)*b^2*c^2)*b^2*c^2)*a^4+(18*b^12+18*c^12-(16*b^8+16*c^8-5*(b^2+c^2)^2*b^2*c^2)*b^2*c^2)*(b^2-c^2)^4*a^2-(b^6+c^6)*(b^2-c^2)^6*(2*b^4-b^2*c^2+2*c^4)) : :
Barycentrics    (SB+SC)*((SA^2-SB*SC)*(48*R^6-2*(SA+29*SW)*R^4+3*(SA^2-SB*SC+7*SW^2)*R^2-(SA^2-SB*SC+2*SW^2)*SW)+(44*R^6+12*(7*SA-5*SW)*R^4-(62*SA^2-57*SB*SC-25*SW^2)*R^2+(13*SA^2-10*SB*SC-3*SW^2)*SW)*S^2-2*(34*R^2-7*SW+9*SA)*S^4) : :
X(52741) = 3*X(3)-X(46604)

See Kadir Altintas and César Lozada, euclid 5602.

X(52741) lies on these lines: {3, 46604}, {20, 50210}

X(52741) = midpoint of X(20) and X(50210)

---

X(52742) = TRIPOLAR CENTROID OF X(96)

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

X(52742) = lies on these lines: {115, 125}, {523, 2623}, {647, 35441}, {14391, 14582}, {20184, 52317}

X(52742) = tripolar centroid of X(96)
X(52742) = X(662)-isoconjugate of X(2383)
X(52742) = X(i)-Dao conjugate of X(j) for these (i,j): {128, 648}, {1084, 2383}
X(52742) = crosssum of X(186) and X(6753)
X(52742) = crossdifference of every pair of points on line {52, 110}
X(52742) = barycentric product X(i)*X(j) for these {i,j}: {231, 525}, {523, 539}, {6368, 40631}
X(52742) = barycentric quotient X(i)/X(j) for these {i,j}: {231, 648}, {512, 2383}, {539, 99}, {40631, 18831}

---

X(52743) = TRIPOLAR CENTROID OF X(186)

Barycentrics    a^2*(b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) : :
X(52743) = X[2081] - 4 X[47230]

X(52743) lies on the cubic K215 and these lines: {6, 647}, {50, 2436}, {110, 14998}, {112, 32712}, {184, 512}, {351, 51335}, {523, 14582}, {526, 2081}, {826, 1640}, {879, 47250}, {1636, 1637}, {2088, 16186}, {2624, 21828}, {3051, 9210}, {3167, 8673}, {3292, 41167}, {3569, 9408}, {5306, 9209}, {6753, 51936}, {9175, 15920}, {9213, 14355}, {11002, 47263}, {14389, 18312}, {14611, 23968}, {34417, 42654}, {45801, 47122}, {46983, 50979}

X(52743) = isogonal conjugate of X(39290)
X(52743) = isogonal conjugate of the isotomic conjugate of X(5664)
X(52743) = tripolar centroid of X(186)
X(52743) = X(i)-Ceva conjugate of X(j) for these (i,j): {6, 2088}, {110, 1495}, {112, 50}, {523, 9409}, {647, 2081}, {9404, 2624}, {14920, 3258}, {15470, 14270}, {39176, 47414}
X(52743) = X(47414)-cross conjugate of X(39176)
X(52743) = X(i)-isoconjugate of X(j) for these (i,j): {1, 39290}, {74, 32680}, {94, 36034}, {328, 36131}, {476, 2349}, {662, 5627}, {799, 40355}, {811, 11079}, {823, 50464}, {1494, 32678}, {1577, 15395}, {2159, 35139}, {2166, 44769}, {14560, 33805}, {14919, 36129}, {16080, 36061}, {35200, 46456}, {36047, 46788}, {36096, 51227}
X(52743) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 39290}, {94, 3258}, {99, 3284}, {133, 46456}, {328, 39008}, {850, 1637}, {1084, 5627}, {1494, 18334}, {3163, 35139}, {3267, 8552}, {6331, 14918}, {9033, 18557}, {10413, 46751}, {11079, 17423}, {11597, 44769}, {16080, 16221}, {35581, 46788}, {36308, 38994}, {36311, 38993}, {38996, 40355}
X(52743) = cevapoint of X(14398) and X(42656)
X(52743) = crosspoint of X(i) and X(j) for these (i,j): {6, 2420}, {110, 323}, {112, 1990}, {523, 44427}, {2407, 15454}, {6137, 6138}, {14591, 52557}
X(52743) = crosssum of X(i) and X(j) for these (i,j): {2, 2394}, {6, 46608}, {110, 32662}, {265, 18558}, {476, 41392}, {523, 1989}, {525, 14919}, {647, 21650}, {2433, 14264}, {23895, 23896}
X(52743) = crossdifference of every pair of points on line {30, 74}
X(52743) = barycentric product X(i)*X(j) for these {i,j}: {6, 5664}, {30, 526}, {50, 41079}, {110, 3258}, {113, 15470}, {186, 9033}, {323, 1637}, {340, 9409}, {512, 6148}, {523, 1511}, {525, 39176}, {647, 14920}, {648, 47414}, {656, 35201}, {1495, 3268}, {1636, 14165}, {1990, 8552}, {2081, 43768}, {2088, 2407}, {2173, 32679}, {2605, 6739}, {2624, 14206}, {2631, 52414}, {3260, 14270}, {3284, 44427}, {3471, 8562}, {4240, 16186}, {5642, 9213}, {6137, 41888}, {6138, 41887}, {6149, 36035}, {7799, 14398}, {9214, 44814}, {11064, 47230}, {14397, 37802}, {14399, 42701}, {14581, 45792}, {18557, 36423}, {41077, 52418}, {46002, 46114}
X(52743) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 39290}, {30, 35139}, {50, 44769}, {186, 16077}, {512, 5627}, {526, 1494}, {669, 40355}, {1495, 476}, {1511, 99}, {1576, 15395}, {1637, 94}, {1990, 46456}, {2088, 2394}, {2173, 32680}, {2420, 39295}, {2624, 2349}, {2682, 51479}, {3049, 11079}, {3258, 850}, {5664, 76}, {6137, 36311}, {6138, 36308}, {6148, 670}, {8562, 46751}, {9033, 328}, {9406, 32678}, {9407, 14560}, {9408, 41392}, {9409, 265}, {14270, 74}, {14397, 18883}, {14398, 1989}, {14920, 6331}, {15470, 40423}, {16186, 34767}, {19627, 32640}, {32679, 33805}, {34397, 1304}, {35201, 811}, {39008, 18557}, {39176, 648}, {39201, 50464}, {39371, 18878}, {41079, 20573}, {42656, 14993}, {44814, 36890}, {47230, 16080}, {47414, 525}, {52418, 15459}
X(52743) = {X(1636),X(14397)}-harmonic conjugate of X(1637)

---

X(52744) = TRIPOLAR CENTROID OF X(264)

Barycentrics    (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 - 2*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 - 4*b^4*c^4 - a^2*c^6 + 2*b^2*c^6) : :
X(52744) = 3 X[1636] - 4 X[45325], 2 X[14417] - 3 X[52720], 2 X[850] + X[17434], 3 X[14401] - 4 X[44564], 4 X[30476] - X[32320]

X(52744) lies on these lines: {2, 1636}, {122, 125}, {297, 525}, {520, 31174}, {1637, 39473}, {1853, 2881}, {1899, 47205}, {1972, 34767}, {2799, 14391}, {3268, 38240}, {3564, 13303}, {9140, 9517}, {9148, 39469}, {9391, 14430}, {14401, 15595}, {26958, 46425}, {30476, 32320}

X(52744) = midpoint of X(14391) and X(23616)
X(52744) = reflection of X(i) in X(j) for these {i,j}: {1636, 2}, {3268, 38240}
X(52744) = anticomplement of X(45325)
X(52744) = tripolar centroid of X(264)
X(52744) = X(i)-isoconjugate of X(j) for these (i,j): {3, 36139}, {63, 32725}, {162, 26717}
X(52744) = X(i)-Dao conjugate of X(j) for these (i,j): {125, 26717}, {852, 33571}, {3162, 32725}, {36103, 36139}
X(52744) = crossdifference of every pair of points on line {112, 184}
X(52744) = barycentric product X(i)*X(j) for these {i,j}: {850, 852}, {3267, 3331}
X(52744) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 36139}, {25, 32725}, {647, 26717}, {852, 110}, {3331, 112}, {33571, 32320}

---

X(52745) = TRIPOLAR CENTROID OF X(291)

Barycentrics    a*(b - c)*(a^2*b - 2*a*b^2 + a^2*c + b^2*c - 2*a*c^2 + b*c^2) : :
X(52745) = 2 X[37] + X[21143], X[192] + 2 X[21211], 4 X[665] - X[21832], X[18080] + 2 X[52602]

X(52745) lies on these lines: {1, 649}, {2, 514}, {37, 513}, {45, 23345}, {192, 21211}, {244, 665}, {354, 4083}, {512, 1962}, {650, 16604}, {661, 764}, {1027, 48572}, {3661, 25381}, {3766, 4928}, {3835, 17244}, {4024, 22011}, {4169, 23354}, {4375, 16826}, {4776, 45658}, {4813, 44008}, {4979, 6161}, {5592, 49293}, {6008, 51057}, {6085, 14408}, {7180, 45208}, {8632, 25569}, {14405, 35123}, {17367, 31286}, {18080, 52602}, {20295, 29569}, {20909, 33933}, {20952, 33943}, {28147, 47130}, {30656, 30665}, {48275, 50351}

X(52745) = midpoint of X(14437) and X(21143)
X(52745) = reflection of X(i) in X(j) for these {i,j}: {1635, 665}, {3766, 4928}, {4776, 45658}, {14433, 2}, {14437, 37}, {21832, 1635}, {36848, 45657}
X(52745) = tripolar centroid of X(291)
X(52745) = X(i)-isoconjugate of X(j) for these (i,j): {101, 18822}, {190, 2382}, {898, 51923}, {901, 46797}, {5378, 52226}
X(52745) = X(i)-Dao conjugate of X(j) for these (i,j): {668, 35123}, {1015, 18822}, {14434, 46782}, {38979, 46797}
X(52745) = crossdifference of every pair of points on line {100, 238}
X(52745) = barycentric product X(i)*X(j) for these {i,j}: {1, 36848}, {104, 42765}, {513, 537}, {514, 20331}, {1635, 46795}, {1646, 46780}, {30571, 45657}
X(52745) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 18822}, {537, 668}, {667, 2382}, {1635, 46797}, {1646, 46782}, {3768, 51923}, {20331, 190}, {27846, 47070}, {36848, 75}, {42765, 3262}
X(52745) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1635, 14413, 14438}, {1646, 2087, 27846}, {14421, 14474, 27846}

   Tripolar Centroidal Conjugates: X(52746) - X(52768)  

This preamble is based on a definition and notes by Bernard Gibert; see Tripolar Centroidal Cubics Suppose that P = p : q : r is a point in the plane of a triangle ABC. The tripolar centroidal conjugate of P is given by

TCC(P) = (-2 p + q + r) (p q - 2 p r + q r) (-2 p q + p r + q r) : :

and TCC(P) = P if and only if P lies on the cubic K015.

Let U = the infinite point of the line GP, and let P' = tripolar centroid of P. Then TCCP is the trilinear pole of the line UP'.

X(52746) = TRIPOLAR CENTROIDAL CONJUGATE OF X(8)

Barycentrics    (2*a - b - c)*(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) : :

X(52746) lies on these lines: {2, 522}, {7, 18821}, {8, 190}, {596, 45700}, {900, 46791}, {2291, 9059}, {2325, 17780}, {2757, 14733}, {4076, 5423}, {4441, 19640}, {4723, 24004}, {36916, 41798}, {36944, 50843}, {50841, 51975}

X(52746) = isotomic conjugate of X(36887)
X(52746) = X(6174)-cross conjugate of X(519)
X(52746) = X(i)-isoconjugate of X(j) for these (i,j): {31, 36887}, {88, 1055}, {106, 1155}, {527, 9456}, {901, 14413}, {1417, 6745}, {1638, 32665}, {2316, 6610}, {6510, 8752}, {23346, 23838}, {23710, 36058}, {32659, 37805}
X(52746) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 36887}, {214, 1155}, {519, 6174}, {527, 4370}, {1323, 52659}, {1638, 35092}, {1647, 30573}, {6366, 51402}, {14413, 38979}, {20619, 23710}
X(52746) = cevapoint of X(519) and X(6174)
X(52746) = trilinear pole of line {519, 1639}
X(52746) = tripolar centroidal conjugate of X(8)
X(52746) = barycentric product X(i)*X(j) for these {i,j}: {519, 1121}, {1156, 4358}, {1639, 35157}, {2291, 3264}, {4723, 34056}, {4768, 37139}, {24004, 35348}
X(52746) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 36887}, {44, 1155}, {519, 527}, {900, 1638}, {902, 1055}, {1121, 903}, {1156, 88}, {1319, 6610}, {1635, 14413}, {1639, 6366}, {2291, 106}, {2325, 6745}, {3689, 6603}, {3911, 1323}, {4120, 30574}, {4358, 30806}, {4370, 6174}, {4432, 24685}, {4434, 6647}, {4845, 2316}, {5440, 6510}, {6174, 35110}, {6544, 30573}, {8756, 23710}, {14418, 14414}, {14427, 14392}, {14439, 35293}, {23703, 23890}, {23757, 42762}, {23893, 23838}, {34068, 9456}, {35348, 1022}, {37790, 38461}, {38462, 37805}, {41553, 15730}, {41798, 1320}

---

X(52747) = TRIPOLAR CENTROIDAL CONJUGATE OF X(10)

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

X(52747) lies on these lines: {2, 523}, {10, 190}, {86, 892}, {111, 9059}, {662, 9884}, {691, 2758}, {966, 52450}, {2640, 9875}, {2643, 12258}, {3264, 4141}, {3943, 17780}, {3992, 24004}, {6535, 6632}, {6998, 48983}, {8756, 46541}, {13740, 14246}, {14609, 48864}, {16052, 51258}, {17964, 24275}, {24229, 37756}, {30761, 30786}, {31090, 31125}

X(52747) = X(i)-isoconjugate of X(j) for these (i,j): {88, 187}, {106, 896}, {351, 4622}, {468, 36058}, {524, 9456}, {901, 14419}, {903, 922}, {1417, 3712}, {2316, 51653}, {2642, 4591}, {3292, 36125}, {4750, 32665}, {14567, 20568}
X(52747) = X(i)-Dao conjugate of X(j) for these (i,j): {106, 15899}, {214, 896}, {468, 20619}, {524, 4370}, {903, 39061}, {4750, 35092}, {4933, 36912}, {7181, 52659}, {14419, 38979}, {14432, 51402}
X(52747) = trilinear pole of line {519, 4120}
X(52747) = tripolar centroidal conjugate of X(10)
X(52747) = barycentric product X(i)*X(j) for these {i,j}: {44, 46277}, {111, 3264}, {519, 671}, {892, 4120}, {895, 46109}, {897, 4358}, {902, 18023}, {3762, 5380}, {3977, 17983}, {4141, 18818}, {8756, 30786}, {14977, 46541}, {22356, 46111}
X(52747) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 896}, {111, 106}, {519, 524}, {671, 903}, {691, 4591}, {892, 4615}, {895, 1797}, {897, 88}, {900, 4750}, {902, 187}, {923, 9456}, {1319, 51653}, {1635, 14419}, {1639, 14432}, {2251, 922}, {2325, 3712}, {3264, 3266}, {3911, 7181}, {3943, 4062}, {3977, 6390}, {3992, 42713}, {4120, 690}, {4141, 39785}, {4358, 14210}, {4432, 4760}, {4434, 7267}, {4700, 4831}, {4727, 4938}, {4730, 2642}, {4908, 4933}, {4958, 30595}, {5380, 3257}, {5466, 4049}, {5547, 2316}, {8753, 8752}, {8756, 468}, {9459, 14567}, {14407, 351}, {14429, 14417}, {14908, 32659}, {16704, 6629}, {17983, 6336}, {21805, 21839}, {22356, 3292}, {23757, 42760}, {24004, 42721}, {30939, 16741}, {31011, 31013}, {36060, 36058}, {36085, 4622}, {36128, 36125}, {46109, 44146}, {46154, 46150}, {46277, 20568}, {46541, 4235}, {52680, 16702}

---

X(52748) = TRIPOLAR CENTROIDAL CONJUGATE OF X(13)

Barycentrics    (a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(Sqrt[3]*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) + 2*(2*a^2 - b^2 - c^2)*S) : :

X(52748) lies on these lines: {2, 523}, {13, 531}, {298, 892}, {303, 52551}, {470, 17983}, {691, 35931}, {1080, 48983}, {6772, 17964}, {11304, 14246}, {11537, 45331}, {31693, 51258}, {36310, 52040}, {37640, 52450}, {51019, 51980}

X(52748) = X(i)-isoconjugate of X(j) for these (i,j): {896, 2378}, {922, 43091}
X(52748) = X(i)-Dao conjugate of X(j) for these (i,j): {2378, 15899}, {39061, 43091}
X(52748) = trilinear pole of line {530, 9200}
X(52748) = tripolar centroidal conjugate of X(13)
X(52748) = barycentric product X(i)*X(j) for these {i,j}: {530, 671}, {892, 9200}, {23712, 30786}
X(52748) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 2378}, {530, 524}, {671, 43091}, {9200, 690}, {11537, 52039}, {18776, 52040}, {23712, 468}, {36307, 36316}

---

X(52749) = TRIPOLAR CENTROIDAL CONJUGATE OF X(14)

Barycentrics    (a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(Sqrt[3]*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) - 2*(2*a^2 - b^2 - c^2)*S) : :

X(52749) lies on these lines: {2, 523}, {14, 530}, {299, 892}, {302, 52551}, {383, 48983}, {471, 17983}, {691, 35932}, {6775, 17964}, {11303, 14246}, {11549, 45331}, {31694, 51258}, {36307, 52039}, {37641, 52450}, {51017, 51980}

X(52749) = X(i)-isoconjugate of X(j) for these (i,j): {896, 2379}, {922, 43092}
X(52749) = X(i)-Dao conjugate of X(j) for these (i,j): {2379, 15899}, {39061, 43092}
X(52749) = trilinear pole of line {531, 9201}
X(52749) = tripolar centroidal conjugate of X(14)
X(52749) = barycentric product X(i)*X(j) for these {i,j}: {531, 671}, {892, 9201}, {23713, 30786}
X(52749) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 2379}, {531, 524}, {671, 43092}, {9201, 690}, {11549, 52040}, {18777, 52039}, {23713, 468}, {36310, 36317}

---

X(52750) = TRIPOLAR CENTROIDAL CONJUGATE OF X(17)

Barycentrics    (a^2 + b^2 - 2*c^2)*(a^2 - 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*Sqrt[3]*(2*a^2 - b^2 - c^2)*S) : :

X(52750) lies on these lines: {2, 523}, {17, 671}, {299, 30468}, {302, 892}, {396, 35314}, {691, 11299}, {11119, 36307}, {11290, 14246}, {11304, 52483}, {11488, 52450}, {37352, 51258}, {37463, 48983}

X(52750) = X(i)-isoconjugate of X(j) for these (i,j): {896, 2380}, {922, 11117}
X(52750) = X(i)-Dao conjugate of X(j) for these (i,j): {618, 52040}, {2380, 15899}, {11117, 39061}
X(52750) = trilinear pole of line {532, 14446}
X(52750) = tripolar centroidal conjugate of X(17)
X(52750) = barycentric product X(i)*X(j) for these {i,j}: {532, 671}, {892, 14446}, {14922, 36307}, {23714, 30786}
X(52750) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 2380}, {396, 52040}, {532, 524}, {671, 11117}, {9115, 30455}, {14446, 690}, {23714, 468}, {30462, 9115}, {43085, 43084}

---

X(52751) = TRIPOLAR CENTROIDAL CONJUGATE OF X(18)

Barycentrics    (a^2 + b^2 - 2*c^2)*(a^2 - 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*Sqrt[3]*(2*a^2 - b^2 - c^2)*S) : :

X(52751) lies on these lines: {2, 523}, {18, 671}, {298, 30465}, {303, 892}, {395, 35315}, {691, 11300}, {11120, 36310}, {11289, 14246}, {11303, 52483}, {11489, 52450}, {37351, 51258}, {37464, 48983}

X(52751) = X(i)-isoconjugate of X(j) for these (i,j): {896, 2381}, {922, 11118}
X(52751) = X(i)-Dao conjugate of X(j) for these (i,j): {619, 52039}, {2381, 15899}, {11118, 39061}
X(52751) = trilinear pole of line {533, 14447}
X(52751) = tripolar centroidal conjugate of X(18)
X(52751) = barycentric product X(i)*X(j) for these {i,j}: {533, 671}, {892, 14447}, {14921, 36310}, {23715, 30786}
X(52751) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 2381}, {395, 52039}, {533, 524}, {671, 11118}, {9117, 30454}, {14447, 690}, {23715, 468}, {30459, 9117}, {43086, 43084}

---

X(52752) = TRIPOLAR CENTROIDAL CONJUGATE OF X(25)

Barycentrics    (2*a^2*b^2 - a^2*c^2 - b^2*c^2)*(a^2*b^2 - 2*a^2*c^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(52752) lies on these lines: {2, 512}, {25, 648}, {305, 886}, {729, 1302}, {842, 6035}, {1495, 2407}, {3440, 5981}, {3441, 5980}, {4240, 14581}, {7757, 11332}, {9211, 14458}, {30528, 32717}, {36839, 37775}, {36840, 37776}, {37132, 38340}, {39290, 40355}

X(52752) = X(i)-isoconjugate of X(j) for these (i,j): {74, 2234}, {538, 2159}, {2349, 3231}, {9148, 36034}, {33805, 33875}
X(52752) = X(i)-Dao conjugate of X(j) for these (i,j): {538, 3163}, {3258, 9148}
X(52752) = trilinear pole of line {30, 14398}
X(52752) = tripolar centroidal conjugate of X(25)
X(52752) = barycentric product X(i)*X(j) for these {i,j}: {30, 3228}, {729, 3260}, {886, 14398}, {1495, 34087}, {1637, 9150}, {9214, 14608}, {14206, 37132}, {32717, 41079}, {36035, 36133}
X(52752) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 538}, {729, 74}, {1495, 3231}, {1637, 9148}, {2173, 2234}, {2407, 23342}, {2420, 5118}, {3228, 1494}, {3260, 30736}, {5642, 45672}, {9407, 33875}, {14398, 888}, {14581, 46522}, {14608, 36890}, {32717, 44769}, {35906, 36822}, {37132, 2349}, {41309, 9717}, {46156, 46147}

---

X(52753) = TRIPOLAR CENTROIDAL CONJUGATE OF X(27)

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

X(52753) lies on these lines: {2, 514}, {27, 648}, {88, 26723}, {106, 1302}, {226, 26743}, {306, 4555}, {553, 40215}, {901, 2688}, {1320, 51077}, {1999, 4080}, {2226, 24177}, {2407, 18653}, {3058, 14190}, {3257, 17781}, {3452, 40594}, {3656, 14260}, {3782, 51908}, {4591, 30528}, {4615, 6035}, {5249, 52553}, {6549, 40940}, {14206, 42716}, {19801, 20568}, {30575, 33133}, {36590, 50796}, {50824, 52478}

X(52753) = X(i)-isoconjugate of X(j) for these (i,j): {44, 74}, {519, 2159}, {902, 2349}, {1319, 15627}, {1404, 44693}, {1494, 2251}, {4120, 36034}, {4358, 40352}, {4730, 44769}, {5440, 8749}, {8756, 35200}, {9459, 33805}, {14429, 36131}, {16080, 23202}, {18877, 38462}, {22356, 36119}
X(52753) = X(i)-Dao conjugate of X(j) for these (i,j): {74, 40595}, {133, 8756}, {519, 3163}, {1494, 9460}, {1511, 22356}, {2325, 6739}, {2349, 40594}, {3258, 4120}, {14429, 39008}
X(52753) = trilinear pole of line {30, 11125}
X(52753) = tripolar centroidal conjugate of X(27)
X(52753) = barycentric product X(i)*X(j) for these {i,j}: {30, 903}, {88, 14206}, {106, 3260}, {1022, 42716}, {1637, 4615}, {1797, 46106}, {2173, 20568}, {2407, 4049}, {4080, 18653}, {4555, 11125}, {4591, 41079}, {4622, 36035}, {4997, 6357}, {6336, 11064}, {9456, 46234}
X(52753) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 519}, {88, 2349}, {106, 74}, {903, 1494}, {1320, 44693}, {1495, 902}, {1637, 4120}, {1784, 38462}, {1797, 14919}, {1990, 8756}, {2173, 44}, {2316, 15627}, {3260, 3264}, {3284, 22356}, {4049, 2394}, {4240, 46541}, {4591, 44769}, {6336, 16080}, {6357, 3911}, {7359, 2325}, {8752, 8749}, {9033, 14429}, {9406, 2251}, {9407, 9459}, {9456, 2159}, {11064, 3977}, {11125, 900}, {13857, 4141}, {14206, 4358}, {14395, 14418}, {14398, 14407}, {14399, 1635}, {14400, 1639}, {18653, 16704}, {20568, 33805}, {32659, 18877}, {36058, 35200}, {36125, 36119}, {42716, 24004}, {42750, 23757}, {46106, 46109}, {46150, 46147}, {51420, 52680}, {51654, 1319}, {52640, 36944}

---

X(52754) = TRIPOLAR CENTROIDAL CONJUGATE OF X(28)

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

X(52754) lies on these lines: {2, 513}, {28, 648}, {30, 42716}, {739, 1302}, {889, 20336}, {898, 2687}, {2407, 51420}, {4664, 37018}, {10308, 36798}, {37129, 38340}

X(52754) = X(i)-isoconjugate of X(j) for these (i,j): {74, 899}, {536, 2159}, {2349, 3230}, {6381, 40352}, {14431, 36034}
X(52754) = X(i)-Dao conjugate of X(j) for these (i,j): {536, 3163}, {3258, 14431}, {4009, 6739}
X(52754) = trilinear pole of line {30, 14399}
X(52754) = tripolar centroidal conjugate of X(28)
X(52754) = barycentric product X(i)*X(j) for these {i,j}: {30, 3227}, {739, 3260}, {889, 14399}, {2173, 31002}, {2407, 35353}, {4607, 11125}, {6357, 36798}, {14206, 37129}, {18653, 41683}, {42716, 43928}
X(52754) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 536}, {739, 74}, {1495, 3230}, {1637, 14431}, {2173, 899}, {3227, 1494}, {3260, 35543}, {6357, 43037}, {7359, 4009}, {11125, 4728}, {14206, 6381}, {14398, 14404}, {14399, 891}, {14400, 14430}, {31002, 33805}, {35353, 2394}, {37129, 2349}, {42716, 41314}, {42750, 42764}

---

X(52755) = TRIPOLAR CENTROIDAL CONJUGATE OF X(75)

Barycentrics    (a + b - 2*c)*(a - 2*b + c)*(a*b + a*c - 2*b*c) : :
X(52755) = X[23891] + 2 X[52626]

X(52755) lies on these lines: {1, 4555}, {2, 514}, {10, 6549}, {75, 537}, {88, 4384}, {106, 9067}, {536, 23891}, {3257, 50127}, {3661, 4080}, {4363, 51908}, {4937, 6381}, {4997, 17284}, {6633, 24407}, {9458, 34762}, {9460, 25055}, {10436, 52553}, {11679, 47056}, {14190, 48805}, {19875, 40095}, {21385, 46797}, {26727, 32097}, {31183, 31227}, {34284, 39264}, {40833, 52654}

X(52755) = isotomic conjugate of X(36872)
X(52755) = X(30583)-cross conjugate of X(23891)
X(52755) = X(i)-isoconjugate of X(j) for these (i,j): {31, 36872}, {44, 739}, {898, 1960}, {900, 32718}, {902, 37129}, {1023, 23892}, {1635, 34075}, {2251, 3227}, {9459, 31002}, {17780, 23349}, {23344, 43928}
X(52755) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 36872}, {44, 40614}, {519, 13466}, {739, 40595}, {1635, 39011}, {1646, 14437}, {3227, 9460}, {37129, 40594}
X(52755) = cevapoint of X(30583) and X(52626)
X(52755) = trilinear pole of line {536, 4728}
X(52755) = tripolar centroidal conjugate of X(75)
X(52755) = barycentric product X(i)*X(j) for these {i,j}: {88, 6381}, {106, 35543}, {536, 903}, {899, 20568}, {1022, 41314}, {4555, 4728}, {4615, 14431}, {4937, 40833}, {4997, 43037}, {6548, 23891}
X(52755) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 36872}, {88, 37129}, {106, 739}, {536, 519}, {891, 1635}, {899, 44}, {901, 34075}, {903, 3227}, {1022, 43928}, {3230, 902}, {3257, 898}, {3768, 1960}, {3994, 3943}, {4009, 2325}, {4049, 35353}, {4080, 41683}, {4465, 4432}, {4526, 4895}, {4555, 4607}, {4706, 4700}, {4728, 900}, {4937, 4908}, {4997, 36798}, {6381, 4358}, {14404, 14407}, {14426, 14408}, {14430, 1639}, {14431, 4120}, {14433, 4448}, {14434, 14437}, {14437, 3251}, {19945, 2087}, {20568, 31002}, {23343, 1023}, {23345, 23892}, {23891, 17780}, {30583, 6544}, {30592, 4984}, {32665, 32718}, {35543, 3264}, {41314, 24004}, {42764, 23757}, {43037, 3911}, {52626, 1647}
X(52755) = {X(4555),X(27922)}-harmonic conjugate of X(1)

---

X(52756) = TRIPOLAR CENTROIDAL CONJUGATE OF X(76)

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

X(52756) lies on these lines: {2, 523}, {3, 48952}, {6, 892}, {69, 52450}, {76, 338}, {111, 183}, {538, 14609}, {691, 1003}, {895, 41238}, {1634, 11152}, {3314, 31125}, {3734, 17964}, {3763, 52551}, {5108, 34760}, {5182, 52035}, {7610, 52141}, {7770, 14246}, {7778, 30786}, {8370, 52483}, {8860, 39576}, {9182, 35606}, {10097, 44155}, {11185, 34169}, {11317, 25225}, {11331, 17983}, {16093, 44526}, {18818, 52660}, {19221, 32740}, {23348, 36207}, {33184, 51258}, {35279, 48540}, {36877, 52713}, {39061, 47352}, {40879, 50941}, {42008, 44558}

X(52756) = isogonal conjugate of X(41309)
X(52756) = isotomic conjugate of X(14608)
X(52756) = isotomic conjugate of the isogonal conjugate of X(14609)
X(52756) = X(45672)-cross conjugate of X(538)
X(52756) = X(i)-isoconjugate of X(j) for these (i,j): {1, 41309}, {31, 14608}, {187, 37132}, {351, 36133}, {729, 896}, {922, 3228}, {2642, 32717}
X(52756) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 14608}, {3, 41309}, {187, 38998}, {351, 39010}, {524, 35073}, {538, 45672}, {729, 15899}, {3228, 39061}
X(52756) = cevapoint of X(538) and X(45672)
X(52756) = trilinear pole of line {538, 9148}
X(52756) = tripolar centroidal conjugate of X(76)
X(52756) = barycentric product X(i)*X(j) for these {i,j}: {76, 14609}, {111, 30736}, {538, 671}, {892, 9148}, {2234, 46277}, {3231, 18023}, {5118, 52632}, {5466, 23342}, {40362, 41294}
X(52756) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14608}, {6, 41309}, {111, 729}, {538, 524}, {671, 3228}, {691, 32717}, {888, 351}, {892, 9150}, {897, 37132}, {2234, 896}, {3231, 187}, {5118, 5467}, {6786, 9155}, {9148, 690}, {14609, 6}, {18023, 34087}, {23342, 5468}, {30736, 3266}, {30938, 16741}, {33875, 14567}, {35073, 45672}, {36085, 36133}, {36822, 5967}, {41294, 1501}, {45672, 2482}, {46154, 46156}, {46522, 44102}, {52625, 21906}

---

X(52757) = TRIPOLAR CENTROIDAL CONJUGATE OF X(81)

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

X(52757) lies on these lines: {2, 513}, {81, 99}, {321, 889}, {524, 42721}, {898, 2721}, {5468, 16702}, {17012, 34916}, {43926, 46799}

X(52757) = X(i)-isoconjugate of X(j) for these (i,j): {111, 899}, {536, 923}, {897, 3230}, {3768, 5380}, {6381, 32740}, {14404, 36085}, {14431, 36142}
X(52757) = X(i)-Dao conjugate of X(j) for these (i,j): {536, 2482}, {3230, 6593}, {14404, 38988}, {14431, 23992}
X(52757) = trilinear pole of line {524, 14419}
X(52757) = tripolar centroidal conjugate of X(81)
X(52757) = crossdifference of every pair of points on line {3230, 14404}
X(52757) = barycentric product X(i)*X(j) for these {i,j}: {524, 3227}, {739, 3266}, {889, 14419}, {896, 31002}, {4607, 4750}, {5468, 35353}, {6629, 41683}, {7181, 36798}, {14210, 37129}, {42721, 43928}
X(52757) = barycentric quotient X(i)/X(j) for these {i,j}: {187, 3230}, {351, 14404}, {524, 536}, {690, 14431}, {739, 111}, {896, 899}, {898, 5380}, {3227, 671}, {3266, 35543}, {3712, 4009}, {4062, 3994}, {4750, 4728}, {4760, 4465}, {4831, 4706}, {4933, 4937}, {7181, 43037}, {14210, 6381}, {14419, 891}, {14432, 14430}, {30605, 28603}, {31002, 46277}, {35353, 5466}, {37129, 897}, {42721, 41314}, {42760, 42764}

---

X(52758) = TRIPOLAR CENTROIDAL CONJUGATE OF X(83)

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

X(52758) lies on these lines: {2, 523}, {5, 48983}, {83, 597}, {111, 7792}, {141, 892}, {524, 22254}, {691, 8356}, {3329, 31125}, {3589, 52551}, {3618, 52450}, {3815, 30786}, {4045, 17964}, {6656, 14246}, {7790, 34169}, {7803, 14263}, {7841, 52483}, {14041, 38393}, {17983, 52289}, {24975, 35297}, {33265, 35345}, {34760, 41939}

X(52758) = X(i)-isoconjugate of X(j) for these (i,j): {755, 896}, {922, 43098}
X(52758) = X(i)-Dao conjugate of X(j) for these (i,j): {755, 15899}, {39061, 43098}
X(52758) = trilinear pole of line {754, 14420}
X(52758) = tripolar centroidal conjugate of X(83)
X(52758) = barycentric product X(i)*X(j) for these {i,j}: {111, 35549}, {671, 754}, {892, 14420}, {2244, 46277}, {8627, 18023}, {14977, 46543}
X(52758) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 755}, {671, 43098}, {754, 524}, {2244, 896}, {4156, 4062}, {4157, 3712}, {5380, 5389}, {7214, 7181}, {8627, 187}, {14420, 690}, {14428, 351}, {33907, 14424}, {35549, 3266}, {46543, 4235}

---

X(52759) = TRIPOLAR CENTROIDAL CONJUGATE OF X(86)

Barycentrics    (a + b - 2*c)*(a - 2*b + c)*(2*a^2 - b^2 - c^2) : :
X(52759) = X[24428] - 3 X[25055]

X(52759) lies on these lines: {2, 514}, {10, 4555}, {86, 99}, {88, 17023}, {519, 23816}, {901, 2729}, {1125, 6549}, {1266, 14028}, {3257, 50093}, {3624, 52574}, {4080, 16826}, {4357, 52553}, {4364, 51908}, {4510, 4997}, {4582, 50118}, {4615, 9170}, {4933, 14210}, {5468, 6629}, {6628, 31614}, {6788, 34342}, {14190, 49740}, {24428, 25055}, {30571, 40833}, {31191, 31227}, {34230, 47358}, {42026, 44572}

X(52759) = X(i)-isoconjugate of X(j) for these (i,j): {44, 111}, {519, 923}, {671, 2251}, {691, 4730}, {897, 902}, {1319, 5547}, {1960, 5380}, {3689, 7316}, {4120, 36142}, {4358, 32740}, {5440, 8753}, {8756, 36060}, {9459, 46277}, {14407, 36085}, {14908, 38462}, {17983, 23202}, {22356, 36128}
X(52759) = X(i)-Dao conjugate of X(j) for these (i,j): {111, 40595}, {519, 2482}, {671, 9460}, {897, 40594}, {902, 6593}, {1560, 8756}, {4120, 23992}, {14407, 38988}
X(52759) = trilinear pole of line {524, 4750}
X(52759) = tripolar centroidal conjugate of X(86)
X(52759) = crossdifference of every pair of points on line {902, 14407}
X(52759) = barycentric product X(i)*X(j) for these {i,j}: {88, 14210}, {106, 3266}, {524, 903}, {690, 4615}, {896, 20568}, {1022, 42721}, {1797, 44146}, {2642, 4634}, {4049, 5468}, {4080, 6629}, {4555, 4750}, {4591, 35522}, {4674, 16741}, {4933, 40833}, {4997, 7181}, {6336, 6390}
X(52759) = barycentric quotient X(i)/X(j) for these {i,j}: {88, 897}, {106, 111}, {187, 902}, {351, 14407}, {468, 8756}, {524, 519}, {690, 4120}, {896, 44}, {903, 671}, {922, 2251}, {1797, 895}, {2316, 5547}, {2642, 4730}, {3257, 5380}, {3266, 3264}, {3292, 22356}, {3712, 2325}, {4049, 5466}, {4062, 3943}, {4235, 46541}, {4591, 691}, {4615, 892}, {4622, 36085}, {4750, 900}, {4760, 4432}, {4831, 4700}, {4933, 4908}, {4938, 4727}, {6336, 17983}, {6390, 3977}, {6629, 16704}, {7181, 3911}, {7267, 4434}, {8752, 8753}, {9456, 923}, {14210, 4358}, {14417, 14429}, {14419, 1635}, {14432, 1639}, {14567, 9459}, {16702, 52680}, {16741, 30939}, {20568, 46277}, {21839, 21805}, {30595, 4958}, {31013, 31011}, {32659, 14908}, {36058, 36060}, {36125, 36128}, {39785, 4141}, {42713, 3992}, {42721, 24004}, {42760, 23757}, {44146, 46109}, {46150, 46154}, {51653, 1319}
X(52759) = {X(1125),X(6549)}-harmonic conjugate of X(27922)

---

X(52760) = TRIPOLAR CENTROIDAL CONJUGATE OF X(96)

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

X(52760) lies on these lines: {2, 523}, {96, 671}, {691, 7576}, {892, 39113}, {895, 3519}, {6091, 44261}, {10018, 17983}, {14246, 14788}

X(52760) = tripolar centroidal conjugate of X(96)
X(52760) = X(896)-isoconjugate of X(2383)
X(52760) = X(i)-Dao conjugate of X(j) for these (i,j): {128, 468}, {2383, 15899}
X(52760) = barycentric product X(i)*X(j) for these {i,j}: {231, 30786}, {539, 671}
X(52760) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 2383}, {231, 468}, {539, 524}

---

X(52761) = TRIPOLAR CENTROIDAL CONJUGATE OF X(105)

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

X(52761) lies on these lines: {2, 513}, {105, 666}, {528, 42722}, {889, 3263}, {1156, 4607}, {2161, 51562}, {7292, 26273}, {20045, 41683}

X(52761) = X(i)-isoconjugate of X(j) for these (i,j): {840, 899}, {3230, 37131}
X(52761) = X(536)-Dao conjugate of X(35113)
X(52761) = trilinear pole of line {528, 1643}
X(52761) = tripolar centroidal conjugate of X(105)
X(52761) = barycentric product X(i)*X(j) for these {i,j}: {528, 3227}, {889, 1643}, {2246, 31002}, {5723, 36798}, {36872, 46790}, {42722, 43928}
X(52761) = barycentric quotient X(i)/X(j) for these {i,j}: {528, 536}, {739, 840}, {1643, 891}, {2246, 899}, {3227, 18821}, {5723, 43037}, {36872, 46791}, {37129, 37131}, {42722, 41314}, {42763, 42764}, {45322, 45338}

---

X(52762) = TRIPOLAR CENTROIDAL CONJUGATE OF X(111)

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

X(52762) lies on these lines: {2, 512}, {111, 385}, {729, 9080}, {886, 3266}, {1641, 52700}, {2502, 9182}, {5971, 9187}, {17948, 17993}, {17964, 34760}, {34087, 43535}

X(52762) = X(843)-isoconjugate of X(2234)
X(52762) = X(538)-Dao conjugate of X(35087)
X(52762) = trilinear pole of line {543, 9171}
X(52762) = tripolar centroidal conjugate of X(111)
X(52762) = barycentric product X(i)*X(j) for these {i,j}: {543, 3228}, {729, 45809}, {886, 9171}, {2502, 34087}, {8371, 9150}, {14608, 17948}
X(52762) = barycentric quotient X(i)/X(j) for these {i,j}: {543, 538}, {729, 843}, {1641, 45672}, {2502, 3231}, {3228, 18823}, {8371, 9148}, {9150, 9170}, {9171, 888}, {9181, 5118}, {9182, 23342}, {14608, 51226}, {17964, 14609}, {41309, 48450}, {45809, 30736}

---

X(52763) = TRIPOLAR CENTROIDAL CONJUGATE OF X(186)

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

X(52763) lies on these lines: {2, 526}, {74, 39290}, {186, 648}, {1302, 32730}, {1464, 38340}, {1511, 2407}, {4240, 39176}, {11081, 36839}, {11086, 36840}, {15462, 30528}

X(52763) = tripolar centroidal conjugate of X(186)
X(52763) = X(2349)-isoconjugate of X(3016)
X(52763) = barycentric product X(3260)*X(32730)
X(52763) = barycentric quotient X(i)/X(j) for these {i,j}: {1495, 3016}, {32730, 74}

---

X(52764) = TRIPOLAR CENTROIDAL CONJUGATE OF X(226)

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

X(52764) lies on these lines: {2, 523}, {111, 9086}, {226, 664}, {333, 892}, {691, 35935}, {5712, 52450}, {7413, 48983}, {14246, 37086}

X(52764) = X(i)-isoconjugate of X(j) for these (i,j): {187, 1156}, {524, 34068}, {896, 2291}, {922, 1121}, {4845, 51653}, {7181, 18889}, {14432, 36141}
X(52764) = X(i)-Dao conjugate of X(j) for these (i,j): {524, 35110}, {1121, 39061}, {2291, 15899}, {4750, 40629}, {14432, 35091}
X(52764) = trilinear pole of line {527, 30574}
X(52764) = tripolar centroidal conjugate of X(226)
X(52764) = barycentric product X(i)*X(j) for these {i,j}: {527, 671}, {892, 30574}, {897, 30806}, {1055, 18023}, {1155, 46277}, {23710, 30786}
X(52764) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 2291}, {527, 524}, {671, 1121}, {897, 1156}, {923, 34068}, {1055, 187}, {1155, 896}, {1323, 7181}, {1638, 4750}, {5547, 4845}, {6366, 14432}, {6610, 51653}, {6647, 7267}, {6745, 3712}, {14413, 14419}, {23710, 468}, {24685, 4760}, {30574, 690}, {30806, 14210}, {42762, 42760}

---

X(52765) = TRIPOLAR CENTROIDAL CONJUGATE OF X(263)

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

X(52765) lies on the cubic K295 and these lines: {2, 512}, {4, 6331}, {32, 110}, {194, 11002}, {237, 2421}, {263, 1992}, {511, 2396}, {543, 36886}, {805, 47047}, {886, 20023}, {2211, 4230}, {2698, 9150}, {5360, 42717}, {5649, 32717}, {5967, 34238}, {7998, 38527}, {9155, 14251}, {9292, 43188}, {14509, 35606}, {14916, 34095}, {30530, 36163}, {34214, 35922}, {36213, 51980}, {37132, 37137}

X(52765) = isogonal conjugate of X(36822)
X(52765) = X(6786)-cross conjugate of X(511)
X(52765) = X(i)-isoconjugate of X(j) for these (i,j): {1, 36822}, {98, 2234}, {336, 46522}, {538, 1910}, {888, 36036}, {1821, 3231}, {9148, 36084}, {33875, 46273}
X(52765) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 36822}, {511, 6786}, {538, 11672}, {888, 2679}, {3231, 40601}, {5976, 30736}, {9148, 38987}
X(52765) = cevapoint of X(511) and X(6786)
X(52765) = trilinear pole of line {511, 2491}
X(52765) = tripolar centroidal conjugate of X(263)
X(52765) = crossdifference of every pair of points on line {3231, 9148}
X(52765) = barycentric product X(i)*X(j) for these {i,j}: {237, 34087}, {325, 729}, {511, 3228}, {886, 2491}, {1959, 37132}, {2799, 32717}, {3569, 9150}, {5968, 14608}, {20022, 46156}, {40810, 51510}
X(52765) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 36822}, {237, 3231}, {325, 30736}, {511, 538}, {729, 98}, {1755, 2234}, {2211, 46522}, {2421, 23342}, {2491, 888}, {3228, 290}, {3569, 9148}, {6786, 35073}, {9150, 43187}, {9155, 45672}, {9418, 33875}, {11672, 6786}, {14608, 52145}, {14966, 5118}, {32717, 2966}, {34087, 18024}, {36133, 36036}, {37132, 1821}, {41309, 5967}, {46156, 20021}, {51369, 30938}, {51510, 14382}, {51980, 14609}

---

X(52766) = TRIPOLAR CENTROIDAL CONJUGATE OF X(264)

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

X(52766) lies on these lines: {2, 525}, {3, 16077}, {264, 339}, {458, 14919}, {5054, 9410}, {5055, 31621}, {14380, 44155}, {16080, 52251}, {41009, 52646}

X(52766) = tripolar centroidal conjugate of X(264)
X(52766) = X(i)-isoconjugate of X(j) for these (i,j): {1636, 36139}, {2173, 26717}
X(52766) = X(26717)-Dao conjugate of X(36896)
X(52766) = barycentric quotient X(i)/X(j) for these {i,j}: {74, 26717}, {852, 3284}, {3331, 1495}, {32695, 32725}

---

X(52767) = TRIPOLAR CENTROIDAL CONJUGATE OF X(275)

Barycentrics    (a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(2*a^8 - 4*a^6*b^2 + a^4*b^4 + 2*a^2*b^6 - b^8 - 4*a^6*c^2 + 4*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 2*b^6*c^2 + a^4*c^4 - 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(52767) lies on these lines: {2, 523}, {275, 671}, {297, 14246}, {343, 892}, {427, 48983}, {691, 35937}, {11064, 52551}, {11427, 52450}, {52282, 52483}

X(52767) = tripolar centroidal conjugate of X(275)
X(52767) = X(896)-isoconjugate of X(51222)
X(52767) = X(i)-Dao conjugate of X(j) for these (i,j): {138, 468}, {15899, 51222}
X(52767) = barycentric product X(892)*X(42731)
X(52767) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 51222}, {42731, 690}

---

X(52768) = TRIPOLAR CENTROIDAL CONJUGATE OF X(291)

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

X(52768) lies on the cubic K296 and these lines: {2, 513}, {36, 898}, {291, 519}, {350, 889}, {536, 3999}, {739, 33854}, {899, 4607}, {1575, 39982}, {4871, 30997}, {20331, 35123}, {30942, 35043}

X(52768) = tripolar centroidal conjugate of X(291)
X(52768) = X(i)-isoconjugate of X(j) for these (i,j): {6, 51923}, {101, 46782}, {899, 2382}
X(52768) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 51923}, {536, 35123}, {1015, 46782}
X(52768) = barycentric product X(i)*X(j) for these {i,j}: {513, 46780}, {537, 3227}, {4607, 36848}, {20331, 31002}, {36872, 46795}
X(52768) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 51923}, {513, 46782}, {537, 536}, {739, 2382}, {3227, 18822}, {20331, 899}, {36848, 4728}, {36872, 46797}, {42765, 42764}, {46780, 668}
X(52768) = {X(889),X(31002)}-harmonic conjugate of X(350)

---

X(52769) = X(7)X(36)∩X(9)X(48)

Barycentrics    a*(a^5-2*(b+c)*a^4+(b+c)*(2*b^2-b*c+2*c^2)*a^2-(b^2+c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)*b*c) : :
X(52769) = X(1)+3*X(21153), 3*X(2)+X(43161), 3*X(3)-X(11495), X(3)+3*X(38031), X(9)+3*X(3576), X(20)+3*X(38037), X(40)+3*X(38316), X(142)-3*X(10165), 3*X(165)+X(43166), X(376)+3*X(38025), X(390)+7*X(3523), 3*X(1001)+X(11495), X(1001)-3*X(38031), X(3062)+3*X(5732), X(3062)+15*X(7987), X(4297)+3*X(38059), X(5732)-5*X(7987), X(5880)-3*X(38122), 3*X(10164)+X(30331), X(11495)+9*X(38031)

See César Lozada, euclid 5606.

X(52769) lies on the cubic K735 and these lines: {1, 1170}, {2, 15931}, {3, 142}, {7, 36}, {9, 48}, {10, 43175}, {11, 34879}, {20, 5259}, {21, 3062}, {24, 1890}, {30, 42356}, {35, 390}, {40, 5883}, {55, 3911}, {56, 954}, {100, 5231}, {103, 52155}, {105, 9746}, {140, 3826}, {165, 1621}, {182, 518}, {226, 37578}, {238, 991}, {376, 38025}, {404, 38052}, {405, 4297}, {411, 3624}, {474, 38204}, {480, 956}, {515, 6666}, {517, 42819}, {519, 6600}, {527, 10269}, {528, 549}, {550, 38043}, {551, 3428}, {573, 16503}, {631, 2550}, {758, 18443}, {927, 2717}, {943, 3333}, {944, 38057}, {958, 12447}, {971, 5450}, {1319, 15837}, {1350, 38048}, {1478, 6992}, {1479, 37112}, {1617, 13405}, {1699, 5284}, {1742, 15485}, {1754, 3720}, {2077, 3524}, {2078, 5218}, {2646, 5728}, {2951, 6909}, {2975, 3984}, {3059, 5440}, {3091, 25542}, {3149, 19862}, {3295, 6744}, {3305, 15064}, {3358, 6261}, {3525, 38149}, {3530, 26285}, {3579, 13374}, {3583, 7678}, {3612, 7675}, {3634, 11500}, {3636, 22770}, {3651, 8227}, {3653, 22765}, {3655, 38067}, {3683, 10167}, {3746, 8236}, {3814, 6947}, {3817, 4423}, {3822, 6827}, {3825, 6825}, {3841, 6989}, {3871, 9588}, {3877, 13253}, {3957, 15104}, {4015, 5534}, {4184, 17188}, {4189, 52653}, {4293, 8232}, {4301, 5584}, {4305, 5809}, {4312, 7280}, {4316, 30311}, {4326, 30282}, {4421, 50829}, {4428, 6244}, {4512, 10857}, {4640, 11227}, {4666, 41338}, {4679, 21635}, {4973, 21165}, {5010, 7676}, {5047, 5691}, {5119, 12736}, {5204, 30424}, {5251, 5731}, {5253, 5735}, {5258, 5686}, {5260, 38154}, {5267, 43177}, {5400, 17125}, {5428, 17768}, {5563, 11038}, {5572, 24929}, {5603, 7688}, {5657, 25439}, {5692, 18444}, {5698, 6875}, {5759, 11012}, {5762, 25557}, {5851, 38602}, {5853, 6684}, {5882, 24393}, {6008, 39227}, {6009, 44805}, {6173, 21161}, {6253, 17529}, {6594, 11715}, {6601, 45700}, {6681, 6954}, {6690, 37364}, {6712, 8299}, {6767, 14563}, {6851, 12558}, {6865, 10198}, {6889, 25639}, {6905, 20195}, {6906, 11372}, {6913, 28164}, {6914, 15726}, {6915, 34595}, {6918, 19878}, {6988, 10200}, {7489, 31672}, {7673, 11010}, {7674, 34625}, {7742, 12573}, {7988, 36002}, {7989, 17536}, {8166, 26105}, {8167, 10171}, {8257, 37611}, {8581, 37605}, {8726, 12514}, {9441, 16484}, {9708, 28236}, {9779, 35986}, {9812, 41853}, {10172, 18491}, {10177, 50371}, {10246, 23344}, {10265, 12331}, {10303, 40333}, {10392, 22760}, {10394, 37616}, {10434, 19649}, {10882, 39553}, {10884, 31803}, {11108, 19925}, {11230, 18482}, {11362, 16202}, {11491, 31423}, {12005, 26921}, {12053, 37601}, {12520, 31435}, {12560, 15803}, {12571, 37411}, {13464, 35239}, {13615, 25893}, {14100, 37600}, {14110, 30143}, {14793, 30379}, {14799, 45043}, {15178, 15570}, {15298, 37618}, {15481, 32153}, {15622, 16286}, {15624, 27473}, {15720, 38121}, {15721, 38092}, {16020, 24779}, {16203, 42885}, {16370, 43182}, {16371, 51100}, {16440, 32582}, {16825, 29016}, {16861, 34628}, {16865, 36991}, {17009, 28443}, {17542, 34648}, {17549, 50836}, {17571, 43181}, {17718, 41341}, {17917, 37441}, {18412, 30284}, {18450, 29007}, {18480, 38318}, {18481, 38108}, {18518, 31399}, {19536, 38076}, {19861, 51717}, {20330, 38028}, {20835, 40998}, {22758, 51705}, {23961, 28534}, {24466, 38060}, {24541, 37301}, {29024, 36674}, {29181, 48930}, {29668, 37619}, {30147, 31786}, {30264, 38061}, {30265, 36018}, {31793, 51715}, {31806, 37615}, {31838, 40257}, {31871, 41854}, {32622, 40566}, {32623, 40565}, {34471, 41712}, {34773, 38113}, {34890, 37719}, {35258, 46684}, {36999, 44217}, {37285, 41012}, {37426, 51118}, {37535, 38030}, {37571, 41861}, {37727, 38126}, {38859, 52511}, {40256, 40296}, {40269, 41700}, {40292, 44675}

X(52769) = midpoint of X(i) and X(j) for these {i, j}: {3, 1001}, {10, 43175}, {1385, 31658}, {3358, 6261}, {5882, 24393}, {6594, 11715}, {8257, 37611}, {11372, 43178}, {11714, 28345}, {43174, 43179}, {43177, 51090}
X(52769) = reflection of X(i) in X(j) for these (i, j): (3826, 140), (15570, 15178)
X(52769) = complement of the complement of X(43161)
X(52769) = circumnormal-isogonal conjugate of X(47641)
X(52769) = circumtangential-isogonal conjugate of X(165)
X(52769) = crossdifference of every pair of points on line {X(1769), X(6586)}
X(52769) = X(21)-beth conjugate of-X(1471
X(52769) = perspector of the circumconic {{A, B, C, X(36037), X(43190)}}
X(52769) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 1445, 30329), (3, 946, 12511), (3, 11496, 12512), (3, 31394, 24309), (3, 38031, 1001), (55, 42884, 30331), (56, 954, 5542), (405, 8273, 4297), (631, 10902, 25440), (1006, 3576, 993), (3612, 15299, 7675), (4423, 7580, 3817), (5259, 35202, 20), (5284, 7411, 1699), (5657, 34486, 25439), (5882, 38130, 24393), (6684, 10267, 8715), (6989, 48482, 3841), (8167, 19541, 10171), (18412, 37525, 30284), (30284, 37787, 18412)

---

X(52770) = X(32)X(631)∩X(182)X(524)

Barycentrics    a^8-8*(b^2+c^2)*a^6+(9*b^4+8*b^2*c^2+9*c^4)*a^4-2*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^2-2*(b^2-c^2)^2*b^2*c^2 : :
X(52770) = 5*X(631)-X(773316), 7*X(3523)+X(15589), 3*X(3524)+X(42850), 4*X(3530)+X(15598), 3*X(5054)-X(42849)

See César Lozada, euclid 5606.

X(52770) lies on these lines: {2, 8722}, {3, 3734}, {32, 631}, {83, 10303}, {98, 2482}, {114, 7865}, {140, 5171}, {182, 524}, {183, 21163}, {511, 15482}, {574, 22712}, {576, 40108}, {620, 12177}, {1078, 3523}, {1351, 44562}, {2080, 5054}, {3203, 37515}, {3398, 15720}, {3525, 12110}, {3526, 10358}, {3530, 10104}, {5034, 10519}, {5188, 11285}, {7709, 17131}, {7751, 13334}, {7761, 37451}, {7771, 37455}, {7781, 49111}, {7798, 11171}, {7810, 9744}, {7818, 43461}, {7824, 30270}, {7908, 40107}, {7913, 38227}, {7914, 37466}, {7935, 37446}, {8588, 35925}, {10165, 10800}, {10334, 33259}, {10350, 33206}, {10788, 15702}, {11623, 34229}, {11842, 15701}, {12042, 17508}, {12150, 15721}, {12203, 15717}, {14676, 18876}, {14880, 15712}, {14981, 16990}, {16187, 44215}, {18502, 46219}, {33008, 38738}

X(52770) = midpoint of X(3) and X(15271)
X(52770) = reflection of X(15491) in X(140)
X(52770) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 15819, 3734), (140, 5171, 7808), (549, 50977, 7622), (1078, 3523, 37479)

---

X(52771) = X(3)X(6)∩X(183)X(7709)

Barycentrics    a^2*(a^6-9*(b^2+c^2)*a^4+(11*b^4+14*b^2*c^2+11*c^4)*a^2-(b^2-3*c^2)*(3*b^2-c^2)*(b^2+c^2)) : :

See César Lozada, euclid 5606.

X(52771) lies on these lines: lies on these lines: {3, 6}, {30, 42849}, {114, 11287}, {183, 7709}, {237, 3066}, {549, 7610}, {631, 1975}, {1352, 8359}, {2482, 5054}, {2549, 37451}, {2782, 15271}, {3329, 39656}, {3523, 6392}, {3524, 22329}, {3526, 7820}, {3796, 37457}, {4045, 37071}, {5077, 13449}, {5921, 32990}, {6721, 33240}, {6776, 33215}, {7464, 52692}, {7771, 9755}, {7810, 11898}, {7824, 32522}, {7841, 43461}, {8182, 50979}, {8356, 9744}, {8719, 35930}, {8721, 39884}, {8724, 21358}, {9888, 33813}, {10303, 17128}, {10519, 40925}, {10601, 41275}, {11165, 50977}, {11174, 11676}, {11257, 11285}, {11472, 35934}, {14650, 33900}, {15810, 50955}, {15980, 31489}, {16043, 40330}, {16431, 37527}, {19911, 49102}, {22112, 37344}, {22712, 31859}, {27088, 38064}, {31860, 37914}, {34511, 48876}, {35302, 43650}, {36990, 37345}, {37348, 44526}, {37461, 47352}, {40916, 44420}

X(52771) = midpoint of X(3) and X(5024)
X(52771) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 1351, 8722), (3, 3398, 5023), (3, 5050, 187), (3, 9605, 5171), (3, 10983, 5188), (3, 11171, 6), (3, 35002, 31884), (3, 38225, 5585), (39, 8722, 1351), (574, 21163, 3), (2021, 18860, 2080), (5092, 9734, 3), (7824, 32522, 39646), (13349, 13350, 17508), (37479, 37512, 3), (43118, 43119, 10541)

---

X(52772) = X(2)X(98)∩X(3)X(523)

Barycentrics    2*a^12-5*(b^2+c^2)*a^10+2*(b^2+2*c^2)*(2*b^2+c^2)*a^8-(b^2+c^2)*(2*b^4+3*b^2*c^2+2*c^4)*a^6+(2*b^4-3*b^2*c^2+2*c^4)*(b^2+c^2)^2*a^4-(b^6-c^6)*(b^4-c^4)*a^2-(b^2-c^2)^4*b^2*c^2 : :
X(52772) = 5*X(631)-X(36875), 3*X(34473)+X(52035)

See César Lozada, euclid 5606.

X(52772) lies on these lines: {2, 98}, {3, 523}, {4, 250}, {6, 5915}, {20, 31990}, {30, 5467}, {376, 477}, {385, 32599}, {511, 2407}, {631, 14385}, {1316, 15928}, {1503, 24975}, {2452, 14687}, {2549, 3018}, {2793, 15566}, {3014, 50008}, {5092, 47044}, {5663, 14559}, {5968, 36166}, {7493, 47220}, {8550, 18122}, {10653, 18776}, {10654, 18777}, {14611, 33927}, {15063, 52488}, {15760, 16934}, {16237, 41204}, {18312, 34156}, {18531, 38971}, {34473, 52035}, {35520, 44134}, {48983, 50941}

X(52772) = midpoint of X(i) and X(j) for these {i, j}: {3, 34810}, {20, 52472}, {376, 9214}, {2407, 7422}
X(52772) = reflection of X(i) in X(j) for these (i, j): (4, 14356), (5967, 182)
X(52772) = crossdifference of every pair of points on the line {X(3003), X(3569)}
X(52772) = perspector of the circumconic {{A, B, C, X(2966), X(2986)}}
X(52772) = intersection, other than A, B, C, of circumconics {{A, B, C, X(98), X(15328)}} and {{A, B, C, X(125), X(14356)}}

---

X(52773) = X(2)X(99)∩X(3)X(512)

Barycentrics    a^2*((b^2+c^2)*a^8-2*(2*b^4+b^2*c^2+2*c^4)*a^6+(b^2+c^2)*(5*b^4-b^2*c^2+5*c^4)*a^4-(2*b^8+2*c^8+(b^4+8*b^2*c^2+c^4)*b^2*c^2)*a^2+2*(b^2+c^2)*b^4*c^4) : :
X(52773) = 5*X(631)-X(36874)

See César Lozada, euclid 5606.

X(52773) lies on these lines: {2, 99}, {3, 512}, {32, 249}, {182, 9145}, {187, 2421}, {237, 5118}, {538, 46777}, {631, 36874}, {2088, 36212}, {2698, 2709}, {7418, 18860}, {7815, 14382}, {9177, 15329}, {9734, 35934}, {11184, 50673}, {14981, 52451}, {15035, 15920}, {15915, 38704}, {21163, 47046}, {37184, 52765}

X(52773) = crossdifference of every pair of points on line {X(230), X(351)}
X(52773) = perspector of the circumconic {{A, B, C, X(892), X(2987)}}
X(52773) = intersection, other than A, B, C, of circumconics {{A, B, C, X(115), X(18872)}} and {{A, B, C, X(187), X(35606)}}
X(52773) = {X(574), X(5106)}-harmonic conjugate of X(14609)

---

X(52774) = X(107)X(284)∩X(109)X(577)

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

See Kadir Altintas and César Lozada, euclid 5607.

X(52774) lies on the circumcircle and these lines: {48, 108}, {99, 6514}, {100, 2289}, {101, 6056}, {107, 284}, {109, 577}, {934, 7125}, {2188, 40117}, {2713, 5060}, {28291, 37480}

X(52774) = isogonal conjugate of the polar conjugate of X(8764)
X(52774) = crosspoint of X(284) and X(2202)
X(52774) = X(i)-isoconjugate-of-X(j) for these {i, j}: {4, 44360}, {92, 8763}
X(52774) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (48, 44360), (184, 8763), {8764, 264}
X(52774) = intersection, other than A, B, C, of circumconics {{A, B, C, X(19), X(7106)}} and {{A, B, C, X(48), X(284)}}
X(52774) = barycentric product X(3)*X(8764)
X(52774) = barycentric quotient X(i)/X(j) for these (i, j): (48, 44360), (184, 8763)
X(52774) = trilinear product X(48)*X(8764)
X(52774) = trilinear quotient X(i)/X(j) for these (i, j): (3, 44360), (48, 8763)

---

X(52775) = X(107)X(42381)∩X(759)X(8747)

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

See Kadir Altintas and César Lozada, euclid 5607.

X(52775) lies on the circumcircle and these lines: {107, 42381}, {759, 8747}, {2222, 36127}, {39426, 40436}

X(52775) = isogonal conjugate of the polar conjugate of X(42381)
X(52775) = X(608)-cross conjugate of-X(23984)
X(52775) = X(42381)-reciprocal conjugate of-X(264)
X(52775) = barycentric product X(3)*X(42381)
X(52775) = trilinear product X(48)*X(42381)

---

X(52776) = X(29)X(26702)∩X(972)X(7412)

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

See Kadir Altintas and César Lozada, euclid 5607.

X(52776) lies on the circumcircle and these lines: {29, 26702}, {104, 37380}, {107, 42389}, {972, 7412}, {1295, 37741}, {2739, 37420}, {7501, 14987}, {26700, 36127}, {34398, 43363}

X(52776) = isogonal conjugate of the polar conjugate of X(42389)
X(52776) = X(607)-cross conjugate of-X(23984)
X(52776) = X(i)-isoconjugate-of-X(j) for these {i, j}: {219, 23727}, {326, 2520}, {520, 17188}, {521, 20277}, {652, 17073}
X(52776) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (34, 23727), (108, 17073), {42389, 264}
X(52776) = barycentric product X(i)*X(j) for these {i, j}: {3, 42389}, {1783, 34398}
X(52776) = barycentric quotient X(i)/X(j) for these (i, j): (34, 23727), (108, 17073)
X(52776) = trilinear product X(48)*X(42389)
X(52776) = trilinear quotient X(i)/X(j) for these (i, j): (107, 17188), (108, 20277), (278, 23727), (653, 17073), (1096, 2520)

---

X(52777) = X(42389)-RECIPROCAL CONJUGATE OF-X(264)

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

See Kadir Altintas and César Lozada, euclid 5607.

X(52777) lies on the circumcircle and this line: {107, 42393}

X(52777) = isogonal conjugate of the polar conjugate of X(42393)
X(52777) = X(42389)-reciprocal conjugate of-X(264)
X(52777) = barycentric product X(3)*X(42393)
X(52777) = trilinear product X(48)*X(42393)

---

X(52778) = X(8)X(105)∩X(106)X(997)

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

See Kadir Altintas and César Lozada, euclid 5607.

X(52778) lies on the circumcircle and these lines: {8, 105}, {106, 997}, {107, 42384}, {108, 3699}, {109, 1026}, {112, 7259}, {190, 1292}, {644, 919}, {662, 907}, {668, 927}, {692, 29289}, {727, 7084}, {739, 7123}, {741, 40403}, {883, 934}, {1018, 28847}, {1037, 8686}, {1290, 4767}, {1308, 4568}, {1310, 4557}, {1477, 7131}, {1633, 6012}, {2377, 30705}, {2737, 4076}, {2756, 51620}, {3952, 13397}, {4033, 26705}, {6065, 35185}, {8817, 15728}, {9058, 17780}, {9097, 38460}

X(52778) = isogonal conjugate of the polar conjugate of X(42384)
X(52778) = crosspoint of X(i) and X(j) for these (i, j): {1, 48329}, {59, 37577}, {1252, 12329}, {1275, 38876}
X(52778) = X(i)-cross conjugate of-X(j) for these (i, j): (63, 765), (100, 8269), (220, 1016)
X(52778) = X(i)-Dao conjugate of-X(j) for these (i, j): (9, 48398), (10, 48403)
X(52778) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 48398}, {58, 48403}, {86, 50490}, {244, 1633}, {269, 17115}
X(52778) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 48398), (37, 48403), (72, 21107), (99, 16750), (100, 4000), {42384, 264}
X(52778) = intersection, other than A, B, C, of circumconics {{A, B, C, X(8), X(644)}} and circumcircle
X(52778) = trilinear pole of the line {6, 3692}
X(52778) = Collings transform of X(i) for these i: {17597, 40609}
X(52778) = barycentric product X(i)*X(j) for these {i, j}: {100, 30701}, {346, 8269}, {644, 8817}, {646, 1037}, {668, 7123}, {765, 48070}
X(52778) = barycentric quotient X(i)/X(j) for these (i, j): (1, 48398), (37, 48403), (72, 21107), (99, 16750), (100, 4000), (101, 614)
X(52778) = trilinear product X(i)*X(j) for these {i, j}: {48, 42384}, {101, 30701}, {190, 7123}, {200, 8269}, {644, 7131}, {668, 7084}
X(52778) = trilinear quotient X(i)/X(j) for these (i, j): (2, 48398), (10, 48403), (42, 50490), (100, 614), (101, 16502), (162, 4211)

---

X(52779) = X(4)X(130)∩X(74)X(8795)

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

See Kadir Altintas and César Lozada, euclid 5607.

X(52779) lies on the circumcircle and these lines: {2, 38976}, {4, 130}, {74, 8795}, {95, 1294}, {98, 8884}, {107, 42401}, {110, 6528}, {111, 8794}, {112, 15352}, {186, 32439}, {264, 18401}, {275, 26717}, {276, 1297}, {648, 6570}, {1093, 5966}, {2052, 51222}, {2693, 43752}, {6529, 26714}, {9091, 15422}, {15412, 34538}, {19189, 48259}, {22456, 42369}, {26701, 40440}

X(52779) = anticomplement of X(38976)
X(52779) = isogonal conjugate of the polar conjugate of X(42401)
X(52779) = polar conjugate of X(17434)
X(52779) = Gibert-circumtangential conjugate of the polar conjugate of X(42369)
X(52779) = crosspoint of X(275) and X(23286)
X(52779) = X(i)-cross conjugate of-X(j) for these (i, j): (4, 34538), (107, 16813)
X(52779) = X(i)-Dao conjugate of-X(j) for these (i, j): (125, 41219), (137, 41212)
X(52779) = X(i)-isoconjugate-of-X(j) for these {i, j}: {48, 17434}, {63, 42293}, {162, 41219}, {216, 822}, {217, 24018}
X(52779) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (4, 17434), (25, 42293), (53, 34983), (54, 32320), (95, 52613), {42401, 264}
X(52779) = inverse of X(130) in polar circle
X(52779) = intersection, other than A, B, C, of circumconics circumcircle and {{A, B, C, X(130), X(15451)}}
X(52779) = trilinear pole of the line {6, 275}
X(52779) = Collings transform of X(52280)
X(52779) = orthoassociate of X(130)
X(52779) = barycentric product X(i)*X(j) for these {i, j}: {3, 42401}, {4, 42405}, {95, 15352}, {99, 8794}, {107, 276}, {184, 42369}
X(52779) = barycentric quotient X(i)/X(j) for these (i, j): (4, 17434), (25, 42293), (53, 34983), (54, 32320), (95, 52613), (107, 216)
X(52779) = trilinear product X(i)*X(j) for these {i, j}: {19, 42405}, {48, 42401}, {92, 16813}, {95, 36126}, {107, 40440}, {158, 18831}
X(52779) = trilinear quotient X(i)/X(j) for these (i, j): (19, 42293), (92, 17434), (158, 15451), (162, 418), (275, 822), (276, 24018)
X(52779) = {X(6528), X(18831)}-harmonic conjugate of X(42405)

---

X(52780) = X(2)X(196)∩X(29)X(102)

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

See Kadir Altintas and César Lozada, euclid 5607.

X(52780) lies on the circumconic with perspector X(4) and these lines: {2, 196}, {8, 1897}, {27, 19607}, {29, 102}, {85, 13149}, {124, 36127}, {158, 7020}, {189, 278}, {243, 14203}, {281, 50442}, {297, 17947}, {312, 6335}, {333, 648}, {415, 685}, {1311, 36067}, {2399, 10015}, {5928, 32677}, {6331, 28660}, {15629, 40435}, {15633, 44901}, {17917, 43043}, {17923, 34050}, {25968, 52167}

X(52780) = polar conjugate of X(515)
X(52780) = crosspoint of X(i) and X(j) for these (i, j): {4, 8755}, {7, 51616}, {273, 51359}, {653, 10015}
X(52780) = X(102)-cross conjugate of-X(34393)
X(52780) = X(i)-Dao conjugate of-X(j) for these (i, j): (9, 46974), (11, 46391), (281, 51375), (1146, 39471), (1214, 51368)
X(52780) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3, 2182}, {6, 46974}, {48, 515}, {109, 46391}, {212, 34050}
X(52780) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 46974), (4, 515), (19, 2182), (33, 51361), (34, 1455)
X(52780) = X(1745)-Zayin conjugate of-X(2182)
X(52780) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(8)}} and {{A, B, C, X(27), X(2052)}}
X(52780) = trilinear pole of the line {4, 522}
X(52780) = barycentric product X(i)*X(j) for these {i, j}: {4, 34393}, {75, 36121}, {92, 36100}, {102, 264}, {331, 15629}, {653, 2399}
X(52780) = barycentric quotient X(i)/X(j) for these (i, j): (1, 46974), (4, 515), (19, 2182), (33, 51361), (34, 1455), (102, 3)
X(52780) = trilinear product X(i)*X(j) for these {i, j}: {2, 36121}, {4, 36100}, {19, 34393}, {92, 102}, {108, 2399}, {264, 32677}
X(52780) = trilinear quotient X(i)/X(j) for these (i, j): (2, 46974), (4, 2182), (92, 515), (102, 48), (108, 2425), (158, 8755)

---

X(52781) = X(7)X(281)∩X(27)X(103)

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

See Kadir Altintas and César Lozada, euclid 5607.

X(52781) lies on the circumconic with perspector X(4), the circumhyperbola dual of Yff parabola and these lines: {2, 1897}, {4, 42073}, {7, 281}, {27, 103}, {63, 34402}, {75, 6335}, {86, 648}, {92, 1088}, {273, 4858}, {278, 36620}, {297, 6650}, {310, 6331}, {423, 685}, {673, 1861}, {675, 40116}, {677, 2989}, {1146, 36118}, {1440, 7003}, {2338, 40942}, {4242, 25954}, {4373, 52283}, {6336, 6548}, {7452, 24619}, {11109, 14621}, {11331, 27494}, {15466, 15467}, {16813, 32657}, {17555, 39721}, {17917, 38254}, {18026, 31640}, {18815, 37805}, {20028, 31909}, {21453, 52412}

X(52781) = isotomic conjugate of X(26006)
X(52781) = polar conjugate of X(516)
X(52781) = crosspoint of X(i) and X(j) for these (i, j): {4, 1886}, {92, 5236}
X(52781) = X(i)-cross conjugate of-X(j) for these (i, j): (103, 18025), (676, 36118)
X(52781) = X(i)-Dao conjugate of-X(j) for these (i, j): (1, 51376), (37, 51366), (118, 47407), (1086, 39470)
X(52781) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3, 910}, {48, 516}, {56, 51376}, {184, 30807}, {212, 43035}
X(52781) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (4, 516), (9, 51376), (10, 51366), (19, 910), (27, 14953)
X(52781) = intersection, other than A, B, C, of circumconics circumhyperbola dual of Yff parabola and {{A, B, C, X(10), X(25935)}}
X(52781) = trilinear pole of the line {4, 514}
X(52781) = barycentric product X(i)*X(j) for these {i, j}: {4, 18025}, {75, 36122}, {92, 36101}, {103, 264}, {281, 52156}, {318, 43736}
X(52781) = barycentric quotient X(i)/X(j) for these (i, j): (4, 516), (9, 51376), (10, 51366), (19, 910), (27, 14953), (33, 41339)
X(52781) = trilinear product X(i)*X(j) for these {i, j}: {2, 36122}, {4, 36101}, {19, 18025}, {33, 52156}, {92, 103}, {158, 1815}
X(52781) = trilinear quotient X(i)/X(j) for these (i, j): (4, 910), (8, 51376), (92, 516), (103, 48), (158, 1886), (264, 30807)

---

X(52782) = X(2)X(58)∩X(3)X(17327)

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

X(52782) lies on the on Kiepert circumhyperbola of the medial triangle and these lines: {1, 41820}, {2, 58}, {3, 17327}, {8, 31514}, {10, 3666}, {39, 1213}, {121, 1054}, {141, 1125}, {386, 5224}, {404, 40592}, {631, 44736}, {993, 37255}, {1329, 3634}, {2049, 48835}, {2885, 3828}, {3159, 4364}, {3216, 41809}, {3624, 18139}, {3626, 48847}, {3678, 4260}, {3763, 16844}, {3821, 15349}, {3841, 47514}, {4205, 50605}, {4472, 24470}, {4658, 37653}, {4708, 5044}, {4725, 52495}, {4745, 48845}, {4778, 23814}, {5030, 17582}, {5248, 8299}, {5737, 20083}, {5743, 20108}, {6292, 17698}, {7483, 24938}, {11110, 17307}, {13725, 48863}, {15309, 27929}, {16825, 25499}, {16828, 32781}, {16848, 17052}, {17300, 28620}, {17385, 31445}, {17758, 39580}, {19270, 24931}, {19858, 32784}, {20532, 39016}, {20985, 25512}, {21251, 51073}, {23157, 45990}, {24206, 50418}, {25354, 27784}, {25440, 37329}, {28619, 32863}, {31253, 50432}, {33067, 41812}, {34573, 50205}, {37339, 51579}, {48836, 49734}, {48866, 49728}

X(52782) = complement of X(43531)
X(52782) = complement of the isogonal conjugate of X(386)
X(52782) = complement of the isotomic conjugate of X(5224)
X(52782) = medial-isogonal conjugate of X(50605)
X(52782) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 50605}, {3, 41340}, {6, 31993}, {31, 17398}, {48, 7536}, {56, 5439}, {100, 834}, {163, 23879}, {386, 10}, {469, 20305}, {692, 6590}, {834, 11}, {1018, 52586}, {3876, 1329}, {5224, 2887}, {8637, 1015}, {14349, 116}, {23879, 21253}, {28606, 141}, {33935, 626}, {33948, 21260}, {33949, 17046}, {34819, 41850}, {42664, 8287}, {44103, 226}, {45746, 21252}, {47842, 125}, {50488, 115}, {52615, 17761}
X(52782) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 17398}, {99, 23879}, {8050, 834}
X(52782) = X(2)-Dao conjugate of X(17398)
X(52782) = barycentric product X(i)*X(j) for these {i,j}: {5224, 17398}, {33948, 50522}
X(52782) = barycentric quotient X(i)/X(j) for these {i,j}: {17398, 43531}, {50522, 43927}
X(52782) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 13728, 48843}, {141, 50409, 1125}, {25498, 37594, 1125}, {49728, 50318, 48866}

---

X(52783) = X(7)X(21)∩X(12)X(57)

Barycentrics    (a + b - c)*(a - b + c)*(2*a^2 + 2*a*b + b^2 + 2*a*c + 2*b*c + c^2) : :
X(52783) = 2 X[1] + 3 X[11246], X[1] + 4 X[24470], 3 X[11246] - 8 X[24470], X[65] - 6 X[553], X[65] + 4 X[4298], 2 X[65] + 3 X[5434], 3 X[65] + 2 X[10106], 4 X[65] + X[10944], 3 X[553] + 2 X[4298], 4 X[553] + X[5434], 9 X[553] + X[10106], 24 X[553] + X[10944], 2 X[3555] + 3 X[34612], 2 X[3874] + 3 X[11112], 8 X[4298] - 3 X[5434], 6 X[4298] - X[10106], 16 X[4298] - X[10944], 9 X[5434] - 4 X[10106], 6 X[5434] - X[10944], 8 X[10106] - 3 X[10944], 2 X[10914] + 3 X[34749], 2 X[17647] + 3 X[24473], 3 X[210] - 8 X[12436], 3 X[354] + 2 X[4292], 6 X[354] - X[6284], 4 X[4292] + X[6284], 4 X[942] + X[7354], 6 X[942] - X[10572], 3 X[942] + 2 X[31776], 3 X[7354] + 2 X[10572], 3 X[7354] - 8 X[31776], X[10572] + 4 X[31776], X[1317] + 4 X[24465], 2 X[1770] + 3 X[3058], X[1770] + 4 X[5045], 3 X[3058] - 8 X[5045], 8 X[3812] - 3 X[34606], 6 X[5902] - X[10950], 3 X[5902] + 2 X[18990], X[10950] + 4 X[18990], 3 X[5919] - 8 X[12577], X[6253] + 4 X[12675], 6 X[10202] - X[11827], X[10483] + 4 X[12433], 4 X[12005] + X[37468], 2 X[15171] - 7 X[50190], 4 X[24471] + X[39897], 4 X[31794] + X[45287]

X(52783) lies on these lines: {1, 550}, {4, 4860}, {5, 3337}, {7, 21}, {10, 4031}, {11, 3338}, {12, 57}, {30, 18398}, {35, 37703}, {36, 6147}, {46, 15888}, {65, 519}, {79, 496}, {84, 7965}, {210, 12436}, {226, 5433}, {278, 31901}, {354, 4292}, {388, 3617}, {495, 3336}, {498, 37545}, {516, 17609}, {527, 25917}, {528, 3889}, {549, 37731}, {938, 12943}, {942, 7354}, {982, 49745}, {1056, 37567}, {1086, 1468}, {1106, 6354}, {1125, 3982}, {1155, 21620}, {1193, 17365}, {1317, 3340}, {1319, 3671}, {1329, 27003}, {1385, 11551}, {1388, 37299}, {1420, 51105}, {1451, 52023}, {1463, 17771}, {1471, 7299}, {1478, 5708}, {1479, 18541}, {1482, 35249}, {1770, 3058}, {1788, 11237}, {1836, 3333}, {2099, 3600}, {2975, 26842}, {3218, 25466}, {3254, 7091}, {3296, 4294}, {3303, 3474}, {3304, 4295}, {3339, 4668}, {3361, 4654}, {3475, 5217}, {3487, 5204}, {3522, 30340}, {3584, 50825}, {3636, 39782}, {3715, 17582}, {3748, 31730}, {3782, 37607}, {3812, 34606}, {3813, 20292}, {3876, 5852}, {3911, 31253}, {3916, 51706}, {4003, 5717}, {4005, 5850}, {4032, 28516}, {4297, 44840}, {4299, 10543}, {4305, 50819}, {4308, 36005}, {4312, 12701}, {4315, 11011}, {4340, 17599}, {4641, 24178}, {4816, 41687}, {4870, 51109}, {4973, 7483}, {4999, 31019}, {5196, 39751}, {5219, 7294}, {5220, 37462}, {5247, 40688}, {5249, 24953}, {5253, 17483}, {5425, 34773}, {5432, 13407}, {5441, 15935}, {5536, 37424}, {5542, 37080}, {5558, 30332}, {5563, 39542}, {5719, 7280}, {5722, 44286}, {5762, 7987}, {5804, 37001}, {5805, 10085}, {5901, 37587}, {5902, 10950}, {5905, 25524}, {5919, 12577}, {6253, 12675}, {6691, 31053}, {7173, 9612}, {7263, 27368}, {7330, 7958}, {7680, 26877}, {7951, 34753}, {8581, 41538}, {9578, 51066}, {9579, 10980}, {9657, 18391}, {9670, 10580}, {10072, 50806}, {10202, 11827}, {10483, 12433}, {10826, 50799}, {10910, 13388}, {10911, 13389}, {11509, 36003}, {11544, 18393}, {11552, 22791}, {12005, 37468}, {12053, 30424}, {12699, 51816}, {13161, 37520}, {14022, 31249}, {15171, 50190}, {15803, 17718}, {16137, 37525}, {16152, 27197}, {17602, 37522}, {17724, 37603}, {18421, 37738}, {20614, 39792}, {24231, 37539}, {24471, 39897}, {24954, 28609}, {25681, 31164}, {25914, 26223}, {28146, 50191}, {28628, 31157}, {29590, 41245}, {30264, 37615}, {30282, 41870}, {30350, 41864}, {31260, 31266}, {31794, 45287}, {33103, 37608}, {34860, 50289}, {35010, 37364}, {35576, 42289}, {36487, 51552}, {37534, 50031}, {37600, 43180}, {41011, 52541}, {50307, 50620}

X(52783) = barycentric product X(i)*X(j) for these {i,j}: {7, 17398}, {664, 50522}
X(52783) = barycentric quotient X(i)/X(j) for these {i,j}: {17398, 8}, {50522, 522}
X(52783) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 15228, 10386}, {1, 24470, 11246}, {7, 56, 3649}, {7, 1434, 3665}, {56, 3649, 15950}, {57, 4355, 10404}, {57, 5290, 24914}, {57, 10404, 12}, {65, 4298, 5434}, {65, 5434, 10944}, {65, 20615, 1401}, {226, 32636, 5433}, {354, 4292, 6284}, {388, 5221, 40663}, {388, 21454, 5221}, {553, 4298, 65}, {942, 31776, 10572}, {1056, 37567, 45081}, {1770, 5045, 3058}, {1836, 3333, 37722}, {3361, 4654, 11375}, {3361, 11375, 5298}, {3474, 11037, 3303}, {4299, 15934, 10543}, {5290, 24914, 12}, {5902, 18990, 10950}, {9612, 17728, 7173}, {10404, 24914, 5290}, {10572, 31776, 7354}, {13407, 37582, 5432}

---

X(52784) = X(10)X(244)∩X(58)X(100)

Barycentrics    a*(b + c)*(a^2*b^2 + a*b^3 - 4*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(52784) lies on these lines: {10, 244}, {38, 49524}, {42, 1386}, {58, 100}, {291, 4651}, {661, 21893}, {678, 50587}, {756, 25354}, {1054, 31855}, {2254, 22313}, {3121, 35309}, {3214, 37582}, {3315, 31514}, {5096, 20999}, {14752, 40521}, {20331, 21858}, {21035, 25322}, {22199, 40585}, {26115, 33115}

X(52784) = X(1019)-Ceva conjugate of X(37)
X(52784) = X(4033)-Dao conjugate of X(4103)

---

X(52785) = X(2)X(3824)∩X(58)X(748)

Barycentrics    5*a^3*b + 10*a^2*b^2 + 7*a*b^3 + 2*b^4 + 5*a^3*c + 17*a^2*b*c + 18*a*b^2*c + 6*b^3*c + 10*a^2*c^2 + 18*a*b*c^2 + 8*b^2*c^2 + 7*a*c^3 + 6*b*c^3 + 2*c^4 : :
X(52785) = 2 X[10] - 3 X[31321]

X(52785) llies on these lines: {2, 3824}, {10, 4970}, {58, 748}, {69, 1386}, {1125, 25056}, {2899, 19808}, {3632, 31514}, {3868, 37039}, {3876, 4260}, {4261, 52706}, {4708, 19767}, {5257, 25082}, {5284, 52018}, {11015, 13725}, {17549, 43281}, {34772, 50410}

---

X(52786) = X(2)X(1386)∩X(58)X(750)

Barycentrics    3*a^2*b + 3*a*b^2 + 2*b^3 + 3*a^2*c + 3*a*b*c + 4*b^2*c + 3*a*c^2 + 4*b*c^2 + 2*c^3 : :

X(52786) lies on these lines: {2, 1386}, {10, 4850}, {58, 750}, {100, 17308}, {443, 3436}, {1054, 10713}, {3120, 32784}, {3240, 17239}, {3315, 48851}, {3679, 31514}, {3681, 4260}, {3773, 28606}, {3828, 25057}, {3952, 17250}, {4708, 9330}, {5047, 52018}, {5224, 26251}, {5297, 17327}, {9352, 19808}, {17126, 17385}, {17228, 29822}, {17238, 46897}, {17240, 27811}, {17303, 33086}, {17307, 26227}, {24542, 29613}, {24988, 29576}, {28220, 31992}, {32776, 48641}

---

X(52787) = X(4)X(1369)∩X(25)X(183)

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

X(52787) lies on these lines: {4, 1369}, {25, 183}, {76, 5064}, {83, 8792}, {338, 1194}, {339, 5133}, {427, 1235}, {428, 7767}, {2970, 42394}, {7499, 30737}, {7539, 41009}, {7667, 26166}, {37439, 41005}, {42554, 46026}, {44146, 52285}

X(52787) = isotomic conjugate of the isogonal conjugate of X(46026)
X(52787) = polar conjugate of the isotomic conjugate of X(42554)
X(52787) = polar conjugate of the isogonal conjugate of X(6292)
X(52787) = X(i)-Ceva conjugate of X(j) for these (i,j): {264, 44142}, {44142, 46026}
X(52787) = X(i)-isoconjugate of X(j) for these (i,j): {9247, 40425}, {41435, 46289}
X(52787) = cevapoint of X(6292) and X(46026)
X(52787) = barycentric product X(i)*X(j) for these {i,j}: {4, 42554}, {76, 46026}, {92, 20898}, {141, 44142}, {264, 6292}, {308, 28666}, {427, 39998}, {428, 8024}, {1235, 3589}, {1969, 17457}, {11205, 18022}, {18027, 22078}, {21038, 44129}, {44091, 52568}
X(52787) = barycentric quotient X(i)/X(j) for these {i,j}: {141, 41435}, {264, 40425}, {427, 3108}, {428, 251}, {1235, 10159}, {3589, 1176}, {5007, 10547}, {6292, 3}, {7767, 28724}, {11205, 184}, {17193, 1790}, {17457, 48}, {20898, 63}, {21038, 71}, {21126, 1459}, {21817, 228}, {22078, 577}, {28666, 39}, {39998, 1799}, {41676, 7953}, {42554, 69}, {44091, 46288}, {44142, 83}, {46026, 6}
X(52787) = {X(39998),X(44142)}-harmonic conjugate of X(428)

---

X(52788) = X(2)X(46288)∩X(141)X(1194)

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

X(52788) lies on these lines: {2, 46288}, {141, 1194}, {206, 7914}, {626, 3589}, {3096, 16285}, {3763, 7664}, {6292, 8265}, {6676, 34573}, {10292, 16890}

X(52788) = complement of the isogonal conjugate of X(1180)
X(52788) = complement of the isotomic conjugate of X(3096)
X(52788) = medial-isogonal conjugate of X(8891)
X(52788) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 8891}, {31, 7889}, {163, 50552}, {1180, 10}, {3096, 2887}, {16285, 37}
X(52788) = X(2)-Ceva conjugate of X(7889)
X(52788) = barycentric product X(3096)*X(7889)

---

X(52789) = X(4)X(83)∩X(25)X(3763)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 + c^4) : :
X(52789) = 6 X[428] - X[1843], X[5095] + 4 X[46682], 4 X[6756] + X[12294], 2 X[11574] + 3 X[34603], X[11750] - 6 X[38136], 2 X[13419] + 3 X[14853], 6 X[13490] - X[37511]

X(52789) lies on these lines: {4, 83}, {25, 3763}, {66, 34417}, {141, 10301}, {265, 18535}, {427, 44091}, {428, 524}, {3589, 52285}, {3620, 6995}, {3867, 44102}, {5059, 45816}, {5095, 12167}, {6756, 10625}, {7391, 19137}, {7394, 19126}, {7408, 8541}, {11574, 34603}, {11750, 38136}, {12140, 19140}, {13419, 14853}, {13490, 37511}, {13596, 48892}, {14483, 39874}, {15004, 43726}, {19118, 51185}, {19121, 37349}, {19125, 21659}, {20806, 48901}, {24206, 34484}, {44803, 48889}

X(52789) = polar conjugate of the isotomic conjugate of X(7889)
X(52789) = barycentric product X(4)*X(7889)
X(52789) = barycentric quotient X(7889)/X(69)
X(52789) = {X(4),X(1974)}-harmonic conjugate of X(46026)

---

X(52790) = X(1)X(3307)∩X(9)X(80)

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

See Ivan Pavlov, euclid 5616.

X(52790) lies on these lines: {1,3307}, {9,80}, {516,2449}, {1699,14504}, {3308,5223}, {12699,23517}

X(52790) = reflection of X(1) in X(24647)
X(52790) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3307, 24647, 1}

---

X(52791) = X(1)X(3308)∩X(9)X(80)

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

See Ivan Pavlov, euclid 5616.

X(52791) lies on these lines: {1,3308}, {9,80}, {516,2448}, {1699,14503}, {3307,5223}, {12699,23477}

X(52791) = reflection of X(1) in X(24646)
X(52791) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3308, 24646, 1}

---

X(52792) = ISOGONAL CONJUGATE OF X(52792)

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

See Angel Montesdeoca and Ivan Pavlov, euclid 5622.

X(52792) lies on the circumconics {{A,B,C,X(1),X(14483)}}, {{A,B,C,X(4),X(60)}}, {{A,B,C,X(6),X(5556)}}, {{A,B,C,X(7),X(1059)}}, {{A,B,C,X(8),X(52518)}} and these lines: { }

X(52792) = isogonal conjugate of X(52793)
X(52792) = isotomic conjugate of X(52794)

---

X(52793) = X(3)X(12)∩X(11)X(35)

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

See Angel Montesdeoca and Ivan Pavlov, euclid 5622.

X(52793) lies on these lines: {1,549}, {2,3847}, {3,12}, {5,5010}, {8,31157}, {10,6174}, {11,35}, {21,3035}, {30,3614}, {36,3530}, {40,15950}, {55,631}, {56,3523}, {57,18217}, {65,10164}, {100,4999}, {165,11375}, {182,9666}, {187,31460}, {371,13958}, {372,13901}, {376,10895}, {388,15717}, {404,6690}, {452,31246}, {474,37601}, {484,37737}, {495,7280}, {496,14869}, {497,10303}, {499,3058}, {529,5303}, {546,4324}, {548,3585}, {550,7951}, {551,4004}, {632,7741}, {952,37616}, {993,21031}, {999,31452}, {1001,6921}, {1125,37568}, {1145,51111}, {1151,19027}, {1152,19028}, {1155,3649}, {1317,1385}, {1319,45081}, {1329,4189}, {1361,38784}, {1376,6910}, {1387,37563}, {1469,21167}, {1479,3526}, {1621,6691}, {1656,4302}, {1737,10543}, {1836,35242}, {1837,30282}, {2066,18966}, {2077,52265}, {2093,31425}, {2307,42945}, {2475,6668}, {2646,6684}, {2886,31260}, {3022,38772}, {3023,38748}, {3024,38793}, {3027,38737}, {3028,38727}, {3053,31497}, {3057,10165}, {3085,3524}, {3090,12953}, {3149,7965}, {3295,15720}, {3299,35256}, {3301,35255}, {3303,5281}, {3311,9648}, {3336,5719}, {3338,37703}, {3522,10588}, {3525,4294}, {3528,10590}, {3533,10591}, {3576,10944}, {3582,11812}, {3583,3628}, {3584,12100}, {3601,24914}, {3612,10950}, {3619,39892}, {3624,12701}, {3634,15670}, {3683,6700}, {3746,12108}, {3816,17566}, {3884,34123}, {3897,8256}, {3911,6744}, {3925,7483}, {4187,31235}, {4188,25466}, {4293,10299}, {4304,17606}, {4316,33923}, {4330,16239}, {4413,6857}, {4421,10527}, {4423,17567}, {4512,24954}, {4640,27385}, {4855,26066}, {4861,13996}, {5023,9596}, {5085,39897}, {5122,13407}, {5131,24470}, {5160,44214}, {5172,6986}, {5206,31501}, {5219,16192}, {5221,5703}, {5229,10304}, {5248,13747}, {5251,47742}, {5252,7987}, {5259,52264}, {5267,17757}, {5332,9606}, {5414,18965}, {5418,19030}, {5420,19029}, {5440,21677}, {5441,12019}, {5443,28174}, {5444,5901}, {5445,37730}, {5499,8068}, {5520,47401}, {5552,34606}, {5559,12735}, {5657,34471}, {5690,37525}, {5697,38028}, {5718,37603}, {5842,6952}, {5844,24926}, {6019,38804}, {6036,15452}, {6147,34502}, {6154,24390}, {6200,9649}, {6221,13963}, {6253,6833}, {6285,10192}, {6396,9646}, {6398,13905}, {6409,44622}, {6410,31472}, {6459,13954}, {6460,13897}, {6696,26888}, {6713,37621}, {6738,50829}, {6796,34879}, {6842,24466}, {6853,34474}, {6863,11826}, {6880,11496}, {6882,33862}, {6897,10953}, {6911,7958}, {6940,37564}, {6942,7680}, {6950,18242}, {6954,10310}, {6956,36999}, {6977,11500}, {7031,31406}, {7127,16772}, {7355,23328}, {7485,9672}, {7676,17535}, {7824,26629}, {7907,26590}, {8144,34477}, {8164,9657}, {8273,11501}, {8359,30104}, {8567,12940}, {8588,9650}, {8703,10483}, {8722,10797}, {8819,48389}, {9342,15674}, {9540,19037}, {9580,34595}, {9588,13384}, {9598,37637}, {9652,10984}, {9656,21734}, {9660,10577}, {9662,42215}, {9663,18996}, {9668,46219}, {9669,15694}, {9670,10589}, {9945,47033}, {10039,13624}, {10053,38750}, {10056,15693}, {10058,38762}, {10065,38794}, {10072,15701}, {10086,38739}, {10088,38728}, {10198,16371}, {10267,10949}, {10385,15721}, {10386,37720}, {10406,35203}, {10523,44222}, {10528,11194}, {10529,34699}, {10572,11231}, {10573,37606}, {10587,40726}, {10593,11539}, {10902,10957}, {10955,14803}, {10956,37561}, {11011,43174}, {11237,15692}, {11238,15702}, {11246,11374}, {11392,15750}, {11399,35486}, {11681,17548}, {12047,31663}, {12743,38133}, {12764,37162}, {12896,34128}, {12903,15051}, {13077,15819}, {13273,37163}, {13367,26956}, {13405,32636}, {13935,19038}, {13995,41542}, {14101,38615}, {14986,15708}, {15513,31476}, {15704,18513}, {15803,17718}, {15908,26285}, {16370,26364}, {17502,45287}, {17530,20104}, {17549,27529}, {17595,36573}, {17603,41538}, {17605,31730}, {18982,21163}, {19872,50202}, {21620,52638}, {24851,37691}, {25681,35258}, {26201,27778}, {26363,34612}, {26481,37364}, {26561,33004}, {26686,33259}, {27020,35297}, {31142,51576}, {31397,37605}, {31447,50194}, {31508,50443}, {35004,38033}, {37572,39542}, {37573,37634}, {37706,38112}, {37719,44682}, {44254,46816}, {45885,48897}, {46683,48375}

X(52793) = isogonal conjugate of X(52792)
X(52793) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5442, 34753}, {2, 5217, 6284}, {2, 6284, 7173}, {3, 12, 15326}, {3, 31479, 4299}, {3, 498, 7354}, {3, 5432, 12}, {5, 5010, 15338}, {11, 140, 7294}, {35, 140, 11}, {55, 5433, 37722}, {55, 631, 5433}, {100, 37291, 4999}, {372, 31499, 13901}, {495, 15712, 7280}, {498, 7354, 12}, {549, 4995, 5298}, {1155, 13411, 3649}, {1376, 6910, 24953}, {2646, 6684, 40663}, {3085, 3524, 5204}, {3085, 5204, 5434}, {3522, 10588, 12943}, {3523, 5218, 56}, {3612, 26446, 10950}, {3624, 35445, 12701}, {4861, 32157, 13996}, {5131, 37731, 24470}, {5251, 47742, 50038}, {5281, 7288, 3303}, {5326, 15338, 5}, {5432, 7354, 498}, {5444, 11010, 5901}, {5690, 37525, 37734}, {6842, 26086, 24466}, {7483, 25440, 3925}, {8703, 10592, 10483}, {9588, 13384, 41687}, {30282, 31423, 1837}, {37720, 51817, 10386}

---

X(52794) = X(8)X(44149)∩X(319)X(1232)

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

See Angel Montesdeoca and Ivan Pavlov, euclid 5622.

X(52794) lies on these lines: {8,44149}, {69,10522}, {75,44148}, {76,32099}, {311,17360}, {319,1232}, {17270,26541}, {17287,51481}, {17296,26592}

X(52794) = isotomic conjugate of X(52792)
X(52794) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {319, 1232, 34387}

---

X(52795) = X(5)X(993)∩X(10)X(547)

Barycentrics    2*a^4-7*a^2*b^2+5*b^4-2*a*b^2*c-7*a^2*c^2-2*a*b*c^2-10*b^2*c^2+5*c^4 : :

See Angel Montesdeoca and Ivan Pavlov, euclid 5622.

X(52795) lies on these lines: {1,38027}, {2,3847}, {5,993}, {10,547}, {11,6668}, {442,6667}, {498,3829}, {529,3614}, {958,5056}, {960,10171}, {1125,12019}, {1329,3090}, {1376,5067}, {1656,2886}, {2475,7294}, {2476,6691}, {2550,46936}, {3035,3628}, {3036,9956}, {3813,10528}, {3814,35018}, {3850,5267}, {4187,31262}, {5046,31260}, {5055,26363}, {5087,18253}, {5141,5433}, {5154,24953}, {5326,52367}, {5443,5855}, {5836,10172}, {6690,7741}, {6860,42356}, {6931,31245}, {6933,10958}, {6981,15843}, {7486,9710}, {7988,26066}, {9688,23275}, {9711,31493}, {10593,49736}, {12433,38062}, {15022,30478}, {15171,20104}, {15699,49732}, {16922,26582}, {20172,32998}, {26687,33009}, {37718,38410}, {37735,38058}

X(52795) = isogonal conjugate of the 1st Saragossa point of X(52792)
X(52795) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11, 7504, 6668}

---

X(52796) = X(389)X(3634)∩X(515)X(3819)

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

See Angel Montesdeoca and Ivan Pavlov, euclid 5622.

X(52796) lies on these lines: {10,11793}, {165,15030}, {185,31423}, {389,3634}, {511,10175}, {515,3819}, {517,10170}, {1216,9956}, {1698,5562}, {2807,5891}, {2818,10176}, {3576,5650}, {3690,5535}, {3917,5587}, {5447,18480}, {5651,15177}, {5818,7999}, {5889,19877}, {5907,6684}, {5943,10172}, {6000,10164}, {7989,45186}, {9780,11444}, {10110,31737}, {11231,13754}, {11381,35242}, {11695,31732}, {12512,13474}, {13348,31673}, {14831,19876}, {15067,38042}, {15644,19925}, {18357,32142}, {22299,37281}, {23841,31399}, {28150,46847}, {31730,44870}

X(52796) = midpoint of X(i) in X(j) for these {i,j}: {165, 15030}, {3917, 5587}, {5891, 26446}, {15067, 38042}
X(52796) = reflection of X(5943) in X(10172)
X(52796) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3634, 31752, 389}, {5891, 26446, 2807}, {31399, 31738, 23841}, {31732, 51073, 11695}

---

X(52797) = X(1)X(166)∩X(40)X(164)

Barycentrics    (2*a-b-c)*sin(A/2)-a*(sin(B/2)+sin(C/2) ) : :
X(52797) = 3*X(164)+X(12879), 3*X(188)-X(12879), 3*X(3576)-X(9837)

See César Lozada, euclid 5623.

X(52797) lies on these lines: {1, 166}, {3, 18258}, {40, 164}, {104, 3659}, {258, 32183}, {515, 2090}, {1128, 1130}, {1158, 31790}, {3576, 7028}, {3651, 12694}, {5731, 7048}, {8078, 12908}, {8092, 18291}, {8108, 8372}, {10234, 12844}, {10493, 10495}, {11495, 12518}, {12513, 12523}, {13443, 15495}

X(52797) = midpoint of X(i) and X(j) for these {i, j}: {40, 9836}, {164, 188}
X(52797) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (164, 363, 40), (164, 505, 20183), (1128, 1130, 16015)

---

X(52798) = ISOGONAL CONJUGATE OF X(52799)

Barycentrics    (a^2-b^2+c^2-2*(a+b+c)*(a*sin(C/2)+c*sin(A/2)))*(a^2+b^2-c^2-2*(a+b+c)*(a*sin(B/2)+b*sin(A/2))) : :

See César Lozada, euclid 5623.

X(52798) lies on this line: {46, 173}

X(52798) = isogonal conjugate of X(52799)

---

X(52799) = X(1)X(164)∩X(35)X(259)

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

See César Lozada, euclid 5623.

X(52799) lies on these lines: {1, 164}, {3, 16012}, {35, 259}, {90, 7707}, {173, 46370}, {8112, 12705}, {8351, 43192}, {11923, 30408}, {15803, 45086}

X(52799) = isogonal conjugate of X(52798)

---

X(52800) = X(1)X(188)∩X(166)X(8089)

Barycentrics    a*(4*b*sin(C/2)+4*c*sin(B/2)-3*a+b+c) : :

See César Lozada, euclid 5623.

X(52800) lies on these lines: {1, 188}, {8, 10505}, {164, 10215}, {166, 8089}, {236, 3973}, {266, 13092}, {363, 8078}, {519, 5430}, {1743, 16016}, {10231, 12879}

X(52800) = {X(164), X(24242)}-harmonic conjugate of X(20114)

---

X(52801) = ISOGONAL CONJUGATE OF X(52802)

Barycentrics    a*(1+4*sin(B/2))*(1+4*sin(C/2)) : :

See César Lozada, euclid 5623.

X(52801) lies on this line: {188, 3632}

X(52801) = isogonal conjugate of X(52802)

---

X(52802) = X(1)X(164)∩X(36)X(42622)

Barycentrics    a*(1+4*sin(A/2)) : :

See César Lozada, euclid 5623.

X(52802) lies on these lines: {1, 164}, {36, 42622}, {188, 30408}, {361, 49997}

X(52802) = isogonal conjugate of X(52801)
X(52802) = {X(266), X(1130)}-harmonic conjugate of X(1)

---

X(52803) = CENTER AND PERSPECTOR OF THE LOZADA-SODDY CONIC

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

Let S_a, S_b, S_c be the A-, B-, C- Soddy circles of ABC, respectively, cutting BC, CA, AB in A', B', C', respectively (A'B'C' is the intouch triangle of ABC). Let A_b be the intersection, other than C', of B'C' and S_b, and let A_c be the intersection, other than B', of B'C' and S_c. Build B_c, B_a, C_a, C_b cyclically. Then these six points lie on conic, here named the Lozada-Soddy conic.

The Lozada-Soddy conic is an ellipse, parabola or hyperbola accordingly (16*R^2-r^2)*(4*R-r)*r^3-R^2*S^2 is positive, zero or negative, respectively. It has coinciding center and perspector X(52803). No remarkable points were found on it. The polar triangle of ABC with respect to this conic is homothetic to ABC.

X(52803) lies on these lines: {7, 13601}, {57, 45204}, {144, 43760}, {1014, 7419}, {3057, 42861}, {3663, 14261}, {8051, 18228}, {15728, 30236}

X(52803) = isogonal conjugate of X(52804)
X(52803) = isotomic conjugate of X(42020)
X(52803) = cyclocevian conjugate of X(36606)
X(52803) = crosspoint of X(i) and X(j) for these (i, j): {2, 3445}, {57, 3057}
X(52803) = X(223)-Dao conjugate of-X(36846)
X(52803) = X(i)-isoconjugate-of-X(j) for these {i, j}: {41, 1997}, {55, 36846}, {200, 41426}
X(52803) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (7, 1997), (57, 36846), (1407, 41426)
X(52803) = intersection, other than A, B, C, of circumconics {A, B, C, X(2), X(3663)} and {A, B, C, X(4), X(106)}
X(52803) = trilinear pole of the line {2487, 3669}
X(52803) = barycentric quotient X(i)/X(j) for these (i, j): (7, 1997), (57, 36846), (1407, 41426)
X(52803) = trilinear quotient X(i)/X(j) for these (i, j): (7, 36846), (85, 1997), (269, 41426)

---

X(52804) = ISOGONAL CONJUGATE OF X(52803)

Barycentrics    a^2*(-a+b+c)*(a^3-(b+c)*a^2-(b^2-8*b*c+c^2)*a+(b+c)*(b^2-4*b*c+c^2)) : :
X(52804) = X(200)-3*X(3158) = 3*X(24392)-5*X(31249)\

X(52804) lies on these lines: {1, 11505}, {3, 519}, {6, 51476}, {9, 55}, {25, 23858}, {56, 2136}, {100, 1617}, {145, 1476}, {197, 36641}, {346, 1261}, {404, 12632}, {518, 6244}, {528, 19541}, {997, 3295}, {999, 3880}, {1001, 20103}, {1012, 34619}, {1376, 5853}, {1604, 17314}, {2192, 3190}, {2801, 13205}, {3052, 3939}, {3169, 5120}, {3189, 5687}, {3304, 3680}, {3474, 5856}, {3633, 40293}, {3746, 16293}, {3754, 7373}, {3811, 6001}, {3813, 16408}, {3830, 13272}, {3870, 17625}, {3893, 22768}, {4097, 52139}, {4413, 24392}, {5253, 12541}, {5534, 12330}, {5537, 30304}, {6765, 10310}, {6913, 45701}, {7580, 34607}, {8069, 48696}, {8301, 29649}, {11248, 34862}, {12127, 13370}, {12536, 37248}, {12625, 37244}, {12642, 20849}, {13257, 25568}, {15347, 36846}, {16057, 35224}, {17660, 41711}, {17784, 35990}, {18237, 49163}, {18755, 41276}, {19589, 20996}, {20841, 20871}, {20918, 39600}, {21627, 25524}, {23089, 23845}, {23853, 39594}, {30283, 38455}, {31146, 33925}, {34612, 37240}, {37271, 49732}, {44455, 48667}

X(52804) = midpoint of X(3189) and X(18391)
X(52804) = isogonal conjugate X(52803)
X(52804) = cevapoint of X(i) and X(j) for these (i, j): {6, 145}, {9, 1476}
X(52804) = crossdifference of every pair of points on line {X(2487), X(3669)}
X(52804) = crosssum of X(i) and X(j) for these (i, j): {6, 3445}, {9, 3057}
X(52804) = X(i)-Ceva conjugate of-X(j) for these (i, j): (145, 6), (1476, 9)
X(52804) = intersection, other than A, B, C, of circumconics {A, B, C, X(55), X(41426)} and {A, B, C, X(200), X(36846)}
X(52804) = barycentric product X(i)*X(j) for these {i, j}: {6, 42020}, {9, 36846}, {55, 1997}, {346, 41426}, {644, 30198}
X(52804) = barycentric quotient X(1997)/X(6063)
X(52804) = trilinear product X(i)*X(j) for these {i, j}: {31, 42020}, {41, 1997}, {55, 36846}, {200, 41426}
X(52804) = trilinear quotient X(1997)/X(85)
X(52804) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (55, 3158, 6600), (55, 3689, 1260), (1864, 3689, 200), (2900, 3158, 3689), (3913, 12513, 12640)

   Miyamoto-Moses Points: X(52805)-X(52834)  

Part 1 of this preamble is based on notes and figures by Keita Miyamoto. Part 2 is based on notes from Keita Miyamoto and Peter Moses.

Part 1.

Proposition 1.1 (inner case).

In a scalene acute triangle ABC, let MaMbMc be its medial triangle. Let Ωa be the circle centered at Ma and passing through B and C, and define Ωb and Ωc cyclically. Inside ABC, let γa be the circle externally tangent to lines CA, AB,Ωa, and define γb and γc cyclically. Inside ABC, let γ be the circle internally tangent to Ωa, Ωb, Ωc. Then there exists a circle Γ that is tangent to the four circles γ, γa, γb, γc.

Let Z₁ denote the touchpoint of Γ and γ, and let Z₂ denote the center of Γ. The points Z₁ and Z₂ are the triangle centers X(52807) and X(52805), respectively. Here, the circle Γ is named the 1st Miyamoto-Moses-Apollonius circle, and the circle internally tangent to γa, γb and γc is named the 2nd Miyamoto-Moses-Apollonius circle. Let Ta be the touchpoint of Γ and γa, and define Tb and Tc cyclically. Here, the triangle TaTbTc is named the 1st-Miyamoto-Moses-Apollonius triangle. See Figure 1.

Proposition 1.1 also holds when ABC is an obtuse triangle. In this case, suppose that angle A is obtuse. Let Hb be the point of intersection, other than A, of Ωc and line CA. Let Hc be the point of intersection, other than A, of Ωb and line AB. Let γa be the circle internally tangent to Ωa and segments AHb and AHc. This circle lies outside ABC, as in Figure 2.

Proposition 1.2 (outer case).

In a scalene acute triangle ABC, let MaMbMc be its medial triangle. Let Ωa be the circle centered at Ma and passing through B and C, and define Ωb and Ωc cyclically. Let γa be the larger of the two circles internally tangent to lines CA, AB, Ωa, and define γb and γc cyclically. Outside ABC, let γ be the circle internally tangent to Ωa, Ωb, Ωc. Then there exists a circle Γ that is tangent to the four circles γ, γa, γb, γc.

Let Z₃ denote the touchpoint of Γ and γ, and let Z₄ denote the center of Γ, as in Figure 3. The points Z₃ and Z₄ are the triangle centers X(52810) and X(52808), respectively. Here, the circle Γ is named the 3rd Miyamoto-Moses-Apollonius circle, and the circle internally tangent to γa, γb, γc other than Γ, is named the 4th Miyamoto-Moses-Apollonius circle. Let Ta be the touchpoint of Γ and γa, and define Tb and Tc cyclically. Here, the triangle TaTbTc is named the 2nd Miyamoto-Moses-Apollonius triangle.

Proposition 1.2 also holds when ABC is an obtuse triangle. In this case, suppose that angle A is obtuse. Let γa; be the smaller of the two circles, etc., as in Figure 4.

It appears that Propositions 1.1 and 1.2 hold for any three circles Ωa, Ωb, Ωc, each of which passes through two of the three points A,B,C. Next, Proposition 1* treats an example of this sort.

Proposition 1*.

In a scalene acute triangle ABC, let A'B'C' be the tangential triangle. Let Ωa be the circle centered at A' and pssing through B and C, and define Ωb and Ωc cyclically. Let γa be a circle tangent (internally or externally) to lines CA, AB, Ωa, and define γb and γc cyclically. Let γ be the circle tangent to Ωa, Ωb, Ωc. Then there exists a circle Γ that is tangent to the four circles γ, γa, γb, γc.

For Proposition 1*, there are 4 cases, as indicated by these figures: Figure 5, Figure 6, Figure 7, Figure 8. The 4 cases depend on internal and external tangencies.

Proposition 2.

Let γ be the incircle of ABC. Let γa be the circle through B and C and internally tangent to γ. Define γb and γc cyclically. Outside ABC, let Γa be the circle externally tangent to CA, AB, γa, and let A' be the center of Γa. Define Γb and Γc cyclically, and define B' and C' cyclically. Let La be the external common tangent, other than BC, of Γb and Γc, so that La is the reflection of BC in B'C'. Define Lb and Lc cyclically. Let A''=Lb∩Lc, and define B" and C" cyclically. The triangle A"B"C" is perspective to ABC. The perspector is denoted by Z₅ in Figure 9. The point Z₅ is the triangle center X(52817). Here, the triangle A'B'C' is named the Miyamoto-Moses triangle.

Part 2.

Barycentrics for points defined in Part 1 are shown below and in a sequel to Figure 1 in GeoGebra: Figure 10. This file has a slider that can be used to observe the effect of changing S to - S in the barycentrics. In particular, the transformation S -> - S takes X(14121) to X(7090).

The point Z₅ shown in Figure 9, appears as P in Figure 11, and P = X(52817).

The circles γa, γb;, γc have a Chinese name that translates roughly as pseudo-circumcircles. For example, let U denote the circular hull of the pseudo-circumcircles that pass through 2 vertices and are tangent to the incircle. Their circular hull has center O = X(3) and radius R + r/2. In Figure 12, the points labeled tA and tA' are given by barycentrics as follows:

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

tA'= 2*a^2 : a^2 - 2*a*b - b^2 - 2*a*c + c^2 : a^2 - 2*a*b + b^2 - 2*a*c - c^2

If P = p : q : r is a point on the circumcircle, then the point

P' = (a^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4)*p + a^2*(a^2 - b^2 - c^2)*(q + r) : :

lies on the the circular hull. The point P' is here named the Moses-Apollonius transform of P. Examples are X(52820) to X(52834).

X(52805) = CENTER OF 1ST MIYAMOTO-MOSES-APOLLONIUS CIRCLE

Barycentrics    2*a^3 - a^2*b - b^3 - a^2*c + b^2*c + b*c^2 - c^3 + 2*a*S : :
X(52805) = 3 X[5603] - 2 X[7596]

X(52805) lies on these lines: {1, 7}, {3, 30277}, {4, 1336}, {5, 30307}, {8, 9907}, {9, 31413}, {10, 1131}, {40, 6204}, {55, 1659}, {65, 30376}, {105, 31546}, {165, 5393}, {489, 49470}, {490, 24723}, {497, 13389}, {527, 3640}, {528, 45714}, {585, 3177}, {942, 30347}, {946, 30381}, {1146, 3070}, {1587, 2082}, {1697, 31533}, {1699, 5405}, {1703, 31562}, {1836, 13390}, {2550, 30557}, {3303, 10910}, {3474, 13388}, {3641, 5853}, {3685, 11293}, {3755, 18991}, {3946, 11371}, {4648, 45704}, {5414, 30324}, {5603, 7596}, {5604, 17276}, {5698, 6460}, {5903, 18411}, {6213, 40131}, {7090, 46835}, {7982, 30320}, {7991, 30397}, {8091, 30369}, {8092, 30419}, {8351, 30407}, {9579, 31532}, {9840, 30361}, {9856, 30289}, {10025, 31548}, {13883, 51955}, {13888, 49632}, {17768, 45713}, {23249, 44038}, {24695, 45426}, {31583, 44431}, {31591, 51400}

X(52805) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 7, 30342}, {1, 482, 30341}, {1, 1371, 31570}, {1, 1373, 5542}, {1, 1374, 31569}, {1, 4312, 481}, {1, 30355, 30401}, {1, 30401, 18460}, {1, 30425, 175}, {1, 30426, 7}, {1, 30432, 30334}, {1, 31564, 18458}, {1, 31574, 30400}, {1, 51763, 31539}, {1, 51764, 482}, {7, 30334, 1}, {176, 390, 1}, {482, 31567, 1}, {17805, 30332, 30333}, {17805, 30333, 1}, {30297, 30386, 3}, {30307, 30314, 5}, {30331, 31570, 1}, {30334, 30426, 30342}, {30424, 31569, 1374}, {30424, 31570, 20121}, {30426, 30432, 1}, {31538, 31568, 1}, {31567, 51764, 30341}

---

X(52806) = CENTER OF 2ND MIYAMOTO-MOSES-APOLLONIUS CIRCLE

Barycentrics    2*a^3 - a^2*b - b^3 - a^2*c + b^2*c + b*c^2 - c^3 - 6*a*S : :
X(52806) = X[944] + 2 X[7596]

X(52806) lies on these lines: {1, 7}, {10, 3591}, {944, 7596}, {4654, 32082}, {5393, 7988}, {5587, 36690}, {5852, 45714}, {10389, 32083}, {13390, 17718}, {30556, 38057}, {44431, 49614}

X(52806) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 175, 30342}, {1, 481, 30341}, {1, 1372, 5542}, {1, 1374, 31570}, {1, 4312, 31538}, {1, 17803, 31569}, {1, 30425, 176}, {1, 30426, 17805}, {1, 30431, 390}, {1, 51763, 482}, {17802, 30333, 1}

---

X(52807) = 1ST MIYAMOTO-MOSES POINT

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

In a scalene acute triangle ABC, let Γa be the circle with diameter BC. Define Γb and Γc cyclically. Inside ABC, let γ be the circle internally tangent to Γa, Γb, Γc. Let γa be the circle externally tangent to Γa and internally tangent to Γb and Γc. Define γb and γc cyclically. Then there exists a circle ω that is tangent to the four circles γ, γa, γb, γc. The touchpoint of ω and γ is X(52807). The center of ω is X(3070). See X(52807). (Keita Miyamoto, February 16, 2023)

See also the GeoGebra sketch X(52807), in which the triangle a1b1c1 is perspective to ABC at X(1336), to the medial triangle at X(14121), and to the excentral triangle at X(6212).

X(52807) lies on the cubic K1051 and these lines: {4, 218}, {7, 22106}, {80, 30432}, {372, 30314}, {485, 30386}, {1146, 3070}, {1336, 30334}, {5540, 6212}, {5723, 13390}, {6560, 30297}, {6564, 30307}, {16232, 30425}

---

X(52808) = CENTER OF 3RD MIYAMOTO-MOSES-APOLLONIUS CIRCLE

Barycentrics    2*a^3 - a^2*b - b^3 - a^2*c + b^2*c + b*c^2 - c^3 - 2*a*S : :

X(52808) lies on these lines: {1, 7}, {3, 30276}, {4, 1123}, {5, 30306}, {8, 9906}, {10, 1132}, {40, 6203}, {55, 13390}, {65, 30375}, {105, 8225}, {165, 5405}, {489, 24723}, {490, 49470}, {497, 13388}, {527, 3641}, {528, 45713}, {586, 3177}, {942, 30346}, {946, 30380}, {1146, 3071}, {1588, 2082}, {1659, 1836}, {1697, 31532}, {1699, 5393}, {1702, 31561}, {2066, 30325}, {2550, 30556}, {3303, 10911}, {3474, 13389}, {3640, 5853}, {3685, 11294}, {3755, 18992}, {3946, 11370}, {4000, 45704}, {5605, 17276}, {5698, 6459}, {5903, 18410}, {6212, 40131}, {7982, 30319}, {7991, 30396}, {8091, 30368}, {8092, 30418}, {8351, 30406}, {9579, 31533}, {9840, 30360}, {9856, 30288}, {10025, 31547}, {12566, 23537}, {13936, 51957}, {13942, 49633}, {14121, 46835}, {17768, 45714}, {24695, 45427}, {31575, 45501}, {31582, 44431}, {31590, 51400}

X(52808) = reflection of X(4) in X(7596)

X(52808) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 7, 30341}, {1, 481, 30342}, {1, 1372, 31569}, {1, 1373, 31570}, {1, 1374, 5542}, {1, 4312, 482}, {1, 30354, 30400}, {1, 30400, 18458}, {1, 30425, 7}, {1, 30426, 176}, {1, 30431, 30333}, {1, 31563, 18460}, {1, 31573, 30401}, {1, 51763, 481}, {1, 51764, 31538}, {7, 30333, 1}, {175, 390, 1}, {481, 31568, 1}, {17802, 30332, 30334}, {17802, 30334, 1}, {30296, 30385, 3}, {30306, 30313, 5}, {30331, 31569, 1}, {30333, 30425, 30341}, {30424, 31569, 20121}, {30424, 31570, 1373}, {30425, 30431, 1}, {31539, 31567, 1}, {31568, 51763, 30342}

---

X(52809) = CENTER OF 4TH MIYAMOTO-MOSES-APOLLONIUS CIRCLE

Barycentrics    2*a^3 - a^2*b - b^3 - a^2*c + b^2*c + b*c^2 - c^3 + 6*a*S : :
X(52809) = 2 X[7596] - 5 X[10595]

X(52809) lies on these lines: {1, 7}, {10, 3590}, {1659, 17718}, {4654, 32083}, {5405, 7988}, {5587, 36691}, {5852, 45713}, {7596, 10595}, {10389, 32082}, {30557, 38057}, {44431, 49616}

X(52809) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 176, 30341}, {1, 482, 30342}, {1, 1371, 5542}, {1, 1373, 31569}, {1, 4312, 31539}, {1, 17806, 31570}, {1, 30425, 17802}, {1, 30426, 175}, {1, 30432, 390}, {1, 51764, 481}, {17805, 30334, 1}

---

X(52810) = 2ND MIYAMOTO-MOSES POINT

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

In a scalene acute triangle ABC, let Γa be the circle with diameter BC. Define Γb and Γc cyclically. Outside ABC, let γ be the circular hull internally tangent to Γa, Γb, Γc. Let γa be the circle internally tangent to Γa and externally tangent to Γb and Γc. Define γb and γc cyclically. Then, there exists a circle ω that is tangent to the four circles γ, γa, γb, γc. The touchpoint of ω and γ is X(52810). The center of ω is X(3071). See X(52810). (Keita Miyamoto, February 16, 2023)

X(52810) lies on the cubic K1051 and these lines: {4, 218}, {7, 22107}, {80, 7133}, {371, 30313}, {486, 30385}, {1123, 30333}, {1146, 3071}, {1659, 5723}, {2362, 30426}, {5540, 6213}, {6561, 30296}, {6565, 30306}

---

X(52811) = PERSPECTOR OF THESE TRIANGLES: 1ST MIYAMOTO-MOSES-APOLLONIUS AND ANTICOMPLEMENTARY

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

X(52811) lies on the Feuerbach circumhyperbola of the anticomplementary triangle abd these lines: {2, 42013}, {8, 9907}, {20, 585}, {192, 17784}, {30426, 49592}, {32793, 45962}

X(52811) = X(6204)-anticomplementary conjugate of X(69)

---

X(52812) = PERSPECTOR OF THESE TRIANGLES: 1ST MIYAMOTO-MOSES-APOLLONIUS AND ORTHIC

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

X(52812) lies on these lines: {4, 30426}, {7, 13389}, {19, 1659}, {33, 1721}, {65, 30376}, {176, 16232}, {482, 6212}, {1901, 30325}, {7071, 30296}, {14121, 30694}, {24328, 34125}, {30397, 51764}

X(52812) = orthic-isogonal conjugate of X(13390)
X(52812) = X(4)-Ceva conjugate of X(13390)
X(52812) = cevapoint of X(1721) and X(6204)
X(52812) = {X(7),X(42013)}-harmonic conjugate of X(13390)

---

X(52813) = PERSPECTOR OF THESE TRIANGLES: 2ND MIYAMOTO-MOSES-APOLLONIUS AND ANTICOMPLEMENTARY

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

X(52813) lies on the Feuerbach circumhyperbola of the anticomplementary triangle and these lines: {2, 7133}, {8, 9906}, {20, 586}, {192, 17784}, {30425, 49593}, {32794, 45962}

X(52813) = X(6203)-anticomplementary conjugate of X(69)

---

X(52814) = PERSPECTOR OF THESE TRIANGLES: 2ND MIYAMOTO-MOSES-APOLLONIUS AND ORTHIC

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

X(52814) lies on these lines: {4, 30425}, {7, 1659}, {19, 3069}, {33, 1721}, {65, 30375}, {175, 2362}, {481, 6213}, {1901, 30324}, {7071, 30297}, {7090, 30694}, {24328, 34121}, {30396, 51763}

X(52814) = orthic-isogonal conjugate of X(1659)
X(52814) = X(4)-Ceva conjugate of X(1659)
X(52814) = cevapoint of X(1721) and X(6203)
X(52814) = {X(7),X(7133)}-harmonic conjugate of X(1659)

---

X(52815) = RADICAL TRACE OF 1ST AND 2ND MIYAMOTO-MOSES-APOLLONIUS CIRCLES

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

X(52815) lies on these lines: {1, 7}, {1336, 31583}, {2362, 51775}, {3177, 31594}, {26658, 31548}, {31535, 46835}

X(52815) = incircle-inverse of X(481)
X(52815) = Conway-circle-inverse of X(31553)
X(52815) = inner-Soddy-circle-inverse of X(482)
X(52815) = crossdifference of every pair of points on line {657, 5414}
X(52815) = {X(1),X(1371)}-harmonic conjugate of X(31566)

---

X(52816) = RADICAL TRACE OF 3RD AND 4TH MIYAMOTO-MOSES-APOLLONIUS CIRCLES

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

X(52816) lies on these lines: {1, 7}, {1123, 31582}, {3177, 31595}, {16232, 51775}, {26658, 31547}, {31534, 46835}

X(52816) = incircle-inverse of X(482)
X(52816) = Conway-circle-inverse of X(31554)
X(52816) = outer-Soddy-cicrle-inverse of X(481)
X(52816) = crossdifference of every pair of points on line {657, 2066}
X(52816) = {X(1),X(1372)}-harmonic conjugate of X(31565)

---

X(52817) = MIYAMOTO-MOSES PERSPECTOR

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

X(52817) lies on these lines: {1, 5759}, {226, 45227}

X(52817) = X(15853)-Dao conjugate of X(8)

---

X(52818) = PERSPECTOR OF THESE TRIANGLES: MIYAMOTO-MOSES AND MEDIAL

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

X(52818) lies on these lines: {2, 220}, {9, 1699}, {10, 1536}, {37, 11019}, {118, 38973}, {1146, 25006}, {1212, 4847}, {1213, 40869}, {1834, 21838}, {3136, 38930}, {3683, 43959}, {3700, 6608}, {3730, 8727}, {3740, 13609}, {3925, 8012}, {5231, 24771}, {5745, 16593}, {8580, 15856}, {20173, 27481}, {20307, 38015}, {32578, 34612}, {34522, 36845}, {34822, 44798}, {37315, 40181}, {40606, 51424}

X(52818) = complement of X(21453)
X(52818) = complement of the isogonal conjugate of X(2293)
X(52818) = complement of the isotomic conjugate of X(4847)
X(52818) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 6706}, {31, 13405}, {41, 6666}, {55, 3740}, {56, 5572}, {142, 17046}, {354, 2886}, {604, 52542}, {692, 6362}, {934, 6607}, {1212, 141}, {1229, 626}, {1418, 21258}, {1475, 142}, {1827, 5}, {1855, 20305}, {2175, 16601}, {2293, 10}, {2488, 11}, {3059, 1329}, {3063, 1111}, {4847, 2887}, {6362, 21252}, {6607, 5514}, {6608, 124}, {8012, 3452}, {8551, 6554}, {8641, 3119}, {10581, 26932}, {16713, 21240}, {17194, 3741}, {18164, 17050}, {20229, 2}, {20880, 17047}, {21039, 3454}, {21127, 116}, {21795, 1211}, {21808, 17052}, {22053, 34822}, {22079, 3}, {35326, 4885}, {35338, 17072}, {35341, 3835}, {40983, 1210}, {48151, 17059}, {51972, 21244}, {52020, 442}
X(52818) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 13405}, {190, 6362}
X(52818) = X(1170)-isoconjugate of X(13404)
X(52818) = X(13405)-Dao conjugate of X(2)
X(52818) = barycentric product X(i)*X(j) for these {i,j}: {1212, 25001}, {4847, 13405}, {15837, 20880}
X(52818) = barycentric quotient X(i)/X(j) for these {i,j}: {2293, 13404}, {13405, 21453}, {15837, 2346}, {25001, 31618}

---

X(52819) = PERSPECTOR OF THESE TRIANGLES: MIYAMOTO-MOSES AND INTOUCH

Barycentrics    (a + b - c)*(a - b + c)*(2*a^3 - 3*a^2*b + b^3 - 3*a^2*c - b^2*c - b*c^2 + c^3)::
X(52819) = 2 X[7] - 3 X[553], 3 X[553] + 2 X[41572], 3 X[11246] - X[31391], 3 X[7671] - 2 X[15006]

X(52819) lies on these lines: {1, 5759}, {2, 7}, {4, 3062}, {6, 3668}, {10, 41712}, {12, 38217}, {44, 52023}, {56, 954}, {65, 516}, {72, 4298}, {77, 4667}, {85, 4416}, {193, 9312}, {218, 6180}, {241, 3664}, {269, 4644}, {279, 1419}, {347, 1449}, {388, 5223}, {390, 3340}, {405, 3671}, {443, 5785}, {452, 5665}, {497, 30330}, {498, 38130}, {515, 18412}, {518, 4032}, {528, 41558}, {942, 5762}, {946, 15299}, {948, 1743}, {962, 10384}, {971, 4292}, {1020, 2260}, {1155, 43151}, {1170, 52542}, {1174, 10509}, {1210, 5805}, {1418, 14564}, {1420, 5766}, {1434, 23618}, {1441, 3686}, {1466, 43177}, {1490, 36996}, {1770, 10399}, {1788, 38052}, {1864, 11246}, {1901, 8287}, {2003, 34028}, {2078, 2346}, {2093, 35514}, {2099, 30331}, {2391, 23839}, {2550, 4848}, {2949, 6684}, {2951, 3474}, {3059, 41539}, {3086, 38036}, {3256, 7676}, {3333, 5758}, {3361, 3487}, {3488, 14563}, {3553, 4341}, {3600, 11523}, {3663, 5228}, {3674, 41239}, {3681, 10865}, {3729, 6604}, {3755, 4331}, {3946, 22464}, {4295, 10396}, {4328, 4419}, {4461, 32003}, {4480, 32007}, {4488, 32098}, {4860, 15841}, {4888, 51302}, {5083, 5856}, {5173, 5572}, {5221, 5729}, {5436, 52653}, {5686, 9578}, {5698, 12560}, {5708, 5812}, {5722, 31671}, {5732, 10393}, {5776, 10402}, {5777, 5843}, {5809, 41824}, {5817, 9612}, {5845, 24471}, {5851, 24465}, {5853, 7672}, {5883, 12572}, {6354, 40940}, {6600, 41570}, {6610, 7277}, {6889, 38123}, {6987, 11529}, {7175, 9454}, {7195, 52511}, {7201, 51052}, {7580, 43182}, {7671, 15006}, {7717, 40983}, {8261, 17768}, {9579, 36991}, {10445, 41004}, {10592, 38179}, {10895, 38158}, {11011, 43179}, {11019, 36971}, {11375, 38059}, {11495, 37541}, {11662, 37566}, {12432, 14054}, {13405, 15837}, {13411, 31658}, {13569, 29057}, {15008, 15171}, {15298, 21620}, {15733, 41577}, {15803, 21151}, {16601, 45227}, {17014, 36640}, {17363, 25719}, {17668, 41566}, {20330, 44675}, {20662, 39063}, {23062, 34521}, {24914, 38204}, {31657, 37582}, {31789, 31794}, {32636, 43180}, {34032, 52424}, {35617, 38478}, {41006, 45738}, {41564, 47387}, {42309, 51190}

X(52819) = midpoint of X(7) and X(41572)
X(52819) = reflection of X(i) in X(j) for these {i,j}: {950, 5728}, {8581, 4298}, {10106, 12573}, {15171, 15008}
X(52819) = X(43672)-Ceva conjugate of X(43035)
X(52819) = X(i)-isoconjugate of X(j) for these (i,j): {9, 13404}, {650, 43344}
X(52819) = X(i)-Dao conjugate of X(j) for these (i,j): {478, 13404}, {13405, 51972}
X(52819) = barycentric product X(i)*X(j) for these {i,j}: {7, 13405}, {57, 25001}, {1088, 15837}
X(52819) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 13404}, {109, 43344}, {13405, 8}, {15837, 200}, {25001, 312}
X(52819) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 10398, 10392}, {6, 3668, 43035}, {7, 9, 226}, {7, 1445, 142}, {7, 8732, 6173}, {7, 12848, 9}, {7, 29007, 41857}, {7, 37787, 21617}, {7, 41563, 8545}, {142, 1445, 3911}, {1418, 14564, 17365}, {4312, 10398, 4}, {17950, 41246, 4357}, {20059, 21454, 7}, {21617, 37787, 6666}, {41857, 50573, 29007}

---

X(52820) = MOSES-APOLLONIUS TRANSFORM OF X(74)

Barycentrics    a^2*((a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)*(a^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4) + (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)*(b^2*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4) + c^2*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4))) : :
X(52820) = 5 X[74] - X[38497], X[7724] - 5 X[35242]

X(52820) lies on these lines: {3, 74}, {28, 12133}, {57, 3024}, {113, 8728}, {125, 8727}, {146, 443}, {265, 6851}, {1112, 4219}, {1539, 44229}, {2771, 9945}, {2777, 20420}, {2781, 4260}, {2948, 37551}, {3028, 3601}, {3587, 12778}, {5787, 13211}, {6282, 33535}, {6675, 6699}, {6824, 15061}, {6826, 7728}, {6841, 20304}, {6861, 34128}, {6869, 20127}, {6989, 14643}, {7501, 12292}, {7724, 35242}, {7727, 15803}, {7978, 15934}, {8674, 13226}, {11018, 44403}, {11670, 17603}, {11709, 24929}, {12244, 50701}, {19470, 30282}

X(52820) = midpoint of X(3024) and X(9904)

---

X(52821) = MOSES-APOLLONIUS TRANSFORM OF X(98)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(-2*a^4 + a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) + (a^4 + b^4 - a^2*c^2 - b^2*c^2)*(a^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4)*(a^4 - a^2*b^2 - b^2*c^2 + c^4) : :
X(52821) = 5 X[98] - X[38498]

X(52821) lies on these lines: {3, 76}, {28, 12131}, {57, 3023}, {114, 8728}, {115, 8727}, {142, 21636}, {147, 443}, {2783, 9945}, {2787, 13226}, {2794, 20420}, {2795, 38747}, {3027, 3601}, {4219, 5186}, {4260, 34454}, {5787, 13178}, {5984, 6904}, {6033, 6826}, {6036, 6675}, {6321, 6851}, {6824, 38224}, {6847, 14651}, {6861, 34127}, {6869, 38741}, {6989, 15561}, {7970, 15934}, {9862, 50701}, {11710, 24929}, {13174, 37551}, {22505, 44229}, {24472, 37544}

X(52821) = midpoint of X(3023) and X(9860)

---

X(52822) = MOSES-APOLLONIUS TRANSFORM OF X(99)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(b^2 - c^2)^2 + (a^2 - b^2)*(a^2 - c^2)*(a^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4) : :
X(52822) = 5 X[99] - X[38499]

X(52822) lies on these lines: {3, 76}, {28, 5186}, {57, 3027}, {114, 8727}, {115, 8728}, {142, 11599}, {148, 443}, {620, 2795}, {1281, 8731}, {2783, 13226}, {2787, 9945}, {2796, 12014}, {3023, 3601}, {3029, 4260}, {4219, 12131}, {5026, 5138}, {5745, 51578}, {5787, 9864}, {6033, 6851}, {6321, 6826}, {6824, 15561}, {6869, 38730}, {6904, 20094}, {6989, 38224}, {7983, 15934}, {9860, 37551}, {11018, 24472}, {11711, 24929}, {13172, 50701}, {14061, 50726}, {14651, 37407}, {20420, 23698}, {22515, 44229}, {35612, 38481}

X(52822) = midpoint of X(3027) and X(13174)

---

X(52823) = MOSES-APOLLONIUS TRANSFORM OF X(101)

Barycentrics    a^2*((a - b)*(a - c)*(a^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4) - (a^2 - b^2 - c^2)*((a - b)*b^2*(b - c) - (a - c)*c^2*(b - c))) : :
X(52823) = 5 X[101] - X[38500]

X(52823) lies on these lines: {3, 101}, {28, 5185}, {43, 57}, {55, 9306}, {109, 20761}, {116, 8728}, {118, 8727}, {142, 28600}, {150, 443}, {165, 3781}, {169, 14520}, {511, 910}, {926, 9508}, {942, 2809}, {971, 40869}, {991, 16588}, {1190, 37492}, {1350, 1615}, {1818, 2272}, {2801, 3035}, {2807, 22276}, {3022, 3601}, {3887, 9945}, {5787, 50903}, {6675, 6710}, {6678, 11018}, {6824, 38764}, {6826, 10739}, {6851, 10741}, {6904, 20096}, {10695, 15934}, {11712, 15746}, {15288, 37474}, {31273, 50726}, {34381, 34855}, {35612, 38479}, {37551, 39156}

X(52823) = midpoint of X(i) and X(j) for these {i,j}: {295, 3033}, {1282, 1362}
X(52823) = reflection of X(3041) in X(28346)
X(52823) = crossdifference of every pair of points on line {676, 4435}

---

X(52824) = MOSES-APOLLONIUS TRANSFORM OF X(102)

Barycentrics    a^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^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^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) + (a^2 - b^2 - c^2)*(-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)*(b^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) + c^2*(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(52824) = 5 X[102] - X[38501]

X(52824) lies on these lines: {3, 102}, {57, 1364}, {117, 8728}, {124, 8727}, {151, 443}, {942, 2817}, {1361, 3601}, {2800, 9942}, {2807, 22276}, {3042, 5745}, {3738, 13226}, {4260, 34455}, {4617, 7215}, {5787, 13532}, {6675, 6711}, {6824, 38776}, {6826, 10740}, {6851, 10747}, {7412, 34956}, {10696, 15934}, {11227, 46330}, {11713, 24929}, {12016, 37544}, {30282, 52129}

---

X(52825) = MOSES-APOLLONIUS TRANSFORM OF 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^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4) + (a^2 - b^2 - c^2)*(-2*a^3 + a^2*b + b^3 + a^2*c - b^2*c - b*c^2 + c^3)*(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(52825) = 5 X[103] - X[38502]

X(52825) lies on these lines: {3, 101}, {57, 3022}, {116, 8727}, {118, 8728}, {152, 443}, {934, 3270}, {942, 14760}, {1282, 37551}, {1362, 3601}, {2801, 9945}, {2807, 11700}, {2809, 31793}, {2810, 6282}, {2823, 9944}, {3887, 13226}, {4219, 5185}, {4260, 34457}, {5787, 50896}, {6675, 6712}, {6826, 10741}, {6851, 10739}, {6869, 38765}, {6989, 38764}, {10697, 15934}, {11028, 37544}

X(52825) = midpoint of X(3022) and X(39156)

---

X(52826) = MOSES-APOLLONIUS TRANSFORM OF X(105)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(-(a*b) + b^2 - a*c + c^2)*(a^2*b + b^3 + a^2*c - 2*a*b*c - b^2*c - b*c^2 + c^3) + a*(a^2 + b^2 - a*c - b*c)*(a^2 - a*b - b*c + c^2)*(a^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4) : :
X(52826) = 5 X[105] - X[38503], X[1358] - 3 X[34578], X[5540] + 3 X[34578], X[20344] - 3 X[34124]

X(52826) lies on these lines: {3, 105}, {5, 20269}, {30, 51400}, {55, 20328}, {57, 1358}, {101, 1086}, {116, 12019}, {120, 8728}, {140, 24774}, {142, 214}, {443, 20344}, {620, 2795}, {673, 1565}, {934, 37771}, {942, 2809}, {952, 4904}, {999, 4000}, {1054, 24795}, {1111, 26007}, {1385, 24181}, {1462, 22144}, {2140, 37737}, {2170, 43057}, {2283, 15253}, {2826, 13226}, {3021, 3601}, {3034, 4260}, {3039, 5745}, {3428, 24779}, {3576, 4859}, {3628, 24784}, {3946, 5049}, {3960, 6084}, {5126, 17067}, {5511, 8727}, {5719, 30949}, {6282, 32486}, {6826, 10743}, {6851, 15521}, {6904, 20097}, {10609, 26140}, {10699, 15934}, {15171, 34847}, {15325, 51775}, {21258, 37730}, {24390, 27006}, {24790, 37609}

X(52826) = midpoint of X(i) and X(j) for these {i,j}: {1358, 5540}, {4904, 9317}
X(52826) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5540, 34578, 1358}, {17044, 17761, 1387}

---

X(52827) = MOSES-APOLLONIUS TRANSFORM OF X(106)

Barycentrics    a^2*((a + b - 2*c)*(a - 2*b + c)*(a^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4) + (-2*a + b + c)*(a^2 - b^2 - c^2)*(b^2*(a + b - 2*c) + c^2*(a - 2*b + c))) : :
X(52827) = 5 X[106] - X[38504], 3 X[1054] - X[3030], 3 X[1357] + X[3030]

X(52827) lies on these lines: {3, 106}, {43, 57}, {88, 3937}, {121, 8728}, {142, 11814}, {443, 21290}, {942, 2802}, {1015, 25577}, {2796, 12014}, {2827, 13226}, {2841, 37582}, {3038, 5745}, {3060, 26745}, {3601, 6018}, {3756, 29349}, {5510, 8727}, {6675, 6715}, {6826, 10744}, {6851, 15522}, {6904, 20098}, {9052, 18201}, {9519, 11227}, {9776, 17777}, {10700, 15934}, {11518, 13541}, {11717, 24929}, {35612, 38478}

X(52827) = midpoint of X(1054) and X(1357)
X(52827) = crossdifference of every pair of points on line {4435, 14425}

---

X(52828) = MOSES-APOLLONIUS TRANSFORM OF X(107)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^2 + b^2 - c^2)^2*(a^2 - b^2 + c^2)^2*(a^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4) - a^2*(a^2 - b^2 - c^2)*(b^2 - c^2)*(-a^2 + b^2 + c^2)^2*((a^2 - b^2)*(a^2 + b^2 - c^2)^2 - (a^2 - c^2)*(a^2 - b^2 + c^2)^2) : :
X(52828) = 5 X[107] - X[38505]

X(52828) lies on these lines: {3, 107}, {57, 3324}, {122, 8728}, {133, 8727}, {443, 34186}, {2777, 20420}, {2803, 9945}, {2828, 13226}, {3601, 7158}, {5667, 50701}, {6675, 6716}, {6826, 10745}, {6851, 22337}, {6869, 23240}, {10701, 15934}, {11718, 24929}, {44229, 49117}

---

X(52829) = MOSES-APOLLONIUS TRANSFORM OF X(108)

Barycentrics    a*((a - b)*(a - c)*(a + b - c)*(a - b + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4) + a*(b - c)*(-a + b + c)*(a^2 - b^2 - c^2)*(-a^2 + b^2 + c^2)*(b*(-a + b)*(a + b - c)*(a^2 + b^2 - c^2) - c*(-a + c)*(a - b + c)*(a^2 - b^2 + c^2))) : :
X(52829) = 5 X[108] - X[38506]

X(52829) lies on these lines: {3, 108}, {30, 51359}, {57, 1359}, {123, 8728}, {443, 34188}, {653, 38554}, {942, 2817}, {2804, 9945}, {2823, 9944}, {2829, 6245}, {3318, 3601}, {4260, 34456}, {6087, 21180}, {6675, 6717}, {6826, 10746}, {6851, 33566}, {8727, 25640}, {10702, 15934}, {11719, 24929}

---

X(52830) = MOSES-APOLLONIUS TRANSFORM OF X(109)

Barycentrics    a^2*((a - b)*(a - c)*(a + b - c)*(a - b + c)*(a^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4) + (-a + b + c)*(-a^2 + b^2 + c^2)*((a - b)*b^2*(b - c)*(a + b - c) - (a - c)*(b - c)*c^2*(a - b + c))) : :
X(52830) = 5 X[109] - X[38507], 3 X[109] + X[51896], 3 X[38507] + 5 X[51896]

X(52830) lies on these lines: {3, 102}, {57, 1361}, {117, 8727}, {124, 8728}, {443, 33650}, {517, 34050}, {942, 1387}, {1020, 33810}, {1364, 3601}, {1425, 6906}, {1807, 2808}, {1845, 5723}, {2807, 11700}, {2817, 31793}, {2841, 37582}, {3040, 5745}, {3738, 9945}, {3960, 8677}, {4260, 34459}, {5787, 50899}, {6000, 46974}, {6282, 34488}, {6675, 6718}, {6826, 10747}, {6851, 10740}, {6869, 38777}, {6950, 19368}, {6989, 38776}, {7078, 37480}, {9957, 35655}, {10703, 15934}, {11018, 12016}, {15251, 37544}, {15803, 52129}, {24220, 39542}, {43043, 45022}

X(52830) = {X(1425),X(6906)}-harmonic conjugate of X(34956)

---

X(52831) = MOSES-APOLLONIUS TRANSFORM OF X(110)

Barycentrics    a^2*((a^2 - b^2)*(a^2 - c^2)*(a^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4) - (a^2 - b^2 - c^2)*(b^2 - c^2)*(b^2*(a^2 - b^2) - c^2*(a^2 - c^2))) : :
X(52831) = 5 X[110] - X[38508], 3 X[3576] + X[7724]

X(52831) lies on these lines: {3, 74}, {28, 1112}, {57, 2948}, {113, 8727}, {125, 8728}, {142, 13605}, {265, 6826}, {443, 3448}, {912, 44898}, {942, 35063}, {1986, 7501}, {2771, 5972}, {2854, 3031}, {3024, 3601}, {3576, 7724}, {4219, 12133}, {5138, 6593}, {5709, 12778}, {5787, 12368}, {6824, 14643}, {6851, 7728}, {6869, 12121}, {6881, 20304}, {6885, 23236}, {6904, 14683}, {6935, 20125}, {6989, 15061}, {7727, 30282}, {7984, 15934}, {8674, 9945}, {8726, 33535}, {9904, 37551}, {10113, 44229}, {11720, 24929}, {12383, 50701}, {14708, 44220}, {15059, 50726}, {15803, 19470}, {16428, 45237}, {17563, 24981}, {17702, 20420}, {18254, 40539}, {19504, 37245}, {32423, 37281}, {35612, 38482}

X(52831) = midpoint of X(2948) and X(3028)

---

X(52832) = MOSES-APOLLONIUS TRANSFORM OF X(111)

Barycentrics    a^2*((a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(a^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4) + (a^2 - b^2 - c^2)*(-2*a^2 + b^2 + c^2)*(b^2*(a^2 + b^2 - 2*c^2) + c^2*(a^2 - 2*b^2 + c^2))) : :
X(52832) = 5 X[111] - X[38509]

X(52832) lies on these lines: {3, 111}, {57, 3325}, {126, 8728}, {443, 14360}, {2805, 9945}, {2830, 13226}, {2854, 3031}, {3601, 6019}, {5138, 28662}, {5512, 8727}, {6675, 6719}, {6824, 38796}, {6826, 10748}, {6851, 22338}, {6881, 40340}, {6904, 20099}, {10704, 15934}, {11721, 24929}, {14654, 50701}, {20420, 23699}, {28452, 32424}

---

X(52833) = MOSES-APOLLONIUS TRANSFORM OF X(112)

Barycentrics    a^2*((a^2 - b^2)*(a^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4) + (b^2 - c^2)*(-a^2 + b^2 + c^2)^2*(b^2*(a^2 - b^2)*(a^2 + b^2 - c^2) - c^2*(a^2 - c^2)*(a^2 - b^2 + c^2))) : :
X(52833) = 5 X[112] - X[38510]

X(52833) lies on these lines: {3, 112}, {28, 13166}, {57, 3320}, {127, 8728}, {132, 8727}, {443, 13219}, {2781, 4260}, {2794, 20420}, {2806, 9945}, {2831, 13226}, {3601, 6020}, {4219, 12145}, {5138, 28343}, {5787, 12784}, {6675, 6720}, {6826, 10749}, {6851, 12918}, {10705, 15934}, {11722, 24929}, {12408, 37551}, {13200, 50701}, {19163, 44229}

X(52833) = midpoint of X(3320) and X(13221)

---

X(52834) = MOSES-APOLLONIUS TRANSFORM OF X(759)

Barycentrics    a*((a + b)*(a + c)*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2*b^2 - b^4 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 - c^4) + a*(b + c)*(a^2 - b^2 - c^2)*(-a^2 + b^2 - b*c + c^2)*(b*(a + b)*(a^2 - a*b + b^2 - c^2) + c*(a + c)*(a^2 - b^2 - a*c + c^2))) : :
X(52834) = 3 X[2] + X[19642], 5 X[759] - X[38511]

X(52834) lies on these lines: {2, 19642}, {3, 759}, {11, 5972}, {57, 1365}, {125, 24624}, {542, 8286}, {942, 35063}, {1155, 1738}, {1283, 8731}, {3035, 3634}, {3601, 34194}, {3911, 44908}, {4260, 6044}, {5620, 37582}, {5745, 51578}, {6089, 9508}, {6667, 40539}, {6678, 11018}, {6723, 8287}, {6841, 7687}, {8727, 42425}, {8728, 25448}, {34196, 37551}, {35612, 38480}

X(52834) = midpoint of X(1365) and X(21381)
X(52834) = {X(24624),X(24916)}-harmonic conjugate of X(125)

---

X(52835) = REFLECTION OF X(9) IN X(4)

Barycentrics    a*S^2*sa-2*SB*(a*sa+b*sb+c*sc)*SC : :

See Ivan Pavlov, euclid 5635.

X(52835) lies on these lines: {1,52023}, {3,18482}, {4,9}, {5,21153}, {7,950}, {20,142}, {30,5732}, {57,10431}, {72,9589}, {144,5175}, {165,3826}, {226,390}, {376,38093}, {381,31658}, {382,971}, {405,11495}, {442,42356}, {497,12573}, {515,3243}, {518,5691}, {527,3543}, {528,1750}, {546,38108}, {548,38171}, {550,38122}, {553,10430}, {946,38316}, {954,3746}, {962,5853}, {1001,1699}, {1445,6895}, {1449,3332}, {1490,12699}, {1503,51194}, {1657,38107}, {1708,45043}, {1770,10396}, {1836,10382}, {2801,10725}, {2829,3254}, {2951,5880}, {3062,6598}, {3091,6666}, {3174,6253}, {3419,5223}, {3487,30331}, {3488,5542}, {3529,21151}, {3583,15299}, {3585,15298}, {3586,4312}, {3601,21617}, {3627,5762}, {3651,8227}, {3830,5779}, {3832,18230}, {3845,38075}, {3850,38113}, {3851,38318}, {3860,38082}, {3861,38139}, {4297,38053}, {4321,7354}, {4326,6284}, {4331,4907}, {4338,10399}, {4512,7965}, {4872,42309}, {5066,38067}, {5128,10395}, {5177,10248}, {5219,36002}, {5438,50700}, {5528,5840}, {5603,43175}, {5715,10267}, {5727,7672}, {5777,48661}, {5809,41824}, {5845,51163}, {5851,12690}, {6172,50687}, {6601,6762}, {6832,35242}, {6835,37551}, {6840,8257}, {6846,31730}, {6908,18483}, {6913,28146}, {6987,28150}, {6990,31423}, {7411,41867}, {7677,50443}, {7957,40659}, {8232,30332}, {8545,51790}, {8581,12943}, {9614,42884}, {9778,40333}, {9955,38031}, {10392,12848}, {10442,29181}, {10895,15837}, {11001,38073}, {11522,42819}, {12512,16845}, {12571,38059}, {12953,14100}, {13727,17306}, {15682,36996}, {15685,38065}, {15704,38137}, {15733,36999}, {16208,24644}, {16593,19542}, {17532,50836}, {18357,38126}, {18446,31162}, {18480,38154}, {18481,20330}, {19130,38117}, {19541,30827}, {20059,50690}, {28194,51102}, {31190,37374}, {31822,34339}, {33703,43177}, {35258,52255}, {37787,51792}, {38025,50802}, {38088,50959}, {38097,50796}, {38101,50803}, {38143,48905}, {38151,43151}, {38186,44882}, {38205,38759}, {51190,51538}

X(52835) = midpoint of X(i) and X(j) for these {i,j}: {7, 3146}, {382, 31671}
X(52835) = reflection of X(i) in X(j) for these {i,j}: {3, 18482}, {9, 4}, {20, 142}, {2951, 5880}, {5732, 5805}, {5735, 31671}, {6762, 6601}, {7957, 40659}, {18481, 20330}, {31672, 3627}, {43161, 946}, {43166, 12699}
X(52835) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 18482, 38150}, {3, 38150, 20195}, {4, 516, 9}, {30, 5805, 5732}, {382, 31671, 971}, {971, 31671, 5735}, {1699, 7580, 25525}, {3586, 4312, 5728}, {3627, 5762, 31672}, {5732, 5805, 6173}, {5819, 8804, 9}, {9812, 50696, 226}, {22793, 37411, 5715}

---

X(52836) = REFLECTION OF X(11) IN X(4)

Barycentrics    (b-c)^2*S^2*sa-2*SB*((b-c)^2*sa+(a-c)^2*sb+(a-b)^2*sc)*SC : :

See Ivan Pavlov, euclid 5635.

X(52836) lies on these lines: {2,38759}, {3,31235}, {4,11}, {5,21154}, {20,3035}, {30,119}, {80,2093}, {100,3146}, {149,17578}, {153,528}, {214,28164}, {381,6713}, {382,5840}, {515,1317}, {516,1145}, {517,12665}, {546,23513}, {550,38760}, {900,42755}, {952,3627}, {962,5854}, {971,11570}, {1387,1699}, {1478,33898}, {1479,30283}, {1484,15687}, {1490,12739}, {1503,51198}, {1532,15326}, {1538,21578}, {1597,9913}, {1656,38754}, {1657,38752}, {1709,12515}, {1750,6326}, {1768,12019}, {1836,3577}, {2771,12295}, {2783,39809}, {2787,39838}, {2800,12688}, {2801,12690}, {2802,13227}, {2803,38956}, {3036,16112}, {3058,12115}, {3091,6667}, {3149,48695}, {3529,20400}, {3534,38762}, {3586,12678}, {3614,6906}, {3830,10738}, {3832,31272}, {3839,45310}, {3845,38077}, {3850,34126}, {3851,38319}, {3853,22938}, {3860,38084}, {3861,38141}, {3925,6923}, {4081,13532}, {4297,34123}, {4301,25416}, {4996,36002}, {5059,38758}, {5066,38069}, {5073,38755}, {5076,12773}, {5083,12680}, {5176,18802}, {5326,6950}, {5432,6938}, {5434,26333}, {5450,7173}, {5533,18514}, {5660,9945}, {5787,10073}, {5848,36990}, {5851,9803}, {5927,18254}, {6245,20118}, {6256,6284}, {6564,13913}, {6565,13977}, {6925,11495}, {7580,51506}, {7957,14740}, {7989,50240}, {8068,8727}, {8674,13202}, {9024,51163}, {9579,24465}, {9856,12758}, {9955,38032}, {10090,19541}, {10483,39692}, {10572,22792}, {10609,21635}, {10707,50687}, {10711,13199}, {10732,38950}, {10755,51538}, {10993,11698}, {11500,12775}, {11715,18483}, {11729,18481}, {11826,21031}, {12102,51529}, {12571,32557}, {12611,28160}, {12650,20586}, {12667,12763}, {12691,33519}, {12736,31391}, {12751,13996}, {13226,37718}, {13922,42258}, {13991,42259}, {15338,18242}, {15704,38763}, {16205,31162}, {18357,38128}, {18480,38156}, {18482,38152}, {19081,23259}, {19082,23249}, {19130,38119}, {19907,28186}, {19925,34122}, {20095,50690}, {28194,50842}, {28204,50846}, {29181,51007}, {30264,37406}, {33810,45885}, {34256,38307}, {34697,37820}, {35015,51422}, {35023,49135}, {38026,50802}, {38060,42356}, {38090,50959}, {38099,50796}, {38104,50803}, {38669,50688}, {40663,52116}, {42271,48714}, {42272,48715}, {42283,48701}, {42284,48700}

X(52836) = midpoint of X(i) and X(j) for these {i,j}: {4, 10728}, {100, 3146}, {153, 10724}, {382, 10742}, {5691, 34789}, {10711, 15682}, {10738, 38756}, {12751, 41869}
X(52836) = reflection of X(i) in X(j) for these {i,j}: {11, 4}, {20, 3035}, {100, 38757}, {119, 22799}, {1317, 1537}, {1768, 12019}, {6154, 37725}, {7957, 14740}, {10609, 21635}, {10993, 11698}, {11715, 18483}, {12248, 20418}, {12680, 5083}, {12758, 9856}, {12767, 9952}, {13996, 12751}, {15326, 1532}, {18481, 11729}, {21578, 1538}, {22938, 3853}, {24466, 119}, {25416, 4301}, {37725, 10742}, {37726, 22938}, {38602, 546}, {38753, 6713}, {38761, 5}, {46684, 19925}
X(52836) = anticomplement of X(38759)
X(52836) = X(i)-Dao conjugate of X(j) for these {i, j}: {38759, 38759}
X(52836) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 10728, 2829}, {4, 2829, 11}, {4, 37002, 10893}, {5, 38761, 21154}, {30, 119, 24466}, {30, 22799, 119}, {80, 12767, 9952}, {119, 24466, 6174}, {153, 10724, 528}, {153, 3543, 10724}, {382, 10742, 5840}, {515, 1537, 1317}, {546, 38602, 23513}, {2829, 20418, 12248}, {3830, 38756, 10738}, {5691, 34789, 952}, {5840, 10742, 37725}, {5840, 37725, 6154}, {11715, 18483, 38038}, {12743, 12831, 1317}

---

X(52837) = REFLECTION OF X(12) IN X(4)

Barycentrics    (b+c)^2*S^2*sb*sc-2*SB*((a+b)^2*sa*sb+(a+c)^2*sa*sc+(b+c)^2*sb*sc)*SC : :

See Ivan Pavlov, euclid 5635.

X(52837) lies on these lines: {3,31260}, {4,12}, {5,21155}, {11,37468}, {20,4999}, {30,11012}, {381,31659}, {382,5841}, {515,11011}, {529,3543}, {546,38109}, {758,12688}, {946,10543}, {952,3627}, {962,5855}, {1699,37737}, {2975,3146}, {3058,26332}, {3091,6668}, {3583,20420}, {3614,6796}, {3845,38078}, {3850,38114}, {3853,41698}, {3860,38085}, {3861,38142}, {3925,31789}, {5066,38070}, {5217,6844}, {5433,6934}, {5434,12116}, {5771,16113}, {5849,36990}, {5852,36991}, {6690,7548}, {6831,15338}, {6839,7958}, {6840,50031}, {6868,24953}, {6905,7173}, {6942,7294}, {7354,10959}, {7491,18407}, {7987,50240}, {7995,41869}, {8068,18514}, {9955,38033}, {10526,18499}, {10724,37433}, {10896,50701}, {11661,18395}, {11827,37820}, {12571,38062}, {12672,31673}, {16204,31162}, {17578,20060}, {18357,38129}, {18480,38157}, {18482,38153}, {18483,38039}, {19130,38120}, {19925,38058}, {24466,37356}, {28164,51111}, {29181,51009}, {34746,37821}, {37722,45977}, {38027,50802}, {38061,42356}, {38091,50959}, {38100,50796}, {38105,50803}

X(52837) = midpoint of X(2975) and X(3146)
X(52837) = reflection of X(i) in X(j) for these {i,j}: {12, 4}, {20, 4999}, {15338, 6831}, {30264, 26470}
X(52837) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 36999, 6253}, {4, 37000, 10894}, {4, 5842, 12}, {30, 26470, 30264}

---

X(52838) = REFLECTION OF X(17) IN X(4)

Barycentrics    sqrt(3)*a^2*S^2+3*S^3-19*S*SB*SC-4*sqrt(3)*SB*SC*SW : :

See Ivan Pavlov, euclid 5635.

X(52838) lies on these lines: {3,22795}, {4,15}, {20,629}, {30,16626}, {381,49106}, {382,5864}, {515,22912}, {516,22896}, {532,3543}, {627,3146}, {1503,51208}, {1593,22657}, {1699,11739}, {2794,11602}, {3070,19070}, {3071,19071}, {3091,6673}, {3412,5478}, {3583,22930}, {3585,22929}, {3627,36962}, {3830,16629}, {5339,7747}, {5349,22512}, {5868,19106}, {5965,36990}, {6253,22903}, {6256,22931}, {6284,22904}, {7354,22905}, {8259,42164}, {10611,22236}, {12173,22482}, {12943,18973}, {12953,22910}, {15682,36386}, {16652,37463}, {17578,22113}, {22921,31412}, {22922,42561}, {22932,48482}, {23004,41021}, {23251,49238}, {23261,49239}, {28164,51116}, {29181,51020}, {33415,44223}, {33465,50688}, {37007,41022}, {41016,43632}, {41038,42158}, {41698,49175}, {42814,44463}, {50687,51486}

X(52838) = midpoint of X(i) and X(j) for these {i,j}: {382, 48666}, {627, 3146}
X(52838) = reflection of X(i) in X(j) for these {i,j}: {3, 22795}, {17, 4}, {20, 629}, {22532, 22832}, {22890, 16626}
X(52838) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 22532, 22832}, {4, 42150, 41036}, {4, 44666, 17}, {30, 16626, 22890}, {22532, 22832, 17}, {22832, 44666, 22532}

---

X(52839) = REFLECTION OF X(18) IN X(4)

Barycentrics    sqrt(3)*a^2*S^2-3*S^3+19*S*SB*SC-4*sqrt(3)*SB*SC*SW : :

See Ivan Pavlov, euclid 5635.

X(52839) lies on these lines: {3,22794}, {4,16}, {20,630}, {30,16627}, {381,49105}, {382,5865}, {515,22867}, {516,22851}, {533,3543}, {628,3146}, {1503,51209}, {1593,22656}, {1699,11740}, {2794,11603}, {3070,19072}, {3071,19069}, {3091,6674}, {3411,5479}, {3583,22885}, {3585,22884}, {3627,36961}, {3830,16628}, {5340,7747}, {5350,22513}, {5869,19107}, {5965,36990}, {6253,22858}, {6256,22886}, {6284,22859}, {7354,22860}, {8260,42165}, {10612,22238}, {12173,22481}, {12943,18972}, {12953,22865}, {15682,36388}, {16653,37464}, {17578,22114}, {22876,31412}, {22877,42561}, {22887,48482}, {23005,41020}, {23251,49236}, {23261,49237}, {28164,51117}, {29181,51021}, {33386,44250}, {33414,52650}, {33464,50688}, {37008,41023}, {41017,43633}, {41039,42157}, {41698,49173}, {42813,44459}, {50687,51487}

X(52839) = midpoint of X(i) and X(j) for these {i,j}: {382, 48665}, {628, 3146}
X(52839) = reflection of X(i) in X(j) for these {i,j}: {3, 22794}, {18, 4}, {20, 630}, {22531, 22831}, {22843, 16627}
X(52839) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 22531, 22831}, {4, 42151, 41037}, {4, 44667, 18}, {30, 16627, 22843}, {16627, 22843, 50860}, {22531, 22831, 18}, {22831, 44667, 22531}

---

X(52840) = REFLECTION OF X(19) IN X(4)

Barycentrics    SB*SC*(a*S^2-2*(c*SA*SB+b*SA*SC+a*SB*SC)) : :

See Ivan Pavlov, euclid 5635.

X(52840) lies on these lines: {3,31261}, {4,9}, {5,21160}, {20,18589}, {30,30265}, {33,1836}, {34,1885}, {65,5895}, {165,37372}, {204,3914}, {235,23305}, {278,9580}, {378,39475}, {390,5236}, {427,7965}, {497,1435}, {517,15942}, {534,3543}, {1096,33094}, {1486,1593}, {1503,51210}, {1699,4219}, {1844,4338}, {1848,9812}, {1871,48661}, {1876,14100}, {1888,12953}, {1891,3146}, {1902,3827}, {2876,12294}, {3091,40530}, {3668,9579}, {4331,40950}, {5174,45738}, {5691,44661}, {7497,28146}, {7521,12512}, {7537,35242}, {8680,51063}, {8765,33137}, {10735,44970}, {12511,30733}, {17578,20061}, {23052,24248}, {28150,37395}, {30808,44922}, {35258,37371}, {35445,37799}, {36999,44438}

X(52840) = midpoint of X(3146) and X(4329)
X(52840) = reflection of X(i) in X(j) for these {i,j}: {19, 4}, {20, 18589}
X(52840) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 516, 19}, {9812, 37104, 1848}

---

X(52841) = REFLECTION OF X(21) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52841) lies on these lines: {2,3}, {79,3671}, {515,34195}, {516,47033}, {758,5691}, {962,36999}, {1699,35016}, {2771,4018}, {2795,10723}, {2829,11604}, {3219,16139}, {3485,10543}, {3486,3649}, {3586,10122}, {3869,31938}, {4297,26725}, {4312,10394}, {5086,11684}, {5260,18406}, {5427,10896}, {5441,12047}, {5731,11281}, {6223,14450}, {6245,41557}, {6253,12607}, {6598,24391}, {9528,10152}, {9581,41547}, {9655,16137}, {9668,15174}, {9812,41571}, {9960,39772}, {10724,34789}, {11263,28164}, {11500,31660}, {16125,16132}, {16155,45287}, {17009,31272}, {17768,36991}, {18481,31019}, {21740,22793}, {25406,51747}, {28160,33858}

X(52841) = midpoint of X(i) and X(j) for these {i,j}: {2475, 3146}, {5073, 16117}
X(52841) = reflection of X(i) in X(j) for these {i,j}: {20, 442}, {21, 4}, {15704, 11277}, {16132, 16125}, {16139, 18480}, {16160, 3853}, {18481, 33592}, {34195, 49177}
X(52841) = anticomplement of X(44238)
X(52841) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 20, 6828}, {4, 30, 21}, {4, 3529, 6824}, {4, 376, 6866}, {4, 411, 7548}, {4, 6853, 546}, {4, 6856, 3839}, {4, 6868, 10883}, {4, 6869, 2476}, {4, 6874, 3843}, {4, 6876, 381}, {4, 6985, 17577}, {4, 6988, 3832}, {30, 11277, 15704}, {30, 3853, 16160}, {30, 442, 20}, {431, 13473, 4}, {515, 49177, 34195}, {2475, 3146, 30}, {2476, 6869, 411}, {3830, 6985, 4}, {3853, 6842, 4}, {5073, 16117, 30}, {6841, 11276, 6852}

---

X(52842) = REFLECTION OF X(22) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52842) lies on these lines: {2,3}, {1176,48905}, {1568,35264}, {1699,51692}, {1993,18400}, {2781,10733}, {3060,18396}, {5731,51718}, {5890,40909}, {6800,18388}, {7592,11750}, {9019,15305}, {9539,9668}, {10721,15102}, {10722,44974}, {11456,44407}, {11572,46730}, {12160,34799}, {12161,15800}, {12278,37498}, {12289,36747}, {13202,19140}, {13470,36753}, {15107,18392}, {17845,34148}, {18405,33586}, {19357,41482}, {23324,32269}, {25406,51744}, {25739,37489}, {31815,44076}, {32062,48884}, {34786,45186}

X(52842) = midpoint of X(3146) and X(7391)
X(52842) = reflection of X(i) in X(j) for these {i,j}: {20, 427}, {22, 4}, {1657, 18570}, {48905, 51739}
X(52842) = anticomplement of X(44239)
X(52842) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 7564, 7569}, {4, 12225, 7503}, {4, 20, 13160}, {4, 30, 22}, {4, 3529, 3547}, {4, 7503, 7566}, {30, 18570, 1657}, {30, 427, 20}, {858, 18533, 15078}, {3146, 7391, 30}, {3627, 12605, 4}, {3830, 9818, 4}, {3853, 7403, 4}, {6240, 14790, 11413}, {7576, 18531, 1995}, {11819, 18404, 10594}, {18405, 33586, 50435}

---

X(52843) = REFLECTION OF X(26) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52843) lies on these lines: {2,3}, {143,40909}, {155,30522}, {156,17845}, {1147,40276}, {1154,12293}, {1531,10539}, {1699,51696}, {1853,32138}, {5448,34785}, {5449,18376}, {5895,44544}, {6102,18396}, {7689,18383}, {7699,51033}, {8144,12953}, {9968,11645}, {10605,34798}, {10721,12270}, {10733,12281}, {11265,23251}, {11266,23261}, {11267,42094}, {11268,42093}, {11432,43575}, {12161,21659}, {12163,18405}, {12164,32423}, {12289,18445}, {12943,32047}, {13321,43835}, {13403,39522}, {13568,18952}, {13754,34786}, {14852,18379}, {14915,41725}, {15305,32338}, {16000,45788}, {16266,17702}, {18400,32139}, {18945,43588}, {22802,44407}, {23324,44158}, {29012,34117}, {32063,32354}, {32137,36990}, {32140,41362}, {32210,40686}, {34148,40242}, {35602,51391}, {41726,48675}

X(52843) = midpoint of X(i) and X(j) for these {i,j}: {3146, 14790}, {5073, 12085}
X(52843) = reflection of X(i) in X(j) for these {i,j}: {3, 18377}, {20, 13371}, {26, 4}, {1657, 11250}, {1658, 18567}, {7689, 18383},
X(52843) = anticomplement of X(44242)
X(52843) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 18386, 5}, {4, 14118, 381}, {4, 18563, 7526}, {4, 20, 10024}, {4, 30, 26}, {4, 7526, 7564}, {20, 7577, 3}, {30, 11250, 1657}, {30, 13371, 20}, {30, 18377, 3}, {30, 18566, 14070}, {30, 18567, 1658}, {30, 18568, 18324}, {30, 18569, 12084}, {1658, 18567, 381}, {3146, 14790, 30}, {3146, 18323, 7530}, {3843, 14070, 13406}, {5073, 12085, 30}, {6240, 18404, 6644}, {7574, 18565, 11413}, {10224, 15332, 10212}, {13406, 18566, 3843}

---

X(52844) = REFLECTION OF X(27) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52844) lies on the circumconic {{A,B,C,X(30),X(3668)}} and these lines: {2,3}, {92,15942}, {273,3586}, {1699,51697}, {1762,11471}, {1834,44698}, {2322,15946}, {2822,3668}, {2905,13202}, {3732,10733}, {5731,51721}, {7283,40445}, {8680,51063}, {10735,44968}

X(52844) = midpoint of X(3146) and X(3151)
X(52844) = reflection of X(i) in X(j) for these {i,j}: {20, 440}, {27, 4}
X(52844) = anticomplement of X(44243)
X(52844) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 30, 27}, {4, 4213, 10151}, {4, 4219, 7541}, {4, 7513, 7563}, {30, 440, 20}, {430, 13473, 4}, {3146, 3151, 30}

---

X(52845) = REFLECTION OF X(28) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52845) lies on these lines: {2,3}, {278,12953}, {1699,51698}, {1844,16118}, {2828,10152}, {2838,10729}, {2906,13202}, {5090,31672}, {5691,44661}, {6197,28146}, {10731,44970}, {10733,40263}, {15338,37799}, {15942,41869}

X(52845) = reflection of X(28) in X(4)
X(52845) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 30, 28}, {4, 4219, 7559}, {4, 451, 10151}, {429, 13473, 4}

---

X(52846) = REFLECTION OF X(29) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52846) lies on the circumconic {{A,B,C,X(30),X(39130)}} and these lines: {2,3}, {92,41869}, {318,14206}, {653,1770}, {1699,51699}, {1784,16118}, {1895,9579}, {1897,10733}, {1940,42387}, {2349,2816}, {2907,13202}, {6198,7100}, {7701,52414}, {7952,18625}, {17923,18483}, {28150,39574}, {31730,52412}

X(52846) = midpoint of X(3146) and X(3152)
X(52846) = reflection of X(i) in X(j) for these {i,j}: {20, 18641}, {29, 4}
X(52846) = anticomplement of X(44244)
X(52846) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 30, 29}, {4, 3144, 10151}, {4, 412, 7541}, {4, 4219, 7563}, {30, 18641, 20}, {407, 13473, 4}, {3146, 3152, 30}

---

X(52847) = REFLECTION OF X(31) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52847) lies on these lines: {3,31237}, {4,31}, {20,2887}, {30,30269}, {515,49454}, {516,4680}, {674,36990}, {744,51063}, {752,3543}, {758,5691}, {766,36997}, {1699,49480}, {2390,37001}, {2835,10729}, {3091,6679}, {3146,6327}, {3845,20575}, {9355,49500}, {10724,44938}, {10733,44940}, {17578,20064}, {42058,50687}

X(52847) = midpoint of X(3146) and X(6327)
X(52847) = reflection of X(i) in X(j) for these {i,j}: {20, 2887}, {31, 4}

---

X(52848) = REFLECTION OF X(33) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52848) lies on these lines: {1,4}, {19,1146}, {20,34822}, {30,36984}, {65,12136}, {84,1452}, {197,1593}, {208,7354}, {235,23304}, {355,1753}, {378,44425}, {406,4297}, {451,7987}, {475,19925}, {971,1905}, {1012,52427}, {1828,12173}, {1829,5895}, {1845,10732}, {1872,18525}, {1875,12943}, {1885,5130}, {1890,36991}, {1892,12678}, {2823,10727}, {3146,52365}, {4293,51359}, {5136,39531}, {5252,40971}, {5338,37395}, {5587,37305}, {7989,52252}, {36999,44438}, {37197,40985}

X(52848) = midpoint of X(3146) and X(52365)
X(52848) = reflection of X(i) in X(j) for these {i,j}: {20, 34822}, {33, 4}
X(52848) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 12667, 11392}, {4, 1870, 1699}, {4, 515, 33}, {1885, 5130, 11471}

---

X(52849) = REFLECTION OF X(34) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52849) lies on these lines: {1,4}, {19,3575}, {20,1861}, {24,44425}, {30,1753}, {108,9581}, {208,1837}, {355,31832}, {382,1872}, {406,19925}, {451,7989}, {475,4297}, {1593,22654}, {1698,37441}, {1824,12173}, {1845,37711}, {1871,18494}, {1876,12680}, {1885,5101}, {1887,12943}, {1902,3827}, {2182,17845}, {2261,19467}, {2840,10730}, {3146,52366}, {5090,11471}, {5338,7487}, {5587,7412}, {5895,40953}, {5928,41362}, {6284,40971}, {7987,52252}, {11399,19541}, {11500,52427}, {31423,37289}, {37001,44438}

X(52849) = midpoint of X(3146) and X(52366)
X(52849) = reflection of X(i) in X(j) for these {i,j}: {20, 34823}, {34, 4}
X(52849) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 515, 34}, {4, 6198, 1699}

---

X(52850) = REFLECTION OF X(35) IN X(4)

Barycentrics    -a^2*(a^2-b^2-b*c-c^2)*S^2-2*(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)*SB*SC : :

See Ivan Pavlov, euclid 5635.

X(52850) lies on these lines: {3,31262}, {4,35}, {20,25639}, {30,11012}, {381,33862}, {382,517}, {515,11009}, {516,5086}, {946,5441}, {1012,14794}, {1071,16118}, {1699,2646}, {1900,12173}, {2475,15931}, {2779,10721}, {3146,52367}, {3149,18514}, {3543,34619}, {3583,7681}, {3585,5842}, {3586,13750}, {3627,6253}, {3830,11849}, {4314,11218}, {4324,6831}, {4330,7680}, {5251,7491}, {5258,37820}, {5259,6917}, {5288,5841}, {6901,25542}, {6934,7741}, {6936,41859}, {6951,35202}, {7971,11280}, {7987,50239}, {9047,36990}, {9668,11522}, {10483,37002}, {10526,48696}, {10724,34789}, {11010,12705}, {11011,12943}, {11499,31160}, {11500,18513}, {12616,15228}, {13253,48680}, {17578,20066}, {18406,37290}, {37000,37719}

X(52850) = midpoint of X(3146) and X(52367)
X(52850) = reflection of X(i) in X(j) for these {i,j}: {20, 25639}, {35, 4}, {4324, 6831}
X(52850) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {382, 36999, 5691}, {3627, 6253, 41698}

---

X(52851) = REFLECTION OF X(36) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52851) lies on these lines: {3,31263}, {4,36}, {20,3814}, {30,119}, {104,24042}, {165,5123}, {381,23961}, {382,517}, {484,1709}, {515,7972}, {516,5176}, {519,10724}, {535,3543}, {1012,18513}, {1319,1699}, {1512,15228}, {1519,36975}, {1532,4316}, {1750,5538}, {1878,12173}, {2078,7965}, {2800,37006}, {2829,3583}, {3062,28534}, {3091,6681}, {3146,5080}, {3245,31673}, {3585,7680}, {3586,5570}, {3627,10943}, {3830,22765}, {4324,18242}, {4325,7681}, {5048,12953}, {5073,35000}, {5087,45036}, {5193,7354}, {5251,6923}, {5259,37290}, {5288,10525}, {5840,48696}, {6256,37000}, {6925,41853}, {6938,7951}, {7686,16118}, {7989,50239}, {8227,18857}, {9037,36990}, {9579,18838}, {9655,11522}, {10085,15239}, {10730,44973}, {10732,44979}, {11114,15931}, {11531,33956}, {11813,28164}, {12114,18514}, {13253,28204}, {14217,41702}, {17533,38759}, {17578,20067}, {18526,23960}, {22758,31159}, {28146,35460}, {30981,50694}, {37002,37720}, {37625,41704}, {44970,44982}

X(52851) = midpoint of X(i) and X(j) for these {i,j}: {3146, 5080}, {5073, 35000}
X(52851) = reflection of X(i) in X(j) for these {i,j}: {20, 3814}, {36, 4}, {104, 24042}, {4316, 1532}, {5537, 5080}, {15228, 1512}, {18526, 23960}, {36975, 1519}, {41702, 14217}, {44425, 41698}
X(52851) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {30, 41698, 44425}, {382, 37001, 5691}, {12688, 33697, 5691}

---

X(52852) = REFLECTION OF X(37) IN X(4)

Barycentrics    5*a^5*b - 2*a^3*b^3 - 3*a*b^5 + 5*a^5*c + 4*a^4*b*c - 2*a^3*b^2*c - 3*a*b^4*c - 4*b^5*c - 2*a^3*b*c^2 + 6*a*b^3*c^2 - 2*a^3*c^3 + 6*a*b^2*c^3 + 8*b^3*c^3 - 3*a*b*c^4 - 3*a*c^5 - 4*b*c^5 : : (Peter Moses, January 31, 2023)

See Ivan Pavlov, euclid 5635.

X(52852) lies on these lines: {3,31238}, {4,37}, {20,3739}, {30,4688}, {75,3146}, {192,17578}, {376,51041}, {381,51042}, {382,4686}, {515,49478}, {516,3686}, {518,5691}, {536,3543}, {537,50862}, {740,51118}, {742,51163}, {962,28581}, {1278,50690}, {1699,15569}, {2805,10724}, {3091,4698}, {3522,4751}, {3627,4718}, {3830,20430}, {3832,4687}, {3839,4755}, {4301,49475}, {4664,50687}, {4681,50688}, {4699,5059}, {4726,50691}, {4732,5493}, {4739,49135}, {4772,50692}, {7957,22271}, {9589,49459}, {9812,49470}, {10725,44970}, {11997,12953}, {12680,13476}, {12688,20718}, {14893,51045}, {14956,25939}, {15681,51049}, {15684,51040}, {15687,51038}, {17225,51026}, {17357,36670}, {24325,28164}, {25498,36693}, {27268,50689}, {29054,31673}, {31162,50778}, {31993,50694}, {34628,51061}, {34632,51036}, {34648,51034}, {35434,51039}, {36999,44438}, {41869,49468}, {49496,51538}

X(52852) = midpoint of X(i) and X(j) for these {i,j}: {75, 3146}, {3543, 51065}, {9589, 49459}, {15684, 51040}
X(52852) = reflection of X(i) in X(j) for these {i,j}: {20, 3739}, {37, 4}, {376, 51041}, {5493, 4732}, {7957, 22271}, {12680, 13476}, {15681, 51049}, {34628, 51061}, {34632, 51036}, {49475, 4301}, {50778, 31162}, {51034, 34648}, {51038, 15687}, {51042, 381}, {51045, 14893}

---

X(52853) = REFLECTION OF X(38) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52853) lies on these lines: {3,31264}, {4,38}, {20,1215}, {30,30272}, {515,41011}, {516,4692}, {537,3543}, {714,51063}, {758,5691}, {3091,6682}, {3146,17165}, {4722,5767}, {9020,36990}, {17578,20068}

X(52853) = midpoint of X(3146) in X(17165)
X(52853) = reflection of X(i) in X(j) for these {i,j}: {20, 1215}, {38, 4}

---

X(52854) = REFLECTION OF X(39) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52854) lies on these lines: {3,31239}, {4,39}, {5,21163}, {20,3934}, {30,5188}, {76,3146}, {98,35007}, {140,22681}, {147,7843}, {185,27375}, {194,17578}, {381,13334}, {382,511}, {538,3543}, {550,15819}, {575,18502}, {730,51118}, {732,51163}, {1350,17130}, {1352,7873}, {1503,5052}, {1513,39565}, {1657,7697}, {2782,3627}, {2794,46321}, {3091,6683}, {3095,3830}, {3202,26883}, {3529,22712}, {3767,46034}, {3832,7786}, {3839,32522}, {3843,11171}, {3845,11272}, {3849,9863}, {3850,40108}, {3853,14881}, {3861,32516}, {5007,39646}, {5008,12110}, {5059,22676}, {5073,9821}, {5097,13111}, {5206,9756}, {5480,7765}, {5691,14839}, {5969,51022}, {5999,7816}, {6194,49135}, {6321,35436}, {7757,50687}, {7804,12203}, {7805,38664}, {7861,13862}, {7935,10516}, {7976,9812}, {8589,37334}, {8719,15515}, {8992,42258}, {11676,15513}, {12251,14711}, {12263,28164}, {13354,29012}, {13357,44518}, {13860,37512}, {13983,42259}, {14994,29181}, {15687,32448}, {20081,50690}, {32450,50688}, {32451,51538}, {33238,51537}, {33250,38747}, {35438,37348}, {35439,48901}, {37243,48889}, {37446,39601}, {42271,49252}, {42272,49253}

X(52854) = midpoint of X(i) and X(j) for these {i,j}: {76, 3146}, {5073, 9821}
X(52854) = reflection of X(i) in X(j) for these {i,j}: {20, 3934}, {39, 4}, {185, 27375}, {5188, 6248}, {14881, 3853}, {32516, 3861}, {35439, 48901}, {44422, 15687}
X(52854) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 39, 22682}, {4, 8721, 5475}, {30, 6248, 5188}, {382, 36990, 36997}, {5059, 31276, 22676}, {5188, 6248, 9466}, {36992, 36994, 48942}

---

X(52855) = REFLECTION OF X(41) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52855) lies on these lines: {3,31240}, {4,41}, {20,17046}, {30,31135}, {766,36997}, {2389,36999}, {2809,5691}, {3091,31284}, {3146,21285}, {8679,36990}, {17578,20071}

X(52855) = midpoint of X(3146) and X(21285)
X(52855) = reflection of X(i) in X(j) for these {i,j}: {20, 17046}, {41, 4}

---

X(52856) = REFLECTION OF X(42) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52856) lies on these lines: {3,31241}, {4,42}, {20,3741}, {30,31136}, {517,4365}, {519,962}, {674,36990}, {1385,50415}, {2308,5767}, {2813,10725}, {3091,6685}, {3146,17135}, {4300,10454}, {7957,14973}, {10724,44940}, {17578,20011}, {19262,30970}

X(52856) = midpoint of X(3146) and X(17135)
X(52856) = reflection of X(i) in X(j) for these {i,j}: {20, 3741}, {42, 4}, {7957, 14973}

---

X(52857) = REFLECTION OF X(43) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52857) lies on these lines: {3,31242}, {4,43}, {20,3840}, {30,31137}, {519,962}, {995,1699}, {1401,9579}, {3091,6686}, {3146,10453}, {3586,24248}, {8227,50415}, {9025,36990}, {9812,20037}, {10724,44938}, {17578,20012}, {29827,37331

X(52857) = midpoint of X(3146) and X(10453)
X(52857) = reflection of X(i) in X(j) for these {i,j}: {20, 3840}, {43, 4}

---

X(52858) = REFLECTION OF X(44) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52858) lies on these lines: {3,31243}, {4,44}, {20,3834}, {30,31138}, {320,3146}, {377,25891}, {516,2325}, {518,5691}, {752,51118}, {944,4864}, {946,1279}, {1699,3246}, {3091,6687}, {3543,4715}, {3836,12512}, {4301,49699}, {5881,49703}, {9812,49709}, {15971,28362}, {17578,20072}, {20070,32850}

X(52858) = midpoint of X(320) and X(3146)
X(52858) = reflection of X(i) in X(j) for these {i,j}: {20, 3834}, {44, 4}, {49699, 4301}, {49703, 5881}

---

X(52859) = REFLECTION OF X(45) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52859) lies on these lines: {3,31244}, {4,45}, {20,34824}, {30,31139}, {516,3707}, {518,5691}, {545,3543}, {3091,31285}, {3146,42697}, {5752,29309}, {17578,20073}, {28580,51118}

X(52859) = midpoint of X(3146) and X(42697)
X(52859) = reflection of X(i) in X(j) for these {i,j}: {20, 34824}, {45, 4}

---

X(52860) = REFLECTION OF X(46) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52860) lies on these lines: {1,1537}, {4,46}, {20,21616}, {30,12679}, {40,37821}, {55,22792}, {56,1699}, {84,3583}, {165,1329}, {382,517}, {515,30323}, {516,3436}, {529,50865}, {912,41704}, {946,4317}, {971,12953}, {1479,10085}, {1519,37618}, {1538,5204}, {1768,9581}, {2098,9580}, {2800,37711}, {3057,40267}, {3062,6598}, {3146,11415}, {3149,4333}, {3338,26333}, {3585,12705}, {3586,15071}, {3612,6938}, {4187,16209}, {4297,50244}, {4301,36977}, {4302,6260}, {4324,52026}, {4338,7686}, {4855,21635}, {5119,6256}, {5225,12246}, {5450,23708}, {5840,17857}, {6259,6284}, {6691,7988}, {6906,37692}, {7491,50528}, {7681,16118}, {7741,52027}, {8069,9612}, {8227,13743}, {8256,37714}, {9578,17662}, {9614,18237}, {9668,12680}, {9812,20076}, {9856,12943}, {10310,37411}, {10531,51816}, {10680,22793}, {10724,12528}, {10728,37708}, {10896,34862}, {11114,12520}, {11522,24928}, {11531,38455}, {11826,37822}, {12514,37437}, {12678,15171}, {15430,40950}, {16128,37700}, {17101,34464}, {17637,37723}, {18443,49178}, {28146,35448}, {37252,40293}, {37567,41700}, {39990,42464}, {41560,41686}

X(52860) = midpoint of X(3146) and X(11415)
X(52860) = reflection of X(i) in X(j) for these {i,j}: {20, 21616}, {40, 37821}, {46, 4}, {1768, 12764}, {4333, 3149}, {5691, 37001}, {10085, 1479}, {10680, 22793}, {34464, 17101}, {36977, 4301}, {37002, 946}, {41686, 41560}
X(52860) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1158, 10826}, {382, 12688, 5691}, {517, 37001, 5691}, {9668, 48664, 12680}

---

X(52861) = REFLECTION OF X(47) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52861) lies on these lines: {4,47}, {20,34825}, {758,5691}

X(52861) = reflection of X(i) in X(j) for these {i,j}: {20, 34825}, {47, 4}

---

X(52862) = REFLECTION OF X(48) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52862) lies on these lines: {3,31265}, {4,48}, {20,20305}, {30,31163}, {382,916}, {2173,39531}, {2801,10725}, {3146,21270}, {5691,44661}, {8679,36990}, {17578,20074}, {29219,51118}

X(52862) = midpoint of X(3146) in X(21270)
X(52862) = reflection of X(i) in X(j) for these {i,j}: {20, 20305}, {48, 4}

---

X(52863) = REFLECTION OF X(49) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52863) lies on these lines: {3,18383}, {4,49}, {5,51033}, {20,34826}, {30,11440}, {52,12902}, {110,18567}, {186,18379}, {265,6240}, {381,13367}, {382,6243}, {567,21659}, {568,12173}, {1658,18392}, {3448,34798}, {3518,10113}, {3521,34224}, {3543,31815}, {3581,9927}, {3627,7728}, {3830,15811}, {3843,11425}, {3845,15806}, {5073,17834}, {5076,43844}, {5876,10296}, {5890,50006}, {6247,20127}, {6288,18563}, {7527,22804}, {10112,46027}, {10254,34785}, {10255,18376}, {11477,48662}, {11597,32365}, {12121,13371}, {12134,18323}, {12162,48675}, {12278,18377}, {12289,44263}, {12295,13419}, {12300,32137}, {12897,32340}, {13561,13619}, {13595,43865}, {13851,45735}, {17702,31724}, {18350,18403}, {18381,18565}, {18394,37814}, {18396,37481}, {18474,18562}, {18569,37477}, {18570,40242}, {22660,23236}, {34783,35480}, {38321,43821}

X(52863) = reflection of X(i) in X(j) for these {i,j}: {20, 34826}, {49, 4}, {37495, 31724}
X(52863) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 30522, 49}, {382, 12293, 6243}, {17702, 31724, 37495}

---

X(52864) = REFLECTION OF X(50) IN X(4)

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

See Ivan Pavlov, euclid 5635.

X(52864) lies on these lines: {4,50}, {20,34827}, {381,22463}, {382,511}, {3447,37924}, {10735,14989}, {18325,39838}

X(52864) = reflection of X(i) in X(j) for these {i,j}: {20, 34827}, {50, 4}

---

X(52865) = X(6)X(18753)∩X(81)X(367)

Barycentrics    a^3 (Sqrt[b] + Sqrt[c]) : :

See Ivan Pavlov, euclid 5638.

X(52865) lies on the circumconic {{A,B,C,X(6),X(31)}} and these lines: {6,18753}, {81,367}, {2214,20682}, {2221,20751}, {14621,20527}

X(52865) = barycentric product X(i)*X(j) for these (i, j): {1, 20664}, {6, 367}, {19, 20751}, {31, 20527}, {58, 20682}, {604, 4181}, {18753, 40378}
X(52865) = barycentric quotient X(i)*X(j) for these (i, j): {367, 76}, {4181, 28659}, {20527, 561}, {20664, 75}, {20682, 313}, {20751, 304}
X(52865) = X(i)-Dao conjugate of X(j) for these {i, j}: {40378, 561}

---

X(52866) = X(6)X(365)∩X(81)X(2069)

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

See Ivan Pavlov, euclid 5638.

X(52866) lies on the circumconic {{A,B,C,X(6),X(31)}} and these lines: {6,365}, {31,18753}, {81,2069}, {1333,20779}, {2298,4180}

X(52866) = barycentric product X(i)*X(j) for these (i, j): {4, 20779}, {6, 40378}, {56, 4180}, {365, 367}, {366, 20664}, {18753, 20527}
X(52866) = barycentric quotient X(i)*X(j) for these (i, j): {4180, 3596}, {20779, 69}, {40378, 76}
X(52866) = X(i)-Dao conjugate of X(j) for these {i, j}: {20527, 76}

---

X(52867) = X(2)X(17)∩X(13)X(2981)

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

See Kadir Altintas and César Lozada, euclid 5648.

X(52867) lies on these lines: {2, 17}, {13, 2981}, {30, 11555}, {396, 8014}, {3412, 44466}, {8029, 9201}, {8929, 16962}, {11542, 30465}, {22892, 30462}, {36186, 42992}, {42496, 52039}

X(52867) = reflection of X(2) in X(12097)
X(52867) = X(13)-Ceva conjugate of-X(34325)
X(52867) = X(396)-waw conjugate of-X(22892)
X(52867) = barycentric product X(i)*X(j) for these {i, j}: {396, 6669}, {618, 34325}

---

X(52868) = X(2)X(18)∩X(14)X(5675)

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

See Kadir Altintas and César Lozada, euclid 5648.

X(52868) lies on these lines: {2, 18}, {14, 5675}, {30, 11556}, {395, 8015}, {3411, 44462}, {8029, 9200}, {8930, 16963}, {11543, 30468}, {22848, 30459}, {36185, 42993}, {42497, 52040}

X(52868) = reflection of X(2) in X(12098)
X(52868) = X(14)-Ceva conjugate of-X(34326)
X(52868) = X(395)-waw conjugate of-X(22848)
X(52868) = barycentric product X(i)*X(j) for these {i, j}: {395, 6670}, {619, 34326}

---

X(52869) = X(389)X(546)∩X(2972)X(3628)

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

See Ivan Pavlov, euclid 5649.

X(52869) lies on these lines: {5,35360}, {30,34334}, {140,47204}, {389,546}, {547,14767}, {1656,44003}, {2972,3628}, {3018,44233}, {6344,34093}, {9033,10272}, {13364,42862}, {14363,15912}, {38605,47087}

X(52869) = midpoint of X(5) and X(35360)
X(52869) = reflection of X(2972) in X(3628)

---

X(52870) = X(7)X(11)∩X(3160)X(4626)

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

See Ivan Pavlov, euclid 5649.

X(52870) lies on these lines: {7,11}, {9,40537}, {142,13609}, {1086,10481}, {3160,4626}, {6173,21258}, {6366,10427}, {7195,40617}, {8255,47374}, {30379,40629}

X(52870) = midpoint of X(7) and X(658)
X(52870) = reflection of X(i) in X(j) for these {i,j}: {9, 40537}, {13609, 142}
X(52870) = X(i)-isoconjugate-of-X(j) for these {i, j}: {4845, 15731}
X(52870) = X(i)-Dao conjugate of X(j) for these {i, j}: {1323, 2}
X(52870) = barycentric product X(i)*X(j) for these (i, j): {15726, 37780}, {30806, 43064}
X(52870) = barycentric quotient X(i)/X(j) for these (i, j): {6610, 15731}, {15726, 41798}, {43064, 1156}
X(52870) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 658, 5851}

---

X(52871) = X(8)X(11)∩X(121)X(519)

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

See Ivan Pavlov, euclid 5649.

X(52871) lies on the circumconic {{A,B,C,X(80),X(4927)}} and these lines: {1,2885}, {2,1120}, {8,11}, {10,3756}, {121,519}, {141,3679}, {528,21290}, {900,1145}, {952,50914}, {1146,2321}, {1213,16613}, {1317,17780}, {2802,21087}, {2968,6736}, {3039,3161}, {3454,3626}, {3617,24988}, {3704,6741}, {3717,4542}, {3880,6018}, {3913,4571}, {3952,13996}, {4085,34832}, {4111,4711}, {4370,36936}, {4487,40663}, {4534,30730}, {4543,14430}, {4662,38992}, {4669,21251}, {6735,51402}, {9457,43055}, {16593,23891}, {16594,17460}, {20333,49772}

X(52871) = midpoint of X(8) and X(3699)
X(52871) = reflection of X(3756) in X(10)
X(52871) = complement of X(1120)
X(52871) = X(i)-isoconjugate-of-X(j) for these {i, j}: {106, 8686}
X(52871) = X(i)-Dao conjugate of X(j) for these {i, j}: {214, 8686}, {2087, 3669}, {2325, 2}, {3880, 45247}, {14425, 40617}, {16610, 7}, {38979, 37627}, {51402, 23836}
X(52871) = barycentric product X(i)*X(j) for these (i, j): {8, 16594}, {9, 20900}, {312, 17460}, {333, 21041}, {345, 5151}, {1266, 2325}, {2321, 17195}, {3596, 20972}, {3699, 21129}, {3762, 23705}, {3880, 4358}, {4152, 52574}, {4723, 16610}, {4738, 52140}, {4927, 30731}, {7017, 22082}, {36791, 45247}
X(52871) = barycentric quotient X(i)/X(j) for these (i, j): {44, 8686}, {1635, 37627}, {1639, 23836}, {2325, 1120}, {3689, 40400}, {3880, 88}, {4152, 52556}, {4723, 36805}, {5151, 278}, {5516, 40617}, {6018, 52206}, {16594, 7}, {17195, 1434}, {17460, 57}, {20900, 85}, {20972, 56}, {21041, 226}, {21129, 3676}, {22082, 222}, {23705, 3257}, {30731, 6079}, {45247, 2226}, {52140, 679}
X(52871) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 3699, 5854}, {17460, 21041, 16594}

---

X(52872) = X(10)X(3120)∩X(37)X(4103)

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

See Ivan Pavlov, euclid 5649.

X(52872) lies on the circumconics {{A,B,C,X(10),X(4738)}}, {{A,B,C,X(37),X(34587)}} and these lines: {2,39697}, {10,3120}, {37,4103}, {210,22306}, {244,3634}, {519,3992}, {960,2802}, {1125,17724}, {1698,17154}, {3293,14752}, {3697,22313}, {3932,16597}, {3956,4732}, {4043,4125}, {4058,21090}, {4075,4868}, {4145,21714}, {4169,21821}, {17449,49993}, {17461,27538}, {17757,31845}, {21081,23936}, {21705,21709}, {28522,42083}, {42056,42285}

X(52872) = midpoint of X(i) and X(j) for these {i,j}: {10, 3952}, {4738, 34587}
X(52872) = reflection of X(i) in X(j) for these {i,j}: {244, 3634}, {1125, 24003}
X(52872) = complement of X(39697)
X(52872) = X(i)-Dao conjugate of X(j) for these {i, j}: {1635, 16726}, {3943, 2}
X(52872) = barycentric product X(i)*X(j) for these (i, j): {3943, 17160}, {3992, 37680}, {4145, 24004}, {4169, 21297}, {4358, 31855}, {18145, 21805}
X(52872) = barycentric quotient X(i)/X(j) for these (i, j): {3943, 39697}, {3992, 39994}, {4145, 1022}, {21714, 4049}, {21805, 39982}, {31855, 88}, {38979, 16726}
X(52872) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 21087, 4013}, {4738, 34587, 519}

---

X(52873) = X(11)X(885)∩X(514)X(1387)

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

See Ivan Pavlov, euclid 5649.

X(52873) lies on these lines: {11,885}, {514,1387}, {528,45322}, {676,2826}, {1086,23838}, {1146,23893}, {3035,40551}, {3036,3900}, {3126,6667}, {3716,6366}, {3887,12019}, {4885,45310}

X(52873) = midpoint of X(11) and X(885)
X(52873) = reflection of X(i) in X(j) for these {i,j}: {3035, 40551}, {3126, 6667}

---

X(52874) = X(3)X(6723)∩X(20)X(122)

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

See Ivan Pavlov, euclid 5649.

X(52874) lies on these lines: {3,6723}, {20,122}, {113,18508}, {133,15774}, {216,44241}, {376,6389}, {548,10600}, {3184,9033}, {4240,38956}, {15526,37853}, {15905,39020}, {16111,44715}, {16386,44436}, {20725,34147}, {32459,44248}, {34851,44244}

X(52874) = reflection of X(13611) in X(3)
X(52874) = X(i)-isoconjugate-of-X(j) for these {i, j}: {5896, 36119}
X(52874) = X(i)-Dao conjugate of X(j) for these {i, j}: {1511, 5896}, {47296, 253}
X(52874) = barycentric product X(i)*X(j) for these (i, j): {13202, 37669}
X(52874) = barycentric quotient X(i)/X(j) for these (i, j): {3284, 5896}, {132
X(52874) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {12113, 16163, 3184}

---

X(52875) = X(10)X(2486)∩X(536)X(899)

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

See Ivan Pavlov, euclid 5649.

X(52875) lies on the circumconic and {{A,B,C,X(37),X(42083)}} these lines: {10,2486}, {37,3121}, {244,4698}, {518,34587}, {536,899}, {537,1125}, {872,14752}, {3293,4069}, {3701,49468}, {3739,24003}, {3943,35068}, {4687,17154}, {4738,28581}

X(52875) = midpoint of X(37) and X(3952)
X(52875) = reflection of X(i) in X(j) for these {i,j}: {244, 4698}, {3739, 24003}
X(52875) = X(i)-Dao conjugate of X(j) for these {i, j}: {4728, 17205}
X(52875) = barycentric product X(i)*X(j) for these (i, j): {536, 19998}, {4033, 38349}
X(52875) = barycentric quotient X(i)/X(j) for these (i, j): {19998, 3227}, {38349, 1019}

---

X(52876) = X(39)X(4576)∩X(538)X(3231)

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

See Ivan Pavlov, euclid 5649.

X(52876) lies on the circumconic {{A,B,C,X(39),X(52067)}} and these lines: {39,4576}, {538,3231}, {620,3589}, {3124,6683}, {3934,15082}, {7786,25047}

X(52876) = midpoint of X(39) and X(4576)
X(52876) = reflection of X(3124) in X(6683)
X(52876) = X(i)-Dao conjugate of X(j) for these {i, j}: {9148, 34294}
X(52876) = barycentric quotient X(38366)/X(18105)

---

X(52877) = X(42)X(3122)∩X(3822)X(4085)

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

See Ivan Pavlov, euclid 5649.

X(52877) lies on these lines: {42,3122}, {2177,23386}, {3822,4085}, {35069,51377}

X(52877) = midpoint of X(42) and X(4577)
X(52877) = X(i)-Dao conjugate of X(9148) for these {i, j}: {9148, 34294}
X(52877) = barycentric product X(44)*X(22294)
X(52877) = barycentric quotient X(22294)/X(20568)

---

X(52878) = X(140)X(6720)∩X(232)X(237)

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

See Ivan Pavlov, euclid 5649.

X(52878) lies on these lines: {5,15897}, {32,51906}, {140,6720}, {216,52604}, {232,237}, {389,40645}, {2871,9969}, {3613,6697}, {11574,23583}

---

X(52879) = X(57)X(934)∩X(223)X(6614)

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

See Ivan Pavlov, euclid 5649.

X(52879) lies on these lines: {57,934}, {223,6614}, {527,28344}, {946,20418}, {972,21164}, {3452,40555}, {5514,6692}

X(52879) = midpoint of X(57) and X(934)
X(52879) = reflection of X(i) in X(j) for these {i,j}: {3452, 40555}, {5514, 6692}
X(52879) = X(i)-Dao conjugate of X(j) for these {i, j}: {6610, 2}
X(52879) = barycentric product X(32625)*X(37780)
X(52879) = barycentric quotient X(32625)/X(41798)

---

X(52880) = X(63)X(6516)∩X(5745)X(6506)

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

See Ivan Pavlov, euclid 5649.

X(52880) lies on these lines: {63,6516}, {3735,16579}, {3916,26932}, {4999,15254}, {5745,6506}, {6139,6366}, {6505,6517}, {22097,39006}

X(52880) = midpoint of X(63) and X(6516)
X(52880) = reflection of X(6506) in X(5745)
X(52880) = X(i)-Dao conjugate of X(j) for these {i, j}: {6510, 2}

---

X(52881) = X(69)X(125)∩X(599)X(626)

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

See Ivan Pavlov, euclid 5649.

X(52881) lies on the circumconics {{A,B,C,X(67),X(9134)}}, {{A,B,C,X(69),X(52629)}} and these lines: {2,39238}, {69,125}, {114,23342}, {141,6388}, {599,626}, {690,5181}, {2482,15390}, {2930,14928}, {3933,15526}, {4576,32114}, {5095,5468}, {6337,36895}, {6593,38020}, {9035,15595}, {17983,35179}, {32244,38940}

X(52881) = midpoint of X(69) and X(4563)
X(52881) = reflection of X(6388) in X(141)
X(52881) = complement of X(41909)
X(52881) = X(i)-isoconjugate-of-X(j) for these {i, j}: {19, 15387}, {923, 2374}, {1973, 44182}
X(52881) = X(i)-Dao conjugate of X(j) for these {i, j}: {6, 15387}, {126, 8753}, {2482, 2374}, {3291, 4}, {6337, 44182}, {6390, 2}, {21906, 2489}, {34158, 41936}
X(52881) = X(i) cross conjugate of X(j) for these {i, j}: {47412, 126}
X(52881) = barycentric product X(i)*X(j) for these (i, j): {69, 126}, {76, 47412}, {304, 17466}, {3266, 8681}, {6390, 47286}, {11634, 45807}, {21905, 52608}
X(52881) = barycentric quotient X(i)/X(j) for these (i, j): {3, 15387}, {69, 44182}, {126, 4}, {524, 2374}, {3291, 8753}, {6390, 41909}, {8681, 111}, {17466, 19}, {21905, 2489}, {47286, 17983}, {47412, 6}
X(52881) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5181, 36792, 50567}

---

X(52882) = X(75)X(244)∩X(1930)X(3061)

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

See Ivan Pavlov, euclid 5649.

X(52882) lies on these lines: {37,40562}, {75,244}, {714,35532}, {1086,20888}, {1930,3061}, {3739,6377}, {4688,16604}, {4699,21224}, {6376,6386}, {14296,51583}, {20891,40619}, {21422,40624}, {41314,42083}

X(52882) = midpoint of X(75) and X(1978)
X(52882) = reflection of X(i) in X(j) for these {i,j}: {37, 40562}, {6377, 3739}
X(52882) = X(i)-Dao conjugate of X(j) for these {i, j}: {2229, 1}, {6381, 2}, {14426, 38986}

---

X(52883) = X(99)X(1649)∩X(5912)X(5976)

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

See Ivan Pavlov, euclid 5649.

X(52883) lies on these lines: {2,42344}, {3,45772}, {99,1649}, {2482,14588}, {5912,5976}, {9181,9182}

X(52883) = X(i)-Dao conjugate of X(j) for these {i, j}: {9182, 2}
X(52883) = barycentric product X(9182)*X(44373)
X(52883) = barycentric quotient X(44373)/X(9180)

---

X(52884) = X(1)X(46409)∩X(100)X(3126)

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

See Ivan Pavlov, euclid 5649.

X(52884) lies on these lines: {1,46409}, {100,3126}, {6174,17060}, {6184,14589}, {6594,46973}

X(52884) = midpoint of X(100) and X(5377)

---

X(52885) = X(2)X(45)∩X(3633)X(4432)

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

See Ivan Pavlov, euclid 5650.

X(52885) lies on these lines: {2,45}, {1268,16814}, {2325,49761}, {3633,4432}, {3731,30598}, {4764,17755}, {12812,24833}, {15712,24813}, {17227,25728}, {17272,17285}, {17273,17339}, {17340,32025}, {24828,33703

X(52885) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {190, 41138, 27191}, {190, 4422, 903}, {1997, 26791, 37684}, {4409, 4422, 2}, {4440, 36522, 190}, {30855, 32933, 2}

---

X(52886) = X(2)X(99)∩X(1285)X(6337)

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

See Ivan Pavlov, euclid 5650.

X(52886) lies on these lines: {2,99}, {98,15712}, {114,33703}, {548,21166}, {1285,6337}, {1657,10722}, {3625,51578}, {3627,38730}, {3633,11711}, {3843,10723}, {3850,15561}, {5013,43527}, {5026,6144}, {5072,38733}, {5182,32455}, {6033,15686}, {6054,15689}, {6321,12812}, {6777,42929}, {6778,42928}, {7782,7784}, {7863,32027}, {7888,45017}, {7891,7911}, {7899,44526}, {8724,45759}, {9112,30471}, {9113,30472}, {9164,39061}, {9862,21735}, {10159,37512}, {12042,15706}, {12108,23235}, {12117,38335}, {12188,15718}, {14893,23234}, {15480,32459}, {15602,16988}, {17538,38736}, {32458,32876}, {38738,49140}, {39091,51585}

X(52886) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {99, 41134, 14061}, {99, 620, 671}, {148, 36521, 99}, {620, 52695, 99}

---

X(52887) = X(2)X(3)∩X(2081)X(2600)

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

See Ivan Pavlov, euclid 5650.

X(52887) lies on these lines: {2,3}, {2081,2600}, {3917,34985}, {4993,42350}, {6750,31388}, {14129,23607}, {14918,32428}, {19167,32438}, {32078,34836}, {34951,45186}

X(52887) = reflection of X(35442) in X(14918)
X(52887) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2167, 51222}
X(52887) = X(i)-Dao conjugate of X(j) for these {i, j}: {138, 275}, {40588, 51222}
X(52887) = intersection, other than A, B, C, of circumconics {{A,B,C,X(2),X(6368)}}, {{A,B,C,X(3),X(17434)}}, {{A,B,C,X(4),X(12077)}}, {{A,B,C,X(24),X(52317)}}, {{A,B,C,X(27),X(21102)}} and {{A,B,C,X(30),X(14391)}}
X(52887) = barycentric product X(i)*X(j) for these (i, j): {14570, 42731}, {41586, 52767}
X(52887) = barycentric quotient X(i)/X(j) for these (i, j): {51, 51222}, {138, 51939}, {42731, 15412}
X(52887) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14918, 32428, 35442}

---

X(52888) = X(2)X(7)∩X(650)X(663)

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

See Ivan Pavlov, euclid 5650.

X(52888) lies on these lines: {2,7}, {38,20310}, {101,44425}, {220,4413}, {497,1200}, {518,3119}, {650,663}, {657,10006}, {1001,32578}, {1146,51463}, {1155,51418}, {1193,9367}, {1202,11019}, {1334,5218}, {1376,6602}, {1465,9502}, {1475,6554}, {1736,25069}, {2177,6181}, {2183,51406}, {2272,17747}, {2635,35326}, {3681,41796}, {3691,46835}, {3720,16588}, {3848,42438}, {3870,41795}, {5199,45751}, {6605,9342}, {6745,35341}, {7123,25938}, {8608,16777}, {14547,46839}, {14943,41798}, {17435,17449}, {17474,41006}, {18839,38375}, {21384,27541}, {30695,34497}, {33299,46830}, {34522,43046}

X(52888) = intersection, other than A, B, C, of circumconics {{A,B,C,X(2),X(3900)}}, {{A,B,C,X(7),X(650)}}, {{A,B,C,X(9),X(4105)}} and {{A,B,C,X(57),X(663)}}
X(52888) = barycentric product X(i)*X(j) for these (i, j): {8, 3000}, {9, 44664}
X(52888) = barycentric quotient X(i)*X(j) for these (i, j): {3000, 7}, {44664, 85}
X(52888) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {30414, 30415, 26125}

---

X(52889) = X(2)X(3)∩X(521)X(650)

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

See Ivan Pavlov, euclid 5650.

X(52889) lies on these lines: {2,3}, {55,17188}, {92,23171}, {521,650}, {1259,27412}, {1260,27398}, {1376,2328}, {1896,20764}, {2194,11502}, {2222,26702}, {2360,11500}, {2655,52610}, {3190,3913}, {3286,3911}, {4658,41344}, {5327,37541}, {5437,17194}, {6708,23207}, {10006,21789}, {37139,37142}, {38288,41083}

X(52889) = X(i)-isoconjugate-of-X(j) for these {i, j}: {65, 23707}, {226, 32726}, {525, 36140}, {1441, 34078}, {14208, 32727}
X(52889) = X(i)-Dao conjugate of X(j) for these {i, j}: {40602, 23707}
X(52889) = intersection, other than A, B, C, of circumconics {{A,B,C,X(2),X(521)}, {{A,B,C,X(3),X(36054)}}, {{A,B,C,X(4),X(650)}}, {{A,B,C,X(21),X(23090)}}, {{A,B,C,X(25),X(3063)}}, {{A,B,C,X(27),X(3737)}}, {{A,B,C,X(28),X(7252)}}, {{A,B,C,X(29),X(1021)}} and {{A,B,C,X(30),X(14395)}}
X(52889) = barycentric product X(i)*X(j) for these (i, j): {333, 2635}, {645, 30691}, {811, 2637}, {4573, 30692}
X(52889) = barycentric quotient X(i)/X(j) for these (i, j): {284, 23707}, {2194, 32726}, {2635, 226}, {2637, 656}, {30691, 7178}, {30692, 3700}, {32676, 36140}
X(52889) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {29, 1816, 3}

---

X(52890) = X(2)X(3)∩X(112)X(9081)

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

See Ivan Pavlov, euclid 5650.

X(52890) lies on these lines: {2,3}, {107,29348}, {112,9081}, {513,1430}, {4267,50065}, {11363,46883}, {15742,16085}, {18185,50068}

X(52890) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3, 41683}, {71, 3227}, {72, 37129}, {73, 36798}, {228, 31002}, {306, 739}, {525, 34075}, {647, 4607}, {656, 898}, {810, 889}, {1331, 35353}, {14208, 32718}, {23892, 52609}
X(52890) = X(i)-Dao conjugate of X(j) for these {i, j}: {5521, 35353}, {13466, 20336}, {14434, 18210}, {36103, 41683}, {39011, 525}, {39052, 4607}, {39062, 889}, {40596, 898}, {40614, 306}
X(52890) = intersection, other than A, B, C, of circumconics {{A,B,C,X(1),X(13735)}}, {{A,B,C,X(2),X(513)}}, {{A,B,C,X(3),X(22383)}}, {{A,B,C,X(4),X(6591)}}, {{A,B,C,X(6),X(11322)}}, {{A,B,C,X(21),X(7252)}}, {{A,B,C,X(25),X(15742)}}, {{A,B,C,X(28),X(43925)}} and {{A,B,C,X(30),X(891)}}
X(52890) = barycentric product X(i)*X(j) for these (i, j): {27, 899}, {28, 536}, {162, 4728}, {286, 3230}, {648, 891}, {811, 3768}, {890, 6331}, {1172, 43037}, {1396, 4009}, {1474, 6381}, {2203, 35543}, {5379, 52626}, {17925, 23343}, {41314, 43925}
X(52890) = barycentric quotient X(i)/X(j) for these (i, j): {19, 41683}, {27, 31002}, {28, 3227}, {112, 898}, {162, 4607}, {536, 20336}, {648, 889}, {890, 647}, {891, 525}, {899, 306}, {1172, 36798}, {1474, 37129}, {1646, 18210}, {2203, 739}, {3230, 72}, {3768, 656}, {3994, 52369}, {4526, 52355}, {4728, 14208}, {5379, 5381}, {6381, 40071}, {6591, 35353}, {14437, 14429}, {19945, 4466}, {23343, 52609}, {32676, 34075}, {43037, 1231}, {43925, 43928}

---

X(52891) = X(2)X(3)∩X(243)X(522)

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

See Ivan Pavlov, euclid 5650.

X(52891) lies on these lines: {2,3}, {33,17194}, {58,7952}, {108,3286}, {243,522}, {278,2328}, {281,4877}, {283,1068}, {333,7046}, {527,23710}, {1119,8822}, {1155,37805}, {1172,34919}, {1249,1778}, {1275,5379}, {1758,52607}, {1785,52680}, {1897,16704}, {3193,38295}, {4512,5307}, {4640,40149}, {4653,34231}, {5057,17923}, {7009,17185}, {17188,17917}, {18163,40971}, {31445,41013}, {37142,41207}, {37780,38461}

X(52891) = X(i)-isoconjugate-of-X(j) for these {i, j}: {71, 34056}, {73, 1156}, {307, 34068}, {525, 36141}, {647, 37139}, {656, 14733}, {810, 35157}, {1121, 1409}, {1214, 2291}, {1439, 4845}, {14208, 32728}, {23067, 35348}, {23893, 52610}, {41798, 52373}
X(52891) = X(i)-Dao conjugate of X(j) for these {i, j}: {6594, 72}, {35091, 525}, {35110, 307}, {39052, 37139}, {39062, 35157}, {40596, 14733}, {40629, 17094}
X(52891) = intersection, other than A, B, C, of circumconics {{A,B,C,X(1),X(6912)}}, {{A,B,C,X(2),X(522)}}, {{A,B,C,X(3),X(652)}}, {{A,B,C,X(4),X(3064)}}, {{A,B,C,X(7),X(10883)}}, {{A,B,C,X(8),X(11114)}}, {{A,B,C,X(9),X(1005)}}, {{A,B,C,X(20),X(1323)}}, {{A,B,C,X(21),X(1021)}}, {{A,B,C,X(29),X(17926)}}, {{A,B,C,X(30),X(6366)}}, {{A,B,C,X(55),X(36014)}, {{A,B,C,X(79),X(52269)}} and {{A,B,C,X(90),X(411)}}
X(52891) = barycentric product X(i)*X(j) for these (i, j): {21, 37805}, {27, 6745}, {29, 527}, {286, 6603}, {333, 23710}, {648, 6366}, {823, 14414}, {1055, 44130}, {1155, 31623}, {1172, 30806}, {1323, 2322}, {1638, 36797}, {1896, 6510}, {2287, 38461}, {4183, 37780}, {6139, 6331}
X(52891) = barycentric quotient X(i)/X(j) for these (i, j): {28, 34056}, {29, 1121}, {112, 14733}, {162, 37139}, {527, 307}, {648, 35157}, {1055, 73}, {1155, 1214}, {1172, 1156}, {1638, 17094}, {2204, 34068}, {2299, 2291}, {2332, 4845}, {4183, 41798}, {6139, 647}, {6366, 525}, {6510, 52385}, {6603, 72}, {6610, 1439}, {6745, 306}, {14392, 8611}, {14413, 51640}, {14414, 24018}, {23346, 52610}, {23710, 226}, {30806, 1231}, {32676, 36141}, {37805, 1441}, {38461, 1446}, {51408, 51368}
X(52891) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21, 1005, 8021}, {21, 3559, 4}, {412, 6912, 4}, {415, 15150, 15149}, {5125, 11114, 4}

---

X(52892) = X(2)X(31)∩X(667)X(788)

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

See Ivan Pavlov, euclid 5650.

X(52892) lies on the circumconic {A,B,C,X(2),X(716)}} and these lines: {2,31}, {667,788}, {722,30874}, {1918,3230}, {2232,3266}

X(52892) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2, 43095}, {76, 717}
X(52892) = X(i)-Dao conjugate of X(j) for these {i, j}: {716, 30875}, {32664, 43095}
X(52892) = barycentric product X(i)*X(j) for these (i, j): {1, 8621}, {6, 2230}, {31, 716}, {560, 35533}, {1501, 30875}
X(52892) = barycentric quotient X(i)/X(j) for these (i, j): {31, 43095}, {560, 717}, {716, 561}, {2230, 76}, {8621, 75}, {30875, 40362}, {35533, 1928}

---

X(52893) = X(2)X(37)∩X(512)X(661)

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

See Ivan Pavlov, euclid 5650.

X(52893) lies on these lines: {2,37}, {38,22206}, {39,52579}, {100,5970}, {238,24052}, {512,661}, {740,2229}, {1574,21816}, {2234,3231}, {3706,23632}, {3896,16606}, {3952,21897}, {3954,41440}, {3969,18905}, {3994,21830}, {4062,16592}, {4365,16584}, {4442,18904}, {4651,21902}, {5695,30647}, {6535,16587}, {16369,37680}, {19998,21893}, {20691,46897}, {21020,21838}, {21888,22294}, {22202,24443}, {40874,44168}

X(52893) = reflection of X(312) in X(2229)
X(52893) = X(i)-isoconjugate-of-X(j) for these {i, j}: {58, 3228}, {81, 37132}, {86, 729}, {513, 36133}, {514, 32717}, {649, 9150}, {886, 1919}, {2206, 34087}, {46156, 52394}
X(52893) = X(i)-Dao conjugate of X(j) for these {i, j}: {10, 3228}, {538, 30938}, {5375, 9150}, {9296, 886}, {35073, 274}, {38998, 81}, {39026, 36133}, {40586, 37132}, {40600, 729}, {40603, 34087}
X(52893) = intersection, other than A, B, C, of circumconics {{A,B,C,X(2),X(512)}}, {{A,B,C,X(37),X(50487)}} and {{A,B,C,X(75),X(661)}}
X(52893) = barycentric product X(i)*X(j) for these (i, j): {10, 2234}, {37, 538}, {100, 9148}, {213, 30736}, {321, 3231}, {668, 888}, {887, 6386}, {1500, 30938}, {4036, 5118}, {4601, 52625}, {4705, 23342}, {14609, 42713}, {20336, 46522}, {21839, 52756}, {27801, 33875}
X(52893) = barycentric quotient X(i)/X(j) for these (i, j): {37, 3228}, {42, 37132}, {100, 9150}, {101, 36133}, {213, 729}, {321, 34087}, {538, 274}, {668, 886}, {692, 32717}, {887, 667}, {1645, 3121}, {2234, 86}, {3231, 81}, {5360, 52765}, {6786, 51369}, {9148, 693}, {14406, 50521}, {21814, 46156}, {21839, 14608}, {23342, 4623}, {30736, 6385}, {33875, 1333}, {35073, 30938}, {45672, 16741}, {46522, 28}, {52625, 3125}
X(52893) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 21883, 21820}, {321, 21877, 21814}, {740, 2229, 3121}

---

X(52894) = X(1)X(2)∩X(512)X(798)

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

See Ivan Pavlov, euclid 5650.

X(52894) lies on these lines: {1,2}, {39,21700}, {101,5970}, {512,798}, {538,2234}, {762,40936}, {1573,1964}, {2388,3122}, {2669,44168}, {3125,4093}, {4116,37673}, {4128,21839}, {18904,39688}, {21035,41440}

X(52894) = X(i)-isoconjugate-of-X(j) for these {i, j}: {81, 3228}, {86, 37132}, {274, 729}, {513, 9150}, {514, 36133}, {667, 886}, {693, 32717}, {1333, 34087}
X(52894) = X(i)-Dao conjugate of X(j) for these {i, j}: {37, 34087}, {6631, 886}, {35073, 310}, {38998, 86}, {39026, 9150}, {40586, 3228}, {40600, 37132}
X(52894) = intersection, other than A, B, C, of circumconics {{A,B,C,X(1),X(798)}}; {{A,B,C,X(2),X(512)}}, {{A,B,C,X(8),X(3709)}} and {{A,B,C,X(10),X(4079)}}
X(52894) = barycentric product X(i)*X(j) for these (i, j): {10, 3231}, {37, 2234}, {42, 538}, {190, 888}, {306, 46522}, {313, 33875}, {872, 30938}, {887, 1978}, {1918, 30736}, {4024, 5118}, {4062, 14609}, {4079, 23342}, {4600, 52625}
X(52894) = barycentric quotient X(i)/X(j) for these (i, j): {10, 34087}, {42, 3228}, {190, 886}, {213, 37132}, {538, 310}, {692, 36133}, {1645, 3122}, {1918, 729}, {2234, 274}, {3231, 86}, {5118, 4610}, {6786, 51370}, {9148, 3261}, {14406, 21123}, {23342, 52612}, {32739, 32717}, {33875, 58}, {41267, 46156}, {46522, 27}, {52625, 3120}
X(52894) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3635, 15519, 3635}, {4691, 21267}, {4691, 21267, 21267}, {15519, 3635}, {22166, 22266, 22166}, {22266, 22166}

---

X(52895) = X(1)X(2)∩X(3880)X(20356)

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

See Ivan Pavlov, euclid 5650.

X(52895) lies on these lines: {1,2}, {3880,20356}, {4083,14408}, {14823,40598}, {23493,24524}, {23508,49450}

X(52895) = intersection, other than A, B, C, of circumconics {{A,B,C,X(1),X(20979)}}, {{A,B,C,X(2),X(4083)}} and {{A,B,C,X(10),X(21834)}}
X(52895) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3635, 15519}, {3635, 15519, 15519}, {4691, 21267}, {4691, 21267, 21267}, {22166, 22266, 22166}, {22266, 22166}

---

X(52896) = X(2)X(7)∩X(513)X(663)

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

See Ivan Pavlov, euclid 5650.

X(52896) lies on these lines: {1,29353}, {2,7}, {6,7225}, {37,1122}, {38,40961}, {41,24328}, {65,49515}, {71,17276}, {109,9081}, {256,17598}, {484,32857}, {513,663}, {536,43037}, {573,4862}, {604,6180}, {651,1404}, {674,1469}, {1018,17132}, {1042,36873}, {1086,2183}, {1111,29069}, {1266,3882}, {1334,4419}, {1401,3720}, {1405,5228}, {1475,4644}, {1633,8647}, {1756,24231}, {2170,34371}, {2171,24471}, {2223,3000}, {2260,17365}, {2269,3663}, {2340,21320}, {2347,4000}, {2356,46152}, {3008,21362}, {3009,46153}, {3169,4452}, {3214,38286}, {3501,4454}, {3665,4364}, {3691,4643}, {3748,4343}, {3782,22097}, {3915,28037}, {3942,8609}, {4310,6210}, {4334,13462}, {4346,37555}, {4503,10459}, {4552,43040}, {4667,17474}, {4670,24739}, {4708,24751}, {4887,20367}, {5119,24248}, {5122,19335}, {6510,17439}, {7248,24494}, {7289,40968}, {9295,10027}, {10030,31625}, {18162,21748}, {19624,38530}, {20880,24336}, {21361,24177}, {22370,31598}, {24929,37331}

X(52896) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 36798}, {8, 739}, {9, 37129}, {41, 31002}, {55, 3227}, {284, 41683}, {522, 34075}, {644, 43928}, {646, 23349}, {650, 898}, {663, 4607}, {889, 3063}, {3271, 5381}, {3699, 23892}, {4391, 32718}, {5546, 35353}, {5547, 52757}, {15627, 52754}
X(52896) = X(i)-Dao conjugate of X(j) for these {i, j}: {9, 36798}, {223, 3227}, {478, 37129}, {1646, 14430}, {3160, 31002}, {10001, 889}, {13466, 312}, {14434, 2170}, {39011, 522}, {40590, 41683}, {40614, 8}
X(52896) = X(i) cross conjugate of X(j) for these {i, j}: {3230, 899}
X(52896) = intersection, other than A, B, C, of circumconics {{A,B,C,X(1),X(50127)}}, {{A,B,C,X(2),X(513)}}; {{A,B,C,X(7),X(3669)}} and {{A,B,C,X(9),X(663)}}
X(52896) = barycentric product X(i)*X(j) for these (i, j): {1, 43037}, {7, 899}, {56, 6381}, {57, 536}, {85, 3230}, {241, 36816}, {269, 4009}, {604, 35543}, {651, 4728}, {658, 4526}, {664, 891}, {890, 4572}, {934, 14430}, {1014, 3994}, {1319, 52755}, {1414, 14431}, {3669, 23891}, {3676, 23343}, {3768, 4554}, {4564, 52626}, {4625, 14404}, {4998, 19945}, {22464, 45145}, {37136, 42764}, {41314, 43924}
X(52896) = barycentric quotient X(i)/X(j) for these (i, j): {1, 36798}, {7, 31002}, {56, 37129}, {57, 3227}, {65, 41683}, {109, 898}, {536, 312}, {604, 739}, {651, 4607}, {664, 889}, {890, 663}, {891, 522}, {899, 8}, {1319, 36872}, {1415, 34075}, {1646, 2170}, {3230, 9}, {3768, 650}, {3994, 3701}, {4009, 341}, {4017, 35353}, {4465, 3975}, {4526, 3239}, {4564, 5381}, {4706, 4673}, {4728, 4391}, {6381, 3596}, {14404, 4041}, {14426, 4147}, {14430, 4397}, {14431, 4086}, {14434, 14430}, {14437, 1639}, {19945, 11}, {23343, 3699}, {23891, 646}, {30583, 4768}, {30592, 4985}, {35543, 28659}, {36816, 36796}, {42083, 4009}, {43037, 75}, {43924, 43928}, {45145, 51565}, {47016, 19945}, {51655, 52757}, {51656, 52754}, {52626, 4858}
X(52896) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 1423, 1400}, {7, 28015, 28081}, {63, 28039, 36570}, {651, 1429, 1404}, {1284, 1463, 1458}, {1319, 6610, 51655}, {2223, 4014, 3000}, {20059, 27624, 44421}

---

X(52897) = X(2)X(6)∩X(36)X(238)

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

See Ivan Pavlov, euclid 5650.

X(52897) lies on these lines: {1,24450}, {2,6}, {9,16696}, {36,238}, {43,18185}, {44,16726}, {45,40773}, {56,27660}, {57,16736}, {58,25524}, {63,16700}, {75,16685}, {110,9081}, {213,4670}, {219,17189}, {239,30939}, {274,2176}, {310,4713}, {314,4361}, {527,17205}, {536,3230}, {545,16711}, {662,3285}, {742,18157}, {748,17187}, {894,16709}, {978,4267}, {995,1001}, {1010,1191}, {1014,1778}, {1016,4601}, {1043,1616}, {1045,4068}, {1086,16752}, {1376,38832}, {1412,34048}, {1580,35327}, {1722,18178}, {1724,19241}, {1740,8053}, {1743,18164}, {1958,5301}, {1964,16684}, {2209,9350}, {2234,3747}, {2295,4472}, {2300,3739}, {2664,4557}, {2669,9265}, {3008,17197}, {3052,13588}, {3216,19249}, {3218,16753}, {3219,18601}, {3242,3786}, {3263,9022}, {3264,40886}, {3290,34377}, {3752,17185}, {3772,17182}, {3973,18186}, {4000,17183}, {4245,37507}, {4274,31198}, {4286,24598}, {4364,16705}, {4410,4721}, {4419,18600}, {4423,10458}, {4503,4708}, {4643,16887}, {4644,17169}, {4715,17179}, {4795,17180}, {4798,17750}, {5132,16374}, {5145,19243}, {5272,18165}, {5710,14005}, {7181,51647}, {7202,49760}, {8610,28371}, {8692,19247}, {10457,28352}, {10461,52541}, {16466,25526}, {16468,38302}, {16483,48805}, {16502,33953}, {16552,18171}, {16583,18189}, {16669,17207}, {16686,16876}, {16706,17202}, {16710,17350}, {16712,24441}, {16714,17334}, {16716,39248}, {16744,21214}, {16746,18613}, {16748,24330}, {16750,24352}, {16969,17318}, {17123,18169}, {17167,24789}, {17173,26724}, {17174,33129}, {17195,41140}, {17203,25345}, {17208,24690}, {17348,20228}, {17929,17946}, {18163,23511}, {18172,21384}, {18179,40941}, {19244,52564}, {19261,37502}, {20669,40508}, {21246,46879}, {21760,36957}, {23396,33782}, {23407,36289}, {24667,51370}, {25523,46882}, {26237,46898}, {30109,50028}, {35983,37540}, {39981,46018}

X(52897) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 41683}, {10, 739}, {37, 37129}, {42, 3227}, {213, 31002}, {512, 4607}, {523, 34075}, {661, 898}, {798, 889}, {1018, 43928}, {1400, 36798}, {1577, 32718}, {3122, 5381}, {3952, 23892}, {4033, 23349}
X(52897) = X(i)-Dao conjugate of X(j) for these {i, j}: {9, 41683}, {1015, 35353}, {1646, 14431}, {6381, 35532}, {6626, 31002}, {13466, 321}, {14434, 3125}, {31998, 889}, {36830, 898}, {39054, 4607}, {40582, 36798}, {40589, 37129}, {40592, 3227}, {40614, 10}
X(52897) = X(i) cross conjugate of X(j) for these {i, j}: {3768, 23343}, {40614, 1}
X(52897) = trilinear pole of line {890,891}
X(52897) = intersection, other than A, B, C, of circumconics {{A,B,C,X(1),X(46922)}}, {{A,B,C,X(2),X(513)}}, {{A,B,C,X(6),X(667)}}, {{A,B,C,X(36),X(4585)}}, {{A,B,C,X(69),X(905)}}; {{A,B,C,X(81),X(3733)}}, {A,B,C,X(86),X(1019)}}, and {{A,B,C,X(87),X(4040)}}
X(52897) = barycentric product X(i)*X(j) for these (i, j): {21, 43037}, {58, 6381}, {81, 536}, {86, 899}, {99, 891}, {274, 3230}, {662, 4728}, {670, 890}, {757, 3994}, {799, 3768}, {1014, 4009}, {1019, 23891}, {1333, 35543}, {1414, 14430}, {1646, 4601}, {3733, 41314}, {4465, 37128}, {4526, 4573}, {4567, 52626}, {4584, 14433}, {4596, 30592}, {4600, 19945}, {4615, 14437}, {4622, 30583}, {4623, 14404}, {7192, 23343}, {17139, 45145}, {18206, 36816}, {52680, 52755}
X(52897) = barycentric quotient X(i)/X(j) for these (i, j): {1, 41683}, {21, 36798}, {58, 37129}, {81, 3227}, {86, 31002}, {99, 889}, {110, 898}, {163, 34075}, {513, 35353}, {536, 321}, {662, 4607}, {899, 10}, {1333, 739}, {1576, 32718}, {1646, 3125}, {3230, 37}, {3733, 43928}, {3768, 661}, {3994, 1089}, {4009, 3701}, {4465, 3948}, {4526, 3700}, {4567, 5381}, {4728, 1577}, {4937, 4125}, {6381, 313}, {14404, 4705}, {14426, 21051}, {14430, 4086}, {14431, 4036}, {14434, 14431}, {14437, 4120}, {16702, 52757}, {19945, 3120}, {23343, 3952}, {23891, 4033}, {30592, 30591}, {33917, 8034}, {35543, 27801}, {41314, 27808}, {42083, 3994}, {43037, 1441}, {45145, 38955}, {51420, 52754}, {52626, 16732}, {52680, 36872}
X(52897) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 86, 18166}, {44, 16726, 18206}, {81, 25507, 37674}, {81, 27643, 4383}, {86, 17277, 16738}, {86, 25508, 15668}, {86, 27644, 6}, {86, 29767, 17178}, {86, 46922, 8025}, {238, 18792, 3286}, {992, 28350, 141}, {1740, 16690, 8053}, {2234, 3747, 4436}, {16752, 17139, 1086}, {17178, 17349, 29767}, {17379, 19743, 46922}, {27623, 28365, 6}, {28251, 28368, 1211}, {28252, 28369, 1213}, {32911, 46922, 6}

---

X(52898) = X(2)X(32)∩X(23)X(385)

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

See Ivan Pavlov, euclid 5650.

X(52898) lies on these lines: {2,32}, {3,31088}, {6,42286}, {22,7754}, {23,385}, {69,9516}, {111,38278}, {148,20063}, {187,3266}, {193,1176}, {194,7492}, {230,40511}, {308,1383}, {325,36953}, {384,31078}, {468,3793}, {524,14567}, {689,5970}, {691,5189}, {732,8627}, {733,9150}, {827,2770}, {1180,7894}, {2076,4576}, {2996,7500}, {3060,10551}, {3291,26276}, {3552,9464}, {4577,18823}, {4590,7779}, {4760,42713}, {5007,15822}, {5169,7823}, {5354,26257}, {5967,41586}, {5986,46518}, {5987,39652}, {6031,7766}, {6179,9465}, {6636,7783}, {7493,10547}, {7495,7762}, {7665,13574}, {7750,31107}, {7771,15302}, {7777,15464}, {7780,26235}, {7806,16890}, {7813,31068}, {7816,8024}, {7840,9164}, {9463,43977}, {14360,51459}, {15080,32451}, {15107,25047}, {17008,17500}, {18710,31102}, {19570,37901}, {20080,33632}, {20099,38946}, {21284,41676}, {22329,34294}, {30749,35007}, {32085,43981}, {32581,52284}, {33175,46289}, {35511,46228}, {37766,42396}, {37876,39955}, {37900,47286}, {41413,46900}

X(52898) = isogonal conjugate of X(46154)
X(52898) = reflection of X(10330) in X(8627)
X(52898) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 46154}, {31, 31125}, {38, 111}, {39, 897}, {75, 41272}, {141, 923}, {427, 36060}, {661, 36827}, {671, 1964}, {691, 8061}, {826, 36142}, {892, 2084}, {895, 17442},{1634, 23894}, {1923, 18023}, {1930, 32740}, {3005, 36085}, {3051, 46277}, {3404, 5968}, {3917, 36128}, {4020, 17983}, {5380, 21123}, {7316, 33299}, {14908, 20883}, {35309, 43926}
X(52898) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2157, 1369}, {3455, 21289}, {9076, 6327}, {37221, 315}, {46289, 11061}
X(52898) = X(i) cross conjugate of X(j) for these {i, j}: {351, 5468}, {7664, 2}, {35522, 4235}
X(52898) = X(i)-Dao conjugate of X(j) for these {i, j}: {2, 31125}, {3, 46154}, {187, 9019}, {206, 41272}, {524, 7813}, {1560, 427}, {1648, 14424}, {1649, 39691}, {2482, 141}, {6593, 39}, {23992, 826}, {36830, 36827}, {38988, 3005}, {41884, 671}
X(52898) = trilinear pole of line {690,5026}
X(52898) = intersection, other than A, B, C, of circumconics {{A,B,C,X(2),X(468)}}, {{A,B,C,X(4),X(7812)}}, {{A,B,C,X(23),X(691)}}, {{A,B,C,X(25),X(1627)}}, {{A,B,C,X(32),X(187)}}, {{A,B,C,X(76),X(7883)}} and {A,B,C,X(98),X(9147)}}
X(52898) = barycentric product X(i)*X(j) for these (i, j): {82, 14210}, {83, 524}, {99, 22105}, {187, 308}, {251, 3266}, {351, 689}, {468, 1799}, {690, 4577}, {827, 35522}, {896, 3112}, {922, 18833}, {1176, 44146}, {2642, 4593}, {3292, 46104}, {4062, 52394}, {4235, 4580}, {5026, 14970}, {5467, 52618}, {5967, 20022}, {6390, 32085}, {6629, 18082}, {7664, 9076}, {7813, 52395}, {10130, 51541}, {11183, 41209}, {14417, 42396}, {14567, 40016}, {16741, 18098}, {18070, 23889}, {18108, 42721}, {28724, 37778}, {31128, 38278}, {39287, 41586}, {42713, 52376}, {51862, 52145}
X(52898) = barycentric quotient X(i)/X(j) for these (i, j): {2, 31125}, {6, 46154}, {32, 41272}, {82, 897}, {83, 671}, {110, 36827}, {187, 39}, {251, 111}, {308, 18023}, {351, 3005}, {468, 427}, {524, 141}, {690, 826}, {827, 691}, {896, 38}, {922, 1964}, {1176, 895}, {1648, 39691}, {1649, 14424}, {1799, 30786}, {2482, 7813}, {2642, 8061}, {3112, 46277}, {3266, 8024}, {3292, 3917}, {3712, 3703}, {3793, 8362}, {4062, 15523}, {4235, 41676}, {4577, 892}, {4580, 14977}, {4599, 36085}, {4630, 32729}, {4750, 16892}, {5026, 732}, {5203, 47730}, {5467, 1634}, {5468, 4576}, {5642, 51360}, {5967, 20021}, {6390, 3933}, {6593, 9019}, {6629, 16887}, {7181, 3665}, {7267, 16720}, {7813, 7794}, {9076, 10415}, {9717, 46147}, {10130, 42008}, {10547, 14908}, {14210, 1930}, {14417, 2525}, {14419, 2530}, {14424, 2528}, {14432, 48278}, {14559, 46155}, {14567, 3051}, {15303, 41583}, {15471, 41585}, {16702, 16696}, {16741, 16703}, {18105, 9178}, {21839, 3954}, {22105, 523}, {23200, 20775}, {32085, 17983}, {34072, 36142}, {35356, 35359}, {35522, 23285}, {41309, 46156}, {44102, 1843}, {44146, 1235}, {46104, 46111}, {46288, 32740}, {46289, 923}, {50567, 51371}, {51541, 23297}, {51862, 5968}, {52618, 52632}
X(52898) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2896, 31124}, {2, 315, 31076}, {32, 26233, 2}, {83, 10130, 2}, {83, 1799, 10130}, {251, 10130, 83}, {251, 1799, 2}, {251, 39668, 42037}, {732, 8627, 10330}, {1627, 33651, 2}, {5189, 19577, 31125}, {14712, 19577, 5189}

---

X(52899) = X(2)X(87)∩X(902)X(932)

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

See Ivan Pavlov, euclid 5650.

X(52899) lies on these lines: {2,87}, {649,4083}, {899,4598}, {902,932}, {2162,3720}, {3231,51864}, {14996,40753}, {17105,17126}, {30950,40720}

X(52899) = intersection, other than A, B, C, of circumconics {{A,B,C,X(2),X(4083)}} and {{A,B,C,X(87),X(649)}}

---

X(52900) = X(1)X(513)∩X(2)X(45)

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

See Ivan Pavlov, euclid 5650.

X(52900) lies on these lines: {1,513}, {2,45}, {76,31625}, {106,1001}, {536,23891}, {899,19945}, {940,40215}, {984,4674}, {3666,52031}, {3752,52140}, {3768,46782}, {3821,4013}, {4492,4792}, {4555,17318}, {4557,4947},{4582,17262}, {4664,46795}, {5091,39154}, {6548,24403}, {16505,27846}, {16777,52553}, {17139,18198}, {17301,46790}, {18073,24004}, {19636,24248}, {20182,47058}, {24416,24428}, {30950,43922}, {41312,52759}, {50068,52753}

X(52900) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 36872}, {44, 37129}, {519, 739}, {898, 1635}, {900, 34075}, {902, 3227}, {1023, 43928}, {1404, 36798}, {1960, 4607}, {2251, 31002}, {3285, 41683}, {3762, 32718}, {17780, 23892}, {23349, 24004}
X(52900) = X(i)-Dao conjugate of X(j) for these {i, j}: {9, 36872}, {1646, 30583}, {9460, 31002}, {13466, 4358}, {14434, 2087}, {39011, 900}, {40594, 3227}, {40595, 37129}, {40614, 519}
X(52900) = trilinear pole of line {891,899}
X(52900) = intersection, other than A, B, C, of circumconics {{A,B,C,X(1),X(190)}}. {{A,B,C,X(2),X(513)}}, {{A,B,C,X(6),X(3768)}}, {{A,B,C,X(45),X(3230)}}, {{A,B,C,X(76),X(764)}} and {{A,B,C,X(88),X(23345)}}
X(52900) = barycentric product X(i)*X(j) for these (i, j): {1, 52755}, {88, 536}, {106, 6381}, {891, 4555}, {899, 903}, {1022, 23891}, {1320, 43037}, {3230, 20568}, {3257, 4728}, {4618, 30583}, {4622, 14431}, {4634, 14404}, {5376, 52626}, {6548, 23343}, {9456, 35543}, {23345, 41314}, {46795, 51923}
X(52900) = barycentric quotient X(i)/X(j) for these (i, j): {1, 36872}, {88, 3227}, {106, 37129}, {536, 4358}, {890, 1960}, {891, 900}, {899, 519}, {901, 898}, {903, 31002}, {1320, 36798}, {1646, 2087}, {3230, 44}, {3257, 4607}, {3768, 1635}, {3994, 3992}, {4009, 4723}, {4526, 1639}, {4555, 889}, {4674, 41683}, {4706, 4742}, {4728, 3762}, {5376, 5381}, {6381, 3264}, {9456, 739}, {14190, 52761}, {14404, 4730}, {14430, 4768}, {14434, 30583}, {14437, 6544}, {19945, 1647}, {23343, 17780}, {23345, 43928}, {23891, 24004}, {32665, 34075}, {32719, 32718}, {33917, 8661}, {45145, 36944}, {51923, 46797}, {52755, 75}
X(52900) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {23352, 24405, 34230}, {24338, 24406, 24405}, {24338, 36814, 23352}

---

X(52901) = X(1)X(4492)∩X(2)X(44)

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

See Ivan Pavlov, euclid 5650.

X(52901) lies on these lines: {1,4492}, {2,44}, {238,2163}, {292,51908}, {513,1960}, {1279,2320}, {2364,16786}, {4506,31625}, {4588,9081}

X(52901) = X(i)-isoconjugate-of-X(j) for these {i, j}: {45, 37129}, {739, 3679}, {898, 4893}, {1405, 36798}, {2177, 3227}, {4273, 41683}, {4607, 4775}, {4752, 43928}, {4767, 23892}, {4777, 34075}, {4791, 32718}
X(52901) = X(i)-Dao conjugate of X(j) for these {i, j}: {1646, 28603}, {13466, 4671}, {39011, 4777}, {40614, 3679}
X(52901) = intersection, other than A, B, C, of circumconics {{A,B,C,X(1),X(3758)}}, {A,B,C,X(2),X(513)}} and {{A,B,C,X(44),X(292)}}
X(52901) = barycentric product X(i)*X(j) for these (i, j): {89, 536}, {891, 4597}, {899, 39704}, {2163, 6381}, {2320, 43037}, {3230, 20569}, {4604, 4728}, {5385, 52626}, {23343, 52620}, {28607, 35543}
X(52901) = barycentric quotient X(i)/X(j) for these (i, j): {89, 3227}, {536, 4671}, {890, 4775}, {891, 4777}, {899, 3679}, {2163, 37129}, {2320, 36798}, {3230, 45}, {3768, 4893}, {3994, 4125}, {4526, 4944}, {4588, 898}, {4597, 889}, {4604, 4607}, {4728, 4791}, {5385, 5381}, {14404, 4770}, {14434, 28603}, {23343, 4767}, {28607, 739}, {34073, 34075}, {39704, 31002}, {42083, 4937}, {45145, 36921}, {52626, 4957}

---

X(52902) = X(2)X(11)∩X(6)X(513)

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

See Ivan Pavlov, euclid 5650.

X(52902) lies on these lines: {1,43921}, {2,11}, {6,513}, {45,18785}, {182,29349}, {536,23343}, {884,52242}, {919,9081}, {1438,2278}, {3242,46149}, {3286,4236}, {3596,4076}, {3913,36802}, {14665,22520}, {17301,24405}, {34230,52030}, {35026,37555}

X(52902) = X(i)-isoconjugate-of-X(j) for these {i, j}: {518, 37129}, {665, 4607}, {672, 3227}, {739, 3912}, {898, 2254}, {918, 34075}, {1026, 43928}, {1458, 36798}, {2223, 31002}, {3286, 41683}, {23892, 42720}, {34230, 36872}
X(52902) = X(i)-Dao conjugate of X(j) for these {i, j}: {13466, 3263}, {14434, 3675}, {39011, 918}, {40614, 3912}
X(52902) = trilinear pole of line {891,3230}
X(52902) = intersection, other than A, B, C, of circumconics {{A,B,C,X(2),X(513)}}, {{A,B,C,X(6),X(100)}}, {{A,B,C,X(11),X(3596)}}, {{A,B,C,X(55),X(3063)}}, {{A,B,C,X(86),X(25531)}} and {{A,B,C,X(105),X(43929)}}
X(52902) = barycentric product X(i)*X(j) for these (i, j): {1, 36816}, {105, 536}, {294, 43037}, {666, 891}, {673, 899}, {890, 36803}, {927, 4526}, {1027, 23891}, {1438, 6381}, {1462, 4009}, {2481, 3230}, {3768, 51560}, {4465, 52030}, {4728, 36086}, {5377, 52626}, {14430, 36146}, {41314, 43929}
X(52902) = barycentric quotient X(i)/X(j) for these (i, j): {105, 3227}, {294, 36798}, {536, 3263}, {666, 889}, {673, 31002}, {890, 665}, {891, 918}, {899, 3912}, {919, 898}, {1438, 37129}, {1646, 3675}, {3230, 518}, {3768, 2254}, {4526, 50333}, {5377, 5381}, {14404, 24290}, {18785, 41683}, {23343, 42720}, {32666, 34075}, {36086, 4607}, {36816, 75}, {43037, 40704}, {43929, 43928}, {51922, 52761}

---

X(52903) = X(2)X(108)∩X(6)X(19)

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

See Ivan Pavlov, euclid 5650.

X(52903) lies on these lines: {2,108}, {6,19}, {427,7337}, {1037,10106}, {1062,37415}, {1861,36741}, {14257,37800}

X(52903) = intersection, other than A, B, C, of circumconics {{A,B,C,X(2),X(3827}} and {{A,B,C,X(6),X(26703}}

---

X(52904) = X(2)X(109)∩X(6)X(41)

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

See Ivan Pavlov, euclid 5650.

X(52904) lies on these lines: {2,109}, {6,41}, {31,37366}, {222,27639}, {601,3149}, {603,27627}, {1479,5264}, {11682,21371}, {37502,52024}

X(52904) = intersection, other than A, B, C, of circumconics {{A,B,C,X(2),X(8679)}} and {{A,B,C,X(6),X(1311)}}

---

X(52905) = X(2)X(112)∩X(6)X(25)

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

See Ivan Pavlov, euclid 5650.

X(52905) lies on these lines: {2,112}, {5,51509}, {6,25}, {22,14961}, {23,39575}, {32,468}, {251,6353}, {373,51437}, {427,5475}, {428,11648}, {577,44210}, {1112,5028}, {1627,38282}, {1968,5094}, {1995,8743}, {3162,5020}, {3172,11284}, {3520,15302}, {4232,10312}, {5306,47144}, {5354,37962}, {5640,41363}, {5651,35325}, {6997,13854}, {7392,8879}, {7493,10316}, {7505,41399}, {9465,37777}, {9699,21213}, {10314,16318}, {10422,34158}, {21313,43136}, {23964,34235}, {24855,52297}, {37187,37645}, {37453,41336}, {44212,47187}

X(52905) = intersection, other than A, B, C, of circumconics {{A,B,C,X(2),X(2393)}} and {{A,B,C,X(6),X(2373)}}
X(52905) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1995, 8743, 14580}

---

X(52906) = X(2)X(6)∩X(826)X(2474)

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

See Ivan Pavlov, euclid 5650.

X(52906) lies on these lines: {2,6}, {251,41749}, {732,39691}, {754,8627}, {826,2474}, {1369,10329}, {1495,7845}, {5207,13519}, {5651,7903}, {7821,40130}, {7871,35275}, {8878,10328}, {11205,21248}, {17949,31125}, {31076,51848}, {31107,32449}, {48440,51371}

X(52906) = X(i)-isoconjugate-of-X(j) for these {i, j}: {82, 755}, {43098, 46289}
X(52906) = X(i)-Dao conjugate of X(j) for these {i, j}: {39, 43098}, {141, 755}
X(52906) = intersection, other than A, B, C, of circumconics {{A,B,C,X(2),X(754)}}, {{A,B,C,X(6),X(3005)}}, {{A,B,C,X(69),X(2525)}}, {{A,B,C,X(81),X(2530)}} and {{A,B,C,X(86),X(16892)}}
X(52906) = barycentric product X(i)*X(j) for these (i, j): {39, 35549}, {99, 33907}, {141, 754}, {1930, 2244}, {2525, 46543}, {3665, 4157}, {3703, 7214}, {4156, 16887}, {4576, 14420}, {4609, 14403}, {7813, 52758}, {8024, 8627}
X(52906) = barycentric quotient X(i)/X(j) for these (i, j): {39, 755}, {141, 43098}, {754, 83}, {2244, 82}, {4156, 18082}, {4553, 5389}, {8627, 251}, {14403, 669}, {14428, 18105}, {35549, 308}, {46543, 42396}
X(52906) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {193, 28675, 141}

---

X(52907) = X(1)X(2)∩X(1266)X(4076)

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

See Ivan Pavlov, euclid 5650.

X(52907) lies on these lines: {1,2}, {1266,4076}, {2976,3667}, {3021,16185}, {5265,44301}, {5698,11067}, {6552,7963}, {8056,15590}, {12035,17132}, {39589,47742}

X(52907) = midpoint of X(37743) in X(46932)
X(52907) = complement of X(51615)
X(52907) = X(i)-isoconjugate-of-X(j) for these {i, j}: {56, 31316}, {8056, 17222}
X(52907) = X(i)-Dao conjugate of X(j) for these {i, j}: {1, 31316}
X(52907) = intersection, other than A, B, C, of circumconics {{A,B,C,X(1),X(4394)}}, {{A,B,C,X(2),X(3667)}}, {{A,B,C,X(8),X(4521)}} and {{A,B,C,X(10),X(14321)}}
X(52907) = barycentric product X(i)*X(j) for these (i, j): {145, 17132}, {12035, 31227}, {43290, 45677}
X(52907) = barycentric quotient X(i)/X(j) for these (i, j): {9, 31316}, {3052, 17222}
X(52907) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3616, 20057}, {1, 3622, 3241}, {1, 3636, 145}, {1, 38314, 8}, {1, 51103, 3622}, {1, 51105, 3244}, {1, 51106, 20014}, {1, 551, 3623}, {2, 145, 4668}, {2, 15519, 20053}, {2, 1961, 29852}, {2, 21267, 9780}, {2, 29854, 29849}, {2, 31137, 29865}, {2, 31189, 29600}, {2, 33140, 31136}, {2, 3635, 20053}, {2, 4691, 9780}, {10, 15519, 3633}, {10, 19872, 46932}, {10, 19878, 19876}, {10, 20053, 31183}, {10, 3616, 5550}, {10, 3635, 3633}, {10, 4677, 4678}, {145, 19877, 8}, {145, 3616, 19877}, {145, 4668, 20053}, {551, 3623, 9780}, {1125, 36573, 145}, {1125, 4677, 46931}, {1125, 51104, 1}, {1647, 20057, 614}, {3241, 25055, 51068}, {3241, 3616, 10}, {3241, 3622, 5550}, {3241, 3633, 0}, {3241, 51103, 38314}, {3241, 5550, 8}, {3244, 51105, 46934}, {3616, 20053, 2}, {3616, 20057, 8}, {3621, 29680, 3636}, {3621, 3622, 25055}, {3621, 51068, 8}, {3622, 3623, 34595}, {3623, 17733, 51092}, {3632, 15519, 29607}, {3632, 29679, 31136}, {3632, 34595, 10}, {3632, 3635, 29607}, {3633, 34595, 21267}, {3633, 4677, 5272}, {3634, 20014, 51072}, {3635, 15519}, {3635, 15519, 15519}, {3635, 20053, 0}, {3635, 4668, 145}, {3636, 31253, 551}, {3679, 11019, 15519}, {3828, 10580, 29621}, {4668, 15519, 145}, {4678, 46931, 10}, {4691, 21267}, {4691, 21267, 21267}, {4691, 29607, 20053}, {4691, 31189, 3623}, {4746, 29582, 20053}, {5268, 29851, 29848}, {5550, 38314, 3622}, {9780, 31253, 19877}, {15519, 20053, 0}, {15808, 51093, 46933}, {19877, 20057, 145}, {20020, 29843, 10}, {20052, 29626, 26531}, {20053, 29600, 29607}, {20057, 39587, 46934}, {21267, 29607, 20053}, {21267, 31189, 3623}, {22166, 22266, 22166}, {22266, 22166}, {25961, 44307, 32776}, {29629, 31191, 5550}, {29629, 50114, 3244}, {37743, 46932, 519}, {51071, 51091, 50282}, {51103, 51104, 4677}

---

X(52908) = X(1)X(2)∩X(350)X(874)

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

See Ivan Pavlov, euclid 5650.

X(52908) lies on these lines: {1,2}, {106,31002}, {192,24427}, {214,4107}, {238,3570}, {291,24841}, {350,874}, {537,20331}, {659,812}, {726,42720}, {740,27918}, {2108,17794}, {2293,34832}, {3685,24428}, {3758,24487}, {3923,9318}, {3993,24403}, {4363,24327}, {4366,27931}, {4702,41144}, {5247,37012}, {6544,21832}, {17756,49455}, {24003,44304}, {24318,49511}, {24406,36222}, {24502,24514}, {24693,30958}, {24709,30990}, {24715,30997}, {27851,40548}

X(52908) = midpoint of X(1) in X(23891)
X(52908) = X(i)-isoconjugate-of-X(j) for these {i, j}: {291, 2382}, {1911, 18822}
X(52908) = X(i)-Dao conjugate of X(j) for these {i, j}: {6651, 18822}, {35123, 335}, {39029, 2382}
X(52908) = intersection, other than A, B, C, of circumconics {{A,B,C,X(1),X(659)}}, {{A,B,C,X(2),X(537)}}, {{A,B,C,X(8),X(3716)};}, {{A,B,C,X(10),X(4010)}} and {{A,B,C,X(42),X(4455)}}
X(52908) = barycentric product X(i)*X(j) for these (i, j): {239, 537}, {350, 20331}, {874, 52745}, {3570, 36848}, {4432, 46795
X(52908) = barycentric quotient X(i)/X(j) for these (i, j): {239, 18822}, {537, 335}, {1914, 2382}, {4375, 47070}, {4432, 46797}, {14405, 2530}, {20331, 291}, {36848, 4444}, {52745, 876}
X(52908) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 23891, 519}, {2, 17780, 899}, {2, 27920, 27919}, {2, 29824, 1647}, {3635, 15519, 3635}, {3685, 27912, 24428}, {4432, 17793, 4465}, {4432, 4465, 4368}, {4465, 8299, 4432}, {4691, 21267}, {4691, 21267, 21267}, {8299, 17793, 4368}, {15519, 3635}, {22166, 22266, 22166}, {22266, 22166}

---

X(52909) = X(3)X(54)∩X(5)X(15)

Barycentrics    a^2*(a^8-4*(b^2+c^2)*a^6+2*(3*b^4+2*b^2*c^2+3*c^4)*a^4-2*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2+(b^2-c^2)^2*(b^4+c^4)+8*S^3*sqrt(3)*(-a^2+b^2+c^2)) : :
Barycentrics    (SB+SC)*(S^2+(2*R^2+2*sqrt(3)*S-SA)*SA) : :

See César Lozada, euclid 5673.

X(52909) lies on these lines: {3, 54}, {5, 15}, {140, 52349}, {143, 3132}, {155, 11480}, {156, 3131}, {1216, 13350}, {1353, 11515}, {2937, 14170}, {3526, 11130}, {5070, 41477}, {5611, 10263}, {10634, 13292}, {11267, 32585}, {11793, 21401}, {15047, 41478}, {15068, 42116}, {17814, 36836}, {18350, 38431}

---

X(52910) = X(3)X(54)∩X(5)X(16)

Barycentrics    a^2*(a^8-4*(b^2+c^2)*a^6+2*(3*b^4+2*b^2*c^2+3*c^4)*a^4-2*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2+(b^2-c^2)^2*(b^4+c^4)-8*S^3*sqrt(3)*(-a^2+b^2+c^2)) : :
Barycentrics    (SB+SC)*(S^2+(2*R^2-2*sqrt(3)*S-SA)*SA) : :

See César Lozada, euclid 5673.

X(52910) lies on these lines: {3, 54}, {5, 16}, {140, 52348}, {143, 3131}, {155, 11481}, {156, 3132}, {1216, 13349}, {1353, 11516}, {2937, 14169}, {3526, 11131}, {5070, 41478}, {5615, 10263}, {10635, 13292}, {11268, 32586}, {11793, 21402}, {15047, 41477}, {15068, 42115}, {17814, 36843}, {18350, 38432}

---

X(52911) = REFLECTION OF X(15295) IN X(14560)

Barycentrics    (a^16-5*(b^2+c^2)*a^14+2*(5*b^4+14*b^2*c^2+5*c^4)*a^12-(b^2+c^2)*(11*b^4+34*b^2*c^2+11*c^4)*a^10+(10*b^8+10*c^8+(10*b^4+93*b^2*c^2+10*c^4)*b^2*c^2)*a^8-(b^2+c^2)*(11*b^8+11*c^8-(45*b^4-107*b^2*c^2+45*c^4)*b^2*c^2)*a^6+(10*b^12+10*c^12-(27*b^8+27*c^8+(15*b^4-82*b^2*c^2+15*c^4)*b^2*c^2)*b^2*c^2)*a^4-(b^4-c^4)*(b^2-c^2)*(5*b^8+5*c^8+(b^4-21*b^2*c^2+c^4)*b^2*c^2)*a^2+(b^6-c^6)*(b^2-c^2)^3*(b^4+4*b^2*c^2+c^4))*((a^2-b^2+c^2)^2-c^2*a^2)*((a^2+b^2-c^2)^2-a^2*b^2)*a^2 : :
X(52911) = 3*X(14560)-X(40355), 3*X(15295)-2*X(40355)

See Kadir Altintas and César Lozada, euclid 5673.

X(52911) lies on this line: {1495, 3003}

X(52911) = reflection of X(15295) in X(14560)

---

X(52912) = (name pending)

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

See Kadir Altintas and César Lozada, euclid 5673.

X(52912) lies on this line: {42, 181}

---

X(52913) = X(20)X(122)∩X(107)X(110)

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

See Ivan Pavlov, euclid 5663.

X(52913) lies on the circumconic {{A,B,C,X(20),X(925)}} and these lines: {2,13611}, {3,47384}, {4,5972}, {20,122}, {99,1301}, {107,110}, {114,6353}, {154,15466}, {162,658}, {250,3233}, {264,35259}, {275,13595}, {297,15448}, {317,4232}, {340,32269}, {403,44990}, {436,39530}, {450,1495}, {476,33640}, {662,7452}, {925,1304}, {933,1302}, {1624,4230}, {1995,36794}, {2052,35264}, {2972,37926}, {3079,14944}, {3448,16080}, {4563,18020}, {5667,15063}, {6053,52057}, {6529,32661}, {6531,20998}, {6800,52147}, {7473,39062}, {10421,25641}, {10556,17983}, {10565,34841}, {11206,46927}, {11251,12121}, {14918,37760}, {15774,23240}, {16386,51892}, {17907,35260}, {23181,30441}, {23239,40948}, {24981,47204}, {32237,35474}, {32605,34286}, {32609,34334}, {35265,46106}, {35266,37765}, {35602,52578}, {37200,41424}, {38700,52493}, {44802,51031}, {46818,51939}, {47391,47392}

X(52913) = anticomplement of X(13611)
X(52913) = X(i)-isoconjugate-of-X(j) for these {i, j}: {64, 656}, {253, 810}, {459, 822}, {523, 19614}, {525, 2155}, {647, 2184}, {661, 1073}, {798, 34403}, {1301, 2632}, {1577, 14642}, {1973, 14638}, {2616, 8798}, {3708, 46639}, {14208, 33581}, {14379, 24006}, {24018, 41489}, {30457, 51640}
X(52913) = X(i)-Dao conjugate of X(j) for these {i, j}: {122, 125}, {6337, 14638}, {6587, 23616}, {13611, 13611}, {31998, 34403}, {36830, 1073}, {39020, 15526}, {39052, 2184}, {39054, 19611}, {39062, 253}, {40596, 64}, {40616, 4466}, {45245, 525}, {45248, 520}, {45249, 6368}
X(52913) = X(i) cross conjugate of X(j) for these {i, j}: {1895, 44699}, {6587, 15466}, {6616, 32230}, {8057, 20}, {35602, 250}, {36413, 23582}, {44705, 1249}
X(52913) = trilinear pole of line {20,1249}
X(52913) = barycentric product X(i)*X(j) for these (i, j): {4, 36841}, {20, 648}, {99, 1249}, {107, 37669}, {110, 15466}, {112, 14615}, {154, 6331}, {162, 18750}, {190, 44698}, {204, 799}, {610, 811}, {643, 44697}, {645, 44696}, {662, 1895}, {670, 3172}, {2407, 10152}, {2966, 44704}, {3079, 44326}, {3213, 7257}, {4558, 14249}, {4560, 44699}, {4563, 6525}, {4573, 44695}, {4590, 44705}, {4625, 7156}, {6528, 15905}, {6587, 18020}, {8057, 23582}, {14345, 42308}, {14570, 38808}, {14944, 34211}, {15352, 35602}, {17402, 44702}, {17403, 44703}, {18623, 36797}, {18831, 42459}, {20580, 32230}, {23895, 44700}, {23896, 44701}, {46639, 52578}
X(52913) = barycentric quotient X(i)/X(j) for these (i, j): {20, 525}, {69, 14638}, {99, 34403}, {107, 459}, {110, 1073}, {112, 64}, {122, 23616}, {154, 647}, {162, 2184}, {163, 19614}, {204, 661}, {250, 46639}, {610, 656}, {648, 253}, {662, 19611}, {1394, 51640}, {1562, 5489}, {1576, 14642}, {1625, 8798}, {1895, 1577}, {2420, 11589}, {3079, 6587}, {3172, 512}, {3213, 4017}, {4558, 15394}, {6331, 41530}, {6525, 2501}, {6528, 52581}, {6529, 6526}, {6587, 125}, {7070, 8611}, {7156, 4041}, {8057, 15526}, {8804, 4064}, {10152, 2394}, {14249, 14618}, {14343, 12037}, {14345, 1650}, {14615, 3267}, {14944, 43673}, {15291, 14380}, {15466, 850}, {15905, 520}, {17898, 20902}, {18020, 44326}, {18623, 17094}, {18750, 14208}, {21172, 4466}, {23964, 1301}, {27382, 52355}, {32661, 14379}, {32676, 2155}, {32713, 41489}, {33629, 23286}, {34211, 16096}, {35360, 13157}, {35602, 52613}, {36413, 8057}, {36841, 69}, {37669, 3265}, {38808, 15412}, {42459, 6368}, {42658, 3269}, {44695, 3700}, {44696, 7178}, {44697, 4077}, {44699, 4552}, {44700, 23870}, {44701, 23871}, {44704, 2799}, {44705, 115}, {46639, 52559}
X(52913) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {107, 110, 648}, {110, 4240, 107}, {1304, 7471, 30716}, {3233, 31510, 250}

---

X(52914) = X(107)X(110)∩X(112)X(931)

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

See Ivan Pavlov, euclid 5663.

X(52914) lies on these lines: {4,27083}, {107,110}, {109,41206}, {112,931}, {162,662}, {250,37966}, {645,36797}, {1897,5379}, {1982,9275}, {2194,31623}, {2651,14192}, {41629,44293}

X(52914) = X(i)-isoconjugate-of-X(j) for these {i, j}: {12, 1459}, {37, 51640}, {42, 17094}, {56, 4064}, {59, 21134}, {65, 656}, {71, 7178}, {72, 4017}, {73, 523}, {77, 4705}, {108, 2632}, {109, 125}, {115, 1813}, {181, 4025}, {201, 513}, {222, 4024}, {225, 520}, {226, 647}, {228, 4077}, {306, 7180}, {307, 512}, {338, 32660}, {348, 4079}, {349, 3049}, {514, 2197}, {521, 1254}, {522, 1425}, {525, 1400}, {603, 4036}, {649, 26942}, {650, 37755}, {651, 3708}, {652, 6354}, {653, 3269}, {657, 20618}, {661, 1214}, {663, 6356}, {664, 20975}, {798, 1231}, {810, 1441}, {822, 40149}, {905, 2171}, {1042, 52355}, {1109, 36059}, {1331, 1365}, {1367, 8750}, {1402, 14208}, {1409, 1577}, {1410, 4086}, {1415, 20902}, {1427, 8611}, {1439, 4041}, {1807, 51645}, {1880, 24018}, {1937, 9391}, {2489, 52565}, {2501, 40152}, {2643, 6516}, {2972, 36127}, {3120, 23067}, {3669, 3949}, {3676, 3690}, {3694, 7216}, {3695, 43924}, {3700, 52373}, {3710, 7250}, {3942, 21859}, {4091, 8736}, {4466, 4559}, {4551, 18210}, {4565, 21046}, {4605, 7117}, {6358, 22383}, {6517, 8754}, {7066, 7649}, {7138, 44426}, {7182, 50487}, {10397, 13853}, {15526, 32674}, {20336, 51642}, {21044, 52610}, {21131, 44717}, {22341, 24006}, {40160, 52310}, {41013, 51641}, {42666, 52392}, {43923, 52387}, {52411, 52623}
X(52914) = X(i)-Dao conjugate of X(j) for these {i, j}: {1, 4064}, {11, 125}, {1146, 20902}, {5375, 26942}, {5521, 1365}, {6615, 21134}, {7358, 7068}, {7952, 4036}, {20620, 1109}, {26932, 1367}, {31998, 1231}, {34961, 72}, {35072, 15526}, {36830, 1214}, {38966, 4092}, {38983, 2632}, {38991, 3708}, {39025, 20975}, {39026, 201}, {39037, 9391}, {39052, 226}, {39054, 307}, {39062, 1441}, {40582, 525}, {40589, 51640}, {40592, 17094}, {40596, 65}, {40602, 656}, {40605, 14208}, {40624, 339}, {40626, 17879}
X(52914) = X(i) cross conjugate of X(j) for these {i, j}: {29, 5379}, {60, 250}, {110, 4636}, {521, 21}, {650, 31623}, {7054, 23582}, {18344, 1172}
X(52914) = intersection, other than A, B, C, of circumconics {{A,B,C,X(7),X(44063)}, {{A,B,C,X(21),X(4240)}} and {{A,B,C,X(29),X(1897)}}
X(52914) = trilinear pole of line {21,270}
X(52914) = barycentric product X(i)*X(j) for these (i, j): {4, 4612}, {21, 648}, {25, 4631}, {27, 643}, {28, 645}, {29, 662}, {33, 4610}, {60, 6335}, {81, 36797}, {92, 4636}, {99, 1172}, {100, 46103}, {107, 1812}, {108, 7058}, {110, 31623}, {112, 314}, {162, 333}, {163, 44130}, {190, 270}, {249, 44426}, {250, 4391}, {261, 1783}, {283, 823}, {284, 811}, {286, 5546}, {318, 4556}, {332, 24019}, {521, 23582}, {607, 4623}, {650, 18020}, {652, 23999}, {653, 1098}, {663, 46254}, {664, 2326}, {668, 2189}, {670, 2204}, {799, 2299}, {931, 44734}, {1101, 46110}, {1396, 7256}, {1414, 2322}, {1474, 7257}, {1896, 4558}, {1897, 2185}, {2193, 6528}, {2194, 6331}, {2212, 52612}, {2332, 4625}, {3064, 24041}, {4183, 4573}, {4560, 5379}, {4571, 36419}, {4584, 14024}, {4590, 18344}, {4592, 8748}, {4603, 14006}, {5081, 37140}, {6061, 13149}, {6064, 6591}, {6332, 24000}, {6514, 36126}, {6516, 36421}, {7054, 18026}, {8750, 52379}, {14395, 42308}, {15146, 41206}, {17515, 47318}, {23964, 35518}, {28660, 32676}
X(52914) = barycentric quotient X(i)/X(j) for these (i, j): {9, 4064}, {21, 525}, {27, 4077}, {28, 7178}, {29, 1577}, {33, 4024}, {58, 51640}, {60, 905}, {81, 17094}, {100, 26942}, {107, 40149}, {108, 6354}, {109, 37755}, {110, 1214}, {112, 65}, {162, 226}, {163, 73}, {249, 6516}, {250, 651}, {261, 15413}, {281, 4036}, {283, 24018}, {284, 656}, {314, 3267}, {318, 52623}, {333, 14208}, {521, 15526}, {522, 20902}, {607, 4705}, {643, 306}, {644, 3695}, {645, 20336}, {648, 1441}, {650, 125}, {651, 6356}, {652, 2632}, {662, 307}, {663, 3708}, {692, 2197}, {811, 349}, {905, 1367}, {906, 7066}, {934, 20618}, {1098, 6332}, {1101, 1813}, {1172, 523}, {1415, 1425}, {1474, 4017}, {1576, 1409}, {1783, 12}, {1812, 3265}, {1896, 14618}, {1897, 6358}, {1946, 3269}, {1951, 9391}, {2150, 1459}, {2170, 21134}, {2185, 4025}, {2189, 513}, {2193, 520}, {2194, 647}, {2203, 7180}, {2204, 512}, {2212, 4079}, {2287, 52355}, {2299, 661}, {2322, 4086}, {2326, 522}, {2328, 8611}, {2332, 4041}, {3063, 20975}, {3064, 1109}, {3699, 52369}, {3737, 4466}, {3939, 3949}, {4041, 21046}, {4183, 3700}, {4391, 339}, {4556, 77}, {4558, 52385}, {4565, 1439}, {4575, 40152}, {4587, 52387}, {4592, 52565}, {4610, 7182}, {4612, 69}, {4631, 305}, {4636, 63}, {5324, 21107}, {5379, 4552}, {5546, 72}, {6332, 17879}, {6335, 34388}, {6528, 52575}, {6591, 1365}, {7012, 4605}, {7054, 521}, {7058, 35518}, {7252, 18210}, {7257, 40071}, {7259, 3710}, {7435, 51365}, {8748, 24006}, {8750, 2171}, {11107, 7265}, {14395, 1650}, {17515, 4707}, {18020, 4554}, {18344, 115}, {23357, 36059}, {23582, 18026}, {23609, 23090}, {23964, 108}, {23995, 32660}, {23999, 46404}, {24000, 653}, {24019, 225}, {31623, 850}, {32660, 7138}, {32661, 22341}, {32674, 1254}, {32676, 1400}, {32713, 1880}, {32714, 6046}, {35518, 36793}, {36054, 2972}, {36420, 43923}, {36421, 44426}, {36797, 321}, {37140, 52392}, {37141, 6355}, {37908, 24290}, {44130, 20948}, {44426, 338}, {46110, 23994}, {46254, 4572}, {46884, 23752}, {52413, 51645}, {52427, 2610}

---

X(52915) = X(107)X(110)∩X(112)X(827)

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

See Ivan Pavlov, euclid 5663.

X(52915) lies on the circumconic {{A,B,C,X(22),X(4240)}} and these lines: {4,15462}, {22,47413}, {49,42873}, {99,39417}, {107,110}, {112,827}, {206,17907}, {250,4558}, {297,18374}, {317,1974}, {338,38851}, {419,44138}, {645,5379}, {670,18020}, {685,30450}, {1576,4230}, {1632,16237}, {1634,23347}, {2409,14570}, {4577,6331}, {9306,44134}, {9407,15143}, {9544,40138}, {11331,19127}, {34990,36176}, {35264,47405}, {37124,43811}

X(52915) = X(i)-isoconjugate-of-X(j) for these {i, j}: {66, 656}, {525, 2156}, {661, 14376}, {810, 18018}, {822, 43678}, {1289, 2632}, {2353, 14208}, {2616, 41168}, {3049, 46244}, {3708, 44766}, {8061, 40404}, {13854, 24018}
X(52915) = X(i)-Dao conjugate of X(j) for these {i, j}: {32, 647}, {127, 125}, {427, 826}, {3265, 23107}, {36830, 14376}, {39062, 18018}, {40596, 66}
X(52915) = X(i) cross conjugate of X(j) for these {i, j}: {2485, 17907}, {8673, 22}, {20806, 250}, {36414, 23582}
X(52915) = trilinear pole of line {22,8743}
X(52915) = barycentric product X(i)*X(j) for these (i, j): {4, 4611}, {22, 648}, {107, 20806}, {110, 17907}, {112, 315}, {162, 1760}, {206, 6331}, {250, 33294}, {670, 17409}, {811, 2172}, {877, 11610}, {2485, 18020}, {3313, 42396}, {4230, 31636}, {4558, 52448}, {4577, 40938}, {5379, 16757}, {6528, 10316}, {8673, 23582}, {11605, 52630}, {14396, 42308}, {20641, 32676}, {32713, 34254}
X(52915) = barycentric quotient X(i)/X(j) for these (i, j): {22, 525}, {107, 43678}, {110, 14376}, {112, 66}, {206, 647}, {250, 44766}, {315, 3267}, {648, 18018}, {811, 46244}, {827, 40404}, {1625, 41168}, {1760, 14208}, {2172, 656}, {2485, 125}, {3313, 2525}, {4230, 34138}, {4456, 4064}, {4611, 69}, {4630, 46765}, {6331, 40421}, {8673, 15526}, {10316, 520}, {11610, 879}, {14396, 1650}, {16715, 48084}, {17186, 1459}, {17409, 512}, {17453, 810}, {17907, 850}, {20806, 3265}, {20968, 3049}, {22075, 39201}, {23964, 1289}, {32676, 2156}, {32713, 13854}, {33294, 339}, {34254, 52617}, {36414, 8673}, {38356, 5489}, {40938, 826}, {47413, 23616}, {52448, 14618}
X(52915) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k):{110, 32713, 648}

---

X(52916) = X(4)X(83)∩X(107)X(110)

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

See Ivan Pavlov, euclid 5663.

X(52916) lies on these lines: {4,83}, {24,14687}, {107,110}, {112,11636}, {250,4230}, {297,32217}, {685,10412}, {691,10423}, {877,10411}, {1304,9060}, {2409,16237}, {2435,44769}, {3518,18114}, {5158,14002}, {8744,14246}, {12106,30258}, {18374,37765}, {21460,44102}, {32235,47204}, {33752,52630}, {35357,52604}, {38851,48540}

X(52916) = X(i)-isoconjugate-of-X(j) for these {i, j}: {67, 656}, {525, 2157}, {661, 34897}, {810, 18019}, {822, 46105}, {935, 2632}, {3455, 14208}, {3708, 17708}, {8791, 24018}
X(52916) = X(i)-Dao conjugate of X(j) for these {i, j}: {187, 14417}, {5099, 125}, {36830, 34897}, {39062, 18019}, {40583, 525}, {40596, 67}
X(52916) = X(i) cross conjugate of X(j) for these {i, j}: {2492, 37765}, {6593, 250}, {9517, 23}, {20410, 23964}, {36415, 23582}
X(52916) = intersection, other than A, B, C, of circumconics {{A,B,C,X(4),X(46151)}} and {{A,B,C,X(23),X(4240)}}
X(52916) = trilinear pole of line {23,8744}
X(52916) = barycentric product X(i)*X(j) for these (i, j): {4, 52630}, {23, 648}, {99, 8744}, {107, 22151}, {110, 37765}, {112, 316}, {162, 16568}, {250, 9979}, {687, 12824}, {2492, 18020}, {4235, 14246}, {6331, 18374}, {6528, 10317}, {9019, 42396}, {9517, 23582}, {14590, 52449}, {15459, 16165}, {20944, 32676}, {32713, 37804}
X(52916) = barycentric quotient X(i)/X(j) for these (i, j): {23, 525}, {107, 46105}, {110, 34897}, {112, 67}, {250, 17708}, {316, 3267}, {648, 18019}, {2492, 125}, {6593, 14417}, {7664, 45807}, {8744, 523}, {9019, 2525}, {9517, 15526}, {9979, 339}, {10317, 520}, {10561, 51258}, {12824, 6334}, {14246, 14977}, {16165, 41077}, {16568, 14208}, {18374, 647}, {20410, 47138}, {22151, 3265}, {23964, 935}, {32676, 2157}, {32713, 8791}, {36415, 9517}, {37765, 850}, {37804, 52617}, {42659, 3269}, {52142, 10097}, {52449, 14592}, {52630, 69}
X(52916) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2409, 16237, 30716}, {4230, 23347, 250}

---

X(52917) = X(107)X(110)∩X(112)X(933)

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

See Ivan Pavlov, euclid 5663.

X(52917) lies on these lines: {4,9934}, {107,110}, {112,933}, {133,6759}, {136,11547}, {184,41371}, {250,40049}, {925,16237}, {1301,1304}, {1614,14249}, {1624,23347}, {4230,4611}, {6525,9544}, {6530,34397}, {9064,33640}, {10002,11003}, {10540,51385}, {14157,34170}, {14165,19128}, {15139,51358}, {15352,32734}, {20976,51334}

X(52917) = X(i)-isoconjugate-of-X(j) for these {i, j}: {68, 656}, {91, 520}, {525, 1820}, {661, 52350}, {810, 20563}, {822, 5392}, {925, 2632}, {2165, 24018}, {2351, 14208}, {15526, 36145}, {16391, 24006}, {17879, 32734}, {20571, 39201}, {30450, 37754}
X(52917) = X(i)-Dao conjugate of X(j) for these {i, j}: {135, 125}, {577, 52613}, {34116, 520}, {36830, 52350}, {39013, 15526}, {39062, 20563}, {40596, 68}
X(52917) = X(i) cross conjugate of X(j) for these {i, j}: {924, 24}, {6753, 11547}, {35603, 250}, {36416, 23582}
X(52917) = intersection, other than A, B, C, of circumconics {{A,B,C,X(4),X(46963)}} and {{A,B,C,X(24),X(1301)}}
X(52917) = trilinear pole of line {24,571}
X(52917) = barycentric product X(i)*X(j) for these (i, j): {4, 41679}, {24, 648}, {47, 823}, {52, 16813}, {99, 8745}, {107, 1993}, {110, 11547}, {112, 317}, {136, 47443}, {162, 1748}, {467, 933}, {571, 6528}, {687, 52000}, {924, 23582}, {1147, 15352}, {6331, 44077}, {6529, 9723}, {6563, 23964}, {6753, 18020}, {7763, 32713}, {14397, 42308}, {14576, 18831}, {14590, 52415}, {15459, 51393}, {20031, 51439}, {24019, 44179}, {30450, 52432}, {32230, 52584}, {36416, 46134}, {46456, 52416}
X(52917) = barycentric quotient X(i)/X(j) for these (i, j): {24, 525}, {47, 24018}, {107, 5392}, {110, 52350}, {112, 68}, {317, 3267}, {571, 520}, {648, 20563}, {823, 20571}, {924, 15526}, {1147, 52613}, {1748, 14208}, {1993, 3265}, {6529, 847}, {6563, 36793}, {6753, 125}, {7763, 52617}, {8745, 523}, {9723, 4143}, {11547, 850}, {14397, 1650}, {14576, 6368}, {16813, 34385}, {18605, 4131}, {23582, 46134}, {23964, 925}, {24019, 91}, {30451, 2972}, {32230, 30450}, {32661, 16391}, {32676, 1820}, {32713, 2165}, {34952, 3269}, {36416, 924}, {41679, 69}, {41937, 32734}, {44077, 647}, {47421, 5489}, {51393, 41077}, {52000, 6334}, {52317, 35442}, {52415, 14592}, {52416, 8552}, {52432, 52584}, {52435, 32320}, {52436, 39201}
X(52917) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 32713, 107}

---

X(52918) = X(107)X(110)∩X(7468)X(10423)

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

See Ivan Pavlov, euclid 5663.

X(52918) lies on these lines: {107,110}, {7468,10423}, {7480,40596}, {9060,44060}, {23181,23347}

X(52918) = X(i)-isoconjugate-of-X(j) for these {i, j}: {70, 656}, {525, 2158}, {810, 20564}, {1288, 2632}
X(52918) = X(i)-Dao conjugate of X(j) for these {i, j}: {571, 52584}, {39062, 20564}, {40596, 70}, {52120, 125}
X(52918) = X(i) cross conjugate of X(j) for these {i, j}: {36418, 23582}
X(52918) = trilinear pole of line {26,8746}
X(52918) = barycentric product X(i)*X(j) for these (i, j): {26, 648}, {99, 8746}, {112, 44128}, {6331, 44078}, {30450, 34116}
X(52918) = barycentric quotient X(i)/X(j) for these (i, j): {26, 525}, {112, 70}, {648, 20564}, {8746, 523}, {23964, 1288}, {32676, 2158}, {34116, 52584}, {44078, 647}, {44128, 3267}
X(52918) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 32713, 35360}

---

X(52919) = X(107)X(110)∩X(662)X(811)

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

See Ivan Pavlov, euclid 5663.

X(52919) lies on these lines: {27,4466}, {107,110}, {250,7479}, {662,811}, {4556,21205}, {4591,23582}, {26856,36421}

X(52919) = X(i)-isoconjugate-of-X(j) for these {i, j}: {10, 822}, {12, 36054}, {37, 520}, {42, 24018}, {48, 4064}, {71, 656}, {72, 647}, {73, 8611}, {100, 3269}, {101, 2632}, {125, 906}, {201, 652}, {213, 3265}, {228, 525}, {255, 4024}, {306, 810}, {321, 39201}, {326, 4079}, {394, 4705}, {513, 52386}, {521, 2197}, {523, 3990}, {577, 4036}, {594, 23224}, {649, 52387}, {650, 7066}, {661, 3682}, {692, 15526}, {756, 4091}, {798, 52396}, {872, 30805}, {879, 42702}, {905, 3690}, {1331, 3708}, {1332, 20975}, {1364, 21859}, {1409, 52355}, {1415, 7068}, {1459, 3949}, {1500, 4131}, {1577, 4055}, {1783, 2972}, {1824, 52613}, {1897, 37754}, {1946, 26942}, {2200, 14208}, {2205, 52617}, {2318, 51640}, {2321, 51641}, {2638, 4605}, {3049, 20336}, {3239, 7138}, {3695, 22383}, {3700, 22341}, {3709, 52385}, {3926, 50487}, {4041, 40152}, {4158, 6591}, {4574, 18210}, {4575, 21046}, {6335, 34980}, {17094, 52370}, {17879, 32739}, {20902, 32656}, {32320, 41013}, {52430, 52623}
X(52919) = X(i)-Dao conjugate of X(j) for these {i, j}: {136, 21046}, {1015, 2632}, {1086, 15526}, {1146, 7068}, {1249, 4064}, {4988, 5489}, {5190, 125}, {5375, 52387}, {5521, 3708}, {6523, 4024}, {6626, 3265}, {8054, 3269}, {15259, 4079}, {31998, 52396}, {34467, 37754}, {36830, 3682}, {39006, 2972}, {39026, 52386}, {39052, 72}, {39053, 26942}, {39054, 3998}, {39062, 306}, {40589, 520}, {40592, 24018}, {40596, 71}, {40615, 1367}, {40616, 122}, {40619, 17879}, {40620, 17216}
X(52919) = X(i) cross conjugate of X(j) for these {i, j}: {21172, 86}, {36419, 23582}, {36421, 32230}
X(52919) = trilinear pole of line {27,58}
X(52919) = barycentric product X(i)*X(j) for these (i, j): {27, 648}, {28, 811}, {58, 6528}, {81, 823}, {86, 107}, {112, 44129}, {162, 286}, {250, 46107}, {261, 36127}, {270, 18026}, {274, 24019}, {310, 32713}, {393, 4610}, {513, 23999}, {514, 23582}, {653, 46103}, {658, 36421}, {693, 24000}, {799, 5317}, {1096, 4623}, {1414, 1896}, {1444, 36126}, {1474, 6331}, {1790, 15352}, {1978, 36420}, {2052, 4556}, {2189, 46404}, {2207, 52612}, {2326, 13149}, {3261, 23964}, {4025, 32230}, {4091, 34538}, {4131, 24021}, {4573, 8748}, {4589, 34856}, {6529, 17206}, {6591, 46254}, {7649, 18020}, {11125, 42308}, {15459, 18653}, {16813, 17167}, {17171, 42396}, {20031, 51370}, {21172, 44181}, {23590, 30805}, {46151, 52394}
X(52919) = barycentric quotient X(i)/X(j) for these (i, j): {4, 4064}, {27, 525}, {28, 656}, {29, 52355}, {58, 520}, {81, 24018}, {86, 3265}, {100, 52387}, {107, 10}, {108, 201}, {109, 7066}, {110, 3682}, {112, 71}, {158, 4036}, {162, 72}, {163, 3990}, {250, 1331}, {261, 52616}, {270, 521}, {286, 14208}, {310, 52617}, {393, 4024}, {513, 2632}, {514, 15526}, {522, 7068}, {593, 4091}, {648, 306}, {653, 26942}, {662, 3998}, {693, 17879}, {757, 4131}, {811, 20336}, {823, 321}, {849, 23224}, {1096, 4705}, {1172, 8611}, {1331, 4158}, {1333, 822}, {1396, 51640}, {1408, 51641}, {1414, 52385}, {1430, 9391}, {1459, 2972}, {1474, 647}, {1509, 30805}, {1576, 4055}, {1783, 3949}, {1790, 52613}, {1896, 4086}, {1897, 3695}, {2052, 52623}, {2150, 36054}, {2189, 652}, {2203, 810}, {2206, 39201}, {2207, 4079}, {2501, 21046}, {2969, 21134}, {3120, 5489}, {3261, 36793}, {3676, 1367}, {4131, 24020}, {4241, 51366}, {4466, 23616}, {4556, 394}, {4565, 40152}, {4573, 52565}, {4610, 3926}, {4612, 3719}, {4636, 1259}, {5317, 661}, {6331, 40071}, {6335, 52369}, {6528, 313}, {6529, 1826}, {6591, 3708}, {7192, 17216}, {7649, 125}, {8748, 3700}, {8750, 3690}, {11125, 1650}, {17171, 2525}, {17206, 4143}, {17924, 20902}, {17925, 4466}, {18020, 4561}, {18653, 41077}, {21102, 35442}, {21172, 122}, {22383, 37754}, {23582, 190}, {23964, 101}, {23984, 4605}, {23999, 668}, {24000, 100}, {24019, 37}, {30805, 23974}, {31905, 24459}, {32230, 1897}, {32674, 2197}, {32676, 228}, {32713, 42}, {32714, 37755}, {34856, 4010}, {36118, 6356}, {36126, 41013}, {36127, 12}, {36421, 3239}, {36797, 3710}, {37168, 14429}, {41937, 32739}, {44129, 3267}, {44698, 8057}, {46103, 6332}, {46107, 339}, {46151, 15523}

---

X(52920) = X(107)X(110)∩X(162)X(163)

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

See Ivan Pavlov, euclid 5663.

X(52920) lies on these lines: {4,43700}, {28,18210}, {107,110}, {112,36077}, {162,163}, {250,7477}, {8059,36068}

X(52920) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3, 4064}, {10, 520}, {37, 24018}, {42, 3265}, {71, 525}, {72, 656}, {73, 52355}, {100, 2632}, {101, 15526}, {109, 7068}, {125, 1331}, {181, 52616}, {190, 3269}, {201, 521}, {228, 14208}, {255, 4036}, {306, 647}, {313, 39201}, {321, 822}, {326, 4705}, {339, 32656}, {394, 4024}, {512, 52396}, {513, 52387}, {514, 52386}, {522, 7066}, {523, 3682}, {577, 52623}, {594, 4091}, {652, 26942}, {661, 3998}, {692, 17879}, {756, 4131}, {810, 20336}, {850, 4055}, {905, 3949}, {906, 20902}, {1089, 23224}, {1214, 8611}, {1332, 3708}, {1367, 3939}, {1459, 3695}, {1500, 30805}, {1577, 3990}, {1826, 52613}, {1897, 2972}, {1918, 52617}, {2197, 6332}, {2200, 3267}, {2318, 17094}, {2333, 4143}, {3049, 40071}, {3690, 4025}, {3694, 51640}, {3700, 40152}, {3701, 51641}, {3709, 52565}, {3926, 4079}, {4041, 52385}, {4086, 22341}, {4092, 6517}, {4158, 7649}, {4397, 7138}, {4466, 4574}, {4557, 17216}, {4558, 21046}, {4561, 20975}, {4570, 5489}, {4605, 35072}, {6335, 37754}, {6358, 36054}, {22383, 52369}, {32739, 36793}
X(52920) = X(i)-Dao conjugate of X(j) for these {i, j}: {11, 7068}, {1015, 15526}, {1086, 17879}, {5190, 20902}, {5521, 125}, {6523, 4036}, {8054, 2632}, {15259, 4705}, {34021, 52617}, {34467, 2972}, {36103, 4064}, {36830, 3998}, {39026, 52387}, {39052, 306}, {39054, 52396}, {39062, 20336}, {40589, 24018}, {40592, 3265}, {40596, 72}, {40617, 1367}, {40619, 36793}, {50330, 5489}
X(52920) = X(i) cross conjugate of X(j) for these {i, j}: {36420, 23582}
X(52920) = trilinear pole of line {28,1104}
X(52920) = barycentric product X(i)*X(j) for these (i, j): {27, 162}, {28, 648}, {58, 823}, {81, 107}, {86, 24019}, {99, 5317}, {100, 36419}, {108, 46103}, {112, 286}, {158, 4556}, {250, 17924}, {270, 653}, {274, 32713}, {513, 23582}, {514, 24000}, {649, 23999}, {662, 8747}, {668, 36420}, {693, 23964}, {811, 1474}, {905, 32230}, {934, 36421}, {1096, 4610}, {1118, 4612}, {1333, 6528}, {1396, 36797}, {1414, 8748}, {1437, 15352}, {1444, 6529}, {1790, 36126}, {1896, 4565}, {2185, 36127}, {2189, 18026}, {2203, 6331}, {2207, 4623}, {2326, 36118}, {4091, 24021}, {4131, 23590}, {4584, 34856}, {4631, 7337}, {5379, 17925}, {6591, 18020}, {14399, 42308}, {15459, 51420}, {16813, 18180}, {20031, 51369}, {23224, 34538}, {24022, 30805}, {32676, 44129}, {40495, 41937}, {46151, 52376}
X(52920) = barycentric quotient X(i)/X(j) for these (i, j): {19, 4064}, {27, 14208}, {28, 525}, {58, 24018}, {81, 3265}, {107, 321}, {108, 26942}, {110, 3998}, {112, 72}, {158, 52623}, {162, 306}, {163, 3682}, {250, 1332}, {270, 6332}, {274, 52617}, {286, 3267}, {393, 4036}, {513, 15526}, {514, 17879}, {593, 4131}, {648, 20336}, {649, 2632}, {650, 7068}, {662, 52396}, {667, 3269}, {692, 52386}, {693, 36793}, {757, 30805}, {811, 40071}, {823, 313}, {849, 4091}, {906, 4158}, {1019, 17216}, {1096, 4024}, {1172, 52355}, {1333, 520}, {1396, 17094}, {1414, 52565}, {1415, 7066}, {1437, 52613}, {1444, 4143}, {1474, 656}, {1576, 3990}, {1783, 3695}, {1897, 52369}, {2185, 52616}, {2189, 521}, {2203, 647}, {2206, 822}, {2207, 4705}, {2299, 8611}, {3125, 5489}, {3669, 1367}, {4091, 24020}, {4131, 23974}, {4211, 21107}, {4246, 51367}, {4556, 326}, {4565, 52385}, {4612, 1264}, {4636, 3719}, {5317, 523}, {5379, 52609}, {6528, 27801}, {6529, 41013}, {6591, 125}, {7649, 20902}, {8747, 1577}, {8748, 4086}, {8750, 3949}, {14399, 1650}, {16947, 51641}, {17924, 339}, {18210, 23616}, {22383, 2972}, {23582, 668}, {23964, 100}, {23999, 1978}, {24019, 10}, {24033, 4605}, {32230, 6335}, {32674, 201}, {32676, 71}, {32713, 37}, {32714, 6356}, {34859, 5360}, {35360, 42698}, {36127, 6358}, {36417, 50487}, {36421, 4397}, {41937, 692}, {43925, 18210}, {46103, 35518}, {51420, 41077}

---

X(52921) = X(107)X(110)∩X(162)X(823)

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

See Ivan Pavlov, euclid 5663.

X(52921) lies on these lines: {107,110}, {108,41207}, {162,823}, {250,50403}, {1896,52380}, {2659,23692}, {5546,36797}

X(52921) = X(i)-isoconjugate-of-X(j) for these {i, j}: {10, 51641}, {12, 23224}, {65, 520}, {71, 51640}, {73, 656}, {108, 2972}, {109, 2632}, {125, 36059}, {181, 4131}, {201, 1459}, {226, 822}, {228, 17094}, {296, 9391}, {307, 810}, {512, 52385}, {513, 7066}, {521, 1425}, {522, 7138}, {523, 22341}, {525, 1409}, {603, 4064}, {647, 1214}, {651, 3269}, {652, 37755}, {653, 37754}, {661, 40152}, {692, 1367}, {798, 52565}, {905, 2197}, {1231, 3049}, {1400, 24018}, {1402, 3265}, {1410, 52355}, {1415, 15526}, {1441, 39201}, {1804, 4705}, {1813, 3708}, {1880, 52613}, {1946, 6356}, {2171, 4091}, {2643, 6517}, {3669, 52386}, {3682, 4017}, {3990, 7178}, {3998, 7180}, {4024, 7125}, {4036, 7335}, {4055, 4077}, {4079, 7183}, {4158, 43923}, {6354, 36054}, {6516, 20975}, {7055, 50487}, {8611, 52373}, {18026, 34980}, {18210, 23067}, {20902, 32660}, {22383, 26942}, {32320, 40149}, {43924, 52387}, {51642, 52396}
X(52921) = X(i)-Dao conjugate of X(j) for these {i, j}: {11, 2632}, {1086, 1367}, {1146, 15526}, {2968, 7068}, {7952, 4064}, {20620, 125}, {31998, 52565}, {34961, 3682}, {36830, 40152}, {38983, 2972}, {38991, 3269}, {39026, 7066}, {39052, 1214}, {39053, 6356}, {39054, 52385}, {39062, 307}, {40582, 24018}, {40596, 73}, {40602, 520}, {40605, 3265}, {40624, 17879}, {40625, 17216}
X(52921) = X(i) cross conjugate of X(j) for these {i, j}: {270, 24000}, {522, 29}, {14331, 333}, {36421, 23582}
X(52921) = intersection, other than A, B, C, of circumconics {{A,B,C,X(21),X(13138}}} and {{A,B,C,X(29),X(4240)}}
X(52921) = trilinear pole of line {29,284}
X(52921) = barycentric product X(i)*X(j) for these (i, j): {21, 823}, {27, 36797}, {29, 648}, {99, 8748}, {107, 333}, {112, 44130}, {158, 4612}, {162, 31623}, {250, 46110}, {270, 6335}, {283, 15352}, {284, 6528}, {314, 24019}, {332, 6529}, {522, 23582}, {645, 8747}, {650, 23999}, {662, 1896}, {664, 36421}, {811, 1172}, {1096, 4631}, {1812, 36126}, {1857, 4610}, {1897, 46103}, {2052, 4636}, {2299, 6331}, {2326, 18026}, {3064, 18020}, {3699, 36419}, {4391, 24000}, {5317, 7257}, {6059, 52612}, {6332, 32230}, {7058, 36127}, {14331, 44181}, {14400, 42308}, {15146, 41207}, {15459, 51382}, {18344, 46254}, {23590, 52616}, {23964, 35519}, {28660, 32713}
X(52921) = barycentric quotient X(i)/X(j) for these (i, j): {21, 24018}, {27, 17094}, {28, 51640}, {29, 525}, {60, 4091}, {101, 7066}, {107, 226}, {108, 37755}, {110, 40152}, {112, 73}, {162, 1214}, {163, 22341}, {249, 6517}, {250, 1813}, {261, 30805}, {270, 905}, {281, 4064}, {283, 52613}, {284, 520}, {332, 4143}, {333, 3265}, {522, 15526}, {643, 3998}, {644, 52387}, {645, 52396}, {648, 307}, {650, 2632}, {652, 2972}, {653, 6356}, {662, 52385}, {663, 3269}, {811, 1231}, {823, 1441}, {1172, 656}, {1333, 51641}, {1415, 7138}, {1783, 201}, {1857, 4024}, {1896, 1577}, {1897, 26942}, {1946, 37754}, {2150, 23224}, {2185, 4131}, {2189, 1459}, {2194, 822}, {2202, 9391}, {2204, 810}, {2299, 647}, {2322, 52355}, {2326, 521}, {3064, 125}, {3239, 7068}, {3939, 52386}, {4183, 8611}, {4391, 17879}, {4556, 1804}, {4560, 17216}, {4587, 4158}, {4610, 7055}, {4612, 326}, {4636, 394}, {5317, 4017}, {5546, 3682}, {5931, 14638}, {6059, 4079}, {6528, 349}, {6529, 225}, {7058, 52616}, {7452, 51368}, {8735, 21134}, {8747, 7178}, {8748, 523}, {8750, 2197}, {14024, 24459}, {14331, 122}, {14400, 1650}, {18344, 3708}, {21044, 5489}, {23582, 664}, {23590, 36127}, {23964, 109}, {23999, 4554}, {24000, 651}, {24019, 65}, {28660, 52617}, {31623, 14208}, {32230, 653}, {32674, 1425}, {32676, 1409}, {32713, 1400}, {34856, 7212}, {35519, 36793}, {36118, 20618}, {36126, 40149}, {36127, 6354}, {36419, 3676}, {36420, 43924}, {36421, 522}, {36797, 306}, {44130, 3267}, {44426, 20902}, {46103, 4025}, {46110, 339}, {51382, 41077}, {52616, 23974}

---

X(52922) = X(660)X(662)∩X(813)X(831)

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

See Ivan Pavlov, euclid 5663.

X(52922) lies on these lines: {660,662}, {813,831}, {876,4562}, {2530,4568}, {4553,21123}, {4583,4602}, {7794,22116}, {7953,36081}

X(52922) = X(i)-isoconjugate-of-X(j) for these {i, j}: {82, 659}, {83, 8632}, {238, 18108}, {251, 812}, {1914, 10566}, {2238, 39179}, {3766, 46289}, {4455, 52394}, {4599, 39786}, {4628, 27918}, {18098, 50456}, {18105, 33295}, {18107, 51321}, {21832, 52376}, {22384, 32085}, {46387, 52395}
X(52922) = X(i)-Dao conjugate of X(j) for these {i, j}: {39, 3766}, {141, 659}, {1215, 804}, {3124, 39786}, {9470, 18108}, {36906, 10566}, {40585, 812}
X(52922) = trilinear pole of line {38,8041}
X(52922) = barycentric product X(i)*X(j) for these (i, j): {38, 4562}, {39, 4583}, {141, 660}, {291, 4568}, {334, 46148}, {335, 4553}, {813, 1930}, {3954, 4589}, {3963, 46161}, {4033, 46159}, {4584, 15523}, {4639, 21035}, {5378, 16892}, {7794, 36081}, {8024, 34067}, {16587, 18829}, {18827, 35309}, {35333, 40217}
X(52922) = barycentric quotient X(i)/X(j) for these (i, j): {38, 812}, {39, 659}, {141, 3766}, {291, 10566}, {292, 18108}, {660, 83}, {741, 39179}, {813, 82}, {1964, 8632}, {2530, 27918}, {3005, 39786}, {3688, 4435}, {3954, 4010}, {4020, 22384}, {4553, 239}, {4562, 3112}, {4568, 350}, {4576, 30940}, {4583, 308}, {4584, 52394}, {16587, 804}, {16720, 14296}, {17187, 50456}, {18787, 18111}, {21035, 21832}, {21123, 27846}, {21752, 5027}, {21814, 4455}, {33299, 3716}, {34067, 251}, {35309, 740}, {35333, 6654}, {36081, 52395}, {41531, 18107}, {43534, 18070}, {46148, 238}, {46153, 1429}, {46159, 1019}, {46161, 40432}

---

X(52923) = ISOGONAL CONJUGATE OF X(43931)

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

See Ivan Pavlov, euclid 5663.

X(52923) lies on the circumconic {{A,B,C,X(43),X(17780)}} and these lines: {2,21320}, {6,23560}, {8,15507}, {9,23407}, {10,25634}, {11,27290}, {37,22209}, {43,3123}, {45,1621}, {99,29199}, {100,190}, {101,932}, {210,11688}, {404,32935}, {644,3903}, {651,660}, {692,765}, {993,51297}, {1026,3888}, {2176,21762}, {2975,5220}, {3573,3939}, {3799,3882}, {4447,20072}, {4473,8299}, {4499,35338}, {4595,25312}, {8701,43350}, {13588,32938}, {15624,17336}, {17154,27666}, {17350,34247}, {20760,27538}, {20964,40749}, {21835,51973}, {23831,35281}, {24405,27191}, {27644,41531}, {27834,37138}, {28196,34594}, {36294,42079}, {36860,36863}, {38869,50995}

X(52923) = isogonal Conjugate of X(43931)
X(52923) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 43931}, {87, 513}, {244, 932}, {330, 649}, {650, 7153}, {667, 6384}, {693, 7121}, {876, 34252}, {1015, 4598}, {1019, 16606}, {1086, 34071}, {1919, 6383}, {2053, 3676}, {2319, 3669}, {3063, 7209}, {3248, 18830}, {3572, 39914}, {3733, 42027}, {4369, 51974}, {4444, 51321}, {4817, 52655}, {5383, 21143}, {6377, 32039}, {7155, 43924}, {7192, 23493}, {7199, 21759}, {7649, 23086}, {15373, 17924}, {20980, 27498}, {20981, 27447}
X(52923) = X(i)-Dao conjugate of X(j) for these {i, j}: {3, 43931}, {75, 3261}, {798, 21143}, {3061, 3776}, {3835, 764}, {5375, 330}, {6377, 1111}, {6631, 6384}, {9296, 6383}, {10001, 7209}, {39026, 87}, {40598, 693}, {40610, 1086}
X(52923) = X(i) cross conjugate of X(j) for these {i, j}: {4083, 43}, {4595, 100}, {8640, 2176}, {20979, 27644}
X(52923) = trilinear pole of line {43,2176}
X(52923) = barycentric product X(i)*X(j) for these (i, j): {1, 4595}, {6, 36863}, {42, 36860}, {43, 190}, {99, 20691}, {100, 192}, {109, 4110}, {644, 3212}, {646, 1403}, {651, 27538}, {662, 3971}, {664, 3208}, {668, 2176}, {692, 6382}, {765, 3835}, {874, 51973}, {1016, 4083}, {1018, 33296}, {1252, 20906}, {1423, 3699}, {1897, 22370}, {1978, 2209}, {3123, 6632}, {3570, 41531}, {3573, 40848}, {3799, 52136}, {3903, 17752}, {3939, 30545}, {3952, 27644}, {4033, 38832}, {4076, 43051}, {4147, 4564}, {4557, 31008}, {4567, 21051}, {4600, 21834}, {4601, 50491}, {4606, 4734}, {4621, 41886}, {4970, 37212}, {5381, 14426}, {5383, 25142}, {6335, 20760}, {7035, 20979}, {7304, 40521}, {8026, 34071}, {8640, 31625}, {15742, 25098}, {27805, 51902}, {29227, 40598}
X(52923) = barycentric quotient X(i)/X(j) for these (i, j): {6, 43931}, {100, 330}, {109, 7153}, {190, 6384}, {192, 693}, {644, 7155}, {664, 7209}, {668, 6383}, {692, 2162}, {765, 4598}, {906, 23086}, {1016, 18830}, {1018, 42027}, {1110, 34071}, {1252, 932}, {1403, 3669}, {1423, 3676}, {2176, 513}, {3123, 6545}, {3208, 522}, {3212, 24002}, {3573, 39914}, {3699, 27424}, {3799, 51837}, {3835, 1111}, {3903, 27447}, {3939, 2319}, {3971, 1577}, {4083, 1086}, {4110, 35519}, {4147, 4858}, {4557, 16606}, {4595, 75}, {4734, 4801}, {4941, 48415}, {4970, 4978}, {6376, 3261}, {6377, 764}, {6382, 40495}, {8640, 1015}, {14408, 1647}, {14426, 52626}, {16695, 16726}, {17217, 16727}, {17459, 48406}, {17752, 4374}, {18197, 17205}, {20284, 3777}, {20691, 523}, {20760, 905}, {20906, 23989}, {20979, 244}, {21051, 16732}, {21762, 8027}, {21834, 3120}, {21835, 8034}, {22090, 3942}, {22370, 4025}, {23845, 27499}, {24533, 7200}, {24749, 21139}, {25098, 1565}, {25142, 21138}, {25312, 20892}, {27538, 4391}, {27644, 7192}, {30545, 52621}, {31008, 52619}, {32656, 15373}, {32739, 7121}, {33296, 7199}, {36860, 310}, {36863, 76}, {38832, 1019}, {38986, 21143}, {41526, 43924}, {41531, 4444}, {41886, 3776}, {43051, 1358}, {43290, 27496}, {50491, 3125}, {51319, 4367}, {51902, 4369}, {51973, 876}
X(52923) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {190, 4557, 100}, {1026, 21362, 3888}, {3882, 4069, 3799}, {4557, 23343, 190}, {23845, 43290, 100}

---

X(52924) = ISOGONAL CONJUGATE OF X(23352)

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

See Ivan Pavlov, euclid 5663.

X(52924) lies on the circumconic {{A,B,C,X(1),X(100}} and these lines: {1,89}, {100,4585}, {149,19640}, {519,36818}, {3573,5385}, {4597,4781}, {4752,34073}

X(52924) = isogonal Conjugate of X(23352)
X(52924) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 23352}, {6, 23598}, {88, 4893}, {106, 4777}, {649, 4945}, {1960, 36594}, {2099, 23838}, {2177, 6548}, {3679, 23345}, {4049, 4273}, {4674, 4833}, {4767, 43922}, {4791, 9456}, {4957, 32665}, {8752, 49280}
X(52924) = X(i)-Dao conjugate of X(j) for these {i, j}: {3, 23352}, {9, 23598}, {44, 23884}, {214, 4777}, {4370, 4791}, {5375, 4945}, {35092, 4957}, {39026, 4792}
X(52924) = trilinear pole of line {44,214}
X(52924) = barycentric product X(i)*X(j) for these (i, j): {44, 4597}, {89, 17780}, {519, 4604}, {900, 5385}, {1023, 39704}, {2163, 24004}, {3264, 34073}, {4358, 4588}, {20569, 23344}, {23703, 30608}
X(52924) = barycentric quotient X(i)/X(j) for these (i, j): {1, 23598}, {6, 23352}, {44, 4777}, {89, 6548}, {100, 4945}, {214, 23884}, {519, 4791}, {900, 4957}, {902, 4893}, {1023, 3679}, {1319, 43052}, {2163, 1022}, {2251, 4775}, {2364, 23838}, {3257, 36594}, {3285, 4833}, {3689, 4944}, {4169, 4125}, {4588, 88}, {4597, 20568}, {4604, 903}, {5385, 4555}, {5440, 49280}, {5549, 1320}, {17780, 4671}, {21805, 4931}, {23344, 45}, {23703, 5219}, {28607, 23345}, {29908, 4411}, {34073, 106}, {52680, 47683}

---

X(52925) = X(88)X(518)∩X(100)X(513)

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

See Ivan Pavlov, euclid 5663.

X(52925) lies on the circumconic {{A,B,C,X(45),X(660} and these lines: {88,518}, {100,513}, {106,899}, {149,19634}, {200,52031}, {1023,5548}, {1376,40215}, {3218,14193}, {3689,14190}, {3870,52140}, {3952,4582}, {4413,34230}, {4555,4618}, {4604,36091}, {4674,5524}, {4752,4893}, {4767,4777}, {4792,4867}, {4945,4954}, {14410,23352}, {16504,52553}, {17757,36590}, {27757,36919}, {39148,48696}

X(52925) = reflection of X(36919) in X(51362) for these {i,j}: {36919, 51362}
X(52925) = X(i)-isoconjugate-of-X(j) for these {i, j}: {89, 1635}, {900, 2163}, {902, 52620}, {1647, 4588}, {1960, 39704}, {2087, 4604}, {2364, 30725}, {3762, 28607}
X(52925) = X(i)-Dao conjugate of X(j) for these {i, j}: {36911, 3762}, {40587, 900}, {40594, 52620
X(52925) = X(i) cross conjugate of X(j) for these {i, j}: {23352, 4792}
X(52925) = trilinear pole of line {45,4752}
X(52925) = barycentric product X(i)*X(j) for these (i, j): {45, 4555}, {88, 4767}, {100, 4945}, {190, 4792}, {765, 23598}, {901, 4671}, {903, 4752}, {1016, 23352}, {1023, 36594}, {2099, 4582}, {3257, 3679}, {4125, 4591}, {4618, 4908}, {4777, 5376}, {4791, 9268}, {4957, 6551}
X(52925) = barycentric quotient X(i)/X(j) for these (i, j): {45, 900}, {88, 52620}, {901, 89}, {2099, 30725}, {2177, 1635}, {3257, 39704}, {3679, 3762}, {3711, 1639}, {4555, 20569}, {4618, 40833}, {4752, 519}, {4767, 4358}, {4775, 2087}, {4792, 514}, {4814, 4530}, {4873, 4768}, {4893, 1647}, {4945, 693}, {5376, 4597}, {5548, 2320}, {6551, 5385}, {9268, 4604}, {17461, 23888}, {23352, 1086}, {23598, 1111}, {32665, 2163}, {32719, 28607}, {36091, 40426}

---

X(52926) = X(262)X(5890)∩X(263)X(1989)

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

See Ivan Pavlov, euclid 5663.

X(52926) lies on these lines: {112,26714}, {262,5890}, {263,1989}, {1987,9475}, {4630,32716}, {15352,46151}, {35360,35362}

X(52926) = X(i)-isoconjugate-of-X(j) for these {i, j}: {183, 2616}, {2167, 23878}, {2623, 3403}, {15412, 52134}
X(52926) = X(i)-Dao conjugate of X(j) for these {i, j}: {40588, 23878}
X(52926) = trilinear pole of line {51,40588}
X(52926) = barycentric product X(i)*X(j) for these (i, j): {5, 26714}, {262, 1625}, {263, 14570}, {2186, 2617}, {35319, 42299}, {35360, 43718}, {42313, 52604}
X(52926) = barycentric quotient X(i)/X(j) for these (i, j): {51, 23878}, {263, 15412}, {1625, 183}, {2617, 3403}, {3402, 2616}, {14570, 20023}, {26714, 95}, {35319, 14994}, {35360, 44144}, {40981, 3288}, {42288, 39182}, {46319, 2623}, {52604, 458}, {52631, 8901}

---

X(52927) = X(59)X(513)∩X(101)X(663)

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

See Ivan Pavlov, euclid 5663.

X(52927) lies on these lines: {2,42834}, {59,513}, {101,663}, {105,517}, {110,32724}, {294,2348}, {644,3900}, {645,7253}, {657,3939}, {666,885}, {672,1438}, {884,5548}, {927,4566}, {971,40567}, {1149,1416}, {1252,8641}, {1292,15402}, {1783,18344}, {1814,9004}, {2283,2424}, {2338,2340}, {2428,2440}, {5375,17115}, {5546,21789}, {7671,47373}, {8638,34067}, {10006,38310}, {15733,28071}, {23344,23351}, {23696,35333}, {35273,51987}

X(52927) = midpoint of X(2348) in X(41339)
X(52927) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 43042}, {7, 2254}, {11, 41353}, {57, 918}, {65, 23829}, {85, 665}, {241, 514}, {244, 883}, {269, 50333}, {513, 9436}, {518, 3676}, {522, 34855}, {649, 40704}, {658, 17435}, {664, 3675}, {672, 24002}, {693, 1458}, {876, 39775}, {905, 5236}, {926, 1088}, {1014, 4088}, {1025, 1086}, {1026, 1358}, {1111, 2283}, {1434, 24290}, {1876, 4025}, {2223, 52621}, {3261, 52635}, {3263, 43924}, {3286, 4077}, {3323, 36086}, {3669, 3912}, {3717, 43932}, {3930, 17096}, {3932, 7203}, {4017, 30941}, {4394, 10029}, {4444, 34253}, {4712, 43930}, {4925, 19604}, {5723, 52228}, {7045, 52305}, {7178, 18206}, {7180, 18157}, {15149, 51640}, {16593, 37626}, {22116, 43041}, {23062, 52614}, {34085, 35505}, {35094, 36146}, {37136, 42770}
X(52927) = X(i)-Dao conjugate of X(j) for these {i, j}: {3, 43042}, {5375, 40704}, {5452, 918}, {6600, 50333}, {17115, 52305}, {34961, 30941}, {38989, 3323}, {39014, 35094}, {39025, 3675}, {39026, 9436}, {40602, 23829}
X(52927) = X(i) cross conjugate of X(j) for these {i, j}: {884, 2195}, {926, 55}X(52927)
X(52927) = intersection, other than A, B, C, of circumconics {{A,B,C,X(33),X(51562)}}, {{A,B,C,X(41),X(34067)}}, {{A,B,C,X(55),X(901)}}, {{A,B,C,X(59),X(1110)}}, {{A,B,C,X(101),X(294)}} and {{A,B,C,X(109),X(35280)}}
X(52927) = trilinear pole of line {55,2195}
X(52927) = barycentric product X(i)*X(j) for these (i, j): {6, 36802}, {8, 919}, {9, 36086}, {41, 51560}, {55, 666}, {59, 28132}, {100, 294}, {101, 14942}, {105, 644}, {109, 6559}, {190, 2195}, {200, 36146}, {220, 927}, {312, 32666}, {346, 32735}, {643, 18785}, {650, 5377}, {651, 28071}, {657, 39293}, {673, 3939}, {692, 36796}, {765, 1024}, {884, 1016}, {885, 1252}, {1253, 34085}, {1416, 6558}, {1438, 3699}, {1462, 4578}, {2175, 36803}, {2348, 39272}, {3886, 36138}, {4076, 43929}, {4571, 8751}, {4587, 36124}, {5546, 13576}, {14727, 16283}, {14827, 46135}, {28809, 32724}
X(52927) = barycentric quotient X(i)/X(j) for these (i, j): {6, 43042}, {41, 2254}, {55, 918}, {100, 40704}, {101, 9436}, {105, 24002}, {220, 50333}, {284, 23829}, {294, 693}, {643, 18157}, {644, 3263}, {665, 3323}, {666, 6063}, {673, 52621}, {692, 241}, {884, 1086}, {885, 23989}, {919, 7}, {926, 35094}, {1024, 1111}, {1110, 1025}, {1252, 883}, {1293, 10029}, {1334, 4088}, {1415, 34855}, {1438, 3676}, {2149, 41353}, {2175, 665}, {2195, 514}, {2426, 39063}, {2440, 40615}, {3063, 3675}, {3939, 3912}, {4258, 50357}, {5377, 4554}, {5546, 30941}, {6065, 42720}, {6066, 2284}, {6559, 35519}, {8638, 35505}, {8641, 17435}, {8750, 5236}, {14827, 926}, {14936, 52305}, {14942, 3261}, {16283, 42341}, {18785, 4077}, {23990, 2283}, {28071, 4391}, {28132, 34387}, {32642, 52213}, {32644, 40154}, {32666, 57}, {32724, 42290}, {32735, 279}, {32739, 1458}, {36086, 85}, {36146, 1088}, {36796, 40495}, {36802, 76}, {36803, 41283}, {39293, 46406}, {41934, 43930}, {43929, 1358}, {46163, 3665}, {51560, 20567}

---

X(52928) = X(109)X(8687)∩X(2298)X(9372)

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

See Ivan Pavlov, euclid 5663.

X(52928) lies on these lines: {2,42828}, {109,8687}, {651,32736}, {961,34051}, {2298,9372}, {4566,4573}

X(52928) = X(i)-isoconjugate-of-X(j) for these {i, j}: {8, 17420}, {9, 3910}, {200, 3004}, {220, 4509}, {312, 52326}, {341, 6371}, {346, 48131}, {514, 3965}, {521, 46878}, {522, 960}, {523, 46877}, {650, 3687}, {657, 20911}, {1021, 1211}, {1043, 50330}, {1146, 3882}, {1193, 4397}, {1577, 46889}, {2269, 4391}, {2287, 21124}, {2292, 7253}, {2300, 52622}, {2354, 15416}, {3239, 3666}, {3674, 4130}, {3700, 17185}, {3704, 3737}, {3900, 4357}, {4086, 4267}, {4163, 24471}, {4171, 16705}, {4524, 16739}, {4560, 21033}, {18155, 40966}, {18697, 21789}, {20967, 35519}, {22074, 46110}, {35518, 40976}
X(52928) = X(i)-Dao conjugate of X(j) for these {i, j}: {478, 3910}, {6609, 3004}
X(52928) = X(i) cross conjugate of X(j) for these {i, j}: {1408, 24027}, {6371, 56}
X(52928) = trilinear pole of line {56,478}
X(52928) = barycentric product X(i)*X(j) for these (i, j): {7, 8687}, {56, 6648}, {57, 36098}, {269, 36147}, {279, 32736}, {651, 961}, {934, 2298}, {1020, 2363}, {1169, 4566}, {1220, 1461}, {1262, 4581}, {1407, 8707}, {1415, 31643}, {1791, 32714}, {1798, 52607}, {2359, 36118}
X(52928) = barycentric quotient X(i)/X(j) for these (i, j): {56, 3910}, {109, 3687}, {163, 46877}, {269, 4509}, {604, 17420}, {692, 3965}, {934, 20911}, {961, 4391}, {1020, 18697}, {1042, 21124}, {1106, 48131}, {1169, 7253}, {1220, 52622}, {1397, 52326}, {1407, 3004}, {1415, 960}, {1461, 4357}, {1576, 46889}, {1791, 15416}, {1798, 15411}, {2298, 4397}, {4559, 3704}, {4566, 1228}, {4581, 23978}, {4637, 16739}, {6614, 3674}, {6648, 3596}, {8687, 8}, {23981, 51407}, {24027, 3882}, {32674, 46878}, {32736, 346}, {36098, 312}, {36147, 341}, {52410, 6371}

---

X(52929) = X(110)X(16807)∩X(930)X(5994)

Barycentrics    a^2*(a^2-b^2)*(a^2-c^2)*(sqrt(3)*S+SA)*(sqrt(3)*S-SB)*(sqrt(3)*S-SC) : :

See Ivan Pavlov, euclid 5663.

X(52929) lies on the circumconic {{A,B,C,X(110),X(32037)}} and these lines: {110,16807}, {930,5994}, {3448,52204}, {5467,35332}, {14705,21462}, {17402,32037}, {32737,35329}, {35360,36306}, {36305,36760}

X(52929) = X(i)-isoconjugate-of-X(j) for these {i, j}: {661, 11144}, {1577, 51547}
X(52929) = X(i)-Dao conjugate of X(j) for these {i, j}: {10640, 23873}, {36830, 11144}
X(52929) = X(i) cross conjugate of X(j) for these {i, j}: {1510, 61}
X(52929) = trilinear pole of line {61,1493
X(52929) = barycentric product X(i)*X(j) for these (i, j): {18, 52605}, {61, 32037}, {99, 51546}, {110, 11143}, {302, 16807}, {10678, 23896}
X(52929) = barycentric quotient X(i)/X(j) for these (i, j): {61, 23873}, {110, 11144}, {1576, 51547}, {10678, 23871}, {11143, 850}, {16807, 17}, {32037, 34389}, {52605, 303}

---

X(52930) = X(110)X(16806)∩X(930)X(5995)

Barycentrics    a^2*(a^2-b^2)*(a^2-c^2)*(sqrt(3)*S-SA)*(sqrt(3)*S+SB)*(sqrt(3)*S+SC) : :

See Ivan Pavlov, euclid 5663.

X(52930) lies on these lines: {110,16806}, {930,5995}, {3448,52203}, {5467,35331}, {14704,21461}, {17403,32036}, {32737,35330}, {35360,36309}, {36304,36759}

X(52930) = X(i)-isoconjugate-of-X(j) for these {i, j}: {661, 11143}, {1577, 51546}
X(52930) = X(i)-Dao conjugate of X(j) for these {i, j}: {10639, 23872}, {36830, 11143}
X(52930) = X(i) cross conjugate of X(j) for these {i, j}: {1510, 62}}
X(52930) = trilinear pole of line {62,1493}
X(52930) = barycentric product X(i)*X(j) for these (i, j): {17, 52606}, {62, 32036}, {99, 51547}, {110, 11144}, {303, 16806}, {10677, 23895}
X(52930) = barycentric quotient X(i)/X(j) for these (i, j): {62, 23872}, {110, 11143}, {1576, 51546}, {10677, 23870}, {11144, 850}, {16806, 18}, {32036, 34390}, {52606, 302}

---

X(52931) = X(941)X(1937)∩X(959)X(2006)

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

See Ivan Pavlov, euclid 5663.

X(52931) lies on these lines: {108,32693}, {941,1937}, {959,2006}, {3869,15267}, {4554,4566}

X(52931) = X(i)-isoconjugate-of-X(j) for these {i, j}: {21, 17418}, {284, 23880}, {652, 44734}, {940, 1021}, {958, 3737}, {1019, 3713}, {1098, 8672}, {1468, 7253}, {2268, 4560}, {2287, 48144}, {2328, 43067}, {5307, 23090}, {7054, 50457}, {7252, 11679}, {10436, 21789}
X(52931) = X(i)-Dao conjugate of X(j) for these {i, j}: {15267, 8672}, {36908, 43067}, {40590, 23880}, {40611, 17418}
X(52931) = X(i) cross conjugate of X(j) for these {i, j}: {8672, 65}
X(52931) = trilinear pole of line {65,2092}
X(52931) = barycentric product X(i)*X(j) for these (i, j): {65, 32038}, {931, 6354}, {941, 4566}, {959, 4552}, {1020, 31359}, {1441, 32693}, {4551, 44733}, {4605, 5331}, {34259, 52607}
X(52931) = barycentric quotient X(i)/X(j) for these (i, j): {65, 23880}, {108, 44734}, {931, 7058}, {941, 7253}, {959, 4560}, {1020, 10436}, {1042, 48144}, {1254, 50457}, {1400, 17418}, {1427, 43067}, {2258, 1021}, {4551, 11679}, {4557, 3713}, {4559, 958}, {4566, 34284}, {21859, 3714}, {32038, 314}, {32693, 21}, {34259, 15411}, {44733, 18155}

---

X(52932) = X(110)X(30450)∩X(925)X(32661)

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

See Ivan Pavlov, euclid 5663.

X(52932) lies on the circumconic {{A,B,C,X(3),X(46965)}} and these lines: {96,12359}, {110,30450}, {925,32661}, {8901,42065}

X(52932) = X(i)-isoconjugate-of-X(j) for these {i, j}: {811, 41213}, {1748, 52317}, {1953, 15423}, {2617, 34338}, {2618, 52432}, {3133, 24006}
X(52932) = X(i)-Dao conjugate of X(j) for these {i, j}: {15450, 41222}, {17423, 41213}
X(52932) = X(i) cross conjugate of X(j) for these {i, j}: {6368, 68}
X(52932) = trilinear pole of line {68,577}
X(52932) = barycentric product X(i)*X(j) for these (i, j): {20563, 32692}
X(52932) = barycentric quotient X(i)/X(j) for these (i, j): {54, 15423}, {925, 467}, {2351, 52317}, {2623, 34338}, {3049, 41213}, {14586, 52432}, {15451, 41222}, {32661, 3133}, {32692, 24}, {32734, 14576}, {41271, 6753}

---

X(52933) = X(1302)X(39290)∩X(1304)X(32681)

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

See Ivan Pavlov, euclid 5663.

X(52933) lies on these lines: {1302,39290}, {1304,32681}, {14264,14685}, {15329,44769}, {32738,34568}

X(52933) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1099, 8675}, {2173, 46229}, {10564, 36035}, {30474, 42074}
X(52933) = X(i)-Dao conjugate of X(j) for these {i, j}: {36896, 46229}
X(52933) = X(i) cross conjugate of X(j) for these {i, j}: {8675, 74}, {32738, 32681}
X(52933) = intersection, other than A, B, C, of circumconics {{A,B,C,X(4),X(110)}} and {{A,B,C,X(6),X(14685}}}
X(52933) = trilinear pole of line {74,3003}
X(52933) = barycentric product X(i)*X(j) for these (i, j): {1302, 40384}, {1494, 32681}, {2349, 36083}, {4846, 34568}, {31621, 32738}
X(52933) = barycentric quotient X(i)/X(j) for these (i, j): {74, 46229}, {1302, 36789}, {4846, 52624}, {32640, 10564}, {32681, 30}, {32738, 3163}, {34568, 44134}, {36083, 14206}, {36149, 1099}, {40353, 8675}, {40384, 30474}

---

X(52934) = X(655)X(23703)∩X(1023)X(51562)

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

See Ivan Pavlov, euclid 5663.

X(52934) lies on the circumconic {{A,B,C,X(1),X(100)}} and these lines: {655,23703}, {1023,51562}, {2320,36944}, {10707,52479}, {14191,14204}, {17780,36804}

X(52934) = X(i)-isoconjugate-of-X(j) for these {i, j}: {4791, 52059}, {7113, 23884}, {8648, 36589}, {21758, 27757}, {34544, 43052}
X(52934) = X(i) cross conjugate of X(j) for these {i, j}: {4777, 80}, {24929, 52377}
X(52934) = trilinear pole of line {44,80}
X(52934) = barycentric quotient X(i)/X(j) for these (i, j): {80, 23884}, {655, 36589}, {5549, 4996}, {51562, 27757}

---

X(52935) = ISOGONAL CONJUGATE OF X(4705)

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

See Ivan Pavlov, euclid 5663.

X(52935) lies on these lines: {58,763}, {60,1509}, {69,43700}, {81,3125}, {86,26725}, {99,110}, {100,4596}, {101,4567}, {163,662}, {274,1437}, {643,4614}, {668,4631}, {757,52553}, {1414,4636}, {2106,5006}, {3733,17929}, {4164,4623}, {4579,17934}, {4604,37140}, {4622,24041}, {9275,51356}, {11634,16680}, {13396,36069}, {17103,17104}, {17935,27808}, {17946,40153}, {51369,51420}

X(52935) = isogonal conjugate of X(4705)
X(52935) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 4705}, {2, 4079}, {6, 4024}, {10, 512}, {12, 663}, {25, 4064}, {31, 4036}, {32, 52623}, {37, 661}, {42, 523}, {65, 4041}, {71, 2501}, {75, 50487}, {80, 42666}, {100, 2643}, {101, 115}, {109, 4092}, {110, 21043}, {112, 21046}, {125, 8750}, {181, 522}, {190, 3124}, {201, 18344}, {210, 4017}, {213, 1577}, {226, 3709}, {228, 24006}, {244, 40521}, {291, 4155}, {306, 2489}, {313, 669}, {321, 798}, {335, 46390}, {338, 32739}, {502, 42653}, {513, 756}, {525, 2333}, {594, 649}, {647, 1826}, {650, 2171}, {652, 8736}, {656, 1824}, {657, 6354}, {662, 21833}, {667, 1089}, {692, 1109}, {693, 872}, {762, 1019}, {810, 41013}, {850, 1918}, {882, 4039}, {1015, 4103}, {1018, 3125}, {1020, 36197}, {1084, 1978}, {1126, 6367}, {1220, 42661}, {1245, 48395}, {1252, 21131}, {1254, 3900}, {1268, 8663}, {1331, 8754}, {1334, 7178}, {1365, 3939}, {1400, 3700}, {1402, 4086}, {1427, 4171}, {1459, 7140}, {1783, 3708}, {1880, 8611}, {1897, 20975}, {1919, 28654}, {1924, 27801}, {1960, 4013}, {2054, 18004}, {2161, 2610}, {2170, 21859}, {2197, 3064}, {2200, 14618}, {2205, 20948}, {2321, 7180}, {2616, 21807}, {2623, 21011}, {2970, 32656}, {2971, 4561}, {3005, 18082}, {3063, 6358}, {3120, 4557}, {3121, 4033}, {3122, 3952}, {3224, 21056}, {3261, 7109}, {3572, 4037}, {3668, 4524}, {3676, 7064}, {3690, 7649}, {3701, 51642}, {3733, 6535}, {3747, 35352}, {3835, 6378}, {3903, 21725}, {3949, 6591}, {4062, 9178}, {4080, 14407}, {4082, 7250}, {4083, 7148}, {4105, 6046}, {4117, 6386}, {4130, 7147}, {4163, 7143}, {4455, 43534}, {4515, 7216}, {4516, 4551}, {4559, 21044}, {4568, 51906}, {4570, 8029}, {4572, 7063}, {4600, 22260}, {4602, 52065}, {4605, 14936}, {4628, 39691}, {4674, 4730}, {4838, 28625}, {4931, 28658}, {4988, 52555}, {5029, 6543}, {6057, 43924}, {6187, 6370}, {6538, 50512}, {6541, 18001}, {8013, 50344}, {8061, 18098}, {9292, 21050}, {9396, 21899}, {11599, 17990}, {13476, 21727}, {14404, 41683}, {15523, 18105}, {16606, 21834}, {17411, 34527}, {18070, 21814}, {18785, 24290}, {20970, 31010}, {21051, 23493}, {21699, 50520}, {21720, 39964}, {21813, 48070}, {21816, 47947}, {21823, 27805}, {21839, 23894}, {34294, 46148}, {40433, 50538}, {40515, 52592}, {41267, 52618}, {42027, 50491}, {42471, 50493}, {46772, 50483}, {51645, 52371}
X(52935) = X(i)-Dao conjugate of X(j) for these {i, j}: {2, 4036}, {3, 4705}, {9, 4024}, {11, 4092}, {206, 50487}, {244, 21043}, {661, 21131}, {662, 21098}, {1015, 115}, {1084, 21833}, {1086, 1109}, {3647, 6367}, {5375, 594}, {5521, 8754}, {6376, 52623}, {6505, 4064}, {6626, 1577}, {6631, 1089}, {8054, 2643}, {8287, 21054}, {9296, 28654}, {9428, 27801}, {10001, 6358}, {26932, 125}, {31998, 321}, {32664, 4079}, {34021, 850}, {34467, 20975}, {34591, 21046}, {34961, 210}, {36830, 37}, {39006, 3708}, {39026, 756}, {39029, 4155}, {39042, 18004}, {39052, 1826}, {39054, 10}, {39062, 41013}, {40582, 3700}, {40584, 2610}, {40589, 661}, {40592, 523}, {40596, 1824}, {40602, 4041}, {40605, 4086}, {40612, 6370}, {40617, 1365}, {40618, 20902}, {40619, 338}, {40620, 16732}, {50330, 8029}, {50497, 22260}
X(52935) = X(i) cross conjugate of X(j) for these {i, j}: {58, 4567}, {60, 249}, {110, 4556}, {513, 81}, {593, 4590}, {662, 4610}, {667, 52376}, {757, 24041}, {905, 274}, {1980, 1333}, {4131, 1444}, {4636, 4612}, {5253, 4564}, {31947, 2}
X(52935) = intersection, other than A, B, C, of circumconics {{A,B,C,X(28),X(4226)}}, {{A,B,C,X(56),X(29299)}}, {{A,B,C,X(58),X(101)}}, {{A,B,C,X(81),X(5468)}}, {{A,B,C,X(86),X(664)}}, {{A,B,C,X(99),X(662)}} and {{A,B,C,X(100),X(1929)}}
X(52935) = trilinear pole of line {81,593}
X(52935) = barycentric product X(i)*X(j) for these (i, j): {1, 4610}, {6, 4623}, {7, 4612}, {21, 4573}, {27, 4592}, {28, 4563}, {31, 52612}, {56, 4631}, {58, 799}, {60, 4554}, {75, 4556}, {81, 99}, {85, 4636}, {86, 662}, {100, 1509}, {109, 52379}, {110, 274}, {162, 17206}, {163, 310}, {190, 757}, {239, 36066}, {249, 693}, {250, 15413}, {261, 651}, {284, 4625}, {286, 4558}, {314, 4565}, {320, 37140}, {333, 1414}, {513, 4590}, {514, 24041}, {552, 644}, {593, 668}, {643, 1434}, {645, 1014}, {646, 7341}, {648, 1444}, {649, 24037}, {650, 7340}, {658, 1098}, {664, 2185}, {667, 34537}, {670, 1333}, {691, 16741}, {763, 3952}, {811, 1790}, {827, 16703}, {849, 1978}, {892, 16702}, {905, 18020}, {932, 7304}, {934, 7058}, {1018, 6628}, {1019, 4600}, {1043, 4637}, {1101, 3261}, {1412, 7257}, {1415, 18021}, {1428, 36806}, {1437, 6331}, {1459, 46254}, {1576, 6385}, {1931, 17930}, {1980, 44168}, {2150, 4572}, {2203, 52608}, {2206, 4602}, {2287, 4616}, {2328, 4635}, {2966, 51369}, {3125, 31614}, {3285, 4634}, {3669, 6064}, {3733, 4601}, {3737, 4620}, {4091, 23999}, {4131, 23582}, {4164, 39292}, {4359, 6578}, {4555, 30576}, {4567, 7192}, {4569, 7054}, {4570, 7199}, {4575, 44129}, {4576, 52376}, {4577, 16696}, {4584, 33295}, {4591, 30939}, {4593, 17187}, {4596, 8025}, {4599, 16887}, {4603, 17103}, {4614, 42028}, {4615, 52680}, {4622, 16704}, {4629, 16709}, {4639, 5009}, {5379, 15419}, {6061, 36838}, {6516, 46103}, {6528, 18604}, {6540, 30581}, {6591, 47389}, {6629, 36085}, {7303, 18047}, {7305, 33946}, {7953, 16707}, {13486, 34016}, {16697, 18831}, {16733, 34574}, {17209, 36036}, {17929, 19623}, {17940, 52137}, {18155, 52378}, {18605, 46134}, {18609, 18878}, {20924, 36069}, {23357, 40495}, {24000, 30805}, {30593, 37212}, {30938, 32717}, {32671, 40075}, {36084, 51370}
X(52935) = barycentric quotient X(i)/X(j) for these (i, j): {1, 4024}, {2, 4036}, {6, 4705}, {21, 3700}, {27, 24006}, {28, 2501}, {31, 4079}, {32, 50487}, {36, 2610}, {58, 661}, {59, 21859}, {60, 650}, {63, 4064}, {75, 52623}, {81, 523}, {86, 1577}, {99, 321}, {100, 594}, {101, 756}, {108, 8736}, {109, 2171}, {110, 37}, {112, 1824}, {162, 1826}, {163, 42}, {190, 1089}, {244, 21131}, {249, 100}, {250, 1783}, {261, 4391}, {270, 3064}, {274, 850}, {283, 8611}, {284, 4041}, {286, 14618}, {310, 20948}, {333, 4086}, {404, 21721}, {512, 21833}, {513, 115}, {514, 1109}, {552, 24002}, {593, 513}, {643, 2321}, {644, 6057}, {645, 3701}, {648, 41013}, {649, 2643}, {650, 4092}, {651, 12}, {656, 21046}, {661, 21043}, {662, 10}, {664, 6358}, {667, 3124}, {668, 28654}, {670, 27801}, {692, 1500}, {757, 514}, {763, 7192}, {765, 4103}, {799, 313}, {827, 18098}, {849, 649}, {873, 3261}, {905, 125}, {906, 3690}, {934, 6354}, {1014, 7178}, {1018, 6535}, {1019, 3120}, {1021, 52335}, {1098, 3239}, {1100, 6367}, {1252, 40521}, {1331, 3949}, {1332, 3695}, {1408, 7180}, {1412, 4017}, {1414, 226}, {1415, 181}, {1434, 4077}, {1437, 647}, {1444, 525}, {1459, 3708}, {1461, 1254}, {1509, 693}, {1576, 213}, {1625, 21807}, {1627, 22322}, {1634, 3954}, {1740, 21056}, {1783, 7140}, {1790, 656}, {1812, 52355}, {1813, 201}, {1914, 4155}, {1931, 18004}, {1958, 21050}, {1980, 1084}, {2150, 663}, {2185, 522}, {2189, 18344}, {2194, 3709}, {2203, 2489}, {2206, 798}, {2210, 46390}, {2300, 42661}, {2303, 48395}, {2328, 4171}, {2396, 42703}, {2530, 39691}, {2605, 21824}, {2617, 21011}, {2644, 21089}, {3121, 22260}, {3125, 8029}, {3216, 21720}, {3218, 6370}, {3257, 4013}, {3261, 23994}, {3285, 4730}, {3286, 24290}, {3337, 17422}, {3573, 4037}, {3669, 1365}, {3733, 3125}, {3737, 21044}, {3798, 17876}, {3882, 20653}, {3888, 16886}, {3909, 20654}, {3942, 21134}, {4025, 20902}, {4091, 2632}, {4131, 15526}, {4236, 21956}, {4251, 21727}, {4273, 4770}, {4436, 52579}, {4554, 34388}, {4556, 1}, {4557, 762}, {4558, 72}, {4561, 52369}, {4563, 20336}, {4565, 65}, {4567, 3952}, {4570, 1018}, {4573, 1441}, {4575, 71}, {4579, 21021}, {4584, 43534}, {4590, 668}, {4591, 4674}, {4592, 306}, {4596, 6539}, {4599, 18082}, {4600, 4033}, {4601, 27808}, {4605, 1091}, {4610, 75}, {4611, 4463}, {4612, 8}, {4616, 1446}, {4617, 6046}, {4622, 4080}, {4623, 76}, {4625, 349}, {4631, 3596}, {4636, 9}, {4637, 3668}, {4653, 4931}, {4658, 4838}, {5006, 17989}, {5009, 21832}, {5467, 21839}, {5468, 42713}, {5546, 210}, {6003, 21961}, {6061, 4130}, {6064, 646}, {6335, 7141}, {6385, 44173}, {6516, 26942}, {6578, 1255}, {6591, 8754}, {6614, 7147}, {6628, 7199}, {7045, 4605}, {7054, 3900}, {7058, 4397}, {7113, 42666}, {7192, 16732}, {7199, 21207}, {7252, 4516}, {7254, 18210}, {7257, 30713}, {7259, 4082}, {7304, 20906}, {7340, 4554}, {7341, 3669}, {8025, 30591}, {9218, 21899}, {9426, 52065}, {10411, 42701}, {13397, 41508}, {13486, 8818}, {13588, 21958}, {14419, 1648}, {14574, 2205}, {14838, 21054}, {14966, 5360}, {15413, 339}, {16696, 826}, {16697, 6368}, {16702, 690}, {16703, 23285}, {16716, 12075}, {16725, 41167}, {16732, 23105}, {16733, 52629}, {16735, 47126}, {16741, 35522}, {16751, 21045}, {16755, 17886}, {16756, 9134}, {16947, 51642}, {16948, 14321}, {17136, 42708}, {17167, 2618}, {17187, 8061}, {17206, 14208}, {17924, 2970}, {17929, 11611}, {17940, 9278}, {17943, 20693}, {18020, 6335}, {18108, 34294}, {18166, 48393}, {18180, 12077}, {18206, 4088}, {18601, 21121}, {18604, 520}, {18605, 924}, {18653, 36035}, {18792, 21053}, {19623, 18003}, {20963, 50538}, {20981, 21725}, {21383, 21728}, {21789, 36197}, {21859, 6058}, {22383, 20975}, {23224, 3269}, {23357, 692}, {23889, 4062}, {23995, 32739}, {24037, 1978}, {26856, 42455}, {27644, 21051}, {28606, 23282}, {30234, 6791}, {30576, 900}, {30581, 4977}, {30593, 4978}, {30606, 4768}, {30805, 17879}, {31614, 4601}, {32661, 228}, {32671, 6187}, {32676, 2333}, {32739, 872}, {33628, 4729}, {33955, 27712}, {34071, 7148}, {34537, 6386}, {34594, 40085}, {34948, 47421}, {35327, 21816}, {35342, 8013}, {36059, 2197}, {36066, 335}, {36069, 2161}, {36841, 52345}, {37128, 35352}, {37135, 6543}, {37140, 80}, {37212, 6538}, {37680, 21714}, {38832, 21834}, {39054, 21098}, {40153, 50330}, {40438, 31010}, {40495, 23962}, {40773, 4122}, {41629, 4404}, {42028, 4815}, {46103, 44426}, {47318, 15065}, {47390, 906}, {47443, 5379}, {51311, 4824}, {51369, 2799}, {51420, 1637}, {52378, 4551}, {52379, 35519}, {52394, 18070}, {52558, 47947}, {52612, 561}, {52680, 4120}
X(52935) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {662, 4556, 4612}

---

X(52936) = X(2)X(9483)∩X(689)X(827)

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

See Ivan Pavlov, euclid 5663.

X(52936) lies on these lines: {2,9483}, {83,39691}, {110,41209}, {689,827}, {1501,9497}, {4577,4630}, {52395,52551}

X(52936) = X(i)-isoconjugate-of-X(j) for these {i, j}: {31, 2528}, {38, 3005}, {39, 8061}, {75, 2531}, {141, 2084}, {163, 15449}, {661, 8041}, {688, 1930}, {798, 7794}, {826, 1964}, {1923, 23285}, {2530, 21035}, {3954, 21123}, {15523, 50521}, {16892, 21814}, {41267, 48084}
X(52936) = X(i)-Dao conjugate of X(j) for these {i, j}: {2, 2528}, {115, 15449}, {206, 2531}, {31998, 7794}, {36830, 8041}, {41884, 826}
X(52936) = X(i) cross conjugate of X(j) for these {i, j}: {523, 83}, {33294, 40016}
X(52936) = trilinear pole of line {83,316}
X(52936) = barycentric product X(i)*X(j) for these (i, j): {82, 4593}, {83, 4577}, {99, 52395}, {251, 689}, {308, 827}, {645, 41284}, {1799, 42396}, {3112, 4599}, {4630, 40016}, {16890, 33515}, {18833, 34072}, {37204, 46289}, {42371, 46288}
X(52936) = barycentric quotient X(i)/X(j) for these (i, j): {2, 2528}, {32, 2531}, {82, 8061}, {83, 826}, {110, 8041}, {251, 3005}, {308, 23285}, {523, 15449}, {689, 8024}, {827, 39}, {1799, 2525}, {4563, 4175}, {4573, 41285}, {4577, 141}, {4593, 1930}, {4599, 38}, {4628, 21035}, {4630, 3051}, {34072, 1964}, {40425, 31067}, {41284, 7178}, {42037, 3806}, {42371, 52568}, {42396, 427}, {46288, 688}, {46289, 2084}, {52376, 2530}, {52394, 16892}, {52395, 523}

---

X(52937) = X(85)X(3119)∩X(4625)X(4626)

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

See Ivan Pavlov, euclid 5663.

X(52937) lies on these lines: {85,3119}, {279,34084}, {658,34085}, {934,34083}, {4554,36838}, {4569,30704}, {4625,4626}

X(52937) = X(i)-isoconjugate-of-X(j) for these {i, j}: {31, 4105}, {32, 4130}, {41, 657}, {55, 8641}, {109, 24012}, {220, 3063}, {480, 667}, {560, 4163}, {649, 6602}, {650, 14827}, {663, 1253}, {692, 3022}, {728, 1919}, {1415, 35508}, {1461, 52064}, {1946, 7071}, {1980, 5423}, {2175, 3900}, {2194, 4524}, {3119, 32739}, {3239, 9447}, {4397, 9448}, {6061, 50487}, {8638, 28071}
X(52937) = X(i)-Dao conjugate of X(j) for these {i, j}: {2, 4105}, {11, 24012}, {223, 8641}, {1086, 3022}, {1146, 35508}, {1212, 6607}, {1214, 4524}, {3160, 657}, {5375, 6602}, {6374, 4163}, {6376, 4130}, {6631, 480}, {9296, 728}, {10001, 220}, {17113, 663}, {35508, 52064}, {36905, 52614}, {39053, 7071}, {39060, 7079}, {40593, 3900}, {40615, 14936}, {40619, 3119}, {40624, 24010}
X(52937) = X(i) cross conjugate of X(j) for these {i, j}: {522, 85}, {4569, 46406}
X(52937) = trilinear pole of line {85,142}
X(52937) = barycentric product X(i)*X(j) for these (i, j): {7, 46406}, {75, 36838}, {76, 4626}, {85, 4569}, {279, 4572}, {349, 4616}, {479, 1978}, {561, 4617}, {658, 6063}, {668, 23062}, {738, 6386}, {934, 20567}, {1088, 4554}, {1275, 52621}, {1441, 4635}, {1446, 4625}, {1461, 41283}, {1502, 6614}, {4391, 24011}, {6046, 52612}, {7056, 46404}, {7182, 13149}, {17860, 42388}, {23586, 35519}
X(52937) = barycentric quotient X(i)/X(j) for these (i, j): {2, 4105}, {7, 657}, {57, 8641}, {75, 4130}, {76, 4163}, {85, 3900}, {100, 6602}, {109, 14827}, {142, 6607}, {226, 4524}, {269, 3063}, {279, 663}, {479, 649}, {522, 35508}, {650, 24012}, {651, 1253}, {653, 7071}, {658, 55}, {664, 220}, {668, 728}, {738, 667}, {934, 41}, {1088, 650}, {1275, 3939}, {1434, 21789}, {1441, 4171}, {1446, 4041}, {1461, 2175}, {1847, 18344}, {1978, 5423}, {3261, 4081}, {3668, 3709}, {3676, 14936}, {3900, 52064}, {4077, 36197}, {4391, 24010}, {4554, 200}, {4566, 1334}, {4569, 9}, {4572, 346}, {4573, 2328}, {4605, 7064}, {4610, 6061}, {4616, 284}, {4617, 31}, {4619, 6066}, {4625, 2287}, {4626, 6}, {4635, 21}, {4637, 2194}, {6046, 4079}, {6063, 3239}, {6386, 30693}, {6516, 1802}, {6614, 32}, {7023, 1919}, {7056, 652}, {7147, 50487}, {7177, 1946}, {7196, 4477}, {7197, 2484}, {7205, 4529}, {7339, 32739}, {7366, 1980}, {9436, 52614}, {10481, 10581}, {13149, 33}, {18026, 7079}, {18033, 4148}, {20567, 4397}, {21314, 17425}, {23062, 513}, {23100, 5532}, {23586, 109}, {23989, 23615}, {24002, 2310}, {24011, 651}, {24013, 1415}, {24015, 41339}, {30682, 1459}, {32714, 2212}, {34085, 28071}, {34855, 46388}, {35312, 8012}, {35338, 8551}, {35519, 23970}, {36118, 607}, {36838, 1}, {37780, 14392}, {41283, 52622}, {41292, 21143}, {46135, 6559}, {46404, 7046}, {46406, 8}, {52621, 1146}

---

X(52938) = X(107)X(1305)∩X(653)X(823)

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

See Ivan Pavlov, euclid 5663.

X(52938) lies on these lines: {85,47372}, {92,34591}, {107,1305}, {331,34387}, {653,823}, {664,811}, {1952,40149}, {4552,6335}, {4566,18026}, {14249,40701}, {24032,36118}

X(52938) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3, 1946}, {6, 36054}, {21, 39201}, {41, 4091}, {48, 652}, {55, 23224}, {109, 2638}, {184, 521}, {212, 1459}, {219, 22383}, {228, 23189}, {255, 663}, {283, 810}, {284, 822}, {394, 3063}, {513, 6056}, {520, 2194}, {522, 52430}, {560, 52616}, {577, 650}, {647, 2193}, {649, 2289}, {651, 39687}, {657, 7125}, {667, 1259}, {692, 1364}, {798, 6514}, {905, 52425}, {906, 7117}, {1092, 18344}, {1172, 32320}, {1264, 1980}, {1409, 23090}, {1415, 35072}, {1804, 8641}, {1809, 23220}, {1812, 3049}, {1919, 3719}, {2175, 4131}, {2204, 52613}, {2328, 51641}, {3064, 4100}, {3270, 36059}, {3709, 18604}, {3737, 4055}, {3900, 7335}, {3990, 7252}, {4391, 14585}, {4571, 22096}, {6332, 9247}, {7004, 32656}, {7107, 22382}, {7254, 52370}, {8606, 23226}, {9447, 30805}, {14395, 18877}, {14418, 32659}, {14575, 35518}, {14578, 52307}, {21789, 22341}, {23606, 44426}, {32660, 34591}
X(52938) = X(i)-Dao conjugate of X(j) for these {i, j}: {9, 36054}, {11, 2638}, {223, 23224}, {281, 10397}, {656, 23614}, {1086, 1364}, {1146, 35072}, {1214, 520}, {1249, 652}, {3160, 4091}, {5190, 7117}, {5375, 2289}, {6374, 52616}, {6523, 663}, {6631, 1259}, {9296, 3719}, {10001, 394}, {20620, 3270}, {31998, 6514}, {36103, 1946}, {36908, 51641}, {38991, 39687}, {39026, 6056}, {39052, 2193}, {39053, 3}, {39060, 63}, {39062, 283}, {40590, 822}, {40593, 4131}, {40611, 39201}, {40624, 24031}, {40837, 1459}, {47345, 647}
X(52938) = X(i) cross conjugate of X(j) for these {i, j}: {273, 24032}, {522, 92}, {653, 46404}, {1737, 39294}, {5125, 46102}, {14302, 312}, {14837, 85}, {46110, 331}
X(52938) = polar conjugate of X(652)
X(52938) = trilinear pole of line {92,226}
X(52938) = barycentric product X(i)*X(j) for these (i, j): {4, 46404}, {76, 36127}, {92, 18026}, {107, 349}, {108, 1969}, {109, 18027}, {158, 4554}, {162, 52575}, {225, 6331}, {226, 6528}, {264, 653}, {273, 6335}, {307, 15352}, {318, 13149}, {331, 1897}, {393, 4572}, {664, 2052}, {811, 40149}, {823, 1441}, {1118, 1978}, {1231, 36126}, {1857, 46406}, {4391, 24032}, {6516, 6521}, {7017, 36118}, {18022, 32674}, {23984, 35519}, {44130, 52607}, {46102, 46107}
X(52938) = barycentric quotient X(i)/X(j) for these (i, j): {1, 36054}, {4, 652}, {7, 4091}, {19, 1946}, {27, 23189}, {29, 23090}, {34, 22383}, {57, 23224}, {65, 822}, {73, 32320}, {76, 52616}, {85, 4131}, {92, 521}, {99, 6514}, {100, 2289}, {107, 284}, {108, 48}, {109, 577}, {158, 650}, {162, 2193}, {225, 647}, {226, 520}, {264, 6332}, {273, 905}, {278, 1459}, {307, 52613}, {331, 4025}, {349, 3265}, {393, 663}, {514, 1364}, {522, 35072}, {648, 283}, {650, 2638}, {651, 255}, {653, 3}, {658, 1804}, {663, 39687}, {664, 394}, {668, 3719}, {811, 1812}, {823, 21}, {934, 7125}, {1020, 22341}, {1093, 3064}, {1096, 3063}, {1275, 6517}, {1400, 39201}, {1414, 18604}, {1415, 52430}, {1427, 51641}, {1441, 24018}, {1461, 7335}, {1783, 212}, {1784, 14395}, {1785, 52307}, {1813, 1092}, {1838, 52306}, {1857, 657}, {1877, 22086}, {1880, 810}, {1882, 46382}, {1896, 1021}, {1897, 219}, {1940, 22382}, {1969, 35518}, {1978, 1264}, {2052, 522}, {3064, 3270}, {4391, 24031}, {4551, 3990}, {4552, 3682}, {4554, 326}, {4559, 4055}, {4566, 40152}, {4569, 7183}, {4572, 3926}, {4605, 7066}, {6063, 30805}, {6331, 332}, {6335, 78}, {6516, 6507}, {6520, 18344}, {6521, 44426}, {6528, 333}, {6529, 2299}, {7012, 906}, {7115, 32656}, {7128, 36059}, {7337, 1919}, {7649, 7117}, {7952, 10397}, {8747, 7252}, {8748, 21789}, {8750, 52425}, {13149, 77}, {14249, 14331}, {15352, 29}, {15742, 4587}, {16082, 37628}, {16813, 35196}, {17906, 22072}, {17924, 7004}, {18026, 63}, {18027, 35519}, {18155, 16731}, {21186, 47410}, {23582, 4636}, {23984, 109}, {23999, 4612}, {24019, 2194}, {24032, 651}, {24033, 1415}, {24035, 46974}, {32660, 23606}, {32674, 184}, {32714, 603}, {34591, 23614}, {35519, 23983}, {36059, 4100}, {36110, 14578}, {36118, 222}, {36126, 1172}, {36127, 6}, {36797, 2327}, {37778, 14432}, {37805, 14414}, {38462, 14418}, {40117, 2188}, {40149, 656}, {41013, 8611}, {44130, 15411}, {44426, 34591}, {46102, 1331}, {46107, 26932}, {46110, 2968}, {46152, 4020}, {46404, 69}, {46406, 7055}, {47372, 14298}, {52575, 14208}, {52607, 73}, {52623, 7068}, {52661, 14400}

---

X(52939) = X(95)X(35442)∩X(110)X(41208)

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

See Ivan Pavlov, euclid 5663.

X(52939) lies on the circumconic {A,B,C,X(97),X(110)} and these lines: {95,35442}, {110,41208}, {15958,18831}, {18315,42405}

X(52939) = X(i)-isoconjugate-of-X(j) for these {i, j}: {19, 34983}, {163, 24862}, {217, 2618}, {810, 36412}, {1087, 3049}, {1953, 15451}, {2179, 6368}, {2181, 17434}, {24006, 46394}, {24019, 41212}, {32676, 39019}
X(52939) = X(i)-Dao conjugate of X(j) for these {i, j}: {6, 34983}, {115, 24862}, {15526, 39019}, {35071, 41212}, {39062, 36412}
X(52939) = X(i) cross conjugate of X(j) for these {i, j}: {525, 95}
X(52939) = trilinear pole of line {95,140}
X(52939) = barycentric product X(i)*X(j) for these (i, j): {95, 18831}, {97, 42405}, {276, 18315}, {933, 34384}, {16813, 34386}
X(52939) = barycentric quotient X(i)/X(j) for these (i, j): {3, 34983}, {54, 15451}, {95, 6368}, {97, 17434}, {275, 12077}, {276, 18314}, {520, 41212}, {523, 24862}, {525, 39019}, {648, 36412}, {811, 1087}, {933, 51}, {6331, 45793}, {8795, 23290}, {8884, 51513}, {14533, 42293}, {14586, 217}, {15423, 41222}, {15958, 418}, {16813, 53}, {18315, 216}, {18831, 5}, {32661, 46394}, {35311, 3078}, {35360, 23607}, {40440, 2618}, {42405, 324}, {43768, 14391}, {46089, 39201}, {46724, 34979}, {46966, 52153}

---

X(52940) = IOGONAL CONJUGATE OF X(21906

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

See Ivan Pavlov, euclid 5663.

X(52940) lies on the Kiepert circumhyperbola, the ccircumconics {{A,B,C,X(2),X(4)}}, {{A,B,C,X(6),X(5108)}}, {{A,B,C,X(99),X(34760)}} and these lines: {2,4590}, {4,18020}, {10,4600}, {76,5108}, {83,41939}, {94,18023}, {98,30786}, {99,1649}, {111,1916}, {226,4620}, {249,598}, {321,4601}, {671,1641}, {691,9150}, {892,5466}, {2394,2396}, {2996,52450}, {4049,4615}, {4444,36085}, {4563,6035}, {5485,47389}, {5503,52141}, {8371,31998}, {9168,33799}, {11053,42370}, {11123,37880}, {11606,31125}, {17948,38239}, {23342,23348}, {32740,34087}, {34245,43674}, {36827,41209}, {42008,43535}, {43187,43665}

X(52940) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 21906}, {31, 1648}, {115, 922}, {163, 33919}, {187, 2643}, {249, 45775}, {351, 661}, {512, 2642}, {560, 52628}, {690, 798}, {810, 14273}, {896, 3124}, {923, 23992}, {1084, 14210}, {1101, 42344}, {1109, 14567}, {1924, 35522}, {2084, 22105}, {2159, 2682}, {3121, 4062}, {3122, 21839}, {3266, 4117}, {3708, 44102}, {4079, 14419}, {4750, 50487}, {14443, 36142}, {22260, 23889}, {23099, 24039}
X(52940) = X(i)-Dao conjugate of X(j) for these {i, j}: {2, 1648}, {3, 21906}, {115, 33919}, {523, 42344}, {524, 14444}, {1648, 46049}, {2482, 23992}, {3163, 2682}, {5976, 51429}, {6374, 52628}, {9428, 35522}, {15477, 1084}, {15899, 3124}, {23992, 14443}, {31998, 690}, {36830, 351}, {39054, 2642}, {39061, 115}, {39062, 14273}
X(52940) = X(i) cross conjugate of X(j) for these {i, j}: {316, 670}, {524, 99}, {597, 39296}, {671, 892}, {858, 4609}, {1641, 9170}, {5468, 31614}, {8352, 35179}, {9225, 110}, {10754, 2966}, {11053, 2}, {11054, 35138}, {11161, 46144}, {11646, 18829}, {22254, 6035}, {32740, 691}, {35279, 9150}, {37803, 6331}, {38239, 5468}, {42344, 523}, {47286, 648}, {48540, 18878}
X(52940) = trilinear pole of line {99,523}
X(52940) = barycentric product X(i)*X(j) for these (i, j): {99, 892}, {111, 34537}, {115, 42370}, {249, 18023}, {670, 691}, {671, 4590}, {689, 36827}, {799, 36085}, {850, 45773}, {897, 24037}, {3266, 34539}, {4602, 36142}, {4609, 32729}, {4623, 5380}, {5466, 31614}, {6035, 50941}, {9170, 34760}, {14588, 14728}, {17983, 47389}, {18020, 30786}, {24041, 46277}, {32740, 44168}
X(52940) = barycentric quotient X(i)/X(j) for these (i, j): {2, 1648}, {6, 21906}, {30, 2682}, {76, 52628}, {99, 690}, {110, 351}, {111, 3124}, {115, 42344}, {249, 187}, {250, 44102}, {316, 5099}, {325, 51429}, {523, 33919}, {524, 23992}, {648, 14273}, {662, 2642}, {670, 35522}, {671, 115}, {690, 14443}, {691, 512}, {892, 523}, {895, 20975}, {897, 2643}, {1101, 922}, {1641, 41176}, {1649, 46049}, {2482, 14444}, {2643, 45775}, {2966, 52038}, {4563, 14417}, {4567, 21839}, {4576, 14424}, {4577, 22105}, {4590, 524}, {4600, 4062}, {4601, 42713}, {4610, 4750}, {5380, 4705}, {5466, 8029}, {5468, 1649}, {5968, 44114}, {6035, 50942}, {6064, 3712}, {6189, 46462}, {6190, 46463}, {7340, 7181}, {8371, 14423}, {8753, 2971}, {9154, 51441}, {9170, 34763}, {9178, 22260}, {9182, 33921}, {10411, 44814}, {11634, 21905}, {14588, 33906}, {14609, 52625}, {14728, 42345}, {16077, 52475}, {16092, 51428}, {16093, 16278}, {17941, 11183}, {17983, 8754}, {18020, 468}, {18023, 338}, {19626, 9427}, {22151, 47415}, {23348, 9171}, {23357, 14567}, {24037, 14210}, {24041, 896}, {30786, 125}, {31125, 39691}, {31614, 5468}, {31632, 11053}, {32583, 17414}, {32729, 669}, {32740, 1084}, {34537, 3266}, {34539, 111}, {34574, 9178}, {34760, 8371}, {35138, 23287}, {35139, 51479}, {36085, 661}, {36142, 798}, {36307, 30452}, {36310, 30453}, {36827, 3005}, {42008, 8288}, {42370, 4590}, {43926, 8034}, {44372, 46464}, {45773, 110}, {46111, 2970}, {46277, 1109}, {47389, 6390}, {47390, 23200}, {50941, 1640}, {51478, 14270}, {51541, 20382}, {52141, 6791}, {52608, 45807}, {52632, 23105}

---

X(52941) = X(59)X(5012)∩X(765)X(3681)

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

See Ivan Pavlov, euclid 5663.

XX(52941) lies on these lines: {2,43979}, {59,5012}, {675,39293}, {765,3681}, {1252,3730}, {4184,4570}, {4600,33297}, {13589,36086}

X(52941) = X(i)-isoconjugate-of-X(j) for these {i, j}: {244, 3006}, {513, 23887}, {674, 1111}, {2225, 23989}, {4858, 43039}, {6545, 42723}, {14964, 16732}, {34387, 51659}
X(52941) = X(i)-Dao conjugate of X(j) for these {i, j}: {39026, 23887}
X(52941) = X(i) cross conjugate of X(j) for these {i, j}: {674, 101}
X(52941) = intersection, other than A, B, C, of circumconics {{A,B,C,X(4),X(103)}} and {{A,B,C,,X(6),X(35365}}
X(52941) = trilinear pole of line {101,6586}
X(52941) = barycentric product X(i)*X(j) for these (i, j): {100, 36087}, {190, 32682}, {675, 1252}, {765, 2224}, {1110, 37130}, {23990, 43093
X(52941) = barycentric quotient X(i)/X(j) for these (i, j): {675, 23989}, {1252, 3006}, {2224, 1111}, {23990, 674}, {36087, 693}

---

X(52942) = EULER LINE INTERCEPT OF X(148)X(1992)

Barycentrics    11 a^4-2 a^2 (b^2+c^2)+16 b^2 c^2-7 (c^4 +b^4) : :

See Kadir Altintas and Angel Montesdeoca, euclid 5685.

X(52942) lies on these lines: {2,3}, {32,36523}, {148,1992}, {385,7620}, {543,7774}, {598,2549}, {671,7737}, {3849,11185}, {5475,32479}, {5485,44367}, {6781,7617}, {7615,17008}, {7622,43457}, {7735,41135}, {7736,32480}, {7763,36521}, {7775,15300}, {7840,23334}, {7898,21356}, {8591,9770}, {8593,20423}, {8594,22492}, {8595,22491}, {8860,20112}, {9166,37809}, {9741,20094}, {9753,9880}, {10722,11161}, {10734,11162}, {11164,22110}, {12154,36969}, {12155,36970}, {18800,39809}, {20065,34505}, {26613,43620}, {32817,41136}, {34504,39590}, {43451,52022}, {43452,52021}

X(52942) = reflection of X(i) in X(j), for these {i, j}: {2,11317}, {376,37348}, {8591,35705}, {33008,33016}, {33207,44543}, {35955,3363}

---

X(52943) = EULER LINE INTERCEPT OF X(315)X(15300)

Barycentrics    25 a^4-16 a^2 (b^2+c^2)+20 b^2 c^2-11 (c^4+b^4) : :

See Kadir Altintas and Angel Montesdeoca, euclid 5685.

X(52943) lies on these lines: {2,3}, {315,15300}, {8596,9740}, {9830,11160}, {11054,47102}, {11184,11742}, {14907,32479}, {43619,51224}

---

X(52944) = EULER LINE INTERCEPT OF X(9830)X(35369)

Barycentrics    47 a^4-20 a^2 (b^2+c^2)+52 b^2 c^2-25 (c^4+b^4) : :

See Kadir Altintas and Angel Montesdeoca, euclid 5685.

X(52944) lies on these lines: {2,3}, {9830,35369}

---

X(52945) = X(5)X(53)∩X(6)X(382)

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

See Ivan Pavlov, euclid 5663.

X(52945) lies on the circumconic {{A,B,C,X(5),X(30)}} and these lines: {2,36430}, {4,5158}, {5,53}, {6,382}, {20,393}, {23,6103}, {30,1990}, {50,3018}, {97,46924}, {112,9380}, {115,3003}, {187,16310}, {230,37897}, {231,1989}, {232,858}, {237,8754}, {297,15526}, {340,40885}, {381,52703}, {401,23583}, {548,22052}, {566,1506}, {570,9698}, {631,10979}, {648,40853}, {1249,33703}, {1273,14570}, {1494,44577}, {1609,9714}, {1637,18558}, {1865,37447}, {2081,2600}, {2549,33871}, {2963,18367}, {2965,40136}, {3087,17578}, {3129,8737}, {3130,8738}, {3146,40138}, {3260,51389}, {3331,15063}, {3526,36751}, {3530,36422}, {3627,6749}, {3853,6748}, {5007,7553}, {5063,7756}, {5169,22240}, {5206,46257}, {5309,18534}, {5355,33872}, {5368,13338}, {5702,15682}, {5899,52166}, {6128,31726}, {7517,7755}, {7519,10311}, {7753,31723}, {8749,52403}, {9220,18573}, {9407,51431}, {9530,35474}, {10296,52418}, {10313,20063}, {11060,13556}, {11648,34288}, {13202,15816}, {14129,34836}, {14537,14836}, {14981,45921}, {15109,35498}, {15166,20408}, {15167,20409}, {15696,36748}, {15905,17800}, {16303,40135}, {16318,37899}, {16328,44961}, {18400,38943}, {18591,37401}, {22257,41523}, {23607,32078}, {23712,37975}, {23713,37974}, {23714,32460}, {23715,32461}, {30716,37760}, {31724,41335}, {33630,49138}, {36413,36431}, {37900,41358}, {37973,44533}, {38292,49134}, {40477,44575}, {40665,41023}, {40666,41022}, {40996,45312}, {47144,47335}, {47383,52247}, {47405,51403}, {49135,52707}

X(52945) = midpoint of X(i) and X(j) for these {i,j}: {648, 40853}, {38943, 38944}
X(52945) = reflection of X(i) in X(j) for these {i,j}: {401, 23583}, {3163, 18487}, {3284, 1990}, {15526, 297}
X(52945) = X(i)-isoconjugate-of-X(j) for these {i, j}: {54, 2349}, {74, 2167}, {92, 46090}, {95, 2159}, {97, 36119}, {275, 35200}, {1494, 2148}, {2169, 16080}, {2190, 14919}, {2394, 36134}, {2616, 44769}, {15412, 36034}, {18877, 40440}
X(52945) = X(i)-Dao conjugate of X(j) for these {i, j}: {5, 14919}, {30, 43768}, {133, 275}, {137, 2394}, {216, 1494}, {1511, 97}, {3163, 95}, {3258, 15412}, {14363, 16080}, {15450, 14380}, {22391, 46090}, {39019, 34767}, {40588, 74}
X(52945) = barycentric product X(i)*X(j) for these (i, j): {4, 1568}, {5, 30}, {51, 3260}, {53, 11064}, {216, 46106}, {311, 1495}, {324, 3284}, {343, 1990}, {648, 14391}, {1154, 14254}, {1263, 10272}, {1273, 14583}, {1625, 41079}, {1637, 14570}, {1784, 44706}, {1953, 14206}, {2173, 14213}, {2179, 46234}, {2407, 12077}, {2420, 18314}, {2617, 36035}, {4240, 6368}, {5562, 52661}, {6110, 44713}, {6111, 44714}, {8800, 51425}, {9033, 35360}, {9214, 41586}, {13450, 51394}, {14581, 28706}, {17500, 51360}, {18653, 21011}, {27353, 46817}, {33529, 36299}, {33530, 36298}, {34334, 44715}, {35912, 39569}, {36412, 43768}, {40449, 51392}, {41078, 41392}
X(52945) = barycentric quotient X(i)/X(j) for these (i, j): {5, 1494}, {30, 95}, {51, 74}, {53, 16080}, {184, 46090}, {216, 14919}, {217, 18877}, {1495, 54}, {1568, 69}, {1625, 44769}, {1637, 15412}, {1784, 40440}, {1953, 2349}, {1990, 275}, {2173, 2167}, {2179, 2159}, {2181, 36119}, {2420, 18315}, {3163, 43768}, {3199, 8749}, {3260, 34384}, {3284, 97}, {4240, 18831}, {6368, 34767}, {7069, 44693}, {9406, 2148}, {9409, 23286}, {11064, 34386}, {12077, 2394}, {14213, 33805}, {14254, 46138}, {14391, 525}, {14398, 2623}, {14581, 8882}, {14583, 1141}, {15451, 14380}, {18487, 4993}, {23347, 933}, {34334, 43752}, {35360, 16077}, {36298, 51268}, {36299, 51275}, {40981, 40352}, {41077, 15414}, {41221, 12079}, {41586, 36890}, {46106, 276}, {51403, 19166}, {51513, 18808}, {52604, 1304}, {52661, 8795}
{X(52945) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {30, 18487, 3163}, {30, 1990, 3284}, {53, 216, 36412}, {53, 42459, 216}, {216, 36412, 233}, {401, 37765, 23583}, {1989, 11063, 231}, {1990, 3284, 3163}, {3018, 47275, 6781}, {3284, 18487, 1990}, {16964, 16965, 13403}, {38943, 38944, 18400}, {52670, 52671, 6750}

---

X(52946) = X(11)X(650)∩X(44)X(516)

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

See Ivan Pavlov, euclid 5663.

X(52946) lies on the circumconic {{A,B,C,X(11),X(528)}} and these lines: {11,650}, {44,516}, {45,14358}, {115,35125}, {294,20119}, {514,1086}, {522,1146}, {528,35113}, {1252,20095}, {1323,17067}, {1635,6075}, {1639,51442}, {1731,2245}, {1738,43065}, {2170,3328}, {2246,3322}, {2310,4041}, {2325,5199}, {3125,23752}, {3259,4893}, {3826,5701}, {4076,27546}, {4530,14393}, {4762,35094}, {5853,6603}, {14300,33647}, {15612,38963}, {15734,34522}, {40540,40865}

X(52946) = midpoint of X(52334) and X(52338)
X(52946) = reflection of X(i) in X(j) for these {i,j}: {1323, 17067}, {2325, 5199}, {40865, 40540}
X(52946) = X(i)-isoconjugate-of-X(j) for these {i, j}: {59, 37131}, {840, 4564}, {2149, 18821}
X(52946) = X(i)-Dao conjugate of X(j) for these {i, j}: {650, 18821}, {6615, 37131}, {35113, 4998}
X(52946) = barycentric product X(i)*X(j) for these (i, j): {11, 528}, {666, 14393}, {1146, 5723}, {1643, 4391}, {2246, 4858}, {4530, 46790}, {42763, 43728}
X(52946) = barycentric quotient X(i)/X(j) for these (i, j): {11, 18821}, {528, 4998}, {1643, 651}, {2170, 37131}, {2246, 4564}, {3271, 840}, {4530, 46791}, {5723, 1275}, {14393, 918}, {14411, 2284}
X(52946) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {52334, 52338, 52305}

---

X(52947) = X(19)X(2207)∩X(661)X(663)

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

See Ivan Pavlov, euclid 5663.

X(52947) lies on the circumconic {{A,B,C,X(19),X(534)}} and these lines:{19,2207}, {661,663}, {910,14581}, {1104,3199}, {1990,5179}

X(52947) = X(i)-isoconjugate-of-X(j) for these {i, j}: {69, 38883}
X(52947) = barycentric product X(19)*X(534)
X(52947) = barycentric quotient X(i)/X(j) for these (i, j): {534, 304}, {1973, 38883}
X(52947) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2207, 36103, 16583}

---

X(52948) = X(5)X(6)∩X(20)X(1249)

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

See Ivan Pavlov, euclid 5663.

X(52948) lies on the circumconic {{A,B,C,X(20),X(30)}} and these lines: {2,5702}, {3,40138}, {5,6}, {20,1249}, {30,1990}, {44,15252}, {50,16303}, {53,3853}, {69,20204}, {115,34569}, {140,5158}, {216,3530}, {230,37911}, {231,44911}, {232,37897}, {340,44216}, {382,393}, {441,648}, {524,23583}, {546,6749}, {548,577}, {549,52703}, {858,13573}, {1172,37447}, {1989,23323}, {1992,52251}, {2173,6357}, {2407,51937}, {2420,15774}, {2452,51611}, {3087,3843}, {3526,15851}, {3534,36427}, {3628,15860}, {3832,40065}, {3856,36412}, {3861,6748}, {4658,52260}, {5007,9825}, {5065,9607}, {5159,6103}, {5304,16317}, {5306,6677}, {5309,44920}, {5523,47339}, {6329,14767}, {6510,40535}, {6587,8057}, {6793,11064}, {7129,9643}, {7493,45141}, {8749,10257}, {10313,37899}, {10314,33881}, {11063,15646}, {11799,52418}, {12101,36430}, {13473,34570}, {14836,18570}, {15048,49669}, {15063,15341}, {15526,44335}, {16328,22249}, {17800,33636}, {18365,47322}, {20207,37669}, {33630,33703}, {34152,52166}, {36748,46853}, {36751,44682}, {39358,44578}, {40856,52229}, {41358,47315}, {44961,47144}, {47162,47309}

X(52948) = midpoint of X(i) and X(j) for these {i,j}: {441, 648}, {1990, 3284}
X(52948) = reflection of X(i) in X(j) for these {i,j}: {15526, 44335}, {44334, 23583}
X(52948) = complement of X(40996)
X(52948) = X(i)-isoconjugate-of-X(j) for these {i, j}: {64, 2349}, {74, 2184}, {253, 2159}, {459, 35200}, {1073, 36119}, {1494, 2155}, {8749, 19611}, {8809, 15627}, {16080, 19614}, {33581, 33805}
X(52948) = X(i)-Dao conjugate of X(j) for these {i, j}: {4, 16080}, {122, 2394}, {133, 459}, {1511, 1073}, {3163, 253}, {39020, 34767}, {45245, 1494}, {45248, 14919}
X(52948) = barycentric product X(i)*X(j) for these (i, j): {20, 30}, {154, 3260}, {610, 14206}, {648, 14345}, {1249, 11064}, {1495, 14615}, {1568, 38808}, {1637, 36841}, {1990, 37669}, {2173, 18750}, {2407, 6587}, {3284, 15466}, {4240, 8057}, {5930, 51382}, {6357, 27382}, {7359, 18623}, {8804, 18653}, {10152, 16163}, {11589, 52578}, {14249, 51394}, {15291, 36789}, {15905, 46106}, {35602, 52661}, {35912, 44704}, {42459, 43768}, {51420, 52345}, {51656, 52346}
X(52948) = barycentric quotient X(i)/X(j) for these (i, j): {20, 1494}, {30, 253}, {154, 74}, {204, 36119}, {610, 2349}, {1249, 16080}, {1495, 64}, {1990, 459}, {2173, 2184}, {2407, 44326}, {2420, 46639}, {3079, 10152}, {3081, 38956}, {3172, 8749}, {3260, 41530}, {3284, 1073}, {6587, 2394}, {7070, 44693}, {8057, 34767}, {9406, 2155}, {9407, 33581}, {11064, 34403}, {11589, 52559}, {14345, 525}, {14581, 41489}, {15291, 40384}, {15905, 14919}, {18750, 33805}, {23347, 1301}, {41077, 14638}, {42658, 14380}, {44705, 18808}, {46106, 52581}, {51382, 5931}, {51394, 15394}, {51656, 8809}
X(52948) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {524, 23583, 44334}, {1249, 15905, 42459}, {1249, 36413, 15905}, {1990, 3284, 30}, {3163, 3284, 1990}, {36413, 45245, 1249}

---

X(52949) = X(21)X(270)∩X(284)X(501)

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

See Ivan Pavlov, euclid 5663.

X(52949) lies on the circumconic {{A,B,C,X(21),X(30)}} and these lines: {6,6985}, {21,270}, {30,1990}, {37,35192}, {44,2341}, {81,1443}, {112,2694}, {163,2182}, {284,501}, {448,648}, {521,650}, {662,6510}, {1576,44670}, {2150,2264}, {2173,9406}, {3109,8756}, {4565,43044}, {5089,7469}, {5777,15796}, {6357,18653}, {6708,46103}, {6749,37406}, {6869,40138}, {7359,51382}, {8609,19622}, {15526,44336}, {15945,44095}, {37405,41502}

X(52949) = midpoint of X(448) and X(64)
X(52949) = reflection of X(15526) in X(44336)
X(52949) = X(i)-isoconjugate-of-X(j) for these {i, j}: {65, 2349}, {73, 16080}, {74, 226}, {109, 2394}, {225, 14919}, {307, 8749}, {349, 40352}, {653, 14380}, {664, 2433}, {1214, 36119}, {1400, 1494}, {1402, 33805}, {1427, 44693}, {1441, 2159}, {1813, 18808}, {3668, 15627}, {7265, 36064}, {12079, 52378}, {18097, 46147}, {32674, 34767}, {35200, 40149}
X(52949) = X(i)-Dao conjugate of X(j) for these {i, j}: {11, 2394}, {133, 40149}, {1511, 1214}, {3163, 1441}, {6739, 321}, {35072, 34767}, {39025, 2433}, {40582, 1494}, {40602, 2349}, {40605, 33805}
X(52949) = barycentric product X(i)*X(j) for these (i, j): {1, 51382}, {8, 51420}, {9, 18653}, {21, 30}, {81, 7359}, {283, 1784}, {284, 14206}, {314, 1495}, {333, 2173}, {521, 4240}, {643, 11125}, {645, 14399}, {648, 14395}, {650, 2407}, {652, 24001}, {662, 14400}, {1043, 51656}, {1172, 11064}, {1637, 4612}, {1812, 1990}, {1896, 51394}, {2193, 46106}, {2194, 3260}, {2287, 6357}, {2420, 4391}, {3284, 31623}, {4631, 14398}, {4636, 36035}, {6739, 52380}, {7252, 42716}, {9406, 28660}, {9407, 40072}, {11604, 16164}, {23347, 35518}
X(52949) = barycentric quotient X(i)/X(j) for these (i, j): {21, 1494}, {30, 1441}, {284, 2349}, {333, 33805}, {521, 34767}, {650, 2394}, {1172, 16080}, {1495, 65}, {1946, 14380}, {1990, 40149}, {2173, 226}, {2193, 14919}, {2194, 74}, {2204, 8749}, {2299, 36119}, {2328, 44693}, {2407, 4554}, {2420, 651}, {3063, 2433}, {3284, 1214}, {4240, 18026}, {4516, 12079}, {6357, 1446}, {7359, 321}, {9406, 1400}, {9407, 1402}, {11064, 1231}, {11125, 4077}, {14206, 349}, {14395, 525}, {14399, 7178}, {14400, 1577}, {14581, 1880}, {18344, 18808}, {18653, 85}, {23347, 108}, {24001, 46404}, {46106, 52575}, {51382, 75}, {51394, 52385}, {51420, 7}, {51656, 3668}
X(52949) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7054, 40582, 40937}

X(52950) = X(5)X(5007)∩X(6)X(4550)

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

See Ivan Pavlov, euclid 5663.

X(52950) lies on the circumconic {{A,B,C,X(22),X(30)}} and these lines: {5,5007}, {6,4550}, {22,8743}, {30,1990}, {32,6644}, {39,18570}, {112,2071}, {115,23323}, {187,15646}, {230,44911}, {232,2070}, {325,6720}, {524,44340}, {538,40856}, {648,15013}, {754,23583}, {1184,3291}, {1568,6793}, {1576,2386}, {1625,8779}, {1968,22120}, {2072,6103}, {2207,18534}, {2420,51394}, {2485,8673}, {3172,21312}, {5039,9813}, {5158,7514}, {5305,44920}, {5332,37697}, {7296,37696}, {7526,7772}, {7574,41358}, {7739,49669}, {7765,52070}, {8744,10313}, {9300,52262}, {9408,10564}, {10298,39575}, {10312,13595}, {11416,41363}, {12225,41366}, {14537,44263}, {14580,37980}, {14929,20204}, {14965,51437}, {15526,44337}, {15860,49671}, {18860,38608}, {21177,44102}, {22121,35452}, {34569,44468}, {35007,37814}, {39371,48453}

X(52950) = midpoint of X(i) and X(j) for these {i,j}: {648, 15013}, {3284, 14581}
X(52950) = reflection of X(15526) in X(44337)
X(52950) = X(i)-isoconjugate-of-X(j) for these {i, j}: {66, 2349}, {1494, 2156}, {2159, 18018}, {2353, 33805}, {14376, 36119}, {35200, 43678}, {40352, 46244}
X(52950) = X(i)-Dao conjugate of X(j) for these {i, j}: {32, 74}, {127, 2394}, {133, 43678}, {1511, 14376}, {3163, 18018}
X(52950) = barycentric product X(i)*X(j) for these (i, j): {22, 30}, {206, 3260}, {315, 1495}, {648, 14396}, {1637, 4611}, {1760, 2173}, {1990, 20806}, {2172, 14206}, {2407, 2485}, {2420, 33294}, {3284, 17907}, {4123, 51656}, {4240, 8673}, {4456, 18653}, {4463, 51420}, {8743, 11064}, {9406, 20641}, {9407, 40073}, {10316, 46106}, {11605, 16165}, {11610, 51389}, {14581, 34254}, {17453, 46234}, {51394, 52448}
X(52950) = barycentric quotient X(i)/X(j) for these (i, j): {22, 1494}, {30, 18018}, {206, 74}, {1495, 66}, {1760, 33805}, {1990, 43678}, {2172, 2349}, {2420, 44766}, {2485, 2394}, {3260, 40421}, {3284, 14376}, {4548, 15627}, {8673, 34767}, {8743, 16080}, {9406, 2156}, {9407, 2353}, {10316, 14919}, {14206, 46244}, {14396, 525}, {14581, 13854}, {17409, 8749}, {17453, 2159}, {20968, 40352}, {22075, 18877}, {23208, 46147}, {23347, 1289}
X(52950) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {112, 52058, 14961}, {3284, 14581, 30}

---

X(52951) = X(3)X(112)∩X(5)X(6103)

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

See Ivan Pavlov, euclid 5663.

X(52951) lies on the circumconic {{A,B,C,X(23),X(30)}} and these lines: {3,112}, {5,6103}, {6,5663}, {23,8744}, {30,1990}, {32,2493}, {111,21309}, {113,6793}, {115,546}, {187,16308}, {232,7575}, {648,40856}, {1511,2420}, {1560,5159}, {1625,5609}, {1637,51430}, {1989,11818}, {1995,44467}, {2079,22331}, {2207,7530}, {2492,6593}, {2548,44529}, {2794,18121}, {3018,18907}, {3269,51522}, {3627,27376}, {5158,47228}, {5306,46030}, {5523,18323}, {6794,7728}, {7464,52058}, {7514,52166}, {7545,10312}, {7753,39504}, {7755,44235}, {8749,9818}, {9300,44236}, {9826,40354}, {10297,16318}, {10313,37924}, {10766,48679}, {13195,14675}, {13861,44533}, {14094,22146}, {14254,35906}, {14961,37950}, {15526,44338}, {17409,44260}, {18449,41363}, {18563,41366}, {18572,41358}, {20410,46340}, {22121,35001}, {38632,39849}, {40138,49669}, {40996,44340}

X(52951) = midpoint of X(648) and X(40856)
X(52951) = reflection of X(15526) in X(44338)
X(52951) = X(i)-isoconjugate-of-X(j) for these {i, j}: {67, 2349}, {1494, 2157}, {2159, 18019}, {3455, 33805}, {34897, 36119}, {35200, 46105}, {37221, 46147}
X(52951) = X(i)-Dao conjugate of X(j) for these {i, j}: {133, 46105}, {187, 36890}, {1511, 34897}, {3163, 18019}, {5099, 2394}, {40583, 1494}
X(52951) = barycentric product X(i)*X(j) for these (i, j): {4, 16165}, {23, 30}, {316, 1495}, {1511, 52449}, {1637, 52630}, {1990, 22151}, {2173, 16568}, {2407, 2492}, {2420, 9979}, {3260, 18374}, {3284, 37765}, {4240, 9517}, {5642, 14246}, {6593, 9214}, {8744, 11064}, {9406, 20944}, {9407, 40074}, {10317, 46106}, {12824, 15454}, {14581, 37804}
X(52951) = barycentric quotient X(i)/X(j) for these (i, j): {23, 1494}, {30, 18019}, {1495, 67}, {1990, 46105}, {2420, 17708}, {2492, 2394}, {3284, 34897}, {6593, 36890}, {8744, 16080}, {9406, 2157}, {9407, 3455}, {9517, 34767}, {10317, 14919}, {14581, 8791}, {16165, 69}, {16568, 33805}, {18374, 74}, {23347, 935}, {42659, 14380}, {52142, 9139}

---

X(52952) = X(6)X(64)∩X(24)X(571)

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

See Ivan Pavlov, euclid 5663.

X(52952) lies on the circumconic {{A,B,C,X(24),X(30)}} and these lines: {6,64}, {24,571}, {30,1990}, {50,232}, {53,7747}, {112,3003}, {187,11062}, {231,403}, {570,3520}, {577,8746}, {648,44328}, {924,6753}, {1576,34854}, {1595,5007}, {5063,8743}, {5158,7526}, {5305,6748}, {6103,37981}, {8882,34484}, {11063,37954}, {15262,46429}, {15526,44339}, {37777,38861}, {40135,47228}, {44102,52604}

X(52952) = midpoint of X(648) and X(44328)
X(52952) = reflection of X(15526) in X(44339)
X(52952) = X(i)-isoconjugate-of-X(j) for these {i, j}: {68, 2349}, {91, 14919}, {1494, 1820}, {2159, 20563}, {2351, 33805}, {5392, 35200}, {18877, 20571}, {34767, 36145}, {36119, 52350}
X(52952) = X(i)-Dao conjugate of X(j) for these {i, j}: {133, 5392}, {135, 2394}, {1511, 52350}, {3163, 20563}, {34116, 14919}, {39013, 34767}
X(52952) = barycentric product X(i)*X(j) for these (i, j): {4, 51393}, {24, 30}, {47, 1784}, {317, 1495}, {571, 46106}, {648, 14397}, {924, 4240}, {1147, 52661}, {1511, 52415}, {1637, 41679}, {1748, 2173}, {1990, 1993}, {2407, 6753}, {3260, 44077}, {3284, 11547}, {6563, 23347}, {7763, 14581}, {8745, 11064}, {14254, 52416}, {14576, 43768}, {15454, 52000}, {18883, 39176}, {34756, 51425}
X(52952) = barycentric quotient X(i)/X(j) for these (i, j): {24, 1494}, {30, 20563}, {571, 14919}, {924, 34767}, {1495, 68}, {1748, 33805}, {1784, 20571}, {1990, 5392}, {3284, 52350}, {4240, 46134}, {6753, 2394}, {8745, 16080}, {9406, 1820}, {9407, 2351}, {14397, 525}, {14581, 2165}, {23347, 925}, {34952, 14380}, {39176, 37802}, {44077, 74}, {51393, 69}, {52436, 18877}
X(52952) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {112, 52418, 3003}, {571, 8745, 14576}, {1990, 39176, 3284}, {3284, 14581, 1990}

---

X(52953) = X(26)X(8746)∩X(30)X(1990)

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

See Ivan Pavlov, euclid 5663.

X(52953) lies on the circumconic {{A,B,C,X(26),X(30)}} and these lines: {6,34783}, {26,8746}, {30,1990}, {232,9380}, {577,8743}, {648,44329}, {2965,45735}, {5007,7403}, {5158,7503}, {7745,36412}, {15526,44341}

X(52953) = midpoint of X(648) and X(44329)
X(52953) = reflection of X(15526) in X(44341)
X(52953) = X(i)-isoconjugate-of-X(j) for these {i, j}: {70, 2349}, {1494, 2158}, {2159, 20564}
X(52953) = X(i)-Dao conjugate of X(j) for these {i, j}: {3163, 20564}, {52120, 2394}
X(52953) = barycentric product X(i)*X(j) for these (i, j): {26, 30}, {1495, 44128}, {3260, 44078}, {8746, 11064}
X(52953) = barycentric quotient X(i)/X(j) for these (i, j): {26, 1494}, {30, 20564}, {1495, 70}, {8746, 16080}, {9406, 2158}, {23347, 1288}, {44078, 74}

---

X(52954) = X(4)X(3017)∩X(27)X(58)

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

See Ivan Pavlov, euclid 5663.

X(52954) lies on the circumconic {{A,B,C,X(27),X(30)}} and these lines: {4,3017}, {27,58}, {28,52374}, {29,4658}, {30,1990}, {112,1886}, {162,1785}, {242,514}, {264,48866}, {286,24208}, {297,540}, {445,49744}, {447,519}, {458,48867}, {551,1982}, {811,6381}, {1099,1784}, {1249,48837}, {1354,6357}, {1877,26743}, {2074,23710}, {2906,3194}, {4240,52753}, {4653,41083}, {9308,48863}, {15526,44342}, {17168,44709}, {17200,44129}, {17555,48839}, {17907,48835}, {25986,50226}, {37448,48868}, {40138,48857}, {48834,52283}

X(52954) = midpoint of X(447) and X(648)
X(52954) = reflection of X(15526) in X(44342)
X(52954) = X(i)-isoconjugate-of-X(j) for these {i, j}: {10, 35200}, {37, 14919}, {71, 2349}, {72, 74}, {73, 44693}, {100, 14380}, {228, 1494}, {306, 2159}, {321, 18877}, {692, 34767}, {906, 2394}, {1214, 15627}, {1332, 2433}, {2200, 33805}, {3682, 36119}, {3990, 16080}, {3998, 8749}, {4064, 36034}, {11079, 42701}, {20336, 40352}
X(52954) = X(i)-Dao conjugate of X(j) for these {i, j}: {133, 10}, {1086, 34767}, {1511, 3682}, {3163, 306}, {3258, 4064}, {5190, 2394}, {6739, 3710}, {8054, 14380}, {40589, 14919}
X(52954) = X(i) cross conjugate of X(j) for these {i, j}: {51656, 51420}
X(52954) = barycentric product X(i)*X(j) for these (i, j): {4, 18653}, {27, 30}, {28, 14206}, {29, 6357}, {58, 46106}, {81, 1784}, {86, 1990}, {92, 51420}, {278, 51382}, {286, 2173}, {310, 14581}, {513, 24001}, {514, 4240}, {648, 11125}, {811, 14399}, {1474, 3260}, {1495, 44129}, {1790, 52661}, {2203, 46234}, {2407, 7649}, {2420, 46107}, {3261, 23347}, {8747, 11064}, {31623, 51656}, {37168, 52753}
X(52954) = barycentric quotient X(i)/X(j) for these (i, j): {27, 1494}, {28, 2349}, {30, 306}, {58, 14919}, {286, 33805}, {1172, 44693}, {1333, 35200}, {1474, 74}, {1495, 71}, {1637, 4064}, {1784, 321}, {1990, 10}, {2173, 72}, {2203, 2159}, {2206, 18877}, {2299, 15627}, {2407, 4561}, {2420, 1331}, {3260, 40071}, {3284, 3682}, {4240, 190}, {5317, 36119}, {6357, 307}, {7359, 3710}, {7649, 2394}, {8747, 16080}, {9406, 228}, {9407, 2200}, {11064, 52396}, {11125, 525}, {14206, 20336}, {14399, 656}, {14400, 52355}, {14581, 42}, {18653, 69}, {23347, 101}, {24001, 668}, {35201, 42701}, {46106, 313}, {51382, 345}, {51420, 63}, {51656, 1214}
X(52954) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {447, 648, 519}, {8747, 44698, 58}

---

X(52955) = X(28)X(1104)∩X(44)X(162)

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

See Ivan Pavlov, euclid 5663.

X(52955) lies on these lines: {28,1104}, {30,1990}, {37,11107}, {44,162}, {112,2687}, {513,1430}, {536,648}, {1100,2326}, {1172,10308}, {1474,52372}, {1875,34079}, {1982,50064}, {2173,42074}, {2630,24000}, {3194,4273}, {3290,37963}, {4240,52754}, {15526,44343}

X(52955) = midpoint of X(648) and X(44330)
X(52955) = reflection of X(15526) in X(44343)
X(52955) = X(i)-isoconjugate-of-X(j) for these {i, j}: {10, 14919}, {71, 1494}, {72, 2349}, {74, 306}, {101, 34767}, {190, 14380}, {228, 33805}, {307, 15627}, {313, 18877}, {321, 35200}, {1214, 44693}, {1331, 2394}, {2159, 20336}, {2433, 4561}, {3682, 16080}, {3998, 36119}, {4064, 44769}, {8749, 52396}, {40071, 40352}
X(52955) = X(i)-Dao conjugate of X(j) for these {i, j}: {133, 321}, {1015, 34767}, {1511, 3998}, {3163, 20336}, {5521, 2394}
X(52955) = barycentric product X(i)*X(j) for these (i, j): {4, 51420}, {19, 18653}, {27, 2173}, {28, 30}, {29, 51656}, {34, 51382}, {58, 1784}, {81, 1990}, {162, 11125}, {274, 14581}, {286, 1495}, {513, 4240}, {648, 14399}, {649, 24001}, {693, 23347}, {1172, 6357}, {1333, 46106}, {1396, 7359}, {1437, 52661}, {1474, 14206}, {2203, 3260}, {2407, 6591}, {2420, 17924}, {5317, 11064}, {9406, 44129}, {18609, 51965}, {42716, 43925}
X(52955) = barycentric quotient X(i)/X(j) for these (i, j): {27, 33805}, {28, 1494}, {30, 20336}, {513, 34767}, {667, 14380}, {1333, 14919}, {1474, 2349}, {1495, 72}, {1784, 313}, {1990, 321}, {2173, 306}, {2203, 74}, {2204, 15627}, {2206, 35200}, {2299, 44693}, {2420, 1332}, {3284, 3998}, {4240, 668}, {5317, 16080}, {6357, 1231}, {6591, 2394}, {9406, 71}, {9407, 228}, {11125, 14208}, {14206, 40071}, {14399, 525}, {14581, 37}, {18653, 304}, {23347, 100}, {24001, 1978}, {39176, 42701}, {46106, 27801}, {51382, 3718}, {51420, 69}, {51656, 307}

---

X(52956) = X(27)X(553)∩X(29)X(284)

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

See Ivan Pavlov, euclid 5663.

X(52956) lies on the circumconic {{A,B,C,X(29),X(30)}} and these lines: {27,553}, {29,284}, {30,1990}, {79,1172}, {112,2695}, {243,522}, {527,648}, {1784,2173}, {1785,2341}, {1826,41502}, {2074,8756}, {2322,4877}, {2325,36797}, {6062,7359}, {15526,44344}, {31900,46884}

X(52956) = midpoint of X(648) and X(44331)
X(52956) = reflection of X(15526) in X(44344)
X(52956) = X(i)-isoconjugate-of-X(j) for these {i, j}: {65, 14919}, {73, 2349}, {74, 1214}, {226, 35200}, {307, 2159}, {651, 14380}, {1231, 40352}, {1409, 1494}, {1415, 34767}, {1439, 15627}, {1441, 18877}, {2394, 36059}, {2433, 6516}, {8749, 52385}, {16080, 22341}, {36119, 40152}, {44693, 52373}
X(52956) = X(i)-Dao conjugate of X(j) for these {i, j}: {133, 226}, {1146, 34767}, {1511, 40152}, {3163, 307}, {6739, 306}, {20620, 2394}, {38991, 14380}, {40602, 14919}
X(52956) = barycentric product X(i)*X(j) for these (i, j): {4, 51382}, {21, 1784}, {27, 7359}, {29, 30}, {281, 18653}, {283, 52661}, {284, 46106}, {318, 51420}, {333, 1990}, {522, 4240}, {648, 14400}, {650, 24001}, {823, 14395}, {1172, 14206}, {1495, 44130}, {2173, 31623}, {2204, 46234}, {2299, 3260}, {2322, 6357}, {2407, 3064}, {2420, 46110}, {8748, 11064}, {11125, 36797}, {14581, 28660}, {23347, 35519}
X(52956) = barycentric quotient X(i)/X(j) for these (i, j): {29, 1494}, {30, 307}, {284, 14919}, {522, 34767}, {663, 14380}, {1172, 2349}, {1495, 73}, {1784, 1441}, {1990, 226}, {2173, 1214}, {2194, 35200}, {2204, 2159}, {2299, 74}, {2332, 15627}, {2420, 1813}, {3064, 2394}, {3284, 40152}, {4183, 44693}, {4240, 664}, {7359, 306}, {8748, 16080}, {9406, 1409}, {11064, 52565}, {11125, 17094}, {14206, 1231}, {14395, 24018}, {14399, 51640}, {14400, 525}, {14581, 1400}, {18653, 348}, {23347, 109}, {24001, 4554}, {31623, 33805}, {46106, 349}, {51382, 69}, {51420, 77}, {51656, 1439}
X(52956) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {648, 44331, 527}

---

X(52957) = X(31)X(1501)∩X(32)X(7113)

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

See Ivan Pavlov, euclid 5663.

X(52957) lies on the circumconic {{A,B,C,X(31),X(752)}} and these lines: {31,1501}, {32,7113}, {213,9459}, {667,788}, {716,4586}, {2225,14567}, {16514,28860}, {20666,35069}

X(52957) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2, 43097}, {76, 753}, {693, 5386}
X(52957) = X(i)-Dao conjugate of X(j) for these {i, j}: {752, 30874}, {32664, 43097}
X(52957) = barycentric product X(i)*X(j) for these (i, j): {1, 8626}, {6, 2243}, {31, 752}, {101, 14438}, {560, 35548}, {604, 4070}, {692, 4809}, {1333, 4144}, {1501, 30874}, {4586, 14402}, {33904, 34069}
X(52957) = barycentric quotient X(i)/X(j) for these (i, j): {31, 43097}, {560, 753}, {752, 561}, {2243, 76}, {4070, 28659}, {4144, 27801}, {4809, 40495}, {8626, 75}, {14402, 824}, {14438, 3261}, {30874, 40362}, {32739, 5386}, {33904, 30870}, {35548, 1928}
X(52957) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1501, 32664, 16584}

---

X(52958) = X(32)X(206)∩X(669)X(688)

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

See Ivan Pavlov, euclid 5663.

X(52958) lies on the circumconic {{A,B,C,X(32),X(688)}} and these lines: {32,206}, {50,11672}, {669,688}, {702,4577}, {1084,14602}

X(52958) = midpoint of X(4577) and X(16985)
X(52958) = X(i)-isoconjugate-of-X(j) for these {i, j}: {75, 43098}, {561, 755}, {3261, 5389}
X(52958) = X(i)-Dao conjugate of X(j) for these {i, j}: {206, 43098}, {40368, 755}
X(52958) = barycentric product X(i)*X(j) for these (i, j): {6, 8627}, {31, 2244}, {32, 754}, {110, 14428}, {1397, 4157}, {1501, 35549}, {1576, 14420}, {2175, 7214}, {2206, 4156}, {3049, 46543}, {4577, 14403}, {4630, 33907}, {14567, 52758}
X(52958) = barycentric quotient X(i)/X(j) for these (i, j): {32, 43098}, {754, 1502}, {1501, 755}, {2244, 561}, {4157, 40363}, {7214, 41283}, {8627, 76}, {14403, 826}, {14420, 44173}, {14428, 850}, {35549, 40362}
X(52958) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {206, 9233, 8265}, {9407, 14567, 33875}

---

X(52959) = X(8)X(39)∩X(10)X(37)

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

See Ivan Pavlov, euclid 5663.

X(52959) lies on these lines: {1,1574}, {8,39}, {9,31433}, {10,37}, {32,5687}, {76,1278}, {100,187}, {115,17757}, {200,9620}, {213,3214}, {292,50016}, {314,44418}, {316,6653}, {350,27076}, {512,661}, {519,1015}, {536,6381}, {538,668}, {574,956}, {672,49984}, {730,20671}, {758,20693}, {762,2292}, {899,3230}, {958,31451}, {978,4050}, {1016,9264}, {1018,2238}, {1107,3626}, {1145,6184}, {1194,33091}, {1196,10327}, {1376,2242}, {1377,31471}, {1449,17750}, {1506,24390}, {1573,2276}, {1739,3726}, {1743,3501}, {1757,5184}, {1783,14581}, {1914,48696}, {2229,19998}, {2241,3913}, {2275,3632}, {2277,4007}, {2295,3293}, {2345,4263}, {2548,5082}, {2549,3421}, {2550,31409}, {2886,31476}, {2975,37512}, {3125,3930}, {3208,6048}, {3244,16604}, {3434,5475}, {3436,7748}, {3617,5283}, {3625,17448}, {3644,6376}, {3661,4850}, {3693,49758}, {3701,7230}, {3721,3987}, {3730,4271}, {3752,29594}, {3767,7080}, {3780,16549}, {3783,14839}, {3896,21827}, {3912,6547}, {3934,17143}, {3954,4642}, {3970,21951}, {3992,4037}, {3997,21904}, {4015,21879}, {4046,16587}, {4111,21035}, {4403,30806}, {4426,8715}, {4441,9466}, {4478,16696}, {4526,28603}, {4568,35101}, {4595,40859}, {4647,21021}, {4651,21838}, {4685,21877}, {4726,20888}, {4727,8610}, {4751,27255}, {4807,14991}, {4847,31398}, {4853,9619}, {4882,9593}, {4915,9592}, {4919,5529}, {5153,16884}, {5231,31441}, {5380,17964}, {5552,7746}, {5844,34460}, {6683,26801}, {7603,11680}, {7737,17784}, {7845,20553}, {9331,19875}, {9708,31477}, {9710,31478}, {10527,31455}, {11681,39565}, {12782,49448}, {13006,41684}, {14537,49719}, {16602,29600}, {16969,17749}, {17053,17299}, {17144,27091}, {17160,19973}, {17316,31198}, {17388,46838}, {17497,42720}, {18591,21855}, {20040,28245}, {20055,24598}, {20255,40006}, {20331,45751}, {20544,51411}, {21226,32450}, {21232,50025}, {21814,22184}, {21859,40663}, {24036,50014}, {24625,40891}, {24944,32089}, {25264,25280}, {26363,31501}, {26757,26978}, {28604,50163}, {29571,31197}, {31402,31416}, {31426,31442}, {31448,31456}, {31459,31482}, {31460,31488}, {31461,31490}, {31462,31491}, {32025,50271}, {33853,38462}, {33908,41142}, {36283,50995}, {36920,43039}, {39590,52367}, {40773,51353}, {46032,49490}

X(52959) = midpoint of X(i) and X(j) for these {i,j}: {668, 17759}, {20693, 21888}
X(52959) = reflection of X(i) in X(j) for these {i,j}: {350, 27076}, {1015, 1575}, {20688, 21830}, {21830, 21897}
X(52959) = X(i)-isoconjugate-of-X(j) for these {i, j}: {58, 3227}, {81, 37129}, {86, 739}, {593, 41683}, {662, 43928}, {799, 23349}, {898, 1019}, {1333, 31002}, {1412, 36798}, {3733, 4607}, {4556, 35353}, {7192, 34075}, {7199, 32718}
X(52959) = X(i)-Dao conjugate of X(j) for these {i, j}: {10, 3227}, {37, 31002}, {1084, 43928}, {13466, 274}, {14434, 16726}, {38986, 23892}, {38996, 23349}, {39011, 7192}, {40586, 37129}, {40599, 36798}, {40600, 739}, {40614, 86}
X(52959) = intersection, other than A, B, C, of circumconics {{A,B,C,X(10),X(661)}} and {{A,B,C,X(37),X(512)}}
X(52959) = barycentric product X(i)*X(j) for these (i, j): {1, 3994}, {10, 899}, {37, 536}, {42, 6381}, {65, 4009}, {100, 14431}, {210, 43037}, {213, 35543}, {321, 3230}, {512, 41314}, {523, 23343}, {661, 23891}, {668, 14404}, {890, 27808}, {891, 3952}, {1018, 4728}, {3768, 4033}, {3930, 36816}, {4526, 4552}, {4551, 14430}, {17757, 45145}, {21805, 52755}, {41683, 42083}
X(52959) = barycentric quotient X(i)/X(j) for these (i, j): {10, 31002}, {37, 3227}, {42, 37129}, {210, 36798}, {213, 739}, {512, 43928}, {536, 274}, {669, 23349}, {756, 41683}, {798, 23892}, {890, 3733}, {891, 7192}, {899, 86}, {1018, 4607}, {1646, 16726}, {3230, 81}, {3768, 1019}, {3952, 889}, {3994, 75}, {4009, 314}, {4465, 30940}, {4526, 4560}, {4557, 898}, {4705, 35353}, {4728, 7199}, {6381, 310}, {14426, 17217}, {14430, 18155}, {14431, 693}, {19945, 17205}, {21805, 36872}, {21839, 52757}, {23891, 799}, {35543, 6385}, {41314, 670}, {52626, 16727}
X(52959) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 17756, 16975}, {10, 1500, 16589}, {10, 20691, 1500}, {10, 21070, 21025}, {100, 5291, 187}, {519, 1575, 1015}, {594, 21858, 2092}, {668, 17759, 538}, {740, 21830, 20688}, {740, 21897, 21830}, {1018, 31855, 2238}, {2276, 3679, 1573}, {2295, 3293, 20970}, {2321, 21857, 21796}, {3930, 4695, 3125}, {3950, 21892, 21826}, {3987, 4006, 3721}, {4515, 21896, 16583}, {16975, 17756, 39}, {17143, 26752, 3934}, {17757, 21956, 115}, {20691, 21868, 10}, {20693, 21888, 758}

---

X(52960) = X(38)X(8041)∩X(39)X(4553)

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

See Ivan Pavlov, euclid 5663.

X(52960) lies on the circumconic {{A,B,C,X(38),X(537)}} and these lines: {38,8041}, {39,4553}, {141,4568}, {537,35123}, {2084,2530}, {3836,21331}, {4562,40857}, {17245,21208}

X(52960) = midpoint of X(4562) and X(40857)
X(52960) = X(i)-isoconjugate-of-X(j) for these {i, j}: {83, 2382}, {251, 18822}, {36081, 52226}
X(52960) = X(i)-Dao conjugate of X(j) for these {i, j}: {35123, 3112}, {40585, 18822}
X(52960) = barycentric product X(i)*X(j) for these (i, j): {38, 537}, {141, 20331}, {4553, 36848}, {4562, 14405}, {4568, 52745}
X(52960) = barycentric quotient X(i)/X(j) for these (i, j): {38, 18822}, {537, 3112}, {1964, 2382}, {14405, 812}, {20331, 83}, {46387, 52226}, {52745, 10566}
X(52960) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8041, 40585, 16587}

---

X(52961) = X(39)X(141)∩X(69)X(8265)

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

See Ivan Pavlov, euclid 5663.

X(52961) lies on the circumconic {{A,B,C,X(39),X(538)}} and these lines: {39,141}, {69,8265}, {160,5206}, {193,46948}, {325,3291}, {352,46298}, {385,32526}, {524,1084}, {538,30736}, {599,3117}, {670,702}, {688,3005}, {1502,20081}, {1634,8623}, {3003,5181}, {3231,6786}, {3314,9465}, {3629,6375}, {3631,45210}, {3815,52704}, {3978,36950}, {4969,6377}, {5106,5201}, {6378,7277}, {7777,39576}, {15302,16986}, {15533,34811}, {16776,46313}, {46154,46161}

X(52961) = midpoint of X(670) and X(4085)
X(52961) = reflection of X(i) in X(j) for these {i,j}: {1084, 3229}, {3978, 36950}
X(52961) = X(i)-isoconjugate-of-X(j) for these {i, j}: {82, 3228}, {83, 37132}, {729, 3112}, {18070, 32717}, {34087, 46289}, {43763, 51510}
X(52961) = X(i)-Dao conjugate of X(j) for these {i, j}: {39, 34087}, {141, 3228}, {34452, 729}, {35073, 308}, {36213, 51510}, {38998, 83}, {52042, 46156}
X(52961) = barycentric product X(i)*X(j) for these (i, j): {38, 2234}, {39, 538}, {141, 3231}, {670, 14406}, {826, 5118}, {888, 4576}, {1634, 9148}, {3005, 23342}, {3051, 30736}, {3933, 46522}, {6786, 20021}, {7813, 14609}, {8024, 33875}, {21814, 30938}, {35073, 46156}, {45672, 46154}
X(52961) = barycentric quotient X(i)/X(j) for these (i, j): {39, 3228}, {141, 34087}, {538, 308}, {887, 18105}, {1634, 9150}, {1645, 51906}, {1964, 37132}, {2234, 3112}, {3051, 729}, {3231, 83}, {4576, 886}, {5118, 4577}, {6786, 20022}, {8623, 51510}, {9148, 52618}, {23342, 689}, {30736, 40016}, {33875, 251}, {46522, 32085}, {52625, 34294}
X(52961) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {524, 3229, 1084}, {670, 40858, 702}, {1634, 38303, 8623}

---

X(52962) = X(6)X(47299)∩X(40)X(40943)

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

See Ivan Pavlov, euclid 5663.

X(52962) lies on the circumconic {{A,B,C,X(40),X(28194)}} and these lines: {6,47299}, {40,40943}, {6129,10397}, {6610,19297}

X(52962) = X(i)-isoconjugate-of-X(j) for these {i, j}: {189, 28193}
X(52962) = barycentric product X(40)*X(28194)
X(52962) = barycentric quotient X(i)/X(j) for these (i, j): {2187, 28193}, {28194, 309}

---

X(52963) = X(32)X(220)∩X(37)X(758)

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

See Ivan Pavlov, euclid 5663.

X(52963) lies on these lines: {6,6767}, {9,1572}, {32,220}, {36,8649}, {37,758}, {39,995}, {42,213}, {44,519}, {45,4274}, {56,9351}, {71,21796}, {101,187}, {115,17747}, {190,538}, {218,2241}, {238,14839}, {511,51921}, {512,798}, {517,49758}, {574,42316}, {595,5007}, {625,41324}, {644,5291}, {672,1015}, {760,49692}, {813,9264}, {902,1017}, {1018,2238}, {1023,52680}, {1191,7772}, {1266,27637}, {1400,21826}, {1464,4559}, {1574,3501}, {1575,49992}, {1914,5526}, {1918,7064}, {1960,14436}, {2256,5042}, {2276,5313}, {2292,21802}, {2295,3294}, {2549,41325}, {2802,50014}, {3199,41320}, {3207,5206}, {3930,21839}, {3991,21874}, {4253,16969}, {4389,48844}, {4403,49777}, {4419,48840}, {4422,30109}, {4424,46907}, {4641,29574}, {4713,9466}, {4754,29383}, {4887,28362}, {5008,21793}, {5030,9259},{5127,5170}, {5165,16672}, {6184,8624}, {7748,17732}, {8750,14581}, {9620,16970}, {10027,33908}, {16583,21872}, {16611,21888}, {17332,25432}, {17354,48860}, {17750,26102}, {20073,48838}, {20331,49997}, {20703,21816}, {21008,24047}, {21805,21821}, {21879,28594}, {22191,22294}, {24045,39565}, {24254,49516}, {24326,46899}, {25349,30106}, {32450,34063}, {33889,33948}, {35102,36226}, {41416,51817}

X(52963) = midpoint of X(190) and X(40859)
X(52963) = reflection of X(i) in X(j) for these {i,j}: {4403, 49777}, {30109, 4422}
X(52963) = X(i)-isoconjugate-of-X(j) for these {i, j}: {58, 20568}, {86, 88}, {106, 274}, {310, 9456}, {513, 4615}, {649, 4634}, {662, 6548}, {757, 4080}, {763, 4013}, {799, 23345}, {1014, 4997}, {1019, 4555}, {1444, 6336}, {1509, 4674}, {4567, 6549}, {4573, 23838}, {4582, 7203}, {4601, 43922}, {4653, 40833}, {5376, 17205}, {6635, 8042}, {9268, 16727}, {14616, 40215}, {17206, 36125}, {24624, 52553}, {27922, 37128}, {32665, 52619}, {36058, 44129}
X(52963) = X(i)-Dao conjugate of X(j) for these {i, j}: {10, 20568}, {214, 274}, {1084, 6548}, {3943, 40089}, {4370, 310}, {5375, 4634}, {20619, 44129}, {35092, 52619}, {38979, 7199}, {38996, 23345}, {39026, 4615}, {40600, 88}, {40607, 4080}, {40627, 6549}
X(52963) = intersection, other than A, B, C, of circumconics {{A,B,C,X(37),X(4908)}}, {{A,B,C,X(42),X(512)}}, {{A,B,C,X(44),X(213)}} and {{A,B,C,X(65),X(49702)}}
X(52963) = barycentric product X(i)*X(j) for these (i, j): {1, 21805}, {6, 3943}, {10, 902}, {31, 3992}, {37, 44}, {42, 519}, {55, 40663}, {65, 3689}, {71, 8756}, {88, 21821}, {100, 4730}, {101, 4120}, {190, 14407}, {210, 1319}, {213, 4358}, {214, 34857}, {228, 38462}, {313, 9459}, {321, 2251}, {512, 17780}, {523, 23344}, {594, 3285}, {649, 4169}, {661, 1023}, {678, 4674}, {756, 52680}, {758, 40172}, {762, 30576}, {798, 24004}, {872, 30939}, {900, 4557}, {909, 21942}, {1017, 4080}, {1018, 1635}, {1020, 14427}, {1334, 3911}, {1400, 2325}, {1402, 4723}, {1404, 2321}, {1500, 16704}, {1639, 4559}, {1824, 5440}, {1826, 22356}, {1877, 2318}, {1911, 4783}, {1918, 3264}, {1960, 3952}, {2161, 40988}, {2200, 46109}, {2333, 3977}, {2334, 4819}, {2429, 14321}, {3690, 37168}, {3724, 51975}, {3939, 30572}, {4041, 23703}, {4145, 40522}, {4551, 4895}, {4727, 28625}, {4908, 28658}, {4969, 52555}, {7180, 30731}, {8750, 14429}, {14439, 18785}, {17747, 45144}, {20970, 31011}, {23202, 41013}, {36944, 51377}, {37790, 52370}
X(52963) = barycentric quotient X(i)/X(j) for these (i, j): {37, 20568}, {42, 903}, {44, 274}, {100, 4634}, {213, 88}, {512, 6548}, {519, 310}, {669, 23345}, {678, 30939}, {692, 4622}, {798, 1022}, {872, 4674}, {900, 52619}, {902, 86}, {1017, 16704}, {1023, 799}, {1334, 4997}, {1404, 1434}, {1500, 4080}, {1635, 7199}, {1918, 106}, {1960, 7192}, {2087, 16727}, {2200, 1797}, {2205, 9456}, {2251, 81}, {2325, 28660}, {2333, 6336}, {3122, 6549}, {3285, 1509}, {3689, 314}, {3724, 52553}, {3747, 27922}, {3943, 76}, {3992, 561}, {4079, 4049}, {4120, 3261}, {4169, 1978}, {4358, 6385}, {4557, 4555}, {4723, 40072}, {4783, 18891}, {4895, 18155}, {4969, 52572}, {8756, 44129}, {9459, 58}, {14436, 4481}, {14439, 18157}, {17780, 670}, {20972, 16711}, {21805, 75}, {21821, 4358}, {22086, 15419}, {22356, 17206}, {23202, 1444}, {23344, 99}, {23703, 4625}, {24004, 4602}, {28658, 40833}, {30572, 52621}, {32739, 4591}, {40172, 14616}, {40663, 6063}, {40988, 20924}, {41267, 46150}, {52680, 873}
X(52963) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {101, 17735, 187}, {190, 40859, 538}, {213, 1334, 1500}, {213, 1500, 20970}, {220, 14974, 32}, {672, 3230, 1015}, {2176, 3730, 39}, {2295, 3294, 16589}, {7109, 40586, 21838}, {21008, 24047, 37512}

---

X(52964) = X(37)X(517)∩X(44)X(519)

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

See Ivan Pavlov, euclid 5663.

X(52964) lies on these lines: {37,517}, {43,2176}, {44,519}, {190,10027}, {536,40859}, {644,1914}, {728,39248}, {978,36647}, {1018,1575}, {1023,2251}, {1100,3997}, {1107,1334}, {1149,20331}, {1266,28362}, {1279,14839}, {1573,16814}, {2238,49984}, {2295,3720}, {2802,49758}, {3290,21888}, {3501,16604}, {3730,17448}, {3809,4689}, {3880,50014}, {4029,4274}, {4083,14408}, {4119,21711}, {4422,49774}, {4426,4513}, {4520,21879}, {4641,17389}, {4721,29699}, {4752,40091}, {5165,48858}, {9351,25440}, {16610,21885}, {17160,27637}, {20363,39258}, {21342,21872}, {21886,49448}, {30109,41310}, {31393,36404}, {35652,41232}, {39260,48855}

X(52964) = midpoint of X(190) and X(10027)
X(52964) = reflection of X(i) in X(j) for these {i,j}: {21331, 37}, {49774, 4422}
X(52964) = X(i)-isoconjugate-of-X(j) for these {i, j}: {87, 88}, {106, 330}, {4598, 23345}, {5383, 43922}, {6336, 23086}, {6384, 9456}, {6548, 34071}, {7121, 20568}
X(52964) = X(i)-Dao conjugate of X(j) for these {i, j}: {214, 330}, {798, 43922}, {4370, 6384}, {40598, 20568}, {40610, 6548}, {52659, 7209}
X(52964) = barycentric product X(i)*X(j) for these (i, j): {43, 519}, {44, 192}, {190, 14408}, {902, 6376}, {1023, 3835}, {1319, 27538}, {1403, 4723}, {1404, 4110}, {1423, 2325}, {1635, 4595}, {1960, 36863}, {2176, 4358}, {2209, 3264}, {2251, 6382}, {3208, 3911}, {3212, 3689}, {3943, 27644}, {3971, 52680}, {3992, 38832}, {4083, 17780}, {4147, 23703}, {4169, 18197}, {4432, 41531}, {4759, 40780}, {8756, 22370}, {14407, 36860}, {16704, 20691}, {20760, 38462}, {20906, 23344}, {20979, 24004}, {21805, 33296}, {30731, 43051}
X(52964) = barycentric quotient X(i)/X(j) for these (i, j): {43, 903}, {44, 330}, {192, 20568}, {519, 6384}, {902, 87}, {1023, 4598}, {1404, 7153}, {1960, 43931}, {2176, 88}, {2209, 106}, {2251, 2162}, {2325, 27424}, {3123, 6549}, {3208, 4997}, {3689, 7155}, {3911, 7209}, {4083, 6548}, {4358, 6383}, {8640, 23345}, {9459, 7121}, {17780, 18830}, {20691, 4080}, {20979, 1022}, {21805, 42027}, {21834, 4049}, {23202, 23086}, {23344, 932}, {38986, 43922}
X(52964) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 517, 21331}, {1018, 3230, 1575}, {2176, 3208, 20691}, {3501, 16969, 16604}

---

X(52965) = X(44)X(214)∩X(45)X(101)

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

See Ivan Pavlov, euclid 5663.

X(52965) lies on these lines: {1,8297}, {44,214}, {45,101}, {88,27950}, {678,1635}, {1015,16666}, {3207,36283}, {3809,30644}, {4597,40860}, {4715,25398}, {16672,21781}, {17461,21782}, {21886,51570}, {35092,50843}

X(52965) = midpoint of X(4597) and X(40860)
X(52965) = X(i)-isoconjugate-of-X(j) for these {i, j}: {106, 35170}
X(52965) = X(i)-Dao conjugate of X(j) for these {i, j}: {214, 35170}, {35124, 20568}
X(52965) = barycentric product X(i)*X(j) for these (i, j): {44, 4715}, {902, 49780}, {4597, 14409}, {14422, 17780}
X(52965) = barycentric quotient X(i)/X(j) for these (i, j): {44, 35170}, {2251, 28317}, {4715, 20568}, {14409, 4777}, {14422, 6548}

---

X(52966) = X(1)X(39)∩X(44)X(2802)

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

See Ivan Pavlov, euclid 5663.

X(52966) lies on the circumconic {{A,B,C,X(45),X(545)}} and these lines: {1,39}, {44,2802}, {45,4752}, {214,21885}, {545,6633}, {1017,5541}, {4513,36283}, {4770,4775}

X(52966) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2163, 35168}, {2384, 39704}
X(52966) = X(i)-Dao conjugate of X(j) for these {i, j}: {35121, 20569}, {40587, 35168}
X(52966) = barycentric product X(i)*X(j) for these (i, j): {45, 545}, {1644, 4792}, {3711, 43038}, {4555, 14410}, {4671, 8649}, {4752, 14475}, {4767, 14421}, {4893, 6633}, {4908, 51908}
X(52966) = barycentric quotient X(i)/X(j) for these (i, j): {45, 35168}, {545, 20569}, {8649, 89}, {14410, 900}, {14421, 52620}, {51908, 40833}
X(52966) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4752, 4792, 45}

---

X(52967) = X(5)X(53)∩X(32)X(206)

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

See Ivan Pavlov, euclid 5663.

X(52967) lies on the circumconic {{A,B,C,X(5),X(217)}} and these lines: {2,46104}, {5,53}, {23,23584}, {30,33874}, {32,206}, {39,5480}, {51,40588}, {132,232}, {217,27374}, {237,2211}, {381,34096}, {446,39073}, {511,11672}, {864,42068}, {1084,1692}, {1506,3613}, {3117,41266}, {3164,40822}, {5103,34990}, {5661,29012}, {13862,22240}, {14600,19558}, {14853,43718}, {14912,51880}, {15451,42293}, {32223,45215}, {35071,38624}, {35319,41586}, {35325,44890}, {37123,51324}

X(52967) = X(i)-isoconjugate-of-X(j) for these {i, j}: {54, 46273}, {95, 1821}, {275, 336}, {276, 293}, {287, 40440}, {290, 2167}, {1910, 34384}, {2148, 18024}, {2616, 43187}, {3404, 41488}, {15412, 36036}, {34386, 36120}
X(52967) = X(i)-Dao conjugate of X(j) for these {i, j}: {132, 276}, {216, 18024}, {2679, 15412}, {11672, 34384}, {40588, 290}, {40601, 95}, {46094, 34386}
X(52967) = barycentric product X(i)*X(j) for these (i, j): {5, 237}, {25, 44716}, {51, 511}, {53, 3289}, {184, 39569}, {216, 232}, {217, 297}, {311, 9418}, {325, 40981}, {343, 2211}, {418, 6530}, {684, 52604}, {1625, 3569}, {1755, 1953}, {1959, 2179}, {2491, 14570}, {3199, 36212}, {4230, 15451}, {5360, 18180}, {5562, 34854}, {7069, 51653}, {9417, 14213}, {12077, 14966}, {17994, 23181}, {20022, 27374}, {35360, 39469}, {36412, 41270}, {41586, 51980}
X(52967) = barycentric quotient X(i)/X(j) for these (i, j): {45, 35168}, {545, 20569}, {8649, 89}, {14410, 900}, {14421, 52620}, {51908, 40833}
X(52967) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 18024}, {51, 290}, {217, 287}, {232, 276}, {237, 95}, {418, 6394}, {511, 34384}, {1625, 43187}, {1953, 46273}, {2179, 1821}, {2211, 275}, {2491, 15412}, {3199, 16081}, {3289, 34386}, {9417, 2167}, {9418, 54}, {27374, 20021}, {34854, 8795}, {34859, 16813}, {36425, 41270}, {39569, 18022}, {40981, 98}, {44088, 17974}, {44716, 305}, {51363, 51257}, {51862, 41488}, {52604, 22456}

---

X(52968) = X(50)X(647)∩X(54)X(570)

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

See Ivan Pavlov, euclid 5663.

X(52968) lies on the circumconic {{A,B,C,C,X(54),X(539}} and these lines: {50,647}, {54,570}, {97,52437}, {231,40631}, {1216,14533}, {3003,14586}, {3284,11077}

X(52968) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2383, 14213}
X(52968) = X(i)-Dao conjugate of X(j) for these {i, j}: {128, 324}
X(52968) = barycentric product X(i)*X(j) for these (i, j): {3, 40631}, {54, 539}, {97, 231}, {252, 45083}, {18315, 52742}, {27423, 43704}
X(52968) = barycentric quotient X(i)/X(j) for these (i, j): {231, 324}, {539, 311}, {40631, 264}, {52742, 18314}

---

X(52969) = X(6)X(101)∩X(41)X(3271)

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

See Ivan Pavlov, euclid 5663.

X(52969) lies these lines: {6,101}, {36,51633}, {37,28345}, {39,34928}, {41,3271}, {44,2801}, {55,2195}, {116,17337}, {150,37650}, {218,34931}, {220,3939}, {528,35113}, {657,663}, {666,40861}, {910,2835}, {991,38599}, {1212,34930}, {1253,3022}, {1279,2809}, {1282,5332}, {1362,1471}, {2293,38365}, {2808,13329}, {2876,51436}, {6710,17245}, {11712,49478}, {16670,51766}, {20096,37681}, {35094,44355}, {38690,50677}

X(52969) = midpoint of X(666) and X(40861)
X(52969) = reflection of X(2383) in X(44355)
X(52969) = X(i)-isoconjugate-of-X(j) for these {i, j}: {7, 37131}, {57, 18821}, {85, 840}, {927, 52228}
X(52969) = X(i)-Dao conjugate of X(j) for these {i, j}: {5452, 18821}, {35113, 6063}
X(52969) = barycentric product X(i)*X(j) for these (i, j): {9, 2246}, {55, 528}, {220, 5723}, {294, 1642}, {644, 1643}, {666, 14411}, {3063, 42722}, {3689, 14190}, {3693, 51922}
X(52969) = barycentric quotient X(i)/X(j) for these (i, j): {41, 37131}, {55, 18821}, {528, 6063}, {1642, 40704}, {1643, 24002}, {2175, 840}, {2246, 85}, {14411, 918}, {46388, 52228}, {51922, 34018}, {52227, 34085}
X(52969) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5452, 14827, 16588}

---

X(52970) = X(56)X(478)∩X(649)X(854)

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

See Ivan Pavlov, euclid 5663.

X(52970) lies on these lines: {56,478}, {222,5114}, {603,4274}, {649,854}, {19297,23980}

X(52970) = X(i)-isoconjugate-of-X(j) for these {i, j}: {312, 38882}
X(52970) = barycentric product X(i)*X(j) for these (i, j): {6, 43036}, {56, 529}, {6648, 14412}
X(52970) = barycentric quotient X(i)/X(j) for these (i, j): {529, 3596}, {1397, 38882}, {14412, 3910}, {43036, 76}
X(52970) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {478, 52410, 17053}

---

X(52971) = X(61)X(1493)∩X(395)X(462)

Barycentrics    a^2*(sqrt(3)*S+SA)*(2*a^2*SA-sqrt(3)*S*(-a^2+2*SA)-b^2*SB-c^2*SC) : :

See Ivan Pavlov, euclid 5663.

X(52971) lies on the circumconic {{A,B,C,X(61),X(533)}} and these lines: {16,30486}, {61,1493}, {62,10095}, {110,11063}, {395,462}, {533,18804}, {1154,5616}, {3129,36978}, {6117,42163}, {10594,10633}, {11086,19294}, {11126,11141}, {11127,51891}, {11136,40580}, {11137,11146}, {11138,51264}, {34328,34425}, {34394,40695}

X(52971) = midpoint of X(5616) and X(6105)
X(52971) = X(i)-Dao conjugate of X(j) for these {i, j}: {10640, 11118}
X(52971) = barycentric product X(i)*X(j) for these (i, j): {61, 533}, {395, 11146}, {1994, 40668}, {11137, 41001}, {11141, 14921}, {14447, 52605}, {16771, 19295}, {23715, 52348}
X(52971) = barycentric quotient X(i)/X(j) for these (i, j): {61, 11118}, {533, 34389}, {11137, 6151}, {11146, 40706}, {40668, 11140}
X(52971) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11126, 11141, 36980}

---

X(52972) = X(62)X(1493)∩X(396)X(463)

Barycentrics    a^2*(sqrt(3)*S-SA)*(2*a^2*(sqrt(3)*S-SA)+b^2*(-(sqrt(3)*S)+SB)+c^2*(-(sqrt(3)*S)+SC)) : :

See Ivan Pavlov, euclid 5663.

X(52972) lies on these lines: {15,30485}, {61,10095}, {62,1493}, {110,11063}, {396,463}, {532,18803}, {1154,5612}, {3130,36980}, {6116,42166}, {10594,10632}, {11081,19295}, {11126,51890}, {11127,11142}, {11134,11145}, {11135,40581}, {11139,51271}, {34327,34424}, {34395,40696}

X(52972) = midpoint of X(5612) and X(6104)
X(52972) = X(i)-Dao conjugate of X(j) for these {i, j}: {10639, 11117}
X(52972) = barycentric product X(i)*X(j) for these (i, j): {62, 532}, {396, 11145}, {1994, 40667}, {11134, 41000}, {11142, 14922}, {14446, 52606}, {16770, 19294}, {23714, 52349}
X(52972) = barycentric quotient X(i)/X(j) for these (i, j): {62, 11117}, {532, 34390}, {11134, 2981}, {11145, 40707}, {40667, 11140}
X(52972) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11127, 11142, 36978}

---

X(52973) = X(32)X(66)∩X(647)X(826)

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

See Ivan Pavlov, euclid 5663.

X(52973) lies on the circumconic {{A,B,C,X(66),X(826)}} and these lines: {32,66}, {647,826}, {2353,3456}, {6680,40421}, {7755,43678}, {7772,14376}, {7813,44766}

X(52973) = barycentric quotient X(14416)/X(23881)

---

X(52974) = X(39)X(647)∩X(67)X(187)

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

See Ivan Pavlov, euclid 5663.

X(52974) lies on the circumconic {{A,B,C,,X(67),X(3906)}} and these lines: {39,647}, {67,187}, {115,46338}, {3284,34897}, {17708,39785}

---

X(52975) = X(68)X(577)∩X(647)X(6368)

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

See Ivan Pavlov, euclid 5663.

X(52975) lies on the circumconic {{A,B,C,X(68),X(539}} and these lines: {68,577}, {647,6368}, {2165,7749}, {5158,34853

X(52975) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1748, 2383}
X(52975) = X(i)-Dao conjugate of X(j) for these {i, j}: {128, 11547}
X(52975) = barycentric product X(i)*X(j) for these (i, j): {68, 539}, {231, 52350}
X(52975) = barycentric quotient X(i)/X(j) for these (i, j): {231, 11547}, {539, 317}, {2351, 2383}

---

X(52976) = X(6)X(647)∩X(74)X(3003)

Barycentrics    a^2*(a^4-2*b^4+b^2*c^2+c^4+a^2*(b^2-2*c^2))*(a^4+b^4+b^2*c^2-2*c^4+a^2*(-2*b^2+c^2))*(2*a^10-(b^2-c^2)^4*(b^2+c^2)+a^6*(-11*b^4+16*b^2*c^2-11*c^4)-a^2*(b^2-c^2)^2*(3*b^4+14*b^2*c^2+3*c^4)+a^4*(13*b^6-11*b^4*c^2-11*b^2*c^4+13*c^6)) : :

See Ivan Pavlov, euclid 5663.

X(52976) lies on these lines: {6,647}, {74,3003}, {187,18877}, {6128,34150}, {8744,8749}, {11060,11079}, {14919,41617}, {15291,46203}, {34288,52488}, {45723,51544}

X(52976) = X(i)-isoconjugate-of-X(j) for these {i, j}: {841, 14206}
X(52976) = intersection, other than A, B, C, of circumconics {{A,B,C,X(6),X(32681)}} and {{A,B,C,X(74),X(541)}}
X(52976) = barycentric product X(74)*X(541)
X(52976) = barycentric quotient X(i)/X(j) for these (i, j): {541, 3260}, {40352, 841}

---

X(52977) = X(77)X(1433)∩X(652)X(905)

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

See Ivan Pavlov, euclid 5663.

X(52977) lies on these lines: {77,1433}, {652,905}, {3338,7271}, {5228,43064}, {6001,6614}, {6212,10252}, {6213,10253}

---

X(52978) = X(44)X(519)∩X(78)X(219)

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

See Ivan Pavlov, euclid 5663.

X(52978) lies on these lines: {6,3991}, {9,3057}, {37,30147}, {44,519}, {71,3916}, {78,219}, {101,2756}, {190,40863}, {521,652}, {1018,2182}, {1145,8756}, {1332,6510}, {1404,14439}, {1731,3880}, {2183,41391}, {2323,3693}, {3927,4047}, {3965,52405}, {4266,30618}, {4855,23073}, {4861,40937}, {5440,22356}, {6735,7359}, {17067,25891}, {23135,46974}

X(52978) = midpoint of X(190) and X(40863)
X(52978) = X(i)-isoconjugate-of-X(j) for these {i, j}: {7, 8752}, {34, 88}, {56, 6336}, {57, 36125}, {106, 278}, {273, 9456}, {653, 23345}, {1395, 20568}, {1396, 4674}, {1398, 4997}, {6548, 32674}, {6549, 7115}, {23838, 32714}, {43922, 46102}
X(52978) = X(i)-Dao conjugate of X(j) for these {i, j}: {1, 6336}, {214, 278}, {519, 37790}, {4370, 273}, {5440, 33129}, {5452, 36125}, {11517, 88}, {35072, 6548}, {40628, 6549}, {51402, 17924}, {52659, 1847}
X(52978) = intersection, other than A, B, C, of circumconics {{A,B,C,X(44),X(219)}}, {{A,B,C,X(69),X(49702)}} and {{A,B,C,X(78),X(519)}}
X(52978) = barycentric product X(i)*X(j) for these (i, j): {3, 4723}, {8, 5440}, {9, 3977}, {44, 345}, {63, 2325}, {69, 3689}, {78, 519}, {190, 14418}, {212, 3264}, {219, 4358}, {283, 3992}, {312, 22356}, {332, 21805}, {521, 17780}, {643, 14429}, {646, 22086}, {652, 24004}, {900, 4571}, {902, 3718}, {905, 30731}, {1023, 6332}, {1145, 1809}, {1259, 38462}, {1265, 1319}, {1331, 4768}, {1332, 1639}, {1404, 52406}, {1792, 40663}, {1808, 4783}, {1812, 3943}, {2289, 46109}, {2318, 30939}, {3596, 23202}, {3692, 3911}, {3694, 16704}, {3710, 52680}, {3719, 8756}, {3762, 4587}, {3949, 30606}, {4528, 6516}, {4561, 4895}, {23344, 35518}, {31343, 39472}, {36944, 51379}
X(52978) = barycentric quotient X(i)/X(j) for these (i, j): {9, 6336}, {41, 8752}, {44, 278}, {55, 36125}, {78, 903}, {184, 1417}, {212, 106}, {219, 88}, {345, 20568}, {519, 273}, {521, 6548}, {652, 1022}, {678, 1877}, {902, 34}, {1023, 653}, {1260, 1320}, {1319, 1119}, {1404, 1435}, {1639, 17924}, {1802, 2316}, {1946, 23345}, {1960, 43923}, {2251, 608}, {2289, 1797}, {2318, 4674}, {2325, 92}, {3285, 1396}, {3689, 4}, {3692, 4997}, {3694, 4080}, {3911, 1847}, {3943, 40149}, {3977, 85}, {4152, 38462}, {4358, 331}, {4370, 37790}, {4528, 44426}, {4571, 4555}, {4587, 3257}, {4723, 264}, {4768, 46107}, {4895, 7649}, {5440, 7}, {6056, 36058}, {6174, 38461}, {7004, 6549}, {8611, 4049}, {9459, 1395}, {14418, 514}, {14427, 3064}, {14429, 4077}, {14439, 5236}, {17780, 18026}, {21805, 225}, {22086, 3669}, {22356, 57}, {22371, 1319}, {23202, 56}, {23344, 108}, {23703, 36118}, {24004, 46404}, {30731, 6335}, {52425, 9456}
X(52978) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {219, 3692, 3694}, {1332, 25083, 6510}

---

X(52979) = X(23)X(385)∩X(83)X(316)

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

See Ivan Pavlov, euclid 5663.

X(52979) lies on the circumconic {{A,B,C,X(23),X(46543)}} and these lines: {23,385}, {83,316}, {325,46228}, {524,4577}, {689,30736}, {1990,42396}, {7762,14247}, {7792,41295}, {8352,38888}, {8627,35549}, {8928,29181}, {34573,40000}, {38946,47286}, {40425,51127}

X(52979) = midpoint of X(4577) and X(40850)
X(52979) = X(i)-isoconjugate-of-X(j) for these {i, j}: {38, 755}, {1964, 43098}, {5389, 21123}
X(52979) = X(i)-Dao conjugate of X(j) for these {i, j}: {41884, 43098}
X(52979) = barycentric product X(i)*X(j) for these (i, j): {83, 754}, {251, 35549}, {308, 8627}, {689, 14428}, {2244, 3112}, {4156, 52394}, {4577, 14420}, {4580, 46543}
X(52979) = barycentric quotient X(i)/X(j) for these (i, j): {83, 43098}, {251, 755}, {754, 141}, {2244, 38}, {4156, 15523}, {4157, 3703}, {7214, 3665}, {8627, 39}, {14403, 2531}, {14420, 826}, {14428, 3005}, {33907, 2528}, {35549, 8024}, {46543, 41676}, {52758, 31125}
X(52979) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4577, 40850, 524}, {41884, 52395, 3589}

---

X(52980) = X(85)X(142)∩X(522)X(693)

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

See Ivan Pavlov, euclid 5663.

X(52980) lies on these lines: {9,34019}, {85,142}, {522,693}, {527,4569}, {2321,50560}, {2325,4554}, {17067,34018}

X(52980) = midpoint of X(4569) and X(40864)
X(52980) = barycentric product X(i)*X(j) for these (i, j): {85, 44664}, {3000, 6063}
X(52980) = barycentric quotient X(i)/X(j) for these (i, j): {3000, 55}, {44664, 9}

---

X(52981) = X(44)X(4604)∩X(89)X(4850)

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

See Ivan Pavlov, euclid 5663.

X(52981) lies on these lines: {44,4604}, {89,4850}, {513,1960}, {536,4597}, {4715,25398}

X(52981) = midpoint of X(4597) and X(29908)
X(52981) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2177, 35170}, {3679, 28317}
X(52981) = X(i)-Dao conjugate of X(j) for these {i, j}: {35124, 4671}
X(52981) = barycentric product X(i)*X(j) for these (i, j): {89, 4715}, {2163, 49780}, {4597, 14422}
X(52981) = barycentric quotient X(i)/X(j) for these (i, j): {89, 35170}, {4715, 4671}, {14422, 4777}, {28607, 28317}
X(52981) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4597, 29908, 536}

---

X(52982) = X(92)X(226)∩X(142)X(264)

Barycentrics    b c SB SC (-a b SA SB - a c SA SC + 2 b c SB SC) : :

See Ivan Pavlov, euclid 5663.

X(52982) lies on these lines: {53,16608}, {92,226}, {142,264}, {240,522}, {324,5249}, {527,1948}, {908,46106}, {1093,6260}, {1947,5745}, {1990,36949}, {2325,6335}, {3452,15466}, {5316,52147}, {17917,18678}, {33971,51687}

X(52982) = midpoint of X(1948) and X(18026)
X(52982) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3, 32726}, {48, 23707}, {63, 34078}
X(52982) = X(i)-Dao conjugate of X(j) for these {i, j}: {1249, 23707}, {3162, 34078}, {36103, 32726}
X(52982) = barycentric product X(264)*X(2635)
X(52982) = barycentric quotient X(i)/X(j) for these (i, j): {89, 35170}, {4715, 4671}, {14422, 4777}, {28607, 28317}
X(52982) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1948, 18026, 527}
{4, 23707}, {19, 32726}, {25, 34078}, {2635, 3}, {2637, 36054}, {30691, 1459}

---

X(52983) = X(5)X(523)∩X(94)X(3580)

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

See Ivan Pavlov, euclid 5663.

X(52983) lies on these lines: {5,523}, {94,3580}, {297,18817}, {524,35139}, {1990,46456}, {40427,47296}

X(52983) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6149, 32730}
X(52983) = X(i)-Dao conjugate of X(j) for these {i, j}: {14993, 32730}
X(52983) = barycentric product X(i)*X(j) for these (i, j): {3016, 20573}
X(52983) = barycentric quotient X(i)/X(j) for these (i, j): {1989, 32730}, {3016, 50}, {14254, 52763}

---

X(52984) = X(95)X(140)∩X(323)X(401)

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

See Ivan Pavlov, euclid 5663.

X(52984) lies on these lines: {30,18831}, {95,140}, {323,401}, {16239,31617}, {16266,20477}

X(52984) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1953, 51222}
X(52984) = X(i)-Dao conjugate of X(j) for these {i, j}: {138, 53}
X(52984) = barycentric quotient X(i)*X(j) for these (i, j): {54, 51222}, {42731, 12077}

---

X(52985) = X(1)X(6)∩X(100)X(650)

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

See Ivan Pavlov, euclid 5663.

X(52985) lies on these lines: {1,6}, {100,650}, {101,513}, {190,28898}, {528,35113}, {644,765}, {666,4762}, {1018,46973}, {2246,14190}, {3900,3939}, {4040,35338}, {4394,41405}, {4944,51562}, {4998,31287}, {6066,11934}, {6667,46101}, {14513,47777}, {31150,51357}, {31209,43986}

X(52985) = midpoint of X(i) and X(j) for these {i,j}: {44, 6603}, {666, 40865}
X(52985) = X(i)-isoconjugate-of-X(j) for these {i, j}: {105, 52228}, {513, 37131}, {514, 840}, {1022, 14191}, {23345, 46791}
X(52985) = X(i)-Dao conjugate of X(j) for these {i, j}: {5375, 18821}, {39026, 37131}, {39046, 52228}
X(52985) = X(i) cross conjugate of X(j) for these {i, j}: {1643, 2246}
X(52985) = intersection, other than A, B, C, of circumconics {{A,B,C,X(1),X(1308)}}, {{A,B,C,X(6),X(919)}} and {{A,B,C,X(9),X(3257)}}
X(52985) = trilinear pole of line {1642,2246}
X(52985) = barycentric product X(i)*X(j) for these (i, j): {6, 42722}, {100, 528}, {190, 2246}, {644, 5723}, {666, 1642}, {1016, 1643}, {1023, 46790}, {3912, 52227}, {14190, 17780}, {23343, 52761}, {42720, 51922}
X(52985) = barycentric quotient X(i)/X(j) for these (i, j): {54, 51222}, {42731, 12077}
X(52985) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {100, 18821}, {101, 37131}, {672, 52228}, {692, 840}, {1023, 46791}, {1642, 918}, {1643, 1086}, {5723, 24002}, {14190, 6548}, {14411, 17435}, {23344, 14191}, {42722, 76}, {52227, 673}

---

X(52986) = X(6)X(31)∩X(101)X(6586)

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

See Ivan Pavlov, euclid 5663.

X(52986) lies on the circumconic {{A,B,C,X(6),X(32682)}} and these lines: {6,31}, {101,6586}, {906,39189}, {4559,21007}

X(52986) = X(i)-isoconjugate-of-X(j) for these {i, j}: {693, 38884}
X(52986) = barycentric product X(101)*X(544)
X(52986) = barycentric quotient X(i)/X(j) for these (i, j): {544, 3261}, {32739, 38884}

---

X(52987) = X(3)X(6)∩X(20)X(542)

Barycentrics    a^2*(a^4 + 3*a^2*b^2 - 4*b^4 + 3*a^2*c^2 - 2*b^2*c^2 - 4*c^4) : :
X(52987) = 5 X[3] - 3 X[6], 4 X[3] - 3 X[182], 3 X[3] - 2 X[575], X[3] - 3 X[1350], 7 X[3] - 3 X[1351], 2 X[3] - 3 X[3098], 13 X[3] - 9 X[5050], 11 X[3] - 9 X[5085], 7 X[3] - 6 X[5092], 17 X[3] - 9 X[5093], 11 X[3] - 6 X[5097], 19 X[3] - 9 X[5102], 9 X[3] - 7 X[10541], 3 X[3] - X[11477], 9 X[3] - 5 X[11482], 19 X[3] - 15 X[12017], 5 X[3] - 6 X[14810], 19 X[3] - 12 X[15516], 16 X[3] - 9 X[15520], 10 X[3] - 9 X[17508], 5 X[3] - 4 X[20190], 8 X[3] - 5 X[22234], 7 X[3] - 4 X[22330], 7 X[3] - 9 X[31884], X[3] + 3 X[33878], 8 X[3] - 3 X[37517], 14 X[3] - 9 X[39561], 11 X[3] - 3 X[44456], 17 X[3] - 12 X[50664], 4 X[6] - 5 X[182], 9 X[6] - 10 X[575], 6 X[6] - 5 X[576], X[6] - 5 X[1350], 7 X[6] - 5 X[1351], 2 X[6] - 5 X[3098] (and many more)

See Manuel Aguilera, euclid 5696.

X(52987) lies on these lines: {2, 44300}, {3, 6}, {4, 7883}, {5, 20582}, {20, 542}, {22, 3292}, {23, 2979}, {30, 22165}, {51, 40916}, {69, 3529}, {74, 33638}, {140, 5476}, {141, 546}, {184, 6030}, {186, 11470}, {194, 12122}, {315, 50567}, {323, 35268}, {343, 46517}, {373, 21766}, {381, 51186}, {382, 599}, {524, 550}, {548, 8550}, {597, 3530}, {631, 20423}, {632, 21850}, {633, 51010}, {634, 51013}, {895, 11270}, {1092, 7556}, {1147, 7555}, {1204, 15073}, {1216, 7530}, {1352, 3146}, {1469, 3746}, {1503, 15704}, {1657, 11645}, {1658, 6593}, {1843, 35502}, {1974, 44879}, {1992, 3528}, {1995, 3917}, {2393, 3357}, {2781, 5609}, {2810, 51528}, {2854, 51522}, {2930, 18436}, {2937, 15039}, {3056, 5563}, {3060, 7496}, {3066, 15082}, {3090, 7944}, {3091, 7938}, {3506, 38873}, {3520, 8541}, {3522, 11179}, {3523, 10168}, {3525, 14561}, {3534, 51188}, {3544, 3619}, {3564, 12103}, {3589, 14869}, {3620, 50688}, {3627, 3818}, {3628, 5480}, {3631, 39884}, {3763, 5079}, {3792, 7301}, {3819, 11284}, {3843, 25561}, {3851, 21358}, {3853, 47354}, {4655, 29032}, {4663, 31663}, {5010, 19369}, {5059, 11180}, {5067, 25565}, {5070, 38072}, {5073, 47353}, {5076, 48889}, {5182, 33014}, {5343, 16002}, {5344, 16001}, {5363, 7186}, {5562, 12082}, {5643, 11002}, {5646, 12045}, {5651, 14002}, {5965, 12254}, {5969, 7751}, {5999, 33706}, {6000, 34787}, {6090, 32237}, {6102, 8546}, {6248, 44774}, {6403, 14865}, {6636, 11422}, {6688, 14924}, {6776, 48892}, {7280, 8540}, {7387, 15606}, {7464, 11649}, {7484, 21849}, {7485, 21969}, {7526, 9813}, {7689, 14984}, {7691, 9972}, {7748, 15993}, {7750, 51438}, {7763, 51396}, {7776, 51397}, {7782, 39099}, {7793, 10754}, {7824, 22486}, {7865, 40250}, {7877, 12252}, {7998, 16042}, {8537, 35473}, {8542, 9019}, {8584, 34200}, {8681, 12163}, {8703, 41149}, {9024, 51529}, {9037, 43146}, {9047, 43149}, {9925, 13754}, {9973, 44753}, {9976, 12041}, {10110, 52163}, {10226, 11255}, {10300, 13567}, {10303, 14853}, {10516, 48895}, {10594, 12294}, {10627, 12106}, {10752, 15020}, {10989, 38397}, {11202, 34117}, {11204, 34788}, {11261, 14881}, {11459, 37946}, {11579, 15021}, {11898, 48905}, {12088, 41716}, {12102, 18358}, {12108, 18583}, {12177, 45017}, {12251, 38664}, {12307, 16010}, {12605, 47558}, {12812, 42786}, {13192, 39576}, {13248, 25564}, {13382, 32621}, {13391, 49671}, {13488, 41585}, {14269, 50993}, {14791, 32273}, {15004, 15246}, {15019, 43650}, {15035, 25556}, {15083, 35707}, {15533, 15681}, {15534, 15688}, {15577, 34779}, {15579, 44668}, {15581, 34146}, {15687, 50991}, {15693, 46267}, {15696, 43273}, {15700, 51185}, {15717, 38064}, {15720, 47352}, {15988, 17574}, {16063, 41586}, {16239, 51166}, {17800, 50955}, {18440, 29323}, {18800, 33208}, {19124, 35475}, {19140, 33851}, {19149, 50414}, {19782, 48939}, {21166, 32135}, {21243, 31099}, {22679, 40253}, {23039, 37924}, {23042, 35228}, {23049, 32767}, {23325, 47341}, {23327, 25563}, {25406, 33751}, {32534, 44102}, {33524, 45187}, {33750, 51170}, {33923, 51737}, {34118, 34786}, {34380, 44245}, {34573, 38136}, {34616, 34623}, {34795, 38688}, {35021, 37667}, {35499, 41398}, {36990, 43150}, {37338, 52658}, {37967, 46261}, {38071, 51143}, {38079, 50984}, {38689, 41367}, {40330, 50689}, {40341, 44748}, {43806, 51179}, {44682, 50983}, {46853, 50979}, {47447, 47468}, {47451, 47569}, {49138, 51023}, {50972, 50986}, {50976, 51174}, {50981, 51130}

X(52987) = midpoint of X(i) and X(j) for these {i,j}: {69, 48873}, {1350, 33878}, {1657, 15069}, {11898, 48905}, {15533, 15681}, {18440, 48872}, {37473, 37484}
X(52987) = reflection of X(i) in X(j) for these {i,j}: {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 20190}, {3, 576, 182}, {3, 11477, 575}, {3, 11482, 10541}, {3, 20190, 17508}, {4, 40107, 11178}, {6, 14810, 17508}, {6, 17508, 182}, {182, 576, 22234}, {182, 37517, 15520}, {371, 372, 5008}, {382, 599, 18553}, {575, 11477, 576}, {576, 3098, 3}, {576, 22234, 15520}, {576, 39561, 22330}, {631, 20423, 25555}, {1351, 5092, 39561}, {1351, 22330, 576}, {1351, 31884, 5092}, {2076, 5028, 41412}, {3098, 17508, 14810}, {5013, 11173, 44500}, {5013, 44500, 42852}, {5085, 44456, 5097}, {5092, 39561, 182}, {5102, 12017, 15516}, {5104, 44453, 32}, {5237, 5238, 5206}, {6453, 6454, 35007}, {7492, 23061, 184}, {7690, 44472, 182}, {7692, 44471, 182}, {7998, 34417, 16187}, {8550, 50965, 548}, {8722, 35002, 9734}, {9738, 11825, 7690}, {9738, 18860, 7692}, {9739, 11824, 7692}, {9739, 18860, 7690}, {9821, 14540, 47066}, {9821, 14541, 47068}, {9821, 30270, 5171}, {9968, 15582, 6759}, {10519, 31670, 24206}, {10541, 11477, 11482}, {10541, 11482, 575}, {10625, 46728, 13346}, {11002, 41462, 22112}, {11438, 50649, 44489}, {11824, 11825, 18860}, {12974, 12975, 47113}, {12974, 44486, 182}, {12975, 44485, 182}, {13340, 37478, 37480}, {14540, 14541, 30270}, {14810, 20190, 3}, {15107, 33884, 5651}, {15644, 37486, 46730}, {22234, 37517, 576}, {47066, 47068, 5171}
X(52987) = Brocard-circle-inverse of X(20190)
X(52987) = crossdifference of every pair of points on line {523, 47458}
X(52987) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 20190}, {3, 576, 182}, {3, 11477, 575}, {3, 11482, 10541}, {3, 20190, 17508}, {4, 40107, 11178}, {6, 14810, 17508}, {6, 17508, 182}, {182, 576, 22234}, {182, 37517, 15520}, {371, 372, 5008}, {382, 599, 18553}, {575, 11477, 576}, {576, 3098, 3}, {576, 22234, 15520}, {576, 39561, 22330}, {631, 20423, 25555}, {1351, 5092, 39561}, {1351, 22330, 576}, {1351, 31884, 5092}, {2076, 5028, 41412}, {3098, 17508, 14810}, {5013, 11173, 44500}, {5013, 44500, 42852}, {5085, 44456, 5097}, {5092, 39561, 182}, {5102, 12017, 15516}, {5104, 44453, 32}, {5237, 5238, 5206}, {6453, 6454, 35007}, {7492, 23061, 184}, {7690, 44472, 182}, {7692, 44471, 182}, {7998, 34417, 16187}, {8550, 50965, 548}, {8722, 35002, 9734}, {9738, 11825, 7690}, {9738, 18860, 7692}, {9739, 11824, 7692}, {9739, 18860, 7690}, {9821, 14540, 47066}, {9821, 14541, 47068}, {9821, 30270, 5171}, {9968, 15582, 6759}, {10519, 31670, 24206}, {10541, 11477, 11482}, {10541, 11482, 575}, {10625, 46728, 13346}, {11002, 41462, 22112}, {11438, 50649, 44489}, {11824, 11825, 18860}, {12974, 12975, 47113}, {12974, 44486, 182}, {12975, 44485, 182}, {13340, 37478, 37480}, {14540, 14541, 30270}, {14810, 20190, 3}, {15107, 33884, 5651}, {15644, 37486, 46730}, {22234, 37517, 576}, {47066, 47068, 5171}

---

X(52988) = X(3)X(6)∩X(1352)X(4201)

Barycentrics    a^2*(a^6*b + a^3*b^4 - 2*a^2*b^5 - a*b^6 + b^7 + a^6*c - a^5*b*c + a^4*b^2*c - a^3*b^3*c - a^2*b^4*c + 2*a*b^5*c - b^6*c + a^4*b*c^2 - 3*a^2*b^3*c^2 - 3*a*b^4*c^2 - b^5*c^2 - a^3*b*c^3 - 3*a^2*b^2*c^3 + 2*a*b^3*c^3 - b^4*c^3 + a^3*c^4 - a^2*b*c^4 - 3*a*b^2*c^4 - b^3*c^4 - 2*a^2*c^5 + 2*a*b*c^5 - b^2*c^5 - a*c^6 - b*c^6 + c^7) : :
X(52988) = 5 X[631] - X[50636], 3 X[3576] + X[50576], 3 X[10165] - X[50611], 3 X[10519] + X[20018], 4 X[20108] - 3 X[38317], 3 X[38029] - X[50635]

See Manuel Aguilera, euclid 5696.

X(52988) lies on these lines: {3,6}, {140, 50609}, {519, 50977}, {631, 50636}, {986, 50615}, {1352, 4201}, {3576, 50576}, {3818, 48836}, {10165, 50611}, {10519, 20018}, {11178, 11359}, {13732, 50618}, {16062, 24206}, {17596, 50612}, {20108, 38317}, {31394, 41886}, {38029, 50635}

X(52988) = midpoint of X(3) and X(50591)
X(52988) = reflection of X(i) in X(j) for these {i,j}: {576, 50595}, {50600, 575}, {50609, 140}

---

X(52989) = X(3)X(6)∩X(125)X(5169)

Barycentrics    a^2*(a^8*b^2 - 2*a^6*b^4 + 2*a^2*b^8 - b^10 + a^8*c^2 + a^6*b^2*c^2 - 3*a^4*b^4*c^2 + b^8*c^2 - 2*a^6*c^4 - 3*a^4*b^2*c^4 - 6*a^2*b^4*c^4 + 2*a^2*c^8 + b^2*c^8 - c^10) : :
X(52989) = 5 X[182] - 2 X[9967], X[182] + 2 X[19161], 4 X[389] - X[576], 2 X[575] + X[37473], 2 X[575] - 5 X[37481], 2 X[5092] - 3 X[40280], 5 X[9730] - X[9967], X[9967] + 5 X[19161], 4 X[15012] - X[44479], 4 X[15516] - X[44439], 4 X[16836] - 3 X[17508], X[18438] - 4 X[50664], 5 X[22234] - 2 X[50649], X[37473] + 5 X[37481], 2 X[37511] + X[37517], 3 X[5640] - 2 X[19130], 3 X[5890] + X[11188], 3 X[11002] - X[31670], X[5889] + 2 X[40107], 2 X[6102] + X[34507], 4 X[9969] - X[48884], 2 X[10168] - 3 X[15045], 5 X[11451] - 4 X[25565], 4 X[13363] - 3 X[38317], 2 X[13382] + X[43130], 7 X[15043] - 4 X[25555], X[15073] - 4 X[33749], 2 X[18553] + X[34783], 8 X[21852] + X[48879], 4 X[32191] - X[48901]

See Manuel Aguilera, euclid 5696.

X(52989) lies on these lines: {3, 6}, {4, 45835}, {51, 31133}, {125, 5169}, {542, 5890}, {1154, 50977}, {1503, 38322}, {1995, 52098}, {2781, 5476}, {2854, 14708}, {2930, 40258}, {3060, 19924}, {3818, 5663}, {5189, 11002}, {5889, 40107}, {6102, 34507}, {8537, 43810}, {8705, 48906}, {9969, 14915}, {9970, 43578}, {9971, 11645}, {10168, 15045}, {10546, 32235}, {11178, 13754}, {11179, 11649}, {11451, 25565}, {11459, 14789}, {11562, 18440}, {13363, 38317}, {13382, 43130}, {15043, 25555}, {15053, 15462}, {15060, 40670}, {15072, 29012}, {15073, 33749}, {16261, 38791}, {18435, 25561}, {18553, 34783}, {20300, 33332}, {21852, 48879}, {22802, 51756}, {23325, 34146}, {32191, 48901}, {32273, 45237}

X(52989) = midpoint of X(9730) and X(19161)
X(52989) = reflection of X(i) in X(j) for these {i,j}: {182, 9730}, {3818, 16776}, {5476, 5946}, {11459, 24206}, {13340, 14810}, {15060, 40670}, {18435, 25561}, {32273, 45237}
X(52989) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {182, 37517, 11511}, {9730, 32110, 40280}, {37473, 37481, 575}

---

X(52990) = X(3)X(6)∩X(381)X(2916)

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

See Manuel Aguilera, euclid 5696.

X(52990) lies on these lines: {3, 6}, {22, 19130}, {26, 38317}, {159, 11178}, {378, 48892}, {381, 2916}, {539, 50977}, {549, 23300}, {1352, 37126}, {1593, 48896}, {1899, 15246}, {2070, 47355}, {3524, 41256}, {3589, 7502}, {3818, 7514}, {5480, 7525}, {6636, 31670}, {6697, 7516}, {7485, 21243}, {7503, 29012}, {7509, 24206}, {7512, 14561}, {7526, 48898}, {7568, 20300}, {9818, 48884}, {10168, 44837}, {10323, 29317}, {11413, 33751}, {11414, 48904}, {12083, 48895}, {12106, 51126}, {12359, 21167}, {13171, 52098}, {14070, 31521}, {18358, 35707}, {32600, 34776}, {34006, 38072}, {34864, 36990}, {35243, 48879}, {35921, 46264}, {44457, 48943}

X(52990) = midpoint of X(1350) and X(36749)
X(52990) = reflection of X(i) in X(j) for these {i,j}: {5157, 5092}, {44494, 182}
X(52990) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5085, 44469, 182}, {5092, 11574, 182}

---

X(52991) = X(3)X(6)∩X(141)X(517)

Barycentrics    a^2*(a^5*b^2 + a^4*b^3 - a*b^6 - b^7 + 2*a^5*b*c + a^4*b^2*c + 2*a^3*b^3*c - 4*a*b^5*c - b^6*c + a^5*c^2 + a^4*b*c^2 + 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 - a*b^4*c^2 - b^5*c^2 + a^4*c^3 + 2*a^3*b*c^3 + 2*a^2*b^2*c^3 - 4*a*b^3*c^3 - b^4*c^3 - a*b^2*c^4 - b^3*c^4 - 4*a*b*c^5 - b^2*c^5 - a*c^6 - b*c^6 - c^7) : :
X(52991) = 3 X[31884] - X[37482], 2 X[5482] - 3 X[21167]

See Manuel Aguilera, euclid 5696.

X(52991) lies on these lines: {3, 6}, {141, 517}, {392, 13728}, {1843, 4227}, {3917, 26637}, {5241, 19130}, {5480, 34466}, {5482, 21167}, {15488, 24206}, {17740, 51377}, {26118, 31670}

X(52991) = midpoint of X(1350) and X(5752)
X(52991) = reflection of X(i) in X(j) for these {i,j}: {182, 15489}, {5480, 34466}, {15488, 24206}

---

X(52992) = X(3)X(6)∩X(140)X(531)

Barycentrics    a^2*(2*a^8 - 3*a^6*b^2 + a^4*b^4 + a^2*b^6 - b^8 - 3*a^6*c^2 - 6*a^4*b^2*c^2 + 5*a^2*b^4*c^2 - 2*b^6*c^2 + a^4*c^4 + 5*a^2*b^2*c^4 + 2*b^4*c^4 + a^2*c^6 - 2*b^2*c^6 - c^8) : :
X(52992) = 5 X[3] - X[35456], X[182] + 2 X[47113], X[187] + 2 X[5092], X[1350] + 3 X[35006], 5 X[1691] + X[35456], 2 X[2030] + X[3098], X[2076] + 3 X[5085], X[2076] - 3 X[38225], X[2456] - 3 X[5085], X[2456] + 3 X[38225], 3 X[5050] - X[5111], X[5104] + 5 X[12017], X[5107] - 4 X[50664], 2 X[8590] + X[14810], 5 X[631] - X[5207], X[12215] + 3 X[21445], 2 X[14693] + X[44882], 3 X[38748] - X[51371]

See Manuel Aguilera, euclid 5696.

X(52992) lies on these lines: {3, 6}, {140, 5031}, {512, 25644}, {542, 35297}, {631, 5207}, {698, 33813}, {1352, 16925}, {1503, 12042}, {1513, 6036}, {1976, 35296}, {3564, 5026}, {3818, 37466}, {3972, 14561}, {4048, 10104}, {5215, 11645}, {5965, 6393}, {6656, 13449}, {6776, 32964}, {7804, 37242}, {7807, 24206}, {8356, 10168}, {11178, 11288}, {11272, 42421}, {12215, 21445}, {13367, 48445}, {14693, 44882}, {22352, 47638}, {33008, 38064}, {33751, 44251}, {36213, 52144}, {37182, 38227}, {37450, 37647}, {38748, 51371}, {46893, 50977}

X(52992) = midpoint of X(i) and X(j) for these {i,j}: {3, 1691}, {6, 35383}, {182, 35375}, {2076, 2456}, {2080, 47619}, {3098, 35377}, {5085, 38225}, {13355, 35374}
X(52992) = reflection of X(i) in X(j) for these {i,j}: {1570, 575}, {5031, 140}, {35375, 47113}, {35377, 2030}, {35431, 2032}
X(52992) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2076, 5085, 2456}, {2456, 38225, 2076}, {2459, 2460, 2021}, {3398, 5116, 50652}, {5092, 13335, 182}, {9737, 41412, 35431}, {12017, 44508, 182}, {13335, 47113, 187}, {13349, 13350, 18860}, {43120, 43121, 39}

---

X(52993) = X(3)X(6)∩X(99)X(5965)

Barycentrics    a^2*(a^8 - 4*a^4*b^4 + 5*a^2*b^6 - 2*b^8 - 9*a^4*b^2*c^2 + 4*a^2*b^4*c^2 + 2*b^6*c^2 - 4*a^4*c^4 + 4*a^2*b^2*c^4 - 2*b^4*c^4 + 5*a^2*c^6 + 2*b^2*c^6 - 2*c^8) : :
X(52993) = 5 X[182] - 8 X[47113], 4 X[187] - X[37517], 4 X[1570] - 5 X[22234], 4 X[1692] - 3 X[15520], 2 X[2456] - 3 X[17508], X[3098] + 2 X[5104], 5 X[3098] - 2 X[35002], 2 X[5097] - 3 X[35006], 5 X[5104] + X[35002], X[5111] - 3 X[38225], 2 X[5111] - 3 X[39561], 3 X[31884] - X[35458], X[35002] - 5 X[35383], 5 X[35375] - 4 X[47113]

See Manuel Aguilera, euclid 5696.

X(52993) lies on these lines: {3, 6}, {99, 5965}, {542, 33265}, {805, 5966}, {1352, 6658}, {5207, 40107}, {9166, 19924}, {9863, 33257}, {11178, 11361}, {14658, 39639}, {16044, 24206}, {29317, 39809}

X(52993) = midpoint of X(5104) and X(35383)
X(52993) = reflection of X(i) in X(j) for these {i,j}: {182, 35375}, {576, 1691}, {3098, 35383}, {5207, 40107}, {15514, 575}, {35377, 187}, {37517, 35377}, {39561, 38225}, {47619, 14810}
X(52993) = reflection of X(35377) in the Lemoine axis
X(52993) = {X(12974),X(12975)}-harmonic conjugate of X(5206)

---

X(52994) = X(3)X(6)∩X(23)X(11673)

Barycentrics    a^2*(a^8 - a^6*b^2 - 3*a^4*b^4 + 4*a^2*b^6 - b^8 - a^6*c^2 - 9*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 + 4*a^2*c^6 + 2*b^2*c^6 - c^8) : :
X(52994) = X[3] - 3 X[2076], 2 X[3] - 3 X[35375], 7 X[3] - 3 X[35458], 5 X[3] - 3 X[47619], 2 X[575] - 3 X[1691], 2 X[576] - 3 X[35377], 6 X[1692] - 5 X[22234], 4 X[2030] - 3 X[39561], 7 X[2076] - X[35458], 5 X[2076] - X[47619], 3 X[2456] - 4 X[20190], 2 X[5007] - 3 X[35376], 3 X[5050] - 4 X[8590], 2 X[5092] - 3 X[38225], 2 X[5107] - 3 X[15520], 3 X[5111] - 4 X[22330], 5 X[11482] - 3 X[15514], 3 X[17508] - 4 X[47113], 3 X[31884] - X[47618], 7 X[35375] - 2 X[35458], 5 X[35375] - 2 X[47619], 5 X[35458] - 7 X[47619], 2 X[3589] - 3 X[38230], 4 X[3628] - 3 X[5103], 2 X[10168] - 3 X[26613], 3 X[11261] + X[32430], 4 X[14693] - 3 X[38317], 2 X[19130] - 3 X[38227]

See Manuel Aguilera, euclid 5696.

X(52994) lies on these lines: {3, 6}, {23, 11673}, {69, 35951}, {140, 32172}, {316, 24206}, {384, 40107}, {542, 11676}, {732, 51524}, {1352, 14712}, {1995, 47638}, {3589, 38230}, {3628, 5103}, {3734, 44774}, {3849, 11178}, {3972, 22677}, {5476, 5569}, {5999, 6055}, {7496, 33873}, {7824, 25555}, {8594, 35918}, {8595, 35917}, {8860, 13860}, {10168, 26613}, {10594, 52460}, {10796, 11261}, {10997, 22564}, {11675, 15562}, {13586, 18800}, {14671, 49122}, {14693, 38317}, {16661, 31989}, {17006, 19130}, {29317, 38734}, {32135, 39099}, {35298, 36213}, {37914, 52162}, {43453, 48873}

X(52994) = midpoint of X(i) and X(j) for these {i,j}: {1350, 9301}, {1352, 14712}, {2080, 5104}, {43453, 48873}
X(52994) = reflection of X(i) in X(j) for these {i,j}: {182, 187}, {316, 24206}, {8586, 5097}, {35002, 14810}, {35375, 2076}, {39099, 32135}
X(52994) = reflection of X(182) in the Lemoine axis
X(52994) = circumcircle inverse of X(11171)
X(52994) = Brocard circle inverse of X(39498)
X(52994) = 1st Lemoine circle inverse of X(39515)
X(52994) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 39498}, {3, 576, 43157}, {187, 5052, 1691}, {1379, 1380, 11171}, {1662, 1663, 39515}, {2076, 5017, 187}, {2076, 5104, 5162}, {3098, 35424, 35422}, {38720, 38721, 38225}, {39498, 43152, 3}

---

X(52995) = X(3)X(6)∩X(30)X(5103)

Barycentrics    a^2*(a^8 - a^6*b^2 + a^4*b^4 - b^8 - a^6*c^2 - 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 - 2*b^2*c^6 - c^8) : :
X(52995) = 3 X[3] - X[2076], 3 X[3] + X[35458], 3 X[182] - 2 X[1692], X[182] + 2 X[18860], X[1692] + 3 X[18860], 4 X[1692] - 3 X[35377], 2 X[2076] - 3 X[35375], X[2076] + 3 X[47619], 3 X[2456] - X[5111], 2 X[5092] + X[35002], 5 X[12017] - 3 X[35006], 4 X[18860] + X[35377], 3 X[35375] + 2 X[35458], X[35375] + 2 X[47619], X[35458] - 3 X[47619], X[316] + 2 X[48892], 4 X[625] - X[48884], 2 X[13449] + X[48896]

See Manuel Aguilera, euclid 5696.

X(52995) lies on these lines: {3, 6}, {30, 5103}, {98, 8295}, {114, 5999}, {316, 7470}, {378, 52460}, {384, 19130}, {512, 7630}, {542, 5152}, {625, 48884}, {732, 12042}, {1352, 7836}, {1503, 51872}, {2071, 32529}, {3506, 36212}, {3552, 31670}, {3589, 44224}, {3788, 3818}, {5207, 7763}, {5989, 51373}, {6054, 22498}, {6636, 33873}, {7485, 47638}, {7711, 35265}, {7803, 43453}, {7832, 24206}, {7874, 24273}, {7880, 11178}, {8149, 14880}, {8841, 34130}, {10352, 10997}, {11673, 15246}, {11676, 29317}, {13449, 48896}, {13586, 19924}, {14134, 37126}, {14561, 35925}, {14660, 32526}, {16984, 37455}, {32432, 45543}, {32435, 45542}, {35951, 51538}, {36213, 37183}, {37457, 52658}, {38661, 52098}

X(52995) = midpoint of X(i) and X(j) for these {i,j}: {3, 47619}, {6, 35456}, {1691, 35002}, {2076, 35458}, {5207, 46264}, {15514, 33878}
X(52995) = reflection of X(i) in X(j) for these {i,j}: {1691, 5092}, {3818, 5031}, {35375, 3}, {35377, 182}, {35379, 13335}, {35383, 14810}, {37517, 1570}
X(52995) = circumcircle inverse of X(9821)
X(52995) = 1st Neuberg cirlce inverse of X(14880)
X(52995) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 182, 35422}, {3, 1350, 43152}, {3, 3095, 35385}, {3, 5116, 5092}, {3, 35002, 5162}, {3, 35458, 2076}, {39, 5092, 182}, {39, 18860, 35002}, {1379, 1380, 9821}, {2076, 47619, 35458}, {2076, 50659, 1692}, {3095, 39750, 35426}, {5999, 8290, 43460}, {7690, 7692, 9737}, {35422, 43157, 182}, {36755, 36756, 35002}, {38720, 38721, 35002}

---

X(52996) = X(3)X(6)∩X(542)X(7833)

Barycentrics    a^2*(2*a^4*b^4 - 3*a^2*b^6 + b^8 + 3*a^4*b^2*c^2 - 4*a^2*b^4*c^2 - 2*b^6*c^2 + 2*a^4*c^4 - 4*a^2*b^2*c^4 - 3*a^2*c^6 - 2*b^2*c^6 + c^8) : :
X(52996) = 3 X[182] - 4 X[13334], 2 X[575] - 3 X[11171], 3 X[3094] - X[3095], X[3095] - 6 X[43147], 3 X[3098] - 2 X[5188], 2 X[5052] - 3 X[39561], 2 X[5097] - 3 X[13331], 3 X[11171] - X[13330], X[11477] - 3 X[32447], 2 X[13354] - 3 X[17508], 5 X[22234] - 4 X[44500], 3 X[39561] - 4 X[50652], 2 X[5] - 3 X[11261], X[76] - 3 X[22677], 2 X[76] - 3 X[44774], 3 X[22677] - 2 X[40107], 4 X[40107] - 3 X[44774], 3 X[599] - X[13108], 5 X[631] - 3 X[31958], 7 X[3526] - 6 X[32149], 3 X[5476] - 4 X[11272], 2 X[49111] - 3 X[50977], 2 X[6248] - 3 X[11178], 5 X[7786] - 4 X[25555], 4 X[10007] - 3 X[38317], 3 X[11179] - 5 X[32522]

See Manuel Aguilera, euclid 5696.

X(52996) lies on these lines: {3, 6}, {5, 11261}, {76, 22677}, {524, 32448}, {542, 7833}, {598, 3399}, {599, 13108}, {631, 31958}, {698, 48876}, {1352, 6655}, {1916, 6036}, {2782, 7761}, {3526, 32149}, {3564, 32429}, {5025, 24206}, {5476, 11272}, {5969, 49102}, {6248, 7841}, {6776, 33260}, {7697, 7853}, {7709, 14907}, {7786, 25555}, {8550, 32516}, {10007, 38317}, {10168, 22486}, {10519, 45803}, {11179, 32522}, {11412, 14133}, {11649, 37991}, {18906, 32832}, {19130, 37446}, {23018, 51265}, {23024, 51272}, {32449, 34380}, {36790, 52658}, {39689, 40251}, {40279, 48901}

X(52996) = midpoint of X(3) and X(44453)
X(52996) = reflection of X(i) in X(j) for these {i,j}: {76, 40107}, {576, 39}, {1351, 44423}, {3094, 43147}, {5052, 50652}, {8550, 32516}, {13330, 575}, {22486, 10168}, {44774, 22677}
X(52996) = 2nd Brocard Circle inverse of X(2080)
X(52996) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 22677, 40107}, {76, 40107, 44774}, {182, 3098, 35375}, {576, 3098, 5171}, {1670, 1671, 2080}, {1689, 1690, 5038}, {3094, 44453, 32452}, {3102, 3103, 574}, {5052, 50652, 39561}, {8160, 8161, 47113}, {11171, 13330, 575}, {11477, 52771, 44508}, {13325, 13326, 3095}, {43120, 43121, 26316}

---

X(52997) = X(3)X(6)∩X(5)X(698)

Barycentrics    a^2*(a^2*b^6 - b^8 + a^4*b^2*c^2 + 4*a^2*b^4*c^2 + 4*a^2*b^2*c^4 + a^2*c^6 - c^8) : :
X(52997) = X[6] - 3 X[32447], 3 X[39] - X[13354], 3 X[39] - 2 X[50652], 3 X[182] - 2 X[13354], 3 X[182] - 4 X[50652], 2 X[575] - 3 X[13331], 3 X[3094] - 2 X[43147], 3 X[3095] + 2 X[43147], 2 X[5052] - 3 X[15520], 2 X[5092] - 3 X[11171], 4 X[13334] - 3 X[17508], 3 X[32447] - 2 X[44423], 3 X[262] - 2 X[19130], 2 X[141] - 3 X[11261], 4 X[141] - 3 X[44774], 5 X[3618] - 3 X[31958], 5 X[3618] - 6 X[51829], 4 X[11272] - 3 X[38317], 2 X[24256] - 3 X[38317], 3 X[7709] - X[46264], 3 X[14561] - X[18906], 3 X[10516] - X[13108], X[18440] + 3 X[32519], X[20081] - 5 X[40330], 3 X[22728] - X[48910], 6 X[32149] - 7 X[47355], X[41747] + 2 X[43150]

See Manuel Aguilera, euclid 5696.

X(52997) lies on these lines: {3, 6}, {5, 698}, {76, 3399}, {114, 262}, {141, 11261}, {147, 33693}, {194, 1352}, {302, 23018}, {303, 23024}, {538, 11178}, {542, 7757}, {732, 34507}, {736, 37242}, {882, 32473}, {1503, 32429}, {2782, 3818}, {3148, 3506}, {3564, 32449}, {3618, 31958}, {3788, 11272}, {3864, 31395}, {5149, 10796}, {5476, 5969}, {5965, 32451}, {7709, 46264}, {7763, 14561}, {7803, 12251}, {7806, 22712}, {7834, 10007}, {7874, 40332}, {7876, 40107}, {7892, 25555}, {8704, 22260}, {9744, 31670}, {9774, 19924}, {10516, 13108}, {11163, 44114}, {11257, 29012}, {11649, 37896}, {12111, 31989}, {14881, 48901}, {18440, 32519}, {20081, 40330}, {22486, 33246}, {22728, 48910}, {30499, 52091}, {31960, 52637}, {32149, 47355}, {32516, 44882}, {36384, 37785}, {36385, 37786}, {37466, 49792}, {40708, 41259}, {41747, 43150}

X(52997) = midpoint of X(i) and X(j) for these {i,j}: {194, 1352}, {1350, 48673}, {1351, 44453}, {3094, 3095}
X(52997) = reflection of X(i) in X(j) for these {i,j}: {6, 44423}, {76, 24206}, {182, 39}, {9821, 14810}, {13330, 5097}, {13354, 50652}, {24256, 11272}, {31958, 51829}, {32429, 32448}, {37517, 35439}, {44774, 11261}, {44882, 32516}, {48901, 14881}, {49111, 10007}
X(52997) = Brocard circle inverse of X(39750)
X(52997) = 2nd Brocard Circle inverse of X(3398)
X(52997) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 39750}, {6, 32447, 44423}, {39, 5052, 2024}, {39, 13354, 50652}, {39, 18860, 11171}, {182, 576, 35377}, {182, 3098, 35422}, {1670, 1671, 3398}, {1689, 1690, 1691}, {2021, 35432, 41412}, {3102, 3103, 32}, {3104, 3107, 36759}, {3105, 3106, 36760}, {11171, 13330, 8590}, {11272, 24256, 38317}, {13354, 50652, 182}, {35375, 35426, 32}, {35377, 43157, 182}

---

X(52998) = X(98)X(9381)∩X(110)X(6328)

Barycentrics   1/((b^2-c^2)(a^2-b^2-c^2) (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))) : :

See Kadir Altintas and Angel Montesdeoca, euclid 5705.

X(52998) lies on these lines: {2,46664}, {4,14979}, {30,18401}, {74,10421}, {98,9381}, {107,23290}, {110,6368}, {112,12077}, {186,1141}, {403,2383}, {427,842}, {468,5966}, {477,3520}, {523,933}, {550,2693}, {827,7473}, {925,39198}, {1287,4230}, {1291,14590}, {1294,18859}, {1297,5189}, {1300,37970}, {2070,39431}, {2073,26708}, {2074,26707}, {2697,6636}, {4226,11635}, {4240,16166}, {6240,32710}, {6242,15907}, {6353,23096}, {7476,26712}, {7482,44061}, {26711,37966}, {33643,37943}

X(52998)= reflection of X(933) in Euler line

---

X(52999) = X(1)X(167)∩X(9)X(18885)

Barycentrics    a*(cos(B/2)+cos(C/2)-cos(A/2))^2 : :
Barycentrics    Sin[A]*(Sec[A/2] + Tan[A/2])^2 : :

See César Lozada, euclid 5707.

X(52999) lies on these lines: {1, 167}, {9, 18885}, {57, 12809}, {258, 11923}, {1699, 8140}, {5666, 12523}, {7057, 10233}, {7271, 7371}, {7589, 45086}, {8056, 8078}, {9793, 10499}, {10504, 18886}, {16015, 45706}

X(52999) = Cevapoint of X(173) and X(2089)
X(52999) = X(2089)-Ceva conjugate of-X(173)
X(52999) = X(223)-Dao conjugate of-X(7002)
X(52999) = X(i)-isoconjugate-of-X(j) for these {i, j}: {55, 7002}, {258, 258}, {289, 7028}
X(52999) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (57, 7002), (173, 7048)
X(52999) = intersection, other than A, B, C, of circumconics {{A, B, C, X(8), X(8422)}} and {{A, B, C, X(173), X(174)}}
X(52999) = trilinear square of X(173)
X(52999) = barycentric product X(i)*X(j) for these {i, j}: {9, 7022}, {173, 7057}, {236, 2089}
X(52999) = barycentric quotient X(i)/X(j) for these (i, j): (57, 7002), (173, 7048)
X(52999) = trilinear product X(i)*X(j) for these {i, j}: {55, 7022}, {173, 173}
X(52999) = trilinear quotient X(i)/X(j) for these (i, j): (7, 7002), (173, 258), (236, 7028), (2089, 1488)
X(52999) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 53000, 8241), (1, 53001, 8242), (177, 13385, 8241), (177, 13443, 1), (6585, 12908, 1)

---

X(53000) = (X(8241), X(11924))-HARMONIC CONJUGATE OF X(11234)

Barycentrics    a*(4*b*c*(b-c)*(3*a^6+3*(b+c)*a^5-(6*b^2+11*b*c+6*c^2)*a^4-(b+c)*(6*b^2+7*b*c+6*c^2)*a^3+3*(b^2+3*b*c+c^2)*(b+c)^2*a^2+(b+c)*(3*b^4+3*c^4+b*c*(7*b^2-24*b*c+7*c^2))*a-4*(b^2-c^2)^2*b*c)*sin(A/2)+4*a*b*c*(a^6+(b-7*c)*a^5-(2*b^2-16*b*c+c^2)*a^4-(2*b^2-23*b*c+33*c^2)*a^3*b+(b^4+c^4-b*c*(23*b^2-32*b*c+7*c^2))*a^2+(b^2-c^2)*(b^3-7*c^3-16*b*c*(b-c))*a+(b^2-c^2)^2*(7*b-c)*c)*sin(B/2)-4*a*b*c*(a^6-(7*b-c)*a^5-(b^2-16*b*c+2*c^2)*a^4-(33*b^2-23*b*c+2*c^2)*a^3*c+(b^4+c^4-b*c*(7*b^2-32*b*c+23*c^2))*a^2+(b^2-c^2)*(7*b^3-c^3-16*b*c*(b-c))*a-(b^2-c^2)^2*(b-7*c)*b)*sin(C/2)+(b-c)*(a^8+(b+c)*a^7-3*(b^2-5*b*c+c^2)*a^6-(b+c)*(3*b^2+32*b*c+3*c^2)*a^5+(3*b^4+3*c^4+b*c*(3*b^2+116*b*c+3*c^2))*a^4+(b+c)*(3*b^4+3*c^4+2*b*c*(16*b^2-79*b*c+16*c^2))*a^3-(b^4+c^4+b*c*(17*b^2-64*b*c+17*c^2))*(b+c)^2*a^2-(b^2-c^2)*(b-c)*(b^4+c^4+2*b*c*(b^2-15*b*c+c^2))*a+(b^2-c^2)^2*b*c*(b^2-14*b*c+c^2))) : :

See César Lozada, euclid 5707.

X(53000) lies on this line: {1, 167}

X(53000) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (8241, 11924, 11234), (8241, 52999 , 1), (30374, 46370, 1)

---

X(53001) = X(1)X(167)∩X(18291)X(24242)

Barycentrics   a*(-4*b*c*(a^5+(b+c)*a^4-(5*b^2-23*b*c+5*c^2)*a^3-(b+c)*(b^2+16*b*c+c^2)*a^2+(4*b^2-3*b*c+4*c^2)*(b+c)^2*a+4*(b^2-c^2)*(b-c)*b*c)*sin(A/2)+4*a*b*c*(b-c)*(5*a^3+(b+18*c)*a^2-(5*b^2+12*b*c+11*c^2)*a-(b+c)*(b^2-11*b*c+8*c^2))*sin(B/2)-4*a*b*c*(b-c)*(5*a^3+(18*b+c)*a^2-(11*b^2+12*b*c+5*c^2)*a-(b+c)*(8*b^2-11*b*c+c^2))*sin(C/2)+(b+c)*a^6+(b+c)^2*a^5-(b+c)*(2*b^2-17*b*c+2*c^2)*a^4-2*(b^4+c^4+2*b*c*(9*b^2-b*c+9*c^2))*a^3+(b+c)*(b^4+c^4+2*b*c*(7*b^2-5*b*c+7*c^2))*a^2+(b^2-c^2)^2*(b+c)^2*a+(b^2-c^2)*(b-c)*(b^2-14*b*c+c^2)*b*c) : :

See César Lozada, euclid 5707.

X(53001) lies on these lines: {1, 167}, {18291, 24242}

X(53001) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (8242, 52999 , 1), (30423, 46370, 1)

---

X(53002) = CENTER OF 1st MIYAMOTO-LOZADA CIRCLE

Barycentrics    a^2*((b^2-4*b*c+c^2)*a^3+(b^2-c^2)*(b-c)*a^2-(b^4+c^4-4*(b^2+b*c+c^2)*b*c)*a-(b+c)*(b^4+c^4-2*(b-c)^2*b*c)) : :
X(53002) = 3*X(165)-X(970) = 3*X(376)+X(31785) = X(6361)+3*X(37521) = 3*X(9778)+X(10441) = X(20070)+3*X(39550)

In a scalene triangle ABC with circumcircle (O) and incircle (I), let A' be the midpoint of arc BC of (O) containing A. Define B' and C' cyclically.
Let o'_a be the circle centered at A' and tangent to BC, and denote o'_b, o'_c cyclically.
Let o"_a be the circle internally tangent to (O) at A and tangent to BC, and cyclically o"_b and o"_c.
Then there exists a circle o₁ simultaneously tangent to the seven circles (I), o'_a, o'_b, o'_c, o"_a, o"_b, o"_c, and touching (I) at X(1357). (Keita Miyamoto, Feb. 17, 2023).

The above circle o₁ is named here as the 1st Miyamoto-Lozada circle. It's center is X(53002) and it has radius (a^3+b^3-5*a*b*c+c^3)/(8*S). (César Lozada, Feb. 18, 2023)

X(53002) lies on these lines: {1, 1357}, {3, 595}, {5, 29349}, {40, 32913}, {46, 12109}, {100, 29958}, {165, 970}, {376, 31785}, {511, 3579}, {517, 548}, {1158, 2808}, {1293, 33771}, {2594, 23832}, {2810, 8715}, {2818, 26285}, {3145, 35281}, {3293, 48921}, {3667, 4075}, {3871, 3937}, {4256, 38620}, {4650, 50583}, {5482, 28174}, {5493, 35631}, {5901, 14131}, {6361, 37521}, {6684, 15310}, {9778, 10441}, {9957, 37743}, {11499, 44865}, {15171, 35059}, {15489, 31663}, {17613, 46850}, {18483, 29229}, {20070, 39550}, {27529, 38389}, {29309, 37536}, {34583, 37722}, {37619, 48893}

X(53002) = midpoint of X(5493) and X(35631)
X(53002) = reflection of X(i) in X(j) for these (i, j): (5901, 14131), (15489, 31663)

---

X(53003) = CENTER OF 2nd MIYAMOTO-LOZADA CIRCLE

Barycentrics    a^2*((b^2-c^2)^2*a^14-2*(b+c)*(b^4+c^4-(b^2-b*c+c^2)*b*c)*a^13-4*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^12-2*(4*b^6+4*c^6-(11*b^2+2*b*c+11*c^2)*b^2*c^2)*a^8*b^2*c^2+2*(b+c)*(5*b^6+5*c^6-(5*b^4+5*c^4-4*(b^2-b*c+c^2)*b*c)*b*c)*a^11+(5*b^8+5*c^8-8*(b^4+3*b^2*c^2+c^4)*b^2*c^2)*a^10-4*(b+c)*(5*b^8+5*c^8-(5*b^6+5*c^6-(3*b^4+3*c^4-(3*b^2-2*b*c+3*c^2)*b*c)*b*c)*b*c)*a^9-2*(b^2-c^2)^4*(b-c)^2*b^4*c^4+4*(b^2-c^2)*(b-c)*(5*b^8+5*c^8+(5*b^6+5*c^6+(7*b^4+7*c^4+(7*b^2+6*b*c+7*c^2)*b*c)*b*c)*b*c)*a^7-(b^2-c^2)^2*(5*b^8+5*c^8-2*(6*b^4+7*b^2*c^2+6*c^4)*b^2*c^2)*a^6-2*(b^4-c^4)*(b^2-c^2)*(b+c)*(5*b^6+5*c^6-(5*b^4+5*c^4-6*(b^2-b*c+c^2)*b*c)*b*c)*a^5+4*(b^2-c^2)^2*(b^10+c^10-2*(b^2-b*c+c^2)*(b^4+c^4+(b^2+b*c+c^2)*b*c)*b^2*c^2)*a^4+2*(b^4-c^4)^2*(b^2-c^2)*(b-c)*(b^4+c^4+(b^2+b*c+c^2)*b*c)*a^3-(b^2-c^2)^4*(b^8-4*b^4*c^4+c^8)*a^2) : :

In a scalene triangle ABC with circumcenter O, let A'B'C' be the circumcevian triangle of O. Let A_b=BC∩C'A' and A_c=BC∩A'B' and denote B_c, C_a, B_a, C_b cyclically. Let (I), i_a, i_b and i_c be the incircles of ABC, A'A_bA_c, B'B_cB_a and C'C_aC_b, respectively. Then, there exists a circle o₂ that is simultaneously tangent to the four circles (I), i_a, i_b and i_c and touching (I) at X(15616).. (Keita Miyamoto, Feb. 17, 2023).

The above circle o₂ is named here the 2nd Miyamoto-Lozada circle. It's center is X(53003) and it has radius:
|(S^6-4*(8*R^2+8*R*r-3*r^2)*r^2*S^4-16*(12*R^2+4*R*r-3*r^2)*(2*R+r)^2*r^4*S^2+64*(4*R+r)^2*(2*R+r)^4*r^6)/(64*S^2*R*r^2*(4*r^2*(2*R+r)^2-S^2))|. (César Lozada, Feb. 18, 2023)

X(53003) lies on the line {1, 15616}

---

X(53004) = MIYAMOTO-LOZADA PERSPECTOR

Barycentrics    (b+c)*(3*a^3+(3*b+5*c)*a^2+(b^2+10*b*c+5*c^2)*a+(b+c)*(b^2+3*c^2))*(3*a^3+(5*b+3*c)*a^2+(5*b^2+10*b*c+c^2)*a+(b+c)*(3*b^2+c^2)) : :

Let (I_a), (I_b) and (I_c) be the A-, B-, C- excircles of a given triangle ABC, respectively, and let (N) be its nine-point circle. Let (J_a) be the circle externally tangent to (I_b) and (I_c), and internally tangent to (I_a). Define (J_b) and (J_c) cyclically. Inside the triangle ABC, let (K_a) be the circle tangent to BC, externally tangent to (J_b) (J_c) (this circle results also to be internally tangent to (N)) and define (K_b) and (K_c) cyclically. Let A'B'C' be the triangle bounded by the pairwise tangent lines to (K_a),(K_b) and (K_c), and such that these circles lie all in the interior region of A'B'C'. Then, A'B'C' is perspective with ABC.. (Keita Miyamoto, Feb. 17, 2023).

The above perspector is named here the Miyamoto-Lozada perspector and it is X(53004). (César Lozada, Feb. 18, 2023)

X(53004) lies on the Kiepert circumhyperbola, the cubics K1219, K1234, K1235, the curve Q172 and these lines: {2, 6}, {4052, 49598}, {4673, 34258}, {13478, 16458}

---

X(53005) = X(10)X(354) ∩ X(165)X(970)

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

Continuing with construction in X(53002), let T'_a, T'_b, T'_c be the touchpoints of o₁ and o'_a, o'_b, o'_c. Triangle T'_aT'_bT'_c is perspective to the following triangles with indicated perspector: Apollonius with homothetic center X(53005), 1st-circumperp with perspector X(53002) and tangential with perspector X(595).
Let T"_a, T"_b, T"_c be the touchpoints of o₁ and o"_a, o"_b, o"_c. Triangle T"_aT"_bT"_c is perspective to ABC with perspector X(595). (César Lozada, Feb. 18, 2023)

X(53005) lies on these lines: {2, 35892}, {10, 354}, {40, 10823}, {42, 28272}, {43, 57}, {55, 386}, {165, 970}, {672, 2092}, {899, 40952}, {1155, 2392}, {1200, 23637}, {1738, 39793}, {2051, 7965}, {2999, 3779}, {3216, 28250}, {3271, 32911}, {3666, 20683}, {3688, 5256}, {3690, 46904}, {3752, 22277}, {3819, 4649}, {3873, 24988}, {3937, 4722}, {4259, 23638}, {4383, 21746}, {4890, 44307}, {5273, 9564}, {5432, 37993}, {5530, 11018}, {5918, 10443}, {6007, 27064}, {6048, 10980}, {6244, 9567}, {7064, 28606}, {7994, 9549}, {9052, 29821}, {9534, 26040}, {9548, 10857}, {10460, 50032}, {17123, 39543}, {35645, 50282}

X(53005) = crossdifference of every pair of points on line {X(4435), X(17166)}
X(53005) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (43, 181, 3030), (43, 4260, 181), (165, 970, 10824), (386, 10822, 1682)

   Miyamoto-Moses Points: X(53006)-X(53007)  

This preamble continues the preamble just before X(52805). It is based on proposition 3 by Keita Miyamoto, with barycentrics found by Peter Moses.

Proposition 3.

In a scalene acute triangle ABC, let T = A'B'C' be the intouch triangle of ABC. Let γ be the incircle of T, and let Γa be the A-excircle of AB'C'. Define Γb and Γc cyclically. Let γa be the circle tangent to B'C' and internally tangent to the incircle of ABC at A'. Define γb and γc cyclically. Then there exists a circle ω that is tangent to all seven of the circles γ, Γa, Γb, Γc, γa, γb, γc. Here the circle ω is named the Miyamoto-Moses circle.

The center of Γa, which is the A-vertex of the 2nd midarc triangle (see X(10491), given by

a*b - b^2 + a*c + 2*b*c - c^2 - 2*x : b*(-a + b + c) : -((a - b - c)*c : : , where

x = sqrt[b*c*(a + b - c)*(a - b + c)] = 2*b*c sin(A/2), and y and z are defined cyclically.

The radius of Γa, is

((2*b*c + x)*S)/(2*b*c*(a + b + c)) = r*(1 + Sin[A/2])

The touch-point of Γa and ω, denoted by a2 in Proposition 3 (GeoGebra)., is given by

(a + b - c)*(a - b + c)*(2*a^3 - 3*a^2*b + 2*a*b^2 - b^3 - 3*a^2*c - 24*a*b*c + b^2*c + 2*a*c^2 + b*c^2 - c^3) + 4*(2*b*c*(a^3 + a^2*b - 3*a*b^2 + b^3 + a^2*c + 6*a*b*c - b^2*c - 3*a*c^2 - b*c^2 + c^3)*Sin[A/2] - a*(c*(a - b + c)*(a^2 + 4*a*b + 7*b^2 - 2*a*c - 8*b*c + c^2)*Sin[B/2] + b*(a + b - c)*(a^2 - 2*a*b + b^2 + 4*a*c - 8*b*c + 7*c^2)*Sin[C/2]))
: -((a - b - c)*(a + b - c)*(a + b + c)*(a^2 - a*b - 2*a*c - b*c + c^2 - 4*a*c*Sin[B/2]))
: -((a - b - c)*(a - b + c)*(a + b + c)*(a^2 - 2*a*b + b^2 - a*c - b*c - 4*a*b*Sin[C/2]))

The center of the circle γa is given by

2*a*x : 2*b*(a + b - c)*c + x*(-a + b + c) : 2*b*c*(a - b + c) + x*(-a + b + c)

Let Z₆ be the touch-point of the circles ω and γ. Then Z₆ = X(53006), and Z₆-of-T = X(1357).

Let Z₇ by the center of circle ω. Then Z₇ = X(53007), and Z₇-of-T = X(53002). The triangle a2b2c2 is perspective to ABC at X(10489) and to the 2nd mid-arc triangle at X(53007). The triangle a3b3c3 is perspective to the intouch triangle at X(10489).

X(53006) = 3RD MIYAMOTO-MOSES POINT

Barycentrics    (a + b - c)*(a - b + c)*(2*a^4 - 9*a^3*b + 13*a^2*b^2 - 7*a*b^3 + b^4 - 9*a^3*c + 2*a^2*b*c + 7*a*b^2*c + 13*a^2*c^2 + 7*a*b*c^2 - 2*b^2*c^2 - 7*a*c^3 + c^4 + 2*z*(2*a^2 - 4*a*b + 2*b^2 - a*c - b*c - c^2) + 2*y*(2*a^2 - a*b - b^2 - 4*a*c - b*c + 2*c^2)) : :
Barycentrics    (a + b - c)*(a - b + c)*((a - b - c)*(2*a^3 - 7*a^2*b + 6*a*b^2 - b^3 - 7*a^2*c - 12*a*b*c + b^2*c + 6*a*c^2 + b*c^2 - c^3) + 4*a*(c*(2*a^2 - a*b - b^2 - 4*a*c - b*c + 2*c^2)*Sin[B/2] + b*(2*a^2 - 4*a*b + 2*b^2 - a*c - b*c - c^2)*Sin[C/2])) : :

X(53006) lies on this line: {2089, 10215}

X(53006) = X(1357)-of-intouch-triangle

---

X(53007) = CENTER OF MIYAMOTO-MOSES CIRCLE

Barycentrics    (a^2 - 2*a*b + b^2 - 2*a*c - 2*b*c + c^2)*(2*a^3 - a^2*b - b^3 - a^2*c + b^2*c + b*c^2 - c^3) + 4*(b*c*(4*a^3 - 5*a^2*b + b^3 - 5*a^2*c - b^2*c - b*c^2 + c^3)*Sin[A/2] - a^2*(c*(a^2 + 4*a*b - 5*b^2 - 2*a*c + 4*b*c + c^2)*Sin[B/2] + b*(a^2 - 2*a*b + b^2 + 4*a*c + 4*b*c - 5*c^2)*Sin[C/2])) : :

X(53007) lies on these lines: {1, 10489}, {516, 32183}

---

X(53008) = X(10)X(201)∩X(33)X(200)

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

X(53008) lies on the cubic K1311 and these lines: {2, 23710}, {4, 3679}, {8, 28950}, {10, 201}, {19, 5282}, {25, 8756}, {29, 4102}, {33, 200}, {34, 9623}, {42, 52335}, {55, 23529}, {65, 21924}, {72, 1825}, {73, 39130}, {92, 1861}, {210, 430}, {226, 21911}, {318, 341}, {406, 45701}, {429, 21031}, {451, 3584}, {519, 5136}, {612, 8755}, {862, 43265}, {1068, 1698}, {1096, 23050}, {1146, 1864}, {1334, 1857}, {1426, 3698}, {1738, 17890}, {1783, 3997}, {1835, 3753}, {1842, 5090}, {1848, 32778}, {1855, 1859}, {1867, 1869}, {1897, 52412}, {2299, 2322}, {2356, 17911}, {3064, 28143}, {3624, 38295}, {4183, 36910}, {4515, 6057}, {5258, 37117}, {5325, 52891}, {7004, 20205}, {7009, 17927}, {7069, 20262}, {7076, 8750}, {7628, 7649}, {7952, 31434}, {9578, 14257}, {11323, 17281}, {21016, 52577}, {23661, 34823}

X(53008) = polar conjugate of X(1434)
X(53008) = polar conjugate of the isotomic conjugate of X(2321)
X(53008) = polar conjugate of the isogonal conjugate of X(1334)
X(53008) = X(i)-Ceva conjugate of X(j) for these (i,j): {7046, 210}, {41013, 1826}
X(53008) = X(i)-isoconjugate of X(j) for these (i,j): {3, 1014}, {7, 1437}, {21, 7053}, {27, 7125}, {28, 1804}, {48, 1434}, {56, 1444}, {57, 1790}, {58, 77}, {60, 1439}, {63, 1412}, {69, 1408}, {72, 7341}, {73, 757}, {81, 222}, {86, 603}, {228, 552}, {261, 1410}, {269, 283}, {274, 52411}, {278, 18604}, {279, 2193}, {284, 7177}, {286, 7335}, {304, 16947}, {307, 849}, {332, 1106}, {333, 7099}, {348, 1333}, {394, 1396}, {593, 1214}, {604, 17206}, {651, 7254}, {652, 4637}, {738, 2327}, {763, 2197}, {905, 4565}, {906, 17096}, {934, 23189}, {1019, 1813}, {1331, 7203}, {1407, 1812}, {1409, 1509}, {1414, 1459}, {1415, 15419}, {1435, 6514}, {1474, 7183}, {1792, 7023}, {1798, 24471}, {1803, 18164}, {1946, 4616}, {2185, 52373}, {2194, 7056}, {2203, 7055}, {2206, 7182}, {3669, 4558}, {3676, 4575}, {3733, 6516}, {3942, 52378}, {4556, 51664}, {4573, 22383}, {4592, 43924}, {4617, 23090}, {7192, 36059}, {7199, 32660}, {7342, 20336}, {16726, 44717}, {17219, 24027}, {24002, 32661}, {27832, 33628}
X(53008) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 1444}, {10, 77}, {37, 348}, {136, 3676}, {522, 17219}, {1146, 15419}, {1214, 7056}, {1249, 1434}, {3161, 17206}, {3162, 1412}, {4075, 307}, {5139, 43924}, {5190, 17096}, {5452, 1790}, {5521, 7203}, {6552, 332}, {6600, 283}, {6741, 4025}, {7952, 86}, {14714, 23189}, {20620, 7192}, {23050, 21}, {24771, 1812}, {36103, 1014}, {38966, 3737}, {38991, 7254}, {39053, 4616}, {39060, 4635}, {40586, 222}, {40590, 7177}, {40591, 1804}, {40599, 63}, {40600, 603}, {40603, 7182}, {40607, 73}, {40608, 1459}, {40611, 7053}, {47345, 279}, {51574, 7183}
X(53008) = cevapoint of X(i) and X(j) for these (i,j): {756, 7140}, {4041, 52335}
X(53008) = barycentric product X(i)*X(j) for these {i,j}: {4, 2321}, {8, 1826}, {9, 41013}, {10, 281}, {12, 2322}, {19, 3701}, {25, 30713}, {27, 6057}, {29, 594}, {33, 321}, {37, 318}, {42, 7017}, {65, 7101}, {92, 210}, {158, 3694}, {200, 40149}, {225, 346}, {226, 7046}, {264, 1334}, {273, 4515}, {278, 4082}, {284, 7141}, {306, 1857}, {312, 1824}, {313, 607}, {333, 7140}, {341, 1880}, {349, 7071}, {393, 3710}, {430, 4102}, {644, 24006}, {756, 31623}, {860, 36910}, {1018, 44426}, {1043, 8736}, {1089, 1172}, {1253, 52575}, {1426, 30693}, {1441, 7079}, {1500, 44130}, {1783, 4086}, {1840, 4451}, {1896, 3949}, {1897, 3700}, {2052, 2318}, {2212, 27801}, {2299, 28654}, {2332, 34388}, {2333, 3596}, {2501, 3699}, {3064, 3952}, {3695, 8748}, {3939, 14618}, {4024, 36797}, {4033, 18344}, {4041, 6335}, {4069, 17924}, {4163, 52607}, {4171, 18026}, {4183, 6358}, {4524, 46404}, {4557, 46110}, {6059, 40071}, {6535, 46103}, {7003, 21075}, {7020, 21871}, {7064, 44129}, {7649, 30730}, {14624, 46878}, {15742, 21044}, {21074, 43742}, {33635, 44143}, {40447, 40967}, {46102, 52335}
X(53008) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 1434}, {8, 17206}, {9, 1444}, {10, 348}, {19, 1014}, {25, 1412}, {27, 552}, {29, 1509}, {33, 81}, {37, 77}, {41, 1437}, {42, 222}, {55, 1790}, {65, 7177}, {71, 1804}, {72, 7183}, {108, 4637}, {181, 52373}, {200, 1812}, {210, 63}, {212, 18604}, {213, 603}, {220, 283}, {225, 279}, {226, 7056}, {228, 7125}, {270, 763}, {281, 86}, {306, 7055}, {318, 274}, {321, 7182}, {346, 332}, {429, 3674}, {430, 553}, {461, 42028}, {480, 2327}, {522, 15419}, {594, 307}, {607, 58}, {644, 4592}, {653, 4616}, {657, 23189}, {663, 7254}, {728, 1792}, {756, 1214}, {762, 201}, {860, 17078}, {862, 1429}, {872, 1409}, {1018, 6516}, {1089, 1231}, {1096, 1396}, {1146, 17219}, {1172, 757}, {1253, 2193}, {1260, 6514}, {1334, 3}, {1400, 7053}, {1402, 7099}, {1426, 738}, {1474, 7341}, {1500, 73}, {1783, 1414}, {1824, 57}, {1826, 7}, {1827, 18164}, {1840, 7176}, {1855, 17169}, {1857, 27}, {1880, 269}, {1893, 42309}, {1897, 4573}, {1918, 52411}, {1973, 1408}, {1974, 16947}, {2171, 1439}, {2200, 7335}, {2204, 849}, {2212, 1333}, {2299, 593}, {2318, 394}, {2321, 69}, {2322, 261}, {2332, 60}, {2333, 56}, {2489, 43924}, {2501, 3676}, {3064, 7192}, {3668, 30682}, {3690, 40152}, {3694, 326}, {3695, 52565}, {3699, 4563}, {3700, 4025}, {3701, 304}, {3709, 1459}, {3710, 3926}, {3939, 4558}, {3949, 52385}, {4024, 17094}, {4041, 905}, {4046, 4001}, {4069, 1332}, {4082, 345}, {4086, 15413}, {4092, 4466}, {4105, 23090}, {4163, 15411}, {4171, 521}, {4183, 2185}, {4433, 20769}, {4515, 78}, {4516, 3942}, {4524, 652}, {4557, 1813}, {4574, 6517}, {4705, 51664}, {4878, 23144}, {6057, 306}, {6059, 1474}, {6335, 4625}, {6535, 26942}, {6591, 7203}, {7017, 310}, {7046, 333}, {7064, 71}, {7069, 16697}, {7071, 284}, {7079, 21}, {7101, 314}, {7140, 226}, {7141, 349}, {7368, 1819}, {7649, 17096}, {8611, 4131}, {8735, 17205}, {8736, 3668}, {8750, 4565}, {14618, 52621}, {15742, 4620}, {18026, 4635}, {18344, 1019}, {20684, 3784}, {21016, 3665}, {21044, 1565}, {21046, 1367}, {21060, 50559}, {21795, 22053}, {21807, 44708}, {21871, 7013}, {24006, 24002}, {28120, 35935}, {30713, 305}, {30730, 4561}, {31623, 873}, {36197, 7004}, {36797, 4610}, {39130, 34400}, {40149, 1088}, {40965, 7289}, {40966, 22097}, {40967, 18607}, {40971, 1817}, {40976, 40153}, {41013, 85}, {42069, 17197}, {44426, 7199}, {46103, 6628}, {46110, 52619}, {46878, 16705}, {52335, 26932}, {52355, 30805}, {52370, 255}, {52577, 7195}, {52607, 4626}
X(53008) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 41013, 225}, {281, 7046, 33}, {1824, 7140, 1826}

---

X(53009) = X(9)X(8755)∩X(12)X(37)

Barycentrics    (b + c)*(-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) : :

X(53009) lies on the cubic K1311 and these lines: {9, 8755}, {12, 37}, {53, 1855}, {198, 46836}, {281, 1785}, {393, 7079}, {1103, 2324}, {1783, 52405}, {1903, 51421}, {3663, 37805}, {3914, 21924}, {4656, 40149}, {5514, 40943}, {5930, 41087}, {6335, 17787}, {7101, 17555}, {17353, 17918}, {20263, 22063}, {21935, 24005}, {23710, 52033}

X(53009) = polar conjugate of the isotomic conjugate of X(21075)
X(53009) = X(i)-isoconjugate of X(j) for these (i,j): {58, 41081}, {60, 52037}, {81, 1433}, {84, 1790}, {189, 1437}, {222, 285}, {268, 1014}, {271, 1412}, {283, 1422}, {593, 52389}, {757, 41087}, {1408, 44189}, {1413, 1812}, {1434, 2188}, {1436, 1444}, {1440, 2193}, {1792, 6612}, {2194, 34400}, {2208, 17206}, {7254, 13138}, {15419, 32652}, {18604, 40836}, {23189, 37141}
X(53009) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 41081}, {281, 86}, {1214, 34400}, {16596, 15419}, {40586, 1433}, {40599, 271}, {40607, 41087}, {47345, 1440}
X(53009) = barycentric product X(i)*X(j) for these {i,j}: {4, 21075}, {10, 7952}, {40, 41013}, {72, 47372}, {92, 21871}, {196, 2321}, {208, 3701}, {210, 342}, {225, 7080}, {227, 318}, {313, 3195}, {321, 2331}, {322, 1824}, {329, 1826}, {594, 41083}, {1089, 3194}, {1334, 40701}, {1441, 40971}, {2324, 40149}, {2360, 7141}, {3209, 30713}, {7140, 8822}, {8736, 27398}
X(53009) = barycentric quotient X(i)/X(j) for these {i,j}: {33, 285}, {37, 41081}, {40, 1444}, {42, 1433}, {196, 1434}, {198, 1790}, {208, 1014}, {210, 271}, {225, 1440}, {226, 34400}, {227, 77}, {329, 17206}, {756, 52389}, {1334, 268}, {1500, 41087}, {1824, 84}, {1826, 189}, {1880, 1422}, {2171, 52037}, {2187, 1437}, {2321, 44189}, {2324, 1812}, {2331, 81}, {2333, 1436}, {3194, 757}, {3195, 58}, {3209, 1412}, {7074, 283}, {7080, 332}, {7140, 39130}, {7368, 2327}, {7952, 86}, {8736, 8808}, {14837, 15419}, {21075, 69}, {21871, 63}, {38357, 17219}, {38362, 17205}, {40971, 21}, {41013, 309}, {41083, 1509}, {47372, 286}

---

X(53010) = X(71)X(210)∩X(201)X(594)

Barycentrics    a*(b + 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(53010) lies on the cubic K1311 and these lines: {71, 210}, {201, 594}, {282, 1794}, {1436, 2933}, {2331, 44692}, {2983, 7129}, {3682, 3694}, {3949, 7066}, {6355, 26942}, {42699, 52396}

X(53010) = X(i)-isoconjugate of X(j) for these (i,j): {27, 2360}, {28, 1817}, {58, 41083}, {60, 196}, {81, 3194}, {208, 2185}, {221, 46103}, {223, 270}, {249, 38362}, {261, 3209}, {342, 2150}, {347, 2189}, {593, 7952}, {757, 2331}, {1474, 8822}, {1509, 3195}, {7078, 36419}
X(53010) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 41083}, {3341, 46103}, {40586, 3194}, {40591, 1817}, {40607, 2331}, {51574, 8822}
X(53010) = barycentric product X(i)*X(j) for these {i,j}: {10, 52389}, {12, 271}, {72, 39130}, {84, 3695}, {189, 3949}, {200, 6355}, {201, 280}, {268, 6358}, {282, 26942}, {306, 1903}, {309, 3690}, {321, 41087}, {594, 41081}, {1089, 1433}, {1436, 52369}, {2171, 44189}, {2188, 34388}, {2197, 34404}, {2321, 52037}, {2357, 20336}, {3692, 13853}, {3694, 8808}, {3710, 52384}, {4064, 13138}, {7020, 7066}, {40836, 52387}
X(53010) = barycentric quotient X(i)/X(j) for these {i,j}: {12, 342}, {37, 41083}, {42, 3194}, {71, 1817}, {72, 8822}, {181, 208}, {201, 347}, {228, 2360}, {268, 2185}, {271, 261}, {282, 46103}, {756, 7952}, {872, 3195}, {1433, 757}, {1500, 2331}, {1903, 27}, {2171, 196}, {2188, 60}, {2192, 270}, {2197, 223}, {2357, 28}, {2643, 38362}, {3690, 40}, {3694, 27398}, {3695, 322}, {3949, 329}, {4064, 17896}, {6355, 1088}, {6358, 40701}, {7064, 40971}, {7066, 7013}, {7118, 2189}, {7129, 36419}, {7140, 47372}, {7367, 2326}, {13853, 1847}, {26942, 40702}, {37755, 14256}, {39130, 286}, {40117, 52919}, {41081, 1509}, {41086, 44698}, {41087, 81}, {44189, 52379}, {52037, 1434}, {52389, 86}

---

X(53011) = X(10)X(459)∩X(33)X(42)

Barycentrics    (b + c)*(-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) : :

X(53011) lies on the cubic K1312 and these lines: {10, 459}, {25, 40128}, {33, 42}, {55, 8755}, {65, 225}, {71, 8802}, {204, 1249}, {281, 37553}, {306, 1897}, {1785, 18397}, {1895, 15466}, {3755, 40149}, {3772, 23710}, {4055, 8750}, {6525, 7156}, {9118, 19614}, {13405, 17902}, {18678, 33993}, {30456, 41086}

X(53011) = polar conjugate of the isotomic conjugate of X(8804)
X(53011) = X(i)-Ceva conjugate of X(j) for these (i,j): {10, 1826}, {7046, 1824}, {7952, 37}, {44695, 3198}
X(53011) = X(i)-isoconjugate of X(j) for these (i,j): {28, 15394}, {58, 19611}, {64, 1444}, {77, 52158}, {81, 1073}, {86, 19614}, {253, 1437}, {274, 14642}, {283, 8809}, {286, 14379}, {459, 18604}, {603, 5931}, {905, 46639}, {1301, 4131}, {1333, 34403}, {1433, 41082}, {1790, 2184}, {2155, 17206}, {22383, 44326}
X(53011) = X(i)-Dao conjugate of X(j) for these (i,j): {4, 86}, {10, 19611}, {37, 34403}, {122, 4025}, {1427, 7056}, {7952, 5931}, {39020, 30805}, {40586, 1073}, {40591, 15394}, {40600, 19614}, {40616, 15419}, {45245, 17206}
X(53011) = crossdifference of every pair of points on line {4091, 23090}
X(53011) = barycentric product X(i)*X(j) for these {i,j}: {4, 8804}, {10, 1249}, {19, 52345}, {20, 1826}, {37, 1895}, {42, 15466}, {71, 14249}, {92, 3198}, {190, 44705}, {204, 321}, {210, 44697}, {225, 27382}, {226, 44695}, {281, 5930}, {306, 6525}, {313, 3172}, {318, 30456}, {594, 44698}, {610, 41013}, {653, 14308}, {1096, 42699}, {1441, 7156}, {1783, 17898}, {1824, 18750}, {1880, 52346}, {1897, 6587}, {2321, 44696}, {2333, 14615}, {3213, 3701}, {4024, 52913}, {7046, 36908}, {7070, 40149}, {7101, 40933}, {21011, 38808}, {21044, 44699}
X(53011) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 34403}, {20, 17206}, {37, 19611}, {42, 1073}, {71, 15394}, {154, 1790}, {204, 81}, {213, 19614}, {281, 5931}, {607, 52158}, {610, 1444}, {1249, 86}, {1562, 17216}, {1824, 2184}, {1826, 253}, {1880, 8809}, {1895, 274}, {1897, 44326}, {1918, 14642}, {2200, 14379}, {2331, 41082}, {2333, 64}, {3172, 58}, {3198, 63}, {3213, 1014}, {4064, 14638}, {5930, 348}, {6525, 27}, {6587, 4025}, {7070, 1812}, {7156, 21}, {8057, 30805}, {8750, 46639}, {8804, 69}, {14249, 44129}, {14308, 6332}, {15466, 310}, {17898, 15413}, {21172, 15419}, {27382, 332}, {30456, 77}, {36908, 7056}, {40933, 7177}, {41086, 41081}, {42658, 4091}, {44695, 333}, {44696, 1434}, {44698, 1509}, {44699, 4620}, {44704, 51370}, {44705, 514}, {52078, 34400}, {52345, 304}, {52913, 4610}
X(53011) = {X(1249),X(44695)}-harmonic conjugate of X(204)

---

X(53012) = X(10)X(459)∩X(64)X(71)

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

X(53012) lies on the cubic K1312 and these lines: {10, 459}, {42, 41088}, {64, 71}, {201, 210}, {1073, 1260}, {1265, 19611}, {1745, 2184}, {1794, 52158}, {2318, 7066}

X(53012) = isogonal conjugate of X(44698)
X(53012) = X(i)-isoconjugate of X(j) for these (i,j): {1, 44698}, {20, 28}, {21, 44696}, {27, 610}, {29, 1394}, {58, 1895}, {81, 1249}, {86, 204}, {154, 286}, {162, 21172}, {270, 5930}, {274, 3172}, {284, 44697}, {333, 3213}, {513, 52913}, {1014, 44695}, {1172, 18623}, {1333, 15466}, {1396, 27382}, {1434, 7156}, {1437, 14249}, {1444, 6525}, {1474, 18750}, {2203, 14615}, {2299, 33673}, {2326, 36908}, {3194, 41084}, {5317, 37669}, {6591, 36841}, {8057, 52920}, {10152, 51420}, {16747, 51508}, {18180, 38808}, {18191, 44699}, {30456, 46103}, {36420, 42699}, {44705, 52935}
X(53012) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 44698}, {10, 1895}, {37, 15466}, {125, 21172}, {226, 33673}, {3343, 86}, {14092, 27}, {14390, 1790}, {39026, 52913}, {40586, 1249}, {40590, 44697}, {40591, 20}, {40600, 204}, {40611, 44696}, {51574, 18750}
X(53012) = barycentric product X(i)*X(j) for these {i,j}: {10, 1073}, {37, 19611}, {42, 34403}, {64, 306}, {71, 253}, {72, 2184}, {307, 30457}, {313, 14642}, {321, 19614}, {459, 3682}, {1214, 44692}, {1826, 15394}, {2155, 20336}, {2197, 5931}, {2200, 41530}, {3694, 8809}, {4055, 52581}, {4064, 46639}, {8750, 14638}, {8804, 52559}, {26942, 52158}, {33581, 40071}, {41489, 52396}
X(53012) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 44698}, {10, 15466}, {37, 1895}, {42, 1249}, {64, 27}, {65, 44697}, {71, 20}, {72, 18750}, {73, 18623}, {101, 52913}, {213, 204}, {228, 610}, {253, 44129}, {306, 14615}, {647, 21172}, {1073, 86}, {1214, 33673}, {1301, 52919}, {1331, 36841}, {1334, 44695}, {1400, 44696}, {1402, 3213}, {1409, 1394}, {1425, 36908}, {1826, 14249}, {1918, 3172}, {2155, 28}, {2184, 286}, {2197, 5930}, {2200, 154}, {2318, 27382}, {2333, 6525}, {3682, 37669}, {3690, 8804}, {3694, 52346}, {3949, 52345}, {4055, 15905}, {4079, 44705}, {8798, 17167}, {8804, 52578}, {11589, 18653}, {14379, 1790}, {14642, 58}, {15394, 17206}, {19611, 274}, {19614, 81}, {30457, 29}, {33581, 1474}, {34403, 310}, {41087, 41084}, {41088, 41083}, {41489, 8747}, {44692, 31623}, {52158, 46103}, {52370, 7070}, {52387, 42699}

---

X(53013) = X(2)X(20221)∩X(10)X(227)

Barycentrics    a*(a - b - c)*(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) : :

X(53013) lies on the cubic K1312 and these lines: {2, 20221}, {10, 227}, {33, 30457}, {42, 41086}, {55, 7008}, {71, 210}, {84, 165}, {197, 1436}, {200, 219}, {280, 341}, {309, 30758}, {612, 2336}, {1854, 44692}, {1856, 5514}, {2318, 4515}, {2335, 7003}, {2358, 43214}, {3085, 40836}, {3694, 4082}, {3925, 13853}, {3992, 40161}, {5777, 44074}, {41539, 52037}

X(53013) = complement of X(20221)
X(53013) = X(i)-Ceva conjugate of X(j) for these (i,j): {39130, 1903}, {52389, 37}
X(53013) = X(i)-isoconjugate of X(j) for these (i,j): {7, 2360}, {27, 7011}, {28, 7013}, {40, 1014}, {56, 8822}, {57, 1817}, {58, 347}, {77, 3194}, {81, 223}, {86, 221}, {196, 1790}, {198, 1434}, {208, 1444}, {222, 41083}, {227, 757}, {274, 2199}, {284, 14256}, {286, 7114}, {322, 1408}, {329, 1412}, {333, 6611}, {342, 1437}, {1119, 1819}, {1333, 40702}, {1394, 41082}, {1407, 27398}, {1414, 6129}, {3209, 17206}, {4565, 14837}, {4570, 38374}, {4637, 14298}, {7341, 21075}
X(53013) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 8822}, {10, 347}, {37, 40702}, {3341, 86}, {5452, 1817}, {6741, 17896}, {24771, 27398}, {40586, 223}, {40590, 14256}, {40591, 7013}, {40599, 329}, {40600, 221}, {40607, 227}, {40608, 6129}, {50330, 38374}, {52389, 33673}
X(53013) = cevapoint of X(42) and X(2357)
X(53013) = barycentric product X(i)*X(j) for these {i,j}: {8, 1903}, {9, 39130}, {10, 282}, {37, 280}, {42, 34404}, {71, 7020}, {72, 7003}, {84, 2321}, {189, 210}, {200, 8808}, {268, 41013}, {271, 1826}, {281, 52389}, {285, 594}, {306, 7008}, {309, 1334}, {312, 2357}, {313, 7118}, {318, 41087}, {321, 2192}, {346, 52384}, {1265, 2358}, {1422, 4082}, {1436, 3701}, {1440, 4515}, {1441, 7367}, {1824, 44189}, {2208, 30713}, {3694, 40836}, {3700, 13138}, {3710, 7129}, {4041, 44327}, {4086, 36049}, {7046, 52037}, {7154, 20336}, {21871, 46355}, {40117, 52355}
X(53013) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 8822}, {10, 40702}, {33, 41083}, {37, 347}, {41, 2360}, {42, 223}, {55, 1817}, {65, 14256}, {71, 7013}, {84, 1434}, {200, 27398}, {210, 329}, {213, 221}, {228, 7011}, {268, 1444}, {271, 17206}, {280, 274}, {282, 86}, {285, 1509}, {607, 3194}, {1334, 40}, {1402, 6611}, {1436, 1014}, {1500, 227}, {1802, 1819}, {1824, 196}, {1826, 342}, {1903, 7}, {1918, 2199}, {2188, 1790}, {2192, 81}, {2200, 7114}, {2208, 1412}, {2321, 322}, {2333, 208}, {2357, 57}, {2358, 1119}, {3125, 38374}, {3700, 17896}, {3709, 6129}, {4041, 14837}, {4171, 8058}, {4515, 7080}, {4524, 14298}, {7003, 286}, {7008, 27}, {7020, 44129}, {7064, 21871}, {7118, 58}, {7151, 1396}, {7154, 28}, {7367, 21}, {8059, 4637}, {8808, 1088}, {13138, 4573}, {30457, 41082}, {32652, 4565}, {34404, 310}, {36049, 1414}, {36197, 38357}, {37141, 4616}, {39130, 85}, {41013, 40701}, {41086, 18623}, {41087, 77}, {44327, 4625}, {52037, 7056}, {52370, 7078}, {52384, 279}, {52389, 348}
X(53013) = {X(39130),X(52389)}-harmonic conjugate of X(52384)

---

X(53014) = X(1)X(7)∩X(2)X(28849)

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

See Ivan Pavlov, euclid 5721.

X(53014) lies on these lines: {1,7}, {2,28849}, {3,28905}, {4,28901}, {8,28870}, {218,5817}, {220,38057}, {277,38107}, {329,28125}, {517,48856}, {750,5308}, {944,48944}, {1002,2808}, {1478,15730}, {1699,50114}, {1962,9778}, {2550,6603}, {2784,50282}, {3008,7988}, {3241,28850}, {3576,28881}, {3616,28874}, {4644,5851}, {4649,36991}, {4845,18391}, {5222,7384}, {5587,36695}, {5603,44431}, {5657,6998}, {5698,34522}, {5886,16020}, {5903,18425}, {6764,17772}, {9318,28234}, {9746,50291}, {9801,17379}, {9812,28845}, {10186,28854}, {10247,28915}, {10647,30301}, {10648,30300}, {16601,21168}, {17014,33134}, {17718,41339}, {17768,42050}, {30309,30317}, {30310,30316}, {51090,52705}

X(53014) = reflection of X(i) in X(j) for these {i,j}: {11200, 1}, {44431, 5603}
X(53014) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 4312, 1323}, {1, 516, 11200}, {1, 990, 18461}, {37830, 37833, 7}, {52806, 52809, 11038}

---

X(53015) = X(2)X(154) ∩ X(4)X(32)

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

See Ivan Pavlov, euclid 5721.

X(53015) lies on these lines: {2,154}, {3,46311}, {4,32}, {5,14535}, {20,183}, {25,42373}, {30,7620}, {64,47435}, {69,5999}, {147,1007}, {230,36990}, {262,14912}, {325,5921}, {376,9466}, {383,43404}, {385,51212}, {542,9770}, {631,7822}, {682,1593}, {1080,43403}, {1350,15589}, {1992,11177}, {2777,9769}, {3090,7852}, {3091,7792}, {3146,37667}, {3522,17128}, {3524,15428}, {3543,22329}, {3545,7694}, {3828,49631}, {5020,20207}, {5304,5480}, {5306,9748}, {5870,6813}, {5871,6811}, {5984,7774}, {5989,6337}, {6776,7736}, {6816,26224}, {7000,13748}, {7374,13638}, {7390,26244}, {7612,14458}, {7738,39646}, {8550,37665}, {8667,29181}, {8719,10304}, {9744,39874}, {9755,14853}, {9863,32006}, {10002,16318}, {10845,36709}, {10846,36714}, {11172,51224}, {12007,14930}, {12203,16043}, {13862,51537}, {14230,26330}, {14233,26331}, {14484,47586}, {14927,34229}, {15069,37668}, {15258,45141}, {15271,44882}, {15484,40927}, {17008,40236}, {18440,37690}, {18860,32817}, {20079,45198}, {22491,41022}, {22492,41023}, {26243,50698}, {32960,37479}, {32985,34473}, {37071,39884}, {37450,40330}, {39887,45510}, {39888,45511}, {42671,52288}, {47382,51963}, {50774,51163}

X(53015) = midpoint of X(i) in X(j) for these {i,j}: {2, 3424}
X(53015) = reflection of X(i) in X(j) for these {i,j}: {2, 9756}, {7710, 2}
X(53015) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1503, 7710}, {2, 3424, 1503}, {4, 98, 7735}, {98, 47737, 34156}, {1503, 9756, 2}, {3424, 9756, 7710}, {6776, 13860, 7736}, {14927, 34229, 37182}

---

X(53016) = X(2)X(2794)∩X(20)X(325)

Barycentrics    15*a^8+2*a^4*(b^2-c^2)^2-12*a^6*(b^2+c^2)+4*a^2*(b^2-c^2)^2*(b^2+c^2)-(b^2-c^2)^2*(9*b^4-2*b^2*c^2+9*c^4) : :

See Ivan Pavlov, euclid 5721.

X(53016) lies on these lines: {2,2794}, {3,39142}, {4,3172}, {20,325}, {30,9741}, {194,3146}, {262,40923}, {297,38918}, {1503,1992}, {1513,41400}, {1562,6776}, {3091,9756}, {3413,35914}, {3414,35913}, {3522,7898}, {5059,7900}, {5870,35794}, {5871,35795}, {5921,18906}, {7709,38383}, {9744,40925}, {9752,36998}, {9754,39647}, {9863,32834}, {10653,36961}, {10654,36962}, {10735,41370}, {17578,20088}, {22676,51373}, {25406,33210}, {39656,40926}, {39838,43448}, {46034,50687}, {49488,51118}

X(53016) = reflection of X(i) in X(j) for these {i,j}: {20, 7710}, {3424, 4}

---

X(53017) = X(3)X(625)∩X(4)X(6)

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

See Ivan Pavlov, euclid 5721.

X(53017) lies on these lines: {3,625}, {4,6}, {30,7618}, {154,52282}, {316,1350}, {381,2794}, {598,38072}, {1316,46988}, {1853,52281}, {2452,46982}, {3053,38227}, {3146,7777}, {3543,7710}, {3564,34505}, {3627,8721}, {3830,32447}, {3832,7806}, {3843,11842}, {5023,37446}, {5077,21163}, {5085,7841}, {5102,7812}, {5140,19161}, {5475,39838}, {8352,43273}, {8370,10516}, {8716,23698}, {9744,44526}, {9755,14639}, {9924,32002}, {10722,13860}, {10723,31859}, {10991,18424}, {11161,11317}, {11185,15069}, {11288,36519}, {13881,36998}, {15271,37348}, {22505,35930}, {31861,33980}, {35260,37174}, {35282,52251}, {36997,39590}

X(53017) = midpoint of X(i) in X(j) for these {i,j}: {3543, 7710}
X(53017) = reflection of X(i) in X(j) for these {i,j}: {9756, 381}
X(53017) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {381, 2794, 9756}, {41038, 41039, 6}

---

X(53018) = X(4)X(28849)∩X(516)X(3543)

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

See Ivan Pavlov, euclid 5721.

X(53018) lies on these lines: {4,28849}, {381,28901}, {515,10186}, {516,3543}, {1699,2784}, {4319,20289}, {5587,9746}, {5790,28897}, {7989,48932}, {9812,29617}, {11200,50864}, {15687,28905}, {18492,48900}, {28164,48809}, {28870,31162}, {44431,50286}

X(53018) = midpoint of X(i) in X(j) for these {i,j}: {11200, 50864}
X(53018) = reflection of X(i) in X(j) for these {i,j}: {9746, 5587}

---

X(53019) = X(3)X(524)∩X(22)X(193)

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

See Ivan Pavlov, euclid 5721.

X(53019) lies on these lines: {3,524}, {6,1196}, {22,193}, {23,47541}, {25,1992}, {69,7484}, {155,32284}, {159,3629}, {195,32368}, {378,50974}, {394,40673}, {542,1597}, {575,43908}, {576,1598}, {599,16419}, {895,11405}, {1351,2393}, {1353,6644}, {1993,10602}, {1995,5032}, {2854,5093}, {2930,15303}, {2936,5477}, {3519,16511}, {3527,43130}, {3564,9818}, {3631,31521}, {5050,9027}, {5655,16510}, {6144,37485}, {6642,9925}, {6776,21312}, {7396,18935}, {7485,11160}, {7502,50986}, {7529,11482}, {8263,11433}, {8573,20794}, {8705,37949}, {8780,19136}, {9004,10246}, {9019,44456}, {9714,51885}, {9755,40888}, {9777,11188}, {9909,15534}, {9924,41580}, {10112,11479}, {11402,41614}, {11477,39568}, {11511,37672}, {12083,50962}, {12160,15073}, {12161,12309}, {12164,50649}, {13595,51170}, {15246,20080}, {15471,21313}, {17814,44495}, {18535,20423}, {19125,40318}, {19347,44492}, {19504,41743}, {19924,44750}, {32245,41612}, {37928,47546}, {37972,47280}, {37973,47549}, {37980,47277}, {44832,51179}

X(53019) = reflection of X(i) in X(j) for these {i,j}: {3, 32621}
X(53019) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {193, 19459, 37491}, {524, 32621, 3}, {1993, 15531, 10602}

---

X(53020) = X(1)X(527)∩X(81)X(329)

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

See Ivan Pavlov, euclid 5721.

X(53020) lies on these lines: {1,527}, {6,52457}, {57,573}, {81,329}, {193,44735}, {278,3782}, {499,4748}, {517,4307}, {524,18391}, {1419,3663}, {1743,3452}, {1905,24476}, {3085,4670}, {3086,4643}, {3421,3751}, {3476,29069}, {3945,12848}, {4470,10039}, {4648,8257}, {7133,13096}, {9309,12915}, {31142,39595}, {36279,48917}

X(53020) = reflection of X(i) in X(j) for these {i,j}: {7961, 1}
X(53020) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 527, 7961}

---

X(53021) = X(2)X(6)∩X(20)X(2393)

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

See Ivan Pavlov, euclid 5721.

X(53021) lies on these lines: {2,6}, {20,2393}, {76,10604}, {1351,18537}, {1352,6623}, {1368,18919}, {1568,9813}, {1660,10565}, {2883,15069}, {2996,19221}, {3087,14615}, {3089,44492}, {3091,8542}, {3523,5486}, {3546,8548}, {3785,22401}, {5622,10519}, {5921,41735}, {6353,8263}, {6391,18935}, {6776,8681}, {6995,11188}, {7386,10602}, {7398,29959}, {9818,34380}, {11898,15760}, {12827,32244}, {14826,44079}, {19119,19588}, {32220,37962}, {32973,40320}, {37077,51028}, {37488,45173}, {37855,52713}, {39874,44458}, {40911,41673}

X(53021) = barycentric product X(i)*X(j) for these (i, j): {69, 40132}
X(53021) = barycentric quotient X(i)*X(j) for these (i, j): {40132, 4}
X(53021) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 1992, 394}, {69, 41614, 2}, {1992, 18928, 6}

---

X(53022) = X(6)X(1368)∩X(182)X(524)

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

See Ivan Pavlov, euclid 5721.

X(53022) lies on these lines: {5,34117}, {6,1368}, {69,7499}, {182,524}, {184,8263}, {193,7485}, {550,9019}, {632,32154}, {1176,41584}, {1503,9813}, {1992,43957}, {2393,48906}, {3618,46444}, {3858,25488}, {5020,41719}, {5050,10257}, {6101,44479}, {6677,19153}, {6723,25329}, {8550,8681}, {9822,34774}, {10154,19127}, {11179,37475}, {11245,41614}, {11442,26926}, {12039,39884}, {15135,22151}, {16266,44503}, {18374,44212}, {25406,37931}, {30744,51171}, {32220,47097}, {34128,38110}, {37450,44363}, {37451,44375}, {37648,44102}

---

X(53023) = X(4)X(6)∩X(5)X(1350)

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

See Ivan Pavlov, euclid 5721.

X(53023) lies on the circumconic {A,B,C,X(66),X(40065)} and these lines: {2,21167}, {3,7889}, {4,6}, {5,1350}, {12,10387}, {13,41025}, {14,41024}, {20,3589}, {30,5085}, {51,1853}, {64,1907}, {66,52518}, {67,3531}, {69,3832}, {98,14488}, {113,2930}, {126,37751}, {140,48873}, {141,3091}, {146,25328}, {154,428}, {159,3574}, {182,382}, {193,50689}, {206,11424}, {235,7716}, {262,14492}, {265,25335}, {371,36712}, {372,36711}, {373,31152}, {381,511}, {383,16644}, {394,7394}, {399,32273}, {403,47450}, {427,17810}, {468,31860}, {485,36657}, {486,36658}, {515,38035}, {516,36721}, {517,38144}, {518,1699}, {524,3839}, {542,5093}, {546,1352}, {550,43621}, {575,5076}, {576,18440}, {590,7000}, {597,3543}, {611,3583}, {613,3585}, {615,7374}, {631,48881}, {698,9766}, {732,34505}, {858,3066}, {946,3242}, {962,49524}, {971,38143}, {1012,5096}, {1080,16645}, {1151,36714}, {1152,36709}, {1192,3088}, {1213,7407}, {1312,15162}, {1313,15163}, {1351,3818}, {1370,17825}, {1386,5691}, {1428,12943}, {1469,10896}, {1513,31489}, {1539,11579}, {1562,15433}, {1595,9786}, {1597,2777}, {1598,20987}, {1656,3098}, {1657,5092}, {1843,37197}, {1899,52285}, {1906,9924}, {1974,12173}, {1993,37349}, {2076,9993}, {2097,7682}, {2330,12953}, {2781,14644}, {2794,6034}, {2829,38147}, {2916,7387}, {3056,10895}, {3146,3618}, {3149,4265}, {3313,13598}, {3416,19925}, {3426,22336}, {3517,10182}, {3518,35228}, {3523,51126}, {3526,14810}, {3527,15321}, {3534,17508}, {3545,10519}, {3564,3845}, {3619,5068}, {3620,3854}, {3627,18583}, {3628,48874}, {3629,5921}, {3656,50790}, {3796,34603}, {3830,5050}, {3844,7989}, {3850,48876}, {3851,24206}, {3853,48906}, {3855,40330}, {3858,18358}, {3860,50956}, {3861,39884}, {4045,14532}, {4301,49688}, {4383,37456}, {5017,13881}, {5026,10723}, {5028,39590}, {5055,19924}, {5056,34573}, {5066,50964}, {5073,12017}, {5094,34417}, {5097,39899}, {5103,7778}, {5116,44519}, {5133,33586}, {5169,37638}, {5188,40332}, {5244,5807}, {5306,9748}, {5640,31133}, {5663,25330}, {5806,24476}, {5842,38148}, {5881,49679}, {5895,11403}, {5925,35502}, {5943,34609}, {6225,22334}, {6249,24273}, {6329,50688}, {6403,35488}, {6410,21736}, {6459,13910}, {6460,13972}, {6593,10733}, {6756,11425}, {6811,8252}, {6813,8253}, {6871,26543}, {6995,23292}, {6997,17811}, {7378,13567}, {7379,17259}, {7385,15668}, {7390,17398}, {7391,10601}, {7403,17834}, {7408,11427}, {7409,11433}, {7486,51128}, {7500,37649}, {7507,9969}, {7519,14389}, {7528,37498}, {7533,15066}, {7540,37506}, {7553,37476}, {7566,11743}, {7576,23041}, {7684,41041}, {7685,41040}, {7710,9300}, {7714,10192}, {7728,16010}, {7736,14484}, {7747,40825}, {7773,18906}, {7784,24256}, {7788,44434}, {7794,40268}, {7982,49690}, {8540,39891}, {8584,51023}, {9581,24471}, {9753,9756}, {9777,11550}, {9781,32191}, {9970,10113}, {9973,50649}, {10019,41584}, {10110,19161}, {10168,15681}, {10295,47453}, {10296,32217}, {10301,41424}, {10303,51127}, {10304,48310}, {10358,44000}, {10594,15577}, {10620,20301}, {10724,51157}, {10734,28662}, {10735,28343}, {10752,32274}, {10991,21309}, {11001,33750}, {11173,18424}, {11174,40236}, {11179,15687}, {11188,22971}, {11284,51360}, {11426,13419}, {11479,37485}, {11480,41034}, {11481,41035}, {11482,48662}, {11522,49465}, {11645,14848}, {11676,44541}, {11818,44413}, {11898,18553}, {12100,50968}, {12101,50979}, {12177,22515}, {12242,14530}, {12293,19139}, {12295,32233}, {12305,37343}, {12306,37342}, {12571,49511}, {12594,26333}, {12595,26332}, {12902,19140}, {13111,35431}, {13202,15118}, {13337,44437}, {13364,31181}, {13473,47459}, {13490,47391}, {14042,39141}, {14490,38005}, {14688,44987}, {14865,15578}, {14927,17578}, {14982,46686}, {15109,52276}, {15448,52301}, {15472,38851}, {15520,51173}, {15533,41099}, {15559,20300}, {15582,26863}, {15640,51026}, {15682,51164}, {15694,25565}, {15696,48879}, {15697,51213}, {16198,39571}, {17702,52697}, {17800,48892}, {17809,31383}, {17821,37122}, {18323,47581}, {18374,18494}, {18383,34779}, {18388,18535}, {18501,39750}, {19124,19136}, {19125,21659}, {19128,35480}, {19132,36989}, {19145,35821}, {19146,35820}, {19150,48675}, {19369,39892}, {19459,43831}, {19541,36740}, {19709,50977}, {19710,50976}, {20190,48896}, {22165,50960}, {23327,51745}, {26118,37679}, {26206,34007}, {28146,38167}, {28150,38118}, {28160,38029}, {28164,38049}, {28174,38116}, {28186,38040}, {28194,38087}, {28212,38165}, {28228,38191}, {30308,51003}, {31099,37648}, {31521,37198}, {31724,44480}, {31725,44503}, {31861,40909}, {32113,37984}, {32305,38790}, {32423,39522}, {32621,51403}, {33748,51022}, {34507,44456}, {34648,47356}, {35490,51730}, {35786,35841}, {35787,35840}, {35930,47619}, {36655,42265}, {36656,42262}, {36716,37474}, {36757,36970}, {36758,36969}, {36991,51150}, {36992,42126}, {36994,42127}, {36999,47373}, {37453,44106}, {37463,43028}, {37464,43029}, {37486,50137}, {38057,44431}, {40927,43620}, {41106,50967}, {42153,51754}, {42154,44666}, {42155,44667}, {42156,51753}, {42263,45545}, {42264,45544}, {42490,52689}, {42491,52688}, {44526,50659}, {45106,45107}, {47280,47474}, {47296,52284}, {47341,52163}, {47358,50802}, {48891,49137}, {48942,50664}, {49481,51063}, {50779,51065}, {50781,50803}, {50782,50799}, {50783,50796}, {50785,51078}, {50787,51076}, {50789,50801}, {50791,51074}, {50955,51188}, {50958,50992}, {50970,51211}, {50971,51029}, {50991,51131}, {51041,51051}, {51075,51089}, {51132,51187}, {51133,51189}

X(53023) = midpoint of X(i) in X(j) for these {i,j}: {2, 51538}, {4, 14853}, {3543, 25406}, {3830, 5050}, {5102, 47353}, {13473, 47459}, {14848, 38335}, {31884, 51024}, {38317, 48901}
X(53023) = reflection of X(i) in X(j) for these {i,j}: {3, 38317}, {6, 14853}, {599, 10516}, {2076, 38227}, {3534, 17508}, {5050, 5476}, {5085, 14561}, {5102, 20423}, {10304, 48310}, {10516, 381}, {14561, 38136}, {14853, 5480}, {15534, 5102}, {21358, 3545}, {25406, 597}, {31884, 2}, {38047, 38146}, {38315, 38035}, {38317, 19130}, {43273, 5050}, {47352, 38072}, {47450, 403}, {51024, 51538}, {52028, 23327}
X(53023) = anticomplement of X(21167)
X(53023) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 29181, 31884}, {2, 51538, 29181}, {3, 48901, 48910}, {3, 48910, 48872}, {4, 10002, 53}, {4, 12233, 15811}, {4, 14853, 1503}, {4, 16657, 18405}, {4, 41371, 16264}, {4, 45089, 1498}, {4, 45440, 23251}, {4, 45441, 23261}, {4, 5480, 6}, {4, 6, 36990}, {4, 6201, 3070}, {4, 6202, 3071}, {5, 1350, 3763}, {5, 31670, 1350}, {30, 14561, 5085}, {30, 38072, 47352}, {30, 38136, 14561}, {51, 5064, 1853}, {182, 382, 48905}, {182, 48895, 382}, {193, 50689, 51537}, {265, 32271, 51941}, {265, 51941, 25335}, {381, 511, 10516}, {427, 17810, 26958}, {511, 10516, 599}, {515, 38035, 38315}, {516, 38146, 38047}, {546, 21850, 1352}, {1351, 15069, 6144}, {1351, 3818, 15069}, {1351, 3843, 3818}, {1352, 21850, 11477}, {1503, 14853, 6}, {1503, 5480, 14853}, {3088, 11745, 1192}, {3146, 3618, 44882}, {3564, 20423, 5102}, {3564, 5102, 15534}, {3589, 51163, 20}, {3618, 44882, 10541}, {3627, 18583, 46264}, {3830, 5050, 29012}, {3830, 50963, 5476}, {3830, 5476, 43273}, {3845, 20423, 47353}, {3851, 33878, 24206}, {5050, 29012, 43273}, {5073, 12017, 48898}, {5085, 14561, 47352}, {5085, 38072, 14561}, {5092, 48904, 1657}, {5102, 47353, 3564}, {5476, 43273, 51185}, {5480, 18382, 10982}, {5893, 15583, 41735}, {7684, 41041, 42098}, {7685, 41040, 42095}, {9969, 12294, 37473}, {10982, 19149, 6}, {14561, 38136, 38072}, {14848, 38335, 11645}, {15534, 47353, 51027}, {17578, 51171, 14927}, {19130, 29317, 38317}, {19130, 48901, 3}, {19130, 48910, 47355}, {20190, 48943, 48896}, {20300, 34778, 40686}, {20423, 47353, 15534}, {22693, 22694, 22728}, {25555, 48898, 12017}, {29181, 51538, 51024}, {29317, 38317, 3}, {31884, 51024, 29181}, {38317, 48901, 29317}, {41036, 41037, 381}, {42645, 42646, 7745}, {42785, 48898, 25555}, {47355, 48872, 3}, {48896, 48943, 49136}

---

X(53024) = X(30)X(3589)∩X(3635)X(15519)

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

See Ivan Pavlov, euclid 5721.

X(53024) lies on the circumconics {A,B,C,X(2),X(30)}, {A,B,C,X(3),X(5092)}, {A,B,C,X(69),X(3589)}, {A,B,C,X(76),X(18550)}, {A,B,C,X(83),X(265)} and these lines: {30,3589}, {7767,11064}, {14389,46809}, {44142,46106}

X(53024) = X(i)-isoconjugate-of-X(j) for these {i, j}: {19, 41462}
X(53024) = X(i)-Dao conjugate of X(j) for these {i, j}: {6, 41462}
X(53024) = barycentric quotient X(i)*X(j) for these (i, j): {3, 41462}

---

X(53025) = X(2)X(648)∩X(4)X(3096)

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

See Ivan Pavlov, euclid 5721.

X(53025) lies on these lines: {2,648}, {4,3096}, {141,340}, {232,16988}, {264,3763}, {297,20582}, {378,47005}, {458,21358}, {1990,34573}, {5094,7868}, {7831,10295}, {7832,37118}, {15066,43530}, {17907,52710}, {31268,39575}, {37638,44133}

X(53025) = barycentric product X(i)*X(j) for these (i, j): {264, 41462}
X(53025) = barycentric quotient X(i)*X(j) for these (i, j): {41462, 3}
X(53025) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {141, 52289, 340}, {340, 52289, 36794}

---

X(53026) = X(6)X(74)∩X(39)X(186)

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

See Ivan Pavlov, euclid 5721.

X(53026) lies on the circumconic {A,B,C,X(74),X(3108)} and these lines: {4,7739}, {6,74}, {25,1180}, {32,35473}, {39,186}, {216,44832}, {232,52294}, {376,5158}, {403,9300}, {1235,7839}, {1285,46432}, {1595,41366}, {1597,8743}, {3172,35501}, {3284,35921}, {3329,44146}, {3520,5007}, {4230,15019}, {4235,12150}, {5013,35472}, {5041,13596}, {5306,37118}, {5309,7577}, {5319,37119}, {6143,7755}, {6240,9607}, {6623,37665}, {6749,7576}, {6794,18388}, {7378,41361}, {7464,15860}, {7485,23115}, {7502,10313}, {7514,22120}, {7760,37125}, {8744,33843}, {9606,10018}, {9698,14940}, {10295,41335}, {10311,47485}, {11410,30435}, {11416,35924}, {12083,22240}, {13331,19128}, {14482,40065}, {14865,41940}, {14930,41370}, {17506,31652}, {18535,22246}, {22332,32534}, {31406,52297}, {36794,41676}, {38292,41891}

X(53026) = barycentric product X(i)*X(j) for these (i, j): {4, 41462}
X(53026) = barycentric quotient X(i)*X(j) for these (i, j): {41462, 69}

---

X(53027) = X(4)X(74)∩X(427)X(1629)

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

See Ivan Pavlov, euclid 5721.

X(53027) lies on the circumconic {A,B,C,X(74),X(3108)} and these lines: {4,74}, {264,31133}, {275,11550}, {378,16263}, {381,52147}, {393,3108}, {427,1629}, {450,48889}, {2052,5064}, {3146,14860}, {5480,51939}, {7409,52448}, {26864,43530}, {32064,40065}

X(53027) = barycentric product X(i)*X(j) for these (i, j): {2052, 41462}
X(53027) = barycentric quotient X(i)*X(j) for these (i, j): {41462, 394}
X(53027) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14165, 16264, 1629}

---

X(53028) = X(2)-CROSS CONJUGATE OF-X(324)

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

See Kadir Altintas and César Lozada, euclid 5722.

X(53028) lies on the line {324,32002}

X(53028) = crosspoint of X(i) and X(j) for these (i, j): {5, 34520}, {311, 25043}
X(53028) = X(2)-cross conjugate of-X(324)
X(53028) = X(216)-Dao conjugate of-X(195)
X(53028) = X(195)-isoconjugate-of-X(2148)
X(53028) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (5, 195), (143, 15787), (311, 45799), (1263, 14367)
X(53028) = intersection, other than A, B, C, of circumconics {A, B, C, X(2), X(25043)} and {A, B, C, X(5), X(11538)}
X(53028) = trilinear pole of the line {16336, 20577}
X(53028) = barycentric product X(311)*X(3459)
X(53028) = barycentric quotient X(i)/X(j) for these (i, j): (5, 195), (143, 15787), (311, 45799), (1263, 14367)

---

X(53029) = ISOGONAL CONJUGATE OF X(3170)

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

See Kadir Altintas and César Lozada, euclid 5722.

X(53029) lies on the cubics K278, K342a and these lines: {13,533}, {14,14372}, {298,11118}, {303,11119}, {395,11080}, {396,11082}, {1989,40578}, {3629,22826}, {8737,23715}, {8919,38932}, {16242,46072}, {36307,52751}, {36515,36967}

X(53029) = isogonal conjugate of X(3170)
X(53029) = isotomic conjugate of X(30471)
X(53029) = crosspoint of X(2) and X(22847)
X(53029) = X(2)-cross conjugate of-X(13)
X(53029) = X(2151)-isoconjugate-of-X(3180)
X(53029) = X(13)-reciprocal conjugate of-X(3180)
X(53029) = intersection, other than A, B, C, of circumconics {A, B, C, X(2), X(3180)} and {A, B, C, X(13), X(300)}
X(53029) = trilinear pole of the line {14447, 20578}
X(53029) = barycentric product X(13)*X(11121)
X(53029) = barycentric quotient X(13)/X(3180)
X(53029) = trilinear product X(2153)*X(11121)
X(53029) = trilinear quotient X(2153)/X(19780)

---

X(53030) = ISOGONAL CONJUGATE OF X(3171)

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

See Kadir Altintas and César Lozada, euclid 5722.

X(53030) lies on the cubics K278, K342b and these lines: {13,14373}, {14,532}, {299,11117}, {302,11120}, {395,11087}, {396,11085}, {3629,22827}, {8738,23714}, {8918,38931}, {16241,46076}, {36310,52750}, {36514,36968}

X(53030) = isogonal conjugate of X(3171)
X(53030) = isotomic conjugate of X(30472)
X(53030) = crosspoint of X(2) and X(22893)
X(53030) = X(2)-cross conjugate of-X(14)
X(53030) = X(2152)-isoconjugate-of-X(3181)
X(53030) = X(14)-reciprocal conjugate of-X(3181)
X(53030) = trilinear pole of the line {14446, 20579}
X(53030) = barycentric product X(14)*X(11122)
X(53030) = barycentric quotient X(14)/X(3181)
X(53030) = trilinear product X(2154)*X(11122)
X(53030) = trilinear quotient X(2154)/X(19781)

---

X(53031) = X(14)X(148) ∩ X(16)X(3440)

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

See Kadir Altintas and César Lozada, euclid 5722.

X(53031) lies on the cubic K129a and these lines: {6,3170}, {14,148}, {16,3440}, {18,38931}, {62,16459}, {395,11080}, {2378,10646}, {3165,3439}, {8739,16538}

X(53031) = crosspoint of X(15) and X(11131)
X(53031) = crosssum of X(15) and X(3180)
X(53031) = X(2153)-isoconjugate-of-X(3180)
X(53031) = X(15)-reciprocal conjugate of-X(3180)
X(53031) = barycentric product X(15)*X(11121)
X(53031) = barycentric quotient X(15)/X(3180)
X(53031) = trilinear product X(2151)*X(11121)
X(53031) = trilinear quotient X(i)/X(j) for these (i, j): (1094, 3170), (2151, 19780)

---

X(53032) = X(13)X(148) ∩ X(15)X(3441)

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

See Kadir Altintas and César Lozada, euclid 5722.

X(53032) lies on the cubic K129b and these lines: {6,3171}, {13,148}, {15,3441}, {17,38932}, {61,16460}, {396,11085}, {2379,10645}, {3166,3438}, {8740,16539}

X(53032) = crosspoint of X(16) and X(11130)
X(53032) = crosssum of X(16) and X(3181)
X(53032) = X(2154)-isoconjugate-of-X(3181)
X(53032) = X(16)-reciprocal conjugate of-X(3181)
X(53032) = barycentric product X(16)*X(11122)
X(53032) = barycentric quotient X(16)/X(3181)
X(53032) = trilinear product X(2152)*X(11122)
X(53032) = trilinear quotient X(i)/X(j) for these (i, j): (1095, 3171), (2152, 19781)

---

X(53033) = X(2)X(39)∩X(20)X(636)

Barycentrics    3 a^4-2 a^2 (b^2+c^2)+3 b^4+2 b^2 c^2+3 c^4 : :
X(53033) = (r^2+4rR-s^2)^2X[141]-5r^2s^2X[631]

See Angel Montesdeoca, euclid 5727.

X(53033) lies on these lines: {2,39}, {3,7710}, {4,7778}, {5,37690}, {6,14069}, {20,626}, {32,33181}, {69,7807}, {99,32974}, {115,33199}, {141,631}, {147,620}, {148,33283}, {183,32970}, {187,33205}, {193,7796}, {230,33189}, {315,7835}, {316,32981}, {325,14001}, {376,7784}, {384,32816}, {439,7922}, {487,5590}, {488,5591}, {524,33197}, {574,33202}, {625,3832}, {639,49039}, {640,49038}, {1003,32006}, {1007,7770}, {1078,3620}, {1587,45473}, {1588,45472}, {1975,14064}, {1992,8366}, {2482,7935}, {2548,7820}, {2549,7863}, {2896,32964}, {3053,33191}, {3090,39663}, {3091,3734}, {3096,32990}, {3146,7816}, {3314,3785}, {3522,7761}, {3525,7612}, {3528,32459}, {3538,34828}, {3541,44145}, {3543,7825}, {3546,14376}, {3618,33217}, {3619,11285}, {3763,32960}, {3815,16045}, {3933,7735}, {4027,33206}, {4869,25663}, {5013,32956}, {5024,8364}, {5025,32815}, {5056,7862}, {5059,7842}, {5254,32817}, {5304,6680}, {5319,7813}, {6179,20080}, {6337,6656}, {6390,7738}, {7620,11318}, {7736,7819}, {7737,7821}, {7745,14039}, {7747,23334}, {7748,33200}, {7750,32985}, {7751,37689}, {7752,32971}, {7756,33210}, {7762,33220}, {7764,37665}, {7767,11288}, {7773,14033}, {7774,7892}, {7776,8369}, {7777,16898}, {7781,33182}, {7782,33023}, {7785,14037}, {7788,33224}, {7791,7891}, {7792,32821}, {7793,10334}, {7794,15589}, {7802,35927}, {7815,10303}, {7823,33255}, {7830,10304}, {7839,14067}, {7846,51171}, {7849,15717}, {7851,32820}, {7853,33025}, {7854,21843}, {7857,37667}, {7865,15692}, {7868,16043}, {7879,35297}, {7883,35287}, {7885,33007}, {7887,39143}, {7895,10513}, {7897,20065}, {7898,33244}, {7899,11185}, {7901,32824}, {7902,14148}, {7905,51170}, {7906,14043}, {7907,16990}, {7911,33272}, {7912,14035}, {7925,16924}, {7928,33008}, {7934,32982}, {7938,32965}, {7939,33246}, {7941,14036}, {8363,31859}, {8368,30435}, {8667,33231}, {8716,33196}, {9605,33185}, {9606,47355}, {9752,12251}, {9770,33237}, {12177,38751}, {13881,32955}, {14063,32826}, {15484,19697}, {15705,40344}, {16041,32819}, {16196,20208}, {16986,33001}, {17008,33245}, {17128,32961}, {17130,43620}, {17811,39643}, {18907,33242}, {22401,28425}, {30747,52284}, {31489,32957}, {32456,50693}, {32521,37466}, {32822,33285}, {32959,37637}, {32975,37647}, {32977,37688}, {32992,34803}, {33213,51122}, {33218,47286}, {33233,34229}, {33236,46453}, {37334,40330}, {41922,45799}

---

X(53034) = X(1)X(5235)∩X(2)X(740)

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

See Ivan Pavlov, euclid 5732.

X(53034) lies on these lines: {1,5235}, {2,740}, {10,21806}, {37,2229}, {86,896}, {191,28620}, {244,1125}, {351,4728}, {442,27560}, {512,14474}, {523,6544}, {551,46909}, {714,51488}, {750,3747}, {756,4090}, {758,15671}, {846,5333}, {899,2667}, {1001,3724}, {1211,8040}, {1213,4062}, {1635,11176}, {1654,4938}, {1707,41930}, {2177,39586}, {2308,37869}, {2650,3616}, {3121,16589}, {3624,3743}, {3712,6707}, {3720,3725}, {3722,16830}, {3723,50756}, {3728,4687}, {3842,21805}, {3936,25354}, {3989,28582}, {4026,21026}, {4065,19878}, {4068,4413}, {4205,27577}, {4414,15668}, {4418,25507}, {4647,19862}, {4672,19740}, {4683,26109}, {4928,9147}, {5218,42446}, {5253,12567}, {5429,17553}, {5550,49598}, {5625,16704}, {5695,19749}, {6089,9125}, {6186,37327}, {6536,17056}, {11203,29301}, {15569,30970}, {16826,32917}, {17322,29632}, {17514,20653}, {17733,19334}, {24697,37635}, {25124,27268}, {29654,41820}, {33114,48822}, {33143,41312}

X(53034) = midpoint of X(i) in X(j) for these {i,j}: {2, 27811}
X(53034) = reflection of X(i) in X(j) for these {i,j}: {1962, 27811}, {27811, 10180}
X(53034) = complement of X(27812)
X(53034) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 10180, 1962}, {2, 1962, 21020}, {2, 27804, 27798}, {2, 27811, 740}, {740, 10180, 27811}, {740, 27811, 1962}, {3842, 29822, 21805}

---

X(53035) = X(3)X(42440)∩X(354)X(1962)

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

See Ivan Pavlov, euclid 5732.

X(53035) lies on these lines: {3,42440}, {140,23555}, {354,1962}, {513,11203}, {523,11124}, {758,4930}, {999,3743}, {1319,2292}, {1617,25080}, {5432,17874}, {8053,23171}, {8680,10180}, {8731,34977}, {16577,52139}, {16579,20470}

---

X(53036) = X(2)X(8680)∩X(12)X(2292)

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

See Ivan Pavlov, euclid 5732.

X(53036) lies on these lines: {2,8680}, {4,18673}, {12,2292}, {27,2173}, {48,5307}, {73,1882}, {92,1953}, {210,20718}, {226,1826}, {278,18675}, {321,3949}, {329,3958}, {442,21671}, {523,23615}, {740,25568}, {758,5587}, {857,25361}, {1211,21675}, {1699,44661}, {1837,2650}, {1848,18674}, {1962,17718}, {2551,49598}, {2654,2658}, {2797,14414}, {2939,31902}, {3120,40973}, {3136,40967}, {3452,18698}, {3772,40977}, {4647,21075}, {5219,25080}, {5226,25255}, {5327,26000}, {5777,45038}, {6358,21801}, {6678,7359}, {7140,21911}, {10158,38052}, {14206,17167}, {18635,50197}, {19645,24315}, {21011,26942}, {21033,42708}, {24435,27398}, {25081,25525}, {27798,38057}, {31163,31164}, {37168,51697}

X(53036) = X(i)-Dao conjugate of X(j) for these {i, j}: {6708, 63}, {18592, 333}
X(53036) = X(i)-Ceva conjugate of X(j) for these {i, j}: {6708, 18592}, {52938, 656}
X(53036) = barycentric product X(i)*X(j) for these (i, j): {92, 18592}, {226, 6708}, {264, 2658}, {307, 42385}, {1441, 2654
X(53036) = barycentric quotient X(i)*X(j) for these (i, j): {408, 255}, {2654, 21}, {2658, 3}, {6708, 333}, {18592, 63}, {40946, 283}, {42385, 29}
X(53036) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {226, 40149, 37755}, {6358, 22000, 21801}

---

X(53037) = X(1)X(4273)∩X(37)X(42)

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

See Ivan Pavlov, euclid 5732.

X(53037) lies on on the circumconic {A,B,C,X(42),X(29833)} and these lines: {1,4273}, {2,27490}, {6,758}, {19,2203}, {37,42}, {44,2292}, {45,3743}, {81,16568}, {523,1643}, {740,17281}, {986,5165}, {1100,2170}, {1333,5429}, {2092,40988}, {2245,40978}, {2325,4065}, {2345,17163}, {2650,16666}, {3589,18697}, {3742,40941}, {3745,40973}, {3898,16685}, {3943,7206}, {3958,16669}, {4053,20970}, {4647,17369}, {5256,30903}, {6370,14398}, {8680,17301}, {16777,25081}, {16972,47373}, {17303,25453}, {17366,18698}, {17367,35550}, {20896,30892}, {38315,44661}

---

X(53038) = X(6)X(1154)∩X(51)X(216)

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

See Ivan Pavlov, euclid 5732.

X(53038) lies on these lines: {6,1154}, {51,216}, {231,6689}, {570,5892}, {1640,6368}, {1990,14978}, {3284,31388}, {5158,42441}, {5647,11402}, {11063,34292}, {14561,32428}, {14576,14845}, {19153,19156}, {46025,52703}

X(53038) = barycentric product X(i)*X(j) for these (i, j): {5, 37513}
X(53038) = barycentric quotient X(i)*X(j) for these (i, j): {37513, 95}

---

X(53039) = X(2)X(740)∩X(10)X(21870)

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

See Ivan Pavlov, euclid 5732.

X(53039) lies on these lines: {2,740}, {10,21870}, {1213,4892}, {1268,29862}, {1698,30832}, {2650,46932}, {3634,24003}, {3743,31253}, {3986,48641}, {4135,31993}, {4457,43223}, {4647,19872}, {4763,45689}, {6707,50755}, {9148,45675}, {19749,49488}, {21242,39580}, {23812,49730}, {25124,31238}, {30818,51073}

X(53039) = midpoint of X(i) in X(j) for these {i,j}: {21020, 27811}
X(53039) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 27798, 10180}, {21020, 27811, 740}

---

X(53040) = X(758)X(38042)∩X(1656)X(23555)

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

See Ivan Pavlov, euclid 5732.

X(53040) lies on these lines: {758,38042}, {1656,23555}, {3090,42440}, {3740,20718}, {5123,49598}

---

X(53041) = X(4)X(23555)∩X(497)X(17874)

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

See Ivan Pavlov, euclid 5732.

X(53041) lies on these lines: {4,23555}, {497,17874}, {740,25568}, {3091,42440}, {3421,4647}, {3681,17163}, {3743,8164}, {4552,44411}, {5176,17164}, {8680,21020}, {19797,27812}, {27471,32915}

---

X(53042) = X(2)X(8680)∩X(828)X(1214)

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

See Ivan Pavlov, euclid 5732.

X(53042) lies on these lines: {2,8680}, {57,25523}, {63,31631}, {241,25132}, {758,10165}, {828,1214}, {2292,7288}, {2294,5435}, {3523,18673}, {3742,10180}, {3911,25080}, {4999,49598}, {6692,25081}, {10164,44661}, {19542,24317}, {24682,37419}, {24684,26934}

---

X(53043) = X(2)X(8680)∩X(7)X(25080)

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

See Ivan Pavlov, euclid 5732.

X(53043) lies on these lines: {2,8680}, {7,25080}, {27,18661}, {57,25255}, {63,17134}, {347,18633}, {440,41804}, {758,5731}, {1762,14953}, {1790,39767}, {1812,17136}, {1817,14543}, {1962,11038}, {2292,3600}, {2294,21454}, {2975,17164}, {3218,25254}, {3522,18673}, {3743,11037}, {3873,20718}, {3952,42705}, {4466,31045}, {4566,40152}, {5273,18698}, {6360,21271}, {9778,44661}, {19797,27812}, {22001,52358}, {24316,50697}

---

X(53044) = X(29)X(73)∩X(48)X(2326)

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

See Ivan Pavlov, euclid 5732.

X(53044) lies on the circumconics {A,B,C,X(1),X(48)}, {A,B,C,X(4),X(1982)}, {A,B,C,X(21),X(29)}, {A,B,C,X(92),X(30588)} and these lines: {21,22341}, {29,73}, {48,2326}, {255,1098}, {410,823}, {1043,3682}, {2287,3990}, {15146,40430}, {26701,51699}

X(53044) = trilinear pole of line {822,1021}
X(53044) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 2658}, {4, 408}, {6, 18592}, {65, 40946}, {73, 2654}, {1409, 6708}, {22341, 42385}
X(53044) = X(i)-Dao conjugate of X(j) for these {i, j}: {3, 2658}, {9, 18592}, {36033, 408}, {40602, 40946}
X(53044) = X(i) cross conjugate of X(j) for these {i, j}: {652, 823}, {2655, 37142}, {21173, 162}
X(53044) = barycentric quotient X(i)*X(j) for these (i, j): {1, 18592}, {6, 2658}, {29, 6708}, {48, 408}, {284, 40946}, {1172, 2654}, {8748, 42385}
X(53044) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {410, 2654, 823}

---

X(53045) = X(2)X(2399)∩X(8)X(522)

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

See Ivan Pavlov, euclid 5735.

X(53045) lies on the circumconic {A,B,C,X(8),X(51583)} and these lines: {2,2399}, {8,522}, {63,514}, {321,4391}, {1281,2785}, {2397,24029}, {2401,4025}, {3239,21198}, {3887,38371}, {3895,8058}, {3904,3960}, {3910,40166}, {3977,50943}, {4712,13259}, {6003,48890}, {10015,26611}, {18623,30719}, {31605,34059}, {42756,51423}

X(53045) = reflection of X(i) in X(j) for these {i,j}: {36038, 10015}
X(53045) = X(i)-isoconjugate-of-X(j) for these {i, j}: {80, 32669}, {104, 32675}, {655, 34858}, {909, 2222}, {1411, 32641}, {1415, 40437}, {1807, 32702}, {2161, 2720}, {2423, 52377}, {6187, 37136}, {36110, 52431}
X(53045) = X(i)-Dao conjugate of X(j) for these {i, j}: {908, 651}, {1146, 40437}, {2245, 109}, {16586, 655}, {23980, 2222}, {35128, 104}, {35204, 32641}, {38981, 2161}, {38984, 909}, {39004, 52431}, {40584, 2720}, {40612, 37136}, {40613, 32675}, {42761, 226}, {46398, 2006}
X(53045) = X(i)-Ceva conjugate of X(j) for these {i, j}: {190, 908}, {664, 4511}
X(53045) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {47645, 150}
X(53045) = barycentric product X(i)*X(j) for these (i, j): {190, 46398}, {314, 42768}, {320, 2804}, {908, 3904}, {1845, 35518}, {3262, 3738}, {4391, 16586}, {4453, 6735}, {4511, 36038}, {10015, 32851}, {20924, 46393}, {34586, 35519}, {38353, 46404}
X(53045) = barycentric quotient X(i)*X(j) for these (i, j): {36, 2720}, {517, 2222}, {522, 40437}, {654, 909}, {908, 655}, {1769, 1411}, {1845, 108}, {1870, 36110}, {2183, 32675}, {2323, 32641}, {2804, 80}, {3218, 37136}, {3262, 35174}, {3738, 104}, {3904, 34234}, {3960, 34051}, {4511, 36037}, {5081, 1309}, {6735, 51562}, {7113, 32669}, {8648, 34858}, {10015, 2006}, {16586, 651}, {23757, 14584}, {32851, 13136}, {34586, 109}, {36038, 18815}, {38353, 652}, {42768, 65}, {44428, 36123}, {46393, 2161}, {52307, 52431}, {52413, 32702}, {52427, 14776}

---

X(53046) = X(6)X(652)∩X(37)X(650)

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

See Ivan Pavlov, euclid 5735.

X(53046) lies on the circumconics {A,B,C,X(48),X(47408)}, {A,B,C,X(55),X(17455)} and these lines: {6,652}, {37,650}, {48,649}, {55,663}, {647,2294}, {654,17455}, {657,27780}, {1459,2423}, {1465,42762}, {1643,10581}, {1769,3310}, {2427,23981}, {5075,19561}, {6332,28936}, {7658,26742}, {21786,51648}, {42078,42079}

X(53046) = X(i)-isoconjugate-of-X(j) for these {i, j}: {80, 37136}, {104, 655}, {651, 40437}, {909, 35174}, {1411, 13136}, {2006, 36037}, {2222, 34234}, {2401, 52377}, {2720, 18359}, {18815, 32641}, {18816, 32675}, {20566, 32669}, {34051, 51562}, {34858, 46405}, {36110, 52351}
X(53046) = X(i)-Dao conjugate of X(j) for these {i, j}: {908, 4554}, {1145, 36804}, {2245, 664}, {3259, 2006}, {10015, 3261}, {13999, 16082}, {16586, 46405}, {23980, 35174}, {35128, 18816}, {35204, 13136}, {38981, 18359}, {38984, 34234}, {38991, 40437}, {39004, 52351}, {40613, 655}, {42761, 349}
X(53046) = X(i)-Ceva conjugate of X(j) for these {i, j}: {101, 2183}, {109, 2361}, {650, 46393}
X(53046) = barycentric product X(i)*X(j) for these (i, j): {21, 42768}, {36, 2804}, {101, 46398}, {517, 3738}, {521, 1845}, {522, 34586}, {650, 16586}, {653, 38353}, {654, 908}, {1769, 4511}, {2183, 3904}, {2323, 10015}, {2361, 36038}, {3218, 46393}, {3262, 8648}, {3310, 32851}, {4242, 35014}, {5081, 8677}, {17923, 52307}, {22350, 44428}
X(53046) = barycentric quotient X(i)*X(j) for these (i, j): {517, 35174}, {654, 34234}, {663, 40437}, {908, 46405}, {1769, 18815}, {1845, 18026}, {2183, 655}, {2323, 13136}, {2361, 36037}, {2804, 20566}, {3310, 2006}, {3738, 18816}, {7113, 37136}, {8648, 104}, {8677, 52392}, {16586, 4554}, {21758, 34051}, {34586, 664}, {38353, 6332}, {42768, 1441}, {46393, 18359}, {46398, 3261}, {52307, 52351}, {52426, 32641}, {52427, 1309}, {52434, 2720}
X(53046) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3310, 52307, 46393}

---

X(53047) = X(1)X(7649)∩X(4)X(522)

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

See Ivan Pavlov, euclid 5735.

X(53047) lies on the circumconic {A,B,C,X(1),X(119)} and these lines: {1,7649}, {4,522}, {119,2804}, {281,3064}, {514,37628}, {521,24474}, {1021,46884}, {1785,23757}, {1845,3738}, {3811,8058}, {14010,35014}, {41013,44426}

X(53047) = X(i)-isoconjugate-of-X(j) for these {i, j}: {655, 14578}, {1795, 2222}, {1807, 2720}, {32669, 52351}, {36059, 40437}, {37136, 52431}
X(53047) = X(i)-Dao conjugate of X(j) for these {i, j}: {908, 6516}, {2245, 1813}, {10015, 4025}, {13999, 104}, {20620, 40437}, {25640, 2222}, {38981, 1807}, {38984, 1795}, {42761, 307}, {46398, 52392}
X(53047) = X(i)-Ceva conjugate of X(j) for these {i, j}: {29, 35015}, {1897, 1785}
X(53047) = barycentric product X(i)*X(j) for these (i, j): {908, 44428}, {1785, 3904}, {1845, 4391}, {1897, 46398}, {2804, 17923}, {5081, 10015}, {16586, 44426}, {31623, 42768}, {32851, 39534}, {34586, 46110}, {38353, 52938}
X(53047) = barycentric quotient X(i)*X(j) for these (i, j): {654, 1795}, {1785, 655}, {1845, 651}, {1870, 37136}, {2804, 52351}, {3064, 40437}, {5081, 13136}, {8648, 14578}, {10015, 52392}, {14571, 2222}, {16586, 6516}, {34586, 1813}, {39534, 2006}, {42768, 1214}, {44428, 34234}, {46393, 1807}, {46398, 4025}, {52413, 2720}, {52427, 32641}

---

X(53048) = X(13)X(3060) ∩ X(51)X(530)

Barycentrics    a^2*(-2*((b^2+c^2)*a^4+6*b^2*c^2*a^2-b^6-c^6)*S+sqrt(3)*((b^2+c^2)*a^6-(3*b^4+4*b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(3*b^4-5*b^2*c^2+3*c^4)*a^2-(b^4-3*b^2*c^2+c^4)*(b^2-c^2)^2)) : :
X(53048) = 2*X(143)+X(16001) = X(2979)-3*X(22489) = 2*X(3819)-3*X(48311) = X(5463)-3*X(5640) = X(5473)-3*X(15045) = X(10263)+2*X(20415) = 3*X(11002)+X(51482) = X(13103)+3*X(13321) = 2*X(21849)+X(47865)

See Kadir Altintas and César Lozada, euclid 5736.

X(53048) lies on these lines: {13,3060}, {51,530}, {115,36978}, {143,16001}, {511,5459}, {568,25154}, {618,5943}, {2979,22489}, {3819,48311}, {3917,6669}, {5463,5640}, {5472,36980}, {5473,15045}, {5478,13754}, {6771,13391}, {9971,22580}, {10263,20415}, {11002,51482}, {11624,33957}, {13103,13321}, {21849,47865}

X(53048) = midpoint of X(i) and X(j) for these {i, j}: {13, 3060}, {568, 25154}, {9971, 22580}
X(53048) = reflection of X(i) in X(j) for these (i, j): (618, 5943), (3917, 6669)

---

X(53049) = X(14)X(3060) ∩ X(51)X(531)

Barycentrics    a^2*(2*((b^2+c^2)*a^4+6*b^2*c^2*a^2-b^6-c^6)*S+sqrt(3)*((b^2+c^2)*a^6-(3*b^4+4*b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(3*b^4-5*b^2*c^2+3*c^4)*a^2-(b^4-3*b^2*c^2+c^4)*(b^2-c^2)^2)) : :
X(53049) = 2*X(143)+X(16002) = X(2979)-3*X(22490) = 2*X(3819)-3*X(48312) = X(5464)-3*X(5640) = X(5474)-3*X(15045) = X(10263)+2*X(20416) = 3*X(11002)+X(51483) = X(13102)+3*X(13321) = 2*X(21849)+X(47866)

See Kadir Altintas and César Lozada, euclid 5736.

X(53049) lies on these lines: {14,3060}, {51,531}, {115,36980}, {143,16002}, {511,5460}, {542,53014}, {568,25164}, {619,5943}, {2979,22490}, {3819,48312}, {3917,6670}, {5464,5640}, {5471,36978}, {5474,15045}, {5479,13754}, {6774,13391}, {9971,22579}, {10263,20416}, {11002,51483}, {11626,33958}, {13102,13321}, {21849,47866}

X(53049) = midpoint of X(i) and X(j) for these {i, j}: {14, 3060}, {568, 25164}, {9971, 22579}
X(53049) = reflection of X(i) in X(j) for these (i, j): (619, 5943), (3917, 6670)

---

X(53050) = ISOGONAL CONJUGATE OF X(31942)

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

See Ercole Suppa and César Lozada, euclid 5738.

X(53050) lies on the inconic with perspector X(18020) and these lines: {3,69}, {4,18418}, {20,154}, {99,15324}, {110,6225}, {125,15077}, {184,15740}, {193,1192}, {343,15717}, {376,1092}, {394,3522}, {524,1620}, {567,45073}, {578,43815}, {631,26917}, {648,3183}, {1097,6060}, {1105,6618}, {1370,41482}, {1498,2063}, {1503,41427}, {1531,11541}, {1568,33703}, {1974,13346}, {1992,9786}, {2060,14365}, {2071,12324}, {3043,35503}, {3060,15887}, {3079,52578}, {3088,28408}, {3146,11064}, {3147,15035}, {3167,44247}, {3516,14826}, {3524,43808}, {3528,5562}, {3529,10721}, {3548,18918}, {3618,11425}, {5095,46730}, {5921,6696}, {6616,52913}, {6803,11430}, {6908,28754}, {7396,17845}, {7487,28708}, {9729,40673}, {9833,25712}, {10282,35513}, {10564,34938}, {10996,13367}, {11008,37487}, {11206,11413}, {11432,44273}, {11433,22467}, {11449,35260}, {11487,18570}, {12160,44268}, {12429,16976}, {13348,41716}, {14927,28419}, {15131,37444}, {16051,21659}, {16196,18945}, {17928,18928}, {18951,43615}, {20427,45771}, {22151,37460}, {22555,22951}, {33522,38444}, {34286,44704}, {36212,52543}, {41465,44242}, {43574,52432}

X(53050) = isogonal conjugate of X(31942)
X(53050) = isotomic conjugate of the polar conjugate of X(36413)
X(53050) = X(4)-Dao conjugate of-X(6526)
X(53050) = X(i)-isoconjugate-of-X(j) for these {i, j}: {459, 2155}, {1096, 52559}, {2184, 41489}
X(53050) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (20, 459), (154, 41489), (394, 52559), (1097, 92), (1249, 6526)
X(53050) = perspector of the circumconic {A, B, C, X(4563), X(36841)}
X(53050) = intersection, other than A, B, C, of circumconics {A, B, C, X(3), X(154)} and {A, B, C, X(4), X(5895)}
X(53050) = barycentric product X(i)*X(j) for these {i, j}: {20, 37669}, {63, 1097}, {69, 36413}, {345, 7338}, {348, 6060}, {394, 52578}
X(53050) = barycentric quotient X(i)/X(j) for these (i, j): (20, 459), (154, 41489), (394, 52559), (1097, 92), (1249, 6526), (2060, 40839)
X(53050) = trilinear product X(i)*X(j) for these {i, j}: {3, 1097}, {63, 36413}, {77, 6060}, {78, 7338}, {255, 52578}, {326, 3079}
X(53050) = trilinear quotient X(i)/X(j) for these (i, j): (326, 52559), (610, 41489), (1097, 4), (1895, 6526), (3079, 1096)
X(53050) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6193, 18931), (20, 2883, 40196), (20, 32605, 5895), (20, 35602, 37669), (110, 30552, 6225), (487, 488, 40995), (6060, 7338, 1097), (11449, 37201, 35260), (27082, 37669, 20)

---

X(53051) = X(5)X(3462)∩X(275)X(1971)

Barycentrics    (a^2+b^2-c^2)*(a^2-b^2+c^2)*(3*a^18*(b^2+c^2)-2*a^16*(10*b^4+13*b^2*c^2+10*c^4)-a^2*(b^2-c^2)^6*(8*b^6+13*b^4*c^2+13*b^2*c^4+8*c^6)+a^14*(55*b^6+78*b^4*c^2+78*b^2*c^4+55*c^6)+(b^2-c^2)^6*(b^8-b^6*c^2-2*b^4*c^4-b^2*c^6+c^8)+a^8*(b^2-c^2)^2*(7*b^8-9*b^6*c^2-18*b^4*c^4-9*b^2*c^6+7*c^8)-7*a^12*(11*b^8+15*b^6*c^2+14*b^4*c^4+15*b^2*c^6+11*c^8)+a^4*(b^2-c^2)^4*(25*b^8+37*b^6*c^2+42*b^4*c^4+37*b^2*c^6+25*c^8)-a^6*(b^2-c^2)^2*(35*b^10+24*b^8*c^2+21*b^6*c^4+21*b^4*c^6+24*b^2*c^8+35*c^10)+a^10*(49*b^10+62*b^8*c^2+33*b^6*c^4+33*b^4*c^6+62*b^2*c^8+49*c^10)) : :

See Ivan Pavlov, euclid 5753.

X(53051) lies on these lines: {5,3462}, {233,51358}, {275,1971}

---

X(53052) = X(1)X(3) ∩ X(9)X(4711)

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

See Tran Viet Hung and César Lozada, euclid 5763.

X(53052) lies on these lines: {1,3}, {9,4711}, {11,19876}, {390,3679}, {495,50865}, {497,19875}, {950,4668}, {1058,9588}, {1479,31436}, {1698,5274}, {1699,8164}, {1706,8167}, {2136,5234}, {2269,3973}, {3058,51066}, {3421,50836}, {3586,5817}, {3624,9785}, {3632,4314}, {3633,4313}, {3895,4512}, {4326,40269}, {4342,5281}, {4345,51105}, {4355,5493}, {4677,10385}, {4882,5250}, {5326,50443}, {5542,34632}, {5726,9580}, {5881,10386}, {8275,34747}, {9576,30145}, {9669,30315}, {10578,28228}, {10589,51785}, {10590,10624}, {11519,31424}, {11525,16418}, {12053,34595}, {12513,51576}, {12632,18249}, {15171,37714}, {15254,51781}, {30308,31479}, {30332,51782}, {39156,44858}, {41166,51767}

X(53052) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 30337, 1), (40, 6767, 10980), (55, 3057, 13384), (55, 5048, 3601), (55, 9819, 1), (165, 31393, 1), (3295, 7991, 1), (3303, 3339, 1), (3303, 5183, 44841), (3361, 37556, 1), (3895, 4512, 4915), (4342, 5281, 25055), (5183, 44841, 3339), (5584, 8171, 36), (5919, 13462, 1), (5919, 35445, 13462), (6767, 10980, 1), (7987, 9957, 1), (10389, 18421, 1), (11224, 24929, 1), (37556, 37568, 3361)

---

X(53053) = X(1)X(3) ∩ X(9)X(3913)

Barycentrics    a*(-a+b+c)*(3*a^2+4*(b+c)*a+(b-c)^2) : :
X(53053) = 3*X(165)-4*X(10268)

See Tran Viet Hung and César Lozada, euclid 5763.

X(53053) lies on these lines: {1,3}, {2,12575}, {4,5726}, {7,5493}, {8,4314}, {9,3913}, {10,390}, {11,51559}, {12,9580}, {21,3895}, {84,9851}, {100,8583}, {140,37704}, {145,8275}, {169,3731}, {191,9898}, {200,3871}, {221,9899}, {226,9589}, {355,10386}, {452,6736}, {495,41869}, {496,31423}, {497,1698}, {498,6964}, {516,5290}, {519,4313}, {595,1253}, {936,8715}, {938,21153}, {944,9948}, {950,3679}, {956,11519}, {958,2136}, {960,3158}, {962,13405}, {993,12629}, {1000,5882}, {1001,1706}, {1015,31421}, {1056,31730}, {1058,6684}, {1125,5281}, {1203,7074}, {1210,9588}, {1479,6939}, {1699,3085}, {1702,35809}, {1703,35808}, {1742,8915}, {1743,2269}, {1837,41864}, {2066,19004}, {2241,9593}, {2268,16667}, {2346,12560}, {2551,47375}, {2999,3915}, {3058,9581}, {3062,7160}, {3091,51783}, {3100,30145}, {3146,51782}, {3189,5837}, {3241,4652}, {3298,9616}, {3474,4355}, {3486,3632}, {3487,28194}, {3488,11362}, {3522,4315}, {3584,30308}, {3586,4309}, {3600,12512}, {3616,4342}, {3624,5218}, {3634,5274}, {3636,4345}, {3646,9709}, {3654,12433}, {3671,10578}, {3672,10521}, {3681,4917}, {3743,42446}, {3751,10387}, {3812,38316}, {3870,12526}, {3890,4855}, {3928,34791}, {3947,9812}, {4189,36846}, {4254,7368}, {4294,5691}, {4298,9778}, {4301,5703}, {4302,9613}, {4304,9799}, {4312,6361}, {4321,7676}, {4326,5223}, {4421,5438}, {4428,5436}, {4640,6762}, {4646,7290}, {4668,5727}, {4677,10950}, {4857,30315}, {4995,11376}, {5011,16673}, {5044,9848}, {5219,12701}, {5248,9623}, {5261,30332}, {5273,12632}, {5302,8168}, {5414,19003}, {5432,34595}, {5435,21625}, {5441,37708}, {5531,10087}, {5542,5586}, {5587,15171}, {5687,8580}, {5690,30286}, {5766,12572}, {5790,31795}, {5815,51090}, {5904,12711}, {6048,33551}, {6284,9578}, {6744,8236}, {7071,7713}, {7080,40998}, {7086,36754}, {7162,38271}, {7218,10482}, {7672,12564}, {7966,7990}, {7993,10058}, {8164,18483}, {9574,16781}, {9575,31477}, {9579,15888}, {9582,35768}, {9592,31451}, {9612,10056}, {9668,18492}, {9845,34862}, {9850,31805}, {10164,14986}, {10384,15837}, {10572,37712}, {10589,19872}, {10827,34746}, {10889,25590}, {10953,18514}, {11036,34632}, {11238,19876}, {11374,31162}, {11522,13411}, {12127,12513}, {12577,50808}, {12675,24645}, {12710,18412}, {12767,37736}, {14094,51794}, {14100,34790}, {14872,52665}, {15006,38057}, {15015,15558}, {15054,51793}, {15104,44547}, {15156,51812}, {15157,51813}, {15172,26446}, {16502,31426}, {17022,24590}, {17054,35227}, {18250,52653}, {18518,18529}, {20075,24987}, {21627,30478}, {23235,51796}, {23681,28027}, {24392,26066}, {25917,46917}, {30384,31452}, {30393,48696}, {31231,37722}, {34628,45287}, {37709,45081}, {37723,40663}, {37828,49736}, {38664,51795}, {38665,51768}, {38666,51809}, {38668,51766}, {38669,51767}, {38670,51769}, {38671,51811}, {38674,51808}, {38675,51815}, {38684,51770}, {38685,51765}, {38688,51814}, {42043,50621}, {45776,52026}

X(53053) = intersection, other than A, B, C, of circumconics {A, B, C, X(4), X(10980)} and {A, B, C, X(8), X(3339)}
X(53053) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 11010, 2093), (1, 16192, 56), (1, 31508, 3), (3, 31393, 1), (40, 3295, 1), (55, 1697, 1), (56, 37556, 1), (57, 3303, 1), (65, 10389, 1), (1420, 5919, 1), (2098, 13384, 1), (2646, 7962, 1), (3057, 3601, 1), (3085, 10624, 1699), (3303, 37568, 57), (3304, 51779, 1), (3333, 6767, 1), (3340, 37080, 1), (3576, 9957, 1), (3746, 5119, 1), (3748, 11518, 1), (3749, 37598, 1), (3871, 5250, 200), (4309, 10039, 3586), (5217, 5919, 1420), (5218, 12053, 3624), (5269, 37548, 1), (5710, 37553, 1), (6361, 21620, 4312), (7982, 24929, 1), (7987, 30337, 1), (7991, 16208, 165), (10578, 20070, 3671), (13411, 30305, 11522), (17594, 37588, 1), (30323, 37571, 1), (30331, 43174, 938), (35445, 37556, 56)

---

X(53054) = X(1)X(3) ∩ X(390)X(511)

Barycentrics    a*(-a+b+c)*(7*a^2+4*(b+c)*a-3*(b-c)^2) : :
X(53054) = 3*X(5726)-4*X(8164)

See Tran Viet Hung and César Lozada, euclid 5763.

X(53054) lies on these lines: {1,3}, {78,5234}, {101,2268}, {214,4326}, {376,4312}, {390,551}, {495,50811}, {497,25055}, {498,37714}, {515,5726}, {519,5281}, {934,28163}, {943,5450}, {944,51784}, {950,3624}, {956,33595}, {958,4866}, {993,5223}, {995,2293}, {997,5785}, {1006,10398}, {1012,3062}, {1056,51705}, {1125,4208}, {1282,34930}, {1478,34628}, {1698,3486}, {1699,4304}, {1837,5326}, {2320,3872}, {2886,34701}, {3058,51110}, {3158,4915}, {3241,8275}, {3243,11194}, {3488,10165}, {3522,3671}, {3523,6738}, {3583,30308}, {3586,6843}, {3616,4314}, {3622,12575}, {3636,9785}, {3679,5218}, {3897,4853}, {4134,40269}, {4189,12526}, {4294,11522}, {4297,5290}, {4302,50865}, {4305,5691}, {4315,10578}, {4323,5493}, {4342,38314}, {4345,51103}, {4511,4512}, {4654,15326}, {4666,4881}, {4677,4995}, {4999,12625}, {5226,28164}, {5251,30393}, {5265,6744}, {5284,8583}, {5303,11520}, {5312,22072}, {5313,14547}, {5426,10382}, {5429,16667}, {5432,5727}, {5433,37723}, {5436,8167}, {5440,8580}, {5441,37692}, {5586,12563}, {5692,10391}, {5720,30291}, {5731,13405}, {5732,9814}, {6855,7989}, {6906,7992}, {7074,16474}, {7963,44842}, {9580,15950}, {9581,10543}, {9624,15171}, {9668,38021}, {9851,12114}, {10176,10394}, {10384,35272}, {10385,51105}, {10387,16491}, {10582,35262}, {11260,12127}, {11376,41864}, {11714,44858}, {12437,30478}, {12447,17558}, {12560,30295}, {12629,51111}, {15735,51809}, {16469,37817}, {18594,37504}, {21214,44843}, {22836,31424}, {25524,45036}, {26446,30286}, {30330,52769}, {31423,37730}, {31434,37712}, {34043,37501}, {37704,38028}, {37724,52793}, {37737,41869}, {50238,50443}, {50742,51090}

X(53054) = midpoint of X(1) and X(31508)
X(53054) = intersection, other than A, B, C, of circumconics {A, B, C, X(9), X(18421)} and {A, B, C, X(21), X(3339)}
X(53054) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 3, 3339), (1, 35, 7991), (1, 55, 9819), (1, 3612, 7987), (1, 5010, 2093), (1, 7987, 3361), (1, 16192, 65), (1, 30282, 165), (36, 10980, 3361), (55, 5048, 1697), (55, 13384, 1), (55, 34471, 5048), (999, 6282, 2093), (1319, 10389, 1), (1388, 37556, 1), (1420, 37080, 1), (1697, 34471, 1), (2093, 5010, 165), (2093, 30282, 5010), (2646, 3601, 1), (3576, 24929, 1), (3601, 13384, 55), (3612, 37571, 1), (5538, 7987, 165), (7987, 10980, 36), (7987, 30337, 8071), (7991, 22766, 3361), (10246, 31393, 1), (24929, 37606, 3576), (37606, 50371, 3612)

---

X(53055) = X(2)X(11) ∩ X(7)X(104)

Barycentrics    a*(-a+b+c)*(a^4-(b+c)*a^3-(b^2-3*b*c+c^2)*a^2+(b^2-c^2)*(b-c)*a+3*(b-c)^2*b*c) : :
X(53055) = 2*X(1)+X(1156) = X(7)-4*X(1387) = 2*X(9)+X(1320) = 2*X(11)+X(390) = 4*X(11)-X(20119) = 2*X(36)-3*X(7677) = 2*X(80)+X(12730) = X(80)+2*X(30331) = X(100)-4*X(1001) = 2*X(390)+X(20119) = 2*X(1145)-5*X(18230) = 2*X(2550)-5*X(31272) = 5*X(3616)-2*X(10427) = 4*X(5572)-X(12755) = 2*X(5572)+X(17638) = 8*X(6667)-5*X(40333) = 3*X(7677)-X(30295) = X(10707)+2*X(47357) = X(12730)-4*X(30331) = X(14151)+2*X(51768) = X(16133)+2*X(46816)

See Tran Viet Hung and César Lozada, euclid 5763.

X(53055) lies on the apollonian circle of mixtilinear incircles and these lines: {1,651}, {2,11}, {3,30332}, {5,47744}, {7,104}, {8,4578}, {9,644}, {21,3254}, {36,516}, {56,18220}, {57,41166}, {59,30206}, {80,2346}, {119,6939}, {153,8232}, {214,4326}, {294,52969}, {404,50443}, {405,5766}, {411,9614}, {480,3036}, {495,10711}, {499,30312}, {517,37787}, {518,5048}, {527,51423}, {901,33646}, {943,15172}, {952,954}, {956,4345}, {958,6068}, {971,25405}, {993,50836}, {1058,37726}, {1145,18230}, {1253,15485}, {1279,3100}, {1317,8162}, {1319,15726}, {1420,8544}, {1445,2093}, {1478,30311}, {1479,6932}, {1482,5729}, {1617,9812}, {1697,5047}, {1768,10980}, {1776,18839}, {1837,17662}, {1864,3957}, {2078,36002}, {2098,5220}, {2293,16484}, {2800,11529}, {2802,9623}, {2975,5698}, {3057,5260}, {3086,6966}, {3219,17642}, {3246,41339}, {3295,5818}, {3304,30340}, {3305,10388}, {3315,7004}, {3475,12831}, {3601,5528}, {3616,10427}, {3679,38216}, {3748,41701}, {3871,7705}, {3873,30223}, {3895,47375}, {4312,37587}, {4313,10609}, {4321,9814}, {4342,5251}, {4511,15733}, {4861,17622}, {4907,35227}, {5010,7676}, {5083,11020}, {5225,11510}, {5248,48713}, {5253,5880}, {5259,12575}, {5261,38757}, {5289,42014}, {5531,30291}, {5540,28345}, {5542,10074}, {5563,30424}, {5572,12755}, {5657,38131}, {5660,13405}, {5686,5854}, {5696,30144}, {5704,10306}, {5727,50890}, {5728,10698}, {5790,38180}, {5825,25416}, {5840,6916}, {5851,11038}, {5853,6735}, {5856,52653}, {6601,45393}, {6702,25438}, {6713,35238}, {6797,31658}, {6905,7743}, {6906,11373}, {6907,10738}, {6920,9957}, {6925,10724}, {6935,10596}, {6945,7678}, {6957,10590}, {6986,10624}, {7292,9371}, {7373,51529}, {7411,9580}, {7672,12758}, {7675,10384}, {7679,39692}, {7688,14217}, {7972,43179}, {8103,11191}, {8255,15950}, {9669,11491}, {9848,51715}, {9963,15006}, {10087,31434}, {10269,38124}, {10389,41553}, {10391,29817}, {10398,11526}, {10580,41556}, {10591,11508}, {10595,12776}, {11019,11219}, {11025,11570}, {11248,47743}, {11372,11715}, {11496,14986}, {11571,20116}, {12531,12648}, {12532,15185}, {12573,34789}, {12686,15528}, {12737,29007}, {12740,14100}, {12764,42356}, {13253,30330}, {13462,30353}, {13464,48694}, {14190,37131}, {15251,37771}, {15837,17636}, {15931,51783}, {17100,32558}, {17613,37789}, {18254,34784}, {21151,38032}, {21630,51506}, {21669,24928}, {22753,38152}, {22760,42886}, {23513,38149}, {25440,50444}, {30305,36976}, {32557,38052}, {33148,38357}, {34126,38121}, {34474,38031}, {37618,43178}, {37633,52428}, {38026,51636}, {38043,38752}, {38044,38107}, {38095,40726}, {40269,42871}, {51639,52015}

X(53055) = midpoint of X(i) and X(j) for these {i, j}: {1, 51768}, {390, 45043}, {1156, 14151}
X(53055) = reflection of X(i) in X(j) for these (i, j): (2, 38060), (7, 38055), (8, 38211), (1156, 51768), (3679, 38216), (5657, 38131), (5790, 38180), (14151, 1), (18450, 1319), (20119, 45043), (21151, 38032), (30295, 36), (30379, 44675), (34474, 38031), (38052, 32557), (38055, 1387), (38107, 38044), (38121, 34126), (38149, 23513), (38202, 45310), (38752, 38043), (45043, 11)
X(53055) = crosssum of X(650) and X(1643)
X(53055) = X(21)-beth conjugate of-X(44858)
X(53055) = perspector of the circumconic {A, B, C, X(666), X(31628)}
X(53055) = inverse of X(100) in Feuerbach circumhyperbola
X(53055) = inverse of X(934) in mixtilinear incircles radical circle
X(53055) = intersection, other than A, B, C, of circumconics {A, B, C, X(104), X(4845)} and {A, B, C, X(105), X(2316)}
X(53055) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (80, 30331, 12730), (1621, 10707, 100), (3295, 12019, 38665), (5572, 17638, 12755), (5698, 42842, 2975), (7677, 30295, 36), (10384, 38316, 7675), (14100, 42819, 30284), (40565, 40566, 100)

---

X(53056) = X(1)X(3) ∩ X(2)X(4312)

Barycentrics    a*(5*a^2-2*(b+c)*a-3*(b-c)^2) : :
X(53056) = X(5274)-3*X(5435)

See Tran Viet Hung and César Lozada, euclid 5763.

X(53056) lies on these lines: {1,3}, {2,4312}, {7,10164}, {9,8169}, {11,50865}, {43,5756}, {63,5785}, {105,8699}, {144,20103}, {200,3218}, {218,1615}, {226,21168}, {238,8056}, {371,8831}, {372,8833}, {388,9588}, {390,50808}, {404,12526}, {405,51576}, {515,30286}, {516,5274}, {553,5218}, {579,2272}, {672,3973}, {910,1743}, {938,12512}, {1044,3216}, {1054,1707}, {1190,17745}, {1200,4253}, {1203,41422}, {1323,9533}, {1376,3928}, {1445,2951}, {1465,34033}, {1478,19875}, {1698,4208}, {1699,3474}, {1706,8170}, {1708,1750}, {1788,5691}, {1836,7988}, {2308,2999}, {2348,52180}, {3035,28609}, {3052,5573}, {3062,19541}, {3085,4355}, {3086,9589}, {3149,7992}, {3243,4421}, {3306,4512}, {3475,4031}, {3487,5586}, {3522,6738}, {3523,3671}, {3598,4862}, {3599,10481}, {3600,43174}, {3624,4295}, {3632,4311}, {3679,4293}, {3731,17122}, {3752,16469}, {3870,23958}, {3916,5234}, {3929,4413}, {3980,18229}, {4018,19537}, {4298,51784}, {4301,5265}, {4303,5312}, {4342,34632}, {4414,17022}, {4640,5437}, {4649,39980}, {4654,5432}, {4668,45287}, {4845,8917}, {4866,9709}, {4915,4973}, {5022,11051}, {5044,30290}, {5057,31224}, {5219,5326}, {5226,30424}, {5281,5542}, {5290,6684}, {5313,10460}, {5393,51764}, {5405,51763}, {5442,37692}, {5445,50238}, {5493,14986}, {5531,27778}, {5556,15022}, {5698,6692}, {5704,51118}, {5726,26446}, {5727,15326}, {5728,10178}, {5745,38052}, {5779,30291}, {5974,44852}, {6173,6690}, {6361,51785}, {6737,37267}, {6745,9965}, {6762,8168}, {7288,11522}, {7354,37714}, {7580,10398}, {7741,7965}, {7951,19876}, {7967,16236}, {7989,9579}, {8275,50810}, {8830,43064}, {8916,16572}, {9315,20459}, {9316,13329}, {9446,52511}, {9580,17728}, {9778,11019}, {10090,12767}, {10157,31391}, {10167,18412}, {10171,31188}, {10582,27003}, {10860,15299}, {10895,30315}, {11041,51705}, {11231,18541}, {11495,30330}, {12047,34595}, {12527,26062}, {12699,50444}, {13257,41707}, {13405,21454}, {15104,17625}, {15325,31162}, {15338,37723}, {15601,16602}, {15855,52705}, {16143,41697}, {16558,46947}, {17580,18249}, {17768,30827}, {20075,31146}, {21000,35227}, {21060,28610}, {24703,31190}, {25264,41833}, {25502,52155}, {25590,32916}, {26102,35270}, {26105,50836}, {28174,37704}, {28272,44849}, {30304,44425}, {30353,37787}, {32932,35613}, {33633,38293}, {34043,36745}, {37712,40663}, {37789,43166}, {44845,53005}

X(53056) = reflection of X(1) in X(13462)
X(53056) = intersection, other than A, B, C, of circumconics {A, B, C, X(9), X(31508)} and {A, B, C, X(55), X(38293)}
X(53056) = barycentric product X(i)*X(j) for these {i, j}: {1, 20059}, {8, 33633}, {85, 38293}
X(53056) = barycentric quotient X(i)/X(j) for these (i, j): (1, 36605), (9, 36625), (55, 36627), (57, 38254)
X(53056) = trilinear product X(i)*X(j) for these {i, j}: {6, 20059}, {7, 38293}, {9, 33633}
X(53056) = trilinear quotient X(i)/X(j) for these (i, j): (2, 36605), (7, 38254), (8, 36625), (9, 36627)
X(53056) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 3339, 1), (36, 46, 2093), (36, 2093, 1), (40, 3361, 1), (46, 15803, 1), (55, 10980, 1), (56, 7991, 1), (57, 165, 1), (57, 1155, 165), (57, 35445, 354), (65, 7987, 1), (165, 10980, 55), (999, 9819, 1), (1319, 11224, 1), (1402, 35621, 1), (1420, 11531, 1), (2093, 15803, 36), (2099, 30392, 1), (3303, 30343, 1), (3304, 30337, 1), (3340, 30389, 1), (3361, 9819, 999), (3474, 3911, 1699), (3550, 18193, 1), (3576, 18421, 1), (3579, 37545, 3333), (5902, 30282, 1), (6244, 8171, 55), (9441, 51302, 1), (9579, 24914, 7989), (10389, 30350, 1), (11407, 37541, 1), (27003, 35258, 10582)

---

X(53057) = X(1)X(3) ∩ X(404)X(5223)

Barycentrics    a*(9*a^3+3*(b+c)*a^2-(9*b^2-10*b*c+9*c^2)*a-3*(b^2-c^2)*(b-c)) : :

See Tran Viet Hung and César Lozada, euclid 5763.

X(53057) lies on these lines: {1,3}, {404,5223}, {474,30393}, {499,30308}, {631,4355}, {936,4973}, {1125,51576}, {1478,30315}, {1707,45047}, {1770,50444}, {1788,37712}, {3062,7285}, {3086,50865}, {3600,9588}, {3614,31231}, {3624,43733}, {3634,5234}, {3817,5556}, {3911,5229}, {4292,7988}, {4293,37714}, {4312,7288}, {4512,46934}, {4652,5550}, {4853,9352}, {5265,11522}, {5302,8169}, {5435,5691}, {5542,15717}, {6744,10304}, {7173,9579}, {7317,11362}, {8166,18483}, {9654,19876}, {9851,44425}, {10592,44847}, {16143,41547}, {16408,36835}, {18990,19875}, {19537,41863}, {19862,31424}, {21153,43180}, {27624,27627}

---

X(53058) = X(1)X(3) ∩ X(515)X(8166)

Barycentrics    a*(7*a^3-3*(b+c)*a^2-(7*b^2-22*b*c+7*c^2)*a+3*(b^2-c^2)*(b-c)) : :

See Tran Viet Hung and César Lozada, euclid 5763.

X(53058) lies on these lines: {1,3}, {495,44847}, {497,34628}, {499,30315}, {515,8166}, {936,51714}, {956,30393}, {1056,25055}, {1125,8165}, {1149,16487}, {1476,7995}, {1478,30308}, {1699,4315}, {1743,47623}, {2320,45834}, {3062,42884}, {3086,37714}, {3476,37712}, {3600,11522}, {3616,46873}, {3622,12577}, {3624,8164}, {3636,11037}, {3679,44848}, {3816,34716}, {4293,50865}, {4308,5274}, {4311,51785}, {4321,9814}, {4355,13464}, {4866,5288}, {4882,17614}, {5265,9588}, {5438,8168}, {5542,38314}, {5904,17624}, {6049,6738}, {7988,10590}, {7989,10106}, {7993,41554}, {9613,50444}, {9708,36835}, {10179,10569}, {11525,16417}, {12128,25917}, {15325,19875}, {16469,44858}, {17728,30286}, {31190,38455}

X(53058) = inverse of X(5537) in mixtilinear incircles radical circle
X(53058) = intersection, other than A, B, C, of circumconics {A, B, C, X(7), X(11224)} and {A, B, C, X(21), X(30337)}
X(53058) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 3, 30337), (1, 36, 9819), (1, 56, 7991), (1, 57, 11224), (1, 999, 10980), (1, 1319, 30392), (1, 1420, 7987), (1, 3304, 30343), (1, 3339, 16189), (1, 3361, 11531), (1, 5563, 3339), (1, 13462, 165), (1, 16192, 9957), (1, 31508, 5919), (1, 37587, 2093), (36, 9819, 165), (999, 8171, 56), (999, 25405, 11529), (999, 50194, 3333), (1420, 20323, 1), (1420, 37556, 37605), (2093, 37587, 3361), (3339, 8069, 165), (3576, 51788, 1), (9819, 13462, 36), (11529, 25405, 1)

---

X(53059) = X(2)X(40405) ∩ X(3)X(40322)

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

X(53059) lies on the cubics K260 and K1313 and on these lines: {2, 40405}, {3, 40322}, {6, 1196}, {32, 11326}, {69, 6387}, {83, 2996}, {182, 3224}, {206, 32740}, {213, 38252}, {577, 1084}, {729, 3565}, {1249, 6531}, {1691, 36615}, {1692, 2207}, {1974, 3080}, {3225, 35136}, {3291, 40318}, {3589, 9516}, {3618, 6340}, {5028, 26206}, {5042, 45785}, {5065, 40799}, {5138, 40770}, {6467, 34481}, {8265, 33871}, {8769, 40747}, {8911, 26461}, {9924, 40350}, {17710, 28662}, {19136, 46288}, {21637, 39764}, {26454, 26920}, {33578, 46319}, {45199, 46432}

X(53059) = complement of the isotomic conjugate of X(15591)
X(53059) = isogonal conjugate of the isotomic conjugate of X(8770)
X(53059) = isogonal conjugate of the polar conjugate of X(14248)
X(53059) = polar conjugate of the isotomic conjugate of X(40319)
X(53059) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 15261}, {15591, 2887}
X(53059) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 15261}, {8770, 40319}
X(53059) = X(i)-isoconjugate of X(j) for these (i,j): {2, 18156}, {75, 193}, {76, 1707}, {92, 6337}, {274, 4028}, {304, 6353}, {310, 21874}, {312, 17081}, {326, 21447}, {561, 3053}, {668, 3798}, {799, 3566}, {1821, 51374}, {1969, 3167}, {3787, 18833}, {4590, 17876}, {4602, 8651}, {6388, 24037}, {17149, 47733}, {19118, 40364}, {30558, 33787}, {32459, 46277}
X(53059) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 193}, {512, 6388}, {15259, 21447}, {15261, 2}, {22391, 6337}, {32664, 18156}, {38996, 3566}, {40368, 3053}, {40601, 51374}
X(53059) = cevapoint of X(i) and X(j) for these (i,j): {184, 40319}, {1084, 3049}
X(53059) = crossdifference of every pair of points on line {3566, 51374}
X(53059) = barycentric product X(i)*X(j) for these {i,j}: {1, 38252}, {3, 14248}, {4, 40319}, {6, 8770}, {25, 6391}, {31, 8769}, {32, 2996}, {184, 34208}, {512, 3565}, {669, 35136}, {1974, 6340}, {5203, 14908}, {10547, 47730}, {15261, 15591}
X(53059) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 18156}, {32, 193}, {184, 6337}, {237, 51374}, {560, 1707}, {669, 3566}, {1084, 6388}, {1397, 17081}, {1501, 3053}, {1918, 4028}, {1919, 3798}, {1974, 6353}, {2205, 21874}, {2207, 21447}, {2996, 1502}, {3565, 670}, {6340, 40050}, {6391, 305}, {8769, 561}, {8770, 76}, {9426, 8651}, {9427, 47430}, {14248, 264}, {14567, 32459}, {14575, 3167}, {14585, 10607}, {27369, 41584}, {34208, 18022}, {34416, 21970}, {35136, 4609}, {38252, 75}, {40319, 69}, {40981, 41588}, {41331, 3787}, {42068, 5139}, {44162, 19118}, {51951, 47733}
X(53059) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 8770, 6391}, {69, 6387, 6388}

---

X(53060) = X(3)X(6413) ∩ X(6)X(8948)

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

X(53060) lies on the cubic K1313 and these lines: {3, 6413}, {6, 8948}, {184, 5062}, {371, 39383}, {485, 6776}, {488, 24246}, {1321, 5870}, {1504, 19032}, {1587, 41515}, {10237, 26886}, {15073, 35840}, {21637, 39764}

X(53060) = isogonal conjugate of the isotomic conjugate of X(24246)
X(53060) = X(6)-Ceva conjugate of X(8577)
X(53060) = X(i)-isoconjugate of X(j) for these (i,j): {492, 19218}, {1969, 8950}
X(53060) = X(485)-Dao conjugate of X(76)
X(53060) = barycentric product X(i)*X(j) for these {i,j}: {6, 24246}, {485, 10132}, {488, 8577}, {3068, 6413}, {6423, 11090}, {17431, 39383}, {41515, 51946}
X(53060) = barycentric quotient X(i)/X(j) for these {i,j}: {488, 45805}, {6413, 5490}, {6423, 1585}, {8577, 24244}, {10132, 492}, {14575, 8950}, {24246, 76}

---

X(53061) = X(3)X(6414) ∩ X(6)X(8946)

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

X(53061) lies on the cubic K1313 and these lines: {3, 6414}, {6, 8946}, {184, 5058}, {372, 39384}, {486, 6776}, {487, 24245}, {1322, 5871}, {1505, 19033}, {1588, 41516}, {15073, 35841}, {21637, 39764}

X(53061) = isogonal conjugate of the isotomic conjugate of X(24245)
X(53061) = X(6)-Ceva conjugate of X(8576)
X(53061) = X(491)-isoconjugate of X(19217)
X(53061) = X(486)-Dao conjugate of X(76)
X(53061) = barycentric product X(i)*X(j) for these {i,j}: {6, 24245}, {486, 10133}, {487, 8576}, {3069, 6414}, {6424, 11091}, {17432, 39384}, {26922, 52291}, {41516, 51905}
X(53061) = barycentric quotient X(i)/X(j) for these {i,j}: {487, 45806}, {6414, 5491}, {6424, 1586}, {8576, 24243}, {10133, 491}, {24245, 76}

---

X(53062) = X(6)X(8946) ∩ X(32)X(26461)

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

X(53062) lies on the cubic K1313 and these lines: {6, 8946}, {32, 26461}, {494, 3167}, {1587, 24243}, {5093, 49377}, {5491, 39875}, {13024, 26293}

X(53062) = X(92)-isoconjugate of X(24245)
X(53062) = X(i)-Dao conjugate of X(j) for these (i,j): {5408, 46743}, {22391, 24245}
X(53062) = cevapoint of X(184) and X(26461)
X(53062) = barycentric product X(i)*X(j) for these {i,j}: {372, 494}, {491, 26461}, {5409, 8946}, {24243, 26920}
X(53062) = barycentric quotient X(i)/X(j) for these {i,j}: {184, 24245}, {494, 34392}, {5409, 46743}, {26461, 486}, {26920, 487}

---

X(53063) = X(1)X(13332) ∩ X(31)X(184)

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

X(53063) lies on the cubic K1313 and these lines: {1, 13332}, {3, 1335}, {6, 34125}, {28, 2362}, {31, 184}, {48, 606}, {104, 7133}, {1400, 8577}, {1436, 19000}, {1444, 13388}, {1686, 3301}, {1791, 30557}, {2047, 44622}, {2048, 44623}, {2217, 7969}, {2252, 6413}, {5415, 19350}, {6502, 16466}

X(53063) = isogonal conjugate of the isotomic conjugate of X(13388)
X(53063) = isogonal conjugate of the polar conjugate of X(2362)
X(53063) = X(i)-isoconjugate of X(j) for these (i,j): {2, 14121}, {8, 13390}, {75, 42013}, {92, 30556}, {264, 2066}, {312, 16232}, {318, 13389}, {6502, 7017}, {7090, 13386}, {7133, 46744}
X(53063) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 42013}, {13389, 76}, {22391, 30556}, {32664, 14121}
X(53063) = barycentric product X(i)*X(j) for these {i,j}: {1, 2067}, {3, 2362}, {6, 13388}, {48, 1659}, {56, 30557}, {57, 5414}, {65, 1805}, {222, 7133}, {603, 7090}, {606, 13390}, {1335, 16232}, {6213, 6502}, {13389, 34121}, {46377, 51842}
X(53063) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 14121}, {32, 42013}, {184, 30556}, {604, 13390}, {1397, 16232}, {1659, 1969}, {1805, 314}, {2067, 75}, {2362, 264}, {5414, 312}, {6502, 46744}, {7133, 7017}, {9247, 2066}, {13388, 76}, {30557, 3596}, {52411, 13389}

---

X(53064) = X(1)X(13333) ∩ X(31)X(184)

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

X(53064) lies on the cubic K1313 and these lines: {1, 13333}, {3, 1124}, {6, 34121}, {28, 16232}, {31, 184}, {48, 605}, {104, 18460}, {1400, 8576}, {1436, 18999}, {1444, 13389}, {1685, 3299}, {1791, 30556}, {2067, 16466}, {2217, 7968}, {2252, 6414}, {5416, 19350}

X(53064) = isogonal conjugate of the isotomic conjugate of X(13389)
X(53064) = isogonal conjugate of the polar conjugate of X(16232)
X(53064) = X(i)-isoconjugate of X(j) for these (i,j): {2, 7090}, {8, 1659}, {75, 7133}, {92, 30557}, {264, 5414}, {312, 2362}, {318, 13388}, {2067, 7017}, {13387, 14121}, {42013, 46745}
X(53064) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 7133}, {13388, 76}, {22391, 30557}, {32664, 7090}
X(53064) = barycentric product X(i)*X(j) for these {i,j}: {1, 6502}, {3, 16232}, {6, 13389}, {48, 13390}, {56, 30556}, {57, 2066}, {65, 1806}, {222, 42013}, {603, 14121}, {605, 1659}, {1124, 2362}, {2067, 6212}, {13388, 34125}, {46376, 51841}
X(53064) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 7090}, {32, 7133}, {184, 30557}, {604, 1659}, {1397, 2362}, {1806, 314}, {2066, 312}, {2067, 46745}, {6502, 75}, {9247, 5414}, {13389, 76}, {13390, 1969}, {16232, 264}, {30556, 3596}, {42013, 7017}, {52411, 13388}

---

X(53065) = X(31)X(32) ∩ X(48)X(605)

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

X(53065) lies on the cubic K1313 and these lines: {6, 34125}, {31, 32}, {42, 8576}, {48, 605}, {219, 2066}, {222, 6502}, {1124, 3167}, {1172, 42013}, {1617, 2067}, {1806, 1812}, {1814, 13389}, {2362, 46379}, {2982, 16232}, {4254, 5414}, {7082, 13455}

X(53065) = isogonal conjugate of the isotomic conjugate of X(30556)
X(53065) = isogonal conjugate of the polar conjugate of X(42013)
X(53065) = X(1806)-Ceva conjugate of X(2066)
X(53065) = X(i)-isoconjugate of X(j) for these (i,j): {2, 1659}, {7, 7090}, {75, 2362}, {85, 7133}, {92, 13388}, {264, 2067}, {273, 30557}, {331, 5414}, {13387, 13390}, {16232, 46745}
X(53065) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 2362}, {13388, 6063}, {22391, 13388}, {32664, 1659}
X(53065) = barycentric product X(i)*X(j) for these {i,j}: {1, 2066}, {3, 42013}, {6, 30556}, {9, 6502}, {37, 1806}, {48, 14121}, {55, 13389}, {212, 13390}, {219, 16232}, {605, 7090}, {1124, 7133}, {5414, 6212}, {30557, 34125}
X(53065) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 1659}, {32, 2362}, {41, 7090}, {184, 13388}, {1806, 274}, {2066, 75}, {2175, 7133}, {5414, 46745}, {6502, 85}, {9247, 2067}, {13389, 6063}, {14121, 1969}, {16232, 331}, {30556, 76}, {42013, 264}, {52425, 30557}

---

X(53066) = X(6)X(34121) ∩ X(31)X(32)

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

X(53066) lies on the cubic K1313 and these lines: {6, 34121}, {31, 32}, {42, 8577}, {48, 606}, {219, 5414}, {222, 2067}, {1172, 7133}, {1335, 3167}, {1617, 6502}, {1805, 1812}, {1814, 13388}, {2066, 4254}, {2362, 2982}, {16232, 46378}

X(53066) = isogonal conjugate of the isotomic conjugate of X(30557)
X(53066) = isogonal conjugate of the polar conjugate of X(7133)
X(53066) = X(1805)-Ceva conjugate of X(5414)
X(53066) = X(i)-isoconjugate of X(j) for these (i,j): {2, 13390}, {7, 14121}, {75, 16232}, {85, 42013}, {92, 13389}, {264, 6502}, {273, 30556}, {331, 2066}, {1659, 13386}, {2362, 46744}
X(53066) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 16232}, {13389, 6063}, {22391, 13389}, {32664, 13390}
X(53066) = barycentric product X(i)*X(j) for these {i,j}: {1, 5414}, {3, 7133}, {6, 30557}, {9, 2067}, {37, 1805}, {48, 7090}, {55, 13388}, {212, 1659}, {219, 2362}, {606, 14121}, {1335, 42013}, {2066, 6213}, {30556, 34121}
X(53066) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 13390}, {32, 16232}, {41, 14121}, {184, 13389}, {1805, 274}, {2066, 46744}, {2067, 85}, {2175, 42013}, {2362, 331}, {5414, 75}, {7090, 1969}, {7133, 264}, {9247, 6502}, {13388, 6063}, {30557, 76}, {52425, 30556}

---

X(53067) = X(6)X(15369) ∩ X(32)X(3167)

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

X(53067) lies on the cubic K1313 and these lines: {6, 15369}, {32, 3167}, {1992, 6339}, {3053, 30558}

X(53067) = isogonal conjugate of the isotomic conjugate of X(30558)
X(53067) = X(15369)-Ceva conjugate of X(40322)
X(53067) = X(i)-isoconjugate of X(j) for these (i,j): {2128, 34208}, {2996, 33781}, {6392, 8769}, {8770, 33787}
X(53067) = barycentric product X(i)*X(j) for these {i,j}: {6, 30558}, {193, 40322}, {3053, 6339}, {6337, 15369}
X(53067) = barycentric quotient X(i)/X(j) for these {i,j}: {1707, 33787}, {3053, 6392}, {3167, 19583}, {10607, 6338}, {15369, 34208}, {30558, 76}, {40322, 2996}

---

X(53068) = X(6)-CEVA CONJUGATE OF X(53059)

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

X(53068) lies on the cubic K1313 and these lines: {3, 6391}, {6, 15369}, {184, 53059}, {193, 3565}, {6340, 18935}, {8770, 19459}, {8950, 53062}

X(53068) = X(6)-Ceva conjugate of X(53059)
X(53068) = X(i)-isoconjugate of X(j) for these (i,j): {92, 30558}, {1969, 53067}
X(53068) = X(i)-Dao conjugate of X(j) for these (i,j): {8770, 76}, {22391, 30558}
X(53068) = barycentric product X(i)*X(j) for these {i,j}: {1611, 6391}, {2128, 38252}, {2519, 3565}, {6392, 40319}, {6461, 14248}, {8770, 19588}, {19583, 53059}
X(53068) = barycentric quotient X(i)/X(j) for these {i,j}: {184, 30558}, {14575, 53067}, {40319, 6339}

---

X(53069) = X(6)-CEVA CONJUGATE OF X(53065)

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

X(53069) lies on the cubic K1313 and these lines: {3, 1124}, {42, 8577}, {184, 53065}, {213, 38252}, {605, 26920}, {8950, 53063}

X(53069) = X(6)-Ceva conjugate of X(53065)
X(53069) = X(30556)-Dao conjugate of X(76)
X(53069) = barycentric product X(2066)*X(6204)

---

X(53070) = X(6)-CEVA CONJUGATE OF X(53066)

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

X(53070) lies on the cubic K1313 and these lines: {3, 1335}, {42, 8576}, {184, 53066}, {213, 38252}, {606, 8911}, {53062, 53064}

X(53070) = X(6)-Ceva conjugate of X(53066)
X(53070) = X(30557)-Dao conjugate of X(76)
X(53070) = barycentric product X(5414)*X(6203)

---

X(53071) = X(289)-ISOCONJUGATE-OF-X(558)

Barycentrics    a*(-a+b+c)*(S*(2*b*c+S)+b*c*(2*S+(-a+b+c)*(a+b+c))*sin(A/2)) : :
Barycentrics    Sin[A]*Csc[A/4]^2*(1 + Sin[A/2]) : : (Peter Moses, March 18, 2023)

Contributed by César Eliud Lozada, March 6, 2023.

X(53071) lies on the line {258, 3082}

X(53071) = X(289)-isoconjugate-of-X(558)
X(53071) = barycentric product X(i)*X(j) for these {i, j}: {9, 53077}, {236, 3082}
X(53071) = trilinear product X(55)*X(53077)
X(53071) = trilinear quotient X(i)/X(j) for these (i, j): (236, 558), (1274, 21456)

---

X(53072) = CENTER OF THE OUTER-APOLLONIUS CIRCLE OF THE INNER-MALFATTI CIRCLES

Barycentrics    (-a+b+c)*(4*(b-c)*(2*S-(-a+b+c)*(a+b+c))*S^2-(b-c)*(-a+b+c)*(4*(a+b+c)*S^2+(2*a^3-2*a*(b+c)^2-8*b*c*(b+c))*S)*sin(A/2)+(a-b+c)*((2*a^4-6*a^3*c-2*(b^2+c^2)*a^2-2*(b+c)*(-3*c+b)*c*a-4*(b^2-c^2)*b*c)*S-(-a+b+c)*(a+b-c)*(a^2-a*b-(b-c)*c)*(a+b+c)^2)*sin(B/2)-(a+b-c)*((2*a^4-6*a^3*b-2*(b^2+c^2)*a^2+2*(b+c)*(-c+3*b)*b*a+4*(b^2-c^2)*b*c)*S-(a-b+c)*(-a+b+c)*(a^2-c*a+(b-c)*b)*(a+b+c)^2)*sin(C/2)) : :

Contributed by César Eliud Lozada, March 6, 2023.

X(53072) lies on these lines: {1, 53076}, {483, 1127}

X(53072) = {X(483), X(31495)}-harmonic conjugate of X(53074)

---

X(53073) = CENTER OF THE INNER-APOLLONIUS CIRCLE OF THE OUTER-MALFATTI CIRCLES

Barycentrics    (-a+b+c)*(4*(b-c)*(-2*S-(-a+b+c)*(a+b+c))*S^2-(b-c)*(-a+b+c)*(4*(a+b+c)*S^2-(2*a^3-2*a*(b+c)^2-8*b*c*(b+c))*S)*sin(A/2)+(a-b+c)*(-(2*a^4-6*a^3*c-2*(b^2+c^2)*a^2-2*(b+c)*(-3*c+b)*c*a-4*(b^2-c^2)*b*c)*S-(-a+b+c)*(a+b-c)*(a^2-a*b-(b-c)*c)*(a+b+c)^2)*sin(B/2)-(a+b-c)*(-(2*a^4-6*a^3*b-2*(b^2+c^2)*a^2+2*(b+c)*(-c+3*b)*b*a+4*(b^2-c^2)*b*c)*S-(a-b+c)*(-a+b+c)*(a^2-c*a+(b-c)*b)*(a+b+c)^2)*sin(C/2)) : :

Contributed by César Eliud Lozada, March 6, 2023.

X(53073) lies on these lines: {1, 53077}, {3082, 8092}

---

X(53074) = CENTER OF THE INNER-APOLLONIUS CIRCLE OF THE INNER-MALFATTI CIRCLES

Barycentrics    b*c*(b-c)*(a+b+c)*(-a+b+c)*(2*(a+b+c)*(151*a^3+29*(b+c)*a^2-(11*b^2+34*b*c+11*c^2)*a-(b^2-c^2)*(b-c))*S-a^6-147*(b+c)*a^5-2*(31*b^2+188*b*c+31*c^2)*a^4+2*(b+c)*(75*b^2-146*b*c+75*c^2)*a^3+(63*b^4+63*c^4-2*b*c*(6*b^2+59*b*c+6*c^2))*a^2-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a+4*(b^2-c^2)^2*b*c)*sin(A/2)-a*c*(a+b+c)*(2*(a+b+c)*(3*a^5-(24*b-c)*a^4+2*(64*b^2-9*b*c-31*c^2)*a^3-2*(45*b^3-15*c^3-b*c*(55*b-39*c))*a^2-(b-c)*(19*b^3+27*c^3-3*b*c*(25*b+9*c))*a+(b^2-c^2)*(b-c)*(2*b^2-17*b*c+c^2))*S+(a-b+c)*(-a+b+c)*(3*a^6-(46*b-55*c)*a^5-2*(134*b^2-89*b*c-31*c^2)*a^4-2*(152*b^3+25*c^3-b*c*(36*b+121*c))*a^3-(b+c)*(77*b^3+65*c^3-b*c*(87*b+83*c))*a^2+(b^2-c^2)*(14*b^3+5*c^3+b*c*(25*b+12*c))*a+2*(b^2-c^2)*(b-c)*b*(3*b^2+9*b*c+2*c^2)))*sin(B/2)+a*(a+b+c)*b*(2*(a+b+c)*(3*a^5+(b-24*c)*a^4-2*(31*b^2+9*b*c-64*c^2)*a^3+2*(15*b^3-45*c^3-b*c*(39*b-55*c))*a^2+(b-c)*(27*b^3+19*c^3-3*b*c*(9*b+25*c))*a+(b^2-c^2)*(b-c)*(b^2-17*b*c+2*c^2))*S+(-a+b+c)*(a+b-c)*(3*a^6+(55*b-46*c)*a^5+2*(31*b^2+89*b*c-134*c^2)*a^4-2*(25*b^3+152*c^3-b*c*(121*b+36*c))*a^3-(b+c)*(65*b^3+77*c^3-b*c*(83*b+87*c))*a^2-(b^2-c^2)*(5*b^3+14*c^3+b*c*(12*b+25*c))*a+2*(b^2-c^2)*(b-c)*(2*b^2+9*b*c+3*c^2)*c))*sin(C/2)+2*S*(b-c)*((10*a^6-28*(b+c)*a^5-8*(8*b^2+29*b*c+8*c^2)*a^4+2*(b+c)*(10*b^2-77*b*c+10*c^2)*a^3+2*(27*b^4+27*c^4+b*c*(17*b^2-28*b*c+17*c^2))*a^2+2*(b+c)*(4*b^4+4*c^4-b*c*(3*b+5*c)*(5*b+3*c))*a-2*(b^2-c^2)^2*b*c)*S+(-a+b+c)*(10*a^5+30*(b+c)*a^4-(26*b^2-55*b*c+26*c^2)*a^3-(b+c)*(14*b^2-27*b*c+14*c^2)*a^2-3*(b-c)^2*b*c*a+(b^2-c^2)*(b-c)*b*c)*(a+b+c)^2) : :

Contributed by César Eliud Lozada, March 6, 2023.

X(53074) lies on these lines: {1, 53078}, {483, 1127}, {1142, 31957}

X(53074) = {X(483), X(31495)}-harmonic conjugate of X(53072)

---

X(53075) = CENTER OF THE OUTER-APOLLONIUS CIRCLE OF THE OUTER-MALFATTI CIRCLES

Barycentrics    b*c*(b-c)*(a+b+c)*(-a+b+c)*(-2*(a+b+c)*(151*a^3+29*(b+c)*a^2-(11*b^2+34*b*c+11*c^2)*a-(b^2-c^2)*(b-c))*S-a^6-147*(b+c)*a^5-2*(31*b^2+188*b*c+31*c^2)*a^4+2*(b+c)*(75*b^2-146*b*c+75*c^2)*a^3+(63*b^4+63*c^4-2*b*c*(6*b^2+59*b*c+6*c^2))*a^2-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a+4*(b^2-c^2)^2*b*c)*sin(A/2)-a*c*(a+b+c)*(-2*(a+b+c)*(3*a^5-(24*b-c)*a^4+2*(64*b^2-9*b*c-31*c^2)*a^3-2*(45*b^3-15*c^3-b*c*(55*b-39*c))*a^2-(b-c)*(19*b^3+27*c^3-3*b*c*(25*b+9*c))*a+(b^2-c^2)*(b-c)*(2*b^2-17*b*c+c^2))*S+(a-b+c)*(-a+b+c)*(3*a^6-(46*b-55*c)*a^5-2*(134*b^2-89*b*c-31*c^2)*a^4-2*(152*b^3+25*c^3-b*c*(36*b+121*c))*a^3-(b+c)*(77*b^3+65*c^3-b*c*(87*b+83*c))*a^2+(b^2-c^2)*(14*b^3+5*c^3+b*c*(25*b+12*c))*a+2*(b^2-c^2)*(b-c)*b*(3*b^2+9*b*c+2*c^2)))*sin(B/2)+a*(a+b+c)*b*(-2*(a+b+c)*(3*a^5+(b-24*c)*a^4-2*(31*b^2+9*b*c-64*c^2)*a^3+2*(15*b^3-45*c^3-b*c*(39*b-55*c))*a^2+(b-c)*(27*b^3+19*c^3-3*b*c*(9*b+25*c))*a+(b^2-c^2)*(b-c)*(b^2-17*b*c+2*c^2))*S+(-a+b+c)*(a+b-c)*(3*a^6+(55*b-46*c)*a^5+2*(31*b^2+89*b*c-134*c^2)*a^4-2*(25*b^3+152*c^3-b*c*(121*b+36*c))*a^3-(b+c)*(65*b^3+77*c^3-b*c*(83*b+87*c))*a^2-(b^2-c^2)*(5*b^3+14*c^3+b*c*(12*b+25*c))*a+2*(b^2-c^2)*(b-c)*(2*b^2+9*b*c+3*c^2)*c))*sin(C/2)-2*S*(b-c)*(-(10*a^6-28*(b+c)*a^5-8*(8*b^2+29*b*c+8*c^2)*a^4+2*(b+c)*(10*b^2-77*b*c+10*c^2)*a^3+2*(27*b^4+27*c^4+b*c*(17*b^2-28*b*c+17*c^2))*a^2+2*(b+c)*(4*b^4+4*c^4-b*c*(3*b+5*c)*(5*b+3*c))*a-2*(b^2-c^2)^2*b*c)*S+(-a+b+c)*(10*a^5+30*(b+c)*a^4-(26*b^2-55*b*c+26*c^2)*a^3-(b+c)*(14*b^2-27*b*c+14*c^2)*a^2-3*(b-c)^2*b*c*a+(b^2-c^2)*(b-c)*b*c)*(a+b+c)^2) : :

Contributed by César Eliud Lozada, March 6, 2023.

X(53075) lies on these lines: {1, 53079}, {3082, 8092}

---

X(53076) = PERSPECTOR (ABC, TOUCHPOINTS OF INNER-MALFATTI CIRCLES AND THEIR OUTER-APOLLONIUS CIRCLE)

Barycentrics    b*c*(1+sin(A/2))*(1-cos(A/2)) : :

Contributed by César Eliud Lozada, March 6, 2023.
This center is the exsimilcenter of the incircle and the outer-Apollonius circle of the inner-Malfatti circles (Keita Miyamoto, March 3, 2023).

X(53076) lies on these lines: {1, 53072}, {174, 175}, {177, 46876}

X(53076) = X(188)-Dao conjugate of-X(3082)
X(53076) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (173, 7001), (236, 3082), (483, 7028), (558, 1488), (1143, 7048)
X(53076) = barycentric product X(i)*X(j) for these {i, j}: {85, 32576}, {1143, 7057}
X(53076) = barycentric quotient X(i)/X(j) for these (i, j): (173, 7001), (236, 3082), (483, 7028), (558, 1488), (1143, 7048)
X(53076) = trilinear product X(i)*X(j) for these {i, j}: {7, 32576}, {173, 1143}, {236, 558}, {483, 2089}
X(53076) = trilinear quotient X(i)/X(j) for these (i, j): (558, 289), (1143, 258)

---

X(53077) = PERSPECTOR (ABC, TOUCHPOINTS OF OUTER-MALFATTI CIRCLES AND THEIR INNER-APOLLONIUS CIRCLE)

Barycentrics    b*c*(1+sin(A/2))*(1+cos(A/2)) : :

Contributed by César Eliud Lozada, March 6, 2023.
This center is the insimilcenter of the incircle and the inner-Apollonius circle of the outer-Malfatti circles (Keita Miyamoto, March 3, 2023).

X(53077) lies on these lines: {1, 53073}, {174, 176}

X(53077) = X(188)-Dao conjugate of-X(483)
X(53077) = X(289)-isoconjugate-of-X(7014)
X(53077) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (173, 7010), (236, 483), (557, 1488), (1274, 7048)
X(53077) = barycentric product X(i)*X(j) for these {i, j}: {85, 53071}, {1274, 7057}
X(53077) = barycentric quotient X(i)/X(j) for these (i, j): (173, 7010), (236, 483), (557, 1488), (1274, 7048), (2089, 558)
X(53077) = trilinear product X(i)*X(j) for these {i, j}: {7, 53071}, {173, 1274}, {236, 557}, {2089, 3082}
X(53077) = trilinear quotient X(i)/X(j) for these (i, j): (236, 7014), (557, 289), (1274, 258)

---

X(53078) = PERSPECTOR (ABC, TOUCHPOINTS OF INNER-MALFATTI CIRCLES AND THEIR INNER-APOLLONIUS CIRCLE)

Barycentrics    (-a+b+c)*(b*c*(b-c)*((24*a^5+272*(b+c)*a^4-40*(4*b^2-7*b*c+4*c^2)*a^3-8*(b+c)*(18*b^2-37*b*c+18*c^2)*a^2+8*(b^2+3*b*c+c^2)*(b-c)^2*a-8*(b^2-c^2)*(b-c)*b*c)*S-4*S^2*(155*a^3+33*(b+c)*a^2-(11*b^2+42*b*c+11*c^2)*a-(b^2-c^2)*(b-c)))*sin(A/2)+(a-b+c)*c*a*((8*a^5-4*(27*b-28*c)*a^4-4*(110*b-89*c)*b*a^3-4*(48*b^3+28*c^3-b*c*(53*b+41*c))*a^2+4*(b-c)*(4*b^3+2*c^3+b*c*(15*b+11*c))*a+4*(3*b^2+9*b*c+2*c^2)*(b-c)^2*b)*S+(a+b+c)*(a+b-c)*(3*a^5-(20*b+7*c)*a^4+2*(64*b^2-3*b*c-31*c^2)*a^3-2*(47*b^3-19*c^3-5*b*c*(11*b-9*c))*a^2-(b-c)*(19*b^3+27*c^3-b*c*(79*b+31*c))*a+(b^2-c^2)*(b-c)*(2*b^2-17*b*c+c^2)))*sin(B/2)-(a+b-c)*a*b*((8*a^5+4*(28*b-27*c)*a^4+4*(89*b-110*c)*c*a^3-4*(28*b^3+48*c^3-b*c*(41*b+53*c))*a^2-4*(b-c)*(2*b^3+4*c^3+b*c*(11*b+15*c))*a+4*c*(2*b^2+9*b*c+3*c^2)*(b-c)^2)*S+(a-b+c)*(a+b+c)*(3*a^5-(7*b+20*c)*a^4-2*(31*b^2+3*b*c-64*c^2)*a^3+2*(19*b^3-47*c^3-5*b*c*(9*b-11*c))*a^2+(b-c)*(27*b^3+19*c^3-b*c*(31*b+79*c))*a+(b^2-c^2)*(b-c)*(b^2-17*b*c+2*c^2)))*sin(C/2)-2*S*(b-c)*((24*a^5+56*(b+c)*a^4-2*(28*b^2-61*b*c+28*c^2)*a^3-2*(b+c)*(12*b^2-23*b*c+12*c^2)*a^2-10*(b-c)^2*b*c*a+2*(b^2-c^2)*(b-c)*b*c)*S+(a+b-c)*(a-b+c)*(4*a^5-24*(b+c)*a^4-(b+8*c)*(8*b+c)*a^3+3*(b+c)*(8*b^2-5*b*c+8*c^2)*a^2+(b^2-5*b*c-4*c^2)*(4*b^2+5*b*c-c^2)*a-(b^2-c^2)*(b-c)*b*c))) : :

Contributed by César Eliud Lozada, March 6, 2023.
This center is the insimilcenter of the incircle and the inner-Apollonius circle of the inner-Malfatti circles (Keita Miyamoto, March 3, 2023).

X(53078) lies on these lines: {1, 53074}, {174, 175}, {179, 31957}

---

X(53079) = PERSPECTOR (ABC, TOUCHPOINTS OF OUTER-MALFATTI CIRCLES AND THEIR OUTER-APOLLONIUS CIRCLE)

Barycentrics    (-a+b+c)*(b*c*(b-c)*(-(24*a^5+272*(b+c)*a^4-40*(4*b^2-7*b*c+4*c^2)*a^3-8*(b+c)*(18*b^2-37*b*c+18*c^2)*a^2+8*(b^2+3*b*c+c^2)*(b-c)^2*a-8*(b^2-c^2)*(b-c)*b*c)*S-4*S^2*(155*a^3+33*(b+c)*a^2-(11*b^2+42*b*c+11*c^2)*a-(b^2-c^2)*(b-c)))*sin(A/2)+(a-b+c)*c*a*(-(8*a^5-4*(27*b-28*c)*a^4-4*(110*b-89*c)*b*a^3-4*(48*b^3+28*c^3-b*c*(53*b+41*c))*a^2+4*(b-c)*(4*b^3+2*c^3+b*c*(15*b+11*c))*a+4*(3*b^2+9*b*c+2*c^2)*(b-c)^2*b)*S+(a+b+c)*(a+b-c)*(3*a^5-(20*b+7*c)*a^4+2*(64*b^2-3*b*c-31*c^2)*a^3-2*(47*b^3-19*c^3-5*b*c*(11*b-9*c))*a^2-(b-c)*(19*b^3+27*c^3-b*c*(79*b+31*c))*a+(b^2-c^2)*(b-c)*(2*b^2-17*b*c+c^2)))*sin(B/2)-(a+b-c)*a*b*(-(8*a^5+4*(28*b-27*c)*a^4+4*(89*b-110*c)*c*a^3-4*(28*b^3+48*c^3-b*c*(41*b+53*c))*a^2-4*(b-c)*(2*b^3+4*c^3+b*c*(11*b+15*c))*a+4*c*(2*b^2+9*b*c+3*c^2)*(b-c)^2)*S+(a-b+c)*(a+b+c)*(3*a^5-(7*b+20*c)*a^4-2*(31*b^2+3*b*c-64*c^2)*a^3+2*(19*b^3-47*c^3-5*b*c*(9*b-11*c))*a^2+(b-c)*(27*b^3+19*c^3-b*c*(31*b+79*c))*a+(b^2-c^2)*(b-c)*(b^2-17*b*c+2*c^2)))*sin(C/2)+2*S*(b-c)*(-(24*a^5+56*(b+c)*a^4-2*(28*b^2-61*b*c+28*c^2)*a^3-2*(b+c)*(12*b^2-23*b*c+12*c^2)*a^2-10*(b-c)^2*b*c*a+2*(b^2-c^2)*(b-c)*b*c)*S+(a+b-c)*(a-b+c)*(4*a^5-24*(b+c)*a^4-(b+8*c)*(8*b+c)*a^3+3*(b+c)*(8*b^2-5*b*c+8*c^2)*a^2+(b^2-5*b*c-4*c^2)*(4*b^2+5*b*c-c^2)*a-(b^2-c^2)*(b-c)*b*c))) : :

Contributed by César Eliud Lozada, March 6, 2023.
This center is the exsimilcenter of the incircle and the outer-Apollonius circle of the outer-Malfatti circles (Keita Miyamoto, March 3, 2023).

X(53079) lies on these lines: {1, 53075}, {174, 176}

---

X(53080) = ISOTOMIC CONJUGATE OF X(351)

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

X(53080) lies on these lines: {670, 850}, {671, 886}, {689, 691}, {804, 9150}, {880, 34760}, {1502, 52551}, {1978, 52623}, {3266, 18023}, {4609, 44168}, {6331, 14618}, {9148, 18829}, {14603, 46783}, {17948, 30736}, {18024, 30786}, {22456, 45773}, {35146, 38988}, {36085, 37133}, {43187, 43665}

X(53080) = isotomic conjugate of X(351)
X(53080) = isotomic conjugate of the anticomplement of X(45689)
X(53080) = isotomic conjugate of the isogonal conjugate of X(892)
X(53080) = cevapoint of X(i) and X(j) for these (i,j): {76, 35522}, {523, 47286}, {671, 9178}, {850, 3266}, {5466, 31125}, {18023, 52632}
X(53080) = trilinear pole of line {76, 338}
X(53080) = X(i)-isoconjugate of X(j) for these (i,j): {31, 351}, {32, 2642}, {163, 21906}, {187, 798}, {512, 922}, {524, 1924}, {560, 690}, {661, 14567}, {669, 896}, {810, 44102}, {1084, 23889}, {1918, 14419}, {1919, 21839}, {1923, 22105}, {1927, 11183}, {1980, 4062}, {2205, 4750}, {4117, 5468}, {9247, 14273}, {9417, 52038}, {23995, 33919}
X(53080) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 351}, {115, 21906}, {6374, 690}, {6376, 2642}, {9296, 21839}, {9428, 524}, {15477, 9426}, {15899, 669}, {18314, 33919}, {31998, 187}, {34021, 14419}, {36830, 14567}, {36901, 1648}, {39054, 922}, {39058, 52038}, {39062, 44102}
X(53080) = barycentric product X(i)*X(j) for these {i,j}: {76, 892}, {99, 18023}, {111, 4609}, {561, 36085}, {670, 671}, {689, 31125}, {691, 1502}, {799, 46277}, {850, 52940}, {886, 52756}, {897, 4602}, {1928, 36142}, {4563, 46111}, {4590, 52632}, {5380, 6385}, {5466, 34537}, {6331, 30786}, {9178, 44168}, {17983, 52608}, {23105, 42370}, {23962, 45773}, {32729, 40362}, {36827, 40016}, {42371, 46154}
X(53080) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 351}, {75, 2642}, {76, 690}, {99, 187}, {110, 14567}, {111, 669}, {264, 14273}, {274, 14419}, {290, 52038}, {305, 14417}, {308, 22105}, {310, 4750}, {338, 33919}, {523, 21906}, {648, 44102}, {662, 922}, {668, 21839}, {670, 524}, {671, 512}, {689, 52898}, {691, 32}, {799, 896}, {850, 1648}, {880, 5026}, {886, 14608}, {892, 6}, {895, 3049}, {897, 798}, {923, 1924}, {1502, 35522}, {1978, 4062}, {2396, 9155}, {3266, 1649}, {3978, 11183}, {4558, 23200}, {4563, 3292}, {4590, 5467}, {4602, 14210}, {4609, 3266}, {4623, 16702}, {4625, 51653}, {5380, 213}, {5466, 3124}, {5468, 39689}, {5968, 2491}, {6331, 468}, {6386, 42713}, {7799, 44814}, {8024, 14424}, {9150, 41309}, {9154, 2422}, {9170, 48450}, {9178, 1084}, {9214, 14398}, {11059, 9125}, {14609, 887}, {14977, 20975}, {16092, 6041}, {17948, 9171}, {17983, 2489}, {18023, 523}, {18818, 46001}, {18829, 18872}, {20573, 51479}, {23105, 42344}, {24037, 23889}, {24039, 42081}, {28660, 14432}, {30786, 647}, {31125, 3005}, {32729, 1501}, {32740, 9426}, {34537, 5468}, {34539, 32729}, {34574, 32740}, {34760, 2502}, {35522, 23992}, {36085, 31}, {36142, 560}, {36792, 33915}, {36821, 9429}, {36827, 3051}, {40050, 45807}, {40074, 18311}, {40826, 23287}, {41079, 2682}, {41259, 45680}, {41272, 9494}, {42008, 17414}, {43187, 5967}, {43926, 1977}, {44173, 52628}, {45773, 23357}, {45809, 33921}, {46111, 2501}, {46144, 51927}, {46154, 688}, {46277, 661}, {47286, 21905}, {50941, 5191}, {51478, 19627}, {52035, 1692}, {52141, 8644}, {52450, 42663}, {52551, 2492}, {52608, 6390}, {52612, 6629}, {52628, 14443}, {52629, 14444}, {52632, 115}, {52747, 14407}, {52756, 888}, {52758, 14428}, {52940, 110}

---

X(53081) = X(1)X(1333)∩X(9)X(45095)

Barycentrics    Cot[A + ArcTan[Tan[A/2]*Tan[B/2]*Tan[C/2]]]*Sin[A] : :
Barycentrics    a*(a^2 + a*b + b^2 + a*c + b*c)*(a^2 + a*b + a*c + b*c + c^2)*(a^2*b - b^3 + a^2*c - a*b*c - c^3) : :

X(53081) lies on the cubic K457 and these lines: {1, 1333}, {9, 45095}, {19, 1759}, {835, 29068}, {1764, 42467}, {1766, 43739}, {2285, 12514}, {3185, 21078}, {40456, 40457}

X(53081) = X(i)-isoconjugate of X(j) for these (i,j): {386, 13478}, {834, 44765}, {2217, 28606}, {14349, 36050}, {32653, 45746}
X(53081) = X(i)-Dao conjugate of X(j) for these (i,j): {124, 14349}, {34588, 23879}
X(53081) = barycentric product X(i)*X(j) for these {i,j}: {835, 21189}, {2214, 4417}, {3869, 43531}, {6589, 37218}
X(53081) = barycentric quotient X(i)/X(j) for these {i,j}: {573, 28606}, {2214, 13478}, {3185, 386}, {3869, 5224}, {4417, 33935}, {6589, 14349}, {17080, 33949}, {21189, 45746}, {43531, 2995}

---

X(53082) = X(1)X(1437)∩X(9)X(19608)

Barycentrics    Cot[A - ArcTan[Cot[A/2]*Cot[B/2]*Cot[C/2]]]*Sin[A] : :
Barycentrics    a*(a*b + b^2 + a*c + b*c + 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(53082) lies on the cubic K457 and these lines: {1, 1437}, {9, 19608}, {57, 40160}, {63, 321}, {15232, 41229}, {17185, 19607}, {42467, 43739}

X(53082) = X(i)-isoconjugate of X(j) for these (i,j): {573, 43531}, {835, 6589}, {2214, 3869}
X(53082) = X(39016)-Dao conjugate of X(21189)
X(53082) = barycentric product X(i)*X(j) for these {i,j}: {386, 2995}, {2217, 5224}, {13478, 28606}, {14349, 44765}, {36050, 45746}
X(53082) = barycentric quotient X(i)/X(j) for these {i,j}: {386, 3869}, {834, 21189}, {2217, 43531}, {28606, 4417}, {36050, 835}, {44765, 37218}, {52615, 16754}
X(53082) = {X(2217),X(42550)}-harmonic conjugate of X(1)

---

X(53083) = X(1)X(859)∩X(2)X(573)

Barycentrics    Cot[A + ArcTan[Cot[A/2]*Cot[B/2]*Cot[C/2]]]*Sin[A] : :
Barycentrics    a*(a + b)*(a + c)*(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(53083) lies on the conic {{A,B,C,X(1),X(7)}}, the cubic K457, and on these lines: {1, 859}, {2, 573}, {6, 34280}, {27, 16082}, {40, 19259}, {57, 16700}, {58, 961}, {105, 38832}, {274, 20367}, {314, 32017}, {330, 18206}, {333, 21061}, {959, 995}, {1019, 2401}, {1219, 10461}, {1224, 47515}, {1255, 25058}, {1280, 5208}, {1390, 35623}, {1402, 18191}, {1412, 34051}, {1999, 39698}, {3218, 39747}, {3227, 41629}, {3928, 36871}, {4216, 10470}, {4245, 10441}, {6553, 20037}, {8056, 52897}, {10434, 17194}, {11521, 41723}, {13478, 43739}, {16704, 35058}, {18164, 39980}, {19607, 29068}, {24310, 25590}, {34267, 37642}

X(53083) = isogonal conjugate of X(21061)
X(53083) = cevapoint of X(i) and X(j) for these (i,j): {649, 18191}, {17420, 38345}
X(53083) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21061}, {2, 52139}, {6, 17751}, {9, 37558}, {10, 572}, {37, 2975}, {42, 14829}, {55, 52358}, {71, 11109}, {81, 14973}, {210, 17074}, {284, 52357}, {321, 20986}, {644, 51662}, {1018, 21173}, {1220, 52087}, {2287, 20617}, {4557, 17496}, {22118, 41013}, {26115, 34278}
X(53083) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 21061}, {9, 17751}, {223, 52358}, {478, 37558}, {2051, 22020}, {32664, 52139}, {40586, 14973}, {40589, 2975}, {40590, 52357}, {40592, 14829}
X(53083) = barycentric product X(i)*X(j) for these {i,j}: {1, 20028}, {57, 46880}, {75, 52150}, {81, 2051}, {86, 34434}, {757, 51870}, {4357, 40453}
X(53083) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17751}, {6, 21061}, {28, 11109}, {31, 52139}, {42, 14973}, {56, 37558}, {57, 52358}, {58, 2975}, {65, 52357}, {81, 14829}, {1019, 17496}, {1042, 20617}, {1333, 572}, {1412, 17074}, {2051, 321}, {2206, 20986}, {2300, 52087}, {3733, 21173}, {16726, 24237}, {17197, 40624}, {18191, 34589}, {18197, 27346}, {20028, 75}, {34434, 10}, {40453, 1220}, {43924, 51662}, {46880, 312}, {51870, 1089}, {52150, 1}
X(53083) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 34262, 2051}, {20028, 46880, 2051}

---

X(53084) = 4th MIYAMOTO-MOSES POINT

Barycentrics    a*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 - 5*a^4*c + 8*a^3*b*c - 6*a^2*b^2*c + 8*a*b^3*c - 5*b^4*c + 10*a^3*c^2 + 18*a^2*b*c^2 + 18*a*b^2*c^2 + 10*b^3*c^2 - 10*a^2*c^3 - 32*a*b*c^3 - 10*b^2*c^3 + 5*a*c^4 + 5*b*c^4 - c^5)*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5 + a^4*c + 8*a^3*b*c + 18*a^2*b^2*c - 32*a*b^3*c + 5*b^4*c - 2*a^3*c^2 - 6*a^2*b*c^2 + 18*a*b^2*c^2 - 10*b^3*c^2 - 2*a^2*c^3 + 8*a*b*c^3 + 10*b^2*c^3 + a*c^4 - 5*b*c^4 + c^5) : :

Let (I), (Ia), (Ib) and (Ic) be the incircle, A-, B-, C-excircles of a given triangle ABC, respectively. Let (Oa) be the circle passing through B and C, and internally tangent to (I), and define (Ob) and (Oc) cyclically. Let La be the radical axis of (Oa) and (Ia), and define Lb and Lc cyclically. Let A′=Lb∩Lc, B′=Lc∩La and C′=La∩Lb. Then, the lines AA′, BB′, CC′ concur in X(53084) (Keita Miyamoto, Peter Moses, March 16, 2023).

X(53084) lies on these lines: { }

---

X(53085) = 5th MIYAMOTO-MOSES POINT

Barycentrics    a*(a^9*b + 3*a^8*b^2 - 8*a^6*b^4 - 6*a^5*b^5 + 6*a^4*b^6 + 8*a^3*b^7 - 3*a*b^9 - b^10 + a^9*c + 4*a^8*b*c - 2*a^7*b^2*c - 6*a^6*b^3*c + 16*a^5*b^4*c + 26*a^4*b^5*c + 2*a^3*b^6*c - 18*a^2*b^7*c - 17*a*b^8*c - 6*b^9*c + 3*a^8*c^2 - 2*a^7*b*c^2 + 28*a^6*b^2*c^2 + 70*a^5*b^3*c^2 + 18*a^4*b^4*c^2 - 38*a^3*b^5*c^2 - 36*a^2*b^6*c^2 - 30*a*b^7*c^2 - 13*b^8*c^2 - 6*a^6*b*c^3 + 70*a^5*b^2*c^3 + 124*a^4*b^3*c^3 - 68*a^3*b^4*c^3 + 18*a^2*b^5*c^3 - 2*a*b^6*c^3 - 8*b^7*c^3 - 8*a^6*c^4 + 16*a^5*b*c^4 + 18*a^4*b^2*c^4 - 68*a^3*b^3*c^4 + 72*a^2*b^4*c^4 + 52*a*b^5*c^4 + 14*b^6*c^4 - 6*a^5*c^5 + 26*a^4*b*c^5 - 38*a^3*b^2*c^5 + 18*a^2*b^3*c^5 + 52*a*b^4*c^5 + 28*b^5*c^5 + 6*a^4*c^6 + 2*a^3*b*c^6 - 36*a^2*b^2*c^6 - 2*a*b^3*c^6 + 14*b^4*c^6 + 8*a^3*c^7 - 18*a^2*b*c^7 - 30*a*b^2*c^7 - 8*b^3*c^7 - 17*a*b*c^8 - 13*b^2*c^8 - 3*a*c^9 - 6*b*c^9 - c^10) : :

Let (Ia) be the A-excircle of ABC, and define (Ib) and (Ic) cyclically. Let (Oa) be the circle passing through B and C, and internally tangent to (Ia), and define (Ob) and (Oc) cyclically. Denote by Ab the intersection point of (Oa) and AB, other than B, and define Bc and Ca cyclically. Denote by Ac the intersection point of (Oa) and CA, other than C, and define Ba and Cb cyclically. Let A′=BcBa∩CaCb, B′=CaCb∩AbAc and C′=AbAc∩BcBa. Let A″ be the intersection point of (Ob) and (Oc), other than A, and define B″ and C″ cyclically. Then, the lines A′A″, B′B″, C′C″ concur in X(53085) (Keita Miyamoto, Peter Moses, March 16, 2023).

X(53085) lies on these lines: {2297,4512}, {5044,15489}

---

X(53086) = 6th MIYAMOTO-MOSES POINT

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

Let (O) be the circumcircle of ABC. Let (Ab) be the circle tangent to ray AB, ray BC and externally tangent to (O), and define (Bc) and (Ca) cyclically. Let (Ac) be the circle tangent to ray AC, ray CB and externally tangent to (O), and define (Ba) and (Cb) cyclically. Let A′=BaBc∩CbCa, B′=CbCa∩AcAb and C′=AcAb∩BaBc. Then, the lines AA′, BB′, CC′ concur in X(53086). Let IaIbIc be the excentral triangle of ABC. Then, the lines IaA′, IbB′, IcC′ concur in X(53087) (Keita Miyamoto, Peter Moses, March 17, 2023).

X(53086) lies on these lines: {40,144}, {57,14522}, {165,198}, {221,1419}, {346,10860}, {2360,37402}, {3672,9533}

X(53086) = isogonal conjugate of X(10860)
X(53086) = X(i)-isoconjugate of X(j) for these (i,j): {1, 10860}, {9, 34488}, {55, 34060}
X(53086) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 10860}, {223, 34060}, {478, 34488}
X(53086) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 10860}, {56, 34488}, {57, 34060}

---

X(53087) = 7th MIYAMOTO-MOSES POINT

Barycentrics    a*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 8*a^6*b*c + 8*a^5*b^2*c + 16*a^4*b^3*c - 16*a^3*b^4*c - 8*a^2*b^5*c + 8*a*b^6*c - 4*a^6*c^2 + 8*a^5*b*c^2 - 44*a^4*b^2*c^2 + 16*a^3*b^3*c^2 + 20*a^2*b^4*c^2 + 8*a*b^5*c^2 - 4*b^6*c^2 + 16*a^4*b*c^3 + 16*a^3*b^2*c^3 - 16*a^2*b^3*c^3 - 16*a*b^4*c^3 + 6*a^4*c^4 - 16*a^3*b*c^4 + 20*a^2*b^2*c^4 - 16*a*b^3*c^4 + 6*b^4*c^4 - 8*a^2*b*c^5 + 8*a*b^2*c^5 - 4*a^2*c^6 + 8*a*b*c^6 - 4*b^2*c^6 + c^8) : :

See X(53086).

X(53087) lies on the Kiepert circumhyperbola of the excentral triangle and these lines: {1,84}, {43,1721}, {165,198}, {1103,6223}, {1282,2823}, {2324,10860}, {2939,2941}, {2947,2951}, {5732,25941}, {7133,8833}, {8831,42013}, {10857,52155}, {24341,30326}

X(53087) = excentral isogonal conjugate of X(2270).
X(53087) = X(i)-Ceva conjugate of X(j) for these (i,j): {2324, 1}, {10860, 165}

---

X(53088) = X(37)X(1697) ∩ X(1191)X(1400)

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

Continuing from X(53086), Let tAb be the touchpoint of (O) and (Ab), and define tAc, tBc, tBa, tCa, tCb similarly. Let A₅=tBatBc∩tCbtCa, and define B₅ and C₅ cyclically. Then, the lines AA₅, BB₅, CC₅ concur in X(53088). Let A₆=tCbtAb∩tBctAc, and define B₆ and C₆ cyclically. Then, the lines AA₆, BB₆ CC₆ concur in X(53089) (Peter Moses, March 17, 2023) .

X(53088) lies on the circumconic {{A,B,C,X(2),X(6)}} and these lines: {2,37499}, {6,37260}, {37,1697}, {42,17810}, {1191,1400}, {5022,39956}, {6554,14624}, {37500,39798}, {41489,44103}

X(53088) = isogonal conjugate of X(37655)
X(53088) = isogonal conjugate of the isotomic conjugate of X(45100)
X(53088) = X(1)-isoconjugate of X(37655)
X(53088) = X(3)-Dao conjugate of X(37655)
X(53088) = barycentric product X(6)*X(45100)
X(53088) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 37655}, {45100, 76}

---

X(53089) = X(1)X(1350) ∩ X(33)X(1829)

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

See X(53088).

X(53089) lies on these lines: {1,1350}, {3,51476}, {6,50621}, {33,1829}, {55,1191}, {56,7050}, {200,960}, {220,2269}, {221,7169}, {390,1043}, {959,5710}, {963,4300}, {1042,52013}, {2098,7073}, {2192,3556}, {2328,4267}, {2342,26357}, {3058,48862}, {5646,28352}, {9785,14942}, {10459,17810}, {10482,37502}, {11477,50617}, {12575,32941}

X(53089) = isogonal conjugate of X(3600)
X(53089) = isogonal conjugate of the anticomplement of X(2551)
X(53089) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3600}, {7, 5269}, {57, 5749}, {269, 7172}, {664, 50517}
X(53089) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 3600}, {5452, 5749}, {6600, 7172}, {39025, 50517}
X(53089) = trilinear pole of line {657, 52326}
X(53089) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 3600}, {41, 5269}, {55, 5749}, {220, 7172}, {3063, 50517}

---

X(53090) = X(1)X(1696) ∩ X(42)X(3304)

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

Continuing from X(53086), let La be the external common tangent to (Ab) and (Ac), other than BC, and define Lb and Lc cyclically. Let A″=Lb∩Lc, B″=Lc∩La and C″=La∩Lb. Then, the lines AA″, BB″, CC″ concur in X(51341) (Keita Miyamoto, March 16, 2023). The triangle A″B″C″ is also perspective to the 3rd mixtilinear triangle at X(53090) (Peter Moses, March 17, 2023).

X(53090) lies on these lines: {1,1696}, {42,3304}, {55,22344}, {56,3158}, {100,6553}, {678,5204}, {8299,12513}

X(53090) = X(1)-Ceva conjugate of X(1616)
X(53090) = X(8051)-isoconjugate of X(51341)
X(53090) = X(23511)-Dao conjugate of X(75)
X(53090) = barycentric product X(6762)*X(23511)

---

X(53091) = X(2)X(1353)∩X(3)X(6)

Barycentrics    a^2*(5*a^4 - 8*a^2*b^2 + 3*b^4 - 8*a^2*c^2 - 10*b^2*c^2 + 3*c^4) : :
X(53091) = 3 X[2] + 2 X[1353], 6 X[2] - X[11898], 3 X[2] - 8 X[51732], 4 X[1353] + X[11898], X[1353] + 4 X[51732], X[11898] - 16 X[51732], X[3] + 4 X[6], 3 X[3] - 8 X[182], X[3] - 16 X[575], 7 X[3] + 8 X[576], 9 X[3] - 4 X[1350], 3 X[3] + 2 X[1351], 13 X[3] - 8 X[3098], X[3] - 6 X[5050], and many others

X(53091) lies on these lines: {2, 1353}, {3, 6}, {4, 18845}, {5, 5921}, {25, 11003}, {30, 47461}, {51, 20850}, {54, 11443}, {69, 3526}, {110, 5020}, {140, 193}, {141, 46219}, {143, 12220}, {159, 13621}, {184, 3066}, {195, 20806}, {381, 6776}, {382, 14853}, {394, 22112}, {399, 52699}, {518, 37624}, {524, 15694}, {542, 19709}, {546, 39874}, {547, 50974}, {549, 5032}, {597, 1352}, {611, 7373}, {613, 6767}, {631, 34380}, {632, 3620}, {895, 32609}, {1147, 6391}, {1199, 7395}, {1386, 10247}, {1482, 16475}, {1503, 3843}, {1598, 19118}, {1656, 3564}, {1657, 21850}, {1992, 5054}, {1993, 16419}, {1994, 7484}, {2104, 28447}, {2105, 28448}, {3167, 5544}, {3329, 9755}, {3517, 12167}, {3525, 20080}, {3531, 43697}, {3534, 51212}, {3541, 46444}, {3543, 51173}, {3567, 16195}, {3589, 5070}, {3751, 10246}, {3763, 5965}, {3796, 15004}, {3818, 33749}, {3830, 5480}, {3851, 6329}, {4663, 38029}, {5012, 9777}, {5071, 50954}, {5073, 46264}, {5079, 18358}, {5095, 15061}, {5182, 13188}, {5304, 37451}, {5462, 6467}, {5476, 14269}, {5477, 38224}, {5622, 10620}, {5640, 12283}, {5645, 9544}, {5646, 37672}, {5899, 32217}, {5943, 8780}, {5946, 6403}, {6090, 11422}, {6144, 40107}, {6771, 49948}, {6774, 49947}, {6800, 15019}, {7403, 19119}, {7506, 19459}, {7528, 18935}, {7529, 19125}, {7539, 45968}, {7574, 41257}, {7579, 20300}, {7592, 11479}, {7666, 43815}, {7697, 14535}, {7754, 10359}, {7878, 39646}, {8547, 51519}, {8548, 15047}, {8549, 41593}, {8584, 15701}, {8705, 37923}, {8717, 44413}, {9039, 45729}, {9654, 39901}, {9669, 39900}, {9703, 12039}, {9863, 51860}, {9924, 23042}, {9955, 39878}, {9976, 32254}, {10124, 50986}, {10168, 15534}, {10250, 32063}, {10264, 25321}, {10516, 25555}, {10519, 15720}, {10752, 15041}, {10765, 52698}, {11004, 40916}, {11160, 11539}, {11180, 38079}, {11216, 15577}, {11258, 36696}, {11284, 15018}, {11286, 39141}, {11423, 19122}, {11427, 30771}, {11579, 12308}, {11645, 35403}, {11799, 47459}, {11911, 51741}, {12160, 43651}, {12161, 26206}, {12272, 15024}, {12294, 35501}, {12410, 16472}, {12834, 35264}, {13093, 34779}, {13363, 15531}, {13394, 21970}, {13665, 49229}, {13785, 49228}, {14093, 51172}, {14389, 26869}, {14530, 19132}, {14643, 32300}, {14645, 38750}, {14763, 15432}, {14926, 15087}, {14984, 15040}, {15002, 43725}, {15038, 18534}, {15069, 38317}, {15118, 38724}, {15576, 39530}, {15681, 20423}, {15684, 43273}, {15685, 48898}, {15688, 48874}, {15689, 48873}, {15692, 50987}, {15700, 50967}, {15702, 50978}, {15703, 24206}, {15707, 20583}, {15714, 50966}, {15718, 50983}, {15721, 51179}, {15723, 21356}, {15988, 16408}, {16010, 25556}, {16187, 17825}, {16434, 37685}, {16669, 46475}, {16989, 37071}, {17800, 31670}, {17810, 32237}, {18325, 47460}, {18350, 19137}, {18510, 39876}, {18512, 39875}, {18525, 39870}, {18535, 44102}, {18584, 20398}, {18906, 32519}, {19153, 39879}, {19347, 21637}, {20304, 32234}, {20415, 42095}, {20416, 42098}, {21313, 44080}, {21358, 46267}, {21554, 37677}, {22115, 32127}, {23327, 41729}, {23515, 32272}, {26446, 51196}, {26958, 32068}, {28453, 51729}, {30535, 34396}, {31479, 39897}, {31958, 32449}, {32244, 34128}, {32987, 33684}, {33586, 34565}, {34155, 51941}, {34200, 51028}, {34507, 47355}, {34718, 51005}, {34748, 47359}, {34774, 34780}, {34778, 43807}, {34787, 39125}, {35259, 44109}, {35268, 44107}, {35400, 51165}, {37609, 38296}, {37922, 51733}, {37924, 52238}, {38040, 39898}, {38072, 48889}, {38167, 39885}, {39893, 42262}, {39894, 42265}, {43588, 46442}, {43650, 44111}, {44214, 47277}, {44903, 51177}, {47465, 47569}, {48896, 51024}, {48905, 49134}, {48910, 49139}, {49136, 51538}, {49524, 51515}, {50988, 51214}, {51150, 51514}

X(53091) = midpoint of X(i) and X(j) for these {i,j}: {631, 51170}, {6776, 51537}, {11482, 12017}, {14093, 51172}
X(53091) = reflection of X(i) in X(j) for these {i,j}: {3, 12017}, {6, 22234}, {1656, 3618}, {3620, 632}, {11482, 6}, {14530, 19132}, {15692, 50987}, {35403, 50963}, {50954, 5071}, {50966, 15714}
X(53091) = isogonal conjugate of X(10155)
X(53091) = Brocard-circle-inverse of X(5093)
X(53091) = crossdifference of every pair of points on line {523, 47278}
X(53091) = barycentric quotient X(6)/X(10155)
X(53091) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1353, 11898}, {3, 6, 5093}, {3, 5093, 44456}, {3, 22246, 32447}, {5, 14912, 39899}, {6, 182, 1351}, {6, 575, 5050}, {6, 1350, 5097}, {6, 1692, 30435}, {6, 5013, 1570}, {6, 5034, 9605}, {6, 5050, 3}, {6, 5085, 576}, {6, 5102, 22330}, {6, 10485, 11173}, {6, 10541, 5102}, {6, 11477, 15520}, {6, 19131, 37493}, {6, 37506, 18449}, {6, 50664, 33878}, {61, 62, 22332}, {69, 38110, 3526}, {182, 576, 14810}, {182, 1351, 3}, {182, 5097, 1350}, {182, 14810, 5085}, {182, 15516, 6}, {182, 39561, 15516}, {371, 372, 15815}, {381, 6776, 48662}, {549, 5032, 50962}, {569, 11432, 3}, {575, 15516, 182}, {575, 39561, 6}, {576, 5085, 33878}, {576, 50664, 5085}, {597, 12007, 1352}, {1350, 5097, 1351}, {1351, 5050, 182}, {1353, 51732, 2}, {1384, 11171, 3}, {1668, 1669, 574}, {3098, 22330, 5102}, {3167, 5544, 5651}, {3311, 3312, 39}, {3311, 45410, 3}, {3312, 45411, 3}, {3398, 9605, 3}, {5012, 9777, 9909}, {5020, 5422, 5644}, {5050, 11432, 19129}, {5050, 11482, 12017}, {5085, 33878, 3}, {5092, 15520, 11477}, {5102, 10541, 3098}, {5422, 11402, 5020}, {5611, 5615, 10983}, {5651, 10601, 5544}, {5943, 17809, 8780}, {6221, 6398, 8589}, {6329, 8550, 14561}, {6419, 43118, 12313}, {6420, 43119, 12314}, {6427, 6428, 41940}, {6427, 26341, 45489}, {6428, 26348, 45488}, {6776, 18583, 381}, {8550, 14561, 18440}, {9605, 30435, 13356}, {9976, 52697, 32254}, {10601, 13366, 3167}, {11179, 14848, 3830}, {11402, 52719, 5422}, {11426, 36752, 3}, {11427, 45298, 30771}, {11485, 11486, 574}, {11579, 45016, 12308}, {12167, 19128, 3517}, {12313, 43118, 3}, {12314, 43119, 3}, {13353, 37493, 3}, {14561, 18440, 3851}, {14810, 50664, 182}, {14853, 48906, 382}, {14912, 51171, 5}, {15723, 51175, 21356}, {18583, 50979, 6776}, {19118, 39588, 1598}, {20190, 37517, 31884}, {21850, 25406, 1657}, {26341, 45489, 3}, {26348, 45488, 3}, {34779, 52028, 13093}, {35423, 35458, 3}, {35770, 45551, 9733}, {35771, 45550, 9732}, {44505, 44506, 44507}, {46267, 51140, 21358}, {47352, 50955, 15703}

---

X(53092) = X(2)X(52719)∩X(3)X(6)

Barycentrics    a^2*(7*a^4 - 12*a^2*b^2 + 5*b^4 - 12*a^2*c^2 - 14*b^2*c^2 + 5*c^4) : :
X(53092) = X[3] + 6 X[6], 5 X[3] - 12 X[182], X[3] - 8 X[575], 3 X[3] + 4 X[576], 13 X[3] - 6 X[1350], 4 X[3] + 3 X[1351], 19 X[3] - 12 X[3098], 2 X[3] - 9 X[5050], 11 X[3] - 18 X[5085], 17 X[3] - 24 X[5092], 5 X[3] + 9 X[5093], 11 X[3] + 24 X[5097], 17 X[3] + 18 X[5102], and many others

X(53092) lies on these lines: {2, 52719}, {3, 6}, {4, 14848}, {5, 11180}, {23, 9777}, {24, 11405}, {25, 15019}, {69, 632}, {110, 30734}, {140, 1992}, {193, 3525}, {381, 8550}, {382, 11179}, {394, 44111}, {524, 3526}, {542, 3851}, {546, 6776}, {597, 1656}, {599, 46219}, {631, 5032}, {895, 6642}, {1199, 12164}, {1352, 5079}, {1353, 3618}, {1498, 10250}, {1598, 44102}, {1657, 20423}, {1993, 44299}, {1994, 40916}, {1995, 8780}, {2070, 15826}, {2854, 15039}, {3066, 44109}, {3090, 3564}, {3091, 7920}, {3146, 33748}, {3167, 5422}, {3292, 10601}, {3515, 8537}, {3516, 15021}, {3517, 8541}, {3518, 12167}, {3523, 50988}, {3527, 7530}, {3529, 21850}, {3533, 11160}, {3544, 5921}, {3567, 15074}, {3589, 11898}, {3627, 14853}, {3751, 15178}, {3796, 34565}, {3843, 5476}, {3858, 51023}, {4663, 10246}, {5020, 13366}, {5054, 8584}, {5055, 15069}, {5056, 38079}, {5070, 34507}, {5072, 12007}, {5073, 43273}, {5076, 5480}, {5095, 20397}, {5159, 11427}, {5182, 51524}, {5198, 39588}, {5462, 40673}, {5477, 20398}, {5544, 6090}, {5609, 52699}, {5622, 48679}, {5644, 9306}, {5946, 15073}, {5965, 47355}, {6391, 9925}, {6593, 32254}, {7387, 36153}, {7484, 23061}, {7506, 32621}, {7545, 14530}, {7550, 12160}, {7581, 12602}, {7582, 12601}, {7592, 14094}, {7878, 38664}, {8542, 53019}, {8549, 32063}, {8718, 45034}, {9707, 11458}, {9715, 11416}, {9833, 23326}, {9909, 15004}, {10124, 50992}, {10222, 16475}, {10282, 17813}, {10303, 48876}, {10519, 12108}, {10594, 19118}, {10602, 12106}, {11188, 15026}, {11255, 14070}, {11799, 47460}, {12103, 51212}, {12308, 25556}, {12315, 34117}, {13861, 19125}, {14002, 26864}, {14869, 34380}, {14924, 16187}, {14984, 15020}, {15022, 18358}, {15024, 15531}, {15025, 32234}, {15027, 15118}, {15533, 46267}, {15534, 15694}, {15579, 35450}, {15582, 34777}, {15696, 51737}, {15704, 25406}, {15712, 50967}, {15720, 20583}, {15723, 22165}, {15988, 16862}, {16010, 34155}, {16051, 45298}, {16239, 21356}, {16671, 46475}, {17821, 34788}, {17845, 23048}, {18553, 19709}, {18919, 31804}, {19130, 48662}, {19132, 50414}, {19357, 21639}, {20379, 41720}, {20415, 42153}, {20416, 42156}, {20582, 51175}, {21735, 51028}, {23041, 39125}, {25565, 51027}, {31670, 49137}, {31859, 35950}, {31958, 32520}, {32300, 38795}, {34566, 43650}, {34780, 41719}, {35407, 48904}, {36696, 51535}, {37958, 51733}, {37967, 52238}, {38136, 39874}, {38224, 41672}, {39879, 41593}, {41991, 51537}, {43810, 51739}, {44214, 47464}, {46261, 52163}, {46264, 49136}, {47598, 50990}, {49139, 51024}, {51138, 51172}

X(53092) = reflection of X(3) in X(10541)
X(53092) = Brocard-circle-inverse of X(11482)
X(53092) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 11482}, {3, 575, 5050}, {3, 5093, 11477}, {3, 11477, 33878}, {3, 11482, 1351}, {3, 44456, 52987}, {6, 182, 5093}, {6, 575, 3}, {6, 1350, 15520}, {6, 5050, 1351}, {6, 5085, 5097}, {6, 11477, 22330}, {6, 44469, 14627}, {6, 44481, 6501}, {6, 44482, 6500}, {6, 44503, 36749}, {6, 44504, 32447}, {6, 44656, 6417}, {6, 44657, 6418}, {61, 62, 5013}, {182, 5093, 33878}, {182, 11477, 3}, {182, 22330, 11477}, {575, 15516, 22234}, {575, 22234, 6}, {575, 22330, 182}, {575, 41940, 2456}, {1351, 5050, 12017}, {3311, 3312, 5024}, {5050, 11482, 3}, {5050, 33878, 182}, {5085, 5097, 44456}, {5085, 52987, 3}, {5093, 22330, 11482}, {5093, 33878, 1351}, {5422, 11422, 11284}, {6090, 15018, 5544}, {6417, 6418, 22246}, {6417, 45410, 12313}, {6418, 45411, 12314}, {6593, 39562, 32254}, {11284, 11422, 3167}, {11477, 22330, 5093}, {12007, 14561, 39899}, {14912, 18583, 18440}, {15069, 25555, 5055}, {15069, 51185, 25555}, {15516, 39561, 6}, {15520, 50664, 1350}, {20583, 38064, 50962}, {21401, 36843, 3}, {21402, 36836, 3}, {22234, 39561, 575}, {22246, 45410, 12314}, {22246, 45411, 12313}, {34507, 47352, 5070}

---

X(53093) = X(2)X(8550)∩X(3)X(6)

Barycentrics    a^2*(5*a^4 - 6*a^2*b^2 + b^4 - 6*a^2*c^2 - 10*b^2*c^2 + c^4) : :
X(53093) = 3 X[2] + 2 X[8550], 6 X[2] - X[15069], 4 X[8550] + X[15069], 2 X[3] + 3 X[6], X[3] - 6 X[182], X[3] + 4 X[575], 3 X[3] + 2 X[576], 8 X[3] - 3 X[1350], 7 X[3] + 3 X[1351], 11 X[3] - 6 X[3098], X[3] + 9 X[5050], 4 X[3] - 9 X[5085], 7 X[3] - 12 X[5092], 11 X[3] + 9 X[5093], and many others

X(53093) lies on these lines: {2, 8550}, {3, 6}, {4, 597}, {5, 11179}, {20, 51737}, {22, 15019}, {23, 3796}, {30, 47458}, {54, 32154}, {64, 7527}, {67, 20397}, {69, 10303}, {110, 14924}, {140, 599}, {141, 3525}, {154, 1995}, {155, 38402}, {156, 43811}, {184, 11284}, {186, 15826}, {264, 15576}, {373, 26864}, {381, 25555}, {382, 5476}, {394, 11422}, {524, 631}, {542, 1656}, {546, 14561}, {549, 15534}, {550, 20423}, {611, 5563}, {613, 3746}, {632, 3564}, {895, 14528}, {1176, 51730}, {1181, 5622}, {1352, 3628}, {1353, 14869}, {1386, 7982}, {1428, 3304}, {1498, 19153}, {1503, 3091}, {1593, 44102}, {1611, 14153}, {1657, 14848}, {1853, 37649}, {1899, 37454}, {1974, 5198}, {1992, 3523}, {1993, 7496}, {2330, 3303}, {2393, 17821}, {2781, 8567}, {2836, 15016}, {2854, 15034}, {2930, 15462}, {3066, 6800}, {3090, 3589}, {3146, 5480}, {3242, 15178}, {3292, 11402}, {3416, 38118}, {3515, 8541}, {3516, 11470}, {3518, 7716}, {3522, 50968}, {3524, 8584}, {3526, 10168}, {3529, 6329}, {3533, 20582}, {3544, 39874}, {3567, 9019}, {3576, 4663}, {3620, 44787}, {3627, 18583}, {3629, 10519}, {3751, 30389}, {3818, 5072}, {3843, 11645}, {3850, 38079}, {3972, 8719}, {4027, 33997}, {5026, 23235}, {5032, 15717}, {5054, 15533}, {5055, 18553}, {5056, 47354}, {5067, 11180}, {5068, 51023}, {5070, 11178}, {5076, 29012}, {5079, 18440}, {5182, 7770}, {5204, 19369}, {5217, 8540}, {5462, 9971}, {5477, 31455}, {5609, 11579}, {5621, 7526}, {5640, 31860}, {5646, 9716}, {5733, 19512}, {5882, 47359}, {5921, 46936}, {6034, 38734}, {6090, 22112}, {6144, 12108}, {6247, 41719}, {6403, 35479}, {6688, 8780}, {6698, 32234}, {7410, 49731}, {7484, 13366}, {7485, 23061}, {7492, 33586}, {7506, 19596}, {7516, 36153}, {7550, 7592}, {7709, 35950}, {7784, 44380}, {7790, 53017}, {7829, 52669}, {7834, 38745}, {7942, 50641}, {7991, 16475}, {8537, 32534}, {8542, 32621}, {8546, 19357}, {8548, 47391}, {8556, 12151}, {8705, 37953}, {8716, 35925}, {9704, 32254}, {9755, 15271}, {9756, 11174}, {9766, 37450}, {9777, 22352}, {9924, 15582}, {9968, 41593}, {9976, 32609}, {10192, 18928}, {10222, 38315}, {10250, 10282}, {10264, 25336}, {10299, 20583}, {10359, 39646}, {10488, 49102}, {10594, 19128}, {10602, 13367}, {10620, 25556}, {10982, 12082}, {10984, 19136}, {11003, 16042}, {11004, 21766}, {11188, 15028}, {11255, 18324}, {11305, 51015}, {11306, 51012}, {11315, 48734}, {11316, 48735}, {11362, 47356}, {11403, 19118}, {11405, 15750}, {11416, 38444}, {11465, 40670}, {11539, 51186}, {11541, 51163}, {11623, 18800}, {11646, 20398}, {11649, 37958}, {11695, 29959}, {11812, 51188}, {12006, 15074}, {12102, 38136}, {12103, 21850}, {12161, 32599}, {12177, 51523}, {12188, 32135}, {12220, 32191}, {12283, 41579}, {12584, 39562}, {12811, 39884}, {13154, 32136}, {13434, 51739}, {13464, 38023}, {13472, 34817}, {13910, 39875}, {13972, 39876}, {14355, 33928}, {14614, 37455}, {14688, 36696}, {14789, 25328}, {14927, 50688}, {14981, 33237}, {15004, 52719}, {15024, 16776}, {15025, 32274}, {15029, 41737}, {15045, 15073}, {15054, 51941}, {15057, 41720}, {15118, 32233}, {15258, 52288}, {15303, 20417}, {15682, 41153}, {15694, 50993}, {15696, 19924}, {15701, 51187}, {15702, 22165}, {15704, 31670}, {15709, 50991}, {15712, 50987}, {15713, 50989}, {15719, 41149}, {15720, 50977}, {15721, 50992}, {15988, 17572}, {15993, 44535}, {16003, 34319}, {16051, 23292}, {16176, 49116}, {16189, 16491}, {16419, 34986}, {16885, 46475}, {17538, 29181}, {17813, 34787}, {18911, 34118}, {19121, 33524}, {19149, 52028}, {20399, 32954}, {20418, 51008}, {21167, 32455}, {21669, 51729}, {21735, 50965}, {22486, 33235}, {23042, 39879}, {24206, 39899}, {26958, 45298}, {28343, 38676}, {28662, 38675}, {31958, 32448}, {32138, 43814}, {32217, 37946}, {32247, 41595}, {32300, 38791}, {32305, 45016}, {33532, 44413}, {33979, 39576}, {34155, 48679}, {34486, 45728}, {35602, 41614}, {35944, 41945}, {35945, 41946}, {37527, 37679}, {38047, 39870}, {38117, 50995}, {38665, 51157}, {38737, 41672}, {40330, 51126}, {43174, 51005}, {43879, 49229}, {43880, 49228}, {44214, 47276}, {44245, 48873}, {46219, 50955}, {47465, 47468}, {48898, 49137}, {48901, 49136}, {49133, 51173}, {49140, 51538}, {50693, 51212}, {51130, 51177}, {51139, 51179}, {51141, 51174}, {51797, 52163}

X(53093) = midpoint of X(3) and X(11482)
X(53093) = reflection of X(i) in X(j) for these {i,j}: {11482, 22234}, {12017, 182}, {22234, 575}, {40330, 51126}, {50993, 15694}
X(53093) = Brocard-circle-inverse of X(11477)
X(53093) = crossdifference of every pair of points on line {523, 47446}
X(53093) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 8550, 15069}, {3, 6, 11477}, {3, 182, 10541}, {3, 575, 6}, {3, 1351, 52987}, {3, 5050, 575}, {3, 10541, 5085}, {3, 11477, 1350}, {3, 52987, 31884}, {6, 182, 5085}, {6, 1350, 5102}, {6, 5013, 10542}, {6, 5023, 13330}, {6, 5085, 1350}, {6, 10541, 3}, {6, 31884, 1351}, {6, 39560, 3053}, {61, 62, 9605}, {69, 33748, 12007}, {182, 575, 3}, {182, 576, 20190}, {182, 1692, 35423}, {182, 3398, 39560}, {182, 5050, 6}, {182, 39561, 5092}, {182, 44470, 13336}, {182, 44483, 26348}, {182, 44484, 26341}, {182, 44489, 37515}, {182, 44490, 13339}, {182, 44491, 37471}, {182, 44503, 37476}, {182, 44504, 52771}, {182, 44509, 43119}, {182, 44510, 43118}, {182, 44656, 45553}, {182, 44657, 45552}, {182, 50664, 5050}, {371, 372, 5024}, {569, 37514, 11425}, {575, 5092, 22330}, {575, 10541, 11477}, {575, 20190, 576}, {575, 22330, 39561}, {576, 20190, 3}, {597, 43273, 38072}, {1342, 1343, 8722}, {1351, 5092, 31884}, {1351, 39561, 6}, {1352, 38110, 47355}, {3066, 6800, 41424}, {3098, 15516, 5093}, {3311, 3312, 22246}, {3311, 44657, 6}, {3311, 45552, 12306}, {3312, 44656, 6}, {3312, 45553, 12305}, {3398, 44507, 6}, {3526, 34507, 21358}, {3589, 6776, 10516}, {3796, 5422, 17810}, {5012, 10601, 154}, {5034, 40825, 6}, {5050, 13353, 44503}, {5085, 11477, 3}, {5092, 22330, 52987}, {5092, 39561, 1351}, {5092, 52987, 3}, {5093, 15516, 6}, {5097, 17508, 33878}, {5480, 25406, 48905}, {5622, 6593, 16010}, {6329, 44882, 14853}, {6417, 44481, 6}, {6418, 44482, 6}, {6800, 15018, 3066}, {7484, 13366, 37672}, {7527, 43697, 34117}, {9730, 44479, 37473}, {10168, 33749, 34507}, {10168, 34507, 3526}, {10249, 34117, 64}, {11171, 43118, 1152}, {11171, 43119, 1151}, {11179, 47352, 47353}, {11402, 43650, 17811}, {11422, 40916, 394}, {11426, 44489, 6}, {11482, 12017, 3}, {11482, 22234, 6}, {13353, 36752, 37476}, {14561, 48906, 36990}, {14810, 15520, 44456}, {14853, 44882, 48910}, {18583, 46264, 53023}, {22112, 44109, 6090}, {22236, 22238, 5013}, {22246, 45552, 12305}, {22246, 45553, 12306}, {22330, 31884, 11477}, {22330, 52987, 1351}, {25406, 51171, 5480}, {36752, 37476, 9786}, {36752, 44503, 6}, {36753, 44469, 6}, {38064, 50979, 599}, {39561, 52987, 22330}, {43118, 45411, 1151}, {43119, 45410, 1152}, {44509, 45410, 6}, {44510, 45411, 6}, {44513, 44514, 44508}, {45410, 45411, 11171}, {45550, 45551, 3}, {48906, 51732, 14561}

---

X(53094) = X(2)X(14927)∩X(3)X(6)

Barycentrics    a^2*(5*a^4 - 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 - 10*b^2*c^2 - 3*c^4) : :
X(53094) = 9 X[2] + X[14927], 6 X[2] - X[36990], 3 X[2] + 2 X[44882], 2 X[14927] + 3 X[36990], X[14927] - 6 X[44882], X[14927] + 3 X[51537], X[36990] + 4 X[44882], 2 X[44882] + X[51537], 4 X[3] + X[6], 3 X[3] + 2 X[182], 11 X[3] + 4 X[575], 13 X[3] + 2 X[576], 6 X[3] - X[1350], and many others

X(53094) lies on these lines: {2, 14927}, {3, 6}, {4, 33750}, {5, 48905}, {20, 3589}, {22, 3066}, {24, 2916}, {25, 22112}, {30, 47453}, {40, 38315}, {64, 206}, {67, 38727}, {69, 15717}, {74, 52697}, {98, 8556}, {110, 3796}, {140, 10516}, {141, 3523}, {154, 5646}, {159, 52028}, {165, 1386}, {353, 46949}, {373, 31860}, {376, 5480}, {381, 48898}, {382, 38317}, {394, 11003}, {426, 26909}, {518, 7987}, {524, 15692}, {542, 15040}, {547, 50988}, {548, 31670}, {549, 1352}, {550, 14561}, {597, 10304}, {599, 3524}, {611, 7280}, {613, 5010}, {631, 1503}, {698, 32522}, {1176, 3532}, {1204, 19125}, {1353, 17504}, {1428, 5217}, {1498, 7509}, {1511, 16010}, {1587, 36717}, {1588, 36702}, {1593, 16936}, {1611, 10329}, {1656, 29012}, {1657, 19130}, {1843, 15750}, {1853, 7499}, {1974, 3516}, {1992, 15705}, {2071, 32217}, {2097, 21164}, {2330, 5204}, {2854, 15051}, {2930, 15035}, {3070, 36701}, {3071, 36703}, {3091, 51126}, {3207, 22390}, {3242, 3576}, {3416, 10164}, {3515, 7716}, {3522, 3618}, {3526, 3818}, {3528, 14853}, {3530, 15069}, {3534, 10168}, {3543, 48310}, {3564, 15712}, {3579, 38029}, {3830, 48896}, {3843, 29323}, {3851, 48884}, {3917, 17809}, {4048, 15271}, {4220, 37679}, {4297, 38047}, {5012, 37672}, {5020, 32237}, {5026, 34473}, {5032, 51138}, {5054, 24206}, {5055, 48889}, {5056, 51127}, {5071, 50975}, {5073, 48891}, {5181, 48375}, {5476, 15688}, {5544, 9909}, {5596, 6696}, {5622, 15036}, {5650, 26864}, {5731, 49524}, {5882, 49690}, {6034, 38738}, {6144, 14912}, {6194, 32449}, {6403, 35472}, {6459, 13972}, {6460, 13910}, {6593, 15055}, {6636, 10601}, {6688, 20850}, {6699, 32233}, {6795, 47284}, {6800, 7496}, {7390, 17337}, {7393, 46261}, {7395, 15811}, {7494, 26958}, {7503, 52093}, {7514, 11472}, {7516, 17814}, {7525, 15805}, {7558, 34775}, {7804, 14532}, {8273, 12329}, {8547, 37283}, {8549, 35228}, {8550, 10299}, {8567, 19132}, {8703, 18583}, {8705, 37952}, {8717, 9818}, {8719, 35925}, {9039, 22769}, {9743, 9751}, {9976, 15042}, {10192, 41735}, {10249, 15577}, {10303, 34573}, {10606, 15578}, {10902, 12595}, {10984, 46207}, {11178, 15701}, {11179, 12100}, {11180, 15719}, {11202, 39879}, {11227, 24476}, {11284, 35268}, {11362, 49679}, {11403, 44091}, {11410, 12294}, {11416, 38441}, {11495, 38048}, {11645, 15694}, {11646, 38737}, {11898, 15700}, {12007, 15534}, {12041, 15462}, {12103, 38136}, {12220, 38438}, {12383, 25330}, {12512, 38049}, {12588, 52793}, {12594, 37561}, {13394, 46336}, {14093, 19924}, {14269, 48942}, {14528, 34817}, {14535, 22682}, {14688, 38698}, {14869, 18358}, {14891, 50979}, {14982, 38793}, {15041, 19140}, {15045, 32191}, {15080, 35259}, {15162, 38708}, {15163, 38709}, {15681, 48904}, {15683, 50959}, {15689, 48920}, {15690, 38079}, {15696, 29317}, {15702, 47354}, {15708, 20582}, {15714, 50987}, {15715, 50967}, {15716, 51187}, {15718, 50955}, {15720, 18440}, {15721, 51023}, {15988, 37307}, {16187, 16419}, {16192, 16475}, {16434, 37682}, {16675, 46475}, {17800, 48895}, {17810, 43650}, {17845, 23300}, {18405, 20300}, {18481, 38144}, {19128, 35477}, {19357, 43617}, {19649, 37674}, {19708, 51185}, {20423, 34200}, {20791, 41716}, {21000, 37619}, {21153, 50995}, {21356, 50984}, {21487, 37527}, {21734, 51171}, {21737, 23261}, {21844, 39588}, {21850, 33923}, {25331, 32247}, {25555, 48880}, {25563, 34776}, {25565, 38335}, {28343, 38717}, {28662, 37751}, {30389, 49465}, {30739, 32125}, {31255, 36201}, {31730, 38035}, {31861, 33534}, {32271, 38788}, {32305, 32609}, {32455, 33748}, {33273, 39141}, {34632, 51006}, {34773, 38116}, {35237, 49671}, {35404, 50964}, {37948, 51733}, {38023, 50808}, {38087, 50811}, {38089, 50815}, {38185, 43161}, {38402, 46945}, {38693, 51157}, {39899, 40107}, {43174, 49681}, {44837, 51739}, {47445, 47569}, {48943, 49139}, {50693, 51538}

X(53094) = midpoint of X(i) and X(j) for these {i,j}: {3, 12017}, {3098, 22234}, {3522, 3618}, {5071, 50975}, {8567, 19132}, {15714, 50987}
X(53094) = reflection of X(i) in X(j) for these {i,j}: {3091, 51126}, {3763, 631}, {15694, 51137}, {36990, 51537}, {50968, 14093}
X(53094) = isogonal conjugate of X(43951)
X(53094) = complement of X(51537)
X(53094) = Brocard-circle-inverse of X(31884)
X(53094) = crossdifference of every pair of points on line {523, 47451}
X(53094) = barycentric quotient X(6)/X(43951)
X(53094) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 44882, 36990}, {3, 6, 31884}, {3, 182, 1350}, {3, 572, 50677}, {3, 1351, 14810}, {3, 5050, 3098}, {3, 5085, 6}, {3, 5092, 5085}, {3, 13329, 37499}, {3, 13336, 17834}, {3, 13339, 37489}, {3, 16836, 37487}, {3, 20190, 11477}, {3, 26341, 11825}, {3, 26348, 11824}, {3, 35423, 2076}, {3, 37471, 37486}, {3, 37479, 5013}, {3, 37513, 37497}, {3, 37515, 9786}, {3, 43118, 12305}, {3, 43119, 12306}, {3, 45552, 1152}, {3, 45553, 1151}, {6, 5085, 10541}, {20, 3589, 53023}, {69, 15717, 21167}, {140, 46264, 10516}, {154, 5646, 5651}, {182, 1350, 6}, {182, 3098, 5097}, {182, 5097, 5050}, {182, 8722, 12212}, {182, 14810, 1351}, {376, 5480, 48872}, {376, 47352, 51024}, {376, 50983, 47352}, {548, 38110, 31670}, {549, 43273, 21358}, {550, 14561, 48910}, {575, 5102, 6}, {575, 33878, 5102}, {1151, 1152, 39}, {1160, 45550, 6432}, {1161, 45551, 6431}, {1350, 5085, 182}, {1351, 14810, 1350}, {1428, 5217, 10387}, {1691, 5013, 6}, {1692, 10542, 6}, {3053, 5013, 13356}, {3053, 50659, 6}, {3098, 5050, 11477}, {3098, 20190, 5050}, {3515, 19124, 7716}, {3523, 25406, 141}, {3524, 51737, 599}, {3528, 14853, 48881}, {3534, 10168, 38072}, {3592, 3594, 41940}, {3796, 7485, 17811}, {5050, 11477, 6}, {5092, 17508, 3}, {5097, 20190, 182}, {5480, 48872, 51024}, {5622, 15036, 33851}, {5651, 7484, 5646}, {6409, 6410, 15815}, {6411, 6412, 8589}, {7484, 22352, 154}, {8550, 10519, 40341}, {8589, 9675, 9600}, {8703, 18583, 48873}, {10168, 33751, 48901}, {10541, 31884, 6}, {11003, 15246, 21766}, {11003, 21766, 394}, {11284, 35268, 41424}, {11480, 11481, 574}, {11824, 26348, 3592}, {11825, 26341, 3594}, {12041, 15462, 51941}, {12103, 38136, 43621}, {12305, 43118, 6426}, {12306, 43119, 6425}, {13339, 37489, 37514}, {15080, 40916, 35259}, {15578, 19149, 10606}, {22236, 22238, 7772}, {23041, 44883, 1498}, {28662, 38716, 37751}, {33751, 48901, 3534}, {34200, 51732, 48874}, {36836, 36843, 22332}, {38064, 48873, 18583}, {38317, 48892, 382}, {45578, 45579, 3095}, {47352, 48872, 5480}, {48310, 50971, 3543}, {48874, 51732, 20423}, {50664, 52987, 5093}

---

X(53095) = X(2)X(11147)∩X(3)X(6)

Barycentrics    a^2*(5*a^2 - 7*b^2 - 7*c^2) : :
X(53095) = X[11742] + 2 X[31415], X[11742] + 3 X[31489], X[11742] - 3 X[44541], 2 X[31415] - 3 X[31489], 2 X[31415] + 3 X[44541]

X(53095) lies on these lines: {2, 11147}, {3, 6}, {4, 3055}, {5, 43619}, {30, 11742}, {35, 9336}, {36, 9331}, {99, 15271}, {111, 40916}, {115, 5054}, {140, 43620}, {141, 33215}, {154, 37457}, {183, 8716}, {230, 3524}, {232, 11410}, {237, 31860}, {316, 11184}, {325, 33008}, {352, 21766}, {353, 5866}, {376, 3815}, {378, 33885}, {382, 31455}, {524, 47061}, {548, 2548}, {549, 2549}, {550, 31401}, {599, 6390}, {620, 11287}, {625, 5077}, {631, 3054}, {980, 21524}, {988, 3723}, {1184, 38862}, {1285, 15710}, {1383, 6636}, {1385, 31422}, {1506, 1657}, {1571, 13624}, {1597, 33880}, {1656, 7756}, {1975, 33004}, {2079, 9609}, {2207, 35477}, {2482, 11646}, {3066, 35298}, {3148, 41424}, {3331, 51880}, {3515, 10985}, {3522, 7745}, {3523, 5254}, {3526, 7748}, {3528, 31400}, {3530, 3767}, {3534, 5475}, {3576, 31443}, {3589, 32985}, {3618, 35287}, {3619, 7789}, {3620, 6337}, {3630, 3785}, {3631, 3926}, {3763, 8359}, {3793, 8182}, {3830, 7603}, {4045, 11288}, {4188, 37675}, {4383, 35276}, {5010, 16784}, {5126, 31433}, {5217, 10987}, {5275, 13587}, {5283, 19537}, {5286, 10299}, {5304, 15705}, {5305, 44682}, {5306, 15698}, {5309, 15700}, {5337, 21507}, {5355, 15716}, {5646, 20998}, {5968, 41404}, {6184, 21781}, {6644, 44521}, {6748, 37460}, {6781, 15484}, {7280, 16785}, {7393, 44528}, {7484, 8770}, {7496, 20481}, {7514, 34866}, {7610, 47286}, {7735, 15692}, {7736, 10304}, {7737, 8703}, {7738, 15717}, {7739, 17504}, {7746, 15720}, {7747, 15696}, {7753, 14093}, {7754, 43459}, {7769, 33234}, {7771, 8667}, {7773, 33260}, {7778, 8356}, {7782, 11285}, {7783, 20105}, {7784, 32965}, {7786, 33235}, {7798, 46893}, {7813, 11165}, {7847, 33233}, {7851, 33259}, {7868, 9878}, {7904, 32821}, {7908, 40344}, {7937, 41134}, {8369, 47355}, {8556, 15301}, {8567, 32445}, {8598, 42849}, {8719, 9743}, {8744, 35473}, {8860, 32480}, {9155, 46276}, {9300, 19708}, {9412, 9475}, {9597, 52793}, {9603, 10984}, {9606, 21734}, {9619, 31663}, {9620, 17502}, {9655, 31501}, {9681, 44648}, {9700, 43809}, {9766, 14907}, {10097, 44814}, {10313, 38446}, {10630, 35936}, {10765, 14060}, {10986, 32534}, {11151, 11317}, {11163, 14712}, {11168, 52713}, {11174, 13586}, {11286, 15482}, {11291, 32789}, {11292, 32790}, {11614, 15694}, {11648, 15701}, {12100, 15048}, {13192, 35296}, {14001, 51126}, {14033, 15491}, {14096, 33979}, {14537, 15695}, {14650, 45012}, {14912, 40925}, {15051, 47406}, {15326, 31497}, {15598, 32836}, {15703, 39601}, {15715, 46453}, {15810, 50993}, {16043, 34573}, {16431, 37674}, {16436, 37679}, {16589, 17573}, {16644, 35303}, {16645, 35304}, {16777, 37599}, {16884, 37589}, {17005, 33264}, {17131, 51122}, {17259, 21937}, {17538, 31404}, {17800, 39590}, {17810, 41275}, {17825, 35302}, {17928, 44525}, {18907, 34200}, {21157, 36772}, {21508, 37680}, {21537, 37633}, {21539, 37682}, {21736, 42283}, {23302, 37173}, {23303, 37172}, {24855, 46336}, {25440, 31490}, {27088, 47352}, {28160, 31441}, {30739, 49123}, {31274, 33240}, {31406, 33923}, {31450, 46853}, {31461, 46846}, {32494, 32786}, {32497, 32785}, {32620, 45723}, {32819, 33001}, {32829, 33226}, {32839, 33238}, {32986, 44377}, {33017, 37647}, {33272, 34803}, {33813, 44531}, {33842, 35501}, {33999, 51927}, {34417, 52277}, {34505, 37688}, {34511, 40341}, {35921, 38463}, {37067, 41254}, {37340, 43028}, {37341, 43029}, {37461, 38072}, {37809, 51185}, {39565, 46219}, {40921, 42121}, {40922, 42124}, {41034, 42094}, {41035, 42093}, {43461, 53017}, {43713, 43718}

X(53095) = midpoint of X(31489) and X(44541)
X(53095) = Brocard-circle-inverse of X(5210)
X(53095) = Schoutte-circle-inverse of X(5050)
X(53095) = isotomic conjugate of the polar conjugate of X(11405)
X(53095) = barycentric product X(69)*X(11405)
X(53095) = barycentric quotient X(11405)/X(4)
X(53095) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 5210}, {3, 39, 5023}, {3, 187, 5585}, {3, 574, 6}, {3, 1384, 8588}, {3, 5013, 3053}, {3, 5024, 187}, {3, 5050, 47113}, {3, 9605, 5206}, {3, 15815, 5013}, {3, 18860, 31884}, {3, 22246, 15603}, {3, 30435, 15513}, {3, 31652, 22331}, {3, 37512, 15815}, {3, 52771, 5085}, {4, 3055, 18584}, {6, 574, 5013}, {6, 5023, 1384}, {6, 5210, 3053}, {6, 5585, 187}, {6, 6200, 9602}, {6, 10541, 10485}, {6, 15815, 574}, {6, 22331, 5008}, {6, 31884, 5104}, {15, 16, 5050}, {39, 1384, 6}, {39, 8588, 1384}, {187, 574, 5024}, {187, 5024, 6}, {187, 5585, 5210}, {187, 8588, 15603}, {187, 15602, 574}, {187, 15603, 5023}, {187, 37512, 15602}, {549, 2549, 37637}, {574, 8588, 39}, {574, 8589, 3}, {574, 15515, 8589}, {574, 15655, 22332}, {631, 43448, 3054}, {1384, 5024, 22246}, {1384, 8588, 5023}, {1384, 15603, 187}, {3054, 43448, 13881}, {3523, 5254, 44535}, {5008, 9605, 6}, {5013, 5210, 6}, {5023, 8588, 5210}, {5024, 22246, 39}, {5050, 5107, 6}, {5104, 50659, 6}, {5206, 9605, 22331}, {5206, 31652, 9605}, {6200, 6396, 12017}, {6396, 9600, 6}, {7747, 31457, 31467}, {7771, 31859, 8667}, {8589, 15602, 187}, {8589, 15815, 5210}, {8589, 37512, 574}, {11480, 11481, 5085}, {12100, 15048, 21843}, {15482, 32456, 11286}, {15484, 15688, 6781}, {15515, 37512, 3}, {15603, 22246, 1384}, {15696, 31467, 7747}

---

X(53096) = X(2)X(7765)∩X(3)X(6)

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

X(53096) lies on these lines: {2, 7765}, {3, 6}, {4, 9698}, {5, 11648}, {20, 7753}, {30, 9606}, {76, 15482}, {83, 45017}, {99, 7808}, {115, 3090}, {140, 5309}, {183, 32450}, {194, 7815}, {230, 14869}, {232, 35502}, {325, 7935}, {381, 31470}, {538, 11285}, {543, 16924}, {546, 3815}, {550, 9300}, {598, 19696}, {620, 7803}, {631, 7739}, {632, 7746}, {671, 33002}, {754, 32965}, {940, 21532}, {980, 21540}, {988, 16676}, {1015, 3303}, {1078, 7798}, {1180, 7496}, {1194, 40916}, {1500, 3304}, {1506, 2549}, {1569, 38664}, {1571, 7991}, {1572, 31421}, {1574, 31449}, {1575, 31456}, {1656, 18362}, {1657, 14537}, {1968, 35475}, {1975, 6683}, {2023, 51524}, {2273, 22357}, {2275, 3746}, {2276, 5563}, {2482, 14001}, {2548, 3146}, {2896, 7916}, {3096, 7908}, {3148, 44110}, {3199, 11403}, {3329, 7782}, {3516, 14581}, {3523, 5319}, {3525, 3767}, {3529, 7736}, {3530, 5306}, {3543, 31407}, {3627, 5475}, {3628, 5254}, {3679, 31431}, {3734, 7783}, {3785, 7890}, {3788, 7913}, {3851, 39563}, {3926, 6292}, {3934, 31859}, {4045, 7763}, {4383, 21510}, {5054, 39593}, {5072, 7603}, {5076, 39590}, {5079, 31489}, {5198, 33843}, {5283, 17531}, {5286, 7749}, {5305, 12108}, {5434, 31462}, {5461, 32998}, {6179, 33273}, {6337, 7820}, {6390, 7822}, {6655, 7775}, {6656, 7888}, {6781, 50693}, {7492, 38862}, {7617, 16922}, {7618, 32973}, {7622, 7827}, {7737, 17538}, {7745, 15704}, {7751, 7757}, {7752, 7872}, {7758, 7810}, {7759, 8356}, {7760, 33004}, {7761, 7903}, {7764, 7791}, {7766, 43459}, {7769, 7844}, {7770, 44562}, {7771, 7839}, {7774, 7830}, {7777, 7825}, {7780, 41748}, {7789, 51127}, {7790, 7862}, {7794, 16043}, {7796, 7865}, {7799, 7869}, {7800, 7813}, {7801, 8362}, {7811, 13571}, {7812, 33260}, {7814, 7924}, {7816, 11174}, {7817, 33233}, {7819, 48310}, {7821, 11287}, {7829, 16925}, {7831, 7896}, {7833, 7858}, {7836, 7914}, {7838, 14907}, {7840, 7936}, {7843, 11163}, {7849, 32821}, {7854, 8359}, {7856, 33259}, {7859, 7891}, {7870, 7948}, {7871, 7928}, {7873, 9766}, {7878, 13586}, {7884, 33245}, {7887, 10150}, {7904, 7905}, {7910, 7941}, {7918, 7925}, {7923, 7940}, {7937, 7947}, {7982, 9574}, {9155, 41273}, {9167, 33189}, {9575, 31422}, {9593, 30389}, {9597, 31476}, {9651, 31460}, {9744, 36997}, {10311, 44879}, {10356, 51872}, {10359, 21166}, {11185, 33261}, {11284, 34481}, {12040, 33186}, {12150, 33014}, {12251, 52770}, {12811, 18424}, {14023, 33215}, {14568, 33015}, {14865, 39575}, {14971, 32976}, {15022, 43448}, {15054, 46301}, {15302, 16042}, {15484, 44519}, {16589, 16862}, {16835, 41367}, {16921, 18546}, {17574, 33854}, {19687, 34504}, {19692, 52695}, {20897, 44106}, {21516, 37687}, {21844, 53026}, {22112, 46906}, {30734, 40350}, {31404, 43457}, {31469, 49732}, {32480, 33018}, {33012, 34506}, {33242, 47352}, {33247, 44678}, {37172, 49812}, {37173, 49813}, {39785, 51186}, {43620, 46936}

X(53096) = Brocard-circle-inverse of X(35007)
X(53096) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7781, 17130}, {3, 6, 35007}, {3, 39, 7772}, {3, 5013, 31652}, {3, 5024, 22332}, {3, 6427, 39865}, {3, 6428, 39866}, {3, 7772, 32}, {3, 22332, 39}, {3, 31652, 574}, {3, 35007, 5206}, {6, 5206, 32}, {6, 37512, 5206}, {32, 574, 15515}, {32, 15515, 8588}, {39, 187, 9605}, {39, 574, 32}, {39, 5013, 574}, {39, 8589, 5041}, {39, 31652, 3}, {39, 37512, 6}, {61, 62, 39561}, {140, 9607, 5309}, {194, 7815, 17131}, {371, 372, 50664}, {574, 5206, 37512}, {574, 7772, 3}, {631, 7739, 7755}, {2549, 31400, 1506}, {4045, 7763, 7867}, {5013, 5024, 39}, {5013, 22332, 3}, {5041, 8589, 3053}, {5237, 5238, 17508}, {5309, 31457, 140}, {5421, 10979, 13342}, {6453, 6454, 20190}, {7738, 31401, 115}, {7757, 7824, 7751}, {7758, 32990, 7810}, {7764, 7791, 7818}, {7769, 7864, 7844}, {7777, 7847, 7825}, {7783, 7786, 3734}, {7796, 33021, 7865}, {7799, 7876, 7869}, {7831, 7906, 7896}, {9605, 15815, 187}, {9605, 41940, 7772}, {9737, 11171, 37479}, {10983, 52771, 5188}, {11163, 33234, 7843}, {16043, 34511, 7794}, {22332, 31652, 7772}, {31467, 44518, 7603}, {31652, 41940, 15815}, {35007, 37512, 3}

---

X(53097) = X(3)X(6)∩X(4)X(599)

Barycentrics    a^2*(a^4 + 6*a^2*b^2 - 7*b^4 + 6*a^2*c^2 - 2*b^2*c^2 - 7*c^4) : :
X(53097) = 4 X[3] - 3 X[6], 7 X[3] - 6 X[182], 5 X[3] - 4 X[575], 3 X[3] - 2 X[576], 2 X[3] - 3 X[1350], 5 X[3] - 3 X[1351], 5 X[3] - 6 X[3098], 11 X[3] - 9 X[5050], 10 X[3] - 9 X[5085], 13 X[3] - 12 X[5092], 13 X[3] - 9 X[5093], 17 X[3] - 12 X[5097], 14 X[3] - 9 X[5102], and many others

X(53097) lies on these lines: {3, 6}, {4, 599}, {5, 21358}, {20, 524}, {22, 23061}, {23, 394}, {26, 9970}, {30, 15069}, {64, 2393}, {69, 3146}, {140, 20423}, {141, 3091}, {154, 3292}, {159, 9968}, {165, 4663}, {185, 16936}, {193, 44882}, {206, 22333}, {315, 51438}, {343, 31099}, {373, 5646}, {376, 8550}, {381, 40107}, {382, 19924}, {516, 49460}, {518, 2136}, {538, 14532}, {542, 1657}, {546, 10516}, {548, 11179}, {550, 43273}, {597, 3523}, {631, 47352}, {632, 14561}, {852, 33924}, {895, 3532}, {1092, 18374}, {1154, 33532}, {1204, 10602}, {1352, 3627}, {1386, 30389}, {1469, 3303}, {1498, 2781}, {1503, 3529}, {1593, 8542}, {1598, 15606}, {1656, 25565}, {1843, 11403}, {1853, 46517}, {1992, 3522}, {1993, 7492}, {1995, 2979}, {2071, 15826}, {2810, 38668}, {2854, 15054}, {3056, 3304}, {3060, 5643}, {3066, 7998}, {3090, 3763}, {3242, 7982}, {3515, 11470}, {3516, 8541}, {3524, 51185}, {3525, 14853}, {3526, 5476}, {3528, 51737}, {3534, 51187}, {3543, 22165}, {3545, 51186}, {3564, 15704}, {3567, 38402}, {3589, 10303}, {3618, 21167}, {3619, 15022}, {3620, 50689}, {3628, 21850}, {3629, 25406}, {3630, 5921}, {3631, 50688}, {3696, 43173}, {3796, 11422}, {3818, 5076}, {3830, 18553}, {3832, 21356}, {3839, 50991}, {3843, 11178}, {3917, 11284}, {3984, 43216}, {4301, 47358}, {5032, 21734}, {5054, 25555}, {5056, 20582}, {5059, 11160}, {5068, 50959}, {5072, 24206}, {5073, 50955}, {5079, 19130}, {5181, 38791}, {5198, 7716}, {5204, 8540}, {5217, 19369}, {5220, 18788}, {5537, 9037}, {5562, 15811}, {5609, 51941}, {5648, 15063}, {5651, 31860}, {5663, 33534}, {5881, 50783}, {5882, 51000}, {5890, 41463}, {5965, 48880}, {5969, 38664}, {5999, 8667}, {6034, 38740}, {6090, 41424}, {6101, 7530}, {6144, 6776}, {6194, 15271}, {6391, 43691}, {6403, 35502}, {6593, 15020}, {6698, 15025}, {6800, 9716}, {7379, 17251}, {7385, 17313}, {7387, 19596}, {7390, 17392}, {7464, 8549}, {7484, 21969}, {7485, 15019}, {7496, 10601}, {7527, 40929}, {7550, 10982}, {7555, 16266}, {7556, 17821}, {7710, 50771}, {7776, 38745}, {7788, 40236}, {7957, 9004}, {8537, 35477}, {8548, 37950}, {8556, 13860}, {8567, 17813}, {8584, 10304}, {8705, 34795}, {9019, 15062}, {9024, 38669}, {9766, 37182}, {9924, 34146}, {9969, 34817}, {9971, 45186}, {9976, 15041}, {10299, 50983}, {10300, 41588}, {10442, 52852}, {10488, 12117}, {10605, 15073}, {10752, 15034}, {11001, 51188}, {11002, 21766}, {11174, 44434}, {11180, 33703}, {11285, 22486}, {11518, 24471}, {11522, 51003}, {11645, 17800}, {11649, 35001}, {11799, 47448}, {11898, 29012}, {11999, 32305}, {12088, 15582}, {12103, 34380}, {12163, 14984}, {12220, 22528}, {12383, 25336}, {12584, 48679}, {12812, 38136}, {13115, 39849}, {13154, 16982}, {13192, 46949}, {13391, 31861}, {13409, 26909}, {13598, 29959}, {13630, 33543}, {13857, 52292}, {14002, 15066}, {14448, 16510}, {14449, 15805}, {14531, 32621}, {14848, 15720}, {14852, 47341}, {14869, 18583}, {14881, 22677}, {14927, 20080}, {15027, 49116}, {15039, 19140}, {15040, 25556}, {15068, 37967}, {15107, 35259}, {15156, 15163}, {15157, 15162}, {15316, 34437}, {15682, 50989}, {15683, 50992}, {15707, 46267}, {15712, 38064}, {15750, 44102}, {15993, 44518}, {16042, 33884}, {16051, 26958}, {16176, 32233}, {16189, 49465}, {16195, 45248}, {16419, 21849}, {17824, 32367}, {18405, 34118}, {18440, 29317}, {18451, 37924}, {19127, 34148}, {19136, 43652}, {19782, 48936}, {21735, 50966}, {22658, 32262}, {22676, 31859}, {23048, 25563}, {23292, 33522}, {24728, 49486}, {25335, 32247}, {29323, 48662}, {30734, 34417}, {31152, 41586}, {31666, 38029}, {32217, 37957}, {33923, 50979}, {34573, 46936}, {34777, 52028}, {34778, 44668}, {34779, 50414}, {35253, 46850}, {35446, 43602}, {35475, 39588}, {35950, 39656}, {37454, 43653}, {37521, 37682}, {37665, 46944}, {38757, 51007}, {39884, 43621}, {39899, 48898}, {42433, 51203}, {42434, 51200}, {43150, 48904}, {43174, 47359}, {44245, 48906}, {44280, 47549}, {44413, 49671}, {46935, 51130}, {47450, 47468}, {47453, 47571}, {49135, 51023}, {50626, 53089}, {50687, 50990}, {50690, 50958}, {50691, 51022}, {50960, 51211}, {50962, 50968}, {50964, 51184}, {50971, 51214}, {50976, 51140}

X(53097) = midpoint of X(i) and X(j) for these {i,j}: {14927, 20080}, {15683, 50992}, {40341, 48872}
X(53097) = reflection of X(i) in X(j) for these {i,j}: {3, 52987}, {6, 1350}, {193, 44882}, {382, 34507}, {597, 50970}, {599, 50967}, {1350, 33878}, {1351, 3098}, {1498, 34787}, {1992, 50965}, {3543, 22165}, {5921, 3630}, {6144, 6776}, {6776, 48881}, {8586, 18860}, {10488, 12117}, {10752, 33851}, {11477, 3}, {13330, 5188}, {15534, 376}, {16176, 32233}, {25335, 32247}, {25336, 12383}, {31670, 48876}, {36990, 69}, {37517, 14810}, {39899, 48898}, {43621, 39884}, {44439, 3313}, {44456, 182}, {46264, 48874}, {48673, 52996}, {48679, 12584}, {48904, 43150}, {48905, 48873}, {48910, 1352}, {49486, 24728}, {50649, 15644}, {51024, 599}, {51027, 11160}, {51028, 597}, {51163, 3631}, {51166, 20582}, {51212, 141}
X(53097) = isogonal conjugate of X(47586)
X(53097) = Brocard-circle-inverse of X(10541)
X(53097) = crossdifference of every pair of points on line {523, 47460}
X(53097) = barycentric quotient X(6)/X(47586)
X(53097) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 10541}, {3, 575, 5085}, {3, 1351, 575}, {3, 11477, 6}, {3, 11482, 182}, {3, 33878, 52987}, {3, 44456, 11482}, {3, 52987, 1350}, {6, 1350, 31884}, {32, 10542, 6}, {61, 62, 43136}, {141, 51212, 53023}, {182, 5102, 6}, {182, 44456, 5102}, {187, 1504, 6424}, {187, 1505, 6423}, {371, 372, 21309}, {382, 34507, 47353}, {575, 3098, 3}, {1151, 1152, 187}, {1160, 11824, 1152}, {1161, 11825, 1151}, {1350, 5085, 3098}, {1350, 11477, 3}, {1351, 3098, 5085}, {1351, 5085, 6}, {2979, 33586, 17811}, {3592, 3594, 5007}, {5013, 13330, 6}, {5210, 8586, 6}, {5480, 10519, 3763}, {6409, 6410, 5585}, {6425, 6426, 22331}, {9732, 12305, 6409}, {9732, 35002, 12306}, {9733, 12306, 6410}, {9733, 35002, 12305}, {9786, 50649, 6}, {10541, 31884, 3}, {10752, 33851, 52697}, {11480, 11481, 8588}, {11916, 45553, 6431}, {11917, 45552, 6432}, {14810, 37517, 5050}, {19161, 50649, 46363}, {22236, 22238, 32}, {25565, 51173, 38072}, {31670, 48876, 10516}, {36836, 36843, 5023}, {37484, 37486, 37498}

---

X(53098) = X(2)X(11482)∩X(4)X(3055)

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

X(53098) lies on the Kiepert circumhyperbola and these lines: {2, 11482}, {4, 3055}, {5, 41895}, {83, 3533}, {140, 5395}, {275, 52290}, {376, 45103}, {598, 631}, {671, 3090}, {1656, 2996}, {3523, 18845}, {3525, 18842}, {3545, 17503}, {3855, 33698}, {5056, 38259}, {5067, 5485}, {5071, 32532}, {5094, 8796}, {6811, 43567}, {6813, 43566}, {7607, 33554}, {7612, 31489}, {7735, 11668}, {8889, 39284}, {11538, 16063}, {14912, 43537}, {37463, 43541}, {37464, 43540}, {43681, 46935}

---

X(53099) = X(2)X(11477)∩X(3)X(18842)

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

X(53099) lies on the Kiepert circumhyperbola and these lines: {2, 11477}, {3, 18842}, {4, 5024}, {5, 5485}, {6, 43537}, {20, 598}, {76, 5056}, {83, 3523}, {98, 37665}, {140, 18841}, {275, 4232}, {381, 32532}, {383, 33602}, {459, 5094}, {550, 18843}, {671, 3091}, {1080, 33603}, {1327, 7000}, {1328, 7374}, {1656, 18840}, {1657, 18844}, {2052, 52284}, {2996, 5068}, {3424, 7736}, {3522, 5395}, {3543, 45103}, {3815, 14484}, {3832, 41895}, {3839, 17503}, {3854, 38259}, {5059, 18845}, {5304, 7612}, {5476, 42011}, {6811, 14226}, {6813, 14241}, {7378, 39284}, {7486, 10302}, {7533, 13579}, {7607, 37689}, {7608, 14853}, {8550, 47586}, {9748, 10155}, {9753, 11669}, {10159, 46935}, {11167, 34507}, {14485, 31400}, {37463, 43543}, {37464, 43542}

X(53099) = cevapoint of X(11) and X(48193)
X(53099) = trilinear pole of line {523, 47446}

---

X(53100) = X(2)X(18553)∩X(3)X(10302)

Barycentrics    (4*a^4 + 2*a^2*b^2 + 4*b^4 - 3*a^2*c^2 - 3*b^2*c^2 - c^4)*(4*a^4 - 3*a^2*b^2 - b^4 + 2*a^2*c^2 - 3*b^2*c^2 + 4*c^4) : :
X(53100) = 16 X[3530] - 15 X[11149], 4 X[15687] - 5 X[17503], 11 X[15715] - 10 X[51584]

X(53100) lies on the Kiepert circumhyperbola and these lines: {2, 18553}, {3, 10302}, {4, 5008}, {76, 550}, {83, 3851}, {262, 8550}, {382, 671}, {383, 33606}, {542, 42010}, {546, 598}, {1080, 33607}, {1503, 7607}, {2052, 10301}, {2996, 14712}, {3529, 5485}, {3530, 11149}, {3855, 7817}, {5392, 37900}, {5939, 8781}, {6811, 43568}, {6813, 43569}, {7612, 43460}, {9744, 10155}, {10033, 25555}, {10159, 15720}, {10299, 18840}, {10991, 43532}, {14269, 45103}, {14494, 53015}, {15687, 17503}, {15715, 51584}, {31856, 44877}, {35018, 43527}, {37463, 43544}, {37464, 43545}, {38227, 43537}, {41895, 50688}, {43676, 49139}

X(53100) = isogonal conjugate of X(52987)
X(53100) = trilinear pole of line {523, 47458}
X(53100) = barycentric quotient X(6)/X(52987)

---

X(53101) = X(2)X(5210)∩X(4)X(14848)

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

X(53101) lies on the Kiepert circumhyperbola and these lines: {2, 5210}, {4, 14848}, {6, 41895}, {20, 7608}, {30, 14494}, {76, 11160}, {98, 3839}, {193, 5485}, {262, 3543}, {376, 10155}, {381, 7612}, {459, 52281}, {597, 5395}, {598, 51171}, {599, 32979}, {671, 5032}, {1916, 8596}, {1992, 2996}, {2482, 8781}, {3091, 7607}, {3424, 53017}, {3620, 10302}, {3832, 43537}, {5056, 10185}, {5304, 43535}, {5503, 8591}, {7388, 43564}, {7389, 43565}, {7745, 38259}, {7747, 32883}, {7812, 43676}, {7841, 18841}, {8352, 18842}, {8370, 18840}, {10159, 32971}, {10304, 11669}, {11167, 37667}, {11303, 43446}, {11304, 43447}, {11361, 40824}, {11668, 13449}, {14035, 43529}, {14063, 43528}, {14484, 50687}, {15687, 52519}, {17503, 43448}, {32897, 33013}, {32974, 43527}, {37174, 43530}, {37809, 39590}, {47586, 50689}

X(53101) = X(63)-isoconjugate of X(11405)
X(53101) = X(3162)-Dao conjugate of X(11405)
X(53101) = barycentric quotient X(25)/X(11405)

---

X(53102) = X(2)X(7843)∩X(4)X(42785)

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

X(53102) lies on the Kiepert circumhyperbola and these lines: {2, 7843}, {4, 42785}, {6, 43676}, {76, 3629}, {98, 3851}, {140, 11669}, {262, 550}, {316, 18841}, {382, 14492}, {546, 14458}, {598, 33229}, {671, 7878}, {1916, 14034}, {3407, 14045}, {3618, 18843}, {7388, 43568}, {7389, 43569}, {7607, 32151}, {7608, 15720}, {7745, 43527}, {7768, 18840}, {7770, 10302}, {7803, 18845}, {7827, 41895}, {7879, 10159}, {10299, 14494}, {10484, 33276}, {11289, 43545}, {11290, 43544}, {11303, 33606}, {11304, 33607}, {14484, 49135}

---

X(53103) = X(2)X(1353)∩X(4)X(5023)

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

X(53103 lies on the Kiepert circumhyperbola and these lines: {2, 1353}, {3, 38259}, {4, 5023}, {5, 18845}, {6, 10155}, {76, 3525}, {83, 5067}, {94, 2974}, {140, 43681}, {147, 8587}, {230, 14494}, {275, 52299}, {376, 41895}, {383, 43553}, {598, 5071}, {631, 2996}, {671, 3524}, {1080, 43552}, {1370, 13585}, {2052, 38282}, {3054, 7612}, {3090, 5395}, {3545, 53101}, {3815, 53098}, {5485, 8716}, {6353, 8796}, {6811, 43560}, {6813, 43561}, {6997, 11538}, {7386, 13579}, {7410, 43533}, {7608, 7735}, {7736, 11669}, {8781, 34229}, {9752, 52519}, {11001, 17503}, {12820, 36994}, {12821, 36992}, {13582, 46336}, {13860, 43951}, {14488, 38227}, {14940, 52583}, {18842, 44401}, {19708, 32532}, {37463, 43556}, {37464, 43557}, {37688, 40824}, {41106, 45103}

X(53103) = isogonal conjugate of X(5093)
X(53103) = isotomic conjugate of X(34803)
X(53103) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5093}, {31, 34803}
X(53103) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 34803}, {3, 5093}
X(53103) = trilinear pole of line {523, 47281}
X(53103) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 34803}, {6, 5093}

---

X(53104) = X(2)X(5965)∩X(4)X(5206)

Barycentrics    (3*a^4 - 4*a^2*b^2 + 3*b^4 - 5*a^2*c^2 - 5*b^2*c^2 + 2*c^4)*(3*a^4 - 5*a^2*b^2 + 2*b^4 - 4*a^2*c^2 - 5*b^2*c^2 + 3*c^4) : :
X(53104) = X[4] - 10 X[12815], 2 X[4] - 5 X[38228], 4 X[12815] - X[38228], 8 X[140] + X[43676], 2 X[549] - 5 X[38226], 2 X[5055] - 5 X[38223], 13 X[10303] + 5 X[50570], 11 X[3525] - 2 X[51587], 2 X[5066] - 5 X[38231]

X(53104) lies on the Kiepert hyperbola and these lines: {2, 5965}, {4, 5206}, {6, 11669}, {13, 16653}, {14, 16652}, {30, 33698}, {76, 3526}, {83, 3628}, {98, 3054}, {140, 43676}, {230, 7608}, {262, 37637}, {383, 12821}, {549, 671}, {598, 5055}, {1080, 12820}, {1656, 53102}, {1916, 15819}, {2052, 37453}, {2996, 10303}, {3090, 18843}, {3525, 51587}, {3534, 17503}, {3590, 45525}, {3591, 45524}, {5066, 38230}, {5395, 7486}, {5485, 7622}, {5503, 15597}, {6036, 9751}, {6054, 8587}, {6811, 12818}, {6813, 12819}, {7607, 9755}, {7610, 42011}, {7612, 43461}, {7735, 10155}, {7736, 53098}, {8781, 37688}, {9744, 43537}, {9752, 14484}, {9756, 14458}, {9993, 43951}, {10302, 47598}, {10304, 41895}, {12816, 44667}, {12817, 44666}, {13860, 14488}, {14492, 38227}, {15022, 18845}, {15698, 32532}, {15717, 38259}, {22531, 43556}, {22532, 43557}, {37463, 43546}, {37464, 43547}

X(53104) = isogonal conjugate of X(5097)
X(53104) = isotomic conjugate of X(37647)
X(53104) = isotomic conjugate of the complement of X(17004)
X(53104) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5097}, {31, 37647}
X(53104) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 37647}, {3, 5097}
X(53104) = cevapoint of X(2) and X(17004)
X(53104) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 37647}, {6, 5097}

---

X(53105) = X(2)X(7748)∩X(4)X(5097)

Barycentrics    (2*a^2 + 2*b^2 - 3*c^2)*(2*a^2 - 3*b^2 + 2*c^2) : :
X(53105) = 9 X[2] - 5 X[45017], 5 X[45017] - 6 X[51581]

X(53105) lies on the Kiepert hyperbola and these lines: {2, 7748}, {4, 5097}, {5, 11669}, {10, 17336}, {76, 3631}, {83, 44518}, {98, 382}, {115, 33257}, {148, 7814}, {262, 546}, {315, 5485}, {316, 2996}, {550, 7607}, {598, 5254}, {671, 7754}, {1916, 14062}, {3407, 14042}, {3424, 50688}, {3529, 7612}, {3530, 11668}, {3544, 10155}, {3851, 7608}, {3855, 14494}, {5286, 18845}, {5395, 7878}, {6329, 53102}, {7388, 43559}, {7389, 43558}, {7752, 51123}, {7768, 43681}, {7771, 33253}, {7790, 43527}, {7812, 17503}, {7827, 18842}, {7841, 10302}, {7860, 40341}, {7871, 14041}, {7910, 18546}, {7926, 14044}, {7937, 10159}, {7942, 33242}, {10185, 15720}, {11185, 18840}, {11289, 43443}, {11290, 43442}, {11303, 43544}, {11304, 43545}, {14034, 43528}, {14045, 43529}, {14269, 14492}, {14458, 15687}, {33235, 44531}, {38734, 43532}, {43537, 49135}

X(53105) = isogonal conjugate of X(5206)
X(53105) = isotomic conjugate of X(40341)
X(53105) = anticomplement of X(51581)
X(53105) = polar conjugate of X(37453)
X(53105) = isotomic conjugate of the anticomplement of X(3629)
X(53105) = isotomic conjugate of the complement of X(11008)
X(53105) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5206}, {31, 40341}, {48, 37453}
X(53105) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 40341}, {3, 5206}, {1249, 37453}
X(53105) = cevapoint of X(i) and X(j) for these (i,j): {2, 11008}, {115, 32478}
X(53105) = trilinear pole of line {523, 31209}
X(53105) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 40341}, {4, 37453}, {6, 5206}, {3629, 51581}

---

X(53106) = X(2)X(7756)∩X(4)X(15520)

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

X(53106) lies on the Kiepert hyperbola and these lines: {2, 7756}, {3, 11668}, {4, 15520}, {98, 3627}, {262, 3843}, {315, 43681}, {316, 43676}, {598, 44518}, {671, 7762}, {1657, 7607}, {1916, 14044}, {3407, 14066}, {3850, 7608}, {5072, 11669}, {5286, 53101}, {5466, 50548}, {5485, 32006}, {7612, 33703}, {7745, 45103}, {7790, 18841}, {7812, 41895}, {7911, 10302}, {9302, 9880}, {10185, 15712}, {11289, 43441}, {11290, 43440}, {11303, 43548}, {11304, 43549}, {14458, 38335}, {14492, 14893}, {18845, 43448}, {33267, 43459}, {33560, 43544}, {33561, 43545}, {43537, 50691}

X(53106) = isogonal conjugate of X(15513)
X(53106) = isotomic conjugate of X(3630)
X(53106) = anticomplement of X(51585)
X(53106) = polar conjugate of X(52297)
X(53106) = isotomic conjugate of the anticomplement of X(32455)
X(53106) = isotomic conjugate of the complement of X(6144)
X(53106) = X(i)-isoconjugate of X(j) for these (i,j): {1, 15513}, {31, 3630}, {48, 52297}
X(53106) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 3630}, {3, 15513}, {1249, 52297}
X(53106) = cevapoint of X(2) and X(6144)
X(53106) = trilinear pole of line {523, 27115}
X(53106) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3630}, {4, 52297}, {6, 15513}, {32455, 51585}

---

X(53107) = X(2)X(7842)∩X(4)X(15516)

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

X(53107) lies on the Kiepert hyperbola and these lines: {2, 7842}, {4, 15516}, {5, 11668}, {76, 3630}, {98, 3843}, {262, 3627}, {316, 18840}, {548, 11669}, {671, 7894}, {1657, 7608}, {1916, 14066}, {3407, 14044}, {3850, 7607}, {5254, 17503}, {7754, 43676}, {7784, 10159}, {7790, 53102}, {7827, 53101}, {8781, 52886}, {10155, 17538}, {11289, 43440}, {11290, 43441}, {11303, 43549}, {11304, 43548}, {14458, 14893}, {14492, 38335}, {14494, 33703}, {21735, 53098}, {50691, 53099}

X(53107) = isogonal conjugate of X(15515)
X(53107) = polar conjugate of X(52298)
X(53107) = X(i)-isoconjugate of X(j) for these (i,j): {1, 15515}, {48, 52298}
X(53107) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 15515}, {1249, 52298}
X(53107) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 52298}, {6, 15515}

---

X(53108) = X(2)X(15520)∩X(4)X(15515)

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

X(53108) lies on the Kiepert hyperbola and these lines: {2, 15520}, {4, 15515}, {6, 11668}, {76, 5070}, {83, 632}, {98, 3055}, {230, 10185}, {275, 52297}, {547, 671}, {598, 5054}, {631, 18844}, {2052, 52298}, {2996, 46936}, {3424, 43461}, {7607, 31489}, {8703, 45103}, {9744, 47586}, {9752, 53099}, {9754, 14494}, {9756, 53100}, {9993, 52519}, {15692, 53101}, {17503, 19709}, {23234, 43535}, {33698, 38071}, {37463, 43551}, {37464, 43550}

X(53108) = isogonal conjugate of X(15516)
X(53108) = isotomic conjugate of the complement of X(17005)
X(53108) = X(1)-isoconjugate of X(15516)
X(53108) = cevapoint of X(2) and X(17005)
X(53108) = barycentric quotient X(6)/X(15516)

---

X(53109) = X(2)X(5206)∩X(4)X(39561)

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

X(53109) lies on the Kiepert hyperbola and these lines: {2, 5206}, {3, 11669}, {4, 39561}, {76, 40341}, {83, 33229}, {98, 546}, {99, 35005}, {262, 382}, {316, 10159}, {550, 7608}, {671, 7745}, {1916, 14042}, {2996, 7760}, {3399, 43147}, {3407, 14062}, {3528, 10155}, {3529, 14494}, {3629, 43676}, {3851, 7607}, {3855, 7612}, {5079, 11668}, {5286, 41895}, {5395, 7790}, {5475, 33257}, {5485, 7812}, {6671, 10188}, {6672, 10187}, {7388, 43558}, {7389, 43559}, {7803, 18842}, {7827, 45103}, {7899, 33242}, {8370, 10302}, {8781, 19687}, {10185, 35018}, {10299, 53098}, {11121, 41745}, {11122, 41746}, {11170, 12203}, {11185, 43681}, {11289, 43442}, {11290, 43443}, {11303, 43545}, {11304, 43544}, {14034, 43529}, {14045, 43528}, {14061, 39590}, {14269, 14458}, {14484, 50688}, {14492, 15687}, {14537, 15031}, {17503, 44518}, {18366, 41254}, {18840, 32006}, {44142, 46105}, {49135, 53099}

X(53109) = isogonal conjugate of X(37512)
X(53109) = isotomic conjugate of X(3631)
X(53109) = isotomic conjugate of the anticomplement of X(6329)
X(53109) = isotomic conjugate of the complement of X(3629)
X(53109) = X(i)-isoconjugate of X(j) for these (i,j): {1, 37512}, {31, 3631}
X(53109) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 3631}, {3, 37512}, {51581, 51587}
X(53109) = cevapoint of X(i) and X(j) for these (i,j): {2, 3629}, {6, 13595}
X(53109) = trilinear pole of line {523, 26777}
X(53109) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3631}, {6, 37512}, {3629, 51587}

---

X(53110) = 10TH MIYAMOTO-LOZADA CENTER

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

Let (I), (Ia), (Ib) and (Ic) be the incircle and A-, B-, C-excircles of a given triangle ABC, respectively. Let (Oa) be the circle externally tangent to (Ib) and (Ic), and internally tangent to (I), other than the nine-point circle of ABC. Define (Ob) and (Oc) cyclically. Let Ab be the touchpoint of (Oa) and (Ib), and define Bc and Ca cyclically. Let Ac be the touchpoint of (Oa) and (Ic), and define Ba and Cb cyclically. Let A'=BcBa∩CaCb, B'=CaCb∩AbAc, C'=AbAc∩BcBa. Then, the lines AA', BB', CC' concur at X(53110). The lines IaA', IbB', IcC' concur at X(53111).
Remark: Let Ta be the touchpoint of (Oa) and (I), and define Tb and Tc cyclically. Then, the lines ATa, BTb, CTc concur in X(3271).
Remark: The centers of (Oa), (Ob) and (Oc) lie on the line X(3)X(6).
Keita Miyamoto, March 15, 2023.

(Oa) has center:
Oa = -((b+c)^2*a^3-(b+c)*(b^2+c^2)*a^2-(b^4+c^4+2*(b^2+c^2)*b*c)*a+(b+c)*(b^4+c^4))*a^2 :
(a^5-(b+c)*a^4-(b-2*c)*b*a^3+(b+c)*b^2*a^2-(b^2-c^2)*(2*b+c)*c*a+(b^2-c^2)*(b+c)*c^2)*b^2 :
(a^5-(b+c)*a^4+(2*b-c)*c*a^3+(b+c)*c^2*a^2+(b^2-c^2)*(b+2*c)*b*a-(b^2-c^2)*(b+c)*b^2)*c^2
and radius:
ra = |(b+c)*(a^2+b*c)-(b^2+b*c+c^2)*a|/(4*S)
Radical center of (Oa), (Ob), (Oc) is X(512).
Construction of Oa: Let N be the nine-point center X(5) of ABC and A"B"C" the excentral triangle-of the medial triangle-of ABC (A"B"C" is the Wasat triangle of ABC). Then Oa = X(3)X(6) ∩ NA", and similarly Ob, Oc.
César Eliud Lozada, March 20, 2023.

X(53110) lies on these lines: {1, 53111}, {3059, 4111}, {4642, 52020}

X(53110) = intersection, other than A, B, C, of circumconics {A, B, C, X(1), X(3709)} and {A, B, C, X(8), X(4111)}
X(53110) = trilinear quotient X(2170)/X(26845)

---

X(53111) = 11TH MIYAMOTO-LOZADA CENTER

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

See X(53110). César Eliud Lozada, March 20, 2023.

X(53111) lies on these lines: {1, 53110}, {2122, 8614}, {3688, 3871}

X(53111) = eigentransform of X(4573)

---

X(53112) = 12TH MIYAMOTO-LOZADA CENTER

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

Let (I), (Ia), (Ib) and (Ic) be the incircle, A-excircle, B-excircle, C-excircle of a given triangle ABC, respectively. Let (Oa') be the circle externally tangent to (I), (Ib) and (Ic), and define (Ob') and (Oc') cyclically. Let Ab' be the touchpoint of (Oa') and (Ib), and define Bc' and Ca' cyclically. Let Ac' be the touchpoint of (Oa') and (Ic), and define Ba' and Cb' cyclically. Let A"=Bc'Ba'∩Ca'Cb', B"=Ca'Cb'∩Ab'Ac', C"=Ab'Ac'∩Bc'Ba'. Then, the lines AA", BB", CC" concur at X(53112). The lines IaA", IbB", IcC" concur at X(53113).
Remark: Let Ta' be the touchpoint of (Oa') and (I), and define Tb' and Tc' cyclically. Then the lines ATa', BTb' CTc' concur in X(6063)
Keita Miyamoto, March 15, 2023.

Circle (Oa') has center:
Oa' = 2*(b+c)*a^4-(3*b^2+4*b*c+3*c^2)*a^3-(b+c)*(b^2+c^2)*a^2+(b^2-c^2)^2*a+(b^2-c^2)^2*(b+c) :
-(c^2-2*c*a+2*a^2)*b^3-(c-a)*(c^2+2*a^2)*b^2+(c^2-a^2)*(c-2*a)*b*c+(c^2-a^2)*(c-a)*c^2 :
-(2*a^2-2*a*b+b^2)*c^3+(a-b)*(2*a^2+b^2)*c^2+(a^2-b^2)*(2*a-b)*c*b+(a^2-b^2)*(a-b)*b^2
and radius:
ra' = a*|(b+c)*(a^2+b*c)-(b^2+b*c+c^2)*a|/(4*S*(b+c))
César Eliud Lozada, March 20, 2023.

X(53112) lies on these lines: {1, 53113}, {1441, 39793}, {4059, 6063}

X(53112) = X(1214)-Dao conjugate of-X(26037)
X(53112) = X(i)-isoconjugate-of-X(j) for these {i, j}: {41, 25508}, {2194, 26037}
X(53112) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (7, 25508), (226, 26037)
X(53112) = intersection, other than A, B, C, of circumconics {A, B, C, X(7), X(226)} and {A, B, C, X(65), X(4059)}
X(53112) = barycentric quotient X(i)/X(j) for these (i, j): (7, 25508), (226, 26037)
X(53112) = trilinear quotient X(i)/X(j) for these (i, j): (85, 25508), (1441, 26037)

---

X(53113) = 13TH MIYAMOTO-LOZADA CENTER

Barycentrics    (b+c)*(3*b^2+8*b*c+3*c^2)*a^7+(3*b^4+3*c^4+b*c*(21*b^2+38*b*c+21*c^2))*a^6-(b+c)*(b^4+c^4-5*b*c*(b^2+3*b*c+c^2))*a^5-(b^2+b*c+c^2)*(3*b^4+3*c^4+b*c*(8*b^2-3*b*c+8*c^2))*a^4-(b+c)*(2*b^6+2*c^6+(15*b^4+15*c^4+b*c*(5*b^2-21*b*c+5*c^2))*b*c)*a^3-(8*b^6+8*c^6+(27*b^4+27*c^4-b*c*(7*b^2+48*b*c+7*c^2))*b*c)*b*c*a^2-(b^2-c^2)*(b-c)*(8*b^2+17*b*c+8*c^2)*b^2*c^2*a-2*(b^2-c^2)^2*b^3*c^3 : :

See X(53112). César Eliud Lozada, March 20, 2023.

X(53113) lies on the line {1, 53112}

---

X(53114) = 14TH MIYAMOTO-LOZADA CENTER

Barycentrics    a*(b+c)*(2*a+2*b-c)*(2*a+2*c-b) : :
X(53114) = 3*X(1)-X(17461) = 3*X(551)-2*X(42285) = X(994)-3*X(5902) = 3*X(30116)-X(49448)

Continuing with constructions in X(53110) and X(53112), let La be the radical axis of (Oa) and (Oa'), and define Lb and Lc cyclically. Let A*=Lb∩Lc, B*=Lc∩La, C*=La∩Lb. Then, the lines AA*, BB*, CC* concur at X(53114) and the lines IaA*, IbB*, IcC* concur at X(53115).
Keita Miyamoto, March 15, 2023.

Barycentric coordinates of A* are:
(-(b+c)*a+b*c-b^2-c^2)*a : (a+c)*(2*a+2*b-c)*b : (a+b)*(2*a+2*c-b)*c]
César Eliud Lozada, March 20, 2023.

X(53114) lies on these lines: {1, 89}, {8, 30589}, {10, 2650}, {19, 1449}, {37, 758}, {42, 3919}, {58, 40430}, {65, 4868}, {75, 519}, {77, 18421}, {81, 759}, {145, 39711}, {158, 5342}, {214, 37520}, {225, 3671}, {267, 17016}, {386, 33815}, {514, 4667}, {517, 13476}, {518, 52901}, {535, 17365}, {551, 17450}, {596, 3244}, {756, 4525}, {876, 14421}, {897, 4604}, {942, 34434}, {994, 4850}, {1125, 30608}, {1159, 19654}, {1320, 16490}, {1581, 15953}, {1910, 34073}, {2153, 39153}, {2154, 39152}, {2166, 30690}, {2214, 16884}, {2363, 4658}, {2802, 49478}, {2841, 39543}, {3241, 17146}, {3635, 34860}, {3743, 4018}, {3753, 21870}, {3754, 50587}, {3833, 31197}, {3898, 4883}, {3899, 29814}, {3914, 5620}, {3931, 4757}, {3968, 4849}, {3993, 41683}, {4022, 39737}, {4090, 52872}, {4351, 17015}, {4424, 4744}, {4597, 18827}, {4732, 46772}, {4792, 41434}, {4867, 37633}, {5313, 24168}, {5883, 16610}, {6015, 8695}, {7232, 48808}, {11518, 30148}, {11520, 30145}, {11700, 50194}, {12559, 30142}, {12609, 41501}, {16474, 22128}, {17038, 30116}, {17057, 26738}, {17097, 34043}, {17460, 39697}, {21251, 25386}, {21839, 52708}, {24692, 48836}, {34195, 37559}, {44840, 49480}{1, 89}, {8, 30589}, {10, 2650}, {19, 1449}, {37, 758}, {42, 3919}, {58, 40430}, {65, 4868}, {75, 519}, {77, 18421}, {81, 759}, {145, 39711}, {158, 5342}, {214, 37520}, {225, 3671}, {267, 17016}, {386, 33815}, {514, 4667}, {517, 13476}, {518, 52901}, {535, 17365}, {551, 17450}, {596, 3244}, {756, 4525}, {876, 14421}, {897, 4604}, {942, 34434}, {994, 4850}, {1125, 30608}, {1159, 19654}, {1320, 16490}, {1581, 15953}, {1910, 34073}, {2153, 39153}, {2154, 39152}, {2166, 30690}, {2214, 16884}, {2363, 4658}, {2802, 49478}, {2841, 39543}, {3241, 17146}, {3635, 34860}, {3743, 4018}, {3753, 21870}, {3754, 50587}, {3833, 31197}, {3898, 4883}, {3899, 29814}, {3914, 5620}, {3931, 4757}, {3968, 4849}, {3993, 41683}, {4022, 39737}, {4090, 52872}, {4351, 17015}, {4424, 4744}, {4597, 18827}, {4732, 46772}, {4792, 41434}, {4867, 37633}, {5313, 24168}, {5883, 16610}, {6015, 8695}, {7232, 48808}, {11518, 30148}, {11520, 30145}, {11700, 50194}, {12559, 30142}, {12609, 41501}, {16474, 22128}, {17038, 30116}, {17057, 26738}, {17097, 34043}, {17460, 39697}, {21251, 25386}, {21839, 52708}, {24692, 48836}, {34195, 37559}, {44840, 49480}

X(53114) = isogonal conjugate of X(4653)
X(53114) = crosspoint of X(i) and X(j) for these (i, j): {10, 4424}, {37, 21806}
X(53114) = X(89)-Ceva conjugate of-X(28658)
X(53114) = X(i)-Dao conjugate of-X(j) for these (i, j): (1, 4720), (9, 5235), (10, 3679), (37, 4671), (115, 4791)
X(53114) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2, 4273}, {6, 5235}, {21, 2099}, {28, 3940}, {45, 81}
X(53114) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 5235), (9, 4720), (10, 4671), (31, 4273), (37, 3679)
X(53114) = X(661)-Zayin conjugate of-X(4893)
X(53114) = intersection, other than A, B, C, of circumconics {A, B, C, X(1), X(10)} and {A, B, C, X(2), X(14996)}
X(53114) = trilinear pole of the line {661, 14407}
X(53114) = barycentric product X(i)*X(j) for these {i, j}: {1, 30588}, {10, 89}, {37, 39704}, {42, 20569}, {65, 30608}, {75, 28658}
X(53114) = barycentric quotient X(i)/X(j) for these (i, j): (1, 5235), (9, 4720), (10, 4671), (31, 4273), (37, 3679), (42, 45)
X(53114) = trilinear product X(i)*X(j) for these {i, j}: {2, 28658}, {6, 30588}, {10, 2163}, {37, 89}, {42, 39704}, {65, 2320}
X(53114) = trilinear quotient X(i)/X(j) for these (i, j): (2, 5235), (6, 4273), (8, 4720), (10, 3679), (37, 45), (42, 2177)
X(53114) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 2163, 2320), (10, 21088, 21042), (37, 21886, 21822), (81, 5425, 49682), (89, 2320, 2163)

---

X(53115) = 15TH MIYAMOTO-LOZADA CENTER

Barycentrics    a*(5*(b+c)*a^2+(b^2-b*c+c^2)*a-(b+c)*(4*b^2-7*b*c+4*c^2)) : :
X(53115) = 3*X(1)-2*X(17461) = 3*X(1)-4*X(53114) = 3*X(4677)-4*X(4793) = 9*X(25055)-8*X(42285)

See X(53114). César Eliud Lozada, March 20, 2023.

X(53115) lies on these lines: {1, 89}, {37, 44663}, {43, 994}, {45, 21886}, {65, 978}, {238, 1159}, {514, 16236}, {517, 1742}, {519, 1278}, {758, 3728}, {995, 4744}, {1046, 3340}, {1449, 1572}, {1743, 18421}, {1745, 5903}, {2802, 49498}, {2943, 7982}, {3339, 7963}, {3577, 9355}, {3632, 17164}, {3679, 31025}, {3873, 17460}, {3877, 17450}, {3899, 26102}, {3919, 16569}, {4083, 23345}, {4650, 50194}, {4677, 4793}, {4888, 21273}, {5425, 8616}, {8915, 11531}, {11041, 24695}, {16676, 21822}, {17147, 51093}, {25055, 42285}, {25415, 32913}, {40587, 49712}

X(53115) = reflection of X(17461) in X(53114)
X(53115) = Cevapoint of X(1) and X(2099)
X(53115) = crosssum of X(1) and X(2320)
X(53115) = X(i)-aleph conjugate of-X(j) for these (i, j): (1, 3576), (174, 3306), (508, 21373), (509, 16670)
X(53115) = X(2099)-Ceva conjugate of-X(1)
X(53115) = {X(17461), X(53114)}-harmonic conjugate of X(1)

---

X(53116) = ISOGONAL CONJUGATE OF X(483)

Barycentrics    a*(1+cos(A/2)) : :
X(53116) = 3*X(1)-2*X(17461) = 3*X(1)-4*X(53114) = 3*X(4677)-4*X(4793) = 9*X(25055)-8*X(42285)

Contributed by César Eliud Lozada, March 20, 2023.

X(53116) lies on the cubic K632 and these lines: {1, 168}, {36, 34201}, {266, 6502}, {289, 7001}, {557, 3082}, {16012, 30336}

X(53116) = isogonal conjugate of X(483)
X(53116) = X(478)-Dao conjugate of-X(558)
X(53116) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2, 7014}, {8, 53117}, {9, 558}, {188, 7010}
X(53116) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (31, 7014), (56, 558), (266, 1143), (557, 75), (604, 53117)
X(53116) = intersection, other than A, B, C, of circumconics {A, B, C, X(56), X(289)} and {A, B, C, X(84), X(3645)}
X(53116) = barycentric product X(i)*X(j) for these {i, j}: {1, 557}, {57, 3082}, {174, 7001}, {266, 1274}
X(53116) = barycentric quotient X(i)/X(j) for these (i, j): (31, 7014), (56, 558), (266, 1143), (557, 75), (604, 53117)
X(53116) = trilinear product X(i)*X(j) for these {i, j}: {6, 557}, {56, 3082}, {266, 7001}
X(53116) = trilinear quotient X(i)/X(j) for these (i, j): (6, 7014), (56, 53117), (57, 558), (174, 1143), (266, 7010), (483, 179)
X(53116) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 259, 53117), (1, 1129, 259)

---

X(53117) = ISOGONAL CONJUGATE OF X(3082)

Barycentrics    a*(1-cos(A/2)) : :

Contributed by César Eliud Lozada, March 20, 2023.

X(53117) lies on the cubic K632 and these lines: {1, 168}, {266, 2067}, {289, 7010}, {483, 558}, {3746, 34201}, {16012, 30335}, {18887, 38487}

X(53117) = isogonal conjugate of X(3082)
X(53117) = X(483)-beth conjugate of-X(483)
X(53117) = X(558)-Ceva conjugate of-X(7014)
X(53117) = X(478)-Dao conjugate of-X(557)
X(53117) = X(i)-isoconjugate-of-X(j) for these {i, j}: {8, 53116}, {9, 557}, {188, 7001}, {259, 1274}
X(53117) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (56, 557), (266, 1274), (483, 312), (558, 75)
X(53117) = barycentric product X(i)*X(j) for these {i, j}: {1, 558}, {7, 7014}, {57, 483}, {174, 7010}, {266, 1143}
X(53117) = barycentric quotient X(i)/X(j) for these (i, j): (56, 557), (266, 1274), (483, 312), (558, 75), (604, 53116)
X(53117) = trilinear product X(i)*X(j) for these {i, j}: {6, 558}, {56, 483}, {57, 7014}, {266, 7010}
X(53117) = trilinear quotient X(i)/X(j) for these (i, j): (56, 53116), (57, 557), (174, 1274), (266, 7001), (483, 8)
X(53117) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 259, 53116), (1, 3645, 7707), (1, 10232, 259)

---

X(53118) = ISOGONAL CONJUGATE OF X(1488)

Barycentrics    a*(1+sin(A/2)) : :

Contributed by César Eliud Lozada, March 20, 2023.

X(53118) lies on the cubic K761 and these lines: {1, 164}, {9, 6726}, {55, 259}, {165, 45086}, {173, 52999}, {174, 6732}, {188, 11924}, {289, 3659}, {503, 8076}, {3445, 16011}, {5563, 42614}, {6724, 8079}, {8133, 12908}, {10490, 13443}

X(53118) = isogonal conjugate of X(1488)
X(53118) = crosssum of X(173) and X(15997)
X(53118) = X(1)-Ceva conjugate of-X(259)
X(53118) = X(188)-Dao conjugate of-X(75)
X(53118) = X(503)-Hirst inverse of-X(8076)
X(53118) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2, 289}, {7, 53119}, {57, 7028}, {77, 8121}, {174, 258}
X(53118) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (31, 289), (41, 53119), (55, 7028), (173, 4146), (236, 75)
X(53118) = intersection, other than A, B, C, of circumconics {A, B, C, X(1), X(52999)} and {A, B, C, X(9), X(173)}
X(53118) = barycentric product X(i)*X(j) for these {i, j}: {1, 236}, {9, 2089}, {63, 8122}, {92, 8139}, {173, 188}
X(53118) = barycentric quotient X(i)/X(j) for these (i, j): (31, 289), (41, 53119), (55, 7028), (173, 4146), (236, 75), (259, 7048)
X(53118) = trilinear product X(i)*X(j) for these {i, j}: {3, 8122}, {4, 8139}, {6, 236}, {55, 2089}, {173, 259}
X(53118) = trilinear quotient X(i)/X(j) for these (i, j): (6, 289), (9, 7028), (33, 8121), (55, 53119), (173, 174), (174, 21456)
X(53118) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 266, 53119), (1, 1130, 266), (1, 8078, 15997), (1, 52802, 10231), (55, 16012, 259), (174, 43192, 11923), (1130, 10231, 52802), (10231, 52802, 266)

---

X(53119) = ISOGONAL CONJUGATE OF X(2089)

Barycentrics    a*(1-sin(A/2)) : :

Contributed by César Eliud Lozada, March 20, 2023.

X(53119) lies on the cubic K761 and these lines: {1, 164}, {6, 259}, {35, 42614}, {167, 45087}, {188, 6732}, {260, 289}, {361, 50581}, {2089, 11923}, {3158, 6726}, {3304, 42622}, {6724, 8092}, {7597, 45874}, {10490, 20183}

X(53119) = isogonal conjugate of X(2089)
X(53119) = Cevapoint of X(289) and X(1488)
X(53119) = crosssum of X(289) and X(53118)
X(53119) = X(i)-Ceva conjugate of-X(j) for these (i, j): (258, 259), (1488, 289)
X(53119) = X(55)-cross conjugate of-X(259)
X(53119) = X(i)-isoconjugate-of-X(j) for these {i, j}: {7, 53118}, {57, 236}, {77, 8122}, {173, 174}
X(53119) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (41, 53118), (55, 236), (258, 4146), (259, 7057)
X(53119) = perspector of the circumconic {A, B, C, X(3659), X(45875)}
X(53119) = intersection, other than A, B, C, of circumconics {A, B, C, X(1), X(259)} and {A, B, C, X(6), X(266)}
X(53119) = barycentric product X(i)*X(j) for these {i, j}: {1, 7028}, {8, 289}, {9, 1488}, {63, 8121}, {92, 8134}, {188, 258}
X(53119) = barycentric quotient X(i)/X(j) for these (i, j): (41, 53118), (55, 236), (258, 4146), (259, 7057), (266, 18886)
X(53119) = trilinear product X(i)*X(j) for these {i, j}: {3, 8121}, {4, 8134}, {6, 7028}, {9, 289}, {55, 1488}, {258, 259}
X(53119) = trilinear quotient X(i)/X(j) for these (i, j): (9, 236), (33, 8122), (55, 53118), (174, 18886), (188, 7057)
X(53119) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 258, 15997), (1, 266, 53118), (1, 10231, 266), (1, 18291, 8078), (188, 8242, 6732)

---

X(53120) = ISOTOMIC CONJUGATE OF X(483)

Barycentrics    b*c*(1+cos(A/2)) : :

Contributed by César Eliud Lozada, March 20, 2023.

X(53120) lies on these lines: {7, 1274}, {75, 234}, {1143, 32087}, {4146, 46892}, {21456, 53077}

X(53120) = isotomic conjugate of X(483)
X(53120) = X(i)-Dao conjugate of-X(j) for these (i, j): (9, 7014), (188, 32576), (223, 53117), (557, 3645)
X(53120) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 7014}, {41, 558}, {55, 53117}
X(53120) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 7014), (7, 558), (57, 53117), (174, 7010), (236, 32576)
X(53120) = intersection, other than A, B, C, of circumconics {A, B, C, X(7), X(555)} and {A, B, C, X(234), X(557)}
X(53120) = barycentric product X(i)*X(j) for these {i, j}: {75, 557}, {76, 53116}, {85, 3082}, {1274, 4146}
X(53120) = barycentric quotient X(i)/X(j) for these (i, j): (1, 7014), (7, 558), (57, 53117), (174, 7010), (236, 32576)
X(53120) = trilinear product X(i)*X(j) for these {i, j}: {2, 557}, {7, 3082}, {75, 53116}, {174, 1274}
X(53120) = trilinear quotient X(i)/X(j) for these (i, j): (2, 7014), (7, 53117), (85, 558), (557, 6), (1274, 259)

---

X(53121) = ISOTOMIC CONJUGATE OF X(3082)

Barycentrics    b*c*(1-cos(A/2)) : :

Contributed by César Eliud Lozada, March 20, 2023.

X(53121) lies on these lines: {7, 1143}, {75, 234}, {1274, 32087}, {4146, 46891}, {21456, 53076}

X(53121) = isotomic conjugate of X(3082)
X(53121) = X(i)-Dao conjugate of-X(j) for these (i, j): (188, 53071), (223, 53116)
X(53121) = X(i)-isoconjugate-of-X(j) for these {i, j}: {41, 557}, {55, 53116}
X(53121) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (7, 557), (57, 53116), (174, 7001), (236, 53071)
X(53121) = intersection, other than A, B, C, of circumconics {A, B, C, X(7), X(555)} and {A, B, C, X(234), X(558)}
X(53121) = barycentric product X(i)*X(j) for these {i, j}: {75, 558}, {76, 53117}, {85, 483}, {1143, 4146}
X(53121) = barycentric quotient X(i)/X(j) for these (i, j): (7, 557), (57, 53116), (174, 7001), (236, 53071), (483, 9)
X(53121) = trilinear product X(i)*X(j) for these {i, j}: {2, 558}, {7, 483}, {75, 53117}, {85, 7014}, {174, 1143}
X(53121) = trilinear quotient X(i)/X(j) for these (i, j): (7, 53116), (85, 557), (483, 55), (558, 6), (1143, 259)

---

X(53122) = ISOTOMIC CONJUGATE OF X(1488)

Barycentrics    b*c*(1+sin(A/2)) : :

Contributed by César Eliud Lozada, March 20, 2023.

X(53122) lies on these lines: {2, 5430}, {8, 177}, {10, 10504}, {75, 4146}, {178, 312}, {7048, 31995}, {53076, 53077}

X(53122) = isotomic conjugate of X(1488)
X(53122) = X(75)-Ceva conjugate of-X(556)
X(53122) = X(178)-cross conjugate of-X(7057)
X(53122) = X(i)-Dao conjugate of-X(j) for these (i, j): (1, 53119), (9, 289), (178, 15997), (188, 1), (236, 258)
X(53122) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 289}, {34, 8134}, {56, 53119}, {603, 8121}, {604, 7028}
X(53122) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 289), (8, 7028), (9, 53119), (173, 266), (177, 41799)
X(53122) = intersection, other than A, B, C, of circumconics {A, B, C, X(1), X(8093)} and {A, B, C, X(7), X(9793)}
X(53122) = barycentric product X(i)*X(j) for these {i, j}: {75, 236}, {76, 53118}, {304, 8122}, {312, 2089}, {556, 7057}, {1969, 8139}
X(53122) = barycentric quotient X(i)/X(j) for these (i, j): (1, 289), (8, 7028), (9, 53119), (173, 266), (177, 41799), (178, 16015)
X(53122) = trilinear product X(i)*X(j) for these {i, j}: {2, 236}, {8, 2089}, {69, 8122}, {75, 53118}, {173, 556}, {188, 7057}
X(53122) = trilinear quotient X(i)/X(j) for these (i, j): (2, 289), (8, 53119), (78, 8134), (178, 16011), (236, 6), (312, 7028)
X(53122) = {X(75), X(4146)}-harmonic conjugate of X(53123)

---

X(53123) = ISOTOMIC CONJUGATE OF X(2089)

Barycentrics    b*c*(1-sin(A/2)) : :

Contributed by César Eliud Lozada, March 20, 2023.

X(53123) lies on these lines: {2, 556}, {8, 7048}, {75, 4146}, {7027, 44720}, {7057, 31995}

X(53123) = isotomic conjugate of X(2089)
X(53123) = X(8)-cross conjugate of-X(556)
X(53123) = X(i)-Dao conjugate of-X(j) for these (i, j): (1, 53118), (188, 52999), (236, 173)
X(53123) = X(i)-isoconjugate-of-X(j) for these {i, j}: {34, 8139}, {56, 53118}, {236, 604}, {266, 42622}
X(53123) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (8, 236), (9, 53118), (188, 173), (219, 8139)
X(53123) = barycentric product X(i)*X(j) for these {i, j}: {75, 7028}, {76, 53119}, {289, 3596}, {304, 8121}, {312, 1488}, {556, 7048}
X(53123) = barycentric quotient X(i)/X(j) for these (i, j): (8, 236), (9, 53118), (188, 173), (219, 8139), (236, 52999)
X(53123) = trilinear product X(i)*X(j) for these {i, j}: {2, 7028}, {8, 1488}, {69, 8121}, {75, 53119}, {188, 7048}, {258, 556}
X(53123) = trilinear quotient X(i)/X(j) for these (i, j): (8, 53118), (78, 8139), (188, 42622), (289, 604), (312, 236)
X(53123) = {X(75), X(4146)}-harmonic conjugate of X(53122)

---

X(53124) = X(3)X(5645)∩X(381)X(14094)

Barycentrics    a^2*(4*a^8 - 20*a^6*b^2 + 36*a^4*b^4 - 28*a^2*b^6 + 8*b^8 - 20*a^6*c^2 + 23*a^4*b^2*c^2 + 39*a^2*b^4*c^2 - 42*b^6*c^2 + 36*a^4*c^4 + 39*a^2*b^2*c^4 + 68*b^4*c^4 - 28*a^2*c^6 - 42*b^2*c^6 + 8*c^8) : : X(53124) = X[3] - 6 X[5645], X[3] + 6 X[14491]

X(53124) lies on these lines: {3, 5645}, {381, 14094}, {382, 51993}, {399, 52163}, {575, 51797}, {576, 599}, {3066, 15039}, {3526, 5643}, {5072, 14852}, {5079, 13565}, {5476, 15027}, {5640, 11935}, {5692, 10222}, {7545, 14530}, {12834, 15693}, {15023, 15033}, {15038, 47391}, {18383, 34780}, {45034, 53093}, {51522, 52055}

X(53124) = midpoint of X(5645) and X(14491)

---

X(53125) = X(2)X(1341)∩X(5)X(52096)

Barycentrics    a^4 - 4*a^2*b^2 + 3*b^4 - 4*a^2*c^2 - 8*b^2*c^2 + 3*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(43125) lies on the Steiner major axis and these lines: {2, 1341}, {5, 52096}, {69, 5071}, {5055, 6189}, {32832, 52095}, {47367, 51523}

---

X(53126) = X(2)X(1340)∩X(5)X(52095)

Barycentrics    a^4 - 4*a^2*b^2 + 3*b^4 - 4*a^2*c^2 - 8*b^2*c^2 + 3*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(43126) lies on the Steiner major axis and these lines: {2, 1340}, {5, 52095}, {69, 5071}, {5055, 6190}, {32832, 52096}, {47368, 51523}

---

X(53127) = X(2)X(99)∩X(5)X(183)

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

X(53127) lies on these lines: {2, 99}, {4, 7771}, {5, 183}, {20, 15031}, {32, 32962}, {39, 32999}, {69, 5071}, {76, 1007}, {83, 32987}, {187, 33016}, {230, 44543}, {274, 6931}, {302, 42910}, {303, 42911}, {316, 3545}, {317, 7577}, {325, 5055}, {381, 14907}, {385, 31415}, {547, 32833}, {598, 1285}, {625, 16990}, {626, 32963}, {1003, 3054}, {1078, 3091}, {1506, 7798}, {1656, 7763}, {1975, 3628}, {2548, 7766}, {2896, 33011}, {3055, 31859}, {3096, 32972}, {3146, 43459}, {3329, 3767}, {3363, 8860}, {3523, 32883}, {3525, 7782}, {3526, 32819}, {3544, 32006}, {3785, 5068}, {3788, 32998}, {3832, 7802}, {3851, 7750}, {3855, 52718}, {3926, 7486}, {3933, 35018}, {3934, 32961}, {3972, 32983}, {5056, 7752}, {5067, 7769}, {5206, 14068}, {5475, 17008}, {5569, 52942}, {5939, 38224}, {6031, 7533}, {6292, 33283}, {6390, 15699}, {6680, 33269}, {6933, 18140}, {7571, 34254}, {7603, 7774}, {7736, 14568}, {7737, 17004}, {7746, 7804}, {7747, 32995}, {7748, 33001}, {7749, 14035}, {7756, 33012}, {7760, 31404}, {7761, 33006}, {7785, 33010}, {7786, 32975}, {7788, 10109}, {7791, 39565}, {7792, 14535}, {7793, 33024}, {7795, 32967}, {7796, 32834}, {7799, 34803}, {7800, 32966}, {7803, 13881}, {7808, 33261}, {7809, 15589}, {7811, 32827}, {7812, 37667}, {7815, 14063}, {7816, 33000}, {7822, 33248}, {7828, 32968}, {7830, 32996}, {7831, 16041}, {7832, 32969}, {7845, 8176}, {7847, 32978}, {7857, 32971}, {7859, 32957}, {7862, 33270}, {7867, 33277}, {7872, 33258}, {7886, 16898}, {7899, 32988}, {7911, 32980}, {7918, 32960}, {7921, 31417}, {7930, 32955}, {7934, 32984}, {7935, 33290}, {7937, 33285}, {7940, 32976}, {7942, 16045}, {7944, 33199}, {8370, 37637}, {8588, 33193}, {8667, 18584}, {8797, 14615}, {9744, 12188}, {9754, 35930}, {10303, 32826}, {11057, 41106}, {11163, 16509}, {11174, 43291}, {11261, 18906}, {11317, 15597}, {11361, 17006}, {12042, 37348}, {12150, 37689}, {12815, 14037}, {14561, 39099}, {15022, 32816}, {15271, 33228}, {15484, 22329}, {15513, 33280}, {15515, 33188}, {16589, 33052}, {16922, 31401}, {16992, 17533}, {17005, 34511}, {18424, 33017}, {19687, 44535}, {20112, 35955}, {21358, 51438}, {21395, 47485}, {23053, 26613}, {31489, 47286}, {32829, 46936}, {32839, 46935}, {32956, 39143}, {33003, 37512}, {33220, 44381}, {33273, 43619}, {35705, 49102}

X(53127) = isotomic conjugate of X(44658)
X(53127) = barycentric quotient X(2)/X(44658)
X(53127) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 32832, 315}, {183, 7773, 14929}, {381, 37688, 14907}, {3091, 32838, 1078}, {3545, 34229, 316}, {5056, 32828, 7752}, {5068, 32870, 3785}, {7761, 39601, 33006}, {11361, 17006, 21843}, {13881, 32992, 7803}, {17004, 33013, 7737}, {17008, 33005, 5475}, {34803, 52713, 7799}

---

X(53128) = X(2)X(2112)∩X(41)X(43)

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

X(53128) lies on the cubic K131 and these lines: {2, 2112}, {21, 22230}, {31, 893}, {35, 20684}, {41, 43}, {42, 172}, {48, 4386}, {55, 19561}, {57, 77}, {171, 19554}, {182, 20665}, {444, 1973}, {612, 2187}, {748, 18904}, {750, 19559}, {846, 8847}, {1468, 22199}, {2268, 2312}, {2298, 21840}, {4224, 17451}, {4645, 40597}, {7293, 23636}, {27963, 30660}, {27982, 30661}

X(53128) = isogonal conjugate of the isotomic conjugate of X(17739)
X(53128) = X(i)-Ceva conjugate of X(j) for these (i,j): {171, 31}, {19554, 2112}
X(53128) = X(i)-isoconjugate of X(j) for these (i,j): {6, 18760}, {7, 40792}, {75, 18784}, {335, 16366}, {7179, 40771}
X(53128) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 18760}, {206, 18784}, {893, 7018}
X(53128) = crossdifference of every pair of points on line {4041, 21196}
X(53128) = barycentric product X(i)*X(j) for these {i,j}: {1, 8424}, {6, 17739}, {9, 40765}, {31, 30660}, {55, 40723}, {75, 18759}, {893, 27963}, {2344, 40797}, {17798, 39920}
X(53128) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 18760}, {32, 18784}, {41, 40792}, {2210, 16366}, {8424, 75}, {17739, 76}, {18759, 1}, {27963, 1920}, {30660, 561}, {39920, 18036}, {40723, 6063}, {40765, 85}
X(53128) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 51947, 2112}, {893, 1691, 31}
/p>

---

X(53129) = X(1)X(20460)∩X(2)X(7)

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

X(53129) lies on the cubic K131 and these lines: {1, 20460}, {2, 7}, {6, 20284}, {8, 21387}, {31, 172}, {37, 1755}, {38, 1107}, {41, 11328}, {42, 694}, {43, 19587}, {55, 19586}, {81, 2311}, {171, 8932}, {190, 41318}, {192, 7075}, {213, 3229}, {218, 22149}, {228, 237}, {511, 20684}, {604, 22163}, {649, 7641}, {846, 1334}, {982, 20459}, {1193, 3117}, {1200, 1282}, {1201, 30646}, {1424, 27340}, {1475, 32913}, {1696, 21001}, {2209, 21775}, {2227, 21877}, {2300, 3051}, {2309, 3116}, {2329, 11688}, {3290, 24511}, {3501, 16557}, {3508, 7081}, {3666, 40972}, {3741, 21369}, {3819, 25061}, {3840, 20372}, {3955, 19554}, {4640, 39258}, {8844, 17792}, {9575, 17474}, {11031, 21808}, {14096, 22060}, {17439, 17469}, {17443, 17471}, {17459, 17475}, {17752, 30661}, {20995, 34247}, {21757, 23533}, {21838, 23439}, {23543, 23579}, {23636, 23988}, {29818, 38346}, {36263, 36808}, {51319, 51921}

X(53129) = isogonal conjugate of the isotomic conjugate of X(17760)
X(53129) = X(i)-Ceva conjugate of X(j) for these (i,j): {171, 42}, {28391, 41350}, {41531, 3009}, {51858, 672}
X(53129) = X(i)-isoconjugate of X(j) for these (i,j): {6, 18299}, {57, 39924}, {40758, 41527}
X(53129) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 18299}, {5452, 39924}, {17792, 3061}, {52651, 7018}
X(53129) = crossdifference of every pair of points on line {663, 3835}
X(53129) = barycentric product X(i)*X(j) for these {i,j}: {1, 17792}, {6, 17760}, {8, 41350}, {9, 28391}, {31, 51861}, {43, 52211}, {75, 18758}, {190, 45902}, {291, 8844}, {518, 52146}, {894, 45240}, {2176, 27436}, {6382, 18269}, {7077, 39919}, {17754, 40785}
X(53129) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 18299}, {55, 39924}, {8844, 350}, {17760, 76}, {17792, 75}, {18269, 2162}, {18758, 1}, {27436, 6383}, {28391, 85}, {39919, 18033}, {41350, 7}, {45240, 257}, {45902, 514}, {51861, 561}, {52146, 2481}, {52211, 6384}
X(53129) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {63, 5364, 672}, {2176, 52655, 3009}, {21757, 23533, 23578}

---

X(53130) = X(2)X(1328)∩X(371)X(376)

Barycentrics    8*a^4 - 7*a^2*b^2 - b^4 - 7*a^2*c^2 + 2*b^2*c^2 - c^4 - 6*a^2*S : :
X(53130) = X[485] - 4 X[1151], 3 X[485] - 2 X[1327], and many more

X(53130) lies on the cubic K1192 and these lines: {2, 1328}, {3, 9681}, {4, 9680}, {5, 43254}, {6, 8703}, {15, 36468}, {16, 36450}, {20, 6453}, {30, 485}, {371, 376}, {372, 10304}, {381, 5418}, {382, 41963}, {486, 549}, {491, 11057}, {524, 13712}, {541, 10819}, {543, 11835}, {546, 6488}, {547, 23261}, {548, 3592}, {550, 6425}, {590, 3830}, {591, 7618}, {615, 6451}, {631, 10194}, {1124, 9662}, {1131, 43515}, {1152, 34200}, {1335, 9649}, {1587, 9543}, {1588, 15692}, {1657, 6519}, {1991, 47102}, {2043, 42158}, {2044, 42157}, {3068, 6480}, {3069, 15698}, {3070, 6407}, {3071, 5054}, {3146, 35812}, {3241, 35610}, {3311, 15688}, {3312, 14093}, {3522, 6419}, {3524, 5420}, {3526, 43381}, {3528, 6420}, {3529, 8960}, {3533, 43433}, {3534, 6221}, {3543, 6484}, {3545, 6486}, {3594, 33923}, {3627, 10147}, {3655, 49226}, {3679, 9582}, {3839, 10576}, {3845, 6433}, {3850, 43885}, {5023, 19102}, {5055, 42268}, {5059, 6482}, {5066, 8253}, {5073, 43879}, {5861, 26615}, {6199, 15695}, {6278, 13692}, {6337, 13701}, {6396, 19053}, {6410, 45759}, {6411, 12100}, {6412, 15759}, {6426, 46853}, {6429, 7583}, {6432, 41982}, {6437, 15690}, {6438, 46332}, {6447, 15696}, {6454, 21735}, {6462, 13678}, {6468, 19710}, {6476, 23267}, {6487, 42523}, {6496, 15700}, {6564, 15682}, {7582, 15710}, {7584, 17504}, {7585, 15697}, {7739, 12963}, {7753, 9674}, {7811, 32809}, {8252, 11812}, {8591, 35824}, {8972, 15640}, {8976, 15684}, {9143, 35826}, {9300, 9600}, {9542, 23249}, {9602, 19100}, {9615, 31162}, {9616, 50811}, {9647, 11237}, {9660, 11238}, {9678, 49732}, {9682, 9909}, {9683, 14070}, {9690, 13665}, {9693, 17538}, {9694, 35776}, {10141, 43258}, {10299, 35813}, {10385, 35768}, {10514, 13812}, {10577, 15702}, {10653, 42201}, {10654, 15764}, {10817, 49222}, {11155, 12158}, {11160, 39893}, {11177, 35878}, {11539, 42262}, {12123, 43120}, {12124, 12313}, {12818, 31412}, {12819, 47478}, {13644, 38425}, {13651, 49260}, {13674, 48735}, {13785, 15701}, {13834, 49261}, {13903, 42272}, {13911, 28208}, {13935, 15705}, {13951, 15707}, {13966, 14891}, {14241, 43791}, {14269, 42271}, {14537, 31463}, {14651, 35699}, {14830, 49266}, {14869, 43571}, {14890, 43559}, {14893, 42568}, {15683, 35820}, {15687, 42265}, {15689, 42259}, {15703, 42270}, {15704, 43342}, {15706, 43431}, {15709, 42561}, {15711, 35256}, {15712, 43212}, {15713, 18762}, {15714, 19116}, {15716, 18510}, {15719, 23273}, {15722, 42573}, {18289, 31152}, {18538, 33699}, {19101, 53095}, {19709, 42283}, {22806, 49790}, {32785, 41099}, {34551, 42150}, {34552, 42151}, {34628, 49601}, {34632, 35641}, {35018, 42642}, {35771, 42637}, {35774, 50808}, {35775, 51705}, {35786, 42413}, {35787, 43558}, {35788, 50864}, {35810, 50872}, {35815, 46333}, {36436, 42245}, {36437, 41944}, {36445, 52214}, {36446, 42219}, {36449, 41100}, {36454, 42243}, {36455, 41943}, {36463, 52217}, {36464, 42220}, {36467, 41101}, {38335, 42273}, {38730, 49214}, {41106, 52666}, {41134, 50719}, {41957, 42527}, {42227, 42934}, {42229, 42935}, {42254, 43100}, {42255, 43107}, {42267, 43386}, {42284, 43887}, {42601, 43792}, {42643, 43384}, {43432, 49139}, {43513, 43522}, {44526, 49262}, {45023, 48780}, {45420, 51224}

X(53130) = reflection of X(i) in X(j) for these {i,j}: {1327, 13846}, {6278, 13692}, {13674, 48735}, {32810, 13701}
X(53130) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1328, 42274}, {2, 6561, 1328}, {371, 51910, 7581}, {381, 6449, 52045}, {381, 52045, 5418}, {486, 549, 43255}, {590, 43210, 3830}, {1151, 42260, 485}, {1327, 13846, 485}, {1657, 6519, 31454}, {3071, 5054, 42603}, {3311, 15688, 41946}, {3524, 6459, 35823}, {3524, 35823, 5420}, {3529, 43521, 42538}, {3534, 6221, 32787}, {3534, 32787, 6560}, {3830, 43210, 42275}, {5418, 42258, 22615}, {6200, 9541, 6561}, {6411, 13847, 12100}, {6433, 42263, 35255}, {6449, 42258, 5418}, {6484, 42266, 9540}, {6564, 15682, 43503}, {7585, 15697, 43256}, {8703, 52047, 6}, {8981, 22644, 485}, {9540, 42266, 42269}, {12100, 42215, 13847}, {15687, 43211, 42265}, {19053, 19708, 6396}, {35255, 42263, 42277}, {41100, 51728, 36449}, {42258, 52045, 381}, {43337, 43568, 33699}

---

X(53131) = X(2)X(1327)∩X(30)X(486)

Barycentrics    8*a^4 - 7*a^2*b^2 - b^4 - 7*a^2*c^2 + 2*b^2*c^2 - c^4 + 6*a^2*S : :
X(53131) = X[486] - 4 X[1152], 3 X[486] - 2 X[1328], 3 X[486] - 4 X[13847], 5 X[486] - 8 X[13966], 13 X[486] - 16 X[13993], 5 X[486] - 2 X[22615], 7 X[486] - 4 X[23261], X[486] + 2 X[42261], 41 X[486] - 44 X[42579], 6 X[1152] - X[1328], 3 X[1152] - X[13847],and many more

X(53131) lies on the cubic K1192 and these lines: {2, 1327}, {3, 9680}, {4, 42603}, {5, 43255}, {6, 8703}, {15, 36449}, {16, 36467}, {20, 6454}, {30, 486}, {371, 10304}, {372, 376}, {381, 5420}, {382, 41964}, {485, 549}, {492, 11057}, {524, 13835}, {541, 10820}, {543, 11836}, {546, 6489}, {547, 23251}, {548, 3594}, {550, 6426}, {590, 6452}, {591, 47102}, {615, 3830}, {631, 10195}, {1132, 43516}, {1151, 34200}, {1587, 15692}, {1588, 51910}, {1657, 6522}, {1991, 7618}, {2043, 42157}, {2044, 42158}, {3068, 15698}, {3069, 6481}, {3070, 5054}, {3071, 6408}, {3146, 35813}, {3241, 35611}, {3311, 14093}, {3312, 15688}, {3522, 6420}, {3524, 5418}, {3526, 43380}, {3528, 6419}, {3529, 42537}, {3533, 43432}, {3534, 6398}, {3543, 6485}, {3545, 6487}, {3592, 33923}, {3627, 10148}, {3655, 49227}, {3839, 10577}, {3845, 6434}, {3850, 43886}, {5023, 19105}, {5055, 42269}, {5059, 6483}, {5066, 8252}, {5073, 43880}, {5860, 26616}, {6200, 19054}, {6281, 13812}, {6337, 13821}, {6395, 15695}, {6409, 45759}, {6411, 15759}, {6412, 12100}, {6425, 46853}, {6430, 7584}, {6431, 41982}, {6437, 46332}, {6438, 15690}, {6448, 15696}, {6453, 21735}, {6463, 13798}, {6469, 19710}, {6477, 23273}, {6486, 42522}, {6497, 15700}, {6565, 15682}, {7581, 15710}, {7583, 17504}, {7586, 15697}, {7739, 12968}, {7811, 32808}, {8253, 11812}, {8591, 35825}, {8960, 15717}, {8976, 15707}, {8981, 14891}, {9143, 35827}, {9540, 15705}, {10142, 43259}, {10299, 35812}, {10385, 35769}, {10515, 13692}, {10576, 15702}, {10653, 15764}, {10654, 42200}, {10818, 49223}, {11155, 12159}, {11160, 39894}, {11177, 35879}, {11539, 42265}, {12123, 12314}, {12124, 43121}, {12818, 47478}, {12819, 42561}, {13665, 15701}, {13711, 49262}, {13763, 38426}, {13770, 49263}, {13785, 15685}, {13794, 48734}, {13941, 15640}, {13951, 15684}, {13961, 42271}, {13973, 28208}, {14226, 43792}, {14269, 42272}, {14651, 35698}, {14830, 49267}, {14869, 43570}, {14890, 43558}, {14893, 42569}, {15683, 35821}, {15687, 42262}, {15689, 42258}, {15703, 42273}, {15704, 17852}, {15706, 43430}, {15709, 31412}, {15711, 35255}, {15712, 43211}, {15713, 18538}, {15714, 19117}, {15716, 18512}, {15719, 23267}, {15722, 42572}, {17525, 31473}, {18290, 31152}, {18762, 33699}, {19709, 42284}, {22541, 53095}, {22807, 49791}, {32786, 41099}, {34551, 42151}, {34552, 42150}, {34628, 49602}, {34632, 35642}, {35018, 42641}, {35739, 36455}, {35770, 42638}, {35774, 51705}, {35775, 50808}, {35786, 43559}, {35787, 42414}, {35789, 50864}, {35811, 50872}, {35814, 46333}, {36436, 42242}, {36437, 41943}, {36445, 52216}, {36447, 42218}, {36450, 41101}, {36454, 42244}, {36463, 52215}, {36465, 42217}, {36468, 41100}, {38335, 42270}, {38730, 49215}, {41106, 52667}, {41134, 50720}, {41958, 42526}, {42228, 42934}, {42230, 42935}, {42256, 43100}, {42257, 43107}, {42266, 43387}, {42283, 43888}, {42600, 43791}, {42644, 43385}, {43433, 49139}, {43514, 43521}, {44526, 49261}, {45024, 48781}, {45421, 51224}

X(53131) = reflection of X(i) in X(j) for these {i,j}: {1328, 13847}, {6281, 13812}, {13794, 48734}, {32811, 13821}
X(53131) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1327, 42277}, {2, 6560, 1327}, {372, 51911, 7582}, {381, 6450, 52046}, {381, 52046, 5420}, {485, 549, 43254}, {615, 43209, 3830}, {1152, 42261, 486}, {1328, 13847, 486}, {3070, 5054, 42602}, {3312, 15688, 41945}, {3522, 6420, 9681}, {3524, 6460, 35822}, {3524, 35822, 5418}, {3529, 43522, 42537}, {3534, 6398, 32788}, {3534, 32788, 6561}, {3830, 43209, 42276}, {5420, 42259, 22644}, {6412, 13846, 12100}, {6434, 42264, 35256}, {6450, 42259, 5420}, {6485, 42267, 13935}, {6565, 15682, 43504}, {7586, 15697, 43257}, {8703, 52048, 6}, {12100, 42216, 13846}, {13935, 42267, 42268}, {13966, 22615, 486}, {15687, 43212, 42262}, {19054, 19708, 6200}, {35256, 42264, 42274}, {42259, 52046, 381}, {43336, 43569, 33699}

---

X(53132) = X(30)X(74)∩X(50)X(67)

Barycentrics    (b - c)^2*(b + c)^2*(-a^2 + b^2 - b*c + c^2)*(-a^2 + b^2 + b*c + 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(53132) lies on the cubic K508 and these lines: {2, 18331}, {30, 74}, {50, 67}, {110, 51872}, {115, 125}, {131, 40948}, {542, 5191}, {1316, 11005}, {1650, 6334}, {1989, 48988}, {3231, 14901}, {3448, 4226}, {5913, 34366}, {11006, 36194}, {12827, 38551}, {14559, 24975}, {15106, 22146}, {18122, 46129}, {32305, 37916}, {32313, 35582}, {34174, 41254}, {35235, 44427}, {35909, 36189}, {38393, 45147}

X(53132) = midpoint of X(3448) and X(4226)
X(53132) = reflection of X(i) in X(j) for these {i,j}: {868, 125}, {14559, 24975}
X(53132) = X(51227)-Ceva conjugate of X(1640)
X(53132) = X(i)-isoconjugate of X(j) for these (i,j): {110, 36096}, {662, 23969}, {5649, 32678}
X(53132) = X(i)-Dao conjugate of X(j) for these (i,j): {244, 36096}, {526, 52179}, {1084, 23969}, {1637, 51228}, {5664, 5641}, {18334, 5649}, {23967, 39295}, {35582, 14559}
X(53132) = crossdifference of every pair of points on line {110, 14998}
X(53132) = X(51872)-lineconjugate of X(110)
X(53132) = barycentric product X(i)*X(j) for these {i,j}: {526, 18312}, {1640, 3268}, {3258, 51227}, {7799, 51428}
X(53132) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 23969}, {526, 5649}, {542, 39295}, {661, 36096}, {1640, 476}, {2088, 842}, {3258, 51228}, {3268, 6035}, {6041, 14560}, {18312, 35139}, {18334, 52179}, {44114, 34370}, {48451, 15395}, {51428, 1989}, {52743, 51263}

---

X(53133) = X(4)X(514)∩X(69)X(144)

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

X(53133) lies on the cubic K616 and these lines: {2, 45144}, {4, 514}, {69, 144}, {103, 376}, {911, 5773}, {2338, 30809}, {4237, 51607}

X(53133) = X(5513)-Dao conjugate of X(516)
X(53133) = trilinear pole of line {2504, 3011}
X(53133) = barycentric product X(i)*X(j) for these {i,j}: {3011, 18025}, {9028, 52781}
X(53133) = barycentric quotient X(i)/X(j) for these {i,j}: {677, 29241}, {2424, 35365}, {2504, 39470}, {3011, 516}, {9028, 26006}, {29240, 676}

---

X(53134) = X(1)X(6)∩X(2)X(24152)

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

X(53134) lies on the cubics K314, K365, K631, K1082 and these lines: {1, 6}, {2, 24152}, {7, 24155}, {145, 24313}, {3513, 6204}, {3514, 6203}, {3870, 24153}, {7133, 40565}, {40566, 42013}

X(53134) = X(i)-isoconjugate of X(j) for these (i,j): {6, 24155}, {17107, 24152}
X(53134) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 24155}, {220, 24152}, {24152, 8}
X(53134) = barycentric product X(i)*X(j) for these {i,j}: {7, 24153}, {3870, 24154}
X(53134) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 24155}, {6600, 24152}, {24153, 8}

---

X(53135) = X(1)X(6)∩X(2)X(24153)

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

X(53135) lies on the cubics K314, K365, K631, K1082 and these lines: {1, 6}, {2, 24153}, {7, 24154}, {145, 24314}, {3513, 6203}, {3514, 6204}, {3870, 24152}, {7133, 40566}, {40565, 42013}

X(53135) = X(i)-isoconjugate of X(j) for these (i,j): {6, 24154}, {17107, 24153}
X(53135) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 24154}, {220, 24153}, {24153, 8}
X(53135) = barycentric product X(i)*X(j) for these {i,j}: {7, 24152}, {3870, 24155}
X(53135) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 24154}, {6600, 24153}, {24152, 8}

---

X(53136) = X(2)X(523)∩X(30)X(99)

Barycentrics    2*a^10 - 4*a^8*b^2 + 5*a^6*b^4 + 2*a^4*b^6 - 7*a^2*b^8 + 2*b^10 - 4*a^8*c^2 + 2*a^6*b^2*c^2 - 6*a^4*b^4*c^2 + 11*a^2*b^6*c^2 - b^8*c^2 + 5*a^6*c^4 - 6*a^4*b^2*c^4 - 6*a^2*b^4*c^4 - b^6*c^4 + 2*a^4*c^6 + 11*a^2*b^2*c^6 - b^4*c^6 - 7*a^2*c^8 - b^2*c^8 + 2*c^10 : :
X(53136) = 3 X[16092] - 4 X[46980], X[16092] - 4 X[46986], X[46980] - 3 X[46986], X[7840] + 2 X[46992], X[325] + 2 X[16316], 2 X[115] + X[47289], X[385] - 3 X[37907], X[385] - 4 X[47171], 3 X[37907] - 2 X[46998], 3 X[37907] - 4 X[47171], 2 X[5099] + X[47293], and many others

X(53136) lies on the cubic K794 and these lines: {2, 523}, {23, 1634}, {30, 99}, {110, 524}, {111, 44398}, {114, 1551}, {115, 47289}, {376, 46987}, {385, 37907}, {468, 648}, {525, 9144}, {542, 36166}, {543, 5099}, {549, 46633}, {599, 9832}, {620, 47291}, {671, 14120}, {691, 41134}, {858, 10717}, {1316, 11163}, {1499, 11006}, {1992, 47550}, {2453, 11184}, {2482, 7472}, {3524, 46981}, {3543, 46988}, {3839, 46982}, {3849, 47326}, {3906, 5465}, {5642, 14999}, {5912, 10418}, {6055, 16760}, {7664, 9182}, {7665, 44373}, {7736, 48721}, {7779, 37909}, {7801, 36165}, {7812, 36156}, {7870, 38526}, {8591, 36174}, {8705, 33873}, {8859, 47239}, {9164, 13574}, {9166, 51258}, {9167, 40544}, {9717, 47597}, {11160, 47565}, {14355, 50979}, {15993, 47169}, {16308, 32526}, {23234, 36170}, {32112, 51227}, {32515, 44266}, {36180, 51224}, {37760, 44367}, {37901, 41136}, {40879, 47596}, {41133, 47097}, {44369, 47551}, {44401, 47242}, {45311, 51428}, {47049, 50977}, {47496, 50254}

X(53136) = midpoint of X(i) and X(j) for these {i,j}: {23, 7840}, {671, 47288}, {842, 6054}, {8591, 36174}, {22110, 47245}, {36196, 47293}, {47097, 47154}
X(53136) = reflection of X(i) in X(j) for these {i,j}: {2, 46986}, {23, 46992}, {376, 46987}, {385, 46998}, {599, 47557}, {671, 14120}, {858, 22110}, {1551, 114}, {1992, 47550}, {3543, 46988}, {6055, 16760}, {7426, 16320}, {7472, 2482}, {11160, 47565}, {14999, 5642}, {15993, 47556}, {16092, 2}, {22329, 468}, {36196, 5099}, {44369, 47551}, {46633, 549}, {46998, 47171}, {47242, 44401}, {50254, 47496}, {51224, 36180}, {51428, 45311}
X(53136) = anticomplement of X(46980)
X(53136) = reflection of X(16092) in the Euler line
X(53136) = orthoptic-circle-of-the-Steiner-inellipse-inverse of X(1649)
X(53136) = orthoptic-circle-of-the-Steiner-circumellipse-inverse of X(44010)
X(53136) = crossdifference of every pair of points on line {187, 6041}
X(53136) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {385, 37907, 46998}, {10418, 23992, 5912}, {46998, 47171, 37907}

---

X(53137) = X(3)X(5627)∩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 - 3*a^2*b^4*c^2 - b^6*c^2 - 6*a^4*c^4 - 3*a^2*b^2*c^4 + 6*b^4*c^4 + 6*a^2*c^6 - b^2*c^6 - 2*c^8) : :

X(53137) lies on the cubic K850 and these lines: {3, 5627}, {4, 14993}, {5, 523}, {20, 265}, {382, 476}, {546, 14583}, {548, 51254}, {1658, 52153}, {3146, 51345}, {3855, 52449}, {5067, 51835}, {7814, 35139}, {8918, 10217}, {8919, 10218}, {33703, 52056}

X(53137) = X(6149)-isoconjugate of X(20480)
X(53137) = X(14993)-Dao conjugate of X(20480)
X(53137) = barycentric quotient X(1989)/X(20480)
X(53137) = {X(34209),X(39170)}-harmonic conjugate of X(14254)

---

X(53138) = X(4)X(1689)∩X(232)X(511)

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

X(53138) lies on the cubics K337, K1096, K1300, and these lines: {4, 1689}, {112, 1687}, {232, 511}, {1688, 19128}

X(53138) = polar-circle-inverse of X(2010)
X(53138) = X(293)-isoconjugate of X(2010)
X(53138) = X(132)-Dao conjugate of X(2010)
X(53138) = barycentric product X(297)*X(1688)
X(53138) = barycentric quotient X(i)/X(j) for these {i,j}: {232, 2010}, {1688, 287}

---

X(53139) = X(4)X(1690)∩X(232)X(511)

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

X(53139) lies on the cubics K337, K1096, K1300, and these lines: {4, 1690}, {112, 1688}, {232, 511}, {1687, 19128}
on K337, K1096, K1300

X(53139) = polar-circle-inverse of X(2009)
X(53139) = X(293)-isoconjugate of X(2009)
X(53139) = X(132)-Dao conjugate of X(2009)
X(53139) = barycentric product X(297)*X(1687)
X(53139) = barycentric quotient X(i)/X(j) for these {i,j}: {232, 2009}, {1687, 287}

---

X(53140) = X(4)X(80)∩X(109)X(191)

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

X(53140) lis on the cubic K720 and these lines: {4, 80}, {12, 10693}, {109, 191}, {219, 478}, {758, 51421}, {2817, 11827}, {6358, 13532}, {11700, 16577}

X(53140) = X(5080)-Ceva conjugate of X(12)
X(53140) = barycentric product X(321)*X(51970)
X(53140) = barycentric quotient X(51970)/X(81)

---

X(53141) = X(2)X(99) ∩ X(20)X(524)

Barycentrics    19*a^4-22*(b^2+c^2)*a^2-(5*b^2-c^2)*(b^2-5*c^2) : :
X(53141) = 5*X(2)-4*X(7615) = 9*X(2)-8*X(7617) = 3*X(2)-4*X(7618) = 15*X(2)-16*X(7619) = 7*X(2)-8*X(7622) = 11*X(2)-8*X(18546) = 9*X(7615)-10*X(7617) = 3*X(7615)-5*X(7618) = 3*X(7615)-4*X(7619) = 6*X(7615)-5*X(7620) = 7*X(7615)-10*X(7622) = 11*X(7615)-10*X(18546) = 2*X(7617)-3*X(7618) = 5*X(7617)-6*X(7619) = 4*X(7617)-3*X(7620) = 7*X(7617)-9*X(7622) = 11*X(7617)-9*X(18546) = 5*X(7618)-4*X(7619) = 7*X(7618)-6*X(7622) = 11*X(7618)-6*X(18546)

See Kadir Altintas and César Lozada, euclid 5847.

X(53141) lies on these lines: {2, 99}, {3, 5485}, {4, 11165}, {20, 524}, {30, 9741}, {193, 9855}, {376, 9740}, {530, 36761}, {531, 41458}, {538, 22676}, {631, 16509}, {1975, 21356}, {2996, 33274}, {3146, 7843}, {3522, 7780}, {3523, 34505}, {3524, 40727}, {3543, 8716}, {3545, 12040}, {3832, 8176}, {3839, 11184}, {3849, 15683}, {4297, 17132}, {5032, 7839}, {5059, 7781}, {5077, 32817}, {5304, 19661}, {5335, 36775}, {5503, 23698}, {5569, 15705}, {5860, 43210}, {5861, 43209}, {5921, 9830}, {6337, 41133}, {6392, 33208}, {6776, 12117}, {7610, 15692}, {7738, 47352}, {7756, 32840}, {7764, 50690}, {7775, 17578}, {7779, 52943}, {7827, 33201}, {7830, 32880}, {7833, 32830}, {7870, 33200}, {7883, 32824}, {8359, 32822}, {8859, 35287}, {9766, 15640}, {10992, 11179}, {11001, 51122}, {11159, 37665}, {11180, 23235}, {13666, 26288}, {13786, 26289}, {15022, 47617}, {15589, 35955}, {15597, 15708}, {15682, 51123}, {21309, 33690}, {27088, 37689}, {32835, 33006}, {32874, 33008}, {32893, 33273}, {33192, 41136}, {38675, 52484}, {39785, 43619}

X(53141) = midpoint of X(20) and X(11148)
X(53141) = reflection of X(i) in X(j) for these (i, j): (4, 11165), (3146, 23334), (3543, 9770), (5485, 3), (7620, 7618), (8182, 34504), (9740, 376), (9770, 8716), (23334, 34511)
X(53141) = anticomplement of X(7620)
X(53141) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2482, 43448, 2), (7615, 7618, 7619), (7618, 7620, 2)

---

X(53142) = X(2)X(99) ∩ X(376)X(524)

Barycentrics    11*a^4-14*(b^2+c^2)*a^2-b^4+10*b^2*c^2-c^4 : :
X(53142) = 5*X(2)-4*X(7617) = 7*X(2)-8*X(7619) = 3*X(2)-4*X(7622) = 7*X(2)-4*X(18546) = 5*X(7615)-6*X(7617) = X(7615)-3*X(7618) = 7*X(7615)-12*X(7619) = 4*X(7615)-3*X(7620) = 7*X(7615)-6*X(18546) = 2*X(7617)-5*X(7618) = 7*X(7617)-10*X(7619) = 8*X(7617)-5*X(7620) = 3*X(7617)-5*X(7622) = 7*X(7617)-5*X(18546) = 7*X(7618)-4*X(7619) = 4*X(7618)-X(7620) = 3*X(7618)-2*X(7622) = 7*X(7618)-2*X(18546) = 16*X(7619)-7*X(7620) = 6*X(7619)-7*X(7622)

See Kadir Altintas and César Lozada, euclid 5847.

X(53142) lies on these lines: {2, 99}, {3, 33850}, {4, 11184}, {20, 3849}, {30, 7710}, {69, 35955}, {183, 47061}, {193, 51224}, {194, 22564}, {376, 524}, {381, 12040}, {538, 6194}, {549, 40727}, {591, 26615}, {599, 32817}, {631, 15597}, {698, 33750}, {1007, 8352}, {1153, 15708}, {1285, 8584}, {1975, 33215}, {1991, 26616}, {1992, 8598}, {3146, 7775}, {3522, 7781}, {3524, 5485}, {3534, 51123}, {3543, 32479}, {3545, 9771}, {3618, 35954}, {3767, 5215}, {3839, 8176}, {3926, 7833}, {4235, 40138}, {5032, 5052}, {5054, 16509}, {5056, 47617}, {5059, 7764}, {5071, 20112}, {5077, 6390}, {5286, 32985}, {5304, 32456}, {5334, 9761}, {5335, 9763}, {5503, 9744}, {5569, 15692}, {5969, 7709}, {6337, 7841}, {7735, 27088}, {7736, 11159}, {7738, 8369}, {7751, 21734}, {7756, 32831}, {7758, 50693}, {7765, 33205}, {7774, 9855}, {7777, 52942}, {7782, 26613}, {7783, 33007}, {7789, 33230}, {7799, 33272}, {7827, 32973}, {7828, 51579}, {7830, 32840}, {7843, 49140}, {7863, 33025}, {7870, 32974}, {7883, 33023}, {7896, 32879}, {7946, 33243}, {8356, 21356}, {8370, 31400}, {8597, 32827}, {8667, 19708}, {8703, 51122}, {9766, 11001}, {9830, 11180}, {10008, 32833}, {10513, 14148}, {10653, 36775}, {11054, 37667}, {11147, 35297}, {11160, 14907}, {11164, 14033}, {11168, 52713}, {13085, 32522}, {13468, 15698}, {13571, 33214}, {14039, 47352}, {15515, 32834}, {15533, 44541}, {15697, 47102}, {15717, 34506}, {15815, 32822}, {16041, 41133}, {16508, 33813}, {22110, 44526}, {23055, 47286}, {30227, 40884}, {31173, 43619}, {32474, 49792}, {32816, 33192}, {32818, 44519}, {32820, 33226}, {32821, 33247}, {32824, 32965}, {32825, 32997}, {32826, 33013}, {32829, 33006}, {32836, 33008}, {32837, 33017}, {33190, 53033}, {33244, 34604}, {33264, 41136}, {33270, 51238}, {33273, 46951}, {33458, 49922}, {33459, 49921}, {34229, 47287}, {35931, 42119}, {35932, 42120}, {35940, 37765}, {37668, 39785}, {39662, 52281}, {41139, 50571}

X(53142) = midpoint of X(i) and X(j) for these {i, j}: {376, 9741}, {5503, 12117}, {9740, 11148}
X(53142) = reflection of X(i) in X(j) for these (i, j): (2, 7618), (4, 11184), (381, 12040), (5485, 7610), (7615, 7622), (7620, 2), (9740, 8182), (9741, 8716), (9770, 11165), (9877, 2482), (16508, 33813), (18546, 7619), (23334, 9770), (34505, 15597), (40727, 549)
X(53142) = anticomplement of X(7615)
X(53142) = circumperp conjugate of X(34014)
X(53142) = inverse of X(32815) in Steiner-Wallace hyperbola
X(53142) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 8591, 32815), (2482, 2549, 2), (3524, 5485, 7610), (7615, 7618, 7622), (7615, 7622, 2), (8598, 31859, 1992), (9740, 10304, 8182), (10304, 11148, 9740), (32480, 52695, 2), (34504, 34511, 20)

---

X(53143) = X(2)X(99) ∩ X(382)X(524)

Barycentrics    17*a^4-14*(b^2+c^2)*a^2-13*b^4+46*b^2*c^2-13*c^4 : :
X(53143) = 4*X(2)-5*X(7615) = 9*X(2)-10*X(7617) = 6*X(2)-5*X(7618) = 3*X(2)-5*X(7620) = 11*X(2)-10*X(7622) = 7*X(2)-10*X(18546) = 9*X(7615)-8*X(7617) = 3*X(7615)-2*X(7618) = 3*X(7615)-4*X(7620) = 11*X(7615)-8*X(7622) = 7*X(7615)-8*X(18546) = 4*X(7617)-3*X(7618) = 7*X(7617)-6*X(7619) = 2*X(7617)-3*X(7620) = 11*X(7617)-9*X(7622) = 7*X(7617)-9*X(18546) = 7*X(7618)-8*X(7619) = 11*X(7618)-12*X(7622) = 7*X(7618)-12*X(18546) = 4*X(7619)-7*X(7620) = 2*X(7619)-3*X(18546)

See Kadir Altintas and César Lozada, euclid 5847.

X(53143) lies on these lines: {2, 99}, {382, 524}, {546, 34511}, {550, 8182}, {1352, 12355}, {3529, 5485}, {3530, 16509}, {3544, 47617}, {3767, 11164}, {3851, 11165}, {3855, 8176}, {3926, 41895}, {4912, 49168}, {5032, 32826}, {5569, 15710}, {7610, 34200}, {7738, 14762}, {7746, 11147}, {7748, 21356}, {7756, 32886}, {7758, 23334}, {7775, 11148}, {7810, 32868}, {8716, 11737}, {10299, 34504}, {11054, 43618}, {11184, 38071}, {12040, 47478}, {15597, 15707}, {15687, 52229}, {15688, 40727}, {43515, 45023}, {43516, 45024}

X(53143) = reflection of X(i) in X(j) for these (i, j): (7618, 7620), (7758, 23334), (8182, 34505), (11148, 7775)
X(53143) = {X(7618), X(7620)}-harmonic conjugate of X(7615)

---

X(53144) = X(2)X(99) ∩ X(524)X(546)

Barycentrics    4*a^4+2*(b^2+c^2)*a^2-11*b^4+32*b^2*c^2-11*c^4 : :
X(53144) = X(2)-5*X(7615) = 3*X(2)-5*X(7617) = 9*X(2)-5*X(7618) = 6*X(2)-5*X(7619) = 3*X(2)+5*X(7620) = 7*X(2)-5*X(7622) = X(2)+5*X(18546) = 3*X(7615)-X(7617) = 9*X(7615)-X(7618) = 6*X(7615)-X(7619) = 3*X(7615)+X(7620) = 7*X(7615)-X(7622) = 3*X(7617)-X(7618) = 7*X(7617)-3*X(7622) = X(7617)+3*X(18546) = 2*X(7618)-3*X(7619) = X(7618)+3*X(7620) = 7*X(7618)-9*X(7622) = X(7618)+9*X(18546) = X(7619)+2*X(7620) = 7*X(7619)-6*X(7622) = X(7619)+6*X(18546)

See Kadir Altintas and César Lozada, euclid 5847.

X(53144) lies on these lines: {2, 99}, {524, 546}, {538, 20112}, {550, 16509}, {754, 14269}, {1153, 3530}, {3363, 32457}, {3529, 8182}, {3544, 34511}, {3849, 15687}, {3851, 7764}, {3855, 5485}, {5079, 11165}, {5254, 14762}, {5569, 15688}, {5965, 40277}, {7610, 15681}, {7751, 23334}, {7816, 41139}, {7825, 32868}, {11054, 43457}, {11737, 52229}, {15597, 17504}, {15720, 34504}, {32828, 41895}, {39565, 41133}

X(53144) = midpoint of X(i) and X(j) for these {i, j}: {5485, 7775}, {7615, 18546}, {7617, 7620}, {7751, 23334}, {8176, 34505}
X(53144) = reflection of X(i) in X(j) for these (i, j): (7619, 7617), (7764, 8176), (34506, 16509)
X(53144) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i, j, k): (7615, 7620, 7617), (7617, 18546, 7620)

---

X(53145) = BARYCENTRIC SQUARE OF X(43)

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

X(53145) lies on these lines: {2, 16969}, {6, 31}, {43, 2176}, {44, 39250}, {75, 25111}, {100, 1613}, {190, 41840}, {200, 16514}, {213, 42043}, {346, 2238}, {594, 37673}, {612, 16515}, {899, 36647}, {1015, 39966}, {1086, 39741}, {1376, 21001}, {1403, 51973}, {1500, 39967}, {1575, 21769}, {1977, 36614}, {2076, 37577}, {2162, 3550}, {3230, 16569}, {3240, 7109}, {3501, 16606}, {3981, 51377}, {4366, 4383}, {4421, 21792}, {6384, 10027}, {9259, 16059}, {17032, 37674}, {17267, 29976}, {17754, 21785}, {18755, 20855}, {20255, 27264}, {21775, 51921}

X(53145) = X(i)-Ceva conjugate of X(j) for these (i,j): {1016, 52923}, {2209, 2176}
X(53145) = X(i)-isoconjugate of X(j) for these (i,j): {87, 330}, {513, 32039}, {2053, 7209}, {2162, 6384}, {4598, 43931}, {6383, 7121}, {7153, 7155}
X(53145) = X(i)-Dao conjugate of X(j) for these (i,j): {4083, 1086}, {39026, 32039}, {40598, 6383}
X(53145) = crossdifference of every pair of points on line {514, 21128}
X(53145) = barycentric product X(i)*X(j) for these {i,j}: {31, 8026}, {43, 43}, {100, 25142}, {101, 23886}, {192, 2176}, {1016, 40610}, {1403, 27538}, {1423, 3208}, {2209, 6376}, {3971, 38832}, {4083, 52923}, {4110, 41526}, {4595, 20979}, {8640, 36863}, {20691, 27644}
X(53145) = barycentric quotient X(i)/X(j) for these {i,j}: {43, 6384}, {101, 32039}, {192, 6383}, {1423, 7209}, {2176, 330}, {2209, 87}, {3208, 27424}, {8026, 561}, {8640, 43931}, {20971, 52573}, {23886, 3261}, {25142, 693}, {40610, 1086}, {41526, 7153}, {52923, 18830}
X(53145) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 41397, 23538}, {42, 672, 24528}, {100, 1613, 21780}, {902, 23538, 41397}, {1376, 21788, 21001}, {3550, 21760, 2162}

---

X(53146) = BARYCENTRIC SQUARE OF X(87)

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

X(53146) lies on these lines: {1, 2162}, {6, 7121}, {56, 51321}, {86, 26143}, {87, 14823}, {292, 2053}, {330, 3226}, {932, 16969}, {979, 16606}, {2176, 17105}, {2319, 39969}, {20669, 36598}, {21759, 39972}

X(53146) = isotomic conjugate of the complement of X(32033)
X(53146) = X(i)-isoconjugate of X(j) for these (i,j): {6, 8026}, {43, 192}, {100, 23886}, {190, 25142}, {1403, 4110}, {1423, 27538}, {2176, 6376}, {2209, 6382}, {3208, 3212}, {3835, 52923}, {3971, 27644}, {4083, 4595}, {7035, 40610}, {20691, 33296}, {20979, 36863}, {36860, 50491}
X(53146) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 8026}, {8054, 23886}
X(53146) = cevapoint of X(i) and X(j) for these (i,j): {2, 32033}, {6, 41397}, {649, 40610}
X(53146) = crossdifference of every pair of points on line {23886, 25142}
X(53146) = barycentric product X(i)*X(j) for these {i,j}: {87, 87}, {330, 2162}, {649, 32039}, {932, 43931}, {2319, 7153}, {6384, 7121}
X(53146) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 8026}, {87, 6376}, {330, 6382}, {649, 23886}, {667, 25142}, {932, 36863}, {1977, 40610}, {2053, 27538}, {2162, 192}, {2319, 4110}, {6383, 40367}, {7121, 43}, {7153, 30545}, {15373, 22370}, {21759, 20691}, {23493, 3971}, {32039, 1978}, {34071, 4595}, {43931, 20906}
X(53146) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {932, 41396, 16969}, {17105, 51864, 2176}

---

X(53147) = BARYCENTRIC SQUARE OF X(194)

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

X(53147) lies on these line: {2, 6}, {42, 27188}, {159, 10997}, {160, 33014}, {194, 6374}, {670, 8264}, {1084, 38262}, {1502, 20081}, {2998, 3229}, {3552, 32543}, {3613, 33011}, {4027, 52016}, {6379, 36648}, {9306, 36432}, {11333, 22152}, {20105, 30736}, {22343, 27105}, {25273, 51907}, {40858, 40907}

X(53147) = X(1613)-Ceva conjugate of X(194)
X(53147) = X(i)-isoconjugate of X(j) for these (i,j): {2998, 34248}, {3223, 3224}, {18832, 51951}
X(53147) = X(i)-Dao conjugate of X(j) for these (i,j): {76, 40162}, {3221, 1084}, {32746, 2998}
X(53147) = barycentric product X(i)*X(j) for these {i,j}: {194, 194}, {1613, 6374}, {1740, 17149}, {20794, 51843}
X(53147) = barycentric quotient X(i)/X(j) for these {i,j}: {194, 2998}, {1613, 3224}, {1740, 3223}, {6374, 40162}, {17149, 18832}, {20794, 3504}
X(53147) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {194, 32746, 6374}, {670, 8264, 32747}

---

X(53148) = BARYCENTRIC SQUARE OF X(3224)

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

X(53148) lies on these lines: {6, 51951}, {2998, 3225}, {3224, 6375}, {9468, 15389}, {21759, 34248}

X(53148) = X(i)-isoconjugate of X(j) for these (i,j): {194, 17149}, {1613, 18837}, {1740, 6374}
X(53148) = barycentric product X(i)*X(j) for these {i,j}: {2998, 51951}, {3223, 34248}, {3224, 3224}
X(53148) = barycentric quotient X(i)/X(j) for these {i,j}: {3223, 18837}, {3224, 6374}, {34248, 17149}, {51951, 194}

---

X(53149) = X(4)X(512)∩X(25)X(669)

Barycentrics    (b^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(53149) lies on the cubic K406 and these lines: {4, 512}, {25, 669}, {98, 3563}, {287, 9134}, {297, 34290}, {393, 22260}, {458, 5652}, {523, 9756}, {685, 4240}, {690, 45031}, {805, 877}, {850, 11442}, {875, 36120}, {881, 17980}, {1824, 50487}, {1990, 9178}, {2207, 2422}, {2393, 15328}, {2869, 20021}, {6531, 8753}, {9171, 40138}, {11182, 52283}, {11183, 52288}, {14248, 44705}, {15475, 32696}, {18808, 33919}, {20031, 23977}

X(53149) = polar conjugate of X(2396)
X(53149) = polar conjugate of the isotomic conjugate of X(2395)
X(53149) = polar conjugate of the isogonal conjugate of X(2422)
X(53149) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {8773, 14721}, {36105, 147}
X(53149) = X(685)-Ceva conjugate of X(6531)
X(53149) = X(i)-isoconjugate of X(j) for these (i,j): {48, 2396}, {63, 2421}, {69, 23997}, {162, 51386}, {163, 6393}, {255, 877}, {293, 15631}, {304, 14966}, {325, 4575}, {326, 4230}, {511, 4592}, {662, 36212}, {684, 24041}, {799, 3289}, {906, 51370}, {1101, 6333}, {1331, 51369}, {1332, 17209}, {1755, 4563}, {1790, 42717}, {1959, 4558}, {4610, 42702}, {9417, 52608}, {17932, 23996}, {24037, 39469}, {32661, 46238}, {36061, 51383}
X(53149) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 6393}, {125, 51386}, {132, 15631}, {135, 51439}, {136, 325}, {512, 39469}, {523, 6333}, {1084, 36212}, {1249, 2396}, {3005, 684}, {3162, 2421}, {5139, 511}, {5190, 51370}, {5521, 51369}, {6523, 877}, {15259, 4230}, {16221, 51383}, {36899, 4563}, {38970, 32458}, {38996, 3289}, {39058, 52608}, {48317, 50567}
X(53149) = trilinear pole of line {1084, 2489}
X(53149) = crossdifference of every pair of points on line {3289, 36212}
X(53149) = barycentric product X(i)*X(j) for these {i,j}: {4, 2395}, {25, 43665}, {98, 2501}, {107, 51404}, {115, 685}, {125, 20031}, {264, 2422}, {290, 2489}, {338, 32696}, {393, 879}, {512, 16081}, {523, 6531}, {648, 51441}, {661, 36120}, {878, 2052}, {1109, 36104}, {1910, 24006}, {1976, 14618}, {2715, 2970}, {2966, 8754}, {2971, 43187}, {3124, 22456}, {6331, 15630}, {8791, 52076}, {9154, 14273}, {14998, 52491}, {16230, 41932}, {17983, 52038}, {17994, 34536}, {18808, 35906}, {22260, 41174}, {34212, 52641}
X(53149) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 2396}, {25, 2421}, {98, 4563}, {115, 6333}, {232, 15631}, {290, 52608}, {393, 877}, {512, 36212}, {523, 6393}, {647, 51386}, {669, 3289}, {685, 4590}, {878, 394}, {879, 3926}, {1084, 39469}, {1824, 42717}, {1910, 4592}, {1973, 23997}, {1974, 14966}, {1976, 4558}, {2207, 4230}, {2395, 69}, {2422, 3}, {2489, 511}, {2501, 325}, {2966, 47389}, {2971, 3569}, {3124, 684}, {6531, 99}, {6591, 51369}, {6753, 51439}, {7649, 51370}, {8754, 2799}, {14273, 50567}, {14601, 32661}, {15630, 647}, {16081, 670}, {16230, 32458}, {17994, 36790}, {20031, 18020}, {22260, 41172}, {22456, 34537}, {24006, 46238}, {32696, 249}, {36104, 24041}, {36120, 799}, {41932, 17932}, {42068, 2491}, {42663, 47406}, {43665, 305}, {43920, 15419}, {47230, 51383}, {50487, 42702}, {51404, 3265}, {51441, 525}, {52038, 6390}, {52076, 37804}

---

X(53150) = X(4)X(514)∩X(27)X(2400)

Barycentrics    (b - c)*(-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(53150) lies on the cubic K406 and these lines: {4, 514}, {27, 2400}, {92, 693}, {103, 917}, {278, 3064}, {393, 21133}, {877, 17930}, {927, 40116}, {1119, 42462}, {1249, 21202}, {1847, 17924}, {2424, 8747}, {6336, 6548}, {36101, 37203}, {36122, 36124}, {44129, 52619}

X(53150) = polar conjugate of X(2398)
X(53150) = polar conjugate of the isotomic conjugate of X(2400)
X(53150) = polar conjugate of the isogonal conjugate of X(2424)
X(53150) = X(36107)-anticomplementary conjugate of X(152)
X(53150) = X(i)-isoconjugate of X(j) for these (i,j): {48, 2398}, {63, 2426}, {109, 51376}, {163, 51366}, {184, 42719}, {255, 41321}, {516, 906}, {692, 26006}, {910, 1331}, {1110, 39470}, {1456, 4587}, {1802, 23973}, {1813, 41339}, {3234, 36056}, {3990, 4241}, {4575, 17747}, {4592, 51436}, {30807, 32656}, {36059, 40869}
X(53150) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 51376}, {115, 51366}, {136, 17747}, {514, 39470}, {1086, 26006}, {1249, 2398}, {3162, 2426}, {5139, 51436}, {5190, 516}, {5521, 910}, {6523, 41321}, {20620, 40869}, {20622, 3234}, {38966, 51418}
X(53150) = cevapoint of X(1830) and X(5089)
X(53150) = trilinear pole of line {1086, 7649}
X(53150) = crossdifference of every pair of points on line {47407, 47422}
X(53150) = barycentric product X(i)*X(j) for these {i,j}: {4, 2400}, {103, 46107}, {264, 2424}, {514, 52781}, {677, 2973}, {693, 36122}, {1897, 15634}, {3064, 52156}, {7649, 18025}, {17924, 36101}, {20901, 36109}, {21666, 24016}, {23989, 40116}, {43736, 44426}
X(53150) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 2398}, {25, 2426}, {92, 42719}, {103, 1331}, {393, 41321}, {514, 26006}, {523, 51366}, {650, 51376}, {911, 906}, {1086, 39470}, {1119, 23973}, {1847, 24015}, {1886, 3234}, {2338, 4587}, {2400, 69}, {2424, 3}, {2489, 51436}, {2501, 17747}, {2969, 676}, {3064, 40869}, {6591, 910}, {7649, 516}, {8747, 4241}, {15634, 4025}, {17924, 30807}, {17925, 14953}, {18025, 4561}, {18344, 41339}, {32701, 15378}, {36101, 1332}, {36122, 100}, {40116, 1252}, {43736, 6516}, {43923, 1456}, {46107, 35517}, {52781, 190}

---

X(53151) = X(4)X(8)∩X(100)X(108)

Barycentrics    (a - b)*(a - c)*(a^2 + b^2 - c^2)*(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(53151) lies on the cubic K406 and these lines: {2, 35014}, {4, 8}, {20, 38517}, {100, 108}, {877, 17935}, {901, 1309}, {1145, 21664}, {1320, 36123}, {1785, 51433}, {1845, 39776}, {1846, 18802}, {2397, 4246}, {2406, 43737}, {4240, 5379}, {7012, 23987}, {15742, 17780}, {18026, 21272}, {23706, 53047}, {30857, 52781}, {35360, 36797}, {36106, 52377}

X(53151) = anticomplement of X(35014)
X(53151) = polar-circle-inverse of X(44013)
X(53151) = polar conjugate of X(2401)
X(53151) = polar conjugate of the isotomic conjugate of X(2397)
X(53151) = polar conjugate of the isogonal conjugate of X(2427)
X(53151) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1309, 33650}, {7012, 153}, {7128, 36918}, {14776, 39351}, {32702, 4440}, {36037, 34188}, {36110, 149}, {39294, 69}
X(53151) = X(i)-isoconjugate of X(j) for these (i,j): {48, 2401}, {56, 37628}, {63, 2423}, {104, 1459}, {255, 43933}, {513, 1795}, {514, 14578}, {603, 43728}, {652, 34051}, {905, 909}, {1331, 15635}, {1364, 36110}, {1809, 43924}, {2250, 7254}, {2720, 7004}, {3937, 36037}, {3942, 32641}, {4025, 34858}, {7117, 37136}, {22379, 40437}, {22383, 34234}, {23224, 36123}, {26932, 32669}
X(53151) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 37628}, {517, 8677}, {1145, 521}, {1249, 2401}, {3162, 2423}, {3259, 3937}, {5521, 15635}, {6523, 43933}, {7952, 43728}, {16586, 4025}, {23980, 905}, {25640, 513}, {38981, 7004}, {39004, 1364}, {39026, 1795}, {40613, 1459}, {46398, 1565}
X(53151) = cevapoint of X(i) and X(j) for these (i,j): {517, 2804}, {1785, 39534}
X(53151) = trilinear pole of line {1785, 14571}
X(53151) = crossdifference of every pair of points on line {7117, 22383}
X(53151) = barycentric product X(i)*X(j) for these {i,j}: {4, 2397}, {107, 51367}, {190, 1785}, {264, 2427}, {312, 23706}, {318, 24029}, {321, 4246}, {517, 6335}, {646, 1875}, {648, 17757}, {653, 6735}, {668, 14571}, {811, 21801}, {908, 1897}, {1016, 39534}, {1309, 26611}, {1783, 3262}, {1845, 36804}, {1846, 4582}, {2804, 46102}, {6331, 51377}, {7017, 23981}, {10015, 15742}, {13136, 21664}, {13149, 51380}, {15632, 16082}
X(53151) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 2401}, {9, 37628}, {25, 2423}, {101, 1795}, {108, 34051}, {281, 43728}, {393, 43933}, {517, 905}, {644, 1809}, {692, 14578}, {859, 7254}, {908, 4025}, {1769, 3942}, {1783, 104}, {1785, 514}, {1845, 3960}, {1846, 30725}, {1875, 3669}, {1897, 34234}, {2183, 1459}, {2397, 69}, {2427, 3}, {2804, 26932}, {3262, 15413}, {3310, 3937}, {4246, 81}, {6335, 18816}, {6591, 15635}, {6735, 6332}, {7012, 37136}, {7115, 2720}, {8750, 909}, {10015, 1565}, {14571, 513}, {14776, 41933}, {15742, 13136}, {17139, 15419}, {17757, 525}, {21664, 10015}, {21801, 656}, {21942, 14429}, {22350, 4091}, {23706, 57}, {23980, 8677}, {23981, 222}, {24029, 77}, {32698, 15381}, {34922, 47317}, {39534, 1086}, {42072, 3310}, {46393, 7004}, {51362, 49280}, {51367, 3265}, {51377, 647}, {51378, 24562}, {52307, 1364}

---

X(53152) = X(4)X(522)∩X(29)X(2399)

Barycentrics    (a - b - c)*(b - c)*(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(53152) lies on the cubic K406 and these lines: {4, 522}, {29, 2399}, {102, 32706}, {158, 44426}, {273, 693}, {281, 3239}, {318, 4397}, {877, 17931}, {1309, 2405}, {1870, 37628}, {2432, 8748}, {7649, 40836}, {15633, 35015}, {36100, 43764}, {36121, 36123}

X(53152) = polar conjugate of X(2406)
X(53152) = polar conjugate of the isotomic conjugate of X(2399)
X(53152) = polar conjugate of the isogonal conjugate of X(2432)
X(53152) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {15379, 34188}, {36113, 151}
X(53152) = X(i)-isoconjugate of X(j) for these (i,j): {48, 2406}, {63, 2425}, {109, 46974}, {163, 51368}, {255, 23987}, {515, 36059}, {577, 24035}, {906, 34050}, {1262, 46391}, {1331, 1455}, {1813, 2182}, {4575, 51421}, {7452, 22341}, {24027, 39471}, {36040, 38554}, {42718, 52411}
X(53152) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 46974}, {115, 51368}, {136, 51421}, {522, 39471}, {1249, 2406}, {3162, 2425}, {5190, 34050}, {5521, 1455}, {6523, 23987}, {10017, 38554}, {13999, 11700}, {20620, 515}, {38966, 51361}
X(53152) = cevapoint of X(42069) and X(52316)
X(53152) = trilinear pole of line {1146, 3064}
X(53152) = barycentric product X(i)*X(j) for these {i,j}: {4, 2399}, {102, 46110}, {264, 2432}, {522, 52780}, {653, 15633}, {3064, 34393}, {4391, 36121}, {15629, 46107}, {23978, 36067}, {36100, 44426}
X(53152) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 2406}, {25, 2425}, {102, 1813}, {158, 24035}, {318, 42718}, {393, 23987}, {523, 51368}, {650, 46974}, {1146, 39471}, {2310, 46391}, {2399, 69}, {2432, 3}, {2501, 51421}, {3064, 515}, {6591, 1455}, {7649, 34050}, {8748, 7452}, {15629, 1331}, {15633, 6332}, {18344, 2182}, {32667, 24027}, {32677, 36059}, {32700, 15386}, {36067, 1262}, {36100, 6516}, {36121, 651}, {46110, 35516}, {52316, 10017}, {52780, 664}

---

X(53153) = X(4)X(2574)∩X(107)X(110)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)^2*(a^2 + b^2 - c^2)^2*(a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8 - 4*a^8*c^2 + 11*a^6*b^2*c^2 - 9*a^4*b^4*c^2 + a^2*b^6*c^2 + b^8*c^2 + 6*a^6*c^4 - 9*a^4*b^2*c^4 + 6*a^2*b^4*c^4 - b^6*c^4 - 4*a^4*c^6 + a^2*b^2*c^6 - b^4*c^6 + a^2*c^8 + b^2*c^8 + 2*a^2*b^2*c^2*(-(b^2*c^2) + (a^2 - b^2 - c^2)^2)*J) : :

X(53153) lies on the X-parabola of ABC (see X(12065)), the cubics K027, K186, K210, K406, and these lines: {2, 16070}, {4, 2574}, {107, 110}, {523, 1114}, {850, 15165}, {1313, 12079}, {2395, 41942}, {2501, 52132}, {4226, 39299}, {4230, 50944}, {7735, 8105}, {15328, 15460}

X(53153) = polar conjugate of X(50944)
X(53153) = polar conjugate of the isotomic conjugate of X(50945)
X(53153) = polar conjugate of the isogonal conjugate of X(52132)
X(53153) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {162, 14808}, {1823, 34186}, {2577, 39352}, {2581, 13219}, {2587, 3448}, {24000, 2574}, {39299, 4329}, {46812, 21294}
X(53153) = X(i)-Ceva conjugate of X(j) for these (i,j): {107, 1114}, {6528, 46812}
X(53153) = X(i)-isoconjugate of X(j) for these (i,j): {48, 50944}, {63, 52131}, {656, 15461}, {662, 15167}, {1113, 2585}, {1312, 4575}, {1822, 2575}, {2576, 46811}, {2579, 8115}, {4592, 44125}, {14500, 36034}, {24018, 41941}
X(53153) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 1312}, {1084, 15167}, {1249, 50944}, {1313, 2575}, {2574, 520}, {3162, 52131}, {3258, 14500}, {5139, 44125}, {8105, 525}, {15166, 46811}, {40596, 15461}, {46814, 3265}
X(53153) = cevapoint of X(i) and X(j) for these (i,j): {512, 8105}, {523, 2574}
X(53153) = trilinear pole of line {115, 1313}
X(53153) = crossdifference of every pair of points on line {3269, 15167}
X(53153) = barycentric product X(i)*X(j) for these {i,j}: {4, 50945}, {264, 52132}, {648, 1313}, {850, 41942}, {1114, 2592}, {2574, 46812}, {2581, 2588}, {2582, 2587}, {6331, 44126}, {6528, 15166}, {8105, 15165}, {14499, 15459}, {14618, 15460}, {39240, 39299}
X(53153) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 50944}, {25, 52131}, {112, 15461}, {512, 15167}, {1114, 8115}, {1313, 525}, {1637, 14500}, {2489, 44125}, {2501, 1312}, {2574, 46811}, {2577, 1822}, {2578, 2585}, {2587, 2580}, {2588, 2583}, {2592, 22340}, {8105, 2575}, {14499, 41077}, {15165, 46813}, {15166, 520}, {15460, 4558}, {32713, 41941}, {41942, 110}, {44126, 647}, {46812, 15164}, {50945, 69}, {52132, 3}

---

X(53154) = X(4)X(2575)∩X(107)X(110)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)^2*(a^2 + b^2 - c^2)^2*(a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8 - 4*a^8*c^2 + 11*a^6*b^2*c^2 - 9*a^4*b^4*c^2 + a^2*b^6*c^2 + b^8*c^2 + 6*a^6*c^4 - 9*a^4*b^2*c^4 + 6*a^2*b^4*c^4 - b^6*c^4 - 4*a^4*c^6 + a^2*b^2*c^6 - b^4*c^6 + a^2*c^8 + b^2*c^8 - 2*a^2*b^2*c^2*(-(b^2*c^2) + (a^2 - b^2 - c^2)^2)*J) : :

X(53154) lies on the X-parabola of ABC (see X(12065)), the cubics K027, K186, K210, K406, and these lines: {2, 16071}, {4, 2575}, {107, 110}, {523, 1113}, {850, 15164}, {1312, 12079}, {2395, 41941}, {2501, 52131}, {4226, 39298}, {4230, 50945}, {7735, 8106}, {15328, 15461}

X(53154) = polar conjugate of X(50945)
X(53154) = polar conjugate of the isotomic conjugate of X(50944)
X(53154) = polar conjugate of the isogonal conjugate of X(52131)
X(53154) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {162, 14807}, {1822, 34186}, {2576, 39352}, {2580, 13219}, {2586, 3448}, {24000, 2575}, {39298, 4329}, {46815, 21294}
X(53154) = X(i)-Ceva conjugate of X(j) for these (i,j): {107, 1113}, {6528, 46815}
X(53154) = X(i)-isoconjugate of X(j) for these (i,j): {48, 50945}, {63, 52132}, {656, 15460}, {662, 15166}, {1114, 2584}, {1313, 4575}, {1823, 2574}, {2577, 46814}, {2578, 8116}, {4592, 44126}, {14499, 36034}, {24018, 41942}
X(53154) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 1313}, {1084, 15166}, {1249, 50945}, {1312, 2574}, {2575, 520}, {3162, 52132}, {3258, 14499}, {5139, 44126}, {8106, 525}, {15167, 46814}, {40596, 15460}, {46811, 3265}
X(53154) = cevapoint of X(i) and X(j) for these (i,j): {512, 8106}, {523, 2575}
X(53154) = trilinear pole of line {115, 1312}
X(53154) = crossdifference of every pair of points on line {3269, 15166}
X(53154) = barycentric product X(i)*X(j) for these {i,j}: {4, 50944}, {264, 52131}, {648, 1312}, {850, 41941}, {1113, 2593}, {2575, 46815}, {2580, 2589}, {2583, 2586}, {6331, 44125}, {6528, 15167}, {8106, 15164}, {14500, 15459}, {14618, 15461}, {39241, 39298}
X(53154) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 50945}, {25, 52132}, {112, 15460}, {512, 15166}, {1113, 8116}, {1312, 525}, {1637, 14499}, {2489, 44126}, {2501, 1313}, {2575, 46814}, {2576, 1823}, {2579, 2584}, {2586, 2581}, {2589, 2582}, {2593, 22339}, {8106, 2574}, {14500, 41077}, {15164, 46810}, {15167, 520}, {15461, 4558}, {32713, 41942}, {41941, 110}, {44125, 647}, {46815, 15165}, {50944, 69}, {52131, 3}

---

X(53155) = X(4)X(542)∩X(685)X(4240)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^2 + b^2 - 2*c^2)*(a^2 + b^2 - c^2)*(a^2 - 2*b^2 + 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(53155) lies on the cubic K406 and these lines: {4, 542}, {685, 4240}, {691, 935}, {877, 892}, {7473, 50941}, {16092, 17986}, {18020, 34760}, {47105, 51405}, {50437, 52756}

X(53155) = polar conjugate of X(50942)
X(53155) = polar conjugate of the isotomic conjugate of X(50941)
X(53155) = X(i)-isoconjugate of X(j) for these (i,j): {48, 50942}, {810, 52094}, {896, 35909}
X(53155) = X(i)-Dao conjugate of X(j) for these (i,j): {542, 39474}, {1249, 50942}, {15899, 35909}, {23967, 14417}, {39062, 52094}, {42426, 690}
X(53155) = trilinear pole of line {6103, 7473}
X(53155) = barycentric product X(i)*X(j) for these {i,j}: {4, 50941}, {107, 51405}, {648, 16092}, {671, 7473}, {892, 6103}, {14999, 17983}, {30786, 35907}
X(53155) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 50942}, {111, 35909}, {542, 14417}, {648, 52094}, {895, 35911}, {6103, 690}, {7473, 524}, {8753, 14998}, {14999, 6390}, {16092, 525}, {17983, 14223}, {23967, 39474}, {35907, 468}, {50941, 69}, {51405, 3265}

---

X(53156) = X(4)X(690)∩X(468)X(1649)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-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(53156) lies on the cubic K406 and these lines: {4, 690}, {468, 1649}, {842, 40118}, {877, 4240}, {1648, 2501}, {14697, 37645}, {18808, 33919}, {44146, 52629}

X(53156) = polar conjugate of X(50941)
X(53156) = polar conjugate of the isotomic conjugate of X(50942)
X(53156) = X(i)-isoconjugate of X(j) for these (i,j): {48, 50941}, {163, 51405}, {4575, 16092}, {14999, 36060}
X(53156) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 51405}, {136, 16092}, {690, 39474}, {1249, 50941}, {1560, 14999}, {48317, 542}
X(53156) = trilinear pole of line {14273, 23992}
X(53156) = barycentric product X(i)*X(j) for these {i,j}: {4, 50942}, {468, 14223}, {2501, 52094}, {5641, 14273}, {14998, 44146}, {35909, 37778}, {51228, 52475}
X(53156) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 50941}, {468, 14999}, {523, 51405}, {2501, 16092}, {14223, 30786}, {14273, 542}, {14998, 895}, {23992, 39474}, {50942, 69}, {52094, 4563}, {52475, 51227}

---

X(53157) = X(4)X(900)∩X(519)X(53047)

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

X(53157) lies on the cubic K406 and these lines: {4, 900}, {519, 53047}, {1647, 7649}, {1877, 39771}, {6544, 8756}, {6550, 43933}, {15742, 17780}, {37168, 50943}

X(53157) = polar conjugate of the isotomic conjugate of X(50943)
X(53157) = X(1331)-isoconjugate of X(52478)
X(53157) = X(5521)-Dao conjugate of X(52478)
X(53157) = barycentric product X(i)*X(j) for these {i,j}: {4, 50943}, {17924, 52479}, {37790, 46041}
X(53157) = barycentric quotient X(i)/X(j) for these {i,j}: {6591, 52478}, {50943, 69}, {52479, 1332}

---

X(53158) = X(4)X(526)∩X(186)X(2411)

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

X(53158) lies on the cubic K406 and these lines: {4, 526}, {186, 2411}, {250, 4240}, {477, 32710}, {2436, 52418}, {15328, 43707}, {34210, 38936}

X(53158) = polar conjugate of X(2410)
X(53158) = polar conjugate of the isotomic conjugate of X(2411)
X(53158) = polar conjugate of the isogonal conjugate of X(2436)
X(53158) = X(36116)-anticomplementary conjugate of X(34193)
X(53158) = X(i)-isoconjugate of X(j) for these (i,j): {48, 2410}, {63, 2437}, {4575, 34209}, {5663, 36061}
X(53158) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 34209}, {1249, 2410}, {3162, 2437}, {16221, 5663}
X(53158) = trilinear pole of line {18334, 47230}
X(53158) = barycentric product X(i)*X(j) for these {i,j}: {4, 2411}, {264, 2436}, {477, 44427}, {14165, 14220}, {14618, 34210}, {30528, 35235}, {32679, 36130}
X(53158) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 2410}, {25, 2437}, {2411, 69}, {2436, 3}, {2501, 34209}, {32712, 15395}, {34210, 4558}, {36130, 32680}, {36151, 36061}, {39176, 42742}, {44427, 35520}, {47230, 5663}, {52418, 7480}

---

X(53159) = X(4)X(9033)∩X(477)X(2693)

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

X(53159) lies on the cubic K406 and these lines: {4, 9033}, {477, 2693}, {523, 1650}, {1990, 14401}, {4240, 32230}, {5627, 43083}, {8057, 18507}, {9214, 15354}, {18808, 43701}, {34288, 47138}

X(53159) = X(i)-isoconjugate of X(j) for these (i,j): {1552, 4575}, {2777, 36034}, {31510, 35200}
X(53159) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 31510}, {136, 1552}, {1650, 12113}, {3258, 2777}
X(53159) = trilinear pole of line {1637, 39008}
X(53159) = barycentric product X(i)*X(j) for these {i,j}: {525, 47111}, {2693, 41079}
X(53159) = barycentric quotient X(i)/X(j) for these {i,j}: {1637, 2777}, {1990, 31510}, {2501, 1552}, {2693, 44769}, {14401, 12113}, {47111, 648}, {52743, 7740}

---

X(53160) = X(2)X(3)∩X(100)X(658)

Barycentrics    a*(a - b)*(a - c)*(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(53160) lies on these lines: {2, 3}, {100, 658}, {108, 46964}, {677, 41353}, {901, 2728}, {1983, 2610}, {3126, 3573}, {13397, 40117}, {17073, 37741}, {41906, 52776}

X(53160) = crossdifference of every pair of points on line {647, 2611}
X(53160) = barycentric product X(i)*X(j) for these {i,j}: {109, 45798}, {664, 8558}
X(53160) = barycentric quotient X(i)/X(j) for these {i,j}: {8558, 522}, {45798, 35519}
X(53160) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {866, 20839, 46586}, {4242, 7451, 13589}

---

X(53161) = X(2)X(3)∩X(125)X(9880)

Barycentrics    2*a^8 - 3*a^6*b^2 + 2*a^4*b^4 + 3*a^2*b^6 - 4*b^8 - 3*a^6*c^2 + 2*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + 9*b^6*c^2 + 2*a^4*c^4 - 4*a^2*b^2*c^4 - 10*b^4*c^4 + 3*a^2*c^6 + 9*b^2*c^6 - 4*c^8 : :
X(53161) = 4 X[868] - X[4226], 3 X[868] - X[45662], 3 X[4226] - 4 X[45662], X[2407] - 4 X[38393]

X(53161) lies on these lines: {2, 3}, {125, 9880}, {247, 11656}, {316, 5468}, {524, 9214}, {542, 34174}, {625, 47047}, {671, 690}, {1494, 18808}, {1648, 52450}, {2396, 7809}, {2407, 14995}, {3448, 12243}, {5641, 34765}, {5967, 11645}, {6321, 31127}, {6792, 43448}, {7703, 40877}, {9155, 22566}, {9168, 34312}, {10723, 30789}, {11054, 41724}, {11161, 48983}, {13169, 48540}, {13449, 13857}, {14246, 23061}, {17983, 32244}, {30465, 31709}, {30468, 31710}, {31173, 45330}, {43453, 52449}, {43535, 45815}, {45291, 47286}

X(53161) = reflection of X(i) in X(j) for these {i,j}: {2, 868}, {2407, 14995}, {4226, 2}, {14995, 38393}
X(53161) = anticomplement of X(45662)
X(53161) = X(46986)-Dao conjugate of X(542)
X(53161) = crossdifference of every pair of points on line {647, 39689}
X(53161) = barycentric product X(671)*X(46986)
X(53161) = barycentric quotient X(46986)/X(524)
X(53161) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 297, 4240}, {381, 36194, 2}, {3545, 35922, 2}, {14694, 47097, 2}, {36185, 52267, 2}, {36186, 52268, 2}

---

X(53162) = X(2)X(3)∩X(110)X(3414)

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

X(53162) lies on these lines: {2, 3}, {110, 3414}, {476, 1380}, {523, 6189}, {1341, 9159}, {3233, 30508}, {3413, 9140}, {5640, 46023}, {6142, 9158}, {14611, 45296}, {18911, 52052}, {34312, 51826}

X(53162) = circumcircle-inverse of X(46600)
X(53162) = antigonal image of X(31863)
X(53162) = symgonal image of X(47088)
X(53162) = psi-transform of X(47365)
X(53162) = crossdifference of every pair of points on line {647, 2029}
X(53162) = barycentric product X(6189)*X(31863)
X(53162) = barycentric quotient X(31863)/X(3414)
X(53162) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1113, 1114, 46600}

---

X(53163) = X(2)X(3)∩X(110)X(3413)

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

X(53163) lies on these lines: {2, 3}, {110, 3413}, {476, 1379}, {523, 6190}, {1340, 9159}, {3233, 30509}, {3414, 9140}, {5640, 46024}, {6141, 9158}, {14611, 45297}, {18911, 52051}, {34312, 51825}

X(53163) = circumcircle-inverse of X(46601)
X(53163) = antigonal image of X(31862)
X(53163) = symgonal image of X(47089)
X(53163) = psi-transform of X(47366)
X(53163) = crossdifference of every pair of points on line {647, 2028}
X(53163) = barycentric product X(6190)*X(31862)
X(53163) = barycentric quotient X(31862)/X(3413)
X(53163) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1113, 1114, 46601}

---

X(53164) = X(3)X(6)∩X(384)X(21001)

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

X(53164) lies on these lines: {3, 6}, {384, 21001}, {1003, 36650}, {1184, 10997}, {1613, 3360}, {3051, 33014}, {3499, 33235}, {20965, 33022}, {53145, 53146}

X(53164) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1613, 3552, 3360}, {2076, 5116, 37485}

---

X(53165) = X(1)X(6)∩X(32)X(2170)

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

See HGT2023,140

X(53165) llies on these lines: {1, 6}, {32, 2170}, {36, 3959}, {39, 49487}, {56, 3125}, {574, 4642}, {993, 3727}, {996, 21021}, {1015, 3924}, {1319, 16583}, {1385, 41015}, {1914, 4291}, {1953, 5019}, {2082, 2251}, {2098, 14974}, {2099, 5021}, {2171, 5042}, {2238, 30144}, {2242, 17451}, {2271, 34471}, {2276, 15955}, {2292, 31456}, {2650, 9346}, {2975, 3735}, {3120, 9651}, {3290, 24928}, {3721, 8666}, {3780, 22836}, {5563, 20271}, {5697, 17735}, {5724, 31466}, {5903, 33863}, {7756, 33094}, {9310, 49758}, {16605, 17614}, {17736, 21331}, {18755, 37525}, {20691, 49494}, {21008, 21842}, {23638, 45891}, {24512, 30147}, {24524, 30113}, {30128, 30945}, {30136, 37686}, {30140, 37678}, {31449, 37614}, {40958, 45932}

X(53165) = barycentric product X(1)*X(29662)
X(53165) = barycentric quotient X(29662)/X(75)
X(53165) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 16968, 3230}, {3554, 5336, 20228}

---

X(53166) = X(2)X(804)∩X(115)X(511)

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

See HGT2023,310323

X(53166) llies on these lines: {2, 46039}, {3, 804}, {115, 511}, {12042, 17423}

X(53166) = complement of X(46039)
X(53166) = complement of the isogonal conjugate of X(51455)
X(53166) = X(i)-complementary conjugate of X(j) for these (i,j): {36051, 2782}, {51455, 10}
X(53166) = X(98)-Ceva conjugate of X(2782)

---

X(53167) = X(2)X(32042)∩X(11)X(35076)

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

See HGT2023,270323

X(53167) llies on the Steiner inellipse and these lines: {2, 32042}, {11, 35076}, {1015, 3120}, {4370, 5257}, {4958, 30595}, {6184, 41862}, {13466, 27184}, {20532, 22035}, {21138, 35134}, {30561, 35124}

X(53167) = complement of X(32042)
X(53167) = complement of the isogonal conjugate of X(4834)
X(53167) = complement of the isotomic conjugate of X(4802)
X(53167) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 4932}, {31, 4802}, {667, 19862}, {1698, 21260}, {1919, 28606}, {3063, 5325}, {4658, 512}, {4802, 2887}, {4810, 20542}, {4813, 141}, {4820, 21244}, {4823, 626}, {4826, 1211}, {4834, 10}, {4838, 21245}, {4840, 3741}, {4960, 21240}, {5221, 17072}, {5333, 42327}, {16777, 3835}, {28605, 21262}, {31902, 21259}, {36074, 21232}, {48005, 3454}
X(53167) = X(2)-Ceva conjugate of X(4802)
X(53167) = X(i)-isoconjugate of X(j) for these (i,j): {1252, 30597}, {8652, 37211}
X(53167) = X(i)-Dao conjugate of X(j) for these (i,j): {661, 30597}, {4802, 2}
X(53167) = barycentric product X(i)*X(j) for these {i,j}: {4802, 4802}, {4813, 4823}, {4838, 4960}
X(53167) = barycentric quotient X(i)/X(j) for these {i,j}: {244, 30597}, {4802, 32042}, {4813, 37211}, {4834, 8652}

---

X(53168) = X(3)X(18127)∩X(4)X(12028)

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

X(53168) lies on the cubic K1318 and these lines: {3, 18127}, {4, 12028}, {5, 523}, {24, 136}, {155, 265}, {476, 7505}, {18404, 42424}

X(53168) = X(12028)-Ceva conjugate of X(5961)
X(53168) = X(6149)-isoconjugate of X(45781)
X(53168) = X(14993)-Dao conjugate of X(45781)
X(53168) = barycentric product X(i)*X(j) for these {i,j}: {94, 45780}, {2072, 18883}
X(53168) = barycentric quotient X(i)/X(j) for these {i,j}: {1989, 45781}, {2072, 37802}, {45780, 323}

---

X(53169) = X(3)X(125)∩X(52)X(924)

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

X(53169) lies on the cubic K1318 and these lines: {3, 125}, {52, 924}, {94, 847}, {578, 14356}, {2904, 34756}, {6070, 25738}, {12028, 12118}, {12370, 14254}

X(53169) = X(94)-Ceva conjugate of X(16310)
X(53169) = X(131)-Dao conjugate of X(5962)
X(53169) = barycentric product X(i)*X(j) for these {i,j}: {94, 12095}, {18883, 44665}
X(53169) = barycentric quotient X(i)/X(j) for these {i,j}: {5961, 43756}, {12095, 323}, {16310, 5962}, {32662, 46969}, {44665, 37802}

---

X(53170) = X(3)X(1986)∩X(4)X(12028)

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

X(53170) lies on the cubic K1318 and these lines: {3, 1986}, {4, 12028}, {24, 18127}, {477, 35471}, {12092, 45781}, {37119, 46087}

X(53170) = X(2072)-isoconjugate of X(2166)
X(53170) = X(11597)-Dao conjugate of X(2072)
X(53170) = barycentric product X(323)*X(38534)
X(53170) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 2072}, {38534, 94}

---

X(53171) = X(3)X(15317)∩X(4)X(15478)

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

X(53171) lies on the cubic K1318 and these lines: {3, 15317}, {4, 15478}, {24, 12095}, {577, 7746}, {924, 10539}, {4558, 34853}

X(53171) = orthic-isogonal conjugate of X(1147)
X(53171) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 1147}, {15478, 12095}

---

X(53172) = X(3)X(16172)∩X(4)X(15478)

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

X(53172) lies on the cubic K1318 and these lines: {3, 16172}, {4, 15478}, {24, 18126}, {155, 403}, {7505, 34853}, {40697, 44138}

---

X(53173) = X(2)X(647)∩X(3)X(525)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 + c^2)^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(53173) lies on the cubics K734 and K1319 and these lines: {2, 647}, {3, 525}, {97, 2525}, {98, 1297}, {248, 6334}, {276, 15412}, {287, 14417}, {394, 3265}, {523, 9756}, {524, 14380}, {684, 14941}, {1073, 38240}, {2394, 2966}, {2715, 2867}, {2848, 45031}, {3268, 46806}, {3926, 23107}, {4580, 42293}, {5481, 9751}, {6287, 15451}, {6394, 41077}, {8552, 34897}, {9409, 14830}, {14376, 52584}, {14966, 34211}, {15526, 35911}, {17708, 34761}, {17974, 39473}, {20021, 42665}, {20081, 38256}, {23870, 33388}, {23871, 33389}, {23872, 33421}, {23873, 33420}, {31635, 44427}, {34579, 41079}, {35906, 40856}, {39647, 42658}
on K734, K1319

X(53173) = isotomic conjugate of the polar conjugate of X(879)
X(53173) = X(i)-Ceva conjugate of X(j) for these (i,j): {2966, 287}, {17932, 17974}, {40428, 125}
X(53173) = trilinear pole of line {520, 15526}
X(53173) = crossdifference of every pair of points on line {232, 237}
X(53173) = X(i)-isoconjugate of X(j) for these (i,j): {19, 4230}, {75, 34859}, {107, 1755}, {112, 240}, {132, 36046}, {158, 14966}, {162, 232}, {163, 6530}, {237, 823}, {297, 32676}, {393, 23997}, {511, 24019}, {662, 34854}, {811, 2211}, {877, 1973}, {1096, 2421}, {1959, 32713}, {2491, 23999}, {2967, 36104}, {3289, 36126}, {3569, 24000}, {5360, 52919}, {6528, 9417}, {9475, 36092}, {20031, 23996}, {36084, 51334}
X(53173) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 4230}, {115, 6530}, {125, 232}, {130, 52967}, {206, 34859}, {525, 2799}, {647, 16230}, {1084, 34854}, {1147, 14966}, {6337, 877}, {6338, 2396}, {6503, 2421}, {15526, 297}, {17423, 2211}, {17434, 684}, {33504, 132}, {34591, 240}, {35071, 511}, {35088, 36426}, {36899, 107}, {38985, 1755}, {38987, 51334}, {39000, 2967}, {39019, 39569}, {39020, 44704}, {39058, 6528}, {39085, 112}, {41181, 114}, {46093, 3289}, {51404, 12131}
X(53173) = barycentric product X(i)*X(j) for these {i,j}: {69, 879}, {98, 3265}, {125, 17932}, {248, 3267}, {287, 525}, {290, 520}, {293, 14208}, {305, 878}, {336, 656}, {339, 43754}, {394, 43665}, {523, 6394}, {822, 46273}, {850, 17974}, {1821, 24018}, {1976, 52617}, {2395, 3926}, {2419, 34156}, {2632, 36036}, {2715, 36793}, {2966, 15526}, {2972, 22456}, {3269, 43187}, {4143, 6531}, {4176, 53149}, {4563, 51404}, {6333, 47388}, {9476, 39473}, {16081, 52613}, {17879, 36084}, {18024, 39201}, {20031, 23974}, {34767, 35912}, {35911, 46786}, {36893, 43701}
X(53173) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 4230}, {32, 34859}, {69, 877}, {98, 107}, {125, 16230}, {248, 112}, {255, 23997}, {287, 648}, {290, 6528}, {293, 162}, {336, 811}, {394, 2421}, {512, 34854}, {520, 511}, {523, 6530}, {525, 297}, {577, 14966}, {647, 232}, {656, 240}, {684, 2967}, {685, 32230}, {822, 1755}, {878, 25}, {879, 4}, {1821, 823}, {1910, 24019}, {1976, 32713}, {2395, 393}, {2422, 2207}, {2435, 39265}, {2715, 23964}, {2799, 36426}, {2966, 23582}, {2972, 684}, {3049, 2211}, {3265, 325}, {3267, 44132}, {3269, 3569}, {3569, 51334}, {3926, 2396}, {3998, 42717}, {4091, 17209}, {4131, 51369}, {4143, 6393}, {5489, 868}, {6368, 39569}, {6394, 99}, {6531, 6529}, {8057, 44704}, {14208, 40703}, {14380, 35908}, {15407, 44770}, {15526, 2799}, {16081, 15352}, {17932, 18020}, {17974, 110}, {20021, 46151}, {20031, 23590}, {20975, 17994}, {22089, 15143}, {23286, 19189}, {24018, 1959}, {24284, 39931}, {30476, 40887}, {30805, 51370}, {32320, 3289}, {34156, 2409}, {34369, 35907}, {34980, 39469}, {35909, 52492}, {35911, 46787}, {35912, 4240}, {36036, 23999}, {36084, 24000}, {36120, 36126}, {39201, 237}, {39473, 15595}, {40079, 37937}, {41077, 51389}, {41932, 20031}, {42293, 52967}, {43083, 14356}, {43665, 2052}, {43754, 250}, {46088, 41270}, {47194, 1513}, {47388, 685}, {51386, 15631}, {51404, 2501}, {51640, 51651}, {51776, 41679}, {51963, 23977}, {52613, 36212}, {53149, 6524}

---

X(53174) = X(2)X(98)∩X(5)X(217)

Barycentrics    (a^2 - b^2 - c^2)*(a^4 + b^4 - a^2*c^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^2*c^2 + c^4) : :
X(53174) = 3 X[51] - 2 X[45123]

X(53174) lies on Johnson circumconic (see K714), the cubic K1319, these lines: {2, 98}, {3, 36952}, {4, 290}, {5, 217}, {51, 324}, {52, 27372}, {68, 248}, {185, 6248}, {265, 879}, {311, 6751}, {343, 418}, {511, 16083}, {684, 14941}, {685, 41203}, {686, 2395}, {1154, 35362}, {1298, 22456}, {2782, 3269}, {3289, 3564}, {3785, 6394}, {5562, 28706}, {6146, 51869}, {6531, 18855}, {10104, 14585}, {10739, 35097}, {11433, 16081}, {11610, 51963}, {12188, 18338}, {12384, 31670}, {13157, 41588}, {13449, 13851}, {14216, 45031}, {15069, 40805}, {15073, 51259}, {15630, 48445}, {18304, 39837}, {18381, 52641}, {18396, 36822}, {18912, 31636}, {18918, 36874}, {20574, 31617}, {21659, 32152}, {23217, 26874}, {25051, 34980}, {25738, 34156}, {25739, 52491}, {32064, 32319}, {34507, 52145}, {35098, 43665}, {35902, 52694}, {36831, 41586}, {39569, 51363}, {41724, 46786}

X(53174) = reflection of X(1625) in X(5)
X(53174) = isogonal conjugate of X(19189)
X(53174) = anticomplement of X(52128)
X(53174) = X(1956)-anticomplementary conjugate of X(147)
X(53174) = X(43754)-Ceva conjugate of X(879)
X(53174) = cevapoint of X(i) and X(j) for these (i,j): {5, 32428}, {51, 51363}
X(53174) = trilinear pole of line {216, 6368}
X(53174) = X(i)-isoconjugate of X(j) for these (i,j): {1, 19189}, {54, 240}, {92, 41270}, {232, 2167}, {237, 40440}, {275, 1755}, {276, 9417}, {297, 2148}, {511, 2190}, {1959, 8882}, {2169, 6530}, {2616, 4230}, {16230, 36134}
X(53174) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 19189}, {5, 511}, {130, 39469}, {137, 16230}, {216, 297}, {343, 51439}, {2972, 684}, {6663, 39569}, {14363, 6530}, {15450, 3569}, {22391, 41270}, {36899, 275}, {39019, 2799}, {39058, 276}, {39085, 54}, {40588, 232}, {45249, 44704}, {52032, 325}
X(53174) = barycentric product X(i)*X(j) for these {i,j}: {5, 287}, {53, 6394}, {98, 343}, {216, 290}, {217, 18024}, {248, 311}, {293, 14213}, {324, 17974}, {336, 1953}, {879, 14570}, {1821, 44706}, {1910, 18695}, {1976, 28706}, {2966, 6368}, {4580, 35362}, {5562, 16081}, {6531, 52347}, {12077, 17932}, {15451, 43187}, {17434, 22456}, {18314, 43754}, {23181, 43665}, {31636, 41168}, {34536, 44716}
X(53174) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 297}, {6, 19189}, {51, 232}, {53, 6530}, {98, 275}, {184, 41270}, {216, 511}, {217, 237}, {248, 54}, {287, 95}, {290, 276}, {293, 2167}, {311, 44132}, {343, 325}, {418, 3289}, {685, 16813}, {878, 2623}, {879, 15412}, {1568, 51389}, {1625, 4230}, {1821, 40440}, {1910, 2190}, {1953, 240}, {1976, 8882}, {2715, 933}, {2966, 18831}, {3199, 34854}, {5562, 36212}, {6368, 2799}, {6394, 34386}, {6531, 8884}, {12077, 16230}, {14213, 40703}, {14570, 877}, {15451, 3569}, {16081, 8795}, {16697, 51369}, {17434, 684}, {17974, 97}, {18695, 46238}, {22456, 42405}, {23181, 2421}, {30493, 43034}, {35362, 41676}, {35912, 43768}, {36412, 39569}, {39569, 36426}, {40981, 2211}, {41168, 34138}, {42293, 39469}, {42459, 44704}, {42698, 42703}, {43754, 18315}, {44706, 1959}, {44709, 17209}, {44715, 35910}, {44716, 36790}, {51363, 132}, {51404, 8901}, {52032, 51439}, {52347, 6393}, {53149, 15422}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {98, 287, 17974}, {98, 17974, 35912}, {11442, 46806, 20021}, {32618, 32619, 20021}, {52190, 52451, 20021}

---

X(53175) = X(4)X(15412)∩X(51)X(647)

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

X(53175) lies on the cubics K027 and K1319, and on these lines: {4, 15412}, {51, 647}, {184, 46088}, {217, 39201}, {418, 23613}, {684, 14941}, {1987, 9409}, {5562, 23103}

X(53175) = X(i)-isoconjugate of X(j) for these (i,j): {162, 16089}, {401, 823}, {811, 41204}, {1955, 6528}, {2313, 42405}, {6130, 23999}, {24019, 44137}
X(53175) = X(i)-Dao conjugate of X(j) for these (i,j): {125, 16089}, {130, 32428}, {17423, 41204}, {35071, 44137}
X(53175) = trilinear pole of line {34980, 42293}
X(53175) = crossdifference of every pair of points on line {401, 16089}
X(53175) = barycentric product X(i)*X(j) for these {i,j}: {520, 1987}, {525, 52177}, {647, 14941}, {822, 1956}, {1298, 17434}, {1972, 39201}, {41210, 41219}
X(53175) = barycentric quotient X(i)/X(j) for these {i,j}: {520, 44137}, {647, 16089}, {1298, 42405}, {1987, 6528}, {3049, 41204}, {14941, 6331}, {39201, 401}, {42293, 32428}, {52177, 648}

---

X(53176) = X(4)X(54)∩X(107)X(110)

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

X(53176) lies on the cubic K027 and these lines: {4, 54}, {107, 110}, {156, 14249}, {186, 7740}, {250, 15329}, {685, 43665}, {933, 46062}, {1301, 33640}, {1304, 5502}, {1844, 17104}, {2914, 38897}, {3043, 34210}, {4230, 47443}, {6000, 38937}, {6761, 18279}, {7480, 30510}, {9544, 52448}, {9934, 10152}, {10540, 11251}, {14165, 14355}, {14560, 15459}, {14590, 52603}, {14591, 52743}, {15139, 51939}, {16237, 30512}, {32695, 32708}, {38936, 52416}, {39689, 51334}, {52418, 52668}

X(53176) = isogonal conjugate of X(43083)
X(53176) = polar conjugate of X(14592)
X(53176) = polar conjugate of the isotomic conjugate of X(14590)
X(53176) = polar conjugate of the isogonal conjugate of X(14591)
X(53176) = X(i)-Ceva conjugate of X(j) for these (i,j): {15459, 112}, {23582, 36423}
X(53176) = cevapoint of X(i) and X(j) for these (i,j): {186, 526}, {34397, 47230}
X(53176) = trilinear pole of line {50, 186}
X(53176) = crossdifference of every pair of points on line {3269, 17434}
X(53176) = X(i)-isoconjugate of X(j) for these (i,j): {1, 43083}, {48, 14592}, {63, 14582}, {94, 822}, {125, 36061}, {255, 10412}, {265, 656}, {326, 15475}, {328, 810}, {476, 2632}, {520, 2166}, {1577, 50433}, {1989, 24018}, {2159, 18557}, {2349, 18558}, {2618, 50463}, {2972, 36129}, {3269, 32680}, {3682, 43082}, {14208, 52153}, {14560, 17879}, {15526, 32678}, {20902, 32662}, {36035, 50464}, {37754, 46456}
X(53176) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 43083}, {1249, 14592}, {3162, 14582}, {3163, 18557}, {3284, 41077}, {6523, 10412}, {11597, 520}, {15259, 15475}, {16221, 125}, {17433, 35442}, {18334, 15526}, {18402, 6368}, {34544, 24018}, {38993, 41998}, {38994, 41997}, {39062, 328}, {40596, 265}, {40604, 3265}
X(53176) = barycentric product X(i)*X(j) for these {i,j}: {4, 14590}, {50, 6528}, {99, 52418}, {107, 323}, {110, 14165}, {112, 340}, {162, 52414}, {186, 648}, {250, 44427}, {264, 14591}, {393, 10411}, {526, 23582}, {687, 1986}, {823, 6149}, {933, 14918}, {1154, 16813}, {1304, 14920}, {1511, 15459}, {2052, 52603}, {2624, 23999}, {3043, 46456}, {3268, 23964}, {5962, 41679}, {6148, 32695}, {6331, 34397}, {6529, 52437}, {7799, 32713}, {8552, 32230}, {11062, 18831}, {15352, 22115}, {16077, 39176}, {16237, 38936}, {18020, 47230}, {20031, 51383}, {24000, 32679}, {30450, 52416}, {35139, 36423}, {35235, 47443}, {37778, 51478}, {37802, 52917}, {38342, 52417}, {42308, 52743}, {42701, 52920}, {44067, 46812}, {44068, 46815}
X(53176) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 14592}, {6, 43083}, {25, 14582}, {30, 18557}, {50, 520}, {107, 94}, {112, 265}, {186, 525}, {323, 3265}, {340, 3267}, {393, 10412}, {526, 15526}, {648, 328}, {1495, 18558}, {1511, 41077}, {1576, 50433}, {1986, 6334}, {2081, 35442}, {2088, 5489}, {2207, 15475}, {2420, 51254}, {2624, 2632}, {3043, 8552}, {3268, 36793}, {5317, 43082}, {6137, 41998}, {6138, 41997}, {6149, 24018}, {6528, 20573}, {6529, 6344}, {7799, 52617}, {8745, 43088}, {10411, 3926}, {11062, 6368}, {14165, 850}, {14270, 3269}, {14586, 50463}, {14590, 69}, {14591, 3}, {15352, 18817}, {16186, 23616}, {16813, 46138}, {19627, 39201}, {22115, 52613}, {23582, 35139}, {23964, 476}, {23977, 43089}, {24000, 32680}, {24019, 2166}, {32230, 46456}, {32640, 50464}, {32679, 17879}, {32695, 5627}, {32708, 12028}, {32713, 1989}, {32715, 11079}, {34397, 647}, {35907, 43087}, {36423, 526}, {38936, 15421}, {39176, 9033}, {41937, 14560}, {44067, 46811}, {44068, 46814}, {44427, 339}, {47230, 125}, {52179, 35911}, {52414, 14208}, {52416, 52584}, {52418, 523}, {52437, 4143}, {52603, 394}, {52743, 1650}, {52917, 18883}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 52917, 107}, {110, 52918, 52913}, {5502, 23347, 46587}, {5502, 46587, 1304}

---

X(53177) = X(4)X(526)∩X(23)X(33752)

Barycentrics    a^2*(b^2 - c^2)*(a^4 - b^4 + b^2*c^2 - c^4)*(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(53177) lies on the cubic K027 and these lines: {4, 690}, {23, 33752}, {523, 52192}, {842, 7418}, {850, 18867}, {1383, 14998}, {9138, 34417}, {9517, 14246}, {9979, 52449}, {10097, 10630}, {10412, 43665}, {13485, 34765}, {13574, 44010}

X(53177) = X(i)-isoconjugate of X(j) for these (i,j): {2157, 14999}, {2247, 17708}
X(53177) = X(i)-Dao conjugate of X(j) for these (i,j): {5099, 542}, {40583, 14999}
X(53177) = trilinear pole of line {2492, 47415}
X(53177) = barycentric product X(i)*X(j) for these {i,j}: {23, 14223}, {316, 14998}, {842, 9979}, {2492, 5641}, {10561, 52094}, {14246, 50942}, {35909, 37765}, {46787, 52076}
X(53177) = barycentric quotient X(i)/X(j) for these {i,j}: {23, 14999}, {842, 17708}, {2492, 542}, {8744, 7473}, {10561, 16092}, {14223, 18019}, {14246, 50941}, {14998, 67}, {35909, 34897}, {47415, 39474}, {52076, 46786}

---

X(53178) = X(4)X(526)∩X(523)X(3134)

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

X(53178) lies on the cubic K027 and these lines: {4, 526}, {477, 16171}, {523, 3134}, {1138, 2411}, {1990, 52743}, {2492, 34288}, {5627, 10412}, {9033, 14254}, {9214, 46789}, {23105, 46081}, {35522, 36889}

X(53178) = midpoint of X(14220) and X(39985)
X(53178) = X(i)-isoconjugate of X(j) for these (i,j): {163, 46788}, {4575, 52493}, {5663, 36034}, {7480, 35200}
X(53178) = X(i)-Dao conjugate of X(j) for these (i,j): {30, 42742}, {115, 46788}, {133, 7480}, {136, 52493}, {3258, 5663}
X(53178) = trilinear pole of line {1637, 47414}
X(53178) = barycentric product X(i)*X(j) for these {i,j}: {477, 41079}, {523, 46789}, {525, 52494}, {2411, 14254}, {5664, 43707}, {14220, 46106}, {36035, 36102}
X(53178) = barycentric quotient X(i)/X(j) for these {i,j}: {477, 44769}, {523, 46788}, {1637, 5663}, {1990, 7480}, {2436, 14385}, {2501, 52493}, {3163, 42742}, {14220, 14919}, {14254, 2410}, {14583, 2437}, {32650, 15395}, {36151, 36034}, {41079, 35520}, {43707, 39290}, {46789, 99}, {52494, 648}

---

X(53179) = X(99)X(527)∩X(109)X(187)

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

X(53179) lies on the circumcircle and these lines: {6, 2701}, {98, 28292}, {99, 527}, {100, 41322}, {109, 187}, {110, 1055}, {111, 663}, {284, 691}, {511, 28291}, {512, 2291}, {573, 2705}, {934, 51653}, {1400, 14733}, {2030, 26715}, {2702, 4262}, {8694, 50361}

X(53179) = reflection of X(2291) in the Brocard axis
X(53179) = Schoutte-circle-inverse of X(109)
X(53179) = X(524)-isoconjugate of X(14202)
X(53179) = trilinear pole of line {6, 5075}
X(53179) = barycentric quotient X(923)/X(14202)

---

X(53180) = X(99)X(528)∩X(110)X(3110)

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

X(53180) lies on the circumcircle and these lines: {98, 2826}, {99, 528}, {101, 24436}, {105, 14419}, {110, 3110}, {111, 665}, {187, 919}, {214, 8691}, {511, 2742}, {512, 840}, {691, 3286}, {927, 7181}, {2702, 5030}, {2703, 33844}

X(53180) = reflection of X(840) in the Brocard axis
X(53180) = Schoutte-circle-inverse of X(919)
X(53180) = trilinear pole of line {6, 5098}

---

X(53181) = X(100)X(527)∩X(101)X(1155)

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

X(53180) lies on the circumcircle and these lines: {36, 14074}, {55, 1308}, {57, 14733}, {100, 527}, {101, 1155}, {104, 28292}, {105, 47757}, {109, 6610}, {165, 2742}, {484, 20219}, {513, 2291}, {517, 28291}, {901, 35445}, {934, 2078}, {1292, 5537}, {2077, 30237}, {2222, 37541}

X(53181) = reflection of X(2291) in the OI line
X(53181) = trilinear pole of line {6, 14413}

---

X(53182) = X(101)X(525)∩X(103)X(1503)

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

X(53182) lies on the circumcircle and these lines: {3, 2741}, {100, 14208}, {101, 525}, {103, 1503}, {107, 46107}, {108, 4077}, {109, 17094}, {110, 4025}, {112, 514}, {516, 1297}, {5088, 39435}

X(53182) = reflection of X(2741) in X(3)
X(53182) = isogonal conjugate of X(9518)
X(53182) = isogonal conjugate of the anticomplement of X(9518)
X(53182) = isogonal conjugate of the complement of X(9518)
X(53182) = Thomson-isogonal conjugate of X(2825)
X(53182) = X(i)-isoconjugate of X(j) for these (i,j): {1, 9518}, {692, 46535}
X(53182) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 9518}, {1086, 46535}
X(53182) = trilinear pole of line {6, 4466}
X(53182) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 9518}, {514, 46535}

---

X(53183) = X(101)X(527)∩X(103)X(28292)

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

X(53183) lies on the circumcircle and these lines: {7, 14733}, {100, 5088}, {101, 527}, {103, 28292}, {108, 38461}, {109, 1323}, {514, 2291}, {516, 28291}, {901, 36887}, {1019, 2249}, {2701, 3663}

X(53183) = Collings transform of X(40629)
X(53183) = trilinear pole of line {6, 1638}

---

X(53184) = X(104)X(527)∩X(513)X(28291)

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

X(53184) lies on the circumcircle and these lines: {36, 15728}, {40, 43080}, {100, 28292}, {103, 50371}, {104, 527}, {513, 28291}, {517, 2291}, {840, 3576}, {1477, 5126}, {2077, 15731}, {2716, 38531}, {2717, 3428}, {2720, 23890}, {14733, 24029}

X(53183) = reflection of X(28291) in the OI line

---

X(53185) = X(110)X(527)∩X(523)X(2291)

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

X(53185) lies on the circumcircle and these lines: {23, 9086}, {30, 28291}, {74, 28292}, {110, 527}, {111, 47800}, {112, 23710}, {226, 14733}, {523, 2291}, {691, 35935}, {1005, 1290}, {2453, 2689}, {2690, 36014}

X(53185) = isogonal conjugate of X(44764)
X(53185) = reflection of X(2291) in the Euler line
X(53185) = isogonal conjugate of the anticomplement of X(44764)
X(53185) = isogonal conjugate of the complement of X(44764)
X(53185) = X(1)-isoconjugate of X(44764)
X(53185) = X(3)-Dao conjugate of X(44764)
X(53185) = trilinear pole of line {6, 30574}
X(53185) = barycentric quotient X(6)/X(44764)

---

X(53186) = X(110)X(6390)∩X(111)X(525)

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

X(53186) lies on the circumcircle and these lines: {69, 691}, {107, 44146}, {110, 6390}, {111, 525}, {112, 524}, {1296, 1503}, {1297, 1499}, {1302, 5971}, {1304, 36890}, {5108, 9091}, {10098, 11180}, {10420, 38940}, {14916, 35188}, {45773, 47389}

X(53186) = isotomic conjugate of the anticomplement of X(41359)
X(53186) = trilinear pole of line {6, 14417}

---

X(53187) = X(112)X(526)∩X(476)X(525)

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

X(53187) lies on the circumcircle and these lines: {74, 7669}, {98, 15357}, {99, 45792}, {107, 44427}, {110, 8552}, {112, 526}, {476, 525}, {477, 1503}, {842, 12112}, {1294, 12383}, {1297, 5663}, {5191, 9184}, {23969, 35909}

X(53187) = trilinear pole of line {6, 16186}

---

X(53188) = X(112)X(5663)∩X(476)X(1503)

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

X(53188) lies on the circumcircle and these lines: {74, 44810}, {107, 3448}, {112, 5663}, {476, 1503}, {477, 525}, {526, 1297}, {1304, 35265}, {2857, 30474}, {11579, 23969}, {18337, 39447}, {32687, 52418}, {32690, 34802}

---

X(53189) = X(30)X(675)∩X(74)X(674)

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

X(53189) lies on the circumcircle and these lines: {30, 675}, {74, 674}, {98, 36026}, {103, 7464}, {110, 36032}, {186, 9085}, {376, 2688}, {476, 4237}, {523, 44876}, {842, 7430}, {917, 10295}, {1006, 2752}, {1302, 7479}, {1304, 4249}, {2697, 30266}, {2770, 6998}, {4243, 9060}, {7453, 10102}, {9058, 36031}

X(53189) = reflection of X(44876) in the Euler line

---

X(53190) = X(23)X(101)∩X(110)X(674)

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

X(53190) lies on the circumcircle and these lines: {2, 2690}, {23, 101}, {27, 935}, {30, 44876}, {42, 32682}, {98, 47771}, {99, 3006}, {100, 16568}, {110, 674}, {111, 6586}, {112, 2073}, {468, 26705}, {476, 7474}, {477, 7433}, {523, 675}, {691, 4184}, {841, 7440}, {842, 7432}, {858, 1305}, {1290, 7465}, {2222, 5297}, {2373, 20294}, {2687, 7445}, {2689, 50404}, {2691, 7411}, {2696, 4229}, {2702, 46407}, {2759, 6790}, {2766, 7466}, {4219, 10101}, {4511, 8691}, {7426, 9057}, {7431, 10098}, {9076, 10566}, {26710, 37760}, {37166, 43660}

X(53190) = reflection of X(675) in the Euler line
X(53190) = trilinear pole of line {6, 2774}

---

X(53191) = X(99)X(521)∩X(648)X(650)

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

X(53191) lies on the Steiner circumellipse and these lines: {99, 521}, {190, 8611}, {523, 18026}, {648, 650}, {656, 664}, {668, 52355}, {670, 35518}, {2481, 43746}, {4569, 17094}, {6528, 44426}, {35149, 35960}

X(53191) = isotomic conjugate of X(2798)
X(53191) = isotomic conjugate of the isogonal conjugate of X(2714)
X(53191) = X(i)-isoconjugate of X(j) for these (i,j): {31, 2798}, {425, 810}, {512, 23695}, {663, 41349}
X(53191) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 2798}, {39054, 23695}, {39062, 425}
X(53191) = cevapoint of X(2) and X(2798)
X(53191) = barycentric product X(i)*X(j) for these {i,j}: {76, 2714}, {4554, 43746}
X(53191) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2798}, {648, 425}, {651, 41349}, {662, 23695}, {2714, 6}, {43746, 650}

---

X(53192) = X(99)X(526)∩X(523)X(35139)

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

X(53192) lies on the Steiner circumellipse and these lines: {99, 526}, {290, 35520}, {523, 35139}, {648, 47230}, {670, 3268}, {892, 9213}, {2611, 14616}, {4590, 15470}, {5641, 52149}, {6137, 23896}, {6138, 23895}, {8901, 46138}, {16077, 33799}, {18823, 44555}

X(53192) = isotomic conjugate of the isogonal conjugate of X(9160)
X(53192) = X(i)-isoconjugate of X(j) for these (i,j): {661, 32761}, {798, 40879}
X(53192) = X(i)-Dao conjugate of X(j) for these (i,j): {31998, 40879}, {36830, 32761}
X(53192) = cevapoint of X(523) and X(18122)
X(53192) = trilinear pole of line {2, 2088}
X(53192) = barycentric product X(76)*X(9160)
X(53192) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 40879}, {110, 32761}, {9160, 6}

---

X(53193) = X(99)X(527)∩X(523)X(1121)

Barycentrics    (a^4 + a^3*b - 4*a^2*b^2 + a*b^3 + b^4 - 2*a^3*c + a^2*b*c + a*b^2*c - 2*b^3*c + 2*a^2*c^2 - 2*a*b*c^2 + 2*b^2*c^2 + a*c^3 + b*c^3 - 2*c^4)*(a^4 - 2*a^3*b + 2*a^2*b^2 + a*b^3 - 2*b^4 + a^3*c + a^2*b*c - 2*a*b^2*c + b^3*c - 4*a^2*c^2 + a*b*c^2 + 2*b^2*c^2 + a*c^3 - 2*b*c^3 + c^4) : :
X(53193) = 4 X[35086] - X[35154]

X(53193) lies on the Steiner circumellipse and these lines: {2, 35086}, {99, 527}, {190, 3712}, {226, 35157}, {333, 892}, {415, 648}, {522, 671}, {523, 1121}, {524, 664}, {666, 50093}, {903, 45669}, {4417, 46143}, {17346, 35148}

X(53193) = reflection of X(i) in X(j) for these {i,j}: {2, 35086}, {35154, 2}
X(53193) = X(187)-isoconjugate of X(14202)
X(53193) = trilinear pole of line {2, 2785}
X(53193) = barycentric quotient X(897)/X(14202)

---

X(53194) = X(190)X(511)∩X(290)X(514)

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

X(53194) lies on the Steiner circumellipse and these lines: {58, 2966}, {99, 17209}, {190, 511}, {290, 514}, {664, 43034}, {668, 1959}, {670, 51370}, {903, 23878}, {18829, 33297}, {24281, 35169}, {35148, 36205}

trilinear pole of line {2, 24353}

---

X(53195) = X(99)X(649)∩X(190)X(512)

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

X(53195) lies on the Steiner circumellipse and these lines: {99, 649}, {190, 512}, {239, 18826}, {514, 670}, {519, 3228}, {538, 903}, {661, 668}, {664, 7180}, {1573, 24505}, {3978, 43093}, {4569, 7216}, {14616, 49755}, {16829, 18822}, {18001, 35148}, {21352, 35172}, {24275, 24502}, {30229, 31036}, {31136, 35168}, {34362, 35153}, {35173, 35957}

X(53195) = X(i)-isoconjugate of X(j) for these (i,j): {667, 40859}, {810, 15147}
X(53195) = X(i)-Dao conjugate of X(j) for these (i,j): {6631, 40859}, {39062, 15147}
X(53195) = cevapoint of X(514) and X(30109)
X(53195) = trilinear pole of line {2, 3122}
X(53195) = barycentric quotient X(i)/X(j) for these {i,j}: {190, 40859}, {648, 15147}

---

X(53196) = X(99)X(6037)∩X(290)X(511)

Barycentrics    b^2*(a^2 - b^2)*(a^2 - c^2)*c^2*(-(a^2*b^2) + b^4 - 2*a^2*c^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)*(2*a^2*b^2 + a^2*c^2 + b^2*c^2 - c^4) : :
X(53196) = X[290] - 3 X[39058]

X(53196) lies on the Steiner circumellipse and these lines: {2, 39009}, {98, 43664}, {99, 6037}, {262, 46142}, {263, 30226}, {290, 511}, {327, 5641}, {576, 14382}, {648, 22456}, {670, 15631}, {685, 18831}, {850, 36885}, {877, 35362}, {1494, 16083}, {2966, 14966}, {3225, 39941}, {4577, 17932}, {4586, 36132}, {17984, 35142}, {34359, 42313}, {35140, 44137}, {46786, 46807}

X(53196) = isogonal conjugate of X(9420)
X(53196) = anticomplement of X(39009)
X(53196) = isotomic conjugate of the isogonal conjugate of X(6037)
X(53196) = X(i)-isoconjugate of X(j) for these (i,j): {1, 9420}, {1755, 3288}, {1910, 33569}, {1924, 51373}, {2491, 52134}, {6784, 23997}, {9417, 23878}, {36132, 39009}
X(53196) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 9420}, {9428, 51373}, {11672, 33569}, {36899, 3288}, {39058, 23878}
X(53196) = cevapoint of X(i) and X(j) for these (i,j): {511, 23878}, {2799, 24206}
X(53196) = trilinear pole of line {2, 290}
X(53196) = barycentric product X(i)*X(j) for these {i,j}: {76, 6037}, {262, 43187}, {327, 2966}, {561, 36132}, {1502, 32716}, {18024, 26714}, {22456, 42313}
X(53196) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 9420}, {98, 3288}, {262, 3569}, {263, 2491}, {290, 23878}, {327, 2799}, {511, 33569}, {670, 51373}, {685, 10311}, {2395, 6784}, {2715, 34396}, {2966, 182}, {6037, 6}, {22456, 458}, {26714, 237}, {32716, 32}, {36036, 52134}, {36132, 31}, {36897, 39680}, {39681, 36213}, {41173, 51542}, {42313, 684}, {43187, 183}, {43718, 39469}, {46786, 45321}, {46807, 41167}, {52926, 52967}

---

X(53197) = X(99)X(237)∩X(290)X(512)

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

X(53197) lies on the Steiner circumellipse and these lines: {32, 2966}, {76, 14251}, {99, 237}, {263, 30226}, {290, 512}, {316, 52446}, {511, 670}, {648, 2211}, {668, 5360}, {886, 20023}, {892, 44155}, {1237, 4562}, {3228, 23878}, {6528, 34854}, {14970, 52618}, {20022, 42371}, {30229, 40814}, {35136, 44137}

X(53197) = isogonal conjugate of X(21444)
X(53197) = isotomic conjugate of X(34383)
X(53197) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21444}, {31, 34383}
X(53197) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 34383}, {3, 21444}
X(53197) = cevapoint of X(2) and X(34383)
X(53197) = trilinear pole of line {2, 2491}
X(53197) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 34383}, {6, 21444}

---

X(53198) = X(290)X(513)∩X(511)X(668)

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

X(53198) lies on the Steiner circumellipse and these lines: {190, 1755}, {290, 513}, {314, 18829}, {511, 668}, {664, 1966}, {670, 51369}, {889, 42711}, {1333, 2966}, {3227, 23878}, {4562, 17787}, {17984, 18026}, {35156, 44155}

---

X(53199) = X(290)X(524)∩X(511)X(671)

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

X(53199) lies on the Steiner circumellipse and these lines: {2, 46142}, {99, 23878}, {290, 524}, {385, 43664}, {511, 671}, {599, 5641}, {892, 2421}, {1494, 7840}, {1992, 35146}, {2966, 5467}, {3226, 37792}, {3228, 22329}, {7788, 46145}, {11163, 18823}, {14607, 35179}, {14999, 35138}, {23342, 46144}, {35165, 41629}, {44137, 46140}

X(53199) = reflection of X(46142) in X(2)
X(53199) = X(i)-isoconjugate of X(j) for these (i,j): {661, 2080}, {798, 39099}, {2642, 21460}
X(53199) = X(i)-Dao conjugate of X(j) for these (i,j): {31998, 39099}, {36830, 2080}
X(53199) = cevapoint of X(523) and X(15993)
X(53199) = trilinear pole of line {2, 2782}
X(53199) = barycentric product X(i)*X(j) for these {i,j}: {99, 43532}, {670, 46316}
X(53199) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 39099}, {110, 2080}, {691, 21460}, {805, 45146}, {43532, 523}, {46316, 512}

---

X(53200) = X(290)X(525)∩X(511)X(648)

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

X(53200) lies on the Steiner circumellipse and these lines: {3, 2966}, {4, 40804}, {30, 39682}, {99, 401}, {290, 525}, {297, 6528}, {315, 18829}, {458, 16077}, {511, 648}, {670, 6393}, {1494, 23878}, {3269, 34536}, {17984, 34138}, {22456, 39000}, {30227, 37190}, {34359, 42313}, {34538, 51334}, {37200, 39265}, {40802, 41074}

X(53200) = polar conjugate of X(47202)
X(53200) = X(i)-isoconjugate of X(j) for these (i,j): {48, 47202}, {9417, 16083}
X(53200) = X(i)-Dao conjugate of X(j) for these (i,j): {1249, 47202}, {39058, 16083}
X(53200) = trilinear pole of line {2, 684}
X(53200) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 47202}, {290, 16083}

---

X(53201) = X(30)X(648)∩X(525)X(1494)

Barycentrics    (a^8 + 2*a^6*b^2 - 6*a^4*b^4 + 2*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 + 3*a^4*c^4 - 8*a^2*b^2*c^4 + 3*b^4*c^4 + 2*a^2*c^6 + 2*b^2*c^6 - 2*c^8)*(a^8 - 4*a^6*b^2 + 3*a^4*b^4 + 2*a^2*b^6 - 2*b^8 + 2*a^6*c^2 + 4*a^4*b^2*c^2 - 8*a^2*b^4*c^2 + 2*b^6*c^2 - 6*a^4*c^4 + 4*a^2*b^2*c^4 + 3*b^4*c^4 + 2*a^2*c^6 - 4*b^2*c^6 + c^8) : :
X(53201) = X[16077] - 4 X[39008], 4 X[3163] - 3 X[23582], 3 X[23582] - 2 X[44653], 3 X[15351] + X[39358], 4 X[15526] - 3 X[31621], 5 X[39062] - 8 X[46115], X[39352] + 3 X[46270]

X(53201) lies on the Steiner circumellipse and these lines: {2, 16077}, {30, 648}, {99, 11064}, {376, 2966}, {401, 46809}, {525, 1494}, {671, 42738}, {3163, 23582}, {3543, 52485}, {4846, 41433}, {6528, 40885}, {11050, 47071}, {15351, 39358}, {15526, 31621}, {18831, 43768}, {31153, 35169}, {39062, 46115}, {39352, 46270}

X(53201) = reflection of X(i) in X(j) for these {i,j}: {2, 39008}, {16077, 2}, {44653, 3163}
X(53201) = polar conjugate of X(47204)
X(53201) = X(i)-isoconjugate of X(j) for these (i,j): {48, 47204}, {163, 42733}, {1651, 2159}, {9406, 16076}, {32676, 52720}
X(53201) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 42733}, {1249, 47204}, {3163, 1651}, {9410, 16076}, {15526, 52720}
X(53201) = trilinear pole of line {2, 9033}
X(53201) = barycentric product X(i)*X(j) for these {i,j}: {1494, 16075}, {3260, 41433}, {16077, 47071}
X(53201) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 47204}, {30, 1651}, {523, 42733}, {525, 52720}, {1494, 16076}, {16075, 30}, {41433, 74}, {47071, 9033}
X(53201) = {X(3163),X(44653)}-harmonic conjugate of X(23582)

---

X(53202) = X(30)X(3228)∩X(525)X(670)

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

X(53202) lies on the Steiner circumellipse and these lines: {30, 3228}, {99, 647}, {290, 47286}, {512, 648}, {525, 670}, {538, 1494}, {878, 2966}, {892, 10097}, {2433, 16077}, {2501, 6528}, {2549, 46142}, {2623, 18831}, {3978, 46140}, {5641, 7841}, {11333, 34360}, {14582, 35139}, {30227, 37190}, {30229, 40814}, {30491, 35138}, {34344, 35146}

X(53202) = isotomic conjugate of X(9035)
X(53202) = polar conjugate of X(47206)
X(53202) = isotomic conjugate of the isogonal conjugate of X(9091)
X(53202) = X(i)-isoconjugate of X(j) for these (i,j): {31, 9035}, {48, 47206}, {662, 865}, {810, 15014}, {1924, 16084}
X(53202) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 9035}, {1084, 865}, {1249, 47206}, {9428, 16084}, {39062, 15014}
X(53202) = cevapoint of X(2) and X(9035)
X(53202) = trilinear pole of line {2, 16098}
X(53202) = barycentric product X(i)*X(j) for these {i,j}: {76, 9091}, {670, 16098}
X(53202) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 9035}, {4, 47206}, {512, 865}, {648, 15014}, {670, 16084}, {9091, 6}, {16098, 512}

---

X(53203) = X(99)X(905)∩X(513)X(648)

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

X(53203) lies on the Steiner circumellipse and these lines: {30, 3227}, {99, 905}, {190, 656}, {513, 648}, {525, 668}, {536, 1494}, {664, 51664}, {666, 10099}, {670, 15413}, {889, 42716}, {2966, 4236}, {4872, 18827}, {6528, 17924}, {7178, 18026}, {17678, 35161}, {18822, 19796}, {34343, 35155}, {35169, 46499}

X(53203) = X(i)-isoconjugate of X(j) for these (i,j): {101, 866}, {810, 44330}, {1919, 16085}
X(53203) = X(i)-Dao conjugate of X(j) for these (i,j): {1015, 866}, {9296, 16085}, {39062, 44330}
X(53203) = trilinear pole of line {2, 16100}
X(53203) = barycentric product X(668)*X(16100)
X(53203) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 866}, {648, 44330}, {668, 16085}, {16100, 513}

---

X(53204) = X(99)X(448)∩X(518)X(648)

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

X(53204) lies on the Steiner circumellipse and these lines: {21, 2966}, {30, 32041}, {72, 666}, {99, 448}, {190, 44694}, {290, 4391}, {297, 18026}, {518, 648}, {525, 2481}, {664, 1959}, {1231, 46135}, {1494, 4762}, {4562, 7270}, {6528, 46108}, {16077, 31926}, {35169, 37086}

X(53204) = polar conjugate of X(47212)
X(53204) = X(i)-isoconjugate of X(j) for these (i,j): {48, 47212}, {9454, 16087}
X(53204) = X(i)-Dao conjugate of X(j) for these (i,j): {1249, 47212}, {33675, 16087}
X(53204) = trilinear pole of line {2, 47203}
X(53204) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 47212}, {2481, 16087}

---

X(53205) = X(290)X(297)∩X(520)X(648)

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

X(53205) lies on the Steiner circumellipse and these lines: {4, 40804}, {30, 14941}, {99, 39062}, {290, 297}, {317, 39359}, {401, 46841}, {520, 648}, {525, 6528}, {670, 4143}, {1494, 1972}, {1625, 18831}, {1956, 35145}, {2966, 4230}, {5641, 52282}, {7473, 35178}, {35142, 40887}, {35474, 39683}, {46138, 46456}

X(53205) = reflection of X(i) in X(j) for these {i,j}: {401, 46841}, {16089, 297}
X(53205) = polar conjugate of X(6130)
X(53205) = X(i)-isoconjugate of X(j) for these (i,j): {48, 6130}, {401, 810}, {647, 1955}, {656, 1971}, {822, 41204}, {2313, 23286}, {36084, 38974}
X(53205) = X(i)-Dao conjugate of X(j) for these (i,j): {1249, 6130}, {38987, 38974}, {39052, 1955}, {39062, 401}, {40596, 1971}
X(53205) = cevapoint of X(i) and X(j) for these (i,j): {4, 3569}, {297, 525}, {520, 46841}, {1625, 4230}, {17434, 32428}
X(53205) = trilinear pole of line {2, 1972}
X(53205) = barycentric product X(i)*X(j) for these {i,j}: {5, 41208}, {343, 41210}, {648, 1972}, {811, 1956}, {1987, 6331}, {6528, 14941}, {22456, 40804}
X(53205) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 6130}, {107, 41204}, {112, 1971}, {162, 1955}, {648, 401}, {685, 32545}, {1298, 23286}, {1956, 656}, {1972, 525}, {1987, 647}, {3569, 38974}, {4230, 52128}, {6331, 44137}, {6528, 16089}, {14941, 520}, {35360, 32428}, {40804, 684}, {41208, 95}, {41210, 275}, {52177, 39201}, {53175, 34980}

---

X(53206) = X(30)X(1121)∩X(522)X(648)

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

X(53206) lies on the Steiner circumellipse and these lines: {30, 1121}, {99, 6332}, {190, 52355}, {290, 43694}, {522, 648}, {525, 664}, {527, 1494}, {1577, 18026}, {2966, 7462}, {6528, 46110}, {8558, 44331}, {10538, 35145}, {17579, 35161}

X(53206) = reflection of X(44331) in X(8558)
X(53206) = polar conjugate of X(47210)
X(53206) = X(i)-isoconjugate of X(j) for these (i,j): {48, 47210}, {647, 14192}, {810, 44331}, {3063, 16091}
X(53206) = X(i)-Dao conjugate of X(j) for these (i,j): {1249, 47210}, {10001, 16091}, {39052, 14192}, {39062, 44331}
X(53206) = cevapoint of X(i) and X(j) for these (i,j): {522, 8558}, {652, 3100}
X(53206) = trilinear pole of line {2, 14544}
X(53206) = barycentric product X(6331)*X(43694)
X(53206) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 47210}, {162, 14192}, {648, 44331}, {664, 16091}, {43694, 647}

---

X(53207) = X(30)X(664)∩X(525)X(1121)

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

X(53207) lies on the Steiner circumellipse and these lines: {29, 16077}, {30, 664}, {99, 51382}, {190, 7359}, {307, 35157}, {522, 1494}, {525, 1121}, {527, 648}, {666, 17781}, {1784, 18026}, {2966, 35935}, {11114, 35169}, {30228, 35154}

X(53207) = trilinear pole of line {2, 14400}

---

X(53208) = X(190)X(650)∩X(513)X(664)

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

X(53208) lies on the Steiner circumellipse and these lines: {99, 3737}, {190, 650}, {350, 18816}, {513, 664}, {522, 668}, {527, 3227}, {536, 1121}, {666, 1024}, {670, 18155}, {903, 19634}, {1266, 2481}, {1275, 6613}, {2397, 4562}, {3226, 9432}, {3509, 35143}, {3676, 4569}, {4389, 35175}, {4419, 35167}, {4555, 23838}, {7649, 18026}, {17139, 18827}, {17274, 18821}, {17759, 35144}, {17950, 35176}, {18822, 50101}, {35157, 35348}, {40862, 43062}, {40875, 40880}

X(53208) = reflection of X(40875) in X(40880)
X(53208) = X(i)-isoconjugate of X(j) for these (i,j): {663, 9364}, {667, 5205}, {810, 15150}, {1919, 40875}, {3063, 40862}
X(53208) = X(i)-Dao conjugate of X(j) for these (i,j): {6631, 5205}, {9296, 40875}, {10001, 40862}, {39062, 15150}
X(53208) = cevapoint of X(i) and X(j) for these (i,j): {514, 5121}, {522, 40880}
X(53208) = trilinear pole of line {2, 2170}
X(53208) = barycentric product X(i)*X(j) for these {i,j}: {664, 52517}, {1978, 9432}, {4554, 9365}
X(53208) = barycentric quotient X(i)/X(j) for these {i,j}: {190, 5205}, {648, 15150}, {651, 9364}, {664, 40862}, {668, 40875}, {9365, 650}, {9432, 649}, {52517, 522}

---

X(53209) = X(190)X(1944)∩X(517)X(664)

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

X(53209) lies on the Steiner circumellipse and these lines: {63, 666}, {190, 1944}, {517, 664}, {522, 18816}, {648, 18206}, {668, 6735}, {1785, 9436}, {2481, 4025}, {4560, 35145}, {4569, 22464}, {7182, 46135}, {33298, 35174}, {37629, 46136}

X(53209) = X(2223)-isoconjugate of X(52480)
X(53209) = trilinear pole of line {2, 46393}
X(53209) = barycentric quotient X(i)/X(j) for these {i,j}: {673, 52480}, {18816, 46794}, {34234, 14198}

---

X(53210) = X(190)X(3693)∩X(518)X(664)

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

X(53210) lies on the Steiner circumellipse and these lines: {9, 666}, {69, 4562}, {75, 46135}, {99, 12032}, {190, 3693}, {518, 664}, {522, 2481}, {527, 32041}, {648, 33295}, {668, 3717}, {1121, 4762}, {1861, 10030}, {4569, 9436}, {15419, 18827}, {24411, 35157}, {39959, 41075}

X(53210) = isotomic conjugate of X(28850)
X(53210) = isotomic conjugate of the isogonal conjugate of X(12032)
X(53210) = X(i)-isoconjugate of X(j) for these (i,j): {31, 28850}, {1415, 28143}, {2223, 14197}, {9454, 46792}
X(53210) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 28850}, {1146, 28143}, {33675, 46792}
X(53210) = cevapoint of X(2) and X(28850)
X(53210) = trilinear pole of line {2, 46793}
X(53210) = barycentric product X(i)*X(j) for these {i,j}: {76, 12032}, {2481, 46793}, {18031, 52004}
X(53210) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 28850}, {522, 28143}, {673, 14197}, {2481, 46792}, {12032, 6}, {46793, 518}, {52004, 672}

---

X(53211) = X(99)X(1275)∩X(521)X(664)

Barycentrics    (a - b)*(a - c)*(a + b - c)^2*(a - b + 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(53211) lies on the Steiner circumellipse and these lines: {99, 1275}, {521, 664}, {522, 18026}, {527, 40843}, {648, 1020}, {666, 24029}, {1121, 1952}, {1937, 2481}, {1944, 44360}, {4025, 4569}, {6354, 35086}, {9436, 18816}, {14616, 16090}, {17950, 35145}, {37628, 41353}

X(53211) = midpoint of X(17950) and X(44354)
X(53211) = reflection of X(1944) in X(44360)
X(53211) = X(i)-isoconjugate of X(j) for these (i,j): {243, 1946}, {521, 51726}, {650, 1951}, {652, 2202}, {663, 1936}, {667, 7360}, {810, 15146}, {851, 21789}, {1021, 42669}, {1944, 3063}, {3270, 23353}, {3900, 26884}, {5088, 8641}, {7253, 44112}
X(53211) = X(i)-Dao conjugate of X(j) for these (i,j): {6631, 7360}, {10001, 1944}, {39053, 243}, {39060, 1948}, {39062, 15146}
X(53211) = cevapoint of X(i) and X(j) for these (i,j): {521, 44360}, {656, 8680}, {6354, 18006}
X(53211) = trilinear pole of line {2, 1952}
X(53211) = barycentric product X(i)*X(j) for these {i,j}: {296, 46404}, {307, 41207}, {664, 1952}, {1441, 41206}, {1937, 4554}, {1945, 4572}, {4566, 35145}, {18026, 40843}
X(53211) = barycentric quotient X(i)/X(j) for these {i,j}: {108, 2202}, {109, 1951}, {190, 7360}, {296, 652}, {648, 15146}, {651, 1936}, {653, 243}, {658, 5088}, {664, 1944}, {1020, 851}, {1021, 1984}, {1461, 26884}, {1937, 650}, {1945, 663}, {1949, 1946}, {1952, 522}, {2249, 21789}, {4566, 8680}, {6516, 6518}, {7128, 23353}, {18026, 1948}, {32674, 51726}, {32714, 1430}, {35145, 7253}, {37142, 1021}, {37755, 9391}, {40843, 521}, {41206, 21}, {41207, 29}, {52222, 3270}

---

X(53212) = X(522)X(1121)∩X(527)X(664)

Barycentrics    (a^4 + 2*a^3*b - 6*a^2*b^2 + 2*a*b^3 + b^4 - 4*a^3*c + 4*a^2*b*c + 4*a*b^2*c - 4*b^3*c + 3*a^2*c^2 - 8*a*b*c^2 + 3*b^2*c^2 + 2*a*c^3 + 2*b*c^3 - 2*c^4)*(a^4 - 4*a^3*b + 3*a^2*b^2 + 2*a*b^3 - 2*b^4 + 2*a^3*c + 4*a^2*b*c - 8*a*b^2*c + 2*b^3*c - 6*a^2*c^2 + 4*a*b*c^2 + 3*b^2*c^2 + 2*a*c^3 - 4*b*c^3 + c^4) : :
X(53212) = 4 X[35091] - X[35157], 3 X[1275] - 4 X[35110], 5 X[10001] - 8 X[36956]

X(53212) lies on the Steiner circumellipse and these lines: {2, 35091}, {190, 6745}, {522, 1121}, {527, 664}, {648, 52891}, {666, 6172}, {903, 44551}, {1275, 35110}, {4569, 37780}, {10001, 36956}, {18026, 37805}

X(53212) = reflection of X(i) in X(j) for these {i,j}: {2, 35091}, {35157, 2}
X(53212) = X(i)-isoconjugate of X(j) for these (i,j): {1415, 14476}, {14477, 34068}
X(53212) = X(i)-Dao conjugate of X(j) for these (i,j): {1146, 14476}, {35110, 14477}
X(53212) = trilinear pole of line {2, 6366}
X(53212) = barycentric quotient X(i)/X(j) for these {i,j}: {522, 14476}, {527, 14477}

---

X(53213) = X(99)X(40865)∩X(190)X(2254)

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

X(53213) lies on the Steiner circumellipse and these lines: {99, 40865}, {190, 2254}, {513, 666}, {528, 3227}, {536, 18821}, {668, 918}, {889, 42722}, {1086, 2481}, {17759, 35152}, {18822, 37756}, {24002, 46135}, {35155, 37857}

X(53213) = X(649)-isoconjugate of X(1083)
X(53213) = X(5375)-Dao conjugate of X(1083)
X(53213) = trilinear pole of line {2, 3675}
X(53213) = barycentric quotient X(100)/X(1083)

---

X(53214) = X(99)X(16728)∩X(190)X(2310)

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

X(53214) lies on the Steiner circumellipse and these lines: {99, 16728}, {190, 2310}, {518, 666}, {528, 14947}, {664, 673}, {668, 1146}, {812, 34905}, {918, 2481}, {1086, 4569}, {4555, 46798}, {4562, 32850}, {4762, 18821}, {5701, 40865}, {8735, 18026}, {33674, 35148}, {35174, 46792}, {40704, 46135}

X(53214) = reflection of X(40865) in X(5701)
X(53214) = X(i)-isoconjugate of X(j) for these (i,j): {672, 5091}, {2223, 9318}
X(53214) = cevapoint of X(518) and X(5701)
X(53214) = trilinear pole of line {2, 885}
X(53214) = barycentric product X(i)*X(j) for these {i,j}: {2481, 14947}, {9319, 18031}
X(53214) = barycentric quotient X(i)/X(j) for these {i,j}: {105, 5091}, {666, 40865}, {673, 9318}, {9319, 672}, {14947, 518}

---

X(53215) = X(30)X(668)∩X(190)X(2173)

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

X(53215) lies on the Steiner circumellipse and these lines: {28, 16077}, {30, 668}, {99, 51420}, {190, 2173}, {513, 1494}, {525, 3227}, {536, 648}, {664, 51654}, {671, 45664}, {889, 20336}, {903, 45686}, {2966, 16046}, {4562, 42033}, {13735, 35169}, {35147, 36216}

X(53215) = trilinear pole of line {2, 14399}

---

X(53216) = X(99)X(667)∩X(190)X(798)

Barycentrics    (a - b)*b*(a - c)*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)*(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(53216) lies on the Steiner circumellipse and these lines: {75, 35166}, {99, 667}, {190, 798}, {350, 18827}, {512, 668}, {513, 670}, {536, 3228}, {538, 3227}, {664, 51641}, {671, 43685}, {2107, 18826}, {2481, 3978}, {2665, 3226}, {4569, 7250}, {4602, 9428}, {17143, 35173}, {18002, 35147}, {18140, 35040}, {18825, 51333}, {32041, 36863}, {34363, 35155}, {35144, 40875}

X(53216) = reflection of X(40874) in X(350)
X(53216) = X(i)-isoconjugate of X(j) for these (i,j): {649, 21788}, {667, 2664}, {669, 2669}, {798, 2106}, {810, 15148}, {1919, 17759}, {1922, 27854}, {1924, 40874}, {1980, 52049}, {3572, 51331}, {9426, 41535}
X(53216) = X(i)-Dao conjugate of X(j) for these (i,j): {5375, 21788}, {6631, 2664}, {9296, 17759}, {9428, 40874}, {31998, 2106}, {39028, 27854}, {39062, 15148}
X(53216) = cevapoint of X(i) and X(j) for these (i,j): {75, 4010}, {350, 513}, {514, 46843}
X(53216) = trilinear pole of line {2, 3121}
X(53216) = barycentric product X(i)*X(j) for these {i,j}: {99, 43685}, {668, 39925}, {1978, 2665}, {2107, 4602}, {6386, 51333}
X(53216) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 2106}, {100, 21788}, {190, 2664}, {350, 27854}, {648, 15148}, {668, 17759}, {670, 40874}, {799, 2669}, {874, 39916}, {1332, 20796}, {1978, 52049}, {2107, 798}, {2665, 649}, {3573, 51331}, {3952, 21897}, {4562, 40796}, {4602, 41535}, {21832, 38978}, {27853, 39028}, {37207, 40772}, {39925, 513}, {40769, 8632}, {41072, 40742}, {43685, 523}, {51333, 667}

---

X(53217) = X(190)X(910)∩X(516)X(668)

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

X(5317) lies on the Steiner circumellipse and these lines: {75, 14727}, {190, 910}, {346, 4562}, {350, 4569}, {513, 18025}, {516, 668}, {536, 32040}, {664, 1456}, {666, 3729}, {2481, 20907}, {7253, 18827}

X(53217) = trilinear pole of line {2, 4148}

---

X(53218) = X(350)X(664)∩X(517)X(668)

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

X(53218) lies on the Steiner circumellipse and these lines: {76, 4555}, {99, 859}, {190, 2183}, {312, 4562}, {350, 664}, {513, 18816}, {517, 668}, {536, 46805}, {666, 51987}, {670, 17139}, {889, 36216}, {903, 3261}, {1875, 18026}, {4569, 18033}, {6635, 31625}, {18155, 18827}

X(53218) = isotomic conjugate of X(2810)
X(53218) = isotomic conjugate of the isogonal conjugate of X(2726)
X(53218) = X(31)-isoconjugate of X(2810)
X(53218) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 2810}, {46398, 45919}
X(53218) = cevapoint of X(2) and X(2810)
X(53218) = trilinear pole of line {2, 3310}
X(53218) = barycentric product X(i)*X(j) for these {i,j}: {76, 2726}, {18816, 46805}
X(53218) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2810}, {2726, 6}, {10015, 45919}, {18816, 46804}, {46805, 517}

---

X(53219) = X(6)X(666)∩X(190)X(350)

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

X(53219) lies on the Steiner circumellipse and these lines: {6, 666}, {7, 46135}, {75, 3252}, {99, 3286}, {190, 350}, {513, 2481}, {518, 668}, {536, 32041}, {664, 1458}, {670, 30941}, {889, 4441}, {903, 4406}, {1002, 36221}, {1086, 52205}, {1876, 18026}, {3227, 4762}, {4555, 34230}, {4569, 34855}, {7199, 18827}, {9309, 14727}, {26234, 35147}, {35961, 51055}, {36803, 38989}, {41075, 52013}

X(53219) = isotomic conjugate of X(14839)
X(53219) = isotomic conjugate of the isogonal conjugate of X(14665)
X(53219) = X(i)-isoconjugate of X(j) for these (i,j): {31, 14839}, {41, 43063}, {9454, 46798}
X(53219) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 14839}, {3160, 43063}, {33675, 46798}
X(53219) = cevapoint of X(2) and X(14839)
X(53219) = trilinear pole of line {2, 665}
X(53219) = barycentric product X(i)*X(j) for these {i,j}: {76, 14665}, {2481, 46802}
X(53219) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14839}, {7, 43063}, {2481, 46798}, {14665, 6}, {46802, 518}

---

X(53220) = X(513)X(1121)∩X(527)X(668)

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

X(53220) lies on the Steiner circumellipse and these lines: {57, 35157}, {190, 1155}, {312, 889}, {513, 1121}, {522, 3227}, {527, 668}, {536, 664}, {648, 15150}, {666, 24411}, {903, 45320}, {4555, 4659}, {4562, 36222}, {35171, 49722}

X(53220) = trilinear pole of line {2, 14413}

---

X(53221) = X(30)X(670)∩X(512)X(1494)

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

X(53221) lies on the Steiner circumellipse and these lines: {25, 16077}, {30, 670}, {99, 1495}, {305, 886}, {512, 1494}, {525, 3228}, {538, 648}, {671, 31174}, {1003, 2966}, {14583, 35139}, {18829, 32833}

X(53221) = trilinear pole of line {2, 9035}

---

X(53222) = X(99)X(2223)∩X(213)X(666)

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

X(53222) lies on the Steiner circumellipse and these lines: {65, 46135}, {99, 2223}, {190, 1966}, {213, 666}, {274, 18829}, {512, 2481}, {518, 670}, {538, 32041}, {668, 3978}, {1909, 4562}, {3228, 4762}, {16737, 18827}

X(53222) = trilinear pole of line {2, 14296}

---

X(53223) = X(190)X(538)∩X(512)X(903)

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

X(53223) lies on the Steiner circumellipse and these lines: {42, 4555}, {99, 902}, {190, 538}, {310, 886}, {512, 903}, {514, 3228}, {519, 670}, {648, 15147}, {668, 21805}, {671, 31147}, {3227, 31148}, {4562, 36226}, {16712, 18829}, {24281, 35148}

X(53223) = trilinear pole of line {2, 14407}

---

X(53224) = X(516)X(671)∩X(523)X(32040)

Barycentrics    (a - b)*(a - c)*(3*a^4 + a^3*b + a*b^3 + 3*b^4 - a^3*c - a^2*b*c - a*b^2*c - b^3*c - 2*a^2*c^2 + a*b*c^2 - 2*b^2*c^2 - a*c^3 - b*c^3 + c^4)*(3*a^4 - a^3*b - 2*a^2*b^2 - a*b^3 + b^4 + 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 + 3*c^4) : :
X(53224) = 4 X[35082] - X[35150]

X(53224) lies on the Steiner circumellipse and these lines: {2, 35082}, {516, 671}, {523, 32040}, {524, 18025}, {903, 22329}, {1992, 35153}, {3570, 46143}, {5641, 31144}, {14614, 35165}, {17346, 35149}, {18827, 37792}, {35151, 46922}

X(53224) = reflection of X(i) in X(j) for these {i,j}: {2, 35082}, {35150, 2}
X(53224) = trilinear pole of line {2, 2784}

---

X(53225) = X(514)X(32040)∩X(516)X(903)

Barycentrics    (a - b)*(a - c)*(3*a^3 + a^2*b + a*b^2 + 3*b^3 - 4*a^2*c - 4*a*b*c - 4*b^2*c + 3*a*c^2 + 3*b*c^2 - 2*c^3)*(3*a^3 - 4*a^2*b + 3*a*b^2 - 2*b^3 + a^2*c - 4*a*b*c + 3*b^2*c + a*c^2 - 4*b*c^2 + 3*c^3) : :
X(53225) = 4 X[35093] - X[35158]

X(53225) lies on the Steiner circumellipse and these lines: {2, 35093}, {514, 32040}, {516, 903}, {519, 18025}, {1121, 41140}, {2398, 4555}, {3241, 18821}, {16834, 35167}, {17078, 35160}, {29574, 35163}, {32041, 44550}, {34342, 35168}, {35150, 49630}, {39293, 41075}, {45276, 46136}

X(53225) = reflection of X(i) in X(j) for these {i,j}: {2, 35093}, {35158, 2}
X(53225) = trilinear pole of line {2, 5845}

---

X(53226) = X(518)X(903)∩X(519)X(2481)

Barycentrics    (a - b)*(a - c)*(2*a^2*b - 3*a*b^2 + b^3 + a^2*c - a*b*c - 3*b^2*c + a*c^2 + 2*b*c^2)*(a^2*b + a*b^2 + 2*a^2*c - a*b*c + 2*b^2*c - 3*a*c^2 - 3*b*c^2 + c^3) : :
4 X[35120] - X[35167]

X(53226) lies on the Steiner circumellipse and these lines: {2, 35120}, {190, 4762}, {514, 32041}, {518, 903}, {519, 2481}, {664, 36236}, {666, 1023}, {1026, 4555}, {1121, 17310}, {3227, 41140}, {3241, 18822}, {3679, 18821}, {16711, 18827}, {16834, 35172}, {17294, 35158}, {18816, 33677}, {29574, 35173}, {29615, 35163}, {35168, 35962}, {35961, 51055}

X(53226) = reflection of X(i) in X(j) for these {i,j}: {2, 35120}, {35167, 2}
X(53226) = cevapoint of X(514) and X(49772)
X(53226) = trilinear pole of line {2, 14439}

---

X(53227) = X(2)X(39012)∩X(99)X(36802)

Barycentrics    (a - b)*b*(a - c)*c*(a^2 + b^2 - a*c - b*c)*(a*b - b^2 + 2*a*c + b*c)*(2*a*b + a*c + b*c - c^2)*(a^2 - a*b - b*c + c^2) : :
X(53227) = X[2481] - 3 X[33675]

X(53227) lies on the Steiner circumellipse and these lines: {2, 39012}, {99, 36802}, {190, 51560}, {518, 2481}, {664, 34085}, {666, 2284}, {668, 36803}, {883, 46135}, {927, 4557}, {1002, 36221}, {1121, 46792}, {1494, 16087}, {3226, 33674}, {3227, 46798}, {4577, 32724}, {4586, 36138}, {4762, 32041}, {10030, 35160}, {18025, 33677}, {18827, 33676}, {27475, 35167}

X(53227) = anticomplement of X(39012)
X(53227) = X(i)-isoconjugate of X(j) for these (i,j): {665, 2280}, {926, 1471}, {1438, 33570}, {2223, 4724}, {4762, 9454}, {5228, 46388}, {8638, 40719}, {36138, 39012}, {45755, 52635}
X(53227) = X(i)-Dao conjugate of X(j) for these (i,j): {6184, 33570}, {33675, 4762}
X(53227) = cevapoint of X(i) and X(j) for these (i,j): {518, 4762}, {918, 3826}
X(53227) = trilinear pole of line {2, 2481}
X(53227) = barycentric product X(i)*X(j) for these {i,j}: {561, 36138}, {1002, 36803}, {1502, 32724}, {2481, 32041}, {18031, 37138}, {27475, 51560}, {40779, 46135}
X(53227) = barycentric quotient X(i)/X(j) for these {i,j}: {518, 33570}, {666, 1001}, {673, 4724}, {927, 5228}, {1002, 665}, {2481, 4762}, {8693, 2223}, {14942, 45755}, {27475, 2254}, {32041, 518}, {32724, 32}, {34085, 40719}, {36086, 2280}, {36138, 31}, {36146, 1471}, {36802, 37658}, {36803, 4441}, {37138, 672}, {40779, 926}, {51560, 4384}, {51563, 18206}

---

X(53228) = X(69)X(46135)∩X(99)X(2724)

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

X(53228) lies on the Steiner circumellipse and these lines: {69, 46135}, {99, 2724}, {190, 35517}, {219, 666}, {322, 4562}, {518, 18026}, {521, 2481}, {664, 1818}, {3261, 18025}, {6606, 16090}, {43093, 46042}

X(53228) = isotomic conjugate of X(2808)
X(53228) = isotomic conjugate of the isogonal conjugate of X(2724)
X(53228) = X(i)-isoconjugate of X(j) for these (i,j): {31, 2808}, {2223, 23694}
X(53228) = X(2)-Dao conjugate of X(2808)
X(53228) = cevapoint of X(2) and X(2808)
X(53228) = barycentric product X(i)*X(j) for these {i,j}: {76, 2724}, {31624, 46042}
X(53228) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2808}, {673, 23694}, {2724, 6}, {23989, 14505}, {46042, 6586}

---

X(53229) = X(98)X(648)∩X(99)X(287)

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

X(53229) lies on the Steiner circumellipse and these lines: {98, 648}, {99, 287}, {115, 6528}, {290, 2799}, {511, 2966}, {542, 9513}, {670, 15526}, {804, 43112}, {5641, 23878}, {5661, 40866}, {18831, 39843}, {46142, 52694}, {46786, 46807}, {46787, 46806}

X(53229) = reflection of X(40866) in X(5661)
X(53229) = X(i)-isoconjugate of X(j) for these (i,j): {163, 31953}, {1316, 1755}, {1959, 44127}, {9417, 44155}, {23997, 47229}
X(53229) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 31953}, {36899, 1316}, {39058, 44155}
X(53229) = cevapoint of X(511) and X(5661)
X(53229) = trilinear pole of line {2, 879}
X(53229) = barycentric product X(i)*X(j) for these {i,j}: {290, 9513}, {2966, 46245}
X(53229) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 1316}, {290, 44155}, {523, 31953}, {1976, 44127}, {2395, 47229}, {2715, 46249}, {2966, 40866}, {9154, 48983}, {9513, 511}, {36897, 38947}, {40077, 36213}, {41173, 43113}, {46245, 2799}

---

X(53230) = X(99)X(3569)∩X(115)X(290)

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

X(53230) lies on the Steiner circumellipse and these lines: {99, 3569}, {115, 290}, {512, 2966}, {538, 5641}, {542, 3228}, {648, 17994}, {670, 2799}, {892, 8430}, {4577, 40866}, {14568, 35146}, {14970, 38947}

X(53230) = X(810)-isoconjugate of X(50437)
X(53230) = X(39062)-Dao conjugate of X(50437)
X(53230) = trilinear pole of line {2, 44114}
X(53230) = barycentric quotient X(648)/X(50437)

---

X(53231) = X(99)X(698)∩X(523)X(3225)

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

X(53231) lies on the Steiner circumellipse and these lines: {6, 18829}, {99, 698}, {385, 670}, {523, 3225}, {648, 44089}, {671, 25423}, {886, 3734}, {892, 7766}, {1084, 51992}, {1502, 35078}, {2966, 33768}, {3407, 41073}, {4562, 20964}, {4590, 51322}, {5970, 14931}, {14970, 18105}, {18827, 20981}, {35146, 46778}

X(53231) = trilinear pole of line {2, 5027}

---

X(53232) = X(3)X(67)∩X(691)X(935)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^4 - a^2*b^2 + b^4 - c^4)*(a^4 - b^4 - a^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(53232) lies on the cubic K1320 and these lines: {3, 67}, {691, 935}, {7473, 18312}, {14966, 23968}, {17708, 34761}, {18019, 52145}

X(53232) = isogonal conjugate of X(53177)
X(53232) = X(i)-isoconjugate of X(j) for these (i,j): {1, 53177}, {14998, 16568}
X(53232) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 53177}, {15900, 14223}, {23967, 9979}, {35582, 5099}
X(53232) = crossdifference of every pair of points on line {2492, 47415}
X(53232) = barycentric product X(i)*X(j) for these {i,j}: {67, 14999}, {542, 17708}, {7473, 34897}, {14357, 50941}
X(53232) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 53177}, {67, 14223}, {542, 9979}, {3455, 14998}, {5191, 2492}, {7473, 37765}, {14357, 50942}, {14999, 316}, {17708, 5641}, {23968, 52449}, {34369, 52076}, {45662, 18311}, {50941, 52551}

---

X(53233) = X(3)X(74)∩X(476)X(1304)

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

X(53233) lies on the cubic K1320 and these lines: {3, 74}, {476, 1304}, {5502, 46608}, {14380, 15329}, {32162, 46632}, {39986, 52493}

X(53233) = isogonal conjugate of X(53178)
X(53233) = X(i)-isoconjugate of X(j) for these (i,j): {1, 53178}, {477, 36035}, {656, 52494}, {661, 46789}, {1637, 36102}, {1784, 14220}, {3258, 36047}, {9033, 36130}, {36151, 41079}
X(53233) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 53178}, {35581, 3258}, {36830, 46789}, {40596, 52494}
X(53233) = crossdifference of every pair of points on line {1637, 47414}
X(53233) = barycentric product X(i)*X(j) for these {i,j}: {110, 46788}, {2410, 14385}, {4558, 52493}, {5663, 44769}, {7480, 14919}, {32640, 35520}, {40384, 42742}
X(53233) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 53178}, {110, 46789}, {112, 52494}, {2437, 14254}, {5663, 41079}, {7480, 46106}, {14385, 2411}, {18877, 14220}, {32640, 477}, {36034, 36102}, {36131, 36130}, {42742, 36789}, {46788, 850}, {52493, 14618}
X(53233) = {X(14380),X(15329)}-harmonic conjugate of X(36831)

---

X(53234) = X(3)X(526)∩X(477)X(32710)

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

X(53234) lies on the cubic K1320 and these lines: {3, 526}, {477, 32710}, {12028, 15328}, {14385, 44808}, {15470, 16186}, {16170, 35189}, {47390, 52603}

X(53234) = X(i)-isoconjugate of X(j) for these (i,j): {2166, 7471}, {3018, 32680}, {17702, 36129}, {25641, 36047}
X(53234) = X(i)-Dao conjugate of X(j) for these (i,j): {11597, 7471}, {35581, 25641}
X(53234) = barycentric product X(i)*X(j) for these {i,j}: {323, 15453}, {8552, 32710}
X(53234) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 7471}, {14270, 3018}, {15453, 94}, {15470, 52498}, {32710, 46456}

---

X(53235) = X(3)X(9033)∩X(30)X(53178)

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

X(53235) lies on the cubic K1320 and these lines: {3, 9033}, {30, 53178}, {250, 4240}, {477, 2693}, {520, 1650}, {2411, 11845}, {3284, 14401}, {14314, 15354}, {15404, 43701}, {32162, 45681}, {43083, 50464}

X(53235) = X(i)-isoconjugate of X(j) for these (i,j): {162, 52493}, {1304, 36063}, {7480, 36119}, {24019, 46788}
X(53235) = X(i)-Dao conjugate of X(j) for these (i,j): {125, 52493}, {1511, 7480}, {1650, 11251}, {35071, 46788}, {38999, 5663}
X(53235) = trilinear pole of line {1636, 39008}
X(53235) = crossdifference of every pair of points on line {47228, 52493}
X(53235) = barycentric product X(i)*X(j) for these {i,j}: {394, 53178}, {477, 41077}, {520, 46789}, {1650, 30528}, {2411, 51254}, {11064, 14220}, {18557, 34210}, {52494, 52613}
X(53235) = barycentric quotient X(i)/X(j) for these {i,j}: {477, 15459}, {520, 46788}, {647, 52493}, {1636, 5663}, {2631, 36063}, {3284, 7480}, {9409, 47228}, {14220, 16080}, {14401, 11251}, {18558, 34209}, {30528, 42308}, {32663, 1304}, {41077, 35520}, {46789, 6528}, {51254, 2410}, {52494, 15352}, {53178, 2052}

---

X(53236) = X(7)X(310)∩X(76)X(17234)

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

X(53236) lies on these lines: {7, 310}, {76, 17234}, {142, 1233}, {261, 4623}, {274, 277}, {3596, 6385}, {3662, 16727}, {14377, 14964}, {16748, 27170}, {17065, 23824}, {23989, 26836}

X(53236) = isotomic conjugate of the isogonal conjugate of X(17169)
X(53236) = X(799)-Ceva conjugate of X(52619)
X(53236) = X(i)-isoconjugate of X(j) for these (i,j): {213, 1174}, {1402, 10482}, {1918, 2346}, {2205, 32008}
X(53236) = X(i)-Dao conjugate of X(j) for these (i,j): {1111, 661}, {1212, 42}, {6626, 1174}, {34021, 2346}, {40605, 10482}, {40606, 213}
X(53236) = cevapoint of X(1233) and X(20880)
X(53236) = barycentric product X(i)*X(j) for these {i,j}: {75, 16708}, {76, 17169}, {86, 1233}, {142, 310}, {274, 20880}, {354, 6385}, {561, 18164}, {670, 21104}, {1418, 40072}, {4602, 48151}, {6063, 16713}, {7257, 23599}, {10481, 28660}, {17194, 20567}, {18021, 52023}
X(53236) = barycentric quotient X(i)/X(j) for these {i,j}: {86, 1174}, {142, 42}, {274, 2346}, {310, 32008}, {314, 6605}, {333, 10482}, {354, 213}, {1229, 210}, {1233, 10}, {1418, 1402}, {1475, 1918}, {3925, 1500}, {4847, 1334}, {6067, 21795}, {6362, 3709}, {10481, 1400}, {13156, 2357}, {16708, 1}, {16713, 55}, {17169, 6}, {17194, 41}, {17206, 47487}, {18164, 31}, {20880, 37}, {21104, 512}, {21808, 872}, {22053, 2200}, {23599, 4017}, {35312, 4559}, {48151, 798}, {51384, 20683}, {51463, 52963}, {52020, 7109}, {52023, 181}

---

X(53237) = X(4)X(7)∩X(28)X(34018)

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

X(53237) lies on these lines: {4, 7}, {28, 34018}, {57, 14377}, {142, 1855}, {241, 277}, {281, 6706}, {331, 41013}, {347, 37407}, {1441, 8728}, {1708, 1729}, {1886, 52542}, {3219, 26003}, {4566, 24474}, {5236, 17758}, {7079, 37805}, {18443, 34059}, {27186, 37448}, {36009, 38859}

X(53237) = polar conjugate of X(6605)
X(53237) = polar conjugate of the isogonal conjugate of X(1418)
X(53237) = X(i)-isoconjugate of X(j) for these (i,j): {3, 10482}, {48, 6605}, {55, 47487}, {212, 2346}, {219, 1174}, {220, 1803}, {1170, 1802}, {1253, 40443}, {32008, 52425}
X(53237) = X(i)-Dao conjugate of X(j) for these (i,j): {142, 1260}, {223, 47487}, {1111, 6332}, {1212, 78}, {1249, 6605}, {17113, 40443}, {36103, 10482}, {40606, 219}, {40837, 2346}
X(53237) = barycentric product X(i)*X(j) for these {i,j}: {34, 1233}, {92, 10481}, {142, 273}, {225, 16708}, {264, 1418}, {278, 20880}, {286, 52023}, {331, 354}, {342, 13156}, {1088, 1855}, {1119, 1229}, {1847, 4847}, {1897, 23599}, {6362, 13149}, {17169, 40149}, {17924, 35312}, {18026, 21104}, {20567, 40983}, {46404, 48151}
X(53237) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 6605}, {19, 10482}, {34, 1174}, {57, 47487}, {142, 78}, {269, 1803}, {273, 32008}, {278, 2346}, {279, 40443}, {354, 219}, {1119, 1170}, {1212, 1260}, {1229, 1265}, {1233, 3718}, {1418, 3}, {1475, 212}, {1827, 220}, {1847, 21453}, {1855, 200}, {2293, 1802}, {3925, 3694}, {4847, 3692}, {10481, 63}, {13149, 6606}, {13156, 271}, {16708, 332}, {16713, 1792}, {17169, 1812}, {17194, 2327}, {18164, 283}, {20880, 345}, {21104, 521}, {21808, 2318}, {22053, 2289}, {23599, 4025}, {35312, 1332}, {35338, 4587}, {40983, 41}, {48151, 652}, {51463, 52978}, {52020, 52370}, {52023, 72}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1847, 38461}, {273, 1847, 4}

---

X(53238) = X(7)X(27)∩X(28)X(277)

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

X(53238) lies on these lines: {4, 991}, {7, 27}, {19, 31922}, {28, 277}, {58, 4000}, {86, 31926}, {142, 17194}, {269, 17197}, {273, 17092}, {281, 286}, {284, 948}, {579, 1730}, {1119, 46884}, {1777, 1780}, {1839, 3664}, {2189, 31906}, {2193, 14953}, {3668, 40979}, {4259, 14018}, {4303, 34830}, {7952, 31902}, {10481, 18164}, {15762, 15937}, {16713, 20880}, {16752, 31910}, {17086, 26802}, {17917, 25521}, {18643, 37447}, {24779, 52680}, {26978, 37398}

X(53238) = polar conjugate of the isotomic conjugate of X(17169)
X(53238) = X(162)-Ceva conjugate of X(17925)
X(53238) = X(i)-isoconjugate of X(j) for these (i,j): {37, 47487}, {71, 2346}, {72, 1174}, {73, 6605}, {210, 1803}, {228, 32008}, {1170, 2318}, {1214, 10482}, {1334, 40443}, {21453, 52370}
X(53238) = X(i)-Dao conjugate of X(j) for these (i,j): {142, 3694}, {1111, 14208}, {1212, 306}, {40589, 47487}, {40606, 72}
X(53238) = barycentric product X(i)*X(j) for these {i,j}: {4, 17169}, {19, 16708}, {27, 142}, {28, 20880}, {29, 10481}, {92, 18164}, {273, 17194}, {278, 16713}, {286, 354}, {310, 40983}, {648, 21104}, {811, 48151}, {1229, 1396}, {1233, 1474}, {1418, 31623}, {1434, 1855}, {1475, 44129}, {13156, 41083}, {17171, 18087}, {46103, 52023}
X(53238) = barycentric quotient X(i)/X(j) for these {i,j}: {27, 32008}, {28, 2346}, {58, 47487}, {142, 306}, {354, 72}, {1014, 40443}, {1172, 6605}, {1212, 3694}, {1233, 40071}, {1396, 1170}, {1412, 1803}, {1418, 1214}, {1474, 1174}, {1475, 71}, {1827, 210}, {1855, 2321}, {2293, 2318}, {2299, 10482}, {3925, 3695}, {4847, 3710}, {6362, 52355}, {10481, 307}, {16708, 304}, {16713, 345}, {17169, 69}, {17194, 78}, {18164, 63}, {20229, 52370}, {20880, 20336}, {21104, 525}, {21127, 8611}, {21808, 3949}, {22053, 3682}, {35326, 4574}, {40983, 42}, {48151, 656}, {52020, 3690}, {52023, 26942}
X(53238) = {X(286),X(15149)}-harmonic conjugate of X(281)

---

X(53239) = X(7)X(292)∩X(277)X(291)

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

X(53239) lies on these lines: {7, 192}, {277, 291}, {292, 24333}, {295, 2809}, {334, 40217}, {666, 2311}, {3721, 7200}, {3868, 9317}, {4051, 9263}, {4075, 17758}, {17169, 21808}

X(53239) = X(660)-Ceva conjugate of X(4444)
X(53239) = X(i)-isoconjugate of X(j) for these (i,j): {238, 1174}, {1428, 6605}, {1429, 10482}, {1914, 2346}, {2201, 47487}, {2210, 32008}
X(53239) = X(i)-Dao conjugate of X(j) for these (i,j): {142, 3684}, {1111, 3766}, {1212, 239}, {9470, 1174}, {36906, 2346}, {40606, 238}
X(53239) = trilinear pole of line {3925, 21104}
X(53239) = barycentric product X(i)*X(j) for these {i,j}: {142, 335}, {291, 20880}, {292, 1233}, {334, 354}, {1475, 18895}, {3925, 18827}, {4518, 10481}, {4562, 21104}, {4583, 48151}, {4847, 7233}, {17169, 43534}, {21808, 40017}, {36800, 52023}, {51384, 52209}
X(53239) = barycentric quotient X(i)/X(j) for these {i,j}: {142, 239}, {291, 2346}, {292, 1174}, {295, 47487}, {335, 32008}, {354, 238}, {1212, 3684}, {1229, 3975}, {1233, 1921}, {1418, 1429}, {1475, 1914}, {3925, 740}, {4847, 3685}, {4876, 6605}, {6362, 3716}, {7077, 10482}, {7233, 21453}, {10481, 1447}, {16708, 30940}, {17169, 33295}, {20880, 350}, {21039, 4433}, {21104, 812}, {21127, 4435}, {21808, 2238}, {22053, 7193}, {35338, 3573}, {48151, 659}, {51384, 17755}, {51463, 4432}, {52020, 3747}, {52023, 16609}
X(53239) = {X(335),X(7233)}-harmonic conjugate of X(4876)

---

X(53240) = X(7)X(528)∩X(88)X(277)

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

X(53240) lies on these lines: {7, 528}, {88, 277}, {106, 43349}, {1111, 6702}, {1797, 5773}, {2320, 40833}, {3218, 21372}, {4080, 17244}, {4089, 21630}, {4674, 6549}, {10481, 35312}, {17780, 32097}

X(53240) = X(3257)-Ceva conjugate of X(6548)
X(53240) = X(i)-isoconjugate of X(j) for these (i,j): {44, 1174}, {902, 2346}, {1319, 10482}, {1404, 6605}, {2251, 32008}
X(53240) = X(i)-Dao conjugate of X(j) for these (i,j): {142, 3689}, {1111, 3762}, {1212, 519}, {3119, 14427}, {9460, 32008}, {40594, 2346}, {40595, 1174}, {40606, 44}
X(53240) = cevapoint of X(142) and X(51463)
X(53240) = trilinear pole of line {142, 21104}
X(53240) = barycentric product X(i)*X(j) for these {i,j}: {88, 20880}, {106, 1233}, {142, 903}, {354, 20568}, {4080, 17169}, {4555, 21104}, {4674, 16708}, {4997, 10481}
X(53240) = barycentric quotient X(i)/X(j) for these {i,j}: {88, 2346}, {106, 1174}, {142, 519}, {354, 44}, {903, 32008}, {1212, 3689}, {1229, 4723}, {1233, 3264}, {1320, 6605}, {1418, 1319}, {1475, 902}, {1797, 47487}, {2316, 10482}, {3925, 3943}, {4847, 2325}, {6362, 1639}, {6608, 14427}, {10481, 3911}, {16708, 30939}, {17169, 16704}, {18164, 52680}, {20880, 4358}, {21104, 900}, {21127, 4895}, {21808, 21805}, {22053, 22356}, {35326, 23344}, {35338, 1023}, {48151, 1635}, {51463, 4370}, {52020, 52963}, {52023, 40663}

---

X(53241) = X(3)X(105)∩X(6)X(7)

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

X(53241) lies on the cubic K1243 and these lines: {1, 2140}, {2, 2481}, {3, 105}, {6, 7}, {142, 2293}, {279, 34018}, {377, 26140}, {666, 29590}, {672, 3008}, {885, 2254}, {1193, 52542}, {1212, 20880}, {1438, 52210}, {1475, 10481}, {1742, 4859}, {1777, 14377}, {2340, 20335}, {3000, 17067}, {3616, 14942}, {3779, 46149}, {3960, 34578}, {6185, 14953}, {7200, 40133}, {8751, 45786}, {9318, 52084}, {15970, 24181}, {16705, 27189}, {18031, 27304}, {25242, 27109}, {26059, 36796}, {26145, 34173}, {26626, 31637}, {31183, 36816}, {36124, 37396}, {37756, 46798}, {50114, 50257}, {51775, 52480}
on K1243

X(53241) = X(52209)-Ceva conjugate of X(21808)
X(53241) = X(i)-isoconjugate of X(j) for these (i,j): {241, 10482}, {518, 1174}, {672, 2346}, {1170, 2340}, {1458, 6605}, {2223, 32008}, {5089, 47487}, {6606, 46388}
X(53241) = X(i)-Dao conjugate of X(j) for these (i,j): {142, 3693}, {1212, 3912}, {40606, 518}
X(53241) = trilinear pole of line {354, 2488}
X(53241) = barycentric product X(i)*X(j) for these {i,j}: {105, 20880}, {142, 673}, {354, 2481}, {666, 21104}, {885, 35312}, {927, 6362}, {1212, 34018}, {1229, 1462}, {1233, 1438}, {1418, 36796}, {1475, 18031}, {2488, 46135}, {6185, 51384}, {10481, 14942}, {13576, 17169}, {16708, 18785}, {21127, 34085}, {48151, 51560}
X(53241) = barycentric quotient X(i)/X(j) for these {i,j}: {105, 2346}, {142, 3912}, {294, 6605}, {354, 518}, {673, 32008}, {927, 6606}, {1212, 3693}, {1418, 241}, {1438, 1174}, {1462, 1170}, {1475, 672}, {2195, 10482}, {2293, 2340}, {2488, 926}, {3925, 3932}, {4847, 3717}, {6362, 50333}, {10481, 9436}, {10581, 52614}, {16708, 18157}, {17169, 30941}, {18164, 18206}, {20880, 3263}, {21104, 918}, {21808, 3930}, {22053, 1818}, {34018, 31618}, {35312, 883}, {35326, 2284}, {35338, 1026}, {36057, 47487}, {40983, 2356}, {48151, 2254}, {51384, 4437}, {52020, 20683}

---

X(53242) = X(7)X(354)∩X(277)X(279)

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

X(53242) lies on these lines: {7, 354}, {9, 37780}, {85, 34019}, {142, 42449}, {277, 279}, {658, 10509}, {1014, 4616}, {1446, 5179}, {2346, 9446}, {6067, 20880}, {7177, 14377}, {7676, 14189}, {15185, 35312}, {18230, 31627}, {20905, 52980}, {23748, 24002}, {41857, 50562}

X(53242) = X(1088)-Ceva conjugate of X(10481)
X(53242) = X(i)-isoconjugate of X(j) for these (i,j): {41, 6605}, {55, 10482}, {220, 1174}, {1170, 6602}, {1253, 2346}, {7071, 47487}, {14827, 32008}
X(53242) = X(i)-Dao conjugate of X(j) for these (i,j): {142, 480}, {223, 10482}, {1111, 3239}, {1212, 200}, {3160, 6605}, {17113, 2346}, {40606, 220}
X(53242) = cevapoint of X(142) and X(41573)
X(53242) = trilinear pole of line {21104, 23599}
X(53242) = barycentric product X(i)*X(j) for these {i,j}: {85, 10481}, {142, 1088}, {269, 1233}, {279, 20880}, {479, 1229}, {664, 23599}, {1418, 6063}, {1446, 17169}, {3668, 16708}, {4569, 21104}, {4847, 23062}, {6362, 36838}, {21127, 52937}, {24002, 35312}, {46406, 48151}
X(53242) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 6605}, {57, 10482}, {142, 200}, {269, 1174}, {279, 2346}, {354, 220}, {479, 1170}, {1088, 32008}, {1212, 480}, {1229, 5423}, {1233, 341}, {1418, 55}, {1475, 1253}, {2293, 6602}, {3925, 4515}, {4847, 728}, {6067, 45791}, {6362, 4130}, {7177, 47487}, {10481, 9}, {16708, 1043}, {17106, 33634}, {17169, 2287}, {18164, 2328}, {20880, 346}, {21104, 3900}, {21127, 4105}, {22053, 1802}, {23062, 21453}, {23599, 522}, {30682, 40443}, {35312, 644}, {36838, 6606}, {41573, 24771}, {48151, 657}, {52023, 210}
X(53242) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 31526, 11025}, {1088, 23062, 7}

---

X(53243) = X(35)X(103)∩X(65)X(105)

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

X(53243) lies on the circumcircle and these lines:{35, 103}, {55, 38451}, {59, 1308}, {65, 105}, {99, 883}, {102, 7688}, {104, 2346}, {108, 35280}, {110, 2283}, {484, 2717}, {651, 1292}, {653, 26705}, {664, 43349}, {675, 21453}, {692, 934}, {767, 31618}, {840, 5172}, {919, 4559}, {927, 4566}, {972, 3579}, {999, 28219}, {1174, 1200}, {1311, 32008}, {1415, 8693}, {1477, 37583}, {1618, 14733}, {2369, 10509}, {3939, 6575}, {4565, 43076}, {14882, 28471}, {24016, 32642}, {28899, 36074}, {40443, 43363}

X(53243) = isogonal conjugate of X(6362)
X(53243) = isogonal conjugate of the isotomic conjugate of X(6606)
X(53243) = Collings transform of X(5173)
X(53243) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 6362}, {206, 2488}, {478, 21104}, {5375, 1229}, {10001, 1233}, {32664, 21127}, {36830, 16713}, {39026, 4847}
X(53243) = cevapoint of X(i) and X(j) for these (i,j): {6, 2488}, {109, 692}, {513, 5173}
X(53243) = trilinear pole of line {6, 1174}
aa X(53243) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6362}, {2, 21127}, {7, 6608}, {8, 48151}, {9, 21104}, {11, 35338}, {75, 2488}, {85, 10581}, {142, 650}, {220, 23599}, {354, 522}, {513, 4847}, {514, 1212}, {523, 17194}, {649, 1229}, {661, 16713}, {663, 20880}, {693, 2293}, {905, 1855}, {1021, 52023}, {1024, 51384}, {1086, 35341}, {1088, 6607}, {1233, 3063}, {1418, 3239}, {1475, 4391}, {1827, 4025}, {2310, 35312}, {3059, 3676}, {3261, 20229}, {3669, 51972}, {3700, 18164}, {3709, 16708}, {3737, 3925}, {3900, 10481}, {4041, 17169}, {4105, 53242}, {4435, 53239}, {4560, 21808}, {4858, 35326}, {4895, 53240}, {5231, 46003}, {7192, 21039}, {7199, 21795}, {8012, 24002}, {8611, 53238}, {13156, 14298}, {17197, 35310}, {18101, 35335}, {18155, 52020}, {22053, 44426}, {22079, 46107}, {23838, 51463}, {35518, 40983}
X(53243) = barycentric product X(i)*X(j) for these {i,j}: {6, 6606}, {100, 1170}, {101, 21453}, {109, 32008}, {651, 2346}, {653, 47487}, {658, 10482}, {664, 1174}, {692, 31618}, {934, 6605}, {1783, 40443}, {1803, 1897}, {3939, 10509}
X(53243) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6362}, {31, 21127}, {32, 2488}, {41, 6608}, {56, 21104}, {100, 1229}, {101, 4847}, {109, 142}, {110, 16713}, {163, 17194}, {269, 23599}, {604, 48151}, {651, 20880}, {664, 1233}, {692, 1212}, {1110, 35341}, {1170, 693}, {1174, 522}, {1262, 35312}, {1414, 16708}, {1415, 354}, {1461, 10481}, {1803, 4025}, {2149, 35338}, {2175, 10581}, {2283, 51384}, {2346, 4391}, {2425, 51424}, {2427, 51416}, {3939, 51972}, {4559, 3925}, {4565, 17169}, {4573, 53236}, {4617, 53242}, {6605, 4397}, {6606, 76}, {8059, 13156}, {8750, 1855}, {10482, 3239}, {10509, 52621}, {14827, 6607}, {21453, 3261}, {31618, 40495}, {32008, 35519}, {32660, 22053}, {32714, 53237}, {32735, 53241}, {32739, 2293}, {35326, 6067}, {40443, 15413}, {47487, 6332}

---

X(53244) = X(99)X(2414)∩X(110)X(2428)

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

X(53244) lies on the circumcircle and these lines: lines {99, 2414}, {110, 2428}, {1174, 1477}, {2291, 33634}, {2346, 9061}, {14827, 40154}, {37206, 43349}

X(53244) = X(i)-isoconjugate of X(j) for these (i,j): {142, 3309}, {344, 48151}, {354, 4468}, {514, 15185}, {1212, 31605}, {1229, 51652}, {1233, 8642}, {1418, 44448}, {1445, 6362}, {2488, 21609}, {3870, 21104}, {4847, 43049}, {4904, 35338}, {6600, 23599}, {6604, 21127}, {6608, 17093}, {35312, 38375}, {35341, 40615}
X(53244) = cevapoint of X(2488) and X(14827)
X(53244) = barycentric product X(i)*X(j) for these {i,j}: {1174, 37206}, {1292, 2346}
X(53244) = barycentric quotient X(i)/X(j) for these {i,j}: {692, 15185}, {1174, 4468}, {1292, 20880}, {2428, 51384}, {10482, 44448}, {17107, 23599}, {32644, 53241}, {37206, 1233}

---

X(53245) = X(6)X(264)∩X(22)X(98)

Barycentrics    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)*(-(a^2*b^2) + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4) : :
X(53245) = 2 X[36790] - 3 X[42313]

X(53245) = lies on these lines: {2, 11794}, {6, 264}, {22, 98}, {51, 324}, {76, 18024}, {94, 9979}, {216, 311}, {511, 14957}, {655, 1821}, {1972, 2799}, {1976, 14265}, {2966, 11077}, {11610, 44176}, {22456, 51222}, {40867, 44155}, {41205, 43754}, {42355, 51776}, {44420, 51259}, {45237, 52190}

X(53245) = reflection of X(i) in X(j) for these {i,j}: {264, 338}, {14570, 216}
X(53245) = isogonal conjugate of X(41270)
X(53245) = polar conjugate of X(19189)
X(53245) = polar conjugate of the isogonal conjugate of X(53174)
X(53245) = X(i)-Ceva conjugate of X(j) for these (i,j): {290, 53174}, {2966, 43665}
X(53245) = X(i)-isoconjugate of X(j) for these (i,j): {1, 41270}, {48, 19189}, {54, 1755}, {95, 9417}, {232, 2169}, {237, 2167}, {240, 14533}, {511, 2148}, {2190, 3289}, {2616, 14966}, {2623, 23997}, {3569, 36134}, {14573, 46238}
X(53245) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 41270}, {5, 3289}, {137, 3569}, {216, 511}, {338, 2799}, {1249, 19189}, {14363, 232}, {15450, 39469}, {36899, 54}, {39019, 684}, {39058, 95}, {39085, 14533}, {40588, 237}, {52032, 36212}, {52878, 9419}
X(53245) = trilinear pole of line {5, 15451}
X(53245) = barycentric product X(i)*X(j) for these {i,j}: {5, 290}, {51, 18024}, {98, 311}, {264, 53174}, {287, 324}, {343, 16081}, {1821, 14213}, {1953, 46273}, {2618, 36036}, {2715, 15415}, {2966, 18314}, {6368, 22456}, {6394, 13450}, {6531, 28706}, {12077, 43187}, {14570, 43665}, {17932, 23290}, {18695, 36120}, {35362, 52618}
X(53245) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 19189}, {5, 511}, {6, 41270}, {51, 237}, {53, 232}, {98, 54}, {216, 3289}, {248, 14533}, {287, 97}, {290, 95}, {293, 2169}, {311, 325}, {324, 297}, {343, 36212}, {685, 933}, {879, 23286}, {1273, 51383}, {1393, 51651}, {1625, 14966}, {1821, 2167}, {1910, 2148}, {1953, 1755}, {2179, 9417}, {2395, 2623}, {2617, 23997}, {2715, 14586}, {2966, 18315}, {3199, 2211}, {6368, 684}, {6531, 8882}, {12077, 3569}, {13450, 6530}, {14213, 1959}, {14569, 34854}, {14570, 2421}, {14601, 14573}, {15451, 39469}, {16081, 275}, {17167, 17209}, {17500, 51862}, {17974, 19210}, {18024, 34384}, {18314, 2799}, {20021, 16030}, {21807, 5360}, {22456, 18831}, {23290, 16230}, {28706, 6393}, {32428, 52128}, {35360, 4230}, {35362, 1634}, {36084, 36134}, {36120, 2190}, {39113, 51439}, {39569, 2967}, {40981, 9418}, {41221, 44114}, {41586, 9155}, {43665, 15412}, {43754, 15958}, {51363, 9475}, {51513, 17994}, {52347, 51386}, {52967, 9419}, {53174, 3}
X(53245) = {X(290),X(16081)}-harmonic conjugate of X(287)

   Intersections of lines tangent to conics: X(53246)-X(53580) and others  

Contributed by Clark Kimberling (definitions and presentation) and Peter Moses (formulas, data, and properties), April 25-May 1, 2023.

Suppose that U and X are distinct points on a conic sΓ. Let (U) be the line tangent to Γ at U, and Let (X) be the line tangent to Γ at X. Define F(U,X) = (U)∩(X).

List here are conics for which F(U,X) appears for selected pairs of points U and X:

circumcircle: X(53246)-X(53330), X(53384), X(52385)
Steiner circumellipse: X(53331)-X(53383)
Bevan circle: X(53389)-X(53412)
Stevanovich circel: X(53389), X(43413)
other circumconics: X(53386)-X(533898)
Kiepert circumhyperbola: X(54314)-X(54520)
incircle: X(54521)-X(54563)
nine-point circle: X(53564)-X(53577)
Yff parabala: X(53588)-X(535602)
dual of Yff parabola; i.e., {A,B,C,X(2), X(7)}: X(53588)-X(535602)

Suppose P = p: q : r is a point and Γ is the circumconic with perspector P. If U = u : v : w and X = x : y : z, then the point f : g : h : F(U,X) is given by the following barycentrics:

f : g : h : p(wy+vz)( (2qu+pv+qv-rv)x + (2pv+pu+qu-ru)y )( (2ru+pw+rw-qw)x + (2pw+pu+ru-qu)z ) : :

In this case, F(U,X) = crossdifference of every pair of points on the line bcfα + gcaβ + habγ = 0, so that F(U,X) is the trilinear pole of the line ghα + hfβ + fgγ = 0 .

X(53246) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(74) AND X(98)

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^4*b^4*c^2 + 2*a^2*b^6*c^2 - 3*b^8*c^2 - 3*a^6*c^4 - 2*a^4*b^2*c^4 - 2*a^2*b^4*c^4 + 3*b^6*c^4 + 3*a^4*c^6 + 2*a^2*b^2*c^6 + 3*b^4*c^6 - a^2*c^8 - 3*b^2*c^8) : :
X(53246) = 3 X[5085] - 2 X[36213], X[25046] - 3 X[25406], 3 X[25317] - X[51212]

X(53246) lies on these lines: {3, 67}, {4, 7668}, {6, 32444}, {20, 25051}, {25, 47204}, {30, 5201}, {64, 1942}, {74, 526}, {98, 804}, {104, 7444}, {125, 1624}, {160, 39874}, {182, 35357}, {186, 17986}, {237, 1503}, {338, 2790}, {523, 50401}, {524, 47620}, {541, 46129}, {669, 47252}, {1141, 13597}, {1297, 29180}, {1300, 45138}, {1350, 34383}, {1576, 5622}, {1593, 14900}, {1597, 6128}, {1625, 5661}, {2697, 9076}, {2781, 20975}, {3001, 13754}, {3003, 6000}, {3448, 23181}, {3818, 35222}, {5085, 36213}, {5191, 46130}, {5467, 11579}, {5502, 40352}, {5663, 46127}, {5899, 14993}, {7687, 43919}, {8266, 46264}, {9140, 15329}, {9149, 11177}, {9407, 51733}, {9862, 37991}, {10620, 14687}, {11179, 35934}, {11328, 47353}, {11457, 31381}, {11477, 31952}, {13171, 13558}, {13596, 15358}, {14096, 51737}, {14673, 24930}, {15151, 40948}, {16080, 46587}, {16186, 46147}, {25046, 25406}, {25317, 51212}, {25324, 29181}, {29011, 29316}, {32274, 44895}, {34098, 53015}, {34396, 51739}, {34519, 46589}, {35325, 38352}, {37338, 47354}, {37465, 51023}, {40947, 44883}, {41328, 48906}

X(53246) = midpoint of X(20) and X(25051)
X(53246) = reflection of X(i) in X(j) for these {i,j}: {4, 7668}, {1634, 3}
X(53246) = crossdifference of every pair of points on line {2492, 3163}
X(53246) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5621, 7669, 3}, {11177, 15915, 9149}

---

X(53247) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(74) AND X(99)

Barycentrics    a^2*(b - c)*(b + c)*(a^6 - 2*a^4*b^2 + a^2*b^4 - 2*a^4*c^2 + 7*a^2*b^2*c^2 - 3*b^4*c^2 + a^2*c^4 - 3*b^2*c^4) : :
X(53247) = 3 X[3] - 2 X[14270], 5 X[3] - 4 X[39477], 5 X[14270] - 6 X[39477], X[14270] - 3 X[44826], 2 X[39477] - 5 X[44826], X[669] - 3 X[22089], 5 X[1656] - 4 X[39509], 4 X[8552] - 3 X[34291], 2 X[21731] - 3 X[34291], 2 X[6140] - 3 X[44814], 4 X[14271] - 5 X[53094]

X(53247) lies on these lines: {3, 690}, {74, 526}, {99, 670}, {186, 52475}, {378, 44427}, {512, 35002}, {523, 7464}, {525, 42660}, {669, 3265}, {924, 15138}, {1593, 16230}, {1656, 39509}, {1975, 14295}, {1995, 9191}, {2491, 5013}, {2780, 8552}, {2799, 39812}, {2935, 41077}, {3725, 22437}, {6140, 44814}, {6642, 44921}, {9003, 16010}, {9185, 40916}, {9479, 44821}, {12567, 25380}, {14271, 53094}, {14417, 32121}, {20186, 46953}, {41254, 45689}

X(53247) = reflection of X(i) in X(j) for these {i,j}: {3, 44826}, {21731, 8552}
X(53247) = X(14398)-Dao conjugate of X(1637)
X(53247) = crossdifference of every pair of points on line {1084, 1196}
X(53247) = {X(8552),X(21731)}-harmonic conjugate of X(34291)

---

X(53248) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(74) AND X(100)

Barycentrics    a^2*(b - c)*(a^5 - 2*a^3*b^2 + a*b^4 - 3*a^3*b*c + 3*a^2*b^2*c + 3*a*b^3*c - 3*b^4*c - 2*a^3*c^2 + 3*a^2*b*c^2 + 4*a*b^2*c^2 - 3*b^3*c^2 + 3*a*b*c^3 - 3*b^2*c^3 + a*c^4 - 3*b*c^4) : :
X(53248) = 2 X[667] - 3 X[48383]

X(53248) lies on these lines: {3, 8674}, {74, 526}, {100, 190}, {513, 35000}, {521, 48390}, {523, 36001}, {667, 15313}, {2773, 44827}, {3733, 48389}, {3887, 39200}, {4057, 50355}, {6003, 48391}, {11509, 51643}, {24290, 36744}, {35057, 48382}

X(53248) = reflection of X(3733) in X(48389)
X(53248) = crossdifference of every pair of points on line {1015, 3163}

---

X(53249) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(74) AND X(101)

Barycentrics    a^2*(a - b - c)*(b - c)*(a^3 - 2*a^2*b - a*b^2 + 2*b^3 - 2*a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2 + 2*c^3) : :
X(53249) = 2 X[649] - 3 X[48387]

X(53249) lies on these lines: {3, 2774}, {6, 42662}, {55, 42657}, {74, 526}, {101, 692}, {522, 22037}, {523, 36026}, {649, 8676}, {2512, 4477}, {2785, 12635}, {3738, 23345}, {3887, 12738}, {3900, 4705}, {4895, 46393}, {6326, 8674}, {6366, 21343}, {9404, 21789}, {35057, 47842}

X(53249) = reflection of X(3) in X(44827)
X(53249) = crossdifference of every pair of points on line {553, 1086}
X(53249) = barycentric product X(55)*X(49274)
X(53249) = barycentric quotient X(49274)/X(6063)

---

X(53250) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(74) AND X(102)

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

X(53250) lies on these lines: {3, 2779}, {74, 526}, {102, 8677}, {103, 28163}, {104, 28159}, {953, 28185}, {1464, 21663}, {15622, 34935}, {28145, 28167}, {28149, 28171}, {28173, 28189}, {28203, 28207}

X(53250) = crossdifference of every pair of points on line {3163, 23986}

---

X(53251) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(74) AND X(103)

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

X(53251) lies on these lines: {3, 2772}, {74, 526}, {102, 28149}, {103, 926}, {104, 28145}, {523, 37166}, {953, 28181}, {2822, 21045}, {28153, 28159}, {28157, 28163}, {28173, 28177}, {28197, 28201}, {29023, 29049}

X(53251) = reflection of X(35327) in X(3)
X(53251) = crossdifference of every pair of points on line {3163, 23972}

---

X(53252) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(74) AND X(104)

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

X(53252) lies on these lines: {1, 46419}, {3, 191}, {74, 526}, {102, 103}, {104, 900}, {106, 28193}, {953, 28173}, {1464, 6000}, {1807, 2283}, {2778, 18210}, {2800, 15626}, {8235, 20849}, {12515, 23832}, {13724, 18243}, {15624, 18446}, {28145, 28159}, {28149, 28163}, {28153, 28167}, {28157, 28171}, {28177, 28185}, {28181, 28189}, {28197, 28203}, {28201, 28207}, {28211, 28219}, {28215, 28223}, {28227, 28233}, {28231, 28235}, {31849, 33811}

X(53252) = crossdifference of every pair of points on line {3163, 23980}

---

X(53253) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(74) AND X(105)

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

X(53253) lies on these lines: {3, 2836}, {36, 18735}, {56, 7202}, {74, 526}, {105, 659}, {523, 50402}, {1464, 3003}, {2223, 3827}, {7291, 20470}, {15904, 18210}

X(53253) = crossdifference of every pair of points on line {3163, 6184}

---

X(53254) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(74) AND X(106)

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

X(53254) lies on these lines: {3, 2842}, {6, 43693}, {74, 526}, {102, 28193}, {104, 28203}, {106, 1960}, {840, 38882}, {902, 2390}, {1464, 1495}, {13205, 38590}, {15621, 41455}, {28159, 28201}, {28197, 28207}, {30269, 41454}, {35327, 41192}

X(53254) = crossdifference of every pair of points on line {3163, 4370}

---

X(53255) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(74) AND X(107)

Barycentrics    a^2*(b - c)*(b + c)*(a^2 - b^2 - c^2)*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 - 3*a^6*c^2 - a^4*b^2*c^2 + a^2*b^4*c^2 + 3*b^6*c^2 + 3*a^4*c^4 + a^2*b^2*c^4 - 6*b^4*c^4 - a^2*c^6 + 3*b^2*c^6) : :
X(53255) = 2 X[9409] + X[14380], 3 X[23286] - 4 X[39201], 3 X[34291] - 4 X[44816]

X(53255) lies on these lines: {3, 9033}, {74, 526}, {107, 1624}, {186, 523}, {684, 14060}, {1609, 47138}, {1637, 52166}, {5489, 45147}, {5621, 9003}, {7669, 10117}, {8675, 42658}, {9007, 22089}, {9411, 32444}, {34291, 42665}, {35243, 46229}

X(53255) = isogonal conjugate of the anticomplement of X(38999)
vX(i)-Ceva conjugate of X(j) for these (i,j): {13597, 125}, {15459, 6}
X(53255) = X(1636)-Dao conjugate of X(41077)
X(53255) = crossdifference of every pair of points on line {216, 549}
X(53255) = barycentric product X(i)*X(j) for these {i,j}: {525, 14157}, {15459, 38999}
X(53255) = barycentric quotient X(i)/X(j) for these {i,j}: {14157, 648}, {38999, 41077}

---

X(53256) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(74) AND X(109)

Barycentrics    a^2*(b - c)*(a^5 + 2*a^4*b - 2*a^3*b^2 - 4*a^2*b^3 + a*b^4 + 2*b^5 + 2*a^4*c - 3*a^3*b*c - a^2*b^2*c + 3*a*b^3*c - b^4*c - 2*a^3*c^2 - a^2*b*c^2 + 4*a*b^2*c^2 - b^3*c^2 - 4*a^2*c^3 + 3*a*b*c^3 - b^2*c^3 + a*c^4 - b*c^4 + 2*c^5) : :
X(53256) = 2 X[663] - 3 X[39199]

X(53256) lies on these lines: {3, 2773}, {74, 526}, {109, 692}, {484, 513}, {663, 39199}, {1768, 8674}, {3738, 12515}, {5221, 11125}, {12778, 35053}, {21180, 36279}

X(53256) = crossdifference of every pair of points on line {1100, 1146}

---

X(53257) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(98) AND X(100)

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

X(53257) lies on these lines: {2, 16158}, {3, 2787}, {21, 30709}, {55, 4010}, {56, 4922}, {98, 804}, {100, 190}, {183, 14296}, {405, 14431}, {474, 14419}, {523, 4477}, {650, 15313}, {814, 48387}, {884, 13576}, {1376, 9508}, {2605, 24782}, {2775, 6985}, {3733, 24533}, {4057, 6133}, {4428, 45342}, {4730, 5687}, {4874, 23865}, {8642, 47803}, {13245, 25380}, {21005, 48383}, {21051, 21789}, {21260, 22160}, {25926, 30792}, {29324, 44408}, {29344, 48386}, {43051, 48283}

X(53257) = crossdifference of every pair of points on line {942, 1015}
X(53257) = {X(14431),X(42670)}-harmonic conjugate of X(405)

---

X(53258) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(98) AND X(101)

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

X(53258) lies on these lines: {3, 2786}, {98, 804}, {101, 692}, {522, 1324}, {667, 3508}, {669, 4477}, {1011, 4120}, {2352, 21894}, {3700, 23864}, {3709, 21789}, {3835, 23093}, {4191, 4750}, {6586, 8676}, {16058, 45661}, {16059, 45674}, {22388, 48269}, {23394, 41190}, {28846, 44408}, {28906, 39476}, {29078, 48383}

X(53258) = crossdifference of every pair of points on line {1086, 11672}

---

X(53259) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(98) AND X(103)

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

X(53259) lies on these lines: {3, 2784}, {98, 804}, {103, 926}, {104, 15323}, {516, 20475}, {29027, 29084}, {29031, 29096}, {29100, 29104}, {29308, 29348}, {29310, 29352}

X(53259) = crossdifference of every pair of points on line {11672, 23972}

---

X(53260) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(98) AND X(104)

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

X(53260) lies on these lines: {3, 2783}, {4, 2486}, {56, 4459}, {98, 804}, {103, 29310}, {104, 900}, {523, 50402}, {912, 35552}, {958, 8235}, {1284, 1503}, {2791, 16732}, {2831, 4516}, {3286, 29057}, {5132, 24257}, {6001, 8609}, {14686, 33535}, {16132, 42842}, {16684, 31394}

X(53260) = reflection of X(i) in X(j) for these {i,j}: {4, 2486}, {4436, 3}
X(53260) = crossdifference of every pair of points on line {11672, 23980}

---

X(53261) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(98) AND X(105)

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

X(53261) lies on these lines: {3, 2795}, {25, 47212}, {56, 7200}, {98, 804}, {105, 659}, {183, 4485}, {230, 1284}, {614, 18191}, {675, 1311}, {1281, 4436}, {1447, 20470}, {2726, 9108}, {3290, 3827}, {4518, 23343}, {5985, 9149}, {7289, 39954}, {9083, 9105}, {9093, 9103}, {9746, 15624}, {16684, 52133}, {26241, 26281}

X(53261) = crossdifference of every pair of points on line {6184, 11672}

---

X(53262) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(98) AND X(109)

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

X(53262) lies on these lines: {3, 2785}, {25, 47210}, {98, 804}, {109, 692}, {513, 3510}, {1758, 23224}, {4458, 23853}, {13738, 30574}, {23865, 39199}, {23866, 39200}

X(53262) = crossdifference of every pair of points on line {1107, 1146}

---

X(53263) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(98) AND X(110)

Barycentrics    a^2*(b - c)*(b + c)*(a^6 - 2*a^4*b^2 + a^2*b^4 - 2*a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4) : :
X(53263) = 3 X[3] - 4 X[39477], 3 X[3] - 2 X[44826], 3 X[14270] - 2 X[39477], 3 X[14270] - X[44826], 3 X[351] - X[684], 3 X[351] - 2 X[6132], 2 X[684] - 3 X[34291], 4 X[6132] - 3 X[34291], 3 X[381] - 4 X[39509], 2 X[5926] - 3 X[34952], 3 X[5085] - 4 X[14271], 3 X[32193] - 2 X[45259]

X(53263) lies on these lines: {3, 690}, {6, 2491}, {23, 385}, {24, 44427}, {25, 16230}, {50, 647}, {98, 804}, {110, 351}, {183, 14295}, {248, 2422}, {381, 39509}, {512, 2080}, {520, 8651}, {525, 5926}, {758, 52726}, {879, 51869}, {900, 13265}, {1177, 14380}, {1499, 42660}, {1510, 3005}, {1637, 44533}, {1649, 40604}, {1995, 9185}, {2254, 3724}, {2292, 8648}, {2574, 42667}, {2575, 42668}, {2780, 44810}, {2793, 33900}, {2799, 11616}, {2881, 19164}, {2930, 9003}, {3511, 5027}, {3566, 39201}, {3725, 22384}, {5085, 14271}, {5113, 19576}, {6370, 39200}, {7669, 10117}, {8644, 8675}, {8664, 20188}, {9033, 12310}, {9189, 11284}, {9191, 40916}, {9208, 11328}, {9479, 44822}, {9517, 11615}, {9818, 44921}, {11141, 23284}, {11142, 23283}, {11620, 33752}, {14417, 44808}, {14424, 44809}, {14652, 14809}, {15470, 47159}, {16678, 50342}, {18004, 52139}, {22089, 44680}, {22159, 52590}, {22160, 42653}, {32193, 45259}, {35909, 44895}, {39469, 52170}, {39905, 52275}, {44821, 45261}, {45321, 52728}, {46602, 46608}

X(53263) = reflection of X(i) in X(j) for these {i,j}: {3, 14270}, {684, 6132}, {21731, 11615}, {22089, 44680}, {33752, 11620}, {34291, 351}, {35364, 3569}, {41167, 5113}, {44826, 39477}, {46953, 8651}
X(53263) = isogonal conjugate of the anticomplement of X(38987)
X(53263) = X(2966)-Ceva conjugate of X(6)
X(53263) = X(3569)-Dao conjugate of X(2799)
X(53263) = crossdifference of every pair of points on line {5, 39}
X(53263) = barycentric product X(i)*X(j) for these {i,j}: {525, 19128}, {2966, 38987}, {16081, 38354}
X(53263) = barycentric quotient X(i)/X(j) for these {i,j}: {19128, 648}, {38354, 36212}, {38987, 2799}
X(53263) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {351, 684, 6132}, {684, 6132, 34291}, {14270, 44826, 39477}, {39477, 44826, 3}

---

X(53264) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(98) AND X(111)

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

X(53264) lies on these lines: {2, 1634}, {3, 543}, {6, 6784}, {23, 8859}, {25, 1989}, {98, 804}, {105, 7448}, {111, 351}, {230, 237}, {338, 47211}, {669, 5914}, {671, 11634}, {755, 14659}, {1141, 40118}, {1995, 14995}, {2079, 39832}, {2393, 3291}, {2421, 46303}, {2493, 20975}, {2871, 3124}, {2930, 5642}, {3054, 20775}, {3511, 15271}, {3563, 43662}, {5020, 23583}, {5201, 22329}, {5915, 42660}, {7467, 13468}, {7468, 46980}, {7606, 8546}, {7735, 34098}, {7806, 35222}, {8266, 17008}, {8770, 16098}, {9756, 32444}, {11168, 14096}, {14651, 37991}, {14684, 15916}, {22735, 36822}, {23055, 37184}, {34374, 34376}, {35324, 39834}, {36182, 41135}, {37688, 41328}, {39201, 47003}, {39427, 52618}, {44420, 46127}, {44468, 47426}, {44889, 47200}

X(53264) = X(9154)-Ceva conjugate of X(6)
X(53264) = X(9155)-Dao conjugate of X(50567)
X(53264) = crossdifference of every pair of points on line {2482, 8552}
X(53264) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 9149, 1634}, {25, 6103, 52604}, {7669, 44533, 25}, {13233, 34010, 3}

---

X(53265) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(98) AND X(112)

Barycentrics    a^2*(b - c)*(b + c)*(a^8 - a^6*b^2 - a^4*b^4 + a^2*b^6 - 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 + a^2*c^6 + b^2*c^6) : :
X(53265) = 2 X[2492] + X[9409], X[684] - 4 X[44820], X[35522] - 4 X[44818]

X(53265) lies on these lines: {3, 2799}, {6, 2507}, {22, 9979}, {25, 1637}, {98, 804}, {112, 1576}, {186, 523}, {351, 46425}, {512, 1691}, {526, 5622}, {669, 6587}, {684, 44820}, {690, 5621}, {1634, 40866}, {1976, 3569}, {2079, 7669}, {2489, 42658}, {2623, 21646}, {3005, 16040}, {3268, 7485}, {3406, 46040}, {5020, 44564}, {7484, 14417}, {7530, 44204}, {8105, 42667}, {8106, 42668}, {8770, 14606}, {9003, 32621}, {9178, 14908}, {9185, 34519}, {9479, 44821}, {9517, 15462}, {14675, 33752}, {16230, 19165}, {18534, 44203}, {21006, 34952}, {32472, 52585}, {35522, 44818}, {37123, 52076}, {42665, 47230}

X(53265) = midpoint of X(i) and X(j) for these {i,j}: {2489, 42658}, {9409, 17994}
X(53265) = reflection of X(i) in X(j) for these {i,j}: {3, 25644}, {17994, 2492}, {21006, 34952}
X(53265) = isogonal conjugate of the anticomplement of X(39000)
X(53265) = X(i)-Ceva conjugate of X(j) for these (i,j): {685, 6}, {32649, 25}
X(53265) = X(i)-Dao conjugate of X(j) for these (i,j): {684, 6333}, {3150, 325}
X(53265) = crossdifference of every pair of points on line {141, 216}
X(53265) = barycentric product X(i)*X(j) for these {i,j}: {112, 3150}, {523, 10313}, {685, 39000}, {2966, 38368}
X(53265) = barycentric quotient X(i)/X(j) for these {i,j}: {3150, 3267}, {10313, 99}, {38368, 2799}, {39000, 6333}
X(53265) = {X(1637),X(42659)}-harmonic conjugate of X(25)

---

X(53266) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(98) AND X(476)

Barycentrics    (b - c)*(b + c)*(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 + 2*b^6*c^2 + a^4*c^4 - a^2*b^2*c^4 - 4*b^4*c^4 + 2*b^2*c^6) : :
X(53266) = X[3] + 2 X[23105], 3 X[381] - 4 X[39482], 2 X[10412] + X[46608], 2 X[879] + X[35364], X[2394] + 2 X[15543], X[14380] + 2 X[15328], 3 X[5054] - 2 X[44814], X[12188] + 2 X[51232]

X(53266) lies on these lines: {2, 523}, {3, 23105}, {6, 47229}, {98, 804}, {99, 52632}, {115, 10097}, {183, 850}, {230, 2395}, {378, 14618}, {381, 512}, {476, 10412}, {526, 9140}, {599, 8675}, {647, 37637}, {684, 30789}, {690, 11632}, {826, 1116}, {868, 879}, {924, 1853}, {1499, 39491}, {1550, 52451}, {1593, 46371}, {1637, 1989}, {1995, 4108}, {2394, 15543}, {2453, 44889}, {3134, 12079}, {3314, 31072}, {3566, 42733}, {3906, 44751}, {5054, 44814}, {7577, 34964}, {7610, 9175}, {7778, 30476}, {8860, 36900}, {9033, 45688}, {10257, 15421}, {10557, 17993}, {11176, 45682}, {11183, 40550}, {12028, 14592}, {12065, 18808}, {12073, 39494}, {12188, 51232}, {14420, 45147}, {14993, 15475}, {15915, 44821}, {15928, 18312}, {16229, 44438}, {17004, 31296}, {23350, 36897}, {34810, 47049}, {35934, 42660}, {37453, 47221}, {41167, 45692}

X(53266) = midpoint of X(i) and X(j) for these {i,j}: {879, 34290}, {4108, 30735}
X(53266) = reflection of X(i) in X(j) for these {i,j}: {11176, 45682}, {11183, 40550}, {34291, 2}, {35364, 34290}, {41167, 45692}
X(53266) = reflection of X(34291) in the Euler line
X(53266) = X(i)-Dao conjugate of X(j) for these (i,j): {34810, 4226}, {47049, 15329}
X(53266) = crossdifference of every pair of points on line {187, 1511}
X(53266) = barycentric product X(i)*X(j) for these {i,j}: {2394, 34810}, {43665, 47049}
X(53266) = barycentric quotient X(i)/X(j) for these {i,j}: {34810, 2407}, {47049, 2421}
X(53266) = {X(5466),X(16092)}-harmonic conjugate of X(9178)

---

X(53267) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(98) AND X(477)

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

X(53267) lies on these lines: {3, 2453}, {4, 23347}, {30, 5467}, {74, 13530}, {98, 804}, {182, 381}, {376, 46994}, {378, 33971}, {476, 46602}, {477, 16171}, {511, 34810}, {523, 7422}, {868, 1503}, {1300, 43660}, {1350, 36207}, {1651, 15448}, {2697, 2770}, {2777, 3018}, {3014, 17702}, {3134, 5502}, {3534, 15356}, {5063, 44526}, {5968, 5999}, {6033, 32233}, {6776, 47076}, {6795, 46127}, {7417, 8371}, {12188, 16010}, {14356, 29012}, {23288, 48983}, {34473, 46599}, {35908, 41204}, {44096, 44438}, {44216, 47474}

X(53267) = reflection of X(i) in X(j) for these {i,j}: {4, 38393}, {5467, 52772}

---

X(53268) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(99) AND X(101)

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

X(53268) lies on these lines: {1, 16683}, {3, 2784}, {6, 4116}, {8, 23851}, {99, 670}, {100, 932}, {101, 692}, {110, 43359}, {237, 4433}, {519, 8618}, {523, 36032}, {643, 3573}, {650, 22280}, {667, 35342}, {859, 23398}, {898, 8708}, {993, 8053}, {1631, 9912}, {2110, 9259}, {2176, 18756}, {2242, 20990}, {2388, 14964}, {2701, 29119}, {2975, 23370}, {3010, 20785}, {4062, 21522}, {4068, 4653}, {4367, 17136}, {4427, 23390}, {6013, 29351}, {8424, 23079}, {8666, 23393}, {8844, 35101}, {16681, 23361}, {17135, 23371}, {18758, 20691}, {20663, 38986}, {20855, 21085}, {21024, 23212}, {21252, 30017}, {23360, 32929}, {23396, 32853}, {23862, 37311}, {29026, 29083}, {29030, 29095}, {29099, 29103}, {29159, 29305}, {29185, 29273}, {29199, 29227}, {34067, 36147}, {46148, 46177}

X(53268) = midpoint of X(3010) and X(20785)
X(53268) = reflection of X(20475) in X(8618)
X(53268) = isogonal conjugate of the anticomplement of X(40627)
X(53268) = X(1101)-anticomplementary conjugate of X(23371)
X(53268) = X(4600)-Ceva conjugate of X(6)
X(53268) = X(i)-isoconjugate of X(j) for these (i,j): {513, 40418}, {514, 1258}, {649, 1221}, {661, 40409}, {799, 40525}
X(53268) = X(i)-Dao conjugate of X(j) for these (i,j): {3122, 3120}, {3741, 523}, {5375, 1221}, {21838, 3261}, {36830, 40409}, {38996, 40525}, {39026, 40418}, {51575, 693}
X(53268) = cevapoint of X(2309) and X(50510)
X(53268) = trilinear pole of line {1197, 2309}
X(53268) = crossdifference of every pair of points on line {1084, 1086}
X(53268) = barycentric product X(i)*X(j) for these {i,j}: {99, 21838}, {100, 1107}, {101, 3741}, {110, 21024}, {190, 2309}, {643, 45208}, {645, 39780}, {662, 3728}, {668, 1197}, {692, 20891}, {1016, 50510}, {1018, 18169}, {1897, 22065}, {3939, 30097}, {4556, 21713}, {4557, 16738}, {4598, 45216}, {4600, 40627}, {4603, 27880}, {4610, 21700}, {4621, 23473}, {6331, 23212}, {6335, 22389}, {18091, 46148}, {22206, 52935}, {32641, 51411}, {36860, 45209}
X(53268) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 1221}, {101, 40418}, {110, 40409}, {669, 40525}, {692, 1258}, {1107, 693}, {1197, 513}, {2309, 514}, {3728, 1577}, {3741, 3261}, {16738, 52619}, {18169, 7199}, {20891, 40495}, {21024, 850}, {21700, 4024}, {21713, 52623}, {21838, 523}, {22065, 4025}, {22206, 4036}, {22389, 905}, {23212, 647}, {23473, 3776}, {30097, 52621}, {39780, 7178}, {40627, 3120}, {45208, 4077}, {45216, 3835}, {50510, 1086}
X(53268) = {X(1634),X(4436)}-harmonic conjugate of X(23363)

---

X(53269) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(99) AND X(103)

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

X(53269) lies on these lines: {3, 2786}, {99, 670}, {103, 926}, {522, 44408}, {1011, 4750}, {3566, 23399}, {4120, 4191}, {4897, 23864}, {15411, 16695}, {16058, 45674}, {16059, 45661}, {28846, 48387}, {28906, 48386}, {29078, 48390}

X(53269) = crossdifference of every pair of points on line {1084, 23972}

---

X(53270) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(99) AND X(104)

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

X(53270) lies on these lines: {3, 2787}, {55, 4922}, {56, 4010}, {99, 670}, {104, 900}, {404, 30709}, {405, 14419}, {474, 14431}, {523, 37960}, {814, 44408}, {956, 4730}, {958, 9508}, {1975, 14296}, {2975, 50343}, {3566, 4367}, {4189, 16158}, {16370, 42670}, {29156, 48388}, {29324, 48387}, {29344, 39476}, {40726, 45342}

X(53270) = crossdifference of every pair of points on line {1084, 23980}

---

X(53271) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(99) AND X(205)

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

X(53271) lies on these lines: {1, 50456}, {3, 2788}, {6, 4164}, {56, 7212}, {99, 670}, {105, 659}, {513, 49706}, {523, 1325}, {667, 4762}, {693, 21005}, {812, 21003}, {940, 5040}, {1001, 4455}, {1919, 4449}, {2837, 14419}, {3227, 18825}, {3287, 9040}, {4057, 29362}, {4083, 21007}, {4155, 24286}, {4378, 28894}, {4383, 14404}, {7192, 16874}, {7252, 50524}, {8635, 47672}, {8646, 43067}, {16692, 16695}, {18108, 26824}, {43929, 46149}, {48136, 48297}

X(53271) = reflection of X(6) in X(4164)
X(53271) = crossdifference of every pair of points on line {1084, 2092}

---

X(53272) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(99) AND X(111)

Barycentrics    a^2*(b - c)*(b + c)*(a^4 + a^2*b^2 + a^2*c^2 - 3*b^2*c^2) : :
X(53272) = 4 X[669] - 3 X[18105], 2 X[669] - 3 X[21006], 3 X[9147] - 2 X[22105], 3 X[351] - 2 X[2492], 4 X[2492] - 3 X[9178], 3 X[1649] - 2 X[14279], X[14279] - 3 X[24976], 4 X[45680] - 3 X[47352]

X(53272) lies on these lines: {3, 2793}, {6, 888}, {23, 385}, {25, 14273}, {99, 670}, {111, 351}, {512, 5104}, {526, 42663}, {690, 2930}, {1649, 14279}, {1995, 9123}, {2079, 7669}, {2395, 51869}, {2667, 8632}, {2799, 11641}, {2872, 3569}, {2916, 13232}, {3050, 3221}, {3225, 3228}, {3288, 9009}, {3747, 3768}, {4068, 4491}, {4155, 21003}, {5113, 22260}, {9023, 9135}, {9125, 11284}, {9402, 21007}, {9489, 23878}, {9855, 25423}, {11615, 14669}, {11620, 15564}, {11643, 23288}, {14278, 44826}, {14610, 18311}, {17997, 51906}, {19596, 33919}, {24354, 25124}, {33900, 33998}, {45680, 47352}

X(53272) = reflection of X(i) in X(j) for these {i,j}: {3, 11616}, {6, 5027}, {1649, 24976}, {9178, 351}, {14277, 47139}, {18105, 21006}, {18311, 14610}, {22260, 5113}, {35522, 6131}
X(53272) = isogonal conjugate of the anticomplement of X(38988)
X(53272) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 31128}, {892, 6}
X(53272) = X(i)-isoconjugate of X(j) for these (i,j): {662, 25322}, {36085, 40517}
X(53272) = X(i)-Dao conjugate of X(j) for these (i,j): {351, 690}, {1084, 25322}, {38988, 40517}
X(53272) = crossdifference of every pair of points on line {39, 597}
X(53272) = barycentric product X(i)*X(j) for these {i,j}: {798, 18075}, {892, 38988}, {9178, 31128}
X(53272) = barycentric quotient X(i)/X(j) for these {i,j}: {351, 40517}, {512, 25322}, {18075, 4602}, {18105, 38278}, {38303, 4576}, {38988, 690}

---

X(53273) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(99) AND X(112)

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

X(53273) lies on these lines: {3, 114}, {6, 2882}, {20, 15270}, {22, 7664}, {25, 1560}, {99, 670}, {110, 3565}, {112, 1576}, {148, 9149}, {157, 378}, {159, 2935}, {160, 376}, {186, 47000}, {187, 21177}, {230, 46522}, {512, 1625}, {523, 7482}, {550, 23208}, {669, 3233}, {682, 5254}, {691, 827}, {759, 23392}, {907, 1296}, {1289, 30247}, {1325, 20875}, {1624, 4230}, {2353, 12084}, {2386, 14961}, {2420, 14574}, {2493, 2971}, {2781, 38551}, {2871, 3269}, {3053, 11325}, {3080, 40326}, {3203, 48262}, {3972, 35222}, {4057, 36032}, {4226, 23181}, {4611, 5467}, {5020, 6719}, {5186, 47211}, {5201, 37927}, {5206, 11360}, {5210, 20885}, {5938, 18859}, {7418, 10722}, {7472, 21006}, {7475, 21005}, {7481, 23854}, {7669, 34866}, {7745, 11326}, {8053, 17512}, {8266, 14907}, {8356, 41328}, {8716, 20794}, {9862, 37991}, {14015, 18610}, {14712, 37896}, {15567, 18369}, {15652, 31152}, {35923, 39842}, {37937, 47253}, {38523, 39836}, {41272, 51906}, {41336, 44089}, {45900, 51869}

X(53273) = X(18020)-Ceva conjugate of X(6)
X(53273) = X(i)-isoconjugate of X(j) for these (i,j): {656, 40413}, {661, 40405}, {683, 810}
X(53273) = X(i)-Dao conjugate of X(j) for these (i,j): {1196, 3267}, {1368, 523}, {20975, 125}, {36830, 40405}, {39062, 683}, {40596, 40413}
X(53273) = cevapoint of X(512) and X(11326)
X(53273) = trilinear pole of line {1196, 6467}
X(53273) = crossdifference of every pair of points on line {1084, 6388}
X(53273) = barycentric product X(i)*X(j) for these {i,j}: {99, 1196}, {100, 16716}, {107, 22401}, {110, 5254}, {112, 1368}, {162, 18671}, {249, 12075}, {648, 6467}, {662, 17872}, {682, 6331}, {3080, 52608}, {3565, 40326}, {4563, 40325}, {8750, 18648}, {21406, 32676}, {45207, 52913}
X(53273) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 40405}, {112, 40413}, {648, 683}, {682, 647}, {1196, 523}, {1368, 3267}, {3080, 2489}, {5254, 850}, {6467, 525}, {12075, 338}, {16716, 693}, {17872, 1577}, {18671, 14208}, {22401, 3265}, {40325, 2501}
X(53273) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 11641, 39857}, {4230, 35278, 1624}, {7669, 34866, 39832}, {14712, 37896, 51862}, {14907, 35924, 8266}, {34217, 39860, 3}

---

X(53274) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(99) AND X(476)

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(2*a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - 4*b^2*c^2 + 2*c^4) : :
X(53274) = X[2407] - 3 X[4226], 2 X[2407] - 3 X[5467], 3 X[4226] - 2 X[35345], 3 X[5467] - 4 X[35345], 3 X[14995] - 4 X[24975], 2 X[24975] - 3 X[45662]

X(53274) lies on these lines: {2, 38393}, {3, 2453}, {22, 157}, {67, 38741}, {99, 670}, {338, 12042}, {376, 50146}, {476, 10412}, {523, 2407}, {599, 3098}, {691, 23288}, {925, 1302}, {935, 2696}, {1316, 46127}, {2420, 23968}, {2930, 13188}, {3001, 51372}, {3233, 5502}, {4230, 30716}, {5181, 38738}, {5201, 36822}, {5468, 11123}, {6781, 41583}, {7473, 16237}, {7669, 12188}, {7778, 31152}, {9178, 50941}, {10278, 46495}, {11594, 37916}, {13586, 52756}, {14995, 24975}, {17941, 33799}, {23698, 41359}, {23895, 35329}, {23896, 35330}, {33927, 36188}, {35278, 35357}, {36207, 47284}, {39290, 51262}, {40879, 47285}

X(53274) = reflection of X(i) in X(j) for these {i,j}: {2407, 35345}, {5467, 4226}, {14995, 45662}
X(53274) = anticomplement of X(38393)
X(53274) = reflection of X(5467) in the Euler line
X(53274) = crossdifference of every pair of points on line {1084, 18334}
X(53274) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2407, 4226, 35345}, {2407, 35345, 5467}, {7473, 16237, 23347}, {11634, 46609, 41337}

---

X(53275) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(99) AND X(477)

Barycentrics    (b - c)*(b + c)*(-5*a^8 + 10*a^6*b^2 - 5*a^4*b^4 + 10*a^6*c^2 - 15*a^4*b^2*c^2 + 5*a^2*b^4*c^2 + 2*b^6*c^2 - 5*a^4*c^4 + 5*a^2*b^2*c^4 - 4*b^4*c^4 + 2*b^2*c^6) : :
X(53275) = 5 X[3] - 2 X[23105], 5 X[1656] - 4 X[39482]

X(53275) lies on these lines: {3, 23105}, {22, 4108}, {30, 34291}, {99, 670}, {376, 523}, {381, 44814}, {477, 16171}, {512, 3534}, {647, 44526}, {1656, 39482}, {3566, 11123}, {5210, 47229}, {8675, 43273}, {14380, 47003}, {14618, 35472}, {15421, 44249}, {15750, 46371}, {16229, 37453}, {31101, 34964}, {37991, 44822}, {44438, 47221}

X(53275) = reflection of X(381) in X(44814)

---

X(53276) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(99) AND X(675)

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

X(53276) lies on these lines: {6, 4107}, {99, 670}, {523, 4467}, {667, 4411}, {802, 21007}, {903, 3226}, {2824, 49130}, {3261, 4057}, {3733, 4374}, {3766, 4491}, {3768, 4508}, {3875, 4145}, {4361, 21832}, {4762, 48320}, {5029, 15668}, {6084, 21222}, {14407, 17259}, {17327, 21053}, {21006, 23395}

X(53276) = reflection of X(6) in X(4107)
X(53276) = crossdifference of every pair of points on line {1084, 20970}

---

X(53277) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(100) AND X(102)

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

X(53277) lies on these lines: {3, 3738}, {36, 23087}, {55, 1769}, {100, 190}, {102, 8677}, {513, 2077}, {521, 39199}, {2803, 42455}, {2804, 25438}, {2815, 52739}, {2827, 12332}, {3733, 48383}, {4057, 15313}, {4768, 5687}, {5204, 22379}, {8674, 39200}, {9001, 44408}, {9525, 10306}, {17524, 35055}, {23187, 39226}

X(53277) = reflection of X(i) in X(j) for these {i,j}: {3733, 48383}, {23187, 39226}
X(53277) = crossdifference of every pair of points on line {1015, 1108}

---

X(53278) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(100) AND X(103)

Barycentrics    a^2*(b - c)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c - a^2*b*c + 5*a*b^2*c - 3*b^3*c - a^2*c^2 + 5*a*b*c^2 - 2*b^2*c^2 + a*c^3 - 3*b*c^3) : :
X(53278) = 3 X[3] - 2 X[52726], 3 X[14414] - X[38329]

X(53278) lies on these lines: {3, 3887}, {55, 2254}, {56, 4895}, {100, 190}, {103, 926}, {513, 5537}, {522, 15599}, {595, 23141}, {676, 9511}, {1001, 25380}, {1376, 3716}, {1734, 21789}, {2814, 10306}, {3052, 22384}, {3295, 3960}, {3303, 14413}, {3304, 23057}, {3309, 7634}, {3733, 8646}, {3738, 13205}, {3762, 5687}, {3837, 14942}, {3871, 21222}, {3900, 44408}, {4057, 4394}, {4105, 9000}, {4162, 22091}, {4428, 45328}, {5217, 8648}, {8674, 15626}, {14414, 38329}, {20871, 23866}, {22160, 48018}, {42316, 46388}

X(53278) = crossdifference of every pair of points on line {1015, 16583}
X(53278) = {X(23865),X(50336)}-harmonic conjugate of X(3733)

---

X(53279) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(100) AND X(108)

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

X(53279) lies on these lines: {1, 15906}, {2, 1626}, {3, 119}, {4, 23843}, {5, 23850}, {11, 20999}, {12, 3145}, {20, 2933}, {22, 26231}, {25, 5521}, {30, 1324}, {55, 4415}, {56, 3756}, {100, 190}, {101, 6011}, {108, 676}, {109, 513}, {110, 33637}, {157, 1013}, {162, 1576}, {197, 7580}, {198, 51406}, {411, 23361}, {523, 1897}, {661, 32739}, {675, 40619}, {692, 4551}, {833, 8707}, {835, 9070}, {851, 20989}, {855, 5172}, {899, 20780}, {958, 19548}, {1005, 8053}, {1283, 17719}, {1290, 26711}, {1376, 24320}, {1415, 32653}, {1473, 11502}, {1478, 11334}, {1602, 1995}, {1603, 11413}, {1618, 14513}, {1623, 31272}, {1624, 4246}, {1631, 35988}, {1632, 36797}, {1745, 14529}, {1826, 47209}, {1846, 4186}, {1936, 8679}, {2201, 8608}, {2217, 5691}, {2385, 16870}, {2551, 36510}, {2822, 28346}, {2834, 15252}, {2840, 11700}, {2947, 10537}, {2969, 45946}, {3072, 42450}, {3120, 6187}, {3149, 22654}, {3185, 51435}, {3220, 44425}, {3242, 36509}, {3430, 22299}, {3556, 11500}, {3585, 10260}, {3658, 23181}, {3822, 36011}, {3827, 51361}, {4220, 52139}, {4222, 23383}, {4242, 39478}, {4299, 20842}, {4999, 36558}, {5020, 6714}, {5080, 37311}, {5348, 26892}, {5432, 16064}, {6012, 8706}, {6154, 23858}, {6690, 20834}, {6718, 44313}, {6985, 9798}, {7354, 37259}, {7412, 15622}, {7413, 44411}, {7466, 18610}, {9058, 13397}, {9059, 43348}, {9639, 20254}, {10058, 13744}, {10728, 14127}, {10895, 13733}, {11185, 38906}, {11491, 23844}, {12943, 37397}, {13329, 38472}, {13621, 35220}, {14004, 23339}, {14956, 23369}, {16305, 17927}, {16678, 35996}, {16686, 28353}, {17798, 46579}, {20470, 33849}, {20872, 46549}, {21382, 38358}, {21935, 40980}, {22321, 52371}, {23380, 23512}, {25882, 40560}, {26095, 37009}, {26704, 30250}, {26706, 40097}, {27504, 40462}, {28164, 51637}, {37366, 37578}, {37411, 39600}, {43190, 47666}

X(53279) = reflection of X(44313) in X(6718)
X(53279) = X(46102)-Ceva conjugate of X(6)
X(53279) = X(i)-isoconjugate of X(j) for these (i,j): {513, 40436}, {663, 34399}, {1459, 34406}, {24031, 52775}
X(53279) = X(i)-Dao conjugate of X(j) for these (i,j): {3772, 35518}, {7117, 26932}, {39026, 40436}
X(53279) = trilinear pole of line {3924, 40968}
X(53279) = crossdifference of every pair of points on line {1015, 34588}
X(53279) = barycentric product X(i)*X(j) for these {i,j}: {100, 3772}, {101, 17861}, {190, 3924}, {651, 1837}, {662, 21935}, {664, 40968}, {1018, 17189}, {1783, 41004}, {1897, 26934}, {3699, 36570}, {4552, 40980}, {4557, 16749}
X(53279) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 40436}, {651, 34399}, {1783, 34406}, {1837, 4391}, {3772, 693}, {3924, 514}, {4076, 42380}, {16749, 52619}, {17189, 7199}, {17861, 3261}, {21935, 1577}, {23985, 52775}, {26934, 4025}, {36570, 3676}, {40968, 522}, {40980, 4560}, {41004, 15413}
X(53279) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {56, 28083, 3756}, {100, 1633, 23845}, {20999, 52242, 11}

---

X(53280) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(100) AND X(110)

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

X(53280) lies on these lines: {1, 18174}, {3, 191}, {6, 3121}, {8, 12746}, {9, 33848}, {11, 15507}, {31, 16687}, {36, 23169}, {47, 42463}, {55, 846}, {56, 1046}, {63, 3185}, {88, 27666}, {99, 931}, {100, 190}, {101, 109}, {110, 351}, {197, 21376}, {198, 1761}, {210, 37619}, {228, 4640}, {333, 11688}, {523, 7477}, {643, 3573}, {651, 52931}, {662, 23363}, {692, 1331}, {758, 859}, {851, 17768}, {855, 44669}, {896, 3286}, {901, 8701}, {960, 22345}, {997, 23206}, {999, 23170}, {1086, 27628}, {1211, 18235}, {1259, 3556}, {1260, 37577}, {1284, 35466}, {1293, 8694}, {1376, 4418}, {1402, 4641}, {1403, 4383}, {1617, 23089}, {1707, 2352}, {1710, 11248}, {1757, 5143}, {1760, 23381}, {1781, 37541}, {1962, 18185}, {1979, 52127}, {2175, 23167}, {2284, 46148}, {2292, 4267}, {2361, 7193}, {2594, 29958}, {2801, 15626}, {2836, 18210}, {2939, 10310}, {2941, 6244}, {2960, 35448}, {2975, 23846}, {3218, 20470}, {3219, 52139}, {3336, 16414}, {3647, 17524}, {3649, 28258}, {3666, 20967}, {3681, 15621}, {3725, 40153}, {3756, 28393}, {3827, 45916}, {3833, 19265}, {3868, 23383}, {3869, 23361}, {3873, 18613}, {3929, 10434}, {3941, 36277}, {4225, 11684}, {4230, 52914}, {4245, 5902}, {4413, 24342}, {4414, 5132}, {4551, 21362}, {4567, 17939}, {4581, 50039}, {4588, 8652}, {4854, 13097}, {5204, 23085}, {5506, 16291}, {5730, 15654}, {5883, 19241}, {6014, 28226}, {6097, 19919}, {6099, 43345}, {6690, 21319}, {7419, 34195}, {7428, 22836}, {8054, 16493}, {8697, 28196}, {8699, 28230}, {8758, 34381}, {9840, 21677}, {10176, 16374}, {11203, 40967}, {11246, 16056}, {12331, 47270}, {12528, 15622}, {12635, 28348}, {13205, 23858}, {14882, 38903}, {15175, 36013}, {15494, 37581}, {15571, 32919}, {15624, 35258}, {16408, 41812}, {16570, 16778}, {16585, 53035}, {16874, 23390}, {16877, 17799}, {17126, 20990}, {17724, 21320}, {17798, 20796}, {18042, 18614}, {18253, 37225}, {20769, 20878}, {21010, 23194}, {22148, 36942}, {22149, 23853}, {22321, 24433}, {23541, 45920}, {24405, 33148}, {28148, 28162}, {28152, 28166}, {28156, 28170}, {28176, 28184}, {28180, 28188}, {28200, 28206}, {28210, 28218}, {28214, 28222}, {28239, 43055}, {28250, 40688}, {31165, 37620}, {31847, 34465}, {34247, 37540}, {35000, 45828}, {36098, 43069}, {37579, 42461}, {43346, 43347}

X(53280) = reflection of X(18210) in X(34977)
X(53280) = isogonal conjugate of X(4581)
X(53280) = isogonal conjugate of the anticomplement of X(50330)
X(53280) = X(i)-Ceva conjugate of X(j) for these (i,j): {59, 1682}, {4567, 6}, {29143, 692}, {43069, 4559}, {50039, 2427}
X(53280) = cevapoint of X(i) and X(j) for these (i,j): {1193, 6371}, {2092, 42661}, {2292, 17420}, {20967, 52326}
X(53280) = trilinear pole of line {1193, 1682}
X(53280) = crossdifference of every pair of points on line {11, 115}
X(53280) = X(15507)-line-conjugate of X(11)
X(53280) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4581}, {11, 36098}, {19, 15420}, {244, 8707}, {513, 1220}, {514, 2298}, {522, 961}, {523, 2363}, {649, 30710}, {661, 14534}, {663, 31643}, {667, 1240}, {798, 40827}, {1019, 14624}, {1086, 36147}, {1111, 32736}, {1169, 1577}, {1791, 7649}, {1798, 24006}, {2170, 6648}, {2359, 17924}, {4858, 8687}, {21186, 40454}, {24026, 52928}
X(53280) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 4581}, {6, 15420}, {960, 523}, {1211, 693}, {2092, 4391}, {3125, 16732}, {3666, 850}, {5375, 30710}, {6631, 1240}, {17419, 4858}, {31998, 40827}, {36830, 14534}, {38992, 11}, {39015, 1086}, {39026, 1220}, {52087, 514}
X(53280) = barycentric product X(i)*X(j) for these {i,j}: {1, 3882}, {59, 3910}, {99, 2092}, {100, 3666}, {101, 4357}, {109, 3687}, {110, 1211}, {163, 18697}, {190, 1193}, {429, 4558}, {644, 24471}, {648, 22076}, {651, 960}, {662, 2292}, {664, 2269}, {668, 2300}, {692, 20911}, {765, 48131}, {799, 3725}, {934, 3965}, {1016, 6371}, {1020, 46877}, {1110, 4509}, {1228, 1576}, {1252, 3004}, {1331, 1848}, {1332, 1829}, {1414, 21033}, {1634, 27067}, {1682, 6648}, {1813, 46878}, {1897, 22097}, {2354, 4561}, {2720, 51407}, {3674, 3939}, {3704, 4565}, {3903, 28369}, {3952, 40153}, {4267, 4552}, {4551, 17185}, {4554, 20967}, {4556, 20653}, {4557, 16705}, {4563, 44092}, {4564, 17420}, {4566, 46889}, {4567, 50330}, {4570, 21124}, {4573, 40966}, {4590, 42661}, {4606, 4719}, {4612, 52567}, {4998, 52326}, {5546, 41003}, {6010, 39774}, {6335, 22345}, {18026, 22074}, {18235, 37137}, {21810, 52935}, {27455, 52923}, {29143, 51571}, {41600, 46640}
X(53280) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 15420}, {6, 4581}, {59, 6648}, {99, 40827}, {100, 30710}, {101, 1220}, {110, 14534}, {163, 2363}, {190, 1240}, {429, 14618}, {651, 31643}, {692, 2298}, {906, 1791}, {960, 4391}, {1110, 36147}, {1193, 514}, {1211, 850}, {1228, 44173}, {1252, 8707}, {1415, 961}, {1576, 1169}, {1682, 3910}, {1829, 17924}, {1848, 46107}, {2092, 523}, {2149, 36098}, {2269, 522}, {2292, 1577}, {2300, 513}, {2354, 7649}, {3004, 23989}, {3666, 693}, {3674, 52621}, {3687, 35519}, {3725, 661}, {3882, 75}, {3910, 34387}, {3965, 4397}, {4267, 4560}, {4357, 3261}, {4503, 4411}, {4557, 14624}, {4612, 52550}, {4719, 4801}, {6371, 1086}, {16705, 52619}, {17185, 18155}, {17420, 4858}, {18697, 20948}, {20653, 52623}, {20911, 40495}, {20967, 650}, {21033, 4086}, {21124, 21207}, {21810, 4036}, {22074, 521}, {22076, 525}, {22097, 4025}, {22345, 905}, {23979, 52928}, {23990, 32736}, {24471, 24002}, {27067, 52618}, {28369, 4374}, {32656, 2359}, {32661, 1798}, {40153, 7192}, {40966, 3700}, {40976, 3064}, {41581, 26546}, {42661, 115}, {44092, 2501}, {46878, 46110}, {46889, 7253}, {48131, 1111}, {50330, 16732}, {52326, 11}
X(53280) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {63, 3185, 16678}, {100, 4427, 4436}, {100, 23845, 23832}, {100, 52923, 3699}, {109, 23067, 2283}, {896, 3724, 3286}, {4557, 23845, 100}, {21320, 28353, 17724}

---

X(53281) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(100) AND X(111)

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

X(53281) lies on these lines: {3, 2830}, {6, 5040}, {25, 47235}, {43, 50456}, {55, 4455}, {100, 190}, {111, 351}, {650, 667}, {661, 16874}, {884, 6187}, {1635, 8650}, {2788, 19544}, {3004, 48323}, {3733, 47827}, {4057, 48226}, {4164, 4383}, {4367, 47784}, {4378, 47880}, {4477, 4843}, {5075, 14407}, {6373, 21786}, {7252, 50491}, {18108, 26777}, {26249, 31150}, {48338, 50503}, {50506, 50509}

X(53281) = isogonal conjugate of the isotomic conjugate of X(30709)
X(53281) = X(5380)-Ceva conjugate of X(6)
X(53281) = crossdifference of every pair of points on line {1015, 2482}
X(53281) = barycentric product X(6)*X(30709)
X(53281) = barycentric quotient X(30709)/X(76)
X(53281) = {X(5040),X(14404)}-harmonic conjugate of X(6)

---

X(53282) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(100) AND X(112)

Barycentrics    a^2*(a - b)*(a - c)*(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(53282) lies on these lines: {3, 2831}, {19, 18610}, {25, 47231}, {48, 18612}, {55, 24444}, {100, 190}, {112, 1576}, {159, 18598}, {523, 7476}, {610, 18611}, {1376, 24335}, {1486, 1781}, {1761, 12329}, {2173, 18622}, {2939, 3556}, {15586, 20857}, {16545, 18616}, {16546, 18617}, {16568, 16876}, {16680, 35327}, {18594, 18615}, {18595, 18618}, {18596, 18619}, {18597, 18620}, {18599, 18621}, {21376, 37577}, {29018, 29201}, {32735, 50344}

X(53282) = X(5379)-Ceva conjugate of X(6)
X(53282) = X(514)-isoconjugate of X(40406)
X(53282) = X(i)-Dao conjugate of X(j) for these (i,j): {21530, 693}, {40941, 3267}
X(53282) = crossdifference of every pair of points on line {1015, 15526}
X(53282) = barycentric product X(i)*X(j) for these {i,j}: {100, 40941}, {101, 23537}, {112, 21530}, {162, 18674}, {662, 40973}, {1783, 18732}, {8750, 18651}, {18709, 35325}
X(53282) = barycentric quotient X(i)/X(j) for these {i,j}: {692, 40406}, {18674, 14208}, {18732, 15413}, {21530, 3267}, {23537, 3261}, {40941, 693}, {40973, 1577}

---

X(53283) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(100) AND X(476)

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

X(53283) lies on these lines: {3, 47270}, {55, 2607}, {100, 190}, {191, 210}, {476, 10412}, {523, 3658}, {1048, 8614}, {1324, 2915}, {1329, 27553}, {2948, 12331}, {2975, 37405}, {3925, 37165}, {5690, 47749}, {6690, 46555}, {23703, 50346}

X(53283) = crossdifference of every pair of points on line {1015, 18334}
X(53283) = {X(13589),X(46611)}-harmonic conjugate of X(23832)

---

X(53284) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(100) AND X(675)

Barycentrics    a*(b - c)*(a^4 - 2*a^3*b + a^2*b^2 - 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(53284) lies on these lines: {55, 812}, {100, 190}, {171, 21786}, {650, 1734}, {667, 47761}, {673, 884}, {693, 23865}, {814, 7255}, {1001, 4728}, {1376, 1635}, {1491, 4833}, {1621, 21297}, {1638, 45695}, {2291, 29352}, {2517, 4057}, {2788, 40166}, {2820, 7580}, {3149, 38324}, {3733, 47824}, {3757, 20950}, {4106, 8641}, {4367, 47891}, {4413, 4763}, {4423, 4928}, {4455, 27916}, {4477, 6362}, {4885, 8642}, {14304, 33848}, {16158, 46403}, {20954, 23400}, {21301, 21789}

X(53284) = crossdifference of every pair of points on line {354, 1015}
X(53284) = {X(4728),X(8645)}-harmonic conjugate of X(1001)

---

X(53285) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(101) AND X(102)

Barycentrics    a^2*(a - b - c)^2*(b - c)*(a^2 - b^2 + b*c - c^2) : :
X(53285) = 3 X[3] - 2 X[52730], X[52730] - 3 X[52739], 3 X[663] + X[4105], X[4895] + 3 X[14392], X[2254] - 3 X[14414], X[4730] - 3 X[11124]

X(53285) lies on these lines: {1, 676}, {3, 928}, {78, 50333}, {101, 692}, {102, 8677}, {200, 4528}, {214, 3738}, {220, 52614}, {512, 48387}, {513, 50371}, {520, 39199}, {521, 2605}, {522, 4794}, {526, 39478}, {650, 663}, {654, 8648}, {667, 8676}, {677, 14733}, {678, 42078}, {900, 6326}, {924, 48383}, {1639, 52371}, {1807, 2804}, {1984, 2310}, {2254, 14414}, {2338, 4845}, {2774, 52726}, {3185, 23220}, {3716, 42455}, {3887, 38324}, {3904, 4511}, {4040, 6362}, {4449, 43052}, {4724, 23745}, {4730, 11124}, {4827, 17412}, {6370, 44427}, {8058, 48302}, {9521, 37531}, {15584, 29188}, {18443, 44819}, {21789, 23090}, {22836, 23887}, {34772, 47695}, {42337, 48307}

X(53285) = reflection of X(i) in X(j) for these {i,j}: {3, 52739}, {42455, 3716}
X(53285) = X(i)-Ceva conjugate of X(j) for these (i,j): {3738, 654}, {4242, 2245}, {5548, 220}, {6099, 219}, {6596, 2170}, {6740, 1146}, {51562, 9}, {52356, 9404}
X(53285) = crossdifference of every pair of points on line {57, 1020}
X(53285) = X(i)-isoconjugate of X(j) for these (i,j): {7, 2222}, {56, 35174}, {57, 655}, {80, 934}, {85, 32675}, {108, 52392}, {109, 18815}, {269, 51562}, {604, 46405}, {651, 2006}, {658, 2161}, {664, 1411}, {759, 4566}, {1020, 24624}, {1407, 36804}, {1414, 52383}, {1427, 47318}, {1461, 18359}, {1807, 36118}, {3676, 52377}, {3960, 23592}, {4569, 6187}, {4616, 34857}, {4617, 36910}, {4626, 52371}, {6354, 37140}, {6614, 52409}, {7339, 52356}, {13149, 52431}, {32714, 52351}, {36048, 45926}, {37136, 52212}
X(53285) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 35174}, {11, 18815}, {2968, 20566}, {3161, 46405}, {3738, 4453}, {5452, 655}, {6600, 51562}, {6608, 52356}, {13999, 273}, {14714, 80}, {15607, 45926}, {24771, 36804}, {34586, 4566}, {35128, 85}, {35204, 664}, {35508, 18359}, {38983, 52392}, {38984, 7}, {38991, 2006}, {39025, 1411}, {40584, 658}, {40608, 52383}, {40612, 4569}
X(53285) = barycentric product X(i)*X(j) for these {i,j}: {8, 654}, {9, 3738}, {36, 3239}, {55, 3904}, {200, 3960}, {219, 44428}, {220, 4453}, {312, 8648}, {320, 657}, {341, 21758}, {522, 2323}, {650, 4511}, {652, 5081}, {663, 32851}, {758, 1021}, {765, 46384}, {860, 23090}, {1043, 21828}, {1098, 2610}, {1309, 38353}, {1443, 4130}, {1983, 24026}, {2245, 7253}, {2310, 4585}, {2328, 4707}, {2342, 53045}, {2361, 4391}, {2600, 44687}, {3218, 3900}, {3936, 21789}, {4086, 4282}, {4105, 17078}, {4242, 34591}, {4397, 7113}, {4528, 40215}, {5548, 51402}, {6332, 52427}, {6370, 7054}, {7058, 42666}, {7101, 22379}, {8611, 17515}, {8641, 20924}, {14427, 52553}, {15411, 44113}, {34544, 52356}, {35128, 51562}, {35519, 52426}, {51565, 53046}, {52434, 52622}
X(53285) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 46405}, {9, 35174}, {36, 658}, {41, 2222}, {55, 655}, {200, 36804}, {220, 51562}, {320, 46406}, {650, 18815}, {652, 52392}, {654, 7}, {657, 80}, {663, 2006}, {1021, 14616}, {1443, 36838}, {1870, 13149}, {1983, 7045}, {2175, 32675}, {2245, 4566}, {2323, 664}, {2328, 47318}, {2361, 651}, {3063, 1411}, {3119, 52356}, {3218, 4569}, {3239, 20566}, {3709, 52383}, {3724, 1020}, {3738, 85}, {3900, 18359}, {3904, 6063}, {3960, 1088}, {4105, 36910}, {4130, 52409}, {4171, 15065}, {4282, 1414}, {4511, 4554}, {4895, 14628}, {5081, 46404}, {7113, 934}, {8641, 2161}, {8648, 57}, {14427, 51975}, {17078, 52937}, {21758, 269}, {21789, 24624}, {21828, 3668}, {22379, 7177}, {32851, 4572}, {33525, 45926}, {35128, 4453}, {42666, 6354}, {44113, 52607}, {44428, 331}, {46384, 1111}, {52413, 36118}, {52426, 109}, {52427, 653}, {52434, 1461}, {52440, 4617}, {53046, 22464}

---

X(53286) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(101) AND X(104)

Barycentrics    a^2*(a - b - c)*(b - c)*(a^3 - a*b^2 + 3*a*b*c - b^2*c - a*c^2 - b*c^2) : :
X(53286) = X[4105] - 5 X[8656], 3 X[14431] - 4 X[33528]

X(53286) lies on these lines: {3, 3887}, {55, 4895}, {56, 2254}, {101, 692}, {104, 900}, {220, 46388}, {294, 884}, {521, 4057}, {522, 3733}, {652, 663}, {654, 2342}, {659, 6366}, {667, 3900}, {898, 36802}, {956, 3762}, {958, 3716}, {995, 23141}, {999, 3960}, {1191, 22384}, {1635, 23858}, {1946, 4162}, {2787, 42455}, {2812, 38324}, {2814, 22770}, {2826, 48694}, {3251, 42670}, {3303, 23057}, {3304, 14413}, {3309, 44408}, {3667, 23187}, {3738, 4491}, {4105, 8656}, {4367, 6362}, {4782, 9366}, {4843, 8639}, {4953, 22096}, {8674, 39200}, {13256, 23866}, {14431, 33528}, {15313, 39199}, {17989, 29278}, {20999, 38325}, {22160, 48294}, {23189, 48307}, {23864, 50518}, {25380, 25524}, {35128, 42771}, {40726, 45328}

X(53286) = reflection of X(i) in X(j) for these {i,j}: {3, 52726}, {48387, 667}
X(53286) = X(36037)-Ceva conjugate of X(6)
X(53286) = X(i)-isoconjugate of X(j) for these (i,j): {57, 50039}, {651, 14554}
X(53286) = X(i)-Dao conjugate of X(j) for these (i,j): {1769, 36038}, {5452, 50039}, {34590, 1441}, {38991, 14554}
X(53286) = crossdifference of every pair of points on line {226, 1086}
X(53286) = barycentric product X(i)*X(j) for these {i,j}: {8, 21786}, {21, 21894}, {55, 21222}, {281, 23087}, {522, 5053}, {4570, 52341}, {5548, 34590}
X(53286) = barycentric quotient X(i)/X(j) for these {i,j}: {55, 50039}, {663, 14554}, {5053, 664}, {21222, 6063}, {21786, 7}, {21894, 1441}, {23087, 348}, {52341, 21207}
X(53286) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {667, 48327, 23865}, {4895, 8648, 55}

---

X(53287) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(101) AND X(105)

Barycentrics    a^2*(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(53287) = 2 X[1960] + X[22108], X[657] + 3 X[8643]

X(53287) lies on these lines: {3, 2820}, {31, 21786}, {36, 238}, {55, 1635}, {101, 692}, {105, 659}, {650, 8642}, {657, 1919}, {663, 6586}, {665, 2195}, {669, 2512}, {812, 1001}, {909, 23351}, {918, 45695}, {1376, 4763}, {1617, 43050}, {1621, 47776}, {2110, 8632}, {2490, 4477}, {2720, 32728}, {4068, 4155}, {4394, 8641}, {4401, 22160}, {4423, 4728}, {4928, 8167}, {5029, 19561}, {5284, 21297}, {5584, 38329}, {6050, 6182}, {8648, 16686}, {8654, 21005}, {9443, 48330}, {10306, 38327}, {16823, 20950}, {20476, 23866}, {23349, 40400}

X(53287) = isogonal conjugate of the anticomplement of X(38980)
X(53287) = X(36086)-Ceva conjugate of X(6)
X(53287) = X(673)-isoconjugate of X(40526)
X(53287) = crossdifference of every pair of points on line {37, 142}
X(53287) = barycentric product X(i)*X(j) for these {i,j}: {650, 7677}, {36086, 38980}
X(53287) = barycentric quotient X(i)/X(j) for these {i,j}: {2223, 40526}, {7677, 4554}, {38379, 3717}
X(53287) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {650, 8642, 23865}, {667, 4455, 21003}, {1635, 8645, 55}, {4455, 21003, 4491}

---

X(53288) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(101) AND X(108)

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

X(53288) lies on these lines: {3, 2823}, {101, 692}, {108, 676}, {198, 9502}, {513, 1020}, {1415, 8750}, {1486, 20990}, {1576, 14776}, {1633, 2283}, {1864, 23204}, {2356, 8607}, {4552, 7437}, {11028, 17463}, {11399, 23843}, {16577, 20853}, {23067, 23703}, {24019, 52604}, {35341, 40521}

X(53288) = isogonal conjugate of the anticomplement of X(40628)
X(53288) = X(7012)-Ceva conjugate of X(6)
X(53288) = X(i)-isoconjugate of X(j) for these (i,j): {513, 40424}, {514, 40399}, {653, 40527}, {693, 1167}, {905, 40444}, {6332, 40397}
X(53288) = X(i)-Dao conjugate of X(j) for these (i,j): {1210, 35518}, {6260, 693}, {7004, 17880}, {39026, 40424}
X(53288) = trilinear pole of line {3611, 40958}
X(53288) = crossdifference of every pair of points on line {1086, 16596}
X(53288) = barycentric product X(i)*X(j) for these {i,j}: {100, 1108}, {101, 1210}, {110, 21933}, {190, 40958}, {644, 37566}, {648, 3611}, {651, 1864}, {692, 17862}, {1071, 1783}, {1226, 32739}, {1532, 32641}, {4551, 40979}, {6260, 36049}, {6335, 23204}, {7012, 40628}
X(53288) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 40424}, {692, 40399}, {1071, 15413}, {1108, 693}, {1210, 3261}, {1864, 4391}, {1946, 40527}, {3611, 525}, {8750, 40444}, {17862, 40495}, {21933, 850}, {23204, 905}, {32739, 1167}, {37566, 24002}, {40628, 17880}, {40958, 514}, {40979, 18155}, {41562, 18160}

---

X(53289) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(101) AND X(111)

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

X(53289) lies on these lines: {3, 2824}, {6, 5029}, {101, 692}, {111, 351}, {292, 3572}, {649, 4057}, {665, 4491}, {667, 2515}, {3247, 4145}, {3709, 3733}, {4107, 17259}, {16777, 21832}

X(53289) = crossdifference of every pair of points on line {1086, 1125}
X(53289) = {X(5029),X(14407)}-harmonic conjugate of X(6)

---

X(53290) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(101) AND X(112)

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

X(53290) lies on these lines: {3, 2825}, {6, 43693}, {101, 692}, {112, 1576}, {607, 23843}, {649, 36075}, {906, 23845}, {919, 8687}, {1415, 2425}, {1973, 23383}, {2172, 23361}, {2301, 20986}, {3573, 7259}, {4559, 32739}, {6574, 30555}, {9406, 42669}, {23844, 52425}, {29014, 29221}, {29041, 29044}

X(53290) = isogonal conjugate of the isotomic conjugate of X(14543)
X(53290) = X(i)-isoconjugate of X(j) for these (i,j): {514, 1257}, {525, 40431}, {656, 40414}, {693, 2983}, {905, 40445}, {951, 4391}, {1111, 29163}
X(53290) = X(i)-Dao conjugate of X(j) for these (i,j): {440, 3261}, {1834, 52622}, {40596, 40414}, {40940, 3267}
X(53290) = trilinear pole of line {40984, 44093}
X(53290) = crossdifference of every pair of points on line {1086, 2968}
X(53290) = barycentric product X(i)*X(j) for these {i,j}: {6, 14543}, {99, 40984}, {100, 1104}, {101, 40940}, {109, 950}, {110, 1834}, {112, 440}, {162, 18673}, {648, 44093}, {651, 2264}, {662, 40977}, {692, 17863}, {1252, 29162}, {1331, 1842}, {8750, 18650}, {32641, 51410}
X(53290) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 40414}, {440, 3267}, {692, 1257}, {950, 35519}, {1104, 693}, {1834, 850}, {1842, 46107}, {2264, 4391}, {8750, 40445}, {14543, 76}, {17863, 40495}, {18673, 14208}, {23990, 29163}, {29162, 23989}, {32676, 40431}, {32739, 2983}, {40940, 3261}, {40977, 1577}, {40984, 523}, {44093, 525}

---

X(53291) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(102) AND X(103)

Barycentrics    a^2*(a^7 - a^6*b - 3*a^3*b^4 + 3*a^2*b^5 + 2*a*b^6 - 2*b^7 - a^6*c + a^5*b*c + 2*a^3*b^3*c - a^2*b^4*c - 3*a*b^5*c + 2*b^6*c + 2*a^3*b^2*c^2 - 2*a^2*b^3*c^2 + 2*a*b^4*c^2 - 2*b^5*c^2 + 2*a^3*b*c^3 - 2*a^2*b^2*c^3 - 2*a*b^3*c^3 + 2*b^4*c^3 - 3*a^3*c^4 - a^2*b*c^4 + 2*a*b^2*c^4 + 2*b^3*c^4 + 3*a^2*c^5 - 3*a*b*c^5 - 2*b^2*c^5 + 2*a*c^6 + 2*b*c^6 - 2*c^7) : :
X(53291) = X[20468] - 3 X[31884]

X(53291) lies on these lines: {1, 34935}, {3, 692}, {4, 21252}, {20, 21293}, {56, 1204}, {64, 22654}, {74, 104}, {102, 8677}, {103, 926}, {106, 28163}, {840, 15731}, {953, 28159}, {1350, 2876}, {1385, 49151}, {2699, 29314}, {2870, 30271}, {2875, 3428}, {2877, 30269}, {2975, 11440}, {3098, 11495}, {3357, 12114}, {4131, 43363}, {5253, 43601}, {7689, 11249}, {8567, 34046}, {9441, 33844}, {9786, 14717}, {11438, 22753}, {12032, 28844}, {20468, 31884}, {22775, 38600}, {28145, 28173}, {28149, 28193}, {28153, 28211}, {28157, 28227}, {28167, 28219}, {28171, 28233}, {28177, 28197}, {28185, 28203}, {28201, 28215}, {28207, 28223}, {28865, 28884}, {29009, 29019}, {29011, 29072}, {29035, 29320}, {32138, 32153}

X(53291) = midpoint of X(20) and X(21293)
X(53291) = reflection of X(i) in X(j) for these {i,j}: {4, 21252}, {692, 3}
X(53291) = crossdifference of every pair of points on line {23972, 23986}

---

X(53292) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(102) AND X(104)

Barycentrics    a^2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7 + a^7*c - 4*a^6*b*c + 4*a^5*b^2*c + 5*a^4*b^3*c - 11*a^3*b^4*c + 2*a^2*b^5*c + 6*a*b^6*c - 3*b^7*c + 4*a^5*b*c^2 - 10*a^4*b^2*c^2 + 8*a^3*b^3*c^2 + 8*a^2*b^4*c^2 - 12*a*b^5*c^2 + 2*b^6*c^2 - 3*a^5*c^3 + 5*a^4*b*c^3 + 8*a^3*b^2*c^3 - 20*a^2*b^3*c^3 + 7*a*b^4*c^3 + 3*b^5*c^3 - 11*a^3*b*c^4 + 8*a^2*b^2*c^4 + 7*a*b^3*c^4 - 4*b^4*c^4 + 3*a^3*c^5 + 2*a^2*b*c^5 - 12*a*b^2*c^5 + 3*b^3*c^5 + 6*a*b*c^6 + 2*b^2*c^6 - a*c^7 - 3*b*c^7) : :
X(53292) = 3 X[3576] - X[33810]

X(53292) lies on these lines: {1, 15622}, {3, 214}, {40, 8683}, {55, 51236}, {56, 774}, {74, 953}, {102, 8677}, {103, 106}, {104, 900}, {999, 2823}, {1593, 1830}, {2807, 34586}, {3428, 12329}, {3576, 8053}, {4557, 6326}, {6261, 23361}, {7416, 37525}, {9910, 18237}, {10703, 23981}, {11014, 15623}, {13734, 15950}, {13744, 34789}, {28159, 28203}, {28163, 28235}, {28173, 28219}, {28185, 28223}, {28193, 28233}, {34471, 37195}

X(53292) = midpoint of X(1) and X(33811)
X(53292) = reflection of X(23845) in X(3)
X(53292) = crossdifference of every pair of points on line {23980, 23986}
X(53292) = {X(3),X(40257)}-harmonic conjugate of X(23846)

---

X(53293) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(102) AND X(105)

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

X(53293) lies on these lines: {3, 2835}, {56, 2097}, {102, 8677}, {105, 659}, {241, 3827}, {1001, 24315}, {1457, 8607}, {4471, 22769}, {10571, 52610}, {27628, 43055}, {33848, 43048}, {39200, 41343}

X(53293) = crossdifference of every pair of points on line {6184, 23986}

---

X(53294) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(102) AND X(106)

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

X(53294) lies on these lines: {1, 15906}, {3, 2841}, {36, 2390}, {56, 3937}, {74, 28203}, {100, 38568}, {102, 8677}, {103, 28235}, {104, 513}, {105, 51642}, {106, 1960}, {214, 2835}, {517, 25438}, {692, 34586}, {1361, 51236}, {1387, 2834}, {1411, 42753}, {2810, 22560}, {2818, 22775}, {2840, 11715}, {3259, 12764}, {8686, 38452}, {10571, 36059}, {12531, 14513}, {17100, 38512}, {21307, 51565}, {28159, 28223}

X(53294) = crossdifference of every pair of points on line {4370, 23986}
X(53294) = barycentric product X(57)*X(10703)
X(53294) = barycentric quotient X(10703)/X(312)

---

X(53295) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(102) AND X(110)

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

X(53295) lies on these lines: {3, 2773}, {102, 8677}, {110, 351}, {520, 2605}, {523, 4833}, {663, 8999}, {924, 3733}, {1510, 50344}, {2776, 44827}, {3738, 6265}, {6326, 8674}, {6366, 39534}, {7252, 52310}, {9001, 48136}, {35057, 39212}

X(53295) = crossdifference of every pair of points on line {115, 117}

---

X(53296) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(103) AND X(104)

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

X(53296) lies on these lines: {3, 2801}, {56, 2310}, {74, 28173}, {84, 23383}, {98, 29308}, {102, 28193}, {103, 926}, {104, 900}, {106, 28227}, {840, 38451}, {953, 28211}, {963, 3556}, {971, 20470}, {1208, 42450}, {1350, 9016}, {1709, 18613}, {1768, 15626}, {2182, 2250}, {2823, 17463}, {3185, 30304}, {5851, 15507}, {8683, 46684}, {10085, 23361}, {10167, 52139}, {11220, 16678}, {12032, 28842}, {12114, 43160}, {15071, 23846}, {23843, 26927}, {28145, 28197}, {28149, 28231}, {28177, 28215}, {28844, 28861}, {28853, 28884}, {29031, 29104}, {42079, 50677}

X(53296) = reflection of X(4557) in X(3)
X(53296) = crossdifference of every pair of points on line {23972, 23980}
X(53296) = {X(1768),X(15626)}-harmonic conjugate of X(23845)

---

X(53297) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(103) AND X(105)

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

X(53297) lies on these lines: {3, 2809}, {6, 51329}, {38, 55}, {56, 2170}, {103, 926}, {104, 972}, {105, 659}, {165, 8683}, {354, 18210}, {672, 8679}, {840, 28471}, {909, 23346}, {910, 20470}, {958, 21232}, {999, 15746}, {1001, 51435}, {1155, 15635}, {1282, 4557}, {1458, 2272}, {1477, 2291}, {1617, 10829}, {2426, 36057}, {2975, 21272}, {3941, 18725}, {4068, 18162}, {5845, 8299}, {6244, 13205}, {7289, 8053}, {11028, 17463}, {11051, 42314}, {16679, 18161}, {16694, 18735}, {17448, 37575}, {35326, 39046}

X(53297) = isogonal conjugate of the anticomplement of X(39077)
X(53297) = X(9503)-Ceva conjugate of X(6)
X(53297) = crossdifference of every pair of points on line {6184, 23972}
X(53297) = barycentric product X(9503)*X(39077)

---

X(53298) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(103) AND X(106)

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

X(53298) lies on these lines: {1, 1633}, {3, 2810}, {4, 17059}, {36, 909}, {56, 1461}, {58, 1576}, {74, 28193}, {101, 24484}, {102, 104}, {103, 926}, {105, 1477}, {106, 1960}, {109, 3937}, {513, 37815}, {522, 24813}, {595, 23388}, {649, 840}, {675, 7192}, {692, 44858}, {727, 28485}, {741, 28476}, {953, 28233}, {991, 2876}, {1086, 2834}, {1293, 13205}, {1412, 5322}, {1458, 2195}, {1623, 1797}, {2222, 15635}, {2390, 40091}, {2712, 28563}, {2726, 8689}, {2801, 16560}, {2975, 3888}, {3207, 52969}, {4306, 22654}, {4334, 51687}, {7291, 18450}, {7987, 33587}, {8679, 13329}, {12032, 28848}, {12033, 35000}, {14839, 24826}, {17223, 28539}, {28145, 28231}, {28149, 28173}, {28163, 28227}, {28171, 28211}, {28513, 28523}, {43149, 48908}

X(53298) = reflection of X(i) in X(j) for these {i,j}: {4, 17059}, {3939, 3}
X(53298) = crossdifference of every pair of points on line {4370, 23757}
X(53298) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3937, 20999, 109}, {4306, 22654, 41401}

---

X(53299) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(103) AND X(108)

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

X(53299) lies on these lines: {3, 2812}, {103, 926}, {108, 676}, {198, 46392}, {513, 2078}, {650, 39199}, {2254, 20999}, {8273, 14414}, {8678, 44408}, {41341, 45884}

X(53299) = crossdifference of every pair of points on line {1212, 17102}

---

X(53300) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(103) AND X(109)

Barycentrics    a^2*(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(53300) = 3 X[3] - 2 X[52739], 3 X[52730] - X[52739]

X(53300) lies on these lines: {3, 928}, {40, 6366}, {44, 513}, {46, 10015}, {57, 676}, {63, 50333}, {103, 926}, {109, 692}, {512, 44408}, {665, 20672}, {876, 2196}, {884, 36057}, {900, 1768}, {924, 48390}, {2488, 39199}, {2605, 8641}, {3218, 47695}, {3738, 46684}, {3900, 4091}, {5221, 30691}, {5709, 9521}, {7004, 17463}, {8642, 34948}, {8713, 48064}, {14413, 42657}, {23684, 32939}, {23737, 49296}, {25380, 51435}, {37534, 44819}, {42316, 52614}, {42649, 48151}, {43932, 51648}

X(53300) = reflection of X(i) in X(j) for these {i,j}: {3, 52730}, {2605, 23224}, {48306, 39199}
X(53300) = X(13138)-isoconjugate of X(48357)
X(53300) = crossdifference of every pair of points on line {1, 1146}
X(53300) = barycentric product X(14837)*X(39558)
X(53300) = barycentric quotient X(39558)/X(44327)

---

X(53301) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(103) AND X(110)

Barycentrics    a^2*(b - c)*(a^4 + a^3*b - 3*a^2*b^2 - a*b^3 + 2*b^4 + 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 + 2*c^4) : :
X(53301) = 3 X[3] - 2 X[44827]

X(53301) lies on these lines: {3, 2774}, {103, 926}, {110, 351}, {513, 5536}, {520, 3733}, {523, 4467}, {649, 9000}, {900, 13243}, {1282, 9508}, {1768, 8674}, {1797, 23345}, {2424, 32657}, {3303, 42657}, {4091, 8676}, {4252, 42662}, {5221, 30574}, {5737, 24353}, {21789, 44410}

X(53301) = reflection of X(44408) in X(4091)
X(53301) = crossdifference of every pair of points on line {115, 118}

---

X(53302) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(104) AND X(105)

Barycentrics    a*(a^5 - a^4*b + a^2*b^3 - a*b^4 - a^4*c + a^3*b*c - a^2*b^2*c + 2*a*b^3*c - b^4*c - a^2*b*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(53302) = 3 X[11194] - X[24826]

X(53302) lies on these lines: {1, 692}, {2, 45920}, {3, 528}, {11, 20999}, {21, 1634}, {25, 23711}, {28, 52604}, {36, 24715}, {56, 1086}, {101, 5701}, {104, 900}, {105, 659}, {106, 759}, {190, 2975}, {238, 8679}, {496, 23850}, {497, 1626}, {513, 37815}, {524, 15981}, {537, 8666}, {545, 11194}, {551, 36011}, {643, 38478}, {673, 7677}, {675, 693}, {761, 14665}, {812, 52726}, {840, 50359}, {851, 41341}, {909, 1319}, {958, 4422}, {993, 4432}, {1001, 4364}, {1201, 40980}, {1279, 2195}, {1324, 15325}, {1420, 2217}, {1486, 42884}, {1623, 10707}, {1633, 53055}, {1647, 6187}, {1935, 41682}, {2218, 28011}, {2291, 4394}, {2716, 28233}, {2796, 15952}, {2834, 15251}, {2933, 7288}, {3058, 16064}, {3086, 23843}, {3145, 37722}, {3246, 34371}, {3304, 13733}, {4224, 18613}, {4511, 35552}, {4966, 5849}, {5253, 27191}, {5563, 37227}, {6174, 23858}, {6691, 38903}, {7668, 37041}, {9026, 23693}, {9028, 49768}, {9041, 12513}, {9078, 9109}, {9710, 16422}, {10072, 11334}, {11235, 49127}, {11249, 29243}, {12773, 14686}, {14987, 28219}, {15621, 19649}, {15624, 27473}, {16484, 18162}, {16487, 18725}, {17369, 36476}, {20834, 49736}, {20881, 43135}, {22758, 24828}, {22765, 24833}, {23855, 47357}, {24315, 24331}, {25524, 40480}, {30385, 45436}, {30386, 45437}, {31157, 52273}, {34139, 38602}, {34583, 35281}, {37646, 41346}

X(53302) = midpoint of X(i) and X(j) for these {i,j}: {1, 16560}, {12513, 24820}
X(53302) = crossdifference of every pair of points on line {6184, 23980}
X(53302) = barycentric product X(1)*X(24618)
X(53302) = barycentric quotient X(24618)/X(75)
X(53302) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1001, 36529, 4364}, {1623, 10707, 13589}

---

X(53303) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(104) AND X(106)

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

X(53303) lies on these lines: {1, 23845}, {3, 2802}, {36, 4674}, {55, 17460}, {56, 244}, {102, 28233}, {104, 900}, {105, 8686}, {106, 1960}, {214, 4557}, {513, 32486}, {537, 11194}, {759, 2718}, {953, 28219}, {956, 4738}, {958, 24003}, {993, 34587}, {999, 1486}, {1149, 2390}, {1319, 15635}, {1320, 23832}, {1388, 23844}, {1420, 23383}, {1626, 22767}, {2975, 3952}, {3576, 15624}, {5048, 23205}, {5126, 20470}, {5563, 20840}, {6075, 52383}, {10090, 39758}, {13744, 16173}, {14026, 32198}, {18170, 37617}, {19335, 38455}, {21842, 23846}, {22306, 52139}, {22313, 37605}, {22654, 41426}, {23361, 37618}, {38452, 43081}, {39200, 41343}

X(53303) = reflection of X(8683) in X(3)
X(53303) = crossdifference of every pair of points on line {4370, 23980}

---

X(53304) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(104) AND X(108)

Barycentrics    a*(b - c)*(a^6 - 2*a^4*b^2 + a^2*b^4 - a^4*b*c + 2*a^3*b^2*c - 2*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 + 2*a*b^3*c^2 + 2*a*b^2*c^3 - 2*b^3*c^3 + a^2*c^4 - 2*a*b*c^4 + b*c^5) : :
X(53304) = 3 X[11125] - X[42755]

X(53304) lies on these lines: {1, 8677}, {3, 2804}, {11, 45922}, {36, 35013}, {104, 900}, {108, 676}, {186, 523}, {404, 23678}, {513, 663}, {522, 23224}, {526, 5494}, {659, 14667}, {851, 36035}, {2178, 47137}, {2849, 11700}, {3738, 11713}, {7004, 17463}, {8648, 30572}, {11125, 42755}, {23220, 43933}, {31667, 35050}, {44410, 48307}, {45945, 52427}

X(53304) = reflection of X(i) in X(j) for these {i,j}: {3, 44807}, {659, 39200}, {35050, 31667}, {39534, 676}
X(53304) = isogonal conjugate of the anticomplement of X(39004)
X(53304) = crossdifference of every pair of points on line {9, 216}
X(53304) = barycentric product X(i)*X(j) for these {i,j}: {905, 45766}, {32641, 38372}
X(53304) = barycentric quotient X(45766)/X(6335)

---

X(53305) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(104) AND X(109)

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

X(53305) lies on these lines: {1, 23087}, {3, 3738}, {36, 238}, {56, 1769}, {104, 900}, {109, 692}, {514, 44812}, {522, 23187}, {654, 909}, {956, 4768}, {2254, 20999}, {2815, 52730}, {2827, 22775}, {2849, 3960}, {3900, 34143}, {4091, 8999}, {8648, 51236}, {8674, 15626}, {9001, 48387}, {9525, 22770}, {20989, 45884}, {23345, 36052}

X(53305) = reflection of X(i) in X(j) for these {i,j}: {4057, 34948}, {4491, 39200}, {39199, 23224}
X(53305) = isogonal conjugate of the anticomplement of X(38981)
X(53305) = X(i)-Ceva conjugate of X(j) for these (i,j): {36040, 56}, {37136, 6}
X(53305) = crossdifference of every pair of points on line {37, 1146}
X(53305) = barycentric product X(i)*X(j) for these {i,j}: {905, 37305}, {37136, 38981}
X(53305) = barycentric quotient X(37305)/X(6335)
X(53305) = {X(1769),X(22379)}-harmonic conjugate of X(56)

---

X(53306) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(104) AND X(110)

Barycentrics    a^2*(b - c)*(a^5 - 2*a^3*b^2 + a*b^4 + a^3*b*c - a^2*b^2*c - a*b^3*c + b^4*c - 2*a^3*c^2 - a^2*b*c^2 + b^3*c^2 - a*b*c^3 + b^2*c^3 + a*c^4 + b*c^4) : :
X(53306) = 3 X[34948] - 2 X[39227], 4 X[39227] - 3 X[48383]

X(53306) lies on these lines: {3, 8674}, {56, 51643}, {104, 900}, {110, 351}, {513, 22765}, {521, 34948}, {523, 1325}, {654, 2423}, {667, 9001}, {924, 2605}, {2254, 22379}, {2991, 23349}, {3738, 39200}, {6003, 48382}, {13277, 20999}, {14380, 43700}, {15313, 23224}, {20188, 50344}, {24290, 36743}, {35057, 39210}

X(53306) = reflection of X(i) in X(j) for these {i,j}: {39200, 52726}, {48383, 34948}, {48390, 23224}, {48391, 39210}
X(53306) = crossdifference of every pair of points on line {115, 119}

---

X(53307) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(105) AND X(106)

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

X(53307) lies on these lines: {1, 4557}, {3, 9519}, {36, 14190}, {56, 16686}, {88, 23832}, {105, 659}, {106, 1960}, {111, 9097}, {244, 23404}, {614, 18613}, {1001, 3923}, {1054, 8683}, {1149, 2183}, {1201, 3122}, {1279, 2223}, {1486, 15287}, {1621, 4781}, {3248, 16492}, {3445, 9432}, {3756, 28353}, {3941, 16487}, {4068, 16484}, {4423, 31264}, {5272, 15621}, {5563, 24436}, {9109, 28479}, {11512, 15625}, {15507, 24405}, {15571, 50023}, {15624, 35227}, {17724, 28393}, {23343, 24841}, {23361, 28011}, {23846, 28082}, {28491, 28505}, {35326, 38346}

X(53307) = crossdifference of every pair of points on line {4370, 6184}
X(53307) = {X(244),X(23404)}-harmonic conjugate of X(23845)

---

X(53308) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(105) AND X(109)

Barycentrics    a^2*(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(53308) lies on these lines: {3, 2814}, {6, 22384}, {21, 21222}, {55, 2254}, {56, 8648}, {105, 659}, {109, 692}, {386, 23141}, {405, 3762}, {513, 2078}, {514, 21789}, {521, 28984}, {649, 834}, {652, 9000}, {665, 911}, {667, 8712}, {764, 42670}, {812, 13245}, {905, 48387}, {999, 52726}, {1001, 3716}, {1376, 25380}, {1769, 4491}, {1946, 3669}, {2488, 2605}, {2881, 32116}, {3295, 3887}, {3303, 4895}, {3776, 48388}, {4057, 6129}, {4401, 52596}, {4421, 45328}, {7202, 18181}, {16158, 46403}, {18199, 23189}, {23224, 43932}, {39199, 51648}, {40166, 47203}

X(53308) = reflection of X(21789) in X(22160)
X(53308) = X(36146)-Ceva conjugate of X(6)
X(53308) = X(100)-isoconjugate of X(43672)
X(53308) = X(8054)-Dao conjugate of X(43672)
X(53308) = crossdifference of every pair of points on line {10, 1146}
X(53308) = barycentric product X(i)*X(j) for these {i,j}: {514, 13329}, {1459, 26003}
X(53308) = barycentric quotient X(i)/X(j) for these {i,j}: {649, 43672}, {13329, 190}
X(53308) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1946, 3669, 44408}, {8648, 14413, 56}

---

X(53309) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(105) AND X(110)

Barycentrics    a^2*(b - c)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c - 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(53309) lies on these lines: {2, 16158}, {3, 2775}, {55, 9508}, {56, 51642}, {105, 659}, {110, 351}, {405, 2787}, {513, 8642}, {523, 7469}, {647, 2605}, {665, 2440}, {667, 22160}, {669, 2106}, {958, 4922}, {1001, 4010}, {1491, 23865}, {1621, 50343}, {2112, 5029}, {2254, 8645}, {3295, 4730}, {3716, 13245}, {4057, 8654}, {4367, 21789}, {4477, 47807}, {4491, 14315}, {5047, 30709}, {8641, 50336}, {8651, 23189}, {8664, 50344}, {11108, 14431}, {14296, 16992}, {15313, 24562}, {25926, 48182}, {28374, 48297}, {48227, 48388}

X(53309) = crossdifference of every pair of points on line {115, 120}
X(53309) = {X(14419),X(42670)}-harmonic conjugate of X(3)

---

X(53310) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(105) AND X(111)

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

X(53310) lies on these lines: {2, 2486}, {3, 9522}, {23, 21009}, {25, 47232}, {105, 659}, {111, 351}, {614, 16726}, {675, 9094}, {1001, 4760}, {2223, 3290}, {3125, 16680}, {3291, 39688}, {4068, 37675}, {4516, 47231}

X(53310) = crossdifference of every pair of points on line {2482, 6184}

---

X(53311) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(105) AND X(112)

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

X(53311) lies on these lines: {3, 9523}, {6, 2878}, {25, 47227}, {105, 659}, {112, 1576}, {523, 2074}, {667, 51648}, {1001, 24459}, {4762, 22160}, {9034, 37492}, {14399, 37538}

X(53311) = crossdifference of every pair of points on line {6184, 15526}

---

X(53312) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(105) AND X(675)

Barycentrics    a*(a^5*b - a^4*b^2 + a^3*b^3 - a^2*b^4 + a^5*c - 2*a^4*b*c + 2*a^2*b^3*c - a*b^4*c - a^4*c^2 - 2*a^2*b^2*c^2 + a*b^3*c^2 - 2*b^4*c^2 + a^3*c^3 + 2*a^2*b*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(53312) lies on these lines: {1, 33985}, {2, 4557}, {98, 9108}, {105, 659}, {111, 9110}, {1111, 23392}, {1311, 9105}, {1447, 20875}, {2368, 52619}, {3290, 16782}, {9098, 9099}, {16680, 24203}, {16684, 37670}, {18613, 26246}, {23845, 24447}, {24331, 24685}, {26240, 26241}, {32486, 48136}

---

X(53313) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(106) AND X(108)

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

X(53313) lies on these lines: {3, 9525}, {56, 1769}, {106, 1960}, {108, 676}, {474, 4768}, {513, 5193}, {999, 3738}, {1946, 6129}, {4057, 51648}, {5563, 23087}

X(53313) = crossdifference of every pair of points on line {4370, 35072}

---

X(53314) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(106) AND X(109)

Barycentrics    a^2*(b - c)*(a^2 - b^2 + b*c - c^2) : :
X(53314) = X[8] - 3 X[26078], 2 X[1960] + X[23345], X[663] - 3 X[1459], 2 X[663] - 3 X[2605], X[663] + 3 X[43924], 4 X[663] - 3 X[48306], 5 X[663] - 3 X[48340], 4 X[1459] - X[48306], 5 X[1459] - X[48340], X[1769] - 3 X[14413], X[2605] + 2 X[43924], 5 X[2605] - 2 X[48340], 4 X[43924] + X[48306], 5 X[43924] + X[48340], 5 X[48306] - 4 X[48340], 3 X[48283] - 2 X[48287], 4 X[48287] - 3 X[48292], 3 X[21173] + X[48282], 3 X[48281] - X[48282], 4 X[1125] - 3 X[48168], 5 X[3616] - 3 X[26144], 7 X[3622] - 3 X[27545], 3 X[3733] - 2 X[50512], 3 X[3737] - X[47970], 2 X[48065] - 3 X[48297], 3 X[11125] - X[21132], 2 X[14315] + X[14812], X[20293] - 3 X[48246], 2 X[20316] - 3 X[48230]

X(53314) lies on these lines: {1, 900}, {3, 2815}, {6, 665}, {8, 26078}, {34, 39534}, {36, 39478}, {42, 48244}, {43, 48229}, {56, 39200}, {58, 42741}, {77, 43042}, {86, 3766}, {106, 1960}, {109, 692}, {214, 3738}, {244, 659}, {513, 663}, {514, 21112}, {521, 7629}, {522, 48283}, {523, 21173}, {526, 6126}, {612, 48182}, {614, 26275}, {654, 17455}, {657, 39521}, {667, 9002}, {676, 1421}, {834, 4834}, {875, 9297}, {876, 1911}, {905, 9001}, {909, 2423}, {918, 24358}, {926, 44858}, {928, 36942}, {995, 14422}, {1100, 4435}, {1125, 48168}, {1193, 28284}, {1201, 28396}, {1386, 45695}, {1411, 30725}, {1718, 10015}, {1870, 44428}, {2254, 3722}, {2424, 23351}, {2483, 20981}, {2530, 9013}, {2642, 4272}, {2787, 14288}, {2827, 11715}, {3025, 38984}, {3216, 24920}, {3315, 13266}, {3616, 26144}, {3622, 27545}, {3667, 48302}, {3720, 4800}, {3733, 5009}, {3737, 4960}, {3768, 5029}, {3811, 39472}, {3837, 17719}, {3904, 6370}, {3920, 31131}, {3924, 24097}, {3941, 8638}, {4040, 28209}, {4057, 6363}, {4145, 21343}, {4261, 22092}, {4408, 17215}, {4449, 4777}, {4526, 16777}, {4724, 28220}, {4778, 48065}, {4802, 17418}, {4833, 6372}, {4926, 48303}, {4928, 24673}, {5268, 30792}, {6006, 48294}, {6550, 30117}, {6586, 20980}, {7191, 44433}, {7250, 39199}, {8632, 21143}, {8648, 22379}, {8702, 50338}, {11125, 21132}, {14315, 14812}, {14407, 23650}, {16816, 27344}, {17244, 28779}, {17420, 31947}, {17929, 36066}, {17954, 18002}, {19860, 25996}, {19861, 25923}, {20293, 48246}, {20316, 48230}, {21106, 23752}, {21606, 41847}, {22383, 43060}, {23655, 50335}, {24188, 27846}, {24754, 45675}, {26102, 48183}, {27074, 29569}, {27292, 29595}, {28082, 28114}, {28175, 50346}, {28183, 48293}, {28195, 46385}, {28217, 48307}, {33148, 46403}, {37696, 44929}, {37697, 44815}, {42079, 42082}

X(53314) = midpoint of X(i) and X(j) for these {i,j}: {1459, 43924}, {4491, 23345}, {6129, 51656}, {17418, 48342}, {21106, 23752}, {21173, 48281}
X(53314) = reflection of X(i) in X(j) for these {i,j}: {2605, 1459}, {4491, 1960}, {17420, 31947}, {21112, 21180}, {48292, 48283}, {48306, 2605}
X(53314) = isogonal conjugate of X(51562)
X(53314) = isogonal conjugate of the anticomplement of X(51402)
X(53314) = isogonal conjugate of the isotomic conjugate of X(4453)
X(53314) = X(i)-Ceva conjugate of X(j) for these (i,j): {901, 16944}, {1022, 649}, {2720, 56}, {3960, 654}, {4565, 52059}, {4585, 2245}
X(53314) = cevapoint of X(i) and X(j) for these (i,j): {1960, 3310}, {8648, 21758}
X(53314) = crossdifference of every pair of points on line {9, 80}
X(53314) = X(i)-isoconjugate of X(j) for these (i,j): {1, 51562}, {6, 36804}, {8, 2222}, {9, 655}, {37, 47318}, {41, 46405}, {55, 35174}, {59, 52356}, {80, 100}, {99, 34857}, {101, 18359}, {109, 52409}, {110, 15065}, {190, 2161}, {312, 32675}, {476, 3678}, {522, 52377}, {594, 37140}, {643, 52383}, {644, 2006}, {651, 36910}, {660, 36815}, {664, 52371}, {668, 6187}, {692, 20566}, {759, 3952}, {901, 51975}, {1018, 24624}, {1089, 36069}, {1168, 17780}, {1411, 3699}, {1783, 52351}, {1807, 1897}, {2341, 4552}, {3738, 46649}, {3939, 18815}, {3969, 32678}, {4033, 34079}, {4551, 6740}, {4555, 40172}, {4557, 14616}, {4867, 52934}, {5548, 14628}, {6335, 52431}, {14147, 17484}, {23703, 36590}, {28654, 32671}, {36797, 52391}
X(53314) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 51562}, {9, 36804}, {11, 52409}, {44, 24004}, {223, 35174}, {244, 15065}, {478, 655}, {1015, 18359}, {1086, 20566}, {2245, 2397}, {3160, 46405}, {6615, 52356}, {8054, 80}, {13999, 318}, {18334, 3969}, {34467, 1807}, {34586, 3952}, {35069, 4033}, {35128, 312}, {35204, 3699}, {38979, 51975}, {38982, 1089}, {38984, 8}, {38986, 34857}, {38991, 36910}, {39006, 52351}, {39025, 52371}, {40584, 190}, {40589, 47318}, {40612, 668}, {40617, 18815}, {51583, 27808}
X(53314) = barycentric product X(i)*X(j) for these {i,j}: {1, 3960}, {6, 4453}, {7, 654}, {36, 514}, {56, 3904}, {57, 3738}, {58, 4707}, {75, 21758}, {85, 8648}, {86, 21828}, {92, 22379}, {101, 4089}, {214, 1022}, {222, 44428}, {244, 4585}, {320, 649}, {513, 3218}, {526, 52393}, {593, 6370}, {650, 1443}, {655, 3025}, {663, 17078}, {667, 20924}, {693, 7113}, {757, 2610}, {758, 1019}, {860, 7254}, {876, 27950}, {900, 40215}, {905, 1870}, {1111, 1983}, {1459, 17923}, {1464, 4560}, {1509, 42666}, {1635, 52553}, {1919, 40075}, {2163, 23884}, {2185, 51663}, {2245, 7192}, {2323, 3676}, {2361, 24002}, {2401, 34586}, {2720, 46398}, {3052, 27836}, {3261, 52434}, {3669, 4511}, {3724, 7199}, {3733, 3936}, {3737, 18593}, {3762, 16944}, {3792, 4817}, {3942, 4242}, {4025, 52413}, {4077, 4282}, {4391, 52440}, {4880, 48074}, {4973, 47947}, {6548, 17455}, {7045, 46384}, {7252, 41804}, {7649, 22128}, {15419, 44113}, {17515, 51664}, {17924, 52407}, {23345, 51583}, {32679, 52375}, {32851, 43924}, {39166, 47680}, {52426, 52621}
X(53314) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 36804}, {6, 51562}, {7, 46405}, {36, 190}, {56, 655}, {57, 35174}, {58, 47318}, {214, 24004}, {320, 1978}, {513, 18359}, {514, 20566}, {526, 3969}, {604, 2222}, {649, 80}, {650, 52409}, {654, 8}, {661, 15065}, {663, 36910}, {667, 2161}, {758, 4033}, {798, 34857}, {849, 37140}, {1019, 14616}, {1397, 32675}, {1415, 52377}, {1443, 4554}, {1459, 52351}, {1464, 4552}, {1635, 51975}, {1870, 6335}, {1919, 6187}, {1983, 765}, {2170, 52356}, {2245, 3952}, {2323, 3699}, {2361, 644}, {2423, 40437}, {2605, 41226}, {2610, 1089}, {2624, 3678}, {3025, 3904}, {3063, 52371}, {3218, 668}, {3669, 18815}, {3724, 1018}, {3733, 24624}, {3738, 312}, {3792, 3807}, {3904, 3596}, {3936, 27808}, {3960, 75}, {4089, 3261}, {4282, 643}, {4453, 76}, {4511, 646}, {4585, 7035}, {4707, 313}, {6370, 28654}, {7113, 100}, {7180, 52383}, {7252, 6740}, {8632, 36815}, {8648, 9}, {11700, 42718}, {16944, 3257}, {17078, 4572}, {17455, 17780}, {20924, 6386}, {21758, 1}, {21828, 10}, {22128, 4561}, {22379, 63}, {22383, 1807}, {27950, 874}, {32675, 46649}, {34586, 2397}, {40215, 4555}, {42666, 594}, {43924, 2006}, {44428, 7017}, {46384, 24026}, {51663, 6358}, {52375, 32680}, {52393, 35139}, {52407, 1332}, {52413, 1897}, {52426, 3939}, {52434, 101}, {52440, 651}, {53046, 6735}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 665, 22108}, {20981, 21123, 2483}, {21758, 21828, 654}

---

X(53315) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(106) AND X(110)

Barycentrics    a^2*(b - c)*(3*a^2 + a*b - 2*b^2 + a*c - b*c - 2*c^2) : :
X(53315) = 4 X[1960] - X[4491], 8 X[1960] + X[23345], 2 X[4491] + X[23345], 2 X[2605] + X[3733], 8 X[2605] + X[50344], 4 X[3733] - X[50344], 2 X[1459] + X[4057], X[1482] + 2 X[44812], 2 X[4367] + X[4833], 2 X[48328] + X[50349]

X(53315) lies on these lines: {1, 4145}, {3, 2776}, {6, 5029}, {56, 51646}, {106, 1960}, {110, 351}, {512, 1326}, {523, 7478}, {659, 3315}, {900, 25569}, {1388, 30572}, {1459, 4057}, {1482, 44812}, {2441, 21786}, {4107, 15668}, {4367, 4833}, {6370, 30580}, {8645, 22379}, {8674, 14419}, {16884, 21832}, {24959, 30709}, {29198, 48297}, {29226, 48283}, {48328, 50349}

X(53315) = midpoint of X(1459) and X(8643)
X(53315) = reflection of X(i) in X(j) for these {i,j}: {4057, 8643}, {30709, 24959}
X(53315) = X(4591)-Ceva conjugate of X(6)
X(53315) = crossdifference of every pair of points on line {115, 121}
X(53315) = barycentric product X(i)*X(j) for these {i,j}: {58, 22037}, {101, 23816}
X(53315) = barycentric quotient X(i)/X(j) for these {i,j}: {22037, 313}, {23816, 3261}

---

X(53316) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(106) AND X(111)

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

X(53316) lies on these lines: {3, 9526}, {6, 2054}, {105, 28539}, {106, 1960}, {111, 351}, {187, 21009}, {739, 28326}, {902, 8610}, {2712, 50344}, {8700, 28338}, {9097, 28479}, {33589, 41454}

X(53316) = crossdifference of every pair of points on line {2482, 2786}
X(53316) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 5168, 35327}, {3122, 5168, 6}, {21009, 39688, 40096}

---

X(53317) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(107) AND X(108)

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

X(53317) lies on these lines: {3, 9528}, {25, 47212}, {92, 18613}, {107, 1624}, {108, 676}, {158, 23383}, {162, 23353}, {243, 20470}, {653, 23845}, {859, 1784}, {1148, 23844}, {1895, 23361}, {1897, 4557}, {4436, 36797}, {15622, 47372}, {23347, 52920}, {26704, 26705}, {52775, 52776}

X(53317) = X(520)-isoconjugate of X(53044)
X(53317) = X(i)-Dao conjugate of X(j) for these (i,j): {6708, 3265}, {18592, 35518}
X(53317) = crossdifference of every pair of points on line {35071, 35072}
X(53317) = barycentric product X(i)*X(j) for these {i,j}: {107, 18592}, {108, 6708}, {162, 53036}, {408, 15352}, {651, 42385}, {653, 2654}, {823, 2658}
X(53317) = barycentric quotient X(i)/X(j) for these {i,j}: {408, 52613}, {2654, 6332}, {2658, 24018}, {6708, 35518}, {18592, 3265}, {24019, 53044}, {42385, 4391}, {53036, 14208}

---

X(53318) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(107) AND X(111)

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

X(53318) lies on these lines: {3, 9529}, {25, 1637}, {107, 1624}, {111, 351}, {523, 37962}, {1995, 9979}, {2799, 5020}, {5466, 34519}, {5621, 42736}, {6587, 39201}, {6644, 44204}, {9909, 25644}, {11284, 14417}, {17810, 39469}, {17994, 46425}, {18534, 44202}, {32193, 45259}, {41254, 45689}

X(53318) = crossdifference of every pair of points on line {2482, 6509}

---

X(53319) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(107) AND X(476)

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

X(53319) lies on these lines: {24, 13450}, {25, 47207}, {107, 1624}, {382, 1853}, {476, 10412}, {523, 4240}, {2935, 38577}, {4226, 10278}, {4232, 47238}, {5467, 30512}, {11657, 35235}, {18121, 44891}

X(53319) = reflection of X(5502) in X(4240)
X(53319) = reflection of X(5502) in the Euler line
X(53319) = crossdifference of every pair of points on line {18334, 35071}
X(53319) = barycentric product X(648)*X(13851)
X(53319) = barycentric quotient X(13851)/X(525)

---

X(53320) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(107) AND X(477)

Barycentrics    (b - c)*(b + c)*(-a^12 + 4*a^10*b^2 - 6*a^8*b^4 + 4*a^6*b^6 - a^4*b^8 + 4*a^10*c^2 - a^8*b^2*c^2 - a^6*b^4*c^2 - 9*a^4*b^6*c^2 + 5*a^2*b^8*c^2 + 2*b^10*c^2 - 6*a^8*c^4 - a^6*b^2*c^4 + 20*a^4*b^4*c^4 - 5*a^2*b^6*c^4 - 8*b^8*c^4 + 4*a^6*c^6 - 9*a^4*b^2*c^6 - 5*a^2*b^4*c^6 + 12*b^6*c^6 - a^4*c^8 + 5*a^2*b^2*c^8 - 8*b^4*c^8 + 2*b^2*c^10) : :

X(53320) lies on these lines: {4, 523}, {107, 1624}, {382, 520}, {477, 16171}, {526, 10721}, {924, 5895}, {1510, 43599}, {2453, 44889}, {2777, 14220}, {3134, 34291}, {7728, 9033}, {8675, 36990}, {8884, 23286}, {9007, 51212}, {15328, 36179}, {18039, 46608}

X(53320) = reflection of X(i) in X(j) for these {i,j}: {14380, 4}, {46585, 53178}, {46608, 18039}
X(53320) = reflection of X(14380) in the Euler line
X(53320) = crossdifference of every pair of points on line {3284, 35071}

---

X(53321) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(108) AND X(109)

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

X(53321) lies on these lines: {1, 15622}, {3, 2817}, {4, 34969}, {6, 1945}, {12, 27685}, {34, 23383}, {42, 52373}, {55, 6611}, {56, 244}, {65, 1410}, {100, 658}, {108, 676}, {109, 692}, {162, 1624}, {197, 7011}, {221, 7138}, {223, 3185}, {227, 11214}, {651, 52931}, {999, 11715}, {1020, 4551}, {1035, 3556}, {1214, 52139}, {1398, 37579}, {1402, 1427}, {1403, 1407}, {1415, 2425}, {1425, 2594}, {1435, 2352}, {1465, 20470}, {1576, 32651}, {1758, 3286}, {1763, 45739}, {1876, 8758}, {2222, 26700}, {2390, 51660}, {3733, 24027}, {3827, 43058}, {4559, 21828}, {4565, 52928}, {4636, 17942}, {4637, 7045}, {5018, 5143}, {5723, 27628}, {6244, 6609}, {7053, 37541}, {7099, 9316}, {7114, 14529}, {7580, 24014}, {8270, 15621}, {8683, 23703}, {10074, 39758}, {10571, 23846}, {11500, 20764}, {15267, 21147}, {16678, 17080}, {18613, 34036}, {22276, 40152}, {33848, 43047}, {36098, 37137}

X(53321) = isogonal conjugate of X(7253)
X(53321) = isogonal conjugate of the anticomplement of X(656)
X(53321) = isogonal conjugate of the isotomic conjugate of X(4566)
X(53321) = isogonal conjugate of the polar conjugate of X(52607)
X(53321) = polar conjugate of the isotomic conjugate of X(52610)
X(53321) = X(i)-Ceva conjugate of X(j) for these (i,j): {934, 1020}, {1020, 4559}, {4566, 52610}, {7045, 1407}, {7128, 6}, {24027, 56}, {24033, 221}, {32651, 1415}
X(53321) = cevapoint of X(i) and X(j) for these (i,j): {42, 647}, {65, 51662}, {512, 1400}, {649, 40956}, {669, 21750}, {1042, 7250}, {1254, 4017}, {1402, 7180}, {1409, 39201}, {1946, 21748}
X(53321) = trilinear pole of line {213, 1042}
X(53321) = crossdifference of every pair of points on line {1146, 2968}
X(53321) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7253}, {2, 1021}, {8, 3737}, {9, 4560}, {11, 643}, {19, 15411}, {21, 522}, {29, 521}, {55, 18155}, {58, 4397}, {60, 4086}, {63, 17926}, {75, 21789}, {81, 3239}, {86, 3900}, {92, 23090}, {99, 2310}, {107, 24031}, {110, 24026}, {162, 2968}, {163, 23978}, {200, 7192}, {220, 7199}, {244, 7256}, {261, 4041}, {270, 52355}, {274, 657}, {283, 44426}, {284, 4391}, {285, 8058}, {310, 8641}, {312, 7252}, {314, 663}, {318, 23189}, {332, 18344}, {333, 650}, {341, 3733}, {346, 1019}, {513, 1043}, {514, 2287}, {523, 1098}, {525, 2326}, {644, 17197}, {645, 2170}, {648, 34591}, {652, 31623}, {661, 7058}, {662, 1146}, {693, 2328}, {728, 17096}, {799, 14936}, {811, 3270}, {823, 35072}, {873, 4524}, {905, 2322}, {1014, 4163}, {1015, 7258}, {1018, 26856}, {1086, 7259}, {1172, 6332}, {1252, 40213}, {1253, 52619}, {1333, 52622}, {1414, 4081}, {1434, 4130}, {1474, 15416}, {1509, 4171}, {1577, 7054}, {1792, 7649}, {1793, 44428}, {1812, 3064}, {1946, 44130}, {1984, 53211}, {2185, 3700}, {2193, 46110}, {2194, 35519}, {2299, 35518}, {2319, 27527}, {2327, 17924}, {2332, 15413}, {2341, 3904}, {2638, 6528}, {3022, 4625}, {3063, 28660}, {3119, 4573}, {3271, 7257}, {3615, 35057}, {3692, 17925}, {3699, 18191}, {3709, 52379}, {3738, 6740}, {4025, 4183}, {4073, 7255}, {4077, 6061}, {4148, 37128}, {4435, 36800}, {4477, 32010}, {4529, 40432}, {4567, 42462}, {4570, 42455}, {4575, 21666}, {4578, 17205}, {4581, 46877}, {4592, 42069}, {4610, 36197}, {4612, 21044}, {4616, 24010}, {4635, 35508}, {4637, 23970}, {4858, 5546}, {4990, 40438}, {5423, 7203}, {6558, 16726}, {7004, 36797}, {7079, 15419}, {7101, 7254}, {8611, 46103}, {14010, 36037}, {18206, 28132}, {23829, 28071}, {23983, 24019}, {24018, 36421}, {43925, 52406}, {52335, 52935}
X(53321) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 7253}, {6, 15411}, {10, 4397}, {37, 52622}, {115, 23978}, {125, 2968}, {136, 21666}, {206, 21789}, {223, 18155}, {226, 35518}, {244, 24026}, {478, 4560}, {661, 40213}, {1084, 1146}, {1214, 35519}, {3162, 17926}, {3259, 14010}, {5139, 42069}, {6609, 7192}, {10001, 28660}, {15267, 523}, {17113, 52619}, {17423, 3270}, {22391, 23090}, {32664, 1021}, {35071, 23983}, {36830, 7058}, {36908, 693}, {38985, 24031}, {38986, 2310}, {38996, 14936}, {39026, 1043}, {39053, 44130}, {40586, 3239}, {40590, 4391}, {40600, 3900}, {40608, 4081}, {40611, 522}, {40622, 34387}, {40627, 42462}, {47345, 46110}, {50330, 42455}, {51574, 15416}, {52877, 4528}
X(53321) = barycentric product X(i)*X(j) for these {i,j}: {1, 1020}, {3, 52607}, {4, 52610}, {6, 4566}, {7, 4559}, {10, 1461}, {12, 4565}, {37, 934}, {42, 658}, {56, 4552}, {57, 4551}, {58, 4605}, {59, 7178}, {65, 651}, {71, 36118}, {72, 32714}, {73, 653}, {100, 1427}, {101, 3668}, {108, 1214}, {109, 226}, {110, 6354}, {112, 6356}, {162, 37755}, {181, 4573}, {190, 1042}, {210, 4617}, {213, 4569}, {225, 1813}, {227, 37141}, {228, 13149}, {269, 1018}, {278, 23067}, {279, 4557}, {307, 32674}, {442, 32651}, {512, 1275}, {520, 23984}, {523, 1262}, {643, 7147}, {645, 7143}, {648, 1425}, {655, 1464}, {656, 7128}, {661, 7045}, {662, 1254}, {664, 1400}, {692, 1446}, {738, 4069}, {756, 4637}, {765, 7216}, {822, 24032}, {823, 7138}, {850, 23979}, {872, 4635}, {940, 52931}, {1014, 21859}, {1016, 7250}, {1106, 4033}, {1119, 4574}, {1211, 52928}, {1310, 8898}, {1332, 1426}, {1334, 4626}, {1398, 52609}, {1402, 4554}, {1407, 3952}, {1409, 18026}, {1410, 6335}, {1414, 2171}, {1415, 1441}, {1439, 1783}, {1500, 4616}, {1577, 24027}, {1880, 6516}, {1897, 52373}, {1918, 46406}, {2149, 4077}, {2222, 18593}, {2294, 36048}, {2321, 6614}, {2594, 38340}, {3120, 4619}, {3265, 23985}, {3700, 7339}, {4017, 4564}, {4032, 29055}, {4171, 24013}, {4524, 23586}, {4848, 38828}, {4998, 7180}, {5546, 6046}, {6613, 21796}, {7012, 51664}, {7023, 30730}, {7115, 17094}, {8269, 16583}, {8685, 16888}, {8687, 41003}, {8804, 36079}, {16577, 26700}, {17747, 24016}, {18097, 46153}, {18785, 41353}, {24018, 24033}, {27808, 52410}, {32675, 41804}, {36059, 40149}, {36067, 51368}, {36127, 40152}, {42669, 53211}
X(53321) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 15411}, {6, 7253}, {10, 52622}, {25, 17926}, {31, 1021}, {32, 21789}, {37, 4397}, {42, 3239}, {56, 4560}, {57, 18155}, {59, 645}, {65, 4391}, {72, 15416}, {73, 6332}, {101, 1043}, {108, 31623}, {109, 333}, {110, 7058}, {163, 1098}, {181, 3700}, {184, 23090}, {213, 3900}, {225, 46110}, {226, 35519}, {244, 40213}, {269, 7199}, {279, 52619}, {512, 1146}, {520, 23983}, {523, 23978}, {604, 3737}, {647, 2968}, {651, 314}, {653, 44130}, {658, 310}, {661, 24026}, {664, 28660}, {669, 14936}, {692, 2287}, {765, 7258}, {798, 2310}, {810, 34591}, {822, 24031}, {872, 4171}, {906, 1792}, {934, 274}, {1018, 341}, {1020, 75}, {1042, 514}, {1106, 1019}, {1110, 7259}, {1214, 35518}, {1252, 7256}, {1254, 1577}, {1262, 99}, {1275, 670}, {1334, 4163}, {1397, 7252}, {1398, 17925}, {1400, 522}, {1402, 650}, {1403, 27527}, {1407, 7192}, {1409, 521}, {1410, 905}, {1414, 52379}, {1415, 21}, {1425, 525}, {1426, 17924}, {1427, 693}, {1439, 15413}, {1446, 40495}, {1461, 86}, {1464, 3904}, {1576, 7054}, {1813, 332}, {1880, 44426}, {1918, 657}, {2149, 643}, {2171, 4086}, {2197, 52355}, {2205, 8641}, {2489, 42069}, {2501, 21666}, {3049, 3270}, {3122, 42462}, {3125, 42455}, {3310, 14010}, {3668, 3261}, {3709, 4081}, {3733, 26856}, {3747, 4148}, {4017, 4858}, {4069, 30693}, {4079, 52335}, {4524, 23970}, {4551, 312}, {4552, 3596}, {4554, 40072}, {4557, 346}, {4559, 8}, {4564, 7257}, {4565, 261}, {4566, 76}, {4569, 6385}, {4573, 18021}, {4574, 1265}, {4605, 313}, {4619, 4600}, {4637, 873}, {6354, 850}, {6356, 3267}, {6614, 1434}, {7023, 17096}, {7045, 799}, {7053, 15419}, {7109, 4524}, {7115, 36797}, {7128, 811}, {7138, 24018}, {7143, 7178}, {7147, 4077}, {7178, 34387}, {7180, 11}, {7216, 1111}, {7250, 1086}, {7339, 4573}, {7366, 7203}, {8750, 2322}, {8898, 2517}, {20964, 4529}, {20970, 4990}, {21044, 23104}, {21741, 35057}, {21750, 17115}, {21796, 42337}, {21859, 3701}, {23067, 345}, {23224, 16731}, {23971, 4616}, {23979, 110}, {23984, 6528}, {23985, 107}, {24013, 4635}, {24027, 662}, {24033, 823}, {32651, 40412}, {32656, 2327}, {32660, 283}, {32674, 29}, {32675, 6740}, {32676, 2326}, {32713, 36421}, {32714, 286}, {32739, 2328}, {36059, 1812}, {36118, 44129}, {37755, 14208}, {39201, 35072}, {40152, 52616}, {41353, 18157}, {42658, 40616}, {42752, 52316}, {43924, 17197}, {43932, 16727}, {50487, 36197}, {51641, 2170}, {51658, 17878}, {51662, 40624}, {51664, 17880}, {52373, 4025}, {52410, 3733}, {52411, 23189}, {52607, 264}, {52610, 69}, {52928, 14534}, {52931, 34258}, {52963, 4528}
X(53321) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {109, 23981, 23845}, {162, 1624, 21789}, {4551, 23067, 4557}

---

X(53322) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(108) AND X(110)

Barycentrics    a^2*(a - b)*(a - c)*(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(53322) lies on these lines: {3, 2778}, {108, 676}, {109, 36082}, {110, 351}, {523, 37966}, {692, 23067}, {1415, 2443}, {2283, 8638}, {3185, 23171}, {3556, 20764}, {8021, 53035}, {15904, 18210}, {16687, 18613}

X(53322) = crossdifference of every pair of points on line {115, 123}
X(53322) = barycentric product X(i)*X(j) for these {i,j}: {109, 45206}, {163, 18692}, {431, 4558}, {651, 1858}, {664, 1195}
X(53322) = barycentric quotient X(i)/X(j) for these {i,j}: {431, 14618}, {1195, 522}, {1858, 4391}, {18692, 20948}, {45206, 35519}

---

X(53323) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(108) AND X(112)

Barycentrics    a^2*(a - b)*(a - c)*(a^2 + b^2 - c^2)*(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(53323) lies on these lines: {4, 2486}, {19, 3941}, {25, 47232}, {92, 16687}, {107, 36077}, {108, 676}, {112, 1576}, {204, 3185}, {240, 3286}, {281, 20990}, {523, 37965}, {653, 2283}, {692, 2498}, {1096, 2352}, {1486, 3209}, {1783, 4557}, {1859, 40956}, {2178, 6059}, {2223, 14571}, {2331, 15624}, {4238, 4436}, {7113, 51726}, {16684, 17913}, {16732, 47212}

X(53323) = X(i)-isoconjugate of X(j) for these (i,j): {326, 14775}, {656, 40412}, {693, 1794}, {905, 40435}, {943, 4025}, {1175, 14208}, {1459, 40422}, {2259, 15413}, {2968, 36048}, {2982, 6332}, {4091, 40447}, {15439, 17880}, {24018, 40395}
X(53323) = X(i)-Dao conjugate of X(j) for these (i,j): {442, 35518}, {942, 3265}, {15259, 14775}, {15607, 2968}, {18591, 15413}, {40596, 40412}, {40937, 3267}, {52119, 339}
X(53323) = crossdifference of every pair of points on line {15526, 16595}
X(53323) = barycentric product X(i)*X(j) for these {i,j}: {100, 1841}, {101, 1838}, {107, 18591}, {108, 40937}, {110, 1865}, {112, 442}, {162, 2294}, {648, 40952}, {651, 1859}, {653, 14547}, {811, 40978}, {942, 1783}, {1018, 46883}, {1110, 23595}, {1897, 2260}, {3952, 46890}, {4551, 46884}, {5249, 8750}, {6335, 40956}, {6734, 32674}, {6742, 44095}, {8021, 52607}
X(53323) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 40412}, {442, 3267}, {942, 15413}, {1783, 40422}, {1838, 3261}, {1841, 693}, {1844, 18160}, {1859, 4391}, {1865, 850}, {2207, 14775}, {2260, 4025}, {2294, 14208}, {4303, 30805}, {8021, 15411}, {8750, 40435}, {14547, 6332}, {14597, 4131}, {18591, 3265}, {32713, 40395}, {32739, 1794}, {33525, 2968}, {40937, 35518}, {40952, 525}, {40956, 905}, {40978, 656}, {44095, 4467}, {46883, 7199}, {46884, 18155}, {46890, 7192}
X(53323) = {X(8750),X(32674)}-harmonic conjugate of X(692)

---

X(53324) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(109) AND X(110)

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

X(53324) lies on these lines: {1, 47483}, {3, 2779}, {6, 17963}, {31, 18613}, {47, 23383}, {100, 4585}, {101, 28162}, {109, 692}, {110, 351}, {162, 23353}, {191, 16164}, {197, 22117}, {222, 1626}, {255, 14529}, {523, 14544}, {643, 4436}, {859, 6149}, {901, 28184}, {1155, 23202}, {1293, 28192}, {1324, 23071}, {1331, 4557}, {2361, 20470}, {2392, 20918}, {2425, 4559}, {2720, 15439}, {2933, 7078}, {3145, 8614}, {3157, 23843}, {3955, 52139}, {4427, 50342}, {5132, 5197}, {5848, 25968}, {8652, 28166}, {8683, 35281}, {8697, 28206}, {8701, 28188}, {14963, 22162}, {15622, 47371}, {20999, 51235}, {22361, 37836}, {22654, 23072}, {23070, 23850}, {23131, 23380}, {28148, 28170}, {32739, 35326}

X(53324) = isogonal conjugate of the isotomic conjugate of X(17136)
X(53324) = X(52378)-Ceva conjugate of X(6)
X(53324) = X(i)-isoconjugate of X(j) for these (i,j): {522, 17097}, {523, 40430}, {40442, 44426}
X(53324) = X(i)-Dao conjugate of X(j) for these (i,j): {5745, 850}, {17056, 35519}, {37836, 523}
X(53324) = crossdifference of every pair of points on line {115, 124}
X(53324) = barycentric product X(i)*X(j) for these {i,j}: {6, 17136}, {58, 22003}, {101, 3664}, {109, 5745}, {110, 17056}, {163, 18698}, {407, 4558}, {651, 2646}, {653, 22361}, {662, 2650}, {664, 21748}, {1332, 40985}, {1414, 21811}, {1461, 6737}, {1813, 40950}, {4556, 21674}, {4565, 21677}, {4570, 23755}, {37836, 44765}
X(53324) = barycentric quotient X(i)/X(j) for these {i,j}: {163, 40430}, {407, 14618}, {1415, 17097}, {2646, 4391}, {2650, 1577}, {3664, 3261}, {5745, 35519}, {6737, 52622}, {17056, 850}, {17136, 76}, {18698, 20948}, {21674, 52623}, {21748, 522}, {21811, 4086}, {22003, 313}, {22361, 6332}, {23755, 21207}, {32660, 40442}, {40950, 46110}, {40985, 17924}
X(53324) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {109, 692, 23845}, {255, 14529, 23361}, {2361, 26884, 20470}

---

X(53325) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(109) AND X(112)

Barycentrics    a^2*(a - b)*(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(53325) lies on these lines: {3, 9532}, {6, 43694}, {109, 692}, {112, 1576}, {601, 15622}, {906, 4557}, {1415, 8750}, {4559, 32656}, {21784, 23861}, {32642, 32651}

X(53325) = isogonal conjugate of the isotomic conjugate of X(14544)
X(53325) = X(4391)-isoconjugate of X(40407)
X(53325) = X(i)-Dao conjugate of X(j) for these (i,j): {18641, 35519}, {40942, 3267}
X(53325) = crossdifference of every pair of points on line {1146, 8287}
X(53325) = barycentric product X(i)*X(j) for these {i,j}: {6, 14544}, {101, 4292}, {109, 40942}, {110, 1901}, {112, 18641}, {162, 18675}, {1415, 23661}, {8750, 18652}
X(53325) = barycentric quotient X(i)/X(j) for these {i,j}: {1901, 850}, {4292, 3261}, {14544, 76}, {18641, 3267}, {18675, 14208}, {40942, 35519}

---

X(53326) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(110) AND X(675)

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

X(53326) lies on these lines: {3, 4750}, {110, 351}, {647, 44410}, {649, 23093}, {669, 4367}, {1011, 2786}, {3798, 22388}, {4057, 47798}, {4120, 16058}, {4191, 45674}, {4467, 23864}, {4809, 39200}, {16373, 45661}, {21758, 23188}, {27486, 48387}, {44408, 47755}

X(53326) = crossdifference of every pair of points on line {115, 3136}

---

X(53327) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(111) AND X(476)

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

X(53327) lies on these lines: {2, 46609}, {3, 8371}, {22, 10278}, {23, 5466}, {24, 41357}, {25, 669}, {111, 351}, {373, 8723}, {476, 10412}, {512, 34417}, {523, 1995}, {1637, 14270}, {1649, 11284}, {2422, 3124}, {2433, 40355}, {3148, 42660}, {3830, 31176}, {5020, 11123}, {6090, 30219}, {7485, 10189}, {7517, 10279}, {7530, 16220}, {9168, 16042}, {9189, 11616}, {11634, 50941}, {13233, 18007}, {14420, 24978}, {23105, 44895}, {23301, 31133}, {40916, 44822}, {44212, 47173}

X(53327) = circumcircle-inverse of X(9179)}
X(53327) = X(44814)-Dao conjugate of X(45808)
X(53327) = crossdifference of every pair of points on line {2482, 18334}
X(53327) = barycentric product X(512)*X(48540)
X(53327) = barycentric quotient X(48540)/X(670)
X(53327) = {X(23),X(5466)}-harmonic conjugate of X(44823)

---

X(53328) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(111) AND X(477)

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

X(53328) lies on these lines: {23, 9214}, {25, 1990}, {111, 351}, {237, 47322}, {477, 16171}, {523, 7418}, {1995, 50149}, {2393, 51980}, {2871, 52199}, {9158, 14995}

---

X(53329) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(112) AND X(476)

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

X(53329) lies on these lines: {3, 18121}, {25, 53}, {112, 1576}, {186, 52472}, {237, 47322}, {476, 10412}, {523, 4230}, {2407, 7468}, {2420, 14560}, {2421, 35346}, {2453, 44895}, {4240, 46607}, {7418, 38393}, {11634, 35345}, {13310, 15141}, {19136, 52471}, {23327, 31670}, {38861, 52916}

X(53329) = X(14208)-isoconjugate of X(38534)
X(53329) = X(i)-Dao conjugate of X(j) for these (i,j): {2072, 3268}, {46085, 3265}
X(53329) = crossdifference of every pair of points on line {15526, 18334}
X(53329) = barycentric product X(i)*X(j) for these {i,j}: {112, 2072}, {32708, 46085}
X(53329) = barycentric quotient X(2072)/X(3267)

---

X(53330) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(112) AND X(477)

Barycentrics    a^2*(b - c)*(b + c)*(a^12 - 3*a^10*b^2 + 2*a^8*b^4 + 2*a^6*b^6 - 3*a^4*b^8 + a^2*b^10 - 3*a^10*c^2 + 13*a^8*b^2*c^2 - 12*a^6*b^4*c^2 - 2*a^4*b^6*c^2 + 3*a^2*b^8*c^2 + b^10*c^2 + 2*a^8*c^4 - 12*a^6*b^2*c^4 + 18*a^4*b^4*c^4 - 4*a^2*b^6*c^4 - 4*b^8*c^4 + 2*a^6*c^6 - 2*a^4*b^2*c^6 - 4*a^2*b^4*c^6 + 6*b^6*c^6 - 3*a^4*c^8 + 3*a^2*b^2*c^8 - 4*b^4*c^8 + a^2*c^10 + b^2*c^10) : :

X(53330) lies on these lines: {3, 45681}, {25, 9209}, {112, 1576}, {184, 512}, {378, 523}, {477, 16171}, {878, 35906}, {1593, 5489}, {1597, 42733}, {2394, 13596}, {3148, 42660}, {7514, 18556}, {16230, 52737}, {37991, 44823}, {44814, 47225}

X(53330) = X(37985)-Dao conjugate of X(35520)
X(53330) = crossdifference of every pair of points on line {3580, 15526}
X(53330) = barycentric product X(112)*X(37985)
X(53330) = barycentric quotient X(37985)/X(3267)

---

X(53331) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(99) AND X(290)

Barycentrics    (b^2 - c^2)*(a^6 - 2*a^4*b^2 + a^2*b^4 - 2*a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4) : :
X(53331) = 3 X[2] - 4 X[24284], 5 X[2] - 4 X[45336], 5 X[3569] - 6 X[45336], 5 X[24284] - 3 X[45336], 3 X[3268] - 2 X[6333], 2 X[5027] - 3 X[5652], 4 X[5027] - 3 X[9147], 3 X[1640] - 2 X[14316], 4 X[2492] - 5 X[3618], 2 X[5113] - 3 X[11183], 3 X[10519] - 4 X[44813], 6 X[14398] - 7 X[51171]

X(53331) lies on these lines: {2, 3569}, {23, 32120}, {69, 526}, {74, 2373}, {99, 110}, {141, 39232}, {146, 2780}, {193, 9035}, {287, 2395}, {290, 879}, {316, 512}, {323, 401}, {385, 52038}, {523, 39099}, {669, 3265}, {804, 25046}, {826, 41298}, {924, 3267}, {1499, 5971}, {1510, 23285}, {1640, 14316}, {1649, 39100}, {2492, 3618}, {2574, 22340}, {2575, 22339}, {3050, 4580}, {3448, 9517}, {3738, 18697}, {5113, 11183}, {6031, 32228}, {7664, 9138}, {9479, 19571}, {9979, 41254}, {10097, 31125}, {10519, 44813}, {14318, 32478}, {14398, 51171}, {14417, 39905}, {14977, 42007}, {21395, 45839}, {33798, 50542}, {38354, 53263}, {41253, 44427}

X(53331) = reflection of X(i) in X(j) for these {i,j}: {23, 32120}, {69, 35522}, {385, 52038}, {3569, 24284}, {4580, 3050}, {9147, 5652}, {31296, 3288}, {39232, 141}, {39905, 14417}
X(53331) = anticomplement of X(3569)
X(53331) = anticomplement of the isogonal conjugate of X(2966)
X(53331) = anticomplement of the isotomic conjugate of X(43187)
X(53331) = isotomic conjugate of the anticomplement of X(38987)
X(53331) = isotomic conjugate of the isogonal conjugate of X(53263)
X(53331) = anticomplementary isogonal conjugate of X(39359)
X(53331) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 39359}, {63, 14721}, {98, 21221}, {162, 40867}, {163, 39355}, {290, 21294}, {293, 39352}, {336, 13219}, {662, 147}, {685, 5905}, {1821, 3448}, {1910, 148}, {1976, 21220}, {2715, 192}, {2966, 8}, {3404, 39346}, {4599, 25046}, {17932, 4329}, {22456, 21270}, {32696, 21216}, {36036, 69}, {36084, 2}, {36104, 193}, {36132, 7774}, {41174, 21300}, {43187, 6327}, {43754, 6360}
X(53331) = X(i)-Ceva conjugate of X(j) for these (i,j): {10425, 69}, {43187, 2}
X(53331) = crossdifference of every pair of points on line {51, 1196}
X(53331) = barycentric product X(i)*X(j) for these {i,j}: {76, 53263}, {3267, 19128}, {38987, 43187}
X(53331) = barycentric quotient X(i)/X(j) for these {i,j}: {19128, 112}, {38354, 3289}, {38987, 3569}, {53263, 6}
X(53331) = {X(3569),X(24284)}-harmonic conjugate of X(2)

---

X(53332) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(99) AND X(668)

Barycentrics    (a - b)*(a - c)*(a*b + b^2 + a*c + c^2) : :
X(53332) = 5 X[3616] - 4 X[9507]

X(53332) lies on these lines: {1, 17141}, {2, 3125}, {7, 41873}, {8, 7261}, {39, 25248}, {69, 2836}, {75, 3877}, {76, 25253}, {86, 39769}, {99, 110}, {100, 1310}, {101, 33952}, {190, 644}, {194, 25270}, {257, 27040}, {274, 17164}, {304, 3869}, {329, 20929}, {344, 18714}, {392, 26234}, {514, 4115}, {517, 3263}, {645, 14543}, {668, 891}, {712, 3230}, {726, 25302}, {758, 14210}, {789, 815}, {874, 3903}, {927, 29163}, {960, 20911}, {986, 27162}, {1018, 4568}, {1228, 22076}, {1264, 4329}, {1265, 18719}, {1334, 17760}, {1930, 3878}, {2170, 17755}, {2176, 17489}, {2238, 17497}, {2292, 16705}, {2295, 25263}, {2802, 4986}, {3061, 27109}, {3212, 18135}, {3436, 21595}, {3570, 30729}, {3596, 21271}, {3616, 9507}, {3721, 27097}, {3727, 26965}, {3807, 4595}, {3868, 18156}, {3890, 39731}, {3905, 3915}, {3954, 26759}, {4103, 23891}, {4436, 16680}, {4465, 21138}, {4554, 4566}, {4555, 4608}, {4581, 4601}, {4597, 32042}, {5195, 16086}, {5692, 33936}, {5697, 33937}, {5903, 33942}, {6377, 28285}, {6633, 47681}, {7985, 18906}, {8682, 21839}, {8707, 29279}, {9067, 28474}, {9263, 33888}, {9317, 17738}, {9560, 27698}, {9802, 42696}, {11684, 17206}, {16583, 26689}, {16614, 28400}, {17139, 35550}, {17169, 41875}, {17751, 33939}, {18055, 28742}, {18059, 30547}, {18157, 20718}, {18600, 41805}, {20244, 33933}, {20347, 20924}, {20535, 27523}, {21067, 29699}, {21208, 34587}, {21609, 23839}, {22011, 29383}, {24291, 32930}, {25282, 32925}, {25305, 35538}, {25307, 33296}, {26242, 35274}, {26274, 35286}, {27133, 27805}, {27538, 51863}, {30939, 39765}, {31023, 34542}, {33295, 39766}, {46369, 53268}

X(53332) = reflection of X(i) in X(j) for these {i,j}: {9263, 46460}, {17497, 2238}, {30941, 14210}
X(53332) = isotomic conjugate of X(4581)
X(53332) = anticomplement of X(3125)
X(53332) = anticomplement of the isogonal conjugate of X(4567)
X(53332) = anticomplement of the isotomic conjugate of X(4601)
X(53332) = isotomic conjugate of the anticomplement of X(50330)
X(53332) = isotomic conjugate of the isogonal conjugate of X(53280)
X(53332) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {59, 17778}, {99, 150}, {100, 21221}, {101, 148}, {110, 4440}, {163, 9263}, {190, 3448}, {249, 1}, {250, 3187}, {284, 17036}, {643, 37781}, {645, 33650}, {662, 149}, {668, 21294}, {692, 21220}, {765, 2895}, {799, 21293}, {1016, 1330}, {1101, 17147}, {1110, 1655}, {1252, 1654}, {1331, 39352}, {1576, 21224}, {4556, 17154}, {4561, 13219}, {4564, 2475}, {4567, 8}, {4570, 2}, {4577, 25049}, {4590, 17135}, {4591, 20042}, {4599, 25048}, {4600, 69}, {4601, 6327}, {4620, 3434}, {4628, 25047}, {4629, 44006}, {4998, 2893}, {5379, 5905}, {5546, 39351}, {6064, 20245}, {7035, 21287}, {7340, 20244}, {9273, 39766}, {17943, 39368}, {18020, 17220}, {23357, 17148}, {24037, 17137}, {24041, 75}, {31614, 17159}, {32739, 25054}, {34537, 17138}, {44717, 3152}, {46148, 39346}, {46254, 20242}, {47389, 18659}, {52378, 145}
X(53332) = X(4601)-Ceva conjugate of X(2)
X(53332) = X(i)-isoconjugate of X(j) for these (i,j): {31, 4581}, {244, 32736}, {512, 2363}, {649, 2298}, {661, 1169}, {663, 961}, {667, 1220}, {798, 14534}, {1015, 36147}, {1240, 1980}, {1919, 30710}, {1924, 40827}, {1973, 15420}, {2170, 8687}, {2310, 52928}, {2359, 6591}, {3248, 8707}, {3271, 36098}
X(53332) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 4581}, {960, 512}, {1211, 513}, {2092, 650}, {3666, 523}, {4357, 4369}, {5375, 2298}, {5750, 48276}, {6337, 15420}, {6631, 1220}, {9296, 30710}, {9428, 40827}, {17197, 18191}, {17419, 2170}, {31998, 14534}, {36830, 1169}, {38992, 3271}, {39015, 1015}, {39054, 2363}, {52087, 649}
X(53332) = cevapoint of X(i) and X(j) for these (i,j): {513, 6703}, {514, 49598}, {523, 44417}, {525, 37613}, {960, 3910}, {2292, 48131}, {3004, 4357}, {3666, 6371}
X(53332) = trilinear pole of line {1211, 2092}
X(53332) = crossdifference of every pair of points on line {1977, 3124}
X(53332) = barycentric product X(i)*X(j) for these {i,j}: {75, 3882}, {76, 53280}, {99, 1211}, {100, 20911}, {110, 1228}, {190, 4357}, {429, 4563}, {643, 45196}, {645, 41003}, {646, 24471}, {662, 18697}, {664, 3687}, {668, 3666}, {670, 2092}, {765, 4509}, {799, 2292}, {960, 4554}, {1016, 3004}, {1018, 16739}, {1193, 1978}, {1848, 4561}, {2269, 4572}, {2300, 6386}, {3674, 3699}, {3704, 4573}, {3725, 4602}, {3910, 4998}, {3952, 16705}, {3965, 4569}, {4576, 27067}, {4594, 27697}, {4600, 21124}, {4601, 50330}, {4610, 20653}, {4623, 21810}, {4625, 21033}, {4631, 52567}, {6331, 22076}, {6371, 31625}, {7035, 48131}, {27455, 36863}, {27808, 40153}, {34537, 42661}, {36860, 45197}, {44092, 52608}
X(53332) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 4581}, {59, 8687}, {69, 15420}, {99, 14534}, {100, 2298}, {110, 1169}, {190, 1220}, {429, 2501}, {651, 961}, {662, 2363}, {668, 30710}, {670, 40827}, {765, 36147}, {960, 650}, {1016, 8707}, {1193, 649}, {1211, 523}, {1228, 850}, {1252, 32736}, {1262, 52928}, {1331, 2359}, {1332, 1791}, {1682, 52326}, {1829, 6591}, {1848, 7649}, {1978, 1240}, {2092, 512}, {2269, 663}, {2292, 661}, {2300, 667}, {3004, 1086}, {3666, 513}, {3674, 3676}, {3687, 522}, {3704, 3700}, {3725, 798}, {3882, 1}, {3910, 11}, {3952, 14624}, {3965, 3900}, {4267, 7252}, {4357, 514}, {4503, 4378}, {4509, 1111}, {4554, 31643}, {4558, 1798}, {4564, 36098}, {4631, 52550}, {4719, 4790}, {4918, 14321}, {4998, 6648}, {6371, 1015}, {16705, 7192}, {16739, 7199}, {17185, 3737}, {17420, 2170}, {18235, 3287}, {18697, 1577}, {20653, 4024}, {20911, 693}, {20967, 3063}, {21033, 4041}, {21124, 3120}, {21810, 4705}, {22074, 1946}, {22076, 647}, {22097, 1459}, {22345, 22383}, {24471, 3669}, {27455, 43931}, {27697, 2533}, {28369, 4367}, {39774, 6002}, {40153, 3733}, {40966, 3709}, {41003, 7178}, {41600, 6588}, {42661, 3124}, {44092, 2489}, {45196, 4077}, {46640, 40454}, {46877, 1021}, {46878, 3064}, {46889, 21789}, {48131, 244}, {50330, 3125}, {51407, 2804}, {51571, 29142}, {52326, 3271}, {53280, 6}
X(53332) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {304, 3869, 17137}, {668, 33948, 3952}, {1018, 4568, 42720}, {1930, 3878, 17152}, {3212, 19582, 18135}, {3807, 4595, 30730}, {3952, 21272, 668}, {4427, 17136, 99}, {5195, 16086, 20553}

---

X(53333) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(99) AND X(903)

Barycentrics    (b - c)*(-3*a^2 - a*b + 2*b^2 - a*c + b*c + 2*c^2) : :
X(53333) = 5 X[2] - 4 X[45661], 3 X[2] - 4 X[45674], X[4120] - 3 X[4750], 5 X[4120] - 6 X[45661], 5 X[4750] - 2 X[45661], 3 X[4750] - 2 X[45674], 3 X[45661] - 5 X[45674], 3 X[47755] - X[51317], 3 X[47780] - 2 X[51317], 8 X[4467] + X[4608], X[4467] + 2 X[4897], 2 X[4467] + X[7192], 4 X[4467] - X[17161], X[4608] - 16 X[4897], X[4608] - 4 X[7192], X[4608] + 2 X[17161], 4 X[4897] - X[7192], 8 X[4897] + X[17161], 2 X[7192] + X[17161], 2 X[50342] + X[50343], 4 X[649] - X[49273], 2 X[21115] - 3 X[48571], and many others

X(53333) lies on these lines: {2, 2786}, {99, 110}, {103, 675}, {513, 47894}, {514, 4984}, {522, 47755}, {523, 4467}, {649, 30519}, {693, 4926}, {812, 21115}, {824, 47763}, {900, 903}, {918, 4773}, {1635, 31349}, {1797, 2403}, {2787, 30595}, {2826, 13243}, {3004, 39386}, {3667, 4025}, {3798, 25259}, {3906, 50556}, {4184, 53269}, {4210, 53258}, {4369, 4931}, {4380, 49302}, {4394, 49272}, {4707, 20092}, {4778, 45746}, {4785, 48156}, {4786, 47771}, {4790, 47677}, {4850, 21894}, {4893, 28906}, {4928, 4958}, {4932, 47659}, {4962, 21183}, {4979, 47653}, {6006, 48543}, {6008, 48422}, {6546, 45679}, {13174, 44010}, {14425, 30565}, {14435, 44009}, {16892, 26853}, {17069, 44449}, {17154, 23829}, {17487, 46457}, {17494, 28851}, {17496, 28468}, {21116, 26824}, {21130, 29148}, {21196, 31290}, {21212, 26798}, {23879, 48580}, {23883, 48568}, {26777, 48082}, {26985, 48266}, {27115, 48270}, {27138, 44432}, {27486, 28846}, {28195, 48107}, {28199, 47657}, {28205, 43067}, {28217, 48550}, {28221, 47891}, {28319, 47729}, {28840, 46915}, {28867, 47759}, {28886, 47878}, {28898, 47762}, {29078, 47824}, {29090, 47836}, {29216, 47796}, {29328, 48241}, {29370, 48254}, {30579, 49274}, {31148, 47792}, {44429, 50452}, {44551, 47786}, {45669, 47774}, {47758, 47790}, {47769, 47785}, {47791, 48574}, {47923, 48016}, {48080, 48211}, {48104, 48427}, {48187, 50336}, {48247, 49275}, {48423, 48563}, {48435, 49281}
reflection of X(i) in X(j) for these {i,j}: {2, 4750}, {2403, 21222}, {4120, 45674}, {4931, 4369}, {4958, 4928}, {6546, 45679}, {17487, 46457}, {20295, 44435}, {21297, 4453}, {25259, 47766}, {26824, 21116}, {44009, 14435}, {44435, 4025}, {47759, 47886}, {47766, 3798}, {47769, 47785}, {47771, 4786}, {47772, 1635}, {47773, 649}, {47774, 47782}, {47775, 27486}, {47780, 47755}, {47782, 45669}, {47786, 44551}, {47790, 47758}, {47791, 48574}, {47792, 31148}, {47870, 47762}, {47892, 4773}, {48080, 48211}, {48187, 50336}, {48269, 44432}, {48423, 48563}, {49273, 47773}, {49275, 48247}

X(53333) = anticomplement of X(4120)
X(53333) = anticomplement of the isogonal conjugate of X(4591)
X(53333) = anticomplement of the isotomic conjugate of X(4615)
X(53333) = isotomic conjugate of the isogonal conjugate of X(53315)
X(53333) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {88, 3448}, {106, 21221}, {110, 30578}, {163, 17487}, {662, 21290}, {901, 2895}, {903, 21294}, {1333, 39349}, {3257, 1330}, {4555, 21287}, {4591, 8}, {4615, 6327}, {4622, 69}, {4634, 315}, {9456, 148}, {32665, 1654}, {32719, 1655}, {36058, 39352}
X(53333) = X(4615)-Ceva conjugate of X(2)
X(53333) = X(23816)-Dao conjugate of X(4395)
X(53333) = crossdifference of every pair of points on line {1017, 3124}
X(53333) = barycentric product X(i)*X(j) for these {i,j}: {76, 53315}, {86, 22037}, {190, 23816}
X(53333) = barycentric quotient X(i)/X(j) for these {i,j}: {22037, 10}, {23816, 514}, {53315, 6}
X(53333) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1635, 47772, 31992}, {3798, 25259, 27013}, {4120, 4750, 45674}, {4120, 45674, 2}, {4453, 21297, 6548}, {4467, 4897, 7192}, {4467, 7192, 17161}, {4773, 47892, 47776}, {7192, 17161, 4608}, {53269, 53326, 4184}

---

X(53334) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(99) AND X(1121)

Barycentrics    (b - c)*(3*a^3 - 4*a^2*b - a*b^2 + 2*b^3 - 4*a^2*c + a*b*c + b^2*c - a*c^2 + b*c^2 + 2*c^3) : :
X(53334) = 2 X[1] + X[49274], 3 X[14432] - X[30574], X[145] + 2 X[4088], 5 X[3616] - 2 X[4707], 7 X[3622] - 4 X[4458], 4 X[6332] - X[21302], X[7192] - 4 X[48290], X[17161] - 4 X[48288], X[20295] + 2 X[47728], 2 X[21181] - 3 X[25055], X[21222] + 2 X[49276], 14 X[21952] - 17 X[46932], X[31145] - 4 X[45344], X[47729] + 2 X[49280], 2 X[48298] + X[49273], X[48298] + 2 X[49279], X[49273] - 4 X[49279], 4 X[48332] - X[49302]

X(53334) lies on these lines: {1, 49274}, {2, 2785}, {99, 110}, {102, 1311}, {145, 4088}, {514, 48172}, {523, 4833}, {663, 48239}, {1121, 6366}, {2787, 30605}, {2789, 4120}, {2826, 3904}, {3616, 4707}, {3622, 4458}, {3667, 31291}, {4391, 28537}, {6332, 21302}, {7192, 48290}, {7650, 28195}, {11684, 53301}, {14413, 48571}, {14431, 18061}, {17161, 48288}, {20295, 47728}, {21181, 25055}, {21222, 49276}, {21297, 29240}, {21952, 46932}, {28468, 47805}, {28573, 48565}, {29094, 47840}, {29172, 48158}, {29304, 47796}, {31145, 45344}, {47729, 49280}, {48136, 48174}, {48250, 48299}, {48298, 49273}, {48332, 49302}

X(53334) = reflection of X(i) in X(j) for these {i,j}: {2, 14432}, {21302, 47808}, {47808, 6332}, {48174, 48136}, {48239, 663}, {48250, 48299}, {48571, 14413}
X(53334) = anticomplement of X(30574)
X(53334) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {163, 39357}, {1121, 21294}, {1156, 3448}, {2291, 21221}, {14733, 2475}, {34068, 148}, {36141, 17778}, {37139, 2893}
X(53334) = {X(48298),X(49279)}-harmonic conjugate of X(49273)

---

X(53335) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(99) AND X(2481)

Barycentrics    (b - c)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c - 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(53335) = 3 X[2] - 4 X[24285], 4 X[8659] - 3 X[47776]

X(53335) lies on these lines: {1, 23829}, {2, 24285}, {7, 51643}, {69, 8674}, {99, 110}, {104, 43363}, {448, 525}, {512, 7192}, {663, 4025}, {693, 3309}, {826, 17161}, {885, 2481}, {900, 53276}, {918, 1814}, {1444, 53306}, {2605, 15419}, {2786, 4107}, {2821, 38478}, {3004, 48336}, {3315, 4453}, {3566, 4367}, {4374, 7253}, {4468, 45755}, {4581, 17159}, {4608, 7927}, {4750, 5029}, {6004, 20295}, {8638, 53326}, {8659, 47776}, {15313, 15413}, {21196, 50541}, {21301, 50452}, {23785, 48307}, {24287, 40459}, {26248, 47836}, {28478, 48013}, {38348, 50342}, {43067, 48301}, {47995, 48367}

X(53335) = reflection of X(i) in X(j) for these {i,j}: {7192, 50556}, {24290, 24285}, {50541, 21196}
X(53335) = anticomplement of X(24290)
X(53335) = isotomic conjugate of the isogonal conjugate of X(53309)
X(53335) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {58, 39353}, {99, 20552}, {105, 21221}, {110, 20533}, {163, 39350}, {284, 14732}, {662, 20344}, {666, 1330}, {673, 3448}, {919, 1654}, {927, 2893}, {1438, 148}, {2481, 21294}, {4565, 52164}, {4599, 25050}, {31637, 13219}, {32666, 1655}, {32735, 17778}, {36057, 39352}, {36086, 2895}, {36146, 2475}, {51560, 21287}
X(53335) = crossdifference of every pair of points on line {3124, 20455}
X(53335) = barycentric product X(76)*X(53309)
X(53335) = barycentric quotient X(53309)/X(6)
X(53335) = {X(24285),X(24290)}-harmonic conjugate of X(2)

---

X(53336) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(190) AND X(290)

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

X(53336) lies on these lines: {2, 24353}, {100, 190}, {290, 879}, {513, 21611}, {834, 20293}, {1043, 53249}, {2774, 10449}, {3287, 3700}, {3716, 4518}, {4064, 20294}, {4088, 32937}, {4195, 42662}, {4397, 48080}, {4458, 24349}, {8676, 52622}, {11688, 53262}, {14432, 19582}, {14829, 53301}, {20979, 48269}, {35518, 50452}

X(53336) = isotomic conjugate of the isogonal conjugate of X(53258)
X(53336) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {37, 39359}, {72, 14721}, {98, 149}, {100, 147}, {290, 21293}, {685, 3868}, {692, 39355}, {1783, 40867}, {1821, 150}, {1910, 4440}, {1976, 9263}, {2715, 17147}, {2966, 75}, {15628, 37781}, {17932, 20243}, {22456, 20242}, {36036, 17135}, {36065, 29840}, {36084, 1}, {36104, 3187}, {43187, 17137}, {43754, 20222}
X(53336) = crossdifference of every pair of points on line {1015, 9419}
X(53336) = barycentric product X(76)*X(53258)
X(53336) = barycentric quotient X(53258)/X(6)

---

X(53337) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(190) AND X(666)

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

X(53337) lies on the cubic K090 and these lines: {2, 7}, {6, 24403}, {41, 25237}, {44, 24407}, {72, 37009}, {100, 190}, {101, 17136}, {169, 20244}, {218, 20247}, {219, 20248}, {239, 36816}, {514, 1023}, {644, 3732}, {645, 4576}, {651, 2428}, {658, 27834}, {666, 885}, {693, 42719}, {883, 927}, {1121, 12531}, {1332, 30626}, {1743, 24398}, {1757, 24428}, {2348, 52210}, {2397, 43991}, {2403, 5376}, {2415, 6632}, {2975, 32024}, {3257, 6548}, {3758, 16482}, {3876, 16376}, {4552, 46725}, {4561, 25272}, {4568, 30729}, {4712, 51435}, {6554, 30616}, {7035, 41315}, {13226, 19515}, {14439, 24685}, {16468, 24427}, {16494, 46922}, {16552, 24455}, {16670, 24408}, {17165, 26241}, {21362, 35341}, {21384, 24460}, {23891, 30731}, {24582, 51406}, {24694, 31031}, {24712, 31058}, {25261, 41239}, {26563, 30618}, {27132, 30617}, {27948, 30807}, {30806, 41391}, {34578, 53240}

X(53337) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {666, 21293}, {692, 39353}, {765, 20552}, {919, 149}, {1110, 20533}, {1252, 20344}, {2149, 52164}, {5377, 69}, {23990, 39350}, {32666, 4440}, {36086, 150}, {39293, 21285}, {52927, 37781}
X(53337) = X(i)-Ceva conjugate of X(j) for these (i,j): {883, 2398}, {927, 100}
X(53337) = X(i)-isoconjugate of X(j) for these (i,j): {6, 35355}, {55, 37626}, {244, 6078}, {649, 1280}, {650, 1477}, {663, 43760}, {667, 36807}, {1810, 6591}, {3063, 35160}
X(53337) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 35355}, {223, 37626}, {3008, 918}, {3693, 50333}, {5375, 1280}, {6631, 36807}, {10001, 35160}, {16593, 514}, {35111, 522}, {39048, 513}
X(53337) = cevapoint of X(i) and X(j) for these (i,j): {1279, 8659}, {3008, 6084}
X(53337) = trilinear pole of line {1279, 3008}
X(53337) = crossdifference of every pair of points on line {663, 1015}
X(53337) = barycentric product X(i)*X(j) for these {i,j}: {85, 23704}, {190, 3008}, {664, 5853}, {666, 16593}, {668, 1279}, {927, 40609}, {1016, 6084}, {2348, 4554}, {4572, 8647}, {7035, 48032}, {8659, 31625}, {13136, 51419}, {20662, 36803}, {42720, 52210}
X(53337) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 35355}, {57, 37626}, {100, 1280}, {109, 1477}, {190, 36807}, {651, 43760}, {664, 35160}, {1252, 6078}, {1279, 513}, {1331, 1810}, {2348, 650}, {2976, 3756}, {3008, 514}, {5377, 39272}, {5853, 522}, {6084, 1086}, {8647, 663}, {8659, 1015}, {16593, 918}, {20662, 665}, {20680, 24290}, {20780, 1459}, {23704, 9}, {40609, 50333}, {48032, 244}, {51419, 10015}
X(53337) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9, 9318, 2}, {190, 3570, 42720}, {644, 3732, 21272}, {3570, 42720, 17780}, {14439, 24685, 31020}, {23891, 32094, 30731}

---

X(53338) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(190) AND X(670)

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

X(53338) lies on these lines: {1, 16722}, {2, 3122}, {6, 25277}, {8, 4783}, {11, 30019}, {69, 20351}, {75, 17142}, {81, 25294}, {86, 25295}, {99, 43359}, {100, 190}, {192, 869}, {256, 26772}, {645, 3573}, {646, 3799}, {660, 4581}, {668, 3888}, {670, 888}, {674, 3264}, {714, 2234}, {846, 27538}, {874, 3903}, {889, 50520}, {982, 27107}, {1740, 17148}, {2228, 27044}, {2886, 29979}, {3123, 17793}, {3596, 21278}, {3688, 28597}, {3728, 16738}, {3778, 27102}, {3948, 6007}, {4022, 27017}, {4033, 4553}, {4363, 17165}, {4418, 24451}, {4440, 17794}, {4568, 20525}, {5695, 25253}, {7155, 17349}, {9025, 25298}, {16571, 17157}, {17138, 20245}, {17178, 24437}, {17786, 25279}, {20340, 23633}, {20456, 26975}, {20512, 33946}, {21035, 26764}, {21865, 29705}, {22279, 29388}, {24004, 40521}, {24342, 24349}, {24463, 26976}, {24575, 26963}, {24688, 31026}, {24696, 31036}, {24717, 31060}, {27095, 41886}, {28604, 28642}, {30939, 44671}

X(53338) = reflection of X(i) in X(j) for these {i,j}: {8, 4783}, {192, 20681}, {20352, 3264}, {23633, 20340}
X(53338) = anticomplement of X(3122)
X(53338) = anticomplement of the isogonal conjugate of X(4600)
X(53338) = isotomic conjugate of the anticomplement of X(40627)
X(53338) = isotomic conjugate of the isogonal conjugate of X(53268)
X(53338) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {21, 17036}, {99, 149}, {100, 148}, {101, 21220}, {110, 9263}, {163, 21224}, {190, 21221}, {249, 17147}, {643, 39351}, {645, 37781}, {662, 4440}, {668, 3448}, {670, 21293}, {692, 25054}, {765, 1654}, {799, 150}, {1016, 2895}, {1101, 17148}, {1252, 1655}, {1332, 39352}, {1978, 21294}, {4553, 39346}, {4564, 17778}, {4567, 2}, {4570, 192}, {4577, 25048}, {4590, 75}, {4593, 25049}, {4596, 44006}, {4600, 8}, {4601, 69}, {4620, 7}, {4622, 20042}, {4998, 2475}, {5379, 193}, {5380, 45291}, {5384, 40721}, {5385, 37635}, {6064, 3869}, {7035, 1330}, {7257, 33650}, {7340, 3873}, {18020, 3868}, {24037, 17135}, {24041, 1}, {31614, 17166}, {31625, 21287}, {34537, 17137}, {42717, 39359}, {44717, 18667}, {46254, 17220}, {47389, 20243}, {52378, 3210}, {52935, 17154}
X(53338) = X(i)-isoconjugate of X(j) for these (i,j): {649, 1258}, {662, 40525}, {667, 40418}, {798, 40409}, {1221, 1919}
X(53338) = X(i)-Dao conjugate of X(j) for these (i,j): {1084, 40525}, {1107, 4369}, {3741, 512}, {5375, 1258}, {6631, 40418}, {9296, 1221}, {16742, 23824}, {21024, 3835}, {21838, 514}, {31998, 40409}, {51575, 513}
X(53338) = cevapoint of X(i) and X(j) for these (i,j): {512, 6685}, {513, 25124}, {1107, 50510}
X(53338) = trilinear pole of line {1107, 3741}
X(53338) = crossdifference of every pair of points on line {1015, 9427}
X(53338) = barycentric product X(i)*X(j) for these {i,j}: {76, 53268}, {99, 21024}, {100, 20891}, {190, 3741}, {668, 1107}, {670, 21838}, {799, 3728}, {1197, 6386}, {1978, 2309}, {3699, 30097}, {3952, 16738}, {4033, 18169}, {4568, 18091}, {4610, 21713}, {4623, 22206}, {7257, 45208}, {7260, 27880}, {13136, 51411}, {21700, 52612}, {27805, 51575}, {31625, 50510}
X(53338) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 40409}, {100, 1258}, {190, 40418}, {512, 40525}, {668, 1221}, {1107, 513}, {1197, 667}, {2309, 649}, {3728, 661}, {3741, 514}, {16738, 7192}, {18091, 10566}, {18169, 1019}, {20891, 693}, {21024, 523}, {21700, 4079}, {21713, 4024}, {21838, 512}, {22065, 1459}, {22206, 4705}, {22389, 22383}, {23212, 3049}, {30097, 3676}, {39780, 7180}, {40627, 3122}, {45208, 4017}, {45216, 20979}, {50510, 1015}, {51411, 10015}, {51575, 4369}, {53268, 6}
X(53338) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {190, 3699, 52923}, {190, 4436, 4427}, {3728, 51575, 16738}, {4033, 4553, 23354}

---

X(53339) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(190) AND X(671)

Barycentrics    (b - c)*(-a^2 - 3*a*b + 2*b^2 - 3*a*c + 3*b*c + 2*c^2) : :
X(53339) = 3 X[2] - 4 X[45661], 5 X[2] - 4 X[45674], 3 X[4120] - X[4750], 3 X[4120] - 2 X[45661], 5 X[4120] - 2 X[45674], 5 X[4750] - 6 X[45674], 5 X[45661] - 3 X[45674], 4 X[18004] - X[50343], 4 X[30565] - 3 X[31992], 3 X[30565] - 2 X[47884], 3 X[31992] - 2 X[47776], 9 X[31992] - 8 X[47884], 3 X[47776] - 4 X[47884], 4 X[4728] - 3 X[6548], 3 X[6548] - 2 X[48571], 4 X[4024] - X[4608], 2 X[4024] + X[31290], X[4608] + 2 X[31290], 2 X[4813] + X[47659], X[20295] + 2 X[25259], X[20295] - 4 X[48269], and many others

> X(53339) lies on these lines: {2, 2786}, {100, 190}, {335, 4080}, {513, 47870}, {514, 4024}, {522, 47769}, {523, 47774}, {661, 17161}, {671, 690}, {812, 4958}, {824, 47759}, {918, 21297}, {2787, 30605}, {3239, 4786}, {3667, 47771}, {3700, 4789}, {4025, 27138}, {4106, 49272}, {4184, 53258}, {4210, 53269}, {4379, 28906}, {4453, 45677}, {4467, 14321}, {4500, 48076}, {4776, 28898}, {4777, 4824}, {4785, 47773}, {4802, 48437}, {4820, 47666}, {4838, 47991}, {4926, 31150}, {4931, 28840}, {4940, 47677}, {4944, 47762}, {4949, 48079}, {4951, 48254}, {4984, 10196}, {6008, 48557}, {6544, 41841}, {16892, 26798}, {17494, 48266}, {21894, 28606}, {23883, 48551}, {26985, 47971}, {27486, 47765}, {28209, 49275}, {28217, 48567}, {28846, 47780}, {28851, 47869}, {28855, 48416}, {28867, 47763}, {28878, 51317}, {29078, 47821}, {29090, 47840}, {29216, 47793}, {29328, 48171}, {29370, 48158}, {30519, 31147}, {44435, 47786}, {47653, 48049}, {47655, 47952}, {47665, 48026}, {47755, 47787}, {47764, 47781}, {47939, 48397}, {48112, 49289}, {48133, 48424}, {48434, 48558}, {49282, 49284}

X(53339) = midpoint of X(4789) and X(44449)
X(53339) = reflection of X(i) in X(j) for these {i,j}: {2, 4120}, {4467, 47784}, {4750, 45661}, {4786, 3239}, {4789, 3700}, {4984, 10196}, {7192, 4789}, {17161, 46915}, {27486, 47765}, {44435, 47786}, {46915, 661}, {47755, 47787}, {47762, 4944}, {47763, 47874}, {47775, 47769}, {47776, 30565}, {47780, 47790}, {47781, 47764}, {47784, 14321}, {47792, 4931}, {47894, 4776}, {48156, 31147}, {48254, 4951}, {48434, 48558}, {48571, 4728}
X(53339) = anticomplement of X(4750)
X(53339) = isotomic conjugate of the isogonal conjugate of X(53289)
X(53339) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {100, 14360}, {111, 149}, {213, 39356}, {671, 21293}, {691, 75}, {692, 8591}, {892, 17137}, {897, 150}, {923, 4440}, {5380, 69}, {5547, 37781}, {6335, 34518}, {32729, 17147}, {32740, 9263}, {36085, 17135}, {36142, 1}
X(53339) = crossdifference of every pair of points on line {1015, 2308}
X(53339) = barycentric product X(76)*X(53289)
X(53339) = barycentric quotient X(53289)/X(6)
X(53339) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3700, 44449, 7192}, {4024, 31290, 4608}, {4106, 49272, 49302}, {4120, 4750, 45661}, {4728, 48571, 6548}, {4750, 45661, 2}, {4949, 48271, 48079}, {20295, 25259, 49273}, {25259, 48269, 20295}, {30565, 47776, 31992}, {48266, 48270, 17494}

---

X(53340) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(190) AND X(889)

Barycentrics    (a - b)*(a - c)*(a*b^2 - 4*a*b*c + b^2*c + a*c^2 + b*c^2) : :
X(53340) = 2 X[16507] - 3 X[36872]

X(53340) lies on these lines: {1, 87}, {2, 19945}, {75, 16482}, {100, 190}, {312, 34583}, {513, 23354}, {536, 16507}, {646, 4499}, {660, 23836}, {889, 41314}, {1278, 16495}, {2345, 19964}, {3759, 16467}, {4363, 24403}, {4414, 9458}, {5695, 16506}, {9296, 27853}, {16501, 32922}, {16505, 17160}, {17165, 49721}, {17487, 17794}, {24722, 31061}, {36294, 49450}

X(53340) = reflection of X(23354) in X(24004)
X(53340) = anticomplement of X(19945)
X(53340) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {889, 21293}, {898, 149}, {1252, 39360}, {4607, 150}, {5381, 69}, {32718, 9263}, {34075, 4440}
X(53340) = X(4871)-Dao conjugate of X(891)
X(53340) = crossdifference of every pair of points on line {1015, 20979}
X(53340) = barycentric product X(190)*X(4871)
X(53340) = barycentric quotient X(4871)/X(514)

---

X(53341) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(190 AND X(892)

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

X(53341) lies on these lines: {2, 24345}, {8, 2607}, {10, 894}, {100, 190}, {524, 52747}, {645, 21295}, {892, 5466}, {1026, 50346}, {2640, 4440}, {2948, 21290}, {6790, 47270}, {13174, 17487}, {17135, 24346}, {17934, 50351}, {24711, 31057}, {24714, 31059}

X(53341) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {691, 149}, {692, 39356}, {892, 21293}, {4567, 14360}, {5380, 3448}, {32729, 9263}, {36085, 150}, {36142, 4440}, {45773, 17166}, {52940, 17137}
X(53341) = X(50755)-Dao conjugate of X(690)
X(53341) = barycentric product X(190)*X(50755)
X(53341) = barycentric quotient X(50755)/X(514)
X(53341) = {X(3699),X(14985)}-harmonic conjugate of X(4427)

---

X(53342) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(190) AND X(1494)

Barycentrics    (a - b - c)*(b - c)*(a^3 - 2*a^2*b - a*b^2 + 2*b^3 - 2*a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2 + 2*c^3) : :
X(53342) = X[11125] - 3 X[14429], X[20293] + 2 X[20294], X[7253] - 4 X[52355], 7 X[9780] - 4 X[21180]

X(53342) lies on these lines: {2, 11125}, {100, 190}, {514, 16086}, {522, 27545}, {1494, 3268}, {3543, 9524}, {4036, 4397}, {4163, 28169}, {4768, 52356}, {6089, 30709}, {6740, 44693}, {7253, 15776}, {9780, 21180}, {11684, 53256}, {35518, 47894}, {39472, 45341}, {47755, 52616}

X(53342) = reflection of X(2) in X(14429)
X(53342) = anticomplement of X(11125)
X(53342) = isotomic conjugate of the isogonal conjugate of X(53249)
X(53342) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {74, 149}, {100, 146}, {692, 39358}, {1304, 3868}, {1494, 21293}, {2159, 4440}, {2349, 150}, {15627, 37781}, {16077, 20242}, {32640, 17147}, {36034, 1}, {36131, 3187}, {40352, 9263}, {44693, 33650}, {44769, 75}
X(53342) = crossdifference of every pair of points on line {1015, 9408}
X(53342) = barycentric product X(i)*X(j) for these {i,j}: {8, 49274}, {76, 53249}
X(53342) = barycentric quotient X(i)/X(j) for these {i,j}: {49274, 7}, {53249, 6}

---

X(53343) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(190) AND X(2481)

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(53343) = 3 X[2] - 4 X[3716], 9 X[2] - 8 X[25380], 5 X[2] - 4 X[45328], 7 X[2] - 8 X[45337], 3 X[2254] - 4 X[25380], 5 X[2254] - 6 X[45328], 7 X[2254] - 12 X[45337], 3 X[3716] - 2 X[25380], 5 X[3716] - 3 X[45328], 7 X[3716] - 6 X[45337], 10 X[25380] - 9 X[45328], 7 X[25380] - 9 X[45337], and many others

X(53343) lies on these lines: {1, 21222}, {2, 2254}, {8, 3762}, {100, 190}, {103, 1311}, {145, 4895}, {149, 3738}, {152, 2820}, {320, 350}, {514, 48304}, {521, 25301}, {522, 3935}, {523, 47969}, {649, 3239}, {650, 48242}, {656, 27045}, {663, 17496}, {676, 4453}, {690, 17164}, {784, 48351}, {812, 17794}, {824, 47972}, {885, 2481}, {918, 1280}, {962, 2814}, {1125, 23795}, {1156, 34234}, {1281, 2786}, {1491, 47821}, {1577, 42325}, {1621, 53308}, {1639, 4925}, {1734, 47793}, {1769, 27545}, {2526, 4776}, {2530, 47840}, {2785, 21132}, {2787, 6161}, {2826, 3904}, {2827, 9809}, {2975, 53286}, {3004, 48161}, {3120, 23821}, {3309, 4391}, {3616, 3960}, {3617, 14430}, {3622, 14413}, {3623, 23057}, {3700, 47687}, {3798, 47801}, {3835, 48164}, {3837, 4800}, {3900, 4462}, {4017, 26854}, {4025, 47798}, {4040, 4560}, {4080, 23838}, {4151, 47970}, {4189, 8648}, {4379, 48073}, {4448, 9508}, {4458, 48571}, {4467, 50347}, {4522, 48169}, {4581, 50520}, {4707, 21201}, {4750, 13246}, {4762, 47974}, {4775, 48298}, {4778, 48142}, {4784, 28217}, {4785, 48072}, {4794, 48321}, {4804, 26824}, {4806, 50328}, {4813, 48037}, {4817, 28867}, {4874, 47824}, {4893, 48017}, {4905, 47796}, {4913, 26777}, {4926, 48240}, {4932, 48578}, {4962, 48008}, {6002, 31291}, {6004, 21301}, {6006, 47763}, {6085, 38478}, {6363, 50524}, {6372, 17166}, {7659, 47762}, {8062, 27167}, {8645, 13245}, {13265, 15626}, {15313, 20293}, {16892, 48203}, {17127, 22384}, {17140, 50342}, {20353, 21303}, {21189, 27293}, {23770, 49301}, {23800, 48173}, {23836, 36798}, {23887, 49274}, {24720, 26985}, {26798, 31095}, {27013, 47804}, {27115, 47828}, {27138, 44429}, {28147, 47927}, {28161, 47926}, {28221, 50339}, {28225, 48141}, {28623, 48340}, {28840, 48153}, {28863, 47702}, {28882, 48105}, {28890, 47705}, {29013, 48111}, {29051, 48264}, {29102, 49303}, {29144, 47693}, {29148, 48324}, {29198, 48301}, {29246, 48392}, {29302, 47977}, {29328, 50358}, {30520, 47692}, {30795, 48183}, {30835, 48547}, {31147, 48042}, {31207, 48575}, {31290, 48021}, {39386, 48251}, {44435, 48015}, {44444, 50329}, {45746, 48006}, {47123, 47676}, {47653, 47701}, {47663, 48061}, {47686, 49295}, {47688, 48349}, {47689, 48271}, {47690, 47870}, {47691, 49302}, {47698, 48040}, {47703, 47792}, {47759, 48023}, {47769, 48039}, {47771, 48069}, {47773, 48106}, {47774, 47945}, {47775, 47975}, {47790, 49285}, {47794, 48018}, {47795, 48075}, {47797, 50348}, {47815, 50501}, {47822, 50335}, {47825, 50341}, {47826, 48010}, {47838, 48066}, {47869, 48119}, {47908, 47980}, {47909, 47986}, {47934, 48001}, {47940, 48026}, {47973, 48156}, {47982, 48543}, {47985, 48544}, {47991, 48583}, {48004, 48407}, {48020, 48049}, {48030, 48157}, {48055, 48408}, {48058, 48409}, {48077, 48270}, {48099, 48410}, {48115, 49289}, {48148, 49292}, {48165, 50350}, {48254, 48405}, {49278, 49288}

X(53343) = reflection of X(i) in X(j) for these {i,j}: {8, 3762}, {145, 4895}, {649, 48063}, {2254, 3716}, {4467, 50347}, {4560, 4040}, {4707, 21201}, {4784, 48248}, {4813, 48037}, {7192, 47694}, {17166, 48305}, {17494, 4724}, {17496, 663}, {20295, 48080}, {21222, 1}, {21301, 48267}, {21302, 4391}, {23795, 1125}, {26824, 4804}, {31290, 48021}, {31291, 48150}, {44444, 50329}, {45746, 48006}, {46403, 4010}, {47653, 47701}, {47663, 48061}, {47676, 47123}, {47685, 4106}, {47686, 49295}, {47687, 3700}, {47688, 48349}, {47689, 48271}, {47690, 49286}, {47698, 48040}, {47908, 47980}, {47909, 47986}, {47926, 48009}, {47934, 48001}, {47940, 48026}, {47945, 48024}, {47975, 48029}, {48020, 48049}, {48023, 48043}, {48077, 48270}, {48108, 7662}, {48115, 49289}, {48119, 48394}, {48148, 49292}, {48298, 4775}, {48304, 48339}, {48321, 4794}, {48407, 48004}, {48408, 48055}, {48409, 48058}, {48410, 48099}, {48583, 47991}, {49273, 49275}, {49274, 49276}, {49278, 49288}, {49301, 23770}, {49302, 47691}, {50328, 4806}, {50343, 659}, {50356, 650}, {50357, 676}, {50359, 4874}
X(53343) = anticomplement of X(2254)
X(53343) = anticomplement of the isogonal conjugate of X(36086)
X(53343) = anticomplement of the isotomic conjugate of X(51560)
X(53343) = isotomic conjugate of the anticomplement of X(38980)
X(53343) = isotomic conjugate of the isogonal conjugate of X(53287)
X(53343) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 39353}, {55, 14732}, {100, 20344}, {101, 20533}, {105, 149}, {109, 52164}, {190, 20552}, {294, 37781}, {666, 69}, {673, 150}, {692, 39350}, {884, 17036}, {919, 2}, {927, 3434}, {1438, 4440}, {2195, 39351}, {2481, 21293}, {5377, 513}, {13576, 3448}, {14942, 33650}, {18785, 21221}, {32666, 192}, {32735, 145}, {34085, 21285}, {35185, 20075}, {35333, 21289}, {36086, 8}, {36138, 17316}, {36146, 7}, {36802, 3436}, {36803, 315}, {39272, 32850}, {39293, 21302}, {46135, 21280}, {46163, 2896}, {51560, 6327}, {52927, 144}
X(53343) = X(51560)-Ceva conjugate of X(2)
X(53343) = X(1438)-isoconjugate of X(40526)
X(53343) = X(6184)-Dao conjugate of X(40526)
X(53343) = crossdifference of every pair of points on line {213, 1015}
X(53343) = barycentric product X(i)*X(j) for these {i,j}: {76, 53287}, {4391, 7677}, {38980, 51560}
X(53343) = barycentric quotient X(i)/X(j) for these {i,j}: {518, 40526}, {7677, 651}, {38379, 2340}, {38980, 2254}, {53287, 6}
X(53343) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 48063, 47805}, {650, 50356, 48242}, {659, 50343, 47776}, {676, 50357, 4453}, {2254, 3716, 2}, {4010, 46403, 21297}, {4874, 50359, 47824}, {4913, 47811, 26777}, {7662, 48108, 47780}, {24720, 47832, 26985}, {47690, 49286, 47870}, {47804, 50336, 27013}, {47945, 48024, 47774}, {47975, 48029, 47775}, {48023, 48043, 47759}, {48119, 48394, 47869}

---

X(53344) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(190) AND X(2966)

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

X(53344) lies on these lines: {2, 24347}, {71, 1654}, {100, 190}, {150, 16562}, {525, 4237}, {879, 2966}, {2938, 50118}, {17134, 24400}, {17221, 24415}, {23902, 34612}, {27958, 32939}

X(53344) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {692, 39359}, {906, 14721}, {2715, 149}, {2966, 21293}, {36084, 150}

---

X(53345) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(290) AND X(648)

Barycentrics    (b^2 - c^2)*(a^8 - a^6*b^2 - a^4*b^4 + a^2*b^6 - 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 + a^2*c^6 + b^2*c^6) : :
X(53345) = 3 X[2] - 4 X[6130], 5 X[2] - 4 X[45319], 7 X[2] - 8 X[45682], 5 X[684] - 6 X[45319], 7 X[684] - 12 X[45682], 5 X[6130] - 3 X[45319], 7 X[6130] - 6 X[45682], 7 X[45319] - 10 X[45682], 3 X[9147] - 4 X[53263], 3 X[9979] - 2 X[16230], 3 X[376] - 4 X[44810], 5 X[631] - 4 X[8552], 7 X[3523] - 8 X[44818], 5 X[37760] - 4 X[47214]

X(53345) lies on these lines: {2, 684}, {4, 9517}, {20, 2797}, {23, 385}, {98, 1297}, {107, 110}, {147, 31953}, {290, 879}, {340, 520}, {376, 44810}, {450, 2451}, {511, 52076}, {525, 35474}, {526, 3448}, {631, 8552}, {690, 15054}, {804, 5984}, {895, 2986}, {924, 30735}, {2574, 2593}, {2575, 2592}, {2804, 13265}, {2848, 38672}, {2881, 12384}, {3523, 44818}, {6103, 13114}, {6563, 47194}, {8673, 14618}, {8675, 41721}, {8901, 35442}, {14220, 46789}, {24978, 25714}, {31065, 50946}, {31127, 53266}, {35098, 43665}, {37760, 47214}, {39469, 53149}

X(53345) = reflection of X(i) in X(j) for these {i,j}: {4, 41079}, {20, 9409}, {110, 32119}, {147, 31953}, {684, 6130}, {6563, 47194}, {44445, 30735}
X(53345) = anticomplement of X(684)
X(53345) = anticomplement of the isogonal conjugate of X(685)
X(53345) = anticomplement of the isotomic conjugate of X(22456)
X(53345) = isotomic conjugate of the anticomplement of X(39000)
X(53345) = isotomic conjugate of the isogonal conjugate of X(53265)
X(53345) = anticomplementary isogonal conjugate of X(14721)
X(53345) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 14721}, {19, 39359}, {162, 147}, {293, 34186}, {685, 8}, {1821, 13219}, {1910, 39352}, {2715, 6360}, {2966, 4329}, {6531, 21221}, {16081, 21294}, {20031, 5905}, {22456, 6327}, {24019, 40867}, {32676, 39355}, {32696, 192}, {36036, 1370}, {36084, 20}, {36104, 2}, {36120, 3448}, {41174, 17217}
X(53345) = X(i)-Ceva conjugate of X(j) for these (i,j): {22456, 2}, {44770, 4}
X(53345) = X(3150)-Dao conjugate of X(511)
X(53345) = crossdifference of every pair of points on line {39, 185}
X(53345) = barycentric product X(i)*X(j) for these {i,j}: {76, 53265}, {648, 3150}, {850, 10313}, {22456, 39000}, {38368, 43187}
X(53345) = barycentric quotient X(i)/X(j) for these {i,j}: {3150, 525}, {10313, 110}, {38368, 3569}, {39000, 684}, {53265, 6}
X(53345) = {X(684),X(6130)}-harmonic conjugate of X(2)

---

X(53346) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(290) AND X(671)

Barycentrics    a^6*b^2 - a^2*b^6 + a^6*c^2 - 4*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - 3*b^6*c^2 + 2*a^2*b^2*c^4 + 6*b^4*c^4 - a^2*c^6 - 3*b^2*c^6 : :
X(53346) = 2 X[338] + X[25051], X[14957] + 2 X[51481], 4 X[7668] - X[14570]

X(53346) lies on these lines: {2, 2782}, {4, 94}, {69, 45809}, {76, 4576}, {98, 4226}, {110, 38664}, {148, 36163}, {264, 15531}, {290, 879}, {316, 41724}, {338, 2854}, {373, 6248}, {419, 35265}, {511, 14957}, {542, 46124}, {671, 690}, {858, 31125}, {868, 31127}, {895, 48540}, {1316, 12188}, {3260, 9027}, {5050, 41231}, {5189, 38953}, {5191, 51523}, {5254, 39691}, {5467, 49006}, {5640, 40814}, {6090, 41238}, {6175, 24808}, {6800, 39646}, {7668, 14570}, {9862, 36181}, {11050, 46229}, {11185, 18911}, {11188, 41760}, {11646, 52450}, {14360, 46336}, {15535, 53132}, {21445, 35933}, {31128, 40916}, {32447, 37988}, {33884, 37190}, {35311, 41253}, {45968, 52281}

X(53346) = reflection of X(i) in X(j) for these {i,j}: {14570, 46127}, {46127, 7668}
X(53346) = anticomplement of X(9155)
X(53346) = anticomplement of the isogonal conjugate of X(9154)
X(53346) = isotomic conjugate of the isogonal conjugate of X(53264)
X(53346) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {897, 147}, {923, 39355}, {1821, 14360}, {1910, 8591}, {9154, 8}, {23894, 39359}, {36084, 44010}, {36128, 40867}
X(53346) = crossdifference of every pair of points on line {9419, 39689}
X(53346) = barycentric product X(76)*X(53264)
X(53346) = barycentric quotient X(53264)/X(6)
X(53346) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 41254, 46512}, {671, 9140, 53161}, {38664, 41254, 110}

---

X(53347) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(290) AND X(892)

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

X(53347) lies on these lines: {2, 47229}, {69, 850}, {111, 2373}, {183, 523}, {290, 879}, {315, 23105}, {316, 52632}, {325, 53266}, {385, 2395}, {512, 11185}, {647, 17008}, {805, 877}, {892, 5466}, {2501, 9308}, {7897, 31072}, {9060, 9080}, {10097, 47286}, {34291, 37688}, {34765, 40428}, {52076, 52145}

X(53347) = isotomic conjugate of the isogonal conjugate of X(44823)
X(53347) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {897, 39359}, {1910, 39356}, {9154, 21221}, {36036, 14360}, {36084, 8591}, {36085, 147}, {36104, 7665}, {36142, 39355}
X(53347) = crossdifference of every pair of points on line {2021, 9419}
X(53347) = barycentric product X(76)*X(44823)
X(53347) = barycentric quotient X(44823)/X(6)

---

X(53348) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(290) AND X(2494)

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

X(53348) lies on these lines: {67, 69}, {76, 15072}, {99, 15055}, {183, 6800}, {264, 5640}, {287, 34211}, {290, 879}, {325, 31127}, {339, 5663}, {385, 34369}, {511, 16083}, {1235, 9730}, {1494, 3268}, {3580, 46788}, {5976, 5984}, {6793, 41145}, {11459, 41009}, {12384, 51538}, {14915, 44146}, {16836, 26166}, {18911, 44134}, {20126, 36890}, {25053, 45237}, {35442, 41005}, {38641, 39874}, {40996, 46147}

X(53348) = isotomic conjugate of the isogonal conjugate of X(53246)
X(53348) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1821, 146}, {1910, 39358}, {2159, 39355}, {2349, 147}, {36104, 45292}, {36119, 40867}
X(53348) = crossdifference of every pair of points on line {9408, 9419}
X(53348) = barycentric product X(76)*X(53246)
X(53348) = barycentric quotient X(53246)/X(6)

---

X(53349) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(648) AND X(668)

Barycentrics    (a - b)*(a - c)*(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(53349) lies on these lines: {2, 16100}, {20, 2828}, {92, 20242}, {100, 13397}, {107, 110}, {242, 37782}, {523, 7477}, {668, 891}, {851, 18668}, {927, 4608}, {1897, 3732}, {3436, 18719}, {4427, 6089}, {4566, 13149}, {7493, 18721}, {8677, 53151}, {10327, 20929}, {12649, 28102}, {14206, 14956}, {14985, 53324}, {15343, 49274}, {17479, 20760}, {18750, 20243}, {23772, 27628}, {34381, 48380}, {36239, 38470}

X(53349) = reflection of X(i) in X(j) for these {i,j}: {14956, 14206}, {18668, 851}
X(53349) = anticomplement of X(18210)
X(53349) = anticomplement of the isogonal conjugate of X(5379)
X(53349) = isotomic conjugate of the isogonal conjugate of X(53282)
X(53349) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {59, 3152}, {101, 39352}, {112, 4440}, {162, 149}, {190, 13219}, {249, 17134}, {250, 1}, {643, 34188}, {648, 150}, {765, 52364}, {811, 21293}, {1101, 20222}, {1110, 18666}, {1252, 3151}, {1331, 34186}, {1783, 21221}, {1897, 3448}, {2149, 18667}, {2299, 17036}, {4564, 2897}, {4567, 4329}, {4570, 20}, {4590, 18659}, {4600, 1370}, {5379, 8}, {6335, 21294}, {7012, 2475}, {7115, 17778}, {8750, 148}, {15742, 1330}, {18020, 17135}, {23582, 17220}, {23964, 3187}, {23999, 20242}, {24000, 3868}, {24041, 20243}, {32676, 9263}, {36797, 33650}, {42396, 25049}, {44183, 18656}, {46102, 2893}, {46254, 17137}, {47443, 17161}, {52378, 347}
X(53349) = X(649)-isoconjugate of X(40406)
X(53349) = X(i)-Dao conjugate of X(j) for these (i,j): {5375, 40406}, {21530, 513}, {23537, 8062}, {40941, 525}
X(53349) = cevapoint of X(513) and X(52259)
X(53349) = trilinear pole of line {21530, 23537}
X(53349) = crossdifference of every pair of points on line {1977, 3269}
X(53349) = barycentric product X(i)*X(j) for these {i,j}: {76, 53282}, {190, 23537}, {648, 21530}, {668, 40941}, {799, 40973}, {811, 18674}, {1897, 18651}, {6335, 18732}, {18709, 41676}
X(53349) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 40406}, {18651, 4025}, {18674, 656}, {18709, 4580}, {18732, 905}, {21530, 525}, {21678, 4064}, {23537, 514}, {40941, 513}, {40973, 661}, {53282, 6}
X(53349) = {X(14543),X(14544)}-harmonic conjugate of X(110)

---

X(53350) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(648) AND X(670)

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

X(53350) lies on these lines: {2, 16098}, {4, 14984}, {20, 2790}, {69, 2871}, {76, 16175}, {99, 3565}, {107, 110}, {147, 39352}, {193, 3186}, {264, 12272}, {317, 40317}, {338, 2854}, {419, 37784}, {512, 14221}, {523, 1634}, {662, 47844}, {670, 888}, {877, 39469}, {892, 4577}, {895, 46512}, {1316, 22143}, {1576, 2407}, {1632, 4226}, {2393, 3260}, {2979, 44133}, {4230, 41678}, {5214, 21362}, {5648, 9214}, {7669, 40879}, {8681, 51481}, {8754, 32114}, {9512, 36207}, {10602, 41238}, {10607, 30549}, {11160, 12251}, {11442, 44134}, {12220, 14615}, {13188, 47283}, {14772, 47430}, {15531, 40814}, {17500, 41579}, {18315, 44768}, {25052, 45809}, {27365, 30506}, {33314, 45279}, {34294, 46154}, {34382, 44145}, {35278, 36841}, {36239, 47694}, {37183, 40888}, {40138, 47405}, {41359, 46184}, {41676, 46592}

X(53350) = reflection of X(i) in X(j) for these {i,j}: {193, 51335}, {14570, 1634}, {14957, 3260}, {25051, 338}
X(53350) = anticomplement of X(20975)
X(53350) = anticomplement of the isogonal conjugate of X(18020)
X(53350) = isotomic conjugate of the isogonal conjugate of X(53273)
X(53350) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {112, 21220}, {162, 148}, {249, 6360}, {250, 192}, {270, 17036}, {648, 21221}, {662, 39352}, {799, 13219}, {811, 3448}, {1101, 3164}, {4567, 3151}, {4570, 18666}, {4590, 4329}, {4592, 34186}, {4593, 25053}, {4600, 52364}, {4620, 2897}, {5379, 1654}, {6064, 52366}, {6331, 21294}, {7340, 52365}, {18020, 8}, {23582, 5905}, {23964, 21216}, {23999, 4}, {24000, 193}, {24037, 1370}, {24041, 20}, {32676, 25054}, {44183, 17481}, {46254, 69}, {47443, 4560}, {52378, 18667}, {52914, 39351}
X(53350) = X(i)-isoconjugate of X(j) for these (i,j): {798, 40405}, {810, 40413}
X(53350) = X(i)-Dao conjugate of X(j) for these (i,j): {1196, 525}, {1368, 512}, {5254, 30476}, {31998, 40405}, {39062, 40413}
X(53350) = cevapoint of X(i) and X(j) for these (i,j): {512, 6677}, {523, 14913}, {5254, 12075}
X(53350) = trilinear pole of line {1196, 1368}
X(53350) = crossdifference of every pair of points on line {3269, 9427}
X(53350) = X(14772)-line-conjugate of X(47430)
X(53350) = barycentric product X(i)*X(j) for these {i,j}: {76, 53273}, {99, 5254}, {162, 21406}, {648, 1368}, {668, 16716}, {670, 1196}, {799, 17872}, {811, 18671}, {1897, 18648}, {4590, 12075}, {6331, 6467}, {6528, 22401}, {35136, 40326}, {40325, 52608}
X(53350) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 40405}, {648, 40413}, {682, 3049}, {1196, 512}, {1368, 525}, {5254, 523}, {6331, 683}, {6467, 647}, {12075, 115}, {16716, 513}, {17872, 661}, {18648, 4025}, {18671, 656}, {21406, 14208}, {22401, 520}, {40325, 2489}, {40326, 3566}, {53273, 6}
X(53350) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {648, 46151, 35360}, {1632, 4558, 4226}

---

X(53351) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(648) AND X(892)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(2*a^4 - a^2*b^2 - 3*b^4 - a^2*c^2 + 6*b^2*c^2 - 3*c^4) : :
X(53351) = 3 X[2407] - 2 X[5467], 7 X[2407] - 4 X[35345], 5 X[2407] - 2 X[53274], 3 X[4226] - 4 X[5467], 7 X[4226] - 8 X[35345], 5 X[4226] - 4 X[53274], 7 X[5467] - 6 X[35345], 5 X[5467] - 3 X[53274], 10 X[35345] - 7 X[53274]

X(53351) lies on these lines: {2, 2452}, {4, 193}, {6, 46512}, {107, 110}, {147, 39358}, {264, 11443}, {385, 7417}, {523, 2407}, {524, 9214}, {690, 52035}, {892, 5466}, {895, 48540}, {1494, 30789}, {2396, 31998}, {4235, 47293}, {4558, 48539}, {7422, 34810}, {9145, 14570}, {9181, 14221}, {10754, 48983}, {11054, 52483}, {11634, 47292}, {14977, 23348}, {15454, 32110}, {18808, 44768}, {32127, 51481}, {32217, 46493}, {37784, 38294}, {39474, 53155}

X(53351) = reflection of X(i) in X(j) for these {i,j}: {4226, 2407}, {7422, 34810}, {53161, 9214}
X(53351) = vX(i)-anticomplementary conjugate of X(j) for these (i,j): {23999, 34518}, {32676, 39356}, {36085, 13219}, {36142, 39352}
X(53351) = X(5159)-Dao conjugate of X(690)
X(53351) = cevapoint of X(690) and X(37911)
X(53351) = trilinear pole of line {5159, 40349}
X(53351) = barycentric product X(i)*X(j) for these {i,j}: {648, 5159}, {670, 40350}, {6331, 21639}, {6528, 40349}
X(53351) = barycentric quotient X(i)/X(j) for these {i,j}: {5159, 525}, {21639, 647}, {40349, 520}, {40350, 512}
X(53351) = {X(2452),X(36207)}-harmonic conjugate of X(2)

---

X(53352) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(648) AND X(903)

Barycentrics    (b - c)*(-3*a^4 + a^3*b + a^2*b^2 - a*b^3 + 2*b^4 + 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 + 2*c^4) : :
X(53352) = 3 X[11125] - X[14429], X[8] - 4 X[21180], X[7253] - 4 X[44409], 4 X[7649] - X[20293], X[20294] - 4 X[21172]

X(53352) lies on these lines: {2, 11125}, {8, 21180}, {20, 9524}, {69, 21205}, {106, 2370}, {107, 110}, {447, 525}, {522, 47796}, {523, 7478}, {676, 25923}, {900, 903}, {966, 47234}, {3315, 47695}, {4581, 7927}, {7649, 9031}, {10015, 39472}, {20294, 21172}, {21178, 46402}, {34195, 53295}

X(53352) = reflection of X(i) in X(j) for these {i,j}: {2, 11125}, {25923, 676}
X(53352) = anticomplement of X(14429)
X(53352) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {88, 13219}, {112, 30578}, {162, 21290}, {901, 52364}, {2203, 39349}, {4591, 4329}, {4622, 1370}, {6336, 21294}, {8752, 21221}, {9456, 39352}, {32665, 3151}, {32676, 17487}, {32719, 18666}, {36058, 34186}, {36125, 3448}
X(53352) = crossdifference of every pair of points on line {1017, 3269}

---

X(53353) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(648) AND X(2481)

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

X(53353) lies on these lines: {2, 47203}, {20, 9520}, {105, 26703}, {107, 110}, {377, 2806}, {425, 2501}, {521, 26546}, {523, 7469}, {659, 53326}, {850, 7253}, {885, 2481}, {2798, 6872}, {2990, 8759}, {17166, 48288}, {17498, 33294}, {26277, 45746}, {43991, 53308}

X(53353) = isotomic conjugate of the isogonal conjugate of X(53311)
X(53353) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {112, 20533}, {162, 20344}, {648, 20552}, {673, 13219}, {919, 3151}, {1438, 39352}, {1474, 39353}, {2299, 14732}, {8751, 21221}, {32666, 18666}, {32676, 39350}, {32735, 3152}, {36057, 34186}, {36086, 52364}, {36124, 3448}, {36146, 2897}
X(53353) = crossdifference of every pair of points on line {3269, 39686}
X(53353) = barycentric product X(76)*X(53311)
X(53353) = barycentric quotient X(53311)/X(6)

---

X(53354) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(648) AND X(3225)

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

X(53354) lies on these lines: {6, 14295}, {107, 110}, {385, 2491}, {690, 7760}, {850, 2451}, {2799, 4027}, {3225, 5027}, {6179, 14270}, {7757, 44826}, {14568, 39509}, {14614, 53263}

X(53354) = X(43761)-anticomplementary conjugate of X(13219)
X(53354) = crossdifference of every pair of points on line {3269, 32452}

---

X(53355) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(664) AND X(670)

Barycentrics    (a - b)*(a - 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(53355) lies on these lines: {1, 23824}, {100, 658}, {670, 888}, {693, 3909}, {1111, 38484}, {1146, 30016}, {2481, 38478}, {2810, 21404}, {3210, 17082}, {3212, 37684}, {3699, 25310}, {4427, 7192}, {20448, 50362}, {20561, 40075}, {20935, 25308}

X(53355) = anticomplement of the isogonal conjugate of X(4620)
X(53355) = isotomic conjugate of the isogonal conjugate of X(23363)
X(53355) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {59, 1655}, {81, 17036}, {99, 37781}, {109, 21220}, {651, 148}, {662, 39351}, {664, 21221}, {799, 33650}, {1275, 2475}, {1414, 4440}, {1415, 25054}, {4554, 3448}, {4563, 34188}, {4564, 1654}, {4565, 9263}, {4567, 144}, {4570, 3177}, {4572, 21294}, {4573, 149}, {4590, 3869}, {4600, 329}, {4601, 3436}, {4620, 8}, {4625, 150}, {4998, 2895}, {5379, 30694}, {6064, 18750}, {6516, 39352}, {7045, 17778}, {7340, 75}, {18020, 92}, {24037, 20245}, {24041, 63}, {31615, 31290}, {44717, 18666}, {52378, 192}
X(53355) = X(i)-Dao conjugate of X(j) for these (i,j): {21246, 512}, {24214, 17066}
X(53355) = trilinear pole of line {21246, 24214}
X(53355) = crossdifference of every pair of points on line {9427, 14936}
X(53355) = barycentric product X(i)*X(j) for these {i,j}: {76, 23363}, {190, 24214}, {651, 21422}, {664, 21246}, {4554, 21334}, {4572, 23640}, {22421, 46404}
X(53355) = barycentric quotient X(i)/X(j) for these {i,j}: {21246, 522}, {21334, 650}, {21422, 4391}, {22421, 652}, {23363, 6}, {23640, 663}, {24214, 514}

---

X(53356) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(664) AND X(671)

Barycentrics    (b - c)*(a^3 - 4*a^2*b + a*b^2 + 2*b^3 - 4*a^2*c + 3*a*b*c - b^2*c + a*c^2 - b*c^2 + 2*c^3) : :
X(53356) = X[14432] - 3 X[30574], X[8] + 2 X[4707], 4 X[10] - X[49274], X[145] - 4 X[4458], 5 X[3617] - 2 X[4088], 2 X[4730] + X[49303], 14 X[21952] - 11 X[46933], 5 X[27013] - 2 X[47728], 2 X[43052] + X[50356]

X(53356) lies on these lines: {1, 21181}, {2, 2785}, {8, 4707}, {10, 49274}, {100, 658}, {145, 4458}, {514, 48242}, {522, 17950}, {671, 690}, {2787, 30595}, {2789, 4750}, {2826, 44553}, {3617, 4088}, {4225, 53262}, {4730, 49303}, {7518, 47210}, {14430, 47772}, {21952, 46933}, {27013, 47728}, {28292, 47798}, {28468, 48164}, {28537, 44550}, {29082, 47885}, {29094, 47836}, {29172, 48254}, {29240, 47776}, {29304, 47793}, {34195, 53249}, {43052, 50356}

X(53356) = reflection of X(i) in X(j) for these {i,j}: {1, 21181}, {2, 30574}, {47772, 14430}
X(53356) = anticomplement of X(14432)
X(53356) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {111, 37781}, {651, 14360}, {691, 3869}, {895, 34188}, {897, 33650}, {923, 39351}, {1402, 39356}, {1415, 8591}, {5380, 3436}, {7316, 149}, {18026, 34518}, {36085, 20245}, {36142, 63}
X(53356) = crossdifference of every pair of points on line {14936, 21748}

---

X(53357) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(664) AND X(2481)

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 - 2*b^2*c^2 + a*c^3 + b*c^3) : :
X(53357) = 3 X[4453] - 2 X[43042]

X(53357) lies on these lines: {1, 20520}, {2, 46793}, {7, 3738}, {100, 658}, {102, 675}, {149, 150}, {239, 514}, {521, 24002}, {673, 918}, {693, 3900}, {812, 41845}, {885, 2481}, {926, 43930}, {1014, 23087}, {1320, 6548}, {1769, 3672}, {2254, 43041}, {2826, 13243}, {3004, 53284}, {3261, 7253}, {3309, 17896}, {3676, 4105}, {4435, 20507}, {4467, 6362}, {4468, 14330}, {4768, 32087}, {8058, 31605}, {8713, 47995}, {14077, 47780}, {15413, 20293}, {17096, 48281}, {28292, 44435}

X(53357) = reflection of X(i) in X(j) for these {i,j}: {4468, 14330}, {17494, 45755}, {20295, 30804}, {46402, 24002}
X(53357) = anticomplement of the isogonal conjugate of X(36146)
X(53357) = anticomplement of the isotomic conjugate of X(34085)
X(53357) = isotomic conjugate of the isogonal conjugate of X(53308)
X(53357) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 14732}, {56, 39353}, {105, 37781}, {109, 20533}, {651, 20344}, {664, 20552}, {666, 3436}, {673, 33650}, {919, 144}, {927, 69}, {1415, 39350}, {1416, 4440}, {1438, 39351}, {1461, 52164}, {1462, 149}, {1814, 34188}, {5377, 4462}, {32644, 20111}, {32666, 3177}, {32735, 2}, {34018, 21293}, {34085, 6327}, {36086, 329}, {36146, 8}, {39293, 20295}, {43929, 17036}, {46135, 315}, {51560, 21286}, {52927, 30695}
X(53357) = X(34085)-Ceva conjugate of X(2)
X(53357) = X(692)-isoconjugate of X(43672)
X(53357) = X(1086)-Dao conjugate of X(43672)
X(53357) = crossdifference of every pair of points on line {42, 1200}
X(53357) = barycentric product X(i)*X(j) for these {i,j}: {76, 53308}, {3261, 13329}, {4025, 26003}
X(53357) = barycentric quotient X(i)/X(j) for these {i,j}: {514, 43672}, {13329, 101}, {26003, 1897}, {53308, 6}

---

X(53358) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(666) AND X(668)

Barycentrics    (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(53358) lies on these lines: {2, 3675}, {4, 8}, {100, 1292}, {239, 52210}, {514, 1026}, {518, 4124}, {523, 2397}, {660, 883}, {666, 885}, {668, 891}, {899, 21140}, {2407, 17944}, {3699, 33946}, {4226, 4567}, {4427, 33951}, {4511, 47043}, {4552, 52923}, {4579, 14543}, {4712, 51381}, {4756, 30731}, {5380, 5466}, {5905, 18343}, {6550, 6634}, {9004, 37788}, {16504, 24346}, {17794, 27844}, {20075, 38503}, {34381, 51832}

X(53358) = reflection of X(2397) in X(23343)
X(53358) = anticomplement of X(3675)
X(53358) = anticomplement of the isogonal conjugate of X(5377)
X(53358) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {59, 52164}, {101, 39353}, {666, 150}, {765, 20344}, {919, 4440}, {1016, 20552}, {1110, 39350}, {1252, 20533}, {2195, 17036}, {3939, 14732}, {5377, 8}, {32666, 9263}, {36086, 149}, {36802, 33650}, {39293, 3434}, {51560, 21293}, {52927, 39351}
X(53358) = X(i)-isoconjugate of X(j) for these (i,j): {649, 2991}, {1027, 34159}, {1459, 15344}, {2254, 15382}, {3248, 35574}
X(53358) = X(i)-Dao conjugate of X(j) for these (i,j): {120, 513}, {1738, 3716}, {3290, 918}, {5375, 2991}
X(53358) = cevapoint of X(i) and X(j) for these (i,j): {513, 6714}, {518, 2977}, {1738, 23770}
X(53358) = trilinear pole of line {120, 1738}
X(53358) = crossdifference of every pair of points on line {1977, 22383}
X(53358) = barycentric product X(i)*X(j) for these {i,j}: {99, 21956}, {120, 666}, {190, 1738}, {321, 4236}, {668, 3290}, {1016, 23770}, {2397, 51832}, {3952, 16752}, {6335, 34381}, {14267, 42720}, {17464, 51560}, {20431, 36086}, {20455, 36803}
X(53358) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 2991}, {120, 918}, {919, 15382}, {1016, 35574}, {1738, 514}, {1783, 15344}, {2284, 34159}, {3290, 513}, {4236, 81}, {16752, 7192}, {17464, 2254}, {20455, 665}, {20482, 4088}, {20702, 24290}, {21956, 523}, {23770, 1086}, {34381, 905}, {51832, 2401}
X(53358) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3573, 36236, 2398}, {3952, 21272, 3799}

---

X(53359) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(666) AND X(671)

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

X(53359) lies on these lines: {321, 4391}, {335, 918}, {514, 29574}, {522, 50093}, {523, 4664}, {666, 885}, {671, 690}, {824, 14433}, {3661, 3700}, {4384, 4467}, {4800, 33904}, {17069, 29628}, {23887, 30565}, {26580, 47790}, {29312, 47870}, {45322, 47798}

X(53359) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {923, 39353}, {5380, 20552}, {32666, 8591}, {36086, 14360}
X(53359) = crossdifference of every pair of points on line {35505, 39689}

---

X(53360) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(666) AND X(889)

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

X(53360) lies on these lines: {2, 36238}, {100, 47805}, {105, 32922}, {190, 47694}, {513, 42722}, {536, 52761}, {666, 885}, {889, 41314}, {901, 4427}, {4576, 4601}, {9266, 17496}, {17166, 32028}, {29824, 36798}

X(53360) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {5381, 20552}, {34075, 39353}

---

X(53361) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(666) AND X(903)

Barycentrics    (b - c)*(-3*a^3 + 4*a^2*b - 3*a*b^2 + 2*b^3 + 4*a^2*c - a*b*c - b^2*c - 3*a*c^2 - b*c^2 + 2*c^3) : :
X(53361) = 2 X[47785] - 3 X[47798], 4 X[47785] - 3 X[48242], 2 X[47790] - 3 X[48172], X[8] - 4 X[21201], X[145] + 2 X[21132], 5 X[3623] - 2 X[21105], 4 X[4448] - 3 X[31992], 2 X[6161] + X[49303], X[17161] - 4 X[50340], 4 X[21185] - X[21302], X[21222] - 4 X[48286], 4 X[47131] - X[49302]

X(53361) lies on these lines: {2, 522}, {8, 21201}, {145, 21132}, {513, 3873}, {514, 3241}, {523, 27804}, {666, 885}, {900, 903}, {918, 1280}, {2397, 17780}, {2403, 6550}, {3623, 21105}, {3667, 6545}, {3676, 51351}, {4391, 4737}, {4448, 4664}, {4560, 4653}, {4926, 36848}, {4962, 21204}, {6089, 9147}, {6161, 49303}, {6366, 50894}, {6546, 28161}, {9539, 42312}, {10072, 47796}, {14475, 44551}, {17161, 50340}, {18391, 21185}, {20042, 23838}, {21222, 48286}, {28169, 44009}, {28205, 45666}, {29144, 47694}, {29166, 48305}, {29204, 49273}, {36534, 48339}, {44433, 47776}, {45701, 47793}, {47131, 49302}, {47894, 48223}, {48157, 48177}, {48234, 48254}

X(53361) = reflection of X(i) in X(j) for these {i,j}: {47776, 44433}, {47894, 48223}, {48157, 48177}, {48169, 47832}, {48242, 47798}, {48254, 48234}
X(53361) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {901, 20344}, {919, 30578}, {3257, 20552}, {9456, 39353}, {32665, 20533}, {32666, 17487}, {32719, 39350}, {36086, 21290}
X(53361) = crossdifference of every pair of points on line {1017, 1055}

---

X(53362) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(666) AND X(1121)

Barycentrics    (b - c)*(3*a^4 - 5*a^3*b + 3*a^2*b^2 - 3*a*b^3 + 2*b^4 - 5*a^3*c + 5*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 - 3*a*c^3 - b*c^3 + 2*c^4) : :
X(53362) = X[144] + 2 X[42462], X[2400] - 4 X[28132]

X(53362) lies on these lines: {2, 514}, {144, 42462}, {522, 6172}, {650, 24635}, {661, 31043}, {666, 885}, {673, 918}, {693, 30854}, {1121, 6366}, {2826, 47776}, {3239, 29616}, {3681, 3900}, {3800, 27806}, {3877, 14077}, {4025, 24599}, {4040, 28125}, {5744, 14330}, {5813, 20295}, {6175, 47774}, {7192, 16054}, {14476, 45670}, {21133, 37650}, {21202, 37681}, {23893, 45293}, {26003, 53150}, {28131, 44448}, {28292, 47769}, {28898, 51053}, {31018, 47790}, {31150, 31169}, {31209, 31225}, {31226, 43042}, {37139, 37143}, {46402, 52457}

X(53362) = reflection of X(47785) in X(14330)
X(53362) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {18889, 14732}, {32666, 39357}, {34068, 39353}, {36141, 52164}
X(53362) = crossdifference of every pair of points on line {902, 35505}

---

X(53363) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(668) AND X(670)

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

X(53363) lies on these lines: {2, 3121}, {75, 16727}, {99, 43356}, {149, 20345}, {274, 27812}, {310, 17163}, {518, 35543}, {561, 17135}, {668, 891}, {670, 888}, {799, 874}, {1909, 31025}, {1920, 4651}, {1921, 29824}, {1966, 16704}, {2895, 30660}, {3187, 18056}, {3807, 41315}, {4033, 42720}, {4087, 20347}, {4495, 32919}, {4572, 35312}, {4583, 4608}, {4631, 5468}, {4671, 17762}, {6327, 20560}, {6374, 25277}, {6376, 31035}, {6382, 17165}, {7018, 31037}, {7244, 32864}, {7257, 17136}, {16584, 27104}, {16748, 21020}, {17140, 40087}, {17141, 40034}, {17143, 39744}, {17145, 40089}, {17147, 17149}, {17150, 18064}, {17156, 18068}, {17162, 18075}, {18066, 52151}, {21443, 31136}, {21814, 26766}, {24732, 26756}, {25286, 41318}, {27804, 31008}, {30632, 31017}, {30941, 35544}

X(53363) = isotomic conjugate of X(50520)
X(53363) = anticomplement of X(3121)
X(53363) = anticomplement of the isogonal conjugate of X(4601)
X(53363) = isotomic conjugate of the anticomplement of X(50497)
X(53363) = isotomic conjugate of the isogonal conjugate of X(4436)
X(53363) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {99, 4440}, {100, 21220}, {101, 25054}, {110, 21224}, {190, 148}, {249, 17148}, {333, 17036}, {645, 39351}, {662, 9263}, {668, 21221}, {670, 150}, {689, 25049}, {765, 1655}, {799, 149}, {1016, 1654}, {1978, 3448}, {4561, 39352}, {4567, 192}, {4568, 39346}, {4570, 194}, {4590, 1}, {4593, 25048}, {4600, 2}, {4601, 8}, {4602, 21293}, {4610, 17154}, {4615, 20042}, {4620, 145}, {4632, 44006}, {4998, 17778}, {5379, 21216}, {6064, 63}, {6386, 21294}, {6632, 31290}, {7035, 2895}, {7257, 37781}, {7340, 3875}, {17934, 39368}, {18020, 3187}, {24037, 75}, {24041, 17147}, {31614, 17161}, {31625, 1330}, {34537, 17135}, {44168, 17138}, {46254, 3868}, {47389, 17134}, {52940, 17162}
X(53363) = X(i)-isoconjugate of X(j) for these (i,j): {31, 50520}, {667, 40433}, {669, 40439}, {798, 40408}, {1919, 32009}, {3248, 8708}
X(53363) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 50520}, {3739, 512}, {6631, 40433}, {9296, 32009}, {16589, 513}, {17205, 16726}, {20888, 42327}, {31998, 40408}
X(53363) = cevapoint of X(i) and X(j) for these (i,j): {512, 4698}, {513, 36812}, {3739, 6372}, {21020, 47672}
X(53363) = trilinear pole of line {3739, 16589}
X(53363) = crossdifference of every pair of points on line {1977, 9427}
X(53363) = barycentric product X(i)*X(j) for these {i,j}: {76, 4436}, {190, 20888}, {646, 4059}, {668, 3739}, {670, 16589}, {799, 21020}, {1978, 3720}, {2667, 4602}, {3691, 4572}, {3706, 4554}, {3952, 16748}, {4033, 17175}, {4601, 48393}, {4609, 21753}, {4623, 52579}, {6372, 31625}, {6386, 20963}, {7035, 47672}, {18166, 27808}, {21699, 52612}, {34537, 50538}
X(53363) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 50520}, {99, 40408}, {190, 40433}, {668, 32009}, {799, 40439}, {1016, 8708}, {2667, 798}, {3691, 663}, {3706, 650}, {3720, 649}, {3739, 513}, {4059, 3669}, {4111, 3709}, {4436, 6}, {4754, 4367}, {4891, 4394}, {6372, 1015}, {16589, 512}, {16748, 7192}, {17175, 1019}, {18089, 18108}, {18166, 3733}, {20888, 514}, {20963, 667}, {21020, 661}, {21699, 4079}, {21753, 669}, {21820, 50487}, {22060, 22383}, {22369, 3049}, {29773, 4040}, {39793, 7180}, {47672, 244}, {48264, 2170}, {48393, 3125}, {50497, 3121}, {50538, 3124}, {52579, 4705}
X(53363) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {668, 1978, 3952}, {799, 874, 4427}, {1978, 3952, 41314}, {18059, 51863, 2}

---

X(53364) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(668) AND X(1121)

Barycentrics    (b - c)*(-a^3 + 2*a^2*b - a*b^2 + 2*a^2*c - 7*a*b*c + 3*b^2*c - a*c^2 + 3*b*c^2) : :
X(53364) = X[14413] - 3 X[14430], X[8] + 2 X[3762], 4 X[10] - X[21222], X[145] - 4 X[3716], 2 X[4462] + X[21302], 2 X[2254] - 5 X[3617], X[2403] - 4 X[48182], 4 X[30583] - X[47776], X[3621] + 2 X[4895], 4 X[3960] - 7 X[9780], 4 X[4147] - X[17496], 2 X[4474] + X[17494], 4 X[4691] - X[23795], 2 X[4774] + X[47969], 4 X[4791] - X[48304], 8 X[25380] - 11 X[46933], 7 X[27115] - 4 X[48325], 7 X[27138] - 4 X[48332]

X(53364) lies on these lines: {2, 14413}, {8, 3762}, {10, 21222}, {145, 3716}, {513, 4397}, {514, 47808}, {668, 891}, {690, 17163}, {1121, 6366}, {1635, 4148}, {2254, 3617}, {2403, 48182}, {2785, 47772}, {2787, 30583}, {2789, 6546}, {2832, 20344}, {3621, 4895}, {3907, 47811}, {3960, 9780}, {4147, 17496}, {4391, 14077}, {4449, 47831}, {4474, 17494}, {4546, 4962}, {4691, 23795}, {4774, 47969}, {4791, 48304}, {4800, 25574}, {5260, 53308}, {25380, 46933}, {27115, 48325}, {27138, 48332}, {29236, 48240}, {30574, 48571}, {48162, 48401}, {48233, 48323}

X(53364) = reflection of X(i) in X(j) for these {i,j}: {2, 14430}, {145, 23057}, {4449, 47831}, {17496, 47828}, {21297, 30709}, {23057, 3716}, {47828, 4147}, {48162, 48401}, {48172, 4391}, {48323, 48233}, {48571, 30574}
X(53364) = anticomplement of X(14413)
X(53364) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {59, 45290}, {101, 39357}, {1121, 150}, {1156, 149}, {2291, 4440}, {4845, 39351}, {14733, 145}, {23351, 17036}, {34068, 9263}, {35157, 3434}, {36141, 3210}, {37139, 7}, {41798, 37781}
X(53364) = crossdifference of every pair of points on line {1977, 20228}

---

X(53365) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(670) AND X(671)

Barycentrics    (b^2 - c^2)*(-a^4 - a^2*b^2 - a^2*c^2 + 3*b^2*c^2) : :
X(53365) = 5 X[2] - 4 X[11176], 3 X[2] - 4 X[45689], 4 X[351] - 3 X[9147], X[351] - 3 X[9148], 5 X[351] - 6 X[11176], X[9147] - 4 X[9148], 5 X[9147] - 8 X[11176], 3 X[9147] - 8 X[45689], 5 X[9148] - 2 X[11176], 3 X[9148] - 2 X[45689], 3 X[11176] - 5 X[45689], 2 X[850] + X[44445], and many others

X(53365) lies on these lines: {2, 351}, {4, 2780}, {69, 9023}, {125, 9138}, {126, 9156}, {316, 512}, {523, 7840}, {526, 3448}, {623, 9162}, {624, 9163}, {631, 9126}, {669, 31072}, {670, 888}, {671, 690}, {686, 23291}, {740, 4728}, {812, 21020}, {858, 9213}, {1635, 27798}, {1649, 8786}, {1962, 4928}, {2793, 6054}, {3090, 11615}, {3524, 16235}, {3545, 19912}, {3618, 9188}, {4108, 31174}, {4155, 17163}, {4365, 22043}, {5189, 20403}, {5996, 23878}, {6088, 14360}, {6995, 47206}, {7378, 17994}, {7665, 11631}, {8029, 9479}, {8371, 32193}, {8599, 8785}, {8644, 30476}, {8663, 26798}, {8675, 30735}, {8889, 47230}, {9135, 24284}, {9178, 31125}, {9185, 44564}, {9279, 47780}, {10278, 14420}, {11182, 13309}, {13242, 34512}, {14277, 14977}, {14279, 18311}, {17414, 23301}, {17989, 31025}, {18010, 34294}, {18867, 45147}, {20021, 43665}, {25332, 44007}, {26824, 50538}, {27550, 50855}, {27551, 50858}, {27812, 47776}, {31128, 53272}, {36829, 44826}, {42659, 49004}, {45678, 53034}, {47671, 47715}

X(53365) = midpoint of X(17163) and X(21297)
X(53365) = reflection of X(i) in X(j) for these {i,j}: {2, 9148}, {351, 45689}, {1635, 27798}, {1962, 4928}, {4108, 31174}, {8644, 30476}, {9131, 14417}, {9135, 24284}, {9138, 125}, {9147, 2}, {9156, 126}, {9162, 623}, {9163, 624}, {9168, 9191}, {9213, 858}, {9485, 1649}, {9979, 9134}, {13242, 34512}, {14420, 10278}, {14977, 14277}, {17414, 23301}, {18311, 14279}, {31296, 17414}
X(53365) = anticomplement of X(351)
X(53365) = anticomplement of the isogonal conjugate of X(892)
X(53365) = anticomplement of the isotomic conjugate of X(53080)
X(53365) = isotomic conjugate of the anticomplement of X(38988)
X(53365) = isotomic conjugate of the isogonal conjugate of X(53272)
X(53365) = anticomplementary isogonal conjugate of X(39356)
X(53365) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 39356}, {111, 21220}, {162, 7665}, {662, 8591}, {671, 21221}, {691, 192}, {799, 14360}, {892, 8}, {897, 148}, {923, 25054}, {4593, 25052}, {5380, 1654}, {18023, 21294}, {24041, 44010}, {32729, 17486}, {34574, 17497}, {36085, 2}, {36142, 194}, {36827, 21217}, {45773, 6758}, {46277, 3448}, {52940, 7192}, {53080, 6327}
X(53365) = X(53080)-Ceva conjugate of X(2)
X(53365) = X(i)-isoconjugate of X(j) for these (i,j): {163, 25322}, {36142, 40517}
X(53365) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 25322}, {23992, 40517}, {31128, 10330}
X(53365) = crossdifference of every pair of points on line {3051, 5008}
X(53365) = barycentric product X(i)*X(j) for these {i,j}: {76, 53272}, {661, 18075}, {5466, 31128}, {38303, 52618}, {38988, 53080}
X(53365) = barycentric quotient X(i)/X(j) for these {i,j}: {523, 25322}, {690, 40517}, {18075, 799}, {31128, 5468}, {38303, 1634}, {38988, 351}, {53272, 6}
X(53365) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {351, 9148, 45689}, {351, 45689, 2}, {9131, 9191, 14417}, {9131, 14417, 9168}, {9134, 9979, 5466}

---

X(53366) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(670) AND X(889)

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

X(53366) lies on the cubic K635 and these lines: {1, 75}, {190, 4598}, {350, 19945}, {513, 27853}, {670, 888}, {714, 36817}, {876, 4639}, {885, 36803}, {889, 41314}, {903, 32035}, {3573, 4601}, {3888, 6386}, {4436, 36860}, {7035, 23343}, {9296, 24004}, {23354, 31625}

X(53366) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {889, 21221}, {898, 21220}, {4600, 39360}, {4607, 148}, {5381, 1654}, {34075, 25054}
X(53366) = X(4607)-Ceva conjugate of X(668)
X(53366) = X(i)-isoconjugate of X(j) for these (i,j): {512, 715}, {669, 18826}
X(53366) = X(i)-Dao conjugate of X(j) for these (i,j): {2229, 891}, {6381, 4728}, {39054, 715}
X(53366) = crossdifference of every pair of points on line {798, 9427}
X(53366) = barycentric product X(i)*X(j) for these {i,j}: {662, 35532}, {670, 2229}, {714, 799}, {4607, 52882}, {27853, 36817}
X(53366) = barycentric quotient X(i)/X(j) for these {i,j}: {662, 715}, {714, 661}, {799, 18826}, {2229, 512}, {35532, 1577}, {36817, 3572}, {52882, 4728}

---

X(53367) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(670) AND X(892)

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

X(53367) lies on these lines: {2, 14948}, {4, 69}, {99, 1296}, {523, 2396}, {524, 52756}, {648, 35136}, {670, 888}, {879, 43187}, {880, 14509}, {892, 5466}, {1975, 47285}, {2407, 17941}, {2854, 45809}, {4226, 4590}, {5467, 9182}, {8681, 36874}, {8782, 39361}, {9214, 22254}, {14263, 47286}, {14588, 53274}, {17008, 39576}, {18829, 53199}

X(53367) = reflection of X(2396) in X(23342)
X(53367) = anticomplement of X(21906)
X(53367) = anticomplement of the isogonal conjugate of X(52940)
X(53367) = isotomic conjugate of the anticomplement of X(21905)
X(53367) = isotomic conjugate of the isogonal conjugate of X(11634)
X(53367) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {662, 39356}, {691, 21220}, {892, 21221}, {24037, 14360}, {24041, 8591}, {34539, 17497}, {36085, 148}, {36142, 25054}, {42370, 21295}, {45773, 4560}, {52940, 8}, {53080, 21294}
X(53367) = X(i)-isoconjugate of X(j) for these (i,j): {798, 41909}, {810, 2374}, {2642, 15387}
X(53367) = X(i)-Dao conjugate of X(j) for these (i,j): {126, 512}, {3291, 690}, {6390, 14417}, {31998, 41909}, {39062, 2374}, {47286, 45689}
X(53367) = cevapoint of X(i) and X(j) for these (i,j): {512, 6719}, {524, 6131}, {9134, 47286}
X(53367) = trilinear pole of line {126, 3291}
X(53367) = crossdifference of every pair of points on line {3049, 9427}
X(53367) = barycentric product X(i)*X(j) for these {i,j}: {76, 11634}, {99, 47286}, {126, 892}, {668, 16756}, {670, 3291}, {2396, 36874}, {4590, 9134}, {5140, 52608}, {6331, 8681}
X(53367) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 41909}, {126, 690}, {648, 2374}, {691, 15387}, {892, 44182}, {2396, 36892}, {3291, 512}, {5140, 2489}, {5468, 34161}, {8681, 647}, {9134, 115}, {11634, 6}, {14263, 9178}, {16756, 513}, {17466, 2642}, {21905, 21906}, {36874, 2395}, {47286, 523}, {52881, 14417}
X(53367) = {X(17941),X(31998)}-harmonic conjugate of X(2407)

---

X(53368) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(670) AND X(903)

Barycentrics    (b - c)*(-(a^3*b) - a^2*b^2 - a^3*c + a^2*b*c - a^2*c^2 + 3*b^2*c^2) : :
X(53368) = 4 X[4374] - X[17159], 2 X[4374] + X[17217], X[17159] + 2 X[17217], 2 X[1635] - 3 X[27344], 2 X[1639] - 3 X[30020], 2 X[4406] + X[20295], 5 X[4699] - 2 X[21832], 4 X[4928] - 3 X[27292], 2 X[14408] - 3 X[27292], X[20979] - 4 X[21206], 4 X[21191] - X[21225], 3 X[30091] - X[47892]

X(53368) lies on these lines: {2, 14407}, {75, 4145}, {86, 53276}, {512, 4374}, {513, 4828}, {670, 888}, {891, 21433}, {900, 903}, {1635, 27344}, {1639, 30020}, {2776, 10446}, {3261, 9002}, {3766, 6085}, {4107, 17379}, {4406, 20295}, {4699, 21832}, {4928, 14408}, {8643, 17215}, {9029, 21304}, {9461, 32032}, {14296, 18149}, {17238, 21053}, {20906, 29226}, {20979, 21206}, {21191, 21225}, {30091, 47892}

X(53368) = reflection of X(14408) in X(4928)
X(53368) = anticomplement of X(14407)
X(53368) = anticomplement of the isogonal conjugate of X(4615)
X(53368) = isotomic conjugate of the isogonal conjugate of X(23394)
X(53368) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {81, 39349}, {88, 148}, {99, 30578}, {106, 21220}, {662, 17487}, {799, 21290}, {901, 1655}, {903, 21221}, {1414, 30577}, {3257, 1654}, {4555, 2895}, {4567, 44009}, {4591, 192}, {4615, 8}, {4622, 2}, {4634, 69}, {5376, 31290}, {9456, 25054}, {20568, 3448}, {52935, 30579}
X(53368) = crossdifference of every pair of points on line {1017, 9427}
X(53368) = barycentric product X(76)*X(23394)
X(53368) = barycentric quotient X(23394)/X(6)
X(53368) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4374, 17217, 17159}, {4928, 14408, 27292}

---

X(53369) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(670) AND X(2494)

Barycentrics    (b^2 - c^2)*(-a^6 + 2*a^4*b^2 - a^2*b^4 + 2*a^4*c^2 - 7*a^2*b^2*c^2 + 3*b^4*c^2 - a^2*c^4 + 3*b^2*c^4) : :
X(53369) = X[69] + 2 X[35522], X[193] - 4 X[24284], 4 X[2492] - 7 X[3619], 2 X[3569] - 5 X[3620], 4 X[3631] - X[39232], X[6776] - 4 X[44813]

X(53369) lies on these lines: {2, 9035}, {69, 526}, {193, 24284}, {512, 16084}, {670, 888}, {1494, 3268}, {2492, 3619}, {2780, 45807}, {3267, 8675}, {3569, 3620}, {3631, 39232}, {5207, 20403}, {6776, 44813}, {45147, 45808}

X(53369) = anticomplement of X(14398)
X(53369) = isotomic conjugate of the isogonal conjugate of X(53247)
X(53369) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {74, 21220}, {662, 39358}, {799, 146}, {1304, 21216}, {1494, 21221}, {2159, 25054}, {2349, 148}, {4593, 25045}, {16077, 5905}, {32640, 17486}, {33805, 3448}, {36034, 194}, {44769, 192}
X(53369) = crossdifference of every pair of points on line {9408, 9427}
X(53369) = barycentric product X(76)*X(53247)
X(53369) = barycentric quotient X(53247)/X(6)

---

X(53370) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(670) AND X(2481)

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

X(53370) lies on these lines: {2, 14296}, {75, 50343}, {76, 30709}, {649, 20907}, {659, 53276}, {670, 888}, {693, 2526}, {850, 4374}, {885, 2481}, {2787, 34284}, {3261, 47694}, {4010, 4441}, {4107, 24356}, {14431, 18135}, {16158, 16992}, {17159, 50520}, {17166, 52619}, {17217, 44445}, {17494, 20906}, {17496, 23807}, {20949, 47969}, {21301, 40495}, {21433, 47776}, {25301, 46402}, {27712, 50451}, {37670, 53257}

X(53370) = isotomic conjugate of the isogonal conjugate of X(53271)
X(53370) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {86, 39353}, {99, 20533}, {105, 21220}, {333, 14732}, {662, 39350}, {666, 1654}, {670, 20552}, {673, 148}, {799, 20344}, {927, 17778}, {1438, 25054}, {2481, 21221}, {4573, 52164}, {4593, 25050}, {18031, 3448}, {31637, 39352}, {34085, 2475}, {36086, 1655}, {36803, 1330}, {46135, 2893}, {51560, 2895}
X(53370) = crossdifference of every pair of points on line {9427, 39686}
X(53370) = barycentric product X(76)*X(53271)
X(53370) = barycentric quotient X(53271)/X(6)

---

X(53371) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(670) AND X(2966)

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

X(53371) lies on these lines: {2, 44114}, {66, 69}, {110, 925}, {385, 51820}, {511, 51259}, {523, 2421}, {525, 11634}, {670, 888}, {805, 877}, {826, 5118}, {879, 2966}, {2445, 7473}, {4235, 9218}, {5468, 31632}, {14977, 36827}, {14999, 53274}, {15066, 36192}, {36874, 51481}

X(53371) = anticomplement of X(44114)
X(53371) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {662, 39359}, {1101, 39355}, {2715, 21220}, {2966, 21221}, {4592, 14721}, {24041, 147}, {36036, 3448}, {36084, 148}, {41174, 21270}, {43187, 21294}
X(53371) = crossdifference of every pair of points on line {9427, 14113}
X(53371) = barycentric product X(662)*X(17901)
X(53371) = barycentric quotient X(17901)/X(1577)

---

X(53372) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(671) AND X(903)

Barycentrics    2*a^3 - a^2*b - a*b^2 - 4*b^3 - a^2*c + 5*b^2*c - a*c^2 + 5*b*c^2 - 4*c^3 : :
X(53372) = 4 X[3120] - X[4427], 2 X[3120] + X[44006], X[4427] + 2 X[44006], 4 X[4442] - X[17162], 2 X[4442] + X[17491], X[17162] + 2 X[17491], X[31301] - 4 X[50755]

X(53372) lies on these lines: {2, 846}, {79, 50234}, {519, 17953}, {524, 4442}, {671, 690}, {740, 31177}, {900, 903}, {1836, 17150}, {3006, 17132}, {3227, 29340}, {4054, 49630}, {4080, 17780}, {4141, 21241}, {4608, 35153}, {4693, 31029}, {4892, 4933}, {4956, 29824}, {5057, 37756}, {5196, 37792}, {16704, 28558}, {16816, 24710}, {17163, 31143}, {17164, 17677}, {17679, 25253}, {20045, 28562}, {21283, 50999}, {24851, 50165}, {27759, 32845}, {27812, 31144}, {29829, 35578}, {31301, 50755}, {33108, 49748}, {33134, 50128}, {33151, 49720}, {48646, 50104}

X(53372) = midpoint of X(2) and X(44006)
X(53372) = reflection of X(i) in X(j) for these {i,j}: {2, 3120}, {4141, 21241}, {4427, 2}, {4933, 4892}
X(53372) = isotomic conjugate of the isogonal conjugate of X(53316)
X(53372) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {88, 14360}, {111, 30578}, {897, 21290}, {923, 17487}, {9456, 8591}
X(53372) = crossdifference of every pair of points on line {1017, 5029}
X(53372) = barycentric product X(76)*X(53316)
X(53372) = barycentric quotient X(53316)/X(6)
X(53372) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3120, 44006, 4427}, {4080, 24715, 17780}, {4442, 17491, 17162}, {6650, 11599, 34766}

---

X(53373) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(671) AND X(2481)

Barycentrics    a^4*b - a^3*b^2 + a^2*b^3 - a*b^4 + a^4*c - 2*a^2*b^2*c - 3*b^4*c - a^3*c^2 - 2*a^2*b*c^2 + 4*a*b^2*c^2 + 3*b^3*c^2 + a^2*c^3 + 3*b^2*c^3 - a*c^4 - 3*b*c^4 : :
X(53373) = 4 X[20544] - X[25257]

X(53373) lies on these lines: {2, 2795}, {4, 2771}, {316, 17491}, {518, 20556}, {671, 690}, {885, 2481}, {2805, 16732}, {2826, 10707}, {3673, 16727}, {4236, 53261}, {4442, 17497}, {15015, 16376}, {20544, 25257}

X(53373) = isotomic conjugate of the isogonal conjugate of X(53310)
X(53373) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {111, 20533}, {671, 20552}, {673, 14360}, {897, 20344}, {923, 39350}, {1438, 8591}, {7316, 52164}
X(53373) = crossdifference of every pair of points on line {39686, 39689}
X(53373) = barycentric product X(76)*X(53310)
X(53373) = barycentric quotient X(53310)/X(6)

---

X(53374) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(671) AND X(2966)

Barycentrics    (b^2 - c^2)*(-3*a^6 + 4*a^4*b^2 - 3*a^2*b^4 + 2*b^6 + 4*a^4*c^2 - a^2*b^2*c^2 - b^4*c^2 - 3*a^2*c^4 - b^2*c^4 + 2*c^6) : :
X(53374) = 3 X[2] - 4 X[45327], 3 X[1640] - 2 X[45327], X[69] - 4 X[45801], 3 X[5466] - 2 X[34290], 3 X[9168] - 4 X[11183], 4 X[18310] - 3 X[21356], 3 X[21356] - 2 X[45808]

X(53374) lies on these lines: {2, 525}, {69, 45801}, {287, 2395}, {512, 3060}, {523, 1992}, {524, 14977}, {648, 18808}, {671, 690}, {826, 5652}, {850, 40814}, {878, 37183}, {879, 2966}, {1499, 3543}, {1637, 39905}, {2501, 37174}, {3288, 11004}, {3566, 8029}, {3906, 7757}, {5468, 17708}, {5890, 30209}, {7417, 22265}, {8371, 44552}, {9143, 32313}, {9476, 34765}, {10097, 45291}, {12073, 44554}, {18310, 21356}, {30227, 36163}, {43665, 51481}

X(53374) = reflection of X(i) in X(j) for these {i,j}: {2, 1640}, {9143, 32313}, {39905, 1637}, {45808, 18310}
X(53374) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {923, 39359}, {9154, 21294}, {36060, 14721}, {36084, 14360}, {36142, 147}
X(53374) = crossdifference of every pair of points on line {1495, 5107}
X(53374) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2394, 51227, 34767}, {18310, 45808, 21356}

---

X(53375) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(671) AND X(3225)

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

X(53375) lies on these lines: {2, 10290}, {76, 3124}, {83, 39024}, {110, 7760}, {194, 5106}, {290, 14316}, {316, 20977}, {523, 3228}, {671, 690}, {694, 698}, {3225, 5027}, {4581, 35143}, {5191, 6179}, {7751, 41273}, {7754, 20998}, {7757, 9155}, {7812, 11002}, {7827, 46906}, {7894, 20976}, {17941, 36849}, {18023, 25322}, {32515, 41143}

X(53375) = X(43761)-anticomplementary conjugate of X(14360)

---

X(53376) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(889) AND X(903)

Barycentrics    (b - c)*(-(a^3*b) - a^2*b^2 - a^3*c + 9*a^2*b*c - 4*a*b^2*c - a^2*c^2 - 4*a*b*c^2 + 3*b^2*c^2) : :
X(53376) = X[192] - 4 X[21211], X[1278] + 2 X[21143], 2 X[14408] - 3 X[27344], 3 X[14408] - 4 X[45675], 9 X[27344] - 8 X[45675]

X(53376) lies on these lines: {2, 14437}, {75, 513}, {192, 21211}, {514, 4740}, {889, 41314}, {900, 903}, {1278, 21143}, {4406, 4441}, {14408, 27344}, {17160, 23345}, {20295, 42697}, {24349, 29350}, {30998, 47779}

X(53376) = reflection of X(i) in X(j) for these {i,j}: {192, 52745}, {47780, 4406}, {52745, 21211}
X(53376) = anticomplement of X(14437)
X(53376) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {739, 39349}, {898, 30578}, {901, 39360}, {4607, 21290}, {5376, 44008}, {34075, 17487}
X(53376) = crossdifference of every pair of points on line {1017, 21760}

---

X(53377) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(889) AND X(2481)

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

X(53377) lies on these lines: {314, 7192}, {513, 4441}, {649, 3729}, {885, 2481}, {889, 41314}, {1278, 17494}, {4406, 47694}, {4671, 47763}, {6161, 34284}, {11185, 21301}, {31130, 47805}

X(53377) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {889, 20552}, {898, 20533}, {4607, 20344}, {34075, 39350}, {36086, 39360}, {37129, 39353}

---

X(53378) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(892) AND X(2494)

Barycentrics    (b^2 - c^2)*(5*a^6 - 8*a^4*b^2 + a^2*b^4 + 2*b^6 - 8*a^4*c^2 + 15*a^2*b^2*c^2 - 5*b^4*c^2 + a^2*c^4 - 5*b^2*c^4 + 2*c^6) : :
X(53378) = 3 X[69] - 2 X[45808], 3 X[5032] - 4 X[45327], X[11008] - 4 X[45801], 2 X[18311] - 3 X[21356]

X(53378) lies on these lines: {69, 523}, {193, 1640}, {340, 18808}, {524, 14977}, {525, 11160}, {690, 16093}, {850, 44133}, {892, 5466}, {1494, 3268}, {1499, 5921}, {1637, 35279}, {2394, 12066}, {2799, 11161}, {5032, 45327}, {9209, 37667}, {11008, 45801}, {18311, 21356}, {30474, 37668}, {46608, 52437}

X(53378) = reflection of X(193) in X(1640)
X(53378) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2159, 39356}, {9139, 21221}, {36034, 8591}, {36085, 146}, {36131, 7665}, {36142, 39358}
X(53378) = crossdifference of every pair of points on line {1692, 9408}

---

X(53379) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(892) AND X(2966)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^6 - a^4*b^2 - b^6 - a^4*c^2 + a^2*b^2*c^2 + b^4*c^2 + b^2*c^4 - c^6) : :
X(53379) = X[41724] - 4 X[47242], 3 X[249] - X[47288], 2 X[7472] - 3 X[9218], 3 X[8859] - 2 X[32225], 3 X[9144] - 2 X[38395], 3 X[15035] - 2 X[46634], 5 X[15051] - 4 X[46987], 3 X[15055] - 4 X[46981], 3 X[21445] - 2 X[32110], 2 X[31854] + X[38679]

X(53379) lies on these lines: {2, 51429}, {30, 22265}, {67, 524}, {74, 46633}, {98, 385}, {99, 3906}, {110, 476}, {249, 826}, {325, 30789}, {450, 38294}, {512, 15342}, {525, 7472}, {542, 36173}, {648, 53199}, {690, 691}, {879, 2966}, {892, 5466}, {1078, 45284}, {1499, 47291}, {1550, 3564}, {2421, 53266}, {2452, 52693}, {2770, 46131}, {2782, 46298}, {3448, 51428}, {3580, 16315}, {4563, 14607}, {4576, 4590}, {7760, 35146}, {7779, 31127}, {7840, 13857}, {7927, 33803}, {7998, 40879}, {8859, 32225}, {9144, 38395}, {9146, 9182}, {10330, 33799}, {10425, 14734}, {10557, 17948}, {11005, 16188}, {14295, 43187}, {14916, 35087}, {15035, 46634}, {15051, 46987}, {15055, 46981}, {15066, 36207}, {15360, 22329}, {15993, 41721}, {16278, 36174}, {18304, 35930}, {21445, 32110}, {23105, 35139}, {31854, 38679}, {32515, 37477}, {32662, 39138}, {32729, 33919}

X(53379) = midpoint of X(15342) and X(47290)
X(53379) = reflection of X(i) in X(j) for these {i,j}: {74, 46633}, {99, 9181}, {110, 14999}, {3448, 51428}, {3580, 16315}, {7840, 13857}, {9140, 16092}, {11005, 16188}, {15360, 22329}, {36174, 16278}, {41721, 15993}
X(53379) = anticomplement of X(51429)
X(53379) = X(36142)-anticomplementary conjugate of X(39359)
X(53379) = X(i)-Dao conjugate of X(j) for these (i,j): {5026, 11183}, {11646, 45327}
X(53379) = barycentric product X(99)*X(11646)
X(53379) = barycentric quotient X(11646)/X(523)
X(53379) = {X(4563),X(31998)}-harmonic conjugate of X(14607)

---

X(53380) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(903) AND X(1494)

Barycentrics    2*a^5 - a^4*b - a^3*b^2 + 5*a^2*b^3 - a*b^4 - 4*b^5 - a^4*c - 4*a^2*b^2*c + 5*b^4*c - a^3*c^2 - 4*a^2*b*c^2 + 2*a*b^2*c^2 - b^3*c^2 + 5*a^2*c^3 - b^2*c^3 - a*c^4 + 5*b*c^4 - 4*c^5 : :
X(53380) = 4 X[4466] - X[14543], X[18661] - 4 X[41804]

X(53380) lies on these lines: {2, 1762}, {7, 3065}, {30, 18661}, {150, 41803}, {519, 3007}, {900, 903}, {1494, 3268}, {1565, 41801}, {2398, 24713}, {2822, 3543}, {3655, 17221}, {3668, 20289}, {3945, 15903}, {4329, 34632}, {15936, 41003}, {17134, 34628}, {17183, 17195}, {17220, 31162}, {20291, 34638}, {21270, 34627}, {21271, 34718}, {31153, 53043}, {39470, 45341}

X(53380) = reflection of X(i) in X(j) for these {i,j}: {2, 4466}, {14543, 2}
X(53380) = isotomic conjugate of the isogonal conjugate of X(53254)
X(53380) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {74, 30578}, {88, 146}, {2159, 17487}, {2349, 21290}, {9456, 39358}
X(53380) = crossdifference of every pair of points on line {1017, 9408}
X(53380) = barycentric product X(76)*X(53254)
X(53380) = barycentric quotient X(53254)/X(6)

---

X(53381) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(903) AND X(2481)

Barycentrics    a^3*b - a*b^3 + a^3*c - 4*a^2*b*c + 2*a*b^2*c - 3*b^3*c + 2*a*b*c^2 + 6*b^2*c^2 - a*c^3 - 3*b*c^3 : :
X(53381) = 4 X[1111] - X[21272]

X(53381) lies on these lines: {2, 14439}, {7, 149}, {75, 3952}, {320, 17145}, {392, 20880}, {518, 13576}, {885, 2481}, {900, 903}, {908, 1266}, {1086, 24403}, {1111, 2802}, {1642, 53241}, {3120, 3663}, {3672, 33148}, {3673, 20244}, {4089, 21630}, {4346, 4392}, {4389, 33108}, {4441, 4485}, {4887, 17449}, {5883, 7264}, {17050, 25261}, {17136, 24203}, {17147, 30985}, {17160, 39744}, {17164, 33940}, {17320, 27811}, {17753, 20247}, {17777, 31995}, {17895, 51378}, {18061, 25272}, {24352, 24596}, {28516, 43534}, {30578, 52709}, {30997, 42720}, {36887, 50891}

X(53381) = anticomplement of X(14439)
X(53381) = isotomic conjugate of the isogonal conjugate of X(53307)
X(53381) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {88, 20344}, {105, 30578}, {106, 20533}, {673, 21290}, {903, 20552}, {919, 44009}, {1416, 30577}, {1438, 17487}, {9456, 39350}, {23345, 39353}, {43929, 39349}
X(53381) = crossdifference of every pair of points on line {1017, 39686}
X(53381) = barycentric product X(76)*X(53307)
X(53381) = barycentric quotient X(53307)/X(6)

---

X(53382) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(1121) AND X(2481)

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

X(53382) lies on these lines: {2, 10186}, {8, 80}, {85, 35312}, {518, 30807}, {885, 2481}, {1121, 6366}, {1441, 10177}, {2398, 14942}, {3912, 31058}, {4124, 13576}, {4384, 4781}, {7253, 14616}, {14004, 17150}, {17163, 50095}, {17755, 41845}, {21029, 41006}, {25935, 31043}, {29824, 30806}

X(53382) = anticomplement of X(35293)
X(53382) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1121, 20552}, {1156, 20344}, {1438, 39357}, {2291, 20533}, {23351, 14732}, {32735, 45290}, {34068, 39350}
X(53382) = crossdifference of every pair of points on line {21758, 39686}

---

X(53383) = INTERSECTION OF LINES TANGENT TO STEINER CIRCUMELLIPSE AT X(1494) AND X(2966)

Barycentrics    (b^2 - c^2)*(3*a^8 - 5*a^6*b^2 + 3*a^4*b^4 - 3*a^2*b^6 + 2*b^8 - 5*a^6*c^2 + 5*a^4*b^2*c^2 + a^2*b^4*c^2 - b^6*c^2 + 3*a^4*c^4 + a^2*b^2*c^4 - 2*b^4*c^4 - 3*a^2*c^6 - b^2*c^6 + 2*c^8) : :
X(53383) = 3 X[5466] - 4 X[53266], 3 X[4] - 4 X[39491], X[20] + 2 X[5489], X[43673] - 4 X[53173], X[18808] - 4 X[43083], 5 X[631] - 4 X[45681], 3 X[3524] - 2 X[5664], 3 X[3545] - 4 X[14566]

X(53383) lies on these lines: {2, 523}, {4, 39491}, {20, 5489}, {30, 2394}, {98, 1297}, {376, 525}, {476, 1304}, {512, 15305}, {520, 2979}, {631, 45681}, {647, 22240}, {684, 31127}, {879, 2966}, {1494, 3268}, {1550, 36875}, {1650, 12079}, {2395, 36899}, {3265, 37668}, {3524, 5664}, {3543, 42733}, {3545, 14566}, {6030, 6368}, {6055, 14223}, {7493, 47194}, {7527, 53330}, {8057, 23616}, {9003, 13169}, {9007, 11160}, {9123, 14697}, {9774, 23878}, {10298, 39201}, {10419, 15421}, {14380, 45289}, {33294, 37667}, {34765, 40428}, {36900, 47383}, {44202, 44427}, {44891, 47248}

X(53383) = reflection of X(i) in X(j) for these {i,j}: {376, 18556}, {3543, 42733}, {14223, 6055}, {44427, 44202}
X(53383) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2159, 39359}, {35200, 14721}, {36034, 147}, {36084, 146}, {36131, 40867}
X(53383) = crossdifference of every pair of points on line {187, 9408}
X(53383) = {X(14977),X(16092)}-harmonic conjugate of X(5466)

---

X(53384) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(110) AND X(1113)

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

X(53384) lies on the Kiepert parabola, the cubics K658, K723, and K1320, and these lines: {3, 2575}, {25, 46629}, {110, 351}, {183, 22340}, {523, 1113}, {647, 52131}, {1114, 3233}, {1344, 15928}, {1822, 23189}, {4230, 39298}, {9174, 11284}, {13415, 44889}, {14380, 15461}, {14966, 52132}, {15154, 38581}, {34212, 41941}, {44123, 46953}

X(53384) = isogonal conjugate of X(53153)
X(53384) = isotomic conjugate of the polar conjugate of X(52131)
X(53384) = isogonal conjugate of the polar conjugate of X(50944)
X(53384) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 8115}, {39298, 6}, {50944, 52131}
X(53384) = X(i)-isoconjugate of X(j) for these (i,j): {1, 53153}, {19, 50945}, {92, 52132}, {162, 1313}, {811, 44126}, {823, 15166}, {1114, 2588}, {1577, 41942}, {2574, 2587}, {2577, 2592}, {2578, 46812}, {2581, 8105}, {15460, 24006}
X(53384) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 53153}, {6, 50945}, {125, 1313}, {2575, 523}, {8106, 14618}, {15167, 2592}, {17423, 44126}, {22391, 52132}, {38999, 14499}, {46811, 850}
X(53384) = cevapoint of X(647) and X(42667)
X(53384) = trilinear pole of line {3269, 15167}
X(53384) = crossdifference of every pair of points on line {115, 1313}
X(53384) = barycentric product X(i)*X(j) for these {i,j}: {3, 50944}, {69, 52131}, {99, 15167}, {394, 53154}, {525, 15461}, {1113, 46811}, {1312, 4558}, {1822, 2583}, {2575, 8115}, {2580, 2585}, {3265, 41941}, {4563, 44125}, {14500, 44769}, {23110, 39299}, {42667, 46813}
X(53384) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 50945}, {6, 53153}, {184, 52132}, {647, 1313}, {1113, 46812}, {1312, 14618}, {1576, 41942}, {1636, 14499}, {1822, 2581}, {2575, 2592}, {2576, 2587}, {2579, 2588}, {2585, 2582}, {3049, 44126}, {8115, 15165}, {14500, 41079}, {15167, 523}, {15461, 648}, {32661, 15460}, {39201, 15166}, {41941, 107}, {42667, 8105}, {44125, 2501}, {46811, 22339}, {50944, 264}, {52131, 4}, {53154, 2052}

---

X(53385) = INTERSECTION OF LINES TANGENT TO CIRCUMCIRCLE AT X(110) AND X(1114)

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

X(53385) lies on the Kiepert parabola, the cubics K658, K723, and K1320, and these lines: {3, 2574}, {25, 46628}, {110, 351}, {183, 22339}, {523, 1114}, {647, 52132}, {1113, 3233}, {1345, 15928}, {1823, 23189}, {4230, 39299}, {9173, 11284}, {13414, 44889}, {14380, 15460}, {14966, 52131}, {15155, 38581}, {34212, 41942}, {44124, 46953}

X(53385) = isogonal conjugate of X(53154)
X(53385) = isotomic conjugate of the polar conjugate of X(52132)
X(53385) = isogonal conjugate of the polar conjugate of X(50945)
X(53385) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 8116}, {39299, 6}, {50945, 52132}
X(53385) = X(i)-isoconjugate of X(j) for these (i,j): {1, 53154}, {19, 50944}, {92, 52131}, {162, 1312}, {811, 44125}, {823, 15167}, {1113, 2589}, {1577, 41941}, {2575, 2586}, {2576, 2593}, {2579, 46815}, {2580, 8106}, {15461, 24006}
X(53385) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 53154}, {6, 50944}, {125, 1312}, {2574, 523}, {8105, 14618}, {15166, 2593}, {17423, 44125}, {22391, 52131}, {38999, 14500}, {46814, 850}
X(53385) = cevapoint of X(647) and X(42668)
X(53385) = trilinear pole of line {3269, 15166}
X(53385) = crossdifference of every pair of points on line {115, 1312}
X(53385) = barycentric product X(i)*X(j) for these {i,j}: {3, 50945}, {69, 52132}, {99, 15166}, {394, 53153}, {525, 15460}, {1114, 46814}, {1313, 4558}, {1823, 2582}, {2574, 8116}, {2581, 2584}, {3265, 41942}, {4563, 44126}, {14499, 44769}, {23109, 39298}, {42668, 46810}
X(53385) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 50944}, {6, 53154}, {184, 52131}, {647, 1312}, {1114, 46815}, {1313, 14618}, {1576, 41941}, {1636, 14500}, {1823, 2580}, {2574, 2593}, {2577, 2586}, {2578, 2589}, {2584, 2583}, {3049, 44125}, {8116, 15164}, {14499, 41079}, {15166, 523}, {15460, 648}, {32661, 15461}, {39201, 15167}, {41942, 107}, {42668, 8106}, {44126, 2501}, {46814, 22340}, {50945, 264}, {52132, 4}, {53153, 2052}

---

X(53386) = INTERSECTION OF LINES TANGENT TO CIRCUMCONIC {{A,B,C,X(4),X(5)}} AT X(4) AND X(53)

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

X(5386) lies on these lines: {4, 54}, {5, 12012}, {25, 2934}, {51, 53}, {93, 41598}, {107, 38833}, {125, 42400}, {132, 137}, {140, 31505}, {233, 3078}, {324, 41586}, {381, 26898}, {393, 15004}, {418, 36412}, {467, 39530}, {1209, 35719}, {1493, 35887}, {1598, 52014}, {1859, 3611}, {1990, 34565}, {3199, 40588}, {5650, 37873}, {5651, 37192}, {6525, 34417}, {6748, 13366}, {6749, 44111}, {6756, 52144}, {6819, 22112}, {6995, 9752}, {7408, 9993}, {7576, 16337}, {8254, 15557}, {8754, 47328}, {11197, 52887}, {14129, 30506}, {16240, 24862}, {16265, 18402}, {32428, 34836}, {42650, 51513}

X(53386) = polar conjugate of X(31617)
X(53386) = isogonal conjugate of the isotomic conjugate of X(14978)
X(53386) = polar conjugate of the isotomic conjugate of X(233)
X(53386) = orthic-isogonal conjugate of X(6748)
X(53386) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 6748}, {107, 12077}, {14978, 233}
X(53386) = X(i)-isoconjugate of X(j) for these (i,j): {48, 31617}, {63, 288}, {75, 20574}, {255, 39286}, {662, 39181}, {2167, 31626}, {2169, 40410}
X(53386) = X(i)-Dao conjugate of X(j) for these (i,j): {140, 69}, {206, 20574}, {233, 34386}, {1084, 39181}, {1249, 31617}, {3162, 288}, {6523, 39286}, {14363, 40410}, {33549, 95}, {35442, 3265}, {40588, 31626}
X(53386) = crossdifference of every pair of points on line {17434, 39181}
X(53386) = barycentric product X(i)*X(j) for these {i,j}: {4, 233}, {5, 6748}, {6, 14978}, {51, 40684}, {53, 140}, {107, 35441}, {216, 44732}, {275, 3078}, {324, 13366}, {523, 35318}, {1232, 3199}, {2052, 32078}, {2181, 20879}, {3087, 31505}, {12077, 35311}, {13450, 22052}, {23290, 35324}
X(53386) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 31617}, {25, 288}, {32, 20574}, {51, 31626}, {53, 40410}, {140, 34386}, {233, 69}, {393, 39286}, {512, 39181}, {3078, 343}, {3199, 1173}, {6748, 95}, {13366, 97}, {14569, 39284}, {14978, 76}, {32078, 394}, {35318, 99}, {35441, 3265}, {40684, 34384}, {44732, 276}, {51513, 39183}
X(53386) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 275, 35884}, {4, 6750, 3574}, {4, 8884, 35717}, {53, 6755, 51}, {3078, 32078, 233}, {8884, 35717, 10619}, {14129, 30506, 39569}, {42400, 52280, 125}

---

X(53387) = INTERSECTION OF LINES TANGENT TO CIRCUMCONIC {{A,B,C,X(4),X(5)}} AT X(25) AND X(37)

Barycentrics    a^2*(b + c)*(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(53387) lies on these lines: {10, 37}, {25, 32}, {30, 16716}, {39, 405}, {115, 429}, {187, 2915}, {213, 21813}, {232, 30733}, {406, 3767}, {442, 3290}, {574, 37246}, {1015, 1104}, {1194, 37325}, {1333, 20831}, {1334, 20966}, {1506, 37315}, {1574, 32777}, {1901, 20227}, {2176, 10974}, {2238, 17742}, {2242, 27802}, {2245, 14974}, {2333, 23664}, {3122, 40978}, {3230, 22076}, {3291, 4239}, {4204, 21838}, {4261, 11108}, {4426, 23447}, {5051, 26242}, {5283, 37314}, {5336, 7106}, {5452, 20970}, {7747, 37398}, {16502, 39690}, {17053, 18591}, {21530, 23537}, {33843, 37318}, {35075, 40610}, {40728, 40954}, {40969, 40982}

X(53387) = X(i)-complementary conjugate of X(j) for these (i,j): {1918, 15487}, {36907, 626}, {40188, 21240}
X(53387) = X(i)-Ceva conjugate of X(j) for these (i,j): {1783, 512}, {40941, 40973}
X(53387) = X(86)-isoconjugate of X(40406)
X(53387) = X(i)-Dao conjugate of X(j) for these (i,j): {18210, 15413}, {21530, 274}, {40600, 40406}, {40941, 305}
X(53387) = crossdifference of every pair of points on line {3265, 3733}
X(53387) = barycentric product X(i)*X(j) for these {i,j}: {1, 40973}, {19, 18674}, {25, 21530}, {37, 40941}, {42, 23537}, {512, 53349}, {523, 53282}, {1474, 21678}, {1824, 18732}, {1843, 18709}, {2333, 18651}
X(53387) = barycentric quotient X(i)/X(j) for these {i,j}: {213, 40406}, {18674, 304}, {21530, 305}, {21678, 40071}, {23537, 310}, {40941, 274}, {40973, 75}, {53282, 99}, {53349, 670}
X(53387) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 4205, 16589}, {3230, 23639, 22076}

---

X(53388) = INTERSECTION OF LINES TANGENT TO CIRCUMCONIC WITH PERSPECTOR X(9) AT X(25) AND X(37)

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

X(53388) lies on these lines: {1, 6596}, {9, 2648}, {55, 17194}, {78, 40081}, {100, 109}, {165, 2836}, {522, 4427}, {601, 8715}, {643, 4612}, {902, 5853}, {906, 1018}, {1054, 46409}, {1292, 1293}, {1707, 2900}, {1771, 11517}, {2743, 39635}, {3158, 3550}, {3576, 34139}, {4069, 4571}, {4421, 45729}, {4432, 4939}, {5856, 17724}, {6600, 37540}, {6737, 22361}, {8616, 24392}, {13486, 35339}, {17059, 24542}, {21362, 53279}, {23693, 44425}, {23704, 35341}, {24388, 35263}, {32486, 48713}, {44669, 52680}

X(53388) = X(4567)-Ceva conjugate of X(9)
X(53388) = X(i)-isoconjugate of X(j) for these (i,j): {513, 17097}, {4017, 40430}, {7649, 40442}
X(53388) = X(i)-Dao conjugate of X(j) for these (i,j): {5745, 4077}, {17056, 693}, {21044, 16732}, {34961, 40430}, {37836, 4017}, {39026, 17097}
X(53388) = trilinear pole of line {2646, 21748}
X(53388) = crossdifference of every pair of points on line {2170, 51642}
X(53388) = barycentric product X(i)*X(j) for these {i,j}: {9, 17136}, {21, 22003}, {99, 21811}, {100, 5745}, {190, 2646}, {312, 53324}, {643, 17056}, {644, 3664}, {645, 2650}, {651, 6737}, {662, 21677}, {668, 21748}, {1332, 40950}, {4612, 21674}, {4636, 42708}, {5546, 18698}, {6335, 22361}
X(53388) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 17097}, {906, 40442}, {2646, 514}, {2650, 7178}, {3664, 24002}, {5546, 40430}, {5745, 693}, {6737, 4391}, {17056, 4077}, {17136, 85}, {21677, 1577}, {21748, 513}, {21811, 523}, {22003, 1441}, {22361, 905}, {40950, 17924}, {53324, 57}
X(53388) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {100, 109, 35338}, {100, 1331, 4551}

---

X(53389) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1054) AND X(1768)

Barycentrics    a*(a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + a^4*c - 6*a^3*b*c + 2*a^2*b^2*c + 4*a*b^3*c - b^4*c + a^3*c^2 + 2*a^2*b*c^2 - 6*a*b^2*c^2 + b^3*c^2 - a^2*c^3 + 4*a*b*c^3 + b^2*c^3 - a*c^4 - b*c^4) : :
X(53389) = 3 X[165] - 2 X[8683]

X(53389) lies on these lines: {1, 23845}, {3, 53388}, {9, 24003}, {31, 57}, {40, 104}, {46, 2217}, {63, 3952}, {100, 53298}, {165, 8683}, {513, 5400}, {537, 3928}, {659, 1054}, {900, 1768}, {1155, 22313}, {1357, 28353}, {1697, 17460}, {2390, 49997}, {2841, 32486}, {2948, 13868}, {3035, 21362}, {3218, 17154}, {3220, 9364}, {3916, 22306}, {3937, 4551}, {3942, 24025}, {4414, 14752}, {5437, 51435}, {7155, 16574}, {12514, 34587}, {17596, 18163}, {20999, 23703}, {22344, 37558}, {23169, 35059}, {24237, 26095}, {31849, 33810}, {34139, 41343}, {34583, 53280}

X(53389) = reflection of X(1) in X(53303)
X(53389) = X(21362)-Dao conjugate of X(21272)

---

X(53390) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1054) AND X(2948)

Barycentrics    a*(b - c)*(3*a^3 + 2*a^2*b - a*b^2 + 2*a^2*c + a*b*c - 3*b^2*c - a*c^2 - 3*b*c^2) : :
X(53390) = X[40] - 4 X[44812], X[1019] + 2 X[50349], 2 X[3733] + X[50346], X[21385] + 2 X[53314]

X(53390) lies on these lines: {1, 4145}, {9, 14407}, {40, 2776}, {512, 2959}, {513, 5131}, {522, 8643}, {526, 2948}, {659, 1054}, {900, 15015}, {1019, 50349}, {1420, 30572}, {1449, 21832}, {1635, 3738}, {3247, 5029}, {3340, 51646}, {3731, 53289}, {3733, 50346}, {4151, 47845}, {9002, 21173}, {10436, 53368}, {21385, 53314}, {25590, 53276}, {29226, 48281}

X(53390) = reflection of X(1) in X(53315)
X(53390) = crossdifference of every pair of points on line {3723, 28282}

---

X(53391) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1054) AND X(5540)

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

X(53991) lies on these lines: {1, 4557}, {2, 3882}, {6, 16726}, {9, 75}, {19, 17906}, {40, 9519}, {44, 20367}, {57, 88}, {83, 18700}, {105, 3939}, {142, 2347}, {192, 29380}, {193, 29552}, {374, 37597}, {573, 7397}, {579, 24604}, {659, 1054}, {1018, 4422}, {1020, 5723}, {1026, 25048}, {1052, 2640}, {1086, 21362}, {1654, 29492}, {1697, 5047}, {1706, 13740}, {1724, 37241}, {1730, 4383}, {1731, 37787}, {1764, 37679}, {2082, 8257}, {2170, 16578}, {2183, 3008}, {2269, 6666}, {2802, 25097}, {3030, 28353}, {3187, 39698}, {3210, 29531}, {3216, 4259}, {3219, 30579}, {3227, 39970}, {3271, 35338}, {3333, 44858}, {3663, 29740}, {4069, 14839}, {4271, 17337}, {4279, 7290}, {4416, 29812}, {4436, 16482}, {4551, 27628}, {4674, 15906}, {4965, 28850}, {5053, 11349}, {5272, 53312}, {5278, 16729}, {5400, 33811}, {5437, 24685}, {5540, 6084}, {9317, 21222}, {9623, 19243}, {10436, 27311}, {10472, 17259}, {16574, 17349}, {16610, 51413}, {17067, 52896}, {17300, 29439}, {17302, 29429}, {17348, 21061}, {17364, 29749}, {17490, 29529}, {18150, 29483}, {21361, 24789}, {21384, 33792}, {22313, 23404}, {23638, 28250}, {24004, 29541}, {24029, 37771}, {26007, 36949}, {29395, 29484}, {29446, 29509}, {29698, 48627}, {46196, 49731}

X(53391) = reflection of X(1) in X(53307)
X(53391) = X(7192)-Ceva conjugate of X(1)
X(53391) = X(1018)-Dao conjugate of X(3952)
X(53391) = barycentric product X(i)*X(j) for these {i,j}: {75, 23404}, {81, 22031}, {86, 22313}, {100, 23810}
X(53391) = barycentric quotient X(i)/X(j) for these {i,j}: {22031, 321}, {22313, 10}, {23404, 1}, {23810, 693}

---

X(53392) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1054) AND X(5541)

Barycentrics    a*(b - c)*(a^3 + 2*a^2*b + a*b^2 + 2*a^2*c - 5*a*b*c - b^2*c + a*c^2 - b*c^2) : :
X(53392) = 3 X[48281] - 4 X[48330], 3 X[48307] - 2 X[48333]

X(53392) lies on these lines: {1, 4491}, {9, 3768}, {40, 2827}, {484, 513}, {659, 1054}, {900, 5541}, {1019, 28209}, {1027, 17954}, {1449, 8632}, {1769, 2832}, {3737, 6363}, {3738, 13256}, {3751, 47330}, {3882, 4427}, {4040, 9002}, {4498, 6006}, {4778, 48144}, {6371, 48336}, {9001, 48111}, {14408, 16676}, {16670, 23650}, {17270, 21303}, {28220, 48320}, {28225, 48064}, {48281, 48330}, {48307, 48333}

X(53392) = reflection of X(1) in X(4491)
X(53392) = X(21385)-Dao conjugate of X(21297)
X(53392) = barycentric product X(57)*X(27545)
X(53392) = barycentric quotient X(27545)/X(312)

---

X(53393) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1054) AND X(21381)

Barycentrics    a*(a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + a^4*c - 4*a^3*b*c + 2*a*b^3*c - b^4*c + a^3*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(53393) lies on these lines: {1, 18174}, {40, 14663}, {57, 1929}, {63, 17155}, {654, 1768}, {659, 1054}, {846, 18163}, {1283, 53388}, {1730, 4650}, {1756, 35466}, {1764, 7262}, {2810, 28353}, {3338, 6126}, {3937, 27628}, {4362, 29529}, {5119, 11525}, {5272, 53261}, {6089, 12078}, {7202, 34977}, {7289, 16793}, {9355, 33811}, {16570, 24310}, {17719, 21362}, {22313, 23832}, {29531, 29649}

X(53393) = X(4581)-Ceva conjugate of X(1)
X(53393) = X(3882)-Dao conjugate of X(53332)
X(53393) = {X(18191),X(53280)}-harmonic conjugate of X(1)

---

X(53394) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1054) AND X(39156)

Barycentrics    a*(a^4 - a^3*b + 2*a^2*b^2 - a*b^3 - b^4 - a^3*c - 3*a^2*b*c + a*b^2*c + 3*b^3*c + 2*a^2*c^2 + a*b*c^2 - 4*b^2*c^2 - a*c^3 + 3*b*c^3 - c^4) : :
X(53394) = 3 X[165] - 2 X[3939], 3 X[1699] - 4 X[17059]

X(53394) lies on these lines: {1, 1633}, {9, 19593}, {40, 2810}, {57, 3271}, {63, 3888}, {165, 3939}, {269, 2195}, {513, 2957}, {527, 18788}, {659, 1054}, {926, 39156}, {1046, 4259}, {1282, 35338}, {1699, 17059}, {1721, 18725}, {1742, 2876}, {1768, 3738}, {2316, 53056}, {2834, 24715}, {3000, 7291}, {3220, 5018}, {3361, 52879}, {4266, 17596}, {7077, 44421}, {7083, 41777}, {7202, 38530}, {9441, 34371}, {18735, 44670}

X(53394) = reflection of X(i) in X(j) for these {i,j}: {1, 53298}, {9355, 16560}
X(53394) = excentral isogonal conjugate of X(28589)
X(53394) = {X(1633),X(3942)}-harmonic conjugate of X(1)

---

X(53395) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1282) AND X(1768)

Barycentrics    a*(a - b - c)*(b - c)*(a^4 + a^3*b - a^2*b^2 - a*b^3 + a^3*c + 3*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(53395) = 3 X[165] - 2 X[53278], 3 X[3576] - 4 X[52726]

X(53395) lies on these lines: {1, 53286}, {9, 3716}, {40, 3887}, {57, 2254}, {63, 53343}, {165, 53278}, {513, 5536}, {522, 649}, {659, 926}, {900, 1768}, {1019, 6362}, {1021, 50347}, {1697, 4895}, {2488, 3737}, {2648, 23838}, {3333, 3960}, {3576, 52726}, {3601, 8648}, {3738, 13256}, {3803, 3900}, {4040, 8676}, {5437, 25380}, {6366, 21385}, {7290, 22384}, {8641, 48307}, {15280, 24719}, {23057, 37556}, {50342, 52305}, {53280, 53388}

X(53395) = reflection of X(i) in X(j) for these {i,j}: {1, 53286}, {24719, 15280}

---

X(53396) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1282) AND X(5540)

Barycentrics    a*(b - c)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c - 5*a^2*b*c + 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(53396) = X[21385] + 2 X[22108]

X(53396) lies on these lines: {1, 53287}, {9, 812}, {40, 2820}, {57, 1635}, {63, 47776}, {200, 53284}, {484, 513}, {649, 28878}, {657, 4498}, {659, 926}, {667, 9443}, {1021, 47890}, {1024, 3512}, {3305, 21297}, {3882, 14543}, {4384, 20950}, {4728, 7308}, {4762, 21390}, {4763, 5437}, {4928, 51780}, {5540, 6084}, {8645, 10389}, {37560, 38327}

X(53396) = midpoint of X(657) and X(4498)
X(53396) = reflection of X(1) in X(53287)

---

X(53397) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1282) AND X(5541)

Barycentrics    a*(a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + a^4*c + 2*a^3*b*c - 6*a^2*b^2*c + 4*a*b^3*c - b^4*c + a^3*c^2 - 6*a^2*b*c^2 + 2*a*b^2*c^2 + b^3*c^2 - a^2*c^3 + 4*a*b*c^3 + b^2*c^3 - a*c^4 - b*c^4) : :
X(53397) = 3 X[165] - 2 X[53296]

X(53397) lies on these lines: {1, 4557}, {8, 3882}, {9, 4432}, {40, 2801}, {63, 4781}, {100, 53298}, {165, 53296}, {518, 20367}, {519, 2183}, {528, 21362}, {659, 926}, {900, 5541}, {1697, 2310}, {1730, 41711}, {2250, 36921}, {2274, 42079}, {2810, 35338}, {3294, 51034}, {3751, 9016}, {3961, 18163}, {4659, 4692}, {4863, 21361}, {5531, 33811}, {7174, 52029}, {16549, 47359}, {16676, 31393}, {17435, 20692}, {23703, 23858}, {29812, 49505}

X(53397) = reflection of X(1) in X(4557)

---

X(53398) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1282) AND X(9904)

Barycentrics    a*(a - b - c)*(b - c)*(a^4 - 3*a^3*b - a^2*b^2 + 3*a*b^3 - 3*a^3*c - 5*a^2*b*c + 3*a*b^2*c + b^3*c - a^2*c^2 + 3*a*b*c^2 - 2*b^2*c^2 + 3*a*c^3 + b*c^3) : :
X(53398) = 3 X[165] - 2 X[53301], 3 X[3576] - 4 X[44827]

X(53398) lies on these lines: {1, 53249}, {40, 2774}, {165, 53301}, {521, 2526}, {522, 47698}, {526, 9904}, {659, 926}, {661, 35057}, {2785, 11523}, {3243, 4458}, {3576, 44827}, {3737, 4524}, {3886, 53336}, {3900, 47959}, {3984, 53334}, {4707, 41863}, {5531, 8674}, {6366, 47680}, {11518, 30574}, {11520, 53356}

X(53398) = reflection of X(1) in X(53249)

---

X(53399) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1282) AND X(13174)

Barycentrics    a*(a^5*b^2 + a^4*b^3 - a^3*b^4 - a^2*b^5 + 2*a^4*b^2*c - 2*a^3*b^3*c + a^5*c^2 + 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 - b^5*c^2 + a^4*c^3 - 2*a^3*b*c^3 - 2*a^2*b^2*c^3 + 4*a*b^3*c^3 + b^4*c^3 - a^3*c^4 + 2*a*b^2*c^4 + b^3*c^4 - a^2*c^5 - b^2*c^5) : :
X(53399) = 3 X[165] - 2 X[53259]

X(53399) lies on these lines: {1, 16683}, {9, 4368}, {40, 2784}, {63, 3886}, {165, 53259}, {295, 35338}, {516, 20785}, {659, 926}, {740, 1755}, {804, 13174}, {846, 18163}, {968, 14752}, {1281, 3882}, {3685, 20610}, {3755, 53129}, {5541, 20375}, {8844, 35104}, {13576, 46148}

X(53399) = reflection of X(1) in X(53268)

---

X(53400) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1282) AND X(21381)

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

X(53400) lies on these lines: {1, 53309}, {9, 4010}, {40, 4730}, {57, 9508}, {63, 50343}, {200, 53257}, {523, 1021}, {659, 926}, {676, 22108}, {832, 50358}, {3333, 14419}, {3738, 13258}, {3870, 16158}, {4063, 8678}, {4784, 8672}, {4922, 6762}, {6089, 12078}, {7662, 21390}, {21385, 29240}

---

X(53401) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1282) AND X(34464)

Barycentrics    a*(a - b - c)*(b - c)*(3*a^4 - a^3*b - 3*a^2*b^2 + a*b^3 - a^3*c + a^2*b*c + a*b^2*c + 3*b^3*c - 3*a^2*c^2 + a*b*c^2 - 6*b^2*c^2 + a*c^3 + 3*b*c^3) : :
X(53401) = X[47970] + 2 X[48387], X[21385] + 2 X[53285]

X(53401) lies on these lines: {165, 513}, {514, 3576}, {522, 3158}, {659, 926}, {663, 7962}, {1420, 21105}, {1459, 5573}, {1635, 3738}, {2832, 14414}, {3601, 21132}, {3737, 18163}, {4040, 5119}, {4724, 35445}, {5010, 47970}, {21385, 53285}, {36636, 43924}

---

X(53402) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1768) AND X(9860)

Barycentrics    a*(a^6*b - a^5*b^2 - 2*a^4*b^3 + 2*a^3*b^4 + a^2*b^5 - a*b^6 + a^6*c + a^4*b^2*c + 2*a^3*b^3*c - 5*a^2*b^4*c + 2*a*b^5*c - b^6*c - a^5*c^2 + a^4*b*c^2 - 6*a^3*b^2*c^2 + 4*a^2*b^3*c^2 + 3*a*b^4*c^2 - b^5*c^2 - 2*a^4*c^3 + 2*a^3*b*c^3 + 4*a^2*b^2*c^3 - 8*a*b^3*c^3 + 2*b^4*c^3 + 2*a^3*c^4 - 5*a^2*b*c^4 + 3*a*b^2*c^4 + 2*b^3*c^4 + a^2*c^5 + 2*a*b*c^5 - b^2*c^5 - a*c^6 - b*c^6) : :
X(53402) = 3 X[165] - 2 X[4436], 3 X[1699] - 4 X[2486]

X(53402) lies on these lines: {1, 53260}, {9, 8245}, {40, 2783}, {57, 4459}, {84, 2648}, {165, 4436}, {804, 9860}, {900, 1768}, {1503, 1756}, {1633, 2250}, {1699, 2486}, {1709, 8557}, {2310, 51329}, {2784, 3882}, {9355, 9359}, {10085, 16143}, {16010, 41229}, {16574, 24728}

X(53402) = reflection of X(1) in X(53260)

---

X(53403) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1768) AND X(20375)

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

X(53403) lies on these lines: {57, 3837}, {63, 659}, {513, 48388}, {649, 4088}, {900, 1768}, {1158, 2821}, {1960, 12514}, {2826, 24467}, {3218, 46403}, {3306, 30795}, {3929, 45314}, {4063, 29324}, {5250, 25569}, {6762, 25574}, {6763, 21385}, {20375, 21391}, {26921, 44805}

---

X(53404) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(1768) AND X(21381)

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

X(53404) lies on these lines: {1, 47483}, {40, 12247}, {57, 3120}, {63, 3886}, {896, 44661}, {900, 1768}, {1054, 48244}, {2265, 24025}, {2796, 3928}, {3218, 44006}, {3576, 5494}, {3647, 18673}, {6089, 12078}, {13243, 53298}, {22321, 23703}, {26934, 33536}

---

X(53405) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(2948) AND X(13408)

Barycentrics    a*(b - c)*(b + c)*(a^7 - a^6*b + 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5 + 2*a*b^6 - a^6*c + 2*a^5*b*c - 2*a^3*b^3*c + a^2*b^4*c - a^3*b^2*c^2 - a^2*b^3*c^2 + a*b^4*c^2 + b^5*c^2 + 2*a^4*c^3 - 2*a^3*b*c^3 - a^2*b^2*c^3 + 2*a*b^3*c^3 - b^4*c^3 - 3*a^3*c^4 + a^2*b*c^4 + a*b^2*c^4 - b^3*c^4 - a^2*c^5 + b^2*c^5 + 2*a*c^6) : :
X(53405) = 3 X[165] - 2 X[9409], 5 X[1698] - 4 X[6130], 3 X[3576] - 4 X[8552], 3 X[5587] - 2 X[41079], 3 X[25055] - 4 X[45319], 5 X[35242] - 4 X[44810]

X(53405) lies on these lines: {1, 684}, {10, 53345}, {40, 9517}, {165, 9409}, {520, 21173}, {523, 47680}, {526, 2948}, {1698, 6130}, {2797, 5691}, {2799, 9864}, {2881, 12408}, {3576, 8552}, {5587, 41079}, {9033, 13211}, {25055, 45319}, {35242, 44810}

X(53405) = reflection of X(i) in X(j) for these {i,j}: {1, 684}, {53345, 10}

---

X(53406) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(2948) AND X(34464)

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 + a*b^4*c - b^5*c - 2*a^4*c^2 + 2*a^2*b^2*c^2 - 2*a*b^3*c^2 + 2*a^3*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(53406) = 3 X[3576] - 2 X[46610]

X(53406) lies on on the Feuerbach circumhyperbola of the tangential triangle and these lines: {1, 523}, {3, 513}, {6, 650}, {40, 46611}, {155, 521}, {159, 21005}, {399, 8674}, {514, 24315}, {522, 22836}, {526, 2948}, {656, 3215}, {661, 2267}, {663, 23757}, {900, 6326}, {1459, 3924}, {1498, 15313}, {1718, 10015}, {2457, 43924}, {2773, 44812}, {2850, 2931}, {2935, 18862}, {3216, 14812}, {3309, 35237}, {3576, 46610}, {3733, 23361}, {3738, 34465}, {4040, 15175}, {4833, 8053}, {6003, 6796}, {8648, 14299}, {9051, 19588}, {16143, 28217}, {16554, 22108}, {21132, 46385}, {23345, 34583}, {23758, 48282}, {24457, 48297}, {25005, 48204}, {26285, 48391}, {32612, 48382}, {34880, 51662}

X(53406) = reflection of X(40) in X(46611)
X(53406) = circumcircle-inverse of X(47081)
X(53406) = X(3658)-Ceva conjugate of X(3)
X(53406) = crossdifference of every pair of points on line {517, 2245}
X(53406) = barycentric product X(514)*X(51506)
X(53406) = barycentric quotient X(51506)/X(190)

---

X(53407) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(3464) AND X(5540)

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

X(53407) lies on these lines: {1, 53249}, {9, 4707}, {40, 30574}, {649, 14837}, {905, 47959}, {1019, 2487}, {2774, 11518}, {2785, 31435}, {3305, 49274}, {3464, 35055}, {3646, 14432}, {3960, 46393}, {5250, 53356}, {5540, 6084}, {7178, 9404}, {16485, 42662}

---

X(53408) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(5540) AND X(21381)

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

X(53408) lies on these lines: {1, 35327}, {9, 1654}, {19, 31922}, {57, 18625}, {63, 30579}, {1731, 7291}, {1762, 18163}, {1781, 18164}, {2161, 21362}, {2246, 16578}, {2264, 18726}, {5540, 6084}, {6089, 12078}, {7297, 20367}, {14543, 17197}, {16568, 18206}, {22031, 53337}, {26998, 29812}

X(53408) = X(4608)-Ceva conjugate of X(1)
X(53408) = X(35342)-Dao conjugate of X(4427)
X(53408) = barycentric product X(81)*X(24086)
X(53408) = barycentric quotient X(24086)/X(321)

---

X(53409) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(5540) AND X(39156)

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

X(53409) lies on these lines: {1, 53297}, {9, 3732}, {40, 103}, {57, 934}, {63, 21272}, {672, 2391}, {926, 39156}, {1018, 5845}, {1323, 2272}, {1768, 6366}, {2093, 18725}, {3245, 18735}, {5119, 7289}, {5540, 6084}, {5902, 18161}, {18163, 26934}, {18726, 50195}

X(53409) = reflection of X(1) in X(53297)

---

X(53410) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(5541) AND X(13174)

Barycentrics    a*(a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + a^4*c + 4*a^3*b*c - 4*a^2*b^2*c - 2*a*b^3*c - b^4*c + a^3*c^2 - 4*a^2*b*c^2 + 5*b^3*c^2 - a^2*c^3 - 2*a*b*c^3 + 5*b^2*c^3 - a*c^4 - b*c^4) : :
X(53410) = 3 X[165] - 2 X[53260], 5 X[1698] - 4 X[2486]

X(53410) lies on these lines: {1, 4436}, {40, 2783}, {165, 53260}, {740, 18206}, {804, 13174}, {900, 5541}, {1018, 24715}, {1697, 4459}, {1698, 2486}, {1756, 28530}, {2796, 3882}, {3501, 50080}, {4050, 50950}, {4659, 5119}, {4693, 20367}, {16552, 50086}, {16676, 40988}, {37555, 50126}

X(53410) = reflection of X(1) in X(4436)

---

X(53411) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(5541) AND X(21381)

Barycentrics    a*(b - c)*(a^3 + 2*a^2*b + a*b^2 + 2*a^2*c + 3*a*b*c - 5*b^2*c + a*c^2 - 5*b*c^2) : :
X(53411) = 5 X[1] - 6 X[53315], 3 X[3576] - 4 X[44812], 3 X[3737] - 2 X[4879], 2 X[4833] - 3 X[50346], 3 X[48293] - 4 X[48328], 3 X[48307] - 4 X[48331]

X(53411) lies on these lines: {1, 4145}, {9, 21832}, {57, 30572}, {522, 4498}, {523, 1019}, {649, 28169}, {812, 4768}, {900, 5541}, {3576, 44812}, {3737, 4139}, {4063, 4777}, {4132, 4833}, {6089, 12078}, {14407, 16676}, {16673, 53289}, {17151, 53276}, {18197, 50339}, {28147, 48341}, {28151, 48320}, {28161, 48011}, {48293, 48328}, {48307, 48331}, {48337, 50349}

X(53411) = reflection of X(48337) in X(50349)
X(53411) = crossdifference of every pair of points on line {4272, 16668}

---

X(53412) = INTERSECTION OF LINES TANGENT TO BEVAN CIRCLE AT X(21381) AND X(52679)

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

X(53412) lies on these lines: {1, 351}, {81, 1635}, {86, 4763}, {239, 514}, {333, 812}, {645, 3570}, {846, 4155}, {1757, 17989}, {1961, 17990}, {3737, 48226}, {4394, 18200}, {4728, 5235}, {5547, 43926}, {6089, 12078}, {17212, 45313}, {18185, 53287}, {25507, 45675}

X(53412) = X(892)-Ceva conjugate of X(1)
X(53412) = X(1018)-isoconjugate of X(7312)
X(53412) = X(2642)-Dao conjugate of X(690)
X(53412) = crossdifference of every pair of points on line {42, 4128}
X(53412) = barycentric product X(i)*X(j) for these {i,j}: {81, 45661}, {5524, 7192}
X(53412) = barycentric quotient X(i)/X(j) for these {i,j}: {3733, 7312}, {5524, 3952}, {45661, 321}

---

X(53413) = INTERSECTION OF LINES TANGENT TO STEVANOVIC CIRCLE AT X(32753) AND X(32754)

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

X(53413) lies on these lines: {19, 25}, {241, 514}, {478, 1743}, {518, 35326}, {919, 9061}, {971, 38358}, {1146, 51424}, {1212, 4413}, {1458, 38375}, {1566, 20623}, {1836, 20310}, {2284, 51378}, {2635, 3119}, {5452, 8270}, {8758, 14936}, {17092, 17093}, {20262, 40606}, {21839, 32126}, {28052, 31140}, {31197, 31203}

X(53413) = Stevanovic-circle-inverse of X(51775)
X(53413) = crossdifference of every pair of points on line {55, 905}
X(53413) = X(i)-lineconjugate of X(j) for these (i,j): {19, 55}, {241, 905}
X(53413) = barycentric product X(1)*X(45281)
X(53413) = barycentric quotient X(45281)/X(75)

---

X(53414) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(2) AND X(96)

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

X(53414) lies on these lines: {2, 6}, {3, 2165}, {5, 571}, {24, 53}, {30, 1879}, {32, 7405}, {50, 252}, {96, 6146}, {115, 22052}, {140, 231}, {187, 36412}, {216, 7542}, {338, 34828}, {393, 3147}, {468, 14576}, {549, 14806}, {577, 7746}, {647, 31364}, {1147, 41523}, {1609, 6642}, {1990, 10018}, {2965, 7745}, {3003, 16238}, {3053, 7401}, {3767, 7393}, {5023, 7487}, {5157, 37451}, {5254, 7509}, {5305, 5421}, {5562, 50647}, {6036, 11574}, {6641, 15508}, {6643, 13881}, {6749, 52296}, {7505, 8745}, {7576, 9220}, {7612, 18935}, {8588, 44261}, {10011, 13562}, {11063, 44802}, {14070, 21843}, {15109, 37126}, {21479, 36743}, {23333, 45838}, {25739, 33629}, {28723, 44530}, {33233, 42406}, {34351, 42459}, {34835, 41627}, {36422, 37512}, {36751, 44535}, {37940, 47275}, {42445, 44201}, {44234, 47167}, {44452, 47168}

X(53414) = complement of X(39113)
X(53414) = complement of the isogonal conjugate of X(41271)
X(53414) = complement of the isotomic conjugate of X(96)
X(53414) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 34835}, {96, 2887}, {661, 46655}, {2168, 141}, {32692, 4369}, {34385, 21235}, {41271, 10}
X(53414) = X(2)-Ceva conjugate of X(34835)
X(53414) = X(34835)-Dao conjugate of X(2)
X(53414) = crossdifference of every pair of points on line {512, 32762}
X(53414) = barycentric product X(96)*X(34835)
X(53414) = barycentric quotient X(i)/X(j) for these {i,j}: {34835, 39113}, {41627, 41628}
X(53414) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {577, 7746, 9722}, {590, 615, 343}, {1594, 8882, 6748}

---

X(53415) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(2) AND X(801)

Barycentrics    2*a^6 - 3*a^4*b^2 + b^6 - 3*a^4*c^2 + 8*a^2*b^2*c^2 - b^4*c^2 - b^2*c^4 + c^6 : :
X(53415) = 3 X[2] + X[394], 9 X[2] - X[6515], 3 X[394] + X[6515], X[6515] - 3 X[13567], 3 X[154] + X[34944], 5 X[631] - X[10605], X[1370] + 3 X[35259], X[1899] + 3 X[6090], X[1899] - 5 X[31255], 3 X[6090] + 5 X[31255], 3 X[31152] + X[31383]

X(53415) lies on these lines: {2, 6}, {3, 1661}, {5, 13346}, {22, 15448}, {25, 29181}, {30, 43586}, {140, 9729}, {154, 7386}, {182, 41619}, {184, 30739}, {219, 20266}, {235, 43652}, {345, 26658}, {427, 5651}, {428, 51360}, {441, 7789}, {468, 3917}, {511, 6677}, {541, 10272}, {547, 46114}, {577, 45200}, {626, 44334}, {631, 10605}, {1073, 6389}, {1092, 12241}, {1125, 11018}, {1146, 1943}, {1214, 17044}, {1216, 16238}, {1350, 6353}, {1352, 23332}, {1368, 1503}, {1370, 35259}, {1375, 1764}, {1407, 27509}, {1495, 7667}, {1514, 44458}, {1596, 37480}, {1853, 14826}, {1885, 22966}, {1899, 6090}, {1944, 6354}, {1990, 15466}, {2781, 3819}, {2979, 32269}, {3060, 40929}, {3098, 10154}, {3167, 8550}, {3292, 11245}, {3452, 36949}, {3523, 3532}, {3530, 32210}, {3546, 6247}, {3666, 26006}, {3782, 26651}, {3796, 46336}, {4641, 25019}, {5020, 5480}, {5092, 7734}, {5133, 35283}, {5159, 21243}, {5254, 41235}, {5294, 25067}, {5447, 13383}, {5642, 22352}, {5650, 7499}, {5891, 10257}, {5893, 31829}, {5907, 6696}, {6180, 27540}, {6509, 34828}, {6510, 45206}, {6554, 18623}, {6642, 11745}, {6643, 34782}, {6678, 24220}, {6688, 18583}, {6804, 11425}, {6816, 35602}, {6995, 51163}, {7365, 27382}, {7396, 36990}, {7398, 53023}, {7485, 13394}, {7494, 21167}, {7714, 48910}, {7784, 52283}, {7807, 35294}, {7880, 44346}, {7999, 10018}, {8263, 11511}, {8542, 10169}, {8703, 46817}, {8780, 46264}, {8889, 10516}, {8968, 15236}, {9308, 46927}, {9909, 48881}, {10020, 32142}, {10128, 19130}, {10170, 14156}, {10219, 25555}, {10519, 38282}, {10546, 34603}, {10565, 31884}, {10627, 44232}, {11574, 15585}, {11585, 18474}, {12007, 34986}, {12134, 37452}, {13568, 17928}, {13857, 50959}, {14128, 23336}, {15030, 47090}, {15060, 15122}, {15067, 44201}, {15069, 23291}, {15644, 21841}, {15812, 34774}, {15818, 35228}, {15873, 37498}, {16063, 35264}, {16319, 47509}, {16579, 17043}, {16655, 43598}, {16657, 43574}, {17188, 37363}, {17810, 40132}, {17917, 52457}, {18324, 35254}, {18537, 37497}, {18553, 47629}, {20850, 48873}, {21015, 26884}, {21258, 37543}, {22466, 23308}, {24537, 49734}, {25017, 49745}, {25091, 44416}, {25563, 40247}, {25930, 32777}, {25935, 37595}, {25941, 25968}, {26881, 35266}, {26885, 26933}, {28272, 43053}, {30744, 45303}, {31152, 31383}, {32062, 47091}, {33533, 34477}, {34608, 41424}, {34664, 51394}, {34990, 46832}, {36987, 47093}, {37453, 43653}, {37478, 44211}, {37495, 50139}, {37536, 52259}, {37911, 40107}, {40323, 40802}, {40330, 52299}, {44234, 44324}, {44673, 44683}

X(53415) = midpoint of X(i) and X(j) for these {i,j}: {394, 13567}, {1368, 9306}, {1596, 37480}, {8263, 11511}
X(53415) = complement of X(13567)
X(53415) = complement of the isogonal conjugate of X(41890)
X(53415) = complement of the isotomic conjugate of X(801)
X(53415) = isotomic conjugate of the polar conjugate of X(1885)
X(53415) = X(i)-complementary conjugate of X(j) for these (i,j): {661, 46658}, {775, 141}, {801, 2887}, {1105, 20305}, {40830, 21235}, {41890, 10}
X(53415) = X(22966)-Dao conjugate of X(6)
X(53415) = barycentric product X(i)*X(j) for these {i,j}: {69, 1885}, {6331, 33968}
X(53415) = barycentric quotient X(i)/X(j) for these {i,j}: {1885, 4}, {22966, 22467}, {22970, 235}, {33968, 647}
X(53415) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 69, 26958}, {2, 81, 25964}, {2, 343, 47296}, {2, 394, 13567}, {2, 1993, 37648}, {2, 11064, 23292}, {2, 11427, 17825}, {2, 15066, 343}, {2, 17811, 141}, {2, 23292, 3589}, {2, 24553, 19701}, {2, 26668, 4383}, {2, 37645, 10601}, {2, 37659, 1211}, {2, 37669, 6}, {6, 599, 53021}, {140, 11591, 44158}, {154, 7386, 44882}, {1352, 30771, 23332}, {2883, 5894, 43695}, {3523, 32605, 15740}, {3546, 17814, 6247}, {3819, 5972, 6676}, {5907, 16196, 6696}, {6090, 31255, 1899}, {10170, 14156, 52262}, {11427, 17825, 597}, {11433, 37672, 3629}, {14826, 16051, 1853}, {15066, 47296, 3631}, {15067, 44452, 44201}, {34986, 45298, 12007}, {39022, 39023, 40888}

---

X(53416) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(4) AND X(94)

Barycentrics    a^6*b^2 - a^4*b^4 - a^2*b^6 + b^8 + a^6*c^2 + a^2*b^4*c^2 - 4*b^6*c^2 - a^4*c^4 + a^2*b^2*c^4 + 6*b^4*c^4 - a^2*c^6 - 4*b^2*c^6 + c^8 : :
X(53416) =X(53416) = X[50] - 3 X[1989], 3 X[1989] - 2 X[16310]

X(53416) lies on these lines: {4, 6}, {5, 566}, {20, 46262}, {23, 230}, {26, 2165}, {30, 50}, {32, 7540}, {94, 3580}, {115, 3003}, {141, 44135}, {161, 14593}, {187, 231}, {216, 1879}, {232, 37981}, {237, 41221}, {297, 338}, {340, 48540}, {403, 11062}, {460, 18374}, {468, 44529}, {546, 41335}, {570, 1506}, {577, 18563}, {686, 2501}, {858, 2493}, {1263, 11071}, {1609, 7517}, {2079, 45171}, {2088, 39235}, {2393, 8754}, {2450, 34981}, {2549, 31861}, {2782, 45921}, {2965, 11819}, {3001, 15980}, {3018, 3284}, {3054, 52300}, {3549, 36751}, {3767, 7530}, {3815, 5169}, {3845, 14836}, {5063, 7748}, {5305, 13338}, {5306, 33886}, {5309, 33872}, {5341, 8736}, {5346, 13345}, {6128, 18487}, {7297, 8735}, {7493, 37637}, {7519, 7735}, {7527, 50660}, {8800, 17834}, {8818, 48903}, {9722, 15761}, {10296, 18365}, {11079, 34150}, {11648, 33871}, {13337, 15048}, {13351, 31406}, {14118, 15109}, {14570, 44388}, {14957, 18371}, {15993, 46154}, {16303, 19656}, {16619, 43291}, {19220, 38321}, {20987, 41762}, {20998, 52125}, {39601, 52704}, {40853, 44375}, {41725, 50647}, {44526, 49669}

X(53416) =reflection of X(i) in X(j) for these {i,j}: {50, 16310}, {2450, 34981}
X(53416) =reflection of X(50) in the Orthic axis
X(53416) =polar conjugate of the isotomic conjugate of X(2072)
X(53416) =X(i)-Ceva conjugate of X(j) for these (i,j): {32708, 2501}, {46456, 523}
X(53416) =X(63)-isoconjugate of X(38534)
X(53416) =X(i)-Dao conjugate of X(j) for these (i,j): {2072, 323}, {3162, 38534}, {16186, 8552}, {46085, 394}
X(53416) =crossdifference of every pair of points on line {520, 1147}
X(53416) =barycentric product X(i)*X(j) for these {i,j}: {4, 2072}, {847, 45780}, {850, 53329}, {1300, 46085}, {5962, 53168}
X(53416) =barycentric quotient X(i)/X(j) for these {i,j}: {25, 38534}, {2072, 69}, {14593, 45781}, {34397, 53170}, {45780, 9723}, {53329, 110}
X(53416) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {50, 1989, 16310}, {115, 52945, 3003}, {393, 15262, 1990}, {566, 9220, 5}, {3003, 52945, 47322}, {3070, 3071, 12022}, {18487, 39563, 6128}, {23251, 23261, 18405}, {42645, 42646, 18396}

---

X(53417) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(4) AND X(94)

Barycentrics    (b + c)*(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(53417) lies on these lines: {4, 6}, {5, 4261}, {30, 1333}, {37, 442}, {45, 1213}, {65, 8736}, {71, 21935}, {75, 26601}, {115, 119}, {141, 33736}, {230, 1030}, {313, 321}, {440, 3772}, {524, 21287}, {536, 21245}, {740, 21076}, {851, 2178}, {857, 4000}, {964, 17398}, {1086, 17863}, {1100, 12690}, {1761, 24851}, {1766, 2245}, {1778, 24883}, {1826, 3914}, {2220, 5305}, {2264, 8735}, {2276, 3136}, {2277, 3142}, {2286, 18961}, {2292, 21675}, {2294, 3120}, {2298, 11604}, {2303, 2475}, {2321, 3454}, {2549, 36743}, {3285, 48890}, {3583, 16470}, {3663, 17052}, {3672, 31043}, {3704, 20654}, {3767, 36744}, {3770, 47286}, {3936, 17314}, {3948, 46738}, {4150, 19791}, {4187, 46838}, {4205, 17303}, {4272, 8818}, {4642, 21011}, {4653, 24937}, {4857, 16488}, {4877, 24880}, {4918, 21689}, {5016, 17362}, {5019, 7748}, {5042, 11648}, {5069, 15048}, {5124, 37399}, {5301, 6284}, {5309, 16946}, {5716, 16884}, {5747, 48837}, {6882, 50650}, {7069, 20966}, {7735, 50698}, {8609, 18591}, {9598, 37225}, {10974, 21853}, {16052, 17281}, {16600, 21065}, {16777, 17056}, {17299, 41014}, {17452, 22073}, {17737, 38871}, {17757, 21858}, {17861, 18642}, {17862, 26603}, {18046, 50319}, {18147, 37096}, {19645, 37646}, {19785, 27052}, {20136, 33030}, {20174, 26558}, {20970, 29093}, {21018, 21965}, {21530, 23537}, {21678, 40973}, {24530, 26019}, {25457, 33034}, {36035, 47137}, {37241, 46010}, {42047, 50087}

X(53417) = isotomic conjugate of the isogonal conjugate of X(53387)
X(53417) = polar conjugate of the isotomic conjugate of X(21530)
X(53417) = X(i)-Ceva conjugate of X(j) for these (i,j): {6335, 523}, {23537, 40973}
X(53417) = X(58)-isoconjugate of X(40406)
X(53417) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 40406}, {18210, 905}, {21530, 81}, {40941, 69}
X(53417) = crossdifference of every pair of points on line {520, 53306}
X(53417) = barycentric product X(i)*X(j) for these {i,j}: {4, 21530}, {10, 23537}, {27, 21678}, {75, 40973}, {76, 53387}, {92, 18674}, {321, 40941}, {427, 18709}, {523, 53349}, {850, 53282}, {1826, 18651}, {18732, 41013}
X(53417) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 40406}, {18651, 17206}, {18674, 63}, {18709, 1799}, {18732, 1444}, {21530, 69}, {21678, 306}, {23537, 86}, {40941, 81}, {40973, 1}, {53282, 110}, {53349, 99}, {53387, 6}
X(53417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 18685, 5317}, {115, 2092, 50036}, {1834, 1901, 6}, {1834, 32580, 1865}, {2345, 5051, 1213}, {5317, 18685, 1990}, {17863, 26605, 18635}, {33736, 44140, 141}

---

X(53418) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(4) AND X(598)

Barycentrics    4*a^4 + a^2*b^2 - 3*b^4 + a^2*c^2 + 6*b^2*c^2 - 3*c^4 : :
X(53418) =2 X[574] - 3 X[3815], X[574] - 3 X[5475], X[183] - 3 X[33016], X[11185] - 3 X[11317], X[14907] - 3 X[44543]

X(53418) lies on these lines: {2, 5210}, {3, 3055}, {4, 6}, {5, 187}, {12, 10987}, {15, 46079}, {16, 46080}, {20, 53095}, {30, 574}, {32, 546}, {34, 9595}, {39, 3627}, {51, 41911}, {76, 3630}, {83, 33229}, {112, 18429}, {115, 3845}, {140, 8588}, {141, 316}, {148, 41624}, {183, 33016}, {193, 34505}, {194, 14066}, {230, 381}, {235, 10985}, {315, 3631}, {325, 11361}, {376, 31489}, {382, 2548}, {403, 10986}, {458, 47296}, {460, 34417}, {485, 8375}, {486, 8376}, {524, 11185}, {548, 31455}, {549, 6781}, {550, 1506}, {597, 598}, {599, 23334}, {625, 8369}, {631, 5585}, {632, 15513}, {671, 8584}, {1003, 44377}, {1285, 41099}, {1352, 11173}, {1383, 5169}, {1593, 9608}, {1648, 20192}, {1656, 15655}, {1657, 31401}, {1692, 38136}, {1968, 23047}, {1975, 14068}, {1995, 24855}, {2023, 22682}, {2030, 19130}, {2079, 13595}, {2482, 50280}, {2549, 3830}, {3053, 3091}, {3090, 5023}, {3094, 51163}, {3146, 5013}, {3363, 3849}, {3529, 11742}, {3543, 7736}, {3545, 37637}, {3583, 16785}, {3585, 16784}, {3589, 7841}, {3619, 7784}, {3620, 32006}, {3628, 5206}, {3629, 7812}, {3767, 3843}, {3832, 13881}, {3839, 7735}, {3850, 7746}, {3853, 7748}, {3857, 35007}, {3858, 39565}, {3860, 18362}, {3861, 5305}, {3933, 7843}, {3972, 33228}, {4262, 36654}, {5033, 10358}, {5046, 37675}, {5052, 39884}, {5055, 21843}, {5056, 44535}, {5070, 15603}, {5076, 9605}, {5097, 38734}, {5107, 21850}, {5140, 9969}, {5229, 16781}, {5309, 14075}, {5354, 37349}, {5355, 39563}, {5395, 18845}, {5461, 19661}, {5471, 31709}, {5472, 31710}, {5477, 9880}, {5486, 47060}, {6032, 7426}, {6036, 14160}, {6199, 39660}, {6321, 12830}, {6390, 7775}, {6395, 39661}, {6412, 21737}, {6421, 22644}, {6422, 22615}, {6423, 42268}, {6424, 42269}, {6620, 31860}, {6656, 51126}, {6661, 7934}, {6823, 22052}, {7388, 32789}, {7389, 32790}, {7408, 15437}, {7533, 11580}, {7620, 15534}, {7738, 17578}, {7739, 38335}, {7750, 16044}, {7752, 19687}, {7753, 15048}, {7756, 31406}, {7764, 15301}, {7769, 33250}, {7770, 34573}, {7772, 12102}, {7773, 7789}, {7778, 14033}, {7785, 14042}, {7786, 19695}, {7787, 14062}, {7792, 14041}, {7797, 14044}, {7802, 32992}, {7804, 33184}, {7808, 8357}, {7811, 15598}, {7819, 7825}, {7820, 31173}, {7823, 33018}, {7835, 48913}, {7842, 8362}, {7844, 37350}, {7850, 22165}, {7851, 32996}, {7867, 19697}, {7912, 14034}, {7925, 19686}, {7937, 51128}, {7941, 32820}, {7945, 14032}, {8176, 27088}, {8354, 15482}, {8356, 15491}, {8598, 9771}, {8627, 41237}, {8972, 9602}, {9112, 41108}, {9113, 41107}, {9466, 14929}, {9596, 12953}, {9599, 12943}, {9600, 42275}, {9604, 14157}, {9609, 18534}, {9650, 15171}, {9665, 18990}, {9668, 31409}, {9675, 18538}, {9699, 18570}, {9766, 32815}, {10024, 18472}, {10151, 10311}, {10314, 44920}, {10583, 14045}, {10788, 39663}, {10979, 12362}, {11001, 44541}, {11114, 37661}, {11159, 22110}, {11163, 52942}, {11174, 33017}, {11303, 23303}, {11304, 23302}, {11648, 12101}, {12103, 15515}, {12963, 42273}, {12968, 42270}, {13567, 41254}, {13586, 37647}, {13596, 34866}, {14712, 33013}, {14832, 15358}, {14907, 44543}, {15052, 45769}, {15271, 32983}, {15597, 51224}, {15704, 37512}, {15820, 44212}, {15980, 26316}, {15993, 47354}, {16001, 44497}, {16002, 44498}, {16308, 18323}, {16808, 41407}, {16809, 41406}, {17005, 33265}, {17398, 17677}, {17800, 31417}, {18843, 18844}, {19780, 42107}, {19781, 42110}, {20112, 22329}, {22338, 45012}, {22512, 41016}, {22513, 41017}, {22575, 41746}, {22576, 41745}, {23292, 52282}, {28146, 31398}, {28150, 31443}, {31400, 33703}, {31403, 52666}, {31450, 49134}, {31463, 42263}, {31492, 49138}, {32459, 33007}, {33190, 47355}, {34484, 44523}, {34803, 35927}, {35786, 49220}, {35787, 49221}, {37665, 50687}, {37809, 41139}, {37925, 44521}, {37946, 50660}, {38071, 39601}, {39809, 44422}, {40341, 52713}, {41106, 46453}, {41620, 47866}, {41621, 47865}, {41939, 53161}, {42852, 48884}, {44538, 47485}, {46988, 48721}

X(53418) =midpoint of X(11163) and X(52942)
X(53418) =reflection of X(i) in X(j) for these {i,j}: {3815, 5475}, {11168, 3363}
X(53418) =X(24018)-isoconjugate of X(52777)
X(53418) =barycentric quotient X(i)/X(j) for these {i,j}: {6529, 42393}, {32713, 52777}
X(53418) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 31415, 3055}, {4, 7745, 5254}, {5, 187, 3054}, {32, 18424, 43291}, {115, 14537, 18907}, {115, 18907, 5306}, {187, 39590, 43457}, {187, 43457, 5}, {316, 8370, 141}, {381, 1384, 43620}, {381, 7737, 230}, {382, 5024, 43619}, {546, 43291, 18424}, {598, 8352, 597}, {1384, 43620, 230}, {2548, 43619, 5024}, {2549, 15484, 9300}, {3529, 31404, 15815}, {3543, 7736, 44526}, {3830, 15484, 2549}, {3845, 14537, 5306}, {3845, 18907, 115}, {5210, 18584, 2}, {5318, 5321, 5480}, {6781, 7603, 549}, {7737, 43620, 1384}, {7747, 39590, 5}, {7747, 43457, 187}, {7773, 14035, 7789}, {7785, 14042, 32819}, {7812, 47286, 3629}, {14033, 32827, 7778}, {14712, 33013, 37688}, {31400, 33703, 44519}, {31415, 43618, 3}

---

X(53419) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(4) AND X(671)

Barycentrics    2*a^4 - a^2*b^2 - 3*b^4 - a^2*c^2 + 6*b^2*c^2 - 3*c^4 : :
X(53419) = 3 X[115] - X[187], 5 X[115] - X[6781], X[115] - 3 X[39563], 3 X[115] - 2 X[43291], 2 X[187] - 3 X[230], 5 X[187] - 3 X[6781], X[187] - 9 X[39563], 5 X[230] - 2 X[6781], X[230] - 6 X[39563], 3 X[230] - 4 X[43291], X[6781] - 15 X[39563], 3 X[6781] - 10 X[43291], 9 X[39563] - 2 X[43291], X[99] - 3 X[33228], 3 X[33228] - 2 X[44377], and many others

X(53419) lies on these lines: {2, 11147}, {3, 3054}, {4, 6}, {5, 574}, {20, 5210}, {23, 47298}, {24, 44528}, {30, 115}, {32, 3627}, {39, 546}, {67, 52477}, {69, 34505}, {76, 3631}, {83, 53106}, {99, 33228}, {111, 858}, {140, 7756}, {141, 7841}, {148, 325}, {183, 33017}, {186, 34866}, {194, 14062}, {232, 10151}, {297, 39062}, {315, 3630}, {316, 524}, {323, 13580}, {338, 44146}, {353, 14389}, {376, 5585}, {378, 44524}, {381, 2549}, {382, 1384}, {395, 51485}, {396, 51484}, {427, 47187}, {460, 1495}, {495, 9664}, {496, 9651}, {511, 38734}, {523, 14832}, {538, 50771}, {542, 44496}, {543, 625}, {547, 15602}, {548, 7749}, {550, 7746}, {597, 11317}, {598, 17503}, {599, 7620}, {631, 44519}, {632, 15515}, {698, 39266}, {754, 15480}, {868, 35606}, {1007, 8716}, {1078, 19695}, {1194, 52285}, {1213, 17677}, {1506, 3850}, {1513, 14639}, {1562, 13851}, {1598, 9608}, {1611, 44442}, {1648, 52450}, {1657, 15655}, {1879, 34664}, {1975, 14063}, {2030, 29012}, {2031, 2794}, {2071, 2079}, {2393, 5140}, {2475, 37675}, {2482, 8355}, {2493, 10297}, {2501, 3566}, {2502, 11064}, {2548, 3843}, {2996, 20080}, {3011, 47098}, {3018, 46203}, {3053, 3146}, {3090, 15815}, {3091, 5013}, {3143, 14609}, {3153, 38463}, {3199, 44226}, {3231, 14957}, {3291, 46517}, {3520, 44523}, {3522, 44535}, {3524, 44541}, {3529, 5023}, {3534, 21843}, {3543, 7735}, {3545, 31489}, {3564, 5107}, {3575, 10985}, {3580, 13192}, {3583, 16784}, {3585, 16785}, {3589, 7790}, {3619, 32974}, {3620, 7784}, {3628, 37512}, {3651, 44517}, {3712, 21057}, {3723, 13161}, {3734, 33184}, {3763, 33190}, {3793, 3849}, {3830, 5306}, {3832, 7738}, {3839, 7736}, {3845, 5475}, {3851, 31401}, {3853, 5008}, {3855, 31400}, {3856, 9698}, {3857, 53096}, {3858, 31406}, {3861, 7765}, {3933, 7825}, {3934, 8357}, {5007, 12102}, {5017, 51163}, {5025, 7789}, {5028, 39884}, {5030, 36654}, {5034, 38136}, {5066, 7603}, {5076, 30435}, {5077, 7615}, {5080, 21956}, {5099, 47245}, {5104, 29181}, {5106, 21531}, {5111, 38664}, {5159, 40349}, {5189, 11580}, {5206, 15704}, {5225, 16781}, {5304, 50687}, {5309, 15687}, {5355, 12101}, {5461, 27088}, {5471, 43417}, {5472, 43416}, {5485, 15533}, {5912, 36174}, {5913, 10989}, {6128, 16303}, {6200, 35831}, {6221, 39660}, {6240, 10986}, {6248, 18358}, {6253, 10988}, {6284, 10987}, {6321, 15980}, {6329, 7827}, {6337, 32980}, {6392, 11008}, {6396, 35830}, {6398, 39661}, {6411, 21737}, {6421, 42268}, {6422, 42269}, {6423, 22644}, {6424, 22615}, {6444, 31463}, {6560, 8376}, {6561, 8375}, {6620, 41424}, {6656, 34573}, {6661, 7919}, {6722, 32456}, {6772, 22576}, {6775, 22575}, {6823, 10979}, {6876, 44520}, {7388, 32790}, {7389, 32789}, {7396, 8770}, {7399, 14806}, {7464, 44533}, {7470, 44530}, {7550, 15109}, {7687, 44228}, {7739, 14269}, {7750, 33019}, {7753, 14893}, {7761, 18546}, {7767, 7842}, {7770, 51126}, {7771, 8353}, {7773, 32996}, {7778, 16041}, {7782, 33249}, {7783, 32993}, {7785, 14044}, {7787, 14066}, {7792, 11361}, {7797, 14042}, {7799, 47287}, {7812, 32455}, {7813, 31173}, {7816, 8361}, {7819, 7861}, {7820, 8360}, {7828, 19687}, {7833, 37688}, {7836, 14045}, {7844, 8369}, {7847, 15031}, {7851, 14035}, {7852, 19697}, {7857, 33250}, {7864, 33018}, {7868, 33251}, {7872, 8362}, {7898, 37671}, {7912, 32820}, {7925, 20094}, {7932, 14034}, {7937, 20582}, {8288, 44569}, {8359, 47617}, {8553, 12082}, {8571, 44158}, {8585, 30739}, {8591, 41133}, {8597, 14712}, {8598, 9166}, {8703, 18362}, {8859, 40246}, {9112, 42973}, {9113, 42972}, {9301, 38732}, {9594, 37697}, {9597, 10896}, {9598, 10895}, {9600, 42277}, {9604, 15033}, {9609, 9818}, {9675, 42225}, {9745, 31105}, {9766, 32827}, {10011, 23514}, {10175, 31443}, {10256, 33813}, {10295, 44529}, {10317, 18323}, {10418, 47097}, {10561, 53365}, {10590, 31477}, {10592, 31451}, {10723, 44534}, {10748, 45012}, {11063, 37946}, {11173, 31670}, {11174, 33016}, {11303, 23302}, {11304, 23303}, {11542, 36251}, {11543, 36252}, {11676, 39663}, {12084, 44527}, {12086, 44537}, {12103, 15513}, {12362, 22052}, {12811, 31652}, {12829, 39838}, {12830, 22505}, {12963, 42271}, {12968, 42272}, {13468, 14907}, {13473, 16318}, {13488, 27371}, {13492, 38951}, {13567, 52282}, {13651, 43791}, {13711, 42275}, {13770, 43792}, {13834, 42276}, {14061, 35297}, {14064, 32826}, {14118, 44525}, {14120, 16320}, {14538, 43277}, {14539, 43276}, {14568, 50774}, {14915, 50387}, {14929, 17131}, {15230, 35133}, {15271, 32986}, {15491, 44543}, {15534, 23334}, {15597, 35955}, {15603, 15681}, {15993, 43453}, {16001, 44498}, {16002, 44497}, {16052, 24275}, {16063, 20481}, {16092, 41404}, {16306, 47309}, {16308, 44468}, {16310, 47339}, {16315, 36204}, {16317, 39602}, {16984, 19686}, {16990, 33278}, {17004, 33264}, {17008, 33192}, {17577, 37661}, {17964, 47242}, {18403, 22121}, {18472, 18563}, {18487, 46211}, {18538, 48773}, {18762, 48772}, {19106, 41406}, {19107, 41407}, {19570, 50251}, {19780, 42109}, {19781, 42108}, {20398, 47113}, {20977, 53346}, {21166, 40336}, {21448, 31152}, {21536, 41273}, {21965, 24851}, {22242, 31862}, {22243, 31863}, {22332, 31404}, {23292, 52281}, {23897, 49728}, {23903, 49745}, {29323, 38010}, {31398, 38140}, {31418, 31490}, {32269, 39691}, {32488, 32494}, {32489, 32497}, {32828, 33238}, {32832, 33234}, {32838, 33226}, {32989, 39143}, {33272, 34229}, {33292, 53033}, {33698, 45103}, {34175, 51404}, {34366, 44972}, {34609, 40326}, {35266, 41939}, {35473, 44538}, {35820, 49221}, {35821, 49220}, {35921, 44521}, {36180, 47243}, {36196, 44398}, {37118, 50718}, {37174, 37643}, {37403, 44542}, {38005, 47060}, {39022, 51899}, {39023, 51898}, {41359, 47467}, {44395, 51438}, {44595, 52667}, {44596, 52666}, {44961, 47186}, {47093, 47297}, {47169, 47332}, {47246, 47326}

X(53419) = midpoint of X(i) and X(j) for these {i,j}: {148, 325}, {316, 47286}, {671, 8352}, {5318, 5321}, {6321, 15980}, {8597, 22329}, {31709, 31710}, {33517, 33518}, {46517, 47295}
X(53419) = reflection of X(i) in X(j) for these {i,j}: {99, 44377}, {187, 43291}, {230, 115}, {2482, 8355}, {6390, 625}, {8598, 44401}, {16320, 14120}, {22110, 37350}, {27088, 5461}, {32456, 6722}, {47113, 20398}, {47242, 51258}, {47245, 5099}, {51438, 44395}
X(53419) = anticomplement of X(32459)
X(53419) = isotomic conjugate of the isogonal conjugate of X(40350)
X(53419) = polar conjugate of the isotomic conjugate of X(5159)
X(53419) = polar conjugate of the isogonal conjugate of X(21639)
X(53419) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {897, 19583}, {8769, 14360}, {38252, 8591}
X(53419) = X(i)-Dao conjugate of X(j) for these (i,j): {5159, 524}, {40349, 5866}
X(53419) = cevapoint of X(21639) and X(40350)
X(53419) = crossdifference of every pair of points on line {520, 3167}
X(53419) = barycentric product X(i)*X(j) for these {i,j}: {4, 5159}, {76, 40350}, {264, 21639}, {523, 53351}, {2052, 40349}
X(53419) = barycentric quotient X(i)/X(j) for these {i,j}: {5159, 69}, {21639, 3}, {40349, 394}, {40350, 6}, {53351, 99}
X(53419) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 43620, 3054}, {4, 5254, 7745}, {4, 6776, 53017}, {4, 43448, 6}, {4, 44518, 5254}, {4, 46034, 36990}, {5, 574, 3055}, {6, 43448, 5254}, {6, 44518, 43448}, {99, 33228, 44377}, {111, 858, 24855}, {115, 187, 43291}, {148, 14041, 325}, {187, 43291, 230}, {297, 41254, 47296}, {316, 671, 47286}, {381, 2549, 3815}, {381, 5024, 31415}, {382, 1384, 43618}, {574, 18424, 5}, {625, 6390, 22110}, {2549, 31415, 5024}, {3070, 3071, 8550}, {3767, 43618, 1384}, {3845, 11648, 9300}, {3845, 15048, 5475}, {3853, 5305, 7747}, {5024, 31415, 3815}, {5025, 32819, 7789}, {5077, 7615, 11168}, {5334, 5335, 14912}, {5461, 27088, 41139}, {5475, 11648, 15048}, {5475, 15048, 9300}, {5585, 11742, 376}, {6390, 37350, 625}, {6748, 42391, 4}, {7748, 18424, 574}, {7756, 39565, 140}, {7790, 8370, 3589}, {7841, 11185, 141}, {7847, 15031, 32992}, {8352, 47286, 316}, {8597, 41135, 22329}, {8598, 9166, 44401}, {11742, 37637, 5585}, {14061, 35297, 44381}, {16041, 32815, 7778}, {16317, 47311, 39602}, {23249, 23259, 39874}, {43619, 43620, 3}, {52450, 53161, 1648}

---

X(53420) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(4) AND X(671)

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

X(53420) lies on these lines: {2, 10607}, {4, 6}, {5, 5065}, {30, 800}, {39, 9825}, {115, 44920}, {141, 41235}, {216, 31829}, {230, 577}, {233, 3055}, {317, 13567}, {419, 15585}, {460, 6467}, {524, 14615}, {570, 6128}, {1609, 21312}, {2165, 16072}, {3543, 52223}, {3767, 15905}, {3815, 5020}, {5063, 9722}, {5306, 34609}, {6620, 9924}, {7396, 7735}, {7398, 7736}, {7491, 50653}, {7748, 42459}, {10602, 41762}, {13342, 18907}, {14913, 34383}, {23292, 36794}, {27377, 40814}, {34809, 40326}, {36616, 46952}, {41489, 44438}

X(53420) = polar conjugate of the isotomic conjugate of X(16196)
X(53420) = X(16196)-Dao conjugate of X(13567)
X(53420) = barycentric product X(4)*X(16196)
X(53420) = barycentric quotient X(16196)/X(69)
X(53420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 6748, 7745}, {6, 44518, 393}, {3070, 3071, 2883}, {5286, 40065, 6}, {7748, 46432, 42459}

---

X(53421) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(4) AND X(1029)

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

X(53421) lies on these lines: {4, 6}, {5, 5124}, {9, 18513}, {21, 5949}, {30, 1030}, {37, 3585}, {80, 21863}, {81, 1029}, {115, 1333}, {230, 37456}, {284, 8818}, {286, 338}, {316, 3770}, {381, 36743}, {382, 36744}, {594, 5080}, {1012, 8553}, {1100, 3583}, {1213, 2475}, {1400, 13273}, {1449, 18514}, {1478, 16777}, {1479, 16884}, {1761, 21018}, {1826, 5341}, {1839, 7297}, {2160, 21044}, {2178, 12943}, {2220, 7747}, {2245, 32431}, {2305, 46704}, {3723, 5270}, {3830, 4254}, {3843, 5120}, {4227, 34866}, {4261, 7748}, {5036, 5816}, {5046, 17398}, {5069, 5475}, {5076, 37503}, {5189, 37675}, {5229, 16672}, {5275, 7391}, {6905, 15109}, {6923, 37499}, {7110, 15586}, {7354, 21773}, {7381, 37674}, {7382, 37679}, {8727, 9722}, {11063, 21669}, {13442, 44517}, {16049, 35212}, {17362, 52367}, {17388, 20060}, {17685, 25508}, {18406, 21866}, {18480, 21853}, {19721, 50697}, {20472, 34119}, {21245, 24271}, {21873, 47033}, {26118, 37637}, {33854, 37349}, {37500, 44229}, {37508, 47032}

X(53421) = reflection of X(1030) in X(50036)
X(53421) = {X(42645),X(42646)}-harmonic conjugate of X(5706)

---

X(53422) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(4) AND X(1446)

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

X(53422) lies on these lines: {4, 6}, {115, 44993}, {169, 2245}, {407, 39690}, {442, 1212}, {594, 52345}, {857, 948}, {1030, 30267}, {1211, 26605}, {1213, 6554}, {1446, 18635}, {3002, 6841}, {4292, 52530}, {5124, 37275}, {5179, 14873}, {5712, 31042}, {7100, 8818}, {7535, 36743}, {8736, 30456}, {15951, 36744}, {18641, 46835}, {21024, 44150}, {26601, 38298}, {31936, 40937}

X(53422) = X(13149)-Ceva conjugate of X(523)
X(53422) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {857, 948, 17056}, {6554, 25015, 1213}, {26605, 30807, 1211}

---

X(53423) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(10) AND X(83)

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

X(53423) lies on these lines: {2, 18755}, {6, 1330}, {10, 37}, {32, 20083}, {39, 48843}, {83, 316}, {86, 17673}, {141, 17034}, {172, 2240}, {187, 6693}, {239, 1211}, {274, 24366}, {284, 8728}, {429, 2201}, {442, 19557}, {754, 17200}, {966, 37164}, {1100, 49564}, {1125, 31488}, {1506, 20108}, {1962, 21718}, {2176, 32773}, {2238, 5051}, {2245, 3496}, {2292, 21711}, {2295, 4972}, {3136, 21034}, {3454, 10026}, {3948, 33938}, {4201, 33863}, {4202, 24512}, {4272, 34528}, {4425, 21879}, {4426, 25453}, {5021, 11359}, {5277, 25441}, {7747, 48866}, {7756, 48836}, {9278, 34920}, {9956, 49637}, {16886, 21840}, {16906, 37632}, {17023, 17056}, {17103, 33831}, {17175, 25468}, {17243, 30173}, {17330, 50058}, {17362, 41814}, {17514, 19856}, {20715, 40952}, {23897, 36478}, {23903, 27040}, {24614, 37634}, {26965, 27067}, {29663, 53128}, {30618, 38930}, {31090, 33296}, {33834, 37678}, {33838, 47355}, {36568, 41269}, {37673, 52258}, {41014, 49477}, {50063, 50095}

X(53423) =complement of X(33954)
X(53423) =X(i)-complementary conjugate of X(j) for these (i,j): {7194, 3741}, {39724, 21240}
X(53423) =X(10)-Ceva conjugate of X(21703)
X(53423) =X(29654)-Dao conjugate of X(141)
X(53423) =barycentric product X(i)*X(j) for these {i,j}: {10, 29654}, {21703, 52394}
X(53423) =barycentric quotient X(i)/X(j) for these {i,j}: {21703, 15523}, {29654, 86}
X(53423) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1213, 1834, 21024}, {1213, 23905, 16589}, {3454, 20970, 10026}

---

X(53424) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(10) AND X(98)

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

X(53424) lies on these lines: {5, 4279}, {6, 7380}, {8, 27688}, {10, 37}, {12, 20964}, {30, 51619}, {98, 230}, {115, 516}, {125, 3011}, {145, 27704}, {519, 51417}, {595, 14873}, {902, 33329}, {1211, 7081}, {1279, 8287}, {2245, 6211}, {3416, 34528}, {3685, 23947}, {3883, 46826}, {3915, 27555}, {3936, 37764}, {4039, 27713}, {4255, 17327}, {4645, 20337}, {4657, 24383}, {4660, 37159}, {4682, 17056}, {5224, 50591}, {5293, 52544}, {5846, 44396}, {5847, 10026}, {8669, 41014}, {9053, 50773}, {17259, 25446}, {17337, 17527}, {17514, 19879}, {17734, 39026}, {17766, 20546}, {23897, 50314}, {26075, 37791}, {27556, 51192}, {28350, 37050}, {31897, 49758}, {32557, 50757}, {34119, 38832}

X(53424) = X(i)-complementary conjugate of X(j) for these (i,j): {1402, 19557}, {3512, 21246}, {8852, 960}
X(53424) = crossdifference of every pair of points on line {3733, 40589}
X(53424) = barycentric product X(523)*X(53344)
X(53424) = barycentric quotient X(53344)/X(99)

---

X(53425) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(10) AND X(262)

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

X(53425) lies on these lines: {2, 50591}, {6, 6998}, {10, 37}, {12, 3778}, {39, 48888}, {262, 1513}, {386, 50418}, {407, 1874}, {442, 20338}, {978, 52544}, {1211, 3705}, {1716, 26066}, {2245, 6210}, {3120, 3136}, {3454, 51417}, {3742, 3756}, {3821, 16052}, {4255, 11110}, {4260, 37662}, {5051, 27688}, {5132, 15981}, {6675, 17337}, {12618, 31398}, {17054, 25521}, {17065, 25466}, {17277, 19312}, {20861, 31460}, {21674, 22174}, {41014, 49613}, {49489, 50775}, {49511, 49554}

X(53425) = X(1402)-complementary conjugate of X(19584)

---

X(53426) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(10) AND X(671)

Barycentrics    (b + c)*(2*a^3 + a^2*b - a*b^2 + b^3 + a^2*c - 2*b^2*c - a*c^2 - 2*b*c^2 + c^3) : :
X(53426) = 5 X[14061] - 4 X[44399]

X(53426) lies on these lines: {1, 23897}, {8, 23903}, {10, 37}, {30, 50252}, {42, 23917}, {43, 23918}, {80, 6543}, {99, 44379}, {115, 519}, {145, 23942}, {148, 17731}, {239, 23947}, {316, 524}, {495, 5949}, {543, 6629}, {644, 2238}, {758, 20720}, {1211, 29615}, {1575, 16613}, {2245, 41322}, {2533, 4024}, {2802, 5164}, {3017, 24275}, {3626, 6537}, {3936, 6542}, {4037, 21711}, {4062, 21057}, {4971, 44396}, {6625, 20090}, {14061, 44399}, {17056, 29574}, {17299, 34528}, {20461, 21888}, {27081, 51353}, {28309, 50773}, {30860, 41140}, {31057, 40891}, {33329, 50756}, {35068, 49772}

X(53426) = midpoint of X(148) and X(17731)
X(53426) = reflection of X(i) in X(j) for these {i,j}: {99, 44379}, {10026, 115}
X(53426) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {13610, 14360}, {18757, 8591}
X(53426) = X(7312)-complementary conjugate of X(3741)
X(53426) = X(50755)-Dao conjugate of X(524)
X(53426) = barycentric product X(i)*X(j) for these {i,j}: {10, 50755}, {523, 53341}
X(53426) = barycentric quotient X(i)/X(j) for these {i,j}: {50755, 86}, {53341, 99}

---

X(53427) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(10) AND X(1029)

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

X(53427) lies on these lines: {1, 5949}, {6, 5046}, {10, 37}, {80, 11069}, {81, 1029}, {115, 1100}, {338, 17863}, {1211, 4478}, {1449, 8818}, {2298, 11604}, {3589, 23947}, {3946, 8287}, {4360, 44396}, {4852, 46826}, {10026, 32844}, {16679, 46536}, {17019, 17056}, {17160, 27707}, {17318, 27688}, {17388, 34528}, {17390, 20337}, {17398, 23897}, {17737, 23917}, {21353, 36250}, {32101, 41809}, {38814, 44378}, {38930, 52405}

X(53427) = X(i)-complementary conjugate of X(j) for these (i,j): {512, 5952}, {1402, 13089}, {5606, 512}

---

X(53428) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(13) AND X(76)

Barycentrics    a^4*b^2 + 4*a^2*b^4 + b^6 + 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 + 2*Sqrt[3]*(a^2*b^2 + b^4 + a^2*c^2 + c^4)*S : :
X(53428) = 3 X[37351] - X[52194]

X(53428) lies on these lines: {2, 11122}, {3, 10616}, {5, 3106}, {6, 622}, {13, 15}, {39, 624}, {61, 20429}, {76, 141}, {115, 623}, {182, 397}, {230, 5980}, {298, 19570}, {299, 7924}, {395, 3643}, {398, 3564}, {511, 36251}, {574, 23302}, {626, 33466}, {630, 31652}, {636, 7765}, {736, 41630}, {3642, 11648}, {3818, 5321}, {3933, 22913}, {5334, 11180}, {5469, 22894}, {6108, 36755}, {8259, 16629}, {9735, 16772}, {10654, 48656}, {11133, 22847}, {11300, 16644}, {14568, 14904}, {15048, 23000}, {18582, 37332}, {22236, 41039}, {22510, 52650}, {22512, 41753}, {23303, 43291}, {24206, 36252}, {31695, 45879}, {33475, 52691}, {35917, 44532}, {36997, 42147}, {41129, 42511}

X(53428) = {X(3643),X(5309)}-harmonic conjugate of X(395)

---

X(53429) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(13) AND X(83)

Barycentrics    4*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 + 2*Sqrt[3]*(a^2*b^2 + b^4 + a^2*c^2 + c^4)*S : :
X(53429) = X[621] - 3 X[11303], 4 X[6671] - 3 X[37340], 2 X[623] - 3 X[37352]

X(53429) lies on these lines: {2, 10617}, {5, 36759}, {6, 621}, {13, 15}, {17, 20429}, {83, 316}, {115, 31713}, {140, 39554}, {187, 624}, {302, 7785}, {303, 384}, {385, 51272}, {395, 623}, {397, 511}, {398, 18583}, {524, 39404}, {531, 39593}, {622, 2076}, {1506, 6672}, {3934, 33468}, {5321, 19130}, {5340, 48872}, {5611, 40693}, {5978, 9300}, {6109, 22797}, {6661, 44382}, {7684, 42166}, {7767, 22915}, {7809, 44383}, {8259, 10613}, {9117, 47857}, {10611, 41061}, {11299, 16644}, {12205, 46054}, {13350, 16772}, {14138, 21401}, {14538, 42148}, {14712, 19781}, {16044, 51265}, {16627, 42598}, {18582, 37333}, {20252, 22510}, {21158, 42945}, {22236, 36993}, {22687, 41752}, {22894, 22998}, {23303, 40334}, {30459, 52867}, {31711, 52193}, {33475, 35942}, {36368, 43229}, {36755, 42943}, {36992, 42164}, {41039, 42156}, {49947, 51484}

X(53429) = reflection of X(42147) in X(10613)
X(53429) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 19780, 10617}, {20428, 36757, 398}

---

X(53430) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(13) AND X(98)

Barycentrics    Sqrt[3]*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(3*a^4 + b^4 - 2*b^2*c^2 + c^4) + 2*(4*a^6 - a^4*b^2 - b^6 - a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + b^2*c^4 - c^6)*S : :
X(53430) = 3 X[13] - X[23005], 3 X[16] - X[25235], 3 X[6778] + X[25235], 3 X[395] - 2 X[6782], 3 X[6108] - X[6782], 3 X[5469] - 2 X[43417], 3 X[5470] - X[36970], X[6777] - 3 X[22511], 2 X[11543] - 3 X[22511], 2 X[33517] + X[42088]

X(53440) lies on these lines: {2, 51159}, {5, 46054}, {6, 383}, {13, 15}, {14, 44289}, {16, 5613}, {39, 397}, {98, 230}, {115, 5321}, {140, 36766}, {141, 5980}, {187, 41023}, {395, 542}, {398, 5305}, {511, 6783}, {524, 5978}, {530, 619}, {616, 11481}, {622, 35918}, {2784, 50439}, {2794, 41045}, {3003, 40666}, {3053, 5869}, {3480, 23283}, {3564, 22509}, {3830, 16943}, {5092, 6115}, {5340, 44519}, {5459, 31694}, {5469, 43417}, {5470, 36970}, {5471, 7685}, {5477, 41071}, {5478, 42102}, {5617, 18358}, {5868, 13881}, {5979, 44382}, {6109, 11645}, {6582, 52194}, {6672, 32553}, {6777, 11543}, {6781, 41070}, {9763, 35932}, {9862, 19781}, {10616, 12042}, {13103, 42086}, {14181, 46818}, {14904, 35297}, {16001, 42165}, {16310, 40665}, {16808, 20252}, {16809, 22846}, {18581, 22847}, {20415, 42166}, {22489, 43104}, {22512, 41407}, {22687, 37341}, {22796, 42107}, {22998, 42913}, {23006, 42148}, {23013, 41753}, {30468, 32461}, {30471, 42120}, {31710, 42940}, {32552, 52193}, {32596, 44453}, {33388, 41035}, {33517, 36756}, {35751, 42792}, {36251, 42147}, {36318, 49948}, {36344, 49906}, {36383, 43229}, {36771, 42598}, {36961, 42101}, {41746, 43274}, {42117, 46855}, {42118, 47859}

X(53430) = midpoint of X(i) and X(j) for these {i,j}: {16, 6778}, {383, 6770}, {35932, 51482}
X(53430) = reflection of X(i) in X(j) for these {i,j}: {395, 6108}, {5321, 115}, {5617, 52263}, {5979, 44382}, {6777, 11543}, {22998, 42913}, {31694, 5459}, {32553, 6672}, {42940, 31710}, {52193, 32552}
X(53430) = crossdifference of every pair of points on line {40580, 41167}
X(53430) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 22513, 5318}, {6115, 6771, 23302}, {6777, 22511, 11543}

---

X(53431) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(13) AND X(262)

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

X(53431) lies on these lines: {2, 51162}, {5, 3105}, {6, 1080}, {13, 15}, {14, 52649}, {16, 52650}, {32, 397}, {39, 51753}, {115, 7684}, {262, 1513}, {325, 51161}, {395, 5476}, {511, 6115}, {617, 9763}, {623, 51388}, {624, 7853}, {1506, 51754}, {3098, 23302}, {3642, 31693}, {5321, 5475}, {5335, 19781}, {5472, 41023}, {5479, 43457}, {5613, 16808}, {5617, 51206}, {6114, 19130}, {6298, 52194}, {6782, 44498}, {7603, 7685}, {9301, 20425}, {9762, 51010}, {14541, 42598}, {15048, 22708}, {16943, 22907}, {22686, 48876}, {22796, 33518}, {23303, 52266}, {24206, 33462}, {35304, 42943}, {35931, 42155}, {41754, 47860}, {42166, 47068}, {44667, 47861}

---

X(53432) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(13) AND X(485)

Barycentrics    (-2 + Sqrt[3])*a^4 + (b^2 - c^2)^2 - a^2*((-1 + Sqrt[3])*(b^2 + c^2) + 2*S) : :

X(53432) lies on these lines: {2, 42220}, {3, 485}, {5, 3390}, {6, 2044}, {13, 15}, {16, 34551}, {61, 42215}, {115, 31699}, {140, 3366}, {371, 42147}, {372, 397}, {381, 42190}, {382, 42255}, {395, 35822}, {398, 3364}, {486, 42279}, {546, 3367}, {550, 3389}, {615, 18582}, {639, 33448}, {1152, 2042}, {2041, 23251}, {2043, 11480}, {2045, 6410}, {2046, 42265}, {3068, 42119}, {3069, 35732}, {3071, 40693}, {3312, 42254}, {3365, 42148}, {3391, 42157}, {3392, 13966}, {3543, 42218}, {3627, 42236}, {3830, 42188}, {3845, 42182}, {5321, 6564}, {5334, 36445}, {6306, 32421}, {6395, 42195}, {6396, 15765}, {6398, 42198}, {6561, 11485}, {6565, 42166}, {7584, 42235}, {7586, 42217}, {8981, 42434}, {10645, 34552}, {10653, 41946}, {10654, 32787}, {11488, 52400}, {13785, 42187}, {13846, 42626}, {14538, 36762}, {14814, 35820}, {15687, 42180}, {15764, 42170}, {16241, 50246}, {16644, 36437}, {16772, 42241}, {16773, 42565}, {16808, 51853}, {16809, 34562}, {16966, 34559}, {18510, 42246}, {18538, 23303}, {18587, 42085}, {22236, 42263}, {22611, 49220}, {22644, 42278}, {31412, 52401}, {32788, 42191}, {34754, 42225}, {35256, 35738}, {35731, 42118}, {35733, 42924}, {35823, 42211}, {36448, 41119}, {36450, 43386}, {36454, 43403}, {36455, 42154}, {36457, 42169}, {36468, 43257}, {36968, 52214}, {37832, 52215}, {41034, 45544}, {42088, 42202}, {42116, 42276}, {42117, 51728}, {42128, 42283}, {42130, 42192}, {42132, 52046}, {42150, 42230}, {42152, 42244}, {42162, 42251}, {42163, 42564}, {42173, 43209}, {42242, 42273}, {42271, 42988}, {42282, 52667}, {42637, 52402}

X(53432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {485, 42249, 3070}, {485, 42256, 43879}, {485, 42257, 42253}, {615, 42194, 18586}, {3366, 35739, 140}, {3390, 42238, 5}, {11480, 42264, 2043}, {18582, 42171, 42194}, {18585, 42214, 6564}, {34551, 42216, 16}, {40693, 42245, 3071}, {42253, 42259, 42257}

---

X(53433) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(13) AND X(486)

Barycentrics    (2 + Sqrt[3])*a^4 - (b^2 - c^2)^2 - a^2*((1 + Sqrt[3])*b^2 + (1 + Sqrt[3])*c^2 + 2*S) : :

X(53433) lies on these lines: {2, 42218}, {3, 486}, {5, 3389}, {6, 2043}, {13, 15}, {16, 34552}, {61, 42216}, {115, 31697}, {140, 3367}, {371, 397}, {372, 42147}, {381, 42188}, {382, 42257}, {395, 35823}, {398, 3365}, {485, 42278}, {546, 3366}, {548, 35739}, {550, 3390}, {590, 18582}, {640, 33449}, {1151, 2041}, {2042, 23261}, {2044, 11480}, {2045, 42262}, {2046, 6409}, {3068, 42282}, {3069, 42119}, {3070, 40693}, {3311, 42256}, {3364, 42148}, {3391, 8981}, {3392, 42157}, {3543, 42220}, {3627, 42238}, {3830, 42190}, {3845, 42180}, {5321, 6565}, {5334, 36463}, {5335, 9541}, {6199, 42197}, {6200, 18585}, {6221, 42196}, {6302, 32419}, {6560, 11485}, {6564, 42166}, {7583, 42237}, {7585, 42219}, {10645, 34551}, {10653, 41945}, {10654, 32788}, {11488, 52399}, {13665, 42189}, {13847, 42626}, {13966, 42434}, {14813, 35821}, {15687, 42182}, {15764, 42211}, {16644, 36455}, {16772, 35740}, {16773, 42564}, {16808, 51855}, {16809, 34559}, {16966, 34562}, {18512, 42248}, {18586, 42085}, {18762, 23303}, {22236, 42264}, {22615, 42279}, {22640, 49221}, {32787, 42193}, {34754, 42226}, {35255, 42598}, {35731, 42124}, {35732, 52666}, {35738, 42164}, {35822, 42213}, {36436, 43403}, {36437, 42154}, {36439, 42170}, {36449, 43256}, {36466, 41119}, {36467, 43387}, {36968, 52215}, {37832, 52214}, {41034, 45545}, {42088, 42201}, {42116, 42275}, {42118, 51728}, {42128, 42284}, {42130, 42194}, {42132, 52045}, {42150, 42229}, {42152, 42245}, {42162, 42253}, {42163, 42565}, {42169, 42943}, {42174, 43210}, {42243, 42270}, {42272, 42988}, {42561, 52402}, {42638, 52401}

X(53433) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {486, 42247, 3071}, {486, 42254, 43880}, {486, 42255, 42251}, {590, 42192, 18587}, {3389, 42236, 5}, {11480, 42263, 2044}, {15765, 42212, 6565}, {18582, 42172, 42192}, {34552, 42215, 16}, {40693, 42244, 3070}, {42237, 51727, 7583}, {42251, 42258, 42255}

---

X(53434) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(13) AND X(598)

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

X(53434) lies on these lines: {6, 51483}, {13, 15}, {148, 37786}, {187, 5459}, {382, 8259}, {395, 5475}, {397, 576}, {531, 5472}, {597, 598}, {623, 9115}, {1080, 9756}, {3849, 40671}, {5321, 5476}, {5460, 43457}, {5615, 10653}, {6781, 45879}, {6783, 32907}, {7603, 45880}, {7684, 41061}, {8588, 23302}, {8598, 33475}, {9117, 31695}, {9736, 42943}, {10616, 18582}, {11549, 16179}, {11648, 43228}, {16644, 35931}, {16808, 52649}, {20428, 51200}, {20429, 30559}, {23004, 41746}, {33623, 49905}, {35943, 44382}, {41035, 42166}, {41745, 50855}, {41754, 46854}, {42102, 48884}, {43104, 52266}

---

X(53435) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(13) AND X(671)

Barycentrics    Sqrt[3]*(3*a^4*b^2 - 4*a^2*b^4 - b^6 + 3*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) + 2*(4*a^4 - a^2*b^2 - 5*b^4 - a^2*c^2 + 8*b^2*c^2 - 5*c^4)*S : :
X(53435) = X[16] - 3 X[5470], 3 X[5470] + X[25156], 3 X[115] - X[9115], 3 X[395] - 2 X[9115], X[298] - 3 X[14041], X[5321] + 2 X[33517], 3 X[5469] - 2 X[11543], 3 X[5469] - X[22998], X[6779] - 3 X[22511], 3 X[22511] - 2 X[42913], X[8595] - 3 X[9166], 3 X[33228] - 2 X[44383], 3 X[16268] - X[25235], 3 X[16530] - 4 X[42497], X[20426] - 3 X[38732], X[37785] - 3 X[41135]

X(53435) lies on these lines: {6, 51482}, {13, 15}, {14, 11603}, {16, 5470}, {99, 44382}, {115, 395}, {148, 299}, {298, 14041}, {316, 524}, {397, 575}, {398, 51200}, {511, 25222}, {532, 39563}, {542, 5321}, {547, 36766}, {549, 46054}, {616, 16645}, {625, 33460}, {2549, 22492}, {3849, 22573}, {5459, 8589}, {5461, 52021}, {5463, 23303}, {5469, 11543}, {5472, 14537}, {5478, 22682}, {6036, 6108}, {6115, 44219}, {6390, 22570}, {6770, 41039}, {6771, 44250}, {6775, 50858}, {6777, 43417}, {6778, 36970}, {6779, 22511}, {7748, 34509}, {7804, 40671}, {8595, 9166}, {8597, 37786}, {9763, 44526}, {10616, 42088}, {10653, 13103}, {11121, 40898}, {11122, 11606}, {11304, 51159}, {14447, 20578}, {14904, 33228}, {16242, 22846}, {16268, 25235}, {16530, 42497}, {16644, 35932}, {16772, 20415}, {16808, 44289}, {20252, 37832}, {20426, 38732}, {22574, 36523}, {22575, 51019}, {22687, 37351}, {22892, 35019}, {25164, 51207}, {32479, 52022}, {35749, 49948}, {35750, 49906}, {35752, 41745}, {36962, 41753}, {37785, 41135}, {37975, 52040}, {39838, 41022}, {41020, 42164}, {41034, 42166}, {42102, 48895}, {43104, 52263}

X(53435) = midpoint of X(i) and X(j) for these {i,j}: {13, 23005}, {16, 25156}, {148, 299}, {6778, 36970}, {8597, 37786}, {31710, 33517}
X(53435) = reflection of X(i) in X(j) for these {i,j}: {99, 44382}, {395, 115}, {5321, 31710}, {6777, 43417}, {6779, 42913}, {22574, 36523}, {22998, 11543}, {32553, 33561}, {35303, 5459}, {41017, 5478}, {42943, 6108}, {44223, 20252}, {44250, 6771}, {52021, 5461}
X(53435) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 6772, 396}, {13, 25154, 5318}, {5469, 22998, 11543}, {5470, 25156, 16}, {6779, 22511, 42913}, {16001, 36251, 397}

---

X(53436) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(13) AND X(1131)

Barycentrics    (-8 + 5*Sqrt[3])*a^4 - (-4 + 3*Sqrt[3])*(b^2 - c^2)^2 + a^2*(-2*(-2 + Sqrt[3])*b^2 - 2*(-2 + Sqrt[3])*c^2 + 2*S) : :

X(53436) lies on these lines: {13, 15}, {382, 42240}, {395, 36454}, {397, 3312}, {398, 19117}, {590, 42088}, {615, 35732}, {1131, 3068}, {1587, 42250}, {2042, 3070}, {2044, 42110}, {3391, 3627}, {5073, 42230}, {5321, 13665}, {5335, 42194}, {5340, 42259}, {6412, 23302}, {6445, 42113}, {6561, 42164}, {6564, 42163}, {10653, 42206}, {16772, 42244}, {18582, 42167}, {18586, 23303}, {23249, 42097}, {32787, 42940}, {35255, 42237}, {36437, 42170}, {36448, 42190}, {42108, 42263}, {42118, 51855}, {42128, 42186}, {42138, 51853}, {42147, 42271}, {42165, 42253}, {42174, 42943}, {42178, 42214}, {42216, 42238}, {42241, 42945}, {42247, 43516}, {42270, 42775}, {42283, 42693}

X(53436) = {X(3070),X(35740)}-harmonic conjugate of X(42148)

---

X(53437) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(13) AND X(113f2)

Barycentrics    (8 + 5*Sqrt[3])*a^4 - (4 + 3*Sqrt[3])*(b^2 - c^2)^2 - 2*a^2*((2 + Sqrt[3])*b^2 + (2 + Sqrt[3])*c^2 - S) : :

X(53437) lies on these lines: {13, 15}, {382, 42239}, {395, 36436}, {397, 3311}, {398, 19116}, {590, 42166}, {615, 42088}, {1132, 3069}, {1588, 42252}, {2041, 3071}, {2043, 42110}, {3392, 3627}, {5073, 42229}, {5321, 13785}, {5335, 42192}, {5340, 42258}, {6411, 23302}, {6446, 42113}, {6560, 42164}, {6565, 42163}, {10653, 42205}, {16772, 42245}, {18582, 42168}, {18587, 23303}, {23259, 42097}, {32788, 42940}, {35256, 42235}, {35740, 42945}, {36455, 42169}, {36466, 42188}, {42108, 42264}, {42118, 51853}, {42128, 42185}, {42138, 51855}, {42147, 42272}, {42165, 42251}, {42173, 42943}, {42177, 42212}, {42215, 42236}, {42249, 43515}, {42273, 42775}, {42284, 42693}, {43316, 51727}

X(53437) = {X(3071),X(42241)}-harmonic conjugate of X(42148)

---

X(53438) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(13) AND X(1327)

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

X(53438) lies on these lines: {13, 15}, {115, 31715}, {381, 42187}, {382, 42249}, {395, 6564}, {397, 6420}, {590, 42086}, {615, 42128}, {1327, 3830}, {2042, 5340}, {2043, 6409}, {2044, 42262}, {3070, 10653}, {3071, 42974}, {5321, 35822}, {5335, 36454}, {10576, 15765}, {14813, 42166}, {15682, 36468}, {15764, 23302}, {16242, 34559}, {16965, 51855}, {18585, 35787}, {18587, 42271}, {23259, 36449}, {23267, 36467}, {32788, 36448}, {33441, 44383}, {34551, 37832}, {35734, 42950}, {35738, 42148}, {35821, 42940}, {36437, 42155}, {36439, 42178}, {36450, 49873}, {36455, 42094}, {36457, 42088}, {36466, 43210}, {36469, 42201}, {36970, 42215}, {37640, 52666}, {41107, 42216}, {41945, 42105}, {41946, 42194}, {42161, 42253}, {42165, 42234}, {42193, 52045}, {42198, 53131}, {42235, 51854}, {42239, 43418}, {42250, 42989}, {42254, 42273}, {42813, 51853}

---

X(53439) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(13) AND X(1328)

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

X(53439) lies on these lines: {13, 15}, {115, 31717}, {381, 42189}, {382, 42247}, {395, 6565}, {397, 6419}, {590, 42128}, {615, 42086}, {1328, 3830}, {2041, 5340}, {2043, 42265}, {2044, 6410}, {3070, 42974}, {3071, 10653}, {5321, 35823}, {5335, 36436}, {10577, 18585}, {14814, 42166}, {15682, 36449}, {15765, 35786}, {16242, 34562}, {16965, 51853}, {18586, 42272}, {23249, 36468}, {23273, 36450}, {32787, 36466}, {33440, 44383}, {34552, 37832}, {35820, 42940}, {36437, 42094}, {36439, 42088}, {36448, 43209}, {36453, 42202}, {36455, 42155}, {36457, 42177}, {36467, 49873}, {36970, 42216}, {37640, 52667}, {41107, 42215}, {41945, 42192}, {41946, 42105}, {42161, 42251}, {42165, 42233}, {42191, 52046}, {42196, 53130}, {42237, 51852}, {42240, 43418}, {42252, 42989}, {42256, 42270}, {42813, 51855}

---

X(53440) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(14) AND X(76)

Barycentrics    a^4*b^2 + 4*a^2*b^4 + b^6 + 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 - 2*Sqrt[3]*(a^2*b^2 + b^4 + a^2*c^2 + c^4)*S : :
X(53440) = 3 X[37352] - X[52193]

X(53440) lies on these lines: {2, 11121}, {3, 10617}, {5, 3107}, {6, 621}, {14, 16}, {39, 623}, {62, 20428}, {76, 141}, {115, 624}, {182, 398}, {230, 5981}, {298, 7924}, {299, 19570}, {396, 3642}, {397, 3564}, {511, 36252}, {574, 23303}, {626, 33467}, {629, 31652}, {635, 7765}, {736, 41640}, {3643, 11648}, {3818, 5318}, {3933, 22868}, {5335, 11180}, {5470, 22850}, {6109, 36756}, {8260, 16628}, {9736, 16773}, {10653, 48655}, {11132, 22893}, {11299, 16645}, {14568, 14905}, {15048, 23009}, {18581, 37333}, {22238, 41038}, {22511, 44223}, {22513, 41751}, {23302, 43291}, {24206, 36251}, {31696, 45880}, {33474, 52691}, {35918, 44532}, {36997, 42148}, {41128, 42510}

X(53440) = {X(3642),X(5309)}-harmonic conjugate of X(396)

---

X(53441) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(14) AND X(83)

Barycentrics    4*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 - 2*Sqrt[3]*(a^2*b^2 + b^4 + a^2*c^2 + c^4)*S : :
X(53441) = X[622] - 3 X[11304], 4 X[6672] - 3 X[37341], 2 X[624] - 3 X[37351]

X(53441) lies on these lines: {2, 10616}, {5, 36760}, {6, 622}, {14, 16}, {18, 20428}, {83, 316}, {115, 31714}, {140, 39555}, {187, 623}, {302, 384}, {303, 7785}, {385, 51265}, {396, 624}, {397, 18583}, {398, 511}, {524, 39405}, {530, 39593}, {621, 2076}, {1506, 6671}, {3934, 33469}, {5318, 19130}, {5339, 48872}, {5615, 40694}, {5979, 9300}, {6108, 22796}, {6661, 44383}, {7685, 42163}, {7767, 22870}, {7809, 44382}, {8260, 10614}, {9115, 47858}, {10612, 41060}, {11300, 16645}, {12204, 46053}, {13349, 16773}, {14139, 21402}, {14539, 42147}, {14712, 19780}, {16044, 51272}, {16626, 42599}, {18581, 37332}, {20253, 22511}, {21159, 42944}, {22238, 36995}, {22689, 41754}, {22850, 22997}, {23302, 40335}, {30462, 52868}, {31712, 52194}, {33474, 35943}, {36366, 43228}, {36756, 42942}, {36994, 42165}, {41038, 42153}, {49948, 51485}

X(53441) = reflection of X(42148) in X(10614)
X(53441) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 19781, 10616}, {20429, 36758, 397}

---

X(53442) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(14) AND X(98)

Barycentrics    Sqrt[3]*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(3*a^4 + b^4 - 2*b^2*c^2 + c^4) - 2*(4*a^6 - a^4*b^2 - b^6 - a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + b^2*c^4 - c^6)*S : :
X(53442) = 3 X[14] - X[23004], 3 X[15] - X[25236], 3 X[6777] + X[25236], 3 X[396] - 2 X[6783], 3 X[6109] - X[6783], 3 X[5469] - X[36969], 3 X[5470] - 2 X[43416], X[6778] - 3 X[22510], 2 X[11542] - 3 X[22510], 2 X[33518] + X[42087]

X(53442) lies on these lines: {2, 51160}, {5, 46053}, {6, 1080}, {13, 52649}, {14, 16}, {15, 5617}, {39, 398}, {98, 230}, {115, 5318}, {141, 5981}, {187, 41022}, {396, 542}, {397, 5305}, {511, 6782}, {524, 5979}, {531, 618}, {617, 11480}, {621, 35917}, {2784, 50438}, {2794, 41044}, {3003, 40665}, {3053, 5868}, {3479, 23284}, {3564, 22507}, {3830, 16942}, {5092, 6114}, {5339, 44519}, {5460, 31693}, {5469, 36969}, {5470, 43416}, {5472, 7684}, {5477, 41070}, {5479, 42101}, {5613, 18358}, {5869, 13881}, {5978, 44383}, {6108, 11645}, {6295, 52193}, {6671, 32552}, {6778, 11542}, {6781, 41071}, {9761, 35931}, {9862, 19780}, {10617, 12042}, {13102, 42085}, {14177, 46818}, {14905, 35297}, {16002, 42164}, {16310, 40666}, {16808, 22891}, {16809, 20253}, {18582, 22893}, {20416, 42163}, {22490, 43101}, {22513, 41406}, {22689, 37340}, {22797, 42110}, {22997, 42912}, {23006, 41751}, {23013, 42147}, {30465, 32460}, {30472, 42119}, {31709, 42941}, {32553, 52194}, {32597, 44453}, {33389, 41034}, {33518, 36755}, {36252, 42148}, {36319, 49905}, {36320, 49947}, {36329, 42791}, {36382, 43228}, {36962, 42102}, {41745, 43275}, {42117, 47860}, {42118, 46854}

X(53442) = midpoint of X(i) and X(j) for these {i,j}: {15, 6777}, {1080, 6773}, {35931, 51483}
X(53442) = reflection of X(i) in X(j) for these {i,j}: {396, 6109}, {5318, 115}, {5613, 52266}, {5978, 44383}, {6778, 11542}, {22997, 42912}, {31693, 5460}, {32552, 6671}, {42941, 31709}, {52194, 32553}
X(53442) = crossdifference of every pair of points on line {40581, 41167}
X(53442) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 22512, 5321}, {6114, 6774, 23303}, {6778, 22510, 11542}

---

X(53443) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(14) AND X(262)

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

X(53443) lies on these lines: {2, 51161}, {5, 3104}, {6, 383}, {13, 44289}, {14, 16}, {15, 44223}, {32, 398}, {39, 51754}, {115, 7685}, {262, 1513}, {325, 51162}, {396, 5476}, {511, 6114}, {616, 9761}, {623, 7853}, {624, 51387}, {1506, 51753}, {3098, 23303}, {3643, 31694}, {5318, 5475}, {5334, 19780}, {5471, 41022}, {5478, 43457}, {5613, 51207}, {5617, 16809}, {6115, 19130}, {6299, 52193}, {6783, 44497}, {7603, 7684}, {9301, 20426}, {9760, 51013}, {14540, 42599}, {15048, 22707}, {16942, 22861}, {22684, 48876}, {22797, 33517}, {23302, 52263}, {24206, 33463}, {35303, 42942}, {35932, 42154}, {41752, 47859}, {42087, 44250}, {42163, 47066}, {44666, 47862}

---

X(53444) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(14) AND X(485)

Barycentrics    (2 + Sqrt[3])*a^4 - (b^2 - c^2)^2 - a^2*((1 + Sqrt[3])*b^2 + (1 + Sqrt[3])*c^2 - 2*S) : :

X(53444) lies on these lines: {2, 42219}, {3, 485}, {5, 3365}, {6, 2043}, {14, 16}, {15, 34552}, {18, 35739}, {62, 42215}, {115, 31700}, {140, 3391}, {371, 42148}, {372, 398}, {381, 42189}, {382, 42254}, {396, 35822}, {397, 3389}, {486, 42278}, {546, 3392}, {550, 3364}, {615, 18581}, {639, 33450}, {1152, 2041}, {2042, 23251}, {2044, 11481}, {2045, 42265}, {2046, 6410}, {3068, 42120}, {3069, 42282}, {3071, 40694}, {3312, 42255}, {3366, 35740}, {3367, 13966}, {3390, 42147}, {3543, 42217}, {3627, 42235}, {3830, 42187}, {3845, 42181}, {5318, 6564}, {5335, 36463}, {6307, 32421}, {6395, 42196}, {6396, 18585}, {6398, 42197}, {6561, 11486}, {6565, 42163}, {7584, 42236}, {7586, 42218}, {8981, 42433}, {10646, 34551}, {10653, 32787}, {10654, 41946}, {11489, 52399}, {13785, 42188}, {13846, 42625}, {14813, 35820}, {15687, 42179}, {15764, 42214}, {16645, 36455}, {16772, 42563}, {16773, 42239}, {16808, 34559}, {16809, 51852}, {16967, 34562}, {18510, 42247}, {18538, 23302}, {18586, 42086}, {19117, 51727}, {22238, 42263}, {22612, 49220}, {22644, 42279}, {31412, 52402}, {32788, 42192}, {34755, 42225}, {35256, 42599}, {35731, 43106}, {35732, 52667}, {35738, 42165}, {35823, 42212}, {36436, 43404}, {36437, 42155}, {36439, 42167}, {36450, 43257}, {36466, 41120}, {36468, 43386}, {36967, 52217}, {37835, 52216}, {41035, 45544}, {42087, 42200}, {42115, 42276}, {42125, 42283}, {42129, 52046}, {42131, 42191}, {42149, 42242}, {42151, 42228}, {42159, 42250}, {42166, 42562}, {42168, 42942}, {42171, 43209}, {42244, 42273}, {42271, 42989}, {42637, 52401}

X(53444) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {485, 42248, 3070}, {485, 42256, 42252}, {485, 42257, 43879}, {615, 42193, 18587}, {3365, 42237, 5}, {11481, 42264, 2044}, {15765, 42213, 6564}, {18581, 42173, 42193}, {34552, 42216, 15}, {40694, 42243, 3071}, {42252, 42259, 42256}

---

X(53445) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(14) AND X(486)

Barycentrics    (-2 + Sqrt[3])*a^4 + (b^2 - c^2)^2 - a^2*((-1 + Sqrt[3])*(b^2 + c^2) - 2*S) : :

X(53445) lies on these lines: {2, 42217}, {3, 486}, {5, 3364}, {6, 2044}, {14, 16}, {15, 34551}, {62, 42216}, {115, 31698}, {140, 3392}, {371, 398}, {372, 42148}, {381, 42187}, {382, 42256}, {396, 35823}, {397, 3390}, {485, 42279}, {511, 35759}, {546, 3391}, {550, 3365}, {590, 18581}, {640, 33451}, {1151, 2042}, {2041, 23261}, {2043, 11481}, {2045, 6409}, {2046, 42262}, {3068, 35732}, {3069, 42120}, {3070, 40694}, {3311, 42257}, {3366, 8981}, {3367, 42158}, {3389, 42147}, {3543, 42219}, {3627, 42237}, {3830, 42189}, {3845, 42179}, {5318, 6565}, {5334, 9541}, {5335, 36445}, {6199, 42198}, {6200, 15765}, {6221, 42195}, {6303, 32419}, {6560, 11486}, {6564, 42163}, {7583, 42238}, {7585, 42220}, {10646, 34552}, {10653, 32788}, {10654, 41945}, {11489, 52400}, {13665, 42190}, {13847, 42625}, {13966, 35739}, {14814, 35821}, {15687, 42181}, {15764, 42167}, {16645, 36437}, {16772, 42562}, {16773, 42240}, {16808, 34562}, {16809, 51854}, {16967, 34559}, {18512, 42249}, {18587, 42086}, {18762, 23302}, {22238, 42264}, {22615, 42278}, {22641, 49221}, {32787, 42194}, {34755, 42226}, {35255, 35738}, {35733, 43009}, {35822, 42214}, {36448, 41120}, {36449, 43387}, {36454, 43404}, {36455, 42155}, {36457, 42168}, {36467, 43256}, {36967, 52216}, {37835, 52217}, {41035, 45545}, {42087, 42199}, {42115, 42275}, {42125, 42284}, {42129, 52045}, {42131, 42193}, {42149, 42243}, {42151, 42227}, {42159, 42252}, {42166, 42563}, {42172, 43210}, {42245, 42270}, {42272, 42989}, {42282, 52666}, {42561, 52401}, {42638, 52402}

X(53445) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {486, 42246, 3071}, {486, 42254, 42250}, {486, 42255, 43880}, {590, 42191, 18586}, {3364, 42235, 5}, {11481, 42263, 2043}, {18581, 42174, 42191}, {18585, 42211, 6565}, {34551, 42215, 15}, {40694, 42242, 3070}, {42250, 42258, 42254}

---

X(53446) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(14) AND X(598)

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

X(53446) lies on these lines: {6, 51482}, {14, 16}, {148, 37785}, {187, 5460}, {382, 8260}, {383, 9756}, {396, 5475}, {398, 576}, {530, 5471}, {597, 598}, {624, 9117}, {3849, 40672}, {5318, 5476}, {5459, 43457}, {5611, 10654}, {6781, 45880}, {6782, 32909}, {7603, 45879}, {7685, 41060}, {8588, 23303}, {8598, 33474}, {9115, 31696}, {9735, 42942}, {10617, 18581}, {11537, 16180}, {11648, 43229}, {16645, 35932}, {16809, 44289}, {20428, 30560}, {20429, 51203}, {23005, 41745}, {33625, 49906}, {35942, 44383}, {41034, 42163}, {41746, 50858}, {41752, 46855}, {42101, 48884}, {43101, 52263}

---

X(53447) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(14) AND X(671)

Barycentrics    Sqrt[3]*(3*a^4*b^2 - 4*a^2*b^4 - b^6 + 3*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) - 2*(4*a^4 - a^2*b^2 - 5*b^4 - a^2*c^2 + 8*b^2*c^2 - 5*c^4)*S : :
X(53447) = X[15] - 3 X[5469], 3 X[5469] + X[25166], 3 X[115] - X[9117], 3 X[396] - 2 X[9117], X[299] - 3 X[14041], X[5318] + 2 X[33518], 3 X[5470] - 2 X[11542], 3 X[5470] - X[22997], X[6780] - 3 X[22510], 3 X[22510] - 2 X[42912], X[8594] - 3 X[9166], 3 X[33228] - 2 X[44382], 3 X[16267] - X[25236], 3 X[16529] - 4 X[42496], X[20425] - 3 X[38732], X[37786] - 3 X[41135]

X(53447) lies on these lines: {6, 51483}, {13, 11602}, {14, 16}, {15, 5469}, {99, 44383}, {115, 396}, {148, 298}, {299, 14041}, {316, 524}, {397, 51203}, {398, 575}, {511, 25221}, {533, 39563}, {542, 5318}, {549, 46053}, {617, 16644}, {625, 33461}, {2549, 22491}, {3849, 22574}, {5460, 8589}, {5461, 52022}, {5464, 23302}, {5470, 11542}, {5471, 14537}, {5479, 22682}, {6036, 6109}, {6390, 22568}, {6772, 50855}, {6773, 41038}, {6777, 36969}, {6778, 43416}, {6780, 22510}, {7748, 34508}, {7804, 40672}, {8594, 9166}, {8597, 37785}, {9761, 44526}, {10617, 42087}, {10654, 13102}, {11121, 11606}, {11122, 40899}, {11303, 51160}, {14446, 20579}, {14905, 33228}, {16241, 22891}, {16267, 25236}, {16529, 42496}, {16645, 35931}, {16773, 20416}, {16809, 52649}, {20253, 37835}, {20425, 38732}, {22573, 36523}, {22576, 51017}, {22689, 37352}, {22848, 35020}, {25154, 51206}, {32479, 52021}, {36327, 49947}, {36330, 41746}, {36331, 49905}, {36961, 41751}, {37786, 41135}, {37974, 52039}, {39838, 41023}, {41021, 42165}, {41035, 42163}, {42101, 48895}, {43101, 52266}

X(53447) = midpoint of X(i) and X(j) for these {i,j}: {14, 23004}, {15, 25166}, {148, 298}, {6777, 36969}, {8597, 37785}, {31709, 33518}
X(53447) = reflection of X(i) in X(j) for these {i,j}: {99, 44383}, {396, 115}, {5318, 31709}, {6778, 43416}, {6780, 42912}, {22573, 36523}, {22997, 11542}, {32552, 33560}, {35304, 5460}, {41016, 5479}, {42942, 6109}, {52022, 5461}, {52650, 20253}
X(53447) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 6775, 395}, {14, 25164, 5321}, {5469, 25166, 15}, {5470, 22997, 11542}, {6780, 22510, 42912}, {16002, 36252, 398}

---

X(53448) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(14) AND X(1131)

Barycentrics    (8 + 5*Sqrt[3])*a^4 - (4 + 3*Sqrt[3])*(b^2 - c^2)^2 - 2*a^2*((2 + Sqrt[3])*b^2 + (2 + Sqrt[3])*c^2 + S) : :

X(53448) lies on these lines: {14, 16}, {382, 35740}, {396, 36436}, {397, 19117}, {398, 3312}, {590, 42087}, {615, 42163}, {1131, 3068}, {1587, 42251}, {2041, 3070}, {2043, 42107}, {3366, 3627}, {5073, 42228}, {5318, 13665}, {5334, 42193}, {5339, 42259}, {6412, 23303}, {6445, 42112}, {6561, 42165}, {6564, 42166}, {10654, 42204}, {16773, 42242}, {18581, 42169}, {18587, 23302}, {23249, 42096}, {32787, 42941}, {35255, 42238}, {36455, 42168}, {36466, 42189}, {42109, 42263}, {42117, 51854}, {42125, 42184}, {42135, 51852}, {42148, 42271}, {42164, 42252}, {42172, 42942}, {42176, 42213}, {42216, 42237}, {42239, 42944}, {42246, 43516}, {42270, 42776}, {42283, 42692}, {43340, 51727}

X(53448) = {X(3070),X(42240)}-harmonic conjugate of X(42147)

---

X(53449) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(14) AND X(1132)

Barycentrics    (-8 + 5*Sqrt[3])*a^4 - (-4 + 3*Sqrt[3])*(b^2 - c^2)^2 - 2*a^2*((-2 + Sqrt[3])*b^2 + (-2 + Sqrt[3])*c^2 + S) : :

X(53449) lies on these lines: {14, 16}, {382, 42241}, {396, 36454}, {397, 19116}, {398, 3311}, {590, 35732}, {615, 42087}, {1132, 3069}, {1588, 42253}, {2042, 3071}, {2044, 42107}, {3367, 3627}, {5073, 42227}, {5318, 13785}, {5334, 42191}, {5339, 42258}, {6411, 23303}, {6446, 42112}, {6560, 42165}, {6565, 42166}, {10654, 42203}, {16773, 42243}, {18581, 42170}, {18586, 23302}, {23259, 42096}, {32788, 42941}, {35256, 42236}, {36437, 42167}, {36448, 42187}, {42109, 42264}, {42117, 51852}, {42125, 42183}, {42135, 51854}, {42148, 42272}, {42164, 42250}, {42171, 42942}, {42175, 42211}, {42215, 42235}, {42240, 42944}, {42248, 43515}, {42273, 42776}, {42284, 42692}

X(53449) = {X(3071),X(42239)}-harmonic conjugate of X(42147)

---

X(53450) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(14) AND X(1327)

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

X(53450) lies on these lines: {14, 16}, {115, 31718}, {381, 42188}, {382, 42248}, {396, 6564}, {398, 6420}, {590, 42085}, {615, 42125}, {1327, 3830}, {2041, 5339}, {2043, 42262}, {2044, 6409}, {3070, 10654}, {3071, 42975}, {5318, 35822}, {5334, 36436}, {10576, 18585}, {12101, 50246}, {14814, 42163}, {15682, 36450}, {15765, 35787}, {16241, 34562}, {16964, 51854}, {18586, 42271}, {23259, 36467}, {23267, 36449}, {32788, 36466}, {33443, 44382}, {34552, 37835}, {35821, 42941}, {36437, 42093}, {36439, 42087}, {36448, 43210}, {36452, 42199}, {36455, 42154}, {36457, 42176}, {36468, 49874}, {36969, 42215}, {37641, 52666}, {41108, 42216}, {41945, 42104}, {41946, 42193}, {42160, 42252}, {42164, 42232}, {42194, 52045}, {42197, 53131}, {42236, 51855}, {42241, 43419}, {42251, 42988}, {42255, 42273}, {42814, 51852}

---

X(53451) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(14) AND X(1328)

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

X(53451) lies on these lines: {14, 16}, {115, 31716}, {381, 42190}, {382, 42246}, {396, 6565}, {398, 6419}, {590, 42125}, {615, 42085}, {1328, 3830}, {2042, 5339}, {2043, 6410}, {2044, 42265}, {3070, 42975}, {3071, 10654}, {5066, 50246}, {5318, 35823}, {5334, 36454}, {10577, 15765}, {14813, 42163}, {15682, 36467}, {15764, 23303}, {16241, 34559}, {16964, 51852}, {18585, 35786}, {18587, 42272}, {23249, 36450}, {23273, 36468}, {32787, 36448}, {33442, 44382}, {34551, 37835}, {35734, 42951}, {35738, 42147}, {35740, 43419}, {35820, 42941}, {36437, 42154}, {36439, 42175}, {36449, 49874}, {36455, 42093}, {36457, 42087}, {36466, 43209}, {36470, 42200}, {36969, 42216}, {37641, 52667}, {41108, 42215}, {41945, 42191}, {41946, 42104}, {42160, 42250}, {42164, 42231}, {42192, 52046}, {42195, 53130}, {42238, 51853}, {42253, 42988}, {42257, 42270}, {42814, 51854}

---

X(53452) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(17) AND X(76)

Barycentrics    a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4 + 2*Sqrt[3]*(b^2 + c^2)*S : :

X(53452) lies on these lines: {5, 3104}, {6, 634}, {13, 42787}, {15, 32151}, {16, 17}, {32, 396}, {39, 636}, {76, 141}, {115, 635}, {298, 17129}, {299, 2896}, {303, 7824}, {395, 5305}, {398, 34507}, {616, 16644}, {618, 7749}, {619, 37512}, {622, 2076}, {623, 39565}, {624, 1506}, {626, 33483}, {1078, 14904}, {3098, 5318}, {3642, 7748}, {3933, 6581}, {5321, 18358}, {5869, 11480}, {7746, 23303}, {7811, 33458}, {7847, 14905}, {8260, 11312}, {8360, 42036}, {9982, 47610}, {11128, 22893}, {11485, 22861}, {11488, 19780}, {13881, 22847}, {14540, 42166}, {16940, 42153}, {22494, 43228}, {22512, 42147}, {22531, 43238}, {33021, 34541}, {36251, 40107}, {39560, 51159}, {42107, 43276}, {42598, 47066}

---

X(53453) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(17) AND X(83)

Barycentrics    2*a^4 + a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4 + 2*Sqrt[3]*(2*a^2 + b^2 + c^2)*S : :

X(53453) lies on these lines: {2, 34533}, {5, 46053}, {6, 633}, {15, 36959}, {16, 17}, {39, 396}, {83, 316}, {187, 6694}, {302, 7941}, {303, 7836}, {398, 25555}, {636, 5472}, {639, 33395}, {640, 33393}, {1506, 6695}, {3934, 33485}, {5092, 5318}, {5869, 42098}, {6036, 6115}, {6296, 7767}, {6680, 10617}, {7824, 10616}, {7832, 44382}, {10583, 19780}, {11303, 51160}, {16241, 52648}, {16966, 37825}, {19781, 33021}, {20394, 37512}, {36766, 42488}, {37832, 47611}

X(53453) = {X(1506),X(6695)}-harmonic conjugate of X(23303)

---

X(53454) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(17) AND X(98)

Barycentrics    2*a^6 - a^4*b^2 - b^6 - a^4*c^2 + b^4*c^2 + b^2*c^4 - c^6 + 2*Sqrt[3]*(2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*S : :
X(53454) = X[16] + 2 X[11542], 2 X[187] + X[5318], X[396] + 2 X[6108], 4 X[6672] - X[52194], X[5321] - 4 X[43291]

X(53454) lies on these lines: {3, 10616}, {4, 19781}, {5, 36760}, {6, 37464}, {13, 38230}, {16, 17}, {30, 5470}, {98, 230}, {115, 44666}, {187, 5318}, {231, 40665}, {395, 3564}, {396, 511}, {398, 7685}, {530, 5215}, {622, 11307}, {624, 7889}, {636, 6672}, {3053, 41039}, {5321, 41037}, {5340, 36995}, {5350, 36994}, {5615, 40693}, {5965, 6783}, {6109, 29012}, {6770, 35006}, {6772, 21158}, {10611, 33959}, {10617, 14693}, {13349, 42148}, {13350, 36251}, {13881, 41038}, {14539, 16772}, {16965, 30559}, {18582, 47517}, {20429, 42166}, {21159, 42943}, {22512, 41025}, {22513, 41036}, {23303, 24206}, {36756, 42945}, {43100, 48314}, {43539, 44223}

X(53454) = midpoint of X(13) and X(39554)
X(53454) = crossdifference of every pair of points on line {10640, 41167}
X(53454) = {X(6672),X(10614)}-harmonic conjugate of X(16773)

---

X(53455) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(17) AND X(262)

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

X(53455) lies on these lines: {2, 51202}, {5, 3107}, {6, 37463}, {13, 44223}, {16, 17}, {39, 7684}, {182, 396}, {262, 1513}, {303, 12215}, {395, 18583}, {574, 5318}, {1506, 7685}, {6115, 24206}, {6774, 14136}, {7777, 51161}, {7822, 42598}, {9698, 51753}, {9736, 42148}, {11308, 42156}, {18582, 40921}, {20252, 46854}, {22666, 51020}, {22688, 22708}, {22715, 48876}, {22847, 43455}, {23303, 38317}, {30261, 37851}, {37352, 42672}, {41025, 42102}, {43539, 51872}

---

X(53456) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(17) AND X(485)

Barycentrics    (-1 + Sqrt[3])*a^4 + (1 + Sqrt[3])*(b^2 - c^2)^2 - 2*a^2*(Sqrt[3]*(b^2 + c^2) + (1 + Sqrt[3])*S) : :

X(53456) lies on these lines: {3, 485}, {5, 3364}, {6, 2046}, {13, 34552}, {15, 8981}, {16, 17}, {30, 3391}, {62, 13966}, {371, 396}, {372, 16773}, {395, 3390}, {398, 8960}, {524, 33352}, {549, 3365}, {550, 42237}, {615, 40693}, {639, 6306}, {641, 52194}, {642, 37341}, {1151, 2041}, {1656, 42254}, {1657, 42189}, {2042, 8253}, {2043, 6409}, {2044, 42265}, {2045, 11481}, {3068, 52401}, {3071, 18582}, {3366, 16242}, {3367, 42215}, {3389, 16772}, {3392, 3628}, {3522, 42219}, {3851, 42187}, {3858, 42179}, {5056, 42217}, {5237, 35730}, {5318, 6200}, {5321, 18538}, {5420, 11486}, {6221, 42255}, {6445, 42196}, {6451, 42197}, {6480, 42222}, {6564, 42240}, {10576, 14813}, {10577, 42598}, {10646, 42170}, {13846, 36455}, {13925, 42147}, {16267, 51852}, {18581, 42194}, {18586, 42582}, {18587, 42245}, {31412, 52399}, {32421, 33393}, {32788, 36456}, {32789, 42089}, {32790, 42132}, {35731, 42943}, {35739, 42216}, {36439, 42239}, {36452, 43100}, {36470, 43229}, {41108, 52217}, {41963, 42152}, {42087, 42214}, {42092, 42229}, {42112, 42172}, {42121, 42565}, {42124, 42169}, {42128, 42192}, {42158, 50245}, {42166, 42236}, {42167, 42224}, {42206, 42283}, {42213, 42267}, {42218, 43512}, {42260, 42278}, {42266, 42280}, {42277, 42279}, {42282, 42638}

X(53456) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 8976, 42257}, {16, 42563, 397}, {17, 42563, 23302}, {140, 11542, 42563}, {485, 42257, 3070}, {590, 42253, 485}, {5418, 42257, 590}, {6306, 33394, 639}, {18582, 42230, 3071}, {42089, 42174, 32789}

---

X(53457) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(17) AND X(486)

Barycentrics    (1 + Sqrt[3])*a^4 + (-1 + Sqrt[3])*(b^2 - c^2)^2 - 2*a^2*(Sqrt[3]*b^2 + Sqrt[3]*c^2 + S - Sqrt[3]*S) : :

X(53457) lies on these lines: {3, 486}, {5, 3365}, {6, 2045}, {13, 34551}, {15, 13966}, {16, 17}, {30, 3392}, {62, 8981}, {371, 16773}, {372, 396}, {395, 3389}, {524, 33351}, {549, 3364}, {550, 42235}, {590, 40693}, {640, 6302}, {641, 37341}, {642, 52194}, {1152, 2042}, {1656, 42256}, {1657, 42187}, {2041, 8252}, {2043, 42262}, {2044, 6410}, {2046, 11481}, {3069, 52402}, {3070, 18582}, {3366, 42216}, {3367, 16242}, {3390, 16772}, {3391, 3628}, {3522, 42217}, {3851, 42189}, {3858, 42181}, {5056, 42219}, {5318, 6396}, {5321, 18762}, {5418, 11486}, {6398, 42257}, {6446, 42198}, {6452, 42195}, {6481, 42224}, {6565, 42239}, {10576, 42598}, {10577, 14814}, {10646, 42169}, {13847, 36437}, {13993, 42147}, {16267, 51854}, {18581, 42192}, {18586, 42244}, {18587, 42583}, {19116, 51727}, {32419, 33395}, {32787, 36438}, {32789, 42132}, {32790, 42089}, {35731, 52215}, {35732, 42637}, {35738, 42166}, {36452, 43229}, {36457, 42240}, {36470, 43100}, {41108, 52216}, {41964, 42152}, {42087, 42212}, {42092, 42230}, {42112, 42171}, {42121, 42564}, {42124, 42170}, {42128, 42194}, {42168, 42222}, {42205, 42284}, {42211, 42266}, {42220, 43511}, {42261, 42279}, {42267, 42281}, {42274, 42278}, {42561, 52400}

X(53457) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 13951, 42255}, {16, 42562, 397}, {17, 42562, 23302}, {140, 11542, 42562}, {486, 42255, 3071}, {615, 42251, 486}, {5420, 42255, 615}, {6302, 33392, 640}, {18582, 42229, 3070}, {42089, 42173, 32790}

---

X(53458) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(17) AND X(671)

Barycentrics    2*a^4 - a^2*b^2 - 3*b^4 - a^2*c^2 + 6*b^2*c^2 - 3*c^4 + 2*Sqrt[3]*(2*a^2 - b^2 - c^2)*S : :
X(53458) = 2 X[187] - 3 X[396], 3 X[395] - 4 X[43291], X[6779] - 3 X[16267], X[9301] - 3 X[20425], 3 X[14041] - X[44362], X[14712] - 3 X[37786], 5 X[16960] - 3 X[39554], 3 X[22510] - X[25235], 3 X[33228] - 2 X[44385]

X(53458) lies on these lines: {6, 622}, {16, 17}, {30, 6778}, {99, 44384}, {115, 532}, {148, 44361}, {187, 396}, {230, 5979}, {316, 524}, {395, 22490}, {398, 44497}, {511, 5318}, {574, 34509}, {624, 23303}, {625, 33477}, {3564, 5321}, {5335, 10519}, {5355, 43229}, {5615, 18582}, {5859, 44526}, {5869, 42096}, {5965, 33518}, {6390, 9886}, {6772, 22495}, {6775, 22494}, {6779, 16267}, {7685, 42110}, {9301, 20425}, {9763, 53095}, {10616, 11488}, {11480, 36995}, {14041, 44362}, {14539, 42088}, {14712, 37786}, {16644, 51485}, {16809, 37825}, {16960, 39554}, {18424, 34508}, {22492, 31415}, {22510, 25235}, {32993, 40900}, {33228, 44385}, {33256, 46710}, {34540, 51272}, {36251, 44498}, {36994, 42108}, {42087, 44667}

X(53458) = midpoint of X(i) and X(j) for these {i,j}: {148, 44361}, {22895, 22900}, {22997, 25156}
X(53458) = reflection of X(i) in X(j) for these {i,j}: {16, 11542}, {99, 44384}, {5318, 33517}, {52194, 624}
X(53458) = {X(20429),X(51207)}-harmonic conjugate of X(5321)

---

X(53459) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(17) AND X(1131)

Barycentrics    (-2 + Sqrt[3])*a^4 + (2 + Sqrt[3])*(b^2 - c^2)^2 - 2*a^2*(Sqrt[3]*(b^2 + c^2) + (1 + 2*Sqrt[3])*S) : :

X(53459) lies on these lines: {16, 17}, {371, 42147}, {395, 485}, {396, 1151}, {590, 42148}, {1131, 3068}, {1587, 22238}, {3070, 42088}, {3071, 42166}, {3364, 43228}, {3843, 42245}, {5318, 6221}, {5340, 41963}, {7583, 42238}, {8981, 42165}, {9680, 42241}, {18510, 40693}, {41945, 42240}, {42104, 42228}, {42107, 42211}, {42170, 42257}, {42191, 42987}, {42249, 42260}, {42250, 42265}, {42259, 42625}

---

X(53460) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(17) AND X(1132)

Barycentrics    (2 + Sqrt[3])*a^4 + (-2 + Sqrt[3])*(b^2 - c^2)^2 - 2*a^2*(Sqrt[3]*(b^2 + c^2) - (-1 + 2*Sqrt[3])*S) : :

X(53460) lies on these lines: {16, 17}, {372, 42147}, {395, 486}, {396, 1152}, {615, 42148}, {1132, 3069}, {1588, 22238}, {3070, 42166}, {3071, 42088}, {3365, 43228}, {3843, 42244}, {5318, 6398}, {5340, 41964}, {7584, 42236}, {13966, 42165}, {16772, 35739}, {18512, 40693}, {41946, 42239}, {42104, 42227}, {42107, 42213}, {42169, 42255}, {42193, 42987}, {42247, 42261}, {42252, 42262}, {42258, 42625}

---

X(53461) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(17) AND X(1327)

Barycentrics    (-3 + Sqrt[3])*a^4 + (3 + Sqrt[3])*(b^2 - c^2)^2 - 2*a^2*(Sqrt[3]*(b^2 + c^2) + (1 + 3*Sqrt[3])*S) : :

X(53461) lies on these lines: {6, 2044}, {13, 42215}, {16, 17}, {62, 35740}, {371, 5318}, {395, 18538}, {396, 6200}, {398, 42214}, {485, 11486}, {590, 10653}, {615, 36456}, {1327, 3830}, {3070, 42086}, {3071, 42128}, {3311, 42249}, {3365, 42170}, {3389, 42088}, {5321, 7583}, {5335, 9540}, {6221, 42196}, {6398, 35734}, {6459, 42140}, {9541, 36468}, {16960, 42169}, {18582, 42583}, {23267, 43481}, {31454, 42193}, {33440, 45871}, {34755, 51854}, {36470, 43104}, {41100, 42216}, {42130, 42258}, {42148, 42202}, {42159, 42245}, {42174, 42253}, {42187, 42816}, {42194, 42254}, {42206, 42250}, {42246, 42273}, {42575, 52399}, {42943, 52214}

---

X(53462) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(17) AND X(1328)

Barycentrics    (3 + Sqrt[3])*a^4 + (-3 + Sqrt[3])*(b^2 - c^2)^2 - 2*a^2*(Sqrt[3]*(b^2 + c^2) - (-1 + 3*Sqrt[3])*S) : :

X(53462) lies on these lines: {6, 2043}, {13, 42216}, {16, 17}, {62, 42241}, {372, 5318}, {395, 18762}, {396, 6396}, {398, 42212}, {486, 11486}, {590, 36438}, {615, 10653}, {1328, 3830}, {3070, 42128}, {3071, 42086}, {3312, 42247}, {3364, 42169}, {3390, 42088}, {5321, 7584}, {5335, 13935}, {6398, 42198}, {6460, 42140}, {16960, 35739}, {18582, 42582}, {23273, 43481}, {33441, 45872}, {34755, 51852}, {36452, 43104}, {41100, 42215}, {42130, 42259}, {42148, 42201}, {42159, 42244}, {42173, 42251}, {42189, 42816}, {42192, 42256}, {42205, 42252}, {42248, 42270}, {42574, 52400}, {42943, 51728}

---

X(53463) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(18) AND X(76)

Barycentrics    a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4 - 2*Sqrt[3]*(b^2 + c^2)*S : :

X(53463) lies on these lines: {5, 3105}, {6, 633}, {14, 42787}, {15, 18}, {16, 32151}, {32, 395}, {39, 635}, {76, 141}, {115, 636}, {298, 2896}, {299, 17129}, {302, 7824}, {396, 5305}, {397, 34507}, {617, 16645}, {618, 37512}, {619, 7749}, {621, 2076}, {623, 1506}, {624, 39565}, {626, 33482}, {1078, 14905}, {3098, 5321}, {3643, 7748}, {3933, 6294}, {5318, 18358}, {5868, 11481}, {7746, 23302}, {7811, 33459}, {7847, 14904}, {8259, 11311}, {8360, 42035}, {9981, 47611}, {11129, 22847}, {11486, 22907}, {11489, 19781}, {13881, 22893}, {14541, 42163}, {16941, 42156}, {22493, 43229}, {22513, 42148}, {22532, 43239}, {33021, 34540}, {36252, 40107}, {39560, 51160}, {42110, 43277}, {42599, 47068}

---

X(53464) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(18) AND X(83)

Barycentrics    2*a^4 + a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4 - 2*Sqrt[3]*(2*a^2 + b^2 + c^2)*S : :

X(53464) lies on these lines: {2, 34534}, {5, 46054}, {6, 634}, {15, 18}, {16, 36958}, {39, 395}, {83, 316}, {187, 6695}, {302, 7836}, {303, 7941}, {397, 25555}, {635, 5471}, {639, 33392}, {640, 33394}, {1506, 6694}, {3934, 33484}, {5092, 5321}, {5868, 42095}, {6036, 6114}, {6297, 7767}, {6680, 10616}, {7824, 10617}, {7832, 44383}, {10583, 19781}, {11304, 51159}, {16242, 52647}, {16967, 37824}, {19780, 33021}, {20395, 37512}, {37835, 47610}

X(53464) = {X(1506),X(6694)}-harmonic conjugate of X(23302)

---

X(53465) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(18) AND X(98)

Barycentrics    2*a^6 - a^4*b^2 - b^6 - a^4*c^2 + b^4*c^2 + b^2*c^4 - c^6 - 2*Sqrt[3]*(2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*S : :
X(53465) = X[15] + 2 X[11543], 2 X[187] + X[5321], X[395] + 2 X[6109], 4 X[6671] - X[52193], X[5318] - 4 X[43291]

X(53465) lies on these lines: {3, 10617}, {4, 19780}, {5, 36759}, {6, 37463}, {14, 38230}, {15, 18}, {30, 5469}, {98, 230}, {115, 44667}, {187, 5321}, {231, 40666}, {395, 511}, {396, 3564}, {397, 7684}, {531, 5215}, {621, 11308}, {623, 7889}, {635, 6671}, {3053, 41038}, {5318, 41036}, {5339, 36993}, {5349, 36992}, {5611, 40694}, {5965, 6782}, {6108, 29012}, {6773, 35006}, {6775, 21159}, {10612, 33960}, {10616, 14693}, {13349, 36252}, {13350, 42147}, {13881, 41039}, {14538, 16773}, {16964, 30560}, {18581, 47519}, {20428, 42163}, {21158, 42942}, {22512, 41037}, {22513, 41024}, {23302, 24206}, {36755, 42944}, {43107, 48313}, {43538, 52650}

X(53465) = midpoint of X(14) and X(39555)
X(53465) = crossdifference of every pair of points on line {10639, 41167}
X(53465) = {X(6671),X(10613)}-harmonic conjugate of X(16772)

---

X(53466) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(18) AND X(262)

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

X(53466) lies on these lines: {2, 51205}, {5, 3106}, {6, 37464}, {14, 52650}, {15, 18}, {39, 7685}, {182, 395}, {262, 1513}, {302, 12215}, {396, 18583}, {574, 5321}, {1506, 7684}, {6114, 24206}, {6771, 14137}, {6772, 36765}, {7777, 51162}, {7822, 42599}, {9698, 51754}, {9735, 42147}, {11307, 42153}, {18581, 40922}, {20253, 46855}, {22665, 51021}, {22690, 22707}, {22714, 48876}, {22893, 43454}, {23302, 38317}, {30260, 37852}, {37351, 42673}, {41024, 42101}, {43538, 51872}

---

X(53467) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(18) AND X(485)

Barycentrics    (1 + Sqrt[3])*a^4 + (-1 + Sqrt[3])*(b^2 - c^2)^2 - 2*a^2*(Sqrt[3]*(b^2 + c^2) + (-1 + Sqrt[3])*S) : :

X(53467) lies on these lines: {3, 485}, {5, 3389}, {6, 2045}, {14, 34551}, {15, 18}, {16, 8981}, {30, 3366}, {61, 13966}, {371, 395}, {372, 16772}, {396, 3365}, {397, 8960}, {524, 33350}, {549, 3390}, {550, 42238}, {615, 40694}, {639, 6307}, {641, 52193}, {642, 37340}, {1151, 2042}, {1656, 42255}, {1657, 42190}, {2041, 8253}, {2043, 42265}, {2044, 6409}, {2046, 11480}, {3068, 52402}, {3071, 18581}, {3364, 16773}, {3367, 3628}, {3391, 16241}, {3392, 42215}, {3522, 42220}, {3530, 35739}, {3851, 42188}, {3858, 42180}, {5056, 42218}, {5318, 18538}, {5321, 6200}, {5420, 11485}, {6221, 42254}, {6445, 42195}, {6451, 42198}, {6480, 42221}, {6564, 35740}, {7584, 51727}, {10576, 14814}, {10577, 42599}, {10645, 42168}, {13846, 36437}, {13925, 42148}, {16268, 51853}, {18582, 42193}, {18586, 42243}, {18587, 42582}, {31412, 52400}, {31700, 35744}, {32421, 33394}, {32788, 36438}, {32789, 42092}, {32790, 42129}, {35732, 42638}, {35738, 42163}, {36453, 43228}, {36457, 42241}, {36469, 43107}, {41107, 52214}, {41963, 42149}, {42088, 42213}, {42089, 42227}, {42113, 42174}, {42121, 42167}, {42124, 42563}, {42125, 42191}, {42169, 42223}, {42204, 42283}, {42214, 42267}, {42217, 43512}, {42260, 42279}, {42266, 42281}, {42277, 42278}

X(53467) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 8976, 42256}, {15, 42565, 398}, {18, 42565, 23303}, {140, 11543, 42565}, {485, 42256, 3070}, {590, 42252, 485}, {5418, 42256, 590}, {6307, 33393, 639}, {18581, 42228, 3071}, {42092, 42172, 32789}

---

X(53468) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(18) AND X(486)

Barycentrics    (-1 + Sqrt[3])*a^4 + (1 + Sqrt[3])*(b^2 - c^2)^2 - 2*a^2*(Sqrt[3]*(b^2 + c^2) - (1 + Sqrt[3])*S) : :

X(53468) lies on these lines: {3, 486}, {5, 3390}, {6, 2046}, {14, 34552}, {15, 18}, {16, 13966}, {30, 3367}, {61, 8981}, {371, 16772}, {372, 395}, {396, 3364}, {524, 33353}, {549, 3389}, {550, 42236}, {590, 40694}, {640, 6303}, {641, 37340}, {642, 52193}, {1152, 2041}, {1656, 42257}, {1657, 42188}, {2042, 8252}, {2043, 6410}, {2044, 42262}, {2045, 11480}, {3069, 52401}, {3070, 18581}, {3365, 16773}, {3366, 3628}, {3391, 42216}, {3392, 16241}, {3522, 42218}, {3851, 42190}, {3858, 42182}, {5056, 42220}, {5318, 18762}, {5321, 6396}, {5418, 11485}, {6398, 42256}, {6446, 42197}, {6452, 42196}, {6481, 42223}, {6565, 42241}, {10576, 42599}, {10577, 14813}, {10645, 42167}, {13847, 36455}, {13993, 42148}, {16268, 51855}, {18582, 42191}, {18586, 42583}, {18587, 42242}, {32419, 33392}, {32787, 36456}, {32789, 42129}, {32790, 42092}, {35255, 51727}, {35731, 42640}, {35740, 36439}, {35759, 43141}, {36453, 43107}, {36469, 43228}, {41107, 52215}, {41964, 42149}, {42088, 42211}, {42089, 42228}, {42113, 42173}, {42121, 42168}, {42124, 42562}, {42125, 42193}, {42163, 42237}, {42170, 42221}, {42203, 42284}, {42212, 42266}, {42219, 43511}, {42261, 42278}, {42267, 42280}, {42274, 42279}, {42282, 42637}, {42561, 52399}

X(53468) = midpoint of X(3367) and X(35739)
X(53468) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 13951, 42254}, {15, 42564, 398}, {18, 42564, 23303}, {140, 11543, 42564}, {486, 42254, 3071}, {615, 42250, 486}, {5420, 42254, 615}, {6303, 33395, 640}, {18581, 42227, 3070}, {42092, 42171, 32790}

---

X(53469) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(18) AND X(671)

Barycentrics    2*a^4 - a^2*b^2 - 3*b^4 - a^2*c^2 + 6*b^2*c^2 - 3*c^4 - 2*Sqrt[3]*(2*a^2 - b^2 - c^2)*S : :
X(53469) = 2 X[187] - 3 X[395], 3 X[396] - 4 X[43291], X[6780] - 3 X[16268], X[9301] - 3 X[20426], 3 X[14041] - X[44361], X[14712] - 3 X[37785], 5 X[16961] - 3 X[39555], 3 X[22511] - X[25236], 3 X[33228] - 2 X[44384]

X(53469) lies on these lines: {6, 621}, {15, 18}, {30, 6777}, {99, 44385}, {115, 533}, {148, 44362}, {187, 395}, {230, 5978}, {316, 524}, {396, 22489}, {397, 44498}, {511, 5321}, {574, 34508}, {623, 23302}, {625, 33476}, {3564, 5318}, {5334, 10519}, {5355, 43228}, {5611, 18581}, {5858, 44526}, {5868, 42097}, {5965, 33517}, {6390, 9885}, {6772, 22493}, {6775, 22496}, {6780, 16268}, {7684, 42107}, {9301, 20426}, {9761, 53095}, {10617, 11489}, {11481, 36993}, {14041, 44361}, {14538, 42087}, {14712, 37785}, {16645, 51484}, {16808, 37824}, {16961, 39555}, {18424, 34509}, {22491, 31415}, {22511, 25236}, {32993, 40901}, {33228, 44384}, {33256, 46711}, {34541, 51265}, {36252, 44497}, {36992, 42109}, {42088, 44666}

X(53469) = midpoint of X(i) and X(j) for these {i,j}: {148, 44362}, {22849, 22856}, {22998, 25166}
X(53469) = reflection of X(i) in X(j) for these {i,j}: {15, 11543}, {99, 44385}, {5321, 33518}, {52193, 623}
X(53469) = {X(20428),X(51206)}-harmonic conjugate of X(5318)

---

X(53470) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(18) AND X(1131)

Barycentrics    (2 + Sqrt[3])*a^4 + (-2 + Sqrt[3])*(b^2 - c^2)^2 - 2*a^2*(Sqrt[3]*(b^2 + c^2) + (-1 + 2*Sqrt[3])*S) : :

X(53470) lies on these lines: {15, 18}, {371, 42148}, {395, 1151}, {396, 485}, {590, 42147}, {1131, 3068}, {1587, 22236}, {3070, 42087}, {3071, 42163}, {3389, 43229}, {3843, 42243}, {5321, 6221}, {5339, 41963}, {7583, 42237}, {8981, 42164}, {9680, 42239}, {15687, 50246}, {18510, 40694}, {35740, 41945}, {42105, 42230}, {42110, 42212}, {42168, 42256}, {42192, 42986}, {42248, 42260}, {42251, 42265}, {42259, 42626}

---

X(53471) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(18) AND X(1132)

Barycentrics    (-2 + Sqrt[3])*a^4 + (2 + Sqrt[3])*(b^2 - c^2)^2 - 2*a^2*(Sqrt[3]*(b^2 + c^2) - (1 + 2*Sqrt[3])*S) : :

X(53471) lies on these lines: {15, 18}, {372, 42148}, {395, 1152}, {396, 486}, {615, 42147}, {1132, 3069}, {1588, 22236}, {3070, 42163}, {3071, 42087}, {3390, 43229}, {3843, 42242}, {5321, 6398}, {5339, 41964}, {7584, 42235}, {13966, 42164}, {18512, 40694}, {41946, 42241}, {42105, 42229}, {42110, 42214}, {42167, 42254}, {42194, 42986}, {42246, 42261}, {42253, 42262}, {42258, 42626}

---

X(53472) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(18) AND X(1327)

Barycentrics    (3 + Sqrt[3])*a^4 + (-3 + Sqrt[3])*(b^2 - c^2)^2 - 2*a^2*(Sqrt[3]*(b^2 + c^2) + (-1 + 3*Sqrt[3])*S) : :

X(53472) lies on these lines: {6, 2043}, {14, 42215}, {15, 18}, {61, 42240}, {371, 5321}, {395, 6200}, {396, 18538}, {397, 42213}, {485, 11485}, {590, 10654}, {615, 36438}, {1327, 3830}, {3070, 42085}, {3071, 42125}, {3311, 42248}, {3364, 42087}, {3390, 42168}, {5318, 7583}, {5334, 9540}, {6221, 42195}, {6459, 42141}, {9541, 36450}, {16961, 42167}, {18581, 42583}, {23267, 43482}, {31454, 42194}, {33442, 45871}, {34754, 51855}, {36453, 43101}, {41101, 42216}, {42117, 51728}, {42131, 42258}, {42147, 42200}, {42162, 42243}, {42163, 51727}, {42172, 42252}, {42188, 42815}, {42193, 42255}, {42204, 42251}, {42247, 42273}, {42575, 52400}, {42942, 52217}

---

X(53473) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(18) AND X(1328)

Barycentrics    (-3 + Sqrt[3])*a^4 + (3 + Sqrt[3])*(b^2 - c^2)^2 - 2*a^2*(Sqrt[3]*(b^2 + c^2) - (1 + 3*Sqrt[3])*S) : :

X(53473) lies on these lines: {6, 2044}, {14, 42216}, {15, 18}, {61, 42239}, {372, 5321}, {395, 6396}, {396, 18762}, {397, 42211}, {486, 11485}, {590, 36456}, {615, 10654}, {1328, 3830}, {3070, 42125}, {3071, 42085}, {3312, 42246}, {3365, 42087}, {3389, 42167}, {5318, 7584}, {5334, 13935}, {6221, 35734}, {6398, 42197}, {6460, 42141}, {16961, 42168}, {18581, 42582}, {23273, 43482}, {33443, 45872}, {34754, 51853}, {36469, 43101}, {41101, 42215}, {42131, 42259}, {42147, 42199}, {42162, 42242}, {42171, 42250}, {42190, 42815}, {42191, 42257}, {42203, 42253}, {42249, 42270}, {42574, 52399}, {42942, 52216}

---

X(53474) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(76) AND X(94)

Barycentrics    b^2*c^2*(2*a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - 4*b^2*c^2 + 2*c^4) : :
X(53474) = 3 X[338] - X[3260], X[3260] + 3 X[51481]

X(53474) lies on these lines: {2, 52154}, {6, 44135}, {30, 43087}, {76, 141}, {94, 3580}, {115, 34827}, {264, 3629}, {290, 34369}, {300, 396}, {301, 395}, {311, 3589}, {338, 524}, {401, 44376}, {538, 44468}, {597, 40814}, {1990, 44138}, {3266, 40511}, {3630, 44133}, {3631, 44148}, {4395, 34387}, {5201, 11594}, {5392, 13567}, {6144, 44136}, {6248, 16776}, {7227, 34388}, {7778, 9464}, {8705, 53346}, {9870, 15271}, {11542, 43085}, {11543, 43086}, {14767, 32450}, {15415, 33294}, {16310, 24975}, {18365, 41626}, {25328, 45279}, {26541, 49738}, {26592, 49731}, {30737, 50774}, {34391, 45871}, {34392, 45872}, {34989, 44388}, {35549, 44380}, {35707, 39646}, {36789, 47296}, {36792, 44395}, {41000, 44383}, {41001, 44382}

X(53474) = midpoint of X(338) and X(51481)
X(53474) = crossdifference of every pair of points on line {3202, 9426}
X(53474) = barycentric product X(850)*X(53274)
X(53474) = barycentric quotient X(53274)/X(110)

---

X(53475) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(76) AND X(98)

Barycentrics    a^4*b^2 + b^6 + a^4*c^2 - 2*a^2*b^2*c^2 - b^4*c^2 - b^2*c^4 + c^6 : :
X(53475) = X[6] - 4 X[43291], 2 X[141] + X[47286], 2 X[230] + X[11646], 2 X[115] + X[15993], 2 X[5103] - 3 X[33228], X[5111] - 3 X[6034], X[2456] - 3 X[38224], 5 X[3763] - 2 X[6390], 4 X[5461] - X[41146], 5 X[14061] - 2 X[44380], X[32113] + 2 X[51258], 2 X[32457] + X[50567], 4 X[44401] - X[51798]

X(53475) lies on these lines: {2, 12215}, {4, 5017}, {5, 6}, {30, 2076}, {32, 3818}, {39, 24206}, {53, 43976}, {69, 5025}, {76, 141}, {98, 230}, {114, 2024}, {115, 511}, {125, 3291}, {140, 5116}, {147, 39095}, {182, 7746}, {187, 29012}, {193, 32966}, {232, 51434}, {297, 694}, {325, 732}, {343, 3981}, {384, 46323}, {385, 5207}, {403, 2211}, {427, 1613}, {460, 52162}, {468, 20998}, {524, 5103}, {525, 23285}, {538, 51371}, {542, 1692}, {599, 33184}, {626, 14994}, {639, 48769}, {640, 48768}, {755, 1287}, {858, 3231}, {1196, 21243}, {1225, 23642}, {1350, 44518}, {1368, 21001}, {1570, 5965}, {1611, 1853}, {2056, 23292}, {2072, 14965}, {2353, 20960}, {2450, 20021}, {2456, 38224}, {2882, 52460}, {3003, 52967}, {3051, 5133}, {3053, 36990}, {3098, 7748}, {3124, 3580}, {3589, 5038}, {3618, 16921}, {3619, 7738}, {3620, 7933}, {3763, 5013}, {3815, 13331}, {3852, 5167}, {3933, 31981}, {4048, 7807}, {5023, 48905}, {5028, 34507}, {5034, 38317}, {5039, 5475}, {5052, 19130}, {5085, 37451}, {5092, 7749}, {5104, 29181}, {5152, 35297}, {5166, 25320}, {5169, 9463}, {5206, 48898}, {5286, 40330}, {5306, 47354}, {5309, 11178}, {5461, 41146}, {5476, 18362}, {5480, 13330}, {5913, 8288}, {5949, 15989}, {5969, 51374}, {6103, 51437}, {6321, 35383}, {6661, 10000}, {6776, 37446}, {6781, 29323}, {7499, 10329}, {7735, 13862}, {7745, 12212}, {7747, 41413}, {7752, 32451}, {7753, 25561}, {7755, 18553}, {7756, 14810}, {7764, 41622}, {7792, 42534}, {7819, 24273}, {7832, 12055}, {7930, 51128}, {7942, 51126}, {8770, 26958}, {9225, 11064}, {9873, 18547}, {9969, 27374}, {10519, 43448}, {10997, 11606}, {11442, 42295}, {11648, 50977}, {14061, 44380}, {14153, 37649}, {14567, 46818}, {14820, 22416}, {15513, 48892}, {15514, 34380}, {16608, 20271}, {18440, 40825}, {19905, 35378}, {19924, 39563}, {20859, 37636}, {20965, 37990}, {21358, 52229}, {23332, 40326}, {26156, 26175}, {31884, 44526}, {32113, 46154}, {32223, 40350}, {32457, 50567}, {32740, 51405}, {33002, 51171}, {33011, 51170}, {35325, 37981}, {39024, 41724}, {39560, 48906}, {40379, 45201}, {43150, 44499}, {44347, 46286}, {44401, 51798}, {44453, 48876}, {44531, 47619}, {44535, 53094}

X(53475) = midpoint of X(i) and X(j) for these {i,j}: {76, 45803}, {385, 5207}, {1691, 11646}, {6321, 35383}, {6393, 47286}
X(53475) = reflection of X(i) in X(j) for these {i,j}: {325, 5031}, {1691, 230}, {6393, 141}, {13196, 3589}
X(53475) = complement of X(12215)
X(53475) = complement of the isogonal conjugate of X(17980)
X(53475) = X(i)-complementary conjugate of X(j) for these (i,j): {19, 39080}, {25, 19563}, {694, 18589}, {881, 16573}, {882, 34846}, {1581, 1368}, {1927, 216}, {1967, 3}, {1973, 5976}, {9468, 1214}, {17980, 10}, {37134, 52598}, {43763, 11574}
X(53475) = X(43187)-Ceva conjugate of X(523)
X(53475) = X(44114)-Dao conjugate of X(3569)
X(53475) = crossdifference of every pair of points on line {206, 924}
X(53475) = barycentric product X(i)*X(j) for these {i,j}: {1, 17901}, {523, 53371}
X(53475) = barycentric quotient X(i)/X(j) for these {i,j}: {17901, 75}, {53371, 99}
X(53475) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {141, 5254, 3094}, {141, 51848, 6656}, {1352, 3767, 6}, {3231, 39691, 858}, {5052, 39565, 19130}, {41413, 48889, 7747}

---

X(53476) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(76) AND X(226)

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(53476) lies on these lines: {2, 45988}, {6, 37086}, {11, 4022}, {37, 226}, {38, 3136}, {65, 22298}, {75, 1211}, {76, 141}, {115, 17052}, {142, 16589}, {192, 3936}, {335, 26601}, {442, 984}, {518, 1834}, {537, 16052}, {594, 4377}, {726, 3454}, {740, 41014}, {851, 34247}, {862, 28106}, {940, 50400}, {982, 28087}, {1086, 1213}, {1107, 34830}, {1230, 17184}, {1231, 3721}, {1278, 31037}, {1423, 2245}, {2092, 3663}, {2238, 4000}, {2277, 30961}, {2305, 8822}, {2478, 28078}, {2667, 4854}, {2887, 21080}, {3056, 44151}, {3120, 3728}, {3123, 23444}, {3662, 3948}, {3736, 50177}, {3770, 15985}, {3943, 22008}, {3946, 20970}, {4016, 16732}, {4199, 33144}, {4205, 24325}, {4272, 17301}, {4346, 27039}, {4389, 27042}, {4419, 26125}, {4425, 25124}, {4699, 33146}, {4751, 40688}, {4772, 27081}, {4859, 46196}, {5051, 24349}, {11997, 38357}, {15668, 33821}, {16752, 28252}, {17139, 28369}, {17157, 25760}, {17306, 52538}, {17390, 25434}, {17470, 23772}, {17514, 40328}, {18137, 18739}, {18179, 21138}, {18635, 40593}, {18698, 21810}, {20335, 21838}, {20358, 40954}, {20486, 21035}, {21246, 24214}, {24169, 25106}, {25660, 31008}, {27017, 30019}, {27471, 50036}, {28081, 37225}, {29964, 31036}, {34528, 49518}, {40025, 52651}, {40773, 41877}, {52087, 52896}

X(53476) = X(i)-complementary conjugate of X(j) for these (i,j): {213, 52657}, {3500, 3741}
X(53476) = X(4572)-Ceva conjugate of X(523)
X(53476) = X(21246)-Dao conjugate of X(6)
X(53476) = crossdifference of every pair of points on line {9426, 21789}
X(53476) = barycentric product X(i)*X(j) for these {i,j}: {10, 24214}, {65, 21422}, {226, 21246}, {349, 23640}, {523, 53355}, {850, 23363}, {1441, 21334}
X(53476) = barycentric quotient X(i)/X(j) for these {i,j}: {21246, 333}, {21334, 21}, {21422, 314}, {22421, 283}, {23363, 110}, {23640, 284}, {24214, 86}, {53355, 99}
X(53476) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3120, 3728, 21926}, {10381, 13161, 1834}

---

X(53477) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(76) AND X(275)

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

X(53477) lies on these lines: {2, 6503}, {3, 41770}, {4, 159}, {6, 297}, {53, 21447}, {66, 39646}, {69, 41237}, {76, 141}, {115, 14767}, {160, 1513}, {275, 1971}, {467, 11427}, {570, 45198}, {1232, 37636}, {1585, 44599}, {1586, 44598}, {1993, 44128}, {2450, 20775}, {2549, 6389}, {5013, 26155}, {5112, 40981}, {5117, 23300}, {6467, 45279}, {6530, 50649}, {7738, 26154}, {7778, 34254}, {7864, 26175}, {8553, 35937}, {8801, 46034}, {12241, 43995}, {13567, 52147}, {19459, 41761}, {28408, 41238}, {37450, 45030}, {41008, 44375}, {44439, 44704}

X(53477) = crossdifference of every pair of points on line {9426, 34983}
X(53477) = {X(141),X(5254)}-harmonic conjugate of X(41760)

---

X(53478) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(76) AND X(121)

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

X(53478) lies on these lines: {2, 39983}, {6, 44140}, {10, 46772}, {37, 4044}, {72, 22272}, {75, 1213}, {76, 141}, {115, 21245}, {192, 27042}, {274, 6707}, {312, 17056}, {313, 321}, {314, 524}, {338, 1234}, {339, 18642}, {349, 52023}, {350, 17045}, {429, 20883}, {442, 1089}, {536, 2092}, {538, 16696}, {599, 44147}, {668, 4478}, {1030, 26243}, {1086, 1269}, {1229, 18635}, {1278, 26772}, {1333, 24271}, {1655, 27164}, {1834, 4385}, {1865, 46108}, {1909, 17390}, {2238, 4361}, {2245, 3729}, {2292, 21730}, {2321, 4377}, {3120, 21713}, {3136, 3703}, {3286, 50156}, {3454, 3773}, {3596, 4665}, {3629, 34283}, {3630, 34282}, {3631, 44139}, {3661, 30596}, {3664, 4410}, {3714, 10381}, {3739, 16589}, {3760, 4657}, {3761, 4851}, {3765, 17362}, {3875, 4272}, {3936, 4671}, {3943, 3963}, {3975, 20174}, {4000, 27040}, {4399, 17143}, {4436, 22369}, {4461, 27039}, {4479, 50112}, {4643, 10447}, {4647, 4733}, {4754, 18166}, {4852, 20970}, {4957, 21442}, {5224, 31060}, {6381, 17239}, {7227, 17790}, {10026, 17788}, {15668, 34284}, {16732, 18697}, {17245, 18137}, {17259, 28809}, {17327, 18135}, {17398, 18147}, {17757, 40564}, {17786, 27792}, {18089, 21753}, {18698, 42713}, {19810, 44416}, {20234, 20337}, {20236, 44396}, {20491, 21412}, {20683, 22289}, {20910, 31946}, {20923, 34824}, {21020, 21699}, {21264, 21838}, {25964, 26592}, {26543, 51481}, {27569, 35550}, {28605, 41809}, {28634, 32104}, {28640, 52716}, {30473, 48636}, {30819, 46838}, {42031, 50312}, {48635, 52043}

X(53478) = midpoint of X(314) and X(3770)
X(53478) = isotomic conjugate of X(40408)
X(53478) = isotomic conjugate of the isogonal conjugate of X(16589)
X(53478) = X(i)-Ceva conjugate of X(j) for these (i,j): {321, 52579}, {6386, 523}, {20888, 21020}
X(53478) = X(i)-isoconjugate of X(j) for these (i,j): {31, 40408}, {32, 40439}, {163, 50520}, {1333, 40433}, {2206, 32009}
X(53478) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 40408}, {37, 40433}, {115, 50520}, {2486, 21007}, {3121, 667}, {3720, 4251}, {3739, 6}, {6376, 40439}, {16589, 81}, {40603, 32009}
X(53478) = trilinear pole of line {48393, 50538}
X(53478) = barycentric product X(i)*X(j) for these {i,j}: {10, 20888}, {75, 21020}, {76, 16589}, {274, 52579}, {310, 21699}, {313, 3720}, {321, 3739}, {349, 3691}, {523, 53363}, {561, 2667}, {594, 16748}, {668, 48393}, {670, 50538}, {850, 4436}, {1089, 17175}, {1441, 3706}, {1502, 21753}, {3596, 39793}, {3701, 4059}, {4033, 47672}, {4111, 6063}, {6372, 27808}, {6385, 21820}, {6386, 50497}, {18022, 22369}, {18166, 28654}, {20963, 27801}, {40071, 40975}
X(53478) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 40408}, {10, 40433}, {75, 40439}, {321, 32009}, {523, 50520}, {2667, 31}, {3691, 284}, {3706, 21}, {3720, 58}, {3739, 81}, {3952, 8708}, {4059, 1014}, {4111, 55}, {4436, 110}, {4891, 16948}, {6372, 3733}, {16589, 6}, {16748, 1509}, {17175, 757}, {18089, 52376}, {18166, 593}, {20888, 86}, {20963, 1333}, {21020, 1}, {21699, 42}, {21753, 32}, {21820, 213}, {22060, 1437}, {22369, 184}, {39793, 56}, {40975, 1474}, {47672, 1019}, {48264, 3737}, {48393, 513}, {50497, 667}, {50538, 512}, {52579, 37}, {53363, 99}
X(53478) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {75, 3948, 1213}, {274, 25660, 6707}, {313, 321, 594}, {321, 1230, 1211}, {1269, 20891, 1086}, {3963, 4043, 3943}, {18137, 20913, 17245}

---

X(53479) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(76) AND X(485)

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

X(53479) lies on these lines: {3, 485}, {5, 3102}, {6, 638}, {32, 45576}, {39, 640}, {76, 141}, {115, 639}, {325, 49352}, {371, 44394}, {490, 12968}, {491, 7791}, {574, 642}, {615, 3767}, {641, 7746}, {1151, 13749}, {1352, 3071}, {1513, 45861}, {1691, 13910}, {2460, 8981}, {3053, 45486}, {3068, 11294}, {3526, 49791}, {3815, 23312}, {3933, 6314}, {5013, 45473}, {5062, 32421}, {5306, 44605}, {5309, 45577}, {5491, 16043}, {5591, 7738}, {6221, 48659}, {6228, 44392}, {6230, 49266}, {6281, 8396}, {6409, 35946}, {6424, 31411}, {6564, 40274}, {6813, 42265}, {7750, 45435}, {7761, 9994}, {7792, 45434}, {7795, 32494}, {7807, 45871}, {7828, 45872}, {7834, 9995}, {8253, 39388}, {8997, 15885}, {9674, 41491}, {9680, 45023}, {10515, 42270}, {10576, 45565}, {11316, 32789}, {13663, 35305}, {13711, 13934}, {13846, 35949}, {13881, 45472}, {19145, 49087}, {23251, 48872}, {36655, 42273}, {42284, 43621}

X(53479) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {490, 13638, 12968}, {1352, 39660, 3071}, {3068, 11294, 12963}, {6421, 11314, 615}, {8976, 13882, 590}

---

X(53480) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(76) AND X(486)

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

X(53480) lies on these lines: {3, 486}, {5, 3103}, {6, 637}, {32, 45577}, {39, 639}, {76, 141}, {115, 640}, {325, 49351}, {372, 44392}, {489, 12963}, {492, 7791}, {574, 641}, {590, 3767}, {642, 7746}, {1152, 13748}, {1352, 3070}, {1513, 45860}, {1691, 13972}, {2459, 13966}, {3053, 45487}, {3069, 11293}, {3526, 49790}, {3815, 23311}, {3933, 6318}, {5013, 45472}, {5058, 32419}, {5306, 44604}, {5309, 45576}, {5490, 16043}, {5590, 7738}, {6229, 44394}, {6231, 49267}, {6278, 8416}, {6398, 48660}, {6410, 35947}, {6423, 32788}, {6565, 40275}, {6811, 42262}, {7750, 45434}, {7761, 9995}, {7792, 45435}, {7795, 32497}, {7807, 45872}, {7828, 45871}, {7834, 9994}, {8252, 39387}, {9600, 11315}, {10514, 42273}, {10577, 45564}, {13783, 35306}, {13834, 13882}, {13847, 35948}, {13879, 31483}, {13881, 45473}, {13989, 15886}, {19146, 49086}, {23261, 48872}, {36656, 42270}, {42283, 43621}

X(53480) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {489, 13758, 12963}, {1352, 39661, 3070}, {3069, 11293, 12968}, {6422, 11313, 590}, {13934, 13951, 615}

---

X(53481) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(76) AND X(801)

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

X(53481) lies on these lines: {2, 3964}, {20, 33582}, {53, 14615}, {69, 297}, {76, 141}, {99, 34828}, {317, 394}, {343, 15466}, {458, 28419}, {524, 22468}, {1249, 53021}, {1975, 6389}, {3087, 37669}, {3186, 8263}, {3620, 41237}, {3631, 48540}, {3926, 26155}, {7399, 28441}, {7778, 8770}, {7807, 41770}, {8024, 26172}, {11064, 36794}, {15069, 34808}, {15812, 39646}, {16713, 26529}, {20081, 26175}, {20806, 27377}, {26154, 32830}, {26156, 51481}, {26167, 26541}, {26173, 32869}, {26527, 27507}, {28408, 52289}, {36212, 45198}, {46741, 52283}

---

X(53482) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(76) AND X(1131)

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

X(53482) lies on these lines: {6, 12222}, {20, 45486}, {30, 45576}, {39, 23312}, {76, 141}, {115, 23311}, {371, 14230}, {485, 15883}, {590, 11293}, {640, 15048}, {641, 43291}, {1131, 3068}, {1503, 39660}, {1587, 53093}, {1991, 33210}, {3070, 13910}, {3102, 45860}, {7388, 13972}, {7738, 45473}, {8399, 42262}, {8975, 31414}, {9601, 13846}, {11315, 13711}, {13638, 42259}, {13644, 22644}, {13848, 43211}, {13921, 35255}, {14233, 35831}, {23261, 51537}, {32497, 45871}, {35841, 45870}

---

X(53483) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(76) AND X(1132)

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

X(53483) lies on these lines: {6, 12221}, {20, 45487}, {30, 45577}, {39, 23311}, {76, 141}, {115, 23312}, {372, 14233}, {486, 15884}, {591, 33210}, {615, 11294}, {639, 15048}, {642, 43291}, {1132, 3069}, {1503, 39661}, {1588, 53093}, {3071, 13972}, {3103, 45861}, {7389, 13910}, {7738, 45472}, {8419, 42265}, {11316, 13834}, {13758, 42258}, {13763, 22615}, {13880, 35256}, {13988, 43212}, {14230, 35830}, {23251, 51537}, {32494, 45872}

---

X(53484) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(83) AND X(262)

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

X(53484) lies on these lines: {2, 5017}, {4, 50659}, {5, 6}, {30, 5116}, {32, 38317}, {39, 19130}, {69, 7941}, {83, 316}, {140, 2076}, {141, 7752}, {182, 5475}, {193, 33002}, {230, 12212}, {262, 1513}, {325, 24256}, {428, 10329}, {511, 1506}, {570, 52967}, {574, 48901}, {597, 7884}, {626, 35432}, {1350, 31489}, {1503, 5038}, {1594, 2211}, {1613, 37439}, {1692, 25555}, {1915, 37649}, {3051, 37990}, {3053, 8362}, {3055, 5104}, {3098, 31455}, {3329, 43450}, {3580, 13410}, {3618, 5025}, {3818, 5034}, {4048, 8370}, {5013, 53023}, {5028, 5476}, {5032, 32994}, {5039, 7746}, {5052, 7603}, {5092, 7747}, {5133, 20965}, {5254, 13331}, {5477, 15516}, {5943, 6388}, {6034, 6054}, {7694, 36990}, {7736, 13862}, {7749, 41413}, {7756, 48895}, {7762, 8177}, {7767, 31982}, {7777, 18906}, {7792, 42535}, {7846, 51126}, {7899, 34573}, {8589, 48885}, {9225, 35283}, {9478, 12829}, {10024, 14965}, {10168, 14537}, {12215, 16044}, {14853, 31404}, {15484, 40825}, {15515, 48880}, {15815, 48910}, {18371, 37347}, {18424, 42852}, {19661, 49788}, {20301, 46301}, {21850, 44453}, {23300, 32445}, {29012, 39590}, {29317, 37512}, {31401, 31670}, {31406, 38136}, {31958, 48876}, {32449, 47286}, {32966, 51171}, {33010, 51170}, {33184, 47352}, {38110, 39560}, {43457, 48889}, {43843, 50137}, {48872, 53095}

X(53484) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2548, 14561, 6}, {3589, 5103, 6656}, {3589, 7745, 1691}, {3815, 5480, 3094}, {3815, 51851, 1513}, {5052, 7603, 24206}, {5052, 24206, 15993}

---

X(53485) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(83) AND X(275)

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

X(53485) lies on these lines: {2, 571}, {4, 206}, {6, 264}, {50, 95}, {83, 316}, {98, 34845}, {262, 1485}, {275, 1971}, {323, 1232}, {401, 570}, {419, 9969}, {566, 46724}, {842, 45108}, {1176, 14957}, {1300, 15033}, {1576, 3613}, {1632, 23635}, {2052, 8746}, {2548, 28405}, {2986, 7578}, {3050, 39182}, {5157, 37190}, {7786, 9608}, {8882, 44893}, {10358, 19137}, {14575, 37988}, {14767, 44375}, {14806, 51350}, {15484, 37073}, {18374, 32085}, {19161, 37124}, {20806, 41231}, {28704, 32968}

X(53485) = crossdifference of every pair of points on line {34983, 39469}
X(53485) = {X(6),X(458)}-harmonic conjugate of X(41760)

---

X(53486) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(83) AND X(321)

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

X(53486) lies on these lines: {2, 1030}, {6, 21287}, {37, 25370}, {83, 316}, {141, 314}, {313, 321}, {442, 4026}, {528, 16052}, {740, 3454}, {1213, 4422}, {1834, 5015}, {2092, 21245}, {2245, 16566}, {2292, 20657}, {2305, 25445}, {3936, 4360}, {3948, 46747}, {4972, 18082}, {5051, 5263}, {5132, 37025}, {5949, 27042}, {17045, 17056}, {17243, 26590}, {17259, 17550}, {17303, 24335}, {17398, 37096}, {17669, 27111}, {18755, 25649}, {36743, 50057}

X(53486) = barycentric product X(21702)*X(52394)
X(53486) = barycentric quotient X(21702)/X(15523)
X(53486) = {X(2092),X(21245)}-harmonic conjugate of X(44396)

---

X(53487) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(83) AND X(485)

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

X(53487) lies on these lines: {2, 12968}, {3, 485}, {6, 637}, {83, 316}, {140, 2459}, {230, 44587}, {489, 12962}, {615, 2548}, {639, 5062}, {642, 31481}, {1151, 45440}, {1152, 45522}, {1504, 45574}, {1506, 50375}, {1587, 10519}, {3068, 7738}, {3071, 14561}, {3094, 19090}, {3103, 7583}, {5254, 13910}, {5305, 49254}, {6118, 7749}, {6119, 7603}, {6313, 7767}, {6316, 49220}, {6409, 35947}, {6422, 7739}, {6424, 13644}, {6811, 9756}, {7791, 12963}, {8253, 39387}, {8960, 45564}, {11291, 31463}, {11315, 32789}, {12323, 32785}, {13846, 35948}, {13879, 41491}, {15883, 31454}, {35822, 40275}, {36656, 42273}, {42216, 42787}

X(53487) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {590, 42259, 32497}, {639, 5062, 44392}, {6423, 11313, 615}

---

X(53488) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(83) AND X(486)

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

X(53488) lies on these lines: {2, 12963}, {3, 486}, {6, 638}, {83, 316}, {140, 2460}, {230, 44586}, {490, 12969}, {590, 2548}, {640, 5058}, {642, 9675}, {1151, 45523}, {1152, 45441}, {1505, 45575}, {1506, 50374}, {1588, 10519}, {3069, 7738}, {3070, 14561}, {3094, 19089}, {3102, 7584}, {5254, 13972}, {5305, 49255}, {6118, 7603}, {6119, 7749}, {6312, 49221}, {6317, 7767}, {6410, 35946}, {6421, 7739}, {6423, 13763}, {6813, 9756}, {7791, 12968}, {8252, 39388}, {11316, 32790}, {12322, 32786}, {13847, 35949}, {13933, 41490}, {16925, 32807}, {35823, 40274}, {36655, 42270}, {42215, 42787}

X(53488) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {615, 42258, 32494}, {640, 5058, 44394}, {6424, 11314, 590}

---

X(53489) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(83) AND X(598)

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

X(53489) lies on these lines: {2, 1285}, {4, 5050}, {5, 7787}, {6, 8370}, {30, 3329}, {32, 32992}, {39, 19687}, {69, 7762}, {76, 3629}, {83, 316}, {99, 9300}, {141, 7812}, {230, 12150}, {297, 14389}, {315, 3763}, {325, 6661}, {381, 16989}, {384, 6390}, {458, 41370}, {546, 7797}, {574, 8598}, {597, 598}, {1003, 7736}, {1007, 33220}, {1513, 10796}, {2548, 7807}, {3096, 51128}, {3407, 11170}, {3552, 31406}, {3618, 7841}, {3627, 7864}, {3734, 41624}, {3793, 8367}, {3815, 3972}, {3818, 10358}, {3853, 51860}, {3933, 7921}, {4045, 14537}, {5008, 22329}, {5013, 33250}, {5024, 33007}, {5210, 42849}, {5254, 7878}, {5304, 32983}, {5305, 16044}, {5422, 52281}, {5475, 7792}, {6329, 7827}, {6677, 34518}, {6680, 31275}, {6704, 7873}, {6781, 44562}, {7388, 8976}, {7389, 13951}, {7735, 44543}, {7737, 8356}, {7747, 19695}, {7750, 7808}, {7754, 32971}, {7757, 47287}, {7767, 20088}, {7771, 15491}, {7772, 32819}, {7773, 8363}, {7774, 11286}, {7776, 16898}, {7777, 8369}, {7782, 9606}, {7785, 7819}, {7789, 7858}, {7803, 33229}, {7817, 43457}, {7823, 8362}, {7829, 39590}, {7832, 19702}, {7836, 19697}, {7843, 7889}, {7875, 33184}, {7879, 16045}, {7881, 33198}, {7883, 34573}, {7885, 8364}, {7900, 16895}, {7912, 33185}, {7920, 33018}, {7925, 8368}, {7937, 51126}, {7941, 19689}, {7947, 19692}, {8359, 14712}, {8361, 10583}, {8366, 37690}, {8753, 12834}, {9466, 50251}, {9605, 14035}, {10348, 51851}, {10788, 37451}, {11054, 20583}, {11163, 35954}, {11303, 11543}, {11304, 11542}, {11317, 43448}, {11361, 15048}, {12007, 38664}, {13449, 25555}, {14030, 51123}, {14033, 31859}, {14881, 44251}, {14929, 16986}, {16924, 30435}, {17008, 21309}, {18843, 18845}, {24256, 50253}, {31400, 33235}, {31404, 33233}, {31467, 32964}, {32816, 33217}, {32827, 33219}, {33013, 43291}, {37644, 41231}

X(53489) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 8370, 47286}, {83, 316, 3589}, {83, 7745, 6656}, {316, 3589, 6656}, {325, 7804, 6661}, {597, 598, 8352}, {3589, 7745, 316}, {3815, 3972, 35297}, {5475, 7792, 33228}, {7737, 11174, 8356}, {7753, 7804, 325}, {14033, 37665, 31859}, {32971, 51170, 52713}, {51170, 52713, 7754}

---

X(53490) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(83) AND X(801)

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

X(53490) lies on these lines: {2, 95}, {4, 19137}, {6, 14615}, {83, 316}, {141, 22468}, {206, 12203}, {264, 26206}, {311, 41254}, {340, 26156}, {419, 11574}, {670, 40405}, {1974, 37190}, {3186, 11511}, {6816, 51031}, {8745, 15466}, {10152, 52069}, {14957, 32085}, {15066, 44149}, {20204, 33184}, {20806, 40814}, {26212, 32974}, {40318, 44133}, {41464, 46493}

X(53490) = {X(26206),X(41238)}-harmonic conjugate of X(264)

---

X(53491) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(83) AND X(1131)

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

X(53491) lies on these lines: {4, 13910}, {5, 43124}, {6, 12221}, {83, 316}, {230, 26429}, {371, 45862}, {485, 13644}, {590, 11294}, {1131, 3068}, {1587, 11477}, {2548, 13934}, {3053, 45871}, {3070, 29181}, {5491, 14033}, {6423, 23311}, {6459, 45484}, {13638, 42273}, {14233, 19145}, {22617, 45870}, {31412, 45486}, {32491, 41411}, {33198, 45473}, {39648, 45861}, {42215, 45574}

---

X(53492) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(83) AND X(1132)

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

X(53492) lies on these lines: {4, 13972}, {5, 43125}, {6, 12222}, {83, 316}, {230, 26430}, {372, 45863}, {486, 13763}, {615, 11293}, {1132, 3069}, {1588, 11477}, {2548, 13882}, {3053, 45872}, {3071, 29181}, {5490, 14033}, {6424, 23312}, {6460, 45485}, {13758, 42270}, {14230, 19146}, {32490, 41410}, {33198, 45472}, {39679, 45860}, {42216, 45575}, {42561, 45487}

---

X(53493) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(94) AND X(98)

Barycentrics    a^8*b^2 - 2*a^6*b^4 + 2*a^4*b^6 - 2*a^2*b^8 + b^10 + a^8*c^2 - a^4*b^4*c^2 + a^2*b^6*c^2 - 3*b^8*c^2 - 2*a^6*c^4 - a^4*b^2*c^4 + 2*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + a^2*b^2*c^6 + 2*b^4*c^6 - 2*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(53493) = 3 X[26613] - X[35706]

X(53493) lies on these lines: {5, 3016}, {6, 3448}, {32, 34514}, {53, 2970}, {94, 3580}, {98, 230}, {110, 44529}, {112, 44795}, {115, 5663}, {125, 2493}, {570, 35319}, {1989, 9140}, {2081, 14582}, {2088, 10264}, {2502, 3054}, {3815, 8288}, {5640, 9220}, {6240, 51233}, {6800, 37637}, {8705, 15993}, {11456, 13881}, {15107, 49007}, {16310, 49006}, {26613, 35706}, {32423, 32761}

X(53493) = crossdifference of every pair of points on line {11597, 14675}

---

X(53494) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(94) AND X(262)

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

X(53494) lies on these lines: {6, 23}, {30, 2088}, {50, 15107}, {51, 51735}, {94, 3580}, {230, 3124}, {232, 1112}, {262, 1513}, {511, 2493}, {566, 5640}, {1989, 15360}, {3981, 5306}, {6792, 47275}, {7575, 32761}, {9300, 20859}, {11062, 44084}, {16310, 47582}, {19154, 39764}, {19627, 37936}, {21850, 47579}, {22110, 36790}, {47169, 47581}

X(53494) = crossdifference of every pair of points on line {3906, 8723}

---

X(53495) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(94) AND X(671)

Barycentrics    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 + 2*a^2*b^4*c^2 - 6*b^6*c^2 - a^4*c^4 + 2*a^2*b^2*c^4 + 10*b^4*c^4 - a^2*c^6 - 6*b^2*c^6 + c^8 : :
X(53495) = 3 X[671] + X[48540], X[9142] - 3 X[11632], X[14712] - 3 X[44376], 3 X[15061] - X[48957]

X(53495) lies on these lines: {4, 51941}, {30, 15364}, {94, 3580}, {99, 44386}, {115, 18122}, {140, 48974}, {141, 311}, {148, 40879}, {265, 48988}, {316, 524}, {395, 21468}, {396, 21469}, {648, 6748}, {1263, 10264}, {1989, 24975}, {3589, 41254}, {8584, 50187}, {9142, 11632}, {14570, 34989}, {14712, 44376}, {15061, 48957}, {25328, 53346}

X(53495) = midpoint of X(i) and X(j) for these {i,j}: {148, 40879}, {265, 48988}
X(53495) = reflection of X(i) in X(j) for these {i,j}: {99, 44386}, {18122, 115}, {48974, 140}
X(53495) = X(51804)-anticomplementary conjugate of X(14360)
X(53495) = X(13162)-Dao conjugate of X(323)
X(53495) = barycentric product X(671)*X(13162)
X(53495) = barycentric quotient X(13162)/X(524)

---

X(53496) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(98) AND X(459)

Barycentrics    2*a^10 - a^8*b^2 - 4*a^6*b^4 + 6*a^4*b^6 - 6*a^2*b^8 + 3*b^10 - a^8*c^2 + 8*a^6*b^2*c^2 - 6*a^4*b^4*c^2 + 8*a^2*b^6*c^2 - 9*b^8*c^2 - 4*a^6*c^4 - 6*a^4*b^2*c^4 - 4*a^2*b^4*c^4 + 6*b^6*c^4 + 6*a^4*c^6 + 8*a^2*b^2*c^6 + 6*b^4*c^6 - 6*a^2*c^8 - 9*b^2*c^8 + 3*c^10 : :
X(53496) = 3 X[230] - X[1971]

X(53496) lies on these lines: {6, 8889}, {98, 230}, {115, 15311}, {125, 16318}, {232, 47296}, {393, 459}, {468, 51363}, {1853, 7735}, {2549, 23328}, {2883, 13881}, {3053, 41362}, {3269, 44909}, {3767, 6247}, {5254, 6696}, {5286, 40686}, {5305, 20299}, {5894, 44518}, {6000, 43291}, {7737, 23324}, {7746, 16252}, {9609, 15578}, {10192, 37637}, {10606, 43448}, {11745, 27371}, {15048, 23329}, {15526, 16096}, {18907, 23325}, {32064, 37689}

X(53496) = crossdifference of every pair of points on line {41167, 45248}

---

X(53497) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(98) AND X(485)

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

X(53497) lies on these lines: {3, 485}, {4, 12963}, {6, 6813}, {30, 2460}, {32, 48467}, {98, 230}, {115, 50374}, {372, 45555}, {490, 16925}, {511, 44394}, {615, 1352}, {638, 1152}, {1151, 45486}, {1504, 48735}, {1587, 31400}, {2459, 37459}, {3053, 13749}, {3068, 51212}, {3071, 3767}, {3102, 7583}, {3103, 48773}, {3564, 6230}, {5033, 48743}, {6281, 13934}, {6290, 39679}, {6398, 49791}, {6421, 31411}, {6460, 32806}, {6566, 32421}, {7000, 44596}, {7388, 42265}, {7745, 44586}, {7746, 48466}, {8960, 40274}, {9675, 45545}, {10515, 42583}, {11294, 31412}, {11314, 42582}, {12968, 45407}, {13638, 37182}, {13663, 35949}, {13711, 42283}, {13748, 13881}, {13847, 51027}, {20423, 32787}, {35297, 44393}, {35822, 45565}, {35946, 42264}, {36714, 49220}, {36719, 41945}, {39660, 42258}, {40825, 49228}, {42284, 43618}, {43125, 45543}

X(53497) = midpoint of X(2459) and X(50720)
X(53497) = crossdifference of every pair of points on line {10962, 41167}
X(53497) = {X(6424),X(36655)}-harmonic conjugate of X(3071)

---

X(53498) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(98) AND X(486)

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

X(53498) lies on these lines: {3, 486}, {4, 12968}, {6, 6811}, {30, 2459}, {32, 48466}, {98, 230}, {115, 50375}, {371, 45554}, {489, 16925}, {511, 44392}, {590, 1352}, {637, 1151}, {1152, 45487}, {1505, 48734}, {1588, 31400}, {2460, 37459}, {3053, 13748}, {3069, 51212}, {3070, 3767}, {3102, 48772}, {3103, 7584}, {3564, 6231}, {5033, 48742}, {6221, 49790}, {6278, 13882}, {6289, 39648}, {6459, 32805}, {6567, 32419}, {7374, 44595}, {7389, 42262}, {7745, 44587}, {7746, 48467}, {10514, 42582}, {11293, 42561}, {11313, 42583}, {12963, 45406}, {13749, 13881}, {13758, 37182}, {13783, 35948}, {13834, 42284}, {13846, 51027}, {20423, 32788}, {31463, 39876}, {35297, 44400}, {35823, 45564}, {35947, 42263}, {36709, 49221}, {36733, 41946}, {39661, 42259}, {40825, 49229}, {42283, 43618}, {43124, 45542}

X(53498) = midpoint of X(2460) and X(50719)
X(53498) = crossdifference of every pair of points on line {10960, 41167}
X(53498) = {X(6423),X(36656)}-harmonic conjugate of X(3070)

---

X(53499) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(98) AND X(598)

Barycentrics    4*a^6 - a^4*b^2 - b^6 - a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + b^2*c^4 - c^6 : :
X(53499) = X[69] - 3 X[13586], 2 X[230] - 3 X[1691], 3 X[1691] - X[11646], 3 X[8598] - X[51438], 2 X[141] - 3 X[35297], X[193] + 3 X[33265], X[316] - 3 X[5182], 3 X[5182] - 2 X[44380], 2 X[14928] + X[50253], 3 X[2482] - 2 X[51397], 4 X[3589] - 3 X[33228], 5 X[3618] - 3 X[14041], 3 X[18800] - X[51396], 3 X[41146] - 2 X[51396], 3 X[5207] - 5 X[7925], 3 X[5215] - 2 X[19662], X[5999] - 3 X[25406], X[7779] - 3 X[12215], 4 X[10011] - 3 X[10516], X[11161] - 3 X[26613], 2 X[14120] - 3 X[47455], X[23004] - 3 X[36758], X[23005] - 3 X[36757], X[36174] - 3 X[52238], 2 X[37350] - 3 X[47352]

X(53499) lies on these lines: {5, 39560}, {6, 30}, {23, 6792}, {50, 32233}, {69, 13586}, {98, 230}, {99, 524}, {115, 2030}, {141, 7771}, {182, 5475}, {187, 542}, {193, 33265}, {316, 5182}, {325, 5026}, {352, 9143}, {511, 1569}, {538, 14928}, {550, 44453}, {575, 7747}, {597, 598}, {599, 5210}, {698, 47287}, {732, 50251}, {804, 52038}, {858, 14567}, {1352, 21843}, {1495, 10418}, {1499, 23287}, {1506, 20190}, {1570, 29317}, {1625, 18371}, {1648, 7426}, {1692, 29012}, {1992, 9855}, {2076, 3564}, {2079, 19596}, {2482, 51397}, {2502, 5913}, {3053, 8721}, {3054, 47354}, {3055, 50983}, {3094, 44882}, {3231, 46818}, {3580, 8627}, {3589, 7919}, {3618, 14041}, {3815, 51737}, {3818, 5033}, {3849, 18800}, {5017, 6776}, {5023, 15069}, {5028, 48898}, {5038, 7745}, {5085, 31489}, {5107, 19924}, {5111, 29181}, {5112, 5967}, {5206, 34507}, {5207, 7925}, {5215, 19662}, {5304, 40236}, {5306, 20194}, {5642, 39602}, {5999, 7736}, {6388, 32237}, {6800, 9745}, {7603, 10168}, {7749, 18553}, {7756, 44499}, {7779, 10997}, {7898, 39141}, {8355, 18584}, {8550, 11257}, {8588, 50977}, {9830, 22329}, {9890, 10488}, {10011, 10516}, {10542, 44519}, {11063, 16010}, {11161, 26613}, {12216, 13196}, {13192, 37901}, {13640, 44394}, {13760, 44392}, {14120, 47455}, {14712, 39099}, {15513, 40107}, {15655, 50955}, {16317, 20998}, {20977, 37900}, {23004, 36758}, {23005, 36757}, {24855, 35266}, {25555, 39590}, {31415, 38064}, {32113, 36180}, {32456, 50567}, {32740, 34169}, {36166, 47169}, {36174, 52238}, {36196, 47544}, {37350, 47352}, {37637, 47353}, {39689, 40112}, {39764, 48901}, {39874, 46453}

X(53499) = midpoint of X(i) and X(j) for these {i,j}: {1992, 9855}, {5477, 6781}, {6776, 11676}, {8593, 51224}, {14712, 39099}
X(53499) = reflection of X(i) in X(j) for these {i,j}: {115, 2030}, {316, 44380}, {325, 5026}, {599, 27088}, {1352, 37459}, {5107, 41672}, {8352, 597}, {11646, 230}, {15980, 182}, {15993, 187}, {32113, 36180}, {36196, 47544}, {41146, 18800}, {50567, 32456}
X(53499) = crossdifference of every pair of points on line {7625, 8542}
X(53499) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {316, 5182, 44380}, {1691, 11646, 230}, {7737, 11179, 6}

---

X(53500) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(98) AND X(801)

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

X(53500) lies on these lines: {5, 12193}, {6, 1196}, {98, 230}, {232, 20998}, {577, 21001}, {1501, 37689}, {1613, 10311}, {1915, 7735}, {1970, 13881}, {3053, 15811}, {3231, 10313}, {3289, 9225}, {3291, 8779}, {9418, 33983}, {14585, 18945}, {17811, 51386}, {32661, 43291}, {33581, 33582}, {38297, 40320}, {39849, 50387}, {43620, 52438}, {44099, 52162}

X(53500) = crossdifference of every pair of points on line {2883, 3566}
X(53500) = {X(230),X(1971)}-harmonic conjugate of X(1691)

---

X(53501) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(226) AND X(671)

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

X(53501) lies on these lines: {37, 226}, {99, 44387}, {115, 527}, {148, 40882}, {190, 20337}, {316, 524}, {320, 23947}, {536, 10026}, {545, 44396}, {1213, 17351}, {1834, 4663}, {3729, 34528}, {3943, 22047}, {4527, 41014}, {4643, 23897}, {4644, 23903}, {4670, 23905}, {4887, 17058}, {5949, 17334}, {8818, 17276}, {10001, 17950}, {21320, 46536}, {23917, 24690}, {23918, 24691}, {24530, 37663}, {28604, 41809}, {32856, 33329}, {32935, 37159}

X(53501) = midpoint of X(148) and X(40882)
X(53501) = reflection of X(99) in X(44387)
X(53501) = X(1247)-anticomplementary conjugate of X(14360)

---

X(53502) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(262) AND X(485)

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

X(53502) lies on these lines: {3, 485}, {4, 31463}, {5, 3103}, {6, 6811}, {262, 1513}, {372, 20576}, {574, 45544}, {615, 14561}, {639, 7821}, {640, 6292}, {1152, 39387}, {1504, 48466}, {1587, 12968}, {2459, 38230}, {2548, 3071}, {3068, 25406}, {3933, 13877}, {5013, 45440}, {5254, 45861}, {5475, 45545}, {6201, 31400}, {6202, 31404}, {6250, 7748}, {6564, 45555}, {6656, 23312}, {7374, 31403}, {7389, 42265}, {7752, 23311}, {9600, 42284}, {10576, 40275}, {11179, 19145}, {11293, 31412}, {11313, 42582}, {12962, 45406}, {13873, 49213}, {14848, 32788}, {15883, 42258}, {22596, 37243}, {22727, 48876}, {23251, 44519}, {31481, 48467}, {35840, 44392}, {35947, 42264}, {35953, 44393}, {36733, 41945}, {45462, 48772}

X(53502) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6422, 36656, 3071}, {35840, 45554, 44392}

---

X(53503) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(262) AND X(486)

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

X(53503) lies on these lines: {3, 486}, {5, 3102}, {6, 6813}, {262, 1513}, {371, 20576}, {574, 45545}, {590, 14561}, {639, 6292}, {640, 7821}, {1151, 39388}, {1505, 48467}, {1588, 12963}, {2460, 38230}, {2548, 3070}, {3069, 25406}, {3933, 13930}, {5013, 45441}, {5254, 45860}, {5475, 45544}, {6201, 31404}, {6202, 31400}, {6251, 7748}, {6565, 45554}, {6656, 23311}, {7388, 42262}, {7752, 23312}, {8396, 31454}, {10577, 40274}, {11179, 19146}, {11294, 32805}, {11314, 42583}, {12969, 45407}, {13926, 49212}, {14848, 32787}, {14853, 31463}, {15884, 42259}, {22625, 37243}, {22726, 48876}, {23261, 44519}, {35841, 44394}, {35946, 42263}, {36719, 41946}, {42283, 43619}, {45463, 48773}

X(53503) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6421, 36655, 3070}, {35841, 45555, 44394}

---

X(53504) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(262) AND X(598)

Barycentrics    3*a^4*b^2 - a^2*b^4 - b^6 + 3*a^4*c^2 + 5*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 + b^2*c^4 - c^6 : :
X(53504) = X[6] + 2 X[43457], 2 X[5480] + X[43461]

X(53504) lies on these lines: {2, 5104}, {4, 5038}, {5, 13330}, {6, 13}, {69, 33005}, {141, 22486}, {262, 1513}, {373, 39602}, {511, 7603}, {524, 7926}, {575, 39590}, {597, 598}, {599, 8176}, {1506, 44453}, {1648, 5640}, {1691, 7737}, {1992, 42536}, {2030, 14537}, {2056, 6997}, {2076, 38317}, {3314, 24256}, {3363, 41146}, {3589, 3972}, {3618, 33017}, {4045, 5077}, {5017, 37637}, {5024, 50963}, {5026, 11361}, {5052, 39601}, {5055, 11173}, {5103, 7934}, {5111, 14853}, {5116, 48901}, {5169, 8288}, {5943, 15820}, {5969, 7777}, {6249, 44518}, {6781, 10168}, {6792, 11640}, {7394, 14153}, {7606, 51224}, {7735, 12212}, {7747, 25555}, {7785, 50249}, {8370, 44380}, {8586, 20423}, {11317, 51798}, {11648, 42852}, {12151, 33016}, {13331, 15048}, {31489, 40248}, {32740, 52189}, {33013, 39099}, {35424, 38739}, {37665, 43450}, {38064, 43618}, {39565, 44500}, {44526, 50659}, {44541, 48910}, {51024, 53095}

X(53504) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 381, 11646}, {5475, 5476, 6}, {5640, 6032, 1648}, {7747, 25555, 39560}

---

X(53505) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(262) AND X(671)

Barycentrics    3*a^4*b^2 - 4*a^2*b^4 - b^6 + 3*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(53505) = X[69] - 3 X[14041], 2 X[141] - 3 X[33228], 3 X[33228] - X[51438], 2 X[230] - 3 X[6034], X[5104] - 3 X[6034], 3 X[1570] - 2 X[41672], 3 X[2456] - X[38730], 4 X[3589] - 3 X[35297], 5 X[3618] - 3 X[13586], 3 X[5032] + X[40246], 4 X[8355] - 3 X[21358], X[8591] - 3 X[41137], X[11676] - 3 X[14853], 3 X[12215] - X[20094], 3 X[14561] - 2 X[37459], 2 X[27088] - 3 X[47352], 3 X[33265] - 7 X[51171], 3 X[35383] - 5 X[38739], 2 X[36180] - 3 X[47455]

X(53505) lies on these lines: {2, 13192}, {5, 44453}, {6, 30}, {69, 14041}, {99, 44380}, {115, 511}, {141, 7934}, {148, 39099}, {187, 19924}, {230, 5104}, {262, 1513}, {316, 524}, {325, 5969}, {538, 51396}, {542, 5107}, {543, 41146}, {550, 39560}, {574, 5476}, {575, 7756}, {576, 7748}, {597, 3972}, {599, 7615}, {625, 50567}, {690, 8430}, {754, 50253}, {858, 1648}, {1350, 37637}, {1503, 5111}, {1570, 29012}, {1691, 29181}, {1692, 29317}, {1992, 8597}, {2030, 6781}, {2387, 52471}, {2393, 2682}, {2456, 38730}, {2502, 40112}, {3124, 5913}, {3314, 18906}, {3564, 15514}, {3580, 8288}, {3589, 35297}, {3618, 13586}, {5017, 5999}, {5028, 48901}, {5031, 51374}, {5032, 40246}, {5033, 48880}, {5038, 12110}, {5085, 44541}, {5103, 6393}, {5112, 46124}, {5116, 18583}, {5254, 13330}, {5477, 11645}, {6390, 9890}, {6772, 51017}, {6775, 51019}, {6791, 39602}, {6792, 10989}, {7426, 41939}, {7603, 19130}, {7746, 52987}, {7747, 44499}, {7765, 44500}, {7790, 22486}, {8355, 21358}, {8589, 10168}, {8591, 41137}, {9877, 22110}, {10418, 13857}, {10485, 51737}, {11178, 18424}, {11477, 44518}, {11676, 14853}, {12215, 20094}, {13881, 53097}, {14120, 32113}, {14561, 37459}, {14567, 37900}, {15534, 44678}, {18800, 32479}, {20403, 52038}, {23004, 51207}, {23005, 51206}, {24206, 39601}, {25555, 37512}, {27088, 47352}, {31489, 38072}, {32452, 44230}, {33265, 51171}, {35383, 38739}, {36174, 48945}, {36180, 47455}, {37665, 40236}, {39565, 40107}, {39764, 48898}, {41172, 44231}, {44519, 53093}, {44532, 47619}, {51404, 52694}

X(53505) = midpoint of X(i) and X(j) for these {i,j}: {148, 39099}, {316, 10754}, {1992, 8597}, {5999, 51212}, {8586, 11646}, {23004, 51207}, {23005, 51206}
X(53505) = reflection of X(i) in X(j) for these {i,j}: {99, 44380}, {599, 37350}, {1513, 5480}, {5104, 230}, {5477, 44496}, {6393, 5103}, {6781, 2030}, {8598, 597}, {15993, 115}, {32113, 14120}, {37461, 5476}, {50567, 625}, {51374, 5031}, {51438, 141}
X(53505) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2549, 20423, 6}, {5104, 6034, 230}, {6791, 51360, 39602}, {33228, 51438, 141}

---

X(53506) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(275) AND X(459)

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

X(53506) lies on these lines: {2, 53}, {4, 10192}, {140, 6750}, {235, 17510}, {275, 1971}, {393, 459}, {428, 42874}, {441, 27358}, {467, 11064}, {468, 6747}, {470, 42945}, {471, 42944}, {472, 42941}, {473, 42940}, {1503, 52249}, {1585, 42258}, {1586, 42259}, {1990, 11547}, {2052, 47296}, {3199, 44334}, {3628, 8887}, {6509, 14576}, {6619, 36990}, {6677, 39569}, {6749, 11427}, {7714, 42854}, {11062, 46832}, {14577, 44436}, {14715, 47187}, {15576, 18950}, {23332, 33971}, {42391, 52281}

X(53506) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11547, 13567, 1990}, {14165, 52280, 23292}, {23292, 52280, 6748}

---

X(53507) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(275) AND X(671)

Barycentrics    a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + 3*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^2*b^2*c^4 - 4*b^4*c^4 + 2*a^2*c^6 + 3*b^2*c^6 - c^8 : :
X(53507) = 3 X[14041] - X[40888], 3 X[14568] - 2 X[44376], 3 X[33228] - 2 X[44389], 3 X[38227] - 2 X[39231]

X(53507) lies on these lines: {4, 2393}, {6, 32002}, {99, 44388}, {115, 44375}, {148, 44363}, {275, 1971}, {316, 524}, {317, 41760}, {338, 340}, {511, 39120}, {570, 17035}, {924, 5962}, {1232, 15108}, {1503, 20774}, {3003, 40853}, {5063, 52247}, {5392, 13579}, {6128, 40885}, {8667, 16275}, {8681, 13449}, {11433, 37192}, {11550, 35709}, {14041, 40888}, {14568, 44376}, {17983, 47280}, {33228, 44389}, {38227, 39231}

X(53507) = midpoint of X(148) and X(44363)
X(53507) = reflection of X(i) in X(j) for these {i,j}: {99, 44388}, {44375, 115}
X(53507) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {91, 14360}, {897, 40697}, {36128, 6193}, {36145, 44010}

---

X(53508) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(321) AND X(671)

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

X(53508) lies on these lines: {99, 44378}, {115, 536}, {148, 19623}, {192, 5949}, {313, 321}, {316, 524}, {543, 16702}, {740, 20499}, {1029, 20086}, {1086, 23947}, {1213, 17261}, {1266, 8287}, {2238, 37857}, {3454, 4535}, {3875, 8818}, {3943, 20337}, {4036, 7265}, {4363, 23903}, {4364, 23897}, {4419, 23942}, {4472, 23905}, {4686, 46826}, {4971, 10026}, {9296, 17790}, {20546, 28516}, {22034, 46828}, {23917, 25349}, {23918, 25350}, {28297, 50773}, {28555, 51417}, {37049, 49453}

X(53508) = midpoint of X(148) and X(19623)
X(53508) = reflection of X(i) in X(j) for these {i,j}: {99, 44378}, {44396, 115}
X(53508) = X(267)-anticomplementary conjugate of X(14360)

---

X(53509) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(321) AND X(1029)

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

X(53509) lies on these lines: {81, 1029}, {115, 3666}, {313, 321}, {502, 36250}, {671, 14534}, {940, 23903}, {1213, 33761}, {1224, 4205}, {1255, 17056}, {1834, 5080}, {3175, 46828}, {3703, 37159}, {5256, 8818}, {5949, 28606}, {6656, 23947}, {16687, 46536}, {17061, 33329}, {17147, 44396}, {17599, 37049}, {18089, 34294}, {40592, 44378}, {42051, 46826}

---

X(53510) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(321) AND X(1446)

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

X(53510) lies on these lines: {1, 20279}, {2, 15474}, {6, 17863}, {9, 17861}, {11, 17447}, {37, 1441}, {44, 23521}, {45, 17895}, {48, 24268}, {69, 20171}, {71, 16609}, {73, 1874}, {75, 966}, {76, 40364}, {85, 4648}, {92, 393}, {141, 1229}, {142, 1111}, {190, 26665}, {192, 3262}, {226, 1826}, {238, 23689}, {242, 41230}, {273, 948}, {286, 2303}, {307, 24005}, {313, 321}, {322, 17314}, {350, 17788}, {497, 1851}, {536, 3965}, {693, 21133}, {857, 16580}, {984, 23690}, {1086, 20905}, {1109, 23928}, {1231, 3721}, {1234, 26601}, {1334, 21231}, {1400, 8680}, {1446, 18635}, {1633, 22363}, {1834, 52345}, {1901, 5244}, {2178, 17134}, {2268, 24315}, {2269, 24424}, {2275, 28087}, {3008, 24208}, {3120, 21931}, {3662, 37788}, {3663, 4858}, {3665, 21239}, {3673, 4000}, {3731, 17885}, {3739, 20880}, {3765, 4812}, {3772, 28796}, {3782, 13567}, {3936, 4150}, {3943, 22040}, {3945, 20082}, {3946, 7264}, {3948, 20234}, {4008, 5698}, {4044, 18697}, {4280, 36022}, {4357, 20236}, {4358, 17267}, {4359, 19732}, {4361, 24352}, {4363, 24993}, {4364, 24547}, {4389, 26538}, {4424, 40564}, {4466, 16888}, {4641, 39767}, {4644, 44735}, {4656, 6358}, {4851, 30806}, {4957, 17246}, {5254, 26169}, {5257, 18698}, {5283, 17866}, {5342, 5716}, {8557, 45738}, {9318, 18162}, {13161, 20883}, {13576, 21867}, {16603, 21011}, {16706, 18151}, {17019, 30690}, {17261, 28974}, {17315, 17791}, {17316, 20930}, {17317, 46749}, {17451, 34830}, {18046, 21591}, {18082, 46897}, {18147, 20444}, {18726, 24220}, {18750, 37642}, {18815, 29007}, {19786, 20919}, {20245, 34377}, {20305, 21044}, {20643, 25660}, {20718, 22298}, {20896, 42707}, {20911, 21615}, {20912, 42724}, {20923, 33930}, {21246, 46180}, {23661, 50065}, {23774, 49537}, {23944, 48643}, {24627, 28930}, {25237, 27514}, {25254, 42700}, {25995, 26076}, {26130, 28081}, {26165, 41760}, {27039, 45744}, {30854, 37650}, {33151, 48380}

X(53510) = isotomic conjugate of X(40403)
X(53510) = isotomic conjugate of the isogonal conjugate of X(16583)
X(53510) = polar conjugate of the isotomic conjugate of X(20235)
X(53510) = polar conjugate of the isogonal conjugate of X(17441)
X(53510) = X(i)-Ceva conjugate of X(j) for these (i,j): {92, 52577}, {3673, 3914}, {31624, 693}
X(53510) = X(i)-isoconjugate of X(j) for these (i,j): {31, 40403}, {58, 7123}, {81, 7084}, {184, 40411}, {284, 1037}, {1041, 2193}, {1576, 48070}, {2194, 7131}, {2206, 30701}, {14935, 52378}
X(53510) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 40403}, {10, 7123}, {1214, 7131}, {4000, 2287}, {4858, 48070}, {6554, 81}, {14936, 21789}, {15487, 58}, {16583, 63}, {17463, 6586}, {18589, 6}, {40586, 7084}, {40590, 1037}, {40603, 30701}, {47345, 1041}
X(53510) = cevapoint of X(i) and X(j) for these (i,j): {4808, 36197}, {16583, 17441}
X(53510) = barycentric product X(i)*X(j) for these {i,j}: {4, 20235}, {10, 3673}, {75, 3914}, {76, 16583}, {92, 18589}, {264, 17441}, {286, 21015}, {304, 52577}, {313, 614}, {321, 4000}, {349, 2082}, {497, 1441}, {561, 40934}, {594, 16750}, {668, 48403}, {850, 1633}, {1446, 6554}, {1502, 21750}, {1577, 3732}, {1851, 20336}, {1969, 23620}, {3596, 40961}, {3701, 7195}, {4033, 48398}, {5324, 34388}, {6063, 40965}, {6335, 21107}, {6385, 21813}, {6386, 50490}, {7124, 52575}, {8020, 40364}, {16502, 27801}, {17170, 41013}, {18022, 22363}, {18084, 20883}, {21073, 41788}, {27509, 40149}, {28017, 30713}
X(53510) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 40403}, {37, 7123}, {42, 7084}, {65, 1037}, {92, 40411}, {225, 1041}, {226, 7131}, {321, 30701}, {497, 21}, {614, 58}, {1040, 283}, {1441, 8817}, {1446, 30705}, {1473, 1437}, {1577, 48070}, {1633, 110}, {1851, 28}, {1863, 4183}, {2082, 284}, {3673, 86}, {3732, 662}, {3914, 1}, {3952, 52778}, {4000, 81}, {4319, 2328}, {4516, 14935}, {4566, 8269}, {5286, 2303}, {5324, 60}, {6554, 2287}, {7083, 2194}, {7124, 2193}, {7195, 1014}, {7289, 1790}, {8020, 1973}, {16502, 1333}, {16583, 6}, {16750, 1509}, {17060, 16728}, {17115, 21789}, {17170, 1444}, {17441, 3}, {18084, 34055}, {18589, 63}, {20235, 69}, {21015, 72}, {21107, 905}, {21750, 32}, {21813, 213}, {22057, 255}, {22363, 184}, {23620, 48}, {27509, 1812}, {28017, 1412}, {40934, 31}, {40961, 56}, {40965, 55}, {40987, 2299}, {41785, 41610}, {48398, 1019}, {48403, 513}, {50490, 667}, {51400, 18206}, {52577, 19}
X(53510) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 16732, 1441}, {322, 20173, 17314}, {1229, 26563, 141}, {3948, 20234, 20336}, {17134, 26267, 2178}, {17863, 30807, 6}, {21049, 52023, 18635}

---

X(53511) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(485) AND X(598)

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

X(53511) lies on these lines: {3, 485}, {597, 598}, {615, 8376}, {2549, 13644}, {3069, 32489}, {5210, 13663}, {7585, 12962}, {7748, 45574}, {8411, 42270}, {8418, 41964}, {8982, 42265}, {9674, 13879}, {9974, 39661}, {11158, 13669}, {13783, 18584}, {13846, 44541}, {15484, 32788}, {22716, 35702}, {35822, 44394}, {35953, 44391}, {42263, 45440}, {42284, 46264}, {44526, 45484}

---

X(53512) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(485) AND X(671)

Barycentrics    2*a^4 - a^2*b^2 - 3*b^4 - a^2*c^2 + 6*b^2*c^2 - 3*c^4 + 2*(2*a^2 - b^2 - c^2)*S : :
X(53512) = 3 X[14041] - X[44364], 2 X[32459] - 3 X[44393], 3 X[33228] - 2 X[44390]

X(53512) lies on these lines: {3, 485}, {30, 44394}, {99, 44391}, {115, 32421}, {148, 44365}, {187, 13908}, {230, 13653}, {316, 524}, {325, 49367}, {492, 33006}, {615, 43620}, {1991, 44526}, {3069, 12323}, {3071, 9975}, {5206, 13921}, {6390, 9894}, {6565, 49355}, {7737, 8375}, {8252, 45508}, {13650, 45023}, {13749, 26441}, {14041, 44364}, {18440, 42283}, {31670, 42284}, {32459, 44393}, {32492, 48659}, {32788, 40287}, {33228, 44390}, {42215, 45463}, {42258, 50680}, {43291, 49214}, {44655, 49028}

X(53512) = midpoint of X(148) and X(44365)
X(53512) = reflection of X(i) in X(j) for these {i,j}: {99, 44391}, {44392, 115}

---

X(53513) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(485) AND X(1131)

Barycentrics    a^4 + 2*a^2*b^2 - 3*b^4 + 2*a^2*c^2 + 6*b^2*c^2 - 3*c^4 + 6*a^2*S : :
X(53513) = X[6453] - 3 X[8960], 2 X[6453] - 3 X[31454]

X(53513) lies on the cubic K1233 and these lines: {2, 6426}, {3, 485}, {4, 3592}, {5, 6420}, {6, 3091}, {13, 42280}, {14, 42281}, {20, 13846}, {30, 6453}, {115, 44647}, {140, 41946}, {371, 3627}, {372, 3628}, {381, 6427}, {382, 1327}, {395, 35732}, {396, 42282}, {486, 5072}, {490, 45871}, {524, 32489}, {546, 3071}, {547, 35813}, {548, 43209}, {550, 35812}, {575, 49229}, {597, 32488}, {615, 1587}, {632, 6454}, {641, 33425}, {1131, 3068}, {1151, 3529}, {1152, 3525}, {3069, 15022}, {3303, 31472}, {3304, 44623}, {3311, 42269}, {3312, 5079}, {3316, 41948}, {3364, 42214}, {3389, 42213}, {3526, 6522}, {3534, 9680}, {3543, 42577}, {3544, 7581}, {3545, 41951}, {3590, 15717}, {3830, 31487}, {3832, 19054}, {3850, 35823}, {3857, 6565}, {5007, 49220}, {5054, 10195}, {5056, 13847}, {5059, 10141}, {5068, 19053}, {5073, 9681}, {5076, 6561}, {5159, 8854}, {5420, 6448}, {5609, 49222}, {6199, 22615}, {6200, 12103}, {6221, 22644}, {6248, 49230}, {6396, 14869}, {6409, 8972}, {6410, 32785}, {6411, 43407}, {6417, 42268}, {6418, 42274}, {6431, 23259}, {6434, 43885}, {6437, 43408}, {6449, 42276}, {6452, 42566}, {6459, 50688}, {6460, 8253}, {6471, 13941}, {6482, 41961}, {6483, 11539}, {6488, 43509}, {6519, 13903}, {7584, 12811}, {7585, 23261}, {7765, 49262}, {7982, 49232}, {7991, 13911}, {8252, 46936}, {8981, 15704}, {9540, 17538}, {9541, 11541}, {9543, 42540}, {10142, 41970}, {10147, 42638}, {10148, 43511}, {10222, 49601}, {10299, 43256}, {10577, 12812}, {10991, 13908}, {11238, 31475}, {11241, 41362}, {11542, 42177}, {11543, 42175}, {11648, 31483}, {12102, 35821}, {12222, 45472}, {12971, 30531}, {13915, 51522}, {13935, 43506}, {14981, 49214}, {15029, 19110}, {15044, 19111}, {15083, 49224}, {15484, 19105}, {15687, 42417}, {15701, 42606}, {15707, 42526}, {15712, 42639}, {15720, 53131}, {15765, 41107}, {16239, 52048}, {17800, 53130}, {18585, 41108}, {18762, 35770}, {19709, 42608}, {20397, 49217}, {20398, 49213}, {20399, 49267}, {20400, 48715}, {22236, 42240}, {22238, 35740}, {22332, 31463}, {22883, 22928}, {23253, 42263}, {31440, 50865}, {31481, 53096}, {32421, 32491}, {32521, 35838}, {32786, 43884}, {33923, 43211}, {35255, 42267}, {35738, 42234}, {35771, 41991}, {35786, 42215}, {38734, 50722}, {38791, 46688}, {41955, 43508}, {41957, 42522}, {42159, 42245}, {42162, 42243}, {42168, 52079}, {42170, 52080}, {42232, 42925}, {42488, 51855}, {42489, 51854}, {42525, 44903}, {43291, 45515}, {43786, 49138}, {45407, 45861}, {45511, 45862}, {49135, 51850}, {49140, 52667}, {50720, 51523}

X(53513) = reflection of X(31454) in X(8960)
X(53513) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 43376, 31414}, {3, 485, 43879}, {3, 43879, 590}, {5, 6420, 43880}, {6, 31412, 42273}, {6, 42273, 42270}, {20, 13846, 41963}, {371, 42284, 42271}, {372, 18538, 42582}, {372, 42582, 32790}, {485, 3070, 590}, {485, 5418, 45384}, {485, 6560, 8976}, {485, 13665, 3070}, {546, 6419, 3071}, {546, 7583, 6419}, {550, 35812, 52045}, {590, 3070, 42259}, {632, 42216, 6454}, {1131, 3068, 23251}, {1151, 23249, 42272}, {1587, 3090, 3594}, {1587, 42265, 615}, {3068, 3146, 6425}, {3068, 23251, 42258}, {3068, 42570, 1131}, {3070, 43879, 3}, {3071, 35787, 43790}, {3090, 3594, 615}, {3146, 6425, 42258}, {3311, 42269, 42283}, {3312, 42277, 42583}, {3594, 42265, 3090}, {5072, 6428, 486}, {5072, 18512, 6428}, {5073, 9681, 43210}, {6419, 6564, 546}, {6420, 43880, 32788}, {6425, 23251, 3146}, {6454, 10576, 632}, {6519, 49137, 42260}, {6564, 7583, 3071}, {9540, 23269, 42264}, {13886, 23249, 1151}, {13903, 49137, 6519}, {22644, 43430, 6221}, {31414, 41952, 41964}, {42163, 42166, 42273}, {42248, 42249, 13665}, {42252, 42253, 590}, {43430, 43791, 22644}

---

X(53514) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(486) AND X(598)

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

X(53514) lies on these lines: {3, 486}, {590, 8375}, {597, 598}, {2549, 13763}, {3068, 32488}, {5210, 13783}, {7586, 12969}, {7748, 45575}, {8398, 41963}, {8403, 42273}, {9975, 39660}, {11157, 13789}, {13663, 18584}, {13847, 44541}, {15484, 32787}, {22718, 35703}, {26441, 42262}, {35823, 44392}, {42264, 45441}, {42283, 46264}, {44526, 45485}

---

X(53515) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(486) AND X(671)

Barycentrics    2*a^4 - a^2*b^2 - 3*b^4 - a^2*c^2 + 6*b^2*c^2 - 3*c^4 - 2*(2*a^2 - b^2 - c^2)*S : :
X(53515) = 3 X[14041] - X[44365], 2 X[32459] - 3 X[44400], 3 X[33228] - 2 X[44391]

X(53515) lies on these lines: {3, 486}, {30, 44392}, {99, 44390}, {115, 32419}, {148, 44364}, {187, 13968}, {230, 13773}, {316, 524}, {325, 49368}, {491, 33006}, {590, 43620}, {591, 44526}, {3068, 12322}, {3070, 9974}, {5206, 13880}, {6390, 9892}, {6564, 49356}, {7737, 8376}, {8253, 45509}, {8982, 13748}, {13771, 45024}, {14041, 44365}, {18440, 42284}, {31670, 42283}, {32459, 44400}, {32495, 48660}, {32787, 40286}, {33228, 44391}, {42216, 45462}, {42259, 50681}, {43291, 49215}, {44654, 49029}

X(53515) = midpoint of X(148) and X(44364)
X(53515) = reflection of X(i) in X(j) for these {i,j}: {99, 44390}, {44394, 115}

---

X(53516) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(486) AND X(1132)

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

X(53516) lies on the cubic K1233 and these lines: {2, 6425}, {3, 486}, {4, 3594}, {5, 6419}, {6, 3091}, {13, 42281}, {14, 42280}, {20, 13847}, {30, 6454}, {115, 44648}, {140, 41945}, {371, 3628}, {372, 3627}, {381, 6428}, {382, 1328}, {395, 42282}, {396, 35732}, {485, 5072}, {489, 45872}, {524, 32488}, {546, 3070}, {547, 35812}, {548, 43210}, {550, 35813}, {575, 49228}, {590, 1588}, {597, 32489}, {632, 6453}, {642, 33424}, {1132, 3069}, {1151, 3525}, {1152, 3529}, {1656, 31454}, {3068, 15022}, {3303, 44622}, {3304, 44624}, {3311, 5079}, {3312, 42268}, {3317, 9541}, {3365, 42212}, {3390, 42211}, {3526, 6519}, {3534, 43786}, {3543, 42576}, {3544, 7582}, {3545, 41952}, {3591, 15717}, {3832, 19053}, {3850, 35822}, {3854, 31414}, {3857, 6564}, {5007, 49221}, {5054, 9681}, {5056, 13846}, {5059, 10142}, {5068, 19054}, {5073, 43209}, {5076, 6560}, {5159, 8855}, {5418, 6447}, {5609, 49223}, {6200, 14869}, {6248, 49231}, {6395, 22644}, {6396, 12103}, {6398, 22615}, {6409, 32786}, {6410, 13941}, {6412, 43408}, {6417, 42277}, {6418, 42269}, {6432, 23249}, {6433, 43886}, {6438, 43407}, {6450, 42275}, {6451, 42567}, {6459, 8252}, {6460, 50688}, {6470, 8972}, {6482, 11539}, {6483, 41962}, {6489, 43510}, {6522, 13961}, {6811, 45870}, {7583, 12811}, {7586, 23251}, {7765, 49261}, {7982, 49233}, {7991, 13973}, {8253, 46936}, {9540, 43505}, {9680, 46219}, {10141, 41969}, {10147, 43512}, {10148, 42637}, {10222, 49602}, {10299, 43257}, {10576, 12812}, {10991, 13968}, {11242, 41362}, {11541, 41970}, {11542, 42178}, {11543, 42176}, {12102, 35820}, {12221, 45473}, {12965, 30531}, {13935, 17538}, {13966, 15704}, {13979, 51522}, {14981, 49215}, {15029, 19111}, {15044, 19110}, {15083, 49225}, {15484, 19102}, {15687, 42418}, {15701, 42607}, {15707, 42527}, {15712, 42640}, {15720, 53130}, {15765, 41108}, {16239, 52047}, {17800, 53131}, {17852, 49140}, {18538, 35771}, {18585, 41107}, {19709, 42609}, {20397, 49216}, {20398, 49212}, {20399, 49266}, {20400, 48714}, {22236, 42239}, {22238, 42241}, {22882, 22927}, {23263, 42264}, {30315, 31440}, {31487, 42602}, {32419, 32490}, {32521, 35839}, {32785, 43883}, {33923, 43212}, {35256, 42266}, {35730, 36439}, {35738, 42231}, {35770, 41991}, {35787, 42216}, {38734, 50721}, {38791, 46689}, {41956, 43507}, {41958, 42523}, {42159, 42244}, {42162, 42242}, {42167, 52079}, {42169, 52080}, {42233, 42924}, {42488, 51853}, {42489, 51852}, {42524, 44903}, {42562, 51727}, {43291, 45514}, {43785, 49138}, {45406, 45860}, {45510, 45863}, {49135, 51849}, {50719, 51523}

X(53516) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 486, 43880}, {3, 43880, 615}, {5, 6419, 43879}, {6, 42270, 42273}, {6, 42561, 42270}, {20, 13847, 41964}, {371, 18762, 42583}, {371, 42583, 32789}, {372, 42283, 42272}, {486, 3071, 615}, {486, 5420, 45385}, {486, 6561, 13951}, {486, 13785, 3071}, {546, 6420, 3070}, {546, 7584, 6420}, {550, 35813, 52046}, {615, 3071, 42258}, {632, 42215, 6453}, {1132, 3069, 23261}, {1152, 23259, 42271}, {1588, 3090, 3592}, {1588, 42262, 590}, {3069, 3146, 6426}, {3069, 23261, 42259}, {3069, 42571, 1132}, {3070, 35786, 43789}, {3071, 43880, 3}, {3090, 3592, 590}, {3146, 6426, 42259}, {3311, 42274, 42582}, {3312, 42268, 42284}, {3592, 42262, 3090}, {5072, 6427, 485}, {5072, 18510, 6427}, {6419, 43879, 32787}, {6420, 6565, 546}, {6426, 23261, 3146}, {6453, 10577, 632}, {6522, 49137, 42261}, {6565, 7584, 3070}, {9681, 10194, 5054}, {13935, 23275, 42263}, {13939, 23259, 1152}, {13961, 49137, 6522}, {22615, 43431, 6398}, {42163, 42166, 42270}, {42246, 42247, 13785}, {42250, 42251, 615}, {43431, 43792, 22615}

---

X(53517) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(1131) AND X(1327)

Barycentrics    5*a^4 + 2*a^2*b^2 - 7*b^4 + 2*a^2*c^2 + 14*b^2*c^2 - 7*c^4 + 10*a^2*S : :

X(53517) lies on these lines: {2, 6434}, {4, 6431}, {5, 372}, {6, 3839}, {20, 42568}, {30, 6480}, {376, 590}, {485, 1657}, {1131, 3068}, {1151, 49138}, {1327, 3830}, {1588, 43790}, {1656, 43514}, {3071, 6427}, {3523, 8253}, {3525, 23269}, {3533, 43338}, {3627, 43340}, {3628, 6483}, {3832, 42571}, {3854, 7586}, {3855, 6432}, {3858, 35770}, {3860, 6565}, {3861, 35771}, {5054, 6452}, {5071, 6438}, {5073, 43430}, {5076, 12818}, {6200, 19710}, {6396, 10124}, {6429, 49135}, {6430, 7486}, {6433, 15683}, {6435, 14893}, {6437, 15682}, {6442, 23267}, {6450, 42601}, {6460, 17852}, {6481, 15699}, {6485, 48154}, {6490, 9541}, {6493, 13935}, {6494, 22615}, {6499, 42268}, {6501, 13785}, {7583, 12102}, {7585, 41955}, {8972, 42540}, {9691, 22644}, {10576, 12108}, {11541, 43339}, {12100, 18538}, {12103, 35255}, {12811, 35814}, {13846, 52667}, {14093, 43336}, {14269, 43342}, {15685, 43887}, {15695, 43568}, {15703, 41946}, {15705, 32785}, {15707, 43513}, {15718, 42602}, {17800, 43515}, {23253, 42271}, {23261, 31414}, {32790, 43510}, {35256, 42418}, {35404, 41945}, {36439, 43200}, {36457, 43199}, {38072, 49017}, {41099, 43381}, {41948, 41950}, {41957, 43376}, {41961, 42260}, {42226, 45759}, {42266, 43434}, {42276, 45384}, {42582, 46219}, {47599, 52046}

X(53517) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {485, 42272, 41963}, {615, 6564, 42273}, {1131, 3146, 42570}, {1327, 13665, 42284}, {1327, 42284, 43789}, {1327, 43791, 13665}, {3070, 6564, 615}, {3830, 42572, 32787}, {6460, 46936, 42569}, {12100, 18538, 42558}, {12100, 42558, 42566}, {13665, 42284, 32787}, {32787, 43789, 42284}, {42276, 45384, 52045}

---

X(53518) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(1131) AND X(1328)

Barycentrics    7*a^4 - 2*a^2*b^2 - 5*b^4 - 2*a^2*c^2 + 10*b^2*c^2 - 5*c^4 + 2*a^2*S : :

X(53518) = 5 X[590] - 4 X[6200], 3 X[590] - 4 X[6564], 7 X[590] - 8 X[18538], 9 X[590] - 8 X[35255], 25 X[590] - 24 X[43211], 7 X[590] - 6 X[52045], 3 X[6200] - 5 X[6564], 7 X[6200] - 10 X[18538], 9 X[6200] - 10 X[35255], 2 X[6200] - 5 X[42284], 5 X[6200] - 6 X[43211], 14 X[6200] - 15 X[52045], 7 X[6564] - 6 X[18538], 3 X[6564] - 2 X[35255], 2 X[6564] - 3 X[42284], and many others

X(53518) lies on these lines: {2, 42538}, {4, 615}, {6, 3543}, {20, 8253}, {30, 590}, {140, 43785}, {372, 3853}, {376, 32789}, {381, 6452}, {382, 3070}, {485, 5073}, {486, 5076}, {546, 42267}, {550, 35786}, {1131, 3068}, {1151, 23253}, {1327, 6221}, {1328, 3830}, {1657, 6496}, {2043, 42194}, {2044, 42193}, {3069, 42574}, {3071, 3627}, {3529, 42265}, {3534, 42277}, {3545, 6412}, {3592, 23269}, {3594, 23263}, {3628, 51910}, {3832, 6410}, {3839, 8252}, {3843, 42261}, {3845, 6396}, {3860, 42567}, {3861, 10577}, {5055, 42601}, {5059, 6409}, {5413, 13473}, {5418, 17800}, {6398, 38335}, {6411, 11001}, {6432, 23275}, {6435, 33699}, {6438, 43256}, {6439, 9542}, {6442, 23259}, {6445, 35400}, {6446, 42603}, {6449, 49133}, {6451, 42602}, {6459, 43411}, {6460, 50688}, {6469, 42605}, {6471, 7586}, {6474, 9681}, {6477, 12101}, {6478, 8960}, {6488, 9540}, {6501, 22615}, {6565, 15687}, {8404, 45440}, {8972, 41952}, {8976, 49134}, {8981, 41961}, {8982, 14239}, {9541, 43789}, {9680, 41965}, {10576, 15704}, {12102, 35787}, {12818, 41950}, {13665, 15684}, {13713, 44456}, {13847, 43405}, {14269, 42274}, {14893, 35256}, {15682, 23249}, {15683, 32785}, {15695, 43562}, {15700, 42600}, {15765, 42182}, {17851, 35403}, {18510, 35434}, {18585, 42181}, {18586, 42105}, {18587, 42104}, {19053, 43508}, {19106, 42211}, {19107, 42212}, {28178, 35788}, {28190, 35763}, {31412, 41963}, {31454, 41954}, {31673, 49233}, {35404, 35822}, {35813, 41947}, {36436, 42096}, {36437, 42102}, {36454, 42097}, {36455, 42101}, {41869, 49232}, {41949, 43313}, {41955, 42641}, {41958, 43406}, {41968, 41988}, {42108, 52399}, {42109, 52400}, {42130, 42192}, {42131, 42191}, {42159, 42252}, {42162, 42253}, {42260, 43879}, {42268, 45385}, {42280, 42563}, {42281, 42565}, {42525, 42639}, {42537, 42540}, {42561, 42579}, {42572, 52047}, {42637, 50689}, {42638, 50692}, {43415, 43888}, {43509, 43536}, {45384, 53130}, {48901, 49229}

X(53518) = reflection of X(590) in X(42284)
X(53518) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 42259, 42270}, {4, 42264, 615}, {4, 42272, 42259}, {4, 43407, 42262}, {382, 3070, 42271}, {382, 22644, 3070}, {550, 35786, 42582}, {615, 42264, 42259}, {615, 42272, 42264}, {1131, 3146, 42413}, {1131, 42413, 6425}, {1328, 6560, 6395}, {1328, 42418, 32788}, {3068, 43507, 23251}, {3146, 23251, 42258}, {3146, 43507, 3068}, {3543, 52667, 6}, {3627, 35820, 3071}, {3830, 6560, 42283}, {3832, 42414, 6410}, {3843, 42261, 42583}, {6395, 6560, 42418}, {6425, 23251, 1131}, {6425, 42413, 42258}, {6560, 42283, 32788}, {6565, 42226, 41946}, {13665, 15684, 42275}, {13665, 42275, 41945}, {13785, 42283, 43790}, {15682, 23249, 42263}, {15687, 42226, 6565}, {18538, 52045, 590}, {23249, 42263, 32787}, {23253, 33703, 1151}, {32788, 43790, 13785}, {42259, 42270, 41964}, {42262, 43510, 615}

---

X(53519) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(1132) AND X(1327)

Barycentrics    7*a^4 - 2*a^2*b^2 - 5*b^4 - 2*a^2*c^2 + 10*b^2*c^2 - 5*c^4 - 2*a^2*S : :
X(53519) = 5 X[615] - 4 X[6396], 3 X[615] - 4 X[6565], 7 X[615] - 8 X[18762], 9 X[615] - 8 X[35256], 25 X[615] - 24 X[43212], 7 X[615] - 6 X[52046], 3 X[6396] - 5 X[6565], 7 X[6396] - 10 X[18762], 9 X[6396] - 10 X[35256], 2 X[6396] - 5 X[42283], 5 X[6396] - 6 X[43212], 14 X[6396] - 15 X[52046], 7 X[6565] - 6 X[18762], 3 X[6565] - 2 X[35256], 2 X[6565] - 3 X[42283], and many others

X(53519) lies on these lines: {2, 42537}, {4, 590}, {6, 3543}, {20, 8252}, {30, 615}, {140, 43786}, {371, 3853}, {376, 32790}, {381, 6451}, {382, 3071}, {485, 5076}, {486, 5073}, {546, 42266}, {550, 35787}, {1132, 3069}, {1152, 23263}, {1327, 3830}, {1328, 6398}, {1657, 6497}, {2043, 42191}, {2044, 42192}, {3068, 42575}, {3070, 3627}, {3529, 42262}, {3534, 42274}, {3545, 6411}, {3592, 23253}, {3594, 23275}, {3628, 51911}, {3832, 6409}, {3839, 8253}, {3843, 42260}, {3845, 6200}, {3860, 42566}, {3861, 10576}, {5055, 42600}, {5059, 6410}, {5412, 13473}, {5420, 17800}, {6221, 38335}, {6412, 11001}, {6431, 23269}, {6436, 33699}, {6437, 43257}, {6440, 13847}, {6441, 23249}, {6445, 42602}, {6446, 35400}, {6450, 49133}, {6452, 42603}, {6459, 50688}, {6460, 43412}, {6468, 42604}, {6470, 7585}, {6476, 12101}, {6489, 11541}, {6500, 22644}, {6564, 15687}, {8412, 45441}, {9647, 18514}, {9660, 18513}, {9690, 43887}, {10577, 15704}, {12102, 35786}, {12819, 41949}, {13785, 15684}, {13836, 44456}, {13846, 43406}, {13941, 41951}, {13951, 49134}, {13966, 41962}, {14235, 26441}, {14269, 42277}, {14893, 35255}, {15682, 23259}, {15683, 32786}, {15695, 43563}, {15700, 42601}, {15765, 42179}, {17852, 42579}, {18512, 35434}, {18585, 42180}, {18586, 42104}, {18587, 42105}, {19054, 43507}, {19106, 42213}, {19107, 42214}, {28178, 35789}, {28190, 35762}, {31412, 42578}, {31454, 42269}, {31673, 49232}, {35404, 35823}, {35812, 41948}, {36436, 42097}, {36437, 42101}, {36454, 42096}, {36455, 42102}, {41869, 49233}, {41950, 43312}, {41953, 41970}, {41956, 42642}, {41957, 43405}, {41964, 42561}, {41967, 41988}, {42108, 52400}, {42109, 52399}, {42130, 42194}, {42131, 42193}, {42159, 42250}, {42162, 42251}, {42261, 43880}, {42280, 42564}, {42281, 42562}, {42524, 42640}, {42538, 42539}, {42573, 52048}, {42637, 50692}, {42638, 50689}, {45385, 53131}, {48901, 49228}

X(53519) = reflection of X(615) in X(42283)
X(53519) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 42258, 42273}, {4, 42263, 590}, {4, 42271, 42258}, {4, 43408, 42265}, {382, 3071, 42272}, {382, 22615, 3071}, {550, 35787, 42583}, {590, 42263, 42258}, {590, 42271, 42263}, {1132, 3146, 42414}, {1132, 42414, 6426}, {1327, 6561, 6199}, {1327, 42417, 32787}, {3069, 43508, 23261}, {3146, 23261, 42259}, {3146, 43508, 3069}, {3543, 52666, 6}, {3627, 35821, 3070}, {3830, 6561, 42284}, {3832, 42413, 6409}, {3843, 42260, 42582}, {6199, 6561, 42417}, {6426, 23261, 1132}, {6426, 42414, 42259}, {6470, 23251, 31414}, {6561, 42284, 32787}, {6564, 42225, 41945}, {7585, 42540, 31414}, {13665, 42284, 43789}, {13785, 15684, 42276}, {13785, 42276, 41946}, {15682, 23259, 42264}, {15687, 42225, 6564}, {18762, 52046, 615}, {23259, 42264, 32788}, {23263, 33703, 1152}, {32787, 43789, 13665}, {42258, 42273, 41963}, {42265, 43509, 590}

---

X(53520) = INTERSECTION OF LINES TANGENT TO KIEPERT CIRCUMHYPERBOLA AT X(1132) AND X(1328)

Barycentrics    5*a^4 + 2*a^2*b^2 - 7*b^4 + 2*a^2*c^2 + 14*b^2*c^2 - 7*c^4 - 10*a^2*S : :

X(53520) lies on these lines: {2, 6433}, {4, 6432}, {5, 371}, {6, 3839}, {20, 42569}, {30, 6481}, {376, 615}, {486, 1657}, {1132, 3069}, {1152, 49138}, {1328, 3830}, {1587, 43789}, {1656, 43513}, {3070, 6428}, {3523, 8252}, {3525, 9541}, {3533, 43339}, {3627, 43341}, {3628, 6482}, {3832, 42570}, {3854, 7585}, {3855, 6431}, {3858, 35771}, {3860, 6564}, {3861, 35770}, {5054, 6451}, {5071, 6437}, {5073, 43431}, {5076, 12819}, {6200, 10124}, {6396, 19710}, {6429, 7486}, {6430, 49135}, {6434, 15683}, {6436, 14893}, {6438, 15682}, {6441, 23273}, {6449, 42600}, {6459, 41955}, {6480, 15699}, {6484, 48154}, {6491, 14226}, {6492, 9540}, {6495, 22644}, {6498, 42269}, {6500, 13665}, {7584, 12102}, {7586, 41956}, {9690, 15703}, {10577, 12108}, {11541, 43338}, {12100, 18762}, {12103, 35256}, {12811, 35815}, {13847, 52666}, {13941, 42539}, {14093, 43337}, {14269, 43343}, {15685, 43888}, {15695, 43569}, {15705, 32786}, {15707, 43514}, {15718, 42603}, {17800, 43516}, {22615, 49134}, {23263, 42272}, {32789, 43509}, {35255, 42417}, {35404, 41946}, {36439, 43199}, {36457, 43200}, {38072, 49016}, {41099, 43380}, {41947, 41949}, {41958, 43377}, {41962, 42261}, {42225, 45759}, {42267, 43435}, {42275, 45385}, {42583, 46219}, {47599, 52045}

X(53520) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {486, 42271, 41964}, {590, 6565, 42270}, {1132, 3146, 42571}, {1328, 13785, 42283}, {1328, 42283, 43790}, {1328, 43792, 13785}, {3071, 6565, 590}, {3830, 42573, 32788}, {6459, 46936, 42568}, {12100, 18762, 42557}, {12100, 42557, 42567}, {13785, 42283, 32788}, {32788, 43790, 42283}, {42275, 45385, 52046}

---

X(53521) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(11) AND X(1355)

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

X(53521) lies on these lines: {2, 24353}, {3, 42662}, {6, 53301}, {11, 244}, {38, 4088}, {386, 2774}, {513, 43051}, {522, 20521}, {523, 20508}, {649, 834}, {654, 22384}, {656, 17069}, {684, 2491}, {810, 44410}, {982, 4458}, {986, 2785}, {1355, 42751}, {1403, 53262}, {2292, 14432}, {2424, 17962}, {3261, 4025}, {3670, 4707}, {3798, 21172}, {3915, 42657}, {4079, 45882}, {4255, 53249}, {4256, 44827}, {5075, 53309}, {6129, 50336}, {8676, 22090}, {14419, 42666}, {16892, 21102}, {17899, 21050}, {21103, 48101}, {21109, 21134}, {21146, 23740}, {21176, 21203}, {21181, 24167}, {21347, 21957}, {23996, 50440}, {24443, 30574}, {24462, 25380}, {24622, 24718}, {51329, 53300}

X(53521) = reflection of X(22090) in X(52595)
X(53521) = complement of X(53336)
X(53521) = X(i)-isoconjugate of X(j) for these (i,j): {10, 36084}, {37, 2966}, {42, 36036}, {72, 685}, {98, 100}, {101, 1821}, {190, 1910}, {213, 43187}, {228, 22456}, {248, 6335}, {287, 1783}, {290, 692}, {293, 1897}, {306, 36104}, {321, 2715}, {336, 8750}, {651, 15628}, {668, 1976}, {879, 5379}, {906, 16081}, {1331, 36120}, {1332, 6531}, {1824, 17932}, {2395, 4567}, {2422, 4601}, {3998, 20031}, {5380, 5967}, {6386, 14601}, {7081, 36065}, {20336, 32696}, {32716, 42711}, {32739, 46273}, {41013, 43754}, {41932, 42717}
X(53521) = X(i)-Dao conjugate of X(j) for these (i,j): {132, 1897}, {1015, 1821}, {1086, 290}, {2679, 42}, {4988, 43665}, {5190, 16081}, {5521, 36120}, {5976, 1978}, {6626, 43187}, {8054, 98}, {11672, 190}, {26932, 336}, {34467, 293}, {35088, 313}, {38987, 10}, {38991, 15628}, {39000, 306}, {39006, 287}, {39039, 6335}, {39040, 668}, {40589, 2966}, {40592, 36036}, {40601, 101}, {40619, 46273}, {40627, 2395}, {41167, 4064}, {46094, 1331}, {50440, 3699}
X(53521) = crossdifference of every pair of points on line {10, 98}
X(53521) = barycentric product X(i)*X(j) for these {i,j}: {27, 684}, {58, 2799}, {86, 3569}, {232, 4025}, {237, 3261}, {240, 905}, {244, 42717}, {297, 1459}, {310, 2491}, {325, 649}, {511, 514}, {512, 51370}, {513, 1959}, {522, 43034}, {523, 17209}, {661, 51369}, {667, 46238}, {693, 1755}, {868, 4556}, {1474, 6333}, {1790, 16230}, {2396, 3122}, {2421, 3120}, {2530, 3405}, {3289, 46107}, {3669, 44694}, {4091, 6530}, {4107, 40810}, {4230, 4466}, {4391, 51651}, {4610, 44114}, {4750, 5968}, {5360, 7199}, {6629, 8430}, {7649, 36212}, {9417, 40495}, {11125, 35910}, {14966, 21207}, {16732, 23997}, {16892, 51862}, {17206, 17994}, {18653, 32112}, {20022, 21123}, {21109, 36823}, {22383, 40703}, {30805, 34854}, {34234, 42751}, {39469, 44129}
X(53521) = barycentric quotient X(i)/X(j) for these {i,j}: {27, 22456}, {58, 2966}, {81, 36036}, {86, 43187}, {232, 1897}, {237, 101}, {240, 6335}, {325, 1978}, {511, 190}, {513, 1821}, {514, 290}, {649, 98}, {663, 15628}, {667, 1910}, {684, 306}, {693, 46273}, {868, 52623}, {905, 336}, {1333, 36084}, {1459, 287}, {1474, 685}, {1755, 100}, {1790, 17932}, {1919, 1976}, {1959, 668}, {2203, 36104}, {2206, 2715}, {2211, 8750}, {2421, 4600}, {2491, 42}, {2799, 313}, {3120, 43665}, {3122, 2395}, {3261, 18024}, {3289, 1331}, {3569, 10}, {4091, 6394}, {4107, 14382}, {4750, 52145}, {5360, 1018}, {6333, 40071}, {6591, 36120}, {7649, 16081}, {9417, 692}, {9418, 32739}, {14966, 4570}, {17209, 99}, {17994, 1826}, {21102, 53245}, {21122, 11610}, {21123, 20021}, {21143, 43920}, {22383, 293}, {23996, 42717}, {23997, 4567}, {33752, 21094}, {36212, 4561}, {39469, 71}, {41172, 4064}, {42717, 7035}, {42751, 908}, {43034, 664}, {44114, 4024}, {44694, 646}, {46238, 6386}, {50521, 3404}, {51369, 799}, {51370, 670}, {51651, 651}
X(53521) = {X(17899),X(21259)}-harmonic conjugate of X(21050)

---

X(53522) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(11) AND X(1359)

Barycentrics    (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(53522) = X[8] - 3 X[23678], 2 X[676] - 3 X[11125], X[1769] - 3 X[11125], 3 X[6129] - 4 X[52596], 3 X[21172] - 2 X[52596], X[40500] + 2 X[48283], X[47695] - 3 X[53352], 2 X[4925] - 3 X[26078], 2 X[4990] - 3 X[45686], 3 X[14413] - X[30572], 3 X[25996] - X[50333]

X(53522) lies on these lines: {1, 2804}, {6, 47137}, {8, 23678}, {11, 244}, {36, 44807}, {56, 53304}, {65, 8677}, {513, 1835}, {521, 21186}, {522, 905}, {523, 1459}, {650, 52587}, {652, 6587}, {885, 1462}, {1319, 35013}, {1359, 6087}, {1411, 30725}, {1637, 22086}, {1834, 36035}, {2501, 22383}, {2522, 3700}, {2827, 21201}, {3287, 47134}, {3315, 47695}, {3667, 7661}, {3738, 10015}, {4491, 30930}, {4925, 26078}, {4976, 6589}, {4977, 21102}, {4990, 45686}, {9001, 21120}, {14304, 24034}, {14413, 30572}, {17094, 21184}, {17724, 24353}, {21103, 28175}, {21202, 43042}, {23838, 46435}, {23986, 46391}, {24025, 24030}, {28083, 28114}, {32714, 36127}, {36121, 36123}, {39540, 42337}

X(53522) = midpoint of X(43728) and X(44428)
X(53522) = reflection of X(i) in X(j) for these {i,j}: {1769, 676}, {6129, 21172}, {10015, 21180}, {30725, 53314}, {48303, 39540}
X(53522) = X(i)-complementary conjugate of X(j) for these (i,j): {604, 46398}, {655, 21244}, {1106, 51402}, {1397, 35128}, {1411, 124}, {2222, 1329}, {6187, 5514}, {32675, 3452}, {53321, 31845}
X(53522) = X(i)-Ceva conjugate of X(j) for these (i,j): {2406, 2182}, {14304, 6087}, {23987, 1455}, {36123, 11}, {43728, 513}
X(53522) = X(i)-isoconjugate of X(j) for these (i,j): {8, 36040}, {40, 6081}, {78, 36067}, {100, 102}, {101, 36100}, {190, 32677}, {312, 32643}, {345, 32667}, {651, 15629}, {692, 34393}, {906, 52780}, {1331, 36121}, {1897, 36055}, {2149, 2399}, {2432, 4564}, {3869, 35183}
X(53522) = X(i)-Dao conjugate of X(j) for these (i,j): {650, 2399}, {1015, 36100}, {1086, 34393}, {5190, 52780}, {5521, 36121}, {8054, 102}, {10017, 8}, {23986, 190}, {34467, 36055}, {38991, 15629}, {51221, 1897}
X(53522) = crossdifference of every pair of points on line {101, 102}
X(53522) = barycentric product X(i)*X(j) for these {i,j}: {11, 2406}, {57, 14304}, {189, 6087}, {244, 42718}, {273, 46391}, {278, 39471}, {514, 515}, {522, 34050}, {649, 35516}, {693, 2182}, {1359, 2399}, {1455, 4391}, {2425, 34387}, {4025, 8755}, {4466, 7452}, {4560, 51421}, {4581, 51414}, {7004, 24035}, {17924, 46974}, {23987, 26932}, {24002, 51361}, {34234, 42755}
X(53522) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 2399}, {513, 36100}, {514, 34393}, {515, 190}, {604, 36040}, {608, 36067}, {649, 102}, {663, 15629}, {667, 32677}, {1359, 2406}, {1395, 32667}, {1397, 32643}, {1436, 6081}, {1455, 651}, {2182, 100}, {2406, 4998}, {2425, 59}, {3271, 2432}, {6087, 329}, {6591, 36121}, {7649, 52780}, {8735, 53152}, {8755, 1897}, {11700, 4585}, {14304, 312}, {21123, 46359}, {22383, 36055}, {23987, 46102}, {24034, 42718}, {34050, 664}, {35516, 1978}, {39471, 345}, {42462, 15633}, {42718, 7035}, {42755, 908}, {46391, 78}, {46974, 1332}, {51361, 644}, {51414, 53332}, {51421, 4552}
X(53522) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1769, 11125, 676}, {21184, 23724, 17094}

---

X(53523) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(11) AND X(3021)

Barycentrics    (a - b - c)*(b - c)*(2*a^2 - a*b + b^2 - a*c - 2*b*c + c^2) : :
X(53523) = 4 X[676] - 3 X[1638], 3 X[1638] - 2 X[2254], 2 X[2505] - 3 X[45677], 3 X[23729] - 4 X[49295], 3 X[1639] - 4 X[3716], 3 X[1639] - 2 X[50333], X[47695] - 3 X[53361], X[53343] + 3 X[53361], 2 X[1491] - 3 X[48179], 4 X[2496] - 3 X[47801], 2 X[4394] - 3 X[47801], 2 X[2526] - 3 X[47756], 2 X[2977] - 3 X[4448], X[4467] - 3 X[48239], and many others

X(535) lies on these lines: {1, 2826}, {2, 4925}, {11, 244}, {35, 44805}, {55, 659}, {65, 2821}, {294, 885}, {390, 6009}, {497, 46403}, {513, 11934}, {514, 4162}, {521, 11927}, {522, 650}, {523, 4724}, {649, 5583}, {654, 53395}, {663, 6362}, {891, 3057}, {918, 1280}, {1491, 48179}, {1697, 21385}, {1960, 2646}, {2098, 21343}, {2496, 4394}, {2526, 47756}, {2804, 15914}, {2820, 11028}, {2827, 5083}, {2976, 3021}, {2977, 4448}, {3254, 23838}, {3309, 7178}, {3667, 3676}, {3699, 25268}, {3887, 10015}, {3900, 21120}, {4105, 40500}, {4122, 28183}, {4170, 28481}, {4423, 25926}, {4458, 50357}, {4467, 48239}, {4528, 14430}, {4762, 48014}, {4773, 44433}, {4777, 48056}, {4841, 48006}, {4874, 48232}, {4895, 6366}, {4905, 30724}, {4926, 47800}, {4928, 4962}, {4940, 48035}, {4977, 47704}, {4990, 48278}, {4995, 45314}, {5048, 6550}, {6004, 48403}, {6161, 29240}, {6363, 40467}, {6608, 6615}, {7655, 7661}, {9508, 26275}, {10058, 19916}, {11238, 48167}, {14321, 48077}, {14392, 40520}, {17069, 47798}, {17724, 43050}, {20317, 44448}, {21105, 23057}, {21138, 36639}, {21146, 47132}, {21321, 27675}, {22942, 34789}, {23761, 48326}, {24093, 25574}, {24719, 28217}, {25569, 34471}, {27542, 27543}, {28161, 48271}, {28221, 47828}, {28292, 43052}, {28882, 48072}, {29126, 48324}, {29142, 48305}, {29162, 48150}, {29278, 48264}, {39540, 43924}, {41800, 48018}, {42462, 46392}, {47687, 48172}, {47694, 48276}, {47767, 48069}, {47784, 48017}, {47799, 50335}, {47876, 47975}, {47890, 48063}, {47891, 48073}

X(53523) = midpoint of X(i) and X(j) for these {i,j}: {4895, 21132}, {21105, 23745}, {47695, 53343}
X(53523) = reflection of X(i) in X(j) for these {i,j}: {2254, 676}, {4394, 2496}, {4773, 44433}, {4841, 48006}, {4905, 34958}, {4976, 50347}, {7178, 21185}, {7655, 7661}, {10015, 21201}, {21104, 47123}, {21146, 47132}, {30725, 1}, {43924, 39540}, {44448, 20317}, {47890, 48063}, {48032, 2976}, {48035, 4940}, {48077, 14321}, {48276, 47694}, {48278, 4990}, {50333, 3716}, {50356, 17069}, {50357, 4458}
X(53523) = anticomplement of X(4925)
X(53523) = tripolar centroid of X(42318)
X(53523) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {105, 34548}, {919, 8055}, {1293, 20344}, {27834, 20552}, {34080, 20533}, {36086, 42020}, {38266, 39353}
X(53523) = X(i)-Ceva conjugate of X(j) for these (i,j): {3717, 4124}, {14942, 11}, {36807, 4534}, {53337, 2348}
X(53523) = X(i)-isoconjugate of X(j) for these (i,j): {57, 6078}, {59, 35355}, {100, 1477}, {101, 43760}, {108, 1810}, {109, 1280}, {692, 35160}, {1252, 37626}, {1415, 36807}, {1458, 39272}
X(53523) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 1280}, {661, 37626}, {1015, 43760}, {1086, 35160}, {1146, 36807}, {3008, 883}, {3693, 42720}, {5452, 6078}, {6615, 35355}, {8054, 1477}, {16593, 664}, {35111, 190}, {38983, 1810}, {39048, 651}
X(53523) = crossdifference of every pair of points on line {56, 101}
X(53523) = barycentric product X(i)*X(j) for these {i,j}: {8, 6084}, {11, 53337}, {312, 48032}, {514, 5853}, {522, 3008}, {693, 2348}, {885, 16593}, {1111, 23704}, {1279, 4391}, {2976, 6557}, {3261, 8647}, {3596, 8659}, {20780, 46110}, {43728, 51419}, {50333, 52210}
X(53523) = barycentric quotient X(i)/X(j) for these {i,j}: {55, 6078}, {244, 37626}, {294, 39272}, {513, 43760}, {514, 35160}, {522, 36807}, {649, 1477}, {650, 1280}, {652, 1810}, {1279, 651}, {2170, 35355}, {2348, 100}, {2976, 5435}, {3008, 664}, {3021, 53337}, {5853, 190}, {6084, 7}, {8647, 101}, {8659, 56}, {16593, 883}, {20662, 2283}, {20780, 1813}, {23704, 765}, {40609, 42720}, {48032, 57}, {52210, 927}, {53337, 4998}
X(53523) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {676, 2254, 1638}, {2496, 4394, 47801}, {3716, 50333, 1639}, {4905, 34958, 30724}, {20317, 44448, 44729}, {23057, 23745, 21105}, {47798, 50356, 17069}, {53343, 53361, 47695}

---

X(53524) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(11) AND X(3024)

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

X(53524) lies on these lines: {1, 399}, {11, 244}, {21, 4636}, {30, 1725}, {33, 5348}, {35, 35194}, {37, 35326}, {38, 3058}, {47, 8144}, {55, 846}, {56, 53252}, {57, 15430}, {58, 16141}, {65, 52524}, {81, 7073}, {90, 1062}, {100, 24433}, {201, 15338}, {394, 7072}, {497, 4392}, {513, 18210}, {526, 2611}, {650, 36197}, {756, 4995}, {774, 7354}, {896, 5160}, {928, 1364}, {971, 8758}, {982, 11238}, {1040, 7082}, {1155, 1736}, {1399, 6198}, {1421, 51768}, {1621, 24434}, {1710, 20831}, {1772, 12019}, {1776, 2361}, {1807, 10058}, {1854, 22760}, {1898, 17102}, {1935, 9627}, {1936, 9629}, {2087, 11998}, {2292, 10543}, {2594, 41562}, {3072, 9628}, {3073, 9630}, {3220, 23848}, {3464, 5221}, {3465, 5172}, {3712, 44694}, {3937, 17463}, {4081, 24031}, {4354, 52408}, {4516, 18191}, {4653, 33857}, {4760, 24454}, {4854, 11031}, {5218, 9330}, {5432, 7069}, {5494, 14127}, {6284, 44706}, {6357, 16272}, {7226, 10385}, {7290, 30223}, {8287, 22094}, {10122, 11553}, {10308, 52372}, {10391, 15569}, {12953, 37591}, {14936, 38358}, {15310, 45916}, {17435, 38347}, {20872, 21368}, {22321, 23845}, {23821, 24235}, {24025, 45885}, {35193, 52126}, {38390, 42753}, {51889, 52383}

X(53524) = X(i)-Ceva conjugate of X(j) for these (i,j): {81, 650}, {1442, 14838}, {3219, 9404}, {4420, 35057}, {6198, 2605}, {10308, 513}
X(53524) = X(i)-isoconjugate of X(j) for these (i,j): {3, 34922}, {37, 35049}, {59, 79}, {100, 26700}, {101, 38340}, {109, 6742}, {765, 52372}, {1252, 52374}, {1262, 7110}, {1415, 15455}, {2149, 30690}, {2160, 4564}, {2427, 47317}, {4551, 13486}, {4570, 52382}, {4998, 6186}, {5379, 52390}, {7012, 7100}, {7045, 7073}, {7115, 52381}, {8818, 52378}, {24027, 52344}
X(53524) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 6742}, {513, 52372}, {522, 52344}, {650, 30690}, {661, 52374}, {1015, 38340}, {1146, 15455}, {1577, 20565}, {3700, 321}, {4988, 43682}, {6615, 79}, {8054, 26700}, {8287, 664}, {14838, 1441}, {17115, 7073}, {35057, 4420}, {36103, 34922}, {40589, 35049}, {40628, 52381}, {50330, 52382}
X(53524) = cevapoint of X(2611) and X(22094)
X(53524) = crossdifference of every pair of points on line {101, 26700}
X(53524) = barycentric product X(i)*X(j) for these {i,j}: {8, 7202}, {11, 3219}, {21, 8287}, {35, 4858}, {81, 6741}, {244, 42033}, {261, 21824}, {284, 17886}, {314, 20982}, {319, 2170}, {333, 2611}, {514, 35057}, {522, 14838}, {643, 21141}, {650, 4467}, {663, 18160}, {693, 9404}, {1086, 4420}, {1111, 52405}, {1146, 1442}, {1399, 23978}, {2003, 24026}, {2174, 34387}, {2185, 21054}, {2310, 17095}, {2605, 4391}, {3024, 30690}, {3271, 33939}, {3678, 17197}, {3737, 7265}, {3969, 18191}, {4041, 16755}, {4466, 11107}, {4516, 34016}, {6198, 26932}, {7004, 52412}, {7110, 7266}, {7282, 34591}, {7332, 52126}, {14936, 52421}, {16732, 35193}, {21207, 35192}, {22094, 31623}, {23226, 46110}
X(53524) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 30690}, {19, 34922}, {35, 4564}, {58, 35049}, {244, 52374}, {513, 38340}, {522, 15455}, {649, 26700}, {650, 6742}, {1015, 52372}, {1146, 52344}, {1399, 1262}, {1442, 1275}, {2003, 7045}, {2170, 79}, {2174, 59}, {2310, 7110}, {2605, 651}, {2611, 226}, {3024, 3219}, {3120, 43682}, {3125, 52382}, {3219, 4998}, {3271, 2160}, {4420, 1016}, {4467, 4554}, {4516, 8818}, {4858, 20565}, {6198, 46102}, {6741, 321}, {7004, 52381}, {7117, 7100}, {7202, 7}, {7252, 13486}, {7266, 17095}, {8287, 1441}, {9404, 100}, {14838, 664}, {14936, 7073}, {14975, 7115}, {16755, 4625}, {17104, 52378}, {17886, 349}, {18160, 4572}, {18191, 52393}, {20982, 65}, {21044, 6757}, {21054, 6358}, {21141, 4077}, {21824, 12}, {22094, 1214}, {23226, 1813}, {35057, 190}, {35192, 4570}, {35193, 4567}, {41502, 5379}, {42033, 7035}, {52380, 39295}, {52405, 765}, {52408, 44717}
X(53524) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {55, 24430, 24431}, {90, 1062, 7299}, {1776, 3100, 2361}, {2310, 7004, 11}, {16141, 38336, 58}

---

X(53525) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(11) AND X(3024)

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

X(53525) lies on these lines: {1, 3}, {2, 24431}, {11, 244}, {38, 5432}, {88, 14191}, {104, 1411}, {106, 10703}, {116, 38964}, {320, 24490}, {335, 28798}, {497, 33102}, {513, 42753}, {515, 14584}, {650, 17435}, {756, 5326}, {758, 52537}, {912, 52005}, {950, 24167}, {952, 1772}, {1015, 38345}, {1210, 51889}, {1317, 24028}, {1357, 1364}, {1393, 7354}, {1399, 26877}, {1421, 1768}, {1464, 34913}, {1725, 15325}, {1776, 7292}, {1807, 10090}, {1837, 6788}, {1870, 52440}, {2006, 11219}, {2330, 18183}, {2361, 3218}, {2594, 12005}, {2801, 43048}, {3022, 5577}, {3058, 42040}, {3125, 11998}, {3141, 50329}, {3326, 14027}, {4124, 21132}, {4392, 5218}, {4551, 17660}, {4642, 37734}, {4674, 17636}, {4679, 24498}, {4995, 42038}, {5083, 24025}, {5272, 7082}, {5433, 44706}, {5443, 5492}, {5573, 30223}, {6075, 42759}, {7248, 11436}, {7299, 24467}, {8286, 22094}, {8562, 38982}, {9335, 10589}, {10017, 15608}, {10265, 52383}, {10950, 24443}, {11570, 34586}, {12016, 39763}, {12053, 23869}, {12831, 52659}, {13226, 15253}, {17054, 22760}, {17063, 24430}, {17638, 32486}, {18188, 18210}, {18412, 26742}, {21578, 34232}, {23757, 34590}, {34589, 44311}, {35128, 53046}, {35365, 43921}, {43043, 45946}, {45950, 51402}

X(53525) = reflection of X(45885) in X(43048)
X(53525) = isogonal conjugate of X(52377)
X(53525) = incircle-inverse of X(14115)
X(53525) = X(i)-complementary conjugate of X(j) for these (i,j): {244, 15608}, {915, 20316}, {1459, 42423}, {2990, 3835}, {3657, 3454}, {6099, 24003}, {32655, 514}, {36052, 513}
X(53525) = X(i)-Ceva conjugate of X(j) for these (i,j): {88, 650}, {104, 513}, {1443, 3960}, {1870, 53314}, {3218, 654}, {3911, 46393}, {3960, 46384}, {4511, 3738}, {5620, 21106}, {18815, 514}, {22464, 30725}
X(53525) = X(i)-isoconjugate of X(j) for these (i,j): {1, 52377}, {36, 46649}, {59, 80}, {100, 2222}, {101, 655}, {109, 51562}, {190, 32675}, {692, 35174}, {765, 1411}, {1110, 18815}, {1252, 2006}, {1262, 36910}, {1415, 36804}, {1807, 7012}, {2149, 18359}, {2161, 4564}, {2323, 23592}, {4559, 47318}, {4570, 52383}, {4998, 6187}, {5379, 52391}, {7045, 52371}, {7115, 52351}, {9268, 14584}, {21794, 39295}, {21859, 37140}, {24027, 52409}, {32739, 46405}, {46102, 52431}
X(53525) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 52377}, {11, 51562}, {513, 1411}, {514, 18815}, {522, 52409}, {650, 18359}, {661, 2006}, {1015, 655}, {1086, 35174}, {1146, 36804}, {1577, 20566}, {1639, 4358}, {3738, 4511}, {6544, 14628}, {6615, 80}, {8054, 2222}, {10015, 3262}, {13999, 1897}, {15898, 46649}, {17115, 52371}, {35128, 190}, {35204, 765}, {38984, 100}, {40584, 4564}, {40612, 4998}, {40619, 46405}, {40628, 52351}, {50330, 52383}
X(53525) = crossdifference of every pair of points on line {101, 650}
X(53525) = X(17435)-line-conjugate of X(650)
X(53525) = barycentric product X(i)*X(j) for these {i,j}: {9, 4089}, {11, 3218}, {36, 4858}, {88, 51402}, {104, 46398}, {244, 32851}, {320, 2170}, {513, 3904}, {514, 3738}, {522, 3960}, {650, 4453}, {654, 693}, {664, 46384}, {758, 17197}, {905, 44428}, {1086, 4511}, {1111, 2323}, {1146, 1443}, {1870, 26932}, {2310, 17078}, {2361, 23989}, {3025, 18359}, {3261, 8648}, {3271, 20924}, {3737, 4707}, {3936, 18191}, {3942, 5081}, {4162, 27836}, {4282, 21207}, {4391, 53314}, {4466, 17515}, {4530, 52553}, {4585, 21132}, {7004, 17923}, {7113, 34387}, {17880, 52413}, {18155, 21828}, {18815, 35128}, {21758, 35519}, {22379, 46110}, {23978, 52440}, {24002, 53285}
X(53525) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 52377}, {11, 18359}, {36, 4564}, {244, 2006}, {513, 655}, {514, 35174}, {522, 36804}, {649, 2222}, {650, 51562}, {654, 100}, {667, 32675}, {693, 46405}, {1015, 1411}, {1086, 18815}, {1146, 52409}, {1411, 23592}, {1443, 1275}, {1647, 14628}, {1870, 46102}, {2087, 14584}, {2161, 46649}, {2170, 80}, {2310, 36910}, {2323, 765}, {2361, 1252}, {3025, 3218}, {3125, 52383}, {3218, 4998}, {3271, 2161}, {3737, 47318}, {3738, 190}, {3904, 668}, {3942, 52392}, {3960, 664}, {4089, 85}, {4282, 4570}, {4453, 4554}, {4511, 1016}, {4530, 51975}, {4858, 20566}, {7004, 52351}, {7113, 59}, {7117, 1807}, {8648, 101}, {14936, 52371}, {17197, 14616}, {18191, 24624}, {21044, 15065}, {21758, 109}, {21828, 4551}, {22379, 1813}, {32851, 7035}, {35128, 4511}, {38353, 51379}, {42462, 52356}, {42666, 21859}, {42753, 52212}, {44428, 6335}, {46384, 522}, {46398, 3262}, {51402, 4358}, {51663, 4605}, {52407, 44717}, {52413, 7012}, {52426, 1110}, {52434, 2149}, {52440, 1262}, {53285, 644}, {53314, 651}
X(53525) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 41343, 1319}, {1, 46820, 36}, {106, 10703, 12740}, {244, 7004, 11}, {1647, 35015, 11}, {2446, 2447, 14115}, {3756, 38357, 11}, {3999, 9371, 18839}, {4694, 45269, 5048}, {6788, 18340, 1837}

---

X(53526) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(11) AND X(3026)

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

X(53526) lies on these lines: {2, 24336}, {7, 7384}, {11, 244}, {75, 17452}, {86, 17440}, {748, 22400}, {950, 50171}, {1111, 24224}, {1229, 20258}, {1837, 4675}, {2098, 17119}, {2170, 3904}, {2171, 24220}, {2268, 10436}, {2269, 3739}, {3057, 4688}, {3217, 27384}, {3942, 16732}, {4359, 30089}, {4475, 23676}, {4644, 27471}, {4957, 7202}, {6173, 9581}, {7225, 24179}, {10950, 17392}, {11238, 31139}, {11376, 17301}, {11998, 16726}, {13384, 36834}, {17182, 20882}, {17183, 21233}, {17237, 17606}, {17282, 30820}, {17419, 17761}, {17863, 30097}, {17871, 21328}, {17895, 43040}, {20171, 30035}, {21020, 21334}, {21033, 21246}, {21044, 26932}, {24319, 24999}, {24470, 38330}, {24633, 24705}

X(53526) = X(i)-Ceva conjugate of X(j) for these (i,j): {940, 50457}, {5307, 48144}, {10435, 513}, {10436, 17418}, {11679, 23880}, {44733, 514}
X(53526) = X(i)-isoconjugate of X(j) for these (i,j): {59, 941}, {100, 32693}, {692, 32038}, {931, 4559}, {959, 1252}, {1110, 44733}, {2149, 31359}, {2258, 4564}, {5546, 52931}, {7115, 34259}
X(53526) = X(i)-Dao conjugate of X(j) for these (i,j): {514, 44733}, {650, 31359}, {661, 959}, {1086, 32038}, {1577, 34258}, {6615, 941}, {8054, 32693}, {17417, 100}, {23880, 11679}, {34261, 765}, {40628, 34259}, {50457, 5226}
X(53526) = crossdifference of every pair of points on line {101, 32693}
X(53526) = barycentric product X(i)*X(j) for these {i,j}: {11, 10436}, {514, 23880}, {522, 43067}, {693, 17418}, {940, 4858}, {958, 1111}, {1086, 11679}, {1468, 34387}, {1867, 17219}, {2170, 34284}, {2268, 23989}, {3026, 31359}, {3714, 17205}, {4185, 17880}, {4391, 48144}, {4466, 44734}, {4560, 50457}, {5307, 26932}, {8672, 18155}, {17197, 31993}
X(53526) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 31359}, {244, 959}, {514, 32038}, {649, 32693}, {940, 4564}, {958, 765}, {1086, 44733}, {1468, 59}, {2170, 941}, {2268, 1252}, {3026, 10436}, {3271, 2258}, {3737, 931}, {4017, 52931}, {4185, 7012}, {4858, 34258}, {5019, 2149}, {5307, 46102}, {7004, 34259}, {8672, 4551}, {10436, 4998}, {11679, 1016}, {17197, 37870}, {17418, 100}, {18191, 5331}, {23880, 190}, {43067, 664}, {48144, 651}, {50457, 4552}
X(53526) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11, 4459, 2310}, {3942, 16732, 21139}, {4858, 17197, 2170}, {24224, 24237, 1111}

---

X(53527) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(11) AND X(3028)

Barycentrics    a*(b - c)*(b + c)*(a^2 - b^2 + b*c - c^2) : :
X(53527) = X[2254] + 2 X[14315], 2 X[2254] + X[24457], 4 X[14315] - X[24457], X[4905] + 3 X[21189], X[4905] - 3 X[23800], X[4983] - 3 X[50330], 4 X[14353] - X[48283], 2 X[48018] - 3 X[50350], 3 X[656] - X[4041], 3 X[4017] + X[4041], X[30572] + 3 X[30574], X[659] - 3 X[28284], X[693] + 2 X[23809], 2 X[3716] - 3 X[48168], X[4088] - 3 X[14429], and many others

X(53527) lies on these lines: {1, 8674}, {11, 244}, {36, 238}, {37, 24290}, {55, 53248}, {56, 51643}, {88, 39155}, {214, 3738}, {512, 48350}, {520, 7250}, {521, 14353}, {522, 4823}, {523, 656}, {526, 3028}, {659, 27628}, {661, 14407}, {662, 2613}, {693, 23809}, {918, 24459}, {1020, 4551}, {1054, 2640}, {1227, 4453}, {1734, 4777}, {2250, 21894}, {2605, 6003}, {2610, 21828}, {2616, 23286}, {2642, 21832}, {2773, 35050}, {2786, 24417}, {2827, 19916}, {2849, 53300}, {2850, 22586}, {3065, 23838}, {3464, 53256}, {3669, 9001}, {3716, 25493}, {3777, 9002}, {4088, 14429}, {4145, 4674}, {4162, 6129}, {4357, 23827}, {4367, 9013}, {4486, 25356}, {4589, 35147}, {4636, 13486}, {4657, 24285}, {4707, 4736}, {4959, 48303}, {4977, 17420}, {5620, 10265}, {6006, 48075}, {6615, 28217}, {7253, 48209}, {8043, 50346}, {8062, 48207}, {8648, 39478}, {8672, 17990}, {8702, 48293}, {10015, 21112}, {10117, 14667}, {14838, 50349}, {16591, 41167}, {17321, 53335}, {18004, 27728}, {23226, 48384}, {23687, 47131}, {23785, 48084}, {23790, 48152}, {23792, 23813}, {23795, 23808}, {24885, 24920}, {24901, 24910}, {25923, 50357}, {26144, 53343}, {28183, 50338}, {28209, 48151}, {28393, 28396}, {28601, 48571}, {28623, 50327}, {32486, 53292}, {35057, 48292}, {42662, 43693}, {42757, 45022}, {50359, 50497}, {51236, 53406}, {51642, 53295}, {53277, 53308}, {53286, 53313}

X(53527) = midpoint of X(i) and X(j) for these {i,j}: {656, 4017}, {1769, 2254}, {6129, 7655}, {17420, 50354}, {21189, 23800}
X(53527) = reflection of X(i) in X(j) for these {i,j}: {1769, 14315}, {3737, 31947}, {4486, 25356}, {14287, 2505}, {14288, 3837}, {21112, 10015}, {24457, 1769}, {48283, 51648}, {48302, 6129}, {50334, 47843}, {50346, 8043}, {50349, 14838}, {51648, 14353}, {53314, 3960}
X(53527) = X(i)-Ceva conjugate of X(j) for these (i,j): {88, 3125}, {104, 18210}, {1290, 1}, {3960, 21828}, {4049, 661}, {4453, 4707}, {5620, 3120}, {18815, 16732}
X(53527) = X(i)-isoconjugate of X(j) for these (i,j): {6, 47318}, {10, 36069}, {21, 2222}, {35, 476}, {37, 37140}, {58, 51562}, {80, 110}, {99, 6187}, {100, 759}, {101, 24624}, {108, 1793}, {109, 6740}, {112, 52351}, {162, 1807}, {163, 18359}, {190, 34079}, {284, 655}, {319, 14560}, {321, 32671}, {333, 32675}, {643, 1411}, {648, 52431}, {651, 2341}, {662, 2161}, {692, 14616}, {1333, 36804}, {1414, 52371}, {1576, 20566}, {2006, 5546}, {2174, 32680}, {2194, 35174}, {3219, 32678}, {3737, 52377}, {4024, 9273}, {4036, 9274}, {4551, 52380}, {4565, 36910}, {4622, 40172}, {4636, 52383}, {6198, 36061}, {32662, 52412}, {34857, 52935}, {36129, 52408}, {52391, 52914}
X(53527) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 47318}, {10, 51562}, {11, 6740}, {37, 36804}, {115, 18359}, {125, 1807}, {244, 80}, {1015, 24624}, {1084, 2161}, {1086, 14616}, {1214, 35174}, {4858, 20566}, {6741, 52409}, {7359, 42716}, {8054, 759}, {13999, 29}, {16221, 6198}, {17433, 35194}, {18334, 3219}, {34586, 100}, {34591, 52351}, {35069, 190}, {35128, 333}, {35204, 643}, {35583, 34772}, {38982, 10}, {38983, 1793}, {38984, 21}, {38986, 6187}, {38991, 2341}, {38993, 46077}, {38994, 46073}, {40584, 662}, {40589, 37140}, {40590, 655}, {40608, 52371}, {40611, 2222}, {40612, 99}, {40622, 18815}, {42761, 3262}, {51583, 668}
X(53527) = cevapoint of X(2610) and X(42666)
X(53527) = crossdifference of every pair of points on line {37, 101}
X(53527) = X(24290)-line-conjugate of X(37)
X(53527) = barycentric product X(i)*X(j) for these {i,j}: {1, 4707}, {10, 3960}, {36, 1577}, {37, 4453}, {65, 3904}, {75, 21828}, {79, 32679}, {81, 6370}, {86, 2610}, {214, 4049}, {226, 3738}, {274, 42666}, {313, 21758}, {320, 661}, {321, 53314}, {333, 51663}, {349, 8648}, {512, 20924}, {513, 3936}, {514, 758}, {522, 18593}, {523, 3218}, {525, 1870}, {526, 30690}, {649, 35550}, {650, 41804}, {654, 1441}, {656, 17923}, {693, 2245}, {798, 40075}, {850, 7113}, {860, 905}, {1018, 4089}, {1214, 44428}, {1443, 3700}, {1446, 53285}, {1464, 4391}, {1835, 6332}, {1983, 21207}, {2160, 3268}, {2323, 4077}, {2624, 20565}, {3120, 4585}, {3261, 3724}, {4017, 32851}, {4041, 17078}, {4053, 7192}, {4120, 52553}, {4242, 4466}, {4511, 7178}, {4849, 27836}, {4973, 31010}, {5081, 51664}, {6548, 40988}, {7100, 44427}, {14208, 52413}, {14618, 52407}, {15413, 44113}, {20948, 52434}, {21192, 39149}, {22128, 24006}, {23870, 39153}, {23871, 39152}, {23884, 53114}, {27950, 35352}, {34234, 42768}
X(53527) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 47318}, {10, 36804}, {36, 662}, {37, 51562}, {58, 37140}, {65, 655}, {79, 32680}, {226, 35174}, {320, 799}, {512, 2161}, {513, 24624}, {514, 14616}, {523, 18359}, {526, 3219}, {647, 1807}, {649, 759}, {650, 6740}, {652, 1793}, {654, 21}, {656, 52351}, {661, 80}, {663, 2341}, {667, 34079}, {758, 190}, {798, 6187}, {810, 52431}, {860, 6335}, {1333, 36069}, {1400, 2222}, {1402, 32675}, {1441, 46405}, {1443, 4573}, {1464, 651}, {1577, 20566}, {1835, 653}, {1870, 648}, {1983, 4570}, {2081, 35194}, {2160, 476}, {2206, 32671}, {2245, 100}, {2323, 643}, {2361, 5546}, {2610, 10}, {2624, 35}, {3218, 99}, {3268, 33939}, {3700, 52409}, {3709, 52371}, {3724, 101}, {3738, 333}, {3904, 314}, {3936, 668}, {3960, 86}, {4017, 2006}, {4024, 15065}, {4041, 36910}, {4053, 3952}, {4079, 34857}, {4089, 7199}, {4120, 51975}, {4282, 4636}, {4453, 274}, {4511, 645}, {4559, 52377}, {4585, 4600}, {4707, 75}, {6137, 46077}, {6138, 46073}, {6186, 32678}, {6370, 321}, {6739, 42716}, {7113, 110}, {7178, 18815}, {7180, 1411}, {7252, 52380}, {8648, 284}, {13486, 39295}, {14270, 2174}, {14407, 40172}, {16944, 4591}, {17078, 4625}, {17923, 811}, {18593, 664}, {20924, 670}, {21044, 52356}, {21123, 46160}, {21758, 58}, {21828, 1}, {21832, 36815}, {22128, 4592}, {22379, 1790}, {30572, 14628}, {30690, 35139}, {32679, 319}, {32851, 7257}, {35550, 1978}, {39152, 23896}, {39153, 23895}, {40075, 4602}, {40215, 4622}, {40988, 17780}, {41804, 4554}, {42666, 37}, {42768, 908}, {44113, 1783}, {44428, 31623}, {47230, 6198}, {51663, 226}, {51664, 52392}, {52407, 4558}, {52413, 162}, {52434, 163}, {52440, 4565}, {52553, 4615}, {53285, 2287}, {53314, 81}
X(53527) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {244, 2254, 13277}, {2254, 14315, 24457}, {4357, 23827, 23829}

---

X(53528) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1317) AND X(1337)

Barycentrics    a*(2*a - b - c)*(b - c)*(a + b - c)*(a - b + c) : :
X(53528) = X[2254] + 2 X[14812], 2 X[1769] - 3 X[14413], 4 X[3669] - 3 X[4017], 2 X[3669] - 3 X[43924], X[3669] - 3 X[51656], X[4017] - 4 X[51656], 3 X[14413] - 4 X[53314], 2 X[30725] + X[39771], 4 X[14838] - 3 X[17420], 2 X[14838] - 3 X[21173], 3 X[17418] - 2 X[47965]

X(53528) lies on these lines: {1, 2827}, {7, 43041}, {56, 4491}, {57, 53392}, {100, 109}, {106, 8686}, {244, 1357}, {513, 663}, {604, 8632}, {659, 6164}, {900, 1317}, {1027, 1462}, {1400, 3768}, {1405, 23650}, {1411, 23345}, {1469, 47330}, {1635, 20972}, {2423, 21758}, {2650, 6089}, {2804, 21105}, {2815, 24028}, {2820, 44858}, {3676, 47981}, {3716, 26139}, {3887, 51766}, {4041, 9001}, {4498, 9048}, {4778, 21180}, {4977, 21112}, {4979, 7180}, {6006, 30719}, {7178, 28209}, {8648, 51236}, {14027, 38979}, {14151, 23057}, {14838, 17420}, {17195, 41801}, {17418, 47965}, {20980, 21127}, {21786, 46393}, {22379, 39200}, {28220, 43052}, {30198, 42312}, {43051, 50358}

X(53528) = midpoint of X(30572) and X(39771)
X(53528) = reflection of X(i) in X(j) for these {i,j}: {1769, 53314}, {4017, 43924}, {6615, 1459}, {17420, 21173}, {30572, 30725}, {43924, 51656}
X(53528) = X(i)-Ceva conjugate of X(j) for these (i,j): {1411, 244}, {1877, 1647}, {2222, 56}, {14584, 14027}, {23703, 1319}, {30725, 1635}, {34051, 2170}, {37136, 604}, {37627, 43924}
X(53528) = X(i)-isoconjugate of X(j) for these (i,j): {2, 5548}, {6, 4582}, {8, 901}, {9, 3257}, {11, 6551}, {55, 4555}, {88, 644}, {100, 1320}, {101, 4997}, {106, 3699}, {190, 2316}, {210, 4622}, {312, 32665}, {522, 9268}, {643, 4674}, {646, 9456}, {650, 5376}, {765, 23838}, {903, 3939}, {1318, 17780}, {1334, 4615}, {2226, 30731}, {2320, 52925}, {2321, 4591}, {2325, 4638}, {2720, 51984}, {3271, 6635}, {3596, 32719}, {3689, 4618}, {3877, 36091}, {4013, 4636}, {4076, 23345}, {4080, 5546}, {4152, 39414}, {4571, 36125}, {4587, 6336}, {4945, 5549}, {5233, 32686}, {6065, 6548}, {6079, 45247}, {32645, 44723}, {34230, 36802}, {35281, 36596}, {36042, 44720}
X(53528) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 4582}, {214, 3699}, {223, 4555}, {478, 3257}, {513, 23838}, {900, 4768}, {1015, 4997}, {1647, 4723}, {4370, 646}, {5516, 44720}, {6544, 4391}, {8054, 1320}, {32664, 5548}, {35092, 312}, {38979, 8}, {38981, 51984}, {40615, 20568}, {40617, 903}, {51402, 341}, {52659, 668}, {52877, 4069}
X(53528) = cevapoint of X(1319) and X(14122)
X(53528) = crossdifference of every pair of points on line {9, 644}
X(53528) = barycentric product X(i)*X(j) for these {i,j}: {1, 30725}, {7, 1635}, {44, 3676}, {56, 3762}, {57, 900}, {81, 30572}, {85, 1960}, {88, 39771}, {269, 1639}, {273, 22086}, {279, 4895}, {479, 14427}, {513, 3911}, {514, 1319}, {519, 3669}, {651, 1647}, {664, 2087}, {693, 1404}, {738, 4528}, {902, 24002}, {905, 1877}, {934, 4530}, {1014, 4120}, {1019, 40663}, {1022, 1317}, {1023, 1358}, {1086, 23703}, {1119, 14418}, {1357, 24004}, {1396, 14429}, {1407, 4768}, {1417, 52627}, {1432, 4922}, {1434, 4730}, {1459, 37790}, {1769, 40218}, {2251, 52621}, {2325, 43932}, {3257, 14027}, {3259, 37136}, {3285, 4077}, {3943, 7203}, {3960, 14584}, {3977, 43923}, {4017, 16704}, {4358, 43924}, {4564, 6550}, {5298, 47947}, {7045, 52338}, {7178, 52680}, {7180, 30939}, {8686, 21129}, {14122, 42555}, {14425, 19604}, {14439, 43930}, {14628, 53314}, {16594, 37627}, {17096, 21805}, {23757, 34051}, {24188, 31615}, {30573, 34056}, {37168, 51664}, {43038, 52225}
X(53528) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4582}, {31, 5548}, {44, 3699}, {56, 3257}, {57, 4555}, {109, 5376}, {513, 4997}, {519, 646}, {604, 901}, {649, 1320}, {667, 2316}, {678, 30731}, {900, 312}, {902, 644}, {1014, 4615}, {1015, 23838}, {1023, 4076}, {1317, 24004}, {1319, 190}, {1357, 1022}, {1397, 32665}, {1404, 100}, {1405, 52925}, {1408, 4591}, {1412, 4622}, {1415, 9268}, {1417, 4638}, {1434, 4634}, {1635, 8}, {1639, 341}, {1647, 4391}, {1877, 6335}, {1960, 9}, {2087, 522}, {2149, 6551}, {2251, 3939}, {3251, 2325}, {3285, 643}, {3669, 903}, {3676, 20568}, {3689, 6558}, {3762, 3596}, {3911, 668}, {4017, 4080}, {4120, 3701}, {4448, 3975}, {4528, 30693}, {4530, 4397}, {4564, 6635}, {4730, 2321}, {4773, 4673}, {4895, 346}, {4922, 17787}, {4984, 3702}, {6085, 52140}, {6544, 4723}, {6550, 4858}, {7180, 4674}, {8661, 2170}, {14027, 3762}, {14122, 6631}, {14407, 210}, {14408, 27538}, {14418, 1265}, {14425, 44720}, {14427, 5423}, {14435, 3902}, {14436, 4517}, {14437, 4009}, {14584, 36804}, {16704, 7257}, {20972, 23705}, {21805, 30730}, {22086, 78}, {22356, 4571}, {23202, 4587}, {23703, 1016}, {24188, 40166}, {30572, 321}, {30725, 75}, {31615, 42372}, {35092, 4768}, {39771, 4358}, {40663, 4033}, {42084, 1639}, {43923, 6336}, {43924, 88}, {46393, 51984}, {51422, 42718}, {51656, 31227}, {52338, 24026}, {52680, 645}, {52963, 4069}
X(53528) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1769, 53314, 14413}, {43924, 51646, 53314}, {51641, 51659, 4017}, {51650, 51662, 4017}

---

X(53529) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1317) AND X(1360)

Barycentrics    (2*a - b - c)*(a + b - c)*(a - b + c)*(2*a^3 - a^2*b - b^3 - a^2*c + b^2*c + b*c^2 - c^3) : :
X(53529) = 3 X[1456] - 2 X[43035]

X(53529) lies on these lines: {11, 109}, {44, 1877}, {56, 1633}, {65, 2835}, {221, 5895}, {222, 3058}, {516, 1456}, {528, 651}, {603, 37722}, {676, 1360}, {900, 1317}, {1155, 43068}, {1359, 1361}, {1362, 2820}, {1394, 12701}, {1421, 24465}, {1457, 3000}, {1537, 11700}, {1836, 45946}, {2099, 4644}, {2283, 15507}, {2823, 3057}, {3246, 30379}, {3321, 3322}, {4081, 24410}, {4318, 17768}, {4551, 6154}, {4675, 15950}, {5228, 50303}, {5432, 34029}, {5433, 40560}, {5723, 24715}, {6174, 23703}, {7354, 34040}, {9580, 34033}, {10571, 15338}, {11246, 34036}, {12832, 13539}, {15171, 34043}, {15252, 34789}, {15601, 24914}, {17074, 49736}, {22464, 28534}, {24466, 34586}, {24708, 24806}, {25968, 33650}, {34048, 34612}, {35015, 51408}, {39692, 46819}, {40862, 49709}

X(53529) = reflection of X(4081) in X(24410)
X(53529) = X(i)-isoconjugate of X(j) for these (i,j): {88, 2338}, {103, 1320}, {677, 23838}, {911, 4997}, {2316, 36101}
X(53529) = X(i)-Dao conjugate of X(j) for these (i,j): {23972, 4997}, {52659, 18025}
X(53529) = crossdifference of every pair of points on line {2316, 2338}
X(53529) = barycentric product X(i)*X(j) for these {i,j}: {7, 51406}, {516, 3911}, {519, 43035}, {1319, 30807}, {1404, 35517}, {1456, 4358}, {1639, 23973}, {1877, 26006}, {2398, 30725}, {4895, 24015}, {14953, 40663}
X(53529) = barycentric quotient X(i)/X(j) for these {i,j}: {516, 4997}, {902, 2338}, {910, 1320}, {1319, 36101}, {1404, 103}, {1456, 88}, {1877, 52781}, {2398, 4582}, {2426, 5548}, {3911, 18025}, {30725, 2400}, {43035, 903}, {51406, 8}
X(53529) = {X(23703),X(52659)}-harmonic conjugate of X(6174)

---

X(53530) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1317) AND X(1361)

Barycentrics    a*(2*a - b - c)*(a + b - c)*(a - b + c)*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3) : :
X(53530) = 3 X[1457] - 2 X[1465]

X(53530) lies on the De Longchamps ellipse and these lines: {1, 104}, {6, 1411}, {34, 1830}, {55, 51236}, {56, 20843}, {57, 106}, {65, 244}, {73, 3057}, {201, 3878}, {214, 23703}, {221, 2098}, {222, 1480}, {225, 4301}, {226, 24222}, {500, 9957}, {517, 1457}, {519, 1877}, {651, 1320}, {665, 53046}, {678, 41541}, {900, 1317}, {902, 1319}, {995, 26742}, {1145, 52659}, {1149, 18838}, {1361, 1769}, {1362, 2821}, {1387, 43043}, {1393, 1772}, {1415, 17439}, {1419, 2823}, {1421, 3340}, {1455, 5048}, {1456, 42082}, {1482, 3157}, {1537, 35015}, {1647, 12832}, {1697, 33811}, {1718, 25415}, {1807, 48667}, {1845, 23706}, {1935, 4861}, {1950, 17438}, {2293, 5919}, {2310, 17638}, {2611, 2650}, {2654, 45776}, {2802, 4551}, {3028, 4017}, {3215, 3915}, {3485, 17719}, {3722, 12739}, {3877, 24806}, {3884, 37558}, {3890, 37523}, {4016, 18675}, {4318, 52368}, {4674, 43048}, {4860, 15306}, {5193, 47622}, {5252, 33104}, {5289, 25934}, {5399, 10284}, {5697, 10571}, {6126, 51654}, {6603, 9502}, {9370, 10912}, {11009, 35197}, {12647, 34029}, {12736, 32486}, {13541, 51766}, {14260, 15906}, {14923, 37694}, {16110, 34242}, {17636, 45885}, {18340, 34789}, {21147, 30323}, {21805, 36920}, {22342, 23846}, {24410, 51565}, {37566, 45219}, {38513, 52129}, {40663, 51415}

X(53530) = reflection of X(i) in X(j) for these {i,j}: {7004, 1}, {24028, 34586}
X(53530) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 1319}, {57, 2183}, {651, 46393}, {7012, 23703}
X(53530) = X(i)-isoconjugate of X(j) for these (i,j): {8, 10428}, {88, 52663}, {104, 1320}, {106, 51565}, {901, 43728}, {903, 2342}, {909, 4997}, {1318, 36944}, {1809, 36125}, {2316, 34234}, {2401, 5548}, {2423, 4582}, {9456, 36795}, {23838, 36037}
X(53530) = X(i)-Dao conjugate of X(j) for these (i,j): {214, 51565}, {3259, 23838}, {3310, 4858}, {3911, 75}, {4370, 36795}, {23980, 4997}, {38979, 43728}, {40613, 1320}, {52659, 18816}
crossdifference of every pair of points on line {2316, 3738}
X(53530) = barycentric product X(i)*X(j) for these {i,j}: {1, 52659}, {44, 22464}, {57, 1145}, {63, 1846}, {214, 52212}, {517, 3911}, {519, 1465}, {651, 23757}, {900, 24029}, {908, 1319}, {1014, 21942}, {1317, 52031}, {1404, 3262}, {1457, 4358}, {1875, 3977}, {3259, 4564}, {3762, 23981}, {10015, 23703}, {14584, 16586}, {14628, 34586}, {22350, 37790}, {24028, 40218}
X(53530) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 51565}, {517, 4997}, {519, 36795}, {604, 10428}, {902, 52663}, {1145, 312}, {1319, 34234}, {1361, 52031}, {1404, 104}, {1457, 88}, {1465, 903}, {1635, 43728}, {1846, 92}, {1875, 6336}, {1877, 16082}, {2183, 1320}, {2251, 2342}, {3259, 4858}, {3310, 23838}, {3911, 18816}, {21942, 3701}, {22086, 37628}, {22356, 1809}, {22464, 20568}, {23703, 13136}, {23757, 4391}, {23981, 3257}, {24029, 4555}, {47420, 7004}, {52659, 75}
X(53530) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 13253, 10703}, {11700, 25485, 1}

---

X(53531) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1317) AND X(1362)

Barycentrics    a*(2*a - b - c)*(a + b - c)*(a - b + c)*(a*b - b^2 + a*c - c^2) : :
X(53531) = 2 X[241] - 3 X[1458]

X(53531) lies on these lines: {1, 651}, {7, 24715}, {42, 17625}, {44, 1319}, {55, 53296}, {56, 4557}, {57, 53397}, {72, 4322}, {77, 16496}, {85, 51055}, {109, 3722}, {222, 3938}, {241, 518}, {244, 4551}, {307, 49505}, {348, 50999}, {354, 45885}, {517, 3000}, {519, 41801}, {537, 4552}, {664, 24841}, {678, 14191}, {883, 39775}, {896, 2078}, {899, 3660}, {900, 1317}, {926, 1362}, {948, 51099}, {976, 34046}, {1042, 3555}, {1106, 3811}, {1254, 3874}, {1407, 41711}, {1441, 49479}, {1456, 4864}, {1462, 4327}, {1465, 17449}, {1469, 9016}, {1471, 3751}, {1617, 32912}, {1647, 41556}, {1757, 7677}, {1854, 10964}, {1943, 32923}, {2003, 17469}, {2114, 7174}, {2263, 3243}, {2265, 53302}, {2635, 18839}, {2650, 10106}, {3214, 37566}, {3339, 51765}, {3476, 4644}, {3870, 9316}, {3877, 24708}, {3911, 21805}, {3930, 52635}, {3935, 9364}, {3961, 17074}, {4318, 49675}, {4320, 41863}, {4334, 7672}, {4675, 5252}, {4695, 18838}, {4712, 51329}, {4792, 18421}, {5219, 17450}, {5400, 18240}, {5919, 23056}, {6180, 42871}, {7004, 17660}, {8581, 42289}, {9363, 34772}, {9370, 28082}, {9440, 30284}, {9441, 18450}, {9502, 17435}, {10944, 42837}, {11570, 24028}, {12831, 35015}, {16484, 29007}, {16577, 42039}, {17077, 49457}, {18412, 21346}, {28968, 32941}, {30379, 49772}, {31225, 50075}, {32486, 46681}, {35326, 38375}, {37787, 49712}, {41712, 42314}

X(53531) = reflection of X(2310) in X(1)
X(53531) = X(i)-isoconjugate of X(j) for these (i,j): {88, 294}, {105, 1320}, {106, 14942}, {673, 2316}, {884, 4555}, {885, 901}, {903, 2195}, {1024, 3257}, {1438, 4997}, {4582, 43929}, {6548, 52927}, {9456, 36796}, {23345, 36802}, {23838, 36086}
X(53531) = X(i)-Dao conjugate of X(j) for these (i,j): {214, 14942}, {4370, 36796}, {6184, 4997}, {36905, 20568}, {38979, 885}, {38989, 23838}, {39046, 1320}, {39063, 903}, {52659, 2481}
X(53531) = crossdifference of every pair of points on line {294, 1024}
X(53531) = barycentric product X(i)*X(j) for these {i,j}: {7, 14439}, {44, 9436}, {241, 519}, {518, 3911}, {883, 1635}, {900, 1025}, {902, 40704}, {918, 23703}, {1023, 43042}, {1026, 30725}, {1319, 3912}, {1404, 3263}, {1458, 4358}, {1639, 41353}, {1818, 37790}, {1876, 3977}, {1877, 25083}, {2283, 3762}, {2325, 34855}, {3264, 52635}, {5236, 5440}, {18206, 40663}, {36819, 52659}
X(53531) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 14942}, {241, 903}, {518, 4997}, {519, 36796}, {665, 23838}, {672, 1320}, {902, 294}, {1023, 36802}, {1025, 4555}, {1026, 4582}, {1319, 673}, {1404, 105}, {1458, 88}, {1635, 885}, {1876, 6336}, {1960, 1024}, {2223, 2316}, {2251, 2195}, {2283, 3257}, {3689, 6559}, {3911, 2481}, {4895, 28132}, {9436, 20568}, {14439, 8}, {22086, 23696}, {23703, 666}, {34253, 27922}, {52635, 106}
X(53531) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 9355, 53055}, {1, 51766, 651}, {109, 37736, 3722}, {651, 14151, 1}, {4334, 49498, 7672}, {4551, 5083, 244}, {8581, 49478, 42289}, {23703, 41553, 678}, {41556, 52659, 1647}

---

X(53532) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1317) AND X(1364)

Barycentrics    a*(2*a - b - c)*(b - c)*(a^2 - b^2 - c^2) : :
X(53532) = 3 X[656] - 4 X[905], 2 X[905] - 3 X[1459], X[30572] - 3 X[30573], 3 X[3737] - 2 X[48003], 2 X[4391] - 3 X[45686], 2 X[10015] - 3 X[11125], 3 X[46385] - 2 X[47921]

X(53532) lies on these lines: {1, 1769}, {3, 22379}, {44, 14427}, {55, 53305}, {56, 53277}, {100, 46118}, {142, 24792}, {513, 4162}, {519, 4768}, {521, 656}, {526, 2292}, {663, 9001}, {834, 47935}, {900, 1317}, {1331, 1332}, {1364, 3270}, {1635, 17455}, {1795, 37628}, {1811, 22437}, {1814, 23696}, {2254, 3722}, {2605, 17420}, {2650, 6370}, {2804, 25416}, {2827, 10698}, {2849, 10702}, {3304, 53313}, {3737, 48003}, {3887, 44858}, {3945, 53357}, {4017, 48283}, {4148, 5839}, {4391, 45686}, {4649, 13259}, {4653, 35055}, {4833, 47918}, {6003, 48281}, {6085, 6161}, {6332, 9031}, {6615, 48302}, {7982, 9525}, {8062, 20293}, {8611, 22383}, {8648, 53306}, {9002, 48150}, {9013, 48131}, {9048, 48329}, {10015, 11125}, {14418, 22086}, {14429, 39472}, {15313, 43924}, {17274, 25602}, {21102, 44409}, {21173, 35057}, {22141, 23141}, {23187, 23226}, {28169, 50767}, {28220, 47907}, {36054, 40628}, {45884, 51361}, {46385, 47921}, {46391, 52431}

X(53532) = reflection of X(i) in X(j) for these {i,j}: {656, 1459}, {1769, 1}, {2254, 53314}, {4017, 48283}, {6615, 48302}, {17420, 2605}, {20293, 8062}, {21102, 44409}, {47918, 4833}, {50338, 21173}, {50354, 48281}
X(53532) = isotomic conjugate of the polar conjugate of X(1635)
X(53532) = isogonal conjugate of the polar conjugate of X(3762)
X(53532) = X(i)-Ceva conjugate of X(j) for these (i,j): {1807, 7004}, {3762, 1635}
X(53532) = X(i)-isoconjugate of X(j) for these (i,j): {4, 901}, {19, 3257}, {25, 4555}, {88, 1783}, {92, 32665}, {100, 36125}, {101, 6336}, {106, 1897}, {108, 1320}, {112, 4080}, {162, 4674}, {190, 8752}, {264, 32719}, {278, 5548}, {608, 4582}, {653, 2316}, {903, 8750}, {1168, 4242}, {1309, 14260}, {1824, 4622}, {1826, 4591}, {2333, 4615}, {2969, 6551}, {4638, 8756}, {4997, 32674}, {5376, 6591}, {6335, 9456}, {6635, 42067}, {7012, 23838}, {7649, 9268}, {10428, 53151}, {14923, 36112}, {15742, 23345}, {32085, 46162}, {39414, 42070}
X(53532) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 3257}, {125, 4674}, {214, 1897}, {1015, 6336}, {1647, 38462}, {4370, 6335}, {6505, 4555}, {6544, 17924}, {8054, 36125}, {22391, 32665}, {26932, 903}, {34467, 106}, {34591, 4080}, {35072, 4997}, {35092, 92}, {36033, 901}, {38979, 4}, {38983, 1320}, {39006, 88}, {40618, 20568}, {51402, 318}, {52659, 18026}
X(53532) = crossdifference of every pair of points on line {19, 1743}
X(53532) = barycentric product X(i)*X(j) for these {i,j}: {3, 3762}, {7, 14418}, {44, 4025}, {63, 900}, {69, 1635}, {75, 22086}, {77, 1639}, {78, 30725}, {81, 14429}, {222, 4768}, {304, 1960}, {348, 4895}, {513, 3977}, {514, 5440}, {519, 905}, {521, 3911}, {525, 52680}, {647, 30939}, {656, 16704}, {693, 22356}, {902, 15413}, {1023, 1565}, {1319, 6332}, {1332, 1647}, {1404, 35518}, {1444, 4120}, {1459, 4358}, {1811, 21129}, {1812, 30572}, {2087, 4561}, {3261, 23202}, {3264, 22383}, {3285, 14208}, {3676, 52978}, {3937, 24004}, {3942, 17780}, {3992, 7254}, {4064, 30576}, {4091, 38462}, {4131, 8756}, {4528, 7177}, {4530, 6516}, {4730, 17206}, {7056, 14427}, {8056, 39472}, {15419, 21805}, {23224, 46109}, {23703, 26932}, {24018, 37168}, {36058, 52627}, {37628, 52659}
X(53532) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 3257}, {44, 1897}, {48, 901}, {63, 4555}, {78, 4582}, {184, 32665}, {212, 5548}, {513, 6336}, {519, 6335}, {521, 4997}, {647, 4674}, {649, 36125}, {652, 1320}, {656, 4080}, {667, 8752}, {900, 92}, {902, 1783}, {905, 903}, {906, 9268}, {1023, 15742}, {1319, 653}, {1331, 5376}, {1404, 108}, {1437, 4591}, {1444, 4615}, {1459, 88}, {1635, 4}, {1639, 318}, {1647, 17924}, {1790, 4622}, {1797, 4618}, {1946, 2316}, {1960, 19}, {2087, 7649}, {2251, 8750}, {3251, 8756}, {3285, 162}, {3762, 264}, {3911, 18026}, {3937, 1022}, {3942, 6548}, {3977, 668}, {4020, 46162}, {4025, 20568}, {4120, 41013}, {4528, 7101}, {4530, 44426}, {4730, 1826}, {4768, 7017}, {4773, 5342}, {4895, 281}, {5440, 190}, {6544, 38462}, {7117, 23838}, {8677, 52031}, {9247, 32719}, {14407, 1824}, {14418, 8}, {14427, 7046}, {14429, 321}, {16704, 811}, {17206, 4634}, {17455, 4242}, {18210, 4049}, {22086, 1}, {22092, 36814}, {22356, 100}, {22371, 1023}, {22379, 40215}, {22383, 106}, {23081, 9272}, {23202, 101}, {23224, 1797}, {23703, 46102}, {30572, 40149}, {30573, 37805}, {30725, 273}, {30939, 6331}, {36058, 4638}, {37168, 823}, {37790, 52938}, {39472, 18743}, {39771, 37790}, {47420, 1769}, {51422, 24035}, {52680, 648}, {52978, 3699}
X(53532) = {X(3),X(23087)}-harmonic conjugate of X(22379)

---

X(53533) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1317) AND X(3020)

Barycentrics    (2*a - b - c)*(b - c)*(b^2 - b*c + c^2) : :
X(53533) = 3 X[2530] - 2 X[50453], 2 X[3776] - 3 X[3777], 4 X[3776] - 3 X[3801], 4 X[3960] - 3 X[4809], 3 X[4728] - X[23745], 2 X[10015] - 3 X[36848], 4 X[14321] - 3 X[48265], 3 X[14413] - 4 X[24099], 3 X[14421] - 2 X[48286], 3 X[14430] - 2 X[24093], 3 X[21118] - 5 X[48414], 6 X[48406] - 5 X[48414], X[24128] - 3 X[53364], 3 X[48392] - 4 X[48417]

X(53533) lies on these lines: {513, 3904}, {514, 4774}, {522, 21343}, {764, 23887}, {900, 1317}, {2254, 4642}, {2530, 50453}, {2826, 4010}, {2832, 50351}, {3159, 22037}, {3762, 3992}, {3776, 3777}, {3808, 3888}, {3837, 21132}, {3960, 4809}, {4088, 23764}, {4122, 49278}, {4367, 28487}, {4435, 22383}, {4728, 23745}, {4730, 23888}, {6362, 48279}, {10015, 36848}, {14321, 48265}, {14413, 24099}, {14421, 48286}, {14430, 24093}, {21111, 40086}, {21118, 48406}, {23738, 40471}, {23746, 48082}, {23765, 23877}, {24128, 53364}, {24720, 28565}, {28468, 50359}, {29017, 47677}, {29120, 48122}, {29122, 47686}, {29156, 47685}, {29172, 47973}, {47708, 48137}, {48335, 48349}, {48392, 48417}

X(53533) = midpoint of X(i) and X(j) for these {i,j}: {4088, 23764}, {23746, 48082}
X(53533) = reflection of X(i) in X(j) for these {i,j}: {3801, 3777}, {4122, 49278}, {4922, 30725}, {21111, 40086}, {21118, 48406}, {21132, 3837}, {47708, 48137}, {48326, 764}, {48349, 48335}
X(53533) = X(i)-isoconjugate of X(j) for these (i,j): {901, 983}, {1320, 8685}, {4621, 9456}, {5548, 7132}, {7033, 32719}, {17743, 32665}
X(53533) = X(i)-Dao conjugate of X(j) for these (i,j): {4370, 4621}, {35092, 17743}, {38979, 983}, {41771, 4555}, {52657, 3257}
X(53533) = crossdifference of every pair of points on line {2316, 5332}
X(53533) = barycentric product X(i)*X(j) for these {i,j}: {519, 3776}, {900, 3662}, {982, 3762}, {1635, 33930}, {1639, 7185}, {1647, 33946}, {3705, 30725}, {3777, 4358}, {3801, 16704}, {3810, 3911}, {4120, 33947}, {4768, 41777}, {14429, 31917}
X(53533) = barycentric quotient X(i)/X(j) for these {i,j}: {519, 4621}, {900, 17743}, {982, 3257}, {1404, 8685}, {1635, 983}, {2275, 901}, {3056, 5548}, {3662, 4555}, {3705, 4582}, {3762, 7033}, {3776, 903}, {3777, 88}, {3801, 4080}, {3810, 4997}, {3888, 5376}, {7032, 32665}, {33947, 4615}, {50514, 9456}

---

X(53534) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1317) AND X(3021)

Barycentrics    (2*a - b - c)*(2*a^2 - a*b + b^2 - a*c - 2*b*c + c^2) : :
X(53534) = 3 X[1] - X[24715], 3 X[1086] - 2 X[24715], X[3943] + 2 X[49699], 3 X[4370] - 4 X[4432], X[4969] - 4 X[49700], 3 X[3241] - X[24841], 3 X[551] - 2 X[25351], X[673] - 3 X[8236], 3 X[1279] - 2 X[3008], 8 X[3635] - X[4409], 5 X[3616] - 4 X[40480], X[3621] - 5 X[4473], 7 X[3622] - 5 X[27191], 5 X[3623] - X[4440], X[6542] + 3 X[49704], 3 X[7967] - X[24813], 3 X[10247] - X[24833], 3 X[16211] - X[24830], 5 X[17266] - 3 X[32850], X[24821] + 3 X[51093], X[31145] - 3 X[41138]

X(53534) lies on these lines: {1, 528}, {8, 4422}, {11, 3722}, {37, 30331}, {44, 519}, {45, 47357}, {55, 53302}, {100, 3756}, {106, 9945}, {145, 190}, {149, 17724}, {214, 14028}, {244, 6154}, {346, 49690}, {390, 3242}, {516, 4864}, {524, 49709}, {536, 49771}, {537, 3244}, {545, 3241}, {551, 25351}, {594, 32941}, {673, 5308}, {678, 1647}, {726, 49691}, {740, 49696}, {900, 1317}, {902, 51463}, {903, 52714}, {952, 24828}, {1155, 49989}, {1279, 3008}, {1482, 29243}, {1616, 3189}, {1697, 16560}, {2161, 31393}, {2796, 3635}, {2886, 17715}, {2976, 3021}, {3052, 36845}, {3057, 3270}, {3058, 3938}, {3246, 49772}, {3315, 20095}, {3616, 40480}, {3621, 4473}, {3622, 27191}, {3623, 4440}, {3632, 15601}, {3685, 9053}, {3689, 51415}, {3748, 17056}, {3749, 37646}, {3782, 34611}, {3883, 4690}, {3886, 24358}, {3932, 17765}, {3961, 49736}, {4026, 49473}, {4030, 32943}, {4312, 15600}, {4318, 43066}, {4364, 36534}, {4366, 4437}, {4472, 5263}, {4660, 48632}, {4665, 50310}, {4667, 49478}, {4674, 12732}, {4684, 28566}, {4693, 28503}, {4947, 24418}, {4952, 30568}, {4966, 17766}, {4971, 50015}, {5222, 50839}, {5604, 31568}, {5605, 31567}, {5723, 12730}, {6018, 34194}, {6758, 14757}, {7277, 49490}, {7967, 24813}, {9055, 49470}, {10247, 24833}, {10700, 12735}, {10707, 37691}, {12035, 16594}, {12437, 45219}, {12690, 24222}, {15170, 30115}, {15570, 50307}, {16211, 24830}, {16496, 17334}, {16666, 50294}, {16672, 48856}, {17014, 20533}, {17230, 50949}, {17245, 42819}, {17246, 49465}, {17266, 32850}, {17278, 35227}, {17314, 49679}, {17340, 49688}, {17354, 49524}, {17362, 49460}, {17365, 42871}, {17369, 36479}, {17387, 50289}, {17388, 49681}, {17397, 26582}, {17597, 20075}, {17755, 28581}, {17768, 49675}, {20016, 27949}, {20042, 51583}, {21000, 24477}, {23703, 41556}, {23704, 35111}, {24331, 49725}, {24821, 51093}, {24840, 37734}, {24842, 49232}, {24843, 49233}, {24871, 34123}, {26139, 43290}, {28538, 49763}, {29616, 50783}, {29659, 48810}, {29660, 48821}, {29820, 49732}, {30333, 45476}, {30334, 45477}, {31145, 41138}, {31183, 51102}, {32847, 49708}, {33076, 48635}, {33104, 37703}, {34699, 49487}, {34824, 49720}, {36480, 49740}, {40688, 49719}, {41553, 52659}, {49466, 49484}, {49475, 49684}

X(53534) = midpoint of X(i) and X(j) for these {i,j}: {145, 190}, {3685, 49695}, {4702, 49699}, {32847, 49708}
X(53534) = reflection of X(i) in X(j) for these {i,j}: {8, 4422}, {1086, 1}, {3943, 4702}, {49701, 4759}, {49772, 3246}
X(53534) = X(i)-isoconjugate of X(j) for these (i,j): {106, 1280}, {901, 35355}, {1022, 6078}, {1320, 1477}, {1810, 36125}, {2316, 43760}, {5548, 37626}, {9456, 36807}
X(53534) = X(i)-Dao conjugate of X(j) for these (i,j): {214, 1280}, {4370, 36807}, {16593, 903}, {35111, 4997}, {38979, 35355}, {39048, 88}, {52659, 35160}
X(53534) = crossdifference of every pair of points on line {2316, 22108}
X(53534) = barycentric product X(i)*X(j) for these {i,j}: {519, 3008}, {900, 53337}, {1279, 4358}, {2415, 2976}, {3911, 5853}, {4487, 51839}, {6084, 17780}, {20780, 46109}, {24004, 48032}, {36944, 51419}
X(53534) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 1280}, {519, 36807}, {1279, 88}, {1319, 43760}, {1404, 1477}, {1635, 35355}, {2348, 1320}, {2976, 2403}, {3008, 903}, {3911, 35160}, {5853, 4997}, {6084, 6548}, {8647, 2316}, {8659, 23345}, {20662, 34230}, {20780, 1797}, {22356, 1810}, {23344, 6078}, {48032, 1022}, {53337, 4555}
X(53534) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {678, 1647, 6174}, {3058, 3938, 4415}, {16594, 17780, 12035}, {36479, 48805, 17369}, {36534, 49746, 4364}

---

X(53535) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1317) AND X(3025)

Barycentrics    a*(2*a - b - c)*(b - c)*(a^2 - b^2 + b*c - c^2) : :
X(53535) = 3 X[1] - X[23838], 3 X[14812] + X[23838], 2 X[14812] + X[24457], 2 X[23838] - 3 X[24457], 2 X[3960] - 3 X[53314], 3 X[551] - 2 X[23808], X[23757] - 3 X[30573], 3 X[30573] + X[39771], 2 X[14315] - 3 X[14413], 2 X[23809] - 3 X[44550]

X(53535) lies on these lines: {1, 513}, {6, 650}, {73, 43924}, {100, 9268}, {214, 3738}, {221, 15313}, {500, 6003}, {514, 4667}, {519, 46781}, {521, 3157}, {522, 3244}, {523, 2650}, {551, 23808}, {651, 52377}, {667, 4471}, {678, 2254}, {693, 17378}, {900, 1317}, {1201, 1459}, {1419, 43049}, {1480, 3309}, {1643, 16666}, {1647, 4448}, {2827, 25485}, {3293, 21173}, {3478, 48329}, {3667, 5882}, {3733, 4267}, {4526, 17465}, {4777, 49490}, {4778, 21201}, {4833, 18166}, {4885, 17313}, {4977, 21111}, {4979, 21143}, {6006, 30331}, {6126, 8674}, {7289, 9048}, {9002, 18183}, {9013, 27469}, {10543, 14284}, {14315, 14413}, {16495, 45666}, {17300, 26985}, {20090, 26824}, {21104, 23728}, {22379, 39478}, {23087, 39200}, {23344, 23703}, {23809, 44550}, {27529, 48246}, {34231, 43933}, {42769, 46974}

X(53535) = midpoint of X(i) and X(j) for these {i,j}: {1, 14812}, {23757, 39771}
X(53535) = reflection of X(i) in X(j) for these {i,j}: {23352, 9269}, {24457, 1}
X(53535) = reflection of X(24457) in the OI line
X(53535) = X(i)-Ceva conjugate of X(j) for these (i,j): {81, 2087}, {100, 36}, {514, 1635}, {651, 44}, {16586, 35128}
X(53535) = X(i)-isoconjugate of X(j) for these (i,j): {80, 901}, {100, 1168}, {106, 51562}, {109, 36590}, {655, 2316}, {1320, 2222}, {2006, 5548}, {2161, 3257}, {4013, 36069}, {4555, 6187}, {4618, 40172}, {4622, 34857}, {4997, 32675}, {9456, 36804}, {14147, 47058}, {18359, 32665}, {20566, 32719}, {23838, 52377}
X(53535) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 36590}, {44, 190}, {214, 51562}, {1639, 4391}, {1647, 51975}, {3936, 668}, {3960, 693}, {4370, 36804}, {8054, 1168}, {35092, 18359}, {35128, 4997}, {38979, 80}, {38982, 4013}, {38984, 1320}, {40584, 3257}, {40612, 4555}, {51402, 52409}, {52659, 35174}
X(53535) = crossdifference of every pair of points on line {44, 517}
X(53535) = X(1)-line-conjugate of X(14190)
X(53535) = barycentric product X(i)*X(j) for these {i,j}: {36, 3762}, {44, 4453}, {214, 514}, {320, 1635}, {513, 51583}, {519, 3960}, {523, 17191}, {649, 1227}, {650, 41801}, {651, 51402}, {693, 17455}, {900, 3218}, {1023, 4089}, {1319, 3904}, {1443, 1639}, {1647, 4585}, {1960, 20924}, {3264, 21758}, {3738, 3911}, {4358, 53314}, {4511, 30725}, {4707, 52680}, {4895, 17078}, {6370, 30576}, {6544, 52553}, {7192, 40988}, {16944, 52627}, {21828, 30939}, {22379, 46109}, {30606, 51663}
X(53535) = barycentric quotient X(i)/X(j) for these {i,j}: {36, 3257}, {44, 51562}, {214, 190}, {519, 36804}, {649, 1168}, {650, 36590}, {654, 1320}, {900, 18359}, {1227, 1978}, {1319, 655}, {1404, 2222}, {1635, 80}, {1639, 52409}, {1960, 2161}, {1983, 9268}, {2361, 5548}, {2610, 4013}, {3218, 4555}, {3738, 4997}, {3762, 20566}, {3911, 35174}, {3960, 903}, {4120, 15065}, {4453, 20568}, {4511, 4582}, {4530, 52356}, {4895, 36910}, {6544, 51975}, {7113, 901}, {8648, 2316}, {14407, 34857}, {16944, 4638}, {17191, 99}, {17455, 100}, {21758, 106}, {21828, 4674}, {22086, 1807}, {22379, 1797}, {30725, 18815}, {39771, 14628}, {40215, 4618}, {40988, 3952}, {41801, 4554}, {51402, 4391}, {51583, 668}, {52434, 32665}, {52680, 47318}, {53314, 88}
X(53535) = {X(30573),X(39771)}-harmonic conjugate of X(23757)

---

X(53536) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1317) AND X(3026)

Barycentrics    (2*a - b - c)*(b - c)*(a^2 + a*b + a*c + 2*b*c) : :
X(53536) = 3 X[47935] - 4 X[48016], 2 X[48107] - 3 X[48149], 6 X[905] - 5 X[30835], 3 X[1635] - 2 X[3762], 3 X[1769] - 2 X[14286], 3 X[3669] - 2 X[23813], 2 X[3835] - 3 X[44550], 4 X[3960] - 3 X[4728], 2 X[4010] - 3 X[14413], 3 X[4391] - 4 X[31286], 3 X[4560] - 2 X[48000], 3 X[47918] - 4 X[48000], 3 X[4750] - 2 X[10015], 4 X[4791] - 5 X[24924], 3 X[17496] - X[20295], 2 X[20295] - 3 X[48131], 4 X[9508] - 3 X[14430], 3 X[14432] - 2 X[50326], 3 X[21115] - 2 X[47680], 2 X[43067] - 3 X[48144], 4 X[43067] - 3 X[50457], 4 X[25380] - 3 X[30709], 7 X[31207] - 6 X[45664], 4 X[47984] - 3 X[48582]

X(53536) lies on these lines: {21, 13245}, {514, 4380}, {661, 29148}, {764, 29340}, {812, 4440}, {814, 48151}, {900, 1317}, {905, 30835}, {1635, 3762}, {1769, 14286}, {2254, 2787}, {2530, 29176}, {2786, 3904}, {3667, 47729}, {3669, 23813}, {3777, 29152}, {3835, 44550}, {3960, 4728}, {4010, 14413}, {4041, 29324}, {4367, 48264}, {4378, 4804}, {4391, 31286}, {4474, 50336}, {4560, 47918}, {4750, 10015}, {4773, 21129}, {4791, 24924}, {4822, 29170}, {4905, 29344}, {6002, 17496}, {8648, 53270}, {8714, 48322}, {9508, 14430}, {14432, 50326}, {16892, 29126}, {21115, 47680}, {23738, 29070}, {23765, 29238}, {23880, 43067}, {23882, 48341}, {25380, 30709}, {29013, 48334}, {29033, 48115}, {29132, 47702}, {29178, 48114}, {29212, 47700}, {29236, 50359}, {29807, 47672}, {30519, 47684}, {31207, 45664}, {47683, 47917}, {47721, 48073}, {47905, 48410}, {47984, 48582}, {48021, 48288}, {48080, 48325}

X(53536) = reflection of X(i) in X(j) for these {i,j}: {661, 48321}, {4474, 50336}, {4804, 4378}, {4895, 4922}, {21129, 4773}, {47672, 48320}, {47721, 48073}, {47905, 48410}, {47917, 47683}, {47918, 4560}, {48021, 48288}, {48080, 48325}, {48114, 48335}, {48131, 17496}, {48264, 4367}, {50457, 48144}
X(53536) = X(i)-isoconjugate of X(j) for these (i,j): {901, 941}, {959, 5548}, {1320, 32693}, {2258, 3257}, {31359, 32665}, {32719, 34258}
X(53536) = X(i)-Dao conjugate of X(j) for these (i,j): {17417, 1320}, {35092, 31359}, {38979, 941}, {52659, 32038}
X(53536) = crossdifference of every pair of points on line {2258, 2316}
X(53536) = barycentric product X(i)*X(j) for these {i,j}: {519, 43067}, {900, 10436}, {940, 3762}, {1635, 34284}, {3911, 23880}, {4358, 48144}, {8672, 30939}, {11679, 30725}, {16704, 50457}
X(53536) = barycentric quotient X(i)/X(j) for these {i,j}: {900, 31359}, {940, 3257}, {1404, 32693}, {1468, 901}, {1635, 941}, {1960, 2258}, {2268, 5548}, {3762, 34258}, {3911, 32038}, {5019, 32665}, {8672, 4674}, {10436, 4555}, {11679, 4582}, {17418, 1320}, {23880, 4997}, {30725, 44733}, {43067, 903}, {48144, 88}, {50457, 4080}, {52680, 931}

---

X(53537) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1317) AND X(3028)

Barycentrics    a*(2*a - b - c)*(a + b - c)*(a - b + c)*(b + c)*(a^2 - b^2 + b*c - c^2) : :
X(53537) = 3 X[1464] - 2 X[18593]

X(53537) lies on these lines: {1, 399}, {37, 1409}, {55, 53252}, {56, 1046}, {65, 3293}, {73, 45288}, {109, 12739}, {226, 52383}, {354, 32486}, {526, 3028}, {651, 1411}, {758, 1464}, {896, 5427}, {900, 1317}, {928, 1361}, {1054, 5221}, {1227, 41801}, {1319, 52680}, {1406, 12635}, {1449, 40584}, {1457, 17449}, {1834, 2650}, {2099, 6180}, {2594, 4868}, {3057, 48897}, {4511, 52440}, {4653, 16140}, {5289, 22129}, {5434, 7200}, {5902, 6127}, {10703, 44858}, {11570, 34586}, {12532, 24433}, {12736, 45885}, {12832, 52659}, {14584, 41558}, {15950, 17450}, {16610, 18838}, {18360, 34772}, {21805, 40663}, {23703, 41541}, {24806, 49448}, {33593, 45926}

X(53537) = X(1476)-Ceva conjugate of X(36)
X(53537) = X(i)-isoconjugate of X(j) for these (i,j): {21, 1168}, {58, 36590}, {88, 2341}, {106, 6740}, {759, 1320}, {1793, 36125}, {2316, 24624}, {4674, 52380}, {4997, 34079}
X(53537) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 36590}, {44, 333}, {214, 6740}, {3936, 314}, {34586, 1320}, {35069, 4997}, {40611, 1168}, {52659, 14616}, {52872, 52409}, {52877, 52371}
X(53537) = crossdifference of every pair of points on line {2316, 2341}
X(53537) = barycentric product X(i)*X(j) for these {i,j}: {7, 40988}, {12, 17191}, {37, 41801}, {44, 41804}, {65, 51583}, {214, 226}, {519, 18593}, {758, 3911}, {1227, 1400}, {1319, 3936}, {1404, 35550}, {1441, 17455}, {1443, 3943}, {1464, 4358}, {1835, 3977}, {3218, 40663}, {4585, 30572}, {4707, 23703}, {17078, 21805}, {36913, 53114}
X(53537) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 36590}, {44, 6740}, {214, 333}, {758, 4997}, {902, 2341}, {1227, 28660}, {1319, 24624}, {1400, 1168}, {1404, 759}, {1464, 88}, {1835, 6336}, {2245, 1320}, {3285, 52380}, {3724, 2316}, {3911, 14616}, {3943, 52409}, {4120, 52356}, {17191, 261}, {17455, 21}, {18593, 903}, {21805, 36910}, {21828, 23838}, {22356, 1793}, {23703, 47318}, {40663, 18359}, {40988, 8}, {41801, 274}, {41804, 20568}, {51583, 314}, {51663, 4049}, {52963, 52371}

---

X(53538) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1357) AND X(1358)

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

X(535) lies on these lines: {1, 9519}, {7, 192}, {9, 19604}, {41, 17107}, {56, 16686}, {57, 88}, {77, 7225}, {85, 4572}, {105, 53394}, {226, 30566}, {241, 52896}, {244, 1357}, {269, 604}, {312, 42304}, {320, 24803}, {553, 41801}, {661, 6791}, {942, 48921}, {1086, 1358}, {1111, 24224}, {1122, 1400}, {1254, 17114}, {1404, 6610}, {1407, 7147}, {1423, 17092}, {1429, 1443}, {1435, 7366}, {2285, 7271}, {2310, 3675}, {3119, 26932}, {3123, 4017}, {3248, 43921}, {3339, 51766}, {3340, 5528}, {4051, 27818}, {4059, 45208}, {4675, 21808}, {4778, 4965}, {4858, 21139}, {4862, 17452}, {5219, 30855}, {6173, 17451}, {6358, 21427}, {7004, 51643}, {7209, 20567}, {8056, 40400}, {16727, 17197}, {17237, 24798}, {17276, 21809}, {20958, 42082}, {21140, 23676}, {21233, 26836}, {21320, 35293}, {24471, 34253}, {24797, 33299}, {30044, 30056}, {39244, 50092}, {40622, 51664}

X(53538) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 4017}, {57, 3669}, {85, 3676}, {226, 30724}, {269, 43924}, {738, 43932}, {1358, 244}, {1407, 7216}, {7209, 24002}, {15728, 14413}, {17107, 649}, {19604, 513}, {35160, 2254}, {40151, 48334}, {40154, 48151}, {42304, 514}, {43932, 764}, {44733, 30723}, {52803, 6615}
X(53538) = X(i)-isoconjugate of X(j) for these (i,j): {2, 6065}, {6, 4076}, {8, 1252}, {9, 765}, {41, 7035}, {55, 1016}, {59, 346}, {76, 6066}, {100, 644}, {101, 3699}, {109, 6558}, {110, 30730}, {190, 3939}, {200, 4564}, {210, 4567}, {219, 15742}, {220, 4998}, {249, 6057}, {312, 1110}, {341, 2149}, {480, 1275}, {643, 1018}, {645, 4557}, {646, 692}, {651, 4578}, {662, 4069}, {663, 6632}, {728, 7045}, {901, 30731}, {1260, 46102}, {1262, 5423}, {1265, 7115}, {1293, 30720}, {1334, 4600}, {1500, 6064}, {1639, 6551}, {1783, 4571}, {1897, 4587}, {2175, 31625}, {2284, 36802}, {2321, 4570}, {2325, 9268}, {3158, 5382}, {3596, 23990}, {3684, 5378}, {3689, 5376}, {3692, 7012}, {3693, 5377}, {3694, 5379}, {3711, 5385}, {3900, 31615}, {3952, 5546}, {4082, 52378}, {4103, 4636}, {4163, 4619}, {4551, 7259}, {4559, 7256}, {4574, 36797}, {4582, 23344}, {4590, 7064}, {4612, 40521}, {4621, 40499}, {4767, 5549}, {5548, 17780}, {7046, 44717}, {8694, 30728}, {8701, 30729}, {24027, 30693}, {28210, 30727}, {28218, 30732}, {42720, 52927}
X(53538) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 4076}, {11, 6558}, {223, 1016}, {244, 30730}, {478, 765}, {513, 9}, {514, 312}, {522, 30693}, {649, 3871}, {650, 341}, {656, 30681}, {661, 8}, {905, 52406}, {1015, 3699}, {1084, 4069}, {1086, 646}, {3160, 7035}, {3669, 18743}, {3835, 3208}, {4369, 4095}, {4521, 44720}, {4988, 3701}, {6544, 4723}, {6609, 4564}, {6615, 346}, {8054, 644}, {17115, 728}, {20317, 30568}, {32664, 6065}, {34467, 4587}, {38979, 30731}, {38991, 4578}, {39006, 4571}, {40593, 31625}, {40615, 668}, {40617, 190}, {40620, 7257}, {40622, 4033}, {40625, 7258}, {40627, 210}, {40628, 1265}, {42336, 21769}, {50330, 2321}, {50497, 1334}
X(53538) = cevapoint of X(i) and X(j) for these (i,j): {1015, 1357}, {3125, 21963}
X(53538) = crossdifference of every pair of points on line {644, 3939}
X(53538) = barycentric product X(i)*X(j) for these {i,j}: {1, 1358}, {7, 244}, {11, 269}, {34, 1565}, {56, 1111}, {57, 1086}, {65, 17205}, {75, 1357}, {77, 2969}, {84, 38374}, {85, 1015}, {200, 41292}, {226, 16726}, {273, 3937}, {278, 3942}, {279, 2170}, {479, 2310}, {513, 3676}, {514, 3669}, {522, 43932}, {523, 7203}, {552, 2643}, {603, 2973}, {604, 23989}, {649, 24002}, {651, 6545}, {661, 17096}, {664, 764}, {667, 52621}, {679, 14027}, {693, 43924}, {738, 1146}, {757, 1365}, {876, 43041}, {934, 21132}, {1014, 3120}, {1019, 7178}, {1022, 30725}, {1027, 43042}, {1088, 3271}, {1090, 7339}, {1106, 34387}, {1109, 7341}, {1119, 7004}, {1122, 40451}, {1319, 6549}, {1396, 4466}, {1398, 17880}, {1400, 16727}, {1407, 4858}, {1408, 21207}, {1411, 4089}, {1412, 16732}, {1415, 23100}, {1426, 17219}, {1427, 17197}, {1432, 7200}, {1434, 3125}, {1435, 26932}, {1456, 15634}, {1461, 40166}, {1847, 7117}, {1977, 20567}, {2191, 40615}, {2254, 43930}, {3248, 6063}, {3323, 51838}, {3668, 18191}, {3733, 4077}, {3756, 19604}, {4017, 7192}, {4025, 43923}, {4552, 8042}, {4554, 21143}, {4560, 7216}, {4572, 8027}, {4617, 42462}, {4625, 8034}, {4904, 17107}, {4927, 37627}, {5532, 24013}, {6084, 37626}, {6377, 7209}, {6614, 42455}, {7023, 24026}, {7045, 7336}, {7146, 43266}, {7147, 26856}, {7153, 21138}, {7177, 8735}, {7180, 7199}, {7182, 42067}, {7202, 52374}, {7233, 27846}, {7250, 18155}, {7342, 23994}, {7366, 23978}, {8056, 40617}, {9436, 43921}, {13438, 22107}, {13460, 22106}, {14936, 23062}, {15635, 22464}, {17925, 51664}, {20615, 21208}, {30723, 47915}, {30724, 47947}, {34051, 42754}, {51641, 52619}
X(53538) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4076}, {7, 7035}, {11, 341}, {31, 6065}, {34, 15742}, {56, 765}, {57, 1016}, {85, 31625}, {244, 8}, {269, 4998}, {512, 4069}, {513, 3699}, {514, 646}, {552, 24037}, {560, 6066}, {604, 1252}, {649, 644}, {650, 6558}, {651, 6632}, {661, 30730}, {663, 4578}, {667, 3939}, {738, 1275}, {757, 6064}, {764, 522}, {876, 36801}, {1014, 4600}, {1015, 9}, {1019, 645}, {1022, 4582}, {1027, 36802}, {1086, 312}, {1106, 59}, {1111, 3596}, {1146, 30693}, {1356, 872}, {1357, 1}, {1358, 75}, {1365, 1089}, {1397, 1110}, {1398, 7012}, {1407, 4564}, {1408, 4570}, {1412, 4567}, {1416, 5377}, {1417, 9268}, {1420, 44724}, {1421, 11607}, {1434, 4601}, {1435, 46102}, {1459, 4571}, {1461, 31615}, {1565, 3718}, {1635, 30731}, {1647, 4723}, {1977, 41}, {2087, 2325}, {2170, 346}, {2310, 5423}, {2643, 6057}, {2969, 318}, {3120, 3701}, {3121, 1334}, {3122, 210}, {3123, 27538}, {3125, 2321}, {3248, 55}, {3249, 3063}, {3271, 200}, {3669, 190}, {3675, 3717}, {3676, 668}, {3733, 643}, {3737, 7256}, {3756, 44720}, {3937, 78}, {3942, 345}, {4017, 3952}, {4077, 27808}, {4394, 30720}, {4403, 4494}, {4475, 3790}, {4516, 4082}, {4560, 7258}, {4790, 30728}, {4979, 30729}, {6085, 23705}, {6377, 3208}, {6545, 4391}, {6550, 4768}, {7004, 1265}, {7023, 7045}, {7099, 44717}, {7117, 3692}, {7153, 5383}, {7178, 4033}, {7180, 1018}, {7192, 7257}, {7200, 17787}, {7202, 42033}, {7203, 99}, {7216, 4552}, {7250, 4551}, {7252, 7259}, {7336, 24026}, {7341, 24041}, {7342, 1101}, {7366, 1262}, {8027, 663}, {8034, 4041}, {8042, 4560}, {8054, 3871}, {8659, 23704}, {8661, 4895}, {8735, 7101}, {14027, 4738}, {14936, 728}, {15635, 51565}, {16592, 4095}, {16614, 3169}, {16726, 333}, {16727, 28660}, {16732, 30713}, {17071, 2136}, {17096, 799}, {17205, 314}, {17477, 3913}, {18191, 1043}, {18210, 3710}, {18211, 52352}, {19945, 4009}, {21132, 4397}, {21138, 4110}, {21143, 650}, {22096, 212}, {22383, 4587}, {23777, 20317}, {23989, 28659}, {24002, 1978}, {24193, 4124}, {26932, 52406}, {27846, 3685}, {27918, 3975}, {30725, 24004}, {34591, 30681}, {37627, 6079}, {38374, 322}, {39786, 3985}, {40151, 5382}, {40166, 52622}, {40617, 18743}, {41292, 1088}, {42067, 33}, {42084, 4152}, {42336, 23845}, {42753, 6735}, {43041, 874}, {43051, 4595}, {43266, 52652}, {43921, 14942}, {43922, 1320}, {43923, 1897}, {43924, 100}, {43930, 51560}, {43932, 664}, {47016, 42083}, {48334, 25268}, {50514, 40499}, {51641, 4557}, {51656, 43290}, {51664, 52609}, {52410, 2149}, {52621, 6386}
X(53538) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {269, 5575, 28017}, {269, 28017, 604}, {1086, 3942, 2170}, {1122, 1418, 1400}, {1358, 40617, 1086}, {1429, 1443, 51653}, {3675, 4014, 2310}

---

X(53539) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1357) AND X(1362)

Barycentrics    a^2*(b - c)*(a + b - c)*(a - b + c)*(a*b - b^2 + a*c - c^2) : :
X(53539) = X[21132] - 3 X[30691], 3 X[30704] - X[47695]

X(53539) lies on these lines: {1, 2821}, {7, 46403}, {9, 25926}, {56, 1960}, {57, 659}, {65, 891}, {109, 692}, {226, 3837}, {244, 1357}, {512, 3669}, {513, 676}, {518, 4925}, {649, 854}, {665, 20662}, {667, 51652}, {832, 51644}, {884, 1416}, {900, 5083}, {926, 1362}, {928, 3960}, {942, 2826}, {1400, 27675}, {1420, 25569}, {1423, 28399}, {1438, 2440}, {1459, 8641}, {1463, 6165}, {1469, 9032}, {1787, 53406}, {1876, 52305}, {2099, 48296}, {2530, 51664}, {2815, 24025}, {2827, 18240}, {3309, 39541}, {3339, 21385}, {3340, 21343}, {3667, 34954}, {4077, 21146}, {4083, 30719}, {4524, 9000}, {4654, 48167}, {4905, 44410}, {5219, 30795}, {6005, 30723}, {6363, 43051}, {6372, 7178}, {6550, 18838}, {7216, 48151}, {8645, 51646}, {8672, 49296}, {9432, 37627}, {9508, 43050}, {17094, 50348}, {21132, 30691}, {23793, 44409}, {23829, 39775}, {24002, 48108}, {30704, 47695}, {37582, 44805}, {42341, 50333}, {43049, 50336}

X(53539) = midpoint of X(i) and X(j) for these {i,j}: {65, 30725}, {4905, 44410}
X(53539) = isogonal conjugate of X(36802)
X(53539) = isogonal conjugate of the isotomic conjugate of X(43042)
X(53539) = X(i)-Ceva conjugate of X(j) for these (i,j): {883, 241}, {1462, 1015}, {1876, 3675}, {2283, 1458}, {24016, 1407}, {32735, 56}, {43930, 3669}, {52213, 35505}
X(53539) = X(i)-isoconjugate of X(j) for these (i,j): {1, 36802}, {8, 36086}, {9, 666}, {41, 36803}, {55, 51560}, {75, 52927}, {100, 14942}, {101, 36796}, {105, 3699}, {190, 294}, {200, 927}, {220, 34085}, {312, 919}, {341, 32735}, {346, 36146}, {522, 5377}, {643, 13576}, {644, 673}, {645, 18785}, {646, 1438}, {651, 6559}, {664, 28071}, {668, 2195}, {765, 885}, {884, 7035}, {1016, 1024}, {1027, 4076}, {1253, 46135}, {1462, 6558}, {2481, 3939}, {3573, 33676}, {3596, 32666}, {3900, 39293}, {4564, 28132}, {4571, 36124}, {5853, 39272}, {15742, 23696}, {28809, 36138}
X(53539) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 36802}, {206, 52927}, {223, 51560}, {478, 666}, {513, 885}, {1015, 36796}, {3126, 4397}, {3160, 36803}, {6184, 646}, {6609, 927}, {8054, 14942}, {17113, 46135}, {27918, 4087}, {35094, 3596}, {35509, 23978}, {36905, 1978}, {38980, 312}, {38989, 8}, {38991, 6559}, {39012, 28809}, {39014, 346}, {39025, 28071}, {39046, 3699}, {39063, 668}, {40615, 18031}, {40617, 2481}
X(53539) = crossdifference of every pair of points on line {8, 220}
X(53539) = barycentric product X(i)*X(j) for these {i,j}: {6, 43042}, {7, 665}, {56, 918}, {57, 2254}, {241, 513}, {244, 1025}, {279, 926}, {331, 23225}, {479, 52614}, {514, 1458}, {518, 3669}, {649, 9436}, {650, 34855}, {651, 3675}, {667, 40704}, {672, 3676}, {676, 52213}, {693, 52635}, {876, 34253}, {883, 1015}, {905, 1876}, {919, 3323}, {927, 35505}, {934, 17435}, {1014, 24290}, {1086, 2283}, {1088, 46388}, {1262, 52305}, {1357, 42720}, {1358, 2284}, {1400, 23829}, {1407, 50333}, {1412, 4088}, {1459, 5236}, {1462, 3126}, {1566, 24016}, {2170, 41353}, {2223, 24002}, {2424, 39063}, {2428, 40615}, {2720, 42770}, {3252, 43041}, {3286, 7178}, {3572, 39775}, {3693, 43932}, {3912, 43924}, {3930, 7203}, {4017, 18206}, {4444, 51329}, {4925, 40151}, {6184, 43930}, {7180, 30941}, {8643, 10029}, {9454, 52621}, {14626, 30723}, {17096, 20683}, {18157, 51641}, {25083, 43923}, {30725, 34230}, {32735, 35094}, {34051, 42758}
X(53539) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 36802}, {7, 36803}, {32, 52927}, {56, 666}, {57, 51560}, {241, 668}, {269, 34085}, {279, 46135}, {513, 36796}, {518, 646}, {604, 36086}, {649, 14942}, {663, 6559}, {665, 8}, {667, 294}, {672, 3699}, {883, 31625}, {918, 3596}, {926, 346}, {1015, 885}, {1025, 7035}, {1106, 36146}, {1362, 42720}, {1397, 919}, {1407, 927}, {1415, 5377}, {1458, 190}, {1461, 39293}, {1876, 6335}, {1919, 2195}, {1977, 884}, {2223, 644}, {2254, 312}, {2283, 1016}, {2284, 4076}, {2340, 6558}, {3063, 28071}, {3248, 1024}, {3252, 36801}, {3271, 28132}, {3286, 645}, {3572, 33676}, {3669, 2481}, {3675, 4391}, {3676, 18031}, {4088, 30713}, {4925, 44723}, {7180, 13576}, {8638, 220}, {9436, 1978}, {9454, 3939}, {15615, 52614}, {17435, 4397}, {18206, 7257}, {20683, 30730}, {20752, 4571}, {23225, 219}, {23829, 28660}, {24290, 3701}, {34230, 4582}, {34253, 874}, {34855, 4554}, {35505, 50333}, {39258, 4069}, {39775, 27853}, {40704, 6386}, {42290, 53227}, {43042, 76}, {43924, 673}, {43932, 34018}, {46388, 200}, {51329, 3570}, {51641, 18785}, {51652, 31638}, {52305, 23978}, {52410, 32735}, {52614, 5423}, {52635, 100}
X(53539) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {53300, 53308, 6139}, {53300, 53314, 53308}

---

X(53540) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1357) AND X(1365)

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

X(53540) lies on these lines: {1, 38576}, {7, 11609}, {11, 1111}, {56, 1421}, {57, 1929}, {65, 3293}, {226, 3971}, {244, 1357}, {278, 16100}, {1284, 18593}, {1365, 2611}, {1393, 17114}, {1397, 36570}, {1402, 1427}, {1408, 52372}, {2099, 12653}, {2835, 28353}, {2948, 5221}, {3121, 7180}, {3125, 16592}, {3649, 52567}, {3676, 23824}, {4014, 7004}, {4306, 28109}, {4347, 36508}, {4459, 24235}, {4554, 7233}, {4705, 21944}, {5018, 5061}, {10473, 52161}, {16576, 24174}, {17476, 21963}, {17723, 24796}, {20680, 35310}, {23772, 24026}, {34846, 40622}

X(53540) = X(i)-Ceva conjugate of X(j) for these (i,j): {65, 4017}, {1014, 3669}, {1426, 7250}, {1427, 7180}, {1441, 7178}, {7233, 7212}, {7249, 3676}, {43924, 764}, {52372, 43924}
X(53540) = X(i)-isoconjugate of X(j) for these (i,j): {8, 4570}, {9, 4567}, {21, 765}, {41, 4601}, {42, 6064}, {55, 4600}, {58, 4076}, {59, 1043}, {78, 5379}, {86, 6065}, {99, 3939}, {100, 643}, {101, 645}, {109, 7256}, {110, 3699}, {162, 4571}, {163, 646}, {190, 5546}, {210, 24041}, {220, 4620}, {249, 2321}, {250, 3710}, {283, 15742}, {284, 1016}, {310, 6066}, {314, 1110}, {333, 1252}, {346, 52378}, {644, 662}, {648, 4587}, {651, 7259}, {692, 7257}, {1018, 4612}, {1021, 31615}, {1101, 3701}, {1331, 36797}, {1334, 4590}, {1414, 4578}, {1415, 7258}, {1792, 7012}, {2194, 7035}, {2287, 4564}, {2318, 18020}, {2322, 44717}, {2327, 46102}, {2328, 4998}, {3786, 5384}, {3952, 4636}, {4069, 52935}, {4556, 30730}, {4565, 6558}, {4591, 30731}, {4627, 30728}, {4629, 30729}, {6632, 7252}, {23357, 30713}, {23990, 28660}, {46254, 52370}
X(53540) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 4076}, {11, 7256}, {115, 646}, {125, 4571}, {223, 4600}, {244, 3699}, {478, 4567}, {513, 21}, {514, 314}, {523, 3701}, {661, 333}, {1015, 645}, {1084, 644}, {1086, 7257}, {1146, 7258}, {1214, 7035}, {3005, 210}, {3160, 4601}, {4369, 7081}, {4988, 312}, {5521, 36797}, {6615, 1043}, {7180, 1999}, {8054, 643}, {36908, 4998}, {38986, 3939}, {38991, 7259}, {40590, 1016}, {40592, 6064}, {40600, 6065}, {40608, 4578}, {40611, 765}, {40615, 799}, {40617, 99}, {40620, 4631}, {40622, 668}, {40627, 9}, {50330, 8}, {50497, 55}
X(53540) = crossdifference of every pair of points on line {643, 644}
X(53540) = barycentric product X(i)*X(j) for these {i,j}: {7, 3125}, {11, 1427}, {12, 16726}, {34, 4466}, {37, 1358}, {56, 16732}, {57, 3120}, {65, 1086}, {81, 1365}, {85, 3122}, {115, 1014}, {125, 1396}, {181, 16727}, {225, 3942}, {226, 244}, {269, 21044}, {278, 18210}, {279, 4516}, {321, 1357}, {338, 1408}, {349, 3248}, {479, 36197}, {512, 24002}, {513, 7178}, {514, 4017}, {522, 7216}, {523, 3669}, {525, 43923}, {552, 21833}, {604, 21207}, {649, 4077}, {661, 3676}, {693, 7180}, {738, 52335}, {764, 4552}, {798, 52621}, {876, 7212}, {1015, 1441}, {1020, 21132}, {1022, 30572}, {1042, 4858}, {1109, 1412}, {1111, 1400}, {1214, 2969}, {1231, 42067}, {1254, 17197}, {1356, 6385}, {1367, 5317}, {1402, 23989}, {1409, 2973}, {1414, 21131}, {1426, 26932}, {1434, 2643}, {1439, 8735}, {1446, 3271}, {1565, 1880}, {1577, 43924}, {1903, 38374}, {2170, 3668}, {2171, 17205}, {2611, 52374}, {3121, 6063}, {3261, 51641}, {3700, 43932}, {3937, 40149}, {4024, 7203}, {4391, 7250}, {4515, 41292}, {4551, 6545}, {4554, 8034}, {4705, 17096}, {6354, 18191}, {6591, 17094}, {7202, 52382}, {7233, 39786}, {7249, 16592}, {7649, 51664}, {8287, 52372}, {14027, 30575}, {16947, 23994}, {17107, 21945}, {19604, 21950}, {21141, 26700}, {21963, 42304}, {22096, 52575}, {24290, 43930}, {34051, 42759}, {38362, 52037}, {40166, 53321}
X(53540) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 4601}, {37, 4076}, {56, 4567}, {57, 4600}, {65, 1016}, {81, 6064}, {115, 3701}, {213, 6065}, {226, 7035}, {244, 333}, {269, 4620}, {512, 644}, {513, 645}, {514, 7257}, {522, 7258}, {523, 646}, {604, 4570}, {608, 5379}, {647, 4571}, {649, 643}, {650, 7256}, {661, 3699}, {663, 7259}, {667, 5546}, {764, 4560}, {798, 3939}, {810, 4587}, {1014, 4590}, {1015, 21}, {1042, 4564}, {1086, 314}, {1106, 52378}, {1109, 30713}, {1111, 28660}, {1356, 213}, {1357, 81}, {1358, 274}, {1365, 321}, {1396, 18020}, {1400, 765}, {1402, 1252}, {1408, 249}, {1410, 44717}, {1412, 24041}, {1426, 46102}, {1427, 4998}, {1434, 24037}, {1441, 31625}, {1880, 15742}, {1977, 2194}, {2170, 1043}, {2205, 6066}, {2611, 42033}, {2643, 2321}, {2969, 31623}, {3120, 312}, {3121, 55}, {3122, 9}, {3124, 210}, {3125, 8}, {3248, 284}, {3271, 2287}, {3669, 99}, {3676, 799}, {3708, 3710}, {3709, 4578}, {3733, 4612}, {3937, 1812}, {3942, 332}, {4017, 190}, {4041, 6558}, {4077, 1978}, {4079, 4069}, {4128, 2329}, {4466, 3718}, {4516, 346}, {4551, 6632}, {4705, 30730}, {4729, 30720}, {4730, 30731}, {4822, 30728}, {4983, 30729}, {6545, 18155}, {6591, 36797}, {7117, 1792}, {7178, 668}, {7180, 100}, {7192, 4631}, {7203, 4610}, {7212, 874}, {7216, 664}, {7250, 651}, {8027, 7252}, {8034, 650}, {14027, 16729}, {16592, 7081}, {16726, 261}, {16727, 18021}, {16732, 3596}, {16947, 1101}, {17096, 4623}, {17205, 52379}, {18191, 7058}, {18210, 345}, {20975, 3694}, {20982, 4420}, {21044, 341}, {21131, 4086}, {21143, 3737}, {21207, 28659}, {21725, 4095}, {21755, 2330}, {21833, 6057}, {21950, 44720}, {21963, 30568}, {22096, 2193}, {22210, 27523}, {23989, 40072}, {24002, 670}, {30572, 24004}, {36197, 5423}, {39786, 3685}, {42067, 1172}, {43923, 648}, {43924, 662}, {43925, 52914}, {43932, 4573}, {51641, 101}, {51664, 4561}, {52335, 30693}, {52621, 4602}, {53321, 31615}
X(53540) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1427, 40961, 1402}, {3120, 18210, 4516}

---

X(53541) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1357) AND X(3023)

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

X(53541) lies on these lines: {1, 2796}, {2, 9359}, {7, 7032}, {42, 4667}, {86, 4623}, {87, 3662}, {142, 22343}, {171, 4579}, {190, 24722}, {193, 25570}, {244, 1357}, {291, 3888}, {354, 23634}, {513, 3122}, {524, 2234}, {527, 3009}, {804, 3023}, {869, 4644}, {872, 7277}, {894, 7184}, {904, 1434}, {1015, 3123}, {1045, 20090}, {1086, 3248}, {1738, 23579}, {1740, 17364}, {1959, 21829}, {1964, 17365}, {2086, 16592}, {2309, 3664}, {2643, 7202}, {2664, 20072}, {3120, 17197}, {3720, 17195}, {3778, 49537}, {3942, 4475}, {3946, 23532}, {4000, 23524}, {4033, 25382}, {4754, 27880}, {4778, 24195}, {5150, 37522}, {7122, 7175}, {7228, 17445}, {7303, 40164}, {7321, 18170}, {8772, 51653}, {10436, 20345}, {15953, 50307}, {16571, 17363}, {17050, 23427}, {17067, 23539}, {17213, 17219}, {17247, 24661}, {17248, 25528}, {17378, 24696}, {17379, 18794}, {17770, 18792}, {18150, 24487}, {18169, 23812}, {18194, 48627}, {20257, 23457}, {21100, 31061}, {21139, 23676}, {21140, 21210}, {21352, 50116}, {23493, 24215}, {23633, 29353}, {23943, 50346}, {27191, 37129}, {39354, 48628}, {43920, 43924}, {46458, 53314}, {49710, 49997}

X(53541) = reflection of X(3122) in X(16726)
X(53541) = X(i)-Ceva conjugate of X(j) for these (i,j): {86, 17212}, {171, 4367}, {1434, 649}, {1909, 4369}, {7175, 20981}, {40164, 1019}
X(53541) = X(i)-isoconjugate of X(j) for these (i,j): {59, 4451}, {100, 3903}, {101, 27805}, {256, 765}, {257, 1252}, {644, 37137}, {893, 1016}, {904, 7035}, {1018, 4603}, {1110, 7018}, {1431, 4076}, {3699, 29055}, {3799, 30670}, {4557, 4594}, {4567, 52651}, {4601, 40729}, {5378, 18786}, {6065, 7249}, {7015, 15742}, {7104, 31625}, {23990, 44187}
X(53541) = X(i)-Dao conjugate of X(j) for these (i,j): {513, 256}, {514, 7018}, {661, 257}, {1015, 27805}, {3709, 2321}, {4369, 10}, {6615, 4451}, {8054, 3903}, {16592, 668}, {21051, 3971}, {25666, 25280}, {40597, 1016}, {40620, 7260}, {40627, 52651}
X(53541) = cevapoint of X(4128) and X(16592)
X(53541) = crossdifference of every pair of points on line {644, 3903}
X(53541) = barycentric product X(i)*X(j) for these {i,j}: {1, 7200}, {11, 7175}, {57, 4459}, {86, 16592}, {171, 1086}, {172, 1111}, {244, 894}, {257, 7207}, {274, 4128}, {310, 21755}, {512, 16737}, {513, 4369}, {514, 4367}, {523, 18200}, {649, 4374}, {661, 17212}, {693, 20981}, {764, 18047}, {873, 21823}, {876, 4107}, {1015, 1909}, {1019, 2533}, {1022, 4922}, {1215, 16726}, {1357, 17787}, {1358, 2329}, {1432, 3023}, {1434, 40608}, {1509, 21725}, {1565, 7119}, {1920, 3248}, {2170, 7176}, {2295, 17205}, {2530, 18111}, {3122, 8033}, {3125, 17103}, {3271, 7196}, {3287, 3676}, {3572, 14296}, {3669, 3907}, {3805, 4817}, {3942, 7009}, {4032, 18191}, {4140, 7203}, {4164, 4444}, {4475, 40745}, {4529, 43932}, {4579, 6545}, {7122, 23989}, {7199, 7234}, {16727, 20964}, {17924, 22093}, {18787, 27918}, {22373, 44129}, {27831, 51656}, {27846, 30669}, {40790, 43266}
X(53541) = barycentric quotient X(i)/X(j) for these {i,j}: {171, 1016}, {172, 765}, {244, 257}, {513, 27805}, {649, 3903}, {894, 7035}, {1015, 256}, {1019, 4594}, {1086, 7018}, {1111, 44187}, {1357, 1432}, {1909, 31625}, {1977, 904}, {2170, 4451}, {2329, 4076}, {2533, 4033}, {3023, 17787}, {3122, 52651}, {3248, 893}, {3287, 3699}, {3733, 4603}, {3805, 3807}, {3907, 646}, {3942, 7019}, {4107, 874}, {4128, 37}, {4164, 3570}, {4367, 190}, {4369, 668}, {4374, 1978}, {4459, 312}, {4477, 6558}, {4579, 6632}, {4922, 24004}, {7119, 15742}, {7122, 1252}, {7175, 4998}, {7192, 7260}, {7200, 75}, {7207, 894}, {7234, 1018}, {14296, 27853}, {16592, 10}, {16726, 32010}, {16737, 670}, {17103, 4601}, {17212, 799}, {18200, 99}, {20981, 100}, {21725, 594}, {21755, 42}, {21823, 756}, {22093, 1332}, {22096, 7116}, {22373, 71}, {24533, 4595}, {27846, 17493}, {40608, 2321}, {43924, 37137}, {45882, 3799}
X(53541) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1015, 4014, 3123}, {1086, 3248, 27846}, {7184, 7240, 894}, {16571, 25572, 17363}, {17197, 23823, 3120}

---

X(53542) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1357) AND X(3024)

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

X(53542) lies on these lines: {1, 2842}, {31, 1469}, {42, 23845}, {56, 53254}, {109, 6187}, {110, 24436}, {181, 9340}, {244, 1357}, {511, 896}, {513, 3120}, {526, 2611}, {748, 3784}, {759, 19470}, {902, 8679}, {1109, 4459}, {1155, 20962}, {1201, 53303}, {1364, 2310}, {1412, 6186}, {2088, 2624}, {2392, 52680}, {2810, 3722}, {2979, 7262}, {3056, 36263}, {3060, 4650}, {3122, 4491}, {3219, 7186}, {3220, 52434}, {3794, 4683}, {3888, 33115}, {4414, 37516}, {4778, 24235}, {5194, 8772}, {8616, 23155}, {9025, 32848}, {15310, 33136}, {16727, 48107}, {17063, 26910}, {21833, 50498}, {27577, 49557}, {47007, 47008}, {49676, 50003}

X(53542) = reflection of X(3120) in X(18191)
X(53542) = X(i)-Ceva conjugate of X(j) for these (i,j): {35, 2605}, {319, 14838}, {1412, 649}
X(53542) = X(i)-isoconjugate of X(j) for these (i,j): {59, 52344}, {78, 34922}, {79, 765}, {100, 6742}, {101, 15455}, {644, 38340}, {1016, 2160}, {1110, 20565}, {1252, 30690}, {2321, 35049}, {3699, 26700}, {3952, 13486}, {4053, 39295}, {4076, 52372}, {4564, 7110}, {4567, 8818}, {4570, 6757}, {4998, 7073}, {5379, 52388}, {6186, 7035}, {7100, 15742}
X(53542) = X(i)-Dao conjugate of X(j) for these (i,j): {513, 79}, {514, 20565}, {661, 30690}, {1015, 15455}, {3700, 30713}, {6615, 52344}, {8054, 6742}, {8287, 668}, {14838, 313}, {40627, 8818}, {50330, 6757}
X(53542) = crossdifference of every pair of points on line {644, 4115}
X(53542) = barycentric product X(i)*X(j) for these {i,j}: {1, 7202}, {11, 2003}, {27, 22094}, {35, 1086}, {58, 8287}, {81, 2611}, {86, 20982}, {110, 21141}, {244, 3219}, {319, 1015}, {512, 16755}, {513, 14838}, {514, 2605}, {593, 21054}, {649, 4467}, {667, 18160}, {757, 21824}, {1111, 2174}, {1333, 17886}, {1357, 42033}, {1358, 52405}, {1399, 4858}, {1412, 6741}, {1442, 2170}, {2160, 7266}, {2594, 17197}, {3024, 52374}, {3120, 40214}, {3122, 34016}, {3248, 33939}, {3271, 17095}, {3669, 35057}, {3676, 9404}, {3678, 16726}, {3733, 7265}, {3937, 52412}, {3942, 6198}, {7117, 7282}, {16577, 18191}, {16732, 17104}, {17924, 23226}
X(53542) = barycentric quotient X(i)/X(j) for these {i,j}: {35, 1016}, {244, 30690}, {319, 31625}, {513, 15455}, {608, 34922}, {649, 6742}, {1015, 79}, {1086, 20565}, {1357, 52374}, {1399, 4564}, {1408, 35049}, {1977, 6186}, {2003, 4998}, {2170, 52344}, {2174, 765}, {2605, 190}, {2611, 321}, {3024, 42033}, {3122, 8818}, {3125, 6757}, {3219, 7035}, {3248, 2160}, {3271, 7110}, {3937, 52381}, {4467, 1978}, {6741, 30713}, {7202, 75}, {7265, 27808}, {7266, 33939}, {8287, 313}, {9404, 3699}, {14838, 668}, {16755, 670}, {17104, 4567}, {17886, 27801}, {18160, 6386}, {20982, 10}, {21054, 28654}, {21141, 850}, {21824, 1089}, {22094, 306}, {23226, 1332}, {35057, 646}, {40214, 4600}, {43924, 38340}, {52405, 4076}
X(53542) = {X(3271),X(3937)}-harmonic conjugate of X(244)

---

X(53543) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1357) AND X(3026)

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

X(53543) lies on these lines: {7, 17053}, {11, 23823}, {39, 4675}, {57, 4274}, {244, 1357}, {599, 1574}, {812, 1015}, {941, 30712}, {1418, 20227}, {1500, 17392}, {2092, 3664}, {2275, 6173}, {2277, 4888}, {3122, 4014}, {3125, 3942}, {3707, 16602}, {3752, 4667}, {4403, 16732}, {4643, 31198}, {4858, 7200}, {6388, 16592}, {7184, 17049}, {16604, 50092}, {17334, 21826}, {17365, 21796}, {17366, 46189}, {17374, 52959}, {20972, 53391}, {21138, 24195}, {25534, 40479}, {25570, 35892}, {26076, 30866}, {26142, 27195}, {28365, 29747}, {37596, 50116}, {39798, 48632}

X(53543) = complement of the isotomic conjugate of X(47915)
X(53543) = X(i)-complementary conjugate of X(j) for these (i,j): {2334, 3835}, {5936, 21262}, {8694, 27076}, {25430, 21260}, {34074, 24003}, {47915, 2887}
X(53543) = X(i)-Ceva conjugate of X(j) for these (i,j): {940, 48144}, {10436, 8672}, {21454, 8712}, {30712, 513}, {34284, 43067}
X(53543) = X(i)-isoconjugate of X(j) for these (i,j): {765, 941}, {931, 1018}, {1016, 2258}, {1110, 34258}, {1252, 31359}, {3699, 32693}, {3939, 32038}, {6065, 44733}, {7259, 52931}
X(53543) = X(i)-Dao conjugate of X(j) for these (i,j): {513, 941}, {514, 34258}, {661, 31359}, {17417, 3699}, {34261, 4076}, {40617, 32038}, {50457, 42034}
X(53543) = crossdifference of every pair of points on line {644, 4557}
X(53543) = barycentric product X(i)*X(j) for these {i,j}: {244, 10436}, {513, 43067}, {514, 48144}, {940, 1086}, {958, 1358}, {959, 3026}, {1015, 34284}, {1019, 50457}, {1111, 1468}, {1565, 4185}, {3669, 23880}, {3676, 17418}, {3942, 5307}, {5019, 23989}, {7192, 8672}, {8639, 52619}, {16726, 31993}
X(53543) = barycentric quotient X(i)/X(j) for these {i,j}: {244, 31359}, {940, 1016}, {958, 4076}, {1015, 941}, {1086, 34258}, {1357, 959}, {1468, 765}, {3248, 2258}, {3669, 32038}, {3733, 931}, {3937, 34259}, {4185, 15742}, {5019, 1252}, {7250, 52931}, {8639, 4557}, {8672, 3952}, {10436, 7035}, {16726, 37870}, {17418, 3699}, {23880, 646}, {23989, 40828}, {34284, 31625}, {43067, 668}, {48144, 190}, {50457, 4033}
X(53543) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1086, 16726, 1015}, {17205, 24237, 1086}

---

X(53544) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1358) AND X(1362)

Barycentrics    a*(b - c)*(a + b - c)*(a - b + c)*(a*b - b^2 + a*c - c^2) : :
X(53544) = 3 X[1635] - 2 X[53396]

X(53544) lies on these lines: {1, 2820}, {7, 812}, {56, 53287}, {57, 1635}, {85, 20950}, {101, 651}, {105, 1477}, {222, 21758}, {226, 4728}, {513, 663}, {514, 7216}, {649, 43049}, {650, 43932}, {656, 9000}, {657, 905}, {661, 3676}, {664, 53213}, {918, 16593}, {926, 1362}, {1022, 34056}, {1024, 1462}, {1086, 1358}, {1407, 21786}, {1441, 20908}, {1617, 8645}, {1638, 46393}, {2826, 38055}, {2827, 53055}, {2832, 51765}, {3126, 42341}, {3323, 38980}, {3333, 38324}, {3572, 43041}, {3887, 14151}, {4041, 9443}, {4077, 47672}, {4391, 46399}, {4449, 6182}, {4468, 25900}, {4481, 17096}, {4763, 5435}, {4928, 5226}, {5083, 38325}, {7178, 47918}, {7250, 17420}, {8712, 17410}, {9511, 14392}, {14349, 30723}, {16892, 17094}, {17496, 46402}, {21222, 26140}, {21454, 47776}, {23730, 39470}, {23737, 37755}, {24118, 41777}, {28846, 31605}, {28902, 30724}, {30719, 48334}, {31393, 38328}

X(53544) = midpoint of X(17496) and X(46402)
X(53544) = reflection of X(i) in X(j) for these {i,j}: {657, 905}, {4391, 46399}
X(53544) = X(i)-Ceva conjugate of X(j) for these (i,j): {1025, 241}, {1414, 51329}, {34855, 3675}, {36146, 57}, {37626, 3669}, {41353, 1458}, {43042, 2254}
X(53544) = X(i)-isoconjugate of X(j) for these (i,j): {2, 52927}, {6, 36802}, {8, 919}, {9, 36086}, {41, 51560}, {55, 666}, {59, 28132}, {100, 294}, {101, 14942}, {105, 644}, {109, 6559}, {190, 2195}, {200, 36146}, {220, 927}, {312, 32666}, {346, 32735}, {643, 18785}, {650, 5377}, {651, 28071}, {657, 39293}, {673, 3939}, {692, 36796}, {765, 1024}, {884, 1016}, {885, 1252}, {1253, 34085}, {1416, 6558}, {1438, 3699}, {1462, 4578}, {2175, 36803}, {2348, 39272}, {3886, 36138}, {4076, 43929}, {4571, 8751}, {4587, 36124}, {5546, 13576}, {14727, 16283}, {14827, 46135}, {28809, 32724}
X(53544) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 36802}, {11, 6559}, {223, 666}, {241, 42719}, {478, 36086}, {513, 1024}, {661, 885}, {665, 3716}, {1015, 14942}, {1086, 36796}, {3126, 3239}, {3160, 51560}, {3716, 4148}, {6184, 3699}, {6609, 36146}, {6615, 28132}, {8054, 294}, {17113, 34085}, {17435, 3717}, {17755, 646}, {27918, 3975}, {32664, 52927}, {35094, 312}, {35509, 24026}, {36905, 668}, {38980, 8}, {38989, 9}, {38991, 28071}, {39012, 3886}, {39014, 200}, {39046, 644}, {39063, 190}, {40593, 36803}, {40609, 6558}, {40615, 2481}, {40617, 673}, {52656, 36801}
X(53544) = trilinear pole of line {3675, 38363}
X(53544) = crossdifference of every pair of points on line {9, 294}
X(53544) = barycentric product X(i)*X(j) for these {i,j}: {1, 43042}, {7, 2254}, {11, 41353}, {57, 918}, {65, 23829}, {85, 665}, {241, 514}, {244, 883}, {269, 50333}, {513, 9436}, {518, 3676}, {522, 34855}, {649, 40704}, {658, 17435}, {664, 3675}, {672, 24002}, {693, 1458}, {876, 39775}, {905, 5236}, {926, 1088}, {1014, 4088}, {1025, 1086}, {1026, 1358}, {1111, 2283}, {1434, 24290}, {1876, 4025}, {2223, 52621}, {3261, 52635}, {3263, 43924}, {3286, 4077}, {3323, 36086}, {3669, 3912}, {3717, 43932}, {3930, 17096}, {3932, 7203}, {4017, 30941}, {4394, 10029}, {4444, 34253}, {4712, 43930}, {4925, 19604}, {5723, 52228}, {7045, 52305}, {7178, 18206}, {7180, 18157}, {15149, 51664}, {16593, 37626}, {22116, 43041}, {23062, 52614}, {34085, 35505}, {35094, 36146}, {37136, 42770}
X(53544) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 36802}, {7, 51560}, {31, 52927}, {56, 36086}, {57, 666}, {85, 36803}, {109, 5377}, {241, 190}, {244, 885}, {269, 927}, {279, 34085}, {513, 14942}, {514, 36796}, {518, 3699}, {604, 919}, {649, 294}, {650, 6559}, {663, 28071}, {665, 9}, {667, 2195}, {672, 644}, {876, 33676}, {883, 7035}, {918, 312}, {926, 200}, {934, 39293}, {1015, 1024}, {1025, 1016}, {1026, 4076}, {1088, 46135}, {1106, 32735}, {1357, 1027}, {1362, 1026}, {1397, 32666}, {1401, 35333}, {1407, 36146}, {1458, 100}, {1477, 39272}, {1818, 4571}, {1876, 1897}, {2170, 28132}, {2223, 3939}, {2254, 8}, {2283, 765}, {2340, 4578}, {3126, 3717}, {3248, 884}, {3286, 643}, {3669, 673}, {3675, 522}, {3676, 2481}, {3693, 6558}, {3912, 646}, {3930, 30730}, {3937, 23696}, {4017, 13576}, {4088, 3701}, {4925, 44720}, {5236, 6335}, {7180, 18785}, {8638, 1253}, {9436, 668}, {14439, 30731}, {17435, 3239}, {18206, 645}, {20662, 23704}, {20683, 4069}, {20752, 4587}, {22116, 36801}, {23225, 212}, {23829, 314}, {24002, 18031}, {24290, 2321}, {30941, 7257}, {34253, 3570}, {34855, 664}, {38989, 3716}, {39063, 42719}, {39775, 874}, {40704, 1978}, {41353, 4998}, {42758, 6735}, {43042, 75}, {43049, 31638}, {43923, 36124}, {43924, 105}, {46388, 220}, {50333, 341}, {50357, 4673}, {51329, 3573}, {52305, 24026}, {52614, 728}, {52635, 101}
X(53544) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {57, 43050, 1635}, {3669, 51664, 48131}

---

X(53545) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1358) AND X(1365)

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

X(53545) lies on these lines: {7, 4393}, {57, 18625}, {65, 21889}, {226, 3995}, {244, 2969}, {278, 36570}, {347, 28081}, {523, 21945}, {1020, 1400}, {1086, 1358}, {1365, 2611}, {1404, 43035}, {1412, 52374}, {1441, 16603}, {2171, 52023}, {3122, 4017}, {3270, 4014}, {3669, 16726}, {3676, 24195}, {4089, 24237}, {4466, 8287}, {4654, 41803}, {4688, 24798}, {4858, 23989}, {6173, 47444}, {16596, 33573}, {16609, 41804}, {17058, 21950}, {17301, 24796}, {17895, 26012}, {21139, 26932}, {22464, 52896}, {24803, 37756}, {24805, 41311}, {27565, 41003}, {28015, 36640}, {43066, 51653}, {43920, 43924}

X(53545) = X(i)-Ceva conjugate of X(j) for these (i,j): {226, 7178}, {349, 4077}, {1434, 3676}, {3668, 4017}, {3669, 6545}, {52374, 3669}
X(53545) = X(i)-isoconjugate of X(j) for these (i,j): {9, 4570}, {21, 1252}, {41, 4600}, {55, 4567}, {59, 2287}, {81, 6065}, {100, 5546}, {101, 643}, {109, 7259}, {110, 644}, {112, 4571}, {162, 4587}, {163, 3699}, {200, 52378}, {210, 249}, {213, 6064}, {219, 5379}, {250, 3694}, {274, 6066}, {284, 765}, {314, 23990}, {333, 1110}, {645, 692}, {646, 1576}, {662, 3939}, {906, 36797}, {1016, 2194}, {1018, 4636}, {1043, 2149}, {1101, 2321}, {1253, 4620}, {1333, 4076}, {1334, 24041}, {1415, 7256}, {1792, 7115}, {2175, 4601}, {2193, 15742}, {2327, 7012}, {2328, 4564}, {3701, 23357}, {4069, 4556}, {4183, 44717}, {4557, 4612}, {4565, 4578}, {4574, 52914}, {7257, 32739}, {18020, 52370}, {21789, 31615}, {23995, 30713}
X(53545) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 7259}, {37, 4076}, {115, 3699}, {125, 4587}, {223, 4567}, {244, 644}, {478, 4570}, {513, 284}, {514, 333}, {523, 2321}, {647, 3710}, {650, 1043}, {661, 21}, {1015, 643}, {1084, 3939}, {1086, 645}, {1146, 7256}, {1214, 1016}, {3005, 1334}, {3120, 30729}, {3160, 4600}, {3669, 41629}, {4369, 2329}, {4521, 52352}, {4858, 646}, {4988, 8}, {5190, 36797}, {6609, 52378}, {6615, 2287}, {6626, 6064}, {6741, 6558}, {7178, 33116}, {7180, 5247}, {8054, 5546}, {14838, 42033}, {17113, 4620}, {18314, 30713}, {34591, 4571}, {36908, 4564}, {40586, 6065}, {40590, 765}, {40593, 4601}, {40611, 1252}, {40615, 99}, {40617, 662}, {40619, 7257}, {40622, 190}, {40624, 7258}, {40627, 55}, {40628, 1792}, {47345, 15742}, {50330, 9}, {50497, 41}
X(53545) = cevapoint of X(3120) and X(21950)
X(53545) = crossdifference of every pair of points on line {3939, 4587}
X(53545) = barycentric product X(i)*X(j) for these {i,j}: {7, 3120}, {10, 1358}, {11, 3668}, {12, 17205}, {56, 21207}, {57, 16732}, {65, 1111}, {73, 2973}, {85, 3125}, {86, 1365}, {115, 1434}, {225, 1565}, {226, 1086}, {244, 1441}, {273, 18210}, {278, 4466}, {279, 21044}, {307, 2969}, {313, 1357}, {338, 1412}, {349, 1015}, {479, 52335}, {512, 52621}, {513, 4077}, {514, 7178}, {523, 3676}, {552, 21043}, {661, 24002}, {693, 4017}, {850, 43924}, {1014, 1109}, {1020, 40166}, {1042, 34387}, {1088, 4516}, {1118, 17216}, {1367, 8747}, {1396, 20902}, {1400, 23989}, {1408, 23994}, {1426, 17880}, {1427, 4858}, {1446, 2170}, {1577, 3669}, {2171, 16727}, {3121, 20567}, {3122, 6063}, {3261, 7180}, {3942, 40149}, {4024, 17096}, {4036, 7203}, {4049, 30725}, {4052, 40617}, {4082, 41292}, {4086, 43932}, {4088, 43930}, {4089, 52383}, {4391, 7216}, {4444, 7212}, {4552, 6545}, {4559, 23100}, {4566, 21132}, {4572, 8034}, {4573, 21131}, {6354, 17197}, {6358, 16726}, {6548, 30572}, {6549, 40663}, {7202, 43682}, {7250, 35519}, {7649, 17094}, {8287, 52374}, {14208, 43923}, {16603, 43266}, {16947, 23962}, {17886, 52372}, {17924, 51664}, {21141, 38340}, {21945, 40154}, {21950, 27818}, {23062, 36197}, {30724, 31010}, {35352, 43041}, {37887, 40622}, {38374, 39130}, {40495, 51641}
X(53545) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 4600}, {10, 4076}, {11, 1043}, {34, 5379}, {42, 6065}, {56, 4570}, {57, 4567}, {65, 765}, {85, 4601}, {86, 6064}, {115, 2321}, {125, 3710}, {225, 15742}, {226, 1016}, {244, 21}, {279, 4620}, {338, 30713}, {349, 31625}, {512, 3939}, {513, 643}, {514, 645}, {522, 7256}, {523, 3699}, {647, 4587}, {649, 5546}, {650, 7259}, {656, 4571}, {661, 644}, {693, 7257}, {764, 3737}, {1014, 24041}, {1015, 284}, {1019, 4612}, {1020, 31615}, {1042, 59}, {1086, 333}, {1109, 3701}, {1111, 314}, {1356, 1918}, {1357, 58}, {1358, 86}, {1365, 10}, {1367, 52396}, {1400, 1252}, {1402, 1110}, {1407, 52378}, {1408, 1101}, {1412, 249}, {1426, 7012}, {1427, 4564}, {1434, 4590}, {1441, 7035}, {1565, 332}, {1577, 646}, {1918, 6066}, {2170, 2287}, {2486, 3996}, {2611, 4420}, {2643, 210}, {2969, 29}, {2973, 44130}, {3120, 8}, {3121, 41}, {3122, 55}, {3124, 1334}, {3125, 9}, {3248, 2194}, {3271, 2328}, {3668, 4998}, {3669, 662}, {3676, 99}, {3700, 6558}, {3708, 3694}, {3733, 4636}, {3756, 52352}, {3937, 283}, {3942, 1812}, {4017, 100}, {4024, 30730}, {4041, 4578}, {4049, 4582}, {4077, 668}, {4092, 4082}, {4120, 30731}, {4128, 2330}, {4391, 7258}, {4466, 345}, {4475, 3786}, {4516, 200}, {4552, 6632}, {4705, 4069}, {4841, 30728}, {4848, 44724}, {4988, 30729}, {6545, 4560}, {7004, 1792}, {7117, 2327}, {7178, 190}, {7180, 101}, {7199, 4631}, {7200, 27958}, {7203, 52935}, {7212, 3570}, {7216, 651}, {7250, 109}, {7649, 36797}, {8034, 663}, {8287, 42033}, {8735, 2322}, {8754, 53008}, {14321, 30720}, {16592, 2329}, {16726, 2185}, {16727, 52379}, {16732, 312}, {16947, 23357}, {17094, 4561}, {17096, 4610}, {17197, 7058}, {17205, 261}, {17216, 1264}, {18191, 1098}, {18210, 78}, {20975, 2318}, {20982, 52405}, {21043, 6057}, {21044, 346}, {21131, 3700}, {21132, 7253}, {21134, 52355}, {21143, 7252}, {21207, 3596}, {21950, 3161}, {21963, 3913}, {23989, 28660}, {24002, 799}, {30572, 17780}, {35352, 36801}, {36197, 728}, {38374, 8822}, {39786, 3684}, {40617, 41629}, {40622, 33116}, {42067, 2299}, {42759, 6735}, {43923, 162}, {43924, 110}, {43932, 1414}, {51641, 692}, {51664, 1332}, {52335, 5423}, {52373, 44717}, {52621, 670}
X(53545) = {X(4466),X(16732)}-harmonic conjugate of X(21044)

---

X(53546) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1358) AND X(3025)

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

X(53546) lies on these lines: {1, 38530}, {6, 57}, {88, 4638}, {241, 24029}, {320, 4053}, {513, 3675}, {514, 46781}, {517, 34230}, {1022, 2087}, {1086, 1358}, {1155, 23344}, {1376, 16504}, {1443, 7113}, {1959, 7238}, {3218, 4585}, {3554, 5575}, {3742, 16494}, {3753, 16506}, {3995, 17300}, {4014, 17463}, {4089, 46398}, {4124, 4977}, {4675, 24484}, {4887, 17444}, {4957, 21139}, {5439, 16500}, {6545, 8042}, {6546, 23730}, {7184, 18183}, {7297, 18735}, {15635, 23345}, {16492, 52541}, {16495, 17063}, {16610, 52031}, {16686, 53394}, {16726, 50456}, {16732, 24237}, {17234, 31053}, {17265, 30852}, {17276, 24868}, {17313, 31164}, {25415, 42871}, {37680, 47058}

X(53546) = X(i)-Ceva conjugate of X(j) for these (i,j): {1443, 53314}, {3218, 3960}, {6549, 244}, {20924, 4453}, {34051, 3669}
X(53546) = X(i)-isoconjugate of X(j) for these (i,j): {9, 52377}, {59, 36910}, {80, 1252}, {101, 51562}, {644, 2222}, {655, 3939}, {692, 36804}, {765, 2161}, {1016, 6187}, {1110, 18359}, {2006, 6065}, {2149, 52409}, {2323, 46649}, {3699, 32675}, {4103, 36069}, {4557, 47318}, {4564, 52371}, {4567, 34857}, {5376, 40172}, {6535, 9273}, {15742, 52431}, {20566, 23990}, {37140, 40521}
X(53546) = X(i)-Dao conjugate of X(j) for these (i,j): {478, 52377}, {513, 2161}, {514, 18359}, {650, 52409}, {661, 80}, {1015, 51562}, {1086, 36804}, {1639, 4723}, {3960, 4358}, {4988, 15065}, {6544, 51975}, {6615, 36910}, {35128, 3699}, {38982, 4103}, {38984, 644}, {40584, 765}, {40612, 1016}, {40615, 35174}, {40617, 655}, {40627, 34857}
X(53546) = cevapoint of X(2087) and X(42753)
X(53546) = crossdifference of every pair of points on line {3900, 3939}
X(53546) = barycentric product X(i)*X(j) for these {i,j}: {1, 4089}, {11, 1443}, {36, 1111}, {214, 6549}, {244, 320}, {513, 4453}, {514, 3960}, {654, 24002}, {658, 46384}, {693, 53314}, {758, 17205}, {1015, 20924}, {1019, 4707}, {1086, 3218}, {1227, 43922}, {1358, 4511}, {1565, 1870}, {1647, 52553}, {1835, 17219}, {1983, 23100}, {2170, 17078}, {2245, 16727}, {2973, 52407}, {3025, 18815}, {3248, 40075}, {3261, 21758}, {3669, 3904}, {3676, 3738}, {3936, 16726}, {3942, 17923}, {4394, 27836}, {4585, 6545}, {7113, 23989}, {7199, 21828}, {8648, 52621}, {17197, 18593}, {18191, 41804}, {22379, 46107}, {34051, 46398}, {34387, 52440}
X(53546) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 52409}, {36, 765}, {56, 52377}, {244, 80}, {320, 7035}, {513, 51562}, {514, 36804}, {654, 644}, {1015, 2161}, {1019, 47318}, {1086, 18359}, {1111, 20566}, {1357, 1411}, {1358, 18815}, {1411, 46649}, {1443, 4998}, {1647, 51975}, {1870, 15742}, {2170, 36910}, {2361, 6065}, {2610, 4103}, {3025, 4511}, {3120, 15065}, {3122, 34857}, {3218, 1016}, {3248, 6187}, {3271, 52371}, {3669, 655}, {3676, 35174}, {3738, 3699}, {3904, 646}, {3937, 1807}, {3942, 52351}, {3960, 190}, {4089, 75}, {4453, 668}, {4511, 4076}, {4585, 6632}, {4707, 4033}, {4881, 44724}, {7113, 1252}, {7202, 41226}, {8648, 3939}, {15635, 40437}, {16726, 24624}, {16944, 9268}, {17205, 14616}, {18191, 6740}, {20924, 31625}, {21132, 52356}, {21758, 101}, {21828, 1018}, {22379, 1331}, {24002, 46405}, {27846, 36815}, {40215, 5376}, {42666, 40521}, {43922, 1168}, {43924, 2222}, {46384, 3239}, {51402, 4723}, {52434, 1110}, {52440, 59}, {53285, 4578}, {53314, 100}
X(53546) = {X(1086),X(3942)}-harmonic conjugate of X(7202)

---

X(53547) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1360) AND X(1362)

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

X(53547) lies on the de Longchamps ellipse and these lines: {1, 103}, {7, 43751}, {31, 57}, {55, 6611}, {65, 1475}, {165, 17080}, {221, 7124}, {354, 7004}, {390, 9533}, {516, 24014}, {518, 6168}, {651, 1282}, {658, 14942}, {672, 1876}, {676, 1360}, {910, 1456}, {926, 1362}, {962, 45239}, {1020, 2310}, {1025, 4712}, {1214, 35270}, {1439, 2293}, {1458, 34855}, {1566, 39077}, {1768, 10980}, {2272, 3827}, {2283, 35293}, {2611, 5173}, {3021, 3321}, {3599, 4344}, {3942, 53297}, {4566, 28850}, {7965, 34969}, {9504, 34253}, {11712, 23890}, {14524, 20978}, {18421, 34056}, {18839, 35065}, {39063, 50441}

X(53547) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 1458}, {57, 910}, {658, 46392}, {39063, 9502}
X(53547) = X(i)-isoconjugate of X(j) for these (i,j): {9, 9503}, {103, 14942}, {294, 36101}, {673, 2338}, {677, 885}, {911, 36796}, {2195, 18025}, {2400, 52927}, {2424, 36802}, {28071, 43736}
X(53547) = X(i)-Dao conjugate of X(j) for these (i,j): {241, 75}, {478, 9503}, {676, 24026}, {23972, 36796}, {39063, 18025}, {39077, 8}, {40869, 312}
X(53547) = crossdifference of every pair of points on line {294, 2338}
X(53547) = barycentric product X(i)*X(j) for these {i,j}: {1, 39063}, {7, 9502}, {57, 50441}, {241, 516}, {518, 43035}, {676, 1025}, {910, 9436}, {926, 24015}, {1456, 3912}, {1458, 30807}, {1566, 7045}, {1876, 26006}, {24014, 52213}, {34855, 40869}, {35517, 52635}
X(53547) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 9503}, {241, 18025}, {516, 36796}, {910, 14942}, {1456, 673}, {1458, 36101}, {1566, 24026}, {1876, 52781}, {2223, 2338}, {9502, 8}, {23973, 34085}, {24015, 46135}, {34855, 52156}, {39063, 75}, {41339, 6559}, {43035, 2481}, {47422, 34591}, {50441, 312}, {52635, 103}

---

X(53548) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1361) AND X(1362)

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

X(53548) lies on these lines: {1, 185}, {7, 39898}, {12, 21252}, {34, 16980}, {51, 34036}, {55, 53291}, {56, 692}, {65, 1086}, {104, 52830}, {109, 3937}, {145, 19367}, {221, 8192}, {388, 21293}, {511, 4318}, {517, 22464}, {518, 1861}, {651, 2810}, {926, 1362}, {942, 38055}, {1066, 1410}, {1145, 45122}, {1317, 3028}, {1361, 1769}, {1401, 9316}, {1456, 8679}, {1457, 2183}, {1458, 2223}, {1461, 44858}, {1465, 51377}, {1469, 2263}, {1483, 32143}, {1813, 36942}, {2099, 2875}, {3321, 43932}, {3917, 8270}, {4260, 7672}, {4551, 53397}, {5173, 40952}, {7967, 19368}, {11491, 40644}, {14717, 19366}, {15507, 24029}, {15888, 20617}, {23706, 42072}, {30284, 50658}, {32062, 36985}, {34040, 42448}, {45966, 52029}

X(53548) = reflection of X(3270) in X(1)
X(53548) = X(34230)-Ceva conjugate of X(1458)
X(53548) = X(i)-isoconjugate of X(j) for these (i,j): {104, 14942}, {105, 51565}, {294, 34234}, {673, 52663}, {885, 36037}, {909, 36796}, {1024, 13136}, {1309, 23696}, {1438, 36795}, {1809, 36124}, {2195, 18816}, {2342, 2481}, {6559, 34051}, {28132, 37136}, {36086, 43728}
X(53548) = X(i)-Dao conjugate of X(j) for these (i,j): {3259, 885}, {6184, 36795}, {23980, 36796}, {38989, 43728}, {39046, 51565}, {39063, 18816}, {40613, 14942}
X(53548) = crossdifference of every pair of points on line {294, 28132}
X(53548) = barycentric product X(i)*X(j) for these {i,j}: {56, 51390}, {59, 42770}, {241, 517}, {518, 1465}, {651, 42758}, {672, 22464}, {883, 3310}, {908, 1458}, {918, 23981}, {1025, 1769}, {1275, 42771}, {1457, 3912}, {1875, 25083}, {2183, 9436}, {2254, 24029}, {2283, 10015}, {2427, 43042}, {3262, 52635}, {5236, 22350}, {34230, 52659}, {41353, 46393}
X(53548) = barycentric quotient X(i)/X(j) for these {i,j}: {241, 18816}, {517, 36796}, {518, 36795}, {665, 43728}, {672, 51565}, {1457, 673}, {1458, 34234}, {1465, 2481}, {1876, 16082}, {2183, 14942}, {2223, 52663}, {2283, 13136}, {2427, 36802}, {3310, 885}, {9454, 2342}, {20752, 1809}, {22464, 18031}, {23981, 666}, {24029, 51560}, {42758, 4391}, {42770, 34387}, {42771, 1146}, {51390, 3596}, {52635, 104}

---

X(53549) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1361) AND X(3022)

Barycentrics    a^2*(a - b - c)*(b - c)*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3) : :
X(53549) = 2 X[65] - 3 X[30691], 4 X[676] - 3 X[30691], 3 X[663] - 2 X[2488], 3 X[210] - 2 X[4528], 3 X[3877] - X[3904]

X(53549) lies on these lines: {1, 928}, {35, 52739}, {36, 52730}, {55, 53285}, {56, 53300}, {65, 676}, {187, 237}, {210, 4528}, {513, 30725}, {517, 10015}, {520, 48303}, {654, 2342}, {692, 2498}, {758, 48286}, {885, 52029}, {891, 21132}, {900, 17638}, {924, 48302}, {926, 2170}, {960, 50333}, {1334, 52614}, {1361, 1769}, {1938, 47123}, {2254, 2821}, {2849, 51643}, {3057, 6366}, {3064, 3700}, {3310, 42078}, {3738, 15558}, {3869, 47695}, {3877, 3904}, {3878, 23887}, {4162, 8676}, {4524, 4814}, {6371, 6615}, {9521, 14110}, {14299, 23757}, {20958, 21742}, {21343, 47329}, {44410, 48294}, {53256, 53313}

X(53549) = midpoint of X(i) and X(j) for these {i,j}: {3869, 47695}, {14299, 23757}
X(53549) = reflection of X(i) in X(j) for these {i,j}: {65, 676}, {4814, 4524}, {44410, 48294}, {50333, 960}
X(53549) = isogonal conjugate of the isotomic conjugate of X(2804)
X(53549) = polar conjugate of the isotomic conjugate of X(52307)
X(53549) = X(i)-Ceva conjugate of X(j) for these (i,j): {104, 7117}, {108, 47434}, {1769, 3310}, {2161, 14936}, {2432, 657}, {2720, 6}, {2804, 52307}, {3445, 47420}, {23981, 2183}, {32675, 41}, {32698, 52425}, {32719, 23638}, {43728, 650}
X(53549) = X(i)-isoconjugate of X(j) for these (i,j): {2, 37136}, {7, 36037}, {57, 13136}, {69, 36110}, {75, 2720}, {76, 32669}, {77, 1309}, {85, 32641}, {104, 664}, {109, 18816}, {190, 34051}, {304, 32702}, {651, 34234}, {658, 52663}, {905, 39294}, {909, 4554}, {927, 36819}, {934, 51565}, {1414, 38955}, {1461, 36795}, {1795, 18026}, {1809, 36118}, {1813, 16082}, {2250, 4573}, {2342, 4569}, {2401, 4564}, {3219, 47317}, {3257, 40218}, {4572, 34858}, {6516, 36123}, {7045, 43728}, {7182, 14776}, {14578, 46404}
X(53549) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 18816}, {206, 2720}, {1145, 668}, {3259, 7}, {5452, 13136}, {14714, 51565}, {16586, 4572}, {17115, 43728}, {23757, 35519}, {23980, 4554}, {25640, 18026}, {32664, 37136}, {35014, 3262}, {35508, 36795}, {38981, 75}, {38991, 34234}, {39004, 69}, {39025, 104}, {40608, 38955}, {40613, 664}, {45247, 4555}, {46398, 6063}
X(53549) = crossdifference of every pair of points on line {2, 222}
X(53549) = barycentric product X(i)*X(j) for these {i,j}: {1, 46393}, {4, 52307}, {6, 2804}, {8, 3310}, {9, 1769}, {11, 2427}, {41, 36038}, {55, 10015}, {59, 52316}, {80, 53046}, {101, 35015}, {219, 39534}, {281, 8677}, {294, 42758}, {517, 650}, {521, 14571}, {522, 2183}, {644, 42753}, {645, 42752}, {649, 6735}, {652, 1785}, {657, 22464}, {663, 908}, {666, 42771}, {859, 3700}, {884, 51390}, {1146, 23981}, {1334, 23788}, {1457, 3239}, {1465, 3900}, {1639, 14260}, {1783, 35014}, {2265, 37629}, {2310, 24029}, {2311, 42767}, {2316, 23757}, {2338, 42756}, {2341, 42768}, {2397, 3271}, {2431, 25640}, {3063, 3262}, {3064, 22350}, {3259, 5548}, {3326, 32641}, {3669, 51380}, {3709, 17139}, {3737, 21801}, {3939, 42754}, {4559, 14010}, {4560, 51377}, {4845, 42762}, {4895, 52031}, {5546, 42759}, {5547, 42760}, {6187, 53045}, {6591, 51379}, {7017, 23220}, {7117, 53151}, {7252, 17757}, {15627, 42750}, {15628, 42751}, {15629, 42755}, {23706, 34591}, {23980, 43728}, {32647, 52114}, {42757, 52663}, {42770, 52927}, {43737, 47434}, {50333, 51987}, {52212, 53285}, {52431, 53047}
X(53549) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 37136}, {32, 2720}, {41, 36037}, {55, 13136}, {517, 4554}, {560, 32669}, {607, 1309}, {650, 18816}, {657, 51565}, {663, 34234}, {667, 34051}, {859, 4573}, {908, 4572}, {1457, 658}, {1465, 4569}, {1769, 85}, {1785, 46404}, {1875, 13149}, {1960, 40218}, {1973, 36110}, {1974, 32702}, {2175, 32641}, {2183, 664}, {2427, 4998}, {2804, 76}, {3063, 104}, {3271, 2401}, {3310, 7}, {3709, 38955}, {3900, 36795}, {6186, 47317}, {6735, 1978}, {8641, 52663}, {8677, 348}, {8750, 39294}, {10015, 6063}, {14571, 18026}, {14936, 43728}, {18344, 16082}, {22464, 46406}, {23220, 222}, {23981, 1275}, {35014, 15413}, {35015, 3261}, {36038, 20567}, {39534, 331}, {42078, 24029}, {42752, 7178}, {42753, 24002}, {42754, 52621}, {42758, 40704}, {42771, 918}, {46388, 36819}, {46393, 75}, {51377, 4552}, {51380, 646}, {51987, 927}, {52307, 69}, {52316, 34387}, {53045, 40075}, {53046, 320}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {65, 676, 30691}, {663, 8648, 1960}, {663, 42657, 8648}, {1960, 6139, 8648}, {4775, 8641, 663}, {8648, 42657, 6139}

---

X(53550) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1362) AND X(1364)

Barycentrics    a^2*(b - c)*(a^2 - b^2 - c^2)*(a*b - b^2 + a*c - c^2) : :
X(53550) = 3 X[354] - 2 X[676], 4 X[942] - 3 X[30691], 2 X[10015] - 3 X[30691], 2 X[4458] - 3 X[30704], 3 X[3873] - X[47695], 2 X[4524] - 3 X[47828]

X(53550) lies on these lines: {1, 928}, {35, 52730}, {36, 52739}, {55, 53300}, {56, 53285}, {65, 6366}, {296, 37628}, {354, 676}, {512, 4378}, {513, 11934}, {514, 44410}, {518, 50333}, {520, 647}, {521, 4025}, {525, 23732}, {650, 9000}, {654, 53301}, {659, 47329}, {663, 39541}, {672, 52614}, {834, 4790}, {891, 21105}, {900, 17660}, {924, 48283}, {926, 1362}, {942, 10015}, {1331, 1813}, {1364, 3270}, {1946, 4091}, {2488, 4724}, {2499, 48021}, {2526, 9029}, {2774, 3960}, {2821, 4895}, {3250, 23569}, {3669, 8676}, {3738, 4458}, {3868, 3904}, {3873, 47695}, {3874, 23887}, {3881, 48286}, {4088, 42341}, {4524, 47828}, {6371, 50521}, {6372, 21118}, {10974, 32679}, {14300, 45745}, {20293, 24622}, {20758, 22092}, {20770, 22155}, {20778, 23225}, {20797, 23188}, {21189, 34954}, {22345, 23220}, {22346, 22379}, {22388, 23224}, {22458, 23184}, {23093, 23187}, {23684, 24349}, {36056, 36057}

X(53550) = midpoint of X(3868) and X(3904)
X(53550) = reflection of X(i) in X(j) for these {i,j}: {663, 39541}, {4724, 2488}, {10015, 942}, {21189, 34954}, {42763, 18839}, {48021, 2499}, {48286, 3881}
X(53550) = reflection of X(42763) in the line X(1)X(3)e
X(53550) = isotomic conjugate of the isogonal conjugate of X(23225)
X(53550) = isotomic conjugate of the polar conjugate of X(665)
X(53550) = isogonal conjugate of the polar conjugate of X(918)
X(53550) = X(i)-Ceva conjugate of X(j) for these (i,j): {295, 3937}, {918, 665}, {1331, 20749}, {1332, 20728}, {1814, 7117}, {36056, 1364}
X(53550) = X(i)-isoconjugate of X(j) for these (i,j): {4, 36086}, {19, 666}, {25, 51560}, {33, 927}, {34, 36802}, {92, 919}, {100, 36124}, {105, 1897}, {108, 14942}, {162, 13576}, {190, 8751}, {264, 32666}, {273, 52927}, {281, 36146}, {294, 653}, {318, 32735}, {607, 34085}, {648, 18785}, {673, 1783}, {885, 7012}, {1024, 46102}, {1027, 15742}, {1438, 6335}, {1973, 36803}, {2195, 18026}, {2212, 46135}, {2481, 8750}, {3434, 36111}, {5377, 7649}, {6559, 32714}, {7128, 28132}, {9503, 41321}, {18344, 39293}, {20927, 32703}, {28071, 36118}, {32085, 35333}, {32674, 36796}, {36106, 52456}
X(53550) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 666}, {125, 13576}, {3126, 44426}, {6184, 6335}, {6337, 36803}, {6505, 51560}, {8054, 36124}, {11517, 36802}, {17435, 46108}, {22391, 919}, {26932, 2481}, {27918, 40717}, {34467, 105}, {35072, 36796}, {35094, 264}, {36033, 36086}, {36905, 46404}, {38980, 92}, {38983, 14942}, {38989, 4}, {39002, 52456}, {39006, 673}, {39014, 281}, {39046, 1897}, {39063, 18026}, {40618, 18031}
X(53550) = crossdifference of every pair of points on line {4, 218}
X(53550) = barycentric product X(i)*X(j) for these {i,j}: {3, 918}, {63, 2254}, {69, 665}, {71, 23829}, {76, 23225}, {219, 43042}, {222, 50333}, {241, 521}, {348, 926}, {513, 25083}, {514, 1818}, {518, 905}, {520, 15149}, {525, 3286}, {647, 30941}, {652, 9436}, {656, 18206}, {672, 4025}, {693, 20752}, {810, 18157}, {883, 7117}, {1025, 7004}, {1026, 3942}, {1332, 3675}, {1444, 24290}, {1458, 6332}, {1459, 3912}, {1565, 2284}, {1790, 4088}, {1814, 3126}, {1861, 4091}, {1946, 40704}, {2223, 15413}, {2283, 26932}, {2356, 30805}, {3263, 22383}, {3932, 7254}, {3937, 42720}, {4131, 5089}, {4444, 20778}, {6516, 17435}, {7056, 52614}, {7182, 46388}, {10099, 16728}, {15419, 20683}, {22384, 40217}, {23224, 46108}, {34591, 41353}, {35518, 52635}, {44717, 52305}
X(53550) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 666}, {48, 36086}, {63, 51560}, {69, 36803}, {77, 34085}, {184, 919}, {219, 36802}, {222, 927}, {241, 18026}, {348, 46135}, {518, 6335}, {521, 36796}, {603, 36146}, {647, 13576}, {649, 36124}, {652, 14942}, {665, 4}, {667, 8751}, {672, 1897}, {810, 18785}, {905, 2481}, {906, 5377}, {918, 264}, {926, 281}, {1458, 653}, {1459, 673}, {1813, 39293}, {1818, 190}, {1946, 294}, {2223, 1783}, {2254, 92}, {2283, 46102}, {2284, 15742}, {3126, 46108}, {3270, 28132}, {3286, 648}, {3675, 17924}, {4020, 35333}, {4025, 18031}, {4091, 31637}, {5236, 52938}, {7117, 885}, {8638, 607}, {9247, 32666}, {9436, 46404}, {9454, 8750}, {15149, 6528}, {17435, 44426}, {18206, 811}, {20728, 53358}, {20749, 53337}, {20752, 100}, {20775, 46163}, {20776, 2284}, {20778, 3570}, {22055, 35313}, {22096, 43929}, {22383, 105}, {22384, 6654}, {23220, 51987}, {23224, 1814}, {23225, 6}, {23829, 44129}, {24290, 41013}, {25083, 668}, {30941, 6331}, {34855, 13149}, {43042, 331}, {46388, 33}, {50333, 7017}, {52411, 32735}, {52425, 52927}, {52614, 7046}, {52635, 108}
X(53550) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {942, 10015, 30691}, {53301, 53308, 654}

---

X(53551) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1362) AND X(1365)

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

X(53551) lies on these lines: {1, 51642}, {7, 50343}, {12, 14431}, {56, 14419}, {57, 9508}, {65, 4730}, {226, 4010}, {388, 2787}, {512, 51664}, {513, 2078}, {523, 4077}, {647, 661}, {656, 4524}, {659, 43050}, {764, 29240}, {832, 43924}, {926, 1362}, {1020, 4551}, {1365, 2611}, {1491, 3676}, {1617, 53309}, {2526, 2530}, {2775, 3295}, {2812, 3960}, {3126, 43042}, {3777, 30719}, {4041, 7216}, {4705, 7178}, {4707, 18006}, {4922, 10106}, {5083, 13277}, {5261, 30709}, {6004, 51652}, {6129, 17115}, {7234, 51662}, {8674, 37736}, {9391, 42666}, {20680, 24290}, {20999, 51643}, {24002, 47975}, {30723, 48066}, {31605, 48039}, {34958, 39212}, {37579, 42670}, {47700, 50554}

X(53551) = reflection of X(53400) in X(9508)
X(53551) = X(i)-Ceva conjugate of X(j) for these (i,j): {1458, 3675}, {5236, 17435}
X(53551) = X(i)-isoconjugate of X(j) for these (i,j): {21, 36086}, {58, 36802}, {86, 52927}, {99, 2195}, {105, 643}, {110, 14942}, {163, 36796}, {284, 666}, {294, 662}, {314, 32666}, {333, 919}, {645, 1438}, {673, 5546}, {884, 4600}, {885, 4570}, {927, 2328}, {1024, 4567}, {1043, 32735}, {1414, 28071}, {1416, 7256}, {1462, 7259}, {2194, 51560}, {2287, 36146}, {3737, 5377}, {4565, 6559}, {4612, 18785}, {4636, 13576}, {5379, 23696}, {21789, 39293}, {28132, 52378}, {36057, 36797}
X(53551) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 36802}, {115, 36796}, {244, 14942}, {1084, 294}, {1214, 51560}, {3126, 7253}, {6184, 645}, {17755, 7257}, {20621, 36797}, {35094, 314}, {36905, 799}, {36908, 927}, {38980, 333}, {38986, 2195}, {38989, 21}, {39014, 2287}, {39046, 643}, {39063, 99}, {40590, 666}, {40600, 52927}, {40608, 28071}, {40609, 7256}, {40611, 36086}, {40622, 2481}, {40627, 1024}, {50330, 885}, {50497, 884}
X(53551) = crossdifference of every pair of points on line {21, 294}
X(53551) = barycentric product X(i)*X(j) for these {i,j}: {7, 24290}, {37, 43042}, {57, 4088}, {65, 918}, {226, 2254}, {241, 523}, {512, 40704}, {518, 7178}, {525, 1876}, {656, 5236}, {661, 9436}, {665, 1441}, {672, 4077}, {850, 52635}, {883, 3125}, {926, 1446}, {1025, 3120}, {1427, 50333}, {1458, 1577}, {1861, 51664}, {2171, 23829}, {2283, 16732}, {3263, 7180}, {3669, 3932}, {3675, 4552}, {3676, 3930}, {3700, 34855}, {3717, 7216}, {3912, 4017}, {4566, 17435}, {4729, 10029}, {5089, 17094}, {7153, 21959}, {7212, 22116}, {20683, 24002}, {21044, 41353}, {23225, 52575}, {34253, 35352}, {39258, 52621}
X(53551) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 36802}, {65, 666}, {213, 52927}, {226, 51560}, {241, 99}, {512, 294}, {518, 645}, {523, 36796}, {661, 14942}, {665, 21}, {672, 643}, {798, 2195}, {883, 4601}, {918, 314}, {926, 2287}, {1020, 39293}, {1025, 4600}, {1042, 36146}, {1400, 36086}, {1402, 919}, {1427, 927}, {1441, 36803}, {1446, 46135}, {1458, 662}, {1876, 648}, {2223, 5546}, {2254, 333}, {2283, 4567}, {2340, 7259}, {3121, 884}, {3122, 1024}, {3125, 885}, {3286, 4612}, {3668, 34085}, {3675, 4560}, {3693, 7256}, {3709, 28071}, {3717, 7258}, {3912, 7257}, {3930, 3699}, {3932, 646}, {4017, 673}, {4041, 6559}, {4077, 18031}, {4088, 312}, {4516, 28132}, {4559, 5377}, {5089, 36797}, {5236, 811}, {7178, 2481}, {7180, 105}, {7250, 1462}, {9436, 799}, {17435, 7253}, {20683, 644}, {21959, 4110}, {23225, 2193}, {23829, 52379}, {24290, 8}, {30941, 4631}, {34855, 4573}, {39258, 3939}, {40704, 670}, {41353, 4620}, {43042, 274}, {46388, 2328}, {51641, 1438}, {51664, 31637}, {52635, 110}

---

X(53552) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1362) AND X(3021)

Barycentrics    a*(a*b - b^2 + a*c - c^2)*(2*a^2 - a*b + b^2 - a*c - 2*b*c + c^2) : :
X(53552) = 2 X[4712] - 3 X[14439], 4 X[8299] - 3 X[14439]

X(53552) lies on the de Longchamnps ellipse, the cubic K981, and these lines: {1, 41}, {6, 46178}, {7, 36905}, {8, 21232}, {38, 55}, {42, 244}, {57, 100}, {145, 3212}, {210, 44304}, {291, 1002}, {518, 672}, {604, 8271}, {678, 1155}, {891, 3251}, {910, 3726}, {926, 1362}, {1054, 10980}, {1279, 2348}, {1317, 1360}, {1400, 15185}, {1423, 30628}, {1617, 23067}, {1697, 53409}, {1962, 2611}, {2108, 49498}, {2310, 21320}, {2347, 14523}, {2650, 4128}, {2976, 3021}, {3027, 4804}, {3059, 28351}, {3174, 28017}, {3315, 5256}, {3675, 35293}, {3935, 31073}, {4981, 24542}, {6601, 28081}, {7674, 28079}, {8298, 49675}, {8301, 42871}, {9318, 14942}, {10389, 28606}, {10459, 28394}, {15733, 52896}, {16496, 41423}, {16593, 40609}, {17441, 17642}, {18031, 35961}, {19860, 25920}, {20504, 29240}, {20662, 20680}, {21139, 28850}, {21346, 34247}, {22346, 37578}, {26234, 49466}, {27626, 34784}, {33115, 33171}, {34855, 41355}, {49448, 52155}, {50744, 51583}

X(53552) = midpoint of X(145) and X(21272)
X(53552) = reflection of X(i) in X(j) for these {i,j}: {8, 21232}, {2170, 1}, {4712, 8299}
X(53552) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 1279}, {57, 672}, {100, 2254}, {4564, 23704}, {40526, 665}
X(53552) = X(i)-isoconjugate of X(j) for these (i,j): {105, 1280}, {294, 43760}, {513, 39272}, {1438, 36807}, {1477, 14942}, {1810, 36124}, {2195, 35160}, {35355, 36086}
X(53552) = X(i)-Dao conjugate of X(j) for these (i,j): {3008, 75}, {3693, 312}, {4925, 4939}, {6184, 36807}, {16593, 2481}, {35111, 36796}, {38989, 35355}, {39026, 39272}, {39046, 1280}, {39048, 673}, {39063, 35160}
X(53552) = crossdifference of every pair of points on line {294, 1027}
X(53552) = X(i)-line-conjugate of X(j) for these (i,j): {1, 294}, {926, 2254}
X(53552) = barycentric product X(i)*X(j) for these {i,j}: {1, 16593}, {57, 40609}, {75, 20662}, {86, 20680}, {92, 20749}, {241, 5853}, {518, 3008}, {1026, 6084}, {1279, 3912}, {2254, 53337}, {2348, 9436}, {4712, 52210}, {4899, 51839}, {8647, 40704}, {20780, 46108}, {23704, 43042}, {36819, 51419}, {42720, 48032}
X(53552) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 39272}, {241, 35160}, {518, 36807}, {665, 35355}, {672, 1280}, {1279, 673}, {1458, 43760}, {2348, 14942}, {3008, 2481}, {5853, 36796}, {8647, 294}, {8659, 1027}, {16593, 75}, {20662, 1}, {20680, 10}, {20749, 63}, {20752, 1810}, {20780, 1814}, {23704, 36802}, {40609, 312}, {52635, 1477}, {53337, 51560}
X(53552) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1282, 105}, {100, 1280, 9451}, {105, 1282, 2246}, {3243, 9451, 1280}, {4712, 8299, 14439}

---

X(53553) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1362) AND X(3023)

Barycentrics    a*(b - c)*(a^2 + b*c)*(a*b - b^2 + a*c - c^2) : :
X(53553) = 3 X[14419] - 2 X[17990], 4 X[17066] - 3 X[21052], 2 X[21349] - 3 X[52745]

X(53553) lies on these lines: {1, 2786}, {38, 21828}, {42, 4750}, {43, 45674}, {55, 53269}, {56, 53258}, {63, 5075}, {354, 21894}, {522, 4318}, {663, 28846}, {665, 42341}, {798, 9040}, {804, 3023}, {926, 1362}, {3239, 24666}, {3287, 3805}, {3572, 21832}, {3720, 4120}, {3907, 4374}, {3912, 21959}, {3960, 24462}, {4025, 23655}, {4040, 28855}, {4107, 18787}, {4502, 48136}, {4724, 28878}, {7658, 24749}, {14419, 17990}, {17018, 53333}, {17066, 21052}, {21272, 35338}, {21349, 52745}, {21831, 29090}, {26102, 45661}, {28372, 50494}, {28906, 48294}, {29078, 48283}, {29814, 53339}

X(53553) = reflection of X(i) in X(j) for these {i,j}: {4502, 48136}, {24462, 3960}
X(53553) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2665, 33650}, {51333, 37781}
X(53553) = X(i)-isoconjugate of X(j) for these (i,j): {105, 3903}, {256, 36086}, {257, 919}, {294, 37137}, {666, 893}, {904, 51560}, {1431, 36802}, {1438, 27805}, {4451, 32735}, {4603, 18785}, {7018, 32666}, {7104, 36803}, {7249, 52927}, {14942, 29055}, {30670, 52029}
X(53553) = X(i)-Dao conjugate of X(j) for these (i,j): {6184, 27805}, {16592, 2481}, {35094, 7018}, {38980, 257}, {38989, 256}, {39046, 3903}, {40597, 666}
X(53553) = crossdifference of every pair of points on line {256, 294}
X(53553) = barycentric product X(i)*X(j) for these {i,j}: {171, 918}, {241, 3907}, {514, 4447}, {518, 4369}, {665, 1909}, {672, 4374}, {894, 2254}, {926, 7196}, {1025, 4459}, {1026, 7200}, {2295, 23829}, {2329, 43042}, {2533, 18206}, {3252, 14296}, {3263, 20981}, {3287, 9436}, {3675, 18047}, {3912, 4367}, {3930, 17212}, {3932, 18200}, {4107, 22116}, {4164, 40217}, {4529, 34855}, {6649, 17435}, {7175, 50333}, {7205, 46388}, {7234, 18157}, {16737, 20683}, {17103, 24290}, {22093, 46108}
X(53553) = barycentric quotient X(i)/X(j) for these {i,j}: {171, 666}, {172, 36086}, {518, 27805}, {665, 256}, {672, 3903}, {894, 51560}, {918, 7018}, {1458, 37137}, {1909, 36803}, {2254, 257}, {2329, 36802}, {3286, 4603}, {3287, 14942}, {3907, 36796}, {4164, 6654}, {4367, 673}, {4369, 2481}, {4374, 18031}, {4447, 190}, {4477, 6559}, {7122, 919}, {7175, 927}, {7176, 34085}, {7196, 46135}, {7234, 18785}, {18206, 4594}, {20981, 105}, {22093, 1814}, {23225, 7116}, {30941, 7260}, {45882, 52029}, {52635, 29055}

---

X(53554) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1362) AND X(3024)

Barycentrics    a^2*(b - c)*(a*b - b^2 + a*c - c^2)*(a^2 - b^2 - b*c - c^2) : :
X(53554) = 2 X[72] - 3 X[14432], 4 X[942] - 3 X[30574], 4 X[2488] - 3 X[47811], 8 X[3812] - 7 X[21952], 3 X[3873] - 2 X[4458]

X(53554) lies on these lines: {1, 2774}, {36, 44827}, {55, 53301}, {56, 53249}, {57, 53398}, {72, 14432}, {512, 48149}, {513, 47704}, {518, 4088}, {526, 2611}, {900, 27778}, {926, 1362}, {928, 4895}, {942, 30574}, {1468, 42662}, {2488, 47811}, {2605, 9404}, {2785, 3868}, {3295, 42657}, {3738, 47695}, {3812, 21952}, {3873, 4458}, {3874, 4707}, {4041, 44410}, {4449, 8676}, {4467, 35057}, {4724, 9000}, {8674, 11670}, {9001, 47972}, {9029, 48023}, {10453, 53336}, {13401, 48277}, {15313, 23726}, {24353, 31330}, {37998, 46403}, {47329, 48032}

X(53554) = reflection of X(i) in X(j) for these {i,j}: {4041, 44410}, {4707, 3874}
X(53554) = X(i)-isoconjugate of X(j) for these (i,j): {79, 36086}, {105, 6742}, {294, 38340}, {666, 2160}, {919, 30690}, {927, 7073}, {1438, 15455}, {6186, 51560}, {7110, 36146}, {13486, 13576}, {14942, 26700}, {20565, 32666}, {23696, 34922}, {32735, 52344}, {36802, 52372}
X(53554) = X(i)-Dao conjugate of X(j) for these (i,j): {6184, 15455}, {8287, 2481}, {35094, 20565}, {38980, 30690}, {38989, 79}, {39014, 7110}, {39046, 6742}
X(53554) = crossdifference of every pair of points on line {79, 294}
X(53554) = barycentric product X(i)*X(j) for these {i,j}: {35, 918}, {241, 35057}, {319, 665}, {518, 14838}, {672, 4467}, {926, 17095}, {1026, 7202}, {2003, 50333}, {2223, 18160}, {2254, 3219}, {2605, 3912}, {3286, 7265}, {4088, 40214}, {9404, 9436}, {16755, 20683}, {23226, 46108}, {43042, 52405}, {46388, 52421}
X(53554) = barycentric quotient X(i)/X(j) for these {i,j}: {35, 666}, {319, 36803}, {518, 15455}, {665, 79}, {672, 6742}, {918, 20565}, {926, 7110}, {1399, 36146}, {1442, 34085}, {1458, 38340}, {2003, 927}, {2174, 36086}, {2254, 30690}, {2605, 673}, {3219, 51560}, {4467, 18031}, {9404, 14942}, {14838, 2481}, {17095, 46135}, {23226, 1814}, {24290, 6757}, {35057, 36796}, {46388, 7073}, {52405, 36802}, {52635, 26700}

---

X(53555) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1362) AND X(3025)

Barycentrics    a^2*(b - c)*(a*b - b^2 + a*c - c^2)*(a^2 - b^2 + b*c - c^2) : :
X(53555) = 2 X[65] + X[21105], 4 X[942] - X[21132], X[6615] - 4 X[34954], 5 X[18398] - 2 X[21201], 2 X[44410] + X[48151]

X(53555) lies on these lines: {31, 1459}, {57, 53401}, {65, 21105}, {222, 35365}, {354, 513}, {512, 14421}, {514, 5902}, {517, 30573}, {522, 3873}, {649, 2280}, {654, 17455}, {663, 999}, {834, 38349}, {926, 1362}, {928, 14413}, {942, 21132}, {1468, 8578}, {1475, 14825}, {1635, 47329}, {2099, 4449}, {2821, 23057}, {3738, 4453}, {4724, 4860}, {4977, 42454}, {6006, 7671}, {6371, 8027}, {6615, 34954}, {8676, 32065}, {9000, 47828}, {9001, 47886}, {18398, 21201}, {20316, 25961}, {21173, 32913}, {23087, 51646}, {30613, 35057}, {32856, 42766}, {44410, 48151}, {49999, 52627}

X(53555) = X(i)-isoconjugate of X(j) for these (i,j): {80, 36086}, {105, 51562}, {294, 655}, {666, 2161}, {885, 52377}, {919, 18359}, {927, 52371}, {1411, 36802}, {1438, 36804}, {2195, 35174}, {2222, 14942}, {6187, 51560}, {18785, 47318}, {18815, 52927}, {20566, 32666}, {32675, 36796}, {32735, 52409}, {36146, X(53555) = 36910}
X(53555) = X(i)-Dao conjugate of X(j) for these (i,j): {3126, 52356}, {6184, 36804}, {35094, 20566}, {35128, 36796}, {35204, 36802}, {36905, 46405}, {38980, 18359}, {38984, 14942}, {38989, 80}, {39014, 36910}, {39046, 51562}, {39063, 35174}, {40584, 666}, {40612, 51560}
X(53555) = crossdifference of every pair of points on line {80, 294}
X(53555) = barycentric product X(i)*X(j) for these {i,j}: {36, 918}, {241, 3738}, {320, 665}, {518, 3960}, {654, 9436}, {672, 4453}, {926, 17078}, {1458, 3904}, {2245, 23829}, {2254, 3218}, {2284, 4089}, {2323, 43042}, {3263, 21758}, {3286, 4707}, {3675, 4585}, {3912, 53314}, {8648, 40704}, {21828, 30941}, {22379, 46108}
X(53555) = barycentric quotient X(i)/X(j) for these {i,j}: {36, 666}, {241, 35174}, {320, 36803}, {518, 36804}, {654, 14942}, {665, 80}, {672, 51562}, {918, 20566}, {926, 36910}, {1443, 34085}, {1458, 655}, {1983, 5377}, {2254, 18359}, {2323, 36802}, {3218, 51560}, {3286, 47318}, {3738, 36796}, {3960, 2481}, {4453, 18031}, {7113, 36086}, {8648, 294}, {9436, 46405}, {17078, 46135}, {17435, 52356}, {21758, 105}, {21828, 13576}, {22379, 1814}, {23225, 52431}, {24290, 15065}, {46388, 52371}, {52426, 52927}, {52434, 919}, {52440, 36146}, {52635, 2222}, {53285, 6559}, {53314, 673}

---

X(53556) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1364) AND X(3027)

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

X(53556) lies on these lines: {1, 2785}, {42, 4088}, {55, 53262}, {75, 24353}, {192, 53336}, {513, 23740}, {521, 25098}, {525, 810}, {647, 656}, {659, 4435}, {663, 21124}, {804, 3027}, {1193, 14432}, {1364, 3270}, {1459, 4025}, {1577, 21831}, {2254, 21828}, {3566, 51641}, {3737, 21196}, {4040, 47679}, {4064, 4580}, {4449, 23755}, {4724, 4988}, {6332, 22090}, {7174, 53398}, {7178, 17478}, {8062, 24622}, {8672, 22216}, {14429, 52609}, {19767, 49274}, {20778, 22384}, {21121, 48302}, {23752, 47123}, {23757, 38360}, {30115, 44827}, {42043, 45344}, {45745, 46385}, {48006, 48340}, {48342, 49296}

X(53556) = isotomic conjugate of the polar conjugate of X(21832)
X(53556) = X(i)-Ceva conjugate of X(j) for these (i,j): {3716, 4010}, {10099, 656}, {31637, 4466}
X(53556) = X(i)-isoconjugate of X(j) for these (i,j): {19, 4584}, {25, 4589}, {27, 813}, {28, 660}, {107, 295}, {112, 335}, {162, 291}, {250, 35352}, {286, 34067}, {292, 648}, {334, 32676}, {337, 32713}, {653, 2311}, {741, 1897}, {805, 7009}, {811, 1911}, {823, 2196}, {876, 5379}, {1474, 4562}, {1783, 37128}, {1808, 36127}, {1824, 36066}, {1922, 6331}, {1973, 4639}, {2203, 4583}, {4238, 52030}, {6335, 18268}, {7119, 37134}, {8750, 18827}, {32674, 36800}, {41072, 46503}
X(53556) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 4584}, {125, 291}, {6337, 4639}, {6505, 4589}, {6651, 811}, {8299, 1897}, {15526, 334}, {16591, 18026}, {17423, 1911}, {19557, 648}, {26932, 18827}, {34467, 741}, {34591, 335}, {35068, 6335}, {35072, 36800}, {35119, 286}, {38978, 1824}, {38985, 295}, {39006, 37128}, {39028, 6331}, {39029, 162}, {40591, 660}, {40618, 40017}, {40623, 27}, {51574, 4562}
X(53556) = crossdifference of every pair of points on line {28, 291}
X(53556) = barycentric product X(i)*X(j) for these {i,j}: {1, 24459}, {63, 4010}, {69, 21832}, {71, 3766}, {72, 812}, {78, 7212}, {238, 525}, {239, 656}, {242, 24018}, {256, 24284}, {304, 4455}, {306, 659}, {307, 4435}, {321, 22384}, {350, 647}, {521, 16609}, {523, 20769}, {740, 905}, {810, 1921}, {822, 40717}, {862, 30805}, {1214, 3716}, {1284, 6332}, {1429, 52355}, {1439, 4148}, {1447, 8611}, {1459, 3948}, {1577, 7193}, {1914, 14208}, {2201, 3265}, {2210, 3267}, {2238, 4025}, {3049, 18891}, {3570, 18210}, {3573, 4466}, {3684, 17094}, {3685, 51664}, {3694, 43041}, {3695, 50456}, {3747, 15413}, {4155, 17206}, {4561, 39786}, {7116, 14295}, {8632, 20336}, {10099, 17755}, {22383, 35544}, {27846, 52609}
X(53556) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 4584}, {63, 4589}, {69, 4639}, {71, 660}, {72, 4562}, {228, 813}, {238, 648}, {239, 811}, {242, 823}, {306, 4583}, {332, 36806}, {350, 6331}, {521, 36800}, {525, 334}, {647, 291}, {656, 335}, {659, 27}, {740, 6335}, {810, 292}, {812, 286}, {822, 295}, {905, 18827}, {1284, 653}, {1459, 37128}, {1790, 36066}, {1914, 162}, {1946, 2311}, {2200, 34067}, {2201, 107}, {2210, 112}, {2238, 1897}, {3049, 1911}, {3267, 44172}, {3684, 36797}, {3694, 36801}, {3708, 35352}, {3716, 31623}, {3747, 1783}, {3766, 44129}, {3808, 31917}, {4010, 92}, {4025, 40017}, {4155, 1826}, {4435, 29}, {4455, 19}, {4574, 5378}, {4839, 5342}, {5027, 7119}, {7015, 37134}, {7116, 805}, {7193, 662}, {7212, 273}, {8611, 4518}, {8632, 28}, {10099, 52209}, {14208, 18895}, {14599, 32676}, {16609, 18026}, {18210, 4444}, {20769, 99}, {21832, 4}, {22383, 741}, {22384, 81}, {24018, 337}, {24284, 1909}, {24459, 75}, {27846, 17925}, {30665, 31909}, {36054, 1808}, {38348, 423}, {39201, 2196}, {39786, 7649}, {41333, 8750}, {46390, 1824}, {51664, 7233}

---

X(53557) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1364) AND X(3027)

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

X(535) lies on the de Longchamps ellipse and these lines: {1, 102}, {11, 1146}, {34, 19614}, {40, 7114}, {55, 6611}, {57, 972}, {73, 3057}, {201, 40944}, {212, 22394}, {223, 40971}, {244, 2638}, {497, 6508}, {513, 35065}, {521, 24031}, {656, 2972}, {1035, 7973}, {1214, 40945}, {1364, 3270}, {1936, 8766}, {2098, 17460}, {2184, 7008}, {2262, 41087}, {2269, 18675}, {2310, 2611}, {2969, 35015}, {3318, 6087}, {4466, 26956}, {7068, 8611}, {7358, 16596}, {7962, 7966}, {11396, 20280}, {13138, 36100}, {17441, 17642}, {19354, 26934}, {20902, 23673}, {24018, 34846}, {38981, 38985}

X(53557) = reflection of X(24031) in X(34588)
X(53557) = reflection of X(35065) in the OI line
X(53557) = orthic-isogonal conjugate of X(656)
X(53557) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 6129}, {4, 656}, {57, 652}, {223, 14298}, {342, 14837}, {2184, 650}, {2968, 7004}, {3345, 513}, {7020, 522}, {36100, 46391}, {40527, 7117}
X(i)-isoconjugate of X(j) for these (i,j): {59, 40836}, {84, 7012}, {108, 13138}, {189, 7115}, {268, 23984}, {271, 24033}, {282, 7128}, {651, 40117}, {653, 36049}, {1262, 7003}, {1275, 7154}, {1413, 15742}, {1436, 46102}, {1783, 37141}, {1897, 8059}, {2188, 24032}, {2358, 4567}, {4564, 7129}, {4998, 7151}, {5379, 52384}, {6081, 23987}, {7008, 7045}, {7020, 24027}, {18026, 32652}, {23985, 44189}, {32674, 44327}
X(53557) = X(i)-Dao conjugate of X(j) for these (i,j): {521, 271}, {522, 7020}, {656, 280}, {905, 309}, {3239, 34404}, {5514, 653}, {6129, 318}, {6615, 40836}, {14298, 63}, {14331, 18750}, {14837, 75}, {16596, 18026}, {17115, 7008}, {24018, 69}, {34467, 8059}, {35072, 44327}, {38983, 13138}, {38991, 40117}, {39006, 37141}, {40627, 2358}, {40628, 189}
X(53557) = crossdifference of every pair of points on line {109, 1783}
X(53557) = barycentric product X(i)*X(j) for these {i,j}: {1, 16596}, {40, 26932}, {57, 7358}, {63, 38357}, {77, 5514}, {85, 47432}, {196, 24031}, {198, 17880}, {208, 23983}, {223, 2968}, {322, 7117}, {329, 7004}, {342, 35072}, {347, 34591}, {521, 14837}, {652, 17896}, {693, 10397}, {905, 8058}, {1146, 7013}, {1565, 2324}, {1819, 16732}, {2638, 40701}, {3270, 40702}, {3318, 41081}, {3692, 38374}, {3719, 38362}, {3942, 7080}, {4025, 14298}, {4858, 7078}, {6129, 6332}, {6260, 40527}, {7011, 24026}, {7114, 23978}, {17219, 21871}, {18210, 27398}
X(53557) = barycentric quotient X(i)/X(j) for these {i,j}: {40, 46102}, {196, 24032}, {198, 7012}, {208, 23984}, {221, 7128}, {521, 44327}, {652, 13138}, {663, 40117}, {1146, 7020}, {1364, 41081}, {1459, 37141}, {1819, 4567}, {1946, 36049}, {2170, 40836}, {2187, 7115}, {2310, 7003}, {2324, 15742}, {2638, 268}, {2968, 34404}, {3122, 2358}, {3209, 24033}, {3270, 282}, {3271, 7129}, {3937, 1422}, {3942, 1440}, {5514, 318}, {6087, 24035}, {6129, 653}, {7004, 189}, {7011, 7045}, {7013, 1275}, {7078, 4564}, {7114, 1262}, {7117, 84}, {7358, 312}, {8058, 6335}, {10397, 100}, {14298, 1897}, {14837, 18026}, {14936, 7008}, {15501, 39294}, {16596, 75}, {17880, 44190}, {17896, 46404}, {18210, 8808}, {22383, 8059}, {24031, 44189}, {26932, 309}, {34591, 280}, {35072, 271}, {38357, 92}, {38374, 1847}, {39687, 2188}, {47432, 9}
X(53557) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1364, 35014, 7004}, {3270, 18210, 7004}

---

X(53558) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1365) AND X(3021)

Barycentrics    (b - c)*(b + c)*(2*a^2 - a*b + b^2 - a*c - 2*b*c + c^2) : :
X(53558) = 4 X[4010] - 3 X[4120], 3 X[4010] - 2 X[18004], 2 X[4088] - 3 X[4120], 3 X[4088] - 4 X[18004], 9 X[4120] - 8 X[18004], X[4988] - 4 X[48349], 4 X[676] - 3 X[1635], 2 X[2254] - 3 X[6545], 3 X[6545] - 4 X[23770], 4 X[2976] - 3 X[48032], 8 X[2977] - 9 X[6544], 4 X[3716] - 3 X[6546], 3 X[6546] - 2 X[48408], 3 X[4379] - 2 X[48069], 4 X[4458] - 3 X[4750], and mnay others

X(53558) lies on these lines: {145, 2789}, {512, 23755}, {513, 47704}, {514, 48304}, {522, 4382}, {523, 661}, {649, 47123}, {659, 8645}, {676, 1635}, {812, 47695}, {824, 47692}, {891, 21132}, {900, 4409}, {918, 47705}, {1365, 2611}, {2254, 6545}, {2526, 4777}, {2785, 49303}, {2826, 23764}, {2976, 3021}, {2977, 6544}, {3667, 47676}, {3716, 6546}, {3776, 50356}, {3800, 50457}, {3887, 47680}, {4041, 48403}, {4083, 21118}, {4106, 48077}, {4132, 23752}, {4145, 21889}, {4151, 21124}, {4379, 48069}, {4458, 4750}, {4498, 21185}, {4500, 47689}, {4728, 50333}, {4729, 7178}, {4730, 30574}, {4762, 47972}, {4800, 48056}, {4802, 48083}, {4809, 14435}, {4818, 48174}, {4830, 44433}, {4895, 29240}, {4913, 47797}, {4925, 4927}, {6362, 48334}, {7662, 48106}, {7927, 48393}, {8714, 47716}, {11599, 43669}, {13246, 47776}, {14432, 50351}, {14475, 25380}, {17166, 29118}, {18785, 21832}, {20482, 21053}, {21105, 21343}, {21115, 50357}, {21116, 48108}, {21180, 53411}, {21196, 48203}, {21201, 21385}, {21212, 48242}, {21297, 28161}, {23729, 48020}, {23879, 47713}, {24720, 48414}, {28147, 47699}, {28155, 47667}, {28169, 48554}, {28882, 47697}, {29025, 48301}, {29029, 48291}, {29098, 48305}, {29144, 47703}, {29162, 48322}, {29208, 48392}, {29288, 48264}, {29350, 49300}, {30573, 48296}, {31147, 48039}, {44435, 48017}, {47132, 47813}, {47663, 48063}, {47687, 49289}, {47690, 48394}, {47694, 48101}, {47698, 48043}, {47798, 48008}, {47812, 48412}, {47832, 48062}, {47887, 50336}, {47908, 47979}, {47909, 47983}, {47926, 48006}, {47932, 50347}, {47985, 48543}, {47988, 48583}, {48000, 48161}, {48023, 49295}, {48060, 48578}, {48080, 48082}, {48118, 49286}, {48142, 50522}, {48189, 48405}, {48273, 48278}, {48396, 48418}, {48425, 50348}, {49283, 49292}

X(53558) = reflection of X(i) in X(j) for these {i,j}: {649, 47123}, {2254, 23770}, {4024, 4804}, {4041, 48403}, {4088, 4010}, {4498, 21185}, {4729, 7178}, {4988, 47701}, {16892, 47691}, {21105, 21343}, {21124, 47712}, {21385, 21201}, {47663, 48063}, {47687, 49289}, {47689, 4500}, {47690, 48394}, {47698, 48043}, {47700, 3700}, {47701, 48349}, {47703, 48120}, {47908, 47979}, {47909, 47983}, {47926, 48006}, {47932, 50347}, {47934, 47998}, {48020, 23729}, {48023, 49295}, {48077, 4106}, {48082, 48080}, {48101, 47694}, {48106, 7662}, {48118, 49286}, {48278, 48273}, {48408, 3716}, {48583, 47988}, {49283, 49292}, {50343, 4458}, {50356, 3776}, {50522, 48142}, {53411, 21180}
X(53558) = X(13576)-Ceva conjugate of X(3120)
X(53558) = X(i)-isoconjugate of X(j) for these (i,j): {81, 6078}, {110, 1280}, {162, 1810}, {163, 36807}, {643, 1477}, {3286, 39272}, {4570, 35355}, {5546, 43760}
X(53558) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 36807}, {125, 1810}, {244, 1280}, {16593, 99}, {35111, 645}, {39048, 662}, {40586, 6078}, {40622, 35160}, {50330, 35355}
X(53558) = crossdifference of every pair of points on line {58, 1810}
X(53558) = barycentric product X(i)*X(j) for these {i,j}: {10, 6084}, {313, 8659}, {321, 48032}, {523, 3008}, {1279, 1577}, {2348, 4077}, {2976, 4052}, {3120, 53337}, {4088, 52210}, {4404, 51839}, {5853, 7178}, {14618, 20780}
X(53558) = barycentric quotient X(i)/X(j) for these {i,j}: {42, 6078}, {523, 36807}, {647, 1810}, {661, 1280}, {1279, 662}, {2348, 643}, {2976, 41629}, {3008, 99}, {3125, 35355}, {4017, 43760}, {5853, 645}, {6084, 86}, {7178, 35160}, {7180, 1477}, {8647, 5546}, {8659, 58}, {18785, 39272}, {20680, 1026}, {20780, 4558}, {48032, 81}, {53337, 4600}
X(53558) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2254, 23770, 6545}, {3716, 48408, 6546}, {4010, 4088, 4120}, {4458, 50343, 4750}, {47690, 48394, 48416}

---

X(53559) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1365) AND X(3023)

Barycentrics    (b - c)^2*(b + c)*(a^2 + b*c) : :
X(53559) = 5 X[27191] - 3 X[46912]

X(53559) lies on these lines: {11, 17476}, {75, 670}, {115, 4934}, {190, 21254}, {244, 1109}, {321, 21100}, {512, 4014}, {522, 23823}, {523, 1086}, {651, 21756}, {662, 24345}, {714, 35550}, {740, 49675}, {804, 3023}, {894, 4154}, {1111, 3123}, {1365, 2611}, {1441, 3778}, {2640, 24617}, {2650, 10106}, {3122, 16732}, {3125, 21144}, {3271, 8672}, {3663, 23928}, {3668, 23663}, {3728, 18698}, {3747, 8680}, {3840, 21418}, {3953, 42005}, {4032, 20964}, {4092, 17058}, {4440, 21295}, {4736, 24222}, {5147, 9318}, {6370, 46458}, {6741, 21944}, {6757, 24046}, {6758, 33148}, {8287, 21043}, {13178, 21221}, {16592, 21823}, {16598, 17719}, {16609, 20984}, {17205, 17886}, {17876, 17880}, {18697, 20443}, {20234, 42027}, {20236, 21330}, {20360, 24231}, {20488, 51417}, {20900, 21020}, {22172, 53510}, {23913, 24199}, {23944, 48627}, {26081, 36223}, {27191, 46912}, {32010, 51865}

X(53559) = midpoint of X(4440) and X(21295)
X(53559) = reflection of X(i) in X(j) for these {i,j}: {190, 21254}, {2643, 1086}, {20360, 24231}
X(53559) = isotomic conjugate of the isogonal conjugate of X(4128)
X(53559) = X(i)-Ceva conjugate of X(j) for these (i,j): {75, 4374}, {85, 661}, {1215, 2533}, {6384, 1577}, {7200, 16592}, {17103, 4369}, {18298, 693}, {51865, 514}
X(53559) = X(i)-isoconjugate of X(j) for these (i,j): {101, 4603}, {110, 3903}, {163, 27805}, {249, 52651}, {256, 4570}, {643, 29055}, {692, 4594}, {765, 1178}, {805, 3573}, {874, 17938}, {893, 4567}, {904, 4600}, {1110, 32010}, {1252, 40432}, {4590, 40729}, {4601, 7104}, {5379, 7015}, {5546, 37137}, {7260, 32739}, {39292, 41333}
X(53559) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 27805}, {244, 3903}, {513, 1178}, {514, 32010}, {661, 40432}, {804, 4154}, {1015, 4603}, {1086, 4594}, {3709, 9}, {4367, 6043}, {4369, 1}, {4988, 257}, {16587, 1016}, {16592, 99}, {21051, 43}, {35078, 3570}, {40597, 4567}, {40619, 7260}, {40627, 893}, {50330, 256}, {50497, 904}
X(53559) = cevapoint of X(16592) and X(40608)
X(53559) = barycentric product X(i)*X(j) for these {i,j}: {10, 7200}, {11, 4032}, {75, 16592}, {76, 4128}, {85, 40608}, {115, 17103}, {171, 16732}, {172, 21207}, {226, 4459}, {244, 3963}, {274, 21725}, {310, 21823}, {514, 2533}, {523, 4369}, {561, 21755}, {661, 4374}, {804, 4444}, {826, 18111}, {850, 20981}, {894, 3120}, {1015, 1237}, {1086, 1215}, {1111, 2295}, {1358, 4095}, {1365, 27958}, {1565, 1840}, {1577, 4367}, {1909, 3125}, {1920, 3122}, {1969, 22373}, {2643, 8033}, {2969, 4019}, {2973, 22061}, {3261, 7234}, {3287, 4077}, {3572, 14295}, {3676, 4140}, {3907, 7178}, {4024, 17212}, {4036, 18200}, {4049, 4922}, {4107, 35352}, {4466, 7009}, {4516, 7196}, {4705, 16737}, {7176, 21044}, {7211, 17197}, {14618, 22093}, {16727, 21803}, {17205, 21021}, {20964, 23989}
X(53559) = barycentric quotient X(i)/X(j) for these {i,j}: {171, 4567}, {172, 4570}, {244, 40432}, {513, 4603}, {514, 4594}, {523, 27805}, {661, 3903}, {693, 7260}, {804, 3570}, {876, 37134}, {894, 4600}, {1015, 1178}, {1086, 32010}, {1215, 1016}, {1237, 31625}, {1840, 15742}, {1909, 4601}, {2086, 3747}, {2295, 765}, {2533, 190}, {2643, 52651}, {3023, 27958}, {3120, 257}, {3121, 904}, {3122, 893}, {3125, 256}, {3287, 643}, {3572, 805}, {3907, 645}, {3963, 7035}, {4017, 37137}, {4032, 4998}, {4095, 4076}, {4128, 6}, {4140, 3699}, {4367, 662}, {4369, 99}, {4374, 799}, {4444, 18829}, {4459, 333}, {4466, 7019}, {4477, 7259}, {4529, 7256}, {7119, 5379}, {7176, 4620}, {7180, 29055}, {7200, 86}, {7234, 101}, {8033, 24037}, {14295, 27853}, {16592, 1}, {16726, 7303}, {16732, 7018}, {16737, 4623}, {17103, 4590}, {17212, 4610}, {18111, 4577}, {18200, 52935}, {18827, 39292}, {20964, 1252}, {20981, 110}, {21044, 4451}, {21207, 44187}, {21725, 37}, {21755, 31}, {21823, 42}, {22093, 4558}, {22373, 48}, {27958, 6064}, {35078, 4154}, {39786, 18786}, {40608, 9}, {41178, 4093}
X(53559) = {X(1086),X(23772)}-harmonic conjugate of X(4475)

---

X(53560) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(1367) AND X(3022)

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

X(53560) lies on these lines: {6, 10693}, {11, 31653}, {37, 1409}, {45, 19350}, {65, 44881}, {71, 52391}, {72, 4574}, {115, 125}, {148, 17947}, {185, 17451}, {220, 44782}, {607, 1854}, {756, 3611}, {906, 1807}, {912, 20752}, {926, 2170}, {1146, 8735}, {1334, 3954}, {1367, 4466}, {1425, 21808}, {1725, 3002}, {1880, 1903}, {1899, 3735}, {1951, 8558}, {1952, 18026}, {2631, 2632}, {2870, 16680}, {4415, 26957}, {5089, 6001}, {5227, 23135}, {5514, 38957}, {5693, 25087}, {5884, 25063}, {6332, 17219}, {7004, 7117}, {7108, 31623}, {16573, 22094}, {17442, 42448}, {17880, 23983}, {18671, 23154}, {19354, 34522}, {20117, 25062}, {20707, 20712}, {21049, 26955}, {21810, 21811}, {22070, 44706}, {35014, 47411}, {38344, 39006}, {49509, 51210}

X(53560) = isotomic conjugate of the polar conjugate of X(4516)
X(53560) = X(i)-Ceva conjugate of X(j) for these (i,j): {21, 1946}, {37, 647}, {65, 4041}, {1146, 21044}, {1172, 650}, {1214, 656}, {1812, 521}, {1903, 661}, {3694, 8611}, {4466, 18210}, {7108, 522}, {15232, 4705}, {40149, 523}, {43703, 512}, {51870, 15451}
X(53560) = X(i)-isoconjugate of X(j) for these (i,j): {4, 52378}, {21, 7128}, {25, 4620}, {27, 59}, {28, 4564}, {29, 1262}, {34, 4567}, {57, 5379}, {58, 46102}, {73, 23582}, {81, 7012}, {86, 7115}, {99, 32674}, {107, 1813}, {108, 662}, {109, 648}, {110, 653}, {112, 664}, {162, 651}, {163, 18026}, {225, 249}, {226, 250}, {278, 4570}, {283, 23984}, {286, 2149}, {307, 23964}, {332, 23985}, {608, 4600}, {643, 32714}, {765, 1396}, {811, 1415}, {823, 36059}, {859, 39294}, {1020, 52914}, {1101, 40149}, {1172, 7045}, {1214, 24000}, {1275, 2299}, {1395, 4601}, {1400, 18020}, {1402, 46254}, {1409, 23999}, {1412, 15742}, {1414, 1783}, {1461, 36797}, {1474, 4998}, {1576, 46404}, {1812, 24033}, {1880, 24041}, {1897, 4565}, {2193, 24032}, {2322, 7339}, {2333, 7340}, {4238, 36146}, {4246, 37136}, {4554, 32676}, {4558, 36127}, {4573, 8750}, {4599, 46152}, {4636, 52607}, {5546, 36118}, {6198, 35049}, {6516, 24019}, {6517, 6529}, {6528, 32660}, {8747, 44717}, {23067, 52919}, {23353, 41206}, {23979, 44130}, {23995, 52575}, {24027, 31623}, {32230, 40152}, {32661, 52938}, {34922, 40214}, {42396, 46153}, {52610, 52921}
X(53560) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 46102}, {11, 648}, {115, 18026}, {125, 651}, {226, 1275}, {244, 653}, {513, 1396}, {521, 1812}, {522, 31623}, {523, 40149}, {525, 1231}, {647, 1441}, {650, 286}, {656, 333}, {905, 274}, {1084, 108}, {1146, 811}, {1577, 44129}, {3005, 1880}, {3124, 46152}, {3126, 15149}, {3239, 314}, {3709, 7009}, {4858, 46404}, {4988, 273}, {5452, 5379}, {6505, 4620}, {6608, 2322}, {6615, 27}, {6741, 6335}, {7358, 645}, {8062, 1943}, {11517, 4567}, {15526, 4554}, {17115, 1172}, {17423, 1415}, {17434, 52385}, {18314, 52575}, {20620, 823}, {26932, 4573}, {34467, 4565}, {34591, 664}, {35071, 6516}, {35072, 99}, {35508, 36797}, {36033, 52378}, {38983, 662}, {38985, 1813}, {38986, 32674}, {38991, 162}, {39006, 1414}, {39014, 4238}, {39025, 112}, {40582, 18020}, {40586, 7012}, {40591, 4564}, {40599, 15742}, {40600, 7115}, {40605, 46254}, {40608, 1783}, {40611, 7128}, {40618, 4625}, {40622, 13149}, {40624, 6331}, {40626, 799}, {40627, 34}, {40628, 86}, {47345, 24032}, {50330, 278}, {50497, 608}, {51574, 4998}, {52355, 2064}
X(53560) = cevapoint of X(3269) and X(3708)
X(53560) = crossdifference of every pair of points on line {108, 110}
X(53560) = barycentric product X(i)*X(j) for these {i,j}: {8, 18210}, {9, 4466}, {10, 7004}, {11, 72}, {21, 125}, {28, 7068}, {29, 2632}, {33, 17216}, {37, 26932}, {42, 17880}, {63, 21044}, {65, 2968}, {69, 4516}, {71, 4858}, {73, 24026}, {77, 52335}, {78, 3120}, {115, 1812}, {123, 43703}, {210, 1565}, {212, 21207}, {219, 16732}, {225, 24031}, {226, 34591}, {228, 34387}, {244, 3710}, {283, 1109}, {284, 20902}, {306, 2170}, {307, 2310}, {314, 20975}, {321, 7117}, {332, 2643}, {333, 3708}, {338, 2193}, {339, 2194}, {345, 3125}, {348, 36197}, {512, 35518}, {513, 52355}, {514, 8611}, {520, 44426}, {521, 523}, {522, 656}, {525, 650}, {643, 21134}, {647, 4391}, {652, 1577}, {661, 6332}, {663, 14208}, {756, 17219}, {810, 35519}, {822, 46110}, {850, 1946}, {905, 3700}, {1086, 3694}, {1111, 2318}, {1146, 1214}, {1172, 15526}, {1231, 14936}, {1364, 41013}, {1365, 1792}, {1367, 4183}, {1409, 23978}, {1439, 4081}, {1441, 3270}, {1444, 4092}, {1459, 4086}, {1789, 21054}, {1809, 42759}, {1880, 23983}, {1896, 2972}, {1903, 16596}, {2185, 21046}, {2204, 36793}, {2299, 17879}, {2321, 3942}, {2394, 14395}, {3063, 3267}, {3064, 24018}, {3122, 3718}, {3239, 51664}, {3265, 18344}, {3269, 31623}, {3271, 20336}, {3695, 18191}, {3701, 3937}, {3709, 15413}, {3737, 4064}, {3900, 17094}, {3949, 17197}, {3998, 8735}, {4025, 4041}, {4036, 23189}, {4049, 14418}, {4088, 23696}, {4574, 40166}, {5489, 52914}, {5514, 52037}, {6741, 7100}, {7019, 40608}, {7108, 16573}, {7180, 15416}, {7358, 52384}, {8606, 17886}, {8736, 16731}, {10099, 50333}, {14429, 23838}, {14618, 36054}, {14935, 20235}, {15232, 34588}, {21666, 22341}, {21933, 40527}, {22094, 52344}, {23067, 42455}, {23989, 52370}, {35014, 38955}, {35072, 40149}, {38357, 52389}, {39687, 52575}, {42069, 52385}, {42761, 52663}
X(53560) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 286}, {21, 18020}, {29, 23999}, {37, 46102}, {42, 7012}, {48, 52378}, {55, 5379}, {63, 4620}, {71, 4564}, {72, 4998}, {73, 7045}, {78, 4600}, {115, 40149}, {125, 1441}, {210, 15742}, {212, 4570}, {213, 7115}, {219, 4567}, {225, 24032}, {228, 59}, {283, 24041}, {332, 24037}, {333, 46254}, {338, 52575}, {345, 4601}, {512, 108}, {520, 6516}, {521, 99}, {522, 811}, {523, 18026}, {525, 4554}, {647, 651}, {650, 648}, {652, 662}, {656, 664}, {661, 653}, {663, 162}, {798, 32674}, {810, 109}, {822, 1813}, {905, 4573}, {926, 4238}, {1015, 1396}, {1146, 31623}, {1172, 23582}, {1214, 1275}, {1364, 1444}, {1400, 7128}, {1409, 1262}, {1410, 7339}, {1444, 7340}, {1459, 1414}, {1577, 46404}, {1792, 6064}, {1812, 4590}, {1880, 23984}, {1946, 110}, {2170, 27}, {2193, 249}, {2194, 250}, {2200, 2149}, {2204, 23964}, {2250, 39294}, {2299, 24000}, {2310, 29}, {2318, 765}, {2611, 7282}, {2632, 307}, {2638, 283}, {2643, 225}, {2968, 314}, {2972, 52385}, {3005, 46152}, {3022, 4183}, {3049, 1415}, {3063, 112}, {3064, 823}, {3119, 2322}, {3120, 273}, {3121, 608}, {3122, 34}, {3124, 1880}, {3125, 278}, {3198, 44699}, {3269, 1214}, {3270, 21}, {3271, 28}, {3694, 1016}, {3700, 6335}, {3708, 226}, {3709, 1783}, {3710, 7035}, {3900, 36797}, {3937, 1014}, {3942, 1434}, {3990, 44717}, {4017, 36118}, {4025, 4625}, {4041, 1897}, {4092, 41013}, {4391, 6331}, {4466, 85}, {4516, 4}, {4574, 31615}, {4858, 44129}, {4895, 46541}, {6332, 799}, {7004, 86}, {7068, 20336}, {7117, 81}, {7178, 13149}, {7180, 32714}, {8034, 43923}, {8611, 190}, {10099, 927}, {14208, 4572}, {14395, 2407}, {14400, 24001}, {14936, 1172}, {15411, 4631}, {15526, 1231}, {16573, 1943}, {16732, 331}, {17094, 4569}, {17216, 7182}, {17219, 873}, {17435, 15149}, {17880, 310}, {18187, 17076}, {18210, 7}, {18344, 107}, {20902, 349}, {20975, 65}, {21044, 92}, {21046, 6358}, {21134, 4077}, {21789, 52914}, {21833, 8736}, {22094, 1442}, {22096, 1408}, {22383, 4565}, {23090, 4612}, {23189, 52935}, {24006, 52938}, {24026, 44130}, {24031, 332}, {26932, 274}, {34591, 333}, {34980, 22341}, {35014, 17139}, {35072, 1812}, {35518, 670}, {36054, 4558}, {36197, 281}, {37754, 40152}, {39201, 36059}, {39687, 2193}, {40608, 7009}, {42069, 1896}, {42752, 1875}, {44426, 6528}, {51664, 658}, {52222, 41206}, {52335, 318}, {52355, 668}, {52370, 1252}
X(53560) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1146, 38357, 8735}, {3125, 36197, 21044}, {7004, 34591, 7117}, {8558, 45272, 1951}, {38345, 38358, 1146}

---

X(53561) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(3022) AND X(3026)

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

X(53561) lies on these lines: {11, 115}, {32, 22760}, {39, 1837}, {55, 1573}, {80, 13006}, {497, 16975}, {574, 11502}, {620, 28925}, {926, 2170}, {950, 1107}, {1146, 2968}, {1500, 10950}, {1506, 10958}, {1572, 30223}, {2275, 9581}, {2276, 5727}, {2646, 16589}, {2654, 20963}, {2971, 23468}, {3125, 7004}, {3486, 5283}, {3734, 28934}, {3735, 24430}, {4534, 38347}, {4872, 43059}, {5517, 10017}, {6999, 45890}, {7117, 21044}, {9336, 50444}, {12019, 34460}, {12053, 17448}, {20982, 47411}, {23978, 26856}, {26357, 31456}, {26572, 27009}, {27076, 28798}, {34591, 36197}

X(53561) = X(i)-Ceva conjugate of X(j) for these (i,j): {391, 40137}, {940, 17418}, {941, 650}, {5307, 8672}
X(53561) = X(i)-isoconjugate of X(j) for these (i,j): {59, 44733}, {109, 32038}, {662, 52931}, {664, 32693}, {931, 1020}, {941, 7045}, {959, 4564}, {1262, 31359}, {1275, 2258}, {7128, 34259}, {24027, 34258}
X(53561) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 32038}, {522, 34258}, {1084, 52931}, {6615, 44733}, {17115, 941}, {17417, 664}, {23880, 34284}, {34261, 4998}, {39025, 32693}
X(53561) = crossdifference of every pair of points on line {651, 52931}
X(53561) = barycentric product X(i)*X(j) for these {i,j}: {11, 958}, {522, 17418}, {650, 23880}, {940, 1146}, {1021, 50457}, {1086, 3713}, {1468, 24026}, {2170, 11679}, {2268, 4858}, {2310, 10436}, {2968, 4185}, {3239, 48144}, {3714, 18191}, {3900, 43067}, {5019, 23978}, {5307, 34591}, {7253, 8672}, {14936, 34284}
X(53561) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 52931}, {650, 32038}, {940, 1275}, {958, 4998}, {1146, 34258}, {1468, 7045}, {2170, 44733}, {2268, 4564}, {2310, 31359}, {3063, 32693}, {3270, 34259}, {3271, 959}, {3713, 1016}, {5019, 1262}, {8639, 53321}, {8672, 4566}, {14936, 941}, {17418, 664}, {21789, 931}, {23880, 4554}, {23978, 40828}, {43067, 4569}, {48144, 658}
X(53561) = {X(11),X(11998)}-harmonic conjugate of X(1015)

---

X(53562) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(3022) AND X(3028)

Barycentrics    a^2*(a - b - c)*(b - c)*(b + c)*(a^2 - b^2 + b*c - c^2) : :
X(53562) = 2 X[65] - 3 X[30574], 3 X[4041] - 4 X[4524], 4 X[960] - 3 X[14432], 6 X[3753] - 7 X[21952]

X(53562) lies on these lines: {1, 2774}, {8, 53336}, {31, 42662}, {35, 44827}, {55, 42657}, {56, 53301}, {65, 30574}, {72, 4088}, {512, 661}, {520, 4017}, {526, 3028}, {652, 663}, {654, 8648}, {656, 924}, {758, 4707}, {900, 14299}, {926, 2170}, {928, 2254}, {960, 14432}, {1697, 53398}, {2773, 51896}, {2785, 3869}, {3657, 52391}, {3738, 3904}, {3753, 21952}, {3868, 4458}, {3900, 4820}, {4120, 34857}, {4557, 4559}, {8674, 10693}, {11529, 53407}, {21828, 42666}, {24353, 31339}, {33969, 50499}, {42649, 48387}, {51641, 52310}, {51646, 53295}

X(53562) = reflection of X(i) in X(j) for these {i,j}: {3868, 4458}, {4088, 72}
X(53562) = X(i)-Ceva conjugate of X(j) for these (i,j): {80, 21044}, {522, 53046}, {3657, 661}
X(53562) = X(i)-isoconjugate of X(j) for these (i,j): {57, 47318}, {58, 35174}, {80, 1414}, {81, 655}, {86, 2222}, {99, 1411}, {109, 14616}, {110, 18815}, {162, 52392}, {226, 37140}, {274, 32675}, {349, 32671}, {476, 1442}, {651, 24624}, {658, 2341}, {662, 2006}, {664, 759}, {934, 6740}, {1014, 51562}, {1333, 46405}, {1399, 35139}, {1412, 36804}, {1441, 36069}, {1793, 36118}, {2003, 32680}, {2161, 4573}, {4554, 34079}, {4565, 18359}, {4566, 52380}, {4591, 14628}, {4616, 52371}, {4622, 14584}, {4625, 6187}, {4637, 36910}, {7192, 52377}, {7282, 36061}, {14560, 52421}, {17095, 32678}, {52383, 52935}
X(53562) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 35174}, {11, 14616}, {37, 46405}, {125, 52392}, {244, 18815}, {1084, 2006}, {5452, 47318}, {6741, 20566}, {13999, 286}, {14714, 6740}, {16221, 7282}, {18334, 17095}, {34586, 664}, {35069, 4554}, {35128, 274}, {35204, 99}, {38982, 1441}, {38984, 86}, {38986, 1411}, {38991, 24624}, {39025, 759}, {40584, 4573}, {40586, 655}, {40599, 36804}, {40600, 2222}, {40608, 80}, {40612, 4625}, {51583, 4572}
X(53562) = crossdifference of every pair of points on line {81, 226}
X(53562) = barycentric product X(i)*X(j) for these {i,j}: {8, 21828}, {10, 654}, {21, 2610}, {36, 3700}, {37, 3738}, {42, 3904}, {55, 4707}, {71, 44428}, {210, 3960}, {226, 53285}, {284, 6370}, {320, 3709}, {321, 8648}, {333, 42666}, {512, 32851}, {522, 2245}, {523, 2323}, {525, 52427}, {526, 7110}, {647, 5081}, {650, 758}, {652, 860}, {657, 41804}, {661, 4511}, {663, 3936}, {850, 52426}, {1334, 4453}, {1443, 4171}, {1464, 3239}, {1577, 2361}, {1870, 8611}, {2287, 51663}, {2321, 53314}, {2624, 52344}, {3063, 35550}, {3218, 4041}, {3701, 21758}, {3724, 4391}, {3737, 4053}, {3900, 18593}, {4036, 4282}, {4086, 7113}, {4516, 4585}, {4524, 17078}, {6332, 44113}, {7073, 32679}, {8606, 44427}, {23838, 40988}, {38955, 53046}, {42768, 52663}, {52355, 52413}
X(53562) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 46405}, {36, 4573}, {37, 35174}, {42, 655}, {55, 47318}, {210, 36804}, {213, 2222}, {512, 2006}, {526, 17095}, {647, 52392}, {650, 14616}, {654, 86}, {657, 6740}, {661, 18815}, {663, 24624}, {758, 4554}, {798, 1411}, {860, 46404}, {1334, 51562}, {1443, 4635}, {1464, 658}, {1835, 13149}, {1918, 32675}, {2194, 37140}, {2245, 664}, {2323, 99}, {2361, 662}, {2610, 1441}, {2624, 1442}, {3063, 759}, {3218, 4625}, {3700, 20566}, {3709, 80}, {3724, 651}, {3738, 274}, {3904, 310}, {3936, 4572}, {4041, 18359}, {4079, 52383}, {4171, 52409}, {4282, 52935}, {4511, 799}, {4524, 36910}, {4707, 6063}, {4730, 14628}, {5081, 6331}, {6370, 349}, {7073, 32680}, {7110, 35139}, {7113, 1414}, {8641, 2341}, {8648, 81}, {14270, 2003}, {14407, 14584}, {18593, 4569}, {21758, 1014}, {21828, 7}, {32679, 52421}, {32851, 670}, {36197, 52356}, {41804, 46406}, {42666, 226}, {44113, 653}, {44428, 44129}, {47230, 7282}, {51663, 1446}, {52426, 110}, {52427, 648}, {52434, 4565}, {52440, 4637}, {53046, 17139}, {53285, 333}, {53314, 1434}
X(53562) = {X(654),X(53285)}-harmonic conjugate of X(8648)

---

X(53563) = INTERSECTION OF LINES TANGENT TO INCIRCLE AT X(3024) AND X(3027)

Barycentrics    a*(b^2 - c^2)*(a^2 - b*c)*(a^2 - b^2 - b*c - c^2) : :
X(53563) = 3 X[351] - 2 X[9508], 3 X[1962] - X[2254], X[2650] - 3 X[23057], 3 X[9147] - X[50343], 3 X[10180] - 2 X[25380], 3 X[27804] + X[53343]

X(53563) lies on these lines: {1, 690}, {33, 16230}, {35, 14270}, {36, 44826}, {42, 18004}, {55, 53263}, {56, 53247}, {350, 14295}, {351, 9508}, {512, 47976}, {513, 8663}, {523, 4724}, {526, 2611}, {659, 4155}, {740, 3716}, {804, 3027}, {900, 14752}, {1062, 6334}, {1734, 42653}, {1962, 2254}, {2276, 2491}, {2292, 4895}, {2533, 21831}, {2605, 3268}, {2650, 23057}, {2799, 39822}, {2901, 18003}, {3743, 3887}, {3747, 22384}, {4040, 6367}, {4068, 53308}, {4132, 8664}, {5010, 39477}, {5027, 38348}, {5297, 9185}, {6198, 44427}, {6370, 47695}, {7292, 9191}, {7951, 39509}, {8674, 9811}, {9003, 32286}, {9147, 50343}, {9208, 40790}, {9279, 48024}, {10180, 25380}, {27804, 53343}, {37697, 44921}, {47971, 48283}, {48277, 48297}, {48307, 50340}

X(53563) = midpoint of X(2292) and X(4895)
X(53563) = reflection of X(i) in X(j) for these {i,j}: {1734, 42653}, {42666, 3743}
X(53563) = X(i)-isoconjugate of X(j) for these (i,j): {291, 13486}, {660, 52375}, {741, 6742}, {813, 52393}, {2160, 4584}, {2311, 38340}, {4589, 6186}, {15455, 18268}
X(53563) = X(i)-Dao conjugate of X(j) for these (i,j): {8287, 18827}, {8299, 6742}, {35068, 15455}, {39029, 13486}, {40623, 52393}
X(53563) = crossdifference of every pair of points on line {2311, 2503}
X(53563) = barycentric product X(i)*X(j) for these {i,j}: {238, 7265}, {319, 21832}, {659, 3969}, {740, 14838}, {812, 3678}, {874, 20982}, {2238, 4467}, {2605, 3948}, {2611, 3570}, {3219, 4010}, {3573, 8287}, {3716, 16577}, {3747, 18160}, {4155, 34016}, {4420, 7212}, {4435, 40999}, {4455, 33939}, {6198, 24459}, {7206, 50456}, {16609, 35057}, {32679, 36815}
X(53563) = barycentric quotient X(i)/X(j) for these {i,j}: {35, 4584}, {319, 4639}, {659, 52393}, {740, 15455}, {1284, 38340}, {1914, 13486}, {2238, 6742}, {2605, 37128}, {2611, 4444}, {3219, 4589}, {3678, 4562}, {3969, 4583}, {4010, 30690}, {4155, 8818}, {4435, 3615}, {4455, 2160}, {4467, 40017}, {7265, 334}, {8632, 52375}, {14838, 18827}, {20982, 876}, {21824, 35352}, {21832, 79}, {35057, 36800}, {36815, 32680}, {40214, 36066}

---

X(53564) = INTERSECTION OF LINES TANGENT TO NINE-POINT CIRCLE AT X(11) AND X(116)

Barycentrics    (b - c)^2*(-(a^2*b) + a*b^2 - a^2*c + b^2*c + a*c^2 + b*c^2) : :
X(53564) = 5 X[1698] - X[53397]

X(53564) lies on these lines: {2, 4557}, {4, 53296}, {5, 2801}, {11, 244}, {12, 53531}, {116, 926}, {125, 46660}, {141, 9016}, {149, 27342}, {190, 30993}, {354, 44411}, {499, 20803}, {513, 24237}, {523, 4858}, {692, 24618}, {1125, 42443}, {1566, 39012}, {1698, 53397}, {1985, 4860}, {2250, 17728}, {2886, 34824}, {3337, 37357}, {3675, 16732}, {3733, 26856}, {3816, 4364}, {3834, 20544}, {3942, 4977}, {4124, 7202}, {4363, 30959}, {4579, 29490}, {4934, 7336}, {4957, 23772}, {5196, 50378}, {5433, 22342}, {5773, 35327}, {6532, 24387}, {6533, 24390}, {6685, 52877}, {7668, 8287}, {8053, 17077}, {9004, 34852}, {16560, 24346}, {17153, 29437}, {17198, 40619}, {17245, 42079}, {21239, 40646}, {23386, 24552}, {23774, 53546}, {33149, 37720}, {35552, 37788}, {40625, 47844}

X(53564) = midpoint of X(i) and X(j) for these {i,j}: {4, 53296}, {4858, 17463}
X(53564) = reflection of X(52877) in X(6685)
X(53564) = complement of X(4557)
X(53564) = complement of the isogonal conjugate of X(7192)
X(53564) = complement of the isotomic conjugate of X(52619)
X(53564) = medial-isogonal conjugate of X(661)
X(53564) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 52592}, {8049, 514}, {13476, 523}, {15320, 4977}, {17135, 8714}, {17198, 1565}, {40619, 1111}
X(53564) = X(i)-isoconjugate of X(j) for these (i,j): {100, 6577}, {765, 34444}, {1110, 8049}, {1252, 39797}, {4567, 40147}, {4570, 40504}, {23990, 39735}
X(53564) = X(i)-Dao conjugate of X(j) for these (i,j): {513, 34444}, {514, 8049}, {661, 39797}, {4988, 40515}, {8054, 6577}, {8714, 17135}, {40627, 40147}, {50330, 40504}, {52592, 2}
X(53564) = crossdifference of every pair of points on line {101, 6577}
X(53564) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 661}, {2, 4129}, {21, 4521}, {28, 3239}, {31, 52592}, {57, 1577}, {58, 650}, {63, 52599}, {75, 31946}, {81, 514}, {83, 29512}, {86, 513}, {87, 798}, {99, 24003}, {106, 21894}, {110, 24036}, {163, 23988}, {244, 115}, {269, 656}, {274, 3835}, {286, 20316}, {310, 21260}, {333, 20317}, {513, 1213}, {514, 1211}, {593, 14838}, {643, 3039}, {649, 16589}, {650, 38930}, {659, 35068}, {661, 6537}, {662, 4422}, {667, 21838}, {693, 3454}, {741, 665}, {757, 523}, {759, 1639}, {763, 21196}, {799, 27076}, {812, 46842}, {849, 647}, {873, 512}, {905, 440}, {1014, 522}, {1015, 16592}, {1019, 2}, {1021, 6554}, {1022, 3936}, {1027, 2238}, {1086, 8287}, {1111, 125}, {1171, 48003}, {1178, 3709}, {1333, 6586}, {1357, 16613}, {1358, 8286}, {1396, 14837}, {1408, 6589}, {1412, 905}, {1414, 3035}, {1422, 24018}, {1434, 4885}, {1444, 20315}, {1459, 18591}, {1474, 2509}, {1509, 4369}, {1565, 34846}, {2194, 52594}, {2206, 52589}, {2665, 46390}, {3125, 6627}, {3248, 1084}, {3261, 21245}, {3669, 17056}, {3676, 442}, {3733, 37}, {3737, 9}, {3766, 45162}, {3937, 16573}, {3942, 15526}, {4025, 21530}, {4077, 34829}, {4453, 31845}, {4560, 3452}, {4565, 16578}, {4567, 10196}, {4573, 21232}, {4637, 17044}, {4653, 52593}, {4833, 16590}, {4979, 51586}, {6385, 21262}, {6628, 52601}, {7192, 10}, {7194, 8061}, {7199, 141}, {7203, 1}, {7252, 1212}, {7254, 1214}, {7257, 3038}, {7312, 2642}, {7649, 50036}, {8042, 6547}, {8747, 14298}, {15376, 43060}, {15419, 18589}, {16726, 1086}, {16727, 116}, {17096, 142}, {17175, 50497}, {17197, 26932}, {17205, 11}, {17212, 51575}, {17217, 34832}, {17219, 123}, {17925, 226}, {17954, 2511}, {18155, 1329}, {18169, 40627}, {18184, 40618}, {18191, 1146}, {18197, 6376}, {18211, 40621}, {18826, 14426}, {18827, 3837}, {21173, 52087}, {23788, 119}, {23824, 5518}, {23829, 120}, {23892, 2229}, {23989, 21253}, {24002, 17052}, {25430, 48551}, {30940, 27854}, {32010, 21051}, {32014, 48049}, {37128, 812}, {39179, 3589}, {39734, 50337}, {39948, 50449}, {39949, 649}, {39950, 693}, {39980, 50457}, {40017, 21261}, {40148, 14991}, {40188, 14208}, {40398, 47890}, {40403, 11068}, {40408, 48000}, {40432, 25666}, {40438, 4977}, {40439, 6372}, {42302, 4762}, {43041, 50440}, {43924, 2092}, {43925, 16583}, {43926, 16611}, {43931, 21024}, {43932, 1834}, {47947, 41809}, {50456, 17755}, {51561, 2832}, {52375, 3700}, {52558, 8043}, {52619, 2887}, {52680, 6544}, {53083, 4391}, {53314, 35069}, {53538, 17058}
X(53564) = barycentric product X(i)*X(j) for these {i,j}: {11, 17077}, {244, 18137}, {514, 8714}, {1086, 17135}, {1111, 16552}, {1565, 17911}, {2973, 22126}, {3120, 29767}, {8053, 23989}, {16727, 22271}, {17205, 21070}, {24002, 50518}, {52592, 52619}
X(53564) = barycentric quotient X(i)/X(j) for these {i,j}: {244, 39797}, {649, 6577}, {1015, 34444}, {1086, 8049}, {1111, 39735}, {3120, 40515}, {3122, 40147}, {3125, 40504}, {8053, 1252}, {8714, 190}, {16552, 765}, {17077, 4998}, {17135, 1016}, {17911, 15742}, {18137, 7035}, {23989, 40005}, {29767, 4600}, {50518, 644}, {52592, 4557}
X(53564) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11, 1086, 2486}, {11, 7004, 34969}, {116, 17059, 21252}, {7668, 8287, 31946}

---

X(53565) = INTERSECTION OF LINES TANGENT TO NINE-POINT CIRCLE AT X(11) AND X(121)

Barycentrics    (b - c)*(-(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 + 2*b^2*c^2 - a*c^3 + b*c^3) : :
X(53565) = 3 X[36848] - X[53527], 4 X[21260] - 3 X[31946], 2 X[21260] - 3 X[44316], 3 X[40086] - 2 X[48406], 5 X[1698] - X[53392], 2 X[3716] - 3 X[24959], 3 X[3733] - X[31291], X[31291] + 3 X[44444], X[4057] - 3 X[48246], 2 X[4472] + X[24721], 3 X[24920] - 4 X[25380], 3 X[26078] + X[46403], 5 X[30795] - 3 X[48168], X[48350] - 3 X[48556], X[48392] - 3 X[50334]

X(53565) lies on these lines: {2, 4491}, {5, 2827}, {11, 244}, {12, 53528}, {86, 21303}, {121, 6085}, {141, 21261}, {513, 3814}, {523, 2530}, {659, 24988}, {1213, 3768}, {1698, 53392}, {2533, 4977}, {3716, 24959}, {3733, 31291}, {3762, 21714}, {4057, 48246}, {4364, 25356}, {4472, 24354}, {4777, 23815}, {8632, 17398}, {9002, 17072}, {17066, 48057}, {17330, 23650}, {21051, 28209}, {21053, 21143}, {21112, 53533}, {23818, 50335}, {24222, 53314}, {24920, 25380}, {26078, 46403}, {30795, 48168}, {48350, 48556}, {48392, 50334}

X(53565) = midpoint of X(i) and X(j) for these {i,j}: {676, 14287}, {2254, 14288}, {3733, 44444}, {21112, 53533}, {24354, 24721}
X(53565) = reflection of X(i) in X(j) for these {i,j}: {3762, 21714}, {4364, 25356}, {24354, 4472}, {31946, 44316}
X(53565) = complement of X(4491)
X(53565) = medial-isogonal conjugate of X(38979)
X(53565) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 38979}, {100, 52872}, {39697, 11}, {39982, 1086}, {39994, 116}, {40522, 4370}
X(53565) = crossdifference of every pair of points on line {101, 2220}

---

X(53566) = INTERSECTION OF LINES TANGENT TO NINE-POINT CIRCLE AT X(11) AND X(125)

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

X(53566) lies on these lines: {2, 53261}, {4, 53252}, {5, 2771}, {11, 244}, {12, 53537}, {65, 51870}, {79, 37357}, {116, 124}, {125, 526}, {191, 25648}, {226, 44411}, {427, 47212}, {496, 36250}, {523, 18210}, {693, 17198}, {1042, 15232}, {1484, 25437}, {1565, 21207}, {2783, 34977}, {2886, 20256}, {3142, 3649}, {3259, 46660}, {3816, 4425}, {3827, 26011}, {4557, 26031}, {4966, 51411}, {6173, 10886}, {7232, 30960}, {8286, 40608}, {8679, 26013}, {11246, 37354}, {11281, 15973}, {11604, 36154}, {14321, 36197}, {15507, 25493}, {16732, 53540}, {17463, 24026}, {17768, 37370}, {22032, 35310}, {23845, 26095}, {24237, 34589}, {26725, 47515}, {38957, 52118}, {45916, 48380}, {46816, 51631}, {52139, 52358}, {53260, 53322}

X(53566) = midpoint of X(i) and X(j) for these {i,j}: {4, 53252}, {45916, 48380}
X(53566) = complement of X(53280)
X(53566) = complement of the isogonal conjugate of X(4581)
X(53566) = medial-isogonal conjugate of X(50330)
X(53566) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 50330}, {244, 15611}, {514, 51571}, {961, 522}, {1169, 14838}, {1220, 513}, {1240, 21260}, {1791, 20315}, {2298, 514}, {2363, 523}, {3248, 39015}, {4581, 10}, {6648, 21232}, {8687, 16578}, {8707, 24003}, {14534, 4369}, {14624, 4129}, {15420, X(53566) = 18589}, {30710, 3835}, {31643, 17072}, {32736, 24036}, {36098, 3035}, {36147, 4422}, {40827, 42327}, {52928, 24025}
X(53566) = X(i)-Ceva conjugate of X(j) for these (i,j): {65, 523}, {20028, 514}
X(53566) = X(i)-isoconjugate of X(j) for these (i,j): {765, 52150}, {1101, 51870}, {1110, 20028}, {1252, 53083}, {2149, 46880}, {4570, 34434}
X(53566) = X(i)-Dao conjugate of X(j) for these (i,j): {513, 52150}, {514, 20028}, {523, 51870}, {650, 46880}, {661, 53083}, {4391, 314}, {4988, 2051}, {34589, 100}, {50330, 34434}
X(53566) = crossdifference of every pair of points on line {101, 1625}
X(53566) = barycentric product X(i)*X(j) for these {i,j}: {10, 24237}, {11, 52358}, {65, 40624}, {226, 34589}, {523, 17496}, {572, 21207}, {1086, 17751}, {1111, 21061}, {1441, 11998}, {1577, 21173}, {2975, 16732}, {3120, 14829}, {4391, 51662}, {4466, 11109}, {4858, 37558}, {14618, 23187}, {14973, 16727}, {17197, 52357}, {23989, 52139}
X(53566) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 46880}, {115, 51870}, {244, 53083}, {572, 4570}, {1015, 52150}, {1086, 20028}, {2975, 4567}, {3120, 2051}, {3125, 34434}, {11998, 21}, {14829, 4600}, {17496, 99}, {17751, 1016}, {21061, 765}, {21173, 662}, {23187, 4558}, {24237, 86}, {34589, 333}, {37558, 4564}, {38344, 283}, {40624, 314}, {51662, 651}, {52139, 1252}, {52358, 4998}
X(53566) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11, 3120, 2486}, {11, 38357, 34969}

---

X(53567) = INTERSECTION OF LINES TANGENT TO NINE-POINT CIRCLE AT X(114) AND X(125)

Barycentrics    (b - c)*(b + c)*(-(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(53567) = 3 X[5] - 2 X[39509], X[6130] - 3 X[45689], X[684] + 3 X[9148], 3 X[34964] - 4 X[39511], 3 X[549] - 2 X[39477], 3 X[16235] - X[44810], 3 X[53266] - X[53345]

X(53567) lies on these lines: {2, 53263}, {4, 53247}, {5, 690}, {11, 53563}, {30, 44826}, {114, 804}, {125, 526}, {140, 14270}, {325, 523}, {427, 16230}, {525, 34964}, {549, 39477}, {686, 6587}, {826, 39512}, {924, 30476}, {1594, 44427}, {2491, 3815}, {2799, 39816}, {6334, 11585}, {9003, 25328}, {9479, 31953}, {10276, 14698}, {14769, 45147}, {16235, 44810}, {18004, 44411}, {22037, 39583}, {23181, 30512}, {31127, 53266}, {34291, 53365}, {34989, 44564}, {35364, 53331}, {41624, 53354}

X(53567) = midpoint of X(i) and X(j) for these {i,j}: {4, 53247}, {34291, 53365}, {35364, 53331}
X(53567) = reflection of X(14270) in X(140)
X(53567) = complement of X(53263)
X(53567) = medial-isogonal conjugate of X(38987)
X(53567) = X(1)-complementary conjugate of X(38987)
X(53567) = X(i)-Ceva conjugate of X(j) for these (i,j): {35364, 523}, {53331, 2799}
X(53567) = crossdifference of every pair of points on line {32, 1625}

---

X(53568) = INTERSECTION OF LINES TANGENT TO NINE-POINT CIRCLE AT X(114) AND X(133)

Barycentrics    (2*a^6 - a^4*b^2 - b^6 - a^4*c^2 + b^4*c^2 + b^2*c^4 - c^6)*(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(53568) lies on these lines: {3, 15116}, {4, 1632}, {5, 2790}, {113, 131}, {114, 804}, {122, 23315}, {125, 1624}, {132, 50938}, {133, 6086}, {157, 3818}, {206, 18437}, {403, 3003}, {441, 1503}, {542, 1576}, {647, 1560}, {852, 41603}, {1568, 3001}, {2393, 44231}, {2781, 15526}, {2794, 40856}, {3184, 11598}, {5502, 5642}, {6699, 14703}, {7687, 9142}, {8369, 47354}, {10151, 51611}, {10516, 37344}, {10600, 16252}, {10991, 32274}, {12827, 15329}, {13558, 46682}, {14687, 46686}, {16240, 53319}, {18402, 50939}, {20208, 34778}, {23976, 28343}, {25641, 50935}, {30549, 42854}, {32125, 44436}, {33582, 34775}, {36518, 46127}, {37981, 47195}, {39072, 47082}, {40135, 51742}

X(53568) = midpoint of X(4) and X(1632)
X(53568) = X(i)-isoconjugate of X(j) for these (i,j): {1297, 36053}, {2435, 36114}, {5504, 8767}, {15421, 36046}
X(53568) = X(i)-Dao conjugate of X(j) for these (i,j): {113, 1297}, {23976, 2986}, {33504, 15421}, {34834, 35140}, {39005, 2435}, {39021, 43673}, {39071, 5504}, {50938, 1300}
X(53568) = crossdifference of every pair of points on line {14910, 34212}
X(53568) = barycentric product X(i)*X(j) for these {i,j}: {403, 441}, {1503, 3580}, {2409, 6334}, {3003, 30737}, {8779, 44138}, {12828, 36894}, {15595, 52451}, {34834, 43089}
X(53568) = barycentric quotient X(i)/X(j) for these {i,j}: {403, 6330}, {686, 2435}, {1503, 2986}, {2312, 36053}, {2409, 687}, {2445, 32708}, {3003, 1297}, {3580, 35140}, {6334, 2419}, {6793, 15454}, {8779, 5504}, {16318, 1300}, {21731, 34212}, {30737, 40832}, {34211, 18878}, {42671, 14910}, {43089, 40427}, {44084, 43717}, {52451, 9476}

---

X(53569) = INTERSECTION OF LINES TANGENT TO NINE-POINT CIRCLE AT X(115) AND X(122)

Barycentrics    (b - c)^2*(b + c)^2*(-a^4 + b^4 + c^4) : :
X(53569) = 3 X[35282] - 4 X[40559]

X(53569) lies on these lines: {2, 1632}, {3, 46184}, {4, 1177}, {5, 2790}, {6, 39645}, {53, 23300}, {66, 393}, {114, 34990}, {115, 804}, {122, 6086}, {125, 136}, {127, 47413}, {132, 52604}, {137, 5522}, {141, 14726}, {206, 17907}, {216, 23333}, {264, 6697}, {297, 2393}, {523, 15526}, {868, 20975}, {924, 34980}, {1503, 39569}, {1576, 2794}, {1634, 3014}, {1853, 6747}, {2450, 3003}, {2781, 18121}, {2847, 40484}, {2871, 15595}, {3018, 9512}, {3260, 33314}, {3708, 21947}, {4558, 36163}, {5181, 53350}, {6328, 15357}, {6530, 34146}, {6720, 39857}, {6749, 10169}, {7669, 47200}, {7745, 51744}, {8743, 11605}, {10933, 50329}, {14672, 38971}, {15328, 34290}, {17974, 32734}, {19153, 39604}, {19360, 20422}, {20300, 39530}, {20625, 35969}, {23293, 44135}, {23514, 34989}, {24975, 38749}, {27377, 39125}, {32366, 53477}, {33971, 51756}, {34207, 52439}, {34845, 36412}, {35282, 40559}, {36212, 45921}, {40074, 40876}, {46151, 50188}

X(53569) = midpoint of X(297) and X(45279)
X(53569) = reflection of X(1576) in X(23583)
X(53569) = complement of X(1632)
X(53569) = polar conjugate of X(44183)
X(53569) = polar conjugate of the isotomic conjugate of X(127)
X(53569) = polar conjugate of the isogonal conjugate of X(38356)
X(53569) = X(i)-complementary conjugate of X(j) for these (i,j): {34405, 21259}, {42407, 42327}
X(53569) = X(i)-Ceva conjugate of X(j) for these (i,j): {315, 33294}, {393, 523}, {17907, 2485}, {23962, 115}, {40073, 23881}, {43717, 16230}, {52583, 2501}
X(53569) = X(i)-isoconjugate of X(j) for these (i,j): {48, 44183}, {63, 15388}, {66, 1101}, {163, 44766}, {249, 2156}, {1289, 4575}, {2353, 24041}, {18018, 23995}, {23963, 46244}, {24037, 40146}
X(53569) = X(i)-Dao conjugate of X(j) for these (i,j): {32, 23357}, {115, 44766}, {127, 110}, {136, 1289}, {512, 40146}, {523, 66}, {647, 14376}, {1249, 44183}, {2485, 69}, {3005, 2353}, {3162, 15388}, {3265, 3926}, {18314, 18018}, {33294, 7796}, {47125, 28419}, {47413, 1634}
X(53569) = crossdifference of every pair of points on line {1634, 32661}
X(53569) = X(3014)-lineconjugate of X(1634)
X(53569) = barycentric product X(i)*X(j) for these {i,j}: {4, 127}, {22, 338}, {115, 315}, {125, 17907}, {206, 23962}, {264, 38356}, {339, 8743}, {523, 33294}, {850, 2485}, {868, 31636}, {1109, 1760}, {2052, 47413}, {2172, 23994}, {2643, 20641}, {2970, 20806}, {3120, 4150}, {3124, 40073}, {4024, 21178}, {4036, 16757}, {4092, 17076}, {4456, 21207}, {4463, 16732}, {4611, 23105}, {8673, 14618}, {8754, 34254}, {15526, 52448}, {18187, 41013}
X(53569) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 44183}, {22, 249}, {25, 15388}, {115, 66}, {125, 14376}, {127, 69}, {206, 23357}, {315, 4590}, {338, 18018}, {523, 44766}, {868, 34138}, {1084, 40146}, {1760, 24041}, {2172, 1101}, {2485, 110}, {2501, 1289}, {2643, 2156}, {2970, 43678}, {3124, 2353}, {4150, 4600}, {4456, 4570}, {4463, 4567}, {8673, 4558}, {8743, 250}, {8754, 13854}, {10316, 47390}, {16757, 52935}, {17076, 7340}, {17453, 23995}, {17907, 18020}, {18187, 1444}, {20641, 24037}, {20968, 23963}, {21178, 4610}, {23881, 4576}, {23962, 40421}, {23994, 46244}, {33294, 99}, {34212, 46967}, {34254, 47389}, {34294, 16277}, {38356, 3}, {40073, 34537}, {47413, 394}, {52448, 23582}, {52915, 47443}
X(53569) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {125, 8754, 338}, {7668, 34981, 115}, {7668, 38393, 34981}, {17907, 41761, 206}

---

X(53570) = INTERSECTION OF LINES TANGENT TO NINE-POINT CIRCLE AT X(115) AND X(127)

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

X(53570) lies on these lines: {2, 53273}, {5, 2794}, {99, 36165}, {115, 804}, {125, 5139}, {127, 2881}, {141, 2882}, {338, 2971}, {339, 523}, {381, 34845}, {427, 53418}, {858, 26276}, {868, 17423}, {1596, 7687}, {2072, 46981}, {3140, 39025}, {3150, 13611}, {3269, 3566}, {3613, 5475}, {5099, 46654}, {5254, 52462}, {7745, 51324}, {7789, 51337}, {10722, 36183}, {15667, 44212}, {15760, 23333}, {15761, 52532}, {23301, 36189}, {37938, 38611}, {39857, 40856}, {45161, 46665}

X(53570) = complement of X(53273)
X(53570) = X(i)-complementary conjugate of X(j) for these (i,j): {683, 21259}, {40405, 4369}, {40413, 8062}
X(53570) = X(25)-Ceva conjugate of X(523)
X(53570) = X(3267)-Dao conjugate of X(305)
X(53570) = crossdifference of every pair of points on line {1634, 45215}
X(53570) = barycentric product X(i)*X(j) for these {i,j}: {115, 33651}, {338, 19121}, {1109, 34065}, {15257, 23962}
X(53570) = barycentric quotient X(i)/X(j) for these {i,j}: {15257, 23357}, {19121, 249}, {33651, 4590}, {34065, 24041}

---

X(53571) = INTERSECTION OF LINES TANGENT TO NINE-POINT CIRCLE AT X(116) AND X(121)

Barycentrics    (b-c)*((b+c)*a^2-2*(b^2+b*c+c^2)*a+2*b*c*(b+c)) : :
X(53571) = X[1] - 5 X[30795], X[3835] + 3 X[17072], X[3835] - 3 X[21260], X[649] + 3 X[31149], X[659] - 5 X[1698], X[663] - 5 X[31251], 3 X[667] - 7 X[31207], X[764] + 3 X[14430], 3 X[1577] + X[50341], X[2254] + 3 X[14431], X[2530] + 3 X[21052], X[2533] + 3 X[47816], 3 X[47816] - X[48059], 7 X[3624] - 3 X[25569], 3 X[3679] + X[21343], X[3762] - 3 X[28603], and many others

X(53571) lies on these lines: {1, 30795}, {2, 1960}, {5, 2821}, {8, 48296}, {10, 891}, {12, 53539}, {116, 926}, {121, 6085}, {141, 9032}, {512, 625}, {519, 45340}, {649, 31149}, {659, 1698}, {663, 31251}, {665, 21053}, {667, 31207}, {693, 4770}, {764, 14430}, {900, 6702}, {905, 29268}, {1577, 50341}, {2254, 14431}, {2530, 21052}, {2533, 47816}, {2787, 25380}, {2826, 9956}, {3624, 25569}, {3626, 25574}, {3679, 21343}, {3762, 28603}, {3836, 6165}, {3906, 4522}, {4147, 23815}, {4384, 30865}, {4528, 45677}, {4705, 47672}, {4728, 4730}, {4775, 30835}, {4791, 50335}, {4794, 48197}, {4807, 4992}, {4922, 14422}, {5123, 6550}, {6363, 20316}, {6371, 44316}, {6372, 21051}, {6373, 27076}, {9148, 42666}, {9508, 29340}, {9780, 46403}, {10015, 48182}, {11231, 44805}, {14838, 29182}, {17066, 21262}, {17069, 29058}, {17606, 53523}, {19875, 21385}, {21212, 29110}, {21301, 27013}, {21302, 47839}, {21439, 33932}, {23789, 48401}, {23818, 25627}, {24623, 29610}, {25666, 29188}, {26752, 27194}, {26853, 47836}, {26985, 48291}, {28470, 31288}, {29102, 44314}, {29166, 50453}, {30858, 53285}, {31250, 48327}, {33920, 51362}, {47666, 47814}, {47724, 47827}, {47795, 48328}, {47829, 48284}, {47841, 48347}, {48196, 48331}, {48198, 48295}, {48218, 48330}

X(53571) = midpoint of X(i) and X(j) for these {i,j}: {8, 48296}, {10, 3837}, {693, 4770}, {2533, 48059}, {4147, 23815}, {4791, 50335}, {4807, 4992}, {17072, 21260}, {21051, 50337}, {21301, 50512}, {23789, 48401}, {28603, 36848}, {48005, 50352}
X(53571) = complement of X(1960)
X(53571) = complement of the isogonal conjugate of X(4555)
X(53571) = medial-isogonal conjugate of X(35092)
X(53571) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 35092}, {75, 3259}, {86, 34590}, {88, 1086}, {99, 34587}, {100, 4370}, {106, 1015}, {190, 16594}, {662, 51583}, {664, 1145}, {668, 121}, {679, 1647}, {765, 6544}, {901, 37}, {903, 11}, {1022, 6547}, {1320, 1146}, {1417, 16614}, {3257, 2}, {4080, 8287}, {4555, 10}, {4582, 3452}, {4591, 3666}, {4615, 3739}, {4618, 519}, {4622, 1125}, {4634, 3741}, {4638, 16610}, {4674, 115}, {4997, 26932}, {5376, 514}, {5382, 21129}, {5548, 1212}, {6551, 24036}, {6635, 24003}, {7045, 23757}, {9268, 650}, {9272, 9460}, {9456, 6377}, {20568, 116}, {23838, 46101}, {27922, 38989}, {31227, 40617}, {32665, 39}, {32719, 16584}, {36091, 17369}, {46162, 16587}, {52553, 51402}, {52925, 16590}
X(53571) = crossdifference of every pair of points on line {1613, 21790}
X(53571) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2533, 47816, 48059}, {21301, 47837, 50512}, {47814, 50352, 48005}

---

X(53572) = INTERSECTION OF LINES TANGENT TO NINE-POINT CIRCLE AT X(119) AND X(120)

Barycentrics    a^4*b^2 - a^3*b^3 + a*b^5 - b^6 + 2*a^4*b*c - 3*a^3*b^2*c + b^5*c + a^4*c^2 - 3*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*b^2*c^3 - 2*b^3*c^3 + b^2*c^4 + a*c^5 + b*c^5 - c^6 : :
X(53572) = 5 X[1698] - X[16560], 3 X[11236] + X[24826]

X(53572) lies on these lines: {2, 45920}, {5, 528}, {10, 21252}, {11, 3722}, {12, 1086}, {119, 900}, {120, 2977}, {121, 31845}, {190, 11681}, {375, 25970}, {442, 7668}, {545, 30448}, {650, 5513}, {673, 7679}, {867, 6174}, {1329, 4422}, {1698, 16560}, {2796, 30449}, {3030, 8286}, {3814, 4432}, {3822, 25351}, {3823, 3827}, {3826, 4472}, {3836, 8679}, {3925, 31264}, {5123, 21237}, {7951, 24715}, {9041, 12607}, {11236, 24826}, {24317, 25352}, {25466, 40480}, {26611, 45260}, {27687, 50038}

X(53572) = complement of X(53302)

---

X(53573) = INTERSECTION OF LINES TANGENT TO NINE-POINT CIRCLE AT X(120) AND X(123)

Barycentrics    (b - c)*(2*a^3 + a^2*b - 4*a*b^2 + b^3 + a^2*c - 4*a*b*c + 3*b^2*c - 4*a*c^2 + 3*b*c^2 + c^3) : :
X(53573) = 9 X[2] - X[47695], 3 X[2] + X[50333], 2 X[676] - 3 X[45318], 3 X[676] - X[47695], 9 X[45318] - 2 X[47695], 3 X[45318] + 2 X[50333], X[47695] + 3 X[50333], X[2977] - 3 X[28602], X[2977] + 3 X[30792], X[3837] + 3 X[28602], X[3837] - 3 X[30792], 3 X[210] + X[53550], X[650] + 3 X[47806], 3 X[650] + X[49285], 9 X[47806] - X[49285], and many others

X(53573) lies on these lines: {1, 4528}, {2, 676}, {5, 9521}, {9, 53300}, {10, 6366}, {120, 2977}, {123, 6087}, {140, 44819}, {210, 53550}, {513, 2490}, {522, 31287}, {523, 4885}, {650, 47806}, {659, 14425}, {661, 48232}, {900, 3035}, {918, 25380}, {928, 5044}, {936, 53285}, {1491, 47807}, {1638, 4088}, {1639, 2254}, {1698, 10015}, {1734, 4990}, {2517, 27417}, {2526, 47766}, {2804, 6667}, {2826, 20400}, {2885, 24099}, {2976, 4448}, {3004, 47693}, {3634, 23887}, {3700, 47828}, {3904, 9780}, {4449, 44729}, {4458, 44902}, {4491, 14287}, {4522, 17069}, {4927, 48408}, {4977, 20316}, {6544, 48032}, {6590, 48193}, {7659, 47765}, {13246, 45675}, {14321, 50336}, {14429, 53522}, {14430, 30725}, {14475, 47705}, {14838, 29278}, {18004, 48229}, {19804, 23684}, {19862, 48286}, {19875, 45341}, {20508, 25637}, {21260, 29162}, {23770, 30795}, {25917, 53549}, {25926, 53308}, {27115, 47687}, {28183, 47831}, {30565, 50357}, {31207, 48077}, {31209, 47808}, {31250, 47123}, {31251, 48403}, {31445, 52730}, {34958, 48218}, {36848, 48055}, {41800, 48272}, {44429, 47686}, {44798, 52614}, {45323, 48405}, {46403, 47884}, {47132, 48206}, {47690, 47784}, {47698, 47891}, {47703, 47876}, {47756, 48106}, {47760, 48069}, {47761, 48039}, {47767, 48023}, {47770, 48015}, {47788, 47975}, {47802, 48062}, {47810, 48276}, {47823, 48047}, {47824, 48046}, {47825, 48274}, {47827, 48396}, {47879, 48017}, {47888, 48395}, {47998, 48235}, {48007, 48219}, {48024, 48249}, {48103, 48178}, {48166, 50359}, {48185, 50348}, {48199, 50335}, {48228, 52355}, {48231, 50328}, {48244, 50326}, {48270, 48575}

X(53573) = midpoint of X(i) and X(j) for these {i,j}: {1, 4528}, {676, 50333}, {1734, 4990}, {2977, 3837}, {3716, 4925}, {4491, 14287}, {4522, 17069}, {14321, 50336}, {14425, 48182}, {28602, 30792}
X(53573) = reflection of X(i) in X(j) for these {i,j}: {44819, 140}, {45318, 2}
X(53573) = complement of X(676)
X(53573) = complement of the isogonal conjugate of X(677)
X(53573) = X(i)-complementary conjugate of X(j) for these (i,j): {100, 118}, {101, 39063}, {103, 11}, {165, 35593}, {677, 10}, {692, 23972}, {911, 1086}, {2338, 26932}, {18025, 21252}, {24016, 11019}, {32642, 37}, {32668, 4000}, {35184, 354}, {36039, 2}, {36056, 2968}, {36101, 116}, {36136, 5222}, {40116, 226}, {43736, 17059}
X(53573) = crossdifference of every pair of points on line {1616, 3053}
X(53573) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 50333, 676}, {2977, 30792, 3837}, {3837, 28602, 2977}, {4522, 47830, 17069}, {4925, 45326, 3716}, {23770, 30795, 45677}, {31209, 47808, 50347}

---

X(53574) = INTERSECTION OF LINES TANGENT TO NINE-POINT CIRCLE AT X(121) AND X(124)

Barycentrics    (b - c)*(a^3*b - a*b^3 + a^3*c - 2*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(53574) = X[1] - 3 X[48168], X[8] + 3 X[26144], 2 X[21714] - 3 X[28603], X[17072] - 3 X[20316], X[1459] - 3 X[48181], X[1769] + 3 X[14430], X[2605] - 3 X[48165], X[20293] + 3 X[48165], 5 X[3617] + 3 X[27545], X[4833] - 3 X[48553], 7 X[9780] - 3 X[26078], X[14288] - 3 X[14431], 3 X[14429] + X[21132], X[21173] - 3 X[48205], X[21180] - 3 X[21198], X[24093] + 3 X[27728], X[43924] - 3 X[48230], 3 X[47793] - X[50349], 3 X[48173] - X[48292], 3 X[48186] - X[48283], 3 X[48207] - X[48281]

X(53574) lies on these lines: {1, 48168}, {2, 53314}, {5, 2815}, {8, 26144}, {10, 900}, {121, 6085}, {124, 8677}, {141, 47329}, {513, 3823}, {514, 25356}, {594, 4526}, {659, 25634}, {665, 1213}, {918, 25357}, {958, 39200}, {966, 22108}, {993, 39478}, {1459, 48181}, {1769, 14430}, {2605, 20293}, {3617, 27545}, {3716, 8674}, {3738, 6702}, {3741, 48183}, {3762, 53527}, {3766, 5224}, {3768, 21053}, {3837, 14426}, {4036, 17420}, {4147, 4777}, {4389, 21433}, {4435, 17275}, {4768, 24457}, {4800, 31330}, {4833, 48553}, {4977, 23789}, {6363, 44316}, {6370, 10015}, {6371, 31946}, {9002, 21260}, {9780, 26078}, {10916, 39472}, {14288, 14431}, {14429, 21132}, {17230, 27292}, {17250, 21606}, {17292, 28779}, {21173, 48205}, {21180, 21198}, {21262, 48057}, {22092, 46838}, {24093, 27728}, {24720, 28220}, {24959, 50605}, {24982, 25996}, {24987, 25923}, {25627, 50335}, {26037, 48244}, {27074, 29591}, {28114, 36568}, {28195, 47843}, {28209, 47994}, {28284, 31339}, {29667, 44433}, {29679, 31131}, {31897, 52305}, {39534, 46878}, {43924, 48230}, {47793, 50349}, {48173, 48292}, {48186, 48283}, {48207, 48281}

X(53574) = midpoint of X(i) and X(j) for these {i,j}: {2605, 20293}, {3762, 53527}, {4036, 17420}, {4768, 24457}
X(53574) = complement of X(53314)
X(53574) = complement of the isogonal conjugate of X(51562)
X(53574) = medial-isogonal conjugate of X(51402)
X(53574) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 51402}, {9, 46398}, {55, 35128}, {80, 11}, {100, 214}, {101, 16586}, {655, 142}, {759, 244}, {901, 52537}, {1168, 1647}, {1411, 3756}, {1793, 34588}, {1807, 2968}, {2006, 4904}, {2161, 1086}, {2222, 1}, {2341, 4858}, {2802, 35587}, {3678, 3258}, {3952, 31845}, {4551, 6739}, {4557, 35069}, {6187, 1015}, {6740, 34589}, {14147, 3218}, {15065, 125}, {18359, 116}, {18815, 17059}, {20566, 21252}, {24624, 17761}, {32675, 3752}, {34857, 115}, {35174, 2886}, {36804, 141}, {36815, 38989}, {36910, 26932}, {37140, 17045}, {40172, 35092}, {46405, 17046}, {46649, 3738}, {46821, 51442}, {47318, 3739}, {51562, 10}, {51975, 3259}, {52356, 46100}, {52371, 1146}, {52377, 522}, {52383, 8286}, {52409, 124}
X(53574) = {X(20293),X(48165)}-harmonic conjugate of X(2605)

---

X(53575) = INTERSECTION OF LINES TANGENT TO NINE-POINT CIRCLE AT X(125) AND X(127)

Barycentrics    (b - c)^2*(b + c)^2*(-(a^2*b^2) + b^4 - a^2*c^2 + b^2*c^2 + c^4) : :
X(53575) = 5 X[3763] - X[52162]

X(53575) lies on these lines: {2, 1576}, {5, 2781}, {66, 2980}, {67, 9512}, {115, 9479}, {122, 20625}, {125, 526}, {127, 2881}, {141, 2871}, {338, 868}, {512, 46650}, {523, 15526}, {626, 2882}, {2794, 40484}, {2878, 21252}, {3001, 33314}, {3014, 53350}, {3258, 46661}, {3271, 8287}, {3763, 52162}, {4558, 30789}, {5099, 46657}, {5622, 5877}, {6697, 34845}, {8754, 38393}, {9019, 21536}, {11007, 34990}, {15357, 33967}, {15613, 46658}, {20582, 22566}, {21243, 46185}, {21531, 34827}, {22087, 46562}, {25051, 31127}, {36471, 46656}, {46652, 46653}, {50188, 52604}

X(53575) = reflection of X(1576) in X(40559)
X(53575) = complement of X(1576)
X(53575) = anticomplement of X(40559)
X(53575) = complement of the isogonal conjugate of X(850)
X(53575) = complement of the isotomic conjugate of X(44173)
X(53575) = medial-isogonal conjugate of X(647)
X(53575) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 52590}, {66, 523}, {44175, 525}
X(53575) = X(i)-isoconjugate of X(j) for these (i,j): {1101, 2980}, {23995, 44176}
X(53575) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 647}, {2, 14838}, {4, 16612}, {10, 650}, {19, 2485}, {31, 52590}, {37, 6586}, {38, 52591}, {42, 52589}, {63, 52584}, {65, 6589}, {75, 523}, {76, 4369}, {81, 52597}, {85, 17069}, {86, 31947}, {91, 2501}, {92, 525}, {99, 16598}, {110, 23993}, {115, 16592}, {125, 16573}, {158, 6587}, {163, 23584}, {210, 52594}, {225, 6588}, {226, 905}, {264, 8062}, {274, 21196}, {286, 21187}, {308, 8060}, {310, 52601}, {313, 513}, {321, 514}, {334, 9508}, {338, 8287}, {339, 34846}, {349, 4885}, {512, 16584}, {514, 3666}, {520, 828}, {522, 40937}, {523, 37}, {525, 1214}, {561, 512}, {648, 16599}, {649, 52535}, {656, 216}, {661, 39}, {662, 34990}, {664, 34977}, {670, 21254}, {693, 1125}, {756, 52592}, {798, 8265}, {799, 620}, {811, 5972}, {823, 23583}, {826, 16587}, {850, 10}, {897, 2492}, {921, 6753}, {1018, 23988}, {1020, 23585}, {1089, 661}, {1096, 52588}, {1109, 115}, {1111, 244}, {1268, 8043}, {1365, 16613}, {1427, 52595}, {1441, 522}, {1446, 7658}, {1502, 42327}, {1577, 2}, {1581, 2491}, {1784, 14401}, {1821, 2799}, {1826, 2509}, {1924, 40377}, {1928, 23301}, {1930, 3005}, {1934, 804}, {1969, 30476}, {2166, 1637}, {2184, 52613}, {2190, 16040}, {2349, 8552}, {2394, 18593}, {2481, 4458}, {2501, 16583}, {2582, 46811}, {2583, 46814}, {2588, 8106}, {2589, 8105}, {2616, 570}, {2618, 233}, {2632, 35071}, {2643, 1084}, {2799, 16591}, {2962, 12077}, {3112, 826}, {3120, 1015}, {3125, 6377}, {3261, 3739}, {3267, 18589}, {3668, 6129}, {3700, 1212}, {3701, 4521}, {3762, 51583}, {3948, 27929}, {3952, 24036}, {3992, 6544}, {4013, 21894}, {4017, 17053}, {4024, 16589}, {4025, 37565}, {4033, 4422}, {4036, 1213}, {4041, 16588}, {4049, 16610}, {4052, 3669}, {4064, 18591}, {4077, 1}, {4080, 3960}, {4086, 9}, {4088, 6184}, {4125, 52593}, {4391, 5745}, {4404, 3161}, {4551, 13006}, {4552, 16578}, {4560, 16579}, {4566, 24025}, {4608, 3743}, {4647, 4988}, {4674, 3310}, {4705, 21838}, {4707, 16586}, {4978, 41820}, {5466, 16611}, {5620, 47227}, {6358, 1577}, {6370, 35069}, {6385, 52602}, {6521, 52585}, {6528, 23998}, {6539, 48003}, {6757, 3700}, {7178, 3752}, {7199, 17045}, {7265, 16585}, {7649, 40941}, {8769, 2489}, {8773, 6132}, {9251, 42293}, {14206, 5664}, {14207, 11165}, {14208, 3}, {14210, 1649}, {14213, 18314}, {14295, 19563}, {14616, 6370}, {14618, 226}, {15065, 1639}, {15412, 16577}, {15455, 40539}, {15526, 16595}, {16732, 1086}, {17094, 17102}, {17876, 15525}, {17879, 122}, {17880, 34588}, {17881, 136}, {17886, 6741}, {17898, 1249}, {17924, 40940}, {18022, 21259}, {18031, 24285}, {18070, 3589}, {18155, 4999}, {18697, 50330}, {18815, 21180}, {18832, 44451}, {20336, 20315}, {20567, 17066}, {20568, 45674}, {20571, 924}, {20883, 23285}, {20902, 15526}, {20910, 6374}, {20948, 141}, {20953, 6665}, {21207, 11}, {23105, 24040}, {23285, 21249}, {23604, 6591}, {23894, 3291}, {23962, 21253}, {23989, 17761}, {23994, 125}, {24002, 3946}, {24006, 6}, {24018, 6509}, {24026, 34591}, {24037, 10190}, {27801, 3835}, {27808, 24003}, {28654, 4129}, {30690, 21192}, {30710, 8045}, {30713, 20317}, {31010, 44307}, {31065, 28594}, {32679, 34834}, {32680, 24975}, {33294, 16582}, {34387, 34589}, {35352, 1575}, {35518, 34851}, {35519, 960}, {35522, 16597}, {36035, 3163}, {36038, 34586}, {36053, 47230}, {36084, 34349}, {36102, 45681}, {36119, 46425}, {40014, 2487}, {40149, 14837}, {40216, 4151}, {40362, 21263}, {40364, 52598}, {40440, 6368}, {40495, 3741}, {40703, 41167}, {41013, 3239}, {42027, 21348}, {43534, 665}, {43665, 16609}, {43675, 21188}, {43677, 7180}, {43682, 41800}, {43683, 7178}, {44173, 2887}, {44426, 40942}, {46107, 942}, {46110, 6708}, {46273, 24284}, {46277, 690}, {51844, 45907}, {52335, 35508}, {52356, 7359}, {52369, 52599}, {52575, 46396}, {52618, 1215}, {52621, 3742}, {52623, 1211}, {52632, 4892}, {53540, 16614}
X(53575) = X(i)-Dao conjugate of X(j) for these (i,j): {523, 2980}, {18314, 44176}, {33294, 315}, {52590, 2}
X(53575) = barycentric product X(i)*X(j) for these {i,j}: {115, 7796}, {160, 23962}, {321, 18188}, {338, 2979}, {339, 39575}, {44173, 52590}
X(53575) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 2980}, {160, 23357}, {338, 44176}, {2979, 249}, {3202, 23963}, {7796, 4590}, {18188, 81}, {23962, 44185}, {39575, 250}, {39691, 27366}, {52590, 1576}
X(53575) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1576, 40559}, {338, 868, 34981}

---

X(53576) = INTERSECTION OF LINES TANGENT TO NINE-POINT CIRCLE AT X(125) AND X(130)

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

X(53576) lies on these lines: {2, 19180}, {5, 32438}, {6, 19166}, {53, 1987}, {54, 67}, {95, 141}, {97, 343}, {125, 526}, {127, 38965}, {275, 6749}, {297, 43752}, {338, 3269}, {427, 21638}, {858, 19167}, {1141, 53187}, {1503, 19189}, {1594, 19168}, {1634, 53174}, {1853, 19174}, {1899, 16030}, {2883, 19206}, {3580, 43768}, {3613, 6752}, {4993, 37648}, {5480, 9792}, {6146, 51255}, {6247, 8884}, {6696, 19172}, {8552, 15526}, {10192, 26887}, {12359, 19210}, {13371, 19211}, {14216, 19173}, {15583, 19197}, {16035, 26937}, {19170, 23292}, {19171, 34774}, {19177, 23324}, {19178, 23326}, {19185, 34782}, {19192, 23328}, {19205, 41362}, {19209, 23295}, {24862, 38394}, {36793, 45792}, {39843, 40866}, {42353, 43975}

X(53576) = isotomic conjugate of the polar conjugate of X(8901)
X(53576) = X(i)-complementary conjugate of X(j) for these (i,j): {40402, 8062}, {40448, 4369}, {42333, 42327}
X(53576) = X(i)-Ceva conjugate of X(j) for these (i,j): {95, 23286}, {97, 525}, {275, 15412}, {8795, 523}
X(53576) = X(i)-isoconjugate of X(j) for these (i,j): {53, 1101}, {112, 2617}, {162, 1625}, {163, 35360}, {216, 24000}, {217, 23999}, {249, 2181}, {250, 1953}, {324, 23995}, {662, 52604}, {2179, 18020}, {3199, 24041}, {14570, 32676}, {18695, 41937}, {23181, 24019}, {23964, 44706}, {40981, 46254}
X(53576) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 35360}, {125, 1625}, {523, 53}, {525, 343}, {647, 5}, {1084, 52604}, {3005, 3199}, {3569, 45123}, {5664, 14918}, {11792, 35318}, {14401, 1568}, {15526, 14570}, {17434, 5562}, {18314, 324}, {23285, 311}, {34591, 2617}, {35071, 23181}, {35443, 6117}, {35444, 6116}, {52584, 467}, {52591, 30506}
X(53576) = cevapoint of X(125) and X(3269)
X(53576) = trilinear pole of line {5489, 16186}
X(53576) = crossdifference of every pair of points on line {1625, 52604}
X(53576) = barycentric product X(i)*X(j) for these {i,j}: {54, 339}, {69, 8901}, {95, 125}, {97, 338}, {115, 34386}, {275, 15526}, {276, 3269}, {525, 15412}, {850, 23286}, {2167, 20902}, {2169, 23994}, {2190, 17879}, {2501, 15414}, {2525, 39182}, {2616, 14208}, {2623, 3267}, {2632, 40440}, {2972, 8795}, {3134, 43767}, {4143, 15422}, {5489, 18831}, {8882, 36793}, {14533, 23962}, {16186, 46138}, {16813, 23616}, {20975, 34384}
X(53576) = barycentric quotient X(i)/X(j) for these {i,j}: {54, 250}, {95, 18020}, {97, 249}, {115, 53}, {125, 5}, {275, 23582}, {338, 324}, {339, 311}, {512, 52604}, {520, 23181}, {523, 35360}, {525, 14570}, {647, 1625}, {656, 2617}, {868, 39569}, {1562, 42459}, {1650, 1568}, {2088, 11062}, {2169, 1101}, {2190, 24000}, {2616, 162}, {2623, 112}, {2632, 44706}, {2643, 2181}, {2970, 13450}, {2972, 5562}, {3124, 3199}, {3269, 216}, {3708, 1953}, {4466, 17167}, {5489, 6368}, {7066, 44710}, {7668, 30506}, {8029, 51513}, {8754, 14569}, {8794, 34538}, {8882, 23964}, {8884, 32230}, {8901, 4}, {14380, 36831}, {14533, 23357}, {15412, 648}, {15414, 4563}, {15422, 6529}, {15526, 343}, {16186, 1154}, {17879, 18695}, {18210, 18180}, {18315, 47443}, {19210, 47390}, {20902, 14213}, {20975, 51}, {21046, 21011}, {21134, 21102}, {23105, 23290}, {23286, 110}, {24862, 23607}, {30465, 6117}, {30468, 6116}, {34386, 4590}, {34980, 418}, {36793, 28706}, {38352, 41334}, {38987, 45123}, {39182, 42396}, {39691, 27371}, {40440, 23999}, {41997, 44713}, {41998, 44714}, {46088, 32661}, {46089, 14587}, {47421, 14576}
X(53576) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {125, 34980, 7668}, {275, 26954, 13567}

---

X(53577) = INTERSECTION OF LINES TANGENT TO NINE-POINT CIRCLE AT X(125) AND X(136)

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

X(53577) lies on these lines: {2, 23181}, {4, 1624}, {5, 1511}, {11, 3137}, {115, 122}, {125, 526}, {136, 6132}, {137, 3258}, {338, 2972}, {427, 39071}, {442, 25882}, {523, 2970}, {661, 39003}, {868, 17423}, {1092, 22261}, {1112, 47208}, {1368, 6036}, {1594, 31867}, {1650, 46093}, {2777, 43917}, {3054, 9722}, {3120, 52309}, {3124, 6587}, {3136, 46095}, {3138, 6506}, {3139, 34467}, {3448, 5877}, {5094, 34845}, {7363, 21015}, {8286, 39004}, {8287, 39007}, {9512, 17847}, {12026, 46114}, {13371, 52534}, {14652, 36188}, {14674, 22115}, {15106, 49006}, {15448, 44227}, {16177, 46658}, {23292, 52128}, {23301, 39001}, {31946, 39002}, {34984, 52584}, {35235, 44809}, {36189, 47249}, {37938, 38618}, {37985, 38999}, {46106, 47153}

X(53577) = complement of X(23181)
X(53577) = X(i)-complementary conjugate of X(j) for these (i,j): {19, 18314}, {275, 4369}, {276, 42327}, {656, 10600}, {933, 16598}, {1096, 17434}, {1109, 20625}, {2148, 52584}, {2190, 523}, {2616, 3}, {2623, 1214}, {2643, 39019}, {8795, 21259}, {8882, 14838}, {8884, 8062}, {8901, 34846}, {15412, 18589}, {15422, 226}, {18831, 21254}, {24006, 1209}, {40440, 512}
X(53577) = X(i)-Ceva conjugate of X(j) for these (i,j): {3, 523}, {42355, 525}
X(53577) = X(14618)-Dao conjugate of X(264)
X(53577) = crossdifference of every pair of points on line {1624, 1625}
X(53577) = X(4)-line-conjugate of X(1624)
X(53577) = barycentric product X(338)*X(34148)
X(53577) = barycentric quotient X(34148)/X(249)
X(53577) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {125, 8901, 7668}, {3134, 8901, 125}

---

X(53578) = INTERSECTION OF LINES TANGENT TO YFF PARABOLA AT X(514) AND X(24979)

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

X(53578) lies on these lines: {30, 511}, {44, 52334}, {238, 10015}, {242, 39534}, {676, 1279}, {1566, 35967}, {3904, 4645}, {4448, 42756}, {5018, 53314}, {14189, 43042}, {18329, 36280}, {21105, 23730}, {31151, 45341}, {32850, 50333}, {34805, 52985}, {39200, 51621}, {47695, 49704}, {48286, 49696}

X(53578) =crossdifference of every pair of points on line {6, 39026}
X(53578) =barycentric product X(14074)*X(35038)

---

X(53579) = INTERSECTION OF LINES TANGENT TO YFF PARABOLA AT X(32344) AND X(31182)

Barycentrics    (3*a - b - c)*(2*a^3 - a^2*b - b^3 - a^2*c + b^2*c + b*c^2 - c^3) : :
X(53579) = 3 X[910] + X[17747], X[910] + 3 X[51406], X[17747] - 3 X[40869], X[17747] - 9 X[51406], X[40869] - 3 X[51406]

X(53579) lies on these lines: {9, 2272}, {10, 3424}, {41, 6738}, {101, 519}, {118, 516}, {169, 1125}, {220, 43174}, {515, 5199}, {993, 38902}, {1146, 28236}, {1323, 3732}, {1376, 17355}, {1447, 3008}, {1743, 5435}, {1886, 41321}, {2348, 3911}, {2391, 17044}, {2976, 3667}, {3158, 3950}, {3207, 41006}, {3509, 5850}, {3731, 5281}, {3817, 5819}, {4251, 6744}, {4297, 6554}, {4847, 26258}, {4856, 20818}, {5011, 28228}, {5179, 28164}, {5540, 44675}, {5691, 27541}, {5743, 5745}, {5883, 23840}, {6168, 21362}, {6686, 17754}, {10443, 27382}, {11019, 40127}, {13405, 40131}, {15288, 52769}, {17060, 24685}, {18594, 27508}, {19925, 46835}, {26267, 40128}, {41325, 50808}

X(53579) = midpoint of X(i) and X(j) for these {i,j}: {101, 8074}, {910, 40869}, {1323, 3732}
X(53579) = reflection of X(1146) in X(44897)
X(53579) = X(43035)-Ceva conjugate of X(516)
X(53579) = X(i)-isoconjugate of X(j) for these (i,j): {103, 8056}, {911, 4373}, {2338, 19604}, {2400, 34080}, {2424, 27834}, {3445, 36101}, {18025, 38266}
X(53579) = X(i)-Dao conjugate of X(j) for these (i,j): {241, 10029}, {4521, 15634}, {23972, 4373}, {40621, 2400}, {45036, 36101}, {50441, 6557}
X(53579) = crossdifference of every pair of points on line {2424, 3445}
X(53579) = barycentric product X(i)*X(j) for these {i,j}: {145, 516}, {676, 43290}, {910, 18743}, {1456, 44720}, {1743, 30807}, {2398, 3667}, {3052, 35517}, {3161, 43035}, {3950, 14953}, {4248, 51366}, {4394, 42719}, {4546, 23973}, {5435, 40869}, {17747, 41629}, {31227, 51406}, {39126, 41339}
X(53579) = barycentric quotient X(i)/X(j) for these {i,j}: {145, 18025}, {516, 4373}, {910, 8056}, {1420, 43736}, {1456, 19604}, {1743, 36101}, {2426, 1293}, {3052, 103}, {3667, 2400}, {3756, 15634}, {5435, 52156}, {8643, 2424}, {17747, 4052}, {20818, 1815}, {30807, 40014}, {39063, 10029}, {40869, 6557}, {41339, 3680}, {43035, 27818}
X(53579) = {X(910),X(51406)}-harmonic conjugate of X(40869)

---

X(53580) = INTERSECTION OF LINES TANGENT TO YFF PARABOLA AT X(4375) AND X(31182)

Barycentrics    (3*a - b - c)*(b - c)*(a^2 - b*c) : :
X(53580) = 3 X[2] + X[48032], 2 X[8689] + X[25666], 2 X[25380] - 3 X[45675], 3 X[650] - X[48017], X[48017] + 3 X[48063], 3 X[659] + X[4010], X[659] + 3 X[4448], 5 X[659] + 3 X[4800], 7 X[659] + X[4810], 3 X[659] - X[4830], 3 X[3716] - X[4010], X[3716] - 3 X[4448], 5 X[3716] - 3 X[4800], 7 X[3716] - X[4810], 3 X[3716] + X[4830], X[4010] - 9 X[4448], and many others

X(53580) lies on these lines: {2, 48032}, {10, 28521}, {513, 6687}, {514, 676}, {522, 2977}, {650, 48017}, {659, 812}, {661, 47805}, {663, 47815}, {693, 48572}, {900, 35023}, {918, 13246}, {1125, 2832}, {1491, 48562}, {1635, 53343}, {2254, 4763}, {2526, 47778}, {2820, 28346}, {2827, 3035}, {2976, 3667}, {3737, 29402}, {3756, 18211}, {3776, 47800}, {3837, 45666}, {3907, 48331}, {4040, 4761}, {4088, 44433}, {4106, 48547}, {4147, 48329}, {4148, 8632}, {4162, 4964}, {4369, 4724}, {4379, 47974}, {4401, 6002}, {4458, 26275}, {4462, 4504}, {4468, 47801}, {4790, 48037}, {4804, 48240}, {4809, 48083}, {4824, 48251}, {4874, 48098}, {4893, 47697}, {4913, 48226}, {4928, 46403}, {5592, 10015}, {6184, 35111}, {6546, 47695}, {7659, 45313}, {9508, 45314}, {10196, 50333}, {13245, 53287}, {17072, 48561}, {20317, 28470}, {24720, 47803}, {26985, 48115}, {28225, 44902}, {28840, 45673}, {28871, 48040}, {30835, 47685}, {43067, 48009}, {44435, 48105}, {45315, 48023}, {45316, 48332}, {45674, 50357}, {47666, 48578}, {47694, 47811}, {47700, 48239}, {47701, 48250}, {47702, 47773}, {47757, 48068}, {47760, 48042}, {47761, 48073}, {47766, 48014}, {47771, 47972}, {47775, 48153}, {47777, 47985}, {47779, 48625}, {47780, 47933}, {47793, 48150}, {47794, 48111}, {47795, 47977}, {47796, 47936}, {47797, 48102}, {47798, 48094}, {47813, 47969}, {47818, 47970}, {47820, 47929}, {47821, 48049}, {47822, 48050}, {47826, 47991}, {47831, 48089}, {47832, 49289}, {47879, 49285}, {47882, 48015}, {47884, 53523}, {47892, 53558}, {47910, 48001}, {47926, 48237}, {47932, 48172}, {47992, 48162}, {47998, 48247}, {48002, 48248}, {48096, 48211}, {48101, 48161}, {48113, 48241}, {48118, 48223}, {48130, 48203}, {48139, 48174}, {48146, 48158}, {48156, 48626}, {48214, 50335}, {48220, 48399}, {48224, 48604}, {48234, 49292}, {48270, 48546}, {48367, 48565}

X(53580) = midpoint of X(i) and X(j) for these {i,j}: {650, 48063}, {659, 3716}, {2526, 48072}, {2976, 4925}, {3776, 48061}, {4010, 4830}, {4147, 48329}, {4369, 4724}, {4458, 48055}, {4462, 4504}, {4790, 48037}, {5592, 10015}, {43067, 48009}, {47694, 48000}, {47969, 49291}, {48050, 50358}
X(53580) = reflection of X(45678) in X(45666)
X(53580) = X(43041)-Ceva conjugate of X(812)
X(53580) = X(i)-isoconjugate of X(j) for these (i,j): {291, 1293}, {292, 27834}, {335, 34080}, {660, 3445}, {813, 8056}, {3572, 5382}, {4373, 34067}, {4562, 38266}, {4876, 38828}, {16945, 36801}
X(53580) = X(i)-Dao conjugate of X(j) for these (i,j): {8, 36801}, {3756, 4518}, {4521, 4444}, {19557, 27834}, {35119, 4373}, {39029, 1293}, {40621, 335}, {40623, 8056}, {45036, 660}
X(53580) = crossdifference of every pair of points on line {292, 3445}
X(53580) = barycentric product X(i)*X(j) for these {i,j}: {145, 812}, {238, 4462}, {239, 3667}, {350, 4394}, {659, 18743}, {1447, 4521}, {1743, 3766}, {1921, 8643}, {2403, 4432}, {3161, 43041}, {3570, 3756}, {3685, 30719}, {3716, 5435}, {3975, 51656}, {4010, 41629}, {4162, 10030}, {4248, 24459}, {4435, 39126}, {4448, 31227}, {4504, 17493}, {4729, 30940}, {4925, 6654}, {7212, 52352}, {14321, 33295}, {14425, 27922}, {27918, 43290}, {50456, 52353}
X(53580) = barycentric quotient X(i)/X(j) for these {i,j}: {145, 4562}, {238, 27834}, {659, 8056}, {812, 4373}, {1428, 38828}, {1743, 660}, {1914, 1293}, {2210, 34080}, {3052, 813}, {3161, 36801}, {3573, 5382}, {3667, 335}, {3684, 31343}, {3716, 6557}, {3756, 4444}, {3766, 40014}, {4010, 4052}, {4148, 6556}, {4162, 4876}, {4394, 291}, {4432, 2415}, {4435, 3680}, {4462, 334}, {4504, 30669}, {4521, 4518}, {4925, 40217}, {8632, 3445}, {8643, 292}, {14321, 43534}, {16948, 4584}, {18743, 4583}, {21950, 35352}, {30719, 7233}, {41629, 4589}, {43041, 27818}
X(53580) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {659, 4010, 4830}, {659, 4448, 3716}, {2976, 14425, 4925}, {3716, 4830, 4010}, {4462, 8643, 4504}, {4724, 47804, 4369}, {26275, 48055, 4458}, {47694, 47811, 48000}, {47778, 48072, 2526}, {47800, 48061, 3776}, {47813, 47969, 49291}, {47822, 50358, 48050}

---

X(53581) = X(190)-CEVA CONJUGATE OF X(42)

Barycentrics    a^4*(b - c)*(b + c)^2 : :
X(53581) = 3 X[14407] - X[40627], 3 X[14407] - 2 X[52592]

X(53581) lies on the Yff parabola and these lines: {10, 21056}, {37, 4083}, {101, 805}, {190, 886}, {213, 667}, {512, 798}, {514, 19565}, {649, 2664}, {669, 1924}, {881, 41267}, {2333, 53149}, {3294, 4063}, {3768, 6372}, {4024, 4039}, {4057, 21791}, {4367, 21763}, {4705, 46390}, {6544, 40586}, {6545, 40147}, {14838, 45882}, {21837, 50488}, {22229, 50491}, {32729, 32739}

X(53581) = reflection of X(i) in X(j) for these {i,j}: {21836, 37}, {40627, 52592}
X(53581) = isogonal conjugate of X(52612)
X(53581) = isogonal conjugate of the isotomic conjugate of X(4079)
X(53581) = X(i)-Ceva conjugate of X(j) for these (i,j): {101, 41267}, {190, 42}, {798, 50487}, {872, 4117}, {7109, 1084}, {32739, 1918}, {40147, 3122}
X(53581) = X(i)-isoconjugate of X(j) for these (i,j): {1, 52612}, {2, 4623}, {7, 4631}, {28, 52608}, {58, 4602}, {75, 4610}, {76, 52935}, {81, 670}, {86, 799}, {99, 274}, {110, 6385}, {190, 873}, {249, 40495}, {261, 4554}, {286, 4563}, {310, 662}, {314, 4573}, {333, 4625}, {513, 34537}, {514, 24037}, {552, 646}, {561, 4556}, {593, 6386}, {651, 18021}, {664, 52379}, {667, 44168}, {668, 1509}, {689, 16696}, {693, 4590}, {757, 1978}, {763, 27808}, {811, 17206}, {892, 16741}, {1043, 4635}, {1098, 46406}, {1333, 4609}, {1414, 28660}, {1434, 7257}, {1444, 6331}, {1447, 36806}, {1921, 36066}, {2185, 4572}, {3261, 24041}, {3888, 7307}, {4025, 46254}, {4033, 6628}, {4391, 7340}, {4565, 40072}, {4567, 52619}, {4569, 7058}, {4577, 16703}, {4589, 30940}, {4592, 44129}, {4593, 16887}, {4594, 8033}, {4596, 52572}, {4600, 7199}, {4601, 7192}, {4612, 6063}, {4615, 30939}, {4620, 18155}, {4632, 16709}, {4634, 16704}, {4636, 20567}, {4639, 33295}, {6064, 24002}, {7260, 17103}, {7304, 18830}, {9150, 30938}, {14296, 39292}, {15413, 18020}, {16702, 53080}, {16707, 35137}, {16732, 31614}, {16735, 35567}, {17187, 37204}, {17924, 47389}, {17930, 52137}, {23999, 30805}, {36036, 51370}, {37140, 40075}, {43187, 51369}
X(53581) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 52612}, {10, 4602}, {37, 4609}, {206, 4610}, {244, 6385}, {512, 514}, {1084, 310}, {2679, 51370}, {3005, 3261}, {5139, 44129}, {6631, 44168}, {9494, 23572}, {15267, 46406}, {17423, 17206}, {23301, 21191}, {32664, 4623}, {38978, 1921}, {38986, 274}, {38991, 18021}, {38996, 86}, {39025, 52379}, {39026, 34537}, {40368, 4556}, {40586, 670}, {40591, 52608}, {40600, 799}, {40607, 1978}, {40608, 28660}, {40627, 52619}, {50497, 7199}
X(53581) = trilinear pole of line {1084, 4117}
X(53581) = crossdifference of every pair of points on line {86, 310}
X(53581) = barycentric product X(i)*X(j) for these {i,j}: {1, 50487}, {6, 4079}, {10, 669}, {31, 4705}, {32, 4024}, {37, 798}, {42, 512}, {71, 2489}, {101, 3124}, {115, 32739}, {163, 21833}, {181, 663}, {190, 1084}, {210, 51641}, {213, 661}, {292, 46390}, {313, 9426}, {321, 1924}, {513, 872}, {514, 7109}, {523, 1918}, {560, 4036}, {594, 1919}, {647, 2333}, {649, 1500}, {664, 7063}, {667, 756}, {668, 4117}, {688, 18082}, {692, 2643}, {799, 52065}, {810, 1824}, {881, 4039}, {1018, 3121}, {1042, 4524}, {1089, 1980}, {1126, 8663}, {1245, 50494}, {1254, 8641}, {1331, 2971}, {1334, 7180}, {1356, 3699}, {1400, 3709}, {1402, 4041}, {1501, 52623}, {1576, 21043}, {1577, 2205}, {1826, 3049}, {1911, 4155}, {1974, 4064}, {1977, 4103}, {1978, 9427}, {2054, 17990}, {2084, 18098}, {2171, 3063}, {2200, 2501}, {3122, 4557}, {3248, 40521}, {4105, 7143}, {4561, 42068}, {4570, 22260}, {4600, 23099}, {4826, 28625}, {6187, 42666}, {6378, 20979}, {7064, 43924}, {7148, 8640}, {8750, 20975}, {8754, 32656}, {18105, 21035}, {21056, 51951}, {21131, 23990}, {21759, 21834}, {23493, 50491}, {40147, 52592}, {46148, 51906}
X(53581) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 52612}, {10, 4609}, {31, 4623}, {32, 4610}, {37, 4602}, {41, 4631}, {42, 670}, {71, 52608}, {101, 34537}, {181, 4572}, {190, 44168}, {213, 799}, {512, 310}, {560, 52935}, {661, 6385}, {663, 18021}, {667, 873}, {669, 86}, {688, 16887}, {692, 24037}, {756, 6386}, {798, 274}, {872, 668}, {1084, 514}, {1356, 3676}, {1402, 4625}, {1500, 1978}, {1501, 4556}, {1918, 99}, {1919, 1509}, {1924, 81}, {1980, 757}, {2084, 16703}, {2200, 4563}, {2205, 662}, {2333, 6331}, {2489, 44129}, {2491, 51370}, {2643, 40495}, {2971, 46107}, {3049, 17206}, {3063, 52379}, {3121, 7199}, {3122, 52619}, {3124, 3261}, {3709, 28660}, {4024, 1502}, {4036, 1928}, {4041, 40072}, {4064, 40050}, {4079, 76}, {4117, 513}, {4155, 18891}, {4705, 561}, {7063, 522}, {7109, 190}, {7143, 52937}, {8663, 1269}, {9426, 58}, {9427, 649}, {9447, 4612}, {9448, 4636}, {9494, 17187}, {14598, 36066}, {18082, 42371}, {18098, 37204}, {21043, 44173}, {21727, 40088}, {21755, 16737}, {21815, 33946}, {21833, 20948}, {22260, 21207}, {23099, 3120}, {23216, 1459}, {23610, 3122}, {32656, 47389}, {32739, 4590}, {36417, 52919}, {40729, 7260}, {41267, 4576}, {42068, 7649}, {42666, 40075}, {46390, 1921}, {50487, 75}, {50494, 44154}, {51858, 36806}, {52065, 661}, {52623, 40362}
X(53581) = {X(14407),X(40627)}-harmonic conjugate of X(52592)

---

X(53582) = X(190)-CEVA CONJUGATE OF X(17790)

Barycentrics    (a - b)*(a - c)*(2*a - b - c)^2 : :
X(53582) = 3 X[4370] - X[35092], X[190] + 3 X[1016], 3 X[190] + X[4555], 5 X[190] + 3 X[6631], 7 X[190] - 3 X[32028], X[190] - 3 X[32094], 5 X[190] - 3 X[32106], 7 X[190] + 9 X[34024], 9 X[1016] - X[4555], 5 X[1016] - X[6631], 3 X[1016] - X[6633], 7 X[1016] + X[32028], 5 X[1016] + X[32106], 7 X[1016] - 3 X[34024], 5 X[4555] - 9 X[6631], X[4555] - 3 X[6633], and many others

X(53582) lies on the Yff parabola and these lines: {44, 519}, {101, 6079}, {190, 514}, {545, 6549}, {644, 3239}, {649, 1018}, {728, 36944}, {1023, 4169}, {1145, 3234}, {1743, 52556}, {3008, 16594}, {3230, 17475}, {3912, 24593}, {3952, 48339}, {4024, 4115}, {4375, 23891}, {4422, 40488}, {4546, 6065}, {4791, 42722}, {6547, 36954}, {6632, 42372}, {17262, 49751}, {17369, 25031}, {30578, 46790}, {30579, 46795}

X(53582) = midpoint of X(i) and X(j) for these {i,j}: {190, 6633}, {1016, 32094}, {6631, 32106}
X(53582) =reflection of X(6547) in X(36954)
X(53582) = X(46972)-complementary conjugate of X(21241)
X(53582) =X(i)-Ceva conjugate of X(j) for these (i,j): {190, 17780}, {1016, 519}
X(53582) =X(i)-isoconjugate of X(j) for these (i,j): {88, 23345}, {106, 1022}, {244, 4638}, {513, 2226}, {649, 679}, {693, 41935}, {1015, 4618}, {1318, 3669}, {2087, 39414}, {3257, 43922}, {3733, 30575}, {6548, 9456}, {6549, 32665}
X(53582) =X(i)-Dao conjugate of X(j) for these (i,j): {214, 1022}, {519, 514}, {900, 6545}, {1647, 1086}, {4370, 6548}, {5375, 679}, {17780, 27191}, {35092, 6549}, {36912, 23598}, {39026, 2226}, {52872, 4049}
X(53582) =cevapoint of X(i) and X(j) for these (i,j): {3251, 21821}, {4370, 6544}
X(53582) =trilinear pole of line {678, 4152}
X(53582) =crossdifference of every pair of points on line {23345, 43922}
X(53582) =barycentric product X(i)*X(j) for these {i,j}: {44, 24004}, {100, 4738}, {101, 36791}, {190, 4370}, {519, 17780}, {664, 4152}, {668, 678}, {799, 21821}, {1016, 6544}, {1017, 1978}, {1018, 16729}, {1023, 4358}, {1252, 52627}, {1317, 3699}, {3251, 7035}, {3264, 23344}, {3911, 30731}, {4076, 39771}, {4169, 16704}, {4543, 4998}, {4555, 8028}, {4561, 42070}, {4723, 23703}, {6632, 35092}, {42372, 46050}
X(53582) =barycentric quotient X(i)/X(j) for these {i,j}: {44, 1022}, {100, 679}, {101, 2226}, {519, 6548}, {678, 513}, {765, 4618}, {900, 6549}, {902, 23345}, {1017, 649}, {1018, 30575}, {1023, 88}, {1252, 4638}, {1317, 3676}, {1960, 43922}, {3251, 244}, {3689, 23838}, {3939, 1318}, {3943, 4049}, {4152, 522}, {4169, 4080}, {4370, 514}, {4542, 21132}, {4543, 11}, {4738, 693}, {4767, 36594}, {4908, 23598}, {6544, 1086}, {8028, 900}, {9268, 39414}, {16729, 7199}, {17780, 903}, {21821, 661}, {22371, 1459}, {23344, 106}, {24004, 20568}, {30731, 4997}, {32739, 41935}, {33922, 1647}, {35092, 6545}, {36791, 3261}, {39771, 1358}, {42070, 7649}, {42084, 764}, {46050, 24188}, {52627, 23989}
X(53582) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {190, 1016, 6633}, {1016, 32028, 34024}, {1023, 30731, 4169}, {6633, 32094, 190}

---

X(53583) = X(190)-CEVA CONJUGATE OF X(3912)

Barycentrics    (b - c)*(-(a*b) + b^2 - a*c + c^2)^2 : :
X(53583) = 4 X[4444] - 3 X[6545], 9 X[6544] - 8 X[27929]

X(53583) lies on the Yff parabola and these lines: {2, 2400}, {8, 514}, {63, 649}, {69, 23730}, {75, 42462}, {321, 693}, {522, 4659}, {525, 2292}, {536, 23757}, {594, 21133}, {657, 48070}, {883, 1025}, {905, 25066}, {918, 2254}, {1281, 2786}, {1577, 23100}, {2345, 21202}, {3126, 14506}, {3234, 53337}, {3578, 28840}, {3798, 10196}, {3912, 52228}, {4091, 44178}, {4122, 36848}, {4391, 20880}, {4750, 4763}, {4786, 31992}, {6002, 48890}, {7046, 53150}, {7265, 22011}, {9318, 24400}, {14475, 47787}, {17496, 17741}, {17780, 38376}, {21114, 21958}, {24331, 49288}, {25381, 30519}, {28734, 47795}, {28878, 47698}, {28894, 47703}, {28898, 49483}, {29212, 36480}, {47671, 47923}, {47690, 48015}

X(53583) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2115, 37781}, {9499, 150}, {9500, 149}
X(53583) = X(i)-Ceva conjugate of X(j) for these (i,j): {190, 3912}, {4437, 35094}
X(53583) = X(i)-isoconjugate of X(j) for these (i,j): {100, 41934}, {101, 51838}, {105, 919}, {294, 32735}, {673, 32666}, {692, 6185}, {1438, 36086}, {1462, 52927}, {2195, 36146}, {5377, 43929}
X(53583) = X(i)-Dao conjugate of X(j) for these (i,j): {518, 101}, {918, 514}, {1015, 51838}, {1086, 6185}, {3126, 1024}, {6184, 36086}, {8054, 41934}, {17435, 1}, {17755, 666}, {27918, 6654}, {35094, 673}, {36905, 927}, {38980, 105}, {38989, 1438}, {39014, 2195}, {39046, 919}, {39063, 36146}
X(53583) = crossdifference of every pair of points on line {1438, 2210}
X(53583) = barycentric product X(i)*X(j) for these {i,j}: {75, 3126}, {190, 35094}, {514, 4437}, {693, 4712}, {918, 3912}, {1362, 35519}, {1577, 16728}, {1978, 35505}, {2254, 3263}, {3261, 6184}, {3323, 3699}, {3717, 43042}, {3932, 23829}, {4025, 34337}, {4088, 30941}, {9436, 50333}, {17060, 48070}, {18157, 24290}, {40495, 42079}
X(53583) = barycentric quotient X(i)/X(j) for these {i,j}: {241, 36146}, {513, 51838}, {514, 6185}, {518, 36086}, {649, 41934}, {665, 1438}, {672, 919}, {883, 39293}, {918, 673}, {926, 2195}, {1026, 5377}, {1362, 109}, {1458, 32735}, {1642, 52227}, {2223, 32666}, {2254, 105}, {2340, 52927}, {3126, 1}, {3263, 51560}, {3323, 3676}, {3675, 1027}, {3717, 36802}, {3912, 666}, {4088, 13576}, {4437, 190}, {4712, 100}, {6184, 101}, {9436, 927}, {16728, 662}, {17060, 3732}, {17435, 1024}, {20504, 14267}, {20776, 32656}, {23102, 1026}, {24290, 18785}, {33570, 2280}, {34337, 1897}, {35094, 514}, {35505, 649}, {39686, 32739}, {40704, 34085}, {42071, 8750}, {42079, 692}, {50333, 14942}, {52304, 21132}, {53539, 1416}, {53544, 1462}, {53550, 36057}

---

X(53584) = X(190)-CEVA CONJUGATE OF X(3679)

Barycentrics    (a - 2*b - 2*c)^2*(b - c) : :
X(53584) = 5 X[4120] - 2 X[47764], 2 X[4120] + X[47873], 4 X[4120] - X[48544], X[31147] + 2 X[47870], 2 X[45343] + X[47769], 4 X[47764] + 5 X[47873], 8 X[47764] - 5 X[48544], 2 X[47873] + X[48544], 23 X[649] - 32 X[2527], X[649] + 8 X[3700], 5 X[649] - 8 X[47767], X[649] - 4 X[47874], 5 X[649] + 4 X[48266], 4 X[2527] + 23 X[3700], 20 X[2527] - 23 X[47767], and many others

X(53584) lies on the Yff parabola and these lines: {514, 4120}, {522, 6544}, {649, 900}, {661, 28151}, {3952, 48339}, {4024, 28169}, {4375, 31992}, {4379, 52620}, {4382, 48557}, {4671, 4791}, {4777, 4800}, {4789, 28886}, {4813, 28220}, {4958, 47881}, {6548, 30519}, {8643, 29058}, {14435, 47766}, {14475, 47787}, {27486, 45684}, {29370, 47832}, {30835, 47894}, {31035, 48321}, {45315, 48437}, {47876, 50482}, {48577, 53339}

X(53584) = reflection of X(i) in X(j) for these {i,j}: {14435, 47766}, {14475, 47787}, {23598, 4791}, {27486, 45684}
X(53584) = X(190)-Ceva conjugate of X(3679)
X(53584) = X(i)-isoconjugate of X(j) for these (i,j): {89, 4588}, {109, 30607}, {2163, 4604}, {4597, 28607}, {34073, 39704}
X(53584) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 30607}, {4777, 514}, {36911, 4597}, {40587, 4604}
X(53584) = crossdifference of every pair of points on line {995, 2163}
X(53584) = barycentric product X(i)*X(j) for these {i,j}: {45, 4791}, {75, 4825}, {523, 4803}, {3679, 4777}, {4125, 4833}, {4671, 4893}, {4752, 4957}, {4873, 43052}, {4908, 23598}, {4931, 5235}, {4944, 5219}
X(53584) = barycentric quotient X(i)/X(j) for these {i,j}: {45, 4604}, {650, 30607}, {2177, 4588}, {3679, 4597}, {4752, 5385}, {4770, 53114}, {4775, 2163}, {4777, 39704}, {4791, 20569}, {4803, 99}, {4814, 2320}, {4825, 1}, {4838, 30587}, {4893, 89}, {4931, 30588}, {4944, 30608}, {23598, 40833}
X(53584) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4120, 47873, 48544}, {4931, 4944, 4893}, {47874, 48266, 47767}

---

X(53585) = X(190)-CEVA CONJUGATE OF X(1698)

Barycentrics    (b - c)*(a + 2*b + 2*c)^2 : :
X(53585) = 25 X[649] - 24 X[4773], 9 X[649] - 8 X[4976], 3 X[649] - 4 X[48275], 7 X[649] - 8 X[48276], 5 X[649] - 4 X[48277], 3 X[649] - 2 X[50482], 27 X[4773] - 25 X[4976], 18 X[4773] - 25 X[48275], 21 X[4773] - 25 X[48276], 6 X[4773] - 5 X[48277], 36 X[4773] - 25 X[50482], 2 X[4976] - 3 X[48275], 7 X[4976] - 9 X[48276], 10 X[4976] - 9 X[48277], 4 X[4976] - 3 X[50482], and many others

X(53585) lies on the Yff parabola and these lines: {514, 48429}, {523, 649}, {661, 28151}, {3239, 4988}, {4024, 28147}, {4375, 47659}, {4379, 47657}, {4382, 47655}, {4608, 48141}, {4727, 4802}, {4777, 50525}, {4893, 47669}, {4979, 28165}, {6544, 6590}, {14779, 47996}, {17161, 48577}, {21204, 45746}, {28161, 50522}, {28175, 48266}, {28199, 48019}, {28878, 48438}, {28894, 47670}, {30835, 47792}, {31182, 45745}, {31207, 46915}, {47665, 47908}, {47671, 47923}, {47991, 48437}

X(53585) = reflection of X(i) in X(j) for these {i,j}: {4382, 47655}, {4813, 4838}, {14779, 47996}, {47669, 48397}, {47908, 47665}, {47923, 47671}, {47926, 47659}, {48141, 4608}, {50482, 48275}
X(53585) = X(190)-Ceva conjugate of X(1698)
X(53585) = X(i)-isoconjugate of X(j) for these (i,j): {101, 30597}, {8652, 25417}, {32042, 34819}
X(53585) = X(i)-Dao conjugate of X(j) for these (i,j): {1015, 30597}, {4802, 514}, {51572, 37211}, {53167, 30598}
X(53585) = barycentric product X(i)*X(j) for these {i,j}: {190, 53167}, {1698, 4802}, {4066, 4840}, {4654, 4820}, {4813, 28605}, {4823, 16777}, {4834, 30596}, {4838, 5333}
X(53585) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 30597}, {1698, 32042}, {4802, 30598}, {4813, 25417}, {4820, 42030}, {4826, 28625}, {16777, 37211}, {53167, 514}
X(53585) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {47669, 48397, 4893}, {48275, 50482, 649}

---

X(53586) = X(190)-CEVA CONJUGATE OF X(3616)

Barycentrics    (b - c)*(3*a + b + c)^2 : :
X(53586) = 16 X[2529] - 11 X[3239], 18 X[2529] - 11 X[14321], 9 X[3239] - 8 X[14321], 9 X[4979] - X[47669], 5 X[4979] - X[48277], 5 X[47669] - 9 X[48277], X[47677] - 3 X[48013], 3 X[661] - 4 X[31182], X[4024] - 3 X[49293], X[4024] + 3 X[50525], 5 X[4932] - 3 X[21204], 5 X[4765] - 6 X[4773], 3 X[4765] - 2 X[4841], 3 X[4773] - 5 X[4790], 9 X[4773] - 5 X[4841], and many others

X(53586) lies on the Yff parabola and these lines: {513, 2529}, {514, 4380}, {649, 28225}, {661, 31182}, {3234, 5375}, {3667, 4024}, {4375, 4932}, {4700, 4706}, {4813, 6544}, {4958, 6590}, {4962, 48275}, {4976, 28229}, {6006, 48276}, {7658, 47763}, {14351, 47883}, {23731, 48574}, {26853, 47674}, {28161, 50522}, {30723, 48580}, {44551, 47995}, {47758, 47937}, {47762, 47978}, {47768, 48019}, {48034, 48567}, {48414, 48577}, {48420, 49297}

X(53586) = midpoint of X(i) and X(j) for these {i,j}: {48067, 48107}, {49293, 50525}
X(53586) = reflection of X(i) in X(j) for these {i,j}: {4765, 4790}, {4813, 43061}, {47981, 7658}
X(53586) = X(190)-Ceva conjugate of X(3616)
X(53586) = X(i)-isoconjugate of X(j) for these (i,j): {2334, 4606}, {5936, 34074}, {8694, 25430}
X(53586) = X(i)-Dao conjugate of X(j) for these (i,j): {4778, 514}, {51576, 4606}
X(53586) = crossdifference of every pair of points on line {1191, 2334}
X(53586) = barycentric product X(i)*X(j) for these {i,j}: {99, 52332}, {391, 30723}, {1449, 4801}, {3361, 4811}, {3616, 4778}, {4765, 21454}, {4790, 19804}, {4841, 42028}, {5257, 48580}
X(53586) = barycentric quotient X(i)/X(j) for these {i,j}: {1449, 4606}, {4778, 5936}, {4790, 25430}, {4801, 40023}, {21454, 4624}, {42028, 4633}, {52332, 523}
X(53586) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4813, 48576, 43061}, {47763, 47981, 7658}

---

X(53587) = X(190)-CEVA CONJUGATE OF X(1125)

Barycentrics    (b - c)*(2*a + b + c)^2 : :
X(53587) = 9 X[4024] - 8 X[4820], 5 X[4024] - 4 X[48266], 3 X[4024] - 4 X[48275], 7 X[4024] - 8 X[48397], 10 X[4820] - 9 X[48266], 2 X[4820] - 3 X[48275], 7 X[4820] - 9 X[48397], 4 X[4820] - 9 X[50522], 3 X[48266] - 5 X[48275], 7 X[48266] - 10 X[48397], 2 X[48266] - 5 X[50522], 7 X[48275] - 6 X[48397], 2 X[48275] - 3 X[50522], 4 X[48397] - 7 X[50522], and many others

X(53587) lies on the Yff parabola and these lines: {513, 4024}, {514, 14779}, {649, 4778}, {661, 1213}, {2786, 48438}, {3239, 4813}, {3578, 28840}, {4120, 48019}, {4369, 41818}, {4375, 6545}, {4379, 47981}, {4785, 47671}, {4790, 28220}, {4838, 28217}, {4893, 31182}, {4969, 4976}, {6546, 31290}, {16892, 28859}, {17422, 21132}, {20295, 48418}, {21104, 47900}, {21116, 49298}, {28195, 48277}, {28213, 47669}, {28867, 48429}, {28878, 48138}, {28886, 47662}, {28902, 48130}, {31147, 47978}, {31148, 47988}, {43067, 47937}, {45746, 48071}, {47667, 48016}, {47769, 48588}, {47771, 47984}, {47790, 48592}, {47791, 48041}, {47873, 49284}, {47890, 47903}, {47907, 49296}, {47908, 48060}, {47926, 48067}, {47958, 48425}, {47979, 48578}, {47991, 48567}, {47995, 48577}, {48076, 49281}, {48079, 48416}, {48082, 49282}, {49291, 49297}

X(53587) = reflection of X(i) in X(j) for these {i,j}: {4024, 50522}, {4813, 49293}, {4988, 4979}, {16892, 48107}, {23731, 7192}, {45746, 48071}, {47667, 48016}, {47900, 21104}, {47903, 47890}, {47907, 49296}, {47908, 48060}, {47926, 48067}, {47937, 43067}, {48019, 48276}, {48076, 49281}, {48082, 49282}, {49297, 49291}
X(53587) = X(i)-Ceva conjugate of X(j) for these (i,j): {190, 1125}, {8050, 8013}
X(53587) = X(i)-isoconjugate of X(j) for these (i,j): {110, 30582}, {163, 30594}, {1126, 37212}, {1255, 8701}, {4596, 52555}, {6540, 28615}
X(53587) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 30594}, {244, 30582}, {1213, 6540}, {3120, 6539}, {3647, 37212}, {4977, 514}, {16726, 32014}, {35076, 1268}
X(53587) = crossdifference of every pair of points on line {1126, 1203}
X(53587) = barycentric product X(i)*X(j) for these {i,j}: {190, 35076}, {513, 6533}, {553, 4976}, {1100, 4978}, {1125, 4977}, {1269, 50512}, {3686, 30724}, {4359, 4979}, {4983, 16709}, {4985, 32636}, {4988, 8025}, {6367, 30593}, {7192, 8040}, {41014, 46542}
X(53587) = barycentric quotient X(i)/X(j) for these {i,j}: {523, 30594}, {661, 30582}, {1100, 37212}, {1125, 6540}, {2308, 8701}, {4976, 4102}, {4977, 1268}, {4978, 32018}, {4979, 1255}, {4984, 31011}, {4988, 6539}, {6367, 6538}, {6533, 668}, {8025, 4632}, {8040, 3952}, {35076, 514}, {50512, 1126}
X(53587) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4979, 4988, 4984}, {7192, 23731, 6545}, {48019, 48276, 4120}

---

X(53588) = INTERSECTION OF LINES TANGENT TO YFF DUAL PARABOLA AT X(2) AND X(554)

Barycentrics    (a - b - c)*(a + b - c)*(a - b + c) - 2*Sqrt[3]*a*S : :
Barycentrics    Sqrt[3]*a + 2*r : :
X(53588) = Sqrt[3]*s*X[1] + 3*r*X[2]

X(53588) lies on these lines: {1, 2}, {9, 37640}, {11, 11706}, {13, 226}, {37, 396}, {44, 43228}, {45, 49947}, {55, 21476}, {56, 21475}, {57, 1277}, {81, 5353}, {165, 37830}, {298, 3879}, {299, 4357}, {333, 41638}, {395, 1100}, {465, 46974}, {466, 17102}, {473, 1785}, {516, 51749}, {532, 37631}, {553, 554}, {559, 3911}, {619, 3666}, {940, 49594}, {988, 37173}, {1449, 37641}, {1699, 37833}, {3180, 4416}, {3247, 11488}, {3303, 21481}, {3304, 21480}, {3452, 5240}, {3663, 49605}, {3723, 23302}, {3817, 51750}, {4364, 33458}, {4464, 46176}, {4643, 5859}, {4682, 50438}, {5239, 5745}, {5243, 16644}, {5266, 37340}, {5274, 30338}, {5281, 30339}, {5357, 32911}, {5717, 37145}, {9763, 41312}, {10980, 30344}, {11297, 48824}, {11298, 48819}, {11303, 13161}, {11791, 42976}, {15772, 52375}, {16645, 16884}, {16666, 43229}, {16667, 30414}, {16672, 49905}, {16673, 30415}, {16676, 49813}, {17045, 44382}, {17390, 44383}, {21869, 37607}, {21898, 37573}, {30357, 43151}, {30382, 33653}, {34064, 49571}, {35303, 37599}, {35304, 37589}, {36669, 37634}, {37172, 37552}, {37341, 37592}, {37786, 50093}, {49565, 50860}
X(53588) = midpoint of X(1999) and X(40713)
X(53588) = complement of X(40714)
X(53588) = complement of the isotomic conjugate of X(554)
X(53588) = X(i)-complementary conjugate of X(j) for these (i,j): {554, 2887}, {33653, 1329}, {33654, 141}, {36073, 513}
X(53588) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 37795, 40713}, {2, 40713, 10}, {226, 1082, 3639}, {554, 37772, 553}, {1125, 49590, 10}, {33397, 39153, 3639}

---

X(53589) = INTERSECTION OF LINES TANGENT TO YFF DUAL PARABOLA AT X(2) AND X(1081)

Barycentrics    (a - b - c)*(a + b - c)*(a - b + c) + 2*Sqrt[3]*a*S : :
Barycentrics    Sqrt[3]*a - 2*r : :
X(53589) = Sqrt[3]*s*X[1] - 3*r*X[2]

X(53589) lies on these lines: {1, 2}, {9, 37641}, {11, 11705}, {14, 226}, {37, 395}, {44, 43229}, {45, 49948}, {55, 21475}, {56, 21476}, {57, 1276}, {81, 5357}, {165, 37833}, {298, 4357}, {299, 3879}, {333, 41648}, {396, 1100}, {465, 17102}, {466, 46974}, {472, 1785}, {516, 51750}, {533, 37631}, {553, 1081}, {618, 3666}, {940, 49595}, {988, 37172}, {1082, 3911}, {1251, 30383}, {1449, 37640}, {1699, 37830}, {3181, 4416}, {3247, 11489}, {3303, 21480}, {3304, 21481}, {3452, 5239}, {3663, 49604}, {3723, 23303}, {3817, 51749}, {4364, 33459}, {4464, 46175}, {4643, 5858}, {4682, 50439}, {5240, 5745}, {5242, 16645}, {5266, 37341}, {5274, 30339}, {5281, 30338}, {5353, 32911}, {5717, 37144}, {9761, 41312}, {10980, 30345}, {11297, 48819}, {11298, 48824}, {11304, 13161}, {11790, 42977}, {15771, 52375}, {16644, 16884}, {16666, 43228}, {16667, 30415}, {16672, 49906}, {16673, 30414}, {16676, 49812}, {17045, 44383}, {17390, 44382}, {21869, 37573}, {21898, 37607}, {30356, 43151}, {34064, 49572}, {35303, 37589}, {35304, 37599}, {36668, 37634}, {37173, 37552}, {37340, 37592}, {37785, 50093}, {49567, 50859}

X(53589) = midpoint of X(1999) and X(40714)
X(53589) = complement of X(40713)
X(53589) = complement of the isotomic conjugate of X(1081)
X(53589) = X(i)-complementary conjugate of X(j) for these (i,j): {1081, 2887}, {1251, 1329}, {2306, 141}, {36072, 513}
X(53589) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 37794, 40714}, {2, 40714, 10}, {226, 559, 3638}, {1081, 37773, 553}, {1125, 49591, 10}, {33396, 39152, 3638}

---

X(53590) = INTERSECTION OF LINES TANGENT TO YFF DUAL PARABOLA AT X(27) AND X(335)

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

X(53590) lies on these lines: {10, 37096}, {27, 58}, {39, 226}, {57, 3497}, {239, 38456}, {244, 857}, {335, 726}, {384, 17023}, {516, 28026}, {982, 37445}, {984, 37097}, {1086, 8680}, {1125, 16050}, {1826, 24173}, {1931, 33129}, {3008, 24630}, {3772, 33863}, {3914, 20985}, {4384, 17680}, {5249, 24214}, {6591, 29013}, {8804, 24162}, {9597, 24268}, {17673, 24603}, {17799, 32857}, {27559, 28244}, {28074, 31042}, {28082, 31015}, {28096, 31014}, {33144, 37280}, {50201, 51706}

---

X(53591) = INTERSECTION OF LINES TANGENT TO YFF DUAL PARABOLA AT X(27) AND X(673)

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

X(53591) lies on these lines: {1, 379}, {2, 35}, {10, 37086}, {11, 1375}, {19, 17861}, {27, 58}, {32, 3772}, {36, 14953}, {48, 24202}, {57, 14377}, {63, 1752}, {80, 48381}, {149, 28757}, {214, 24559}, {226, 4251}, {238, 516}, {239, 758}, {284, 34830}, {321, 17744}, {333, 20888}, {448, 759}, {497, 52015}, {499, 24580}, {610, 24179}, {614, 24584}, {857, 3583}, {950, 36019}, {1001, 37075}, {1111, 7291}, {1125, 16054}, {1203, 4295}, {1210, 37389}, {1429, 17761}, {1509, 52361}, {1723, 18655}, {1731, 8680}, {1751, 24310}, {1781, 17863}, {2210, 3120}, {2802, 40863}, {3086, 24604}, {3500, 13478}, {3671, 41245}, {3739, 24335}, {3912, 50200}, {4000, 16470}, {4047, 17348}, {4302, 14021}, {4314, 17682}, {4329, 24779}, {4330, 37111}, {4384, 12514}, {4512, 16832}, {4872, 24781}, {5011, 16609}, {5225, 30809}, {5247, 29743}, {5251, 37076}, {5267, 35935}, {5271, 41229}, {5272, 24605}, {5358, 31926}, {5540, 30807}, {6253, 19542}, {6284, 30810}, {7292, 24585}, {7297, 16732}, {7384, 12558}, {7406, 41785}, {7522, 19763}, {8301, 20544}, {9669, 31184}, {10072, 24608}, {10572, 25935}, {10593, 31186}, {10896, 30808}, {12511, 37416}, {12526, 16833}, {12559, 16834}, {12567, 16819}, {12609, 17023}, {13576, 40910}, {16545, 24204}, {16546, 24205}, {16547, 53510}, {17681, 29604}, {17923, 39052}, {17924, 29013}, {20236, 26998}, {24602, 29456}, {24609, 26363}, {24610, 26959}, {25453, 52243}, {26006, 30384}, {29598, 32772}, {29775, 37607}, {30107, 41236}, {33140, 46574}, {34529, 37797}, {34852, 51376}, {37185, 52018}, {40955, 40975}
X(53591) = {X(673),X(6996)}-harmonic conjugate of X(3008)

---

X(53592) = INTERSECTION OF LINES TANGENT TO YFF DUAL PARABOLA AT X(27) AND X(1440)

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

X(53592) lies on these lines: {1, 9799}, {4, 1394}, {10, 52366}, {27, 58}, {34, 6245}, {35, 37048}, {77, 6837}, {84, 278}, {222, 946}, {223, 6847}, {226, 36746}, {255, 516}, {269, 1256}, {475, 20205}, {991, 1745}, {1012, 5930}, {1076, 12572}, {1125, 4303}, {1210, 1448}, {1465, 6705}, {2956, 4295}, {3008, 15803}, {3182, 7490}, {3338, 7177}, {3664, 12047}, {4306, 44675}, {4311, 37817}, {4340, 9612}, {5691, 51375}, {7365, 10396}, {10374, 51490}, {12688, 51616}, {12699, 23072}, {17102, 43035}, {37252, 41402}, {37447, 40960}, {40950, 51654}

X(53592) = {X(18623),X(37434)}-harmonic conjugate of X(1)

---

X(53593) = INTERSECTION OF LINES TANGENT TO YFF DUAL PARABOLA AT X(75) AND X(1659)

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

X(53593) lies on these lines: {1, 488}, {2, 31583}, {10, 75}, {57, 482}, {65, 8243}, {69, 49593}, {176, 5265}, {388, 481}, {492, 49624}, {519, 32808}, {527, 13883}, {988, 5391}, {1362, 22106}, {3008, 31595}, {3241, 49614}, {3875, 49592}, {3946, 13936}, {4000, 7090}, {4419, 14121}, {4655, 49348}, {13388, 31531}, {13911, 17276}, {13973, 17301}, {25055, 49616}, {32798, 48818}, {32921, 49347}, {45444, 49453}, {49547, 50114}

---

X(53594) = INTERSECTION OF LINES TANGENT TO YFF DUAL PARABOLA AT X(75) AND X(4373)

Barycentrics    a*b + b^2 + a*c - 6*b*c + c^2 : :
X(53594) = X[7] - 9 X[36588], X[17151] + 9 X[36588], 3 X[142] - 2 X[17243], 4 X[142] - 3 X[29600], X[3950] - 4 X[7263], 3 X[3950] - 4 X[17243], 2 X[3950] - 3 X[29600], 3 X[7263] - X[17243], 8 X[7263] - 3 X[29600], 8 X[17243] - 9 X[29600], X[144] - 3 X[16833], 2 X[4078] - 3 X[38204], 3 X[6173] - X[17314]

X(53594) lies on these lines: {1, 4452}, {2, 31326}, {7, 519}, {8, 4373}, {9, 17132}, {10, 75}, {63, 19819}, {69, 3625}, {77, 22837}, {141, 4058}, {142, 536}, {144, 16833}, {145, 4888}, {190, 15828}, {192, 4098}, {193, 50019}, {226, 42051}, {239, 31300}, {244, 22214}, {312, 24175}, {314, 17205}, {319, 903}, {320, 49677}, {321, 24177}, {346, 4859}, {517, 43172}, {522, 24389}, {527, 4361}, {545, 17348}, {551, 4021}, {594, 50092}, {596, 3668}, {599, 4060}, {740, 5542}, {894, 50114}, {1001, 28557}, {1018, 28351}, {1086, 2321}, {1100, 49727}, {1111, 20895}, {1122, 10914}, {1125, 3672}, {1278, 3912}, {1449, 7222}, {1743, 4402}, {2325, 17278}, {3008, 3729}, {3161, 31183}, {3241, 32105}, {3244, 3664}, {3263, 4135}, {3452, 4052}, {3626, 4346}, {3632, 4902}, {3635, 3945}, {3661, 4821}, {3662, 4431}, {3686, 17119}, {3707, 17334}, {3731, 31211}, {3739, 3986}, {3755, 49483}, {3759, 49722}, {3811, 4328}, {3817, 48643}, {3826, 28555}, {3840, 4441}, {3879, 4896}, {3946, 4363}, {3973, 4488}, {4000, 4659}, {4029, 4718}, {4044, 20892}, {4054, 17495}, {4078, 28516}, {4133, 49676}, {4349, 32921}, {4353, 50314}, {4356, 24325}, {4359, 4656}, {4360, 50108}, {4364, 4739}, {4395, 17351}, {4399, 17345}, {4416, 4440}, {4461, 17284}, {4464, 17378}, {4480, 17349}, {4644, 4856}, {4665, 17235}, {4667, 4852}, {4669, 17274}, {4681, 34824}, {4685, 20347}, {4688, 5257}, {4691, 5232}, {4701, 32099}, {4709, 49505}, {4764, 17234}, {4772, 17247}, {4780, 49479}, {4788, 17244}, {4847, 17155}, {4851, 17133}, {4924, 49499}, {4971, 17376}, {5249, 50106}, {5271, 19826}, {5437, 42047}, {5493, 10444}, {5750, 17118}, {5847, 30424}, {6173, 17314}, {6646, 50095}, {6666, 17262}, {6736, 17885}, {6765, 7274}, {7179, 49554}, {7190, 22836}, {7227, 17382}, {7229, 29598}, {7231, 50112}, {7238, 17372}, {7264, 24993}, {7271, 12629}, {9843, 17863}, {10442, 28228}, {10443, 29069}, {10447, 50608}, {10481, 39126}, {10504, 53123}, {10520, 29649}, {10916, 22464}, {11019, 24165}, {11519, 43983}, {15254, 28556}, {16602, 21826}, {16667, 35578}, {16825, 28526}, {17023, 17116}, {17067, 17279}, {17143, 24215}, {17229, 48631}, {17233, 50100}, {17239, 49741}, {17255, 28634}, {17260, 50090}, {17275, 49747}, {17282, 50107}, {17298, 49765}, {17304, 29604}, {17317, 50110}, {17320, 19883}, {17321, 19862}, {17344, 50098}, {17350, 41140}, {17353, 37756}, {17361, 50088}, {17364, 49770}, {17366, 50115}, {17375, 49761}, {17483, 50306}, {17490, 45204}, {17761, 20258}, {17877, 24162}, {19789, 40940}, {19820, 32939}, {20050, 32093}, {20052, 36606}, {20053, 52714}, {20057, 30712}, {20106, 23681}, {21093, 53381}, {21432, 23675}, {21625, 44735}, {24070, 27559}, {24182, 46843}, {24214, 32104}, {24231, 49474}, {24393, 28582}, {25076, 28978}, {25269, 29628}, {25525, 42049}, {25557, 28484}, {26806, 29574}, {26842, 50292}, {28522, 38054}, {28580, 30331}, {28639, 49733}, {28850, 43182}, {29016, 43177}, {30699, 39595}, {31145, 33800}, {33076, 49630}, {36525, 50081}, {49532, 49772}, {49533, 50011}, {49543, 50128}

X(53594) = midpoint of X(7) and X(17151)
X(53594) = reflection of X(i) in X(j) for these {i,j}: {142, 7263}, {3950, 142}, {17262, 6666}, {51090, 16825}
X(53594) = X(16602)-Dao conjugate of X(145)
X(53594) = cevapoint of X(3893) and X(16602)
X(53594) = barycentric product X(i)*X(j) for these {i,j}: {75, 16602}, {85, 3893}, {310, 21826}, {1969, 22376}, {3261, 8683}
X(53594) = barycentric quotient X(i)/X(j) for these {i,j}: {3893, 9}, {8683, 101}, {16602, 1}, {21826, 42}, {22376, 48}
X(53594) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 4373, 4862}, {75, 1266, 3663}, {75, 3663, 10}, {75, 4389, 4967}, {75, 4398, 4357}, {142, 3950, 29600}, {192, 24199, 29571}, {192, 29571, 4098}, {1086, 2321, 21255}, {1086, 4686, 2321}, {1266, 4357, 4398}, {1278, 48627, 3912}, {3632, 4902, 21296}, {3662, 4431, 29594}, {3662, 4740, 4431}, {3663, 24209, 24213}, {3664, 3875, 3244}, {3672, 25590, 1125}, {3672, 52709, 25590}, {3875, 42697, 3664}, {3879, 7321, 4896}, {4000, 4659, 17355}, {4000, 17355, 31191}, {4021, 10436, 551}, {4346, 32087, 17272}, {4357, 4398, 3663}, {4402, 4454, 1743}, {4440, 17117, 4416}, {4452, 31995, 1}, {4488, 24599, 3973}, {4688, 17246, 5257}, {4718, 17245, 4029}, {4772, 17247, 24603}, {4852, 7228, 4667}, {7321, 17160, 3879}, {10436, 50101, 4021}, {17118, 17301, 5750}, {17119, 17276, 3686}, {17272, 32087, 3626}, {50108, 50116, 51071}

---

X(53595) = INTERSECTION OF LINES TANGENT TO YFF DUAL PARABOLA AT X(75) AND X(13390)

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

X(53595) lies on these lines: {1, 487}, {2, 31582}, {10, 75}, {12, 8230}, {57, 481}, {69, 49592}, {175, 5265}, {388, 482}, {491, 49625}, {519, 32809}, {527, 13936}, {988, 1267}, {1362, 22107}, {3008, 31594}, {3241, 49616}, {3875, 49593}, {3946, 13883}, {4000, 14121}, {4419, 7090}, {4655, 49347}, {5393, 8957}, {9808, 12721}, {13389, 31530}, {13911, 17301}, {13973, 17276}, {25055, 49614}, {32797, 48818}, {32921, 49348}, {45445, 49453}, {49548, 50114}

---

X(53596) = INTERSECTION OF LINES TANGENT TO YFF DUAL PARABOLA AT X(86) AND X(273)

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(53596) lies on these lines: {1, 307}, {2, 1723}, {6, 17073}, {7, 90}, {10, 322}, {36, 18650}, {46, 4329}, {56, 41004}, {57, 1848}, {58, 86}, {65, 41007}, {69, 997}, {75, 10916}, {158, 273}, {347, 18391}, {572, 43054}, {579, 18589}, {942, 41003}, {982, 3663}, {1108, 16608}, {1375, 2264}, {1441, 1737}, {1479, 18655}, {1565, 24471}, {1731, 24884}, {1732, 31261}, {1743, 25019}, {2257, 18634}, {2260, 4466}, {2893, 17647}, {3879, 22836}, {3912, 25078}, {4001, 26637}, {4021, 6744}, {4032, 26012}, {4416, 25023}, {5228, 20270}, {5231, 25590}, {5336, 28078}, {6356, 44547}, {6734, 18698}, {8583, 17272}, {9119, 16596}, {10039, 40999}, {10436, 26363}, {10481, 24213}, {10572, 17134}, {17188, 41011}, {17220, 30384}, {17353, 28738}, {17863, 41804}, {17917, 37642}, {17923, 40940}, {18156, 51612}, {18635, 40937}, {21270, 45287}, {23790, 23798}, {24202, 44675}, {24986, 30806}, {25964, 43065}, {27626, 34847}

X(53596) = X(i)-Dao conjugate of X(j) for these (i,j): {45206, 7283}, {53560, 8611}
X(53596) = barycentric product X(i)*X(j) for these {i,j}: {7, 45206}, {81, 18692}, {85, 1858}, {431, 17206}, {1195, 6063}, {3261, 53322}
X(53596) = barycentric quotient X(i)/X(j) for these {i,j}: {431, 1826}, {1195, 55}, {1858, 9}, {18692, 321}, {45206, 8}, {53322, 101}
X(53596) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 3086, 24179}, {1210, 3668, 17861}, {1441, 5740, 1737}, {1731, 24884, 40530}

---

X(53597) = INTERSECTION OF LINES TANGENT TO YFF DUAL PARABOLA AT X(86) AND X(1088)

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

X(53597) lies on these lines: {1, 348}, {2, 16572}, {7, 84}, {10, 16284}, {11, 4059}, {57, 17170}, {58, 86}, {69, 936}, {77, 581}, {81, 18652}, {85, 1210}, {150, 10106}, {200, 25583}, {226, 17181}, {277, 24600}, {279, 938}, {307, 3945}, {326, 3811}, {354, 3665}, {515, 7176}, {553, 33865}, {942, 1565}, {950, 5088}, {1014, 18650}, {1088, 3673}, {1193, 17213}, {1256, 34400}, {1323, 6738}, {1434, 4292}, {1446, 51364}, {1475, 51400}, {1837, 7223}, {2646, 7181}, {3086, 40719}, {3304, 30617}, {3306, 41826}, {3576, 17081}, {3663, 3976}, {3668, 33673}, {3717, 33942}, {3912, 3933}, {4251, 51775}, {4253, 34847}, {4298, 4911}, {4515, 25355}, {4644, 40880}, {4920, 24231}, {5249, 17169}, {5728, 14520}, {7179, 21620}, {7278, 10039}, {7960, 30625}, {9312, 18391}, {10436, 19843}, {10573, 25719}, {12053, 17753}, {13161, 24215}, {13411, 14828}, {16604, 50011}, {17179, 17211}, {17205, 23537}, {17353, 28753}, {17364, 25023}, {17609, 24798}, {17778, 27399}, {20347, 41012}, {20880, 26015}, {21049, 44664}, {21075, 36854}, {21096, 25242}, {21258, 40133}, {23796, 23798}, {24172, 24216}, {24210, 24214}, {24781, 40940}, {24982, 30806}, {25524, 47595}, {31397, 33298}, {45700, 50116}, {48900, 50307}, {51893, 52565}

X(53597) = X(4635)-Ceva conjugate of X(514)
X(53597) = X(i)-Dao conjugate of X(j) for these (i,j): {18635, 200}, {36197, 4171}
X(53597) = barycentric product X(i)*X(j) for these {i,j}: {85, 10391}, {86, 18635}, {1509, 21717}
X(53597) = barycentric quotient X(i)/X(j) for these {i,j}: {10391, 9}, {18635, 10}, {21717, 594}
X(53597) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {348, 14548, 1}, {942, 1565, 3674}, {1434, 4872, 4292}, {10481, 11019, 3673}, {14828, 17095, 13411}, {24215, 24241, 13161}, {25242, 31038, 21096}

---

X(53598) = INTERSECTION OF LINES TANGENT TO YFF DUAL PARABOLA AT X(86) AND X(4373)

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

X(53598) = X[6] - 3 X[50092], 3 X[69] + X[3875], X[69] + 3 X[17274], 5 X[69] + 3 X[50101], 7 X[69] + 3 X[50108], 3 X[3663] - X[3875], X[3663] - 3 X[17274], 5 X[3663] - 3 X[50101], 7 X[3663] - 3 X[50108], X[3875] - 9 X[17274], 5 X[3875] - 9 X[50101], 7 X[3875] - 9 X[50108], X[4780] + 3 X[51004], 5 X[17274] - X[50101], 7 X[17274] - X[50108], and many others

X(53598) lies on these lines: {1, 21296}, {2, 3973}, {4, 43172}, {6, 50092}, {7, 10}, {8, 4373}, {9, 21255}, {37, 29606}, {44, 48632}, {58, 86}, {63, 20106}, {69, 519}, {72, 1122}, {75, 3626}, {77, 30144}, {141, 527}, {142, 4643}, {144, 17284}, {192, 49765}, {193, 17304}, {226, 37660}, {269, 997}, {307, 6700}, {319, 1266}, {391, 4859}, {516, 1350}, {524, 3946}, {536, 3631}, {545, 17229}, {551, 3945}, {553, 1211}, {599, 2321}, {726, 14994}, {758, 24471}, {894, 29604}, {903, 5564}, {936, 7271}, {966, 6173}, {1086, 3686}, {1269, 21207}, {1654, 24199}, {1743, 31191}, {2183, 29812}, {2325, 17231}, {2796, 50567}, {2809, 18252}, {3008, 3662}, {3244, 3672}, {3452, 42887}, {3589, 4715}, {3616, 32093}, {3617, 33800}, {3619, 50127}, {3620, 3729}, {3625, 4346}, {3629, 17382}, {3630, 4852}, {3632, 4452}, {3634, 4896}, {3635, 3879}, {3636, 17321}, {3679, 4902}, {3687, 26840}, {3707, 17278}, {3731, 4869}, {3739, 7238}, {3741, 20347}, {3755, 49680}, {3763, 50115}, {3821, 34379}, {3828, 5224}, {3831, 20245}, {3834, 6666}, {3840, 30946}, {3844, 5852}, {3912, 6646}, {3927, 39589}, {3950, 4419}, {3977, 31017}, {3982, 31993}, {3986, 4648}, {4001, 16704}, {4029, 17311}, {4058, 4659}, {4060, 4686}, {4072, 29616}, {4098, 29573}, {4138, 40719}, {4298, 10401}, {4353, 5847}, {4364, 17376}, {4398, 17360}, {4414, 50753}, {4431, 4440}, {4478, 4726}, {4480, 17280}, {4545, 50098}, {4644, 17306}, {4657, 4667}, {4669, 32087}, {4675, 5257}, {4683, 40998}, {4684, 24723}, {4690, 7263}, {4691, 17270}, {4700, 17366}, {4704, 29601}, {4745, 4967}, {4746, 42696}, {4758, 25498}, {4851, 17255}, {4909, 17249}, {4912, 50991}, {5235, 5249}, {5542, 50295}, {5550, 30712}, {5739, 24177}, {5750, 17237}, {5860, 49621}, {5861, 49620}, {6172, 15828}, {6745, 33065}, {7228, 17239}, {7274, 9623}, {7277, 17384}, {8580, 51351}, {9436, 20103}, {10008, 28526}, {10442, 51118}, {10444, 28164}, {10446, 50608}, {10452, 31964}, {10481, 12447}, {10520, 29655}, {10916, 24213}, {11160, 49543}, {12572, 41004}, {13405, 33064}, {14552, 23681}, {14555, 24175}, {15533, 50109}, {16552, 28351}, {16834, 20080}, {17023, 17236}, {17067, 17348}, {17133, 17372}, {17216, 18650}, {17227, 17347}, {17232, 17333}, {17234, 17329}, {17238, 50128}, {17240, 49748}, {17242, 50090}, {17245, 31138}, {17246, 17374}, {17247, 17375}, {17250, 31253}, {17252, 24603}, {17254, 17300}, {17257, 17298}, {17258, 17297}, {17286, 21356}, {17291, 20072}, {17292, 31300}, {17299, 28313}, {17301, 40341}, {17324, 20090}, {17343, 48627}, {17346, 48629}, {17350, 29596}, {17352, 48637}, {17354, 48638}, {17363, 50019}, {17386, 50110}, {17396, 50133}, {17781, 33172}, {18206, 28402}, {19604, 24797}, {19868, 50307}, {20053, 32105}, {20059, 29611}, {20335, 24690}, {21033, 53538}, {21061, 52896}, {21246, 24237}, {24231, 33082}, {24325, 43180}, {24778, 27108}, {25269, 29577}, {25728, 29579}, {26768, 27017}, {27184, 37684}, {28244, 53543}, {28301, 50081}, {28333, 34573}, {28522, 49518}, {28580, 50315}, {28633, 49733}, {30424, 50314}, {33143, 50754}, {37668, 49554}, {48628, 50119}, {48630, 49722}, {49497, 50091}, {49520, 49766}, {49684, 50285}, {50089, 50990}

X(53598) = midpoint of X(i) and X(j) for these {i,j}: {69, 3663}, {141, 17345}, {2321, 17276}, {3630, 4852}, {4655, 49511}, {4660, 49505}, {11160, 49543}, {15533, 50109}
X(53598) = reflection of X(i) in X(j) for these {i,j}: {3946, 17235}, {4856, 3946}, {17355, 141}
X(53598) = X(21950)-Dao conjugate of X(14321)
X(53598) = barycentric product X(1509)*X(21712)
X(53598) = barycentric quotient X(21712)/X(594)
X(53598) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 5232, 25590}, {7, 17272, 10}, {8, 45789, 4862}, {69, 17274, 3663}, {193, 17304, 50114}, {320, 4357, 3664}, {320, 17273, 4357}, {599, 17276, 2321}, {1086, 17344, 3686}, {3620, 3729, 29594}, {3630, 49741, 4852}, {3662, 4416, 3008}, {3662, 4741, 4416}, {3664, 4357, 1125}, {3679, 4902, 31995}, {3731, 4869, 29600}, {3834, 17332, 6666}, {3879, 4021, 3635}, {3879, 4389, 4021}, {4001, 17184, 40940}, {4346, 32099, 17151}, {4389, 17361, 3879}, {4419, 17296, 3950}, {4440, 17287, 4431}, {4643, 7232, 142}, {4675, 17253, 5257}, {5232, 25590, 10}, {6646, 17288, 3912}, {7321, 17271, 4967}, {17151, 32099, 3625}, {17227, 17347, 17353}, {17231, 17334, 2325}, {17232, 17333, 25101}, {17232, 25101, 41141}, {17234, 17329, 50093}, {17234, 50093, 25072}, {17236, 17364, 17023}, {17237, 17365, 5750}, {17247, 17375, 29574}, {17252, 26806, 24603}, {17257, 17298, 29571}, {17272, 25590, 5232}, {17343, 48627, 50095}, {17348, 48631, 17067}, {17350, 48633, 29596}

---

X(53599) = INTERSECTION OF LINES TANGENT TO YFF DUAL PARABOLA AT X(273) AND X(673)

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

X(53599 lies on these lines: {1, 6835}, {3, 17278}, {4, 990}, {5, 37}, {6, 5805}, {11, 1465}, {40, 14268}, {44, 5762}, {45, 38108}, {75, 12618}, {142, 991}, {158, 273}, {225, 9581}, {238, 516}, {355, 3242}, {381, 17301}, {386, 946}, {511, 50011}, {515, 30117}, {908, 5400}, {942, 52023}, {952, 4864}, {971, 1086}, {975, 6864}, {986, 12617}, {1070, 18216}, {1072, 5587}, {1074, 3586}, {1090, 1737}, {1104, 20420}, {1279, 15251}, {1352, 16973}, {1449, 5733}, {1478, 4327}, {1479, 4319}, {1490, 24159}, {1699, 2999}, {1736, 22464}, {1743, 5735}, {1750, 23681}, {1754, 26723}, {1766, 36670}, {1785, 7541}, {1834, 5806}, {1836, 52424}, {1851, 21370}, {1861, 4858}, {2257, 5292}, {2345, 36682}, {2476, 24554}, {2801, 24231}, {3011, 44425}, {3091, 3672}, {3100, 37771}, {3332, 5222}, {3660, 51424}, {3662, 48878}, {3666, 8226}, {3752, 8727}, {3772, 19541}, {3782, 5927}, {3817, 4356}, {3821, 45305}, {3836, 28850}, {3912, 29016}, {4187, 25067}, {4193, 26669}, {4297, 24178}, {4318, 45043}, {4328, 9612}, {4331, 15299}, {4349, 38151}, {4353, 13161}, {4357, 48888}, {4415, 10157}, {4419, 5817}, {4675, 38107}, {4850, 10883}, {4859, 5732}, {5262, 6894}, {5691, 23536}, {5709, 39943}, {5759, 37650}, {5779, 17276}, {5787, 17054}, {5816, 16517}, {6051, 7958}, {6245, 24046}, {6734, 20236}, {6818, 21062}, {6836, 23604}, {7411, 26724}, {7580, 24789}, {7649, 21188}, {8728, 15852}, {9355, 32857}, {9436, 43672}, {9779, 33134}, {9812, 33131}, {10167, 40688}, {10395, 37591}, {12047, 42289}, {12571, 36250}, {12699, 36745}, {13727, 16706}, {14022, 25939}, {14561, 36404}, {15253, 51361}, {16020, 43161}, {16602, 37364}, {16610, 37374}, {16666, 38137}, {17321, 36660}, {17337, 31658}, {17366, 18482}, {17382, 36722}, {18446, 26728}, {20330, 49478}, {21153, 31183}, {21346, 23689}, {22350, 30384}, {24779, 30265}, {24827, 29085}, {26635, 52255}, {33129, 36002}, {34852, 51366}, {36526, 46475}, {37817, 50701}, {38122, 50677}, {43169, 51212}

X(53599) = midpoint of X(9355) and X(32857)
X(53599) = reflection of X(i) in X(j) for these {i,j}: {1279, 15251}, {13329, 3008}
X(53599) = reflection of X(13329) in the Gergonne line
X(53599) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 4000, 990}, {75, 36652, 12618}

---

X(53600) = INTERSECTION OF LINES TANGENT TO YFF DUAL PARABOLA AT X(335) AND X(673)

Barycentrics    a^3*b - 2*a^2*b^2 - b^4 + a^3*c + a*b^2*c + b^3*c - 2*a^2*c^2 + a*b*c^2 + b*c^3 - c^4 : : X(53600) = 3 X[2] + X[41842], 2 X[4437] - 3 X[29594]

X(53600) lies on these lines: {1, 20533}, {2, 846}, {6, 20748}, {9, 33869}, {10, 6656}, {37, 142}, {88, 30857}, {190, 4357}, {226, 43063}, {238, 516}, {239, 6653}, {321, 20898}, {335, 726}, {442, 25073}, {519, 32029}, {527, 50011}, {528, 1386}, {537, 3773}, {543, 24261}, {545, 17359}, {579, 16560}, {712, 49781}, {812, 39979}, {903, 4582}, {986, 33838}, {1125, 16061}, {1647, 31058}, {1766, 27626}, {2051, 43686}, {2292, 17672}, {2321, 9055}, {2549, 24249}, {2786, 23596}, {2809, 20455}, {3094, 24274}, {3661, 33888}, {3662, 3729}, {3666, 35310}, {3675, 20364}, {4045, 24254}, {4260, 32118}, {4366, 17023}, {4384, 4660}, {4422, 5257}, {4432, 50290}, {4473, 17248}, {4684, 24841}, {4920, 25066}, {5091, 19561}, {5222, 41845}, {5750, 24358}, {6999, 8926}, {7377, 41886}, {7791, 16822}, {7819, 24850}, {12610, 27633}, {16549, 24211}, {17234, 49518}, {17304, 27147}, {17316, 49455}, {17322, 27191}, {17671, 24174}, {17681, 24851}, {17744, 33868}, {17754, 24241}, {17758, 40788}, {20331, 24318}, {20681, 25748}, {21073, 24172}, {21090, 21208}, {21214, 27129}, {23868, 24309}, {24443, 33839}, {24631, 26590}, {24728, 36698}, {24821, 29611}, {25351, 50298}, {25498, 40480}, {27474, 49560}, {27954, 33021}, {28562, 32096}, {29573, 49446}, {29574, 49464}, {29587, 41844}, {30826, 34524}, {30837, 32851}, {32108, 49770}, {35103, 36230}, {38384, 45204}, {49630, 50297}

X(53600) = midpoint of X(i) and X(j) for these {i,j}: {3729, 4440}, {17738, 41842}
X(53600) = reflection of X(i) in X(j) for these {i,j}: {190, 17355}, {3663, 1086}
X(53600) = complement of X(17738)
X(53600) = complement of the isogonal conjugate of X(18783)
X(53600) = polar conjugate of the isogonal conjugate of X(20786)
X(53600) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 9470}, {2113, 141}, {9472, 20333}, {18264, 17755}, {18783, 10}, {41528, 2}
X(53600) = X(27942)-Dao conjugate of X(3912)
X(53600) = crossdifference of every pair of points on line {5029, 19561}
X(53600) = X(5091)-lineconjugate of X(19561)
X(53600) = barycentric product X(i)*X(j) for these {i,j}: {264, 20786}, {335, 27942}
X(53600) = barycentric quotient X(i)/X(j) for these {i,j}: {20786, 3}, {27942, 239}
X(53600) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 41842, 17738}, {3675, 20689, 20364}, {3797, 3912, 6541}, {3821, 24295, 25354}, {4422, 25357, 5257}, {17755, 26582, 10}, {24631, 26590, 29655}

---

X(53601) = INTERSECTION OF LINES TANGENT TO YFF DUAL PARABOLA AT X(335) AND X(903)

Barycentrics    a^2*b - 3*a*b^2 - b^3 + a^2*c + 2*a*b*c + 2*b^2*c - 3*a*c^2 + 2*b*c^2 - c^3 : :
X(53601) = 3 X[10] - 4 X[25351], 3 X[1086] - 2 X[25351], 3 X[903] - X[24715], 3 X[903] + X[24841], 4 X[3912] - 3 X[6541], X[3912] - 3 X[24231], 2 X[3912] - 3 X[49676], X[6541] - 4 X[24231], 3 X[551] - 2 X[4432], 3 X[1757] - 5 X[29590], 3 X[3576] - X[24817], 7 X[3624] - 5 X[4473], 4 X[3634] - 5 X[27191], 3 X[3817] - 2 X[24828], 2 X[4370] - 3 X[19883], and many others

X(53601) lies on these lines: {1, 2796}, {2, 24821}, {7, 49455}, {10, 537}, {75, 4783}, {142, 49520}, {190, 1125}, {226, 24816}, {244, 11814}, {291, 24193}, {320, 519}, {335, 726}, {515, 24833}, {516, 24813}, {527, 49710}, {528, 3244}, {536, 49764}, {545, 551}, {673, 5850}, {899, 24200}, {900, 21630}, {950, 24837}, {984, 25352}, {993, 24826}, {1279, 17767}, {1644, 42026}, {1647, 4080}, {1738, 49697}, {1757, 29590}, {2786, 4010}, {3120, 17154}, {3122, 24399}, {3246, 4912}, {3576, 24817}, {3624, 4473}, {3625, 9041}, {3634, 27191}, {3662, 49532}, {3663, 49479}, {3755, 49535}, {3773, 49525}, {3782, 29655}, {3817, 24828}, {3821, 24349}, {3834, 4439}, {3836, 28582}, {3923, 4310}, {3943, 28554}, {3993, 5542}, {4013, 24188}, {4090, 24177}, {4297, 29243}, {4346, 36479}, {4353, 33682}, {4363, 50285}, {4364, 24325}, {4370, 19883}, {4389, 31178}, {4392, 25385}, {4395, 4753}, {4398, 49490}, {4415, 42053}, {4419, 24331}, {4422, 19862}, {4425, 17140}, {4480, 4759}, {4659, 50311}, {4660, 4862}, {4669, 36525}, {4674, 24131}, {4684, 28522}, {4686, 50315}, {4702, 28542}, {4709, 49505}, {4758, 24358}, {4864, 17764}, {4966, 28516}, {4974, 5852}, {5845, 51196}, {5886, 24844}, {6650, 20016}, {6651, 29592}, {7238, 28503}, {7263, 49457}, {9055, 49511}, {10106, 24836}, {12053, 24840}, {13411, 24845}, {16173, 36237}, {16593, 38054}, {17132, 49768}, {17155, 33064}, {17165, 24169}, {17230, 51056}, {17234, 49517}, {17244, 51035}, {17290, 50313}, {17365, 49472}, {17487, 25055}, {17647, 24834}, {17738, 26626}, {17766, 32857}, {17768, 49705}, {17770, 32922}, {20042, 53372}, {20533, 29602}, {21342, 48643}, {24603, 31349}, {24820, 25440}, {24827, 31673}, {24846, 44675}, {25031, 46458}, {25378, 33151}, {25557, 49456}, {28534, 49700}, {28562, 49771}, {29571, 50777}, {29576, 33888}, {29585, 41842}, {29606, 51098}, {29628, 51297}, {29654, 32940}, {29656, 32939}, {29671, 33103}, {29672, 32933}, {29673, 33146}, {32029, 34379}, {32845, 50748}, {33149, 49499}, {33165, 48629}, {34824, 50094}, {36480, 42697}, {37756, 49712}, {40480, 51073}, {40688, 42054}, {48627, 49448}, {48802, 52709}, {49464, 50307}, {49493, 49560}, {49713, 49772}, {51071, 53534}

X(53601) =midpoint of X(i) and X(j) for these {i,j}: {1, 4440}, {24715, 24841}
X(53601) =reflection of X(i) in X(j) for these {i,j}: {10, 1086}, {190, 1125}, {4439, 3834}, {4480, 4759}, {4753, 4395}, {6541, 49676}, {24692, 4887}, {31673, 24827}, {49676, 24231}, {49697, 1738}, {49710, 50023}, {49713, 49772}
X(53601) =complement of X(24821)
X(53601) =X(i)-anticomplementary conjugate of X(j) for these (i,j): {1929, 21290}, {17962, 30578}
X(53601) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {244, 21093, 11814}, {244, 30566, 25377}, {903, 24841, 24715}, {3782, 42055, 29655}, {4407, 4688, 10}, {21093, 25377, 30566}, {25377, 30566, 11814}

---

X(53602) = INTERSECTION OF LINES TANGENT TO YFF DUAL PARABOLA AT X(673) AND X(14621)

Barycentrics    2*a^4 + a^2*b^2 - a*b^3 - 2*a^2*b*c - b^3*c + a^2*c^2 + 2*b^2*c^2 - a*c^3 - b*c^3 : :
5 X[29590] - X[41842]

X(53602) lies on these lines: {1, 17691}, {2, 902}, {6, 527}, {7, 16779}, {10, 7770}, {86, 33628}, {238, 516}, {239, 726}, {384, 30038}, {519, 4482}, {551, 20131}, {672, 52210}, {748, 24596}, {1001, 21514}, {1125, 16060}, {1572, 24249}, {1707, 24283}, {1759, 24172}, {1914, 20335}, {2243, 27918}, {2796, 20142}, {3244, 20162}, {3452, 16283}, {3664, 16503}, {3729, 3973}, {3759, 49518}, {3821, 14621}, {3840, 24586}, {3911, 24279}, {3912, 4366}, {3923, 4384}, {4000, 33869}, {4209, 21214}, {4279, 37086}, {4353, 4649}, {4357, 20179}, {4393, 49464}, {4426, 20257}, {4437, 17765}, {4657, 24699}, {4785, 23597}, {4871, 24602}, {4887, 16786}, {4888, 17304}, {4967, 17277}, {5222, 16468}, {5255, 17681}, {5263, 29604}, {5299, 24214}, {5853, 49706}, {6244, 19517}, {7745, 17062}, {12206, 49612}, {14953, 18792}, {16502, 24215}, {16834, 49455}, {16916, 30030}, {16918, 30063}, {16920, 30036}, {17000, 24199}, {17026, 24260}, {20367, 20459}, {20992, 24309}, {21443, 52652}, {23605, 37416}, {24259, 24592}, {24280, 24599}, {24295, 24603}, {28550, 29590}, {29594, 32941}, {35466, 41163}, {49477, 49519}, {49497, 49543}, {49630, 50300}, {49704, 52157}

X(53602) = midpoint of X(239) and X(17738)
X(53602) = reflection of X(i) in X(j) for these {i,j}: {3923, 4759}, {24692, 3821}
X(53602) = crossdifference of every pair of points on line {9029, 19586}
X(53602) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {238, 673, 3008}, {14621, 17023, 33682}

   Reflection points on the circumcircle: X(53603)-X(53613) and others  

Contributed by Clark Kimberling and Peter Moses May, 2023.

The appearance of {i,j,k} in the following list means that X(k) = reflection of X(i) in the line X(3)X(j):

{74,98,43654}, {74,99,43654}, {74,100,43655}, {74,104,43655}, {74,110,74}, {74,1113,477}, {74,1114,477}, {74,1379,842}, {74,1380,842}, {74,1381,2687}, {74,1382,2687},

{98,74,9161}, {98,99,98}, {98,110,9161}, {98,1113,842}, {98,1114,842}, {98,1379,2698}, {98,1380,2698}, {98,1381,2699}, {98,1382,2699}, {98,36735,953}, {98,36736,953},

{99,74,9160}, {99,98,99}, {99,100,53606}, {99,104,53606}, {99,110,9160}, {99,1113,691}, {99,1114,691}, {99,1379,805}, {99,1380,805}, {99,1381,2703}, {99,1382,2703}, {99,36735,901}, {99,36736,901},

{100,101,53607}, {100,103,53607}, {100,104,100}, {100,1113,1290}, {100,1114,1290}, {100,1379,2703}, {100,1380,2703}, {100,1381,901}, {100,1382,901}, v {101,100,39444}, {101,103,101}, {101,104,39444}, {101,1113,2690}, {101,1114,2690}, {101,1379,2702}, {101,1380,2702}, {101,1381,1308}, {101,1382,1308},

{102,109,102}, {102,1113,2695}, {102,1114,2695}, {102,1379,2708}, {102,1380,2708}, {102,1381,2716}, {102,1382,2716},

{103,101,103}, {103,1113,2688}, {103,1114,2688}, {103,1379,2700}, {103,1380,2700}, {103,1381,2717}, {103,1382,2717},

{104,100,104}, {104,1113,2687}, {104,1114,2687}, {104,1379,2699}, {104,1380,2699}, {104,1381,953}, {104,1382,953},

{105,100,53608}, {105,104,53608}, {105,1113,2752}, {105,1114,2752}, {105,1292,105}, {105,1379,2711}, {105,1380,2711}, {105,1381,840}, {105,1382,840},

{106,100,39445}, {106,104,39445}, {106,1113,2758}, {106,1114,2758}, {106,1293,106}, {106,1379,2712}, {106,1380,2712}, {106,1381,2718}, {106,1382,2718},

{107,1113,1304}, {107,1114,1304}, {107,1294,107}, {107,1379,2713}, {107,1380,2713}, {107,1381,2719}, {107,1382,2719},

{108,100,53609}, {108,104,53609}, {108,1113,2766}, {108,1114,2766}, {108,1295,108}, {108,1379,2714}, {108,1380,2714}, {108,1381,2720}, {108,1382,2720},

{109,100,53610}, {109,102,109}, {109,104,53610}, {109,1113,2689}, {109,1114,2689}, {109,1379,2701}, {109,1380,2701}, {109,1381,2222}, {109,1382,2222},

{110,74,110}, {110,98,53603}, {110,99,53603}, {110,100,53611}, {110,102,53612}, {110,104,53611}, {110,109,53612}, {110,1113,476}, {110,1114,476}, {110,1379,691}, {110,1380,691}, {110,1381,1290}, {110,1382,1290},

{111,74,9184}, {111,98,53604}, {111,99,53604}, {111,110,9184}, {111,1113,2770}, {111,1114,2770}, {111,1296,111}, {111,1379,843}, {111,1380,843}, {111,1381,2721}, {111,1382,2721},

{112,74,53187}, {112,110,53187}, {112,1113,935}, {112,1114,935}, {112,1297,112}, {112,1379,2715}, {112,1380,2715}, {112,1381,2722}, {112,1382,2722},

{476,74,16170}, {476,98,20404}, {476,99,20404}, {476,110,16170}, {476,477,476}, {476,691,805}, {476,842,805}, {476,935,2867}, {476,1113,110}, {476,1114,110}, {476,1290,901}, {476,1304,6080}, {476,1379,9160}, {476,1380,9160}, {476,2687,901}, {476,2688,927}, {476,2689,1309}, {476,2690,927}, {476,2691,6078}, {476,2692,6079}, {476,2693,6080}, {476,2694,6081}, {476,2695,1309}, {476,2696,6082}, {476,2697,2867}, {476,2752,6078}, {476,2758,6079}, {476,2766,6081}, {476,2770,6082}, {476,12030,6083},

{477,74,16169}, {477,98,53605}, {477,99,53605}, {477,110,16169}, {477,476,477}, {477,691,2698}, {477,842,2698}, {477,1113,74}, {477,1114,74}, {477,1290,953}, {477,1291,15907}, {477,1304,44874}, {477,1379,9161}, {477,1380,9161}, {477,2687,953}, {477,2688,2724}, {477,2689,2734}, {477,2690,2724}, {477,2691,28914}, {477,2692,44873}, {477,2693,44874}, {477,2695,2734}, {477,2696,6093}, {477,2752,28914}, {477,2758,44873}, {477,2770,6093}, {477,14979,15907},

{675,1113,53190}, {675,1114,53190}, {675,44876,675},

{691,111,53613}, {691,842,691}, {691,1113,99}, {691,1114,99}, {691,1296,53613}, {691,1379,110}, {691,1380,110},

{699,30254,699}, {727,28469,727}, {729,1379,5970}, {729,1380,5970}, {729,39639,729}, {737,39629,737}, {739,1379,35107}, {739,1380,35107}, {739,1381,9081}, {739,1382,9081}, {739,28474,739}, {741,1379,12031}, {741,1380,12031}, {741,6010,741}, {753,28467,753}, {759,1113,12030}, {759,1114,12030}, {759,6011,759},

{805,74,20404}, {805,110,20404}, {805,691,476}, {805,842,476}, {805,843,6082}, {805,1113,53603}, {805,1114,53603}, {805,1379,99}, {805,1380,99}, {805,2698,805}, {805,2699,901}, {805,2700,927}, {805,2701,1309}, {805,2702,927}, {805,2703,901}, {805,2704,6078}, {805,2705,6079}, {805,2706,6080}, {805,2707,6081}, {805,2708,1309}, {805,2709,6082}, {805,2710,2867}, {805,2711,6078}, {805,2712,6079}, {805,2713,6080}, {805,2714,6081}, {805,2715,2867}, {805,9160,16170}, {805,9161,16170},

{813,12032,813}, {815,29009,815}, {825,28844,825}, {827,1113,1287}, {827,1114,1287}, {827,1379,46970}, {827,1380,46970}, {827,29011,827}, {835,45136,835},

{840,1379,53180}, {840,1380,53180}, {840,1381,105}, {840,1382,105}, {840,2742,840},

{841,1113,43660}, {841,1114,43660}, {841,9060,841}, {842,691,842}, {842,1113,98}, {842,1114,98}, {842,1379,74}, {842,1380,74}, {843,1379,111}, {843,1380,111}, {843,2709,843}, {898,1381,39443}, {898,1382,39443}, {898,29348,898},

{901,102,35011}, {901,109,35011}, {901,840,6078}, {901,953,901}, {901,1113,53611}, {901,1114,53611}, {901,1290,476}, {901,1308,927}, {901,1379,53606}, {901,1380,53606}, {901,1381,100}, {901,1382,100}, {901,2222,1309}, {901,2687,476}, {901,2699,805}, {901,2703,805}, {901,2716,1309}, {901,2717,927}, {901,2718,6079}, {901,2719,6080}, {901,2720,6081}, {901,2721,6082}, {901,2722,2867}, {901,2742,6078}, {901,2743,6079}, {901,2744,6080}, {901,2745,6081}, {901,2746,6082}, {901,2747,2867}, {901,36735,99}, {901,36736,99},

{907,29180,907}, {915,1381,43078}, {915,1382,43078}, {915,13397,915}, {917,1305,917}, {919,28838,919}, {925,1113,10420}, {925,1114,10420}, {925,1300,925},

{927,929,1309}, {927,1308,901}, {927,1381,53607}, {927,1382,53607}, {927,2688,476}, {927,2690,476}, {927,2700,805}, {927,2702,805}, {927,2717,901}, {927,2723,1309}, {927,2724,927}, {927,2725,6078}, {927,2726,6079}, {927,2727,6080}, {927,2728,6081}, {927,2729,6082}, {927,2736,6078}, {927,2737,6079}, {927,2738,6080}, {927,2739,6081}, {927,2740,6082}, {927,2741,2867}, {927,53182,2867},

{929,2723,929}, {930,1113,1291}, {930,1114,1291}, {930,1141,930}, {932,1381,43362}, {932,1382,43362}, {932,15323,932}, {933,1113,52998}, {933,1114,52998}, {933,18401,933}, {934,972,934}, {934,1381,14733}, {934,1382,14733}, {935,1113,112}, {935,1114,112}, {935,2697,935},

{953,840,28914}, {953,901,953}, {953,1113,43655}, {953,1114,43655}, {953,1290,477}, {953,1308,2724}, {953,1381,104}, {953,1382,104}, {953,2222,2734}, {953,2687,477}, {953,2699,2698}, {953,2703,2698}, {953,2716,2734}, {953,2717,2724}, {953,2718,44873}, {953,2719,44874}, {953,2721,6093}, {953,2742,28914}, {953,2743,44873}, {953,2744,44874}, {953,2746,6093}, {953,36735,98}, {953,36736,98},

{972,934,972}, {1113,1114,1113}, {1114,1113,1114}, {1141,930,1141}, {1141,1113,14979}, {1141,1114,14979}, {1287,1113,827}, {1287,1114,827}, {1289,1113,10423}, {1289,1114,10423}, {1289,34168,1289},

{1290,1113,100}, {1290,1114,100}, {1290,1381,110}, {1290,1382,110}, {1290,2687,1290},

{1291,1113,930}, {1291,1114,930}, {1291,1379,39448}, {1291,1380,39448}, {1291,14979,1291},

{1292,105,1292}, {1292,1113,2691}, {1292,1114,2691}, {1292,1379,2704}, {1292,1380,2704}, {1292,1381,2742}, {1292,1382,2742},

{1293,106,1293}, {1293,1113,2692}, {1293,1114,2692}, {1293,1379,2705}, {1293,1380,2705}, {1293,1381,2743}, {1293,1382,2743},

{1294,107,1294}, {1294,1113,2693}, {1294,1114,2693}, {1294,1379,2706}, {1294,1380,2706}, {1294,1381,2744}, {1294,1382,2744},

{1295,108,1295}, {1295,1113,2694}, {1295,1114,2694}, {1295,1379,2707}, {1295,1380,2707}, {1295,1381,2745}, {1295,1382,2745},

{1296,111,1296}, {1296,1113,2696}, {1296,1114,2696}, {1296,1379,2709}, {1296,1380,2709}, {1296,1381,2746}, {1296,1382,2746},

{1297,74,53188}, {1297,110,53188}, {1297,112,1297}, {1297,1113,2697}, {1297,1114,2697}, {1297,1379,2710}, {1297,1380,2710}, {1297,1381,2747}, {1297,1382,2747},

{1298,1113,32439}, {1298,1114,32439}, {1298,1303,1298}, {1299,13398,1299}, {1300,925,1300}, {1300,1113,32710}, {1300,1114,32710}, {1301,1113,22239}, {1301,1114,22239}, {1301,5897,1301}, {1302,1113,9060}, {1302,1114,9060}, {1302,43660,1302}, {1303,1298,1303}, {1304,1113,107}, {1304,1114,107}, {1304,2693,1304}, {1305,917,1305}, {1308,1381,101}, {1308,1382,101}, {1308,2717,1308},

{1309,100,35011}, {1309,104,35011}, {1309,929,927}, {1309,1113,53612}, {1309,1114,53612}, {1309,2222,901}, {1309,2689,476}, {1309,2695,476}, {1309,2701,805}, {1309,2708,805}, {1309,2716,901}, {1309,2723,927}, {1309,2730,6078}, {1309,2731,6079}, {1309,2732,6080}, {1309,2733,6081}, {1309,2734,1309}, {1309,2735,6082}, {1309,2751,6078}, {1309,2757,6079}, {1309,2762,6080}, {1309,2765,6081}, {1309,2768,6082}, {1309,2769,2867}, {

1310,45137,1310}, {1379,1380,1379}, {1379,1381,36735}, {1379,1382,36735}, {1380,1379,1380}, {1380,1381,36736}, {1380,1382,36736}, {1381,1382,1381}, {1382,1381,1382}, {2222,1381,109}, {2222,1382,109}, {2222,2716,2222}, {2249,36516,2249},

{2291,1113,53185}, {2291,1114,53185}, {2291,1379,53179}, {2291,1380,53179}, {2291,1381,53181}, {2291,1382,53181}, {2291,28291,2291},

{2370,32704,2370}, {2373,30247,2373}, {2374,1113,40119}, {2374,1114,40119}, {2374,20187,2374}, {2378,1379,2379}, {2378,1380,2379}, {2378,9202,2378}, {2379,1379,2378}, {2379,1380,2378}, {2379,9203,2379}, {2380,39636,2380}, {2381,39637,2381}, {2382,28520,2382}, {2383,20185,2383}, {2384,28293,2384}, {2687,1113,104}, {2687,1114,104}, {2687,1290,2687}, {2687,1381,74}, {2687,1382,74}, {2688,1113,103}, {2688,1114,103}, {2688,2690,2688}, {2689,1113,109}, {2689,1114,109}, {2689,2695,2689}, {2690,1113,101}, {2690,1114,101}, {2690,2688,2690}, {2691,1113,1292}, {2691,1114,1292}, {2691,2752,2691}, {2692,1113,1293}, {2692,1114,1293}, {2692,2758,2692}, {2693,1113,1294}, {2693,1114,1294}, {2693,1304,2693}, {2694,1113,1295}, {2694,1114,1295}, {2694,2766,2694}, {2695,1113,102}, {2695,1114,102}, {2695,2689,2695}, {2696,1113,1296}, {2696,1114,1296}, {2696,2770,2696}, {2697,935,2697}, {2697,1113,1297}, {2697,1114,1297},

{2698,74,53605}, {2698,110,53605}, {2698,691,477}, {2698,805,2698}, {2698,842,477}, {2698,843,6093}, {2698,1113,43654}, {2698,1114,43654}, {2698,1379,98}, {2698,1380,98}, {2698,2379,44875}, {2698,2699,953}, {2698,2700,2724}, {2698,2701,2734}, {2698,2702,2724}, {2698,2703,953}, {2698,2704,28914}, {2698,2705,44873}, {2698,2706,44874}, {2698,2708,2734}, {2698,2709,6093}, {2698,2711,28914}, {2698,2712,44873}, {2698,2713,44874}, {2698,9160,16169}, {2698,9161,16169}, {2698,9203,44875},

{2699,1379,104}, {2699,1380,104}, {2699,1381,98}, {2699,1382,98}, {2699,2703,2699}, {2700,1379,103}, {2700,1380,103}, {2700,2702,2700}, {2701,1379,109}, {2701,1380,109}, {2701,2708,2701}, {2702,1379,101}, {2702,1380,101}, {2702,2700,2702}, {2703,1379,100}, {2703,1380,100}, {2703,1381,99}, {2703,1382,99}, {2703,2699,2703}, {2704,1379,1292}, {2704,1380,1292}, {2704,2711,2704}, {2705,1379,1293}, {2705,1380,1293}, {2705,2712,2705}, {2706,1379,1294}, {2706,1380,1294}, {2706,2713,2706}, {2707,1379,1295}, {2707,1380,1295}, {2707,2714,2707}, {2708,1379,102}, {2708,1380,102}, {2708,2701,2708}, {2709,843,2709}, {2709,1379,1296}, {2709,1380,1296}, {2710,1379,1297}, {2710,1380,1297}, {2710,2715,2710}, {2711,1379,105}, {2711,1380,105}, {2711,2704,2711}, {2712,1379,106}, {2712,1380,106}, {2712,2705,2712}, {2713,1379,107}, {2713,1380,107}, {2713,2706,2713}, {2714,1379,108}, {2714,1380,108}, {2714,2707,2714}, {2715,1113,53692}, {2715,1114,53692}, {2715,1379,112}, {2715,1380,112}, {2715,2710,2715}, {2716,1381,102}, {2716,1382,102}, {2716,2222,2716}, {2717,1308,2717}, {2717,1381,103}, {2717,1382,103}, {2718,1381,106}, {2718,1382,106}, {2718,2743,2718}, {2719,1381,107}, {2719,1382,107}, {2719,2744,2719}, {2720,1381,108}, {2720,1382,108}, {2720,2745,2720}, {2721,1381,111}, {2721,1382,111}, {2721,2746,2721}, {2722,1381,112}, {2722,1382,112}, {2722,2747,2722}, {2723,929,2723},

{2724,927,2724}, {2724,929,2734}, {2724,1308,953}, {2724,2688,477}, {2724,2690,477}, {2724,2700,2698}, {2724,2702,2698}, {2724,2717,953}, {2724,2723,2734}, {2724,2725,28914}, {2724,2726,44873}, {2724,2727,44874}, {2724,2729,6093}, {2724,2736,28914}, {2724,2737,44873}, {2724,2738,44874}, {2724,2740,6093},

{2725,2736,2725}, {2726,2737,2726}, {2727,2738,2727}, {2728,2739,2728}, {2729,2740,2729}, {2730,2751,2730}, {2731,2757,2731}, {2732,2762,2732}, {2733,2765,2733},

{2734,929,2724}, {2734,1309,2734}, {2734,2222,953}, {2734,2689,477}, {2734,2695,477}, {2734,2701,2698}, {2734,2708,2698}, {2734,2716,953}, {2734,2723,2724}, {2734,2730,28914}, {2734,2731,44873}, {2734,2732,44874}, {2734,2735,6093}, {2734,2751,28914}, {2734,2757,44873}, {2734,2762,44874}, {2734,2768,6093},

{2735,2768,2735}, {2736,2725,2736}, {2737,2726,2737}, {2738,2727,2738}, {2739,2728,2739}, {2740,2729,2740}, {2741,53182,2741}, {2742,840,2742}, {2742,1381,1292}, {2742,1382,1292}, {2743,1381,1293}, {2743,1382,1293}, {2743,2718,2743}, {2744,1381,1294}, {2744,1382,1294}, {2744,2719,2744}, {2745,1381,1295}, {2745,1382,1295}, {2745,2720,2745}, {2746,1381,1296}, {2746,1382,1296}, {2746,2721,2746}, {2747,1381,1297}, {2747,1382,1297}, {2747,2722,2747}, {2751,2730,2751}, {2752,1113,105}, {2752,1114,105}, {2752,2691,2752}, {2757,2731,2757}, {2758,1113,106}, {2758,1114,106}, {2758,2692,2758}, {2762,2732,2762}, {2765,2733,2765}, {2766,1113,108}, {2766,1114,108}, {2766,2694,2766}, {2768,2735,2768}, {2770,1113,111}, {2770,1114,111}, {2770,2696,2770},

{2867,935,476}, {2867,2697,476}, {2867,2710,805}, {2867,2715,805}, {2867,2722,901}, {2867,2741,927}, {2867,2747,901}, {2867,2754,6078}, {2867,2760,6079}, {2867,2764,6080}, {2867,2769,1309}, {2867,53182,927}, {2867,53186,6082}, {2867,53187,16170}, {2867,53188,16170},

{3563,1113,40118}, {3563,1114,40118}, {3563,1379,23700}, {3563,1380,23700}, {3563,3565,3563}, {3565,1379,10425}, {3565,1380,10425}, {3565,3563,3565}, {3659,7597,3659}, {4588,28159,4588}, {5606,1381,6584}, {5606,1382,6584}, {5606,5951,5606}, {5896,44060,5896}, {5897,1301,5897}, {5951,5606,5951}, {5970,1379,729}, {5970,1380,729}, {5994,1379,5995}, {5994,1380,5995}, {5995,1379,5994}, {5995,1380,5994}, {6010,741,6010}, {6011,759,6011}, {6013,29310,6013}, {6014,28233,6014},

{6078,840,901}, {6078,2691,476}, {6078,2704,805}, {6078,2711,805}, {6078,2725,927}, {6078,2730,1309}, {6078,2736,927}, {6078,2742,901}, {6078,2748,6079}, {6078,2749,6080}, {6078,2750,6081}, {6078,2751,1309}, {6078,2752,476}, {6078,2753,6082}, {6078,2754,2867}, {6078,28914,6078},

{6079,2692,476}, {6079,2705,805}, {6079,2712,805}, {6079,2718,901}, {6079,2726,927}, {6079,2731,1309}, {6079,2737,927}, {6079,2743,901}, {6079,2748,6078}, {6079,2755,6080}, {6079,2756,6081}, {6079,2757,1309}, {6079,2758,476}, {6079,2759,6082}, {6079,2760,2867}, {6079,44873,6079},

{6080,1304,476}, {6080,2693,476}, {6080,2706,805}, {6080,2713,805}, {6080,2719,901}, {6080,2727,927}, {6080,2732,1309}, {6080,2738,927}, {6080,2744,901}, {6080,2749,6078}, {6080,2755,6079}, {6080,2761,6081}, {6080,2762,1309}, {6080,2763,6082}, {6080,2764,2867}, {6080,44874,6080},

{6081,2694,476}, {6081,2707,805}, {6081,2714,805}, {6081,2720,901}, {6081,2728,927}, {6081,2733,1309}, {6081,2739,927}, {6081,2745,901}, {6081,2750,6078}, {6081,2756,6079}, {6081,2761,6080}, {6081,2765,1309}, {6081,2766,476}, {6081,2767,6082},

{6082,843,805}, {6082,2696,476}, {6082,2709,805}, {6082,2721,901}, {6082,2729,927}, {6082,2735,1309}, {6082,2740,927}, {6082,2746,901}, {6082,2753,6078}, {6082,2759,6079}, {6082,2763,6080}, {6082,2767,6081}, {6082,2768,1309}, {6082,2770,476}, {6082,6093,6082}, {6082,9184,16170}, {6082,39446,20404}, {6082,53186,2867},

{6083,12030,476},

{6093,843,2698}, {6093,2696,477}, {6093,2709,2698}, {6093,2721,953}, {6093,2729,2724}, {6093,2735,2734}, {6093,2740,2724}, {6093,2746,953}, {6093,2753,28914}, {6093,2759,44873}, {6093,2763,44874}, {6093,2768,2734}, {6093,2770,477}, {6093,6082,6093}, {6093,9184,16169}, {6093,39446,53605},

{6099,1381,13397}, {6099,1382,13397}, {6099,43078,6099}, {6233,1379,13241}, {6233,1380,13241}, {6233,6323,6233}, {6236,1113,32229}, {6236,1114,32229}, {6236,6325,6236}, {6323,1379,9831}, {6323,1380,9831}, {6323,6233,6323}, {6325,74,11593}, {6325,110,11593}, {6325,6236,6325}, {6584,1381,5606}, {6584,1382,5606}, {7597,3659,7597}, {7953,29316,7953}, {7954,29322,7954}, {8652,28145,8652}, {8686,1381,43081}, {8686,1382,43081}, {8686,30236,8686}, {8687,29206,8687}, {8694,28193,8694}, {8696,28320,8696}, {8697,28203,8697}, {8698,733,2824}, {8699,28235,8699}, {8700,28305,8700}, {8701,28173,8701}, {8706,43348,33435}, {8706,48257,8706}, {8708,29308,8708}, {9060,841,9060}, {9060,1113,1302}, {9060,1114,1302}, {9076,44061,9076}, {9080,48260,9080}, {9081,1381,739}, {9081,1382,739}, {9082,39631,9082}, {9084,1113,10102}, {9084,1114,10102}, {9084,30256,9084}, {9093,100,28535}, {9093,104,28535}, {9136,1379,52230}, {9136,1380,52230}, {9150,137 9,39442}, {9150,1380,39442}, {9160,74,99}, {9160,110,99}, {9160,1379,476}, {9160,1380,476}, {9160,9161,9160}, {9161,74,98}, {9161,110,98}, {9161,1379,477}, {9161,1380,477}, {9161,9160,9161}, {9184,74,111}, {9184,110,111}, {9184,39446,53604}, {9202,1379,9203}, {9202,1380,9203}, {9202,2378,9202}, {9203,1379,9202}, {9203,1380,9202}, {9203,2379,9203}, {9831,1379,6323}, {9831,1380,6323}, {9831,13241,9831}, {10098,1113,30247}, {10098,1114,30247}, {10101,1113,26706}, {10101,1114,26706}, {10102,1113,9084}, {10102,1114,9084}, {10409,1379,36514}, {10409,1380,36514}, {10410,1379,36515}, {10410,1380,36515}, {10420,74,35189}, {10420,110,35189}, {10420,1113,925}, {10420,1114,925}, {10420,32710,10420}, {10423,1113,1289}, {10423,1114,1289}, {10425,74,35191}, {10425,110,35191}, {10425,1379,3565}, {10425,1380,3565}, {10425,23700,10425}, {11568,1113,32425}, {11568,1114,32425}, {11593,74,6325}, {11593,110,6325}, {11636,1379,32694}, {11636,1380,32694}, {11636,14388,11636}, {12030,1113,759}, {12030,1114,759}, {12031,1379,741}, {12031,1380,741}, {12032,813,12032}, {12092,22751,12092}, {12507,13238,12507}, {13238,12507,13238}, {13241,691,32229}, {13241,842,32229}, {13241,1379,6233}, {13241,1380,6233}, {13241,9831,13241}, {13397,915,13397}, {13397,1381,6099}, {13397,1382,6099}, {13398,1299,13398}, {13530,53693,13530}, {13593,13594,13593}, {13594,13593,13594}, {13597,20189,13597}, {13863,1113,30248}, {13863,1114,30248}, {14074,15731,14074}, {14388,11636,14388}, {14719,14720,14719}, {14720,14719,14720}, {14733,1381,934}, {14733,1382,934}, {14979,1113,1141}, {14979,1114,1141}, {14979,1291,14979}, {14987,33637,14987}, {15323,932,15323}, {15728,30237,15728}, {15731,1381,43080}, {15731,1382,43080}, {15731,14074,15731}, {15907,1291,477}, {15907,14979,477}, {16169,74,477}, {16169,110,477}, {16169,9160,2698}, {16169,9161,2698}, {16169,9184,6093}, {16169,16170,16169}, {16170,74,476}, {16170,110,476}, {16170,9160,805}, {16170,9161,805}, {16170,9184,6082}, {16170,16169,16170}, {16170,53187,2867}, {16170,53188,2867}, {17222,28295,17222}, {17223,28307,17223}, {18401,933,18401}, {20185,2383,20185}, {20187,2374,20187}, {20189,1113,53705}, {20189,1114,53705}, {20189,13597,20189}, {20219,38451,20219}, {20404,74,805}, {20404,98,476}, {20404,99,476}, {20404,110,805}, {20404,39446,6082}, {20404,53605,20404}, {22239,1113,1301}, {22239,1114,1301}, {22456,1113,53699}, {22456,1114,53699}, {22456,48259,22456}, {22751,12092,22751},

{23700,1379,3563}, {23700,1380,3563}, {23700,10425,23700}, {23701,35188,23701}, {26700,1113,35056}, {26700,1114,35056}, {26700,1381,34921}, {26700,1382,34921},

{26703,26706,26703}, {26704,41904,26704}, {26705,41905,26705}, {26706,1113,10101}, {26706,1114,10101}, {26706,26703,26706}, {28145,8652,28145}, {28148,28149,28148}, {28149,28148,28149}, {28152,28153,28152}, {28153,28152,28153}, {28156,28157,28156}, {28157,28156,28157}, {28159,4588,28159}, {28162,28163,28162}, {28163,28162,28163}, {28166,28167,28166}, {28167,28166,28167}, {28170,28171,28170}, {28171,28170,28171}, {28173,8701,28173}, {28176,28177,28176}, {28177,28176,28177}, {28180,28181,28180}, {28181,28180,28181}, {28184,28185,28184}, {28185,28184,28185}, {28188,28189,28188}, {28189,28188,28189}, {28193,8694,28193}, {28196,28197,28196}, {28197,28196,28197}, {28200,28201,28200}, {28201,28200,28201}, {28203,8697,28203}, {28206,28207,28206}, {28207,28206,28207}, {28210,28211,28210}, {28211,28210,28211}, {28214,28215,28214}, {28215,28214,28215}, {28218,28219,28218}, {28219,28218,28219}, {28222,28223,28222}, {28223,28222,28223}, {28226,28227,28226}, {28227,28226,28227}, {28230,28231,28230}, {28231,28230,28231}, {28233,6014,28233}, {28235,8699,28235}, {28291,1381,53184}, {28291,1382,53184}, {28291,2291,28291}, {28293,2384,28293}, {28295,17222,28295}, {28298,28299,28298}, {28299,28298,28299}, {28302,28303,28302}, {28303,28302,28303}, {28305,8700,28305}, {28307,17223,28307}, {28310,28311,28310}, {28311,28310,28311}, {28314,28315,28314}, {28315,28314,28315}, {28317,28318,28317}, {28318,28317,28318}, {28320,8696,28320}, {28323,28324,28323}, {28324,28323,28324}, {28326,28327,28326}, {28327,28326,28327}, {28330,28331,28330}, {28331,28330,28331}, {28334,28335,28334}, {28335,28334,28335}, {28338,28339,28338}, {28339,28338,28339}, {28467,753,28467}, {28469,727,28469}, {28474,739,28474}, {28476,28477,28476}, {28477,28476,28477}, {28479,28480,28479}, {28480,28479,28480}, {28482,28483,28482}, {28483,28482,28483}, {28485,28486,28485}, {28486,28485,28486}, {28488,28489,28488}, {28489,28488,28489}, {28491,28492,28491}, {28492,28491,28492}, {28495,28496,28495}, {28496,28495,28496}, {28499,28500,28499}, {28500,28499,28500}, {28505,28506,28505}, {28506,28505,28506}, {28509,28510,28509}, {28510,28509,28510}, {28513,28514,28513}, {28514,28513,28514}, {28517,28518,28517}, {28518,28517,28518}, {28520,2382,28520}, {28523,28524,28523}, {28524,28523,28524}, {28527,28528,28527}, {28528,28527,28528}, {28531,28532,28531}, {28532,28531,28532}, {28535,100,9093}, {28535,104,9093}, {28535,28536,28535}, {28536,28535,28536}, {28539,28540,28539}, {28540,28539,28540}, {28543,28544,28543}, {28544,28543,28544}, {28547,28548,28547}, {28548,28547,28548}, {28551,28552,28551}, {28552,28551,28552}, {28559,28560,28559}, {28560,28559,28560}, {28563,28564,28563}, {28564,28563,28564}, {28567,28568,28567}, {28568,28567,28568}, {28571,28572,28571}, {28572,28571,28572}, {28574,28575,28574}, {28575,28574,28575}, {28577,28578,28577}, {28578,28577,28578}, {28583,28584,28583}, {28584,28583,28584}, {28838,919,28838}, {28841,28842,28841}, {28842,28841,28842}, {28844,825,28844}, {28847,28848,28847}, {28848,28847,28848}, {28852,28853,28852}, {28853,28852,28853}, {28856,28857,28856}, {28857,28856,28857}, {28864,28865,28864}, {28865,28864,28865}, {28868,28869,28868}, {28869,28868,28869}, {28872,28873,28872}, {28873,28872,28873}, {28875,28876,28875}, {28876,28875,28876}, {28879,28880,28879}, {28880,28879,28880}, {28883,28884,28883}, {28884,28883,28884}, {28887,28888,28887}, {28888,28887,28888}, {28891,28892,28891}, {28892,28891,28892}, {28895,28896,28895}, {28896,28895,28896}, {28899,28900,28899}, {28900,28899,28900}, {28903,28904,28903}, {28904,28903,28904}, {28907,28908,28907}, {28908,28907,28908}, {28911,28912,28911}, {28912,28911,28912},

{28914,840,953}, {28914,2691,477}, {28914,2704,2698}, {28914,2711,2698}, {28914,2725,2724}, {28914,2730,2734}, {28914,2736,2724}, {28914,2742,953}, {28914,2748,44873}, {28914,2749,44874}, {28914,2751,2734}, {28914,2752,477}, {28914,2753,6093}, {28914,6078,28914},

{29009,815,29009}, {29011,827,29011}, {29014,29015,29014}, {29015,29014,29015}, {29018,29019,29018}, {29019,29018,29019}, {29022,29023,29022}, {29023,29022,29023}, {29026,29027,29026}, {29027,29026,29027}, {29030,29031,29030}, {29031,29030,29031}, {29034,29035,29034}, {29035,29034,29035}, {29038,29039,29038}, {29039,29038,29039}, {29041,29042,29041}, {29042,29041,29042}, {29044,29045,29044}, {29045,29044,29045}, {29048,29049,29048}, {29049,29048,29049}, {29052,29053,29052}, {29053,29052,29053}, {29055,29056,29055}, {29056,29055,29056}, {29059,29060,29059}, {29060,29059,29060}, {29063,29064,29063}, {29064,29063,29064}, {29067,29068,29067}, {29068,29067,29068}, {29071,29072,29071}, {29072,29071,29072}, {29075,29076,29075}, {29076,29075,29076}, {29079,29080,29079}, {29080,29079,29080}, {29083,29084,29083}, {29084,29083,29084}, {29087,29088,29087}, {29088,29087,29088}, {29091,29092,29091}, {29092,29091,29092}, {29095,29096,29095}, {29096,29095,29096}, {29099,29100,29099}, {29100,29099,29100}, {29103,29104,29103}, {29104,29103,29104}, {29107,29108,29107}, {29108,29107,29108}, {29111,29112,29111}, {29112,29111,29112}, {29180,907,29180}, {29206,8687,29206}, {29209,29210,29209}, {29210,29209,29210}, {29213,29214,29213}, {29214,29213,29214}, {29217,29218,29217}, {29218,29217,29218}, {29221,29222,29221}, {29222,29221,29222}, {29227,29228,29227}, {29228,29227,29228}, {29233,29234,29233}, {29234,29233,29234}, {29241,29242,29241}, {29242,29241,29242}, {29253,29254,29253}, {29254,29253,29254}, {29257,29258,29257}, {29258,29257,29258}, {29261,29262,29261}, {29262,29261,29262}, {29281,29282,29281}, {29282,29281,29282}, {29285,29286,29285}, {29286,29285,29286}, {29289,29290,29289}, {29290,29289,29290}, {29295,29296,29295}, {29296,29295,29296}, {29299,29300,29299}, {29300,29299,29300}, {29305,29306,29305}, {29306,29305,29306}, {29308,8708,29308}, {29310,6013,29310}, {29313,29314,29313}, {29314,29313,29314}, {29316,7953,29316}, {29319,29320,29319}, {29320,29319,29320}, {29322,7954,29322}, {29325,29326,29325}, {29326,29325,29326}, {29329,29330,29329}, {29330,29329,29330}, {29333,29334,29333}, {29334,29333,29334}, {29337,29338,29337}, {29338,29337,29338}, {29341,29342,29341}, {29342,29341,29342}, {29345,29346,29345}, {29346,29345,29346}, {29348,898,29348}, {29351,29352,29351}, {29352,29351,29352}, {29355,29356,29355}, {29356,29355,29356}, {29359,29360,29359}, {29360,29359,29360}, {29363,29364,29363}, {29364,29363,29364}, {29367,29368,29367}, {29368,29367,29368}, {29371,29372,29371}, {29372,29371,29372}, {30236,8686,30236}, {30237,15728,30237}, {30239,19657,38808}, {30247,1113,10098}, {30247,1114,10098}, {30247,2373,30247}, {30248,1113,13863}, {30248,1114,13863}, {30248,39431,30248}, {30249,39434,30249}, {30250,39435,30250}, {30251,39436,30251}, {30254,699,30254}, {30256,9084,30256}, {32229,691,13241}, {32229,842,13241}, {32229,1113,6236}, {32229,1114,6236}, {32425,1113,11568}, {32425,1114,11568}, {32439,1113,1298}, {32439,1114,1298}, {32694,1379,11636}, {32694,1380,11636}, {32704,2370,32704}, {32706,41906,32706}, {32710,1113,1300}, {32710,1114,1300}, {32710,10420,32710}, {33637,1381,43355}, {33637,1382,43355}, {33637,14987,33637}, {33638,43656,33638}, {33639,33643,33639}, {33643,33639,33643}, {34168,1289,34168}, {34921,1381,26700}, {34921,1382,26700}, {35011,100,1309}, {35011,102,901}, {35011,104,1309}, {35011,109,901}, {35056,1113,26700}, {35056,1114,26700}, {35107,1379,739}, {35107,1380,739}, {35188,23701,35188}, {35189,74,10420}, {35189,110,10420}, {35191,74,10425}, {35191,110,10425}, {36514,1379,10409}, {36514,1380,10409}, {36515,1379,10410}, {36515,1380,10410}, {36516,2249,36516}, {36735,1381,1379}, {36735,1382,1379}, {36735,36736,36735}, {36736,1381,1380}, {36736,1382,1380}, {36736,36735,36736}, {38451,20219,38451}, {38452,39628,38452}, {38882,39635,38882}, {38884,39640,38884}, {39422,1379,39423}, {39422,1380,39423}, {39423,1379,39422}, {39423,1380,39422}, {39431,30248,39431}, {39434,30249,39434}, {39435,30250,39435}, {39436,30251,39436}, {39437,44064,39437}, {39439,44065,39439}, {39442,1379,9150}, {39442,1380,9150}, {39443,1381,898}, {39443,1382,898}, {39444,100,101}, {39444,104,101}, {39445,100,106}, {39445,104,106}, {39448,1379,1291}, {39448,1380,1291}, {39628,38452,39628}, {39629,737,39629}, {39631,9082,39631}, {39635,38882,39635}, {39636,2380,39636}, {39637,2381,39637}, {39639,729,39639}, {39640,38884,39640}, {40118,1113,3563}, {40118,1114,3563}, {40119,1113,2374}, {40119,1114,2374}, {41904,26704,41904}, {41905,26705,41905}, {41906,32706,41906}, {43078,1381,915}, {43078,1382,915}, {43078,6099,43078}, {43080,1381,15731}, {43080,1382,15731}, {43081,1381,8686}, {43081,1382,8686}, {43351,45138,43351}, {43354,51760,43354}, {43355,1381,33637}, {43355,1382,33637}, {43356,32630,11940}, {43359,29030,27142}, {43359,29031,27142}, {43362,1381,932}, {43362,1382,932}, {43363,44059,43363}, {43654,98,74}, {43654,99,74}, {43654,1113,2698}, {43654,1114,2698}, {43654,53603,43654}, {43655,100,74}, {43655,104,74}, {43655,1113,953}, {43655,1114,953}, {43655,53611,43655}, {43656,33638,43656}, {43657,44066,43657}, {43659,44063,43659}, {43660,1113,841}, {43660,1114,841}, {43660,1302,43660}, {44059,43363,44059}, {44060,5896,44060}, {44061,9076,44061}, {44062,45781,44062}, {44063,43659,44063}, {44064,39437,44064}, {44065,39439,44065}, {44066,43657,44066}, {44873,2692,477}, {44873,2705,2698}, {44873,2712,2698}, {44873,2718,953}, {44873,2726,2724}, {44873,2731,2734}, {44873,2737,2724}, {44873,2743,953}, {44873,2748,28914}, {44873,2755,44874}, {44873,2757,2734}, {44873,2758,477}, {44873,2759,6093}, {44873,6079,44873},

{44874,1304,477}, {44874,2693,477}, {44874,2706,2698}, {44874,2713,2698}, {44874,2719,953}, {44874,2727,2724}, {44874,2732,2734}, {44874,2738,2724}, {44874,2744,953}, {44874,2749,28914}, {44874,2755,44873}, {44874,2762,2734}, {44874,2763,6093}, {44874,6080,44874},

{44875,2379,2698}, {44875,9203,2698}, {44875,11612,53605}, {44876,675,44876}, {44876,1113,53189}, {44876,1114,53189}, {45135,46965,45135}, {45136,835,45136}, {45137,1310,45137}, {45138,43351,45138}, {45781,44062,45781}, {46963,51761,46963}, {46964,51762,46964}, {46965,45135,46965}, {46970,1379,827}, {46970,1380,827}, {48257,8706,48257}, {48259,22456,48259}, {48260,9080,48260}, {51760,43354,51760}, {51761,46963,51761}, {51762,46964,51762}, {52230,1379,9136}, {52230,1380,9136}, {52998,1113,933}, {52998,1114,933}, {53179,1379,2291}, {53179,1380,2291}, {53180,1379,840}, {53180,1380,840}, {53181,1381,2291}, {53181,1382,2291}, {53181,53184,53181}, {53182,2741,53182}, {53184,1381,28291}, {53184,1382,28291}, {53184,53181,53184}, {53185,1113,2291}, {53185,1114,2291}, {53187,74,112}, {53187,110,112}, {53187,53188,53187}, {53188,74,1297}, {53188,110,1297}, {53188,53187,53188}, {53189,1113,44876}, {53189,1114,44876}, {53189,53190,53189}, {53190,1113,675}, {53190,1114,675}, {53190,53189,53190}, {53603,98,110}, {53603,99,110}, {53603,1113,805}, {53603,1114,805}, {53603,43654,53603}, {53604,98,111}, {53604,99,111}, {53604,39446,9184}, {53605,74,2698}, {53605,98,477}, {53605,99,477}, {53605,110,2698}, {53605,11612,44875}, {53605,20404,53605}, {53605,39446,6093}, {53606,100,99}, {53606,104,99}, {53606,1379,901}, {53606,1380,901}, {53607,101,100}, {53607,103,100}, {53607,1381,927}, {53607,1382,927}, {53608,100,105}, {53608,104,105}, {53609,100,108}, {53609,104,108}, {53610,100,109}, {53610,104,109}, {53611,100,110}, {53611,104,110}, {53611,1113,901}, {53611,1114,901}, {53611,43655,53611}, {53612,102,110}, {53612,109,110}, {53612,1113,1309}, {53612,1114,1309}, {53613,111,691}, {53613,1296,691}, {53692,1113,2715}, {53692,1114,2715}, {53693,13530,53693}, {53699,1113,22456}, {53699,1114,22456}, {53705,1113,20189}, {53705,1114,20189}

X(53603) = REFLECTION OF X(110) IN X(3)X(98)

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

X(53603) lies on the circumcircle and these lines: {3, 43654}, {23, 53704}, {30, 2698}, {74, 2782}, {98, 15915}, {99, 14295}, {110, 804}, {111, 9159}, {477, 47620}, {523, 805}, {542, 9161}, {690, 9160}, {691, 48951}, {733, 34294}, {741, 53559}, {842, 5999}, {2071, 48259}, {2367, 5152}, {2797, 53971}, {3563, 12131}, {4226, 53936}, {4374, 36066}, {5663, 53867}, {6037, 7468}, {9091, 47258}, {9149, 53604}, {9830, 11593}, {9831, 11594}, {11007, 38947}, {32710, 35474}, {36163, 53700}, {43765, 46303}, {47293, 53892}, {47326, 53918}, {53738, 53949}

X(53603) = reflection of X(i) in X(j) for these {i,j}: {38947, 11007}, {43654, 3}
X(53603) = reflection of X(805) in the Euler line
X(53603) = reflection of X(110) in X(3)X(98)
X(53603) = reflection of X(805) in X(3)X(1113)
X(53603) = isotomic conjugate of the anticomplement of X(47229)
X(53603) = Collings transform of X(i) for these i: {5661, 11007}
X(53603) = cevapoint of X(i) and X(j) for these (i,j): {512, 5661}, {523, 11007}
X(53603) = trilinear pole of line {6, 36207}
X(53603) = barycentric product X(43187)*X(52446)
X(53603) = barycentric quotient X(52446)/X(3569)

---

X(53604) = REFLECTION OF X(111) IN X(3)X(98)

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

X(53604) lies on the circumcircle and these lines: {110, 5026}, {111, 804}, {112, 5186}, {385, 691}, {524, 805}, {669, 2770}, {733, 22105}, {842, 48980}, {1296, 2782}, {1499, 2698}, {5912, 5970}, {5913, 9091}, {5939, 9066}, {5971, 9150}

X(53604) = trilinear pole of line {6, 11183}

---

X(53605) = REFLECTION OF X477) IN X(3)X(98)

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

X(53605) lies on the circumcircle and these lines: {3, 20404}, {4, 35582}, {56, 44051}, {99, 12584}, {110, 8724}, {111, 1640}, {112, 32234}, {476, 2782}, {477, 804}, {526, 2698}, {542, 691}, {690, 842}, {805, 5663}, {5967, 18331}, {9830, 32229}, {9831, 32228}, {10553, 45773}, {10556, 12243}, {11005, 23969}, {11636, 12177}, {35191, 51474}

X(53605) = reflection of X(i) in X(j) for these {i,j}: {4, 35582}, {20404, 3}
X(53605) = Thomson-isogonal conjugate of X(20403)
X(53605) = Collings transform of X(35582)

---

X(53606) = REFLECTION OF X477) IN X(3)X(98)

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

X(53606) lies on the circumcircle and these lines: {98, 952}, {99, 900}, {100, 4730}, {101, 14407}, {106, 3122}, {110, 1960}, {111, 8649}, {187, 2384}, {214, 662}, {249, 36069}, {316, 2861}, {511, 953}, {512, 901}, {835, 15343}, {2222, 4564}, {4557, 6551}, {6083, 33803}, {6224, 19628}, {17100, 28471}

X(53606) = reflection of X(901) in the Brocard axis
X(53606) = Schoutte-circle-inverse of X(2384)
X(53606) = X(i)-isoconjugate of X(j) for these (i,j): {88, 46457}, {100, 46458}, {1577, 5170}, {1647, 39154}, {2170, 45273}, {17998, 30939}, {18011, 52680}
X(53606) = X(8054)-Dao conjugate of X(46458)
X(53606) = cevapoint of X(214) and X(41191)
X(53606) = crossdifference of every pair of points on line {46457, 46458}
X(53606) = barycentric product X(5376)*X(39155)
X(53606) = barycentric quotient X(i)/X(j) for these {i,j}: {59, 45273}, {649, 46458}, {902, 46457}, {1576, 5170}

---

X(53607) = REFLECTION OF X100) IN X(3)X(101)

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

X(53607) lies on the circumcircle and these lines: {36, 12032}, {59, 919}, {100, 926}, {101, 34905}, {104, 2808}, {105, 651}, {513, 927}, {517, 2724}, {934, 53539}, {1155, 39421}, {1319, 14665}, {2291, 9319}, {2369, 52508}, {2711, 5172}, {3887, 39444}, {4564, 9320}

X(53607) = reflection of X(927) in the OI line
X(53607) = X(i)-isoconjugate of X(j) for these (i,j): {522, 5091}, {650, 9318}, {2170, 40865}, {17435, 34906}
X(53607) = trilinear pole of line {6, 2283}
X(53607) = barycentric product X(i)*X(j) for these {i,j}: {651, 14947}, {664, 9319}, {2283, 53214}, {34905, 39293}
X(53607) = barycentric quotient X(i)/X(j) for these {i,j}: {59, 40865}, {109, 9318}, {1415, 5091}, {9319, 522}, {14947, 4391}

---

X(53608) = REFLECTION OF X105) IN X(3)X(100)

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

X(53608) lies on the circumcircle and these lines: {44, 919}, {101, 14439}, {105, 900}, {106, 2254}, {108, 1862}, {109, 3722}, {518, 901}, {952, 1292}, {953, 3309}, {1293, 1768}, {17100, 52778}

---

X(53609) = REFLECTION OF X108) IN X(3)X(100)

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

X(53609) lies on the circumcircle and these lines: {101, 14418}, {102, 1768}, {106, 7004}, {108, 900}, {109, 53532}, {521, 901}, {915, 18341}, {952, 1295}, {953, 6001}, {4571, 6551}, {9058, 15343}

---

X(53610) = REFLECTION OF X109) IN X(3)X(100)

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

X(53610) lies on the circumcircle and these lines: {11, 106}, {100, 4768}, {101, 1639}, {102, 952}, {109, 900}, {110, 15343}, {515, 953}, {522, 901}, {759, 14204}, {1324, 2758}, {2370, 17100}, {3699, 6551}

X(53610) = trilinear pole of line {6, 4530}

---

X(53611) = REFLECTION OF X110) IN X(3)X(100)

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

X(53611) lies on the circumcircle and these lines: {3, 43655}, {23, 2726}, {30, 953}, {74, 952}, {101, 4120}, {102, 12119}, {106, 3120}, {109, 30572}, {110, 900}, {523, 901}, {2071, 2734}, {2222, 36167}, {2687, 6909}, {2690, 36030}, {2752, 37449}, {3952, 6551}, {36155, 38950}

X(53611) = reflection of X(i) in X(j) for these {i,j}: {38950, 36155}, {43655, 3}
X(53611) = reflection of X(901) in the Euler line
X(53611) = Collings transform of X(36155)
X(53611) = cevapoint of X(523) and X(36155)
X(53611) = trilinear pole of line {6, 6788}

---

X(53612) = REFLECTION OF X110) IN X(3)X(102)

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

X(53612) lies on the circumcircle and these lines: {4, 43655}, {30, 2734}, {74, 2818}, {104, 18210}, {107, 39534}, {110, 8677}, {112, 3310}, {162, 759}, {186, 953}, {250, 36069}, {468, 2726}, {523, 1309}, {1294, 36158}, {2222, 7012}, {2687, 37305}, {2694, 6909}, {2695, 37420}, {4242, 35050}, {12030, 17515}, {19628, 52167}, {39444, 46418}

X(53612) = reflection of X(1309) in the Euler line
X(53612) = X(i)-isoconjugate of X(j) for these (i,j): {656, 3109}, {36037, 45922}, {38950, 53532}
X(53612) = X(i)-Dao conjugate of X(j) for these (i,j): {3259, 45922}, {40596, 3109}
X(53612) = trilinear pole of line {6, 43692}
X(53612) = barycentric product X(648)*X(43692)
X(53612) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 3109}, {3310, 45922}, {43692, 525}

---

X(53613) = REFLECTION OF X691) IN X(3)X(111)

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

X(53613) lies on the circumcircle and these lines: {98, 14666}, {99, 34206}, {111, 9176}, {542, 6093}, {543, 2770}, {671, 9084}, {690, 6082}, {691, 6088}, {842, 33962}, {843, 2854}, {895, 52230}, {2696, 2793}, {2709, 2780}

X(53613) = trilinear pole of line {6, 32583}
X(53613) = barycentric quotient X(34206)/X(52628)

---

X(53614) = SUPPA-CUCOANES-CIRCLE-INVERSE OF X(2)

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

The circle ((X1), r*sqrt(2)) described in the paper Ercole Suppa and Marian Cucoanes, Solution of Problem 2023-1-6 is named here Suppa Cucoanes circle

See Ivan Pavlov, euclid 5876.

X(53614) lies on the circumconic {{A, B, C, X(2726), X(30741)}} and these lines: {1, 2}, {44, 3039}, {80, 24231}, {150, 4887}, {515, 18201}, {952, 24216}, {3036, 4864}, {5126, 14027}, {5853, 26727}, {14563, 17717}, {17715, 38127}, {33103, 50796}, {37599, 37730}

X(53614) = reflection of X(i) in X(j) for these {i,j}: {1, 51615}, {5121, 6788}, {6790, 50535}
X(52154) = Suppa-Cucoanes-circle-inverse of X(2)
X(53614) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 22166, 25055}, {78, 3621, 1125}, {519, 50535, 6790}, {519, 6788, 5121}, {1125, 34641, 4847}, {1125, 6743, 4668}, {3244, 3679, 3624}, {3616, 20049, 3244}, {3617, 3625, 4677}, {3617, 49765, 145}, {3621, 20057, 12629}, {4701, 19883, 8}, {9534, 51096, 31253}, {15808, 29600, 1}, {20053, 38314, 20014}

---

X(53615) = SUPPA-CUCOANES-CIRCLE-INVERSE OF X(3)

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

See Ivan Pavlov, euclid 5876.

X(53615) lies on these lines: {1, 3}, {5, 45288}, {11, 14988}, {47, 3924}, {72, 5123}, {80, 912}, {499, 3869}, {513, 21111}, {515, 11570}, {518, 10057}, {519, 44311}, {535, 18389}, {758, 908}, {938, 5180}, {1104, 2964}, {1210, 4084}, {1411, 52407}, {1772, 22350}, {1785, 1830}, {1788, 10320}, {1795, 34242}, {1858, 37702}, {1878, 1905}, {2390, 15906}, {2800, 5533}, {3555, 33956}, {3582, 44663}, {3583, 6001}, {3585, 7686}, {3754, 5249}, {3812, 26725}, {3868, 5176}, {3880, 26726}, {3901, 17615}, {3919, 31397}, {4018, 5087}, {4293, 18419}, {4324, 9943}, {4511, 10090}, {4744, 11019}, {5080, 5905}, {5445, 31837}, {5692, 30827}, {5693, 10826}, {5694, 17606}, {5728, 28534}, {5883, 6681}, {5884, 10572}, {5887, 7741}, {6147, 10954}, {6256, 10052}, {6882, 12832}, {7743, 17638}, {8070, 12047}, {10896, 40266}, {10948, 22791}, {10950, 24475}, {11219, 17654}, {12688, 18514}, {13391, 49745}, {13411, 33815}, {15071, 52851}, {16153, 44547}, {17660, 28204}, {17885, 52385}, {18393, 22835}, {18397, 28609}, {24201, 24216}, {24473, 34690}, {30329, 41572}, {38128, 40663}, {45272, 46044}

X(53615) = midpoint of X(i) in X(j) for these {i,j}: {15071, 52851}, {2077, 37625}, {3583, 11571}, {3868, 5176}, {4084, 11813}
X(53615) = reflection of X(i) in X(j) for these {i,j}: {1, 5570}, {1319, 942}, {1737, 12736}, {17638, 7743}, {23961, 5885}, {36, 18838}, {72, 5123}
X(53615) = circumcircle inverse of X(36152)
X(52615) = Suppa-Cucoanes-circle-inverse of X(3)
X(53615) = X(i)-vertex conjugate of X(j) for these {i, j}: {513, 36152}
X(53615) = X(i)-Ceva conjugate of X(j) for these {i, j}: {43355, 513}
X(53615) = intersection, other than A, B, C, of circumconics {{A, B, C, X(46), X(47645)}}, {{A, B, C, X(59), X(36152)}}, {{A, B, C, X(80), X(32760)}}, {{A, B, C, X(1385), X(43947)}}, {{A, B, C, X(5665), X(17700)}} and {{A, B, C, X(29374), X(37531)}}
X(53615) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 14792, 2646}, {1, 31515, 3576}, {1, 3339, 17700}, {1, 484, 32760}, {36, 37525, 18857}, {36, 5902, 18838}, {65, 24474, 5903}, {65, 3057, 35004}, {517, 18838, 36}, {517, 5885, 23961}, {517, 942, 1319}, {758, 12736, 1737}, {942, 15178, 13751}, {1381, 1382, 36152}, {2077, 37625, 517}, {2099, 22767, 1}, {2446, 2447, 1385}, {3057, 10202, 37525}, {3057, 13751, 15178}, {3583, 11571, 6001}, {5425, 5902, 942}, {5902, 5903, 57}, {13151, 37568, 14799}

---

X(53616) = SUPPA-CUCOANES-CIRCLE-INVERSE OF X(5)

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

See Ivan Pavlov, euclid 5876.

X(53616) lies on these lines: {1, 5}, {4, 11571}, {35, 12619}, {36, 20118}, {56, 12747}, {100, 18395}, {149, 10573}, {214, 5086}, {484, 5840}, {499, 6224}, {1479, 12247}, {1621, 10087}, {1737, 17010}, {2067, 35853}, {2099, 51517}, {2800, 3583}, {2802, 41684}, {3035, 38411}, {3086, 20085}, {3299, 49240}, {3301, 49241}, {3336, 12832}, {3585, 6246}, {3884, 15863}, {4324, 46684}, {4857, 12758}, {5083, 5270}, {5330, 12531}, {5444, 34126}, {5445, 33814}, {5563, 18976}, {5697, 13274}, {5902, 13273}, {5903, 10738}, {6502, 35852}, {6690, 34122}, {6702, 35016}, {6713, 37616}, {6738, 33593}, {7280, 12119}, {9956, 41541}, {10265, 10572}, {10896, 48667}, {12047, 41558}, {12736, 18244}, {12761, 15071}, {13272, 41686}, {14793, 18515}, {17606, 22935}, {17660, 18480}, {18514, 34789}, {25485, 40259}, {37567, 48680}, {41698, 41699}

X(53616) = reflection of X(i) in X(j) for these {i,j}: {1, 5533}, {18861, 10265}, {36, 20118}, {45764, 39692}
X(52616) = Suppa-Cucoanes-circle-inverse of X(5)
X(53616) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 31517, 5886}, {11, 1317, 5901}, {11, 19907, 37735}, {80, 10073, 1}, {80, 12750, 12751}, {80, 37702, 11}, {80, 49176, 9897}, {80, 7972, 355}, {952, 39692, 45764}, {6246, 11570, 3585}, {7972, 37735, 19907}, {13274, 19914, 5697}, {37718, 45764, 39692}

---

X(53617) = SUPPA-CUCOANES-CIRCLE-INVERSE OF X(7)

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

See Ivan Pavlov, euclid 5876.

X(53617) lies on these lines: {1, 7}, {10, 3177}, {11, 241}, {36, 105}, {46, 2082}, {65, 14520}, {100, 43288}, {109, 2724}, {514, 1734}, {517, 1362}, {519, 39353}, {527, 34931}, {664, 28849}, {908, 35293}, {927, 28848}, {948, 5218}, {1054, 3008}, {1155, 5723}, {1212, 3826}, {1478, 50291}, {1698, 5199}, {1737, 43672}, {1738, 43065}, {2093, 50282}, {2369, 39640}, {2550, 42050}, {3322, 5126}, {3751, 7960}, {3790, 25242}, {4989, 52542}, {5045, 39790}, {5074, 12047}, {5179, 7951}, {5880, 34522}, {5902, 14519}, {6603, 17768}, {6610, 38454}, {9436, 28850}, {10186, 40719}, {14197, 53241}, {14942, 17078}, {15658, 50190}, {24635, 33108}, {28534, 35110}, {38052, 52705}, {41339, 43066}, {44664, 50441}

X(53617) = midpoint of X(10651) and X(10652)
X(53617) = reflection of X(i) in X(j) for these {i,j}: {1, 1323}, {3322, 5126}
X(52617) = Suppa-Cucoanes-circle-inverse of X(7)
X(53617) = intersection, other than A, B, C, of circumconics {{A, B, C, X(7), X(2724)}}, {{A, B, C, X(105), X(38459)}}, {{A, B, C, X(514), X(42309)}}, {{A, B, C, X(1308), X(41353)}}, {{A, B, C, X(1323), X(43672)}}, {A, B, C, X(1458), X(28848)}} and {{A, B, C, X(4041), X(42289)}}
X(53617) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {279, 14189, 1323}, {4318, 18461, 1}, {10651, 10652, 516}, {52815, 52816, 14189}

---

X(53618) = SUPPA-CUCOANES-CIRCLE-INVERSE OF X(8)

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

See Ivan Pavlov, euclid 5876.

X(53618) lies on these lines: {1, 2}, {36, 38712}, {517, 6018}, {946, 33103}, {1738, 34124}, {1785, 5151}, {2718, 13397}, {3667, 4017}, {3756, 3880}, {3813, 21949}, {3976, 12053}, {4674, 15637}, {5516, 17757}, {9336, 21073}, {10165, 17715}, {11067, 15803}, {13161, 37722}, {13625, 37706}, {15170, 37599}, {17449, 51423}, {17460, 51433}, {18193, 30305}, {18201, 28194}, {21627, 24174}, {33144, 37704}, {50307, 51816}

X(53618) = midpoint of X(5211) and X(38475)
X(53618) = reflection of X(i) in X(j) for these {i,j}: {38471, 5121}, {46932, 10}
X(52618) = Suppa-Cucoanes-circle-inverse of X(8)
X(53618) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2718), X(3811)}}, {{A, B, C, X(3214), X(4017)}} and {{A, B, C, X(6765), X(9282)}}
X(53618) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 19875, 51073}, {1, 19877, 3624}, {1, 21267, 3679}, {1, 31145, 3633}, {1, 34595, 51108}, {1, 3582, 50745}, {1, 4668, 4701}, {1, 4677, 20014}, {2, 1722, 3622}, {2, 37762, 612}, {8, 19862, 34747}, {8, 3617, 38098}, {8, 46931, 51071}, {10, 15519, 9780}, {10, 19878, 46933}, {10, 21267, 4678}, {10, 3616, 19875}, {10, 3626, 3616}, {10, 3632, 1698}, {10, 3679, 3632}, {10, 4669, 19878}, {10, 4678, 1}, {10, 4701, 19877}, {10, 4746, 2}, {10, 519, 46932}, {519, 5121, 38471}, {612, 3625, 36634}, {3244, 49477, 51092}, {3624, 3636, 25055}, {3624, 3679, 4668}, {3625, 29640, 29582}, {3626, 51073, 31145}, {3633, 3679, 3626}, {3828, 4701, 3636}, {3828, 51073, 46931}, {4669, 19878, 20053}, {4678, 20053, 4669}, {4678, 37743, 51615}, {4694, 30384, 24231}, {5211, 38475, 519}, {9780, 20014, 19883}, {19862, 51070, 8}, {19883, 28092, 13411}, {26046, 51072, 28797}, {46932, 51615, 37743}

---

X(53619) = SUPPA-CUCOANES-CIRCLE-INVERSE OF X(10)

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

See Ivan Pavlov, euclid 5876.

X(53619) lies on these lines: {1, 2}, {30, 18201}, {36, 14027}, {80, 4694}, {354, 37717}, {381, 33103}, {484, 51769}, {515, 24216}, {942, 33097}, {982, 5722}, {988, 37723}, {1447, 24240}, {1837, 3976}, {2718, 33637}, {3419, 17063}, {3583, 3667}, {3586, 18193}, {3756, 44669}, {3836, 30721}, {3880, 26727}, {3953, 37702}, {4864, 5123}, {4867, 5516}, {5080, 17449}, {5883, 33109}, {5902, 33106}, {10896, 18326}, {11512, 12625}, {15934, 17717}, {17072, 48209}, {17556, 33101}, {17597, 37716}, {17598, 37715}, {17715, 26446}, {17757, 49675}, {18527, 33095}, {24473, 33096}, {32856, 37375}

X(53619) = midpoint of X(3632) and X(19872)
X(53619) = reflection of X(i) in X(j) for these {i,j}: {36, 14027}, {30721, 3836}, {47622, 23869}, {5529, 5121}
X(52619) = Suppa-Cucoanes-circle-inverse of X(10)
X(53619) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2718), X(22836)}}, {{A, B, C, X(2726), X(29641)}v} and {{A, B, C, X(3811), X(9282)}}
X(53619) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 31520, 2}, {1, 4701, 4677}, {8, 20014, 4669}, {8, 20052, 3244}, {8, 25979, 10}, {8, 3622, 4746}, {8, 3625, 1698}, {519, 23869, 47622}, {519, 5121, 5529}, {1698, 51110, 34595}, {3241, 19875, 51110}, {3241, 3625, 3632}, {3632, 19872, 519}, {4678, 20053, 3636}, {12649, 28074, 978}, {19878, 51093, 1}, {21267, 51091, 51073}, {31145, 51615, 37743}

---

X(53620) = X(1)X(2)∩X(355)X(376)

Barycentrics    a-5*(b+c) : :

See Ivan Pavlov, euclid 5881.

X(53620) lies on these lines: {1, 2}, {3, 34627}, {4, 3654}, {5, 34718}, {6, 50949}, {7, 11237}, {21, 4421}, {23, 47496}, {30, 5657}, {37, 51036}, {40, 3543}, {55, 16858}, {56, 36006}, {69, 5936}, {72, 5828}, {75, 4723}, {80, 50841}, {100, 9708}, {141, 28635}, {148, 9881}, {165, 38155}, {192, 4732}, {193, 50950}, {210, 44663}, {274, 25278}, {291, 39360}, {329, 17532}, {333, 48832}, {341, 42029}, {344, 31311}, {346, 36911}, {354, 4711}, {355, 376}, {377, 19797}, {381, 962}, {388, 26060}, {390, 51102}, {391, 50115}, {404, 11194}, {515, 10304}, {516, 50687}, {517, 3545}, {518, 4731}, {524, 38087}, {527, 5686}, {528, 38057}, {529, 38100}, {542, 38116}, {545, 17251}, {547, 1482}, {549, 944}, {553, 9578}, {594, 5296}, {597, 50783}, {599, 49524}, {620, 9884}, {631, 3655}, {740, 9333}, {758, 27812}, {858, 47488}, {946, 50827}, {952, 5054}, {956, 16417}, {958, 17549}, {964, 19723}, {966, 3161}, {984, 4695}, {986, 42039}, {999, 9342}, {1001, 17547}, {1010, 4921}, {1043, 17553}, {1121, 50441}, {1145, 9802}, {1150, 19290}, {1213, 16674}, {1278, 51035}, {1282, 50895}, {1319, 31188}, {1327, 35611}, {1328, 35610}, {1376, 13587}, {1385, 15702}, {1387, 50894}, {1483, 10124}, {1573, 17756}, {1621, 16857}, {1654, 7229}, {1699, 38076}, {1706, 3929}, {1739, 4392}, {1768, 50906}, {1788, 5434}, {1837, 10385}, {1992, 3416}, {2049, 19738}, {2094, 3421}, {2320, 5440}, {2345, 16885}, {2476, 9710}, {2478, 37829}, {2482, 50885}, {2550, 5080}, {2551, 52367}, {2948, 50919}, {2975, 9709}, {3035, 10031}, {3036, 6174}, {3090, 5734}, {3146, 34648}, {3242, 20582}, {3295, 17542}, {3303, 17536}, {3419, 51787}, {3476, 5298}, {3486, 4995}, {3488, 11545}, {3522, 9588}, {3523, 5881}, {3524, 5731}, {3525, 37727}, {3526, 34748}, {3533, 15178}, {3576, 15708}, {3578, 50169}, {3579, 11001}, {3589, 51000}, {3618, 47356}, {3619, 49688}, {3620, 49529}, {3648, 15679}, {3653, 7967}, {3656, 5071}, {3672, 50099}, {3681, 3753}, {3696, 4664}, {3697, 3869}, {3698, 3868}, {3701, 42034}, {3702, 20942}, {3710, 43533}, {3717, 28301}, {3729, 49630}, {3739, 51055}, {3740, 3877}, {3746, 16859}, {3751, 11160}, {3755, 50089}, {3763, 51193}, {3820, 11680}, {3823, 49707}, {3826, 51099}, {3829, 4193}, {3830, 6361}, {3832, 7991}, {3839, 5587}, {3844, 47358}, {3845, 12702}, {3871, 4428}, {3876, 3983}, {3885, 25917}, {3895, 7308}, {3902, 18743}, {3913, 5047}, {3918, 5904}, {3951, 4866}, {3956, 5692}, {3968, 4661}, {3992, 4671}, {4002, 24473}, {4005, 10107}, {4015, 5903}, {4031, 51789}, {4147, 47929}, {4152, 25529}, {4188, 5258}, {4197, 12607}, {4297, 50801}, {4301, 5068}, {4308, 24914}, {4312, 50834}, {4323, 4870}, {4344, 26083}, {4345, 10589}, {4359, 4737}, {4370, 31722}, {4371, 17327}, {4385, 4980}, {4402, 17382}, {4405, 25503}, {4413, 40726}, {4423, 8168}, {4430, 5883}, {4441, 18145}, {4445, 49738}, {4460, 17322}, {4461, 50090}, {4470, 4690}, {4479, 18135}, {4487, 24589}, {4488, 17333}, {4533, 50193}, {4642, 42041}, {4644, 10022}, {4645, 4715}, {4648, 28633}, {4654, 4848}, {4665, 4748}, {4680, 42058}, {4688, 24349}, {4698, 50778}, {4699, 31178}, {4704, 4709}, {4714, 28605}, {4733, 9791}, {4739, 49501}, {4741, 24693}, {4742, 30829}, {4751, 51061}, {4755, 49470}, {4761, 47774}, {4772, 49448}, {4788, 51037}, {4821, 49520}, {4895, 45337}, {4903, 42056}, {4916, 6707}, {4922, 45691}, {4945, 51390}, {4967, 5232}, {4968, 44720}, {4997, 36593}, {5032, 5847}, {5044, 14923}, {5046, 51573}, {5051, 42047}, {5055, 5603}, {5056, 7982}, {5059, 34638}, {5067, 10222}, {5086, 11111}, {5090, 7714}, {5119, 27065}, {5122, 5176}, {5123, 5328}, {5159, 47494}, {5175, 11113}, {5177, 28609}, {5180, 31018}, {5223, 51100}, {5224, 32087}, {5234, 50738}, {5252, 5435}, {5260, 5687}, {5265, 37709}, {5278, 11354}, {5281, 5727}, {5284, 19536}, {5303, 19705}, {5433, 6049}, {5459, 50848}, {5460, 50851}, {5461, 7983}, {5493, 17578}, {5541, 50889}, {5642, 50920}, {5691, 15683}, {5743, 6557}, {5749, 16669}, {5774, 19277}, {5793, 16393}, {5795, 18231}, {5815, 17528}, {5837, 8165}, {5844, 15699}, {5846, 47352}, {5854, 32558}, {5882, 10303}, {5886, 38083}, {5901, 15703}, {6055, 50880}, {6173, 24393}, {6175, 11236}, {6284, 7319}, {6329, 51148}, {6537, 23942}, {6684, 15692}, {6702, 50891}, {6883, 38665}, {7253, 45660}, {7426, 47562}, {7486, 13464}, {7667, 34668}, {7705, 26129}, {7810, 34715}, {7812, 34738}, {7840, 50254}, {7883, 34686}, {7976, 44562}, {7984, 45311}, {7987, 50829}, {7989, 50802}, {8227, 50817}, {8236, 38025}, {8256, 11415}, {8591, 13178}, {8596, 13174}, {8666, 17572}, {8692, 17277}, {8703, 18525}, {8715, 16865}, {9041, 21358}, {9143, 13211}, {9172, 50925}, {9263, 41836}, {9331, 52708}, {9350, 37617}, {9436, 52715}, {9460, 36887}, {9589, 50689}, {9624, 46936}, {9782, 11024}, {9785, 11238}, {9809, 10711}, {9860, 50879}, {9864, 11177}, {9875, 20094}, {9900, 50850}, {9901, 50847}, {9904, 50876}, {9947, 9961}, {10005, 50092}, {10109, 18493}, {10154, 34713}, {10164, 15705}, {10171, 11224}, {10172, 16200}, {10175, 38021}, {10246, 11539}, {10247, 38022}, {10248, 19925}, {10283, 47599}, {10543, 15675}, {10713, 17777}, {10989, 47321}, {11038, 38093}, {11041, 31479}, {11050, 12438}, {11114, 34612}, {11178, 39898}, {11235, 27870}, {11248, 28461}, {11274, 50893}, {11359, 48806}, {11491, 28466}, {11522, 15022}, {11681, 17530}, {11724, 50883}, {11725, 50888}, {11729, 50910}, {11735, 50923}, {11737, 50806}, {12100, 37705}, {12513, 17531}, {12531, 50843}, {12571, 51120}, {12645, 15694}, {12699, 41099}, {12780, 51483}, {12781, 51482}, {13466, 17794}, {13595, 37546}, {13624, 15719}, {13688, 33456}, {13808, 33457}, {13846, 49233}, {13847, 49232}, {13911, 19054}, {13973, 19053}, {14005, 42028}, {14269, 28174}, {14891, 50826}, {14893, 48661}, {15177, 37940}, {15621, 19245}, {15640, 31673}, {15671, 44669}, {15674, 31452}, {15677, 47033}, {15678, 18253}, {15681, 50797}, {15682, 18480}, {15684, 50867}, {15686, 50813}, {15687, 50822}, {15689, 28186}, {15693, 34773}, {15700, 50825}, {15701, 18526}, {15721, 31423}, {15723, 37624}, {16062, 19819}, {16173, 38104}, {16192, 50815}, {16475, 38089}, {16483, 37687}, {16668, 17303}, {16711, 30966}, {17079, 31994}, {17095, 25718}, {17143, 18146}, {17166, 45332}, {17257, 28542}, {17264, 41848}, {17270, 21296}, {17280, 50297}, {17320, 42696}, {17321, 50088}, {17342, 18230}, {17346, 49725}, {17349, 50300}, {17377, 32089}, {17378, 32025}, {17379, 50283}, {17487, 24715}, {17494, 50764}, {17504, 28224}, {17576, 31446}, {17579, 34606}, {17757, 33108}, {17781, 50736}, {18228, 31140}, {18481, 19708}, {19065, 32787}, {19066, 32788}, {19328, 37580}, {19709, 22791}, {19732, 48862}, {19742, 48867}, {19822, 48816}, {20080, 50952}, {21198, 53361}, {21222, 45328}, {21290, 48799}, {21302, 48065}, {21343, 45340}, {21454, 51782}, {21735, 31447}, {21816, 24075}, {21969, 23841}, {22329, 50247}, {22793, 50799}, {22851, 51487}, {22896, 51486}, {23046, 28212}, {24280, 50118}, {24344, 30564}, {24391, 37436}, {25280, 34284}, {26062, 34610}, {26065, 33074}, {26132, 28595}, {26444, 45420}, {26445, 45421}, {26685, 33076}, {26985, 50760}, {27268, 49459}, {27549, 50093}, {28443, 32141}, {28538, 38047}, {28604, 50074}, {28653, 50132}, {30745, 47492}, {31151, 49693}, {31156, 34607}, {31332, 50121}, {31359, 35652}, {31458, 37291}, {32003, 52422}, {33118, 51591}, {33166, 48839}, {33559, 36977}, {34200, 50819}, {34573, 49690}, {34605, 37545}, {34609, 34730}, {34720, 49736}, {35641, 42602}, {35642, 42603}, {35762, 43255}, {35763, 43254}, {36590, 52746}, {37650, 50130}, {37681, 50294}, {37701, 38105}, {37710, 51113}, {37911, 47493}, {38024, 38204}, {38028, 51515}, {38052, 38210}, {38102, 53055}, {38121, 38175}, {38126, 38149}, {38315, 48310}, {38514, 50145}, {39156, 50902}, {39800, 49730}, {40341, 51124}, {41915, 48804}, {42044, 50083}, {42049, 50321}, {44553, 50333}, {44566, 47695}, {44567, 47729}, {45320, 48304}, {45323, 48298}, {45344, 49274}, {47355, 51006}, {48628, 50080}, {48639, 49714}, {49474, 50777}, {49536, 50787}, {49679, 51126}, {49717, 50155}, {49723, 50172}, {49724, 50171}, {49729, 50165}, {49731, 49746}, {49744, 50277}, {49749, 50279}, {50041, 50058}, {50043, 50047}, {50045, 50048}, {50046, 50052}, {50133, 50299}, {50158, 50184}, {50160, 50271}, {50163, 50257}, {50226, 50256}, {50229, 50272}, {50786, 51196}, {50814, 51118}, {50831, 51700}, {50853, 51484}, {50856, 51485}, {51005, 51169}, {51127, 51145}, {51128, 51154}, {51168, 51170}

X(53620) = midpoint of X(i) in X(j) for these {i,j}: {10, 38098}, {3679, 19875}, {38081, 38112}, {38097, 38200}, {4669, 19883}, {5657, 38074}, {5686, 38092}, {5790, 38066}, {8, 38314}
X(53620) = reflection of X(i) in X(j) for these {i,j}: {1, 19883}, {10246, 11539}, {10247, 38022}, {10283, 47599}, {1699, 38076}, {11038, 38093}, {16173, 38104}, {16475, 38089}, {19875, 10}, {19883, 3828}, {2, 19875}, {3241, 38314}, {3524, 26446}, {3576, 38068}, {3653, 11231}, {3679, 38098}, {3839, 5587}, {37701, 38105}, {38021, 10175}, {38024, 38204}, {38066, 38112}, {38074, 5790}, {38092, 38200}, {38098, 4745}, {38314, 2}, {38315, 48310}, {5055, 38042}, {5603, 5055}, {5657, 38066}, {5686, 38097}, {5731, 3524}, {5790, 38081}, {5886, 38083}, {53055, 38102}, {7426, 47562}, {7967, 3653}, {8236, 38025}, {9812, 3839}
X(53620) = anticomplement of X(25055)
X(53620) = trilinear pole of line {28169,47777}
X(53620) = X(i)-isoconjugate-of-X(j) for these {i, j}:, {513, 28170}
X(53620) = X(i)-Dao conjugate of X(j) for these {i, j}:, {39026, 28170}
X(53620) = X(i)-Zayin conjugate of X(j) for these {i, j}: {47777, 649}
X(53620) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(9333)}}, {{A, B, C, X(2), X(40029)}}, {{A, B, C, X(4), X(3624)}}, {{A, B, C, X(7), X(551)}}, {{A, B, C, X(75), X(3241)}}, {{A, B, C, X(80), X(19875)}}, {{A, B, C, X(519), X(28169)}}, {{A, B, C, X(899), X(47777)}}, {{A, B, C, X(903), X(38314)}}, {{A, B, C, X(996), X(4669)}}, {{A, B, C, X(1000), X(51093)}}, {{A, B, C, X(1219), X(20050)}}, {{A, B, C, X(1220), X(46933)}}, {{A, B, C, X(1222), X(20053)}}, {{A, B, C, X(3616), X(39704)}}, {{A, B, C, X(3622), X(31359)}}, {{A, B, C, X(3633), X(39708)}}, {{A, B, C, X(3634), X(7319)}}, {{A, B, C, X(3679), X(5936)}}, {{A, B, C, X(5556), X(46934)}}, {{A, B, C, X(6745), X(36590)}}, {{A, B, C, X(17012), X(26745)}}, {{A, B, C, X(19878), X(43531)}}, {{A, B, C, X(20014), X(46872)}}, {{A, B, C, X(24857), X(39709)}}, {{A, B, C, X(27760), X(32631)}}, {{A, B, C, X(29601), X(30701)}}, {{A, B, C, X(29602), X(34892)}}, {{A, B, C, X(33696), X(34595)}} and {{A, B, C, X(42285), X(51103)}}
X(53620) = barycentric product X(i)*X(j) for these (i, j):, {190, 28169}, {16676, 75}, {18421, 312}, {47777, 668}
X(53620) = barycentric quotient X(i)/X(j) for these (i, j):, {101, 28170}, {16676, 1}, {18421, 57}, {28169, 514}, {47777, 513}
X(53620) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 10, 46933}, {1, 1698, 19878}, {1, 21267, 4678}, {1, 3679, 4669}, {1, 3828, 2}, {1, 4669, 31145}, {1, 4678, 8}, {1, 46933, 19877}, {1, 4701, 20014}, {1, 8, 20053}, {2, 17310, 5308}, {2, 20049, 3622}, {2, 31137, 26103}, {2, 40891, 26626}, {2, 4745, 51068}, {2, 51068, 51072}, {2, 51092, 51110}, {2, 519, 38314}, {4, 3654, 34632}, {10, 21267, 1}, {10, 3626, 1698}, {10, 38098, 519}, {10, 4668, 46932}, {10, 4669, 3828}, {10, 4745, 3679}, {10, 49772, 29576}, {10, 51067, 551}, {10, 51068, 3241}, {10, 51070, 19876}, {10, 519, 19875}, {10, 551, 51069}, {30, 38066, 5657}, {30, 38081, 5790}, {30, 38112, 38066}, {30, 5790, 38074}, {40, 50796, 3543}, {145, 1698, 5550}, {355, 376, 50864}, {355, 50821, 376}, {381, 50810, 962}, {381, 5690, 50810}, {519, 3828, 19883}, {519, 4745, 38098}, {527, 38097, 5686}, {527, 38200, 38092}, {547, 50823, 1482}, {549, 50798, 944}, {597, 50783, 51192}, {984, 50096, 4740}, {1125, 34641, 51093}, {1125, 3621, 20057}, {1125, 4668, 3621}, {1125, 51070, 34641}, {1698, 3626, 145}, {1698, 3679, 4677}, {3036, 6174, 50890}, {3090, 34631, 51709}, {3244, 46934, 52907}, {3244, 4816, 3912}, {3524, 28204, 5731}, {3576, 38068, 15708}, {3578, 50169, 50234}, {3622, 20049, 51071}, {3622, 30833, 20050}, {3623, 46931, 3624}, {3624, 33091, 50581}, {3624, 34747, 51103}, {3625, 51103, 34747}, {3626, 3828, 51091}, {3632, 51071, 20049}, {3635, 4816, 20054}, {3653, 11231, 15709}, {3656, 9956, 5071}, {3679, 4668, 51070}, {3679, 4745, 3617}, {3679, 51066, 10}, {3679, 51108, 51615}, {3679, 51110, 4746}, {3698, 4662, 3868}, {3751, 50781, 11160}, {3828, 4701, 51108}, {3829, 44847, 4193}, {3829, 9711, 44847}, {3839, 28194, 9812}, {3871, 16861, 4428}, {3983, 5836, 3876}, {4688, 50075, 24349}, {4732, 50094, 50086}, {4745, 51069, 51067}, {4746, 19862, 3633}, {4967, 5232, 31995}, {5071, 12245, 3656}, {5587, 28194, 3839}, {5691, 50808, 15683}, {5818, 50810, 381}, {6173, 24393, 50835}, {6174, 50890, 6224}, {6684, 50811, 15692}, {7308, 51781, 3895}, {7967, 15709, 3653}, {7982, 31399, 5056}, {9710, 21031, 2476}, {10056, 15933, 10578}, {10056, 18391, 15933}, {10588, 41687, 4323}, {12645, 15694, 50824}, {13174, 50884, 8596}, {15702, 50818, 1385}, {15703, 50805, 5901}, {19876, 51093, 1125}, {19877, 20053, 3616}, {19877, 46933, 9780}, {19878, 21267, 3626}, {19878, 22837, 39595}, {19925, 20070, 10248}, {20014, 46933, 51073}, {20014, 51615, 4701}, {20020, 29629, 17284}, {20052, 46934, 3244}, {21267, 22166, 4668}, {22166, 34641, 51106}, {26446, 28204, 3524}, {30116, 31855, 3240}, {31423, 50828, 15721}, {34606, 49732, 17579}, {34631, 51709, 5734}, {34747, 51096, 7292}, {34747, 51103, 3623}, {37714, 43174, 3146}, {37743, 46933, 3634}, {38081, 38112, 30}, {38097, 38200, 527}, {40333, 50835, 6173}, {45310, 50842, 1320}, {48661, 50800, 14893}, {50041, 50058, 50071}, {50046, 50052, 50061}, {50048, 50051, 50055}, {50086, 50094, 192}, {50163, 50275, 50257}

   Touchpoints on the circumcircle: X(53621)-X(53638) and X(53682)-X(53708)  

Contributed by Clark Kimberling (definitions and presentation) and Peter Moses (formulas, data, and properties), May 6, 2023.

A point P, as a function of (a,b,c), is here defined to be an inner point if P lies inside the circumcircle, Γ, for every triangle ABC (that is, for every (a,b,c) satisfying b+c>a and c+a>b and a+b>c); and an outer point otherwise. If X is an outer triangle center (so that its Γ-invers is an inner triangle center), then X lies on two lines that are tangent to Γ, so that there are two touchpoints, which in some cases are triangle centers and in some cases are a bicentric pair. The appearance of X(i) → (X(j),X(k) in the following list means that X(i) is an outer triangle center and X(j) and X(k) are the touchpoints

X(190) → X(100, 932)
X(351) → X(110, 111)
X(659) → X(100, 105)
X(670) → X(99, 3222)
X(676) → X(105, 108)
X(684) → X(110, 1297)
X(692) → X(101, 109)
X(874) → X(99, 8709)
X(875) → X(741, 813)
X(878) → X(98, 2715)
X(880) → X(99, 53621)
X(881) → X(733, 805)
X(884) → X(105, 919)
X(887) → X(99, 729)
X(890) → X(100, 739)
X(1461) → X(109, 53622)
X(1576) → X(110, 112)
X(1624) → X(107, 110)
X(1632) → X(99, 107)
X(1633) → X(100, 934)
X(1634) → X(99, 110)
X(1960) → X(101, 106)
X(1979) → (bicentric pair)
X(2284) → X(101, 813)
X(2421) → X(110, 805)
X(2426) → X(109, 919)
X(2492) → X(111, 112)
X(2976) → X(100, 53623)
X(2977) → X(100, 15344)
X(3570) → X(100, 53624)
X(3699) → X(100, 53625)
X(3807) → X(100, 53626)
X(3939) → X(101, 1293)
X(3952) → X(100, 53627)
X(4427) → X(100, 34594)
X(4436) → X(99, 100)
X(4491) → X(100, 106)
X(4556) → X(110, 53628)
X(4557) → X(100, 101)
X(4558) → X(110, 3565)
X(4571) → X(100, 53629)
X(4574) → X(101, 29014)
X(4578) → X(100, 53630)
X(4623) → X(99, 53631
X(4626) → X(934, 53632
X(4636) → X(110, 53633
X(4638) → X(901, 53634
X(4756) → X(100, 53635
X(4767) → X(100, 53636
X(4781) → X(100, 53637)
X(5381) → X(898, 53638)
X(5467) → X(110, 691)
X(5502) → X(110, 1304)
X(5989) → (bicentric pair)
X(6130) → X(98, 107)
X(6131) → X(99, 2374)
X(6132) → X(110, 3563)
X(6139) → X(109, 2291)
X(7418) → X(98, 842)

In the analogous context for the Steiner circumellipse, instead of the circumcircle, see the preamble just before X(53639).

X(53621) = CEVAPOINT OF X(385) AND X(669)

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

X(53621) lies on the circumcircle and these lines: {99, 9428}, {111, 39939}, {385, 733}, {669, 689}, {703, 3231}, {729, 51326}, {741, 51934}, {827, 46294}, {2698, 51249}

X(53621) = X(i)-isoconjugate of X(j) for these (i,j): {661, 51983}, {798, 40858}
X(53621) = X(i)-Dao conjugate of X(j) for these (i,j): {31998, 40858}, {36830, 51983}
X(53621) = cevapoint of X(385) and X(669)
X(53621) = trilinear pole of line {6, 19585}
X(53621) = barycentric product X(i)*X(j) for these {i,j}: {99, 39939}, {670, 51326}, {799, 51934}, {43187, 51249}
X(53621) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 40858}, {110, 51983}, {2966, 8870}, {17941, 38382}, {39939, 523}, {51249, 3569}, {51326, 512}, {51934, 661}

---

X(53622) = CEVAPOINT OF X(56) AND X(663)

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

X(53622) lies on the circumcircle and these lines: {1, 972}, {56, 103}, {57, 15731}, {102, 999}, {104, 3062}, {105, 1420}, {663, 6614}, {675, 36620}, {840, 5193}, {1292, 23890}, {1293, 2283}, {1295, 3576}, {1311, 10405}, {1319, 2717}, {1415, 26716}, {1429, 39421}, {1477, 41426}, {1617, 2291}, {1813, 43344}, {2078, 43080}, {2716, 5126}, {3660, 53181}, {6014, 23981}, {28148, 36059}, {28471, 37583}, {29056, 37609}, {32674, 40116}, {36118, 52596}

X(53622) = X(i)-isoconjugate of X(j) for these (i,j): {9, 7658}, {144, 650}, {165, 522}, {651, 13609}, {657, 31627}, {663, 16284}, {1419, 3239}, {3160, 3900}, {3207, 4391}, {3737, 21060}, {4105, 50561}, {4130, 9533}, {4163, 17106}, {4560, 21872}, {8641, 50560}, {22117, 44426}
X(53622) = X(i)-Dao conjugate of X(j) for these (i,j): {478, 7658}, {38991, 13609}
X(53622) = cevapoint of X(56) and X(663)
X(53622) = trilinear pole of line {6, 1200}
X(53622) = barycentric product X(i)*X(j) for these {i,j}: {101, 36620}, {109, 10405}, {651, 3062}, {664, 11051}, {934, 19605}, {1415, 44186}, {13138, 42872}
X(53622) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 7658}, {109, 144}, {651, 16284}, {658, 50560}, {663, 13609}, {934, 31627}, {1415, 165}, {1461, 3160}, {3062, 4391}, {4559, 21060}, {4617, 50561}, {6614, 9533}, {10405, 35519}, {11051, 522}, {19605, 4397}, {32660, 22117}, {36620, 3261}, {42872, 17896}, {52610, 50563}

---

X(53623) = CEVAPOINT OF X(56) AND X(1279)

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

X(53623) lies on the circumcircle and these lines: {55, 30236}, {56, 1292}, {57, 1293}, {100, 1617}, {101, 1420}, {901, 3660}, {934, 41426}, {999, 30237}, {1308, 5193}, {1319, 2742}, {2078, 2743}, {2975, 44301}, {6078, 7677}, {13462, 28291}

X(53623) = cevapoint of X(56) and X(1279)
X(53623) = trilinear pole of line {6, 51656}

---

X(53624) = CEVAPOINT OF X(238) AND X(667)

Barycentrics    a*(a - b)*(a - 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)*(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(53624) lies on the circumcircle and these lines: {21, 12031}, {98, 43685}, {99, 667}, {100, 669}, {101, 1924}, {103, 8934}, {104, 15981}, {105, 385}, {106, 2665}, {110, 1980}, {238, 741}, {689, 18108}, {729, 3230}, {739, 3231}, {1016, 8708}, {1083, 2711}, {1621, 25819}, {2375, 5291}, {2382, 16484}, {2721, 5108}, {4455, 4589}, {5970, 37675}, {8693, 52923}, {8937, 28482}

X(53624) = isogonal conjugate of the isotomic conjugate of X(53216)
X(53624) = X(i)-isoconjugate of X(j) for these (i,j): {292, 27854}, {512, 2669}, {513, 2664}, {514, 21788}, {649, 17759}, {656, 15148}, {659, 40796}, {661, 2106}, {667, 52049}, {669, 41535}, {798, 40874}, {875, 39028}, {1019, 21897}, {3572, 39916}, {4444, 51331}, {4589, 38978}, {7649, 20796}, {30665, 40772}
X(53624) = X(i)-Dao conjugate of X(j) for these (i,j): {5375, 17759}, {6631, 52049}, {19557, 27854}, {31998, 40874}, {36830, 2106}, {39026, 2664}, {39054, 2669}, {40596, 15148}
X(53624) = cevapoint of X(i) and X(j) for these (i,j): {1, 4455}, {238, 667}
X(53624) = trilinear pole of line {6, 1045}
X(53624) = barycentric product X(i)*X(j) for these {i,j}: {6, 53216}, {100, 39925}, {110, 43685}, {190, 2665}, {668, 51333}, {799, 2107}, {4562, 40769}, {37207, 40798}
X(53624) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 40874}, {100, 17759}, {101, 2664}, {110, 2106}, {112, 15148}, {190, 52049}, {238, 27854}, {662, 2669}, {692, 21788}, {799, 41535}, {813, 40796}, {906, 20796}, {2107, 661}, {2665, 514}, {3570, 39028}, {3573, 39916}, {4557, 21897}, {8937, 21196}, {30664, 40742}, {39925, 693}, {40769, 812}, {40798, 4486}, {43685, 850}, {51333, 513}, {53216, 76}

---

X(53625) = CEVAPOINT OF X(9) AND X(667)

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

X(53625) lies on the circumcircle and these lines: {99, 36863}, {100, 23833}, {104, 13732}, {105, 39694}, {106, 979}, {109, 23831}, {110, 52923}, {190, 43350}, {644, 6010}, {645, 8690}, {727, 8616}, {739, 38869}, {741, 41531}, {759, 8669}, {932, 3952}, {1621, 28583}, {2975, 8686}, {4756, 34594}, {9082, 26264}

X(53625) = Collings transform of X(3976)
X(53625) = X(i)-isoconjugate of X(j) for these (i,j): {190, 16614}, {513, 978}, {514, 21769}, {649, 3210}, {1019, 21857}, {3169, 3669}, {7649, 20805}, {8643, 27835}, {19582, 43924}
X(53625) = X(i)-Dao conjugate of X(j) for these (i,j): {5375, 3210}, {39026, 978}
X(53625) = cevapoint of X(i) and X(j) for these (i,j): {9, 667}, {513, 3976}
X(53625) = trilinear pole of line {6, 979}
X(53625) = barycentric product X(i)*X(j) for these {i,j}: {100, 39694}, {190, 979}, {27834, 39701}
X(53625) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 3210}, {101, 978}, {644, 19582}, {667, 16614}, {692, 21769}, {906, 20805}, {979, 514}, {3939, 3169}, {4557, 21857}, {27834, 27835}, {39694, 693}, {39701, 4462}

---

X(53626) = CEVAPOINT OF X(667) AND X(984)

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

X(53626) lies on the circumcircle and these lines: {28895, 52923}

X(53626) = X(7649)-isoconjugate of X(23194)
X(53626) = cevapoint of X(667) and X(984)
X(53626) = barycentric quotient X(906)/X(23194)

---

X(53627) = CEVAPOINT OF X(37) AND X(667)

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

X(5362) lies on the circumcircle and these lines: {100, 23866}, {105, 35058}, {106, 2975}, {190, 34594}, {644, 29014}, {675, 40010}, {727, 1621}, {739, 38871}, {759, 42471}, {932, 4756}, {2703, 9266}, {3573, 6577}, {4057, 8050}, {8652, 52923}

X(53627) = Collings transform of X(3953)
X(53627) = X(i)-isoconjugate of X(j) for these (i,j): {58, 31946}, {86, 50493}, {513, 3216}, {514, 16685}, {593, 21720}, {649, 17147}, {667, 18133}, {1019, 21858}, {1919, 40034}, {3159, 3733}, {7649, 22458}
X(53627) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 31946}, {5375, 17147}, {6631, 18133}, {9296, 40034}, {39026, 3216}, {40600, 50493}
X(53627) = cevapoint of X(i) and X(j) for these (i,j): {1, 4057}, {9, 48387}, {37, 667}, {513, 3953}
X(53627) = trilinear pole of line {6, 3293}
X(53627) = barycentric product X(i)*X(j) for these {i,j}: {100, 35058}, {101, 40010}, {190, 39748}, {662, 42471}, {668, 39964}
X(53627) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 31946}, {100, 17147}, {101, 3216}, {190, 18133}, {213, 50493}, {668, 40034}, {692, 16685}, {756, 21720}, {906, 22458}, {1018, 3159}, {3952, 40603}, {4557, 21858}, {15409, 7254}, {35058, 693}, {39748, 514}, {39964, 513}, {40010, 3261}, {42471, 1577}

---

X(53628) = CEVAPOINT OF X(58) AND X(512)

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

X(53628) lies on the circumcircle and these lines: {21, 15168}, {58, 28482}, {98, 6625}, {99, 21295}, {100, 17943}, {105, 229}, {110, 23390}, {111, 2248}, {163, 2702}, {593, 4128}, {675, 40164}, {741, 18757}, {759, 5429}, {4184, 15377}, {4558, 43359}, {4565, 29055}, {5546, 28841}, {35105, 38858}

X(53628) = Collings transform of X(i) for these i: {58, 34119}
X(53628) = X(i)-isoconjugate of X(j) for these (i,j): {37, 21196}, {42, 50451}, {256, 24381}, {512, 17762}, {513, 21085}, {514, 21879}, {523, 846}, {649, 27569}, {650, 27691}, {656, 4213}, {661, 1654}, {662, 6627}, {798, 51857}, {1577, 18755}, {4024, 38814}, {4041, 17084}, {4122, 40751}, {4155, 52207}, {4705, 6626}, {21709, 52935}, {22139, 24006}, {30591, 38836}
X(53628) = X(i)-Dao conjugate of X(j) for these (i,j): {1084, 6627}, {5375, 27569}, {31998, 51857}, {36830, 1654}, {39026, 21085}, {39054, 17762}, {40589, 21196}, {40592, 50451}, {40596, 4213}
X(53628) = cevapoint of X(i) and X(j) for these (i,j): {58, 512}, {523, 34119}
X(53628) = trilinear pole of line {6, 2248}
X(53628) = barycentric product X(i)*X(j) for these {i,j}: {99, 2248}, {101, 40164}, {110, 6625}, {163, 51865}, {662, 13610}, {799, 18757}, {52208, 52935}
X(53628) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 21196}, {81, 50451}, {99, 51857}, {100, 27569}, {101, 21085}, {109, 27691}, {110, 1654}, {112, 4213}, {163, 846}, {172, 24381}, {512, 6627}, {662, 17762}, {692, 21879}, {1576, 18755}, {2248, 523}, {4079, 21709}, {4556, 6626}, {4565, 17084}, {6625, 850}, {13610, 1577}, {15377, 4064}, {17940, 39921}, {18757, 661}, {32661, 22139}, {40164, 3261}, {51865, 20948}, {52208, 4036}

---

X(53629) = CEVAPOINT OF X(219) AND X(667)

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

X(53629) lies on the circumcircle and these lines: {104, 42469}, {105, 37652}, {106, 39946}, {644, 28477}, {739, 38875}, {915, 12245}, {3699, 9104}, {8686, 15375}

X(53629) = X(i)-isoconjugate of X(j) for these (i,j): {513, 1722}, {649, 30699}, {650, 28039}, {663, 31598}, {2899, 43924}, {6591, 8897}, {7649, 42461}
X(53629) = X(i)-Dao conjugate of X(j) for these (i,j): {5375, 30699}, {39026, 1722}
X(53629) = cevapoint of X(i) and X(j) for these (i,j): {9, 50501}, {219, 667}
X(53629) = trilinear pole of line {6, 37552}
X(53629) = barycentric product X(i)*X(j) for these {i,j}: {100, 39696}, {190, 39946}, {646, 15375}, {6335, 42469}
X(53629) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 30699}, {101, 1722}, {109, 28039}, {644, 2899}, {651, 31598}, {906, 42461}, {1331, 8897}, {15375, 3669}, {39696, 693}, {39946, 514}, {42469, 905}

---

X(53630) = CEVAPOINT OF X(220) AND X(667)

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

X(53630) lies on the circumcircle and these lines: {100, 30720}, {105, 6553}, {106, 3973}, {644, 1293}, {651, 6571}, {739, 38876}, {1023, 8699}, {1477, 2137}, {2975, 44301}, {8051, 15728}, {8686, 11194}

X(53630) = X(i)-isoconjugate of X(j) for these (i,j): {190, 17071}, {513, 23511}, {514, 1616}, {649, 4452}, {667, 33780}, {1019, 21896}, {2136, 3669}, {4394, 47636}, {7649, 23089}, {8055, 43924}, {8643, 27828}, {24151, 51656}
X(53630) = X(i)-Dao conjugate of X(j) for these (i,j): {5375, 4452}, {6631, 33780}, {39026, 23511}
X(53630) = cevapoint of X(i) and X(j) for these (i,j): {9, 4394}, {220, 667}
X(53630) = trilinear pole of line {6, 3158}
X(53630) = barycentric product X(i)*X(j) for these {i,j}: {100, 6553}, {644, 8051}, {2137, 3699}, {24150, 27834}, {31343, 44301}
X(53630) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 4452}, {101, 23511}, {190, 33780}, {644, 8055}, {667, 17071}, {692, 1616}, {906, 23089}, {1293, 47636}, {2137, 3676}, {3939, 2136}, {4557, 21896}, {4578, 6552}, {6553, 693}, {8051, 24002}, {24150, 4462}, {27834, 27828}

---

X(53631) = CEVAPOINT OF X(81) AND X(669)

Barycentrics    (a - b)*(a + b)*(a - c)*(a + 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)*(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(53631) lies on the circumcircle and these lines: {98, 43684}, {729, 33770}, {741, 40737}, {759, 18298}, {4573, 29055}, {12031, 39926}

X(53631) = isogonal conjugate of X(9402)
X(53631) = isogonal conjugate of the anticomplement of X(9402)
X(53631) = isogonal conjugate of the complement of X(9402)
X(53631) = Collings transform of X(i) for these i: {17669, 39057}
X(53631) = X(i)-isoconjugate of X(j) for these (i,j): {1, 9402}, {512, 1045}, {523, 18756}, {649, 21883}, {661, 21779}, {669, 51863}, {798, 1655}, {4705, 51330}, {34021, 53581}, {39915, 50487}
X(53631) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 9402}, {5375, 21883}, {31998, 1655}, {36830, 21779}, {39054, 1045}
X(53631) = cevapoint of X(i) and X(j) for these (i,j): {81, 669}, {86, 4367}, {523, 17669}
X(53631) = trilinear pole of line {6, 2669}
X(53631) = barycentric product X(i)*X(j) for these {i,j}: {110, 43684}, {662, 18298}, {670, 40770}, {799, 40737}
X(53631) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 9402}, {99, 1655}, {100, 21883}, {110, 21779}, {163, 18756}, {662, 1045}, {799, 51863}, {4556, 51330}, {4558, 23079}, {4610, 39915}, {4623, 34021}, {18298, 1577}, {40737, 661}, {40770, 512}, {43684, 850}

---

X(53632) = CEVAPOINT OF X(57) AND X(8641)

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

X(53632) lies on the circumcircle and these lines: {103, 4334}, {104, 43750}

X(53632) = X(i)-isoconjugate of X(j) for these (i,j): {220, 21195}, {657, 3177}, {1021, 21856}, {1742, 3900}, {3239, 20995}, {4105, 31526}, {4130, 34497}, {8641, 20935}, {21084, 21789}, {51846, 52614}
X(53632) = cevapoint of X(57) and X(8641)
X(53632) = trilinear pole of line {6, 34497}
X(53632) = barycentric product X(651)*X(43750)
X(53632) = barycentric quotient X(i)/X(j) for these {i,j}: {269, 21195}, {658, 20935}, {934, 3177}, {1020, 21084}, {1461, 1742}, {4617, 31526}, {4626, 40593}, {6614, 34497}, {43750, 4391}, {53321, 21856}

---

X(53633) = CEVAPOINT OF X(284) AND X(512)

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

X(53633) lies on the circumcircle and these lines: {98, 19642}, {106, 501}, {108, 17942}, {662, 38470}, {741, 19655}, {759, 1247}, {1283, 28482}, {2372, 36934}, {2701, 4575}, {5546, 6010}, {26702, 38480}, {28531, 33774}, {38453, 51624}

X(53633) = Collings transform of X(284)
X(53633) = X(i)-isoconjugate of X(j) for these (i,j): {513, 3178}, {523, 1046}, {656, 3144}, {661, 17778}, {1577, 2305}, {36927, 51663}
X(53633) = X(i)-Dao conjugate of X(j) for these (i,j): {36830, 17778}, {39026, 3178}, {40596, 3144}
X(53633) = cevapoint of X(284) and X(512)
X(53633) = trilinear pole of line {6, 2653}
X(53633) = barycentric product X(i)*X(j) for these {i,j}: {662, 1247}, {4556, 36934}
X(53633) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 3178}, {110, 17778}, {112, 3144}, {163, 1046}, {1247, 1577}, {1576, 2305}, {4636, 40605}, {36934, 52623}

---

X(53634) = CEVAPOINT OF X(106) AND X(1960)

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

X(53634) lies on the circumcircle and these lines: {100, 9271}, {1960, 39414}, {2384, 9259}, {2718, 9325}

X(53634) = X(i)-isoconjugate of X(j) for these (i,j): {519, 9269}, {900, 9324}, {1635, 17487}, {3251, 9460}, {3762, 21781}, {6544, 9326}, {9272, 35092}
X(53634) = cevapoint of X(106) and X(1960)
X(53634) = trilinear pole of line {6, 41461}
X(53634) = barycentric product X(i)*X(j) for these {i,j}: {88, 9271}, {3257, 9325}
X(53634) = barycentric quotient X(i)/X(j) for these {i,j}: {901, 17487}, {4638, 9460}, {9271, 4358}, {9325, 3762}, {9456, 9269}, {32665, 9324}, {32719, 21781}

---

X(53635) = CEVAPOINT OF X(667) AND X(16777)

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

X(53635) lies on the circumcircle and these lines: {644, 29271}, {9266, 29341}, {28200, 52923}

X(53635) = cevapoint of X(667) and X(16777)

---

X(53636) = CEVAPOINT OF X(45) AND X(667)

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

X(53636) lies on the circumcircle and these lines: {106, 748}, {644, 29179}, {28152, 52923}

X(53636) = X(7649)-isoconjugate of X(23170)
X(53636) = cevapoint of X(45) and X(667)
X(53636) = trilinear pole of line {6, 48696}
X(53636) = barycentric quotient X(906)/X(23170)

---

X(53637) = CEVAPOINT OF X(667) AND X(16666)

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

X(53637) lies on the circumcircle and these lines: {105, 39706}, {106, 1621}

X(53637) = Collings transform of X(4424)
X(53637) = X(649)-isoconjugate of X(31035)
X(53637) = X(5375)-Dao conjugate of X(31035)
X(53637) = cevapoint of X(i) and X(j) for these (i,j): {513, 4424}, {667, 16666}
X(53637) = trilinear pole of line {6, 49997}
X(53637) = barycentric product X(100)*X(39706)
X(53637) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 31035}, {39706, 693}

---

X(53638) = CEVAPOINT OF X(100) AND X(890)

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

X(53638) lies on the circumcircle and these lines: {100, 9299}, {2382, 9361}, {9081, 9295}, {9267, 39443}

X(53638) = X(i)-isoconjugate of X(j) for these (i,j): {890, 18149}, {891, 9359}, {899, 38238}, {1646, 9362}, {1979, 4728}, {3230, 21211}, {3768, 9263}, {9266, 19945}
X(53638) = cevapoint of X(100) and X(890)
X(53638) = trilinear pole of line {6, 9265}
X(53638) = barycentric product X(i)*X(j) for these {i,j}: {889, 9265}, {898, 9295}, {4607, 9361}, {5381, 9267}
X(53638) = barycentric quotient X(i)/X(j) for these {i,j}: {739, 38238}, {898, 9263}, {4607, 18149}, {5381, 9296}, {9265, 891}, {9267, 52626}, {9299, 1646}, {9361, 4728}, {32718, 1979}, {34075, 9359}, {37129, 21211}

   Touchpoints on the Steiner circumellipse: X(53639)-X(53659)  

Contributed by Clark Kimberling (definitions and presentation) and Peter Moses (formulas, data, and properties), May 7, 2023.

In the manner of the preamble just before X(53621), a point P, as a function of (a,b,c), is defined in this section to be an inner point if P lies inside the Steiner circumellipse, SCE, for every triangle ABC (that is, for every (a,b,c) satisfying b+c>a and c+a>b and a+b>c); and an outer point otherwise. If X is an outer triangle center (so that its SCE-inverse is an inner triangle center), then X lies on two lines that are tangent to SCE, so that there are two touchpoints, which in some cases are triangle centers and in some cases are a bicentric pair. The appearance of X(i) → (X(j),X(k) in the following list means that X(i) is an outer triangle center and X(j) and X(k) are the touchpoints

X(100) → X(190, 664)
X(107) → X(648, 53639)
X(110) → X(99, 648)
X(148) → (bicentric pair)
X(658) → X(664, 53640)
X(659) → X(190, 3226)
X(876) → X(4562, 18827)
X(879) → X(290, 2966)
X(882) → X(14970, 18829)
X(885) → X(666, 2481)
X(890) → X(190, 53641)
X(934) → X(664, 53642)
X(1633) → X(190, 53643)
X(1978) → X(668, 18830)
X(2283) → X(664, 53644)
X(2396) → X(99, 18829)
X(2398) → X(664, 666)
X(2976) → X(190, 53645)
X(2977) → X(190, 53646)
X(3268) → X(99, 1494)
X(3570) → X(190, 35148)
X(3573) → X(99, 666)
X(3699) → X(190, 53647)
X(3766) → X(668, 2481)
X(3799) → X(668, 32041)
X(3807) → X(190, 53648)
X(3952) → X(190, 668)
X(4226) → X(99, 2966)
X(4240) → X(648, 16077)
X(4427) → X(99, 190)
X(4436) → X(190, 53649)
X(4440) → (bicentric pair)
X(4453) → X(664, 903)
X(4491) → X(190, 53650)
X(4557) → X(190, 53651)
X(4563) → X(99, 35136)
X(4566) → X(664, 18026)
X(4571) → X(190, 53652)
X(4576) → X(99, 670)
X(4578) → X(190, 53653)
X(4609) → X(670, 53654)
X(4610) → X(99, 53655)
X(4618) → X(4555, 53656)
X(4630) → X(4577, 53657)
X(4756) → X(190, 53658)
X(4767) → X(190, 53659)
X(4781) → X(190, 4597)
X(4927) → X(903, 53776)

In the analogous context for the circumcircle, instead of the Steiner circumellipse, see the preamble just before X(53621).

X(53639) = CEVAPOINT OF X(4) AND X(525)

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

X(53639) lies on the Steiner circumellipse and these lines: {2, 39020}, {4, 14944}, {20, 47435}, {30, 16096}, {64, 290}, {76, 39268}, {99, 1301}, {112, 35571}, {253, 317}, {264, 15394}, {316, 47109}, {438, 46829}, {458, 40813}, {459, 671}, {525, 6529}, {648, 2404}, {664, 36797}, {811, 4569}, {877, 35136}, {1562, 6330}, {1975, 41085}, {2184, 35145}, {2966, 39062}, {3228, 41489}, {3343, 15466}, {5931, 34393}, {6526, 35142}, {6527, 14362}, {16077, 44181}, {20477, 46351}, {37669, 40839}, {41530, 46140}, {44216, 53201}

X(53639) = reflection of X(14944) in X(4)
X(53639) = isogonal conjugate of X(42658)
X(53639) = isotomic conjugate of X(8057)
X(53639) = anticomplement of X(39020)
X(53639) = polar conjugate of X(6587)
X(53639) = isotomic conjugate of the isogonal conjugate of X(1301)
X(53639) = polar conjugate of the isotomic conjugate of X(44326)
X(53639) = polar conjugate of the isogonal conjugate of X(46639)
X(53639) = X(i)-Ceva conjugate of X(j) for these (i,j): {44181, 253}, {44326, 648}
X(53639) = X(i)-isoconjugate of X(j) for these (i,j): {1, 42658}, {20, 810}, {31, 8057}, {48, 6587}, {122, 32676}, {154, 656}, {163, 1562}, {184, 17898}, {204, 520}, {228, 21172}, {255, 44705}, {603, 14308}, {610, 647}, {652, 30456}, {661, 15905}, {798, 37669}, {822, 1249}, {1409, 14331}, {1459, 3198}, {1895, 39201}, {1919, 42699}, {1946, 5930}, {1973, 20580}, {2159, 14345}, {2631, 15291}, {3049, 18750}, {3172, 24018}, {8804, 22383}, {14380, 52948}, {23224, 53011}, {24019, 47409}, {44695, 51640}
X(53639) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 8057}, {3, 42658}, {115, 1562}, {1249, 6587}, {3163, 14345}, {3343, 520}, {6337, 20580}, {6523, 44705}, {7952, 14308}, {9296, 42699}, {14092, 647}, {14390, 32320}, {15526, 122}, {31998, 37669}, {35071, 47409}, {36830, 15905}, {39052, 610}, {39053, 5930}, {39062, 20}, {40596, 154}, {40839, 523}
X(53639) = cevapoint of X(i) and X(j) for these (i,j): {2, 8057}, {4, 525}, {25, 2451}, {27, 7253}, {520, 46831}, {523, 26958}, {850, 44131}, {1301, 46639}, {3265, 15394}
X(53639) = trilinear pole of line {2, 253}
X(53639) = barycentric product X(i)*X(j) for these {i,j}: {4, 44326}, {64, 6331}, {76, 1301}, {99, 459}, {107, 34403}, {110, 52581}, {112, 41530}, {253, 648}, {264, 46639}, {525, 44181}, {653, 5931}, {670, 41489}, {811, 2184}, {823, 19611}, {1073, 6528}, {3267, 15384}, {4563, 6526}, {8798, 42405}, {13157, 18831}, {14638, 32230}, {15352, 15394}, {35571, 52283}, {46404, 52158}
X(53639) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 8057}, {4, 6587}, {6, 42658}, {27, 21172}, {29, 14331}, {30, 14345}, {64, 647}, {69, 20580}, {92, 17898}, {99, 37669}, {107, 1249}, {108, 30456}, {110, 15905}, {112, 154}, {162, 610}, {253, 525}, {281, 14308}, {393, 44705}, {459, 523}, {520, 47409}, {523, 1562}, {525, 122}, {648, 20}, {653, 5930}, {668, 42699}, {811, 18750}, {823, 1895}, {933, 33629}, {1073, 520}, {1301, 6}, {1304, 15291}, {1783, 3198}, {1897, 8804}, {2155, 810}, {2184, 656}, {2404, 1559}, {4558, 35602}, {5931, 6332}, {6331, 14615}, {6335, 52345}, {6526, 2501}, {6528, 15466}, {6529, 6525}, {7253, 40616}, {8057, 39020}, {8798, 17434}, {8809, 51664}, {11589, 1636}, {13157, 6368}, {14379, 32320}, {14642, 39201}, {15352, 14249}, {15384, 112}, {15394, 52613}, {15459, 10152}, {16096, 39473}, {16813, 38808}, {18020, 36841}, {19611, 24018}, {19614, 822}, {23582, 52913}, {24019, 204}, {32713, 3172}, {32714, 40933}, {33581, 3049}, {33584, 52588}, {34403, 3265}, {35360, 42459}, {35571, 42287}, {36079, 52373}, {36118, 36908}, {36797, 27382}, {36841, 53050}, {38956, 14401}, {40117, 41086}, {41489, 512}, {41530, 3267}, {41678, 2883}, {44181, 648}, {44326, 69}, {44692, 8611}, {46639, 3}, {52158, 652}, {52283, 14343}, {52581, 850}, {52913, 36413}, {52919, 44698}

---

X(53640) = CEVAPOINT OF X(7) AND X(522)

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

X(53640) lies on the Steiner circumellipse and these lines: {7, 4081}, {75, 46137}, {99, 53622}, {522, 4626}, {651, 32040}, {664, 23973}, {666, 10001}, {903, 36620}, {1121, 6604}, {2481, 3062}, {14727, 34085}, {18816, 44186}, {34393, 42697}

X(53640) = isotomic conjugate of the isogonal conjugate of X(53622)
X(53640) = X(i)-isoconjugate of X(j) for these (i,j): {41, 7658}, {144, 3063}, {165, 663}, {650, 3207}, {657, 1419}, {1415, 13609}, {3160, 8641}, {4105, 17106}, {7252, 21872}, {18344, 22117}, {21127, 33634}
X(53640) = X(i)-Dao conjugate of X(j) for these (i,j): {1146, 13609}, {3160, 7658}, {10001, 144}
X(53640) = cevapoint of X(i) and X(j) for these (i,j): {7, 522}, {57, 4449}
X(53640) = trilinear pole of line {2, 3160}
X(53640) = barycentric product X(i)*X(j) for these {i,j}: {76, 53622}, {190, 36620}, {651, 44186}, {664, 10405}, {3062, 4554}, {4569, 19605}, {4572, 11051}
X(53640) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 7658}, {109, 3207}, {522, 13609}, {651, 165}, {658, 3160}, {664, 144}, {934, 1419}, {1813, 22117}, {3062, 650}, {4551, 21872}, {4552, 21060}, {4554, 16284}, {4569, 31627}, {4617, 17106}, {4626, 9533}, {10405, 522}, {11051, 663}, {19605, 3900}, {36620, 514}, {36838, 50561}, {42872, 6129}, {44186, 4391}, {46406, 50560}, {53243, 33634}, {53622, 6}

---

X(53641) = CEVAPOINT OF X(192) AND X(899)

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

X(53641) lies on the Steiner circumellipse and these lines: {1, 18830}, {43, 668}, {99, 38832}, {190, 194}, {330, 3224}, {664, 1403}, {670, 33296}, {2162, 32039}, {3226, 23394}, {3229, 53195}, {3360, 21790}, {4562, 10027}

X(53641) = cevapoint of X(192) and X(899)
X(53641) = trilinear pole of line {2, 20979}

---

X(53642) = CEVAPOINT OF X(57) AND X(522)

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

X(53642) lies on the Steiner circumellipse and these lines: {7, 34393}, {57, 34404}, {84, 2481}, {85, 271}, {99, 8059}, {189, 1121}, {190, 2406}, {280, 6604}, {309, 18816}, {348, 41084}, {522, 6614}, {648, 1414}, {658, 18026}, {664, 13138}, {666, 36049}, {671, 8808}, {903, 1440}, {1413, 3226}, {1422, 3227}, {1903, 35144}, {3341, 40702}, {4025, 36118}, {6183, 40117}, {7013, 47436}, {10001, 53208}, {10030, 53228}, {18827, 52384}, {32939, 44189}, {35140, 52078}, {35141, 39130}, {35145, 52037}, {35164, 38468}

X(53642) = isotomic conjugate of X(8058)
X(53642) = isotomic conjugate of the isogonal conjugate of X(8059)
X(53642) = X(i)-isoconjugate of X(j) for these (i,j): {6, 14298}, {19, 10397}, {31, 8058}, {40, 663}, {41, 14837}, {55, 6129}, {108, 47432}, {198, 650}, {221, 3900}, {223, 657}, {227, 21789}, {329, 3063}, {347, 8641}, {513, 7074}, {521, 3195}, {522, 2187}, {649, 2324}, {652, 2331}, {667, 7080}, {692, 38357}, {798, 27398}, {1415, 5514}, {1459, 40971}, {1817, 3709}, {1946, 7952}, {2175, 17896}, {2199, 3239}, {2360, 4041}, {3318, 32652}, {3669, 7368}, {4130, 6611}, {7078, 18344}, {7252, 21871}, {8750, 53557}, {15501, 53549}
X(53642) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 8058}, {6, 10397}, {9, 14298}, {223, 6129}, {1086, 38357}, {1146, 5514}, {3160, 14837}, {3341, 3900}, {5375, 2324}, {6631, 7080}, {10001, 329}, {16596, 3318}, {26932, 53557}, {31998, 27398}, {38983, 47432}, {39026, 7074}, {39053, 7952}, {40593, 17896}, {40618, 16596}, {40626, 7358}, {52389, 14308}
X(53642) = cevapoint of X(i) and X(j) for these (i,j): {1, 4091}, {2, 8058}, {7, 4025}, {57, 522}, {271, 6332}, {514, 3086}, {905, 51490}, {3341, 8063}, {3900, 46830}, {13138, 37141}
X(53642) = trilinear pole of line {2, 77}
X(53642) = barycentric product X(i)*X(j) for these {i,j}: {7, 44327}, {75, 37141}, {76, 8059}, {84, 4554}, {85, 13138}, {99, 8808}, {109, 44190}, {189, 664}, {190, 1440}, {271, 13149}, {280, 658}, {282, 4569}, {309, 651}, {668, 1422}, {799, 52384}, {811, 52037}, {934, 34404}, {1413, 1978}, {1433, 46404}, {1436, 4572}, {1897, 34400}, {1903, 4625}, {2192, 46406}, {4573, 39130}, {4635, 53013}, {6063, 36049}, {6606, 13156}, {7182, 40117}, {7367, 52937}, {18026, 41081}, {20567, 32652}, {36118, 44189}, {44326, 52078}
X(53642) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 14298}, {2, 8058}, {3, 10397}, {7, 14837}, {57, 6129}, {84, 650}, {85, 17896}, {99, 27398}, {100, 2324}, {101, 7074}, {108, 2331}, {109, 198}, {189, 522}, {190, 7080}, {280, 3239}, {282, 3900}, {285, 1021}, {309, 4391}, {514, 38357}, {522, 5514}, {651, 40}, {652, 47432}, {653, 7952}, {658, 347}, {664, 329}, {905, 53557}, {934, 223}, {1020, 227}, {1413, 649}, {1414, 1817}, {1415, 2187}, {1422, 513}, {1433, 652}, {1436, 663}, {1440, 514}, {1461, 221}, {1783, 40971}, {1813, 7078}, {1903, 4041}, {2192, 657}, {2208, 3063}, {2357, 3709}, {2405, 1528}, {2406, 51375}, {3939, 7368}, {4025, 16596}, {4551, 21871}, {4552, 21075}, {4554, 322}, {4558, 1819}, {4565, 2360}, {4569, 40702}, {4573, 8822}, {4626, 14256}, {5932, 8063}, {6081, 15629}, {6332, 7358}, {6612, 43924}, {6614, 6611}, {7118, 8641}, {7129, 18344}, {7367, 4105}, {8059, 6}, {8063, 13612}, {8064, 7037}, {8808, 523}, {13138, 9}, {13149, 342}, {13156, 6362}, {14837, 3318}, {32652, 41}, {32674, 3195}, {32714, 208}, {34050, 6087}, {34400, 4025}, {34404, 4397}, {36049, 55}, {36118, 196}, {37136, 15501}, {37141, 1}, {39130, 3700}, {40117, 33}, {40836, 3064}, {41081, 521}, {41084, 14331}, {42549, 6615}, {44190, 35519}, {44327, 8}, {52037, 656}, {52078, 6587}, {52384, 661}, {52389, 8611}, {53013, 4171}

---

X(53643) = CEVAPOINT OF X(514) AND X(614)

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)*(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(53643) lies on the Steiner circumellipse and these lines: {190, 25266}, {653, 21178}, {664, 40576}, {671, 36907}, {903, 39732}, {1121, 41791}, {2481, 46740}, {3227, 40188}, {4025, 8750}, {4360, 18025}, {4389, 34393}, {15418, 18830}, {33296, 35145}

X(53643) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2509}, {213, 17498}, {513, 12329}, {649, 17742}, {652, 20613}, {663, 8270}, {667, 10327}, {1459, 23050}, {1919, 46738}, {3063, 28739}
X(53643) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 2509}, {5375, 17742}, {6626, 17498}, {6631, 10327}, {9296, 46738}, {10001, 28739}, {39026, 12329}
X(53643) = cevapoint of X(i) and X(j) for these (i,j): {1, 4025}, {514, 614}
X(53643) = trilinear pole of line {2, 169}
X(53643) = barycentric product X(i)*X(j) for these {i,j}: {99, 36907}, {100, 46740}, {190, 39732}, {664, 41791}, {668, 40188}
X(53643) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2509}, {86, 17498}, {100, 17742}, {101, 12329}, {108, 20613}, {190, 10327}, {651, 8270}, {664, 28739}, {668, 46738}, {1633, 15487}, {1783, 23050}, {3732, 11677}, {4558, 1801}, {36907, 523}, {39732, 514}, {40184, 8678}, {40188, 513}, {41791, 522}, {46740, 693}

---

X(53644) = CEVAPOINT OF X(522) AND X(672)

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

X(53644) lies on the Steiner circumellipse and these lines: {99, 4564}, {648, 7012}, {655, 14616}, {664, 17496}, {2481, 6996}, {4559, 4560}, {5662, 53209}

X(53644) = X(i)-isoconjugate of X(j) for these (i,j): {41, 20520}, {650, 20470}, {663, 20367}, {1024, 39046}, {3063, 20347}, {4250, 7117}, {7252, 20718}, {18344, 20744}
X(53644) = X(i)-Dao conjugate of X(j) for these (i,j): {3160, 20520}, {10001, 20347}
X(53644) = cevapoint of X(i) and X(j) for these (i,j): {522, 672}, {1025, 4552}, {2171, 2254}
X(53644) = trilinear pole of line {2, 4551}
X(53644) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 20520}, {109, 20470}, {651, 20367}, {664, 20347}, {1813, 20744}, {2283, 39046}, {4551, 20718}, {4554, 20448}, {7012, 4250}

---

X(53645) = CEVAPOINT OF X(145) AND X(3008)

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

X(53645) lies on the Steiner circumellipse and these lines: {145, 27828}, {190, 4452}, {664, 6049}, {668, 17158}, {27818, 44301}

X(53645) = X(692)-isoconjugate of X(2505)
X(53645) = X(1086)-Dao conjugate of X(2505)
X(53645) = cevapoint of X(145) and X(3008)
X(53645) = trilinear pole of line {2, 31182}
X(53645) = barycentric quotient X(514)/X(2505)

---

X(53646) = CEVAPOINT OF X(2) AND X(28526)

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

X(53646) lies on the Steiner circumellipse and these lines: {86, 35136}, {99, 28527}, {190, 193}, {230, 35148}, {664, 1788}, {668, 18156}, {44369, 46143}

X(53646) = isotomic conjugate of X(28526)
X(53646) = isotomic conjugate of the isogonal conjugate of X(28527)
X(53646) = X(31)-isoconjugate of X(28526)
X(53646) = X(2)-Dao conjugate of X(28526)
X(53646) = cevapoint of X(2) and X(28526)
X(53646) = trilinear pole of line {2, 3798}
X(53646) = barycentric product X(76)*X(28527)
X(53646) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 28526}, {28527, 6}

---

X(53647) = CEVAPOINT OF X(8) AND X(514)

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

X(53647) lies on the Steiner circumellipse and these lines: {1, 27813}, {2, 40621}, {8, 1358}, {10, 27820}, {42, 27824}, {69, 903}, {75, 19604}, {76, 14261}, {99, 1293}, {145, 27828}, {183, 3226}, {190, 2415}, {200, 27829}, {239, 27830}, {322, 18816}, {325, 35153}, {385, 17967}, {514, 6558}, {523, 46143}, {524, 17951}, {658, 6613}, {664, 31343}, {666, 5382}, {668, 21580}, {671, 4052}, {1121, 4417}, {1339, 5088}, {1494, 44727}, {1909, 35159}, {2481, 3680}, {3227, 8056}, {3262, 35175}, {3570, 32040}, {3676, 6079}, {3699, 4927}, {3875, 47636}, {4360, 27835}, {4534, 36807}, {4555, 4561}, {4586, 34080}, {4595, 32041}, {6556, 18025}, {6648, 38828}, {9362, 27837}, {10029, 32850}, {14727, 51560}, {17277, 27819}, {17295, 27826}, {18743, 24151}, {18825, 38266}, {33677, 53219}, {35165, 39099}, {35168, 41141}, {35517, 53218}

X(53647) = isogonal conjugate of X(8643)
X(53647) = isotomic conjugate of X(3667)
X(53647) = anticomplement of X(40621)
X(53647) = isotomic conjugate of the isogonal conjugate of X(1293)
X(53647) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {5382, 42020}, {27834, 34548}
X(53647) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8643}, {6, 4394}, {31, 3667}, {32, 4462}, {41, 30719}, {44, 2441}, {55, 51656}, {56, 4162}, {58, 4729}, {145, 667}, {163, 21950}, {512, 16948}, {513, 3052}, {604, 4521}, {649, 1743}, {661, 33628}, {663, 1420}, {692, 3756}, {798, 41629}, {810, 4248}, {904, 4504}, {1106, 4546}, {1110, 23764}, {1333, 14321}, {1408, 44729}, {1415, 4534}, {1911, 53580}, {1919, 18743}, {2206, 4404}, {2251, 2403}, {3063, 5435}, {3158, 43924}, {3248, 43290}, {3733, 4849}, {4557, 18211}, {4943, 16945}, {4949, 34819}, {6591, 20818}, {8027, 44724}, {9456, 14425}, {31182, 38266}, {34080, 40621}, {51641, 52352}
X(53647) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 4162}, {2, 3667}, {3, 8643}, {8, 4943}, {9, 4394}, {10, 4729}, {37, 14321}, {115, 21950}, {223, 51656}, {514, 23764}, {1086, 3756}, {1146, 4534}, {2968, 4953}, {3160, 30719}, {3161, 4521}, {3752, 14284}, {4370, 14425}, {5375, 1743}, {6376, 4462}, {6552, 4546}, {6631, 145}, {6651, 53580}, {9296, 18743}, {9460, 2403}, {10001, 5435}, {16593, 2976}, {17755, 4925}, {24151, 513}, {31998, 41629}, {36830, 33628}, {39026, 3052}, {39054, 16948}, {39062, 4248}, {40595, 2441}, {40603, 4404}, {40615, 40617}, {40624, 4939}
X(53647) = cevapoint of X(i) and X(j) for these (i,j): {1, 4498}, {2, 3667}, {8, 514}, {513, 16602}, {519, 21129}, {522, 30827}, {693, 20895}, {3676, 19604}, {27834, 31343}
X(53647) = trilinear pole of line {2, 2415}
X(53647) = barycentric product X(i)*X(j) for these {i,j}: {75, 27834}, {76, 1293}, {85, 31343}, {99, 4052}, {100, 40014}, {190, 4373}, {561, 34080}, {646, 19604}, {658, 6556}, {664, 6557}, {668, 8056}, {693, 5382}, {903, 2415}, {1978, 3445}, {3596, 38828}, {3680, 4554}, {3699, 27818}, {6386, 38266}, {10029, 36802}, {16078, 30720}, {17936, 34899}, {17951, 46143}, {27833, 34409}
X(53647) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4394}, {2, 3667}, {6, 8643}, {7, 30719}, {8, 4521}, {9, 4162}, {10, 14321}, {37, 4729}, {57, 51656}, {75, 4462}, {99, 41629}, {100, 1743}, {101, 3052}, {106, 2441}, {110, 33628}, {145, 31182}, {190, 145}, {239, 53580}, {321, 4404}, {346, 4546}, {514, 3756}, {519, 14425}, {522, 4534}, {523, 21950}, {644, 3158}, {645, 52352}, {646, 44720}, {648, 4248}, {651, 1420}, {662, 16948}, {664, 5435}, {668, 18743}, {894, 4504}, {903, 2403}, {1016, 43290}, {1018, 4849}, {1019, 18211}, {1086, 23764}, {1293, 6}, {1331, 20818}, {1332, 4855}, {1698, 4949}, {2321, 44729}, {2397, 51433}, {2398, 53579}, {2403, 15637}, {2415, 519}, {2429, 902}, {3008, 2976}, {3161, 4943}, {3239, 4953}, {3445, 649}, {3452, 14284}, {3617, 14350}, {3622, 14351}, {3667, 40621}, {3676, 40617}, {3680, 650}, {3699, 3161}, {3912, 4925}, {3952, 3950}, {3977, 39472}, {4033, 52353}, {4052, 523}, {4076, 30720}, {4373, 514}, {4391, 4939}, {4427, 4856}, {4552, 4848}, {4554, 39126}, {4555, 31227}, {4568, 4884}, {4578, 4936}, {4585, 4881}, {4756, 4898}, {5382, 100}, {6556, 3239}, {6557, 522}, {6558, 6555}, {6632, 44724}, {8056, 513}, {10029, 43042}, {10566, 18113}, {17261, 4964}, {17936, 37792}, {17951, 2789}, {18830, 27496}, {19604, 3669}, {21129, 5516}, {21272, 45204}, {21362, 45219}, {24004, 4487}, {25268, 12640}, {27813, 4106}, {27818, 3676}, {27819, 3309}, {27820, 4897}, {27823, 4170}, {27826, 21185}, {27827, 4905}, {27831, 4459}, {27833, 1836}, {27834, 1}, {27836, 4089}, {27837, 4014}, {30720, 15519}, {31343, 9}, {34080, 31}, {36042, 9456}, {38266, 667}, {38828, 56}, {40014, 693}, {40151, 43924}, {42720, 4899}, {51839, 48032}, {52609, 52354}
X(53647) = {X(239),X(27830)}-harmonic conjugate of X(51839)

---

X(53648) = CEVAPOINT OF X(514) AND X(3661)

Barycentrics    (a - b)*(a - c)*(a*b + 2*b^2 - a*c + b*c)*(a*b - a*c - b*c - 2*c^2) : :
X(53648) = X[3226] - 4 X[20532], X[3226] + 2 X[39354], 2 X[20532] + X[39354]

X(53648) lies on the Steiner circumellipse and these lines: {2, 3226}, {99, 4482}, {190, 23354}, {519, 35172}, {524, 35165}, {599, 903}, {671, 34475}, {1018, 32042}, {1121, 31141}, {1978, 31147}, {2481, 4479}, {3227, 3679}, {3228, 31144}, {3661, 43266}, {3699, 35148}, {3807, 48167}, {4033, 18830}, {4505, 28840}, {4597, 23891}, {4598, 45313}, {4664, 31170}, {7245, 18827}, {7788, 18025}, {7840, 35153}, {10713, 35168}, {17271, 43096}, {17297, 53219}, {17310, 18822}, {18825, 40735}

X(53648) = midpoint of X(2) and X(39354)
X(53648) = reflection of X(i) in X(j) for these {i,j}: {2, 20532}, {3226, 2}
X(53648) = isotomic conjugate of X(4785)
X(53648) = isotomic conjugate of the isogonal conjugate of X(43077)
X(53648) = X(i)-isoconjugate of X(j) for these (i,j): {6, 4782}, {31, 4785}, {513, 21793}, {649, 16468}, {661, 34476}, {667, 4393}, {810, 31912}, {1333, 4806}, {1919, 30963}, {1980, 10009}, {3733, 21904}, {6591, 23095}, {8640, 40720}, {9456, 45314}, {20979, 40753}
X(53648) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 4785}, {9, 4782}, {37, 4806}, {4370, 45314}, {5375, 16468}, {6631, 4393}, {9296, 30963}, {36830, 34476}, {39026, 21793}, {39062, 31912}
X(53648) = cevapoint of X(i) and X(j) for these (i,j): {2, 4785}, {514, 3661}
X(53648) = trilinear pole of line {2, 726}
X(53648) = barycentric product X(i)*X(j) for these {i,j}: {76, 43077}, {99, 34475}, {190, 27494}, {668, 52654}, {6386, 40735}, {18830, 40780}, {27808, 51449}
X(53648) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4782}, {2, 4785}, {10, 4806}, {100, 16468}, {101, 21793}, {110, 34476}, {190, 4393}, {519, 45314}, {648, 31912}, {668, 30963}, {932, 40753}, {1018, 21904}, {1331, 23095}, {1978, 10009}, {3799, 3795}, {3807, 27481}, {3952, 3993}, {4427, 4991}, {4598, 40720}, {17780, 4759}, {27494, 514}, {34475, 523}, {40735, 667}, {40780, 4083}, {43077, 6}, {51449, 3733}, {52654, 513}
X(53648) = {X(20532),X(39354)}-harmonic conjugate of X(3226)

---

X(53649) = CEVAPOINT OF X(514) AND X(3720)

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

X(53649) lies on the Steiner circumellipse and these lines: {86, 2481}, {99, 43076}, {190, 4576}, {662, 666}, {671, 17758}, {889, 36860}, {903, 39734}, {2350, 3228}, {2669, 35173}, {3227, 33296}, {4360, 13476}, {4557, 7192}, {4573, 6606}, {4577, 4610}, {4625, 46135}, {7199, 35338}, {14616, 40216}, {17731, 35152}, {18826, 34063}, {51563, 53227}

X(53649) = isotomic conjugate of X(4151)
X(53649) = isotomic conjugate of the isogonal conjugate of X(43076)
X(53649) = X(i)-isoconjugate of X(j) for these (i,j): {31, 4151}, {37, 21007}, {42, 4040}, {58, 21727}, {213, 17494}, {512, 1621}, {649, 3294}, {661, 4251}, {667, 4651}, {669, 17143}, {692, 2486}, {798, 17277}, {810, 14004}, {1018, 38346}, {1824, 22160}, {1918, 20954}, {1919, 4043}, {1924, 18152}, {3733, 40607}, {3996, 51641}, {4524, 38859}, {4551, 38365}, {4559, 38347}, {7252, 20616}, {9426, 40088}
X(53649) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 4151}, {10, 21727}, {1086, 2486}, {5375, 3294}, {6626, 17494}, {6631, 4651}, {9296, 4043}, {9428, 18152}, {31998, 17277}, {34021, 20954}, {36830, 4251}, {39054, 1621}, {39062, 14004}, {40589, 21007}, {40592, 4040}, {40620, 17761}
X(53649) = cevapoint of X(i) and X(j) for these (i,j): {1, 7192}, {2, 4151}, {514, 3720}, {523, 17245}, {1019, 17187}, {4025, 4303}, {4560, 17194}, {7199, 40004}
X(53649) = trilinear pole of line {2, 2350}
X(53649) = barycentric product X(i)*X(j) for these {i,j}: {76, 43076}, {99, 17758}, {100, 40004}, {190, 39734}, {662, 40216}, {668, 39950}, {670, 2350}, {799, 13476}
X(53649) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 4151}, {37, 21727}, {58, 21007}, {81, 4040}, {86, 17494}, {99, 17277}, {100, 3294}, {110, 4251}, {190, 4651}, {274, 20954}, {514, 2486}, {645, 3996}, {648, 14004}, {662, 1621}, {668, 4043}, {670, 18152}, {799, 17143}, {1018, 40607}, {1790, 22160}, {2350, 512}, {3733, 38346}, {3737, 38347}, {4551, 20616}, {4602, 40088}, {4616, 33765}, {4637, 38859}, {4639, 40094}, {7192, 17761}, {7199, 40619}, {7252, 38365}, {13476, 661}, {17197, 42454}, {17758, 523}, {39734, 514}, {39950, 513}, {40004, 693}, {40216, 1577}, {43076, 6}

---

X(53650) = CEVAPOINT OF X(1) AND X(17495)

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

X(53650) lies on the Steiner circumellipse and these lines: {190, 17147}, {668, 4360}, {1255, 25534}, {4597, 34063}

X(53650) = X(692)-isoconjugate of X(53565)
X(53650) = X(1086)-Dao conjugate of X(53565)
X(53650) = cevapoint of X(i) and X(j) for these (i,j): {1, 17495}, {3995, 31855}
X(53650) = trilinear pole of line {2, 4063}
X(53650) = barycentric quotient X(514)/X(53565)

---

X(53651) = CEVAPOINT OF X(42) AND X(514)

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

X(53651) lies on the Steiner circumellipse and these lines: {99, 6577}, {190, 46725}, {671, 40515}, {903, 8049}, {2481, 4360}, {3226, 34444}, {3227, 34063}, {3228, 40147}, {6631, 53195}, {18025, 32025}, {18827, 40504}, {40005, 43093}

X(53651) = isotomic conjugate of X(8714)
X(53651) = isotomic conjugate of the isogonal conjugate of X(6577)
X(53651) = X(i)-isoconjugate of X(j) for these (i,j): {31, 8714}, {56, 50518}, {81, 52592}, {513, 8053}, {649, 16552}, {667, 17135}, {692, 53564}, {798, 29767}, {1019, 40586}, {1919, 18137}, {3063, 17077}, {3733, 22271}, {3737, 52024}, {6591, 22126}, {17911, 22383}
X(53651) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 50518}, {2, 8714}, {1086, 53564}, {5375, 16552}, {6631, 17135}, {9296, 18137}, {10001, 17077}, {31998, 29767}, {39026, 8053}, {40586, 52592}
X(53651) = cevapoint of X(i) and X(j) for these (i,j): {1, 17494}, {2, 8714}, {10, 25259}, {42, 514}
X(53651) = trilinear pole of line {2, 2140}
X(53651) = barycentric product X(i)*X(j) for these {i,j}: {76, 6577}, {99, 40515}, {100, 39735}, {101, 40005}, {190, 8049}, {668, 39797}, {670, 40147}, {799, 40504}, {1978, 34444}
X(53651) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 8714}, {9, 50518}, {42, 52592}, {99, 29767}, {100, 16552}, {101, 8053}, {190, 17135}, {514, 53564}, {664, 17077}, {668, 18137}, {1018, 22271}, {1331, 22126}, {1897, 17911}, {3952, 21070}, {4557, 40586}, {4559, 52024}, {6577, 6}, {8049, 514}, {34444, 649}, {39735, 693}, {39797, 513}, {40005, 3261}, {40147, 512}, {40504, 661}, {40515, 523}

---

X(53652) = CEVAPOINT OF X(78) AND X(514)

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

X(53652) lies on the Steiner circumellipse and these lines: {322, 46133}, {903, 39695}, {3226, 34430}, {3227, 39947}

X(53652) = X(i)-isoconjugate of X(j) for these (i,j): {649, 1723}, {663, 34489}, {667, 12649}, {2900, 43924}, {3211, 6591}
X(53652) = X(i)-Dao conjugate of X(j) for these (i,j): {5375, 1723}, {6631, 12649}
X(53652) = cevapoint of X(78) and X(514)
X(53652) = trilinear pole of line {2, 24179}
X(53652) = barycentric product X(i)*X(j) for these {i,j}: {190, 39695}, {668, 39947}, {1978, 34430}
X(53652) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 1723}, {190, 12649}, {644, 2900}, {651, 34489}, {1331, 3211}, {1332, 224}, {2397, 51432}, {34430, 649}, {39695, 514}, {39947, 513}, {41505, 6591}

---

X(53653) = CEVAPOINT OF X(200) AND X(514)

Barycentrics    (a - b)*(a - c)*(a^3 - a^2*b - a*b^2 + b^3 - 3*a^2*c + 2*a*b*c - 3*b^2*c + 3*a*c^2 + 3*b*c^2 - c^3)*(a^3 - 3*a^2*b + 3*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(53653) lies on the Steiner circumellipse and these lines: {69, 35160}, {322, 2481}, {903, 42361}

X(53653) = X(i)-isoconjugate of X(j) for these (i,j): {513, 21002}, {649, 16572}, {667, 36845}, {1919, 20946}, {3063, 8732}, {3174, 43924}, {6591, 22153}
X(53653) = X(i)-Dao conjugate of X(j) for these (i,j): {5375, 16572}, {6631, 36845}, {9296, 20946}, {10001, 8732}, {39026, 21002}
X(53653) = cevapoint of X(i) and X(j) for these (i,j): {8, 4468}, {200, 514}
X(53653) = trilinear pole of line {2, 24181}
X(53653) = barycentric product X(i)*X(j) for these {i,j}: {190, 42361}, {4554, 42470}
X(53653) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 16572}, {101, 21002}, {190, 36845}, {644, 3174}, {664, 8732}, {668, 20946}, {1331, 22153}, {3952, 21096}, {4578, 24771}, {42361, 514}, {42470, 650}

---

X(53654) = CEVAPOINT OF X(76) AND X(512)

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

X(53654) lies on the Steiner circumellipse and these lines: {76, 3224}, {99, 3222}, {290, 43714}, {308, 19606}, {648, 880}, {670, 9429}, {671, 40162}, {2998, 3228}, {3223, 18826}, {3734, 51951}, {4602, 18830}, {9428, 18829}, {14970, 42551}, {18827, 18832}, {37890, 39932}, {39927, 53231}

X(53654) = reflection of X(39932) in X(37890)
X(53654) = isogonal conjugate of X(9491)
X(53654) = isotomic conjugate of X(3221)
X(53654) = isotomic conjugate of the isogonal conjugate of X(3222)
X(53654) = X(3222)-Ceva conjugate of X(670)
X(53654) = X(i)-isoconjugate of X(j) for these (i,j): {1, 9491}, {6, 23503}, {31, 3221}, {194, 1924}, {213, 23572}, {560, 23301}, {669, 1740}, {798, 1613}, {810, 11325}, {1501, 20910}, {1918, 50516}, {1919, 21877}, {1973, 2524}, {1980, 21080}, {2084, 38834}, {2205, 21191}, {3049, 51913}, {9426, 17149}, {14208, 41293}
X(53654) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 3221}, {3, 9491}, {9, 23503}, {6337, 2524}, {6374, 23301}, {6626, 23572}, {9296, 21877}, {9428, 194}, {31998, 1613}, {34021, 50516}, {39062, 11325}
X(53654) = cevapoint of X(i) and X(j) for these (i,j): {2, 3221}, {76, 512}, {310, 17217}, {669, 19606}, {21300, 28660}
X(53654) = trilinear pole of line {2, 2998}
X(53654) = barycentric product X(i)*X(j) for these {i,j}: {76, 3222}, {99, 40162}, {670, 2998}, {689, 42551}, {799, 18832}, {3223, 4602}, {3224, 4609}, {6331, 43714}
X(53654) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 23503}, {2, 3221}, {6, 9491}, {69, 2524}, {76, 23301}, {86, 23572}, {99, 1613}, {274, 50516}, {310, 21191}, {313, 21056}, {561, 20910}, {648, 11325}, {668, 21877}, {670, 194}, {799, 1740}, {811, 51913}, {1978, 21080}, {2396, 51427}, {2998, 512}, {3222, 6}, {3223, 798}, {3224, 669}, {3261, 21144}, {3504, 3049}, {4563, 20794}, {4577, 38834}, {4602, 17149}, {4609, 6374}, {4625, 1424}, {6331, 3186}, {6385, 23807}, {6386, 22028}, {7257, 7075}, {18829, 47642}, {18832, 661}, {19606, 9494}, {28660, 25128}, {34248, 1924}, {39927, 5027}, {40162, 523}, {42551, 3005}, {43714, 647}, {47733, 8651}, {51951, 9426}

---

X(53655) = CEVAPOINT OF X(86) AND X(523)

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

X(53655) lies on the Steiner circumellipse and these lines: {86, 21043}, {99, 21295}, {190, 22033}, {648, 17914}, {662, 31998}, {668, 17934}, {671, 6625}, {903, 40164}, {1509, 53559}, {2248, 3228}, {13610, 18827}, {14616, 51865}, {18757, 18826}

X(53655) = isotomic conjugate of the isogonal conjugate of X(53628)
X(53655) = X(i)-isoconjugate of X(j) for these (i,j): {163, 6627}, {213, 21196}, {512, 846}, {649, 21879}, {661, 18755}, {667, 21085}, {669, 17762}, {798, 1654}, {810, 4213}, {904, 24381}, {1918, 50451}, {1919, 27569}, {1924, 51857}, {3063, 27691}, {4079, 38814}, {4155, 51867}, {4983, 38836}, {6626, 50487}, {45783, 46390}
X(53655) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 6627}, {5375, 21879}, {6626, 21196}, {6631, 21085}, {9296, 27569}, {9428, 51857}, {10001, 27691}, {31998, 1654}, {34021, 50451}, {36830, 18755}, {39054, 846}, {39062, 4213}
X(53655) = cevapoint of X(i) and X(j) for these (i,j): {81, 4367}, {86, 523}, {33295, 38348}
X(53655) = trilinear pole of line {2, 1931}
X(53655) = barycentric product X(i)*X(j) for these {i,j}: {76, 53628}, {99, 6625}, {190, 40164}, {662, 51865}, {670, 2248}, {799, 13610}, {4602, 18757}, {4623, 52208}
X(53655) = barycentric quotient X(i)/X(j) for these {i,j}: {86, 21196}, {99, 1654}, {100, 21879}, {110, 18755}, {190, 21085}, {274, 50451}, {523, 6627}, {648, 4213}, {662, 846}, {664, 27691}, {668, 27569}, {670, 51857}, {799, 17762}, {894, 24381}, {2248, 512}, {4024, 21709}, {4558, 22139}, {4573, 17084}, {4610, 6626}, {4629, 38836}, {6625, 523}, {13610, 661}, {17930, 39921}, {17940, 51332}, {18757, 798}, {36066, 45783}, {40164, 514}, {51865, 1577}, {52208, 4705}, {52935, 38814}, {53628, 6}

---

X(53656) = CEVAPOINT OF X(88) AND X(900)

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

X(5364) lies on the Steiner circumellipse and these lines: {900, 39414}, {903, 21630}, {8046, 35168}

X(i)-isoconjugate of X(j) for these (i,j): {1635, 3196}, {1960, 5541}, {2251, 21198}, {3251, 39148}
X(9460)-Dao conjugate of X(21198)
cevapoint of X(i) and X(j) for these (i,j): {88, 900}, {903, 4453}
trilinear pole of line {2, 8046}
barycentric product X(4555)*X(8046)
barycentric quotient X(i)/X(j) for these {i,j}: {901, 3196}, {903, 21198}, {3257, 5541}, {4555, 30578}, {4618, 40594}, {4638, 39148}, {8046, 900}, {41529, 6544}

---

X(53657) = CEVAPOINT OF X(32) AND X(826)

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

X(53657) lies on the Steiner circumellipse and these lines: {32, 40421}, {66, 33768}, {83, 35140}, {99, 44766}, {251, 46140}, {290, 11610}, {648, 827}, {670, 4611}, {671, 16277}, {754, 52973}, {1494, 40404}, {1799, 40357}, {2353, 14970}, {6179, 43678}, {6528, 42396}, {10317, 16097}, {40146, 43094}

X(53657) = isotomic conjugate of X(23881)
X(53657) = X(i)-isoconjugate of X(j) for these (i,j): {22, 8061}, {31, 23881}, {38, 2485}, {315, 2084}, {656, 40938}, {661, 3313}, {688, 20641}, {822, 41375}, {826, 2172}, {1577, 23208}, {1760, 3005}, {1964, 33294}, {2530, 4456}, {4150, 50521}, {4463, 21123}, {8673, 17442}, {16757, 21035}, {17453, 23285}, {21034, 48084}, {21178, 21814}, {24018, 27373}
X(53657) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 23881}, {36830, 3313}, {40596, 40938}, {41884, 33294}
X(53657) = cevapoint of X(i) and X(j) for these (i,j): {2, 23881}, {32, 826}, {112, 1289}, {251, 4580}
X(53657) = trilinear pole of line {2, 66}
X(53657) = barycentric product X(i)*X(j) for these {i,j}: {66, 4577}, {83, 44766}, {99, 16277}, {648, 40404}, {689, 2353}, {827, 18018}, {1289, 1799}, {2156, 4593}, {4580, 44183}, {4630, 40421}, {6331, 46765}, {14376, 42396}, {34072, 46244}, {40146, 42371}
X(53657) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 23881}, {66, 826}, {83, 33294}, {107, 41375}, {110, 3313}, {112, 40938}, {251, 2485}, {689, 40073}, {827, 22}, {1176, 8673}, {1289, 427}, {1576, 23208}, {1632, 52532}, {2156, 8061}, {2353, 3005}, {4577, 315}, {4580, 127}, {4593, 20641}, {4599, 1760}, {4628, 4456}, {4630, 206}, {14376, 2525}, {15388, 35325}, {16277, 523}, {18018, 23285}, {32713, 27373}, {34072, 2172}, {40146, 688}, {40357, 47125}, {40404, 525}, {42396, 17907}, {44183, 41676}, {44766, 141}, {46765, 647}, {46766, 52588}, {46967, 46164}, {52376, 16757}, {52394, 21178}, {52973, 14416}

---

X(53658) = CEVAPOINT OF X(514) AND X(1698)

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

X(53658) lies on the Steiner circumellipse and these lines: {99, 3699}, {190, 4606}, {644, 32040}, {664, 3952}, {666, 32094}, {671, 41816}, {903, 5224}, {1121, 5739}, {2334, 3226}, {2481, 4385}, {3227, 25430}, {4115, 35177}, {4518, 18827}, {4555, 33948}, {4561, 4597}, {4562, 47915}, {4586, 34074}, {5545, 9059}, {14616, 52409}, {14626, 53219}, {32042, 35339}, {34024, 35148}

X(53658) = isotomic conjugate of X(4778)
X(53658) = isotomic conjugate of the isogonal conjugate of X(8694)
X(53658) = X(4633)-Ceva conjugate of X(4606)
X(53658) = X(i)-isoconjugate of X(j) for these (i,j): {6, 4790}, {31, 4778}, {32, 4801}, {41, 30723}, {58, 4822}, {81, 4832}, {213, 48580}, {604, 4765}, {649, 1449}, {663, 3361}, {667, 3616}, {798, 42028}, {810, 31903}, {1014, 8653}, {1333, 4841}, {1397, 4811}, {1407, 4827}, {1408, 4843}, {1459, 5338}, {1911, 4830}, {1919, 19804}, {2206, 4815}, {3063, 21454}, {3669, 4258}, {3733, 37593}, {4047, 43925}, {4512, 43924}, {4706, 23349}, {4773, 9456}, {4839, 18268}
X(53658) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 4778}, {9, 4790}, {10, 4822}, {37, 4841}, {3160, 30723}, {3161, 4765}, {4370, 4773}, {5375, 1449}, {6376, 4801}, {6626, 48580}, {6631, 3616}, {6651, 4830}, {9296, 19804}, {10001, 21454}, {17755, 50357}, {24771, 4827}, {27481, 4818}, {31998, 42028}, {35068, 4839}, {39062, 31903}, {40586, 4832}, {40603, 4815}
X(53658) = cevapoint of X(i) and X(j) for these (i,j): {1, 48011}, {2, 4778}, {10, 47678}, {514, 1698}, {522, 18228}, {4843, 38930}
X(53658) = trilinear pole of line {2, 2321}
X(53658) = barycentric product X(i)*X(j) for these {i,j}: {8, 4624}, {10, 4633}, {75, 4606}, {76, 8694}, {100, 40023}, {190, 5936}, {313, 4627}, {321, 4614}, {561, 34074}, {668, 25430}, {1978, 2334}, {4554, 4866}, {4572, 34820}, {5545, 30713}, {7035, 47915}, {14626, 36803}, {32018, 35339}
X(53658) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4790}, {2, 4778}, {7, 30723}, {8, 4765}, {10, 4841}, {37, 4822}, {42, 4832}, {75, 4801}, {86, 48580}, {99, 42028}, {100, 1449}, {190, 3616}, {200, 4827}, {239, 4830}, {312, 4811}, {321, 4815}, {519, 4773}, {644, 4512}, {646, 4673}, {648, 31903}, {651, 3361}, {664, 21454}, {668, 19804}, {740, 4839}, {1018, 37593}, {1332, 4652}, {1334, 8653}, {1783, 5338}, {2321, 4843}, {2334, 649}, {2397, 51423}, {3616, 53586}, {3661, 4818}, {3699, 391}, {3882, 4719}, {3912, 50357}, {3939, 4258}, {3952, 5257}, {4076, 30728}, {4169, 4819}, {4552, 3671}, {4595, 4734}, {4606, 1}, {4614, 81}, {4624, 7}, {4627, 58}, {4633, 86}, {4841, 52332}, {4866, 650}, {5545, 1412}, {5936, 514}, {6335, 5342}, {8694, 6}, {14626, 665}, {17780, 4700}, {23891, 4706}, {24004, 4742}, {25430, 513}, {27805, 4835}, {30730, 4061}, {34074, 31}, {34820, 663}, {35339, 1100}, {40023, 693}, {42720, 4684}, {47915, 244}, {52609, 4101}

---

X(53659) = CEVAPOINT OF X(514) AND X(3679)

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

X(53659) lies on the Steiner circumellipse and these lines: {8, 903}, {85, 20900}, {99, 6014}, {190, 6009}, {664, 17780}, {666, 6633}, {1121, 29616}, {2481, 4900}, {3226, 41436}, {3227, 4384}, {3699, 4555}, {3912, 35168}, {16284, 18816}, {17294, 35170}, {23891, 32041}, {30806, 35175}

X(53659) = isogonal conjugate of X(8656)
X(53659) = isotomic conjugate of X(6006)
X(53659) = isotomic conjugate of the isogonal conjugate of X(6014)
X(53659) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8656}, {31, 6006}, {649, 16670}, {663, 13462}, {667, 3241}, {1919, 30829}, {3733, 21870}, {6591, 23073}, {28607, 52593}
X(53659) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 6006}, {3, 8656}, {5375, 16670}, {6631, 3241}, {9296, 30829}, {36911, 52593}
X(53659) = cevapoint of X(i) and X(j) for these (i,j): {2, 6006}, {513, 31197}, {514, 3679}, {522, 5328}
X(53659) = trilinear pole of line {2, 1266}
X(53659) = barycentric product X(i)*X(j) for these {i,j}: {76, 6014}, {100, 40029}, {190, 36588}, {668, 39963}, {1978, 41436}, {4554, 4900}, {4555, 36915}
X(53659) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 6006}, {6, 8656}, {100, 16670}, {190, 3241}, {651, 13462}, {668, 30829}, {1018, 21870}, {1331, 23073}, {3679, 52593}, {3952, 4029}, {4427, 4982}, {4767, 36911}, {4900, 650}, {6014, 6}, {36588, 514}, {36915, 900}, {36924, 6544}, {39963, 513}, {40029, 693}, {41436, 649}



/

   Strictly inner points: X(53660)-X(53671)  

Contributed by Clark Kimberling (definitions and presentation) and Peter Moses (formulas, data, and properties), May 8, 2023.

In the preamble just before X(53621), a point P is defined to be an inner point if P lies inside the circumcircle for all triangles ABC. Here, a point P is defined to be a strictly inner point point if P lies inside ABC for every nondegenerate triangle ABC. That is, if b+c>a and c+a>b and a+b>c, then P has barycentrics (and trilinears) that are all positive for all such (a,b,c). Many well-known triangle centers have this property (e.g., X(1), X(2), X(6), X(9), X(10), X(37).) A point P is a strictly outer point point if P lies outside ABC for every nondegenerate triangle ABC.

X(53660) = X(2)X(3677)∩X(8)X(4125)

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

X(53660) lies on these lines: {2, 3677}, {3, 53666}, {8, 4125}, {149, 30615}, {3699, 33168}, {3703, 4767}, {3935, 4082}, {3952, 4388}, {3967, 33110}, {4090, 33093}, {4901, 27131}, {5014, 30578}, {10327, 17484}, {17483, 32937}, {27538, 33090}, {33086, 42054}

X(53660) = circumcircle-inverse of 53666
X(53660) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5423, 53672}, {2, 53662, 5423}, {2, 53674, 53673}, {3952, 33091, 26792}, {5423, 53661, 2}, {5423, 53673, 53674}, {53661, 53662, 53672}, {53673, 53674, 53672}

---

X(53661) = X(2)X(3677)∩X(8)X(80)

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

X(53661) lies on these lines: {2, 3677}, {3, 53667}, {8, 80}, {193, 50000}, {312, 49698}, {329, 33091}, {341, 5554}, {346, 3935}, {908, 4901}, {1265, 4696}, {2899, 36500}, {3219, 7172}, {3434, 3967}, {3617, 26580}, {3681, 3974}, {3699, 17740}, {3701, 12649}, {3710, 10528}, {3729, 49991}, {3870, 4082}, {4009, 49688}, {4011, 49696}, {4090, 33088}, {4513, 23970}, {4645, 5905}, {4661, 34255}, {4737, 12648}, {4756, 5698}, {4767, 33089}, {4942, 34612}, {5942, 17615}, {9330, 39581}, {10580, 46938}, {18228, 33090}, {20020, 27064}, {20045, 26685}, {25568, 32862}, {26034, 42054}, {26227, 27549}, {26228, 32927}, {31019, 39570}, {31289, 32920}

X(53661) = circumcircle-inverse of 53667
X(53661) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5423, 53673}, {2, 53660, 5423}, {2, 53662, 53672}, {8, 3952, 31018}, {908, 4901, 31091}, {3967, 30615, 3434}, {10327, 32937, 5905}, {53660, 53672, 53662}, {53662, 53672, 5423}

---

X(53662) = X(2)X(3677)∩X(3)X(53668)

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

X(53662) lies on these lines: {2, 3677}, {3, 53668}

X(53662) = circumcircle-inverse of X(53668)
X(53662) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5423, 53674}, {5423, 53660, 2}, {5423, 53661, 53672}, {53660, 53672, 53661}, {53661, 53672, 2}

---

X(53663) = X(2)X(3677)∩X(10)X(12)

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

X(53663) lies on these lines: {1, 3974}, {2, 3677}, {3, 53669}, {10, 12}, {31, 50115}, {37, 4082}, {42, 2321}, {55, 17355}, {142, 25961}, {200, 2345}, {306, 46897}, {321, 3755}, {346, 37553}, {498, 39559}, {519, 25496}, {527, 26034}, {594, 4061}, {612, 5750}, {756, 3778}, {908, 29667}, {968, 2325}, {1125, 32920}, {1126, 50606}, {1698, 33144}, {1962, 4029}, {2331, 7046}, {2340, 26035}, {3011, 26061}, {3175, 4356}, {3452, 25960}, {3475, 17284}, {3626, 4865}, {3634, 26128}, {3679, 26098}, {3699, 19808}, {3710, 26115}, {3741, 49529}, {3757, 17353}, {3773, 4028}, {3883, 27064}, {3914, 48642}, {3950, 6057}, {3967, 4026}, {3971, 50290}, {4011, 29669}, {4035, 15523}, {4046, 4058}, {4054, 4972}, {4071, 21805}, {4078, 43223}, {4096, 50298}, {4133, 48644}, {4357, 32937}, {4417, 39597}, {4685, 40718}, {4847, 44417}, {5249, 29679}, {5269, 5749}, {5294, 26227}, {5325, 32917}, {5745, 29828}, {5793, 6737}, {5835, 6736}, {7308, 39581}, {9623, 34036}, {9780, 25527}, {10171, 30824}, {11019, 30818}, {13405, 32777}, {14973, 22277}, {17023, 32926}, {17123, 50305}, {17165, 26251}, {17351, 44419}, {17718, 20106}, {17781, 33083}, {19717, 50000}, {19784, 34937}, {20064, 26223}, {20335, 26037}, {21865, 22276}, {24177, 49483}, {24210, 29659}, {24231, 33174}, {24239, 33169}, {24386, 33120}, {24393, 31330}, {25590, 26040}, {25719, 40723}, {26083, 29634}, {26132, 46933}, {29596, 33124}, {29639, 31264}, {29673, 38191}, {30768, 33127}, {31161, 32781}, {32916, 50313}, {32942, 49466}, {33074, 41011}, {33079, 50307}, {33152, 36478}, {36538, 46916}, {38047, 40940}, {42056, 48853}

X(53663) = circumcircle-inverse of 53669
X(53663) = X(18841)-complementary conjugate of X(141)
X(53663) = X(1014)-isoconjugate of X(53089)
X(53663) = barycentric product X(i)*X(j) for these {i,j}: {10, 5749}, {226, 7172}, {321, 5269}, {2321, 3600}, {4033, 50517}
X(53663) = barycentric quotient X(i)/X(j) for these {i,j}: {1334, 53089}, {3600, 1434}, {5269, 81}, {5749, 86}, {7172, 333}, {50517, 1019}
X(53663) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5423, 7322}, {10, 1215, 226}, {10, 4090, 4104}, {10, 4138, 28595}, {10, 21060, 1211}, {594, 4849, 4061}, {3967, 4026, 4656}, {5749, 7172, 5269}, {6057, 37593, 3950}, {29828, 33163, 5745}, {31264, 33162, 29639}, {44417, 49524, 4847}

---

X(53664) = X(2)X(16674)∩X(6)X(519)

Barycentrics    (b + c)^2 + (-a + b + c)^2 : :
X(53664) = X[6] + 4 X[2321], 13 X[6] - 8 X[4856], X[6] - 6 X[17281], 3 X[6] + 2 X[17299], 3 X[6] - 8 X[17355], 2 X[6] + 3 X[50087], 7 X[6] - 12 X[50115], 11 X[6] - 6 X[50131], 13 X[2321] + 2 X[4856], 2 X[2321] + 3 X[17281], 6 X[2321] - X[17299], 3 X[2321] + 2 X[17355], 8 X[2321] - 3 X[50087], 7 X[2321] + 3 X[50115], 22 X[2321] + 3 X[50131], and many others

X(53664) lies on these lines: {2, 16674}, {3, 53670}, {6, 519}, {8, 16885}, {9, 4668}, {10, 16675}, {37, 1698}, {44, 4007}, {45, 346}, {55, 6535}, {69, 28333}, {75, 17265}, {141, 28297}, {190, 4445}, {192, 17293}, {193, 49726}, {321, 30811}, {344, 4665}, {391, 4370}, {536, 3763}, {545, 3620}, {599, 3729}, {894, 17309}, {1086, 4461}, {1213, 16677}, {1278, 17285}, {1449, 4727}, {1575, 31242}, {1766, 28160}, {2276, 31241}, {2325, 4058}, {2345, 3616}, {3161, 17330}, {3589, 50120}, {3618, 4971}, {3623, 16884}, {3629, 50079}, {3632, 16669}, {3633, 16668}, {3644, 17292}, {3661, 17253}, {3663, 21358}, {3679, 16814}, {3686, 15828}, {3713, 17796}, {3723, 51105}, {3731, 4908}, {3773, 4660}, {3875, 17359}, {3912, 17118}, {3923, 4535}, {3946, 50100}, {3950, 16672}, {3969, 31034}, {3973, 50082}, {4029, 31253}, {4034, 15492}, {4072, 5750}, {4102, 37652}, {4133, 38047}, {4361, 17280}, {4363, 17233}, {4398, 29587}, {4399, 26685}, {4409, 45789}, {4419, 48635}, {4422, 42696}, {4431, 17119}, {4484, 22167}, {4527, 49486}, {4659, 17231}, {4664, 17327}, {4681, 17308}, {4686, 17284}, {4690, 25728}, {4699, 31244}, {4718, 17306}, {4726, 17282}, {4740, 17283}, {4764, 17291}, {4788, 17305}, {4821, 27191}, {4852, 47352}, {4869, 49727}, {4942, 33084}, {4967, 41313}, {5232, 49742}, {5341, 17742}, {5564, 17339}, {5737, 42033}, {5743, 42032}, {5749, 17388}, {6144, 17372}, {6329, 50129}, {7227, 17316}, {7229, 17392}, {7232, 17230}, {7241, 21330}, {7263, 29579}, {7321, 29577}, {15533, 50118}, {15534, 50084}, {15668, 17242}, {17054, 50042}, {17116, 17240}, {17117, 17342}, {17132, 50993}, {17143, 29542}, {17151, 17357}, {17160, 17358}, {17228, 17255}, {17234, 31139}, {17238, 24441}, {17246, 29611}, {17251, 17261}, {17254, 48640}, {17257, 48636}, {17259, 17264}, {17271, 25269}, {17276, 29594}, {17289, 17318}, {17294, 17351}, {17336, 29615}, {17337, 32087}, {17362, 20052}, {17365, 29616}, {17768, 50995}, {25101, 28634}, {28309, 51126}, {28329, 51185}, {28337, 51170}, {28582, 49509}, {31187, 50104}, {31249, 44798}, {34573, 50101}, {37650, 50098}, {37660, 50105}, {37674, 50048}, {50123, 51097}

X(53664) = reflection of X(3763) in X(17286)
X(53664) = circumcircle-inverse of 53670
X(53664) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 2321, 50087}, {8, 17340, 16885}, {75, 17268, 17265}, {75, 17269, 17267}, {192, 17293, 17325}, {346, 594, 45}, {1278, 17285, 17290}, {2321, 17281, 6}, {2321, 17355, 17299}, {2325, 4058, 17275}, {2345, 3943, 16777}, {3644, 17292, 17323}, {3661, 17262, 17253}, {3729, 17229, 599}, {3875, 17359, 47355}, {3950, 17303, 16672}, {4363, 17233, 17311}, {4431, 17279, 17119}, {17116, 17240, 17313}, {17261, 48630, 17251}, {17264, 48628, 17259}, {17265, 17268, 17267}, {17265, 17269, 17268}, {17281, 17299, 17355}, {17294, 17351, 40341}, {17299, 17355, 6}, {17314, 17369, 16884}, {17372, 50127, 6144}

---

X(53665) = X(2)X(6)∩X(7)X(3834)

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

X(53665) lies on these lines: {2, 6}, {3, 53671}, {7, 3834}, {8, 3823}, {9, 21255}, {10, 3243}, {37, 29627}, {44, 21296}, {45, 48632}, {75, 29579}, {142, 2345}, {144, 4422}, {145, 17311}, {192, 27487}, {200, 2191}, {320, 17341}, {344, 3662}, {346, 1086}, {497, 25957}, {894, 29629}, {991, 36682}, {996, 1056}, {1449, 31191}, {2321, 4859}, {2325, 4862}, {2550, 3836}, {2887, 26105}, {3008, 5839}, {3090, 48888}, {3161, 17276}, {3242, 39570}, {3247, 29600}, {3315, 31091}, {3416, 16020}, {3454, 17559}, {3596, 30866}, {3616, 17384}, {3617, 48635}, {3663, 41141}, {3672, 17243}, {3686, 31183}, {3730, 29812}, {3731, 50092}, {3739, 5936}, {3844, 39581}, {3875, 3912}, {3932, 4310}, {3943, 4452}, {3946, 29573}, {3974, 29687}, {4035, 23511}, {4346, 17262}, {4360, 29583}, {4361, 29616}, {4371, 17294}, {4395, 17309}, {4402, 17299}, {4454, 17340}, {4461, 7263}, {4470, 17289}, {4643, 18230}, {4644, 17298}, {4657, 5308}, {4660, 47357}, {4675, 5749}, {4699, 29587}, {4772, 36494}, {4849, 26047}, {4851, 5222}, {4871, 24669}, {4873, 53594}, {4888, 50115}, {4898, 50109}, {4916, 16834}, {4928, 24130}, {4966, 49680}, {5218, 29632}, {5296, 17237}, {5324, 50715}, {5437, 20106}, {5695, 7613}, {6172, 17345}, {6173, 7222}, {6389, 25932}, {6554, 41790}, {6601, 17059}, {6666, 17272}, {6740, 51834}, {7229, 17359}, {7238, 20059}, {7277, 32093}, {7321, 17342}, {7402, 24220}, {7822, 37176}, {8557, 25880}, {9776, 32777}, {9780, 31238}, {10385, 32948}, {10436, 26039}, {10589, 30957}, {11038, 49524}, {12618, 21151}, {16435, 44736}, {16593, 41325}, {16706, 17241}, {16726, 39956}, {16777, 29621}, {16815, 48634}, {17014, 17390}, {17045, 29624}, {17067, 17151}, {17117, 29577}, {17227, 17257}, {17229, 32087}, {17230, 42696}, {17233, 27191}, {17235, 41313}, {17240, 37756}, {17242, 50101}, {17244, 17291}, {17248, 29626}, {17256, 48638}, {17258, 48637}, {17260, 48633}, {17264, 48629}, {17268, 48627}, {17274, 25101}, {17280, 42697}, {17281, 31995}, {17286, 24199}, {17287, 29628}, {17288, 17338}, {17292, 27147}, {17293, 34824}, {17302, 29572}, {17306, 29571}, {17312, 17367}, {17317, 17370}, {17319, 29582}, {17326, 29581}, {17334, 45789}, {17348, 32099}, {17358, 26806}, {17362, 24599}, {17363, 29607}, {17373, 29590}, {17374, 31189}, {17380, 29585}, {17383, 29569}, {17386, 50129}, {17391, 29630}, {17396, 29575}, {17554, 49728}, {17567, 24884}, {17758, 18840}, {18135, 18157}, {18144, 28809}, {18743, 26132}, {20335, 30822}, {20946, 53510}, {20992, 44304}, {21454, 44416}, {21554, 40330}, {23681, 42047}, {24151, 30827}, {24177, 42049}, {24654, 30038}, {24678, 26103}, {24789, 34255}, {24924, 47845}, {25961, 26040}, {26034, 29851}, {26997, 27514}, {28641, 29603}, {30821, 30949}, {31244, 46933}, {31252, 33087}, {38057, 49511}, {49529, 51099}, {49775, 51190}

X(53665) = complement of X(37681)
X(53665) = circumcircle-inverse of 53671
X(53665) = barycentric product X(8)*X(24797)
X(53665) = barycentric quotient X(24797)/X(7)
X(53665) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 69, 37650}, {2, 141, 966}, {2, 193, 17352}, {2, 391, 17337}, {2, 3620, 17277}, {2, 3945, 3589}, {2, 4869, 6}, {2, 5232, 17259}, {2, 17232, 69}, {2, 17234, 4648}, {2, 17300, 3618}, {2, 18139, 5712}, {2, 18141, 37642}, {69, 37650, 37654}, {141, 17259, 5232}, {141, 17265, 2}, {142, 17284, 2345}, {320, 17341, 26685}, {344, 3662, 4419}, {346, 30833, 17267}, {599, 17337, 391}, {1086, 17267, 346}, {3008, 17296, 5839}, {3589, 17313, 3945}, {3662, 17266, 344}, {3763, 17245, 2}, {3834, 17279, 7}, {3912, 4000, 17314}, {3912, 17282, 4000}, {4422, 7232, 144}, {4675, 17357, 5749}, {4851, 17356, 5222}, {5232, 17259, 966}, {6173, 17355, 7222}, {7263, 17269, 4461}, {15668, 34573, 2}, {16706, 17241, 17316}, {17227, 17263, 17257}, {17231, 17278, 8}, {17231, 31243, 17278}, {17234, 17283, 2}, {17243, 17290, 3672}, {17244, 17291, 17321}, {17262, 48631, 4346}, {17268, 48627, 50107}, {17276, 41310, 3161}, {17291, 17321, 26104}, {17297, 17352, 193}, {17298, 17353, 4644}, {17311, 17366, 145}, {17317, 17370, 26626}, {25961, 33171, 26040}

---

X(53666) = CIRCUMCIRCLE-INVERSE OF X(53660)

Barycentrics    a^2*(a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10 + 8*a^8*b*c + 6*a^7*b^2*c - 14*a^6*b^3*c - 12*a^5*b^4*c + 12*a^4*b^5*c + 14*a^3*b^6*c - 6*a^2*b^7*c - 8*a*b^8*c - 5*a^8*c^2 + 6*a^7*b*c^2 - 16*a^6*b^2*c^2 + 14*a^5*b^3*c^2 - 3*a^4*b^4*c^2 - 14*a^3*b^5*c^2 + 20*a^2*b^6*c^2 - 6*a*b^7*c^2 + 4*b^8*c^2 - 14*a^6*b*c^3 + 14*a^5*b^2*c^3 + 60*a^4*b^3*c^3 - 80*a^3*b^4*c^3 + 28*a^2*b^5*c^3 - 22*a*b^6*c^3 + 6*b^7*c^3 + 10*a^6*c^4 - 12*a^5*b*c^4 - 3*a^4*b^2*c^4 - 80*a^3*b^3*c^4 + 145*a^2*b^4*c^4 - 60*a*b^5*c^4 + 13*b^6*c^4 + 12*a^4*b*c^5 - 14*a^3*b^2*c^5 + 28*a^2*b^3*c^5 - 60*a*b^4*c^5 + 20*b^5*c^5 - 10*a^4*c^6 + 14*a^3*b*c^6 + 20*a^2*b^2*c^6 - 22*a*b^3*c^6 + 13*b^4*c^6 - 6*a^2*b*c^7 - 6*a*b^2*c^7 + 6*b^3*c^7 + 5*a^2*c^8 - 8*a*b*c^8 + 4*b^2*c^8 - c^10) : :

X(53666) lies on this line: {3, 53660}

X(53666) = circumcircle-inverse of 53660

---

X(53667) = CIRCUMCIRCLE-INVERSE OF X(53661)

Barycentrics    a^2*(a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10 + 10*a^8*b*c + 6*a^7*b^2*c - 10*a^6*b^3*c - 6*a^5*b^4*c + 6*a^4*b^5*c + 10*a^3*b^6*c - 6*a^2*b^7*c - 10*a*b^8*c - 5*a^8*c^2 + 6*a^7*b*c^2 - 25*a^6*b^2*c^2 + 14*a^5*b^3*c^2 - 3*a^4*b^4*c^2 - 14*a^3*b^5*c^2 + 29*a^2*b^6*c^2 - 6*a*b^7*c^2 + 4*b^8*c^2 - 10*a^6*b*c^3 + 14*a^5*b^2*c^3 + 46*a^4*b^3*c^3 - 68*a^3*b^4*c^3 + 22*a^2*b^5*c^3 - 26*a*b^6*c^3 + 6*b^7*c^3 + 10*a^6*c^4 - 6*a^5*b*c^4 - 3*a^4*b^2*c^4 - 68*a^3*b^3*c^4 + 136*a^2*b^4*c^4 - 54*a*b^5*c^4 + 13*b^6*c^4 + 6*a^4*b*c^5 - 14*a^3*b^2*c^5 + 22*a^2*b^3*c^5 - 54*a*b^4*c^5 + 20*b^5*c^5 - 10*a^4*c^6 + 10*a^3*b*c^6 + 29*a^2*b^2*c^6 - 26*a*b^3*c^6 + 13*b^4*c^6 - 6*a^2*b*c^7 - 6*a*b^2*c^7 + 6*b^3*c^7 + 5*a^2*c^8 - 10*a*b*c^8 + 4*b^2*c^8 - c^10) : :

X(53667) lies on this line: {3, 53661}

X(53667) = circumcircle-inverse of 53661

---

X(53668) = CIRCUMCIRCLE-INVERSE OF X(53662)

Barycentrics    a^2*(4*a^10 - 20*a^8*b^2 + 40*a^6*b^4 - 40*a^4*b^6 + 20*a^2*b^8 - 4*b^10 + 28*a^8*b*c + 24*a^7*b^2*c - 64*a^6*b^3*c - 60*a^5*b^4*c + 60*a^4*b^5*c + 64*a^3*b^6*c - 24*a^2*b^7*c - 28*a*b^8*c - 20*a^8*c^2 + 24*a^7*b*c^2 - 43*a^6*b^2*c^2 + 56*a^5*b^3*c^2 - 12*a^4*b^4*c^2 - 56*a^3*b^5*c^2 + 59*a^2*b^6*c^2 - 24*a*b^7*c^2 + 16*b^8*c^2 - 64*a^6*b*c^3 + 56*a^5*b^2*c^3 + 268*a^4*b^3*c^3 - 344*a^3*b^4*c^3 + 124*a^2*b^5*c^3 - 80*a*b^6*c^3 + 24*b^7*c^3 + 40*a^6*c^4 - 60*a^5*b*c^4 - 12*a^4*b^2*c^4 - 344*a^3*b^3*c^4 + 601*a^2*b^4*c^4 - 252*a*b^5*c^4 + 52*b^6*c^4 + 60*a^4*b*c^5 - 56*a^3*b^2*c^5 + 124*a^2*b^3*c^5 - 252*a*b^4*c^5 + 80*b^5*c^5 - 40*a^4*c^6 + 64*a^3*b*c^6 + 59*a^2*b^2*c^6 - 80*a*b^3*c^6 + 52*b^4*c^6 - 24*a^2*b*c^7 - 24*a*b^2*c^7 + 24*b^3*c^7 + 20*a^2*c^8 - 28*a*b*c^8 + 16*b^2*c^8 - 4*c^10) : :

X(53668) lies on this line: {3, 53662}

X(53668) = circumcircle-inverse of 53662

---

X(53669) = CIRCUMCIRCLE-INVERSE OF X(53663)

Barycentrics    a^2*(a^10 + 3*a^9*b + 5*a^8*b^2 + 10*a^7*b^3 + 12*a^6*b^4 - 12*a^4*b^6 - 10*a^3*b^7 - 5*a^2*b^8 - 3*a*b^9 - b^10 + 3*a^9*c + 9*a^8*b*c + 12*a^7*b^2*c + 12*a^6*b^3*c + 6*a^5*b^4*c - 6*a^4*b^5*c - 12*a^3*b^6*c - 12*a^2*b^7*c - 9*a*b^8*c - 3*b^9*c + 5*a^8*c^2 + 12*a^7*b*c^2 + 6*a^6*b^2*c^2 + 12*a^5*b^3*c^2 - 12*a^3*b^5*c^2 - 6*a^2*b^6*c^2 - 12*a*b^7*c^2 - 5*b^8*c^2 + 10*a^7*c^3 + 12*a^6*b*c^3 + 12*a^5*b^2*c^3 + 28*a^4*b^3*c^3 - 10*a^3*b^4*c^3 + 12*a^2*b^5*c^3 - 12*a*b^6*c^3 - 4*b^7*c^3 + 12*a^6*c^4 + 6*a^5*b*c^4 - 10*a^3*b^3*c^4 + 54*a^2*b^4*c^4 - 12*a*b^5*c^4 + 6*b^6*c^4 - 6*a^4*b*c^5 - 12*a^3*b^2*c^5 + 12*a^2*b^3*c^5 - 12*a*b^4*c^5 + 14*b^5*c^5 - 12*a^4*c^6 - 12*a^3*b*c^6 - 6*a^2*b^2*c^6 - 12*a*b^3*c^6 + 6*b^4*c^6 - 10*a^3*c^7 - 12*a^2*b*c^7 - 12*a*b^2*c^7 - 4*b^3*c^7 - 5*a^2*c^8 - 9*a*b*c^8 - 5*b^2*c^8 - 3*a*c^9 - 3*b*c^9 - c^10) : :

X(53669) lies on this line: {3, 53663}

X(53669) = circumcircle-inverse of 53663

---

X(53670) = CIRCUMCIRCLE-INVERSE OF X(53664)

Barycentrics    a^2*(2*a^8 + 2*a^7*b - 2*a^6*b^2 - 2*a^5*b^3 + 2*a^3*b^5 + 2*a^2*b^6 - 2*a*b^7 - 2*b^8 + 2*a^7*c + 6*a^6*b*c - 4*a^5*b^2*c - 8*a^4*b^3*c + 8*a^3*b^4*c + 4*a^2*b^5*c - 6*a*b^6*c - 2*b^7*c - 2*a^6*c^2 - 4*a^5*b*c^2 + 16*a^4*b^2*c^2 - 5*a^3*b^3*c^2 - 6*a^2*b^4*c^2 - 5*a*b^5*c^2 + 3*b^6*c^2 - 2*a^5*c^3 - 8*a^4*b*c^3 - 5*a^3*b^2*c^3 + 14*a^2*b^3*c^3 - 9*a*b^4*c^3 + 8*b^5*c^3 + 8*a^3*b*c^4 - 6*a^2*b^2*c^4 - 9*a*b^3*c^4 + 10*b^4*c^4 + 2*a^3*c^5 + 4*a^2*b*c^5 - 5*a*b^2*c^5 + 8*b^3*c^5 + 2*a^2*c^6 - 6*a*b*c^6 + 3*b^2*c^6 - 2*a*c^7 - 2*b*c^7 - 2*c^8) : :

X(53670) lies on this line: {3, 53664}

X(53670) = circumcircle-inverse of 53664

---

X(53671) = CIRCUMCIRCLE-INVERSE OF X(53665)

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

X(53671) lies on this line: {3, 53665}

X(53671) = circumcircle-inverse of 53665

---

X(53672) = X(2)X(3677)∩X(149)X(4009)

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

X(53672) lies on these lines: {2, 3677}, {149, 4009}, {3699, 32849}, {3932, 4767}, {3952, 4645}, {4096, 33083}, {4358, 49698}, {10327, 26792}, {20095, 49991}, {26842, 32937}, {27538, 33091}, {30578, 32850}, {31252, 33148}, {31289, 32927}

X(53672) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5423, 53660}, {2, 53662, 53661}, {2, 53674, 5423}, {5423, 53661, 53662}, {5423, 53673, 2}, {53661, 53662, 53660}, {53673, 53674, 53660}

---

X(53673) = X(2)X(3677)∩X(8)X(3884)

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

X(53673) lies on these lines: {2, 3677}, {8, 3884}, {145, 26688}, {149, 8055}, {1265, 5554}, {3434, 4009}, {3452, 31091}, {3474, 4756}, {3699, 17776}, {3935, 6555}, {3952, 5905}, {4096, 26034}, {4388, 10327}, {4723, 12648}, {4767, 25568}, {7172, 27065}, {9812, 30578}, {12649, 46937}, {18228, 33091}, {20075, 30568}, {25531, 30614}, {26047, 33150}, {31053, 39570}, {36845, 46938}

X(53673) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5423, 53661}, {2, 53672, 5423}, {2, 53674, 53660}, {1265, 52353, 5554}, {10327, 27538, 31018}, {30568, 49991, 20075}, {53660, 53672, 53674}, {53660, 53674, 5423}

---

X(53674) = {X(2),X(5423)}-HARMONIC CONJUGATE OF X(53662)

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

X(53674) lies on this line:: {2, 3677}

X(53674) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5423, 53662}, {5423, 53672, 2}, {5423, 53673, 53660}, {53660, 53672, 53673}, {53660, 53673, 2}

---

X(53675) = X(1)X(2)∩X(76)X(1278)

Barycentrics    (a*b + a*c - b*c)^2 : :
X(53675) = 3 X[2] - 4 X[27091]

X(53675) lies on these lines: {1, 2}, {4, 6653}, {6, 40720}, {55, 16914}, {75, 21868}, {76, 1278}, {100, 3552}, {192, 4110}, {194, 668}, {319, 26042}, {330, 1575}, {335, 24440}, {341, 3797}, {346, 17493}, {384, 5687}, {388, 17565}, {536, 20943}, {932, 20996}, {956, 7824}, {958, 33063}, {1015, 38247}, {1329, 33061}, {1376, 6645}, {1500, 4704}, {1574, 27318}, {1740, 25311}, {1992, 26076}, {2176, 4595}, {2276, 25280}, {2550, 33030}, {2551, 17685}, {2886, 33060}, {2975, 33004}, {3210, 52043}, {3212, 40848}, {3295, 16918}, {3421, 7791}, {3434, 16044}, {3436, 6655}, {3501, 17350}, {3684, 17743}, {3730, 26797}, {3820, 33046}, {3871, 16916}, {3913, 4366}, {3952, 25270}, {4083, 26776}, {4360, 26143}, {4515, 25994}, {4687, 25614}, {4772, 26817}, {4788, 6381}, {4821, 20888}, {5025, 17757}, {5080, 33019}, {5082, 16924}, {5291, 7793}, {5711, 20145}, {7187, 16284}, {7900, 20553}, {8165, 33057}, {9708, 33047}, {9709, 16917}, {11680, 33002}, {11681, 32966}, {12607, 26582}, {12782, 26778}, {14035, 17784}, {16571, 26135}, {16604, 31999}, {16921, 24390}, {17143, 31276}, {17144, 30998}, {17178, 33297}, {17232, 20255}, {17233, 21025}, {17299, 26107}, {17303, 20146}, {17309, 27111}, {17343, 26764}, {17373, 27102}, {17375, 20561}, {17379, 26077}, {17393, 25535}, {17489, 30730}, {17490, 20917}, {17750, 37677}, {17756, 21226}, {17759, 20081}, {17786, 21857}, {17792, 27429}, {18743, 25125}, {20060, 33823}, {20532, 34063}, {20533, 33822}, {20669, 27136}, {21024, 26772}, {21031, 26590}, {21216, 33889}, {21223, 25286}, {21257, 25635}, {21780, 53146}, {21858, 30473}, {21877, 25287}, {24349, 46032}, {25107, 30963}, {25296, 31298}, {25534, 50120}, {26149, 32087}, {28654, 31060}, {31419, 33045}, {31997, 41836}, {33018, 52367}, {34282, 44418}, {41775, 49755}

X(53675) = isogonal conjugate of X(53146)
X(53675) = isotomic conjugate of the isogonal conjugate of X(53145)
X(53675) = X(i)-Ceva conjugate of X(j) for these (i,j): {43, 192}, {31625, 36863}
X(53675) = X(i)-isoconjugate of X(j) for these (i,j): {1, 53146}, {87, 2162}, {330, 7121}, {667, 32039}, {2053, 7153}, {34071, 43931}
X(53675) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 53146}, {75, 6384}, {4083, 1015}, {6631, 32039}, {16604, 52573}, {40598, 330}, {40610, 43931}, {53145, 35223}
X(53675) = cevapoint of X(25142) and X(40610)
X(53675) = trilinear pole of line {23886, 25142}
X(53675) = barycentric product X(i)*X(j) for these {i,j}: {1, 8026}, {43, 6376}, {76, 53145}, {190, 23886}, {192, 192}, {668, 25142}, {1423, 4110}, {2176, 6382}, {3208, 30545}, {3212, 27538}, {3835, 4595}, {3971, 33296}, {4083, 36863}, {20691, 31008}, {20906, 52923}, {21834, 36860}, {31625, 40610}
X(53675) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 53146}, {43, 87}, {190, 32039}, {192, 330}, {1423, 7153}, {2176, 2162}, {2209, 7121}, {3208, 2319}, {3971, 42027}, {4083, 43931}, {4110, 27424}, {4595, 4598}, {6376, 6384}, {6382, 6383}, {8026, 75}, {20691, 16606}, {20760, 23086}, {23886, 514}, {25142, 513}, {27538, 7155}, {30545, 7209}, {34832, 52573}, {36863, 18830}, {40610, 1015}, {52895, 52899}, {52923, 932}, {53145, 6}
X(53675) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 43, 20971}, {8, 26752, 2}, {192, 40598, 6376}, {194, 668, 21219}, {1376, 6645, 33062}, {1500, 27269, 4704}, {1575, 24524, 330}, {2276, 25280, 41838}, {3293, 29691, 30114}, {3913, 26687, 4366}, {4393, 27044, 2}, {6376, 20691, 192}, {17316, 26048, 2}, {17756, 25278, 21226}, {20971, 52895, 1}, {21868, 25102, 75}, {21877, 25287, 41840}, {41240, 50581, 4393}

---

X(53676) = X(1)X(6)∩X(43)X(192)

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

X(53676) lies on these lines: {1, 6}, {10, 25635}, {31, 40753}, {42, 4704}, {43, 192}, {75, 16569}, {87, 3009}, {144, 7184}, {190, 1740}, {193, 25573}, {200, 4451}, {244, 39742}, {256, 4517}, {341, 740}, {346, 3783}, {536, 36634}, {612, 16995}, {668, 39467}, {726, 978}, {756, 17038}, {869, 17261}, {872, 4096}, {894, 25528}, {899, 1278}, {995, 49520}, {1089, 6048}, {1111, 40025}, {1201, 31302}, {1423, 41531}, {1613, 51864}, {1716, 3507}, {1964, 17336}, {2209, 52923}, {2277, 3097}, {2664, 3729}, {3216, 49445}, {3217, 8300}, {3248, 36598}, {3501, 21830}, {3550, 20676}, {3644, 42083}, {3681, 22167}, {3688, 50613}, {3758, 24661}, {3840, 27291}, {3868, 22220}, {3875, 39044}, {3993, 50581}, {4454, 25571}, {4488, 25570}, {4687, 6682}, {4738, 49469}, {5268, 16999}, {12782, 21796}, {17155, 27636}, {17248, 21803}, {17257, 40790}, {17335, 17445}, {17749, 50117}, {17795, 49521}, {20072, 25572}, {21214, 24349}, {21892, 46032}, {21927, 32865}, {22316, 31855}, {24003, 30090}, {26102, 27268}, {27261, 29827}, {27494, 27646}, {27644, 40780}, {28244, 52654}, {32937, 42027}, {37699, 51046}, {49532, 49997}

X(53676) = isogonal conjugate of the isotomic conjugate of X(8026)
X(53676) = X(i)-Ceva conjugate of X(j) for these (i,j): {2176, 43}, {7035, 4595}
X(53676) = X(i)-isoconjugate of X(j) for these (i,j): {2, 53146}, {330, 2162}, {649, 32039}, {932, 43931}, {2319, 7153}, {6384, 7121}
X(53676) = X(i)-Dao conjugate of X(j) for these (i,j): {75, 6383}, {4083, 244}, {5375, 32039}, {20691, 27438}, {32664, 53146}, {40598, 6384}
X(53676) = barycentric product X(i)*X(j) for these {i,j}: {6, 8026}, {43, 192}, {75, 53145}, {100, 23886}, {190, 25142}, {1403, 4110}, {1423, 27538}, {2176, 6376}, {2209, 6382}, {3208, 3212}, {3835, 52923}, {3971, 27644}, {4083, 4595}, {7035, 40610}, {20691, 33296}, {20979, 36863}, {36860, 50491}
X(53676) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 53146}, {43, 330}, {100, 32039}, {192, 6384}, {1403, 7153}, {2176, 87}, {2209, 2162}, {3208, 7155}, {3212, 7209}, {4595, 18830}, {6376, 6383}, {8026, 76}, {17459, 52573}, {20287, 27498}, {20691, 42027}, {20979, 43931}, {23886, 693}, {25142, 514}, {27538, 27424}, {40610, 244}, {52923, 4598}, {53145, 1}
X(53676) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2663, 3247, 1}, {2664, 3729, 16571}, {3009, 17350, 87}

---

X(53677) = X(2)X(87)∩X(6)X(32011)

Barycentrics    1/(a*b + a*c - b*c)^2 : :

X(53677) lies on the circumconic {{A,B,C,X(2),X(7)}} and these lines: {2, 87}, {6, 32011}, {7, 7153}, {75, 330}, {86, 26143}, {192, 40881}, {310, 17178}, {335, 7155}, {903, 32039}, {932, 20992}, {1278, 18830}, {1740, 25311}, {2162, 17379}, {4598, 17349}, {6384, 30998}, {7249, 41777}, {16606, 41836}, {20530, 40027}, {21759, 37677}, {23493, 24661}, {25535, 30598}, {26149, 28626}, {26821, 39720}, {27494, 42027}, {31000, 31002}

X(53677) = isogonal conjugate of X(53145)
X(53677) = isotomic conjugate of the isogonal conjugate of X(53146)
X(53677) = X(i)-isoconjugate of X(j) for these (i,j): {1, 53145}, {32, 8026}, {43, 2176}, {101, 25142}, {192, 2209}, {692, 23886}, {765, 40610}, {1403, 3208}, {4595, 8640}, {20691, 38832}, {20979, 52923}, {27538, 41526}
X(53677) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 53145}, {513, 40610}, {1015, 25142}, {1086, 23886}, {6376, 8026}
X(53677) = cevapoint of X(i) and X(j) for these (i,j): {6, 35223}, {1015, 43931}
X(53677) = trilinear pole of line {514, 21128}
X(53677) = barycentric product X(i)*X(j) for these {i,j}: {76, 53146}, {87, 6384}, {330, 330}, {514, 32039}, {2162, 6383}, {2319, 7209}, {7153, 27424}, {18830, 43931}
X(53677) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 53145}, {75, 8026}, {87, 43}, {330, 192}, {513, 25142}, {514, 23886}, {932, 52923}, {1015, 40610}, {2162, 2176}, {2319, 3208}, {4598, 4595}, {6383, 6382}, {6384, 6376}, {7121, 2209}, {7153, 1423}, {7155, 27538}, {7209, 30545}, {16606, 20691}, {18830, 36863}, {23086, 20760}, {27424, 4110}, {32039, 190}, {42027, 3971}, {43931, 4083}, {52573, 34832}, {52899, 52895}, {53146, 6}
X(53677) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {87, 15966, 22343}, {18830, 32033, 1278}, {22343, 52899, 15966}

---

X(53678) = X(1)X(2162)∩X(2)X(87)

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

X(53678) lies on the circumconic {{A,B,C,X(2),X(2)}} and these lines: {1, 2162}, {2, 87}, {31, 17105}, {57, 34252}, {81, 40753}, {274, 18192}, {291, 2319}, {330, 24165}, {932, 8616}, {1258, 7121}, {1432, 7153}, {1613, 51864}, {3227, 32039}, {3494, 30701}, {4598, 16569}, {6384, 32020}, {7155, 39703}, {7167, 18193}, {16606, 52654}, {23493, 39738}, {26102, 40720}, {39694, 42027}

X(53678) = isotomic conjugate of X(8026)
X(53678) = X(53146)-Ceva conjugate of X(87)
X(53678) = X(i)-isoconjugate of X(j) for these (i,j): {2, 53145}, {31, 8026}, {100, 25142}, {101, 23886}, {192, 2176}, {1016, 40610}, {1403, 27538}, {1423, 3208}, {2209, 6376}, {3971, 38832}, {4083, 52923}, {4110, 41526}, {4595, 20979}, {8640, 36863}, {20691, 27644}
X(53678) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 8026}, {1015, 23886}, {8054, 25142}, {32664, 53145}
X(53678) = cevapoint of X(75) and X(33789)
X(53678) = barycentric product X(i)*X(j) for these {i,j}: {75, 53146}, {87, 330}, {513, 32039}, {2053, 7209}, {2162, 6384}, {4598, 43931}, {6383, 7121}, {7153, 7155}
X(53678) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 8026}, {31, 53145}, {87, 192}, {330, 6376}, {513, 23886}, {649, 25142}, {932, 4595}, {2053, 3208}, {2162, 43}, {2319, 27538}, {3248, 40610}, {4598, 36863}, {6384, 6382}, {7121, 2176}, {7153, 3212}, {7155, 4110}, {15373, 20760}, {16606, 3971}, {23086, 22370}, {23493, 20691}, {32039, 668}, {34071, 52923}, {43931, 3835}, {52573, 20899}, {53146, 1}
X(53678) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 52899, 43114}, {87, 43114, 2}, {4598, 33784, 16569}

---

X(53679) = X(75)X(330)∩X(76)X(3840)

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

X(53679) lies on these lines: {75, 330}, {76, 3840}, {87, 1221}, {194, 8026}, {274, 18192}, {334, 27424}, {870, 53146}, {2481, 32039}, {4598, 21384}, {7185, 7209}, {16606, 27318}, {17144, 18830}, {20943, 33789}, {25918, 52655}

X(53679) = X(i)-isoconjugate of X(j) for these (i,j): {6, 53145}, {43, 2209}, {560, 8026}, {692, 25142}, {1252, 40610}, {3208, 41526}, {8640, 52923}, {23886, 32739}
X(53679) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 53145}, {661, 40610}, {1086, 25142}, {6374, 8026}, {40619, 23886}
X(53679) = cevapoint of X(i) and X(j) for these (i,j): {21138, 48406}, {27438, 42027}
X(53679) = barycentric product X(i)*X(j) for these {i,j}: {87, 6383}, {330, 6384}, {561, 53146}, {693, 32039}, {7155, 7209}
X(53679) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 53145}, {76, 8026}, {87, 2176}, {244, 40610}, {330, 43}, {514, 25142}, {693, 23886}, {2162, 2209}, {4598, 52923}, {6383, 6376}, {6384, 192}, {7153, 1403}, {7155, 3208}, {7209, 3212}, {18830, 4595}, {27424, 27538}, {27498, 20287}, {32039, 100}, {42027, 20691}, {43931, 20979}, {52573, 17459}, {53146, 31}
X(53679) = {X(76),X(52573)}-harmonic conjugate of X(6384)

---

X(53680) = X(1)X(3972)∩X(55)X(83)

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

X(53680) lies on these lines: {1, 3972}, {2, 9665}, {11, 7857}, {32, 4366}, {35, 7786}, {55, 83}, {76, 1914}, {86, 14377}, {99, 16502}, {192, 5007}, {330, 7816}, {350, 6179}, {384, 2241}, {668, 16920}, {1003, 16781}, {1015, 3552}, {1479, 7828}, {1500, 7787}, {1573, 16914}, {2275, 7782}, {2276, 7878}, {3329, 31451}, {4188, 27195}, {4294, 7803}, {4302, 7847}, {5248, 20179}, {5299, 7757}, {5332, 7894}, {6284, 7790}, {6658, 9651}, {7752, 26629}, {7769, 9599}, {7771, 26959}, {7780, 30998}, {7792, 15171}, {7797, 9664}, {7802, 26561}, {7827, 9598}, {7846, 26590}, {7851, 9668}, {7934, 30104}, {10896, 14061}, {10987, 27020}, {16975, 17692}, {17034, 21793}, {17200, 17380}

X(53680) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {350, 7031, 6179}, {5332, 25264, 7894}

---

X(53681) = X(2)X(2112)∩X(21)X(257)

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

X(53681) lies on these lines: {1, 32115}, {2, 2112}, {8, 8298}, {21, 257}, {31, 19580}, {41, 17743}, {100, 18265}, {171, 7369}, {172, 17752}, {192, 983}, {238, 17493}, {239, 1914}, {292, 4586}, {385, 1580}, {893, 40415}, {894, 2330}, {1428, 39914}, {1468, 9263}, {1691, 1966}, {1921, 14599}, {3212, 3552}, {3573, 39916}, {4594, 7104}, {5150, 24502}, {5773, 16738}, {6652, 39044}, {7200, 17103}, {8932, 17257}, {9317, 16915}, {17741, 21021}, {19563, 51903}, {20333, 27931}, {21221, 26222}, {24436, 24505}, {27913, 27950}, {39921, 40767}, {40744, 41252}

X(53681) = X(i)-Ceva conjugate of X(j) for these (i,j): {171, 27982}, {238, 6652}, {4586, 30654}, {40415, 238}
X(53681) = X(i)-isoconjugate of X(j) for these (i,j): {171, 41517}, {256, 52205}, {291, 694}, {292, 1581}, {334, 9468}, {335, 1967}, {881, 4639}, {882, 4584}, {893, 30663}, {904, 40098}, {1911, 1916}, {1922, 1934}, {1927, 18895}, {7018, 51856}, {8789, 44172}, {14598, 18896}, {18267, 44187}, {18893, 44160}
X(53681) = X(i)-Dao conjugate of X(j) for these (i,j): {1966, 7018}, {6651, 1916}, {8290, 335}, {18277, 18896}, {18904, 2887}, {19557, 1581}, {19576, 292}, {35078, 35352}, {39028, 1934}, {39029, 694}, {39030, 44172}, {39031, 1911}, {39043, 291}, {39044, 334}, {40597, 30663}
X(53681) = cevapoint of X(19557) and X(19580)
X(53681) = barycentric product X(i)*X(j) for these {i,j}: {86, 4154}, {171, 39044}, {238, 1966}, {239, 385}, {242, 12215}, {257, 4027}, {350, 1580}, {874, 4164}, {880, 4455}, {894, 4366}, {1691, 1921}, {1909, 8300}, {1914, 3978}, {1920, 51328}, {1926, 2210}, {1933, 18891}, {3570, 4107}, {3573, 14296}, {4010, 17941}, {4039, 33295}, {4368, 17103}, {4375, 18047}, {4579, 27855}, {6652, 30669}, {7018, 51903}, {7193, 17984}, {14599, 14603}, {14602, 44169}, {17493, 27982}, {18894, 18901}, {19563, 40415}, {44187, 51318}
X(53681) = barycentric quotient X(i)/X(j) for these {i,j}: {171, 30663}, {172, 52205}, {238, 1581}, {239, 1916}, {350, 1934}, {385, 335}, {804, 35352}, {893, 41517}, {894, 40098}, {1580, 291}, {1691, 292}, {1914, 694}, {1921, 18896}, {1926, 44172}, {1933, 1911}, {1966, 334}, {2210, 1967}, {3978, 18895}, {4027, 894}, {4039, 43534}, {4107, 4444}, {4154, 10}, {4164, 876}, {4366, 257}, {4455, 882}, {6652, 17493}, {7193, 36214}, {8300, 256}, {12215, 337}, {12835, 1431}, {14599, 9468}, {14602, 1922}, {14603, 44170}, {17941, 4589}, {18892, 1927}, {18894, 8789}, {18902, 18897}, {19563, 2887}, {27982, 30669}, {30654, 30671}, {39044, 7018}, {44169, 44160}, {51318, 172}, {51328, 893}, {51903, 171}
X(53681) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1691, 1966, 27982}, {39044, 51328, 6652}

---

X(53682) = TRILINEAR POLE OF X(6)X(41405)

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

X(53692) lies on the circumcircle and these lines: {100, 6164}, {101, 9262}, {813, 5548}, {901, 8661}, {1960, 6551}, {2384, 21781}, {2702, 32665}, {2718, 9282}, {2726, 6630}, {39444, 42555}

X(53682) = X(i)-isoconjugate of X(j) for these (i,j): {44, 21204}, {100, 24131}, {522, 14122}, {900, 1054}, {1635, 4440}, {1647, 6163}, {1960, 18159}, {2087, 6631}, {3762, 9259}, {4895, 17089}, {4919, 30725}
X(53682) = X(i)-Dao conjugate of X(j) for these (i,j): {8054, 24131}, {40595, 21204}
X(53682) = cevapoint of X(101) and X(1960)
X(53682) = trilinear pole of line {6, 41405}
X(53682) = barycentric product X(i)*X(j) for these {i,j}: {901, 6630}, {3257, 9282}, {5376, 6164}, {9268, 42555}
X(53682) = barycentric quotient X(i)/X(j) for these {i,j}: {106, 21204}, {649, 24131}, {901, 4440}, {1415, 14122}, {3257, 18159}, {6551, 6634}, {9262, 1647}, {9268, 6631}, {9282, 3762}, {32665, 1054}, {32719, 9259}

---

X(53683) = TRILINEAR POLE OF X(6)X(1754)

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

X(53683) lies on the circumcircle and these lines: {21, 103}, {28, 917}, {74, 1006}, {98, 4223}, {100, 1624}, {104, 36011}, {105, 40980}, {108, 4241}, {163, 35182}, {662, 43344}, {675, 4228}, {972, 36017}, {1290, 7479}, {1292, 4237}, {1294, 36029}, {1297, 36018}, {1300, 36009}, {1325, 2688}, {1414, 24016}, {1621, 2249}, {2691, 36032}, {2975, 26702}, {4233, 9085}, {4249, 26706}, {4250, 30250}, {4566, 21789}, {6011, 7437}, {7259, 29163}, {7469, 53190}, {15323, 36015}, {16049, 41905}, {36077, 52913}

X(53683) = Collings transform of X(8226)
X(53683) = X(i)-isoconjugate of X(j) for these (i,j): {10, 44408}, {42, 46402}, {649, 45744}, {656, 4219}, {661, 37659}, {4566, 14714}
X(53683) = X(i)-Dao conjugate of X(j) for these (i,j): {5375, 45744}, {36830, 37659}, {40592, 46402}, {40596, 4219}
X(53683) = cevapoint of X(i) and X(j) for these (i,j): {1, 21789}, {523, 8226}, {667, 1104}, {3737, 18165}
X(53683) = trilinear pole of line {6, 1754}
X(53683) = barycentric quotient X(i)/X(j) for these {i,j}: {81, 46402}, {100, 45744}, {110, 37659}, {112, 4219}, {1333, 44408}

---

X(53684) = TRILINEAR POLE OF X(6)X(6985)

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

X(53684) lies on the circumcircle and these lines: {21, 74}, {28, 1300}, {98, 4228}, {100, 15329}, {104, 11101}, {108, 4240}, {477, 1325}, {841, 37960}, {842, 7469}, {915, 13739}, {1290, 7471}, {1292, 4226}, {1294, 16049}, {1299, 30733}, {1304, 52914}, {2074, 32710}, {2691, 7468}, {2766, 7480}, {3563, 4233}, {3658, 6011}, {4221, 43660}, {4230, 26706}, {4236, 30257}, {4246, 30250}, {7473, 10101}, {13397, 30512}, {13486, 36064}, {17560, 45138}, {37963, 40118}, {40097, 46587}

X(53684) = Collings transform of X(6841)
X(53684) = X(i)-isoconjugate of X(j) for these (i,j): {10, 48382}, {656, 7414}
X(53684) = X(40596)-Dao conjugate of X(7414)
X(53684) = cevapoint of X(523) and X(6841)
X(53684) = trilinear pole of line {6, 6985}
X(53684) = barycentric product X(648)*X(34800)
X(53684) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 7414}, {1333, 48382}, {34800, 525}

---

X(53685) = TRILINEAR POLE OF X(6)X(1018)

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

X(53685) lies on the circumcircle and these lines: {99, 7035}, {100, 4057}, {105, 15571}, {106, 24625}, {110, 765}, {190, 53637}, {644, 29149}, {660, 741}, {675, 40039}, {759, 51562}, {932, 4767}, {953, 34466}, {1621, 2382}, {2718, 2975}, {3699, 53627}, {3733, 3952}, {4588, 52923}, {4756, 43350}, {5276, 9111}

X(53685) = Collings transform of X(4694)
X(53685) = X(i)-isoconjugate of X(j) for these (i,j): {513, 49997}, {649, 17495}, {667, 39995}, {7649, 23169}, {23345, 34587}
X(53685) = X(i)-Dao conjugate of X(j) for these (i,j): {5375, 17495}, {6631, 39995}, {39026, 49997}
X(53685) = cevapoint of X(i) and X(j) for these (i,j): {1, 4491}, {9, 53286}, {44, 667}, {513, 4694}, {756, 1635}, {1023, 4557}
X(53685) = trilinear pole of line {6, 1018}
X(53685) = barycentric product X(i)*X(j) for these {i,j}: {100, 39698}, {101, 40039}
X(53685) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 17495}, {101, 49997}, {190, 39995}, {906, 23169}, {1023, 34587}, {39698, 693}, {40039, 3261}

---

X(53686) = TRILINEAR POLE OF X(6)X(4705)

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

X(53686) lies on the circumcircle and these lines: {37, 110}, {99, 321}, {100, 594}, {101, 756}, {108, 8736}, {109, 2171}, {112, 1824}, {335, 36066}, {572, 39633}, {827, 18098}, {934, 6354}, {1255, 6578}, {1290, 17737}, {1333, 21353}, {1635, 28482}, {2161, 36069}, {2170, 29038}, {2703, 5164}, {2975, 30241}, {5251, 8691}, {5606, 33148}, {9090, 20989}, {13397, 41508}, {34594, 40085}, {52208, 53628}
X(53686) = X(i)-isoconjugate of X(j) for these (i,j): {58, 44396}, {86, 5164}, {424, 1790}
X(53686) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 44396}, {40600, 5164}
X(53686) = cevapoint of X(37) and X(5291)
X(53686) = trilinear pole of line {6, 4705}
X(53686) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 44396}, {213, 5164}, {1824, 424}

---

X(53687) = X(98)X(10557)∩X(99)X(5466)

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

X(53687) lies on the circumcircle and these lines: {98, 10557}, {99, 5466}, {110, 9178}, {842, 46783}, {843, 17964}, {895, 9184}, {2709, 36827}, {2770, 16092}, {6082, 53379}, {9140, 39446}, {10425, 32583}, {34574, 45773}

X(53687) = X(i)-isoconjugate of X(j) for these (i,j): {896, 9168}, {2642, 41134}, {23889, 44398}
X(53687) = X(15899)-Dao conjugate of X(9168)
X(53687) = cevapoint of X(i) and X(j) for these (i,j): {187, 9171}, {9178, 17964}
X(53687) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 9168}, {691, 41134}, {9171, 41177}, {9178, 44398}

---

X(53688) = TRILINEAR POLE OF X(6)X(50344)

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

X(53688) lies on the circumcircle and these lines: {42, 8701}, {58, 6578}, {99, 1125}, {100, 1255}, {101, 1126}, {110, 1171}, {813, 9506}, {1796, 29329}, {2702, 20472}, {2712, 50344}, {3122, 4629}, {4368, 4632}, {6543, 39415}, {6650, 46961}, {8708, 40767}, {9278, 15322}, {9505, 36066}, {17972, 28162}, {17982, 26705}, {33635, 43077}

X(53688) = X(i)-isoconjugate of X(j) for these (i,j): {423, 3958}, {1100, 6542}, {1125, 1757}, {1213, 1931}, {1269, 18266}, {1326, 4647}, {1962, 17731}, {2308, 20947}, {2786, 35342}, {3916, 17927}, {4359, 17735}, {4427, 9508}, {4974, 40794}, {4983, 17934}, {8025, 20693}, {17943, 30591}, {20970, 52137}
X(53688) = cevapoint of X(i) and X(j) for these (i,j): {2054, 17962}, {3122, 5029}
X(53688) = trilinear pole of line {6, 50344}
X(53688) = barycentric product X(i)*X(j) for these {i,j}: {1126, 6650}, {1171, 11599}, {1255, 1929}, {1268, 17962}, {1796, 17982}, {2054, 32014}, {2702, 4608}, {4629, 18014}, {4632, 18001}, {6543, 52558}, {9278, 40438}, {17940, 31010}, {18032, 28615}, {35148, 50344}, {37135, 47947}
X(53688) = barycentric quotient X(i)/X(j) for these {i,j}: {1126, 6542}, {1171, 17731}, {1255, 20947}, {1929, 4359}, {2054, 1213}, {2702, 4427}, {4629, 17934}, {6543, 52576}, {6650, 1269}, {9278, 4647}, {11599, 1230}, {17962, 1125}, {17972, 4001}, {18001, 4988}, {28615, 1757}, {40438, 52137}, {50344, 2786}, {52555, 6541}

---

X(53689) = X(99)X(3666)∩X(100)X(2092)

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

X(53689) lies on the circumcircle and these lines: {37, 8707}, {99, 3666}, {100, 2092}, {101, 3725}, {108, 17981}, {109, 17954}, {110, 1169}, {171, 831}, {932, 45218}, {1310, 17946}, {1402, 8687}, {2703, 5078}, {17053, 38470}, {28480, 37619}

X(53689) = X(i)-isoconjugate of X(j) for these (i,j): {1193, 17790}, {1848, 17977}, {2092, 5209}, {2292, 19623}, {2787, 3882}, {3666, 17763}, {3687, 5061}, {4357, 5291}, {5006, 18697}, {17944, 21124}, {17987, 22097}
X(53689) = cevapoint of X(3121) and X(5040)
X(53689) = barycentric product X(i)*X(j) for these {i,j}: {961, 11609}, {1169, 11611}, {1220, 17954}, {1791, 17981}, {2298, 17946}, {2703, 4581}, {17961, 30710}
vbarycentric quotient X(i)/X(j) for these {i,j}: {1169, 19623}, {2298, 17790}, {2363, 5209}, {2703, 53332}, {11611, 1228}, {17946, 20911}, {17954, 4357}, {17961, 3666}, {18002, 50330}

---

X(53690) = TRILINEAR POLE OF X(6)X(691)

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

X(53609) lies on the circumcircle and these lines: {98, 14694}, {99, 1649}, {110, 39527}, {111, 21906}, {351, 691}, {476, 9123}, {805, 32583}, {843, 2502}, {892, 9080}, {2770, 18823}, {5467, 45773}, {10102, 44420}, {20404, 34763}, {32694, 32729}, {35279, 39446}

X(53690) = isogonal conjugate of X(33921)
X(53690) = Parry-circle-inverse of X(691)
X(53690) = Parry-isodynamic-circle-inverse of X(843)
X(53690) = X(i)-isoconjugate of X(j) for these (i,j): {1, 33921}, {543, 2642}, {661, 1641}, {896, 8371}, {1649, 17955}, {9171, 14210}, {14423, 24041}, {17993, 24038}, {18007, 42081}, {36085, 41176}
X(53690) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 33921}, {3005, 14423}, {15477, 9171}, {15899, 8371}, {36830, 1641}, {38988, 41176}
X(53690) = cevapoint of X(i) and X(j) for these (i,j): {351, 2502}, {691, 23348}
X(53690) = trilinear pole of line {6, 691}
X(53690) = barycentric product X(i)*X(j) for these {i,j}: {111, 9170}, {691, 18823}, {843, 892}, {34539, 34763}, {34574, 51226}
X(53690) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 33921}, {110, 1641}, {111, 8371}, {351, 41176}, {691, 543}, {843, 690}, {892, 45809}, {3124, 14423}, {9170, 3266}, {9180, 52628}, {10630, 18007}, {18823, 35522}, {23348, 35087}, {32729, 2502}, {32740, 9171}, {34539, 34760}, {34574, 17948}, {41936, 17993}, {48450, 1649}, {51226, 52629}

---

X(53691) = TRILINEAR POLE OF X(6)X(842)

Barycentrics    a^2*(a^2 - b^2)*(a^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)*(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(53691) lies on the circumcircle and these lines: {2, 46416}, {74, 1976}, {98, 868}, {99, 6035}, {110, 5649}, {111, 248}, {112, 14998}, {351, 1304}, {476, 2966}, {685, 935}, {691, 9409}, {842, 5191}, {1297, 40080}, {2373, 31635}, {2715, 3569}, {2857, 5641}, {3563, 52492}, {5967, 18331}, {6037, 43113}, {9161, 52179}, {14223, 35278}, {20404, 34761}, {23969, 53177}, {26714, 46249}, {40118, 50188}, {51542, 52199}

X(53691) = anticomplement of X(46416)
X(53691) = Parry-circle-inverse of X(1304)
X(53691) = X(i)-isoconjugate of X(j) for these (i,j): {1109, 42743}, {1640, 1959}, {1755, 18312}, {2247, 2799}, {6041, 46238}
X(53691) = X(36899)-Dao conjugate of X(18312)
X(53691) = cevapoint of X(i) and X(j) for these (i,j): {842, 23350}, {3569, 5191}, {35909, 40080}
X(53691) = trilinear pole of line {6, 842}
X(53691) = barycentric product X(i)*X(j) for these {i,j}: {98, 5649}, {842, 2966}, {1976, 6035}, {2715, 5641}, {41173, 46787}
X(53691) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 18312}, {842, 2799}, {1976, 1640}, {2422, 51428}, {2715, 542}, {5649, 325}, {14601, 6041}, {14998, 868}, {23350, 35088}, {23357, 42743}, {23969, 14356}, {32696, 6103}, {41173, 46786}, {51263, 51389}, {51542, 45321}, {52199, 41167}

---

X(53692) = TRILINEAR POLE OF X(6)X(868)

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

X(53692) lies on the circumcircle and these lines: {23, 2857}, {30, 2710}, {74, 2794}, {98, 338}, {110, 2799}, {111, 47200}, {112, 16230}, {114, 1297}, {132, 3563}, {476, 47004}, {477, 47326}, {523, 2715}, {542, 53188}, {690, 53187}, {699, 13195}, {842, 1513}, {1294, 21166}, {1304, 47627}, {2693, 44252}, {2770, 47327}, {2781, 9161}, {2797, 39447}, {9090, 47199}, {9160, 9517}, {14223, 35278}, {14659, 16315}, {22456, 30716}

X(53692) = reflection of X(2715) in the Euler line
X(53692) = Collings transform of X(1316)
X(53692) = X(656)-isoconjugate of X(36176)
X(53692) = X(40596)-Dao conjugate of X(36176)
X(53692) = cevapoint of X(523) and X(1316)
X(53692) = trilinear pole of line {6, 868}
X(53692) = barycentric quotient X(112)/X(36176)

---

X(53693) = TRILINEAR POLE OF X(6)X(5055)

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

X(53693) lies on the circumcircle and these lines: {2, 43656}, {3, 13530}, {74, 3534}, {98, 7492}, {110, 53274}, {111, 37637}, {112, 47053}, {477, 37950}, {549, 19307}, {842, 10989}, {1291, 4558}, {1299, 35480}, {1300, 35473}, {3563, 5094}, {4226, 11636}, {6236, 11634}, {7472, 32229}, {9146, 10425}, {31626, 43657}, {37969, 40118}

X(53693) = reflection of X(i) in X(j) for these {i,j}: {13530, 3}, {52173, 549}
X(53693) = Collings transform of X(i) for these i: {549, 566, 22452}
X(53693) = X(i)-isoconjugate of X(j) for these (i,j): {656, 47485}, {661, 11004}, {9703, 24006}
X(53693) = X(i)-Dao conjugate of X(j) for these (i,j): {36830, 11004}, {40596, 47485}
X(53693) = cevapoint of X(i) and X(j) for these (i,j): {512, 566}, {523, 549}
X(53693) = trilinear pole of line {6, 5055}
X(53693) = barycentric product X(99)*X(52154)
X(53693) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 11004}, {112, 47485}, {32661, 9703}, {52154, 523}

---

X(53694) = X(74)X(15329)∩X(477)X(7471)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(3*a^10*b^2 - 10*a^8*b^4 + 10*a^6*b^6 - 5*a^2*b^10 + 2*b^12 + a^10*c^2 + 2*a^8*b^2*c^2 + a^6*b^4*c^2 - 17*a^4*b^6*c^2 + 18*a^2*b^8*c^2 - 5*b^10*c^2 - 4*a^8*c^4 - 5*a^6*b^2*c^4 + 22*a^4*b^4*c^4 - 17*a^2*b^6*c^4 + 6*a^6*c^6 - 5*a^4*b^2*c^6 + a^2*b^4*c^6 + 10*b^6*c^6 - 4*a^4*c^8 + 2*a^2*b^2*c^8 - 10*b^4*c^8 + a^2*c^10 + 3*b^2*c^10)*(a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10 + 3*a^10*c^2 + 2*a^8*b^2*c^2 - 5*a^6*b^4*c^2 - 5*a^4*b^6*c^2 + 2*a^2*b^8*c^2 + 3*b^10*c^2 - 10*a^8*c^4 + a^6*b^2*c^4 + 22*a^4*b^4*c^4 + a^2*b^6*c^4 - 10*b^8*c^4 + 10*a^6*c^6 - 17*a^4*b^2*c^6 - 17*a^2*b^4*c^6 + 10*b^6*c^6 + 18*a^2*b^2*c^8 - 5*a^2*c^10 - 5*b^2*c^10 + 2*c^12) : :

X(53694) lies on the circumcircle and these lines: {74, 15329}, {477, 7471}, {841, 7468}, {1294, 30512}, {1299, 46587}, {1300, 4240}, {2693, 40049}, {4226, 43660}, {7480, 32710}, {14094, 16169}

X(53694) = barycentric quotient X(2420)/X(47084)

---

X(53695) = TRILINEAR POLE OF X(6)X(47049)

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

X(53695) lies on the circumcircle and these lines: {74, 11634}, {98, 4226}, {111, 15329}, {477, 7472}, {842, 7468}, {1299, 46592}, {1300, 4235}, {2373, 30512}, {2374, 4240}, {2409, 40120}, {2770, 7471}, {3563, 4230}, {7473, 40118}, {7480, 40119}, {7482, 32710}

X(53695) = Collings transform of X(34291)
X(53695) = cevapoint of X(511) and X(34291)
X(53695) = trilinear pole of line {6, 47049}
X(53695) = barycentric quotient X(14966)/X(47079)

---

X(53696) = TRILINEAR POLE OF X(6)X(52592)

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

X(53696) lies on the circumcircle and these lines: {27, 26705}, {42, 6577}, {74, 46596}, {99, 17135}, {100, 22271}, {101, 4184}, {109, 52024}, {110, 8053}, {112, 33774}, {2690, 5196}, {4229, 44876}, {7474, 9057}, {15320, 17198}

X(53696) = Collings transform of X(53564)
X(53696) = X(37)-isoconjugate of X(17729)
X(53696) = X(40589)-Dao conjugate of X(17729)
X(53696) = trilinear pole of line {6, 52592}
X(53696) = barycentric quotient X(58)/X(17729)

---

X(53697) = X(74)X(13589)∩X(104)X(3658)

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

X(53697) lies on the circumcircle and these lines: {74, 13589}, {104, 3658}, {110, 53406}, {477, 36167}, {759, 15329}, {915, 4246}, {1299, 46588}, {1300, 4242}, {2687, 7477}, {4240, 39439}, {7450, 14987}, {7471, 12030}, {30512, 39435}, {32710, 37964}

X(53697) = X(10058)-isoconjugate of X(53527)

---

X(53698) = X(74)X(46593)∩X(105)X(4236)

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

X(53698) lies on the circumcircle and these lines: {74, 46593}, {105, 4236}, {111, 13589}, {759, 11634}, {2374, 4242}, {2752, 7475}, {2770, 36167}, {3658, 9061}, {4235, 39439}, {4238, 15344}, {7472, 12030}, {37964, 40119}

---

X(53699) = TRILINEAR POLE OF X(6)X(4230)

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

X(53699) lies on the circumcircle and these lines: {4, 43654}, {30, 48259}, {74, 9513}, {98, 648}, {99, 684}, {107, 17994}, {110, 39469}, {112, 2491}, {186, 2698}, {250, 2715}, {477, 35474}, {523, 22456}, {935, 46245}, {1297, 15915}, {2373, 9832}, {2693, 47620}, {2697, 5999}, {6037, 7473}, {36176, 46253}

X(53699) = reflection of X(22456) in the Euler line
X(53699) = isotomic conjugate of the anticomplement of X(47233)
X(53699) = X(i)-isoconjugate of X(j) for these (i,j): {63, 47229}, {293, 31953}, {656, 1316}, {810, 44155}, {3708, 40866}, {14208, 44127}, {20902, 46249}
X(53699) = X(i)-Dao conjugate of X(j) for these (i,j): {132, 31953}, {3162, 47229}, {39062, 44155}, {40596, 1316}
X(53699) = trilinear pole of line {6, 4230}
X(53699) = barycentric product X(i)*X(j) for these {i,j}: {250, 46245}, {648, 9513}, {4230, 53229}, {41174, 43112}
X(53699) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 47229}, {112, 1316}, {232, 31953}, {250, 40866}, {648, 44155}, {9513, 525}, {40077, 24284}, {43112, 41172}, {46245, 339}

---

X(53700) = TRILINEAR POLE OF X(6)X(878)

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

X(53700) lies on the circumcircle and these lines: {98, 3569}, {99, 287}, {107, 3124}, {110, 248}, {112, 1976}, {237, 2715}, {297, 22456}, {476, 48452}, {694, 18858}, {805, 15391}, {842, 3288}, {2395, 43654}, {2857, 46245}, {5191, 26714}, {6037, 34369}, {36163, 53603}, {36176, 46253}, {46970, 51327}, {51542, 52199}

X(53700) = isogonal conjugate of the isotomic conjugate of X(53229)
X(53700) = X(i)-isoconjugate of X(j) for these (i,j): {662, 31953}, {1316, 1959}, {1755, 44155}, {44127, 46238}
X(53700) = X(i)-Dao conjugate of X(j) for these (i,j): {1084, 31953}, {36899, 44155}
X(53700) = cevapoint of X(9513) and X(40077)
X(53700) = trilinear pole of line {6, 878}
X(53700) = barycentric product X(i)*X(j) for these {i,j}: {6, 53229}, {98, 9513}, {2715, 46245}, {36897, 40077}
X(53700) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 44155}, {512, 31953}, {1976, 1316}, {2422, 47229}, {2715, 40866}, {9513, 325}, {14601, 44127}, {34238, 38947}, {40077, 5976}, {43112, 41167}, {53229, 76}

---

X(53701) = TRILINEAR POLE OF X(6)X(3613)

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

X(53701) lies on the circumcircle and these lines: {74, 52190}, {98, 34133}, {110, 9514}, {111, 47198}, {685, 933}, {733, 36897}, {827, 2966}, {842, 3613}, {1632, 26714}, {2698, 12110}, {2706, 42487}, {2710, 36952}, {19189, 39452}, {39427, 51869}

X(53701) = Collings transform of X(i) for these i: {237, 52591}
X(53701) = X(i)-isoconjugate of X(j) for these (i,j): {1755, 31296}, {1959, 3050}, {2491, 33764}, {3405, 52591}, {3569, 18042}, {7668, 23997}
X(53701) = X(36899)-Dao conjugate of X(31296)
X(53701) = cevapoint of X(i) and X(j) for these (i,j): {237, 523}, {2395, 51869}
X(53701) = trilinear pole of line {6, 3613}
X(53701) = barycentric product X(i)*X(j) for these {i,j}: {98, 11794}, {685, 36952}, {2966, 3613}, {27375, 43187}, {27867, 43665}
X(53701) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 31296}, {685, 36794}, {878, 38352}, {1976, 3050}, {2395, 7668}, {2715, 5012}, {2966, 1078}, {3613, 2799}, {11794, 325}, {20031, 1629}, {27375, 3569}, {27867, 2421}, {32696, 10312}, {36036, 33764}, {36084, 18042}, {36952, 6333}, {43187, 33769}, {43665, 36901}, {51869, 52591}

---

X(53702) = TRILINEAR POLE OF X(6)X(2250)

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

X(53702) lies on the circumcircle and these lines: {101, 35321}, {102, 6905}, {109, 21173}, {110, 36037}, {759, 40437}, {859, 38955}, {901, 7451}, {953, 31849}, {1309, 7461}, {1311, 33849}, {2051, 2717}, {2687, 51870}, {2716, 6906}, {2734, 11491}, {7428, 36944}, {8687, 32641}, {29068, 52663}

X(53702) = X(i)-isoconjugate of X(j) for these (i,j): {517, 21173}, {572, 10015}, {1769, 2975}, {1785, 23187}, {2183, 17496}, {2427, 24237}, {3310, 14829}, {8677, 11109}, {11998, 24029}, {17074, 46393}, {20986, 36038}, {23788, 52139}, {23981, 34589}
X(53702) = trilinear pole of line {6, 2250}
X(53702) = barycentric product X(i)*X(j) for these {i,j}: {2051, 36037}, {13136, 34434}
X(53702) = barycentric quotient X(i)/X(j) for these {i,j}: {104, 17496}, {909, 21173}, {2051, 36038}, {2720, 17074}, {14578, 23187}, {32641, 2975}, {34434, 10015}, {36037, 14829}, {43728, 40624}, {53083, 23788}

---

X(53703) = TRILINEAR POLE OF X(6)X(8677)

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

X(53) lies on the circumcircle and these lines: {1, 2728}, {2, 1309}, {20, 2737}, {22, 901}, {23, 53612}, {25, 35012}, {100, 3100}, {101, 22350}, {102, 663}, {104, 905}, {105, 6129}, {108, 1465}, {109, 3220}, {112, 859}, {198, 919}, {222, 2720}, {347, 927}, {858, 53611}, {934, 1455}, {935, 3109}, {1055, 26715}, {1289, 37168}, {1311, 47798}, {1458, 8059}, {2189, 36071}, {2192, 6081}, {2726, 45945}, {2730, 52026}, {3663, 53182}, {3737, 26702}, {5089, 40117}, {6905, 26706}

X(53703) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(10017)
X(53703) = X(i)-isoconjugate of X(j) for these (i,j): {515, 14203}, {36037, 45945}
X(53703) = X(3259)-Dao conjugate of X(45945)
X(53703) = cevapoint of X(3) and X(2810)
X(53703) = trilinear pole of line {6, 8677}
X(53703) = barycentric quotient X(i)/X(j) for these {i,j}: {3310, 45945}, {32677, 14203}

---

X(53704) = TRILINEAR POLE OF X(6)X(804)

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

X(53704) lies on the circumcircle and these lines: {2, 805}, {23, 53603}, {25, 22456}, {98, 669}, {99, 237}, {101, 4039}, {110, 385}, {111, 4108}, {112, 419}, {183, 9150}, {401, 3565}, {689, 51862}, {691, 1316}, {729, 3288}, {741, 4369}, {813, 1215}, {842, 47218}, {1296, 11676}, {1501, 2715}, {2502, 33758}, {2696, 37991}, {2698, 5652}, {2709, 5108}, {3124, 36897}, {3231, 26714}, {5976, 34537}, {6037, 36822}, {7735, 9091}, {8033, 36066}, {8840, 9063}, {9066, 26276}, {9202, 22689}, {9203, 22687}, {13518, 39639}, {15107, 25424}, {16609, 29055}, {18858, 34536}, {26243, 53624}, {46777, 46778}

X(53704) = isogonal conjugate of X(34383)
X(53704) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(2679)
X(53704) = isogonal conjugate of the isotomic conjugate of X(53197)
X(53704) = psi-transform of X(43765)
X(53704) = Collings transform of X(35078)
X(53704) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34383}, {75, 21444}
X(53704) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 34383}, {206, 21444}
X(53704) = cevapoint of X(i) and X(j) for these (i,j): {6, 21444}, {25, 47202}
X(53704) = trilinear pole of line {6, 804}
X(53704) = barycentric product X(6)*X(53197)
X(53704) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 34383}, {32, 21444}, {53197, 76}

---

X(53705) = TRILINEAR POLE OF X(6)X(21308)

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

X(53705) lies on the circumcircle and these lines: {30, 11703}, {74, 13391}, {98, 20063}, {110, 20188}, {428, 40118}, {477, 548}, {523, 20189}, {842, 15246}, {1141, 52056}, {1300, 35489}, {2070, 43657}, {3563, 37920}, {7468, 7953}, {9218, 39448}, {14865, 32710}, {14979, 34864}

X(53705) = reflection of X(20189) in the Euler line
X(53705) = Collings transform of X(11063)
X(53705) = cevapoint of X(512) and X(11063)
X(53705) = trilinear pole of line {6, 21308}
X(53705) = barycentric quotient X(2420)/X(13392)

---

X(53706) = TRILINEAR POLE OF X(6)X(21308)

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

X(53706) lies on the circumcircle and these lines: {74, 351}, {98, 9189}, {99, 5642}, {110, 40283}, {111, 14398}, {112, 2502}, {647, 843}, {691, 1495}, {842, 42654}, {1296, 9129}, {1304, 44102}, {9138, 9184}

X(53706) = Parry-circle-inverse of X(74)
X(53706) = Moses-radical-circle-inverse of X(843)
X(53706) = Parry-isodynamic-circle-inverse of X(112)
X(53706) = trilinear pole of X(6)X(21308)

---

X(53707) = TRILINEAR POLE OF X(6)X(3737)

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

X(53707) lies on the circumcircle and these lines: {1, 43076}, {21, 101}, {27, 108}, {28, 26705}, {56, 26845}, {58, 6577}, {74, 7442}, {81, 109}, {98, 7445}, {99, 2975}, {100, 333}, {110, 1621}, {112, 270}, {934, 1434}, {1290, 5196}, {1292, 4229}, {1305, 16049}, {1325, 2690}, {1444, 43349}, {2073, 2766}, {2222, 24624}, {2363, 8687}, {3286, 13576}, {4219, 30250}, {4221, 44876}, {4228, 9057}, {5253, 27189}, {5331, 32693}, {6011, 7411}, {7192, 53297}, {7431, 26706}, {7465, 9070}, {7474, 9058}, {26700, 52393}, {28624, 39673}, {29055, 40432}, {37960, 53189}

X(53707) = isogonal conjugate of X(20718)
X(53707) = Collings transform of X(17761)
X(53707) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20718}, {10, 20470}, {37, 20367}, {42, 20347}, {213, 20448}, {656, 4250}, {1826, 20744}, {4557, 20520}, {13576, 39046}
X(53707) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 20718}, {6626, 20448}, {40589, 20367}, {40592, 20347}, {40596, 4250}
X(53707) = cevapoint of X(1) and X(3286)
X(53707) = trilinear pole of line {6, 3737}
X(53707) = barycentric product X(3737)*X(53644)
X(53707) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 20718}, {58, 20367}, {81, 20347}, {86, 20448}, {112, 4250}, {1019, 20520}, {1333, 20470}, {1437, 20744}

---

X(53708) = TRILINEAR POLE OF X(6)X(1987)

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

X(53708) lies on the circumcircle and these lines: {4, 38974}, {24, 2698}, {53, 32439}, {74, 1987}, {98, 232}, {99, 39062}, {107, 647}, {110, 32320}, {112, 39201}, {403, 43654}, {477, 33885}, {933, 23964}, {935, 47214}, {1294, 35474}, {1297, 14941}, {1298, 19189}, {1300, 41368}, {1303, 1625}, {1495, 26717}, {1956, 26702}, {1972, 2373}, {2409, 6037}, {2491, 20031}, {2697, 15355}, {2710, 40804}, {2770, 47223}, {5897, 47620}, {5999, 34168}, {15412, 34538}, {18338, 39575}, {22456, 53173}, {32230, 42293}

X(53708) = polar-circle-inverse of X(38974)
X(53708) = Moses-radical-circle-inverse of X(107)
X(53708) = isogonal conjugate of the isotomic conjugate of X(53205)
X(53708) = X(i)-isoconjugate of X(j) for these (i,j): {63, 6130}, {401, 656}, {525, 1955}, {810, 44137}, {822, 16089}, {1971, 14208}, {24018, 41204}, {36036, 38974}
X(53708) = X(i)-Dao conjugate of X(j) for these (i,j): {2679, 38974}, {3162, 6130}, {39062, 44137}, {40596, 401}
X(53708) = cevapoint of X(i) and X(j) for these (i,j): {25, 2491}, {232, 647}, {6747, 16230}
X(53708) = trilinear pole of line {6, 1987}
X(53708) = barycentric product X(i)*X(j) for these {i,j}: {6, 53205}, {51, 41208}, {107, 14941}, {112, 1972}, {162, 1956}, {216, 41210}, {648, 1987}, {685, 40804}, {1298, 35360}, {6528, 52177}, {44770, 51960}
X(53708) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 6130}, {107, 16089}, {112, 401}, {648, 44137}, {1956, 14208}, {1972, 3267}, {1987, 525}, {2491, 38974}, {14941, 3265}, {32542, 24284}, {32676, 1955}, {32696, 32545}, {32713, 41204}, {40804, 6333}, {41208, 34384}, {41210, 276}, {52177, 520}, {52604, 32428}, {53175, 2972}, {53205, 76}

   Midpoints of points on the circumcircle: X(53709)-X(53763)  

Contributed by Clark Kimberling and Peter Moses, May 10, 2023.

The points all lie inside the circumcircle, Γ, so that, they are "inner points" as defined in the preamble just before X(53621). Conjecture: none of these midpoints is "strictly inner"; i.e., inside triangle ABC for all A,B,C.

Suppose that P is a fixed point on Γ and that X is a variable point on Γ. The locus of the midpoint of P and X is a circle, Γ(P,X). Let O(P,X) be the center of Γ(P,X). For example, O(X(74),X) passes through X(i) for these i: 12041, 12042, 33813, 33814, 38599, 38600, 38601, 38602, 38607, 1511, 14650, 38608, 38609. If P is not fixed but goes around the circle, then the locus of the centers of the circles has center O and radius (1/2)*circumradius. This circle, with perspector X(3431), passes through X(i) for these i: 1511, 12041,12042, 14650, 33813, 33814, 35231, 35232, 38599, 38600, 38601, 38602, 38603, 38604, 38605, 38606, 38607, 38608, 38609, 38610, 38611, 38612, 38613, 38614, 38615, 38616, 38617, 38618, 38619, 38620, 38621, 38622, 38623, 38624, 38625, 49119,

X(53709) = MIDPOINT OF X(74) AND X(98)

Barycentrics    2*a^14 - 4*a^12*b^2 + 7*a^8*b^6 - 11*a^6*b^8 + 9*a^4*b^10 - 3*a^2*b^12 - 4*a^12*c^2 + 12*a^10*b^2*c^2 - 11*a^8*b^4*c^2 + 12*a^6*b^6*c^2 - 12*a^4*b^8*c^2 + 4*a^2*b^10*c^2 - b^12*c^2 - 11*a^8*b^2*c^4 + 3*a^4*b^6*c^4 + a^2*b^8*c^4 + 3*b^10*c^4 + 7*a^8*c^6 + 12*a^6*b^2*c^6 + 3*a^4*b^4*c^6 - 4*a^2*b^6*c^6 - 2*b^8*c^6 - 11*a^6*c^8 - 12*a^4*b^2*c^8 + a^2*b^4*c^8 - 2*b^6*c^8 + 9*a^4*c^10 + 4*a^2*b^2*c^10 + 3*b^4*c^10 - 3*a^2*c^12 - b^2*c^12 : :
X(53709) = X[10991] + 2 X[20417], 3 X[14830] + X[15545], 3 X[14850] - X[52090], X[15545] - 3 X[20126], 3 X[74] + X[22265], 3 X[98] - X[22265], X[99] - 3 X[15055], X[110] - 3 X[34473], 3 X[5465] - 4 X[33511], 3 X[6055] - 2 X[33511], 2 X[620] - 3 X[38727], 3 X[5622] - X[10753], 2 X[5972] - 3 X[38737], X[6033] - 3 X[15061], X[6321] - 3 X[14849],and many others

X(53709) lies on these lines: {3, 67}, {4, 15359}, {30, 15535}, {74, 98}, {99, 15055}, {110, 34473}, {113, 6036}, {114, 6699}, {115, 2777}, {125, 1316}, {187, 34369}, {265, 38741}, {511, 46633}, {512, 36166}, {541, 5465}, {620, 38727}, {842, 12073}, {1499, 47502}, {1503, 36180}, {2781, 31850}, {2782, 12041}, {3111, 5663}, {4226, 9140}, {5118, 11579}, {5622, 10753}, {5972, 38737}, {6033, 15061}, {6321, 14849}, {6722, 36518}, {6723, 36519}, {7417, 34290}, {7687, 39838}, {7728, 38224}, {9003, 9142}, {9181, 46981}, {9862, 11005}, {10053, 10081}, {10065, 10069}, {10620, 18332}, {10721, 14639}, {10722, 14644}, {10990, 11623}, {11006, 11177}, {11676, 51943}, {12121, 38742}, {12188, 15041}, {12244, 14651}, {12308, 38634}, {14643, 38739}, {15021, 38664}, {15054, 15342}, {15463, 39834}, {15561, 38728}, {16111, 23698}, {16163, 38747}, {16181, 41023}, {16182, 41022}, {17702, 38749}, {19055, 19060}, {19056, 19059}, {19457, 39841}, {20304, 22505}, {22515, 34584}, {23514, 46686}, {31854, 51522}, {32110, 46632}, {32274, 36156}, {32607, 39860}, {33512, 38748}, {37853, 38738}, {38729, 38745}, {38730, 38788}, {38740, 38791}, {39835, 46430}, {46272, 46301}, {49212, 49217}, {49213, 49216}

X(53709) = midpoint of X(i) and X(j) for these {i,j}: {74, 98}, {265, 38741}, {6321, 20127}, {9862, 11005}, {10620, 18332}, {10990, 16278}, {10991, 15357}, {11006, 11177}, {14830, 20126}, {15054, 15342}
X(53709) = reflection of X(i) in X(j) for these {i,j}: {4, 15359}, {113, 6036}, {114, 6699}, {5465, 6055}, {9181, 46981}, {15357, 20417}, {16163, 38747}, {16278, 11623}, {22505, 20304}, {38738, 37853}, {39838, 7687}
X(53709) = circumcircle-inverse of X(53246)
X(53709) = {X(14849),X(20127)}-harmonic conjugate of X(6321)

---

X(53710) = MIDPOINT OF X(74) AND X(99)

Barycentrics    2*a^14 - 6*a^12*b^2 + 4*a^10*b^4 + 7*a^8*b^6 - 15*a^6*b^8 + 11*a^4*b^10 - 3*a^2*b^12 - 6*a^12*c^2 + 20*a^10*b^2*c^2 - 23*a^8*b^4*c^2 + 18*a^6*b^6*c^2 - 10*a^4*b^8*c^2 + b^12*c^2 + 4*a^10*c^4 - 23*a^8*b^2*c^4 + 12*a^6*b^4*c^4 - 3*a^4*b^6*c^4 + 11*a^2*b^8*c^4 - 3*b^10*c^4 + 7*a^8*c^6 + 18*a^6*b^2*c^6 - 3*a^4*b^4*c^6 - 16*a^2*b^6*c^6 + 2*b^8*c^6 - 15*a^6*c^8 - 10*a^4*b^2*c^8 + 11*a^2*b^4*c^8 + 2*b^6*c^8 + 11*a^4*c^10 - 3*b^4*c^10 - 3*a^2*c^12 + b^2*c^12 : :
X(53710) = 3 X[3] - X[18332], 4 X[3] - X[31854], 4 X[18332] - 3 X[31854], 3 X[376] + X[18331], 3 X[11006] - X[18331], X[98] - 3 X[15055], X[110] - 3 X[21166], 3 X[2482] - 2 X[33512], 3 X[3524] - X[9144], 3 X[5182] - X[10752], 3 X[5622] - X[10754], 2 X[5972] - 3 X[38748], X[6033] - 3 X[14850], 3 X[14850] + X[20127], 2 X[6036] - 3 X[38727], and many others

X(53710) lies on these lines: {3, 690}, {20, 11005}, {69, 74}, {98, 15055}, {110, 21166}, {113, 620}, {114, 2777}, {115, 6699}, {125, 23698}, {265, 38730}, {511, 7472}, {512, 46634}, {525, 46981}, {541, 2482}, {549, 5465}, {1499, 46987}, {2682, 16760}, {2781, 5026}, {2782, 12041}, {2794, 16111}, {2799, 6795}, {3028, 15452}, {3524, 9144}, {3906, 46633}, {5108, 32121}, {5182, 10752}, {5622, 10754}, {5663, 33813}, {5972, 38748}, {6033, 14850}, {6036, 16278}, {6055, 48982}, {6321, 15061}, {6721, 36518}, {6723, 23514}, {7687, 39809}, {7728, 15561}, {7782, 38520}, {9140, 12117}, {9880, 41254}, {10065, 10089}, {10081, 10086}, {10706, 41134}, {10723, 14644}, {10992, 20417}, {12121, 15545}, {12192, 39652}, {12308, 38635}, {12900, 31274}, {13188, 15041}, {14639, 15059}, {14643, 38750}, {14677, 51872}, {14915, 47326}, {15021, 23235}, {15035, 15342}, {15357, 17702}, {15463, 39805}, {16003, 50711}, {19059, 19109}, {19060, 19108}, {19457, 39812}, {20304, 22515}, {22265, 34473}, {22505, 34584}, {32607, 39831}, {33511, 38737}, {36519, 46686}, {38224, 38728}, {38702, 53379}, {38729, 38734}, {38741, 38788}, {38751, 38791}, {39806, 46430}, {49216, 49267}, {49217, 49266}

X(53710) = midpoint of X(i) and X(j) for these {i,j}: {20, 11005}, {74, 99}, {265, 38730}, {376, 11006}, {6033, 20127}, {9140, 12117}, {12121, 15545}, {14677, 51872}, {15357, 38738}
X(53710) = reflection of X(i) in X(j) for these {i,j}: {113, 620}, {115, 6699}, {2682, 16760}, {5465, 549}, {6321, 15359}, {9880, 45311}, {16163, 38736}, {16278, 6036}, {22515, 20304}, {38749, 37853}, {39809, 7687}
X(53710) = circumcircle-inverse of X(53247)
X(53710) = crossdifference of every pair of points on line {2493, 14398}
X(53710) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6321, 15061, 15359}, {14850, 20127, 6033}, {15545, 38731, 12121}, {16278, 38727, 6036}, {51898, 51899, 36875}

---

X(53711) = MIDPOINT OF X(74) AND X(100)

Barycentrics    a*(2*a^12 - 2*a^11*b - 5*a^10*b^2 + 5*a^9*b^3 + 10*a^6*b^6 - 10*a^5*b^7 - 10*a^4*b^8 + 10*a^3*b^9 + 3*a^2*b^10 - 3*a*b^11 - 2*a^11*c + 4*a^10*b*c + 3*a^9*b^2*c - 6*a^8*b^3*c + 2*a^7*b^4*c - 7*a^6*b^5*c - 4*a^5*b^6*c + 17*a^4*b^7*c - 9*a^2*b^9*c + a*b^10*c + b^11*c - 5*a^10*c^2 + 3*a^9*b*c^2 + 14*a^8*b^2*c^2 - 12*a^7*b^3*c^2 - 13*a^6*b^4*c^2 + 19*a^5*b^5*c^2 + 4*a^4*b^6*c^2 - 14*a^3*b^7*c^2 + 4*a*b^9*c^2 + 5*a^9*c^3 - 6*a^8*b*c^3 - 12*a^7*b^2*c^3 + 26*a^6*b^3*c^3 - a^5*b^4*c^3 - 19*a^4*b^5*c^3 + 6*a^3*b^6*c^3 + 2*a^2*b^7*c^3 + 2*a*b^8*c^3 - 3*b^9*c^3 + 2*a^7*b*c^4 - 13*a^6*b^2*c^4 - a^5*b^3*c^4 + 12*a^4*b^4*c^4 - 2*a^3*b^5*c^4 - 3*a^2*b^6*c^4 + 5*a*b^7*c^4 - 7*a^6*b*c^5 + 19*a^5*b^2*c^5 - 19*a^4*b^3*c^5 - 2*a^3*b^4*c^5 + 14*a^2*b^5*c^5 - 9*a*b^6*c^5 + 2*b^7*c^5 + 10*a^6*c^6 - 4*a^5*b*c^6 + 4*a^4*b^2*c^6 + 6*a^3*b^3*c^6 - 3*a^2*b^4*c^6 - 9*a*b^5*c^6 - 10*a^5*c^7 + 17*a^4*b*c^7 - 14*a^3*b^2*c^7 + 2*a^2*b^3*c^7 + 5*a*b^4*c^7 + 2*b^5*c^7 - 10*a^4*c^8 + 2*a*b^3*c^8 + 10*a^3*c^9 - 9*a^2*b*c^9 + 4*a*b^2*c^9 - 3*b^3*c^9 + 3*a^2*c^10 + a*b*c^10 - 3*a*c^11 + b*c^11) : :
X(53711) = X[104] - 3 X[15055], X[110] - 3 X[34474], 3 X[5622] - X[10755], 2 X[5972] - 3 X[38760], 2 X[6713] - 3 X[38727], 4 X[6723] - 3 X[23513], X[7728] - 3 X[38752], X[9904] + 3 X[15015], X[10724] - 3 X[14644], X[10738] - 3 X[15061], X[10993] + 2 X[20417], 2 X[11723] - 3 X[34123], X[12308] - 9 X[38636], X[12331] + 3 X[15041], 4 X[12900] - 5 X[31235], 3 X[14643] - 5 X[38762], 5 X[15021] + X[38665], 4 X[15088] - 3 X[38141], X[38753] - 3 X[38788], 5 X[38763] - 2 X[38791]

X(53711) lies on these lines: {2, 10767}, {3, 8674}, {11, 6699}, {72, 74}, {104, 15055}, {110, 34474}, {113, 3035}, {119, 2777}, {125, 5840}, {513, 46635}, {517, 31525}, {541, 6174}, {542, 51007}, {952, 12041}, {1385, 31523}, {2775, 46409}, {2781, 51157}, {2802, 11709}, {2829, 16111}, {5622, 10755}, {5663, 33814}, {5972, 38760}, {6713, 38727}, {6723, 23513}, {7728, 38752}, {8702, 46636}, {9904, 15015}, {10065, 10090}, {10081, 10087}, {10724, 14644}, {10738, 15061}, {10742, 20127}, {10778, 13199}, {10993, 20417}, {11698, 14677}, {11723, 34123}, {12119, 13211}, {12308, 38636}, {12331, 15041}, {12900, 31235}, {14643, 38762}, {15021, 38665}, {15088, 38141}, {17702, 24466}, {19059, 19113}, {19060, 19112}, {20304, 22938}, {22799, 34584}, {37853, 38761}, {38753, 38788}, {38763, 38791}, {48714, 49217}, {48715, 49216}

X(53711) = midpoint of X(i) and X(j) for these {i,j}: {74, 100}, {10742, 20127}, {10778, 13199}, {11698, 14677}, {12119, 13211}
X(53711) = reflection of X(i) in X(j) for these {i,j}: {11, 6699}, {113, 3035}, {22938, 20304}, {31523, 1385}, {38761, 37853}
X(53711) = complement of X(10767)
X(53711) = circumcircle-inverse of X(53248)

---

X(53712) = MIDPOINT OF X(74) AND X(101)

Barycentrics    a^2*(2*a^12 - 2*a^11*b - 4*a^10*b^2 + 3*a^9*b^3 + 2*a^7*b^5 + 6*a^6*b^6 - 4*a^5*b^7 - 8*a^4*b^8 + 6*a^2*b^10 + a*b^11 - 2*b^12 - 2*a^11*c + 2*a^10*b*c + 5*a^9*b^2*c - 5*a^8*b^3*c - 10*a^5*b^6*c + 10*a^4*b^7*c + 10*a^3*b^8*c - 10*a^2*b^9*c - 3*a*b^10*c + 3*b^11*c - 4*a^10*c^2 + 5*a^9*b*c^2 + 12*a^8*b^2*c^2 - 12*a^7*b^3*c^2 - 10*a^6*b^4*c^2 - a^5*b^5*c^2 + 11*a^4*b^6*c^2 + 6*a^3*b^7*c^2 - 8*a^2*b^8*c^2 + 2*a*b^9*c^2 - b^10*c^2 + 3*a^9*c^3 - 5*a^8*b*c^3 - 12*a^7*b^2*c^3 + 14*a^6*b^3*c^3 + 19*a^5*b^4*c^3 - 13*a^4*b^5*c^3 - 14*a^3*b^6*c^3 + 4*a^2*b^7*c^3 + 4*a*b^8*c^3 - 10*a^6*b^2*c^4 + 19*a^5*b^3*c^4 - 4*a^4*b^4*c^4 - 2*a^3*b^5*c^4 + 2*a^2*b^6*c^4 - 9*a*b^7*c^4 + 4*b^8*c^4 + 2*a^7*c^5 - a^5*b^2*c^5 - 13*a^4*b^3*c^5 - 2*a^3*b^4*c^5 + 12*a^2*b^5*c^5 + 5*a*b^6*c^5 - 3*b^7*c^5 + 6*a^6*c^6 - 10*a^5*b*c^6 + 11*a^4*b^2*c^6 - 14*a^3*b^3*c^6 + 2*a^2*b^4*c^6 + 5*a*b^5*c^6 - 2*b^6*c^6 - 4*a^5*c^7 + 10*a^4*b*c^7 + 6*a^3*b^2*c^7 + 4*a^2*b^3*c^7 - 9*a*b^4*c^7 - 3*b^5*c^7 - 8*a^4*c^8 + 10*a^3*b*c^8 - 8*a^2*b^2*c^8 + 4*a*b^3*c^8 + 4*b^4*c^8 - 10*a^2*b*c^9 + 2*a*b^2*c^9 + 6*a^2*c^10 - 3*a*b*c^10 - b^2*c^10 + a*c^11 + 3*b*c^11 - 2*c^12) : :
X(53712) = X[103] - 3 X[15055], X[110] - 3 X[38690], 3 X[5622] - X[10756], 2 X[5972] - 3 X[38772], 2 X[6712] - 3 X[38727], X[7728] - 3 X[38764], X[10725] - 3 X[14644], X[10739] - 3 X[15061], 3 X[14643] - 5 X[38774], 5 X[15021] + X[38666], 3 X[15041] + X[38572], 2 X[20417] + X[33520], X[38765] - 3 X[38788], 5 X[38775] - 2 X[38791]

X(53712) lies on these lines: {3, 2774}, {71, 74}, {103, 15055}, {110, 38690}, {113, 6710}, {116, 6699}, {118, 2777}, {2808, 12041}, {2809, 11709}, {5622, 10756}, {5663, 38599}, {5972, 38772}, {6712, 38727}, {7728, 38764}, {10725, 14644}, {10739, 15061}, {10741, 20127}, {14643, 38774}, {15021, 38666}, {15041, 38572}, {20417, 33520}, {37853, 38773}, {38765, 38788}, {38775, 38791}

X(53712) = midpoint of X(i) and X(j) for these {i,j}: {74, 101}, {10741, 20127}
X(53712) = reflection of X(i) in X(j) for these {i,j}: {113, 6710}, {116, 6699}, {38773, 37853}
X(53712) = circumcircle-inverse of X(53249)

---

X(53713) = MIDPOINT OF X(74) AND X(102)

Barycentrics    a^2*(2*a^14 - 2*a^13*b - 4*a^12*b^2 + 7*a^11*b^3 - 6*a^10*b^4 - 5*a^9*b^5 + 20*a^8*b^6 - 10*a^7*b^7 - 10*a^6*b^8 + 20*a^5*b^9 - 12*a^4*b^10 - 13*a^3*b^11 + 14*a^2*b^12 + 3*a*b^13 - 4*b^14 - 2*a^13*c + 6*a^12*b*c - 3*a^11*b^2*c - 11*a^10*b^3*c + 21*a^9*b^4*c - 11*a^8*b^5*c - 14*a^7*b^6*c + 34*a^6*b^7*c - 24*a^5*b^8*c - 16*a^4*b^9*c + 33*a^3*b^10*c - 7*a^2*b^11*c - 11*a*b^12*c + 5*b^13*c - 4*a^12*c^2 - 3*a^11*b*c^2 + 32*a^10*b^2*c^2 - 18*a^9*b^3*c^2 - 36*a^8*b^4*c^2 + 59*a^7*b^5*c^2 - 45*a^6*b^6*c^2 - 39*a^5*b^7*c^2 + 91*a^4*b^8*c^2 - 12*a^3*b^9*c^2 - 39*a^2*b^10*c^2 + 13*a*b^11*c^2 + b^12*c^2 + 7*a^11*c^3 - 11*a^10*b*c^3 - 18*a^9*b^2*c^3 + 56*a^8*b^3*c^3 - 35*a^7*b^4*c^3 - 47*a^6*b^5*c^3 + 97*a^5*b^6*c^3 - 39*a^4*b^7*c^3 - 52*a^3*b^8*c^3 + 50*a^2*b^9*c^3 + a*b^10*c^3 - 9*b^11*c^3 - 6*a^10*c^4 + 21*a^9*b*c^4 - 36*a^8*b^2*c^4 - 35*a^7*b^3*c^4 + 136*a^6*b^4*c^4 - 54*a^5*b^5*c^4 - 83*a^4*b^6*c^4 + 93*a^3*b^7*c^4 - 32*a^2*b^8*c^4 - 25*a*b^9*c^4 + 21*b^10*c^4 - 5*a^9*c^5 - 11*a^8*b*c^5 + 59*a^7*b^2*c^5 - 47*a^6*b^3*c^5 - 54*a^5*b^4*c^5 + 118*a^4*b^5*c^5 - 49*a^3*b^6*c^5 - 43*a^2*b^7*c^5 + 41*a*b^8*c^5 - 9*b^9*c^5 + 20*a^8*c^6 - 14*a^7*b*c^6 - 45*a^6*b^2*c^6 + 97*a^5*b^3*c^6 - 83*a^4*b^4*c^6 - 49*a^3*b^5*c^6 + 114*a^2*b^6*c^6 - 22*a*b^7*c^6 - 18*b^8*c^6 - 10*a^7*c^7 + 34*a^6*b*c^7 - 39*a^5*b^2*c^7 - 39*a^4*b^3*c^7 + 93*a^3*b^4*c^7 - 43*a^2*b^5*c^7 - 22*a*b^6*c^7 + 26*b^7*c^7 - 10*a^6*c^8 - 24*a^5*b*c^8 + 91*a^4*b^2*c^8 - 52*a^3*b^3*c^8 - 32*a^2*b^4*c^8 + 41*a*b^5*c^8 - 18*b^6*c^8 + 20*a^5*c^9 - 16*a^4*b*c^9 - 12*a^3*b^2*c^9 + 50*a^2*b^3*c^9 - 25*a*b^4*c^9 - 9*b^5*c^9 - 12*a^4*c^10 + 33*a^3*b*c^10 - 39*a^2*b^2*c^10 + a*b^3*c^10 + 21*b^4*c^10 - 13*a^3*c^11 - 7*a^2*b*c^11 + 13*a*b^2*c^11 - 9*b^3*c^11 + 14*a^2*c^12 - 11*a*b*c^12 + b^2*c^12 + 3*a*c^13 + 5*b*c^13 - 4*c^14) : :
X(53713) = X[109] - 3 X[15055], X[110] - 3 X[38691], 3 X[5622] - X[10757], 2 X[5972] - 3 X[38784], 2 X[6718] - 3 X[38727], X[7728] - 3 X[38776], X[10726] - 3 X[14644], X[10740] - 3 X[15061], 3 X[14643] - 5 X[38786], 5 X[15021] + X[38667], 3 X[15041] + X[38573], X[38777] - 3 X[38788], 5 X[38787] - 2 X[38791]

X(53713) lies on these lines: {3, 2779}, {74, 102}, {109, 15055}, {110, 38691}, {113, 6711}, {117, 6699}, {124, 2777}, {2817, 11709}, {2818, 12041}, {5622, 10757}, {5663, 38600}, {5972, 38784}, {6718, 38727}, {7728, 38776}, {10726, 14644}, {10740, 15061}, {10747, 20127}, {14643, 38786}, {15021, 38667}, {15041, 38573}, {37853, 38785}, {38777, 38788}, {38787, 38791}

X(53713) = midpoint of X(i) and X(j) for these {i,j}: {74, 102}, {10747, 20127}
X(53713) = reflection of X(i) in X(j) for these {i,j}: {113, 6711}, {117, 6699}, {38785, 37853}
X(53713) = circumcircle-inverse of X(53250)

---

X(53714) = MIDPOINT OF X(74) AND X(103)

Barycentrics    a^2*(2*a^12 - 2*a^11*b - 2*a^10*b^2 - a^9*b^3 - 2*a^8*b^4 + 10*a^7*b^5 + 2*a^6*b^6 - 4*a^5*b^7 - 4*a^4*b^8 - 8*a^3*b^9 + 8*a^2*b^10 + 5*a*b^11 - 4*b^12 - 2*a^11*c + 2*a^10*b*c + 5*a^9*b^2*c - 5*a^8*b^3*c - 10*a^5*b^6*c + 10*a^4*b^7*c + 10*a^3*b^8*c - 10*a^2*b^9*c - 3*a*b^10*c + 3*b^11*c - 2*a^10*c^2 + 5*a^9*b*c^2 + 12*a^8*b^2*c^2 - 12*a^7*b^3*c^2 - 12*a^6*b^4*c^2 - 13*a^5*b^5*c^2 + 11*a^4*b^6*c^2 + 26*a^3*b^7*c^2 - 8*a^2*b^8*c^2 - 6*a*b^9*c^2 - b^10*c^2 - a^9*c^3 - 5*a^8*b*c^3 - 12*a^7*b^2*c^3 + 22*a^6*b^3*c^3 + 27*a^5*b^4*c^3 - 21*a^4*b^5*c^3 - 14*a^3*b^6*c^3 - 4*a^2*b^7*c^3 + 8*b^9*c^3 - 2*a^8*c^4 - 12*a^6*b^2*c^4 + 27*a^5*b^3*c^4 + 8*a^4*b^4*c^4 - 14*a^3*b^5*c^4 - 4*a^2*b^6*c^4 - 9*a*b^7*c^4 + 6*b^8*c^4 + 10*a^7*c^5 - 13*a^5*b^2*c^5 - 21*a^4*b^3*c^5 - 14*a^3*b^4*c^5 + 36*a^2*b^5*c^5 + 13*a*b^6*c^5 - 11*b^7*c^5 + 2*a^6*c^6 - 10*a^5*b*c^6 + 11*a^4*b^2*c^6 - 14*a^3*b^3*c^6 - 4*a^2*b^4*c^6 + 13*a*b^5*c^6 - 2*b^6*c^6 - 4*a^5*c^7 + 10*a^4*b*c^7 + 26*a^3*b^2*c^7 - 4*a^2*b^3*c^7 - 9*a*b^4*c^7 - 11*b^5*c^7 - 4*a^4*c^8 + 10*a^3*b*c^8 - 8*a^2*b^2*c^8 + 6*b^4*c^8 - 8*a^3*c^9 - 10*a^2*b*c^9 - 6*a*b^2*c^9 + 8*b^3*c^9 + 8*a^2*c^10 - 3*a*b*c^10 - b^2*c^10 + 5*a*c^11 + 3*b*c^11 - 4*c^12) : :
X(53714) = X[101] - 3 X[15055], X[110] - 3 X[38692], 3 X[5622] - X[10758], 2 X[6710] - 3 X[38727], X[10727] - 3 X[14644], X[10741] - 3 X[15061], X[12121] - 3 X[38766], 5 X[15021] + X[38668], 3 X[15041] + X[38574], 2 X[20417] + X[33521], 5 X[38728] - 3 X[38764], 5 X[38729] - 2 X[38769]

X(53714) lies on these lines: {3, 2772}, {74, 103}, {101, 15055}, {110, 38692}, {113, 6712}, {116, 2777}, {118, 6699}, {265, 38765}, {2808, 12041}, {5622, 10758}, {5663, 38601}, {6710, 38727}, {10727, 14644}, {10739, 20127}, {10741, 15061}, {12121, 38766}, {15021, 38668}, {15041, 38574}, {16163, 38771}, {17702, 38773}, {20417, 33521}, {38728, 38764}, {38729, 38769}

X(53714) = midpoint of X(i) and X(j) for these {i,j}: {74, 103}, {265, 38765}, {10739, 20127}
X(53714) = reflection of X(i) in X(j) for these {i,j}: {113, 6712}, {118, 6699}, {16163, 38771}
X(53714) = circumcircle-inverse of X(53251)

---

X(53715) = MIDPOINT OF X(74) AND X(104)

Barycentrics    a*(2*a^12 - 2*a^11*b - 5*a^10*b^2 + 5*a^9*b^3 + 10*a^6*b^6 - 10*a^5*b^7 - 10*a^4*b^8 + 10*a^3*b^9 + 3*a^2*b^10 - 3*a*b^11 - 2*a^11*c + 8*a^10*b*c - a^9*b^2*c - 12*a^8*b^3*c + 10*a^7*b^4*c - 11*a^6*b^5*c - 4*a^5*b^6*c + 25*a^4*b^7*c - 8*a^3*b^8*c - 9*a^2*b^9*c + 5*a*b^10*c - b^11*c - 5*a^10*c^2 - a^9*b*c^2 + 22*a^8*b^2*c^2 - 12*a^7*b^3*c^2 - 21*a^6*b^4*c^2 + 27*a^5*b^5*c^2 - 4*a^4*b^6*c^2 - 14*a^3*b^7*c^2 + 8*a^2*b^8*c^2 + 5*a^9*c^3 - 12*a^8*b*c^3 - 12*a^7*b^2*c^3 + 46*a^6*b^3*c^3 - 13*a^5*b^4*c^3 - 29*a^4*b^5*c^3 + 26*a^3*b^6*c^3 - 8*a^2*b^7*c^3 - 6*a*b^8*c^3 + 3*b^9*c^3 + 10*a^7*b*c^4 - 21*a^6*b^2*c^4 - 13*a^5*b^3*c^4 + 36*a^4*b^4*c^4 - 14*a^3*b^5*c^4 - 11*a^2*b^6*c^4 + 13*a*b^7*c^4 - 11*a^6*b*c^5 + 27*a^5*b^2*c^5 - 29*a^4*b^3*c^5 - 14*a^3*b^4*c^5 + 34*a^2*b^5*c^5 - 9*a*b^6*c^5 - 2*b^7*c^5 + 10*a^6*c^6 - 4*a^5*b*c^6 - 4*a^4*b^2*c^6 + 26*a^3*b^3*c^6 - 11*a^2*b^4*c^6 - 9*a*b^5*c^6 - 10*a^5*c^7 + 25*a^4*b*c^7 - 14*a^3*b^2*c^7 - 8*a^2*b^3*c^7 + 13*a*b^4*c^7 - 2*b^5*c^7 - 10*a^4*c^8 - 8*a^3*b*c^8 + 8*a^2*b^2*c^8 - 6*a*b^3*c^8 + 10*a^3*c^9 - 9*a^2*b*c^9 + 3*b^3*c^9 + 3*a^2*c^10 + 5*a*b*c^10 - 3*a*c^11 - b*c^11) : :
X(53715) = X[100] - 3 X[15055], X[110] - 3 X[38693], 2 X[3035] - 3 X[38727], 3 X[5622] - X[10759], 2 X[5972] - 3 X[21154], 4 X[6667] - 3 X[36518], X[10728] - 3 X[14644], X[10742] - 3 X[15061], X[10990] + 2 X[20418], 2 X[11723] - 3 X[38032], X[12121] - 3 X[38754], X[12308] - 9 X[38637], X[12773] + 3 X[15041], 5 X[15021] + X[38669], 3 X[23513] - 2 X[46686], 5 X[38728] - 3 X[38752], 5 X[38729] - 2 X[38757]

X(53715) lies on these lines: {3, 191}, {11, 2777}, {20, 10778}, {74, 104}, {100, 15055}, {110, 38693}, {113, 6713}, {119, 6699}, {125, 2829}, {265, 38753}, {513, 46618}, {517, 46636}, {952, 12041}, {1484, 14677}, {1537, 11735}, {2778, 31849}, {2779, 13868}, {2800, 11709}, {3028, 52830}, {3035, 38727}, {5622, 10759}, {5663, 38602}, {5840, 16111}, {5972, 21154}, {6667, 36518}, {7687, 52836}, {10058, 10081}, {10065, 10074}, {10728, 14644}, {10738, 20127}, {10742, 15061}, {10767, 12244}, {10990, 20418}, {11715, 31523}, {11723, 38032}, {12121, 38754}, {12308, 38637}, {12773, 15041}, {15021, 38669}, {16163, 38759}, {16164, 17009}, {17702, 38761}, {19059, 19082}, {19060, 19081}, {20304, 22799}, {22938, 34584}, {23513, 46686}, {24466, 37853}, {38728, 38752}, {38729, 38757}, {48700, 49217}, {48701, 49216}

X(53715) = midpoint of X(i) and X(j) for these {i,j}: {20, 10778}, {74, 104}, {265, 38753}, {1484, 14677}, {10738, 20127}, {10767, 12244}
X(53715) = reflection of X(i) in X(j) for these {i,j}: {113, 6713}, {119, 6699}, {1537, 11735}, {16163, 38759}, {16164, 17009}, {22799, 20304}, {24466, 37853}, {31523, 11715}, {31525, 11709}, {52836, 7687}
X(53715) = /circumcircle-inverse of X(53252)

---

X(53716) = MIDPOINT OF X(74) AND X(107)

Barycentrics    (a^2 - b^2 - c^2)*(2*a^20 - 4*a^18*b^2 - 8*a^16*b^4 + 23*a^14*b^6 + 2*a^12*b^8 - 47*a^10*b^10 + 44*a^8*b^12 - 7*a^6*b^14 - 8*a^4*b^16 + 3*a^2*b^18 - 4*a^18*c^2 + 28*a^16*b^2*c^2 - 27*a^14*b^4*c^2 - 82*a^12*b^6*c^2 + 166*a^10*b^8*c^2 - 61*a^8*b^10*c^2 - 63*a^6*b^12*c^2 + 52*a^4*b^14*c^2 - 8*a^2*b^16*c^2 - b^18*c^2 - 8*a^16*c^4 - 27*a^14*b^2*c^4 + 162*a^12*b^4*c^4 - 119*a^10*b^6*c^4 - 164*a^8*b^8*c^4 + 231*a^6*b^10*c^4 - 78*a^4*b^12*c^4 - 5*a^2*b^14*c^4 + 8*b^16*c^4 + 23*a^14*c^6 - 82*a^12*b^2*c^6 - 119*a^10*b^4*c^6 + 362*a^8*b^6*c^6 - 161*a^6*b^8*c^6 - 28*a^4*b^10*c^6 + 33*a^2*b^12*c^6 - 28*b^14*c^6 + 2*a^12*c^8 + 166*a^10*b^2*c^8 - 164*a^8*b^4*c^8 - 161*a^6*b^6*c^8 + 124*a^4*b^8*c^8 - 23*a^2*b^10*c^8 + 56*b^12*c^8 - 47*a^10*c^10 - 61*a^8*b^2*c^10 + 231*a^6*b^4*c^10 - 28*a^4*b^6*c^10 - 23*a^2*b^8*c^10 - 70*b^10*c^10 + 44*a^8*c^12 - 63*a^6*b^2*c^12 - 78*a^4*b^4*c^12 + 33*a^2*b^6*c^12 + 56*b^8*c^12 - 7*a^6*c^14 + 52*a^4*b^2*c^14 - 5*a^2*b^4*c^14 - 28*b^6*c^14 - 8*a^4*c^16 - 8*a^2*b^2*c^16 + 8*b^4*c^16 + 3*a^2*c^18 - b^2*c^18) : :
X(53716) = X[10152] - 3 X[14644], X[10990] + 3 X[14847], 2 X[20417] + X[52057], X[110] - 3 X[23239], X[1294] - 3 X[15055], 3 X[5622] - X[10762], 4 X[6723] - 3 X[36520], X[10745] - 3 X[15061], 5 X[15021] + X[38672], 3 X[15041] + X[38577], 2 X[34842] - 3 X[38727]

X(53716) lies on these lines: {3, 9033}, {4, 74}, {110, 23239}, {113, 6716}, {122, 6699}, {265, 23240}, {1294, 15055}, {3184, 17702}, {5622, 10762}, {5663, 38605}, {6000, 31510}, {6723, 36520}, {10745, 15061}, {15021, 38672}, {15041, 38577}, {16003, 43389}, {16270, 31850}, {18338, 32119}, {20127, 22337}, {20304, 49117}, {21663, 36164}, {34842, 38727}

X(53716) = midpoint of X(i) and X(j) for these {i,j}: {74, 107}, {265, 23240}, {20127, 22337}
X(53716) = reflection of X(i) in X(j) for these {i,j}: {113, 6716}, {122, 6699}, {133, 24930}, {49117, 20304}
X(53716) = circumcircle-inverse of X(53255)
X(53716) = crossdifference of every pair of points on line {1636, 47228}

---

X(53717) = MIDPOINT OF X(74) AND X(109)

Barycentrics    a^2*(2*a^14 - 2*a^13*b - 6*a^12*b^2 + 7*a^11*b^3 + 2*a^10*b^4 - 5*a^9*b^5 + 10*a^8*b^6 - 10*a^7*b^7 - 10*a^6*b^8 + 20*a^5*b^9 - 2*a^4*b^10 - 13*a^3*b^11 + 6*a^2*b^12 + 3*a*b^13 - 2*b^14 - 2*a^13*c + 6*a^12*b*c + a^11*b^2*c - 15*a^10*b^3*c + 9*a^9*b^4*c + a^8*b^5*c - 6*a^7*b^6*c + 26*a^6*b^7*c - 16*a^5*b^8*c - 24*a^4*b^9*c + 21*a^3*b^10*c + 5*a^2*b^11*c - 7*a*b^12*c + b^13*c - 6*a^12*c^2 + a^11*b*c^2 + 28*a^10*b^2*c^2 - 14*a^9*b^3*c^2 - 32*a^8*b^4*c^2 + 35*a^7*b^5*c^2 - 15*a^6*b^6*c^2 - 31*a^5*b^7*c^2 + 47*a^4*b^8*c^2 + 8*a^3*b^9*c^2 - 25*a^2*b^10*c^2 + a*b^11*c^2 + 3*b^12*c^2 + 7*a^11*c^3 - 15*a^10*b*c^3 - 14*a^9*b^2*c^3 + 48*a^8*b^3*c^3 - 15*a^7*b^4*c^3 - 43*a^6*b^5*c^3 + 49*a^5*b^6*c^3 + a^4*b^7*c^3 - 32*a^3*b^8*c^3 + 10*a^2*b^9*c^3 + 5*a*b^10*c^3 - b^11*c^3 + 2*a^10*c^4 + 9*a^9*b*c^4 - 32*a^8*b^2*c^4 - 15*a^7*b^3*c^4 + 80*a^6*b^4*c^4 - 22*a^5*b^5*c^4 - 49*a^4*b^6*c^4 + 33*a^3*b^7*c^4 - 4*a^2*b^8*c^4 - 5*a*b^9*c^4 + 3*b^10*c^4 - 5*a^9*c^5 + a^8*b*c^5 + 35*a^7*b^2*c^5 - 43*a^6*b^3*c^5 - 22*a^5*b^4*c^5 + 54*a^4*b^5*c^5 - 17*a^3*b^6*c^5 - 15*a^2*b^7*c^5 + 17*a*b^8*c^5 - 5*b^9*c^5 + 10*a^8*c^6 - 6*a^7*b*c^6 - 15*a^6*b^2*c^6 + 49*a^5*b^3*c^6 - 49*a^4*b^4*c^6 - 17*a^3*b^5*c^6 + 46*a^2*b^6*c^6 - 14*a*b^7*c^6 - 4*b^8*c^6 - 10*a^7*c^7 + 26*a^6*b*c^7 - 31*a^5*b^2*c^7 + a^4*b^3*c^7 + 33*a^3*b^4*c^7 - 15*a^2*b^5*c^7 - 14*a*b^6*c^7 + 10*b^7*c^7 - 10*a^6*c^8 - 16*a^5*b*c^8 + 47*a^4*b^2*c^8 - 32*a^3*b^3*c^8 - 4*a^2*b^4*c^8 + 17*a*b^5*c^8 - 4*b^6*c^8 + 20*a^5*c^9 - 24*a^4*b*c^9 + 8*a^3*b^2*c^9 + 10*a^2*b^3*c^9 - 5*a*b^4*c^9 - 5*b^5*c^9 - 2*a^4*c^10 + 21*a^3*b*c^10 - 25*a^2*b^2*c^10 + 5*a*b^3*c^10 + 3*b^4*c^10 - 13*a^3*c^11 + 5*a^2*b*c^11 + a*b^2*c^11 - b^3*c^11 + 6*a^2*c^12 - 7*a*b*c^12 + 3*b^2*c^12 + 3*a*c^13 + b*c^13 - 2*c^14) : :
X(53717) = X[102] - 3 X[15055], X[110] - 3 X[38697], 3 X[5622] - X[10764], 2 X[6711] - 3 X[38727], X[10732] - 3 X[14644], X[10747] - 3 X[15061], X[12121] - 3 X[38778], 5 X[15021] + X[38674], 3 X[15041] + X[38579], 5 X[38728] - 3 X[38776], 5 X[38729] - 2 X[38781]

X(53717) lies on these lines: {3, 2773}, {35, 73}, {102, 15055}, {110, 38697}, {113, 6718}, {117, 2777}, {124, 6699}, {265, 38777}, {1795, 10065}, {2800, 11709}, {2818, 12041}, {5622, 10764}, {5663, 38607}, {6711, 38727}, {10732, 14644}, {10740, 20127}, {10747, 15061}, {12121, 38778}, {15021, 38674}, {15041, 38579}, {16163, 38783}, {17702, 38785}, {38728, 38776}, {38729, 38781}

X(53717) = midpoint of X(i) and X(j) for these {i,j}: {74, 109}, {265, 38777}, {10740, 20127}
X(53717) = reflection of X(i) in X(j) for these {i,j}: {113, 6718}, {124, 6699}, {16163, 38783}
X(53717) = circumcircle-inverse of X(53256)

---

X(53718) = MIDPOINT OF X(74) AND X(111)

Barycentrics    a^2*(a^14 - 4*a^12*b^2 + 12*a^8*b^6 - 9*a^6*b^8 - 6*a^4*b^10 + 8*a^2*b^12 - 2*b^14 - 4*a^12*c^2 + 22*a^10*b^2*c^2 - 26*a^8*b^4*c^2 - 29*a^6*b^6*c^2 + 64*a^4*b^8*c^2 - 35*a^2*b^10*c^2 + 8*b^12*c^2 - 26*a^8*b^2*c^4 + 93*a^6*b^4*c^4 - 60*a^4*b^6*c^4 + 4*a^2*b^8*c^4 - 3*b^10*c^4 + 12*a^8*c^6 - 29*a^6*b^2*c^6 - 60*a^4*b^4*c^6 + 46*a^2*b^6*c^6 - 3*b^8*c^6 - 9*a^6*c^8 + 64*a^4*b^2*c^8 + 4*a^2*b^4*c^8 - 3*b^6*c^8 - 6*a^4*c^10 - 35*a^2*b^2*c^10 - 3*b^4*c^10 + 8*a^2*c^12 + 8*b^2*c^12 - 2*c^14) : :
X(53718) = X[110] - 3 X[38698], X[1296] - 3 X[15055], 3 X[5622] - X[10765], 2 X[5972] - 3 X[38804], X[7728] - 3 X[38796], X[10620] + 3 X[52698], X[10734] - 3 X[14644], X[10748] - 3 X[15061], X[10752] - 3 X[36696], X[11258] + 3 X[15041], 3 X[15041] - X[35447], 3 X[14643] - 5 X[38806], 5 X[15021] + X[38675], 3 X[34128] - 2 X[40340], 3 X[38727] - 2 X[40556], 3 X[38788] - X[38797], 2 X[38791] - 5 X[38807]

X(53718) lies on these lines: {3, 2854}, {30, 9179}, {74, 111}, {110, 38698}, {113, 6719}, {125, 23699}, {126, 6699}, {541, 9172}, {1296, 15055}, {1499, 36168}, {2777, 5512}, {2781, 28662}, {5112, 32219}, {5622, 10765}, {5663, 9129}, {5972, 38804}, {7728, 38796}, {9126, 9215}, {10620, 52698}, {10734, 14644}, {10748, 15061}, {10752, 36696}, {11258, 15041}, {12041, 33962}, {14643, 38806}, {14653, 15483}, {14666, 19905}, {15021, 38675}, {16220, 48540}, {20127, 22338}, {34128, 40340}, {37470, 47049}, {37853, 38805}, {38727, 40556}, {38788, 38797}, {38791, 38807}

X(53718) = midpoint of X(i) and X(j) for these {i,j}: {74, 111}, {11258, 35447}, {14666, 20126}, {20127, 22338}
X(53718) = reflection of X(i) in X(j) for these {i,j}: {113, 6719}, {126, 6699}, {9129, 14650}, {38805, 37853}
X(53718) = circumcircle-inverse of X(9142)
X(53718) = {X(11258),X(15041)}-harmonic conjugate of X(35447)

---

X(53719) = MIDPOINT OF X(74) AND X(112)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^16 - 2*a^14*b^2 - a^12*b^4 + 5*a^10*b^6 - 4*a^8*b^8 + 3*a^4*b^12 - 3*a^2*b^14 + b^16 - 2*a^14*c^2 + 8*a^12*b^2*c^2 - 7*a^10*b^4*c^2 + 5*a^6*b^8*c^2 - 11*a^4*b^10*c^2 + 8*a^2*b^12*c^2 - b^14*c^2 - a^12*c^4 - 7*a^10*b^2*c^4 + 9*a^8*b^4*c^4 - 5*a^6*b^6*c^4 + 15*a^4*b^8*c^4 - 6*a^2*b^10*c^4 - 5*b^12*c^4 + 5*a^10*c^6 - 5*a^6*b^4*c^6 - 14*a^4*b^6*c^6 + a^2*b^8*c^6 + 13*b^10*c^6 - 4*a^8*c^8 + 5*a^6*b^2*c^8 + 15*a^4*b^4*c^8 + a^2*b^6*c^8 - 16*b^8*c^8 - 11*a^4*b^2*c^10 - 6*a^2*b^4*c^10 + 13*b^6*c^10 + 3*a^4*c^12 + 8*a^2*b^2*c^12 - 5*b^4*c^12 - 3*a^2*c^14 - b^2*c^14 + c^16) : :
X(53719) = 3 X[5622] - X[10766], X[110] - 3 X[38699], 2 X[1112] - 3 X[16224], X[1297] - 3 X[15055], X[10735] - 3 X[14644], X[10749] - 3 X[15061], X[12308] - 9 X[38639], X[13310] + 3 X[15041], X[13417] - 3 X[16225], X[14900] + 2 X[20417], 5 X[15021] + X[38676], 2 X[34841] - 3 X[38727]

X(53719) lies on the Brocard circle and these lines: {3, 684}, {6, 74}, {110, 38699}, {113, 6720}, {125, 1316}, {127, 6699}, {132, 2777}, {246, 13198}, {525, 46637}, {690, 18338}, {1112, 16224}, {1297, 15055}, {1503, 46619}, {1899, 53132}, {2780, 50381}, {2799, 6795}, {3569, 35901}, {5191, 14649}, {5663, 17974}, {6776, 18331}, {10065, 13312}, {10081, 13311}, {10735, 14644}, {10749, 15061}, {12308, 38639}, {12918, 20127}, {13310, 15041}, {13417, 16225}, {14585, 38520}, {14689, 17702}, {14900, 20417}, {15021, 38676}, {15462, 38551}, {18332, 35912}, {19160, 34584}, {19163, 20304}, {34841, 38727}

X(53719) = midpoint of X(i) and X(j) for these {i,j}: {74, 112}, {12918, 20127}
X(53719) = reflection of X(i) in X(j) for these {i,j}: {113, 6720}, {127, 6699}, {19163, 20304}
X(53719) = circumcircle-inverse of X(9409)
X(53719) = psi-transform of X(1304)
X(53719) = crossdifference of every pair of points on line {6103, 9033}
X(53719) = {X(74),X(5622)}-harmonic conjugate of X(3269)

---

X(53720) = MIDPOINT OF X(98) AND X(100)

Barycentrics    2*a^11 - 2*a^10*b - 5*a^9*b^2 + 5*a^8*b^3 + 6*a^7*b^4 - 6*a^6*b^5 - 5*a^5*b^6 + 5*a^4*b^7 + 2*a^3*b^8 - 2*a^2*b^9 - 2*a^10*c + 4*a^9*b*c + 3*a^8*b^2*c - 6*a^7*b^3*c - 4*a^6*b^4*c + 7*a^5*b^5*c + 3*a^4*b^6*c - 6*a^3*b^7*c + a*b^9*c - 5*a^9*c^2 + 3*a^8*b*c^2 + 2*a^7*b^2*c^2 + 2*a^5*b^4*c^2 - 4*a^4*b^5*c^2 - 4*a^3*b^6*c^2 + 6*a^2*b^7*c^2 + a*b^8*c^2 - b^9*c^2 + 5*a^8*c^3 - 6*a^7*b*c^3 - 2*a^5*b^3*c^3 + 4*a^3*b^5*c^3 + 2*a^2*b^6*c^3 - 6*a*b^7*c^3 + b^8*c^3 + 6*a^7*c^4 - 4*a^6*b*c^4 + 2*a^5*b^2*c^4 + 4*a^3*b^4*c^4 - 6*a^2*b^5*c^4 - a*b^6*c^4 + 3*b^7*c^4 - 6*a^6*c^5 + 7*a^5*b*c^5 - 4*a^4*b^2*c^5 + 4*a^3*b^3*c^5 - 6*a^2*b^4*c^5 + 10*a*b^5*c^5 - 3*b^6*c^5 - 5*a^5*c^6 + 3*a^4*b*c^6 - 4*a^3*b^2*c^6 + 2*a^2*b^3*c^6 - a*b^4*c^6 - 3*b^5*c^6 + 5*a^4*c^7 - 6*a^3*b*c^7 + 6*a^2*b^2*c^7 - 6*a*b^3*c^7 + 3*b^4*c^7 + 2*a^3*c^8 + a*b^2*c^8 + b^3*c^8 - 2*a^2*c^9 + a*b*c^9 - b^2*c^9 : :
X(53720) = X[99] - 3 X[34474], X[104] - 3 X[34473], 2 X[620] - 3 X[38760], X[6033] - 3 X[38752], X[6154] + 4 X[35021], 2 X[6713] - 3 X[38737], 4 X[6721] - 5 X[31235], 4 X[6722] - 3 X[23513], X[9860] + 3 X[15015], X[10724] - 3 X[14639], X[10738] - 3 X[38224], X[10769] - 3 X[14651], X[13199] + 3 X[14651], X[10993] + 2 X[11623], 2 X[11724] - 3 X[34123], X[14217] - 3 X[38220], 4 X[15092] - 3 X[38141], 3 X[15561] - 5 X[38762], 3 X[38742] - X[38753], 2 X[38745] - 5 X[38763]

X(53720) lies on these lines: {2, 10768}, {3, 2787}, {11, 6036}, {98, 100}, {99, 34474}, {104, 34473}, {114, 3035}, {115, 5840}, {119, 2794}, {528, 6055}, {542, 6174}, {620, 38760}, {952, 12042}, {2782, 33814}, {2788, 46409}, {2802, 11710}, {2829, 38749}, {6033, 38752}, {6154, 35021}, {6713, 38737}, {6721, 31235}, {6722, 23513}, {9860, 15015}, {10053, 10090}, {10069, 10087}, {10724, 14639}, {10738, 38224}, {10742, 38741}, {10769, 13199}, {10993, 11623}, {11724, 34123}, {12119, 13178}, {14217, 38220}, {15092, 38141}, {15561, 38762}, {19055, 19113}, {19056, 19112}, {23698, 24466}, {38742, 38753}, {38745, 38763}, {38747, 38761}, {48714, 49213}, {48715, 49212}

X(53720) = midpoint of X(i) and X(j) for these {i,j}: {98, 100}, {10742, 38741}, {10769, 13199}, {12119, 13178}
X(53720) = reflection of X(i) in X(j) for these {i,j}: {11, 6036}, {114, 3035}, {38761, 38747}
X(53720) = complement of X(10768)
X(53720) = circumcircle-inverse of X(53257)
X(53720) = {X(13199),X(14651)}-harmonic conjugate of X(10769)

---

X(53721) = MIDPOINT OF X(98) AND X(101)

Barycentrics    2*a^12 - 2*a^11*b - 4*a^10*b^2 + 3*a^9*b^3 + 6*a^8*b^4 - 4*a^7*b^5 - 5*a^6*b^6 + 3*a^5*b^7 + 2*a^4*b^8 - a^2*b^10 - 2*a^11*c + 2*a^10*b*c + 5*a^9*b^2*c - 5*a^8*b^3*c - 6*a^7*b^4*c + 6*a^6*b^5*c + 5*a^5*b^6*c - 5*a^4*b^7*c - 2*a^3*b^8*c + 2*a^2*b^9*c - 4*a^10*c^2 + 5*a^9*b*c^2 + a^6*b^4*c^2 - 5*a^4*b^6*c^2 + 2*a^3*b^7*c^2 + a^2*b^8*c^2 + a*b^9*c^2 - b^10*c^2 + 3*a^9*c^3 - 5*a^8*b*c^3 + 2*a^6*b^3*c^3 - 4*a^5*b^4*c^3 + 2*a^4*b^5*c^3 + 6*a^3*b^6*c^3 - 4*a^2*b^7*c^3 - a*b^8*c^3 + b^9*c^3 + 6*a^8*c^4 - 6*a^7*b*c^4 + a^6*b^2*c^4 - 4*a^5*b^3*c^4 + 8*a^4*b^4*c^4 - 6*a^3*b^5*c^4 - 3*a*b^7*c^4 + 2*b^8*c^4 - 4*a^7*c^5 + 6*a^6*b*c^5 + 2*a^4*b^3*c^5 - 6*a^3*b^4*c^5 + 4*a^2*b^5*c^5 + 3*a*b^6*c^5 - b^7*c^5 - 5*a^6*c^6 + 5*a^5*b*c^6 - 5*a^4*b^2*c^6 + 6*a^3*b^3*c^6 + 3*a*b^5*c^6 - 2*b^6*c^6 + 3*a^5*c^7 - 5*a^4*b*c^7 + 2*a^3*b^2*c^7 - 4*a^2*b^3*c^7 - 3*a*b^4*c^7 - b^5*c^7 + 2*a^4*c^8 - 2*a^3*b*c^8 + a^2*b^2*c^8 - a*b^3*c^8 + 2*b^4*c^8 + 2*a^2*b*c^9 + a*b^2*c^9 + b^3*c^9 - a^2*c^10 - b^2*c^10 : :
X(53721) = X[99] - 3 X[38690], X[103] - 3 X[34473], 2 X[620] - 3 X[38772], X[6033] - 3 X[38764], 2 X[6712] - 3 X[38737], X[10725] - 3 X[14639], X[10739] - 3 X[38224], 2 X[11623] + X[33520], 3 X[15561] - 5 X[38774], 3 X[38742] - X[38765], 2 X[38745] - 5 X[38775]

X(53721) lies on these lines: {3, 2786}, {10, 98}, {99, 38690}, {103, 34473}, {114, 6710}, {116, 6036}, {118, 2794}, {544, 6055}, {620, 38772}, {2782, 38599}, {2785, 31852}, {2808, 12042}, {2809, 11710}, {6033, 38764}, {6712, 38737}, {10725, 14639}, {10739, 38224}, {10741, 38741}, {11623, 33520}, {15561, 38774}, {38742, 38765}, {38745, 38775}, {38747, 38773}

X(53721) = midpoint of X(i) and X(j) for these {i,j}: {98, 101}, {10741, 38741}
X(53721) = reflection of X(i) in X(j) for these {i,j}: {114, 6710}, {116, 6036}, {38773, 38747}
X(53721) = circumcircle-inverse of X(53258)

---

X(53722) = MIDPOINT OF X(98) AND X(104)

Barycentrics    2*a^11 - 2*a^10*b - 5*a^9*b^2 + 5*a^8*b^3 + 6*a^7*b^4 - 6*a^6*b^5 - 5*a^5*b^6 + 5*a^4*b^7 + 2*a^3*b^8 - 2*a^2*b^9 - 2*a^10*c + 8*a^9*b*c - a^8*b^2*c - 12*a^7*b^3*c + 4*a^6*b^4*c + 11*a^5*b^5*c - 5*a^4*b^6*c - 6*a^3*b^7*c + 4*a^2*b^8*c - a*b^9*c - 5*a^9*c^2 - a^8*b*c^2 + 10*a^7*b^2*c^2 - 6*a^5*b^4*c^2 - 4*a^4*b^5*c^2 + 4*a^3*b^6*c^2 + 2*a^2*b^7*c^2 + a*b^8*c^2 - b^9*c^2 + 5*a^8*c^3 - 12*a^7*b*c^3 + 2*a^5*b^3*c^3 + 4*a^4*b^4*c^3 + 2*a^3*b^5*c^3 - 6*a^2*b^6*c^3 + b^8*c^3 + 6*a^7*c^4 + 4*a^6*b*c^4 - 6*a^5*b^2*c^4 + 4*a^4*b^3*c^4 - 4*a^3*b^4*c^4 + 2*a^2*b^5*c^4 - a*b^6*c^4 + 3*b^7*c^4 - 6*a^6*c^5 + 11*a^5*b*c^5 - 4*a^4*b^2*c^5 + 2*a^3*b^3*c^5 + 2*a^2*b^4*c^5 + 2*a*b^5*c^5 - 3*b^6*c^5 - 5*a^5*c^6 - 5*a^4*b*c^6 + 4*a^3*b^2*c^6 - 6*a^2*b^3*c^6 - a*b^4*c^6 - 3*b^5*c^6 + 5*a^4*c^7 - 6*a^3*b*c^7 + 2*a^2*b^2*c^7 + 3*b^4*c^7 + 2*a^3*c^8 + 4*a^2*b*c^8 + a*b^2*c^8 + b^3*c^8 - 2*a^2*c^9 - a*b*c^9 - b^2*c^9 : :
X(53722) = X[99] - 3 X[38693], X[100] - 3 X[34473], 2 X[620] - 3 X[21154], 2 X[3035] - 3 X[38737], 4 X[6667] - 3 X[36519], X[10728] - 3 X[14639], X[10742] - 3 X[38224], X[10991] + 2 X[20418], 2 X[11724] - 3 X[38032], X[12248] + 3 X[14651], X[34789] - 3 X[38220], X[38730] - 3 X[38754], 5 X[38739] - 3 X[38752], 5 X[38740] - 2 X[38757]

X(53722) lies on these lines: {3, 2783}, {11, 2794}, {20, 10769}, {98, 104}, {99, 38693}, {100, 34473}, {114, 6713}, {115, 2829}, {119, 6036}, {620, 21154}, {952, 12042}, {1537, 11725}, {2782, 38602}, {2800, 11710}, {3035, 38737}, {5840, 38749}, {6321, 38753}, {6667, 36519}, {9862, 10768}, {10053, 10074}, {10058, 10069}, {10728, 14639}, {10738, 38741}, {10742, 38224}, {10991, 20418}, {11724, 38032}, {12248, 14651}, {19055, 19082}, {19056, 19081}, {23698, 38761}, {24466, 38747}, {34789, 38220}, {38730, 38754}, {38738, 38759}, {38739, 38752}, {38740, 38757}, {48700, 49213}, {48701, 49212}

X(53722) = midpoint of X(i) and X(j) for these {i,j}: {20, 10769}, {98, 104}, {6321, 38753}, {9862, 10768}, {10738, 38741}
X(53722) = reflection of X(i) in X(j) for these {i,j}: {114, 6713}, {119, 6036}, {1537, 11725}, {24466, 38747}, {38738, 38759}
X(53722) = circumcircle-inverse of X(53260)

---

X(53723) = MIDPOINT OF X(98) AND X(107)

Barycentrics    a^20 - 3*a^18*b^2 + a^16*b^4 + 7*a^14*b^6 - 14*a^12*b^8 + 14*a^10*b^10 - 7*a^8*b^12 - a^6*b^14 + 3*a^4*b^16 - a^2*b^18 - 3*a^18*c^2 + 12*a^16*b^2*c^2 - 15*a^14*b^4*c^2 + 8*a^12*b^6*c^2 - 6*a^10*b^8*c^2 + 13*a^6*b^12*c^2 - 12*a^4*b^14*c^2 + 3*a^2*b^16*c^2 + a^16*c^4 - 15*a^14*b^2*c^4 + 21*a^12*b^4*c^4 - 9*a^10*b^6*c^4 + 15*a^8*b^8*c^4 - 27*a^6*b^10*c^4 + 21*a^4*b^12*c^4 - 5*a^2*b^14*c^4 - 2*b^16*c^4 + 7*a^14*c^6 + 8*a^12*b^2*c^6 - 9*a^10*b^4*c^6 - 16*a^8*b^6*c^6 + 15*a^6*b^8*c^6 - 24*a^4*b^10*c^6 + 7*a^2*b^12*c^6 + 12*b^14*c^6 - 14*a^12*c^8 - 6*a^10*b^2*c^8 + 15*a^8*b^4*c^8 + 15*a^6*b^6*c^8 + 24*a^4*b^8*c^8 - 4*a^2*b^10*c^8 - 30*b^12*c^8 + 14*a^10*c^10 - 27*a^6*b^4*c^10 - 24*a^4*b^6*c^10 - 4*a^2*b^8*c^10 + 40*b^10*c^10 - 7*a^8*c^12 + 13*a^6*b^2*c^12 + 21*a^4*b^4*c^12 + 7*a^2*b^6*c^12 - 30*b^8*c^12 - a^6*c^14 - 12*a^4*b^2*c^14 - 5*a^2*b^4*c^14 + 12*b^6*c^14 + 3*a^4*c^16 + 3*a^2*b^2*c^16 - 2*b^4*c^16 - a^2*c^18 : :
X(53723) = X[99] - 3 X[23239], X[1294] - 3 X[34473], X[5667] + 3 X[14651], 4 X[6722] - 3 X[36520], X[10152] - 3 X[14639], X[10745] - 3 X[38224], 2 X[11623] + X[52057], 2 X[34842] - 3 X[38737]

X(53723) lies on these lines: {3, 2797}, {25, 98}, {99, 23239}, {114, 6716}, {115, 2777}, {122, 6036}, {133, 2794}, {542, 24930}, {1294, 34473}, {1637, 35901}, {2782, 38605}, {3184, 23698}, {5667, 14651}, {6055, 9530}, {6103, 46942}, {6321, 23240}, {6722, 36520}, {9033, 36207}, {10152, 14639}, {10745, 38224}, {11005, 26869}, {11623, 52057}, {16315, 36166}, {18338, 42731}, {22337, 38741}, {31510, 47147}, {34842, 38737}

X(53723) = midpoint of X(i) and X(j) for these {i,j}: {98, 107}, {6321, 23240}, {22337, 38741}
X(53723) = reflection of X(i) in X(j) for these {i,j}: {114, 6716}, {122, 6036}
X(53723) = circumcircle-inverse of X(6130)
X(53723) = Dao-Moses-Telv circle inverse of X(35901)

---

X(53724) = MIDPOINT OF X(98) AND X(109)

Barycentrics    2*a^14 - 2*a^13*b - 6*a^12*b^2 + 7*a^11*b^3 + 8*a^10*b^4 - 11*a^9*b^5 - 7*a^8*b^6 + 11*a^7*b^7 + 3*a^6*b^8 - 7*a^5*b^9 + a^4*b^10 + 2*a^3*b^11 - a^2*b^12 - 2*a^13*c + 6*a^12*b*c + a^11*b^2*c - 15*a^10*b^3*c + 3*a^9*b^4*c + 19*a^8*b^5*c - 5*a^7*b^6*c - 17*a^6*b^7*c + 7*a^5*b^8*c + 7*a^4*b^9*c - 4*a^3*b^10*c - 6*a^12*c^2 + a^11*b*c^2 + 16*a^10*b^2*c^2 - 2*a^9*b^3*c^2 - 15*a^8*b^4*c^2 - 6*a^7*b^5*c^2 + 13*a^6*b^6*c^2 + 11*a^5*b^7*c^2 - 12*a^4*b^8*c^2 - 5*a^3*b^9*c^2 + 5*a^2*b^10*c^2 + a*b^11*c^2 - b^12*c^2 + 7*a^11*c^3 - 15*a^10*b*c^3 - 2*a^9*b^2*c^3 + 12*a^8*b^3*c^3 + 4*a^7*b^4*c^3 - 9*a^5*b^6*c^3 - 7*a^4*b^7*c^3 + 11*a^3*b^8*c^3 + a^2*b^9*c^3 - 3*a*b^10*c^3 + b^11*c^3 + 8*a^10*c^4 + 3*a^9*b*c^4 - 15*a^8*b^2*c^4 + 4*a^7*b^3*c^4 - 2*a^6*b^4*c^4 - 2*a^5*b^5*c^4 + 7*a^4*b^6*c^4 + 7*a^3*b^7*c^4 - 13*a^2*b^8*c^4 + 3*b^10*c^4 - 11*a^9*c^5 + 19*a^8*b*c^5 - 6*a^7*b^2*c^5 - 2*a^5*b^4*c^5 + 8*a^4*b^5*c^5 - 11*a^3*b^6*c^5 - a^2*b^7*c^5 + 8*a*b^8*c^5 - 4*b^9*c^5 - 7*a^8*c^6 - 5*a^7*b*c^6 + 13*a^6*b^2*c^6 - 9*a^5*b^3*c^6 + 7*a^4*b^4*c^6 - 11*a^3*b^5*c^6 + 18*a^2*b^6*c^6 - 6*a*b^7*c^6 - 2*b^8*c^6 + 11*a^7*c^7 - 17*a^6*b*c^7 + 11*a^5*b^2*c^7 - 7*a^4*b^3*c^7 + 7*a^3*b^4*c^7 - a^2*b^5*c^7 - 6*a*b^6*c^7 + 6*b^7*c^7 + 3*a^6*c^8 + 7*a^5*b*c^8 - 12*a^4*b^2*c^8 + 11*a^3*b^3*c^8 - 13*a^2*b^4*c^8 + 8*a*b^5*c^8 - 2*b^6*c^8 - 7*a^5*c^9 + 7*a^4*b*c^9 - 5*a^3*b^2*c^9 + a^2*b^3*c^9 - 4*b^5*c^9 + a^4*c^10 - 4*a^3*b*c^10 + 5*a^2*b^2*c^10 - 3*a*b^3*c^10 + 3*b^4*c^10 + 2*a^3*c^11 + a*b^2*c^11 + b^3*c^11 - a^2*c^12 - b^2*c^12 : :
X(53724) = X[99] - 3 X[38697], X[102] - 3 X[34473], 2 X[6711] - 3 X[38737], X[10732] - 3 X[14639], X[10747] - 3 X[38224], X[38730] - 3 X[38778], 5 X[38739] - 3 X[38776], 5 X[38740] - 2 X[38781]

X(53724) lies on these lines: {3, 2785}, {98, 109}, {99, 38697}, {102, 34473}, {114, 6718}, {117, 2794}, {124, 6036}, {1795, 10053}, {2782, 38607}, {2800, 11710}, {2818, 12042}, {6321, 38777}, {6711, 38737}, {10732, 14639}, {10740, 38741}, {10747, 38224}, {23698, 38785}, {38730, 38778}, {38738, 38783}, {38739, 38776}, {38740, 38781}

X(53724) = midpoint of X(i) and X(j) for these {i,j}: {98, 109}, {6321, 38777}, {10740, 38741}
X(53724) = reflection of X(i) in X(j) for these {i,j}: {114, 6718}, {124, 6036}, {38738, 38783}
X(53724) = circumcircle-inverse of X(53262)

---

X(53725) = MIDPOINT OF X(98) AND X(110)

Barycentrics    2*a^14 - 6*a^12*b^2 + 8*a^10*b^4 - 7*a^8*b^6 + 3*a^6*b^8 + a^4*b^10 - a^2*b^12 - 6*a^12*c^2 + 12*a^10*b^2*c^2 - 9*a^8*b^4*c^2 + 6*a^6*b^6*c^2 - 6*a^4*b^8*c^2 + 4*a^2*b^10*c^2 - b^12*c^2 + 8*a^10*c^4 - 9*a^8*b^2*c^4 + 3*a^4*b^6*c^4 - 7*a^2*b^8*c^4 + 3*b^10*c^4 - 7*a^8*c^6 + 6*a^6*b^2*c^6 + 3*a^4*b^4*c^6 + 8*a^2*b^6*c^6 - 2*b^8*c^6 + 3*a^6*c^8 - 6*a^4*b^2*c^8 - 7*a^2*b^4*c^8 - 2*b^6*c^8 + a^4*c^10 + 4*a^2*b^2*c^10 + 3*b^4*c^10 - a^2*c^12 - b^2*c^12 : :
X(53725) = X[12177] - 3 X[15462], X[24981] + 4 X[35021], 2 X[3] + X[31854], X[74] - 3 X[34473], X[15342] + 3 X[34473], X[99] - 3 X[15035], 3 X[15035] + X[22265], X[265] - 3 X[38224], 2 X[15359] - 3 X[38224], 2 X[620] - 3 X[38793], 5 X[631] - X[18331], 3 X[3524] - X[11006], X[6033] - 3 X[14643], 2 X[6699] - 3 X[38737], X[15357] - 3 X[38737], and many others

X(53725 lies on these lines: {2, 98}, {3, 690}, {30, 5465}, {74, 15342}, {99, 15035}, {113, 2794}, {115, 17702}, {265, 15359}, {376, 9144}, {511, 14999}, {512, 46633}, {524, 47570}, {525, 46987}, {543, 11656}, {620, 38793}, {631, 18331}, {842, 53379}, {1495, 7471}, {1499, 46981}, {1503, 36170}, {1511, 2782}, {1551, 11645}, {2777, 38749}, {2790, 20772}, {2793, 6795}, {2799, 32119}, {3111, 5663}, {3154, 11064}, {3524, 11006}, {3906, 46634}, {5092, 9828}, {5467, 46129}, {5655, 14830}, {5969, 33851}, {6033, 14643}, {6321, 12121}, {6699, 15357}, {6722, 23515}, {7417, 46131}, {7687, 23514}, {7728, 38741}, {8980, 46688}, {9003, 40879}, {9033, 36207}, {9147, 14932}, {9149, 12584}, {9517, 31848}, {10053, 10091}, {10069, 10088}, {10733, 14639}, {10753, 52699}, {10991, 16534}, {11623, 30714}, {11646, 32233}, {12188, 32609}, {12383, 14651}, {12900, 36519}, {13188, 15040}, {13967, 46689}, {14061, 14644}, {14223, 53345}, {14559, 46127}, {14849, 23236}, {14850, 38750}, {14981, 33512}, {15020, 23235}, {15034, 38664}, {15051, 21166}, {15061, 15545}, {15535, 32423}, {15561, 38794}, {15928, 35259}, {16111, 38747}, {16163, 16278}, {16165, 36178}, {16222, 39835}, {16223, 39846}, {16760, 51429}, {17511, 51360}, {19055, 19111}, {19056, 19110}, {20127, 38742}, {20304, 34127}, {22109, 39825}, {29012, 36173}, {33928, 34810}, {34156, 34157}, {36253, 38740}, {38723, 38730}, {38726, 38738}, {38745, 38795}, {38748, 48378}, {39838, 46686}, {40107, 52179}, {49212, 49269}, {49213, 49268}

X(53725) = midpoint of X(i) and X(j) for these {i,j}: {3, 18332}, {74, 15342}, {98, 110}, {99, 22265}, {376, 9144}, {842, 53379}, {5655, 14830}, {6321, 12121}, {7728, 38741}, {11646, 32233}, {14999, 36166}, {16163, 16278}
X(53725) = reflection of X(i) in X(j) for these {i,j}: {114, 5972}, {115, 33511}, {125, 6036}, {265, 15359}, {14981, 33512}, {15357, 6699}, {16111, 38747}, {31854, 18332}, {38738, 38726}, {39838, 46686}, {51429, 16760}
X(53725) = complement of X(11005)
X(53725) = reflection of X(31854) in the Fermat line
X(53725) = circumcircle-inverse of X(53263)
X(53725) = Brocard-circle-inverse of X(47049)
X(53725) = psi-transform of X(4226)
X(53725) = crossdifference of every pair of points on line {2493, 3569}
X(53725) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {265, 38224, 15359}, {13414, 13415, 47049}, {15035, 22265, 99}, {15342, 34473, 74}, {15357, 38737, 6699}, {15545, 38739, 15061}, {47365, 47366, 52772}

---

X(53726) = MIDPOINT OF X(98) AND X(111)

Barycentrics    2*a^14 - 8*a^12*b^2 + 6*a^10*b^4 - a^8*b^6 - 5*a^6*b^8 + 9*a^4*b^10 - 3*a^2*b^12 - 8*a^12*c^2 + 32*a^10*b^2*c^2 - 27*a^8*b^4*c^2 + 36*a^6*b^6*c^2 - 34*a^4*b^8*c^2 + 18*a^2*b^10*c^2 - b^12*c^2 + 6*a^10*c^4 - 27*a^8*b^2*c^4 - 28*a^6*b^4*c^4 + 21*a^4*b^6*c^4 - 45*a^2*b^8*c^4 + 5*b^10*c^4 - a^8*c^6 + 36*a^6*b^2*c^6 + 21*a^4*b^4*c^6 + 60*a^2*b^6*c^6 - 4*b^8*c^6 - 5*a^6*c^8 - 34*a^4*b^2*c^8 - 45*a^2*b^4*c^8 - 4*b^6*c^8 + 9*a^4*c^10 + 18*a^2*b^2*c^10 + 5*b^4*c^10 - 3*a^2*c^12 - b^2*c^12 : :
X(53726) = X[99] - 3 X[38698], 2 X[620] - 3 X[38804], X[1296] - 3 X[34473], X[6033] - 3 X[38796], X[10734] - 3 X[14639], X[10748] - 3 X[38224], X[10753] - 3 X[36696], X[12188] + 3 X[52698], 3 X[14651] + X[14654], 3 X[15561] - 5 X[38806], 3 X[34127] - 2 X[40340], 3 X[38737] - 2 X[40556], 3 X[38742] - X[38797], 2 X[38745] - 5 X[38807]

X(53726) lies on these lines: {3, 543}, {98, 111}, {99, 38698}, {114, 6719}, {115, 23699}, {126, 6036}, {542, 9129}, {620, 38804}, {1296, 34473}, {1499, 5914}, {2782, 14650}, {2794, 5512}, {5912, 8704}, {6033, 38796}, {10734, 14639}, {10748, 38224}, {10753, 36696}, {12042, 33962}, {12188, 52698}, {14651, 14654}, {15561, 38806}, {22338, 38741}, {32424, 49102}, {34127, 40340}, {38737, 40556}, {38742, 38797}, {38745, 38807}, {38747, 38805}

X(53726) = midpoint of X(i) and X(j) for these {i,j}: {98, 111}, {11632, 14666}, {22338, 38741}
X(53726) = reflection of X(i) in X(j) for these {i,j}: {114, 6719}, {126, 6036}, {38805, 38747}
X(53726) = circumcircle-inverse of X(53264)

---

X(53727) = MIDPOINT OF X(98) AND X(112)

Barycentrics    2*a^18 - 6*a^16*b^2 + 8*a^14*b^4 - 5*a^12*b^6 - 3*a^10*b^8 + 8*a^8*b^10 - 6*a^6*b^12 + 3*a^4*b^14 - a^2*b^16 - 6*a^16*c^2 + 12*a^14*b^2*c^2 - 11*a^12*b^4*c^2 + 10*a^10*b^6*c^2 - 9*a^8*b^8*c^2 + 8*a^6*b^10*c^2 - 5*a^4*b^12*c^2 + 2*a^2*b^14*c^2 - b^16*c^2 + 8*a^14*c^4 - 11*a^12*b^2*c^4 + 4*a^10*b^4*c^4 - a^8*b^6*c^4 - 6*a^6*b^8*c^4 + a^4*b^10*c^4 + 2*a^2*b^12*c^4 + 3*b^14*c^4 - 5*a^12*c^6 + 10*a^10*b^2*c^6 - a^8*b^4*c^6 + 8*a^6*b^6*c^6 + a^4*b^8*c^6 - 10*a^2*b^10*c^6 - 3*b^12*c^6 - 3*a^10*c^8 - 9*a^8*b^2*c^8 - 6*a^6*b^4*c^8 + a^4*b^6*c^8 + 14*a^2*b^8*c^8 + b^10*c^8 + 8*a^8*c^10 + 8*a^6*b^2*c^10 + a^4*b^4*c^10 - 10*a^2*b^6*c^10 + b^8*c^10 - 6*a^6*c^12 - 5*a^4*b^2*c^12 + 2*a^2*b^4*c^12 - 3*b^6*c^12 + 3*a^4*c^14 + 2*a^2*b^2*c^14 + 3*b^4*c^14 - a^2*c^16 - b^2*c^16 : :
X(53727) = X[10735] - 3 X[14639], 2 X[11623] + X[14900], X[13200] + 3 X[14651], X[99] - 3 X[38699], X[1297] - 3 X[34473], X[10749] - 3 X[38224], 3 X[16224] - 2 X[39835], 3 X[16225] - X[39846], 2 X[34841] - 3 X[38737]

X(53727) lies on these lines: {3, 2799}, {4, 32}, {99, 38699}, {114, 6720}, {127, 6036}, {512, 41175}, {542, 43389}, {690, 18338}, {1297, 34473}, {2781, 31850}, {2782, 38608}, {2793, 50381}, {5622, 15357}, {9517, 31848}, {10053, 13312}, {10069, 13311}, {10749, 38224}, {12918, 38741}, {14223, 40080}, {14689, 23698}, {14981, 40866}, {16224, 39835}, {16225, 39846}, {18332, 18348}, {19055, 19115}, {19056, 19114}, {34841, 38737}, {49212, 49271}, {49213, 49270}

X(53727) = midpoint of X(i) and X(j) for these {i,j}: {98, 112}, {12918, 38741}
X(53727) = reflection of X(i) in X(j) for these {i,j}: {114, 6720}, {127, 6036}
X(53727) = circumcircle-inverse of X(53265)

---

X(53728) = MIDPOINT OF X(98) AND X(476)

Barycentrics    2*a^20 - 8*a^18*b^2 + 13*a^16*b^4 - 13*a^14*b^6 + 12*a^12*b^8 - 12*a^10*b^10 + 13*a^8*b^12 - 13*a^6*b^14 + 8*a^4*b^16 - 2*a^2*b^18 - 8*a^18*c^2 + 28*a^16*b^2*c^2 - 37*a^14*b^4*c^2 + 24*a^12*b^6*c^2 - 6*a^10*b^8*c^2 - 15*a^8*b^10*c^2 + 32*a^6*b^12*c^2 - 25*a^4*b^14*c^2 + 7*a^2*b^16*c^2 + 13*a^16*c^4 - 37*a^14*b^2*c^4 + 38*a^12*b^4*c^4 - 18*a^10*b^6*c^4 + 22*a^8*b^8*c^4 - 37*a^6*b^10*c^4 + 32*a^4*b^12*c^4 - 10*a^2*b^14*c^4 - 3*b^16*c^4 - 13*a^14*c^6 + 24*a^12*b^2*c^6 - 18*a^10*b^4*c^6 - 14*a^8*b^6*c^6 + 16*a^6*b^8*c^6 - 23*a^4*b^10*c^6 + 8*a^2*b^12*c^6 + 18*b^14*c^6 + 12*a^12*c^8 - 6*a^10*b^2*c^8 + 22*a^8*b^4*c^8 + 16*a^6*b^6*c^8 + 16*a^4*b^8*c^8 - 3*a^2*b^10*c^8 - 45*b^12*c^8 - 12*a^10*c^10 - 15*a^8*b^2*c^10 - 37*a^6*b^4*c^10 - 23*a^4*b^6*c^10 - 3*a^2*b^8*c^10 + 60*b^10*c^10 + 13*a^8*c^12 + 32*a^6*b^2*c^12 + 32*a^4*b^4*c^12 + 8*a^2*b^6*c^12 - 45*b^8*c^12 - 13*a^6*c^14 - 25*a^4*b^2*c^14 - 10*a^2*b^4*c^14 + 18*b^6*c^14 + 8*a^4*c^16 + 7*a^2*b^2*c^16 - 3*b^4*c^16 - 2*a^2*c^18 : :
X(53728) = X[99] - 3 X[38700], X[477] - 3 X[34473], 3 X[14639] - X[44967], 3 X[14849] + X[52056], X[20957] - 3 X[38224], 2 X[31379] - 3 X[38737]

X(53728) lies on these lines: {3, 23105}, {23, 94}, {30, 15535}, {99, 38700}, {114, 22104}, {477, 34473}, {542, 6070}, {690, 46632}, {2782, 38609}, {2793, 9179}, {2794, 25641}, {3258, 6036}, {5466, 52772}, {6055, 16092}, {9140, 36188}, {12042, 16168}, {14639, 44967}, {14849, 52056}, {15359, 36184}, {16978, 39806}, {20957, 38224}, {31379, 38737}, {46633, 47082}

X(53728) = midpoint of X(98) and X(476)
X(53728) = reflection of X(i) in X(j) for these {i,j}: {114, 22104}, {3258, 6036}, {16978, 39806}, {36184, 15359}
X(53728) = circumcircle-inverse of X(53266)

---

X(53729) = MIDPOINT OF X(99) AND X(100)

Barycentrics    (a - b)*(a - c)*(2*a^5 - 3*a^3*b^2 + 2*a*b^4 + 2*a^3*b*c - a*b^3*c + b^4*c - 3*a^3*c^2 - b^3*c^2 - a*b*c^3 - b^2*c^3 + 2*a*c^4 + b*c^4) : :
X(53729) = X[98] - 3 X[34474], X[104] - 3 X[21166], 3 X[5182] - X[10755], 2 X[6036] - 3 X[38760], X[6154] + 4 X[35022], X[6321] - 3 X[38752], 4 X[6667] - 5 X[31274], 2 X[6713] - 3 X[38748], 4 X[6721] - 3 X[23513], 4 X[6722] - 5 X[31235], 3 X[9167] - 2 X[45310], X[10707] - 3 X[41134], X[10738] - 3 X[15561], 2 X[11725] - 3 X[34123], X[13174] + 3 X[15015], 3 X[38224] - 5 X[38762], 3 X[38731] - X[38753], 2 X[38734] - 5 X[38763]

X(53729) lies on these lines: {2, 10769}, {3, 2783}, {11, 620}, {98, 34474}, {99, 100}, {104, 21166}, {114, 5840}, {115, 3035}, {119, 23698}, {528, 2482}, {542, 51007}, {543, 6174}, {952, 33813}, {2782, 33814}, {2794, 24466}, {2795, 35204}, {2796, 50844}, {2802, 11711}, {2829, 38738}, {5026, 9024}, {5091, 35103}, {5182, 10755}, {5848, 50567}, {5969, 51157}, {6036, 38760}, {6154, 35022}, {6321, 38752}, {6667, 31274}, {6713, 38748}, {6721, 23513}, {6722, 31235}, {7782, 38521}, {9167, 45310}, {9830, 51158}, {9864, 12119}, {10086, 10090}, {10087, 10089}, {10707, 41134}, {10711, 12117}, {10738, 15561}, {10742, 38730}, {10768, 13199}, {11725, 34123}, {13174, 15015}, {13194, 39652}, {14645, 51198}, {19108, 19113}, {19109, 19112}, {38224, 38762}, {38731, 38753}, {38734, 38763}, {38736, 38761}, {48714, 49267}, {48715, 49266}

X(53729) = midpoint of X(i) and X(j) for these {i,j}: {99, 100}, {9864, 12119}, {10711, 12117}, {10742, 38730}, {10768, 13199}
X(53729) = reflection of X(i) in X(j) for these {i,j}: {11, 620}, {115, 3035}, {38761, 38736}
X(53729) = complement of X(10769)
X(53729) = circumcircle-inverse of X(4436)

---

X(53730) = MIDPOINT OF X(99) AND X(101)

Barycentrics    (a - b)*(a - c)*(2*a^6 - 2*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 - 2*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 + a^2*c^4 - a*b*c^4 - b^2*c^4 + a*c^5 + b*c^5) : :
X(53730) = X[98] - 3 X[38690], X[103] - 3 X[21166], 3 X[5182] - X[10756], 2 X[6036] - 3 X[38772], X[6321] - 3 X[38764], 2 X[6712] - 3 X[38748], X[10708] - 3 X[41134], X[10739] - 3 X[15561], 3 X[38224] - 5 X[38774], 3 X[38731] - X[38765], 2 X[38734] - 5 X[38775]

X(53730) lies on these lines: {3, 2784}, {98, 38690}, {99, 101}, {103, 21166}, {115, 6710}, {116, 620}, {118, 23698}, {544, 2482}, {2782, 38599}, {2796, 24279}, {2808, 33813}, {2809, 11711}, {2810, 5026}, {3022, 15452}, {5182, 10756}, {6036, 38772}, {6321, 38764}, {6712, 38748}, {7782, 38522}, {10708, 41134}, {10710, 12117}, {10739, 15561}, {10741, 38730}, {38224, 38774}, {38731, 38765}, {38734, 38775}, {38736, 38773}

X(53730) = midpoint of X(i) and X(j) for these {i,j}: {99, 101}, {10710, 12117}, {10741, 38730}
X(53730) = reflection of X(i) in X(j) for these {i,j}: {115, 6710}, {116, 620}, {38773, 38736}
X(53730) = circumcircle-inverse of X(53268)

---

X(53731) = MIDPOINT OF X(99) AND X(102)

Barycentrics    2*a^14 - 2*a^13*b - 6*a^12*b^2 + 9*a^11*b^3 + 4*a^10*b^4 - 17*a^9*b^5 + 7*a^8*b^6 + 17*a^7*b^7 - 15*a^6*b^8 - 9*a^5*b^9 + 11*a^4*b^10 + 2*a^3*b^11 - 3*a^2*b^12 - 2*a^13*c + 6*a^12*b*c - a^11*b^2*c - 17*a^10*b^3*c + 17*a^9*b^4*c + 17*a^8*b^5*c - 31*a^7*b^6*c - 3*a^6*b^7*c + 25*a^5*b^8*c - 7*a^4*b^9*c - 8*a^3*b^10*c + 4*a^2*b^11*c - 6*a^12*c^2 - a^11*b*c^2 + 28*a^10*b^2*c^2 - 10*a^9*b^3*c^2 - 41*a^8*b^4*c^2 + 22*a^7*b^5*c^2 + 35*a^6*b^6*c^2 - 23*a^5*b^7*c^2 - 16*a^4*b^8*c^2 + 13*a^3*b^9*c^2 - a^2*b^10*c^2 - a*b^11*c^2 + b^12*c^2 + 9*a^11*c^3 - 17*a^10*b*c^3 - 10*a^9*b^2*c^3 + 40*a^8*b^3*c^3 - 4*a^7*b^4*c^3 - 32*a^6*b^5*c^3 - 3*a^5*b^6*c^3 + 27*a^4*b^7*c^3 - 3*a^3*b^8*c^3 - 9*a^2*b^9*c^3 + 3*a*b^10*c^3 - b^11*c^3 + 4*a^10*c^4 + 17*a^9*b*c^4 - 41*a^8*b^2*c^4 - 4*a^7*b^3*c^4 + 26*a^6*b^4*c^4 + 10*a^5*b^5*c^4 - 7*a^4*b^6*c^4 - 19*a^3*b^7*c^4 + 17*a^2*b^8*c^4 - 3*b^10*c^4 - 17*a^9*c^5 + 17*a^8*b*c^5 + 22*a^7*b^2*c^5 - 32*a^6*b^3*c^5 + 10*a^5*b^4*c^5 - 16*a^4*b^5*c^5 + 15*a^3*b^6*c^5 + 5*a^2*b^7*c^5 - 8*a*b^8*c^5 + 4*b^9*c^5 + 7*a^8*c^6 - 31*a^7*b*c^6 + 35*a^6*b^2*c^6 - 3*a^5*b^3*c^6 - 7*a^4*b^4*c^6 + 15*a^3*b^5*c^6 - 26*a^2*b^6*c^6 + 6*a*b^7*c^6 + 2*b^8*c^6 + 17*a^7*c^7 - 3*a^6*b*c^7 - 23*a^5*b^2*c^7 + 27*a^4*b^3*c^7 - 19*a^3*b^4*c^7 + 5*a^2*b^5*c^7 + 6*a*b^6*c^7 - 6*b^7*c^7 - 15*a^6*c^8 + 25*a^5*b*c^8 - 16*a^4*b^2*c^8 - 3*a^3*b^3*c^8 + 17*a^2*b^4*c^8 - 8*a*b^5*c^8 + 2*b^6*c^8 - 9*a^5*c^9 - 7*a^4*b*c^9 + 13*a^3*b^2*c^9 - 9*a^2*b^3*c^9 + 4*b^5*c^9 + 11*a^4*c^10 - 8*a^3*b*c^10 - a^2*b^2*c^10 + 3*a*b^3*c^10 - 3*b^4*c^10 + 2*a^3*c^11 + 4*a^2*b*c^11 - a*b^2*c^11 - b^3*c^11 - 3*a^2*c^12 + b^2*c^12 : :
X(53731) = X[98] - 3 X[38691], X[109] - 3 X[21166], 3 X[5182] - X[10757], 2 X[6036] - 3 X[38784], X[6321] - 3 X[38776], 2 X[6718] - 3 X[38748], X[10709] - 3 X[41134], X[10740] - 3 X[15561], 3 X[38224] - 5 X[38786], 3 X[38731] - X[38777], 2 X[38734] - 5 X[38787]

X(53731) lies on these lines: {3, 2785}, {98, 38691}, {99, 102}, {109, 21166}, {115, 6711}, {117, 620}, {124, 23698}, {1361, 15452}, {2782, 38600}, {2817, 11711}, {2818, 33813}, {5182, 10757}, {6036, 38784}, {6321, 38776}, {6718, 38748}, {10709, 41134}, {10716, 12117}, {10740, 15561}, {10747, 38730}, {38224, 38786}, {38731, 38777}, {38734, 38787}, {38736, 38785}

X(53731) = midpoint of X(i) and X(j) for these {i,j}: {99, 102}, {10716, 12117}, {10747, 38730}
X(53731) = reflection of X(i) in X(j) for these {i,j}: {115, 6711}, {117, 620}, {38785, 38736}

---

X(53732) = MIDPOINT OF X(99) AND X(103)

Barycentrics    2*a^12 - 2*a^11*b - 4*a^10*b^2 + a^9*b^3 + 6*a^8*b^4 + 4*a^7*b^5 - 7*a^6*b^6 - 7*a^5*b^7 + 6*a^4*b^8 + 4*a^3*b^9 - 3*a^2*b^10 - 2*a^11*c + 2*a^10*b*c + 7*a^9*b^2*c - 7*a^8*b^3*c - 10*a^7*b^4*c + 10*a^6*b^5*c + 7*a^5*b^6*c - 7*a^4*b^7*c - 2*a^3*b^8*c + 2*a^2*b^9*c - 4*a^10*c^2 + 7*a^9*b*c^2 + 4*a^8*b^2*c^2 - 4*a^7*b^3*c^2 - 9*a^6*b^4*c^2 + 4*a^5*b^5*c^2 + 9*a^4*b^6*c^2 - 6*a^3*b^7*c^2 - a^2*b^8*c^2 - a*b^9*c^2 + b^10*c^2 + a^9*c^3 - 7*a^8*b*c^3 - 4*a^7*b^2*c^3 + 18*a^6*b^3*c^3 - 14*a^4*b^5*c^3 - 2*a^3*b^6*c^3 + 8*a^2*b^7*c^3 + a*b^8*c^3 - b^9*c^3 + 6*a^8*c^4 - 10*a^7*b*c^4 - 9*a^6*b^2*c^4 + 8*a^4*b^4*c^4 + 6*a^3*b^5*c^4 - 4*a^2*b^6*c^4 + 3*a*b^7*c^4 - 2*b^8*c^4 + 4*a^7*c^5 + 10*a^6*b*c^5 + 4*a^5*b^2*c^5 - 14*a^4*b^3*c^5 + 6*a^3*b^4*c^5 - 4*a^2*b^5*c^5 - 3*a*b^6*c^5 + b^7*c^5 - 7*a^6*c^6 + 7*a^5*b*c^6 + 9*a^4*b^2*c^6 - 2*a^3*b^3*c^6 - 4*a^2*b^4*c^6 - 3*a*b^5*c^6 + 2*b^6*c^6 - 7*a^5*c^7 - 7*a^4*b*c^7 - 6*a^3*b^2*c^7 + 8*a^2*b^3*c^7 + 3*a*b^4*c^7 + b^5*c^7 + 6*a^4*c^8 - 2*a^3*b*c^8 - a^2*b^2*c^8 + a*b^3*c^8 - 2*b^4*c^8 + 4*a^3*c^9 + 2*a^2*b*c^9 - a*b^2*c^9 - b^3*c^9 - 3*a^2*c^10 + b^2*c^10 : :
X(53732) = X[98] - 3 X[38692], X[101] - 3 X[21166], 3 X[5182] - X[10758], 2 X[6710] - 3 X[38748], X[10710] - 3 X[41134], X[10741] - 3 X[15561], 3 X[14639] - 5 X[31273], X[38741] - 3 X[38766], 5 X[38750] - 3 X[38764], 5 X[38751] - 2 X[38769]

X(53732) lies on these lines: {3, 2786}, {98, 38692}, {99, 103}, {101, 21166}, {115, 6712}, {116, 23698}, {118, 620}, {1362, 15452}, {2782, 38601}, {2794, 38773}, {2808, 33813}, {5182, 10758}, {6033, 38765}, {6710, 38748}, {10708, 12117}, {10710, 41134}, {10739, 38730}, {10741, 15561}, {14639, 31273}, {38741, 38766}, {38749, 38771}, {38750, 38764}, {38751, 38769}

X(53732) = midpoint of X(i) and X(j) for these {i,j}: {99, 103}, {6033, 38765}, {10708, 12117}, {10739, 38730}
X(53732) = reflection of X(i) in X(j) for these {i,j}: {115, 6712}, {118, 620}, {38749, 38771}
X(53732) = circumcircle-inverse of X(53269)

---

X(53733) = MIDPOINT OF X(99) AND X(104)

Barycentrics    2*a^11 - 2*a^10*b - 7*a^9*b^2 + 7*a^8*b^3 + 10*a^7*b^4 - 10*a^6*b^5 - 7*a^5*b^6 + 7*a^4*b^7 + 2*a^3*b^8 - 2*a^2*b^9 - 2*a^10*c + 8*a^9*b*c + a^8*b^2*c - 18*a^7*b^3*c + 4*a^6*b^4*c + 17*a^5*b^5*c - 7*a^4*b^6*c - 6*a^3*b^7*c + 4*a^2*b^8*c - a*b^9*c - 7*a^9*c^2 + a^8*b*c^2 + 18*a^7*b^2*c^2 - 4*a^6*b^3*c^2 - 14*a^5*b^4*c^2 + 8*a^3*b^6*c^2 - 2*a^2*b^7*c^2 - a*b^8*c^2 + b^9*c^2 + 7*a^8*c^3 - 18*a^7*b*c^3 - 4*a^6*b^2*c^3 + 14*a^5*b^3*c^3 + 4*a^4*b^4*c^3 - 4*a^3*b^5*c^3 - 6*a^2*b^6*c^3 + 6*a*b^7*c^3 - b^8*c^3 + 10*a^7*c^4 + 4*a^6*b*c^4 - 14*a^5*b^2*c^4 + 4*a^4*b^3*c^4 - 4*a^3*b^4*c^4 + 6*a^2*b^5*c^4 + a*b^6*c^4 - 3*b^7*c^4 - 10*a^6*c^5 + 17*a^5*b*c^5 - 4*a^3*b^3*c^5 + 6*a^2*b^4*c^5 - 10*a*b^5*c^5 + 3*b^6*c^5 - 7*a^5*c^6 - 7*a^4*b*c^6 + 8*a^3*b^2*c^6 - 6*a^2*b^3*c^6 + a*b^4*c^6 + 3*b^5*c^6 + 7*a^4*c^7 - 6*a^3*b*c^7 - 2*a^2*b^2*c^7 + 6*a*b^3*c^7 - 3*b^4*c^7 + 2*a^3*c^8 + 4*a^2*b*c^8 - a*b^2*c^8 - b^3*c^8 - 2*a^2*c^9 - a*b*c^9 + b^2*c^9 : :
X(53733) = X[98] - 3 X[38693], X[100] - 3 X[21166], 2 X[3035] - 3 X[38748], 3 X[5182] - X[10759], 2 X[5461] - 3 X[38069], 2 X[6036] - 3 X[21154], 4 X[6667] - 3 X[23514], X[10711] - 3 X[41134], X[10742] - 3 X[15561], X[10992] + 2 X[20418], 2 X[11725] - 3 X[38032], 3 X[14639] - 5 X[31272], X[38741] - 3 X[38754], 5 X[38750] - 3 X[38752], 5 X[38751] - 2 X[38757]

X(53733) lies on these lines: {3, 2787}, {11, 23698}, {20, 10768}, {98, 38693}, {99, 104}, {100, 21166}, {114, 2829}, {115, 6713}, {119, 620}, {952, 33813}, {1317, 15452}, {1537, 11724}, {2782, 38602}, {2794, 38761}, {2800, 11711}, {3035, 38748}, {5182, 10759}, {5461, 38069}, {5840, 38738}, {6033, 38753}, {6036, 21154}, {6667, 23514}, {9880, 45310}, {10058, 10089}, {10074, 10086}, {10707, 12117}, {10711, 41134}, {10738, 38730}, {10742, 15561}, {10769, 13172}, {10992, 20418}, {11725, 38032}, {12199, 39652}, {14639, 31272}, {19081, 19109}, {19082, 19108}, {24466, 38736}, {38741, 38754}, {38749, 38759}, {38750, 38752}, {38751, 38757}, {48700, 49267}, {48701, 49266}

X(53733) = midpoint of X(i) and X(j) for these {i,j}: {20, 10768}, {99, 104}, {6033, 38753}, {10707, 12117}, {10738, 38730}, {10769, 13172}
X(53733) = reflection of X(i) in X(j) for these {i,j}: {115, 6713}, {119, 620}, {1537, 11724}, {9880, 45310}, {24466, 38736}, {38749, 38759}
X(53733) = circumcircle-inverse of X(53270)

---

X(53734) = MIDPOINT OF X(99) AND X(109)

Barycentrics    (a - b)*(a - c)*(2*a^8 - 4*a^6*b^2 + a^5*b^3 + 3*a^4*b^4 - 2*a^3*b^5 - a^2*b^6 + a*b^7 + 4*a^6*b*c - a^5*b^2*c - 5*a^4*b^3*c + 2*a^3*b^4*c + 2*a^2*b^5*c - a*b^6*c + b^7*c - 4*a^6*c^2 - a^5*b*c^2 + 6*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + a*b^5*c^2 - b^6*c^2 + a^5*c^3 - 5*a^4*b*c^3 + 2*a^2*b^3*c^3 - a*b^4*c^3 - b^5*c^3 + 3*a^4*c^4 + 2*a^3*b*c^4 - 2*a^2*b^2*c^4 - a*b^3*c^4 + 2*b^4*c^4 - 2*a^3*c^5 + 2*a^2*b*c^5 + a*b^2*c^5 - b^3*c^5 - a^2*c^6 - a*b*c^6 - b^2*c^6 + a*c^7 + b*c^7) : :
X(53734) = X[98] - 3 X[38697], X[102] - 3 X[21166], 3 X[5182] - X[10764], 2 X[6711] - 3 X[38748], X[10716] - 3 X[41134], X[10747] - 3 X[15561], X[38741] - 3 X[38778], 5 X[38750] - 3 X[38776], 5 X[38751] - 2 X[38781]

X(53734) lies on these lines: {3, 2792}, {98, 38697}, {99, 109}, {102, 21166}, {115, 6718}, {117, 23698}, {124, 620}, {1364, 15452}, {1795, 10086}, {2782, 38607}, {2794, 38785}, {2800, 11711}, {2818, 33813}, {5182, 10764}, {6033, 38777}, {6711, 38748}, {10709, 12117}, {10716, 41134}, {10740, 38730}, {10747, 15561}, {38741, 38778}, {38749, 38783}, {38750, 38776}, {38751, 38781}

X(53734) = midpoint of X(i) and X(j) for these {i,j}: {99, 109}, {6033, 38777}, {10709, 12117}, {10740, 38730}
X(53734) = reflection of X(i) in X(j) for these {i,j}: {115, 6718}, {124, 620}, {38749, 38783}
X(53734) = circumcircle-inverse of X(23363)

---

X(53735) = MIDPOINT OF X(99) AND X(110)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(2*a^6 - 2*a^4*b^2 + b^6 - 2*a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - b^2*c^4 + c^6) : :
X(53735) = 3 X[2482] - X[15357], 3 X[14850] - X[15545], 3 X[14850] + X[23236], X[74] - 3 X[21166], X[98] - 3 X[15035], 3 X[99] + X[15342], 3 X[110] - X[15342], 3 X[249] + X[47288], 3 X[249] - X[53379], X[265] - 3 X[15561], 3 X[5465] - 2 X[16278], 3 X[5642] - X[16278], X[895] - 3 X[5182], 2 X[6036] - 3 X[38793], X[6321] - 3 X[14643], 2 X[6699] - 3 X[38748], and many others

X(53735) lies on these lines: {2, 15359}, {3, 67}, {74, 21166}, {98, 15035}, {99, 110}, {113, 23698}, {114, 17702}, {115, 5972}, {125, 620}, {140, 15535}, {249, 826}, {265, 15561}, {511, 46634}, {512, 7472}, {523, 9181}, {524, 36180}, {530, 16182}, {531, 16181}, {543, 1316}, {671, 46512}, {691, 12073}, {895, 5182}, {1511, 2782}, {1632, 48952}, {2777, 38738}, {2794, 16163}, {2795, 16164}, {2854, 3111}, {3024, 15452}, {3906, 14999}, {3972, 52693}, {5095, 14645}, {5663, 33813}, {5969, 6593}, {6033, 12121}, {6036, 38793}, {6054, 7422}, {6321, 14643}, {6699, 38748}, {6721, 23515}, {6723, 31274}, {7687, 36519}, {7728, 38730}, {7782, 38523}, {7927, 9218}, {8289, 47044}, {8591, 9144}, {8997, 46688}, {9003, 9145}, {9033, 40866}, {9140, 41134}, {9143, 11006}, {9167, 45311}, {10086, 10091}, {10088, 10089}, {10564, 14934}, {10706, 12117}, {10754, 52699}, {10992, 16534}, {11005, 12383}, {11656, 11693}, {12074, 20404}, {12188, 15040}, {12900, 23514}, {13188, 18332}, {13193, 39652}, {13989, 46689}, {14849, 38739}, {14984, 31850}, {15020, 38664}, {15034, 22265}, {15051, 34473}, {15061, 38750}, {15920, 35925}, {16111, 38736}, {16222, 39806}, {16223, 39817}, {19108, 19111}, {19109, 19110}, {20127, 38731}, {22109, 39854}, {24981, 35022}, {31854, 51524}, {32135, 41330}, {32478, 33803}, {34153, 51872}, {36253, 38751}, {38224, 38794}, {38723, 38741}, {38726, 38749}, {38734, 38795}, {38737, 48378}, {39809, 46686}, {40544, 51428}, {41720, 50639}, {49266, 49269}, {49267, 49268}, {52035, 53350}

X(53735) = midpoint of X(i) and X(j) for these {i,j}: {99, 110}, {5648, 51798}, {6033, 12121}, {7728, 38730}, {8591, 9144}, {9143, 11006}, {10706, 12117}, {11005, 12383}, {13188, 18332}, {14999, 47293}, {15545, 23236}, {22265, 23235}, {34153, 51872}, {41720, 50639}, {47288, 53379}
X(53735) = reflection of X(i) in X(j) for these {i,j}: {114, 33512}, {115, 5972}, {125, 620}, {5465, 5642}, {15535, 140}, {16111, 38736}, {38749, 38726}, {39809, 46686}, {51428, 40544}
X(53735) = anticomplement of X(15359)
X(53735) = circumcircle-inverse of X(1634)
X(53735) = crossdifference of every pair of points on line {2492, 3124}
X(53735) = barycentric product X(i)*X(j) for these {i,j}: {99, 41939}, {14999, 36825}
X(53735) = barycentric quotient X(i)/X(j) for these {i,j}: {36825, 14223}, {41939, 523}
X(53735) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {99, 35278, 48951}, {249, 47288, 53379}, {9143, 52695, 11006}, {13188, 32609, 18332}, {14850, 23236, 15545}

---

X(53736) = MIDPOINT OF X(99) AND X(111)

Barycentrics    2*a^10 - 6*a^8*b^2 + 5*a^4*b^6 - 3*a^2*b^8 - 6*a^8*c^2 + 28*a^6*b^2*c^2 - 21*a^4*b^4*c^2 + 6*a^2*b^6*c^2 + b^8*c^2 - 21*a^4*b^2*c^4 + 12*a^2*b^4*c^4 - 3*b^6*c^4 + 5*a^4*c^6 + 6*a^2*b^2*c^6 - 3*b^4*c^6 - 3*a^2*c^8 + b^2*c^8 : :
X(53736) = X[10717] - 3 X[41134], X[98] - 3 X[38698], X[1296] - 3 X[21166], 3 X[5182] - X[10765], 2 X[6036] - 3 X[38804], X[6321] - 3 X[38796], X[10748] - 3 X[15561], X[10754] - 3 X[36696], X[13188] + 3 X[52698], 3 X[38224] - 5 X[38806], 3 X[38731] - X[38797], 2 X[38734] - 5 X[38807], 3 X[38748] - 2 X[40556]

X(53736) lies on these lines: {2, 99}, {3, 2793}, {39, 52883}, {98, 38698}, {114, 23699}, {187, 9182}, {523, 47077}, {538, 5912}, {625, 36196}, {690, 5108}, {804, 47078}, {1296, 21166}, {2782, 14650}, {2799, 50381}, {2854, 3111}, {5182, 10765}, {5465, 11053}, {5512, 23698}, {5969, 28662}, {6019, 15452}, {6036, 38804}, {6094, 16508}, {6321, 38796}, {6390, 44398}, {7472, 32456}, {7782, 38524}, {7813, 44373}, {8724, 14666}, {9147, 15566}, {9149, 34010}, {10748, 15561}, {10754, 25315}, {13188, 52698}, {13586, 34205}, {14120, 44377}, {14645, 48945}, {15921, 35925}, {22338, 38730}, {27088, 44397}, {33813, 33962}, {38224, 38806}, {38731, 38797}, {38734, 38807}, {38736, 38805}, {38748, 40556}

X(53736) = midpoint of X(i) and X(j) for these {i,j}: {99, 111}, {8724, 14666}, {22338, 38730}
X(53736) = reflection of X(i) in X(j) for these {i,j}: {115, 6719}, {126, 620}, {38805, 38736}
X(53736) = circumcircle-inverse of X(53272)
X(53736) = psi-transform of X(5468)
X(53736) = {X(99),X(41134)}-harmonic conjugate of X(31128)

---

X(53737) = MIDPOINT OF X(99) AND X(112)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(2*a^10 - 2*a^8*b^2 + a^4*b^6 - 2*a^2*b^8 + b^10 - 2*a^8*c^2 + 2*a^6*b^2*c^2 - a^4*b^4*c^2 + 4*a^2*b^6*c^2 - b^8*c^2 - a^4*b^2*c^4 - 4*a^2*b^4*c^4 + a^4*c^6 + 4*a^2*b^2*c^6 - 2*a^2*c^8 - b^2*c^8 + c^10) : :
X(53737) = X[10749] - 3 X[15561], 2 X[34841] - 3 X[38748], X[98] - 3 X[38699], X[1297] - 3 X[21166], 3 X[5182] - X[10766], X[10718] - 3 X[41134], 3 X[16224] - 2 X[39806], 3 X[16225] - X[39817]

X(53737) lies on these lines: {3, 114}, {98, 38699}, {99, 112}, {115, 6720}, {132, 23698}, {543, 50381}, {690, 14574}, {1297, 21166}, {2781, 5026}, {2782, 38608}, {5182, 10766}, {5969, 28343}, {6020, 15452}, {6333, 46249}, {7473, 30476}, {7782, 38525}, {10086, 13312}, {10089, 13311}, {10718, 41134}, {12918, 38730}, {13195, 39652}, {14649, 47082}, {16224, 39806}, {16225, 39817}, {19108, 19115}, {19109, 19114}, {32456, 46634}, {49266, 49271}, {49267, 49270}

X(53737) = midpoint of X(i) and X(j) for these {i,j}: {99, 112}, {12918, 38730}
X(53737) = reflection of X(i) in X(j) for these {i,j}: {115, 6720}, {127, 620}
X(53737) = circumcircle-inverse of X(53273)

---

X(53738) = MIDPOINT OF X(99) AND X(476)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(2*a^12 - 4*a^10*b^2 + a^8*b^4 + 2*a^6*b^6 + a^4*b^8 - 4*a^2*b^10 + 2*b^12 - 4*a^10*c^2 + 10*a^8*b^2*c^2 - 6*a^6*b^4*c^2 - 9*a^4*b^6*c^2 + 13*a^2*b^8*c^2 - 7*b^10*c^2 + a^8*c^4 - 6*a^6*b^2*c^4 + 18*a^4*b^4*c^4 - 9*a^2*b^6*c^4 + 10*b^8*c^4 + 2*a^6*c^6 - 9*a^4*b^2*c^6 - 9*a^2*b^4*c^6 - 10*b^6*c^6 + a^4*c^8 + 13*a^2*b^2*c^8 + 10*b^4*c^8 - 4*a^2*c^10 - 7*b^2*c^10 + 2*c^12) : :
X(53738) = X[98] - 3 X[38700], X[477] - 3 X[21166], 3 X[14850] + X[52056], 3 X[15561] - X[20957], 2 X[31379] - 3 X[38748], X[34312] - 3 X[41134]

X(53738) lies on these lines: {3, 2453}, {98, 38700}, {99, 476}, {115, 22104}, {477, 21166}, {523, 9181}, {542, 46632}, {543, 9179}, {620, 3258}, {690, 7471}, {2482, 46634}, {2782, 38609}, {3734, 34512}, {6070, 50711}, {14611, 43969}, {14850, 52056}, {15452, 33965}, {15561, 20957}, {16168, 33813}, {16978, 39835}, {23698, 25641}, {31379, 38748}, {34312, 41134}

X(53738) = midpoint of X(99) and X(476)
X(53738) = reflection of X(i) in X(j) for these {i,j}: {115, 22104}, {3258, 620}, {16978, 39835}
X(53738) = reflection of X(9181) in the Euler line
X(53738) = circumcircle-inverse of X(53274)

---

X(53739) = MIDPOINT OF X(100) AND X(101)

Barycentrics    a*(a - b)*(a - c)*(2*a^4 - 2*a^3*b - a^2*b^2 + b^4 - 2*a^3*c + 4*a^2*b*c - b^3*c - a^2*c^2 - b*c^3 + c^4) : :
X(53739) = X[103] - 3 X[34474], X[104] - 3 X[38690], X[1282] + 3 X[15015], X[6154] + 4 X[35024], 2 X[6712] - 3 X[38760], 2 X[6713] - 3 X[38772], X[10738] - 3 X[38764], X[10739] - 3 X[38752], 2 X[11726] - 3 X[34123]

X(53739) lies on these lines: {2, 10770}, {3, 2801}, {11, 6710}, {100, 101}, {103, 34474}, {104, 38690}, {116, 3035}, {118, 5840}, {214, 2809}, {544, 6174}, {952, 38599}, {1110, 14838}, {1282, 15015}, {2802, 8301}, {2808, 33814}, {2810, 51157}, {6154, 35024}, {6712, 38760}, {6713, 38772}, {9323, 48003}, {10738, 38764}, {10739, 38752}, {10772, 13199}, {11726, 34123}, {12119, 50903}, {14513, 36167}, {41553, 51682}

X(53739) = midpoint of X(i) and X(j) for these {i,j}: {100, 101}, {10772, 13199}, {12119, 50903}
X(53739) = reflection of X(i) in X(j) for these {i,j}: {11, 6710}, {116, 3035}
X(53739) = complement of X(10770)
X(53739) = circumcircle-inverse of X(4557)

---

X(53740) = MIDPOINT OF X(100) AND X(102)

Barycentrics    a*(2*a^12 - 4*a^11*b - 3*a^10*b^2 + 13*a^9*b^3 - 8*a^8*b^4 - 12*a^7*b^5 + 22*a^6*b^6 - 2*a^5*b^7 - 18*a^4*b^8 + 8*a^3*b^9 + 5*a^2*b^10 - 3*a*b^11 - 4*a^11*c + 14*a^10*b*c - 9*a^9*b^2*c - 24*a^8*b^3*c + 47*a^7*b^4*c - 13*a^6*b^5*c - 47*a^5*b^6*c + 43*a^4*b^7*c + 9*a^3*b^8*c - 21*a^2*b^9*c + 4*a*b^10*c + b^11*c - 3*a^10*c^2 - 9*a^9*b*c^2 + 42*a^8*b^2*c^2 - 27*a^7*b^3*c^2 - 57*a^6*b^4*c^2 + 85*a^5*b^5*c^2 - a^4*b^6*c^2 - 53*a^3*b^7*c^2 + 20*a^2*b^8*c^2 + 4*a*b^9*c^2 - b^10*c^2 + 13*a^9*c^3 - 24*a^8*b*c^3 - 27*a^7*b^2*c^3 + 92*a^6*b^3*c^3 - 36*a^5*b^4*c^3 - 71*a^4*b^5*c^3 + 57*a^3*b^6*c^3 + 6*a^2*b^7*c^3 - 7*a*b^8*c^3 - 3*b^9*c^3 - 8*a^8*c^4 + 47*a^7*b*c^4 - 57*a^6*b^2*c^4 - 36*a^5*b^3*c^4 + 94*a^4*b^4*c^4 - 21*a^3*b^5*c^4 - 25*a^2*b^6*c^4 + 2*a*b^7*c^4 + 4*b^8*c^4 - 12*a^7*c^5 - 13*a^6*b*c^5 + 85*a^5*b^2*c^5 - 71*a^4*b^3*c^5 - 21*a^3*b^4*c^5 + 30*a^2*b^5*c^5 + 2*b^7*c^5 + 22*a^6*c^6 - 47*a^5*b*c^6 - a^4*b^2*c^6 + 57*a^3*b^3*c^6 - 25*a^2*b^4*c^6 - 6*b^6*c^6 - 2*a^5*c^7 + 43*a^4*b*c^7 - 53*a^3*b^2*c^7 + 6*a^2*b^3*c^7 + 2*a*b^4*c^7 + 2*b^5*c^7 - 18*a^4*c^8 + 9*a^3*b*c^8 + 20*a^2*b^2*c^8 - 7*a*b^3*c^8 + 4*b^4*c^8 + 8*a^3*c^9 - 21*a^2*b*c^9 + 4*a*b^2*c^9 - 3*b^3*c^9 + 5*a^2*c^10 + 4*a*b*c^10 - b^2*c^10 - 3*a*c^11 + b*c^11) : :
X(53740) = X[104] - 3 X[38691], X[109] - 3 X[34474], 2 X[6713] - 3 X[38784], 2 X[6718] - 3 X[38760], 3 X[10165] - 2 X[29008], X[10738] - 3 X[38776], X[10740] - 3 X[38752], 2 X[11727] - 3 X[34123]

X(53740) lies on these lines: {2, 10771}, {3, 3738}, {11, 6711}, {40, 78}, {104, 38691}, {109, 34474}, {117, 3035}, {124, 5840}, {214, 2817}, {952, 38600}, {2802, 11713}, {2804, 31866}, {2814, 46409}, {2818, 33814}, {6713, 38784}, {6718, 38760}, {10165, 29008}, {10738, 38776}, {10740, 38752}, {10777, 13199}, {11727, 34123}, {12119, 13532}

X(53740) = midpoint of X(i) and X(j) for these {i,j}: {100, 102}, {10777, 13199}, {12119, 13532}
X(53740) = reflection of X(i) in X(j) for these {i,j}: {11, 6711}, {117, 3035}
X(53740) = complement of X(10771)
X(53740) = circumcircle-inverse of X(53277)

---

X(53741) = MIDPOINT OF X(100) AND X(103)

Barycentrics    a*(2*a^10 - 4*a^9*b - a^8*b^2 + 3*a^7*b^3 + 3*a^6*b^4 + 3*a^5*b^5 - 11*a^4*b^6 + a^3*b^7 + 7*a^2*b^8 - 3*a*b^9 - 4*a^9*c + 10*a^8*b*c + a^7*b^2*c - 12*a^6*b^3*c - 8*a^5*b^4*c + 15*a^4*b^5*c + 9*a^3*b^6*c - 14*a^2*b^7*c + 2*a*b^8*c + b^9*c - a^8*c^2 + a^7*b*c^2 - 4*a^6*b^2*c^2 + 13*a^5*b^3*c^2 - 10*a^4*b^4*c^2 - a^3*b^5*c^2 + 3*a*b^7*c^2 - b^8*c^2 + 3*a^7*c^3 - 12*a^6*b*c^3 + 13*a^5*b^2*c^3 + 8*a^4*b^3*c^3 - 9*a^3*b^4*c^3 - 10*a^2*b^5*c^3 + 9*a*b^6*c^3 - 2*b^7*c^3 + 3*a^6*c^4 - 8*a^5*b*c^4 - 10*a^4*b^2*c^4 - 9*a^3*b^3*c^4 + 34*a^2*b^4*c^4 - 11*a*b^5*c^4 + b^6*c^4 + 3*a^5*c^5 + 15*a^4*b*c^5 - a^3*b^2*c^5 - 10*a^2*b^3*c^5 - 11*a*b^4*c^5 + 2*b^5*c^5 - 11*a^4*c^6 + 9*a^3*b*c^6 + 9*a*b^3*c^6 + b^4*c^6 + a^3*c^7 - 14*a^2*b*c^7 + 3*a*b^2*c^7 - 2*b^3*c^7 + 7*a^2*c^8 + 2*a*b*c^8 - b^2*c^8 - 3*a*c^9 + b*c^9) : :
X(53741) = X[101] - 3 X[34474], X[104] - 3 X[38692], 2 X[6710] - 3 X[38760], X[10741] - 3 X[38752], 2 X[11728] - 3 X[34123], 3 X[15015] + X[39156], X[38753] - 3 X[38766], 5 X[38762] - 3 X[38764], 5 X[38763] - 2 X[38769]

X(53741) lies on these lines: {2, 10772}, {3, 3887}, {11, 6712}, {63, 100}, {101, 34474}, {104, 38692}, {116, 5840}, {118, 3035}, {952, 38601}, {2802, 11714}, {2808, 33814}, {2820, 46409}, {2826, 31852}, {2829, 38773}, {6710, 38760}, {10741, 38752}, {10742, 38765}, {10770, 13199}, {11728, 34123}, {12119, 50896}, {15015, 39156}, {38753, 38766}, {38761, 38771}, {38762, 38764}, {38763, 38769}

X(53741) = midpoint of X(i) and X(j) for these {i,j}: {100, 103}, {10742, 38765}, {10770, 13199}, {12119, 50896}
X(53741) = reflection of X(i) in X(j) for these {i,j}: {11, 6712}, {118, 3035}, {38761, 38771}
X(53741) = complement of X(10772)
X(53741) = circumcircle-inverse of X(53278)

---

X(53742) = MIDPOINT OF X(100) AND X(109)

Barycentrics    a*(a - b)*(a - c)*(2*a^6 - 2*a^5*b - 3*a^4*b^2 + 4*a^3*b^3 - 2*a*b^5 + b^6 - 2*a^5*c + 8*a^4*b*c - 4*a^3*b^2*c - 7*a^2*b^3*c + 6*a*b^4*c - b^5*c - 3*a^4*c^2 - 4*a^3*b*c^2 + 14*a^2*b^2*c^2 - 4*a*b^3*c^2 - b^4*c^2 + 4*a^3*c^3 - 7*a^2*b*c^3 - 4*a*b^2*c^3 + 2*b^3*c^3 + 6*a*b*c^4 - b^2*c^4 - 2*a*c^5 - b*c^5 + c^6) : :
X(53742) = X[102] - 3 X[34474], X[104] - 3 X[38697], 2 X[6711] - 3 X[38760], X[10747] - 3 X[38752], 2 X[11734] - 3 X[34123], X[38753] - 3 X[38778], 5 X[38762] - 3 X[38776], 5 X[38763] - 2 X[38781]

X(53742) lies on these lines: {2, 10777}, {3, 214}, {11, 6718}, {80, 37043}, {100, 109}, {102, 34474}, {104, 38697}, {117, 5840}, {124, 3035}, {901, 36167}, {952, 38607}, {1795, 10087}, {2802, 11700}, {2818, 33814}, {2827, 53279}, {2829, 38785}, {2835, 46409}, {6711, 38760}, {7972, 36944}, {10742, 38777}, {10747, 38752}, {10771, 13199}, {11734, 34123}, {12119, 50899}, {21630, 29008}, {25440, 52005}, {35057, 36037}, {38753, 38778}, {38761, 38783}, {38762, 38776}, {38763, 38781}

X(53742) = midpoint of X(i) and X(j) for these {i,j}: {100, 109}, {10742, 38777}, {10771, 13199}, {12119, 50899}
X(53742) = reflection of X(i) in X(j) for these {i,j}: {11, 6718}, {124, 3035}, {21630, 29008}, {38761, 38783}
X(53742) = complement of X(10777)
X(53742) = circumcircle-inverse of X(23845)

---

X(53743) = MIDPOINT OF X(100) AND X(110)

Barycentrics    a*(a - b)*(a - c)*(2*a^6 - 3*a^4*b^2 + b^6 + 2*a^4*b*c - a^2*b^3*c + a*b^4*c - 3*a^4*c^2 + 4*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*b*c^4 - b^2*c^4 + c^6) : :
X(53743) = X[2948] + 3 X[15015], X[74] - 3 X[34474], X[104] - 3 X[15035], X[265] - 3 X[38752], 2 X[6699] - 3 X[38760], 2 X[6713] - 3 X[38793], 4 X[6723] - 5 X[31235], X[10738] - 3 X[14643], X[10755] - 3 X[52699], X[10993] + 2 X[16534], 2 X[11735] - 3 X[34123], X[12331] + 3 X[32609], X[12773] - 5 X[15040], 4 X[12900] - 3 X[23513], 7 X[15020] - X[38669], 5 X[15034] + X[38665], 5 X[15051] - 3 X[38693], 3 X[15061] - 5 X[38762], 3 X[21154] - 4 X[48378], X[24981] + 4 X[35023], 2 X[36253] - 5 X[38763], 3 X[38723] - X[38753]

X(53743) lies on these lines: {2, 10778}, {3, 191}, {11, 5972}, {74, 34474}, {100, 110}, {104, 15035}, {113, 5840}, {119, 17702}, {125, 3035}, {214, 16598}, {265, 38752}, {513, 36167}, {517, 46635}, {528, 5642}, {542, 6174}, {758, 47402}, {952, 1511}, {2777, 24466}, {2802, 11720}, {2805, 9129}, {2829, 16163}, {2836, 34583}, {2842, 13868}, {2854, 51157}, {3109, 16164}, {5181, 5848}, {5663, 33814}, {6265, 12778}, {6593, 9024}, {6699, 38760}, {6713, 38793}, {6723, 31235}, {10087, 10091}, {10088, 10090}, {10738, 14643}, {10742, 12121}, {10755, 52699}, {10767, 13199}, {10956, 46683}, {10993, 16534}, {11698, 34153}, {11735, 34123}, {12119, 12368}, {12331, 32609}, {12773, 15040}, {12896, 39692}, {12900, 23513}, {13922, 46688}, {13991, 46689}, {15020, 38669}, {15034, 38665}, {15051, 38693}, {15061, 38762}, {19110, 19113}, {19111, 19112}, {21154, 48378}, {24465, 40622}, {24981, 35023}, {36253, 38763}, {38723, 38753}, {38726, 38761}, {48714, 49269}, {48715, 49268}

X(53743) = midpoint of X(i) and X(j) for these {i,j}: {100, 110}, {6265, 12778}, {10742, 12121}, {10767, 13199}, {11698, 34153}, {12119, 12368}
X(53743) = reflection of X(i) in X(j) for these {i,j}: {11, 5972}, {125, 3035}, {31523, 11720}, {31525, 214}, {38761, 38726}
X(53743) = complement of X(10778)
X(53743) = circumcircle-inverse of X(53280)
X(53743) = crossdifference of every pair of points on line {3125, 47227}

---

X(53744) = MIDPOINT OF X(100) AND X(111)

Barycentrics    a*(2*a^8 - 2*a^7*b - 5*a^6*b^2 + 5*a^5*b^3 - 4*a^4*b^4 + 4*a^3*b^5 + 3*a^2*b^6 - 3*a*b^7 - 2*a^7*c + 4*a^6*b*c + 3*a^5*b^2*c - 6*a^4*b^3*c + 6*a^3*b^4*c - 9*a^2*b^5*c + a*b^6*c + b^7*c - 5*a^6*c^2 + 3*a^5*b*c^2 + 22*a^4*b^2*c^2 - 20*a^3*b^3*c^2 - 6*a^2*b^4*c^2 + 10*a*b^5*c^2 + 5*a^5*c^3 - 6*a^4*b*c^3 - 20*a^3*b^2*c^3 + 30*a^2*b^3*c^3 - 4*a*b^4*c^3 - 3*b^5*c^3 - 4*a^4*c^4 + 6*a^3*b*c^4 - 6*a^2*b^2*c^4 - 4*a*b^3*c^4 + 4*a^3*c^5 - 9*a^2*b*c^5 + 10*a*b^2*c^5 - 3*b^3*c^5 + 3*a^2*c^6 + a*b*c^6 - 3*a*c^7 + b*c^7) : :
X(53744) = X[104] - 3 X[38698], X[1296] - 3 X[34474], 2 X[6713] - 3 X[38804], X[10738] - 3 X[38796], X[10748] - 3 X[38752], X[10755] - 3 X[36696], X[12331] + 3 X[52698], 3 X[38760] - 2 X[40556]

X(53744) lies on these lines: {2, 10779}, {3, 2830}, {11, 6719}, {37, 100}, {104, 38698}, {119, 23699}, {126, 3035}, {528, 9172}, {543, 6174}, {952, 14650}, {1296, 34474}, {2802, 11721}, {2806, 50381}, {2837, 46409}, {2854, 51157}, {5512, 5840}, {6713, 38804}, {8674, 9129}, {9024, 28662}, {10738, 38796}, {10748, 38752}, {10755, 36696}, {12331, 52698}, {33814, 33962}, {38760, 40556}

X(53744) = midpoint of X(100) and X(111)
X(53744) = reflection of X(i) in X(j) for these {i,j}: {11, 6719}, {126, 3035}
X(53744) = complement of X(10779)
X(53744) = circumcircle-inverse of X(53281)

---

X(53745) = MIDPOINT OF X(100) AND X(112)

Barycentrics    a*(a - b)*(a - c)*(2*a^10 - 3*a^8*b^2 + 2*a^4*b^6 - 2*a^2*b^8 + b^10 + 2*a^8*b*c - a^6*b^3*c + a^5*b^4*c - 2*a^4*b^5*c + a^2*b^7*c - a*b^8*c - 3*a^8*c^2 + 4*a^6*b^2*c^2 - a^5*b^3*c^2 - 2*a^4*b^4*c^2 + 2*a^2*b^6*c^2 + a*b^7*c^2 - b^8*c^2 - a^6*b*c^3 - a^5*b^2*c^3 + 4*a^4*b^3*c^3 - a^2*b^5*c^3 + 3*a*b^6*c^3 + a^5*b*c^4 - 2*a^4*b^2*c^4 - 3*a*b^5*c^4 - 2*a^4*b*c^5 - a^2*b^3*c^5 - 3*a*b^4*c^5 + 2*a^4*c^6 + 2*a^2*b^2*c^6 + 3*a*b^3*c^6 + a^2*b*c^7 + a*b^2*c^7 - 2*a^2*c^8 - a*b*c^8 - b^2*c^8 + c^10) : :
X(53745) = X[104] - 3 X[38699], X[1297] - 3 X[34474], X[10749] - 3 X[38752], X[13221] + 3 X[15015], 2 X[34841] - 3 X[38760]

X(53745) lies on these lines: {2, 10780}, {3, 2831}, {11, 6720}, {100, 112}, {104, 38699}, {119, 2794}, {127, 3035}, {132, 5840}, {952, 38608}, {1297, 34474}, {2781, 51157}, {2802, 11722}, {2805, 50381}, {2829, 14689}, {2838, 46409}, {9024, 28343}, {10087, 13312}, {10090, 13311}, {10749, 38752}, {12119, 12784}, {13221, 15015}, {19112, 19115}, {19113, 19114}, {34841, 38760}, {48714, 49271}, {48715, 49270}

X(53745) = midpoint of X(i) and X(j) for these {i,j}: {100, 112}, {12119, 12784}
X(53745) = reflection of X(i) in X(j) for these {i,j}: {11, 6720}, {127, 3035}
X(53745) = complement of X(10780)
X(53745) = circumcircle-inverse of X(53282)

---

X(53746) = MIDPOINT OF X(101) AND X(104)

Barycentrics    a*(2*a^10 - 4*a^9*b - 3*a^8*b^2 + 9*a^7*b^3 + a^6*b^4 - 7*a^5*b^5 - a^4*b^6 + 3*a^3*b^7 + a^2*b^8 - a*b^9 - 4*a^9*c + 14*a^8*b*c - 5*a^7*b^2*c - 20*a^6*b^3*c + 8*a^5*b^4*c + 17*a^4*b^5*c - 5*a^3*b^6*c - 10*a^2*b^7*c + 6*a*b^8*c - b^9*c - 3*a^8*c^2 - 5*a^7*b*c^2 + 16*a^6*b^2*c^2 + 7*a^5*b^3*c^2 - 18*a^4*b^4*c^2 - 11*a^3*b^5*c^2 + 20*a^2*b^6*c^2 - 7*a*b^7*c^2 + b^8*c^2 + 9*a^7*c^3 - 20*a^6*b*c^3 + 7*a^5*b^2*c^3 + 13*a^3*b^4*c^3 - 6*a^2*b^5*c^3 - 5*a*b^6*c^3 + 2*b^7*c^3 + a^6*c^4 + 8*a^5*b*c^4 - 18*a^4*b^2*c^4 + 13*a^3*b^3*c^4 - 10*a^2*b^4*c^4 + 7*a*b^5*c^4 - b^6*c^4 - 7*a^5*c^5 + 17*a^4*b*c^5 - 11*a^3*b^2*c^5 - 6*a^2*b^3*c^5 + 7*a*b^4*c^5 - 2*b^5*c^5 - a^4*c^6 - 5*a^3*b*c^6 + 20*a^2*b^2*c^6 - 5*a*b^3*c^6 - b^4*c^6 + 3*a^3*c^7 - 10*a^2*b*c^7 - 7*a*b^2*c^7 + 2*b^3*c^7 + a^2*c^8 + 6*a*b*c^8 + b^2*c^8 - a*c^9 - b*c^9) : :
X(53746) = X[100] - 3 X[38690], X[103] - 3 X[38693], 2 X[3035] - 3 X[38772], 2 X[6712] - 3 X[21154], X[10742] - 3 X[38764], 2 X[11726] - 3 X[38032], 2 X[20418] + X[33520], 3 X[38752] - 5 X[38774], 3 X[38754] - X[38765], 2 X[38757] - 5 X[38775]

X(53746) lies on these lines: {3, 3887}, {9, 48}, {20, 10772}, {100, 38690}, {103, 38693}, {116, 6713}, {118, 2829}, {119, 6710}, {952, 38599}, {1537, 11728}, {2800, 11712}, {2808, 38602}, {2809, 11715}, {2820, 41191}, {3035, 38772}, {5773, 11219}, {6366, 31852}, {6712, 21154}, {10741, 38753}, {10742, 38764}, {11726, 38032}, {20418, 33520}, {38752, 38774}, {38754, 38765}, {38757, 38775}, {38759, 38773}

X(53746) = midpoint of X(i) and X(j) for these {i,j}: {20, 10772}, {101, 104}, {10741, 38753}
X(53746) = reflection of X(i) in X(j) for these {i,j}: {116, 6713}, {119, 6710}, {1537, 11728}, {38773, 38759}
X(53746) = circumcircle-inverse of X(53286)

---

X(53747) = MIDPOINT OF X(101) AND X(110)

Barycentrics    a^2*(a - b)*(a - c)*(2*a^6 - 2*a^4*b^2 - a^3*b^3 - a^2*b^4 + a*b^5 + b^6 + a^3*b^2*c - a^2*b^3*c - a*b^4*c + b^5*c - 2*a^4*c^2 + a^3*b*c^2 + 6*a^2*b^2*c^2 - 2*b^4*c^2 - a^3*c^3 - a^2*b*c^3 - a^2*c^4 - a*b*c^4 - 2*b^2*c^4 + a*c^5 + b*c^5 + c^6) : :
X(53747) = X[74] - 3 X[38690], X[103] - 3 X[15035], X[265] - 3 X[38764], 2 X[6699] - 3 X[38772], 2 X[6712] - 3 X[38793], X[10739] - 3 X[14643], X[10756] - 3 X[52699], 7 X[15020] - X[38668], 5 X[15034] + X[38666], 5 X[15040] - X[38574], 5 X[15051] - 3 X[38692], 3 X[15061] - 5 X[38774], 2 X[16534] + X[33520], X[24981] + 4 X[35024], 3 X[32609] + X[38572], 2 X[36253] - 5 X[38775], 3 X[38723] - X[38765]

X(53747) lies on these lines: {3, 2772}, {74, 38690}, {101, 110}, {103, 15035}, {116, 5972}, {118, 17702}, {125, 6710}, {265, 38764}, {544, 5642}, {1511, 2808}, {2809, 11720}, {2810, 6593}, {2813, 9129}, {2842, 17972}, {5663, 38599}, {6699, 38772}, {6712, 38793}, {10739, 14643}, {10741, 12121}, {10756, 52699}, {15020, 38668}, {15034, 38666}, {15040, 38574}, {15051, 38692}, {15061, 38774}, {16534, 33520}, {24981, 35024}, {32609, 38572}, {36253, 38775}, {38723, 38765}, {38726, 38773}

X(53747) = midpoint of X(i) and X(j) for these {i,j}: {101, 110}, {10741, 12121}
X(53747) = reflection of X(i) in X(j) for these {i,j}: {116, 5972}, {125, 6710}, {38773, 38726}
X(53747) = circumcircle-inverse of X(35327)
X(53747) = crossdifference of every pair of points on line {3120, 47234}

---

X(53748) = MIDPOINT OF X(102) AND X(104)

Barycentrics    a*(2*a^12 - 4*a^11*b - 3*a^10*b^2 + 13*a^9*b^3 - 8*a^8*b^4 - 12*a^7*b^5 + 22*a^6*b^6 - 2*a^5*b^7 - 18*a^4*b^8 + 8*a^3*b^9 + 5*a^2*b^10 - 3*a*b^11 - 4*a^11*c + 18*a^10*b*c - 17*a^9*b^2*c - 26*a^8*b^3*c + 65*a^7*b^4*c - 29*a^6*b^5*c - 53*a^5*b^6*c + 63*a^4*b^7*c - a^3*b^8*c - 25*a^2*b^9*c + 10*a*b^10*c - b^11*c - 3*a^10*c^2 - 17*a^9*b*c^2 + 70*a^8*b^2*c^2 - 53*a^7*b^3*c^2 - 79*a^6*b^4*c^2 + 147*a^5*b^5*c^2 - 39*a^4*b^6*c^2 - 67*a^3*b^7*c^2 + 50*a^2*b^8*c^2 - 10*a*b^9*c^2 + b^10*c^2 + 13*a^9*c^3 - 26*a^8*b*c^3 - 53*a^7*b^2*c^3 + 172*a^6*b^3*c^3 - 92*a^5*b^4*c^3 - 115*a^4*b^5*c^3 + 143*a^3*b^6*c^3 - 34*a^2*b^7*c^3 - 11*a*b^8*c^3 + 3*b^9*c^3 - 8*a^8*c^4 + 65*a^7*b*c^4 - 79*a^6*b^2*c^4 - 92*a^5*b^3*c^4 + 218*a^4*b^4*c^4 - 83*a^3*b^5*c^4 - 55*a^2*b^6*c^4 + 38*a*b^7*c^4 - 4*b^8*c^4 - 12*a^7*c^5 - 29*a^6*b*c^5 + 147*a^5*b^2*c^5 - 115*a^4*b^3*c^5 - 83*a^3*b^4*c^5 + 118*a^2*b^5*c^5 - 24*a*b^6*c^5 - 2*b^7*c^5 + 22*a^6*c^6 - 53*a^5*b*c^6 - 39*a^4*b^2*c^6 + 143*a^3*b^3*c^6 - 55*a^2*b^4*c^6 - 24*a*b^5*c^6 + 6*b^6*c^6 - 2*a^5*c^7 + 63*a^4*b*c^7 - 67*a^3*b^2*c^7 - 34*a^2*b^3*c^7 + 38*a*b^4*c^7 - 2*b^5*c^7 - 18*a^4*c^8 - a^3*b*c^8 + 50*a^2*b^2*c^8 - 11*a*b^3*c^8 - 4*b^4*c^8 + 8*a^3*c^9 - 25*a^2*b*c^9 - 10*a*b^2*c^9 + 3*b^3*c^9 + 5*a^2*c^10 + 10*a*b*c^10 + b^2*c^10 - 3*a*c^11 - b*c^11) : :
X(53748) = X[100] - 3 X[38691], X[109] - 3 X[38693], 2 X[3035] - 3 X[38784], 2 X[6718] - 3 X[21154], X[10742] - 3 X[38776], 2 X[11727] - 3 X[38032], 3 X[38752] - 5 X[38786], 3 X[38754] - X[38777], 2 X[38757] - 5 X[38787]

X(53748) lies on these lines: {3, 214}, {20, 10777}, {100, 38691}, {102, 104}, {109, 38693}, {117, 6713}, {119, 6711}, {124, 2829}, {952, 38600}, {953, 46618}, {1387, 52829}, {1537, 11734}, {1795, 39763}, {2817, 11715}, {2818, 38602}, {2849, 53294}, {3035, 38784}, {6718, 21154}, {10703, 23703}, {10742, 38776}, {10747, 38753}, {11727, 38032}, {13532, 36944}, {34789, 37043}, {38752, 38786}, {38754, 38777}, {38757, 38787}, {38759, 38785}

X(53748) = midpoint of X(i) and X(j) for these {i,j}: {20, 10777}, {102, 104}, {10747, 38753}
X(53748) = reflection of X(i) in X(j) for these {i,j}: {117, 6713}, {119, 6711}, {1537, 11734}, {38785, 38759}
X(53748) = circumcircle-inverse of X(53292)

---

X(53749) = MIDPOINT OF X(102) AND X(110)

Barycentrics    a^2*(2*a^14 - 2*a^13*b - 6*a^12*b^2 + 9*a^11*b^3 + 2*a^10*b^4 - 15*a^9*b^5 + 10*a^8*b^6 + 10*a^7*b^7 - 10*a^6*b^8 - 2*a^4*b^10 - 3*a^3*b^11 + 6*a^2*b^12 + a*b^13 - 2*b^14 - 2*a^13*c + 6*a^12*b*c - a^11*b^2*c - 17*a^10*b^3*c + 19*a^9*b^4*c + 11*a^8*b^5*c - 26*a^7*b^6*c + 6*a^6*b^7*c + 4*a^5*b^8*c - 4*a^4*b^9*c + 11*a^3*b^10*c - 5*a^2*b^11*c - 5*a*b^12*c + 3*b^13*c - 6*a^12*c^2 - a^11*b*c^2 + 32*a^10*b^2*c^2 - 14*a^9*b^3*c^2 - 44*a^8*b^4*c^2 + 41*a^7*b^5*c^2 - a^6*b^6*c^2 - 29*a^5*b^7*c^2 + 37*a^4*b^8*c^2 - 4*a^3*b^9*c^2 - 19*a^2*b^10*c^2 + 7*a*b^11*c^2 + b^12*c^2 + 9*a^11*c^3 - 17*a^10*b*c^3 - 14*a^9*b^2*c^3 + 52*a^8*b^3*c^3 - 21*a^7*b^4*c^3 - 41*a^6*b^5*c^3 + 51*a^5*b^6*c^3 - 13*a^4*b^7*c^3 - 28*a^3*b^8*c^3 + 26*a^2*b^9*c^3 + 3*a*b^10*c^3 - 7*b^11*c^3 + 2*a^10*c^4 + 19*a^9*b*c^4 - 44*a^8*b^2*c^4 - 21*a^7*b^3*c^4 + 88*a^6*b^4*c^4 - 26*a^5*b^5*c^4 - 47*a^4*b^6*c^4 + 43*a^3*b^7*c^4 - 8*a^2*b^8*c^4 - 15*a*b^9*c^4 + 9*b^10*c^4 - 15*a^9*c^5 + 11*a^8*b*c^5 + 41*a^7*b^2*c^5 - 41*a^6*b^3*c^5 - 26*a^5*b^4*c^5 + 58*a^4*b^5*c^5 - 19*a^3*b^6*c^5 - 21*a^2*b^7*c^5 + 11*a*b^8*c^5 + b^9*c^5 + 10*a^8*c^6 - 26*a^7*b*c^6 - a^6*b^2*c^6 + 51*a^5*b^3*c^6 - 47*a^4*b^4*c^6 - 19*a^3*b^5*c^6 + 42*a^2*b^6*c^6 - 2*a*b^7*c^6 - 8*b^8*c^6 + 10*a^7*c^7 + 6*a^6*b*c^7 - 29*a^5*b^2*c^7 - 13*a^4*b^3*c^7 + 43*a^3*b^4*c^7 - 21*a^2*b^5*c^7 - 2*a*b^6*c^7 + 6*b^7*c^7 - 10*a^6*c^8 + 4*a^5*b*c^8 + 37*a^4*b^2*c^8 - 28*a^3*b^3*c^8 - 8*a^2*b^4*c^8 + 11*a*b^5*c^8 - 8*b^6*c^8 - 4*a^4*b*c^9 - 4*a^3*b^2*c^9 + 26*a^2*b^3*c^9 - 15*a*b^4*c^9 + b^5*c^9 - 2*a^4*c^10 + 11*a^3*b*c^10 - 19*a^2*b^2*c^10 + 3*a*b^3*c^10 + 9*b^4*c^10 - 3*a^3*c^11 - 5*a^2*b*c^11 + 7*a*b^2*c^11 - 7*b^3*c^11 + 6*a^2*c^12 - 5*a*b*c^12 + b^2*c^12 + a*c^13 + 3*b*c^13 - 2*c^14) : :
X(53749) = X[74] - 3 X[38691], X[109] - 3 X[15035], X[265] - 3 X[38776], 2 X[6699] - 3 X[38784], 2 X[6718] - 3 X[38793], X[10740] - 3 X[14643], X[10757] - 3 X[52699], 7 X[15020] - X[38674], 5 X[15034] + X[38667], 5 X[15040] - X[38579], 5 X[15051] - 3 X[38697], 3 X[15061] - 5 X[38786], 3 X[32609] + X[38573], 2 X[36253] - 5 X[38787], 3 X[38723] - X[38777]

X(53749) lies on these lines: {3, 2773}, {74, 38691}, {102, 110}, {109, 15035}, {117, 5972}, {124, 17702}, {125, 6711}, {265, 38776}, {1511, 2818}, {2817, 11720}, {2819, 9129}, {5663, 38600}, {6699, 38784}, {6718, 38793}, {7471, 51382}, {10740, 14643}, {10747, 12121}, {10757, 52699}, {15020, 38674}, {15034, 38667}, {15040, 38579}, {15051, 38697}, {15061, 38786}, {32609, 38573}, {36253, 38787}, {38723, 38777}, {38726, 38785}

X(53749) = midpoint of X(i) and X(j) for these {i,j}: {102, 110}, {10747, 12121}
X(53749) = reflection of X(i) in X(j) for these {i,j}: {117, 5972}, {125, 6711}, {38785, 38726}
X(53749) = circumcircle-inverse of X(53295)

---

X(53750) = MIDPOINT OF X(103) AND X(104)

Barycentrics    a*(2*a^10 - 4*a^9*b - a^8*b^2 + 3*a^7*b^3 + 3*a^6*b^4 + 3*a^5*b^5 - 11*a^4*b^6 + a^3*b^7 + 7*a^2*b^8 - 3*a*b^9 - 4*a^9*c + 14*a^8*b*c - 7*a^7*b^2*c - 10*a^6*b^3*c - 6*a^5*b^4*c + 17*a^4*b^5*c + 9*a^3*b^6*c - 20*a^2*b^7*c + 8*a*b^8*c - b^9*c - a^8*c^2 - 7*a^7*b*c^2 + 16*a^6*b^2*c^2 + 3*a^5*b^3*c^2 - 16*a^4*b^4*c^2 - 5*a^3*b^5*c^2 + 16*a^2*b^6*c^2 - 7*a*b^7*c^2 + b^8*c^2 + 3*a^7*c^3 - 10*a^6*b*c^3 + 3*a^5*b^2*c^3 + 20*a^4*b^3*c^3 - 5*a^3*b^4*c^3 - 16*a^2*b^5*c^3 + 3*a*b^6*c^3 + 2*b^7*c^3 + 3*a^6*c^4 - 6*a^5*b*c^4 - 16*a^4*b^2*c^4 - 5*a^3*b^3*c^4 + 26*a^2*b^4*c^4 - a*b^5*c^4 - b^6*c^4 + 3*a^5*c^5 + 17*a^4*b*c^5 - 5*a^3*b^2*c^5 - 16*a^2*b^3*c^5 - a*b^4*c^5 - 2*b^5*c^5 - 11*a^4*c^6 + 9*a^3*b*c^6 + 16*a^2*b^2*c^6 + 3*a*b^3*c^6 - b^4*c^6 + a^3*c^7 - 20*a^2*b*c^7 - 7*a*b^2*c^7 + 2*b^3*c^7 + 7*a^2*c^8 + 8*a*b*c^8 + b^2*c^8 - 3*a*c^9 - b*c^9) : :
X(53750) = X[100] - 3 X[38692], X[101] - 3 X[38693], 2 X[6710] - 3 X[21154], 2 X[11728] - 3 X[38032], 2 X[20418] + X[33521]

X(53750) lies on these lines: {3, 2801}, {20, 10770}, {100, 38692}, {101, 38693}, {103, 104}, {116, 2829}, {118, 6713}, {119, 6712}, {952, 38601}, {1537, 11726}, {2800, 11714}, {2808, 38602}, {2809, 46684}, {5840, 38773}, {6710, 21154}, {10738, 38765}, {10739, 38753}, {11728, 38032}, {20418, 33521}, {24466, 38771}, {41166, 51682}

X(53750) = midpoint of X(i) and X(j) for these {i,j}: {20, 10770}, {103, 104}, {10738, 38765}, {10739, 38753}
X(53750) = reflection of X(i) in X(j) for these {i,j}: {118, 6713}, {119, 6712}, {1537, 11726}, {24466, 38771}
X(53750) = circumcircle-inverse of X(53296)

---

X(53751) = MIDPOINT OF X(103) AND X(110)

Barycentrics    a^2*(2*a^12 - 2*a^11*b - 4*a^10*b^2 + a^9*b^3 + 4*a^8*b^4 + 6*a^7*b^5 - 6*a^6*b^6 - 4*a^5*b^7 + 4*a^4*b^8 - 4*a^3*b^9 + 2*a^2*b^10 + 3*a*b^11 - 2*b^12 - 2*a^11*c + 2*a^10*b*c + 7*a^9*b^2*c - 7*a^8*b^3*c - 8*a^7*b^4*c + 8*a^6*b^5*c + 2*a^5*b^6*c - 2*a^4*b^7*c + 2*a^3*b^8*c - 2*a^2*b^9*c - a*b^10*c + b^11*c - 4*a^10*c^2 + 7*a^9*b*c^2 + 8*a^8*b^2*c^2 - 8*a^7*b^3*c^2 - 10*a^6*b^4*c^2 - 7*a^5*b^5*c^2 + 9*a^4*b^6*c^2 + 14*a^3*b^7*c^2 - 4*a^2*b^8*c^2 - 6*a*b^9*c^2 + b^10*c^2 + a^9*c^3 - 7*a^8*b*c^3 - 8*a^7*b^2*c^3 + 22*a^6*b^3*c^3 + 13*a^5*b^4*c^3 - 19*a^4*b^5*c^3 - 6*a^3*b^6*c^3 + 4*b^9*c^3 + 4*a^8*c^4 - 8*a^7*b*c^4 - 10*a^6*b^2*c^4 + 13*a^5*b^3*c^4 + 12*a^4*b^4*c^4 - 6*a^3*b^5*c^4 - 6*a^2*b^6*c^4 + a*b^7*c^4 + 6*a^7*c^5 + 8*a^6*b*c^5 - 7*a^5*b^2*c^5 - 19*a^4*b^3*c^5 - 6*a^3*b^4*c^5 + 20*a^2*b^5*c^5 + 3*a*b^6*c^5 - 5*b^7*c^5 - 6*a^6*c^6 + 2*a^5*b*c^6 + 9*a^4*b^2*c^6 - 6*a^3*b^3*c^6 - 6*a^2*b^4*c^6 + 3*a*b^5*c^6 + 2*b^6*c^6 - 4*a^5*c^7 - 2*a^4*b*c^7 + 14*a^3*b^2*c^7 + a*b^4*c^7 - 5*b^5*c^7 + 4*a^4*c^8 + 2*a^3*b*c^8 - 4*a^2*b^2*c^8 - 4*a^3*c^9 - 2*a^2*b*c^9 - 6*a*b^2*c^9 + 4*b^3*c^9 + 2*a^2*c^10 - a*b*c^10 + b^2*c^10 + 3*a*c^11 + b*c^11 - 2*c^12) : :
X(53751) = X[74] - 3 X[38692], X[101] - 3 X[15035], 2 X[6710] - 3 X[38793], X[10741] - 3 X[14643], X[10758] - 3 X[52699], 3 X[14644] - 5 X[31273], 7 X[15020] - X[38666], 5 X[15034] + X[38668], 5 X[15040] - X[38572], 5 X[15051] - 3 X[38690], 2 X[16534] + X[33521], X[20127] - 3 X[38766], 3 X[32609] + X[38574], 3 X[38764] - 5 X[38794], 2 X[38769] - 5 X[38795], 3 X[38772] - 4 X[48378]

X(53751) lies on these lines: {3, 2774}, {74, 38692}, {101, 15035}, {103, 110}, {116, 17702}, {118, 5972}, {125, 6712}, {1511, 2808}, {2777, 38773}, {2810, 33851}, {2824, 9129}, {5663, 38601}, {6710, 38793}, {7471, 18653}, {7728, 38765}, {10739, 12121}, {10741, 14643}, {10758, 52699}, {14644, 31273}, {15020, 38666}, {15034, 38668}, {15040, 38572}, {15051, 38690}, {16111, 38771}, {16534, 33521}, {20127, 38766}, {32609, 38574}, {38764, 38794}, {38769, 38795}, {38772, 48378}

X(53751) = midpoint of X(i) and X(j) for these {i,j}: {103, 110}, {7728, 38765}, {10739, 12121}
X(53751) = reflection of X(i) in X(j) for these {i,j}: {118, 5972}, {125, 6712}, {16111, 38771}
X(53751) = circumcircle-inverse of X(53301)

---

X(53752) = MIDPOINT OF X(104) AND X(109)

Barycentrics    a*(2*a^12 - 4*a^11*b - 5*a^10*b^2 + 15*a^9*b^3 - 20*a^7*b^5 + 10*a^6*b^6 + 10*a^5*b^7 - 10*a^4*b^8 + 3*a^2*b^10 - a*b^11 - 4*a^11*c + 18*a^10*b*c - 11*a^9*b^2*c - 40*a^8*b^3*c + 53*a^7*b^4*c + 13*a^6*b^5*c - 53*a^5*b^6*c + 21*a^4*b^7*c + 11*a^3*b^8*c - 11*a^2*b^9*c + 4*a*b^10*c - b^11*c - 5*a^10*c^2 - 11*a^9*b*c^2 + 58*a^8*b^2*c^2 - 25*a^7*b^3*c^2 - 83*a^6*b^4*c^2 + 79*a^5*b^5*c^2 + 13*a^4*b^6*c^2 - 39*a^3*b^7*c^2 + 16*a^2*b^8*c^2 - 4*a*b^9*c^2 + b^10*c^2 + 15*a^9*c^3 - 40*a^8*b*c^3 - 25*a^7*b^2*c^3 + 116*a^6*b^3*c^3 - 36*a^5*b^4*c^3 - 73*a^4*b^5*c^3 + 51*a^3*b^6*c^3 - 6*a^2*b^7*c^3 - 5*a*b^8*c^3 + 3*b^9*c^3 + 53*a^7*b*c^4 - 83*a^6*b^2*c^4 - 36*a^5*b^3*c^4 + 98*a^4*b^4*c^4 - 23*a^3*b^5*c^4 - 19*a^2*b^6*c^4 + 14*a*b^7*c^4 - 4*b^8*c^4 - 20*a^7*c^5 + 13*a^6*b*c^5 + 79*a^5*b^2*c^5 - 73*a^4*b^3*c^5 - 23*a^3*b^4*c^5 + 34*a^2*b^5*c^5 - 8*a*b^6*c^5 - 2*b^7*c^5 + 10*a^6*c^6 - 53*a^5*b*c^6 + 13*a^4*b^2*c^6 + 51*a^3*b^3*c^6 - 19*a^2*b^4*c^6 - 8*a*b^5*c^6 + 6*b^6*c^6 + 10*a^5*c^7 + 21*a^4*b*c^7 - 39*a^3*b^2*c^7 - 6*a^2*b^3*c^7 + 14*a*b^4*c^7 - 2*b^5*c^7 - 10*a^4*c^8 + 11*a^3*b*c^8 + 16*a^2*b^2*c^8 - 5*a*b^3*c^8 - 4*b^4*c^8 - 11*a^2*b*c^9 - 4*a*b^2*c^9 + 3*b^3*c^9 + 3*a^2*c^10 + 4*a*b*c^10 + b^2*c^10 - a*c^11 - b*c^11) : :
X(53752) = X[100] - 3 X[38697], X[102] - 3 X[38693], 2 X[6711] - 3 X[21154], 2 X[11734] - 3 X[38032]

X(53752) lies on these lines: {1, 104}, {3, 3738}, {20, 10771}, {100, 38697}, {102, 38693}, {117, 2829}, {119, 6718}, {124, 6713}, {900, 31866}, {946, 29008}, {952, 38607}, {1537, 11727}, {2716, 36037}, {2779, 13868}, {2802, 14690}, {2817, 46684}, {2818, 38602}, {3109, 10265}, {5840, 38785}, {6711, 21154}, {10738, 38777}, {10740, 38753}, {11734, 38032}, {24466, 38783}

X(53752) = midpoint of X(i) and X(j) for these {i,j}: {20, 10771}, {104, 109}, {2716, 36037}, {10738, 38777}, {10740, 38753}
X(53752) = reflection of X(i) in X(j) for these {i,j}: {119, 6718}, {124, 6713}, {946, 29008}, {1537, 11727}, {24466, 38783}, {25485, 47115}
X(53752) = circumcircle-inverse of X(53305)

---

X(53753) = MIDPOINT OF X(104) AND X(110)

Barycentrics    a*(2*a^12 - 2*a^11*b - 7*a^10*b^2 + 7*a^9*b^3 + 8*a^8*b^4 - 8*a^7*b^5 - 2*a^6*b^6 + 2*a^5*b^7 - 2*a^4*b^8 + 2*a^3*b^9 + a^2*b^10 - a*b^11 - 2*a^11*c + 8*a^10*b*c + a^9*b^2*c - 18*a^8*b^3*c + 6*a^7*b^4*c + 7*a^6*b^5*c - 4*a^5*b^6*c + 7*a^4*b^7*c - 4*a^3*b^8*c - 3*a^2*b^9*c + 3*a*b^10*c - b^11*c - 7*a^10*c^2 + a^9*b*c^2 + 22*a^8*b^2*c^2 - 8*a^7*b^3*c^2 - 19*a^6*b^4*c^2 + 13*a^5*b^5*c^2 - 6*a^3*b^7*c^2 + 4*a^2*b^8*c^2 + 7*a^9*c^3 - 18*a^8*b*c^3 - 8*a^7*b^2*c^3 + 34*a^6*b^3*c^3 - 7*a^5*b^4*c^3 - 17*a^4*b^5*c^3 + 14*a^3*b^6*c^3 - 2*a^2*b^7*c^3 - 6*a*b^8*c^3 + 3*b^9*c^3 + 8*a^8*c^4 + 6*a^7*b*c^4 - 19*a^6*b^2*c^4 - 7*a^5*b^3*c^4 + 20*a^4*b^4*c^4 - 6*a^3*b^5*c^4 - 5*a^2*b^6*c^4 + 3*a*b^7*c^4 - 8*a^7*c^5 + 7*a^6*b*c^5 + 13*a^5*b^2*c^5 - 17*a^4*b^3*c^5 - 6*a^3*b^4*c^5 + 10*a^2*b^5*c^5 + a*b^6*c^5 - 2*b^7*c^5 - 2*a^6*c^6 - 4*a^5*b*c^6 + 14*a^3*b^3*c^6 - 5*a^2*b^4*c^6 + a*b^5*c^6 + 2*a^5*c^7 + 7*a^4*b*c^7 - 6*a^3*b^2*c^7 - 2*a^2*b^3*c^7 + 3*a*b^4*c^7 - 2*b^5*c^7 - 2*a^4*c^8 - 4*a^3*b*c^8 + 4*a^2*b^2*c^8 - 6*a*b^3*c^8 + 2*a^3*c^9 - 3*a^2*b*c^9 + 3*b^3*c^9 + a^2*c^10 + 3*a*b*c^10 - a*c^11 - b*c^11) : :
X(53753) = X[74] - 3 X[38693], X[100] - 3 X[15035], 2 X[3035] - 3 X[38793], 4 X[6667] - 3 X[23515], 2 X[6699] - 3 X[21154], 2 X[7687] - 3 X[23513], X[10742] - 3 X[14643], X[10759] - 3 X[52699], 2 X[11735] - 3 X[38032], X[12331] - 5 X[15040], X[12773] + 3 X[32609], 3 X[14644] - 5 X[31272], 7 X[15020] - X[38665], 5 X[15034] + X[38669], 5 X[15051] - 3 X[34474], 2 X[15118] - 3 X[38119], X[20127] - 3 X[38754], 2 X[20304] - 3 X[34126], 2 X[20418] + X[30714], 3 X[38069] - 2 X[45311], 3 X[38752] - 5 X[38794], 2 X[38757] - 5 X[38795], 3 X[38760] - 4 X[48378]

X(53753) lies on these lines: {3, 8674}, {11, 17702}, {20, 10767}, {21, 104}, {74, 38693}, {100, 15035}, {113, 2829}, {119, 5972}, {125, 6713}, {513, 46636}, {517, 31523}, {952, 1511}, {1484, 34153}, {1537, 11723}, {2777, 38761}, {2800, 11720}, {2830, 9129}, {2850, 31847}, {3035, 38793}, {5533, 12896}, {5663, 38602}, {5840, 16163}, {6667, 23515}, {6699, 21154}, {7471, 51420}, {7687, 23513}, {7728, 38753}, {8068, 18968}, {8702, 46635}, {9024, 33851}, {10058, 10091}, {10074, 10088}, {10738, 12121}, {10742, 14643}, {10759, 52699}, {10778, 12383}, {11735, 38032}, {12331, 15040}, {12737, 12778}, {12773, 32609}, {12898, 19914}, {13913, 46688}, {13977, 46689}, {14644, 31272}, {15020, 38665}, {15034, 38669}, {15051, 34474}, {15118, 38119}, {16111, 38759}, {19081, 19111}, {19082, 19110}, {20127, 38754}, {20304, 34126}, {20418, 30714}, {24466, 38726}, {38069, 45311}, {38752, 38794}, {38757, 38795}, {38760, 48378}, {46686, 52836}, {48700, 49269}, {48701, 49268}

X(53753) = midpoint of X(i) and X(j) for these {i,j}: {20, 10767}, {104, 110}, {1484, 34153}, {7728, 38753}, {10738, 12121}, {10778, 12383}, {12737, 12778}, {12898, 19914}
X(53753) = reflection of X(i) in X(j) for these {i,j}: {119, 5972}, {125, 6713}, {1537, 11723}, {16111, 38759}, {24466, 38726}, {31525, 1385}, {52836, 46686}
X(53753) = circumcircle-inverse of X(53306)

---

X(53754) = MIDPOINT OF X(104) AND X(111)

Barycentrics    a*(2*a^12 - 2*a^11*b - 9*a^10*b^2 + 9*a^9*b^3 + 8*a^8*b^4 - 8*a^7*b^5 + 6*a^6*b^6 - 6*a^5*b^7 - 10*a^4*b^8 + 10*a^3*b^9 + 3*a^2*b^10 - 3*a*b^11 - 2*a^11*c + 8*a^10*b*c + 3*a^9*b^2*c - 28*a^8*b^3*c + 14*a^7*b^4*c - a^6*b^5*c - 8*a^5*b^6*c + 29*a^4*b^7*c - 12*a^3*b^8*c - 7*a^2*b^9*c + 5*a*b^10*c - b^11*c - 9*a^10*c^2 + 3*a^9*b*c^2 + 46*a^8*b^2*c^2 - 24*a^7*b^3*c^2 - 61*a^6*b^4*c^2 + 59*a^5*b^5*c^2 + 12*a^4*b^6*c^2 - 34*a^3*b^7*c^2 - 4*a^2*b^8*c^2 + 12*a*b^9*c^2 + 9*a^9*c^3 - 28*a^8*b*c^3 - 24*a^7*b^2*c^3 + 122*a^6*b^3*c^3 - 37*a^5*b^4*c^3 - 81*a^4*b^5*c^3 + 58*a^3*b^6*c^3 + 14*a^2*b^7*c^3 - 22*a*b^8*c^3 + 5*b^9*c^3 + 8*a^8*c^4 + 14*a^7*b*c^4 - 61*a^6*b^2*c^4 - 37*a^5*b^3*c^4 + 92*a^4*b^4*c^4 - 22*a^3*b^5*c^4 - 15*a^2*b^6*c^4 - 11*a*b^7*c^4 - 8*a^7*c^5 - a^6*b*c^5 + 59*a^5*b^2*c^5 - 81*a^4*b^3*c^5 - 22*a^3*b^4*c^5 + 18*a^2*b^5*c^5 + 19*a*b^6*c^5 - 4*b^7*c^5 + 6*a^6*c^6 - 8*a^5*b*c^6 + 12*a^4*b^2*c^6 + 58*a^3*b^3*c^6 - 15*a^2*b^4*c^6 + 19*a*b^5*c^6 - 6*a^5*c^7 + 29*a^4*b*c^7 - 34*a^3*b^2*c^7 + 14*a^2*b^3*c^7 - 11*a*b^4*c^7 - 4*b^5*c^7 - 10*a^4*c^8 - 12*a^3*b*c^8 - 4*a^2*b^2*c^8 - 22*a*b^3*c^8 + 10*a^3*c^9 - 7*a^2*b*c^9 + 12*a*b^2*c^9 + 5*b^3*c^9 + 3*a^2*c^10 + 5*a*b*c^10 - 3*a*c^11 - b*c^11) : :
X(53754) =X[100] - 3 X[38698], X[1296] - 3 X[38693], 2 X[3035] - 3 X[38804], X[10742] - 3 X[38796], X[10759] - 3 X[36696], X[12773] + 3 X[52698], 3 X[21154] - 2 X[40556], 3 X[34126] - 2 X[40340], 3 X[38752] - 5 X[38806], 3 X[38754] - X[38797], 2 X[38757] - 5 X[38807]

X(53754) lies on these lines: {3, 2805}, {11, 23699}, {100, 38698}, {104, 111}, {119, 6719}, {126, 6713}, {952, 14650}, {1296, 38693}, {2771, 9129}, {2800, 11721}, {2829, 5512}, {2831, 50381}, {3035, 38804}, {10742, 38796}, {10759, 36696}, {10779, 14654}, {12773, 52698}, {21154, 40556}, {22338, 38753}, {33962, 38602}, {34126, 40340}, {38752, 38806}, {38754, 38797}, {38757, 38807}, {38759, 38805}

X(53754) = midpoint of X(i) and X(j) for these {i,j}: {104, 111}, {10779, 14654}, {22338, 38753}
X(53754) = reflection of X(i) in X(j) for these {i,j}: {119, 6719}, {126, 6713}, {38805, 38759}

---

X(53755) = MIDPOINT OF X(104) AND X(112)

Barycentrics    a*(2*a^16 - 2*a^15*b - 7*a^14*b^2 + 7*a^13*b^3 + 8*a^12*b^4 - 8*a^11*b^5 - a^10*b^6 + a^9*b^7 - 6*a^8*b^8 + 6*a^7*b^9 + 7*a^6*b^10 - 7*a^5*b^11 - 4*a^4*b^12 + 4*a^3*b^13 + a^2*b^14 - a*b^15 - 2*a^15*c + 8*a^14*b*c + a^13*b^2*c - 18*a^12*b^3*c + 6*a^11*b^4*c + 9*a^10*b^5*c - 7*a^9*b^6*c + 7*a^8*b^7*c + 2*a^7*b^8*c - 14*a^6*b^9*c + 3*a^5*b^10*c + 12*a^4*b^11*c - 6*a^3*b^12*c - 3*a^2*b^13*c + 3*a*b^14*c - b^15*c - 7*a^14*c^2 + a^13*b*c^2 + 22*a^12*b^2*c^2 - 8*a^11*b^3*c^2 - 20*a^10*b^4*c^2 + 12*a^9*b^5*c^2 + 8*a^8*b^6*c^2 - 12*a^7*b^7*c^2 - 3*a^6*b^8*c^2 + 13*a^5*b^9*c^2 - 6*a^4*b^10*c^2 - 4*a^3*b^11*c^2 + 6*a^2*b^12*c^2 - 2*a*b^13*c^2 + 7*a^13*c^3 - 18*a^12*b*c^3 - 8*a^11*b^2*c^3 + 30*a^10*b^3*c^3 - 2*a^9*b^4*c^3 - 17*a^8*b^5*c^3 + 4*a^7*b^6*c^3 + 16*a^6*b^7*c^3 - 9*a^5*b^8*c^3 - 8*a^4*b^9*c^3 + 12*a^3*b^10*c^3 - 6*a^2*b^11*c^3 - 4*a*b^12*c^3 + 3*b^13*c^3 + 8*a^12*c^4 + 6*a^11*b*c^4 - 20*a^10*b^2*c^4 - 2*a^9*b^3*c^4 + 12*a^8*b^4*c^4 - 4*a^6*b^6*c^4 - 8*a^5*b^7*c^4 + 12*a^4*b^8*c^4 - 6*a^3*b^9*c^4 - 8*a^2*b^10*c^4 + 10*a*b^11*c^4 - 8*a^11*c^5 + 9*a^10*b*c^5 + 12*a^9*b^2*c^5 - 17*a^8*b^3*c^5 - 4*a^6*b^5*c^5 + 8*a^5*b^6*c^5 - 4*a^4*b^7*c^5 - 8*a^3*b^8*c^5 + 19*a^2*b^9*c^5 - 4*a*b^10*c^5 - 3*b^11*c^5 - a^10*c^6 - 7*a^9*b*c^6 + 8*a^8*b^2*c^6 + 4*a^7*b^3*c^6 - 4*a^6*b^4*c^6 + 8*a^5*b^5*c^6 - 4*a^4*b^6*c^6 + 8*a^3*b^7*c^6 + a^2*b^8*c^6 - 13*a*b^9*c^6 + a^9*c^7 + 7*a^8*b*c^7 - 12*a^7*b^2*c^7 + 16*a^6*b^3*c^7 - 8*a^5*b^4*c^7 - 4*a^4*b^5*c^7 + 8*a^3*b^6*c^7 - 20*a^2*b^7*c^7 + 11*a*b^8*c^7 + b^9*c^7 - 6*a^8*c^8 + 2*a^7*b*c^8 - 3*a^6*b^2*c^8 - 9*a^5*b^3*c^8 + 12*a^4*b^4*c^8 - 8*a^3*b^5*c^8 + a^2*b^6*c^8 + 11*a*b^7*c^8 + 6*a^7*c^9 - 14*a^6*b*c^9 + 13*a^5*b^2*c^9 - 8*a^4*b^3*c^9 - 6*a^3*b^4*c^9 + 19*a^2*b^5*c^9 - 13*a*b^6*c^9 + b^7*c^9 + 7*a^6*c^10 + 3*a^5*b*c^10 - 6*a^4*b^2*c^10 + 12*a^3*b^3*c^10 - 8*a^2*b^4*c^10 - 4*a*b^5*c^10 - 7*a^5*c^11 + 12*a^4*b*c^11 - 4*a^3*b^2*c^11 - 6*a^2*b^3*c^11 + 10*a*b^4*c^11 - 3*b^5*c^11 - 4*a^4*c^12 - 6*a^3*b*c^12 + 6*a^2*b^2*c^12 - 4*a*b^3*c^12 + 4*a^3*c^13 - 3*a^2*b*c^13 - 2*a*b^2*c^13 + 3*b^3*c^13 + a^2*c^14 + 3*a*b*c^14 - a*c^15 - b*c^15) : :
X(53755) = X[100] - 3 X[38699], X[1297] - 3 X[38693], X[14900] + 2 X[20418], 3 X[21154] - 2 X[34841]

X(53755) lies on these lines: {3, 2806}, {11, 2794}, {100, 38699}, {104, 112}, {119, 6720}, {127, 6713}, {132, 2829}, {952, 38608}, {1297, 38693}, {2800, 11722}, {2830, 50381}, {5840, 14689}, {10058, 13312}, {10074, 13311}, {10780, 13200}, {12918, 38753}, {14900, 20418}, {19081, 19115}, {19082, 19114}, {21154, 34841}, {48700, 49271}, {48701, 49270}

X(53755) = midpoint of X(i) and X(j) for these {i,j}: {104, 112}, {10780, 13200}, {12918, 38753}
X(53755) = reflection of X(i) in X(j) for these {i,j}: {119, 6720}, {127, 6713}

---

X(53756) = MIDPOINT OF X(105) AND X(110)

Barycentrics    a*(2*a^10 - 2*a^9*b - a^8*b^2 + a^7*b^3 - 3*a^6*b^4 + 3*a^5*b^5 + a^4*b^6 - a^3*b^7 + a^2*b^8 - a*b^9 - 2*a^9*c - 4*a^8*b*c + 7*a^7*b^2*c - 5*a^5*b^4*c + 5*a^4*b^5*c - a^3*b^6*c + a*b^8*c - b^9*c - a^8*c^2 + 7*a^7*b*c^2 + 8*a^6*b^2*c^2 - 8*a^5*b^3*c^2 - 4*a^4*b^4*c^2 - 4*a^3*b^5*c^2 - 3*a^2*b^6*c^2 + 3*a*b^7*c^2 + 2*b^8*c^2 + a^7*c^3 - 8*a^5*b^2*c^3 + 2*a^4*b^3*c^3 + 10*a^3*b^4*c^3 - 4*a^2*b^5*c^3 - 3*a*b^6*c^3 - 3*a^6*c^4 - 5*a^5*b*c^4 - 4*a^4*b^2*c^4 + 10*a^3*b^3*c^4 + 8*a^2*b^4*c^4 - 2*b^6*c^4 + 3*a^5*c^5 + 5*a^4*b*c^5 - 4*a^3*b^2*c^5 - 4*a^2*b^3*c^5 + 2*b^5*c^5 + a^4*c^6 - a^3*b*c^6 - 3*a^2*b^2*c^6 - 3*a*b^3*c^6 - 2*b^4*c^6 - a^3*c^7 + 3*a*b^2*c^7 + a^2*c^8 + a*b*c^8 + 2*b^2*c^8 - a*c^9 - b*c^9) : :
X(53756) = X[74] - 3 X[38694], X[1292] - 3 X[15035], X[10743] - 3 X[14643], X[10760] - 3 X[52699], 7 X[15020] - X[38684], 5 X[15034] + X[38670], 5 X[15040] - X[38589], 5 X[15051] - 3 X[38712], 3 X[32609] + X[38575]

X(53756) lies on these lines: {3, 2775}, {74, 38694}, {81, 105}, {120, 5972}, {125, 6714}, {528, 5642}, {1292, 15035}, {1511, 28915}, {2795, 16164}, {2809, 11720}, {2834, 20772}, {2837, 9129}, {5511, 17702}, {5663, 38603}, {8674, 46409}, {9181, 51420}, {10743, 14643}, {10760, 52699}, {12121, 15521}, {15020, 38684}, {15034, 38670}, {15040, 38589}, {15051, 38712}, {32609, 38575}

X(53756) = midpoint of X(i) and X(j) for these {i,j}: {105, 110}, {12121, 15521}
X(53756) = reflection of X(i) in X(j) for these {i,j}: {120, 5972}, {125, 6714}
X(53756) = circumcircle-inverse of X(53309)
X(53756) = crossdifference of every pair of points on line {24290, 47231}

---

X(53757) = MIDPOINT OF X(107) AND X(110)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(2*a^14 - 2*a^12*b^2 - 8*a^10*b^4 + 13*a^8*b^6 - 2*a^6*b^8 - 4*a^4*b^10 + b^14 - 2*a^12*c^2 + 18*a^10*b^2*c^2 - 13*a^8*b^4*c^2 - 24*a^6*b^6*c^2 + 20*a^4*b^8*c^2 + 6*a^2*b^10*c^2 - 5*b^12*c^2 - 8*a^10*c^4 - 13*a^8*b^2*c^4 + 52*a^6*b^4*c^4 - 16*a^4*b^6*c^4 - 24*a^2*b^8*c^4 + 9*b^10*c^4 + 13*a^8*c^6 - 24*a^6*b^2*c^6 - 16*a^4*b^4*c^6 + 36*a^2*b^6*c^6 - 5*b^8*c^6 - 2*a^6*c^8 + 20*a^4*b^2*c^8 - 24*a^2*b^4*c^8 - 5*b^6*c^8 - 4*a^4*c^10 + 6*a^2*b^2*c^10 + 9*b^4*c^10 - 5*b^2*c^12 + c^14) : :
X(53757) = X[10745] - 3 X[14643], 2 X[34842] - 3 X[38793], X[74] - 3 X[23239], X[1294] - 3 X[15035], X[10762] - 3 X[52699], 4 X[12900] - 3 X[36520], 3 X[14847] + X[24981], 7 X[15020] - X[38686], 5 X[15034] + X[38672], 5 X[15040] - X[38591], 5 X[15051] - 3 X[38714], 2 X[16534] + X[52057], 3 X[32609] + X[38577]

X(53757) lies on these lines: {3, 113}, {74, 23239}, {107, 110}, {112, 14345}, {125, 6716}, {133, 17702}, {520, 31510}, {542, 24930}, {1294, 15035}, {1301, 39447}, {1304, 6368}, {2790, 20772}, {2847, 9129}, {5642, 9530}, {5663, 38605}, {9528, 16164}, {10706, 11845}, {10762, 52699}, {12121, 22337}, {12900, 36520}, {13202, 36162}, {14847, 24981}, {14934, 51393}, {15020, 38686}, {15034, 38672}, {15040, 38591}, {15051, 38714}, {16534, 52057}, {32609, 38577}

X(53757) = midpoint of X(i) and X(j) for these {i,j}: {107, 110}, {7728, 23240}, {12121, 22337}
X(53757) = reflection of X(i) in X(j) for these {i,j}: {122, 5972}, {125, 6716}
X(53757) = circumcircle-inverse of X(1624)
X(53757) = crossdifference of every pair of points on line {3269, 46425}

---

X(53758) = MIDPOINT OF X(109) AND X(110)

Barycentrics    a^2*(a - b)*(a - c)*(2*a^8 - 4*a^6*b^2 + a^5*b^3 + a^4*b^4 - 2*a^3*b^5 + 2*a^2*b^6 + a*b^7 - b^8 + 4*a^6*b*c - a^5*b^2*c - 5*a^4*b^3*c + 2*a^3*b^4*c - 2*a^2*b^5*c - a*b^6*c + 3*b^7*c - 4*a^6*c^2 - a^5*b*c^2 + 10*a^4*b^2*c^2 - 5*a^2*b^4*c^2 + a*b^5*c^2 - b^6*c^2 + a^5*c^3 - 5*a^4*b*c^3 + 10*a^2*b^3*c^3 - a*b^4*c^3 - 3*b^5*c^3 + a^4*c^4 + 2*a^3*b*c^4 - 5*a^2*b^2*c^4 - a*b^3*c^4 + 4*b^4*c^4 - 2*a^3*c^5 - 2*a^2*b*c^5 + a*b^2*c^5 - 3*b^3*c^5 + 2*a^2*c^6 - a*b*c^6 - b^2*c^6 + a*c^7 + 3*b*c^7 - c^8) : :
X(53758) = X[74] - 3 X[38697], X[102] - 3 X[15035], 2 X[6711] - 3 X[38793], X[10747] - 3 X[14643], X[10764] - 3 X[52699], 7 X[15020] - X[38667], 5 X[15034] + X[38674], 5 X[15040] - X[38573], 5 X[15051] - 3 X[38691], X[20127] - 3 X[38778], 3 X[32609] + X[38579], 3 X[38776] - 5 X[38794], 2 X[38781] - 5 X[38795], 3 X[38784] - 4 X[48378]

X(53758) lies on these lines: {3, 2779}, {74, 38697}, {102, 15035}, {109, 110}, {117, 17702}, {124, 5972}, {125, 6718}, {1511, 2818}, {1795, 10088}, {2777, 38785}, {2800, 11720}, {2852, 9129}, {5663, 38607}, {6126, 43692}, {6711, 38793}, {7728, 38777}, {10740, 12121}, {10747, 14643}, {10764, 52699}, {15020, 38667}, {15034, 38674}, {15040, 38573}, {15051, 38691}, {16111, 38783}, {17973, 44764}, {20127, 38778}, {32609, 38579}, {38776, 38794}, {38781, 38795}, {38784, 48378}

X(53758) = midpoint of X(i) and X(j) for these {i,j}: {109, 110}, {7728, 38777}, {10740, 12121}
X(53758) = reflection of X(i) in X(j) for these {i,j}: {124, 5972}, {125, 6718}, {16111, 38783}
X(53758) = circumcircle-inverse of X(53324)

---

X(53759) = MIDPOINT OF X(109) AND X(111)

Barycentrics    a^2*(2*a^10 - 2*a^9*b - 6*a^8*b^2 + 7*a^7*b^3 - 2*a^6*b^4 - a^5*b^5 + 8*a^4*b^6 - 7*a^3*b^7 + 3*a*b^9 - 2*b^10 - 2*a^9*c + 6*a^8*b*c + a^7*b^2*c - 15*a^6*b^3*c + 13*a^5*b^4*c - 11*a^4*b^5*c + 3*a^3*b^6*c + 11*a^2*b^7*c - 7*a*b^8*c + b^9*c - 6*a^8*c^2 + a^7*b*c^2 + 36*a^6*b^2*c^2 - 22*a^5*b^3*c^2 - 30*a^4*b^4*c^2 + 38*a^3*b^5*c^2 - 21*a^2*b^6*c^2 - 5*a*b^7*c^2 + 9*b^8*c^2 + 7*a^7*c^3 - 15*a^6*b*c^3 - 22*a^5*b^2*c^3 + 72*a^4*b^3*c^3 - 30*a^3*b^4*c^3 - 28*a^2*b^5*c^3 + 23*a*b^6*c^3 - 7*b^7*c^3 - 2*a^6*c^4 + 13*a^5*b*c^4 - 30*a^4*b^2*c^4 - 30*a^3*b^3*c^4 + 72*a^2*b^4*c^4 - 14*a*b^5*c^4 - 11*b^6*c^4 - a^5*c^5 - 11*a^4*b*c^5 + 38*a^3*b^2*c^5 - 28*a^2*b^3*c^5 - 14*a*b^4*c^5 + 20*b^5*c^5 + 8*a^4*c^6 + 3*a^3*b*c^6 - 21*a^2*b^2*c^6 + 23*a*b^3*c^6 - 11*b^4*c^6 - 7*a^3*c^7 + 11*a^2*b*c^7 - 5*a*b^2*c^7 - 7*b^3*c^7 - 7*a*b*c^8 + 9*b^2*c^8 + 3*a*c^9 + b*c^9 - 2*c^10) : :
X(53759) = X[102] - 3 X[38698], X[1296] - 3 X[38697], 2 X[6711] - 3 X[38804], X[10747] - 3 X[38796], X[10764] - 3 X[36696], X[38579] + 3 X[52698], 3 X[38776] - 5 X[38806], 3 X[38778] - X[38797], 2 X[38781] - 5 X[38807]

X(53759) lies on these lines: {3, 2819}, {102, 38698}, {109, 111}, {117, 23699}, {124, 6719}, {126, 6718}, {1296, 38697}, {2773, 9129}, {2800, 11721}, {2818, 14650}, {2853, 50381}, {6711, 38804}, {10747, 38796}, {10764, 36696}, {22338, 38777}, {33962, 38607}, {38579, 52698}, {38776, 38806}, {38778, 38797}, {38781, 38807}, {38783, 38805}

X(53759) = midpoint of X(i) and X(j) for these {i,j}: {109, 111}, {22338, 38777}
X(53759) = reflection of X(i) in X(j) for these {i,j}: {124, 6719}, {126, 6718}, {38805, 38783}

---

X(53760) = MIDPOINT OF X(110) AND X(112)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a^10 - a^8*b^2 - a^6*b^4 + a^4*b^6 - a^8*c^2 + 3*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 + 4*a^2*b^4*c^4 - b^6*c^4 + a^4*c^6 - 2*a^2*b^2*c^6 - b^4*c^6 + b^2*c^8) : :
X(53760) = X[74] - 3 X[38699], X[1297] - 3 X[15035], X[10749] - 3 X[14643], X[10766] - 3 X[52699], 2 X[12236] - 3 X[16224], X[13115] - 5 X[15040], X[13310] + 3 X[32609], X[14900] + 2 X[16534], 7 X[15020] - X[38689], 5 X[15034] + X[38676], 5 X[15051] - 3 X[38717], 3 X[16225] - X[21649], 2 X[34841] - 3 X[38793]

X(53760) lies on these lines: {3, 1177}, {67, 41255}, {74, 38699}, {110, 112}, {113, 2794}, {125, 6720}, {127, 5972}, {132, 17702}, {525, 46619}, {542, 43389}, {690, 14574}, {1297, 15035}, {1503, 46637}, {2777, 14689}, {2854, 28343}, {5649, 11636}, {5663, 17974}, {9129, 20772}, {9181, 34968}, {9513, 13193}, {10088, 13312}, {10091, 13311}, {10749, 14643}, {10766, 52699}, {12121, 12918}, {12236, 16224}, {12824, 36176}, {13115, 15040}, {13310, 32609}, {13923, 46688}, {13992, 46689}, {14900, 16534}, {15020, 38689}, {15034, 38676}, {15051, 38717}, {16165, 37921}, {16225, 21649}, {16278, 44127}, {18556, 35278}, {19110, 19115}, {19111, 19114}, {34841, 38793}, {49268, 49271}, {49269, 49270}

X(53760) = midpoint of X(i) and X(j) for these {i,j}: {110, 112}, {12121, 12918}
X(53760) = reflection of X(i) in X(j) for these {i,j}: {125, 6720}, {127, 5972}
X(53760) = circumcircle-inverse of X(1576)
X(53760) = crossdifference of every pair of points on line {125, 47138}
X(53760) = X(6720)-lineconjugate of X(125)
X(53760) = barycentric product X(i)*X(j) for these {i,j}: {110, 40856}, {249, 47004}
X(53760) = barycentric quotient X(i)/X(j) for these {i,j}: {40856, 850}, {47004, 338}

---

X(53761) = X(3)X(2828)∩X(100)X(190)

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

X(53761) lies on these lines: {3, 2828}, {100, 190}, {107, 1624}, {108, 40117}, {162, 53325}, {221, 1047}, {523, 37966}, {653, 53321}, {1376, 1762}, {1897, 53288}, {2939, 11500}, {3658, 53349}, {44661, 52889}, {53253, 53566}

X(53761) = crossdifference of every pair of points on line {1015, 35071}
X(53761) = circumcircle-inverse of the midpoint of X(100) and X(107)
X(53761) = {X(100),X(14543)}-harmonic conjugate of X(53280)

---

X(53762) = CIRCUMCIRCLE-INVERSE OF X(53755)

Barycentrics    a^2*(b - c)*(a^7 - a^5*b^2 - a^3*b^4 + a*b^6 + 3*a^5*b*c - 3*a^4*b^2*c - 2*a^3*b^3*c + 2*a^2*b^4*c - a*b^5*c + b^6*c - a^5*c^2 - 3*a^4*b*c^2 + 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 - a*b^4*c^2 + b^5*c^2 - 2*a^3*b*c^3 + 2*a^2*b^2*c^3 + 2*a*b^3*c^3 - 2*b^4*c^3 - a^3*c^4 + 2*a^2*b*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(53762) lies on these lines: {3, 2806}, {104, 900}, {112, 1576}, {512, 5006}, {523, 37961}, {667, 17115}, {884, 32658}, {4185, 47203}, {16049, 53353}, {21789, 39201}, {34858, 53549}

X(53762) = crossdifference of every pair of points on line {1211, 15526}
X(53762) = circumcircle-inverse of X(53755)

---

X(53763) = CIRCUMCIRCLE-INVERSE OF X(53759)

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

X(53763) lies on these lines: {3, 2819}, {6, 5075}, {55, 21828}, {109, 692}, {111, 351}, {663, 6589}, {968, 21894}, {1945, 23351}, {3310, 53287}, {5040, 53315}, {42666, 53249}

X(53763) = isogonal conjugate of the isotomic conjugate of X(53356)
X(53763) = crossdifference of every pair of points on line {1146, 2482}
X(53763) = circumcircle-inverse of X(53759)
X(53763) = barycentric product X(6)*X(53356)
X(53763) = barycentric quotient X(53356)/X(76)
X(53763) = {X(351),X(53281)}-harmonic conjugate of X(53289)

---

X(53764) = X(3)X(524)∩X(4)X(5166)

Barycentrics    (a^4 - 4*a^2*b^2 + b^4 - c^4)*(a^4 - b^4 - 4*a^2*c^2 + c^4)*(4*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 4*b^2*c^2 + c^4) : :
X(53764) = X[5486] - 3 X[51239], X[4] - 3 X[5166]

X((53764) lies on the cubics K1281 and K1321 and also on these lines: {3, 524}, {4, 5166}, {30, 6094}, {98, 5913}, {352, 6776}, {542, 9127}, {598, 38951}, {843, 30247}, {1499, 23287}, {2030, 46659}, {7737, 46959}, {8593, 37860}, {11336, 32133}, {12506, 16511}

X(53764) = midpoint of X(352) and X(6776)
X(53764) = reflection of X(46659) in X(2030)
X(53764) = isogonal conjugate of X(34241)
X(53764) = barycentric product X(i)*X(j) for these {i,j}: {5486, 22329}, {13608, 17952}, {18775, 39157}
X(53764) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 34241}, {2030, 1995}, {5486, 5503}, {17968, 14262}, {18775, 34166}, {22329, 11185}

---

X(53765) = X(3)X(76)∩X(4)X(12177)

Barycentrics    a^8 + a^6*b^2 - a^4*b^4 - a^2*b^6 + 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 + 4*b^4*c^4 - a^2*c^6 - b^2*c^6 : :
X(53765) = 3 X[98] - 4 X[14880], 3 X[99] - 2 X[1975], X[1975] - 3 X[5989], 3 X[34473] - 2 X[38654], 3 X[671] - 4 X[5254], 3 X[5182] - 4 X[19120], 8 X[7789] - 9 X[41134], 4 X[7805] - 3 X[36849], 10 X[7851] - 9 X[9166]

X((53765) lies on the cubic K1321 and these lines: {3, 76}, {4, 12177}, {5, 35705}, {32, 543}, {83, 597}, {115, 7803}, {147, 32816}, {148, 4027}, {182, 11185}, {194, 14931}, {315, 542}, {316, 1503}, {325, 52090}, {384, 8289}, {620, 33001}, {698, 5104}, {1569, 8178}, {1691, 47286}, {1916, 7839}, {2482, 7815}, {2930, 45809}, {3398, 32819}, {3785, 11177}, {4048, 40332}, {5026, 7770}, {5171, 10992}, {5969, 7754}, {5986, 20023}, {6054, 7752}, {6308, 17129}, {6321, 18502}, {6392, 20094}, {6656, 11646}, {7470, 50640}, {7748, 10350}, {7760, 10754}, {7763, 14981}, {7783, 9888}, {7789, 41134}, {7793, 8591}, {7805, 36849}, {7812, 8593}, {7824, 15483}, {7841, 9830}, {7851, 9166}, {7859, 14061}, {7883, 11161}, {8352, 12151}, {8592, 33013}, {8596, 12191}, {8598, 11054}, {10330, 53346}, {10349, 44518}, {10358, 32135}, {10991, 14907}, {11159, 12150}, {11623, 32832}, {11632, 12054}, {12213, 31709}, {12214, 31710}, {12215, 39266}, {12355, 18501}, {13196, 53419}, {19911, 33274}, {20398, 53127}, {31513, 41429}, {31859, 42535}, {32818, 46236}, {34505, 39560}, {35356, 46512}, {39141, 43448}, {39603, 51932}, {41377, 44132}

X(53765) = reflection of X(i) in X(j) for these {i,j}: {99, 5989}, {38664, 39646}
X(53765) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {99, 38664, 76}, {99, 52034, 98}, {1975, 14880, 1078}, {5038, 8370, 83}, {12203, 38664, 98}, {32135, 38734, 10358}, {32751, 32752, 38664}

---

X(53766) = X(3)X(249)∩X(23)X(1976)

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

X(53766) lies on the cubic K1321 and these lines: {3, 249}, {23, 1976}, {98, 230}, {248, 2021}, {1692, 11610}, {2065, 51862}, {2080, 14601}, {11653, 32305}, {15545, 37459}, {17974, 52992}, {47195, 51776}

X(53766) = barycentric quotient X(14113)/X(868)
X(53766) = {X(1691),X(14600)}-harmonic conjugate of X(2715)

---

X(53767) = X(3)X(1177)∩X(4)X(32)

Barycentrics    a^2*(a^12 - 3*a^10*b^2 + 2*a^8*b^4 + 2*a^6*b^6 - 3*a^4*b^8 + a^2*b^10 - 3*a^10*c^2 + a^8*b^2*c^2 + 2*a^4*b^6*c^2 - a^2*b^8*c^2 + b^10*c^2 + 2*a^8*c^4 - 2*a^4*b^4*c^4 + 2*a^6*c^6 + 2*a^4*b^2*c^6 - 2*b^6*c^6 - 3*a^4*c^8 - a^2*b^2*c^8 + a^2*c^10 + b^2*c^10) : :
X(53767) = 2 X[3] - 3 X[14649], X[3] - 3 X[51240], 2 X[14676] + X[19164], 3 X[9157] - 2 X[15562], 3 X[9157] + X[38676], 2 X[15562] + X[38676]

X(53767) lies on the cubic K1321 and these lines: {3, 1177}, {4, 32}, {23, 9157}, {26, 19165}, {39, 10766}, {127, 3549}, {147, 4611}, {237, 51458}, {577, 41719}, {827, 1297}, {1503, 10317}, {1594, 11605}, {1976, 31850}, {2967, 39857}, {3542, 34163}, {3785, 13219}, {4558, 11061}, {5191, 13558}, {5938, 44668}, {6240, 14983}, {7517, 11641}, {7526, 38608}, {7527, 11638}, {7530, 33900}, {7552, 7810}, {7556, 38689}, {7669, 40949}, {8721, 10316}, {9517, 11615}, {10024, 10749}, {10594, 20410}, {12384, 31304}, {14118, 38699}, {14574, 17974}, {23208, 32379}, {28343, 30435}, {38525, 51818}, {39648, 49372}, {39679, 49371}

X(53767) = midpoint of X(i) and X(j) for these {i,j}: {112, 19164}, {11641, 13310}
X(53767) = reflection of X(i) in X(j) for these {i,j}: {112, 14676}, {1297, 34217}, {14649, 51240}, {17974, 14574}, {19165, 40121}
X(53767) = circumcircle-inverse of X(15462)
X(53767) = crossdifference of every pair of points on line {684, 47138}

---

X(53768) = X(3)X(43084)∩X(4)X(250)

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^10 - 2*a^8*b^2 + 2*a^4*b^6 - a^2*b^8 - 2*a^8*c^2 + a^6*b^2*c^2 + a^4*b^4*c^2 + a^2*b^6*c^2 - b^8*c^2 + a^4*b^2*c^4 - 4*a^2*b^4*c^4 + b^6*c^4 + 2*a^4*c^6 + a^2*b^2*c^6 + b^4*c^6 - a^2*c^8 - b^2*c^8) : :
X(53768) = 3 X[1989] - 2 X[51847]

X(53768) lies on the cubic K1321 and these lines: {3, 43084}, {4, 250}, {23, 94}, {26, 30715}, {30, 50}, {186, 328}, {265, 1177}, {691, 1141}, {858, 18883}, {2410, 7422}, {2697, 23969}, {3581, 48988}, {5189, 30529}, {5961, 45171}, {6795, 14880}, {7517, 10688}, {7519, 52449}, {7530, 14254}, {9833, 53169}, {9970, 14559}, {10412, 44823}, {14790, 53168}, {15462, 36189}, {15475, 21732}, {16619, 34209}, {17984, 20573}, {18576, 47336}, {46633, 52056}

X(53768) = barycentric product X(i)*X(j) for these {i,j}: {94, 15462}, {265, 41253}, {36189, 39295}
X(53768) = barycentric quotient X(i)/X(j) for these {i,j}: {15462, 323}, {41253, 340}
X(53768) = {X(23),X(94)}-harmonic conjugate of X(476)

---

X(53769) = X(20)X(112)∩X(24)X(98)

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

X(53769) lies on the cubic K1321 and these lines: {3, 51260}, {4, 1177}, {20, 112}, {24, 98}, {1235, 22467}, {3089, 50938}, {3448, 50188}, {8743, 12203}, {12082, 38672}, {18560, 33695}, {34107, 41377}, {38885, 46151}, {44427, 53345}

X(53769) = reflection of X(1289) in X(34131)

---

X(53770) = X(3)X(52668)∩X(67)X(524)

Barycentrics    a^2*(a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(a^6 - 5*a^4*b^2 + 5*a^2*b^4 - b^6 - 5*a^4*c^2 + 7*a^2*b^2*c^2 - 3*b^4*c^2 + 5*a^2*c^4 - 3*b^2*c^4 - c^6) : :
X(53770) = 3 X[34320] - 2 X[51938]

X(53770) lies on the cubic K474 and these lines: {3, 52668}, {67, 524}, {111, 352}, {315, 6328}, {323, 691}, {394, 52142}, {576, 6792}, {843, 9181}, {1297, 35188}, {1351, 52152}, {1499, 14094}, {2979, 10559}, {3292, 32729}, {3917, 10558}, {5181, 30718}, {5643, 32525}, {7998, 21460}, {9127, 15360}, {9169, 15019}, {10560, 33884}, {10562, 38676}, {14916, 51980}, {30209, 38688}

X(53770) = reflection of X(i) in X(j) for these {i,j}: {111, 32583}, {843, 9181}, {15360, 9127}
X(53770) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {352, 52198, 111}, {895, 36827, 15899}, {23061, 36827, 895}

---

X(53771) = X(3)X(3447)∩X(30)X(50)

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^10 - a^8*b^2 - 2*a^6*b^4 + 2*a^4*b^6 + a^2*b^8 - b^10 - a^8*c^2 - 3*a^6*b^2*c^2 + 6*a^4*b^4*c^2 - 3*a^2*b^6*c^2 + b^8*c^2 - 2*a^6*c^4 + 6*a^4*b^2*c^4 - 4*a^2*b^4*c^4 + 2*a^4*c^6 - 3*a^2*b^2*c^6 + a^2*c^8 + b^2*c^8 - c^10) : :
X(53771) =3 X[1989] - 4 X[51847], 3 X[1989] - 2 X[53768]

X(53771) lies on the cubic K474 and these lines: 3, 3447}, {4, 43084}, {23, 18883}, {30, 50}, {67, 265}, {94, 5189}, {328, 3153}, {476, 858}, {6321, 14980}, {7387, 53168}, {10295, 52415}, {12918, 41392}, {13573, 37444}, {14254, 14791}, {14559, 32233}, {14583, 31152}, {14731, 34834}, {16063, 52449}, {16188, 44529}, {18323, 52864}, {18572, 18576}, {20063, 30529}, {34209, 47341}, {34370, 52464}, {37498, 53169}

X(53771) =reflection of X(i) in X(j) for these {i,j}: {52864, 18323}, {53768, 51847}
X(53771) ={X(51847),X(53768)}-harmonic conjugate of X(1989)

---

X(53772) = X(3)X(112)∩X(4)X(67)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^12*b^2 - 2*a^10*b^4 - a^8*b^6 + 4*a^6*b^8 - a^4*b^10 - 2*a^2*b^12 + b^14 + a^12*c^2 - 2*a^10*b^2*c^2 + 2*a^6*b^6*c^2 - a^4*b^8*c^2 - 2*a^10*c^4 + 2*a^6*b^4*c^4 - 2*a^4*b^6*c^4 + 2*b^10*c^4 - a^8*c^6 + 2*a^6*b^2*c^6 - 2*a^4*b^4*c^6 + 4*a^2*b^6*c^6 - 3*b^8*c^6 + 4*a^6*c^8 - a^4*b^2*c^8 - 3*b^6*c^8 - a^4*c^10 + 2*b^4*c^10 - 2*a^2*c^12 + c^14) : :
X(53772) = 2 X[4] - 3 X[20410]

X(53772) lies on the cubic K474 and these lines: {3, 112}, {4, 67}, {24, 53767}, {127, 13160}, {132, 1594}, {511, 5523}, {648, 53769}, {2794, 6240}, {3320, 9630}, {3515, 51240}, {7507, 12145}, {9517, 44427}, {10295, 14900}, {10510, 15262}, {10735, 12173}, {12384, 37444}, {12918, 18569}, {13200, 35471}, {14649, 32534}, {15141, 32713}, {17928, 18876}, {19160, 31724}, {34146, 51940}, {40281, 53026}, {41377, 44668}

X(53772) = reflection of X(10735) in X(13166)

---

X(53773) = X(23)X(7669)∩X(1503)X(53764)

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

X(53773) lies on the cubic K1321 and these lines: {23, 7669}, {1503, 53764}, {2492, 2793}, {6593, 18800}, {14366, 53765}, {41498, 52628}

---

X(53774) = X(23)X(9157)∩X(316)X(1503)

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

X(53774) lies on the cubic K1321 and these lines: {23, 9157}, {316, 1503}, {2030, 8744}, {3800, 52076}, {14246, 53093}, {53764, 53769}

X(53774) = isogonal conjugate of X(14981)
X(53774) = isogonal conjugate of the anticomplement of X(11623)
X(53774) = isogonal conjugate of the complement of X(38664)
X(53774) = trilinear pole of line {2492, 53263}
X(53774) = barycentric quotient X(6)/X(14981)

---

X(53775) = X(4)X(12177)∩X(23)X(1976)

Barycentrics    a^2*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 2*a^4*c^2 - 2*a^2*b^2*c^2 - 2*b^4*c^2 + 3*a^2*c^4 + 3*b^2*c^4)*(a^6 - a^4*b^2 + a^2*b^4 - b^6 - a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6)*(a^6 - 2*a^4*b^2 + 3*a^2*b^4 - a^4*c^2 - 2*a^2*b^2*c^2 + 3*b^4*c^2 - a^2*c^4 - 2*b^2*c^4 + c^6) : :
X(53775) = 2 X[13517] - 3 X[14561]

X(53775) lies on the cubics K289, K298, and K1321, and also on these lines: {4, 12177}, {23, 1976}, {2353, 53767}, {9970, 14906}, {13517, 14561}, {53764, 53768}

X(53775) = antigonal image of X(1352)
X(53775) = symgonal image of X(182)
X(53775) = barycentric quotient X(3148)/X(15993)

---

X(53776) = X(110)X(924)∩X(186)X(3003)

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

X(53776) lies on the cubics K147 and K1322 and these lines: {110, 924}, {186, 3003}, {523, 687}, {2986, 47348}, {5467, 18879}

---

X(53777) = X(2)X(19510)∩X(3)X(6)

Barycentrics    a^2*(2*a^2 - b^2 - c^2)*(a^4 - b^4 + 4*b^2*c^2 - c^4) : :
X(53777) = 3 X[6] - X[10510], 3 X[5050] - X[37477], 5 X[11482] - 3 X[18449], X[37496] - 5 X[53091], X[23] + 3 X[37784], X[895] - 3 X[37784], X[110] - 3 X[52238], X[41617] + 3 X[52238], X[323] - 3 X[52699], X[3292] - 3 X[44102], 2 X[6593] - 3 X[44102], 2 X[8262] - 3 X[32225], 3 X[15303] - 4 X[15471], 2 X[15471] - 3 X[47545], and many others

X(53777) lies on the cubic K1323 and these lines: {2, 19510}, {3, 6}, {23, 895}, {51, 37827}, {110, 9027}, {112, 12593}, {113, 47581}, {193, 9716}, {206, 40318}, {323, 52699}, {351, 8675}, {468, 524}, {542, 11799}, {543, 50150}, {597, 13857}, {858, 15118}, {1176, 22829}, {1495, 2854}, {1503, 1533}, {1531, 5480}, {1568, 51742}, {1976, 6096}, {1992, 5486}, {1995, 8542}, {2882, 37927}, {2892, 15126}, {2930, 8681}, {3260, 46512}, {3291, 32740}, {3529, 18919}, {3564, 5609}, {3580, 32220}, {3629, 21637}, {3906, 23287}, {4558, 35298}, {5166, 53770}, {5622, 7464}, {5640, 12039}, {5651, 9176}, {5972, 47455}, {6000, 16010}, {6128, 15980}, {6153, 9972}, {6467, 35707}, {7495, 16511}, {7530, 8548}, {7545, 43130}, {7555, 15074}, {7556, 15073}, {7575, 14984}, {8546, 19127}, {8547, 35268}, {8584, 13366}, {9019, 15826}, {9192, 40352}, {9516, 30749}, {9968, 36982}, {9970, 13754}, {10540, 32254}, {11008, 19122}, {11061, 41724}, {11064, 47457}, {11188, 14002}, {11216, 33586}, {11416, 37929}, {11579, 14915}, {11580, 32583}, {11645, 18325}, {12367, 32237}, {12584, 51393}, {15034, 19128}, {15054, 34146}, {15139, 32276}, {15362, 50955}, {15534, 34986}, {16324, 51389}, {17430, 52692}, {18583, 51391}, {19121, 32366}, {19596, 37973}, {20423, 49669}, {20975, 37916}, {22143, 37914}, {22151, 23061}, {23327, 31099}, {23698, 47322}, {25321, 37779}, {32113, 32223}, {32269, 41583}, {32300, 47458}, {33921, 47139}, {34319, 41731}, {37924, 39562}, {38734, 53416}, {38745, 45921}, {40350, 46154}, {41720, 44555}, {47277, 47582}, {51394, 51733}

X(53777) = midpoint of X(i) and X(j) for these {i,j}: {23, 895}, {110, 41617}, {193, 41721}, {1351, 3581}, {1495, 32127}, {1992, 15360}, {3580, 32220}, {5095, 41586}, {9970, 32599}, {11061, 41724}, {15139, 32276}, {41720, 44555}, {47277, 47582}, {47546, 47558}
X(53777) = reflection of X(i) in X(j) for these {i,j}: {113, 47581}, {858, 15118}, {1495, 32217}, {1531, 5480}, {1568, 51742}, {2892, 15126}, {3292, 6593}, {5095, 47549}, {5181, 468}, {5642, 47544}, {9181, 2030}, {10564, 182}, {11064, 47457}, {12367, 32237}, {13857, 597}, {15303, 47545}, {32113, 32223}, {41583, 32269}, {51389, 16324}, {51391, 18583}, {51394, 51733}
X(53777) = anticomplement of X(19510)
X(53777) = isogonal conjugate of the polar conjugate of X(37855)
X(53777) = X(i)-Ceva conjugate of X(j) for these (i,j): {43697, 6593}, {51541, 187}
X(53777) = X(i)-isoconjugate of X(j) for these (i,j): {525, 36115}, {897, 5486}, {1577, 35188}, {10097, 37217}, {14208, 32709}
X(53777) = X(i)-Dao conjugate of X(j) for these (i,j): {574, 42008}, {5512, 5466}, {6593, 5486}, {10354, 5485}
X(53777) = crossdifference of every pair of points on line {523, 2549}
X(53777) = X(1992)-line conjugate of X(5486)
X(53777) = barycentric product X(i)*X(j) for these {i,j}: {3, 37855}, {187, 11185}, {468, 41614}, {524, 1995}, {3266, 19136}, {4235, 30209}, {8542, 51541}, {14262, 27088}, {18800, 34241}, {29959, 52898}
X(53777) = barycentric quotient X(i)/X(j) for these {i,j}: {187, 5486}, {1576, 35188}, {1995, 671}, {8542, 42008}, {11185, 18023}, {19136, 111}, {29959, 31125}, {30209, 14977}, {32676, 36115}, {37855, 264}, {41614, 30786}
X(53777) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 576, 50649}, {3, 53092, 44503}, {23, 37784, 895}, {468, 41616, 44102}, {576, 44470, 3}, {1992, 7493, 5486}, {1992, 43697, 11422}, {1995, 8542, 29959}, {1995, 41614, 8542}, {3292, 44102, 6593}, {8542, 19136, 1995}, {19136, 41614, 29959}, {41612, 41618, 15303}, {41612, 47545, 44102}, {41617, 52238, 110}, {44490, 44493, 9730}

---

X(53778) = X(67)X(524)∩X(193)X(1503)

Barycentrics    (a^2 - b^2 - c^2)*(5*a^2 - b^2 - c^2)*(2*a^4 - a^2*b^2 - 3*b^4 - a^2*c^2 + 6*b^2*c^2 - 3*c^4) : :
X(53778) = X[10510] - 3 X[47280], 3 X[1992] - 2 X[15471], 4 X[15471] - 3 X[35266], 2 X[5159] - 3 X[21639], 2 X[5972] - 3 X[47463], 5 X[32269] - 4 X[41583], 4 X[32300] - 5 X[47462]

X(53777) lies on the cubic K1323 and these lines: {67, 524}, {193, 1503}, {235, 576}, {511, 20725}, {1351, 1514}, {1531, 3564}, {1992, 4232}, {2393, 47546}, {3629, 20987}, {5159, 21639}, {5181, 47464}, {5972, 47463}, {6391, 32533}, {8538, 20303}, {11064, 47277}, {15118, 47552}, {15531, 37473}, {17813, 31099}, {30552, 53097}, {32223, 47278}, {32269, 41583}, {32300, 47462}, {44665, 44791}, {53351, 53419}

X(53778) = reflection of X(i) in X(j) for these {i,j}: {1514, 1351}, {5181, 47464}, {11064, 47277}, {35266, 1992}, {47278, 32223}, {47552, 15118}
X(53778) = X(5159)-Dao conjugate of X(52477)
X(53778) = barycentric product X(i)*X(j) for these {i,j}: {1992, 5159}, {11059, 21639}
X(53778) = barycentric quotient X(i)/X(j) for these {i,j}: {5159, 5485}, {21639, 21448}

---

X(53779) = X(3)X(7699)∩X(30)X(1493)

Barycentrics    (2*a^4 + a^2*b^2 - 3*b^4 + a^2*c^2 + 6*b^2*c^2 - 3*c^4)*(5*a^6 - 8*a^4*b^2 + a^2*b^4 + 2*b^6 - 8*a^4*c^2 + 7*a^2*b^2*c^2 - 2*b^4*c^2 + a^2*c^4 - 2*b^2*c^4 + 2*c^6) : :
X(53779) = 4 X[34563] - 3 X[43585], 5 X[34563] - 3 X[44866], 5 X[43585] - 4 X[44866], 3 X[382] - X[16835], 2 X[546] - 3 X[46027], 5 X[3091] - 3 X[18442], 3 X[3521] - X[3529], 4 X[3628] - 3 X[35240], 5 X[5076] - 3 X[15062], 3 X[8718] - X[49137], 4 X[12102] - 3 X[18488]

X(53779) lies on the cubic K1323 and these lines: {3, 7699}, {30, 1493}, {382, 16835}, {546, 44106}, {3091, 18442}, {3146, 10263}, {3521, 3529}, {3627, 12359}, {3628, 35240}, {3629, 29012}, {5073, 43599}, {5076, 15062}, {8718, 49137}, {11541, 31815}, {12102, 18488}, {15704, 18475}

X(53779) = midpoint of X(5073) and X(43599)

---

X(53780) = X(30)X(1351)∩X(381)X(1531)

Barycentrics    (a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4)*(17*a^6 - 29*a^4*b^2 + 7*a^2*b^4 + 5*b^6 - 29*a^4*c^2 + 18*a^2*b^2*c^2 - 5*b^4*c^2 + 7*a^2*c^4 - 5*b^2*c^4 + 5*c^6) : :
X(53780) = 3 X[11820] - 5 X[44750], 5 X[381] - 4 X[4550], 3 X[381] - 4 X[51993], 2 X[4550] - 5 X[40909], 3 X[4550] - 5 X[51993], 3 X[40909] - 2 X[51993], 2 X[4549] - 3 X[5055], 3 X[5054] - 4 X[7706], 2 X[11472] - 3 X[38335], 5 X[15694] - 4 X[35254], 3 X[18494] - 2 X[47353]

X(53780) lies on the cubic K1323 and these lines: {30, 1351}, {381, 1531}, {382, 541}, {549, 41465}, {2777, 51024}, {3426, 3543}, {3534, 3796}, {4549, 5055}, {4846, 15681}, {5054, 7706}, {5655, 45082}, {10938, 14831}, {11001, 11402}, {11405, 49670}, {11472, 32608}, {14915, 21969}, {15087, 35237}, {15640, 45968}, {15694, 35254}, {17702, 50962}, {18494, 47353}, {26944, 34725}, {34613, 48672}

X(53780) = reflection of X(i) in X(j) for these {i,j}: {381, 40909}, {3426, 3543}, {10938, 14831}, {15681, 4846}, {41465, 549}

---

X(53781) = X(3)X(54)∩X(30)X(17854)

Barycentrics    a^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)*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + 5*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + b^6*c^2 - 4*a^2*b^2*c^4 + 2*a^2*c^6 + b^2*c^6 - c^8) : :
X(53781) = 3 X[5890] - X[43574], 3 X[12824] - 4 X[52000], 2 X[2072] - 3 X[46430], 3 X[9730] - 2 X[14156], 2 X[13148] + X[41724], 2 X[13851] - 3 X[45237], 3 X[37941] - 2 X[41673]

X(53781) lies on the cubic K1323 and these lines: {3, 54}, {30, 17854}, {52, 12897}, {110, 37917}, {113, 403}, {185, 10112}, {389, 1568}, {511, 16386}, {539, 11562}, {974, 2071}, {1531, 32411}, {1992, 44439}, {2072, 46430}, {2781, 37784}, {3060, 44438}, {3146, 34751}, {5562, 44673}, {5663, 31726}, {6243, 34350}, {6293, 41736}, {6515, 44440}, {9730, 14156}, {10295, 45780}, {10706, 38898}, {11250, 43904}, {11744, 52403}, {12111, 37197}, {12236, 18403}, {12284, 14157}, {12359, 45177}, {13148, 41724}, {13445, 39562}, {13851, 45237}, {14708, 22115}, {14831, 52069}, {18400, 21649}, {22467, 22966}, {37511, 40316}, {37941, 41673}, {38320, 41714}, {41589, 43605}, {41628, 44458}, {44755, 45957}

X(53781) = midpoint of X(12284) and X(14157)
X(53781) = reflection of X(i) in X(j) for these {i,j}: {1531, 32411}, {1568, 389}, {2071, 974}, {5562, 44673}, {12825, 403}, {18403, 12236}, {22115, 14708}
X(53781) = X(22466)-isoconjugate of X(36053)
X(53781) = X(i)-Dao conjugate of X(j) for these (i,j): {113, 22466}, {46658, 15328}
X(53781) = barycentric product X(i)*X(j) for these {i,j}: {3003, 22468}, {3580, 22467}
X(53781) = barycentric quotient X(i)/X(j) for these {i,j}: {3003, 22466}, {22467, 2986}, {22468, 40832}
X(53781) = {X(5889),X(5890)}-harmonic conjugate of X(1993)

---

X(53782) = X(3)X(647)∩X(20)X(111)

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

X(53782) lies on the cubic K1324 and these lines: {3, 647}, {20, 111}, {32, 39169}, {691, 1968}, {3291, 11634}, {3767, 14609}, {3926, 15526}, {5206, 6091}, {10316, 14908}, {22401, 51253}

X(53782) = X(44182)-Ceva conjugate of X(895)
X(53782) = X(158)-isoconjugate of X(34161)
X(53782) = X(i)-Dao conjugate of X(j) for these (i,j): {126, 37778}, {1147, 34161}, {8681, 126}, {34158, 4}
X(53782) = barycentric product X(i)*X(j) for these {i,j}: {394, 14263}, {895, 8681}, {3926, 51819}, {15398, 47412}
X(53782) = barycentric quotient X(i)/X(j) for these {i,j}: {577, 34161}, {3291, 37778}, {8681, 44146}, {14263, 2052}, {14908, 2374}, {47412, 34336}, {51819, 393}

---

X(53783) = X(2)X(32545)∩X(3)X(525)

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

X(53783) lies on the cubic K1324 and these lines: {2, 32545}, {3, 525}, {4, 2966}, {5, 35906}, {20, 98}, {68, 248}, {69, 47388}, {193, 47741}, {647, 52006}, {1092, 3926}, {1352, 51963}, {1976, 44470}, {2715, 18337}, {4226, 14265}, {5967, 43754}, {6337, 17932}, {7612, 40428}, {9545, 14355}, {13335, 52145}, {15069, 34369}, {20021, 51776}, {31381, 51869}, {39138, 52451}, {40330, 52081}

X(53783) = X(i)-Ceva conjugate of X(j) for these (i,j): {40428, 287}, {47388, 17974}
X(53783) = X(i)-isoconjugate of X(j) for these (i,j): {158, 34157}, {240, 3563}, {1096, 52091}, {6530, 36051}, {8773, 34854}, {17994, 36105}
X(53783) = X(i)-Dao conjugate of X(j) for these (i,j): {114, 6530}, {230, 36426}, {1147, 34157}, {3564, 114}, {6503, 52091}, {34156, 4}, {35067, 297}, {39001, 17994}, {39072, 34854}, {39085, 3563}, {41181, 2799}
X(53783) = barycentric product X(i)*X(j) for these {i,j}: {230, 6394}, {287, 3564}, {394, 14265}, {3926, 51820}, {4226, 53173}, {17974, 51481}, {35067, 40428}
X(53783) = barycentric quotient X(i)/X(j) for these {i,j}: {114, 36426}, {230, 6530}, {248, 3563}, {287, 35142}, {394, 52091}, {577, 34157}, {1692, 34854}, {3564, 297}, {6394, 8781}, {14265, 2052}, {17974, 2987}, {35067, 114}, {43754, 32697}, {47406, 2967}, {50433, 39374}, {51335, 51334}, {51820, 393}, {52038, 52476}, {52144, 232}
X(53783) = {X(3),X(34156)}-harmonic conjugate of X(35912)

---

X(53784) = X(3)X(3265)∩X(20)X(1296)

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

X(53784) lies on the cubic K1324 and these lines: {3, 3265}, {20, 1296}, {3266, 4235}, {3926, 4558}, {5467, 6390}, {5649, 7763}, {35178, 46140}

X(53784) = X(i)-isoconjugate of X(j) for these (i,j): {92, 51962}, {158, 34158}, {897, 14580}, {923, 5523}, {2393, 36128}, {8753, 18669}, {23894, 46592}
X(53784) = X(i)-Dao conjugate of X(j) for these (i,j): {187, 20410}, {524, 1560}, {1147, 34158}, {2482, 5523}, {6593, 14580}, {22391, 51962}, {52881, 858}
X(53784) = barycentric product X(i)*X(j) for these {i,j}: {2373, 6390}, {3266, 18876}, {3292, 46140}, {3926, 51823}, {36792, 41511}
X(53784) = barycentric quotient X(i)/X(j) for these {i,j}: {184, 51962}, {187, 14580}, {524, 5523}, {577, 34158}, {1177, 8753}, {2373, 17983}, {2482, 1560}, {3292, 2393}, {3964, 51253}, {5467, 46592}, {6390, 858}, {6593, 20410}, {14417, 47138}, {18876, 111}, {41511, 10630}, {46140, 46111}, {51823, 393}, {52898, 21459}

---

X(53785) = X(3)X(520)∩X(4)X(36831)

Barycentrics    a^4*(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)*(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(53785) lies on the cubic K1324 and these lines: {3, 520}, {4, 36831}, {20, 68}, {52, 35908}, {155, 9717}, {525, 52010}, {1147, 14385}, {1304, 10539}, {3470, 15083}, {5877, 11438}, {8675, 51895}, {9033, 14933}, {9730, 35910}, {9927, 34150}, {10316, 18877}, {11411, 36875}, {11459, 46788}, {12079, 12359}, {12111, 52493}, {12162, 52646}, {12893, 15468}, {13352, 44769}, {13754, 14264}, {16077, 18027}, {18436, 44715}, {23128, 48451}, {50464, 51254}

X(53785) = X(i)-Ceva conjugate of X(j) for these (i,j): {16077, 6334}, {40423, 14919}
X(53785) = X(i)-isoconjugate of X(j) for these (i,j): {92, 51965}, {158, 15454}, {1096, 52552}, {1300, 1784}, {36053, 52661}
X(53785) = X(i)-Dao conjugate of X(j) for these (i,j): {113, 52661}, {1147, 15454}, {6503, 52552}, {13754, 113}, {22391, 51965}, {39174, 4}
X(53785) = crossdifference of every pair of points on line {1990, 14397}
X(53785) = barycentric product X(i)*X(j) for these {i,j}: {394, 14264}, {3926, 51821}, {13754, 14919}, {34834, 50464}
X(53785) = barycentric quotient X(i)/X(j) for these {i,j}: {184, 51965}, {394, 52552}, {577, 15454}, {2315, 1784}, {3003, 52661}, {13754, 46106}, {14264, 2052}, {18877, 1300}, {47405, 34334}, {50433, 39375}, {50464, 40427}, {51821, 393}

---

X(53786) = X(3)X(521)∩X(20)X(104)

Barycentrics    a^2*(a^2 - b^2 - c^2)^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)*(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(53786) lies on the cubic K1324 and these lines: {3, 521}, {20, 104}, {905, 52005}, {1069, 1795}, {1092, 1259}, {3560, 52640}, {3658, 14266}, {10316, 14578}, {11248, 36037}

X(53786) = X(i)-isoconjugate of X(j) for these (i,j): {158, 39173}, {915, 1785}, {14571, 37203}, {36106, 39534}
X(53786) = X(i)-Dao conjugate of X(j) for these (i,j): {912, 119}, {1147, 39173}, {39002, 39534}, {39175, 4}
X(53786) = barycentric product X(i)*X(j) for these {i,j}: {394, 14266}, {914, 1795}, {3926, 51824}
X(53786) = barycentric quotient X(i)/X(j) for these {i,j}: {577, 39173}, {1795, 37203}, {2252, 1785}, {14266, 2052}, {14578, 915}, {47408, 21664}, {51824, 393}

---

X(53787) = X(3)X(512)∩X(20)X(3563)

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

X(53787) lies on the cubic K1324 and these lines: {3, 512}, {20, 3563}, {68, 3926}, {1092, 40319}, {2987, 8538}, {6643, 36891}, {9967, 52091}, {10316, 32654}

X(53787) = X(1733)-isoconjugate of X(40120)
X(53787) = X(i)-Dao conjugate of X(j) for these (i,j): {31842, 44145}, {34157, 4}, {34382, 31842}
X(53787) = barycentric product X(2987)*X(34382)
X(53787) = barycentric quotient X(i)/X(j) for these {i,j}: {32654, 40120}, {34382, 51481}

---

X(53788) = X(2)X(38936)∩X(3)X(523)

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

X(53788) lies on the cubic K1324 and these lines: {2, 38936}, {3, 523}, {4, 10420}, {20, 254}, {68, 125}, {2072, 39371}, {7763, 18878}, {10316, 14910}, {11585, 16934}, {12028, 18531}, {12095, 30512}, {18404, 22261}

X(53788) = X(1299)-isoconjugate of X(1725)
X(53788) = X(i)-Dao conjugate of X(j) for these (i,j): {131, 403}, {12095, 1986}, {15454, 4}, {44665, 131}
X(53788) = barycentric product X(i)*X(j) for these {i,j}: {2986, 44665}, {15421, 30512}
X(53788) = barycentric quotient X(i)/X(j) for these {i,j}: {2314, 1725}, {5504, 43756}, {14910, 1299}, {16310, 403}, {30512, 16237}, {44665, 3580}, {50433, 39373}

---

X(53789) = X(3)X(2416)∩X20)X(110)

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^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(53789) lies on the cubics K041 and K1324 and these lines: {3, 2416}, {20, 110}, {2420, 15774}, {3233, 35602}, {4240, 23097}, {13573, 14216}, {15404, 51254}, {20427, 46968}, {47390, 53050}

X(53789) = isogonal conjugate of X(52646)
X(53789) = X(i)-isoconjugate of X(j) for these (i,j): {1, 52646}, {92, 51964}, {158, 39174}, {2159, 51358}, {6000, 36119}, {35200, 51385}
X(53789) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 52646}, {30, 133}, {133, 51385}, {1147, 39174}, {1511, 6000}, {3163, 51358}, {22391, 51964}, {52874, 1559}
X(53789) = cevapoint of X(i) and X(j) for these (i,j): {30, 38605}, {16163, 51394}
X(53789) = trilinear pole of line {3284, 14345}
X(53789) = barycentric product X(i)*X(j) for these {i,j}: {1294, 11064}, {2407, 43701}, {2416, 4240}, {15404, 36789}
X(53789) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 52646}, {30, 51358}, {184, 51964}, {577, 39174}, {1294, 16080}, {1990, 51385}, {2416, 34767}, {2420, 46587}, {2430, 14380}, {3163, 133}, {3284, 6000}, {4240, 2404}, {15404, 40384}, {23347, 2442}, {43701, 2394}, {50433, 39376}, {51394, 44436}

---

X(53790) = X(3)X(106)∩X30)X(511)

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

X(53790) lies on these lines: {1, 1357}, {3, 106}, {4, 10744}, {5, 121}, {8, 38389}, {10, 3038}, {20, 20098}, {30, 511}, {36, 51626}, {40, 1054}, {140, 6715}, {182, 1480}, {355, 50914}, {381, 10713}, {382, 10730}, {392, 15082}, {946, 11814}, {962, 2899}, {970, 3030}, {1320, 3937}, {1351, 10761}, {1385, 11717}, {1482, 10700}, {3036, 38390}, {3579, 14664}, {3656, 50915}, {3877, 5650}, {4677, 45829}, {5901, 11731}, {5903, 12109}, {7982, 13541}, {8752, 23135}, {9432, 12652}, {9730, 25413}, {10441, 38478}, {10738, 10774}, {11249, 34139}, {11531, 46362}, {12014, 13464}, {12672, 46847}, {12702, 17749}, {13205, 53294}, {14810, 41455}, {14923, 29958}, {15520, 44414}, {16836, 37562}, {22791, 25652}, {31393, 51765}, {34461, 51525}

X(53790) = isogonal conjugate of X(44873)
X(53790) = Thomson-isogonal conjugate of X(6079)
X(53790) = crossdifference of every pair of points on line {6, 14425}
X(53790) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 106, 38604}, {3, 1293, 38620}, {3, 38576, 106}, {3, 38590, 1293}, {3, 38671, 51531}, {4, 21290, 10744}, {4, 34548, 15522}, {106, 1293, 3}, {106, 38576, 51531}, {106, 38590, 38620}, {106, 38671, 38576}, {106, 38685, 1293}, {106, 38713, 38695}, {121, 5510, 5}, {1293, 38576, 38604}, {1293, 38671, 106}, {1293, 38685, 38590}, {1293, 38695, 38713}, {1320, 38512, 3937}, {1357, 6018, 1}, {10730, 44984, 382}, {10744, 15522, 4}, {21290, 34548, 4}, {38504, 38515, 3}, {38576, 38590, 3}, {38576, 38685, 38620}, {38590, 38671, 38604}, {38604, 38620, 3}, {38604, 51531, 106}, {38620, 51531, 38604}, {38671, 38685, 3}, {38695, 38713, 3}, {51765, 51811, 31393}

---

X(53791) = X(3)X(729)∩X30)X(511)

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

X(53791) lies on these lines: {1, 6022}, {3, 729}, {5, 44949}, {30, 511}, {351, 19913}, {382, 44939}, {1350, 40122}, {6234, 11654}, {9466, 20326}, {9821, 9998}, {12525, 49111}, {13518, 14881}, {35399, 52987}

X(53791) = barycentric quotient X(5351)/X(1881)
X(53791) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {729, 39639, 3}, {6022, 7333, 1}

---

X(53792) = X(3)X(36735)∩X30)X(511)

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

X(53792) lies on these lines: {3, 36735}, {30, 511}, {36, 1284}, {98, 901}, {99, 953}, {114, 1281}, {115, 2245}, {147, 30579}, {182, 5091}, {190, 24808}, {238, 7336}, {484, 1756}, {576, 24695}, {894, 19890}, {970, 31847}, {1155, 5988}, {1319, 24231}, {1352, 24280}, {1936, 3326}, {2720, 41349}, {2957, 5535}, {3023, 13756}, {3025, 3027}, {3218, 6075}, {3286, 22765}, {3923, 24206}, {4645, 39185}, {4655, 40107}, {4672, 25555}, {4736, 12770}, {4887, 5126}, {5087, 6721}, {5695, 34507}, {5790, 49721}, {6033, 40100}, {6073, 48363}, {6321, 38954}, {10016, 39828}, {10246, 49747}, {10722, 44973}, {10723, 44979}, {10768, 31512}, {11246, 34583}, {11724, 41193}, {12042, 38614}, {12188, 38584}, {13174, 34464}, {13188, 38586}, {14027, 18201}, {14513, 17484}, {15488, 31849}, {19636, 39154}, {21166, 38707}, {23235, 38682}, {24201, 24472}, {24728, 48892}, {31785, 38498}, {32115, 32135}, {32117, 32152}, {33100, 37527}, {33813, 38617}, {34151, 51377}, {34473, 38705}, {35459, 38738}, {38028, 49741}, {38042, 49726}, {38499, 38513}, {38556, 38568}, {38557, 38569}, {38736, 50371}, {39479, 39825}

X(53792) = Thomson-isogonal conjugate of X(53606)
X(53792) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {36, 32857, 43909}, {36735, 36736, 3}, {38530, 38531, 3}

---

X(53793) = X(3)X(691)∩X30)X(511)

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

X(53793) lies on these lines: {1, 6023}, {3, 691}, {4, 38552}, {5, 5099}, {23, 2080}, {30, 511}, {98, 47290}, {114, 52106}, {140, 16760}, {186, 14565}, {187, 2493}, {249, 32609}, {316, 7574}, {373, 34094}, {382, 44969}, {468, 14693}, {476, 1316}, {477, 805}, {1511, 9181}, {1551, 22566}, {2021, 16308}, {2453, 35930}, {2679, 25641}, {2967, 7482}, {2979, 13170}, {3095, 36182}, {3111, 15536}, {3258, 5650}, {4230, 47223}, {5112, 47324}, {5140, 37984}, {5611, 14175}, {5615, 14176}, {6033, 36173}, {6321, 36174}, {6772, 25222}, {6775, 25221}, {6787, 9996}, {7464, 35002}, {7468, 9155}, {7471, 51430}, {7472, 33813}, {7998, 34312}, {9137, 40282}, {9149, 9301}, {9158, 11673}, {9213, 39528}, {9218, 15035}, {9831, 32229}, {11171, 37991}, {11188, 12157}, {11272, 36157}, {11295, 25226}, {11296, 25225}, {11459, 18321}, {11799, 47211}, {12042, 36166}, {12093, 53136}, {12833, 51383}, {13188, 47288}, {13449, 18572}, {14113, 48721}, {14687, 31861}, {14731, 33884}, {14989, 44971}, {15067, 31848}, {15122, 47570}, {15535, 51428}, {15544, 18907}, {15919, 33532}, {16092, 49102}, {16175, 48657}, {16320, 37459}, {16461, 25151}, {16462, 25161}, {16619, 47584}, {16978, 16981}, {18332, 53379}, {18571, 47113}, {18860, 37950}, {22103, 31379}, {25174, 30439}, {25179, 30440}, {32662, 34370}, {33964, 44042}, {35001, 47618}, {36165, 49111}, {36822, 38580}, {37906, 47327}, {38701, 38703}, {47292, 51523}, {47293, 51524}

X(53793) = isogonal conjugate of X(53605)
X(53793) = Thomson-isogonal conjugate of X(20404)
X(53793) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 691, 38611}, {3, 842, 38613}, {3, 38582, 691}, {3, 38583, 842}, {691, 842, 3}, {691, 38583, 38613}, {691, 38679, 38582}, {691, 38680, 842}, {691, 38704, 38702}, {842, 38582, 38611}, {842, 38679, 691}, {842, 38680, 38583}, {842, 38702, 38704}, {5099, 16188, 5}, {6023, 6027, 1}, {7472, 46634, 33813}, {16760, 40544, 140}, {36166, 46633, 12042}, {36166, 47291, 46633}, {38526, 38528, 3}, {38582, 38583, 3}, {38582, 38680, 38613}, {38583, 38679, 38611}, {38611, 38613, 3}, {38679, 38680, 3}, {38702, 38704, 3}, {44969, 44972, 382}

---

X(53794) = X(3)X(759)∩X30)X(511)

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

X(53794) lies on these lines: {1, 1365}, {3, 759}, {4, 25650}, {5, 25652}, {20, 19642}, {30, 511}, {40, 21381}, {110, 36171}, {125, 6740}, {149, 38568}, {382, 44970}, {950, 24235}, {970, 6044}, {1283, 9840}, {1385, 5620}, {3109, 5972}, {3746, 24222}, {4297, 52685}, {5138, 48837}, {5497, 48903}, {5691, 46976}, {6723, 36155}, {7984, 38514}, {10441, 38480}, {13161, 17724}, {13199, 38569}, {13442, 39566}, {14680, 18480}, {24299, 48894}, {26286, 47749}, {31393, 51771}, {32223, 47346}, {36158, 37853}

X(53794) = Thomson-isogonal conjugate of X(6083)
X(53794) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 759, 38612}, {3, 14663, 759}, {759, 6011, 3}, {1365, 34194, 1}, {3109, 6739, 5972}, {6011, 14663, 38612}, {6740, 36154, 125}, {21381, 34196, 40}, {31845, 42425, 5}

---

X(53795) = X(3)X(112)∩X30)X(511)

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

X(53795) lies on these lines: {1, 3320}, {3, 112}, {4, 339}, {5, 127}, {6, 40281}, {20, 12253}, {26, 19165}, {30, 511}, {40, 12408}, {55, 13116}, {56, 13117}, {110, 37921}, {140, 6720}, {143, 11437}, {182, 28343}, {355, 12784}, {371, 35828}, {372, 35829}, {381, 10718}, {382, 10735}, {550, 14689}, {1112, 1316}, {1160, 12806}, {1161, 12805}, {1351, 10766}, {1385, 11722}, {1478, 12945}, {1479, 12955}, {1482, 10705}, {1511, 53760}, {1561, 16105}, {1658, 5171}, {2453, 40949}, {2549, 44439}, {3150, 47202}, {3311, 19094}, {3312, 19093}, {3313, 42459}, {3398, 12207}, {5562, 14929}, {5946, 16224}, {6403, 16983}, {6644, 18876}, {7387, 11641}, {7737, 37473}, {8981, 13918}, {9157, 9909}, {9730, 16225}, {9732, 49316}, {9733, 49315}, {9737, 11250}, {9738, 48733}, {9739, 48732}, {9821, 12503}, {9826, 36177}, {10525, 12925}, {10526, 12935}, {10669, 12996}, {10673, 12997}, {10679, 13118}, {10680, 13119}, {10738, 10780}, {11007, 13416}, {11248, 12340}, {11249, 19159}, {11251, 12796}, {11252, 12478}, {11253, 12479}, {11605, 31723}, {12041, 53719}, {12042, 53727}, {12220, 14532}, {13417, 51431}, {13966, 13985}, {14070, 51240}, {14649, 18324}, {14650, 50381}, {15048, 50649}, {15562, 17714}, {18531, 34163}, {18907, 19161}, {25641, 35594}, {33330, 46413}, {33813, 53737}, {33814, 53745}, {34579, 52926}, {34854, 44334}, {37466, 51454}, {38602, 53755}, {41673, 51389}, {44920, 46186}, {46619, 46631}, {46620, 46637}, {47413, 52604}, {48460, 48474}, {48461, 48475}, {49038, 49046}, {49039, 49047}, {49377, 49385}, {49378, 49386}

X(53795) = Thomson-isogonal conjugate of X(2867)
X(53795) = X(40281)-line conjugate of X(6)
X(53795) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 112, 38608}, {3, 1297, 38624}, {3, 13115, 1297}, {3, 13310, 112}, {3, 38676, 51536}, {4, 10749, 19163}, {4, 12384, 12918}, {4, 12918, 19160}, {4, 13219, 10749}, {112, 1297, 3}, {112, 13115, 38624}, {112, 13310, 51536}, {112, 38525, 38652}, {112, 38676, 13310}, {112, 38689, 1297}, {112, 38717, 38699}, {127, 132, 5}, {371, 35828, 49218}, {371, 35880, 49270}, {372, 35829, 49219}, {372, 35881, 49271}, {382, 48658, 44988}, {382, 48681, 10735}, {1297, 13310, 38608}, {1297, 38676, 112}, {1297, 38689, 13115}, {1297, 38699, 38717}, {2967, 38553, 3}, {3320, 6020, 1}, {6720, 34841, 140}, {10705, 13099, 1482}, {10735, 44988, 382}, {10749, 12384, 19160}, {10749, 12918, 4}, {11641, 12413, 7387}, {11722, 12265, 1385}, {12145, 13166, 4}, {12207, 13195, 3398}, {12253, 13200, 20}, {12340, 13206, 11248}, {12384, 13219, 4}, {12408, 13221, 40}, {12478, 13229, 11252}, {12479, 13231, 11253}, {12503, 13236, 9821}, {12784, 13280, 355}, {12796, 13281, 11251}, {12805, 13282, 1161}, {12806, 13283, 1160}, {12918, 13219, 19163}, {12925, 13294, 10525}, {12935, 13295, 10526}, {12945, 13296, 1478}, {12955, 13297, 1479}, {12996, 13298, 10669}, {12997, 13299, 10673}, {13115, 13310, 3}, {13115, 38676, 38608}, {13116, 13311, 55}, {13117, 13312, 56}, {13118, 13313, 10679}, {13119, 13314, 10680}, {13310, 38689, 38624}, {13918, 13923, 8981}, {13985, 13992, 13966}, {19093, 19114, 3312}, {19094, 19115, 3311}, {19159, 19162, 11249}, {19160, 19163, 4}, {19165, 53767, 40121}, {35828, 35880, 371}, {35829, 35881, 372}, {38510, 38519, 3}, {38525, 38529, 3}, {38567, 38571, 3}, {38608, 38624, 3}, {38608, 51536, 112}, {38624, 51536, 38608}, {38676, 38689, 3}, {38699, 38717, 3}, {48474, 48537, 48460}, {48475, 48538, 48461}, {48658, 48681, 382}, {48732, 48788, 9739}, {48733, 48789, 9738}, {49046, 49100, 49038}, {49047, 49101, 49039}, {49153, 49205, 11249}, {49154, 49206, 11248}, {49218, 49270, 371}, {49219, 49271, 372}, {49315, 49371, 9733}, {49316, 49372, 9732}, {49385, 49443, 49377}, {49386, 49444, 49378}

---

X(53796) = X(3)X(2971)∩X30)X(511)

Barycentrics    a^2*(a^10*b^2 - a^8*b^4 - 2*a^6*b^6 + 2*a^4*b^8 + a^2*b^10 - b^12 + a^10*c^2 - 10*a^8*b^2*c^2 + 15*a^6*b^4*c^2 - 11*a^4*b^6*c^2 + 5*b^10*c^2 - a^8*c^4 + 15*a^6*b^2*c^4 - 6*a^4*b^4*c^4 + 3*a^2*b^6*c^4 - 15*b^8*c^4 - 2*a^6*c^6 - 11*a^4*b^2*c^6 + 3*a^2*b^4*c^6 + 22*b^6*c^6 + 2*a^4*c^8 - 15*b^4*c^8 + a^2*c^10 + 5*b^2*c^10 - c^12) : :

X(53796) lies on these lines: {3, 2971}, {4, 2974}, {5, 5139}, {26, 2079}, {30, 511}, {143, 7737}, {1112, 36181}, {1658, 51460}, {5140, 10011}, {6091, 14070}, {6239, 12224}, {6291, 12604}, {6400, 12223}, {6406, 12603}, {7387, 39653}, {7761, 32142}, {7804, 32205}, {9909, 39907}, {12085, 35453}, {13416, 36163}, {15074, 44526}, {18531, 41521}, {33532, 50529}

X(53796) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3563, 3565, 3}, {5139, 31842, 5}

---

X(53797) = X(3)X(805)∩X30)X(511)

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

X(53797) lies on these lines: {1, 44042}, {3, 805}, {5, 2679}, {30, 511}, {76, 18321}, {140, 22103}, {262, 41330}, {382, 44971}, {2023, 14113}, {2080, 21444}, {3111, 40108}, {5188, 52006}, {5976, 12833}, {6072, 51872}, {6321, 31513}, {6787, 7697}, {10263, 16979}, {12188, 14510}, {13188, 14509}, {13340, 22679}, {14178, 22714}, {14182, 22715}, {14881, 31850}, {22688, 25151}, {22690, 25161}, {22691, 25178}, {22692, 25173}, {22693, 25180}, {22694, 25175}, {22695, 25182}, {22696, 25177}, {22701, 22999}, {22702, 23008}, {22707, 23017}, {22708, 23023}, {31701, 31707}, {31702, 31708}, {31848, 49111}, {33478, 33481}, {33479, 33480}, {35745, 35761}, {36321, 36345}, {36322, 36328}, {36323, 36325}, {36347, 36354}, {36364, 36367}, {36365, 36369}, {36384, 36387}, {36385, 36389}, {36780, 36783}, {38383, 46046}, {41052, 41058}, {41053, 41059}

X(53797) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {805, 2698, 3}, {2679, 33330, 5}

---

X(53798) = X(3)X(843)∩X30)X(511)

Barycentrics    a^2*(a^10*b^2 - 9*a^8*b^4 + 26*a^6*b^6 - 26*a^4*b^8 + 9*a^2*b^10 - b^12 + a^10*c^2 + 2*a^8*b^2*c^2 - 4*a^6*b^4*c^2 + 10*a^4*b^6*c^2 - 4*a^2*b^8*c^2 - b^10*c^2 - 9*a^8*c^4 - 4*a^6*b^2*c^4 - 24*a^4*b^4*c^4 + 12*a^2*b^6*c^4 + 8*b^8*c^4 + 26*a^6*c^6 + 10*a^4*b^2*c^6 + 12*a^2*b^4*c^6 - 20*b^6*c^6 - 26*a^4*c^8 - 4*a^2*b^2*c^8 + 8*b^4*c^8 + 9*a^2*c^10 - b^2*c^10 - c^12) : :

X(53798) lies on these lines: {1, 47020}, {3, 843}, {5, 44956}, {30, 511}, {352, 2080}, {382, 44946}, {805, 6093}, {1351, 45143}, {2679, 6092}, {2698, 6082}, {5640, 13170}, {12093, 12833}, {13225, 41330}, {16341, 49102}, {31654, 33330}

X(53798) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {843, 2709, 3}, {44956, 46659, 5}

---

X(53799) = X(3)X(843)∩X30)X(511)

Barycentrics    2*a^7 - 6*a^6*b + 2*a^5*b^2 + 12*a^4*b^3 - 9*a^3*b^4 - 5*a^2*b^5 + 5*a*b^6 - b^7 - 6*a^6*c + 24*a^5*b*c - 28*a^4*b^2*c - 12*a^3*b^3*c + 33*a^2*b^4*c - 12*a*b^5*c + b^6*c + 2*a^5*c^2 - 28*a^4*b*c^2 + 60*a^3*b^2*c^2 - 30*a^2*b^3*c^2 - 5*a*b^4*c^2 + 3*b^5*c^2 + 12*a^4*c^3 - 12*a^3*b*c^3 - 30*a^2*b^2*c^3 + 24*a*b^3*c^3 - 3*b^4*c^3 - 9*a^3*c^4 + 33*a^2*b*c^4 - 5*a*b^2*c^4 - 3*b^3*c^4 - 5*a^2*c^5 - 12*a*b*c^5 + 3*b^2*c^5 + 5*a*c^6 + b*c^6 - c^7 : :

X(53799) lies on these lines: {1, 14026}, {3, 2718}, {8, 3259}, {30, 511}, {145, 901}, {953, 6079}, {1320, 6075}, {1482, 5516}, {1483, 38614}, {3699, 14507}, {5048, 53618}, {5690, 6789}, {6073, 10698}, {7967, 38705}, {8148, 38938}, {10016, 12410}, {10222, 23869}, {12531, 31512}, {12619, 16338}, {12645, 18326}, {12735, 39752}, {13756, 44046}, {17101, 36972}, {25405, 37743}, {25416, 52478}, {38617, 51991}

X(53799) = reflection of X(952) in the Nagel line
X(53799) = {X(2718),X(2743)}-harmonic conjugate of X(3)

---

X(53800) = X(3)X(901)∩X30)X(511)

Barycentrics    a^2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 - 4*a^5*b^2*c + 4*a^4*b^3*c + 8*a^3*b^4*c - 8*a^2*b^5*c - 4*a*b^6*c + 4*b^7*c + a^6*c^2 - 4*a^5*b*c^2 + 10*a^4*b^2*c^2 - 12*a^3*b^3*c^2 - 6*a^2*b^4*c^2 + 16*a*b^5*c^2 - 5*b^6*c^2 + 4*a^4*b*c^3 - 12*a^3*b^2*c^3 + 24*a^2*b^3*c^3 - 12*a*b^4*c^3 - 4*b^5*c^3 - 3*a^4*c^4 + 8*a^3*b*c^4 - 6*a^2*b^2*c^4 - 12*a*b^3*c^4 + 12*b^4*c^4 - 8*a^2*b*c^5 + 16*a*b^2*c^5 - 4*b^3*c^5 + 3*a^2*c^6 - 4*a*b*c^6 - 5*b^2*c^6 + 4*b*c^7 - c^8) : :

X(53800) lies on these lines: {1, 3025}, {3, 901}, {4, 38954}, {5, 3259}, {26, 10016}, {30, 511}, {40, 34464}, {65, 23152}, {140, 22102}, {143, 5903}, {382, 44973}, {942, 24201}, {962, 14266}, {1145, 34151}, {1290, 13868}, {1387, 14115}, {1484, 6075}, {1537, 46044}, {1658, 39479}, {3245, 45885}, {3636, 34923}, {3754, 32205}, {3878, 32142}, {3937, 52478}, {5603, 47045}, {5690, 31847}, {5885, 13752}, {6073, 11698}, {10738, 31512}, {12006, 35004}, {12331, 14513}, {12773, 14511}, {17101, 37821}, {18342, 51562}, {22791, 31849}, {34583, 38028}, {38722, 53294}

X(53800) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 23153, 13756}, {3, 901, 38614}, {3, 953, 38617}, {3, 38584, 901}, {3, 38586, 953}, {901, 953, 3}, {901, 38586, 38617}, {901, 38682, 953}, {901, 38707, 38705}, {953, 38584, 38614}, {953, 38682, 38586}, {953, 38705, 38707}, {3025, 13756, 1}, {3259, 31841, 5}, {13752, 13753, 5885}, {24201, 33645, 942}, {38512, 38513, 3}, {38568, 38569, 3}, {38584, 38586, 3}, {38584, 38682, 38617}, {38614, 38617, 3}, {38705, 38707, 3}, {38954, 40100, 4}, {44973, 44979, 382}

---

X(53801) = X(3)X(1308)∩X30)X(511)

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

X(53801) lies on these lines: {1, 3322}, {3, 1308}, {30, 511}, {901, 2724}, {927, 953}, {1155, 5723}, {1159, 34232}, {1323, 5126}, {1566, 31841}, {3259, 9779}, {5532, 37718}, {13756, 44043}, {20328, 40554}, {22102, 34464}, {34805, 51406}, {35459, 47621}, {38682, 47043}, {44975, 44979}

X(53801) = barycentric quotient X(40862)/X(33953)
X(53801) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1308, 2717, 3}, {3322, 3328, 1}

---

X(53802) = X(3)X(847)∩X30)X(511)

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

X(53802) lies on these lines: {3, 847}, {4, 13556}, {5, 131}, {26, 13558}, {30, 511}, {110, 15112}, {125, 51451}, {140, 34840}, {382, 44974}, {1112, 36160}, {1658, 5961}, {6146, 44209}, {8800, 43917}, {8905, 23335}, {9826, 14894}, {10224, 23702}, {11250, 13496}, {12091, 36184}, {12118, 34757}, {13371, 15665}, {13553, 16238}, {15035, 15111}, {18377, 22823}, {18569, 39118}, {25641, 46414}, {36179, 44573}

X(53802) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {131, 136, 5}, {131, 21667, 136}, {925, 1300, 3}, {34840, 34844, 140}, {44974, 44990, 382}

---

X(53803) = X(3)X(107)∩X30)X(511)

Barycentrics    a^14*b^2 - 5*a^12*b^4 + 10*a^10*b^6 - 10*a^8*b^8 + 5*a^6*b^10 - a^4*b^12 + a^14*c^2 + 6*a^12*b^2*c^2 - 9*a^10*b^4*c^2 - 13*a^8*b^6*c^2 + 19*a^6*b^8*c^2 - 3*a^2*b^12*c^2 - b^14*c^2 - 5*a^12*c^4 - 9*a^10*b^2*c^4 + 46*a^8*b^4*c^4 - 24*a^6*b^6*c^4 - 23*a^4*b^8*c^4 + 9*a^2*b^10*c^4 + 6*b^12*c^4 + 10*a^10*c^6 - 13*a^8*b^2*c^6 - 24*a^6*b^4*c^6 + 48*a^4*b^6*c^6 - 6*a^2*b^8*c^6 - 15*b^10*c^6 - 10*a^8*c^8 + 19*a^6*b^2*c^8 - 23*a^4*b^4*c^8 - 6*a^2*b^6*c^8 + 20*b^8*c^8 + 5*a^6*c^10 + 9*a^2*b^4*c^10 - 15*b^6*c^10 - a^4*c^12 - 3*a^2*b^2*c^12 + 6*b^4*c^12 - b^2*c^14 : :

X(53803) lies on these lines: {1, 3324}, {3, 107}, {4, 2972}, {5, 122}, {20, 1075}, {26, 14703}, {30, 511}, {74, 36162}, {140, 6716}, {355, 50916}, {381, 10714}, {382, 10152}, {548, 52869}, {550, 3184}, {1112, 36179}, {1351, 10762}, {1385, 11718}, {1482, 10701}, {1511, 53757}, {3146, 44003}, {3534, 51877}, {3627, 8798}, {5901, 11732}, {6662, 32137}, {6699, 24930}, {7387, 14673}, {10738, 10775}, {12041, 53716}, {12042, 53723}, {14644, 18870}, {14847, 38727}, {14855, 42453}, {15035, 37926}, {17800, 23241}, {23515, 28144}, {31510, 38625}, {32162, 47087}, {37814, 40082}, {41202, 43576}, {42329, 52066}

X(53803) = isogonal conjugate of X(44874)
X(53803) = Thomson-isogonal conjugate of X(6080)
X(53803) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 107, 38605}, {3, 1294, 38621}, {3, 38577, 107}, {3, 38591, 1294}, {3, 38672, 51532}, {4, 10745, 49117}, {4, 34186, 10745}, {4, 34549, 22337}, {20, 5667, 23240}, {20, 35360, 40948}, {107, 1294, 3}, {107, 38577, 51532}, {107, 38591, 38621}, {107, 38672, 38577}, {107, 38686, 1294}, {107, 38714, 23239}, {122, 133, 5}, {1294, 23239, 38714}, {1294, 38577, 38605}, {1294, 38672, 107}, {1294, 38686, 38591}, {3324, 7158, 1}, {6716, 34842, 140}, {10152, 44985, 382}, {10745, 22337, 4}, {22337, 34186, 49117}, {23239, 38714, 3}, {34186, 34549, 4}, {38505, 38516, 3}, {38577, 38591, 3}, {38577, 38686, 38621}, {38591, 38672, 38605}, {38605, 38621, 3}, {38605, 51532, 107}, {38621, 51532, 38605}, {38672, 38686, 3}

---

X(53804) = X(3)X(934)∩X30)X(511)

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

X(53804) lies on these lines: {1, 1360}, {3, 934}, {4, 10405}, {5, 5514}, {30, 511}, {40, 2124}, {140, 40555}, {382, 44978}, {946, 32446}, {1565, 52156}, {10004, 28344}, {12699, 48357}, {15934, 37028}, {17044, 31852}, {18329, 38767}, {24470, 51490}, {35110, 38690}

X(53804) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {934, 972, 3}, {5514, 44993, 5}, {44978, 44980, 382}

---

X(53805) = X(3)X(2696)∩X30)X(511)

Barycentrics    a^14*b^2 - a^12*b^4 - 2*a^10*b^6 + 2*a^8*b^8 + a^6*b^10 - a^4*b^12 + a^14*c^2 - 14*a^12*b^2*c^2 + 24*a^10*b^4*c^2 + 18*a^8*b^6*c^2 - 32*a^6*b^8*c^2 - 3*a^4*b^10*c^2 + 7*a^2*b^12*c^2 - b^14*c^2 - a^12*c^4 + 24*a^10*b^2*c^4 - 96*a^8*b^4*c^4 + 48*a^6*b^6*c^4 + 48*a^4*b^8*c^4 - 21*a^2*b^10*c^4 + 2*b^12*c^4 - 2*a^10*c^6 + 18*a^8*b^2*c^6 + 48*a^6*b^4*c^6 - 96*a^4*b^6*c^6 + 14*a^2*b^8*c^6 + b^10*c^6 + 2*a^8*c^8 - 32*a^6*b^2*c^8 + 48*a^4*b^4*c^8 + 14*a^2*b^6*c^8 - 4*b^8*c^8 + a^6*c^10 - 3*a^4*b^2*c^10 - 21*a^2*b^4*c^10 + b^6*c^10 - a^4*c^12 + 7*a^2*b^2*c^12 + 2*b^4*c^12 - b^2*c^14 : :

X(53805) lies on these lines: {3, 2696}, {5, 31655}, {30, 511}, {381, 34320}, {476, 6093}, {477, 6082}, {3258, 6092}, {5077, 9159}, {5912, 46633}, {5913, 11799}, {5971, 7464}, {14650, 36168}, {16339, 49116}, {25641, 31654}, {33964, 44048}

X(53805) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 38598, 2696}, {2696, 2770, 3}

---

X(53806) = X(3)X(2378)∩X30)X(511)

Barycentrics    a^2*(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 - 8*a^2*b^4*c^2 - 5*b^6*c^2 - 5*a^4*c^4 - 8*a^2*b^2*c^4 + 16*b^4*c^4 + 5*a^2*c^6 - 5*b^2*c^6 - c^8 + 2*Sqrt[3]*(a^4*b^2 - b^6 + a^4*c^2 - 4*a^2*b^2*c^2 + 2*b^4*c^2 + 2*b^2*c^4 - c^6)*S) : :

X(537) lies on these lines: {1, 47009}, {3, 2378}, {15, 2502}, {30, 511}, {182, 14174}, {373, 40672}, {5610, 5611}, {5640, 25176}, {5981, 16259}, {6138, 40283}, {11626, 25174}, {14169, 14176}, {16260, 25210}, {16462, 25172}, {30439, 47860}

X(53806) = isogonal conjugate of X(44875)
X(53806) = crossdifference of every pair of points on line {6, 9195}
X(53806) = barycentric quotient X(i)/X(j) for these {i,j}: {2346, 34286}, {17230, 15325}, {24056, 44953}
X(53806) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2378, 9202, 3}, {14176, 14184, 14169}, {25172, 25215, 16462}, {25174, 25221, 11626}, {25176, 25225, 5640}, {25210, 25212, 16260}

---

X(53807) = X(3)X(2379)∩X30)X(511)

Barycentrics    a^2*(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 - 8*a^2*b^4*c^2 - 5*b^6*c^2 - 5*a^4*c^4 - 8*a^2*b^2*c^4 + 16*b^4*c^4 + 5*a^2*c^6 - 5*b^2*c^6 - c^8 - 2*Sqrt[3]*(a^4*b^2 - b^6 + a^4*c^2 - 4*a^2*b^2*c^2 + 2*b^4*c^2 + 2*b^2*c^4 - c^6)*S) : :

X(53807) lies on these lines: {1, 47010}, {3, 2379}, {16, 2502}, {30, 511}, {182, 14180}, {373, 40671}, {805, 44875}, {5614, 5615}, {5640, 25181}, {5980, 16260}, {6137, 40283}, {11624, 25179}, {14170, 14175}, {16259, 25209}, {16461, 25171}, {30440, 47859}

X(53807) = crossdifference of every pair of points on line {6, 9194}
X(53807) = barycentric product X(34732)*X(39509)
X(53807) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2379, 9203, 3}, {14175, 14183, 14170}, {25171, 25218, 16461}, {25179, 25222, 11624}, {25181, 25226, 5640}, {25209, 25211, 16259}

---

X(53808) = X(3)X(2379)∩X30)X(511)

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

X(53808) lies on these lines: {3, 933}, {5, 18402}, {30, 511}, {382, 44977}, {546, 10214}, {1658, 8157}, {1986, 36161}, {5498, 13856}, {5562, 8439}, {5576, 45997}, {5889, 34304}, {7512, 14152}, {10125, 11701}, {10226, 15345}, {11587, 14118}, {13506, 34783}, {25043, 44279}

X(53808) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 933, 38616}, {3, 38585, 933}, {933, 18401, 3}, {18401, 38585, 38616}, {18402, 20625, 5}

---

X(53809) = X(3)X(1290)∩X30)X(511)

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

X(53809) lies on these lines: {1, 31522}, {3, 1290}, {5, 5520}, {12, 51886}, {30, 511}, {36, 39751}, {40, 11749}, {355, 47273}, {382, 44982}, {476, 953}, {477, 901}, {1325, 22765}, {1482, 36171}, {1483, 47274}, {3025, 24470}, {3258, 31841}, {3259, 25641}, {5690, 14731}, {6742, 52056}, {7575, 11809}, {10738, 36175}, {13756, 33965}, {14989, 44973}, {15171, 23152}, {17404, 19630}, {18319, 40273}, {18357, 20957}, {21664, 37964}, {22102, 31379}, {22791, 38580}, {33814, 36167}, {35000, 36001}, {38581, 38584}, {38602, 46618}, {38609, 38617}, {38610, 38614}, {38677, 38682}, {38700, 38707}, {38701, 38705}, {44967, 44979}, {47324, 47346}

X(53809) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 38588, 1290}, {1290, 2687, 3}, {3109, 52200, 5901}, {5520, 42422, 5}, {31522, 31524, 1}, {36167, 46635, 33814}, {46618, 46636, 38602}

---

X(53810) = X(3)X(7597)∩X30)X(511)

Barycentrics    a*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3 - (a + b - c)*(a - b + c)*(b + c)*Csc[A/2] - b*(a - b - c)*(a + b - c)*Csc[B/2] - (a - b - c)*c*(a - b + c)*Csc[C/2]) : :

X(53810) lies on these lines: {1, 8099}, {3, 164}, {4, 9807}, {5, 21633}, {30, 511}, {40, 167}, {72, 11691}, {177, 942}, {258, 8081}, {355, 17657}, {1071, 12539}, {1385, 12523}, {1482, 8111}, {3579, 12518}, {5044, 18258}, {5045, 5571}, {5049, 11191}, {5728, 7670}, {5777, 12694}, {7588, 18448}, {8077, 18456}, {8078, 8082}, {8084, 8092}, {8091, 8094}, {8103, 12772}, {8104, 12771}, {8148, 11528}, {8351, 11032}, {8422, 9957}, {9940, 12443}, {9947, 12450}, {9955, 12614}, {9956, 12622}, {9959, 13091}, {10441, 12554}, {10500, 30408}, {11033, 11044}, {12488, 12879}, {12489, 12884}, {12490, 13090}, {12491, 13092}, {18399, 30420}, {18409, 30370}, {31735, 31776}, {31766, 31792}, {31768, 31794}, {31769, 31795}, {31780, 31783}, {31787, 31790}, {31797, 31800}, {35631, 35644}

X(53810) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {164, 12844, 3}, {177, 942, 12813}, {3659, 7597, 3}, {5571, 12908, 5045}, {8084, 8093, 8092}, {8094, 10967, 8091}, {8099, 8100, 1}, {10501, 10506, 1}, {31766, 32183, 31792}

---

X(53811) = TRILINEAR POLE OF LINE X(36)X(80)

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

See César Lozada, euclid 5728.

X(53811) lies on these lines: {7,40215}, {651,23592}, {655,2401}, {2006,34234}, {4585,13136}, {4588,52934}, {18815,52663}, {34232,40437}, {36037,43728}

X(53811) = X(i)-cross conjugate of-X(j) for these (i, j): (226, 39294), (514, 34234), (522, 18815), (651, 37136)
X(53811) = X(1086)-Dao conjugate of-X(46398)
X(53811) = X(i)-isoconjugate-of-X(j) for these {i, j}: {36, 46393}, {284, 42768}, {517, 654}, {650, 34586}, {652, 1845}
X(53811) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (65, 42768), (80, 2804), (104, 3738), (108, 1845), (109, 34586)
X(53811) = intersection, other than A, B, C, of circumconics {A, B, C, X(1), X(24029)} and {A, B, C, X(2), X(653)}
X(53811) = trilinear pole of the line {36, 80}
X(53811) = barycentric product X(i)*X(j) for these {i, j}: {104, 35174}, {655, 34234}, {664, 40437}, {909, 46405}, {2006, 13136}, {2222, 18816}
X(53811) = barycentric quotient X(i)/X(j) for these (i, j): (65, 42768), (80, 2804), (104, 3738), (108, 1845), (109, 34586), (514, 46398)
X(53811) = trilinear product X(i)*X(j) for these {i, j}: {80, 37136}, {104, 655}, {651, 40437}, {909, 35174}, {1411, 13136}, {2006, 36037}
X(53811) = trilinear quotient X(i)/X(j) for these (i, j): (80, 46393), (104, 654), (226, 42768), (521, 38353), (651, 34586), (653, 1845)

---

X(53812) = X(3)X(47)∩X(1)X(24029)

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

See Ivan Pavlov, euclid 5735.

X(53812) lies on these lines: {1,24029}, {3,47}, {36,34913}, {46,38507}, {102,36052}, {517,14584}, {580,38565}, {950,5697}, {1718,51896}, {1725,35014}, {1772,2818}, {3293,11010}, {3583,38938}, {3887,53562}, {13744,42450}

---

X(53813) = X(4)X(2182)∩X(92)X(515)

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

See Ivan Pavlov, euclid 5739.

X(53813) lies on circumconics: {A,B,C,X(4),X(27)}, {A,B,C,X(7),X(1065) and these lines: {4,2182}, {40,40445}, {92,515}, {278,1455}, {412,37203}, {37420,40444}

X(53813) = X(i)-isoconjugate-of-X(j) for these {i, j}: {255, 5587}
X(53813) = X(i)-Dao conjugate of X(j) for these {i, j}: {6523, 5587}
X(53813) = barycentric quotient X(i)*X(j) for these (i, j): {393, 5587}

---

X(53814) = X(63)X(69)∩X(322)X(668)

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

See Ivan Pavlov, euclid 5739.

X(53814) lies on circumconics: {A,B,C,X(63),X(34393)}, {A,B,C,X(71),X(5587)} and these lines: {63,69}, {322,668}, {326,52347}, {10432,32099}, {20928,33672}, {35516,40071}

X(53814) = barycentric product X(i)*X(j) for these (i, j): {3926, 5587}
X(53814) = barycentric quotient X(i)*X(j) for these (i, j): {5587, 393}

---

X(53815) = X(3)X(48)∩X(40)X(78)

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

See Ivan Pavlov, euclid 5739.

X(53815) lies on circumconics {A,B,C,X(3),X(36100)}, {A,B,C,X(48),X(102)}, {A,B,C,X(64),X(19350)}, {A,B,C,X(71),X(5587)} and these lines: {1,7549}, {3,48}, {19,5720}, {40,78}, {55,1064}, {64,10310}, {255,1364}, {306,515}, {581,2268}, {912,26934}, {1060,37755}, {1490,1766}, {1715,25440}, {1782,5693}, {2187,15177}, {2947,36029}, {6001,30269}, {6905,24310}, {8251,18673}, {10165,26006}, {10319,18446}, {10902,16452}, {11471,37531}, {12528,21368}, {37694,41227}

X(53815) = barycentric product X(i)*X(j) for these (i, j): {394, 5587}
X(53815) = barycentric quotient X(i)*X(j) for these (i, j): {5587, 2052}
X(53815) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8251, 37700, 18673}

---

X(53816) = X(1)X(2)∩X(63)X(343)

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

See Ivan Pavlov, euclid 5739.

X(53816) lies on circumconics {A,B,C,X(1),X(52780)}, {A,B,C,X(2),X(52392)}, {A,B,C,X(8),X(52351)}, {A,B,C,X(10),X(5587)} and these lines: {1,2}, {63,343}, {77,914}, {281,908}, {307,6350}, {312,6335}, {394,22123}, {440,8251}, {1817,6796}, {2327,5235}, {3306,16608}, {3416,25968}, {3692,51367}, {3694,37638}, {3886,23541}, {5294,11433}, {6708,30852}, {7013,51368}, {11347,11499}, {13567,32777}, {17231,25934}, {17279,26005}, {17286,26591}, {17306,26635}, {17720,21933}, {17861,48380}, {17862,25527}, {18642,30675}, {23681,24209}, {24632,26645}, {25019,37643}, {25091,26958}, {26011,30811}, {26540,32779}, {26651,31017}, {27413,52412}

X(53816) = barycentric product X(i)*X(j) for these (i, j): {69, 5587}
X(53816) = barycentric quotient X(i)*X(j) for these (i, j): {5587, 4}
X(53816) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3635, 15519}, {3635, 15519, 15519}, {4691, 21267}, {4691, 21267, 21267}, {22166, 22266, 22166}, {22266, 22166}

---

X(53817) = X(4)X(48)∩X(6)X(158)

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

See Ivan Pavlov, euclid 5739.

X(53817) lies on circumconics {A,B,C,X(4),X(29)}, {A,B,C,X(6),X(48)}, {A,B,C,X(9),X(275)}, {A,B,C,X(19),X(40402)} and these lines: {4,48}, {6,158}, {29,2193}, {212,281}, {219,318}, {222,273}, {681,32726}, {1812,44130}, {2194,8748}, {14578,36123}, {17073,37279}

X(53817) = trilinear pole of line {1946,3064}
X(53817) = X(i)-isoconjugate-of-X(j) for these {i, j}: {63, 19366}, {653, 680}, {26941, 30493}, {32674, 35521}
X(53817) = X(i)-Dao conjugate of X(j) for these {i, j}: {3162, 19366}, {35072, 35521}
X(53817) = barycentric product X(i)*X(j) for these (i, j): {521, 681}
X(53817) = barycentric quotient X(i)*X(j) for these (i, j): {25, 19366}, {521, 35521}, {681, 18026}, {1946, 680}

---

X(53818) = X(69)X(73)∩X(76)X(331)

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

See Ivan Pavlov, euclid 5739.

X(53818) lies on circumconics {A,B,C,X(69),X(1969)}, {A,B,C,X(73),X(19366)} and these lines: {69,73}, {76,331}, {85,343}, {394,17095}, {5224,52385}, {5249,52422}, {5562,17181}

X(53818) = X(i)-Dao conjugate of X(j) for these {i, j}: {38969, 3063}, {39060, 681}
X(53818) = barycentric product X(i)*X(j) for these (i, j): {305, 19366}, {18026, 35521}
X(53818) = barycentric quotient X(i)*X(j) for these (i, j): {680, 1946}, {18026, 681}, {19366, 25}, {35521, 521}

---

X(53819) = X(2)X(92)∩X(3)X(73)

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

See Ivan Pavlov, euclid 5739.

X(53819) lies on circumconics {A,B,C,X(2),X(255)}, {A,B,C,X(3),X(92)} and these lines: {2,92}, {3,73}, {9,6509}, {57,216}, {63,46832}, {223,18591}, {386,22341}, {418,26892}, {426,26890}, {577,2003}, {856,1038}, {1364,26904}, {1407,36751}, {1473,26898}, {1708,3002}, {1947,3164}, {2260,7011}, {2954,47848}, {3220,26880}, {3305,44436}, {3487,17102}, {3917,26901}, {3929,35072}, {3937,26900}, {5158,52423}, {5219,18592}, {6641,26884}, {7308,46831}, {7536,43043}, {13388,32593}, {13389,32591}, {13409,26893}, {16577,25063}, {21482,34048}, {26870,26929}, {26903,30493}, {26905,26933}, {26906,26932}, {26908,39796}, {26942,41005}, {34032,40590}, {37551,52543}

X(53819) = X(i)-isoconjugate-of-X(j) for these {i, j}: {652, 681}
X(53819) = X(i)-Dao conjugate of X(j) for these {i, j}: {38969, 650}
X(53819) = barycentric product X(i)*X(j) for these (i, j): {69, 19366}, {108, 35521}, {680, 18026}, {26941, 44708}
X(53819) = barycentric quotient X(i)*X(j) for these (i, j): {108, 681}, {680, 521}, {19366, 4}, {35521, 35518}
X(53819) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3937, 26907, 26900}

---

X(53820) = X(53)X(65)∩X(185)X(1826)

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

See Ivan Pavlov, euclid 5739.

X(53820) lies on these lines: {53, 65}, {185, 1826}, {1856, 42447}

---

X(53821) = X(2)X(7)∩X(5)X(273)

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

See Ivan Pavlov, euclid 5739.

X(53821) lies on circumconics {A,B,C,X(2),X(7331)}, {A,B,C,X(9),X(3469)}, {A,B,C,X(63),X(264)} and these lines: {2, 7}, {3, 7282}, {5, 273}, {75, 45198}, {77, 1745}, {85, 18749}, {223, 18631}, {264, 1441}, {318, 41005}, {342, 442}, {347, 1838}, {469, 1214}, {581, 1442}, {857, 18591}, {1038, 26120}, {1119, 3090}, {1532, 41007}, {1804, 17095}, {1876, 7380}, {1892, 6998}, {1893, 36687}, {3614, 6046}, {3673, 7399}, {3876, 7066}, {4329, 6838}, {4911, 7395}, {6840, 17134}, {7279, 22467}, {7567, 41004}, {10538, 40680}, {17073, 26003}

X(53821) = X(i)-Dao conjugate of X(j) for these {i, j}: {38969, 1946}
X(53821) = barycentric product X(i)*X(j) for these (i, j): {76, 19366}
X(53821) = barycentric quotient X(i)*X(j) for these (i, j): {680, 36054}, {19366, 6}, {26941, 44687}
X(53821) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 6356, 273}

---

X(53822) = X(2)X(1289)∩X(3)X(132)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53822) lies on the nine-point circle, the circumconic {A,B,C,X(127),X(34129)} and these lines: {2, 1289}, {3, 132}, {4, 34168}, {5, 50938}, {113, 206}, {114, 11585}, {125, 339}, {126, 30771}, {127, 47413}, {133, 15760}, {135, 868}, {136, 3150}, {381, 50937}, {1368, 1560}, {2072, 42426}, {3548, 31842}, {10257, 16188}, {10297, 18809}, {12605, 44953}, {13854, 19615}, {15526, 46654}, {16178, 36189}, {16221, 37987}, {20621, 21530}, {25640, 30445}, {44249, 50934}

X(53822) = midpoint of X(i) in X(j) for these {i,j}: {4, 34168}
X(53822) = reflection of X(i) in X(j) for these {i,j}: {50938, 5}
X(53822) = X(i)-Dao conjugate of X(j) for these {i, j}: {127, 39417}, {2485, 13575}, {8673, 39172}, {47125, 2}, {52588, 14376}
X(53822) = X(i)-Ceva conjugate of X(j) for these {i, j}: {2, 47125}, {4, 8673}, {18018, 23881}
X(53822) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 8673}, {22, 8062}, {31, 47125}, {48, 3265}, {127, 21253}, {163, 6720}, {206, 16612}, {315, 21259}, {525, 16607}, {647, 16580}, {656, 427}, {810, 32}, {1459, 40959}, {1760, 30476}, {2172, 525}, {2485, 226}, {4463, 20316}, {8673, 10}, {10316, 14838}, {14208, 6697}, {16757, 34830}, {17453, 2485}, {18187, 116}, {20806, 4369}, {21034, 2509}, {21122, 40941}, {33294, 20305}, {34254, 42327}, {38356, 8287}, {47413, 34846}, {51640, 18636}, {52915, 23998}
X(53822) = barycentric product X(i)*X(j) for these (i, j): {127, 1370}
X(53822) = barycentric quotient X(i)*X(j) for these (i, j): {127, 13575}, {159, 15388}, {1370, 44183}, {2485, 39417}, {38356, 34207}, {47125, 1289}, {47413, 52041}
X(53822) = complement of X(1289)
X(53822) = complement of the isogonal conjugate of the perspector of the circumconic {A,B,C,X(2),X(22)}

---

X(53823) = X(2)X(743)∩X(11)X(6377)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53823) lies on the nine-point circle and these lines: {2, 743}, {5, 44951}, {11, 6377}, {20, 44941}, {116, 1015}, {118, 21796}, {120, 37661}, {121, 25352}, {125, 16592}, {127, 16573}, {1213, 45162}, {1376, 29943}, {2679, 3271}, {3125, 5509}, {8265, 44412}, {16583, 44952}, {16613, 38968}

X(53823) = midpoint of X(i) in X(j) for these {i,j}: {20, 44941}
X(53823) = reflection of X(i) in X(j) for these {i,j}: {789, 40545}, {44951, 5}
X(53823) = X(i)-Dao conjugate of X(j) for these {i, j}: {4874, 2}, {40545, 40545}
X(53823) = X(i)-Ceva conjugate of X(j) for these {i, j}: {2, 4874}, {4, 788}, {330, 824}, {6650, 30665}
X(53823) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 788}, {31, 4874}, {32, 824}, {649, 21264}, {667, 24325}, {788, 10}, {824, 626}, {869, 513}, {904, 3805}, {984, 21260}, {1469, 17072}, {1491, 2887}, {1911, 30665}, {1919, 17023}, {2276, 3835}, {3250, 141}, {3661, 21262}, {3736, 512}, {3774, 4129}, {3862, 21261}, {4475, 21252}, {4481, 21240}, {8630, 37}, {16514, 27854}, {18900, 650}, {29956, 20335}, {30654, 39080}, {30665, 20542}, {30671, 20541}, {30966, 23301}, {40728, 514}, {40736, 31286}, {40773, 42327}, {46386, 2}, {46503, 8062}, {52655, 21191}
X(53823) = barycentric product X(i)*X(j) for these (i, j): {1491, 4874}, {3923, 4475}
X(53823) = barycentric quotient X(i)*X(j) for these (i, j): {3765, 5388}, {4874, 789}
X(53823) = complement of X(789)
X(53823) = anticomplement X(40545)
X(53823) = complement of the isogonal conjugate of the perspector of the circumconic {A,B,C,X(2),X(31)}
X(53823) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 789, 40545}

---

X(53824) = X(2)X(6183)∩X(9)X(120)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53824) lies on the nine-point circle and these lines: {2, 6183}, {9, 120}, {11, 17435}, {116, 6506}, {123, 13609}, {1146, 5511}, {1566, 35508}, {1826, 20622}, {3119, 5521}, {5179, 33331}, {34530, 35967}, {44993, 45250}

X(53824) = X(i)-Ceva conjugate of X(j) for these {i, j}: {4, 6182}
X(53824) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 6182}, {663, 1001}, {948, 46399}, {2263, 3900}, {2550, 17072}, {3063, 37597}, {6182, 10}, {16054, 17066}, {37580, 522}, {40131, 4885}, {47123, 2886}
X(53824) = complement of X(6183)
X(53824) = complement of the isogonal conjugate of the perspector of the circumconic {A,B,C,X(2),X(33)}

---

X(53825) = X(120)X(5268)∩X(123)X(1086)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53825) lies on the nine-point circle and these lines: {120, 5268}, {123, 1086}, {127, 8286}, {3120, 5521}, {3125, 5514}, {15611, 17071}

X(53825) = X(i)-complementary conjugate of X(j) for these (i,j): {43924, 25524}, {47136, 1329}, {52082, 20316}
X(53825) = complement of the isogonal conjugate of the perspector of the circumconic {A,B,C,X(2),X(34)}

---

X(53826) = X(116)X(47882)∩X(124)X(46398)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53826) lies on the nine-point circle and these lines: {116, 47882}, {124, 46398}, {650, 38963}, {11231, 31841}

X(53826) = X(i)-Ceva conjugate of X(j) for these {i, j}: {36917, 23876}
X(53826) = X(i)-complementary conjugate of X(j) for these (i,j): {7113, 23876}, {23876, 21237}
X(53826) = complement of the isogonal conjugate of the perspector of the circumconic {A,B,C,X(2),X(36)}

---

X(53827) = X(114)X(233)∩X(127)X(39019)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53827) lies on the nine-point circle and these lines: {114, 233}, {127, 39019}, {7668, 38974}, {15819, 31843}, {18322, 33330}

X(53827) = X(i)-complementary conjugate of X(j) for these (i,j): {2179, 23878}, {3288, 21231}, {39530, 21259}
X(53827) = complement of the isogonal conjugate of the perspector of the circumconic {A,B,C,X(2),X(51)}

---

X(53828) = X(125)X(4858)∩X(127)X(34588)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53828) lies on the nine-point circle and these lines: {119, 39566}, {125, 4858}, {127, 34588}, {7336, 15608}

X(53828) = complement of the isogonal conjugate of the perspector of the circumconic {A,B,C,X(2),X(60)}

---

X(53829) = X(2)X(931)∩X(11)X(17058)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53829) lies on the nine-point circle and these lines: {2, 931}, {5, 50928}, {11, 17058}, {116, 8286}, {123, 15526}, {124, 8287}, {126, 10472}, {3120, 38967}, {3841, 31845}

X(53829) = reflection of X(i) in X(j) for these {i,j}: {50928, 5}
X(53829) = X(i)-Ceva conjugate of X(j) for these {i, j}: {4, 8672}
X(53829) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 8672}, {512, 5257}, {661, 5743}, {798, 5283}, {940, 4369}, {1400, 23880}, {1867, 20316}, {4017, 25466}, {4185, 8062}, {5019, 14838}, {5307, 30476}, {8639, 37}, {8672, 10}, {17418, 960}, {23880, 21246}, {31993, 3835}, {34284, 42327}, {43067, 3741}, {43924, 4719}, {48144, 3739}, {50457, 141}, {51642, 2277}
X(53829) = complement of X(931)
X(53829) = complement of the isogonal conjugate of the perspector of the circumconic {A,B,C,X(2),X(65)}

---

X(53830) = X(114)X(7499)∩X(137)X(338)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53830) lies on the nine-point circle and these lines: {113, 13160}, {114, 7499}, {128, 14652}, {131, 12362}, {133, 23047}, {137, 338}, {3153, 25641}, {15526, 35969}, {21284, 42426}, {35442, 38971}

X(53830) = X(i)-complementary conjugate of X(j) for these (i,j): {570, 14838}, {656, 7542}, {661, 37649}, {1109, 8901}, {1594, 8062}, {2616, 6689}, {2618, 52526}, {16698, 21196}, {24006, 5462}, {37636, 4369}, {47328, 16612}, {50947, 16598}
X(53830) = complement of the isogonal conjugate of the perspector of the circumconic {A,B,C,X(2),X(70)}

---

X(53831) = X(5)X(50929)∩X(11)X(16573)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53831) lies on the nine-point circle and these lines: {5, 50929}, {11, 16573}, {116, 15526}, {132, 7380}

X(53831) = reflection of X(i) in X(j) for these {i,j}: {50929, 5}
X(53831) = X(i)-complementary conjugate of X(j) for these (i,j): {228, 28623}, {656, 47514}, {810, 386}, {1011, 8062}, {2304, 525}, {10449, 21259}, {44120, 16612}
X(53831) = complement of the isogonal conjugate of the perspector of the circumconic {A,B,C,X(2),X(71)}

---

X(53832) = X(5)X(50929)∩X(11)X(16573)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53832) lies on the nine-point circle and these lines: {2, 74}, {4, 9064}, {5, 50935}, {25, 50937}, {114, 30739}, {115, 3134}, {125, 9003}, {126, 3014}, {127, 1650}, {131, 1368}, {132, 5094}, {133, 427}, {136, 38393}, {376, 52447}, {468, 18809}, {523, 46436}, {858, 25641}, {868, 5512}, {1560, 3815}, {1561, 44889}, {2972, 35968}, {3138, 25642}, {3139, 5511}, {3150, 14672}, {3154, 5099}, {5139, 35235}, {5159, 42424}, {9191, 16280}, {10749, 37072}, {12079, 15526}, {13611, 20625}, {14702, 40916}, {16051, 30789}, {16188, 47097}, {30745, 45180}, {30775, 46339}, {34980, 35579}, {37439, 50927}, {37454, 44953}, {37985, 38971}, {38975, 39602}, {42426, 47215}

X(53832) = midpoint of X(i) in X(j) for these {i,j}: {4, 43660}, {376, 52447}
X(53832) = reflection of X(i) in X(j) for these {i,j}: {50935, 5}
X(53832) = X(i)-Dao conjugate of X(j) for these {i, j}: {9209, 2}
X(53832) = X(i)-Ceva conjugate of X(j) for these {i, j}: {2, 9209}, {4, 8675}, {1494, 46229}
X(53832) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 8675}, {31, 9209}, {378, 8062}, {656, 15760}, {661, 37648}, {798, 5309}, {810, 5158}, {2159, 46229}, {2616, 5892}, {5063, 14838}, {8675, 10}, {15066, 4369}, {30474, 2887}, {32833, 42327}, {42660, 37}, {42704, 3835}, {44080, 16612}, {44134, 21259}
X(53832) = barycentric product X(i)*X(j) for these (i, j): {9209, 30474}
X(53832) = barycentric quotient X(i)*X(j) for these (i, j): {9209, 1302}
X(53832) = complement of X(1302)
X(53832) = complement of the isogonal conjugate of the perspector of the circumconic {A,B,C,X(2),X(74)}

---

X(53833) = X(2)X(40117)∩X(3)X(40555)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53833) lies on the nine-point circle and these lines: {2, 40117}, {3, 40555}, {5, 50930}, {11, 31893}, {117, 18589}, {118, 6260}, {120, 30779}, {125, 7358}, {223, 20623}, {4466, 31653}, {5514, 16596}, {6907, 17073}, {7952, 20621}, {20622, 37160}

X(53833) = reflection of X(i) in X(j) for these {i,j}: {50930, 5}
X(53833) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 14331}, {73, 24018}, {78, 20318}, {109, 40535}, {198, 3239}, {221, 14837}, {222, 8058}, {223, 521}, {329, 20316}, {347, 46396}, {521, 20205}, {647, 1901}, {650, 20263}, {652, 281}, {822, 46837}, {905, 946}, {1459, 57}, {1817, 8062}, {1946, 46830}, {2187, 2509}, {2199, 6588}, {2331, 14298}, {2360, 525}, {3063, 20311}, {3209, 52587}, {4025, 21239}, {6129, 226}, {6332, 20306}, {6611, 21172}, {7011, 522}, {7013, 4885}, {7078, 514}, {7114, 905}, {8058, 41883}, {8822, 30476}, {10397, 9}, {14298, 20262}, {14837, 5}, {16596, 124}, {17896, 20305}, {22383, 1108}, {41083, 520}, {47432, 13609}, {52373, 17898}
X(53833) = complement of X(40117)
X(53833) = complement of the isogonal conjugate of the perspector of the circumconic {A,B,C,X(2),X(77)}

---

X(53834) = X(2)X(8652)∩X(4)X(28145)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53834) lies on the nine-point circle and these lines: {2, 8652}, {4, 28145}, {5, 45167}, {20, 45157}, {119, 15016}, {120, 29679}, {5520, 26933}, {8287, 38967}, {21252, 46660}

X(53834) = midpoint of X(i) in X(j) for these {i,j}: {4, 28145}, {20, 45157}
X(53834) = reflection of X(i) in X(j) for these {i,j}: {45167, 5}
X(53834) = X(i)-Ceva conjugate of X(j) for these {i, j}: {4, 4802}
X(53834) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 4802}, {2, 4932}, {513, 19862}, {649, 28606}, {650, 5325}, {2160, 23883}, {3715, 4521}, {3927, 20315}, {4007, 20317}, {4066, 31946}, {4654, 4885}, {4658, 523}, {4756, 24003}, {4802, 10}, {4810, 17793}, {4813, 2}, {4820, 3452}, {4823, 141}, {4826, 16589}, {4834, 37}, {4838, 1211}, {4840, 1125}, {4893, 30563}, {4960, 3739}, {5221, 522}, {5333, 4369}, {7241, 4961}, {7649, 6147}, {28605, 3835}, {30589, 47779}, {30595, 16597}, {30596, 21260}, {31902, 8062}, {36074, 16578}, {43260, 48049}, {48005, 1213}
X(53834) = complement of X(8652)
X(53834) = complement of the isogonal conjugate of the perspector of the circumconic {A,B,C,X(2),X(79)}

---

X(53835) = X(2)X(831)∩X(11)X(21208)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53835) lies on the nine-point circle and these lines: {2, 831}, {11, 21208}, {120, 3634}, {8287, 46654}, {45162, 52529}

X(53835) = X(i)-Ceva conjugate of X(j) for these {i, j}: {4, 830}
X(53835) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 830}, {6, 3004}, {251, 23885}, {649, 17384}, {830, 10}, {2483, 2}, {5314, 20315}, {7247, 17072}, {8635, 37}, {17289, 3835}, {23885, 21248}, {28594, 4129}, {33941, 21260}, {47660, 141}, {47711, 3454}, {50496, 1213}
X(53835) = complement of X(831)
X(53835) = complement of the isogonal conjugate of the perspector of the circumconic {A,B,C,X(2),X(82)}

---

X(53836) = X(5)X(50932)∩X(11)X(21945)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53836) lies on the nine-point circle and these lines: {5, 50932}, {11, 21945}, {119, 8728}, {3139, 38967}, {5142, 25640}, {5514, 26933}, {7004, 13612}, {7490, 50930}, {26932, 46663}, {26956, 35580}

X(53836) = X(i)-Ceva conjugate of X(j) for these {i, j}: {4, 9000}
X(53836) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 9000}, {7431, 8062}, {9000, 10}
X(53836) = complement of X(9057)

---

X(53837) = X(2)X(104)∩X(4)X(9107)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53837) lies on the nine-point circle and these lines: {2, 104}, {4, 9107}, {11, 17888}, {113, 37360}, {115, 3139}, {118, 37315}, {120, 29857}, {124, 26933}, {133, 37362}, {427, 25640}, {858, 42422}, {867, 5511}, {1368, 42423}, {2170, 5517}, {3140, 5512}, {3270, 35580}, {5094, 20621}, {6075, 46415}, {6841, 50935}, {15608, 21252}, {15635, 26932}, {16051, 30787}, {16067, 50936}, {20623, 40131}, {26020, 39535}, {37432, 50930}

X(53837) = X(i)-Dao conjugate of X(j) for these {i, j}: {40134, 2}
X(53837) = X(i)-Ceva conjugate of X(j) for these {i, j}: {2, 40134}, {4, 9001}
X(53837) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 9001}, {31, 40134}, {649, 17720}, {656, 30445}, {1459, 1060}, {1470, 522}, {4227, 8062}, {9001, 10}, {11383, 3239}, {17740, 3835}, {26637, 4369}
X(53837) = barycentric quotient X(i)*X(j) for these (i, j): {40134, 9058}
X(53837) = complement of X(9058)

---

X(53838) = X(2)X(106)∩X(124)X(244)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53838) lies on the nine-point circle and these lines: {2, 106}, {4, 9088}, {116, 1647}, {117, 614}, {120, 29639}, {124, 244}, {125, 3756}, {427, 20619}, {1015, 38963}, {1086, 3259}, {1125, 11796}, {3011, 31841}, {3120, 5510}, {3815, 5513}, {5272, 52050}, {24239, 31845}

X(53838) = reflection of X(i) in X(j) for these {i,j}: {11796, 1125}
X(53838) = X(i)-Dao conjugate of X(j) for these {i, j}: {47766, 2}
X(53838) = X(i)-Ceva conjugate of X(j) for these {i, j}: {2, 47766}, {4, 9002}, {903, 23888}
X(53838) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 9002}, {31, 47766}, {649, 30818}, {667, 17369}, {4247, 8062}, {4266, 20317}, {4389, 21260}, {4424, 31946}, {4850, 3835}, {9002, 10}, {16712, 42327}, {23206, 20315}, {33934, 21262}, {43924, 3753}, {44435, 2887}, {48335, 141}, {48350, 3454}, {50453, 21245}
X(53838) = barycentric product X(i)*X(j) for these (i, j): {44435, 47766}, {47845, 50453}
X(53838) = barycentric quotient X(i)*X(j) for these (i, j): {47766, 9059}
X(53838) = complement of X(9059)

---

X(53839) = X(1)X(116)∩X(2)X(109)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53839) lies on the nine-point circle and these lines: {1, 116}, {2, 109}, {11, 3752}, {115, 3142}, {117, 30574}, {118, 2254}, {121, 14430}, {123, 1214}, {125, 17056}, {127, 18641}, {132, 47210}, {223, 30757}, {407, 5139}, {427, 20620}, {1054, 1699}, {1155, 3259}, {1212, 5514}, {1368, 38977}, {3011, 15608}, {3160, 30740}, {4551, 6505}, {5099, 36195}, {5511, 6831}, {5552, 27135}, {24918, 27622}, {34049, 52659}, {34144, 40916}, {34337, 47806}, {35110, 46415}, {36905, 51364}, {39035, 46670}, {47800, 50940}

X(53839) = X(i)-Ceva conjugate of X(j) for these {i, j}: {4, 8679}
X(53839) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 8679}, {7463, 8062}, {8679, 10}, {33864, 2887}
X(53839) = complement of X(1311)

---

X(53840) = X(2)X(45773)∩X(115)X(9293)

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

See Ivan Pavlov and Elias Hagos, euclid 5794.

X(53840) lies on the nine-point circle and these lines: {2, 45773}, {115, 9293}, {1648, 35582}, {3258, 23992}, {5912, 23991}

X(53840) = X(i)-Ceva conjugate of X(j) for these {i, j}: {4, 33919}
X(53840) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 33919}, {351, 16598}, {661, 11053}, {690, 21254}, {896, 10190}, {1109, 45689}, {1648, 4369}, {2642, 620}, {2643, 690}, {8029, 4892}, {14443, 16597}, {21833, 45661}, {21906, 14838}, {22260, 16611}, {23105, 21256}, {23894, 40553}, {33919, 10}, {42344, 8287}, {45775, 115}, {52628, 42327}
X(53840) = complement of X(45773)

---

X(53841) = (name pending)

Barycentrics    (a*(b+c))/(a*b*c*(b+c)+a*Sqrt[-a^2+2*(b+c)^2]*S+2*S^2) : :

See Angel Montesdeoca, euclid 5796.

X(53841) lies on these lines: {}

---

X(53842) = (name pending)

Barycentrics    (a*(b+c))/(a*b*c*(b+c)-a*Sqrt[-a^2+2*(b+c)^2]*S+2*S^2) : :

See Angel Montesdeoca, euclid 5796.

X(53842) lies on these lines: {}

---

X(53843) = EULER LINE INTERCEPT OF X(230)X(6032)

Barycentrics    2*a^6-5*(b^2+c^2)*a^4-2*(b^4+3*b^2*c^2+c^4)*a^2+5*(b^4-c^4)*(b^2-c^2) : :
X(53843) = 5*X(2)-X(7492)

See César Lozada and Kadir Altintas, euclid 5799.

X(53843) lies on these lines: {2, 3}, {110, 47354}, {114, 30685}, {230, 6032}, {373, 12824}, {542, 14389}, {597, 9140}, {1989, 3815}, {3055, 9220}, {3448, 50979}, {3580, 5476}, {3589, 7703}, {5461, 10162}, {5480, 15360}, {5640, 44569}, {5642, 25561}, {5888, 51128}, {7792, 23297}, {7809, 11056}, {7998, 9971}, {8584, 41724}, {9300, 47298}, {9745, 43620}, {9771, 42008}, {10418, 39601}, {11178, 40112}, {13857, 24206}, {14848, 37644}, {15018, 38079}, {15448, 50960}, {16092, 18122}, {16317, 40103}, {18374, 48310}, {18911, 47352}, {19130, 32225}, {21243, 34565}, {22165, 23061}, {32269, 50959}, {34512, 44380}, {37638, 38072}, {46818, 47353}, {47187, 50718}

X(53843) = midpoint of X(2) and X(5169)
X(53843) = reflection of X(i) in X(j) for these (i, j): (2, 37454), (7495, 2)
X(53843) = inverse of X(47596) in: orthocentroidal circle, Yff hyperbola
X(53843) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 381, 7426), (2, 3545, 1995), (2, 3839, 7493), (2, 7496, 11539), (2, 10989, 549), (2, 16063, 5054), (2, 31099, 3524), (2, 31105, 3), (381, 549, 7576), (381, 15703, 10201), (381, 48411, 549), (427, 549, 10989), (547, 47097, 2), (1656, 32216, 2), (3090, 30775, 2), (5055, 5094, 2), (5133, 7426, 381), (5169, 37454, 7495), (11548, 43957, 2), (15699, 30739, 2), (40916, 52284, 858)

---

X(53844) = X(3)X(64) ∩ X(5)X(122)

Barycentrics    a^2*(-a^2+b^2+c^2)^2*((b^2+c^2)*a^8-4*(b^2-c^2)^2*a^6+6*(b^4-c^4)*(b^2-c^2)*a^4-4*(b^4-c^4)^2*a^2+(b^4-c^4)*(b^2-c^2)^3) : :
X(53844) = 2*X(3)+X(8798) = 4*X(140)-X(14363) = 5*X(631)-X(1075) = 5*X(631)+X(15318) = 10*X(631)-X(22257) = 13*X(10303)-3*X(51877) = 2*X(15318)+X(22257)

See César Lozada and Kadir Altintas, euclid 5799.

X(53844) lies on these lines: {2, 3346}, {3, 64}, {4, 46831}, {5, 122}, {20, 16253}, {39, 23976}, {140, 14363}, {185, 2972}, {216, 631}, {417, 3917}, {426, 43652}, {520, 23103}, {549, 33549}, {1214, 6684}, {2954, 47848}, {3184, 5894}, {3523, 46832}, {5650, 26897}, {6389, 53033}, {6638, 13348}, {6700, 25078}, {6889, 18592}, {10257, 47164}, {10303, 51877}, {13598, 38283}, {14919, 43605}, {15238, 20208}, {15259, 45188}, {15404, 38933}, {18560, 33553}, {25563, 34842}, {33580, 34815}, {36162, 47215}, {37526, 47850}, {40082, 43604}

X(53844) = midpoint of X(i) and X(j) for these {i, j}: {3, 14059}, {1075, 15318}
X(53844) = reflection of X(i) in X(j) for these (i, j): (8798, 14059), (22257, 1075)
X(53844) = complement of X(14249)
X(53844) = X(i)-Ceva conjugate of-X(j) for these (i, j): (2, 20265), (107, 520)
X(53844) = X(i)-complementary conjugate of-X(j) for these (i, j): (3, 20308), (31, 20265), (48, 20207), (255, 2883), (560, 20313)
X(53844) = intersection, other than A, B, C, of circumconics {A, B, C, X(3), X(46353)} and {A, B, C, X(64), X(6247)}
X(53844) = eigentransform of X(107)
X(53844) = barycentric product X(394)*X(6247)
X(53844) = trilinear product X(255)*X(6247)
X(53844) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 1073, 1498), (3, 3357, 11589), (3, 6760, 10282), (3, 14379, 12096), (3, 40675, 10606), (46831, 52543, 4)

---

X(53845) = X(3)X(6) ∩ X(141)X(11614)

Barycentrics    a^2*(8*a^4-11*(b^2+c^2)*a^2+5*b^4-14*b^2*c^2+5*c^4) : :

See César Lozada and Kadir Altintas, euclid 5799.

X(53845) lies on these lines: {3, 6}, {115, 50979}, {141, 11614}, {373, 39689}, {542, 39601}, {597, 5477}, {1353, 3054}, {2502, 44109}, {3055, 51732}, {3787, 44111}, {5012, 13192}, {5032, 21843}, {5471, 31693}, {5472, 31694}, {6776, 18424}, {7708, 11422}, {7769, 20080}, {7817, 31415}, {8550, 39565}, {10765, 15560}, {14912, 43620}, {18583, 43457}, {33748, 43448}, {33842, 44102}, {45018, 47617}

X(53845) = midpoint of X(6) and X(10485)
X(53845) = isogonal conjugate of the isotomic conjugate of X(15597)
X(53845) = barycentric product X(6)*X(15597)
X(53845) = trilinear product X(31)*X(15597)
X(53845) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 182, 5107), (6, 574, 1570), (6, 1692, 5008), (6, 2030, 5052), (6, 5050, 574), (6, 5104, 5097), (6, 5210, 5093), (182, 5107, 8589), (597, 5477, 7603), (2030, 15516, 6), (5008, 8589, 10631), (5033, 11173, 187), (8375, 8376, 5585), (14630, 14631, 15815)

---

X(53846) = X(212)X(8606) ∩ X(3157)X(5452)

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

See César Lozada and Kadir Altintas, euclid 5805.

X(53846) lies on these lines: {212, 8606}, {651, 34398}, {3157, 5452}

X(53846) = crosssum of X(3) and X(14756)

---

X(53847) = X(1)X(3) ∩ X(199)X(204)

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

See César Lozada and Kadir Altintas, euclid 5805.

X(53847) lies on these lines: {1, 3}, {34, 20838}, {48, 39796}, {199, 204}, {255, 51394}, {408, 23196}, {426, 22060}, {577, 22053}, {580, 40407}, {856, 993}, {1364, 7125}, {1817, 14192}, {1836, 17073}, {2238, 9254}, {2360, 7355}, {3215, 13367}, {3474, 6349}, {5432, 51368}, {6056, 40152}, {6284, 44244}, {7114, 40944}, {7354, 18641}, {20780, 26880}, {33098, 42761}

X(53847) = isogonal conjugate of the polar conjugate of X(17073)
X(53847) = crosssum of X(3) and X(42387)
X(53847) = X(1147)-Dao conjugate of-X(37741)
X(53847) = X(i)-isoconjugate-of-X(j) for these {i, j}: {33, 34398}, {158, 37741}, {522, 52776}, {652, 42389}, {1096, 34409}
X(53847) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (108, 42389), (222, 34398), (394, 34409), (577, 37741), (1415, 52776)
X(53847) = intersection, other than A, B, C, of circumconics {A, B, C, X(1), X(17188)} and {A, B, C, X(35), X(255)}
X(53847) = barycentric product X(i)*X(j) for these {i, j}: {3, 17073}, {63, 20277}, {394, 1836}, {1331, 23727}, {1790, 21912}, {1804, 46835}
X(53847) = trilinear product X(i)*X(j) for these {i, j}: {3, 20277}, {48, 17073}, {255, 1836}, {906, 23727}, {1437, 21912}, {1804, 4336}
X(53847) = trilinear quotient X(i)/X(j) for these (i, j): (77, 34398), (109, 52776), (255, 37741), (326, 34409), (653, 42389), (1836, 158)
X(53847) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 56, 40946), (3, 7011, 55), (3, 20764, 35)

---

X(53848) = X(155)X(3162) ∩ X(184)X(216)

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

See César Lozada and Kadir Altintas, euclid 5805.

X(53848) lies on these lines: {2, 23128}, {155, 3162}, {184, 216}, {426, 39643}, {19139, 40601}, {32320, 34966}, {32661, 44208}, {39034, 52249}

X(53848) = isogonal conjugate of the polar conjugate of X(41767)
X(53848) = crosssum of X(3) and X(22391)
X(53848) = X(2)-Ceva conjugate of-X(426)
X(53848) = X(31)-complementary conjugate of-X(426)
X(53848) = X(426)-Dao conjugate of-X(2)
X(53848) = barycentric product X(3)*X(41767)
X(53848) = trilinear product X(48)*X(41767)

---

X(53849) = X(212)X(8606) ∩ X(3157)X(5452)

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

See César Lozada and Kadir Altintas, euclid 5805.

X(53849) lies on these lines: {48, 18591}, {478, 3211}, {3217, 23980}

X(53849) = crosssum of X(3) and X(14737)

---

X(53850) = X(1)X(3) ∩ X(11)X(2218)

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

See César Lozada and Kadir Altintas, euclid 5805.

X(53850) lies on these lines: {1, 3}, {11, 2218}, {19, 37259}, {33, 3145}, {58, 11435}, {71, 577}, {109, 7355}, {117, 6796}, {185, 3215}, {212, 22076}, {255, 1364}, {386, 11428}, {408, 22368}, {497, 27407}, {580, 11436}, {581, 11429}, {603, 39796}, {856, 25440}, {1259, 3719}, {1782, 45272}, {1802, 35072}, {1837, 40980}, {1842, 28077}, {1852, 19542}, {1859, 11334}, {1869, 37397}, {1951, 41320}, {2268, 18591}, {2933, 3198}, {3486, 6350}, {3682, 6056}, {5218, 37180}, {5432, 18641}, {6905, 41227}, {7354, 51368}, {13733, 40950}, {13738, 52427}, {20728, 22401}, {22057, 22072}, {23843, 51361}

X(53850) = crosssum of X(3) and X(42379)
X(53850) = X(1332)-Ceva conjugate of-X(36054)
X(53850) = X(i)-isoconjugate-of-X(j) for these {i, j}: {34, 34406}, {522, 52775}, {652, 42381}, {1096, 34399}, {1118, 40436}
X(53850) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (108, 42381), (219, 34406), (394, 34399), (1415, 52775)
X(53850) = intersection, other than A, B, C, of circumconics {A, B, C, X(36), X(255)} and {A, B, C, X(56), X(1259)}
X(53850) = barycentric product X(i)*X(j) for these {i, j}: {78, 26934}, {219, 41004}, {326, 40968}, {394, 1837}, {1259, 3772}
X(53850) = trilinear product X(i)*X(j) for these {i, j}: {212, 41004}, {219, 26934}, {255, 1837}, {394, 40968}, {1259, 3924}
X(53850) = trilinear quotient X(i)/X(j) for these (i, j): (78, 34406), (109, 52775), (326, 34399), (653, 42381), (1259, 40436), (1837, 158)
X(53850) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 55, 40946), (3, 7011, 5204), (3, 20764, 36)

---

X(53851) = X(2)X(44766) ∩ X(6)X(25)

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

See César Lozada and Kadir Altintas, euclid 5805.

X(53851) lies on these lines: {2, 44766}, {6, 25}, {578, 39524}, {1853, 46829}, {3162, 34146}, {3343, 17825}, {5305, 13567}, {5359, 19161}, {6525, 43717}, {8743, 41580}, {14092, 15661}, {15341, 41602}, {17409, 38356}, {22391, 39071}, {34427, 44885}

X(53851) = complement of the isotomic conjugate of X(52448)
X(53851) = X(i)-complementary conjugate of-X(j) for these (i, j): (158, 6697), (393, 16607), (1096, 427), (2172, 6389), (2207, 16580)
X(53851) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (22, 34412), (1853, 18018)
X(53851) = barycentric product X(22)*X(1853)
X(53851) = barycentric quotient X(i)/X(j) for these (i, j): (22, 34412), (1853, 18018), (2156, 34412)
X(53851) = trilinear product X(1853)*X(2172)
X(53851) = trilinear quotient X(1760)/X(34412)

---

X(53852) = X(2)X(107) ∩ X(3)X(64)

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

See César Lozada and Kadir Altintas, euclid 5805.

X(53852) lies on the cubic K1265 and these lines: {2, 107}, {3, 64}, {20, 35711}, {22, 44436}, {25, 46831}, {127, 14725}, {182, 39045}, {184, 2972}, {216, 1033}, {426, 577}, {441, 30270}, {574, 35071}, {631, 3183}, {1350, 6617}, {1473, 35072}, {1593, 52543}, {1853, 20208}, {1899, 15526}, {3091, 45037}, {3098, 6638}, {3523, 31377}, {5158, 13409}, {5418, 22838}, {5420, 22839}, {5650, 6641}, {6389, 7386}, {7485, 46832}, {9544, 14919}, {10193, 34842}, {10691, 42353}, {10979, 40588}, {15258, 20213}, {15781, 37480}, {17810, 34815}, {22057, 22072}, {22410, 47410}, {26874, 41462}, {26905, 43957}, {26958, 37072}

X(53852) = complement of X(52448)
X(53852) = isogonal conjugate of the polar conjugate of X(20208)
X(53852) = crosssum of X(i) and X(j) for these (i, j): {6, 41369}, {520, 525}
X(53852) = X(112)-Ceva conjugate of-X(520)
X(53852) = X(i)-complementary conjugate of-X(j) for these (i, j): (255, 206), (326, 52532), (394, 21247), (577, 16582)
X(53852) = X(i)-isoconjugate-of-X(j) for these {i, j}: {204, 34407}, {1096, 34412}
X(53852) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (394, 34412), (1073, 34407), (1853, 2052)
X(53852) = X(3)-waw conjugate of-X(577)
X(53852) = intersection, other than A, B, C, of circumconics {A, B, C, X(3), X(14944)} and {A, B, C, X(64), X(1853)}
X(53852) = barycentric product X(i)*X(j) for these {i, j}: {3, 20208}, {394, 1853}
X(53852) = barycentric quotient X(i)/X(j) for these (i, j): (394, 34412), (1073, 34407), (1853, 2052)
X(53852) = trilinear product X(i)*X(j) for these {i, j}: {48, 20208}, {255, 1853}
X(53852) = trilinear quotient X(i)/X(j) for these (i, j): (326, 34412), (1853, 158)
X(53852) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 1073, 154), (3, 6509, 26880), (3, 6760, 11202), (3, 10606, 11589), (3, 14059, 6759), (3, 33924, 34147), (3, 40675, 64), (154, 1073, 34147), (154, 33924, 1073), (426, 3917, 577), (13409, 43650, 5158)

---

X(53853) = X(3)X(9) ∩ X(71)X(577)

Barycentrics    a^2*(-a^2+b^2+c^2)^2*(a^5-(b-c)^2*a^3+(b^2-c^2)*(b-c)*a^2-(b^2-c^2)^2*(b+c)) : :

See César Lozada and Kadir Altintas, euclid 5805.

X(53853) lies on these lines: {3, 9}, {48, 35072}, {71, 577}, {216, 2267}, {426, 22060}, {572, 15629}, {836, 7335}, {1011, 51971}, {1033, 40946}, {1826, 37397}, {3213, 52273}, {21871, 46974}, {36748, 42316}

X(53853) = X(1096)-isoconjugate-of-X(34411)
X(53853) = X(394)-reciprocal conjugate of-X(34411)
X(53853) = trilinear quotient X(326)/X(34411)
X(53853) = {X(3), X(268)}-harmonic conjugate of X(198)

---

X(53854) = X(7)X(5550)∩X(329)X(42030)

Barycentrics    (2*a+2*b+c)*(2*a+b+2*c)*(a^3+2*a^2*(b+c)-2*(b-c)^2*(b+c)-a*(b^2+4*b*c+c^2)) : :

See Ivan Pavlov, euclid 5807.

X(53854) lies on circumconics {A,B,C,X(4),X(12702)} and {A,B,C,X(7),X(36889)} and these lines: {7, 5550}, {329, 42030}, {17484, 25417}

X(53854) = barycentric quotient X(i)*X(j) for these (i, j): {12702, 16777}

---

X(53855) = X(3)X(10308)∩X(56)X(1203)

Barycentrics    a^2*(2*a+2*b+c)*(2*a+b+2*c)*(a^3+2*a^2*(b+c)-2*(b-c)^2*(b+c)-a*(b^2+4*b*c+c^2)) : :

See Ivan Pavlov, euclid 5807.

X(53855) lies on the circumconic {A,B,C,X(56),X(3426)} and these lines: {3, 10308}, {56, 1203}, {7373, 25417}, {8652, 28203}, {19297, 34819}

X(53855) = barycentric product X(i)*X(j) for these (i, j): {12702, 25417}
X(53855) = barycentric quotient X(i)/X(j) for these (i, j): {12702, 28605}

---

X(53856) = X(2)X(32532)∩X(20)X(15533)

Barycentrics    203*a^4-49*b^4+226*b^2*c^2-49*c^4-170*a^2*(b^2+c^2) : :

See Ivan Pavlov, euclid 5807.

X(53856) lies on these lines: {2, 32532}, {20, 15533}, {147, 15300}, {148, 10153}, {376, 47586}, {3534, 46944}, {6194, 14711}, {7616, 10304}, {9742, 15640}, {39785, 49140}

X(53856) = anticomplement of X(32532)
X(53856) = reflection of X(i) in X(j) for these {i,j}: {148, 10153}, {32532, 51589}, {47586, 376}
X(53856) = X(i)-Dao conjugate of X(j) for these {i, j}: {32532, 32532}
X(53856) = X(i)-Ceva conjugate of X(j) for these {i, j}: {50992, 2}
X(53856) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {15655, 8}, {50992, 6327}

---

X(53857) = X(2)X(3)∩X(69)X(15471)

Barycentrics    (a^2+b^2-c^2)*(a^2-b^2+c^2)*(11*a^2-7*(b^2+c^2)) : :

See Ivan Pavlov, euclid 5807.

X(53857) lies on circumconics {A,B,C,X(2),X(50992)}, {A,B,C,X(3),X(15655)} and these lines: {2, 3}, {69, 15471}, {232, 39576}, {2207, 20481}, {2211, 8617}, {2501, 44010}, {2996, 7665}, {5090, 46930}, {5095, 11160}, {5466, 47217}, {5643, 8541}, {5656, 20417}, {7718, 46931}, {8550, 37643}, {8743, 11580}, {8796, 10185}, {9168, 41357}, {11363, 46932}, {12294, 33879}, {15258, 16080}, {17006, 43981}, {22235, 37776}, {22237, 37775}, {32234, 51215}, {32825, 37804}, {33884, 44084}, {35260, 47296}, {37688, 52710}, {37689, 40138}, {38940, 52467}

X(53857) = polar conjugate of X(32532)
X(53857) = X(i)-isoconjugate-of-X(j) for these {i, j}: {48, 32532}
X(53857) = X(i)-Dao conjugate of X(j) for these {i, j}: {1249, 32532}
X(53857) = X(i) cross conjugate of X(j) for these {i, j}: {15655, 50992}
X(53857) = barycentric product X(i)*X(j) for these (i, j): {4, 50992}, {264, 15655}
X(53857) = barycentric quotient X(i)/X(j) for these (i, j): {4, 32532}, {15655, 3}, {50992, 69}
X(53857) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 16042, 7486}, {2, 468, 4232}, {2, 6353, 7378}, {140, 7400, 3523}, {468, 5094, 6353}

---

X(53858) = X(3)X(6)∩X(4)X(8584)

Barycentrics    a^2*(7*a^4+11*b^4-14*b^2*c^2+11*c^4-18*a^2*(b^2+c^2)) : :

See Ivan Pavlov, euclid 5807.

X(53858) lies on circumconics {A,B,C,X(3),X(32532)}, {A,B,C,X(4),X(15655)} and these lines: {3, 6}, {4, 8584}, {5, 15534}, {23, 17809}, {69, 32897}, {140, 51185}, {154, 11422}, {193, 15022}, {394, 15019}, {524, 3090}, {542, 51173}, {546, 47353}, {547, 51188}, {597, 3525}, {599, 3628}, {632, 47352}, {895, 52518}, {1352, 12811}, {1498, 11216}, {1503, 50688}, {1656, 15533}, {1657, 33749}, {1992, 3091}, {1993, 16042}, {1994, 14002}, {2854, 9781}, {2930, 13861}, {3066, 11004}, {3146, 5032}, {3292, 9777}, {3303, 8540}, {3304, 19369}, {3518, 37827}, {3529, 20583}, {3544, 3629}, {3545, 41149}, {3564, 3857}, {3627, 20423}, {3751, 16189}, {4663, 7982}, {5055, 51187}, {5067, 22165}, {5070, 50993}, {5072, 5476}, {5079, 14848}, {5095, 7507}, {5198, 8541}, {5480, 50689}, {5643, 14924}, {5646, 15018}, {6090, 44107}, {6144, 12812}, {7486, 50992}, {7487, 15471}, {7592, 15826}, {8537, 10594}, {8546, 12088}, {8716, 35951}, {9716, 35259}, {9968, 39125}, {9971, 32284}, {10169, 40686}, {10601, 23061}, {11002, 12061}, {11179, 15704}, {11284, 15004}, {11403, 11405}, {11416, 33524}, {11541, 14912}, {11799, 47466}, {12007, 48905}, {12102, 36990}, {12103, 50979}, {12106, 51933}, {14537, 37839}, {14853, 32455}, {14869, 50981}, {15068, 52163}, {15074, 16982}, {15582, 19132}, {15699, 50989}, {15703, 51189}, {15709, 41153}, {16010, 35502}, {17813, 34117}, {17825, 34565}, {18583, 40341}, {19357, 37953}, {21358, 25555}, {33748, 48881}, {34380, 47355}, {34788, 50414}, {38144, 51196}, {39571, 47546}, {40107, 50962}, {41991, 51027}, {41992, 50978}, {46935, 50990}, {49136, 51024}, {50693, 50976}

X(53858) = reflection of X(i) in X(j) for these {i,j}: {51189, 15703}
X(53858) = Cundy-Parry-Psi of X(32532)
X(53858) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 11482, 22330}, {3, 22330, 6}, {6, 1351, 5085}, {6, 5102, 1350}, {6, 576, 11477}, {61, 62, 1384}, {371, 372, 15655}, {575, 576, 1351}, {576, 11477, 5102}, {576, 11482, 6}, {576, 15520, 575}, {576, 22330, 3}, {576, 5097, 11482}, {1351, 15520, 6}, {3592, 3594, 3053}, {5093, 11482, 576}, {5093, 5097, 6}, {6419, 6420, 30435}, {14848, 51175, 25565}, {22236, 22238, 5210}

---

X(53859) = X(4)X(15655)∩X(3)X(32532)

Barycentrics    (11*a^4+7*b^4-18*b^2*c^2+11*c^4-2*a^2*(9*b^2+7*c^2))*(11*a^4+11*b^4-18*b^2*c^2+7*c^4-2*a^2*(7*b^2+9*c^2)) : :

See Ivan Pavlov, euclid 5807.

X(53859) lies on the Kiepert Hyperbola, circumconics {A,B,C,X(3),X(15655)}, {A,B,C,X(20),X(52292)} and these lines: {3, 32532}, {4, 15655}, {20, 17503}, {83, 46935}, {140, 5485}, {459, 52292}, {598, 5056}, {671, 3523}, {1656, 18842}, {2996, 17006}, {3054, 43537}, {3091, 45103}, {3522, 41895}, {4232, 39284}, {5304, 10155}, {7000, 43563}, {7374, 43562}, {7608, 37689}, {11669, 37665}, {14484, 37637}, {18840, 46219}, {18843, 35018}, {33602, 37463}, {33603, 37464}, {33698, 49135}

X(53859) = Cundy-Parry-Phi of X(32532)

---

X(53860) = X(3)X(1199)∩X(4)X(34986)

Barycentrics    a^2*(4*a^8-14*a^6*(b^2+c^2)+(b^2-c^2)^2*(2*b^4-5*b^2*c^2+2*c^4)+3*a^4*(6*b^4+7*b^2*c^2+6*c^4)-2*a^2*(5*b^6-b^4*c^2-b^2*c^4+5*c^6)) : :

See Ivan Pavlov, euclid 5807.

X(53860) lies on these lines: {3, 1199}, {4, 34986}, {20, 32136}, {52, 3431}, {546, 9716}, {569, 41462}, {576, 19128}, {1493, 3146}, {1993, 35500}, {3090, 37645}, {3292, 3544}, {3525, 11433}, {3528, 13366}, {3529, 11422}, {5944, 16981}, {6102, 11270}, {6800, 12088}, {9545, 37440}, {10095, 14491}, {11004, 37472}, {11264, 31857}, {11423, 17538}, {11803, 18562}, {12086, 12161}, {14831, 23040}, {15692, 36153}, {16881, 38942}, {36749, 44802}

---

X(53861) = X(4)X(1000)∩X(65)X(225)

Barycentrics    a*(a+b-c)*(a-b+c)*(b+c)*(a^2+b^2-c^2)*(a^2-b^2-4*b*c-c^2)*(a^2-b^2+c^2) : :

See Ivan Pavlov, euclid 5807.

X(53861) lies on circumconics {A,B,C,X(4),X(18535)} and these lines: {4, 1000}, {12, 39579}, {33, 37080}, {34, 1900}, {65, 225}, {210, 1882}, {227, 42440}, {278, 1887}, {354, 1068}, {860, 3698}, {1319, 4185}, {1598, 11508}, {1785, 1871}, {1838, 1872}, {1859, 7952}, {1865, 21871}, {1880, 52555}, {3295, 18535}, {5919, 40950}, {14016, 18191}, {14018, 37566}, {17609, 23710}, {37117, 37605}, {37194, 37568}

X(53861) = Zosma transformation of X(631)
X(53861) = X(i)-isoconjugate-of-X(j) for these {i, j}: {283, 3296}, {284, 30679}
X(53861) = X(i)-Dao conjugate of X(j) for these {i, j}: {40590, 30679}
X(53861) = barycentric product X(i)*X(j) for these (i, j): {225, 3305}, {278, 3697}, {1426, 42032}, {1824, 52422}, {1826, 7190}, {1880, 42696}, {3295, 40149}, {41013, 52424}
X(53861) = barycentric quotient X(i)/X(j) for these (i, j): {65, 30679}, {1880, 3296}, {3295, 1812}, {3305, 332}, {3697, 345}, {7190, 17206}, {52424, 1444}
X(53861) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {225, 1824, 65}, {225, 1825, 1426}, {1426, 1824, 1825}, {1426, 1825, 65}

---

X(53862) = X(111)X(10979)∩X(74)X(17538)

Barycentrics    (a-b)*(a+b)*(a-c)*(a+c)*(3*a^4+3*(b^2-c^2)^2-2*a^2*(3*b^2+c^2))*(3*a^4+3*(b^2-c^2)^2-2*a^2*(b^2+3*c^2)) : :

See Ivan Pavlov, euclid 5807.

X(53862) lies on circumconic {A,B,C,X(74),X(98)} and these lines: {74, 17538}, {111, 10979}, {842, 46517}, {933, 48539}, {1300, 35477}, {3563, 8889}, {35446, 45138}

X(53862) = trilinear pole of line {6,3090}
X(53862) = Collings transformation of X(631)
X(53862) = X(i)-isoconjugate-of-X(j) for these {i, j}: {656, 3517}, {798, 32829}
X(53862) = X(i)-Dao conjugate of X(j) for these {i, j}: {31998, 32829}, {40596, 3517}
X(53862) = barycentric quotient X(i)/X(j) for these (i, j): {99, 32829}, {112, 3517}

---

X(53863) = X(6)X(22)∩X(2)X(576)

Barycentrics    a^2*(a^4+2*b^4-3*b^2*c^2+2*c^4-3*a^2*(b^2+c^2)) : :

See Ivan Pavlov, euclid 5807.

X(53863) lies on these lines: {2, 576}, {3, 13421}, {4, 10116}, {5, 1173}, {6, 22}, {23, 13366}, {25, 8537}, {49, 38848}, {51, 110}, {52, 7691}, {54, 143}, {61, 21461}, {62, 21462}, {111, 2056}, {184, 11002}, {193, 9813}, {195, 10095}, {216, 39243}, {265, 11805}, {275, 35360}, {323, 5943}, {382, 43585}, {389, 2071}, {394, 11451}, {399, 14483}, {427, 9140}, {467, 42873}, {511, 15246}, {524, 37990}, {542, 37349}, {550, 43600}, {568, 15033}, {575, 6636}, {578, 10298}, {588, 3102}, {589, 3103}, {648, 30506}, {895, 8584}, {1154, 15038}, {1199, 5446}, {1351, 2979}, {1369, 51396}, {1493, 9705}, {1570, 34945}, {1627, 13330}, {1976, 35426}, {1992, 6997}, {1993, 5020}, {3051, 39024}, {3124, 45843}, {3167, 10546}, {3410, 19130}, {3448, 11225}, {3518, 9706}, {3527, 11441}, {3567, 6644}, {3627, 43602}, {3845, 14094}, {3855, 15083}, {3917, 5888}, {3920, 8540}, {4274, 33774}, {5007, 37183}, {5032, 6995}, {5038, 38862}, {5102, 7998}, {5111, 11673}, {5133, 41724}, {5158, 26874}, {5182, 35929}, {5476, 37353}, {5480, 45968}, {5889, 9818}, {5890, 13445}, {5946, 43574}, {6102, 15062}, {6243, 43651}, {6676, 15360}, {7191, 19369}, {7378, 11470}, {7391, 20423}, {7396, 18911}, {7485, 11477}, {7488, 37505}, {7492, 22234}, {7494, 8538}, {7517, 11423}, {7527, 14831}, {7539, 14848}, {7592, 18534}, {7605, 15108}, {7747, 32464}, {7772, 37184}, {8041, 15514}, {8550, 34603}, {8718, 37949}, {9218, 15544}, {9306, 10545}, {9544, 34417}, {9781, 12161}, {9909, 11255}, {10203, 44056}, {10540, 13451}, {10574, 11432}, {10575, 43612}, {10627, 15047}, {10982, 12111}, {11179, 20062}, {11402, 20850}, {11424, 11440}, {11425, 38446}, {11433, 26913}, {11438, 35493}, {11442, 14853}, {11536, 52675}, {11801, 43580}, {12160, 15056}, {12212, 38873}, {12316, 14128}, {13338, 14060}, {13352, 15053}, {13353, 14449}, {13364, 50461}, {13391, 15037}, {13564, 16982}, {13567, 15059}, {13596, 15054}, {13630, 35452}, {14118, 16625}, {14153, 20977}, {14157, 15087}, {14389, 41588}, {14480, 34093}, {14561, 45794}, {15024, 16266}, {15043, 36747}, {15072, 44413}, {15516, 22352}, {15646, 16881}, {15800, 43575}, {15826, 47313}, {16949, 22486}, {16981, 39561}, {17825, 44299}, {18114, 47053}, {18191, 32911}, {18378, 32136}, {18583, 37636}, {20063, 33749}, {20115, 22101}, {20424, 43821}, {22233, 23039}, {22750, 43823}, {23293, 37644}, {23607, 42350}, {26910, 52424}, {32068, 51360}, {32142, 46084}, {32184, 43813}, {32467, 46518}, {34482, 44500}, {35265, 44106}, {37495, 43597}, {37517, 41462}, {37938, 43836}, {39099, 39998}, {39125, 41580}, {42990, 48794}, {42991, 48796}, {43573, 46450}, {43812, 45034}, {43838, 44829}

X(53863) = reflection of X(i) in X(j) for these {i,j}: {34545, 34565}
X(53863) = X(i)-Ceva conjugate of X(j) for these {i, j}: {39289, 2}
X(53863) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 2889}, {1173, 21289}, {39289, 6327}
X(53863) = barycentric product X(i)*X(j) for these (i, j): {249, 38394}
X(53863) = barycentric quotient X(i)/X(j) for these (i, j): {38394, 338}
X(53863) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 12834, 5643}, {2, 15004, 15019}, {2, 15019, 12834}, {6, 3060, 5012}, {51, 1994, 110}, {51, 34986, 13595}, {51, 5097, 1994}, {52, 13434, 7691}, {195, 10095, 43598}, {511, 34565, 34545}, {575, 21969, 6636}, {576, 15004, 2}, {576, 15019, 23061}, {1199, 5446, 52525}, {1351, 5422, 2979}, {1493, 13621, 9705}, {1993, 9777, 5640}, {1994, 13595, 34986}, {3060, 15080, 33586}, {3060, 5012, 15107}, {3567, 36749, 34148}, {5012, 15107, 6030}, {5093, 9777, 1993}, {12834, 23061, 2}, {13366, 21849, 23}, {13595, 34986, 110}, {15019, 23061, 5643}, {16982, 36153, 13564}

---

X(53864) = X(5)X(11810)∩X(141)X(575)

Barycentrics    (2*a^4+2*b^4-3*b^2*c^2+c^4-3*a^2*(b^2+c^2))*(2*a^4+b^4-3*b^2*c^2+2*c^4-3*a^2*(b^2+c^2)) : :

See Ivan Pavlov, euclid 5807.

X(53864) lies on circumconics {A,B,C,X(2),X(66)}, {A,B,C,X(3),X(10018)} and these lines: {5, 11810}, {141, 575}, {427, 3054}, {2963, 38394}, {3613, 7749}, {7496, 17006}, {8024, 37688}

X(53864) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1101, 38394}
X(53864) = X(i)-Dao conjugate of X(j) for these {i, j}: {523, 38394}
X(53864) = barycentric quotient X(i)/X(j) for these (i, j): {115, 38394}

---

X(53865) = X(6)X(98)∩X(263)X(2698)

Barycentrics    (a^4 + b^4 - a^2*c^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)*(a^4 - a^2*b^2 - b^2*c^2 + c^4)*(a^6 - 3*a^4*b^2 + 2*a^2*b^4 - 3*a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + 2*a^2*c^4 + b^2*c^4) : :

X(53865) lies on the cubics K055 and K1325 and on these lines: {6, 98}, {263, 2698}, {287, 10796}, {511, 39681}, {576, 14382}, {5039, 47388}, {5093, 39941}, {9154, 51980}, {12212, 47741}, {19136, 32716}, {39141, 40803}

X(53865) = X(39100)-Dao conjugate of X(51373)
X(53865) = barycentric quotient X(i)/X(j) for these {i,j}: {6037, 53199}, {39099, 51373}
X(53865) = {X(263),X(5967)}-harmonic conjugate of X(6037)

---

X(53866) = X(4)X(685)∩X(30)X(51228)

Barycentrics    (a^4 + b^4 - a^2*c^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^4 - a^2*b^2 - b^2*c^2 + c^4)*(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(53866) lies on the cubic K1325 and these lines: {4, 685}, {30, 51228}, {98, 523}, {477, 53691}, {542, 5641}, {1503, 34174}, {5649, 52772}, {5999, 46787}, {6034, 34175}, {7422, 50942}, {9154, 47388}, {9214, 35912}, {14355, 14560}, {14382, 14387}, {23350, 53267}, {35906, 52472}, {41204, 52492}

X(53866) = X(i)-isoconjugate of X(j) for these (i,j): {1755, 51227}, {1959, 48451}, {2247, 35910}
X(53866) = X(36899)-Dao conjugate of X(51227)
X(53866) = trilinear pole of line {1637, 35906}
X(53866) = barycentric product X(i)*X(j) for these {i,j}: {98, 51228}, {290, 48453}, {5641, 35906}, {41079, 53691}, {43665, 51263}
X(53866) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 51227}, {842, 35910}, {1976, 48451}, {2420, 42743}, {2715, 51262}, {6531, 17986}, {14998, 32112},* {35906, 542}, {48453, 511}, {51228, 325}, {51263, 2421}, {53691, 44769}

---

X(53867) = REFLECTION OF X(74) IN X(3)X(9161)

Barycentrics    a^2*(a^14*b^6 - 5*a^12*b^8 + 10*a^10*b^10 - 10*a^8*b^12 + 5*a^6*b^14 - a^4*b^16 + a^12*b^6*c^2 - 3*a^10*b^8*c^2 + 3*a^8*b^10*c^2 - a^6*b^12*c^2 + a^16*c^4 - a^12*b^4*c^4 + 2*a^10*b^6*c^4 - 3*a^6*b^10*c^4 + 2*a^4*b^12*c^4 - b^16*c^4 - 6*a^14*c^6 + a^10*b^4*c^6 - 3*a^8*b^6*c^6 + 6*a^6*b^8*c^6 - 3*a^4*b^10*c^6 - a^2*b^12*c^6 + 5*b^14*c^6 + 15*a^12*c^8 - 3*a^6*b^6*c^8 + 3*a^2*b^10*c^8 - 10*b^12*c^8 - 20*a^10*c^10 + a^6*b^4*c^10 + 2*a^4*b^6*c^10 - 3*a^2*b^8*c^10 + 10*b^10*c^10 + 15*a^8*c^12 - a^4*b^4*c^12 + a^2*b^6*c^12 - 5*b^8*c^12 - 6*a^6*c^14 + b^6*c^14 + a^4*c^16)*(a^16*b^4 - 6*a^14*b^6 + 15*a^12*b^8 - 20*a^10*b^10 + 15*a^8*b^12 - 6*a^6*b^14 + a^4*b^16 - a^12*b^4*c^4 + a^10*b^6*c^4 + a^6*b^10*c^4 - a^4*b^12*c^4 + a^14*c^6 + a^12*b^2*c^6 + 2*a^10*b^4*c^6 - 3*a^8*b^6*c^6 - 3*a^6*b^8*c^6 + 2*a^4*b^10*c^6 + a^2*b^12*c^6 + b^14*c^6 - 5*a^12*c^8 - 3*a^10*b^2*c^8 + 6*a^6*b^6*c^8 - 3*a^2*b^10*c^8 - 5*b^12*c^8 + 10*a^10*c^10 + 3*a^8*b^2*c^10 - 3*a^6*b^4*c^10 - 3*a^4*b^6*c^10 + 3*a^2*b^8*c^10 + 10*b^10*c^10 - 10*a^8*c^12 - a^6*b^2*c^12 + 2*a^4*b^4*c^12 - a^2*b^6*c^12 - 10*b^8*c^12 + 5*a^6*c^14 + 5*b^6*c^14 - a^4*c^16 - b^4*c^16) : :

Centers X(53867), X(53868), X(53869), and seven others lie equally spaced on the circumcircle. Specifically, the points are ordered as follows, starting at X(53867): X(53867), X(9160), X(74), X(99), X(43654), X(53868), X(9161), X(110), X(989), X(53603), X(53867),... The central angle (e.g., X(74)-to-X(3)-to-X(99)) is π/5 = 72 °.

X(53867) lies on the circumcircle and these lines: {526, 43654}, {5663, 53603}

---

X(53868) = REFLECTION OF X(98) IN X(3)X(43654)

Barycentrics    (a^14*b^6 - 4*a^12*b^8 + 6*a^10*b^10 - 4*a^8*b^12 + a^6*b^14 + a^16*c^4 - a^14*b^2*c^4 + a^12*b^4*c^4 + 3*a^10*b^6*c^4 - 6*a^8*b^8*c^4 + 3*a^6*b^10*c^4 + a^4*b^12*c^4 - a^2*b^14*c^4 + b^16*c^4 - 5*a^14*c^6 + 3*a^12*b^2*c^6 - 6*a^10*b^4*c^6 + 3*a^8*b^6*c^6 + 3*a^6*b^8*c^6 - 6*a^4*b^10*c^6 + 3*a^2*b^12*c^6 - 5*b^14*c^6 + 10*a^12*c^8 - 3*a^10*b^2*c^8 + 6*a^8*b^4*c^8 - 6*a^6*b^6*c^8 + 6*a^4*b^8*c^8 - 3*a^2*b^10*c^8 + 10*b^12*c^8 - 10*a^10*c^10 + a^8*b^2*c^10 - a^6*b^4*c^10 - a^4*b^6*c^10 + a^2*b^8*c^10 - 10*b^10*c^10 + 5*a^8*c^12 + 5*b^8*c^12 - a^6*c^14 - b^6*c^14)*(a^16*b^4 - 5*a^14*b^6 + 10*a^12*b^8 - 10*a^10*b^10 + 5*a^8*b^12 - a^6*b^14 - a^14*b^4*c^2 + 3*a^12*b^6*c^2 - 3*a^10*b^8*c^2 + a^8*b^10*c^2 + a^12*b^4*c^4 - 6*a^10*b^6*c^4 + 6*a^8*b^8*c^4 - a^6*b^10*c^4 + a^14*c^6 + 3*a^10*b^4*c^6 + 3*a^8*b^6*c^6 - 6*a^6*b^8*c^6 - a^4*b^10*c^6 - b^14*c^6 - 4*a^12*c^8 - 6*a^8*b^4*c^8 + 3*a^6*b^6*c^8 + 6*a^4*b^8*c^8 + a^2*b^10*c^8 + 5*b^12*c^8 + 6*a^10*c^10 + 3*a^6*b^4*c^10 - 6*a^4*b^6*c^10 - 3*a^2*b^8*c^10 - 10*b^10*c^10 - 4*a^8*c^12 + a^4*b^4*c^12 + 3*a^2*b^6*c^12 + 10*b^8*c^12 + a^6*c^14 - a^2*b^4*c^14 - 5*b^6*c^14 + b^4*c^16) : :

X(53868) lies on the circumcircle and these lines: {804, 9161}, {2782, 9160}

---

X(53869) = REFLECTION OF X(99) IN X(3)X(43654)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^12*b^4 - 4*a^10*b^6 + 6*a^8*b^8 - 4*a^6*b^10 + a^4*b^12 + a^10*b^4*c^2 - 3*a^8*b^6*c^2 + 4*a^6*b^8*c^2 - 3*a^4*b^10*c^2 + a^2*b^12*c^2 + 2*a^8*b^4*c^4 - 5*a^6*b^6*c^4 + 6*a^4*b^8*c^4 - 3*a^2*b^10*c^4 + b^12*c^4 - a^10*c^6 - a^8*b^2*c^6 + 3*a^6*b^4*c^6 - 5*a^4*b^6*c^6 + 4*a^2*b^8*c^6 - 4*b^10*c^6 + 2*a^8*c^8 - a^6*b^2*c^8 + 2*a^4*b^4*c^8 - 3*a^2*b^6*c^8 + 6*b^8*c^8 - a^6*c^10 + a^2*b^4*c^10 - 4*b^6*c^10 + b^4*c^12)*(-(a^10*b^6) + 2*a^8*b^8 - a^6*b^10 - a^8*b^6*c^2 - a^6*b^8*c^2 + a^12*c^4 + a^10*b^2*c^4 + 2*a^8*b^4*c^4 + 3*a^6*b^6*c^4 + 2*a^4*b^8*c^4 + a^2*b^10*c^4 + b^12*c^4 - 4*a^10*c^6 - 3*a^8*b^2*c^6 - 5*a^6*b^4*c^6 - 5*a^4*b^6*c^6 - 3*a^2*b^8*c^6 - 4*b^10*c^6 + 6*a^8*c^8 + 4*a^6*b^2*c^8 + 6*a^4*b^4*c^8 + 4*a^2*b^6*c^8 + 6*b^8*c^8 - 4*a^6*c^10 - 3*a^4*b^2*c^10 - 3*a^2*b^4*c^10 - 4*b^6*c^10 + a^4*c^12 + a^2*b^2*c^12 + b^4*c^12) : :

X(53869) lies on the circumcircle and these lines: {3, 53868}, {542, 53867}, {804, 9160}, {2782, 9161}

X(53869) = reflection of X(53868) in X(3)

---

X(53870) = REFLECTION OF X(74) IN X(3)X(102)

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 - 2*a^9*c + 2*a^8*b*c + 4*a^7*b^2*c - 4*a^6*b^3*c - 4*a^3*b^6*c + 4*a^2*b^7*c + 2*a*b^8*c - 2*b^9*c + a^8*c^2 - 6*a^7*b*c^2 + 6*a^6*b^2*c^2 + 6*a^5*b^3*c^2 - 14*a^4*b^4*c^2 + 6*a^3*b^5*c^2 + 6*a^2*b^6*c^2 - 6*a*b^7*c^2 + b^8*c^2 + 4*a^7*c^3 + 2*a^6*b*c^3 - 14*a^5*b^2*c^3 + 8*a^4*b^3*c^3 + 8*a^3*b^4*c^3 - 14*a^2*b^5*c^3 + 2*a*b^6*c^3 + 4*b^7*c^3 - 7*a^6*c^4 + 8*a^5*b*c^4 + 7*a^4*b^2*c^4 - 16*a^3*b^3*c^4 + 7*a^2*b^4*c^4 + 8*a*b^5*c^4 - 7*b^6*c^4 - 8*a^4*b*c^5 + 8*a^3*b^2*c^5 + 8*a^2*b^3*c^5 - 8*a*b^4*c^5 + 5*a^4*c^6 + 2*a^3*b*c^6 - 12*a^2*b^2*c^6 + 2*a*b^3*c^6 + 5*b^4*c^6 - 4*a^3*c^7 + 2*a^2*b*c^7 + 2*a*b^2*c^7 - 4*b^3*c^7 + 2*a^2*c^8 - 4*a*b*c^8 + 2*b^2*c^8 + 2*a*c^9 + 2*b*c^9 - 2*c^10)*(a^10 - 2*a^9*b + a^8*b^2 + 4*a^7*b^3 - 7*a^6*b^4 + 5*a^4*b^6 - 4*a^3*b^7 + 2*a^2*b^8 + 2*a*b^9 - 2*b^10 + 2*a^8*b*c - 6*a^7*b^2*c + 2*a^6*b^3*c + 8*a^5*b^4*c - 8*a^4*b^5*c + 2*a^3*b^6*c + 2*a^2*b^7*c - 4*a*b^8*c + 2*b^9*c - 3*a^8*c^2 + 4*a^7*b*c^2 + 6*a^6*b^2*c^2 - 14*a^5*b^3*c^2 + 7*a^4*b^4*c^2 + 8*a^3*b^5*c^2 - 12*a^2*b^6*c^2 + 2*a*b^7*c^2 + 2*b^8*c^2 - 4*a^6*b*c^3 + 6*a^5*b^2*c^3 + 8*a^4*b^3*c^3 - 16*a^3*b^4*c^3 + 8*a^2*b^5*c^3 + 2*a*b^6*c^3 - 4*b^7*c^3 + 2*a^6*c^4 - 14*a^4*b^2*c^4 + 8*a^3*b^3*c^4 + 7*a^2*b^4*c^4 - 8*a*b^5*c^4 + 5*b^6*c^4 + 6*a^3*b^2*c^5 - 14*a^2*b^3*c^5 + 8*a*b^4*c^5 + 2*a^4*c^6 - 4*a^3*b*c^6 + 6*a^2*b^2*c^6 + 2*a*b^3*c^6 - 7*b^4*c^6 + 4*a^2*b*c^7 - 6*a*b^2*c^7 + 4*b^3*c^7 - 3*a^2*c^8 + 2*a*b*c^8 + b^2*c^8 - 2*b*c^9 + c^10) : :

X(53870) lies on the circumcircle and these lines: {3, 53612}, {20, 53611}, {30, 1309}, {74, 8677}, {107, 3109}, {110, 2818}, {186, 36067}, {523, 2734}, {859, 1304}, {901, 2071}, {2766, 6905}, {11700, 26700}, {22239, 37168}

X(53870) = reflection of X(53612) in X(3)
X(53870) = reflection of X(2734) in the Euler line

---

X(53871) = REFLECTION OF X(74) IN X(3)X(112)

Barycentrics    a^2*(a^14 - a^12*b^2 - 3*a^10*b^4 + 3*a^8*b^6 + 3*a^6*b^8 - 3*a^4*b^10 - a^2*b^12 + b^14 - 2*a^12*c^2 + a^10*b^2*c^2 + 6*a^8*b^4*c^2 - 10*a^6*b^6*c^2 + 6*a^4*b^8*c^2 + a^2*b^10*c^2 - 2*b^12*c^2 + 5*a^10*c^4 - 5*a^6*b^4*c^4 - 5*a^4*b^6*c^4 + 5*b^10*c^4 - 10*a^8*c^6 + 5*a^6*b^2*c^6 + 10*a^4*b^4*c^6 + 5*a^2*b^6*c^6 - 10*b^8*c^6 + 7*a^6*c^8 - 6*a^4*b^2*c^8 - 6*a^2*b^4*c^8 + 7*b^6*c^8 - 2*a^4*c^10 - 2*a^2*b^2*c^10 - 2*b^4*c^10 + 3*a^2*c^12 + 3*b^2*c^12 - 2*c^14)*(a^14 - 2*a^12*b^2 + 5*a^10*b^4 - 10*a^8*b^6 + 7*a^6*b^8 - 2*a^4*b^10 + 3*a^2*b^12 - 2*b^14 - a^12*c^2 + a^10*b^2*c^2 + 5*a^6*b^6*c^2 - 6*a^4*b^8*c^2 - 2*a^2*b^10*c^2 + 3*b^12*c^2 - 3*a^10*c^4 + 6*a^8*b^2*c^4 - 5*a^6*b^4*c^4 + 10*a^4*b^6*c^4 - 6*a^2*b^8*c^4 - 2*b^10*c^4 + 3*a^8*c^6 - 10*a^6*b^2*c^6 - 5*a^4*b^4*c^6 + 5*a^2*b^6*c^6 + 7*b^8*c^6 + 3*a^6*c^8 + 6*a^4*b^2*c^8 - 10*b^6*c^8 - 3*a^4*c^10 + a^2*b^2*c^10 + 5*b^4*c^10 - a^2*c^12 - 2*b^2*c^12 + c^14) : :

X(53871) lies on the circumcircle and these lines: {30, 2867}, {74, 2881}, {99, 12253}, {110, 37921}, {186, 32687}, {441, 476}, {1297, 8552}, {2781, 53187}, {9517, 53188}, {9530, 39447}, {23969, 40080}

---

X(53872) = REFLECTION OF X(691) IN X(3)X(74)

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^8 + a^4*b^4 - 6*a^2*b^6 + 4*b^8 - 3*a^6*c^2 - 2*a^4*b^2*c^2 + 10*a^2*b^4*c^2 - 6*b^6*c^2 + 4*a^4*c^4 - 2*a^2*b^2*c^4 + b^4*c^4 - 3*a^2*c^6 + c^8)*(a^8 - 3*a^6*b^2 + 4*a^4*b^4 - 3*a^2*b^6 + b^8 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + a^4*c^4 + 10*a^2*b^2*c^4 + b^4*c^4 - 6*a^2*c^6 - 6*b^2*c^6 + 4*c^8) : :

X(53872) lies on the circumcircle and these lines: {98, 20126}, {99, 45808}, {110, 44814}, {111, 2088}, {476, 690}, {477, 542}, {526, 691}, {842, 5663}, {1302, 15342}, {2437, 23969}, {10411, 45773}, {11006, 53605}, {16255, 39424}, {16256, 39425}

X(53872) = X(661)-isoconjugate of X(45331)
X(53872) = X(36830)-Dao conjugate of X(45331)
X(53872) = trilinear pole of line {6, 33927}
X(53872) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 45331}, {2378, 9201}, {2379, 9200}

---

X(53873) = REFLECTION OF X(98) IN X(3)X(100)

Barycentrics    a^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^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^3*b*c^2 + 2*a*b^3*c^2 - b^4*c^2 - 2*a^3*c^3 - 2*b^3*c^3 + 2*a^2*c^4 - 4*a*b*c^4 + 2*b^2*c^4 + 2*a*c^5 + 2*b*c^5 - 2*c^6)*(a^6 - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + 2*a*b^5 - 2*b^6 - 2*a^5*c + 2*a^4*b*c + 2*a^3*b^2*c - 4*a*b^4*c + 2*b^5*c - a^4*c^2 - 2*a^3*b*c^2 + 2*b^4*c^2 + 4*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 + 2*a*b*c^4 - b^2*c^4 - 2*a*c^5 + c^6) : :

X(53873) lies on the circumcircle and these lines: {3, 53606}, {98, 900}, {99, 952}, {106, 53521}, {112, 5170}, {511, 901}, {512, 953}, {2715, 3285}, {6011, 46684}, {18860, 28293}, {19628, 50343}

X(53873) = reflection of X(53606) in X(3)
X(53873) = isogonal conjugate of X(53792)
X(53873) = reflection of X(953) in the Brocard axis
X(53873) = barycentric quotient X(6)/X(53792)

---

X(53874) = REFLECTION OF X(100) IN X(3)X(98)

Barycentrics    (a - b)*(a - c)*(-(a^3*b^3) + a^5*c + a^4*b*c + a*b^4*c + b^5*c - a^4*c^2 - b^4*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^2*c^4)*(a^5*b - a^4*b^2 - a^3*b^3 + a^2*b^4 + a^4*b*c - a^2*b^3*c + a*b^4*c - a*b^3*c^2 + b^4*c^2 - a^3*c^3 - b^3*c^3 + a*b*c^4 - b^2*c^4 + b*c^5) : :

X(53874) lies on the circumcircle and these lines: {99, 14296}, {100, 804}, {104, 2782}, {105, 5985}, {110, 4164}, {513, 805}, {517, 2698}, {741, 53541}, {767, 5152}, {813, 39185}, {2699, 53792}, {2787, 53606}, {2795, 53180}, {17212, 36066}

X(53874) = reflection of X(805) in the OI line
X(53874) = trilinear pole of line {6, 24345}

---

X(53875) = REFLECTION OF X(112) IN X(3)X(98)

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(a^8*b^2 - a^6*b^4 - a^4*b^6 + a^2*b^8 - a^2*b^6*c^2 + b^8*c^2 - a^6*c^4 + 3*a^4*b^2*c^4 - b^6*c^4 - a^4*c^6 - b^4*c^6 + b^2*c^8)*(-(a^6*b^4) - a^4*b^6 + a^8*c^2 + 3*a^4*b^4*c^2 + b^8*c^2 - a^6*c^4 - b^6*c^4 - a^4*c^6 - a^2*b^2*c^6 - b^4*c^6 + a^2*c^8 + b^2*c^8) : :

X(53875) lies on the circumcircle and these lines: {110, 24284}, {112, 804}, {525, 805}, {733, 38947}, {827, 40866}, {1297, 2782}, {1503, 2698}, {2373, 5986}, {9830, 13238}, {20021, 53700}, {41676, 53699}

---

X(53876) = REFLECTION OF X(691) IN X(3)X(98)

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(a^8*b^2 - a^6*b^4 - a^4*b^6 + a^2*b^8 - 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 + 4*a^4*b^2*c^4 + a^2*b^4*c^4 - b^6*c^4 - a^4*c^6 - a^2*b^2*c^6 - b^4*c^6 + b^2*c^8)*(-(a^6*b^4) - a^4*b^6 + a^8*c^2 - a^6*b^2*c^2 + 4*a^4*b^4*c^2 - a^2*b^6*c^2 + b^8*c^2 - a^6*c^4 + a^4*b^2*c^4 + a^2*b^4*c^4 - b^6*c^4 - a^4*c^6 - 2*a^2*b^2*c^6 - b^4*c^6 + a^2*c^8 + b^2*c^8) : :

X(53876) lies on the circumcircle and these lines: {99, 44822}, {110, 11183}, {542, 2698}, {690, 805}, {691, 804}, {842, 2782}, {2770, 5987}, {2868, 5152}, {15342, 26714}, {17941, 45773}, {43357, 53735}

X(53876) = trilinear pole of line {6, 53379}

---

X(53877) = REFLECTION OF X(102) IN X(3)X(100)

Barycentrics    (a^8 - 3*a^7*b + 3*a^6*b^2 + 3*a^5*b^3 - 8*a^4*b^4 + 3*a^3*b^5 + 3*a^2*b^6 - 3*a*b^7 + b^8 + 3*a^6*b*c - 10*a^5*b^2*c + 7*a^4*b^3*c + 7*a^3*b^4*c - 10*a^2*b^5*c + 3*a*b^6*c - 3*a^6*c^2 + 6*a^5*b*c^2 + 4*a^4*b^2*c^2 - 14*a^3*b^3*c^2 + 4*a^2*b^4*c^2 + 6*a*b^5*c^2 - 3*b^6*c^2 - 6*a^4*b*c^3 + 7*a^3*b^2*c^3 + 7*a^2*b^3*c^3 - 6*a*b^4*c^3 + 3*a^4*c^4 - 3*a^3*b*c^4 - 6*a^2*b^2*c^4 - 3*a*b^3*c^4 + 3*b^4*c^4 + 3*a^2*b*c^5 + 3*a*b^2*c^5 - a^2*c^6 - b^2*c^6)*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 - 3*a^7*c + 3*a^6*b*c + 6*a^5*b^2*c - 6*a^4*b^3*c - 3*a^3*b^4*c + 3*a^2*b^5*c + 3*a^6*c^2 - 10*a^5*b*c^2 + 4*a^4*b^2*c^2 + 7*a^3*b^3*c^2 - 6*a^2*b^4*c^2 + 3*a*b^5*c^2 - b^6*c^2 + 3*a^5*c^3 + 7*a^4*b*c^3 - 14*a^3*b^2*c^3 + 7*a^2*b^3*c^3 - 3*a*b^4*c^3 - 8*a^4*c^4 + 7*a^3*b*c^4 + 4*a^2*b^2*c^4 - 6*a*b^3*c^4 + 3*b^4*c^4 + 3*a^3*c^5 - 10*a^2*b*c^5 + 6*a*b^2*c^5 + 3*a^2*c^6 + 3*a*b*c^6 - 3*b^2*c^6 - 3*a*c^7 + c^8) : :

X(53877) lies on the circumcircle and these lines: {3, 53610}, {102, 900}, {106, 53522}, {108, 10090}, {109, 952}, {515, 901}, {522, 953}, {1293, 24466}, {1877, 36067}, {18861, 32704}, {26711, 45767}, {35011, 51565}

X(53877) = reflection of X(53610) in X(3)

---

X(53878) = REFLECTION OF X(103) IN X(3)X(100)

Barycentrics    (a^7 - 2*a^6*b + a^5*b^2 + a^2*b^5 - 2*a*b^6 + b^7 - a^6*c + 4*a^5*b*c - 3*a^4*b^2*c - 3*a^2*b^4*c + 4*a*b^5*c - b^6*c - 2*a^5*c^2 + 3*a^3*b^2*c^2 + 3*a^2*b^3*c^2 - 2*b^5*c^2 + 2*a^4*c^3 - 4*a^3*b*c^3 - 2*a^2*b^2*c^3 - 4*a*b^3*c^3 + 2*b^4*c^3 + a^3*c^4 + 2*a^2*b*c^4 + 2*a*b^2*c^4 + b^3*c^4 - a^2*c^5 - b^2*c^5)*(a^7 - a^6*b - 2*a^5*b^2 + 2*a^4*b^3 + a^3*b^4 - a^2*b^5 - 2*a^6*c + 4*a^5*b*c - 4*a^3*b^3*c + 2*a^2*b^4*c + a^5*c^2 - 3*a^4*b*c^2 + 3*a^3*b^2*c^2 - 2*a^2*b^3*c^2 + 2*a*b^4*c^2 - b^5*c^2 + 3*a^2*b^2*c^3 - 4*a*b^3*c^3 + b^4*c^3 - 3*a^2*b*c^4 + 2*b^3*c^4 + a^2*c^5 + 4*a*b*c^5 - 2*b^2*c^5 - 2*a*c^6 - b*c^6 + c^7) : :

X(53878) lies on the circumcircle and these lines: {3, 39444}, {11, 109}, {100, 24026}, {101, 952}, {103, 900}, {104, 52726}, {106, 676}, {514, 953}, {516, 901}, {934, 1111}, {2222, 14204}, {2801, 53607}, {3319, 5532}, {8756, 40116}, {15626, 53610}, {15634, 24016}, {26712, 45767}, {34234, 35011}, {43079, 47766}

X(53878) = reflection of X(39444) in X(3)
X(53878) = trilinear pole of line {6, 42462}

---

X(53879) = REFLECTION OF X(104) IN X(3)X(101)

Barycentrics    a^2*(a^5*b^3 - 2*a^4*b^4 + 2*a^2*b^6 - a*b^7 + a^7*c - a^6*b*c - 3*a^4*b^3*c + 5*a^3*b^4*c - 3*a^2*b^5*c + 2*a*b^6*c - b^7*c - 2*a^6*c^2 + 3*a^5*b*c^2 + 2*a^4*b^2*c^2 + 2*a^3*b^3*c^2 - 4*a^2*b^4*c^2 - 3*a*b^5*c^2 + 2*b^6*c^2 - a^5*c^3 - 2*a^4*b*c^3 - 4*a^3*b^2*c^3 + 2*a^2*b^3*c^3 + 5*a*b^4*c^3 + 4*a^4*c^4 - 2*a^3*b*c^4 + 2*a^2*b^2*c^4 - 3*a*b^3*c^4 - 2*b^4*c^4 - a^3*c^5 + 3*a^2*b*c^5 + b^3*c^5 - 2*a^2*c^6 - a*b*c^6 + a*c^7)*(a^7*b - 2*a^6*b^2 - a^5*b^3 + 4*a^4*b^4 - a^3*b^5 - 2*a^2*b^6 + a*b^7 - a^6*b*c + 3*a^5*b^2*c - 2*a^4*b^3*c - 2*a^3*b^4*c + 3*a^2*b^5*c - a*b^6*c + 2*a^4*b^2*c^2 - 4*a^3*b^3*c^2 + 2*a^2*b^4*c^2 + a^5*c^3 - 3*a^4*b*c^3 + 2*a^3*b^2*c^3 + 2*a^2*b^3*c^3 - 3*a*b^4*c^3 + b^5*c^3 - 2*a^4*c^4 + 5*a^3*b*c^4 - 4*a^2*b^2*c^4 + 5*a*b^3*c^4 - 2*b^4*c^4 - 3*a^2*b*c^5 - 3*a*b^2*c^5 + 2*a^2*c^6 + 2*a*b*c^6 + 2*b^2*c^6 - a*c^7 - b*c^7) : :

X(53879) lies on the circumcircle and these lines: {3, 53607}, {100, 2808}, {104, 926}, {105, 53549}, {108, 5091}, {513, 2724}, {517, 927}, {813, 2077}, {934, 53548}, {2222, 9441}, {2223, 2720}, {2801, 39444}

X(53879) = reflection of X(53607) in X(3)
X(53879) = reflection of X(2724) in the OI line

---

X(53880) = REFLECTION OF X(110) IN X(3)X(101)

Barycentrics    a^2*(a - b)*(a - c)*(a^5 - a^3*b^2 - 2*a*b^4 + 2*b^5 - a^4*c + a^2*b^2*c + 2*a*b^3*c - 2*b^4*c + a*b^2*c^2 - b^2*c^3 - a*c^4 + c^5)*(a^5 - a^4*b - a*b^4 + b^5 - a^3*c^2 + a^2*b*c^2 + a*b^2*c^2 - b^3*c^2 + 2*a*b*c^3 - 2*a*c^4 - 2*b*c^4 + 2*c^5) : :

X(53880) lies on the circumcircle and these lines: {23, 43079}, {30, 2724}, {74, 2808}, {99, 50333}, {101, 44827}, {105, 4516}, {110, 926}, {250, 36071}, {523, 927}, {662, 26702}, {919, 3709}, {934, 53551}
/p>

X(53880) = reflection of X(927) in the Euler line
X(53880) = X(656)-isoconjugate of X(14119)
X(53880) = X(40596)-Dao conjugate of X(14119)
X(53880) = trilinear pole of line {6, 30437}
X(53880) = barycentric quotient X(112)/X(14119)

---

X(53881) = REFLECTION OF X(476) IN X(3)X(107)

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(a^8 - a^7*b + a^6*b^2 + a^5*b^3 - 4*a^4*b^4 + a^3*b^5 + a^2*b^6 - a*b^7 + b^8 - 2*a^6*c^2 + 4*a^4*b^2*c^2 - 4*a^3*b^3*c^2 + 4*a^2*b^4*c^2 - 2*b^6*c^2 + 3*a^3*b*c^4 - 7*a^2*b^2*c^4 + 3*a*b^3*c^4 + 2*a^2*c^6 - 2*a*b*c^6 + 2*b^2*c^6 - c^8)*(a^8 + a^7*b + a^6*b^2 - a^5*b^3 - 4*a^4*b^4 - a^3*b^5 + a^2*b^6 + a*b^7 + b^8 - 2*a^6*c^2 + 4*a^4*b^2*c^2 + 4*a^3*b^3*c^2 + 4*a^2*b^4*c^2 - 2*b^6*c^2 - 3*a^3*b*c^4 - 7*a^2*b^2*c^4 - 3*a*b^3*c^4 + 2*a^2*c^6 + 2*a*b*c^6 + 2*b^2*c^6 - c^8)*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - a^7*c + 3*a^3*b^4*c - 2*a*b^6*c + a^6*c^2 + 4*a^4*b^2*c^2 - 7*a^2*b^4*c^2 + 2*b^6*c^2 + a^5*c^3 - 4*a^3*b^2*c^3 + 3*a*b^4*c^3 - 4*a^4*c^4 + 4*a^2*b^2*c^4 + a^3*c^5 + a^2*c^6 - 2*b^2*c^6 - a*c^7 + c^8)*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 + a^7*c - 3*a^3*b^4*c + 2*a*b^6*c + a^6*c^2 + 4*a^4*b^2*c^2 - 7*a^2*b^4*c^2 + 2*b^6*c^2 - a^5*c^3 + 4*a^3*b^2*c^3 - 3*a*b^4*c^3 - 4*a^4*c^4 + 4*a^2*b^2*c^4 - a^3*c^5 + a^2*c^6 - 2*b^2*c^6 + a*c^7 + c^8) : :

X(53881) lies on the circumcircle and these lines: {74, 1650}, {112, 14401}, {476, 6086}, {477, 53803}, {526, 6080}, {933, 53757}, {1294, 13289}, {1304, 9033}, {2693, 2777}, {5663, 44874}, {13494, 15329}

X(53881) = Collings transform of X(i) for these i: {2777, 9033}
X(53881) = cevapoint of X(2777) and X(9033)
X(53881) = trilinear pole of line {6, 39008}

---

X(53882) = REFLECTION OF X(111) IN X(3)X(691)

Barycentrics    a^2*(a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(a^10 - 5*a^8*b^2 + 4*a^6*b^4 + 4*a^4*b^6 - 5*a^2*b^8 + b^10 + a^8*c^2 + a^6*b^2*c^2 - 3*a^4*b^4*c^2 + a^2*b^6*c^2 + b^8*c^2 + a^6*c^4 - 3*a^4*b^2*c^4 - 3*a^2*b^4*c^4 + b^6*c^4 - 2*a^4*c^6 + 10*a^2*b^2*c^6 - 2*b^4*c^6 - 2*a^2*c^8 - 2*b^2*c^8 + c^10)*(a^10 + a^8*b^2 + a^6*b^4 - 2*a^4*b^6 - 2*a^2*b^8 + b^10 - 5*a^8*c^2 + a^6*b^2*c^2 - 3*a^4*b^4*c^2 + 10*a^2*b^6*c^2 - 2*b^8*c^2 + 4*a^6*c^4 - 3*a^4*b^2*c^4 - 3*a^2*b^4*c^4 - 2*b^6*c^4 + 4*a^4*c^6 + a^2*b^2*c^6 + b^4*c^6 - 5*a^2*c^8 + b^2*c^8 + c^10) : :

X(53882) lies on the circumcircle and these lines: {23, 53690}, {110, 53770}, {111, 20403}, {476, 34320}, {524, 20404}, {691, 9177}, {1296, 53793}, {1499, 53605}, {1649, 2770}, {2715, 52198}, {15899, 53687}, {17964, 39413}

X(53882) = reflection of X(111) in X(3)X(691)

---

X(53883) = REFLECTION OF X(691) IN X(3)X(112)

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^12 - a^10*b^2 - a^8*b^4 + 2*a^6*b^6 - a^4*b^8 - a^2*b^10 + b^12 - 2*a^10*c^2 + 2*a^8*b^2*c^2 + 2*a^2*b^8*c^2 - 2*b^10*c^2 + 2*a^8*c^4 - 3*a^6*b^2*c^4 - 2*a^4*b^4*c^4 - 3*a^2*b^6*c^4 + 2*b^8*c^4 + 6*a^4*b^2*c^6 + 6*a^2*b^4*c^6 - 3*a^4*c^8 - 6*a^2*b^2*c^8 - 3*b^4*c^8 + 2*a^2*c^10 + 2*b^2*c^10)*(a^12 - 2*a^10*b^2 + 2*a^8*b^4 - 3*a^4*b^8 + 2*a^2*b^10 - a^10*c^2 + 2*a^8*b^2*c^2 - 3*a^6*b^4*c^2 + 6*a^4*b^6*c^2 - 6*a^2*b^8*c^2 + 2*b^10*c^2 - a^8*c^4 - 2*a^4*b^4*c^4 + 6*a^2*b^6*c^4 - 3*b^8*c^4 + 2*a^6*c^6 - 3*a^2*b^4*c^6 - a^4*c^8 + 2*a^2*b^2*c^8 + 2*b^4*c^8 - a^2*c^10 - 2*b^2*c^10 + c^12) : :

X(53883) lies on the circumcircle and these lines: {98, 10749}, {112, 33752}, {690, 2867}, {691, 2881}, {842, 53795}, {935, 2799}, {1297, 39854}, {2697, 2794}, {2710, 2781}, {2715, 9517}, {26714, 53760}

---

X(53884) = REFLECTION OF X(5966) IN X(3)

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^2 + b^2 - 4*c^2)*(a^2 - 4*b^2 + c^2) : :

X(53884) lies on the circumcircle and these lines: {2, 45161}, {3, 5966}, {4, 31843}, {30, 45151}, {74, 36987}, {98, 550}, {111, 6636}, {186, 23096}, {187, 14658}, {376, 1141}, {378, 2383}, {427, 2374}, {827, 11634}, {842, 18859}, {1287, 7472}, {2373, 52397}, {2380, 10646}, {2381, 10645}, {2770, 5189}, {3520, 3563}, {4221, 26707}, {4229, 26708}, {4236, 26711}, {4237, 26710}, {6240, 40120}, {7462, 26709}, {7464, 14979}, {7468, 16166}, {7495, 9084}, {7953, 53273}, {13619, 40118}, {14671, 43662}, {18401, 21312}, {21284, 40119}, {33643, 35921}, {39431, 44239}, {43657, 44832}

X(53884) = reflection of X(i) in X(j) for these {i,j}: {4, 31843}, {5966, 3}
X(53884) = isogonal conjugate of X(32478)
X(53884) = anticomplement of X(45161)
X(53884) = Schoutte-circle-inverse of X(14658)
X(53884) = isogonal conjugate of the anticomplement of X(32478)
X(53884) = isogonal conjugate of the complement of X(32478)
X(53884) = Thomson-isogonal conjugate of X(5965)
X(53884) = Collings transform of X(i) for these i: {31843, 37512}
X(53884) = X(i)-isoconjugate of X(j) for these (i,j): {1, 32478}, {661, 3629}, {1577, 35007}
X(53884) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 32478}, {36830, 3629}
X(53884) = cevapoint of X(512) and X(37512)
X(53884) = trilinear pole of line {6, 11451}
X(53884) = barycentric product X(110)*X(43676)
X(53884) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 32478}, {110, 3629}, {1576, 35007}, {43676, 850}

---

X(53885) = REFLECTION OF X(755) IN X(3)

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^4 + b^4 - 3*a^2*c^2 - 3*b^2*c^2 - 2*c^4)*(a^4 - 3*a^2*b^2 - 2*b^4 - 3*b^2*c^2 + c^4) : :

X(53885) lies on the circumcircle and these lines: {3, 755}, {74, 5188}, {98, 13468}, {107, 46543}, {111, 5092}, {182, 733}, {729, 3398}, {842, 47619}, {843, 9301}, {1350, 29011}, {2858, 4576}, {5008, 52230}, {9069, 10330}, {9076, 46264}, {9145, 39639}, {11636, 45722}, {12203, 39427}

X(53885) = reflection of X(755) in X(3)
X(53885) = isogonal conjugate of X(32473)
X(53885) = Thomson-isogonal conjugate of X(754)
X(53885) = X(i)-isoconjugate of X(j) for these (i,j): {1, 32473}, {661, 41624}, {1577, 41413}
X(53885) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 32473}, {36830, 41624}
X(53885) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 32473}, {110, 41624}, {1576, 41413}

---

X(53886) = REFLECTION OF X(15324) IN X(3)

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 - 3*c^4)^2*(a^4 + 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4)^2 : :

X(53886) lies on the circumcircle and these lines: {2, 13613}, {3, 15324}, {64, 1297}, {69, 39434}, {74, 52559}, {107, 53639}, {112, 46639}, {253, 1294}, {1073, 47409}, {3563, 31942}, {5897, 10606}, {5931, 41904}, {33583, 34168}

X(53886) = reflection of X(15324) in X(3)
X(53886) = anticomplement of X(13613)
X(53886) = Thomson-isogonal conjugate of X(15312)
X(53886) = Collings transform of X(i) for these i: {1073, 3343, 6247, 20207}
X(53886) = X(i)-isoconjugate of X(j) for these (i,j): {154, 17898}, {204, 8057}, {512, 1097}, {610, 6587}, {656, 3079}, {661, 36413}, {810, 52578}, {1394, 14308}, {1895, 42658}, {3198, 21172}, {4017, 6060}, {4041, 7338}, {14331, 30456}, {24019, 39020}
X(53886) = X(i)-Dao conjugate of X(j) for these (i,j): {3343, 8057}, {14092, 6587}, {34961, 6060}, {35071, 39020}, {36830, 36413}, {39054, 1097}, {39062, 52578}, {40596, 3079}
X(53886) = cevapoint of X(i) and X(j) for these (i,j): {520, 1073}, {525, 6247}
X(53886) = trilinear pole of line {6, 1073}
X(53886) = barycentric product X(i)*X(j) for these {i,j}: {64, 44326}, {253, 46639}, {648, 52559}, {1073, 53639}, {1301, 34403}, {4563, 31942}, {14638, 15384}, {35571, 40813}
X(53886) = barycentric quotient X(i)/X(j) for these {i,j}: {64, 6587}, {110, 36413}, {112, 3079}, {520, 39020}, {648, 52578}, {662, 1097}, {1073, 8057}, {1301, 1249}, {2184, 17898}, {4558, 53050}, {4565, 7338}, {5546, 6060}, {11589, 14345}, {14642, 42658}, {15394, 20580}, {30457, 14308}, {31942, 2501}, {36079, 36908}, {40813, 14343}, {41489, 44705}, {44326, 14615}, {46639, 20}, {52158, 14331}, {52559, 525}, {53639, 15466}

---

X(53887) = REFLECTION OF X(43080) IN X(3)

Barycentrics    a^2*(a - b)*(a - c)*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3 - a^2*c + 6*a*b*c - 7*b^2*c - a*c^2 + 2*b*c^2 + c^3)*(a^3 - a^2*b - a*b^2 + b^3 + 2*a^2*c + 6*a*b*c + 2*b^2*c - 7*a*c^2 - 7*b*c^2 + 4*c^3) : :

X(53887) lies on the circumcircle and these lines: {1, 53181}, {3, 43080}, {36, 2291}, {100, 14077}, {104, 15726}, {105, 5126}, {513, 14074}, {517, 15731}, {840, 999}, {972, 50371}, {1319, 15728}, {2222, 23890}, {2717, 3576}, {2742, 23344}, {22765, 38451}, {47357, 53183}

X(53887) = reflection of X(43080) in X(3)
X(53887) = reflection of X(14074) in the OI line
X(53887) = Thomson-isogonal conjugate of X(5851)
X(53887) = X(650)-isoconjugate of X(50573)
X(53887) = barycentric quotient X(109)/X(50573)

---

X(53888) = REFLECTION OF X(1477) IN X(3)

Barycentrics    a^2*(a - b)*(a - c)*(a^3 - a^2*b - a*b^2 + b^3 - 3*a^2*c + 2*a*b*c - 3*b^2*c + 3*a*c^2 + 3*b*c^2 - c^3)*(a^3 - 3*a^2*b + 3*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(53888) lies on the circumcircle and these lines: {1, 53623}, {3, 1477}, {40, 105}, {99, 53653}, {103, 10310}, {104, 6282}, {106, 3428}, {165, 15728}, {675, 42361}, {840, 2077}, {1253, 2377}, {1292, 3939}, {2283, 8059}, {2291, 6244}, {2717, 13528}, {2718, 50371}, {3576, 8686}, {4587, 6078}, {21272, 43349}

X(53888) = reflection of X(1477) in X(3)
X(53888) = isogonal conjugate of the isotomic conjugate of X(53653)
X(53888) = Thomson-isogonal conjugate of X(5853)
X(53888) = Collings transform of X(17642)
X(53888) = X(i)-isoconjugate of X(j) for these (i,j): {513, 36845}, {514, 16572}, {649, 20946}, {650, 8732}, {693, 21002}, {1019, 21096}, {3174, 3676}, {17924, 22153}
X(53888) = X(i)-Dao conjugate of X(j) for these (i,j): {5375, 20946}, {39026, 36845}
X(53888) = cevapoint of X(i) and X(j) for these (i,j): {513, 17642}, {649, 1253}
X(53888) = barycentric product X(i)*X(j) for these {i,j}: {6, 53653}, {101, 42361}, {651, 42470}
X(53888) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 20946}, {101, 36845}, {109, 8732}, {692, 16572}, {4557, 21096}, {32656, 22153}, {32739, 21002}, {35326, 41573}, {42361, 3261}, {42470, 4391}, {53653, 76}

---

X(53889) = REFLECTION OF X(689) IN X(3)

Barycentrics    (2*a^4*b^4 + a^6*c^2 - a^4*b^2*c^2 - a^2*b^4*c^2 + b^6*c^2 - a^4*c^4 - b^4*c^4)*(a^6*b^2 - a^4*b^4 - a^4*b^2*c^2 + 2*a^4*c^4 - a^2*b^2*c^4 - b^4*c^4 + b^2*c^6) : :

X(53889) lies on the circumcircle and these lines: {2, 44947}, {3, 689}, {4, 35971}, {30, 37888}, {55, 7334}, {56, 6026}, {74, 14718}, {99, 9230}, {107, 27369}, {110, 384}, {112, 41331}, {476, 37896}, {783, 14970}, {805, 14712}, {827, 3972}, {1304, 37912}, {12122, 39639}, {22456, 51869}

X(53889) = reflection of X(i) in X(j) for these {i,j}: {4, 35971}, {689, 3}
X(53889) = anticomplement of X(44947)
X(53889) = Thomson-isogonal conjugate of X(688)
X(53889) = Collings transform of X(35971)
X(53889) = cevapoint of X(55) and X(2231)

---

X(53890) = REFLECTION OF X(12074) IN X(3)

Barycentrics    a^2*(a^6 - a^4*b^2 - a^2*b^4 + b^6 + 5*a^4*c^2 + 5*b^4*c^2 - 4*a^2*c^4 - 4*b^2*c^4 - 2*c^6)*(a^6 + 5*a^4*b^2 - 4*a^2*b^4 - 2*b^6 - a^4*c^2 - 4*b^4*c^2 - a^2*c^4 + 5*b^2*c^4 + c^6) : :

X(53890) lies on the circumcircle and these lines: {3, 12074}, {4, 20389}, {99, 8703}, {107, 10301}, {110, 5092}, {112, 5008}, {182, 31951}, {476, 37901}, {691, 9301}, {729, 45723}, {842, 53246}, {1296, 5188}, {2709, 47619}, {3398, 11636}, {9142, 14388}, {14811, 43357}, {30247, 35492}, {53605, 53709}

X(53890) = reflection of X(i) in X(j) for these {i,j}: {4, 20389}, {12074, 3}
X(53890) = isogonal conjugate of X(19924)
X(53890) = Thomson-isogonal conjugate of X(12073)
X(53890) = Collings transform of X(20389)
X(53890) = X(1)-isoconjugate of X(19924)
X(53890) = X(3)-Dao conjugate of X(19924)
X(53890) = barycentric quotient X(6)/X(19924)

---

X(53891) = REFLECTION OF X(14665) IN X(3)

Barycentrics    a*(a - b)*(a - c)*(a^3*b - 2*a^2*b^2 + a*b^3 + a^2*b*c + b^3*c - 2*a^2*c^2 + a*b*c^2 - 2*b^2*c^2 + b*c^3)*(-2*a^2*b^2 + a^3*c + a^2*b*c + a*b^2*c + b^3*c - 2*a^2*c^2 - 2*b^2*c^2 + a*c^3 + b*c^3) : :

X(53891) lies on the circumcircle and these lines: {3, 14665}, {40, 12032}, {100, 4380}, {101, 4401}, {727, 13329}, {741, 8924}, {840, 3098}, {1110, 8685}, {1292, 8671}, {1293, 15599}, {2711, 3579}, {2712, 41430}, {2724, 10310}, {2726, 47048}, {6244, 39421}, {11495, 29348}, {28838, 30270}, {35002, 53180}

X(53891) = reflection of X(14665) in X(3)
X(53891) = Thomson-isogonal conjugate of X(14839)
X(53891) = cevapoint of X(55) and X(659)
X(53891) = trilinear pole of line {6, 21320}

---

X(53892) = REFLECTION OF X(831) IN X(3)

Barycentrics    a*(a^5 - a^3*b^2 - a^2*b^3 + b^5 + 2*a^3*b*c + 2*a*b^3*c - a^2*b*c^2 - a*b^2*c^2 - a*c^4 - b*c^4)*(a^5 - a*b^4 + 2*a^3*b*c - a^2*b^2*c - b^4*c - a^3*c^2 - a*b^2*c^2 - a^2*c^3 + 2*a*b*c^3 + c^5) : :

X(53892) lies on the circumcircle and these lines: {1, 8687}, {3, 831}, {40, 28480}, {100, 3687}, {101, 960}, {105, 13246}, {108, 1848}, {109, 3666}, {110, 17185}, {312, 8707}, {934, 3674}, {1292, 12512}, {2222, 5078}, {2728, 51638}, {29023, 53291}, {29310, 53302}

X(53892) = reflection of X(831) in X(3)
X(53892) = Thomson-isogonal conjugate of X(830)
X(53892) = Collings transform of X(17419)
X(53892) = trilinear pole of line {6, 17420}

---

X(53893) = REFLECTION OF X(5970) IN X(3)

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(2*a^4*b^2 - a^2*b^4 + 3*b^6 - a^4*c^2 - 3*a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 + 2*b^2*c^4)*(a^4*b^2 + a^2*b^4 - 2*a^4*c^2 + 3*a^2*b^2*c^2 - 2*b^4*c^2 + a^2*c^4 + b^2*c^4 - 3*c^6) : :

X(53893) lies on the circumcircle and these lines: {3, 5970}, {4, 9152}, {98, 538}, {99, 32472}, {111, 32526}, {511, 729}, {512, 39639}, {574, 9136}, {699, 2080}, {733, 47618}, {842, 30270}, {843, 3098}, {1350, 2698}, {2396, 9150}, {2715, 5118}, {2770, 38613}, {5104, 52230}, {9737, 14659}, {14509, 25424}, {47047, 53704}, {47293, 53603}

X(53893) = reflection of X(i) in X(j) for these {i,j}: {4, 9152}, {5970, 3}
X(53893) = reflection of X(39639) in the Brocard axis
X(53893) = Thomson-isogonal conjugate of X(5969)
X(53893) = Collings transform of X(9152)
X(53893) = trilinear pole of line {6, 6786}

---

X(53894) = REFLECTION OF X(46970) IN X(3)

Barycentrics    a^2*(a^8 - a^6*b^2 - a^2*b^6 + b^8 - a^4*b^2*c^2 - a^2*b^4*c^2 + 2*a^2*b^2*c^4 + a^2*c^6 + b^2*c^6 - 2*c^8)*(a^8 + a^2*b^6 - 2*b^8 - a^6*c^2 - a^4*b^2*c^2 + 2*a^2*b^4*c^2 + b^6*c^2 - a^2*b^2*c^4 - a^2*c^6 + c^8) : :

X(53894) lies on the circumcircle and these lines: {3, 46970}, {4, 46669}, {39, 2715}, {98, 826}, {99, 5207}, {107, 420}, {110, 6660}, {112, 2076}, {476, 11007}, {511, 827}, {512, 29011}, {691, 14810}, {805, 3493}, {1235, 22456}, {1287, 24206}, {30255, 47619}, {30270, 36517}, {35002, 43357}, {46157, 53691}

X(53894) = reflection of X(i) in X(j) for these {i,j}: {4, 46669}, {46970, 3}
X(53894) = reflection of X(29011) in the Brocard axis
X(53894) = Thomson-isogonal conjugate of X(9479)
X(53894) = Collings transform of X(46669)

---

X(53895) = REFLECTION OF X(40118) IN X(3)

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 2*a^4*c^2 + 5*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6)*(a^6 - 2*a^4*b^2 - 2*a^2*b^4 + b^6 - a^4*c^2 + 5*a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6) : :

X(53895) lies on the circumcircle and these lines: {2, 40119}, {3, 40118}, {5, 23096}, {20, 842}, {22, 2770}, {23, 2374}, {30, 3563}, {74, 3564}, {98, 2071}, {107, 7468}, {108, 7475}, {110, 3566}, {111, 858}, {112, 7472}, {186, 40120}, {376, 32710}, {477, 21312}, {523, 3565}, {690, 35191}, {691, 53351}, {915, 37960}, {935, 11634}, {1289, 7482}, {1296, 47293}, {1297, 16386}, {1299, 10295}, {1300, 7464}, {1301, 7473}, {1302, 40049}, {1304, 4226}, {1325, 15344}, {2373, 3266}, {2697, 11413}, {2752, 16049}, {2766, 4236}, {3153, 5966}, {3767, 14659}, {4230, 22239}, {4235, 10423}, {5159, 5203}, {5196, 9085}, {6776, 23700}, {7471, 9064}, {7476, 40097}, {7477, 9107}, {7481, 40101}, {7493, 10102}, {9060, 30512}, {9084, 37980}, {10296, 43656}, {14979, 44239}, {15075, 38971}, {26705, 36032}, {30249, 37937}, {34866, 46087}, {37918, 53704}, {39417, 46619}, {43662, 47090}

X(53895) = reflection of X(i) in X(j) for these {i,j}: {5203, 5159}, {40118, 3}
X(53895) = anticomplement of X(48317)
X(53895) = reflection of X(3565) in the Euler line
X(53895) = isotomic conjugate of the anticomplement of X(14273)
X(53895) = Thomson-isogonal conjugate of X(14984)
X(53895) = Collings transform of X(i) for these i: {5159, 14961}
X(53895) = X(i)-isoconjugate of X(j) for these (i,j): {656, 37777}, {661, 37784}, {798, 37803}, {1577, 41336}, {24006, 41615}
X(53895) = X(i)-Dao conjugate of X(j) for these (i,j): {31998, 37803}, {36830, 37784}, {40596, 37777}
X(53895) = cevapoint of X(i) and X(j) for these (i,j): {3, 690}, {512, 14961}, {523, 5159}
X(53895) = trilinear pole of line {6, 5181}
X(53895) = barycentric product X(i)*X(j) for these {i,j}: {99, 40347}, {4563, 41521}
X(53895) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 37803}, {110, 37784}, {112, 37777}, {1576, 41336}, {2420, 20772}, {4558, 5866}, {14273, 48317}, {32661, 41615}, {40347, 523}, {41521, 2501}

---

X(53896) = REFLECTION OF X(8691) IN X(3)

Barycentrics    a*(a^4 + a^3*b - 4*a^2*b^2 + a*b^3 + b^4 - a^3*c + 2*a^2*b*c + 2*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)*(a^4 - a^3*b + a^2*b^2 - a*b^3 + a^3*c + 2*a^2*b*c - 2*a*b^2*c - b^3*c - 4*a^2*c^2 + 2*a*b*c^2 + b^2*c^2 + a*c^3 - b*c^3 + c^4) : :
X(53896) = 3 X[165] - X[52679]

X(53896) lies on the circumcircle and these lines: {3, 8691}, {40, 1296}, {55, 1366}, {100, 4416}, {101, 4640}, {105, 45674}, {109, 4689}, {110, 35258}, {111, 9811}, {165, 52679}, {1292, 50808}, {2077, 2746}, {6011, 11495}, {9090, 44425}, {36060, 36070}

X(53896) = reflection of X(8691) in X(3)
X(53896) = Thomson-isogonal conjugate of X(4160)
X(53896) = cevapoint of X(55) and X(896)

---

X(53897) = REFLECTION OF X(43081) IN X(3)

Barycentrics    a^2*(a - b)*(a - c)*(a^4 - 2*a^2*b^2 + b^4 - 5*a^3*c + 7*a^2*b*c + 7*a*b^2*c - 5*b^3*c + 3*a^2*c^2 - 16*a*b*c^2 + 3*b^2*c^2 + 5*a*c^3 + 5*b*c^3 - 4*c^4)*(a^4 - 5*a^3*b + 3*a^2*b^2 + 5*a*b^3 - 4*b^4 + 7*a^2*b*c - 16*a*b^2*c + 5*b^3*c - 2*a^2*c^2 + 7*a*b*c^2 + 3*b^2*c^2 - 5*b*c^3 + c^4) : :

X(53897) lies on the circumcircle and these lines: {3, 43081}, {40, 2718}, {100, 30198}, {104, 3880}, {106, 2077}, {513, 30236}, {517, 8686}, {840, 6244}, {944, 44873}, {953, 10310}, {1155, 53623}, {1477, 5537}, {2720, 23832}, {8683, 39628}

X(53897) = reflection of X(43081) in X(3)
X(53897) = reflection of X(30236) in the OI line
X(53897) = Thomson-isogonal conjugate of X(5854)
X(53897) = barycentric quotient X(2427)/X(18802)

---

X(53898) = REFLECTION OF X(1311) IN X(3)

Barycentrics    (a - b)*(a - c)*(a^4 - a^3*b - 2*a^2*b^2 - a*b^3 + b^4 - a^3*c + 3*a^2*b*c + 3*a*b^2*c - b^3*c - a^2*c^2 - 3*a*b*c^2 - b^2*c^2 + a*c^3 + b*c^3)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + 3*a^2*b*c - 3*a*b^2*c + b^3*c - 2*a^2*c^2 + 3*a*b*c^2 - b^2*c^2 - a*c^3 - b*c^3 + c^4) : :

X(53898) lies on the circumcircle and these lines: {3, 1311}, {98, 7421}, {102, 376}, {103, 944}, {105, 6906}, {106, 5603}, {107, 7463}, {110, 7462}, {111, 7413}, {378, 32706}, {675, 7416}, {901, 47676}, {1297, 30268}, {1302, 7450}, {2695, 7464}, {4224, 9061}, {7428, 9083}, {7436, 39440}, {7449, 9084}, {7451, 9058}, {7452, 9064}, {7460, 9057}, {7461, 9107}, {9060, 50403}, {15344, 37117}, {21312, 41904}, {26703, 37404}, {32660, 32689}

X(53898) = reflection of X(1311) in X(3)
X(53898) = Thomson-isogonal conjugate of X(8679)
X(53898) = X(649)-isoconjugate of X(28978)
X(53898) = X(5375)-Dao conjugate of X(28978)
X(53898) = trilinear pole of line {6, 17728}
X(53898) = barycentric quotient X(100)/X(28978)

---

X(53899) = REFLECTION OF X(6012) IN X(3)

Barycentrics    a*(a^5 - a^4*b - a*b^4 + b^5 - a^4*c + 2*a^3*b*c + 2*a*b^3*c - b^4*c - 2*a^2*b*c^2 - 2*a*b^2*c^2 + a^2*c^3 + 2*a*b*c^3 + b^2*c^3 - a*c^4 - b*c^4)*(a^5 - a^4*b + a^2*b^3 - a*b^4 - a^4*c + 2*a^3*b*c - 2*a^2*b^2*c + 2*a*b^3*c - b^4*c - 2*a*b^2*c^2 + b^3*c^2 + 2*a*b*c^3 - a*c^4 - b*c^4 + c^5) : :

X(53899) lies on the circumcircle and these lines: {1, 8685}, {3, 6012}, {40, 28575}, {56, 3020}, {98, 53302}, {99, 15952}, {100, 3705}, {101, 3061}, {106, 7427}, {109, 982}, {110, 3794}, {934, 7185}, {1292, 37403}, {1385, 29095}, {3865, 29055}, {4518, 8684}, {9083, 46586}

X(53899) = reflection of X(6012) in X(3)
X(53899) = Thomson-isogonal conjugate of X(6004)

---

X(53900) = REFLECTION OF X(43077) IN X(3)

Barycentrics    a^2*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4 - a^4*c + 2*a^2*b^2*c - b^4*c - 2*a^3*c^2 - a^2*b*c^2 - a*b^2*c^2 - 2*b^3*c^2 + 3*a^2*c^3 + 3*b^2*c^3)*(a^4*b + 2*a^3*b^2 - 3*a^2*b^3 - a^4*c + a^2*b^2*c + a^3*c^2 - 2*a^2*b*c^2 + a*b^2*c^2 - 3*b^3*c^2 + a^2*c^3 + 2*b^2*c^3 - a*c^4 + b*c^4) : :

X(53900) lies on the circumcircle and these lines: {3, 43077}, {32, 26716}, {58, 26714}, {100, 52134}, {101, 182}, {107, 31912}, {109, 21010}, {112, 34476}, {934, 4334}, {1292, 47641}, {1293, 8722}, {1350, 28469}, {2080, 2702}, {2700, 53298}, {4279, 32722}, {8693, 9310}, {11842, 30554}

X(53900) = reflection of X(43077) in X(3)
X(53900) = Thomson-isogonal conjugate of X(4785)

---

X(53901) = REFLECTION OF X(9061) IN X(3)

Barycentrics    a*(a - b)*(a - c)*(a^5 - a^4*b - a*b^4 + b^5 - a^4*c - 4*a^3*b*c + 10*a^2*b^2*c - 4*a*b^3*c - b^4*c - 2*a^2*b*c^2 + 10*a*b^2*c^2 - 4*a*b*c^3 - a*c^4 - b*c^4 + c^5)*(a^5 - a^4*b - a*b^4 + b^5 - a^4*c - 4*a^3*b*c - 2*a^2*b^2*c - 4*a*b^3*c - b^4*c + 10*a^2*b*c^2 + 10*a*b^2*c^2 - 4*a*b*c^3 - a*c^4 - b*c^4 + c^5) : :

X(53901) lies on the circumcircle and these lines: {3, 9061}, {100, 46593}, {105, 376}, {111, 3651}, {378, 15344}, {915, 35485}, {1302, 4236}, {2373, 30267}, {2374, 7414}, {2752, 7464}, {2770, 36001}, {4220, 9084}, {4238, 9064}, {7475, 9060}, {10102, 37959}, {11634, 53684}, {21312, 26703}, {37979, 40119}

X(53901) = reflection of X(9061) in X(3)
X(53901) = Thomson-isogonal conjugate of X(9004)

---

X(53902) = REFLECTION OF X(8707) IN X(3)

Barycentrics    (a^5 - a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - a*b^4 + b^5 + a^4*c - 4*a^2*b^2*c + b^4*c - a^3*c^2 + a^2*b*c^2 + a*b^2*c^2 - b^3*c^2 - a^2*c^3 - b^2*c^3)*(a^5 + a^4*b - a^3*b^2 - a^2*b^3 - a^4*c + a^2*b^2*c + 2*a^3*c^2 - 4*a^2*b*c^2 + a*b^2*c^2 - b^3*c^2 + 2*a^2*c^3 - b^2*c^3 - a*c^4 + b*c^4 + c^5) : :

X(53902) lies on the circumcircle and these lines: {3, 8707}, {4, 15611}, {100, 4696}, {101, 17355}, {102, 24813}, {109, 5255}, {110, 11115}, {376, 28480}, {831, 24309}, {901, 20067}, {944, 39635}, {1293, 12512}, {8687, 52411}

X(53902) = reflection of X(i) in X(j) for these {i,j}: {4, 15611}, {8707, 3}
X(53902) = Thomson-isogonal conjugate of X(6371)
X(53902) = Collings transform of X(15611)

---

X(53903) = REFLECTION OF X(9070) IN X(3)

Barycentrics    a*(a + b)*(a + c)*(a^5 - a^4*b - a*b^4 + b^5 + 3*a^3*b*c - 4*a^2*b^2*c + 3*a*b^3*c - a^3*c^2 + a^2*b*c^2 + a*b^2*c^2 - b^3*c^2 + a^2*c^3 - 3*a*b*c^3 + b^2*c^3 - c^5)*(a^5 - a^3*b^2 + a^2*b^3 - b^5 - a^4*c + 3*a^3*b*c + a^2*b^2*c - 3*a*b^3*c - 4*a^2*b*c^2 + a*b^2*c^2 + b^3*c^2 + 3*a*b*c^3 - b^2*c^3 - a*c^4 + c^5) : :

X(53903) lies on the circumcircle and these lines: {3, 9070}, {21, 9058}, {28, 9107}, {98, 14127}, {100, 4221}, {107, 14015}, {108, 4227}, {109, 4424}, {110, 3877}, {111, 7427}, {376, 6011}, {378, 30250}, {842, 46618}, {1290, 37960}, {1302, 11101}, {2766, 37961}, {3701, 9059}, {4226, 53697}, {7415, 33637}, {9056, 37227}, {9057, 36011}, {9064, 13739}, {9084, 46586}, {21312, 44065}

X(53903) = reflection of X(9070) in X(3)
X(53903) = Thomson-isogonal conjugate of X(9013)

---

X(53904) = REFLECTION OF X(9059) IN X(3)

Barycentrics    (a^5 - 2*a^4*b + 3*a^3*b^2 + 3*a^2*b^3 - 2*a*b^4 + b^5 + a^4*c - 6*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 - a^2*c^3 - b^2*c^3)*(a^5 + a^4*b - a^3*b^2 - a^2*b^3 - 2*a^4*c + 2*a^2*b^2*c + 3*a^3*c^2 - 6*a^2*b*c^2 + 2*a*b^2*c^2 - b^3*c^2 + 3*a^2*c^3 - b^2*c^3 - 2*a*c^4 + b*c^4 + c^5) : :
X(53904) = 3 X[3576] - 2 X[11796]

X(53904) lies on the circumcircle and these lines: {3, 9059}, {4, 9088}, {98, 7444}, {100, 4737}, {105, 7447}, {107, 4247}, {109, 3476}, {110, 4234}, {111, 7434}, {376, 1293}, {378, 32704}, {404, 9058}, {476, 7481}, {675, 7446}, {953, 24813}, {1302, 7419}, {1311, 7457}, {2692, 7464}, {3576, 11796}, {4221, 34594}, {4222, 9107}, {4245, 9057}, {4248, 9064}, {7428, 9056}, {7448, 9084}, {7459, 9061}, {7478, 9060}, {26713, 35921}, {32659, 32686}

X(53904) = reflection of X(9059) in X(3)
X(53904) = Thomson-isogonal conjugate of X(9002)
X(53904) = trilinear pole of line {6, 47766}

---

X(53905) = REFLECTION OF X(34594) IN X(3)

Barycentrics    a*(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 - a^4*c^2 + a^3*b*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 + a^2*c^4 - 2*a*b*c^4 + b^2*c^4)*(a^5*b + a^4*b^2 - a^3*b^3 - a^2*b^4 - a^5*c + a^4*b*c - a^3*b^2*c - a^2*b^3*c + 2*a*b^4*c - a*b^3*c^2 - b^4*c^2 + 2*a^3*c^3 - a*b^2*c^3 - b^3*c^3 + a*b*c^4 + b^2*c^4 - a*c^5 + b*c^5) : :

X(53905) lies on the circumcircle and these lines: {3, 34594}, {99, 44139}, {104, 7444}, {105, 7434}, {107, 4222}, {110, 404}, {112, 2220}, {476, 37919}, {759, 7447}, {925, 35998}, {1293, 3651}, {2692, 36001}, {4220, 9059}, {4231, 9088}, {4245, 53683}, {6011, 11491}, {7414, 32704}, {7419, 53684}, {7446, 53707}, {7448, 9061}

X(53905) = reflection of X(34594) in X(3)
X(53905) = Thomson-isogonal conjugate of X(4132)

---

X(53906) = REFLECTION OF X(9028) IN X(3)

Barycentrics    a^2*(a - b)*(a - c)*(a^6 - 5*a^4*b^2 + 4*a^3*b^3 + 3*a^2*b^4 - 4*a*b^5 + b^6 - 4*a^3*b^2*c + 4*a^2*b^3*c + 4*a*b^4*c - 4*b^5*c - a^4*c^2 + 2*a^2*b^2*c^2 + 4*a*b^3*c^2 + 3*b^4*c^2 - 4*a*b^2*c^3 + 4*b^3*c^3 - a^2*c^4 - 5*b^2*c^4 + c^6)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 5*a^4*c^2 - 4*a^3*b*c^2 + 2*a^2*b^2*c^2 - 4*a*b^3*c^2 - 5*b^4*c^2 + 4*a^3*c^3 + 4*a^2*b*c^3 + 4*a*b^2*c^3 + 4*b^3*c^3 + 3*a^2*c^4 + 4*a*b*c^4 + 3*b^2*c^4 - 4*a*c^5 - 4*b*c^5 + c^6) : :

X(53906) lies on the circumcircle and these lines: {3, 9085}, {20, 675}, {98, 30266}, {103, 21312}, {104, 30265}, {105, 36029}, {107, 4237}, {376, 917}, {1006, 15344}, {1301, 4249}, {1304, 36032}, {2071, 53190}, {2374, 6998}, {3563, 7430}, {4229, 39438}, {4243, 9064}, {7437, 9107}, {9061, 36018}, {9083, 36024}, {9084, 26252}, {26708, 44239}, {36026, 40118}

X(53906) = reflection of X(9085) in X(3)
X(53906) = Thomson-isogonal conjugate of X(9028)

---

X(53907) = REFLECTION OF X(9058) IN X(3)

Barycentrics    a*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^5*c + 5*a^4*b*c - 4*a^3*b^2*c - 4*a^2*b^3*c + 5*a*b^4*c - b^5*c - 2*a^4*c^2 + 8*a^2*b^2*c^2 - 2*b^4*c^2 + 2*a^3*c^3 - 4*a^2*b*c^3 - 4*a*b^2*c^3 + 2*b^3*c^3 + a^2*c^4 + b^2*c^4 - a*c^5 - b*c^5)*(a^6 - a^5*b - 2*a^4*b^2 + 2*a^3*b^3 + a^2*b^4 - a*b^5 + 5*a^4*b*c - 4*a^2*b^3*c - 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 - 4*a^2*b*c^3 + 2*b^3*c^3 - a^2*c^4 + 5*a*b*c^4 - 2*b^2*c^4 - b*c^5 + c^6) : :

X(53907) lies on the circumcircle and these lines: {3, 9058}, {4, 9107}, {21, 1302}, {28, 9064}, {98, 7429}, {100, 376}, {105, 14127}, {107, 4227}, {108, 378}, {109, 5119}, {110, 4221}, {111, 7425}, {476, 37960}, {675, 7442}, {1006, 9057}, {1290, 7464}, {1304, 37961}, {1311, 7456}, {1325, 9060}, {2752, 46618}, {2766, 10295}, {2770, 50402}, {3651, 9070}, {4236, 53697}, {6906, 9056}, {7423, 9084}, {7427, 9061}, {7447, 9083}, {9059, 37403}, {13397, 21312}, {14578, 32685}, {17512, 53684}, {18533, 40097}, {26706, 35485}, {26711, 35921}, {43078, 53291}

X(53907) = reflection of X(9058) in X(3)
X(53907) = Thomson-isogonal conjugate of X(9001)
X(53907) = Collings transform of X(47051)
X(53907) = cevapoint of X(517) and X(47051)
X(53907) = trilinear pole of line {6, 40134}

---

X(53908) = REFLECTION OF X(9057) IN X(3)

Barycentrics    (a^7 + 4*a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + 4*a^2*b^5 + b^7 - a^6*c + a^4*b^2*c + a^2*b^4*c - b^6*c - 2*a^5*c^2 + 4*a^3*b^2*c^2 + 4*a^2*b^3*c^2 - 2*b^5*c^2 + 2*a^4*c^3 - 8*a^2*b^2*c^3 + 2*b^4*c^3 + a^3*c^4 + b^3*c^4 - a^2*c^5 - b^2*c^5)*(a^7 - a^6*b - 2*a^5*b^2 + 2*a^4*b^3 + a^3*b^4 - a^2*b^5 + 4*a^5*c^2 + a^4*b*c^2 + 4*a^3*b^2*c^2 - 8*a^2*b^3*c^2 - b^5*c^2 - 5*a^4*c^3 + 4*a^2*b^2*c^3 + b^4*c^3 - 5*a^3*c^4 + a^2*b*c^4 + 2*b^3*c^4 + 4*a^2*c^5 - 2*b^2*c^5 - b*c^6 + c^7) : :

X(53908) lies on the circumcircle and these lines: {3, 9057}, {27, 9064}, {98, 7440}, {101, 376}, {105, 7442}, {107, 7431}, {110, 4229}, {111, 7433}, {378, 26705}, {675, 46596}, {842, 37166}, {1302, 4184}, {1305, 21312}, {1311, 7455}, {2690, 7464}, {4219, 9107}, {4221, 53683}, {5196, 9060}, {7411, 9058}, {7416, 9056}, {7432, 9084}, {7445, 9061}, {7446, 9083}, {26710, 35921}, {32657, 32684}

X(53908) = reflection of X(9057) in X(3)
X(53908) = Thomson-isogonal conjugate of X(9000)

---

X(53909) = REFLECTION OF X(9064) IN X(3)

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 + 4*a^8*c^2 + 12*a^6*b^2*c^2 - 32*a^4*b^4*c^2 + 12*a^2*b^6*c^2 + 4*b^8*c^2 - 16*a^6*c^4 + 16*a^4*b^2*c^4 + 16*a^2*b^4*c^4 - 16*b^6*c^4 + 14*a^4*c^6 - 24*a^2*b^2*c^6 + 14*b^4*c^6 - a^2*c^8 - b^2*c^8 - 2*c^10)*(a^10 + 4*a^8*b^2 - 16*a^6*b^4 + 14*a^4*b^6 - a^2*b^8 - 2*b^10 - 3*a^8*c^2 + 12*a^6*b^2*c^2 + 16*a^4*b^4*c^2 - 24*a^2*b^6*c^2 - b^8*c^2 + 2*a^6*c^4 - 32*a^4*b^2*c^4 + 16*a^2*b^4*c^4 + 14*b^6*c^4 + 2*a^4*c^6 + 12*a^2*b^2*c^6 - 16*b^4*c^6 - 3*a^2*c^8 + 4*b^2*c^8 + c^10) : :

X(53909) lies on the circumcircle and these lines: {3, 9064}, {20, 1302}, {107, 376}, {110, 11820}, {378, 1301}, {1289, 35485}, {1304, 7464}, {2071, 9060}, {2374, 7422}, {3563, 46585}, {3651, 9107}, {7429, 15344}, {7440, 9085}, {9056, 30268}, {9057, 30266}, {9058, 30267}, {10295, 22239}, {15291, 32681}, {16167, 16386}, {18533, 30249}, {36164, 40118}, {40119, 50401}

X(53909) = reflection of X(9064) in X(3)
X(53909) = Thomson-isogonal conjugate of X(9007)
X(53909) = cevapoint of X(3) and X(14915)

---

X(53910) = REFLECTION OF X(53243) IN X(3)

Barycentrics    a^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 + 2*a^4*c^2 - 2*a^3*b*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^2*c^3 - 2*b^3*c^3 - a^2*c^4 - 4*a*b*c^4 - b^2*c^4 + 4*a*c^5 + 4*b*c^5 - 2*c^6)*(a^6 - 2*a^5*b + 2*a^4*b^2 - 2*a^3*b^3 - a^2*b^4 + 4*a*b^5 - 2*b^6 - 2*a^5*c + 2*a^4*b*c - 2*a^3*b^2*c + 2*a^2*b^3*c - 4*a*b^4*c + 4*b^5*c - a^4*c^2 + 2*a*b^3*c^2 - b^4*c^2 + 4*a^3*c^3 - 2*a*b^2*c^3 - 2*b^3*c^3 - a^2*c^4 + 2*a*b*c^4 + 2*b^2*c^4 - 2*a*c^5 - 2*b*c^5 + c^6) : :

X(53910) lies on the circumcircle and these lines: {3, 53243}, {101, 11012}, {108, 354}, {109, 15931}, {934, 1385}, {972, 53291}, {1292, 14110}, {2222, 5536}, {2720, 41341}, {3428, 20219}, {5563, 53622}, {6244, 28218}, {14733, 22765}

X(53910) = reflection of X(53243) in X(3)
X(53910) = Thomson-isogonal conjugate of X(6362)

---

X(53911) = REFLECTION OF X(14733) IN X(3)

Barycentrics    a^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 + 4*a^4*b*c - 4*a^3*b^2*c - 4*a^2*b^3*c + 4*a*b^4*c - 2*a^4*c^2 + 2*a^3*b*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^2*c^3 - 2*b^3*c^3 + 3*a^2*c^4 - 8*a*b*c^4 + 3*b^2*c^4 + 2*a*c^5 + 2*b*c^5 - 2*c^6)*(a^6 - 2*a^4*b^2 - 2*a^3*b^3 + 3*a^2*b^4 + 2*a*b^5 - 2*b^6 - 2*a^5*c + 4*a^4*b*c + 2*a^3*b^2*c + 2*a^2*b^3*c - 8*a*b^4*c + 2*b^5*c - a^4*c^2 - 4*a^3*b*c^2 + 2*a*b^3*c^2 + 3*b^4*c^2 + 4*a^3*c^3 - 4*a^2*b*c^3 + 2*a*b^2*c^3 - 2*b^3*c^3 - a^2*c^4 + 4*a*b*c^4 - 2*b^2*c^4 - 2*a*c^5 + c^6) : :

X(53911) lies on the circumcircle and these lines: {2, 50940}, {3, 14733}, {4, 46415}, {36, 53622}, {40, 1308}, {55, 2720}, {57, 35065}, {100, 971}, {101, 2077}, {104, 3900}, {107, 52891}, {108, 1155}, {109, 5537}, {165, 2222}, {513, 972}, {517, 934}, {652, 2291}, {901, 6244}, {927, 35514}, {929, 5657}, {1145, 43353}, {1292, 13528}, {1309, 7046}, {1311, 47808}, {2078, 8059}, {2717, 45884}, {2742, 10310}, {3660, 30239}, {5536, 26700}, {6282, 53184}, {9086, 33864}, {14074, 50371}, {15931, 34921}, {35000, 53243}

X(53911) = reflection of X(i) in X(j) for these {i,j}: {4, 46415}, {14733, 3}
X(53911) = anticomplement of X(50940)
X(53911) = reflection of X(972) in the OI line
X(53911) = Thomson-isogonal conjugate of X(6366)
X(53911) = Collings transform of X(46415)
X(53911) = cevapoint of X(35065) and X(45884)

---

X(53912) = REFLECTION OF X(2867) IN X(3)

Barycentrics    (a^12 - a^10*b^2 + 3*a^8*b^4 - 6*a^6*b^6 + 3*a^4*b^8 - a^2*b^10 + b^12 - a^10*c^2 + a^6*b^4*c^2 + a^4*b^6*c^2 - b^10*c^2 - 2*a^8*c^4 + 3*a^6*b^2*c^4 - 2*a^4*b^4*c^4 + 3*a^2*b^6*c^4 - 2*b^8*c^4 + 2*a^6*c^6 - 3*a^4*b^2*c^6 - 3*a^2*b^4*c^6 + 2*b^6*c^6 + a^4*c^8 + 2*a^2*b^2*c^8 + b^4*c^8 - a^2*c^10 - b^2*c^10)*(a^12 - a^10*b^2 - 2*a^8*b^4 + 2*a^6*b^6 + a^4*b^8 - a^2*b^10 - a^10*c^2 + 3*a^6*b^4*c^2 - 3*a^4*b^6*c^2 + 2*a^2*b^8*c^2 - b^10*c^2 + 3*a^8*c^4 + a^6*b^2*c^4 - 2*a^4*b^4*c^4 - 3*a^2*b^6*c^4 + b^8*c^4 - 6*a^6*c^6 + a^4*b^2*c^6 + 3*a^2*b^4*c^6 + 2*b^6*c^6 + 3*a^4*c^8 - 2*b^4*c^8 - a^2*c^10 - b^2*c^10 + c^12) : :

X(53912) lies on the circumcircle and these lines: {3, 2867}, {4, 32687}, {56, 44049}, {107, 41377}, {110, 441}, {111, 50642}, {112, 1503}, {476, 37921}, {525, 1297}, {691, 14927}, {935, 15577}, {1301, 15448}, {1304, 35260}, {2715, 8721}, {6776, 10423}, {13509, 26714}, {14376, 18337}, {18338, 39575}, {19128, 39417}, {22456, 51257}

X(53912) = reflection of X(i) in X(j) for these {i,j}: {4, 33504}, {2867, 3}
X(53912) = isogonal conjugate of X(53795)
X(53912) = Thomson-isogonal conjugate of X(2881)
X(53912) = Collings transform of X(33504)
X(53912) = X(1)-isoconjugate of X(53795)
X(53912) = barycentric quotient X(6)/X(53795)

---

X(53913) = REFLECTION OF X(9088) IN X(3)

Barycentrics    a^2*(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 - 2*a^6*c + 2*a^4*b^2*c + 2*a^2*b^4*c - 2*b^6*c + 3*a^5*c^2 - 9*a^4*b*c^2 + 6*a^3*b^2*c^2 + 6*a^2*b^3*c^2 - 9*a*b^4*c^2 + 3*b^5*c^2 + 6*a^4*c^3 - 12*a^2*b^2*c^3 + 6*b^4*c^3 - 5*a^3*c^4 + 7*a^2*b*c^4 + 7*a*b^2*c^4 - 5*b^3*c^4 - 2*a^2*c^5 - 2*b^2*c^5 + a*c^6 + b*c^6 - 2*c^7)*(a^7 - 2*a^6*b + 3*a^5*b^2 + 6*a^4*b^3 - 5*a^3*b^4 - 2*a^2*b^5 + a*b^6 - 2*b^7 + a^6*c - 9*a^4*b^2*c + 7*a^2*b^4*c + b^6*c - a^5*c^2 + 2*a^4*b*c^2 + 6*a^3*b^2*c^2 - 12*a^2*b^3*c^2 + 7*a*b^4*c^2 - 2*b^5*c^2 - a^4*c^3 + 6*a^2*b^2*c^3 - 5*b^4*c^3 - a^3*c^4 + 2*a^2*b*c^4 - 9*a*b^2*c^4 + 6*b^3*c^4 - a^2*c^5 + 3*b^2*c^5 + a*c^6 - 2*b*c^6 + c^7) : :

X(53913) lies on the circumcircle and these lines: {3, 9088}, {20, 9059}, {107, 4234}, {376, 32704}, {404, 9107}, {1293, 21312}, {1301, 4247}, {1304, 7481}, {2374, 7434}, {3563, 7444}, {7419, 9064}, {7446, 9085}, {7447, 15344}, {9057, 36024}, {9058, 35998}, {26713, 44239}

X(53913) = reflection of X(9088) in X(3)
X(53913) = Thomson-isogonal conjugate of X(9031)

---

X(53914) = REFLECTION OF X(2764) IN X(3)

Barycentrics    a^2*(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 + 2*a^10*c^2 + 2*a^8*b^2*c^2 - 4*a^6*b^4*c^2 - 4*a^4*b^6*c^2 + 2*a^2*b^8*c^2 + 2*b^10*c^2 - 8*a^8*c^4 + 8*a^6*b^2*c^4 + 8*a^2*b^6*c^4 - 8*b^8*c^4 + 4*a^6*c^6 - 4*a^4*b^2*c^6 - 4*a^2*b^4*c^6 + 4*b^6*c^6 + a^4*c^8 - 4*a^2*b^2*c^8 + b^4*c^8 + 2*a^2*c^10 + 2*b^2*c^10 - 2*c^12)*(a^12 + 2*a^10*b^2 - 8*a^8*b^4 + 4*a^6*b^6 + a^4*b^8 + 2*a^2*b^10 - 2*b^12 - 4*a^10*c^2 + 2*a^8*b^2*c^2 + 8*a^6*b^4*c^2 - 4*a^4*b^6*c^2 - 4*a^2*b^8*c^2 + 2*b^10*c^2 + 7*a^8*c^4 - 4*a^6*b^2*c^4 - 4*a^2*b^6*c^4 + b^8*c^4 - 8*a^6*c^6 - 4*a^4*b^2*c^6 + 8*a^2*b^4*c^6 + 4*b^6*c^6 + 7*a^4*c^8 + 2*a^2*b^2*c^8 - 8*b^4*c^8 - 4*a^2*c^10 + 2*b^2*c^10 + c^12) : :

X(53914) lies on the circumcircle and these lines: {3, 2764}, {6, 32687}, {74, 42658}, {107, 1503}, {110, 34147}, {112, 6000}, {154, 1304}, {520, 1297}, {525, 1294}, {1073, 46968}, {1301, 1495}, {1899, 22239}, {10423, 11456}, {51222, 53246}

X(53914) = reflection of X(2764) in X(3)
X(53914) = isogonal conjugate of X(9530)
X(53914) = Thomson-isogonal conjugate of X(2848)
X(53914) = X(1)-isoconjugate of X(9530)
X(53914) = barycentric quotient X(6)/X(9530)

---

X(53915) = REFLECTION OF X(8059) IN X(3)

Barycentrics    a^2*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7 + 4*a^5*b*c - 8*a^3*b^3*c + 4*a*b^5*c - a^5*c^2 - a^4*b*c^2 + 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 - a*b^4*c^2 - b^5*c^2 - 2*a^4*c^3 + 4*a^3*b*c^3 - 4*a^2*b^2*c^3 + 4*a*b^3*c^3 - 2*b^4*c^3 - a^3*c^4 + a^2*b*c^4 + a*b^2*c^4 - b^3*c^4 + 4*a^2*c^5 - 8*a*b*c^5 + 4*b^2*c^5 + a*c^6 + b*c^6 - 2*c^7)*(a^7 - a^5*b^2 - 2*a^4*b^3 - a^3*b^4 + 4*a^2*b^5 + a*b^6 - 2*b^7 - a^6*c + 4*a^5*b*c - a^4*b^2*c + 4*a^3*b^3*c + a^2*b^4*c - 8*a*b^5*c + b^6*c - 3*a^5*c^2 + 2*a^3*b^2*c^2 - 4*a^2*b^3*c^2 + a*b^4*c^2 + 4*b^5*c^2 + 3*a^4*c^3 - 8*a^3*b*c^3 + 2*a^2*b^2*c^3 + 4*a*b^3*c^3 - b^4*c^3 + 3*a^3*c^4 - a*b^2*c^4 - 2*b^3*c^4 - 3*a^2*c^5 + 4*a*b*c^5 - b^2*c^5 - a*c^6 + c^7) : :

X(53915) lies on the circumcircle and these lines: {1, 30239}, {3, 8059}, {4, 46663}, {9, 40117}, {34, 3342}, {40, 108}, {100, 1490}, {101, 1604}, {109, 1035}, {110, 1819}, {271, 52027}, {934, 6282}, {971, 34902}, {1436, 8064}, {2077, 2720}, {2222, 13528}, {2765, 17613}, {3428, 53622}, {5657, 26704}, {14110, 26700}, {18283, 32704}, {26705, 35514}, {41906, 52366}

X(53915) = reflection of X(i) in X(j) for these {i,j}: {4, 46663}, {8059, 3}
X(53915) = Thomson-isogonal conjugate of X(8058)
X(53915) = Collings transform of X(46663)
X(53915) = X(4)-isoconjugate of X(15524)
X(53915) = X(36033)-Dao conjugate of X(15524)
X(53915) = trilinear pole of line {6, 10397}
X(53915) = barycentric quotient X(48)/X(15524)

---

X(53916) = REFLECTION OF X(9056) IN X(3)

Barycentrics    (a^8 - a^7*b + 4*a^6*b^2 + a^5*b^3 - 10*a^4*b^4 + a^3*b^5 + 4*a^2*b^6 - a*b^7 + b^8 + a^6*b*c - 5*a^5*b^2*c + 4*a^4*b^3*c + 4*a^3*b^4*c - 5*a^2*b^5*c + a*b^6*c - 3*a^6*c^2 + 2*a^5*b*c^2 + 5*a^4*b^2*c^2 - 8*a^3*b^3*c^2 + 5*a^2*b^4*c^2 + 2*a*b^5*c^2 - 3*b^6*c^2 - 2*a^4*b*c^3 + 4*a^3*b^2*c^3 + 4*a^2*b^3*c^3 - 2*a*b^4*c^3 + 3*a^4*c^4 - a^3*b*c^4 - 8*a^2*b^2*c^4 - a*b^3*c^4 + 3*b^4*c^4 + a^2*b*c^5 + a*b^2*c^5 - a^2*c^6 - b^2*c^6)*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 - a^7*c + a^6*b*c + 2*a^5*b^2*c - 2*a^4*b^3*c - a^3*b^4*c + a^2*b^5*c + 4*a^6*c^2 - 5*a^5*b*c^2 + 5*a^4*b^2*c^2 + 4*a^3*b^3*c^2 - 8*a^2*b^4*c^2 + a*b^5*c^2 - b^6*c^2 + a^5*c^3 + 4*a^4*b*c^3 - 8*a^3*b^2*c^3 + 4*a^2*b^3*c^3 - a*b^4*c^3 - 10*a^4*c^4 + 4*a^3*b*c^4 + 5*a^2*b^2*c^4 - 2*a*b^3*c^4 + 3*b^4*c^4 + a^3*c^5 - 5*a^2*b*c^5 + 2*a*b^2*c^5 + 4*a^2*c^6 + a*b*c^6 - 3*b^2*c^6 - a*c^7 + c^8) : :

X(53916) lies on the circumcircle and these lines: {3, 9056}, {29, 9064}, {30, 53939}, {98, 7454}, {105, 7456}, {107, 7436}, {109, 376}, {110, 7415}, {111, 7441}, {378, 26704}, {411, 9058}, {675, 7455}, {1302, 4225}, {2075, 53944}, {2689, 7464}, {7412, 9107}, {7420, 9057}, {7424, 9060}, {7439, 9084}, {7443, 9061}, {7457, 9083}, {10295, 53965}, {21312, 41906}, {26709, 35921}

X(53916) = reflection of X(9056) in X(3)
X(53916) = Thomson-isogonal conjugate of X(8999)

---

X(53917) = REFLECTION OF X(9057) IN X(3)

Barycentrics    a*(a^9 - a^8*b - 2*a^7*b^2 + 2*a^6*b^3 + 2*a^3*b^6 - 2*a^2*b^7 - a*b^8 + b^9 + 6*a^7*b*c - 6*a^5*b^3*c - 6*a^3*b^5*c + 6*a*b^7*c - 2*a^7*c^2 - 4*a^6*b*c^2 + 16*a^5*b^2*c^2 - 10*a^4*b^3*c^2 - 10*a^3*b^4*c^2 + 16*a^2*b^5*c^2 - 4*a*b^6*c^2 - 2*b^7*c^2 - 10*a^5*b*c^3 + 20*a^3*b^3*c^3 - 10*a*b^5*c^3 + 10*a^4*b*c^4 - 10*a^3*b^2*c^4 - 10*a^2*b^3*c^4 + 10*a*b^4*c^4 + 2*a^3*b*c^5 + 2*a*b^3*c^5 + 2*a^3*c^6 - 4*a^2*b*c^6 - 4*a*b^2*c^6 + 2*b^3*c^6 + 2*a*b*c^7 - a*c^8 - b*c^8)*(a^9 - 2*a^7*b^2 + 2*a^3*b^6 - a*b^8 - a^8*c + 6*a^7*b*c - 4*a^6*b^2*c - 10*a^5*b^3*c + 10*a^4*b^4*c + 2*a^3*b^5*c - 4*a^2*b^6*c + 2*a*b^7*c - b^8*c - 2*a^7*c^2 + 16*a^5*b^2*c^2 - 10*a^3*b^4*c^2 - 4*a*b^6*c^2 + 2*a^6*c^3 - 6*a^5*b*c^3 - 10*a^4*b^2*c^3 + 20*a^3*b^3*c^3 - 10*a^2*b^4*c^3 + 2*a*b^5*c^3 + 2*b^6*c^3 - 10*a^3*b^2*c^4 + 10*a*b^4*c^4 - 6*a^3*b*c^5 + 16*a^2*b^2*c^5 - 10*a*b^3*c^5 + 2*a^3*c^6 - 4*a*b^2*c^6 - 2*a^2*c^7 + 6*a*b*c^7 - 2*b^2*c^7 - a*c^8 + c^9) : :

X(53917) lies on the circumcircle and these lines: {3, 9107}, {20, 9058}, {21, 9064}, {30, 53948}, {100, 21312}, {107, 4221}, {108, 376}, {378, 40097}, {1301, 4227}, {1302, 16049}, {1304, 37960}, {1325, 53944}, {2071, 53941}, {2374, 7425}, {2766, 7464}, {3563, 7429}, {7442, 9085}, {9056, 37404}, {9057, 36029}, {9070, 30267}, {9088, 37403}, {14127, 15344}, {22239, 37961}, {26711, 44239}, {40119, 50402}, {46618, 53956}

X(53917) = reflection of X(9107) in X(3)
X(53917) = Thomson-isogonal conjugate of X(9051)

---

X(53918) = REFLECTION OF X(689) IN EULER LINE

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(-b^6 + a^4*c^2 - a^2*b^2*c^2 + a^2*c^4)*(a^4*b^2 + a^2*b^4 - a^2*b^2*c^2 - c^6) : :

X(53918) lies on the circumcircle and these lines: {23, 733}, {30, 37888}, {99, 3005}, {110, 688}, {385, 9076}, {477, 7470}, {523, 689}, {669, 827}, {729, 8627}, {755, 3231}, {842, 7467}, {2770, 26257}, {5354, 5970}, {7792, 53704}, {32710, 35476}, {34537, 35567}, {39449, 46228}

X(53918) = reflection of X(689) in the Euler line
X(53918) = X(656)-isoconjugate of X(37912)
X(53918) = X(40596)-Dao conjugate of X(37912)
X(53918) = trilinear pole of line {6, 13210}
X(53918) = barycentric quotient X(112)/X(37912)

---

X(53919) = REFLECTION OF X(729) IN EULER LINE

Barycentrics    (a^6*b^2 + a^2*b^6 - 2*a^6*c^2 - 2*b^6*c^2 + 2*a^4*c^4 - 2*a^2*b^2*c^4 + 2*b^4*c^4)*(2*a^6*b^2 - 2*a^4*b^4 - a^6*c^2 + 2*a^2*b^4*c^2 - 2*b^4*c^4 - a^2*c^6 + 2*b^2*c^6) : :

X(53919) lies on the circumcircle and these lines: {23, 9066}, {30, 39639}, {74, 32472}, {76, 9150}, {98, 5926}, {99, 30736}, {110, 538}, {476, 37465}, {523, 729}, {691, 1003}, {805, 3734}, {843, 46778}, {2715, 36822}, {5652, 32730}, {9087, 32526}, {9091, 14700}, {47326, 53603}

X(53919) =reflection of X(729) in the Euler line
X(53919) =trilinear pole of line {6, 9148}

---

X(53920) = REFLECTION OF X(741) IN EULER LINE

Barycentrics    (a + b)*(a + c)*(-(a^4*b^2) + a^3*b^3 - a^2*b^4 + a^5*c - a^4*b*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^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^2*c^4)*(a^5*b + a^4*b^2 - 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 - a^4*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 - b^3*c^3 - a^2*c^4 - a*b*c^4 + b^2*c^4 + b*c^5) : :

X(53920) lies on the circumcircle and these lines: {30, 6010}, {74, 6002}, {75, 36066}, {99, 35544}, {101, 4037}, {109, 7235}, {110, 740}, {523, 741}, {594, 813}, {691, 11104}, {932, 1325}, {1290, 13588}, {2690, 36015}, {5196, 29329}, {5209, 13396}, {7424, 29325}, {15150, 53612}, {36069, 36815}, {37960, 39631}, {49127, 53936}

X(53920) = reflection of X(741) in the Euler line
X(53920) = trilinear pole of line {6, 24506}

---

X(53921) = REFLECTION OF X(915) IN EULER LINE

Barycentrics    a*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7 + a^5*b*c - 2*a^3*b^3*c + a*b^5*c - a^5*c^2 + a^3*b^2*c^2 + a^2*b^3*c^2 - b^5*c^2 + a^3*b*c^3 + a*b^3*c^3 - a^3*c^4 - b^3*c^4 - 2*a*b*c^5 + a*c^6 + b*c^6)*(a^7 - a^5*b^2 - a^3*b^4 + a*b^6 - a^6*c + a^5*b*c + a^3*b^3*c - 2*a*b^5*c + b^6*c - 3*a^5*c^2 + a^3*b^2*c^2 + 3*a^4*c^3 - 2*a^3*b*c^3 + a^2*b^2*c^3 + a*b^3*c^3 - b^4*c^3 + 3*a^3*c^4 - 3*a^2*c^5 + a*b*c^5 - b^2*c^5 - a*c^6 + c^7) : :

X(53921) lies on the circumcircle and these lines: {3, 53952}, {4, 1290}, {21, 10420}, {24, 2766}, {25, 53941}, {28, 476}, {30, 13397}, {72, 6099}, {74, 15313}, {99, 37961}, {100, 186}, {108, 403}, {109, 1725}, {110, 912}, {112, 8609}, {378, 2691}, {468, 9058}, {523, 915}, {691, 4227}, {925, 1325}, {1006, 53925}, {1292, 10295}, {1295, 46618}, {1297, 50402}, {1300, 44426}, {1302, 37963}, {1304, 30733}, {2689, 37117}, {2690, 36009}, {2693, 7429}, {2694, 14127}, {2697, 7425}, {3565, 37960}, {4221, 53895}, {4228, 16167}, {4233, 9060}, {6011, 37979}, {6353, 53948}, {7414, 53936}, {7423, 53929}, {7427, 53964}, {7447, 53928}, {7469, 53958}, {9107, 37777}, {10101, 18533}, {16049, 53953}, {26711, 37943}, {36001, 44065}, {37305, 53611}, {37951, 40097}, {37966, 53697}

X(53921) = reflection of X(i) in X(j) for these {i,j}: {4, 53988}, {53952, 3}
X(53921) = reflection of X(915) in the Euler line
X(53921) = polar-circle-inverse of X(42422)
X(53921) = Collings transform of X(53988)
X(53921) = X(i)-isoconjugate of X(j) for these (i,j): {656, 7477}, {22350, 38952}
X(53921) = X(40596)-Dao conjugate of X(7477)
X(53921) = trilinear pole of line {6, 47235}
X(53921) = barycentric quotient X(112)/X(7477)

---

X(53922) = REFLECTION OF X(53925) IN EULER LINE

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^7 - a^4*b^3 - a^2*b^5 - a*b^6 + 2*b^7 - a^6*c + a^4*b^2*c + a^2*b^4*c - b^6*c - 3*a^5*c^2 - a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a*b^4*c^2 - b^5*c^2 + 3*a^4*c^3 - a^2*b^2*c^3 + 3*a^3*c^4 + a*b^2*c^4 - b^3*c^4 - 3*a^2*c^5 - a*c^6 + c^7)*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7 + a^4*b*c^2 - a^3*b^2*c^2 - a^2*b^3*c^2 + a*b^4*c^2 - a^4*c^3 + 2*a^2*b^2*c^3 - b^4*c^3 + a^2*b*c^4 + a*b^2*c^4 - a^2*c^5 - b^2*c^5 - a*c^6 - b*c^6 + 2*c^7) : :

X(53922) lies on the circumcircle and these lines: {3, 53925}, {4, 2690}, {25, 53940}, {27, 476}, {30, 1305}, {71, 35182}, {74, 8676}, {100, 52414}, {101, 186}, {110, 916}, {112, 8608}, {378, 53189}, {403, 26705}, {468, 9057}, {523, 917}, {691, 7431}, {925, 5196}, {1290, 4219}, {1294, 37166}, {2074, 53683}, {2693, 7440}, {2694, 7442}, {2697, 7433}, {4184, 10420}, {4229, 53895}, {7411, 53952}, {7432, 53929}, {7445, 53964}, {7446, 53928}, {7464, 53906}, {7466, 53941}, {7474, 16167}, {10295, 44876}, {26710, 37943}

X(53922) = reflection of X(53925) in X(3)
X(53922) = reflection of X(917) in the Euler line
X(53922) = X(656)-isoconjugate of X(7479)
X(53922) = X(40596)-Dao conjugate of X(7479)
X(53922) = barycentric quotient X(112)/X(7479)

---

X(53923) = REFLECTION OF X(1288) IN EULER LINE

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^2 + b^2 - c^2)^2*(a^2 - b^2 + c^2)^2*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 - b^6*c^2 - a^4*c^4 - b^4*c^4 + a^2*c^6 + b^2*c^6)*(a^8 - a^6*b^2 - a^4*b^4 + a^2*b^6 - 4*a^6*c^2 + a^4*b^2*c^2 + b^6*c^2 + 6*a^4*c^4 + a^2*b^2*c^4 - b^4*c^4 - 4*a^2*c^6 - b^2*c^6 + c^8) : :

X(53923) lies on the circumcircle and these lines: {74, 38534}, {98, 37981}, {107, 38861}, {186, 51761}, {250, 925}, {403, 45781}, {477, 35471}, {523, 1288}, {842, 21213}, {1286, 7473}, {1294, 44246}, {1297, 37978}, {2693, 12084}, {2697, 7391}, {10420, 53176}, {16237, 53953}, {26284, 53929}, {46963, 52917}

X(53923) = reflection of X(1288) in the Euler line
X(53923) = X(i)-isoconjugate of X(j) for these (i,j): {656, 2072}, {17879, 53329}, {24018, 53416}
X(53923) = X(40596)-Dao conjugate of X(2072)
X(53923) = trilinear pole of line {6, 38534}
X(53923) = barycentric product X(i)*X(j) for these {i,j}: {648, 38534}, {41679, 45781}, {46456, 53170}
X(53923) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 2072}, {32713, 53416}, {38534, 525}, {41937, 53329}, {53170, 8552}

---

X(53924) = REFLECTION OF X(1288) IN EULER LINE

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^12 - 3*a^10*b^2 + 3*a^8*b^4 - 2*a^6*b^6 + 3*a^4*b^8 - 3*a^2*b^10 + b^12 - 3*a^10*c^2 + 8*a^8*b^2*c^2 - 5*a^6*b^4*c^2 - 5*a^4*b^6*c^2 + 8*a^2*b^8*c^2 - 3*b^10*c^2 + 2*a^8*c^4 - 7*a^6*b^2*c^4 + 6*a^4*b^4*c^4 - 7*a^2*b^6*c^4 + 2*b^8*c^4 + 2*a^6*c^6 + 3*a^4*b^2*c^6 + 3*a^2*b^4*c^6 + 2*b^6*c^6 - 3*a^4*c^8 - 2*a^2*b^2*c^8 - 3*b^4*c^8 + a^2*c^10 + b^2*c^10)*(a^12 - 3*a^10*b^2 + 2*a^8*b^4 + 2*a^6*b^6 - 3*a^4*b^8 + a^2*b^10 - 3*a^10*c^2 + 8*a^8*b^2*c^2 - 7*a^6*b^4*c^2 + 3*a^4*b^6*c^2 - 2*a^2*b^8*c^2 + b^10*c^2 + 3*a^8*c^4 - 5*a^6*b^2*c^4 + 6*a^4*b^4*c^4 + 3*a^2*b^6*c^4 - 3*b^8*c^4 - 2*a^6*c^6 - 5*a^4*b^2*c^6 - 7*a^2*b^4*c^6 + 2*b^6*c^6 + 3*a^4*c^8 + 8*a^2*b^2*c^8 + 2*b^4*c^8 - 3*a^2*c^10 - 3*b^2*c^10 + c^12) : :

X(53924) lies on the circumcircle and these lines: {3, 53953}, {4, 10420}, {24, 476}, {25, 16167}, {30, 13398}, {68, 16221}, {74, 40048}, {99, 44138}, {107, 37951}, {110, 403}, {112, 16310}, {136, 12028}, {186, 925}, {378, 53895}, {468, 53958}, {523, 1299}, {691, 18533}, {930, 37970}, {1290, 31384}, {1291, 6240}, {1302, 37777}, {1304, 3542}, {2071, 44064}, {2766, 31385}, {3520, 53960}, {3565, 10295}, {5897, 36164}, {6353, 9060}, {7414, 53952}, {7576, 11635}, {12118, 53172}, {13397, 37979}, {13619, 20185}, {31510, 53694}, {34168, 50401}, {35485, 53961}, {46585, 53934}, {46619, 53695}

X(53924) = reflection of X(i) in X(j) for these {i,j}: {4, 16178}, {53953, 3}
X(53924) = reflection of X(1299) in the Euler line
X(53924) = polar-circle-inverse of X(42424)
X(53924) = Collings transform of X(16178)
X(53924) = X(656)-isoconjugate of X(40049)
X(53924) = X(i)-Dao conjugate of X(j) for these (i,j): {25641, 42424}, {40596, 40049}
X(53924) = cevapoint of X(25) and X(3018)
X(53924) = trilinear pole of line {6, 40048}
X(53924) = barycentric product X(648)*X(40048)
X(53924) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 40049}, {3018, 42424}, {40048, 525}

---

X(53925) = REFLECTION OF X(53922) IN EULER LINE

Barycentrics    a^2*(a - b)*(a - c)*(a^6 - 2*a^4*b^2 + a^3*b^3 - a*b^5 + b^6 - a^3*b^2*c + a^2*b^3*c + a*b^4*c - b^5*c - a^4*c^2 + 2*a^2*b^2*c^2 + a*b^3*c^2 - a*b^2*c^3 + b^3*c^3 - a^2*c^4 - 2*b^2*c^4 + c^6)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 2*a^4*c^2 - a^3*b*c^2 + 2*a^2*b^2*c^2 - a*b^3*c^2 - 2*b^4*c^2 + a^3*c^3 + a^2*b*c^3 + a*b^2*c^3 + b^3*c^3 + a*b*c^4 - a*c^5 - b*c^5 + c^6) : :

X(53925) lies on the circumcircle and these lines: {2, 53947}, {3, 53922}, {20, 2688}, {22, 53190}, {23, 9085}, {30, 917}, {74, 916}, {99, 20294}, {103, 2071}, {107, 7479}, {108, 36031}, {110, 8676}, {112, 36032}, {477, 30266}, {523, 1305}, {675, 858}, {759, 3100}, {934, 51658}, {935, 4237}, {1006, 53921}, {1300, 36026}, {1304, 4243}, {2373, 3006}, {2687, 36029}, {2752, 36018}, {2758, 36024}, {2760, 6790}, {2766, 7437}, {2770, 26252}, {3153, 26708}, {4223, 53956}, {4241, 22239}, {4249, 10423}, {4511, 26702}, {5196, 39438}, {6998, 40118}, {7430, 32710}, {7453, 40119}, {7460, 53965}, {16386, 41905}, {36023, 53185}, {36030, 53612}

X(53925) = reflection of X(53922) in X(3)
X(53925) = reflection of X(1305) in the Euler line
X(53925) = cevapoint of X(3) and X(2774)
X(53925) = trilinear pole of line {6, 41164}

---

X(53926) = REFLECTION OF X(1311) IN EULER LINE

Barycentrics    a^2*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^5*c + a^4*b*c + a*b^4*c - b^5*c - a^4*c^2 + a^3*b*c^2 + a*b^3*c^2 - b^4*c^2 - a^2*b*c^3 - a*b^2*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)*(a^6 - a^5*b - a^4*b^2 + 2*a^2*b^4 + a*b^5 - 2*b^6 + a^4*b*c + a^3*b^2*c - a^2*b^3*c - 2*a*b^4*c + b^5*c - a^4*c^2 - a*b^3*c^2 + 2*b^4*c^2 + a*b^2*c^3 - a^2*c^4 + a*b*c^4 - b^2*c^4 - b*c^5 + c^6) : :

X(53926) lies on the circumcircle and these lines: {2, 2689}, {23, 109}, {25, 53965}, {29, 935}, {30, 53898}, {98, 47798}, {99, 7424}, {104, 39212}, {110, 8679}, {111, 6589}, {112, 2075}, {411, 2691}, {468, 26704}, {477, 7441}, {523, 1311}, {691, 4225}, {841, 7454}, {842, 7439}, {858, 41906}, {1290, 35996}, {1400, 32689}, {1995, 53939}, {2687, 7443}, {2690, 36007}, {2696, 7415}, {2701, 46410}, {2720, 20122}, {2766, 35973}, {7412, 10101}, {7420, 53189}, {7426, 9056}, {7436, 10098}, {26709, 37760}

X(53926) = reflection of X(1311) in the Euler line
X(53926) = trilinear pole of line {6, 2773}

---

X(53927) = REFLECTION OF X(53932) IN EULER LINE

Barycentrics    a*(a - b)*(a - c)*(a^6 - a^5*b - a^4*b^2 + 2*a^3*b^3 - a^2*b^4 - a*b^5 + b^6 + a^4*b*c - 2*a^3*b^2*c + 2*a*b^4*c - b^5*c - a^4*c^2 + a^3*b*c^2 + a^2*b^2*c^2 - b^4*c^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 - b*c^5 + c^6)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^5*c + a^4*b*c + a^3*b^2*c + a^2*b^3*c + a*b^4*c - b^5*c - a^4*c^2 - 2*a^3*b*c^2 + a^2*b^2*c^2 - 2*a*b^3*c^2 - b^4*c^2 + 2*a^3*c^3 + 2*b^3*c^3 - a^2*c^4 + 2*a*b*c^4 - b^2*c^4 - a*c^5 - b*c^5 + c^6) : :

X(53927) lies on the circumcircle and these lines: {3, 53932}, {11, 759}, {30, 2716}, {74, 2800}, {102, 6265}, {104, 25437}, {109, 53527}, {110, 3738}, {523, 2222}, {859, 12030}, {901, 36167}, {1309, 37964}, {1633, 53609}, {2616, 36078}, {2687, 6905}, {2752, 33849}, {3737, 36069}, {4242, 35050}, {4996, 39435}, {13589, 53611}, {43655, 47270}, {53279, 53610}

X(53927) = reflection of X(53932) in X(3)
X(53927) = reflection of X(2222) in the Euler line
X(53927) = Collings transform of X(47400)
X(53927) = cevapoint of X(523) and X(47400)

---

X(53928) = REFLECTION OF X(2370) IN EULER LINE

Barycentrics    a^2*(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 - 2*a^6*c + 2*a^4*b^2*c + 2*a^2*b^4*c - 2*b^6*c - 3*a^4*b*c^2 + 3*a^3*b^2*c^2 + 3*a^2*b^3*c^2 - 3*a*b^4*c^2 + 3*a^4*c^3 - 6*a^2*b^2*c^3 + 3*b^4*c^3 - 2*a^3*c^4 + a^2*b*c^4 + a*b^2*c^4 - 2*b^3*c^4 + a^2*c^5 + b^2*c^5 + a*c^6 + b*c^6 - 2*c^7)*(a^7 - 2*a^6*b + 3*a^4*b^3 - 2*a^3*b^4 + a^2*b^5 + a*b^6 - 2*b^7 + a^6*c - 3*a^4*b^2*c + a^2*b^4*c + b^6*c - a^5*c^2 + 2*a^4*b*c^2 + 3*a^3*b^2*c^2 - 6*a^2*b^3*c^2 + a*b^4*c^2 + b^5*c^2 - a^4*c^3 + 3*a^2*b^2*c^3 - 2*b^4*c^3 - a^3*c^4 + 2*a^2*b*c^4 - 3*a*b^2*c^4 + 3*b^3*c^4 - a^2*c^5 + a*c^6 - 2*b*c^6 + c^7) : :

X(53928) lies on the circumcircle and these lines: {20, 2692}, {22, 53942}, {23, 9088}, {30, 32704}, {74, 32475}, {99, 3007}, {107, 7478}, {108, 37919}, {110, 2390}, {112, 7481}, {404, 2766}, {523, 2370}, {858, 9059}, {935, 4234}, {1290, 35998}, {1293, 2071}, {1304, 7419}, {2222, 4296}, {2373, 44435}, {2689, 38514}, {2690, 36024}, {3153, 26713}, {4247, 10423}, {4248, 22239}, {7428, 53965}, {7434, 40118}, {7444, 32710}, {7446, 53922}, {7447, 53921}, {7448, 40119}, {7459, 53956}, {38941, 53182}

X(53928) = reflection of X(2370) in the Euler line
X(53928) = cevapoint of X(3) and X(2842)

---

X(53929) = REFLECTION OF X(2373) IN EULER LINE

Barycentrics    a^2*(a^8 - 2*a^4*b^4 + b^8 - 2*a^6*c^2 + 2*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - 2*b^6*c^2 + a^4*c^4 - 4*a^2*b^2*c^4 + b^4*c^4 + 2*a^2*c^6 + 2*b^2*c^6 - 2*c^8)*(a^8 - 2*a^6*b^2 + a^4*b^4 + 2*a^2*b^6 - 2*b^8 + 2*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + 2*b^6*c^2 - 2*a^4*c^4 + 2*a^2*b^2*c^4 + b^4*c^4 - 2*b^2*c^6 + c^8) : :

X(53929) lies on the circumcircle and these lines: {2, 935}, {3, 10098}, {20, 2696}, {22, 691}, {23, 112}, {25, 10423}, {30, 30247}, {74, 30209}, {98, 47258}, {99, 858}, {107, 7426}, {110, 2393}, {111, 2485}, {186, 39382}, {403, 30251}, {468, 1289}, {476, 7493}, {523, 2373}, {925, 16387}, {1290, 26253}, {1294, 46620}, {1296, 2071}, {1300, 36166}, {1301, 37962}, {1302, 47188}, {1304, 1995}, {1370, 53895}, {2374, 36168}, {2689, 26254}, {2690, 26252}, {2693, 46594}, {2715, 6800}, {2766, 4239}, {3153, 44061}, {3565, 37929}, {4220, 10101}, {4232, 22239}, {4580, 9076}, {6236, 10296}, {7417, 40118}, {7418, 32710}, {7423, 53921}, {7432, 53922}, {7449, 53965}, {7458, 53956}, {7495, 52998}, {9832, 22456}, {10298, 32229}, {10420, 26283}, {11413, 53961}, {15398, 39413}, {16386, 20187}, {26255, 53944}, {26284, 53923}, {26706, 37959}, {37777, 39417}, {40049, 53695}, {40119, 46589}, {44420, 53691}, {47324, 53692}

X(53929) = reflection of X(10098) in X(3)
X(53929) = reflection of X(2373) in the Euler line
X(53929) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(38971)
X(53929) = isotomic conjugate of the anticomplement of X(44467)
X(53929) = X(656)-isoconjugate of X(46619)
X(53929) = X(40596)-Dao conjugate of X(46619)
X(53929) = cevapoint of X(3) and X(2854)
X(53929) = trilinear pole of line {6, 9517}
X(53929) = barycentric quotient X(112)/X(46619)

---

X(53930) = REFLECTION OF X(2383) IN EULER LINE

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(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 - 3*a^10*c^2 + 8*a^8*b^2*c^2 - 5*a^6*b^4*c^2 - 5*a^4*b^6*c^2 + 8*a^2*b^8*c^2 - 3*b^10*c^2 + 2*a^8*c^4 - 6*a^6*b^2*c^4 + 2*a^4*b^4*c^4 - 6*a^2*b^6*c^4 + 2*b^8*c^4 + 2*a^6*c^6 + 5*a^4*b^2*c^6 + 5*a^2*b^4*c^6 + 2*b^6*c^6 - 3*a^4*c^8 - 4*a^2*b^2*c^8 - 3*b^4*c^8 + a^2*c^10 + b^2*c^10)*(a^12 - 3*a^10*b^2 + 2*a^8*b^4 + 2*a^6*b^6 - 3*a^4*b^8 + a^2*b^10 - 4*a^10*c^2 + 8*a^8*b^2*c^2 - 6*a^6*b^4*c^2 + 5*a^4*b^6*c^2 - 4*a^2*b^8*c^2 + b^10*c^2 + 7*a^8*c^4 - 5*a^6*b^2*c^4 + 2*a^4*b^4*c^4 + 5*a^2*b^6*c^4 - 3*b^8*c^4 - 8*a^6*c^6 - 5*a^4*b^2*c^6 - 6*a^2*b^4*c^6 + 2*b^6*c^6 + 7*a^4*c^8 + 8*a^2*b^2*c^8 + 2*b^4*c^8 - 4*a^2*c^10 - 3*b^2*c^10 + c^12) : :

X(53930) lies on the circumcircle and these lines: {3, 53960}, {4, 1291}, {5, 10420}, {24, 52998}, {30, 20185}, {74, 20184}, {110, 539}, {112, 231}, {186, 930}, {403, 933}, {476, 3518}, {523, 2383}, {691, 7576}, {925, 2070}, {1300, 23290}, {3153, 13398}, {3542, 53962}, {6240, 13863}, {7488, 53953}, {10295, 53884}, {13595, 16167}, {13619, 33639}, {20626, 37951}, {30248, 37970}, {35489, 44066}, {35921, 53895}, {37760, 53958}

X(53930) = reflection of X(i) in X(j) for these {i,j}: {4, 53989}, {53960, 3}
X(53930) = reflection of X(2383) in the Euler line
X(53930) = polar-circle-inverse of X(45180)
X(53930) = Collings transform of X(53989)
X(53930) = X(63)-isoconjugate of X(47226)
X(53930) = X(3162)-Dao conjugate of X(47226)
X(53930) = trilinear pole of line {6, 52742}
X(53930) = barycentric quotient X(25)/X(47226)

---

X(53931) = REFLECTION OF X(53692) IN EULER LINE

Barycentrics    (a^12 - 3*a^8*b^4 + 4*a^6*b^6 - 3*a^4*b^8 + b^12 - 3*a^10*c^2 + 2*a^8*b^2*c^2 + a^6*b^4*c^2 + a^4*b^6*c^2 + 2*a^2*b^8*c^2 - 3*b^10*c^2 + 4*a^8*c^4 - 2*a^6*b^2*c^4 - 2*a^4*b^4*c^4 - 2*a^2*b^6*c^4 + 4*b^8*c^4 - 4*a^6*c^6 + a^4*b^2*c^6 + a^2*b^4*c^6 - 4*b^6*c^6 + 3*a^4*c^8 + 3*b^4*c^8 - a^2*c^10 - b^2*c^10)*(a^12 - 3*a^10*b^2 + 4*a^8*b^4 - 4*a^6*b^6 + 3*a^4*b^8 - a^2*b^10 + 2*a^8*b^2*c^2 - 2*a^6*b^4*c^2 + a^4*b^6*c^2 - b^10*c^2 - 3*a^8*c^4 + a^6*b^2*c^4 - 2*a^4*b^4*c^4 + a^2*b^6*c^4 + 3*b^8*c^4 + 4*a^6*c^6 + a^4*b^2*c^6 - 2*a^2*b^4*c^6 - 4*b^6*c^6 - 3*a^4*c^8 + 2*a^2*b^2*c^8 + 4*b^4*c^8 - 3*b^2*c^10 + c^12) : :

X(53931) lies on the circumcircle and these lines: {3, 53692}, {30, 2715}, {74, 2799}, {98, 41079}, {107, 34473}, {110, 2794}, {112, 38749}, {297, 1304}, {476, 37183}, {523, 2710}, {542, 53187}, {690, 53188}, {1301, 36191}, {2781, 9160}, {2790, 39447}, {9161, 9517}, {9530, 53972}, {38526, 53699}, {51228, 53691}

X(53931) = reflection of X(53692) in X(3)
X(53931) = reflection of X(2710) in the Euler line

---

X(53932) = REFLECTION OF X(2716) IN EULER LINE

Barycentrics    a*(a^9 - a^8*b - 2*a^7*b^2 + 2*a^6*b^3 + 2*a^3*b^6 - 2*a^2*b^7 - a*b^8 + b^9 - a^8*c + 4*a^7*b*c - a^6*b^2*c - 4*a^5*b^3*c + 4*a^4*b^4*c - 4*a^3*b^5*c - a^2*b^6*c + 4*a*b^7*c - b^8*c - 2*a^7*c^2 - 2*a^6*b*c^2 + 9*a^5*b^2*c^2 - 5*a^4*b^3*c^2 - 5*a^3*b^4*c^2 + 9*a^2*b^5*c^2 - 2*a*b^6*c^2 - 2*b^7*c^2 + 3*a^6*c^3 - 6*a^5*b*c^3 - 2*a^4*b^2*c^3 + 10*a^3*b^3*c^3 - 2*a^2*b^4*c^3 - 6*a*b^5*c^3 + 3*b^6*c^3 + 6*a^4*b*c^4 - 5*a^3*b^2*c^4 - 5*a^2*b^3*c^4 + 6*a*b^4*c^4 - 3*a^4*c^5 + 2*a^2*b^2*c^5 - 3*b^4*c^5 + 2*a^3*c^6 - 2*a^2*b*c^6 - 2*a*b^2*c^6 + 2*b^3*c^6 + a^2*c^7 + 2*a*b*c^7 + b^2*c^7 - a*c^8 - b*c^8)*(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 + 4*a^7*b*c - 2*a^6*b^2*c - 6*a^5*b^3*c + 6*a^4*b^4*c - 2*a^2*b^6*c + 2*a*b^7*c - b^8*c - 2*a^7*c^2 - a^6*b*c^2 + 9*a^5*b^2*c^2 - 2*a^4*b^3*c^2 - 5*a^3*b^4*c^2 + 2*a^2*b^5*c^2 - 2*a*b^6*c^2 + b^7*c^2 + 2*a^6*c^3 - 4*a^5*b*c^3 - 5*a^4*b^2*c^3 + 10*a^3*b^3*c^3 - 5*a^2*b^4*c^3 + 2*b^6*c^3 + 4*a^4*b*c^4 - 5*a^3*b^2*c^4 - 2*a^2*b^3*c^4 + 6*a*b^4*c^4 - 3*b^5*c^4 - 4*a^3*b*c^5 + 9*a^2*b^2*c^5 - 6*a*b^3*c^5 + 2*a^3*c^6 - a^2*b*c^6 - 2*a*b^2*c^6 + 3*b^3*c^6 - 2*a^2*c^7 + 4*a*b*c^7 - 2*b^2*c^7 - a*c^8 - b*c^8 + c^9) : :

X(53932) lies on the circumcircle and these lines: {3, 53927}, {30, 2222}, {74, 3738}, {100, 13532}, {108, 10058}, {109, 12515}, {110, 2800}, {523, 2716}, {953, 46618}, {1290, 6909}, {1304, 17515}, {1793, 53697}, {2766, 37305}, {6011, 24466}, {7427, 53933}, {14127, 43655}, {26700, 38602}, {36067, 47185}, {37449, 53941}

X(53932) = reflection of X(53927) in X(3)
X(53932) = reflection of X(2716) in the Euler line

---

X(53933) = REFLECTION OF X(2726) IN EULER LINE

Barycentrics    a^2*(a^6 - a^4*b^2 - a^2*b^4 + 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^3*b*c^2 + 2*a*b^3*c^2 - b^4*c^2 - 2*a^2*b*c^3 - 2*a*b^2*c^3 + 2*a^2*c^4 - 4*a*b*c^4 + 2*b^2*c^4 + 2*a*c^5 + 2*b*c^5 - 2*c^6)*(a^6 - 2*a^5*b - a^4*b^2 + 2*a^2*b^4 + 2*a*b^5 - 2*b^6 + 2*a^4*b*c + 2*a^3*b^2*c - 2*a^2*b^3*c - 4*a*b^4*c + 2*b^5*c - a^4*c^2 - 2*a*b^3*c^2 + 2*b^4*c^2 + 2*a*b^2*c^3 - a^2*c^4 + 2*a*b*c^4 - b^2*c^4 - 2*b*c^5 + c^6) : :

X(53933) lies on the circumcircle and these lines: {2, 53611}, {23, 901}, {25, 53612}, {30, 2737}, {74, 2821}, {98, 26275}, {99, 3109}, {104, 14419}, {109, 24436}, {110, 2810}, {111, 3310}, {468, 1309}, {523, 2726}, {691, 859}, {935, 37168}, {1290, 33849}, {2691, 6905}, {7427, 53932}, {8691, 11712}, {36067, 37777}

X(53933) = reflection of X(2726) in the Euler line

---

X(53934) = REFLECTION OF X(5897) IN EULER LINE

Barycentrics    (a^14 + 2*a^12*b^2 - 12*a^10*b^4 + 9*a^8*b^6 + 9*a^6*b^8 - 12*a^4*b^10 + 2*a^2*b^12 + b^14 - 4*a^12*c^2 + 8*a^10*b^2*c^2 + 20*a^8*b^4*c^2 - 48*a^6*b^6*c^2 + 20*a^4*b^8*c^2 + 8*a^2*b^10*c^2 - 4*b^12*c^2 + 5*a^10*c^4 - 29*a^8*b^2*c^4 + 24*a^6*b^4*c^4 + 24*a^4*b^6*c^4 - 29*a^2*b^8*c^4 + 5*b^10*c^4 + 20*a^6*b^2*c^6 - 40*a^4*b^4*c^6 + 20*a^2*b^6*c^6 - 5*a^6*c^8 + 4*a^4*b^2*c^8 + 4*a^2*b^4*c^8 - 5*b^6*c^8 + 4*a^4*c^10 - 4*a^2*b^2*c^10 + 4*b^4*c^10 - a^2*c^12 - b^2*c^12)*(a^14 - 4*a^12*b^2 + 5*a^10*b^4 - 5*a^6*b^8 + 4*a^4*b^10 - a^2*b^12 + 2*a^12*c^2 + 8*a^10*b^2*c^2 - 29*a^8*b^4*c^2 + 20*a^6*b^6*c^2 + 4*a^4*b^8*c^2 - 4*a^2*b^10*c^2 - b^12*c^2 - 12*a^10*c^4 + 20*a^8*b^2*c^4 + 24*a^6*b^4*c^4 - 40*a^4*b^6*c^4 + 4*a^2*b^8*c^4 + 4*b^10*c^4 + 9*a^8*c^6 - 48*a^6*b^2*c^6 + 24*a^4*b^4*c^6 + 20*a^2*b^6*c^6 - 5*b^8*c^6 + 9*a^6*c^8 + 20*a^4*b^2*c^8 - 29*a^2*b^4*c^8 - 12*a^4*c^10 + 8*a^2*b^2*c^10 + 5*b^4*c^10 + 2*a^2*c^12 - 4*b^2*c^12 + c^14) : :

X(53934) lies on the circumcircle and these lines: {3, 22239}, {20, 1304}, {22, 53944}, {30, 1301}, {74, 8057}, {107, 2071}, {110, 15311}, {111, 46620}, {186, 30249}, {376, 10423}, {476, 11413}, {523, 5897}, {550, 53962}, {858, 9064}, {935, 21312}, {1289, 7464}, {1299, 36164}, {1302, 37929}, {1370, 9060}, {2766, 30267}, {2770, 46594}, {6080, 12111}, {6526, 16177}, {10295, 39417}, {10420, 30552}, {12225, 16166}, {18859, 20626}, {20427, 46968}, {30268, 53965}, {36001, 40097}, {36067, 36158}, {40120, 50401}, {46585, 53924}

X(53934) = reflection of X(22239) in X(3)
X(53934) = reflection of X(5897) in the Euler line
X(53934) = cevapoint of X(3) and X(2777)
X(53934) = trilinear pole of line {6, 14345}

---

X(53935) = REFLECTION OF X(5966) IN EULER LINE

Barycentrics    (a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 2*a^8*c^2 + 5*a^6*b^2*c^2 + 5*a^2*b^6*c^2 - 2*b^8*c^2 - 7*a^4*b^2*c^4 - 7*a^2*b^4*c^4 + 2*a^4*c^6 + 6*a^2*b^2*c^6 + 2*b^4*c^6 - a^2*c^8 - b^2*c^8)*(a^10 - 2*a^8*b^2 + 2*a^4*b^6 - a^2*b^8 - 3*a^8*c^2 + 5*a^6*b^2*c^2 - 7*a^4*b^4*c^2 + 6*a^2*b^6*c^2 - b^8*c^2 + 2*a^6*c^4 - 7*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 5*a^2*b^2*c^6 - 3*a^2*c^8 - 2*b^2*c^8 + c^10) : :

X(53935) lies on the circumcircle and these lines: {2, 1291}, {5, 691}, {22, 53960}, {23, 930}, {25, 52998}, {30, 45151}, {74, 32478}, {99, 2070}, {110, 5965}, {111, 12077}, {112, 37943}, {427, 13863}, {468, 933}, {476, 13595}, {523, 5966}, {827, 10096}, {858, 20185}, {935, 3518}, {1141, 51479}, {1287, 13621}, {2696, 35921}, {3153, 3565}, {5189, 33639}, {5994, 51274}, {5995, 51267}, {6353, 53962}, {7417, 53945}, {7488, 53895}, {7576, 10098}, {10409, 52193}, {10410, 52194}, {12074, 44264}, {16806, 22998}, {16807, 22997}, {20063, 44066}, {20626, 37777}, {21284, 30248}, {21308, 53951}, {29011, 36166}, {39183, 43657}, {44239, 53961}

X(53935) = reflection of X(5966) in the Euler line
X(53935) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(46439)
X(53935) = trilinear pole of line {6, 45147}

---

X(53936) = REFLECTION OF X(6011) IN EULER LINE

Barycentrics    a*(a - b)*(a - c)*(a^6 - a^5*b - a^4*b^2 + 2*a^3*b^3 - a^2*b^4 - a*b^5 + b^6 - a^4*b*c + 2*a^2*b^3*c - b^5*c - a^4*c^2 - a^3*b*c^2 + 3*a^2*b^2*c^2 + 2*a*b^3*c^2 - b^4*c^2 - a^2*b*c^3 + 2*b^3*c^3 - a^2*c^4 - a*b*c^4 - b^2*c^4 - b*c^5 + c^6)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^5*c - a^4*b*c - a^3*b^2*c - a^2*b^3*c - a*b^4*c - b^5*c - a^4*c^2 + 3*a^2*b^2*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 - b^2*c^4 - a*c^5 - b*c^5 + c^6) : :

X(53936) lies on the circumcircle and these lines: {3, 12030}, {30, 759}, {74, 758}, {104, 36001}, {105, 37959}, {107, 37964}, {110, 6003}, {186, 39439}, {476, 13589}, {523, 6011}, {915, 37979}, {953, 51717}, {1304, 4242}, {2071, 39435}, {2687, 3651}, {2694, 30267}, {2696, 46593}, {2716, 30264}, {2752, 4220}, {4231, 53956}, {4239, 53943}, {7414, 53921}, {7464, 53903}, {7477, 53684}, {22239, 46588}, {35056, 53283}, {36031, 53683}, {49127, 53920}

X(53936) = reflection of X(12030) in X(3)
X(53936) = reflection of X(6011) in the Euler line
X(53936) = cevapoint of X(55) and X(2610)

---

X(53937) = REFLECTION OF X(6037) IN EULER LINE

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(2*a^8*b^2 - 4*a^6*b^4 + 3*a^4*b^6 - 2*a^2*b^8 + b^10 + a^8*c^2 - a^6*b^2*c^2 - a^4*b^4*c^2 + 3*a^2*b^6*c^2 - 2*b^8*c^2 - a^6*c^4 + 2*a^4*b^2*c^4 - a^2*b^4*c^4 + 3*b^6*c^4 - a^4*c^6 - a^2*b^2*c^6 - 4*b^4*c^6 + a^2*c^8 + 2*b^2*c^8)*(a^8*b^2 - a^6*b^4 - a^4*b^6 + a^2*b^8 + 2*a^8*c^2 - a^6*b^2*c^2 + 2*a^4*b^4*c^2 - a^2*b^6*c^2 + 2*b^8*c^2 - 4*a^6*c^4 - a^4*b^2*c^4 - a^2*b^4*c^4 - 4*b^6*c^4 + 3*a^4*c^6 + 3*a^2*b^2*c^6 + 3*b^4*c^6 - 2*a^2*c^8 - 2*b^2*c^8 + c^10) : :

X(53937) lies on the circumcircle and these lines: {6, 53700}, {74, 37991}, {98, 1316}, {99, 42743}, {237, 842}, {401, 2697}, {419, 40118}, {477, 11676}, {523, 6037}, {729, 9408}, {805, 7468}, {1297, 37918}, {1576, 53691}, {2715, 3288}, {3233, 9150}, {3569, 26714}, {4226, 53603}, {4230, 53699}, {7473, 22456}, {26717, 39689}, {37937, 53708}

X(53937) = reflection of X(6037) in the Euler line
X(53937) = Collings transform of X(36183)
X(53937) = cevapoint of X(523) and X(36183)
X(53937) = trilinear pole of line {6, 6785}

---

X(53938) = REFLECTION OF X(6325) IN EULER LINE

Barycentrics    a^2*(2*a^8 - 4*a^4*b^4 + 2*b^8 - 3*a^6*c^2 + 2*a^4*b^2*c^2 + 2*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 + 3*a^2*c^6 + 3*b^2*c^6 - 4*c^8)*(2*a^8 - 3*a^6*b^2 + 2*a^4*b^4 + 3*a^2*b^6 - 4*b^8 + 2*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + 3*b^6*c^2 - 4*a^4*c^4 + 2*a^2*b^2*c^4 + 2*b^4*c^4 - 3*b^2*c^6 + 2*c^8) : :

X(53938) lies on the circumcircle and these lines: {2, 53950}, {3, 32229}, {23, 11636}, {30, 6236}, {99, 10989}, {110, 8705}, {112, 37969}, {353, 2715}, {476, 11628}, {523, 6325}, {691, 7492}, {935, 5094}, {1296, 37950}, {2696, 3534}, {7418, 53954}, {9830, 20404}, {9831, 20403}, {10098, 35473}, {13241, 53793}, {13530, 36166}, {15080, 32694}, {35499, 38338}

X(53938) = reflection of X(32229) in X(3)
X(53938) = reflection of X(6325) in the Euler line
X(53938) = Thomson-isogonal conjugate of X(32228)
X(53938) = X(16568)-isoconjugate of X(52974)
X(53938) = trilinear pole of line {6, 14699}
X(53938) = barycentric quotient X(3455)/X(52974)

---

X(53939) = REFLECTION OF X(9056) IN EULER LINE

Barycentrics    a^2*(a - b)*(a - c)*(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 + a^4*b^2*c + a^2*b^4*c - b^6*c + 2*a^5*c^2 - a^4*b*c^2 + 3*a^3*b^2*c^2 + 3*a^2*b^3*c^2 - a*b^4*c^2 + 2*b^5*c^2 + a^4*c^3 + 3*a^3*b*c^3 + 3*a*b^3*c^3 + b^4*c^3 - 7*a^3*c^4 - 4*a^2*b*c^4 - 4*a*b^2*c^4 - 7*b^3*c^4 + a^2*c^5 - 3*a*b*c^5 + b^2*c^5 + 4*a*c^6 + 4*b*c^6 - c^7)*(a^7 - a^6*b + 2*a^5*b^2 + a^4*b^3 - 7*a^3*b^4 + a^2*b^5 + 4*a*b^6 - b^7 + a^6*c - a^4*b^2*c + 3*a^3*b^3*c - 4*a^2*b^4*c - 3*a*b^5*c + 4*b^6*c - a^5*c^2 + a^4*b*c^2 + 3*a^3*b^2*c^2 - 4*a*b^4*c^2 + b^5*c^2 - a^4*c^3 + 3*a^2*b^2*c^3 + 3*a*b^3*c^3 - 7*b^4*c^3 - a^3*c^4 + a^2*b*c^4 - a*b^2*c^4 + b^3*c^4 - a^2*c^5 + 2*b^2*c^5 + a*c^6 - b*c^6 + c^7) : :

X(53939) lies on the circumcircle and these lines: {2, 2695}, {23, 102}, {30, 53916}, {99, 50403}, {110, 8999}, {111, 8607}, {468, 32706}, {477, 7413}, {523, 9056}, {691, 7450}, {841, 7421}, {842, 7449}, {858, 41904}, {935, 7452}, {1302, 47798}, {1311, 7426}, {1995, 53926}, {2687, 4224}, {2688, 50404}, {2691, 7451}, {2696, 7462}, {2697, 26254}, {7460, 53189}, {7461, 10101}, {7463, 10098}

X(53939) = reflection of X(9056) in the Euler line
X(53939) = trilinear pole of line {6, 2779}

---

X(53940) = REFLECTION OF X(9057) IN EULER LINE

Barycentrics    a^2*(a - b)*(a - c)*(a^6 + 2*a^4*b^2 - 3*a^3*b^3 - 4*a^2*b^4 + 3*a*b^5 + b^6 + 3*a^3*b^2*c - 3*a^2*b^3*c - 3*a*b^4*c + 3*b^5*c - a^4*c^2 + 6*a^2*b^2*c^2 - 3*a*b^3*c^2 - 4*b^4*c^2 + 3*a*b^2*c^3 - 3*b^3*c^3 - a^2*c^4 + 2*b^2*c^4 + c^6)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 + 2*a^4*c^2 + 3*a^3*b*c^2 + 6*a^2*b^2*c^2 + 3*a*b^3*c^2 + 2*b^4*c^2 - 3*a^3*c^3 - 3*a^2*b*c^3 - 3*a*b^2*c^3 - 3*b^3*c^3 - 4*a^2*c^4 - 3*a*b*c^4 - 4*b^2*c^4 + 3*a*c^5 + 3*b*c^5 + c^6) : :

X(53940) lies on the circumcircle and these lines: {2, 2688}, {23, 103}, {25, 53922}, {30, 53908}, {99, 7479}, {110, 9000}, {111, 8608}, {468, 917}, {477, 6998}, {523, 9057}, {675, 7426}, {691, 4243}, {841, 7430}, {842, 7453}, {858, 41905}, {935, 4241}, {1292, 36031}, {1296, 36032}, {1302, 47771}, {1995, 53190}, {2687, 4223}, {2691, 7437}, {2694, 36018}, {2695, 36007}, {2696, 4237}, {2697, 26252}, {4232, 53947}, {4249, 10098}, {4250, 10101}, {7469, 53707}, {9085, 37962}, {26708, 37760}, {36026, 43660}, {46595, 53189}

X(53940) = reflection of X(9057) in the Euler line
X(53940) = trilinear pole of line {6, 2772}

---

X(53941) = REFLECTION OF X(9058) IN EULER LINE

Barycentrics    a*(a - b)*(a - c)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 + 3*a^4*b*c + 3*a^3*b^2*c + 3*a^2*b^3*c + 3*a*b^4*c - a^4*c^2 - 3*a^3*b*c^2 - 3*a*b^3*c^2 - b^4*c^2 - 3*a^2*b*c^3 - 3*a*b^2*c^3 - a^2*c^4 + 3*a*b*c^4 - b^2*c^4 + c^6)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 + 3*a^4*b*c - 3*a^3*b^2*c - 3*a^2*b^3*c + 3*a*b^4*c - a^4*c^2 + 3*a^3*b*c^2 - 3*a*b^3*c^2 - b^4*c^2 + 3*a^2*b*c^3 - 3*a*b^2*c^3 - a^2*c^4 + 3*a*b*c^4 - b^2*c^4 + c^6) : :

X(53941) lies on the circumcircle and these lines: {2, 2687}, {22, 2694}, {23, 104}, {25, 53921}, {30, 53907}, {74, 37959}, {99, 7477}, {105, 7426}, {110, 9001}, {111, 8609}, {112, 37966}, {468, 915}, {477, 4220}, {523, 9058}, {691, 3658}, {759, 7469}, {841, 3651}, {842, 4239}, {858, 1295}, {935, 4246}, {1292, 36167}, {1296, 7475}, {1302, 47804}, {1325, 53903}, {1995, 2752}, {2071, 53917}, {2688, 7465}, {2691, 13589}, {2695, 35996}, {2696, 4236}, {2697, 26253}, {2718, 5322}, {4228, 12030}, {4231, 32710}, {4232, 53956}, {4238, 10098}, {4242, 10101}, {7435, 10423}, {7437, 53189}, {7438, 40118}, {7466, 53922}, {7476, 30247}, {7493, 53964}, {15344, 37962}, {26255, 53943}, {26703, 37980}, {26706, 37964}, {26707, 37760}, {33849, 43655}, {36001, 43660}, {36031, 44876}, {37449, 53932}, {37919, 53904}, {37963, 39439}, {37965, 39382}

X(53941) = reflection of X(9058) in the Euler line
X(53941) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(42422)
X(53941) = X(656)-isoconjugate of X(37961)
X(53941) = X(40596)-Dao conjugate of X(37961)
X(53941) = trilinear pole of line {6, 2771}
X(53941) = barycentric quotient X(112)/X(37961)

---

X(53942) = REFLECTION OF X(9059) IN EULER LINE

Barycentrics    a^2*(a - b)*(a - c)*(a^4 + 2*a^3*b + 2*a^2*b^2 + 2*a*b^3 + 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 - 3*c^4)*(a^4 - 2*a^3*b + a^2*b^2 + a*b^3 - 3*b^4 + 2*a^3*c - 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*a*c^3 - 2*b*c^3 + c^4) : :

X(53942) lies on the circumcircle and these lines: {2, 2758}, {22, 53928}, {23, 106}, {30, 53904}, {99, 44435}, {100, 48350}, {110, 9002}, {111, 8610}, {468, 40101}, {523, 9059}, {759, 7292}, {858, 2370}, {935, 46541}, {1995, 53946}, {2071, 53913}, {2372, 37764}, {2373, 3007}, {2687, 19649}, {2718, 7191}, {2726, 26230}, {2729, 38941}, {5211, 39445}, {6012, 36167}, {7426, 9083}, {37959, 53905}

X(53942) = reflection of X(9059) in the Euler line
X(53942) = trilinear pole of line {6, 2842}

---

X(53943) = REFLECTION OF X(9061) IN EULER LINE

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 - 2*a^6*c - a^5*b*c + 4*a^4*b^2*c - 2*a^3*b^3*c + 4*a^2*b^4*c - a*b^5*c - 2*b^6*c + a^5*c^2 + a^3*b^2*c^2 + a^2*b^3*c^2 + b^5*c^2 + a^3*b*c^3 - 8*a^2*b^2*c^3 + a*b^3*c^3 - a^3*c^4 - b^3*c^4 + 2*a^2*c^5 + 2*a*b*c^5 + 2*b^2*c^5 - a*c^6 - b*c^6)*(a^7 - 2*a^6*b + a^5*b^2 - a^3*b^4 + 2*a^2*b^5 - a*b^6 - a^6*c - a^5*b*c + a^3*b^3*c + 2*a*b^5*c - b^6*c + a^5*c^2 + 4*a^4*b*c^2 + a^3*b^2*c^2 - 8*a^2*b^3*c^2 + 2*b^5*c^2 - a^4*c^3 - 2*a^3*b*c^3 + a^2*b^2*c^3 + a*b^3*c^3 - b^4*c^3 - a^3*c^4 + 4*a^2*b*c^4 + a^2*c^5 - a*b*c^5 + b^2*c^5 - a*c^6 - 2*b*c^6 + c^7) : :

X(53943) lies on the circumcircle and these lines: {2, 2691}, {21, 2696}, {23, 1292}, {25, 10101}, {28, 10098}, {30, 53901}, {99, 7469}, {100, 7426}, {108, 37962}, {110, 9004}, {112, 37963}, {468, 26706}, {477, 7423}, {523, 9061}, {691, 4228}, {841, 7425}, {842, 7458}, {935, 4233}, {1290, 1995}, {1296, 1325}, {2074, 30247}, {2687, 46586}, {2766, 4232}, {4223, 53189}, {4239, 53936}, {7477, 53698}, {7493, 53952}, {13397, 37980}, {16049, 53961}, {26255, 53941}, {26712, 37760}, {30256, 37960}, {30257, 37959}, {43660, 50402}

X(53943) = reflection of X(9061) in the Euler line
X(53943) = trilinear pole of line {6, 2775}

---

X(53944) = REFLECTION OF X(9064) IN EULER LINE

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8 + 3*a^6*b^2 - 8*a^4*b^4 + 3*a^2*b^6 + b^8 - 2*a^6*c^2 + 6*a^4*b^2*c^2 + 6*a^2*b^4*c^2 - 2*b^6*c^2 - 11*a^2*b^2*c^4 + 2*a^2*c^6 + 2*b^2*c^6 - c^8)*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 + 3*a^6*c^2 + 6*a^4*b^2*c^2 - 11*a^2*b^4*c^2 + 2*b^6*c^2 - 8*a^4*c^4 + 6*a^2*b^2*c^4 + 3*a^2*c^6 - 2*b^2*c^6 + c^8) : :

X(53944) lies on the circumcircle and these lines: {2, 2693}, {4, 841}, {22, 53934}, {23, 1294}, {25, 477}, {30, 53909}, {74, 468}, {98, 37962}, {99, 7480}, {104, 37963}, {110, 9007}, {111, 1990}, {112, 9209}, {186, 43660}, {523, 9064}, {691, 4240}, {842, 4232}, {858, 5897}, {935, 46587}, {1292, 37966}, {1295, 7469}, {1296, 7473}, {1297, 7426}, {1300, 37777}, {1325, 53917}, {1995, 2697}, {2073, 53908}, {2074, 53907}, {2075, 53916}, {2409, 10098}, {2687, 4233}, {2691, 4246}, {2694, 4228}, {2696, 4230}, {3563, 47148}, {3565, 7471}, {4226, 53961}, {4241, 53189}, {6353, 32710}, {7435, 10101}, {7468, 20187}, {7476, 53901}, {7478, 53913}, {7479, 53906}, {7482, 30256}, {13595, 53959}, {15329, 53895}, {18401, 37760}, {26255, 53929}, {30247, 37937}, {30257, 37964}, {34168, 37980}, {37929, 39434}, {42426, 53914}

X(53944) = reflection of X(4) in X(53984)
X(53944) = reflection of X(9064) in the Euler line
X(53944) = polar-circle-inverse of X(46436)
X(53944) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(18809)
X(53944) = Collings transform of X(i) for these i: {37984, 53984}
X(53944) = X(i)-isoconjugate of X(j) for these (i,j): {656, 7464}, {14208, 40114}
X(53944) = X(40596)-Dao conjugate of X(7464)
X(53944) = cevapoint of X(523) and X(37984)
X(53944) = trilinear pole of line {6, 2777}
X(53944) = barycentric product X(648)*X(10293)
X(53944) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 7464}, {9064, 52497}, {10293, 525}

---

X(53945) = REFLECTION OF X(9076) IN EULER LINE

Barycentrics    a^2*(a^8 - 2*a^4*b^4 + b^8 - a^6*c^2 - b^6*c^2 + a^4*c^4 + b^4*c^4 + a^2*c^6 + b^2*c^6 - 2*c^8)*(a^8 - a^6*b^2 + a^4*b^4 + a^2*b^6 - 2*b^8 + b^6*c^2 - 2*a^4*c^4 + b^4*c^4 - b^2*c^6 + c^8) : :

X(53945) lies on the circumcircle and these lines: {2, 1287}, {22, 11635}, {23, 827}, {30, 44061}, {99, 5189}, {110, 9019}, {112, 21284}, {427, 935}, {476, 7495}, {523, 9076}, {550, 2696}, {689, 40074}, {691, 6636}, {858, 23318}, {1141, 36166}, {1296, 18859}, {1995, 16166}, {3520, 10098}, {7417, 53935}, {7418, 14979}, {7426, 53957}, {13619, 30247}, {26712, 37959}, {37970, 39382}, {52397, 53895}

X(53945) = reflection of X(9076) in the Euler line

---

X(53946) = REFLECTION OF X(9083) IN EULER LINE

Barycentrics    a^2*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 3*a^5*c + 3*a^4*b*c + 3*a*b^4*c - 3*b^5*c - a^4*c^2 + 3*a^3*b*c^2 + 3*a*b^3*c^2 - b^4*c^2 - 3*a^2*b*c^3 - 3*a*b^2*c^3 + 2*a^2*c^4 - 6*a*b*c^4 + 2*b^2*c^4 + 3*a*c^5 + 3*b*c^5 - 2*c^6)*(a^6 - 3*a^5*b - a^4*b^2 + 2*a^2*b^4 + 3*a*b^5 - 2*b^6 + 3*a^4*b*c + 3*a^3*b^2*c - 3*a^2*b^3*c - 6*a*b^4*c + 3*b^5*c - a^4*c^2 - 3*a*b^3*c^2 + 2*b^4*c^2 + 3*a*b^2*c^3 - a^2*c^4 + 3*a*b*c^4 - b^2*c^4 - 3*b*c^5 + c^6) : :

X(53946) lies on the circumcircle and these lines: {2, 2692}, {23, 1293}, {99, 7478}, {110, 9026}, {404, 2691}, {468, 32704}, {477, 7434}, {523, 9083}, {691, 7419}, {841, 7444}, {842, 7448}, {935, 4248}, {1292, 37919}, {1296, 7481}, {1995, 53942}, {2687, 7459}, {2696, 4234}, {2748, 4420}, {4222, 10101}, {4245, 53189}, {4247, 10098}, {7292, 26700}, {7426, 9059}, {7469, 34594}, {9088, 37962}, {26713, 37760}

X(53946) = reflection of X(9083) in the Euler line
X(53946) = trilinear pole of line {6, 2776}

---

X(53947) = REFLECTION OF X(9085) IN EULER LINE

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^7 - 3*a^5*b^2 + 2*a^4*b^3 + 2*a^3*b^4 - 3*a^2*b^5 + b^7 - a^6*c + a^4*b^2*c + a^2*b^4*c - b^6*c + a^3*b^2*c^2 + a^2*b^3*c^2 - 2*a^2*b^2*c^3 - a^3*c^4 - b^3*c^4 + a^2*c^5 + b^2*c^5)*(a^7 - a^6*b - a^3*b^4 + a^2*b^5 - 3*a^5*c^2 + a^4*b*c^2 + a^3*b^2*c^2 - 2*a^2*b^3*c^2 + b^5*c^2 + 2*a^4*c^3 + a^2*b^2*c^3 - b^4*c^3 + 2*a^3*c^4 + a^2*b*c^4 - 3*a^2*c^5 - b*c^6 + c^7) : :

X(53947) lies on the circumcircle and these lines: {2, 53925}, {4, 53189}, {23, 1305}, {25, 2690}, {27, 691}, {30, 53906}, {99, 2073}, {101, 468}, {110, 9028}, {111, 7649}, {112, 3011}, {186, 44876}, {306, 29241}, {523, 9085}, {1290, 7466}, {2691, 4219}, {2693, 7433}, {2694, 7445}, {2696, 7431}, {2697, 7432}, {3565, 5196}, {4184, 53895}, {4229, 53961}, {4232, 53940}, {7465, 53952}, {7474, 10420}, {9057, 37962}, {26705, 37777}, {37166, 53909}, {37963, 53683}

X(53947) = reflection of X(9085) in the Euler line
X(53947) = X(656)-isoconjugate of X(36032)
X(53947) = X(40596)-Dao conjugate of X(36032)
X(53947) = barycentric quotient X(112)/X(36032)

---

X(53948) = REFLECTION OF X(9085) IN EULER LINE

Barycentrics    a*(a - b)*(a - c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(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 + 3*a^5*b*c + a^4*b^2*c - 6*a^3*b^3*c + a^2*b^4*c + 3*a*b^5*c - b^6*c - a^5*c^2 - a^4*b*c^2 + 6*a^3*b^2*c^2 + 6*a^2*b^3*c^2 - a*b^4*c^2 - b^5*c^2 + a^4*c^3 - 6*a^2*b^2*c^3 + b^4*c^3 - a^3*c^4 - a^2*b*c^4 - a*b^2*c^4 - b^3*c^4 + a^2*c^5 - 3*a*b*c^5 + b^2*c^5 + a*c^6 + b*c^6 - c^7)*(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 + 3*a^5*b*c - a^4*b^2*c - a^2*b^4*c - 3*a*b^5*c + b^6*c - a^5*c^2 + a^4*b*c^2 + 6*a^3*b^2*c^2 - 6*a^2*b^3*c^2 - a*b^4*c^2 + b^5*c^2 - a^4*c^3 - 6*a^3*b*c^3 + 6*a^2*b^2*c^3 - b^4*c^3 - a^3*c^4 + a^2*b*c^4 - a*b^2*c^4 + b^3*c^4 - a^2*c^5 + 3*a*b*c^5 - b^2*c^5 + a*c^6 - b*c^6 + c^7) : :

X(53948) lies on the circumcircle and these lines: {2, 2694}, {23, 1295}, {25, 2687}, {30, 53917}, {99, 37966}, {104, 468}, {105, 37962}, {110, 9051}, {111, 14571}, {112, 40134}, {186, 53907}, {477, 4231}, {523, 9107}, {691, 4246}, {759, 37963}, {841, 7414}, {842, 7438}, {915, 37777}, {935, 7435}, {1292, 37964}, {1294, 37959}, {1296, 7476}, {1995, 53964}, {2074, 53903}, {2688, 7466}, {2691, 4242}, {2693, 4220}, {2695, 35973}, {2696, 4238}, {2697, 4239}, {2752, 4232}, {3565, 7477}, {3658, 53895}, {4233, 12030}, {4236, 53961}, {4244, 10098}, {4250, 53189}, {6353, 53921}, {7426, 26703}, {7469, 39435}, {7475, 20187}, {10101, 46588}, {30247, 37965}, {33849, 53870}, {36001, 53909}, {36031, 53906}, {37919, 53913}, {37979, 43660}

X(53948) = reflection of X(9107) in the Euler line
X(53948) = X(656)-isoconjugate of X(37960)
X(53948) = X(40596)-Dao conjugate of X(37960)
X(53948) = trilinear pole of line {6, 2778}
X(53948) = barycentric quotient X(112)/X(37960)

---

X(53949) = REFLECTION OF X(11635) IN EULER LINE

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^2*b^2*c^2 - b^4*c^2 - b^2*c^4 + c^6)*(a^6 + b^6 - a^4*c^2 - a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6) : :

X(53949) lies on the circumcircle and these lines: {2, 53983}, {5, 3563}, {20, 29011}, {22, 9076}, {74, 41464}, {98, 7488}, {523, 11635}, {733, 22240}, {842, 3153}, {858, 23318}, {933, 4226}, {1297, 12225}, {1299, 7576}, {1300, 35921}, {2070, 40118}, {2374, 13595}, {3518, 40120}, {4230, 20626}, {7468, 52998}, {7473, 53962}, {10096, 23096}, {13621, 53963}, {37760, 40119}, {39193, 53895}

X(53949) = anticomplement of X(53983)
X(53949) = reflection of X(11635) in the Euler line
X(53949) = Collings transform of X(6676)
X(53949) = cevapoint of X(i) and X(j) for these (i,j): {3, 826}, {523, 6676}
X(53949) = trilinear pole of line {6, 8280}

---

X(53950) = REFLECTION OF X(11635) IN EULER LINE

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(2*a^6 + a^4*b^2 + a^2*b^4 + 2*b^6 - 2*a^4*c^2 - 2*b^4*c^2 - 2*a^2*c^4 - 2*b^2*c^4 + 2*c^6)*(2*a^6 - 2*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 - 2*b^2*c^4 + 2*c^6) : :

X(53950) lies on the circumcircle and these lines: {2, 53938}, {3, 53955}, {23, 6325}, {30, 14388}, {74, 11645}, {98, 7575}, {110, 3906}, {111, 37907}, {381, 842}, {523, 11636}, {691, 23288}, {1297, 10296}, {2697, 10298}, {2770, 14002}, {6323, 11646}, {7468, 53693}, {7472, 33638}, {7954, 47289}, {10989, 52105}, {11178, 53973}, {30247, 30716}, {40118, 47485}, {53603, 53738}

X(53950) = reflection of X(i) in X(j) for these {i,j}: {10989, 52105}, {53955, 3}
X(53950) = reflection of X(11636) in the Euler line
X(53950) = Collings transform of X(i) for these i: {7426, 17436, 36904, 52105}
X(53950) = X(656)-isoconjugate of X(37969)
X(53950) = X(40596)-Dao conjugate of X(37969)
X(53950) = cevapoint of X(i) and X(j) for these (i,j): {523, 7426}, {3906, 52105}
X(53950) = trilinear pole of line {6, 6032}
X(53950) = barycentric quotient X(112)/X(37969)

---

X(53951) = REFLECTION OF X(12074) IN EULER LINE

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(a^6 + 5*a^4*b^2 + 5*a^2*b^4 + b^6 - a^4*c^2 - 9*a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 + 5*a^4*c^2 - 9*a^2*b^2*c^2 - b^4*c^2 + 5*a^2*c^4 - b^2*c^4 + c^6) : :

X(53951) lies on the circumcircle and these lines: {30, 53890}, {74, 19924}, {98, 37924}, {110, 12073}, {111, 37909}, {523, 12074}, {549, 842}, {691, 35345}, {827, 47291}, {1287, 53274}, {1291, 52630}, {2709, 35358}, {2770, 16042}, {5966, 44264}, {12150, 38582}, {21308, 53935}, {32710, 35484}, {40118, 52294}

X(53951) = reflection of X(12074) in the Euler line
X(53951) = Collings transform of X(10989)
X(53951) = cevapoint of X(523) and X(10989)
X(53951) = trilinear pole of line {6, 7693}

---

X(53952) = REFLECTION OF X(53921) IN X(3)

Barycentrics    a*(a - b)*(a - 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 + 4*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)*(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 + 4*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(53952) lies on the circumcircle and these lines: {2, 53956}, {3, 53921}, {20, 2687}, {22, 2752}, {23, 15344}, {30, 915}, {74, 912}, {104, 2071}, {105, 858}, {107, 7477}, {108, 36167}, {110, 15313}, {112, 7475}, {477, 30267}, {523, 13397}, {935, 4236}, {1071, 43078}, {1289, 7476}, {1295, 16386}, {1299, 37979}, {1300, 36001}, {1301, 37966}, {1304, 3658}, {1325, 39439}, {1370, 53964}, {2373, 3263}, {2694, 11413}, {2758, 35998}, {2766, 13589}, {2770, 26253}, {3153, 26707}, {3563, 37959}, {3651, 32710}, {4220, 40118}, {4238, 10423}, {4239, 40119}, {4246, 22239}, {7411, 53922}, {7414, 53924}, {7451, 53965}, {7465, 53947}, {7493, 53943}, {9061, 37980}, {12030, 16049}, {26703, 37929}, {26705, 36031}, {32849, 43659}, {37919, 40101}, {37964, 40097}, {37965, 39417}, {40049, 53684}

X(53952) = reflection of X(53921) in X(3)
X(53952) = anticomplement of X(53988)
X(53952) = reflection of X(13397) in the Euler line
X(53952) = isotomic conjugate of the anticomplement of X(47235)
X(53952) = Collings transform of X(18455)
X(53952) = X(7649)-isoconjugate of X(51632)
X(53952) = cevapoint of X(i) and X(j) for these (i,j): {3, 8674}, {513, 18455}
X(53952) = barycentric quotient X(i)/X(j) for these {i,j}: {906, 51632}, {47235, 53988}

---

X(53953) = REFLECTION OF X(13398) IN EULER LINE

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(a^10 - 2*a^8*b^2 + a^6*b^4 + a^4*b^6 - 2*a^2*b^8 + b^10 - 3*a^8*c^2 + 5*a^6*b^2*c^2 + 5*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 7*a^4*b^2*c^4 - 7*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 7*a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10)*(a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 2*a^8*c^2 + 5*a^6*b^2*c^2 - 7*a^4*b^4*c^2 + 7*a^2*b^6*c^2 - 3*b^8*c^2 + a^6*c^4 - 7*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(53953) lies on the circumcircle and these lines: {2, 16178}, {3, 53924}, {20, 32710}, {22, 40118}, {23, 40120}, {30, 1299}, {74, 16386}, {98, 37929}, {107, 40049}, {111, 16387}, {186, 39437}, {477, 11413}, {523, 13398}, {842, 1370}, {858, 3563}, {925, 39193}, {1289, 7468}, {1300, 2071}, {1301, 7471}, {1304, 30512}, {2374, 37980}, {2383, 3153}, {2693, 30552}, {2770, 26283}, {4226, 10423}, {7472, 39382}, {7473, 39417}, {7477, 40097}, {7480, 30249}, {7482, 30251}, {7488, 53930}, {7493, 40119}, {10257, 16172}, {12225, 14979}, {15329, 22239}, {16049, 53921}, {16237, 53923}, {45781, 52504}

X(53953) = reflection of X(i) in X(j) for these {i,j}: {16172, 10257}, {53924, 3}
X(53953) = anticomplement of X(16178)
X(53953) = reflection of X(13398) in the Euler line
X(53953) = isotomic conjugate of the anticomplement of X(47236)
X(53953) = Collings transform of X(10257)
X(53953) = X(656)-isoconjugate of X(37951)
X(53953) = X(40596)-Dao conjugate of X(37951)
X(53953) = cevapoint of X(523) and X(10257)
X(53953) = trilinear pole of line {6, 12900}
X(53953) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 37951}, {2420, 20771}, {47236, 16178}

---

X(53954) = REFLECTION OF X(13398) IN EULER LINE

Barycentrics    a^2*(2*a^10 - 6*a^8*b^2 + 4*a^6*b^4 + 4*a^4*b^6 - 6*a^2*b^8 + 2*b^10 - 3*a^8*c^2 + 7*a^6*b^2*c^2 - 8*a^4*b^4*c^2 + 7*a^2*b^6*c^2 - 3*b^8*c^2 + a^6*c^4 + 4*a^4*b^2*c^4 + 4*a^2*b^4*c^4 + b^6*c^4 - 5*a^4*c^6 - 14*a^2*b^2*c^6 - 5*b^4*c^6 + 9*a^2*c^8 + 9*b^2*c^8 - 4*c^10)*(2*a^10 - 3*a^8*b^2 + a^6*b^4 - 5*a^4*b^6 + 9*a^2*b^8 - 4*b^10 - 6*a^8*c^2 + 7*a^6*b^2*c^2 + 4*a^4*b^4*c^2 - 14*a^2*b^6*c^2 + 9*b^8*c^2 + 4*a^6*c^4 - 8*a^4*b^2*c^4 + 4*a^2*b^4*c^4 - 5*b^6*c^4 + 4*a^4*c^6 + 7*a^2*b^2*c^6 + b^4*c^6 - 6*a^2*c^8 - 3*b^2*c^8 + 2*c^10) : :

X(53954) lies on the circumcircle and these lines: {4, 46438}, {30, 53693}, {99, 52149}, {110, 3581}, {112, 16328}, {381, 476}, {523, 13530}, {925, 10296}, {1302, 37907}, {1304, 39239}, {2715, 41414}, {6325, 36166}, {7418, 53938}, {7422, 53955}, {7464, 33638}, {9060, 14002}, {10298, 10420}, {14388, 50401}, {32694, 41330}

X(53954) = reflection of X(4) in X(46438)
X(53954) = reflection of X(13530) in the Euler line
X(53954) = Collings transform of X(46438)

---

X(53955) = REFLECTION OF X(53950) IN EULER LINE

Barycentrics    (2*a^10 + a^8*b^2 - 3*a^6*b^4 - 3*a^4*b^6 + a^2*b^8 + 2*b^10 - 4*a^8*c^2 + 3*a^6*b^2*c^2 + 3*a^2*b^6*c^2 - 4*b^8*c^2 + 4*a^4*c^6 - 2*a^2*b^2*c^6 + 4*b^4*c^6 - 2*a^2*c^8 - 2*b^2*c^8)*(2*a^10 - 4*a^8*b^2 + 4*a^4*b^6 - 2*a^2*b^8 + a^8*c^2 + 3*a^6*b^2*c^2 - 2*a^2*b^6*c^2 - 2*b^8*c^2 - 3*a^6*c^4 + 4*b^6*c^4 - 3*a^4*c^6 + 3*a^2*b^2*c^6 + a^2*c^8 - 4*b^2*c^8 + 2*c^10) : :

X(53955) lies on the circumcircle and these lines: {3, 53950}, {30, 11636}, {74, 3906}, {99, 37950}, {107, 37969}, {110, 10989}, {112, 41358}, {376, 32229}, {476, 7492}, {523, 14388}, {691, 3534}, {842, 53267}, {935, 35473}, {1304, 5094}, {6236, 7464}, {7422, 53954}, {9060, 47596}, {10423, 35480}, {11648, 44972}, {13530, 50401}, {36166, 43656}

X(53955) = reflection of X(53950) in X(3)
X(53955) = reflection of X(14388) in the Euler line

---

X(53956) = REFLECTION OF X(15344) IN EULER LINE

Barycentrics    a*(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^5*c - 2*a^4*b*c + 3*a^3*b^2*c + 3*a^2*b^3*c - 2*a*b^4*c - b^5*c + 2*a^2*b^2*c^2 - a^2*b*c^3 - a*b^2*c^3 - a^2*c^4 - b^2*c^4 + a*c^5 + b*c^5)*(a^6 - a^5*b - a^2*b^4 + a*b^5 - 2*a^4*b*c - a^2*b^3*c + b^5*c - a^4*c^2 + 3*a^3*b*c^2 + 2*a^2*b^2*c^2 - a*b^3*c^2 - b^4*c^2 + 3*a^2*b*c^3 - a^2*c^4 - 2*a*b*c^4 - b*c^5 + c^6) : :

X(53956) lies on the circumcircle and these lines: {2, 53952}, {4, 2691}, {21, 53895}, {23, 13397}, {24, 10101}, {25, 1290}, {28, 691}, {99, 2074}, {100, 468}, {108, 37777}, {110, 34381}, {111, 6591}, {112, 3290}, {186, 1292}, {403, 26706}, {476, 4233}, {523, 15344}, {925, 7469}, {935, 30733}, {1294, 50402}, {1296, 37961}, {1325, 3565}, {2693, 7425}, {2694, 7427}, {2696, 4227}, {2697, 7423}, {2766, 6353}, {4221, 53961}, {4223, 53925}, {4228, 10420}, {4231, 53936}, {4232, 53941}, {7458, 53929}, {7459, 53928}, {7476, 53698}, {9058, 37962}, {10295, 53901}, {20187, 37960}, {20336, 35574}, {26712, 37943}, {30257, 37979}, {36009, 53189}, {37959, 44065}, {46586, 53964}, {46618, 53917}

X(53956) = reflection of X(15344) in the Euler line
X(53956) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(53988)
X(53956) = X(i)-isoconjugate of X(j) for these (i,j): {63, 47231}, {656, 7475}, {1818, 34173}
X(53956) = X(i)-Dao conjugate of X(j) for these (i,j): {3162, 47231}, {40596, 7475}
X(53956) = cevapoint of X(25) and X(47232)
X(53956) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 47231}, {112, 7475}, {8751, 34173}, {40084, 25083}

---

X(53957) = REFLECTION OF X(16166) IN EULER LINE

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(a^4 + 3*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 + 3*a^2*c^2 - 2*b^2*c^2 + c^4) : :
X(53957) = 3 X[381] - X[48682]

X(53957) lies on the circumcircle and these lines: {2, 29011}, {4, 48683}, {5, 74}, {98, 13595}, {110, 32193}, {381, 48682}, {476, 1624}, {477, 2070}, {523, 16166}, {648, 33640}, {755, 3767}, {842, 37760}, {930, 15329}, {933, 4240}, {1141, 11815}, {1291, 7471}, {1294, 7488}, {1300, 3518}, {1304, 14480}, {1576, 12092}, {1995, 9076}, {2373, 26235}, {2693, 3153}, {4226, 53884}, {5897, 12225}, {7426, 53945}, {7480, 52998}, {10096, 14979}, {13597, 18369}, {20185, 30512}, {20189, 23181}, {20626, 46587}, {30551, 33643}, {31510, 53962}, {32710, 37943}, {35921, 43660}, {36829, 43351}, {40049, 53960}, {44239, 53909}

X(53957) = midpoint of X(4) and X(48683)
X(53957) = reflection of X(16166) in the Euler line
X(53957) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(44953)
X(53957) = Collings transform of X(i) for these i: {546, 52869}
X(53957) = X(i)-isoconjugate of X(j) for these (i,j): {656, 3520}, {2616, 11591}
X(53957) = X(40596)-Dao conjugate of X(3520)
X(53957) = cevapoint of X(523) and X(546)
X(53957) = trilinear pole of line {6, 382}
X(53957) = barycentric product X(i)*X(j) for these {i,j}: {5, 30527}, {648, 3521}, {4558, 15424}, {11815, 14570}
X(53957) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 3520}, {1625, 11591}, {3521, 525}, {11815, 15412}, {15424, 14618}, {30527, 95}

---

X(53958) = REFLECTION OF X(16167) IN EULER LINE

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^6 + a^4*b^2 - 5*a^2*b^4 + 3*b^6 - a^4*c^2 + 2*a^2*b^2*c^2 - 5*b^4*c^2 - a^2*c^4 + b^2*c^4 + c^6)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - 5*a^2*c^4 - 5*b^2*c^4 + 3*c^6) : :

X(53958) lies on the circumcircle and these lines: {2, 131}, {20, 43660}, {22, 74}, {23, 32710}, {25, 1299}, {98, 7493}, {99, 30512}, {107, 16237}, {111, 22240}, {112, 15329}, {468, 53924}, {477, 858}, {523, 16167}, {691, 40049}, {841, 2071}, {842, 37980}, {915, 4228}, {917, 7474}, {935, 7471}, {1289, 4240}, {1294, 1370}, {1297, 26283}, {1304, 52915}, {1995, 3563}, {2373, 35520}, {2374, 26255}, {2383, 13595}, {2409, 30251}, {2693, 37929}, {2697, 16387}, {2855, 6563}, {3658, 26706}, {4226, 30247}, {4230, 39382}, {4232, 40120}, {4558, 10420}, {6353, 39437}, {7426, 40118}, {7468, 10098}, {7469, 53921}, {7477, 10101}, {7480, 10423}, {10313, 32730}, {10565, 45138}, {11413, 53909}, {13398, 23181}, {16049, 53907}, {20480, 30744}, {32734, 46969}, {37760, 53930}, {39417, 46587}

X(53958) = anticomplement of X(53993)
X(53958) = reflection of X(16167) in the Euler line
X(53958) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(131)
X(53958) = Collings transform of X(i) for these i: {5158, 15760}
X(53958) = X(i)-isoconjugate of X(j) for these (i,j): {656, 18533}, {661, 37645}, {24006, 47391}
X(53958) = X(i)-Dao conjugate of X(j) for these (i,j): {36830, 37645}, {40596, 18533}
X(53958) = cevapoint of X(i) and X(j) for these (i,j): {3, 8675}, {512, 5158}, {523, 15760}
X(53958) = trilinear pole of line {6, 4550}
X(53958) = barycentric product X(i)*X(j) for these {i,j}: {648, 34801}, {4558, 52487}
X(53958) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 37645}, {112, 18533}, {32661, 47391}, {32681, 40387}, {32738, 52165}, {34801, 525}, {52487, 14618}

---

X(53959) = REFLECTION OF X(18401) IN EULER LINE

Barycentrics    (a^14 - a^12*b^2 - 3*a^10*b^4 + 3*a^8*b^6 + 3*a^6*b^8 - 3*a^4*b^10 - a^2*b^12 + b^14 - 4*a^12*c^2 + 5*a^10*b^2*c^2 + 2*a^8*b^4*c^2 - 6*a^6*b^6*c^2 + 2*a^4*b^8*c^2 + 5*a^2*b^10*c^2 - 4*b^12*c^2 + 5*a^10*c^4 - 8*a^8*b^2*c^4 + 3*a^6*b^4*c^4 + 3*a^4*b^6*c^4 - 8*a^2*b^8*c^4 + 5*b^10*c^4 + 5*a^6*b^2*c^6 - 4*a^4*b^4*c^6 + 5*a^2*b^6*c^6 - 5*a^6*c^8 - 2*a^4*b^2*c^8 - 2*a^2*b^4*c^8 - 5*b^6*c^8 + 4*a^4*c^10 + 2*a^2*b^2*c^10 + 4*b^4*c^10 - a^2*c^12 - b^2*c^12)*(a^14 - 4*a^12*b^2 + 5*a^10*b^4 - 5*a^6*b^8 + 4*a^4*b^10 - a^2*b^12 - a^12*c^2 + 5*a^10*b^2*c^2 - 8*a^8*b^4*c^2 + 5*a^6*b^6*c^2 - 2*a^4*b^8*c^2 + 2*a^2*b^10*c^2 - b^12*c^2 - 3*a^10*c^4 + 2*a^8*b^2*c^4 + 3*a^6*b^4*c^4 - 4*a^4*b^6*c^4 - 2*a^2*b^8*c^4 + 4*b^10*c^4 + 3*a^8*c^6 - 6*a^6*b^2*c^6 + 3*a^4*b^4*c^6 + 5*a^2*b^6*c^6 - 5*b^8*c^6 + 3*a^6*c^8 + 2*a^4*b^2*c^8 - 8*a^2*b^4*c^8 - 3*a^4*c^10 + 5*a^2*b^2*c^10 + 5*b^4*c^10 - a^2*c^12 - 4*b^2*c^12 + c^14) : :

X(53959) lies on the circumcircle and these lines: {3, 52998}, {4, 46664}, {5, 1304}, {20, 1291}, {30, 933}, {74, 6368}, {107, 2070}, {110, 1568}, {112, 9380}, {186, 18284}, {476, 7488}, {523, 18401}, {550, 13863}, {691, 44239}, {930, 2071}, {935, 35921}, {1301, 37943}, {3518, 22239}, {6799, 13619}, {7576, 10423}, {9064, 37760}, {10420, 12225}, {11413, 53960}, {13595, 53944}, {14980, 46966}, {16166, 18279}, {16386, 20185}, {18859, 30248}, {44057, 52403}

X(53959) = reflection of X(i) in X(j) for these {i,j}: {4, 46664}, {52403, 44057}, {52998, 3}
X(53959) = isogonal conjugate of X(10628)
X(53959) = reflection of X(18401) in the Euler line
X(53959) = Collings transform of X(46664)
X(53959) = X(1)-isoconjugate of X(10628)
X(53959) = cevapoint of X(3) and X(32423)
X(53959) = trilinear pole of line {6, 14391}
X(53959) = barycentric quotient X(6)/X(10628)

---

X(53960) = REFLECTION OF X(18401) IN EULER LINE

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 + 7*a^6*b^2*c^2 - 2*a^4*b^4*c^2 + 7*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 8*a^4*b^2*c^4 - 8*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 7*a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10)*(a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 + 7*a^6*b^2*c^2 - 8*a^4*b^4*c^2 + 7*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 2*a^4*b^2*c^4 - 8*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 7*a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10) : :

X(53960) lies on the circumcircle and these lines: {2, 23096}, {3, 53930}, {20, 14979}, {22, 53935}, {23, 53963}, {30, 2383}, {74, 539}, {110, 20184}, {523, 20185}, {550, 32710}, {842, 52397}, {858, 5966}, {1141, 2071}, {1299, 13619}, {1300, 18859}, {3153, 33643}, {3520, 53924}, {3563, 5189}, {6636, 40118}, {7495, 40119}, {11413, 53959}, {16166, 30512}, {16386, 18401}, {21284, 40120}, {37970, 39437}, {40049, 53957}, {43657, 44450}, {46590, 52998}

X(53960) = reflection of X(53930) in X(3)
X(53960) = anticomplement of X(53989)
X(53960) = reflection of X(20185) in the Euler line
X(53960) = cevapoint of X(3) and X(45147)

---

X(53961) = REFLECTION OF X(20187) IN EULER LINE

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^8 - 2*a^4*b^4 + b^8 - 7*a^6*c^2 + 11*a^4*b^2*c^2 + 11*a^2*b^4*c^2 - 7*b^6*c^2 + 5*a^4*c^4 - 26*a^2*b^2*c^4 + 5*b^4*c^4 + 7*a^2*c^6 + 7*b^2*c^6 - 6*c^8)*(a^8 - 7*a^6*b^2 + 5*a^4*b^4 + 7*a^2*b^6 - 6*b^8 + 11*a^4*b^2*c^2 - 26*a^2*b^4*c^2 + 7*b^6*c^2 - 2*a^4*c^4 + 11*a^2*b^2*c^4 + 5*b^4*c^4 - 7*b^2*c^6 + c^8) : :

X(53961) lies on the circumcircle and these lines: {3, 40119}, {20, 2770}, {22, 10102}, {30, 2374}, {74, 8681}, {107, 7472}, {110, 20186}, {111, 2071}, {376, 40118}, {523, 20187}, {842, 21312}, {858, 9084}, {1301, 7482}, {1304, 11634}, {2373, 16386}, {3563, 7464}, {4221, 53956}, {4226, 53944}, {4229, 53947}, {4235, 22239}, {4236, 53948}, {7468, 9064}, {7475, 9107}, {10295, 40120}, {11413, 53929}, {15344, 37960}, {16049, 53943}, {23096, 35921}, {30249, 46619}, {35485, 53924}, {44239, 53935}

X(53961) = reflection of X(40119) in X(3)
X(53961) = reflection of X(20187) in the Euler line
X(53961) = cevapoint of X(3) and X(2780)

---

X(53962) = REFLECTION OF X(20626) IN EULER LINE

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^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 - 4*a^4*b^4*c^2 + 3*a^2*b^6*c^2 - b^8*c^2 - 2*a^6*c^4 - 2*b^6*c^4 + 2*a^4*c^6 - 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 - 2*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 - a^2*b^6*c^2 + b^8*c^2 + 2*a^6*c^4 - 4*a^4*b^2*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 3*a^2*b^2*c^6 - 2*b^4*c^6 - 3*a^2*c^8 - b^2*c^8 + c^10) : :

X(53962) lies on the circumcircle and these lines: {4, 46664}, {24, 14979}, {25, 24977}, {74, 32349}, {107, 40596}, {112, 16040}, {186, 18401}, {403, 1141}, {427, 2697}, {477, 6240}, {523, 20626}, {550, 53934}, {827, 37937}, {1287, 2409}, {1291, 46591}, {1294, 13619}, {1297, 21284}, {2383, 37951}, {2693, 3520}, {3542, 53930}, {4230, 11635}, {5189, 34168}, {5897, 18859}, {5966, 37777}, {6353, 53935}, {7473, 53949}, {16166, 46587}, {26712, 37965}, {31510, 53957}, {37943, 39431}, {44061, 46619}, {52998, 53176}

X(53962) = reflection of X(20626) in the Euler line
X(53962) = polar-circle-inverse of X(46664)
X(53962) = X(656)-isoconjugate of X(3153)
X(53962) = X(i)-Dao conjugate of X(j) for these (i,j): {40596, 3153}, {46439, 46664}
X(53962) = cevapoint of X(647) and X(34397)
X(53962) = barycentric quotient X(112)/X(3153)

---

X(53963) = REFLECTION OF X(23096) IN EULER LINE

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8 - 5*a^6*b^2 + 8*a^4*b^4 - 5*a^2*b^6 + b^8 - a^6*c^2 + 4*a^4*b^2*c^2 + 4*a^2*b^4*c^2 - b^6*c^2 - a^4*c^4 - 6*a^2*b^2*c^4 - b^4*c^4 + a^2*c^6 + b^2*c^6)*(a^8 - a^6*b^2 - a^4*b^4 + a^2*b^6 - 5*a^6*c^2 + 4*a^4*b^2*c^2 - 6*a^2*b^4*c^2 + b^6*c^2 + 8*a^4*c^4 + 4*a^2*b^2*c^4 - b^4*c^4 - 5*a^2*c^6 - b^2*c^6 + c^8) : :

X(53963) lies on the circumcircle and these lines: {2, 20185}, {4, 31843}, {5, 3565}, {23, 53960}, {25, 930}, {99, 3518}, {110, 41588}, {427, 33639}, {428, 44066}, {468, 1291}, {523, 23096}, {691, 37943}, {925, 13595}, {933, 6353}, {1296, 7576}, {2070, 53895}, {10096, 11635}, {10420, 37760}, {13621, 53949}, {20187, 35921}, {32692, 40633}, {37777, 52998}

X(53963) = reflection of X(23096) in the Euler line
X(53963) = polar-circle-inverse of X(31843)
X(53963) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(53986)
X(53963) = trilinear pole of line {6, 20184}

---

X(53964) = REFLECTION OF X(26703) IN EULER LINE

Barycentrics    a*(a^8 - 2*a^4*b^4 + b^8 - a^7*c + a^4*b^3*c + a^3*b^4*c - b^7*c - a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 - b^6*c^2 + a^5*c^3 + a^4*b*c^3 - 2*a^3*b^2*c^3 - 2*a^2*b^3*c^3 + a*b^4*c^3 + b^5*c^3 - a^4*c^4 - b^4*c^4 + a^3*c^5 + b^3*c^5 + a^2*c^6 + b^2*c^6 - a*c^7 - b*c^7)*(a^8 - a^7*b - a^6*b^2 + a^5*b^3 - a^4*b^4 + a^3*b^5 + a^2*b^6 - a*b^7 + a^4*b^3*c - b^7*c + a^4*b^2*c^2 - 2*a^3*b^3*c^2 + b^6*c^2 + a^4*b*c^3 - 2*a^2*b^3*c^3 + b^5*c^3 - 2*a^4*c^4 + a^3*b*c^4 + a^2*b^2*c^4 + a*b^3*c^4 - b^4*c^4 + b^3*c^5 - b^2*c^6 - b*c^7 + c^8) : :

X(53964) lies on the circumcircle and these lines: {2, 2766}, {3, 10101}, {20, 2691}, {21, 935}, {22, 1290}, {23, 108}, {28, 10423}, {30, 26706}, {100, 858}, {101, 18669}, {107, 7469}, {110, 3827}, {111, 6588}, {112, 1325}, {468, 40097}, {523, 26703}, {691, 16049}, {693, 2373}, {1289, 2074}, {1292, 2071}, {1300, 50402}, {1301, 37963}, {1304, 4228}, {1370, 53952}, {1880, 32688}, {1995, 53948}, {2690, 36018}, {3153, 26712}, {4221, 10098}, {4224, 53965}, {4233, 22239}, {7423, 40118}, {7425, 32710}, {7426, 9107}, {7427, 53921}, {7445, 53922}, {7458, 40119}, {7493, 53941}, {9058, 37980}, {13397, 37929}, {30247, 37960}, {30250, 37959}, {36029, 53189}, {37449, 53612}, {37961, 39382}, {46586, 53956}

X(53964) = reflection of X(10101) in X(3)
X(53964) = reflection of X(26703) in the Euler line
X(53964) = isotomic conjugate of the anticomplement of X(47232)
X(53964) = X(656)-isoconjugate of X(37965)
X(53964) = X(40596)-Dao conjugate of X(37965)
X(53964) = cevapoint of X(3) and X(2836)
X(53964) = trilinear pole of line {6, 2850}
X(53964) = barycentric quotient X(112)/X(37965)

---

X(53965) = REFLECTION OF X(26704) IN EULER LINE

Barycentrics    a^2*(a - b)*(a - c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + 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 + 2*a^2*b^2*c - b^4*c + a^3*c^2 + b^3*c^2 + a*b*c^3 - 2*a*c^4 - 2*b*c^4 + c^5)*(a^5 - a^4*b + a^3*b^2 - 2*a*b^4 + b^5 + a^4*c + a*b^3*c - 2*b^4*c - 2*a^3*c^2 + 2*a^2*b*c^2 - 2*a^2*c^3 + b^2*c^3 + a*c^4 - b*c^4 + c^5) : :

X(53965) lies on the circumcircle and these lines: {4, 2695}, {25, 53926}, {30, 41904}, {101, 52310}, {102, 186}, {112, 6589}, {403, 32706}, {468, 1311}, {476, 7452}, {523, 26704}, {691, 7463}, {759, 1870}, {925, 50403}, {1290, 7461}, {2687, 37117}, {2693, 7421}, {2694, 6906}, {2697, 7413}, {2733, 21740}, {4224, 53964}, {7428, 53928}, {7449, 53929}, {7450, 10420}, {7451, 53952}, {7460, 53925}, {7462, 53895}, {10295, 53916}, {30268, 53934}, {33637, 37964}

X(53965) = reflection of X(26704) in the Euler line
X(53965) = X(i)-isoconjugate of X(j) for these (i,j): {521, 38945}, {656, 7424}
X(53965) = X(40596)-Dao conjugate of X(7424)
X(53965) = barycentric product X(653)*X(40081)
X(53965) = +barycentric quotient X(i)/X(j) for these {i,j}: {112, 7424}, {32674, 38945}, {40081, 6332}

---

X(53966) = REFLECTION OF X(699) IN BROCARD AXIS

Barycentrics    a^2*(a^6*b^2 + a^2*b^6 - a^6*c^2 - a^4*b^2*c^2 - a^2*b^4*c^2 - b^6*c^2 + 2*a^2*b^2*c^4)*(a^6*b^2 - a^6*c^2 + a^4*b^2*c^2 - 2*a^2*b^4*c^2 + a^2*b^2*c^4 - a^2*c^6 + b^2*c^6) : :

X(53966) lies on the circumcircle and these lines: {32, 805}, {76, 32544}, {98, 30217}, {99, 698}, {110, 3229}, {112, 16385}, {182, 53893}, {187, 25424}, {511, 30254}, {512, 699}, {691, 36821}, {2080, 39639}, {2709, 11842}, {2715, 32540}, {3398, 36517}, {9427, 41517}, {46970, 51248}

X(53966) = reflection of X(699) in the Brocard axis
X(53966) = Schoutte-circle-inverse of X(25424)
X(53966) = isogonal conjugate of the isotomic conjugate of X(53231)
X(53966) = barycentric product X(6)*X(53231)
X(53966) = barycentric quotient X(53231)/X(76)

---

X(53967) = REFLECTION OF X(727) IN BROCARD AXIS

Barycentrics    a^2*(a^4*b + a*b^4 - a^4*c - b^4*c - a^3*c^2 - b^3*c^2 + a^2*c^3 + b^2*c^3)*(a^4*b + a^3*b^2 - a^2*b^3 - a^4*c - b^3*c^2 + b^2*c^3 - a*c^4 + b*c^4) : :

X(53967) lies on the circumcircle and these lines: {10, 8709}, {32, 2702}, {58, 805}, {98, 28470}, {99, 726}, {100, 1580}, {101, 1691}, {110, 3009}, {172, 813}, {182, 2705}, {187, 43077}, {511, 28469}, {512, 727}, {691, 34476}, {1293, 2080}, {1428, 29055}, {2456, 28528}, {2703, 4279}, {12197, 53891}

X(53967) = reflection of X(727) in the Brocard axis
X(53967) = Schoutte-circle-inverse of X(43077)
X(53967) = isogonal conjugate of the isotomic conjugate of X(35165)
X(53967) = barycentric product X(6)*X(35165)
X(53967) = barycentric quotient X(35165)/X(76)

---

X(53968) = REFLECTION OF X(733) IN BROCARD AXIS

Barycentrics    a^2*(-2*a^4*b^4 + a^6*c^2 + b^6*c^2 + a^4*c^4 + b^4*c^4 - a^2*c^6 - b^2*c^6)*(a^6*b^2 + a^4*b^4 - a^2*b^6 - b^6*c^2 - 2*a^4*c^4 + b^4*c^4 + b^2*c^6) : :

X(53968) lies on the circumcircle and these lines: {3, 36517}, {32, 46970}, {39, 805}, {99, 732}, {110, 8623}, {111, 14318}, {112, 41429}, {187, 43357}, {512, 733}, {689, 3978}, {691, 12212}, {695, 783}, {813, 40936}, {827, 1691}, {1078, 53621}, {2080, 53885}, {2295, 36081}, {8870, 53701}, {14810, 53893}, {14822, 39397}, {39639, 47618}

X(53968) = reflection of X(36517) in X(3)
X(53968) = reflection of X(733) in the Brocard axis
X(53968) = Schoutte-circle-inverse of X(43357)

---

X(53969) = REFLECTION OF X(755) IN BROCARD AXIS

Barycentrics    a^2*(a^6 - 2*a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - 2*c^6)*(a^6 + a^4*b^2 + a^2*b^4 - 2*b^6 - 2*a^4*c^2 + b^4*c^2 - 2*a^2*c^4 + b^2*c^4 + c^6) : :
X(53969) = 4 X[187] - 3 X[52696]

a X(53969) lies on the circumcircle and these lines: {6, 24973}, {39, 691}, {98, 32473}, {99, 754}, {110, 2076}, {111, 3005}, {112, 44090}, {187, 827}, {511, 53885}, {512, 755}, {689, 3266}, {1287, 7747}, {1296, 47618}, {2709, 14810}, {2715, 41413}, {3098, 36517}, {5104, 43357}, {9301, 30255}, {12212, 32694}, {14658, 14898}, {40517, 43459}

X(53969) = reflection of X(755) in the Brocard axis
X(53969) = Schoutte-circle-inverse of X(827)
X(53969) = Collings transform of X(39079)
X(53969) = trilinear pole of line {6, 5113}

---

X(53970) = REFLECTION OF X(759) IN BROCARD AXIS

Barycentrics    a^2*(a + b)*(a + c)*(a^2*b^3 - b^5 + a^4*c - a^3*c^2 - a^2*b*c^2 + b^3*c^2 - a^2*c^3 + a*c^4)*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4 - a^2*b^2*c + a^2*c^3 + b^2*c^3 - c^5) : :

X(53970) lies on the circumcircle and these lines: {31, 36069}, {58, 2701}, {81, 1290}, {98, 6003}, {99, 758}, {100, 2651}, {101, 5060}, {109, 1326}, {110, 3724}, {181, 2222}, {284, 2702}, {511, 6011}, {512, 759}, {691, 9275}, {1308, 3110}, {2703, 4276}, {5006, 29044}, {15322, 50361}

X(53970) = reflection of X(759) in the Brocard axis
X(53970) = X(i)-isoconjugate of X(j) for these (i,j): {75, 5202}, {3936, 48449}
X(53970) = X(206)-Dao conjugate of X(5202)
X(53970) = cevapoint of X(6) and X(5202)
X(53970) = trilinear pole of line {6, 42741}
X(53970) = barycentric quotient X(32)/X(5202)

---

X(53971) = REFLECTION OF X(919) IN BROCARD AXIS

Barycentrics    a^2*(a - b)*(a - c)*(a^4 + b^4 - a^3*c - a^2*b*c - a*b^2*c - b^3*c + 2*a*b*c^2)*(a^4 - a^3*b - a^2*b*c + 2*a*b^2*c - a*b*c^2 - b*c^3 + c^4) : :

X(53971) lies on the circumcircle and these lines: {6, 53180}, {98, 43671}, {99, 918}, {100, 24290}, {105, 3125}, {110, 665}, {163, 741}, {187, 840}, {249, 36066}, {511, 28838}, {512, 919}, {813, 1110}, {927, 7178}, {1415, 53607}, {1691, 14665}, {2700, 13329}, {5009, 12031}

X(53971) = reflection of X(919) in the Brocard axis
X(53971) = Schoutte-circle-inverse of X(840)
X(53971) = X(i)-isoconjugate of X(j) for these (i,j): {75, 5098}, {1577, 3110}
X(53971) = X(206)-Dao conjugate of X(5098)
X(53971) = cevapoint of X(6) and X(5098)
X(53971) = trilinear pole of line {6, 24488}
X(53971) = barycentric product X(110)*X(43671)
X(53971) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 5098}, {1576, 3110}, {43671, 850}

---

X(53972) = REFLECTION OF X(1304) IN BROCARD AXIS

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^8 + a^6*b^2 - 4*a^4*b^4 + a^2*b^6 + b^8 - 3*a^6*c^2 + 5*a^4*b^2*c^2 + 5*a^2*b^4*c^2 - 3*b^6*c^2 - 11*a^2*b^2*c^4 + 5*a^2*c^6 + 5*b^2*c^6 - 3*c^8)*(a^8 - 3*a^6*b^2 + 5*a^2*b^6 - 3*b^8 + a^6*c^2 + 5*a^4*b^2*c^2 - 11*a^2*b^4*c^2 + 5*b^6*c^2 - 4*a^4*c^4 + 5*a^2*b^2*c^4 + a^2*c^6 - 3*b^2*c^6 + c^8) : :

X(53972) lies on the circumcircle and these lines: {74, 20975}, {98, 2777}, {99, 9033}, {110, 9409}, {111, 46253}, {112, 14398}, {511, 2693}, {512, 1304}, {1294, 37853}, {1297, 15055}, {1300, 5667}, {2790, 43654}, {2797, 53603}, {2848, 53692}, {5191, 53706}, {5896, 46639}, {9218, 10420}, {9530, 53931}, {15054, 53188}

X(53972) = reflection of X(1304) in the Brocard axis
X(53972) = X(i)-isoconjugate of X(j) for these (i,j): {75, 42654}, {162, 46459}
X(53972) = X(i)-Dao conjugate of X(j) for these (i,j): {125, 46459}, {206, 42654}
X(53972) = cevapoint of X(6) and X(42654)
X(53972) = trilinear pole of line {6, 5502}
X(53972) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 42654}, {647, 46459}

---

X(53973) = REFLECTION OF X(32694) IN BROCARD AXIS

Barycentrics    a^2*(2*a^8 - 3*a^6*b^2 + 2*a^4*b^4 - 3*a^2*b^6 + 2*b^8 - a^4*b^2*c^2 - a^2*b^4*c^2 - a^4*c^4 + 2*a^2*b^2*c^4 - b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 4*c^8)*(2*a^8 - a^4*b^4 + 3*a^2*b^6 - 4*b^8 - 3*a^6*c^2 - a^4*b^2*c^2 + 2*a^2*b^4*c^2 + 3*b^6*c^2 + 2*a^4*c^4 - a^2*b^2*c^4 - b^4*c^4 - 3*a^2*c^6 + 2*c^8) : :

X(53973) lies on the circumcircle and these lines: {3, 32694}, {98, 3906}, {99, 11645}, {110, 35002}, {111, 9210}, {112, 5104}, {476, 36194}, {511, 11636}, {512, 14388}, {574, 2715}, {691, 3098}, {1350, 13241}, {2709, 30270}, {6233, 18860}, {10546, 38613}, {11178, 53950}, {11674, 12507}, {31951, 47618}

X(53973) = reflection of X(32694) in X(3)
X(53973) = reflection of X(14388) in the Brocard axis

---

X(53974) = REFLECTION OF X(14659) IN BROCARD AXIS

Barycentrics    a^2*(a^4 - 6*a^2*b^2 + b^4 + 3*a^2*c^2 + 3*b^2*c^2 - 2*c^4)*(a^4 + 3*a^2*b^2 - 2*b^4 - 6*a^2*c^2 + 3*b^2*c^2 + c^4) : :

X(53974) lies on the circumcircle and these lines: {6, 3565}, {99, 193}, {100, 21874}, {110, 3053}, {111, 42663}, {112, 19118}, {187, 10425}, {512, 14659}, {691, 1692}, {827, 33632}, {925, 21463}, {1296, 1351}, {2709, 47113}, {3222, 47733}, {9737, 39639}, {28528, 37508}, {32220, 53895}, {35383, 53893}, {39383, 41410}, {39384, 41411}, {47735, 52454}

X(53974) = reflection of X(14659) in the Brocard axis
X(53974) = Schoutte-circle-inverse of X(10425)
X(53974) = X(662)-isoconjugate of X(45688)
X(53974) = X(1084)-Dao conjugate of X(45688)
X(53974) = trilinear pole of line {6, 8651}
X(53974) = barycentric quotient X(512)/X(45688)

---

X(53975) = REFLECTION OF X(17222) IN BROCARD AXIS

Barycentrics    a^2*(a^4 + 3*a^3*b - 4*a^2*b^2 + 3*a*b^3 + b^4 - 6*a^3*c + 3*a^2*b*c + 3*a*b^2*c - 6*b^3*c + 2*a^2*c^2 - 6*a*b*c^2 + 2*b^2*c^2 + 3*a*c^3 + 3*b*c^3 - 2*c^4)*(a^4 - 6*a^3*b + 2*a^2*b^2 + 3*a*b^3 - 2*b^4 + 3*a^3*c + 3*a^2*b*c - 6*a*b^2*c + 3*b^3*c - 4*a^2*c^2 + 3*a*b*c^2 + 2*b^2*c^2 + 3*a*c^3 - 6*b*c^3 + c^4) : :

X(53975) lies on the circumcircle and these lines: {6, 2705}, {58, 2709}, {98, 28296}, {99, 17132}, {101, 2030}, {111, 8643}, {187, 1293}, {511, 28295}, {512, 17222}, {691, 33628}, {1296, 1326}, {1384, 2702}, {1691, 28564}, {5104, 28469}, {5107, 28528}, {13241, 34476}, {28307, 51619}

X(53975) = reflection of X(17222) in the Brocard axis
X(53975) = Schoutte-circle-inverse of X(1293)

---

X(53976) = (name pending)

Barycentrics    (a - b - c) (a + b + c - x)^2*y^2*z^2 : :
where x = 2*(a + b + c)*(1 + Tan[A / 4]) / ((1 + Tan[B / 4])*(1 + Tan[C / 4])) , y and z are defined cyclically.

See Thanos Kalogerakis, Antreas Hatzipolakis and Peter Moses euclid 5919.

X(53976) lies on this line: {7, 1488}

---

X(53977) = X(2)X(18256)∩X(174)X(175)

Barycentrics    (a + b + c - x)*((a + b + c)*((a + b - c)*y + (a - b + c)*z) - 2*a*y*z) : :
where x = 2*(a + b + c)*(1 + Tan[A / 4]) / ((1 + Tan[B / 4])*(1 + Tan[C / 4])) , y and z are defined cyclically.

See Thanos Kalogerakis, Antreas Hatzipolakis and Peter Moses euclid 5919.

X(53977) lies on these lines: {2, 18256}, {174, 175}, {3621, 7057}

X(53977) = anticomplement of X(18256)
X(53977) = X(7)-Ceva conjugate of X(46891)
X(53977) = cevapoint of X(21455) and X(53078)
X(53977) = {X(174),X(1143)}-harmonic conjugate of X(46891)

---

X(53978) = (name pending)

Barycentrics    (a - b - c) (a + b + c - x)^2*y^2*z^2 : :
where x = 2*(a + b + c)*(1 + Cot[A / 4]) / ((1 + Cot[B / 4])*(1 + Cot[C / 4])), and y and z are defined cyclically.

See Thanos Kalogerakis, Antreas Hatzipolakis and Peter Moses euclid 5927.

X(53978) lies on this line: {7, 1488}

---

X(53979) = X(174)X(176)∩X(3621)X(7057)

Barycentrics    (a + b + c - x)*((a + b + c)*((a + b - c)*y + (a - b + c)*z) - 2*a*y*z) : :
where x = 2*(a + b + c)*(1 + Cot[A / 4]) / ((1 + Cot[B / 4])*(1 + Cot[C / 4])), and y and z are defined cyclically.

See Thanos Kalogerakis, Antreas Hatzipolakis and Peter Moses euclid 5927.

X(53979) lies on these lines: {174, 176}, {3621, 7057}

X(53979) = X(7)-Ceva conjugate of X(46892)
X(53979) = {X(174),X(1274)}-harmonic conjugate of X(46892)

---

X(53980) = POLAR-CIRCLE-INVERSE OF X(675)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a*b^2 - b^3 + a*c^2 - c^3)*(2*a^5 - a^4*b - b^5 - a^4*c + b^4*c + b*c^4 - c^5) : :

See Antreas Hatzipolakis and Peter Moses euclid 5924.

X(53980) lies on the nine-point-circle and these lines: {2, 26705}, {4, 675}, {11, 1827}, {25, 5190}, {115, 430}, {116, 427}, {123, 37315}, {125, 40954}, {127, 3136}, {440, 53822}, {1560, 47234}, {1566, 42073}, {33329, 38971}, {37330, 38977}, {38972, 51361}, {42426, 47224}

X(53980) = polar-circle-inverse of X(675)
X(53980) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(26705)

---

X(53981) = POLAR-CIRCLE-INVERSE OF X(733)

Barycentrics    a^2*(a^2 - b*c)*(a^2 + b*c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(b^2 + c^2)*(a^4*b^4 - b^8 + a^4*c^4 - c^8) : :

See Antreas Hatzipolakis and Peter Moses euclid 5924.

X(53981) lies on the nine-point-circle and these lines: {4, 733}, {39, 46654}, {115, 52462}, {125, 1194}, {127, 6656}, {232, 46669}, {427, 35971}, {1691, 2679}, {2967, 33330}, {5099, 46560}, {40379, 40938}

X(53981) = polar-circle-inverse of X(733)
X(53981) = complement of the isogonal conjugate of X(3852)
X(53981) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 3852}, {3852, 10}, {40876, 21235}
X(53981) = X(4)-Ceva conjugate of X(3852)

---

X(53982) = POLAR-CIRCLE-INVERSE OF X(759)

Barycentrics    (b + c)^2*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-a^2 + b^2 - b*c + c^2)*(-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) : :

See Antreas Hatzipolakis and Peter Moses euclid 5924.

X(53982) lies on the nine-point-circle and these lines: {2, 39435}, {4, 759}, {11, 429}, {36, 860}, {65, 125}, {115, 1826}, {122, 27687}, {123, 442}, {124, 1845}, {127, 37346}, {136, 225}, {431, 1842}, {451, 37816}, {1841, 5190}, {1861, 46670}, {5514, 50036}, {5520, 37982}, {20620, 42385}, {27553, 53850}, {30447, 46974}, {31845, 42768}, {37565, 38977}

X(53982) = midpoint of X(4) and X(30250)
X(53982) = complement of X(39435)
X(53982) = polar-circle-inverse of X(759)
X(53982) = X(i)-complementary conjugate of X(j) for these (i,j): {42, 2182}, {65, 50368}, {45272, 960}, {46588, 8062}
X(53982) = barycentric quotient X(46588)/X(37140)

---

X(53983) = POLAR-CIRCLE-INVERSE OF X(827)

Barycentrics    (b - c)^2*(b + c)^2*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(b^2 + c^2)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*c^2 - a^2*b^2*c^2 - a^2*c^4 + c^6) : :

See Antreas Hatzipolakis and Peter Moses euclid 5924.

X(53983) lies on the nine-point-circle and these lines: {2, 53949}, {4, 827}, {113, 46026}, {114, 1594}, {127, 8754}, {131, 37347}, {132, 3575}, {140, 31842}, {186, 16188}, {427, 23318}, {868, 20625}, {2970, 53830}, {2971, 5099}, {5139, 34981}, {6143, 31843}, {7574, 42424}, {36189, 46664}, {36472, 47421}

X(53983) = complement of X(53949)
X(53983) = polar-circle-inverse of X(827)

X(53984) = POLAR-CIRCLE-INVERSE OF X(841)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^10 - 11*a^6*b^4 + 13*a^4*b^6 - 3*a^2*b^8 - b^10 + 16*a^6*b^2*c^2 - 11*a^4*b^4*c^2 - 8*a^2*b^6*c^2 + 3*b^8*c^2 - 11*a^6*c^4 - 11*a^4*b^2*c^4 + 22*a^2*b^4*c^4 - 2*b^6*c^4 + 13*a^4*c^6 - 8*a^2*b^2*c^6 - 2*b^4*c^6 - 3*a^2*c^8 + 3*b^2*c^8 - c^10)*(a^12*b^2 - 4*a^10*b^4 + 5*a^8*b^6 - 5*a^4*b^10 + 4*a^2*b^12 - b^14 + a^12*c^2 + 8*a^10*b^2*c^2 - 8*a^8*b^4*c^2 - 30*a^6*b^6*c^2 + 43*a^4*b^8*c^2 - 10*a^2*b^10*c^2 - 4*b^12*c^2 - 4*a^10*c^4 - 8*a^8*b^2*c^4 + 64*a^6*b^4*c^4 - 38*a^4*b^6*c^4 - 32*a^2*b^8*c^4 + 18*b^10*c^4 + 5*a^8*c^6 - 30*a^6*b^2*c^6 - 38*a^4*b^4*c^6 + 76*a^2*b^6*c^6 - 13*b^8*c^6 + 43*a^4*b^2*c^8 - 32*a^2*b^4*c^8 - 13*b^6*c^8 - 5*a^4*c^10 - 10*a^2*b^2*c^10 + 18*b^4*c^10 + 4*a^2*c^12 - 4*b^2*c^12 - c^14) : :

See Antreas Hatzipolakis and Peter Moses euclid 5924.

X(53984) lies on the nine-point-circle and these lines: {4, 841}, {113, 9007}, {122, 11799}, {125, 1514}, {127, 47332}, {381, 16177}, {403, 53832}, {1596, 3258}, {6623, 16221}, {10297, 35968}, {11251, 16188}, {31842, 36169}

X(53984) = midpoint of X(4) and X(53944)
X(53984) = polar-circle-inverse of X(841)

---

X(53985) = POLAR-CIRCLE-INVERSE OF X(901)

Barycentrics    (2*a - b - c)*(b - c)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4*b - 2*a^2*b^3 + b^5 + a^4*c - 2*a^3*b*c + 2*a*b^3*c - b^4*c + 4*a*b^2*c^2 - 2*b^3*c^2 - 2*a^2*c^3 + 2*a*b*c^3 - 2*b^2*c^3 - b*c^4 + c^5) : :

827 See Antreas Hatzipolakis and Peter Moses euclid 5924.

X(53985) lies on the nine-point-circle and these lines: {4, 901}, {11, 51648}, {116, 3798}, {118, 42070}, {119, 5151}, {125, 50329}, {235, 39535}, {1532, 42423}, {2969, 5510}, {5521, 16228}, {8229, 31842}, {8735, 38962}, {20623, 21956}

X(53985) = polar-circle-inverse of X(901)

---

X(53986) = POLAR-CIRCLE-INVERSE OF X(930)

Barycentrics    (b - c)^2*(b + 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 - 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) : :

See Antreas Hatzipolakis and Peter Moses euclid 5924.

X(53986) lies on the nine-point-circle and these lines: {2, 20185}, {4, 128}, {11, 47017}, {25, 14656}, {113, 5446}, {114, 428}, {131, 546}, {235, 18402}, {403, 45180}, {427, 31843}, {2970, 11792}, {5133, 31842}, {5522, 8902}, {11563, 42424}, {16178, 34338}, {30446, 42423}

X(53986) = midpoint of X(4) and X(2383)
X(53986) = complement of X(20185)
X(53986) = polar-circle-inverse of X(930)
X(53986) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(53963)
X(53986) = complement of the isogonal conjugate of X(20184)
X(53986) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 20184}, {20184, 10}, {41628, 4369}
X(53986) = X(4)-Ceva conjugate of X(20184)
X(53986) = X(20184)-Dao conjugate of X(39171)

---

X(53987) = POLAR-CIRCLE-INVERSE OF X(1287)

Barycentrics    (b - c)^2*(b + 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 - a^2*b^2*c^2 + b^4*c^2 - 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 + a^2*b^2*c^4 - 2*b^4*c^4 - 2*a^2*c^6 + c^8) : :

See Antreas Hatzipolakis and Peter Moses euclid 5924.

X(53987) lies on the nine-point-circle and these lines: {2, 11635}, {4, 1287}, {114, 186}, {131, 7574}, {403, 44953}, {868, 46664}, {1560, 41336}, {1594, 16188}, {3575, 42426}, {20625, 36189}, {31842, 37938}, {37347, 42424}

X(53987) = complement of X(11635)
X(53987) = polar-circle-inverse of X(1287)
X(53987) = X(661)-complementary conjugate of X(22151)

---

X(53988) = POLAR-CIRCLE-INVERSE OF X(1290)

Barycentrics    (b - 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^2*c - a*b*c + b^2*c - a*c^2 + b*c^2 + c^3)*(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 + 4*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) : :

See Antreas Hatzipolakis and Peter Moses euclid 5924.

X(53988) lies on the nine-point-circle and these lines: {2, 53952}, {4, 1290}, {30, 42423}, {113, 912}, {115, 6591}, {119, 403}, {120, 468}, {123, 47399}, {125, 15313}, {127, 37986}, {136, 16228}, {523, 5521}, {1560, 3290}, {1902, 31841}, {2074, 47106}, {3139, 16177}, {3140, 38971}, {6841, 42424}, {10151, 25640}, {15763, 25641}, {16188, 37362}, {20621, 37981}, {31845, 37982}

X(53988) = midpoint of X(i) and X(j) for these {i,j}: {4, 53921}, {2074, 47106}
X(53988) = complement of X(53952)
X(53988) = reflection of X(5521) in the Euler line
X(53988) = polar-circle-inverse of X(1290)
X(53988) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(53956)
X(53988) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 47235}, {51632, 20315}
X(53988) = X(2)-Ceva conjugate of X(47235)
X(53988) = X(47235)-Dao conjugate of X(2)
X(53988) = barycentric quotient X(47235)/X(53952)

---

X(53989) = POLAR-CIRCLE-INVERSE OF X(1291)

Barycentrics    (b - c)^2*(b + c)^2*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^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^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 + 7*a^6*b^2*c^2 - 8*a^4*b^4*c^2 + 7*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 8*a^4*b^2*c^4 - 2*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 7*a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10) : :

See Antreas Hatzipolakis and Peter Moses euclid 5924.

X(53989) lies on the nine-point-circle and these lines: {2, 23096}, {4, 1291}, {23, 31842}, {113, 539}, {125, 20184}, {128, 403}, {131, 11563}, {428, 16188}, {468, 31843}, {546, 42424}, {10151, 18402}, {12079, 46658}, {12300, 33333}, {35235, 46437}

X(53989) = midpoint of X(4) and X(53930)
X(53989) = complement of X(53960)
X(53989) = polar-circle-inverse of X(1291)
X(53989) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(23096)

---

X(53990) = POLAR-CIRCLE-INVERSE OF X(1292)

Barycentrics    (b - c)^2*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - 2*a*b + b^2 - 2*a*c + c^2)*(a^5 - a^4*b - a*b^4 + b^5 - a^4*c - 2*a^3*b*c + 6*a^2*b^2*c - 2*a*b^3*c - b^4*c + 6*a^2*b*c^2 - 2*a*b^2*c^2 - 2*a*b*c^3 - a*c^4 - b*c^4 + c^5) : :

See Antreas Hatzipolakis and Peter Moses euclid 5924.

X(53990) lies on the nine-point-circle and these lines: {4, 120}, {114, 15763}, {118, 21077}, {119, 1596}, {122, 3140}, {126, 37362}, {235, 20621}, {381, 42423}, {6623, 25640}, {6841, 31842}, {8735, 38959}, {16177, 37986}, {37368, 53839}, {37984, 42422}

X(53990) = midpoint of X(4) and X(15344)
X(53990) = polar-circle-inverse of X(1292)

---

X(53991) = POLAR-CIRCLE-INVERSE OF X(1295)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(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 + 2*a^3*b^2*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 + 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)*(a^8*b - 2*a^6*b^3 + 2*a^2*b^7 - b^9 + a^8*c - 2*a^7*b*c + 2*a^6*b^2*c - 6*a^4*b^4*c + 6*a^3*b^5*c + 2*a^2*b^6*c - 4*a*b^7*c + b^8*c + 2*a^6*b*c^2 + 6*a^4*b^3*c^2 - 10*a^2*b^5*c^2 + 2*b^7*c^2 - 2*a^6*c^3 + 6*a^4*b^2*c^3 - 12*a^3*b^3*c^3 + 6*a^2*b^4*c^3 + 4*a*b^5*c^3 - 2*b^6*c^3 - 6*a^4*b*c^4 + 6*a^2*b^3*c^4 + 6*a^3*b*c^5 - 10*a^2*b^2*c^5 + 4*a*b^3*c^5 + 2*a^2*b*c^6 - 2*b^3*c^6 + 2*a^2*c^7 - 4*a*b*c^7 + 2*b^2*c^7 + b*c^8 - c^9) : :

See Antreas Hatzipolakis and Peter Moses euclid 5924.

X(53991) lies on the nine-point-circle and these lines: {4, 123}, {11, 34}, {12, 13612}, {19, 5514}, {25, 53837}, {122, 429}, {125, 431}, {442, 35968}, {5511, 6623}, {5520, 10151}, {5521, 37197}, {14312, 25640}, {16177, 37982}, {30444, 53822}, {37376, 53836}

X(53991) = midpoint of X(4) and X(40097)
X(53991) = polar-circle-inverse of X(1295)
a

X(53992) = POLAR-CIRCLE-INVERSE OF X(1296)

Barycentrics    (b - c)^2*(b + c)^2*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-5*a^2 + b^2 + c^2)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*c^2 + 14*a^2*b^2*c^2 - 5*b^4*c^2 - a^2*c^4 - 5*b^2*c^4 + c^6) : :

See Antreas Hatzipolakis and Peter Moses euclid 5924.

X(53992) lies on the nine-point-circle and these lines: {2, 20187}, {4, 126}, {113, 1351}, {114, 1596}, {122, 3143}, {132, 6623}, {235, 1560}, {381, 31842}, {403, 31655}, {10151, 47350}, {14120, 16177}, {16188, 37984}, {42424, 47332}

X(53992) = midpoint of X(4) and X(2374)
X(53992) = complement of X(20187)
X(53992) = polar-circle-inverse of X(1296)
X(53992) = complement of the isogonal conjugate of X(20186)
X(53992) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 20186}, {20186, 10}
X(53992) = X(4)-Ceva conjugate of X(20186)

---

X(53993) = POLAR-CIRCLE-INVERSE OF X(1302)

Barycentrics    (b - c)^2*(b + 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)*(3*a^6 - 5*a^4*b^2 + a^2*b^4 + b^6 - 5*a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 + a^2*c^4 - b^2*c^4 + c^6) : :

See Antreas Hatzipolakis and Peter Moses euclid 5924.

X(53993) lies on the nine-point-circle and these lines: {2, 131}, {4, 1302}, {25, 133}, {113, 427}, {114, 5094}, {115, 35235}, {122, 2970}, {127, 3134}, {132, 47208}, {135, 53577}, {468, 25641}, {858, 42424}, {868, 14672}, {1560, 3018}, {1650, 53822}, {3154, 38971}, {3258, 8754}, {5064, 50927}, {16178, 34981}, {16319, 42426}, {18809, 37981}, {30739, 31842}

X(53993) = complement of X(53958)
X(53993) = polar-circle-inverse of X(1302)
X(53993) = orthoptic-circle-of-the-Steiner-inellipse-inverse of X(1300)
X(53993) = X(i)-complementary conjugate of X(j) for these (i,j): {656, 18531}, {661, 37638}, {2616, 44201}, {18533, 8062}, {37645, 4369}

---

X(53994) = X(2)X(914)∩X(6)X(281)

Barycentrics    (-a+b+c)*(a^4-2*a^2*(b-c)^2+(b^2-c^2)^2) : :

See Thanassis Gakopoulos, Antreas Hatzipolakis and Ivan Pavlov, euclid 5842.

X(53994) lies on these lines: {1,20262}, {2,914}, {4,1903}, {6,281}, {7,4858}, {8,9}, {11,15849}, {20,1741}, {34,10365}, {51,7102}, {57,189}, {69,37788}, {77,26001}, {92,11433}, {145,27508}, {193,1944}, {198,944}, {219,1067}, {220,17362}, {239,27509}, {242,6776}, {268,2968}, {278,13567}, {282,1210}, {319,30854}, {387,46878}, {515,2270}, {519,2324}, {579,52366}, {610,5768}, {836,3086}, {938,1449}, {948,16608}, {966,40937}, {1100,46835}, {1172,43742}, {1212,17275}, {1213,34522}, {1264,3975}, {1419,44356}, {1440,34492}, {1443,20082}, {1474,45748}, {1587,14121}, {1588,7090}, {1696,10944}, {1743,10573}, {1788,5120}, {1851,1899}, {1857,11436}, {1863,3270}, {1864,7046}, {1999,27539}, {2261,8756}, {2289,3684}, {2293,28118}, {2323,12649}, {2551,5814}, {2969,26869}, {2975,15817}, {2999,20205}, {3187,27540}, {3452,34255}, {3486,4254}, {3731,12647}, {3759,37774}, {3875,40880}, {3879,27384}, {3943,34524}, {3950,49169}, {3973,41684}, {4000,26932}, {4012,15733}, {4035,30827}, {4124,12589}, {4264,10570}, {4305,36744}, {4393,27547}, {4511,27522}, {4851,34852}, {4856,5199}, {5273,33168}, {5554,5749}, {5712,6708}, {5813,21270}, {5905,37644}, {5932,18634}, {6601,30620}, {6906,11434}, {7129,51359}, {7140,9777}, {7982,21068}, {8558,52365}, {9581,20263}, {10396,39130}, {11427,52412}, {12245,21871}, {15830,16552}, {16596,40995}, {16704,28921}, {17363,27420}, {17869,18909}, {17917,26958}, {17923,37643}, {18228,37656}, {18623,20266}, {18912,31387}, {26540,37800}, {27507,28922}, {28739,48381}, {34413,44327}, {36845,40869}, {36916,50087}

X(53994) = isogonal conjugate of X(53995)
X(53994) = isotomic conjugate of X(34401)
X(53994) = perspector of circumconic {{A, B, C, X(1309), X(3699)}}
X(53994) = X(i)-isoconjugate-of-X(j) for these {i, j}: {31, 34401}, {57, 42019}, {837, 22341}, {2199, 34413}
X(53994) = X(i)-Dao conjugate of X(j) for these {i, j}: {2, 34401}, {3086, 329}, {5452, 42019}, {24005, 52385}, {38015, 7}, {38357, 14837}, {40650, 13436}, {49171, 57}
X(53994) = X(i)-Ceva conjugate of X(j) for these {i, j}: {44327, 522}
X(53994) = X(i)-complementary conjugate of X(j) for these {i, j}: {45818, 2886}
X(53994) = X(i)-cross conjugate of X(j) for these {i, j}: {30223, 3086}
X(53994) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(15501)}}, {{A, B, C, X(8), X(1067)}} and {{A, B, C, X(9), X(3554)}}
X(53994) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 1146, 281}, {1449, 23058, 40942}, {3554, 24005, 3086}, {3686, 41006, 9}, {5839, 6554, 219}

---

X(53995) = X(56)X(1066)∩X(198)X(6612)

Barycentrics    a^2*(a+b-c)*(a-b+c)*(a^4+4*a*b^2*c+(b^2-c^2)^2-2*a^2*(b^2+c^2))*(a^4+4*a*b*c^2+(b^2-c^2)^2-2*a^2*(b^2+c^2)) : :

See Thanassis Gakopoulos, Antreas Hatzipolakis and Ivan Pavlov, euclid 5842.

X(53995) lies on these lines: {3,40971}, {56,1066}, {198,6612}, {222,23585}, {738,40212}, {837,1817}, {905,34048}, {1407,2178}, {1434,31631}, {1435,1465}, {6511,26611}, {7073,11509}, {7091,37282}

X(53995) = isogonal conjugate of X(53994)
X(53995) = trilinear pole of the line {8677,43924}
X(53995) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2, 30223}, {8, 3554}, {9, 3086}, {21, 24005}, {33, 26871}, {84, 38015}, {92, 19354}, {284, 17869}, {836, 1896}, {1519, 52663}, {2326, 26955}, {13454, 38003}, {13456, 40650}
X(53995) = X(i)-vertex conjugate of X(j) for these {i, j}: {200, 1422}
X(53995) = X(i)-Dao conjugate of X(j) for these {i, j}: {478, 3086}, {22391, 19354}, {32664, 30223}, {40590, 17869}, {40611, 24005}
X(53995) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(222)}}, {{A, B, C, X(3), X(1817)}}, {{A, B, C, X(6), X(913)}}, {{A, B, C, X(9), X(198)}}, {{A, B, C, X(21), X(37269)}}, {{A, B, C, X(55), X(2149)}}, {{A, B, C, X(56), X(57)}} and {{A, B, C, X(58), X(36745)}}

---

X(53996) = X(1)X(142)∩X(9)X(77)

Barycentrics    a*(a^4+2*a*b*c*(b+c)-2*a^2*(b^2+c^2)+(b-c)^2*(b^2+c^2)) : :

See Thanassis Gakopoulos, Antreas Hatzipolakis and Ivan Pavlov, euclid 5842.

X(53996) lies on the circumconic {{A, B, C, X(77), X(36956)}} and these lines: {1,142}, {2,914}, {3,12335}, {6,6510}, {9,77}, {57,2289}, {63,15066}, {78,17296}, {101,7289}, {141,997}, {144,1the 443}, {169,18161}, {219,241}, {222,25091}, {223,3452}, {269,527}, {306,20266}, {326,3912}, {346,28982}, {347,52457}, {355,21239}, {394,1708}, {521,2424}, {1158,17814}, {1214,17811}, {1253,35293}, {1323,34526}, {1418,6603}, {1445,2323}, {1459,28590}, {1465,25934}, {1741,52385}, {2256,37597}, {2262,37272}, {2321,44356}, {2331,17906}, {2340,8271}, {2397,3729}, {2428,7123}, {2999,6692}, {3008,3554}, {3306,26742}, {3553,3664}, {3811,4851}, {4319,35338}, {4402,38460}, {4511,4869}, {4552,26651}, {4641,23140}, {4657,17043}, {5256,5437}, {5287,25525}, {5336,28350}, {6173,7190}, {6261,18589}, {6600,30621}, {6700,20206}, {6706,28639}, {7225,17439}, {7308,47057}, {8270,25941}, {9312,27384}, {10436,26665}, {11712,24309}, {15668,50317}, {16826,25521}, {17234,44179}, {17279,36949}, {17306,19861}, {17390,21258}, {18726,40131}, {20335,34036}, {21147,21246}, {21233,24806}, {21244,37694}, {21255,30144}, {25243,28968}, {25524,37594}, {25878,40937}, {26130,34847}, {26658,27509}, {26672,26685}, {27381,30806}, {27514,28961}, {33633,36973}, {34048,34052}, {37805,52033}

X(53996) = midpoint of X(269) and X(2324)
X(53996) = complement of X(53994)
X(53996) = X(i)-complementary conjugate of X(j) for these {i, j}: {34401, 2887}, {42019, 3452}
X(53996) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6505, 45126}, {77, 25930, 9}, {141, 17044, 17073}, {269, 2324, 527}, {651, 26669, 9}

---

X(53997) = X(2)X(914)∩X(8)X(77)

Barycentrics    (a+b-c)*(a-b+c)*(3*a^3-a^2*(b+c)+(b+c)^3-a*(3*b^2+2*b*c+3*c^2)) : :

See Thanassis Gakopoulos, Antreas Hatzipolakis and Ivan Pavlov, euclid 5842.

X(53997) lies on these lines: {1,24213}, {2,914}, {7,145}, {8,77}, {20,52385}, {57,20043}, {69,347}, {78,5932}, {100,1804}, {144,4552}, {193,12848}, {223,34255}, {239,8732}, {241,5839}, {269,519}, {273,30806}, {281,6510}, {306,18623}, {307,3160}, {319,348}, {326,1440}, {329,20211}, {346,651}, {948,4851}, {1122,37738}, {1214,14552}, {1419,2321}, {1441,3945}, {1443,3621}, {3241,7190}, {3244,4328}, {3476,24471}, {3598,19993}, {3620,17086}, {3623,7269}, {3633,7271}, {3662,25726}, {3694,43044}, {4350,6764}, {4357,25716}, {4402,30379}, {4461,28968}, {4869,37800}, {5687,7053}, {5933,7176}, {5942,27508}, {6180,17314}, {6604,17377}, {6610,17299}, {7274,51093}, {7674,30619}, {8232,17316}, {9436,20015}, {10436,25719}, {16284,33673}, {17079,50132}, {17080,37655}, {17296,43035}, {17300,30275}, {17394,52422}, {17950,20080}, {20212,27382}, {21296,22464}, {26125,29585}, {28079,43040}, {28739,29616}, {36854,52673}

X(53997) = anticomplement of X(53994)
X(53997) = (i)-anticomplementary conjugate of X(j) for these {i, j}: {34401, 6327}, {42019, 329}
X(53997) = X(i)-isoconjugate-of-X(j) for these {i, j}: {55, 45818}
X(53997) = X(i)-Dao conjugate of X(j) for these {i, j}: {223, 45818}, {23681, 9581}
X(53997) = X(i)-Ceva conjugate of X(j) for these {i, j}: {34401, 2}
X(53997) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 664, 347}, {3160, 32099, 307}, {3879, 9312, 7}

---

X(53998) = X(19)X(44)∩X(198)X(1426)

Barycentrics    a*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^4*(b-c)^2+a^5*(b+c)+(b-c)^4*(b+c)^2+a*(b-c)^2*(b+c)^3-2*a^3*(b^3+b^2*c+b*c^2+c^3)-2*a^2*(b^4-2*b^3*c-2*b^2*c^2-2*b*c^3+c^4)) : :

See Thanassis Gakopoulos, Antreas Hatzipolakis and Ivan Pavlov, euclid 5842.

X(53998) lies on these lines: {4,31965}, {19,44}, {48,11214}, {198,1426}, {278,51413}, {1146,1826}, {1824,14593}, {1868,44545}, {1875,2331}, {1880,2183}, {2817,20263}, {3057,53009}, {14524,52607}, {14557,40149}

X(53998) = Zosma transform of X(53994)

---

X(53999) = {X(2)X(5)}-HARMONIC CONJUGATE OF X(26)

Barycentrics    a^10 - 4*a^8*b^2 + 4*a^6*b^4 + 2*a^4*b^6 - 5*a^2*b^8 + 2*b^10 - 4*a^8*c^2 + 12*a^4*b^4*c^2 - 2*a^2*b^6*c^2 - 6*b^8*c^2 + 4*a^6*c^4 + 12*a^4*b^2*c^4 + 14*a^2*b^4*c^4 + 4*b^6*c^4 + 2*a^4*c^6 - 2*a^2*b^2*c^6 + 4*b^4*c^6 - 5*a^2*c^8 - 6*b^2*c^8 + 2*c^10 : :

X(53999) lies on these lines: {2, 3}, {156, 10516}, {206, 32767}, {3167, 8254}, {3589, 18952}, {3763, 32142}, {5050, 18356}, {5449, 19139}, {10601, 34826}, {11178, 41597}, {13561, 15805}, {13565, 14852}, {15026, 37638}, {15069, 32136}, {20396, 25335}, {20564, 40410}, {21230, 37493}, {25561, 50414}, {26958, 32205}, {32165, 53091}, {43811, 47355}

X(53999) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5, 26}, {2, 5576, 7516}, {2, 7528, 7568}, {2, 13371, 13154}, {2, 14790, 140}, {5, 140, 11818}, {5, 632, 6756}, {5, 7514, 7564}, {5, 7568, 7528}, {5, 44288, 3851}, {1656, 7539, 5}, {3090, 14786, 5}, {3628, 13371, 2}, {5055, 7395, 5}, {5056, 18531, 5}, {5079, 7507, 5}, {5576, 7516, 31181}, {7528, 7568, 26}, {7558, 50137, 7530}, {7569, 7571, 1656}, {13490, 47525, 26}, {14782, 14783, 44831}

---

X(54000) = {X(2),X(5)}-HARMONIC CONJUGATE OF X(186)

Barycentrics    a^10 - 4*a^8*b^2 + 4*a^6*b^4 + 2*a^4*b^6 - 5*a^2*b^8 + 2*b^10 - 4*a^8*c^2 + 3*a^6*b^2*c^2 + 2*a^4*b^4*c^2 + 5*a^2*b^6*c^2 - 6*b^8*c^2 + 4*a^6*c^4 + 2*a^4*b^2*c^4 + 4*b^6*c^4 + 2*a^4*c^6 + 5*a^2*b^2*c^6 + 4*b^4*c^6 - 5*a^2*c^8 - 6*b^2*c^8 + 2*c^10 : :

X(54000) lies on these lines: {2, 3}, {328, 40410}, {1199, 5449}, {5892, 11562}, {6236, 10173}, {9221, 42410}, {11597, 20304}, {11695, 43866}, {11793, 32352}, {15032, 23293}, {18392, 39242}, {22151, 24206}, {34545, 41730}

X(54000) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5, 186}, {2, 2072, 7550}, {2, 3153, 140}, {2, 5056, 18420}, {2, 7577, 35921}, {5, 632, 45971}, {5, 18563, 3091}, {5, 18567, 3851}, {427, 7552, 37925}, {1594, 6676, 46450}, {1656, 7569, 3090}, {2070, 13413, 7565}, {2072, 3628, 2}, {3549, 52295, 12088}, {5070, 7514, 2}, {5169, 10201, 52294}, {6676, 46450, 7512}, {10024, 44236, 52403}, {25402, 44904, 5}, {35921, 49674, 7577}, {44236, 52403, 14865}

---

This is the end of PART 27: Centers X(52001) - X(54000)

+

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)